LMFDB - Embedded newform 684.3.e.a.305.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(305,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("684.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 156x^{10} + 8721x^{8} + 208784x^{6} + 2024760x^{4} + 7117056x^{2} + 6533136 $ x^12 + 156x^10 + 8721x^8 + 208784x^6 + 2024760x^4 + 7117056*x^2 + 6533136
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			305.5
Root			$-2.16068i$ of defining polynomial
Character	$\chi$	$=$	684.305
Dual form			684.3.e.a.305.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16068i q^{5} -9.11429 q^{7} +O(q^{10})$	$q-2.16068i q^{5} -9.11429 q^{7} -1.10046i q^{11} +2.28570 q^{13} +19.2684i q^{17} +4.35890 q^{19} +1.28620i q^{23} +20.3315 q^{25} +37.0190i q^{29} +31.3785 q^{31} +19.6930i q^{35} +34.9511 q^{37} +43.5088i q^{41} +0.553740 q^{43} -9.35139i q^{47} +34.0703 q^{49} +75.0713i q^{53} -2.37773 q^{55} -38.1174i q^{59} -62.4816 q^{61} -4.93866i q^{65} +85.6623 q^{67} -15.9252i q^{71} +67.1233 q^{73} +10.0299i q^{77} -58.5499 q^{79} +115.564i q^{83} +41.6328 q^{85} -6.59543i q^{89} -20.8325 q^{91} -9.41818i q^{95} -14.0228 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{7}+O(q^{10}) $ 12 * q + 16 * q^7	$ 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) $ 12 * q + 16 * q^7 - 16 * q^13 - 12 * q^25 - 40 * q^31 - 32 * q^37 + 92 * q^43 - 84 * q^55 - 48 * q^61 - 88 * q^67 + 148 * q^73 - 56 * q^79 + 228 * q^85 - 8 * q^91 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.16068i	− 0.432136i	−0.976378	−	0.216068i	$-0.930677\pi$
$5$	− 2.16068i	− 0.432136i	0.976378	−	0.216068i	$-0.0693232\pi$
$6$	0	0
$7$	−9.11429	−1.30204	−0.651021	−	0.759060i	$-0.725659\pi$
$7$	−9.11429	−1.30204	−0.651021	+	0.759060i	$0.725659\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.10046i	− 0.100041i	−0.998748	−	0.0500207i	$-0.984071\pi$
$11$	− 1.10046i	− 0.100041i	0.998748	−	0.0500207i	$-0.0159287\pi$
$12$	0	0
$13$	2.28570	0.175823	0.0879116	−	0.996128i	$-0.471981\pi$
$13$	2.28570	0.175823	0.0879116	+	0.996128i	$0.471981\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	19.2684i	1.13343i	0.823913	+	0.566717i	$0.191787\pi$
$17$	19.2684i	1.13343i	−0.823913	+	0.566717i	$0.808213\pi$
$18$	0	0
$19$	4.35890	0.229416
$19$	4.35890	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.28620i	0.0559216i	0.999609	+	0.0279608i	$0.00890136\pi$
$23$	1.28620i	0.0559216i	−0.999609	+	0.0279608i	$0.991099\pi$
$24$	0	0
$25$	20.3315	0.813259
$26$	0	0
$27$	0	0
$28$	0	0
$29$	37.0190i	1.27652i	0.769821	+	0.638259i	$0.220345\pi$
$29$	37.0190i	1.27652i	−0.769821	+	0.638259i	$0.779655\pi$
$30$	0	0
$31$	31.3785	1.01221	0.506104	−	0.862472i	$-0.331085\pi$
$31$	31.3785	1.01221	0.506104	+	0.862472i	$0.331085\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	19.6930i	0.562659i
$36$	0	0
$37$	34.9511	0.944625	0.472312	−	0.881431i	$-0.343419\pi$
$37$	34.9511	0.944625	0.472312	+	0.881431i	$0.343419\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	43.5088i	1.06119i	0.847626	+	0.530595i	$0.178031\pi$
$41$	43.5088i	1.06119i	−0.847626	+	0.530595i	$0.821969\pi$
$42$	0	0
$43$	0.553740	0.0128777	0.00643884	−	0.999979i	$-0.497950\pi$
$43$	0.553740	0.0128777	0.00643884	+	0.999979i	$0.497950\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.35139i	− 0.198966i	−0.995039	−	0.0994828i	$-0.968281\pi$
$47$	− 9.35139i	− 0.198966i	0.995039	−	0.0994828i	$-0.0317188\pi$
$48$	0	0
$49$	34.0703	0.695312
$50$	0	0
$51$	0	0
$52$	0	0
$53$	75.0713i	1.41644i	0.705992	+	0.708220i	$0.250501\pi$
$53$	75.0713i	1.41644i	−0.705992	+	0.708220i	$0.749499\pi$
$54$	0	0
$55$	−2.37773	−0.0432315
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 38.1174i	− 0.646057i	−0.946389	−	0.323028i	$-0.895299\pi$
$59$	− 38.1174i	− 0.646057i	0.946389	−	0.323028i	$-0.104701\pi$
$60$	0	0
$61$	−62.4816	−1.02429	−0.512145	−	0.858899i	$-0.671149\pi$
$61$	−62.4816	−1.02429	−0.512145	+	0.858899i	$0.671149\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 4.93866i	− 0.0759794i
$66$	0	0
$67$	85.6623	1.27854	0.639271	−	0.768982i	$-0.279236\pi$
$67$	85.6623	1.27854	0.639271	+	0.768982i	$0.279236\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 15.9252i	− 0.224298i	−0.993691	−	0.112149i	$-0.964227\pi$
$71$	− 15.9252i	− 0.224298i	0.993691	−	0.112149i	$-0.0357734\pi$
$72$	0	0
$73$	67.1233	0.919497	0.459748	−	0.888049i	$-0.347940\pi$
$73$	67.1233	0.919497	0.459748	+	0.888049i	$0.347940\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.0299i	0.130258i
$78$	0	0
$79$	−58.5499	−0.741138	−0.370569	−	0.928805i	$-0.620837\pi$
$79$	−58.5499	−0.741138	−0.370569	+	0.928805i	$0.620837\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	115.564i	1.39233i	0.717881	+	0.696166i	$0.245112\pi$
$83$	115.564i	1.39233i	−0.717881	+	0.696166i	$0.754888\pi$
$84$	0	0
$85$	41.6328	0.489797
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.59543i	− 0.0741060i	−0.999313	−	0.0370530i	$-0.988203\pi$
$89$	− 6.59543i	− 0.0741060i	0.999313	−	0.0370530i	$-0.0117970\pi$
$90$	0	0
$91$	−20.8325	−0.228929
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 9.41818i	− 0.0991387i
$96$	0	0
$97$	−14.0228	−0.144565	−0.0722825	−	0.997384i	$-0.523028\pi$
$97$	−14.0228	−0.144565	−0.0722825	+	0.997384i	$0.523028\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	99.4086i	0.984243i	0.870526	+	0.492122i	$0.163779\pi$
$101$	99.4086i	0.984243i	−0.870526	+	0.492122i	$0.836221\pi$
$102$	0	0
$103$	10.8374	0.105217	0.0526087	−	0.998615i	$-0.483246\pi$
$103$	10.8374	0.105217	0.0526087	+	0.998615i	$0.483246\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	133.136i	1.24426i	0.782913	+	0.622131i	$0.213733\pi$
$107$	133.136i	1.24426i	−0.782913	+	0.622131i	$0.786267\pi$
$108$	0	0
$109$	8.56068	0.0785384	0.0392692	−	0.999229i	$-0.487497\pi$
$109$	8.56068	0.0785384	0.0392692	+	0.999229i	$0.487497\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	190.600i	1.68672i	0.537345	+	0.843362i	$0.319427\pi$
$113$	190.600i	1.68672i	−0.537345	+	0.843362i	$0.680573\pi$
$114$	0	0
$115$	2.77906	0.0241657
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 175.618i	− 1.47578i
$120$	0	0
$121$	119.789	0.989992
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 97.9467i	− 0.783574i
$126$	0	0
$127$	−66.8407	−0.526305	−0.263153	−	0.964754i	$-0.584762\pi$
$127$	−66.8407	−0.526305	−0.263153	+	0.964754i	$0.584762\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	75.5187i	0.576479i	0.957558	+	0.288239i	$0.0930699\pi$
$131$	75.5187i	0.576479i	−0.957558	+	0.288239i	$0.906930\pi$
$132$	0	0
$133$	−39.7283	−0.298709
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 169.943i	− 1.24046i	−0.784420	−	0.620230i	$-0.787039\pi$
$137$	− 169.943i	− 1.24046i	0.784420	−	0.620230i	$-0.212961\pi$
$138$	0	0
$139$	130.111	0.936052	0.468026	−	0.883715i	$-0.344965\pi$
$139$	130.111	0.936052	0.468026	+	0.883715i	$0.344965\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 2.51531i	− 0.0175896i
$144$	0	0
$145$	79.9863	0.551629
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 63.5185i	− 0.426298i	−0.977020	−	0.213149i	$-0.931628\pi$
$149$	− 63.5185i	− 0.426298i	0.977020	−	0.213149i	$-0.0683720\pi$
$150$	0	0
$151$	−5.44525	−0.0360613	−0.0180306	−	0.999837i	$-0.505740\pi$
$151$	−5.44525	−0.0360613	−0.0180306	+	0.999837i	$0.505740\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 67.7988i	− 0.437412i
$156$	0	0
$157$	−111.650	−0.711143	−0.355572	−	0.934649i	$-0.615714\pi$
$157$	−111.650	−0.711143	−0.355572	+	0.934649i	$0.615714\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 11.7228i	− 0.0728122i
$162$	0	0
$163$	−234.008	−1.43563	−0.717817	−	0.696232i	$-0.754858\pi$
$163$	−234.008	−1.43563	−0.717817	+	0.696232i	$0.754858\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 146.835i	− 0.879253i	−0.898181	−	0.439627i	$-0.855111\pi$
$167$	− 146.835i	− 0.879253i	0.898181	−	0.439627i	$-0.144889\pi$
$168$	0	0
$169$	−163.776	−0.969086
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 41.9823i	− 0.242672i	−0.992611	−	0.121336i	$-0.961282\pi$
$173$	− 41.9823i	− 0.242672i	0.992611	−	0.121336i	$-0.0387179\pi$
$174$	0	0
$175$	−185.307	−1.05890
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 234.866i	− 1.31210i	−0.754717	−	0.656051i	$-0.772226\pi$
$179$	− 234.866i	− 1.31210i	0.754717	−	0.656051i	$-0.227774\pi$
$180$	0	0
$181$	95.8804	0.529726	0.264863	−	0.964286i	$-0.414673\pi$
$181$	95.8804	0.529726	0.264863	+	0.964286i	$0.414673\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 75.5181i	− 0.408206i
$186$	0	0
$187$	21.2040	0.113390
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 164.583i	− 0.861691i	−0.902426	−	0.430845i	$-0.858215\pi$
$191$	− 164.583i	− 0.861691i	0.902426	−	0.430845i	$-0.141785\pi$
$192$	0	0
$193$	185.715	0.962255	0.481128	−	0.876651i	$-0.340227\pi$
$193$	185.715	0.962255	0.481128	+	0.876651i	$0.340227\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	159.624i	0.810275i	0.914256	+	0.405137i	$0.132776\pi$
$197$	159.624i	0.810275i	−0.914256	+	0.405137i	$0.867224\pi$
$198$	0	0
$199$	−291.623	−1.46544	−0.732721	−	0.680529i	$-0.761750\pi$
$199$	−291.623	−1.46544	−0.732721	+	0.680529i	$0.761750\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 337.402i	− 1.66208i
$204$	0	0
$205$	94.0084	0.458578
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.79678i	− 0.0229511i
$210$	0	0
$211$	−139.147	−0.659464	−0.329732	−	0.944075i	$-0.606958\pi$
$211$	−139.147	−0.659464	−0.329732	+	0.944075i	$0.606958\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.19645i	− 0.00556490i
$216$	0	0
$217$	−285.993	−1.31794
$218$	0	0
$219$	0	0
$220$	0	0
$221$	44.0417i	0.199284i
$222$	0	0
$223$	98.6258	0.442268	0.221134	−	0.975243i	$-0.429024\pi$
$223$	98.6258	0.442268	0.221134	+	0.975243i	$0.429024\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	103.162i	0.454460i	0.973841	+	0.227230i	$0.0729669\pi$
$227$	103.162i	0.454460i	−0.973841	+	0.227230i	$0.927033\pi$
$228$	0	0
$229$	−8.27807	−0.0361488	−0.0180744	−	0.999837i	$-0.505754\pi$
$229$	−8.27807	−0.0361488	−0.0180744	+	0.999837i	$0.505754\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	164.236i	0.704874i	0.935835	+	0.352437i	$0.114647\pi$
$233$	164.236i	0.704874i	−0.935835	+	0.352437i	$0.885353\pi$
$234$	0	0
$235$	−20.2053	−0.0859802
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 306.024i	− 1.28044i	−0.768194	−	0.640218i	$-0.778844\pi$
$239$	− 306.024i	− 1.28044i	0.768194	−	0.640218i	$-0.221156\pi$
$240$	0	0
$241$	159.742	0.662829	0.331414	−	0.943485i	$-0.392474\pi$
$241$	159.742	0.662829	0.331414	+	0.943485i	$0.392474\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 73.6149i	− 0.300469i
$246$	0	0
$247$	9.96314	0.0403366
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 311.540i	− 1.24120i	−0.784129	−	0.620598i	$-0.786890\pi$
$251$	− 311.540i	− 1.24120i	0.784129	−	0.620598i	$-0.213110\pi$
$252$	0	0
$253$	1.41540	0.00559448
$254$	0	0
$255$	0	0
$256$	0	0
$257$	299.046i	1.16360i	0.813331	+	0.581802i	$0.197652\pi$
$257$	299.046i	1.16360i	−0.813331	+	0.581802i	$0.802348\pi$
$258$	0	0
$259$	−318.555	−1.22994
$260$	0	0
$261$	0	0
$262$	0	0
$263$	256.742i	0.976207i	0.872786	+	0.488104i	$0.162311\pi$
$263$	256.742i	0.976207i	−0.872786	+	0.488104i	$0.837689\pi$
$264$	0	0
$265$	162.205	0.612094
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 101.739i	− 0.378212i	−0.981957	−	0.189106i	$-0.939441\pi$
$269$	− 101.739i	− 0.378212i	0.981957	−	0.189106i	$-0.0605589\pi$
$270$	0	0
$271$	−413.450	−1.52565	−0.762823	−	0.646608i	$-0.776187\pi$
$271$	−413.450	−1.52565	−0.762823	+	0.646608i	$0.776187\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 22.3739i	− 0.0813596i
$276$	0	0
$277$	−52.8385	−0.190753	−0.0953764	−	0.995441i	$-0.530405\pi$
$277$	−52.8385	−0.190753	−0.0953764	+	0.995441i	$0.530405\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	249.276i	0.887102i	0.896249	+	0.443551i	$0.146281\pi$
$281$	249.276i	0.887102i	−0.896249	+	0.443551i	$0.853719\pi$
$282$	0	0
$283$	264.801	0.935691	0.467846	−	0.883810i	$-0.345030\pi$
$283$	264.801	0.935691	0.467846	+	0.883810i	$0.345030\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 396.551i	− 1.38171i
$288$	0	0
$289$	−82.2702	−0.284672
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 140.716i	− 0.480261i	−0.970741	−	0.240130i	$-0.922810\pi$
$293$	− 140.716i	− 0.480261i	0.970741	−	0.240130i	$-0.0771902\pi$
$294$	0	0
$295$	−82.3594	−0.279184
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.93986i	0.00983231i
$300$	0	0
$301$	−5.04695	−0.0167673
$302$	0	0
$303$	0	0
$304$	0	0
$305$	135.003i	0.442632i
$306$	0	0
$307$	−224.226	−0.730379	−0.365189	−	0.930933i	$-0.618996\pi$
$307$	−224.226	−0.730379	−0.365189	+	0.930933i	$0.618996\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 190.561i	− 0.612737i	−0.951913	−	0.306369i	$-0.900886\pi$
$311$	− 190.561i	− 0.612737i	0.951913	−	0.306369i	$-0.0991140\pi$
$312$	0	0
$313$	346.633	1.10745	0.553727	−	0.832698i	$-0.313205\pi$
$313$	346.633	1.10745	0.553727	+	0.832698i	$0.313205\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 394.319i	− 1.24391i	−0.783054	−	0.621954i	$-0.786339\pi$
$317$	− 394.319i	− 1.24391i	0.783054	−	0.621954i	$-0.213661\pi$
$318$	0	0
$319$	40.7378	0.127705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	83.9889i	0.260028i
$324$	0	0
$325$	46.4716	0.142990
$326$	0	0
$327$	0	0
$328$	0	0
$329$	85.2313i	0.259062i
$330$	0	0
$331$	−207.167	−0.625882	−0.312941	−	0.949773i	$-0.601314\pi$
$331$	−207.167	−0.625882	−0.312941	+	0.949773i	$0.601314\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 185.089i	− 0.552504i
$336$	0	0
$337$	288.916	0.857319	0.428659	−	0.903466i	$-0.358986\pi$
$337$	288.916	0.857319	0.428659	+	0.903466i	$0.358986\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 34.5306i	− 0.101263i
$342$	0	0
$343$	136.074	0.396717
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 111.771i	− 0.322106i	−0.986946	−	0.161053i	$-0.948511\pi$
$347$	− 111.771i	− 0.322106i	0.986946	−	0.161053i	$-0.0514891\pi$
$348$	0	0
$349$	−364.709	−1.04501	−0.522505	−	0.852636i	$-0.675003\pi$
$349$	−364.709	−1.04501	−0.522505	+	0.852636i	$0.675003\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	227.805i	0.645341i	0.946511	+	0.322671i	$0.104581\pi$
$353$	227.805i	0.645341i	−0.946511	+	0.322671i	$0.895419\pi$
$354$	0	0
$355$	−34.4091	−0.0969272
$356$	0	0
$357$	0	0
$358$	0	0
$359$	125.832i	0.350506i	0.984523	+	0.175253i	$0.0560744\pi$
$359$	125.832i	0.350506i	−0.984523	+	0.175253i	$0.943926\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 145.032i	− 0.397347i
$366$	0	0
$367$	372.047	1.01375	0.506876	−	0.862019i	$-0.330800\pi$
$367$	372.047	1.01375	0.506876	+	0.862019i	$0.330800\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 684.221i	− 1.84426i
$372$	0	0
$373$	122.215	0.327655	0.163828	−	0.986489i	$-0.447616\pi$
$373$	122.215	0.327655	0.163828	+	0.986489i	$0.447616\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	84.6144i	0.224442i
$378$	0	0
$379$	−76.5257	−0.201915	−0.100957	−	0.994891i	$-0.532191\pi$
$379$	−76.5257	−0.201915	−0.100957	+	0.994891i	$0.532191\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	494.820i	1.29196i	0.763355	+	0.645979i	$0.223551\pi$
$383$	494.820i	1.29196i	−0.763355	+	0.645979i	$0.776449\pi$
$384$	0	0
$385$	21.6713	0.0562892
$386$	0	0
$387$	0	0
$388$	0	0
$389$	685.225i	1.76150i	0.473578	+	0.880752i	$0.342962\pi$
$389$	685.225i	1.76150i	−0.473578	+	0.880752i	$0.657038\pi$
$390$	0	0
$391$	−24.7829	−0.0633834
$392$	0	0
$393$	0	0
$394$	0	0
$395$	126.508i	0.320272i
$396$	0	0
$397$	−171.558	−0.432136	−0.216068	−	0.976378i	$-0.569323\pi$
$397$	−171.558	−0.432136	−0.216068	+	0.976378i	$0.569323\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	704.126i	1.75593i	0.478729	+	0.877963i	$0.341098\pi$
$401$	704.126i	1.75593i	−0.478729	+	0.877963i	$0.658902\pi$
$402$	0	0
$403$	71.7218	0.177970
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 38.4622i	− 0.0945016i
$408$	0	0
$409$	223.677	0.546889	0.273444	−	0.961888i	$-0.411837\pi$
$409$	223.677	0.546889	0.273444	+	0.961888i	$0.411837\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	347.413i	0.841193i
$414$	0	0
$415$	249.696	0.601676
$416$	0	0
$417$	0	0
$418$	0	0
$419$	527.562i	1.25910i	0.776961	+	0.629549i	$0.216760\pi$
$419$	527.562i	1.25910i	−0.776961	+	0.629549i	$0.783240\pi$
$420$	0	0
$421$	239.059	0.567836	0.283918	−	0.958849i	$-0.408366\pi$
$421$	239.059	0.567836	0.283918	+	0.958849i	$0.408366\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	391.754i	0.921775i
$426$	0	0
$427$	569.476	1.33367
$428$	0	0
$429$	0	0
$430$	0	0
$431$	165.894i	0.384905i	0.981306	+	0.192452i	$0.0616441\pi$
$431$	165.894i	0.384905i	−0.981306	+	0.192452i	$0.938356\pi$
$432$	0	0
$433$	266.348	0.615123	0.307561	−	0.951528i	$-0.400487\pi$
$433$	266.348	0.615123	0.307561	+	0.951528i	$0.400487\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.60640i	0.0128293i
$438$	0	0
$439$	216.344	0.492811	0.246406	−	0.969167i	$-0.420750\pi$
$439$	216.344	0.492811	0.246406	+	0.969167i	$0.420750\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	307.303i	0.693686i	0.937923	+	0.346843i	$0.112746\pi$
$443$	307.303i	0.693686i	−0.937923	+	0.346843i	$0.887254\pi$
$444$	0	0
$445$	−14.2506	−0.0320238
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 652.614i	− 1.45348i	−0.686911	−	0.726741i	$-0.741034\pi$
$449$	− 652.614i	− 1.45348i	0.686911	−	0.726741i	$-0.258966\pi$
$450$	0	0
$451$	47.8795	0.106163
$452$	0	0
$453$	0	0
$454$	0	0
$455$	45.0124i	0.0989284i
$456$	0	0
$457$	141.214	0.309002	0.154501	−	0.987993i	$-0.450623\pi$
$457$	141.214	0.309002	0.154501	+	0.987993i	$0.450623\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 43.3889i	− 0.0941190i	−0.998892	−	0.0470595i	$-0.985015\pi$
$461$	− 43.3889i	− 0.0941190i	0.998892	−	0.0470595i	$-0.0149850\pi$
$462$	0	0
$463$	640.051	1.38240	0.691199	−	0.722664i	$-0.257083\pi$
$463$	640.051	1.38240	0.691199	+	0.722664i	$0.257083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 80.7873i	− 0.172992i	−0.996252	−	0.0864961i	$-0.972433\pi$
$467$	− 80.7873i	− 0.172992i	0.996252	−	0.0864961i	$-0.0275670\pi$
$468$	0	0
$469$	−780.751	−1.66471
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.609367i	− 0.00128830i
$474$	0	0
$475$	88.6228	0.186574
$476$	0	0
$477$	0	0
$478$	0	0
$479$	585.566i	1.22248i	0.791447	+	0.611238i	$0.209328\pi$
$479$	585.566i	1.22248i	−0.791447	+	0.611238i	$0.790672\pi$
$480$	0	0
$481$	79.8878	0.166087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.2988i	0.0624717i
$486$	0	0
$487$	−314.449	−0.645686	−0.322843	−	0.946453i	$-0.604639\pi$
$487$	−314.449	−0.645686	−0.322843	+	0.946453i	$0.604639\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	422.460i	0.860408i	0.902732	+	0.430204i	$0.141558\pi$
$491$	422.460i	0.860408i	−0.902732	+	0.430204i	$0.858442\pi$
$492$	0	0
$493$	−713.297	−1.44685
$494$	0	0
$495$	0	0
$496$	0	0
$497$	145.147i	0.292045i
$498$	0	0
$499$	975.125	1.95416	0.977079	−	0.212877i	$-0.0682833\pi$
$499$	975.125	1.95416	0.977079	+	0.212877i	$0.0682833\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	675.102i	1.34215i	0.741389	+	0.671076i	$0.234168\pi$
$503$	675.102i	1.34215i	−0.741389	+	0.671076i	$0.765832\pi$
$504$	0	0
$505$	214.790	0.425327
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.87656i	− 0.00761603i	−0.999993	−	0.00380801i	$-0.998788\pi$
$509$	− 3.87656i	− 0.00761603i	0.999993	−	0.00380801i	$-0.00121213\pi$
$510$	0	0
$511$	−611.781	−1.19722
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 23.4161i	− 0.0454682i
$516$	0	0
$517$	−10.2908	−0.0199048
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 890.215i	− 1.70867i	−0.519726	−	0.854333i	$-0.673966\pi$
$521$	− 890.215i	− 1.70867i	0.519726	−	0.854333i	$-0.326034\pi$
$522$	0	0
$523$	−786.449	−1.50373	−0.751864	−	0.659319i	$-0.770845\pi$
$523$	−786.449	−1.50373	−0.751864	+	0.659319i	$0.770845\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	604.612i	1.14727i
$528$	0	0
$529$	527.346	0.996873
$530$	0	0
$531$	0	0
$532$	0	0
$533$	99.4480i	0.186582i
$534$	0	0
$535$	287.664	0.537690
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 37.4928i	− 0.0695600i
$540$	0	0
$541$	462.603	0.855088	0.427544	−	0.903995i	$-0.359379\pi$
$541$	462.603	0.855088	0.427544	+	0.903995i	$0.359379\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 18.4969i	− 0.0339392i
$546$	0	0
$547$	−980.369	−1.79227	−0.896133	−	0.443786i	$-0.853635\pi$
$547$	−980.369	−1.79227	−0.896133	+	0.443786i	$0.853635\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	161.362i	0.292853i
$552$	0	0
$553$	533.641	0.964993
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.4835i	− 0.0403654i	−0.999796	−	0.0201827i	$-0.993575\pi$
$557$	− 22.4835i	− 0.0403654i	0.999796	−	0.0201827i	$-0.00642479\pi$
$558$	0	0
$559$	1.26568	0.00226419
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 620.228i	− 1.10165i	−0.834621	−	0.550824i	$-0.814313\pi$
$563$	− 620.228i	− 1.10165i	0.834621	−	0.550824i	$-0.185687\pi$
$564$	0	0
$565$	411.825	0.728894
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 160.563i	− 0.282184i	−0.989997	−	0.141092i	$-0.954939\pi$
$569$	− 160.563i	− 0.282184i	0.989997	−	0.141092i	$-0.0450613\pi$
$570$	0	0
$571$	541.216	0.947840	0.473920	−	0.880568i	$-0.342839\pi$
$571$	541.216	0.947840	0.473920	+	0.880568i	$0.342839\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	26.1503i	0.0454787i
$576$	0	0
$577$	−43.5123	−0.0754112	−0.0377056	−	0.999289i	$-0.512005\pi$
$577$	−43.5123	−0.0754112	−0.0377056	+	0.999289i	$0.512005\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1053.28i	− 1.81287i
$582$	0	0
$583$	82.6126	0.141703
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 64.7751i	− 0.110349i	−0.998477	−	0.0551747i	$-0.982428\pi$
$587$	− 64.7751i	− 0.110349i	0.998477	−	0.0551747i	$-0.0175716\pi$
$588$	0	0
$589$	136.776	0.232217
$590$	0	0
$591$	0	0
$592$	0	0
$593$	702.719i	1.18502i	0.805562	+	0.592512i	$0.201864\pi$
$593$	702.719i	1.18502i	−0.805562	+	0.592512i	$0.798136\pi$
$594$	0	0
$595$	−379.453	−0.637736
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 858.099i	− 1.43255i	−0.697817	−	0.716276i	$-0.745845\pi$
$599$	− 858.099i	− 1.43255i	0.697817	−	0.716276i	$-0.254155\pi$
$600$	0	0
$601$	−1116.90	−1.85840	−0.929200	−	0.369577i	$-0.879502\pi$
$601$	−1116.90	−1.85840	−0.929200	+	0.369577i	$0.879502\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 258.825i	− 0.427811i
$606$	0	0
$607$	1135.10	1.87002	0.935012	−	0.354616i	$-0.115388\pi$
$607$	1135.10	1.87002	0.935012	+	0.354616i	$0.115388\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 21.3745i	− 0.0349828i
$612$	0	0
$613$	228.738	0.373145	0.186573	−	0.982441i	$-0.440262\pi$
$613$	228.738	0.373145	0.186573	+	0.982441i	$0.440262\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1084.17i	1.75716i	0.477591	+	0.878582i	$0.341510\pi$
$617$	1084.17i	1.75716i	−0.477591	+	0.878582i	$0.658490\pi$
$618$	0	0
$619$	−479.659	−0.774893	−0.387446	−	0.921892i	$-0.626643\pi$
$619$	−479.659	−0.774893	−0.387446	+	0.921892i	$0.626643\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	60.1127i	0.0964890i
$624$	0	0
$625$	296.655	0.474649
$626$	0	0
$627$	0	0
$628$	0	0
$629$	673.451i	1.07067i
$630$	0	0
$631$	−39.9974	−0.0633873	−0.0316937	−	0.999498i	$-0.510090\pi$
$631$	−39.9974	−0.0633873	−0.0316937	+	0.999498i	$0.510090\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	144.421i	0.227435i
$636$	0	0
$637$	77.8745	0.122252
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1121.58i	1.74973i	0.484364	+	0.874866i	$0.339051\pi$
$641$	1121.58i	1.74973i	−0.484364	+	0.874866i	$0.660949\pi$
$642$	0	0
$643$	653.314	1.01604	0.508020	−	0.861345i	$-0.330378\pi$
$643$	653.314	1.01604	0.508020	+	0.861345i	$0.330378\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	475.997i	0.735698i	0.929885	+	0.367849i	$0.119906\pi$
$647$	475.997i	0.735698i	−0.929885	+	0.367849i	$0.880094\pi$
$648$	0	0
$649$	−41.9465	−0.0646325
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 949.403i	− 1.45391i	−0.686685	−	0.726955i	$-0.740935\pi$
$653$	− 949.403i	− 1.45391i	0.686685	−	0.726955i	$-0.259065\pi$
$654$	0	0
$655$	163.172	0.249117
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 462.803i	− 0.702281i	−0.936323	−	0.351140i	$-0.885794\pi$
$659$	− 462.803i	− 0.702281i	0.936323	−	0.351140i	$-0.114206\pi$
$660$	0	0
$661$	395.105	0.597738	0.298869	−	0.954294i	$-0.403391\pi$
$661$	395.105	0.597738	0.298869	+	0.954294i	$0.403391\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	85.8400i	0.129083i
$666$	0	0
$667$	−47.6138	−0.0713849
$668$	0	0
$669$	0	0
$670$	0	0
$671$	68.7583i	0.102471i
$672$	0	0
$673$	−718.411	−1.06748	−0.533738	−	0.845650i	$-0.679213\pi$
$673$	−718.411	−1.06748	−0.533738	+	0.845650i	$0.679213\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1162.70i	− 1.71743i	−0.512457	−	0.858713i	$-0.671265\pi$
$677$	− 1162.70i	− 1.71743i	0.512457	−	0.858713i	$-0.328735\pi$
$678$	0	0
$679$	127.808	0.188230
$680$	0	0
$681$	0	0
$682$	0	0
$683$	419.297i	0.613905i	0.951725	+	0.306952i	$0.0993092\pi$
$683$	419.297i	0.613905i	−0.951725	+	0.306952i	$0.900691\pi$
$684$	0	0
$685$	−367.192	−0.536047
$686$	0	0
$687$	0	0
$688$	0	0
$689$	171.590i	0.249043i
$690$	0	0
$691$	−767.513	−1.11073	−0.555364	−	0.831607i	$-0.687421\pi$
$691$	−767.513	−1.11073	−0.555364	+	0.831607i	$0.687421\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 281.129i	− 0.404502i
$696$	0	0
$697$	−838.343	−1.20279
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1227.38i	− 1.75089i	−0.483314	−	0.875447i	$-0.660567\pi$
$701$	− 1227.38i	− 1.75089i	0.483314	−	0.875447i	$-0.339433\pi$
$702$	0	0
$703$	152.348	0.216712
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 906.039i	− 1.28153i
$708$	0	0
$709$	302.852	0.427154	0.213577	−	0.976926i	$-0.431489\pi$
$709$	302.852	0.427154	0.213577	+	0.976926i	$0.431489\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	40.3589i	0.0566043i
$714$	0	0
$715$	−5.43478	−0.00760109
$716$	0	0
$717$	0	0
$718$	0	0
$719$	834.235i	1.16027i	0.814520	+	0.580136i	$0.197001\pi$
$719$	834.235i	1.16027i	−0.814520	+	0.580136i	$0.802999\pi$
$720$	0	0
$721$	−98.7751	−0.136997
$722$	0	0
$723$	0	0
$724$	0	0
$725$	752.652i	1.03814i
$726$	0	0
$727$	−760.433	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$727$	−760.433	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10.6697i	0.0145960i
$732$	0	0
$733$	−486.420	−0.663602	−0.331801	−	0.943349i	$-0.607656\pi$
$733$	−486.420	−0.663602	−0.331801	+	0.943349i	$0.607656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 94.2676i	− 0.127907i
$738$	0	0
$739$	542.728	0.734408	0.367204	−	0.930140i	$-0.380315\pi$
$739$	542.728	0.734408	0.367204	+	0.930140i	$0.380315\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	250.265i	0.336830i	0.985716	+	0.168415i	$0.0538649\pi$
$743$	250.265i	0.336830i	−0.985716	+	0.168415i	$0.946135\pi$
$744$	0	0
$745$	−137.243	−0.184219
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1213.44i	− 1.62008i
$750$	0	0
$751$	−1194.76	−1.59090	−0.795448	−	0.606022i	$-0.792764\pi$
$751$	−1194.76	−1.59090	−0.795448	+	0.606022i	$0.792764\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.7654i	0.0155834i
$756$	0	0
$757$	−1179.94	−1.55871	−0.779355	−	0.626583i	$-0.784453\pi$
$757$	−1179.94	−1.55871	−0.779355	+	0.626583i	$0.784453\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 523.434i	− 0.687823i	−0.939002	−	0.343912i	$-0.888248\pi$
$761$	− 523.434i	− 0.687823i	0.939002	−	0.343912i	$-0.111752\pi$
$762$	0	0
$763$	−78.0246	−0.102260
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 87.1249i	− 0.113592i
$768$	0	0
$769$	1050.26	1.36574	0.682872	−	0.730538i	$-0.260731\pi$
$769$	1050.26	1.36574	0.682872	+	0.730538i	$0.260731\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1231.05i	− 1.59256i	−0.604928	−	0.796280i	$-0.706798\pi$
$773$	− 1231.05i	− 1.59256i	0.604928	−	0.796280i	$-0.293202\pi$
$774$	0	0
$775$	637.970	0.823188
$776$	0	0
$777$	0	0
$778$	0	0
$779$	189.650i	0.243453i
$780$	0	0
$781$	−17.5249	−0.0224391
$782$	0	0
$783$	0	0
$784$	0	0
$785$	241.239i	0.307310i
$786$	0	0
$787$	−151.612	−0.192646	−0.0963228	−	0.995350i	$-0.530708\pi$
$787$	−151.612	−0.192646	−0.0963228	+	0.995350i	$0.530708\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1737.18i	− 2.19619i
$792$	0	0
$793$	−142.814	−0.180094
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1199.06i	− 1.50446i	−0.658899	−	0.752232i	$-0.728977\pi$
$797$	− 1199.06i	− 1.50446i	0.658899	−	0.752232i	$-0.271023\pi$
$798$	0	0
$799$	180.186	0.225514
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 73.8662i	− 0.0919878i
$804$	0	0
$805$	−25.3291	−0.0314648
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 21.7239i	− 0.0268528i	−0.999910	−	0.0134264i	$-0.995726\pi$
$809$	− 21.7239i	− 0.0268528i	0.999910	−	0.0134264i	$-0.00427389\pi$
$810$	0	0
$811$	−1050.05	−1.29476	−0.647381	−	0.762166i	$-0.724136\pi$
$811$	−1050.05	−1.29476	−0.647381	+	0.762166i	$0.724136\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	505.617i	0.620388i
$816$	0	0
$817$	2.41370	0.00295434
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1316.52i	− 1.60355i	−0.597624	−	0.801776i	$-0.703889\pi$
$821$	− 1316.52i	− 1.60355i	0.597624	−	0.801776i	$-0.296111\pi$
$822$	0	0
$823$	−4.71401	−0.00572784	−0.00286392	−	0.999996i	$-0.500912\pi$
$823$	−4.71401	−0.00572784	−0.00286392	+	0.999996i	$0.500912\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 107.497i	− 0.129985i	−0.997886	−	0.0649924i	$-0.979298\pi$
$827$	− 107.497i	− 0.129985i	0.997886	−	0.0649924i	$-0.0207023\pi$
$828$	0	0
$829$	−1234.04	−1.48859	−0.744293	−	0.667853i	$-0.767213\pi$
$829$	−1234.04	−1.48859	−0.744293	+	0.667853i	$0.767213\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	656.479i	0.788090i
$834$	0	0
$835$	−317.264	−0.379957
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 212.712i	− 0.253530i	−0.991933	−	0.126765i	$-0.959541\pi$
$839$	− 212.712i	− 0.253530i	0.991933	−	0.126765i	$-0.0404595\pi$
$840$	0	0
$841$	−529.410	−0.629500
$842$	0	0
$843$	0	0
$844$	0	0
$845$	353.866i	0.418777i
$846$	0	0
$847$	−1091.79	−1.28901
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.9540i	0.0528249i
$852$	0	0
$853$	−581.654	−0.681892	−0.340946	−	0.940083i	$-0.610747\pi$
$853$	−581.654	−0.681892	−0.340946	+	0.940083i	$0.610747\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 452.387i	− 0.527873i	−0.964540	−	0.263937i	$-0.914979\pi$
$857$	− 452.387i	− 0.527873i	0.964540	−	0.263937i	$-0.0850209\pi$
$858$	0	0
$859$	880.415	1.02493	0.512465	−	0.858708i	$-0.328732\pi$
$859$	880.415	1.02493	0.512465	+	0.858708i	$0.328732\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 470.242i	− 0.544892i	−0.962171	−	0.272446i	$-0.912167\pi$
$863$	− 470.242i	− 0.544892i	0.962171	−	0.272446i	$-0.0878326\pi$
$864$	0	0
$865$	−90.7103	−0.104867
$866$	0	0
$867$	0	0
$868$	0	0
$869$	64.4316i	0.0741445i
$870$	0	0
$871$	195.798	0.224797
$872$	0	0
$873$	0	0
$874$	0	0
$875$	892.715i	1.02025i
$876$	0	0
$877$	64.6856	0.0737579	0.0368789	−	0.999320i	$-0.488258\pi$
$877$	64.6856	0.0737579	0.0368789	+	0.999320i	$0.488258\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	804.020i	0.912622i	0.889820	+	0.456311i	$0.150829\pi$
$881$	804.020i	0.912622i	−0.889820	+	0.456311i	$0.849171\pi$
$882$	0	0
$883$	894.396	1.01291	0.506453	−	0.862267i	$-0.330956\pi$
$883$	894.396	1.01291	0.506453	+	0.862267i	$0.330956\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1450.55i	− 1.63535i	−0.575682	−	0.817674i	$-0.695263\pi$
$887$	− 1450.55i	− 1.63535i	0.575682	−	0.817674i	$-0.304737\pi$
$888$	0	0
$889$	609.206	0.685271
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 40.7618i	− 0.0456459i
$894$	0	0
$895$	−507.470	−0.567006
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1161.60i	1.29210i
$900$	0	0
$901$	−1446.50	−1.60544
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 207.167i	− 0.228914i
$906$	0	0
$907$	1663.35	1.83391	0.916954	−	0.398993i	$-0.130640\pi$
$907$	1663.35	1.83391	0.916954	+	0.398993i	$0.130640\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	150.701i	0.165424i	0.996574	+	0.0827118i	$0.0263581\pi$
$911$	150.701i	0.165424i	−0.996574	+	0.0827118i	$0.973642\pi$
$912$	0	0
$913$	127.173	0.139291
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 688.300i	− 0.750600i
$918$	0	0
$919$	826.616	0.899473	0.449736	−	0.893161i	$-0.351518\pi$
$919$	826.616	0.899473	0.449736	+	0.893161i	$0.351518\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 36.4001i	− 0.0394368i
$924$	0	0
$925$	710.607	0.768224
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1130.62i	− 1.21703i	−0.793541	−	0.608517i	$-0.791765\pi$
$929$	− 1130.62i	− 1.21703i	0.793541	−	0.608517i	$-0.208235\pi$
$930$	0	0
$931$	148.509	0.159515
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 45.8150i	− 0.0490000i
$936$	0	0
$937$	34.1269	0.0364214	0.0182107	−	0.999834i	$-0.494203\pi$
$937$	34.1269	0.0364214	0.0182107	+	0.999834i	$0.494203\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 424.451i	− 0.451063i	−0.974236	−	0.225532i	$-0.927588\pi$
$941$	− 424.451i	− 0.451063i	0.974236	−	0.225532i	$-0.0724119\pi$
$942$	0	0
$943$	−55.9608	−0.0593434
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1653.85i	1.74641i	0.487356	+	0.873203i	$0.337961\pi$
$947$	1653.85i	1.74641i	−0.487356	+	0.873203i	$0.662039\pi$
$948$	0	0
$949$	153.424	0.161669
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 458.801i	− 0.481428i	−0.970596	−	0.240714i	$-0.922618\pi$
$953$	− 458.801i	− 0.481428i	0.970596	−	0.240714i	$-0.0773816\pi$
$954$	0	0
$955$	−355.611	−0.372367
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1548.91i	1.61513i
$960$	0	0
$961$	23.6087	0.0245668
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 401.271i	− 0.415825i
$966$	0	0
$967$	953.711	0.986257	0.493129	−	0.869956i	$-0.335853\pi$
$967$	953.711	0.986257	0.493129	+	0.869956i	$0.335853\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 668.931i	− 0.688909i	−0.938803	−	0.344454i	$-0.888064\pi$
$971$	− 668.931i	− 0.688909i	0.938803	−	0.344454i	$-0.111936\pi$
$972$	0	0
$973$	−1185.87	−1.21878
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1236.97i	− 1.26609i	−0.774116	−	0.633044i	$-0.781805\pi$
$977$	− 1236.97i	− 1.26609i	0.774116	−	0.633044i	$-0.218195\pi$
$978$	0	0
$979$	−7.25798	−0.00741367
$980$	0	0
$981$	0	0
$982$	0	0
$983$	608.856i	0.619385i	0.950837	+	0.309693i	$0.100226\pi$
$983$	608.856i	0.619385i	−0.950837	+	0.309693i	$0.899774\pi$
$984$	0	0
$985$	344.896	0.350149
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.712219i	0 0.000720140i
$990$	0	0
$991$	1705.01	1.72049	0.860246	−	0.509878i	$-0.170310\pi$
$991$	1705.01	1.72049	0.860246	+	0.509878i	$0.170310\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	630.104i	0.633270i
$996$	0	0
$997$	74.8450	0.0750702	0.0375351	−	0.999295i	$-0.488049\pi$
$997$	74.8450	0.0750702	0.0375351	+	0.999295i	$0.488049\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.e.a.305.5	✓	12
3.2	odd	2	inner	684.3.e.a.305.8	yes	12
4.3	odd	2		2736.3.h.c.305.5		12
12.11	even	2		2736.3.h.c.305.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.e.a.305.5	✓	12	1.1	even	1	trivial
684.3.e.a.305.8	yes	12	3.2	odd	2	inner
2736.3.h.c.305.5		12	4.3	odd	2
2736.3.h.c.305.8		12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.16068i q^{5} -9.11429 q^{7} +O(q^{10})\)	\(q-2.16068i q^{5} -9.11429 q^{7} -1.10046i q^{11} +2.28570 q^{13} +19.2684i q^{17} +4.35890 q^{19} +1.28620i q^{23} +20.3315 q^{25} +37.0190i q^{29} +31.3785 q^{31} +19.6930i q^{35} +34.9511 q^{37} +43.5088i q^{41} +0.553740 q^{43} -9.35139i q^{47} +34.0703 q^{49} +75.0713i q^{53} -2.37773 q^{55} -38.1174i q^{59} -62.4816 q^{61} -4.93866i q^{65} +85.6623 q^{67} -15.9252i q^{71} +67.1233 q^{73} +10.0299i q^{77} -58.5499 q^{79} +115.564i q^{83} +41.6328 q^{85} -6.59543i q^{89} -20.8325 q^{91} -9.41818i q^{95} -14.0228 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{7}+O(q^{10}) \) 12 * q + 16 * q^7	\( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) \) 12 * q + 16 * q^7 - 16 * q^13 - 12 * q^25 - 40 * q^31 - 32 * q^37 + 92 * q^43 - 84 * q^55 - 48 * q^61 - 88 * q^67 + 148 * q^73 - 56 * q^79 + 228 * q^85 - 8 * q^91 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more