LMFDB - Embedded newform 3700.2.d.i.149.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(149,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.d (of order $2$, degree $1$, not 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:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{6} + 139x^{4} + 273x^{2} + 144 $ x^8 + 23x^6 + 139x^4 + 273*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.8
Root			$2.24418i$ of defining polynomial
Character	$\chi$	$=$	3700.149
Dual form			3700.2.d.i.149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24418i q^{3} -1.24418i q^{7} -7.52471 q^{9} +O(q^{10})$	$q+3.24418i q^{3} -1.24418i q^{7} -7.52471 q^{9} -6.19063 q^{11} +1.66592i q^{13} -5.66592i q^{17} +4.94645 q^{19} +4.03635 q^{21} -4.48836i q^{23} -14.6790i q^{27} +4.48836 q^{29} +0.385383 q^{31} -20.0835i q^{33} -1.00000i q^{37} -5.40453 q^{39} -0.190629 q^{41} +4.56106i q^{43} +6.08765i q^{47} +5.45201 q^{49} +18.3813 q^{51} +1.70227i q^{53} +16.0472i q^{57} +6.10298 q^{59} +2.00000 q^{61} +9.36211i q^{63} -13.4348i q^{67} +14.5611 q^{69} -16.6790 q^{71} -0.786097i q^{73} +7.70227i q^{77} +10.9464 q^{79} +25.0472 q^{81} +13.8052i q^{83} +14.5611i q^{87} -5.33183 q^{89} +2.07270 q^{91} +1.25025i q^{93} +0.561064i q^{97} +46.5827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 26 q^{9}+O(q^{10}) $ 8 * q - 26 * q^9	$ 8 q - 26 q^{9} + 10 q^{11} + 38 q^{21} - 4 q^{29} - 8 q^{31} - 4 q^{39} + 58 q^{41} - 2 q^{49} + 28 q^{51} + 20 q^{59} + 16 q^{61} + 88 q^{69} - 34 q^{71} + 48 q^{79} + 56 q^{81} + 8 q^{89} + 28 q^{91} + 44 q^{99}+O(q^{100}) $ 8 * q - 26 * q^9 + 10 * q^11 + 38 * q^21 - 4 * q^29 - 8 * q^31 - 4 * q^39 + 58 * q^41 - 2 * q^49 + 28 * q^51 + 20 * q^59 + 16 * q^61 + 88 * q^69 - 34 * q^71 + 48 * q^79 + 56 * q^81 + 8 * q^89 + 28 * q^91 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	3.24418i	1.87303i	0.350629	+	0.936515i	$0.385968\pi$
$3$	3.24418i	1.87303i	−0.350629	+	0.936515i	$0.614032\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.24418i	− 0.470256i	−0.971964	−	0.235128i	$-0.924449\pi$
$7$	− 1.24418i	− 0.470256i	0.971964	−	0.235128i	$-0.0755510\pi$
$8$	0	0
$9$	−7.52471	−2.50824
$10$	0	0
$11$	−6.19063	−1.86654	−0.933272	−	0.359169i	$-0.883060\pi$
$11$	−6.19063	−1.86654	−0.933272	+	0.359169i	$0.883060\pi$
$12$	0	0
$13$	1.66592i	0.462042i	0.972949	+	0.231021i	$0.0742066\pi$
$13$	1.66592i	0.462042i	−0.972949	+	0.231021i	$0.925793\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.66592i	− 1.37419i	−0.726569	−	0.687093i	$-0.758886\pi$
$17$	− 5.66592i	− 1.37419i	0.726569	−	0.687093i	$-0.241114\pi$
$18$	0	0
$19$	4.94645	1.13479	0.567396	−	0.823445i	$-0.307951\pi$
$19$	4.94645	1.13479	0.567396	+	0.823445i	$0.307951\pi$
$20$	0	0
$21$	4.03635	0.880804
$22$	0	0
$23$	− 4.48836i	− 0.935888i	−0.883758	−	0.467944i	$-0.844995\pi$
$23$	− 4.48836i	− 0.935888i	0.883758	−	0.467944i	$-0.155005\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 14.6790i	− 2.82497i
$28$	0	0
$29$	4.48836	0.833468	0.416734	−	0.909028i	$-0.363175\pi$
$29$	4.48836	0.833468	0.416734	+	0.909028i	$0.363175\pi$
$30$	0	0
$31$	0.385383	0.0692169	0.0346084	−	0.999401i	$-0.488982\pi$
$31$	0.385383	0.0692169	0.0346084	+	0.999401i	$0.488982\pi$
$32$	0	0
$33$	− 20.0835i	− 3.49609i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	0	0
$39$	−5.40453	−0.865418
$40$	0	0
$41$	−0.190629	−0.0297712	−0.0148856	−	0.999889i	$-0.504738\pi$
$41$	−0.190629	−0.0297712	−0.0148856	+	0.999889i	$0.504738\pi$
$42$	0	0
$43$	4.56106i	0.695556i	0.937577	+	0.347778i	$0.113064\pi$
$43$	4.56106i	0.695556i	−0.937577	+	0.347778i	$0.886936\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.08765i	0.887975i	0.896033	+	0.443987i	$0.146437\pi$
$47$	6.08765i	0.887975i	−0.896033	+	0.443987i	$0.853563\pi$
$48$	0	0
$49$	5.45201	0.778859
$50$	0	0
$51$	18.3813	2.57389
$52$	0	0
$53$	1.70227i	0.233824i	0.993142	+	0.116912i	$0.0372996\pi$
$53$	1.70227i	0.233824i	−0.993142	+	0.116912i	$0.962700\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.0472i	2.12550i
$58$	0	0
$59$	6.10298	0.794540	0.397270	−	0.917702i	$-0.369958\pi$
$59$	6.10298	0.794540	0.397270	+	0.917702i	$0.369958\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	9.36211i	1.17951i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.4348i	− 1.64132i	−0.571414	−	0.820662i	$-0.693605\pi$
$67$	− 13.4348i	− 1.64132i	0.571414	−	0.820662i	$-0.306395\pi$
$68$	0	0
$69$	14.5611	1.75295
$70$	0	0
$71$	−16.6790	−1.97943	−0.989716	−	0.143046i	$-0.954310\pi$
$71$	−16.6790	−1.97943	−0.989716	+	0.143046i	$0.954310\pi$
$72$	0	0
$73$	− 0.786097i	− 0.0920057i	−0.998941	−	0.0460028i	$-0.985352\pi$
$73$	− 0.786097i	− 0.0920057i	0.998941	−	0.0460028i	$-0.0146483\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.70227i	0.877755i
$78$	0	0
$79$	10.9464	1.23157	0.615786	−	0.787914i	$-0.288839\pi$
$79$	10.9464	1.23157	0.615786	+	0.787914i	$0.288839\pi$
$80$	0	0
$81$	25.0472	2.78302
$82$	0	0
$83$	13.8052i	1.51532i	0.652648	+	0.757661i	$0.273658\pi$
$83$	13.8052i	1.51532i	−0.652648	+	0.757661i	$0.726342\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	14.5611i	1.56111i
$88$	0	0
$89$	−5.33183	−0.565173	−0.282586	−	0.959242i	$-0.591192\pi$
$89$	−5.33183	−0.565173	−0.282586	+	0.959242i	$0.591192\pi$
$90$	0	0
$91$	2.07270	0.217278
$92$	0	0
$93$	1.25025i	0.129645i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.561064i	0.0569674i	0.999594	+	0.0284837i	$0.00906787\pi$
$97$	0.561064i	0.0569674i	−0.999594	+	0.0284837i	$0.990932\pi$
$98$	0	0
$99$	46.5827	4.68174
$100$	0	0
$101$	10.1696	1.01191	0.505957	−	0.862559i	$-0.331139\pi$
$101$	10.1696	1.01191	0.505957	+	0.862559i	$0.331139\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.50751i	− 0.339084i	−0.985523	−	0.169542i	$-0.945771\pi$
$107$	− 3.50751i	− 0.339084i	0.985523	−	0.169542i	$-0.0542288\pi$
$108$	0	0
$109$	14.4540	1.38444	0.692219	−	0.721687i	$-0.256633\pi$
$109$	14.4540	1.38444	0.692219	+	0.721687i	$0.256633\pi$
$110$	0	0
$111$	3.24418	0.307924
$112$	0	0
$113$	− 18.7153i	− 1.76059i	−0.474425	−	0.880296i	$-0.657344\pi$
$113$	− 18.7153i	− 1.76059i	0.474425	−	0.880296i	$-0.342656\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 12.5355i	− 1.15891i
$118$	0	0
$119$	−7.04943	−0.646220
$120$	0	0
$121$	27.3239	2.48399
$122$	0	0
$123$	− 0.618435i	− 0.0557624i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.80525i	− 0.870075i	−0.900412	−	0.435037i	$-0.856735\pi$
$127$	− 9.80525i	− 0.870075i	0.900412	−	0.435037i	$-0.143265\pi$
$128$	0	0
$129$	−14.7969	−1.30280
$130$	0	0
$131$	13.7899	1.20483	0.602415	−	0.798183i	$-0.294205\pi$
$131$	13.7899	1.20483	0.602415	+	0.798183i	$0.294205\pi$
$132$	0	0
$133$	− 6.15428i	− 0.533644i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.53779i	− 0.473125i	−0.971616	−	0.236563i	$-0.923979\pi$
$137$	− 5.53779i	− 0.473125i	0.971616	−	0.236563i	$-0.0760209\pi$
$138$	0	0
$139$	−21.4307	−1.81773	−0.908863	−	0.417094i	$-0.863049\pi$
$139$	−21.4307	−1.81773	−0.908863	+	0.417094i	$0.863049\pi$
$140$	0	0
$141$	−19.7494	−1.66320
$142$	0	0
$143$	− 10.3131i	− 0.862422i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.6873i	1.45883i
$148$	0	0
$149$	−16.9787	−1.39095	−0.695474	−	0.718552i	$-0.744805\pi$
$149$	−16.9787	−1.39095	−0.695474	+	0.718552i	$0.744805\pi$
$150$	0	0
$151$	−6.56106	−0.533932	−0.266966	−	0.963706i	$-0.586021\pi$
$151$	−6.56106	−0.533932	−0.266966	+	0.963706i	$0.586021\pi$
$152$	0	0
$153$	42.6344i	3.44679i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.19063i	− 0.653683i	−0.945079	−	0.326842i	$-0.894016\pi$
$157$	− 8.19063i	− 0.653683i	0.945079	−	0.326842i	$-0.105984\pi$
$158$	0	0
$159$	−5.52246	−0.437960
$160$	0	0
$161$	−5.58434	−0.440108
$162$	0	0
$163$	− 21.7131i	− 1.70070i	−0.526217	−	0.850350i	$-0.676390\pi$
$163$	− 21.7131i	− 1.70070i	0.526217	−	0.850350i	$-0.323610\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.3318i	1.34118i	0.741829	+	0.670589i	$0.233958\pi$
$167$	17.3318i	1.34118i	−0.741829	+	0.670589i	$0.766042\pi$
$168$	0	0
$169$	10.2247	0.786517
$170$	0	0
$171$	−37.2206	−2.84633
$172$	0	0
$173$	− 21.8355i	− 1.66012i	−0.557671	−	0.830062i	$-0.688305\pi$
$173$	− 21.8355i	− 1.66012i	0.557671	−	0.830062i	$-0.311695\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.7992i	1.48820i
$178$	0	0
$179$	8.03028	0.600211	0.300106	−	0.953906i	$-0.402978\pi$
$179$	8.03028	0.600211	0.300106	+	0.953906i	$0.402978\pi$
$180$	0	0
$181$	−2.16961	−0.161266	−0.0806329	−	0.996744i	$-0.525694\pi$
$181$	−2.16961	−0.161266	−0.0806329	+	0.996744i	$0.525694\pi$
$182$	0	0
$183$	6.48836i	0.479634i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	35.0756i	2.56498i
$188$	0	0
$189$	−18.2633	−1.32846
$190$	0	0
$191$	−26.4115	−1.91107	−0.955536	−	0.294875i	$-0.904722\pi$
$191$	−26.4115	−1.91107	−0.955536	+	0.294875i	$0.904722\pi$
$192$	0	0
$193$	− 1.40453i	− 0.101100i	−0.998722	−	0.0505502i	$-0.983903\pi$
$193$	− 1.40453i	− 0.101100i	0.998722	−	0.0505502i	$-0.0160975\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.4992i	− 1.46051i	−0.683177	−	0.730253i	$-0.739402\pi$
$197$	− 20.4992i	− 1.46051i	0.683177	−	0.730253i	$-0.260598\pi$
$198$	0	0
$199$	3.89702	0.276252	0.138126	−	0.990415i	$-0.455892\pi$
$199$	3.89702	0.276252	0.138126	+	0.990415i	$0.455892\pi$
$200$	0	0
$201$	43.5850	3.07425
$202$	0	0
$203$	− 5.58434i	− 0.391944i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	33.7736i	2.34743i
$208$	0	0
$209$	−30.6216	−2.11814
$210$	0	0
$211$	6.05737	0.417007	0.208503	−	0.978022i	$-0.433141\pi$
$211$	6.05737	0.417007	0.208503	+	0.978022i	$0.433141\pi$
$212$	0	0
$213$	− 54.1097i	− 3.70753i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.479487i	− 0.0325497i
$218$	0	0
$219$	2.55024	0.172329
$220$	0	0
$221$	9.43894	0.634932
$222$	0	0
$223$	8.29361i	0.555381i	0.960671	+	0.277691i	$0.0895690\pi$
$223$	8.29361i	0.555381i	−0.960671	+	0.277691i	$0.910431\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.3813i	1.22001i	0.792399	+	0.610003i	$0.208832\pi$
$227$	18.3813i	1.22001i	−0.792399	+	0.610003i	$0.791168\pi$
$228$	0	0
$229$	13.1674	0.870123	0.435062	−	0.900401i	$-0.356727\pi$
$229$	13.1674	0.870123	0.435062	+	0.900401i	$0.356727\pi$
$230$	0	0
$231$	−24.9875	−1.64406
$232$	0	0
$233$	− 10.2480i	− 0.671369i	−0.941975	−	0.335684i	$-0.891032\pi$
$233$	− 10.2480i	− 0.671369i	0.941975	−	0.335684i	$-0.108968\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	35.5123i	2.30677i
$238$	0	0
$239$	10.7666	0.696436	0.348218	−	0.937414i	$-0.386787\pi$
$239$	10.7666	0.696436	0.348218	+	0.937414i	$0.386787\pi$
$240$	0	0
$241$	14.8435	0.956152	0.478076	−	0.878319i	$-0.341334\pi$
$241$	14.8435	0.956152	0.478076	+	0.878319i	$0.341334\pi$
$242$	0	0
$243$	37.2206i	2.38770i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.24036i	0.524322i
$248$	0	0
$249$	−44.7867	−2.83824
$250$	0	0
$251$	24.4842	1.54543	0.772716	−	0.634752i	$-0.218898\pi$
$251$	24.4842	1.54543	0.772716	+	0.634752i	$0.218898\pi$
$252$	0	0
$253$	27.7858i	1.74688i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.07045i	− 0.565799i	−0.959150	−	0.282899i	$-0.908704\pi$
$257$	− 9.07045i	− 0.565799i	0.959150	−	0.282899i	$-0.0912963\pi$
$258$	0	0
$259$	−1.24418	−0.0773097
$260$	0	0
$261$	−33.7736	−2.09054
$262$	0	0
$263$	− 25.3430i	− 1.56272i	−0.624081	−	0.781359i	$-0.714527\pi$
$263$	− 25.3430i	− 1.56272i	0.624081	−	0.781359i	$-0.285473\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 17.2974i	− 1.05859i
$268$	0	0
$269$	13.1310	0.800611	0.400306	−	0.916382i	$-0.368904\pi$
$269$	13.1310	0.800611	0.400306	+	0.916382i	$0.368904\pi$
$270$	0	0
$271$	26.3966	1.60348	0.801739	−	0.597674i	$-0.203908\pi$
$271$	26.3966	1.60348	0.801739	+	0.597674i	$0.203908\pi$
$272$	0	0
$273$	6.72422i	0.406968i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 24.3602i	− 1.46366i	−0.681485	−	0.731832i	$-0.738665\pi$
$277$	− 24.3602i	− 1.46366i	0.681485	−	0.731832i	$-0.261335\pi$
$278$	0	0
$279$	−2.89990	−0.173612
$280$	0	0
$281$	−3.43894	−0.205150	−0.102575	−	0.994725i	$-0.532708\pi$
$281$	−3.43894	−0.205150	−0.102575	+	0.994725i	$0.532708\pi$
$282$	0	0
$283$	26.3813i	1.56820i	0.620633	+	0.784102i	$0.286876\pi$
$283$	26.3813i	1.56820i	−0.620633	+	0.784102i	$0.713124\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.237177i	0.0140001i
$288$	0	0
$289$	−15.1026	−0.888388
$290$	0	0
$291$	−1.82019	−0.106702
$292$	0	0
$293$	− 25.6404i	− 1.49793i	−0.662611	−	0.748964i	$-0.730552\pi$
$293$	− 25.6404i	− 1.49793i	0.662611	−	0.748964i	$-0.269448\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	90.8722i	5.27294i
$298$	0	0
$299$	7.47723	0.432420
$300$	0	0
$301$	5.67479	0.327090
$302$	0	0
$303$	32.9921i	1.89534i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.20978i	0.183192i	0.995796	+	0.0915958i	$0.0291968\pi$
$307$	3.20978i	0.183192i	−0.995796	+	0.0915958i	$0.970803\pi$
$308$	0	0
$309$	−6.48836	−0.369110
$310$	0	0
$311$	31.5337	1.78811	0.894055	−	0.447957i	$-0.147848\pi$
$311$	31.5337	1.78811	0.894055	+	0.447957i	$0.147848\pi$
$312$	0	0
$313$	1.12587i	0.0636380i	0.999494	+	0.0318190i	$0.0101300\pi$
$313$	1.12587i	0.0636380i	−0.999494	+	0.0318190i	$0.989870\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.415662i	− 0.0233459i	−0.999932	−	0.0116729i	$-0.996284\pi$
$317$	− 0.415662i	− 0.0233459i	0.999932	−	0.0116729i	$-0.00371570\pi$
$318$	0	0
$319$	−27.7858	−1.55571
$320$	0	0
$321$	11.3790	0.635114
$322$	0	0
$323$	− 28.0262i	− 1.55942i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	46.8913i	2.59309i
$328$	0	0
$329$	7.57414	0.417576
$330$	0	0
$331$	30.5913	1.68145	0.840726	−	0.541461i	$-0.182128\pi$
$331$	30.5913	1.68145	0.840726	+	0.541461i	$0.182128\pi$
$332$	0	0
$333$	7.52471i	0.412352i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.52697i	− 0.518967i	−0.965748	−	0.259483i	$-0.916448\pi$
$337$	− 9.52697i	− 0.518967i	0.965748	−	0.259483i	$-0.0835523\pi$
$338$	0	0
$339$	60.7160	3.29764
$340$	0	0
$341$	−2.38577	−0.129196
$342$	0	0
$343$	− 15.4926i	− 0.836520i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.46509i	0.0786501i	0.999226	+	0.0393250i	$0.0125208\pi$
$347$	1.46509i	0.0786501i	−0.999226	+	0.0393250i	$0.987479\pi$
$348$	0	0
$349$	−24.8224	−1.32872	−0.664358	−	0.747415i	$-0.731295\pi$
$349$	−24.8224	−1.32872	−0.664358	+	0.747415i	$0.731295\pi$
$350$	0	0
$351$	24.4540	1.30526
$352$	0	0
$353$	11.1221i	0.591971i	0.955192	+	0.295986i	$0.0956480\pi$
$353$	11.1221i	0.591971i	−0.955192	+	0.295986i	$0.904352\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 22.8696i	− 1.21039i
$358$	0	0
$359$	−4.50369	−0.237696	−0.118848	−	0.992912i	$-0.537920\pi$
$359$	−4.50369	−0.237696	−0.118848	+	0.992912i	$0.537920\pi$
$360$	0	0
$361$	5.46734	0.287755
$362$	0	0
$363$	88.6436i	4.65258i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.8888i	− 0.933786i	−0.884314	−	0.466893i	$-0.845373\pi$
$367$	− 17.8888i	− 0.933786i	0.884314	−	0.466893i	$-0.154627\pi$
$368$	0	0
$369$	1.43443	0.0746734
$370$	0	0
$371$	2.11793	0.109957
$372$	0	0
$373$	30.8850i	1.59916i	0.600558	+	0.799581i	$0.294945\pi$
$373$	30.8850i	1.59916i	−0.600558	+	0.799581i	$0.705055\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.47723i	0.385097i
$378$	0	0
$379$	−20.7816	−1.06748	−0.533739	−	0.845649i	$-0.679214\pi$
$379$	−20.7816	−1.06748	−0.533739	+	0.845649i	$0.679214\pi$
$380$	0	0
$381$	31.8100	1.62968
$382$	0	0
$383$	34.6599i	1.77104i	0.464602	+	0.885520i	$0.346197\pi$
$383$	34.6599i	1.77104i	−0.464602	+	0.885520i	$0.653803\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 34.3207i	− 1.74462i
$388$	0	0
$389$	8.73636	0.442951	0.221476	−	0.975166i	$-0.428913\pi$
$389$	8.73636	0.442951	0.221476	+	0.975166i	$0.428913\pi$
$390$	0	0
$391$	−25.4307	−1.28609
$392$	0	0
$393$	44.7370i	2.25668i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6.29773i	− 0.316074i	−0.987433	−	0.158037i	$-0.949483\pi$
$397$	− 6.29773i	− 0.316074i	0.987433	−	0.158037i	$-0.0505166\pi$
$398$	0	0
$399$	19.9656	0.999530
$400$	0	0
$401$	−30.7625	−1.53621	−0.768103	−	0.640326i	$-0.778799\pi$
$401$	−30.7625	−1.53621	−0.768103	+	0.640326i	$0.778799\pi$
$402$	0	0
$403$	0.642016i	0.0319811i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.19063i	0.306858i
$408$	0	0
$409$	−12.4884	−0.617510	−0.308755	−	0.951142i	$-0.599912\pi$
$409$	−12.4884	−0.617510	−0.308755	+	0.951142i	$0.599912\pi$
$410$	0	0
$411$	17.9656	0.886178
$412$	0	0
$413$	− 7.59321i	− 0.373638i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 69.5250i	− 3.40466i
$418$	0	0
$419$	−7.49631	−0.366219	−0.183109	−	0.983093i	$-0.558616\pi$
$419$	−7.49631	−0.366219	−0.183109	+	0.983093i	$0.558616\pi$
$420$	0	0
$421$	1.36249	0.0664038	0.0332019	−	0.999449i	$-0.489430\pi$
$421$	1.36249	0.0664038	0.0332019	+	0.999449i	$0.489430\pi$
$422$	0	0
$423$	− 45.8078i	− 2.22725i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.48836i	− 0.120420i
$428$	0	0
$429$	33.4575	1.61534
$430$	0	0
$431$	−25.1180	−1.20989	−0.604946	−	0.796267i	$-0.706805\pi$
$431$	−25.1180	−1.20989	−0.604946	+	0.796267i	$0.706805\pi$
$432$	0	0
$433$	18.2168i	0.875443i	0.899111	+	0.437721i	$0.144214\pi$
$433$	18.2168i	0.875443i	−0.899111	+	0.437721i	$0.855786\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 22.2015i	− 1.06204i
$438$	0	0
$439$	13.9232	0.664517	0.332258	−	0.943188i	$-0.392189\pi$
$439$	13.9232	0.664517	0.332258	+	0.943188i	$0.392189\pi$
$440$	0	0
$441$	−41.0248	−1.95356
$442$	0	0
$443$	− 0.721415i	− 0.0342754i	−0.999853	−	0.0171377i	$-0.994545\pi$
$443$	− 0.721415i	− 0.0342754i	0.999853	−	0.0171377i	$-0.00545537\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 55.0819i	− 2.60528i
$448$	0	0
$449$	16.8696	0.796127	0.398063	−	0.917358i	$-0.369682\pi$
$449$	16.8696	0.796127	0.398063	+	0.917358i	$0.369682\pi$
$450$	0	0
$451$	1.18011	0.0555694
$452$	0	0
$453$	− 21.2853i	− 1.00007i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.46221i	− 0.115177i	−0.998340	−	0.0575887i	$-0.981659\pi$
$457$	− 2.46221i	− 0.115177i	0.998340	−	0.0575887i	$-0.0183412\pi$
$458$	0	0
$459$	−83.1699	−3.88204
$460$	0	0
$461$	14.5145	0.676008	0.338004	−	0.941145i	$-0.390248\pi$
$461$	14.5145	0.676008	0.338004	+	0.941145i	$0.390248\pi$
$462$	0	0
$463$	− 33.5378i	− 1.55863i	−0.626630	−	0.779317i	$-0.715566\pi$
$463$	− 33.5378i	− 1.55863i	0.626630	−	0.779317i	$-0.284434\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	−16.7153	−0.771843
$470$	0	0
$471$	26.5719	1.22437
$472$	0	0
$473$	− 28.2359i	− 1.29829i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.8091i	− 0.586487i
$478$	0	0
$479$	−21.2550	−0.971166	−0.485583	−	0.874191i	$-0.661393\pi$
$479$	−21.2550	−0.971166	−0.485583	+	0.874191i	$0.661393\pi$
$480$	0	0
$481$	1.66592	0.0759592
$482$	0	0
$483$	− 18.1166i	− 0.824334i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.7176i	1.07475i	0.843344	+	0.537373i	$0.180583\pi$
$487$	23.7176i	1.07475i	−0.843344	+	0.537373i	$0.819417\pi$
$488$	0	0
$489$	70.4412	3.18546
$490$	0	0
$491$	11.3318	0.511398	0.255699	−	0.966756i	$-0.417694\pi$
$491$	11.3318	0.511398	0.255699	+	0.966756i	$0.417694\pi$
$492$	0	0
$493$	− 25.4307i	− 1.14534i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.7517i	0.930841i
$498$	0	0
$499$	3.65666	0.163694	0.0818472	−	0.996645i	$-0.473918\pi$
$499$	3.65666	0.163694	0.0818472	+	0.996645i	$0.473918\pi$
$500$	0	0
$501$	−56.2276	−2.51206
$502$	0	0
$503$	11.1100i	0.495370i	0.968841	+	0.247685i	$0.0796698\pi$
$503$	11.1100i	0.495370i	−0.968841	+	0.247685i	$0.920330\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	33.1709i	1.47317i
$508$	0	0
$509$	−9.62956	−0.426823	−0.213411	−	0.976962i	$-0.568457\pi$
$509$	−9.62956	−0.426823	−0.213411	+	0.976962i	$0.568457\pi$
$510$	0	0
$511$	−0.978047	−0.0432663
$512$	0	0
$513$	− 72.6089i	− 3.20576i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 37.6864i	− 1.65745i
$518$	0	0
$519$	70.8384	3.10946
$520$	0	0
$521$	14.6236	0.640670	0.320335	−	0.947304i	$-0.396205\pi$
$521$	14.6236	0.640670	0.320335	+	0.947304i	$0.396205\pi$
$522$	0	0
$523$	23.4389i	1.02491i	0.858713	+	0.512457i	$0.171264\pi$
$523$	23.4389i	1.02491i	−0.858713	+	0.512457i	$0.828736\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.18355i	− 0.0951169i
$528$	0	0
$529$	2.85460	0.124113
$530$	0	0
$531$	−45.9232	−1.99290
$532$	0	0
$533$	− 0.317572i	− 0.0137556i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	26.0517i	1.12421i
$538$	0	0
$539$	−33.7514	−1.45378
$540$	0	0
$541$	21.0150	0.903506	0.451753	−	0.892143i	$-0.350799\pi$
$541$	21.0150	0.903506	0.451753	+	0.892143i	$0.350799\pi$
$542$	0	0
$543$	− 7.03860i	− 0.302055i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.1753i	0.435064i	0.976053	+	0.217532i	$0.0698007\pi$
$547$	10.1753i	0.435064i	−0.976053	+	0.217532i	$0.930199\pi$
$548$	0	0
$549$	−15.0494	−0.642294
$550$	0	0
$551$	22.2015	0.945814
$552$	0	0
$553$	− 13.6194i	− 0.579154i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.7415i	− 1.21782i	−0.793241	−	0.608908i	$-0.791608\pi$
$557$	− 28.7415i	− 1.21782i	0.793241	−	0.608908i	$-0.208392\pi$
$558$	0	0
$559$	−7.59835	−0.321376
$560$	0	0
$561$	−113.792	−4.80428
$562$	0	0
$563$	36.7281i	1.54791i	0.633243	+	0.773953i	$0.281723\pi$
$563$	36.7281i	1.54791i	−0.633243	+	0.773953i	$0.718277\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 31.1632i	− 1.30873i
$568$	0	0
$569$	−10.2480	−0.429619	−0.214809	−	0.976656i	$-0.568913\pi$
$569$	−10.2480	−0.429619	−0.214809	+	0.976656i	$0.568913\pi$
$570$	0	0
$571$	−21.0602	−0.881344	−0.440672	−	0.897668i	$-0.645260\pi$
$571$	−21.0602	−0.881344	−0.440672	+	0.897668i	$0.645260\pi$
$572$	0	0
$573$	− 85.6838i	− 3.57949i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.14315i	0.214112i	0.994253	+	0.107056i	$0.0341424\pi$
$577$	5.14315i	0.214112i	−0.994253	+	0.107056i	$0.965858\pi$
$578$	0	0
$579$	4.55656	0.189364
$580$	0	0
$581$	17.1762	0.712590
$582$	0	0
$583$	− 10.5381i	− 0.436443i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.86962i	0.200991i	0.994938	+	0.100495i	$0.0320428\pi$
$587$	4.86962i	0.200991i	−0.994938	+	0.100495i	$0.967957\pi$
$588$	0	0
$589$	1.90628	0.0785468
$590$	0	0
$591$	66.5031	2.73557
$592$	0	0
$593$	− 17.5569i	− 0.720974i	−0.932764	−	0.360487i	$-0.882611\pi$
$593$	− 17.5569i	− 0.720974i	0.932764	−	0.360487i	$-0.117389\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.6426i	0.517429i
$598$	0	0
$599$	−45.9764	−1.87855	−0.939273	−	0.343171i	$-0.888499\pi$
$599$	−45.9764	−1.87855	−0.939273	+	0.343171i	$0.888499\pi$
$600$	0	0
$601$	−37.2764	−1.52054	−0.760268	−	0.649609i	$-0.774932\pi$
$601$	−37.2764	−1.52054	−0.760268	+	0.649609i	$0.774932\pi$
$602$	0	0
$603$	101.093i	4.11683i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.96560i	0.160959i	0.996756	+	0.0804793i	$0.0256451\pi$
$607$	3.96560i	0.160959i	−0.996756	+	0.0804793i	$0.974355\pi$
$608$	0	0
$609$	18.1166	0.734122
$610$	0	0
$611$	−10.1415	−0.410282
$612$	0	0
$613$	− 12.3239i	− 0.497757i	−0.968535	−	0.248879i	$-0.919938\pi$
$613$	− 12.3239i	− 0.497757i	0.968535	−	0.248879i	$-0.0800620\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.2551i	− 1.29854i	−0.760558	−	0.649270i	$-0.775074\pi$
$617$	− 32.2551i	− 1.29854i	0.760558	−	0.649270i	$-0.224926\pi$
$618$	0	0
$619$	−8.98755	−0.361240	−0.180620	−	0.983553i	$-0.557810\pi$
$619$	−8.98755	−0.361240	−0.180620	+	0.983553i	$0.557810\pi$
$620$	0	0
$621$	−65.8846	−2.64386
$622$	0	0
$623$	6.63377i	0.265776i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 99.3421i	− 3.96734i
$628$	0	0
$629$	−5.66592	−0.225915
$630$	0	0
$631$	−10.7246	−0.426940	−0.213470	−	0.976950i	$-0.568476\pi$
$631$	−10.7246	−0.426940	−0.213470	+	0.976950i	$0.568476\pi$
$632$	0	0
$633$	19.6512i	0.781065i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.08259i	0.359865i
$638$	0	0
$639$	125.505	4.96489
$640$	0	0
$641$	34.9321	1.37974	0.689868	−	0.723935i	$-0.257669\pi$
$641$	34.9321	1.37974	0.689868	+	0.723935i	$0.257669\pi$
$642$	0	0
$643$	− 25.1483i	− 0.991751i	−0.868394	−	0.495876i	$-0.834847\pi$
$643$	− 25.1483i	− 0.991751i	0.868394	−	0.495876i	$-0.165153\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 39.9452i	− 1.57041i	−0.619237	−	0.785204i	$-0.712558\pi$
$647$	− 39.9452i	− 1.57041i	0.619237	−	0.785204i	$-0.287442\pi$
$648$	0	0
$649$	−37.7813	−1.48305
$650$	0	0
$651$	1.55554	0.0609665
$652$	0	0
$653$	− 39.7392i	− 1.55512i	−0.628811	−	0.777558i	$-0.716458\pi$
$653$	− 39.7392i	− 1.55512i	0.628811	−	0.777558i	$-0.283542\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.91515i	0.230772i
$658$	0	0
$659$	41.4415	1.61433	0.807166	−	0.590325i	$-0.201000\pi$
$659$	41.4415	1.61433	0.807166	+	0.590325i	$0.201000\pi$
$660$	0	0
$661$	21.2210	0.825401	0.412700	−	0.910867i	$-0.364586\pi$
$661$	21.2210	0.825401	0.412700	+	0.910867i	$0.364586\pi$
$662$	0	0
$663$	30.6216i	1.18925i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 20.1454i	− 0.780033i
$668$	0	0
$669$	−26.9060	−1.04024
$670$	0	0
$671$	−12.3813	−0.477973
$672$	0	0
$673$	6.96590i	0.268516i	0.990946	+	0.134258i	$0.0428651\pi$
$673$	6.96590i	0.268516i	−0.990946	+	0.134258i	$0.957135\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16.9315i	− 0.650730i	−0.945588	−	0.325365i	$-0.894513\pi$
$677$	− 16.9315i	− 0.650730i	0.945588	−	0.325365i	$-0.105487\pi$
$678$	0	0
$679$	0.698066	0.0267893
$680$	0	0
$681$	−59.6321	−2.28511
$682$	0	0
$683$	− 49.3963i	− 1.89010i	−0.326931	−	0.945048i	$-0.606015\pi$
$683$	− 49.3963i	− 1.89010i	0.326931	−	0.945048i	$-0.393985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	42.7173i	1.62977i
$688$	0	0
$689$	−2.83583	−0.108037
$690$	0	0
$691$	−24.1026	−0.916906	−0.458453	−	0.888719i	$-0.651596\pi$
$691$	−24.1026	−0.916906	−0.458453	+	0.888719i	$0.651596\pi$
$692$	0	0
$693$	− 57.9573i	− 2.20162i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.08009i	0.0409112i
$698$	0	0
$699$	33.2464	1.25749
$700$	0	0
$701$	−40.3768	−1.52501	−0.762504	−	0.646983i	$-0.776030\pi$
$701$	−40.3768	−1.52501	−0.762504	+	0.646983i	$0.776030\pi$
$702$	0	0
$703$	− 4.94645i	− 0.186559i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.6528i	− 0.475859i
$708$	0	0
$709$	34.8435	1.30857	0.654287	−	0.756246i	$-0.272969\pi$
$709$	34.8435	1.30857	0.654287	+	0.756246i	$0.272969\pi$
$710$	0	0
$711$	−82.3689	−3.08907
$712$	0	0
$713$	− 1.72974i	− 0.0647793i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34.9289i	1.30445i
$718$	0	0
$719$	−1.35541	−0.0505483	−0.0252742	−	0.999681i	$-0.508046\pi$
$719$	−1.35541	−0.0505483	−0.0252742	+	0.999681i	$0.508046\pi$
$720$	0	0
$721$	2.48836	0.0926715
$722$	0	0
$723$	48.1549i	1.79090i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.5566i	− 0.539873i	−0.962878	−	0.269936i	$-0.912997\pi$
$727$	− 14.5566i	− 0.539873i	0.962878	−	0.269936i	$-0.0870027\pi$
$728$	0	0
$729$	−45.6089	−1.68922
$730$	0	0
$731$	25.8426	0.955823
$732$	0	0
$733$	18.0415i	0.666377i	0.942860	+	0.333189i	$0.108125\pi$
$733$	18.0415i	0.666377i	−0.942860	+	0.333189i	$0.891875\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	83.1699i	3.06360i
$738$	0	0
$739$	−7.74431	−0.284879	−0.142439	−	0.989804i	$-0.545495\pi$
$739$	−7.74431	−0.284879	−0.142439	+	0.989804i	$0.545495\pi$
$740$	0	0
$741$	−26.7332	−0.982070
$742$	0	0
$743$	− 30.4689i	− 1.11780i	−0.829236	−	0.558898i	$-0.811224\pi$
$743$	− 30.4689i	− 1.11780i	0.829236	−	0.558898i	$-0.188776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 103.881i	− 3.80079i
$748$	0	0
$749$	−4.36398	−0.159456
$750$	0	0
$751$	−12.9270	−0.471713	−0.235856	−	0.971788i	$-0.575789\pi$
$751$	−12.9270	−0.471713	−0.235856	+	0.971788i	$0.575789\pi$
$752$	0	0
$753$	79.4313i	2.89464i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.4501i	− 0.525197i	−0.964905	−	0.262598i	$-0.915421\pi$
$757$	− 14.4501i	− 0.525197i	0.964905	−	0.262598i	$-0.0845794\pi$
$758$	0	0
$759$	−90.1421	−3.27195
$760$	0	0
$761$	2.33977	0.0848168	0.0424084	−	0.999100i	$-0.486497\pi$
$761$	2.33977	0.0848168	0.0424084	+	0.999100i	$0.486497\pi$
$762$	0	0
$763$	− 17.9833i	− 0.651041i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.1670i	0.367111i
$768$	0	0
$769$	−25.0449	−0.903143	−0.451571	−	0.892235i	$-0.649136\pi$
$769$	−25.0449	−0.903143	−0.451571	+	0.892235i	$0.649136\pi$
$770$	0	0
$771$	29.4262	1.05976
$772$	0	0
$773$	12.7900i	0.460024i	0.973188	+	0.230012i	$0.0738765\pi$
$773$	12.7900i	0.460024i	−0.973188	+	0.230012i	$0.926123\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.03635i	− 0.144803i
$778$	0	0
$779$	−0.942936	−0.0337842
$780$	0	0
$781$	103.253	3.69470
$782$	0	0
$783$	− 65.8846i	− 2.35453i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 32.4224i	− 1.15573i	−0.816132	−	0.577866i	$-0.803886\pi$
$787$	− 32.4224i	− 1.15573i	0.816132	−	0.577866i	$-0.196114\pi$
$788$	0	0
$789$	82.2174	2.92702
$790$	0	0
$791$	−23.2853	−0.827929
$792$	0	0
$793$	3.33183i	0.118317i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18.4972i	− 0.655206i	−0.944816	−	0.327603i	$-0.893759\pi$
$797$	− 18.4972i	− 0.655206i	0.944816	−	0.327603i	$-0.106241\pi$
$798$	0	0
$799$	34.4921	1.22024
$800$	0	0
$801$	40.1205	1.41759
$802$	0	0
$803$	4.86643i	0.171733i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.5994i	1.49957i
$808$	0	0
$809$	6.38126	0.224353	0.112177	−	0.993688i	$-0.464218\pi$
$809$	6.38126	0.224353	0.112177	+	0.993688i	$0.464218\pi$
$810$	0	0
$811$	6.88869	0.241895	0.120947	−	0.992659i	$-0.461407\pi$
$811$	6.88869	0.241895	0.120947	+	0.992659i	$0.461407\pi$
$812$	0	0
$813$	85.6353i	3.00336i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.5611i	0.789312i
$818$	0	0
$819$	−15.5965	−0.544985
$820$	0	0
$821$	22.3921	0.781489	0.390745	−	0.920499i	$-0.372218\pi$
$821$	22.3921	0.781489	0.390745	+	0.920499i	$0.372218\pi$
$822$	0	0
$823$	1.51202i	0.0527057i	0.999653	+	0.0263528i	$0.00838934\pi$
$823$	1.51202i	0.0527057i	−0.999653	+	0.0263528i	$0.991611\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.30568i	− 0.254043i	−0.991900	−	0.127022i	$-0.959458\pi$
$827$	− 7.30568i	− 0.254043i	0.991900	−	0.127022i	$-0.0405418\pi$
$828$	0	0
$829$	47.1438	1.63737	0.818685	−	0.574242i	$-0.194703\pi$
$829$	47.1438	1.63737	0.818685	+	0.574242i	$0.194703\pi$
$830$	0	0
$831$	79.0290	2.74149
$832$	0	0
$833$	− 30.8906i	− 1.07030i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.65704i	− 0.195536i
$838$	0	0
$839$	12.8313	0.442986	0.221493	−	0.975162i	$-0.428907\pi$
$839$	12.8313	0.442986	0.221493	+	0.975162i	$0.428907\pi$
$840$	0	0
$841$	−8.85460	−0.305331
$842$	0	0
$843$	− 11.1565i	− 0.384251i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 33.9959i	− 1.16811i
$848$	0	0
$849$	−85.5856	−2.93729
$850$	0	0
$851$	−4.48836	−0.153859
$852$	0	0
$853$	15.6060i	0.534339i	0.963650	+	0.267169i	$0.0860883\pi$
$853$	15.6060i	0.534339i	−0.963650	+	0.267169i	$0.913912\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.4157i	0.834023i	0.908901	+	0.417012i	$0.136923\pi$
$857$	24.4157i	0.834023i	−0.908901	+	0.417012i	$0.863077\pi$
$858$	0	0
$859$	46.1973	1.57623	0.788116	−	0.615526i	$-0.211057\pi$
$859$	46.1973	1.57623	0.788116	+	0.615526i	$0.211057\pi$
$860$	0	0
$861$	−0.769445	−0.0262226
$862$	0	0
$863$	18.3089i	0.623244i	0.950206	+	0.311622i	$0.100872\pi$
$863$	18.3089i	0.623244i	−0.950206	+	0.311622i	$0.899128\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 48.9956i	− 1.66398i
$868$	0	0
$869$	−67.7654	−2.29878
$870$	0	0
$871$	22.3813	0.758360
$872$	0	0
$873$	− 4.22185i	− 0.142888i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.9963i	0.540155i	0.962839	+	0.270078i	$0.0870494\pi$
$877$	15.9963i	0.540155i	−0.962839	+	0.270078i	$0.912951\pi$
$878$	0	0
$879$	83.1821	2.80566
$880$	0	0
$881$	−43.2719	−1.45787	−0.728934	−	0.684584i	$-0.759984\pi$
$881$	−43.2719	−1.45787	−0.728934	+	0.684584i	$0.759984\pi$
$882$	0	0
$883$	− 47.1438i	− 1.58651i	−0.608887	−	0.793257i	$-0.708384\pi$
$883$	− 47.1438i	− 1.58651i	0.608887	−	0.793257i	$-0.291616\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.0988i	0.708428i	0.935164	+	0.354214i	$0.115251\pi$
$887$	21.0988i	0.708428i	−0.935164	+	0.354214i	$0.884749\pi$
$888$	0	0
$889$	−12.1995	−0.409158
$890$	0	0
$891$	−155.058	−5.19463
$892$	0	0
$893$	30.1122i	1.00767i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.2575i	0.809934i
$898$	0	0
$899$	1.72974	0.0576901
$900$	0	0
$901$	9.64490	0.321318
$902$	0	0
$903$	18.4101i	0.612648i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 37.0756i	− 1.23107i	−0.788108	−	0.615537i	$-0.788939\pi$
$907$	− 37.0756i	− 1.23107i	0.788108	−	0.615537i	$-0.211061\pi$
$908$	0	0
$909$	−76.5234	−2.53812
$910$	0	0
$911$	25.9187	0.858724	0.429362	−	0.903133i	$-0.358739\pi$
$911$	25.9187	0.858724	0.429362	+	0.903133i	$0.358739\pi$
$912$	0	0
$913$	− 85.4632i	− 2.82842i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 17.1572i	− 0.566579i
$918$	0	0
$919$	49.0332	1.61745	0.808727	−	0.588184i	$-0.200157\pi$
$919$	49.0332	1.61745	0.808727	+	0.588184i	$0.200157\pi$
$920$	0	0
$921$	−10.4131	−0.343123
$922$	0	0
$923$	− 27.7858i	− 0.914580i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15.0494i	− 0.494288i
$928$	0	0
$929$	39.2770	1.28864	0.644319	−	0.764757i	$-0.277141\pi$
$929$	39.2770	1.28864	0.644319	+	0.764757i	$0.277141\pi$
$930$	0	0
$931$	26.9681	0.883844
$932$	0	0
$933$	102.301i	3.34918i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 17.3855i	− 0.567958i	−0.958830	−	0.283979i	$-0.908345\pi$
$937$	− 17.3855i	− 0.567958i	0.958830	−	0.283979i	$-0.0916546\pi$
$938$	0	0
$939$	−3.65253	−0.119196
$940$	0	0
$941$	−12.4628	−0.406277	−0.203138	−	0.979150i	$-0.565114\pi$
$941$	−12.4628	−0.406277	−0.203138	+	0.979150i	$0.565114\pi$
$942$	0	0
$943$	0.855612i	0.0278626i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.6478i	1.12590i	0.826490	+	0.562951i	$0.190334\pi$
$947$	34.6478i	1.12590i	−0.826490	+	0.562951i	$0.809666\pi$
$948$	0	0
$949$	1.30957	0.0425105
$950$	0	0
$951$	1.34848	0.0437275
$952$	0	0
$953$	− 28.0880i	− 0.909861i	−0.890527	−	0.454930i	$-0.849664\pi$
$953$	− 28.0880i	− 0.909861i	0.890527	−	0.454930i	$-0.150336\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 90.1421i	− 2.91388i
$958$	0	0
$959$	−6.89002	−0.222490
$960$	0	0
$961$	−30.8515	−0.995209
$962$	0	0
$963$	26.3930i	0.850503i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.9302i	0.801700i	0.916144	+	0.400850i	$0.131285\pi$
$967$	24.9302i	0.801700i	−0.916144	+	0.400850i	$0.868715\pi$
$968$	0	0
$969$	90.9219	2.92083
$970$	0	0
$971$	−15.7858	−0.506590	−0.253295	−	0.967389i	$-0.581514\pi$
$971$	−15.7858	−0.506590	−0.253295	+	0.967389i	$0.581514\pi$
$972$	0	0
$973$	26.6637i	0.854798i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37.6525i	− 1.20461i	−0.798266	−	0.602306i	$-0.794249\pi$
$977$	− 37.6525i	− 1.20461i	0.798266	−	0.602306i	$-0.205751\pi$
$978$	0	0
$979$	33.0074	1.05492
$980$	0	0
$981$	−108.762	−3.47250
$982$	0	0
$983$	− 12.9618i	− 0.413417i	−0.978403	−	0.206708i	$-0.933725\pi$
$983$	− 12.9618i	− 0.413417i	0.978403	−	0.206708i	$-0.0662751\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.5719i	0.782132i
$988$	0	0
$989$	20.4717	0.650963
$990$	0	0
$991$	−38.7666	−1.23146	−0.615731	−	0.787956i	$-0.711139\pi$
$991$	−38.7666	−1.23146	−0.615731	+	0.787956i	$0.711139\pi$
$992$	0	0
$993$	99.2439i	3.14941i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.6478i	0.463900i	0.972728	+	0.231950i	$0.0745105\pi$
$997$	14.6478i	0.463900i	−0.972728	+	0.231950i	$0.925489\pi$
$998$	0	0
$999$	−14.6790	−0.464423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3700.2.d.i.149.8		8
5.2	odd	4		740.2.a.f.1.4	✓	4
5.3	odd	4		3700.2.a.j.1.1		4
5.4	even	2	inner	3700.2.d.i.149.1		8
15.2	even	4		6660.2.a.r.1.4		4
20.7	even	4		2960.2.a.v.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.f.1.4	✓	4	5.2	odd	4
2960.2.a.v.1.1		4	20.7	even	4
3700.2.a.j.1.1		4	5.3	odd	4
3700.2.d.i.149.1		8	5.4	even	2	inner
3700.2.d.i.149.8		8	1.1	even	1	trivial
6660.2.a.r.1.4		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.24418i q^{3} -1.24418i q^{7} -7.52471 q^{9} +O(q^{10})\)	\(q+3.24418i q^{3} -1.24418i q^{7} -7.52471 q^{9} -6.19063 q^{11} +1.66592i q^{13} -5.66592i q^{17} +4.94645 q^{19} +4.03635 q^{21} -4.48836i q^{23} -14.6790i q^{27} +4.48836 q^{29} +0.385383 q^{31} -20.0835i q^{33} -1.00000i q^{37} -5.40453 q^{39} -0.190629 q^{41} +4.56106i q^{43} +6.08765i q^{47} +5.45201 q^{49} +18.3813 q^{51} +1.70227i q^{53} +16.0472i q^{57} +6.10298 q^{59} +2.00000 q^{61} +9.36211i q^{63} -13.4348i q^{67} +14.5611 q^{69} -16.6790 q^{71} -0.786097i q^{73} +7.70227i q^{77} +10.9464 q^{79} +25.0472 q^{81} +13.8052i q^{83} +14.5611i q^{87} -5.33183 q^{89} +2.07270 q^{91} +1.25025i q^{93} +0.561064i q^{97} +46.5827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 26 q^{9}+O(q^{10}) \) 8 * q - 26 * q^9	\( 8 q - 26 q^{9} + 10 q^{11} + 38 q^{21} - 4 q^{29} - 8 q^{31} - 4 q^{39} + 58 q^{41} - 2 q^{49} + 28 q^{51} + 20 q^{59} + 16 q^{61} + 88 q^{69} - 34 q^{71} + 48 q^{79} + 56 q^{81} + 8 q^{89} + 28 q^{91} + 44 q^{99}+O(q^{100}) \) 8 * q - 26 * q^9 + 10 * q^11 + 38 * q^21 - 4 * q^29 - 8 * q^31 - 4 * q^39 + 58 * q^41 - 2 * q^49 + 28 * q^51 + 20 * q^59 + 16 * q^61 + 88 * q^69 - 34 * q^71 + 48 * q^79 + 56 * q^81 + 8 * q^89 + 28 * q^91 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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