LMFDB - Embedded newform 9904.2.a.n.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9904,2,Mod(1,9904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9904, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9904 = 2^{4} \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9904.a (trivial)

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:	yes
Analytic conductor:	$79.0838381619$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 619)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	9904.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.725873 q^{3} -0.225133 q^{5} -2.47282 q^{7} -2.47311 q^{9} +O(q^{10})$	$q+0.725873 q^{3} -0.225133 q^{5} -2.47282 q^{7} -2.47311 q^{9} +4.06256 q^{11} +2.26536 q^{13} -0.163418 q^{15} +7.19086 q^{17} +7.47015 q^{19} -1.79495 q^{21} +0.178059 q^{23} -4.94931 q^{25} -3.97278 q^{27} +10.5609 q^{29} -1.27650 q^{31} +2.94891 q^{33} +0.556714 q^{35} +2.08508 q^{37} +1.64436 q^{39} +4.23367 q^{41} -8.52011 q^{43} +0.556779 q^{45} -9.96677 q^{47} -0.885174 q^{49} +5.21965 q^{51} +0.619984 q^{53} -0.914619 q^{55} +5.42238 q^{57} -11.8540 q^{59} +6.30300 q^{61} +6.11554 q^{63} -0.510008 q^{65} -12.5388 q^{67} +0.129248 q^{69} +9.50401 q^{71} +5.14906 q^{73} -3.59258 q^{75} -10.0460 q^{77} +16.3932 q^{79} +4.53559 q^{81} -4.79273 q^{83} -1.61890 q^{85} +7.66589 q^{87} -1.64933 q^{89} -5.60182 q^{91} -0.926577 q^{93} -1.68178 q^{95} +3.34400 q^{97} -10.0472 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) $ 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9	$ 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) $ 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9 - 23 * q^11 + 9 * q^13 + 2 * q^15 + 4 * q^17 + q^19 + 30 * q^21 - 4 * q^23 + 35 * q^25 + 5 * q^27 + 90 * q^29 - 2 * q^31 - 6 * q^33 - 9 * q^35 + 19 * q^37 - 32 * q^39 + 59 * q^41 + 4 * q^43 + 30 * q^45 - 4 * q^47 + 30 * q^49 + 34 * q^53 + 17 * q^55 - 8 * q^57 - 13 * q^59 + 16 * q^61 + 40 * q^63 + 31 * q^65 + 11 * q^67 + 6 * q^69 - 42 * q^71 - 4 * q^73 + 52 * q^75 + 29 * q^77 - 3 * q^79 + 30 * q^81 + 11 * q^83 + 19 * q^85 + 20 * q^87 + 58 * q^89 + 39 * q^91 - 15 * q^93 - 23 * q^95 - 9 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.725873	0.419083	0.209542	−	0.977800i	$-0.432803\pi$
$3$	0.725873	0.419083	0.209542	+	0.977800i	$0.432803\pi$
$4$	0	0
$5$	−0.225133	−0.100683	−0.0503414	−	0.998732i	$-0.516031\pi$
$5$	−0.225133	−0.100683	−0.0503414	+	0.998732i	$0.516031\pi$
$6$	0	0
$7$	−2.47282	−0.934637	−0.467319	−	0.884089i	$-0.654780\pi$
$7$	−2.47282	−0.934637	−0.467319	+	0.884089i	$0.654780\pi$
$8$	0	0
$9$	−2.47311	−0.824369
$10$	0	0
$11$	4.06256	1.22491	0.612455	−	0.790506i	$-0.290182\pi$
$11$	4.06256	1.22491	0.612455	+	0.790506i	$0.290182\pi$
$12$	0	0
$13$	2.26536	0.628298	0.314149	−	0.949374i	$-0.398281\pi$
$13$	2.26536	0.628298	0.314149	+	0.949374i	$0.398281\pi$
$14$	0	0
$15$	−0.163418	−0.0421944
$16$	0	0
$17$	7.19086	1.74404	0.872019	−	0.489471i	$-0.162810\pi$
$17$	7.19086	1.74404	0.872019	+	0.489471i	$0.162810\pi$
$18$	0	0
$19$	7.47015	1.71377	0.856885	−	0.515507i	$-0.172397\pi$
$19$	7.47015	1.71377	0.856885	+	0.515507i	$0.172397\pi$
$20$	0	0
$21$	−1.79495	−0.391691
$22$	0	0
$23$	0.178059	0.0371279	0.0185639	−	0.999828i	$-0.494091\pi$
$23$	0.178059	0.0371279	0.0185639	+	0.999828i	$0.494091\pi$
$24$	0	0
$25$	−4.94931	−0.989863
$26$	0	0
$27$	−3.97278	−0.764562
$28$	0	0
$29$	10.5609	1.96111	0.980557	−	0.196232i	$-0.0628707\pi$
$29$	10.5609	1.96111	0.980557	+	0.196232i	$0.0628707\pi$
$30$	0	0
$31$	−1.27650	−0.229266	−0.114633	−	0.993408i	$-0.536569\pi$
$31$	−1.27650	−0.229266	−0.114633	+	0.993408i	$0.536569\pi$
$32$	0	0
$33$	2.94891	0.513339
$34$	0	0
$35$	0.556714	0.0941018
$36$	0	0
$37$	2.08508	0.342784	0.171392	−	0.985203i	$-0.445173\pi$
$37$	2.08508	0.342784	0.171392	+	0.985203i	$0.445173\pi$
$38$	0	0
$39$	1.64436	0.263309
$40$	0	0
$41$	4.23367	0.661188	0.330594	−	0.943773i	$-0.392751\pi$
$41$	4.23367	0.661188	0.330594	+	0.943773i	$0.392751\pi$
$42$	0	0
$43$	−8.52011	−1.29930	−0.649652	−	0.760232i	$-0.725085\pi$
$43$	−8.52011	−1.29930	−0.649652	+	0.760232i	$0.725085\pi$
$44$	0	0
$45$	0.556779	0.0829997
$46$	0	0
$47$	−9.96677	−1.45380	−0.726901	−	0.686742i	$-0.759040\pi$
$47$	−9.96677	−1.45380	−0.726901	+	0.686742i	$0.759040\pi$
$48$	0	0
$49$	−0.885174	−0.126453
$50$	0	0
$51$	5.21965	0.730897
$52$	0	0
$53$	0.619984	0.0851614	0.0425807	−	0.999093i	$-0.486442\pi$
$53$	0.619984	0.0851614	0.0425807	+	0.999093i	$0.486442\pi$
$54$	0	0
$55$	−0.914619	−0.123327
$56$	0	0
$57$	5.42238	0.718212
$58$	0	0
$59$	−11.8540	−1.54326	−0.771628	−	0.636074i	$-0.780557\pi$
$59$	−11.8540	−1.54326	−0.771628	+	0.636074i	$0.780557\pi$
$60$	0	0
$61$	6.30300	0.807017	0.403508	−	0.914976i	$-0.367791\pi$
$61$	6.30300	0.807017	0.403508	+	0.914976i	$0.367791\pi$
$62$	0	0
$63$	6.11554	0.770486
$64$	0	0
$65$	−0.510008	−0.0632587
$66$	0	0
$67$	−12.5388	−1.53186	−0.765928	−	0.642927i	$-0.777720\pi$
$67$	−12.5388	−1.53186	−0.765928	+	0.642927i	$0.777720\pi$
$68$	0	0
$69$	0.129248	0.0155597
$70$	0	0
$71$	9.50401	1.12792	0.563959	−	0.825803i	$-0.309277\pi$
$71$	9.50401	1.12792	0.563959	+	0.825803i	$0.309277\pi$
$72$	0	0
$73$	5.14906	0.602652	0.301326	−	0.953521i	$-0.402571\pi$
$73$	5.14906	0.602652	0.301326	+	0.953521i	$0.402571\pi$
$74$	0	0
$75$	−3.59258	−0.414835
$76$	0	0
$77$	−10.0460	−1.14485
$78$	0	0
$79$	16.3932	1.84438	0.922190	−	0.386737i	$-0.126398\pi$
$79$	16.3932	1.84438	0.922190	+	0.386737i	$0.126398\pi$
$80$	0	0
$81$	4.53559	0.503954
$82$	0	0
$83$	−4.79273	−0.526071	−0.263035	−	0.964786i	$-0.584724\pi$
$83$	−4.79273	−0.526071	−0.263035	+	0.964786i	$0.584724\pi$
$84$	0	0
$85$	−1.61890	−0.175595
$86$	0	0
$87$	7.66589	0.821870
$88$	0	0
$89$	−1.64933	−0.174829	−0.0874143	−	0.996172i	$-0.527860\pi$
$89$	−1.64933	−0.174829	−0.0874143	+	0.996172i	$0.527860\pi$
$90$	0	0
$91$	−5.60182	−0.587231
$92$	0	0
$93$	−0.926577	−0.0960816
$94$	0	0
$95$	−1.68178	−0.172547
$96$	0	0
$97$	3.34400	0.339531	0.169766	−	0.985484i	$-0.445699\pi$
$97$	3.34400	0.339531	0.169766	+	0.985484i	$0.445699\pi$
$98$	0	0
$99$	−10.0472	−1.00978
$100$	0	0
$101$	−12.6806	−1.26176	−0.630882	−	0.775878i	$-0.717307\pi$
$101$	−12.6806	−1.26176	−0.630882	+	0.775878i	$0.717307\pi$
$102$	0	0
$103$	4.47808	0.441238	0.220619	−	0.975360i	$-0.429192\pi$
$103$	4.47808	0.441238	0.220619	+	0.975360i	$0.429192\pi$
$104$	0	0
$105$	0.404104	0.0394365
$106$	0	0
$107$	−15.0754	−1.45740	−0.728699	−	0.684834i	$-0.759875\pi$
$107$	−15.0754	−1.45740	−0.728699	+	0.684834i	$0.759875\pi$
$108$	0	0
$109$	3.66599	0.351138	0.175569	−	0.984467i	$-0.443823\pi$
$109$	3.66599	0.351138	0.175569	+	0.984467i	$0.443823\pi$
$110$	0	0
$111$	1.51350	0.143655
$112$	0	0
$113$	6.54316	0.615528	0.307764	−	0.951463i	$-0.400419\pi$
$113$	6.54316	0.615528	0.307764	+	0.951463i	$0.400419\pi$
$114$	0	0
$115$	−0.0400870	−0.00373814
$116$	0	0
$117$	−5.60248	−0.517949
$118$	0	0
$119$	−17.7817	−1.63004
$120$	0	0
$121$	5.50443	0.500403
$122$	0	0
$123$	3.07311	0.277093
$124$	0	0
$125$	2.23992	0.200345
$126$	0	0
$127$	15.0494	1.33542	0.667711	−	0.744420i	$-0.267274\pi$
$127$	15.0494	1.33542	0.667711	+	0.744420i	$0.267274\pi$
$128$	0	0
$129$	−6.18452	−0.544516
$130$	0	0
$131$	−9.57374	−0.836462	−0.418231	−	0.908341i	$-0.637350\pi$
$131$	−9.57374	−0.836462	−0.418231	+	0.908341i	$0.637350\pi$
$132$	0	0
$133$	−18.4723	−1.60175
$134$	0	0
$135$	0.894406	0.0769782
$136$	0	0
$137$	2.15272	0.183919	0.0919595	−	0.995763i	$-0.470687\pi$
$137$	2.15272	0.183919	0.0919595	+	0.995763i	$0.470687\pi$
$138$	0	0
$139$	−5.62106	−0.476772	−0.238386	−	0.971171i	$-0.576618\pi$
$139$	−5.62106	−0.476772	−0.238386	+	0.971171i	$0.576618\pi$
$140$	0	0
$141$	−7.23461	−0.609264
$142$	0	0
$143$	9.20317	0.769608
$144$	0	0
$145$	−2.37762	−0.197450
$146$	0	0
$147$	−0.642524	−0.0529945
$148$	0	0
$149$	15.1382	1.24017	0.620086	−	0.784534i	$-0.287098\pi$
$149$	15.1382	1.24017	0.620086	+	0.784534i	$0.287098\pi$
$150$	0	0
$151$	19.7193	1.60474	0.802369	−	0.596828i	$-0.203573\pi$
$151$	19.7193	1.60474	0.802369	+	0.596828i	$0.203573\pi$
$152$	0	0
$153$	−17.7838	−1.43773
$154$	0	0
$155$	0.287383	0.0230831
$156$	0	0
$157$	15.7111	1.25389	0.626943	−	0.779065i	$-0.284306\pi$
$157$	15.7111	1.25389	0.626943	+	0.779065i	$0.284306\pi$
$158$	0	0
$159$	0.450030	0.0356897
$160$	0	0
$161$	−0.440308	−0.0347011
$162$	0	0
$163$	11.8717	0.929864	0.464932	−	0.885346i	$-0.346079\pi$
$163$	11.8717	0.929864	0.464932	+	0.885346i	$0.346079\pi$
$164$	0	0
$165$	−0.663897	−0.0516843
$166$	0	0
$167$	5.37936	0.416267	0.208134	−	0.978100i	$-0.433261\pi$
$167$	5.37936	0.416267	0.208134	+	0.978100i	$0.433261\pi$
$168$	0	0
$169$	−7.86814	−0.605242
$170$	0	0
$171$	−18.4745	−1.41278
$172$	0	0
$173$	−18.6393	−1.41712	−0.708560	−	0.705651i	$-0.750655\pi$
$173$	−18.6393	−1.41712	−0.708560	+	0.705651i	$0.750655\pi$
$174$	0	0
$175$	12.2388	0.925163
$176$	0	0
$177$	−8.60448	−0.646753
$178$	0	0
$179$	22.6264	1.69117	0.845587	−	0.533837i	$-0.179250\pi$
$179$	22.6264	1.69117	0.845587	+	0.533837i	$0.179250\pi$
$180$	0	0
$181$	12.1002	0.899398	0.449699	−	0.893180i	$-0.351531\pi$
$181$	12.1002	0.899398	0.449699	+	0.893180i	$0.351531\pi$
$182$	0	0
$183$	4.57518	0.338207
$184$	0	0
$185$	−0.469420	−0.0345124
$186$	0	0
$187$	29.2133	2.13629
$188$	0	0
$189$	9.82397	0.714588
$190$	0	0
$191$	0.166765	0.0120667	0.00603336	−	0.999982i	$-0.498080\pi$
$191$	0.166765	0.0120667	0.00603336	+	0.999982i	$0.498080\pi$
$192$	0	0
$193$	−16.5260	−1.18956	−0.594782	−	0.803887i	$-0.702762\pi$
$193$	−16.5260	−1.18956	−0.594782	+	0.803887i	$0.702762\pi$
$194$	0	0
$195$	−0.370201	−0.0265107
$196$	0	0
$197$	9.82274	0.699841	0.349921	−	0.936779i	$-0.386209\pi$
$197$	9.82274	0.699841	0.349921	+	0.936779i	$0.386209\pi$
$198$	0	0
$199$	−11.5691	−0.820113	−0.410056	−	0.912060i	$-0.634491\pi$
$199$	−11.5691	−0.820113	−0.410056	+	0.912060i	$0.634491\pi$
$200$	0	0
$201$	−9.10156	−0.641975
$202$	0	0
$203$	−26.1152	−1.83293
$204$	0	0
$205$	−0.953140	−0.0665702
$206$	0	0
$207$	−0.440359	−0.0306071
$208$	0	0
$209$	30.3480	2.09921
$210$	0	0
$211$	19.2316	1.32396	0.661978	−	0.749523i	$-0.269717\pi$
$211$	19.2316	1.32396	0.661978	+	0.749523i	$0.269717\pi$
$212$	0	0
$213$	6.89871	0.472692
$214$	0	0
$215$	1.91816	0.130817
$216$	0	0
$217$	3.15655	0.214281
$218$	0	0
$219$	3.73757	0.252561
$220$	0	0
$221$	16.2899	1.09578
$222$	0	0
$223$	2.26463	0.151651	0.0758255	−	0.997121i	$-0.475841\pi$
$223$	2.26463	0.151651	0.0758255	+	0.997121i	$0.475841\pi$
$224$	0	0
$225$	12.2402	0.816013
$226$	0	0
$227$	−12.6616	−0.840378	−0.420189	−	0.907437i	$-0.638036\pi$
$227$	−12.6616	−0.840378	−0.420189	+	0.907437i	$0.638036\pi$
$228$	0	0
$229$	27.7524	1.83393	0.916967	−	0.398964i	$-0.130630\pi$
$229$	27.7524	1.83393	0.916967	+	0.398964i	$0.130630\pi$
$230$	0	0
$231$	−7.29211	−0.479786
$232$	0	0
$233$	−10.9164	−0.715157	−0.357579	−	0.933883i	$-0.616398\pi$
$233$	−10.9164	−0.715157	−0.357579	+	0.933883i	$0.616398\pi$
$234$	0	0
$235$	2.24385	0.146373
$236$	0	0
$237$	11.8994	0.772949
$238$	0	0
$239$	−6.41476	−0.414936	−0.207468	−	0.978242i	$-0.566522\pi$
$239$	−6.41476	−0.414936	−0.207468	+	0.978242i	$0.566522\pi$
$240$	0	0
$241$	9.85126	0.634576	0.317288	−	0.948329i	$-0.397228\pi$
$241$	9.85126	0.634576	0.317288	+	0.948329i	$0.397228\pi$
$242$	0	0
$243$	15.2106	0.975761
$244$	0	0
$245$	0.199282	0.0127317
$246$	0	0
$247$	16.9226	1.07676
$248$	0	0
$249$	−3.47892	−0.220467
$250$	0	0
$251$	−5.56947	−0.351542	−0.175771	−	0.984431i	$-0.556242\pi$
$251$	−5.56947	−0.351542	−0.175771	+	0.984431i	$0.556242\pi$
$252$	0	0
$253$	0.723376	0.0454783
$254$	0	0
$255$	−1.17512	−0.0735887
$256$	0	0
$257$	24.0328	1.49913	0.749564	−	0.661932i	$-0.230263\pi$
$257$	24.0328	1.49913	0.749564	+	0.661932i	$0.230263\pi$
$258$	0	0
$259$	−5.15601	−0.320379
$260$	0	0
$261$	−26.1183	−1.61668
$262$	0	0
$263$	−14.9598	−0.922463	−0.461232	−	0.887280i	$-0.652592\pi$
$263$	−14.9598	−0.922463	−0.461232	+	0.887280i	$0.652592\pi$
$264$	0	0
$265$	−0.139579	−0.00857428
$266$	0	0
$267$	−1.19720	−0.0732677
$268$	0	0
$269$	13.4385	0.819357	0.409679	−	0.912230i	$-0.365641\pi$
$269$	13.4385	0.819357	0.409679	+	0.912230i	$0.365641\pi$
$270$	0	0
$271$	−5.72605	−0.347833	−0.173916	−	0.984760i	$-0.555642\pi$
$271$	−5.72605	−0.347833	−0.173916	+	0.984760i	$0.555642\pi$
$272$	0	0
$273$	−4.06621	−0.246098
$274$	0	0
$275$	−20.1069	−1.21249
$276$	0	0
$277$	12.0981	0.726905	0.363452	−	0.931613i	$-0.381598\pi$
$277$	12.0981	0.726905	0.363452	+	0.931613i	$0.381598\pi$
$278$	0	0
$279$	3.15692	0.189000
$280$	0	0
$281$	−6.21778	−0.370921	−0.185461	−	0.982652i	$-0.559378\pi$
$281$	−6.21778	−0.370921	−0.185461	+	0.982652i	$0.559378\pi$
$282$	0	0
$283$	−12.9505	−0.769826	−0.384913	−	0.922953i	$-0.625769\pi$
$283$	−12.9505	−0.769826	−0.384913	+	0.922953i	$0.625769\pi$
$284$	0	0
$285$	−1.22076	−0.0723116
$286$	0	0
$287$	−10.4691	−0.617971
$288$	0	0
$289$	34.7084	2.04167
$290$	0	0
$291$	2.42732	0.142292
$292$	0	0
$293$	14.3490	0.838279	0.419139	−	0.907922i	$-0.362332\pi$
$293$	14.3490	0.838279	0.419139	+	0.907922i	$0.362332\pi$
$294$	0	0
$295$	2.66873	0.155379
$296$	0	0
$297$	−16.1397	−0.936520
$298$	0	0
$299$	0.403368	0.0233274
$300$	0	0
$301$	21.0687	1.21438
$302$	0	0
$303$	−9.20450	−0.528784
$304$	0	0
$305$	−1.41902	−0.0812526
$306$	0	0
$307$	28.7422	1.64040	0.820202	−	0.572075i	$-0.193861\pi$
$307$	28.7422	1.64040	0.820202	+	0.572075i	$0.193861\pi$
$308$	0	0
$309$	3.25052	0.184916
$310$	0	0
$311$	29.9677	1.69931	0.849655	−	0.527339i	$-0.176810\pi$
$311$	29.9677	1.69931	0.849655	+	0.527339i	$0.176810\pi$
$312$	0	0
$313$	−13.6402	−0.770990	−0.385495	−	0.922710i	$-0.625969\pi$
$313$	−13.6402	−0.770990	−0.385495	+	0.922710i	$0.625969\pi$
$314$	0	0
$315$	−1.37681	−0.0775746
$316$	0	0
$317$	12.9433	0.726967	0.363484	−	0.931601i	$-0.381587\pi$
$317$	12.9433	0.726967	0.363484	+	0.931601i	$0.381587\pi$
$318$	0	0
$319$	42.9044	2.40219
$320$	0	0
$321$	−10.9429	−0.610771
$322$	0	0
$323$	53.7168	2.98888
$324$	0	0
$325$	−11.2120	−0.621929
$326$	0	0
$327$	2.66104	0.147156
$328$	0	0
$329$	24.6460	1.35878
$330$	0	0
$331$	−8.48346	−0.466293	−0.233147	−	0.972442i	$-0.574902\pi$
$331$	−8.48346	−0.466293	−0.233147	+	0.972442i	$0.574902\pi$
$332$	0	0
$333$	−5.15662	−0.282581
$334$	0	0
$335$	2.82290	0.154231
$336$	0	0
$337$	−28.4431	−1.54939	−0.774697	−	0.632332i	$-0.782098\pi$
$337$	−28.4431	−1.54939	−0.774697	+	0.632332i	$0.782098\pi$
$338$	0	0
$339$	4.74950	0.257958
$340$	0	0
$341$	−5.18586	−0.280830
$342$	0	0
$343$	19.4986	1.05283
$344$	0	0
$345$	−0.0290981	−0.00156659
$346$	0	0
$347$	−6.37057	−0.341990	−0.170995	−	0.985272i	$-0.554698\pi$
$347$	−6.37057	−0.341990	−0.170995	+	0.985272i	$0.554698\pi$
$348$	0	0
$349$	−12.7281	−0.681317	−0.340659	−	0.940187i	$-0.610650\pi$
$349$	−12.7281	−0.681317	−0.340659	+	0.940187i	$0.610650\pi$
$350$	0	0
$351$	−8.99978	−0.480373
$352$	0	0
$353$	1.48049	0.0787984	0.0393992	−	0.999224i	$-0.487456\pi$
$353$	1.48049	0.0787984	0.0393992	+	0.999224i	$0.487456\pi$
$354$	0	0
$355$	−2.13967	−0.113562
$356$	0	0
$357$	−12.9072	−0.683124
$358$	0	0
$359$	27.5753	1.45537	0.727685	−	0.685911i	$-0.240596\pi$
$359$	27.5753	1.45537	0.727685	+	0.685911i	$0.240596\pi$
$360$	0	0
$361$	36.8032	1.93701
$362$	0	0
$363$	3.99552	0.209710
$364$	0	0
$365$	−1.15923	−0.0606766
$366$	0	0
$367$	−6.08957	−0.317873	−0.158936	−	0.987289i	$-0.550807\pi$
$367$	−6.08957	−0.317873	−0.158936	+	0.987289i	$0.550807\pi$
$368$	0	0
$369$	−10.4703	−0.545063
$370$	0	0
$371$	−1.53311	−0.0795950
$372$	0	0
$373$	22.4949	1.16474	0.582371	−	0.812923i	$-0.302125\pi$
$373$	22.4949	1.16474	0.582371	+	0.812923i	$0.302125\pi$
$374$	0	0
$375$	1.62590	0.0839611
$376$	0	0
$377$	23.9243	1.23216
$378$	0	0
$379$	14.1157	0.725075	0.362538	−	0.931969i	$-0.381910\pi$
$379$	14.1157	0.725075	0.362538	+	0.931969i	$0.381910\pi$
$380$	0	0
$381$	10.9240	0.559653
$382$	0	0
$383$	−24.3617	−1.24482	−0.622412	−	0.782690i	$-0.713847\pi$
$383$	−24.3617	−1.24482	−0.622412	+	0.782690i	$0.713847\pi$
$384$	0	0
$385$	2.26169	0.115266
$386$	0	0
$387$	21.0711	1.07111
$388$	0	0
$389$	−16.7319	−0.848341	−0.424170	−	0.905582i	$-0.639434\pi$
$389$	−16.7319	−0.848341	−0.424170	+	0.905582i	$0.639434\pi$
$390$	0	0
$391$	1.28040	0.0647525
$392$	0	0
$393$	−6.94932	−0.350547
$394$	0	0
$395$	−3.69066	−0.185697
$396$	0	0
$397$	−21.7689	−1.09255	−0.546274	−	0.837607i	$-0.683954\pi$
$397$	−21.7689	−1.09255	−0.546274	+	0.837607i	$0.683954\pi$
$398$	0	0
$399$	−13.4086	−0.671268
$400$	0	0
$401$	20.2269	1.01008	0.505041	−	0.863095i	$-0.331477\pi$
$401$	20.2269	1.01008	0.505041	+	0.863095i	$0.331477\pi$
$402$	0	0
$403$	−2.89173	−0.144047
$404$	0	0
$405$	−1.02111	−0.0507395
$406$	0	0
$407$	8.47075	0.419880
$408$	0	0
$409$	28.5461	1.41151	0.705757	−	0.708454i	$-0.250607\pi$
$409$	28.5461	1.41151	0.705757	+	0.708454i	$0.250607\pi$
$410$	0	0
$411$	1.56260	0.0770773
$412$	0	0
$413$	29.3127	1.44238
$414$	0	0
$415$	1.07900	0.0529662
$416$	0	0
$417$	−4.08018	−0.199807
$418$	0	0
$419$	36.3207	1.77438	0.887192	−	0.461400i	$-0.152653\pi$
$419$	36.3207	1.77438	0.887192	+	0.461400i	$0.152653\pi$
$420$	0	0
$421$	−20.8986	−1.01853	−0.509267	−	0.860609i	$-0.670083\pi$
$421$	−20.8986	−1.01853	−0.509267	+	0.860609i	$0.670083\pi$
$422$	0	0
$423$	24.6489	1.19847
$424$	0	0
$425$	−35.5898	−1.72636
$426$	0	0
$427$	−15.5862	−0.754268
$428$	0	0
$429$	6.68034	0.322530
$430$	0	0
$431$	−5.65795	−0.272534	−0.136267	−	0.990672i	$-0.543510\pi$
$431$	−5.65795	−0.272534	−0.136267	+	0.990672i	$0.543510\pi$
$432$	0	0
$433$	−0.797593	−0.0383299	−0.0191649	−	0.999816i	$-0.506101\pi$
$433$	−0.797593	−0.0383299	−0.0191649	+	0.999816i	$0.506101\pi$
$434$	0	0
$435$	−1.72585	−0.0827481
$436$	0	0
$437$	1.33013	0.0636286
$438$	0	0
$439$	18.2826	0.872581	0.436290	−	0.899806i	$-0.356292\pi$
$439$	18.2826	0.872581	0.436290	+	0.899806i	$0.356292\pi$
$440$	0	0
$441$	2.18913	0.104244
$442$	0	0
$443$	18.4038	0.874391	0.437195	−	0.899367i	$-0.355972\pi$
$443$	18.4038	0.874391	0.437195	+	0.899367i	$0.355972\pi$
$444$	0	0
$445$	0.371319	0.0176022
$446$	0	0
$447$	10.9884	0.519735
$448$	0	0
$449$	2.95767	0.139581	0.0697905	−	0.997562i	$-0.477767\pi$
$449$	2.95767	0.139581	0.0697905	+	0.997562i	$0.477767\pi$
$450$	0	0
$451$	17.1996	0.809896
$452$	0	0
$453$	14.3138	0.672519
$454$	0	0
$455$	1.26116	0.0591240
$456$	0	0
$457$	18.1383	0.848472	0.424236	−	0.905552i	$-0.360543\pi$
$457$	18.1383	0.848472	0.424236	+	0.905552i	$0.360543\pi$
$458$	0	0
$459$	−28.5677	−1.33343
$460$	0	0
$461$	−16.5572	−0.771145	−0.385573	−	0.922677i	$-0.625996\pi$
$461$	−16.5572	−0.771145	−0.385573	+	0.922677i	$0.625996\pi$
$462$	0	0
$463$	−4.01151	−0.186430	−0.0932152	−	0.995646i	$-0.529714\pi$
$463$	−4.01151	−0.186430	−0.0932152	+	0.995646i	$0.529714\pi$
$464$	0	0
$465$	0.208603	0.00967376
$466$	0	0
$467$	4.61263	0.213447	0.106724	−	0.994289i	$-0.465964\pi$
$467$	4.61263	0.213447	0.106724	+	0.994289i	$0.465964\pi$
$468$	0	0
$469$	31.0061	1.43173
$470$	0	0
$471$	11.4043	0.525482
$472$	0	0
$473$	−34.6135	−1.59153
$474$	0	0
$475$	−36.9721	−1.69640
$476$	0	0
$477$	−1.53329	−0.0702044
$478$	0	0
$479$	37.2886	1.70376	0.851879	−	0.523739i	$-0.175463\pi$
$479$	37.2886	1.70376	0.851879	+	0.523739i	$0.175463\pi$
$480$	0	0
$481$	4.72345	0.215371
$482$	0	0
$483$	−0.319607	−0.0145426
$484$	0	0
$485$	−0.752845	−0.0341849
$486$	0	0
$487$	−25.0176	−1.13366	−0.566828	−	0.823836i	$-0.691830\pi$
$487$	−25.0176	−1.13366	−0.566828	+	0.823836i	$0.691830\pi$
$488$	0	0
$489$	8.61736	0.389690
$490$	0	0
$491$	23.7021	1.06966	0.534830	−	0.844960i	$-0.320376\pi$
$491$	23.7021	1.06966	0.534830	+	0.844960i	$0.320376\pi$
$492$	0	0
$493$	75.9421	3.42026
$494$	0	0
$495$	2.26195	0.101667
$496$	0	0
$497$	−23.5017	−1.05419
$498$	0	0
$499$	−27.1748	−1.21651	−0.608255	−	0.793742i	$-0.708130\pi$
$499$	−27.1748	−1.21651	−0.608255	+	0.793742i	$0.708130\pi$
$500$	0	0
$501$	3.90473	0.174451
$502$	0	0
$503$	−12.0746	−0.538381	−0.269191	−	0.963087i	$-0.586756\pi$
$503$	−12.0746	−0.538381	−0.269191	+	0.963087i	$0.586756\pi$
$504$	0	0
$505$	2.85482	0.127038
$506$	0	0
$507$	−5.71128	−0.253647
$508$	0	0
$509$	−43.7176	−1.93775	−0.968873	−	0.247557i	$-0.920372\pi$
$509$	−43.7176	−1.93775	−0.968873	+	0.247557i	$0.920372\pi$
$510$	0	0
$511$	−12.7327	−0.563261
$512$	0	0
$513$	−29.6773	−1.31028
$514$	0	0
$515$	−1.00817	−0.0444251
$516$	0	0
$517$	−40.4906	−1.78078
$518$	0	0
$519$	−13.5298	−0.593891
$520$	0	0
$521$	−35.3638	−1.54932	−0.774659	−	0.632379i	$-0.782078\pi$
$521$	−35.3638	−1.54932	−0.774659	+	0.632379i	$0.782078\pi$
$522$	0	0
$523$	−19.8938	−0.869897	−0.434949	−	0.900455i	$-0.643233\pi$
$523$	−19.8938	−0.869897	−0.434949	+	0.900455i	$0.643233\pi$
$524$	0	0
$525$	8.88378	0.387720
$526$	0	0
$527$	−9.17913	−0.399849
$528$	0	0
$529$	−22.9683	−0.998622
$530$	0	0
$531$	29.3162	1.27221
$532$	0	0
$533$	9.59079	0.415423
$534$	0	0
$535$	3.39399	0.146735
$536$	0	0
$537$	16.4239	0.708743
$538$	0	0
$539$	−3.59608	−0.154894
$540$	0	0
$541$	−15.3119	−0.658311	−0.329156	−	0.944276i	$-0.606764\pi$
$541$	−15.3119	−0.658311	−0.329156	+	0.944276i	$0.606764\pi$
$542$	0	0
$543$	8.78319	0.376923
$544$	0	0
$545$	−0.825337	−0.0353535
$546$	0	0
$547$	−42.3037	−1.80878	−0.904388	−	0.426711i	$-0.859672\pi$
$547$	−42.3037	−1.80878	−0.904388	+	0.426711i	$0.859672\pi$
$548$	0	0
$549$	−15.5880	−0.665280
$550$	0	0
$551$	78.8917	3.36090
$552$	0	0
$553$	−40.5374	−1.72383
$554$	0	0
$555$	−0.340739	−0.0144636
$556$	0	0
$557$	7.44351	0.315392	0.157696	−	0.987488i	$-0.449593\pi$
$557$	7.44351	0.315392	0.157696	+	0.987488i	$0.449593\pi$
$558$	0	0
$559$	−19.3011	−0.816350
$560$	0	0
$561$	21.2052	0.895283
$562$	0	0
$563$	30.6551	1.29196	0.645978	−	0.763356i	$-0.276450\pi$
$563$	30.6551	1.29196	0.645978	+	0.763356i	$0.276450\pi$
$564$	0	0
$565$	−1.47308	−0.0619731
$566$	0	0
$567$	−11.2157	−0.471014
$568$	0	0
$569$	45.0445	1.88836	0.944182	−	0.329425i	$-0.106855\pi$
$569$	45.0445	1.88836	0.944182	+	0.329425i	$0.106855\pi$
$570$	0	0
$571$	20.5533	0.860127	0.430063	−	0.902799i	$-0.358491\pi$
$571$	20.5533	0.860127	0.430063	+	0.902799i	$0.358491\pi$
$572$	0	0
$573$	0.121051	0.00505696
$574$	0	0
$575$	−0.881270	−0.0367515
$576$	0	0
$577$	2.43980	0.101570	0.0507851	−	0.998710i	$-0.483828\pi$
$577$	2.43980	0.101570	0.0507851	+	0.998710i	$0.483828\pi$
$578$	0	0
$579$	−11.9957	−0.498526
$580$	0	0
$581$	11.8516	0.491685
$582$	0	0
$583$	2.51873	0.104315
$584$	0	0
$585$	1.26131	0.0521486
$586$	0	0
$587$	−28.0624	−1.15826	−0.579129	−	0.815236i	$-0.696607\pi$
$587$	−28.0624	−1.15826	−0.579129	+	0.815236i	$0.696607\pi$
$588$	0	0
$589$	−9.53565	−0.392910
$590$	0	0
$591$	7.13007	0.293292
$592$	0	0
$593$	−20.1600	−0.827873	−0.413937	−	0.910306i	$-0.635847\pi$
$593$	−20.1600	−0.827873	−0.413937	+	0.910306i	$0.635847\pi$
$594$	0	0
$595$	4.00325	0.164117
$596$	0	0
$597$	−8.39771	−0.343696
$598$	0	0
$599$	−23.5060	−0.960427	−0.480214	−	0.877152i	$-0.659441\pi$
$599$	−23.5060	−0.960427	−0.480214	+	0.877152i	$0.659441\pi$
$600$	0	0
$601$	−20.9839	−0.855952	−0.427976	−	0.903790i	$-0.640773\pi$
$601$	−20.9839	−0.855952	−0.427976	+	0.903790i	$0.640773\pi$
$602$	0	0
$603$	31.0097	1.26281
$604$	0	0
$605$	−1.23923	−0.0503819
$606$	0	0
$607$	−32.2511	−1.30903	−0.654516	−	0.756048i	$-0.727128\pi$
$607$	−32.2511	−1.30903	−0.654516	+	0.756048i	$0.727128\pi$
$608$	0	0
$609$	−18.9564	−0.768150
$610$	0	0
$611$	−22.5783	−0.913421
$612$	0	0
$613$	−37.2120	−1.50298	−0.751489	−	0.659746i	$-0.770664\pi$
$613$	−37.2120	−1.50298	−0.751489	+	0.659746i	$0.770664\pi$
$614$	0	0
$615$	−0.691859	−0.0278985
$616$	0	0
$617$	38.3479	1.54383	0.771915	−	0.635726i	$-0.219299\pi$
$617$	38.3479	1.54383	0.771915	+	0.635726i	$0.219299\pi$
$618$	0	0
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	0	0
$621$	−0.707390	−0.0283866
$622$	0	0
$623$	4.07849	0.163401
$624$	0	0
$625$	24.2423	0.969692
$626$	0	0
$627$	22.0288	0.879745
$628$	0	0
$629$	14.9935	0.597829
$630$	0	0
$631$	−38.9829	−1.55188	−0.775942	−	0.630804i	$-0.782725\pi$
$631$	−38.9829	−1.55188	−0.775942	+	0.630804i	$0.782725\pi$
$632$	0	0
$633$	13.9597	0.554848
$634$	0	0
$635$	−3.38813	−0.134454
$636$	0	0
$637$	−2.00524	−0.0794504
$638$	0	0
$639$	−23.5044	−0.929822
$640$	0	0
$641$	−0.319675	−0.0126264	−0.00631321	−	0.999980i	$-0.502010\pi$
$641$	−0.319675	−0.0126264	−0.00631321	+	0.999980i	$0.502010\pi$
$642$	0	0
$643$	−0.159017	−0.00627101	−0.00313550	−	0.999995i	$-0.500998\pi$
$643$	−0.159017	−0.00627101	−0.00313550	+	0.999995i	$0.500998\pi$
$644$	0	0
$645$	1.39234	0.0548234
$646$	0	0
$647$	−24.6688	−0.969829	−0.484915	−	0.874562i	$-0.661149\pi$
$647$	−24.6688	−0.969829	−0.484915	+	0.874562i	$0.661149\pi$
$648$	0	0
$649$	−48.1575	−1.89035
$650$	0	0
$651$	2.29126	0.0898014
$652$	0	0
$653$	−28.3461	−1.10927	−0.554635	−	0.832094i	$-0.687142\pi$
$653$	−28.3461	−1.10927	−0.554635	+	0.832094i	$0.687142\pi$
$654$	0	0
$655$	2.15537	0.0842172
$656$	0	0
$657$	−12.7342	−0.496808
$658$	0	0
$659$	−39.5061	−1.53894	−0.769470	−	0.638683i	$-0.779479\pi$
$659$	−39.5061	−1.53894	−0.769470	+	0.638683i	$0.779479\pi$
$660$	0	0
$661$	−15.7041	−0.610818	−0.305409	−	0.952221i	$-0.598793\pi$
$661$	−15.7041	−0.610818	−0.305409	+	0.952221i	$0.598793\pi$
$662$	0	0
$663$	11.8244	0.459221
$664$	0	0
$665$	4.15874	0.161269
$666$	0	0
$667$	1.88047	0.0728120
$668$	0	0
$669$	1.64384	0.0635544
$670$	0	0
$671$	25.6064	0.988523
$672$	0	0
$673$	−23.0243	−0.887522	−0.443761	−	0.896145i	$-0.646356\pi$
$673$	−23.0243	−0.887522	−0.443761	+	0.896145i	$0.646356\pi$
$674$	0	0
$675$	19.6626	0.756812
$676$	0	0
$677$	7.63096	0.293282	0.146641	−	0.989190i	$-0.453154\pi$
$677$	7.63096	0.293282	0.146641	+	0.989190i	$0.453154\pi$
$678$	0	0
$679$	−8.26909	−0.317339
$680$	0	0
$681$	−9.19070	−0.352188
$682$	0	0
$683$	−0.644724	−0.0246697	−0.0123348	−	0.999924i	$-0.503926\pi$
$683$	−0.644724	−0.0246697	−0.0123348	+	0.999924i	$0.503926\pi$
$684$	0	0
$685$	−0.484648	−0.0185175
$686$	0	0
$687$	20.1448	0.768571
$688$	0	0
$689$	1.40449	0.0535067
$690$	0	0
$691$	29.5024	1.12232	0.561162	−	0.827706i	$-0.310354\pi$
$691$	29.5024	1.12232	0.561162	+	0.827706i	$0.310354\pi$
$692$	0	0
$693$	24.8448	0.943776
$694$	0	0
$695$	1.26549	0.0480027
$696$	0	0
$697$	30.4437	1.15314
$698$	0	0
$699$	−7.92393	−0.299710
$700$	0	0
$701$	−30.4332	−1.14944	−0.574722	−	0.818348i	$-0.694890\pi$
$701$	−30.4332	−1.14944	−0.574722	+	0.818348i	$0.694890\pi$
$702$	0	0
$703$	15.5758	0.587453
$704$	0	0
$705$	1.62875	0.0613424
$706$	0	0
$707$	31.3568	1.17929
$708$	0	0
$709$	10.4754	0.393413	0.196707	−	0.980462i	$-0.436975\pi$
$709$	10.4754	0.393413	0.196707	+	0.980462i	$0.436975\pi$
$710$	0	0
$711$	−40.5422	−1.52045
$712$	0	0
$713$	−0.227292	−0.00851217
$714$	0	0
$715$	−2.07194	−0.0774862
$716$	0	0
$717$	−4.65630	−0.173893
$718$	0	0
$719$	−0.815327	−0.0304066	−0.0152033	−	0.999884i	$-0.504840\pi$
$719$	−0.815327	−0.0304066	−0.0152033	+	0.999884i	$0.504840\pi$
$720$	0	0
$721$	−11.0735	−0.412398
$722$	0	0
$723$	7.15077	0.265940
$724$	0	0
$725$	−52.2694	−1.94123
$726$	0	0
$727$	−5.58478	−0.207128	−0.103564	−	0.994623i	$-0.533025\pi$
$727$	−5.58478	−0.207128	−0.103564	+	0.994623i	$0.533025\pi$
$728$	0	0
$729$	−2.56578	−0.0950290
$730$	0	0
$731$	−61.2669	−2.26604
$732$	0	0
$733$	27.7414	1.02465	0.512327	−	0.858791i	$-0.328784\pi$
$733$	27.7414	1.02465	0.512327	+	0.858791i	$0.328784\pi$
$734$	0	0
$735$	0.144654	0.00533563
$736$	0	0
$737$	−50.9396	−1.87638
$738$	0	0
$739$	10.9980	0.404569	0.202285	−	0.979327i	$-0.435163\pi$
$739$	10.9980	0.404569	0.202285	+	0.979327i	$0.435163\pi$
$740$	0	0
$741$	12.2837	0.451251
$742$	0	0
$743$	−27.5706	−1.01147	−0.505733	−	0.862690i	$-0.668778\pi$
$743$	−27.5706	−1.01147	−0.505733	+	0.862690i	$0.668778\pi$
$744$	0	0
$745$	−3.40812	−0.124864
$746$	0	0
$747$	11.8529	0.433677
$748$	0	0
$749$	37.2788	1.36214
$750$	0	0
$751$	−2.39041	−0.0872274	−0.0436137	−	0.999048i	$-0.513887\pi$
$751$	−2.39041	−0.0872274	−0.0436137	+	0.999048i	$0.513887\pi$
$752$	0	0
$753$	−4.04273	−0.147325
$754$	0	0
$755$	−4.43948	−0.161569
$756$	0	0
$757$	22.8502	0.830505	0.415253	−	0.909706i	$-0.363693\pi$
$757$	22.8502	0.830505	0.415253	+	0.909706i	$0.363693\pi$
$758$	0	0
$759$	0.525080	0.0190592
$760$	0	0
$761$	35.2537	1.27795	0.638974	−	0.769229i	$-0.279359\pi$
$761$	35.2537	1.27795	0.638974	+	0.769229i	$0.279359\pi$
$762$	0	0
$763$	−9.06532	−0.328187
$764$	0	0
$765$	4.00372	0.144755
$766$	0	0
$767$	−26.8535	−0.969625
$768$	0	0
$769$	14.0283	0.505875	0.252937	−	0.967483i	$-0.418603\pi$
$769$	14.0283	0.505875	0.252937	+	0.967483i	$0.418603\pi$
$770$	0	0
$771$	17.4448	0.628259
$772$	0	0
$773$	2.05297	0.0738401	0.0369200	−	0.999318i	$-0.488245\pi$
$773$	2.05297	0.0738401	0.0369200	+	0.999318i	$0.488245\pi$
$774$	0	0
$775$	6.31780	0.226942
$776$	0	0
$777$	−3.74261	−0.134265
$778$	0	0
$779$	31.6262	1.13312
$780$	0	0
$781$	38.6107	1.38160
$782$	0	0
$783$	−41.9563	−1.49939
$784$	0	0
$785$	−3.53710	−0.126245
$786$	0	0
$787$	7.63128	0.272026	0.136013	−	0.990707i	$-0.456571\pi$
$787$	7.63128	0.272026	0.136013	+	0.990707i	$0.456571\pi$
$788$	0	0
$789$	−10.8589	−0.386589
$790$	0	0
$791$	−16.1800	−0.575296
$792$	0	0
$793$	14.2786	0.507047
$794$	0	0
$795$	−0.101317	−0.00359333
$796$	0	0
$797$	23.8432	0.844570	0.422285	−	0.906463i	$-0.361228\pi$
$797$	23.8432	0.844570	0.422285	+	0.906463i	$0.361228\pi$
$798$	0	0
$799$	−71.6696	−2.53549
$800$	0	0
$801$	4.07897	0.144123
$802$	0	0
$803$	20.9184	0.738194
$804$	0	0
$805$	0.0991279	0.00349380
$806$	0	0
$807$	9.75462	0.343379
$808$	0	0
$809$	19.9428	0.701152	0.350576	−	0.936534i	$-0.385986\pi$
$809$	19.9428	0.701152	0.350576	+	0.936534i	$0.385986\pi$
$810$	0	0
$811$	24.3626	0.855488	0.427744	−	0.903900i	$-0.359308\pi$
$811$	24.3626	0.855488	0.427744	+	0.903900i	$0.359308\pi$
$812$	0	0
$813$	−4.15639	−0.145771
$814$	0	0
$815$	−2.67272	−0.0936212
$816$	0	0
$817$	−63.6465	−2.22671
$818$	0	0
$819$	13.8539	0.484095
$820$	0	0
$821$	−15.6976	−0.547850	−0.273925	−	0.961751i	$-0.588322\pi$
$821$	−15.6976	−0.547850	−0.273925	+	0.961751i	$0.588322\pi$
$822$	0	0
$823$	7.38747	0.257511	0.128755	−	0.991676i	$-0.458902\pi$
$823$	7.38747	0.257511	0.128755	+	0.991676i	$0.458902\pi$
$824$	0	0
$825$	−14.5951	−0.508135
$826$	0	0
$827$	−11.5262	−0.400806	−0.200403	−	0.979714i	$-0.564225\pi$
$827$	−11.5262	−0.400806	−0.200403	+	0.979714i	$0.564225\pi$
$828$	0	0
$829$	−17.7336	−0.615914	−0.307957	−	0.951400i	$-0.599645\pi$
$829$	−17.7336	−0.615914	−0.307957	+	0.951400i	$0.599645\pi$
$830$	0	0
$831$	8.78169	0.304634
$832$	0	0
$833$	−6.36516	−0.220540
$834$	0	0
$835$	−1.21107	−0.0419109
$836$	0	0
$837$	5.07126	0.175288
$838$	0	0
$839$	28.6447	0.988924	0.494462	−	0.869199i	$-0.335365\pi$
$839$	28.6447	0.988924	0.494462	+	0.869199i	$0.335365\pi$
$840$	0	0
$841$	82.5332	2.84597
$842$	0	0
$843$	−4.51332	−0.155447
$844$	0	0
$845$	1.77138	0.0609374
$846$	0	0
$847$	−13.6115	−0.467695
$848$	0	0
$849$	−9.40041	−0.322621
$850$	0	0
$851$	0.371266	0.0127269
$852$	0	0
$853$	41.6536	1.42619	0.713097	−	0.701066i	$-0.247292\pi$
$853$	41.6536	1.42619	0.713097	+	0.701066i	$0.247292\pi$
$854$	0	0
$855$	4.15922	0.142242
$856$	0	0
$857$	−9.05282	−0.309238	−0.154619	−	0.987974i	$-0.549415\pi$
$857$	−9.05282	−0.309238	−0.154619	+	0.987974i	$0.549415\pi$
$858$	0	0
$859$	−9.47579	−0.323310	−0.161655	−	0.986847i	$-0.551683\pi$
$859$	−9.47579	−0.323310	−0.161655	+	0.986847i	$0.551683\pi$
$860$	0	0
$861$	−7.59924	−0.258981
$862$	0	0
$863$	34.6314	1.17887	0.589434	−	0.807817i	$-0.299351\pi$
$863$	34.6314	1.17887	0.589434	+	0.807817i	$0.299351\pi$
$864$	0	0
$865$	4.19633	0.142679
$866$	0	0
$867$	25.1939	0.855630
$868$	0	0
$869$	66.5985	2.25920
$870$	0	0
$871$	−28.4048	−0.962461
$872$	0	0
$873$	−8.27007	−0.279899
$874$	0	0
$875$	−5.53892	−0.187250
$876$	0	0
$877$	15.5441	0.524886	0.262443	−	0.964947i	$-0.415472\pi$
$877$	15.5441	0.524886	0.262443	+	0.964947i	$0.415472\pi$
$878$	0	0
$879$	10.4156	0.351308
$880$	0	0
$881$	52.4905	1.76845	0.884224	−	0.467063i	$-0.154688\pi$
$881$	52.4905	1.76845	0.884224	+	0.467063i	$0.154688\pi$
$882$	0	0
$883$	32.8989	1.10713	0.553567	−	0.832804i	$-0.313266\pi$
$883$	32.8989	1.10713	0.553567	+	0.832804i	$0.313266\pi$
$884$	0	0
$885$	1.93716	0.0651168
$886$	0	0
$887$	40.9458	1.37483	0.687413	−	0.726267i	$-0.258746\pi$
$887$	40.9458	1.37483	0.687413	+	0.726267i	$0.258746\pi$
$888$	0	0
$889$	−37.2145	−1.24814
$890$	0	0
$891$	18.4261	0.617298
$892$	0	0
$893$	−74.4533	−2.49148
$894$	0	0
$895$	−5.09395	−0.170272
$896$	0	0
$897$	0.292794	0.00977611
$898$	0	0
$899$	−13.4810	−0.449617
$900$	0	0
$901$	4.45822	0.148525
$902$	0	0
$903$	15.2932	0.508925
$904$	0	0
$905$	−2.72415	−0.0905539
$906$	0	0
$907$	22.3371	0.741692	0.370846	−	0.928694i	$-0.379068\pi$
$907$	22.3371	0.741692	0.370846	+	0.928694i	$0.379068\pi$
$908$	0	0
$909$	31.3604	1.04016
$910$	0	0
$911$	−15.9284	−0.527732	−0.263866	−	0.964559i	$-0.584998\pi$
$911$	−15.9284	−0.527732	−0.263866	+	0.964559i	$0.584998\pi$
$912$	0	0
$913$	−19.4708	−0.644389
$914$	0	0
$915$	−1.03003	−0.0340516
$916$	0	0
$917$	23.6741	0.781788
$918$	0	0
$919$	40.6296	1.34025	0.670124	−	0.742249i	$-0.266241\pi$
$919$	40.6296	1.34025	0.670124	+	0.742249i	$0.266241\pi$
$920$	0	0
$921$	20.8632	0.687465
$922$	0	0
$923$	21.5300	0.708669
$924$	0	0
$925$	−10.3197	−0.339309
$926$	0	0
$927$	−11.0748	−0.363743
$928$	0	0
$929$	−38.1089	−1.25031	−0.625155	−	0.780500i	$-0.714964\pi$
$929$	−38.1089	−1.25031	−0.625155	+	0.780500i	$0.714964\pi$
$930$	0	0
$931$	−6.61238	−0.216712
$932$	0	0
$933$	21.7527	0.712152
$934$	0	0
$935$	−6.57689	−0.215087
$936$	0	0
$937$	20.9882	0.685653	0.342827	−	0.939399i	$-0.388616\pi$
$937$	20.9882	0.685653	0.342827	+	0.939399i	$0.388616\pi$
$938$	0	0
$939$	−9.90106	−0.323109
$940$	0	0
$941$	0.271913	0.00886412	0.00443206	−	0.999990i	$-0.498589\pi$
$941$	0.271913	0.00886412	0.00443206	+	0.999990i	$0.498589\pi$
$942$	0	0
$943$	0.753843	0.0245485
$944$	0	0
$945$	−2.21170	−0.0719467
$946$	0	0
$947$	−18.9489	−0.615756	−0.307878	−	0.951426i	$-0.599619\pi$
$947$	−18.9489	−0.615756	−0.307878	+	0.951426i	$0.599619\pi$
$948$	0	0
$949$	11.6645	0.378645
$950$	0	0
$951$	9.39519	0.304660
$952$	0	0
$953$	−2.67357	−0.0866055	−0.0433027	−	0.999062i	$-0.513788\pi$
$953$	−2.67357	−0.0866055	−0.0433027	+	0.999062i	$0.513788\pi$
$954$	0	0
$955$	−0.0375445	−0.00121491
$956$	0	0
$957$	31.1432	1.00672
$958$	0	0
$959$	−5.32327	−0.171897
$960$	0	0
$961$	−29.3705	−0.947437
$962$	0	0
$963$	37.2832	1.20143
$964$	0	0
$965$	3.72054	0.119769
$966$	0	0
$967$	−20.1196	−0.647004	−0.323502	−	0.946227i	$-0.604860\pi$
$967$	−20.1196	−0.647004	−0.323502	+	0.946227i	$0.604860\pi$
$968$	0	0
$969$	38.9916	1.25259
$970$	0	0
$971$	−15.9577	−0.512106	−0.256053	−	0.966663i	$-0.582422\pi$
$971$	−15.9577	−0.512106	−0.256053	+	0.966663i	$0.582422\pi$
$972$	0	0
$973$	13.8998	0.445609
$974$	0	0
$975$	−8.13848	−0.260640
$976$	0	0
$977$	42.1759	1.34933	0.674663	−	0.738126i	$-0.264289\pi$
$977$	42.1759	1.34933	0.674663	+	0.738126i	$0.264289\pi$
$978$	0	0
$979$	−6.70051	−0.214149
$980$	0	0
$981$	−9.06639	−0.289467
$982$	0	0
$983$	31.8772	1.01673	0.508363	−	0.861143i	$-0.330251\pi$
$983$	31.8772	1.01673	0.508363	+	0.861143i	$0.330251\pi$
$984$	0	0
$985$	−2.21143	−0.0704619
$986$	0	0
$987$	17.8899	0.569441
$988$	0	0
$989$	−1.51708	−0.0482404
$990$	0	0
$991$	−28.4135	−0.902584	−0.451292	−	0.892376i	$-0.649037\pi$
$991$	−28.4135	−0.902584	−0.451292	+	0.892376i	$0.649037\pi$
$992$	0	0
$993$	−6.15792	−0.195416
$994$	0	0
$995$	2.60459	0.0825712
$996$	0	0
$997$	−38.9058	−1.23216	−0.616079	−	0.787684i	$-0.711280\pi$
$997$	−38.9058	−1.23216	−0.616079	+	0.787684i	$0.711280\pi$
$998$	0	0
$999$	−8.28355	−0.262080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9904.2.a.n.1.18		30
4.3	odd	2		619.2.a.b.1.8	✓	30
12.11	even	2		5571.2.a.g.1.23		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.8	✓	30	4.3	odd	2
5571.2.a.g.1.23		30	12.11	even	2
9904.2.a.n.1.18		30	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9904 = 2^{4} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9904.a (trivial)

\(f(q)\)	\(=\)	\(q+0.725873 q^{3} -0.225133 q^{5} -2.47282 q^{7} -2.47311 q^{9} +O(q^{10})\)	\(q+0.725873 q^{3} -0.225133 q^{5} -2.47282 q^{7} -2.47311 q^{9} +4.06256 q^{11} +2.26536 q^{13} -0.163418 q^{15} +7.19086 q^{17} +7.47015 q^{19} -1.79495 q^{21} +0.178059 q^{23} -4.94931 q^{25} -3.97278 q^{27} +10.5609 q^{29} -1.27650 q^{31} +2.94891 q^{33} +0.556714 q^{35} +2.08508 q^{37} +1.64436 q^{39} +4.23367 q^{41} -8.52011 q^{43} +0.556779 q^{45} -9.96677 q^{47} -0.885174 q^{49} +5.21965 q^{51} +0.619984 q^{53} -0.914619 q^{55} +5.42238 q^{57} -11.8540 q^{59} +6.30300 q^{61} +6.11554 q^{63} -0.510008 q^{65} -12.5388 q^{67} +0.129248 q^{69} +9.50401 q^{71} +5.14906 q^{73} -3.59258 q^{75} -10.0460 q^{77} +16.3932 q^{79} +4.53559 q^{81} -4.79273 q^{83} -1.61890 q^{85} +7.66589 q^{87} -1.64933 q^{89} -5.60182 q^{91} -0.926577 q^{93} -1.68178 q^{95} +3.34400 q^{97} -10.0472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) \) 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9	\( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) \) 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9 - 23 * q^11 + 9 * q^13 + 2 * q^15 + 4 * q^17 + q^19 + 30 * q^21 - 4 * q^23 + 35 * q^25 + 5 * q^27 + 90 * q^29 - 2 * q^31 - 6 * q^33 - 9 * q^35 + 19 * q^37 - 32 * q^39 + 59 * q^41 + 4 * q^43 + 30 * q^45 - 4 * q^47 + 30 * q^49 + 34 * q^53 + 17 * q^55 - 8 * q^57 - 13 * q^59 + 16 * q^61 + 40 * q^63 + 31 * q^65 + 11 * q^67 + 6 * q^69 - 42 * q^71 - 4 * q^73 + 52 * q^75 + 29 * q^77 - 3 * q^79 + 30 * q^81 + 11 * q^83 + 19 * q^85 + 20 * q^87 + 58 * q^89 + 39 * q^91 - 15 * q^93 - 23 * q^95 - 9 * q^97 + 8 * q^99

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