LMFDB - Embedded newform 7168.2.a.bj.1.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 $ x^12 - 24x^10 + 221x^8 - 968x^6 + 2008x^4 - 1640*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.39251$ 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)

$f(q)$	$=$	$q+0.848497 q^{3} +1.37882 q^{5} +1.00000 q^{7} -2.28005 q^{9} +O(q^{10})$	$q+0.848497 q^{3} +1.37882 q^{5} +1.00000 q^{7} -2.28005 q^{9} +2.43823 q^{11} +2.69538 q^{13} +1.16992 q^{15} -6.71697 q^{17} +4.17063 q^{19} +0.848497 q^{21} -5.29883 q^{23} -3.09887 q^{25} -4.48011 q^{27} +4.28512 q^{29} +1.19996 q^{31} +2.06883 q^{33} +1.37882 q^{35} -3.18200 q^{37} +2.28702 q^{39} +3.94994 q^{41} +9.93190 q^{43} -3.14377 q^{45} +3.06186 q^{47} +1.00000 q^{49} -5.69933 q^{51} +4.26919 q^{53} +3.36187 q^{55} +3.53876 q^{57} +7.02560 q^{59} +13.7065 q^{61} -2.28005 q^{63} +3.71643 q^{65} -5.02889 q^{67} -4.49604 q^{69} +11.5771 q^{71} +10.3271 q^{73} -2.62938 q^{75} +2.43823 q^{77} -4.06883 q^{79} +3.03880 q^{81} +12.9809 q^{83} -9.26146 q^{85} +3.63591 q^{87} +16.9287 q^{89} +2.69538 q^{91} +1.01816 q^{93} +5.75052 q^{95} -2.51522 q^{97} -5.55930 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 + 12 * q^9	$ 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

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
$3$	0.848497	0.489880	0.244940	−	0.969538i	$-0.421232\pi$
$3$	0.848497	0.489880	0.244940	+	0.969538i	$0.421232\pi$
$4$	0	0
$5$	1.37882	0.616625	0.308312	−	0.951285i	$-0.400236\pi$
$5$	1.37882	0.616625	0.308312	+	0.951285i	$0.400236\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.28005	−0.760018
$10$	0	0
$11$	2.43823	0.735155	0.367577	−	0.929993i	$-0.380187\pi$
$11$	2.43823	0.735155	0.367577	+	0.929993i	$0.380187\pi$
$12$	0	0
$13$	2.69538	0.747564	0.373782	−	0.927517i	$-0.378061\pi$
$13$	2.69538	0.747564	0.373782	+	0.927517i	$0.378061\pi$
$14$	0	0
$15$	1.16992	0.302072
$16$	0	0
$17$	−6.71697	−1.62910	−0.814552	−	0.580090i	$-0.803017\pi$
$17$	−6.71697	−1.62910	−0.814552	+	0.580090i	$0.803017\pi$
$18$	0	0
$19$	4.17063	0.956807	0.478404	−	0.878140i	$-0.341216\pi$
$19$	4.17063	0.956807	0.478404	+	0.878140i	$0.341216\pi$
$20$	0	0
$21$	0.848497	0.185157
$22$	0	0
$23$	−5.29883	−1.10488	−0.552441	−	0.833552i	$-0.686303\pi$
$23$	−5.29883	−1.10488	−0.552441	+	0.833552i	$0.686303\pi$
$24$	0	0
$25$	−3.09887	−0.619774
$26$	0	0
$27$	−4.48011	−0.862197
$28$	0	0
$29$	4.28512	0.795726	0.397863	−	0.917445i	$-0.369752\pi$
$29$	4.28512	0.795726	0.397863	+	0.917445i	$0.369752\pi$
$30$	0	0
$31$	1.19996	0.215518	0.107759	−	0.994177i	$-0.465632\pi$
$31$	1.19996	0.215518	0.107759	+	0.994177i	$0.465632\pi$
$32$	0	0
$33$	2.06883	0.360138
$34$	0	0
$35$	1.37882	0.233062
$36$	0	0
$37$	−3.18200	−0.523118	−0.261559	−	0.965187i	$-0.584237\pi$
$37$	−3.18200	−0.523118	−0.261559	+	0.965187i	$0.584237\pi$
$38$	0	0
$39$	2.28702	0.366217
$40$	0	0
$41$	3.94994	0.616877	0.308438	−	0.951244i	$-0.400194\pi$
$41$	3.94994	0.616877	0.308438	+	0.951244i	$0.400194\pi$
$42$	0	0
$43$	9.93190	1.51460	0.757301	−	0.653067i	$-0.226518\pi$
$43$	9.93190	1.51460	0.757301	+	0.653067i	$0.226518\pi$
$44$	0	0
$45$	−3.14377	−0.468646
$46$	0	0
$47$	3.06186	0.446619	0.223309	−	0.974748i	$-0.428314\pi$
$47$	3.06186	0.446619	0.223309	+	0.974748i	$0.428314\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.69933	−0.798066
$52$	0	0
$53$	4.26919	0.586418	0.293209	−	0.956048i	$-0.405277\pi$
$53$	4.26919	0.586418	0.293209	+	0.956048i	$0.405277\pi$
$54$	0	0
$55$	3.36187	0.453315
$56$	0	0
$57$	3.53876	0.468721
$58$	0	0
$59$	7.02560	0.914655	0.457328	−	0.889298i	$-0.348807\pi$
$59$	7.02560	0.914655	0.457328	+	0.889298i	$0.348807\pi$
$60$	0	0
$61$	13.7065	1.75493	0.877467	−	0.479637i	$-0.159232\pi$
$61$	13.7065	1.75493	0.877467	+	0.479637i	$0.159232\pi$
$62$	0	0
$63$	−2.28005	−0.287260
$64$	0	0
$65$	3.71643	0.460967
$66$	0	0
$67$	−5.02889	−0.614377	−0.307188	−	0.951649i	$-0.599388\pi$
$67$	−5.02889	−0.614377	−0.307188	+	0.951649i	$0.599388\pi$
$68$	0	0
$69$	−4.49604	−0.541259
$70$	0	0
$71$	11.5771	1.37395	0.686974	−	0.726682i	$-0.258939\pi$
$71$	11.5771	1.37395	0.686974	+	0.726682i	$0.258939\pi$
$72$	0	0
$73$	10.3271	1.20869	0.604346	−	0.796722i	$-0.293434\pi$
$73$	10.3271	1.20869	0.604346	+	0.796722i	$0.293434\pi$
$74$	0	0
$75$	−2.62938	−0.303615
$76$	0	0
$77$	2.43823	0.277862
$78$	0	0
$79$	−4.06883	−0.457780	−0.228890	−	0.973452i	$-0.573510\pi$
$79$	−4.06883	−0.457780	−0.228890	+	0.973452i	$0.573510\pi$
$80$	0	0
$81$	3.03880	0.337644
$82$	0	0
$83$	12.9809	1.42483	0.712417	−	0.701756i	$-0.247600\pi$
$83$	12.9809	1.42483	0.712417	+	0.701756i	$0.247600\pi$
$84$	0	0
$85$	−9.26146	−1.00455
$86$	0	0
$87$	3.63591	0.389810
$88$	0	0
$89$	16.9287	1.79444	0.897221	−	0.441582i	$-0.145583\pi$
$89$	16.9287	1.79444	0.897221	+	0.441582i	$0.145583\pi$
$90$	0	0
$91$	2.69538	0.282553
$92$	0	0
$93$	1.01816	0.105578
$94$	0	0
$95$	5.75052	0.589991
$96$	0	0
$97$	−2.51522	−0.255382	−0.127691	−	0.991814i	$-0.540757\pi$
$97$	−2.51522	−0.255382	−0.127691	+	0.991814i	$0.540757\pi$
$98$	0	0
$99$	−5.55930	−0.558731
$100$	0	0
$101$	−12.1097	−1.20496	−0.602478	−	0.798135i	$-0.705820\pi$
$101$	−12.1097	−1.20496	−0.602478	+	0.798135i	$0.705820\pi$
$102$	0	0
$103$	−17.8356	−1.75739	−0.878696	−	0.477382i	$-0.841586\pi$
$103$	−17.8356	−1.75739	−0.878696	+	0.477382i	$0.841586\pi$
$104$	0	0
$105$	1.16992	0.114173
$106$	0	0
$107$	−12.0107	−1.16112	−0.580559	−	0.814218i	$-0.697166\pi$
$107$	−12.0107	−1.16112	−0.580559	+	0.814218i	$0.697166\pi$
$108$	0	0
$109$	10.2771	0.984366	0.492183	−	0.870492i	$-0.336199\pi$
$109$	10.2771	0.984366	0.492183	+	0.870492i	$0.336199\pi$
$110$	0	0
$111$	−2.69992	−0.256265
$112$	0	0
$113$	13.8351	1.30150	0.650749	−	0.759293i	$-0.274455\pi$
$113$	13.8351	1.30150	0.650749	+	0.759293i	$0.274455\pi$
$114$	0	0
$115$	−7.30610	−0.681297
$116$	0	0
$117$	−6.14561	−0.568162
$118$	0	0
$119$	−6.71697	−0.615744
$120$	0	0
$121$	−5.05502	−0.459547
$122$	0	0
$123$	3.35151	0.302196
$124$	0	0
$125$	−11.1668	−0.998793
$126$	0	0
$127$	14.1434	1.25502	0.627512	−	0.778607i	$-0.284073\pi$
$127$	14.1434	1.25502	0.627512	+	0.778607i	$0.284073\pi$
$128$	0	0
$129$	8.42719	0.741973
$130$	0	0
$131$	4.31328	0.376853	0.188427	−	0.982087i	$-0.439661\pi$
$131$	4.31328	0.376853	0.188427	+	0.982087i	$0.439661\pi$
$132$	0	0
$133$	4.17063	0.361639
$134$	0	0
$135$	−6.17724	−0.531652
$136$	0	0
$137$	11.5811	0.989440	0.494720	−	0.869052i	$-0.335271\pi$
$137$	11.5811	0.989440	0.494720	+	0.869052i	$0.335271\pi$
$138$	0	0
$139$	3.18322	0.269997	0.134999	−	0.990846i	$-0.456897\pi$
$139$	3.18322	0.269997	0.134999	+	0.990846i	$0.456897\pi$
$140$	0	0
$141$	2.59798	0.218790
$142$	0	0
$143$	6.57197	0.549576
$144$	0	0
$145$	5.90838	0.490665
$146$	0	0
$147$	0.848497	0.0699829
$148$	0	0
$149$	3.25103	0.266335	0.133167	−	0.991094i	$-0.457485\pi$
$149$	3.25103	0.266335	0.133167	+	0.991094i	$0.457485\pi$
$150$	0	0
$151$	−18.1587	−1.47774	−0.738868	−	0.673850i	$-0.764639\pi$
$151$	−18.1587	−1.47774	−0.738868	+	0.673850i	$0.764639\pi$
$152$	0	0
$153$	15.3150	1.23815
$154$	0	0
$155$	1.65452	0.132894
$156$	0	0
$157$	15.2891	1.22021	0.610103	−	0.792322i	$-0.291128\pi$
$157$	15.2891	1.22021	0.610103	+	0.792322i	$0.291128\pi$
$158$	0	0
$159$	3.62239	0.287275
$160$	0	0
$161$	−5.29883	−0.417606
$162$	0	0
$163$	9.09488	0.712366	0.356183	−	0.934416i	$-0.384078\pi$
$163$	9.09488	0.712366	0.356183	+	0.934416i	$0.384078\pi$
$164$	0	0
$165$	2.85254	0.222070
$166$	0	0
$167$	0.661950	0.0512232	0.0256116	−	0.999672i	$-0.491847\pi$
$167$	0.661950	0.0512232	0.0256116	+	0.999672i	$0.491847\pi$
$168$	0	0
$169$	−5.73492	−0.441148
$170$	0	0
$171$	−9.50925	−0.727190
$172$	0	0
$173$	8.98859	0.683390	0.341695	−	0.939811i	$-0.388999\pi$
$173$	8.98859	0.683390	0.341695	+	0.939811i	$0.388999\pi$
$174$	0	0
$175$	−3.09887	−0.234252
$176$	0	0
$177$	5.96120	0.448071
$178$	0	0
$179$	−10.7707	−0.805040	−0.402520	−	0.915411i	$-0.631866\pi$
$179$	−10.7707	−0.805040	−0.402520	+	0.915411i	$0.631866\pi$
$180$	0	0
$181$	−19.7852	−1.47062	−0.735310	−	0.677731i	$-0.762963\pi$
$181$	−19.7852	−1.47062	−0.735310	+	0.677731i	$0.762963\pi$
$182$	0	0
$183$	11.6299	0.859707
$184$	0	0
$185$	−4.38740	−0.322568
$186$	0	0
$187$	−16.3775	−1.19764
$188$	0	0
$189$	−4.48011	−0.325880
$190$	0	0
$191$	−22.4422	−1.62386	−0.811929	−	0.583756i	$-0.801582\pi$
$191$	−22.4422	−1.62386	−0.811929	+	0.583756i	$0.801582\pi$
$192$	0	0
$193$	−6.31408	−0.454498	−0.227249	−	0.973837i	$-0.572973\pi$
$193$	−6.31408	−0.454498	−0.227249	+	0.973837i	$0.572973\pi$
$194$	0	0
$195$	3.15338	0.225818
$196$	0	0
$197$	5.39819	0.384605	0.192302	−	0.981336i	$-0.438405\pi$
$197$	5.39819	0.384605	0.192302	+	0.981336i	$0.438405\pi$
$198$	0	0
$199$	−25.6304	−1.81689	−0.908445	−	0.418005i	$-0.862730\pi$
$199$	−25.6304	−1.81689	−0.908445	+	0.418005i	$0.862730\pi$
$200$	0	0
$201$	−4.26700	−0.300971
$202$	0	0
$203$	4.28512	0.300756
$204$	0	0
$205$	5.44624	0.380382
$206$	0	0
$207$	12.0816	0.839729
$208$	0	0
$209$	10.1690	0.703401
$210$	0	0
$211$	1.04248	0.0717674	0.0358837	−	0.999356i	$-0.488575\pi$
$211$	1.04248	0.0717674	0.0358837	+	0.999356i	$0.488575\pi$
$212$	0	0
$213$	9.82312	0.673069
$214$	0	0
$215$	13.6943	0.933941
$216$	0	0
$217$	1.19996	0.0814583
$218$	0	0
$219$	8.76249	0.592114
$220$	0	0
$221$	−18.1048	−1.21786
$222$	0	0
$223$	−5.76178	−0.385838	−0.192919	−	0.981215i	$-0.561795\pi$
$223$	−5.76178	−0.385838	−0.192919	+	0.981215i	$0.561795\pi$
$224$	0	0
$225$	7.06558	0.471039
$226$	0	0
$227$	25.7640	1.71001	0.855007	−	0.518617i	$-0.173553\pi$
$227$	25.7640	1.71001	0.855007	+	0.518617i	$0.173553\pi$
$228$	0	0
$229$	0.725155	0.0479196	0.0239598	−	0.999713i	$-0.492373\pi$
$229$	0.725155	0.0479196	0.0239598	+	0.999713i	$0.492373\pi$
$230$	0	0
$231$	2.06883	0.136119
$232$	0	0
$233$	14.4425	0.946160	0.473080	−	0.881019i	$-0.343142\pi$
$233$	14.4425	0.946160	0.473080	+	0.881019i	$0.343142\pi$
$234$	0	0
$235$	4.22174	0.275396
$236$	0	0
$237$	−3.45239	−0.224257
$238$	0	0
$239$	7.34716	0.475248	0.237624	−	0.971357i	$-0.423631\pi$
$239$	7.34716	0.475248	0.237624	+	0.971357i	$0.423631\pi$
$240$	0	0
$241$	−3.25347	−0.209575	−0.104787	−	0.994495i	$-0.533416\pi$
$241$	−3.25347	−0.209575	−0.104787	+	0.994495i	$0.533416\pi$
$242$	0	0
$243$	16.0187	1.02760
$244$	0	0
$245$	1.37882	0.0880893
$246$	0	0
$247$	11.2414	0.715275
$248$	0	0
$249$	11.0142	0.697998
$250$	0	0
$251$	2.94262	0.185736	0.0928682	−	0.995678i	$-0.470396\pi$
$251$	2.94262	0.185736	0.0928682	+	0.995678i	$0.470396\pi$
$252$	0	0
$253$	−12.9198	−0.812259
$254$	0	0
$255$	−7.85832	−0.492107
$256$	0	0
$257$	10.9620	0.683788	0.341894	−	0.939738i	$-0.388932\pi$
$257$	10.9620	0.683788	0.341894	+	0.939738i	$0.388932\pi$
$258$	0	0
$259$	−3.18200	−0.197720
$260$	0	0
$261$	−9.77029	−0.604766
$262$	0	0
$263$	10.1931	0.628534	0.314267	−	0.949335i	$-0.398241\pi$
$263$	10.1931	0.628534	0.314267	+	0.949335i	$0.398241\pi$
$264$	0	0
$265$	5.88642	0.361600
$266$	0	0
$267$	14.3640	0.879061
$268$	0	0
$269$	2.48593	0.151570	0.0757850	−	0.997124i	$-0.475854\pi$
$269$	2.48593	0.151570	0.0757850	+	0.997124i	$0.475854\pi$
$270$	0	0
$271$	5.66166	0.343921	0.171961	−	0.985104i	$-0.444990\pi$
$271$	5.66166	0.343921	0.171961	+	0.985104i	$0.444990\pi$
$272$	0	0
$273$	2.28702	0.138417
$274$	0	0
$275$	−7.55576	−0.455630
$276$	0	0
$277$	−30.1692	−1.81269	−0.906346	−	0.422536i	$-0.861140\pi$
$277$	−30.1692	−1.81269	−0.906346	+	0.422536i	$0.861140\pi$
$278$	0	0
$279$	−2.73596	−0.163798
$280$	0	0
$281$	1.60009	0.0954532	0.0477266	−	0.998860i	$-0.484802\pi$
$281$	1.60009	0.0954532	0.0477266	+	0.998860i	$0.484802\pi$
$282$	0	0
$283$	20.3562	1.21005	0.605025	−	0.796206i	$-0.293163\pi$
$283$	20.3562	1.21005	0.605025	+	0.796206i	$0.293163\pi$
$284$	0	0
$285$	4.87930	0.289025
$286$	0	0
$287$	3.94994	0.233158
$288$	0	0
$289$	28.1177	1.65398
$290$	0	0
$291$	−2.13416	−0.125107
$292$	0	0
$293$	0.378816	0.0221307	0.0110653	−	0.999939i	$-0.496478\pi$
$293$	0.378816	0.0221307	0.0110653	+	0.999939i	$0.496478\pi$
$294$	0	0
$295$	9.68700	0.563999
$296$	0	0
$297$	−10.9235	−0.633849
$298$	0	0
$299$	−14.2824	−0.825970
$300$	0	0
$301$	9.93190	0.572465
$302$	0	0
$303$	−10.2750	−0.590284
$304$	0	0
$305$	18.8987	1.08214
$306$	0	0
$307$	−17.5087	−0.999275	−0.499638	−	0.866234i	$-0.666534\pi$
$307$	−17.5087	−0.999275	−0.499638	+	0.866234i	$0.666534\pi$
$308$	0	0
$309$	−15.1334	−0.860911
$310$	0	0
$311$	3.69468	0.209506	0.104753	−	0.994498i	$-0.466595\pi$
$311$	3.69468	0.209506	0.104753	+	0.994498i	$0.466595\pi$
$312$	0	0
$313$	−10.4700	−0.591799	−0.295900	−	0.955219i	$-0.595619\pi$
$313$	−10.4700	−0.591799	−0.295900	+	0.955219i	$0.595619\pi$
$314$	0	0
$315$	−3.14377	−0.177131
$316$	0	0
$317$	22.3362	1.25453	0.627264	−	0.778807i	$-0.284175\pi$
$317$	22.3362	1.25453	0.627264	+	0.778807i	$0.284175\pi$
$318$	0	0
$319$	10.4481	0.584982
$320$	0	0
$321$	−10.1910	−0.568809
$322$	0	0
$323$	−28.0140	−1.55874
$324$	0	0
$325$	−8.35263	−0.463321
$326$	0	0
$327$	8.72008	0.482221
$328$	0	0
$329$	3.06186	0.168806
$330$	0	0
$331$	−9.56376	−0.525672	−0.262836	−	0.964841i	$-0.584658\pi$
$331$	−9.56376	−0.525672	−0.262836	+	0.964841i	$0.584658\pi$
$332$	0	0
$333$	7.25514	0.397579
$334$	0	0
$335$	−6.93391	−0.378840
$336$	0	0
$337$	27.1949	1.48140	0.740700	−	0.671836i	$-0.234494\pi$
$337$	27.1949	1.48140	0.740700	+	0.671836i	$0.234494\pi$
$338$	0	0
$339$	11.7390	0.637578
$340$	0	0
$341$	2.92577	0.158439
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.19920	−0.333754
$346$	0	0
$347$	−5.91364	−0.317461	−0.158730	−	0.987322i	$-0.550740\pi$
$347$	−5.91364	−0.317461	−0.158730	+	0.987322i	$0.550740\pi$
$348$	0	0
$349$	34.7652	1.86094	0.930469	−	0.366371i	$-0.119400\pi$
$349$	34.7652	1.86094	0.930469	+	0.366371i	$0.119400\pi$
$350$	0	0
$351$	−12.0756	−0.644548
$352$	0	0
$353$	33.3458	1.77482	0.887409	−	0.460982i	$-0.152503\pi$
$353$	33.3458	1.77482	0.887409	+	0.460982i	$0.152503\pi$
$354$	0	0
$355$	15.9627	0.847210
$356$	0	0
$357$	−5.69933	−0.301641
$358$	0	0
$359$	−7.89898	−0.416892	−0.208446	−	0.978034i	$-0.566841\pi$
$359$	−7.89898	−0.416892	−0.208446	+	0.978034i	$0.566841\pi$
$360$	0	0
$361$	−1.60588	−0.0845202
$362$	0	0
$363$	−4.28917	−0.225123
$364$	0	0
$365$	14.2391	0.745310
$366$	0	0
$367$	−29.5941	−1.54480	−0.772401	−	0.635136i	$-0.780944\pi$
$367$	−29.5941	−1.54480	−0.772401	+	0.635136i	$0.780944\pi$
$368$	0	0
$369$	−9.00607	−0.468837
$370$	0	0
$371$	4.26919	0.221645
$372$	0	0
$373$	6.10417	0.316062	0.158031	−	0.987434i	$-0.449485\pi$
$373$	6.10417	0.316062	0.158031	+	0.987434i	$0.449485\pi$
$374$	0	0
$375$	−9.47503	−0.489289
$376$	0	0
$377$	11.5500	0.594857
$378$	0	0
$379$	−21.4208	−1.10031	−0.550155	−	0.835062i	$-0.685432\pi$
$379$	−21.4208	−1.10031	−0.550155	+	0.835062i	$0.685432\pi$
$380$	0	0
$381$	12.0006	0.614811
$382$	0	0
$383$	6.62147	0.338341	0.169171	−	0.985587i	$-0.445891\pi$
$383$	6.62147	0.338341	0.169171	+	0.985587i	$0.445891\pi$
$384$	0	0
$385$	3.36187	0.171337
$386$	0	0
$387$	−22.6453	−1.15112
$388$	0	0
$389$	−36.2901	−1.83998	−0.919990	−	0.391942i	$-0.871803\pi$
$389$	−36.2901	−1.83998	−0.919990	+	0.391942i	$0.871803\pi$
$390$	0	0
$391$	35.5921	1.79997
$392$	0	0
$393$	3.65981	0.184613
$394$	0	0
$395$	−5.61017	−0.282278
$396$	0	0
$397$	−13.3940	−0.672226	−0.336113	−	0.941822i	$-0.609112\pi$
$397$	−13.3940	−0.672226	−0.336113	+	0.941822i	$0.609112\pi$
$398$	0	0
$399$	3.53876	0.177160
$400$	0	0
$401$	−17.5138	−0.874598	−0.437299	−	0.899316i	$-0.644065\pi$
$401$	−17.5138	−0.874598	−0.437299	+	0.899316i	$0.644065\pi$
$402$	0	0
$403$	3.23434	0.161114
$404$	0	0
$405$	4.18994	0.208200
$406$	0	0
$407$	−7.75847	−0.384573
$408$	0	0
$409$	10.3755	0.513037	0.256519	−	0.966539i	$-0.417425\pi$
$409$	10.3755	0.513037	0.256519	+	0.966539i	$0.417425\pi$
$410$	0	0
$411$	9.82653	0.484707
$412$	0	0
$413$	7.02560	0.345707
$414$	0	0
$415$	17.8982	0.878589
$416$	0	0
$417$	2.70096	0.132266
$418$	0	0
$419$	−17.8255	−0.870831	−0.435416	−	0.900230i	$-0.643399\pi$
$419$	−17.8255	−0.870831	−0.435416	+	0.900230i	$0.643399\pi$
$420$	0	0
$421$	−24.9733	−1.21712	−0.608561	−	0.793507i	$-0.708253\pi$
$421$	−24.9733	−1.21712	−0.608561	+	0.793507i	$0.708253\pi$
$422$	0	0
$423$	−6.98121	−0.339438
$424$	0	0
$425$	20.8150	1.00968
$426$	0	0
$427$	13.7065	0.663303
$428$	0	0
$429$	5.57630	0.269226
$430$	0	0
$431$	−9.58438	−0.461663	−0.230832	−	0.972994i	$-0.574145\pi$
$431$	−9.58438	−0.461663	−0.230832	+	0.972994i	$0.574145\pi$
$432$	0	0
$433$	0.958633	0.0460690	0.0230345	−	0.999735i	$-0.492667\pi$
$433$	0.958633	0.0460690	0.0230345	+	0.999735i	$0.492667\pi$
$434$	0	0
$435$	5.01325	0.240367
$436$	0	0
$437$	−22.0994	−1.05716
$438$	0	0
$439$	−9.97389	−0.476028	−0.238014	−	0.971262i	$-0.576496\pi$
$439$	−9.97389	−0.476028	−0.238014	+	0.971262i	$0.576496\pi$
$440$	0	0
$441$	−2.28005	−0.108574
$442$	0	0
$443$	−21.8986	−1.04043	−0.520216	−	0.854035i	$-0.674149\pi$
$443$	−21.8986	−1.04043	−0.520216	+	0.854035i	$0.674149\pi$
$444$	0	0
$445$	23.3416	1.10650
$446$	0	0
$447$	2.75849	0.130472
$448$	0	0
$449$	−8.77877	−0.414296	−0.207148	−	0.978310i	$-0.566418\pi$
$449$	−8.77877	−0.414296	−0.207148	+	0.978310i	$0.566418\pi$
$450$	0	0
$451$	9.63087	0.453500
$452$	0	0
$453$	−15.4076	−0.723913
$454$	0	0
$455$	3.71643	0.174229
$456$	0	0
$457$	1.87158	0.0875486	0.0437743	−	0.999041i	$-0.486062\pi$
$457$	1.87158	0.0875486	0.0437743	+	0.999041i	$0.486062\pi$
$458$	0	0
$459$	30.0928	1.40461
$460$	0	0
$461$	18.7685	0.874135	0.437067	−	0.899429i	$-0.356017\pi$
$461$	18.7685	0.874135	0.437067	+	0.899429i	$0.356017\pi$
$462$	0	0
$463$	1.89824	0.0882189	0.0441095	−	0.999027i	$-0.485955\pi$
$463$	1.89824	0.0882189	0.0441095	+	0.999027i	$0.485955\pi$
$464$	0	0
$465$	1.40385	0.0651021
$466$	0	0
$467$	1.53291	0.0709348	0.0354674	−	0.999371i	$-0.488708\pi$
$467$	1.53291	0.0709348	0.0354674	+	0.999371i	$0.488708\pi$
$468$	0	0
$469$	−5.02889	−0.232213
$470$	0	0
$471$	12.9728	0.597754
$472$	0	0
$473$	24.2163	1.11347
$474$	0	0
$475$	−12.9242	−0.593004
$476$	0	0
$477$	−9.73398	−0.445688
$478$	0	0
$479$	−28.2972	−1.29293	−0.646466	−	0.762942i	$-0.723754\pi$
$479$	−28.2972	−1.29293	−0.646466	+	0.762942i	$0.723754\pi$
$480$	0	0
$481$	−8.57672	−0.391065
$482$	0	0
$483$	−4.49604	−0.204577
$484$	0	0
$485$	−3.46803	−0.157475
$486$	0	0
$487$	−6.19712	−0.280818	−0.140409	−	0.990094i	$-0.544842\pi$
$487$	−6.19712	−0.280818	−0.140409	+	0.990094i	$0.544842\pi$
$488$	0	0
$489$	7.71698	0.348974
$490$	0	0
$491$	−30.4986	−1.37638	−0.688191	−	0.725530i	$-0.741595\pi$
$491$	−30.4986	−1.37638	−0.688191	+	0.725530i	$0.741595\pi$
$492$	0	0
$493$	−28.7830	−1.29632
$494$	0	0
$495$	−7.66525	−0.344527
$496$	0	0
$497$	11.5771	0.519303
$498$	0	0
$499$	3.47172	0.155415	0.0777077	−	0.996976i	$-0.475240\pi$
$499$	3.47172	0.155415	0.0777077	+	0.996976i	$0.475240\pi$
$500$	0	0
$501$	0.561663	0.0250932
$502$	0	0
$503$	−13.1803	−0.587680	−0.293840	−	0.955855i	$-0.594933\pi$
$503$	−13.1803	−0.587680	−0.293840	+	0.955855i	$0.594933\pi$
$504$	0	0
$505$	−16.6970	−0.743006
$506$	0	0
$507$	−4.86606	−0.216109
$508$	0	0
$509$	2.17092	0.0962245	0.0481122	−	0.998842i	$-0.484679\pi$
$509$	2.17092	0.0962245	0.0481122	+	0.998842i	$0.484679\pi$
$510$	0	0
$511$	10.3271	0.456843
$512$	0	0
$513$	−18.6849	−0.824957
$514$	0	0
$515$	−24.5920	−1.08365
$516$	0	0
$517$	7.46553	0.328334
$518$	0	0
$519$	7.62680	0.334779
$520$	0	0
$521$	10.0170	0.438854	0.219427	−	0.975629i	$-0.429581\pi$
$521$	10.0170	0.438854	0.219427	+	0.975629i	$0.429581\pi$
$522$	0	0
$523$	23.0919	1.00974	0.504869	−	0.863196i	$-0.331541\pi$
$523$	23.0919	1.00974	0.504869	+	0.863196i	$0.331541\pi$
$524$	0	0
$525$	−2.62938	−0.114756
$526$	0	0
$527$	−8.06007	−0.351102
$528$	0	0
$529$	5.07755	0.220763
$530$	0	0
$531$	−16.0187	−0.695154
$532$	0	0
$533$	10.6466	0.461155
$534$	0	0
$535$	−16.5605	−0.715974
$536$	0	0
$537$	−9.13891	−0.394373
$538$	0	0
$539$	2.43823	0.105022
$540$	0	0
$541$	9.47405	0.407321	0.203661	−	0.979042i	$-0.434716\pi$
$541$	9.47405	0.407321	0.203661	+	0.979042i	$0.434716\pi$
$542$	0	0
$543$	−16.7877	−0.720427
$544$	0	0
$545$	14.1702	0.606985
$546$	0	0
$547$	29.9895	1.28226	0.641130	−	0.767432i	$-0.278466\pi$
$547$	29.9895	1.28226	0.641130	+	0.767432i	$0.278466\pi$
$548$	0	0
$549$	−31.2515	−1.33378
$550$	0	0
$551$	17.8716	0.761357
$552$	0	0
$553$	−4.06883	−0.173024
$554$	0	0
$555$	−3.72269	−0.158020
$556$	0	0
$557$	−23.9826	−1.01617	−0.508087	−	0.861306i	$-0.669647\pi$
$557$	−23.9826	−1.01617	−0.508087	+	0.861306i	$0.669647\pi$
$558$	0	0
$559$	26.7703	1.13226
$560$	0	0
$561$	−13.8963	−0.586702
$562$	0	0
$563$	10.6004	0.446752	0.223376	−	0.974732i	$-0.428292\pi$
$563$	10.6004	0.446752	0.223376	+	0.974732i	$0.428292\pi$
$564$	0	0
$565$	19.0761	0.802536
$566$	0	0
$567$	3.03880	0.127618
$568$	0	0
$569$	44.0529	1.84679	0.923396	−	0.383848i	$-0.125401\pi$
$569$	44.0529	1.84679	0.923396	+	0.383848i	$0.125401\pi$
$570$	0	0
$571$	13.3454	0.558489	0.279244	−	0.960220i	$-0.409916\pi$
$571$	13.3454	0.558489	0.279244	+	0.960220i	$0.409916\pi$
$572$	0	0
$573$	−19.0421	−0.795496
$574$	0	0
$575$	16.4204	0.684777
$576$	0	0
$577$	16.4631	0.685369	0.342685	−	0.939450i	$-0.388664\pi$
$577$	16.4631	0.685369	0.342685	+	0.939450i	$0.388664\pi$
$578$	0	0
$579$	−5.35748	−0.222649
$580$	0	0
$581$	12.9809	0.538537
$582$	0	0
$583$	10.4093	0.431108
$584$	0	0
$585$	−8.47366	−0.350343
$586$	0	0
$587$	30.4695	1.25761	0.628806	−	0.777562i	$-0.283544\pi$
$587$	30.4695	1.25761	0.628806	+	0.777562i	$0.283544\pi$
$588$	0	0
$589$	5.00457	0.206210
$590$	0	0
$591$	4.58034	0.188410
$592$	0	0
$593$	−26.2918	−1.07967	−0.539837	−	0.841770i	$-0.681514\pi$
$593$	−26.2918	−1.07967	−0.539837	+	0.841770i	$0.681514\pi$
$594$	0	0
$595$	−9.26146	−0.379683
$596$	0	0
$597$	−21.7473	−0.890058
$598$	0	0
$599$	27.8214	1.13675	0.568376	−	0.822769i	$-0.307572\pi$
$599$	27.8214	1.13675	0.568376	+	0.822769i	$0.307572\pi$
$600$	0	0
$601$	−18.3631	−0.749047	−0.374523	−	0.927217i	$-0.622194\pi$
$601$	−18.3631	−0.749047	−0.374523	+	0.927217i	$0.622194\pi$
$602$	0	0
$603$	11.4661	0.466937
$604$	0	0
$605$	−6.96994	−0.283368
$606$	0	0
$607$	9.91188	0.402311	0.201155	−	0.979559i	$-0.435530\pi$
$607$	9.91188	0.402311	0.201155	+	0.979559i	$0.435530\pi$
$608$	0	0
$609$	3.63591	0.147334
$610$	0	0
$611$	8.25289	0.333876
$612$	0	0
$613$	43.1350	1.74221	0.871103	−	0.491100i	$-0.163405\pi$
$613$	43.1350	1.74221	0.871103	+	0.491100i	$0.163405\pi$
$614$	0	0
$615$	4.62112	0.186341
$616$	0	0
$617$	−7.78309	−0.313336	−0.156668	−	0.987651i	$-0.550075\pi$
$617$	−7.78309	−0.313336	−0.156668	+	0.987651i	$0.550075\pi$
$618$	0	0
$619$	−39.1049	−1.57176	−0.785880	−	0.618379i	$-0.787790\pi$
$619$	−39.1049	−1.57176	−0.785880	+	0.618379i	$0.787790\pi$
$620$	0	0
$621$	23.7393	0.952626
$622$	0	0
$623$	16.9287	0.678235
$624$	0	0
$625$	0.0973335	0.00389334
$626$	0	0
$627$	8.62833	0.344582
$628$	0	0
$629$	21.3734	0.852215
$630$	0	0
$631$	−6.10120	−0.242885	−0.121442	−	0.992598i	$-0.538752\pi$
$631$	−6.10120	−0.242885	−0.121442	+	0.992598i	$0.538752\pi$
$632$	0	0
$633$	0.884542	0.0351574
$634$	0	0
$635$	19.5011	0.773879
$636$	0	0
$637$	2.69538	0.106795
$638$	0	0
$639$	−26.3964	−1.04422
$640$	0	0
$641$	8.80169	0.347646	0.173823	−	0.984777i	$-0.444388\pi$
$641$	8.80169	0.347646	0.173823	+	0.984777i	$0.444388\pi$
$642$	0	0
$643$	−28.8355	−1.13716	−0.568581	−	0.822627i	$-0.692507\pi$
$643$	−28.8355	−1.13716	−0.568581	+	0.822627i	$0.692507\pi$
$644$	0	0
$645$	11.6195	0.457519
$646$	0	0
$647$	−50.8466	−1.99898	−0.999492	−	0.0318632i	$-0.989856\pi$
$647$	−50.8466	−1.99898	−0.999492	+	0.0318632i	$0.989856\pi$
$648$	0	0
$649$	17.1300	0.672413
$650$	0	0
$651$	1.01816	0.0399048
$652$	0	0
$653$	−2.39100	−0.0935669	−0.0467835	−	0.998905i	$-0.514897\pi$
$653$	−2.39100	−0.0935669	−0.0467835	+	0.998905i	$0.514897\pi$
$654$	0	0
$655$	5.94722	0.232377
$656$	0	0
$657$	−23.5463	−0.918627
$658$	0	0
$659$	−4.25954	−0.165928	−0.0829641	−	0.996553i	$-0.526439\pi$
$659$	−4.25954	−0.165928	−0.0829641	+	0.996553i	$0.526439\pi$
$660$	0	0
$661$	−2.73963	−0.106559	−0.0532796	−	0.998580i	$-0.516967\pi$
$661$	−2.73963	−0.106559	−0.0532796	+	0.998580i	$0.516967\pi$
$662$	0	0
$663$	−15.3619	−0.596606
$664$	0	0
$665$	5.75052	0.222996
$666$	0	0
$667$	−22.7061	−0.879183
$668$	0	0
$669$	−4.88886	−0.189014
$670$	0	0
$671$	33.4196	1.29015
$672$	0	0
$673$	17.6937	0.682041	0.341021	−	0.940056i	$-0.389227\pi$
$673$	17.6937	0.682041	0.341021	+	0.940056i	$0.389227\pi$
$674$	0	0
$675$	13.8833	0.534367
$676$	0	0
$677$	15.8720	0.610010	0.305005	−	0.952351i	$-0.401342\pi$
$677$	15.8720	0.610010	0.305005	+	0.952351i	$0.401342\pi$
$678$	0	0
$679$	−2.51522	−0.0965254
$680$	0	0
$681$	21.8606	0.837702
$682$	0	0
$683$	−9.59924	−0.367305	−0.183652	−	0.982991i	$-0.558792\pi$
$683$	−9.59924	−0.367305	−0.183652	+	0.982991i	$0.558792\pi$
$684$	0	0
$685$	15.9682	0.610113
$686$	0	0
$687$	0.615292	0.0234749
$688$	0	0
$689$	11.5071	0.438385
$690$	0	0
$691$	−48.5742	−1.84785	−0.923926	−	0.382572i	$-0.875038\pi$
$691$	−48.5742	−1.84785	−0.923926	+	0.382572i	$0.875038\pi$
$692$	0	0
$693$	−5.55930	−0.211180
$694$	0	0
$695$	4.38908	0.166487
$696$	0	0
$697$	−26.5316	−1.00496
$698$	0	0
$699$	12.2544	0.463505
$700$	0	0
$701$	−6.62579	−0.250253	−0.125126	−	0.992141i	$-0.539934\pi$
$701$	−6.62579	−0.250253	−0.125126	+	0.992141i	$0.539934\pi$
$702$	0	0
$703$	−13.2709	−0.500523
$704$	0	0
$705$	3.58214	0.134911
$706$	0	0
$707$	−12.1097	−0.455431
$708$	0	0
$709$	24.0605	0.903613	0.451806	−	0.892116i	$-0.350780\pi$
$709$	24.0605	0.903613	0.451806	+	0.892116i	$0.350780\pi$
$710$	0	0
$711$	9.27715	0.347920
$712$	0	0
$713$	−6.35836	−0.238122
$714$	0	0
$715$	9.06153	0.338882
$716$	0	0
$717$	6.23405	0.232815
$718$	0	0
$719$	24.5042	0.913853	0.456927	−	0.889504i	$-0.348950\pi$
$719$	24.5042	0.913853	0.456927	+	0.889504i	$0.348950\pi$
$720$	0	0
$721$	−17.8356	−0.664232
$722$	0	0
$723$	−2.76056	−0.102666
$724$	0	0
$725$	−13.2790	−0.493170
$726$	0	0
$727$	41.6544	1.54488	0.772438	−	0.635090i	$-0.219037\pi$
$727$	41.6544	1.54488	0.772438	+	0.635090i	$0.219037\pi$
$728$	0	0
$729$	4.47546	0.165758
$730$	0	0
$731$	−66.7123	−2.46744
$732$	0	0
$733$	−26.2122	−0.968168	−0.484084	−	0.875021i	$-0.660847\pi$
$733$	−26.2122	−0.968168	−0.484084	+	0.875021i	$0.660847\pi$
$734$	0	0
$735$	1.16992	0.0431532
$736$	0	0
$737$	−12.2616	−0.451662
$738$	0	0
$739$	−6.98514	−0.256952	−0.128476	−	0.991713i	$-0.541009\pi$
$739$	−6.98514	−0.256952	−0.128476	+	0.991713i	$0.541009\pi$
$740$	0	0
$741$	9.53832	0.350399
$742$	0	0
$743$	17.8484	0.654796	0.327398	−	0.944887i	$-0.393828\pi$
$743$	17.8484	0.654796	0.327398	+	0.944887i	$0.393828\pi$
$744$	0	0
$745$	4.48257	0.164229
$746$	0	0
$747$	−29.5970	−1.08290
$748$	0	0
$749$	−12.0107	−0.438861
$750$	0	0
$751$	−18.3471	−0.669495	−0.334747	−	0.942308i	$-0.608651\pi$
$751$	−18.3471	−0.669495	−0.334747	+	0.942308i	$0.608651\pi$
$752$	0	0
$753$	2.49680	0.0909885
$754$	0	0
$755$	−25.0375	−0.911209
$756$	0	0
$757$	−6.93582	−0.252087	−0.126043	−	0.992025i	$-0.540228\pi$
$757$	−6.93582	−0.252087	−0.126043	+	0.992025i	$0.540228\pi$
$758$	0	0
$759$	−10.9624	−0.397909
$760$	0	0
$761$	3.48443	0.126311	0.0631553	−	0.998004i	$-0.479884\pi$
$761$	3.48443	0.126311	0.0631553	+	0.998004i	$0.479884\pi$
$762$	0	0
$763$	10.2771	0.372056
$764$	0	0
$765$	21.1166	0.763473
$766$	0	0
$767$	18.9367	0.683764
$768$	0	0
$769$	−13.5559	−0.488837	−0.244418	−	0.969670i	$-0.578597\pi$
$769$	−13.5559	−0.488837	−0.244418	+	0.969670i	$0.578597\pi$
$770$	0	0
$771$	9.30119	0.334974
$772$	0	0
$773$	28.8219	1.03665	0.518325	−	0.855184i	$-0.326556\pi$
$773$	28.8219	1.03665	0.518325	+	0.855184i	$0.326556\pi$
$774$	0	0
$775$	−3.71851	−0.133573
$776$	0	0
$777$	−2.69992	−0.0968592
$778$	0	0
$779$	16.4737	0.590232
$780$	0	0
$781$	28.2276	1.01006
$782$	0	0
$783$	−19.1978	−0.686073
$784$	0	0
$785$	21.0809	0.752409
$786$	0	0
$787$	39.1989	1.39729	0.698646	−	0.715468i	$-0.253786\pi$
$787$	39.1989	1.39729	0.698646	+	0.715468i	$0.253786\pi$
$788$	0	0
$789$	8.64883	0.307906
$790$	0	0
$791$	13.8351	0.491920
$792$	0	0
$793$	36.9442	1.31193
$794$	0	0
$795$	4.99461	0.177141
$796$	0	0
$797$	4.07063	0.144189	0.0720945	−	0.997398i	$-0.477032\pi$
$797$	4.07063	0.144189	0.0720945	+	0.997398i	$0.477032\pi$
$798$	0	0
$799$	−20.5664	−0.727588
$800$	0	0
$801$	−38.5984	−1.36381
$802$	0	0
$803$	25.1798	0.888576
$804$	0	0
$805$	−7.30610	−0.257506
$806$	0	0
$807$	2.10931	0.0742511
$808$	0	0
$809$	−21.5478	−0.757581	−0.378791	−	0.925482i	$-0.623660\pi$
$809$	−21.5478	−0.757581	−0.378791	+	0.925482i	$0.623660\pi$
$810$	0	0
$811$	−28.9881	−1.01791	−0.508956	−	0.860793i	$-0.669968\pi$
$811$	−28.9881	−1.01791	−0.508956	+	0.860793i	$0.669968\pi$
$812$	0	0
$813$	4.80390	0.168480
$814$	0	0
$815$	12.5402	0.439263
$816$	0	0
$817$	41.4222	1.44918
$818$	0	0
$819$	−6.14561	−0.214745
$820$	0	0
$821$	12.8802	0.449524	0.224762	−	0.974414i	$-0.427840\pi$
$821$	12.8802	0.449524	0.224762	+	0.974414i	$0.427840\pi$
$822$	0	0
$823$	41.4587	1.44516	0.722580	−	0.691287i	$-0.242956\pi$
$823$	41.4587	1.44516	0.722580	+	0.691287i	$0.242956\pi$
$824$	0	0
$825$	−6.41104	−0.223204
$826$	0	0
$827$	20.9493	0.728477	0.364239	−	0.931306i	$-0.381329\pi$
$827$	20.9493	0.728477	0.364239	+	0.931306i	$0.381329\pi$
$828$	0	0
$829$	−24.4586	−0.849482	−0.424741	−	0.905315i	$-0.639635\pi$
$829$	−24.4586	−0.849482	−0.424741	+	0.905315i	$0.639635\pi$
$830$	0	0
$831$	−25.5985	−0.888002
$832$	0	0
$833$	−6.71697	−0.232729
$834$	0	0
$835$	0.912707	0.0315855
$836$	0	0
$837$	−5.37593	−0.185819
$838$	0	0
$839$	−26.6645	−0.920562	−0.460281	−	0.887773i	$-0.652251\pi$
$839$	−26.6645	−0.920562	−0.460281	+	0.887773i	$0.652251\pi$
$840$	0	0
$841$	−10.6378	−0.366820
$842$	0	0
$843$	1.35767	0.0467606
$844$	0	0
$845$	−7.90739	−0.272023
$846$	0	0
$847$	−5.05502	−0.173693
$848$	0	0
$849$	17.2722	0.592779
$850$	0	0
$851$	16.8609	0.577984
$852$	0	0
$853$	17.5464	0.600779	0.300389	−	0.953817i	$-0.402883\pi$
$853$	17.5464	0.600779	0.300389	+	0.953817i	$0.402883\pi$
$854$	0	0
$855$	−13.1115	−0.448404
$856$	0	0
$857$	−2.47165	−0.0844298	−0.0422149	−	0.999109i	$-0.513441\pi$
$857$	−2.47165	−0.0844298	−0.0422149	+	0.999109i	$0.513441\pi$
$858$	0	0
$859$	26.8473	0.916019	0.458010	−	0.888947i	$-0.348563\pi$
$859$	26.8473	0.916019	0.458010	+	0.888947i	$0.348563\pi$
$860$	0	0
$861$	3.35151	0.114219
$862$	0	0
$863$	−35.3848	−1.20451	−0.602255	−	0.798303i	$-0.705731\pi$
$863$	−35.3848	−1.20451	−0.602255	+	0.798303i	$0.705731\pi$
$864$	0	0
$865$	12.3936	0.421395
$866$	0	0
$867$	23.8578	0.810253
$868$	0	0
$869$	−9.92076	−0.336539
$870$	0	0
$871$	−13.5548	−0.459286
$872$	0	0
$873$	5.73484	0.194095
$874$	0	0
$875$	−11.1668	−0.377508
$876$	0	0
$877$	6.11443	0.206470	0.103235	−	0.994657i	$-0.467081\pi$
$877$	6.11443	0.206470	0.103235	+	0.994657i	$0.467081\pi$
$878$	0	0
$879$	0.321424	0.0108414
$880$	0	0
$881$	26.3944	0.889249	0.444625	−	0.895717i	$-0.353337\pi$
$881$	26.3944	0.889249	0.444625	+	0.895717i	$0.353337\pi$
$882$	0	0
$883$	−41.4475	−1.39482	−0.697409	−	0.716673i	$-0.745664\pi$
$883$	−41.4475	−1.39482	−0.697409	+	0.716673i	$0.745664\pi$
$884$	0	0
$885$	8.21940	0.276292
$886$	0	0
$887$	−18.2241	−0.611904	−0.305952	−	0.952047i	$-0.598975\pi$
$887$	−18.2241	−0.611904	−0.305952	+	0.952047i	$0.598975\pi$
$888$	0	0
$889$	14.1434	0.474355
$890$	0	0
$891$	7.40930	0.248221
$892$	0	0
$893$	12.7699	0.427328
$894$	0	0
$895$	−14.8508	−0.496407
$896$	0	0
$897$	−12.1185	−0.404626
$898$	0	0
$899$	5.14195	0.171494
$900$	0	0
$901$	−28.6760	−0.955337
$902$	0	0
$903$	8.42719	0.280439
$904$	0	0
$905$	−27.2801	−0.906821
$906$	0	0
$907$	18.4834	0.613731	0.306866	−	0.951753i	$-0.400720\pi$
$907$	18.4834	0.613731	0.306866	+	0.951753i	$0.400720\pi$
$908$	0	0
$909$	27.6107	0.915788
$910$	0	0
$911$	−21.3908	−0.708708	−0.354354	−	0.935111i	$-0.615299\pi$
$911$	−21.3908	−0.708708	−0.354354	+	0.935111i	$0.615299\pi$
$912$	0	0
$913$	31.6504	1.04747
$914$	0	0
$915$	16.0355	0.530117
$916$	0	0
$917$	4.31328	0.142437
$918$	0	0
$919$	−10.5692	−0.348645	−0.174322	−	0.984689i	$-0.555773\pi$
$919$	−10.5692	−0.348645	−0.174322	+	0.984689i	$0.555773\pi$
$920$	0	0
$921$	−14.8561	−0.489525
$922$	0	0
$923$	31.2047	1.02711
$924$	0	0
$925$	9.86062	0.324215
$926$	0	0
$927$	40.6661	1.33565
$928$	0	0
$929$	−17.5372	−0.575378	−0.287689	−	0.957724i	$-0.592887\pi$
$929$	−17.5372	−0.575378	−0.287689	+	0.957724i	$0.592887\pi$
$930$	0	0
$931$	4.17063	0.136687
$932$	0	0
$933$	3.13493	0.102633
$934$	0	0
$935$	−22.5816	−0.738497
$936$	0	0
$937$	43.7391	1.42890	0.714448	−	0.699689i	$-0.246678\pi$
$937$	43.7391	1.42890	0.714448	+	0.699689i	$0.246678\pi$
$938$	0	0
$939$	−8.88376	−0.289911
$940$	0	0
$941$	10.7701	0.351096	0.175548	−	0.984471i	$-0.443830\pi$
$941$	10.7701	0.351096	0.175548	+	0.984471i	$0.443830\pi$
$942$	0	0
$943$	−20.9300	−0.681576
$944$	0	0
$945$	−6.17724	−0.200946
$946$	0	0
$947$	13.7100	0.445514	0.222757	−	0.974874i	$-0.428494\pi$
$947$	13.7100	0.445514	0.222757	+	0.974874i	$0.428494\pi$
$948$	0	0
$949$	27.8354	0.903575
$950$	0	0
$951$	18.9522	0.614568
$952$	0	0
$953$	−3.19629	−0.103538	−0.0517690	−	0.998659i	$-0.516486\pi$
$953$	−3.19629	−0.103538	−0.0517690	+	0.998659i	$0.516486\pi$
$954$	0	0
$955$	−30.9436	−1.00131
$956$	0	0
$957$	8.86519	0.286571
$958$	0	0
$959$	11.5811	0.373973
$960$	0	0
$961$	−29.5601	−0.953552
$962$	0	0
$963$	27.3850	0.882470
$964$	0	0
$965$	−8.70595	−0.280255
$966$	0	0
$967$	−25.7940	−0.829479	−0.414739	−	0.909940i	$-0.636127\pi$
$967$	−25.7940	−0.829479	−0.414739	+	0.909940i	$0.636127\pi$
$968$	0	0
$969$	−23.7698	−0.763595
$970$	0	0
$971$	16.8297	0.540090	0.270045	−	0.962848i	$-0.412961\pi$
$971$	16.8297	0.540090	0.270045	+	0.962848i	$0.412961\pi$
$972$	0	0
$973$	3.18322	0.102049
$974$	0	0
$975$	−7.08719	−0.226972
$976$	0	0
$977$	−2.74136	−0.0877038	−0.0438519	−	0.999038i	$-0.513963\pi$
$977$	−2.74136	−0.0877038	−0.0438519	+	0.999038i	$0.513963\pi$
$978$	0	0
$979$	41.2762	1.31919
$980$	0	0
$981$	−23.4323	−0.748136
$982$	0	0
$983$	−44.6295	−1.42346	−0.711729	−	0.702454i	$-0.752088\pi$
$983$	−44.6295	−1.42346	−0.711729	+	0.702454i	$0.752088\pi$
$984$	0	0
$985$	7.44310	0.237157
$986$	0	0
$987$	2.59798	0.0826947
$988$	0	0
$989$	−52.6274	−1.67345
$990$	0	0
$991$	1.08451	0.0344508	0.0172254	−	0.999852i	$-0.494517\pi$
$991$	1.08451	0.0344508	0.0172254	+	0.999852i	$0.494517\pi$
$992$	0	0
$993$	−8.11482	−0.257516
$994$	0	0
$995$	−35.3396	−1.12034
$996$	0	0
$997$	41.5652	1.31638	0.658191	−	0.752851i	$-0.271322\pi$
$997$	41.5652	1.31638	0.658191	+	0.752851i	$0.271322\pi$
$998$	0	0
$999$	14.2557	0.451031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bj.1.8		12
4.3	odd	2		7168.2.a.bi.1.5		12
8.3	odd	2		7168.2.a.bi.1.8		12
8.5	even	2	inner	7168.2.a.bj.1.5		12
32.3	odd	8		896.2.m.h.673.2		12
32.5	even	8		112.2.m.d.85.4	yes	12
32.11	odd	8		896.2.m.h.225.2		12
32.13	even	8		112.2.m.d.29.4	✓	12
32.19	odd	8		448.2.m.d.337.5		12
32.21	even	8		896.2.m.g.225.5		12
32.27	odd	8		448.2.m.d.113.5		12
32.29	even	8		896.2.m.g.673.5		12
224.5	odd	24		784.2.x.m.165.5		24
224.13	odd	8		784.2.m.h.589.4		12
224.37	even	24		784.2.x.l.165.5		24
224.45	odd	24		784.2.x.m.765.5		24
224.69	odd	8		784.2.m.h.197.4		12
224.101	odd	24		784.2.x.m.373.1		24
224.109	even	24		784.2.x.l.765.5		24
224.165	even	24		784.2.x.l.373.1		24
224.173	odd	24		784.2.x.m.557.1		24
224.205	even	24		784.2.x.l.557.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.4	✓	12	32.13	even	8
112.2.m.d.85.4	yes	12	32.5	even	8
448.2.m.d.113.5		12	32.27	odd	8
448.2.m.d.337.5		12	32.19	odd	8
784.2.m.h.197.4		12	224.69	odd	8
784.2.m.h.589.4		12	224.13	odd	8
784.2.x.l.165.5		24	224.37	even	24
784.2.x.l.373.1		24	224.165	even	24
784.2.x.l.557.1		24	224.205	even	24
784.2.x.l.765.5		24	224.109	even	24
784.2.x.m.165.5		24	224.5	odd	24
784.2.x.m.373.1		24	224.101	odd	24
784.2.x.m.557.1		24	224.173	odd	24
784.2.x.m.765.5		24	224.45	odd	24
896.2.m.g.225.5		12	32.21	even	8
896.2.m.g.673.5		12	32.29	even	8
896.2.m.h.225.2		12	32.11	odd	8
896.2.m.h.673.2		12	32.3	odd	8
7168.2.a.bi.1.5		12	4.3	odd	2
7168.2.a.bi.1.8		12	8.3	odd	2
7168.2.a.bj.1.5		12	8.5	even	2	inner
7168.2.a.bj.1.8		12	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)

\(f(q)\)	\(=\)	\(q+0.848497 q^{3} +1.37882 q^{5} +1.00000 q^{7} -2.28005 q^{9} +O(q^{10})\)	\(q+0.848497 q^{3} +1.37882 q^{5} +1.00000 q^{7} -2.28005 q^{9} +2.43823 q^{11} +2.69538 q^{13} +1.16992 q^{15} -6.71697 q^{17} +4.17063 q^{19} +0.848497 q^{21} -5.29883 q^{23} -3.09887 q^{25} -4.48011 q^{27} +4.28512 q^{29} +1.19996 q^{31} +2.06883 q^{33} +1.37882 q^{35} -3.18200 q^{37} +2.28702 q^{39} +3.94994 q^{41} +9.93190 q^{43} -3.14377 q^{45} +3.06186 q^{47} +1.00000 q^{49} -5.69933 q^{51} +4.26919 q^{53} +3.36187 q^{55} +3.53876 q^{57} +7.02560 q^{59} +13.7065 q^{61} -2.28005 q^{63} +3.71643 q^{65} -5.02889 q^{67} -4.49604 q^{69} +11.5771 q^{71} +10.3271 q^{73} -2.62938 q^{75} +2.43823 q^{77} -4.06883 q^{79} +3.03880 q^{81} +12.9809 q^{83} -9.26146 q^{85} +3.63591 q^{87} +16.9287 q^{89} +2.69538 q^{91} +1.01816 q^{93} +5.75052 q^{95} -2.51522 q^{97} -5.55930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 + 12 * q^9	\( 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

Introduction

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