LMFDB - Embedded newform 3800.2.d.n.3649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$0.264658 + 1.38923i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.n.3649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.470683i q^{3} -2.71982i q^{7} +2.77846 q^{9} +O(q^{10})$	$q-0.470683i q^{3} -2.71982i q^{7} +2.77846 q^{9} -5.55691 q^{11} +2.02760i q^{13} +3.77846i q^{17} -1.00000 q^{19} -1.28018 q^{21} -5.77846i q^{23} -2.71982i q^{27} +5.66119 q^{29} +7.55691 q^{31} +2.61555i q^{33} +3.75086i q^{37} +0.954357 q^{39} -12.6155 q^{41} -9.43965i q^{43} -11.1138i q^{47} -0.397442 q^{49} +1.77846 q^{51} -8.85170i q^{53} +0.470683i q^{57} -11.4526 q^{59} -10.6155 q^{61} -7.55691i q^{63} +11.5845i q^{67} -2.71982 q^{69} -9.45264i q^{73} +15.1138i q^{77} -8.94137 q^{79} +7.05520 q^{81} -4.94137i q^{83} -2.66463i q^{87} -15.4948 q^{89} +5.51471 q^{91} -3.55691i q^{93} -10.8647i q^{97} -15.4396 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) $ 6 * q - 6 * q^19 - 26 * q^21 + 14 * q^29 + 12 * q^31 - 6 * q^39 - 44 * q^41 - 24 * q^49 - 6 * q^51 - 22 * q^59 - 32 * q^61 + 2 * q^69 - 52 * q^79 - 26 * q^81 + 12 * q^89 + 58 * q^91 - 56 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 0.470683i	− 0.271749i	−0.990726	−	0.135875i	$-0.956616\pi$
$3$	− 0.470683i	− 0.271749i	0.990726	−	0.135875i	$-0.0433844\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.71982i	− 1.02800i	−0.857791	−	0.513998i	$-0.828164\pi$
$7$	− 2.71982i	− 1.02800i	0.857791	−	0.513998i	$-0.171836\pi$
$8$	0	0
$9$	2.77846	0.926152
$10$	0	0
$11$	−5.55691	−1.67547	−0.837736	−	0.546075i	$-0.816121\pi$
$11$	−5.55691	−1.67547	−0.837736	+	0.546075i	$0.816121\pi$
$12$	0	0
$13$	2.02760i	0.562354i	0.959656	+	0.281177i	$0.0907249\pi$
$13$	2.02760i	0.562354i	−0.959656	+	0.281177i	$0.909275\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.77846i	0.916410i	0.888846	+	0.458205i	$0.151508\pi$
$17$	3.77846i	0.916410i	−0.888846	+	0.458205i	$0.848492\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.28018	−0.279357
$22$	0	0
$23$	− 5.77846i	− 1.20489i	−0.798160	−	0.602446i	$-0.794193\pi$
$23$	− 5.77846i	− 1.20489i	0.798160	−	0.602446i	$-0.205807\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.71982i	− 0.523430i
$28$	0	0
$29$	5.66119	1.05126	0.525628	−	0.850714i	$-0.323830\pi$
$29$	5.66119	1.05126	0.525628	+	0.850714i	$0.323830\pi$
$30$	0	0
$31$	7.55691	1.35726	0.678631	−	0.734479i	$-0.262574\pi$
$31$	7.55691	1.35726	0.678631	+	0.734479i	$0.262574\pi$
$32$	0	0
$33$	2.61555i	0.455308i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.75086i	0.616637i	0.951283	+	0.308319i	$0.0997663\pi$
$37$	3.75086i	0.616637i	−0.951283	+	0.308319i	$0.900234\pi$
$38$	0	0
$39$	0.954357	0.152819
$40$	0	0
$41$	−12.6155	−1.97022	−0.985109	−	0.171932i	$-0.944999\pi$
$41$	−12.6155	−1.97022	−0.985109	+	0.171932i	$0.944999\pi$
$42$	0	0
$43$	− 9.43965i	− 1.43953i	−0.694216	−	0.719766i	$-0.744249\pi$
$43$	− 9.43965i	− 1.43953i	0.694216	−	0.719766i	$-0.255751\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.1138i	− 1.62112i	−0.585657	−	0.810559i	$-0.699163\pi$
$47$	− 11.1138i	− 1.62112i	0.585657	−	0.810559i	$-0.300837\pi$
$48$	0	0
$49$	−0.397442	−0.0567775
$50$	0	0
$51$	1.77846	0.249034
$52$	0	0
$53$	− 8.85170i	− 1.21587i	−0.793985	−	0.607937i	$-0.791997\pi$
$53$	− 8.85170i	− 1.21587i	0.793985	−	0.607937i	$-0.208003\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.470683i	0.0623435i
$58$	0	0
$59$	−11.4526	−1.49101	−0.745503	−	0.666502i	$-0.767791\pi$
$59$	−11.4526	−1.49101	−0.745503	+	0.666502i	$0.767791\pi$
$60$	0	0
$61$	−10.6155	−1.35918	−0.679591	−	0.733591i	$-0.737843\pi$
$61$	−10.6155	−1.35918	−0.679591	+	0.733591i	$0.737843\pi$
$62$	0	0
$63$	− 7.55691i	− 0.952082i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.5845i	1.41527i	0.706576	+	0.707637i	$0.250239\pi$
$67$	11.5845i	1.41527i	−0.706576	+	0.707637i	$0.749761\pi$
$68$	0	0
$69$	−2.71982	−0.327428
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 9.45264i	− 1.10635i	−0.833066	−	0.553174i	$-0.813417\pi$
$73$	− 9.45264i	− 1.10635i	0.833066	−	0.553174i	$-0.186583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.1138i	1.72238i
$78$	0	0
$79$	−8.94137	−1.00598	−0.502991	−	0.864292i	$-0.667767\pi$
$79$	−8.94137	−1.00598	−0.502991	+	0.864292i	$0.667767\pi$
$80$	0	0
$81$	7.05520	0.783911
$82$	0	0
$83$	− 4.94137i	− 0.542385i	−0.962525	−	0.271193i	$-0.912582\pi$
$83$	− 4.94137i	− 0.542385i	0.962525	−	0.271193i	$-0.0874181\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.66463i	− 0.285678i
$88$	0	0
$89$	−15.4948	−1.64245	−0.821225	−	0.570604i	$-0.806709\pi$
$89$	−15.4948	−1.64245	−0.821225	+	0.570604i	$0.806709\pi$
$90$	0	0
$91$	5.51471	0.578099
$92$	0	0
$93$	− 3.55691i	− 0.368835i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.8647i	− 1.10314i	−0.834128	−	0.551571i	$-0.814029\pi$
$97$	− 10.8647i	− 1.10314i	0.834128	−	0.551571i	$-0.185971\pi$
$98$	0	0
$99$	−15.4396	−1.55174
$100$	0	0
$101$	4.49828	0.447596	0.223798	−	0.974636i	$-0.428154\pi$
$101$	4.49828	0.447596	0.223798	+	0.974636i	$0.428154\pi$
$102$	0	0
$103$	4.36641i	0.430235i	0.976588	+	0.215117i	$0.0690134\pi$
$103$	4.36641i	0.430235i	−0.976588	+	0.215117i	$0.930987\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.64658i	0.159181i	0.996828	+	0.0795906i	$0.0253613\pi$
$107$	1.64658i	0.159181i	−0.996828	+	0.0795906i	$0.974639\pi$
$108$	0	0
$109$	0.954357	0.0914108	0.0457054	−	0.998955i	$-0.485446\pi$
$109$	0.954357	0.0914108	0.0457054	+	0.998955i	$0.485446\pi$
$110$	0	0
$111$	1.76547	0.167571
$112$	0	0
$113$	5.68879i	0.535156i	0.963536	+	0.267578i	$0.0862233\pi$
$113$	5.68879i	0.535156i	−0.963536	+	0.267578i	$0.913777\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.63359i	0.520826i
$118$	0	0
$119$	10.2767	0.942067
$120$	0	0
$121$	19.8793	1.80721
$122$	0	0
$123$	5.93793i	0.535405i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.3043i	0.914362i	0.889374	+	0.457181i	$0.151141\pi$
$127$	10.3043i	0.914362i	−0.889374	+	0.457181i	$0.848859\pi$
$128$	0	0
$129$	−4.44309	−0.391192
$130$	0	0
$131$	3.11383	0.272056	0.136028	−	0.990705i	$-0.456566\pi$
$131$	3.11383	0.272056	0.136028	+	0.990705i	$0.456566\pi$
$132$	0	0
$133$	2.71982i	0.235839i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.4526i	− 1.14934i	−0.818386	−	0.574668i	$-0.805131\pi$
$137$	− 13.4526i	− 1.14934i	0.818386	−	0.574668i	$-0.194869\pi$
$138$	0	0
$139$	−9.55691	−0.810607	−0.405303	−	0.914182i	$-0.632834\pi$
$139$	−9.55691	−0.810607	−0.405303	+	0.914182i	$0.632834\pi$
$140$	0	0
$141$	−5.23109	−0.440538
$142$	0	0
$143$	− 11.2672i	− 0.942209i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.187070i	0.0154292i
$148$	0	0
$149$	−19.6121	−1.60669	−0.803343	−	0.595516i	$-0.796948\pi$
$149$	−19.6121	−1.60669	−0.803343	+	0.595516i	$0.796948\pi$
$150$	0	0
$151$	−17.0518	−1.38765	−0.693826	−	0.720143i	$-0.744076\pi$
$151$	−17.0518	−1.38765	−0.693826	+	0.720143i	$0.744076\pi$
$152$	0	0
$153$	10.4983i	0.848736i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.4983i	1.79556i	0.440446	+	0.897779i	$0.354820\pi$
$157$	22.4983i	1.79556i	−0.440446	+	0.897779i	$0.645180\pi$
$158$	0	0
$159$	−4.16635	−0.330413
$160$	0	0
$161$	−15.7164	−1.23862
$162$	0	0
$163$	22.4362i	1.75734i	0.477430	+	0.878670i	$0.341568\pi$
$163$	22.4362i	1.75734i	−0.477430	+	0.878670i	$0.658432\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.69223i	0.517860i	0.965896	+	0.258930i	$0.0833699\pi$
$167$	6.69223i	0.517860i	−0.965896	+	0.258930i	$0.916630\pi$
$168$	0	0
$169$	8.88885	0.683758
$170$	0	0
$171$	−2.77846	−0.212474
$172$	0	0
$173$	− 2.86469i	− 0.217798i	−0.994053	−	0.108899i	$-0.965267\pi$
$173$	− 2.86469i	− 0.217798i	0.994053	−	0.108899i	$-0.0347325\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.39057i	0.405180i
$178$	0	0
$179$	−1.38445	−0.103479	−0.0517394	−	0.998661i	$-0.516477\pi$
$179$	−1.38445	−0.103479	−0.0517394	+	0.998661i	$0.516477\pi$
$180$	0	0
$181$	−20.8793	−1.55195	−0.775973	−	0.630766i	$-0.782741\pi$
$181$	−20.8793	−1.55195	−0.775973	+	0.630766i	$0.782741\pi$
$182$	0	0
$183$	4.99656i	0.369357i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.9966i	− 1.53542i
$188$	0	0
$189$	−7.39744	−0.538085
$190$	0	0
$191$	19.2733	1.39457	0.697284	−	0.716795i	$-0.254392\pi$
$191$	19.2733	1.39457	0.697284	+	0.716795i	$0.254392\pi$
$192$	0	0
$193$	− 8.01461i	− 0.576904i	−0.957494	−	0.288452i	$-0.906859\pi$
$193$	− 8.01461i	− 0.576904i	0.957494	−	0.288452i	$-0.0931406\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.8793i	− 1.20260i	−0.799023	−	0.601300i	$-0.794650\pi$
$197$	− 16.8793i	− 1.20260i	0.799023	−	0.601300i	$-0.205350\pi$
$198$	0	0
$199$	−1.28018	−0.0907493	−0.0453746	−	0.998970i	$-0.514448\pi$
$199$	−1.28018	−0.0907493	−0.0453746	+	0.998970i	$0.514448\pi$
$200$	0	0
$201$	5.45264	0.384599
$202$	0	0
$203$	− 15.3974i	− 1.08069i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 16.0552i	− 1.11591i
$208$	0	0
$209$	5.55691	0.384380
$210$	0	0
$211$	−1.54392	−0.106288	−0.0531441	−	0.998587i	$-0.516924\pi$
$211$	−1.54392	−0.106288	−0.0531441	+	0.998587i	$0.516924\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.5535i	− 1.39526i
$218$	0	0
$219$	−4.44920	−0.300649
$220$	0	0
$221$	−7.66119	−0.515347
$222$	0	0
$223$	− 3.92332i	− 0.262725i	−0.991334	−	0.131363i	$-0.958065\pi$
$223$	− 3.92332i	− 0.262725i	0.991334	−	0.131363i	$-0.0419352\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.9069i	− 0.723916i	−0.932194	−	0.361958i	$-0.882108\pi$
$227$	− 10.9069i	− 0.723916i	0.932194	−	0.361958i	$-0.117892\pi$
$228$	0	0
$229$	−10.1725	−0.672215	−0.336108	−	0.941824i	$-0.609111\pi$
$229$	−10.1725	−0.672215	−0.336108	+	0.941824i	$0.609111\pi$
$230$	0	0
$231$	7.11383	0.468056
$232$	0	0
$233$	− 9.11383i	− 0.597067i	−0.954399	−	0.298533i	$-0.903503\pi$
$233$	− 9.11383i	− 0.597067i	0.954399	−	0.298533i	$-0.0964974\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.20855i	0.273375i
$238$	0	0
$239$	−20.1595	−1.30401	−0.652004	−	0.758216i	$-0.726071\pi$
$239$	−20.1595	−1.30401	−0.652004	+	0.758216i	$0.726071\pi$
$240$	0	0
$241$	−4.82410	−0.310748	−0.155374	−	0.987856i	$-0.549658\pi$
$241$	−4.82410	−0.310748	−0.155374	+	0.987856i	$0.549658\pi$
$242$	0	0
$243$	− 11.4802i	− 0.736457i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.02760i	− 0.129013i
$248$	0	0
$249$	−2.32582	−0.147393
$250$	0	0
$251$	−18.2277	−1.15052	−0.575260	−	0.817971i	$-0.695099\pi$
$251$	−18.2277	−1.15052	−0.575260	+	0.817971i	$0.695099\pi$
$252$	0	0
$253$	32.1104i	2.01876i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 28.5941i	− 1.78365i	−0.452382	−	0.891824i	$-0.649426\pi$
$257$	− 28.5941i	− 1.78365i	0.452382	−	0.891824i	$-0.350574\pi$
$258$	0	0
$259$	10.2017	0.633901
$260$	0	0
$261$	15.7294	0.973624
$262$	0	0
$263$	− 27.3776i	− 1.68817i	−0.536206	−	0.844087i	$-0.680143\pi$
$263$	− 27.3776i	− 1.68817i	0.536206	−	0.844087i	$-0.319857\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.29317i	0.446334i
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	15.8337	0.961826	0.480913	−	0.876768i	$-0.340305\pi$
$271$	15.8337	0.961826	0.480913	+	0.876768i	$0.340305\pi$
$272$	0	0
$273$	− 2.59568i	− 0.157098i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.70683i	0.402975i	0.979491	+	0.201487i	$0.0645775\pi$
$277$	6.70683i	0.402975i	−0.979491	+	0.201487i	$0.935423\pi$
$278$	0	0
$279$	20.9966	1.25703
$280$	0	0
$281$	8.38101	0.499969	0.249985	−	0.968250i	$-0.419574\pi$
$281$	8.38101	0.499969	0.249985	+	0.968250i	$0.419574\pi$
$282$	0	0
$283$	− 20.2897i	− 1.20610i	−0.797704	−	0.603050i	$-0.793952\pi$
$283$	− 20.2897i	− 1.20610i	0.797704	−	0.603050i	$-0.206048\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.3121i	2.02538i
$288$	0	0
$289$	2.72326	0.160192
$290$	0	0
$291$	−5.11383	−0.299778
$292$	0	0
$293$	− 6.76041i	− 0.394947i	−0.980308	−	0.197474i	$-0.936726\pi$
$293$	− 6.76041i	− 0.394947i	0.980308	−	0.197474i	$-0.0632737\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	15.1138i	0.876993i
$298$	0	0
$299$	11.7164	0.677576
$300$	0	0
$301$	−25.6742	−1.47984
$302$	0	0
$303$	− 2.11727i	− 0.121634i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.74398i	− 0.441972i	−0.975277	−	0.220986i	$-0.929072\pi$
$307$	− 7.74398i	− 0.441972i	0.975277	−	0.220986i	$-0.0709276\pi$
$308$	0	0
$309$	2.05520	0.116916
$310$	0	0
$311$	−11.0456	−0.626341	−0.313170	−	0.949697i	$-0.601391\pi$
$311$	−11.0456	−0.626341	−0.313170	+	0.949697i	$0.601391\pi$
$312$	0	0
$313$	1.16291i	0.0657315i	0.999460	+	0.0328658i	$0.0104634\pi$
$313$	1.16291i	0.0657315i	−0.999460	+	0.0328658i	$0.989537\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.8742i	1.22858i	0.789080	+	0.614290i	$0.210557\pi$
$317$	21.8742i	1.22858i	−0.789080	+	0.614290i	$0.789443\pi$
$318$	0	0
$319$	−31.4588	−1.76135
$320$	0	0
$321$	0.775019	0.0432574
$322$	0	0
$323$	− 3.77846i	− 0.210239i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 0.449200i	− 0.0248408i
$328$	0	0
$329$	−30.2277	−1.66650
$330$	0	0
$331$	14.6646	0.806041	0.403020	−	0.915191i	$-0.367960\pi$
$331$	14.6646	0.806041	0.403020	+	0.915191i	$0.367960\pi$
$332$	0	0
$333$	10.4216i	0.571100i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.0958i	1.09469i	0.836908	+	0.547344i	$0.184361\pi$
$337$	20.0958i	1.09469i	−0.836908	+	0.547344i	$0.815639\pi$
$338$	0	0
$339$	2.67762	0.145428
$340$	0	0
$341$	−41.9931	−2.27406
$342$	0	0
$343$	− 17.9578i	− 0.969630i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0586i	0.593659i	0.954931	+	0.296829i	$0.0959292\pi$
$347$	11.0586i	0.593659i	−0.954931	+	0.296829i	$0.904071\pi$
$348$	0	0
$349$	−8.11727	−0.434507	−0.217254	−	0.976115i	$-0.569710\pi$
$349$	−8.11727	−0.434507	−0.217254	+	0.976115i	$0.569710\pi$
$350$	0	0
$351$	5.51471	0.294353
$352$	0	0
$353$	12.0130i	0.639387i	0.947521	+	0.319693i	$0.103580\pi$
$353$	12.0130i	0.639387i	−0.947521	+	0.319693i	$0.896420\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.83709i	− 0.256006i
$358$	0	0
$359$	17.9509	0.947413	0.473707	−	0.880683i	$-0.342916\pi$
$359$	17.9509	0.947413	0.473707	+	0.880683i	$0.342916\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 9.35685i	− 0.491108i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.11383i	0.162541i	0.996692	+	0.0812703i	$0.0258977\pi$
$367$	3.11383i	0.162541i	−0.996692	+	0.0812703i	$0.974102\pi$
$368$	0	0
$369$	−35.0518	−1.82472
$370$	0	0
$371$	−24.0751	−1.24991
$372$	0	0
$373$	− 21.8742i	− 1.13261i	−0.824197	−	0.566303i	$-0.808373\pi$
$373$	− 21.8742i	− 1.13261i	0.824197	−	0.566303i	$-0.191627\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.4786i	0.591179i
$378$	0	0
$379$	5.28018	0.271224	0.135612	−	0.990762i	$-0.456700\pi$
$379$	5.28018	0.271224	0.135612	+	0.990762i	$0.456700\pi$
$380$	0	0
$381$	4.85008	0.248477
$382$	0	0
$383$	− 16.1319i	− 0.824300i	−0.911116	−	0.412150i	$-0.864778\pi$
$383$	− 16.1319i	− 0.824300i	0.911116	−	0.412150i	$-0.135222\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 26.2277i	− 1.33323i
$388$	0	0
$389$	−3.43965	−0.174397	−0.0871985	−	0.996191i	$-0.527791\pi$
$389$	−3.43965	−0.174397	−0.0871985	+	0.996191i	$0.527791\pi$
$390$	0	0
$391$	21.8337	1.10418
$392$	0	0
$393$	− 1.46563i	− 0.0739311i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.29317i	0.0649021i	0.999473	+	0.0324511i	$0.0103313\pi$
$397$	1.29317i	0.0649021i	−0.999473	+	0.0324511i	$0.989669\pi$
$398$	0	0
$399$	1.28018	0.0640890
$400$	0	0
$401$	−35.1070	−1.75316	−0.876579	−	0.481258i	$-0.840180\pi$
$401$	−35.1070	−1.75316	−0.876579	+	0.481258i	$0.840180\pi$
$402$	0	0
$403$	15.3224i	0.763262i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 20.8432i	− 1.03316i
$408$	0	0
$409$	24.8793	1.23020	0.615101	−	0.788448i	$-0.289115\pi$
$409$	24.8793	1.23020	0.615101	+	0.788448i	$0.289115\pi$
$410$	0	0
$411$	−6.33193	−0.312331
$412$	0	0
$413$	31.1492i	1.53275i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.49828i	0.220282i
$418$	0	0
$419$	35.4328	1.73100	0.865502	−	0.500905i	$-0.167000\pi$
$419$	35.4328	1.73100	0.865502	+	0.500905i	$0.167000\pi$
$420$	0	0
$421$	14.2147	0.692780	0.346390	−	0.938091i	$-0.387407\pi$
$421$	14.2147	0.692780	0.346390	+	0.938091i	$0.387407\pi$
$422$	0	0
$423$	− 30.8793i	− 1.50140i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.8724i	1.39723i
$428$	0	0
$429$	−5.30328	−0.256045
$430$	0	0
$431$	−24.4983	−1.18004	−0.590020	−	0.807388i	$-0.700880\pi$
$431$	−24.4983	−1.18004	−0.590020	+	0.807388i	$0.700880\pi$
$432$	0	0
$433$	9.45426i	0.454343i	0.973855	+	0.227171i	$0.0729477\pi$
$433$	9.45426i	0.454343i	−0.973855	+	0.227171i	$0.927052\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.77846i	0.276421i
$438$	0	0
$439$	−9.70340	−0.463118	−0.231559	−	0.972821i	$-0.574383\pi$
$439$	−9.70340	−0.463118	−0.231559	+	0.972821i	$0.574383\pi$
$440$	0	0
$441$	−1.10428	−0.0525846
$442$	0	0
$443$	− 6.85008i	− 0.325457i	−0.986671	−	0.162729i	$-0.947971\pi$
$443$	− 6.85008i	− 0.325457i	0.986671	−	0.162729i	$-0.0520295\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.23109i	0.436616i
$448$	0	0
$449$	25.7655	1.21595	0.607974	−	0.793957i	$-0.291983\pi$
$449$	25.7655	1.21595	0.607974	+	0.793957i	$0.291983\pi$
$450$	0	0
$451$	70.1035	3.30105
$452$	0	0
$453$	8.02598i	0.377093i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.400880i	− 0.0187524i	−0.999956	−	0.00937619i	$-0.997015\pi$
$457$	− 0.400880i	− 0.0187524i	0.999956	−	0.00937619i	$-0.00298458\pi$
$458$	0	0
$459$	10.2767	0.479677
$460$	0	0
$461$	23.1070	1.07620	0.538099	−	0.842882i	$-0.319143\pi$
$461$	23.1070	1.07620	0.538099	+	0.842882i	$0.319143\pi$
$462$	0	0
$463$	4.96735i	0.230852i	0.993316	+	0.115426i	$0.0368233\pi$
$463$	4.96735i	0.230852i	−0.993316	+	0.115426i	$0.963177\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 21.6190i	− 1.00041i	−0.865908	−	0.500204i	$-0.833258\pi$
$467$	− 21.6190i	− 1.00041i	0.865908	−	0.500204i	$-0.166742\pi$
$468$	0	0
$469$	31.5078	1.45490
$470$	0	0
$471$	10.5896	0.487942
$472$	0	0
$473$	52.4553i	2.41190i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 24.5941i	− 1.12608i
$478$	0	0
$479$	−2.20855	−0.100911	−0.0504557	−	0.998726i	$-0.516067\pi$
$479$	−2.20855	−0.100911	−0.0504557	+	0.998726i	$0.516067\pi$
$480$	0	0
$481$	−7.60523	−0.346769
$482$	0	0
$483$	7.39744i	0.336595i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.4250i	0.517718i	0.965915	+	0.258859i	$0.0833465\pi$
$487$	11.4250i	0.517718i	−0.965915	+	0.258859i	$0.916654\pi$
$488$	0	0
$489$	10.5604	0.477556
$490$	0	0
$491$	23.6673	1.06809	0.534045	−	0.845456i	$-0.320671\pi$
$491$	23.6673	1.06809	0.534045	+	0.845456i	$0.320671\pi$
$492$	0	0
$493$	21.3906i	0.963383i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.99312	−0.178757	−0.0893784	−	0.995998i	$-0.528488\pi$
$499$	−3.99312	−0.178757	−0.0893784	+	0.995998i	$0.528488\pi$
$500$	0	0
$501$	3.14992	0.140728
$502$	0	0
$503$	0.338809i	0.0151068i	0.999971	+	0.00755338i	$0.00240434\pi$
$503$	0.338809i	0.0151068i	−0.999971	+	0.00755338i	$0.997596\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.18383i	− 0.185811i
$508$	0	0
$509$	28.4914	1.26286	0.631430	−	0.775433i	$-0.282468\pi$
$509$	28.4914	1.26286	0.631430	+	0.775433i	$0.282468\pi$
$510$	0	0
$511$	−25.7095	−1.13732
$512$	0	0
$513$	2.71982i	0.120083i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	61.7586i	2.71614i
$518$	0	0
$519$	−1.34836	−0.0591865
$520$	0	0
$521$	41.4328	1.81520	0.907601	−	0.419833i	$-0.137911\pi$
$521$	41.4328	1.81520	0.907601	+	0.419833i	$0.137911\pi$
$522$	0	0
$523$	− 22.1741i	− 0.969605i	−0.874624	−	0.484802i	$-0.838892\pi$
$523$	− 22.1741i	− 0.969605i	0.874624	−	0.484802i	$-0.161108\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.5535i	1.24381i
$528$	0	0
$529$	−10.3906	−0.451764
$530$	0	0
$531$	−31.8207	−1.38090
$532$	0	0
$533$	− 25.5793i	− 1.10796i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.651639i	0.0281203i
$538$	0	0
$539$	2.20855	0.0951291
$540$	0	0
$541$	−7.61211	−0.327270	−0.163635	−	0.986521i	$-0.552322\pi$
$541$	−7.61211	−0.327270	−0.163635	+	0.986521i	$0.552322\pi$
$542$	0	0
$543$	9.82754i	0.421740i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.3303i	0.783748i	0.920019	+	0.391874i	$0.128173\pi$
$547$	18.3303i	0.783748i	−0.920019	+	0.391874i	$0.871827\pi$
$548$	0	0
$549$	−29.4948	−1.25881
$550$	0	0
$551$	−5.66119	−0.241175
$552$	0	0
$553$	24.3189i	1.03415i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 36.7000i	− 1.55503i	−0.628866	−	0.777514i	$-0.716481\pi$
$557$	− 36.7000i	− 1.55503i	0.628866	−	0.777514i	$-0.283519\pi$
$558$	0	0
$559$	19.1398	0.809528
$560$	0	0
$561$	−9.88273	−0.417249
$562$	0	0
$563$	− 16.1319i	− 0.679877i	−0.940448	−	0.339939i	$-0.889594\pi$
$563$	− 16.1319i	− 0.679877i	0.940448	−	0.339939i	$-0.110406\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 19.1889i	− 0.805858i
$568$	0	0
$569$	19.2051	0.805120	0.402560	−	0.915394i	$-0.368120\pi$
$569$	19.2051	0.805120	0.402560	+	0.915394i	$0.368120\pi$
$570$	0	0
$571$	−2.46907	−0.103327	−0.0516636	−	0.998665i	$-0.516452\pi$
$571$	−2.46907	−0.103327	−0.0516636	+	0.998665i	$0.516452\pi$
$572$	0	0
$573$	− 9.07162i	− 0.378972i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.7233i	− 0.488046i	−0.969769	−	0.244023i	$-0.921533\pi$
$577$	− 11.7233i	− 0.488046i	0.969769	−	0.244023i	$-0.0784672\pi$
$578$	0	0
$579$	−3.77234	−0.156773
$580$	0	0
$581$	−13.4396	−0.557571
$582$	0	0
$583$	49.1881i	2.03716i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.28973i	0.342154i	0.985258	+	0.171077i	$0.0547246\pi$
$587$	8.28973i	0.342154i	−0.985258	+	0.171077i	$0.945275\pi$
$588$	0	0
$589$	−7.55691	−0.311377
$590$	0	0
$591$	−7.94480	−0.326806
$592$	0	0
$593$	− 31.5760i	− 1.29667i	−0.761354	−	0.648336i	$-0.775465\pi$
$593$	− 31.5760i	− 1.29667i	0.761354	−	0.648336i	$-0.224535\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.602558i	0.0246610i
$598$	0	0
$599$	−1.28629	−0.0525564	−0.0262782	−	0.999655i	$-0.508366\pi$
$599$	−1.28629	−0.0525564	−0.0262782	+	0.999655i	$0.508366\pi$
$600$	0	0
$601$	38.8432	1.58445	0.792224	−	0.610231i	$-0.208923\pi$
$601$	38.8432	1.58445	0.792224	+	0.610231i	$0.208923\pi$
$602$	0	0
$603$	32.1871i	1.31076i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 43.6888i	− 1.77327i	−0.462467	−	0.886637i	$-0.653036\pi$
$607$	− 43.6888i	− 1.77327i	0.462467	−	0.886637i	$-0.346964\pi$
$608$	0	0
$609$	−7.24732	−0.293676
$610$	0	0
$611$	22.5344	0.911643
$612$	0	0
$613$	− 30.8172i	− 1.24470i	−0.782741	−	0.622348i	$-0.786179\pi$
$613$	− 30.8172i	− 1.24470i	0.782741	−	0.622348i	$-0.213821\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.4983i	− 0.422645i	−0.977416	−	0.211322i	$-0.932223\pi$
$617$	− 10.4983i	− 0.422645i	0.977416	−	0.211322i	$-0.0677770\pi$
$618$	0	0
$619$	−13.3224	−0.535472	−0.267736	−	0.963492i	$-0.586275\pi$
$619$	−13.3224	−0.535472	−0.267736	+	0.963492i	$0.586275\pi$
$620$	0	0
$621$	−15.7164	−0.630677
$622$	0	0
$623$	42.1432i	1.68843i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.61555i	− 0.104455i
$628$	0	0
$629$	−14.1725	−0.565093
$630$	0	0
$631$	1.21199	0.0482486	0.0241243	−	0.999709i	$-0.492320\pi$
$631$	1.21199	0.0482486	0.0241243	+	0.999709i	$0.492320\pi$
$632$	0	0
$633$	0.726700i	0.0288837i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.805853i	− 0.0319291i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−49.6965	−1.96289	−0.981447	−	0.191732i	$-0.938589\pi$
$641$	−49.6965	−1.96289	−0.981447	+	0.191732i	$0.938589\pi$
$642$	0	0
$643$	− 39.0449i	− 1.53978i	−0.638177	−	0.769890i	$-0.720311\pi$
$643$	− 39.0449i	− 1.53978i	0.638177	−	0.769890i	$-0.279689\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.4232i	1.11743i	0.829359	+	0.558716i	$0.188706\pi$
$647$	28.4232i	1.11743i	−0.829359	+	0.558716i	$0.811294\pi$
$648$	0	0
$649$	63.6413	2.49814
$650$	0	0
$651$	−9.67418	−0.379161
$652$	0	0
$653$	29.0586i	1.13715i	0.822631	+	0.568576i	$0.192506\pi$
$653$	29.0586i	1.13715i	−0.822631	+	0.568576i	$0.807494\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 26.2637i	− 1.02465i
$658$	0	0
$659$	−11.8957	−0.463392	−0.231696	−	0.972788i	$-0.574427\pi$
$659$	−11.8957	−0.463392	−0.231696	+	0.972788i	$0.574427\pi$
$660$	0	0
$661$	30.8923	1.20157	0.600785	−	0.799410i	$-0.294855\pi$
$661$	30.8923	1.20157	0.600785	+	0.799410i	$0.294855\pi$
$662$	0	0
$663$	3.60600i	0.140045i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 32.7129i	− 1.26665i
$668$	0	0
$669$	−1.84664	−0.0713953
$670$	0	0
$671$	58.9897	2.27727
$672$	0	0
$673$	37.5354i	1.44688i	0.690385	+	0.723442i	$0.257441\pi$
$673$	37.5354i	1.44688i	−0.690385	+	0.723442i	$0.742559\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.6466i	1.21628i	0.793831	+	0.608138i	$0.208083\pi$
$677$	31.6466i	1.21628i	−0.793831	+	0.608138i	$0.791917\pi$
$678$	0	0
$679$	−29.5500	−1.13403
$680$	0	0
$681$	−5.13369	−0.196724
$682$	0	0
$683$	22.6233i	0.865656i	0.901477	+	0.432828i	$0.142484\pi$
$683$	22.6233i	0.865656i	−0.901477	+	0.432828i	$0.857516\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.78801i	0.182674i
$688$	0	0
$689$	17.9477	0.683752
$690$	0	0
$691$	17.7655	0.675830	0.337915	−	0.941177i	$-0.390278\pi$
$691$	17.7655	0.675830	0.337915	+	0.941177i	$0.390278\pi$
$692$	0	0
$693$	41.9931i	1.59519i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 47.6673i	− 1.80553i
$698$	0	0
$699$	−4.28973	−0.162252
$700$	0	0
$701$	25.6052	0.967096	0.483548	−	0.875318i	$-0.339348\pi$
$701$	25.6052	0.967096	0.483548	+	0.875318i	$0.339348\pi$
$702$	0	0
$703$	− 3.75086i	− 0.141466i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.2345i	− 0.460127i
$708$	0	0
$709$	−23.2311	−0.872462	−0.436231	−	0.899835i	$-0.643687\pi$
$709$	−23.2311	−0.872462	−0.436231	+	0.899835i	$0.643687\pi$
$710$	0	0
$711$	−24.8432	−0.931693
$712$	0	0
$713$	− 43.6673i	− 1.63535i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.48873i	0.354363i
$718$	0	0
$719$	−24.9544	−0.930640	−0.465320	−	0.885142i	$-0.654061\pi$
$719$	−24.9544	−0.930640	−0.465320	+	0.885142i	$0.654061\pi$
$720$	0	0
$721$	11.8759	0.442280
$722$	0	0
$723$	2.27062i	0.0844454i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.39744i	0.126004i	0.998013	+	0.0630021i	$0.0200675\pi$
$727$	3.39744i	0.126004i	−0.998013	+	0.0630021i	$0.979933\pi$
$728$	0	0
$729$	15.7620	0.583779
$730$	0	0
$731$	35.6673	1.31920
$732$	0	0
$733$	− 9.66730i	− 0.357070i	−0.983934	−	0.178535i	$-0.942864\pi$
$733$	− 9.66730i	− 0.357070i	0.983934	−	0.178535i	$-0.0571358\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 64.3741i	− 2.37125i
$738$	0	0
$739$	−17.0225	−0.626184	−0.313092	−	0.949723i	$-0.601365\pi$
$739$	−17.0225	−0.626184	−0.313092	+	0.949723i	$0.601365\pi$
$740$	0	0
$741$	−0.954357	−0.0350592
$742$	0	0
$743$	8.57496i	0.314585i	0.987552	+	0.157292i	$0.0502765\pi$
$743$	8.57496i	0.314585i	−0.987552	+	0.157292i	$0.949723\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 13.7294i	− 0.502332i
$748$	0	0
$749$	4.47842	0.163638
$750$	0	0
$751$	12.8310	0.468209	0.234104	−	0.972211i	$-0.424784\pi$
$751$	12.8310	0.468209	0.234104	+	0.972211i	$0.424784\pi$
$752$	0	0
$753$	8.57946i	0.312653i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.8827i	1.30418i	0.758142	+	0.652090i	$0.226108\pi$
$757$	35.8827i	1.30418i	−0.758142	+	0.652090i	$0.773892\pi$
$758$	0	0
$759$	15.1138	0.548597
$760$	0	0
$761$	−30.1234	−1.09197	−0.545986	−	0.837794i	$-0.683845\pi$
$761$	−30.1234	−1.09197	−0.545986	+	0.837794i	$0.683845\pi$
$762$	0	0
$763$	− 2.59568i	− 0.0939700i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 23.2213i	− 0.838474i
$768$	0	0
$769$	6.22154	0.224355	0.112177	−	0.993688i	$-0.464218\pi$
$769$	6.22154	0.224355	0.112177	+	0.993688i	$0.464218\pi$
$770$	0	0
$771$	−13.4588	−0.484705
$772$	0	0
$773$	10.2362i	0.368169i	0.982910	+	0.184084i	$0.0589320\pi$
$773$	10.2362i	0.368169i	−0.982910	+	0.184084i	$0.941068\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.80176i	− 0.172262i
$778$	0	0
$779$	12.6155	0.451999
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 15.3974i	− 0.550260i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.34654i	0.154937i	0.996995	+	0.0774687i	$0.0246838\pi$
$787$	4.34654i	0.154937i	−0.996995	+	0.0774687i	$0.975316\pi$
$788$	0	0
$789$	−12.8862	−0.458760
$790$	0	0
$791$	15.4725	0.550139
$792$	0	0
$793$	− 21.5241i	− 0.764342i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.1932i	1.06950i	0.845011	+	0.534749i	$0.179594\pi$
$797$	30.1932i	1.06950i	−0.845011	+	0.534749i	$0.820406\pi$
$798$	0	0
$799$	41.9931	1.48561
$800$	0	0
$801$	−43.0518	−1.52116
$802$	0	0
$803$	52.5275i	1.85366i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.58957i	− 0.231964i
$808$	0	0
$809$	11.8077	0.415136	0.207568	−	0.978221i	$-0.433445\pi$
$809$	11.8077	0.415136	0.207568	+	0.978221i	$0.433445\pi$
$810$	0	0
$811$	0.811111	0.0284820	0.0142410	−	0.999899i	$-0.495467\pi$
$811$	0.811111	0.0284820	0.0142410	+	0.999899i	$0.495467\pi$
$812$	0	0
$813$	− 7.45264i	− 0.261375i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.43965i	0.330251i
$818$	0	0
$819$	15.3224	0.535407
$820$	0	0
$821$	40.6707	1.41942	0.709709	−	0.704495i	$-0.248826\pi$
$821$	40.6707	1.41942	0.709709	+	0.704495i	$0.248826\pi$
$822$	0	0
$823$	− 7.48185i	− 0.260801i	−0.991461	−	0.130401i	$-0.958374\pi$
$823$	− 7.48185i	− 0.260801i	0.991461	−	0.130401i	$-0.0416263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.4346i	1.61469i	0.590080	+	0.807344i	$0.299096\pi$
$827$	46.4346i	1.61469i	−0.590080	+	0.807344i	$0.700904\pi$
$828$	0	0
$829$	10.7267	0.372554	0.186277	−	0.982497i	$-0.440358\pi$
$829$	10.7267	0.372554	0.186277	+	0.982497i	$0.440358\pi$
$830$	0	0
$831$	3.15680	0.109508
$832$	0	0
$833$	− 1.50172i	− 0.0520315i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 20.5535i	− 0.710432i
$838$	0	0
$839$	−10.1465	−0.350295	−0.175148	−	0.984542i	$-0.556040\pi$
$839$	−10.1465	−0.350295	−0.175148	+	0.984542i	$0.556040\pi$
$840$	0	0
$841$	3.04908	0.105141
$842$	0	0
$843$	− 3.94480i	− 0.135866i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 54.0682i	− 1.85780i
$848$	0	0
$849$	−9.55004	−0.327756
$850$	0	0
$851$	21.6742	0.742981
$852$	0	0
$853$	− 26.1104i	− 0.894003i	−0.894533	−	0.447001i	$-0.852492\pi$
$853$	− 26.1104i	− 0.894003i	0.894533	−	0.447001i	$-0.147508\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 20.6922i	− 0.706833i	−0.935466	−	0.353416i	$-0.885020\pi$
$857$	− 20.6922i	− 0.706833i	0.935466	−	0.353416i	$-0.114980\pi$
$858$	0	0
$859$	−3.53093	−0.120474	−0.0602370	−	0.998184i	$-0.519186\pi$
$859$	−3.53093	−0.120474	−0.0602370	+	0.998184i	$0.519186\pi$
$860$	0	0
$861$	16.1501	0.550395
$862$	0	0
$863$	− 3.39906i	− 0.115705i	−0.998325	−	0.0578527i	$-0.981575\pi$
$863$	− 3.39906i	− 0.115705i	0.998325	−	0.0578527i	$-0.0184254\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.28179i	− 0.0435320i
$868$	0	0
$869$	49.6864	1.68550
$870$	0	0
$871$	−23.4887	−0.795885
$872$	0	0
$873$	− 30.1871i	− 1.02168i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.422364i	0.0142622i	0.999975	+	0.00713111i	$0.00226992\pi$
$877$	0.422364i	0.0142622i	−0.999975	+	0.00713111i	$0.997730\pi$
$878$	0	0
$879$	−3.18201	−0.107327
$880$	0	0
$881$	16.5243	0.556716	0.278358	−	0.960477i	$-0.410210\pi$
$881$	16.5243	0.556716	0.278358	+	0.960477i	$0.410210\pi$
$882$	0	0
$883$	24.5535i	0.826290i	0.910665	+	0.413145i	$0.135570\pi$
$883$	24.5535i	0.826290i	−0.910665	+	0.413145i	$0.864430\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.5975i	1.39671i	0.715753	+	0.698354i	$0.246084\pi$
$887$	41.5975i	1.39671i	−0.715753	+	0.698354i	$0.753916\pi$
$888$	0	0
$889$	28.0260	0.939961
$890$	0	0
$891$	−39.2051	−1.31342
$892$	0	0
$893$	11.1138i	0.371910i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.51471i	− 0.184131i
$898$	0	0
$899$	42.7811	1.42683
$900$	0	0
$901$	33.4458	1.11424
$902$	0	0
$903$	12.0844i	0.402144i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 31.9294i	− 1.06020i	−0.847935	−	0.530100i	$-0.822154\pi$
$907$	− 31.9294i	− 1.06020i	0.847935	−	0.530100i	$-0.177846\pi$
$908$	0	0
$909$	12.4983	0.414542
$910$	0	0
$911$	26.9605	0.893240	0.446620	−	0.894724i	$-0.352628\pi$
$911$	26.9605	0.893240	0.446620	+	0.894724i	$0.352628\pi$
$912$	0	0
$913$	27.4588i	0.908752i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.46907i	− 0.279673i
$918$	0	0
$919$	0.394005	0.0129970	0.00649850	−	0.999979i	$-0.497931\pi$
$919$	0.394005	0.0129970	0.00649850	+	0.999979i	$0.497931\pi$
$920$	0	0
$921$	−3.64496	−0.120106
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.1319i	0.398463i
$928$	0	0
$929$	32.9284	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$929$	32.9284	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$930$	0	0
$931$	0.397442	0.0130256
$932$	0	0
$933$	5.19900i	0.170208i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.3388i	0.860451i	0.902721	+	0.430226i	$0.141566\pi$
$937$	26.3388i	0.860451i	−0.902721	+	0.430226i	$0.858434\pi$
$938$	0	0
$939$	0.547362	0.0178625
$940$	0	0
$941$	4.71982	0.153862	0.0769309	−	0.997036i	$-0.475488\pi$
$941$	4.71982	0.153862	0.0769309	+	0.997036i	$0.475488\pi$
$942$	0	0
$943$	72.8984i	2.37390i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 53.9639i	− 1.75359i	−0.480863	−	0.876796i	$-0.659677\pi$
$947$	− 53.9639i	− 1.75359i	0.480863	−	0.876796i	$-0.340323\pi$
$948$	0	0
$949$	19.1661	0.622159
$950$	0	0
$951$	10.2958	0.333866
$952$	0	0
$953$	− 13.0801i	− 0.423707i	−0.977301	−	0.211853i	$-0.932050\pi$
$953$	− 13.0801i	− 0.423707i	0.977301	−	0.211853i	$-0.0679499\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.8071i	0.478646i
$958$	0	0
$959$	−36.5888	−1.18151
$960$	0	0
$961$	26.1070	0.842160
$962$	0	0
$963$	4.57496i	0.147426i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10.4914i	− 0.337381i	−0.985669	−	0.168690i	$-0.946046\pi$
$967$	− 10.4914i	− 0.337381i	0.985669	−	0.168690i	$-0.0539538\pi$
$968$	0	0
$969$	−1.77846	−0.0571323
$970$	0	0
$971$	−12.4691	−0.400151	−0.200076	−	0.979780i	$-0.564119\pi$
$971$	−12.4691	−0.400151	−0.200076	+	0.979780i	$0.564119\pi$
$972$	0	0
$973$	25.9931i	0.833301i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.4699i	1.67866i	0.543621	+	0.839331i	$0.317053\pi$
$977$	52.4699i	1.67866i	−0.543621	+	0.839331i	$0.682947\pi$
$978$	0	0
$979$	86.1035	2.75188
$980$	0	0
$981$	2.65164	0.0846603
$982$	0	0
$983$	14.1939i	0.452717i	0.974044	+	0.226358i	$0.0726820\pi$
$983$	14.1939i	0.452717i	−0.974044	+	0.226358i	$0.927318\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	14.2277i	0.452871i
$988$	0	0
$989$	−54.5466	−1.73448
$990$	0	0
$991$	19.4036	0.616374	0.308187	−	0.951326i	$-0.400278\pi$
$991$	19.4036	0.616374	0.308187	+	0.951326i	$0.400278\pi$
$992$	0	0
$993$	− 6.90240i	− 0.219041i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.14648i	0.258002i	0.991644	+	0.129001i	$0.0411770\pi$
$997$	8.14648i	0.258002i	−0.991644	+	0.129001i	$0.958823\pi$
$998$	0	0
$999$	10.2017	0.322767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.n.3649.3		6
5.2	odd	4		760.2.a.i.1.2	✓	3
5.3	odd	4		3800.2.a.w.1.2		3
5.4	even	2	inner	3800.2.d.n.3649.4		6
15.2	even	4		6840.2.a.bm.1.3		3
20.3	even	4		7600.2.a.bp.1.2		3
20.7	even	4		1520.2.a.q.1.2		3
40.27	even	4		6080.2.a.br.1.2		3
40.37	odd	4		6080.2.a.bx.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.2	✓	3	5.2	odd	4
1520.2.a.q.1.2		3	20.7	even	4
3800.2.a.w.1.2		3	5.3	odd	4
3800.2.d.n.3649.3		6	1.1	even	1	trivial
3800.2.d.n.3649.4		6	5.4	even	2	inner
6080.2.a.br.1.2		3	40.27	even	4
6080.2.a.bx.1.2		3	40.37	odd	4
6840.2.a.bm.1.3		3	15.2	even	4
7600.2.a.bp.1.2		3	20.3	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.470683i q^{3} -2.71982i q^{7} +2.77846 q^{9} +O(q^{10})\)	\(q-0.470683i q^{3} -2.71982i q^{7} +2.77846 q^{9} -5.55691 q^{11} +2.02760i q^{13} +3.77846i q^{17} -1.00000 q^{19} -1.28018 q^{21} -5.77846i q^{23} -2.71982i q^{27} +5.66119 q^{29} +7.55691 q^{31} +2.61555i q^{33} +3.75086i q^{37} +0.954357 q^{39} -12.6155 q^{41} -9.43965i q^{43} -11.1138i q^{47} -0.397442 q^{49} +1.77846 q^{51} -8.85170i q^{53} +0.470683i q^{57} -11.4526 q^{59} -10.6155 q^{61} -7.55691i q^{63} +11.5845i q^{67} -2.71982 q^{69} -9.45264i q^{73} +15.1138i q^{77} -8.94137 q^{79} +7.05520 q^{81} -4.94137i q^{83} -2.66463i q^{87} -15.4948 q^{89} +5.51471 q^{91} -3.55691i q^{93} -10.8647i q^{97} -15.4396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) 6 * q - 6 * q^19 - 26 * q^21 + 14 * q^29 + 12 * q^31 - 6 * q^39 - 44 * q^41 - 24 * q^49 - 6 * q^51 - 22 * q^59 - 32 * q^61 + 2 * q^69 - 52 * q^79 - 26 * q^81 + 12 * q^89 + 58 * q^91 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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