LMFDB - Embedded newform 3800.2.d.j.3649.5

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:	6.0.59105344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 21x^{4} + 116x^{2} + 64 $ x^6 + 21x^4 + 116x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.5
Root			$-0.786802i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.j.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.08387i q^{3} +4.29707i q^{7} -6.51027 q^{9} +O(q^{10})$	$q+3.08387i q^{3} +4.29707i q^{7} -6.51027 q^{9} -1.21320 q^{11} -5.08387i q^{13} -2.29707i q^{17} +1.00000 q^{19} -13.2516 q^{21} -7.67801i q^{23} -10.8252i q^{27} +0.489731 q^{29} -3.74135i q^{33} -2.00000i q^{37} +15.6780 q^{39} -4.16774 q^{41} -12.9545i q^{43} -5.80734i q^{47} -11.4648 q^{49} +7.08387 q^{51} -1.93667i q^{53} +3.08387i q^{57} +11.0839 q^{59} -5.38094 q^{61} -27.9751i q^{63} +2.48973i q^{67} +23.6780 q^{69} -11.7413 q^{71} +8.46482i q^{73} -5.21320i q^{77} -1.83226 q^{79} +13.8528 q^{81} +7.02054i q^{83} +1.51027i q^{87} -13.7413 q^{89} +21.8458 q^{91} -3.57360i q^{97} +7.89825 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{9}+O(q^{10}) $ 6 * q - 24 * q^9	$ 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) $ 6 * q - 24 * q^9 - 10 * q^11 + 6 * q^19 - 18 * q^21 + 18 * q^29 + 38 * q^39 + 16 * q^41 - 10 * q^49 + 22 * q^51 + 46 * q^59 + 6 * q^61 + 86 * q^69 - 24 * q^71 - 52 * q^79 + 94 * q^81 - 36 * q^89 + 34 * q^91 + 102 * 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
$3$	3.08387i	1.78047i	0.455497	+	0.890237i	$0.349462\pi$
$3$	3.08387i	1.78047i	−0.455497	+	0.890237i	$0.650538\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.29707i	1.62414i	0.583560	+	0.812070i	$0.301659\pi$
$7$	4.29707i	1.62414i	−0.583560	+	0.812070i	$0.698341\pi$
$8$	0	0
$9$	−6.51027	−2.17009
$10$	0	0
$11$	−1.21320	−0.365793	−0.182897	−	0.983132i	$-0.558547\pi$
$11$	−1.21320	−0.365793	−0.182897	+	0.983132i	$0.558547\pi$
$12$	0	0
$13$	− 5.08387i	− 1.41001i	−0.709201	−	0.705006i	$-0.750944\pi$
$13$	− 5.08387i	− 1.41001i	0.709201	−	0.705006i	$-0.249056\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.29707i	− 0.557121i	−0.960419	−	0.278561i	$-0.910143\pi$
$17$	− 2.29707i	− 0.557121i	0.960419	−	0.278561i	$-0.0898573\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−13.2516	−2.89174
$22$	0	0
$23$	− 7.67801i	− 1.60098i	−0.599348	−	0.800488i	$-0.704574\pi$
$23$	− 7.67801i	− 1.60098i	0.599348	−	0.800488i	$-0.295426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 10.8252i	− 2.08331i
$28$	0	0
$29$	0.489731	0.0909408	0.0454704	−	0.998966i	$-0.485521\pi$
$29$	0.489731	0.0909408	0.0454704	+	0.998966i	$0.485521\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 3.74135i	− 0.651285i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	15.6780	2.51049
$40$	0	0
$41$	−4.16774	−0.650892	−0.325446	−	0.945561i	$-0.605514\pi$
$41$	−4.16774	−0.650892	−0.325446	+	0.945561i	$0.605514\pi$
$42$	0	0
$43$	− 12.9545i	− 1.97555i	−0.155887	−	0.987775i	$-0.549824\pi$
$43$	− 12.9545i	− 1.97555i	0.155887	−	0.987775i	$-0.450176\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.80734i	− 0.847087i	−0.905876	−	0.423544i	$-0.860786\pi$
$47$	− 5.80734i	− 0.847087i	0.905876	−	0.423544i	$-0.139214\pi$
$48$	0	0
$49$	−11.4648	−1.63783
$50$	0	0
$51$	7.08387	0.991941
$52$	0	0
$53$	− 1.93667i	− 0.266021i	−0.991115	−	0.133011i	$-0.957536\pi$
$53$	− 1.93667i	− 0.266021i	0.991115	−	0.133011i	$-0.0424645\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.08387i	0.408469i
$58$	0	0
$59$	11.0839	1.44300	0.721499	−	0.692416i	$-0.243454\pi$
$59$	11.0839	1.44300	0.721499	+	0.692416i	$0.243454\pi$
$60$	0	0
$61$	−5.38094	−0.688959	−0.344480	−	0.938794i	$-0.611945\pi$
$61$	−5.38094	−0.688959	−0.344480	+	0.938794i	$0.611945\pi$
$62$	0	0
$63$	− 27.9751i	− 3.52453i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.48973i	0.304169i	0.988367	+	0.152085i	$0.0485986\pi$
$67$	2.48973i	0.304169i	−0.988367	+	0.152085i	$0.951401\pi$
$68$	0	0
$69$	23.6780	2.85050
$70$	0	0
$71$	−11.7413	−1.39344	−0.696721	−	0.717342i	$-0.745358\pi$
$71$	−11.7413	−1.39344	−0.696721	+	0.717342i	$0.745358\pi$
$72$	0	0
$73$	8.46482i	0.990732i	0.868684	+	0.495366i	$0.164966\pi$
$73$	8.46482i	0.990732i	−0.868684	+	0.495366i	$0.835034\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.21320i	− 0.594099i
$78$	0	0
$79$	−1.83226	−0.206145	−0.103072	−	0.994674i	$-0.532867\pi$
$79$	−1.83226	−0.206145	−0.103072	+	0.994674i	$0.532867\pi$
$80$	0	0
$81$	13.8528	1.53920
$82$	0	0
$83$	7.02054i	0.770604i	0.922791	+	0.385302i	$0.125903\pi$
$83$	7.02054i	0.770604i	−0.922791	+	0.385302i	$0.874097\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.51027i	0.161918i
$88$	0	0
$89$	−13.7413	−1.45658	−0.728290	−	0.685269i	$-0.759685\pi$
$89$	−13.7413	−1.45658	−0.728290	+	0.685269i	$0.759685\pi$
$90$	0	0
$91$	21.8458	2.29006
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.57360i	− 0.362844i	−0.983405	−	0.181422i	$-0.941930\pi$
$97$	− 3.57360i	− 0.362844i	0.983405	−	0.181422i	$-0.0580701\pi$
$98$	0	0
$99$	7.89825	0.793804
$100$	0	0
$101$	−6.33549	−0.630405	−0.315202	−	0.949024i	$-0.602072\pi$
$101$	−6.33549	−0.630405	−0.315202	+	0.949024i	$0.602072\pi$
$102$	0	0
$103$	5.44693i	0.536702i	0.963321	+	0.268351i	$0.0864788\pi$
$103$	5.44693i	0.536702i	−0.963321	+	0.268351i	$0.913521\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.08387i	0.298129i	0.988827	+	0.149065i	$0.0476262\pi$
$107$	3.08387i	0.298129i	−0.988827	+	0.149065i	$0.952374\pi$
$108$	0	0
$109$	−8.10441	−0.776262	−0.388131	−	0.921604i	$-0.626879\pi$
$109$	−8.10441	−0.776262	−0.388131	+	0.921604i	$0.626879\pi$
$110$	0	0
$111$	6.16774	0.585416
$112$	0	0
$113$	− 14.3355i	− 1.34857i	−0.738471	−	0.674285i	$-0.764452\pi$
$113$	− 14.3355i	− 1.34857i	0.738471	−	0.674285i	$-0.235548\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	33.0974i	3.05985i
$118$	0	0
$119$	9.87067	0.904843
$120$	0	0
$121$	−9.52815	−0.866195
$122$	0	0
$123$	− 12.8528i	− 1.15890i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.74135i	− 0.331991i	−0.986127	−	0.165995i	$-0.946916\pi$
$127$	− 3.74135i	− 0.331991i	0.986127	−	0.165995i	$-0.0530837\pi$
$128$	0	0
$129$	39.9502	3.51742
$130$	0	0
$131$	−12.9545	−1.13184	−0.565922	−	0.824459i	$-0.691480\pi$
$131$	−12.9545	−1.13184	−0.565922	+	0.824459i	$0.691480\pi$
$132$	0	0
$133$	4.29707i	0.372603i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.63256i	− 0.566658i	−0.959023	−	0.283329i	$-0.908561\pi$
$137$	− 6.63256i	− 0.566658i	0.959023	−	0.283329i	$-0.0914388\pi$
$138$	0	0
$139$	4.23374	0.359101	0.179550	−	0.983749i	$-0.442536\pi$
$139$	4.23374	0.359101	0.179550	+	0.983749i	$0.442536\pi$
$140$	0	0
$141$	17.9091	1.50822
$142$	0	0
$143$	6.16774i	0.515773i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 35.3560i	− 2.91612i
$148$	0	0
$149$	9.63959	0.789706	0.394853	−	0.918744i	$-0.370795\pi$
$149$	9.63959	0.789706	0.394853	+	0.918744i	$0.370795\pi$
$150$	0	0
$151$	−5.44693	−0.443265	−0.221633	−	0.975130i	$-0.571139\pi$
$151$	−5.44693	−0.443265	−0.221633	+	0.975130i	$0.571139\pi$
$152$	0	0
$153$	14.9545i	1.20900i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	5.97243	0.473644
$160$	0	0
$161$	32.9930	2.60021
$162$	0	0
$163$	16.3355i	1.27949i	0.768585	+	0.639747i	$0.220961\pi$
$163$	16.3355i	1.27949i	−0.768585	+	0.639747i	$0.779039\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.5736i	− 1.05036i	−0.850992	−	0.525178i	$-0.823999\pi$
$167$	− 13.5736i	− 1.05036i	0.850992	−	0.525178i	$-0.176001\pi$
$168$	0	0
$169$	−12.8458	−0.988135
$170$	0	0
$171$	−6.51027	−0.497853
$172$	0	0
$173$	− 18.3355i	− 1.39402i	−0.717061	−	0.697011i	$-0.754513\pi$
$173$	− 18.3355i	− 1.39402i	0.717061	−	0.697011i	$-0.245487\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	34.1812i	2.56922i
$178$	0	0
$179$	−19.9502	−1.49115	−0.745573	−	0.666424i	$-0.767824\pi$
$179$	−19.9502	−1.49115	−0.745573	+	0.666424i	$0.767824\pi$
$180$	0	0
$181$	−6.85279	−0.509364	−0.254682	−	0.967025i	$-0.581971\pi$
$181$	−6.85279	−0.509364	−0.254682	+	0.967025i	$0.581971\pi$
$182$	0	0
$183$	− 16.5941i	− 1.22667i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.78680i	0.203791i
$188$	0	0
$189$	46.5167	3.38359
$190$	0	0
$191$	19.7798	1.43121	0.715607	−	0.698503i	$-0.246150\pi$
$191$	19.7798	1.43121	0.715607	+	0.698503i	$0.246150\pi$
$192$	0	0
$193$	9.61468i	0.692080i	0.938220	+	0.346040i	$0.112474\pi$
$193$	9.61468i	0.692080i	−0.938220	+	0.346040i	$0.887526\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.16774i	0.296940i	0.988917	+	0.148470i	$0.0474348\pi$
$197$	4.16774i	0.296940i	−0.988917	+	0.148470i	$0.952565\pi$
$198$	0	0
$199$	−5.48535	−0.388846	−0.194423	−	0.980918i	$-0.562283\pi$
$199$	−5.48535	−0.388846	−0.194423	+	0.980918i	$0.562283\pi$
$200$	0	0
$201$	−7.67801	−0.541565
$202$	0	0
$203$	2.10441i	0.147701i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	49.9859i	3.47426i
$208$	0	0
$209$	−1.21320	−0.0839187
$210$	0	0
$211$	−1.89559	−0.130498	−0.0652489	−	0.997869i	$-0.520784\pi$
$211$	−1.89559	−0.130498	−0.0652489	+	0.997869i	$0.520784\pi$
$212$	0	0
$213$	− 36.2088i	− 2.48099i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−26.1044	−1.76397
$220$	0	0
$221$	−11.6780	−0.785548
$222$	0	0
$223$	− 3.74135i	− 0.250539i	−0.992123	−	0.125270i	$-0.960020\pi$
$223$	− 3.74135i	− 0.250539i	0.992123	−	0.125270i	$-0.0399796\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.6780i	1.30608i	0.757325	+	0.653038i	$0.226506\pi$
$227$	19.6780i	1.30608i	−0.757325	+	0.653038i	$0.773494\pi$
$228$	0	0
$229$	−21.7164	−1.43506	−0.717531	−	0.696526i	$-0.754728\pi$
$229$	−21.7164	−1.43506	−0.717531	+	0.696526i	$0.754728\pi$
$230$	0	0
$231$	16.0768	1.05778
$232$	0	0
$233$	10.5692i	0.692413i	0.938158	+	0.346206i	$0.112530\pi$
$233$	10.5692i	0.692413i	−0.938158	+	0.346206i	$0.887470\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5.65044i	− 0.367036i
$238$	0	0
$239$	16.0384	1.03744	0.518720	−	0.854944i	$-0.326409\pi$
$239$	16.0384	1.03744	0.518720	+	0.854944i	$0.326409\pi$
$240$	0	0
$241$	−17.0205	−1.09639	−0.548195	−	0.836351i	$-0.684685\pi$
$241$	−17.0205	−1.09639	−0.548195	+	0.836351i	$0.684685\pi$
$242$	0	0
$243$	10.2446i	0.657190i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.08387i	− 0.323479i
$248$	0	0
$249$	−21.6504	−1.37204
$250$	0	0
$251$	−19.1223	−1.20699	−0.603494	−	0.797367i	$-0.706225\pi$
$251$	−19.1223	−1.20699	−0.603494	+	0.797367i	$0.706225\pi$
$252$	0	0
$253$	9.31495i	0.585626i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 0.553066i	− 0.0344993i	−0.999851	−	0.0172497i	$-0.994509\pi$
$257$	− 0.553066i	− 0.0344993i	0.999851	−	0.0172497i	$-0.00549101\pi$
$258$	0	0
$259$	8.59414	0.534014
$260$	0	0
$261$	−3.18828	−0.197350
$262$	0	0
$263$	− 7.04545i	− 0.434441i	−0.976123	−	0.217221i	$-0.930301\pi$
$263$	− 7.04545i	− 0.434441i	0.976123	−	0.217221i	$-0.0696991\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 42.3766i	− 2.59340i
$268$	0	0
$269$	−12.7619	−0.778106	−0.389053	−	0.921215i	$-0.627198\pi$
$269$	−12.7619	−0.778106	−0.389053	+	0.921215i	$0.627198\pi$
$270$	0	0
$271$	15.6780	0.952371	0.476186	−	0.879345i	$-0.342019\pi$
$271$	15.6780	0.952371	0.476186	+	0.879345i	$0.342019\pi$
$272$	0	0
$273$	67.3695i	4.07739i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.97508i	− 0.359008i	−0.983757	−	0.179504i	$-0.942551\pi$
$277$	− 5.97508i	− 0.359008i	0.983757	−	0.179504i	$-0.0574493\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.2088	1.80211	0.901054	−	0.433708i	$-0.142795\pi$
$281$	30.2088	1.80211	0.901054	+	0.433708i	$0.142795\pi$
$282$	0	0
$283$	1.21320i	0.0721171i	0.999350	+	0.0360586i	$0.0114803\pi$
$283$	1.21320i	0.0721171i	−0.999350	+	0.0360586i	$0.988520\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 17.9091i	− 1.05714i
$288$	0	0
$289$	11.7235	0.689616
$290$	0	0
$291$	11.0205	0.646035
$292$	0	0
$293$	− 0.104410i	− 0.00609969i	−0.999995	−	0.00304984i	$-0.999029\pi$
$293$	− 0.104410i	− 0.00609969i	0.999995	−	0.00304984i	$-0.000970797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13.1331i	0.762062i
$298$	0	0
$299$	−39.0340	−2.25740
$300$	0	0
$301$	55.6666	3.20857
$302$	0	0
$303$	− 19.5378i	− 1.12242i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.852793i	0.0486715i	0.999704	+	0.0243357i	$0.00774707\pi$
$307$	0.852793i	0.0486715i	−0.999704	+	0.0243357i	$0.992253\pi$
$308$	0	0
$309$	−16.7976	−0.955585
$310$	0	0
$311$	−6.12933	−0.347562	−0.173781	−	0.984784i	$-0.555599\pi$
$311$	−6.12933	−0.347562	−0.173781	+	0.984784i	$0.555599\pi$
$312$	0	0
$313$	3.17478i	0.179449i	0.995967	+	0.0897246i	$0.0285987\pi$
$313$	3.17478i	0.179449i	−0.995967	+	0.0897246i	$0.971401\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.6986i	− 1.16255i	−0.813708	−	0.581273i	$-0.802555\pi$
$317$	− 20.6986i	− 1.16255i	0.813708	−	0.581273i	$-0.197445\pi$
$318$	0	0
$319$	−0.594141	−0.0332655
$320$	0	0
$321$	−9.51027	−0.530811
$322$	0	0
$323$	− 2.29707i	− 0.127812i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 24.9930i	− 1.38211i
$328$	0	0
$329$	24.9545	1.37579
$330$	0	0
$331$	9.84576	0.541172	0.270586	−	0.962696i	$-0.412783\pi$
$331$	9.84576	0.541172	0.270586	+	0.962696i	$0.412783\pi$
$332$	0	0
$333$	13.0205i	0.713521i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 18.9296i	− 1.03116i	−0.856841	−	0.515581i	$-0.827576\pi$
$337$	− 18.9296i	− 1.03116i	0.856841	−	0.515581i	$-0.172424\pi$
$338$	0	0
$339$	44.2088	2.40109
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 19.1856i	− 1.03593i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.78680i	− 0.364335i	−0.983268	−	0.182167i	$-0.941689\pi$
$347$	− 6.78680i	− 0.364335i	0.983268	−	0.182167i	$-0.0583112\pi$
$348$	0	0
$349$	−6.23374	−0.333684	−0.166842	−	0.985984i	$-0.553357\pi$
$349$	−6.23374	−0.333684	−0.166842	+	0.985984i	$0.553357\pi$
$350$	0	0
$351$	−55.0340	−2.93750
$352$	0	0
$353$	− 31.7191i	− 1.68824i	−0.536157	−	0.844118i	$-0.680124\pi$
$353$	− 31.7191i	− 1.68824i	0.536157	−	0.844118i	$-0.319876\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	30.4399i	1.61105i
$358$	0	0
$359$	23.3944	1.23471	0.617356	−	0.786684i	$-0.288204\pi$
$359$	23.3944	1.23471	0.617356	+	0.786684i	$0.288204\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 29.3836i	− 1.54224i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 20.3355i	− 1.06150i	−0.847527	−	0.530752i	$-0.821910\pi$
$367$	− 20.3355i	− 1.06150i	0.847527	−	0.530752i	$-0.178090\pi$
$368$	0	0
$369$	27.1331	1.41249
$370$	0	0
$371$	8.32199	0.432056
$372$	0	0
$373$	36.9021i	1.91072i	0.295450	+	0.955358i	$0.404530\pi$
$373$	36.9021i	1.91072i	−0.295450	+	0.955358i	$0.595470\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.48973i	− 0.128228i
$378$	0	0
$379$	15.9367	0.818611	0.409306	−	0.912397i	$-0.365771\pi$
$379$	15.9367	0.818611	0.409306	+	0.912397i	$0.365771\pi$
$380$	0	0
$381$	11.5378	0.591101
$382$	0	0
$383$	19.0205i	0.971904i	0.873985	+	0.485952i	$0.161527\pi$
$383$	19.0205i	0.971904i	−0.873985	+	0.485952i	$0.838473\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	84.3376i	4.28712i
$388$	0	0
$389$	−2.61906	−0.132791	−0.0663957	−	0.997793i	$-0.521150\pi$
$389$	−2.61906	−0.132791	−0.0663957	+	0.997793i	$0.521150\pi$
$390$	0	0
$391$	−17.6369	−0.891938
$392$	0	0
$393$	− 39.9502i	− 2.01522i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 31.1634i	− 1.56404i	−0.623251	−	0.782022i	$-0.714188\pi$
$397$	− 31.1634i	− 1.56404i	0.623251	−	0.782022i	$-0.285812\pi$
$398$	0	0
$399$	−13.2516	−0.663411
$400$	0	0
$401$	21.7413	1.08571	0.542856	−	0.839826i	$-0.317343\pi$
$401$	21.7413	1.08571	0.542856	+	0.839826i	$0.317343\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.42640i	0.120272i
$408$	0	0
$409$	23.2651	1.15039	0.575193	−	0.818018i	$-0.304927\pi$
$409$	23.2651	1.15039	0.575193	+	0.818018i	$0.304927\pi$
$410$	0	0
$411$	20.4540	1.00892
$412$	0	0
$413$	47.6282i	2.34363i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.0563i	0.639370i
$418$	0	0
$419$	−8.33549	−0.407215	−0.203608	−	0.979053i	$-0.565267\pi$
$419$	−8.33549	−0.407215	−0.203608	+	0.979053i	$0.565267\pi$
$420$	0	0
$421$	0.748383	0.0364740	0.0182370	−	0.999834i	$-0.494195\pi$
$421$	0.748383	0.0364740	0.0182370	+	0.999834i	$0.494195\pi$
$422$	0	0
$423$	37.8073i	1.83826i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 23.1223i	− 1.11897i
$428$	0	0
$429$	−19.0205	−0.918320
$430$	0	0
$431$	31.3560	1.51037	0.755183	−	0.655514i	$-0.227548\pi$
$431$	31.3560	1.51037	0.755183	+	0.655514i	$0.227548\pi$
$432$	0	0
$433$	− 9.48270i	− 0.455709i	−0.973695	−	0.227855i	$-0.926829\pi$
$433$	− 9.48270i	− 0.455709i	0.973695	−	0.227855i	$-0.0731711\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.67801i	− 0.367289i
$438$	0	0
$439$	31.9502	1.52490	0.762449	−	0.647048i	$-0.223997\pi$
$439$	31.9502	1.52490	0.762449	+	0.647048i	$0.223997\pi$
$440$	0	0
$441$	74.6390	3.55424
$442$	0	0
$443$	− 2.91878i	− 0.138676i	−0.997593	−	0.0693378i	$-0.977911\pi$
$443$	− 2.91878i	− 0.138676i	0.997593	−	0.0693378i	$-0.0220886\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	29.7273i	1.40605i
$448$	0	0
$449$	−0.0909069	−0.00429016	−0.00214508	−	0.999998i	$-0.500683\pi$
$449$	−0.0909069	−0.00429016	−0.00214508	+	0.999998i	$0.500683\pi$
$450$	0	0
$451$	5.05630	0.238092
$452$	0	0
$453$	− 16.7976i	− 0.789222i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.1499i	0.895793i	0.894085	+	0.447896i	$0.147827\pi$
$457$	19.1499i	0.895793i	−0.894085	+	0.447896i	$0.852173\pi$
$458$	0	0
$459$	−24.8663	−1.16066
$460$	0	0
$461$	33.9253	1.58006	0.790028	−	0.613070i	$-0.210066\pi$
$461$	33.9253	1.58006	0.790028	+	0.613070i	$0.210066\pi$
$462$	0	0
$463$	− 27.3809i	− 1.27250i	−0.771483	−	0.636250i	$-0.780485\pi$
$463$	− 27.3809i	− 1.27250i	0.771483	−	0.636250i	$-0.219515\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.2132i	0.796532i	0.917270	+	0.398266i	$0.130388\pi$
$467$	17.2132i	0.796532i	−0.917270	+	0.398266i	$0.869612\pi$
$468$	0	0
$469$	−10.6986	−0.494013
$470$	0	0
$471$	−43.1742	−1.98936
$472$	0	0
$473$	15.7164i	0.722642i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.6082i	0.577290i
$478$	0	0
$479$	−15.4827	−0.707422	−0.353711	−	0.935355i	$-0.615080\pi$
$479$	−15.4827	−0.707422	−0.353711	+	0.935355i	$0.615080\pi$
$480$	0	0
$481$	−10.1677	−0.463609
$482$	0	0
$483$	101.746i	4.62961i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.1179i	1.36477i	0.730992	+	0.682386i	$0.239058\pi$
$487$	30.1179i	1.36477i	−0.730992	+	0.682386i	$0.760942\pi$
$488$	0	0
$489$	−50.3766	−2.27811
$490$	0	0
$491$	−18.2944	−0.825615	−0.412808	−	0.910818i	$-0.635452\pi$
$491$	−18.2944	−0.825615	−0.412808	+	0.910818i	$0.635452\pi$
$492$	0	0
$493$	− 1.12495i	− 0.0506651i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 50.4534i	− 2.26314i
$498$	0	0
$499$	−10.5282	−0.471305	−0.235652	−	0.971837i	$-0.575723\pi$
$499$	−10.5282	−0.471305	−0.235652	+	0.971837i	$0.575723\pi$
$500$	0	0
$501$	41.8593	1.87013
$502$	0	0
$503$	− 37.2018i	− 1.65875i	−0.558696	−	0.829373i	$-0.688698\pi$
$503$	− 37.2018i	− 1.65875i	0.558696	−	0.829373i	$-0.311302\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 39.6147i	− 1.75935i
$508$	0	0
$509$	20.4264	0.905384	0.452692	−	0.891667i	$-0.350464\pi$
$509$	20.4264	0.905384	0.452692	+	0.891667i	$0.350464\pi$
$510$	0	0
$511$	−36.3739	−1.60909
$512$	0	0
$513$	− 10.8252i	− 0.477945i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.04545i	0.309859i
$518$	0	0
$519$	56.5443	2.48202
$520$	0	0
$521$	−37.8235	−1.65708	−0.828539	−	0.559932i	$-0.810827\pi$
$521$	−37.8235	−1.65708	−0.828539	+	0.559932i	$0.810827\pi$
$522$	0	0
$523$	− 13.0428i	− 0.570322i	−0.958480	−	0.285161i	$-0.907953\pi$
$523$	− 13.0428i	− 0.570322i	0.958480	−	0.285161i	$-0.0920470\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−35.9519	−1.56313
$530$	0	0
$531$	−72.1590	−3.13143
$532$	0	0
$533$	21.1883i	0.917766i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 61.5238i	− 2.65495i
$538$	0	0
$539$	13.9091	0.599107
$540$	0	0
$541$	−27.6754	−1.18986	−0.594928	−	0.803779i	$-0.702820\pi$
$541$	−27.6754	−1.18986	−0.594928	+	0.803779i	$0.702820\pi$
$542$	0	0
$543$	− 21.1331i	− 0.906910i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	35.0314	1.49510
$550$	0	0
$551$	0.489731	0.0208633
$552$	0	0
$553$	− 7.87333i	− 0.334808i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 36.2695i	− 1.53679i	−0.639977	−	0.768394i	$-0.721056\pi$
$557$	− 36.2695i	− 1.53679i	0.639977	−	0.768394i	$-0.278944\pi$
$558$	0	0
$559$	−65.8593	−2.78555
$560$	0	0
$561$	−8.59414	−0.362845
$562$	0	0
$563$	− 13.9091i	− 0.586198i	−0.956082	−	0.293099i	$-0.905313\pi$
$563$	− 13.9091i	− 0.586198i	0.956082	−	0.293099i	$-0.0946866\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	59.5264i	2.49987i
$568$	0	0
$569$	−29.7413	−1.24682	−0.623411	−	0.781894i	$-0.714254\pi$
$569$	−29.7413	−1.24682	−0.623411	+	0.781894i	$0.714254\pi$
$570$	0	0
$571$	−36.6710	−1.53463	−0.767316	−	0.641269i	$-0.778408\pi$
$571$	−36.6710	−1.53463	−0.767316	+	0.641269i	$0.778408\pi$
$572$	0	0
$573$	60.9983i	2.54824i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.4648i	0.851961i	0.904732	+	0.425981i	$0.140071\pi$
$577$	20.4648i	0.851961i	−0.904732	+	0.425981i	$0.859929\pi$
$578$	0	0
$579$	−29.6504	−1.23223
$580$	0	0
$581$	−30.1677	−1.25157
$582$	0	0
$583$	2.34956i	0.0973088i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.2748i	0.754282i	0.926156	+	0.377141i	$0.123093\pi$
$587$	18.2748i	0.754282i	−0.926156	+	0.377141i	$0.876907\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−12.8528	−0.528693
$592$	0	0
$593$	− 31.5238i	− 1.29453i	−0.762267	−	0.647263i	$-0.775914\pi$
$593$	− 31.5238i	− 1.29453i	0.762267	−	0.647263i	$-0.224086\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 16.9161i	− 0.692331i
$598$	0	0
$599$	26.3766	1.07772	0.538859	−	0.842396i	$-0.318856\pi$
$599$	26.3766	1.07772	0.538859	+	0.842396i	$0.318856\pi$
$600$	0	0
$601$	−27.3971	−1.11755	−0.558776	−	0.829319i	$-0.688729\pi$
$601$	−27.3971	−1.11755	−0.558776	+	0.829319i	$0.688729\pi$
$602$	0	0
$603$	− 16.2088i	− 0.660074i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.9859i	1.37945i	0.724073	+	0.689723i	$0.242268\pi$
$607$	33.9859i	1.37945i	−0.724073	+	0.689723i	$0.757732\pi$
$608$	0	0
$609$	−6.48973	−0.262977
$610$	0	0
$611$	−29.5238	−1.19440
$612$	0	0
$613$	− 3.08653i	− 0.124664i	−0.998055	−	0.0623319i	$-0.980146\pi$
$613$	− 3.08653i	− 0.124664i	0.998055	−	0.0623319i	$-0.0198537\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.1018i	− 1.05082i	−0.850850	−	0.525409i	$-0.823913\pi$
$617$	− 26.1018i	− 1.05082i	0.850850	−	0.525409i	$-0.176087\pi$
$618$	0	0
$619$	−29.0616	−1.16808	−0.584042	−	0.811723i	$-0.698530\pi$
$619$	−29.0616	−1.16808	−0.584042	+	0.811723i	$0.698530\pi$
$620$	0	0
$621$	−83.1162	−3.33534
$622$	0	0
$623$	− 59.0475i	− 2.36569i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.74135i	− 0.149415i
$628$	0	0
$629$	−4.59414	−0.183180
$630$	0	0
$631$	20.1018	0.800238	0.400119	−	0.916463i	$-0.368969\pi$
$631$	20.1018	0.800238	0.400119	+	0.916463i	$0.368969\pi$
$632$	0	0
$633$	− 5.84576i	− 0.232348i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	58.2857i	2.30936i
$638$	0	0
$639$	76.4393	3.02389
$640$	0	0
$641$	24.7619	0.978036	0.489018	−	0.872274i	$-0.337355\pi$
$641$	24.7619	0.978036	0.489018	+	0.872274i	$0.337355\pi$
$642$	0	0
$643$	11.8431i	0.467046i	0.972351	+	0.233523i	$0.0750255\pi$
$643$	11.8431i	0.467046i	−0.972351	+	0.233523i	$0.924975\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.4854i	1.47370i	0.676056	+	0.736851i	$0.263688\pi$
$647$	37.4854i	1.47370i	−0.676056	+	0.736851i	$0.736312\pi$
$648$	0	0
$649$	−13.4469	−0.527838
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.8279i	1.36292i	0.731855	+	0.681460i	$0.238655\pi$
$653$	34.8279i	1.36292i	−0.731855	+	0.681460i	$0.761345\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 55.1082i	− 2.14998i
$658$	0	0
$659$	−28.2722	−1.10133	−0.550663	−	0.834727i	$-0.685625\pi$
$659$	−28.2722	−1.10133	−0.550663	+	0.834727i	$0.685625\pi$
$660$	0	0
$661$	2.58064	0.100375	0.0501876	−	0.998740i	$-0.484018\pi$
$661$	2.58064	0.100375	0.0501876	+	0.998740i	$0.484018\pi$
$662$	0	0
$663$	− 36.0135i	− 1.39865i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.76016i	− 0.145594i
$668$	0	0
$669$	11.5378	0.446079
$670$	0	0
$671$	6.52815	0.252016
$672$	0	0
$673$	− 50.5443i	− 1.94834i	−0.225816	−	0.974170i	$-0.572505\pi$
$673$	− 50.5443i	− 1.94834i	0.225816	−	0.974170i	$-0.427495\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.825221i	0.0317158i	0.999874	+	0.0158579i	$0.00504794\pi$
$677$	0.825221i	0.0317158i	−0.999874	+	0.0158579i	$0.994952\pi$
$678$	0	0
$679$	15.3560	0.589310
$680$	0	0
$681$	−60.6845	−2.32543
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 66.9707i	− 2.55509i
$688$	0	0
$689$	−9.84576	−0.375094
$690$	0	0
$691$	0.0249160	0.000947850 0	0.000473925	−	1.00000i	$-0.499849\pi$
$691$	0.0249160	0.000947850 0	0.000473925		1.00000i	$0.499849\pi$
$692$	0	0
$693$	33.9393i	1.28925i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.57360i	0.362626i
$698$	0	0
$699$	−32.5941	−1.23282
$700$	0	0
$701$	29.3560	1.10876	0.554381	−	0.832263i	$-0.312955\pi$
$701$	29.3560	1.10876	0.554381	+	0.832263i	$0.312955\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 27.2240i	− 1.02387i
$708$	0	0
$709$	17.4827	0.656576	0.328288	−	0.944578i	$-0.393528\pi$
$709$	17.4827	0.656576	0.328288	+	0.944578i	$0.393528\pi$
$710$	0	0
$711$	11.9285	0.447353
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.4604i	1.84713i
$718$	0	0
$719$	−18.5915	−0.693345	−0.346673	−	0.937986i	$-0.612689\pi$
$719$	−18.5915	−0.693345	−0.346673	+	0.937986i	$0.612689\pi$
$720$	0	0
$721$	−23.4059	−0.871680
$722$	0	0
$723$	− 52.4892i	− 1.95209i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.50589i	− 0.315466i	−0.987482	−	0.157733i	$-0.949581\pi$
$727$	− 8.50589i	− 0.315466i	0.987482	−	0.157733i	$-0.0504185\pi$
$728$	0	0
$729$	9.96539	0.369089
$730$	0	0
$731$	−29.7575	−1.10062
$732$	0	0
$733$	− 24.5032i	− 0.905048i	−0.891752	−	0.452524i	$-0.850524\pi$
$733$	− 24.5032i	− 0.905048i	0.891752	−	0.452524i	$-0.149476\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.02054i	− 0.111263i
$738$	0	0
$739$	20.9047	0.768992	0.384496	−	0.923127i	$-0.374375\pi$
$739$	20.9047	0.768992	0.384496	+	0.923127i	$0.374375\pi$
$740$	0	0
$741$	15.6780	0.575946
$742$	0	0
$743$	− 21.5238i	− 0.789631i	−0.918761	−	0.394815i	$-0.870809\pi$
$743$	− 21.5238i	− 0.789631i	0.918761	−	0.394815i	$-0.129191\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 45.7056i	− 1.67228i
$748$	0	0
$749$	−13.2516	−0.484204
$750$	0	0
$751$	2.37656	0.0867221	0.0433610	−	0.999059i	$-0.486193\pi$
$751$	2.37656	0.0867221	0.0433610	+	0.999059i	$0.486193\pi$
$752$	0	0
$753$	− 58.9707i	− 2.14901i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.9545i	1.70659i	0.521427	+	0.853296i	$0.325400\pi$
$757$	46.9545i	1.70659i	−0.521427	+	0.853296i	$0.674600\pi$
$758$	0	0
$759$	−28.7261	−1.04269
$760$	0	0
$761$	−5.44428	−0.197355	−0.0986775	−	0.995119i	$-0.531461\pi$
$761$	−5.44428	−0.197355	−0.0986775	+	0.995119i	$0.531461\pi$
$762$	0	0
$763$	− 34.8252i	− 1.26076i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 56.3490i	− 2.03464i
$768$	0	0
$769$	−8.92697	−0.321915	−0.160957	−	0.986961i	$-0.551458\pi$
$769$	−8.92697	−0.321915	−0.160957	+	0.986961i	$0.551458\pi$
$770$	0	0
$771$	1.70559	0.0614252
$772$	0	0
$773$	17.8047i	0.640390i	0.947352	+	0.320195i	$0.103748\pi$
$773$	17.8047i	0.640390i	−0.947352	+	0.320195i	$0.896252\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	26.5032i	0.950798i
$778$	0	0
$779$	−4.16774	−0.149325
$780$	0	0
$781$	14.2446	0.509711
$782$	0	0
$783$	− 5.30145i	− 0.189458i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.3836i	0.477074i	0.971133	+	0.238537i	$0.0766678\pi$
$787$	13.3836i	0.477074i	−0.971133	+	0.238537i	$0.923332\pi$
$788$	0	0
$789$	21.7273	0.773512
$790$	0	0
$791$	61.6006	2.19027
$792$	0	0
$793$	27.3560i	0.971441i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.5666i	1.01188i	0.862569	+	0.505940i	$0.168854\pi$
$797$	28.5666i	1.01188i	−0.862569	+	0.505940i	$0.831146\pi$
$798$	0	0
$799$	−13.3399	−0.471931
$800$	0	0
$801$	89.4599	3.16091
$802$	0	0
$803$	− 10.2695i	− 0.362403i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 39.3560i	− 1.38540i
$808$	0	0
$809$	−45.2625	−1.59134	−0.795672	−	0.605728i	$-0.792882\pi$
$809$	−45.2625	−1.59134	−0.795672	+	0.605728i	$0.792882\pi$
$810$	0	0
$811$	−49.4551	−1.73660	−0.868302	−	0.496036i	$-0.834789\pi$
$811$	−49.4551	−1.73660	−0.868302	+	0.496036i	$0.834789\pi$
$812$	0	0
$813$	48.3490i	1.69567i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 12.9545i	− 0.453222i
$818$	0	0
$819$	−142.222	−4.96963
$820$	0	0
$821$	−41.7933	−1.45860	−0.729298	−	0.684197i	$-0.760153\pi$
$821$	−41.7933	−1.45860	−0.729298	+	0.684197i	$0.760153\pi$
$822$	0	0
$823$	8.68239i	0.302649i	0.988484	+	0.151325i	$0.0483539\pi$
$823$	8.68239i	0.302649i	−0.988484	+	0.151325i	$0.951646\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.1196i	0.456214i	0.973636	+	0.228107i	$0.0732537\pi$
$827$	13.1196i	0.456214i	−0.973636	+	0.228107i	$0.926746\pi$
$828$	0	0
$829$	8.83053	0.306697	0.153349	−	0.988172i	$-0.450994\pi$
$829$	8.83053	0.306697	0.153349	+	0.988172i	$0.450994\pi$
$830$	0	0
$831$	18.4264	0.639205
$832$	0	0
$833$	26.3355i	0.912471i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.80296	−0.165817	−0.0829083	−	0.996557i	$-0.526421\pi$
$839$	−4.80296	−0.165817	−0.0829083	+	0.996557i	$0.526421\pi$
$840$	0	0
$841$	−28.7602	−0.991730
$842$	0	0
$843$	93.1601i	3.20861i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 40.9431i	− 1.40682i
$848$	0	0
$849$	−3.74135	−0.128403
$850$	0	0
$851$	−15.3560	−0.526398
$852$	0	0
$853$	2.51730i	0.0861908i	0.999071	+	0.0430954i	$0.0137219\pi$
$853$	2.51730i	0.0861908i	−0.999071	+	0.0430954i	$0.986278\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 37.3560i	− 1.27606i	−0.770013	−	0.638029i	$-0.779750\pi$
$857$	− 37.3560i	− 1.27606i	0.770013	−	0.638029i	$-0.220250\pi$
$858$	0	0
$859$	−40.6461	−1.38683	−0.693413	−	0.720540i	$-0.743894\pi$
$859$	−40.6461	−1.38683	−0.693413	+	0.720540i	$0.743894\pi$
$860$	0	0
$861$	55.2294	1.88221
$862$	0	0
$863$	7.40586i	0.252098i	0.992024	+	0.126049i	$0.0402297\pi$
$863$	7.40586i	0.252098i	−0.992024	+	0.126049i	$0.959770\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	36.1537i	1.22784i
$868$	0	0
$869$	2.22289	0.0754063
$870$	0	0
$871$	12.6575	0.428882
$872$	0	0
$873$	23.2651i	0.787405i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.5308i	1.70630i	0.521662	+	0.853152i	$0.325312\pi$
$877$	50.5308i	1.70630i	−0.521662	+	0.853152i	$0.674688\pi$
$878$	0	0
$879$	0.321987	0.0108603
$880$	0	0
$881$	33.4631	1.12740	0.563700	−	0.825980i	$-0.309377\pi$
$881$	33.4631	1.12740	0.563700	+	0.825980i	$0.309377\pi$
$882$	0	0
$883$	− 30.0162i	− 1.01012i	−0.863083	−	0.505062i	$-0.831470\pi$
$883$	− 30.0162i	− 1.01012i	0.863083	−	0.505062i	$-0.168530\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.0704i	0.640320i	0.947363	+	0.320160i	$0.103737\pi$
$887$	19.0704i	0.640320i	−0.947363	+	0.320160i	$0.896263\pi$
$888$	0	0
$889$	16.0768	0.539200
$890$	0	0
$891$	−16.8062	−0.563028
$892$	0	0
$893$	− 5.80734i	− 0.194335i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 120.376i	− 4.01924i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−4.44866	−0.148206
$902$	0	0
$903$	171.669i	5.71278i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 48.0135i	− 1.59426i	−0.603806	−	0.797131i	$-0.706350\pi$
$907$	− 48.0135i	− 1.59426i	0.603806	−	0.797131i	$-0.293650\pi$
$908$	0	0
$909$	41.2457	1.36803
$910$	0	0
$911$	−46.2446	−1.53215	−0.766076	−	0.642750i	$-0.777793\pi$
$911$	−46.2446	−1.53215	−0.766076	+	0.642750i	$0.777793\pi$
$912$	0	0
$913$	− 8.51730i	− 0.281882i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 55.6666i	− 1.83827i
$918$	0	0
$919$	−41.5103	−1.36930	−0.684649	−	0.728873i	$-0.740044\pi$
$919$	−41.5103	−1.36930	−0.684649	+	0.728873i	$0.740044\pi$
$920$	0	0
$921$	−2.62990	−0.0866583
$922$	0	0
$923$	59.6915i	1.96477i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 35.4610i	− 1.16469i
$928$	0	0
$929$	−55.4551	−1.81942	−0.909712	−	0.415240i	$-0.863698\pi$
$929$	−55.4551	−1.81942	−0.909712	+	0.415240i	$0.863698\pi$
$930$	0	0
$931$	−11.4648	−0.375744
$932$	0	0
$933$	− 18.9021i	− 0.618826i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 6.08825i	− 0.198894i	−0.995043	−	0.0994472i	$-0.968293\pi$
$937$	− 6.08825i	− 0.198894i	0.995043	−	0.0994472i	$-0.0317075\pi$
$938$	0	0
$939$	−9.79061	−0.319505
$940$	0	0
$941$	−11.3836	−0.371095	−0.185547	−	0.982635i	$-0.559406\pi$
$941$	−11.3836	−0.371095	−0.185547	+	0.982635i	$0.559406\pi$
$942$	0	0
$943$	32.0000i	1.04206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.3766i	− 1.24707i	−0.781795	−	0.623535i	$-0.785696\pi$
$947$	− 38.3766i	− 1.24707i	0.781795	−	0.623535i	$-0.214304\pi$
$948$	0	0
$949$	43.0340	1.39694
$950$	0	0
$951$	63.8317	2.06988
$952$	0	0
$953$	− 30.2857i	− 0.981049i	−0.871427	−	0.490524i	$-0.836805\pi$
$953$	− 30.2857i	− 0.981049i	0.871427	−	0.490524i	$-0.163195\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.83226i	− 0.0592284i
$958$	0	0
$959$	28.5006	0.920332
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 20.0768i	− 0.646967i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 24.6710i	− 0.793365i	−0.917956	−	0.396683i	$-0.870161\pi$
$967$	− 24.6710i	− 0.793365i	0.917956	−	0.396683i	$-0.129839\pi$
$968$	0	0
$969$	7.08387	0.227567
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	18.1927i	0.583230i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.4059i	− 0.684834i	−0.939548	−	0.342417i	$-0.888754\pi$
$977$	− 21.4059i	− 0.684834i	0.939548	−	0.342417i	$-0.111246\pi$
$978$	0	0
$979$	16.6710	0.532807
$980$	0	0
$981$	52.7619	1.68456
$982$	0	0
$983$	− 38.1677i	− 1.21736i	−0.793415	−	0.608681i	$-0.791699\pi$
$983$	− 38.1677i	− 1.21736i	0.793415	−	0.608681i	$-0.208301\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	76.9566i	2.44956i
$988$	0	0
$989$	−99.4652	−3.16281
$990$	0	0
$991$	16.4675	0.523106	0.261553	−	0.965189i	$-0.415765\pi$
$991$	16.4675	0.523106	0.261553	+	0.965189i	$0.415765\pi$
$992$	0	0
$993$	30.3631i	0.963543i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.8431i	0.438415i	0.975678	+	0.219208i	$0.0703472\pi$
$997$	13.8431i	0.438415i	−0.975678	+	0.219208i	$0.929653\pi$
$998$	0	0
$999$	−21.6504	−0.684990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.j.3649.5		6
5.2	odd	4		3800.2.a.r.1.3		3
5.3	odd	4		152.2.a.c.1.1	✓	3
5.4	even	2	inner	3800.2.d.j.3649.2		6
15.8	even	4		1368.2.a.n.1.2		3
20.3	even	4		304.2.a.g.1.3		3
20.7	even	4		7600.2.a.bv.1.1		3
35.13	even	4		7448.2.a.bf.1.3		3
40.3	even	4		1216.2.a.v.1.1		3
40.13	odd	4		1216.2.a.u.1.3		3
60.23	odd	4		2736.2.a.bd.1.2		3
95.18	even	4		2888.2.a.o.1.3		3
380.303	odd	4		5776.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.c.1.1	✓	3	5.3	odd	4
304.2.a.g.1.3		3	20.3	even	4
1216.2.a.u.1.3		3	40.13	odd	4
1216.2.a.v.1.1		3	40.3	even	4
1368.2.a.n.1.2		3	15.8	even	4
2736.2.a.bd.1.2		3	60.23	odd	4
2888.2.a.o.1.3		3	95.18	even	4
3800.2.a.r.1.3		3	5.2	odd	4
3800.2.d.j.3649.2		6	5.4	even	2	inner
3800.2.d.j.3649.5		6	1.1	even	1	trivial
5776.2.a.bp.1.1		3	380.303	odd	4
7448.2.a.bf.1.3		3	35.13	even	4
7600.2.a.bv.1.1		3	20.7	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)

\(f(q)\)	\(=\)	\(q+3.08387i q^{3} +4.29707i q^{7} -6.51027 q^{9} +O(q^{10})\)	\(q+3.08387i q^{3} +4.29707i q^{7} -6.51027 q^{9} -1.21320 q^{11} -5.08387i q^{13} -2.29707i q^{17} +1.00000 q^{19} -13.2516 q^{21} -7.67801i q^{23} -10.8252i q^{27} +0.489731 q^{29} -3.74135i q^{33} -2.00000i q^{37} +15.6780 q^{39} -4.16774 q^{41} -12.9545i q^{43} -5.80734i q^{47} -11.4648 q^{49} +7.08387 q^{51} -1.93667i q^{53} +3.08387i q^{57} +11.0839 q^{59} -5.38094 q^{61} -27.9751i q^{63} +2.48973i q^{67} +23.6780 q^{69} -11.7413 q^{71} +8.46482i q^{73} -5.21320i q^{77} -1.83226 q^{79} +13.8528 q^{81} +7.02054i q^{83} +1.51027i q^{87} -13.7413 q^{89} +21.8458 q^{91} -3.57360i q^{97} +7.89825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{9}+O(q^{10}) \) 6 * q - 24 * q^9	\( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) \) 6 * q - 24 * q^9 - 10 * q^11 + 6 * q^19 - 18 * q^21 + 18 * q^29 + 38 * q^39 + 16 * q^41 - 10 * q^49 + 22 * q^51 + 46 * q^59 + 6 * q^61 + 86 * q^69 - 24 * q^71 - 52 * q^79 + 94 * q^81 - 36 * q^89 + 34 * q^91 + 102 * q^99

Introduction

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