LMFDB - Embedded newform 7168.2.a.bj.1.9

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			1.9
Root			$2.48658$ 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)

$f(q)$	$=$	$q+0.892634 q^{3} +3.31293 q^{5} +1.00000 q^{7} -2.20320 q^{9} +O(q^{10})$	$q+0.892634 q^{3} +3.31293 q^{5} +1.00000 q^{7} -2.20320 q^{9} -3.08737 q^{11} +5.70479 q^{13} +2.95723 q^{15} -0.347931 q^{17} -6.02924 q^{19} +0.892634 q^{21} +6.23788 q^{23} +5.97550 q^{25} -4.64456 q^{27} +1.72479 q^{29} -1.26238 q^{31} -2.75589 q^{33} +3.31293 q^{35} +9.08323 q^{37} +5.09229 q^{39} -2.68519 q^{41} +5.73432 q^{43} -7.29906 q^{45} -4.64498 q^{47} +1.00000 q^{49} -0.310575 q^{51} +11.9375 q^{53} -10.2282 q^{55} -5.38191 q^{57} +7.32246 q^{59} -0.00754709 q^{61} -2.20320 q^{63} +18.8996 q^{65} +4.27107 q^{67} +5.56814 q^{69} -0.828913 q^{71} -6.25173 q^{73} +5.33394 q^{75} -3.08737 q^{77} +0.755891 q^{79} +2.46372 q^{81} +5.18431 q^{83} -1.15267 q^{85} +1.53961 q^{87} -6.24461 q^{89} +5.70479 q^{91} -1.12684 q^{93} -19.9745 q^{95} +2.18393 q^{97} +6.80210 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.892634	0.515363	0.257681	−	0.966230i	$-0.417042\pi$
$3$	0.892634	0.515363	0.257681	+	0.966230i	$0.417042\pi$
$4$	0	0
$5$	3.31293	1.48159	0.740794	−	0.671733i	$-0.234450\pi$
$5$	3.31293	1.48159	0.740794	+	0.671733i	$0.234450\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.20320	−0.734401
$10$	0	0
$11$	−3.08737	−0.930876	−0.465438	−	0.885080i	$-0.654103\pi$
$11$	−3.08737	−0.930876	−0.465438	+	0.885080i	$0.654103\pi$
$12$	0	0
$13$	5.70479	1.58222	0.791112	−	0.611671i	$-0.209502\pi$
$13$	5.70479	1.58222	0.791112	+	0.611671i	$0.209502\pi$
$14$	0	0
$15$	2.95723	0.763555
$16$	0	0
$17$	−0.347931	−0.0843858	−0.0421929	−	0.999109i	$-0.513434\pi$
$17$	−0.347931	−0.0843858	−0.0421929	+	0.999109i	$0.513434\pi$
$18$	0	0
$19$	−6.02924	−1.38320	−0.691602	−	0.722279i	$-0.743095\pi$
$19$	−6.02924	−1.38320	−0.691602	+	0.722279i	$0.743095\pi$
$20$	0	0
$21$	0.892634	0.194789
$22$	0	0
$23$	6.23788	1.30069	0.650344	−	0.759640i	$-0.274625\pi$
$23$	6.23788	1.30069	0.650344	+	0.759640i	$0.274625\pi$
$24$	0	0
$25$	5.97550	1.19510
$26$	0	0
$27$	−4.64456	−0.893846
$28$	0	0
$29$	1.72479	0.320285	0.160143	−	0.987094i	$-0.448805\pi$
$29$	1.72479	0.320285	0.160143	+	0.987094i	$0.448805\pi$
$30$	0	0
$31$	−1.26238	−0.226729	−0.113365	−	0.993553i	$-0.536163\pi$
$31$	−1.26238	−0.226729	−0.113365	+	0.993553i	$0.536163\pi$
$32$	0	0
$33$	−2.75589	−0.479739
$34$	0	0
$35$	3.31293	0.559987
$36$	0	0
$37$	9.08323	1.49327	0.746637	−	0.665232i	$-0.231667\pi$
$37$	9.08323	1.49327	0.746637	+	0.665232i	$0.231667\pi$
$38$	0	0
$39$	5.09229	0.815419
$40$	0	0
$41$	−2.68519	−0.419356	−0.209678	−	0.977770i	$-0.567242\pi$
$41$	−2.68519	−0.419356	−0.209678	+	0.977770i	$0.567242\pi$
$42$	0	0
$43$	5.73432	0.874476	0.437238	−	0.899346i	$-0.355957\pi$
$43$	5.73432	0.874476	0.437238	+	0.899346i	$0.355957\pi$
$44$	0	0
$45$	−7.29906	−1.08808
$46$	0	0
$47$	−4.64498	−0.677540	−0.338770	−	0.940869i	$-0.610011\pi$
$47$	−4.64498	−0.677540	−0.338770	+	0.940869i	$0.610011\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.310575	−0.0434893
$52$	0	0
$53$	11.9375	1.63974	0.819870	−	0.572549i	$-0.194046\pi$
$53$	11.9375	1.63974	0.819870	+	0.572549i	$0.194046\pi$
$54$	0	0
$55$	−10.2282	−1.37917
$56$	0	0
$57$	−5.38191	−0.712851
$58$	0	0
$59$	7.32246	0.953303	0.476652	−	0.879092i	$-0.341850\pi$
$59$	7.32246	0.953303	0.476652	+	0.879092i	$0.341850\pi$
$60$	0	0
$61$	−0.00754709	−0.000966305 0	−0.000483153	−	1.00000i	$-0.500154\pi$
$61$	−0.00754709	−0.000966305 0	−0.000483153		1.00000i	$0.500154\pi$
$62$	0	0
$63$	−2.20320	−0.277578
$64$	0	0
$65$	18.8996	2.34420
$66$	0	0
$67$	4.27107	0.521795	0.260897	−	0.965367i	$-0.415982\pi$
$67$	4.27107	0.521795	0.260897	+	0.965367i	$0.415982\pi$
$68$	0	0
$69$	5.56814	0.670326
$70$	0	0
$71$	−0.828913	−0.0983739	−0.0491869	−	0.998790i	$-0.515663\pi$
$71$	−0.828913	−0.0983739	−0.0491869	+	0.998790i	$0.515663\pi$
$72$	0	0
$73$	−6.25173	−0.731709	−0.365855	−	0.930672i	$-0.619223\pi$
$73$	−6.25173	−0.731709	−0.365855	+	0.930672i	$0.619223\pi$
$74$	0	0
$75$	5.33394	0.615910
$76$	0	0
$77$	−3.08737	−0.351838
$78$	0	0
$79$	0.755891	0.0850443	0.0425222	−	0.999096i	$-0.486461\pi$
$79$	0.755891	0.0850443	0.0425222	+	0.999096i	$0.486461\pi$
$80$	0	0
$81$	2.46372	0.273747
$82$	0	0
$83$	5.18431	0.569052	0.284526	−	0.958668i	$-0.408164\pi$
$83$	5.18431	0.569052	0.284526	+	0.958668i	$0.408164\pi$
$84$	0	0
$85$	−1.15267	−0.125025
$86$	0	0
$87$	1.53961	0.165063
$88$	0	0
$89$	−6.24461	−0.661928	−0.330964	−	0.943643i	$-0.607374\pi$
$89$	−6.24461	−0.661928	−0.330964	+	0.943643i	$0.607374\pi$
$90$	0	0
$91$	5.70479	0.598025
$92$	0	0
$93$	−1.12684	−0.116848
$94$	0	0
$95$	−19.9745	−2.04934
$96$	0	0
$97$	2.18393	0.221745	0.110872	−	0.993835i	$-0.464635\pi$
$97$	2.18393	0.221745	0.110872	+	0.993835i	$0.464635\pi$
$98$	0	0
$99$	6.80210	0.683637
$100$	0	0
$101$	6.72212	0.668876	0.334438	−	0.942418i	$-0.391454\pi$
$101$	6.72212	0.668876	0.334438	+	0.942418i	$0.391454\pi$
$102$	0	0
$103$	15.7259	1.54952	0.774760	−	0.632256i	$-0.217871\pi$
$103$	15.7259	1.54952	0.774760	+	0.632256i	$0.217871\pi$
$104$	0	0
$105$	2.95723	0.288597
$106$	0	0
$107$	−9.44807	−0.913380	−0.456690	−	0.889626i	$-0.650965\pi$
$107$	−9.44807	−0.913380	−0.456690	+	0.889626i	$0.650965\pi$
$108$	0	0
$109$	1.14906	0.110060	0.0550299	−	0.998485i	$-0.482475\pi$
$109$	1.14906	0.110060	0.0550299	+	0.998485i	$0.482475\pi$
$110$	0	0
$111$	8.10800	0.769578
$112$	0	0
$113$	18.5170	1.74193	0.870966	−	0.491343i	$-0.163494\pi$
$113$	18.5170	1.74193	0.870966	+	0.491343i	$0.163494\pi$
$114$	0	0
$115$	20.6656	1.92708
$116$	0	0
$117$	−12.5688	−1.16199
$118$	0	0
$119$	−0.347931	−0.0318948
$120$	0	0
$121$	−1.46816	−0.133469
$122$	0	0
$123$	−2.39689	−0.216121
$124$	0	0
$125$	3.23176	0.289058
$126$	0	0
$127$	13.7063	1.21624	0.608121	−	0.793845i	$-0.291924\pi$
$127$	13.7063	1.21624	0.608121	+	0.793845i	$0.291924\pi$
$128$	0	0
$129$	5.11865	0.450672
$130$	0	0
$131$	−4.99111	−0.436076	−0.218038	−	0.975940i	$-0.569966\pi$
$131$	−4.99111	−0.436076	−0.218038	+	0.975940i	$0.569966\pi$
$132$	0	0
$133$	−6.02924	−0.522802
$134$	0	0
$135$	−15.3871	−1.32431
$136$	0	0
$137$	15.6540	1.33741	0.668704	−	0.743529i	$-0.266849\pi$
$137$	15.6540	1.33741	0.668704	+	0.743529i	$0.266849\pi$
$138$	0	0
$139$	2.15766	0.183010	0.0915050	−	0.995805i	$-0.470832\pi$
$139$	2.15766	0.183010	0.0915050	+	0.995805i	$0.470832\pi$
$140$	0	0
$141$	−4.14627	−0.349179
$142$	0	0
$143$	−17.6128	−1.47286
$144$	0	0
$145$	5.71410	0.474531
$146$	0	0
$147$	0.892634	0.0736232
$148$	0	0
$149$	13.0643	1.07027	0.535136	−	0.844766i	$-0.320260\pi$
$149$	13.0643	1.07027	0.535136	+	0.844766i	$0.320260\pi$
$150$	0	0
$151$	11.7266	0.954297	0.477149	−	0.878823i	$-0.341670\pi$
$151$	11.7266	0.954297	0.477149	+	0.878823i	$0.341670\pi$
$152$	0	0
$153$	0.766564	0.0619730
$154$	0	0
$155$	−4.18216	−0.335919
$156$	0	0
$157$	−14.6821	−1.17176	−0.585880	−	0.810397i	$-0.699251\pi$
$157$	−14.6821	−1.17176	−0.585880	+	0.810397i	$0.699251\pi$
$158$	0	0
$159$	10.6558	0.845061
$160$	0	0
$161$	6.23788	0.491614
$162$	0	0
$163$	−22.1401	−1.73415	−0.867075	−	0.498178i	$-0.834003\pi$
$163$	−22.1401	−1.73415	−0.867075	+	0.498178i	$0.834003\pi$
$164$	0	0
$165$	−9.13007	−0.710775
$166$	0	0
$167$	−2.12023	−0.164068	−0.0820341	−	0.996630i	$-0.526142\pi$
$167$	−2.12023	−0.164068	−0.0820341	+	0.996630i	$0.526142\pi$
$168$	0	0
$169$	19.5446	1.50343
$170$	0	0
$171$	13.2836	1.01583
$172$	0	0
$173$	−3.00738	−0.228647	−0.114324	−	0.993444i	$-0.536470\pi$
$173$	−3.00738	−0.228647	−0.114324	+	0.993444i	$0.536470\pi$
$174$	0	0
$175$	5.97550	0.451705
$176$	0	0
$177$	6.53628	0.491297
$178$	0	0
$179$	−2.00693	−0.150005	−0.0750026	−	0.997183i	$-0.523896\pi$
$179$	−2.00693	−0.150005	−0.0750026	+	0.997183i	$0.523896\pi$
$180$	0	0
$181$	20.2503	1.50519	0.752595	−	0.658484i	$-0.228802\pi$
$181$	20.2503	1.50519	0.752595	+	0.658484i	$0.228802\pi$
$182$	0	0
$183$	−0.00673679	−0.000497998 0
$184$	0	0
$185$	30.0921	2.21242
$186$	0	0
$187$	1.07419	0.0785527
$188$	0	0
$189$	−4.64456	−0.337842
$190$	0	0
$191$	6.22279	0.450265	0.225133	−	0.974328i	$-0.427718\pi$
$191$	6.22279	0.450265	0.225133	+	0.974328i	$0.427718\pi$
$192$	0	0
$193$	1.57618	0.113456	0.0567280	−	0.998390i	$-0.481933\pi$
$193$	1.57618	0.113456	0.0567280	+	0.998390i	$0.481933\pi$
$194$	0	0
$195$	16.8704	1.20811
$196$	0	0
$197$	−15.1580	−1.07996	−0.539981	−	0.841677i	$-0.681568\pi$
$197$	−15.1580	−1.07996	−0.539981	+	0.841677i	$0.681568\pi$
$198$	0	0
$199$	−25.5363	−1.81022	−0.905109	−	0.425180i	$-0.860211\pi$
$199$	−25.5363	−1.81022	−0.905109	+	0.425180i	$0.860211\pi$
$200$	0	0
$201$	3.81251	0.268914
$202$	0	0
$203$	1.72479	0.121056
$204$	0	0
$205$	−8.89584	−0.621313
$206$	0	0
$207$	−13.7433	−0.955226
$208$	0	0
$209$	18.6145	1.28759
$210$	0	0
$211$	−13.0091	−0.895582	−0.447791	−	0.894138i	$-0.647789\pi$
$211$	−13.0091	−0.895582	−0.447791	+	0.894138i	$0.647789\pi$
$212$	0	0
$213$	−0.739916	−0.0506982
$214$	0	0
$215$	18.9974	1.29561
$216$	0	0
$217$	−1.26238	−0.0856956
$218$	0	0
$219$	−5.58051	−0.377096
$220$	0	0
$221$	−1.98488	−0.133517
$222$	0	0
$223$	12.7530	0.854003	0.427001	−	0.904251i	$-0.359570\pi$
$223$	12.7530	0.854003	0.427001	+	0.904251i	$0.359570\pi$
$224$	0	0
$225$	−13.1652	−0.877683
$226$	0	0
$227$	−17.3171	−1.14938	−0.574689	−	0.818372i	$-0.694877\pi$
$227$	−17.3171	−1.14938	−0.574689	+	0.818372i	$0.694877\pi$
$228$	0	0
$229$	14.0459	0.928178	0.464089	−	0.885789i	$-0.346382\pi$
$229$	14.0459	0.928178	0.464089	+	0.885789i	$0.346382\pi$
$230$	0	0
$231$	−2.75589	−0.181324
$232$	0	0
$233$	−18.3143	−1.19981	−0.599905	−	0.800072i	$-0.704795\pi$
$233$	−18.3143	−1.19981	−0.599905	+	0.800072i	$0.704795\pi$
$234$	0	0
$235$	−15.3885	−1.00383
$236$	0	0
$237$	0.674734	0.0438287
$238$	0	0
$239$	18.3443	1.18660	0.593298	−	0.804983i	$-0.297826\pi$
$239$	18.3443	1.18660	0.593298	+	0.804983i	$0.297826\pi$
$240$	0	0
$241$	−25.4147	−1.63710	−0.818552	−	0.574433i	$-0.805223\pi$
$241$	−25.4147	−1.63710	−0.818552	+	0.574433i	$0.805223\pi$
$242$	0	0
$243$	16.1329	1.03492
$244$	0	0
$245$	3.31293	0.211655
$246$	0	0
$247$	−34.3956	−2.18854
$248$	0	0
$249$	4.62769	0.293268
$250$	0	0
$251$	15.4575	0.975670	0.487835	−	0.872936i	$-0.337787\pi$
$251$	15.4575	0.975670	0.487835	+	0.872936i	$0.337787\pi$
$252$	0	0
$253$	−19.2586	−1.21078
$254$	0	0
$255$	−1.02891	−0.0644331
$256$	0	0
$257$	2.29652	0.143253	0.0716265	−	0.997432i	$-0.477181\pi$
$257$	2.29652	0.143253	0.0716265	+	0.997432i	$0.477181\pi$
$258$	0	0
$259$	9.08323	0.564404
$260$	0	0
$261$	−3.80006	−0.235218
$262$	0	0
$263$	11.8508	0.730749	0.365375	−	0.930861i	$-0.380941\pi$
$263$	11.8508	0.730749	0.365375	+	0.930861i	$0.380941\pi$
$264$	0	0
$265$	39.5481	2.42942
$266$	0	0
$267$	−5.57416	−0.341133
$268$	0	0
$269$	−8.07220	−0.492171	−0.246085	−	0.969248i	$-0.579144\pi$
$269$	−8.07220	−0.492171	−0.246085	+	0.969248i	$0.579144\pi$
$270$	0	0
$271$	−26.1234	−1.58688	−0.793441	−	0.608648i	$-0.791712\pi$
$271$	−26.1234	−1.58688	−0.793441	+	0.608648i	$0.791712\pi$
$272$	0	0
$273$	5.09229	0.308200
$274$	0	0
$275$	−18.4486	−1.11249
$276$	0	0
$277$	2.80401	0.168477	0.0842385	−	0.996446i	$-0.473154\pi$
$277$	2.80401	0.168477	0.0842385	+	0.996446i	$0.473154\pi$
$278$	0	0
$279$	2.78127	0.166510
$280$	0	0
$281$	6.52475	0.389234	0.194617	−	0.980879i	$-0.437654\pi$
$281$	6.52475	0.389234	0.194617	+	0.980879i	$0.437654\pi$
$282$	0	0
$283$	19.7393	1.17338	0.586690	−	0.809812i	$-0.300431\pi$
$283$	19.7393	1.17338	0.586690	+	0.809812i	$0.300431\pi$
$284$	0	0
$285$	−17.8299	−1.05615
$286$	0	0
$287$	−2.68519	−0.158502
$288$	0	0
$289$	−16.8789	−0.992879
$290$	0	0
$291$	1.94945	0.114279
$292$	0	0
$293$	3.63576	0.212404	0.106202	−	0.994345i	$-0.466131\pi$
$293$	3.63576	0.212404	0.106202	+	0.994345i	$0.466131\pi$
$294$	0	0
$295$	24.2588	1.41240
$296$	0	0
$297$	14.3395	0.832060
$298$	0	0
$299$	35.5858	2.05798
$300$	0	0
$301$	5.73432	0.330521
$302$	0	0
$303$	6.00039	0.344714
$304$	0	0
$305$	−0.0250030	−0.00143167
$306$	0	0
$307$	−26.6229	−1.51945	−0.759725	−	0.650244i	$-0.774667\pi$
$307$	−26.6229	−1.51945	−0.759725	+	0.650244i	$0.774667\pi$
$308$	0	0
$309$	14.0375	0.798564
$310$	0	0
$311$	23.8918	1.35478	0.677389	−	0.735625i	$-0.263111\pi$
$311$	23.8918	1.35478	0.677389	+	0.735625i	$0.263111\pi$
$312$	0	0
$313$	18.0884	1.02242	0.511208	−	0.859457i	$-0.329198\pi$
$313$	18.0884	1.02242	0.511208	+	0.859457i	$0.329198\pi$
$314$	0	0
$315$	−7.29906	−0.411255
$316$	0	0
$317$	−7.58676	−0.426115	−0.213057	−	0.977040i	$-0.568342\pi$
$317$	−7.58676	−0.426115	−0.213057	+	0.977040i	$0.568342\pi$
$318$	0	0
$319$	−5.32506	−0.298146
$320$	0	0
$321$	−8.43367	−0.470722
$322$	0	0
$323$	2.09776	0.116723
$324$	0	0
$325$	34.0890	1.89092
$326$	0	0
$327$	1.02569	0.0567207
$328$	0	0
$329$	−4.64498	−0.256086
$330$	0	0
$331$	−31.6166	−1.73781	−0.868903	−	0.494983i	$-0.835174\pi$
$331$	−31.6166	−1.73781	−0.868903	+	0.494983i	$0.835174\pi$
$332$	0	0
$333$	−20.0122	−1.09666
$334$	0	0
$335$	14.1498	0.773084
$336$	0	0
$337$	−11.5086	−0.626914	−0.313457	−	0.949602i	$-0.601487\pi$
$337$	−11.5086	−0.626914	−0.313457	+	0.949602i	$0.601487\pi$
$338$	0	0
$339$	16.5289	0.897727
$340$	0	0
$341$	3.89742	0.211057
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	18.4469	0.993146
$346$	0	0
$347$	−17.4217	−0.935247	−0.467623	−	0.883928i	$-0.654890\pi$
$347$	−17.4217	−0.935247	−0.467623	+	0.883928i	$0.654890\pi$
$348$	0	0
$349$	0.355676	0.0190389	0.00951945	−	0.999955i	$-0.496970\pi$
$349$	0.355676	0.0190389	0.00951945	+	0.999955i	$0.496970\pi$
$350$	0	0
$351$	−26.4962	−1.41426
$352$	0	0
$353$	16.6888	0.888255	0.444128	−	0.895964i	$-0.353514\pi$
$353$	16.6888	0.888255	0.444128	+	0.895964i	$0.353514\pi$
$354$	0	0
$355$	−2.74613	−0.145749
$356$	0	0
$357$	−0.310575	−0.0164374
$358$	0	0
$359$	−17.9781	−0.948849	−0.474425	−	0.880296i	$-0.657344\pi$
$359$	−17.9781	−0.948849	−0.474425	+	0.880296i	$0.657344\pi$
$360$	0	0
$361$	17.3518	0.913251
$362$	0	0
$363$	−1.31053	−0.0687850
$364$	0	0
$365$	−20.7115	−1.08409
$366$	0	0
$367$	−21.6270	−1.12892	−0.564461	−	0.825460i	$-0.690916\pi$
$367$	−21.6270	−1.12892	−0.564461	+	0.825460i	$0.690916\pi$
$368$	0	0
$369$	5.91602	0.307976
$370$	0	0
$371$	11.9375	0.619764
$372$	0	0
$373$	−17.1296	−0.886936	−0.443468	−	0.896290i	$-0.646252\pi$
$373$	−17.1296	−0.886936	−0.443468	+	0.896290i	$0.646252\pi$
$374$	0	0
$375$	2.88478	0.148970
$376$	0	0
$377$	9.83956	0.506763
$378$	0	0
$379$	−6.14123	−0.315454	−0.157727	−	0.987483i	$-0.550417\pi$
$379$	−6.14123	−0.315454	−0.157727	+	0.987483i	$0.550417\pi$
$380$	0	0
$381$	12.2348	0.626805
$382$	0	0
$383$	−32.8242	−1.67724	−0.838619	−	0.544718i	$-0.816637\pi$
$383$	−32.8242	−1.67724	−0.838619	+	0.544718i	$0.816637\pi$
$384$	0	0
$385$	−10.2282	−0.521279
$386$	0	0
$387$	−12.6339	−0.642216
$388$	0	0
$389$	−0.332323	−0.0168494	−0.00842471	−	0.999965i	$-0.502682\pi$
$389$	−0.332323	−0.0168494	−0.00842471	+	0.999965i	$0.502682\pi$
$390$	0	0
$391$	−2.17035	−0.109759
$392$	0	0
$393$	−4.45524	−0.224737
$394$	0	0
$395$	2.50421	0.126001
$396$	0	0
$397$	−2.79991	−0.140523	−0.0702617	−	0.997529i	$-0.522383\pi$
$397$	−2.79991	−0.140523	−0.0702617	+	0.997529i	$0.522383\pi$
$398$	0	0
$399$	−5.38191	−0.269432
$400$	0	0
$401$	5.15044	0.257200	0.128600	−	0.991697i	$-0.458952\pi$
$401$	5.15044	0.257200	0.128600	+	0.991697i	$0.458952\pi$
$402$	0	0
$403$	−7.20159	−0.358737
$404$	0	0
$405$	8.16213	0.405579
$406$	0	0
$407$	−28.0433	−1.39005
$408$	0	0
$409$	−0.0729036	−0.00360485	−0.00180243	−	0.999998i	$-0.500574\pi$
$409$	−0.0729036	−0.00360485	−0.00180243	+	0.999998i	$0.500574\pi$
$410$	0	0
$411$	13.9733	0.689250
$412$	0	0
$413$	7.32246	0.360315
$414$	0	0
$415$	17.1752	0.843100
$416$	0	0
$417$	1.92600	0.0943165
$418$	0	0
$419$	−19.4350	−0.949461	−0.474730	−	0.880131i	$-0.657454\pi$
$419$	−19.4350	−0.949461	−0.474730	+	0.880131i	$0.657454\pi$
$420$	0	0
$421$	−5.48852	−0.267494	−0.133747	−	0.991015i	$-0.542701\pi$
$421$	−5.48852	−0.267494	−0.133747	+	0.991015i	$0.542701\pi$
$422$	0	0
$423$	10.2338	0.497586
$424$	0	0
$425$	−2.07906	−0.100849
$426$	0	0
$427$	−0.00754709	−0.000365229 0
$428$	0	0
$429$	−15.7218	−0.759055
$430$	0	0
$431$	33.8033	1.62825	0.814125	−	0.580690i	$-0.197217\pi$
$431$	33.8033	1.62825	0.814125	+	0.580690i	$0.197217\pi$
$432$	0	0
$433$	−18.6931	−0.898334	−0.449167	−	0.893448i	$-0.648279\pi$
$433$	−18.6931	−0.898334	−0.449167	+	0.893448i	$0.648279\pi$
$434$	0	0
$435$	5.10061	0.244555
$436$	0	0
$437$	−37.6097	−1.79911
$438$	0	0
$439$	6.03142	0.287864	0.143932	−	0.989588i	$-0.454025\pi$
$439$	6.03142	0.287864	0.143932	+	0.989588i	$0.454025\pi$
$440$	0	0
$441$	−2.20320	−0.104914
$442$	0	0
$443$	−35.2362	−1.67412	−0.837061	−	0.547109i	$-0.815728\pi$
$443$	−35.2362	−1.67412	−0.837061	+	0.547109i	$0.815728\pi$
$444$	0	0
$445$	−20.6880	−0.980703
$446$	0	0
$447$	11.6617	0.551578
$448$	0	0
$449$	21.9883	1.03769	0.518846	−	0.854868i	$-0.326362\pi$
$449$	21.9883	1.03769	0.518846	+	0.854868i	$0.326362\pi$
$450$	0	0
$451$	8.29017	0.390369
$452$	0	0
$453$	10.4676	0.491809
$454$	0	0
$455$	18.8996	0.886026
$456$	0	0
$457$	26.7381	1.25076	0.625378	−	0.780322i	$-0.284945\pi$
$457$	26.7381	1.25076	0.625378	+	0.780322i	$0.284945\pi$
$458$	0	0
$459$	1.61599	0.0754278
$460$	0	0
$461$	−33.3136	−1.55157	−0.775786	−	0.630997i	$-0.782646\pi$
$461$	−33.3136	−1.55157	−0.775786	+	0.630997i	$0.782646\pi$
$462$	0	0
$463$	9.83629	0.457131	0.228566	−	0.973529i	$-0.426596\pi$
$463$	9.83629	0.457131	0.228566	+	0.973529i	$0.426596\pi$
$464$	0	0
$465$	−3.73314	−0.173120
$466$	0	0
$467$	−2.78025	−0.128655	−0.0643274	−	0.997929i	$-0.520490\pi$
$467$	−2.78025	−0.128655	−0.0643274	+	0.997929i	$0.520490\pi$
$468$	0	0
$469$	4.27107	0.197220
$470$	0	0
$471$	−13.1058	−0.603882
$472$	0	0
$473$	−17.7040	−0.814029
$474$	0	0
$475$	−36.0277	−1.65307
$476$	0	0
$477$	−26.3007	−1.20423
$478$	0	0
$479$	3.78688	0.173027	0.0865136	−	0.996251i	$-0.472427\pi$
$479$	3.78688	0.173027	0.0865136	+	0.996251i	$0.472427\pi$
$480$	0	0
$481$	51.8179	2.36269
$482$	0	0
$483$	5.56814	0.253359
$484$	0	0
$485$	7.23522	0.328534
$486$	0	0
$487$	16.8200	0.762186	0.381093	−	0.924537i	$-0.375548\pi$
$487$	16.8200	0.762186	0.381093	+	0.924537i	$0.375548\pi$
$488$	0	0
$489$	−19.7630	−0.893716
$490$	0	0
$491$	−9.07385	−0.409497	−0.204749	−	0.978815i	$-0.565638\pi$
$491$	−9.07385	−0.409497	−0.204749	+	0.978815i	$0.565638\pi$
$492$	0	0
$493$	−0.600108	−0.0270275
$494$	0	0
$495$	22.5349	1.01287
$496$	0	0
$497$	−0.828913	−0.0371818
$498$	0	0
$499$	6.63693	0.297110	0.148555	−	0.988904i	$-0.452538\pi$
$499$	6.63693	0.297110	0.148555	+	0.988904i	$0.452538\pi$
$500$	0	0
$501$	−1.89259	−0.0845546
$502$	0	0
$503$	−4.37360	−0.195009	−0.0975045	−	0.995235i	$-0.531086\pi$
$503$	−4.37360	−0.195009	−0.0975045	+	0.995235i	$0.531086\pi$
$504$	0	0
$505$	22.2699	0.990998
$506$	0	0
$507$	17.4462	0.774814
$508$	0	0
$509$	2.73818	0.121368	0.0606839	−	0.998157i	$-0.480672\pi$
$509$	2.73818	0.121368	0.0606839	+	0.998157i	$0.480672\pi$
$510$	0	0
$511$	−6.25173	−0.276560
$512$	0	0
$513$	28.0032	1.23637
$514$	0	0
$515$	52.0988	2.29575
$516$	0	0
$517$	14.3408	0.630706
$518$	0	0
$519$	−2.68449	−0.117836
$520$	0	0
$521$	−35.3300	−1.54783	−0.773917	−	0.633287i	$-0.781705\pi$
$521$	−35.3300	−1.54783	−0.773917	+	0.633287i	$0.781705\pi$
$522$	0	0
$523$	−33.0720	−1.44614	−0.723068	−	0.690777i	$-0.757269\pi$
$523$	−33.0720	−1.44614	−0.723068	+	0.690777i	$0.757269\pi$
$524$	0	0
$525$	5.33394	0.232792
$526$	0	0
$527$	0.439220	0.0191327
$528$	0	0
$529$	15.9111	0.691787
$530$	0	0
$531$	−16.1329	−0.700107
$532$	0	0
$533$	−15.3184	−0.663516
$534$	0	0
$535$	−31.3008	−1.35325
$536$	0	0
$537$	−1.79146	−0.0773070
$538$	0	0
$539$	−3.08737	−0.132982
$540$	0	0
$541$	−7.25740	−0.312020	−0.156010	−	0.987755i	$-0.549863\pi$
$541$	−7.25740	−0.312020	−0.156010	+	0.987755i	$0.549863\pi$
$542$	0	0
$543$	18.0761	0.775719
$544$	0	0
$545$	3.80675	0.163063
$546$	0	0
$547$	23.7810	1.01680	0.508402	−	0.861120i	$-0.330236\pi$
$547$	23.7810	1.01680	0.508402	+	0.861120i	$0.330236\pi$
$548$	0	0
$549$	0.0166278	0.000709656 0
$550$	0	0
$551$	−10.3992	−0.443020
$552$	0	0
$553$	0.755891	0.0321437
$554$	0	0
$555$	26.8612	1.14020
$556$	0	0
$557$	15.6800	0.664382	0.332191	−	0.943212i	$-0.392212\pi$
$557$	15.6800	0.664382	0.332191	+	0.943212i	$0.392212\pi$
$558$	0	0
$559$	32.7131	1.38362
$560$	0	0
$561$	0.958861	0.0404831
$562$	0	0
$563$	−31.4035	−1.32350	−0.661751	−	0.749724i	$-0.730186\pi$
$563$	−31.4035	−1.32350	−0.661751	+	0.749724i	$0.730186\pi$
$564$	0	0
$565$	61.3455	2.58082
$566$	0	0
$567$	2.46372	0.103466
$568$	0	0
$569$	0.317171	0.0132965	0.00664825	−	0.999978i	$-0.497884\pi$
$569$	0.317171	0.0132965	0.00664825	+	0.999978i	$0.497884\pi$
$570$	0	0
$571$	11.0359	0.461839	0.230920	−	0.972973i	$-0.425827\pi$
$571$	11.0359	0.461839	0.230920	+	0.972973i	$0.425827\pi$
$572$	0	0
$573$	5.55467	0.232050
$574$	0	0
$575$	37.2744	1.55445
$576$	0	0
$577$	−10.2699	−0.427540	−0.213770	−	0.976884i	$-0.568574\pi$
$577$	−10.2699	−0.427540	−0.213770	+	0.976884i	$0.568574\pi$
$578$	0	0
$579$	1.40695	0.0584710
$580$	0	0
$581$	5.18431	0.215081
$582$	0	0
$583$	−36.8554	−1.52640
$584$	0	0
$585$	−41.6396	−1.72159
$586$	0	0
$587$	−39.5930	−1.63418	−0.817088	−	0.576512i	$-0.804413\pi$
$587$	−39.5930	−1.63418	−0.817088	+	0.576512i	$0.804413\pi$
$588$	0	0
$589$	7.61117	0.313613
$590$	0	0
$591$	−13.5305	−0.556572
$592$	0	0
$593$	34.5902	1.42045	0.710225	−	0.703974i	$-0.248593\pi$
$593$	34.5902	1.42045	0.710225	+	0.703974i	$0.248593\pi$
$594$	0	0
$595$	−1.15267	−0.0472549
$596$	0	0
$597$	−22.7945	−0.932919
$598$	0	0
$599$	−14.0866	−0.575562	−0.287781	−	0.957696i	$-0.592918\pi$
$599$	−14.0866	−0.575562	−0.287781	+	0.957696i	$0.592918\pi$
$600$	0	0
$601$	−7.07501	−0.288595	−0.144298	−	0.989534i	$-0.546092\pi$
$601$	−7.07501	−0.288595	−0.144298	+	0.989534i	$0.546092\pi$
$602$	0	0
$603$	−9.41005	−0.383207
$604$	0	0
$605$	−4.86391	−0.197746
$606$	0	0
$607$	5.99294	0.243246	0.121623	−	0.992576i	$-0.461190\pi$
$607$	5.99294	0.243246	0.121623	+	0.992576i	$0.461190\pi$
$608$	0	0
$609$	1.53961	0.0623880
$610$	0	0
$611$	−26.4986	−1.07202
$612$	0	0
$613$	17.4183	0.703520	0.351760	−	0.936090i	$-0.385583\pi$
$613$	17.4183	0.703520	0.351760	+	0.936090i	$0.385583\pi$
$614$	0	0
$615$	−7.94074	−0.320201
$616$	0	0
$617$	40.3690	1.62519	0.812596	−	0.582827i	$-0.198053\pi$
$617$	40.3690	1.62519	0.812596	+	0.582827i	$0.198053\pi$
$618$	0	0
$619$	43.7523	1.75855	0.879276	−	0.476313i	$-0.158027\pi$
$619$	43.7523	1.75855	0.879276	+	0.476313i	$0.158027\pi$
$620$	0	0
$621$	−28.9722	−1.16261
$622$	0	0
$623$	−6.24461	−0.250185
$624$	0	0
$625$	−19.1709	−0.766836
$626$	0	0
$627$	16.6159	0.663576
$628$	0	0
$629$	−3.16034	−0.126011
$630$	0	0
$631$	23.7329	0.944792	0.472396	−	0.881386i	$-0.343389\pi$
$631$	23.7329	0.944792	0.472396	+	0.881386i	$0.343389\pi$
$632$	0	0
$633$	−11.6124	−0.461550
$634$	0	0
$635$	45.4082	1.80197
$636$	0	0
$637$	5.70479	0.226032
$638$	0	0
$639$	1.82626	0.0722459
$640$	0	0
$641$	−14.9883	−0.592000	−0.296000	−	0.955188i	$-0.595653\pi$
$641$	−14.9883	−0.592000	−0.296000	+	0.955188i	$0.595653\pi$
$642$	0	0
$643$	−16.7517	−0.660621	−0.330311	−	0.943872i	$-0.607153\pi$
$643$	−16.7517	−0.660621	−0.330311	+	0.943872i	$0.607153\pi$
$644$	0	0
$645$	16.9577	0.667710
$646$	0	0
$647$	−4.12471	−0.162159	−0.0810795	−	0.996708i	$-0.525837\pi$
$647$	−4.12471	−0.162159	−0.0810795	+	0.996708i	$0.525837\pi$
$648$	0	0
$649$	−22.6071	−0.887408
$650$	0	0
$651$	−1.12684	−0.0441643
$652$	0	0
$653$	35.9939	1.40855	0.704276	−	0.709926i	$-0.251272\pi$
$653$	35.9939	1.40855	0.704276	+	0.709926i	$0.251272\pi$
$654$	0	0
$655$	−16.5352	−0.646084
$656$	0	0
$657$	13.7738	0.537368
$658$	0	0
$659$	−10.4427	−0.406789	−0.203395	−	0.979097i	$-0.565197\pi$
$659$	−10.4427	−0.406789	−0.203395	+	0.979097i	$0.565197\pi$
$660$	0	0
$661$	19.9296	0.775173	0.387586	−	0.921833i	$-0.373309\pi$
$661$	19.9296	0.775173	0.387586	+	0.921833i	$0.373309\pi$
$662$	0	0
$663$	−1.77177	−0.0688098
$664$	0	0
$665$	−19.9745	−0.774576
$666$	0	0
$667$	10.7590	0.416591
$668$	0	0
$669$	11.3838	0.440121
$670$	0	0
$671$	0.0233006	0.000899511 0
$672$	0	0
$673$	−35.0089	−1.34949	−0.674746	−	0.738050i	$-0.735747\pi$
$673$	−35.0089	−1.34949	−0.674746	+	0.738050i	$0.735747\pi$
$674$	0	0
$675$	−27.7536	−1.06824
$676$	0	0
$677$	−15.9608	−0.613423	−0.306711	−	0.951803i	$-0.599229\pi$
$677$	−15.9608	−0.613423	−0.306711	+	0.951803i	$0.599229\pi$
$678$	0	0
$679$	2.18393	0.0838117
$680$	0	0
$681$	−15.4579	−0.592346
$682$	0	0
$683$	47.2729	1.80885	0.904424	−	0.426635i	$-0.140301\pi$
$683$	47.2729	1.80885	0.904424	+	0.426635i	$0.140301\pi$
$684$	0	0
$685$	51.8604	1.98149
$686$	0	0
$687$	12.5378	0.478348
$688$	0	0
$689$	68.1009	2.59444
$690$	0	0
$691$	7.12339	0.270987	0.135493	−	0.990778i	$-0.456738\pi$
$691$	7.12339	0.270987	0.135493	+	0.990778i	$0.456738\pi$
$692$	0	0
$693$	6.80210	0.258390
$694$	0	0
$695$	7.14816	0.271145
$696$	0	0
$697$	0.934262	0.0353877
$698$	0	0
$699$	−16.3480	−0.618337
$700$	0	0
$701$	2.50783	0.0947193	0.0473596	−	0.998878i	$-0.484919\pi$
$701$	2.50783	0.0947193	0.0473596	+	0.998878i	$0.484919\pi$
$702$	0	0
$703$	−54.7650	−2.06550
$704$	0	0
$705$	−13.7363	−0.517339
$706$	0	0
$707$	6.72212	0.252811
$708$	0	0
$709$	12.5127	0.469924	0.234962	−	0.972005i	$-0.424503\pi$
$709$	12.5127	0.469924	0.234962	+	0.972005i	$0.424503\pi$
$710$	0	0
$711$	−1.66538	−0.0624567
$712$	0	0
$713$	−7.87454	−0.294904
$714$	0	0
$715$	−58.3499	−2.18216
$716$	0	0
$717$	16.3748	0.611527
$718$	0	0
$719$	11.8257	0.441025	0.220512	−	0.975384i	$-0.429227\pi$
$719$	11.8257	0.441025	0.220512	+	0.975384i	$0.429227\pi$
$720$	0	0
$721$	15.7259	0.585663
$722$	0	0
$723$	−22.6860	−0.843702
$724$	0	0
$725$	10.3065	0.382773
$726$	0	0
$727$	27.7703	1.02994	0.514972	−	0.857207i	$-0.327802\pi$
$727$	27.7703	1.02994	0.514972	+	0.857207i	$0.327802\pi$
$728$	0	0
$729$	7.00960	0.259615
$730$	0	0
$731$	−1.99515	−0.0737933
$732$	0	0
$733$	50.6947	1.87245	0.936226	−	0.351399i	$-0.114294\pi$
$733$	50.6947	1.87245	0.936226	+	0.351399i	$0.114294\pi$
$734$	0	0
$735$	2.95723	0.109079
$736$	0	0
$737$	−13.1864	−0.485726
$738$	0	0
$739$	−14.2961	−0.525889	−0.262944	−	0.964811i	$-0.584694\pi$
$739$	−14.2961	−0.525889	−0.262944	+	0.964811i	$0.584694\pi$
$740$	0	0
$741$	−30.7027	−1.12789
$742$	0	0
$743$	−31.4037	−1.15209	−0.576045	−	0.817418i	$-0.695405\pi$
$743$	−31.4037	−1.15209	−0.576045	+	0.817418i	$0.695405\pi$
$744$	0	0
$745$	43.2812	1.58570
$746$	0	0
$747$	−11.4221	−0.417912
$748$	0	0
$749$	−9.44807	−0.345225
$750$	0	0
$751$	−12.6531	−0.461717	−0.230859	−	0.972987i	$-0.574153\pi$
$751$	−12.6531	−0.461717	−0.230859	+	0.972987i	$0.574153\pi$
$752$	0	0
$753$	13.7979	0.502824
$754$	0	0
$755$	38.8494	1.41387
$756$	0	0
$757$	27.7971	1.01030	0.505151	−	0.863031i	$-0.331437\pi$
$757$	27.7971	1.01030	0.505151	+	0.863031i	$0.331437\pi$
$758$	0	0
$759$	−17.1909	−0.623990
$760$	0	0
$761$	12.2754	0.444984	0.222492	−	0.974934i	$-0.428581\pi$
$761$	12.2754	0.444984	0.222492	+	0.974934i	$0.428581\pi$
$762$	0	0
$763$	1.14906	0.0415987
$764$	0	0
$765$	2.53957	0.0918184
$766$	0	0
$767$	41.7731	1.50834
$768$	0	0
$769$	44.0633	1.58896	0.794481	−	0.607289i	$-0.207743\pi$
$769$	44.0633	1.58896	0.794481	+	0.607289i	$0.207743\pi$
$770$	0	0
$771$	2.04995	0.0738272
$772$	0	0
$773$	−43.7257	−1.57271	−0.786353	−	0.617778i	$-0.788033\pi$
$773$	−43.7257	−1.57271	−0.786353	+	0.617778i	$0.788033\pi$
$774$	0	0
$775$	−7.54333	−0.270964
$776$	0	0
$777$	8.10800	0.290873
$778$	0	0
$779$	16.1897	0.580055
$780$	0	0
$781$	2.55916	0.0915739
$782$	0	0
$783$	−8.01088	−0.286286
$784$	0	0
$785$	−48.6408	−1.73607
$786$	0	0
$787$	−18.9496	−0.675482	−0.337741	−	0.941239i	$-0.609663\pi$
$787$	−18.9496	−0.675482	−0.337741	+	0.941239i	$0.609663\pi$
$788$	0	0
$789$	10.5784	0.376601
$790$	0	0
$791$	18.5170	0.658388
$792$	0	0
$793$	−0.0430546	−0.00152891
$794$	0	0
$795$	35.3020	1.25203
$796$	0	0
$797$	−22.7766	−0.806787	−0.403394	−	0.915027i	$-0.632169\pi$
$797$	−22.7766	−0.806787	−0.403394	+	0.915027i	$0.632169\pi$
$798$	0	0
$799$	1.61613	0.0571747
$800$	0	0
$801$	13.7582	0.486121
$802$	0	0
$803$	19.3014	0.681131
$804$	0	0
$805$	20.6656	0.728368
$806$	0	0
$807$	−7.20553	−0.253647
$808$	0	0
$809$	37.6047	1.32211	0.661056	−	0.750336i	$-0.270109\pi$
$809$	37.6047	1.32211	0.661056	+	0.750336i	$0.270109\pi$
$810$	0	0
$811$	9.90100	0.347671	0.173836	−	0.984775i	$-0.444384\pi$
$811$	9.90100	0.347671	0.173836	+	0.984775i	$0.444384\pi$
$812$	0	0
$813$	−23.3186	−0.817819
$814$	0	0
$815$	−73.3487	−2.56929
$816$	0	0
$817$	−34.5736	−1.20958
$818$	0	0
$819$	−12.5688	−0.439190
$820$	0	0
$821$	1.00677	0.0351364	0.0175682	−	0.999846i	$-0.494408\pi$
$821$	1.00677	0.0351364	0.0175682	+	0.999846i	$0.494408\pi$
$822$	0	0
$823$	12.0278	0.419262	0.209631	−	0.977781i	$-0.432774\pi$
$823$	12.0278	0.419262	0.209631	+	0.977781i	$0.432774\pi$
$824$	0	0
$825$	−16.4678	−0.573336
$826$	0	0
$827$	−22.8873	−0.795869	−0.397934	−	0.917414i	$-0.630273\pi$
$827$	−22.8873	−0.795869	−0.397934	+	0.917414i	$0.630273\pi$
$828$	0	0
$829$	4.68909	0.162859	0.0814294	−	0.996679i	$-0.474051\pi$
$829$	4.68909	0.162859	0.0814294	+	0.996679i	$0.474051\pi$
$830$	0	0
$831$	2.50296	0.0868267
$832$	0	0
$833$	−0.347931	−0.0120551
$834$	0	0
$835$	−7.02417	−0.243081
$836$	0	0
$837$	5.86318	0.202661
$838$	0	0
$839$	47.4324	1.63755	0.818774	−	0.574116i	$-0.194654\pi$
$839$	47.4324	1.63755	0.818774	+	0.574116i	$0.194654\pi$
$840$	0	0
$841$	−26.0251	−0.897417
$842$	0	0
$843$	5.82422	0.200597
$844$	0	0
$845$	64.7500	2.22747
$846$	0	0
$847$	−1.46816	−0.0504466
$848$	0	0
$849$	17.6200	0.604716
$850$	0	0
$851$	56.6601	1.94228
$852$	0	0
$853$	5.65121	0.193494	0.0967468	−	0.995309i	$-0.469156\pi$
$853$	5.65121	0.193494	0.0967468	+	0.995309i	$0.469156\pi$
$854$	0	0
$855$	44.0078	1.50503
$856$	0	0
$857$	11.1854	0.382086	0.191043	−	0.981582i	$-0.438813\pi$
$857$	11.1854	0.382086	0.191043	+	0.981582i	$0.438813\pi$
$858$	0	0
$859$	−21.3132	−0.727199	−0.363599	−	0.931555i	$-0.618452\pi$
$859$	−21.3132	−0.727199	−0.363599	+	0.931555i	$0.618452\pi$
$860$	0	0
$861$	−2.39689	−0.0816859
$862$	0	0
$863$	29.6751	1.01015	0.505076	−	0.863075i	$-0.331464\pi$
$863$	29.6751	1.01015	0.505076	+	0.863075i	$0.331464\pi$
$864$	0	0
$865$	−9.96325	−0.338761
$866$	0	0
$867$	−15.0667	−0.511693
$868$	0	0
$869$	−2.33371	−0.0791658
$870$	0	0
$871$	24.3656	0.825596
$872$	0	0
$873$	−4.81165	−0.162850
$874$	0	0
$875$	3.23176	0.109254
$876$	0	0
$877$	−4.03199	−0.136151	−0.0680753	−	0.997680i	$-0.521686\pi$
$877$	−4.03199	−0.136151	−0.0680753	+	0.997680i	$0.521686\pi$
$878$	0	0
$879$	3.24541	0.109465
$880$	0	0
$881$	23.4719	0.790789	0.395394	−	0.918511i	$-0.370608\pi$
$881$	23.4719	0.790789	0.395394	+	0.918511i	$0.370608\pi$
$882$	0	0
$883$	31.1695	1.04894	0.524468	−	0.851430i	$-0.324264\pi$
$883$	31.1695	1.04894	0.524468	+	0.851430i	$0.324264\pi$
$884$	0	0
$885$	21.6542	0.727899
$886$	0	0
$887$	−18.6629	−0.626640	−0.313320	−	0.949648i	$-0.601441\pi$
$887$	−18.6629	−0.626640	−0.313320	+	0.949648i	$0.601441\pi$
$888$	0	0
$889$	13.7063	0.459696
$890$	0	0
$891$	−7.60641	−0.254824
$892$	0	0
$893$	28.0057	0.937175
$894$	0	0
$895$	−6.64882	−0.222246
$896$	0	0
$897$	31.7651	1.06061
$898$	0	0
$899$	−2.17733	−0.0726181
$900$	0	0
$901$	−4.15343	−0.138371
$902$	0	0
$903$	5.11865	0.170338
$904$	0	0
$905$	67.0877	2.23007
$906$	0	0
$907$	−4.17253	−0.138547	−0.0692734	−	0.997598i	$-0.522068\pi$
$907$	−4.17253	−0.138547	−0.0692734	+	0.997598i	$0.522068\pi$
$908$	0	0
$909$	−14.8102	−0.491223
$910$	0	0
$911$	26.5648	0.880132	0.440066	−	0.897965i	$-0.354955\pi$
$911$	26.5648	0.880132	0.440066	+	0.897965i	$0.354955\pi$
$912$	0	0
$913$	−16.0059	−0.529717
$914$	0	0
$915$	−0.0223185	−0.000737827 0
$916$	0	0
$917$	−4.99111	−0.164821
$918$	0	0
$919$	−21.9819	−0.725116	−0.362558	−	0.931961i	$-0.618096\pi$
$919$	−21.9819	−0.725116	−0.362558	+	0.931961i	$0.618096\pi$
$920$	0	0
$921$	−23.7645	−0.783068
$922$	0	0
$923$	−4.72878	−0.155650
$924$	0	0
$925$	54.2769	1.78461
$926$	0	0
$927$	−34.6474	−1.13797
$928$	0	0
$929$	14.0560	0.461163	0.230581	−	0.973053i	$-0.425937\pi$
$929$	14.0560	0.461163	0.230581	+	0.973053i	$0.425937\pi$
$930$	0	0
$931$	−6.02924	−0.197600
$932$	0	0
$933$	21.3266	0.698202
$934$	0	0
$935$	3.55872	0.116383
$936$	0	0
$937$	−57.9965	−1.89466	−0.947331	−	0.320256i	$-0.896231\pi$
$937$	−57.9965	−1.89466	−0.947331	+	0.320256i	$0.896231\pi$
$938$	0	0
$939$	16.1463	0.526915
$940$	0	0
$941$	21.8739	0.713070	0.356535	−	0.934282i	$-0.383958\pi$
$941$	21.8739	0.713070	0.356535	+	0.934282i	$0.383958\pi$
$942$	0	0
$943$	−16.7499	−0.545451
$944$	0	0
$945$	−15.3871	−0.500542
$946$	0	0
$947$	−26.7901	−0.870562	−0.435281	−	0.900295i	$-0.643351\pi$
$947$	−26.7901	−0.870562	−0.435281	+	0.900295i	$0.643351\pi$
$948$	0	0
$949$	−35.6648	−1.15773
$950$	0	0
$951$	−6.77220	−0.219604
$952$	0	0
$953$	−20.2198	−0.654983	−0.327491	−	0.944854i	$-0.606203\pi$
$953$	−20.2198	−0.654983	−0.327491	+	0.944854i	$0.606203\pi$
$954$	0	0
$955$	20.6157	0.667107
$956$	0	0
$957$	−4.75333	−0.153653
$958$	0	0
$959$	15.6540	0.505493
$960$	0	0
$961$	−29.4064	−0.948594
$962$	0	0
$963$	20.8160	0.670787
$964$	0	0
$965$	5.22178	0.168095
$966$	0	0
$967$	33.3933	1.07385	0.536927	−	0.843628i	$-0.319585\pi$
$967$	33.3933	1.07385	0.536927	+	0.843628i	$0.319585\pi$
$968$	0	0
$969$	1.87253	0.0601545
$970$	0	0
$971$	10.6377	0.341381	0.170691	−	0.985325i	$-0.445400\pi$
$971$	10.6377	0.341381	0.170691	+	0.985325i	$0.445400\pi$
$972$	0	0
$973$	2.15766	0.0691713
$974$	0	0
$975$	30.4290	0.974508
$976$	0	0
$977$	−28.2764	−0.904641	−0.452321	−	0.891855i	$-0.649404\pi$
$977$	−28.2764	−0.904641	−0.452321	+	0.891855i	$0.649404\pi$
$978$	0	0
$979$	19.2794	0.616173
$980$	0	0
$981$	−2.53161	−0.0808281
$982$	0	0
$983$	1.94376	0.0619964	0.0309982	−	0.999519i	$-0.490131\pi$
$983$	1.94376	0.0619964	0.0309982	+	0.999519i	$0.490131\pi$
$984$	0	0
$985$	−50.2173	−1.60006
$986$	0	0
$987$	−4.14627	−0.131977
$988$	0	0
$989$	35.7700	1.13742
$990$	0	0
$991$	−8.60542	−0.273360	−0.136680	−	0.990615i	$-0.543643\pi$
$991$	−8.60542	−0.273360	−0.136680	+	0.990615i	$0.543643\pi$
$992$	0	0
$993$	−28.2221	−0.895600
$994$	0	0
$995$	−84.5998	−2.68199
$996$	0	0
$997$	−11.0075	−0.348611	−0.174306	−	0.984692i	$-0.555768\pi$
$997$	−11.0075	−0.348611	−0.174306	+	0.984692i	$0.555768\pi$
$998$	0	0
$999$	−42.1876	−1.33476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bj.1.9		12
4.3	odd	2		7168.2.a.bi.1.4		12
8.3	odd	2		7168.2.a.bi.1.9		12
8.5	even	2	inner	7168.2.a.bj.1.4		12
32.3	odd	8		448.2.m.d.337.3		12
32.5	even	8		896.2.m.g.225.3		12
32.11	odd	8		448.2.m.d.113.3		12
32.13	even	8		896.2.m.g.673.3		12
32.19	odd	8		896.2.m.h.673.4		12
32.21	even	8		112.2.m.d.85.6	yes	12
32.27	odd	8		896.2.m.h.225.4		12
32.29	even	8		112.2.m.d.29.6	✓	12
224.53	even	24		784.2.x.l.373.2		24
224.61	odd	24		784.2.x.m.557.2		24
224.93	even	24		784.2.x.l.557.2		24
224.117	odd	24		784.2.x.m.165.3		24
224.125	odd	8		784.2.m.h.589.6		12
224.149	even	24		784.2.x.l.165.3		24
224.157	odd	24		784.2.x.m.765.3		24
224.181	odd	8		784.2.m.h.197.6		12
224.213	odd	24		784.2.x.m.373.2		24
224.221	even	24		784.2.x.l.765.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.6	✓	12	32.29	even	8
112.2.m.d.85.6	yes	12	32.21	even	8
448.2.m.d.113.3		12	32.11	odd	8
448.2.m.d.337.3		12	32.3	odd	8
784.2.m.h.197.6		12	224.181	odd	8
784.2.m.h.589.6		12	224.125	odd	8
784.2.x.l.165.3		24	224.149	even	24
784.2.x.l.373.2		24	224.53	even	24
784.2.x.l.557.2		24	224.93	even	24
784.2.x.l.765.3		24	224.221	even	24
784.2.x.m.165.3		24	224.117	odd	24
784.2.x.m.373.2		24	224.213	odd	24
784.2.x.m.557.2		24	224.61	odd	24
784.2.x.m.765.3		24	224.157	odd	24
896.2.m.g.225.3		12	32.5	even	8
896.2.m.g.673.3		12	32.13	even	8
896.2.m.h.225.4		12	32.27	odd	8
896.2.m.h.673.4		12	32.19	odd	8
7168.2.a.bi.1.4		12	4.3	odd	2
7168.2.a.bi.1.9		12	8.3	odd	2
7168.2.a.bj.1.4		12	8.5	even	2	inner
7168.2.a.bj.1.9		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)

\(f(q)\)	\(=\)	\(q+0.892634 q^{3} +3.31293 q^{5} +1.00000 q^{7} -2.20320 q^{9} +O(q^{10})\)	\(q+0.892634 q^{3} +3.31293 q^{5} +1.00000 q^{7} -2.20320 q^{9} -3.08737 q^{11} +5.70479 q^{13} +2.95723 q^{15} -0.347931 q^{17} -6.02924 q^{19} +0.892634 q^{21} +6.23788 q^{23} +5.97550 q^{25} -4.64456 q^{27} +1.72479 q^{29} -1.26238 q^{31} -2.75589 q^{33} +3.31293 q^{35} +9.08323 q^{37} +5.09229 q^{39} -2.68519 q^{41} +5.73432 q^{43} -7.29906 q^{45} -4.64498 q^{47} +1.00000 q^{49} -0.310575 q^{51} +11.9375 q^{53} -10.2282 q^{55} -5.38191 q^{57} +7.32246 q^{59} -0.00754709 q^{61} -2.20320 q^{63} +18.8996 q^{65} +4.27107 q^{67} +5.56814 q^{69} -0.828913 q^{71} -6.25173 q^{73} +5.33394 q^{75} -3.08737 q^{77} +0.755891 q^{79} +2.46372 q^{81} +5.18431 q^{83} -1.15267 q^{85} +1.53961 q^{87} -6.24461 q^{89} +5.70479 q^{91} -1.12684 q^{93} -19.9745 q^{95} +2.18393 q^{97} +6.80210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) 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

L-functions

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Database

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