LMFDB - Embedded newform 4704.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.360409$ of defining polynomial
Character	$\chi$	$=$	4704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.13503 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.13503 q^{5} +1.00000 q^{9} +0.298573 q^{11} +6.43361 q^{13} +2.13503 q^{15} +1.11564 q^{17} +1.01939 q^{19} -4.29857 q^{23} -0.441637 q^{25} -1.00000 q^{27} -0.422246 q^{29} +10.6762 q^{31} -0.298573 q^{33} -5.65685 q^{37} -6.43361 q^{39} +8.32600 q^{41} -7.09849 q^{43} -2.13503 q^{45} -8.11788 q^{47} -1.11564 q^{51} -3.44164 q^{53} -0.637463 q^{55} -1.01939 q^{57} -6.46103 q^{59} +5.83646 q^{61} -13.7360 q^{65} +5.05971 q^{67} +4.29857 q^{69} +1.35828 q^{71} -2.85585 q^{73} +0.441637 q^{75} +7.69564 q^{79} +1.00000 q^{81} -6.03878 q^{83} -2.38193 q^{85} +0.422246 q^{87} +15.3696 q^{89} -10.6762 q^{93} -2.17643 q^{95} -0.512705 q^{97} +0.298573 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} + 4 q^{9} - 4 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{17} - 8 q^{19} - 12 q^{23} + 12 q^{25} - 4 q^{27} + 8 q^{31} + 4 q^{33} - 8 q^{39} + 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 4 q^{51} + 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} + 8 q^{67} + 12 q^{69} - 12 q^{71} + 8 q^{73} - 12 q^{75} - 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} - 24 q^{95} + 40 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^17 - 8 * q^19 - 12 * q^23 + 12 * q^25 - 4 * q^27 + 8 * q^31 + 4 * q^33 - 8 * q^39 + 20 * q^41 + 8 * q^43 + 4 * q^45 + 16 * q^47 - 4 * q^51 + 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 + 8 * q^67 + 12 * q^69 - 12 * q^71 + 8 * q^73 - 12 * q^75 - 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 - 24 * q^95 + 40 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.13503	−0.954815	−0.477408	−	0.878682i	$-0.658424\pi$
$5$	−2.13503	−0.954815	−0.477408	+	0.878682i	$0.658424\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.298573	0.0900231	0.0450116	−	0.998986i	$-0.485668\pi$
$11$	0.298573	0.0900231	0.0450116	+	0.998986i	$0.485668\pi$
$12$	0	0
$13$	6.43361	1.78436	0.892181	−	0.451679i	$-0.149175\pi$
$13$	6.43361	1.78436	0.892181	+	0.451679i	$0.149175\pi$
$14$	0	0
$15$	2.13503	0.551263
$16$	0	0
$17$	1.11564	0.270583	0.135291	−	0.990806i	$-0.456803\pi$
$17$	1.11564	0.270583	0.135291	+	0.990806i	$0.456803\pi$
$18$	0	0
$19$	1.01939	0.233864	0.116932	−	0.993140i	$-0.462694\pi$
$19$	1.01939	0.233864	0.116932	+	0.993140i	$0.462694\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.29857	−0.896314	−0.448157	−	0.893955i	$-0.647920\pi$
$23$	−4.29857	−0.896314	−0.448157	+	0.893955i	$0.647920\pi$
$24$	0	0
$25$	−0.441637	−0.0883275
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.422246	−0.0784091	−0.0392045	−	0.999231i	$-0.512482\pi$
$29$	−0.422246	−0.0784091	−0.0392045	+	0.999231i	$0.512482\pi$
$30$	0	0
$31$	10.6762	1.91751	0.958755	−	0.284233i	$-0.0917390\pi$
$31$	10.6762	1.91751	0.958755	+	0.284233i	$0.0917390\pi$
$32$	0	0
$33$	−0.298573	−0.0519749
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.65685	−0.929981	−0.464991	−	0.885316i	$-0.653942\pi$
$37$	−5.65685	−0.929981	−0.464991	+	0.885316i	$0.653942\pi$
$38$	0	0
$39$	−6.43361	−1.03020
$40$	0	0
$41$	8.32600	1.30030	0.650151	−	0.759805i	$-0.274706\pi$
$41$	8.32600	1.30030	0.650151	+	0.759805i	$0.274706\pi$
$42$	0	0
$43$	−7.09849	−1.08251	−0.541255	−	0.840859i	$-0.682051\pi$
$43$	−7.09849	−1.08251	−0.541255	+	0.840859i	$0.682051\pi$
$44$	0	0
$45$	−2.13503	−0.318272
$46$	0	0
$47$	−8.11788	−1.18411	−0.592057	−	0.805896i	$-0.701684\pi$
$47$	−8.11788	−1.18411	−0.592057	+	0.805896i	$0.701684\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.11564	−0.156221
$52$	0	0
$53$	−3.44164	−0.472745	−0.236373	−	0.971662i	$-0.575959\pi$
$53$	−3.44164	−0.472745	−0.236373	+	0.971662i	$0.575959\pi$
$54$	0	0
$55$	−0.637463	−0.0859555
$56$	0	0
$57$	−1.01939	−0.135022
$58$	0	0
$59$	−6.46103	−0.841154	−0.420577	−	0.907257i	$-0.638172\pi$
$59$	−6.46103	−0.841154	−0.420577	+	0.907257i	$0.638172\pi$
$60$	0	0
$61$	5.83646	0.747282	0.373641	−	0.927573i	$-0.378109\pi$
$61$	5.83646	0.747282	0.373641	+	0.927573i	$0.378109\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.7360	−1.70374
$66$	0	0
$67$	5.05971	0.618142	0.309071	−	0.951039i	$-0.399982\pi$
$67$	5.05971	0.618142	0.309071	+	0.951039i	$0.399982\pi$
$68$	0	0
$69$	4.29857	0.517487
$70$	0	0
$71$	1.35828	0.161198	0.0805992	−	0.996747i	$-0.474317\pi$
$71$	1.35828	0.161198	0.0805992	+	0.996747i	$0.474317\pi$
$72$	0	0
$73$	−2.85585	−0.334252	−0.167126	−	0.985936i	$-0.553449\pi$
$73$	−2.85585	−0.334252	−0.167126	+	0.985936i	$0.553449\pi$
$74$	0	0
$75$	0.441637	0.0509959
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.69564	0.865827	0.432913	−	0.901436i	$-0.357486\pi$
$79$	7.69564	0.865827	0.432913	+	0.901436i	$0.357486\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.03878	−0.662843	−0.331421	−	0.943483i	$-0.607528\pi$
$83$	−6.03878	−0.662843	−0.331421	+	0.943483i	$0.607528\pi$
$84$	0	0
$85$	−2.38193	−0.258356
$86$	0	0
$87$	0.422246	0.0452695
$88$	0	0
$89$	15.3696	1.62918	0.814589	−	0.580038i	$-0.196962\pi$
$89$	15.3696	1.62918	0.814589	+	0.580038i	$0.196962\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.6762	−1.10708
$94$	0	0
$95$	−2.17643	−0.223297
$96$	0	0
$97$	−0.512705	−0.0520573	−0.0260287	−	0.999661i	$-0.508286\pi$
$97$	−0.512705	−0.0520573	−0.0260287	+	0.999661i	$0.508286\pi$
$98$	0	0
$99$	0.298573	0.0300077
$100$	0	0
$101$	−4.96346	−0.493883	−0.246941	−	0.969030i	$-0.579425\pi$
$101$	−4.96346	−0.493883	−0.246941	+	0.969030i	$0.579425\pi$
$102$	0	0
$103$	9.48195	0.934285	0.467142	−	0.884182i	$-0.345284\pi$
$103$	9.48195	0.934285	0.467142	+	0.884182i	$0.345284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.8387	−1.62786	−0.813929	−	0.580964i	$-0.802676\pi$
$107$	−16.8387	−1.62786	−0.813929	+	0.580964i	$0.802676\pi$
$108$	0	0
$109$	3.09849	0.296782	0.148391	−	0.988929i	$-0.452591\pi$
$109$	3.09849	0.296782	0.148391	+	0.988929i	$0.452591\pi$
$110$	0	0
$111$	5.65685	0.536925
$112$	0	0
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	0	0
$115$	9.17759	0.855815
$116$	0	0
$117$	6.43361	0.594787
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9109	−0.991896
$122$	0	0
$123$	−8.32600	−0.750730
$124$	0	0
$125$	11.6181	1.03915
$126$	0	0
$127$	17.1373	1.52069	0.760344	−	0.649521i	$-0.225031\pi$
$127$	17.1373	1.52069	0.760344	+	0.649521i	$0.225031\pi$
$128$	0	0
$129$	7.09849	0.624987
$130$	0	0
$131$	−16.1582	−1.41175	−0.705874	−	0.708337i	$-0.749446\pi$
$131$	−16.1582	−1.41175	−0.705874	+	0.708337i	$0.749446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.13503	0.183754
$136$	0	0
$137$	21.7747	1.86034	0.930171	−	0.367127i	$-0.119659\pi$
$137$	21.7747	1.86034	0.930171	+	0.367127i	$0.119659\pi$
$138$	0	0
$139$	−15.3137	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$140$	0	0
$141$	8.11788	0.683649
$142$	0	0
$143$	1.92090	0.160634
$144$	0	0
$145$	0.901508	0.0748662
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.87207	−0.317212	−0.158606	−	0.987342i	$-0.550700\pi$
$149$	−3.87207	−0.317212	−0.158606	+	0.987342i	$0.550700\pi$
$150$	0	0
$151$	−4.75535	−0.386985	−0.193492	−	0.981102i	$-0.561981\pi$
$151$	−4.75535	−0.386985	−0.193492	+	0.981102i	$0.561981\pi$
$152$	0	0
$153$	1.11564	0.0901942
$154$	0	0
$155$	−22.7941	−1.83087
$156$	0	0
$157$	11.5482	0.921644	0.460822	−	0.887493i	$-0.347555\pi$
$157$	11.5482	0.921644	0.460822	+	0.887493i	$0.347555\pi$
$158$	0	0
$159$	3.44164	0.272940
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.9109	−0.932930	−0.466465	−	0.884540i	$-0.654473\pi$
$163$	−11.9109	−0.932930	−0.466465	+	0.884540i	$0.654473\pi$
$164$	0	0
$165$	0.637463	0.0496264
$166$	0	0
$167$	15.2734	1.18189	0.590945	−	0.806712i	$-0.298755\pi$
$167$	15.2734	1.18189	0.590945	+	0.806712i	$0.298755\pi$
$168$	0	0
$169$	28.3913	2.18394
$170$	0	0
$171$	1.01939	0.0779548
$172$	0	0
$173$	−4.00710	−0.304654	−0.152327	−	0.988330i	$-0.548677\pi$
$173$	−4.00710	−0.304654	−0.152327	+	0.988330i	$0.548677\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.46103	0.485641
$178$	0	0
$179$	20.7496	1.55089	0.775447	−	0.631412i	$-0.217524\pi$
$179$	20.7496	1.55089	0.775447	+	0.631412i	$0.217524\pi$
$180$	0	0
$181$	21.7246	1.61478	0.807388	−	0.590021i	$-0.200880\pi$
$181$	21.7246	1.61478	0.807388	+	0.590021i	$0.200880\pi$
$182$	0	0
$183$	−5.83646	−0.431443
$184$	0	0
$185$	12.0776	0.887960
$186$	0	0
$187$	0.333100	0.0243587
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.6720	1.49577	0.747886	−	0.663827i	$-0.231069\pi$
$191$	20.6720	1.49577	0.747886	+	0.663827i	$0.231069\pi$
$192$	0	0
$193$	15.5998	1.12290	0.561450	−	0.827510i	$-0.310244\pi$
$193$	15.5998	1.12290	0.561450	+	0.827510i	$0.310244\pi$
$194$	0	0
$195$	13.7360	0.983652
$196$	0	0
$197$	25.6386	1.82668	0.913338	−	0.407202i	$-0.133496\pi$
$197$	25.6386	1.82668	0.913338	+	0.407202i	$0.133496\pi$
$198$	0	0
$199$	−2.88327	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$199$	−2.88327	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$200$	0	0
$201$	−5.05971	−0.356884
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−17.7763	−1.24155
$206$	0	0
$207$	−4.29857	−0.298771
$208$	0	0
$209$	0.304363	0.0210532
$210$	0	0
$211$	24.5401	1.68941	0.844706	−	0.535230i	$-0.179775\pi$
$211$	24.5401	1.68941	0.844706	+	0.535230i	$0.179775\pi$
$212$	0	0
$213$	−1.35828	−0.0930679
$214$	0	0
$215$	15.1555	1.03360
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.85585	0.192981
$220$	0	0
$221$	7.17759	0.482817
$222$	0	0
$223$	−20.5080	−1.37332	−0.686659	−	0.726980i	$-0.740923\pi$
$223$	−20.5080	−1.37332	−0.686659	+	0.726980i	$0.740923\pi$
$224$	0	0
$225$	−0.441637	−0.0294425
$226$	0	0
$227$	−13.6972	−0.909113	−0.454557	−	0.890718i	$-0.650202\pi$
$227$	−13.6972	−0.909113	−0.454557	+	0.890718i	$0.650202\pi$
$228$	0	0
$229$	16.4175	1.08490	0.542451	−	0.840088i	$-0.317496\pi$
$229$	16.4175	1.08490	0.542451	+	0.840088i	$0.317496\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.8915	−0.713523	−0.356762	−	0.934195i	$-0.616119\pi$
$233$	−10.8915	−0.713523	−0.356762	+	0.934195i	$0.616119\pi$
$234$	0	0
$235$	17.3319	1.13061
$236$	0	0
$237$	−7.69564	−0.499885
$238$	0	0
$239$	−4.29857	−0.278052	−0.139026	−	0.990289i	$-0.544397\pi$
$239$	−4.29857	−0.278052	−0.139026	+	0.990289i	$0.544397\pi$
$240$	0	0
$241$	7.12808	0.459160	0.229580	−	0.973290i	$-0.426265\pi$
$241$	7.12808	0.459160	0.229580	+	0.973290i	$0.426265\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.55836	0.417299
$248$	0	0
$249$	6.03878	0.382692
$250$	0	0
$251$	18.9690	1.19731	0.598657	−	0.801005i	$-0.295701\pi$
$251$	18.9690	1.19731	0.598657	+	0.801005i	$0.295701\pi$
$252$	0	0
$253$	−1.28344	−0.0806890
$254$	0	0
$255$	2.38193	0.149162
$256$	0	0
$257$	20.4842	1.27777	0.638885	−	0.769303i	$-0.279396\pi$
$257$	20.4842	1.27777	0.638885	+	0.769303i	$0.279396\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.422246	−0.0261364
$262$	0	0
$263$	−23.5553	−1.45248	−0.726240	−	0.687441i	$-0.758734\pi$
$263$	−23.5553	−1.45248	−0.726240	+	0.687441i	$0.758734\pi$
$264$	0	0
$265$	7.34801	0.451384
$266$	0	0
$267$	−15.3696	−0.940607
$268$	0	0
$269$	12.9086	0.787052	0.393526	−	0.919313i	$-0.371255\pi$
$269$	12.9086	0.787052	0.393526	+	0.919313i	$0.371255\pi$
$270$	0	0
$271$	19.8705	1.20705	0.603525	−	0.797344i	$-0.293763\pi$
$271$	19.8705	1.20705	0.603525	+	0.797344i	$0.293763\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.131861	−0.00795151
$276$	0	0
$277$	25.9884	1.56149	0.780746	−	0.624848i	$-0.214839\pi$
$277$	25.9884	1.56149	0.780746	+	0.624848i	$0.214839\pi$
$278$	0	0
$279$	10.6762	0.639170
$280$	0	0
$281$	−19.6553	−1.17254	−0.586269	−	0.810116i	$-0.699404\pi$
$281$	−19.6553	−1.17254	−0.586269	+	0.810116i	$0.699404\pi$
$282$	0	0
$283$	−19.0582	−1.13289	−0.566445	−	0.824099i	$-0.691682\pi$
$283$	−19.0582	−1.13289	−0.566445	+	0.824099i	$0.691682\pi$
$284$	0	0
$285$	2.17643	0.128921
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.7553	−0.926785
$290$	0	0
$291$	0.512705	0.0300553
$292$	0	0
$293$	7.84890	0.458538	0.229269	−	0.973363i	$-0.426366\pi$
$293$	7.84890	0.458538	0.229269	+	0.973363i	$0.426366\pi$
$294$	0	0
$295$	13.7945	0.803147
$296$	0	0
$297$	−0.298573	−0.0173250
$298$	0	0
$299$	−27.6553	−1.59935
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.96346	0.285143
$304$	0	0
$305$	−12.4610	−0.713516
$306$	0	0
$307$	−26.5301	−1.51415	−0.757076	−	0.653327i	$-0.773373\pi$
$307$	−26.5301	−1.51415	−0.757076	+	0.653327i	$0.773373\pi$
$308$	0	0
$309$	−9.48195	−0.539410
$310$	0	0
$311$	0.726608	0.0412022	0.0206011	−	0.999788i	$-0.493442\pi$
$311$	0.726608	0.0412022	0.0206011	+	0.999788i	$0.493442\pi$
$312$	0	0
$313$	15.1259	0.854967	0.427484	−	0.904023i	$-0.359400\pi$
$313$	15.1259	0.854967	0.427484	+	0.904023i	$0.359400\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.9496	−1.12048	−0.560242	−	0.828329i	$-0.689292\pi$
$317$	−19.9496	−1.12048	−0.560242	+	0.828329i	$0.689292\pi$
$318$	0	0
$319$	−0.126071	−0.00705863
$320$	0	0
$321$	16.8387	0.939845
$322$	0	0
$323$	1.13727	0.0632797
$324$	0	0
$325$	−2.84132	−0.157608
$326$	0	0
$327$	−3.09849	−0.171347
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.1943	0.945084	0.472542	−	0.881308i	$-0.343337\pi$
$331$	17.1943	0.945084	0.472542	+	0.881308i	$0.343337\pi$
$332$	0	0
$333$	−5.65685	−0.309994
$334$	0	0
$335$	−10.8026	−0.590211
$336$	0	0
$337$	10.0388	0.546847	0.273424	−	0.961894i	$-0.411844\pi$
$337$	10.0388	0.546847	0.273424	+	0.961894i	$0.411844\pi$
$338$	0	0
$339$	−9.31371	−0.505851
$340$	0	0
$341$	3.18764	0.172620
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−9.17759	−0.494105
$346$	0	0
$347$	2.41799	0.129804	0.0649022	−	0.997892i	$-0.479326\pi$
$347$	2.41799	0.129804	0.0649022	+	0.997892i	$0.479326\pi$
$348$	0	0
$349$	−6.26689	−0.335459	−0.167730	−	0.985833i	$-0.553644\pi$
$349$	−6.26689	−0.335459	−0.167730	+	0.985833i	$0.553644\pi$
$350$	0	0
$351$	−6.43361	−0.343400
$352$	0	0
$353$	−10.3099	−0.548742	−0.274371	−	0.961624i	$-0.588470\pi$
$353$	−10.3099	−0.548742	−0.274371	+	0.961624i	$0.588470\pi$
$354$	0	0
$355$	−2.89997	−0.153915
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.8984	−0.944642	−0.472321	−	0.881427i	$-0.656584\pi$
$359$	−17.8984	−0.944642	−0.472321	+	0.881427i	$0.656584\pi$
$360$	0	0
$361$	−17.9608	−0.945307
$362$	0	0
$363$	10.9109	0.572671
$364$	0	0
$365$	6.09733	0.319149
$366$	0	0
$367$	23.0415	1.20276	0.601378	−	0.798965i	$-0.294619\pi$
$367$	23.0415	1.20276	0.601378	+	0.798965i	$0.294619\pi$
$368$	0	0
$369$	8.32600	0.433434
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.68898	0.191008	0.0955042	−	0.995429i	$-0.469554\pi$
$373$	3.68898	0.191008	0.0955042	+	0.995429i	$0.469554\pi$
$374$	0	0
$375$	−11.6181	−0.599955
$376$	0	0
$377$	−2.71656	−0.139910
$378$	0	0
$379$	8.76386	0.450169	0.225085	−	0.974339i	$-0.427734\pi$
$379$	8.76386	0.450169	0.225085	+	0.974339i	$0.427734\pi$
$380$	0	0
$381$	−17.1373	−0.877969
$382$	0	0
$383$	21.7665	1.11222	0.556109	−	0.831109i	$-0.312294\pi$
$383$	21.7665	1.11222	0.556109	+	0.831109i	$0.312294\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.09849	−0.360837
$388$	0	0
$389$	6.38346	0.323654	0.161827	−	0.986819i	$-0.448261\pi$
$389$	6.38346	0.323654	0.161827	+	0.986819i	$0.448261\pi$
$390$	0	0
$391$	−4.79566	−0.242527
$392$	0	0
$393$	16.1582	0.815073
$394$	0	0
$395$	−16.4304	−0.826705
$396$	0	0
$397$	5.53210	0.277648	0.138824	−	0.990317i	$-0.455668\pi$
$397$	5.53210	0.277648	0.138824	+	0.990317i	$0.455668\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.5778	−0.777916	−0.388958	−	0.921256i	$-0.627165\pi$
$401$	−15.5778	−0.777916	−0.388958	+	0.921256i	$0.627165\pi$
$402$	0	0
$403$	68.6867	3.42153
$404$	0	0
$405$	−2.13503	−0.106091
$406$	0	0
$407$	−1.68898	−0.0837198
$408$	0	0
$409$	−16.8625	−0.833797	−0.416899	−	0.908953i	$-0.636883\pi$
$409$	−16.8625	−0.833797	−0.416899	+	0.908953i	$0.636883\pi$
$410$	0	0
$411$	−21.7747	−1.07407
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.8930	0.632892
$416$	0	0
$417$	15.3137	0.749916
$418$	0	0
$419$	27.8942	1.36272	0.681359	−	0.731949i	$-0.261389\pi$
$419$	27.8942	1.36272	0.681359	+	0.731949i	$0.261389\pi$
$420$	0	0
$421$	−31.6774	−1.54386	−0.771931	−	0.635706i	$-0.780709\pi$
$421$	−31.6774	−1.54386	−0.771931	+	0.635706i	$0.780709\pi$
$422$	0	0
$423$	−8.11788	−0.394705
$424$	0	0
$425$	−0.492709	−0.0238999
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.92090	−0.0927419
$430$	0	0
$431$	−24.8066	−1.19489	−0.597445	−	0.801910i	$-0.703817\pi$
$431$	−24.8066	−1.19489	−0.597445	+	0.801910i	$0.703817\pi$
$432$	0	0
$433$	13.4918	0.648374	0.324187	−	0.945993i	$-0.394909\pi$
$433$	13.4918	0.648374	0.324187	+	0.945993i	$0.394909\pi$
$434$	0	0
$435$	−0.901508	−0.0432240
$436$	0	0
$437$	−4.38193	−0.209616
$438$	0	0
$439$	35.3137	1.68543	0.842716	−	0.538359i	$-0.180956\pi$
$439$	35.3137	1.68543	0.842716	+	0.538359i	$0.180956\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.4385	1.44618	0.723089	−	0.690755i	$-0.242721\pi$
$443$	30.4385	1.44618	0.723089	+	0.690755i	$0.242721\pi$
$444$	0	0
$445$	−32.8147	−1.55556
$446$	0	0
$447$	3.87207	0.183143
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	2.48592	0.117057
$452$	0	0
$453$	4.75535	0.223426
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.1943	0.710759	0.355379	−	0.934722i	$-0.384352\pi$
$457$	15.1943	0.710759	0.355379	+	0.934722i	$0.384352\pi$
$458$	0	0
$459$	−1.11564	−0.0520736
$460$	0	0
$461$	38.3935	1.78816	0.894082	−	0.447903i	$-0.147829\pi$
$461$	38.3935	1.78816	0.894082	+	0.447903i	$0.147829\pi$
$462$	0	0
$463$	−5.59984	−0.260247	−0.130123	−	0.991498i	$-0.541537\pi$
$463$	−5.59984	−0.260247	−0.130123	+	0.991498i	$0.541537\pi$
$464$	0	0
$465$	22.7941	1.05705
$466$	0	0
$467$	23.9717	1.10928	0.554639	−	0.832091i	$-0.312856\pi$
$467$	23.9717	1.10928	0.554639	+	0.832091i	$0.312856\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.5482	−0.532111
$472$	0	0
$473$	−2.11942	−0.0974509
$474$	0	0
$475$	−0.450201	−0.0206567
$476$	0	0
$477$	−3.44164	−0.157582
$478$	0	0
$479$	−5.23461	−0.239175	−0.119588	−	0.992824i	$-0.538157\pi$
$479$	−5.23461	−0.239175	−0.119588	+	0.992824i	$0.538157\pi$
$480$	0	0
$481$	−36.3940	−1.65942
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.09464	0.0497051
$486$	0	0
$487$	11.2447	0.509544	0.254772	−	0.967001i	$-0.418000\pi$
$487$	11.2447	0.509544	0.254772	+	0.967001i	$0.418000\pi$
$488$	0	0
$489$	11.9109	0.538627
$490$	0	0
$491$	13.5250	0.610374	0.305187	−	0.952292i	$-0.401281\pi$
$491$	13.5250	0.610374	0.305187	+	0.952292i	$0.401281\pi$
$492$	0	0
$493$	−0.471075	−0.0212161
$494$	0	0
$495$	−0.637463	−0.0286518
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.87207	−0.441935	−0.220967	−	0.975281i	$-0.570921\pi$
$499$	−9.87207	−0.441935	−0.220967	+	0.975281i	$0.570921\pi$
$500$	0	0
$501$	−15.2734	−0.682365
$502$	0	0
$503$	39.0415	1.74077	0.870387	−	0.492369i	$-0.163869\pi$
$503$	39.0415	1.74077	0.870387	+	0.492369i	$0.163869\pi$
$504$	0	0
$505$	10.5971	0.471567
$506$	0	0
$507$	−28.3913	−1.26090
$508$	0	0
$509$	−24.2202	−1.07354	−0.536770	−	0.843729i	$-0.680356\pi$
$509$	−24.2202	−1.07354	−0.536770	+	0.843729i	$0.680356\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.01939	−0.0450072
$514$	0	0
$515$	−20.2443	−0.892069
$516$	0	0
$517$	−2.42378	−0.106598
$518$	0	0
$519$	4.00710	0.175892
$520$	0	0
$521$	−22.1638	−0.971012	−0.485506	−	0.874233i	$-0.661365\pi$
$521$	−22.1638	−0.971012	−0.485506	+	0.874233i	$0.661365\pi$
$522$	0	0
$523$	−35.8053	−1.56566	−0.782829	−	0.622237i	$-0.786224\pi$
$523$	−35.8053	−1.56566	−0.782829	+	0.622237i	$0.786224\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.9109	0.518845
$528$	0	0
$529$	−4.52227	−0.196620
$530$	0	0
$531$	−6.46103	−0.280385
$532$	0	0
$533$	53.5662	2.32021
$534$	0	0
$535$	35.9512	1.55430
$536$	0	0
$537$	−20.7496	−0.895409
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.7941	0.550063	0.275031	−	0.961435i	$-0.411312\pi$
$541$	12.7941	0.550063	0.275031	+	0.961435i	$0.411312\pi$
$542$	0	0
$543$	−21.7246	−0.932292
$544$	0	0
$545$	−6.61538	−0.283372
$546$	0	0
$547$	−28.9675	−1.23856	−0.619280	−	0.785170i	$-0.712576\pi$
$547$	−28.9675	−1.23856	−0.619280	+	0.785170i	$0.712576\pi$
$548$	0	0
$549$	5.83646	0.249094
$550$	0	0
$551$	−0.430434	−0.0183371
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.0776	−0.512664
$556$	0	0
$557$	9.75265	0.413233	0.206617	−	0.978422i	$-0.433755\pi$
$557$	9.75265	0.413233	0.206617	+	0.978422i	$0.433755\pi$
$558$	0	0
$559$	−45.6689	−1.93159
$560$	0	0
$561$	−0.333100	−0.0140635
$562$	0	0
$563$	10.6192	0.447547	0.223774	−	0.974641i	$-0.428162\pi$
$563$	10.6192	0.447547	0.223774	+	0.974641i	$0.428162\pi$
$564$	0	0
$565$	−19.8851	−0.836571
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.1660	1.55808	0.779040	−	0.626974i	$-0.215707\pi$
$569$	37.1660	1.55808	0.779040	+	0.626974i	$0.215707\pi$
$570$	0	0
$571$	23.6956	0.991632	0.495816	−	0.868428i	$-0.334869\pi$
$571$	23.6956	0.991632	0.495816	+	0.868428i	$0.334869\pi$
$572$	0	0
$573$	−20.6720	−0.863585
$574$	0	0
$575$	1.89841	0.0791692
$576$	0	0
$577$	9.84465	0.409838	0.204919	−	0.978779i	$-0.434307\pi$
$577$	9.84465	0.409838	0.204919	+	0.978779i	$0.434307\pi$
$578$	0	0
$579$	−15.5998	−0.648307
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.02758	−0.0425580
$584$	0	0
$585$	−13.7360	−0.567912
$586$	0	0
$587$	−30.4610	−1.25726	−0.628631	−	0.777704i	$-0.716384\pi$
$587$	−30.4610	−1.25726	−0.628631	+	0.777704i	$0.716384\pi$
$588$	0	0
$589$	10.8833	0.448438
$590$	0	0
$591$	−25.6386	−1.05463
$592$	0	0
$593$	−10.0720	−0.413607	−0.206804	−	0.978382i	$-0.566306\pi$
$593$	−10.0720	−0.413607	−0.206804	+	0.978382i	$0.566306\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.88327	0.118005
$598$	0	0
$599$	−31.7220	−1.29612	−0.648062	−	0.761587i	$-0.724420\pi$
$599$	−31.7220	−1.29612	−0.648062	+	0.761587i	$0.724420\pi$
$600$	0	0
$601$	29.7533	1.21366	0.606832	−	0.794830i	$-0.292440\pi$
$601$	29.7533	1.21366	0.606832	+	0.794830i	$0.292440\pi$
$602$	0	0
$603$	5.05971	0.206047
$604$	0	0
$605$	23.2950	0.947077
$606$	0	0
$607$	17.6247	0.715366	0.357683	−	0.933843i	$-0.383567\pi$
$607$	17.6247	0.715366	0.357683	+	0.933843i	$0.383567\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−52.2273	−2.11289
$612$	0	0
$613$	12.7068	0.513224	0.256612	−	0.966514i	$-0.417394\pi$
$613$	12.7068	0.513224	0.256612	+	0.966514i	$0.417394\pi$
$614$	0	0
$615$	17.7763	0.716808
$616$	0	0
$617$	−40.7386	−1.64008	−0.820038	−	0.572309i	$-0.806048\pi$
$617$	−40.7386	−1.64008	−0.820038	+	0.572309i	$0.806048\pi$
$618$	0	0
$619$	46.9407	1.88671	0.943354	−	0.331788i	$-0.107652\pi$
$619$	46.9407	1.88671	0.943354	+	0.331788i	$0.107652\pi$
$620$	0	0
$621$	4.29857	0.172496
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−22.5968	−0.903871
$626$	0	0
$627$	−0.304363	−0.0121551
$628$	0	0
$629$	−6.31102	−0.251637
$630$	0	0
$631$	27.4804	1.09398	0.546989	−	0.837140i	$-0.315774\pi$
$631$	27.4804	1.09398	0.546989	+	0.837140i	$0.315774\pi$
$632$	0	0
$633$	−24.5401	−0.975383
$634$	0	0
$635$	−36.5886	−1.45198
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.35828	0.0537328
$640$	0	0
$641$	−48.6580	−1.92188	−0.960938	−	0.276764i	$-0.910738\pi$
$641$	−48.6580	−1.92188	−0.960938	+	0.276764i	$0.910738\pi$
$642$	0	0
$643$	−17.6856	−0.697452	−0.348726	−	0.937225i	$-0.613386\pi$
$643$	−17.6856	−0.697452	−0.348726	+	0.937225i	$0.613386\pi$
$644$	0	0
$645$	−15.1555	−0.596748
$646$	0	0
$647$	−4.04032	−0.158841	−0.0794206	−	0.996841i	$-0.525307\pi$
$647$	−4.04032	−0.158841	−0.0794206	+	0.996841i	$0.525307\pi$
$648$	0	0
$649$	−1.92909	−0.0757233
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.9690	1.21191	0.605956	−	0.795498i	$-0.292791\pi$
$653$	30.9690	1.21191	0.605956	+	0.795498i	$0.292791\pi$
$654$	0	0
$655$	34.4983	1.34796
$656$	0	0
$657$	−2.85585	−0.111417
$658$	0	0
$659$	−4.81815	−0.187689	−0.0938443	−	0.995587i	$-0.529916\pi$
$659$	−4.81815	−0.187689	−0.0938443	+	0.995587i	$0.529916\pi$
$660$	0	0
$661$	−4.38324	−0.170488	−0.0852442	−	0.996360i	$-0.527167\pi$
$661$	−4.38324	−0.170488	−0.0852442	+	0.996360i	$0.527167\pi$
$662$	0	0
$663$	−7.17759	−0.278755
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.81505	0.0702792
$668$	0	0
$669$	20.5080	0.792885
$670$	0	0
$671$	1.74261	0.0672727
$672$	0	0
$673$	−8.67471	−0.334386	−0.167193	−	0.985924i	$-0.553470\pi$
$673$	−8.67471	−0.334386	−0.167193	+	0.985924i	$0.553470\pi$
$674$	0	0
$675$	0.441637	0.0169986
$676$	0	0
$677$	1.04103	0.0400099	0.0200049	−	0.999800i	$-0.493632\pi$
$677$	1.04103	0.0400099	0.0200049	+	0.999800i	$0.493632\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	13.6972	0.524877
$682$	0	0
$683$	−35.7220	−1.36686	−0.683432	−	0.730014i	$-0.739513\pi$
$683$	−35.7220	−1.36686	−0.683432	+	0.730014i	$0.739513\pi$
$684$	0	0
$685$	−46.4898	−1.77628
$686$	0	0
$687$	−16.4175	−0.626368
$688$	0	0
$689$	−22.1421	−0.843548
$690$	0	0
$691$	16.9221	0.643745	0.321873	−	0.946783i	$-0.395688\pi$
$691$	16.9221	0.643745	0.321873	+	0.946783i	$0.395688\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32.6953	1.24020
$696$	0	0
$697$	9.28882	0.351839
$698$	0	0
$699$	10.8915	0.411953
$700$	0	0
$701$	−22.2052	−0.838678	−0.419339	−	0.907830i	$-0.637738\pi$
$701$	−22.2052	−0.838678	−0.419339	+	0.907830i	$0.637738\pi$
$702$	0	0
$703$	−5.76655	−0.217490
$704$	0	0
$705$	−17.3319	−0.652759
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−7.35435	−0.276198	−0.138099	−	0.990418i	$-0.544099\pi$
$709$	−7.35435	−0.276198	−0.138099	+	0.990418i	$0.544099\pi$
$710$	0	0
$711$	7.69564	0.288609
$712$	0	0
$713$	−45.8926	−1.71869
$714$	0	0
$715$	−4.10118	−0.153376
$716$	0	0
$717$	4.29857	0.160533
$718$	0	0
$719$	27.5495	1.02742	0.513711	−	0.857963i	$-0.328270\pi$
$719$	27.5495	1.02742	0.513711	+	0.857963i	$0.328270\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.12808	−0.265096
$724$	0	0
$725$	0.186480	0.00692568
$726$	0	0
$727$	14.9321	0.553801	0.276901	−	0.960899i	$-0.410693\pi$
$727$	14.9321	0.553801	0.276901	+	0.960899i	$0.410693\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.91937	−0.292908
$732$	0	0
$733$	42.4684	1.56860	0.784302	−	0.620379i	$-0.213021\pi$
$733$	42.4684	1.56860	0.784302	+	0.620379i	$0.213021\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.51069	0.0556470
$738$	0	0
$739$	7.83329	0.288152	0.144076	−	0.989567i	$-0.453979\pi$
$739$	7.83329	0.288152	0.144076	+	0.989567i	$0.453979\pi$
$740$	0	0
$741$	−6.55836	−0.240927
$742$	0	0
$743$	30.4064	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$743$	30.4064	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$744$	0	0
$745$	8.26700	0.302879
$746$	0	0
$747$	−6.03878	−0.220948
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	0	0
$753$	−18.9690	−0.691270
$754$	0	0
$755$	10.1528	0.369499
$756$	0	0
$757$	−19.5289	−0.709791	−0.354895	−	0.934906i	$-0.615484\pi$
$757$	−19.5289	−0.709791	−0.354895	+	0.934906i	$0.615484\pi$
$758$	0	0
$759$	1.28344	0.0465858
$760$	0	0
$761$	34.0746	1.23520	0.617602	−	0.786491i	$-0.288104\pi$
$761$	34.0746	1.23520	0.617602	+	0.786491i	$0.288104\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.38193	−0.0861188
$766$	0	0
$767$	−41.5677	−1.50092
$768$	0	0
$769$	−28.3523	−1.02241	−0.511206	−	0.859458i	$-0.670801\pi$
$769$	−28.3523	−1.02241	−0.511206	+	0.859458i	$0.670801\pi$
$770$	0	0
$771$	−20.4842	−0.737720
$772$	0	0
$773$	1.96615	0.0707175	0.0353588	−	0.999375i	$-0.488743\pi$
$773$	1.96615	0.0707175	0.0353588	+	0.999375i	$0.488743\pi$
$774$	0	0
$775$	−4.71503	−0.169369
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.48745	0.304094
$780$	0	0
$781$	0.405546	0.0145116
$782$	0	0
$783$	0.422246	0.0150898
$784$	0	0
$785$	−24.6557	−0.880000
$786$	0	0
$787$	2.39165	0.0852531	0.0426266	−	0.999091i	$-0.486427\pi$
$787$	2.39165	0.0852531	0.0426266	+	0.999091i	$0.486427\pi$
$788$	0	0
$789$	23.5553	0.838590
$790$	0	0
$791$	0	0
$792$	0	0
$793$	37.5495	1.33342
$794$	0	0
$795$	−7.34801	−0.260607
$796$	0	0
$797$	−30.2674	−1.07213	−0.536064	−	0.844177i	$-0.680089\pi$
$797$	−30.2674	−1.07213	−0.536064	+	0.844177i	$0.680089\pi$
$798$	0	0
$799$	−9.05664	−0.320401
$800$	0	0
$801$	15.3696	0.543060
$802$	0	0
$803$	−0.852680	−0.0300904
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.9086	−0.454405
$808$	0	0
$809$	28.7692	1.01147	0.505736	−	0.862688i	$-0.331221\pi$
$809$	28.7692	1.01147	0.505736	+	0.862688i	$0.331221\pi$
$810$	0	0
$811$	−51.6270	−1.81287	−0.906435	−	0.422345i	$-0.861207\pi$
$811$	−51.6270	−1.81287	−0.906435	+	0.422345i	$0.861207\pi$
$812$	0	0
$813$	−19.8705	−0.696890
$814$	0	0
$815$	25.4301	0.890776
$816$	0	0
$817$	−7.23614	−0.253161
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.1885	1.54219	0.771094	−	0.636721i	$-0.219710\pi$
$821$	44.1885	1.54219	0.771094	+	0.636721i	$0.219710\pi$
$822$	0	0
$823$	33.8151	1.17872	0.589359	−	0.807871i	$-0.299380\pi$
$823$	33.8151	1.17872	0.589359	+	0.807871i	$0.299380\pi$
$824$	0	0
$825$	0.131861	0.00459081
$826$	0	0
$827$	−48.0748	−1.67173	−0.835863	−	0.548938i	$-0.815032\pi$
$827$	−48.0748	−1.67173	−0.835863	+	0.548938i	$0.815032\pi$
$828$	0	0
$829$	−9.26203	−0.321684	−0.160842	−	0.986980i	$-0.551421\pi$
$829$	−9.26203	−0.321684	−0.160842	+	0.986980i	$0.551421\pi$
$830$	0	0
$831$	−25.9884	−0.901528
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−32.6092	−1.12849
$836$	0	0
$837$	−10.6762	−0.369025
$838$	0	0
$839$	18.1567	0.626838	0.313419	−	0.949615i	$-0.398526\pi$
$839$	18.1567	0.626838	0.313419	+	0.949615i	$0.398526\pi$
$840$	0	0
$841$	−28.8217	−0.993852
$842$	0	0
$843$	19.6553	0.676965
$844$	0	0
$845$	−60.6163	−2.08526
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19.0582	0.654075
$850$	0	0
$851$	24.3164	0.833555
$852$	0	0
$853$	−40.7652	−1.39577	−0.697886	−	0.716208i	$-0.745876\pi$
$853$	−40.7652	−1.39577	−0.697886	+	0.716208i	$0.745876\pi$
$854$	0	0
$855$	−2.17643	−0.0744325
$856$	0	0
$857$	12.0559	0.411823	0.205911	−	0.978571i	$-0.433984\pi$
$857$	12.0559	0.411823	0.205911	+	0.978571i	$0.433984\pi$
$858$	0	0
$859$	4.25247	0.145092	0.0725461	−	0.997365i	$-0.476888\pi$
$859$	4.25247	0.145092	0.0725461	+	0.997365i	$0.476888\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3280	−0.385610	−0.192805	−	0.981237i	$-0.561758\pi$
$863$	−11.3280	−0.385610	−0.192805	+	0.981237i	$0.561758\pi$
$864$	0	0
$865$	8.55529	0.290889
$866$	0	0
$867$	15.7553	0.535080
$868$	0	0
$869$	2.29771	0.0779444
$870$	0	0
$871$	32.5522	1.10299
$872$	0	0
$873$	−0.512705	−0.0173524
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.6871	1.07000	0.534999	−	0.844853i	$-0.320312\pi$
$877$	31.6871	1.07000	0.534999	+	0.844853i	$0.320312\pi$
$878$	0	0
$879$	−7.84890	−0.264737
$880$	0	0
$881$	−24.2690	−0.817643	−0.408821	−	0.912614i	$-0.634060\pi$
$881$	−24.2690	−0.817643	−0.408821	+	0.912614i	$0.634060\pi$
$882$	0	0
$883$	47.1439	1.58652	0.793260	−	0.608883i	$-0.208382\pi$
$883$	47.1439	1.58652	0.793260	+	0.608883i	$0.208382\pi$
$884$	0	0
$885$	−13.7945	−0.463697
$886$	0	0
$887$	−14.6647	−0.492391	−0.246196	−	0.969220i	$-0.579181\pi$
$887$	−14.6647	−0.492391	−0.246196	+	0.969220i	$0.579181\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.298573	0.0100026
$892$	0	0
$893$	−8.27530	−0.276922
$894$	0	0
$895$	−44.3010	−1.48082
$896$	0	0
$897$	27.6553	0.923384
$898$	0	0
$899$	−4.50800	−0.150350
$900$	0	0
$901$	−3.83963	−0.127917
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−46.3827	−1.54181
$906$	0	0
$907$	−3.99149	−0.132535	−0.0662676	−	0.997802i	$-0.521109\pi$
$907$	−3.99149	−0.132535	−0.0662676	+	0.997802i	$0.521109\pi$
$908$	0	0
$909$	−4.96346	−0.164628
$910$	0	0
$911$	−18.9260	−0.627046	−0.313523	−	0.949581i	$-0.601509\pi$
$911$	−18.9260	−0.627046	−0.313523	+	0.949581i	$0.601509\pi$
$912$	0	0
$913$	−1.80302	−0.0596711
$914$	0	0
$915$	12.4610	0.411949
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−46.4898	−1.53356	−0.766778	−	0.641912i	$-0.778141\pi$
$919$	−46.4898	−1.53356	−0.766778	+	0.641912i	$0.778141\pi$
$920$	0	0
$921$	26.5301	0.874196
$922$	0	0
$923$	8.73865	0.287636
$924$	0	0
$925$	2.49828	0.0821429
$926$	0	0
$927$	9.48195	0.311428
$928$	0	0
$929$	28.7221	0.942343	0.471171	−	0.882042i	$-0.343831\pi$
$929$	28.7221	0.942343	0.471171	+	0.882042i	$0.343831\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−0.726608	−0.0237881
$934$	0	0
$935$	−0.711179	−0.0232581
$936$	0	0
$937$	−21.2560	−0.694404	−0.347202	−	0.937790i	$-0.612868\pi$
$937$	−21.2560	−0.694404	−0.347202	+	0.937790i	$0.612868\pi$
$938$	0	0
$939$	−15.1259	−0.493616
$940$	0	0
$941$	19.4100	0.632747	0.316373	−	0.948635i	$-0.397535\pi$
$941$	19.4100	0.632747	0.316373	+	0.948635i	$0.397535\pi$
$942$	0	0
$943$	−35.7899	−1.16548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.6123	−0.897279	−0.448639	−	0.893713i	$-0.648091\pi$
$947$	−27.6123	−0.897279	−0.448639	+	0.893713i	$0.648091\pi$
$948$	0	0
$949$	−18.3734	−0.596426
$950$	0	0
$951$	19.9496	0.646911
$952$	0	0
$953$	14.5080	0.469960	0.234980	−	0.972000i	$-0.424497\pi$
$953$	14.5080	0.469960	0.234980	+	0.972000i	$0.424497\pi$
$954$	0	0
$955$	−44.1354	−1.42819
$956$	0	0
$957$	0.126071	0.00407530
$958$	0	0
$959$	0	0
$960$	0	0
$961$	82.9822	2.67685
$962$	0	0
$963$	−16.8387	−0.542620
$964$	0	0
$965$	−33.3062	−1.07216
$966$	0	0
$967$	56.8449	1.82801	0.914005	−	0.405703i	$-0.132973\pi$
$967$	56.8449	1.82801	0.914005	+	0.405703i	$0.132973\pi$
$968$	0	0
$969$	−1.13727	−0.0365345
$970$	0	0
$971$	10.1970	0.327237	0.163618	−	0.986524i	$-0.447683\pi$
$971$	10.1970	0.327237	0.163618	+	0.986524i	$0.447683\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.84132	0.0909951
$976$	0	0
$977$	−8.53628	−0.273100	−0.136550	−	0.990633i	$-0.543601\pi$
$977$	−8.53628	−0.273100	−0.136550	+	0.990633i	$0.543601\pi$
$978$	0	0
$979$	4.58896	0.146664
$980$	0	0
$981$	3.09849	0.0989272
$982$	0	0
$983$	38.1970	1.21829	0.609147	−	0.793057i	$-0.291512\pi$
$983$	38.1970	1.21829	0.609147	+	0.793057i	$0.291512\pi$
$984$	0	0
$985$	−54.7393	−1.74414
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.5134	0.970269
$990$	0	0
$991$	−45.8810	−1.45746	−0.728730	−	0.684802i	$-0.759889\pi$
$991$	−45.8810	−1.45746	−0.728730	+	0.684802i	$0.759889\pi$
$992$	0	0
$993$	−17.1943	−0.545644
$994$	0	0
$995$	6.15588	0.195155
$996$	0	0
$997$	−35.0749	−1.11083	−0.555417	−	0.831572i	$-0.687441\pi$
$997$	−35.0749	−1.11083	−0.555417	+	0.831572i	$0.687441\pi$
$998$	0	0
$999$	5.65685	0.178975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.a.bx.1.1	yes	4
4.3	odd	2		4704.2.a.bz.1.1	yes	4
7.6	odd	2		4704.2.a.by.1.4	yes	4
8.3	odd	2		9408.2.a.ek.1.4		4
8.5	even	2		9408.2.a.em.1.4		4
28.27	even	2		4704.2.a.bw.1.4	✓	4
56.13	odd	2		9408.2.a.el.1.1		4
56.27	even	2		9408.2.a.en.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bw.1.4	✓	4	28.27	even	2
4704.2.a.bx.1.1	yes	4	1.1	even	1	trivial
4704.2.a.by.1.4	yes	4	7.6	odd	2
4704.2.a.bz.1.1	yes	4	4.3	odd	2
9408.2.a.ek.1.4		4	8.3	odd	2
9408.2.a.el.1.1		4	56.13	odd	2
9408.2.a.em.1.4		4	8.5	even	2
9408.2.a.en.1.1		4	56.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.13503 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.13503 q^{5} +1.00000 q^{9} +0.298573 q^{11} +6.43361 q^{13} +2.13503 q^{15} +1.11564 q^{17} +1.01939 q^{19} -4.29857 q^{23} -0.441637 q^{25} -1.00000 q^{27} -0.422246 q^{29} +10.6762 q^{31} -0.298573 q^{33} -5.65685 q^{37} -6.43361 q^{39} +8.32600 q^{41} -7.09849 q^{43} -2.13503 q^{45} -8.11788 q^{47} -1.11564 q^{51} -3.44164 q^{53} -0.637463 q^{55} -1.01939 q^{57} -6.46103 q^{59} +5.83646 q^{61} -13.7360 q^{65} +5.05971 q^{67} +4.29857 q^{69} +1.35828 q^{71} -2.85585 q^{73} +0.441637 q^{75} +7.69564 q^{79} +1.00000 q^{81} -6.03878 q^{83} -2.38193 q^{85} +0.422246 q^{87} +15.3696 q^{89} -10.6762 q^{93} -2.17643 q^{95} -0.512705 q^{97} +0.298573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{9} - 4 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{17} - 8 q^{19} - 12 q^{23} + 12 q^{25} - 4 q^{27} + 8 q^{31} + 4 q^{33} - 8 q^{39} + 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 4 q^{51} + 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} + 8 q^{67} + 12 q^{69} - 12 q^{71} + 8 q^{73} - 12 q^{75} - 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} - 24 q^{95} + 40 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^17 - 8 * q^19 - 12 * q^23 + 12 * q^25 - 4 * q^27 + 8 * q^31 + 4 * q^33 - 8 * q^39 + 20 * q^41 + 8 * q^43 + 4 * q^45 + 16 * q^47 - 4 * q^51 + 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 + 8 * q^67 + 12 * q^69 - 12 * q^71 + 8 * q^73 - 12 * q^75 - 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 - 24 * q^95 + 40 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more