LMFDB - Embedded newform 4624.2.a.bh.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4624,2,Mod(1,4624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4624 = 2^{4} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-6,0,3,0,0,0,6,0,-12,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.9228258946$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 578)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	4624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87939 q^{3} -1.65270 q^{5} -2.41147 q^{7} +5.29086 q^{9} +0.467911 q^{11} -6.22668 q^{13} -4.75877 q^{15} +5.90167 q^{19} -6.94356 q^{21} +1.92127 q^{23} -2.26857 q^{25} +6.59627 q^{27} -4.90167 q^{29} +0.837496 q^{31} +1.34730 q^{33} +3.98545 q^{35} -1.16250 q^{37} -17.9290 q^{39} -4.04189 q^{41} +2.65270 q^{43} -8.74422 q^{45} +3.98545 q^{47} -1.18479 q^{49} +1.49020 q^{53} -0.773318 q^{55} +16.9932 q^{57} -13.8084 q^{59} -4.41147 q^{61} -12.7588 q^{63} +10.2909 q^{65} -2.10607 q^{67} +5.53209 q^{69} -12.2686 q^{71} -13.0496 q^{73} -6.53209 q^{75} -1.12836 q^{77} -8.82295 q^{79} +3.12061 q^{81} -8.88713 q^{83} -14.1138 q^{87} +14.3628 q^{89} +15.0155 q^{91} +2.41147 q^{93} -9.75372 q^{95} -13.0000 q^{97} +2.47565 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{11} - 12 q^{13} - 3 q^{15} + 6 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} + 6 q^{27} - 3 q^{29} + 3 q^{33} - 6 q^{35} - 6 q^{37} - 21 q^{39} - 9 q^{41} + 9 q^{43} + 3 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^11 - 12 * q^13 - 3 * q^15 + 6 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 + 6 * q^27 - 3 * q^29 + 3 * q^33 - 6 * q^35 - 6 * q^37 - 21 * q^39 - 9 * q^41 + 9 * q^43 + 3 * q^45 - 6 * q^47 + 3 * q^53 - 9 * q^55 + 9 * q^57 - 3 * q^59 - 3 * q^61 - 27 * q^63 + 15 * q^65 + 6 * q^67 + 12 * q^69 - 27 * q^71 - 12 * q^73 - 15 * q^75 + 15 * q^77 - 6 * q^79 + 15 * q^81 + 3 * q^83 - 6 * q^87 - 6 * q^89 - 3 * q^91 - 3 * q^93 - 33 * q^95 - 39 * q^97 - 12 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.87939	1.66241	0.831207	−	0.555963i	$-0.187650\pi$
$3$	2.87939	1.66241	0.831207	+	0.555963i	$0.187650\pi$
$4$	0	0
$5$	−1.65270	−0.739112	−0.369556	−	0.929209i	$-0.620490\pi$
$5$	−1.65270	−0.739112	−0.369556	+	0.929209i	$0.620490\pi$
$6$	0	0
$7$	−2.41147	−0.911452	−0.455726	−	0.890120i	$-0.650620\pi$
$7$	−2.41147	−0.911452	−0.455726	+	0.890120i	$0.650620\pi$
$8$	0	0
$9$	5.29086	1.76362
$10$	0	0
$11$	0.467911	0.141081	0.0705403	−	0.997509i	$-0.477528\pi$
$11$	0.467911	0.141081	0.0705403	+	0.997509i	$0.477528\pi$
$12$	0	0
$13$	−6.22668	−1.72697	−0.863485	−	0.504374i	$-0.831723\pi$
$13$	−6.22668	−1.72697	−0.863485	+	0.504374i	$0.831723\pi$
$14$	0	0
$15$	−4.75877	−1.22871
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.90167	1.35394	0.676968	−	0.736012i	$-0.263293\pi$
$19$	5.90167	1.35394	0.676968	+	0.736012i	$0.263293\pi$
$20$	0	0
$21$	−6.94356	−1.51521
$22$	0	0
$23$	1.92127	0.400613	0.200307	−	0.979733i	$-0.435806\pi$
$23$	1.92127	0.400613	0.200307	+	0.979733i	$0.435806\pi$
$24$	0	0
$25$	−2.26857	−0.453714
$26$	0	0
$27$	6.59627	1.26945
$28$	0	0
$29$	−4.90167	−0.910218	−0.455109	−	0.890436i	$-0.650400\pi$
$29$	−4.90167	−0.910218	−0.455109	+	0.890436i	$0.650400\pi$
$30$	0	0
$31$	0.837496	0.150419	0.0752094	−	0.997168i	$-0.476037\pi$
$31$	0.837496	0.150419	0.0752094	+	0.997168i	$0.476037\pi$
$32$	0	0
$33$	1.34730	0.234534
$34$	0	0
$35$	3.98545	0.673664
$36$	0	0
$37$	−1.16250	−0.191114	−0.0955572	−	0.995424i	$-0.530463\pi$
$37$	−1.16250	−0.191114	−0.0955572	+	0.995424i	$0.530463\pi$
$38$	0	0
$39$	−17.9290	−2.87094
$40$	0	0
$41$	−4.04189	−0.631237	−0.315619	−	0.948886i	$-0.602212\pi$
$41$	−4.04189	−0.631237	−0.315619	+	0.948886i	$0.602212\pi$
$42$	0	0
$43$	2.65270	0.404534	0.202267	−	0.979330i	$-0.435169\pi$
$43$	2.65270	0.404534	0.202267	+	0.979330i	$0.435169\pi$
$44$	0	0
$45$	−8.74422	−1.30351
$46$	0	0
$47$	3.98545	0.581338	0.290669	−	0.956824i	$-0.406122\pi$
$47$	3.98545	0.581338	0.290669	+	0.956824i	$0.406122\pi$
$48$	0	0
$49$	−1.18479	−0.169256
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.49020	0.204695	0.102347	−	0.994749i	$-0.467365\pi$
$53$	1.49020	0.204695	0.102347	+	0.994749i	$0.467365\pi$
$54$	0	0
$55$	−0.773318	−0.104274
$56$	0	0
$57$	16.9932	2.25080
$58$	0	0
$59$	−13.8084	−1.79770	−0.898850	−	0.438256i	$-0.855596\pi$
$59$	−13.8084	−1.79770	−0.898850	+	0.438256i	$0.855596\pi$
$60$	0	0
$61$	−4.41147	−0.564831	−0.282416	−	0.959292i	$-0.591136\pi$
$61$	−4.41147	−0.564831	−0.282416	+	0.959292i	$0.591136\pi$
$62$	0	0
$63$	−12.7588	−1.60745
$64$	0	0
$65$	10.2909	1.27642
$66$	0	0
$67$	−2.10607	−0.257297	−0.128649	−	0.991690i	$-0.541064\pi$
$67$	−2.10607	−0.257297	−0.128649	+	0.991690i	$0.541064\pi$
$68$	0	0
$69$	5.53209	0.665985
$70$	0	0
$71$	−12.2686	−1.45601	−0.728006	−	0.685571i	$-0.759553\pi$
$71$	−12.2686	−1.45601	−0.728006	+	0.685571i	$0.759553\pi$
$72$	0	0
$73$	−13.0496	−1.52734	−0.763672	−	0.645605i	$-0.776605\pi$
$73$	−13.0496	−1.52734	−0.763672	+	0.645605i	$0.776605\pi$
$74$	0	0
$75$	−6.53209	−0.754261
$76$	0	0
$77$	−1.12836	−0.128588
$78$	0	0
$79$	−8.82295	−0.992659	−0.496330	−	0.868134i	$-0.665319\pi$
$79$	−8.82295	−0.992659	−0.496330	+	0.868134i	$0.665319\pi$
$80$	0	0
$81$	3.12061	0.346735
$82$	0	0
$83$	−8.88713	−0.975489	−0.487744	−	0.872986i	$-0.662180\pi$
$83$	−8.88713	−0.975489	−0.487744	+	0.872986i	$0.662180\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−14.1138	−1.51316
$88$	0	0
$89$	14.3628	1.52245	0.761226	−	0.648487i	$-0.224598\pi$
$89$	14.3628	1.52245	0.761226	+	0.648487i	$0.224598\pi$
$90$	0	0
$91$	15.0155	1.57405
$92$	0	0
$93$	2.41147	0.250058
$94$	0	0
$95$	−9.75372	−1.00071
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	2.47565	0.248812
$100$	0	0
$101$	−15.3550	−1.52788	−0.763942	−	0.645285i	$-0.776738\pi$
$101$	−15.3550	−1.52788	−0.763942	+	0.645285i	$0.776738\pi$
$102$	0	0
$103$	−4.49525	−0.442930	−0.221465	−	0.975168i	$-0.571084\pi$
$103$	−4.49525	−0.442930	−0.221465	+	0.975168i	$0.571084\pi$
$104$	0	0
$105$	11.4757	1.11991
$106$	0	0
$107$	18.9786	1.83473	0.917367	−	0.398041i	$-0.130310\pi$
$107$	18.9786	1.83473	0.917367	+	0.398041i	$0.130310\pi$
$108$	0	0
$109$	12.0915	1.15816	0.579079	−	0.815272i	$-0.303412\pi$
$109$	12.0915	1.15816	0.579079	+	0.815272i	$0.303412\pi$
$110$	0	0
$111$	−3.34730	−0.317711
$112$	0	0
$113$	−1.27126	−0.119590	−0.0597950	−	0.998211i	$-0.519045\pi$
$113$	−1.27126	−0.119590	−0.0597950	+	0.998211i	$0.519045\pi$
$114$	0	0
$115$	−3.17530	−0.296098
$116$	0	0
$117$	−32.9445	−3.04572
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.7811	−0.980096
$122$	0	0
$123$	−11.6382	−1.04938
$124$	0	0
$125$	12.0128	1.07446
$126$	0	0
$127$	4.24897	0.377035	0.188518	−	0.982070i	$-0.439632\pi$
$127$	4.24897	0.377035	0.188518	+	0.982070i	$0.439632\pi$
$128$	0	0
$129$	7.63816	0.672502
$130$	0	0
$131$	−18.4415	−1.61124	−0.805621	−	0.592432i	$-0.798168\pi$
$131$	−18.4415	−1.61124	−0.805621	+	0.592432i	$0.798168\pi$
$132$	0	0
$133$	−14.2317	−1.23405
$134$	0	0
$135$	−10.9017	−0.938267
$136$	0	0
$137$	−4.79561	−0.409716	−0.204858	−	0.978792i	$-0.565673\pi$
$137$	−4.79561	−0.409716	−0.204858	+	0.978792i	$0.565673\pi$
$138$	0	0
$139$	−4.51249	−0.382744	−0.191372	−	0.981518i	$-0.561294\pi$
$139$	−4.51249	−0.382744	−0.191372	+	0.981518i	$0.561294\pi$
$140$	0	0
$141$	11.4757	0.966424
$142$	0	0
$143$	−2.91353	−0.243642
$144$	0	0
$145$	8.10101	0.672753
$146$	0	0
$147$	−3.41147	−0.281374
$148$	0	0
$149$	11.3696	0.931433	0.465716	−	0.884934i	$-0.345797\pi$
$149$	11.3696	0.931433	0.465716	+	0.884934i	$0.345797\pi$
$150$	0	0
$151$	6.51485	0.530171	0.265086	−	0.964225i	$-0.414600\pi$
$151$	6.51485	0.530171	0.265086	+	0.964225i	$0.414600\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.38413	−0.111176
$156$	0	0
$157$	−9.55169	−0.762308	−0.381154	−	0.924512i	$-0.624473\pi$
$157$	−9.55169	−0.762308	−0.381154	+	0.924512i	$0.624473\pi$
$158$	0	0
$159$	4.29086	0.340287
$160$	0	0
$161$	−4.63310	−0.365140
$162$	0	0
$163$	11.5963	0.908290	0.454145	−	0.890928i	$-0.349945\pi$
$163$	11.5963	0.908290	0.454145	+	0.890928i	$0.349945\pi$
$164$	0	0
$165$	−2.22668	−0.173347
$166$	0	0
$167$	−0.879385	−0.0680489	−0.0340244	−	0.999421i	$-0.510832\pi$
$167$	−0.879385	−0.0680489	−0.0340244	+	0.999421i	$0.510832\pi$
$168$	0	0
$169$	25.7716	1.98243
$170$	0	0
$171$	31.2249	2.38783
$172$	0	0
$173$	−17.1138	−1.30114	−0.650569	−	0.759447i	$-0.725470\pi$
$173$	−17.1138	−1.30114	−0.650569	+	0.759447i	$0.725470\pi$
$174$	0	0
$175$	5.47060	0.413538
$176$	0	0
$177$	−39.7597	−2.98852
$178$	0	0
$179$	14.2567	1.06560	0.532798	−	0.846242i	$-0.321140\pi$
$179$	14.2567	1.06560	0.532798	+	0.846242i	$0.321140\pi$
$180$	0	0
$181$	10.7939	0.802301	0.401150	−	0.916012i	$-0.368610\pi$
$181$	10.7939	0.802301	0.401150	+	0.916012i	$0.368610\pi$
$182$	0	0
$183$	−12.7023	−0.938984
$184$	0	0
$185$	1.92127	0.141255
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−15.9067	−1.15704
$190$	0	0
$191$	9.09926	0.658399	0.329200	−	0.944260i	$-0.393221\pi$
$191$	9.09926	0.658399	0.329200	+	0.944260i	$0.393221\pi$
$192$	0	0
$193$	−7.98545	−0.574805	−0.287403	−	0.957810i	$-0.592792\pi$
$193$	−7.98545	−0.574805	−0.287403	+	0.957810i	$0.592792\pi$
$194$	0	0
$195$	29.6313	2.12194
$196$	0	0
$197$	18.9786	1.35217	0.676086	−	0.736823i	$-0.263675\pi$
$197$	18.9786	1.35217	0.676086	+	0.736823i	$0.263675\pi$
$198$	0	0
$199$	−1.36959	−0.0970873	−0.0485437	−	0.998821i	$-0.515458\pi$
$199$	−1.36959	−0.0970873	−0.0485437	+	0.998821i	$0.515458\pi$
$200$	0	0
$201$	−6.06418	−0.427734
$202$	0	0
$203$	11.8203	0.829620
$204$	0	0
$205$	6.68004	0.466555
$206$	0	0
$207$	10.1652	0.706530
$208$	0	0
$209$	2.76146	0.191014
$210$	0	0
$211$	−12.4338	−0.855976	−0.427988	−	0.903785i	$-0.640777\pi$
$211$	−12.4338	−0.855976	−0.427988	+	0.903785i	$0.640777\pi$
$212$	0	0
$213$	−35.3259	−2.42049
$214$	0	0
$215$	−4.38413	−0.298995
$216$	0	0
$217$	−2.01960	−0.137099
$218$	0	0
$219$	−37.5749	−2.53908
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.37733	0.360092	0.180046	−	0.983658i	$-0.442375\pi$
$223$	5.37733	0.360092	0.180046	+	0.983658i	$0.442375\pi$
$224$	0	0
$225$	−12.0027	−0.800179
$226$	0	0
$227$	−11.0574	−0.733903	−0.366952	−	0.930240i	$-0.619599\pi$
$227$	−11.0574	−0.733903	−0.366952	+	0.930240i	$0.619599\pi$
$228$	0	0
$229$	23.9659	1.58371	0.791854	−	0.610710i	$-0.209116\pi$
$229$	23.9659	1.58371	0.791854	+	0.610710i	$0.209116\pi$
$230$	0	0
$231$	−3.24897	−0.213767
$232$	0	0
$233$	10.4534	0.684823	0.342411	−	0.939550i	$-0.388756\pi$
$233$	10.4534	0.684823	0.342411	+	0.939550i	$0.388756\pi$
$234$	0	0
$235$	−6.58677	−0.429674
$236$	0	0
$237$	−25.4047	−1.65021
$238$	0	0
$239$	−1.86484	−0.120626	−0.0603131	−	0.998180i	$-0.519210\pi$
$239$	−1.86484	−0.120626	−0.0603131	+	0.998180i	$0.519210\pi$
$240$	0	0
$241$	−18.3696	−1.18329	−0.591644	−	0.806199i	$-0.701521\pi$
$241$	−18.3696	−1.18329	−0.591644	+	0.806199i	$0.701521\pi$
$242$	0	0
$243$	−10.8033	−0.693035
$244$	0	0
$245$	1.95811	0.125099
$246$	0	0
$247$	−36.7478	−2.33821
$248$	0	0
$249$	−25.5895	−1.62167
$250$	0	0
$251$	1.30810	0.0825663	0.0412831	−	0.999147i	$-0.486855\pi$
$251$	1.30810	0.0825663	0.0412831	+	0.999147i	$0.486855\pi$
$252$	0	0
$253$	0.898986	0.0565187
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.9736	0.746892	0.373446	−	0.927652i	$-0.378176\pi$
$257$	11.9736	0.746892	0.373446	+	0.927652i	$0.378176\pi$
$258$	0	0
$259$	2.80335	0.174192
$260$	0	0
$261$	−25.9341	−1.60528
$262$	0	0
$263$	14.5253	0.895667	0.447834	−	0.894117i	$-0.352196\pi$
$263$	14.5253	0.895667	0.447834	+	0.894117i	$0.352196\pi$
$264$	0	0
$265$	−2.46286	−0.151292
$266$	0	0
$267$	41.3560	2.53094
$268$	0	0
$269$	−27.9786	−1.70589	−0.852944	−	0.522002i	$-0.825185\pi$
$269$	−27.9786	−1.70589	−0.852944	+	0.522002i	$0.825185\pi$
$270$	0	0
$271$	13.3678	0.812038	0.406019	−	0.913865i	$-0.366917\pi$
$271$	13.3678	0.812038	0.406019	+	0.913865i	$0.366917\pi$
$272$	0	0
$273$	43.2354	2.61672
$274$	0	0
$275$	−1.06149	−0.0640102
$276$	0	0
$277$	−0.595333	−0.0357701	−0.0178850	−	0.999840i	$-0.505693\pi$
$277$	−0.595333	−0.0357701	−0.0178850	+	0.999840i	$0.505693\pi$
$278$	0	0
$279$	4.43107	0.265281
$280$	0	0
$281$	17.8999	1.06782	0.533910	−	0.845541i	$-0.320722\pi$
$281$	17.8999	1.06782	0.533910	+	0.845541i	$0.320722\pi$
$282$	0	0
$283$	−3.95130	−0.234881	−0.117440	−	0.993080i	$-0.537469\pi$
$283$	−3.95130	−0.234881	−0.117440	+	0.993080i	$0.537469\pi$
$284$	0	0
$285$	−28.0847	−1.66359
$286$	0	0
$287$	9.74691	0.575342
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−37.4320	−2.19430
$292$	0	0
$293$	28.0479	1.63857	0.819287	−	0.573383i	$-0.194369\pi$
$293$	28.0479	1.63857	0.819287	+	0.573383i	$0.194369\pi$
$294$	0	0
$295$	22.8212	1.32870
$296$	0	0
$297$	3.08647	0.179095
$298$	0	0
$299$	−11.9632	−0.691848
$300$	0	0
$301$	−6.39693	−0.368713
$302$	0	0
$303$	−44.2131	−2.53997
$304$	0	0
$305$	7.29086	0.417473
$306$	0	0
$307$	−21.5202	−1.22822	−0.614112	−	0.789219i	$-0.710486\pi$
$307$	−21.5202	−1.22822	−0.614112	+	0.789219i	$0.710486\pi$
$308$	0	0
$309$	−12.9436	−0.736334
$310$	0	0
$311$	−4.72874	−0.268142	−0.134071	−	0.990972i	$-0.542805\pi$
$311$	−4.72874	−0.268142	−0.134071	+	0.990972i	$0.542805\pi$
$312$	0	0
$313$	−17.6331	−0.996682	−0.498341	−	0.866981i	$-0.666057\pi$
$313$	−17.6331	−0.996682	−0.498341	+	0.866981i	$0.666057\pi$
$314$	0	0
$315$	21.0865	1.18809
$316$	0	0
$317$	−17.7939	−0.999402	−0.499701	−	0.866198i	$-0.666557\pi$
$317$	−17.7939	−0.999402	−0.499701	+	0.866198i	$0.666557\pi$
$318$	0	0
$319$	−2.29355	−0.128414
$320$	0	0
$321$	54.6468	3.05009
$322$	0	0
$323$	0	0
$324$	0	0
$325$	14.1257	0.783551
$326$	0	0
$327$	34.8161	1.92534
$328$	0	0
$329$	−9.61081	−0.529861
$330$	0	0
$331$	34.9590	1.92152	0.960761	−	0.277376i	$-0.0894648\pi$
$331$	34.9590	1.92152	0.960761	+	0.277376i	$0.0894648\pi$
$332$	0	0
$333$	−6.15064	−0.337053
$334$	0	0
$335$	3.48070	0.190171
$336$	0	0
$337$	15.6631	0.853225	0.426613	−	0.904434i	$-0.359707\pi$
$337$	15.6631	0.853225	0.426613	+	0.904434i	$0.359707\pi$
$338$	0	0
$339$	−3.66044	−0.198808
$340$	0	0
$341$	0.391874	0.0212212
$342$	0	0
$343$	19.7374	1.06572
$344$	0	0
$345$	−9.14290	−0.492237
$346$	0	0
$347$	−10.4466	−0.560801	−0.280400	−	0.959883i	$-0.590467\pi$
$347$	−10.4466	−0.560801	−0.280400	+	0.959883i	$0.590467\pi$
$348$	0	0
$349$	−8.32770	−0.445771	−0.222886	−	0.974845i	$-0.571548\pi$
$349$	−8.32770	−0.445771	−0.222886	+	0.974845i	$0.571548\pi$
$350$	0	0
$351$	−41.0729	−2.19231
$352$	0	0
$353$	7.20439	0.383451	0.191726	−	0.981449i	$-0.438592\pi$
$353$	7.20439	0.383451	0.191726	+	0.981449i	$0.438592\pi$
$354$	0	0
$355$	20.2763	1.07615
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.1557	−0.641553	−0.320777	−	0.947155i	$-0.603944\pi$
$359$	−12.1557	−0.641553	−0.320777	+	0.947155i	$0.603944\pi$
$360$	0	0
$361$	15.8298	0.833145
$362$	0	0
$363$	−31.0428	−1.62933
$364$	0	0
$365$	21.5672	1.12888
$366$	0	0
$367$	36.1702	1.88807	0.944036	−	0.329843i	$-0.106996\pi$
$367$	36.1702	1.88807	0.944036	+	0.329843i	$0.106996\pi$
$368$	0	0
$369$	−21.3851	−1.11326
$370$	0	0
$371$	−3.59358	−0.186569
$372$	0	0
$373$	28.0077	1.45019	0.725093	−	0.688651i	$-0.241797\pi$
$373$	28.0077	1.45019	0.725093	+	0.688651i	$0.241797\pi$
$374$	0	0
$375$	34.5895	1.78619
$376$	0	0
$377$	30.5212	1.57192
$378$	0	0
$379$	0.0709849	0.00364625	0.00182312	−	0.999998i	$-0.499420\pi$
$379$	0.0709849	0.00364625	0.00182312	+	0.999998i	$0.499420\pi$
$380$	0	0
$381$	12.2344	0.626788
$382$	0	0
$383$	16.9341	0.865290	0.432645	−	0.901564i	$-0.357580\pi$
$383$	16.9341	0.865290	0.432645	+	0.901564i	$0.357580\pi$
$384$	0	0
$385$	1.86484	0.0950409
$386$	0	0
$387$	14.0351	0.713443
$388$	0	0
$389$	−30.6364	−1.55333	−0.776664	−	0.629916i	$-0.783089\pi$
$389$	−30.6364	−1.55333	−0.776664	+	0.629916i	$0.783089\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−53.1002	−2.67855
$394$	0	0
$395$	14.5817	0.733686
$396$	0	0
$397$	−0.681799	−0.0342185	−0.0171093	−	0.999854i	$-0.505446\pi$
$397$	−0.681799	−0.0342185	−0.0171093	+	0.999854i	$0.505446\pi$
$398$	0	0
$399$	−40.9786	−2.05150
$400$	0	0
$401$	−16.3797	−0.817963	−0.408981	−	0.912543i	$-0.634116\pi$
$401$	−16.3797	−0.817963	−0.408981	+	0.912543i	$0.634116\pi$
$402$	0	0
$403$	−5.21482	−0.259769
$404$	0	0
$405$	−5.15745	−0.256276
$406$	0	0
$407$	−0.543948	−0.0269625
$408$	0	0
$409$	−1.64084	−0.0811345	−0.0405673	−	0.999177i	$-0.512917\pi$
$409$	−1.64084	−0.0811345	−0.0405673	+	0.999177i	$0.512917\pi$
$410$	0	0
$411$	−13.8084	−0.681118
$412$	0	0
$413$	33.2986	1.63852
$414$	0	0
$415$	14.6878	0.720995
$416$	0	0
$417$	−12.9932	−0.636279
$418$	0	0
$419$	29.4861	1.44049	0.720245	−	0.693720i	$-0.244030\pi$
$419$	29.4861	1.44049	0.720245	+	0.693720i	$0.244030\pi$
$420$	0	0
$421$	20.2550	0.987166	0.493583	−	0.869699i	$-0.335687\pi$
$421$	20.2550	0.987166	0.493583	+	0.869699i	$0.335687\pi$
$422$	0	0
$423$	21.0865	1.02526
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.6382	0.514816
$428$	0	0
$429$	−8.38919	−0.405034
$430$	0	0
$431$	3.18573	0.153451	0.0767255	−	0.997052i	$-0.475553\pi$
$431$	3.18573	0.153451	0.0767255	+	0.997052i	$0.475553\pi$
$432$	0	0
$433$	−5.41653	−0.260302	−0.130151	−	0.991494i	$-0.541546\pi$
$433$	−5.41653	−0.260302	−0.130151	+	0.991494i	$0.541546\pi$
$434$	0	0
$435$	23.3259	1.11839
$436$	0	0
$437$	11.3387	0.542405
$438$	0	0
$439$	2.00505	0.0956959	0.0478480	−	0.998855i	$-0.484764\pi$
$439$	2.00505	0.0956959	0.0478480	+	0.998855i	$0.484764\pi$
$440$	0	0
$441$	−6.26857	−0.298503
$442$	0	0
$443$	−0.0760373	−0.00361264	−0.00180632	−	0.999998i	$-0.500575\pi$
$443$	−0.0760373	−0.00361264	−0.00180632	+	0.999998i	$0.500575\pi$
$444$	0	0
$445$	−23.7374	−1.12526
$446$	0	0
$447$	32.7374	1.54843
$448$	0	0
$449$	−12.1489	−0.573342	−0.286671	−	0.958029i	$-0.592549\pi$
$449$	−12.1489	−0.573342	−0.286671	+	0.958029i	$0.592549\pi$
$450$	0	0
$451$	−1.89124	−0.0890552
$452$	0	0
$453$	18.7588	0.881364
$454$	0	0
$455$	−24.8161	−1.16340
$456$	0	0
$457$	−9.51249	−0.444975	−0.222488	−	0.974935i	$-0.571418\pi$
$457$	−9.51249	−0.444975	−0.222488	+	0.974935i	$0.571418\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.6459	0.495829	0.247914	−	0.968782i	$-0.420255\pi$
$461$	10.6459	0.495829	0.247914	+	0.968782i	$0.420255\pi$
$462$	0	0
$463$	20.2172	0.939572	0.469786	−	0.882780i	$-0.344331\pi$
$463$	20.2172	0.939572	0.469786	+	0.882780i	$0.344331\pi$
$464$	0	0
$465$	−3.98545	−0.184821
$466$	0	0
$467$	18.5936	0.860408	0.430204	−	0.902732i	$-0.358442\pi$
$467$	18.5936	0.860408	0.430204	+	0.902732i	$0.358442\pi$
$468$	0	0
$469$	5.07873	0.234514
$470$	0	0
$471$	−27.5030	−1.26727
$472$	0	0
$473$	1.24123	0.0570718
$474$	0	0
$475$	−13.3884	−0.614300
$476$	0	0
$477$	7.88444	0.361004
$478$	0	0
$479$	−25.1284	−1.14814	−0.574072	−	0.818805i	$-0.694637\pi$
$479$	−25.1284	−1.14814	−0.574072	+	0.818805i	$0.694637\pi$
$480$	0	0
$481$	7.23854	0.330049
$482$	0	0
$483$	−13.3405	−0.607013
$484$	0	0
$485$	21.4851	0.975590
$486$	0	0
$487$	26.6955	1.20969	0.604845	−	0.796343i	$-0.293235\pi$
$487$	26.6955	1.20969	0.604845	+	0.796343i	$0.293235\pi$
$488$	0	0
$489$	33.3901	1.50995
$490$	0	0
$491$	4.79797	0.216529	0.108265	−	0.994122i	$-0.465471\pi$
$491$	4.79797	0.216529	0.108265	+	0.994122i	$0.465471\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.09152	−0.183900
$496$	0	0
$497$	29.5853	1.32708
$498$	0	0
$499$	−7.02734	−0.314587	−0.157293	−	0.987552i	$-0.550277\pi$
$499$	−7.02734	−0.314587	−0.157293	+	0.987552i	$0.550277\pi$
$500$	0	0
$501$	−2.53209	−0.113125
$502$	0	0
$503$	−33.0692	−1.47448	−0.737242	−	0.675629i	$-0.763872\pi$
$503$	−33.0692	−1.47448	−0.737242	+	0.675629i	$0.763872\pi$
$504$	0	0
$505$	25.3773	1.12928
$506$	0	0
$507$	74.2063	3.29562
$508$	0	0
$509$	0.182104	0.00807163	0.00403581	−	0.999992i	$-0.498715\pi$
$509$	0.182104	0.00807163	0.00403581	+	0.999992i	$0.498715\pi$
$510$	0	0
$511$	31.4688	1.39210
$512$	0	0
$513$	38.9290	1.71876
$514$	0	0
$515$	7.42932	0.327375
$516$	0	0
$517$	1.86484	0.0820155
$518$	0	0
$519$	−49.2772	−2.16303
$520$	0	0
$521$	32.8976	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$521$	32.8976	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$522$	0	0
$523$	27.3756	1.19705	0.598525	−	0.801104i	$-0.295754\pi$
$523$	27.3756	1.19705	0.598525	+	0.801104i	$0.295754\pi$
$524$	0	0
$525$	15.7520	0.687472
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.3087	−0.839509
$530$	0	0
$531$	−73.0583	−3.17046
$532$	0	0
$533$	25.1676	1.09013
$534$	0	0
$535$	−31.3661	−1.35607
$536$	0	0
$537$	41.0506	1.77146
$538$	0	0
$539$	−0.554378	−0.0238787
$540$	0	0
$541$	12.3865	0.532537	0.266269	−	0.963899i	$-0.414209\pi$
$541$	12.3865	0.532537	0.266269	+	0.963899i	$0.414209\pi$
$542$	0	0
$543$	31.0797	1.33376
$544$	0	0
$545$	−19.9837	−0.856008
$546$	0	0
$547$	−0.626296	−0.0267785	−0.0133892	−	0.999910i	$-0.504262\pi$
$547$	−0.626296	−0.0267785	−0.0133892	+	0.999910i	$0.504262\pi$
$548$	0	0
$549$	−23.3405	−0.996148
$550$	0	0
$551$	−28.9281	−1.23238
$552$	0	0
$553$	21.2763	0.904761
$554$	0	0
$555$	5.53209	0.234824
$556$	0	0
$557$	−31.1516	−1.31993	−0.659967	−	0.751294i	$-0.729430\pi$
$557$	−31.1516	−1.31993	−0.659967	+	0.751294i	$0.729430\pi$
$558$	0	0
$559$	−16.5175	−0.698618
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.1607	0.428225	0.214112	−	0.976809i	$-0.431314\pi$
$563$	10.1607	0.428225	0.214112	+	0.976809i	$0.431314\pi$
$564$	0	0
$565$	2.10101	0.0883903
$566$	0	0
$567$	−7.52528	−0.316032
$568$	0	0
$569$	−13.3577	−0.559985	−0.279992	−	0.960002i	$-0.590332\pi$
$569$	−13.3577	−0.559985	−0.279992	+	0.960002i	$0.590332\pi$
$570$	0	0
$571$	−5.14971	−0.215509	−0.107754	−	0.994178i	$-0.534366\pi$
$571$	−5.14971	−0.215509	−0.107754	+	0.994178i	$0.534366\pi$
$572$	0	0
$573$	26.2003	1.09453
$574$	0	0
$575$	−4.35855	−0.181764
$576$	0	0
$577$	−6.24628	−0.260036	−0.130018	−	0.991512i	$-0.541504\pi$
$577$	−6.24628	−0.260036	−0.130018	+	0.991512i	$0.541504\pi$
$578$	0	0
$579$	−22.9932	−0.955564
$580$	0	0
$581$	21.4311	0.889111
$582$	0	0
$583$	0.697281	0.0288784
$584$	0	0
$585$	54.4475	2.25113
$586$	0	0
$587$	30.5340	1.26027	0.630136	−	0.776485i	$-0.282999\pi$
$587$	30.5340	1.26027	0.630136	+	0.776485i	$0.282999\pi$
$588$	0	0
$589$	4.94263	0.203657
$590$	0	0
$591$	54.6468	2.24787
$592$	0	0
$593$	−23.7306	−0.974499	−0.487250	−	0.873263i	$-0.662000\pi$
$593$	−23.7306	−0.974499	−0.487250	+	0.873263i	$0.662000\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.94356	−0.161399
$598$	0	0
$599$	−8.16344	−0.333549	−0.166775	−	0.985995i	$-0.553335\pi$
$599$	−8.16344	−0.333549	−0.166775	+	0.985995i	$0.553335\pi$
$600$	0	0
$601$	31.0770	1.26766	0.633828	−	0.773474i	$-0.281483\pi$
$601$	31.0770	1.26766	0.633828	+	0.773474i	$0.281483\pi$
$602$	0	0
$603$	−11.1429	−0.453774
$604$	0	0
$605$	17.8179	0.724400
$606$	0	0
$607$	−5.73885	−0.232933	−0.116466	−	0.993195i	$-0.537157\pi$
$607$	−5.73885	−0.232933	−0.116466	+	0.993195i	$0.537157\pi$
$608$	0	0
$609$	34.0351	1.37917
$610$	0	0
$611$	−24.8161	−1.00395
$612$	0	0
$613$	−33.1762	−1.33998	−0.669988	−	0.742372i	$-0.733701\pi$
$613$	−33.1762	−1.33998	−0.669988	+	0.742372i	$0.733701\pi$
$614$	0	0
$615$	19.2344	0.775607
$616$	0	0
$617$	−14.9736	−0.602814	−0.301407	−	0.953496i	$-0.597456\pi$
$617$	−14.9736	−0.602814	−0.301407	+	0.953496i	$0.597456\pi$
$618$	0	0
$619$	−20.1985	−0.811847	−0.405924	−	0.913907i	$-0.633050\pi$
$619$	−20.1985	−0.811847	−0.405924	+	0.913907i	$0.633050\pi$
$620$	0	0
$621$	12.6732	0.508560
$622$	0	0
$623$	−34.6355	−1.38764
$624$	0	0
$625$	−8.51073	−0.340429
$626$	0	0
$627$	7.95130	0.317544
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−4.05737	−0.161521	−0.0807607	−	0.996734i	$-0.525735\pi$
$631$	−4.05737	−0.161521	−0.0807607	+	0.996734i	$0.525735\pi$
$632$	0	0
$633$	−35.8016	−1.42299
$634$	0	0
$635$	−7.02229	−0.278671
$636$	0	0
$637$	7.37733	0.292300
$638$	0	0
$639$	−64.9113	−2.56785
$640$	0	0
$641$	10.7956	0.426401	0.213200	−	0.977008i	$-0.431611\pi$
$641$	10.7956	0.426401	0.213200	+	0.977008i	$0.431611\pi$
$642$	0	0
$643$	16.5348	0.652068	0.326034	−	0.945358i	$-0.394288\pi$
$643$	16.5348	0.652068	0.326034	+	0.945358i	$0.394288\pi$
$644$	0	0
$645$	−12.6236	−0.497054
$646$	0	0
$647$	−49.4397	−1.94368	−0.971839	−	0.235648i	$-0.924279\pi$
$647$	−49.4397	−1.94368	−0.971839	+	0.235648i	$0.924279\pi$
$648$	0	0
$649$	−6.46110	−0.253621
$650$	0	0
$651$	−5.81521	−0.227916
$652$	0	0
$653$	−31.0479	−1.21500	−0.607499	−	0.794321i	$-0.707827\pi$
$653$	−31.0479	−1.21500	−0.607499	+	0.794321i	$0.707827\pi$
$654$	0	0
$655$	30.4783	1.19089
$656$	0	0
$657$	−69.0438	−2.69365
$658$	0	0
$659$	−45.3783	−1.76769	−0.883843	−	0.467784i	$-0.845053\pi$
$659$	−45.3783	−1.76769	−0.883843	+	0.467784i	$0.845053\pi$
$660$	0	0
$661$	24.3141	0.945708	0.472854	−	0.881141i	$-0.343224\pi$
$661$	24.3141	0.945708	0.472854	+	0.881141i	$0.343224\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.5208	0.912099
$666$	0	0
$667$	−9.41746	−0.364646
$668$	0	0
$669$	15.4834	0.598623
$670$	0	0
$671$	−2.06418	−0.0796867
$672$	0	0
$673$	−16.8658	−0.650128	−0.325064	−	0.945692i	$-0.605386\pi$
$673$	−16.8658	−0.650128	−0.325064	+	0.945692i	$0.605386\pi$
$674$	0	0
$675$	−14.9641	−0.575968
$676$	0	0
$677$	−20.7939	−0.799173	−0.399586	−	0.916696i	$-0.630846\pi$
$677$	−20.7939	−0.799173	−0.399586	+	0.916696i	$0.630846\pi$
$678$	0	0
$679$	31.3492	1.20307
$680$	0	0
$681$	−31.8384	−1.22005
$682$	0	0
$683$	−4.19522	−0.160526	−0.0802628	−	0.996774i	$-0.525576\pi$
$683$	−4.19522	−0.160526	−0.0802628	+	0.996774i	$0.525576\pi$
$684$	0	0
$685$	7.92572	0.302826
$686$	0	0
$687$	69.0069	2.63278
$688$	0	0
$689$	−9.27900	−0.353502
$690$	0	0
$691$	39.9195	1.51861	0.759305	−	0.650735i	$-0.225539\pi$
$691$	39.9195	1.51861	0.759305	+	0.650735i	$0.225539\pi$
$692$	0	0
$693$	−5.96997	−0.226780
$694$	0	0
$695$	7.45781	0.282891
$696$	0	0
$697$	0	0
$698$	0	0
$699$	30.0993	1.13846
$700$	0	0
$701$	−33.6100	−1.26943	−0.634716	−	0.772746i	$-0.718883\pi$
$701$	−33.6100	−1.26943	−0.634716	+	0.772746i	$0.718883\pi$
$702$	0	0
$703$	−6.86072	−0.258757
$704$	0	0
$705$	−18.9659	−0.714295
$706$	0	0
$707$	37.0283	1.39259
$708$	0	0
$709$	−43.6468	−1.63919	−0.819596	−	0.572943i	$-0.805802\pi$
$709$	−43.6468	−1.63919	−0.819596	+	0.572943i	$0.805802\pi$
$710$	0	0
$711$	−46.6810	−1.75067
$712$	0	0
$713$	1.60906	0.0602598
$714$	0	0
$715$	4.81521	0.180079
$716$	0	0
$717$	−5.36959	−0.200531
$718$	0	0
$719$	−43.5458	−1.62398	−0.811992	−	0.583668i	$-0.801617\pi$
$719$	−43.5458	−1.62398	−0.811992	+	0.583668i	$0.801617\pi$
$720$	0	0
$721$	10.8402	0.403710
$722$	0	0
$723$	−52.8931	−1.96712
$724$	0	0
$725$	11.1198	0.412979
$726$	0	0
$727$	−18.3473	−0.680464	−0.340232	−	0.940342i	$-0.610506\pi$
$727$	−18.3473	−0.680464	−0.340232	+	0.940342i	$0.610506\pi$
$728$	0	0
$729$	−40.4688	−1.49885
$730$	0	0
$731$	0	0
$732$	0	0
$733$	34.9495	1.29089	0.645446	−	0.763806i	$-0.276672\pi$
$733$	34.9495	1.29089	0.645446	+	0.763806i	$0.276672\pi$
$734$	0	0
$735$	5.63816	0.207967
$736$	0	0
$737$	−0.985452	−0.0362996
$738$	0	0
$739$	6.58408	0.242199	0.121100	−	0.992640i	$-0.461358\pi$
$739$	6.58408	0.242199	0.121100	+	0.992640i	$0.461358\pi$
$740$	0	0
$741$	−105.811	−3.88707
$742$	0	0
$743$	−8.32232	−0.305316	−0.152658	−	0.988279i	$-0.548783\pi$
$743$	−8.32232	−0.305316	−0.152658	+	0.988279i	$0.548783\pi$
$744$	0	0
$745$	−18.7906	−0.688433
$746$	0	0
$747$	−47.0205	−1.72039
$748$	0	0
$749$	−45.7665	−1.67227
$750$	0	0
$751$	−51.4662	−1.87803	−0.939013	−	0.343881i	$-0.888258\pi$
$751$	−51.4662	−1.87803	−0.939013	+	0.343881i	$0.888258\pi$
$752$	0	0
$753$	3.76651	0.137259
$754$	0	0
$755$	−10.7671	−0.391856
$756$	0	0
$757$	−22.5371	−0.819126	−0.409563	−	0.912282i	$-0.634319\pi$
$757$	−22.5371	−0.819126	−0.409563	+	0.912282i	$0.634319\pi$
$758$	0	0
$759$	2.58853	0.0939575
$760$	0	0
$761$	−8.39599	−0.304354	−0.152177	−	0.988353i	$-0.548628\pi$
$761$	−8.39599	−0.304354	−0.152177	+	0.988353i	$0.548628\pi$
$762$	0	0
$763$	−29.1584	−1.05560
$764$	0	0
$765$	0	0
$766$	0	0
$767$	85.9805	3.10458
$768$	0	0
$769$	−32.0063	−1.15418	−0.577089	−	0.816682i	$-0.695811\pi$
$769$	−32.0063	−1.15418	−0.577089	+	0.816682i	$0.695811\pi$
$770$	0	0
$771$	34.4766	1.24164
$772$	0	0
$773$	−22.4165	−0.806266	−0.403133	−	0.915141i	$-0.632079\pi$
$773$	−22.4165	−0.806266	−0.403133	+	0.915141i	$0.632079\pi$
$774$	0	0
$775$	−1.89992	−0.0682471
$776$	0	0
$777$	8.07192	0.289578
$778$	0	0
$779$	−23.8539	−0.854655
$780$	0	0
$781$	−5.74060	−0.205415
$782$	0	0
$783$	−32.3327	−1.15548
$784$	0	0
$785$	15.7861	0.563430
$786$	0	0
$787$	−33.4133	−1.19106	−0.595528	−	0.803334i	$-0.703057\pi$
$787$	−33.4133	−1.19106	−0.595528	+	0.803334i	$0.703057\pi$
$788$	0	0
$789$	41.8239	1.48897
$790$	0	0
$791$	3.06561	0.109000
$792$	0	0
$793$	27.4688	0.975447
$794$	0	0
$795$	−7.09152	−0.251510
$796$	0	0
$797$	24.2422	0.858701	0.429351	−	0.903138i	$-0.358742\pi$
$797$	24.2422	0.858701	0.429351	+	0.903138i	$0.358742\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	75.9914	2.68503
$802$	0	0
$803$	−6.10607	−0.215478
$804$	0	0
$805$	7.65715	0.269879
$806$	0	0
$807$	−80.5613	−2.83589
$808$	0	0
$809$	−45.0969	−1.58552	−0.792761	−	0.609532i	$-0.791357\pi$
$809$	−45.0969	−1.58552	−0.792761	+	0.609532i	$0.791357\pi$
$810$	0	0
$811$	38.1539	1.33977	0.669883	−	0.742467i	$-0.266344\pi$
$811$	38.1539	1.33977	0.669883	+	0.742467i	$0.266344\pi$
$812$	0	0
$813$	38.4911	1.34994
$814$	0	0
$815$	−19.1652	−0.671327
$816$	0	0
$817$	15.6554	0.547713
$818$	0	0
$819$	79.4448	2.77603
$820$	0	0
$821$	−30.4888	−1.06407	−0.532033	−	0.846724i	$-0.678572\pi$
$821$	−30.4888	−1.06407	−0.532033	+	0.846724i	$0.678572\pi$
$822$	0	0
$823$	42.6269	1.48588	0.742940	−	0.669358i	$-0.233431\pi$
$823$	42.6269	1.48588	0.742940	+	0.669358i	$0.233431\pi$
$824$	0	0
$825$	−3.05644	−0.106411
$826$	0	0
$827$	−13.2585	−0.461042	−0.230521	−	0.973067i	$-0.574043\pi$
$827$	−13.2585	−0.461042	−0.230521	+	0.973067i	$0.574043\pi$
$828$	0	0
$829$	−6.68779	−0.232276	−0.116138	−	0.993233i	$-0.537052\pi$
$829$	−6.68779	−0.232276	−0.116138	+	0.993233i	$0.537052\pi$
$830$	0	0
$831$	−1.71419	−0.0594647
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.45336	0.0502957
$836$	0	0
$837$	5.52435	0.190949
$838$	0	0
$839$	−9.91178	−0.342193	−0.171096	−	0.985254i	$-0.554731\pi$
$839$	−9.91178	−0.342193	−0.171096	+	0.985254i	$0.554731\pi$
$840$	0	0
$841$	−4.97359	−0.171503
$842$	0	0
$843$	51.5408	1.77516
$844$	0	0
$845$	−42.5928	−1.46524
$846$	0	0
$847$	25.9982	0.893310
$848$	0	0
$849$	−11.3773	−0.390469
$850$	0	0
$851$	−2.23349	−0.0765630
$852$	0	0
$853$	−5.57304	−0.190817	−0.0954087	−	0.995438i	$-0.530416\pi$
$853$	−5.57304	−0.190817	−0.0954087	+	0.995438i	$0.530416\pi$
$854$	0	0
$855$	−51.6056	−1.76487
$856$	0	0
$857$	−13.9453	−0.476363	−0.238181	−	0.971221i	$-0.576551\pi$
$857$	−13.9453	−0.476363	−0.238181	+	0.971221i	$0.576551\pi$
$858$	0	0
$859$	36.0401	1.22967	0.614837	−	0.788654i	$-0.289222\pi$
$859$	36.0401	1.22967	0.614837	+	0.788654i	$0.289222\pi$
$860$	0	0
$861$	28.0651	0.956456
$862$	0	0
$863$	23.4088	0.796844	0.398422	−	0.917202i	$-0.369558\pi$
$863$	23.4088	0.796844	0.398422	+	0.917202i	$0.369558\pi$
$864$	0	0
$865$	28.2841	0.961687
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.12836	−0.140045
$870$	0	0
$871$	13.1138	0.444344
$872$	0	0
$873$	−68.7812	−2.32789
$874$	0	0
$875$	−28.9685	−0.979315
$876$	0	0
$877$	46.3988	1.56678	0.783388	−	0.621533i	$-0.213490\pi$
$877$	46.3988	1.56678	0.783388	+	0.621533i	$0.213490\pi$
$878$	0	0
$879$	80.7606	2.72399
$880$	0	0
$881$	−33.8138	−1.13922	−0.569608	−	0.821917i	$-0.692905\pi$
$881$	−33.8138	−1.13922	−0.569608	+	0.821917i	$0.692905\pi$
$882$	0	0
$883$	37.5776	1.26459	0.632293	−	0.774729i	$-0.282114\pi$
$883$	37.5776	1.26459	0.632293	+	0.774729i	$0.282114\pi$
$884$	0	0
$885$	65.7110	2.20885
$886$	0	0
$887$	13.4097	0.450254	0.225127	−	0.974329i	$-0.427720\pi$
$887$	13.4097	0.450254	0.225127	+	0.974329i	$0.427720\pi$
$888$	0	0
$889$	−10.2463	−0.343649
$890$	0	0
$891$	1.46017	0.0489175
$892$	0	0
$893$	23.5208	0.787095
$894$	0	0
$895$	−23.5621	−0.787595
$896$	0	0
$897$	−34.4466	−1.15014
$898$	0	0
$899$	−4.10513	−0.136914
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−18.4192	−0.612953
$904$	0	0
$905$	−17.8390	−0.592990
$906$	0	0
$907$	28.0933	0.932822	0.466411	−	0.884568i	$-0.345547\pi$
$907$	28.0933	0.932822	0.466411	+	0.884568i	$0.345547\pi$
$908$	0	0
$909$	−81.2413	−2.69461
$910$	0	0
$911$	−22.4593	−0.744111	−0.372056	−	0.928210i	$-0.621347\pi$
$911$	−22.4593	−0.744111	−0.372056	+	0.928210i	$0.621347\pi$
$912$	0	0
$913$	−4.15839	−0.137622
$914$	0	0
$915$	20.9932	0.694014
$916$	0	0
$917$	44.4712	1.46857
$918$	0	0
$919$	−51.0215	−1.68304	−0.841521	−	0.540224i	$-0.818340\pi$
$919$	−51.0215	−1.68304	−0.841521	+	0.540224i	$0.818340\pi$
$920$	0	0
$921$	−61.9650	−2.04182
$922$	0	0
$923$	76.3925	2.51449
$924$	0	0
$925$	2.63722	0.0867113
$926$	0	0
$927$	−23.7837	−0.781161
$928$	0	0
$929$	37.3637	1.22586	0.612932	−	0.790136i	$-0.289990\pi$
$929$	37.3637	1.22586	0.612932	+	0.790136i	$0.289990\pi$
$930$	0	0
$931$	−6.99226	−0.229162
$932$	0	0
$933$	−13.6159	−0.445763
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.1480	0.592868	0.296434	−	0.955053i	$-0.404203\pi$
$937$	18.1480	0.592868	0.296434	+	0.955053i	$0.404203\pi$
$938$	0	0
$939$	−50.7725	−1.65690
$940$	0	0
$941$	−27.1385	−0.884689	−0.442344	−	0.896845i	$-0.645853\pi$
$941$	−27.1385	−0.884689	−0.442344	+	0.896845i	$0.645853\pi$
$942$	0	0
$943$	−7.76558	−0.252882
$944$	0	0
$945$	26.2891	0.855185
$946$	0	0
$947$	−53.0934	−1.72530	−0.862652	−	0.505799i	$-0.831198\pi$
$947$	−53.0934	−1.72530	−0.862652	+	0.505799i	$0.831198\pi$
$948$	0	0
$949$	81.2559	2.63768
$950$	0	0
$951$	−51.2354	−1.66142
$952$	0	0
$953$	5.62454	0.182197	0.0910984	−	0.995842i	$-0.470962\pi$
$953$	5.62454	0.182197	0.0910984	+	0.995842i	$0.470962\pi$
$954$	0	0
$955$	−15.0384	−0.486631
$956$	0	0
$957$	−6.60401	−0.213477
$958$	0	0
$959$	11.5645	0.373437
$960$	0	0
$961$	−30.2986	−0.977374
$962$	0	0
$963$	100.413	3.23577
$964$	0	0
$965$	13.1976	0.424845
$966$	0	0
$967$	−6.73111	−0.216458	−0.108229	−	0.994126i	$-0.534518\pi$
$967$	−6.73111	−0.216458	−0.108229	+	0.994126i	$0.534518\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.1465	0.678624	0.339312	−	0.940674i	$-0.389806\pi$
$971$	21.1465	0.678624	0.339312	+	0.940674i	$0.389806\pi$
$972$	0	0
$973$	10.8817	0.348853
$974$	0	0
$975$	40.6732	1.30259
$976$	0	0
$977$	42.3242	1.35407	0.677035	−	0.735951i	$-0.263264\pi$
$977$	42.3242	1.35407	0.677035	+	0.735951i	$0.263264\pi$
$978$	0	0
$979$	6.72050	0.214788
$980$	0	0
$981$	63.9745	2.04255
$982$	0	0
$983$	−7.81883	−0.249382	−0.124691	−	0.992196i	$-0.539794\pi$
$983$	−7.81883	−0.249382	−0.124691	+	0.992196i	$0.539794\pi$
$984$	0	0
$985$	−31.3661	−0.999406
$986$	0	0
$987$	−27.6732	−0.880849
$988$	0	0
$989$	5.09657	0.162062
$990$	0	0
$991$	−5.30304	−0.168457	−0.0842284	−	0.996446i	$-0.526843\pi$
$991$	−5.30304	−0.168457	−0.0842284	+	0.996446i	$0.526843\pi$
$992$	0	0
$993$	100.661	3.19437
$994$	0	0
$995$	2.26352	0.0717584
$996$	0	0
$997$	−32.8949	−1.04179	−0.520895	−	0.853620i	$-0.674402\pi$
$997$	−32.8949	−1.04179	−0.520895	+	0.853620i	$0.674402\pi$
$998$	0	0
$999$	−7.66819	−0.242611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4624.2.a.bh.1.3		3
4.3	odd	2		578.2.a.g.1.1	✓	3
12.11	even	2		5202.2.a.bi.1.2		3
17.16	even	2		4624.2.a.bc.1.1		3
68.3	even	16		578.2.d.i.179.1		24
68.7	even	16		578.2.d.i.423.6		24
68.11	even	16		578.2.d.i.155.6		24
68.15	odd	8		578.2.c.h.327.6		12
68.19	odd	8		578.2.c.h.327.1		12
68.23	even	16		578.2.d.i.155.1		24
68.27	even	16		578.2.d.i.423.1		24
68.31	even	16		578.2.d.i.179.6		24
68.39	even	16		578.2.d.i.399.6		24
68.43	odd	8		578.2.c.h.251.1		12
68.47	odd	4		578.2.b.e.577.1		6
68.55	odd	4		578.2.b.e.577.6		6
68.59	odd	8		578.2.c.h.251.6		12
68.63	even	16		578.2.d.i.399.1		24
68.67	odd	2		578.2.a.h.1.3	yes	3
204.203	even	2		5202.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
578.2.a.g.1.1	✓	3	4.3	odd	2
578.2.a.h.1.3	yes	3	68.67	odd	2
578.2.b.e.577.1		6	68.47	odd	4
578.2.b.e.577.6		6	68.55	odd	4
578.2.c.h.251.1		12	68.43	odd	8
578.2.c.h.251.6		12	68.59	odd	8
578.2.c.h.327.1		12	68.19	odd	8
578.2.c.h.327.6		12	68.15	odd	8
578.2.d.i.155.1		24	68.23	even	16
578.2.d.i.155.6		24	68.11	even	16
578.2.d.i.179.1		24	68.3	even	16
578.2.d.i.179.6		24	68.31	even	16
578.2.d.i.399.1		24	68.63	even	16
578.2.d.i.399.6		24	68.39	even	16
578.2.d.i.423.1		24	68.27	even	16
578.2.d.i.423.6		24	68.7	even	16
4624.2.a.bc.1.1		3	17.16	even	2
4624.2.a.bh.1.3		3	1.1	even	1	trivial
5202.2.a.bg.1.2		3	204.203	even	2
5202.2.a.bi.1.2		3	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4624 = 2^{4} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4624.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87939 q^{3} -1.65270 q^{5} -2.41147 q^{7} +5.29086 q^{9} +0.467911 q^{11} -6.22668 q^{13} -4.75877 q^{15} +5.90167 q^{19} -6.94356 q^{21} +1.92127 q^{23} -2.26857 q^{25} +6.59627 q^{27} -4.90167 q^{29} +0.837496 q^{31} +1.34730 q^{33} +3.98545 q^{35} -1.16250 q^{37} -17.9290 q^{39} -4.04189 q^{41} +2.65270 q^{43} -8.74422 q^{45} +3.98545 q^{47} -1.18479 q^{49} +1.49020 q^{53} -0.773318 q^{55} +16.9932 q^{57} -13.8084 q^{59} -4.41147 q^{61} -12.7588 q^{63} +10.2909 q^{65} -2.10607 q^{67} +5.53209 q^{69} -12.2686 q^{71} -13.0496 q^{73} -6.53209 q^{75} -1.12836 q^{77} -8.82295 q^{79} +3.12061 q^{81} -8.88713 q^{83} -14.1138 q^{87} +14.3628 q^{89} +15.0155 q^{91} +2.41147 q^{93} -9.75372 q^{95} -13.0000 q^{97} +2.47565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{11} - 12 q^{13} - 3 q^{15} + 6 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} + 6 q^{27} - 3 q^{29} + 3 q^{33} - 6 q^{35} - 6 q^{37} - 21 q^{39} - 9 q^{41} + 9 q^{43} + 3 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^11 - 12 * q^13 - 3 * q^15 + 6 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 + 6 * q^27 - 3 * q^29 + 3 * q^33 - 6 * q^35 - 6 * q^37 - 21 * q^39 - 9 * q^41 + 9 * q^43 + 3 * q^45 - 6 * q^47 + 3 * q^53 - 9 * q^55 + 9 * q^57 - 3 * q^59 - 3 * q^61 - 27 * q^63 + 15 * q^65 + 6 * q^67 + 12 * q^69 - 27 * q^71 - 12 * q^73 - 15 * q^75 + 15 * q^77 - 6 * q^79 + 15 * q^81 + 3 * q^83 - 6 * q^87 - 6 * q^89 - 3 * q^91 - 3 * q^93 - 33 * q^95 - 39 * q^97 - 12 * q^99

Introduction

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