LMFDB - Embedded newform 1216.3.d.d.191.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(191,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.191");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not 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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + \cdots + 16384 $ x^14 - 2x^13 + x^12 + 14x^11 - 42x^10 + 28x^9 + 132x^8 - 440x^7 + 528x^6 + 448x^5 - 2688x^4 + 3584x^3 + 1024x^2 - 8192x + 16384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.7
Root			$-1.92254 + 0.551226i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.d.191.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.644704i q^{3} +2.32715 q^{5} +8.62924i q^{7} +8.58436 q^{9} +O(q^{10})$	$q-0.644704i q^{3} +2.32715 q^{5} +8.62924i q^{7} +8.58436 q^{9} -19.2717i q^{11} -13.8067 q^{13} -1.50032i q^{15} -12.5180 q^{17} -4.35890i q^{19} +5.56330 q^{21} -37.4981i q^{23} -19.5844 q^{25} -11.3367i q^{27} +6.36930 q^{29} -5.44851i q^{31} -12.4245 q^{33} +20.0816i q^{35} -20.9250 q^{37} +8.90120i q^{39} +72.8337 q^{41} -10.1553i q^{43} +19.9771 q^{45} -32.4450i q^{47} -25.4638 q^{49} +8.07040i q^{51} +42.9339 q^{53} -44.8482i q^{55} -2.81020 q^{57} -38.9728i q^{59} -25.5611 q^{61} +74.0765i q^{63} -32.1302 q^{65} +65.3183i q^{67} -24.1751 q^{69} -18.7557i q^{71} +72.8251 q^{73} +12.6261i q^{75} +166.300 q^{77} -139.874i q^{79} +69.9504 q^{81} -94.7116i q^{83} -29.1313 q^{85} -4.10631i q^{87} +33.3546 q^{89} -119.141i q^{91} -3.51267 q^{93} -10.1438i q^{95} -150.817 q^{97} -165.435i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 68 q^{9}+O(q^{10}) $ 14 * q - 68 * q^9	$ 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) $ 14 * q - 68 * q^9 - 54 * q^13 + 34 * q^17 + 38 * q^21 - 86 * q^25 - 54 * q^29 + 20 * q^33 - 100 * q^37 + 224 * q^41 + 168 * q^45 - 220 * q^49 - 14 * q^53 + 38 * q^57 - 28 * q^61 - 472 * q^65 - 122 * q^69 + 70 * q^73 - 228 * q^77 + 334 * q^81 - 48 * q^85 + 176 * q^93 + 308 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 0.644704i	− 0.214901i	−0.994210	−	0.107451i	$-0.965731\pi$
$3$	− 0.644704i	− 0.214901i	0.994210	−	0.107451i	$-0.0342688\pi$
$4$	0	0
$5$	2.32715	0.465431	0.232715	−	0.972545i	$-0.425239\pi$
$5$	2.32715	0.465431	0.232715	+	0.972545i	$0.425239\pi$
$6$	0	0
$7$	8.62924i	1.23275i	0.787453	+	0.616374i	$0.211399\pi$
$7$	8.62924i	1.23275i	−0.787453	+	0.616374i	$0.788601\pi$
$8$	0	0
$9$	8.58436	0.953817
$10$	0	0
$11$	− 19.2717i	− 1.75197i	−0.482334	−	0.875987i	$-0.660211\pi$
$11$	− 19.2717i	− 1.75197i	0.482334	−	0.875987i	$-0.339789\pi$
$12$	0	0
$13$	−13.8067	−1.06205	−0.531025	−	0.847356i	$-0.678193\pi$
$13$	−13.8067	−1.06205	−0.531025	+	0.847356i	$0.678193\pi$
$14$	0	0
$15$	− 1.50032i	− 0.100022i
$16$	0	0
$17$	−12.5180	−0.736353	−0.368177	−	0.929756i	$-0.620018\pi$
$17$	−12.5180	−0.736353	−0.368177	+	0.929756i	$0.620018\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	5.56330	0.264919
$22$	0	0
$23$	− 37.4981i	− 1.63035i	−0.579214	−	0.815175i	$-0.696640\pi$
$23$	− 37.4981i	− 1.63035i	0.579214	−	0.815175i	$-0.303360\pi$
$24$	0	0
$25$	−19.5844	−0.783374
$26$	0	0
$27$	− 11.3367i	− 0.419878i
$28$	0	0
$29$	6.36930	0.219631	0.109816	−	0.993952i	$-0.464974\pi$
$29$	6.36930	0.219631	0.109816	+	0.993952i	$0.464974\pi$
$30$	0	0
$31$	− 5.44851i	− 0.175758i	−0.996131	−	0.0878791i	$-0.971991\pi$
$31$	− 5.44851i	− 0.175758i	0.996131	−	0.0878791i	$-0.0280089\pi$
$32$	0	0
$33$	−12.4245	−0.376502
$34$	0	0
$35$	20.0816i	0.573759i
$36$	0	0
$37$	−20.9250	−0.565542	−0.282771	−	0.959187i	$-0.591254\pi$
$37$	−20.9250	−0.565542	−0.282771	+	0.959187i	$0.591254\pi$
$38$	0	0
$39$	8.90120i	0.228236i
$40$	0	0
$41$	72.8337	1.77643	0.888216	−	0.459425i	$-0.151945\pi$
$41$	72.8337	1.77643	0.888216	+	0.459425i	$0.151945\pi$
$42$	0	0
$43$	− 10.1553i	− 0.236171i	−0.993003	−	0.118085i	$-0.962324\pi$
$43$	− 10.1553i	− 0.236171i	0.993003	−	0.118085i	$-0.0376757\pi$
$44$	0	0
$45$	19.9771	0.443936
$46$	0	0
$47$	− 32.4450i	− 0.690318i	−0.938544	−	0.345159i	$-0.887825\pi$
$47$	− 32.4450i	− 0.690318i	0.938544	−	0.345159i	$-0.112175\pi$
$48$	0	0
$49$	−25.4638	−0.519669
$50$	0	0
$51$	8.07040i	0.158243i
$52$	0	0
$53$	42.9339	0.810074	0.405037	−	0.914300i	$-0.367259\pi$
$53$	42.9339	0.810074	0.405037	+	0.914300i	$0.367259\pi$
$54$	0	0
$55$	− 44.8482i	− 0.815423i
$56$	0	0
$57$	−2.81020	−0.0493017
$58$	0	0
$59$	− 38.9728i	− 0.660556i	−0.943884	−	0.330278i	$-0.892858\pi$
$59$	− 38.9728i	− 0.660556i	0.943884	−	0.330278i	$-0.107142\pi$
$60$	0	0
$61$	−25.5611	−0.419034	−0.209517	−	0.977805i	$-0.567189\pi$
$61$	−25.5611	−0.419034	−0.209517	+	0.977805i	$0.567189\pi$
$62$	0	0
$63$	74.0765i	1.17582i
$64$	0	0
$65$	−32.1302	−0.494311
$66$	0	0
$67$	65.3183i	0.974900i	0.873151	+	0.487450i	$0.162073\pi$
$67$	65.3183i	0.974900i	−0.873151	+	0.487450i	$0.837927\pi$
$68$	0	0
$69$	−24.1751	−0.350364
$70$	0	0
$71$	− 18.7557i	− 0.264165i	−0.991239	−	0.132083i	$-0.957834\pi$
$71$	− 18.7557i	− 0.264165i	0.991239	−	0.132083i	$-0.0421665\pi$
$72$	0	0
$73$	72.8251	0.997604	0.498802	−	0.866716i	$-0.333773\pi$
$73$	72.8251	0.997604	0.498802	+	0.866716i	$0.333773\pi$
$74$	0	0
$75$	12.6261i	0.168348i
$76$	0	0
$77$	166.300	2.15974
$78$	0	0
$79$	− 139.874i	− 1.77056i	−0.465060	−	0.885279i	$-0.653967\pi$
$79$	− 139.874i	− 1.77056i	0.465060	−	0.885279i	$-0.346033\pi$
$80$	0	0
$81$	69.9504	0.863585
$82$	0	0
$83$	− 94.7116i	− 1.14110i	−0.821262	−	0.570552i	$-0.806729\pi$
$83$	− 94.7116i	− 1.14110i	0.821262	−	0.570552i	$-0.193271\pi$
$84$	0	0
$85$	−29.1313	−0.342721
$86$	0	0
$87$	− 4.10631i	− 0.0471990i
$88$	0	0
$89$	33.3546	0.374770	0.187385	−	0.982287i	$-0.439999\pi$
$89$	33.3546	0.374770	0.187385	+	0.982287i	$0.439999\pi$
$90$	0	0
$91$	− 119.141i	− 1.30924i
$92$	0	0
$93$	−3.51267	−0.0377707
$94$	0	0
$95$	− 10.1438i	− 0.106777i
$96$	0	0
$97$	−150.817	−1.55482	−0.777408	−	0.628996i	$-0.783466\pi$
$97$	−150.817	−1.55482	−0.777408	+	0.628996i	$0.783466\pi$
$98$	0	0
$99$	− 165.435i	− 1.67106i
$100$	0	0
$101$	−77.7745	−0.770045	−0.385022	−	0.922907i	$-0.625806\pi$
$101$	−77.7745	−0.770045	−0.385022	+	0.922907i	$0.625806\pi$
$102$	0	0
$103$	− 14.9481i	− 0.145127i	−0.997364	−	0.0725637i	$-0.976882\pi$
$103$	− 14.9481i	− 0.145127i	0.997364	−	0.0725637i	$-0.0231180\pi$
$104$	0	0
$105$	12.9467	0.123302
$106$	0	0
$107$	80.5205i	0.752528i	0.926512	+	0.376264i	$0.122792\pi$
$107$	80.5205i	0.752528i	−0.926512	+	0.376264i	$0.877208\pi$
$108$	0	0
$109$	−53.8512	−0.494048	−0.247024	−	0.969009i	$-0.579453\pi$
$109$	−53.8512	−0.494048	−0.247024	+	0.969009i	$0.579453\pi$
$110$	0	0
$111$	13.4905i	0.121536i
$112$	0	0
$113$	105.921	0.937355	0.468677	−	0.883369i	$-0.344731\pi$
$113$	105.921	0.937355	0.468677	+	0.883369i	$0.344731\pi$
$114$	0	0
$115$	− 87.2638i	− 0.758815i
$116$	0	0
$117$	−118.521	−1.01300
$118$	0	0
$119$	− 108.021i	− 0.907739i
$120$	0	0
$121$	−250.399	−2.06941
$122$	0	0
$123$	− 46.9562i	− 0.381758i
$124$	0	0
$125$	−103.755	−0.830037
$126$	0	0
$127$	17.1409i	0.134968i	0.997720	+	0.0674839i	$0.0214971\pi$
$127$	17.1409i	0.134968i	−0.997720	+	0.0674839i	$0.978503\pi$
$128$	0	0
$129$	−6.54719	−0.0507534
$130$	0	0
$131$	− 229.336i	− 1.75066i	−0.483526	−	0.875330i	$-0.660644\pi$
$131$	− 229.336i	− 1.75066i	0.483526	−	0.875330i	$-0.339356\pi$
$132$	0	0
$133$	37.6140	0.282812
$134$	0	0
$135$	− 26.3822i	− 0.195424i
$136$	0	0
$137$	−120.567	−0.880054	−0.440027	−	0.897985i	$-0.645031\pi$
$137$	−120.567	−0.880054	−0.440027	+	0.897985i	$0.645031\pi$
$138$	0	0
$139$	145.864i	1.04938i	0.851292	+	0.524692i	$0.175819\pi$
$139$	145.864i	1.04938i	−0.851292	+	0.524692i	$0.824181\pi$
$140$	0	0
$141$	−20.9174	−0.148350
$142$	0	0
$143$	266.078i	1.86069i
$144$	0	0
$145$	14.8223	0.102223
$146$	0	0
$147$	16.4166i	0.111678i
$148$	0	0
$149$	−19.2547	−0.129226	−0.0646132	−	0.997910i	$-0.520581\pi$
$149$	−19.2547	−0.129226	−0.0646132	+	0.997910i	$0.520581\pi$
$150$	0	0
$151$	− 165.526i	− 1.09620i	−0.836414	−	0.548099i	$-0.815352\pi$
$151$	− 165.526i	− 1.09620i	0.836414	−	0.548099i	$-0.184648\pi$
$152$	0	0
$153$	−107.459	−0.702347
$154$	0	0
$155$	− 12.6795i	− 0.0818033i
$156$	0	0
$157$	−149.449	−0.951907	−0.475954	−	0.879470i	$-0.657897\pi$
$157$	−149.449	−0.951907	−0.475954	+	0.879470i	$0.657897\pi$
$158$	0	0
$159$	− 27.6796i	− 0.174086i
$160$	0	0
$161$	323.580	2.00981
$162$	0	0
$163$	146.763i	0.900384i	0.892932	+	0.450192i	$0.148644\pi$
$163$	146.763i	0.900384i	−0.892932	+	0.450192i	$0.851356\pi$
$164$	0	0
$165$	−28.9138	−0.175235
$166$	0	0
$167$	− 213.493i	− 1.27840i	−0.769040	−	0.639201i	$-0.779265\pi$
$167$	− 213.493i	− 1.27840i	0.769040	−	0.639201i	$-0.220735\pi$
$168$	0	0
$169$	21.6237	0.127951
$170$	0	0
$171$	− 37.4183i	− 0.218821i
$172$	0	0
$173$	280.715	1.62263	0.811314	−	0.584610i	$-0.198753\pi$
$173$	280.715	1.62263	0.811314	+	0.584610i	$0.198753\pi$
$174$	0	0
$175$	− 168.998i	− 0.965704i
$176$	0	0
$177$	−25.1259	−0.141954
$178$	0	0
$179$	− 35.1878i	− 0.196580i	−0.995158	−	0.0982899i	$-0.968663\pi$
$179$	− 35.1878i	− 0.196580i	0.995158	−	0.0982899i	$-0.0313372\pi$
$180$	0	0
$181$	345.539	1.90906	0.954529	−	0.298120i	$-0.0963594\pi$
$181$	345.539	1.90906	0.954529	+	0.298120i	$0.0963594\pi$
$182$	0	0
$183$	16.4793i	0.0900509i
$184$	0	0
$185$	−48.6958	−0.263220
$186$	0	0
$187$	241.244i	1.29007i
$188$	0	0
$189$	97.8271	0.517604
$190$	0	0
$191$	100.233i	0.524782i	0.964962	+	0.262391i	$0.0845110\pi$
$191$	100.233i	0.524782i	−0.964962	+	0.262391i	$0.915489\pi$
$192$	0	0
$193$	168.119	0.871085	0.435543	−	0.900168i	$-0.356557\pi$
$193$	168.119	0.871085	0.435543	+	0.900168i	$0.356557\pi$
$194$	0	0
$195$	20.7145i	0.106228i
$196$	0	0
$197$	−216.705	−1.10002	−0.550012	−	0.835157i	$-0.685377\pi$
$197$	−216.705	−1.10002	−0.550012	+	0.835157i	$0.685377\pi$
$198$	0	0
$199$	− 82.2545i	− 0.413339i	−0.978411	−	0.206670i	$-0.933738\pi$
$199$	− 82.2545i	− 0.413339i	0.978411	−	0.206670i	$-0.0662625\pi$
$200$	0	0
$201$	42.1109	0.209507
$202$	0	0
$203$	54.9623i	0.270750i
$204$	0	0
$205$	169.495	0.826806
$206$	0	0
$207$	− 321.897i	− 1.55506i
$208$	0	0
$209$	−84.0035	−0.401931
$210$	0	0
$211$	16.4195i	0.0778173i	0.999243	+	0.0389087i	$0.0123881\pi$
$211$	16.4195i	0.0778173i	−0.999243	+	0.0389087i	$0.987612\pi$
$212$	0	0
$213$	−12.0919	−0.0567695
$214$	0	0
$215$	− 23.6331i	− 0.109921i
$216$	0	0
$217$	47.0165	0.216666
$218$	0	0
$219$	− 46.9506i	− 0.214386i
$220$	0	0
$221$	172.832	0.782044
$222$	0	0
$223$	195.225i	0.875450i	0.899109	+	0.437725i	$0.144216\pi$
$223$	195.225i	0.875450i	−0.899109	+	0.437725i	$0.855784\pi$
$224$	0	0
$225$	−168.119	−0.747196
$226$	0	0
$227$	− 305.403i	− 1.34539i	−0.739920	−	0.672695i	$-0.765137\pi$
$227$	− 305.403i	− 1.34539i	0.739920	−	0.672695i	$-0.234863\pi$
$228$	0	0
$229$	229.686	1.00300	0.501499	−	0.865158i	$-0.332782\pi$
$229$	229.686	1.00300	0.501499	+	0.865158i	$0.332782\pi$
$230$	0	0
$231$	− 107.214i	− 0.464132i
$232$	0	0
$233$	187.667	0.805436	0.402718	−	0.915324i	$-0.368065\pi$
$233$	187.667	0.805436	0.402718	+	0.915324i	$0.368065\pi$
$234$	0	0
$235$	− 75.5044i	− 0.321295i
$236$	0	0
$237$	−90.1774	−0.380495
$238$	0	0
$239$	69.8504i	0.292261i	0.989265	+	0.146130i	$0.0466819\pi$
$239$	69.8504i	0.292261i	−0.989265	+	0.146130i	$0.953318\pi$
$240$	0	0
$241$	297.561	1.23469	0.617346	−	0.786691i	$-0.288208\pi$
$241$	297.561	1.23469	0.617346	+	0.786691i	$0.288208\pi$
$242$	0	0
$243$	− 147.128i	− 0.605463i
$244$	0	0
$245$	−59.2582	−0.241870
$246$	0	0
$247$	60.1818i	0.243651i
$248$	0	0
$249$	−61.0609	−0.245225
$250$	0	0
$251$	− 36.4667i	− 0.145286i	−0.997358	−	0.0726429i	$-0.976857\pi$
$251$	− 36.4667i	− 0.145286i	0.997358	−	0.0726429i	$-0.0231433\pi$
$252$	0	0
$253$	−722.652	−2.85633
$254$	0	0
$255$	18.7811i	0.0736513i
$256$	0	0
$257$	−175.943	−0.684604	−0.342302	−	0.939590i	$-0.611207\pi$
$257$	−175.943	−0.684604	−0.342302	+	0.939590i	$0.611207\pi$
$258$	0	0
$259$	− 180.567i	− 0.697171i
$260$	0	0
$261$	54.6764	0.209488
$262$	0	0
$263$	444.887i	1.69158i	0.533512	+	0.845792i	$0.320872\pi$
$263$	444.887i	1.69158i	−0.533512	+	0.845792i	$0.679128\pi$
$264$	0	0
$265$	99.9138	0.377033
$266$	0	0
$267$	− 21.5038i	− 0.0805386i
$268$	0	0
$269$	−149.744	−0.556670	−0.278335	−	0.960484i	$-0.589783\pi$
$269$	−149.744	−0.556670	−0.278335	+	0.960484i	$0.589783\pi$
$270$	0	0
$271$	439.685i	1.62245i	0.584731	+	0.811227i	$0.301200\pi$
$271$	439.685i	1.62245i	−0.584731	+	0.811227i	$0.698800\pi$
$272$	0	0
$273$	−76.8106	−0.281358
$274$	0	0
$275$	377.424i	1.37245i
$276$	0	0
$277$	−83.2452	−0.300524	−0.150262	−	0.988646i	$-0.548012\pi$
$277$	−83.2452	−0.300524	−0.150262	+	0.988646i	$0.548012\pi$
$278$	0	0
$279$	− 46.7719i	− 0.167641i
$280$	0	0
$281$	−12.9851	−0.0462105	−0.0231052	−	0.999733i	$-0.507355\pi$
$281$	−12.9851	−0.0462105	−0.0231052	+	0.999733i	$0.507355\pi$
$282$	0	0
$283$	− 111.022i	− 0.392303i	−0.980574	−	0.196152i	$-0.937156\pi$
$283$	− 111.022i	− 0.392303i	0.980574	−	0.196152i	$-0.0628445\pi$
$284$	0	0
$285$	−6.53976	−0.0229465
$286$	0	0
$287$	628.500i	2.18990i
$288$	0	0
$289$	−132.300	−0.457784
$290$	0	0
$291$	97.2324i	0.334132i
$292$	0	0
$293$	141.572	0.483179	0.241590	−	0.970378i	$-0.422331\pi$
$293$	141.572	0.483179	0.241590	+	0.970378i	$0.422331\pi$
$294$	0	0
$295$	− 90.6956i	− 0.307443i
$296$	0	0
$297$	−218.478	−0.735615
$298$	0	0
$299$	517.723i	1.73151i
$300$	0	0
$301$	87.6329	0.291139
$302$	0	0
$303$	50.1415i	0.165484i
$304$	0	0
$305$	−59.4846	−0.195031
$306$	0	0
$307$	− 357.744i	− 1.16529i	−0.812727	−	0.582645i	$-0.802018\pi$
$307$	− 357.744i	− 1.16529i	0.812727	−	0.582645i	$-0.197982\pi$
$308$	0	0
$309$	−9.63710	−0.0311880
$310$	0	0
$311$	− 472.383i	− 1.51892i	−0.650557	−	0.759458i	$-0.725464\pi$
$311$	− 472.383i	− 1.51892i	0.650557	−	0.759458i	$-0.274536\pi$
$312$	0	0
$313$	63.7596	0.203705	0.101852	−	0.994800i	$-0.467523\pi$
$313$	63.7596	0.203705	0.101852	+	0.994800i	$0.467523\pi$
$314$	0	0
$315$	172.387i	0.547261i
$316$	0	0
$317$	−198.721	−0.626880	−0.313440	−	0.949608i	$-0.601481\pi$
$317$	−198.721	−0.626880	−0.313440	+	0.949608i	$0.601481\pi$
$318$	0	0
$319$	− 122.747i	− 0.384788i
$320$	0	0
$321$	51.9119	0.161719
$322$	0	0
$323$	54.5647i	0.168931i
$324$	0	0
$325$	270.394	0.831983
$326$	0	0
$327$	34.7181i	0.106172i
$328$	0	0
$329$	279.975	0.850989
$330$	0	0
$331$	438.736i	1.32549i	0.748846	+	0.662744i	$0.230608\pi$
$331$	438.736i	1.32549i	−0.748846	+	0.662744i	$0.769392\pi$
$332$	0	0
$333$	−179.628	−0.539424
$334$	0	0
$335$	152.006i	0.453748i
$336$	0	0
$337$	−538.699	−1.59851	−0.799257	−	0.600989i	$-0.794773\pi$
$337$	−538.699	−1.59851	−0.799257	+	0.600989i	$0.794773\pi$
$338$	0	0
$339$	− 68.2877i	− 0.201439i
$340$	0	0
$341$	−105.002	−0.307924
$342$	0	0
$343$	203.100i	0.592127i
$344$	0	0
$345$	−56.2593	−0.163070
$346$	0	0
$347$	584.405i	1.68417i	0.539349	+	0.842083i	$0.318671\pi$
$347$	584.405i	1.68417i	−0.539349	+	0.842083i	$0.681329\pi$
$348$	0	0
$349$	300.832	0.861983	0.430991	−	0.902356i	$-0.358164\pi$
$349$	300.832	0.861983	0.430991	+	0.902356i	$0.358164\pi$
$350$	0	0
$351$	156.522i	0.445931i
$352$	0	0
$353$	−316.023	−0.895250	−0.447625	−	0.894221i	$-0.647730\pi$
$353$	−316.023	−0.895250	−0.447625	+	0.894221i	$0.647730\pi$
$354$	0	0
$355$	− 43.6475i	− 0.122951i
$356$	0	0
$357$	−69.6415	−0.195074
$358$	0	0
$359$	− 116.055i	− 0.323272i	−0.986850	−	0.161636i	$-0.948323\pi$
$359$	− 116.055i	− 0.323272i	0.986850	−	0.161636i	$-0.0516770\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	161.433i	0.444720i
$364$	0	0
$365$	169.475	0.464316
$366$	0	0
$367$	− 67.8773i	− 0.184952i	−0.995715	−	0.0924759i	$-0.970522\pi$
$367$	− 67.8773i	− 0.184952i	0.995715	−	0.0924759i	$-0.0294781\pi$
$368$	0	0
$369$	625.231	1.69439
$370$	0	0
$371$	370.487i	0.998617i
$372$	0	0
$373$	267.337	0.716721	0.358361	−	0.933583i	$-0.383336\pi$
$373$	267.337	0.716721	0.358361	+	0.933583i	$0.383336\pi$
$374$	0	0
$375$	66.8910i	0.178376i
$376$	0	0
$377$	−87.9388	−0.233259
$378$	0	0
$379$	372.834i	0.983731i	0.870671	+	0.491866i	$0.163685\pi$
$379$	372.834i	0.983731i	−0.870671	+	0.491866i	$0.836315\pi$
$380$	0	0
$381$	11.0508	0.0290047
$382$	0	0
$383$	19.6246i	0.0512392i	0.999672	+	0.0256196i	$0.00815587\pi$
$383$	19.6246i	0.0512392i	−0.999672	+	0.0256196i	$0.991844\pi$
$384$	0	0
$385$	387.006	1.00521
$386$	0	0
$387$	− 87.1771i	− 0.225264i
$388$	0	0
$389$	−198.727	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$389$	−198.727	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$390$	0	0
$391$	469.401i	1.20051i
$392$	0	0
$393$	−147.854	−0.376219
$394$	0	0
$395$	− 325.509i	− 0.824072i
$396$	0	0
$397$	−295.448	−0.744203	−0.372101	−	0.928192i	$-0.621363\pi$
$397$	−295.448	−0.744203	−0.372101	+	0.928192i	$0.621363\pi$
$398$	0	0
$399$	− 24.2499i	− 0.0607766i
$400$	0	0
$401$	−63.9986	−0.159597	−0.0797987	−	0.996811i	$-0.525428\pi$
$401$	−63.9986	−0.159597	−0.0797987	+	0.996811i	$0.525428\pi$
$402$	0	0
$403$	75.2256i	0.186664i
$404$	0	0
$405$	162.785	0.401939
$406$	0	0
$407$	403.262i	0.990815i
$408$	0	0
$409$	504.473	1.23343	0.616715	−	0.787186i	$-0.288463\pi$
$409$	504.473	1.23343	0.616715	+	0.787186i	$0.288463\pi$
$410$	0	0
$411$	77.7302i	0.189125i
$412$	0	0
$413$	336.306	0.814299
$414$	0	0
$415$	− 220.408i	− 0.531105i
$416$	0	0
$417$	94.0393	0.225514
$418$	0	0
$419$	470.299i	1.12243i	0.827670	+	0.561215i	$0.189666\pi$
$419$	470.299i	1.12243i	−0.827670	+	0.561215i	$0.810334\pi$
$420$	0	0
$421$	470.034	1.11647	0.558235	−	0.829683i	$-0.311479\pi$
$421$	470.034	1.11647	0.558235	+	0.829683i	$0.311479\pi$
$422$	0	0
$423$	− 278.519i	− 0.658437i
$424$	0	0
$425$	245.157	0.576840
$426$	0	0
$427$	− 220.573i	− 0.516564i
$428$	0	0
$429$	171.541	0.399864
$430$	0	0
$431$	− 224.005i	− 0.519733i	−0.965645	−	0.259866i	$-0.916321\pi$
$431$	− 224.005i	− 0.519733i	0.965645	−	0.259866i	$-0.0836785\pi$
$432$	0	0
$433$	180.128	0.415999	0.207999	−	0.978129i	$-0.433305\pi$
$433$	180.128	0.415999	0.207999	+	0.978129i	$0.433305\pi$
$434$	0	0
$435$	− 9.55602i	− 0.0219679i
$436$	0	0
$437$	−163.450	−0.374028
$438$	0	0
$439$	− 590.004i	− 1.34397i	−0.740563	−	0.671987i	$-0.765441\pi$
$439$	− 590.004i	− 1.34397i	0.740563	−	0.671987i	$-0.234559\pi$
$440$	0	0
$441$	−218.590	−0.495670
$442$	0	0
$443$	157.459i	0.355439i	0.984081	+	0.177719i	$0.0568720\pi$
$443$	157.459i	0.355439i	−0.984081	+	0.177719i	$0.943128\pi$
$444$	0	0
$445$	77.6212	0.174430
$446$	0	0
$447$	12.4136i	0.0277709i
$448$	0	0
$449$	−715.434	−1.59339	−0.796697	−	0.604379i	$-0.793421\pi$
$449$	−715.434	−1.59339	−0.796697	+	0.604379i	$0.793421\pi$
$450$	0	0
$451$	− 1403.63i	− 3.11226i
$452$	0	0
$453$	−106.715	−0.235574
$454$	0	0
$455$	− 277.259i	− 0.609361i
$456$	0	0
$457$	31.3497	0.0685988	0.0342994	−	0.999412i	$-0.489080\pi$
$457$	31.3497	0.0685988	0.0342994	+	0.999412i	$0.489080\pi$
$458$	0	0
$459$	141.913i	0.309178i
$460$	0	0
$461$	−368.719	−0.799824	−0.399912	−	0.916554i	$-0.630959\pi$
$461$	−368.719	−0.799824	−0.399912	+	0.916554i	$0.630959\pi$
$462$	0	0
$463$	817.405i	1.76545i	0.469887	+	0.882726i	$0.344295\pi$
$463$	817.405i	1.76545i	−0.469887	+	0.882726i	$0.655705\pi$
$464$	0	0
$465$	−8.17453	−0.0175796
$466$	0	0
$467$	86.9194i	0.186123i	0.995660	+	0.0930614i	$0.0296653\pi$
$467$	86.9194i	0.186123i	−0.995660	+	0.0930614i	$0.970335\pi$
$468$	0	0
$469$	−563.647	−1.20181
$470$	0	0
$471$	96.3506i	0.204566i
$472$	0	0
$473$	−195.711	−0.413765
$474$	0	0
$475$	85.3662i	0.179718i
$476$	0	0
$477$	368.560	0.772662
$478$	0	0
$479$	333.107i	0.695421i	0.937602	+	0.347710i	$0.113041\pi$
$479$	333.107i	0.695421i	−0.937602	+	0.347710i	$0.886959\pi$
$480$	0	0
$481$	288.905	0.600634
$482$	0	0
$483$	− 208.613i	− 0.431911i
$484$	0	0
$485$	−350.975	−0.723659
$486$	0	0
$487$	264.690i	0.543511i	0.962366	+	0.271755i	$0.0876042\pi$
$487$	264.690i	0.543511i	−0.962366	+	0.271755i	$0.912396\pi$
$488$	0	0
$489$	94.6183	0.193494
$490$	0	0
$491$	402.874i	0.820518i	0.911969	+	0.410259i	$0.134562\pi$
$491$	402.874i	0.820518i	−0.911969	+	0.410259i	$0.865438\pi$
$492$	0	0
$493$	−79.7310	−0.161726
$494$	0	0
$495$	− 384.993i	− 0.777764i
$496$	0	0
$497$	161.848	0.325650
$498$	0	0
$499$	350.536i	0.702477i	0.936286	+	0.351238i	$0.114239\pi$
$499$	350.536i	0.702477i	−0.936286	+	0.351238i	$0.885761\pi$
$500$	0	0
$501$	−137.640	−0.274730
$502$	0	0
$503$	339.615i	0.675179i	0.941293	+	0.337589i	$0.109612\pi$
$503$	339.615i	0.675179i	−0.941293	+	0.337589i	$0.890388\pi$
$504$	0	0
$505$	−180.993	−0.358403
$506$	0	0
$507$	− 13.9409i	− 0.0274969i
$508$	0	0
$509$	933.003	1.83301	0.916506	−	0.400020i	$-0.130997\pi$
$509$	933.003	1.83301	0.916506	+	0.400020i	$0.130997\pi$
$510$	0	0
$511$	628.426i	1.22980i
$512$	0	0
$513$	−49.4155	−0.0963266
$514$	0	0
$515$	− 34.7866i	− 0.0675467i
$516$	0	0
$517$	−625.270	−1.20942
$518$	0	0
$519$	− 180.978i	− 0.348705i
$520$	0	0
$521$	−415.284	−0.797091	−0.398545	−	0.917149i	$-0.630485\pi$
$521$	−415.284	−0.797091	−0.398545	+	0.917149i	$0.630485\pi$
$522$	0	0
$523$	− 241.598i	− 0.461947i	−0.972960	−	0.230973i	$-0.925809\pi$
$523$	− 241.598i	− 0.461947i	0.972960	−	0.230973i	$-0.0741911\pi$
$524$	0	0
$525$	−108.954	−0.207531
$526$	0	0
$527$	68.2044i	0.129420i
$528$	0	0
$529$	−877.105	−1.65804
$530$	0	0
$531$	− 334.556i	− 0.630050i
$532$	0	0
$533$	−1005.59	−1.88666
$534$	0	0
$535$	187.384i	0.350250i
$536$	0	0
$537$	−22.6857	−0.0422452
$538$	0	0
$539$	490.731i	0.910447i
$540$	0	0
$541$	304.787	0.563377	0.281689	−	0.959506i	$-0.409106\pi$
$541$	304.787	0.563377	0.281689	+	0.959506i	$0.409106\pi$
$542$	0	0
$543$	− 222.770i	− 0.410259i
$544$	0	0
$545$	−125.320	−0.229945
$546$	0	0
$547$	555.915i	1.01630i	0.861269	+	0.508149i	$0.169670\pi$
$547$	555.915i	1.01630i	−0.861269	+	0.508149i	$0.830330\pi$
$548$	0	0
$549$	−219.425	−0.399682
$550$	0	0
$551$	− 27.7632i	− 0.0503869i
$552$	0	0
$553$	1207.01	2.18265
$554$	0	0
$555$	31.3944i	0.0565664i
$556$	0	0
$557$	192.055	0.344802	0.172401	−	0.985027i	$-0.444848\pi$
$557$	192.055	0.344802	0.172401	+	0.985027i	$0.444848\pi$
$558$	0	0
$559$	140.211i	0.250825i
$560$	0	0
$561$	155.531	0.277238
$562$	0	0
$563$	− 107.771i	− 0.191423i	−0.995409	−	0.0957114i	$-0.969487\pi$
$563$	− 107.771i	− 0.191423i	0.995409	−	0.0957114i	$-0.0305126\pi$
$564$	0	0
$565$	246.495	0.436274
$566$	0	0
$567$	603.619i	1.06458i
$568$	0	0
$569$	−647.934	−1.13872	−0.569362	−	0.822087i	$-0.692810\pi$
$569$	−647.934	−1.13872	−0.569362	+	0.822087i	$0.692810\pi$
$570$	0	0
$571$	− 285.402i	− 0.499828i	−0.968268	−	0.249914i	$-0.919598\pi$
$571$	− 285.402i	− 0.499828i	0.968268	−	0.249914i	$-0.0804023\pi$
$572$	0	0
$573$	64.6208	0.112776
$574$	0	0
$575$	734.376i	1.27717i
$576$	0	0
$577$	−92.4624	−0.160247	−0.0801234	−	0.996785i	$-0.525531\pi$
$577$	−92.4624	−0.160247	−0.0801234	+	0.996785i	$0.525531\pi$
$578$	0	0
$579$	− 108.387i	− 0.187197i
$580$	0	0
$581$	817.289	1.40669
$582$	0	0
$583$	− 827.410i	− 1.41923i
$584$	0	0
$585$	−275.817	−0.471482
$586$	0	0
$587$	654.717i	1.11536i	0.830056	+	0.557681i	$0.188309\pi$
$587$	654.717i	1.11536i	−0.830056	+	0.557681i	$0.811691\pi$
$588$	0	0
$589$	−23.7495	−0.0403217
$590$	0	0
$591$	139.710i	0.236396i
$592$	0	0
$593$	106.297	0.179253	0.0896264	−	0.995975i	$-0.471433\pi$
$593$	106.297	0.179253	0.0896264	+	0.995975i	$0.471433\pi$
$594$	0	0
$595$	− 251.381i	− 0.422489i
$596$	0	0
$597$	−53.0298	−0.0888271
$598$	0	0
$599$	− 478.675i	− 0.799124i	−0.916706	−	0.399562i	$-0.869162\pi$
$599$	− 478.675i	− 0.799124i	0.916706	−	0.399562i	$-0.130838\pi$
$600$	0	0
$601$	808.024	1.34447	0.672233	−	0.740340i	$-0.265335\pi$
$601$	808.024	1.34447	0.672233	+	0.740340i	$0.265335\pi$
$602$	0	0
$603$	560.715i	0.929876i
$604$	0	0
$605$	−582.717	−0.963169
$606$	0	0
$607$	694.694i	1.14447i	0.820089	+	0.572236i	$0.193924\pi$
$607$	694.694i	1.14447i	−0.820089	+	0.572236i	$0.806076\pi$
$608$	0	0
$609$	35.4344	0.0581845
$610$	0	0
$611$	447.956i	0.733153i
$612$	0	0
$613$	496.830	0.810489	0.405245	−	0.914208i	$-0.367186\pi$
$613$	496.830	0.810489	0.405245	+	0.914208i	$0.367186\pi$
$614$	0	0
$615$	− 109.274i	− 0.177682i
$616$	0	0
$617$	387.951	0.628770	0.314385	−	0.949296i	$-0.398202\pi$
$617$	387.951	0.628770	0.314385	+	0.949296i	$0.398202\pi$
$618$	0	0
$619$	− 882.105i	− 1.42505i	−0.701647	−	0.712525i	$-0.747552\pi$
$619$	− 882.105i	− 1.42505i	0.701647	−	0.712525i	$-0.252448\pi$
$620$	0	0
$621$	−425.104	−0.684548
$622$	0	0
$623$	287.824i	0.461998i
$624$	0	0
$625$	248.156	0.397050
$626$	0	0
$627$	54.1574i	0.0863754i
$628$	0	0
$629$	261.940	0.416439
$630$	0	0
$631$	− 502.601i	− 0.796515i	−0.917274	−	0.398258i	$-0.869615\pi$
$631$	− 502.601i	− 0.796515i	0.917274	−	0.398258i	$-0.130385\pi$
$632$	0	0
$633$	10.5857	0.0167230
$634$	0	0
$635$	39.8895i	0.0628181i
$636$	0	0
$637$	351.570	0.551915
$638$	0	0
$639$	− 161.006i	− 0.251966i
$640$	0	0
$641$	385.625	0.601600	0.300800	−	0.953687i	$-0.402746\pi$
$641$	385.625	0.601600	0.300800	+	0.953687i	$0.402746\pi$
$642$	0	0
$643$	− 925.293i	− 1.43902i	−0.694480	−	0.719512i	$-0.744365\pi$
$643$	− 925.293i	− 1.43902i	0.694480	−	0.719512i	$-0.255635\pi$
$644$	0	0
$645$	−15.2363	−0.0236222
$646$	0	0
$647$	− 656.956i	− 1.01539i	−0.861538	−	0.507694i	$-0.830498\pi$
$647$	− 656.956i	− 1.01539i	0.861538	−	0.507694i	$-0.169502\pi$
$648$	0	0
$649$	−751.073	−1.15728
$650$	0	0
$651$	− 30.3117i	− 0.0465617i
$652$	0	0
$653$	−392.897	−0.601680	−0.300840	−	0.953675i	$-0.597267\pi$
$653$	−392.897	−0.601680	−0.300840	+	0.953675i	$0.597267\pi$
$654$	0	0
$655$	− 533.701i	− 0.814811i
$656$	0	0
$657$	625.157	0.951533
$658$	0	0
$659$	− 695.576i	− 1.05550i	−0.849399	−	0.527751i	$-0.823035\pi$
$659$	− 695.576i	− 1.05550i	0.849399	−	0.527751i	$-0.176965\pi$
$660$	0	0
$661$	−1179.05	−1.78373	−0.891866	−	0.452299i	$-0.850604\pi$
$661$	−1179.05	−1.78373	−0.891866	+	0.452299i	$0.850604\pi$
$662$	0	0
$663$	− 111.425i	− 0.168062i
$664$	0	0
$665$	87.5335	0.131629
$666$	0	0
$667$	− 238.837i	− 0.358076i
$668$	0	0
$669$	125.863	0.188135
$670$	0	0
$671$	492.606i	0.734137i
$672$	0	0
$673$	−881.170	−1.30932	−0.654658	−	0.755925i	$-0.727187\pi$
$673$	−881.170	−1.30932	−0.654658	+	0.755925i	$0.727187\pi$
$674$	0	0
$675$	222.022i	0.328921i
$676$	0	0
$677$	126.536	0.186907	0.0934533	−	0.995624i	$-0.470209\pi$
$677$	126.536	0.186907	0.0934533	+	0.995624i	$0.470209\pi$
$678$	0	0
$679$	− 1301.44i	− 1.91670i
$680$	0	0
$681$	−196.895	−0.289126
$682$	0	0
$683$	873.042i	1.27825i	0.769105	+	0.639123i	$0.220702\pi$
$683$	873.042i	1.27825i	−0.769105	+	0.639123i	$0.779298\pi$
$684$	0	0
$685$	−280.579	−0.409604
$686$	0	0
$687$	− 148.080i	− 0.215545i
$688$	0	0
$689$	−592.774	−0.860339
$690$	0	0
$691$	610.841i	0.883996i	0.897016	+	0.441998i	$0.145730\pi$
$691$	610.841i	0.883996i	−0.897016	+	0.441998i	$0.854270\pi$
$692$	0	0
$693$	1427.58	2.06000
$694$	0	0
$695$	339.449i	0.488415i
$696$	0	0
$697$	−911.733	−1.30808
$698$	0	0
$699$	− 120.989i	− 0.173089i
$700$	0	0
$701$	506.393	0.722387	0.361193	−	0.932491i	$-0.382369\pi$
$701$	506.393	0.722387	0.361193	+	0.932491i	$0.382369\pi$
$702$	0	0
$703$	91.2102i	0.129744i
$704$	0	0
$705$	−48.6780	−0.0690467
$706$	0	0
$707$	− 671.135i	− 0.949272i
$708$	0	0
$709$	1036.42	1.46180	0.730902	−	0.682483i	$-0.239100\pi$
$709$	1036.42	1.46180	0.730902	+	0.682483i	$0.239100\pi$
$710$	0	0
$711$	− 1200.73i	− 1.68879i
$712$	0	0
$713$	−204.308	−0.286548
$714$	0	0
$715$	619.204i	0.866020i
$716$	0	0
$717$	45.0328	0.0628072
$718$	0	0
$719$	− 834.943i	− 1.16126i	−0.814169	−	0.580628i	$-0.802807\pi$
$719$	− 834.943i	− 1.16126i	0.814169	−	0.580628i	$-0.197193\pi$
$720$	0	0
$721$	128.991	0.178905
$722$	0	0
$723$	− 191.839i	− 0.265337i
$724$	0	0
$725$	−124.739	−0.172053
$726$	0	0
$727$	− 225.252i	− 0.309838i	−0.987927	−	0.154919i	$-0.950488\pi$
$727$	− 225.252i	− 0.309838i	0.987927	−	0.154919i	$-0.0495116\pi$
$728$	0	0
$729$	534.700	0.733470
$730$	0	0
$731$	127.125i	0.173905i
$732$	0	0
$733$	−697.401	−0.951434	−0.475717	−	0.879598i	$-0.657811\pi$
$733$	−697.401	−0.951434	−0.475717	+	0.879598i	$0.657811\pi$
$734$	0	0
$735$	38.2040i	0.0519782i
$736$	0	0
$737$	1258.80	1.70800
$738$	0	0
$739$	1065.87i	1.44231i	0.692772	+	0.721157i	$0.256389\pi$
$739$	1065.87i	1.44231i	−0.692772	+	0.721157i	$0.743611\pi$
$740$	0	0
$741$	38.7994	0.0523609
$742$	0	0
$743$	− 1050.43i	− 1.41376i	−0.707332	−	0.706881i	$-0.750102\pi$
$743$	− 1050.43i	− 1.41376i	0.707332	−	0.706881i	$-0.249898\pi$
$744$	0	0
$745$	−44.8087	−0.0601459
$746$	0	0
$747$	− 813.038i	− 1.08840i
$748$	0	0
$749$	−694.831	−0.927678
$750$	0	0
$751$	− 26.6606i	− 0.0355002i	−0.999842	−	0.0177501i	$-0.994350\pi$
$751$	− 26.6606i	− 0.0355002i	0.999842	−	0.0177501i	$-0.00565033\pi$
$752$	0	0
$753$	−23.5102	−0.0312221
$754$	0	0
$755$	− 385.204i	− 0.510204i
$756$	0	0
$757$	−445.989	−0.589153	−0.294577	−	0.955628i	$-0.595179\pi$
$757$	−445.989	−0.589153	−0.294577	+	0.955628i	$0.595179\pi$
$758$	0	0
$759$	465.897i	0.613830i
$760$	0	0
$761$	294.799	0.387384	0.193692	−	0.981062i	$-0.437954\pi$
$761$	294.799	0.387384	0.193692	+	0.981062i	$0.437954\pi$
$762$	0	0
$763$	− 464.695i	− 0.609037i
$764$	0	0
$765$	−250.074	−0.326894
$766$	0	0
$767$	538.084i	0.701543i
$768$	0	0
$769$	−279.363	−0.363281	−0.181641	−	0.983365i	$-0.558141\pi$
$769$	−279.363	−0.363281	−0.181641	+	0.983365i	$0.558141\pi$
$770$	0	0
$771$	113.431i	0.147122i
$772$	0	0
$773$	1092.48	1.41329	0.706647	−	0.707566i	$-0.250207\pi$
$773$	1092.48	1.41329	0.706647	+	0.707566i	$0.250207\pi$
$774$	0	0
$775$	106.705i	0.137684i
$776$	0	0
$777$	−116.412	−0.149823
$778$	0	0
$779$	− 317.475i	− 0.407542i
$780$	0	0
$781$	−361.456	−0.462811
$782$	0	0
$783$	− 72.2069i	− 0.0922183i
$784$	0	0
$785$	−347.792	−0.443047
$786$	0	0
$787$	− 231.147i	− 0.293706i	−0.989158	−	0.146853i	$-0.953086\pi$
$787$	− 231.147i	− 0.293706i	0.989158	−	0.146853i	$-0.0469145\pi$
$788$	0	0
$789$	286.820	0.363524
$790$	0	0
$791$	914.019i	1.15552i
$792$	0	0
$793$	352.913	0.445035
$794$	0	0
$795$	− 64.4148i	− 0.0810249i
$796$	0	0
$797$	−453.389	−0.568869	−0.284435	−	0.958695i	$-0.591806\pi$
$797$	−453.389	−0.568869	−0.284435	+	0.958695i	$0.591806\pi$
$798$	0	0
$799$	406.146i	0.508318i
$800$	0	0
$801$	286.327	0.357462
$802$	0	0
$803$	− 1403.47i	− 1.74778i
$804$	0	0
$805$	753.020	0.935429
$806$	0	0
$807$	96.5406i	0.119629i
$808$	0	0
$809$	528.874	0.653738	0.326869	−	0.945070i	$-0.394006\pi$
$809$	528.874	0.653738	0.326869	+	0.945070i	$0.394006\pi$
$810$	0	0
$811$	− 1185.82i	− 1.46217i	−0.682287	−	0.731084i	$-0.739015\pi$
$811$	− 1185.82i	− 1.46217i	0.682287	−	0.731084i	$-0.260985\pi$
$812$	0	0
$813$	283.467	0.348668
$814$	0	0
$815$	341.539i	0.419066i
$816$	0	0
$817$	−44.2661	−0.0541813
$818$	0	0
$819$	− 1022.75i	− 1.24878i
$820$	0	0
$821$	570.471	0.694849	0.347424	−	0.937708i	$-0.387056\pi$
$821$	570.471	0.694849	0.347424	+	0.937708i	$0.387056\pi$
$822$	0	0
$823$	− 1386.72i	− 1.68496i	−0.538728	−	0.842480i	$-0.681095\pi$
$823$	− 1386.72i	− 1.68496i	0.538728	−	0.842480i	$-0.318905\pi$
$824$	0	0
$825$	243.327	0.294942
$826$	0	0
$827$	− 1073.59i	− 1.29818i	−0.760712	−	0.649089i	$-0.775150\pi$
$827$	− 1073.59i	− 1.29818i	0.760712	−	0.649089i	$-0.224850\pi$
$828$	0	0
$829$	−1109.24	−1.33804	−0.669021	−	0.743243i	$-0.733287\pi$
$829$	−1109.24	−1.33804	−0.669021	+	0.743243i	$0.733287\pi$
$830$	0	0
$831$	53.6685i	0.0645830i
$832$	0	0
$833$	318.756	0.382660
$834$	0	0
$835$	− 496.831i	− 0.595007i
$836$	0	0
$837$	−61.7681	−0.0737970
$838$	0	0
$839$	1547.29i	1.84421i	0.386943	+	0.922104i	$0.373531\pi$
$839$	1547.29i	1.84421i	−0.386943	+	0.922104i	$0.626469\pi$
$840$	0	0
$841$	−800.432	−0.951762
$842$	0	0
$843$	8.37157i	0.00993069i
$844$	0	0
$845$	50.3218	0.0595524
$846$	0	0
$847$	− 2160.76i	− 2.55107i
$848$	0	0
$849$	−71.5762	−0.0843065
$850$	0	0
$851$	784.649i	0.922031i
$852$	0	0
$853$	341.108	0.399892	0.199946	−	0.979807i	$-0.435923\pi$
$853$	341.108	0.399892	0.199946	+	0.979807i	$0.435923\pi$
$854$	0	0
$855$	− 87.0782i	− 0.101846i
$856$	0	0
$857$	1535.74	1.79200	0.896000	−	0.444054i	$-0.146460\pi$
$857$	1535.74	1.79200	0.896000	+	0.444054i	$0.146460\pi$
$858$	0	0
$859$	1223.16i	1.42393i	0.702213	+	0.711967i	$0.252196\pi$
$859$	1223.16i	1.42393i	−0.702213	+	0.711967i	$0.747804\pi$
$860$	0	0
$861$	405.196	0.470611
$862$	0	0
$863$	672.603i	0.779377i	0.920947	+	0.389689i	$0.127417\pi$
$863$	672.603i	0.779377i	−0.920947	+	0.389689i	$0.872583\pi$
$864$	0	0
$865$	653.266	0.755221
$866$	0	0
$867$	85.2940i	0.0983783i
$868$	0	0
$869$	−2695.62	−3.10197
$870$	0	0
$871$	− 901.827i	− 1.03539i
$872$	0	0
$873$	−1294.67	−1.48301
$874$	0	0
$875$	− 895.324i	− 1.02323i
$876$	0	0
$877$	1328.68	1.51503	0.757514	−	0.652819i	$-0.226414\pi$
$877$	1328.68	1.51503	0.757514	+	0.652819i	$0.226414\pi$
$878$	0	0
$879$	− 91.2717i	− 0.103836i
$880$	0	0
$881$	372.924	0.423296	0.211648	−	0.977346i	$-0.432117\pi$
$881$	372.924	0.423296	0.211648	+	0.977346i	$0.432117\pi$
$882$	0	0
$883$	− 669.314i	− 0.758000i	−0.925397	−	0.379000i	$-0.876268\pi$
$883$	− 669.314i	− 0.758000i	0.925397	−	0.379000i	$-0.123732\pi$
$884$	0	0
$885$	−58.4718	−0.0660699
$886$	0	0
$887$	874.225i	0.985598i	0.870143	+	0.492799i	$0.164026\pi$
$887$	874.225i	0.985598i	−0.870143	+	0.492799i	$0.835974\pi$
$888$	0	0
$889$	−147.913	−0.166381
$890$	0	0
$891$	− 1348.06i	− 1.51298i
$892$	0	0
$893$	−141.424	−0.158370
$894$	0	0
$895$	− 81.8873i	− 0.0914942i
$896$	0	0
$897$	333.778	0.372105
$898$	0	0
$899$	− 34.7032i	− 0.0386020i
$900$	0	0
$901$	−537.447	−0.596500
$902$	0	0
$903$	− 56.4973i	− 0.0625662i
$904$	0	0
$905$	804.123	0.888534
$906$	0	0
$907$	− 33.1952i	− 0.0365989i	−0.999833	−	0.0182994i	$-0.994175\pi$
$907$	− 33.1952i	− 0.0365989i	0.999833	−	0.0182994i	$-0.00582522\pi$
$908$	0	0
$909$	−667.644	−0.734482
$910$	0	0
$911$	− 25.4273i	− 0.0279114i	−0.999903	−	0.0139557i	$-0.995558\pi$
$911$	− 25.4273i	− 0.0279114i	0.999903	−	0.0139557i	$-0.00444238\pi$
$912$	0	0
$913$	−1825.26	−1.99918
$914$	0	0
$915$	38.3499i	0.0419125i
$916$	0	0
$917$	1979.00	2.15812
$918$	0	0
$919$	445.408i	0.484666i	0.970193	+	0.242333i	$0.0779126\pi$
$919$	445.408i	0.484666i	−0.970193	+	0.242333i	$0.922087\pi$
$920$	0	0
$921$	−230.639	−0.250422
$922$	0	0
$923$	258.954i	0.280557i
$924$	0	0
$925$	409.804	0.443031
$926$	0	0
$927$	− 128.320i	− 0.138425i
$928$	0	0
$929$	−408.473	−0.439691	−0.219846	−	0.975535i	$-0.570555\pi$
$929$	−408.473	−0.439691	−0.219846	+	0.975535i	$0.570555\pi$
$930$	0	0
$931$	110.994i	0.119220i
$932$	0	0
$933$	−304.547	−0.326417
$934$	0	0
$935$	561.411i	0.600439i
$936$	0	0
$937$	−1101.68	−1.17576	−0.587878	−	0.808950i	$-0.700037\pi$
$937$	−1101.68	−1.17576	−0.587878	+	0.808950i	$0.700037\pi$
$938$	0	0
$939$	− 41.1060i	− 0.0437764i
$940$	0	0
$941$	1342.45	1.42662	0.713309	−	0.700850i	$-0.247196\pi$
$941$	1342.45	1.42662	0.713309	+	0.700850i	$0.247196\pi$
$942$	0	0
$943$	− 2731.12i	− 2.89621i
$944$	0	0
$945$	227.659	0.240909
$946$	0	0
$947$	786.294i	0.830300i	0.909753	+	0.415150i	$0.136271\pi$
$947$	786.294i	0.830300i	−0.909753	+	0.415150i	$0.863729\pi$
$948$	0	0
$949$	−1005.47	−1.05951
$950$	0	0
$951$	128.116i	0.134717i
$952$	0	0
$953$	1134.64	1.19060	0.595301	−	0.803503i	$-0.297033\pi$
$953$	1134.64	1.19060	0.595301	+	0.803503i	$0.297033\pi$
$954$	0	0
$955$	233.258i	0.244250i
$956$	0	0
$957$	−79.1357	−0.0826915
$958$	0	0
$959$	− 1040.40i	− 1.08488i
$960$	0	0
$961$	931.314	0.969109
$962$	0	0
$963$	691.217i	0.717775i
$964$	0	0
$965$	391.240	0.405430
$966$	0	0
$967$	963.333i	0.996208i	0.867117	+	0.498104i	$0.165970\pi$
$967$	963.333i	0.996208i	−0.867117	+	0.498104i	$0.834030\pi$
$968$	0	0
$969$	35.1781	0.0363035
$970$	0	0
$971$	1320.90i	1.36035i	0.733050	+	0.680174i	$0.238096\pi$
$971$	1320.90i	1.36035i	−0.733050	+	0.680174i	$0.761904\pi$
$972$	0	0
$973$	−1258.70	−1.29363
$974$	0	0
$975$	− 174.324i	− 0.178794i
$976$	0	0
$977$	−1159.80	−1.18710	−0.593550	−	0.804797i	$-0.702274\pi$
$977$	−1159.80	−1.18710	−0.593550	+	0.804797i	$0.702274\pi$
$978$	0	0
$979$	− 642.800i	− 0.656588i
$980$	0	0
$981$	−462.278	−0.471232
$982$	0	0
$983$	8.37645i	0.00852131i	0.999991	+	0.00426065i	$0.00135621\pi$
$983$	8.37645i	0.00852131i	−0.999991	+	0.00426065i	$0.998644\pi$
$984$	0	0
$985$	−504.305	−0.511985
$986$	0	0
$987$	− 180.501i	− 0.182879i
$988$	0	0
$989$	−380.806	−0.385041
$990$	0	0
$991$	− 854.355i	− 0.862114i	−0.902325	−	0.431057i	$-0.858141\pi$
$991$	− 854.355i	− 0.862114i	0.902325	−	0.431057i	$-0.141859\pi$
$992$	0	0
$993$	282.855	0.284849
$994$	0	0
$995$	− 191.419i	− 0.192381i
$996$	0	0
$997$	77.4664	0.0776995	0.0388497	−	0.999245i	$-0.487631\pi$
$997$	77.4664	0.0776995	0.0388497	+	0.999245i	$0.487631\pi$
$998$	0	0
$999$	237.221i	0.237458i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.d.191.7		14
4.3	odd	2	inner	1216.3.d.d.191.8		14
8.3	odd	2		76.3.b.b.39.2	yes	14
8.5	even	2		76.3.b.b.39.1	✓	14
24.5	odd	2		684.3.g.b.343.14		14
24.11	even	2		684.3.g.b.343.13		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.b.b.39.1	✓	14	8.5	even	2
76.3.b.b.39.2	yes	14	8.3	odd	2
684.3.g.b.343.13		14	24.11	even	2
684.3.g.b.343.14		14	24.5	odd	2
1216.3.d.d.191.7		14	1.1	even	1	trivial
1216.3.d.d.191.8		14	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.644704i q^{3} +2.32715 q^{5} +8.62924i q^{7} +8.58436 q^{9} +O(q^{10})\)	\(q-0.644704i q^{3} +2.32715 q^{5} +8.62924i q^{7} +8.58436 q^{9} -19.2717i q^{11} -13.8067 q^{13} -1.50032i q^{15} -12.5180 q^{17} -4.35890i q^{19} +5.56330 q^{21} -37.4981i q^{23} -19.5844 q^{25} -11.3367i q^{27} +6.36930 q^{29} -5.44851i q^{31} -12.4245 q^{33} +20.0816i q^{35} -20.9250 q^{37} +8.90120i q^{39} +72.8337 q^{41} -10.1553i q^{43} +19.9771 q^{45} -32.4450i q^{47} -25.4638 q^{49} +8.07040i q^{51} +42.9339 q^{53} -44.8482i q^{55} -2.81020 q^{57} -38.9728i q^{59} -25.5611 q^{61} +74.0765i q^{63} -32.1302 q^{65} +65.3183i q^{67} -24.1751 q^{69} -18.7557i q^{71} +72.8251 q^{73} +12.6261i q^{75} +166.300 q^{77} -139.874i q^{79} +69.9504 q^{81} -94.7116i q^{83} -29.1313 q^{85} -4.10631i q^{87} +33.3546 q^{89} -119.141i q^{91} -3.51267 q^{93} -10.1438i q^{95} -150.817 q^{97} -165.435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 68 q^{9}+O(q^{10}) \) 14 * q - 68 * q^9	\( 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) \) 14 * q - 68 * q^9 - 54 * q^13 + 34 * q^17 + 38 * q^21 - 86 * q^25 - 54 * q^29 + 20 * q^33 - 100 * q^37 + 224 * q^41 + 168 * q^45 - 220 * q^49 - 14 * q^53 + 38 * q^57 - 28 * q^61 - 472 * q^65 - 122 * q^69 + 70 * q^73 - 228 * q^77 + 334 * q^81 - 48 * q^85 + 176 * q^93 + 308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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