LMFDB - Embedded newform 6975.2.a.bj.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,9,0,0,0,-9,0,0,6,0,-16,-8,0,11,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.20308.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 12 $ x^4 - x^3 - 8x^2 + 4x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.80027$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80027 q^{2} +5.84153 q^{4} -1.04125 q^{7} -10.7573 q^{8} -4.64180 q^{11} -2.95875 q^{13} +2.91579 q^{14} +18.4404 q^{16} +6.29942 q^{17} -1.11552 q^{19} +12.9983 q^{22} +3.87454 q^{23} +8.28530 q^{26} -6.08251 q^{28} -2.35650 q^{29} +1.00000 q^{31} -30.1234 q^{32} -17.6401 q^{34} -1.30112 q^{37} +3.12376 q^{38} -1.92573 q^{41} +4.29942 q^{43} -27.1152 q^{44} -10.8498 q^{46} -3.31525 q^{47} -5.91579 q^{49} -17.2836 q^{52} +5.10964 q^{53} +11.2011 q^{56} +6.59884 q^{58} +5.07256 q^{59} +5.31525 q^{61} -2.80027 q^{62} +47.4730 q^{64} +8.64180 q^{67} +36.7982 q^{68} +3.88278 q^{71} -12.1740 q^{73} +3.64350 q^{74} -6.51634 q^{76} +4.83329 q^{77} -7.12376 q^{79} +5.39258 q^{82} -7.46614 q^{83} -12.0396 q^{86} +49.9333 q^{88} +12.1598 q^{89} +3.08080 q^{91} +22.6332 q^{92} +9.28360 q^{94} +0.915792 q^{97} +16.5658 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 9 q^{4} - 9 q^{8} + 6 q^{11} - 16 q^{13} - 8 q^{14} + 11 q^{16} + q^{17} + 5 q^{19} + 24 q^{22} + 12 q^{26} - 16 q^{28} - 6 q^{29} + 4 q^{31} - 29 q^{32} - 18 q^{34} - 9 q^{37} - 13 q^{41}+ \cdots + 19 q^{98}+O(q^{100}) $ 4 * q - q^2 + 9 * q^4 - 9 * q^8 + 6 * q^11 - 16 * q^13 - 8 * q^14 + 11 * q^16 + q^17 + 5 * q^19 + 24 * q^22 + 12 * q^26 - 16 * q^28 - 6 * q^29 + 4 * q^31 - 29 * q^32 - 18 * q^34 - 9 * q^37 - 13 * q^41 - 7 * q^43 - 20 * q^44 - 26 * q^46 - 14 * q^47 - 4 * q^49 - 20 * q^52 + 11 * q^53 + 4 * q^56 - 22 * q^58 - 13 * q^59 + 22 * q^61 - q^62 + 47 * q^64 + 10 * q^67 + 30 * q^68 - 3 * q^71 - 9 * q^73 + 18 * q^74 + 14 * q^76 + 8 * q^77 - 16 * q^79 - 6 * q^82 - 17 * q^83 - 16 * q^86 + 44 * q^88 + 12 * q^89 - 24 * q^91 - 10 * q^92 - 12 * q^94 - 16 * q^97 + 19 * q^98

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 $

$2$	−2.80027	−1.98009	−0.990046	−	0.140746i	$-0.955050\pi$
$2$	−2.80027	−1.98009	−0.990046	+	0.140746i	$0.955050\pi$
$3$	0	0
$3$	0	0
$4$	5.84153	2.92076
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.04125	−0.393557	−0.196778	−	0.980448i	$-0.563048\pi$
$7$	−1.04125	−0.393557	−0.196778	+	0.980448i	$0.563048\pi$
$8$	−10.7573	−3.80329
$9$	0	0
$10$	0	0
$11$	−4.64180	−1.39955	−0.699777	−	0.714361i	$-0.746717\pi$
$11$	−4.64180	−1.39955	−0.699777	+	0.714361i	$0.746717\pi$
$12$	0	0
$13$	−2.95875	−0.820609	−0.410304	−	0.911949i	$-0.634578\pi$
$13$	−2.95875	−0.820609	−0.410304	+	0.911949i	$0.634578\pi$
$14$	2.91579	0.779278
$15$	0	0
$16$	18.4404	4.61009
$17$	6.29942	1.52783	0.763917	−	0.645314i	$-0.223274\pi$
$17$	6.29942	1.52783	0.763917	+	0.645314i	$0.223274\pi$
$18$	0	0
$19$	−1.11552	−0.255918	−0.127959	−	0.991779i	$-0.540843\pi$
$19$	−1.11552	−0.255918	−0.127959	+	0.991779i	$0.540843\pi$
$20$	0	0
$21$	0	0
$22$	12.9983	2.77125
$23$	3.87454	0.807897	0.403949	−	0.914782i	$-0.367637\pi$
$23$	3.87454	0.807897	0.403949	+	0.914782i	$0.367637\pi$
$24$	0	0
$25$	0	0
$26$	8.28530	1.62488
$27$	0	0
$28$	−6.08251	−1.14949
$29$	−2.35650	−0.437591	−0.218796	−	0.975771i	$-0.570213\pi$
$29$	−2.35650	−0.437591	−0.218796	+	0.975771i	$0.570213\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−30.1234	−5.32512
$33$	0	0
$34$	−17.6401	−3.02525
$35$	0	0
$36$	0	0
$37$	−1.30112	−0.213903	−0.106952	−	0.994264i	$-0.534109\pi$
$37$	−1.30112	−0.213903	−0.106952	+	0.994264i	$0.534109\pi$
$38$	3.12376	0.506741
$39$	0	0
$40$	0	0
$41$	−1.92573	−0.300749	−0.150375	−	0.988629i	$-0.548048\pi$
$41$	−1.92573	−0.300749	−0.150375	+	0.988629i	$0.548048\pi$
$42$	0	0
$43$	4.29942	0.655656	0.327828	−	0.944737i	$-0.393683\pi$
$43$	4.29942	0.655656	0.327828	+	0.944737i	$0.393683\pi$
$44$	−27.1152	−4.08777
$45$	0	0
$46$	−10.8498	−1.59971
$47$	−3.31525	−0.483579	−0.241789	−	0.970329i	$-0.577734\pi$
$47$	−3.31525	−0.483579	−0.241789	+	0.970329i	$0.577734\pi$
$48$	0	0
$49$	−5.91579	−0.845113
$50$	0	0
$51$	0	0
$52$	−17.2836	−2.39680
$53$	5.10964	0.701862	0.350931	−	0.936401i	$-0.385865\pi$
$53$	5.10964	0.701862	0.350931	+	0.936401i	$0.385865\pi$
$54$	0	0
$55$	0	0
$56$	11.2011	1.49681
$57$	0	0
$58$	6.59884	0.866471
$59$	5.07256	0.660392	0.330196	−	0.943912i	$-0.392885\pi$
$59$	5.07256	0.660392	0.330196	+	0.943912i	$0.392885\pi$
$60$	0	0
$61$	5.31525	0.680548	0.340274	−	0.940326i	$-0.389480\pi$
$61$	5.31525	0.680548	0.340274	+	0.940326i	$0.389480\pi$
$62$	−2.80027	−0.355635
$63$	0	0
$64$	47.4730	5.93413
$65$	0	0
$66$	0	0
$67$	8.64180	1.05576	0.527882	−	0.849318i	$-0.322986\pi$
$67$	8.64180	1.05576	0.527882	+	0.849318i	$0.322986\pi$
$68$	36.7982	4.46244
$69$	0	0
$70$	0	0
$71$	3.88278	0.460801	0.230401	−	0.973096i	$-0.425996\pi$
$71$	3.88278	0.460801	0.230401	+	0.973096i	$0.425996\pi$
$72$	0	0
$73$	−12.1740	−1.42485	−0.712427	−	0.701746i	$-0.752404\pi$
$73$	−12.1740	−1.42485	−0.712427	+	0.701746i	$0.752404\pi$
$74$	3.64350	0.423548
$75$	0	0
$76$	−6.51634	−0.747475
$77$	4.83329	0.550804
$78$	0	0
$79$	−7.12376	−0.801486	−0.400743	−	0.916191i	$-0.631248\pi$
$79$	−7.12376	−0.801486	−0.400743	+	0.916191i	$0.631248\pi$
$80$	0	0
$81$	0	0
$82$	5.39258	0.595511
$83$	−7.46614	−0.819515	−0.409757	−	0.912195i	$-0.634387\pi$
$83$	−7.46614	−0.819515	−0.409757	+	0.912195i	$0.634387\pi$
$84$	0	0
$85$	0	0
$86$	−12.0396	−1.29826
$87$	0	0
$88$	49.9333	5.32291
$89$	12.1598	1.28894	0.644470	−	0.764630i	$-0.277078\pi$
$89$	12.1598	1.28894	0.644470	+	0.764630i	$0.277078\pi$
$90$	0	0
$91$	3.08080	0.322956
$92$	22.6332	2.35968
$93$	0	0
$94$	9.28360	0.957530
$95$	0	0
$96$	0	0
$97$	0.915792	0.0929846	0.0464923	−	0.998919i	$-0.485196\pi$
$97$	0.915792	0.0929846	0.0464923	+	0.998919i	$0.485196\pi$
$98$	16.5658	1.67340
$99$	0	0
$100$	0	0
$101$	4.11382	0.409340	0.204670	−	0.978831i	$-0.434388\pi$
$101$	4.11382	0.409340	0.204670	+	0.978831i	$0.434388\pi$
$102$	0	0
$103$	16.9983	1.67489	0.837446	−	0.546520i	$-0.184048\pi$
$103$	16.9983	1.67489	0.837446	+	0.546520i	$0.184048\pi$
$104$	31.8282	3.12101
$105$	0	0
$106$	−14.3084	−1.38975
$107$	4.84976	0.468844	0.234422	−	0.972135i	$-0.424680\pi$
$107$	4.84976	0.468844	0.234422	+	0.972135i	$0.424680\pi$
$108$	0	0
$109$	9.19803	0.881011	0.440506	−	0.897750i	$-0.354799\pi$
$109$	9.19803	0.881011	0.440506	+	0.897750i	$0.354799\pi$
$110$	0	0
$111$	0	0
$112$	−19.2011	−1.81433
$113$	3.47508	0.326908	0.163454	−	0.986551i	$-0.447736\pi$
$113$	3.47508	0.326908	0.163454	+	0.986551i	$0.447736\pi$
$114$	0	0
$115$	0	0
$116$	−13.7656	−1.27810
$117$	0	0
$118$	−14.2046	−1.30764
$119$	−6.55929	−0.601289
$120$	0	0
$121$	10.5463	0.958753
$122$	−14.8841	−1.34755
$123$	0	0
$124$	5.84153	0.524584
$125$	0	0
$126$	0	0
$127$	−12.8728	−1.14228	−0.571140	−	0.820853i	$-0.693499\pi$
$127$	−12.8728	−1.14228	−0.571140	+	0.820853i	$0.693499\pi$
$128$	−72.6906	−6.42500
$129$	0	0
$130$	0	0
$131$	−15.4668	−1.35134	−0.675672	−	0.737202i	$-0.736147\pi$
$131$	−15.4668	−1.35134	−0.675672	+	0.737202i	$0.736147\pi$
$132$	0	0
$133$	1.16154	0.100718
$134$	−24.1994	−2.09051
$135$	0	0
$136$	−67.7649	−5.81079
$137$	−16.5734	−1.41596	−0.707981	−	0.706231i	$-0.750394\pi$
$137$	−16.5734	−1.41596	−0.707981	+	0.706231i	$0.750394\pi$
$138$	0	0
$139$	21.5259	1.82581	0.912903	−	0.408176i	$-0.133835\pi$
$139$	21.5259	1.82581	0.912903	+	0.408176i	$0.133835\pi$
$140$	0	0
$141$	0	0
$142$	−10.8728	−0.912428
$143$	13.7339	1.14849
$144$	0	0
$145$	0	0
$146$	34.0904	2.82134
$147$	0	0
$148$	−7.60054	−0.624761
$149$	−20.7556	−1.70037	−0.850183	−	0.526487i	$-0.823509\pi$
$149$	−20.7556	−1.70037	−0.850183	+	0.526487i	$0.823509\pi$
$150$	0	0
$151$	−18.4451	−1.50104	−0.750522	−	0.660846i	$-0.770198\pi$
$151$	−18.4451	−1.50104	−0.750522	+	0.660846i	$0.770198\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	−13.5345	−1.09064
$155$	0	0
$156$	0	0
$157$	−5.64350	−0.450400	−0.225200	−	0.974313i	$-0.572304\pi$
$157$	−5.64350	−0.450400	−0.225200	+	0.974313i	$0.572304\pi$
$158$	19.9485	1.58701
$159$	0	0
$160$	0	0
$161$	−4.03438	−0.317953
$162$	0	0
$163$	16.1994	1.26883	0.634417	−	0.772991i	$-0.281240\pi$
$163$	16.1994	1.26883	0.634417	+	0.772991i	$0.281240\pi$
$164$	−11.2492	−0.878417
$165$	0	0
$166$	20.9072	1.62271
$167$	5.14919	0.398456	0.199228	−	0.979953i	$-0.436157\pi$
$167$	5.14919	0.398456	0.199228	+	0.979953i	$0.436157\pi$
$168$	0	0
$169$	−4.24582	−0.326601
$170$	0	0
$171$	0	0
$172$	25.1152	1.91501
$173$	10.7621	0.818226	0.409113	−	0.912484i	$-0.365838\pi$
$173$	10.7621	0.818226	0.409113	+	0.912484i	$0.365838\pi$
$174$	0	0
$175$	0	0
$176$	−85.5965	−6.45208
$177$	0	0
$178$	−34.0509	−2.55222
$179$	10.2740	0.767914	0.383957	−	0.923351i	$-0.374561\pi$
$179$	10.2740	0.767914	0.383957	+	0.923351i	$0.374561\pi$
$180$	0	0
$181$	2.53622	0.188516	0.0942578	−	0.995548i	$-0.469952\pi$
$181$	2.53622	0.188516	0.0942578	+	0.995548i	$0.469952\pi$
$182$	−8.62709	−0.639483
$183$	0	0
$184$	−41.6796	−3.07266
$185$	0	0
$186$	0	0
$187$	−29.2406	−2.13829
$188$	−19.3661	−1.41242
$189$	0	0
$190$	0	0
$191$	4.45031	0.322013	0.161007	−	0.986953i	$-0.448526\pi$
$191$	4.45031	0.322013	0.161007	+	0.986953i	$0.448526\pi$
$192$	0	0
$193$	2.71300	0.195286	0.0976430	−	0.995222i	$-0.468870\pi$
$193$	2.71300	0.195286	0.0976430	+	0.995222i	$0.468870\pi$
$194$	−2.56447	−0.184118
$195$	0	0
$196$	−34.5573	−2.46838
$197$	−5.04125	−0.359174	−0.179587	−	0.983742i	$-0.557476\pi$
$197$	−5.04125	−0.359174	−0.179587	+	0.983742i	$0.557476\pi$
$198$	0	0
$199$	−14.3531	−1.01746	−0.508732	−	0.860925i	$-0.669886\pi$
$199$	−14.3531	−1.01746	−0.508732	+	0.860925i	$0.669886\pi$
$200$	0	0
$201$	0	0
$202$	−11.5198	−0.810531
$203$	2.45371	0.172217
$204$	0	0
$205$	0	0
$206$	−47.5999	−3.31644
$207$	0	0
$208$	−54.5604	−3.78308
$209$	5.17802	0.358171
$210$	0	0
$211$	−11.6288	−0.800559	−0.400280	−	0.916393i	$-0.631087\pi$
$211$	−11.6288	−0.800559	−0.400280	+	0.916393i	$0.631087\pi$
$212$	29.8481	2.04997
$213$	0	0
$214$	−13.5807	−0.928355
$215$	0	0
$216$	0	0
$217$	−1.04125	−0.0706849
$218$	−25.7570	−1.74448
$219$	0	0
$220$	0	0
$221$	−18.6384	−1.25375
$222$	0	0
$223$	−5.45483	−0.365283	−0.182641	−	0.983180i	$-0.558465\pi$
$223$	−5.45483	−0.365283	−0.182641	+	0.983180i	$0.558465\pi$
$224$	31.3661	2.09574
$225$	0	0
$226$	−9.73118	−0.647309
$227$	−7.97352	−0.529221	−0.264611	−	0.964355i	$-0.585243\pi$
$227$	−7.97352	−0.529221	−0.264611	+	0.964355i	$0.585243\pi$
$228$	0	0
$229$	22.0197	1.45510	0.727550	−	0.686054i	$-0.240659\pi$
$229$	22.0197	1.45510	0.727550	+	0.686054i	$0.240659\pi$
$230$	0	0
$231$	0	0
$232$	25.3496	1.66428
$233$	−17.2723	−1.13155	−0.565773	−	0.824561i	$-0.691422\pi$
$233$	−17.2723	−1.13155	−0.565773	+	0.824561i	$0.691422\pi$
$234$	0	0
$235$	0	0
$236$	29.6315	1.92885
$237$	0	0
$238$	18.3678	1.19061
$239$	21.3462	1.38077	0.690386	−	0.723441i	$-0.257441\pi$
$239$	21.3462	1.38077	0.690386	+	0.723441i	$0.257441\pi$
$240$	0	0
$241$	−4.15984	−0.267959	−0.133979	−	0.990984i	$-0.542776\pi$
$241$	−4.15984	−0.267959	−0.133979	+	0.990984i	$0.542776\pi$
$242$	−29.5325	−1.89842
$243$	0	0
$244$	31.0492	1.98772
$245$	0	0
$246$	0	0
$247$	3.30054	0.210008
$248$	−10.7573	−0.683090
$249$	0	0
$250$	0	0
$251$	−5.76896	−0.364134	−0.182067	−	0.983286i	$-0.558279\pi$
$251$	−5.76896	−0.364134	−0.182067	+	0.983286i	$0.558279\pi$
$252$	0	0
$253$	−17.9848	−1.13070
$254$	36.0475	2.26182
$255$	0	0
$256$	108.607	6.78796
$257$	−22.4417	−1.39988	−0.699938	−	0.714203i	$-0.746789\pi$
$257$	−22.4417	−1.39988	−0.699938	+	0.714203i	$0.746789\pi$
$258$	0	0
$259$	1.35480	0.0841831
$260$	0	0
$261$	0	0
$262$	43.3114	2.67579
$263$	10.1904	0.628369	0.314185	−	0.949362i	$-0.398269\pi$
$263$	10.1904	0.628369	0.314185	+	0.949362i	$0.398269\pi$
$264$	0	0
$265$	0	0
$266$	−3.25262	−0.199431
$267$	0	0
$268$	50.4813	3.08364
$269$	−28.0904	−1.71270	−0.856351	−	0.516394i	$-0.827274\pi$
$269$	−28.0904	−1.71270	−0.856351	+	0.516394i	$0.827274\pi$
$270$	0	0
$271$	0.0429548	0.00260932	0.00130466	−	0.999999i	$-0.499585\pi$
$271$	0.0429548	0.00260932	0.00130466	+	0.999999i	$0.499585\pi$
$272$	116.164	7.04346
$273$	0	0
$274$	46.4101	2.80374
$275$	0	0
$276$	0	0
$277$	21.8288	1.31156	0.655782	−	0.754951i	$-0.272339\pi$
$277$	21.8288	1.31156	0.655782	+	0.754951i	$0.272339\pi$
$278$	−60.2785	−3.61526
$279$	0	0
$280$	0	0
$281$	9.20415	0.549074	0.274537	−	0.961577i	$-0.411475\pi$
$281$	9.20415	0.549074	0.274537	+	0.961577i	$0.411475\pi$
$282$	0	0
$283$	6.85317	0.407379	0.203689	−	0.979036i	$-0.434707\pi$
$283$	6.85317	0.407379	0.203689	+	0.979036i	$0.434707\pi$
$284$	22.6813	1.34589
$285$	0	0
$286$	−38.4587	−2.27411
$287$	2.00518	0.118362
$288$	0	0
$289$	22.6827	1.33428
$290$	0	0
$291$	0	0
$292$	−71.1145	−4.16166
$293$	−16.6101	−0.970375	−0.485188	−	0.874410i	$-0.661249\pi$
$293$	−16.6101	−0.970375	−0.485188	+	0.874410i	$0.661249\pi$
$294$	0	0
$295$	0	0
$296$	13.9966	0.813536
$297$	0	0
$298$	58.1214	3.36688
$299$	−11.4638	−0.662968
$300$	0	0
$301$	−4.47679	−0.258038
$302$	51.6514	2.97220
$303$	0	0
$304$	−20.5706	−1.17980
$305$	0	0
$306$	0	0
$307$	−16.9553	−0.967693	−0.483846	−	0.875153i	$-0.660761\pi$
$307$	−16.9553	−0.967693	−0.483846	+	0.875153i	$0.660761\pi$
$308$	28.2338	1.60877
$309$	0	0
$310$	0	0
$311$	0.638734	0.0362193	0.0181096	−	0.999836i	$-0.494235\pi$
$311$	0.638734	0.0362193	0.0181096	+	0.999836i	$0.494235\pi$
$312$	0	0
$313$	18.1141	1.02387	0.511934	−	0.859025i	$-0.328929\pi$
$313$	18.1141	1.02387	0.511934	+	0.859025i	$0.328929\pi$
$314$	15.8033	0.891834
$315$	0	0
$316$	−41.6136	−2.34095
$317$	−5.06433	−0.284441	−0.142220	−	0.989835i	$-0.545424\pi$
$317$	−5.06433	−0.284441	−0.142220	+	0.989835i	$0.545424\pi$
$318$	0	0
$319$	10.9384	0.612433
$320$	0	0
$321$	0	0
$322$	11.2974	0.629577
$323$	−7.02713	−0.391000
$324$	0	0
$325$	0	0
$326$	−45.3627	−2.51241
$327$	0	0
$328$	20.7157	1.14383
$329$	3.45201	0.190316
$330$	0	0
$331$	−20.0344	−1.10119	−0.550594	−	0.834773i	$-0.685599\pi$
$331$	−20.0344	−1.10119	−0.550594	+	0.834773i	$0.685599\pi$
$332$	−43.6136	−2.39361
$333$	0	0
$334$	−14.4191	−0.788979
$335$	0	0
$336$	0	0
$337$	4.86729	0.265138	0.132569	−	0.991174i	$-0.457677\pi$
$337$	4.86729	0.265138	0.132569	+	0.991174i	$0.457677\pi$
$338$	11.8894	0.646700
$339$	0	0
$340$	0	0
$341$	−4.64180	−0.251367
$342$	0	0
$343$	13.4486	0.726157
$344$	−46.2502	−2.49365
$345$	0	0
$346$	−30.1368	−1.62016
$347$	−10.1221	−0.543381	−0.271690	−	0.962385i	$-0.587583\pi$
$347$	−10.1221	−0.543381	−0.271690	+	0.962385i	$0.587583\pi$
$348$	0	0
$349$	−18.3245	−0.980888	−0.490444	−	0.871473i	$-0.663165\pi$
$349$	−18.3245	−0.980888	−0.490444	+	0.871473i	$0.663165\pi$
$350$	0	0
$351$	0	0
$352$	139.827	7.45279
$353$	−14.3248	−0.762435	−0.381217	−	0.924485i	$-0.624495\pi$
$353$	−14.3248	−0.762435	−0.381217	+	0.924485i	$0.624495\pi$
$354$	0	0
$355$	0	0
$356$	71.0320	3.76469
$357$	0	0
$358$	−28.7700	−1.52054
$359$	−12.3197	−0.650207	−0.325104	−	0.945678i	$-0.605399\pi$
$359$	−12.3197	−0.650207	−0.325104	+	0.945678i	$0.605399\pi$
$360$	0	0
$361$	−17.7556	−0.934506
$362$	−7.10210	−0.373278
$363$	0	0
$364$	17.9966	0.943278
$365$	0	0
$366$	0	0
$367$	−24.4706	−1.27735	−0.638676	−	0.769475i	$-0.720518\pi$
$367$	−24.4706	−1.27735	−0.638676	+	0.769475i	$0.720518\pi$
$368$	71.4479	3.72448
$369$	0	0
$370$	0	0
$371$	−5.32042	−0.276223
$372$	0	0
$373$	−26.5276	−1.37355	−0.686775	−	0.726870i	$-0.740974\pi$
$373$	−26.5276	−1.37355	−0.686775	+	0.726870i	$0.740974\pi$
$374$	81.8818	4.23400
$375$	0	0
$376$	35.6632	1.83919
$377$	6.97229	0.359091
$378$	0	0
$379$	12.5528	0.644795	0.322398	−	0.946604i	$-0.395511\pi$
$379$	12.5528	0.644795	0.322398	+	0.946604i	$0.395511\pi$
$380$	0	0
$381$	0	0
$382$	−12.4621	−0.637615
$383$	11.5882	0.592129	0.296064	−	0.955168i	$-0.404326\pi$
$383$	11.5882	0.592129	0.296064	+	0.955168i	$0.404326\pi$
$384$	0	0
$385$	0	0
$386$	−7.59714	−0.386684
$387$	0	0
$388$	5.34962	0.271586
$389$	−32.5790	−1.65182	−0.825909	−	0.563803i	$-0.809338\pi$
$389$	−32.5790	−1.65182	−0.825909	+	0.563803i	$0.809338\pi$
$390$	0	0
$391$	24.4074	1.23433
$392$	63.6381	3.21421
$393$	0	0
$394$	14.1169	0.711198
$395$	0	0
$396$	0	0
$397$	−29.8508	−1.49817	−0.749084	−	0.662475i	$-0.769506\pi$
$397$	−29.8508	−1.49817	−0.749084	+	0.662475i	$0.769506\pi$
$398$	40.1926	2.01467
$399$	0	0
$400$	0	0
$401$	24.9176	1.24432	0.622162	−	0.782889i	$-0.286255\pi$
$401$	24.9176	1.24432	0.622162	+	0.782889i	$0.286255\pi$
$402$	0	0
$403$	−2.95875	−0.147386
$404$	24.0310	1.19559
$405$	0	0
$406$	−6.87107	−0.341005
$407$	6.03955	0.299369
$408$	0	0
$409$	13.8085	0.682787	0.341393	−	0.939920i	$-0.389101\pi$
$409$	13.8085	0.682787	0.341393	+	0.939920i	$0.389101\pi$
$410$	0	0
$411$	0	0
$412$	99.2960	4.89196
$413$	−5.28182	−0.259902
$414$	0	0
$415$	0	0
$416$	89.1276	4.36984
$417$	0	0
$418$	−14.4999	−0.709211
$419$	21.7799	1.06402	0.532009	−	0.846738i	$-0.321437\pi$
$419$	21.7799	1.06402	0.532009	+	0.846738i	$0.321437\pi$
$420$	0	0
$421$	3.23758	0.157790	0.0788949	−	0.996883i	$-0.474861\pi$
$421$	3.23758	0.157790	0.0788949	+	0.996883i	$0.474861\pi$
$422$	32.5638	1.58518
$423$	0	0
$424$	−54.9660	−2.66938
$425$	0	0
$426$	0	0
$427$	−5.53452	−0.267834
$428$	28.3300	1.36938
$429$	0	0
$430$	0	0
$431$	12.6466	0.609167	0.304583	−	0.952486i	$-0.401483\pi$
$431$	12.6466	0.609167	0.304583	+	0.952486i	$0.401483\pi$
$432$	0	0
$433$	−31.9914	−1.53741	−0.768705	−	0.639604i	$-0.779098\pi$
$433$	−31.9914	−1.53741	−0.768705	+	0.639604i	$0.779098\pi$
$434$	2.91579	0.139963
$435$	0	0
$436$	53.7305	2.57322
$437$	−4.32212	−0.206755
$438$	0	0
$439$	−13.8432	−0.660701	−0.330351	−	0.943858i	$-0.607167\pi$
$439$	−13.8432	−0.660701	−0.330351	+	0.943858i	$0.607167\pi$
$440$	0	0
$441$	0	0
$442$	52.1926	2.48255
$443$	−20.3479	−0.966759	−0.483379	−	0.875411i	$-0.660591\pi$
$443$	−20.3479	−0.966759	−0.483379	+	0.875411i	$0.660591\pi$
$444$	0	0
$445$	0	0
$446$	15.2750	0.723293
$447$	0	0
$448$	−49.4314	−2.33542
$449$	−8.71640	−0.411353	−0.205676	−	0.978620i	$-0.565939\pi$
$449$	−8.71640	−0.411353	−0.205676	+	0.978620i	$0.565939\pi$
$450$	0	0
$451$	8.93886	0.420915
$452$	20.2998	0.954822
$453$	0	0
$454$	22.3280	1.04791
$455$	0	0
$456$	0	0
$457$	0.0311855	0.00145879	0.000729397	−	1.00000i	$-0.499768\pi$
$457$	0.0311855	0.00145879	0.000729397		1.00000i	$0.499768\pi$
$458$	−61.6611	−2.88123
$459$	0	0
$460$	0	0
$461$	29.4882	1.37340	0.686700	−	0.726941i	$-0.259059\pi$
$461$	29.4882	1.37340	0.686700	+	0.726941i	$0.259059\pi$
$462$	0	0
$463$	19.1547	0.890196	0.445098	−	0.895482i	$-0.353169\pi$
$463$	19.1547	0.890196	0.445098	+	0.895482i	$0.353169\pi$
$464$	−43.4547	−2.01734
$465$	0	0
$466$	48.3671	2.24056
$467$	−23.9367	−1.10766	−0.553829	−	0.832630i	$-0.686834\pi$
$467$	−23.9367	−1.10766	−0.553829	+	0.832630i	$0.686834\pi$
$468$	0	0
$469$	−8.99830	−0.415503
$470$	0	0
$471$	0	0
$472$	−54.5672	−2.51166
$473$	−19.9570	−0.917626
$474$	0	0
$475$	0	0
$476$	−38.3163	−1.75622
$477$	0	0
$478$	−59.7752	−2.73406
$479$	−21.4039	−0.977968	−0.488984	−	0.872293i	$-0.662632\pi$
$479$	−21.4039	−0.977968	−0.488984	+	0.872293i	$0.662632\pi$
$480$	0	0
$481$	3.84969	0.175531
$482$	11.6487	0.530583
$483$	0	0
$484$	61.6064	2.80029
$485$	0	0
$486$	0	0
$487$	−26.0763	−1.18163	−0.590815	−	0.806807i	$-0.701194\pi$
$487$	−26.0763	−1.18163	−0.590815	+	0.806807i	$0.701194\pi$
$488$	−57.1778	−2.58832
$489$	0	0
$490$	0	0
$491$	11.9367	0.538696	0.269348	−	0.963043i	$-0.413192\pi$
$491$	11.9367	0.538696	0.269348	+	0.963043i	$0.413192\pi$
$492$	0	0
$493$	−14.8446	−0.668567
$494$	−9.24241	−0.415836
$495$	0	0
$496$	18.4404	0.827997
$497$	−4.04295	−0.181351
$498$	0	0
$499$	16.5394	0.740406	0.370203	−	0.928951i	$-0.379288\pi$
$499$	16.5394	0.740406	0.370203	+	0.928951i	$0.379288\pi$
$500$	0	0
$501$	0	0
$502$	16.1547	0.721018
$503$	16.6418	0.742021	0.371011	−	0.928629i	$-0.379011\pi$
$503$	16.6418	0.742021	0.371011	+	0.928629i	$0.379011\pi$
$504$	0	0
$505$	0	0
$506$	50.3624	2.23888
$507$	0	0
$508$	−75.1970	−3.33633
$509$	43.0388	1.90766	0.953831	−	0.300345i	$-0.0971018\pi$
$509$	43.0388	1.90766	0.953831	+	0.300345i	$0.0971018\pi$
$510$	0	0
$511$	12.6762	0.560761
$512$	−158.749	−7.01579
$513$	0	0
$514$	62.8430	2.77188
$515$	0	0
$516$	0	0
$517$	15.3887	0.676795
$518$	−3.79381	−0.166690
$519$	0	0
$520$	0	0
$521$	−37.2138	−1.63037	−0.815184	−	0.579203i	$-0.803364\pi$
$521$	−37.2138	−1.63037	−0.815184	+	0.579203i	$0.803364\pi$
$522$	0	0
$523$	23.4085	1.02358	0.511791	−	0.859110i	$-0.328982\pi$
$523$	23.4085	1.02358	0.511791	+	0.859110i	$0.328982\pi$
$524$	−90.3500	−3.94696
$525$	0	0
$526$	−28.5360	−1.24423
$527$	6.29942	0.274407
$528$	0	0
$529$	−7.98795	−0.347302
$530$	0	0
$531$	0	0
$532$	6.78516	0.294174
$533$	5.69776	0.246797
$534$	0	0
$535$	0	0
$536$	−92.9626	−4.01537
$537$	0	0
$538$	78.6608	3.39131
$539$	27.4599	1.18278
$540$	0	0
$541$	2.38298	0.102452	0.0512261	−	0.998687i	$-0.483687\pi$
$541$	2.38298	0.102452	0.0512261	+	0.998687i	$0.483687\pi$
$542$	−0.120285	−0.00516669
$543$	0	0
$544$	−189.760	−8.13590
$545$	0	0
$546$	0	0
$547$	21.4400	0.916709	0.458355	−	0.888769i	$-0.348439\pi$
$547$	21.4400	0.916709	0.458355	+	0.888769i	$0.348439\pi$
$548$	−96.8140	−4.13569
$549$	0	0
$550$	0	0
$551$	2.62872	0.111987
$552$	0	0
$553$	7.41764	0.315430
$554$	−61.1265	−2.59702
$555$	0	0
$556$	125.744	5.33275
$557$	−12.5276	−0.530813	−0.265407	−	0.964137i	$-0.585506\pi$
$557$	−12.5276	−0.530813	−0.265407	+	0.964137i	$0.585506\pi$
$558$	0	0
$559$	−12.7209	−0.538037
$560$	0	0
$561$	0	0
$562$	−25.7741	−1.08722
$563$	10.7361	0.452472	0.226236	−	0.974073i	$-0.427358\pi$
$563$	10.7361	0.452472	0.226236	+	0.974073i	$0.427358\pi$
$564$	0	0
$565$	0	0
$566$	−19.1907	−0.806647
$567$	0	0
$568$	−41.7683	−1.75256
$569$	−1.00783	−0.0422504	−0.0211252	−	0.999777i	$-0.506725\pi$
$569$	−1.00783	−0.0422504	−0.0211252	+	0.999777i	$0.506725\pi$
$570$	0	0
$571$	−9.88631	−0.413729	−0.206865	−	0.978370i	$-0.566326\pi$
$571$	−9.88631	−0.413729	−0.206865	+	0.978370i	$0.566326\pi$
$572$	80.2270	3.35446
$573$	0	0
$574$	−5.61504	−0.234367
$575$	0	0
$576$	0	0
$577$	−22.1564	−0.922384	−0.461192	−	0.887300i	$-0.652578\pi$
$577$	−22.1564	−0.922384	−0.461192	+	0.887300i	$0.652578\pi$
$578$	−63.5178	−2.64199
$579$	0	0
$580$	0	0
$581$	7.77414	0.322526
$582$	0	0
$583$	−23.7179	−0.982295
$584$	130.959	5.41913
$585$	0	0
$586$	46.5129	1.92143
$587$	16.7526	0.691452	0.345726	−	0.938336i	$-0.387633\pi$
$587$	16.7526	0.691452	0.345726	+	0.938336i	$0.387633\pi$
$588$	0	0
$589$	−1.11552	−0.0459642
$590$	0	0
$591$	0	0
$592$	−23.9932	−0.986114
$593$	−21.0526	−0.864525	−0.432262	−	0.901748i	$-0.642285\pi$
$593$	−21.0526	−0.864525	−0.432262	+	0.901748i	$0.642285\pi$
$594$	0	0
$595$	0	0
$596$	−121.244	−4.96637
$597$	0	0
$598$	32.1017	1.31274
$599$	24.1994	0.988760	0.494380	−	0.869246i	$-0.335395\pi$
$599$	24.1994	0.988760	0.494380	+	0.869246i	$0.335395\pi$
$600$	0	0
$601$	−37.0275	−1.51038	−0.755192	−	0.655504i	$-0.772456\pi$
$601$	−37.0275	−1.51038	−0.755192	+	0.655504i	$0.772456\pi$
$602$	12.5362	0.510938
$603$	0	0
$604$	−107.748	−4.38419
$605$	0	0
$606$	0	0
$607$	10.9011	0.442461	0.221231	−	0.975222i	$-0.428993\pi$
$607$	10.9011	0.442461	0.221231	+	0.975222i	$0.428993\pi$
$608$	33.6033	1.36279
$609$	0	0
$610$	0	0
$611$	9.80898	0.396829
$612$	0	0
$613$	−17.8882	−0.722497	−0.361249	−	0.932469i	$-0.617649\pi$
$613$	−17.8882	−0.722497	−0.361249	+	0.932469i	$0.617649\pi$
$614$	47.4796	1.91612
$615$	0	0
$616$	−51.9932	−2.09487
$617$	−39.3548	−1.58436	−0.792182	−	0.610285i	$-0.791055\pi$
$617$	−39.3548	−1.58436	−0.792182	+	0.610285i	$0.791055\pi$
$618$	0	0
$619$	19.5429	0.785495	0.392747	−	0.919646i	$-0.371525\pi$
$619$	19.5429	0.785495	0.392747	+	0.919646i	$0.371525\pi$
$620$	0	0
$621$	0	0
$622$	−1.78863	−0.0717175
$623$	−12.6615	−0.507271
$624$	0	0
$625$	0	0
$626$	−50.7243	−2.02735
$627$	0	0
$628$	−32.9666	−1.31551
$629$	−8.19632	−0.326809
$630$	0	0
$631$	−33.4486	−1.33157	−0.665784	−	0.746145i	$-0.731903\pi$
$631$	−33.4486	−1.33157	−0.665784	+	0.746145i	$0.731903\pi$
$632$	76.6325	3.04828
$633$	0	0
$634$	14.1815	0.563219
$635$	0	0
$636$	0	0
$637$	17.5033	0.693507
$638$	−30.6305	−1.21267
$639$	0	0
$640$	0	0
$641$	−7.13506	−0.281818	−0.140909	−	0.990023i	$-0.545002\pi$
$641$	−7.13506	−0.281818	−0.140909	+	0.990023i	$0.545002\pi$
$642$	0	0
$643$	−21.1141	−0.832660	−0.416330	−	0.909214i	$-0.636684\pi$
$643$	−21.1141	−0.832660	−0.416330	+	0.909214i	$0.636684\pi$
$644$	−23.5669	−0.928666
$645$	0	0
$646$	19.6779	0.774216
$647$	−1.08185	−0.0425320	−0.0212660	−	0.999774i	$-0.506770\pi$
$647$	−1.08185	−0.0425320	−0.0212660	+	0.999774i	$0.506770\pi$
$648$	0	0
$649$	−23.5458	−0.924254
$650$	0	0
$651$	0	0
$652$	94.6291	3.70596
$653$	47.5112	1.85926	0.929629	−	0.368497i	$-0.120127\pi$
$653$	47.5112	1.85926	0.929629	+	0.368497i	$0.120127\pi$
$654$	0	0
$655$	0	0
$656$	−35.5112	−1.38648
$657$	0	0
$658$	−9.66657	−0.376842
$659$	24.3048	0.946782	0.473391	−	0.880852i	$-0.343030\pi$
$659$	24.3048	0.946782	0.473391	+	0.880852i	$0.343030\pi$
$660$	0	0
$661$	1.23308	0.0479613	0.0239806	−	0.999712i	$-0.492366\pi$
$661$	1.23308	0.0479613	0.0239806	+	0.999712i	$0.492366\pi$
$662$	56.1017	2.18045
$663$	0	0
$664$	80.3156	3.11685
$665$	0	0
$666$	0	0
$667$	−9.13035	−0.353529
$668$	30.0791	1.16380
$669$	0	0
$670$	0	0
$671$	−24.6723	−0.952464
$672$	0	0
$673$	11.0242	0.424951	0.212476	−	0.977166i	$-0.431847\pi$
$673$	11.0242	0.424951	0.212476	+	0.977166i	$0.431847\pi$
$674$	−13.6297	−0.524998
$675$	0	0
$676$	−24.8020	−0.953925
$677$	−39.3994	−1.51424	−0.757122	−	0.653274i	$-0.773395\pi$
$677$	−39.3994	−1.51424	−0.757122	+	0.653274i	$0.773395\pi$
$678$	0	0
$679$	−0.953571	−0.0365947
$680$	0	0
$681$	0	0
$682$	12.9983	0.497731
$683$	42.8213	1.63851	0.819256	−	0.573428i	$-0.194387\pi$
$683$	42.8213	1.63851	0.819256	+	0.573428i	$0.194387\pi$
$684$	0	0
$685$	0	0
$686$	−37.6598	−1.43786
$687$	0	0
$688$	79.2829	3.02263
$689$	−15.1181	−0.575954
$690$	0	0
$691$	−34.1911	−1.30069	−0.650346	−	0.759638i	$-0.725376\pi$
$691$	−34.1911	−1.30069	−0.650346	+	0.759638i	$0.725376\pi$
$692$	62.8670	2.38984
$693$	0	0
$694$	28.3445	1.07594
$695$	0	0
$696$	0	0
$697$	−12.1310	−0.459495
$698$	51.3136	1.94225
$699$	0	0
$700$	0	0
$701$	31.6598	1.19577	0.597886	−	0.801581i	$-0.296007\pi$
$701$	31.6598	1.19577	0.597886	+	0.801581i	$0.296007\pi$
$702$	0	0
$703$	1.45143	0.0547417
$704$	−220.360	−8.30514
$705$	0	0
$706$	40.1135	1.50969
$707$	−4.28353	−0.161099
$708$	0	0
$709$	−19.8198	−0.744349	−0.372174	−	0.928163i	$-0.621388\pi$
$709$	−19.8198	−0.744349	−0.372174	+	0.928163i	$0.621388\pi$
$710$	0	0
$711$	0	0
$712$	−130.807	−4.90221
$713$	3.87454	0.145103
$714$	0	0
$715$	0	0
$716$	60.0158	2.24290
$717$	0	0
$718$	34.4984	1.28747
$719$	−25.7339	−0.959713	−0.479856	−	0.877347i	$-0.659311\pi$
$719$	−25.7339	−0.959713	−0.479856	+	0.877347i	$0.659311\pi$
$720$	0	0
$721$	−17.6995	−0.659165
$722$	49.7206	1.85041
$723$	0	0
$724$	14.8154	0.550610
$725$	0	0
$726$	0	0
$727$	−24.6909	−0.915734	−0.457867	−	0.889021i	$-0.651386\pi$
$727$	−24.6909	−0.915734	−0.457867	+	0.889021i	$0.651386\pi$
$728$	−33.1412	−1.22829
$729$	0	0
$730$	0	0
$731$	27.0839	1.00173
$732$	0	0
$733$	−23.0265	−0.850505	−0.425252	−	0.905075i	$-0.639815\pi$
$733$	−23.0265	−0.850505	−0.425252	+	0.905075i	$0.639815\pi$
$734$	68.5242	2.52928
$735$	0	0
$736$	−116.714	−4.30215
$737$	−40.1135	−1.47760
$738$	0	0
$739$	−27.2176	−1.00121	−0.500607	−	0.865675i	$-0.666890\pi$
$739$	−27.2176	−1.00121	−0.500607	+	0.865675i	$0.666890\pi$
$740$	0	0
$741$	0	0
$742$	14.8986	0.546946
$743$	−23.0434	−0.845380	−0.422690	−	0.906274i	$-0.638914\pi$
$743$	−23.0434	−0.845380	−0.422690	+	0.906274i	$0.638914\pi$
$744$	0	0
$745$	0	0
$746$	74.2846	2.71975
$747$	0	0
$748$	−170.810	−6.24543
$749$	−5.04983	−0.184517
$750$	0	0
$751$	−12.5480	−0.457883	−0.228941	−	0.973440i	$-0.573526\pi$
$751$	−12.5480	−0.457883	−0.228941	+	0.973440i	$0.573526\pi$
$752$	−61.1344	−2.22934
$753$	0	0
$754$	−19.5243	−0.711033
$755$	0	0
$756$	0	0
$757$	0.441361	0.0160415	0.00802077	−	0.999968i	$-0.497447\pi$
$757$	0.441361	0.0160415	0.00802077	+	0.999968i	$0.497447\pi$
$758$	−35.1513	−1.27675
$759$	0	0
$760$	0	0
$761$	23.0903	0.837024	0.418512	−	0.908211i	$-0.362552\pi$
$761$	23.0903	0.837024	0.418512	+	0.908211i	$0.362552\pi$
$762$	0	0
$763$	−9.57747	−0.346728
$764$	25.9966	0.940524
$765$	0	0
$766$	−32.4501	−1.17247
$767$	−15.0084	−0.541923
$768$	0	0
$769$	−16.5906	−0.598272	−0.299136	−	0.954210i	$-0.596699\pi$
$769$	−16.5906	−0.598272	−0.299136	+	0.954210i	$0.596699\pi$
$770$	0	0
$771$	0	0
$772$	15.8481	0.570384
$773$	50.6033	1.82008	0.910038	−	0.414525i	$-0.136052\pi$
$773$	50.6033	1.82008	0.910038	+	0.414525i	$0.136052\pi$
$774$	0	0
$775$	0	0
$776$	−9.85147	−0.353647
$777$	0	0
$778$	91.2300	3.27075
$779$	2.14819	0.0769670
$780$	0	0
$781$	−18.0231	−0.644916
$782$	−68.3472	−2.44409
$783$	0	0
$784$	−109.089	−3.89605
$785$	0	0
$786$	0	0
$787$	−42.9038	−1.52936	−0.764678	−	0.644413i	$-0.777102\pi$
$787$	−42.9038	−1.52936	−0.764678	+	0.644413i	$0.777102\pi$
$788$	−29.4486	−1.04906
$789$	0	0
$790$	0	0
$791$	−3.61844	−0.128657
$792$	0	0
$793$	−15.7265	−0.558463
$794$	83.5903	2.96651
$795$	0	0
$796$	−83.8440	−2.97177
$797$	−33.5707	−1.18913	−0.594567	−	0.804046i	$-0.702676\pi$
$797$	−33.5707	−1.18913	−0.594567	+	0.804046i	$0.702676\pi$
$798$	0	0
$799$	−20.8841	−0.738828
$800$	0	0
$801$	0	0
$802$	−69.7760	−2.46388
$803$	56.5091	1.99416
$804$	0	0
$805$	0	0
$806$	8.28530	0.291837
$807$	0	0
$808$	−44.2536	−1.55684
$809$	−35.3774	−1.24380	−0.621902	−	0.783095i	$-0.713640\pi$
$809$	−35.3774	−1.24380	−0.621902	+	0.783095i	$0.713640\pi$
$810$	0	0
$811$	54.9833	1.93072	0.965362	−	0.260916i	$-0.0840245\pi$
$811$	54.9833	1.93072	0.965362	+	0.260916i	$0.0840245\pi$
$812$	14.3334	0.503005
$813$	0	0
$814$	−16.9124	−0.592779
$815$	0	0
$816$	0	0
$817$	−4.79609	−0.167794
$818$	−38.6676	−1.35198
$819$	0	0
$820$	0	0
$821$	−47.1995	−1.64727	−0.823636	−	0.567118i	$-0.808058\pi$
$821$	−47.1995	−1.64727	−0.823636	+	0.567118i	$0.808058\pi$
$822$	0	0
$823$	31.1661	1.08638	0.543192	−	0.839609i	$-0.317216\pi$
$823$	31.1661	1.08638	0.543192	+	0.839609i	$0.317216\pi$
$824$	−182.856	−6.37009
$825$	0	0
$826$	14.7905	0.514629
$827$	2.62427	0.0912548	0.0456274	−	0.998959i	$-0.485471\pi$
$827$	2.62427	0.0912548	0.0456274	+	0.998959i	$0.485471\pi$
$828$	0	0
$829$	30.1876	1.04846	0.524230	−	0.851577i	$-0.324353\pi$
$829$	30.1876	1.04846	0.524230	+	0.851577i	$0.324353\pi$
$830$	0	0
$831$	0	0
$832$	−140.461	−4.86960
$833$	−37.2661	−1.29119
$834$	0	0
$835$	0	0
$836$	30.2475	1.04613
$837$	0	0
$838$	−60.9897	−2.10685
$839$	9.75555	0.336799	0.168399	−	0.985719i	$-0.446140\pi$
$839$	9.75555	0.336799	0.168399	+	0.985719i	$0.446140\pi$
$840$	0	0
$841$	−23.4469	−0.808514
$842$	−9.06610	−0.312438
$843$	0	0
$844$	−67.9299	−2.33824
$845$	0	0
$846$	0	0
$847$	−10.9814	−0.377324
$848$	94.2236	3.23565
$849$	0	0
$850$	0	0
$851$	−5.04125	−0.172812
$852$	0	0
$853$	−39.4547	−1.35090	−0.675452	−	0.737404i	$-0.736052\pi$
$853$	−39.4547	−1.35090	−0.675452	+	0.737404i	$0.736052\pi$
$854$	15.4982	0.530336
$855$	0	0
$856$	−52.1705	−1.78315
$857$	−10.6006	−0.362110	−0.181055	−	0.983473i	$-0.557951\pi$
$857$	−10.6006	−0.362110	−0.181055	+	0.983473i	$0.557951\pi$
$858$	0	0
$859$	−30.1660	−1.02925	−0.514625	−	0.857416i	$-0.672069\pi$
$859$	−30.1660	−1.02925	−0.514625	+	0.857416i	$0.672069\pi$
$860$	0	0
$861$	0	0
$862$	−35.4140	−1.20621
$863$	11.7962	0.401546	0.200773	−	0.979638i	$-0.435655\pi$
$863$	11.7962	0.401546	0.200773	+	0.979638i	$0.435655\pi$
$864$	0	0
$865$	0	0
$866$	89.5847	3.04421
$867$	0	0
$868$	−6.08251	−0.206454
$869$	33.0670	1.12172
$870$	0	0
$871$	−25.5689	−0.866369
$872$	−98.9461	−3.35074
$873$	0	0
$874$	12.1031	0.409394
$875$	0	0
$876$	0	0
$877$	−12.3531	−0.417134	−0.208567	−	0.978008i	$-0.566880\pi$
$877$	−12.3531	−0.417134	−0.208567	+	0.978008i	$0.566880\pi$
$878$	38.7648	1.30825
$879$	0	0
$880$	0	0
$881$	−28.2740	−0.952575	−0.476288	−	0.879290i	$-0.658018\pi$
$881$	−28.2740	−0.952575	−0.476288	+	0.879290i	$0.658018\pi$
$882$	0	0
$883$	−14.7328	−0.495798	−0.247899	−	0.968786i	$-0.579740\pi$
$883$	−14.7328	−0.495798	−0.247899	+	0.968786i	$0.579740\pi$
$884$	−108.877	−3.66192
$885$	0	0
$886$	56.9797	1.91427
$887$	26.9237	0.904009	0.452005	−	0.892016i	$-0.350709\pi$
$887$	26.9237	0.904009	0.452005	+	0.892016i	$0.350709\pi$
$888$	0	0
$889$	13.4039	0.449552
$890$	0	0
$891$	0	0
$892$	−31.8645	−1.06690
$893$	3.69822	0.123756
$894$	0	0
$895$	0	0
$896$	75.6893	2.52860
$897$	0	0
$898$	24.4083	0.814516
$899$	−2.35650	−0.0785937
$900$	0	0
$901$	32.1877	1.07233
$902$	−25.0313	−0.833450
$903$	0	0
$904$	−37.3826	−1.24333
$905$	0	0
$906$	0	0
$907$	25.7869	0.856239	0.428119	−	0.903722i	$-0.359176\pi$
$907$	25.7869	0.856239	0.428119	+	0.903722i	$0.359176\pi$
$908$	−46.5775	−1.54573
$909$	0	0
$910$	0	0
$911$	−24.4130	−0.808839	−0.404419	−	0.914574i	$-0.632526\pi$
$911$	−24.4130	−0.808839	−0.404419	+	0.914574i	$0.632526\pi$
$912$	0	0
$913$	34.6563	1.14696
$914$	−0.0873278	−0.00288855
$915$	0	0
$916$	128.628	4.25000
$917$	16.1049	0.531831
$918$	0	0
$919$	−6.86800	−0.226554	−0.113277	−	0.993563i	$-0.536135\pi$
$919$	−6.86800	−0.226554	−0.113277	+	0.993563i	$0.536135\pi$
$920$	0	0
$921$	0	0
$922$	−82.5749	−2.71946
$923$	−11.4882	−0.378137
$924$	0	0
$925$	0	0
$926$	−53.6385	−1.76267
$927$	0	0
$928$	70.9858	2.33022
$929$	−12.6470	−0.414934	−0.207467	−	0.978242i	$-0.566522\pi$
$929$	−12.6470	−0.414934	−0.207467	+	0.978242i	$0.566522\pi$
$930$	0	0
$931$	6.59918	0.216279
$932$	−100.897	−3.30498
$933$	0	0
$934$	67.0293	2.19326
$935$	0	0
$936$	0	0
$937$	−42.8873	−1.40107	−0.700534	−	0.713619i	$-0.747055\pi$
$937$	−42.8873	−1.40107	−0.700534	+	0.713619i	$0.747055\pi$
$938$	25.1977	0.822734
$939$	0	0
$940$	0	0
$941$	−24.2254	−0.789725	−0.394863	−	0.918740i	$-0.629208\pi$
$941$	−24.2254	−0.789725	−0.394863	+	0.918740i	$0.629208\pi$
$942$	0	0
$943$	−7.46133	−0.242974
$944$	93.5400	3.04447
$945$	0	0
$946$	55.8852	1.81698
$947$	−16.2682	−0.528647	−0.264323	−	0.964434i	$-0.585149\pi$
$947$	−16.2682	−0.528647	−0.264323	+	0.964434i	$0.585149\pi$
$948$	0	0
$949$	36.0197	1.16925
$950$	0	0
$951$	0	0
$952$	70.5604	2.28688
$953$	19.2891	0.624837	0.312418	−	0.949945i	$-0.398861\pi$
$953$	19.2891	0.624837	0.312418	+	0.949945i	$0.398861\pi$
$954$	0	0
$955$	0	0
$956$	124.694	4.03291
$957$	0	0
$958$	59.9367	1.93647
$959$	17.2571	0.557261
$960$	0	0
$961$	1.00000	0.0322581
$962$	−10.7802	−0.347567
$963$	0	0
$964$	−24.2998	−0.782644
$965$	0	0
$966$	0	0
$967$	−20.9745	−0.674496	−0.337248	−	0.941416i	$-0.609496\pi$
$967$	−20.9745	−0.674496	−0.337248	+	0.941416i	$0.609496\pi$
$968$	−113.450	−3.64641
$969$	0	0
$970$	0	0
$971$	16.2572	0.521718	0.260859	−	0.965377i	$-0.415994\pi$
$971$	16.2572	0.521718	0.260859	+	0.965377i	$0.415994\pi$
$972$	0	0
$973$	−22.4139	−0.718558
$974$	73.0207	2.33973
$975$	0	0
$976$	98.0151	3.13739
$977$	12.3649	0.395587	0.197794	−	0.980244i	$-0.436622\pi$
$977$	12.3649	0.395587	0.197794	+	0.980244i	$0.436622\pi$
$978$	0	0
$979$	−56.4435	−1.80394
$980$	0	0
$981$	0	0
$982$	−33.4260	−1.06667
$983$	39.4586	1.25853	0.629267	−	0.777189i	$-0.283355\pi$
$983$	39.4586	1.25853	0.629267	+	0.777189i	$0.283355\pi$
$984$	0	0
$985$	0	0
$986$	41.5689	1.32382
$987$	0	0
$988$	19.2802	0.613385
$989$	16.6583	0.529702
$990$	0	0
$991$	−55.5390	−1.76425	−0.882127	−	0.471011i	$-0.843889\pi$
$991$	−55.5390	−1.76425	−0.882127	+	0.471011i	$0.843889\pi$
$992$	−30.1234	−0.956420
$993$	0	0
$994$	11.3214	0.359092
$995$	0	0
$996$	0	0
$997$	−13.4120	−0.424762	−0.212381	−	0.977187i	$-0.568122\pi$
$997$	−13.4120	−0.424762	−0.212381	+	0.977187i	$0.568122\pi$
$998$	−46.3149	−1.46607
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bj.1.1		4
3.2	odd	2		775.2.a.g.1.4		4
5.4	even	2		1395.2.a.m.1.4		4
15.2	even	4		775.2.b.e.249.8		8
15.8	even	4		775.2.b.e.249.1		8
15.14	odd	2		155.2.a.d.1.1	✓	4
60.59	even	2		2480.2.a.z.1.2		4
105.104	even	2		7595.2.a.q.1.1		4
120.29	odd	2		9920.2.a.ch.1.2		4
120.59	even	2		9920.2.a.cd.1.3		4
465.464	even	2		4805.2.a.j.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.d.1.1	✓	4	15.14	odd	2
775.2.a.g.1.4		4	3.2	odd	2
775.2.b.e.249.1		8	15.8	even	4
775.2.b.e.249.8		8	15.2	even	4
1395.2.a.m.1.4		4	5.4	even	2
2480.2.a.z.1.2		4	60.59	even	2
4805.2.a.j.1.1		4	465.464	even	2
6975.2.a.bj.1.1		4	1.1	even	1	trivial
7595.2.a.q.1.1		4	105.104	even	2
9920.2.a.cd.1.3		4	120.59	even	2
9920.2.a.ch.1.2		4	120.29	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80027 q^{2} +5.84153 q^{4} -1.04125 q^{7} -10.7573 q^{8} -4.64180 q^{11} -2.95875 q^{13} +2.91579 q^{14} +18.4404 q^{16} +6.29942 q^{17} -1.11552 q^{19} +12.9983 q^{22} +3.87454 q^{23} +8.28530 q^{26} -6.08251 q^{28} -2.35650 q^{29} +1.00000 q^{31} -30.1234 q^{32} -17.6401 q^{34} -1.30112 q^{37} +3.12376 q^{38} -1.92573 q^{41} +4.29942 q^{43} -27.1152 q^{44} -10.8498 q^{46} -3.31525 q^{47} -5.91579 q^{49} -17.2836 q^{52} +5.10964 q^{53} +11.2011 q^{56} +6.59884 q^{58} +5.07256 q^{59} +5.31525 q^{61} -2.80027 q^{62} +47.4730 q^{64} +8.64180 q^{67} +36.7982 q^{68} +3.88278 q^{71} -12.1740 q^{73} +3.64350 q^{74} -6.51634 q^{76} +4.83329 q^{77} -7.12376 q^{79} +5.39258 q^{82} -7.46614 q^{83} -12.0396 q^{86} +49.9333 q^{88} +12.1598 q^{89} +3.08080 q^{91} +22.6332 q^{92} +9.28360 q^{94} +0.915792 q^{97} +16.5658 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 9 q^{4} - 9 q^{8} + 6 q^{11} - 16 q^{13} - 8 q^{14} + 11 q^{16} + q^{17} + 5 q^{19} + 24 q^{22} + 12 q^{26} - 16 q^{28} - 6 q^{29} + 4 q^{31} - 29 q^{32} - 18 q^{34} - 9 q^{37} - 13 q^{41}+ \cdots + 19 q^{98}+O(q^{100}) \) 4 * q - q^2 + 9 * q^4 - 9 * q^8 + 6 * q^11 - 16 * q^13 - 8 * q^14 + 11 * q^16 + q^17 + 5 * q^19 + 24 * q^22 + 12 * q^26 - 16 * q^28 - 6 * q^29 + 4 * q^31 - 29 * q^32 - 18 * q^34 - 9 * q^37 - 13 * q^41 - 7 * q^43 - 20 * q^44 - 26 * q^46 - 14 * q^47 - 4 * q^49 - 20 * q^52 + 11 * q^53 + 4 * q^56 - 22 * q^58 - 13 * q^59 + 22 * q^61 - q^62 + 47 * q^64 + 10 * q^67 + 30 * q^68 - 3 * q^71 - 9 * q^73 + 18 * q^74 + 14 * q^76 + 8 * q^77 - 16 * q^79 - 6 * q^82 - 17 * q^83 - 16 * q^86 + 44 * q^88 + 12 * q^89 - 24 * q^91 - 10 * q^92 - 12 * q^94 - 16 * q^97 + 19 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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