LMFDB - Embedded newform 1216.2.h.d.1215.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1215,1216)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1216.1215");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.h (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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14453810176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 6x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1215.6
Root			$-1.06789 + 0.927153i$ of defining polynomial
Character	$\chi$	$=$	1216.1215
Dual form			1216.2.h.d.1215.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.19935 q^{3} -1.56155 q^{5} +0.868210i q^{7} -1.56155 q^{9} +O(q^{10})$	$q+1.19935 q^{3} -1.56155 q^{5} +0.868210i q^{7} -1.56155 q^{9} -3.09218i q^{11} -4.74990i q^{13} -1.87285 q^{15} -1.00000 q^{17} +(-3.07221 + 3.09218i) q^{19} +1.04129i q^{21} -3.96039i q^{23} -2.56155 q^{25} -5.47091 q^{27} +8.45851i q^{29} -4.27156 q^{31} -3.70861i q^{33} -1.35576i q^{35} -3.70861i q^{37} -5.69681i q^{39} -3.70861i q^{41} -11.0129i q^{43} +2.43845 q^{45} -9.27653i q^{47} +6.24621 q^{49} -1.19935 q^{51} -1.04129i q^{53} +4.82860i q^{55} +(-3.68466 + 3.70861i) q^{57} -11.6153 q^{59} +0.684658 q^{61} -1.35576i q^{63} +7.41722i q^{65} +9.74247 q^{67} -4.74990i q^{69} -10.9418 q^{71} +8.12311 q^{73} -3.07221 q^{75} +2.68466 q^{77} -8.01726 q^{79} -1.87689 q^{81} -9.65719i q^{83} +1.56155 q^{85} +10.1447i q^{87} -5.79119i q^{89} +4.12391 q^{91} -5.12311 q^{93} +(4.79741 - 4.82860i) q^{95} +16.9170i q^{97} +4.82860i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^5 + 4 * q^9	$ 8 q + 4 q^{5} + 4 q^{9} - 8 q^{17} - 4 q^{25} + 36 q^{45} - 16 q^{49} + 20 q^{57} - 44 q^{61} + 32 q^{73} - 28 q^{77} - 48 q^{81} - 4 q^{85} - 8 q^{93}+O(q^{100}) $ 8 * q + 4 * q^5 + 4 * q^9 - 8 * q^17 - 4 * q^25 + 36 * q^45 - 16 * q^49 + 20 * q^57 - 44 * q^61 + 32 * q^73 - 28 * q^77 - 48 * q^81 - 4 * q^85 - 8 * q^93

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
$3$	1.19935			0.692447			0.346223	−	0.938152i	$-0.387464\pi$
$3$	1.19935			0.692447			0.346223	+	0.938152i	$0.387464\pi$
$4$		0			0
$5$	−1.56155			−0.698348			−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155			−0.698348			−0.349174	+	0.937058i	$0.613538\pi$
$6$		0			0
$7$			0.868210i			0.328153i	0.986448	+	0.164076i	$0.0524643\pi$
$7$			0.868210i			0.328153i	−0.986448	+	0.164076i	$0.947536\pi$
$8$		0			0
$9$	−1.56155			−0.520518
$10$		0			0
$11$		−	3.09218i		−	0.932326i	−0.884699	−	0.466163i	$-0.845636\pi$
$11$		−	3.09218i		−	0.932326i	0.884699	−	0.466163i	$-0.154364\pi$
$12$		0			0
$13$		−	4.74990i		−	1.31739i	−0.752412	−	0.658693i	$-0.771110\pi$
$13$		−	4.74990i		−	1.31739i	0.752412	−	0.658693i	$-0.228890\pi$
$14$		0			0
$15$	−1.87285			−0.483569
$16$		0			0
$17$	−1.00000			−0.242536			−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000			−0.242536			−0.121268	+	0.992620i	$0.538696\pi$
$18$		0			0
$19$	−3.07221	+	3.09218i	−0.704812	+	0.709394i
$19$	−3.07221	+	3.09218i	−0.704812	+	0.709394i
$20$		0			0
$21$			1.04129i			0.227228i
$22$		0			0
$23$		−	3.96039i		−	0.825798i	−0.910777	−	0.412899i	$-0.864516\pi$
$23$		−	3.96039i		−	0.825798i	0.910777	−	0.412899i	$-0.135484\pi$
$24$		0			0
$25$	−2.56155			−0.512311
$26$		0			0
$27$	−5.47091			−1.05288
$28$		0			0
$29$			8.45851i			1.57071i	0.619048	+	0.785353i	$0.287519\pi$
$29$			8.45851i			1.57071i	−0.619048	+	0.785353i	$0.712481\pi$
$30$		0			0
$31$	−4.27156			−0.767195			−0.383597	−	0.923500i	$-0.625315\pi$
$31$	−4.27156			−0.767195			−0.383597	+	0.923500i	$0.625315\pi$
$32$		0			0
$33$		−	3.70861i		−	0.645586i
$34$		0			0
$35$		−	1.35576i		−	0.229165i
$36$		0			0
$37$		−	3.70861i		−	0.609692i	−0.952402	−	0.304846i	$-0.901395\pi$
$37$		−	3.70861i		−	0.609692i	0.952402	−	0.304846i	$-0.0986050\pi$
$38$		0			0
$39$		−	5.69681i		−	0.912219i
$40$		0			0
$41$		−	3.70861i		−	0.579188i	−0.957150	−	0.289594i	$-0.906480\pi$
$41$		−	3.70861i		−	0.579188i	0.957150	−	0.289594i	$-0.0935202\pi$
$42$		0			0
$43$		−	11.0129i		−	1.67946i	−0.543005	−	0.839729i	$-0.682714\pi$
$43$		−	11.0129i		−	1.67946i	0.543005	−	0.839729i	$-0.317286\pi$
$44$		0			0
$45$	2.43845			0.363502
$46$		0			0
$47$		−	9.27653i		−	1.35312i	−0.736387	−	0.676560i	$-0.763470\pi$
$47$		−	9.27653i		−	1.35312i	0.736387	−	0.676560i	$-0.236530\pi$
$48$		0			0
$49$	6.24621			0.892316
$50$		0			0
$51$	−1.19935			−0.167943
$52$		0			0
$53$		−	1.04129i		−	0.143032i	−0.997439	−	0.0715161i	$-0.977216\pi$
$53$		−	1.04129i		−	0.143032i	0.997439	−	0.0715161i	$-0.0227837\pi$
$54$		0			0
$55$			4.82860i			0.651088i
$56$		0			0
$57$	−3.68466	+	3.70861i	−0.488045	+	0.491217i
$58$		0			0
$59$	−11.6153			−1.51219			−0.756093	−	0.654464i	$-0.772894\pi$
$59$	−11.6153			−1.51219			−0.756093	+	0.654464i	$0.772894\pi$
$60$		0			0
$61$	0.684658			0.0876615			0.0438308	−	0.999039i	$-0.486044\pi$
$61$	0.684658			0.0876615			0.0438308	+	0.999039i	$0.486044\pi$
$62$		0			0
$63$		−	1.35576i		−	0.170809i
$64$		0			0
$65$			7.41722i			0.919993i
$66$		0			0
$67$	9.74247			1.19023			0.595116	−	0.803640i	$-0.297106\pi$
$67$	9.74247			1.19023			0.595116	+	0.803640i	$0.297106\pi$
$68$		0			0
$69$		−	4.74990i		−	0.571821i
$70$		0			0
$71$	−10.9418			−1.29856			−0.649278	−	0.760551i	$-0.724929\pi$
$71$	−10.9418			−1.29856			−0.649278	+	0.760551i	$0.724929\pi$
$72$		0			0
$73$	8.12311			0.950738			0.475369	−	0.879787i	$-0.342315\pi$
$73$	8.12311			0.950738			0.475369	+	0.879787i	$0.342315\pi$
$74$		0			0
$75$	−3.07221			−0.354748
$76$		0			0
$77$	2.68466			0.305945
$78$		0			0
$79$	−8.01726			−0.902013			−0.451006	−	0.892521i	$-0.648935\pi$
$79$	−8.01726			−0.902013			−0.451006	+	0.892521i	$0.648935\pi$
$80$		0			0
$81$	−1.87689			−0.208544
$82$		0			0
$83$		−	9.65719i		−	1.06001i	−0.847993	−	0.530007i	$-0.822189\pi$
$83$		−	9.65719i		−	1.06001i	0.847993	−	0.530007i	$-0.177811\pi$
$84$		0			0
$85$	1.56155			0.169374
$86$		0			0
$87$			10.1447i			1.08763i
$88$		0			0
$89$		−	5.79119i		−	0.613865i	−0.951731	−	0.306932i	$-0.900697\pi$
$89$		−	5.79119i		−	0.613865i	0.951731	−	0.306932i	$-0.0993026\pi$
$90$		0			0
$91$	4.12391			0.432303
$92$		0			0
$93$	−5.12311			−0.531241
$94$		0			0
$95$	4.79741	−	4.82860i	0.492204	−	0.495404i
$96$		0			0
$97$			16.9170i			1.71766i	0.512258	+	0.858832i	$0.328809\pi$
$97$			16.9170i			1.71766i	−0.512258	+	0.858832i	$0.671191\pi$
$98$		0			0
$99$			4.82860i			0.485292i
$100$		0			0
$101$	−8.24621			−0.820529			−0.410264	−	0.911967i	$-0.634564\pi$
$101$	−8.24621			−0.820529			−0.410264	+	0.911967i	$0.634564\pi$
$102$		0			0
$103$	3.74571			0.369075			0.184538	−	0.982825i	$-0.440921\pi$
$103$	3.74571			0.369075			0.184538	+	0.982825i	$0.440921\pi$
$104$		0			0
$105$		−	1.62603i		−	0.158684i
$106$		0			0
$107$	−7.86962			−0.760785			−0.380392	−	0.924825i	$-0.624211\pi$
$107$	−7.86962			−0.760785			−0.380392	+	0.924825i	$0.624211\pi$
$108$		0			0
$109$			15.8757i			1.52062i	0.649561	+	0.760310i	$0.274953\pi$
$109$			15.8757i			1.52062i	−0.649561	+	0.760310i	$0.725047\pi$
$110$		0			0
$111$		−	4.44793i		−	0.422179i
$112$		0			0
$113$			3.70861i			0.348877i	0.984668	+	0.174438i	$0.0558110\pi$
$113$			3.70861i			0.348877i	−0.984668	+	0.174438i	$0.944189\pi$
$114$		0			0
$115$			6.18435i			0.576694i
$116$		0			0
$117$			7.41722i			0.685722i
$118$		0			0
$119$		−	0.868210i		−	0.0795887i
$120$		0			0
$121$	1.43845			0.130768
$122$		0			0
$123$		−	4.44793i		−	0.401057i
$124$		0			0
$125$	11.8078			1.05612
$126$		0			0
$127$	1.87285			0.166189			0.0830944	−	0.996542i	$-0.473520\pi$
$127$	1.87285			0.166189			0.0830944	+	0.996542i	$0.473520\pi$
$128$		0			0
$129$		−	13.2084i		−	1.16294i
$130$		0			0
$131$			4.82860i			0.421876i	0.977499	+	0.210938i	$0.0676519\pi$
$131$			4.82860i			0.421876i	−0.977499	+	0.210938i	$0.932348\pi$
$132$		0			0
$133$	−2.68466	−	2.66732i	−0.232789	−	0.231286i
$134$		0			0
$135$	8.54312			0.735274
$136$		0			0
$137$	−3.87689			−0.331225			−0.165613	−	0.986191i	$-0.552960\pi$
$137$	−3.87689			−0.331225			−0.165613	+	0.986191i	$0.552960\pi$
$138$		0			0
$139$			0.380664i			0.0322875i	0.999870	+	0.0161438i	$0.00513894\pi$
$139$			0.380664i			0.0322875i	−0.999870	+	0.0161438i	$0.994861\pi$
$140$		0			0
$141$		−	11.1258i		−	0.936964i
$142$		0			0
$143$	−14.6875			−1.22823
$144$		0			0
$145$		−	13.2084i		−	1.09690i
$146$		0			0
$147$	7.49141			0.617881
$148$		0			0
$149$	17.8078			1.45887			0.729434	−	0.684051i	$-0.239783\pi$
$149$	17.8078			1.45887			0.729434	+	0.684051i	$0.239783\pi$
$150$		0			0
$151$	24.2824			1.97607			0.988035	−	0.154231i	$-0.0492900\pi$
$151$	24.2824			1.97607			0.988035	+	0.154231i	$0.0492900\pi$
$152$		0			0
$153$	1.56155			0.126244
$154$		0			0
$155$	6.67026			0.535769
$156$		0			0
$157$	−6.00000			−0.478852			−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000			−0.478852			−0.239426	+	0.970915i	$0.576959\pi$
$158$		0			0
$159$		−	1.24887i		−	0.0990422i
$160$		0			0
$161$	3.43845			0.270988
$162$		0			0
$163$			10.6323i			0.832785i	0.909185	+	0.416392i	$0.136706\pi$
$163$			10.6323i			0.832785i	−0.909185	+	0.416392i	$0.863294\pi$
$164$		0			0
$165$			5.79119i			0.450844i
$166$		0			0
$167$	0.525853			0.0406917			0.0203459	−	0.999793i	$-0.493523\pi$
$167$	0.525853			0.0406917			0.0203459	+	0.999793i	$0.493523\pi$
$168$		0			0
$169$	−9.56155			−0.735504
$170$		0			0
$171$	4.79741	−	4.82860i	0.366867	−	0.369252i
$172$		0			0
$173$		−	16.9170i		−	1.28618i	−0.765792	−	0.643089i	$-0.777653\pi$
$173$		−	16.9170i		−	1.28618i	0.765792	−	0.643089i	$-0.222347\pi$
$174$		0			0
$175$		−	2.22397i		−	0.168116i
$176$		0			0
$177$	−13.9309			−1.04711
$178$		0			0
$179$	−9.06897			−0.677847			−0.338923	−	0.940814i	$-0.610063\pi$
$179$	−9.06897			−0.677847			−0.338923	+	0.940814i	$0.610063\pi$
$180$		0			0
$181$		−	2.08258i		−	0.154797i	−0.997000	−	0.0773985i	$-0.975339\pi$
$181$		−	2.08258i		−	0.154797i	0.997000	−	0.0773985i	$-0.0246614\pi$
$182$		0			0
$183$	0.821147			0.0607009
$184$		0			0
$185$			5.79119i			0.425777i
$186$		0			0
$187$			3.09218i			0.226122i
$188$		0			0
$189$		−	4.74990i		−	0.345504i
$190$		0			0
$191$			8.78898i			0.635948i	0.948099	+	0.317974i	$0.103003\pi$
$191$			8.78898i			0.635948i	−0.948099	+	0.317974i	$0.896997\pi$
$192$		0			0
$193$		−	3.70861i		−	0.266952i	−0.991052	−	0.133476i	$-0.957386\pi$
$193$		−	3.70861i		−	0.266952i	0.991052	−	0.133476i	$-0.0426138\pi$
$194$		0			0
$195$			8.89586i			0.637046i
$196$		0			0
$197$	−10.4924			−0.747554			−0.373777	−	0.927519i	$-0.621937\pi$
$197$	−10.4924			−0.747554			−0.373777	+	0.927519i	$0.621937\pi$
$198$		0			0
$199$			13.2369i			0.938340i	0.883108	+	0.469170i	$0.155447\pi$
$199$			13.2369i			0.938340i	−0.883108	+	0.469170i	$0.844553\pi$
$200$		0			0
$201$	11.6847			0.824172
$202$		0			0
$203$	−7.34376			−0.515431
$204$		0			0
$205$			5.79119i			0.404474i
$206$		0			0
$207$			6.18435i			0.429842i
$208$		0			0
$209$	9.56155	+	9.49980i	0.661386	+	0.657115i
$210$		0			0
$211$	2.02050			0.139097			0.0695485	−	0.997579i	$-0.477844\pi$
$211$	2.02050			0.139097			0.0695485	+	0.997579i	$0.477844\pi$
$212$		0			0
$213$	−13.1231			−0.899180
$214$		0			0
$215$			17.1973i			1.17285i
$216$		0			0
$217$		−	3.70861i		−	0.251757i
$218$		0			0
$219$	9.74247			0.658335
$220$		0			0
$221$			4.74990i			0.319513i
$222$		0			0
$223$	−16.2651			−1.08919			−0.544595	−	0.838699i	$-0.683317\pi$
$223$	−16.2651			−1.08919			−0.544595	+	0.838699i	$0.683317\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$	17.4644			1.15916			0.579578	−	0.814917i	$-0.303217\pi$
$227$	17.4644			1.15916			0.579578	+	0.814917i	$0.303217\pi$
$228$		0			0
$229$	2.43845			0.161137			0.0805686	−	0.996749i	$-0.474326\pi$
$229$	2.43845			0.161137			0.0805686	+	0.996749i	$0.474326\pi$
$230$		0			0
$231$	3.21985			0.211851
$232$		0			0
$233$	−9.80776			−0.642528			−0.321264	−	0.946990i	$-0.604108\pi$
$233$	−9.80776			−0.642528			−0.321264	+	0.946990i	$0.604108\pi$
$234$		0			0
$235$			14.4858i			0.944949i
$236$		0			0
$237$	−9.61553			−0.624596
$238$		0			0
$239$		−	19.4213i		−	1.25626i	−0.778110	−	0.628129i	$-0.783821\pi$
$239$		−	19.4213i		−	1.25626i	0.778110	−	0.628129i	$-0.216179\pi$
$240$		0			0
$241$			15.2910i			0.984979i	0.870318	+	0.492490i	$0.163913\pi$
$241$			15.2910i			0.984979i	−0.870318	+	0.492490i	$0.836087\pi$
$242$		0			0
$243$	14.1617			0.908472
$244$		0			0
$245$	−9.75379			−0.623147
$246$		0			0
$247$	14.6875	+	14.5927i	0.934545	+	0.928509i
$248$		0			0
$249$		−	11.5824i		−	0.734004i
$250$		0			0
$251$		−	22.4066i		−	1.41429i	−0.707069	−	0.707145i	$-0.749983\pi$
$251$		−	22.4066i		−	1.41429i	0.707069	−	0.707145i	$-0.250017\pi$
$252$		0			0
$253$	−12.2462			−0.769913
$254$		0			0
$255$	1.87285			0.117283
$256$		0			0
$257$		−	18.9996i		−	1.18516i	−0.805511	−	0.592581i	$-0.798109\pi$
$257$		−	18.9996i		−	1.18516i	0.805511	−	0.592581i	$-0.201891\pi$
$258$		0			0
$259$	3.21985			0.200072
$260$		0			0
$261$		−	13.2084i		−	0.817580i
$262$		0			0
$263$			6.56502i			0.404816i	0.979301	+	0.202408i	$0.0648768\pi$
$263$			6.56502i			0.404816i	−0.979301	+	0.202408i	$0.935123\pi$
$264$		0			0
$265$			1.62603i			0.0998862i
$266$		0			0
$267$		−	6.94568i		−	0.425069i
$268$		0			0
$269$			3.70861i			0.226118i	0.993588	+	0.113059i	$0.0360649\pi$
$269$			3.70861i			0.226118i	−0.993588	+	0.113059i	$0.963935\pi$
$270$		0			0
$271$		−	15.3540i		−	0.932689i	−0.884603	−	0.466345i	$-0.845571\pi$
$271$		−	15.3540i		−	0.932689i	0.884603	−	0.466345i	$-0.154429\pi$
$272$		0			0
$273$	4.94602			0.299347
$274$		0			0
$275$			7.92077i			0.477641i
$276$		0			0
$277$	24.0540			1.44526			0.722632	−	0.691233i	$-0.242932\pi$
$277$	24.0540			1.44526			0.722632	+	0.691233i	$0.242932\pi$
$278$		0			0
$279$	6.67026			0.399338
$280$		0			0
$281$			11.5824i			0.690947i	0.938429	+	0.345473i	$0.112282\pi$
$281$			11.5824i			0.690947i	−0.938429	+	0.345473i	$0.887718\pi$
$282$		0			0
$283$			4.82860i			0.287030i	0.989648	+	0.143515i	$0.0458406\pi$
$283$			4.82860i			0.287030i	−0.989648	+	0.143515i	$0.954159\pi$
$284$		0			0
$285$	5.75379	−	5.79119i	0.340825	−	0.343041i
$286$		0			0
$287$	3.21985			0.190062
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$			20.2895i			1.18939i
$292$		0			0
$293$			4.74990i			0.277492i	0.990328	+	0.138746i	$0.0443072\pi$
$293$			4.74990i			0.277492i	−0.990328	+	0.138746i	$0.955693\pi$
$294$		0			0
$295$	18.1379			1.05603
$296$		0			0
$297$			16.9170i			0.981625i
$298$		0			0
$299$	−18.8114			−1.08789
$300$		0			0
$301$	9.56155			0.551119
$302$		0			0
$303$	−9.89012			−0.568172
$304$		0			0
$305$	−1.06913			−0.0612182
$306$		0			0
$307$	8.01726			0.457569			0.228785	−	0.973477i	$-0.426525\pi$
$307$	8.01726			0.457569			0.228785	+	0.973477i	$0.426525\pi$
$308$		0			0
$309$	4.49242			0.255565
$310$		0			0
$311$			0.868210i			0.0492317i	0.999697	+	0.0246158i	$0.00783626\pi$
$311$			0.868210i			0.0492317i	−0.999697	+	0.0246158i	$0.992164\pi$
$312$		0			0
$313$	−4.56155			−0.257834			−0.128917	−	0.991655i	$-0.541150\pi$
$313$	−4.56155			−0.257834			−0.128917	+	0.991655i	$0.541150\pi$
$314$		0			0
$315$			2.11708i			0.119284i
$316$		0			0
$317$		−	4.74990i		−	0.266781i	−0.991064	−	0.133390i	$-0.957414\pi$
$317$		−	4.74990i		−	0.266781i	0.991064	−	0.133390i	$-0.0425864\pi$
$318$		0			0
$319$	26.1552			1.46441
$320$		0			0
$321$	−9.43845			−0.526803
$322$		0			0
$323$	3.07221	−	3.09218i	0.170942	−	0.172053i
$324$		0			0
$325$			12.1671i			0.674910i
$326$		0			0
$327$			19.0406i			1.05295i
$328$		0			0
$329$	8.05398			0.444030
$330$		0			0
$331$	−2.25106			−0.123729			−0.0618647	−	0.998085i	$-0.519705\pi$
$331$	−2.25106			−0.123729			−0.0618647	+	0.998085i	$0.519705\pi$
$332$		0			0
$333$			5.79119i			0.317355i
$334$		0			0
$335$	−15.2134			−0.831196
$336$		0			0
$337$		−	26.4168i		−	1.43902i	−0.694484	−	0.719508i	$-0.744367\pi$
$337$		−	26.4168i		−	1.43902i	0.694484	−	0.719508i	$-0.255633\pi$
$338$		0			0
$339$			4.44793i			0.241579i
$340$		0			0
$341$			13.2084i			0.715276i
$342$		0			0
$343$			11.5005i			0.620968i
$344$		0			0
$345$			7.41722i			0.399330i
$346$		0			0
$347$			24.1430i			1.29606i	0.761613	+	0.648032i	$0.224408\pi$
$347$			24.1430i			1.29606i	−0.761613	+	0.648032i	$0.775592\pi$
$348$		0			0
$349$	28.0540			1.50169			0.750847	−	0.660476i	$-0.229645\pi$
$349$	28.0540			1.50169			0.750847	+	0.660476i	$0.229645\pi$
$350$		0			0
$351$			25.9863i			1.38705i
$352$		0			0
$353$	6.31534			0.336132			0.168066	−	0.985776i	$-0.446248\pi$
$353$	6.31534			0.336132			0.168066	+	0.985776i	$0.446248\pi$
$354$		0			0
$355$	17.0862			0.906843
$356$		0			0
$357$		−	1.04129i		−	0.0551109i
$358$		0			0
$359$		−	27.3420i		−	1.44306i	−0.692384	−	0.721529i	$-0.743440\pi$
$359$		−	27.3420i		−	1.44306i	0.692384	−	0.721529i	$-0.256560\pi$
$360$		0			0
$361$	−0.123106	−	18.9996i	−0.00647924	−	0.999979i
$362$		0			0
$363$	1.72521			0.0905498
$364$		0			0
$365$	−12.6847			−0.663945
$366$		0			0
$367$		−	14.1051i		−	0.736282i	−0.929770	−	0.368141i	$-0.879994\pi$
$367$		−	14.1051i		−	0.736282i	0.929770	−	0.368141i	$-0.120006\pi$
$368$		0			0
$369$			5.79119i			0.301477i
$370$		0			0
$371$	0.904059			0.0469364
$372$		0			0
$373$			12.1671i			0.629990i	0.949093	+	0.314995i	$0.102003\pi$
$373$			12.1671i			0.629990i	−0.949093	+	0.314995i	$0.897997\pi$
$374$		0			0
$375$	14.1617			0.731306
$376$		0			0
$377$	40.1771			2.06922
$378$		0			0
$379$	−6.81791			−0.350213			−0.175106	−	0.984550i	$-0.556027\pi$
$379$	−6.81791			−0.350213			−0.175106	+	0.984550i	$0.556027\pi$
$380$		0			0
$381$	2.24621			0.115077
$382$		0			0
$383$	16.2651			0.831107			0.415554	−	0.909569i	$-0.363588\pi$
$383$	16.2651			0.831107			0.415554	+	0.909569i	$0.363588\pi$
$384$		0			0
$385$	−4.19224			−0.213656
$386$		0			0
$387$			17.1973i			0.874188i
$388$		0			0
$389$	5.80776			0.294465			0.147233	−	0.989102i	$-0.452963\pi$
$389$	5.80776			0.294465			0.147233	+	0.989102i	$0.452963\pi$
$390$		0			0
$391$			3.96039i			0.200285i
$392$		0			0
$393$			5.79119i			0.292127i
$394$		0			0
$395$	12.5194			0.629918
$396$		0			0
$397$	−24.9309			−1.25124			−0.625622	−	0.780126i	$-0.715155\pi$
$397$	−24.9309			−1.25124			−0.625622	+	0.780126i	$0.715155\pi$
$398$		0			0
$399$	−3.21985	−	3.19906i	−0.161194	−	0.160153i
$400$		0			0
$401$		−	20.6256i		−	1.02999i	−0.857192	−	0.514997i	$-0.827793\pi$
$401$		−	20.6256i		−	1.02999i	0.857192	−	0.514997i	$-0.172207\pi$
$402$		0			0
$403$			20.2895i			1.01069i
$404$		0			0
$405$	2.93087			0.145636
$406$		0			0
$407$	−11.4677			−0.568432
$408$		0			0
$409$			18.5431i			0.916895i	0.888722	+	0.458447i	$0.151594\pi$
$409$			18.5431i			0.916895i	−0.888722	+	0.458447i	$0.848406\pi$
$410$		0			0
$411$	−4.64976			−0.229356
$412$		0			0
$413$		−	10.0845i		−	0.496228i
$414$		0			0
$415$			15.0802i			0.740259i
$416$		0			0
$417$			0.456551i			0.0223574i
$418$		0			0
$419$		−	27.4489i		−	1.34097i	−0.741924	−	0.670484i	$-0.766087\pi$
$419$		−	27.4489i		−	1.34097i	0.741924	−	0.670484i	$-0.233913\pi$
$420$		0			0
$421$		−	23.2930i		−	1.13523i	−0.823294	−	0.567614i	$-0.807866\pi$
$421$		−	23.2930i		−	1.13523i	0.823294	−	0.567614i	$-0.192134\pi$
$422$		0			0
$423$			14.4858i			0.704323i
$424$		0			0
$425$	2.56155			0.124254
$426$		0			0
$427$			0.594427i			0.0287664i
$428$		0			0
$429$	−17.6155			−0.850486
$430$		0			0
$431$	−17.0862			−0.823015			−0.411507	−	0.911406i	$-0.634998\pi$
$431$	−17.0862			−0.823015			−0.411507	+	0.911406i	$0.634998\pi$
$432$		0			0
$433$			28.0429i			1.34765i	0.738889	+	0.673827i	$0.235351\pi$
$433$			28.0429i			1.34765i	−0.738889	+	0.673827i	$0.764649\pi$
$434$		0			0
$435$		−	15.8415i		−	0.759544i
$436$		0			0
$437$	12.2462	+	12.1671i	0.585816	+	0.582032i
$438$		0			0
$439$	−24.8082			−1.18403			−0.592015	−	0.805927i	$-0.701668\pi$
$439$	−24.8082			−1.18403			−0.592015	+	0.805927i	$0.701668\pi$
$440$		0			0
$441$	−9.75379			−0.464466
$442$		0			0
$443$		−	34.7753i		−	1.65222i	−0.563507	−	0.826111i	$-0.690548\pi$
$443$		−	34.7753i		−	1.65222i	0.563507	−	0.826111i	$-0.309452\pi$
$444$		0			0
$445$			9.04325i			0.428691i
$446$		0			0
$447$	21.3578			1.01019
$448$		0			0
$449$			35.9166i			1.69501i	0.530787	+	0.847505i	$0.321896\pi$
$449$			35.9166i			1.69501i	−0.530787	+	0.847505i	$0.678104\pi$
$450$		0			0
$451$	−11.4677			−0.539992
$452$		0			0
$453$	29.1231			1.36832
$454$		0			0
$455$	−6.43971			−0.301898
$456$		0			0
$457$	12.1231			0.567095			0.283547	−	0.958958i	$-0.408489\pi$
$457$	12.1231			0.567095			0.283547	+	0.958958i	$0.408489\pi$
$458$		0			0
$459$	5.47091			0.255360
$460$		0			0
$461$	−19.3153			−0.899605			−0.449803	−	0.893128i	$-0.648506\pi$
$461$	−19.3153			−0.899605			−0.449803	+	0.893128i	$0.648506\pi$
$462$		0			0
$463$		−	21.6452i		−	1.00594i	−0.864304	−	0.502970i	$-0.832241\pi$
$463$		−	21.6452i		−	1.00594i	0.864304	−	0.502970i	$-0.167759\pi$
$464$		0			0
$465$	8.00000			0.370991
$466$		0			0
$467$		−	3.09218i		−	0.143089i	−0.997437	−	0.0715444i	$-0.977207\pi$
$467$		−	3.09218i		−	0.143089i	0.997437	−	0.0715444i	$-0.0227928\pi$
$468$		0			0
$469$			8.45851i			0.390578i
$470$		0			0
$471$	−7.19612			−0.331580
$472$		0			0
$473$	−34.0540			−1.56580
$474$		0			0
$475$	7.86962	−	7.92077i	0.361083	−	0.363430i
$476$		0			0
$477$			1.62603i			0.0744508i
$478$		0			0
$479$			8.68210i			0.396695i	0.980132	+	0.198348i	$0.0635575\pi$
$479$			8.68210i			0.396695i	−0.980132	+	0.198348i	$0.936442\pi$
$480$		0			0
$481$	−17.6155			−0.803199
$482$		0			0
$483$	4.12391			0.187644
$484$		0			0
$485$		−	26.4168i		−	1.19953i
$486$		0			0
$487$	−9.59482			−0.434783			−0.217391	−	0.976085i	$-0.569755\pi$
$487$	−9.59482			−0.434783			−0.217391	+	0.976085i	$0.569755\pi$
$488$		0			0
$489$			12.7519i			0.576659i
$490$		0			0
$491$		−	2.71151i		−	0.122369i	−0.998126	−	0.0611844i	$-0.980512\pi$
$491$		−	2.71151i		−	0.122369i	0.998126	−	0.0611844i	$-0.0194878\pi$
$492$		0			0
$493$		−	8.45851i		−	0.380952i
$494$		0			0
$495$		−	7.54011i		−	0.338903i
$496$		0			0
$497$		−	9.49980i		−	0.426124i
$498$		0			0
$499$		−	11.0129i		−	0.493007i	−0.969142	−	0.246504i	$-0.920718\pi$
$499$		−	11.0129i		−	0.493007i	0.969142	−	0.246504i	$-0.0792817\pi$
$500$		0			0
$501$	0.630683			0.0281768
$502$		0			0
$503$		−	35.6435i		−	1.58926i	−0.607091	−	0.794632i	$-0.707664\pi$
$503$		−	35.6435i		−	1.58926i	0.607091	−	0.794632i	$-0.292336\pi$
$504$		0			0
$505$	12.8769			0.573014
$506$		0			0
$507$	−11.4677			−0.509297
$508$		0			0
$509$			9.49980i			0.421071i	0.977586	+	0.210536i	$0.0675208\pi$
$509$			9.49980i			0.421071i	−0.977586	+	0.210536i	$0.932479\pi$
$510$		0			0
$511$			7.05256i			0.311987i
$512$		0			0
$513$	16.8078	−	16.9170i	0.742081	−	0.746905i
$514$		0			0
$515$	−5.84912			−0.257743
$516$		0			0
$517$	−28.6847			−1.26155
$518$		0			0
$519$		−	20.2895i		−	0.890609i
$520$		0			0
$521$		−	15.2910i		−	0.669910i	−0.942234	−	0.334955i	$-0.891279\pi$
$521$		−	15.2910i		−	0.669910i	0.942234	−	0.334955i	$-0.108721\pi$
$522$		0			0
$523$	−33.4990			−1.46481			−0.732404	−	0.680871i	$-0.761602\pi$
$523$	−33.4990			−1.46481			−0.732404	+	0.680871i	$0.761602\pi$
$524$		0			0
$525$		−	2.66732i		−	0.116411i
$526$		0			0
$527$	4.27156			0.186072
$528$		0			0
$529$	7.31534			0.318058
$530$		0			0
$531$	18.1379			0.787120
$532$		0			0
$533$	−17.6155			−0.763013
$534$		0			0
$535$	12.2888			0.531292
$536$		0			0
$537$	−10.8769			−0.469373
$538$		0			0
$539$		−	19.3144i		−	0.831929i
$540$		0			0
$541$	12.1922			0.524185			0.262093	−	0.965043i	$-0.415587\pi$
$541$	12.1922			0.524185			0.262093	+	0.965043i	$0.415587\pi$
$542$		0			0
$543$		−	2.49775i		−	0.107189i
$544$		0			0
$545$		−	24.7908i		−	1.06192i
$546$		0			0
$547$	−19.4849			−0.833116			−0.416558	−	0.909109i	$-0.636764\pi$
$547$	−19.4849			−0.833116			−0.416558	+	0.909109i	$0.636764\pi$
$548$		0			0
$549$	−1.06913			−0.0456294
$550$		0			0
$551$	−26.1552	−	25.9863i	−1.11425	−	1.10705i
$552$		0			0
$553$		−	6.96067i		−	0.295998i
$554$		0			0
$555$			6.94568i			0.294828i
$556$		0			0
$557$	18.9309			0.802127			0.401063	−	0.916050i	$-0.368641\pi$
$557$	18.9309			0.802127			0.401063	+	0.916050i	$0.368641\pi$
$558$		0			0
$559$	−52.3104			−2.21249
$560$		0			0
$561$			3.70861i			0.156578i
$562$		0			0
$563$	−33.6466			−1.41804			−0.709018	−	0.705191i	$-0.750861\pi$
$563$	−33.6466			−1.41804			−0.709018	+	0.705191i	$0.750861\pi$
$564$		0			0
$565$		−	5.79119i		−	0.243637i
$566$		0			0
$567$		−	1.62954i		−	0.0684342i
$568$		0			0
$569$		−	12.7519i		−	0.534586i	−0.963615	−	0.267293i	$-0.913871\pi$
$569$		−	12.7519i		−	0.534586i	0.963615	−	0.267293i	$-0.0861291\pi$
$570$		0			0
$571$			17.5780i			0.735615i	0.929902	+	0.367807i	$0.119891\pi$
$571$			17.5780i			0.735615i	−0.929902	+	0.367807i	$0.880109\pi$
$572$		0			0
$573$			10.5411i			0.440360i
$574$		0			0
$575$			10.1447i			0.423065i
$576$		0			0
$577$	27.0000			1.12402			0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000			1.12402			0.562012	+	0.827129i	$0.310027\pi$
$578$		0			0
$579$		−	4.44793i		−	0.184850i
$580$		0			0
$581$	8.38447			0.347847
$582$		0			0
$583$	−3.21985			−0.133353
$584$		0			0
$585$		−	11.5824i		−	0.478873i
$586$		0			0
$587$			4.82860i			0.199297i	0.995023	+	0.0996487i	$0.0317719\pi$
$587$			4.82860i			0.199297i	−0.995023	+	0.0996487i	$0.968228\pi$
$588$		0			0
$589$	13.1231	−	13.2084i	0.540728	−	0.544243i
$590$		0			0
$591$	−12.5841			−0.517641
$592$		0			0
$593$	−36.7386			−1.50867			−0.754337	−	0.656487i	$-0.772042\pi$
$593$	−36.7386			−1.50867			−0.754337	+	0.656487i	$0.772042\pi$
$594$		0			0
$595$			1.35576i			0.0555806i
$596$		0			0
$597$			15.8757i			0.649750i
$598$		0			0
$599$	7.72197			0.315511			0.157756	−	0.987478i	$-0.449574\pi$
$599$	7.72197			0.315511			0.157756	+	0.987478i	$0.449574\pi$
$600$		0			0
$601$		−	39.1687i		−	1.59772i	−0.601514	−	0.798862i	$-0.705436\pi$
$601$		−	39.1687i		−	1.59772i	0.601514	−	0.798862i	$-0.294564\pi$
$602$		0			0
$603$	−15.2134			−0.619537
$604$		0			0
$605$	−2.24621			−0.0913215
$606$		0			0
$607$	13.6358			0.553461			0.276730	−	0.960948i	$-0.410749\pi$
$607$	13.6358			0.553461			0.276730	+	0.960948i	$0.410749\pi$
$608$		0			0
$609$	−8.80776			−0.356909
$610$		0			0
$611$	−44.0626			−1.78258
$612$		0			0
$613$	−24.3002			−0.981475			−0.490738	−	0.871307i	$-0.663273\pi$
$613$	−24.3002			−0.981475			−0.490738	+	0.871307i	$0.663273\pi$
$614$		0			0
$615$			6.94568i			0.280077i
$616$		0			0
$617$	−28.5464			−1.14923			−0.574617	−	0.818422i	$-0.694849\pi$
$617$	−28.5464			−1.14923			−0.574617	+	0.818422i	$0.694849\pi$
$618$		0			0
$619$		−	34.3946i		−	1.38244i	−0.722646	−	0.691218i	$-0.757074\pi$
$619$		−	34.3946i		−	1.38244i	0.722646	−	0.691218i	$-0.242926\pi$
$620$		0			0
$621$			21.6669i			0.869464i
$622$		0			0
$623$	5.02797			0.201441
$624$		0			0
$625$	−5.63068			−0.225227
$626$		0			0
$627$	11.4677	+	11.3936i	0.457975	+	0.455017i
$628$		0			0
$629$			3.70861i			0.147872i
$630$		0			0
$631$		−	12.7494i		−	0.507544i	−0.967264	−	0.253772i	$-0.918329\pi$
$631$		−	12.7494i		−	0.507544i	0.967264	−	0.253772i	$-0.0816713\pi$
$632$		0			0
$633$	2.42329			0.0963172
$634$		0			0
$635$	−2.92456			−0.116058
$636$		0			0
$637$		−	29.6689i		−	1.17552i
$638$		0			0
$639$	17.0862			0.675921
$640$		0			0
$641$			48.6685i			1.92229i	0.276045	+	0.961145i	$0.410976\pi$
$641$			48.6685i			1.92229i	−0.276045	+	0.961145i	$0.589024\pi$
$642$		0			0
$643$		−	6.56502i		−	0.258899i	−0.991586	−	0.129449i	$-0.958679\pi$
$643$		−	6.56502i		−	0.258899i	0.991586	−	0.129449i	$-0.0413210\pi$
$644$		0			0
$645$			20.6256i			0.812133i
$646$		0			0
$647$		−	36.2379i		−	1.42466i	−0.701845	−	0.712329i	$-0.747640\pi$
$647$		−	36.2379i		−	1.42466i	0.701845	−	0.712329i	$-0.252360\pi$
$648$		0			0
$649$			35.9166i			1.40985i
$650$		0			0
$651$		−	4.44793i		−	0.174328i
$652$		0			0
$653$	45.8078			1.79260			0.896298	−	0.443452i	$-0.146246\pi$
$653$	45.8078			1.79260			0.896298	+	0.443452i	$0.146246\pi$
$654$		0			0
$655$		−	7.54011i		−	0.294616i
$656$		0			0
$657$	−12.6847			−0.494876
$658$		0			0
$659$	−3.36750			−0.131179			−0.0655896	−	0.997847i	$-0.520893\pi$
$659$	−3.36750			−0.131179			−0.0655896	+	0.997847i	$0.520893\pi$
$660$		0			0
$661$		−	17.5018i		−	0.680740i	−0.940292	−	0.340370i	$-0.889448\pi$
$661$		−	17.5018i		−	0.680740i	0.940292	−	0.340370i	$-0.110552\pi$
$662$		0			0
$663$			5.69681i			0.221246i
$664$		0			0
$665$	4.19224	+	4.16516i	0.162568	+	0.161518i
$666$		0			0
$667$	33.4990			1.29709
$668$		0			0
$669$	−19.5076			−0.754207
$670$		0			0
$671$		−	2.11708i		−	0.0817291i
$672$		0			0
$673$		−	31.7515i		−	1.22393i	−0.790885	−	0.611964i	$-0.790380\pi$
$673$		−	31.7515i		−	1.22393i	0.790885	−	0.611964i	$-0.209620\pi$
$674$		0			0
$675$	14.0140			0.539400
$676$		0			0
$677$		−	4.29335i		−	0.165007i	−0.996591	−	0.0825034i	$-0.973708\pi$
$677$		−	4.29335i		−	0.165007i	0.996591	−	0.0825034i	$-0.0262915\pi$
$678$		0			0
$679$	−14.6875			−0.563656
$680$		0			0
$681$	20.9460			0.802653
$682$		0			0
$683$	13.6358			0.521760			0.260880	−	0.965371i	$-0.415987\pi$
$683$	13.6358			0.521760			0.260880	+	0.965371i	$0.415987\pi$
$684$		0			0
$685$	6.05398			0.231311
$686$		0			0
$687$	2.92456			0.111579
$688$		0			0
$689$	−4.94602			−0.188429
$690$		0			0
$691$		−	23.3817i		−	0.889480i	−0.895660	−	0.444740i	$-0.853296\pi$
$691$		−	23.3817i		−	0.889480i	0.895660	−	0.444740i	$-0.146704\pi$
$692$		0			0
$693$	−4.19224			−0.159250
$694$		0			0
$695$		−	0.594427i		−	0.0225479i
$696$		0			0
$697$			3.70861i			0.140474i
$698$		0			0
$699$	−11.7630			−0.444916
$700$		0			0
$701$	−29.3693			−1.10926			−0.554632	−	0.832096i	$-0.687141\pi$
$701$	−29.3693			−1.10926			−0.554632	+	0.832096i	$0.687141\pi$
$702$		0			0
$703$	11.4677	+	11.3936i	0.432512	+	0.429718i
$704$		0			0
$705$			17.3736i			0.654327i
$706$		0			0
$707$		−	7.15944i		−	0.269259i
$708$		0			0
$709$	−37.3693			−1.40343			−0.701717	−	0.712456i	$-0.747583\pi$
$709$	−37.3693			−1.40343			−0.701717	+	0.712456i	$0.747583\pi$
$710$		0			0
$711$	12.5194			0.469513
$712$		0			0
$713$			16.9170i			0.633547i
$714$		0			0
$715$	22.9354			0.857733
$716$		0			0
$717$		−	23.2930i		−	0.869891i
$718$		0			0
$719$		−	15.9484i		−	0.594776i	−0.954757	−	0.297388i	$-0.903885\pi$
$719$		−	15.9484i		−	0.594776i	0.954757	−	0.297388i	$-0.0961155\pi$
$720$		0			0
$721$			3.25206i			0.121113i
$722$		0			0
$723$			18.3393i			0.682046i
$724$		0			0
$725$		−	21.6669i		−	0.804689i
$726$		0			0
$727$			23.6554i			0.877332i	0.898650	+	0.438666i	$0.144549\pi$
$727$			23.6554i			0.877332i	−0.898650	+	0.438666i	$0.855451\pi$
$728$		0			0
$729$	22.6155			0.837612
$730$		0			0
$731$			11.0129i			0.407329i
$732$		0			0
$733$	−3.12311			−0.115355			−0.0576773	−	0.998335i	$-0.518369\pi$
$733$	−3.12311			−0.115355			−0.0576773	+	0.998335i	$0.518369\pi$
$734$		0			0
$735$	−11.6982			−0.431496
$736$		0			0
$737$		−	30.1254i		−	1.10968i
$738$		0			0
$739$		−	30.3273i		−	1.11561i	−0.829972	−	0.557804i	$-0.811644\pi$
$739$		−	30.3273i		−	1.11561i	0.829972	−	0.557804i	$-0.188356\pi$
$740$		0			0
$741$	17.6155	+	17.5018i	0.647123	+	0.642943i
$742$		0			0
$743$	−38.4440			−1.41037			−0.705187	−	0.709021i	$-0.749137\pi$
$743$	−38.4440			−1.41037			−0.705187	+	0.709021i	$0.749137\pi$
$744$		0			0
$745$	−27.8078			−1.01880
$746$		0			0
$747$			15.0802i			0.551756i
$748$		0			0
$749$		−	6.83248i		−	0.249653i
$750$		0			0
$751$	−8.24782			−0.300967			−0.150484	−	0.988612i	$-0.548083\pi$
$751$	−8.24782			−0.300967			−0.150484	+	0.988612i	$0.548083\pi$
$752$		0			0
$753$		−	26.8734i		−	0.979320i
$754$		0			0
$755$	−37.9182			−1.37998
$756$		0			0
$757$	24.1922			0.879282			0.439641	−	0.898174i	$-0.355106\pi$
$757$	24.1922			0.879282			0.439641	+	0.898174i	$0.355106\pi$
$758$		0			0
$759$	−14.6875			−0.533123
$760$		0			0
$761$	−10.6155			−0.384813			−0.192406	−	0.981315i	$-0.561629\pi$
$761$	−10.6155			−0.384813			−0.192406	+	0.981315i	$0.561629\pi$
$762$		0			0
$763$	−13.7835			−0.498995
$764$		0			0
$765$	−2.43845			−0.0881622
$766$		0			0
$767$			55.1716i			1.99213i
$768$		0			0
$769$	54.2311			1.95562			0.977811	−	0.209489i	$-0.0671801\pi$
$769$	54.2311			1.95562			0.977811	+	0.209489i	$0.0671801\pi$
$770$		0			0
$771$		−	22.7872i		−	0.820662i
$772$		0			0
$773$		−	25.8321i		−	0.929115i	−0.885543	−	0.464558i	$-0.846213\pi$
$773$		−	25.8321i		−	0.929115i	0.885543	−	0.464558i	$-0.153787\pi$
$774$		0			0
$775$	10.9418			0.393042
$776$		0			0
$777$	3.86174			0.138539
$778$		0			0
$779$	11.4677	+	11.3936i	0.410872	+	0.408219i
$780$		0			0
$781$			33.8340i			1.21068i
$782$		0			0
$783$		−	46.2758i		−	1.65376i
$784$		0			0
$785$	9.36932			0.334405
$786$		0			0
$787$	40.9904			1.46115			0.730575	−	0.682833i	$-0.239252\pi$
$787$	40.9904			1.46115			0.730575	+	0.682833i	$0.239252\pi$
$788$		0			0
$789$			7.87377i			0.280314i
$790$		0			0
$791$	−3.21985			−0.114485
$792$		0			0
$793$		−	3.25206i		−	0.115484i
$794$		0			0
$795$			1.95018i			0.0691659i
$796$		0			0
$797$			23.2930i			0.825079i	0.910940	+	0.412539i	$0.135358\pi$
$797$			23.2930i			0.825079i	−0.910940	+	0.412539i	$0.864642\pi$
$798$		0			0
$799$			9.27653i			0.328180i
$800$		0			0
$801$			9.04325i			0.319528i
$802$		0			0
$803$		−	25.1181i		−	0.886398i
$804$		0			0
$805$	−5.36932			−0.189244
$806$		0			0
$807$			4.44793i			0.156575i
$808$		0			0
$809$	19.6307			0.690178			0.345089	−	0.938570i	$-0.387849\pi$
$809$	19.6307			0.690178			0.345089	+	0.938570i	$0.387849\pi$
$810$		0			0
$811$	−0.378206			−0.0132806			−0.00664030	−	0.999978i	$-0.502114\pi$
$811$	−0.378206			−0.0132806			−0.00664030	+	0.999978i	$0.502114\pi$
$812$		0			0
$813$		−	18.4149i		−	0.645837i
$814$		0			0
$815$		−	16.6029i		−	0.581573i
$816$		0			0
$817$	34.0540	+	33.8340i	1.19140	+	1.18370i
$818$		0			0
$819$	−6.43971			−0.225022
$820$		0			0
$821$	−33.4233			−1.16648			−0.583240	−	0.812300i	$-0.698215\pi$
$821$	−33.4233			−1.16648			−0.583240	+	0.812300i	$0.698215\pi$
$822$		0			0
$823$			22.1328i			0.771500i	0.922603	+	0.385750i	$0.126057\pi$
$823$			22.1328i			0.771500i	−0.922603	+	0.385750i	$0.873943\pi$
$824$		0			0
$825$			9.49980i			0.330741i
$826$		0			0
$827$	−25.1864			−0.875817			−0.437909	−	0.899019i	$-0.644281\pi$
$827$	−25.1864			−0.875817			−0.437909	+	0.899019i	$0.644281\pi$
$828$		0			0
$829$		−	8.91506i		−	0.309633i	−0.987943	−	0.154816i	$-0.950521\pi$
$829$		−	8.91506i		−	0.309633i	0.987943	−	0.154816i	$-0.0494786\pi$
$830$		0			0
$831$	28.8492			1.00077
$832$		0			0
$833$	−6.24621			−0.216418
$834$		0			0
$835$	−0.821147			−0.0284170
$836$		0			0
$837$	23.3693			0.807762
$838$		0			0
$839$	−51.2587			−1.76965			−0.884823	−	0.465926i	$-0.845721\pi$
$839$	−51.2587			−1.76965			−0.884823	+	0.465926i	$0.845721\pi$
$840$		0			0
$841$	−42.5464			−1.46712
$842$		0			0
$843$			13.8914i			0.478444i
$844$		0			0
$845$	14.9309			0.513638
$846$		0			0
$847$			1.24887i			0.0429118i
$848$		0			0
$849$			5.79119i			0.198753i
$850$		0			0
$851$	−14.6875			−0.503482
$852$		0			0
$853$	−6.63068			−0.227030			−0.113515	−	0.993536i	$-0.536211\pi$
$853$	−6.63068			−0.227030			−0.113515	+	0.993536i	$0.536211\pi$
$854$		0			0
$855$	−7.49141	+	7.54011i	−0.256201	+	0.257866i
$856$		0			0
$857$		−	4.16516i		−	0.142279i	−0.997466	−	0.0711396i	$-0.977336\pi$
$857$		−	4.16516i		−	0.142279i	0.997466	−	0.0711396i	$-0.0226636\pi$
$858$		0			0
$859$			22.6203i			0.771795i	0.922542	+	0.385898i	$0.126108\pi$
$859$			22.6203i			0.771795i	−0.922542	+	0.385898i	$0.873892\pi$
$860$		0			0
$861$	3.86174			0.131608
$862$		0			0
$863$	22.7048			0.772880			0.386440	−	0.922315i	$-0.373705\pi$
$863$	22.7048			0.772880			0.386440	+	0.922315i	$0.373705\pi$
$864$		0			0
$865$			26.4168i			0.898199i
$866$		0			0
$867$	−19.1896			−0.651715
$868$		0			0
$869$			24.7908i			0.840970i
$870$		0			0
$871$		−	46.2758i		−	1.56799i
$872$		0			0
$873$		−	26.4168i		−	0.894074i
$874$		0			0
$875$			10.2516i			0.346568i
$876$		0			0
$877$			14.2497i			0.481178i	0.970627	+	0.240589i	$0.0773406\pi$
$877$			14.2497i			0.481178i	−0.970627	+	0.240589i	$0.922659\pi$
$878$		0			0
$879$			5.69681i			0.192149i
$880$		0			0
$881$	35.1771			1.18515			0.592573	−	0.805517i	$-0.298112\pi$
$881$	35.1771			1.18515			0.592573	+	0.805517i	$0.298112\pi$
$882$		0			0
$883$		−	0.594427i		−	0.0200041i	−0.999950	−	0.0100020i	$-0.996816\pi$
$883$		−	0.594427i		−	0.0200041i	0.999950	−	0.0100020i	$-0.00318380\pi$
$884$		0			0
$885$	21.7538			0.731246
$886$		0			0
$887$	31.4785			1.05694			0.528472	−	0.848951i	$-0.322765\pi$
$887$	31.4785			1.05694			0.528472	+	0.848951i	$0.322765\pi$
$888$		0			0
$889$			1.62603i			0.0545353i
$890$		0			0
$891$			5.80369i			0.194431i
$892$		0			0
$893$	28.6847	+	28.4994i	0.959895	+	0.953696i
$894$		0			0
$895$	14.1617			0.473373
$896$		0			0
$897$	−22.5616			−0.753308
$898$		0			0
$899$		−	36.1310i		−	1.20504i
$900$		0			0
$901$			1.04129i			0.0346904i
$902$		0			0
$903$	11.4677			0.381620
$904$		0			0
$905$			3.25206i			0.108102i
$906$		0			0
$907$	36.7188			1.21923			0.609614	−	0.792698i	$-0.291324\pi$
$907$	36.7188			1.21923			0.609614	+	0.792698i	$0.291324\pi$
$908$		0			0
$909$	12.8769			0.427100
$910$		0			0
$911$	−10.6465			−0.352735			−0.176368	−	0.984324i	$-0.556435\pi$
$911$	−10.6465			−0.352735			−0.176368	+	0.984324i	$0.556435\pi$
$912$		0			0
$913$	−29.8617			−0.988279
$914$		0			0
$915$	−1.28226			−0.0423904
$916$		0			0
$917$	−4.19224			−0.138440
$918$		0			0
$919$		−	12.8563i		−	0.424089i	−0.977260	−	0.212044i	$-0.931988\pi$
$919$		−	12.8563i		−	0.424089i	0.977260	−	0.212044i	$-0.0680121\pi$
$920$		0			0
$921$	9.61553			0.316842
$922$		0			0
$923$			51.9726i			1.71070i
$924$		0			0
$925$			9.49980i			0.312352i
$926$		0			0
$927$	−5.84912			−0.192110
$928$		0			0
$929$	−25.6847			−0.842686			−0.421343	−	0.906901i	$-0.638441\pi$
$929$	−25.6847			−0.842686			−0.421343	+	0.906901i	$0.638441\pi$
$930$		0			0
$931$	−19.1896	+	19.3144i	−0.628915	+	0.633003i
$932$		0			0
$933$			1.04129i			0.0340903i
$934$		0			0
$935$		−	4.82860i		−	0.157912i
$936$		0			0
$937$	−22.1231			−0.722730			−0.361365	−	0.932424i	$-0.617689\pi$
$937$	−22.1231			−0.722730			−0.361365	+	0.932424i	$0.617689\pi$
$938$		0			0
$939$	−5.47091			−0.178536
$940$		0			0
$941$			24.9190i			0.812336i	0.913799	+	0.406168i	$0.133135\pi$
$941$			24.9190i			0.812336i	−0.913799	+	0.406168i	$0.866865\pi$
$942$		0			0
$943$	−14.6875			−0.478292
$944$		0			0
$945$			7.41722i			0.241282i
$946$		0			0
$947$		−	2.71151i		−	0.0881123i	−0.999029	−	0.0440561i	$-0.985972\pi$
$947$		−	2.71151i		−	0.0881123i	0.999029	−	0.0440561i	$-0.0140280\pi$
$948$		0			0
$949$		−	38.5839i		−	1.25249i
$950$		0			0
$951$		−	5.69681i		−	0.184732i
$952$		0			0
$953$		−	24.7908i		−	0.803053i	−0.915848	−	0.401526i	$-0.868480\pi$
$953$		−	24.7908i		−	0.803053i	0.915848	−	0.401526i	$-0.131520\pi$
$954$		0			0
$955$		−	13.7245i		−	0.444113i
$956$		0			0
$957$	31.3693			1.01403
$958$		0			0
$959$		−	3.36596i		−	0.108692i
$960$		0			0
$961$	−12.7538			−0.411413
$962$		0			0
$963$	12.2888			0.396002
$964$		0			0
$965$			5.79119i			0.186425i
$966$		0			0
$967$			26.4738i			0.851341i	0.904878	+	0.425670i	$0.139962\pi$
$967$			26.4738i			0.851341i	−0.904878	+	0.425670i	$0.860038\pi$
$968$		0			0
$969$	3.68466	−	3.70861i	0.118368	−	0.119138i
$970$		0			0
$971$	19.4849			0.625301			0.312651	−	0.949868i	$-0.398783\pi$
$971$	19.4849			0.625301			0.312651	+	0.949868i	$0.398783\pi$
$972$		0			0
$973$	−0.330497			−0.0105952
$974$		0			0
$975$			14.5927i			0.467339i
$976$		0			0
$977$			26.8734i			0.859755i	0.902887	+	0.429878i	$0.141443\pi$
$977$			26.8734i			0.859755i	−0.902887	+	0.429878i	$0.858557\pi$
$978$		0			0
$979$	−17.9074			−0.572322
$980$		0			0
$981$		−	24.7908i		−	0.791509i
$982$		0			0
$983$	13.3405			0.425497			0.212748	−	0.977107i	$-0.431759\pi$
$983$	13.3405			0.425497			0.212748	+	0.977107i	$0.431759\pi$
$984$		0			0
$985$	16.3845			0.522053
$986$		0			0
$987$	9.65956			0.307467
$988$		0			0
$989$	−43.6155			−1.38689
$990$		0			0
$991$	54.1833			1.72119			0.860594	−	0.509292i	$-0.170093\pi$
$991$	54.1833			1.72119			0.860594	+	0.509292i	$0.170093\pi$
$992$		0			0
$993$	−2.69981			−0.0856760
$994$		0			0
$995$		−	20.6701i		−	0.655288i
$996$		0			0
$997$	28.5464			0.904073			0.452037	−	0.891999i	$-0.350698\pi$
$997$	28.5464			0.904073			0.452037	+	0.891999i	$0.350698\pi$
$998$		0			0
$999$			20.2895i			0.641931i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.h.d.1215.6		8
4.3	odd	2	inner	1216.2.h.d.1215.3		8
8.3	odd	2		76.2.d.a.75.1	✓	8
8.5	even	2		76.2.d.a.75.7	yes	8
19.18	odd	2	inner	1216.2.h.d.1215.4		8
24.5	odd	2		684.2.f.b.379.2		8
24.11	even	2		684.2.f.b.379.8		8
76.75	even	2	inner	1216.2.h.d.1215.5		8
152.37	odd	2		76.2.d.a.75.2	yes	8
152.75	even	2		76.2.d.a.75.8	yes	8
456.227	odd	2		684.2.f.b.379.1		8
456.341	even	2		684.2.f.b.379.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.d.a.75.1	✓	8	8.3	odd	2
76.2.d.a.75.2	yes	8	152.37	odd	2
76.2.d.a.75.7	yes	8	8.5	even	2
76.2.d.a.75.8	yes	8	152.75	even	2
684.2.f.b.379.1		8	456.227	odd	2
684.2.f.b.379.2		8	24.5	odd	2
684.2.f.b.379.7		8	456.341	even	2
684.2.f.b.379.8		8	24.11	even	2
1216.2.h.d.1215.3		8	4.3	odd	2	inner
1216.2.h.d.1215.4		8	19.18	odd	2	inner
1216.2.h.d.1215.5		8	76.75	even	2	inner
1216.2.h.d.1215.6		8	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.19935 q^{3} -1.56155 q^{5} +0.868210i q^{7} -1.56155 q^{9} +O(q^{10})\)	\(q+1.19935 q^{3} -1.56155 q^{5} +0.868210i q^{7} -1.56155 q^{9} -3.09218i q^{11} -4.74990i q^{13} -1.87285 q^{15} -1.00000 q^{17} +(-3.07221 + 3.09218i) q^{19} +1.04129i q^{21} -3.96039i q^{23} -2.56155 q^{25} -5.47091 q^{27} +8.45851i q^{29} -4.27156 q^{31} -3.70861i q^{33} -1.35576i q^{35} -3.70861i q^{37} -5.69681i q^{39} -3.70861i q^{41} -11.0129i q^{43} +2.43845 q^{45} -9.27653i q^{47} +6.24621 q^{49} -1.19935 q^{51} -1.04129i q^{53} +4.82860i q^{55} +(-3.68466 + 3.70861i) q^{57} -11.6153 q^{59} +0.684658 q^{61} -1.35576i q^{63} +7.41722i q^{65} +9.74247 q^{67} -4.74990i q^{69} -10.9418 q^{71} +8.12311 q^{73} -3.07221 q^{75} +2.68466 q^{77} -8.01726 q^{79} -1.87689 q^{81} -9.65719i q^{83} +1.56155 q^{85} +10.1447i q^{87} -5.79119i q^{89} +4.12391 q^{91} -5.12311 q^{93} +(4.79741 - 4.82860i) q^{95} +16.9170i q^{97} +4.82860i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 + 4 * q^9	\( 8 q + 4 q^{5} + 4 q^{9} - 8 q^{17} - 4 q^{25} + 36 q^{45} - 16 q^{49} + 20 q^{57} - 44 q^{61} + 32 q^{73} - 28 q^{77} - 48 q^{81} - 4 q^{85} - 8 q^{93}+O(q^{100}) \) 8 * q + 4 * q^5 + 4 * q^9 - 8 * q^17 - 4 * q^25 + 36 * q^45 - 16 * q^49 + 20 * q^57 - 44 * q^61 + 32 * q^73 - 28 * q^77 - 48 * q^81 - 4 * q^85 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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