LMFDB - Embedded newform 450.6.f.f.107.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + \cdots + 498214340649 $ x^16 - 252x^14 + 27174x^12 - 1635700x^10 + 60061815x^8 - 1376564028x^6 + 19220200150x^4 - 149540021784*x^2 + 498214340649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{8}\cdot 5^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.3
Root			$-5.20458 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.f.143.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(50.1584 + 50.1584i) q^{7} +(45.2548 + 45.2548i) q^{8} +O(q^{10})$	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(50.1584 + 50.1584i) q^{7} +(45.2548 + 45.2548i) q^{8} +659.741i q^{11} +(-517.605 + 517.605i) q^{13} -283.739 q^{14} -256.000 q^{16} +(62.7710 - 62.7710i) q^{17} +38.1009i q^{19} +(-1866.03 - 1866.03i) q^{22} +(-455.322 - 455.322i) q^{23} -2928.02i q^{26} +(802.535 - 802.535i) q^{28} +4598.69 q^{29} +2690.73 q^{31} +(724.077 - 724.077i) q^{32} +355.086i q^{34} +(-1486.78 - 1486.78i) q^{37} +(-107.766 - 107.766i) q^{38} +194.091i q^{41} +(-9864.36 + 9864.36i) q^{43} +10555.9 q^{44} +2575.69 q^{46} +(-5300.97 + 5300.97i) q^{47} -11775.3i q^{49} +(8281.68 + 8281.68i) q^{52} +(20592.5 + 20592.5i) q^{53} +4539.82i q^{56} +(-13007.1 + 13007.1i) q^{58} +18628.3 q^{59} -44820.8 q^{61} +(-7610.55 + 7610.55i) q^{62} +4096.00i q^{64} +(-25712.6 - 25712.6i) q^{67} +(-1004.34 - 1004.34i) q^{68} -61412.8i q^{71} +(-27244.9 + 27244.9i) q^{73} +8410.49 q^{74} +609.615 q^{76} +(-33091.6 + 33091.6i) q^{77} +73400.0i q^{79} +(-548.974 - 548.974i) q^{82} +(-58926.6 - 58926.6i) q^{83} -55801.3i q^{86} +(-29856.5 + 29856.5i) q^{88} +112776. q^{89} -51924.5 q^{91} +(-7285.16 + 7285.16i) q^{92} -29986.8i q^{94} +(87995.6 + 87995.6i) q^{97} +(33305.5 + 33305.5i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 528 q^{7}+O(q^{10}) $ 16 * q - 528 * q^7	$ 16 q - 528 q^{7} + 192 q^{13} - 4096 q^{16} - 2688 q^{22} - 8448 q^{28} + 13024 q^{31} - 47328 q^{37} - 55440 q^{43} + 44544 q^{46} - 3072 q^{52} - 101184 q^{58} + 28400 q^{61} + 242256 q^{67} - 430944 q^{73} - 7168 q^{76} + 158208 q^{82} - 43008 q^{88} - 185472 q^{91} + 457152 q^{97}+O(q^{100}) $ 16 * q - 528 * q^7 + 192 * q^13 - 4096 * q^16 - 2688 * q^22 - 8448 * q^28 + 13024 * q^31 - 47328 * q^37 - 55440 * q^43 + 44544 * q^46 - 3072 * q^52 - 101184 * q^58 + 28400 * q^61 + 242256 * q^67 - 430944 * q^73 - 7168 * q^76 + 158208 * q^82 - 43008 * q^88 - 185472 * q^91 + 457152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.82843	+	2.82843i	−0.500000	+	0.500000i
$2$	−2.82843	+	2.82843i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	50.1584	+	50.1584i	0.386900	+	0.386900i	0.873580	−	0.486680i	$-0.161792\pi$
$7$	50.1584	+	50.1584i	0.386900	+	0.386900i	−0.486680	+	0.873580i	$0.661792\pi$
$8$	45.2548	+	45.2548i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$			659.741i			1.64396i	0.569514	+	0.821981i	$0.307131\pi$
$11$			659.741i			1.64396i	−0.569514	+	0.821981i	$0.692869\pi$
$12$		0			0
$13$	−517.605	+	517.605i	−0.849454	+	0.849454i	−0.990065	−	0.140610i	$-0.955093\pi$
$13$	−517.605	+	517.605i	−0.849454	+	0.849454i	0.140610	+	0.990065i	$0.455093\pi$
$14$	−283.739			−0.386900
$15$		0			0
$16$	−256.000			−0.250000
$17$	62.7710	−	62.7710i	0.0526789	−	0.0526789i	−0.680277	−	0.732955i	$-0.738140\pi$
$17$	62.7710	−	62.7710i	0.0526789	−	0.0526789i	0.732955	+	0.680277i	$0.238140\pi$
$18$		0			0
$19$			38.1009i			0.0242132i	0.999927	+	0.0121066i	$0.00385374\pi$
$19$			38.1009i			0.0242132i	−0.999927	+	0.0121066i	$0.996146\pi$
$20$		0			0
$21$		0			0
$22$	−1866.03	−	1866.03i	−0.821981	−	0.821981i
$23$	−455.322	−	455.322i	−0.179473	−	0.179473i	0.611653	−	0.791126i	$-0.290505\pi$
$23$	−455.322	−	455.322i	−0.179473	−	0.179473i	−0.791126	+	0.611653i	$0.790505\pi$
$24$		0			0
$25$		0			0
$26$		−	2928.02i		−	0.849454i
$27$		0			0
$28$	802.535	−	802.535i	0.193450	−	0.193450i
$29$	4598.69			1.01540			0.507702	−	0.861533i	$-0.330495\pi$
$29$	4598.69			1.01540			0.507702	+	0.861533i	$0.330495\pi$
$30$		0			0
$31$	2690.73			0.502883			0.251441	−	0.967873i	$-0.419095\pi$
$31$	2690.73			0.502883			0.251441	+	0.967873i	$0.419095\pi$
$32$	724.077	−	724.077i	0.125000	−	0.125000i
$33$		0			0
$34$			355.086i			0.0526789i
$35$		0			0
$36$		0			0
$37$	−1486.78	−	1486.78i	−0.178543	−	0.178543i	0.612178	−	0.790720i	$-0.290294\pi$
$37$	−1486.78	−	1486.78i	−0.178543	−	0.178543i	−0.790720	+	0.612178i	$0.790294\pi$
$38$	−107.766	−	107.766i	−0.0121066	−	0.0121066i
$39$		0			0
$40$		0			0
$41$			194.091i			0.0180321i	0.999959	+	0.00901606i	$0.00286994\pi$
$41$			194.091i			0.0180321i	−0.999959	+	0.00901606i	$0.997130\pi$
$42$		0			0
$43$	−9864.36	+	9864.36i	−0.813575	+	0.813575i	−0.985168	−	0.171593i	$-0.945109\pi$
$43$	−9864.36	+	9864.36i	−0.813575	+	0.813575i	0.171593	+	0.985168i	$0.445109\pi$
$44$	10555.9			0.821981
$45$		0			0
$46$	2575.69			0.179473
$47$	−5300.97	+	5300.97i	−0.350034	+	0.350034i	−0.860122	−	0.510088i	$-0.829613\pi$
$47$	−5300.97	+	5300.97i	−0.350034	+	0.350034i	0.510088	+	0.860122i	$0.329613\pi$
$48$		0			0
$49$		−	11775.3i		−	0.700617i
$50$		0			0
$51$		0			0
$52$	8281.68	+	8281.68i	0.424727	+	0.424727i
$53$	20592.5	+	20592.5i	1.00697	+	1.00697i	0.999976	+	0.00699943i	$0.00222801\pi$
$53$	20592.5	+	20592.5i	1.00697	+	1.00697i	0.00699943	+	0.999976i	$0.497772\pi$
$54$		0			0
$55$		0			0
$56$			4539.82i			0.193450i
$57$		0			0
$58$	−13007.1	+	13007.1i	−0.507702	+	0.507702i
$59$	18628.3			0.696694			0.348347	−	0.937366i	$-0.386743\pi$
$59$	18628.3			0.696694			0.348347	+	0.937366i	$0.386743\pi$
$60$		0			0
$61$	−44820.8			−1.54225			−0.771126	−	0.636682i	$-0.780306\pi$
$61$	−44820.8			−1.54225			−0.771126	+	0.636682i	$0.780306\pi$
$62$	−7610.55	+	7610.55i	−0.251441	+	0.251441i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	−25712.6	−	25712.6i	−0.699776	−	0.699776i	0.264586	−	0.964362i	$-0.414765\pi$
$67$	−25712.6	−	25712.6i	−0.699776	−	0.699776i	−0.964362	+	0.264586i	$0.914765\pi$
$68$	−1004.34	−	1004.34i	−0.0263395	−	0.0263395i
$69$		0			0
$70$		0			0
$71$		−	61412.8i		−	1.44582i	−0.690944	−	0.722908i	$-0.742805\pi$
$71$		−	61412.8i		−	1.44582i	0.690944	−	0.722908i	$-0.257195\pi$
$72$		0			0
$73$	−27244.9	+	27244.9i	−0.598381	+	0.598381i	−0.939882	−	0.341501i	$-0.889065\pi$
$73$	−27244.9	+	27244.9i	−0.598381	+	0.598381i	0.341501	+	0.939882i	$0.389065\pi$
$74$	8410.49			0.178543
$75$		0			0
$76$	609.615			0.0121066
$77$	−33091.6	+	33091.6i	−0.636049	+	0.636049i
$78$		0			0
$79$			73400.0i			1.32321i	0.749853	+	0.661604i	$0.230124\pi$
$79$			73400.0i			1.32321i	−0.749853	+	0.661604i	$0.769876\pi$
$80$		0			0
$81$		0			0
$82$	−548.974	−	548.974i	−0.00901606	−	0.00901606i
$83$	−58926.6	−	58926.6i	−0.938894	−	0.938894i	0.0593440	−	0.998238i	$-0.481099\pi$
$83$	−58926.6	−	58926.6i	−0.938894	−	0.938894i	−0.998238	+	0.0593440i	$0.981099\pi$
$84$		0			0
$85$		0			0
$86$		−	55801.3i		−	0.813575i
$87$		0			0
$88$	−29856.5	+	29856.5i	−0.410991	+	0.410991i
$89$	112776.			1.50918			0.754592	−	0.656194i	$-0.227835\pi$
$89$	112776.			1.50918			0.754592	+	0.656194i	$0.227835\pi$
$90$		0			0
$91$	−51924.5			−0.657308
$92$	−7285.16	+	7285.16i	−0.0897365	+	0.0897365i
$93$		0			0
$94$		−	29986.8i		−	0.350034i
$95$		0			0
$96$		0			0
$97$	87995.6	+	87995.6i	0.949580	+	0.949580i	0.998789	−	0.0492086i	$-0.0156699\pi$
$97$	87995.6	+	87995.6i	0.949580	+	0.949580i	−0.0492086	+	0.998789i	$0.515670\pi$
$98$	33305.5	+	33305.5i	0.350308	+	0.350308i
$99$		0			0
$100$		0			0
$101$		−	12987.2i		−	0.126681i	−0.997992	−	0.0633407i	$-0.979825\pi$
$101$		−	12987.2i		−	0.126681i	0.997992	−	0.0633407i	$-0.0201755\pi$
$102$		0			0
$103$	−147468.	+	147468.i	−1.36963	+	1.36963i	−0.508669	+	0.860962i	$0.669862\pi$
$103$	−147468.	+	147468.i	−1.36963	+	1.36963i	−0.860962	+	0.508669i	$0.830138\pi$
$104$	−46848.3			−0.424727
$105$		0			0
$106$	−116489.			−1.00697
$107$	−65928.5	+	65928.5i	−0.556691	+	0.556691i	−0.928364	−	0.371673i	$-0.878784\pi$
$107$	−65928.5	+	65928.5i	−0.556691	+	0.556691i	0.371673	+	0.928364i	$0.378784\pi$
$108$		0			0
$109$		−	181646.i		−	1.46440i	−0.681091	−	0.732198i	$-0.738494\pi$
$109$		−	181646.i		−	1.46440i	0.681091	−	0.732198i	$-0.261506\pi$
$110$		0			0
$111$		0			0
$112$	−12840.6	−	12840.6i	−0.0967250	−	0.0967250i
$113$	−38222.8	−	38222.8i	−0.281596	−	0.281596i	0.552149	−	0.833745i	$-0.313808\pi$
$113$	−38222.8	−	38222.8i	−0.281596	−	0.281596i	−0.833745	+	0.552149i	$0.813808\pi$
$114$		0			0
$115$		0			0
$116$		−	73579.0i		−	0.507702i
$117$		0			0
$118$	−52688.7	+	52688.7i	−0.348347	+	0.348347i
$119$	6296.99			0.0407630
$120$		0			0
$121$	−274208.			−1.70261
$122$	126772.	−	126772.i	0.771126	−	0.771126i
$123$		0			0
$124$		−	43051.8i		−	0.251441i
$125$		0			0
$126$		0			0
$127$	−72272.3	−	72272.3i	−0.397615	−	0.397615i	0.479776	−	0.877391i	$-0.340718\pi$
$127$	−72272.3	−	72272.3i	−0.397615	−	0.397615i	−0.877391	+	0.479776i	$0.840718\pi$
$128$	−11585.2	−	11585.2i	−0.0625000	−	0.0625000i
$129$		0			0
$130$		0			0
$131$		−	163217.i		−	0.830973i	−0.909599	−	0.415486i	$-0.863611\pi$
$131$		−	163217.i		−	0.830973i	0.909599	−	0.415486i	$-0.136389\pi$
$132$		0			0
$133$	−1911.08	+	1911.08i	−0.00936808	+	0.00936808i
$134$	145453.			0.699776
$135$		0			0
$136$	5681.38			0.0263395
$137$	−30415.8	+	30415.8i	−0.138451	+	0.138451i	−0.772936	−	0.634484i	$-0.781212\pi$
$137$	−30415.8	+	30415.8i	−0.138451	+	0.138451i	0.634484	+	0.772936i	$0.281212\pi$
$138$		0			0
$139$		−	343921.i		−	1.50981i	−0.655835	−	0.754905i	$-0.727683\pi$
$139$		−	343921.i		−	1.50981i	0.655835	−	0.754905i	$-0.272317\pi$
$140$		0			0
$141$		0			0
$142$	173702.	+	173702.i	0.722908	+	0.722908i
$143$	−341485.	−	341485.i	−1.39647	−	1.39647i
$144$		0			0
$145$		0			0
$146$		−	154120.i		−	0.598381i
$147$		0			0
$148$	−23788.4	+	23788.4i	−0.0892713	+	0.0892713i
$149$	174105.			0.642461			0.321230	−	0.947001i	$-0.395904\pi$
$149$	174105.			0.642461			0.321230	+	0.947001i	$0.395904\pi$
$150$		0			0
$151$	−374613.			−1.33703			−0.668513	−	0.743700i	$-0.733069\pi$
$151$	−374613.			−1.33703			−0.668513	+	0.743700i	$0.733069\pi$
$152$	−1724.25	+	1724.25i	−0.00605329	+	0.00605329i
$153$		0			0
$154$		−	187194.i		−	0.636049i
$155$		0			0
$156$		0			0
$157$	−95123.8	−	95123.8i	−0.307992	−	0.307992i	0.536138	−	0.844130i	$-0.319883\pi$
$157$	−95123.8	−	95123.8i	−0.307992	−	0.307992i	−0.844130	+	0.536138i	$0.819883\pi$
$158$	−207606.	−	207606.i	−0.661604	−	0.661604i
$159$		0			0
$160$		0			0
$161$		−	45676.5i		−	0.138876i
$162$		0			0
$163$	−124177.	+	124177.i	−0.366076	+	0.366076i	−0.866044	−	0.499968i	$-0.833345\pi$
$163$	−124177.	+	124177.i	−0.366076	+	0.366076i	0.499968	+	0.866044i	$0.333345\pi$
$164$	3105.46			0.00901606
$165$		0			0
$166$	333339.			0.938894
$167$	−22938.8	+	22938.8i	−0.0636473	+	0.0636473i	−0.738214	−	0.674567i	$-0.764331\pi$
$167$	−22938.8	+	22938.8i	−0.0636473	+	0.0636473i	0.674567	+	0.738214i	$0.264331\pi$
$168$		0			0
$169$		−	164537.i		−	0.443146i
$170$		0			0
$171$		0			0
$172$	157830.	+	157830.i	0.406788	+	0.406788i
$173$	−479359.	−	479359.i	−1.21771	−	1.21771i	−0.968430	−	0.249284i	$-0.919805\pi$
$173$	−479359.	−	479359.i	−1.21771	−	1.21771i	−0.249284	−	0.968430i	$-0.580195\pi$
$174$		0			0
$175$		0			0
$176$		−	168894.i		−	0.410991i
$177$		0			0
$178$	−318979.	+	318979.i	−0.754592	+	0.754592i
$179$	−265461.			−0.619254			−0.309627	−	0.950858i	$-0.600204\pi$
$179$	−265461.			−0.619254			−0.309627	+	0.950858i	$0.600204\pi$
$180$		0			0
$181$	−743564.			−1.68703			−0.843513	−	0.537109i	$-0.819516\pi$
$181$	−743564.			−1.68703			−0.843513	+	0.537109i	$0.819516\pi$
$182$	146865.	−	146865.i	0.328654	−	0.328654i
$183$		0			0
$184$		−	41211.1i		−	0.0897365i
$185$		0			0
$186$		0			0
$187$	41412.6	+	41412.6i	0.0866021	+	0.0866021i
$188$	84815.5	+	84815.5i	0.175017	+	0.175017i
$189$		0			0
$190$		0			0
$191$			539270.i			1.06960i	0.844978	+	0.534801i	$0.179614\pi$
$191$			539270.i			1.06960i	−0.844978	+	0.534801i	$0.820386\pi$
$192$		0			0
$193$	−111105.	+	111105.i	−0.214705	+	0.214705i	−0.806263	−	0.591558i	$-0.798513\pi$
$193$	−111105.	+	111105.i	−0.214705	+	0.214705i	0.591558	+	0.806263i	$0.298513\pi$
$194$	−497778.			−0.949580
$195$		0			0
$196$	−188404.			−0.350308
$197$	112701.	−	112701.i	0.206900	−	0.206900i	−0.596048	−	0.802949i	$-0.703263\pi$
$197$	112701.	−	112701.i	0.206900	−	0.206900i	0.802949	+	0.596048i	$0.203263\pi$
$198$		0			0
$199$		−	328398.i		−	0.587853i	−0.955828	−	0.293926i	$-0.905038\pi$
$199$		−	328398.i		−	0.587853i	0.955828	−	0.293926i	$-0.0949620\pi$
$200$		0			0
$201$		0			0
$202$	36733.4	+	36733.4i	0.0633407	+	0.0633407i
$203$	230663.	+	230663.i	0.392860	+	0.392860i
$204$		0			0
$205$		0			0
$206$		−	834203.i		−	1.36963i
$207$		0			0
$208$	132507.	−	132507.i	0.212364	−	0.212364i
$209$	−25136.8			−0.0398055
$210$		0			0
$211$	−647753.			−1.00162			−0.500811	−	0.865557i	$-0.666965\pi$
$211$	−647753.			−1.00162			−0.500811	+	0.865557i	$0.666965\pi$
$212$	329479.	−	329479.i	0.503487	−	0.503487i
$213$		0			0
$214$		−	372948.i		−	0.556691i
$215$		0			0
$216$		0			0
$217$	134963.	+	134963.i	0.194565	+	0.194565i
$218$	513772.	+	513772.i	0.732198	+	0.732198i
$219$		0			0
$220$		0			0
$221$			64981.2i			0.0894967i
$222$		0			0
$223$	360829.	−	360829.i	0.485892	−	0.485892i	−0.421115	−	0.907007i	$-0.638361\pi$
$223$	360829.	−	360829.i	0.485892	−	0.485892i	0.907007	+	0.421115i	$0.138361\pi$
$224$	72637.2			0.0967250
$225$		0			0
$226$	216221.			0.281596
$227$	−1.01792e6	+	1.01792e6i	−1.31114	+	1.31114i	−0.390572	+	0.920572i	$0.627723\pi$
$227$	−1.01792e6	+	1.01792e6i	−1.31114	+	1.31114i	−0.920572	+	0.390572i	$0.872277\pi$
$228$		0			0
$229$			984516.i			1.24061i	0.784362	+	0.620303i	$0.212991\pi$
$229$			984516.i			1.24061i	−0.784362	+	0.620303i	$0.787009\pi$
$230$		0			0
$231$		0			0
$232$	208113.	+	208113.i	0.253851	+	0.253851i
$233$	−1.03993e6	−	1.03993e6i	−1.25491	−	1.25491i	−0.953491	−	0.301423i	$-0.902539\pi$
$233$	−1.03993e6	−	1.03993e6i	−1.25491	−	1.25491i	−0.301423	−	0.953491i	$-0.597461\pi$
$234$		0			0
$235$		0			0
$236$		−	298052.i		−	0.348347i
$237$		0			0
$238$	−17810.6	+	17810.6i	−0.0203815	+	0.0203815i
$239$	810562.			0.917891			0.458946	−	0.888464i	$-0.348227\pi$
$239$	810562.			0.917891			0.458946	+	0.888464i	$0.348227\pi$
$240$		0			0
$241$	1.35804e6			1.50616			0.753078	−	0.657931i	$-0.228568\pi$
$241$	1.35804e6			1.50616			0.753078	+	0.657931i	$0.228568\pi$
$242$	775576.	−	775576.i	0.851306	−	0.851306i
$243$		0			0
$244$			717133.i			0.771126i
$245$		0			0
$246$		0			0
$247$	−19721.2	−	19721.2i	−0.0205680	−	0.0205680i
$248$	121769.	+	121769.i	0.125721	+	0.125721i
$249$		0			0
$250$		0			0
$251$			176671.i			0.177003i	0.996076	+	0.0885017i	$0.0282079\pi$
$251$			176671.i			0.177003i	−0.996076	+	0.0885017i	$0.971792\pi$
$252$		0			0
$253$	300395.	−	300395.i	0.295047	−	0.295047i
$254$	408834.			0.397615
$255$		0			0
$256$	65536.0			0.0625000
$257$	−1.14874e6	+	1.14874e6i	−1.08490	+	1.08490i	−0.0888531	+	0.996045i	$0.528320\pi$
$257$	−1.14874e6	+	1.14874e6i	−1.08490	+	1.08490i	−0.996045	+	0.0888531i	$0.971680\pi$
$258$		0			0
$259$		−	149149.i		−	0.138156i
$260$		0			0
$261$		0			0
$262$	461647.	+	461647.i	0.415486	+	0.415486i
$263$	837838.	+	837838.i	0.746914	+	0.746914i	0.973898	−	0.226985i	$-0.0728867\pi$
$263$	837838.	+	837838.i	0.746914	+	0.746914i	−0.226985	+	0.973898i	$0.572887\pi$
$264$		0			0
$265$		0			0
$266$		−	10810.7i		−	0.00936808i
$267$		0			0
$268$	−411402.	+	411402.i	−0.349888	+	0.349888i
$269$	−240388.			−0.202550			−0.101275	−	0.994858i	$-0.532292\pi$
$269$	−240388.			−0.202550			−0.101275	+	0.994858i	$0.532292\pi$
$270$		0			0
$271$	−26337.3			−0.0217846			−0.0108923	−	0.999941i	$-0.503467\pi$
$271$	−26337.3			−0.0217846			−0.0108923	+	0.999941i	$0.503467\pi$
$272$	−16069.4	+	16069.4i	−0.0131697	+	0.0131697i
$273$		0			0
$274$		−	172057.i		−	0.138451i
$275$		0			0
$276$		0			0
$277$	−1.78533e6	−	1.78533e6i	−1.39804	−	1.39804i	−0.805665	−	0.592371i	$-0.798192\pi$
$277$	−1.78533e6	−	1.78533e6i	−1.39804	−	1.39804i	−0.592371	−	0.805665i	$-0.701808\pi$
$278$	972756.	+	972756.i	0.754905	+	0.754905i
$279$		0			0
$280$		0			0
$281$			2.09342e6i			1.58158i	0.612087	+	0.790790i	$0.290330\pi$
$281$			2.09342e6i			1.58158i	−0.612087	+	0.790790i	$0.709670\pi$
$282$		0			0
$283$	1.00478e6	−	1.00478e6i	0.745772	−	0.745772i	−0.227910	−	0.973682i	$-0.573189\pi$
$283$	1.00478e6	−	1.00478e6i	0.745772	−	0.745772i	0.973682	+	0.227910i	$0.0731893\pi$
$284$	−982605.			−0.722908
$285$		0			0
$286$	1.93173e6			1.39647
$287$	−9735.33	+	9735.33i	−0.00697663	+	0.00697663i
$288$		0			0
$289$			1.41198e6i			0.994450i
$290$		0			0
$291$		0			0
$292$	435918.	+	435918.i	0.299190	+	0.299190i
$293$	1.52933e6	+	1.52933e6i	1.04072	+	1.04072i	0.999135	+	0.0415819i	$0.0132397\pi$
$293$	1.52933e6	+	1.52933e6i	1.04072	+	1.04072i	0.0415819	+	0.999135i	$0.486760\pi$
$294$		0			0
$295$		0			0
$296$		−	134568.i		−	0.0892713i
$297$		0			0
$298$	−492445.	+	492445.i	−0.321230	+	0.321230i
$299$	471354.			0.304908
$300$		0			0
$301$	−989562.			−0.629545
$302$	1.05956e6	−	1.05956e6i	0.668513	−	0.668513i
$303$		0			0
$304$		−	9753.84i		−	0.00605329i
$305$		0			0
$306$		0			0
$307$	−1.95052e6	−	1.95052e6i	−1.18115	−	1.18115i	−0.979447	−	0.201702i	$-0.935353\pi$
$307$	−1.95052e6	−	1.95052e6i	−1.18115	−	1.18115i	−0.201702	−	0.979447i	$-0.564647\pi$
$308$	529465.	+	529465.i	0.318025	+	0.318025i
$309$		0			0
$310$		0			0
$311$			1.92784e6i			1.13024i	0.825009	+	0.565120i	$0.191170\pi$
$311$			1.92784e6i			1.13024i	−0.825009	+	0.565120i	$0.808830\pi$
$312$		0			0
$313$	64282.4	−	64282.4i	0.0370878	−	0.0370878i	−0.688320	−	0.725407i	$-0.741651\pi$
$313$	64282.4	−	64282.4i	0.0370878	−	0.0370878i	0.725407	+	0.688320i	$0.241651\pi$
$314$	538101.			0.307992
$315$		0			0
$316$	1.17440e6			0.661604
$317$	1.08190e6	−	1.08190e6i	0.604697	−	0.604697i	−0.336858	−	0.941555i	$-0.609364\pi$
$317$	1.08190e6	−	1.08190e6i	0.604697	−	0.604697i	0.941555	+	0.336858i	$0.109364\pi$
$318$		0			0
$319$			3.03395e6i			1.66929i
$320$		0			0
$321$		0			0
$322$	129193.	+	129193.i	0.0694381	+	0.0694381i
$323$	2391.63	+	2391.63i	0.00127552	+	0.00127552i
$324$		0			0
$325$		0			0
$326$		−	702449.i		−	0.366076i
$327$		0			0
$328$	−8783.58	+	8783.58i	−0.00450803	+	0.00450803i
$329$	−531777.			−0.270857
$330$		0			0
$331$	3.12047e6			1.56549			0.782744	−	0.622344i	$-0.213820\pi$
$331$	3.12047e6			1.56549			0.782744	+	0.622344i	$0.213820\pi$
$332$	−942826.	+	942826.i	−0.469447	+	0.469447i
$333$		0			0
$334$		−	129762.i		−	0.0636473i
$335$		0			0
$336$		0			0
$337$	−63979.7	−	63979.7i	−0.0306879	−	0.0306879i	0.691596	−	0.722284i	$-0.256908\pi$
$337$	−63979.7	−	63979.7i	−0.0306879	−	0.0306879i	−0.722284	+	0.691596i	$0.756908\pi$
$338$	465381.	+	465381.i	0.221573	+	0.221573i
$339$		0			0
$340$		0			0
$341$			1.77519e6i			0.826720i
$342$		0			0
$343$	1.43364e6	−	1.43364e6i	0.657969	−	0.657969i
$344$	−892820.			−0.406788
$345$		0			0
$346$	2.71166e6			1.21771
$347$	−1.23074e6	+	1.23074e6i	−0.548711	+	0.548711i	−0.926068	−	0.377357i	$-0.876833\pi$
$347$	−1.23074e6	+	1.23074e6i	−0.548711	+	0.548711i	0.377357	+	0.926068i	$0.376833\pi$
$348$		0			0
$349$		−	847301.i		−	0.372370i	−0.982515	−	0.186185i	$-0.940388\pi$
$349$		−	847301.i		−	0.372370i	0.982515	−	0.186185i	$-0.0596123\pi$
$350$		0			0
$351$		0			0
$352$	477704.	+	477704.i	0.205495	+	0.205495i
$353$	690970.	+	690970.i	0.295136	+	0.295136i	0.839105	−	0.543969i	$-0.183079\pi$
$353$	690970.	+	690970.i	0.295136	+	0.295136i	−0.543969	+	0.839105i	$0.683079\pi$
$354$		0			0
$355$		0			0
$356$		−	1.80442e6i		−	0.754592i
$357$		0			0
$358$	750838.	−	750838.i	0.309627	−	0.309627i
$359$	2.53332e6			1.03742			0.518709	−	0.854951i	$-0.326413\pi$
$359$	2.53332e6			1.03742			0.518709	+	0.854951i	$0.326413\pi$
$360$		0			0
$361$	2.47465e6			0.999414
$362$	2.10312e6	−	2.10312e6i	0.843513	−	0.843513i
$363$		0			0
$364$			830792.i			0.328654i
$365$		0			0
$366$		0			0
$367$	1.44127e6	+	1.44127e6i	0.558573	+	0.558573i	0.928901	−	0.370328i	$-0.120755\pi$
$367$	1.44127e6	+	1.44127e6i	0.558573	+	0.558573i	−0.370328	+	0.928901i	$0.620755\pi$
$368$	116562.	+	116562.i	0.0448683	+	0.0448683i
$369$		0			0
$370$		0			0
$371$			2.06577e6i			0.779198i
$372$		0			0
$373$	3.55217e6	−	3.55217e6i	1.32197	−	1.32197i	0.409793	−	0.912179i	$-0.365601\pi$
$373$	3.55217e6	−	3.55217e6i	1.32197	−	1.32197i	0.912179	−	0.409793i	$-0.134399\pi$
$374$	−234265.			−0.0866021
$375$		0			0
$376$	−479789.			−0.175017
$377$	−2.38030e6	+	2.38030e6i	−0.862540	+	0.862540i
$378$		0			0
$379$			2.00856e6i			0.718267i	0.933286	+	0.359133i	$0.116928\pi$
$379$			2.00856e6i			0.718267i	−0.933286	+	0.359133i	$0.883072\pi$
$380$		0			0
$381$		0			0
$382$	−1.52528e6	−	1.52528e6i	−0.534801	−	0.534801i
$383$	−575760.	−	575760.i	−0.200560	−	0.200560i	0.599680	−	0.800240i	$-0.295295\pi$
$383$	−575760.	−	575760.i	−0.200560	−	0.200560i	−0.800240	+	0.599680i	$0.795295\pi$
$384$		0			0
$385$		0			0
$386$		−	628507.i		−	0.214705i
$387$		0			0
$388$	1.40793e6	−	1.40793e6i	0.474790	−	0.474790i
$389$	−4.43267e6			−1.48522			−0.742612	−	0.669722i	$-0.766413\pi$
$389$	−4.43267e6			−1.48522			−0.742612	+	0.669722i	$0.766413\pi$
$390$		0			0
$391$	−57162.1			−0.0189089
$392$	532888.	−	532888.i	0.175154	−	0.175154i
$393$		0			0
$394$			637531.i			0.206900i
$395$		0			0
$396$		0			0
$397$	−337038.	−	337038.i	−0.107325	−	0.107325i	0.651405	−	0.758730i	$-0.274180\pi$
$397$	−337038.	−	337038.i	−0.107325	−	0.107325i	−0.758730	+	0.651405i	$0.774180\pi$
$398$	928851.	+	928851.i	0.293926	+	0.293926i
$399$		0			0
$400$		0			0
$401$			3.35110e6i			1.04070i	0.853952	+	0.520352i	$0.174199\pi$
$401$			3.35110e6i			1.04070i	−0.853952	+	0.520352i	$0.825801\pi$
$402$		0			0
$403$	−1.39274e6	+	1.39274e6i	−0.427176	+	0.427176i
$404$	−207795.			−0.0633407
$405$		0			0
$406$	−1.30483e6			−0.392860
$407$	980889.	−	980889.i	0.293517	−	0.293517i
$408$		0			0
$409$		−	3.51691e6i		−	1.03957i	−0.854298	−	0.519784i	$-0.826013\pi$
$409$		−	3.51691e6i		−	1.03957i	0.854298	−	0.519784i	$-0.173987\pi$
$410$		0			0
$411$		0			0
$412$	2.35948e6	+	2.35948e6i	0.684816	+	0.684816i
$413$	934364.	+	934364.i	0.269551	+	0.269551i
$414$		0			0
$415$		0			0
$416$			749572.i			0.212364i
$417$		0			0
$418$	71097.5	−	71097.5i	0.0199028	−	0.0199028i
$419$	−6.15249e6			−1.71205			−0.856024	−	0.516937i	$-0.827072\pi$
$419$	−6.15249e6			−1.71205			−0.856024	+	0.516937i	$0.827072\pi$
$420$		0			0
$421$	−2.64403e6			−0.727044			−0.363522	−	0.931586i	$-0.618426\pi$
$421$	−2.64403e6			−0.727044			−0.363522	+	0.931586i	$0.618426\pi$
$422$	1.83212e6	−	1.83212e6i	0.500811	−	0.500811i
$423$		0			0
$424$			1.86382e6i			0.503487i
$425$		0			0
$426$		0			0
$427$	−2.24814e6	−	2.24814e6i	−0.596698	−	0.596698i
$428$	1.05486e6	+	1.05486e6i	0.278345	+	0.278345i
$429$		0			0
$430$		0			0
$431$			823737.i			0.213597i	0.994281	+	0.106799i	$0.0340600\pi$
$431$			823737.i			0.213597i	−0.994281	+	0.106799i	$0.965940\pi$
$432$		0			0
$433$	2.17513e6	−	2.17513e6i	0.557525	−	0.557525i	−0.371077	−	0.928602i	$-0.621011\pi$
$433$	2.17513e6	−	2.17513e6i	0.557525	−	0.557525i	0.928602	+	0.371077i	$0.121011\pi$
$434$	−763466.			−0.194565
$435$		0			0
$436$	−2.90633e6			−0.732198
$437$	17348.2	−	17348.2i	0.00434561	−	0.00434561i
$438$		0			0
$439$			3.59369e6i			0.889978i	0.895536	+	0.444989i	$0.146793\pi$
$439$			3.59369e6i			0.889978i	−0.895536	+	0.444989i	$0.853207\pi$
$440$		0			0
$441$		0			0
$442$	−183795.	−	183795.i	−0.0447483	−	0.0447483i
$443$	−1.31093e6	−	1.31093e6i	−0.317373	−	0.317373i	0.530384	−	0.847757i	$-0.322048\pi$
$443$	−1.31093e6	−	1.31093e6i	−0.317373	−	0.317373i	−0.847757	+	0.530384i	$0.822048\pi$
$444$		0			0
$445$		0			0
$446$			2.04116e6i			0.485892i
$447$		0			0
$448$	−205449.	+	205449.i	−0.0483625	+	0.0483625i
$449$	1.25573e6			0.293954			0.146977	−	0.989140i	$-0.453046\pi$
$449$	1.25573e6			0.293954			0.146977	+	0.989140i	$0.453046\pi$
$450$		0			0
$451$	−128050.			−0.0296441
$452$	−611564.	+	611564.i	−0.140798	+	0.140798i
$453$		0			0
$454$		−	5.75824e6i		−	1.31114i
$455$		0			0
$456$		0			0
$457$	6.00853e6	+	6.00853e6i	1.34579	+	1.34579i	0.890178	+	0.455614i	$0.150580\pi$
$457$	6.00853e6	+	6.00853e6i	1.34579	+	1.34579i	0.455614	+	0.890178i	$0.349420\pi$
$458$	−2.78463e6	−	2.78463e6i	−0.620303	−	0.620303i
$459$		0			0
$460$		0			0
$461$		−	4.47946e6i		−	0.981687i	−0.871248	−	0.490844i	$-0.836689\pi$
$461$		−	4.47946e6i		−	0.981687i	0.871248	−	0.490844i	$-0.163311\pi$
$462$		0			0
$463$	78368.4	−	78368.4i	0.0169898	−	0.0169898i	−0.698561	−	0.715551i	$-0.746176\pi$
$463$	78368.4	−	78368.4i	0.0169898	−	0.0169898i	0.715551	+	0.698561i	$0.246176\pi$
$464$	−1.17726e6			−0.253851
$465$		0			0
$466$	5.88272e6			1.25491
$467$	2.36197e6	−	2.36197e6i	0.501167	−	0.501167i	−0.410633	−	0.911801i	$-0.634692\pi$
$467$	2.36197e6	−	2.36197e6i	0.501167	−	0.501167i	0.911801	+	0.410633i	$0.134692\pi$
$468$		0			0
$469$		−	2.57941e6i		−	0.541487i
$470$		0			0
$471$		0			0
$472$	843019.	+	843019.i	0.174174	+	0.174174i
$473$	−6.50793e6	−	6.50793e6i	−1.33749	−	1.33749i
$474$		0			0
$475$		0			0
$476$		−	100752.i		−	0.0203815i
$477$		0			0
$478$	−2.29261e6	+	2.29261e6i	−0.458946	+	0.458946i
$479$	3.71979e6			0.740763			0.370382	−	0.928880i	$-0.379227\pi$
$479$	3.71979e6			0.740763			0.370382	+	0.928880i	$0.379227\pi$
$480$		0			0
$481$	1.53913e6			0.303328
$482$	−3.84112e6	+	3.84112e6i	−0.753078	+	0.753078i
$483$		0			0
$484$			4.38732e6i			0.851306i
$485$		0			0
$486$		0			0
$487$	−2.63171e6	−	2.63171e6i	−0.502823	−	0.502823i	0.409491	−	0.912314i	$-0.365706\pi$
$487$	−2.63171e6	−	2.63171e6i	−0.502823	−	0.502823i	−0.912314	+	0.409491i	$0.865706\pi$
$488$	−2.02836e6	−	2.02836e6i	−0.385563	−	0.385563i
$489$		0			0
$490$		0			0
$491$		−	5.67262e6i		−	1.06189i	−0.847406	−	0.530946i	$-0.821837\pi$
$491$		−	5.67262e6i		−	1.06189i	0.847406	−	0.530946i	$-0.178163\pi$
$492$		0			0
$493$	288664.	−	288664.i	0.0534904	−	0.0534904i
$494$	111560.			0.0205680
$495$		0			0
$496$	−688828.			−0.125721
$497$	3.08037e6	−	3.08037e6i	0.559387	−	0.559387i
$498$		0			0
$499$		−	155951.i		−	0.0280373i	−0.999902	−	0.0140186i	$-0.995538\pi$
$499$		−	155951.i		−	0.0280373i	0.999902	−	0.0140186i	$-0.00446242\pi$
$500$		0			0
$501$		0			0
$502$	−499702.	−	499702.i	−0.0885017	−	0.0885017i
$503$	185072.	+	185072.i	0.0326152	+	0.0326152i	0.723226	−	0.690611i	$-0.242658\pi$
$503$	185072.	+	185072.i	0.0326152	+	0.0326152i	−0.690611	+	0.723226i	$0.742658\pi$
$504$		0			0
$505$		0			0
$506$			1.69929e6i			0.295047i
$507$		0			0
$508$	−1.15636e6	+	1.15636e6i	−0.198807	+	0.198807i
$509$	−2.07656e6			−0.355263			−0.177632	−	0.984097i	$-0.556844\pi$
$509$	−2.07656e6			−0.355263			−0.177632	+	0.984097i	$0.556844\pi$
$510$		0			0
$511$	−2.73312e6			−0.463027
$512$	−185364.	+	185364.i	−0.0312500	+	0.0312500i
$513$		0			0
$514$		−	6.49825e6i		−	1.08490i
$515$		0			0
$516$		0			0
$517$	−3.49727e6	−	3.49727e6i	−0.575444	−	0.575444i
$518$	421857.	+	421857.i	0.0690781	+	0.0690781i
$519$		0			0
$520$		0			0
$521$		−	4.46917e6i		−	0.721328i	−0.932696	−	0.360664i	$-0.882550\pi$
$521$		−	4.46917e6i		−	0.721328i	0.932696	−	0.360664i	$-0.117450\pi$
$522$		0			0
$523$	−4.83883e6	+	4.83883e6i	−0.773545	+	0.773545i	−0.978724	−	0.205179i	$-0.934222\pi$
$523$	−4.83883e6	+	4.83883e6i	−0.773545	+	0.773545i	0.205179	+	0.978724i	$0.434222\pi$
$524$	−2.61147e6			−0.415486
$525$		0			0
$526$	−4.73952e6			−0.746914
$527$	168900.	−	168900.i	0.0264913	−	0.0264913i
$528$		0			0
$529$		−	6.02171e6i		−	0.935579i
$530$		0			0
$531$		0			0
$532$	30577.3	+	30577.3i	0.00468404	+	0.00468404i
$533$	−100463.	−	100463.i	−0.0153175	−	0.0153175i
$534$		0			0
$535$		0			0
$536$		−	2.32724e6i		−	0.349888i
$537$		0			0
$538$	679919.	−	679919.i	0.101275	−	0.101275i
$539$	7.76863e6			1.15179
$540$		0			0
$541$	−8.63652e6			−1.26866			−0.634330	−	0.773062i	$-0.718724\pi$
$541$	−8.63652e6			−1.26866			−0.634330	+	0.773062i	$0.718724\pi$
$542$	74493.3	−	74493.3i	0.0108923	−	0.0108923i
$543$		0			0
$544$		−	90902.1i		−	0.0131697i
$545$		0			0
$546$		0			0
$547$	6.40411e6	+	6.40411e6i	0.915146	+	0.915146i	0.996671	−	0.0815255i	$-0.0259792\pi$
$547$	6.40411e6	+	6.40411e6i	0.915146	+	0.915146i	−0.0815255	+	0.996671i	$0.525979\pi$
$548$	486652.	+	486652.i	0.0692257	+	0.0692257i
$549$		0			0
$550$		0			0
$551$			175214.i			0.0245862i
$552$		0			0
$553$	−3.68163e6	+	3.68163e6i	−0.511949	+	0.511949i
$554$	1.00993e7			1.39804
$555$		0			0
$556$	−5.50274e6			−0.754905
$557$	4.41693e6	−	4.41693e6i	0.603230	−	0.603230i	−0.337938	−	0.941168i	$-0.609730\pi$
$557$	4.41693e6	−	4.41693e6i	0.603230	−	0.603230i	0.941168	+	0.337938i	$0.109730\pi$
$558$		0			0
$559$		−	1.02117e7i		−	1.38219i
$560$		0			0
$561$		0			0
$562$	−5.92110e6	−	5.92110e6i	−0.790790	−	0.790790i
$563$	6.69896e6	+	6.69896e6i	0.890711	+	0.890711i	0.994590	−	0.103879i	$-0.0331256\pi$
$563$	6.69896e6	+	6.69896e6i	0.890711	+	0.890711i	−0.103879	+	0.994590i	$0.533126\pi$
$564$		0			0
$565$		0			0
$566$			5.68391e6i			0.745772i
$567$		0			0
$568$	2.77923e6	−	2.77923e6i	0.361454	−	0.361454i
$569$	5.55048e6			0.718704			0.359352	−	0.933202i	$-0.382998\pi$
$569$	5.55048e6			0.718704			0.359352	+	0.933202i	$0.382998\pi$
$570$		0			0
$571$	−7.99758e6			−1.02652			−0.513261	−	0.858233i	$-0.671563\pi$
$571$	−7.99758e6			−1.02652			−0.513261	+	0.858233i	$0.671563\pi$
$572$	−5.46377e6	+	5.46377e6i	−0.698236	+	0.698236i
$573$		0			0
$574$		−	55071.3i		−	0.00697663i
$575$		0			0
$576$		0			0
$577$	1.94056e6	+	1.94056e6i	0.242654	+	0.242654i	0.817947	−	0.575293i	$-0.195112\pi$
$577$	1.94056e6	+	1.94056e6i	0.242654	+	0.242654i	−0.575293	+	0.817947i	$0.695112\pi$
$578$	−3.99367e6	−	3.99367e6i	−0.497225	−	0.497225i
$579$		0			0
$580$		0			0
$581$		−	5.91134e6i		−	0.726516i
$582$		0			0
$583$	−1.35857e7	+	1.35857e7i	−1.65543	+	1.65543i
$584$	−2.46592e6			−0.299190
$585$		0			0
$586$	−8.65121e6			−1.04072
$587$	−8.23325e6	+	8.23325e6i	−0.986225	+	0.986225i	−0.999906	−	0.0136812i	$-0.995645\pi$
$587$	−8.23325e6	+	8.23325e6i	−0.986225	+	0.986225i	0.0136812	+	0.999906i	$0.495645\pi$
$588$		0			0
$589$			102520.i			0.0121764i
$590$		0			0
$591$		0			0
$592$	380615.	+	380615.i	0.0446356	+	0.0446356i
$593$	−5.96277e6	−	5.96277e6i	−0.696324	−	0.696324i	0.267292	−	0.963616i	$-0.413871\pi$
$593$	−5.96277e6	−	5.96277e6i	−0.696324	−	0.696324i	−0.963616	+	0.267292i	$0.913871\pi$
$594$		0			0
$595$		0			0
$596$		−	2.78569e6i		−	0.321230i
$597$		0			0
$598$	−1.33319e6	+	1.33319e6i	−0.152454	+	0.152454i
$599$	−1.96839e6			−0.224153			−0.112077	−	0.993700i	$-0.535750\pi$
$599$	−1.96839e6			−0.224153			−0.112077	+	0.993700i	$0.535750\pi$
$600$		0			0
$601$	1.21707e7			1.37445			0.687226	−	0.726443i	$-0.258828\pi$
$601$	1.21707e7			1.37445			0.687226	+	0.726443i	$0.258828\pi$
$602$	2.79890e6	−	2.79890e6i	0.314772	−	0.314772i
$603$		0			0
$604$			5.99380e6i			0.668513i
$605$		0			0
$606$		0			0
$607$	2.50153e6	+	2.50153e6i	0.275571	+	0.275571i	0.831338	−	0.555767i	$-0.187575\pi$
$607$	2.50153e6	+	2.50153e6i	0.275571	+	0.275571i	−0.555767	+	0.831338i	$0.687575\pi$
$608$	27588.0	+	27588.0i	0.00302665	+	0.00302665i
$609$		0			0
$610$		0			0
$611$		−	5.48762e6i		−	0.594677i
$612$		0			0
$613$	−7.43814e6	+	7.43814e6i	−0.799490	+	0.799490i	−0.983015	−	0.183525i	$-0.941249\pi$
$613$	−7.43814e6	+	7.43814e6i	−0.799490	+	0.799490i	0.183525	+	0.983015i	$0.441249\pi$
$614$	1.10338e7			1.18115
$615$		0			0
$616$	−2.99511e6			−0.318025
$617$	−3.13725e6	+	3.13725e6i	−0.331769	+	0.331769i	−0.853258	−	0.521489i	$-0.825377\pi$
$617$	−3.13725e6	+	3.13725e6i	−0.331769	+	0.331769i	0.521489	+	0.853258i	$0.325377\pi$
$618$		0			0
$619$		−	6.26468e6i		−	0.657161i	−0.944476	−	0.328581i	$-0.893430\pi$
$619$		−	6.26468e6i		−	0.657161i	0.944476	−	0.328581i	$-0.106570\pi$
$620$		0			0
$621$		0			0
$622$	−5.45276e6	−	5.45276e6i	−0.565120	−	0.565120i
$623$	5.65667e6	+	5.65667e6i	0.583903	+	0.583903i
$624$		0			0
$625$		0			0
$626$			363636.i			0.0370878i
$627$		0			0
$628$	−1.52198e6	+	1.52198e6i	−0.153996	+	0.153996i
$629$	−186653.			−0.0188109
$630$		0			0
$631$	2.70818e6			0.270772			0.135386	−	0.990793i	$-0.456773\pi$
$631$	2.70818e6			0.270772			0.135386	+	0.990793i	$0.456773\pi$
$632$	−3.32170e6	+	3.32170e6i	−0.330802	+	0.330802i
$633$		0			0
$634$			6.12014e6i			0.604697i
$635$		0			0
$636$		0			0
$637$	6.09494e6	+	6.09494e6i	0.595142	+	0.595142i
$638$	−8.58129e6	−	8.58129e6i	−0.834644	−	0.834644i
$639$		0			0
$640$		0			0
$641$		−	1.42435e7i		−	1.36921i	−0.728914	−	0.684605i	$-0.759975\pi$
$641$		−	1.42435e7i		−	1.36921i	0.728914	−	0.684605i	$-0.240025\pi$
$642$		0			0
$643$	−1.93862e6	+	1.93862e6i	−0.184912	+	0.184912i	−0.793492	−	0.608580i	$-0.791739\pi$
$643$	−1.93862e6	+	1.93862e6i	−0.184912	+	0.184912i	0.608580	+	0.793492i	$0.291739\pi$
$644$	−730824.			−0.0694381
$645$		0			0
$646$	−13529.1			−0.00127552
$647$	−856023.	+	856023.i	−0.0803942	+	0.0803942i	−0.746160	−	0.665766i	$-0.768105\pi$
$647$	−856023.	+	856023.i	−0.0803942	+	0.0803942i	0.665766	+	0.746160i	$0.268105\pi$
$648$		0			0
$649$			1.22898e7i			1.14534i
$650$		0			0
$651$		0			0
$652$	1.98683e6	+	1.98683e6i	0.183038	+	0.183038i
$653$	−5.94490e6	−	5.94490e6i	−0.545584	−	0.545584i	0.379576	−	0.925160i	$-0.376070\pi$
$653$	−5.94490e6	−	5.94490e6i	−0.545584	−	0.545584i	−0.925160	+	0.379576i	$0.876070\pi$
$654$		0			0
$655$		0			0
$656$		−	49687.4i		−	0.00450803i
$657$		0			0
$658$	1.50409e6	−	1.50409e6i	0.135428	−	0.135428i
$659$	2.00924e7			1.80226			0.901130	−	0.433548i	$-0.142739\pi$
$659$	2.00924e7			1.80226			0.901130	+	0.433548i	$0.142739\pi$
$660$		0			0
$661$	−1.49344e7			−1.32949			−0.664745	−	0.747071i	$-0.731460\pi$
$661$	−1.49344e7			−1.32949			−0.664745	+	0.747071i	$0.731460\pi$
$662$	−8.82601e6	+	8.82601e6i	−0.782744	+	0.782744i
$663$		0			0
$664$		−	5.33343e6i		−	0.469447i
$665$		0			0
$666$		0			0
$667$	−2.09389e6	−	2.09389e6i	−0.182238	−	0.182238i
$668$	367021.	+	367021.i	0.0318236	+	0.0318236i
$669$		0			0
$670$		0			0
$671$		−	2.95702e7i		−	2.53541i
$672$		0			0
$673$	−5.94402e6	+	5.94402e6i	−0.505875	+	0.505875i	−0.913257	−	0.407383i	$-0.866441\pi$
$673$	−5.94402e6	+	5.94402e6i	−0.505875	+	0.505875i	0.407383	+	0.913257i	$0.366441\pi$
$674$	361924.			0.0306879
$675$		0			0
$676$	−2.63259e6			−0.221573
$677$	1.68563e7	−	1.68563e7i	1.41348	−	1.41348i	0.684014	−	0.729469i	$-0.260233\pi$
$677$	1.68563e7	−	1.68563e7i	1.41348	−	1.41348i	0.729469	−	0.684014i	$-0.239767\pi$
$678$		0			0
$679$			8.82744e6i			0.734785i
$680$		0			0
$681$		0			0
$682$	−5.02099e6	−	5.02099e6i	−0.413360	−	0.413360i
$683$	1.34538e7	+	1.34538e7i	1.10355	+	1.10355i	0.993979	+	0.109574i	$0.0349486\pi$
$683$	1.34538e7	+	1.34538e7i	1.10355	+	1.10355i	0.109574	+	0.993979i	$0.465051\pi$
$684$		0			0
$685$		0			0
$686$			8.10990e6i			0.657969i
$687$		0			0
$688$	2.52528e6	−	2.52528e6i	0.203394	−	0.203394i
$689$	−2.13175e7			−1.71076
$690$		0			0
$691$	2.73492e6			0.217896			0.108948	−	0.994047i	$-0.465252\pi$
$691$	2.73492e6			0.217896			0.108948	+	0.994047i	$0.465252\pi$
$692$	−7.66974e6	+	7.66974e6i	−0.608857	+	0.608857i
$693$		0			0
$694$		−	6.96214e6i		−	0.548711i
$695$		0			0
$696$		0			0
$697$	12183.3	+	12183.3i	0.000949912	+	0.000949912i
$698$	2.39653e6	+	2.39653e6i	0.186185	+	0.186185i
$699$		0			0
$700$		0			0
$701$			1.80427e7i			1.38678i	0.720563	+	0.693390i	$0.243884\pi$
$701$			1.80427e7i			1.38678i	−0.720563	+	0.693390i	$0.756116\pi$
$702$		0			0
$703$	56647.6	−	56647.6i	0.00432308	−	0.00432308i
$704$	−2.70230e6			−0.205495
$705$		0			0
$706$	−3.90871e6			−0.295136
$707$	651418.	−	651418.i	0.0490130	−	0.0490130i
$708$		0			0
$709$		−	1.11791e7i		−	0.835200i	−0.908631	−	0.417600i	$-0.862871\pi$
$709$		−	1.11791e7i		−	0.835200i	0.908631	−	0.417600i	$-0.137129\pi$
$710$		0			0
$711$		0			0
$712$	5.10366e6	+	5.10366e6i	0.377296	+	0.377296i
$713$	−1.22515e6	−	1.22515e6i	−0.0902539	−	0.0902539i
$714$		0			0
$715$		0			0
$716$			4.24738e6i			0.309627i
$717$		0			0
$718$	−7.16531e6	+	7.16531e6i	−0.518709	+	0.518709i
$719$	6.75900e6			0.487596			0.243798	−	0.969826i	$-0.421607\pi$
$719$	6.75900e6			0.487596			0.243798	+	0.969826i	$0.421607\pi$
$720$		0			0
$721$	−1.47935e7			−1.05982
$722$	−6.99936e6	+	6.99936e6i	−0.499707	+	0.499707i
$723$		0			0
$724$			1.18970e7i			0.843513i
$725$		0			0
$726$		0			0
$727$	−389019.	−	389019.i	−0.0272982	−	0.0272982i	0.693326	−	0.720624i	$-0.256145\pi$
$727$	−389019.	−	389019.i	−0.0272982	−	0.0272982i	−0.720624	+	0.693326i	$0.756145\pi$
$728$	−2.34984e6	−	2.34984e6i	−0.164327	−	0.164327i
$729$		0			0
$730$		0			0
$731$			1.23839e6i			0.0857165i
$732$		0			0
$733$	−268610.	+	268610.i	−0.0184655	+	0.0184655i	−0.716279	−	0.697814i	$-0.754156\pi$
$733$	−268610.	+	268610.i	−0.0184655	+	0.0184655i	0.697814	+	0.716279i	$0.254156\pi$
$734$	−8.15305e6			−0.558573
$735$		0			0
$736$	−659377.			−0.0448683
$737$	1.69637e7	−	1.69637e7i	1.15041	−	1.15041i
$738$		0			0
$739$			8.34926e6i			0.562389i	0.959651	+	0.281195i	$0.0907306\pi$
$739$			8.34926e6i			0.562389i	−0.959651	+	0.281195i	$0.909269\pi$
$740$		0			0
$741$		0			0
$742$	−5.84288e6	−	5.84288e6i	−0.389599	−	0.389599i
$743$	9.91181e6	+	9.91181e6i	0.658690	+	0.658690i	0.955070	−	0.296380i	$-0.0957795\pi$
$743$	9.91181e6	+	9.91181e6i	0.658690	+	0.658690i	−0.296380	+	0.955070i	$0.595780\pi$
$744$		0			0
$745$		0			0
$746$			2.00941e7i			1.32197i
$747$		0			0
$748$	662602.	−	662602.i	0.0433011	−	0.0433011i
$749$	−6.61374e6			−0.430767
$750$		0			0
$751$	1.00908e7			0.652872			0.326436	−	0.945219i	$-0.394152\pi$
$751$	1.00908e7			0.652872			0.326436	+	0.945219i	$0.394152\pi$
$752$	1.35705e6	−	1.35705e6i	0.0875086	−	0.0875086i
$753$		0			0
$754$		−	1.34650e7i		−	0.862540i
$755$		0			0
$756$		0			0
$757$	1.57442e7	+	1.57442e7i	0.998577	+	0.998577i	0.999999	−	0.00142163i	$-0.000452520\pi$
$757$	1.57442e7	+	1.57442e7i	0.998577	+	0.998577i	−0.00142163	+	0.999999i	$0.500453\pi$
$758$	−5.68105e6	−	5.68105e6i	−0.359133	−	0.359133i
$759$		0			0
$760$		0			0
$761$		−	3.65317e6i		−	0.228669i	−0.993442	−	0.114335i	$-0.963526\pi$
$761$		−	3.65317e6i		−	0.228669i	0.993442	−	0.114335i	$-0.0364736\pi$
$762$		0			0
$763$	9.11106e6	−	9.11106e6i	0.566575	−	0.566575i
$764$	8.62831e6			0.534801
$765$		0			0
$766$	3.25699e6			0.200560
$767$	−9.64208e6	+	9.64208e6i	−0.591810	+	0.591810i
$768$		0			0
$769$		−	1.75833e7i		−	1.07222i	−0.844147	−	0.536112i	$-0.819893\pi$
$769$		−	1.75833e7i		−	1.07222i	0.844147	−	0.536112i	$-0.180107\pi$
$770$		0			0
$771$		0			0
$772$	1.77768e6	+	1.77768e6i	0.107352	+	0.107352i
$773$	−1.32600e6	−	1.32600e6i	−0.0798171	−	0.0798171i	0.666071	−	0.745888i	$-0.267975\pi$
$773$	−1.32600e6	−	1.32600e6i	−0.0798171	−	0.0798171i	−0.745888	+	0.666071i	$0.767975\pi$
$774$		0			0
$775$		0			0
$776$			7.96445e6i			0.474790i
$777$		0			0
$778$	1.25375e7	−	1.25375e7i	0.742612	−	0.742612i
$779$	−7395.07			−0.000436615
$780$		0			0
$781$	4.05166e7			2.37687
$782$	161679.	−	161679.i	0.00945444	−	0.00945444i
$783$		0			0
$784$			3.01447e6i			0.175154i
$785$		0			0
$786$		0			0
$787$	−5.03092e6	−	5.03092e6i	−0.289541	−	0.289541i	0.547357	−	0.836899i	$-0.315634\pi$
$787$	−5.03092e6	−	5.03092e6i	−0.289541	−	0.289541i	−0.836899	+	0.547357i	$0.815634\pi$
$788$	−1.80321e6	−	1.80321e6i	−0.103450	−	0.103450i
$789$		0			0
$790$		0			0
$791$		−	3.83439e6i		−	0.217899i
$792$		0			0
$793$	2.31995e7	−	2.31995e7i	1.31007	−	1.31007i
$794$	1.90658e6			0.107325
$795$		0			0
$796$	−5.25438e6			−0.293926
$797$	−1.31915e7	+	1.31915e7i	−0.735611	+	0.735611i	−0.971725	−	0.236114i	$-0.924126\pi$
$797$	−1.31915e7	+	1.31915e7i	−0.735611	+	0.735611i	0.236114	+	0.971725i	$0.424126\pi$
$798$		0			0
$799$			665495.i			0.0368789i
$800$		0			0
$801$		0			0
$802$	−9.47835e6	−	9.47835e6i	−0.520352	−	0.520352i
$803$	−1.79746e7	−	1.79746e7i	−0.983716	−	0.983716i
$804$		0			0
$805$		0			0
$806$		−	7.87852e6i		−	0.427176i
$807$		0			0
$808$	587734.	−	587734.i	0.0316703	−	0.0316703i
$809$	−4.79366e6			−0.257511			−0.128756	−	0.991676i	$-0.541098\pi$
$809$	−4.79366e6			−0.257511			−0.128756	+	0.991676i	$0.541098\pi$
$810$		0			0
$811$	2.44630e7			1.30604			0.653021	−	0.757340i	$-0.273502\pi$
$811$	2.44630e7			1.30604			0.653021	+	0.757340i	$0.273502\pi$
$812$	3.69061e6	−	3.69061e6i	0.196430	−	0.196430i
$813$		0			0
$814$			5.54874e6i			0.293517i
$815$		0			0
$816$		0			0
$817$	−375841.	−	375841.i	−0.0196992	−	0.0196992i
$818$	9.94731e6	+	9.94731e6i	0.519784	+	0.519784i
$819$		0			0
$820$		0			0
$821$			1.42838e7i			0.739581i	0.929115	+	0.369790i	$0.120570\pi$
$821$			1.42838e7i			0.739581i	−0.929115	+	0.369790i	$0.879430\pi$
$822$		0			0
$823$	−1.56583e6	+	1.56583e6i	−0.0805832	+	0.0805832i	−0.746250	−	0.665666i	$-0.768147\pi$
$823$	−1.56583e6	+	1.56583e6i	−0.0805832	+	0.0805832i	0.665666	+	0.746250i	$0.268147\pi$
$824$	−1.33472e7			−0.684816
$825$		0			0
$826$	−5.28556e6			−0.269551
$827$	2.18679e7	−	2.18679e7i	1.11184	−	1.11184i	0.118939	−	0.992902i	$-0.462051\pi$
$827$	2.18679e7	−	2.18679e7i	1.11184	−	1.11184i	0.992902	−	0.118939i	$-0.0379492\pi$
$828$		0			0
$829$		−	5.85804e6i		−	0.296050i	−0.988984	−	0.148025i	$-0.952708\pi$
$829$		−	5.85804e6i		−	0.296050i	0.988984	−	0.148025i	$-0.0472917\pi$
$830$		0			0
$831$		0			0
$832$	−2.12011e6	−	2.12011e6i	−0.106182	−	0.106182i
$833$	−739145.	−	739145.i	−0.0369077	−	0.0369077i
$834$		0			0
$835$		0			0
$836$			402188.i			0.0199028i
$837$		0			0
$838$	1.74019e7	−	1.74019e7i	0.856024	−	0.856024i
$839$	−1.40555e7			−0.689354			−0.344677	−	0.938721i	$-0.612012\pi$
$839$	−1.40555e7			−0.689354			−0.344677	+	0.938721i	$0.612012\pi$
$840$		0			0
$841$	636798.			0.0310464
$842$	7.47844e6	−	7.47844e6i	0.363522	−	0.363522i
$843$		0			0
$844$			1.03641e7i			0.500811i
$845$		0			0
$846$		0			0
$847$	−1.37538e7	−	1.37538e7i	−0.658741	−	0.658741i
$848$	−5.27167e6	−	5.27167e6i	−0.251744	−	0.251744i
$849$		0			0
$850$		0			0
$851$			1.35393e6i			0.0640872i
$852$		0			0
$853$	1.46482e7	−	1.46482e7i	0.689305	−	0.689305i	−0.272773	−	0.962078i	$-0.587941\pi$
$853$	1.46482e7	−	1.46482e7i	0.689305	−	0.689305i	0.962078	+	0.272773i	$0.0879409\pi$
$854$	1.27174e7			0.596698
$855$		0			0
$856$	−5.96717e6			−0.278345
$857$	2.43742e7	−	2.43742e7i	1.13365	−	1.13365i	0.144085	−	0.989565i	$-0.453976\pi$
$857$	2.43742e7	−	2.43742e7i	1.13365	−	1.13365i	0.989565	−	0.144085i	$-0.0460238\pi$
$858$		0			0
$859$			2.36043e7i			1.09146i	0.837961	+	0.545731i	$0.183748\pi$
$859$			2.36043e7i			1.09146i	−0.837961	+	0.545731i	$0.816252\pi$
$860$		0			0
$861$		0			0
$862$	−2.32988e6	−	2.32988e6i	−0.106799	−	0.106799i
$863$	1.61979e7	+	1.61979e7i	0.740343	+	0.740343i	0.972644	−	0.232301i	$-0.0746255\pi$
$863$	1.61979e7	+	1.61979e7i	0.740343	+	0.740343i	−0.232301	+	0.972644i	$0.574626\pi$
$864$		0			0
$865$		0			0
$866$			1.23044e7i			0.557525i
$867$		0			0
$868$	2.15941e6	−	2.15941e6i	0.0972827	−	0.0972827i
$869$	−4.84250e7			−2.17530
$870$		0			0
$871$	2.66180e7			1.18886
$872$	8.22034e6	−	8.22034e6i	0.366099	−	0.366099i
$873$		0			0
$874$			98136.3i			0.00434561i
$875$		0			0
$876$		0			0
$877$	7.08182e6	+	7.08182e6i	0.310918	+	0.310918i	0.845265	−	0.534347i	$-0.179443\pi$
$877$	7.08182e6	+	7.08182e6i	0.310918	+	0.310918i	−0.534347	+	0.845265i	$0.679443\pi$
$878$	−1.01645e7	−	1.01645e7i	−0.444989	−	0.444989i
$879$		0			0
$880$		0			0
$881$			2.30452e7i			1.00032i	0.865932	+	0.500162i	$0.166726\pi$
$881$			2.30452e7i			1.00032i	−0.865932	+	0.500162i	$0.833274\pi$
$882$		0			0
$883$	−1.99788e7	+	1.99788e7i	−0.862317	+	0.862317i	−0.991607	−	0.129290i	$-0.958730\pi$
$883$	−1.99788e7	+	1.99788e7i	−0.862317	+	0.862317i	0.129290	+	0.991607i	$0.458730\pi$
$884$	1.03970e6			0.0447483
$885$		0			0
$886$	7.41573e6			0.317373
$887$	−1.41730e7	+	1.41730e7i	−0.604858	+	0.604858i	−0.941598	−	0.336740i	$-0.890676\pi$
$887$	−1.41730e7	+	1.41730e7i	−0.604858	+	0.604858i	0.336740	+	0.941598i	$0.390676\pi$
$888$		0			0
$889$		−	7.25013e6i		−	0.307675i
$890$		0			0
$891$		0			0
$892$	−5.77327e6	−	5.77327e6i	−0.242946	−	0.242946i
$893$	−201972.	−	201972.i	−0.00847545	−	0.00847545i
$894$		0			0
$895$		0			0
$896$		−	1.16219e6i		−	0.0483625i
$897$		0			0
$898$	−3.55173e6	+	3.55173e6i	−0.146977	+	0.146977i
$899$	1.23739e7			0.510629
$900$		0			0
$901$	2.58522e6			0.106093
$902$	362181.	−	362181.i	0.0148221	−	0.0148221i
$903$		0			0
$904$		−	3.45953e6i		−	0.140798i
$905$		0			0
$906$		0			0
$907$	1.50979e7	+	1.50979e7i	0.609393	+	0.609393i	0.942787	−	0.333395i	$-0.108194\pi$
$907$	1.50979e7	+	1.50979e7i	0.609393	+	0.609393i	−0.333395	+	0.942787i	$0.608194\pi$
$908$	1.62868e7	+	1.62868e7i	0.655572	+	0.655572i
$909$		0			0
$910$		0			0
$911$			4.31847e7i			1.72399i	0.506920	+	0.861993i	$0.330784\pi$
$911$			4.31847e7i			1.72399i	−0.506920	+	0.861993i	$0.669216\pi$
$912$		0			0
$913$	3.88763e7	−	3.88763e7i	1.54351	−	1.54351i
$914$	−3.39894e7			−1.34579
$915$		0			0
$916$	1.57523e7			0.620303
$917$	8.18670e6	−	8.18670e6i	0.321504	−	0.321504i
$918$		0			0
$919$			2.65475e7i			1.03690i	0.855109	+	0.518448i	$0.173490\pi$
$919$			2.65475e7i			1.03690i	−0.855109	+	0.518448i	$0.826510\pi$
$920$		0			0
$921$		0			0
$922$	1.26698e7	+	1.26698e7i	0.490844	+	0.490844i
$923$	3.17876e7	+	3.17876e7i	1.22816	+	1.22816i
$924$		0			0
$925$		0			0
$926$			443318.i			0.0169898i
$927$		0			0
$928$	3.32981e6	−	3.32981e6i	0.126926	−	0.126926i
$929$	−9.56387e6			−0.363575			−0.181788	−	0.983338i	$-0.558188\pi$
$929$	−9.56387e6			−0.363575			−0.181788	+	0.983338i	$0.558188\pi$
$930$		0			0
$931$	448649.			0.0169641
$932$	−1.66389e7	+	1.66389e7i	−0.627457	+	0.627457i
$933$		0			0
$934$			1.33613e7i			0.501167i
$935$		0			0
$936$		0			0
$937$	−3.24994e6	−	3.24994e6i	−0.120928	−	0.120928i	0.644053	−	0.764981i	$-0.277252\pi$
$937$	−3.24994e6	−	3.24994e6i	−0.120928	−	0.120928i	−0.764981	+	0.644053i	$0.777252\pi$
$938$	7.29567e6	+	7.29567e6i	0.270744	+	0.270744i
$939$		0			0
$940$		0			0
$941$			331919.i			0.0122196i	0.999981	+	0.00610982i	$0.00194483\pi$
$941$			331919.i			0.0122196i	−0.999981	+	0.00610982i	$0.998055\pi$
$942$		0			0
$943$	88374.2	−	88374.2i	0.00323628	−	0.00323628i
$944$	−4.76883e6			−0.174174
$945$		0			0
$946$	3.68144e7			1.33749
$947$	−1.33791e6	+	1.33791e6i	−0.0484790	+	0.0484790i	−0.730931	−	0.682452i	$-0.760914\pi$
$947$	−1.33791e6	+	1.33791e6i	−0.0484790	+	0.0484790i	0.682452	+	0.730931i	$0.260914\pi$
$948$		0			0
$949$		−	2.82042e7i		−	1.01659i
$950$		0			0
$951$		0			0
$952$	284969.	+	284969.i	0.0101907	+	0.0101907i
$953$	−1.03531e7	−	1.03531e7i	−0.369265	−	0.369265i	0.497944	−	0.867209i	$-0.334089\pi$
$953$	−1.03531e7	−	1.03531e7i	−0.369265	−	0.369265i	−0.867209	+	0.497944i	$0.834089\pi$
$954$		0			0
$955$		0			0
$956$		−	1.29690e7i		−	0.458946i
$957$		0			0
$958$	−1.05212e7	+	1.05212e7i	−0.370382	+	0.370382i
$959$	−3.05121e6			−0.107134
$960$		0			0
$961$	−2.13891e7			−0.747109
$962$	−4.35331e6	+	4.35331e6i	−0.151664	+	0.151664i
$963$		0			0
$964$		−	2.17287e7i		−	0.753078i
$965$		0			0
$966$		0			0
$967$	−2.31470e7	−	2.31470e7i	−0.796027	−	0.796027i	0.186440	−	0.982466i	$-0.440305\pi$
$967$	−2.31470e7	−	2.31470e7i	−0.796027	−	0.796027i	−0.982466	+	0.186440i	$0.940305\pi$
$968$	−1.24092e7	−	1.24092e7i	−0.425653	−	0.425653i
$969$		0			0
$970$		0			0
$971$			2.76465e7i			0.941004i	0.882399	+	0.470502i	$0.155927\pi$
$971$			2.76465e7i			0.941004i	−0.882399	+	0.470502i	$0.844073\pi$
$972$		0			0
$973$	1.72506e7	−	1.72506e7i	0.584145	−	0.584145i
$974$	1.48872e7			0.502823
$975$		0			0
$976$	1.14741e7			0.385563
$977$	−2.29525e7	+	2.29525e7i	−0.769297	+	0.769297i	−0.977983	−	0.208686i	$-0.933081\pi$
$977$	−2.29525e7	+	2.29525e7i	−0.769297	+	0.769297i	0.208686	+	0.977983i	$0.433081\pi$
$978$		0			0
$979$			7.44031e7i			2.48104i
$980$		0			0
$981$		0			0
$982$	1.60446e7	+	1.60446e7i	0.530946	+	0.530946i
$983$	8.12183e6	+	8.12183e6i	0.268084	+	0.268084i	0.828328	−	0.560244i	$-0.189293\pi$
$983$	8.12183e6	+	8.12183e6i	0.268084	+	0.268084i	−0.560244	+	0.828328i	$0.689293\pi$
$984$		0			0
$985$		0			0
$986$			1.63293e6i			0.0534904i
$987$		0			0
$988$	−315540.	+	315540.i	−0.0102840	+	0.0102840i
$989$	8.98292e6			0.292030
$990$		0			0
$991$	−3.30343e6			−0.106851			−0.0534257	−	0.998572i	$-0.517014\pi$
$991$	−3.30343e6			−0.106851			−0.0534257	+	0.998572i	$0.517014\pi$
$992$	1.94830e6	−	1.94830e6i	0.0628603	−	0.0628603i
$993$		0			0
$994$			1.74252e7i			0.559387i
$995$		0			0
$996$		0			0
$997$	1.23928e7	+	1.23928e7i	0.394851	+	0.394851i	0.876412	−	0.481562i	$-0.159930\pi$
$997$	1.23928e7	+	1.23928e7i	0.394851	+	0.394851i	−0.481562	+	0.876412i	$0.659930\pi$
$998$	441095.	+	441095.i	0.0140186	+	0.0140186i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.f.107.3	✓	16
3.2	odd	2	inner	450.6.f.f.107.7	yes	16
5.2	odd	4		450.6.f.g.143.2	yes	16
5.3	odd	4	inner	450.6.f.f.143.7	yes	16
5.4	even	2		450.6.f.g.107.6	yes	16
15.2	even	4		450.6.f.g.143.6	yes	16
15.8	even	4	inner	450.6.f.f.143.3	yes	16
15.14	odd	2		450.6.f.g.107.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.f.f.107.3	✓	16	1.1	even	1	trivial
450.6.f.f.107.7	yes	16	3.2	odd	2	inner
450.6.f.f.143.3	yes	16	15.8	even	4	inner
450.6.f.f.143.7	yes	16	5.3	odd	4	inner
450.6.f.g.107.2	yes	16	15.14	odd	2
450.6.f.g.107.6	yes	16	5.4	even	2
450.6.f.g.143.2	yes	16	5.2	odd	4
450.6.f.g.143.6	yes	16	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(50.1584 + 50.1584i) q^{7} +(45.2548 + 45.2548i) q^{8} +O(q^{10})\)	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(50.1584 + 50.1584i) q^{7} +(45.2548 + 45.2548i) q^{8} +659.741i q^{11} +(-517.605 + 517.605i) q^{13} -283.739 q^{14} -256.000 q^{16} +(62.7710 - 62.7710i) q^{17} +38.1009i q^{19} +(-1866.03 - 1866.03i) q^{22} +(-455.322 - 455.322i) q^{23} -2928.02i q^{26} +(802.535 - 802.535i) q^{28} +4598.69 q^{29} +2690.73 q^{31} +(724.077 - 724.077i) q^{32} +355.086i q^{34} +(-1486.78 - 1486.78i) q^{37} +(-107.766 - 107.766i) q^{38} +194.091i q^{41} +(-9864.36 + 9864.36i) q^{43} +10555.9 q^{44} +2575.69 q^{46} +(-5300.97 + 5300.97i) q^{47} -11775.3i q^{49} +(8281.68 + 8281.68i) q^{52} +(20592.5 + 20592.5i) q^{53} +4539.82i q^{56} +(-13007.1 + 13007.1i) q^{58} +18628.3 q^{59} -44820.8 q^{61} +(-7610.55 + 7610.55i) q^{62} +4096.00i q^{64} +(-25712.6 - 25712.6i) q^{67} +(-1004.34 - 1004.34i) q^{68} -61412.8i q^{71} +(-27244.9 + 27244.9i) q^{73} +8410.49 q^{74} +609.615 q^{76} +(-33091.6 + 33091.6i) q^{77} +73400.0i q^{79} +(-548.974 - 548.974i) q^{82} +(-58926.6 - 58926.6i) q^{83} -55801.3i q^{86} +(-29856.5 + 29856.5i) q^{88} +112776. q^{89} -51924.5 q^{91} +(-7285.16 + 7285.16i) q^{92} -29986.8i q^{94} +(87995.6 + 87995.6i) q^{97} +(33305.5 + 33305.5i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 528 q^{7}+O(q^{10}) \) 16 * q - 528 * q^7	\( 16 q - 528 q^{7} + 192 q^{13} - 4096 q^{16} - 2688 q^{22} - 8448 q^{28} + 13024 q^{31} - 47328 q^{37} - 55440 q^{43} + 44544 q^{46} - 3072 q^{52} - 101184 q^{58} + 28400 q^{61} + 242256 q^{67} - 430944 q^{73} - 7168 q^{76} + 158208 q^{82} - 43008 q^{88} - 185472 q^{91} + 457152 q^{97}+O(q^{100}) \) 16 * q - 528 * q^7 + 192 * q^13 - 4096 * q^16 - 2688 * q^22 - 8448 * q^28 + 13024 * q^31 - 47328 * q^37 - 55440 * q^43 + 44544 * q^46 - 3072 * q^52 - 101184 * q^58 + 28400 * q^61 + 242256 * q^67 - 430944 * q^73 - 7168 * q^76 + 158208 * q^82 - 43008 * q^88 - 185472 * q^91 + 457152 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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