LMFDB - Embedded newform 225.6.f.b.143.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(107,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 19978x^{16} + 11248353x^{12} + 1386043201x^{8} + 1477627450x^{4} + 332150625 $ x^20 + 19978x^16 + 11248353x^12 + 1386043201x^8 + 1477627450x^4 + 332150625
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{29}\cdot 3^{24}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.3
Root			$-3.15495 + 3.15495i$ of defining polynomial
Character	$\chi$	$=$	225.143
Dual form			225.6.f.b.107.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.49760 - 3.49760i) q^{2} -7.53358i q^{4} +(-24.1207 + 24.1207i) q^{7} +(-138.273 + 138.273i) q^{8} +O(q^{10})$	\(q+(-3.49760 - 3.49760i) q^{2} -7.53358i q^{4} +(-24.1207 + 24.1207i) q^{7} +(-138.273 + 138.273i) q^{8} -173.562i q^{11} +(-597.849 - 597.849i) q^{13} +168.729 q^{14} +726.171 q^{16} +(734.740 + 734.740i) q^{17} +1622.15i q^{19} +(-607.051 + 607.051i) q^{22} +(-667.211 + 667.211i) q^{23} +4182.07i q^{26} +(181.715 + 181.715i) q^{28} +3698.77 q^{29} -9560.23 q^{31} +(1884.87 + 1884.87i) q^{32} -5139.65i q^{34} +(5758.92 - 5758.92i) q^{37} +(5673.62 - 5673.62i) q^{38} +16087.5i q^{41} +(-9780.29 - 9780.29i) q^{43} -1307.54 q^{44} +4667.28 q^{46} +(11110.0 + 11110.0i) q^{47} +15643.4i q^{49} +(-4503.94 + 4503.94i) q^{52} +(1013.67 - 1013.67i) q^{53} -6670.46i q^{56} +(-12936.8 - 12936.8i) q^{58} +46524.8 q^{59} +52196.0 q^{61} +(33437.9 + 33437.9i) q^{62} -36422.5i q^{64} +(31213.8 - 31213.8i) q^{67} +(5535.22 - 5535.22i) q^{68} +22440.4i q^{71} +(3589.53 + 3589.53i) q^{73} -40284.8 q^{74} +12220.6 q^{76} +(4186.44 + 4186.44i) q^{77} -17531.6i q^{79} +(56267.8 - 56267.8i) q^{82} +(71122.6 - 71122.6i) q^{83} +68415.1i q^{86} +(23998.9 + 23998.9i) q^{88} -48621.5 q^{89} +28841.0 q^{91} +(5026.49 + 5026.49i) q^{92} -77716.5i q^{94} +(7908.79 - 7908.79i) q^{97} +(54714.3 - 54714.3i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 152 q^{7}+O(q^{10}) $ 20 * q - 152 * q^7	$ 20 q - 152 q^{7} - 1844 q^{13} - 6280 q^{16} + 6512 q^{22} - 29728 q^{28} + 43120 q^{31} + 37516 q^{37} - 5408 q^{43} + 118720 q^{46} + 285256 q^{52} - 274752 q^{58} + 163360 q^{61} + 302704 q^{67} - 146324 q^{73} + 760800 q^{76} + 305744 q^{82} - 355296 q^{88} + 499120 q^{91} + 362524 q^{97}+O(q^{100}) $ 20 * q - 152 * q^7 - 1844 * q^13 - 6280 * q^16 + 6512 * q^22 - 29728 * q^28 + 43120 * q^31 + 37516 * q^37 - 5408 * q^43 + 118720 * q^46 + 285256 * q^52 - 274752 * q^58 + 163360 * q^61 + 302704 * q^67 - 146324 * q^73 + 760800 * q^76 + 305744 * q^82 - 355296 * q^88 + 499120 * q^91 + 362524 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−3.49760	−	3.49760i	−0.618294	−	0.618294i	0.326799	−	0.945094i	$-0.394030\pi$
$2$	−3.49760	−	3.49760i	−0.618294	−	0.618294i	−0.945094	+	0.326799i	$0.894030\pi$
$3$		0			0
$3$		0			0
$4$		−	7.53358i		−	0.235424i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−24.1207	+	24.1207i	−0.186056	+	0.186056i	−0.793989	−	0.607932i	$-0.791999\pi$
$7$	−24.1207	+	24.1207i	−0.186056	+	0.186056i	0.607932	+	0.793989i	$0.291999\pi$
$8$	−138.273	+	138.273i	−0.763856	+	0.763856i
$9$		0			0
$10$		0			0
$11$		−	173.562i		−	0.432487i	−0.976339	−	0.216244i	$-0.930619\pi$
$11$		−	173.562i		−	0.432487i	0.976339	−	0.216244i	$-0.0693805\pi$
$12$		0			0
$13$	−597.849	−	597.849i	−0.981145	−	0.981145i	0.0186808	−	0.999825i	$-0.494053\pi$
$13$	−597.849	−	597.849i	−0.981145	−	0.981145i	−0.999825	+	0.0186808i	$0.994053\pi$
$14$	168.729			0.230075
$15$		0			0
$16$	726.171			0.709151
$17$	734.740	+	734.740i	0.616611	+	0.616611i	0.944661	−	0.328050i	$-0.106391\pi$
$17$	734.740	+	734.740i	0.616611	+	0.616611i	−0.328050	+	0.944661i	$0.606391\pi$
$18$		0			0
$19$			1622.15i			1.03088i	0.856927	+	0.515438i	$0.172371\pi$
$19$			1622.15i			1.03088i	−0.856927	+	0.515438i	$0.827629\pi$
$20$		0			0
$21$		0			0
$22$	−607.051	+	607.051i	−0.267404	+	0.267404i
$23$	−667.211	+	667.211i	−0.262993	+	0.262993i	−0.826269	−	0.563276i	$-0.809541\pi$
$23$	−667.211	+	667.211i	−0.262993	+	0.262993i	0.563276	+	0.826269i	$0.309541\pi$
$24$		0			0
$25$		0			0
$26$			4182.07i			1.21327i
$27$		0			0
$28$	181.715	+	181.715i	0.0438022	+	0.0438022i
$29$	3698.77			0.816700			0.408350	−	0.912825i	$-0.366104\pi$
$29$	3698.77			0.816700			0.408350	+	0.912825i	$0.366104\pi$
$30$		0			0
$31$	−9560.23			−1.78675			−0.893376	−	0.449310i	$-0.851670\pi$
$31$	−9560.23			−1.78675			−0.893376	+	0.449310i	$0.851670\pi$
$32$	1884.87	+	1884.87i	0.325392	+	0.325392i
$33$		0			0
$34$		−	5139.65i		−	0.762494i
$35$		0			0
$36$		0			0
$37$	5758.92	−	5758.92i	0.691571	−	0.691571i	−0.271007	−	0.962577i	$-0.587357\pi$
$37$	5758.92	−	5758.92i	0.691571	−	0.691571i	0.962577	+	0.271007i	$0.0873565\pi$
$38$	5673.62	−	5673.62i	0.637385	−	0.637385i
$39$		0			0
$40$		0			0
$41$			16087.5i			1.49462i	0.664477	+	0.747309i	$0.268654\pi$
$41$			16087.5i			1.49462i	−0.664477	+	0.747309i	$0.731346\pi$
$42$		0			0
$43$	−9780.29	−	9780.29i	−0.806641	−	0.806641i	0.177483	−	0.984124i	$-0.443205\pi$
$43$	−9780.29	−	9780.29i	−0.806641	−	0.806641i	−0.984124	+	0.177483i	$0.943205\pi$
$44$	−1307.54			−0.101818
$45$		0			0
$46$	4667.28			0.325214
$47$	11110.0	+	11110.0i	0.733615	+	0.733615i	0.971334	−	0.237719i	$-0.0763996\pi$
$47$	11110.0	+	11110.0i	0.733615	+	0.733615i	−0.237719	+	0.971334i	$0.576400\pi$
$48$		0			0
$49$			15643.4i			0.930766i
$50$		0			0
$51$		0			0
$52$	−4503.94	+	4503.94i	−0.230985	+	0.230985i
$53$	1013.67	−	1013.67i	0.0495687	−	0.0495687i	−0.681888	−	0.731457i	$-0.738841\pi$
$53$	1013.67	−	1013.67i	0.0495687	−	0.0495687i	0.731457	+	0.681888i	$0.238841\pi$
$54$		0			0
$55$		0			0
$56$		−	6670.46i		−	0.284240i
$57$		0			0
$58$	−12936.8	−	12936.8i	−0.504961	−	0.504961i
$59$	46524.8			1.74002			0.870010	−	0.493035i	$-0.164112\pi$
$59$	46524.8			1.74002			0.870010	+	0.493035i	$0.164112\pi$
$60$		0			0
$61$	52196.0			1.79602			0.898012	−	0.439971i	$-0.145011\pi$
$61$	52196.0			1.79602			0.898012	+	0.439971i	$0.145011\pi$
$62$	33437.9	+	33437.9i	1.10474	+	1.10474i
$63$		0			0
$64$		−	36422.5i		−	1.11153i
$65$		0			0
$66$		0			0
$67$	31213.8	−	31213.8i	0.849494	−	0.849494i	−0.140576	−	0.990070i	$-0.544895\pi$
$67$	31213.8	−	31213.8i	0.849494	−	0.849494i	0.990070	+	0.140576i	$0.0448954\pi$
$68$	5535.22	−	5535.22i	0.145165	−	0.145165i
$69$		0			0
$70$		0			0
$71$			22440.4i			0.528306i	0.964481	+	0.264153i	$0.0850924\pi$
$71$			22440.4i			0.528306i	−0.964481	+	0.264153i	$0.914908\pi$
$72$		0			0
$73$	3589.53	+	3589.53i	0.0788371	+	0.0788371i	0.745426	−	0.666589i	$-0.232246\pi$
$73$	3589.53	+	3589.53i	0.0788371	+	0.0788371i	−0.666589	+	0.745426i	$0.732246\pi$
$74$	−40284.8			−0.855189
$75$		0			0
$76$	12220.6			0.242693
$77$	4186.44	+	4186.44i	0.0804669	+	0.0804669i
$78$		0			0
$79$		−	17531.6i		−	0.316049i	−0.987435	−	0.158024i	$-0.949488\pi$
$79$		−	17531.6i		−	0.316049i	0.987435	−	0.158024i	$-0.0505125\pi$
$80$		0			0
$81$		0			0
$82$	56267.8	−	56267.8i	0.924113	−	0.924113i
$83$	71122.6	−	71122.6i	1.13322	−	1.13322i	0.143576	−	0.989639i	$-0.454140\pi$
$83$	71122.6	−	71122.6i	1.13322	−	1.13322i	0.989639	−	0.143576i	$-0.0458601\pi$
$84$		0			0
$85$		0			0
$86$			68415.1i			0.997483i
$87$		0			0
$88$	23998.9	+	23998.9i	0.330358	+	0.330358i
$89$	−48621.5			−0.650659			−0.325330	−	0.945601i	$-0.605475\pi$
$89$	−48621.5			−0.650659			−0.325330	+	0.945601i	$0.605475\pi$
$90$		0			0
$91$	28841.0			0.365096
$92$	5026.49	+	5026.49i	0.0619149	+	0.0619149i
$93$		0			0
$94$		−	77716.5i		−	0.907180i
$95$		0			0
$96$		0			0
$97$	7908.79	−	7908.79i	0.0853455	−	0.0853455i	−0.663145	−	0.748491i	$-0.730779\pi$
$97$	7908.79	−	7908.79i	0.0853455	−	0.0853455i	0.748491	+	0.663145i	$0.230779\pi$
$98$	54714.3	−	54714.3i	0.575487	−	0.575487i
$99$		0			0
$100$		0			0
$101$		−	69114.4i		−	0.674163i	−0.941475	−	0.337082i	$-0.890560\pi$
$101$		−	69114.4i		−	0.674163i	0.941475	−	0.337082i	$-0.109440\pi$
$102$		0			0
$103$	66208.5	+	66208.5i	0.614923	+	0.614923i	0.944225	−	0.329302i	$-0.106813\pi$
$103$	66208.5	+	66208.5i	0.614923	+	0.614923i	−0.329302	+	0.944225i	$0.606813\pi$
$104$	165332.			1.49891
$105$		0			0
$106$	−7090.84			−0.0612961
$107$	84771.5	+	84771.5i	0.715798	+	0.715798i	0.967742	−	0.251944i	$-0.0810699\pi$
$107$	84771.5	+	84771.5i	0.715798	+	0.715798i	−0.251944	+	0.967742i	$0.581070\pi$
$108$		0			0
$109$			43319.2i			0.349232i	0.984637	+	0.174616i	$0.0558685\pi$
$109$			43319.2i			0.349232i	−0.984637	+	0.174616i	$0.944132\pi$
$110$		0			0
$111$		0			0
$112$	−17515.7	+	17515.7i	−0.131942	+	0.131942i
$113$	−97783.2	+	97783.2i	−0.720391	+	0.720391i	−0.968685	−	0.248294i	$-0.920130\pi$
$113$	−97783.2	+	97783.2i	−0.720391	+	0.720391i	0.248294	+	0.968685i	$0.420130\pi$
$114$		0			0
$115$		0			0
$116$		−	27865.0i		−	0.192271i
$117$		0			0
$118$	−162725.	−	162725.i	−1.07584	−	1.07584i
$119$	−35444.8			−0.229449
$120$		0			0
$121$	130927.			0.812955
$122$	−182561.	−	182561.i	−1.11047	−	1.11047i
$123$		0			0
$124$			72022.8i			0.420645i
$125$		0			0
$126$		0			0
$127$	−136748.	+	136748.i	−0.752337	+	0.752337i	−0.974915	−	0.222578i	$-0.928553\pi$
$127$	−136748.	+	136748.i	−0.752337	+	0.752337i	0.222578	+	0.974915i	$0.428553\pi$
$128$	−67075.6	+	67075.6i	−0.361859	+	0.361859i
$129$		0			0
$130$		0			0
$131$			8932.22i			0.0454759i	0.999741	+	0.0227379i	$0.00723834\pi$
$131$			8932.22i			0.0454759i	−0.999741	+	0.0227379i	$0.992762\pi$
$132$		0			0
$133$	−39127.3	−	39127.3i	−0.191801	−	0.191801i
$134$	−218347.			−1.05047
$135$		0			0
$136$	−203189.			−0.942004
$137$	155119.	+	155119.i	0.706093	+	0.706093i	0.965711	−	0.259618i	$-0.0835966\pi$
$137$	155119.	+	155119.i	0.706093	+	0.706093i	−0.259618	+	0.965711i	$0.583597\pi$
$138$		0			0
$139$			148665.i			0.652636i	0.945260	+	0.326318i	$0.105808\pi$
$139$			148665.i			0.652636i	−0.945260	+	0.326318i	$0.894192\pi$
$140$		0			0
$141$		0			0
$142$	78487.7	−	78487.7i	0.326649	−	0.326649i
$143$	−103764.	+	103764.i	−0.424332	+	0.424332i
$144$		0			0
$145$		0			0
$146$		−	25109.5i		−	0.0974890i
$147$		0			0
$148$	−43385.3	−	43385.3i	−0.162813	−	0.162813i
$149$	168325.			0.621130			0.310565	−	0.950552i	$-0.399482\pi$
$149$	168325.			0.621130			0.310565	+	0.950552i	$0.399482\pi$
$150$		0			0
$151$	−365709.			−1.30525			−0.652625	−	0.757681i	$-0.726332\pi$
$151$	−365709.			−1.30525			−0.652625	+	0.757681i	$0.726332\pi$
$152$	−224299.	−	224299.i	−0.787440	−	0.787440i
$153$		0			0
$154$		−	29285.0i		−	0.0995045i
$155$		0			0
$156$		0			0
$157$	−336039.	+	336039.i	−1.08803	+	1.08803i	−0.0922993	+	0.995731i	$0.529422\pi$
$157$	−336039.	+	336039.i	−1.08803	+	1.08803i	−0.995731	+	0.0922993i	$0.970578\pi$
$158$	−61318.6	+	61318.6i	−0.195411	+	0.195411i
$159$		0			0
$160$		0			0
$161$		−	32187.2i		−	0.0978629i
$162$		0			0
$163$	425628.	+	425628.i	1.25476	+	1.25476i	0.953561	+	0.301200i	$0.0973873\pi$
$163$	425628.	+	425628.i	1.25476	+	1.25476i	0.301200	+	0.953561i	$0.402613\pi$
$164$	121197.			0.351869
$165$		0			0
$166$	−497517.			−1.40132
$167$	−272794.	−	272794.i	−0.756909	−	0.756909i	0.218849	−	0.975759i	$-0.429770\pi$
$167$	−272794.	−	272794.i	−0.756909	−	0.756909i	−0.975759	+	0.218849i	$0.929770\pi$
$168$		0			0
$169$			343554.i			0.925290i
$170$		0			0
$171$		0			0
$172$	−73680.6	+	73680.6i	−0.189903	+	0.189903i
$173$	−166817.	+	166817.i	−0.423764	+	0.423764i	−0.886498	−	0.462733i	$-0.846869\pi$
$173$	−166817.	+	166817.i	−0.423764	+	0.423764i	0.462733	+	0.886498i	$0.346869\pi$
$174$		0			0
$175$		0			0
$176$		−	126036.i		−	0.306699i
$177$		0			0
$178$	170059.	+	170059.i	0.402299	+	0.402299i
$179$	−651990.			−1.52093			−0.760463	−	0.649381i	$-0.775028\pi$
$179$	−651990.			−1.52093			−0.760463	+	0.649381i	$0.775028\pi$
$180$		0			0
$181$	247026.			0.560461			0.280231	−	0.959933i	$-0.409589\pi$
$181$	247026.			0.560461			0.280231	+	0.959933i	$0.409589\pi$
$182$	−100874.	−	100874.i	−0.225737	−	0.225737i
$183$		0			0
$184$		−	184514.i		−	0.401777i
$185$		0			0
$186$		0			0
$187$	127523.	−	127523.i	0.266676	−	0.266676i
$188$	83697.9	−	83697.9i	0.172711	−	0.172711i
$189$		0			0
$190$		0			0
$191$		−	63445.3i		−	0.125839i	−0.998019	−	0.0629196i	$-0.979959\pi$
$191$		−	63445.3i		−	0.125839i	0.998019	−	0.0629196i	$-0.0200412\pi$
$192$		0			0
$193$	−184478.	−	184478.i	−0.356493	−	0.356493i	0.506026	−	0.862518i	$-0.331114\pi$
$193$	−184478.	−	184478.i	−0.356493	−	0.356493i	−0.862518	+	0.506026i	$0.831114\pi$
$194$	−55323.6			−0.105537
$195$		0			0
$196$	117851.			0.219125
$197$	−153826.	−	153826.i	−0.282399	−	0.282399i	0.551666	−	0.834065i	$-0.313992\pi$
$197$	−153826.	−	153826.i	−0.282399	−	0.282399i	−0.834065	+	0.551666i	$0.813992\pi$
$198$		0			0
$199$			88348.0i			0.158148i	0.996869	+	0.0790740i	$0.0251963\pi$
$199$			88348.0i			0.158148i	−0.996869	+	0.0790740i	$0.974804\pi$
$200$		0			0
$201$		0			0
$202$	−241734.	+	241734.i	−0.416831	+	0.416831i
$203$	−89216.9	+	89216.9i	−0.151952	+	0.151952i
$204$		0			0
$205$		0			0
$206$		−	463142.i		−	0.760407i
$207$		0			0
$208$	−434140.	−	434140.i	−0.695780	−	0.695780i
$209$	281543.			0.445840
$210$		0			0
$211$	220425.			0.340843			0.170422	−	0.985371i	$-0.445487\pi$
$211$	220425.			0.340843			0.170422	+	0.985371i	$0.445487\pi$
$212$	−7636.57	−	7636.57i	−0.0116697	−	0.0116697i
$213$		0			0
$214$		−	592994.i		−	0.885147i
$215$		0			0
$216$		0			0
$217$	230599.	−	230599.i	0.332436	−	0.332436i
$218$	151513.	−	151513.i	0.215928	−	0.215928i
$219$		0			0
$220$		0			0
$221$		−	878527.i		−	1.20997i
$222$		0			0
$223$	550351.	+	550351.i	0.741101	+	0.741101i	0.972790	−	0.231689i	$-0.0744252\pi$
$223$	550351.	+	550351.i	0.741101	+	0.741101i	−0.231689	+	0.972790i	$0.574425\pi$
$224$	−90928.7			−0.121082
$225$		0			0
$226$	684013.			0.890827
$227$	32455.5	+	32455.5i	0.0418045	+	0.0418045i	0.727700	−	0.685896i	$-0.240589\pi$
$227$	32455.5	+	32455.5i	0.0418045	+	0.0418045i	−0.685896	+	0.727700i	$0.740589\pi$
$228$		0			0
$229$			389872.i			0.491285i	0.969360	+	0.245643i	$0.0789990\pi$
$229$			389872.i			0.491285i	−0.969360	+	0.245643i	$0.921001\pi$
$230$		0			0
$231$		0			0
$232$	−511439.	+	511439.i	−0.623841	+	0.623841i
$233$	−737801.	+	737801.i	−0.890326	+	0.890326i	−0.994554	−	0.104227i	$-0.966763\pi$
$233$	−737801.	+	737801.i	−0.890326	+	0.890326i	0.104227	+	0.994554i	$0.466763\pi$
$234$		0			0
$235$		0			0
$236$		−	350498.i		−	0.409643i
$237$		0			0
$238$	123972.	+	123972.i	0.141867	+	0.141867i
$239$	−367547.			−0.416216			−0.208108	−	0.978106i	$-0.566731\pi$
$239$	−367547.			−0.416216			−0.208108	+	0.978106i	$0.566731\pi$
$240$		0			0
$241$	1.10156e6			1.22170			0.610850	−	0.791747i	$-0.290828\pi$
$241$	1.10156e6			1.22170			0.610850	+	0.791747i	$0.290828\pi$
$242$	−457931.	−	457931.i	−0.502645	−	0.502645i
$243$		0			0
$244$		−	393222.i		−	0.422828i
$245$		0			0
$246$		0			0
$247$	969799.	−	969799.i	1.01144	−	1.01144i
$248$	1.32192e6	−	1.32192e6i	1.36482	−	1.36482i
$249$		0			0
$250$		0			0
$251$			1.75554e6i			1.75884i	0.476049	+	0.879419i	$0.342068\pi$
$251$			1.75554e6i			1.75884i	−0.476049	+	0.879419i	$0.657932\pi$
$252$		0			0
$253$	115803.	+	115803.i	0.113741	+	0.113741i
$254$	956581.			0.930331
$255$		0			0
$256$	−696313.			−0.664056
$257$	1.38179e6	+	1.38179e6i	1.30500	+	1.30500i	0.924978	+	0.380020i	$0.124083\pi$
$257$	1.38179e6	+	1.38179e6i	1.30500	+	1.30500i	0.380020	+	0.924978i	$0.375917\pi$
$258$		0			0
$259$			277818.i			0.257342i
$260$		0			0
$261$		0			0
$262$	31241.3	−	31241.3i	0.0281175	−	0.0281175i
$263$	504029.	−	504029.i	0.449330	−	0.449330i	−0.445801	−	0.895132i	$-0.647081\pi$
$263$	504029.	−	504029.i	0.449330	−	0.449330i	0.895132	+	0.445801i	$0.147081\pi$
$264$		0			0
$265$		0			0
$266$			273703.i			0.237179i
$267$		0			0
$268$	−235152.	−	235152.i	−0.199992	−	0.199992i
$269$	−715977.			−0.603279			−0.301640	−	0.953422i	$-0.597534\pi$
$269$	−715977.			−0.603279			−0.301640	+	0.953422i	$0.597534\pi$
$270$		0			0
$271$	−1.50875e6			−1.24794			−0.623969	−	0.781449i	$-0.714481\pi$
$271$	−1.50875e6			−1.24794			−0.623969	+	0.781449i	$0.714481\pi$
$272$	533547.	+	533547.i	0.437270	+	0.437270i
$273$		0			0
$274$		−	1.08509e6i		−	0.873147i
$275$		0			0
$276$		0			0
$277$	1.19043e6	−	1.19043e6i	0.932189	−	0.932189i	−0.0656538	−	0.997842i	$-0.520913\pi$
$277$	1.19043e6	−	1.19043e6i	0.932189	−	0.932189i	0.997842	+	0.0656538i	$0.0209133\pi$
$278$	519970.	−	519970.i	0.403521	−	0.403521i
$279$		0			0
$280$		0			0
$281$			1.32423e6i			1.00045i	0.865894	+	0.500227i	$0.166750\pi$
$281$			1.32423e6i			1.00045i	−0.865894	+	0.500227i	$0.833250\pi$
$282$		0			0
$283$	−13034.8	−	13034.8i	−0.00967472	−	0.00967472i	0.702253	−	0.711928i	$-0.252178\pi$
$283$	−13034.8	−	13034.8i	−0.00967472	−	0.00967472i	−0.711928	+	0.702253i	$0.752178\pi$
$284$	169057.			0.124376
$285$		0			0
$286$	725849.			0.524725
$287$	−388042.	−	388042.i	−0.278083	−	0.278083i
$288$		0			0
$289$		−	340172.i		−	0.239582i
$290$		0			0
$291$		0			0
$292$	27042.0	−	27042.0i	0.0185602	−	0.0185602i
$293$	−776691.	+	776691.i	−0.528542	+	0.528542i	−0.920137	−	0.391596i	$-0.871923\pi$
$293$	−776691.	+	776691.i	−0.528542	+	0.528542i	0.391596	+	0.920137i	$0.371923\pi$
$294$		0			0
$295$		0			0
$296$			1.59260e6i			1.05652i
$297$		0			0
$298$	−588733.	−	588733.i	−0.384041	−	0.384041i
$299$	797783.			0.516068
$300$		0			0
$301$	471814.			0.300161
$302$	1.27910e6	+	1.27910e6i	0.807028	+	0.807028i
$303$		0			0
$304$			1.17796e6i			0.731047i
$305$		0			0
$306$		0			0
$307$	−397374.	+	397374.i	−0.240632	+	0.240632i	−0.817112	−	0.576479i	$-0.804426\pi$
$307$	−397374.	+	397374.i	−0.240632	+	0.240632i	0.576479	+	0.817112i	$0.304426\pi$
$308$	31538.8	−	31538.8i	0.0189439	−	0.0189439i
$309$		0			0
$310$		0			0
$311$			726100.i			0.425692i	0.977086	+	0.212846i	$0.0682732\pi$
$311$			726100.i			0.425692i	−0.977086	+	0.212846i	$0.931727\pi$
$312$		0			0
$313$	−971529.	−	971529.i	−0.560525	−	0.560525i	0.368931	−	0.929457i	$-0.379724\pi$
$313$	−971529.	−	971529.i	−0.560525	−	0.560525i	−0.929457	+	0.368931i	$0.879724\pi$
$314$	2.35066e6			1.34545
$315$		0			0
$316$	−132076.			−0.0744056
$317$	1.02045e6	+	1.02045e6i	0.570352	+	0.570352i	0.932227	−	0.361875i	$-0.117863\pi$
$317$	1.02045e6	+	1.02045e6i	0.570352	+	0.570352i	−0.361875	+	0.932227i	$0.617863\pi$
$318$		0			0
$319$		−	641967.i		−	0.353212i
$320$		0			0
$321$		0			0
$322$	−112578.	+	112578.i	−0.0605081	+	0.0605081i
$323$	−1.19186e6	+	1.19186e6i	−0.635649	+	0.635649i
$324$		0			0
$325$		0			0
$326$		−	2.97735e6i		−	1.55162i
$327$		0			0
$328$	−2.22447e6	−	2.22447e6i	−1.14167	−	1.14167i
$329$	−535960.			−0.272988
$330$		0			0
$331$	640294.			0.321225			0.160613	−	0.987018i	$-0.448653\pi$
$331$	640294.			0.321225			0.160613	+	0.987018i	$0.448653\pi$
$332$	−535808.	−	535808.i	−0.266786	−	0.266786i
$333$		0			0
$334$			1.90825e6i			0.935986i
$335$		0			0
$336$		0			0
$337$	−891757.	+	891757.i	−0.427732	+	0.427732i	−0.887855	−	0.460123i	$-0.847805\pi$
$337$	−891757.	+	891757.i	−0.427732	+	0.427732i	0.460123	+	0.887855i	$0.347805\pi$
$338$	1.20161e6	−	1.20161e6i	0.572102	−	0.572102i
$339$		0			0
$340$		0			0
$341$			1.65929e6i			0.772747i
$342$		0			0
$343$	−782725.	−	782725.i	−0.359231	−	0.359231i
$344$	2.70469e6			1.23232
$345$		0			0
$346$	1.16692e6			0.524022
$347$	2.35342e6	+	2.35342e6i	1.04924	+	1.04924i	0.998723	+	0.0505198i	$0.0160878\pi$
$347$	2.35342e6	+	2.35342e6i	1.04924	+	1.04924i	0.0505198	+	0.998723i	$0.483912\pi$
$348$		0			0
$349$			781439.i			0.343425i	0.985147	+	0.171712i	$0.0549299\pi$
$349$			781439.i			0.343425i	−0.985147	+	0.171712i	$0.945070\pi$
$350$		0			0
$351$		0			0
$352$	327142.	−	327142.i	0.140728	−	0.140728i
$353$	1.35451e6	−	1.35451e6i	0.578558	−	0.578558i	−0.355948	−	0.934506i	$-0.615842\pi$
$353$	1.35451e6	−	1.35451e6i	0.578558	−	0.578558i	0.934506	+	0.355948i	$0.115842\pi$
$354$		0			0
$355$		0			0
$356$			366294.i			0.153181i
$357$		0			0
$358$	2.28040e6	+	2.28040e6i	0.940380	+	0.940380i
$359$	−3.10750e6			−1.27255			−0.636276	−	0.771461i	$-0.719526\pi$
$359$	−3.10750e6			−1.27255			−0.636276	+	0.771461i	$0.719526\pi$
$360$		0			0
$361$	−155264.			−0.0627049
$362$	−863997.	−	863997.i	−0.346530	−	0.346530i
$363$		0			0
$364$		−	217276.i		−	0.0859526i
$365$		0			0
$366$		0			0
$367$	1.25930e6	−	1.25930e6i	0.488050	−	0.488050i	−0.419640	−	0.907690i	$-0.637844\pi$
$367$	1.25930e6	−	1.25930e6i	0.488050	−	0.488050i	0.907690	+	0.419640i	$0.137844\pi$
$368$	−484509.	+	484509.i	−0.186502	+	0.186502i
$369$		0			0
$370$		0			0
$371$			48900.9i			0.0184451i
$372$		0			0
$373$	731112.	+	731112.i	0.272090	+	0.272090i	0.829941	−	0.557851i	$-0.188374\pi$
$373$	731112.	+	731112.i	0.272090	+	0.272090i	−0.557851	+	0.829941i	$0.688374\pi$
$374$	−892049.			−0.329769
$375$		0			0
$376$	−3.07241e6			−1.12075
$377$	−2.21131e6	−	2.21131e6i	−0.801301	−	0.801301i
$378$		0			0
$379$		−	4.11660e6i		−	1.47211i	−0.676921	−	0.736056i	$-0.736686\pi$
$379$		−	4.11660e6i		−	1.47211i	0.676921	−	0.736056i	$-0.263314\pi$
$380$		0			0
$381$		0			0
$382$	−221906.	+	221906.i	−0.0778057	+	0.0778057i
$383$	−998442.	+	998442.i	−0.347797	+	0.347797i	−0.859288	−	0.511491i	$-0.829093\pi$
$383$	−998442.	+	998442.i	−0.347797	+	0.347797i	0.511491	+	0.859288i	$0.329093\pi$
$384$		0			0
$385$		0			0
$386$			1.29046e6i			0.440835i
$387$		0			0
$388$	−59581.5	−	59581.5i	−0.0200924	−	0.0200924i
$389$	−1.81467e6			−0.608027			−0.304014	−	0.952668i	$-0.598327\pi$
$389$	−1.81467e6			−0.608027			−0.304014	+	0.952668i	$0.598327\pi$
$390$		0			0
$391$	−980454.			−0.324329
$392$	−2.16305e6	−	2.16305e6i	−0.710971	−	0.710971i
$393$		0			0
$394$			1.07604e6i			0.349212i
$395$		0			0
$396$		0			0
$397$	−130258.	+	130258.i	−0.0414789	+	0.0414789i	−0.727542	−	0.686063i	$-0.759337\pi$
$397$	−130258.	+	130258.i	−0.0414789	+	0.0414789i	0.686063	+	0.727542i	$0.259337\pi$
$398$	309006.	−	309006.i	0.0977820	−	0.0977820i
$399$		0			0
$400$		0			0
$401$		−	1.71814e6i		−	0.533578i	−0.963755	−	0.266789i	$-0.914037\pi$
$401$		−	1.71814e6i		−	0.533578i	0.963755	−	0.266789i	$-0.0859626\pi$
$402$		0			0
$403$	5.71557e6	+	5.71557e6i	1.75306	+	1.75306i
$404$	−520678.			−0.158714
$405$		0			0
$406$	624090.			0.187902
$407$	−999530.	−	999530.i	−0.299095	−	0.299095i
$408$		0			0
$409$		−	2.89387e6i		−	0.855402i	−0.903920	−	0.427701i	$-0.859324\pi$
$409$		−	2.89387e6i		−	0.855402i	0.903920	−	0.427701i	$-0.140676\pi$
$410$		0			0
$411$		0			0
$412$	498787.	−	498787.i	0.144768	−	0.144768i
$413$	−1.12221e6	+	1.12221e6i	−0.323742	+	0.323742i
$414$		0			0
$415$		0			0
$416$		−	2.25374e6i		−	0.638513i
$417$		0			0
$418$	−984726.	−	984726.i	−0.275661	−	0.275661i
$419$	143217.			0.0398527			0.0199264	−	0.999801i	$-0.493657\pi$
$419$	143217.			0.0398527			0.0199264	+	0.999801i	$0.493657\pi$
$420$		0			0
$421$	1.34045e6			0.368592			0.184296	−	0.982871i	$-0.440999\pi$
$421$	1.34045e6			0.368592			0.184296	+	0.982871i	$0.440999\pi$
$422$	−770959.	−	770959.i	−0.210742	−	0.210742i
$423$		0			0
$424$			280326.i			0.0757267i
$425$		0			0
$426$		0			0
$427$	−1.25900e6	+	1.25900e6i	−0.334162	+	0.334162i
$428$	638633.	−	638633.i	0.168516	−	0.168516i
$429$		0			0
$430$		0			0
$431$		−	6.72542e6i		−	1.74392i	−0.489577	−	0.871960i	$-0.662849\pi$
$431$		−	6.72542e6i		−	1.74392i	0.489577	−	0.871960i	$-0.337151\pi$
$432$		0			0
$433$	493739.	+	493739.i	0.126555	+	0.126555i	0.767547	−	0.640993i	$-0.221477\pi$
$433$	493739.	+	493739.i	0.126555	+	0.126555i	−0.640993	+	0.767547i	$0.721477\pi$
$434$	−1.61309e6			−0.411087
$435$		0			0
$436$	326349.			0.0822178
$437$	−1.08232e6	−	1.08232e6i	−0.271113	−	0.271113i
$438$		0			0
$439$			7.03925e6i			1.74327i	0.490154	+	0.871636i	$0.336941\pi$
$439$			7.03925e6i			1.74327i	−0.490154	+	0.871636i	$0.663059\pi$
$440$		0			0
$441$		0			0
$442$	−3.07274e6	+	3.07274e6i	−0.748117	+	0.748117i
$443$	−1.43616e6	+	1.43616e6i	−0.347691	+	0.347691i	−0.859249	−	0.511558i	$-0.829069\pi$
$443$	−1.43616e6	+	1.43616e6i	−0.347691	+	0.347691i	0.511558	+	0.859249i	$0.329069\pi$
$444$		0			0
$445$		0			0
$446$		−	3.84981e6i		−	0.916437i
$447$		0			0
$448$	878536.	+	878536.i	0.206807	+	0.206807i
$449$	−3.63556e6			−0.851052			−0.425526	−	0.904946i	$-0.639911\pi$
$449$	−3.63556e6			−0.851052			−0.425526	+	0.904946i	$0.639911\pi$
$450$		0			0
$451$	2.79219e6			0.646403
$452$	736657.	+	736657.i	0.169597	+	0.169597i
$453$		0			0
$454$		−	227033.i		−	0.0516950i
$455$		0			0
$456$		0			0
$457$	1.72616e6	−	1.72616e6i	0.386625	−	0.386625i	−0.486857	−	0.873482i	$-0.661857\pi$
$457$	1.72616e6	−	1.72616e6i	0.386625	−	0.386625i	0.873482	+	0.486857i	$0.161857\pi$
$458$	1.36362e6	−	1.36362e6i	0.303759	−	0.303759i
$459$		0			0
$460$		0			0
$461$			1.59109e6i			0.348693i	0.984684	+	0.174347i	$0.0557813\pi$
$461$			1.59109e6i			0.348693i	−0.984684	+	0.174347i	$0.944219\pi$
$462$		0			0
$463$	−5.66635e6	−	5.66635e6i	−1.22843	−	1.22843i	−0.964557	−	0.263874i	$-0.915000\pi$
$463$	−5.66635e6	−	5.66635e6i	−1.22843	−	1.22843i	−0.263874	−	0.964557i	$-0.585000\pi$
$464$	2.68594e6			0.579164
$465$		0			0
$466$	5.16106e6			1.10097
$467$	−2.14280e6	−	2.14280e6i	−0.454662	−	0.454662i	0.442237	−	0.896898i	$-0.354185\pi$
$467$	−2.14280e6	−	2.14280e6i	−0.454662	−	0.454662i	−0.896898	+	0.442237i	$0.854185\pi$
$468$		0			0
$469$			1.50580e6i			0.316107i
$470$		0			0
$471$		0			0
$472$	−6.43310e6	+	6.43310e6i	−1.32912	+	1.32912i
$473$	−1.69749e6	+	1.69749e6i	−0.348862	+	0.348862i
$474$		0			0
$475$		0			0
$476$			267027.i			0.0540178i
$477$		0			0
$478$	1.28553e6	+	1.28553e6i	0.257344	+	0.257344i
$479$	779790.			0.155288			0.0776441	−	0.996981i	$-0.475260\pi$
$479$	779790.			0.155288			0.0776441	+	0.996981i	$0.475260\pi$
$480$		0			0
$481$	−6.88593e6			−1.35706
$482$	−3.85281e6	−	3.85281e6i	−0.755370	−	0.755370i
$483$		0			0
$484$		−	986350.i		−	0.191389i
$485$		0			0
$486$		0			0
$487$	−2.11432e6	+	2.11432e6i	−0.403970	+	0.403970i	−0.879629	−	0.475659i	$-0.842209\pi$
$487$	−2.11432e6	+	2.11432e6i	−0.403970	+	0.403970i	0.475659	+	0.879629i	$0.342209\pi$
$488$	−7.21727e6	+	7.21727e6i	−1.37190	+	1.37190i
$489$		0			0
$490$		0			0
$491$		−	2.20117e6i		−	0.412049i	−0.978547	−	0.206025i	$-0.933947\pi$
$491$		−	2.20117e6i		−	0.412049i	0.978547	−	0.206025i	$-0.0660527\pi$
$492$		0			0
$493$	2.71764e6	+	2.71764e6i	0.503586	+	0.503586i
$494$	−6.78394e6			−1.25073
$495$		0			0
$496$	−6.94236e6			−1.26708
$497$	−541279.	−	541279.i	−0.0982947	−	0.0982947i
$498$		0			0
$499$			287378.i			0.0516656i	0.999666	+	0.0258328i	$0.00822375\pi$
$499$			287378.i			0.0516656i	−0.999666	+	0.0258328i	$0.991776\pi$
$500$		0			0
$501$		0			0
$502$	6.14017e6	−	6.14017e6i	1.08748	−	1.08748i
$503$	−734293.	+	734293.i	−0.129405	+	0.129405i	−0.768843	−	0.639438i	$-0.779167\pi$
$503$	−734293.	+	734293.i	−0.129405	+	0.129405i	0.639438	+	0.768843i	$0.279167\pi$
$504$		0			0
$505$		0			0
$506$		−	810063.i		−	0.140651i
$507$		0			0
$508$	1.03020e6	+	1.03020e6i	0.177118	+	0.177118i
$509$	9.29537e6			1.59028			0.795138	−	0.606429i	$-0.207399\pi$
$509$	9.29537e6			1.59028			0.795138	+	0.606429i	$0.207399\pi$
$510$		0			0
$511$	−173164.			−0.0293363
$512$	4.58184e6	+	4.58184e6i	0.772441	+	0.772441i
$513$		0			0
$514$		−	9.66591e6i		−	1.61375i
$515$		0			0
$516$		0			0
$517$	1.92827e6	−	1.92827e6i	0.317279	−	0.317279i
$518$	971697.	−	971697.i	0.159113	−	0.159113i
$519$		0			0
$520$		0			0
$521$			4.12354e6i			0.665543i	0.943008	+	0.332771i	$0.107984\pi$
$521$			4.12354e6i			0.665543i	−0.943008	+	0.332771i	$0.892016\pi$
$522$		0			0
$523$	−3.87029e6	−	3.87029e6i	−0.618713	−	0.618713i	0.326489	−	0.945201i	$-0.394135\pi$
$523$	−3.87029e6	−	3.87029e6i	−0.618713	−	0.618713i	−0.945201	+	0.326489i	$0.894135\pi$
$524$	67291.6			0.0107061
$525$		0			0
$526$	−3.52578e6			−0.555637
$527$	−7.02428e6	−	7.02428e6i	−1.10173	−	1.10173i
$528$		0			0
$529$			5.54600e6i			0.861670i
$530$		0			0
$531$		0			0
$532$	−294769.	+	294769.i	−0.0451546	+	0.0451546i
$533$	9.61792e6	−	9.61792e6i	1.46644	−	1.46644i
$534$		0			0
$535$		0			0
$536$			8.63204e6i			1.29778i
$537$		0			0
$538$	2.50420e6	+	2.50420e6i	0.373004	+	0.373004i
$539$	2.71510e6			0.402544
$540$		0			0
$541$	−7.61255e6			−1.11825			−0.559123	−	0.829085i	$-0.688862\pi$
$541$	−7.61255e6			−1.11825			−0.559123	+	0.829085i	$0.688862\pi$
$542$	5.27699e6	+	5.27699e6i	0.771593	+	0.771593i
$543$		0			0
$544$			2.76978e6i			0.401280i
$545$		0			0
$546$		0			0
$547$	−8.10895e6	+	8.10895e6i	−1.15877	+	1.15877i	−0.174027	+	0.984741i	$0.555678\pi$
$547$	−8.10895e6	+	8.10895e6i	−1.15877	+	1.15877i	−0.984741	+	0.174027i	$0.944322\pi$
$548$	1.16860e6	−	1.16860e6i	0.166232	−	0.166232i
$549$		0			0
$550$		0			0
$551$			5.99996e6i			0.841917i
$552$		0			0
$553$	422874.	+	422874.i	0.0588029	+	0.0588029i
$554$	−8.32729e6			−1.15273
$555$		0			0
$556$	1.11998e6			0.153646
$557$	7.99279e6	+	7.99279e6i	1.09159	+	1.09159i	0.995359	+	0.0962338i	$0.0306796\pi$
$557$	7.99279e6	+	7.99279e6i	1.09159	+	1.09159i	0.0962338	+	0.995359i	$0.469320\pi$
$558$		0			0
$559$			1.16943e7i			1.58286i
$560$		0			0
$561$		0			0
$562$	4.63163e6	−	4.63163e6i	0.618576	−	0.618576i
$563$	−4.58626e6	+	4.58626e6i	−0.609801	+	0.609801i	−0.942894	−	0.333093i	$-0.891908\pi$
$563$	−4.58626e6	+	4.58626e6i	−0.609801	+	0.609801i	0.333093	+	0.942894i	$0.391908\pi$
$564$		0			0
$565$		0			0
$566$			91181.1i			0.0119637i
$567$		0			0
$568$	−3.10290e6	−	3.10290e6i	−0.403550	−	0.403550i
$569$	3.63459e6			0.470624			0.235312	−	0.971920i	$-0.424389\pi$
$569$	3.63459e6			0.470624			0.235312	+	0.971920i	$0.424389\pi$
$570$		0			0
$571$	3.08250e6			0.395652			0.197826	−	0.980237i	$-0.436612\pi$
$571$	3.08250e6			0.395652			0.197826	+	0.980237i	$0.436612\pi$
$572$	781713.	+	781713.i	0.0998982	+	0.0998982i
$573$		0			0
$574$			2.71443e6i			0.343874i
$575$		0			0
$576$		0			0
$577$	3.08704e6	−	3.08704e6i	0.386014	−	0.386014i	−0.487249	−	0.873263i	$-0.662000\pi$
$577$	3.08704e6	−	3.08704e6i	0.386014	−	0.386014i	0.873263	+	0.487249i	$0.162000\pi$
$578$	−1.18979e6	+	1.18979e6i	−0.148132	+	0.148132i
$579$		0			0
$580$		0			0
$581$			3.43105e6i			0.421684i
$582$		0			0
$583$	−175935.	−	175935.i	−0.0214378	−	0.0214378i
$584$	−992668.			−0.120440
$585$		0			0
$586$	5.43311e6			0.653589
$587$	635529.	+	635529.i	0.0761272	+	0.0761272i	0.744145	−	0.668018i	$-0.232857\pi$
$587$	635529.	+	635529.i	0.0761272	+	0.0761272i	−0.668018	+	0.744145i	$0.732857\pi$
$588$		0			0
$589$		−	1.55081e7i		−	1.84192i
$590$		0			0
$591$		0			0
$592$	4.18196e6	−	4.18196e6i	0.490428	−	0.490428i
$593$	4.12654e6	−	4.12654e6i	0.481892	−	0.481892i	−0.423843	−	0.905735i	$-0.639319\pi$
$593$	4.12654e6	−	4.12654e6i	0.481892	−	0.481892i	0.905735	+	0.423843i	$0.139319\pi$
$594$		0			0
$595$		0			0
$596$		−	1.26809e6i		−	0.146229i
$597$		0			0
$598$	−2.79033e6	−	2.79033e6i	−0.319082	−	0.319082i
$599$	9.76542e6			1.11205			0.556025	−	0.831166i	$-0.312326\pi$
$599$	9.76542e6			1.11205			0.556025	+	0.831166i	$0.312326\pi$
$600$		0			0
$601$	1.09854e7			1.24059			0.620297	−	0.784367i	$-0.287012\pi$
$601$	1.09854e7			1.24059			0.620297	+	0.784367i	$0.287012\pi$
$602$	−1.65022e6	−	1.65022e6i	−0.185588	−	0.185588i
$603$		0			0
$604$			2.75510e6i			0.307287i
$605$		0			0
$606$		0			0
$607$	−3.19843e6	+	3.19843e6i	−0.352343	+	0.352343i	−0.860981	−	0.508638i	$-0.830149\pi$
$607$	−3.19843e6	+	3.19843e6i	−0.352343	+	0.352343i	0.508638	+	0.860981i	$0.330149\pi$
$608$	−3.05754e6	+	3.05754e6i	−0.335438	+	0.335438i
$609$		0			0
$610$		0			0
$611$		−	1.32842e7i		−	1.43957i
$612$		0			0
$613$	−7.58934e6	−	7.58934e6i	−0.815742	−	0.815742i	0.169746	−	0.985488i	$-0.445705\pi$
$613$	−7.58934e6	−	7.58934e6i	−0.815742	−	0.815742i	−0.985488	+	0.169746i	$0.945705\pi$
$614$	2.77971e6			0.297563
$615$		0			0
$616$	−1.15774e6			−0.122930
$617$	7.77526e6	+	7.77526e6i	0.822247	+	0.822247i	0.986430	−	0.164183i	$-0.0524988\pi$
$617$	7.77526e6	+	7.77526e6i	0.822247	+	0.822247i	−0.164183	+	0.986430i	$0.552499\pi$
$618$		0			0
$619$		−	8.12168e6i		−	0.851960i	−0.904733	−	0.425980i	$-0.859929\pi$
$619$		−	8.12168e6i		−	0.851960i	0.904733	−	0.425980i	$-0.140071\pi$
$620$		0			0
$621$		0			0
$622$	2.53961e6	−	2.53961e6i	0.263203	−	0.263203i
$623$	1.17278e6	−	1.17278e6i	0.121059	−	0.121059i
$624$		0			0
$625$		0			0
$626$			6.79604e6i			0.693139i
$627$		0			0
$628$	2.53158e6	+	2.53158e6i	0.256149	+	0.256149i
$629$	8.46262e6			0.852860
$630$		0			0
$631$	6.96100e6			0.695983			0.347991	−	0.937498i	$-0.386864\pi$
$631$	6.96100e6			0.695983			0.347991	+	0.937498i	$0.386864\pi$
$632$	2.42414e6	+	2.42414e6i	0.241416	+	0.241416i
$633$		0			0
$634$		−	7.13824e6i		−	0.705291i
$635$		0			0
$636$		0			0
$637$	9.35238e6	−	9.35238e6i	0.913216	−	0.913216i
$638$	−2.24534e6	+	2.24534e6i	−0.218389	+	0.218389i
$639$		0			0
$640$		0			0
$641$			2.95734e6i			0.284286i	0.989846	+	0.142143i	$0.0453993\pi$
$641$			2.95734e6i			0.284286i	−0.989846	+	0.142143i	$0.954601\pi$
$642$		0			0
$643$	−6.84906e6	−	6.84906e6i	−0.653287	−	0.653287i	0.300496	−	0.953783i	$-0.402848\pi$
$643$	−6.84906e6	−	6.84906e6i	−0.653287	−	0.653287i	−0.953783	+	0.300496i	$0.902848\pi$
$644$	−242485.			−0.0230393
$645$		0			0
$646$	8.33728e6			0.786037
$647$	7.70791e6	+	7.70791e6i	0.723896	+	0.723896i	0.969396	−	0.245501i	$-0.0789524\pi$
$647$	7.70791e6	+	7.70791e6i	0.723896	+	0.723896i	−0.245501	+	0.969396i	$0.578952\pi$
$648$		0			0
$649$		−	8.07493e6i		−	0.752536i
$650$		0			0
$651$		0			0
$652$	3.20650e6	−	3.20650e6i	0.295401	−	0.295401i
$653$	1.24415e7	−	1.24415e7i	1.14180	−	1.14180i	0.153682	−	0.988120i	$-0.450887\pi$
$653$	1.24415e7	−	1.24415e7i	1.14180	−	1.14180i	0.988120	−	0.153682i	$-0.0491133\pi$
$654$		0			0
$655$		0			0
$656$			1.16823e7i			1.05991i
$657$		0			0
$658$	1.87457e6	+	1.87457e6i	0.168787	+	0.168787i
$659$	−2.14806e7			−1.92679			−0.963393	−	0.268092i	$-0.913607\pi$
$659$	−2.14806e7			−1.92679			−0.963393	+	0.268092i	$0.913607\pi$
$660$		0			0
$661$	7.56548e6			0.673493			0.336746	−	0.941595i	$-0.390673\pi$
$661$	7.56548e6			0.673493			0.336746	+	0.941595i	$0.390673\pi$
$662$	−2.23949e6	−	2.23949e6i	−0.198612	−	0.198612i
$663$		0			0
$664$			1.96686e7i			1.73123i
$665$		0			0
$666$		0			0
$667$	−2.46786e6	+	2.46786e6i	−0.214786	+	0.214786i
$668$	−2.05512e6	+	2.05512e6i	−0.178195	+	0.178195i
$669$		0			0
$670$		0			0
$671$		−	9.05924e6i		−	0.776757i
$672$		0			0
$673$	1.45146e7	+	1.45146e7i	1.23529	+	1.23529i	0.961906	+	0.273381i	$0.0881420\pi$
$673$	1.45146e7	+	1.45146e7i	1.23529	+	1.23529i	0.273381	+	0.961906i	$0.411858\pi$
$674$	6.23802e6			0.528929
$675$		0			0
$676$	2.58819e6			0.217836
$677$	−6.19483e6	−	6.19483e6i	−0.519467	−	0.519467i	0.397943	−	0.917410i	$-0.369724\pi$
$677$	−6.19483e6	−	6.19483e6i	−0.519467	−	0.519467i	−0.917410	+	0.397943i	$0.869724\pi$
$678$		0			0
$679$			381531.i			0.0317581i
$680$		0			0
$681$		0			0
$682$	5.80355e6	−	5.80355e6i	0.477785	−	0.477785i
$683$	1.43009e7	−	1.43009e7i	1.17303	−	1.17303i	0.191550	−	0.981483i	$-0.438649\pi$
$683$	1.43009e7	−	1.43009e7i	1.17303	−	1.17303i	0.981483	−	0.191550i	$-0.0613515\pi$
$684$		0			0
$685$		0			0
$686$			5.47532e6i			0.444221i
$687$		0			0
$688$	−7.10216e6	−	7.10216e6i	−0.572031	−	0.572031i
$689$	−1.21204e6			−0.0972682
$690$		0			0
$691$	−1.62663e6			−0.129596			−0.0647982	−	0.997898i	$-0.520640\pi$
$691$	−1.62663e6			−0.129596			−0.0647982	+	0.997898i	$0.520640\pi$
$692$	1.25673e6	+	1.25673e6i	0.0997644	+	0.0997644i
$693$		0			0
$694$		−	1.64627e7i		−	1.29748i
$695$		0			0
$696$		0			0
$697$	−1.18202e7	+	1.18202e7i	−0.921597	+	0.921597i
$698$	2.73316e6	−	2.73316e6i	0.212337	−	0.212337i
$699$		0			0
$700$		0			0
$701$		−	3.74205e6i		−	0.287617i	−0.989606	−	0.143809i	$-0.954065\pi$
$701$		−	3.74205e6i		−	0.287617i	0.989606	−	0.143809i	$-0.0459350\pi$
$702$		0			0
$703$	9.34182e6	+	9.34182e6i	0.712924	+	0.712924i
$704$	−6.32157e6			−0.480721
$705$		0			0
$706$	−9.47510e6			−0.715438
$707$	1.66709e6	+	1.66709e6i	0.125432	+	0.125432i
$708$		0			0
$709$		−	288785.i		−	0.0215754i	−0.999942	−	0.0107877i	$-0.996566\pi$
$709$		−	288785.i		−	0.0215754i	0.999942	−	0.0107877i	$-0.00343389\pi$
$710$		0			0
$711$		0			0
$712$	6.72303e6	−	6.72303e6i	0.497010	−	0.497010i
$713$	6.37870e6	−	6.37870e6i	0.469903	−	0.469903i
$714$		0			0
$715$		0			0
$716$			4.91181e6i			0.358063i
$717$		0			0
$718$	1.08688e7	+	1.08688e7i	0.786812	+	0.786812i
$719$	2.55140e7			1.84059			0.920294	−	0.391228i	$-0.127950\pi$
$719$	2.55140e7			1.84059			0.920294	+	0.391228i	$0.127950\pi$
$720$		0			0
$721$	−3.19399e6			−0.228821
$722$	543050.	+	543050.i	0.0387701	+	0.0387701i
$723$		0			0
$724$		−	1.86099e6i		−	0.131946i
$725$		0			0
$726$		0			0
$727$	1.35802e7	−	1.35802e7i	0.952949	−	0.952949i	−0.0459929	−	0.998942i	$-0.514645\pi$
$727$	1.35802e7	−	1.35802e7i	0.952949	−	0.952949i	0.998942	+	0.0459929i	$0.0146452\pi$
$728$	−3.98793e6	+	3.98793e6i	−0.278881	+	0.278881i
$729$		0			0
$730$		0			0
$731$		−	1.43719e7i		−	0.994768i
$732$		0			0
$733$	−9.25323e6	−	9.25323e6i	−0.636111	−	0.636111i	0.313483	−	0.949594i	$-0.398504\pi$
$733$	−9.25323e6	−	9.25323e6i	−0.636111	−	0.636111i	−0.949594	+	0.313483i	$0.898504\pi$
$734$	−8.80907e6			−0.603517
$735$		0			0
$736$	−2.51521e6			−0.171151
$737$	−5.41754e6	−	5.41754e6i	−0.367395	−	0.367395i
$738$		0			0
$739$		−	1.43488e7i		−	0.966503i	−0.875482	−	0.483251i	$-0.839456\pi$
$739$		−	1.43488e7i		−	0.966503i	0.875482	−	0.483251i	$-0.160544\pi$
$740$		0			0
$741$		0			0
$742$	171036.	−	171036.i	0.0114045	−	0.0114045i
$743$	−1.33920e7	+	1.33920e7i	−0.889966	+	0.889966i	−0.994519	−	0.104553i	$-0.966659\pi$
$743$	−1.33920e7	+	1.33920e7i	−0.889966	+	0.889966i	0.104553	+	0.994519i	$0.466659\pi$
$744$		0			0
$745$		0			0
$746$		−	5.11428e6i		−	0.336463i
$747$		0			0
$748$	−960704.	−	960704.i	−0.0627821	−	0.0627821i
$749$	−4.08949e6			−0.266357
$750$		0			0
$751$	−1.51119e7			−0.977729			−0.488865	−	0.872360i	$-0.662589\pi$
$751$	−1.51119e7			−0.977729			−0.488865	+	0.872360i	$0.662589\pi$
$752$	8.06774e6	+	8.06774e6i	0.520244	+	0.520244i
$753$		0			0
$754$			1.54685e7i			0.990880i
$755$		0			0
$756$		0			0
$757$	−1.61045e7	+	1.61045e7i	−1.02143	+	1.02143i	−0.0216614	+	0.999765i	$0.506896\pi$
$757$	−1.61045e7	+	1.61045e7i	−1.02143	+	1.02143i	−0.999765	+	0.0216614i	$0.993104\pi$
$758$	−1.43982e7	+	1.43982e7i	−0.910198	+	0.910198i
$759$		0			0
$760$		0			0
$761$			2.01897e7i			1.26377i	0.775061	+	0.631886i	$0.217719\pi$
$761$			2.01897e7i			1.26377i	−0.775061	+	0.631886i	$0.782281\pi$
$762$		0			0
$763$	−1.04489e6	−	1.04489e6i	−0.0649769	−	0.0649769i
$764$	−477970.			−0.0296256
$765$		0			0
$766$	6.98430e6			0.430082
$767$	−2.78148e7	−	2.78148e7i	−1.70721	−	1.70721i
$768$		0			0
$769$			6.14899e6i			0.374963i	0.982268	+	0.187481i	$0.0600324\pi$
$769$			6.14899e6i			0.374963i	−0.982268	+	0.187481i	$0.939968\pi$
$770$		0			0
$771$		0			0
$772$	−1.38978e6	+	1.38978e6i	−0.0839270	+	0.0839270i
$773$	−5.16075e6	+	5.16075e6i	−0.310645	+	0.310645i	−0.845159	−	0.534515i	$-0.820494\pi$
$773$	−5.16075e6	+	5.16075e6i	−0.310645	+	0.310645i	0.534515	+	0.845159i	$0.320494\pi$
$774$		0			0
$775$		0			0
$776$			2.18714e6i			0.130383i
$777$		0			0
$778$	6.34698e6	+	6.34698e6i	0.375940	+	0.375940i
$779$	−2.60964e7			−1.54076
$780$		0			0
$781$	3.89481e6			0.228486
$782$	3.42924e6	+	3.42924e6i	0.200530	+	0.200530i
$783$		0			0
$784$			1.13598e7i			0.660054i
$785$		0			0
$786$		0			0
$787$	−2.01433e7	+	2.01433e7i	−1.15929	+	1.15929i	−0.174666	+	0.984628i	$0.555884\pi$
$787$	−2.01433e7	+	2.01433e7i	−1.15929	+	1.15929i	−0.984628	+	0.174666i	$0.944116\pi$
$788$	−1.15886e6	+	1.15886e6i	−0.0664836	+	0.0664836i
$789$		0			0
$790$		0			0
$791$		−	4.71719e6i		−	0.268066i
$792$		0			0
$793$	−3.12053e7	−	3.12053e7i	−1.76216	−	1.76216i
$794$	911180.			0.0512924
$795$		0			0
$796$	665576.			0.0372319
$797$	1.26474e7	+	1.26474e7i	0.705268	+	0.705268i	0.965536	−	0.260269i	$-0.0838111\pi$
$797$	1.26474e7	+	1.26474e7i	0.705268	+	0.705268i	−0.260269	+	0.965536i	$0.583811\pi$
$798$		0			0
$799$			1.63259e7i			0.904711i
$800$		0			0
$801$		0			0
$802$	−6.00937e6	+	6.00937e6i	−0.329908	+	0.329908i
$803$	623006.	−	623006.i	0.0340960	−	0.0340960i
$804$		0			0
$805$		0			0
$806$		−	3.99816e7i		−	2.16782i
$807$		0			0
$808$	9.55663e6	+	9.55663e6i	0.514963	+	0.514963i
$809$	2.22055e7			1.19286			0.596428	−	0.802666i	$-0.296586\pi$
$809$	2.22055e7			1.19286			0.596428	+	0.802666i	$0.296586\pi$
$810$		0			0
$811$	−1.63199e7			−0.871297			−0.435648	−	0.900117i	$-0.643481\pi$
$811$	−1.63199e7			−0.871297			−0.435648	+	0.900117i	$0.643481\pi$
$812$	672123.	+	672123.i	0.0357733	+	0.0357733i
$813$		0			0
$814$			6.99191e6i			0.369858i
$815$		0			0
$816$		0			0
$817$	1.58651e7	−	1.58651e7i	0.831547	−	0.831547i
$818$	−1.01216e7	+	1.01216e7i	−0.528890	+	0.528890i
$819$		0			0
$820$		0			0
$821$			1.30178e7i			0.674030i	0.941499	+	0.337015i	$0.109417\pi$
$821$			1.30178e7i			0.674030i	−0.941499	+	0.337015i	$0.890583\pi$
$822$		0			0
$823$	−1.21922e6	−	1.21922e6i	−0.0627457	−	0.0627457i	0.675038	−	0.737783i	$-0.264127\pi$
$823$	−1.21922e6	−	1.21922e6i	−0.0627457	−	0.0627457i	−0.737783	+	0.675038i	$0.764127\pi$
$824$	−1.83097e7			−0.939425
$825$		0			0
$826$	7.85008e6			0.400335
$827$	−1.43495e7	−	1.43495e7i	−0.729578	−	0.729578i	0.240957	−	0.970536i	$-0.422539\pi$
$827$	−1.43495e7	−	1.43495e7i	−0.729578	−	0.729578i	−0.970536	+	0.240957i	$0.922539\pi$
$828$		0			0
$829$			1.26744e7i			0.640530i	0.947328	+	0.320265i	$0.103772\pi$
$829$			1.26744e7i			0.640530i	−0.947328	+	0.320265i	$0.896228\pi$
$830$		0			0
$831$		0			0
$832$	−2.17752e7	+	2.17752e7i	−1.09057	+	1.09057i
$833$	−1.14938e7	+	1.14938e7i	−0.573921	+	0.573921i
$834$		0			0
$835$		0			0
$836$		−	2.12103e6i		−	0.104962i
$837$		0			0
$838$	−500914.	−	500914.i	−0.0246407	−	0.0246407i
$839$	2.29844e7			1.12727			0.563636	−	0.826023i	$-0.309402\pi$
$839$	2.29844e7			1.12727			0.563636	+	0.826023i	$0.309402\pi$
$840$		0			0
$841$	−6.83023e6			−0.333001
$842$	−4.68837e6	−	4.68837e6i	−0.227898	−	0.227898i
$843$		0			0
$844$		−	1.66059e6i		−	0.0802428i
$845$		0			0
$846$		0			0
$847$	−3.15805e6	+	3.15805e6i	−0.151255	+	0.151255i
$848$	736099.	−	736099.i	0.0351517	−	0.0351517i
$849$		0			0
$850$		0			0
$851$			7.68483e6i			0.363756i
$852$		0			0
$853$	7.90959e6	+	7.90959e6i	0.372204	+	0.372204i	0.868280	−	0.496075i	$-0.165226\pi$
$853$	7.90959e6	+	7.90959e6i	0.372204	+	0.372204i	−0.496075	+	0.868280i	$0.665226\pi$
$854$	8.80697e6			0.413221
$855$		0			0
$856$	−2.34432e7			−1.09353
$857$	4.10620e6	+	4.10620e6i	0.190980	+	0.190980i	0.796120	−	0.605139i	$-0.206883\pi$
$857$	4.10620e6	+	4.10620e6i	0.190980	+	0.190980i	−0.605139	+	0.796120i	$0.706883\pi$
$858$		0			0
$859$		−	8.04118e6i		−	0.371824i	−0.982566	−	0.185912i	$-0.940476\pi$
$859$		−	8.04118e6i		−	0.371824i	0.982566	−	0.185912i	$-0.0595239\pi$
$860$		0			0
$861$		0			0
$862$	−2.35228e7	+	2.35228e7i	−1.07826	+	1.07826i
$863$	−7.70658e6	+	7.70658e6i	−0.352237	+	0.352237i	−0.860941	−	0.508704i	$-0.830125\pi$
$863$	−7.70658e6	+	7.70658e6i	−0.352237	+	0.352237i	0.508704	+	0.860941i	$0.330125\pi$
$864$		0			0
$865$		0			0
$866$		−	3.45380e6i		−	0.156496i
$867$		0			0
$868$	−1.73724e6	−	1.73724e6i	−0.0782636	−	0.0782636i
$869$	−3.04282e6			−0.136687
$870$		0			0
$871$	−3.73223e7			−1.66695
$872$	−5.98986e6	−	5.98986e6i	−0.266763	−	0.266763i
$873$		0			0
$874$			7.57101e6i			0.335255i
$875$		0			0
$876$		0			0
$877$	2.59846e7	−	2.59846e7i	1.14082	−	1.14082i	0.152521	−	0.988300i	$-0.451261\pi$
$877$	2.59846e7	−	2.59846e7i	1.14082	−	1.14082i	0.988300	−	0.152521i	$-0.0487393\pi$
$878$	2.46205e7	−	2.46205e7i	1.07786	−	1.07786i
$879$		0			0
$880$		0			0
$881$		−	2.62840e7i		−	1.14091i	−0.821329	−	0.570455i	$-0.806767\pi$
$881$		−	2.62840e7i		−	1.14091i	0.821329	−	0.570455i	$-0.193233\pi$
$882$		0			0
$883$	2.82514e7	+	2.82514e7i	1.21938	+	1.21938i	0.967850	+	0.251530i	$0.0809336\pi$
$883$	2.82514e7	+	2.82514e7i	1.21938	+	1.21938i	0.251530	+	0.967850i	$0.419066\pi$
$884$	−6.61845e6			−0.284856
$885$		0			0
$886$	1.00462e7			0.429951
$887$	−3.11073e7	−	3.11073e7i	−1.32756	−	1.32756i	−0.907496	−	0.420061i	$-0.862009\pi$
$887$	−3.11073e7	−	3.11073e7i	−1.32756	−	1.32756i	−0.420061	−	0.907496i	$-0.637991\pi$
$888$		0			0
$889$		−	6.59692e6i		−	0.279954i
$890$		0			0
$891$		0			0
$892$	4.14611e6	−	4.14611e6i	0.174473	−	0.174473i
$893$	−1.80220e7	+	1.80220e7i	−0.756266	+	0.756266i
$894$		0			0
$895$		0			0
$896$		−	3.23582e6i		−	0.134652i
$897$		0			0
$898$	1.27158e7	+	1.27158e7i	0.526201	+	0.526201i
$899$	−3.53611e7			−1.45924
$900$		0			0
$901$	1.48957e6			0.0611292
$902$	−9.76595e6	−	9.76595e6i	−0.399667	−	0.399667i
$903$		0			0
$904$		−	2.70415e7i		−	1.10055i
$905$		0			0
$906$		0			0
$907$	−1.56307e7	+	1.56307e7i	−0.630900	+	0.630900i	−0.948294	−	0.317393i	$-0.897192\pi$
$907$	−1.56307e7	+	1.56307e7i	−0.630900	+	0.630900i	0.317393	+	0.948294i	$0.397192\pi$
$908$	244506.	−	244506.i	0.00984181	−	0.00984181i
$909$		0			0
$910$		0			0
$911$			1.64166e7i			0.655371i	0.944787	+	0.327685i	$0.106269\pi$
$911$			1.64166e7i			0.655371i	−0.944787	+	0.327685i	$0.893731\pi$
$912$		0			0
$913$	−1.23442e7	−	1.23442e7i	−0.490101	−	0.490101i
$914$	−1.20748e7			−0.478096
$915$		0			0
$916$	2.93713e6			0.115661
$917$	−215451.	−	215451.i	−0.00846108	−	0.00846108i
$918$		0			0
$919$		−	6.26477e6i		−	0.244690i	−0.992488	−	0.122345i	$-0.960959\pi$
$919$		−	6.26477e6i		−	0.244690i	0.992488	−	0.122345i	$-0.0390414\pi$
$920$		0			0
$921$		0			0
$922$	5.56501e6	−	5.56501e6i	0.215595	−	0.215595i
$923$	1.34160e7	−	1.34160e7i	0.518345	−	0.518345i
$924$		0			0
$925$		0			0
$926$			3.96372e7i			1.51906i
$927$		0			0
$928$	6.97171e6	+	6.97171e6i	0.265748	+	0.265748i
$929$	−2.18132e7			−0.829241			−0.414620	−	0.909995i	$-0.636086\pi$
$929$	−2.18132e7			−0.829241			−0.414620	+	0.909995i	$0.636086\pi$
$930$		0			0
$931$	−2.53759e7			−0.959504
$932$	5.55828e6	+	5.55828e6i	0.209604	+	0.209604i
$933$		0			0
$934$			1.49893e7i			0.562230i
$935$		0			0
$936$		0			0
$937$	1.22575e7	−	1.22575e7i	0.456093	−	0.456093i	−0.441277	−	0.897371i	$-0.645475\pi$
$937$	1.22575e7	−	1.22575e7i	0.456093	−	0.456093i	0.897371	+	0.441277i	$0.145475\pi$
$938$	5.26668e6	−	5.26668e6i	0.195447	−	0.195447i
$939$		0			0
$940$		0			0
$941$		−	1.86115e7i		−	0.685183i	−0.939484	−	0.342592i	$-0.888695\pi$
$941$		−	1.86115e7i		−	0.685183i	0.939484	−	0.342592i	$-0.111305\pi$
$942$		0			0
$943$	−1.07338e7	−	1.07338e7i	−0.393074	−	0.393074i
$944$	3.37849e7			1.23394
$945$		0			0
$946$	1.18743e7			0.431399
$947$	1.15031e7	+	1.15031e7i	0.416811	+	0.416811i	0.884103	−	0.467292i	$-0.154770\pi$
$947$	1.15031e7	+	1.15031e7i	0.416811	+	0.416811i	−0.467292	+	0.884103i	$0.654770\pi$
$948$		0			0
$949$		−	4.29199e6i		−	0.154701i
$950$		0			0
$951$		0			0
$952$	4.90105e6	−	4.90105e6i	0.175266	−	0.175266i
$953$	2.82233e7	−	2.82233e7i	1.00664	−	1.00664i	0.00666540	−	0.999978i	$-0.497878\pi$
$953$	2.82233e7	−	2.82233e7i	1.00664	−	1.00664i	0.999978	−	0.00666540i	$-0.00212168\pi$
$954$		0			0
$955$		0			0
$956$			2.76895e6i			0.0979873i
$957$		0			0
$958$	−2.72739e6	−	2.72739e6i	−0.0960138	−	0.0960138i
$959$	−7.48313e6			−0.262746
$960$		0			0
$961$	6.27689e7			2.19248
$962$	2.40842e7	+	2.40842e7i	0.839064	+	0.839064i
$963$		0			0
$964$		−	8.29866e6i		−	0.287618i
$965$		0			0
$966$		0			0
$967$	2.24829e7	−	2.24829e7i	0.773191	−	0.773191i	−0.205472	−	0.978663i	$-0.565873\pi$
$967$	2.24829e7	−	2.24829e7i	0.773191	−	0.773191i	0.978663	+	0.205472i	$0.0658729\pi$
$968$	−1.81037e7	+	1.81037e7i	−0.620980	+	0.620980i
$969$		0			0
$970$		0			0
$971$		−	3.49863e7i		−	1.19083i	−0.803418	−	0.595416i	$-0.796987\pi$
$971$		−	3.49863e7i		−	1.19083i	0.803418	−	0.595416i	$-0.203013\pi$
$972$		0			0
$973$	−3.58590e6	−	3.58590e6i	−0.121427	−	0.121427i
$974$	1.47901e7			0.499545
$975$		0			0
$976$	3.79032e7			1.27365
$977$	1.51625e7	+	1.51625e7i	0.508200	+	0.508200i	0.913974	−	0.405774i	$-0.132998\pi$
$977$	1.51625e7	+	1.51625e7i	0.508200	+	0.508200i	−0.405774	+	0.913974i	$0.632998\pi$
$978$		0			0
$979$			8.43886e6i			0.281402i
$980$		0			0
$981$		0			0
$982$	−7.69880e6	+	7.69880e6i	−0.254768	+	0.254768i
$983$	−1.33195e7	+	1.33195e7i	−0.439648	+	0.439648i	−0.891894	−	0.452245i	$-0.850623\pi$
$983$	−1.33195e7	+	1.33195e7i	−0.439648	+	0.439648i	0.452245	+	0.891894i	$0.350623\pi$
$984$		0			0
$985$		0			0
$986$		−	1.90104e7i		−	0.622729i
$987$		0			0
$988$	−7.30606e6	−	7.30606e6i	−0.238117	−	0.238117i
$989$	1.30510e7			0.424282
$990$		0			0
$991$	5.61948e6			0.181766			0.0908829	−	0.995862i	$-0.471031\pi$
$991$	5.61948e6			0.181766			0.0908829	+	0.995862i	$0.471031\pi$
$992$	−1.80198e7	−	1.80198e7i	−0.581394	−	0.581394i
$993$		0			0
$994$			3.78635e6i			0.121550i
$995$		0			0
$996$		0			0
$997$	−7.08932e6	+	7.08932e6i	−0.225874	+	0.225874i	−0.810967	−	0.585092i	$-0.801058\pi$
$997$	−7.08932e6	+	7.08932e6i	−0.225874	+	0.225874i	0.585092	+	0.810967i	$0.301058\pi$
$998$	1.00513e6	−	1.00513e6i	0.0319446	−	0.0319446i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.f.b.143.3		20
3.2	odd	2	inner	225.6.f.b.143.8		20
5.2	odd	4	inner	225.6.f.b.107.8		20
5.3	odd	4		45.6.f.a.17.3	yes	20
5.4	even	2		45.6.f.a.8.8	yes	20
15.2	even	4	inner	225.6.f.b.107.3		20
15.8	even	4		45.6.f.a.17.8	yes	20
15.14	odd	2		45.6.f.a.8.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.f.a.8.3	✓	20	15.14	odd	2
45.6.f.a.8.8	yes	20	5.4	even	2
45.6.f.a.17.3	yes	20	5.3	odd	4
45.6.f.a.17.8	yes	20	15.8	even	4
225.6.f.b.107.3		20	15.2	even	4	inner
225.6.f.b.107.8		20	5.2	odd	4	inner
225.6.f.b.143.3		20	1.1	even	1	trivial
225.6.f.b.143.8		20	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.49760 - 3.49760i) q^{2} -7.53358i q^{4} +(-24.1207 + 24.1207i) q^{7} +(-138.273 + 138.273i) q^{8} +O(q^{10})\)	\(q+(-3.49760 - 3.49760i) q^{2} -7.53358i q^{4} +(-24.1207 + 24.1207i) q^{7} +(-138.273 + 138.273i) q^{8} -173.562i q^{11} +(-597.849 - 597.849i) q^{13} +168.729 q^{14} +726.171 q^{16} +(734.740 + 734.740i) q^{17} +1622.15i q^{19} +(-607.051 + 607.051i) q^{22} +(-667.211 + 667.211i) q^{23} +4182.07i q^{26} +(181.715 + 181.715i) q^{28} +3698.77 q^{29} -9560.23 q^{31} +(1884.87 + 1884.87i) q^{32} -5139.65i q^{34} +(5758.92 - 5758.92i) q^{37} +(5673.62 - 5673.62i) q^{38} +16087.5i q^{41} +(-9780.29 - 9780.29i) q^{43} -1307.54 q^{44} +4667.28 q^{46} +(11110.0 + 11110.0i) q^{47} +15643.4i q^{49} +(-4503.94 + 4503.94i) q^{52} +(1013.67 - 1013.67i) q^{53} -6670.46i q^{56} +(-12936.8 - 12936.8i) q^{58} +46524.8 q^{59} +52196.0 q^{61} +(33437.9 + 33437.9i) q^{62} -36422.5i q^{64} +(31213.8 - 31213.8i) q^{67} +(5535.22 - 5535.22i) q^{68} +22440.4i q^{71} +(3589.53 + 3589.53i) q^{73} -40284.8 q^{74} +12220.6 q^{76} +(4186.44 + 4186.44i) q^{77} -17531.6i q^{79} +(56267.8 - 56267.8i) q^{82} +(71122.6 - 71122.6i) q^{83} +68415.1i q^{86} +(23998.9 + 23998.9i) q^{88} -48621.5 q^{89} +28841.0 q^{91} +(5026.49 + 5026.49i) q^{92} -77716.5i q^{94} +(7908.79 - 7908.79i) q^{97} +(54714.3 - 54714.3i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 152 q^{7}+O(q^{10}) \) 20 * q - 152 * q^7	\( 20 q - 152 q^{7} - 1844 q^{13} - 6280 q^{16} + 6512 q^{22} - 29728 q^{28} + 43120 q^{31} + 37516 q^{37} - 5408 q^{43} + 118720 q^{46} + 285256 q^{52} - 274752 q^{58} + 163360 q^{61} + 302704 q^{67} - 146324 q^{73} + 760800 q^{76} + 305744 q^{82} - 355296 q^{88} + 499120 q^{91} + 362524 q^{97}+O(q^{100}) \) 20 * q - 152 * q^7 - 1844 * q^13 - 6280 * q^16 + 6512 * q^22 - 29728 * q^28 + 43120 * q^31 + 37516 * q^37 - 5408 * q^43 + 118720 * q^46 + 285256 * q^52 - 274752 * q^58 + 163360 * q^61 + 302704 * q^67 - 146324 * q^73 + 760800 * q^76 + 305744 * q^82 - 355296 * q^88 + 499120 * q^91 + 362524 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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