LMFDB - Embedded newform 234.6.b.c.181.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("234.181");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	234.b (of order $2$, degree $1$, 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:	$37.5298138362$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 $ x^6 - 2x^5 + 2x^4 - 2946x^3 + 131769x^2 - 1332936*x + 6741792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.3
Root			$9.96927 + 9.96927i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.6.b.c.181.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} +86.8538i q^{5} +98.7774i q^{7} +64.0000i q^{8} +O(q^{10})$	\(q-4.00000i q^{2} -16.0000 q^{4} +86.8538i q^{5} +98.7774i q^{7} +64.0000i q^{8} +347.415 q^{10} -610.754i q^{11} +(592.276 - 143.186i) q^{13} +395.109 q^{14} +256.000 q^{16} +1148.60 q^{17} +2267.14i q^{19} -1389.66i q^{20} -2443.02 q^{22} -433.224 q^{23} -4418.59 q^{25} +(-572.744 - 2369.10i) q^{26} -1580.44i q^{28} -7669.59 q^{29} +7367.13i q^{31} -1024.00i q^{32} -4594.42i q^{34} -8579.19 q^{35} +10575.0i q^{37} +9068.57 q^{38} -5558.65 q^{40} +3699.17i q^{41} -6061.57 q^{43} +9772.07i q^{44} +1732.89i q^{46} -8740.69i q^{47} +7050.03 q^{49} +17674.4i q^{50} +(-9476.42 + 2290.97i) q^{52} -34784.8 q^{53} +53046.3 q^{55} -6321.75 q^{56} +30678.4i q^{58} +11949.6i q^{59} -45400.2 q^{61} +29468.5 q^{62} -4096.00 q^{64} +(12436.2 + 51441.4i) q^{65} -45698.1i q^{67} -18377.7 q^{68} +34316.8i q^{70} +23826.9i q^{71} +37759.1i q^{73} +42299.9 q^{74} -36274.3i q^{76} +60328.7 q^{77} -35307.0 q^{79} +22234.6i q^{80} +14796.7 q^{82} -31484.4i q^{83} +99760.7i q^{85} +24246.3i q^{86} +39088.3 q^{88} -59051.5i q^{89} +(14143.5 + 58503.5i) q^{91} +6931.58 q^{92} -34962.7 q^{94} -196910. q^{95} +4965.78i q^{97} -28200.1i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 96 q^{4}+O(q^{10}) $ 6 * q - 96 * q^4	$ 6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100}) $ 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$.

$n$	$145$	$209$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	4.00000i		−	0.707107i
$2$		−	4.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−16.0000			−0.500000
$5$			86.8538i			1.55369i	0.629693	+	0.776844i	$0.283181\pi$
$5$			86.8538i			1.55369i	−0.629693	+	0.776844i	$0.716819\pi$
$6$		0			0
$7$			98.7774i			0.761925i	0.924590	+	0.380963i	$0.124407\pi$
$7$			98.7774i			0.761925i	−0.924590	+	0.380963i	$0.875593\pi$
$8$			64.0000i			0.353553i
$9$		0			0
$10$	347.415			1.09862
$11$		−	610.754i		−	1.52190i	−0.648813	−	0.760948i	$-0.724734\pi$
$11$		−	610.754i		−	1.52190i	0.648813	−	0.760948i	$-0.275266\pi$
$12$		0			0
$13$	592.276	−	143.186i	0.971999	−	0.234986i
$13$	592.276	−	143.186i	0.971999	−	0.234986i
$14$	395.109			0.538762
$15$		0			0
$16$	256.000			0.250000
$17$	1148.60			0.963936			0.481968	−	0.876189i	$-0.339922\pi$
$17$	1148.60			0.963936			0.481968	+	0.876189i	$0.339922\pi$
$18$		0			0
$19$			2267.14i			1.44077i	0.693574	+	0.720386i	$0.256035\pi$
$19$			2267.14i			1.44077i	−0.693574	+	0.720386i	$0.743965\pi$
$20$		−	1389.66i		−	0.776844i
$21$		0			0
$22$	−2443.02			−1.07614
$23$	−433.224			−0.170763			−0.0853813	−	0.996348i	$-0.527211\pi$
$23$	−433.224			−0.170763			−0.0853813	+	0.996348i	$0.527211\pi$
$24$		0			0
$25$	−4418.59			−1.41395
$26$	−572.744	−	2369.10i	−0.166160	−	0.687307i
$27$		0			0
$28$		−	1580.44i		−	0.380963i
$29$	−7669.59			−1.69347			−0.846734	−	0.532016i	$-0.821434\pi$
$29$	−7669.59			−1.69347			−0.846734	+	0.532016i	$0.821434\pi$
$30$		0			0
$31$			7367.13i			1.37687i	0.725296	+	0.688437i	$0.241703\pi$
$31$			7367.13i			1.37687i	−0.725296	+	0.688437i	$0.758297\pi$
$32$		−	1024.00i		−	0.176777i
$33$		0			0
$34$		−	4594.42i		−	0.681606i
$35$	−8579.19			−1.18379
$36$		0			0
$37$			10575.0i			1.26991i	0.772547	+	0.634957i	$0.218982\pi$
$37$			10575.0i			1.26991i	−0.772547	+	0.634957i	$0.781018\pi$
$38$	9068.57			1.01878
$39$		0			0
$40$	−5558.65			−0.549312
$41$			3699.17i			0.343672i	0.985126	+	0.171836i	$0.0549700\pi$
$41$			3699.17i			0.343672i	−0.985126	+	0.171836i	$0.945030\pi$
$42$		0			0
$43$	−6061.57			−0.499936			−0.249968	−	0.968254i	$-0.580420\pi$
$43$	−6061.57			−0.499936			−0.249968	+	0.968254i	$0.580420\pi$
$44$			9772.07i			0.760948i
$45$		0			0
$46$			1732.89i			0.120747i
$47$		−	8740.69i		−	0.577166i	−0.957455	−	0.288583i	$-0.906816\pi$
$47$		−	8740.69i		−	0.577166i	0.957455	−	0.288583i	$-0.0931841\pi$
$48$		0			0
$49$	7050.03			0.419470
$50$			17674.4i			0.999812i
$51$		0			0
$52$	−9476.42	+	2290.97i	−0.485999	+	0.117493i
$53$	−34784.8			−1.70098			−0.850491	−	0.525990i	$-0.823695\pi$
$53$	−34784.8			−1.70098			−0.850491	+	0.525990i	$0.823695\pi$
$54$		0			0
$55$	53046.3			2.36455
$56$	−6321.75			−0.269381
$57$		0			0
$58$			30678.4i			1.19746i
$59$			11949.6i			0.446915i	0.974714	+	0.223457i	$0.0717344\pi$
$59$			11949.6i			0.446915i	−0.974714	+	0.223457i	$0.928266\pi$
$60$		0			0
$61$	−45400.2			−1.56219			−0.781094	−	0.624414i	$-0.785338\pi$
$61$	−45400.2			−1.56219			−0.781094	+	0.624414i	$0.785338\pi$
$62$	29468.5			0.973597
$63$		0			0
$64$	−4096.00			−0.125000
$65$	12436.2	+	51441.4i	0.365095	+	1.51018i
$66$		0			0
$67$		−	45698.1i		−	1.24369i	−0.783141	−	0.621844i	$-0.786384\pi$
$67$		−	45698.1i		−	1.24369i	0.783141	−	0.621844i	$-0.213616\pi$
$68$	−18377.7			−0.481968
$69$		0			0
$70$			34316.8i			0.837069i
$71$			23826.9i			0.560946i	0.959862	+	0.280473i	$0.0904913\pi$
$71$			23826.9i			0.560946i	−0.959862	+	0.280473i	$0.909509\pi$
$72$		0			0
$73$			37759.1i			0.829305i	0.909980	+	0.414653i	$0.136097\pi$
$73$			37759.1i			0.829305i	−0.909980	+	0.414653i	$0.863903\pi$
$74$	42299.9			0.897965
$75$		0			0
$76$		−	36274.3i		−	0.720386i
$77$	60328.7			1.15957
$78$		0			0
$79$	−35307.0			−0.636492			−0.318246	−	0.948008i	$-0.603094\pi$
$79$	−35307.0			−0.636492			−0.318246	+	0.948008i	$0.603094\pi$
$80$			22234.6i			0.388422i
$81$		0			0
$82$	14796.7			0.243013
$83$		−	31484.4i		−	0.501649i	−0.968033	−	0.250824i	$-0.919298\pi$
$83$		−	31484.4i		−	0.501649i	0.968033	−	0.250824i	$-0.0807017\pi$
$84$		0			0
$85$			99760.7i			1.49766i
$86$			24246.3i			0.353508i
$87$		0			0
$88$	39088.3			0.538071
$89$		−	59051.5i		−	0.790235i	−0.918631	−	0.395117i	$-0.870704\pi$
$89$		−	59051.5i		−	0.790235i	0.918631	−	0.395117i	$-0.129296\pi$
$90$		0			0
$91$	14143.5	+	58503.5i	0.179042	+	0.740590i
$92$	6931.58			0.0853813
$93$		0			0
$94$	−34962.7			−0.408118
$95$	−196910.			−2.23851
$96$		0			0
$97$			4965.78i			0.0535868i	0.999641	+	0.0267934i	$0.00852963\pi$
$97$			4965.78i			0.0535868i	−0.999641	+	0.0267934i	$0.991470\pi$
$98$		−	28200.1i		−	0.296610i
$99$		0			0
$100$	70697.4			0.706974
$101$	125605.			1.22519			0.612596	−	0.790396i	$-0.290125\pi$
$101$	125605.			1.22519			0.612596	+	0.790396i	$0.290125\pi$
$102$		0			0
$103$	−126352.			−1.17351			−0.586756	−	0.809764i	$-0.699595\pi$
$103$	−126352.			−1.17351			−0.586756	+	0.809764i	$0.699595\pi$
$104$	9163.90	+	37905.7i	0.0830801	+	0.343653i
$105$		0			0
$106$			139139.i			1.20278i
$107$	−33067.8			−0.279219			−0.139610	−	0.990207i	$-0.544585\pi$
$107$	−33067.8			−0.279219			−0.139610	+	0.990207i	$0.544585\pi$
$108$		0			0
$109$			147666.i			1.19046i	0.803555	+	0.595230i	$0.202939\pi$
$109$			147666.i			1.19046i	−0.803555	+	0.595230i	$0.797061\pi$
$110$		−	212185.i		−	1.67199i
$111$		0			0
$112$			25287.0i			0.190481i
$113$	−118111.			−0.870152			−0.435076	−	0.900394i	$-0.643278\pi$
$113$	−118111.			−0.870152			−0.435076	+	0.900394i	$0.643278\pi$
$114$		0			0
$115$		−	37627.1i		−	0.265312i
$116$	122713.			0.846734
$117$		0			0
$118$	47798.6			0.316016
$119$			113456.i			0.734447i
$120$		0			0
$121$	−211970.			−1.31616
$122$			181601.i			1.10463i
$123$		0			0
$124$		−	117874.i		−	0.688437i
$125$		−	112353.i		−	0.643147i
$126$		0			0
$127$	322666.			1.77518			0.887592	−	0.460631i	$-0.152377\pi$
$127$	322666.			1.77518			0.887592	+	0.460631i	$0.152377\pi$
$128$			16384.0i			0.0883883i
$129$		0			0
$130$	205766.	−	49745.0i	1.06786	−	0.258161i
$131$	357874.			1.82202			0.911008	−	0.412390i	$-0.135306\pi$
$131$	357874.			1.82202			0.911008	+	0.412390i	$0.135306\pi$
$132$		0			0
$133$	−223942.			−1.09776
$134$	−182792.			−0.879420
$135$		0			0
$136$			73510.7i			0.340803i
$137$			347064.i			1.57982i	0.613221	+	0.789912i	$0.289874\pi$
$137$			347064.i			1.57982i	−0.613221	+	0.789912i	$0.710126\pi$
$138$		0			0
$139$	−103030.			−0.452299			−0.226149	−	0.974093i	$-0.572614\pi$
$139$	−103030.			−0.452299			−0.226149	+	0.974093i	$0.572614\pi$
$140$	137267.			0.591897
$141$		0			0
$142$	95307.4			0.396649
$143$	−87451.4	−	361735.i	−0.357624	−	1.47928i
$144$		0			0
$145$		−	666133.i		−	2.63112i
$146$	151036.			0.586407
$147$		0			0
$148$		−	169199.i		−	0.634957i
$149$		−	90695.3i		−	0.334672i	−0.985900	−	0.167336i	$-0.946484\pi$
$149$		−	90695.3i		−	0.334672i	0.985900	−	0.167336i	$-0.0535164\pi$
$150$		0			0
$151$		−	104288.i		−	0.372215i	−0.982529	−	0.186107i	$-0.940413\pi$
$151$		−	104288.i		−	0.372215i	0.982529	−	0.186107i	$-0.0595873\pi$
$152$	−145097.			−0.509390
$153$		0			0
$154$		−	241315.i		−	0.819940i
$155$	−639863.			−2.13923
$156$		0			0
$157$	−314419.			−1.01803			−0.509014	−	0.860758i	$-0.669990\pi$
$157$	−314419.			−1.01803			−0.509014	+	0.860758i	$0.669990\pi$
$158$			141228.i			0.450068i
$159$		0			0
$160$	88938.3			0.274656
$161$		−	42792.7i		−	0.130108i
$162$		0			0
$163$			140732.i			0.414881i	0.978248	+	0.207441i	$0.0665134\pi$
$163$			140732.i			0.414881i	−0.978248	+	0.207441i	$0.933487\pi$
$164$		−	59186.7i		−	0.171836i
$165$		0			0
$166$	−125937.			−0.354719
$167$		−	108944.i		−	0.302281i	−0.988512	−	0.151141i	$-0.951705\pi$
$167$		−	108944.i		−	0.302281i	0.988512	−	0.151141i	$-0.0482946\pi$
$168$		0			0
$169$	330289.	−	169611.i	0.889563	−	0.456812i
$170$	399043.			1.05900
$171$		0			0
$172$	96985.2			0.249968
$173$	−401188.			−1.01914			−0.509569	−	0.860430i	$-0.670195\pi$
$173$	−401188.			−1.01914			−0.509569	+	0.860430i	$0.670195\pi$
$174$		0			0
$175$		−	436456.i		−	1.07732i
$176$		−	156353.i		−	0.380474i
$177$		0			0
$178$	−236206.			−0.558780
$179$	209535.			0.488792			0.244396	−	0.969676i	$-0.421410\pi$
$179$	209535.			0.488792			0.244396	+	0.969676i	$0.421410\pi$
$180$		0			0
$181$	236277.			0.536075			0.268038	−	0.963408i	$-0.413625\pi$
$181$	236277.			0.536075			0.268038	+	0.963408i	$0.413625\pi$
$182$	234014.	−	56574.1i	0.523676	−	0.126602i
$183$		0			0
$184$		−	27726.3i		−	0.0603737i
$185$	−918476.			−1.97305
$186$		0			0
$187$		−	701515.i		−	1.46701i
$188$			139851.i			0.288583i
$189$		0			0
$190$			787640.i			1.58287i
$191$	−509719.			−1.01099			−0.505496	−	0.862829i	$-0.668690\pi$
$191$	−509719.			−1.01099			−0.505496	+	0.862829i	$0.668690\pi$
$192$		0			0
$193$		−	58146.8i		−	0.112365i	−0.998421	−	0.0561827i	$-0.982107\pi$
$193$		−	58146.8i		−	0.112365i	0.998421	−	0.0561827i	$-0.0178929\pi$
$194$	19863.1			0.0378916
$195$		0			0
$196$	−112801.			−0.209735
$197$		−	91830.0i		−	0.168585i	−0.996441	−	0.0842925i	$-0.973137\pi$
$197$		−	91830.0i		−	0.168585i	0.996441	−	0.0842925i	$-0.0268630\pi$
$198$		0			0
$199$	562291.			1.00653			0.503267	−	0.864131i	$-0.332131\pi$
$199$	562291.			1.00653			0.503267	+	0.864131i	$0.332131\pi$
$200$		−	282790.i		−	0.499906i
$201$		0			0
$202$		−	502421.i		−	0.866341i
$203$		−	757582.i		−	1.29030i
$204$		0			0
$205$	−321287.			−0.533960
$206$			505406.i			0.829798i
$207$		0			0
$208$	151623.	−	36655.6i	0.243000	−	0.0587465i
$209$	1.38467e6			2.19270
$210$		0			0
$211$	650662.			1.00612			0.503059	−	0.864252i	$-0.332208\pi$
$211$	650662.			1.00612			0.503059	+	0.864252i	$0.332208\pi$
$212$	556557.			0.850491
$213$		0			0
$214$			132271.i			0.197438i
$215$		−	526471.i		−	0.776745i
$216$		0			0
$217$	−727706.			−1.04907
$218$	590665.			0.841782
$219$		0			0
$220$	−848741.			−1.18228
$221$	680291.	−	164464.i	0.936944	−	0.226511i
$222$		0			0
$223$			302795.i			0.407743i	0.978998	+	0.203871i	$0.0653525\pi$
$223$			302795.i			0.407743i	−0.978998	+	0.203871i	$0.934648\pi$
$224$	101148.			0.134691
$225$		0			0
$226$			472445.i			0.615290i
$227$		−	723191.i		−	0.931512i	−0.884913	−	0.465756i	$-0.845782\pi$
$227$		−	723191.i		−	0.931512i	0.884913	−	0.465756i	$-0.154218\pi$
$228$		0			0
$229$			1.39378e6i			1.75633i	0.478356	+	0.878166i	$0.341233\pi$
$229$			1.39378e6i			1.75633i	−0.478356	+	0.878166i	$0.658767\pi$
$230$	−150509.			−0.187604
$231$		0			0
$232$		−	490854.i		−	0.598731i
$233$	319385.			0.385411			0.192706	−	0.981257i	$-0.438274\pi$
$233$	319385.			0.385411			0.192706	+	0.981257i	$0.438274\pi$
$234$		0			0
$235$	759162.			0.896736
$236$		−	191194.i		−	0.223457i
$237$		0			0
$238$	453824.			0.519332
$239$			1.52265e6i			1.72427i	0.506682	+	0.862133i	$0.330872\pi$
$239$			1.52265e6i			1.72427i	−0.506682	+	0.862133i	$0.669128\pi$
$240$		0			0
$241$			607839.i			0.674134i	0.941481	+	0.337067i	$0.109435\pi$
$241$			607839.i			0.674134i	−0.941481	+	0.337067i	$0.890565\pi$
$242$			847879.i			0.930669i
$243$		0			0
$244$	726403.			0.781094
$245$			612323.i			0.651726i
$246$		0			0
$247$	324623.	+	1.34277e6i	0.338561	+	1.40043i
$248$	−471496.			−0.486798
$249$		0			0
$250$	−449412.			−0.454773
$251$	1.05049e6			1.05247			0.526234	−	0.850340i	$-0.323604\pi$
$251$	1.05049e6			1.05247			0.526234	+	0.850340i	$0.323604\pi$
$252$		0			0
$253$			264593.i			0.259883i
$254$		−	1.29066e6i		−	1.25524i
$255$		0			0
$256$	65536.0			0.0625000
$257$	−236240.			−0.223111			−0.111555	−	0.993758i	$-0.535583\pi$
$257$	−236240.			−0.223111			−0.111555	+	0.993758i	$0.535583\pi$
$258$		0			0
$259$	−1.04457e6			−0.967580
$260$	−198980.	−	823063.i	−0.182547	−	0.755092i
$261$		0			0
$262$		−	1.43150e6i		−	1.28836i
$263$	−928964.			−0.828151			−0.414076	−	0.910242i	$-0.635895\pi$
$263$	−928964.			−0.828151			−0.414076	+	0.910242i	$0.635895\pi$
$264$		0			0
$265$		−	3.02119e6i		−	2.64280i
$266$			895770.i			0.776233i
$267$		0			0
$268$			731170.i			0.621844i
$269$	−827650.			−0.697374			−0.348687	−	0.937239i	$-0.613372\pi$
$269$	−827650.			−0.697374			−0.348687	+	0.937239i	$0.613372\pi$
$270$		0			0
$271$			660781.i			0.546555i	0.961935	+	0.273278i	$0.0881078\pi$
$271$			660781.i			0.546555i	−0.961935	+	0.273278i	$0.911892\pi$
$272$	294043.			0.240984
$273$		0			0
$274$	1.38826e6			1.11710
$275$			2.69867e6i			2.15188i
$276$		0			0
$277$	−1.38275e6			−1.08279			−0.541395	−	0.840769i	$-0.682104\pi$
$277$	−1.38275e6			−1.08279			−0.541395	+	0.840769i	$0.682104\pi$
$278$			412119.i			0.319823i
$279$		0			0
$280$		−	549068.i		−	0.418534i
$281$		−	1.44893e6i		−	1.09467i	−0.836915	−	0.547333i	$-0.815643\pi$
$281$		−	1.44893e6i		−	1.09467i	0.836915	−	0.547333i	$-0.184357\pi$
$282$		0			0
$283$	1.22044e6			0.905841			0.452921	−	0.891551i	$-0.350382\pi$
$283$	1.22044e6			0.905841			0.452921	+	0.891551i	$0.350382\pi$
$284$		−	381230.i		−	0.280473i
$285$		0			0
$286$	−1.44694e6	+	349806.i	−1.04601	+	0.252878i
$287$	−365394.			−0.261853
$288$		0			0
$289$	−100565.			−0.0708279
$290$	−2.66453e6			−1.86048
$291$		0			0
$292$		−	604146.i		−	0.414653i
$293$			2.19465e6i			1.49347i	0.665123	+	0.746734i	$0.268379\pi$
$293$			2.19465e6i			1.49347i	−0.665123	+	0.746734i	$0.731621\pi$
$294$		0			0
$295$	−1.03787e6			−0.694366
$296$	−676798.			−0.448983
$297$		0			0
$298$	−362781.			−0.236649
$299$	−256588.	+	62031.5i	−0.165981	+	0.0401268i
$300$		0			0
$301$		−	598746.i		−	0.380914i
$302$	−417154.			−0.263196
$303$		0			0
$304$			580389.i			0.360193i
$305$		−	3.94318e6i		−	2.42715i
$306$		0			0
$307$		−	83619.1i		−	0.0506360i	−0.999679	−	0.0253180i	$-0.991940\pi$
$307$		−	83619.1i		−	0.0506360i	0.999679	−	0.0253180i	$-0.00805983\pi$
$308$	−965259.			−0.579785
$309$		0			0
$310$			2.55945e6i			1.51267i
$311$	−491767.			−0.288309			−0.144155	−	0.989555i	$-0.546046\pi$
$311$	−491767.			−0.288309			−0.144155	+	0.989555i	$0.546046\pi$
$312$		0			0
$313$	896969.			0.517508			0.258754	−	0.965943i	$-0.416688\pi$
$313$	896969.			0.517508			0.258754	+	0.965943i	$0.416688\pi$
$314$			1.25768e6i			0.719854i
$315$		0			0
$316$	564912.			0.318246
$317$			3.15449e6i			1.76312i	0.472074	+	0.881559i	$0.343506\pi$
$317$			3.15449e6i			1.76312i	−0.472074	+	0.881559i	$0.656494\pi$
$318$		0			0
$319$			4.68423e6i			2.57728i
$320$		−	355753.i		−	0.194211i
$321$		0			0
$322$	−171171.			−0.0920004
$323$			2.60405e6i			1.38881i
$324$		0			0
$325$	−2.61702e6	+	632679.i	−1.37436	+	0.332258i
$326$	562928.			0.293365
$327$		0			0
$328$	−236747.			−0.121507
$329$	863382.			0.439757
$330$		0			0
$331$		−	597196.i		−	0.299604i	−0.988716	−	0.149802i	$-0.952136\pi$
$331$		−	597196.i		−	0.299604i	0.988716	−	0.149802i	$-0.0478636\pi$
$332$			503750.i			0.250824i
$333$		0			0
$334$	−435775.			−0.213745
$335$	3.96906e6			1.93230
$336$		0			0
$337$	3.61808e6			1.73541			0.867707	−	0.497077i	$-0.165593\pi$
$337$	3.61808e6			1.73541			0.867707	+	0.497077i	$0.165593\pi$
$338$	−678444.	−	1.32115e6i	−0.323015	−	0.629016i
$339$		0			0
$340$		−	1.59617e6i		−	0.748828i
$341$	4.49951e6			2.09546
$342$		0			0
$343$			2.35653e6i			1.08153i
$344$		−	387941.i		−	0.176754i
$345$		0			0
$346$			1.60475e6i			0.720639i
$347$	1.62893e6			0.726236			0.363118	−	0.931743i	$-0.381712\pi$
$347$	1.62893e6			0.726236			0.363118	+	0.931743i	$0.381712\pi$
$348$		0			0
$349$		−	2.60140e6i		−	1.14326i	−0.820513	−	0.571628i	$-0.806312\pi$
$349$		−	2.60140e6i		−	1.14326i	0.820513	−	0.571628i	$-0.193688\pi$
$350$	−1.74583e6			−0.761782
$351$		0			0
$352$	−625412.			−0.269036
$353$		−	1.30441e6i		−	0.557155i	−0.960414	−	0.278578i	$-0.910137\pi$
$353$		−	1.30441e6i		−	0.557155i	0.960414	−	0.278578i	$-0.0898630\pi$
$354$		0			0
$355$	−2.06945e6			−0.871535
$356$			944824.i			0.395117i
$357$		0			0
$358$		−	838140.i		−	0.345628i
$359$		−	213674.i		−	0.0875013i	−0.999042	−	0.0437507i	$-0.986069\pi$
$359$		−	213674.i		−	0.0875013i	0.999042	−	0.0437507i	$-0.0139307\pi$
$360$		0			0
$361$	−2.66384e6			−1.07582
$362$		−	945110.i		−	0.379062i
$363$		0			0
$364$	−226296.	−	936055.i	−0.0895208	−	0.370295i
$365$	−3.27952e6			−1.28848
$366$		0			0
$367$	1.15670e6			0.448285			0.224142	−	0.974556i	$-0.428042\pi$
$367$	1.15670e6			0.448285			0.224142	+	0.974556i	$0.428042\pi$
$368$	−110905.			−0.0426906
$369$		0			0
$370$			3.67390e6i			1.39516i
$371$		−	3.43595e6i		−	1.29602i
$372$		0			0
$373$	1.97983e6			0.736810			0.368405	−	0.929665i	$-0.379904\pi$
$373$	1.97983e6			0.736810			0.368405	+	0.929665i	$0.379904\pi$
$374$	−2.80606e6			−1.03733
$375$		0			0
$376$	559404.			0.204059
$377$	−4.54251e6	+	1.09818e6i	−1.64605	+	0.397941i
$378$		0			0
$379$		−	3.86155e6i		−	1.38090i	−0.723379	−	0.690451i	$-0.757412\pi$
$379$		−	3.86155e6i		−	1.38090i	0.723379	−	0.690451i	$-0.242588\pi$
$380$	3.15056e6			1.11925
$381$		0			0
$382$			2.03888e6i			0.714879i
$383$			1.55518e6i			0.541731i	0.962617	+	0.270866i	$0.0873099\pi$
$383$			1.55518e6i			0.541731i	−0.962617	+	0.270866i	$0.912690\pi$
$384$		0			0
$385$			5.23978e6i			1.80161i
$386$	−232587.			−0.0794544
$387$		0			0
$388$		−	79452.4i		−	0.0267934i
$389$	−2.85249e6			−0.955763			−0.477881	−	0.878424i	$-0.658595\pi$
$389$	−2.85249e6			−0.955763			−0.477881	+	0.878424i	$0.658595\pi$
$390$		0			0
$391$	−497603.			−0.164604
$392$			451202.i			0.148305i
$393$		0			0
$394$	−367320.			−0.119208
$395$		−	3.06655e6i		−	0.988911i
$396$		0			0
$397$		−	1.25222e6i		−	0.398752i	−0.979923	−	0.199376i	$-0.936109\pi$
$397$		−	1.25222e6i		−	0.398752i	0.979923	−	0.199376i	$-0.0638915\pi$
$398$		−	2.24917e6i		−	0.711727i
$399$		0			0
$400$	−1.13116e6			−0.353487
$401$		−	4.20050e6i		−	1.30449i	−0.758009	−	0.652244i	$-0.773828\pi$
$401$		−	4.20050e6i		−	1.30449i	0.758009	−	0.652244i	$-0.226172\pi$
$402$		0			0
$403$	1.05487e6	+	4.36337e6i	0.323546	+	1.33832i
$404$	−2.00968e6			−0.612596
$405$		0			0
$406$	−3.03033e6			−0.912377
$407$	6.45870e6			1.93268
$408$		0			0
$409$		−	650694.i		−	0.192340i	−0.995365	−	0.0961698i	$-0.969341\pi$
$409$		−	650694.i		−	0.192340i	0.995365	−	0.0961698i	$-0.0306592\pi$
$410$			1.28515e6i			0.377567i
$411$		0			0
$412$	2.02163e6			0.586756
$413$	−1.18035e6			−0.340516
$414$		0			0
$415$	2.73454e6			0.779406
$416$	−146622.	−	606491.i	−0.0415400	−	0.171827i
$417$		0			0
$418$		−	5.53867e6i		−	1.55047i
$419$	6.82010e6			1.89782			0.948912	−	0.315541i	$-0.102186\pi$
$419$	6.82010e6			1.89782			0.948912	+	0.315541i	$0.102186\pi$
$420$		0			0
$421$		−	3.11659e6i		−	0.856988i	−0.903545	−	0.428494i	$-0.859044\pi$
$421$		−	3.11659e6i		−	0.856988i	0.903545	−	0.428494i	$-0.140956\pi$
$422$		−	2.60265e6i		−	0.711433i
$423$		0			0
$424$		−	2.22623e6i		−	0.601388i
$425$	−5.07521e6			−1.36296
$426$		0			0
$427$		−	4.48451e6i		−	1.19027i
$428$	529084.			0.139610
$429$		0			0
$430$	−2.10588e6			−0.549241
$431$			4.88213e6i			1.26595i	0.774173	+	0.632974i	$0.218166\pi$
$431$			4.88213e6i			1.26595i	−0.774173	+	0.632974i	$0.781834\pi$
$432$		0			0
$433$	3.64275e6			0.933705			0.466852	−	0.884335i	$-0.345388\pi$
$433$	3.64275e6			0.933705			0.466852	+	0.884335i	$0.345388\pi$
$434$			2.91082e6i			0.741808i
$435$		0			0
$436$		−	2.36266e6i		−	0.595230i
$437$		−	982180.i		−	0.246030i
$438$		0			0
$439$	−1.70122e6			−0.421308			−0.210654	−	0.977561i	$-0.567559\pi$
$439$	−1.70122e6			−0.421308			−0.210654	+	0.977561i	$0.567559\pi$
$440$			3.39497e6i			0.835995i
$441$		0			0
$442$	−657856.	−	2.72116e6i	−0.160168	−	0.662520i
$443$	5.85057e6			1.41641			0.708205	−	0.706007i	$-0.249505\pi$
$443$	5.85057e6			1.41641			0.708205	+	0.706007i	$0.249505\pi$
$444$		0			0
$445$	5.12885e6			1.22778
$446$	1.21118e6			0.288318
$447$		0			0
$448$		−	404592.i		−	0.0952406i
$449$			1.56399e6i			0.366115i	0.983102	+	0.183058i	$0.0585995\pi$
$449$			1.56399e6i			0.366115i	−0.983102	+	0.183058i	$0.941401\pi$
$450$		0			0
$451$	2.25928e6			0.523033
$452$	1.88978e6			0.435076
$453$		0			0
$454$	−2.89276e6			−0.658678
$455$	−5.08125e6	+	1.22842e6i	−1.15065	+	0.278175i
$456$		0			0
$457$			336948.i			0.0754697i	0.999288	+	0.0377349i	$0.0120142\pi$
$457$			336948.i			0.0754697i	−0.999288	+	0.0377349i	$0.987986\pi$
$458$	5.57513e6			1.24191
$459$		0			0
$460$			602034.i			0.132656i
$461$		−	3.89174e6i		−	0.852887i	−0.904514	−	0.426444i	$-0.859766\pi$
$461$		−	3.89174e6i		−	0.852887i	0.904514	−	0.426444i	$-0.140234\pi$
$462$		0			0
$463$		−	6.34214e6i		−	1.37494i	−0.726213	−	0.687470i	$-0.758721\pi$
$463$		−	6.34214e6i		−	1.37494i	0.726213	−	0.687470i	$-0.241279\pi$
$464$	−1.96341e6			−0.423367
$465$		0			0
$466$		−	1.27754e6i		−	0.272527i
$467$	−2.48691e6			−0.527677			−0.263839	−	0.964567i	$-0.584989\pi$
$467$	−2.48691e6			−0.527677			−0.263839	+	0.964567i	$0.584989\pi$
$468$		0			0
$469$	4.51394e6			0.947597
$470$		−	3.03665e6i		−	0.634088i
$471$		0			0
$472$	−764777.			−0.158008
$473$			3.70213e6i			0.760850i
$474$		0			0
$475$		−	1.00176e7i		−	2.03718i
$476$		−	1.81530e6i		−	0.367223i
$477$		0			0
$478$	6.09058e6			1.21924
$479$			5.31079e6i			1.05760i	0.848747	+	0.528798i	$0.177357\pi$
$479$			5.31079e6i			1.05760i	−0.848747	+	0.528798i	$0.822643\pi$
$480$		0			0
$481$	1.51419e6	+	6.26330e6i	0.298412	+	1.23436i
$482$	2.43136e6			0.476685
$483$		0			0
$484$	3.39151e6			0.658082
$485$	−431297.			−0.0832572
$486$		0			0
$487$		−	7.73506e6i		−	1.47789i	−0.673767	−	0.738944i	$-0.735325\pi$
$487$		−	7.73506e6i		−	1.47789i	0.673767	−	0.738944i	$-0.264675\pi$
$488$		−	2.90561e6i		−	0.552317i
$489$		0			0
$490$	2.44929e6			0.460840
$491$	322490.			0.0603689			0.0301844	−	0.999544i	$-0.490391\pi$
$491$	322490.			0.0603689			0.0301844	+	0.999544i	$0.490391\pi$
$492$		0			0
$493$	−8.80932e6			−1.63239
$494$	5.37110e6	−	1.29849e6i	0.990252	−	0.239399i
$495$		0			0
$496$			1.88599e6i			0.344218i
$497$	−2.35355e6			−0.427399
$498$		0			0
$499$			414653.i			0.0745476i	0.999305	+	0.0372738i	$0.0118674\pi$
$499$			414653.i			0.0745476i	−0.999305	+	0.0372738i	$0.988133\pi$
$500$			1.79765e6i			0.321573i
$501$		0			0
$502$		−	4.20197e6i		−	0.744207i
$503$	1.11227e7			1.96016			0.980080	−	0.198604i	$-0.0636408\pi$
$503$	1.11227e7			1.96016			0.980080	+	0.198604i	$0.0636408\pi$
$504$		0			0
$505$			1.09093e7i			1.90357i
$506$	1.05837e6			0.183765
$507$		0			0
$508$	−5.16265e6			−0.887592
$509$			754143.i			0.129021i	0.997917	+	0.0645103i	$0.0205485\pi$
$509$			754143.i			0.129021i	−0.997917	+	0.0645103i	$0.979451\pi$
$510$		0			0
$511$	−3.72974e6			−0.631869
$512$		−	262144.i		−	0.0441942i
$513$		0			0
$514$			944959.i			0.157763i
$515$		−	1.09741e7i		−	1.82327i
$516$		0			0
$517$	−5.33841e6			−0.878386
$518$			4.17827e6i			0.684182i
$519$		0			0
$520$	−3.29225e6	+	795920.i	−0.533930	+	0.129081i
$521$	−6.34188e6			−1.02358			−0.511792	−	0.859109i	$-0.671018\pi$
$521$	−6.34188e6			−1.02358			−0.511792	+	0.859109i	$0.671018\pi$
$522$		0			0
$523$	7.64810e6			1.22264			0.611321	−	0.791382i	$-0.290638\pi$
$523$	7.64810e6			1.22264			0.611321	+	0.791382i	$0.290638\pi$
$524$	−5.72599e6			−0.911008
$525$		0			0
$526$			3.71586e6i			0.585591i
$527$			8.46192e6i			1.32722i
$528$		0			0
$529$	−6.24866e6			−0.970840
$530$	−1.20848e7			−1.86874
$531$		0			0
$532$	3.58308e6			0.548880
$533$	529669.	+	2.19093e6i	0.0807582	+	0.334049i
$534$		0			0
$535$		−	2.87206e6i		−	0.433820i
$536$	2.92468e6			0.439710
$537$		0			0
$538$			3.31060e6i			0.493118i
$539$		−	4.30584e6i		−	0.638390i
$540$		0			0
$541$			5.40473e6i			0.793927i	0.917834	+	0.396964i	$0.129936\pi$
$541$			5.40473e6i			0.793927i	−0.917834	+	0.396964i	$0.870064\pi$
$542$	2.64312e6			0.386473
$543$		0			0
$544$		−	1.17617e6i		−	0.170401i
$545$	−1.28254e7			−1.84960
$546$		0			0
$547$	6.06535e6			0.866737			0.433368	−	0.901217i	$-0.357325\pi$
$547$	6.06535e6			0.866737			0.433368	+	0.901217i	$0.357325\pi$
$548$		−	5.55303e6i		−	0.789912i
$549$		0			0
$550$	1.07947e7			1.52161
$551$		−	1.73881e7i		−	2.43990i
$552$		0			0
$553$		−	3.48753e6i		−	0.484960i
$554$			5.53100e6i			0.765648i
$555$		0			0
$556$	1.64847e6			0.226149
$557$		−	6.81169e6i		−	0.930287i	−0.885235	−	0.465143i	$-0.846003\pi$
$557$		−	6.81169e6i		−	0.930287i	0.885235	−	0.465143i	$-0.153997\pi$
$558$		0			0
$559$	−3.59012e6	+	867932.i	−0.485937	+	0.117478i
$560$	−2.19627e6			−0.295949
$561$		0			0
$562$	−5.79572e6			−0.774046
$563$	−4.93741e6			−0.656491			−0.328245	−	0.944593i	$-0.606457\pi$
$563$	−4.93741e6			−0.656491			−0.328245	+	0.944593i	$0.606457\pi$
$564$		0			0
$565$		−	1.02584e7i		−	1.35194i
$566$		−	4.88178e6i		−	0.640526i
$567$		0			0
$568$	−1.52492e6			−0.198324
$569$	−9.18371e6			−1.18915			−0.594576	−	0.804039i	$-0.702680\pi$
$569$	−9.18371e6			−1.18915			−0.594576	+	0.804039i	$0.702680\pi$
$570$		0			0
$571$	−1.11741e7			−1.43424			−0.717120	−	0.696949i	$-0.754540\pi$
$571$	−1.11741e7			−1.43424			−0.717120	+	0.696949i	$0.754540\pi$
$572$	1.39922e6	+	5.78776e6i	0.178812	+	0.739640i
$573$		0			0
$574$			1.46158e6i			0.185158i
$575$	1.91424e6			0.241449
$576$		0			0
$577$		−	8.62732e6i		−	1.07879i	−0.842053	−	0.539394i	$-0.818653\pi$
$577$		−	8.62732e6i		−	1.07879i	0.842053	−	0.539394i	$-0.181347\pi$
$578$			402262.i			0.0500829i
$579$		0			0
$580$			1.06581e7i			1.31556i
$581$	3.10994e6			0.382219
$582$		0			0
$583$			2.12450e7i			2.58872i
$584$	−2.41658e6			−0.293204
$585$		0			0
$586$	8.77859e6			1.05604
$587$		−	5.86783e6i		−	0.702882i	−0.936210	−	0.351441i	$-0.885692\pi$
$587$		−	5.86783e6i		−	0.702882i	0.936210	−	0.351441i	$-0.114308\pi$
$588$		0			0
$589$	−1.67023e7			−1.98376
$590$			4.15149e6i			0.490991i
$591$		0			0
$592$			2.70719e6i			0.317479i
$593$			1.28187e7i			1.49694i	0.663166	+	0.748472i	$0.269212\pi$
$593$			1.28187e7i			1.49694i	−0.663166	+	0.748472i	$0.730788\pi$
$594$		0			0
$595$	−9.85410e6			−1.14110
$596$			1.45112e6i			0.167336i
$597$		0			0
$598$	248126.	+	1.02635e6i	0.0283739	+	0.117366i
$599$	8.82262e6			1.00469			0.502343	−	0.864668i	$-0.332471\pi$
$599$	8.82262e6			1.00469			0.502343	+	0.864668i	$0.332471\pi$
$600$		0			0
$601$	8.16061e6			0.921587			0.460793	−	0.887507i	$-0.347565\pi$
$601$	8.16061e6			0.921587			0.460793	+	0.887507i	$0.347565\pi$
$602$	−2.39499e6			−0.269347
$603$		0			0
$604$			1.66862e6i			0.186107i
$605$		−	1.84104e7i		−	2.04491i
$606$		0			0
$607$	8.03055e6			0.884654			0.442327	−	0.896854i	$-0.354153\pi$
$607$	8.03055e6			0.884654			0.442327	+	0.896854i	$0.354153\pi$
$608$	2.32155e6			0.254695
$609$		0			0
$610$	−1.57727e7			−1.71626
$611$	−1.25154e6	−	5.17690e6i	−0.135626	−	0.561005i
$612$		0			0
$613$		−	3.99383e6i		−	0.429278i	−0.976693	−	0.214639i	$-0.931143\pi$
$613$		−	3.99383e6i		−	0.429278i	0.976693	−	0.214639i	$-0.0688575\pi$
$614$	−334476.			−0.0358051
$615$		0			0
$616$			3.86104e6i			0.409970i
$617$			1.20933e7i			1.27888i	0.768840	+	0.639442i	$0.220834\pi$
$617$			1.20933e7i			1.27888i	−0.768840	+	0.639442i	$0.779166\pi$
$618$		0			0
$619$		−	1.31953e7i		−	1.38418i	−0.721813	−	0.692088i	$-0.756691\pi$
$619$		−	1.31953e7i		−	1.38418i	0.721813	−	0.692088i	$-0.243309\pi$
$620$	1.02378e7			1.06962
$621$		0			0
$622$			1.96707e6i			0.203865i
$623$	5.83295e6			0.602100
$624$		0			0
$625$	−4.04979e6			−0.414699
$626$		−	3.58788e6i		−	0.365933i
$627$		0			0
$628$	5.03070e6			0.509014
$629$			1.21464e7i			1.22412i
$630$		0			0
$631$			1.16207e7i			1.16188i	0.813947	+	0.580939i	$0.197314\pi$
$631$			1.16207e7i			1.16188i	−0.813947	+	0.580939i	$0.802686\pi$
$632$		−	2.25965e6i		−	0.225034i
$633$		0			0
$634$	1.26180e7			1.24671
$635$			2.80247e7i			2.75808i
$636$		0			0
$637$	4.17557e6	−	1.00947e6i	0.407724	−	0.0985696i
$638$	1.87369e7			1.82241
$639$		0			0
$640$	−1.42301e6			−0.137328
$641$	1.15298e7			1.10835			0.554173	−	0.832402i	$-0.313035\pi$
$641$	1.15298e7			1.10835			0.554173	+	0.832402i	$0.313035\pi$
$642$		0			0
$643$		−	2.90732e6i		−	0.277310i	−0.990341	−	0.138655i	$-0.955722\pi$
$643$		−	2.90732e6i		−	0.277310i	0.990341	−	0.138655i	$-0.0442778\pi$
$644$			684683.i			0.0650541i
$645$		0			0
$646$	1.04162e7			0.982038
$647$	4.58122e6			0.430249			0.215125	−	0.976587i	$-0.430984\pi$
$647$	4.58122e6			0.430249			0.215125	+	0.976587i	$0.430984\pi$
$648$		0			0
$649$	7.29829e6			0.680157
$650$	2.53072e6	+	1.04681e7i	0.234942	+	0.971816i
$651$		0			0
$652$		−	2.25171e6i		−	0.207441i
$653$	−1.95066e7			−1.79019			−0.895096	−	0.445874i	$-0.852893\pi$
$653$	−1.95066e7			−1.79019			−0.895096	+	0.445874i	$0.852893\pi$
$654$		0			0
$655$			3.10827e7i			2.83084i
$656$			946987.i			0.0859181i
$657$		0			0
$658$		−	3.45353e6i		−	0.310955i
$659$	−8.41225e6			−0.754568			−0.377284	−	0.926098i	$-0.623142\pi$
$659$	−8.41225e6			−0.754568			−0.377284	+	0.926098i	$0.623142\pi$
$660$		0			0
$661$			1.64620e7i			1.46548i	0.680510	+	0.732739i	$0.261758\pi$
$661$			1.64620e7i			1.46548i	−0.680510	+	0.732739i	$0.738242\pi$
$662$	−2.38879e6			−0.211852
$663$		0			0
$664$	2.01500e6			0.177360
$665$		−	1.94503e7i		−	1.70558i
$666$		0			0
$667$	3.32265e6			0.289181
$668$			1.74310e6i			0.151141i
$669$		0			0
$670$		−	1.58762e7i		−	1.36634i
$671$			2.77284e7i			2.37748i
$672$		0			0
$673$	9.88194e6			0.841016			0.420508	−	0.907289i	$-0.361852\pi$
$673$	9.88194e6			0.841016			0.420508	+	0.907289i	$0.361852\pi$
$674$		−	1.44723e7i		−	1.22712i
$675$		0			0
$676$	−5.28462e6	+	2.71378e6i	−0.444782	+	0.228406i
$677$	731933.			0.0613761			0.0306881	−	0.999529i	$-0.490230\pi$
$677$	731933.			0.0613761			0.0306881	+	0.999529i	$0.490230\pi$
$678$		0			0
$679$	−490506.			−0.0408291
$680$	−6.38468e6			−0.529501
$681$		0			0
$682$		−	1.79980e7i		−	1.48171i
$683$			6.46151e6i			0.530008i	0.964247	+	0.265004i	$0.0853733\pi$
$683$			6.46151e6i			0.530008i	−0.964247	+	0.265004i	$0.914627\pi$
$684$		0			0
$685$	−3.01439e7			−2.45455
$686$	9.42614e6			0.764757
$687$		0			0
$688$	−1.55176e6			−0.124984
$689$	−2.06022e7	+	4.98069e6i	−1.65335	+	0.399707i
$690$		0			0
$691$			1.16211e7i			0.925877i	0.886390	+	0.462939i	$0.153205\pi$
$691$			1.16211e7i			0.925877i	−0.886390	+	0.462939i	$0.846795\pi$
$692$	6.41901e6			0.509569
$693$		0			0
$694$		−	6.51570e6i		−	0.513526i
$695$		−	8.94852e6i		−	0.702731i
$696$		0			0
$697$			4.24888e6i			0.331278i
$698$	−1.04056e7			−0.808404
$699$		0			0
$700$			6.98330e6i			0.538661i
$701$	1.39904e6			0.107531			0.0537655	−	0.998554i	$-0.482878\pi$
$701$	1.39904e6			0.107531			0.0537655	+	0.998554i	$0.482878\pi$
$702$		0			0
$703$	−2.39750e7			−1.82966
$704$			2.50165e6i			0.190237i
$705$		0			0
$706$	−5.21763e6			−0.393968
$707$			1.24069e7i			0.933504i
$708$		0			0
$709$		−	1.84802e7i		−	1.38067i	−0.723489	−	0.690336i	$-0.757463\pi$
$709$		−	1.84802e7i		−	1.38067i	0.723489	−	0.690336i	$-0.242537\pi$
$710$			8.27781e6i			0.616268i
$711$		0			0
$712$	3.77930e6			0.279390
$713$		−	3.19162e6i		−	0.235118i
$714$		0			0
$715$	3.14181e7	−	7.59549e6i	2.29834	−	0.555636i
$716$	−3.35256e6			−0.244396
$717$		0			0
$718$	−854694.			−0.0618728
$719$	−7.61846e6			−0.549598			−0.274799	−	0.961502i	$-0.588611\pi$
$719$	−7.61846e6			−0.549598			−0.274799	+	0.961502i	$0.588611\pi$
$720$		0			0
$721$		−	1.24807e7i		−	0.894128i
$722$			1.06554e7i			0.760721i
$723$		0			0
$724$	−3.78044e6			−0.268038
$725$	3.38888e7			2.39448
$726$		0			0
$727$	2.55002e7			1.78940			0.894699	−	0.446670i	$-0.147390\pi$
$727$	2.55002e7			1.78940			0.894699	+	0.446670i	$0.147390\pi$
$728$	−3.74422e6	+	905185.i	−0.261838	+	0.0633008i
$729$		0			0
$730$			1.31181e7i			0.911094i
$731$	−6.96235e6			−0.481906
$732$		0			0
$733$			1.27470e7i			0.876293i	0.898903	+	0.438147i	$0.144365\pi$
$733$			1.27470e7i			0.876293i	−0.898903	+	0.438147i	$0.855635\pi$
$734$		−	4.62678e6i		−	0.316985i
$735$		0			0
$736$			443621.i			0.0301868i
$737$	−2.79103e7			−1.89276
$738$		0			0
$739$			1.54752e7i			1.04238i	0.853441	+	0.521190i	$0.174512\pi$
$739$			1.54752e7i			1.04238i	−0.853441	+	0.521190i	$0.825488\pi$
$740$	1.46956e7			0.986526
$741$		0			0
$742$	−1.37438e7			−0.916425
$743$		−	3.52596e6i		−	0.234318i	−0.993113	−	0.117159i	$-0.962621\pi$
$743$		−	3.52596e6i		−	0.234318i	0.993113	−	0.117159i	$-0.0373787\pi$
$744$		0			0
$745$	7.87723e6			0.519976
$746$		−	7.91931e6i		−	0.521003i
$747$		0			0
$748$			1.12242e7i			0.733505i
$749$		−	3.26635e6i		−	0.212744i
$750$		0			0
$751$	8.20700e6			0.530988			0.265494	−	0.964113i	$-0.414465\pi$
$751$	8.20700e6			0.530988			0.265494	+	0.964113i	$0.414465\pi$
$752$		−	2.23762e6i		−	0.144292i
$753$		0			0
$754$	4.39271e6	+	1.81701e7i	0.281387	+	1.16393i
$755$	9.05785e6			0.578306
$756$		0			0
$757$	−1.27038e7			−0.805736			−0.402868	−	0.915258i	$-0.631987\pi$
$757$	−1.27038e7			−0.805736			−0.402868	+	0.915258i	$0.631987\pi$
$758$	−1.54462e7			−0.976446
$759$		0			0
$760$		−	1.26022e7i		−	0.791433i
$761$		−	1.41876e7i		−	0.888073i	−0.896009	−	0.444037i	$-0.853546\pi$
$761$		−	1.41876e7i		−	0.888073i	0.896009	−	0.444037i	$-0.146454\pi$
$762$		0			0
$763$	−1.45861e7			−0.907041
$764$	8.15551e6			0.505496
$765$		0			0
$766$	6.22072e6			0.383062
$767$	1.71102e6	+	7.07748e6i	0.105019	+	0.434401i
$768$		0			0
$769$		−	1.10700e7i		−	0.675045i	−0.941318	−	0.337522i	$-0.890411\pi$
$769$		−	1.10700e7i		−	0.675045i	0.941318	−	0.337522i	$-0.109589\pi$
$770$	2.09591e7			1.27393
$771$		0			0
$772$			930349.i			0.0561827i
$773$		−	5.17639e6i		−	0.311586i	−0.987790	−	0.155793i	$-0.950207\pi$
$773$		−	5.17639e6i		−	0.311586i	0.987790	−	0.155793i	$-0.0497933\pi$
$774$		0			0
$775$		−	3.25523e7i		−	1.94683i
$776$	−317810.			−0.0189458
$777$		0			0
$778$			1.14100e7i			0.675826i
$779$	−8.38655e6			−0.495153
$780$		0			0
$781$	1.45524e7			0.853701
$782$			1.99041e6i			0.116393i
$783$		0			0
$784$	1.80481e6			0.104868
$785$		−	2.73085e7i		−	1.58170i

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:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} +86.8538i q^{5} +98.7774i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} +86.8538i q^{5} +98.7774i q^{7} +64.0000i q^{8} +347.415 q^{10} -610.754i q^{11} +(592.276 - 143.186i) q^{13} +395.109 q^{14} +256.000 q^{16} +1148.60 q^{17} +2267.14i q^{19} -1389.66i q^{20} -2443.02 q^{22} -433.224 q^{23} -4418.59 q^{25} +(-572.744 - 2369.10i) q^{26} -1580.44i q^{28} -7669.59 q^{29} +7367.13i q^{31} -1024.00i q^{32} -4594.42i q^{34} -8579.19 q^{35} +10575.0i q^{37} +9068.57 q^{38} -5558.65 q^{40} +3699.17i q^{41} -6061.57 q^{43} +9772.07i q^{44} +1732.89i q^{46} -8740.69i q^{47} +7050.03 q^{49} +17674.4i q^{50} +(-9476.42 + 2290.97i) q^{52} -34784.8 q^{53} +53046.3 q^{55} -6321.75 q^{56} +30678.4i q^{58} +11949.6i q^{59} -45400.2 q^{61} +29468.5 q^{62} -4096.00 q^{64} +(12436.2 + 51441.4i) q^{65} -45698.1i q^{67} -18377.7 q^{68} +34316.8i q^{70} +23826.9i q^{71} +37759.1i q^{73} +42299.9 q^{74} -36274.3i q^{76} +60328.7 q^{77} -35307.0 q^{79} +22234.6i q^{80} +14796.7 q^{82} -31484.4i q^{83} +99760.7i q^{85} +24246.3i q^{86} +39088.3 q^{88} -59051.5i q^{89} +(14143.5 + 58503.5i) q^{91} +6931.58 q^{92} -34962.7 q^{94} -196910. q^{95} +4965.78i q^{97} -28200.1i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{4}+O(q^{10}) \) 6 * q - 96 * q^4	\( 6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100}) \) 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Introduction

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