LMFDB - Embedded newform 4012.2.b.a.237.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.7
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23721i q^{3} +0.538464i q^{5} -0.0679637i q^{7} -2.00513 q^{9} +O(q^{10})$	$q-2.23721i q^{3} +0.538464i q^{5} -0.0679637i q^{7} -2.00513 q^{9} -1.24868i q^{11} +4.00333 q^{13} +1.20466 q^{15} +(-2.13968 + 3.52445i) q^{17} -3.80395 q^{19} -0.152049 q^{21} +7.87799i q^{23} +4.71006 q^{25} -2.22574i q^{27} -9.03144i q^{29} -7.50639i q^{31} -2.79356 q^{33} +0.0365961 q^{35} +7.34445i q^{37} -8.95631i q^{39} -11.9152i q^{41} -2.82562 q^{43} -1.07969i q^{45} -0.438627 q^{47} +6.99538 q^{49} +(7.88495 + 4.78693i) q^{51} +7.43517 q^{53} +0.672368 q^{55} +8.51025i q^{57} -1.00000 q^{59} -15.0300i q^{61} +0.136276i q^{63} +2.15565i q^{65} -4.19676 q^{67} +17.6248 q^{69} +7.28442i q^{71} -10.1666i q^{73} -10.5374i q^{75} -0.0848647 q^{77} -6.00453i q^{79} -10.9948 q^{81} +14.4552 q^{83} +(-1.89779 - 1.15214i) q^{85} -20.2053 q^{87} +14.5266 q^{89} -0.272081i q^{91} -16.7934 q^{93} -2.04829i q^{95} +6.93571i q^{97} +2.50376i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	2.23721i		−	1.29166i	−0.763483	−	0.645828i	$-0.776512\pi$
$3$		−	2.23721i		−	1.29166i	0.763483	−	0.645828i	$-0.223488\pi$
$4$		0			0
$5$			0.538464i			0.240809i	0.992725	+	0.120404i	$0.0384191\pi$
$5$			0.538464i			0.240809i	−0.992725	+	0.120404i	$0.961581\pi$
$6$		0			0
$7$		−	0.0679637i		−	0.0256879i	−0.999918	−	0.0128439i	$-0.995912\pi$
$7$		−	0.0679637i		−	0.0256879i	0.999918	−	0.0128439i	$-0.00408847\pi$
$8$		0			0
$9$	−2.00513			−0.668377
$10$		0			0
$11$		−	1.24868i		−	0.376490i	−0.982122	−	0.188245i	$-0.939720\pi$
$11$		−	1.24868i		−	0.376490i	0.982122	−	0.188245i	$-0.0602799\pi$
$12$		0			0
$13$	4.00333			1.11032			0.555162	−	0.831742i	$-0.312656\pi$
$13$	4.00333			1.11032			0.555162	+	0.831742i	$0.312656\pi$
$14$		0			0
$15$	1.20466			0.311042
$16$		0			0
$17$	−2.13968	+	3.52445i	−0.518949	+	0.854805i
$17$	−2.13968	+	3.52445i	−0.518949	+	0.854805i
$18$		0			0
$19$	−3.80395			−0.872685			−0.436343	−	0.899781i	$-0.643726\pi$
$19$	−3.80395			−0.872685			−0.436343	+	0.899781i	$0.643726\pi$
$20$		0			0
$21$	−0.152049			−0.0331799
$22$		0			0
$23$			7.87799i			1.64268i	0.570442	+	0.821338i	$0.306772\pi$
$23$			7.87799i			1.64268i	−0.570442	+	0.821338i	$0.693228\pi$
$24$		0			0
$25$	4.71006			0.942011
$26$		0			0
$27$		−	2.22574i		−	0.428343i
$28$		0			0
$29$		−	9.03144i		−	1.67710i	−0.544828	−	0.838548i	$-0.683405\pi$
$29$		−	9.03144i		−	1.67710i	0.544828	−	0.838548i	$-0.316595\pi$
$30$		0			0
$31$		−	7.50639i		−	1.34819i	−0.738646	−	0.674093i	$-0.764535\pi$
$31$		−	7.50639i		−	1.34819i	0.738646	−	0.674093i	$-0.235465\pi$
$32$		0			0
$33$	−2.79356			−0.486296
$34$		0			0
$35$	0.0365961			0.00618586
$36$		0			0
$37$			7.34445i			1.20742i	0.797204	+	0.603710i	$0.206311\pi$
$37$			7.34445i			1.20742i	−0.797204	+	0.603710i	$0.793689\pi$
$38$		0			0
$39$		−	8.95631i		−	1.43416i
$40$		0			0
$41$		−	11.9152i		−	1.86083i	−0.366502	−	0.930417i	$-0.619445\pi$
$41$		−	11.9152i		−	1.86083i	0.366502	−	0.930417i	$-0.380555\pi$
$42$		0			0
$43$	−2.82562			−0.430903			−0.215452	−	0.976515i	$-0.569122\pi$
$43$	−2.82562			−0.430903			−0.215452	+	0.976515i	$0.569122\pi$
$44$		0			0
$45$		−	1.07969i		−	0.160951i
$46$		0			0
$47$	−0.438627			−0.0639803			−0.0319902	−	0.999488i	$-0.510185\pi$
$47$	−0.438627			−0.0639803			−0.0319902	+	0.999488i	$0.510185\pi$
$48$		0			0
$49$	6.99538			0.999340
$50$		0			0
$51$	7.88495	+	4.78693i	1.10411	+	0.670304i
$52$		0			0
$53$	7.43517			1.02130			0.510649	−	0.859789i	$-0.329405\pi$
$53$	7.43517			1.02130			0.510649	+	0.859789i	$0.329405\pi$
$54$		0			0
$55$	0.672368			0.0906621
$56$		0			0
$57$			8.51025i			1.12721i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$		−	15.0300i		−	1.92439i	−0.272357	−	0.962196i	$-0.587803\pi$
$61$		−	15.0300i		−	1.92439i	0.272357	−	0.962196i	$-0.412197\pi$
$62$		0			0
$63$			0.136276i			0.0171692i
$64$		0			0
$65$			2.15565i			0.267376i
$66$		0			0
$67$	−4.19676			−0.512716			−0.256358	−	0.966582i	$-0.582523\pi$
$67$	−4.19676			−0.512716			−0.256358	+	0.966582i	$0.582523\pi$
$68$		0			0
$69$	17.6248			2.12177
$70$		0			0
$71$			7.28442i			0.864502i	0.901753	+	0.432251i	$0.142281\pi$
$71$			7.28442i			0.864502i	−0.901753	+	0.432251i	$0.857719\pi$
$72$		0			0
$73$		−	10.1666i		−	1.18991i	−0.803760	−	0.594954i	$-0.797170\pi$
$73$		−	10.1666i		−	1.18991i	0.803760	−	0.594954i	$-0.202830\pi$
$74$		0			0
$75$		−	10.5374i		−	1.21675i
$76$		0			0
$77$	−0.0848647			−0.00967123
$78$		0			0
$79$		−	6.00453i		−	0.675562i	−0.941225	−	0.337781i	$-0.890324\pi$
$79$		−	6.00453i		−	0.675562i	0.941225	−	0.337781i	$-0.109676\pi$
$80$		0			0
$81$	−10.9948			−1.22165
$82$		0			0
$83$	14.4552			1.58666			0.793330	−	0.608792i	$-0.208345\pi$
$83$	14.4552			1.58666			0.793330	+	0.608792i	$0.208345\pi$
$84$		0			0
$85$	−1.89779	−	1.15214i	−0.205844	−	0.124967i
$86$		0			0
$87$	−20.2053			−2.16623
$88$		0			0
$89$	14.5266			1.53982			0.769908	−	0.638155i	$-0.220302\pi$
$89$	14.5266			1.53982			0.769908	+	0.638155i	$0.220302\pi$
$90$		0			0
$91$		−	0.272081i		−	0.0285219i
$92$		0			0
$93$	−16.7934			−1.74139
$94$		0			0
$95$		−	2.04829i		−	0.210150i
$96$		0			0
$97$			6.93571i			0.704215i	0.935960	+	0.352108i	$0.114535\pi$
$97$			6.93571i			0.704215i	−0.935960	+	0.352108i	$0.885465\pi$
$98$		0			0
$99$			2.50376i			0.251637i
$100$		0			0
$101$	−9.29985			−0.925369			−0.462685	−	0.886523i	$-0.653114\pi$
$101$	−9.29985			−0.925369			−0.462685	+	0.886523i	$0.653114\pi$
$102$		0			0
$103$	−18.3913			−1.81215			−0.906076	−	0.423116i	$-0.860936\pi$
$103$	−18.3913			−1.81215			−0.906076	+	0.423116i	$0.860936\pi$
$104$		0			0
$105$		−	0.0818733i		−	0.00799001i
$106$		0			0
$107$			4.66745i			0.451219i	0.974218	+	0.225610i	$0.0724374\pi$
$107$			4.66745i			0.451219i	−0.974218	+	0.225610i	$0.927563\pi$
$108$		0			0
$109$		−	18.8258i		−	1.80318i	−0.432589	−	0.901591i	$-0.642400\pi$
$109$		−	18.8258i		−	1.80318i	0.432589	−	0.901591i	$-0.357600\pi$
$110$		0			0
$111$	16.4311			1.55957
$112$		0			0
$113$		−	4.38853i		−	0.412838i	−0.978464	−	0.206419i	$-0.933819\pi$
$113$		−	4.38853i		−	0.412838i	0.978464	−	0.206419i	$-0.0661810\pi$
$114$		0			0
$115$	−4.24202			−0.395570
$116$		0			0
$117$	−8.02720			−0.742115
$118$		0			0
$119$	0.239535	+	0.145421i	0.0219581	+	0.0133307i
$120$		0			0
$121$	9.44081			0.858255
$122$		0			0
$123$	−26.6568			−2.40356
$124$		0			0
$125$			5.22852i			0.467653i
$126$		0			0
$127$	−7.89783			−0.700819			−0.350410	−	0.936597i	$-0.613958\pi$
$127$	−7.89783			−0.700819			−0.350410	+	0.936597i	$0.613958\pi$
$128$		0			0
$129$			6.32152i			0.556579i
$130$		0			0
$131$			0.178635i			0.0156074i	0.999970	+	0.00780371i	$0.00248402\pi$
$131$			0.178635i			0.0156074i	−0.999970	+	0.00780371i	$0.997516\pi$
$132$		0			0
$133$			0.258531i			0.0224174i
$134$		0			0
$135$	1.19848			0.103149
$136$		0			0
$137$	−0.423151			−0.0361522			−0.0180761	−	0.999837i	$-0.505754\pi$
$137$	−0.423151			−0.0361522			−0.0180761	+	0.999837i	$0.505754\pi$
$138$		0			0
$139$		−	6.07255i		−	0.515067i	−0.966269	−	0.257533i	$-0.917090\pi$
$139$		−	6.07255i		−	0.515067i	0.966269	−	0.257533i	$-0.0829097\pi$
$140$		0			0
$141$			0.981303i			0.0826406i
$142$		0			0
$143$		−	4.99886i		−	0.418026i
$144$		0			0
$145$	4.86311			0.403859
$146$		0			0
$147$		−	15.6502i		−	1.29080i
$148$		0			0
$149$	−9.23411			−0.756488			−0.378244	−	0.925706i	$-0.623472\pi$
$149$	−9.23411			−0.756488			−0.378244	+	0.925706i	$0.623472\pi$
$150$		0			0
$151$	6.53374			0.531708			0.265854	−	0.964013i	$-0.414346\pi$
$151$	6.53374			0.531708			0.265854	+	0.964013i	$0.414346\pi$
$152$		0			0
$153$	4.29034	−	7.06698i	0.346854	−	0.571332i
$154$		0			0
$155$	4.04192			0.324655
$156$		0			0
$157$	21.2750			1.69793			0.848966	−	0.528447i	$-0.177226\pi$
$157$	21.2750			1.69793			0.848966	+	0.528447i	$0.177226\pi$
$158$		0			0
$159$		−	16.6341i		−	1.31917i
$160$		0			0
$161$	0.535418			0.0421969
$162$		0			0
$163$			4.11572i			0.322368i	0.986924	+	0.161184i	$0.0515312\pi$
$163$			4.11572i			0.322368i	−0.986924	+	0.161184i	$0.948469\pi$
$164$		0			0
$165$		−	1.50423i		−	0.117104i
$166$		0			0
$167$			16.6309i			1.28694i	0.765472	+	0.643469i	$0.222506\pi$
$167$			16.6309i			1.28694i	−0.765472	+	0.643469i	$0.777494\pi$
$168$		0			0
$169$	3.02665			0.232819
$170$		0			0
$171$	7.62741			0.583283
$172$		0			0
$173$		−	14.7350i		−	1.12028i	−0.828398	−	0.560140i	$-0.810748\pi$
$173$		−	14.7350i		−	1.12028i	0.828398	−	0.560140i	$-0.189252\pi$
$174$		0			0
$175$		−	0.320113i		−	0.0241983i
$176$		0			0
$177$			2.23721i			0.168159i
$178$		0			0
$179$	4.37280			0.326838			0.163419	−	0.986557i	$-0.447748\pi$
$179$	4.37280			0.326838			0.163419	+	0.986557i	$0.447748\pi$
$180$		0			0
$181$		−	12.1219i		−	0.901015i	−0.892773	−	0.450507i	$-0.851243\pi$
$181$		−	12.1219i		−	0.901015i	0.892773	−	0.450507i	$-0.148757\pi$
$182$		0			0
$183$	−33.6253			−2.48565
$184$		0			0
$185$	−3.95472			−0.290757
$186$		0			0
$187$	4.40090	+	2.67177i	0.321826	+	0.195379i
$188$		0			0
$189$	−0.151269			−0.0110032
$190$		0			0
$191$	−1.56006			−0.112882			−0.0564410	−	0.998406i	$-0.517975\pi$
$191$	−1.56006			−0.112882			−0.0564410	+	0.998406i	$0.517975\pi$
$192$		0			0
$193$		−	17.6262i		−	1.26876i	−0.773021	−	0.634381i	$-0.781255\pi$
$193$		−	17.6262i		−	1.26876i	0.773021	−	0.634381i	$-0.218745\pi$
$194$		0			0
$195$	4.82265			0.345357
$196$		0			0
$197$		−	18.0448i		−	1.28564i	−0.766018	−	0.642820i	$-0.777764\pi$
$197$		−	18.0448i		−	1.28564i	0.766018	−	0.642820i	$-0.222236\pi$
$198$		0			0
$199$		−	12.8668i		−	0.912100i	−0.889954	−	0.456050i	$-0.849264\pi$
$199$		−	12.8668i		−	0.912100i	0.889954	−	0.456050i	$-0.150736\pi$
$200$		0			0
$201$			9.38905i			0.662253i
$202$		0			0
$203$	−0.613810			−0.0430810
$204$		0			0
$205$	6.41589			0.448105
$206$		0			0
$207$		−	15.7964i		−	1.09793i
$208$		0			0
$209$			4.74990i			0.328557i
$210$		0			0
$211$			22.9691i			1.58126i	0.612297	+	0.790628i	$0.290246\pi$
$211$			22.9691i			1.58126i	−0.612297	+	0.790628i	$0.709754\pi$
$212$		0			0
$213$	16.2968			1.11664
$214$		0			0
$215$		−	1.52150i		−	0.103765i
$216$		0			0
$217$	−0.510162			−0.0346321
$218$		0			0
$219$	−22.7448			−1.53695
$220$		0			0
$221$	−8.56586	+	14.1095i	−0.576202	+	0.949110i
$222$		0			0
$223$	−10.2087			−0.683623			−0.341812	−	0.939768i	$-0.611041\pi$
$223$	−10.2087			−0.683623			−0.341812	+	0.939768i	$0.611041\pi$
$224$		0			0
$225$	−9.44428			−0.629618
$226$		0			0
$227$		−	16.8687i		−	1.11962i	−0.828622	−	0.559809i	$-0.810875\pi$
$227$		−	16.8687i		−	1.11962i	0.828622	−	0.559809i	$-0.189125\pi$
$228$		0			0
$229$	1.50455			0.0994236			0.0497118	−	0.998764i	$-0.484170\pi$
$229$	1.50455			0.0994236			0.0497118	+	0.998764i	$0.484170\pi$
$230$		0			0
$231$			0.189861i			0.0124919i
$232$		0			0
$233$			13.2873i			0.870479i	0.900315	+	0.435240i	$0.143336\pi$
$233$			13.2873i			0.870479i	−0.900315	+	0.435240i	$0.856664\pi$
$234$		0			0
$235$		−	0.236185i		−	0.0154070i
$236$		0			0
$237$	−13.4334			−0.872595
$238$		0			0
$239$	−18.2117			−1.17802			−0.589008	−	0.808127i	$-0.700481\pi$
$239$	−18.2117			−1.17802			−0.589008	+	0.808127i	$0.700481\pi$
$240$		0			0
$241$			2.92906i			0.188678i	0.995540	+	0.0943388i	$0.0300737\pi$
$241$			2.92906i			0.188678i	−0.995540	+	0.0943388i	$0.969926\pi$
$242$		0			0
$243$			17.9206i			1.14961i
$244$		0			0
$245$			3.76676i			0.240650i
$246$		0			0
$247$	−15.2285			−0.968964
$248$		0			0
$249$		−	32.3393i		−	2.04942i
$250$		0			0
$251$	25.2406			1.59317			0.796586	−	0.604525i	$-0.206637\pi$
$251$	25.2406			1.59317			0.796586	+	0.604525i	$0.206637\pi$
$252$		0			0
$253$	9.83707			0.618451
$254$		0			0
$255$	−2.57759	+	4.24577i	−0.161415	+	0.265880i
$256$		0			0
$257$	−22.1033			−1.37877			−0.689384	−	0.724396i	$-0.742119\pi$
$257$	−22.1033			−1.37877			−0.689384	+	0.724396i	$0.742119\pi$
$258$		0			0
$259$	0.499156			0.0310160
$260$		0			0
$261$			18.1092i			1.12093i
$262$		0			0
$263$	−18.6590			−1.15056			−0.575280	−	0.817956i	$-0.695107\pi$
$263$	−18.6590			−1.15056			−0.575280	+	0.817956i	$0.695107\pi$
$264$		0			0
$265$			4.00357i			0.245938i
$266$		0			0
$267$		−	32.4991i		−	1.98891i
$268$		0			0
$269$		−	6.36955i		−	0.388358i	−0.980966	−	0.194179i	$-0.937796\pi$
$269$		−	6.36955i		−	0.388358i	0.980966	−	0.194179i	$-0.0622043\pi$
$270$		0			0
$271$	12.3499			0.750200			0.375100	−	0.926984i	$-0.377608\pi$
$271$	12.3499			0.750200			0.375100	+	0.926984i	$0.377608\pi$
$272$		0			0
$273$	−0.608704			−0.0368405
$274$		0			0
$275$		−	5.88134i		−	0.354658i
$276$		0			0
$277$		−	0.748251i		−	0.0449581i	−0.999747	−	0.0224790i	$-0.992844\pi$
$277$		−	0.748251i		−	0.0449581i	0.999747	−	0.0224790i	$-0.00715590\pi$
$278$		0			0
$279$			15.0513i			0.901097i
$280$		0			0
$281$	−3.61078			−0.215401			−0.107700	−	0.994183i	$-0.534349\pi$
$281$	−3.61078			−0.215401			−0.107700	+	0.994183i	$0.534349\pi$
$282$		0			0
$283$			7.96292i			0.473346i	0.971589	+	0.236673i	$0.0760570\pi$
$283$			7.96292i			0.473346i	−0.971589	+	0.236673i	$0.923943\pi$
$284$		0			0
$285$	−4.58247			−0.271442
$286$		0			0
$287$	−0.809799			−0.0478009
$288$		0			0
$289$	−7.84351	−	15.0824i	−0.461383	−	0.887201i
$290$		0			0
$291$	15.5167			0.909604
$292$		0			0
$293$	−7.19202			−0.420162			−0.210081	−	0.977684i	$-0.567373\pi$
$293$	−7.19202			−0.420162			−0.210081	+	0.977684i	$0.567373\pi$
$294$		0			0
$295$		−	0.538464i		−	0.0313506i
$296$		0			0
$297$	−2.77923			−0.161267
$298$		0			0
$299$			31.5382i			1.82390i
$300$		0			0
$301$			0.192040i			0.0110690i
$302$		0			0
$303$			20.8058i			1.19526i
$304$		0			0
$305$	8.09311			0.463410
$306$		0			0
$307$	−10.8238			−0.617749			−0.308874	−	0.951103i	$-0.599952\pi$
$307$	−10.8238			−0.617749			−0.308874	+	0.951103i	$0.599952\pi$
$308$		0			0
$309$			41.1453i			2.34068i
$310$		0			0
$311$			24.2798i			1.37678i	0.725339	+	0.688392i	$0.241683\pi$
$311$			24.2798i			1.37678i	−0.725339	+	0.688392i	$0.758317\pi$
$312$		0			0
$313$			23.5894i			1.33335i	0.745349	+	0.666675i	$0.232283\pi$
$313$			23.5894i			1.33335i	−0.745349	+	0.666675i	$0.767717\pi$
$314$		0			0
$315$	−0.0733799			−0.00413449
$316$		0			0
$317$			14.7881i			0.830584i	0.909688	+	0.415292i	$0.136321\pi$
$317$			14.7881i			0.830584i	−0.909688	+	0.415292i	$0.863679\pi$
$318$		0			0
$319$	−11.2773			−0.631410
$320$		0			0
$321$	10.4421			0.582820
$322$		0			0
$323$	8.13924	−	13.4068i	0.452880	−	0.745976i
$324$		0			0
$325$	18.8559			1.04594
$326$		0			0
$327$	−42.1173			−2.32909
$328$		0			0
$329$			0.0298107i			0.00164352i
$330$		0			0
$331$	−14.2912			−0.785516			−0.392758	−	0.919642i	$-0.628479\pi$
$331$	−14.2912			−0.785516			−0.392758	+	0.919642i	$0.628479\pi$
$332$		0			0
$333$		−	14.7266i		−	0.807011i
$334$		0			0
$335$		−	2.25981i		−	0.123466i
$336$		0			0
$337$			14.9198i			0.812733i	0.913710	+	0.406367i	$0.133204\pi$
$337$			14.9198i			0.812733i	−0.913710	+	0.406367i	$0.866796\pi$
$338$		0			0
$339$	−9.81808			−0.533245
$340$		0			0
$341$	−9.37305			−0.507579
$342$		0			0
$343$		−	0.951178i		−	0.0513588i
$344$		0			0
$345$			9.49031i			0.510941i
$346$		0			0
$347$			3.68590i			0.197870i	0.995094	+	0.0989348i	$0.0315435\pi$
$347$			3.68590i			0.197870i	−0.995094	+	0.0989348i	$0.968456\pi$
$348$		0			0
$349$	7.68570			0.411406			0.205703	−	0.978614i	$-0.434052\pi$
$349$	7.68570			0.411406			0.205703	+	0.978614i	$0.434052\pi$
$350$		0			0
$351$		−	8.91036i		−	0.475600i
$352$		0			0
$353$	17.6066			0.937105			0.468553	−	0.883436i	$-0.344776\pi$
$353$	17.6066			0.937105			0.468553	+	0.883436i	$0.344776\pi$
$354$		0			0
$355$	−3.92240			−0.208179
$356$		0			0
$357$	0.325338	−	0.535891i	0.0172187	−	0.0283624i
$358$		0			0
$359$	−23.8482			−1.25866			−0.629331	−	0.777138i	$-0.716671\pi$
$359$	−23.8482			−1.25866			−0.629331	+	0.777138i	$0.716671\pi$
$360$		0			0
$361$	−4.52998			−0.238420
$362$		0			0
$363$		−	21.1211i		−	1.10857i
$364$		0			0
$365$	5.47434			0.286540
$366$		0			0
$367$		−	10.2109i		−	0.533004i	−0.963834	−	0.266502i	$-0.914132\pi$
$367$		−	10.2109i		−	0.533004i	0.963834	−	0.266502i	$-0.0858679\pi$
$368$		0			0
$369$			23.8914i			1.24374i
$370$		0			0
$371$		−	0.505322i		−	0.0262350i
$372$		0			0
$373$	−32.8717			−1.70203			−0.851017	−	0.525139i	$-0.824013\pi$
$373$	−32.8717			−1.70203			−0.851017	+	0.525139i	$0.824013\pi$
$374$		0			0
$375$	11.6973			0.604047
$376$		0			0
$377$		−	36.1558i		−	1.86212i
$378$		0			0
$379$		−	38.3714i		−	1.97101i	−0.169658	−	0.985503i	$-0.554266\pi$
$379$		−	38.3714i		−	1.97101i	0.169658	−	0.985503i	$-0.445734\pi$
$380$		0			0
$381$			17.6691i			0.905218i
$382$		0			0
$383$	31.5309			1.61115			0.805576	−	0.592493i	$-0.201856\pi$
$383$	31.5309			1.61115			0.805576	+	0.592493i	$0.201856\pi$
$384$		0			0
$385$		−	0.0456966i		−	0.00232892i
$386$		0			0
$387$	5.66574			0.288006
$388$		0			0
$389$	32.2550			1.63539			0.817697	−	0.575649i	$-0.195251\pi$
$389$	32.2550			1.63539			0.817697	+	0.575649i	$0.195251\pi$
$390$		0			0
$391$	−27.7656	−	16.8564i	−1.40417	−	0.852465i
$392$		0			0
$393$	0.399645			0.0201594
$394$		0			0
$395$	3.23323			0.162681
$396$		0			0
$397$		−	1.17555i		−	0.0589989i	−0.999565	−	0.0294995i	$-0.990609\pi$
$397$		−	1.17555i		−	0.0589989i	0.999565	−	0.0294995i	$-0.00939133\pi$
$398$		0			0
$399$	0.578388			0.0289556
$400$		0			0
$401$			31.5392i			1.57499i	0.616320	+	0.787496i	$0.288623\pi$
$401$			31.5392i			1.57499i	−0.616320	+	0.787496i	$0.711377\pi$
$402$		0			0
$403$		−	30.0505i		−	1.49692i
$404$		0			0
$405$		−	5.92033i		−	0.294184i
$406$		0			0
$407$	9.17084			0.454581
$408$		0			0
$409$	−6.00449			−0.296903			−0.148451	−	0.988920i	$-0.547429\pi$
$409$	−6.00449			−0.296903			−0.148451	+	0.988920i	$0.547429\pi$
$410$		0			0
$411$			0.946679i			0.0466962i
$412$		0			0
$413$			0.0679637i			0.00334428i
$414$		0			0
$415$			7.78359i			0.382081i
$416$		0			0
$417$	−13.5856			−0.665289
$418$		0			0
$419$		−	25.5198i		−	1.24673i	−0.781933	−	0.623363i	$-0.785766\pi$
$419$		−	25.5198i		−	1.24673i	0.781933	−	0.623363i	$-0.214234\pi$
$420$		0			0
$421$	−39.3285			−1.91675			−0.958377	−	0.285505i	$-0.907839\pi$
$421$	−39.3285			−1.91675			−0.958377	+	0.285505i	$0.907839\pi$
$422$		0			0
$423$	0.879505			0.0427630
$424$		0			0
$425$	−10.0780	+	16.6004i	−0.488856	+	0.805236i
$426$		0			0
$427$	−1.02149			−0.0494336
$428$		0			0
$429$	−11.1835			−0.539946
$430$		0			0
$431$		−	1.98939i		−	0.0958254i	−0.998852	−	0.0479127i	$-0.984743\pi$
$431$		−	1.98939i		−	0.0958254i	0.998852	−	0.0479127i	$-0.0152569\pi$
$432$		0			0
$433$	11.2511			0.540692			0.270346	−	0.962763i	$-0.412862\pi$
$433$	11.2511			0.540692			0.270346	+	0.962763i	$0.412862\pi$
$434$		0			0
$435$		−	10.8798i		−	0.521647i
$436$		0			0
$437$		−	29.9675i		−	1.43354i
$438$		0			0
$439$			7.49891i			0.357903i	0.983858	+	0.178952i	$0.0572706\pi$
$439$			7.49891i			0.357903i	−0.983858	+	0.178952i	$0.942729\pi$
$440$		0			0
$441$	−14.0267			−0.667936
$442$		0			0
$443$	3.11941			0.148208			0.0741039	−	0.997251i	$-0.476390\pi$
$443$	3.11941			0.148208			0.0741039	+	0.997251i	$0.476390\pi$
$444$		0			0
$445$			7.82205i			0.370801i
$446$		0			0
$447$			20.6587i			0.977122i
$448$		0			0
$449$		−	17.9491i		−	0.847070i	−0.905880	−	0.423535i	$-0.860789\pi$
$449$		−	17.9491i		−	0.847070i	0.905880	−	0.423535i	$-0.139211\pi$
$450$		0			0
$451$	−14.8782			−0.700586
$452$		0			0
$453$		−	14.6174i		−	0.686784i
$454$		0			0
$455$	0.146506			0.00686831
$456$		0			0
$457$	37.3936			1.74920			0.874599	−	0.484846i	$-0.161124\pi$
$457$	37.3936			1.74920			0.874599	+	0.484846i	$0.161124\pi$
$458$		0			0
$459$	7.84450	+	4.76237i	0.366150	+	0.222288i
$460$		0			0
$461$	13.4195			0.625007			0.312504	−	0.949917i	$-0.398832\pi$
$461$	13.4195			0.625007			0.312504	+	0.949917i	$0.398832\pi$
$462$		0			0
$463$	24.2885			1.12878			0.564391	−	0.825507i	$-0.309111\pi$
$463$	24.2885			1.12878			0.564391	+	0.825507i	$0.309111\pi$
$464$		0			0
$465$		−	9.04265i		−	0.419343i
$466$		0			0
$467$	16.6251			0.769316			0.384658	−	0.923059i	$-0.374319\pi$
$467$	16.6251			0.769316			0.384658	+	0.923059i	$0.374319\pi$
$468$		0			0
$469$			0.285228i			0.0131706i
$470$		0			0
$471$		−	47.5968i		−	2.19315i
$472$		0			0
$473$			3.52829i			0.162231i
$474$		0			0
$475$	−17.9168			−0.822079
$476$		0			0
$477$	−14.9085			−0.682613
$478$		0			0
$479$		−	41.1983i		−	1.88240i	−0.337852	−	0.941199i	$-0.609700\pi$
$479$		−	41.1983i		−	1.88240i	0.337852	−	0.941199i	$-0.390300\pi$
$480$		0			0
$481$			29.4022i			1.34063i
$482$		0			0
$483$		−	1.19785i		−	0.0545038i
$484$		0			0
$485$	−3.73464			−0.169581
$486$		0			0
$487$			34.1704i			1.54841i	0.632935	+	0.774205i	$0.281850\pi$
$487$			34.1704i			1.54841i	−0.632935	+	0.774205i	$0.718150\pi$
$488$		0			0
$489$	9.20774			0.416388
$490$		0			0
$491$	−6.00943			−0.271202			−0.135601	−	0.990764i	$-0.543296\pi$
$491$	−6.00943			−0.271202			−0.135601	+	0.990764i	$0.543296\pi$
$492$		0			0
$493$	31.8309	+	19.3244i	1.43359	+	0.870328i
$494$		0			0
$495$	−1.34819			−0.0605964
$496$		0			0
$497$	0.495077			0.0222072
$498$		0			0
$499$		−	21.5072i		−	0.962796i	−0.876502	−	0.481398i	$-0.840129\pi$
$499$		−	21.5072i		−	0.962796i	0.876502	−	0.481398i	$-0.159871\pi$
$500$		0			0
$501$	37.2069			1.66228
$502$		0			0
$503$			22.7335i			1.01364i	0.862052	+	0.506819i	$0.169179\pi$
$503$			22.7335i			1.01364i	−0.862052	+	0.506819i	$0.830821\pi$
$504$		0			0
$505$		−	5.00764i		−	0.222837i
$506$		0			0
$507$		−	6.77127i		−	0.300723i
$508$		0			0
$509$	−15.3585			−0.680755			−0.340377	−	0.940289i	$-0.610555\pi$
$509$	−15.3585			−0.680755			−0.340377	+	0.940289i	$0.610555\pi$
$510$		0			0
$511$	−0.690958			−0.0305662
$512$		0			0
$513$			8.46659i			0.373809i
$514$		0			0
$515$		−	9.90308i		−	0.436382i
$516$		0			0
$517$			0.547703i			0.0240880i
$518$		0			0
$519$	−32.9653			−1.44702
$520$		0			0
$521$			18.5216i			0.811448i	0.913996	+	0.405724i	$0.132981\pi$
$521$			18.5216i			0.811448i	−0.913996	+	0.405724i	$0.867019\pi$
$522$		0			0
$523$	−10.6431			−0.465391			−0.232695	−	0.972550i	$-0.574755\pi$
$523$	−10.6431			−0.465391			−0.232695	+	0.972550i	$0.574755\pi$
$524$		0			0
$525$	−0.716162			−0.0312559
$526$		0			0
$527$	26.4559	+	16.0613i	1.15244	+	0.699641i
$528$		0			0
$529$	−39.0628			−1.69838
$530$		0			0
$531$	2.00513			0.0870152
$532$		0			0
$533$		−	47.7003i		−	2.06613i
$534$		0			0
$535$	−2.51325			−0.108657
$536$		0			0
$537$		−	9.78288i		−	0.422163i
$538$		0			0
$539$		−	8.73497i		−	0.376242i
$540$		0			0
$541$		−	13.3166i		−	0.572524i	−0.958151	−	0.286262i	$-0.907587\pi$
$541$		−	13.3166i		−	0.572524i	0.958151	−	0.286262i	$-0.0924128\pi$
$542$		0			0
$543$	−27.1193			−1.16380
$544$		0			0
$545$	10.1370			0.434222
$546$		0			0
$547$			11.5686i			0.494636i	0.968934	+	0.247318i	$0.0795493\pi$
$547$			11.5686i			0.494636i	−0.968934	+	0.247318i	$0.920451\pi$
$548$		0			0
$549$			30.1371i			1.28622i
$550$		0			0
$551$			34.3551i			1.46358i
$552$		0			0
$553$	−0.408090			−0.0173538
$554$		0			0
$555$			8.84757i			0.375558i
$556$		0			0
$557$	29.5416			1.25172			0.625859	−	0.779936i	$-0.284748\pi$
$557$	29.5416			1.25172			0.625859	+	0.779936i	$0.284748\pi$
$558$		0			0
$559$	−11.3119			−0.478442
$560$		0			0
$561$	5.97733	−	9.84576i	0.252363	−	0.415688i
$562$		0			0
$563$	8.38299			0.353301			0.176650	−	0.984274i	$-0.443474\pi$
$563$	8.38299			0.353301			0.176650	+	0.984274i	$0.443474\pi$
$564$		0			0
$565$	2.36307			0.0994149
$566$		0			0
$567$			0.747251i			0.0313816i
$568$		0			0
$569$	26.9209			1.12858			0.564292	−	0.825576i	$-0.309149\pi$
$569$	26.9209			1.12858			0.564292	+	0.825576i	$0.309149\pi$
$570$		0			0
$571$			33.6553i			1.40843i	0.709986	+	0.704216i	$0.248701\pi$
$571$			33.6553i			1.40843i	−0.709986	+	0.704216i	$0.751299\pi$
$572$		0			0
$573$			3.49019i			0.145805i
$574$		0			0
$575$			37.1058i			1.54742i
$576$		0			0
$577$	25.0192			1.04156			0.520781	−	0.853690i	$-0.325641\pi$
$577$	25.0192			1.04156			0.520781	+	0.853690i	$0.325641\pi$
$578$		0			0
$579$	−39.4336			−1.63880
$580$		0			0
$581$		−	0.982427i		−	0.0407579i
$582$		0			0
$583$		−	9.28412i		−	0.384509i
$584$		0			0
$585$		−	4.32236i		−	0.178708i
$586$		0			0
$587$	−1.91456			−0.0790224			−0.0395112	−	0.999219i	$-0.512580\pi$
$587$	−1.91456			−0.0790224			−0.0395112	+	0.999219i	$0.512580\pi$
$588$		0			0
$589$			28.5539i			1.17654i
$590$		0			0
$591$	−40.3701			−1.66060
$592$		0			0
$593$	24.0382			0.987131			0.493565	−	0.869709i	$-0.335693\pi$
$593$	24.0382			0.987131			0.493565	+	0.869709i	$0.335693\pi$
$594$		0			0
$595$	−0.0783040	+	0.128981i	−0.00321015	+	0.00528771i
$596$		0			0
$597$	−28.7857			−1.17812
$598$		0			0
$599$	32.8609			1.34266			0.671331	−	0.741158i	$-0.265723\pi$
$599$	32.8609			1.34266			0.671331	+	0.741158i	$0.265723\pi$
$600$		0			0
$601$			7.59752i			0.309909i	0.987922	+	0.154955i	$0.0495231\pi$
$601$			7.59752i			0.309909i	−0.987922	+	0.154955i	$0.950477\pi$
$602$		0			0
$603$	8.41505			0.342687
$604$		0			0
$605$			5.08354i			0.206675i
$606$		0			0
$607$		−	19.0717i		−	0.774096i	−0.922060	−	0.387048i	$-0.873495\pi$
$607$		−	19.0717i		−	0.774096i	0.922060	−	0.387048i	$-0.126505\pi$
$608$		0			0
$609$			1.37323i			0.0556459i
$610$		0			0
$611$	−1.75597			−0.0710389
$612$		0			0
$613$	−0.921264			−0.0372095			−0.0186047	−	0.999827i	$-0.505922\pi$
$613$	−0.921264			−0.0372095			−0.0186047	+	0.999827i	$0.505922\pi$
$614$		0			0
$615$		−	14.3537i		−	0.578798i
$616$		0			0
$617$			31.5651i			1.27076i	0.772198	+	0.635382i	$0.219158\pi$
$617$			31.5651i			1.27076i	−0.772198	+	0.635382i	$0.780842\pi$
$618$		0			0
$619$			22.0416i			0.885927i	0.896540	+	0.442963i	$0.146073\pi$
$619$			22.0416i			0.885927i	−0.896540	+	0.442963i	$0.853927\pi$
$620$		0			0
$621$	17.5343			0.703629
$622$		0			0
$623$		−	0.987281i		−	0.0395546i
$624$		0			0
$625$	20.7349			0.829396
$626$		0			0
$627$	10.6265			0.424383
$628$		0			0
$629$	−25.8851	−	15.7148i	−1.03211	−	0.626590i
$630$		0			0
$631$	47.7630			1.90141			0.950707	−	0.310089i	$-0.100359\pi$
$631$	47.7630			1.90141			0.950707	+	0.310089i	$0.100359\pi$
$632$		0			0
$633$	51.3867			2.04244
$634$		0			0
$635$		−	4.25270i		−	0.168763i
$636$		0			0
$637$	28.0048			1.10959
$638$		0			0
$639$		−	14.6062i		−	0.577813i
$640$		0			0
$641$		−	6.92689i		−	0.273596i	−0.990599	−	0.136798i	$-0.956319\pi$
$641$		−	6.92689i		−	0.273596i	0.990599	−	0.136798i	$-0.0436811\pi$
$642$		0			0
$643$		−	22.1869i		−	0.874965i	−0.899227	−	0.437482i	$-0.855870\pi$
$643$		−	22.1869i		−	0.874965i	0.899227	−	0.437482i	$-0.144130\pi$
$644$		0			0
$645$	−3.40391			−0.134029
$646$		0			0
$647$	−12.7351			−0.500668			−0.250334	−	0.968159i	$-0.580540\pi$
$647$	−12.7351			−0.500668			−0.250334	+	0.968159i	$0.580540\pi$
$648$		0			0
$649$			1.24868i			0.0490148i
$650$		0			0
$651$			1.14134i			0.0447327i
$652$		0			0
$653$			20.8049i			0.814160i	0.913393	+	0.407080i	$0.133453\pi$
$653$			20.8049i			0.814160i	−0.913393	+	0.407080i	$0.866547\pi$
$654$		0			0
$655$	−0.0961887			−0.00375840
$656$		0			0
$657$			20.3853i			0.795306i
$658$		0			0
$659$	45.4677			1.77117			0.885585	−	0.464477i	$-0.153758\pi$
$659$	45.4677			1.77117			0.885585	+	0.464477i	$0.153758\pi$
$660$		0			0
$661$	25.3461			0.985848			0.492924	−	0.870072i	$-0.335928\pi$
$661$	25.3461			0.985848			0.492924	+	0.870072i	$0.335928\pi$
$662$		0			0
$663$	31.5661	+	19.1637i	1.22592	+	0.744255i
$664$		0			0
$665$	−0.139210			−0.00539831
$666$		0			0
$667$	71.1496			2.75492
$668$		0			0
$669$			22.8390i			0.883007i
$670$		0			0
$671$	−18.7676			−0.724515
$672$		0			0
$673$		−	41.0320i		−	1.58167i	−0.612031	−	0.790834i	$-0.709647\pi$
$673$		−	41.0320i		−	1.58167i	0.612031	−	0.790834i	$-0.290353\pi$
$674$		0			0
$675$		−	10.4833i		−	0.403504i
$676$		0			0
$677$			18.9598i			0.728683i	0.931265	+	0.364341i	$0.118706\pi$
$677$			18.9598i			0.728683i	−0.931265	+	0.364341i	$0.881294\pi$
$678$		0			0
$679$	0.471377			0.0180898
$680$		0			0
$681$	−37.7390			−1.44616
$682$		0			0
$683$			7.11620i			0.272294i	0.990689	+	0.136147i	$0.0434719\pi$
$683$			7.11620i			0.272294i	−0.990689	+	0.136147i	$0.956528\pi$
$684$		0			0
$685$		−	0.227852i		−	0.00870576i
$686$		0			0
$687$		−	3.36601i		−	0.128421i
$688$		0			0
$689$	29.7654			1.13397
$690$		0			0
$691$			13.0448i			0.496249i	0.968728	+	0.248124i	$0.0798142\pi$
$691$			13.0448i			0.496249i	−0.968728	+	0.248124i	$0.920186\pi$
$692$		0			0
$693$	0.170165			0.00646403
$694$		0			0
$695$	3.26985			0.124033
$696$		0			0
$697$	41.9944	+	25.4947i	1.59065	+	0.965679i
$698$		0			0
$699$	29.7265			1.12436
$700$		0			0
$701$	30.8484			1.16513			0.582563	−	0.812785i	$-0.302050\pi$
$701$	30.8484			1.16513			0.582563	+	0.812785i	$0.302050\pi$
$702$		0			0
$703$		−	27.9379i		−	1.05370i
$704$		0			0
$705$	−0.528397			−0.0199006
$706$		0			0
$707$			0.632052i			0.0237708i
$708$		0			0
$709$			42.4982i			1.59605i	0.602622	+	0.798027i	$0.294123\pi$
$709$			42.4982i			1.59605i	−0.602622	+	0.798027i	$0.705877\pi$
$710$		0			0
$711$			12.0399i			0.451530i
$712$		0			0
$713$	59.1353			2.21463
$714$		0			0
$715$	2.69171			0.100664
$716$		0			0
$717$			40.7434i			1.52159i
$718$		0			0
$719$			40.2016i			1.49927i	0.661853	+	0.749633i	$0.269770\pi$
$719$			40.2016i			1.49927i	−0.661853	+	0.749633i	$0.730230\pi$
$720$		0			0
$721$			1.24994i			0.0465503i
$722$		0			0
$723$	6.55295			0.243707
$724$		0			0
$725$		−	42.5386i		−	1.57984i
$726$		0			0
$727$	42.5800			1.57920			0.789602	−	0.613619i	$-0.210287\pi$
$727$	42.5800			1.57920			0.789602	+	0.613619i	$0.210287\pi$
$728$		0			0
$729$	7.10774			0.263250
$730$		0			0
$731$	6.04593	−	9.95876i	0.223617	−	0.368338i
$732$		0			0
$733$	34.9773			1.29192			0.645958	−	0.763373i	$-0.276458\pi$
$733$	34.9773			1.29192			0.645958	+	0.763373i	$0.276458\pi$
$734$		0			0
$735$	8.42706			0.310837
$736$		0			0
$737$			5.24039i			0.193032i
$738$		0			0
$739$	−9.18760			−0.337971			−0.168986	−	0.985619i	$-0.554049\pi$
$739$	−9.18760			−0.337971			−0.168986	+	0.985619i	$0.554049\pi$
$740$		0			0
$741$			34.0693i			1.25157i
$742$		0			0
$743$		−	0.293179i		−	0.0107557i	−0.999986	−	0.00537784i	$-0.998288\pi$
$743$		−	0.293179i		−	0.0107557i	0.999986	−	0.00537784i	$-0.00171183\pi$
$744$		0			0
$745$		−	4.97224i		−	0.182169i
$746$		0			0
$747$	−28.9845			−1.06049
$748$		0			0
$749$	0.317217			0.0115909
$750$		0			0
$751$		−	34.6526i		−	1.26449i	−0.774769	−	0.632245i	$-0.782134\pi$
$751$		−	34.6526i		−	1.26449i	0.774769	−	0.632245i	$-0.217866\pi$
$752$		0			0
$753$		−	56.4686i		−	2.05783i
$754$		0			0
$755$			3.51819i			0.128040i
$756$		0			0
$757$	−45.4889			−1.65332			−0.826661	−	0.562701i	$-0.809762\pi$
$757$	−45.4889			−1.65332			−0.826661	+	0.562701i	$0.809762\pi$
$758$		0			0
$759$		−	22.0076i		−	0.798826i
$760$		0			0
$761$	−51.3442			−1.86123			−0.930614	−	0.366003i	$-0.880726\pi$
$761$	−51.3442			−1.86123			−0.930614	+	0.366003i	$0.880726\pi$
$762$		0			0
$763$	−1.27947			−0.0463199
$764$		0			0
$765$	3.80532	+	2.31020i	0.137582	+	0.0835254i
$766$		0			0
$767$	−4.00333			−0.144552
$768$		0			0
$769$	−25.1432			−0.906687			−0.453344	−	0.891336i	$-0.649769\pi$
$769$	−25.1432			−0.906687			−0.453344	+	0.891336i	$0.649769\pi$
$770$		0			0
$771$			49.4499i			1.78089i
$772$		0			0
$773$	0.148731			0.00534947			0.00267473	−	0.999996i	$-0.499149\pi$
$773$	0.148731			0.00534947			0.00267473	+	0.999996i	$0.499149\pi$
$774$		0			0
$775$		−	35.3555i		−	1.27001i
$776$		0			0
$777$		−	1.11672i		−	0.0400621i
$778$		0			0
$779$			45.3246i			1.62392i
$780$		0			0
$781$	9.09588			0.325476
$782$		0			0
$783$	−20.1016			−0.718373
$784$		0			0
$785$			11.4558i			0.408877i
$786$		0			0
$787$		−	49.7135i		−	1.77209i	−0.463595	−	0.886047i	$-0.653441\pi$
$787$		−	49.7135i		−	1.77209i	0.463595	−	0.886047i	$-0.346559\pi$
$788$		0			0
$789$			41.7441i			1.48613i
$790$		0			0
$791$	−0.298261			−0.0106049
$792$		0			0
$793$		−	60.1700i		−	2.13670i
$794$		0			0
$795$	8.95686			0.317667
$796$		0			0
$797$	−14.7889			−0.523851			−0.261926	−	0.965088i	$-0.584358\pi$
$797$	−14.7889			−0.523851			−0.261926	+	0.965088i	$0.584358\pi$
$798$		0			0
$799$	0.938523	−	1.54592i	0.0332026	−	0.0546907i
$800$		0			0
$801$	−29.1277			−1.02918
$802$		0			0
$803$	−12.6948			−0.447988
$804$		0			0
$805$			0.288304i			0.0101614i
$806$		0			0
$807$	−14.2500			−0.501625
$808$		0			0
$809$		−	36.9850i		−	1.30032i	−0.759796	−	0.650161i	$-0.774701\pi$
$809$		−	36.9850i		−	1.30032i	0.759796	−	0.650161i	$-0.225299\pi$
$810$		0			0
$811$		−	27.8127i		−	0.976634i	−0.872666	−	0.488317i	$-0.837611\pi$
$811$		−	27.8127i		−	0.976634i	0.872666	−	0.488317i	$-0.162389\pi$
$812$		0			0
$813$		−	27.6293i		−	0.969001i
$814$		0			0
$815$	−2.21617			−0.0776289
$816$		0			0
$817$	10.7485			0.376043
$818$		0			0
$819$			0.545558i			0.0190634i
$820$		0			0
$821$			49.9216i			1.74228i	0.491038	+	0.871138i	$0.336618\pi$
$821$			49.9216i			1.74228i	−0.491038	+	0.871138i	$0.663382\pi$
$822$		0			0
$823$			0.824220i			0.0287305i	0.999897	+	0.0143653i	$0.00457276\pi$
$823$			0.824220i			0.0287305i	−0.999897	+	0.0143653i	$0.995427\pi$
$824$		0			0
$825$	−13.1578			−0.458096
$826$		0			0
$827$			17.6915i			0.615195i	0.951517	+	0.307597i	$0.0995250\pi$
$827$			17.6915i			0.615195i	−0.951517	+	0.307597i	$0.900475\pi$
$828$		0			0
$829$	−18.2134			−0.632578			−0.316289	−	0.948663i	$-0.602437\pi$
$829$	−18.2134			−0.632578			−0.316289	+	0.948663i	$0.602437\pi$
$830$		0			0
$831$	−1.67400			−0.0580704
$832$		0			0
$833$	−14.9679	+	24.6549i	−0.518607	+	0.854241i
$834$		0			0
$835$	−8.95516			−0.309906
$836$		0			0
$837$	−16.7072			−0.577487
$838$		0			0
$839$		−	21.6765i		−	0.748355i	−0.927357	−	0.374177i	$-0.877925\pi$
$839$		−	21.6765i		−	0.748355i	0.927357	−	0.374177i	$-0.122075\pi$
$840$		0			0
$841$	−52.5669			−1.81265
$842$		0			0
$843$			8.07809i			0.278224i
$844$		0			0
$845$			1.62974i			0.0560649i
$846$		0			0
$847$		−	0.641633i		−	0.0220468i
$848$		0			0
$849$	17.8148			0.611401
$850$		0			0
$851$	−57.8595			−1.98340
$852$		0			0
$853$			28.5140i			0.976302i	0.872759	+	0.488151i	$0.162328\pi$
$853$			28.5140i			0.976302i	−0.872759	+	0.488151i	$0.837672\pi$
$854$		0			0
$855$			4.10709i			0.140460i
$856$		0			0
$857$		−	33.4831i		−	1.14376i	−0.820337	−	0.571881i	$-0.806214\pi$
$857$		−	33.4831i		−	1.14376i	0.820337	−	0.571881i	$-0.193786\pi$
$858$		0			0
$859$	38.6618			1.31912			0.659562	−	0.751650i	$-0.270742\pi$
$859$	38.6618			1.31912			0.659562	+	0.751650i	$0.270742\pi$
$860$		0			0
$861$			1.81169i			0.0617423i
$862$		0			0
$863$	−4.31401			−0.146851			−0.0734253	−	0.997301i	$-0.523393\pi$
$863$	−4.31401			−0.146851			−0.0734253	+	0.997301i	$0.523393\pi$
$864$		0			0
$865$	7.93426			0.269773
$866$		0			0
$867$	−33.7426	+	17.5476i	−1.14596	+	0.595948i
$868$		0			0
$869$	−7.49771			−0.254343
$870$		0			0
$871$	−16.8010			−0.569281
$872$		0			0
$873$		−	13.9070i		−	0.470681i
$874$		0			0
$875$	0.355350			0.0120130
$876$		0			0
$877$			30.6885i			1.03628i	0.855297	+	0.518138i	$0.173375\pi$
$877$			30.6885i			1.03628i	−0.855297	+	0.518138i	$0.826625\pi$
$878$		0			0
$879$			16.0901i			0.542706i
$880$		0			0
$881$			39.1195i			1.31797i	0.752157	+	0.658985i	$0.229014\pi$
$881$			39.1195i			1.31797i	−0.752157	+	0.658985i	$0.770986\pi$
$882$		0			0
$883$	51.0135			1.71674			0.858370	−	0.513032i	$-0.171478\pi$
$883$	51.0135			1.71674			0.858370	+	0.513032i	$0.171478\pi$
$884$		0			0
$885$	−1.20466			−0.0404942
$886$		0			0
$887$			21.9687i			0.737637i	0.929501	+	0.368818i	$0.120238\pi$
$887$			21.9687i			0.737637i	−0.929501	+	0.368818i	$0.879762\pi$
$888$		0			0
$889$			0.536766i			0.0180026i
$890$		0			0
$891$			13.7290i			0.459939i
$892$		0			0
$893$	1.66851			0.0558347
$894$		0			0
$895$			2.35460i			0.0787054i
$896$		0			0
$897$	70.5578			2.35585
$898$		0			0
$899$	−67.7935			−2.26104
$900$		0			0
$901$	−15.9089	+	26.2049i	−0.530002	+	0.873011i
$902$		0			0
$903$	0.429634			0.0142973
$904$		0			0
$905$	6.52722			0.216972
$906$		0			0
$907$		−	46.1535i		−	1.53250i	−0.642542	−	0.766250i	$-0.722120\pi$
$907$		−	46.1535i		−	1.53250i	0.642542	−	0.766250i	$-0.277880\pi$
$908$		0			0
$909$	18.6474			0.618495
$910$		0			0
$911$		−	22.6987i		−	0.752040i	−0.926611	−	0.376020i	$-0.877292\pi$
$911$		−	22.6987i		−	0.752040i	0.926611	−	0.376020i	$-0.122708\pi$
$912$		0			0
$913$		−	18.0498i		−	0.597362i
$914$		0			0
$915$		−	18.1060i		−	0.598567i
$916$		0			0
$917$	0.0121407			0.000400922
$918$		0			0
$919$	36.3839			1.20019			0.600096	−	0.799928i	$-0.295129\pi$
$919$	36.3839			1.20019			0.600096	+	0.799928i	$0.295129\pi$
$920$		0			0
$921$			24.2152i			0.797919i
$922$		0			0
$923$			29.1619i			0.959877i
$924$		0			0
$925$			34.5928i			1.13740i
$926$		0			0
$927$	36.8770			1.21120
$928$		0			0
$929$			4.99217i			0.163788i	0.996641	+	0.0818939i	$0.0260969\pi$
$929$			4.99217i			0.163788i	−0.996641	+	0.0818939i	$0.973903\pi$
$930$		0			0
$931$	−26.6101			−0.872110
$932$		0			0
$933$	54.3192			1.77833
$934$		0			0
$935$	−1.43865	+	2.36973i	−0.0470490	+	0.0774984i
$936$		0			0
$937$	−50.7810			−1.65894			−0.829471	−	0.558550i	$-0.811358\pi$
$937$	−50.7810			−1.65894			−0.829471	+	0.558550i	$0.811358\pi$
$938$		0			0
$939$	52.7745			1.72223
$940$		0			0
$941$			25.8096i			0.841370i	0.907207	+	0.420685i	$0.138210\pi$
$941$			25.8096i			0.841370i	−0.907207	+	0.420685i	$0.861790\pi$
$942$		0			0
$943$	93.8675			3.05675
$944$		0			0
$945$		−	0.0814532i		−	0.00264967i
$946$		0			0
$947$		−	30.3908i		−	0.987570i	−0.869584	−	0.493785i	$-0.835613\pi$
$947$		−	30.3908i		−	0.987570i	0.869584	−	0.493785i	$-0.164387\pi$
$948$		0			0
$949$		−	40.7001i		−	1.32118i
$950$		0			0
$951$	33.0842			1.07283
$952$		0			0
$953$	6.58660			0.213361			0.106681	−	0.994293i	$-0.465978\pi$
$953$	6.58660			0.213361			0.106681	+	0.994293i	$0.465978\pi$
$954$		0			0
$955$		−	0.840037i		−	0.0271830i
$956$		0			0
$957$			25.2298i			0.815565i
$958$		0			0
$959$			0.0287589i			0.000928673i
$960$		0			0
$961$	−25.3458			−0.817608
$962$		0			0
$963$		−	9.35884i		−	0.301584i
$964$		0			0
$965$	9.49108			0.305529
$966$		0			0
$967$	12.8556			0.413409			0.206704	−	0.978403i	$-0.433726\pi$
$967$	12.8556			0.413409			0.206704	+	0.978403i	$0.433726\pi$
$968$		0			0
$969$	−29.9940	−	18.2092i	−0.963545	−	0.584965i
$970$		0			0
$971$	0.175222			0.00562314			0.00281157	−	0.999996i	$-0.499105\pi$
$971$	0.175222			0.00562314			0.00281157	+	0.999996i	$0.499105\pi$
$972$		0			0
$973$	−0.412713			−0.0132310
$974$		0			0
$975$		−	42.1847i		−	1.35099i
$976$		0			0
$977$	−46.8868			−1.50004			−0.750020	−	0.661415i	$-0.769956\pi$
$977$	−46.8868			−1.50004			−0.750020	+	0.661415i	$0.769956\pi$
$978$		0			0
$979$		−	18.1390i		−	0.579725i
$980$		0			0
$981$			37.7481i			1.20521i
$982$		0			0
$983$			40.1430i			1.28036i	0.768224	+	0.640181i	$0.221141\pi$
$983$			40.1430i			1.28036i	−0.768224	+	0.640181i	$0.778859\pi$
$984$		0			0
$985$	9.71649			0.309593
$986$		0			0
$987$	0.0666930			0.00212286
$988$		0			0
$989$		−	22.2602i		−	0.707834i
$990$		0			0
$991$		−	52.6000i		−	1.67089i	−0.549572	−	0.835446i	$-0.685209\pi$
$991$		−	52.6000i		−	1.67089i	0.549572	−	0.835446i	$-0.314791\pi$
$992$		0			0
$993$			31.9725i			1.01462i
$994$		0			0
$995$	6.92829			0.219642
$996$		0			0
$997$			8.61292i			0.272774i	0.990656	+	0.136387i	$0.0435490\pi$
$997$			8.61292i			0.272774i	−0.990656	+	0.136387i	$0.956451\pi$
$998$		0			0
$999$	16.3468			0.517190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.7	✓	40
17.16	even	2	inner	4012.2.b.a.237.34	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.7	✓	40	1.1	even	1	trivial
4012.2.b.a.237.34	yes	40	17.16	even	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23721i q^{3} +0.538464i q^{5} -0.0679637i q^{7} -2.00513 q^{9} +O(q^{10})\)	\(q-2.23721i q^{3} +0.538464i q^{5} -0.0679637i q^{7} -2.00513 q^{9} -1.24868i q^{11} +4.00333 q^{13} +1.20466 q^{15} +(-2.13968 + 3.52445i) q^{17} -3.80395 q^{19} -0.152049 q^{21} +7.87799i q^{23} +4.71006 q^{25} -2.22574i q^{27} -9.03144i q^{29} -7.50639i q^{31} -2.79356 q^{33} +0.0365961 q^{35} +7.34445i q^{37} -8.95631i q^{39} -11.9152i q^{41} -2.82562 q^{43} -1.07969i q^{45} -0.438627 q^{47} +6.99538 q^{49} +(7.88495 + 4.78693i) q^{51} +7.43517 q^{53} +0.672368 q^{55} +8.51025i q^{57} -1.00000 q^{59} -15.0300i q^{61} +0.136276i q^{63} +2.15565i q^{65} -4.19676 q^{67} +17.6248 q^{69} +7.28442i q^{71} -10.1666i q^{73} -10.5374i q^{75} -0.0848647 q^{77} -6.00453i q^{79} -10.9948 q^{81} +14.4552 q^{83} +(-1.89779 - 1.15214i) q^{85} -20.2053 q^{87} +14.5266 q^{89} -0.272081i q^{91} -16.7934 q^{93} -2.04829i q^{95} +6.93571i q^{97} +2.50376i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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