LMFDB - Embedded newform 3872.2.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3872,2,Mod(1,3872)] 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("3872.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2,0,0,0,18,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.9180756626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 55 $ x^4 - 15*x^2 + 55
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.52626$ of defining polynomial
Character	$\chi$	$=$	3872.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52626 q^{3} -2.85410 q^{5} +2.52626 q^{7} +3.38197 q^{9} +4.85410 q^{13} -7.21019 q^{15} -5.61803 q^{17} +2.52626 q^{19} +6.38197 q^{21} +8.17513 q^{23} +3.14590 q^{25} +0.964944 q^{27} -1.38197 q^{29} +6.61382 q^{31} -7.21019 q^{35} -0.618034 q^{37} +12.2627 q^{39} -8.61803 q^{41} +3.12262 q^{43} -9.65248 q^{45} +0.596368 q^{47} -0.618034 q^{49} -14.1926 q^{51} +0.381966 q^{53} +6.38197 q^{57} +10.7014 q^{59} +3.61803 q^{61} +8.54371 q^{63} -13.8541 q^{65} +13.2276 q^{67} +20.6525 q^{69} +11.6663 q^{71} -4.32624 q^{73} +7.94734 q^{75} +3.49120 q^{79} -7.70820 q^{81} -11.6663 q^{83} +16.0344 q^{85} -3.49120 q^{87} -7.23607 q^{89} +12.2627 q^{91} +16.7082 q^{93} -7.21019 q^{95} +17.7984 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 18 q^{9} + 6 q^{13} - 18 q^{17} + 30 q^{21} + 26 q^{25} - 10 q^{29} + 2 q^{37} - 30 q^{41} + 24 q^{45} + 2 q^{49} + 6 q^{53} + 30 q^{57} + 10 q^{61} - 42 q^{65} + 20 q^{69} + 14 q^{73} - 4 q^{81}+ \cdots + 22 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 18 * q^9 + 6 * q^13 - 18 * q^17 + 30 * q^21 + 26 * q^25 - 10 * q^29 + 2 * q^37 - 30 * q^41 + 24 * q^45 + 2 * q^49 + 6 * q^53 + 30 * q^57 + 10 * q^61 - 42 * q^65 + 20 * q^69 + 14 * q^73 - 4 * q^81 + 6 * q^85 - 20 * q^89 + 40 * q^93 + 22 * q^97

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
$3$	2.52626	1.45853	0.729267	−	0.684229i	$-0.239861\pi$
$3$	2.52626	1.45853	0.729267	+	0.684229i	$0.239861\pi$
$4$	0	0
$5$	−2.85410	−1.27639	−0.638197	−	0.769873i	$-0.720319\pi$
$5$	−2.85410	−1.27639	−0.638197	+	0.769873i	$0.720319\pi$
$6$	0	0
$7$	2.52626	0.954835	0.477417	−	0.878677i	$-0.341573\pi$
$7$	2.52626	0.954835	0.477417	+	0.878677i	$0.341573\pi$
$8$	0	0
$9$	3.38197	1.12732
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.85410	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$13$	4.85410	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$14$	0	0
$15$	−7.21019	−1.86166
$16$	0	0
$17$	−5.61803	−1.36257	−0.681287	−	0.732017i	$-0.738579\pi$
$17$	−5.61803	−1.36257	−0.681287	+	0.732017i	$0.738579\pi$
$18$	0	0
$19$	2.52626	0.579563	0.289781	−	0.957093i	$-0.406417\pi$
$19$	2.52626	0.579563	0.289781	+	0.957093i	$0.406417\pi$
$20$	0	0
$21$	6.38197	1.39266
$22$	0	0
$23$	8.17513	1.70463	0.852317	−	0.523026i	$-0.175197\pi$
$23$	8.17513	1.70463	0.852317	+	0.523026i	$0.175197\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	0	0
$27$	0.964944	0.185703
$28$	0	0
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	6.61382	1.18788	0.593939	−	0.804510i	$-0.297572\pi$
$31$	6.61382	1.18788	0.593939	+	0.804510i	$0.297572\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.21019	−1.21874
$36$	0	0
$37$	−0.618034	−0.101604	−0.0508021	−	0.998709i	$-0.516178\pi$
$37$	−0.618034	−0.101604	−0.0508021	+	0.998709i	$0.516178\pi$
$38$	0	0
$39$	12.2627	1.96360
$40$	0	0
$41$	−8.61803	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$41$	−8.61803	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$42$	0	0
$43$	3.12262	0.476196	0.238098	−	0.971241i	$-0.423476\pi$
$43$	3.12262	0.476196	0.238098	+	0.971241i	$0.423476\pi$
$44$	0	0
$45$	−9.65248	−1.43891
$46$	0	0
$47$	0.596368	0.0869892	0.0434946	−	0.999054i	$-0.486151\pi$
$47$	0.596368	0.0869892	0.0434946	+	0.999054i	$0.486151\pi$
$48$	0	0
$49$	−0.618034	−0.0882906
$50$	0	0
$51$	−14.1926	−1.98736
$52$	0	0
$53$	0.381966	0.0524671	0.0262335	−	0.999656i	$-0.491649\pi$
$53$	0.381966	0.0524671	0.0262335	+	0.999656i	$0.491649\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.38197	0.845312
$58$	0	0
$59$	10.7014	1.39320	0.696601	−	0.717459i	$-0.254695\pi$
$59$	10.7014	1.39320	0.696601	+	0.717459i	$0.254695\pi$
$60$	0	0
$61$	3.61803	0.463242	0.231621	−	0.972806i	$-0.425597\pi$
$61$	3.61803	0.463242	0.231621	+	0.972806i	$0.425597\pi$
$62$	0	0
$63$	8.54371	1.07641
$64$	0	0
$65$	−13.8541	−1.71839
$66$	0	0
$67$	13.2276	1.61601	0.808007	−	0.589173i	$-0.200546\pi$
$67$	13.2276	1.61601	0.808007	+	0.589173i	$0.200546\pi$
$68$	0	0
$69$	20.6525	2.48627
$70$	0	0
$71$	11.6663	1.38454	0.692270	−	0.721639i	$-0.256611\pi$
$71$	11.6663	1.38454	0.692270	+	0.721639i	$0.256611\pi$
$72$	0	0
$73$	−4.32624	−0.506348	−0.253174	−	0.967421i	$-0.581474\pi$
$73$	−4.32624	−0.506348	−0.253174	+	0.967421i	$0.581474\pi$
$74$	0	0
$75$	7.94734	0.917680
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.49120	0.392791	0.196395	−	0.980525i	$-0.437076\pi$
$79$	3.49120	0.392791	0.196395	+	0.980525i	$0.437076\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−11.6663	−1.28055	−0.640273	−	0.768147i	$-0.721179\pi$
$83$	−11.6663	−1.28055	−0.640273	+	0.768147i	$0.721179\pi$
$84$	0	0
$85$	16.0344	1.73918
$86$	0	0
$87$	−3.49120	−0.374296
$88$	0	0
$89$	−7.23607	−0.767022	−0.383511	−	0.923536i	$-0.625285\pi$
$89$	−7.23607	−0.767022	−0.383511	+	0.923536i	$0.625285\pi$
$90$	0	0
$91$	12.2627	1.28548
$92$	0	0
$93$	16.7082	1.73256
$94$	0	0
$95$	−7.21019	−0.739750
$96$	0	0
$97$	17.7984	1.80715	0.903576	−	0.428429i	$-0.140933\pi$
$97$	17.7984	1.80715	0.903576	+	0.428429i	$0.140933\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.38197	0.834037	0.417018	−	0.908898i	$-0.363075\pi$
$101$	8.38197	0.834037	0.417018	+	0.908898i	$0.363075\pi$
$102$	0	0
$103$	−5.64888	−0.556601	−0.278300	−	0.960494i	$-0.589771\pi$
$103$	−5.64888	−0.556601	−0.278300	+	0.960494i	$0.589771\pi$
$104$	0	0
$105$	−18.2148	−1.77758
$106$	0	0
$107$	16.9466	1.63829	0.819147	−	0.573584i	$-0.194447\pi$
$107$	16.9466	1.63829	0.819147	+	0.573584i	$0.194447\pi$
$108$	0	0
$109$	−10.4721	−1.00305	−0.501524	−	0.865144i	$-0.667227\pi$
$109$	−10.4721	−1.00305	−0.501524	+	0.865144i	$0.667227\pi$
$110$	0	0
$111$	−1.56131	−0.148193
$112$	0	0
$113$	−8.14590	−0.766302	−0.383151	−	0.923686i	$-0.625161\pi$
$113$	−8.14590	−0.766302	−0.383151	+	0.923686i	$0.625161\pi$
$114$	0	0
$115$	−23.3327	−2.17578
$116$	0	0
$117$	16.4164	1.51770
$118$	0	0
$119$	−14.1926	−1.30103
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−21.7714	−1.96306
$124$	0	0
$125$	5.29180	0.473313
$126$	0	0
$127$	−4.68393	−0.415632	−0.207816	−	0.978168i	$-0.566636\pi$
$127$	−4.68393	−0.415632	−0.207816	+	0.978168i	$0.566636\pi$
$128$	0	0
$129$	7.88854	0.694548
$130$	0	0
$131$	−1.92989	−0.168615	−0.0843075	−	0.996440i	$-0.526868\pi$
$131$	−1.92989	−0.168615	−0.0843075	+	0.996440i	$0.526868\pi$
$132$	0	0
$133$	6.38197	0.553387
$134$	0	0
$135$	−2.75405	−0.237031
$136$	0	0
$137$	9.90983	0.846654	0.423327	−	0.905977i	$-0.360862\pi$
$137$	9.90983	0.846654	0.423327	+	0.905977i	$0.360862\pi$
$138$	0	0
$139$	13.8240	1.17254	0.586269	−	0.810117i	$-0.300596\pi$
$139$	13.8240	1.17254	0.586269	+	0.810117i	$0.300596\pi$
$140$	0	0
$141$	1.50658	0.126877
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.94427	0.327554
$146$	0	0
$147$	−1.56131	−0.128775
$148$	0	0
$149$	−9.32624	−0.764035	−0.382018	−	0.924155i	$-0.624771\pi$
$149$	−9.32624	−0.764035	−0.382018	+	0.924155i	$0.624771\pi$
$150$	0	0
$151$	−10.7014	−0.870867	−0.435433	−	0.900221i	$-0.643405\pi$
$151$	−10.7014	−0.870867	−0.435433	+	0.900221i	$0.643405\pi$
$152$	0	0
$153$	−19.0000	−1.53606
$154$	0	0
$155$	−18.8765	−1.51620
$156$	0	0
$157$	19.5623	1.56124	0.780621	−	0.625005i	$-0.214903\pi$
$157$	19.5623	1.56124	0.780621	+	0.625005i	$0.214903\pi$
$158$	0	0
$159$	0.964944	0.0765250
$160$	0	0
$161$	20.6525	1.62764
$162$	0	0
$163$	−11.6663	−0.913778	−0.456889	−	0.889524i	$-0.651036\pi$
$163$	−11.6663	−0.913778	−0.456889	+	0.889524i	$0.651036\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.61382	0.511793	0.255896	−	0.966704i	$-0.417629\pi$
$167$	6.61382	0.511793	0.255896	+	0.966704i	$0.417629\pi$
$168$	0	0
$169$	10.5623	0.812485
$170$	0	0
$171$	8.54371	0.653354
$172$	0	0
$173$	−17.7426	−1.34895	−0.674474	−	0.738298i	$-0.735630\pi$
$173$	−17.7426	−1.34895	−0.674474	+	0.738298i	$0.735630\pi$
$174$	0	0
$175$	7.94734	0.600763
$176$	0	0
$177$	27.0344	2.03203
$178$	0	0
$179$	2.52626	0.188821	0.0944106	−	0.995533i	$-0.469903\pi$
$179$	2.52626	0.188821	0.0944106	+	0.995533i	$0.469903\pi$
$180$	0	0
$181$	19.0344	1.41482	0.707409	−	0.706804i	$-0.249864\pi$
$181$	19.0344	1.41482	0.707409	+	0.706804i	$0.249864\pi$
$182$	0	0
$183$	9.14008	0.675654
$184$	0	0
$185$	1.76393	0.129687
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.43769	0.177316
$190$	0	0
$191$	0.596368	0.0431517	0.0215758	−	0.999767i	$-0.493132\pi$
$191$	0.596368	0.0431517	0.0215758	+	0.999767i	$0.493132\pi$
$192$	0	0
$193$	−11.3262	−0.815280	−0.407640	−	0.913143i	$-0.633648\pi$
$193$	−11.3262	−0.815280	−0.407640	+	0.913143i	$0.633648\pi$
$194$	0	0
$195$	−34.9990	−2.50633
$196$	0	0
$197$	12.6525	0.901452	0.450726	−	0.892662i	$-0.351165\pi$
$197$	12.6525	0.901452	0.450726	+	0.892662i	$0.351165\pi$
$198$	0	0
$199$	3.12262	0.221357	0.110678	−	0.993856i	$-0.464698\pi$
$199$	3.12262	0.221357	0.110678	+	0.993856i	$0.464698\pi$
$200$	0	0
$201$	33.4164	2.35701
$202$	0	0
$203$	−3.49120	−0.245034
$204$	0	0
$205$	24.5967	1.71791
$206$	0	0
$207$	27.6480	1.92167
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.49120	0.240344	0.120172	−	0.992753i	$-0.461655\pi$
$211$	3.49120	0.240344	0.120172	+	0.992753i	$0.461655\pi$
$212$	0	0
$213$	29.4721	2.01940
$214$	0	0
$215$	−8.91229	−0.607813
$216$	0	0
$217$	16.7082	1.13423
$218$	0	0
$219$	−10.9292	−0.738526
$220$	0	0
$221$	−27.2705	−1.83441
$222$	0	0
$223$	−17.6838	−1.18419	−0.592097	−	0.805867i	$-0.701700\pi$
$223$	−17.6838	−1.18419	−0.592097	+	0.805867i	$0.701700\pi$
$224$	0	0
$225$	10.6393	0.709288
$226$	0	0
$227$	0.596368	0.0395823	0.0197912	−	0.999804i	$-0.493700\pi$
$227$	0.596368	0.0395823	0.0197912	+	0.999804i	$0.493700\pi$
$228$	0	0
$229$	−2.56231	−0.169322	−0.0846610	−	0.996410i	$-0.526981\pi$
$229$	−2.56231	−0.169322	−0.0846610	+	0.996410i	$0.526981\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.3262	1.26610	0.633052	−	0.774109i	$-0.281802\pi$
$233$	19.3262	1.26610	0.633052	+	0.774109i	$0.281802\pi$
$234$	0	0
$235$	−1.70210	−0.111032
$236$	0	0
$237$	8.81966	0.572898
$238$	0	0
$239$	−10.7014	−0.692215	−0.346108	−	0.938195i	$-0.612497\pi$
$239$	−10.7014	−0.692215	−0.346108	+	0.938195i	$0.612497\pi$
$240$	0	0
$241$	−3.41641	−0.220070	−0.110035	−	0.993928i	$-0.535096\pi$
$241$	−3.41641	−0.220070	−0.110035	+	0.993928i	$0.535096\pi$
$242$	0	0
$243$	−22.3677	−1.43489
$244$	0	0
$245$	1.76393	0.112693
$246$	0	0
$247$	12.2627	0.780257
$248$	0	0
$249$	−29.4721	−1.86772
$250$	0	0
$251$	−0.368576	−0.0232643	−0.0116321	−	0.999932i	$-0.503703\pi$
$251$	−0.368576	−0.0232643	−0.0116321	+	0.999932i	$0.503703\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	40.5071	2.53665
$256$	0	0
$257$	−7.27051	−0.453522	−0.226761	−	0.973950i	$-0.572814\pi$
$257$	−7.27051	−0.453522	−0.226761	+	0.973950i	$0.572814\pi$
$258$	0	0
$259$	−1.56131	−0.0970152
$260$	0	0
$261$	−4.67376	−0.289299
$262$	0	0
$263$	6.98240	0.430553	0.215277	−	0.976553i	$-0.430935\pi$
$263$	6.98240	0.430553	0.215277	+	0.976553i	$0.430935\pi$
$264$	0	0
$265$	−1.09017	−0.0669686
$266$	0	0
$267$	−18.2802	−1.11873
$268$	0	0
$269$	−4.09017	−0.249382	−0.124691	−	0.992196i	$-0.539794\pi$
$269$	−4.09017	−0.249382	−0.124691	+	0.992196i	$0.539794\pi$
$270$	0	0
$271$	27.0517	1.64327	0.821636	−	0.570013i	$-0.193062\pi$
$271$	27.0517	1.64327	0.821636	+	0.570013i	$0.193062\pi$
$272$	0	0
$273$	30.9787	1.87492
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−16.2705	−0.977600	−0.488800	−	0.872396i	$-0.662565\pi$
$277$	−16.2705	−0.977600	−0.488800	+	0.872396i	$0.662565\pi$
$278$	0	0
$279$	22.3677	1.33912
$280$	0	0
$281$	−24.2705	−1.44786	−0.723929	−	0.689875i	$-0.757666\pi$
$281$	−24.2705	−1.44786	−0.723929	+	0.689875i	$0.757666\pi$
$282$	0	0
$283$	14.5612	0.865571	0.432786	−	0.901497i	$-0.357531\pi$
$283$	14.5612	0.865571	0.432786	+	0.901497i	$0.357531\pi$
$284$	0	0
$285$	−18.2148	−1.07895
$286$	0	0
$287$	−21.7714	−1.28512
$288$	0	0
$289$	14.5623	0.856606
$290$	0	0
$291$	44.9632	2.63579
$292$	0	0
$293$	−0.798374	−0.0466415	−0.0233207	−	0.999728i	$-0.507424\pi$
$293$	−0.798374	−0.0466415	−0.0233207	+	0.999728i	$0.507424\pi$
$294$	0	0
$295$	−30.5429	−1.77827
$296$	0	0
$297$	0	0
$298$	0	0
$299$	39.6829	2.29492
$300$	0	0
$301$	7.88854	0.454688
$302$	0	0
$303$	21.1750	1.21647
$304$	0	0
$305$	−10.3262	−0.591279
$306$	0	0
$307$	−1.92989	−0.110144	−0.0550722	−	0.998482i	$-0.517539\pi$
$307$	−1.92989	−0.110144	−0.0550722	+	0.998482i	$0.517539\pi$
$308$	0	0
$309$	−14.2705	−0.811821
$310$	0	0
$311$	−7.57877	−0.429752	−0.214876	−	0.976641i	$-0.568935\pi$
$311$	−7.57877	−0.429752	−0.214876	+	0.976641i	$0.568935\pi$
$312$	0	0
$313$	4.09017	0.231190	0.115595	−	0.993296i	$-0.463123\pi$
$313$	4.09017	0.231190	0.115595	+	0.993296i	$0.463123\pi$
$314$	0	0
$315$	−24.3846	−1.37392
$316$	0	0
$317$	−5.03444	−0.282762	−0.141381	−	0.989955i	$-0.545154\pi$
$317$	−5.03444	−0.282762	−0.141381	+	0.989955i	$0.545154\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	42.8115	2.38951
$322$	0	0
$323$	−14.1926	−0.789697
$324$	0	0
$325$	15.2705	0.847055
$326$	0	0
$327$	−26.4553	−1.46298
$328$	0	0
$329$	1.50658	0.0830603
$330$	0	0
$331$	−6.24525	−0.343270	−0.171635	−	0.985161i	$-0.554905\pi$
$331$	−6.24525	−0.343270	−0.171635	+	0.985161i	$0.554905\pi$
$332$	0	0
$333$	−2.09017	−0.114541
$334$	0	0
$335$	−37.7530	−2.06267
$336$	0	0
$337$	−10.0344	−0.546611	−0.273305	−	0.961927i	$-0.588117\pi$
$337$	−10.0344	−0.546611	−0.273305	+	0.961927i	$0.588117\pi$
$338$	0	0
$339$	−20.5786	−1.11768
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.2451	−1.03914
$344$	0	0
$345$	−58.9443	−3.17345
$346$	0	0
$347$	−17.9116	−0.961544	−0.480772	−	0.876846i	$-0.659644\pi$
$347$	−17.9116	−0.961544	−0.480772	+	0.876846i	$0.659644\pi$
$348$	0	0
$349$	−4.14590	−0.221925	−0.110962	−	0.993825i	$-0.535393\pi$
$349$	−4.14590	−0.221925	−0.110962	+	0.993825i	$0.535393\pi$
$350$	0	0
$351$	4.68393	0.250010
$352$	0	0
$353$	−7.70820	−0.410266	−0.205133	−	0.978734i	$-0.565763\pi$
$353$	−7.70820	−0.410266	−0.205133	+	0.978734i	$0.565763\pi$
$354$	0	0
$355$	−33.2969	−1.76722
$356$	0	0
$357$	−35.8541	−1.89760
$358$	0	0
$359$	0.596368	0.0314751	0.0157375	−	0.999876i	$-0.494990\pi$
$359$	0.596368	0.0314751	0.0157375	+	0.999876i	$0.494990\pi$
$360$	0	0
$361$	−12.6180	−0.664107
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.3475	0.646299
$366$	0	0
$367$	3.71899	0.194130	0.0970649	−	0.995278i	$-0.469055\pi$
$367$	3.71899	0.194130	0.0970649	+	0.995278i	$0.469055\pi$
$368$	0	0
$369$	−29.1459	−1.51727
$370$	0	0
$371$	0.964944	0.0500974
$372$	0	0
$373$	−9.12461	−0.472454	−0.236227	−	0.971698i	$-0.575911\pi$
$373$	−9.12461	−0.472454	−0.236227	+	0.971698i	$0.575911\pi$
$374$	0	0
$375$	13.3684	0.690343
$376$	0	0
$377$	−6.70820	−0.345490
$378$	0	0
$379$	−21.0342	−1.08045	−0.540227	−	0.841519i	$-0.681662\pi$
$379$	−21.0342	−1.08045	−0.540227	+	0.841519i	$0.681662\pi$
$380$	0	0
$381$	−11.8328	−0.606214
$382$	0	0
$383$	−32.3320	−1.65209	−0.826043	−	0.563607i	$-0.809413\pi$
$383$	−32.3320	−1.65209	−0.826043	+	0.563607i	$0.809413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.5606	0.536826
$388$	0	0
$389$	−3.27051	−0.165821	−0.0829107	−	0.996557i	$-0.526422\pi$
$389$	−3.27051	−0.165821	−0.0829107	+	0.996557i	$0.526422\pi$
$390$	0	0
$391$	−45.9282	−2.32269
$392$	0	0
$393$	−4.87539	−0.245931
$394$	0	0
$395$	−9.96424	−0.501355
$396$	0	0
$397$	16.2918	0.817662	0.408831	−	0.912610i	$-0.365937\pi$
$397$	16.2918	0.817662	0.408831	+	0.912610i	$0.365937\pi$
$398$	0	0
$399$	16.1225	0.807133
$400$	0	0
$401$	−9.90983	−0.494873	−0.247437	−	0.968904i	$-0.579588\pi$
$401$	−9.90983	−0.494873	−0.247437	+	0.968904i	$0.579588\pi$
$402$	0	0
$403$	32.1042	1.59922
$404$	0	0
$405$	22.0000	1.09319
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−38.2705	−1.89235	−0.946177	−	0.323648i	$-0.895091\pi$
$409$	−38.2705	−1.89235	−0.946177	+	0.323648i	$0.895091\pi$
$410$	0	0
$411$	25.0348	1.23487
$412$	0	0
$413$	27.0344	1.33028
$414$	0	0
$415$	33.2969	1.63448
$416$	0	0
$417$	34.9230	1.71019
$418$	0	0
$419$	8.17513	0.399381	0.199691	−	0.979859i	$-0.436006\pi$
$419$	8.17513	0.399381	0.199691	+	0.979859i	$0.436006\pi$
$420$	0	0
$421$	−23.0902	−1.12535	−0.562673	−	0.826680i	$-0.690227\pi$
$421$	−23.0902	−1.12535	−0.562673	+	0.826680i	$0.690227\pi$
$422$	0	0
$423$	2.01690	0.0980649
$424$	0	0
$425$	−17.6738	−0.857303
$426$	0	0
$427$	9.14008	0.442319
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.0166	−1.34951	−0.674756	−	0.738041i	$-0.735751\pi$
$431$	−28.0166	−1.34951	−0.674756	+	0.738041i	$0.735751\pi$
$432$	0	0
$433$	−12.5066	−0.601028	−0.300514	−	0.953777i	$-0.597158\pi$
$433$	−12.5066	−0.601028	−0.300514	+	0.953777i	$0.597158\pi$
$434$	0	0
$435$	9.96424	0.477749
$436$	0	0
$437$	20.6525	0.987942
$438$	0	0
$439$	−28.3852	−1.35475	−0.677375	−	0.735638i	$-0.736883\pi$
$439$	−28.3852	−1.35475	−0.677375	+	0.735638i	$0.736883\pi$
$440$	0	0
$441$	−2.09017	−0.0995319
$442$	0	0
$443$	−4.45614	−0.211718	−0.105859	−	0.994381i	$-0.533759\pi$
$443$	−4.45614	−0.211718	−0.105859	+	0.994381i	$0.533759\pi$
$444$	0	0
$445$	20.6525	0.979021
$446$	0	0
$447$	−23.5605	−1.11437
$448$	0	0
$449$	−21.0344	−0.992677	−0.496338	−	0.868129i	$-0.665322\pi$
$449$	−21.0344	−0.992677	−0.496338	+	0.868129i	$0.665322\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−27.0344	−1.27019
$454$	0	0
$455$	−34.9990	−1.64078
$456$	0	0
$457$	−4.96556	−0.232279	−0.116140	−	0.993233i	$-0.537052\pi$
$457$	−4.96556	−0.232279	−0.116140	+	0.993233i	$0.537052\pi$
$458$	0	0
$459$	−5.42109	−0.253035
$460$	0	0
$461$	38.1803	1.77824	0.889118	−	0.457678i	$-0.151319\pi$
$461$	38.1803	1.77824	0.889118	+	0.457678i	$0.151319\pi$
$462$	0	0
$463$	−33.4377	−1.55398	−0.776991	−	0.629512i	$-0.783255\pi$
$463$	−33.4377	−1.55398	−0.776991	+	0.629512i	$0.783255\pi$
$464$	0	0
$465$	−47.6869	−2.21143
$466$	0	0
$467$	10.4736	0.484660	0.242330	−	0.970194i	$-0.422088\pi$
$467$	10.4736	0.484660	0.242330	+	0.970194i	$0.422088\pi$
$468$	0	0
$469$	33.4164	1.54303
$470$	0	0
$471$	49.4194	2.27712
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.94734	0.364649
$476$	0	0
$477$	1.29180	0.0591473
$478$	0	0
$479$	24.8940	1.13743	0.568717	−	0.822533i	$-0.307440\pi$
$479$	24.8940	1.13743	0.568717	+	0.822533i	$0.307440\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	52.1734	2.37397
$484$	0	0
$485$	−50.7984	−2.30664
$486$	0	0
$487$	42.2092	1.91268	0.956340	−	0.292255i	$-0.0944056\pi$
$487$	42.2092	1.91268	0.956340	+	0.292255i	$0.0944056\pi$
$488$	0	0
$489$	−29.4721	−1.33278
$490$	0	0
$491$	36.4195	1.64359	0.821795	−	0.569783i	$-0.192973\pi$
$491$	36.4195	1.64359	0.821795	+	0.569783i	$0.192973\pi$
$492$	0	0
$493$	7.76393	0.349670
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.4721	1.32201
$498$	0	0
$499$	−10.7014	−0.479060	−0.239530	−	0.970889i	$-0.576993\pi$
$499$	−10.7014	−0.479060	−0.239530	+	0.970889i	$0.576993\pi$
$500$	0	0
$501$	16.7082	0.746468
$502$	0	0
$503$	−13.8240	−0.616382	−0.308191	−	0.951324i	$-0.599724\pi$
$503$	−13.8240	−0.616382	−0.308191	+	0.951324i	$0.599724\pi$
$504$	0	0
$505$	−23.9230	−1.06456
$506$	0	0
$507$	26.6831	1.18504
$508$	0	0
$509$	20.9787	0.929865	0.464933	−	0.885346i	$-0.346079\pi$
$509$	20.9787	0.929865	0.464933	+	0.885346i	$0.346079\pi$
$510$	0	0
$511$	−10.9292	−0.483479
$512$	0	0
$513$	2.43769	0.107627
$514$	0	0
$515$	16.1225	0.710441
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−44.8225	−1.96749
$520$	0	0
$521$	21.7426	0.952563	0.476281	−	0.879293i	$-0.341984\pi$
$521$	21.7426	0.952563	0.476281	+	0.879293i	$0.341984\pi$
$522$	0	0
$523$	29.2093	1.27724	0.638618	−	0.769524i	$-0.279507\pi$
$523$	29.2093	1.27724	0.638618	+	0.769524i	$0.279507\pi$
$524$	0	0
$525$	20.0770	0.876233
$526$	0	0
$527$	−37.1567	−1.61857
$528$	0	0
$529$	43.8328	1.90577
$530$	0	0
$531$	36.1917	1.57059
$532$	0	0
$533$	−41.8328	−1.81198
$534$	0	0
$535$	−48.3674	−2.09111
$536$	0	0
$537$	6.38197	0.275402
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.9787	0.772965	0.386483	−	0.922297i	$-0.373690\pi$
$541$	17.9787	0.772965	0.386483	+	0.922297i	$0.373690\pi$
$542$	0	0
$543$	48.0859	2.06356
$544$	0	0
$545$	29.8885	1.28028
$546$	0	0
$547$	−11.8941	−0.508556	−0.254278	−	0.967131i	$-0.581838\pi$
$547$	−11.8941	−0.508556	−0.254278	+	0.967131i	$0.581838\pi$
$548$	0	0
$549$	12.2361	0.522223
$550$	0	0
$551$	−3.49120	−0.148730
$552$	0	0
$553$	8.81966	0.375050
$554$	0	0
$555$	4.45614	0.189153
$556$	0	0
$557$	−10.0344	−0.425173	−0.212586	−	0.977142i	$-0.568189\pi$
$557$	−10.0344	−0.425173	−0.212586	+	0.977142i	$0.568189\pi$
$558$	0	0
$559$	15.1575	0.641095
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.4370	−1.78851	−0.894253	−	0.447562i	$-0.852292\pi$
$563$	−42.4370	−1.78851	−0.894253	+	0.447562i	$0.852292\pi$
$564$	0	0
$565$	23.2492	0.978102
$566$	0	0
$567$	−19.4729	−0.817785
$568$	0	0
$569$	−10.7984	−0.452691	−0.226346	−	0.974047i	$-0.572678\pi$
$569$	−10.7984	−0.452691	−0.226346	+	0.974047i	$0.572678\pi$
$570$	0	0
$571$	−4.31536	−0.180592	−0.0902961	−	0.995915i	$-0.528781\pi$
$571$	−4.31536	−0.180592	−0.0902961	+	0.995915i	$0.528781\pi$
$572$	0	0
$573$	1.50658	0.0629382
$574$	0	0
$575$	25.7181	1.07252
$576$	0	0
$577$	28.3820	1.18156	0.590778	−	0.806834i	$-0.298821\pi$
$577$	28.3820	1.18156	0.590778	+	0.806834i	$0.298821\pi$
$578$	0	0
$579$	−28.6130	−1.18911
$580$	0	0
$581$	−29.4721	−1.22271
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−46.8541	−1.93718
$586$	0	0
$587$	−35.2268	−1.45397	−0.726983	−	0.686656i	$-0.759078\pi$
$587$	−35.2268	−1.45397	−0.726983	+	0.686656i	$0.759078\pi$
$588$	0	0
$589$	16.7082	0.688450
$590$	0	0
$591$	31.9634	1.31480
$592$	0	0
$593$	−2.29180	−0.0941128	−0.0470564	−	0.998892i	$-0.514984\pi$
$593$	−2.29180	−0.0941128	−0.0470564	+	0.998892i	$0.514984\pi$
$594$	0	0
$595$	40.5071	1.66063
$596$	0	0
$597$	7.88854	0.322857
$598$	0	0
$599$	−0.368576	−0.0150596	−0.00752980	−	0.999972i	$-0.502397\pi$
$599$	−0.368576	−0.0150596	−0.00752980	+	0.999972i	$0.502397\pi$
$600$	0	0
$601$	−19.2705	−0.786060	−0.393030	−	0.919526i	$-0.628573\pi$
$601$	−19.2705	−0.786060	−0.393030	+	0.919526i	$0.628573\pi$
$602$	0	0
$603$	44.7354	1.82177
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.0518	−0.570345	−0.285173	−	0.958476i	$-0.592051\pi$
$607$	−14.0518	−0.570345	−0.285173	+	0.958476i	$0.592051\pi$
$608$	0	0
$609$	−8.81966	−0.357391
$610$	0	0
$611$	2.89483	0.117112
$612$	0	0
$613$	−36.3262	−1.46720	−0.733601	−	0.679580i	$-0.762162\pi$
$613$	−36.3262	−1.46720	−0.733601	+	0.679580i	$0.762162\pi$
$614$	0	0
$615$	62.1377	2.50563
$616$	0	0
$617$	17.1246	0.689411	0.344705	−	0.938711i	$-0.387979\pi$
$617$	17.1246	0.689411	0.344705	+	0.938711i	$0.387979\pi$
$618$	0	0
$619$	11.8941	0.478065	0.239033	−	0.971012i	$-0.423170\pi$
$619$	11.8941	0.478065	0.239033	+	0.971012i	$0.423170\pi$
$620$	0	0
$621$	7.88854	0.316556
$622$	0	0
$623$	−18.2802	−0.732379
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.47214	0.138443
$630$	0	0
$631$	−11.8941	−0.473498	−0.236749	−	0.971571i	$-0.576082\pi$
$631$	−11.8941	−0.473498	−0.236749	+	0.971571i	$0.576082\pi$
$632$	0	0
$633$	8.81966	0.350550
$634$	0	0
$635$	13.3684	0.530510
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	39.4551	1.56082
$640$	0	0
$641$	−33.7426	−1.33275	−0.666377	−	0.745615i	$-0.732156\pi$
$641$	−33.7426	−1.33275	−0.666377	+	0.745615i	$0.732156\pi$
$642$	0	0
$643$	9.73645	0.383968	0.191984	−	0.981398i	$-0.438508\pi$
$643$	9.73645	0.383968	0.191984	+	0.981398i	$0.438508\pi$
$644$	0	0
$645$	−22.5147	−0.886516
$646$	0	0
$647$	−19.1043	−0.751068	−0.375534	−	0.926809i	$-0.622541\pi$
$647$	−19.1043	−0.751068	−0.375534	+	0.926809i	$0.622541\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	42.2092	1.65431
$652$	0	0
$653$	−10.7984	−0.422573	−0.211287	−	0.977424i	$-0.567765\pi$
$653$	−10.7984	−0.422573	−0.211287	+	0.977424i	$0.567765\pi$
$654$	0	0
$655$	5.50810	0.215219
$656$	0	0
$657$	−14.6312	−0.570817
$658$	0	0
$659$	−1.92989	−0.0751777	−0.0375889	−	0.999293i	$-0.511968\pi$
$659$	−1.92989	−0.0751777	−0.0375889	+	0.999293i	$0.511968\pi$
$660$	0	0
$661$	25.5279	0.992919	0.496459	−	0.868060i	$-0.334633\pi$
$661$	25.5279	0.992919	0.496459	+	0.868060i	$0.334633\pi$
$662$	0	0
$663$	−68.8923	−2.67555
$664$	0	0
$665$	−18.2148	−0.706339
$666$	0	0
$667$	−11.2978	−0.437451
$668$	0	0
$669$	−44.6738	−1.72719
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−39.9787	−1.54107	−0.770533	−	0.637400i	$-0.780010\pi$
$673$	−39.9787	−1.54107	−0.770533	+	0.637400i	$0.780010\pi$
$674$	0	0
$675$	3.03561	0.116841
$676$	0	0
$677$	−40.5623	−1.55894	−0.779468	−	0.626442i	$-0.784511\pi$
$677$	−40.5623	−1.55894	−0.779468	+	0.626442i	$0.784511\pi$
$678$	0	0
$679$	44.9632	1.72553
$680$	0	0
$681$	1.50658	0.0577322
$682$	0	0
$683$	42.8056	1.63791	0.818955	−	0.573858i	$-0.194554\pi$
$683$	42.8056	1.63791	0.818955	+	0.573858i	$0.194554\pi$
$684$	0	0
$685$	−28.2837	−1.08066
$686$	0	0
$687$	−6.47304	−0.246962
$688$	0	0
$689$	1.85410	0.0706357
$690$	0	0
$691$	−29.2093	−1.11118	−0.555588	−	0.831458i	$-0.687507\pi$
$691$	−29.2093	−1.11118	−0.555588	+	0.831458i	$0.687507\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−39.4551	−1.49662
$696$	0	0
$697$	48.4164	1.83390
$698$	0	0
$699$	48.8230	1.84666
$700$	0	0
$701$	−12.0344	−0.454535	−0.227267	−	0.973832i	$-0.572979\pi$
$701$	−12.0344	−0.454535	−0.227267	+	0.973832i	$0.572979\pi$
$702$	0	0
$703$	−1.56131	−0.0588860
$704$	0	0
$705$	−4.29993	−0.161945
$706$	0	0
$707$	21.1750	0.796367
$708$	0	0
$709$	52.1591	1.95887	0.979437	−	0.201749i	$-0.0646626\pi$
$709$	52.1591	1.95887	0.979437	+	0.201749i	$0.0646626\pi$
$710$	0	0
$711$	11.8071	0.442801
$712$	0	0
$713$	54.0689	2.02490
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−27.0344	−1.00962
$718$	0	0
$719$	16.4911	0.615012	0.307506	−	0.951546i	$-0.400506\pi$
$719$	16.4911	0.615012	0.307506	+	0.951546i	$0.400506\pi$
$720$	0	0
$721$	−14.2705	−0.531462
$722$	0	0
$723$	−8.63072	−0.320980
$724$	0	0
$725$	−4.34752	−0.161463
$726$	0	0
$727$	−6.24525	−0.231623	−0.115812	−	0.993271i	$-0.536947\pi$
$727$	−6.24525	−0.231623	−0.115812	+	0.993271i	$0.536947\pi$
$728$	0	0
$729$	−33.3820	−1.23637
$730$	0	0
$731$	−17.5430	−0.648851
$732$	0	0
$733$	16.3262	0.603023	0.301512	−	0.953462i	$-0.402509\pi$
$733$	16.3262	0.603023	0.301512	+	0.953462i	$0.402509\pi$
$734$	0	0
$735$	4.45614	0.164367
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6.61382	−0.243293	−0.121647	−	0.992573i	$-0.538817\pi$
$739$	−6.61382	−0.243293	−0.121647	+	0.992573i	$0.538817\pi$
$740$	0	0
$741$	30.9787	1.13803
$742$	0	0
$743$	10.9292	0.400953	0.200476	−	0.979699i	$-0.435751\pi$
$743$	10.9292	0.400953	0.200476	+	0.979699i	$0.435751\pi$
$744$	0	0
$745$	26.6180	0.975209
$746$	0	0
$747$	−39.4551	−1.44359
$748$	0	0
$749$	42.8115	1.56430
$750$	0	0
$751$	25.1218	0.916706	0.458353	−	0.888770i	$-0.348439\pi$
$751$	25.1218	0.916706	0.458353	+	0.888770i	$0.348439\pi$
$752$	0	0
$753$	−0.931116	−0.0339318
$754$	0	0
$755$	30.5429	1.11157
$756$	0	0
$757$	−38.4508	−1.39752	−0.698760	−	0.715356i	$-0.746264\pi$
$757$	−38.4508	−1.39752	−0.698760	+	0.715356i	$0.746264\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.8541	−0.828461	−0.414230	−	0.910172i	$-0.635949\pi$
$761$	−22.8541	−0.828461	−0.414230	+	0.910172i	$0.635949\pi$
$762$	0	0
$763$	−26.4553	−0.957746
$764$	0	0
$765$	54.2279	1.96062
$766$	0	0
$767$	51.9456	1.87565
$768$	0	0
$769$	−16.1803	−0.583478	−0.291739	−	0.956498i	$-0.594234\pi$
$769$	−16.1803	−0.583478	−0.291739	+	0.956498i	$0.594234\pi$
$770$	0	0
$771$	−18.3672	−0.661477
$772$	0	0
$773$	1.85410	0.0666874	0.0333437	−	0.999444i	$-0.489384\pi$
$773$	1.85410	0.0666874	0.0333437	+	0.999444i	$0.489384\pi$
$774$	0	0
$775$	20.8064	0.747388
$776$	0	0
$777$	−3.94427	−0.141500
$778$	0	0
$779$	−21.7714	−0.780040
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.33352	−0.0476561
$784$	0	0
$785$	−55.8328	−1.99276
$786$	0	0
$787$	−52.5420	−1.87292	−0.936460	−	0.350774i	$-0.885918\pi$
$787$	−52.5420	−1.87292	−0.936460	+	0.350774i	$0.885918\pi$
$788$	0	0
$789$	17.6393	0.627976
$790$	0	0
$791$	−20.5786	−0.731691
$792$	0	0
$793$	17.5623	0.623656
$794$	0	0
$795$	−2.75405	−0.0976760
$796$	0	0
$797$	23.5066	0.832646	0.416323	−	0.909217i	$-0.363319\pi$
$797$	23.5066	0.832646	0.416323	+	0.909217i	$0.363319\pi$
$798$	0	0
$799$	−3.35042	−0.118529
$800$	0	0
$801$	−24.4721	−0.864680
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−58.9443	−2.07751
$806$	0	0
$807$	−10.3328	−0.363732
$808$	0	0
$809$	9.14590	0.321553	0.160776	−	0.986991i	$-0.448600\pi$
$809$	9.14590	0.321553	0.160776	+	0.986991i	$0.448600\pi$
$810$	0	0
$811$	20.8064	0.730612	0.365306	−	0.930888i	$-0.380964\pi$
$811$	20.8064	0.730612	0.365306	+	0.930888i	$0.380964\pi$
$812$	0	0
$813$	68.3394	2.39677
$814$	0	0
$815$	33.2969	1.16634
$816$	0	0
$817$	7.88854	0.275985
$818$	0	0
$819$	41.4720	1.44915
$820$	0	0
$821$	−11.3820	−0.397233	−0.198617	−	0.980077i	$-0.563645\pi$
$821$	−11.3820	−0.397233	−0.198617	+	0.980077i	$0.563645\pi$
$822$	0	0
$823$	−24.8940	−0.867750	−0.433875	−	0.900973i	$-0.642854\pi$
$823$	−24.8940	−0.867750	−0.433875	+	0.900973i	$0.642854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.3144	1.36709	0.683547	−	0.729906i	$-0.260436\pi$
$827$	39.3144	1.36709	0.683547	+	0.729906i	$0.260436\pi$
$828$	0	0
$829$	−33.1591	−1.15166	−0.575831	−	0.817569i	$-0.695321\pi$
$829$	−33.1591	−1.15166	−0.575831	+	0.817569i	$0.695321\pi$
$830$	0	0
$831$	−41.1035	−1.42586
$832$	0	0
$833$	3.47214	0.120302
$834$	0	0
$835$	−18.8765	−0.653249
$836$	0	0
$837$	6.38197	0.220593
$838$	0	0
$839$	30.1743	1.04173	0.520866	−	0.853638i	$-0.325609\pi$
$839$	30.1743	1.04173	0.520866	+	0.853638i	$0.325609\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	0	0
$843$	−61.3135	−2.11175
$844$	0	0
$845$	−30.1459	−1.03705
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36.7852	1.26247
$850$	0	0
$851$	−5.05251	−0.173198
$852$	0	0
$853$	29.9787	1.02645	0.513226	−	0.858254i	$-0.328450\pi$
$853$	29.9787	1.02645	0.513226	+	0.858254i	$0.328450\pi$
$854$	0	0
$855$	−24.3846	−0.833936
$856$	0	0
$857$	−19.7082	−0.673219	−0.336610	−	0.941644i	$-0.609280\pi$
$857$	−19.7082	−0.673219	−0.336610	+	0.941644i	$0.609280\pi$
$858$	0	0
$859$	−26.4553	−0.902643	−0.451321	−	0.892361i	$-0.649047\pi$
$859$	−26.4553	−0.902643	−0.451321	+	0.892361i	$0.649047\pi$
$860$	0	0
$861$	−55.0000	−1.87439
$862$	0	0
$863$	−9.28086	−0.315924	−0.157962	−	0.987445i	$-0.550492\pi$
$863$	−9.28086	−0.315924	−0.157962	+	0.987445i	$0.550492\pi$
$864$	0	0
$865$	50.6393	1.72179
$866$	0	0
$867$	36.7881	1.24939
$868$	0	0
$869$	0	0
$870$	0	0
$871$	64.2083	2.17562
$872$	0	0
$873$	60.1935	2.03724
$874$	0	0
$875$	13.3684	0.451935
$876$	0	0
$877$	23.4508	0.791879	0.395939	−	0.918277i	$-0.370419\pi$
$877$	23.4508	0.791879	0.395939	+	0.918277i	$0.370419\pi$
$878$	0	0
$879$	−2.01690	−0.0680282
$880$	0	0
$881$	−20.6525	−0.695800	−0.347900	−	0.937532i	$-0.613105\pi$
$881$	−20.6525	−0.695800	−0.347900	+	0.937532i	$0.613105\pi$
$882$	0	0
$883$	−49.1916	−1.65543	−0.827714	−	0.561150i	$-0.810359\pi$
$883$	−49.1916	−1.65543	−0.827714	+	0.561150i	$0.810359\pi$
$884$	0	0
$885$	−77.1591	−2.59367
$886$	0	0
$887$	34.4896	1.15805	0.579024	−	0.815310i	$-0.303434\pi$
$887$	34.4896	1.15805	0.579024	+	0.815310i	$0.303434\pi$
$888$	0	0
$889$	−11.8328	−0.396860
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.50658	0.0504157
$894$	0	0
$895$	−7.21019	−0.241010
$896$	0	0
$897$	100.249	3.34722
$898$	0	0
$899$	−9.14008	−0.304839
$900$	0	0
$901$	−2.14590	−0.0714902
$902$	0	0
$903$	19.9285	0.663178
$904$	0	0
$905$	−54.3262	−1.80587
$906$	0	0
$907$	15.5261	0.515536	0.257768	−	0.966207i	$-0.417013\pi$
$907$	15.5261	0.515536	0.257768	+	0.966207i	$0.417013\pi$
$908$	0	0
$909$	28.3475	0.940228
$910$	0	0
$911$	47.4895	1.57340	0.786699	−	0.617337i	$-0.211789\pi$
$911$	47.4895	1.57340	0.786699	+	0.617337i	$0.211789\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−26.0867	−0.862400
$916$	0	0
$917$	−4.87539	−0.161000
$918$	0	0
$919$	30.4021	1.00287	0.501436	−	0.865195i	$-0.332805\pi$
$919$	30.4021	1.00287	0.501436	+	0.865195i	$0.332805\pi$
$920$	0	0
$921$	−4.87539	−0.160650
$922$	0	0
$923$	56.6296	1.86398
$924$	0	0
$925$	−1.94427	−0.0639273
$926$	0	0
$927$	−19.1043	−0.627468
$928$	0	0
$929$	−34.0902	−1.11846	−0.559231	−	0.829012i	$-0.688904\pi$
$929$	−34.0902	−1.11846	−0.559231	+	0.829012i	$0.688904\pi$
$930$	0	0
$931$	−1.56131	−0.0511699
$932$	0	0
$933$	−19.1459	−0.626809
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.8541	−0.746611	−0.373305	−	0.927708i	$-0.621776\pi$
$937$	−22.8541	−0.746611	−0.373305	+	0.927708i	$0.621776\pi$
$938$	0	0
$939$	10.3328	0.337199
$940$	0	0
$941$	19.9098	0.649042	0.324521	−	0.945879i	$-0.394797\pi$
$941$	19.9098	0.649042	0.324521	+	0.945879i	$0.394797\pi$
$942$	0	0
$943$	−70.4536	−2.29428
$944$	0	0
$945$	−6.95743	−0.226325
$946$	0	0
$947$	26.4553	0.859681	0.429841	−	0.902905i	$-0.358570\pi$
$947$	26.4553	0.859681	0.429841	+	0.902905i	$0.358570\pi$
$948$	0	0
$949$	−21.0000	−0.681689
$950$	0	0
$951$	−12.7183	−0.412419
$952$	0	0
$953$	−8.43769	−0.273324	−0.136662	−	0.990618i	$-0.543637\pi$
$953$	−8.43769	−0.273324	−0.136662	+	0.990618i	$0.543637\pi$
$954$	0	0
$955$	−1.70210	−0.0550785
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.0348	0.808415
$960$	0	0
$961$	12.7426	0.411053
$962$	0	0
$963$	57.3129	1.84688
$964$	0	0
$965$	32.3262	1.04062
$966$	0	0
$967$	39.6829	1.27612	0.638059	−	0.769988i	$-0.279738\pi$
$967$	39.6829	1.27612	0.638059	+	0.769988i	$0.279738\pi$
$968$	0	0
$969$	−35.8541	−1.15180
$970$	0	0
$971$	20.0693	0.644053	0.322027	−	0.946731i	$-0.395636\pi$
$971$	20.0693	0.644053	0.322027	+	0.946731i	$0.395636\pi$
$972$	0	0
$973$	34.9230	1.11958
$974$	0	0
$975$	38.5772	1.23546
$976$	0	0
$977$	46.3394	1.48253	0.741264	−	0.671213i	$-0.234227\pi$
$977$	46.3394	1.48253	0.741264	+	0.671213i	$0.234227\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−35.4164	−1.13076
$982$	0	0
$983$	15.0167	0.478960	0.239480	−	0.970901i	$-0.423023\pi$
$983$	15.0167	0.478960	0.239480	+	0.970901i	$0.423023\pi$
$984$	0	0
$985$	−36.1115	−1.15061
$986$	0	0
$987$	3.80600	0.121146
$988$	0	0
$989$	25.5279	0.811739
$990$	0	0
$991$	−12.4905	−0.396774	−0.198387	−	0.980124i	$-0.563570\pi$
$991$	−12.4905	−0.396774	−0.198387	+	0.980124i	$0.563570\pi$
$992$	0	0
$993$	−15.7771	−0.500671
$994$	0	0
$995$	−8.91229	−0.282538
$996$	0	0
$997$	−54.0344	−1.71129	−0.855644	−	0.517565i	$-0.826839\pi$
$997$	−54.0344	−1.71129	−0.855644	+	0.517565i	$0.826839\pi$
$998$	0	0
$999$	−0.596368	−0.0188682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.a.bk.1.3		4
4.3	odd	2	inner	3872.2.a.bk.1.2		4
8.3	odd	2		7744.2.a.dl.1.3		4
8.5	even	2		7744.2.a.dl.1.2		4
11.2	odd	10		352.2.m.d.257.2	yes	8
11.6	odd	10		352.2.m.d.289.2	yes	8
11.10	odd	2		3872.2.a.bj.1.3		4
44.35	even	10		352.2.m.d.257.1	✓	8
44.39	even	10		352.2.m.d.289.1	yes	8
44.43	even	2		3872.2.a.bj.1.2		4
88.13	odd	10		704.2.m.j.257.1		8
88.21	odd	2		7744.2.a.dm.1.2		4
88.35	even	10		704.2.m.j.257.2		8
88.43	even	2		7744.2.a.dm.1.3		4
88.61	odd	10		704.2.m.j.641.1		8
88.83	even	10		704.2.m.j.641.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.d.257.1	✓	8	44.35	even	10
352.2.m.d.257.2	yes	8	11.2	odd	10
352.2.m.d.289.1	yes	8	44.39	even	10
352.2.m.d.289.2	yes	8	11.6	odd	10
704.2.m.j.257.1		8	88.13	odd	10
704.2.m.j.257.2		8	88.35	even	10
704.2.m.j.641.1		8	88.61	odd	10
704.2.m.j.641.2		8	88.83	even	10
3872.2.a.bj.1.2		4	44.43	even	2
3872.2.a.bj.1.3		4	11.10	odd	2
3872.2.a.bk.1.2		4	4.3	odd	2	inner
3872.2.a.bk.1.3		4	1.1	even	1	trivial
7744.2.a.dl.1.2		4	8.5	even	2
7744.2.a.dl.1.3		4	8.3	odd	2
7744.2.a.dm.1.2		4	88.21	odd	2
7744.2.a.dm.1.3		4	88.43	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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52626 q^{3} -2.85410 q^{5} +2.52626 q^{7} +3.38197 q^{9} +4.85410 q^{13} -7.21019 q^{15} -5.61803 q^{17} +2.52626 q^{19} +6.38197 q^{21} +8.17513 q^{23} +3.14590 q^{25} +0.964944 q^{27} -1.38197 q^{29} +6.61382 q^{31} -7.21019 q^{35} -0.618034 q^{37} +12.2627 q^{39} -8.61803 q^{41} +3.12262 q^{43} -9.65248 q^{45} +0.596368 q^{47} -0.618034 q^{49} -14.1926 q^{51} +0.381966 q^{53} +6.38197 q^{57} +10.7014 q^{59} +3.61803 q^{61} +8.54371 q^{63} -13.8541 q^{65} +13.2276 q^{67} +20.6525 q^{69} +11.6663 q^{71} -4.32624 q^{73} +7.94734 q^{75} +3.49120 q^{79} -7.70820 q^{81} -11.6663 q^{83} +16.0344 q^{85} -3.49120 q^{87} -7.23607 q^{89} +12.2627 q^{91} +16.7082 q^{93} -7.21019 q^{95} +17.7984 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 18 q^{9} + 6 q^{13} - 18 q^{17} + 30 q^{21} + 26 q^{25} - 10 q^{29} + 2 q^{37} - 30 q^{41} + 24 q^{45} + 2 q^{49} + 6 q^{53} + 30 q^{57} + 10 q^{61} - 42 q^{65} + 20 q^{69} + 14 q^{73} - 4 q^{81}+ \cdots + 22 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 18 * q^9 + 6 * q^13 - 18 * q^17 + 30 * q^21 + 26 * q^25 - 10 * q^29 + 2 * q^37 - 30 * q^41 + 24 * q^45 + 2 * q^49 + 6 * q^53 + 30 * q^57 + 10 * q^61 - 42 * q^65 + 20 * q^69 + 14 * q^73 - 4 * q^81 + 6 * q^85 - 20 * q^89 + 40 * q^93 + 22 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more