LMFDB - Embedded newform 7168.2.a.x.1.3

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:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.74912$ 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)

$f(q)$	$=$	$q+0.334904 q^{3} +4.22274 q^{5} +1.00000 q^{7} -2.88784 q^{9} +O(q^{10})$	$q+0.334904 q^{3} +4.22274 q^{5} +1.00000 q^{7} -2.88784 q^{9} -3.61706 q^{11} -3.07931 q^{13} +1.41421 q^{15} +0.473626 q^{17} +7.25402 q^{19} +0.334904 q^{21} +0.0840215 q^{23} +12.8316 q^{25} -1.97186 q^{27} +1.91598 q^{29} +0.196182 q^{31} -1.21137 q^{33} +4.22274 q^{35} +2.46696 q^{37} -1.03127 q^{39} +5.81324 q^{41} +10.3267 q^{43} -12.1946 q^{45} -3.52637 q^{47} +1.00000 q^{49} +0.158619 q^{51} +6.91245 q^{53} -15.2739 q^{55} +2.42940 q^{57} -10.6240 q^{59} -8.16647 q^{61} -2.88784 q^{63} -13.0031 q^{65} -13.8249 q^{67} +0.0281391 q^{69} +8.95668 q^{71} +10.7229 q^{73} +4.29734 q^{75} -3.61706 q^{77} -9.51156 q^{79} +8.00313 q^{81} -8.27863 q^{83} +2.00000 q^{85} +0.641669 q^{87} +15.3231 q^{89} -3.07931 q^{91} +0.0657022 q^{93} +30.6318 q^{95} +11.2623 q^{97} +10.4455 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7	$ 4 q + 4 q^{5} + 4 q^{7} - 8 q^{13} - 4 q^{17} - 4 q^{19} - 8 q^{23} + 12 q^{27} + 16 q^{29} + 4 q^{31} - 8 q^{33} + 4 q^{35} + 16 q^{37} + 16 q^{39} + 12 q^{41} + 16 q^{43} - 16 q^{45} - 20 q^{47} + 4 q^{49} - 8 q^{51} + 8 q^{53} - 24 q^{55} + 4 q^{57} - 4 q^{59} + 20 q^{61} - 12 q^{65} - 16 q^{67} + 20 q^{69} + 16 q^{71} + 8 q^{73} + 4 q^{75} - 8 q^{81} + 8 q^{83} + 8 q^{85} - 20 q^{87} + 8 q^{89} - 8 q^{91} + 32 q^{93} + 40 q^{95} + 36 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 - 8 * q^13 - 4 * q^17 - 4 * q^19 - 8 * q^23 + 12 * q^27 + 16 * q^29 + 4 * q^31 - 8 * q^33 + 4 * q^35 + 16 * q^37 + 16 * q^39 + 12 * q^41 + 16 * q^43 - 16 * q^45 - 20 * q^47 + 4 * q^49 - 8 * q^51 + 8 * q^53 - 24 * q^55 + 4 * q^57 - 4 * q^59 + 20 * q^61 - 12 * q^65 - 16 * q^67 + 20 * q^69 + 16 * q^71 + 8 * q^73 + 4 * q^75 - 8 * q^81 + 8 * q^83 + 8 * q^85 - 20 * q^87 + 8 * q^89 - 8 * q^91 + 32 * q^93 + 40 * q^95 + 36 * q^97 + 16 * 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
$3$	0.334904	0.193357	0.0966785	−	0.995316i	$-0.469178\pi$
$3$	0.334904	0.193357	0.0966785	+	0.995316i	$0.469178\pi$
$4$	0	0
$5$	4.22274	1.88847	0.944234	−	0.329275i	$-0.106804\pi$
$5$	4.22274	1.88847	0.944234	+	0.329275i	$0.106804\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.88784	−0.962613
$10$	0	0
$11$	−3.61706	−1.09058	−0.545292	−	0.838246i	$-0.683581\pi$
$11$	−3.61706	−1.09058	−0.545292	+	0.838246i	$0.683581\pi$
$12$	0	0
$13$	−3.07931	−0.854047	−0.427023	−	0.904241i	$-0.640438\pi$
$13$	−3.07931	−0.854047	−0.427023	+	0.904241i	$0.640438\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	0.473626	0.114871	0.0574356	−	0.998349i	$-0.481708\pi$
$17$	0.473626	0.114871	0.0574356	+	0.998349i	$0.481708\pi$
$18$	0	0
$19$	7.25402	1.66419	0.832093	−	0.554637i	$-0.187143\pi$
$19$	7.25402	1.66419	0.832093	+	0.554637i	$0.187143\pi$
$20$	0	0
$21$	0.334904	0.0730820
$22$	0	0
$23$	0.0840215	0.0175197	0.00875985	−	0.999962i	$-0.497212\pi$
$23$	0.0840215	0.0175197	0.00875985	+	0.999962i	$0.497212\pi$
$24$	0	0
$25$	12.8316	2.56631
$26$	0	0
$27$	−1.97186	−0.379485
$28$	0	0
$29$	1.91598	0.355788	0.177894	−	0.984050i	$-0.443072\pi$
$29$	1.91598	0.355788	0.177894	+	0.984050i	$0.443072\pi$
$30$	0	0
$31$	0.196182	0.0352354	0.0176177	−	0.999845i	$-0.494392\pi$
$31$	0.196182	0.0352354	0.0176177	+	0.999845i	$0.494392\pi$
$32$	0	0
$33$	−1.21137	−0.210872
$34$	0	0
$35$	4.22274	0.713774
$36$	0	0
$37$	2.46696	0.405566	0.202783	−	0.979224i	$-0.435001\pi$
$37$	2.46696	0.405566	0.202783	+	0.979224i	$0.435001\pi$
$38$	0	0
$39$	−1.03127	−0.165136
$40$	0	0
$41$	5.81324	0.907876	0.453938	−	0.891033i	$-0.350019\pi$
$41$	5.81324	0.907876	0.453938	+	0.891033i	$0.350019\pi$
$42$	0	0
$43$	10.3267	1.57480	0.787401	−	0.616442i	$-0.211426\pi$
$43$	10.3267	1.57480	0.787401	+	0.616442i	$0.211426\pi$
$44$	0	0
$45$	−12.1946	−1.81786
$46$	0	0
$47$	−3.52637	−0.514375	−0.257187	−	0.966362i	$-0.582796\pi$
$47$	−3.52637	−0.514375	−0.257187	+	0.966362i	$0.582796\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.158619	0.0222111
$52$	0	0
$53$	6.91245	0.949498	0.474749	−	0.880121i	$-0.342539\pi$
$53$	6.91245	0.949498	0.474749	+	0.880121i	$0.342539\pi$
$54$	0	0
$55$	−15.2739	−2.05953
$56$	0	0
$57$	2.42940	0.321782
$58$	0	0
$59$	−10.6240	−1.38313	−0.691564	−	0.722315i	$-0.743078\pi$
$59$	−10.6240	−1.38313	−0.691564	+	0.722315i	$0.743078\pi$
$60$	0	0
$61$	−8.16647	−1.04561	−0.522804	−	0.852453i	$-0.675114\pi$
$61$	−8.16647	−1.04561	−0.522804	+	0.852453i	$0.675114\pi$
$62$	0	0
$63$	−2.88784	−0.363834
$64$	0	0
$65$	−13.0031	−1.61284
$66$	0	0
$67$	−13.8249	−1.68898	−0.844490	−	0.535571i	$-0.820096\pi$
$67$	−13.8249	−1.68898	−0.844490	+	0.535571i	$0.820096\pi$
$68$	0	0
$69$	0.0281391	0.00338756
$70$	0	0
$71$	8.95668	1.06296	0.531481	−	0.847070i	$-0.321636\pi$
$71$	8.95668	1.06296	0.531481	+	0.847070i	$0.321636\pi$
$72$	0	0
$73$	10.7229	1.25502	0.627512	−	0.778607i	$-0.284073\pi$
$73$	10.7229	1.25502	0.627512	+	0.778607i	$0.284073\pi$
$74$	0	0
$75$	4.29734	0.496214
$76$	0	0
$77$	−3.61706	−0.412202
$78$	0	0
$79$	−9.51156	−1.07013	−0.535067	−	0.844810i	$-0.679714\pi$
$79$	−9.51156	−1.07013	−0.535067	+	0.844810i	$0.679714\pi$
$80$	0	0
$81$	8.00313	0.889237
$82$	0	0
$83$	−8.27863	−0.908697	−0.454349	−	0.890824i	$-0.650128\pi$
$83$	−8.27863	−0.908697	−0.454349	+	0.890824i	$0.650128\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0.641669	0.0687941
$88$	0	0
$89$	15.3231	1.62425	0.812124	−	0.583484i	$-0.198311\pi$
$89$	15.3231	1.62425	0.812124	+	0.583484i	$0.198311\pi$
$90$	0	0
$91$	−3.07931	−0.322799
$92$	0	0
$93$	0.0657022	0.00681300
$94$	0	0
$95$	30.6318	3.14276
$96$	0	0
$97$	11.2623	1.14351	0.571755	−	0.820425i	$-0.306263\pi$
$97$	11.2623	1.14351	0.571755	+	0.820425i	$0.306263\pi$
$98$	0	0
$99$	10.4455	1.04981
$100$	0	0
$101$	14.9832	1.49089	0.745444	−	0.666568i	$-0.232238\pi$
$101$	14.9832	1.49089	0.745444	+	0.666568i	$0.232238\pi$
$102$	0	0
$103$	12.6417	1.24562	0.622810	−	0.782373i	$-0.285991\pi$
$103$	12.6417	1.24562	0.622810	+	0.782373i	$0.285991\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	17.1645	1.65936	0.829678	−	0.558242i	$-0.188524\pi$
$107$	17.1645	1.65936	0.829678	+	0.558242i	$0.188524\pi$
$108$	0	0
$109$	10.6483	1.01993	0.509963	−	0.860197i	$-0.329659\pi$
$109$	10.6483	1.01993	0.509963	+	0.860197i	$0.329659\pi$
$110$	0	0
$111$	0.826195	0.0784190
$112$	0	0
$113$	−3.00313	−0.282511	−0.141256	−	0.989973i	$-0.545114\pi$
$113$	−3.00313	−0.282511	−0.141256	+	0.989973i	$0.545114\pi$
$114$	0	0
$115$	0.354801	0.0330854
$116$	0	0
$117$	8.89255	0.822117
$118$	0	0
$119$	0.473626	0.0434172
$120$	0	0
$121$	2.08312	0.189374
$122$	0	0
$123$	1.94688	0.175544
$124$	0	0
$125$	33.0707	2.95793
$126$	0	0
$127$	−18.8562	−1.67321	−0.836607	−	0.547803i	$-0.815464\pi$
$127$	−18.8562	−1.67321	−0.836607	+	0.547803i	$0.815464\pi$
$128$	0	0
$129$	3.45844	0.304499
$130$	0	0
$131$	1.55960	0.136263	0.0681314	−	0.997676i	$-0.478296\pi$
$131$	1.55960	0.136263	0.0681314	+	0.997676i	$0.478296\pi$
$132$	0	0
$133$	7.25402	0.629003
$134$	0	0
$135$	−8.32666	−0.716645
$136$	0	0
$137$	−8.04922	−0.687691	−0.343846	−	0.939026i	$-0.611730\pi$
$137$	−8.04922	−0.687691	−0.343846	+	0.939026i	$0.611730\pi$
$138$	0	0
$139$	−6.64049	−0.563239	−0.281619	−	0.959526i	$-0.590872\pi$
$139$	−6.64049	−0.563239	−0.281619	+	0.959526i	$0.590872\pi$
$140$	0	0
$141$	−1.18100	−0.0994579
$142$	0	0
$143$	11.1380	0.931410
$144$	0	0
$145$	8.09069	0.671895
$146$	0	0
$147$	0.334904	0.0276224
$148$	0	0
$149$	1.42754	0.116949	0.0584744	−	0.998289i	$-0.481376\pi$
$149$	1.42754	0.116949	0.0584744	+	0.998289i	$0.481376\pi$
$150$	0	0
$151$	0.863230	0.0702487	0.0351243	−	0.999383i	$-0.488817\pi$
$151$	0.863230	0.0702487	0.0351243	+	0.999383i	$0.488817\pi$
$152$	0	0
$153$	−1.36776	−0.110576
$154$	0	0
$155$	0.828427	0.0665409
$156$	0	0
$157$	16.4720	1.31461	0.657306	−	0.753624i	$-0.271696\pi$
$157$	16.4720	1.31461	0.657306	+	0.753624i	$0.271696\pi$
$158$	0	0
$159$	2.31501	0.183592
$160$	0	0
$161$	0.0840215	0.00662182
$162$	0	0
$163$	−16.1023	−1.26123	−0.630616	−	0.776095i	$-0.717198\pi$
$163$	−16.1023	−1.26123	−0.630616	+	0.776095i	$0.717198\pi$
$164$	0	0
$165$	−5.11529	−0.398225
$166$	0	0
$167$	−4.47363	−0.346180	−0.173090	−	0.984906i	$-0.555375\pi$
$167$	−4.47363	−0.346180	−0.173090	+	0.984906i	$0.555375\pi$
$168$	0	0
$169$	−3.51785	−0.270604
$170$	0	0
$171$	−20.9484	−1.60197
$172$	0	0
$173$	3.84519	0.292344	0.146172	−	0.989259i	$-0.453305\pi$
$173$	3.84519	0.292344	0.146172	+	0.989259i	$0.453305\pi$
$174$	0	0
$175$	12.8316	0.969975
$176$	0	0
$177$	−3.55802	−0.267437
$178$	0	0
$179$	15.4022	1.15121	0.575606	−	0.817727i	$-0.304766\pi$
$179$	15.4022	1.15121	0.575606	+	0.817727i	$0.304766\pi$
$180$	0	0
$181$	−3.57178	−0.265489	−0.132744	−	0.991150i	$-0.542379\pi$
$181$	−3.57178	−0.265489	−0.132744	+	0.991150i	$0.542379\pi$
$182$	0	0
$183$	−2.73498	−0.202176
$184$	0	0
$185$	10.4173	0.765899
$186$	0	0
$187$	−1.71313	−0.125277
$188$	0	0
$189$	−1.97186	−0.143432
$190$	0	0
$191$	−20.0233	−1.44884	−0.724418	−	0.689361i	$-0.757891\pi$
$191$	−20.0233	−1.44884	−0.724418	+	0.689361i	$0.757891\pi$
$192$	0	0
$193$	5.68943	0.409534	0.204767	−	0.978811i	$-0.434356\pi$
$193$	5.68943	0.409534	0.204767	+	0.978811i	$0.434356\pi$
$194$	0	0
$195$	−4.35480	−0.311854
$196$	0	0
$197$	23.6221	1.68300	0.841501	−	0.540256i	$-0.181673\pi$
$197$	23.6221	1.68300	0.841501	+	0.540256i	$0.181673\pi$
$198$	0	0
$199$	17.7968	1.26158	0.630789	−	0.775954i	$-0.282731\pi$
$199$	17.7968	1.26158	0.630789	+	0.775954i	$0.282731\pi$
$200$	0	0
$201$	−4.63001	−0.326576
$202$	0	0
$203$	1.91598	0.134475
$204$	0	0
$205$	24.5478	1.71449
$206$	0	0
$207$	−0.242641	−0.0168647
$208$	0	0
$209$	−26.2382	−1.81493
$210$	0	0
$211$	−1.33962	−0.0922230	−0.0461115	−	0.998936i	$-0.514683\pi$
$211$	−1.33962	−0.0922230	−0.0461115	+	0.998936i	$0.514683\pi$
$212$	0	0
$213$	2.99963	0.205531
$214$	0	0
$215$	43.6068	2.97396
$216$	0	0
$217$	0.196182	0.0133177
$218$	0	0
$219$	3.59115	0.242668
$220$	0	0
$221$	−1.45844	−0.0981053
$222$	0	0
$223$	9.64037	0.645567	0.322783	−	0.946473i	$-0.395381\pi$
$223$	9.64037	0.645567	0.322783	+	0.946473i	$0.395381\pi$
$224$	0	0
$225$	−37.0555	−2.47037
$226$	0	0
$227$	12.9507	0.859566	0.429783	−	0.902932i	$-0.358590\pi$
$227$	12.9507	0.859566	0.429783	+	0.902932i	$0.358590\pi$
$228$	0	0
$229$	7.76244	0.512957	0.256478	−	0.966550i	$-0.417438\pi$
$229$	7.76244	0.512957	0.256478	+	0.966550i	$0.417438\pi$
$230$	0	0
$231$	−1.21137	−0.0797021
$232$	0	0
$233$	−8.67371	−0.568234	−0.284117	−	0.958790i	$-0.591700\pi$
$233$	−8.67371	−0.568234	−0.284117	+	0.958790i	$0.591700\pi$
$234$	0	0
$235$	−14.8910	−0.971380
$236$	0	0
$237$	−3.18546	−0.206918
$238$	0	0
$239$	−6.83286	−0.441981	−0.220990	−	0.975276i	$-0.570929\pi$
$239$	−6.83286	−0.441981	−0.220990	+	0.975276i	$0.570929\pi$
$240$	0	0
$241$	−18.2985	−1.17871	−0.589356	−	0.807874i	$-0.700618\pi$
$241$	−18.2985	−1.17871	−0.589356	+	0.807874i	$0.700618\pi$
$242$	0	0
$243$	8.59586	0.551425
$244$	0	0
$245$	4.22274	0.269781
$246$	0	0
$247$	−22.3374	−1.42129
$248$	0	0
$249$	−2.77254	−0.175703
$250$	0	0
$251$	21.5596	1.36083	0.680415	−	0.732827i	$-0.261800\pi$
$251$	21.5596	1.36083	0.680415	+	0.732827i	$0.261800\pi$
$252$	0	0
$253$	−0.303911	−0.0191067
$254$	0	0
$255$	0.669808	0.0419450
$256$	0	0
$257$	−18.4588	−1.15143	−0.575715	−	0.817651i	$-0.695276\pi$
$257$	−18.4588	−1.15143	−0.575715	+	0.817651i	$0.695276\pi$
$258$	0	0
$259$	2.46696	0.153290
$260$	0	0
$261$	−5.53304	−0.342486
$262$	0	0
$263$	9.99684	0.616432	0.308216	−	0.951316i	$-0.400268\pi$
$263$	9.99684	0.616432	0.308216	+	0.951316i	$0.400268\pi$
$264$	0	0
$265$	29.1895	1.79310
$266$	0	0
$267$	5.13178	0.314060
$268$	0	0
$269$	27.6530	1.68604	0.843018	−	0.537885i	$-0.180777\pi$
$269$	27.6530	1.68604	0.843018	+	0.537885i	$0.180777\pi$
$270$	0	0
$271$	12.5947	0.765072	0.382536	−	0.923940i	$-0.375051\pi$
$271$	12.5947	0.765072	0.382536	+	0.923940i	$0.375051\pi$
$272$	0	0
$273$	−1.03127	−0.0624155
$274$	0	0
$275$	−46.4125	−2.79878
$276$	0	0
$277$	−1.76363	−0.105966	−0.0529830	−	0.998595i	$-0.516873\pi$
$277$	−1.76363	−0.105966	−0.0529830	+	0.998595i	$0.516873\pi$
$278$	0	0
$279$	−0.566543	−0.0339180
$280$	0	0
$281$	−23.3459	−1.39270	−0.696349	−	0.717703i	$-0.745194\pi$
$281$	−23.3459	−1.39270	−0.696349	+	0.717703i	$0.745194\pi$
$282$	0	0
$283$	−10.8594	−0.645526	−0.322763	−	0.946480i	$-0.604612\pi$
$283$	−10.8594	−0.645526	−0.322763	+	0.946480i	$0.604612\pi$
$284$	0	0
$285$	10.2587	0.607675
$286$	0	0
$287$	5.81324	0.343145
$288$	0	0
$289$	−16.7757	−0.986805
$290$	0	0
$291$	3.77178	0.221105
$292$	0	0
$293$	−20.0820	−1.17321	−0.586603	−	0.809875i	$-0.699535\pi$
$293$	−20.0820	−1.17321	−0.586603	+	0.809875i	$0.699535\pi$
$294$	0	0
$295$	−44.8624	−2.61199
$296$	0	0
$297$	7.13234	0.413860
$298$	0	0
$299$	−0.258728	−0.0149626
$300$	0	0
$301$	10.3267	0.595219
$302$	0	0
$303$	5.01795	0.288273
$304$	0	0
$305$	−34.4849	−1.97460
$306$	0	0
$307$	17.5994	1.00445	0.502225	−	0.864737i	$-0.332515\pi$
$307$	17.5994	1.00445	0.502225	+	0.864737i	$0.332515\pi$
$308$	0	0
$309$	4.23375	0.240849
$310$	0	0
$311$	2.76978	0.157060	0.0785300	−	0.996912i	$-0.474977\pi$
$311$	2.76978	0.157060	0.0785300	+	0.996912i	$0.474977\pi$
$312$	0	0
$313$	−10.8401	−0.612718	−0.306359	−	0.951916i	$-0.599111\pi$
$313$	−10.8401	−0.612718	−0.306359	+	0.951916i	$0.599111\pi$
$314$	0	0
$315$	−12.1946	−0.687088
$316$	0	0
$317$	−19.9715	−1.12171	−0.560855	−	0.827914i	$-0.689527\pi$
$317$	−19.9715	−1.12171	−0.560855	+	0.827914i	$0.689527\pi$
$318$	0	0
$319$	−6.93021	−0.388017
$320$	0	0
$321$	5.74846	0.320848
$322$	0	0
$323$	3.43569	0.191167
$324$	0	0
$325$	−39.5123	−2.19175
$326$	0	0
$327$	3.56617	0.197210
$328$	0	0
$329$	−3.52637	−0.194415
$330$	0	0
$331$	15.5867	0.856722	0.428361	−	0.903608i	$-0.359091\pi$
$331$	15.5867	0.856722	0.428361	+	0.903608i	$0.359091\pi$
$332$	0	0
$333$	−7.12419	−0.390403
$334$	0	0
$335$	−58.3790	−3.18959
$336$	0	0
$337$	−0.329422	−0.0179448	−0.00897239	−	0.999960i	$-0.502856\pi$
$337$	−0.329422	−0.0179448	−0.00897239	+	0.999960i	$0.502856\pi$
$338$	0	0
$339$	−1.00576	−0.0546255
$340$	0	0
$341$	−0.709603	−0.0384271
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.118824	0.00639729
$346$	0	0
$347$	19.8155	1.06375	0.531875	−	0.846823i	$-0.321488\pi$
$347$	19.8155	1.06375	0.531875	+	0.846823i	$0.321488\pi$
$348$	0	0
$349$	6.62900	0.354842	0.177421	−	0.984135i	$-0.443225\pi$
$349$	6.62900	0.354842	0.177421	+	0.984135i	$0.443225\pi$
$350$	0	0
$351$	6.07197	0.324098
$352$	0	0
$353$	−1.89060	−0.100626	−0.0503132	−	0.998733i	$-0.516022\pi$
$353$	−1.89060	−0.100626	−0.0503132	+	0.998733i	$0.516022\pi$
$354$	0	0
$355$	37.8217	2.00737
$356$	0	0
$357$	0.158619	0.00839502
$358$	0	0
$359$	−32.4334	−1.71177	−0.855886	−	0.517165i	$-0.826987\pi$
$359$	−32.4334	−1.71177	−0.855886	+	0.517165i	$0.826987\pi$
$360$	0	0
$361$	33.6208	1.76951
$362$	0	0
$363$	0.697645	0.0366169
$364$	0	0
$365$	45.2802	2.37007
$366$	0	0
$367$	−19.9992	−1.04395	−0.521975	−	0.852961i	$-0.674805\pi$
$367$	−19.9992	−1.04395	−0.521975	+	0.852961i	$0.674805\pi$
$368$	0	0
$369$	−16.7877	−0.873933
$370$	0	0
$371$	6.91245	0.358876
$372$	0	0
$373$	7.21580	0.373620	0.186810	−	0.982396i	$-0.440185\pi$
$373$	7.21580	0.373620	0.186810	+	0.982396i	$0.440185\pi$
$374$	0	0
$375$	11.0755	0.571936
$376$	0	0
$377$	−5.89989	−0.303860
$378$	0	0
$379$	−13.3794	−0.687254	−0.343627	−	0.939106i	$-0.611656\pi$
$379$	−13.3794	−0.687254	−0.343627	+	0.939106i	$0.611656\pi$
$380$	0	0
$381$	−6.31501	−0.323528
$382$	0	0
$383$	−11.9156	−0.608858	−0.304429	−	0.952535i	$-0.598466\pi$
$383$	−11.9156	−0.608858	−0.304429	+	0.952535i	$0.598466\pi$
$384$	0	0
$385$	−15.2739	−0.778431
$386$	0	0
$387$	−29.8217	−1.51592
$388$	0	0
$389$	19.5027	0.988824	0.494412	−	0.869228i	$-0.335383\pi$
$389$	19.5027	0.988824	0.494412	+	0.869228i	$0.335383\pi$
$390$	0	0
$391$	0.0397948	0.00201251
$392$	0	0
$393$	0.522316	0.0263474
$394$	0	0
$395$	−40.1649	−2.02091
$396$	0	0
$397$	22.3118	1.11979	0.559897	−	0.828562i	$-0.310841\pi$
$397$	22.3118	1.11979	0.559897	+	0.828562i	$0.310841\pi$
$398$	0	0
$399$	2.42940	0.121622
$400$	0	0
$401$	−10.0166	−0.500208	−0.250104	−	0.968219i	$-0.580465\pi$
$401$	−10.0166	−0.500208	−0.250104	+	0.968219i	$0.580465\pi$
$402$	0	0
$403$	−0.604106	−0.0300927
$404$	0	0
$405$	33.7952	1.67930
$406$	0	0
$407$	−8.92315	−0.442304
$408$	0	0
$409$	−5.54913	−0.274387	−0.137193	−	0.990544i	$-0.543808\pi$
$409$	−5.54913	−0.274387	−0.137193	+	0.990544i	$0.543808\pi$
$410$	0	0
$411$	−2.69572	−0.132970
$412$	0	0
$413$	−10.6240	−0.522773
$414$	0	0
$415$	−34.9585	−1.71605
$416$	0	0
$417$	−2.22393	−0.108906
$418$	0	0
$419$	−7.58551	−0.370576	−0.185288	−	0.982684i	$-0.559322\pi$
$419$	−7.58551	−0.370576	−0.185288	+	0.982684i	$0.559322\pi$
$420$	0	0
$421$	−27.6393	−1.34706	−0.673528	−	0.739162i	$-0.735222\pi$
$421$	−27.6393	−1.34706	−0.673528	+	0.739162i	$0.735222\pi$
$422$	0	0
$423$	10.1836	0.495144
$424$	0	0
$425$	6.07736	0.294795
$426$	0	0
$427$	−8.16647	−0.395203
$428$	0	0
$429$	3.73018	0.180095
$430$	0	0
$431$	25.6979	1.23783	0.618913	−	0.785460i	$-0.287573\pi$
$431$	25.6979	1.23783	0.618913	+	0.785460i	$0.287573\pi$
$432$	0	0
$433$	4.47599	0.215102	0.107551	−	0.994200i	$-0.465699\pi$
$433$	4.47599	0.215102	0.107551	+	0.994200i	$0.465699\pi$
$434$	0	0
$435$	2.70960	0.129916
$436$	0	0
$437$	0.609494	0.0291560
$438$	0	0
$439$	−21.2404	−1.01375	−0.506874	−	0.862020i	$-0.669199\pi$
$439$	−21.2404	−1.01375	−0.506874	+	0.862020i	$0.669199\pi$
$440$	0	0
$441$	−2.88784	−0.137516
$442$	0	0
$443$	−8.37352	−0.397838	−0.198919	−	0.980016i	$-0.563743\pi$
$443$	−8.37352	−0.397838	−0.198919	+	0.980016i	$0.563743\pi$
$444$	0	0
$445$	64.7057	3.06734
$446$	0	0
$447$	0.478089	0.0226128
$448$	0	0
$449$	7.63095	0.360127	0.180063	−	0.983655i	$-0.442370\pi$
$449$	7.63095	0.360127	0.180063	+	0.983655i	$0.442370\pi$
$450$	0	0
$451$	−21.0268	−0.990115
$452$	0	0
$453$	0.289099	0.0135831
$454$	0	0
$455$	−13.0031	−0.609596
$456$	0	0
$457$	29.2078	1.36628	0.683142	−	0.730286i	$-0.260613\pi$
$457$	29.2078	1.36628	0.683142	+	0.730286i	$0.260613\pi$
$458$	0	0
$459$	−0.933924	−0.0435918
$460$	0	0
$461$	−0.523498	−0.0243817	−0.0121909	−	0.999926i	$-0.503881\pi$
$461$	−0.523498	−0.0243817	−0.0121909	+	0.999926i	$0.503881\pi$
$462$	0	0
$463$	39.2012	1.82183	0.910916	−	0.412592i	$-0.135376\pi$
$463$	39.2012	1.82183	0.910916	+	0.412592i	$0.135376\pi$
$464$	0	0
$465$	0.277444	0.0128661
$466$	0	0
$467$	−14.6075	−0.675955	−0.337978	−	0.941154i	$-0.609743\pi$
$467$	−14.6075	−0.675955	−0.337978	+	0.941154i	$0.609743\pi$
$468$	0	0
$469$	−13.8249	−0.638374
$470$	0	0
$471$	5.51655	0.254189
$472$	0	0
$473$	−37.3522	−1.71745
$474$	0	0
$475$	93.0804	4.27082
$476$	0	0
$477$	−19.9620	−0.913999
$478$	0	0
$479$	21.2528	0.971067	0.485533	−	0.874218i	$-0.338625\pi$
$479$	21.2528	0.971067	0.485533	+	0.874218i	$0.338625\pi$
$480$	0	0
$481$	−7.59654	−0.346372
$482$	0	0
$483$	0.0281391	0.00128038
$484$	0	0
$485$	47.5576	2.15948
$486$	0	0
$487$	−9.74088	−0.441401	−0.220701	−	0.975342i	$-0.570834\pi$
$487$	−9.74088	−0.441401	−0.220701	+	0.975342i	$0.570834\pi$
$488$	0	0
$489$	−5.39274	−0.243868
$490$	0	0
$491$	20.7792	0.937753	0.468876	−	0.883264i	$-0.344659\pi$
$491$	20.7792	0.937753	0.468876	+	0.883264i	$0.344659\pi$
$492$	0	0
$493$	0.907457	0.0408698
$494$	0	0
$495$	44.1086	1.98253
$496$	0	0
$497$	8.95668	0.401762
$498$	0	0
$499$	−41.1349	−1.84145	−0.920725	−	0.390211i	$-0.872402\pi$
$499$	−41.1349	−1.84145	−0.920725	+	0.390211i	$0.872402\pi$
$500$	0	0
$501$	−1.49824	−0.0669362
$502$	0	0
$503$	−3.30038	−0.147157	−0.0735784	−	0.997289i	$-0.523442\pi$
$503$	−3.30038	−0.147157	−0.0735784	+	0.997289i	$0.523442\pi$
$504$	0	0
$505$	63.2704	2.81549
$506$	0	0
$507$	−1.17814	−0.0523232
$508$	0	0
$509$	−16.7611	−0.742925	−0.371462	−	0.928448i	$-0.621144\pi$
$509$	−16.7611	−0.742925	−0.371462	+	0.928448i	$0.621144\pi$
$510$	0	0
$511$	10.7229	0.474355
$512$	0	0
$513$	−14.3039	−0.631533
$514$	0	0
$515$	53.3825	2.35231
$516$	0	0
$517$	12.7551	0.560969
$518$	0	0
$519$	1.28777	0.0565268
$520$	0	0
$521$	−28.9778	−1.26954	−0.634769	−	0.772702i	$-0.718905\pi$
$521$	−28.9778	−1.26954	−0.634769	+	0.772702i	$0.718905\pi$
$522$	0	0
$523$	−24.7130	−1.08062	−0.540312	−	0.841465i	$-0.681694\pi$
$523$	−24.7130	−1.08062	−0.540312	+	0.841465i	$0.681694\pi$
$524$	0	0
$525$	4.29734	0.187551
$526$	0	0
$527$	0.0929169	0.00404753
$528$	0	0
$529$	−22.9929	−0.999693
$530$	0	0
$531$	30.6804	1.33142
$532$	0	0
$533$	−17.9008	−0.775368
$534$	0	0
$535$	72.4813	3.13364
$536$	0	0
$537$	5.15825	0.222595
$538$	0	0
$539$	−3.61706	−0.155798
$540$	0	0
$541$	−7.38831	−0.317648	−0.158824	−	0.987307i	$-0.550770\pi$
$541$	−7.38831	−0.317648	−0.158824	+	0.987307i	$0.550770\pi$
$542$	0	0
$543$	−1.19620	−0.0513340
$544$	0	0
$545$	44.9652	1.92610
$546$	0	0
$547$	−15.8969	−0.679701	−0.339850	−	0.940479i	$-0.610376\pi$
$547$	−15.8969	−0.679701	−0.339850	+	0.940479i	$0.610376\pi$
$548$	0	0
$549$	23.5834	1.00652
$550$	0	0
$551$	13.8985	0.592098
$552$	0	0
$553$	−9.51156	−0.404473
$554$	0	0
$555$	3.48881	0.148092
$556$	0	0
$557$	−23.3603	−0.989808	−0.494904	−	0.868948i	$-0.664797\pi$
$557$	−23.3603	−0.989808	−0.494904	+	0.868948i	$0.664797\pi$
$558$	0	0
$559$	−31.7990	−1.34495
$560$	0	0
$561$	−0.573735	−0.0242231
$562$	0	0
$563$	25.8416	1.08910	0.544548	−	0.838730i	$-0.316701\pi$
$563$	25.8416	1.08910	0.544548	+	0.838730i	$0.316701\pi$
$564$	0	0
$565$	−12.6815	−0.533513
$566$	0	0
$567$	8.00313	0.336100
$568$	0	0
$569$	18.7351	0.785417	0.392708	−	0.919663i	$-0.371538\pi$
$569$	18.7351	0.785417	0.392708	+	0.919663i	$0.371538\pi$
$570$	0	0
$571$	−17.0292	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$571$	−17.0292	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$572$	0	0
$573$	−6.70589	−0.280142
$574$	0	0
$575$	1.07813	0.0449610
$576$	0	0
$577$	−3.96647	−0.165126	−0.0825632	−	0.996586i	$-0.526311\pi$
$577$	−3.96647	−0.165126	−0.0825632	+	0.996586i	$0.526311\pi$
$578$	0	0
$579$	1.90541	0.0791862
$580$	0	0
$581$	−8.27863	−0.343455
$582$	0	0
$583$	−25.0027	−1.03551
$584$	0	0
$585$	37.5510	1.55254
$586$	0	0
$587$	−7.27050	−0.300086	−0.150043	−	0.988679i	$-0.547941\pi$
$587$	−7.27050	−0.300086	−0.150043	+	0.988679i	$0.547941\pi$
$588$	0	0
$589$	1.42311	0.0586382
$590$	0	0
$591$	7.91112	0.325420
$592$	0	0
$593$	−18.3463	−0.753390	−0.376695	−	0.926337i	$-0.622940\pi$
$593$	−18.3463	−0.753390	−0.376695	+	0.926337i	$0.622940\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	5.96021	0.243935
$598$	0	0
$599$	−35.3692	−1.44515	−0.722573	−	0.691295i	$-0.757041\pi$
$599$	−35.3692	−1.44515	−0.722573	+	0.691295i	$0.757041\pi$
$600$	0	0
$601$	34.1275	1.39209	0.696044	−	0.717999i	$-0.254942\pi$
$601$	34.1275	1.39209	0.696044	+	0.717999i	$0.254942\pi$
$602$	0	0
$603$	39.9241	1.62583
$604$	0	0
$605$	8.79648	0.357628
$606$	0	0
$607$	−12.3626	−0.501781	−0.250890	−	0.968016i	$-0.580723\pi$
$607$	−12.3626	−0.501781	−0.250890	+	0.968016i	$0.580723\pi$
$608$	0	0
$609$	0.641669	0.0260017
$610$	0	0
$611$	10.8588	0.439300
$612$	0	0
$613$	−27.7731	−1.12174	−0.560871	−	0.827903i	$-0.689534\pi$
$613$	−27.7731	−1.12174	−0.560871	+	0.827903i	$0.689534\pi$
$614$	0	0
$615$	8.22117	0.331509
$616$	0	0
$617$	1.01019	0.0406689	0.0203344	−	0.999793i	$-0.493527\pi$
$617$	1.01019	0.0406689	0.0203344	+	0.999793i	$0.493527\pi$
$618$	0	0
$619$	−1.15706	−0.0465063	−0.0232532	−	0.999730i	$-0.507402\pi$
$619$	−1.15706	−0.0465063	−0.0232532	+	0.999730i	$0.507402\pi$
$620$	0	0
$621$	−0.165679	−0.00664846
$622$	0	0
$623$	15.3231	0.613908
$624$	0	0
$625$	75.4912	3.01965
$626$	0	0
$627$	−8.78728	−0.350930
$628$	0	0
$629$	1.16842	0.0465878
$630$	0	0
$631$	31.6898	1.26155	0.630775	−	0.775966i	$-0.282737\pi$
$631$	31.6898	1.26155	0.630775	+	0.775966i	$0.282737\pi$
$632$	0	0
$633$	−0.448643	−0.0178319
$634$	0	0
$635$	−79.6248	−3.15981
$636$	0	0
$637$	−3.07931	−0.122007
$638$	0	0
$639$	−25.8654	−1.02322
$640$	0	0
$641$	15.2445	0.602121	0.301061	−	0.953605i	$-0.402659\pi$
$641$	15.2445	0.602121	0.301061	+	0.953605i	$0.402659\pi$
$642$	0	0
$643$	−30.5512	−1.20482	−0.602412	−	0.798186i	$-0.705793\pi$
$643$	−30.5512	−1.20482	−0.602412	+	0.798186i	$0.705793\pi$
$644$	0	0
$645$	14.6041	0.575036
$646$	0	0
$647$	22.8048	0.896547	0.448274	−	0.893896i	$-0.352039\pi$
$647$	22.8048	0.896547	0.448274	+	0.893896i	$0.352039\pi$
$648$	0	0
$649$	38.4277	1.50842
$650$	0	0
$651$	0.0657022	0.00257507
$652$	0	0
$653$	−18.8391	−0.737232	−0.368616	−	0.929582i	$-0.620168\pi$
$653$	−18.8391	−0.737232	−0.368616	+	0.929582i	$0.620168\pi$
$654$	0	0
$655$	6.58579	0.257328
$656$	0	0
$657$	−30.9661	−1.20810
$658$	0	0
$659$	−13.6145	−0.530344	−0.265172	−	0.964201i	$-0.585429\pi$
$659$	−13.6145	−0.530344	−0.265172	+	0.964201i	$0.585429\pi$
$660$	0	0
$661$	25.6405	0.997298	0.498649	−	0.866804i	$-0.333830\pi$
$661$	25.6405	0.997298	0.498649	+	0.866804i	$0.333830\pi$
$662$	0	0
$663$	−0.488437	−0.0189693
$664$	0	0
$665$	30.6318	1.18785
$666$	0	0
$667$	0.160983	0.00623330
$668$	0	0
$669$	3.22860	0.124825
$670$	0	0
$671$	29.5386	1.14032
$672$	0	0
$673$	−4.57745	−0.176448	−0.0882239	−	0.996101i	$-0.528119\pi$
$673$	−4.57745	−0.176448	−0.0882239	+	0.996101i	$0.528119\pi$
$674$	0	0
$675$	−25.3021	−0.973876
$676$	0	0
$677$	25.6022	0.983971	0.491985	−	0.870603i	$-0.336271\pi$
$677$	25.6022	0.983971	0.491985	+	0.870603i	$0.336271\pi$
$678$	0	0
$679$	11.2623	0.432206
$680$	0	0
$681$	4.33723	0.166203
$682$	0	0
$683$	11.2404	0.430101	0.215051	−	0.976603i	$-0.431008\pi$
$683$	11.2404	0.430101	0.215051	+	0.976603i	$0.431008\pi$
$684$	0	0
$685$	−33.9898	−1.29868
$686$	0	0
$687$	2.59967	0.0991837
$688$	0	0
$689$	−21.2856	−0.810916
$690$	0	0
$691$	8.97174	0.341301	0.170651	−	0.985332i	$-0.445413\pi$
$691$	8.97174	0.341301	0.170651	+	0.985332i	$0.445413\pi$
$692$	0	0
$693$	10.4455	0.396791
$694$	0	0
$695$	−28.0411	−1.06366
$696$	0	0
$697$	2.75330	0.104289
$698$	0	0
$699$	−2.90486	−0.109872
$700$	0	0
$701$	−27.6017	−1.04250	−0.521250	−	0.853404i	$-0.674534\pi$
$701$	−27.6017	−1.04250	−0.521250	+	0.853404i	$0.674534\pi$
$702$	0	0
$703$	17.8954	0.674937
$704$	0	0
$705$	−4.98705	−0.187823
$706$	0	0
$707$	14.9832	0.563503
$708$	0	0
$709$	6.08793	0.228637	0.114318	−	0.993444i	$-0.463532\pi$
$709$	6.08793	0.228637	0.114318	+	0.993444i	$0.463532\pi$
$710$	0	0
$711$	27.4679	1.03013
$712$	0	0
$713$	0.0164835	0.000617313 0
$714$	0	0
$715$	47.0331	1.75894
$716$	0	0
$717$	−2.28835	−0.0854601
$718$	0	0
$719$	−26.6322	−0.993215	−0.496608	−	0.867975i	$-0.665421\pi$
$719$	−26.6322	−0.993215	−0.496608	+	0.867975i	$0.665421\pi$
$720$	0	0
$721$	12.6417	0.470800
$722$	0	0
$723$	−6.12825	−0.227912
$724$	0	0
$725$	24.5850	0.913064
$726$	0	0
$727$	−38.4103	−1.42456	−0.712279	−	0.701896i	$-0.752337\pi$
$727$	−38.4103	−1.42456	−0.712279	+	0.701896i	$0.752337\pi$
$728$	0	0
$729$	−21.1306	−0.782615
$730$	0	0
$731$	4.89097	0.180899
$732$	0	0
$733$	13.6451	0.503993	0.251997	−	0.967728i	$-0.418913\pi$
$733$	13.6451	0.503993	0.251997	+	0.967728i	$0.418913\pi$
$734$	0	0
$735$	1.41421	0.0521641
$736$	0	0
$737$	50.0055	1.84198
$738$	0	0
$739$	37.3106	1.37249	0.686245	−	0.727370i	$-0.259258\pi$
$739$	37.3106	1.37249	0.686245	+	0.727370i	$0.259258\pi$
$740$	0	0
$741$	−7.48087	−0.274817
$742$	0	0
$743$	−1.53540	−0.0563284	−0.0281642	−	0.999603i	$-0.508966\pi$
$743$	−1.53540	−0.0563284	−0.0281642	+	0.999603i	$0.508966\pi$
$744$	0	0
$745$	6.02814	0.220854
$746$	0	0
$747$	23.9073	0.874724
$748$	0	0
$749$	17.1645	0.627178
$750$	0	0
$751$	−20.7562	−0.757404	−0.378702	−	0.925519i	$-0.623630\pi$
$751$	−20.7562	−0.757404	−0.378702	+	0.925519i	$0.623630\pi$
$752$	0	0
$753$	7.22040	0.263126
$754$	0	0
$755$	3.64520	0.132662
$756$	0	0
$757$	29.4729	1.07121	0.535605	−	0.844469i	$-0.320084\pi$
$757$	29.4729	1.07121	0.535605	+	0.844469i	$0.320084\pi$
$758$	0	0
$759$	−0.101781	−0.00369442
$760$	0	0
$761$	−42.3705	−1.53593	−0.767965	−	0.640492i	$-0.778730\pi$
$761$	−42.3705	−1.53593	−0.767965	+	0.640492i	$0.778730\pi$
$762$	0	0
$763$	10.6483	0.385496
$764$	0	0
$765$	−5.77568	−0.208820
$766$	0	0
$767$	32.7146	1.18126
$768$	0	0
$769$	−13.4764	−0.485970	−0.242985	−	0.970030i	$-0.578127\pi$
$769$	−13.4764	−0.485970	−0.242985	+	0.970030i	$0.578127\pi$
$770$	0	0
$771$	−6.18193	−0.222637
$772$	0	0
$773$	19.4302	0.698856	0.349428	−	0.936963i	$-0.386376\pi$
$773$	19.4302	0.698856	0.349428	+	0.936963i	$0.386376\pi$
$774$	0	0
$775$	2.51732	0.0904249
$776$	0	0
$777$	0.826195	0.0296396
$778$	0	0
$779$	42.1694	1.51087
$780$	0	0
$781$	−32.3968	−1.15925
$782$	0	0
$783$	−3.77804	−0.135016
$784$	0	0
$785$	69.5572	2.48260
$786$	0	0
$787$	−20.8046	−0.741605	−0.370802	−	0.928712i	$-0.620917\pi$
$787$	−20.8046	−0.741605	−0.370802	+	0.928712i	$0.620917\pi$
$788$	0	0
$789$	3.34798	0.119191
$790$	0	0
$791$	−3.00313	−0.106779
$792$	0	0
$793$	25.1471	0.892999
$794$	0	0
$795$	9.77568	0.346708
$796$	0	0
$797$	4.58907	0.162553	0.0812766	−	0.996692i	$-0.474100\pi$
$797$	4.58907	0.162553	0.0812766	+	0.996692i	$0.474100\pi$
$798$	0	0
$799$	−1.67018	−0.0590868
$800$	0	0
$801$	−44.2507	−1.56352
$802$	0	0
$803$	−38.7855	−1.36871
$804$	0	0
$805$	0.354801	0.0125051
$806$	0	0
$807$	9.26111	0.326007
$808$	0	0
$809$	−43.9611	−1.54559	−0.772795	−	0.634655i	$-0.781142\pi$
$809$	−43.9611	−1.54559	−0.772795	+	0.634655i	$0.781142\pi$
$810$	0	0
$811$	−17.9975	−0.631977	−0.315988	−	0.948763i	$-0.602336\pi$
$811$	−17.9975	−0.631977	−0.315988	+	0.948763i	$0.602336\pi$
$812$	0	0
$813$	4.21801	0.147932
$814$	0	0
$815$	−67.9961	−2.38180
$816$	0	0
$817$	74.9098	2.62076
$818$	0	0
$819$	8.89255	0.310731
$820$	0	0
$821$	21.9979	0.767733	0.383867	−	0.923389i	$-0.374592\pi$
$821$	21.9979	0.767733	0.383867	+	0.923389i	$0.374592\pi$
$822$	0	0
$823$	−26.7572	−0.932698	−0.466349	−	0.884601i	$-0.654431\pi$
$823$	−26.7572	−0.932698	−0.466349	+	0.884601i	$0.654431\pi$
$824$	0	0
$825$	−15.5437	−0.541163
$826$	0	0
$827$	−13.4066	−0.466194	−0.233097	−	0.972453i	$-0.574886\pi$
$827$	−13.4066	−0.466194	−0.233097	+	0.972453i	$0.574886\pi$
$828$	0	0
$829$	15.9100	0.552576	0.276288	−	0.961075i	$-0.410896\pi$
$829$	15.9100	0.552576	0.276288	+	0.961075i	$0.410896\pi$
$830$	0	0
$831$	−0.590646	−0.0204893
$832$	0	0
$833$	0.473626	0.0164102
$834$	0	0
$835$	−18.8910	−0.653749
$836$	0	0
$837$	−0.386844	−0.0133713
$838$	0	0
$839$	−30.4767	−1.05217	−0.526087	−	0.850431i	$-0.676341\pi$
$839$	−30.4767	−1.05217	−0.526087	+	0.850431i	$0.676341\pi$
$840$	0	0
$841$	−25.3290	−0.873415
$842$	0	0
$843$	−7.81863	−0.269288
$844$	0	0
$845$	−14.8550	−0.511027
$846$	0	0
$847$	2.08312	0.0715768
$848$	0	0
$849$	−3.63686	−0.124817
$850$	0	0
$851$	0.207278	0.00710540
$852$	0	0
$853$	42.3182	1.44895	0.724473	−	0.689303i	$-0.242083\pi$
$853$	42.3182	1.44895	0.724473	+	0.689303i	$0.242083\pi$
$854$	0	0
$855$	−88.4599	−3.02526
$856$	0	0
$857$	−17.5515	−0.599547	−0.299774	−	0.954010i	$-0.596911\pi$
$857$	−17.5515	−0.599547	−0.299774	+	0.954010i	$0.596911\pi$
$858$	0	0
$859$	9.50332	0.324249	0.162125	−	0.986770i	$-0.448165\pi$
$859$	9.50332	0.324249	0.162125	+	0.986770i	$0.448165\pi$
$860$	0	0
$861$	1.94688	0.0663494
$862$	0	0
$863$	−52.7588	−1.79593	−0.897965	−	0.440067i	$-0.854955\pi$
$863$	−52.7588	−1.79593	−0.897965	+	0.440067i	$0.854955\pi$
$864$	0	0
$865$	16.2373	0.552083
$866$	0	0
$867$	−5.61824	−0.190805
$868$	0	0
$869$	34.4039	1.16707
$870$	0	0
$871$	42.5711	1.44247
$872$	0	0
$873$	−32.5236	−1.10076
$874$	0	0
$875$	33.0707	1.11799
$876$	0	0
$877$	15.8058	0.533723	0.266862	−	0.963735i	$-0.414013\pi$
$877$	15.8058	0.533723	0.266862	+	0.963735i	$0.414013\pi$
$878$	0	0
$879$	−6.72556	−0.226847
$880$	0	0
$881$	−23.6304	−0.796128	−0.398064	−	0.917358i	$-0.630318\pi$
$881$	−23.6304	−0.796128	−0.398064	+	0.917358i	$0.630318\pi$
$882$	0	0
$883$	26.4656	0.890640	0.445320	−	0.895372i	$-0.353090\pi$
$883$	26.4656	0.890640	0.445320	+	0.895372i	$0.353090\pi$
$884$	0	0
$885$	−15.0246	−0.505047
$886$	0	0
$887$	46.0412	1.54591	0.772957	−	0.634459i	$-0.218777\pi$
$887$	46.0412	1.54591	0.772957	+	0.634459i	$0.218777\pi$
$888$	0	0
$889$	−18.8562	−0.632416
$890$	0	0
$891$	−28.9478	−0.969788
$892$	0	0
$893$	−25.5804	−0.856015
$894$	0	0
$895$	65.0394	2.17403
$896$	0	0
$897$	−0.0866491	−0.00289313
$898$	0	0
$899$	0.375881	0.0125363
$900$	0	0
$901$	3.27391	0.109070
$902$	0	0
$903$	3.45844	0.115090
$904$	0	0
$905$	−15.0827	−0.501367
$906$	0	0
$907$	−28.2180	−0.936964	−0.468482	−	0.883473i	$-0.655199\pi$
$907$	−28.2180	−0.936964	−0.468482	+	0.883473i	$0.655199\pi$
$908$	0	0
$909$	−43.2692	−1.43515
$910$	0	0
$911$	−7.94815	−0.263334	−0.131667	−	0.991294i	$-0.542033\pi$
$911$	−7.94815	−0.263334	−0.131667	+	0.991294i	$0.542033\pi$
$912$	0	0
$913$	29.9443	0.991011
$914$	0	0
$915$	−11.5491	−0.381802
$916$	0	0
$917$	1.55960	0.0515025
$918$	0	0
$919$	30.2767	0.998735	0.499367	−	0.866390i	$-0.333566\pi$
$919$	30.2767	0.998735	0.499367	+	0.866390i	$0.333566\pi$
$920$	0	0
$921$	5.89411	0.194217
$922$	0	0
$923$	−27.5804	−0.907819
$924$	0	0
$925$	31.6550	1.04081
$926$	0	0
$927$	−36.5071	−1.19905
$928$	0	0
$929$	50.9720	1.67234	0.836169	−	0.548473i	$-0.184790\pi$
$929$	50.9720	1.67234	0.836169	+	0.548473i	$0.184790\pi$
$930$	0	0
$931$	7.25402	0.237741
$932$	0	0
$933$	0.927612	0.0303686
$934$	0	0
$935$	−7.23412	−0.236581
$936$	0	0
$937$	−6.62686	−0.216490	−0.108245	−	0.994124i	$-0.534523\pi$
$937$	−6.62686	−0.216490	−0.108245	+	0.994124i	$0.534523\pi$
$938$	0	0
$939$	−3.63039	−0.118473
$940$	0	0
$941$	−17.7719	−0.579346	−0.289673	−	0.957126i	$-0.593547\pi$
$941$	−17.7719	−0.579346	−0.289673	+	0.957126i	$0.593547\pi$
$942$	0	0
$943$	0.488437	0.0159057
$944$	0	0
$945$	−8.32666	−0.270866
$946$	0	0
$947$	39.8906	1.29627	0.648134	−	0.761526i	$-0.275549\pi$
$947$	39.8906	1.29627	0.648134	+	0.761526i	$0.275549\pi$
$948$	0	0
$949$	−33.0192	−1.07185
$950$	0	0
$951$	−6.68852	−0.216890
$952$	0	0
$953$	38.6729	1.25274	0.626369	−	0.779526i	$-0.284540\pi$
$953$	38.6729	1.25274	0.626369	+	0.779526i	$0.284540\pi$
$954$	0	0
$955$	−84.5533	−2.73608
$956$	0	0
$957$	−2.32095	−0.0750258
$958$	0	0
$959$	−8.04922	−0.259923
$960$	0	0
$961$	−30.9615	−0.998758
$962$	0	0
$963$	−49.5684	−1.59732
$964$	0	0
$965$	24.0250	0.773392
$966$	0	0
$967$	−16.4567	−0.529213	−0.264607	−	0.964356i	$-0.585242\pi$
$967$	−16.4567	−0.529213	−0.264607	+	0.964356i	$0.585242\pi$
$968$	0	0
$969$	1.15063	0.0369634
$970$	0	0
$971$	−4.32400	−0.138764	−0.0693818	−	0.997590i	$-0.522103\pi$
$971$	−4.32400	−0.138764	−0.0693818	+	0.997590i	$0.522103\pi$
$972$	0	0
$973$	−6.64049	−0.212884
$974$	0	0
$975$	−13.2328	−0.423790
$976$	0	0
$977$	3.21193	0.102759	0.0513793	−	0.998679i	$-0.483638\pi$
$977$	3.21193	0.102759	0.0513793	+	0.998679i	$0.483638\pi$
$978$	0	0
$979$	−55.4247	−1.77138
$980$	0	0
$981$	−30.7507	−0.981793
$982$	0	0
$983$	−8.33539	−0.265858	−0.132929	−	0.991126i	$-0.542438\pi$
$983$	−8.33539	−0.265858	−0.132929	+	0.991126i	$0.542438\pi$
$984$	0	0
$985$	99.7499	3.17829
$986$	0	0
$987$	−1.18100	−0.0375915
$988$	0	0
$989$	0.867662	0.0275900
$990$	0	0
$991$	−4.05237	−0.128728	−0.0643640	−	0.997926i	$-0.520502\pi$
$991$	−4.05237	−0.128728	−0.0643640	+	0.997926i	$0.520502\pi$
$992$	0	0
$993$	5.22004	0.165653
$994$	0	0
$995$	75.1511	2.38245
$996$	0	0
$997$	−15.5851	−0.493585	−0.246793	−	0.969068i	$-0.579377\pi$
$997$	−15.5851	−0.493585	−0.246793	+	0.969068i	$0.579377\pi$
$998$	0	0
$999$	−4.86451	−0.153906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.x.1.3		4
4.3	odd	2		7168.2.a.w.1.2		4
8.3	odd	2		7168.2.a.s.1.3		4
8.5	even	2		7168.2.a.t.1.2		4
32.3	odd	8		1792.2.m.c.1345.2	yes	8
32.5	even	8		1792.2.m.d.449.2	yes	8
32.11	odd	8		1792.2.m.c.449.2	yes	8
32.13	even	8		1792.2.m.d.1345.2	yes	8
32.19	odd	8		1792.2.m.b.1345.3	yes	8
32.21	even	8		1792.2.m.a.449.3	✓	8
32.27	odd	8		1792.2.m.b.449.3	yes	8
32.29	even	8		1792.2.m.a.1345.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.a.449.3	✓	8	32.21	even	8
1792.2.m.a.1345.3	yes	8	32.29	even	8
1792.2.m.b.449.3	yes	8	32.27	odd	8
1792.2.m.b.1345.3	yes	8	32.19	odd	8
1792.2.m.c.449.2	yes	8	32.11	odd	8
1792.2.m.c.1345.2	yes	8	32.3	odd	8
1792.2.m.d.449.2	yes	8	32.5	even	8
1792.2.m.d.1345.2	yes	8	32.13	even	8
7168.2.a.s.1.3		4	8.3	odd	2
7168.2.a.t.1.2		4	8.5	even	2
7168.2.a.w.1.2		4	4.3	odd	2
7168.2.a.x.1.3		4	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)

\(f(q)\)	\(=\)	\(q+0.334904 q^{3} +4.22274 q^{5} +1.00000 q^{7} -2.88784 q^{9} +O(q^{10})\)	\(q+0.334904 q^{3} +4.22274 q^{5} +1.00000 q^{7} -2.88784 q^{9} -3.61706 q^{11} -3.07931 q^{13} +1.41421 q^{15} +0.473626 q^{17} +7.25402 q^{19} +0.334904 q^{21} +0.0840215 q^{23} +12.8316 q^{25} -1.97186 q^{27} +1.91598 q^{29} +0.196182 q^{31} -1.21137 q^{33} +4.22274 q^{35} +2.46696 q^{37} -1.03127 q^{39} +5.81324 q^{41} +10.3267 q^{43} -12.1946 q^{45} -3.52637 q^{47} +1.00000 q^{49} +0.158619 q^{51} +6.91245 q^{53} -15.2739 q^{55} +2.42940 q^{57} -10.6240 q^{59} -8.16647 q^{61} -2.88784 q^{63} -13.0031 q^{65} -13.8249 q^{67} +0.0281391 q^{69} +8.95668 q^{71} +10.7229 q^{73} +4.29734 q^{75} -3.61706 q^{77} -9.51156 q^{79} +8.00313 q^{81} -8.27863 q^{83} +2.00000 q^{85} +0.641669 q^{87} +15.3231 q^{89} -3.07931 q^{91} +0.0657022 q^{93} +30.6318 q^{95} +11.2623 q^{97} +10.4455 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7	\( 4 q + 4 q^{5} + 4 q^{7} - 8 q^{13} - 4 q^{17} - 4 q^{19} - 8 q^{23} + 12 q^{27} + 16 q^{29} + 4 q^{31} - 8 q^{33} + 4 q^{35} + 16 q^{37} + 16 q^{39} + 12 q^{41} + 16 q^{43} - 16 q^{45} - 20 q^{47} + 4 q^{49} - 8 q^{51} + 8 q^{53} - 24 q^{55} + 4 q^{57} - 4 q^{59} + 20 q^{61} - 12 q^{65} - 16 q^{67} + 20 q^{69} + 16 q^{71} + 8 q^{73} + 4 q^{75} - 8 q^{81} + 8 q^{83} + 8 q^{85} - 20 q^{87} + 8 q^{89} - 8 q^{91} + 32 q^{93} + 40 q^{95} + 36 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 - 8 * q^13 - 4 * q^17 - 4 * q^19 - 8 * q^23 + 12 * q^27 + 16 * q^29 + 4 * q^31 - 8 * q^33 + 4 * q^35 + 16 * q^37 + 16 * q^39 + 12 * q^41 + 16 * q^43 - 16 * q^45 - 20 * q^47 + 4 * q^49 - 8 * q^51 + 8 * q^53 - 24 * q^55 + 4 * q^57 - 4 * q^59 + 20 * q^61 - 12 * q^65 - 16 * q^67 + 20 * q^69 + 16 * q^71 + 8 * q^73 + 4 * q^75 - 8 * q^81 + 8 * q^83 + 8 * q^85 - 20 * q^87 + 8 * q^89 - 8 * q^91 + 32 * q^93 + 40 * q^95 + 36 * q^97 + 16 * q^99

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