LMFDB - Embedded newform 7168.2.a.bf.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$57.2367681689$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.9433055232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 $ x^8 - 4x^7 - 6x^6 + 32x^5 + 9x^4 - 76x^3 - 4x^2 + 48*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.02908$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.242103 q^{3} -0.379610 q^{5} +1.00000 q^{7} -2.94139 q^{9} +O(q^{10})$	$q-0.242103 q^{3} -0.379610 q^{5} +1.00000 q^{7} -2.94139 q^{9} +2.61527 q^{11} +2.31328 q^{13} +0.0919045 q^{15} +7.37134 q^{17} +5.44437 q^{19} -0.242103 q^{21} +6.44892 q^{23} -4.85590 q^{25} +1.43843 q^{27} -5.07278 q^{29} +6.10161 q^{31} -0.633164 q^{33} -0.379610 q^{35} +10.4914 q^{37} -0.560052 q^{39} -0.836588 q^{41} -5.50056 q^{43} +1.11658 q^{45} -6.02070 q^{47} +1.00000 q^{49} -1.78462 q^{51} -0.813824 q^{53} -0.992782 q^{55} -1.31810 q^{57} -7.53795 q^{59} -1.31502 q^{61} -2.94139 q^{63} -0.878144 q^{65} -8.79384 q^{67} -1.56130 q^{69} -11.4285 q^{71} -3.68616 q^{73} +1.17563 q^{75} +2.61527 q^{77} -4.21672 q^{79} +8.47591 q^{81} +16.9913 q^{83} -2.79823 q^{85} +1.22813 q^{87} +9.32780 q^{89} +2.31328 q^{91} -1.47722 q^{93} -2.06673 q^{95} -13.9032 q^{97} -7.69252 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q + 8 * q^5 + 8 * q^7 + 12 * q^9	$ 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9} + 12 q^{11} + 20 q^{13} + 4 q^{17} + 4 q^{19} + 8 q^{23} + 12 q^{25} - 12 q^{27} + 8 q^{29} - 4 q^{31} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 16 q^{39} - 12 q^{41} - 4 q^{43} + 52 q^{45} + 20 q^{47} + 8 q^{49} + 32 q^{51} + 40 q^{53} + 24 q^{55} - 4 q^{57} + 4 q^{59} - 8 q^{61} + 12 q^{63} + 36 q^{65} + 28 q^{67} + 4 q^{69} - 16 q^{71} + 16 q^{73} - 28 q^{75} + 12 q^{77} + 20 q^{81} - 8 q^{83} + 16 q^{85} + 20 q^{87} + 16 q^{89} + 20 q^{91} + 16 q^{93} - 40 q^{95} - 36 q^{97} - 4 q^{99}+O(q^{100}) $ 8 * q + 8 * q^5 + 8 * q^7 + 12 * q^9 + 12 * q^11 + 20 * q^13 + 4 * q^17 + 4 * q^19 + 8 * q^23 + 12 * q^25 - 12 * q^27 + 8 * q^29 - 4 * q^31 + 8 * q^33 + 8 * q^35 + 8 * q^37 - 16 * q^39 - 12 * q^41 - 4 * q^43 + 52 * q^45 + 20 * q^47 + 8 * q^49 + 32 * q^51 + 40 * q^53 + 24 * q^55 - 4 * q^57 + 4 * q^59 - 8 * q^61 + 12 * q^63 + 36 * q^65 + 28 * q^67 + 4 * q^69 - 16 * q^71 + 16 * q^73 - 28 * q^75 + 12 * q^77 + 20 * q^81 - 8 * q^83 + 16 * q^85 + 20 * q^87 + 16 * q^89 + 20 * q^91 + 16 * q^93 - 40 * q^95 - 36 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.242103	−0.139778	−0.0698890	−	0.997555i	$-0.522265\pi$
$3$	−0.242103	−0.139778	−0.0698890	+	0.997555i	$0.522265\pi$
$4$	0	0
$5$	−0.379610	−0.169767	−0.0848833	−	0.996391i	$-0.527052\pi$
$5$	−0.379610	−0.169767	−0.0848833	+	0.996391i	$0.527052\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.94139	−0.980462
$10$	0	0
$11$	2.61527	0.788533	0.394267	−	0.918996i	$-0.370999\pi$
$11$	2.61527	0.788533	0.394267	+	0.918996i	$0.370999\pi$
$12$	0	0
$13$	2.31328	0.641589	0.320794	−	0.947149i	$-0.396050\pi$
$13$	2.31328	0.641589	0.320794	+	0.947149i	$0.396050\pi$
$14$	0	0
$15$	0.0919045	0.0237296
$16$	0	0
$17$	7.37134	1.78781	0.893906	−	0.448255i	$-0.147954\pi$
$17$	7.37134	1.78781	0.893906	+	0.448255i	$0.147954\pi$
$18$	0	0
$19$	5.44437	1.24902	0.624512	−	0.781015i	$-0.285298\pi$
$19$	5.44437	1.24902	0.624512	+	0.781015i	$0.285298\pi$
$20$	0	0
$21$	−0.242103	−0.0528312
$22$	0	0
$23$	6.44892	1.34469	0.672347	−	0.740236i	$-0.265286\pi$
$23$	6.44892	1.34469	0.672347	+	0.740236i	$0.265286\pi$
$24$	0	0
$25$	−4.85590	−0.971179
$26$	0	0
$27$	1.43843	0.276825
$28$	0	0
$29$	−5.07278	−0.941991	−0.470996	−	0.882135i	$-0.656105\pi$
$29$	−5.07278	−0.941991	−0.470996	+	0.882135i	$0.656105\pi$
$30$	0	0
$31$	6.10161	1.09588	0.547941	−	0.836517i	$-0.315412\pi$
$31$	6.10161	1.09588	0.547941	+	0.836517i	$0.315412\pi$
$32$	0	0
$33$	−0.633164	−0.110220
$34$	0	0
$35$	−0.379610	−0.0641657
$36$	0	0
$37$	10.4914	1.72477	0.862386	−	0.506252i	$-0.168969\pi$
$37$	10.4914	1.72477	0.862386	+	0.506252i	$0.168969\pi$
$38$	0	0
$39$	−0.560052	−0.0896801
$40$	0	0
$41$	−0.836588	−0.130653	−0.0653265	−	0.997864i	$-0.520809\pi$
$41$	−0.836588	−0.130653	−0.0653265	+	0.997864i	$0.520809\pi$
$42$	0	0
$43$	−5.50056	−0.838828	−0.419414	−	0.907795i	$-0.637764\pi$
$43$	−5.50056	−0.838828	−0.419414	+	0.907795i	$0.637764\pi$
$44$	0	0
$45$	1.11658	0.166450
$46$	0	0
$47$	−6.02070	−0.878209	−0.439104	−	0.898436i	$-0.644704\pi$
$47$	−6.02070	−0.878209	−0.439104	+	0.898436i	$0.644704\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.78462	−0.249897
$52$	0	0
$53$	−0.813824	−0.111787	−0.0558936	−	0.998437i	$-0.517801\pi$
$53$	−0.813824	−0.111787	−0.0558936	+	0.998437i	$0.517801\pi$
$54$	0	0
$55$	−0.992782	−0.133867
$56$	0	0
$57$	−1.31810	−0.174586
$58$	0	0
$59$	−7.53795	−0.981357	−0.490679	−	0.871341i	$-0.663251\pi$
$59$	−7.53795	−0.981357	−0.490679	+	0.871341i	$0.663251\pi$
$60$	0	0
$61$	−1.31502	−0.168371	−0.0841857	−	0.996450i	$-0.526829\pi$
$61$	−1.31502	−0.168371	−0.0841857	+	0.996450i	$0.526829\pi$
$62$	0	0
$63$	−2.94139	−0.370580
$64$	0	0
$65$	−0.878144	−0.108920
$66$	0	0
$67$	−8.79384	−1.07434	−0.537170	−	0.843474i	$-0.680506\pi$
$67$	−8.79384	−1.07434	−0.537170	+	0.843474i	$0.680506\pi$
$68$	0	0
$69$	−1.56130	−0.187959
$70$	0	0
$71$	−11.4285	−1.35631	−0.678155	−	0.734919i	$-0.737220\pi$
$71$	−11.4285	−1.35631	−0.678155	+	0.734919i	$0.737220\pi$
$72$	0	0
$73$	−3.68616	−0.431433	−0.215716	−	0.976456i	$-0.569209\pi$
$73$	−3.68616	−0.431433	−0.215716	+	0.976456i	$0.569209\pi$
$74$	0	0
$75$	1.17563	0.135750
$76$	0	0
$77$	2.61527	0.298038
$78$	0	0
$79$	−4.21672	−0.474418	−0.237209	−	0.971459i	$-0.576233\pi$
$79$	−4.21672	−0.474418	−0.237209	+	0.971459i	$0.576233\pi$
$80$	0	0
$81$	8.47591	0.941768
$82$	0	0
$83$	16.9913	1.86504	0.932518	−	0.361123i	$-0.117607\pi$
$83$	16.9913	1.86504	0.932518	+	0.361123i	$0.117607\pi$
$84$	0	0
$85$	−2.79823	−0.303511
$86$	0	0
$87$	1.22813	0.131670
$88$	0	0
$89$	9.32780	0.988745	0.494373	−	0.869250i	$-0.335398\pi$
$89$	9.32780	0.988745	0.494373	+	0.869250i	$0.335398\pi$
$90$	0	0
$91$	2.31328	0.242498
$92$	0	0
$93$	−1.47722	−0.153180
$94$	0	0
$95$	−2.06673	−0.212043
$96$	0	0
$97$	−13.9032	−1.41166	−0.705828	−	0.708384i	$-0.749425\pi$
$97$	−13.9032	−1.41166	−0.705828	+	0.708384i	$0.749425\pi$
$98$	0	0
$99$	−7.69252	−0.773127
$100$	0	0
$101$	1.15104	0.114533	0.0572666	−	0.998359i	$-0.481762\pi$
$101$	1.15104	0.114533	0.0572666	+	0.998359i	$0.481762\pi$
$102$	0	0
$103$	−8.39975	−0.827652	−0.413826	−	0.910356i	$-0.635808\pi$
$103$	−8.39975	−0.827652	−0.413826	+	0.910356i	$0.635808\pi$
$104$	0	0
$105$	0.0919045	0.00896896
$106$	0	0
$107$	−0.997270	−0.0964097	−0.0482048	−	0.998837i	$-0.515350\pi$
$107$	−0.997270	−0.0964097	−0.0482048	+	0.998837i	$0.515350\pi$
$108$	0	0
$109$	17.1491	1.64259	0.821295	−	0.570503i	$-0.193252\pi$
$109$	17.1491	1.64259	0.821295	+	0.570503i	$0.193252\pi$
$110$	0	0
$111$	−2.53999	−0.241085
$112$	0	0
$113$	3.51705	0.330856	0.165428	−	0.986222i	$-0.447099\pi$
$113$	3.51705	0.330856	0.165428	+	0.986222i	$0.447099\pi$
$114$	0	0
$115$	−2.44807	−0.228284
$116$	0	0
$117$	−6.80425	−0.629054
$118$	0	0
$119$	7.37134	0.675729
$120$	0	0
$121$	−4.16036	−0.378215
$122$	0	0
$123$	0.202540	0.0182624
$124$	0	0
$125$	3.74139	0.334640
$126$	0	0
$127$	5.86352	0.520303	0.260152	−	0.965568i	$-0.416227\pi$
$127$	5.86352	0.520303	0.260152	+	0.965568i	$0.416227\pi$
$128$	0	0
$129$	1.33170	0.117250
$130$	0	0
$131$	11.0171	0.962572	0.481286	−	0.876564i	$-0.340170\pi$
$131$	11.0171	0.962572	0.481286	+	0.876564i	$0.340170\pi$
$132$	0	0
$133$	5.44437	0.472087
$134$	0	0
$135$	−0.546040	−0.0469957
$136$	0	0
$137$	−6.34879	−0.542414	−0.271207	−	0.962521i	$-0.587423\pi$
$137$	−6.34879	−0.542414	−0.271207	+	0.962521i	$0.587423\pi$
$138$	0	0
$139$	19.9573	1.69275	0.846377	−	0.532584i	$-0.178779\pi$
$139$	19.9573	1.69275	0.846377	+	0.532584i	$0.178779\pi$
$140$	0	0
$141$	1.45763	0.122754
$142$	0	0
$143$	6.04986	0.505914
$144$	0	0
$145$	1.92568	0.159919
$146$	0	0
$147$	−0.242103	−0.0199683
$148$	0	0
$149$	10.0223	0.821061	0.410531	−	0.911847i	$-0.365343\pi$
$149$	10.0223	0.821061	0.410531	+	0.911847i	$0.365343\pi$
$150$	0	0
$151$	−8.28521	−0.674241	−0.337120	−	0.941462i	$-0.609453\pi$
$151$	−8.28521	−0.674241	−0.337120	+	0.941462i	$0.609453\pi$
$152$	0	0
$153$	−21.6819	−1.75288
$154$	0	0
$155$	−2.31623	−0.186044
$156$	0	0
$157$	10.8753	0.867942	0.433971	−	0.900927i	$-0.357112\pi$
$157$	10.8753	0.867942	0.433971	+	0.900927i	$0.357112\pi$
$158$	0	0
$159$	0.197029	0.0156254
$160$	0	0
$161$	6.44892	0.508246
$162$	0	0
$163$	−6.32087	−0.495089	−0.247545	−	0.968877i	$-0.579624\pi$
$163$	−6.32087	−0.495089	−0.247545	+	0.968877i	$0.579624\pi$
$164$	0	0
$165$	0.240355	0.0187116
$166$	0	0
$167$	12.8905	0.997493	0.498747	−	0.866748i	$-0.333794\pi$
$167$	12.8905	0.997493	0.498747	+	0.866748i	$0.333794\pi$
$168$	0	0
$169$	−7.64873	−0.588364
$170$	0	0
$171$	−16.0140	−1.22462
$172$	0	0
$173$	8.78590	0.667980	0.333990	−	0.942577i	$-0.391605\pi$
$173$	8.78590	0.667980	0.333990	+	0.942577i	$0.391605\pi$
$174$	0	0
$175$	−4.85590	−0.367071
$176$	0	0
$177$	1.82496	0.137172
$178$	0	0
$179$	−2.88010	−0.215269	−0.107635	−	0.994191i	$-0.534328\pi$
$179$	−2.88010	−0.215269	−0.107635	+	0.994191i	$0.534328\pi$
$180$	0	0
$181$	23.6338	1.75669	0.878343	−	0.478030i	$-0.158649\pi$
$181$	23.6338	1.75669	0.878343	+	0.478030i	$0.158649\pi$
$182$	0	0
$183$	0.318371	0.0235346
$184$	0	0
$185$	−3.98263	−0.292809
$186$	0	0
$187$	19.2780	1.40975
$188$	0	0
$189$	1.43843	0.104630
$190$	0	0
$191$	−5.11015	−0.369758	−0.184879	−	0.982761i	$-0.559189\pi$
$191$	−5.11015	−0.369758	−0.184879	+	0.982761i	$0.559189\pi$
$192$	0	0
$193$	0.676235	0.0486765	0.0243382	−	0.999704i	$-0.492252\pi$
$193$	0.676235	0.0486765	0.0243382	+	0.999704i	$0.492252\pi$
$194$	0	0
$195$	0.212601	0.0152247
$196$	0	0
$197$	20.2867	1.44537	0.722684	−	0.691178i	$-0.242908\pi$
$197$	20.2867	1.44537	0.722684	+	0.691178i	$0.242908\pi$
$198$	0	0
$199$	−4.94660	−0.350655	−0.175328	−	0.984510i	$-0.556098\pi$
$199$	−4.94660	−0.350655	−0.175328	+	0.984510i	$0.556098\pi$
$200$	0	0
$201$	2.12901	0.150169
$202$	0	0
$203$	−5.07278	−0.356039
$204$	0	0
$205$	0.317577	0.0221805
$206$	0	0
$207$	−18.9688	−1.31842
$208$	0	0
$209$	14.2385	0.984897
$210$	0	0
$211$	−6.61583	−0.455453	−0.227726	−	0.973725i	$-0.573129\pi$
$211$	−6.61583	−0.455453	−0.227726	+	0.973725i	$0.573129\pi$
$212$	0	0
$213$	2.76686	0.189582
$214$	0	0
$215$	2.08807	0.142405
$216$	0	0
$217$	6.10161	0.414204
$218$	0	0
$219$	0.892431	0.0603049
$220$	0	0
$221$	17.0520	1.14704
$222$	0	0
$223$	4.16691	0.279037	0.139518	−	0.990219i	$-0.455445\pi$
$223$	4.16691	0.279037	0.139518	+	0.990219i	$0.455445\pi$
$224$	0	0
$225$	14.2831	0.952204
$226$	0	0
$227$	−17.1150	−1.13596	−0.567982	−	0.823041i	$-0.692276\pi$
$227$	−17.1150	−1.13596	−0.567982	+	0.823041i	$0.692276\pi$
$228$	0	0
$229$	19.1324	1.26431	0.632153	−	0.774844i	$-0.282171\pi$
$229$	19.1324	1.26431	0.632153	+	0.774844i	$0.282171\pi$
$230$	0	0
$231$	−0.633164	−0.0416591
$232$	0	0
$233$	−13.3857	−0.876924	−0.438462	−	0.898750i	$-0.644477\pi$
$233$	−13.3857	−0.876924	−0.438462	+	0.898750i	$0.644477\pi$
$234$	0	0
$235$	2.28551	0.149091
$236$	0	0
$237$	1.02088	0.0663133
$238$	0	0
$239$	20.6475	1.33558	0.667788	−	0.744352i	$-0.267241\pi$
$239$	20.6475	1.33558	0.667788	+	0.744352i	$0.267241\pi$
$240$	0	0
$241$	0.401861	0.0258861	0.0129431	−	0.999916i	$-0.495880\pi$
$241$	0.401861	0.0258861	0.0129431	+	0.999916i	$0.495880\pi$
$242$	0	0
$243$	−6.36732	−0.408464
$244$	0	0
$245$	−0.379610	−0.0242524
$246$	0	0
$247$	12.5944	0.801360
$248$	0	0
$249$	−4.11364	−0.260691
$250$	0	0
$251$	−13.0881	−0.826113	−0.413056	−	0.910705i	$-0.635539\pi$
$251$	−13.0881	−0.826113	−0.413056	+	0.910705i	$0.635539\pi$
$252$	0	0
$253$	16.8657	1.06034
$254$	0	0
$255$	0.677459	0.0424241
$256$	0	0
$257$	16.3273	1.01847	0.509234	−	0.860628i	$-0.329929\pi$
$257$	16.3273	1.01847	0.509234	+	0.860628i	$0.329929\pi$
$258$	0	0
$259$	10.4914	0.651902
$260$	0	0
$261$	14.9210	0.923587
$262$	0	0
$263$	13.3352	0.822284	0.411142	−	0.911571i	$-0.365130\pi$
$263$	13.3352	0.822284	0.411142	+	0.911571i	$0.365130\pi$
$264$	0	0
$265$	0.308935	0.0189777
$266$	0	0
$267$	−2.25829	−0.138205
$268$	0	0
$269$	27.3334	1.66655	0.833275	−	0.552859i	$-0.186463\pi$
$269$	27.3334	1.66655	0.833275	+	0.552859i	$0.186463\pi$
$270$	0	0
$271$	24.5968	1.49415	0.747074	−	0.664741i	$-0.231458\pi$
$271$	24.5968	1.49415	0.747074	+	0.664741i	$0.231458\pi$
$272$	0	0
$273$	−0.560052	−0.0338959
$274$	0	0
$275$	−12.6995	−0.765807
$276$	0	0
$277$	25.4522	1.52928	0.764638	−	0.644460i	$-0.222918\pi$
$277$	25.4522	1.52928	0.764638	+	0.644460i	$0.222918\pi$
$278$	0	0
$279$	−17.9472	−1.07447
$280$	0	0
$281$	13.7357	0.819404	0.409702	−	0.912219i	$-0.365633\pi$
$281$	13.7357	0.819404	0.409702	+	0.912219i	$0.365633\pi$
$282$	0	0
$283$	−14.1905	−0.843540	−0.421770	−	0.906703i	$-0.638591\pi$
$283$	−14.1905	−0.843540	−0.421770	+	0.906703i	$0.638591\pi$
$284$	0	0
$285$	0.500362	0.0296389
$286$	0	0
$287$	−0.836588	−0.0493822
$288$	0	0
$289$	37.3366	2.19627
$290$	0	0
$291$	3.36600	0.197318
$292$	0	0
$293$	−27.9132	−1.63070	−0.815352	−	0.578965i	$-0.803457\pi$
$293$	−27.9132	−1.63070	−0.815352	+	0.578965i	$0.803457\pi$
$294$	0	0
$295$	2.86148	0.166602
$296$	0	0
$297$	3.76187	0.218286
$298$	0	0
$299$	14.9182	0.862741
$300$	0	0
$301$	−5.50056	−0.317047
$302$	0	0
$303$	−0.278671	−0.0160092
$304$	0	0
$305$	0.499195	0.0285838
$306$	0	0
$307$	−4.45200	−0.254089	−0.127045	−	0.991897i	$-0.540549\pi$
$307$	−4.45200	−0.254089	−0.127045	+	0.991897i	$0.540549\pi$
$308$	0	0
$309$	2.03360	0.115688
$310$	0	0
$311$	−32.2711	−1.82993	−0.914964	−	0.403535i	$-0.867781\pi$
$311$	−32.2711	−1.82993	−0.914964	+	0.403535i	$0.867781\pi$
$312$	0	0
$313$	−22.3372	−1.26257	−0.631286	−	0.775550i	$-0.717473\pi$
$313$	−22.3372	−1.26257	−0.631286	+	0.775550i	$0.717473\pi$
$314$	0	0
$315$	1.11658	0.0629121
$316$	0	0
$317$	−0.209793	−0.0117832	−0.00589158	−	0.999983i	$-0.501875\pi$
$317$	−0.209793	−0.0117832	−0.00589158	+	0.999983i	$0.501875\pi$
$318$	0	0
$319$	−13.2667	−0.742792
$320$	0	0
$321$	0.241442	0.0134760
$322$	0	0
$323$	40.1323	2.23302
$324$	0	0
$325$	−11.2331	−0.623098
$326$	0	0
$327$	−4.15186	−0.229598
$328$	0	0
$329$	−6.02070	−0.331932
$330$	0	0
$331$	−28.4861	−1.56574	−0.782868	−	0.622188i	$-0.786244\pi$
$331$	−28.4861	−1.56574	−0.782868	+	0.622188i	$0.786244\pi$
$332$	0	0
$333$	−30.8592	−1.69107
$334$	0	0
$335$	3.33823	0.182387
$336$	0	0
$337$	−6.83335	−0.372236	−0.186118	−	0.982527i	$-0.559591\pi$
$337$	−6.83335	−0.372236	−0.186118	+	0.982527i	$0.559591\pi$
$338$	0	0
$339$	−0.851487	−0.0462464
$340$	0	0
$341$	15.9574	0.864140
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.592685	0.0319091
$346$	0	0
$347$	−28.0706	−1.50691	−0.753454	−	0.657501i	$-0.771614\pi$
$347$	−28.0706	−1.50691	−0.753454	+	0.657501i	$0.771614\pi$
$348$	0	0
$349$	23.4899	1.25738	0.628692	−	0.777654i	$-0.283591\pi$
$349$	23.4899	1.25738	0.628692	+	0.777654i	$0.283591\pi$
$350$	0	0
$351$	3.32748	0.177608
$352$	0	0
$353$	5.41293	0.288101	0.144051	−	0.989570i	$-0.453987\pi$
$353$	5.41293	0.288101	0.144051	+	0.989570i	$0.453987\pi$
$354$	0	0
$355$	4.33836	0.230256
$356$	0	0
$357$	−1.78462	−0.0944522
$358$	0	0
$359$	−29.1561	−1.53880	−0.769399	−	0.638768i	$-0.779444\pi$
$359$	−29.1561	−1.53880	−0.769399	+	0.638768i	$0.779444\pi$
$360$	0	0
$361$	10.6412	0.560061
$362$	0	0
$363$	1.00724	0.0528662
$364$	0	0
$365$	1.39930	0.0732429
$366$	0	0
$367$	6.14461	0.320746	0.160373	−	0.987056i	$-0.448730\pi$
$367$	6.14461	0.320746	0.160373	+	0.987056i	$0.448730\pi$
$368$	0	0
$369$	2.46073	0.128100
$370$	0	0
$371$	−0.813824	−0.0422516
$372$	0	0
$373$	−29.6063	−1.53296	−0.766478	−	0.642270i	$-0.777993\pi$
$373$	−29.6063	−1.53296	−0.766478	+	0.642270i	$0.777993\pi$
$374$	0	0
$375$	−0.905802	−0.0467754
$376$	0	0
$377$	−11.7348	−0.604371
$378$	0	0
$379$	33.4317	1.71727	0.858635	−	0.512587i	$-0.171313\pi$
$379$	33.4317	1.71727	0.858635	+	0.512587i	$0.171313\pi$
$380$	0	0
$381$	−1.41957	−0.0727270
$382$	0	0
$383$	6.99982	0.357674	0.178837	−	0.983879i	$-0.442767\pi$
$383$	6.99982	0.357674	0.178837	+	0.983879i	$0.442767\pi$
$384$	0	0
$385$	−0.992782	−0.0505968
$386$	0	0
$387$	16.1793	0.822439
$388$	0	0
$389$	−21.8974	−1.11024	−0.555121	−	0.831770i	$-0.687328\pi$
$389$	−21.8974	−1.11024	−0.555121	+	0.831770i	$0.687328\pi$
$390$	0	0
$391$	47.5372	2.40406
$392$	0	0
$393$	−2.66728	−0.134546
$394$	0	0
$395$	1.60071	0.0805403
$396$	0	0
$397$	−8.00336	−0.401677	−0.200839	−	0.979624i	$-0.564367\pi$
$397$	−8.00336	−0.401677	−0.200839	+	0.979624i	$0.564367\pi$
$398$	0	0
$399$	−1.31810	−0.0659874
$400$	0	0
$401$	−26.7191	−1.33429	−0.667144	−	0.744929i	$-0.732483\pi$
$401$	−26.7191	−1.33429	−0.667144	+	0.744929i	$0.732483\pi$
$402$	0	0
$403$	14.1147	0.703106
$404$	0	0
$405$	−3.21754	−0.159881
$406$	0	0
$407$	27.4378	1.36004
$408$	0	0
$409$	−20.7456	−1.02581	−0.512903	−	0.858447i	$-0.671430\pi$
$409$	−20.7456	−1.02581	−0.512903	+	0.858447i	$0.671430\pi$
$410$	0	0
$411$	1.53706	0.0758176
$412$	0	0
$413$	−7.53795	−0.370918
$414$	0	0
$415$	−6.45006	−0.316621
$416$	0	0
$417$	−4.83171	−0.236610
$418$	0	0
$419$	−8.65187	−0.422671	−0.211336	−	0.977414i	$-0.567781\pi$
$419$	−8.65187	−0.422671	−0.211336	+	0.977414i	$0.567781\pi$
$420$	0	0
$421$	12.7739	0.622562	0.311281	−	0.950318i	$-0.399242\pi$
$421$	12.7739	0.622562	0.311281	+	0.950318i	$0.399242\pi$
$422$	0	0
$423$	17.7092	0.861050
$424$	0	0
$425$	−35.7945	−1.73629
$426$	0	0
$427$	−1.31502	−0.0636384
$428$	0	0
$429$	−1.46469	−0.0707157
$430$	0	0
$431$	2.81338	0.135516	0.0677579	−	0.997702i	$-0.478415\pi$
$431$	2.81338	0.135516	0.0677579	+	0.997702i	$0.478415\pi$
$432$	0	0
$433$	20.6954	0.994558	0.497279	−	0.867591i	$-0.334333\pi$
$433$	20.6954	0.994558	0.497279	+	0.867591i	$0.334333\pi$
$434$	0	0
$435$	−0.466211	−0.0223531
$436$	0	0
$437$	35.1103	1.67955
$438$	0	0
$439$	−15.6336	−0.746151	−0.373075	−	0.927801i	$-0.621697\pi$
$439$	−15.6336	−0.746151	−0.373075	+	0.927801i	$0.621697\pi$
$440$	0	0
$441$	−2.94139	−0.140066
$442$	0	0
$443$	26.3904	1.25385	0.626923	−	0.779081i	$-0.284314\pi$
$443$	26.3904	1.25385	0.626923	+	0.779081i	$0.284314\pi$
$444$	0	0
$445$	−3.54092	−0.167856
$446$	0	0
$447$	−2.42643	−0.114766
$448$	0	0
$449$	26.8536	1.26730	0.633650	−	0.773620i	$-0.281556\pi$
$449$	26.8536	1.26730	0.633650	+	0.773620i	$0.281556\pi$
$450$	0	0
$451$	−2.18790	−0.103024
$452$	0	0
$453$	2.00587	0.0942441
$454$	0	0
$455$	−0.878144	−0.0411680
$456$	0	0
$457$	0.385896	0.0180514	0.00902572	−	0.999959i	$-0.497127\pi$
$457$	0.385896	0.0180514	0.00902572	+	0.999959i	$0.497127\pi$
$458$	0	0
$459$	10.6031	0.494911
$460$	0	0
$461$	3.02146	0.140723	0.0703617	−	0.997522i	$-0.477585\pi$
$461$	3.02146	0.140723	0.0703617	+	0.997522i	$0.477585\pi$
$462$	0	0
$463$	−32.5560	−1.51301	−0.756503	−	0.653991i	$-0.773094\pi$
$463$	−32.5560	−1.51301	−0.756503	+	0.653991i	$0.773094\pi$
$464$	0	0
$465$	0.560766	0.0260049
$466$	0	0
$467$	−11.3179	−0.523729	−0.261864	−	0.965105i	$-0.584337\pi$
$467$	−11.3179	−0.523729	−0.261864	+	0.965105i	$0.584337\pi$
$468$	0	0
$469$	−8.79384	−0.406062
$470$	0	0
$471$	−2.63294	−0.121319
$472$	0	0
$473$	−14.3855	−0.661444
$474$	0	0
$475$	−26.4373	−1.21303
$476$	0	0
$477$	2.39377	0.109603
$478$	0	0
$479$	−20.2388	−0.924735	−0.462368	−	0.886688i	$-0.653000\pi$
$479$	−20.2388	−0.924735	−0.462368	+	0.886688i	$0.653000\pi$
$480$	0	0
$481$	24.2695	1.10659
$482$	0	0
$483$	−1.56130	−0.0710417
$484$	0	0
$485$	5.27779	0.239652
$486$	0	0
$487$	36.6988	1.66298	0.831492	−	0.555537i	$-0.187487\pi$
$487$	36.6988	1.66298	0.831492	+	0.555537i	$0.187487\pi$
$488$	0	0
$489$	1.53030	0.0692026
$490$	0	0
$491$	19.2355	0.868085	0.434043	−	0.900892i	$-0.357087\pi$
$491$	19.2355	0.868085	0.434043	+	0.900892i	$0.357087\pi$
$492$	0	0
$493$	−37.3932	−1.68410
$494$	0	0
$495$	2.92015	0.131251
$496$	0	0
$497$	−11.4285	−0.512637
$498$	0	0
$499$	−34.2197	−1.53188	−0.765942	−	0.642910i	$-0.777727\pi$
$499$	−34.2197	−1.53188	−0.765942	+	0.642910i	$0.777727\pi$
$500$	0	0
$501$	−3.12081	−0.139428
$502$	0	0
$503$	−13.4123	−0.598026	−0.299013	−	0.954249i	$-0.596657\pi$
$503$	−13.4123	−0.598026	−0.299013	+	0.954249i	$0.596657\pi$
$504$	0	0
$505$	−0.436947	−0.0194439
$506$	0	0
$507$	1.85178	0.0822404
$508$	0	0
$509$	26.8407	1.18969	0.594846	−	0.803840i	$-0.297213\pi$
$509$	26.8407	1.18969	0.594846	+	0.803840i	$0.297213\pi$
$510$	0	0
$511$	−3.68616	−0.163066
$512$	0	0
$513$	7.83132	0.345761
$514$	0	0
$515$	3.18863	0.140508
$516$	0	0
$517$	−15.7457	−0.692497
$518$	0	0
$519$	−2.12709	−0.0933689
$520$	0	0
$521$	−0.00960703	−0.000420892 0	−0.000210446	−	1.00000i	$-0.500067\pi$
$521$	−0.00960703	−0.000420892 0	−0.000210446		1.00000i	$0.500067\pi$
$522$	0	0
$523$	−33.3512	−1.45835	−0.729173	−	0.684329i	$-0.760095\pi$
$523$	−33.3512	−1.45835	−0.729173	+	0.684329i	$0.760095\pi$
$524$	0	0
$525$	1.17563	0.0513085
$526$	0	0
$527$	44.9770	1.95923
$528$	0	0
$529$	18.5886	0.808201
$530$	0	0
$531$	22.1720	0.962183
$532$	0	0
$533$	−1.93526	−0.0838256
$534$	0	0
$535$	0.378573	0.0163671
$536$	0	0
$537$	0.697281	0.0300899
$538$	0	0
$539$	2.61527	0.112648
$540$	0	0
$541$	16.4724	0.708205	0.354102	−	0.935207i	$-0.384786\pi$
$541$	16.4724	0.708205	0.354102	+	0.935207i	$0.384786\pi$
$542$	0	0
$543$	−5.72181	−0.245546
$544$	0	0
$545$	−6.50998	−0.278857
$546$	0	0
$547$	19.8048	0.846792	0.423396	−	0.905945i	$-0.360838\pi$
$547$	19.8048	0.846792	0.423396	+	0.905945i	$0.360838\pi$
$548$	0	0
$549$	3.86799	0.165082
$550$	0	0
$551$	−27.6181	−1.17657
$552$	0	0
$553$	−4.21672	−0.179313
$554$	0	0
$555$	0.964205	0.0409282
$556$	0	0
$557$	34.4796	1.46095	0.730473	−	0.682941i	$-0.239299\pi$
$557$	34.4796	1.46095	0.730473	+	0.682941i	$0.239299\pi$
$558$	0	0
$559$	−12.7244	−0.538183
$560$	0	0
$561$	−4.66727	−0.197052
$562$	0	0
$563$	−37.1639	−1.56627	−0.783136	−	0.621850i	$-0.786381\pi$
$563$	−37.1639	−1.56627	−0.783136	+	0.621850i	$0.786381\pi$
$564$	0	0
$565$	−1.33510	−0.0561683
$566$	0	0
$567$	8.47591	0.355955
$568$	0	0
$569$	36.8053	1.54296	0.771479	−	0.636255i	$-0.219517\pi$
$569$	36.8053	1.54296	0.771479	+	0.636255i	$0.219517\pi$
$570$	0	0
$571$	32.6027	1.36438	0.682190	−	0.731175i	$-0.261028\pi$
$571$	32.6027	1.36438	0.682190	+	0.731175i	$0.261028\pi$
$572$	0	0
$573$	1.23718	0.0516840
$574$	0	0
$575$	−31.3153	−1.30594
$576$	0	0
$577$	−3.51057	−0.146147	−0.0730735	−	0.997327i	$-0.523281\pi$
$577$	−3.51057	−0.146147	−0.0730735	+	0.997327i	$0.523281\pi$
$578$	0	0
$579$	−0.163718	−0.00680390
$580$	0	0
$581$	16.9913	0.704918
$582$	0	0
$583$	−2.12837	−0.0881480
$584$	0	0
$585$	2.58296	0.106792
$586$	0	0
$587$	39.6426	1.63622	0.818112	−	0.575058i	$-0.195021\pi$
$587$	39.6426	1.63622	0.818112	+	0.575058i	$0.195021\pi$
$588$	0	0
$589$	33.2194	1.36878
$590$	0	0
$591$	−4.91147	−0.202031
$592$	0	0
$593$	−6.27767	−0.257793	−0.128896	−	0.991658i	$-0.541143\pi$
$593$	−6.27767	−0.257793	−0.128896	+	0.991658i	$0.541143\pi$
$594$	0	0
$595$	−2.79823	−0.114716
$596$	0	0
$597$	1.19759	0.0490139
$598$	0	0
$599$	10.3600	0.423300	0.211650	−	0.977346i	$-0.432116\pi$
$599$	10.3600	0.423300	0.211650	+	0.977346i	$0.432116\pi$
$600$	0	0
$601$	−23.9656	−0.977576	−0.488788	−	0.872403i	$-0.662561\pi$
$601$	−23.9656	−0.977576	−0.488788	+	0.872403i	$0.662561\pi$
$602$	0	0
$603$	25.8661	1.05335
$604$	0	0
$605$	1.57931	0.0642083
$606$	0	0
$607$	14.4285	0.585633	0.292817	−	0.956169i	$-0.405407\pi$
$607$	14.4285	0.585633	0.292817	+	0.956169i	$0.405407\pi$
$608$	0	0
$609$	1.22813	0.0497665
$610$	0	0
$611$	−13.9276	−0.563449
$612$	0	0
$613$	4.43815	0.179255	0.0896277	−	0.995975i	$-0.471432\pi$
$613$	4.43815	0.179255	0.0896277	+	0.995975i	$0.471432\pi$
$614$	0	0
$615$	−0.0768862	−0.00310035
$616$	0	0
$617$	−15.7644	−0.634651	−0.317325	−	0.948317i	$-0.602785\pi$
$617$	−15.7644	−0.634651	−0.317325	+	0.948317i	$0.602785\pi$
$618$	0	0
$619$	8.72071	0.350515	0.175257	−	0.984523i	$-0.443924\pi$
$619$	8.72071	0.350515	0.175257	+	0.984523i	$0.443924\pi$
$620$	0	0
$621$	9.27630	0.372245
$622$	0	0
$623$	9.32780	0.373711
$624$	0	0
$625$	22.8592	0.914369
$626$	0	0
$627$	−3.44718	−0.137667
$628$	0	0
$629$	77.3355	3.08357
$630$	0	0
$631$	30.5796	1.21736	0.608678	−	0.793417i	$-0.291700\pi$
$631$	30.5796	1.21736	0.608678	+	0.793417i	$0.291700\pi$
$632$	0	0
$633$	1.60171	0.0636623
$634$	0	0
$635$	−2.22585	−0.0883301
$636$	0	0
$637$	2.31328	0.0916556
$638$	0	0
$639$	33.6155	1.32981
$640$	0	0
$641$	−18.6228	−0.735556	−0.367778	−	0.929914i	$-0.619881\pi$
$641$	−18.6228	−0.735556	−0.367778	+	0.929914i	$0.619881\pi$
$642$	0	0
$643$	−39.3632	−1.55233	−0.776166	−	0.630529i	$-0.782838\pi$
$643$	−39.3632	−1.55233	−0.776166	+	0.630529i	$0.782838\pi$
$644$	0	0
$645$	−0.505527	−0.0199051
$646$	0	0
$647$	−29.2607	−1.15036	−0.575178	−	0.818029i	$-0.695067\pi$
$647$	−29.2607	−1.15036	−0.575178	+	0.818029i	$0.695067\pi$
$648$	0	0
$649$	−19.7138	−0.773833
$650$	0	0
$651$	−1.47722	−0.0578967
$652$	0	0
$653$	24.5826	0.961990	0.480995	−	0.876723i	$-0.340276\pi$
$653$	24.5826	0.961990	0.480995	+	0.876723i	$0.340276\pi$
$654$	0	0
$655$	−4.18221	−0.163413
$656$	0	0
$657$	10.8424	0.423004
$658$	0	0
$659$	39.1780	1.52616	0.763079	−	0.646305i	$-0.223687\pi$
$659$	39.1780	1.52616	0.763079	+	0.646305i	$0.223687\pi$
$660$	0	0
$661$	−26.5220	−1.03159	−0.515794	−	0.856713i	$-0.672503\pi$
$661$	−26.5220	−1.03159	−0.515794	+	0.856713i	$0.672503\pi$
$662$	0	0
$663$	−4.12833	−0.160331
$664$	0	0
$665$	−2.06673	−0.0801445
$666$	0	0
$667$	−32.7140	−1.26669
$668$	0	0
$669$	−1.00882	−0.0390032
$670$	0	0
$671$	−3.43914	−0.132767
$672$	0	0
$673$	8.89179	0.342753	0.171377	−	0.985206i	$-0.445179\pi$
$673$	8.89179	0.342753	0.171377	+	0.985206i	$0.445179\pi$
$674$	0	0
$675$	−6.98485	−0.268847
$676$	0	0
$677$	32.5301	1.25023	0.625117	−	0.780531i	$-0.285051\pi$
$677$	32.5301	1.25023	0.625117	+	0.780531i	$0.285051\pi$
$678$	0	0
$679$	−13.9032	−0.533556
$680$	0	0
$681$	4.14360	0.158783
$682$	0	0
$683$	11.4983	0.439969	0.219984	−	0.975503i	$-0.429399\pi$
$683$	11.4983	0.439969	0.219984	+	0.975503i	$0.429399\pi$
$684$	0	0
$685$	2.41006	0.0920838
$686$	0	0
$687$	−4.63201	−0.176722
$688$	0	0
$689$	−1.88260	−0.0717215
$690$	0	0
$691$	1.54743	0.0588671	0.0294335	−	0.999567i	$-0.490630\pi$
$691$	1.54743	0.0588671	0.0294335	+	0.999567i	$0.490630\pi$
$692$	0	0
$693$	−7.69252	−0.292215
$694$	0	0
$695$	−7.57597	−0.287373
$696$	0	0
$697$	−6.16677	−0.233583
$698$	0	0
$699$	3.24071	0.122575
$700$	0	0
$701$	16.3562	0.617764	0.308882	−	0.951100i	$-0.400045\pi$
$701$	16.3562	0.617764	0.308882	+	0.951100i	$0.400045\pi$
$702$	0	0
$703$	57.1189	2.15428
$704$	0	0
$705$	−0.553329	−0.0208396
$706$	0	0
$707$	1.15104	0.0432895
$708$	0	0
$709$	3.81344	0.143217	0.0716085	−	0.997433i	$-0.477187\pi$
$709$	3.81344	0.143217	0.0716085	+	0.997433i	$0.477187\pi$
$710$	0	0
$711$	12.4030	0.465149
$712$	0	0
$713$	39.3488	1.47363
$714$	0	0
$715$	−2.29658	−0.0858873
$716$	0	0
$717$	−4.99882	−0.186684
$718$	0	0
$719$	−6.08527	−0.226942	−0.113471	−	0.993541i	$-0.536197\pi$
$719$	−6.08527	−0.226942	−0.113471	+	0.993541i	$0.536197\pi$
$720$	0	0
$721$	−8.39975	−0.312823
$722$	0	0
$723$	−0.0972917	−0.00361832
$724$	0	0
$725$	24.6329	0.914843
$726$	0	0
$727$	−41.6162	−1.54346	−0.771729	−	0.635951i	$-0.780608\pi$
$727$	−41.6162	−1.54346	−0.771729	+	0.635951i	$0.780608\pi$
$728$	0	0
$729$	−23.8862	−0.884674
$730$	0	0
$731$	−40.5465	−1.49967
$732$	0	0
$733$	−33.3653	−1.23238	−0.616188	−	0.787599i	$-0.711324\pi$
$733$	−33.3653	−1.23238	−0.616188	+	0.787599i	$0.711324\pi$
$734$	0	0
$735$	0.0919045	0.00338995
$736$	0	0
$737$	−22.9983	−0.847152
$738$	0	0
$739$	10.6509	0.391801	0.195900	−	0.980624i	$-0.437237\pi$
$739$	10.6509	0.391801	0.195900	+	0.980624i	$0.437237\pi$
$740$	0	0
$741$	−3.04913	−0.112013
$742$	0	0
$743$	14.4179	0.528943	0.264472	−	0.964393i	$-0.414802\pi$
$743$	14.4179	0.528943	0.264472	+	0.964393i	$0.414802\pi$
$744$	0	0
$745$	−3.80457	−0.139389
$746$	0	0
$747$	−49.9779	−1.82860
$748$	0	0
$749$	−0.997270	−0.0364394
$750$	0	0
$751$	−17.6760	−0.645008	−0.322504	−	0.946568i	$-0.604525\pi$
$751$	−17.6760	−0.645008	−0.322504	+	0.946568i	$0.604525\pi$
$752$	0	0
$753$	3.16866	0.115472
$754$	0	0
$755$	3.14514	0.114464
$756$	0	0
$757$	−30.1724	−1.09663	−0.548317	−	0.836271i	$-0.684731\pi$
$757$	−30.1724	−1.09663	−0.548317	+	0.836271i	$0.684731\pi$
$758$	0	0
$759$	−4.08323	−0.148212
$760$	0	0
$761$	22.2510	0.806597	0.403299	−	0.915068i	$-0.367864\pi$
$761$	22.2510	0.806597	0.403299	+	0.915068i	$0.367864\pi$
$762$	0	0
$763$	17.1491	0.620841
$764$	0	0
$765$	8.23068	0.297581
$766$	0	0
$767$	−17.4374	−0.629628
$768$	0	0
$769$	−30.5537	−1.10179	−0.550897	−	0.834573i	$-0.685714\pi$
$769$	−30.5537	−1.10179	−0.550897	+	0.834573i	$0.685714\pi$
$770$	0	0
$771$	−3.95288	−0.142360
$772$	0	0
$773$	27.0949	0.974534	0.487267	−	0.873253i	$-0.337994\pi$
$773$	27.0949	0.974534	0.487267	+	0.873253i	$0.337994\pi$
$774$	0	0
$775$	−29.6288	−1.06430
$776$	0	0
$777$	−2.53999	−0.0911217
$778$	0	0
$779$	−4.55469	−0.163189
$780$	0	0
$781$	−29.8885	−1.06950
$782$	0	0
$783$	−7.29682	−0.260767
$784$	0	0
$785$	−4.12836	−0.147348
$786$	0	0
$787$	19.5404	0.696540	0.348270	−	0.937394i	$-0.386769\pi$
$787$	19.5404	0.696540	0.348270	+	0.937394i	$0.386769\pi$
$788$	0	0
$789$	−3.22849	−0.114937
$790$	0	0
$791$	3.51705	0.125052
$792$	0	0
$793$	−3.04202	−0.108025
$794$	0	0
$795$	−0.0747941	−0.00265267
$796$	0	0
$797$	−12.4240	−0.440082	−0.220041	−	0.975491i	$-0.570619\pi$
$797$	−12.4240	−0.440082	−0.220041	+	0.975491i	$0.570619\pi$
$798$	0	0
$799$	−44.3806	−1.57007
$800$	0	0
$801$	−27.4367	−0.969427
$802$	0	0
$803$	−9.64032	−0.340199
$804$	0	0
$805$	−2.44807	−0.0862833
$806$	0	0
$807$	−6.61750	−0.232947
$808$	0	0
$809$	−2.11808	−0.0744676	−0.0372338	−	0.999307i	$-0.511855\pi$
$809$	−2.11808	−0.0744676	−0.0372338	+	0.999307i	$0.511855\pi$
$810$	0	0
$811$	−0.287353	−0.0100903	−0.00504517	−	0.999987i	$-0.501606\pi$
$811$	−0.287353	−0.0100903	−0.00504517	+	0.999987i	$0.501606\pi$
$812$	0	0
$813$	−5.95495	−0.208849
$814$	0	0
$815$	2.39946	0.0840496
$816$	0	0
$817$	−29.9471	−1.04772
$818$	0	0
$819$	−6.80425	−0.237760
$820$	0	0
$821$	−29.4076	−1.02633	−0.513166	−	0.858290i	$-0.671527\pi$
$821$	−29.4076	−1.02633	−0.513166	+	0.858290i	$0.671527\pi$
$822$	0	0
$823$	10.1430	0.353563	0.176782	−	0.984250i	$-0.443431\pi$
$823$	10.1430	0.353563	0.176782	+	0.984250i	$0.443431\pi$
$824$	0	0
$825$	3.07458	0.107043
$826$	0	0
$827$	27.7379	0.964542	0.482271	−	0.876022i	$-0.339812\pi$
$827$	27.7379	0.964542	0.482271	+	0.876022i	$0.339812\pi$
$828$	0	0
$829$	−13.0904	−0.454650	−0.227325	−	0.973819i	$-0.572998\pi$
$829$	−13.0904	−0.454650	−0.227325	+	0.973819i	$0.572998\pi$
$830$	0	0
$831$	−6.16205	−0.213759
$832$	0	0
$833$	7.37134	0.255402
$834$	0	0
$835$	−4.89334	−0.169341
$836$	0	0
$837$	8.77672	0.303368
$838$	0	0
$839$	9.57102	0.330428	0.165214	−	0.986258i	$-0.447168\pi$
$839$	9.57102	0.330428	0.165214	+	0.986258i	$0.447168\pi$
$840$	0	0
$841$	−3.26691	−0.112652
$842$	0	0
$843$	−3.32545	−0.114535
$844$	0	0
$845$	2.90353	0.0998845
$846$	0	0
$847$	−4.16036	−0.142952
$848$	0	0
$849$	3.43557	0.117908
$850$	0	0
$851$	67.6581	2.31929
$852$	0	0
$853$	−20.2042	−0.691779	−0.345890	−	0.938275i	$-0.612423\pi$
$853$	−20.2042	−0.691779	−0.345890	+	0.938275i	$0.612423\pi$
$854$	0	0
$855$	6.07907	0.207900
$856$	0	0
$857$	37.1981	1.27066	0.635332	−	0.772239i	$-0.280863\pi$
$857$	37.1981	1.27066	0.635332	+	0.772239i	$0.280863\pi$
$858$	0	0
$859$	−51.1291	−1.74450	−0.872251	−	0.489058i	$-0.837341\pi$
$859$	−51.1291	−1.74450	−0.872251	+	0.489058i	$0.837341\pi$
$860$	0	0
$861$	0.202540	0.00690255
$862$	0	0
$863$	27.6826	0.942325	0.471163	−	0.882046i	$-0.343835\pi$
$863$	27.6826	0.942325	0.471163	+	0.882046i	$0.343835\pi$
$864$	0	0
$865$	−3.33521	−0.113401
$866$	0	0
$867$	−9.03930	−0.306991
$868$	0	0
$869$	−11.0279	−0.374095
$870$	0	0
$871$	−20.3426	−0.689284
$872$	0	0
$873$	40.8947	1.38407
$874$	0	0
$875$	3.74139	0.126482
$876$	0	0
$877$	3.08114	0.104043	0.0520214	−	0.998646i	$-0.483434\pi$
$877$	3.08114	0.104043	0.0520214	+	0.998646i	$0.483434\pi$
$878$	0	0
$879$	6.75785	0.227937
$880$	0	0
$881$	39.9678	1.34655	0.673276	−	0.739392i	$-0.264887\pi$
$881$	39.9678	1.34655	0.673276	+	0.739392i	$0.264887\pi$
$882$	0	0
$883$	−7.82110	−0.263201	−0.131601	−	0.991303i	$-0.542012\pi$
$883$	−7.82110	−0.263201	−0.131601	+	0.991303i	$0.542012\pi$
$884$	0	0
$885$	−0.692772	−0.0232873
$886$	0	0
$887$	47.8800	1.60765	0.803826	−	0.594865i	$-0.202794\pi$
$887$	47.8800	1.60765	0.803826	+	0.594865i	$0.202794\pi$
$888$	0	0
$889$	5.86352	0.196656
$890$	0	0
$891$	22.1668	0.742616
$892$	0	0
$893$	−32.7789	−1.09690
$894$	0	0
$895$	1.09332	0.0365455
$896$	0	0
$897$	−3.61173	−0.120592
$898$	0	0
$899$	−30.9521	−1.03231
$900$	0	0
$901$	−5.99897	−0.199855
$902$	0	0
$903$	1.33170	0.0443163
$904$	0	0
$905$	−8.97162	−0.298227
$906$	0	0
$907$	40.1949	1.33465	0.667325	−	0.744767i	$-0.267439\pi$
$907$	40.1949	1.33465	0.667325	+	0.744767i	$0.267439\pi$
$908$	0	0
$909$	−3.38566	−0.112295
$910$	0	0
$911$	−50.4192	−1.67046	−0.835231	−	0.549899i	$-0.814666\pi$
$911$	−50.4192	−1.67046	−0.835231	+	0.549899i	$0.814666\pi$
$912$	0	0
$913$	44.4368	1.47064
$914$	0	0
$915$	−0.120857	−0.00399539
$916$	0	0
$917$	11.0171	0.363818
$918$	0	0
$919$	41.1782	1.35834	0.679171	−	0.733980i	$-0.262339\pi$
$919$	41.1782	1.35834	0.679171	+	0.733980i	$0.262339\pi$
$920$	0	0
$921$	1.07784	0.0355161
$922$	0	0
$923$	−26.4373	−0.870193
$924$	0	0
$925$	−50.9450	−1.67506
$926$	0	0
$927$	24.7069	0.811482
$928$	0	0
$929$	−57.2128	−1.87709	−0.938546	−	0.345155i	$-0.887826\pi$
$929$	−57.2128	−1.87709	−0.938546	+	0.345155i	$0.887826\pi$
$930$	0	0
$931$	5.44437	0.178432
$932$	0	0
$933$	7.81293	0.255784
$934$	0	0
$935$	−7.31813	−0.239328
$936$	0	0
$937$	22.3565	0.730353	0.365177	−	0.930938i	$-0.381009\pi$
$937$	22.3565	0.730353	0.365177	+	0.930938i	$0.381009\pi$
$938$	0	0
$939$	5.40789	0.176480
$940$	0	0
$941$	21.2245	0.691899	0.345949	−	0.938253i	$-0.387557\pi$
$941$	21.2245	0.691899	0.345949	+	0.938253i	$0.387557\pi$
$942$	0	0
$943$	−5.39509	−0.175688
$944$	0	0
$945$	−0.546040	−0.0177627
$946$	0	0
$947$	−10.2010	−0.331487	−0.165743	−	0.986169i	$-0.553002\pi$
$947$	−10.2010	−0.331487	−0.165743	+	0.986169i	$0.553002\pi$
$948$	0	0
$949$	−8.52714	−0.276803
$950$	0	0
$951$	0.0507915	0.00164703
$952$	0	0
$953$	49.0686	1.58949	0.794744	−	0.606944i	$-0.207605\pi$
$953$	49.0686	1.58949	0.794744	+	0.606944i	$0.207605\pi$
$954$	0	0
$955$	1.93986	0.0627725
$956$	0	0
$957$	3.21190	0.103826
$958$	0	0
$959$	−6.34879	−0.205013
$960$	0	0
$961$	6.22968	0.200957
$962$	0	0
$963$	2.93335	0.0945260
$964$	0	0
$965$	−0.256705	−0.00826364
$966$	0	0
$967$	3.07877	0.0990064	0.0495032	−	0.998774i	$-0.484236\pi$
$967$	3.07877	0.0990064	0.0495032	+	0.998774i	$0.484236\pi$
$968$	0	0
$969$	−9.71614	−0.312127
$970$	0	0
$971$	22.6421	0.726619	0.363310	−	0.931668i	$-0.381647\pi$
$971$	22.6421	0.726619	0.363310	+	0.931668i	$0.381647\pi$
$972$	0	0
$973$	19.9573	0.639801
$974$	0	0
$975$	2.71955	0.0870954
$976$	0	0
$977$	−10.9770	−0.351185	−0.175592	−	0.984463i	$-0.556184\pi$
$977$	−10.9770	−0.351185	−0.175592	+	0.984463i	$0.556184\pi$
$978$	0	0
$979$	24.3947	0.779659
$980$	0	0
$981$	−50.4423	−1.61050
$982$	0	0
$983$	−15.7349	−0.501864	−0.250932	−	0.968005i	$-0.580737\pi$
$983$	−15.7349	−0.501864	−0.250932	+	0.968005i	$0.580737\pi$
$984$	0	0
$985$	−7.70103	−0.245375
$986$	0	0
$987$	1.45763	0.0463968
$988$	0	0
$989$	−35.4727	−1.12797
$990$	0	0
$991$	46.7969	1.48655	0.743276	−	0.668984i	$-0.233271\pi$
$991$	46.7969	1.48655	0.743276	+	0.668984i	$0.233271\pi$
$992$	0	0
$993$	6.89656	0.218856
$994$	0	0
$995$	1.87778	0.0595295
$996$	0	0
$997$	−41.2295	−1.30575	−0.652875	−	0.757466i	$-0.726437\pi$
$997$	−41.2295	−1.30575	−0.652875	+	0.757466i	$0.726437\pi$
$998$	0	0
$999$	15.0911	0.477460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bf.1.4		8
4.3	odd	2		7168.2.a.be.1.5		8
8.3	odd	2		7168.2.a.ba.1.4		8
8.5	even	2		7168.2.a.bb.1.5		8
32.3	odd	8		1792.2.m.g.1345.5	yes	16
32.5	even	8		1792.2.m.h.449.5	yes	16
32.11	odd	8		1792.2.m.g.449.5	yes	16
32.13	even	8		1792.2.m.h.1345.5	yes	16
32.19	odd	8		1792.2.m.f.1345.4	yes	16
32.21	even	8		1792.2.m.e.449.4	✓	16
32.27	odd	8		1792.2.m.f.449.4	yes	16
32.29	even	8		1792.2.m.e.1345.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.4	✓	16	32.21	even	8
1792.2.m.e.1345.4	yes	16	32.29	even	8
1792.2.m.f.449.4	yes	16	32.27	odd	8
1792.2.m.f.1345.4	yes	16	32.19	odd	8
1792.2.m.g.449.5	yes	16	32.11	odd	8
1792.2.m.g.1345.5	yes	16	32.3	odd	8
1792.2.m.h.449.5	yes	16	32.5	even	8
1792.2.m.h.1345.5	yes	16	32.13	even	8
7168.2.a.ba.1.4		8	8.3	odd	2
7168.2.a.bb.1.5		8	8.5	even	2
7168.2.a.be.1.5		8	4.3	odd	2
7168.2.a.bf.1.4		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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-0.242103 q^{3} -0.379610 q^{5} +1.00000 q^{7} -2.94139 q^{9} +O(q^{10})\)	\(q-0.242103 q^{3} -0.379610 q^{5} +1.00000 q^{7} -2.94139 q^{9} +2.61527 q^{11} +2.31328 q^{13} +0.0919045 q^{15} +7.37134 q^{17} +5.44437 q^{19} -0.242103 q^{21} +6.44892 q^{23} -4.85590 q^{25} +1.43843 q^{27} -5.07278 q^{29} +6.10161 q^{31} -0.633164 q^{33} -0.379610 q^{35} +10.4914 q^{37} -0.560052 q^{39} -0.836588 q^{41} -5.50056 q^{43} +1.11658 q^{45} -6.02070 q^{47} +1.00000 q^{49} -1.78462 q^{51} -0.813824 q^{53} -0.992782 q^{55} -1.31810 q^{57} -7.53795 q^{59} -1.31502 q^{61} -2.94139 q^{63} -0.878144 q^{65} -8.79384 q^{67} -1.56130 q^{69} -11.4285 q^{71} -3.68616 q^{73} +1.17563 q^{75} +2.61527 q^{77} -4.21672 q^{79} +8.47591 q^{81} +16.9913 q^{83} -2.79823 q^{85} +1.22813 q^{87} +9.32780 q^{89} +2.31328 q^{91} -1.47722 q^{93} -2.06673 q^{95} -13.9032 q^{97} -7.69252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q + 8 * q^5 + 8 * q^7 + 12 * q^9	\( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9} + 12 q^{11} + 20 q^{13} + 4 q^{17} + 4 q^{19} + 8 q^{23} + 12 q^{25} - 12 q^{27} + 8 q^{29} - 4 q^{31} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 16 q^{39} - 12 q^{41} - 4 q^{43} + 52 q^{45} + 20 q^{47} + 8 q^{49} + 32 q^{51} + 40 q^{53} + 24 q^{55} - 4 q^{57} + 4 q^{59} - 8 q^{61} + 12 q^{63} + 36 q^{65} + 28 q^{67} + 4 q^{69} - 16 q^{71} + 16 q^{73} - 28 q^{75} + 12 q^{77} + 20 q^{81} - 8 q^{83} + 16 q^{85} + 20 q^{87} + 16 q^{89} + 20 q^{91} + 16 q^{93} - 40 q^{95} - 36 q^{97} - 4 q^{99}+O(q^{100}) \) 8 * q + 8 * q^5 + 8 * q^7 + 12 * q^9 + 12 * q^11 + 20 * q^13 + 4 * q^17 + 4 * q^19 + 8 * q^23 + 12 * q^25 - 12 * q^27 + 8 * q^29 - 4 * q^31 + 8 * q^33 + 8 * q^35 + 8 * q^37 - 16 * q^39 - 12 * q^41 - 4 * q^43 + 52 * q^45 + 20 * q^47 + 8 * q^49 + 32 * q^51 + 40 * q^53 + 24 * q^55 - 4 * q^57 + 4 * q^59 - 8 * q^61 + 12 * q^63 + 36 * q^65 + 28 * q^67 + 4 * q^69 - 16 * q^71 + 16 * q^73 - 28 * q^75 + 12 * q^77 + 20 * q^81 - 8 * q^83 + 16 * q^85 + 20 * q^87 + 16 * q^89 + 20 * q^91 + 16 * q^93 - 40 * q^95 - 36 * q^97 - 4 * q^99

Introduction

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