LMFDB - Embedded newform 6975.2.a.cf.1.3

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 = [8,0,0,16,0,0,-4,0,0,0,0,0,-8,0,0,12,0] 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:	$8$
Coefficient field:	8.8.116450197504.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 58x^{4} - 62x^{2} + 1 $ x^8 - 14x^6 + 58x^4 - 62*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34751$ 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)

$f(q)$	$=$	$q-1.77541 q^{2} +1.15208 q^{4} -2.51081 q^{7} +1.50540 q^{8} -3.40053 q^{11} -1.11324 q^{13} +4.45772 q^{14} -4.97687 q^{16} +2.56259 q^{17} +6.79185 q^{19} +6.03733 q^{22} -4.06799 q^{23} +1.97646 q^{26} -2.89267 q^{28} +1.89513 q^{29} -1.00000 q^{31} +5.82519 q^{32} -4.54965 q^{34} -3.62405 q^{37} -12.0583 q^{38} +3.28081 q^{41} -9.35873 q^{43} -3.91770 q^{44} +7.22235 q^{46} +9.51394 q^{47} -0.695831 q^{49} -1.28255 q^{52} -1.72466 q^{53} -3.77977 q^{56} -3.36463 q^{58} -7.07107 q^{59} +13.7903 q^{61} +1.77541 q^{62} -0.388373 q^{64} -5.54375 q^{67} +2.95232 q^{68} +13.8337 q^{71} -1.28255 q^{73} +6.43418 q^{74} +7.82479 q^{76} +8.53808 q^{77} +0.549654 q^{79} -5.82479 q^{82} +9.53412 q^{83} +16.6156 q^{86} -5.11915 q^{88} +9.76575 q^{89} +2.79514 q^{91} -4.68667 q^{92} -16.8912 q^{94} -14.5206 q^{97} +1.23539 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 4 q^{7} - 8 q^{13} + 12 q^{16} - 28 q^{22} - 36 q^{28} - 8 q^{31} - 28 q^{34} - 12 q^{37} - 52 q^{43} - 16 q^{46} + 8 q^{49} - 56 q^{52} - 16 q^{58} - 28 q^{61} + 48 q^{64} - 24 q^{67}+ \cdots - 44 q^{97}+O(q^{100}) $ 8 * q + 16 * q^4 - 4 * q^7 - 8 * q^13 + 12 * q^16 - 28 * q^22 - 36 * q^28 - 8 * q^31 - 28 * q^34 - 12 * q^37 - 52 * q^43 - 16 * q^46 + 8 * q^49 - 56 * q^52 - 16 * q^58 - 28 * q^61 + 48 * q^64 - 24 * q^67 - 56 * q^73 + 4 * q^76 - 4 * q^79 + 12 * q^82 - 52 * q^88 - 8 * q^91 - 64 * q^94 - 44 * q^97

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 $

$2$	−1.77541	−1.25541	−0.627703	−	0.778453i	$-0.716005\pi$
$2$	−1.77541	−1.25541	−0.627703	+	0.778453i	$0.716005\pi$
$3$	0	0
$3$	0	0
$4$	1.15208	0.576042
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.51081	−0.948997	−0.474499	−	0.880256i	$-0.657371\pi$
$7$	−2.51081	−0.948997	−0.474499	+	0.880256i	$0.657371\pi$
$8$	1.50540	0.532239
$9$	0	0
$10$	0	0
$11$	−3.40053	−1.02530	−0.512649	−	0.858598i	$-0.671336\pi$
$11$	−3.40053	−1.02530	−0.512649	+	0.858598i	$0.671336\pi$
$12$	0	0
$13$	−1.11324	−0.308758	−0.154379	−	0.988012i	$-0.549338\pi$
$13$	−1.11324	−0.308758	−0.154379	+	0.988012i	$0.549338\pi$
$14$	4.45772	1.19138
$15$	0	0
$16$	−4.97687	−1.24422
$17$	2.56259	0.621520	0.310760	−	0.950488i	$-0.399416\pi$
$17$	2.56259	0.621520	0.310760	+	0.950488i	$0.399416\pi$
$18$	0	0
$19$	6.79185	1.55816	0.779079	−	0.626926i	$-0.215687\pi$
$19$	6.79185	1.55816	0.779079	+	0.626926i	$0.215687\pi$
$20$	0	0
$21$	0	0
$22$	6.03733	1.28716
$23$	−4.06799	−0.848235	−0.424117	−	0.905607i	$-0.639416\pi$
$23$	−4.06799	−0.848235	−0.424117	+	0.905607i	$0.639416\pi$
$24$	0	0
$25$	0	0
$26$	1.97646	0.387616
$27$	0	0
$28$	−2.89267	−0.546663
$29$	1.89513	0.351917	0.175958	−	0.984398i	$-0.443698\pi$
$29$	1.89513	0.351917	0.175958	+	0.984398i	$0.443698\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.82519	1.02976
$33$	0	0
$34$	−4.54965	−0.780259
$35$	0	0
$36$	0	0
$37$	−3.62405	−0.595790	−0.297895	−	0.954599i	$-0.596285\pi$
$37$	−3.62405	−0.595790	−0.297895	+	0.954599i	$0.596285\pi$
$38$	−12.0583	−1.95612
$39$	0	0
$40$	0	0
$41$	3.28081	0.512376	0.256188	−	0.966627i	$-0.417533\pi$
$41$	3.28081	0.512376	0.256188	+	0.966627i	$0.417533\pi$
$42$	0	0
$43$	−9.35873	−1.42719	−0.713596	−	0.700557i	$-0.752935\pi$
$43$	−9.35873	−1.42719	−0.713596	+	0.700557i	$0.752935\pi$
$44$	−3.91770	−0.590615
$45$	0	0
$46$	7.22235	1.06488
$47$	9.51394	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$47$	9.51394	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$48$	0	0
$49$	−0.695831	−0.0994044
$50$	0	0
$51$	0	0
$52$	−1.28255	−0.177857
$53$	−1.72466	−0.236900	−0.118450	−	0.992960i	$-0.537792\pi$
$53$	−1.72466	−0.236900	−0.118450	+	0.992960i	$0.537792\pi$
$54$	0	0
$55$	0	0
$56$	−3.77977	−0.505093
$57$	0	0
$58$	−3.36463	−0.441798
$59$	−7.07107	−0.920575	−0.460287	−	0.887770i	$-0.652254\pi$
$59$	−7.07107	−0.920575	−0.460287	+	0.887770i	$0.652254\pi$
$60$	0	0
$61$	13.7903	1.76567	0.882836	−	0.469681i	$-0.155631\pi$
$61$	13.7903	1.76567	0.882836	+	0.469681i	$0.155631\pi$
$62$	1.77541	0.225477
$63$	0	0
$64$	−0.388373	−0.0485467
$65$	0	0
$66$	0	0
$67$	−5.54375	−0.677276	−0.338638	−	0.940917i	$-0.609966\pi$
$67$	−5.54375	−0.677276	−0.338638	+	0.940917i	$0.609966\pi$
$68$	2.95232	0.358022
$69$	0	0
$70$	0	0
$71$	13.8337	1.64176	0.820881	−	0.571099i	$-0.193483\pi$
$71$	13.8337	1.64176	0.820881	+	0.571099i	$0.193483\pi$
$72$	0	0
$73$	−1.28255	−0.150111	−0.0750555	−	0.997179i	$-0.523913\pi$
$73$	−1.28255	−0.150111	−0.0750555	+	0.997179i	$0.523913\pi$
$74$	6.43418	0.747958
$75$	0	0
$76$	7.82479	0.897564
$77$	8.53808	0.973005
$78$	0	0
$79$	0.549654	0.0618409	0.0309204	−	0.999522i	$-0.490156\pi$
$79$	0.549654	0.0618409	0.0309204	+	0.999522i	$0.490156\pi$
$80$	0	0
$81$	0	0
$82$	−5.82479	−0.643240
$83$	9.53412	1.04651	0.523253	−	0.852177i	$-0.324718\pi$
$83$	9.53412	1.04651	0.523253	+	0.852177i	$0.324718\pi$
$84$	0	0
$85$	0	0
$86$	16.6156	1.79170
$87$	0	0
$88$	−5.11915	−0.545703
$89$	9.76575	1.03517	0.517584	−	0.855633i	$-0.326832\pi$
$89$	9.76575	1.03517	0.517584	+	0.855633i	$0.326832\pi$
$90$	0	0
$91$	2.79514	0.293010
$92$	−4.68667	−0.488619
$93$	0	0
$94$	−16.8912	−1.74219
$95$	0	0
$96$	0	0
$97$	−14.5206	−1.47435	−0.737173	−	0.675704i	$-0.763840\pi$
$97$	−14.5206	−1.47435	−0.737173	+	0.675704i	$0.763840\pi$
$98$	1.23539	0.124793
$99$	0	0
$100$	0	0
$101$	5.12518	0.509975	0.254987	−	0.966944i	$-0.417929\pi$
$101$	5.12518	0.509975	0.254987	+	0.966944i	$0.417929\pi$
$102$	0	0
$103$	2.51410	0.247722	0.123861	−	0.992300i	$-0.460472\pi$
$103$	2.51410	0.247722	0.123861	+	0.992300i	$0.460472\pi$
$104$	−1.67587	−0.164333
$105$	0	0
$106$	3.06197	0.297405
$107$	7.80749	0.754779	0.377389	−	0.926055i	$-0.376822\pi$
$107$	7.80749	0.754779	0.377389	+	0.926055i	$0.376822\pi$
$108$	0	0
$109$	0.477875	0.0457721	0.0228860	−	0.999738i	$-0.492715\pi$
$109$	0.477875	0.0457721	0.0228860	+	0.999738i	$0.492715\pi$
$110$	0	0
$111$	0	0
$112$	12.4960	1.18076
$113$	−16.2460	−1.52830	−0.764149	−	0.645040i	$-0.776841\pi$
$113$	−16.2460	−1.52830	−0.764149	+	0.645040i	$0.776841\pi$
$114$	0	0
$115$	0	0
$116$	2.18335	0.202719
$117$	0	0
$118$	12.5541	1.15569
$119$	−6.43418	−0.589820
$120$	0	0
$121$	0.563588	0.0512352
$122$	−24.4835	−2.21663
$123$	0	0
$124$	−1.15208	−0.103460
$125$	0	0
$126$	0	0
$127$	−0.925602	−0.0821339	−0.0410669	−	0.999156i	$-0.513076\pi$
$127$	−0.925602	−0.0821339	−0.0410669	+	0.999156i	$0.513076\pi$
$128$	−10.9609	−0.968813
$129$	0	0
$130$	0	0
$131$	17.7820	1.55362	0.776811	−	0.629734i	$-0.216836\pi$
$131$	17.7820	1.55362	0.776811	+	0.629734i	$0.216836\pi$
$132$	0	0
$133$	−17.0530	−1.47869
$134$	9.84243	0.850256
$135$	0	0
$136$	3.85772	0.330797
$137$	1.79362	0.153239	0.0766196	−	0.997060i	$-0.475587\pi$
$137$	1.79362	0.153239	0.0766196	+	0.997060i	$0.475587\pi$
$138$	0	0
$139$	6.32317	0.536324	0.268162	−	0.963374i	$-0.413584\pi$
$139$	6.32317	0.536324	0.268162	+	0.963374i	$0.413584\pi$
$140$	0	0
$141$	0	0
$142$	−24.5606	−2.06108
$143$	3.78561	0.316568
$144$	0	0
$145$	0	0
$146$	2.27705	0.188450
$147$	0	0
$148$	−4.17521	−0.343201
$149$	−15.7107	−1.28707	−0.643534	−	0.765418i	$-0.722532\pi$
$149$	−15.7107	−1.28707	−0.643534	+	0.765418i	$0.722532\pi$
$150$	0	0
$151$	−9.79185	−0.796849	−0.398425	−	0.917201i	$-0.630443\pi$
$151$	−9.79185	−0.796849	−0.398425	+	0.917201i	$0.630443\pi$
$152$	10.2244	0.829312
$153$	0	0
$154$	−15.1586	−1.22151
$155$	0	0
$156$	0	0
$157$	−15.3276	−1.22327	−0.611637	−	0.791139i	$-0.709489\pi$
$157$	−15.3276	−1.22327	−0.611637	+	0.791139i	$0.709489\pi$
$158$	−0.975861	−0.0776354
$159$	0	0
$160$	0	0
$161$	10.2140	0.804972
$162$	0	0
$163$	−1.94544	−0.152379	−0.0761894	−	0.997093i	$-0.524275\pi$
$163$	−1.94544	−0.152379	−0.0761894	+	0.997093i	$0.524275\pi$
$164$	3.77977	0.295150
$165$	0	0
$166$	−16.9270	−1.31379
$167$	14.4322	1.11680	0.558400	−	0.829572i	$-0.311415\pi$
$167$	14.4322	1.11680	0.558400	+	0.829572i	$0.311415\pi$
$168$	0	0
$169$	−11.7607	−0.904669
$170$	0	0
$171$	0	0
$172$	−10.7820	−0.822123
$173$	−11.6915	−0.888885	−0.444442	−	0.895807i	$-0.646598\pi$
$173$	−11.6915	−0.888885	−0.444442	+	0.895807i	$0.646598\pi$
$174$	0	0
$175$	0	0
$176$	16.9240	1.27569
$177$	0	0
$178$	−17.3382	−1.29955
$179$	19.1553	1.43174	0.715868	−	0.698236i	$-0.246031\pi$
$179$	19.1553	1.43174	0.715868	+	0.698236i	$0.246031\pi$
$180$	0	0
$181$	−21.3946	−1.59024	−0.795122	−	0.606449i	$-0.792593\pi$
$181$	−21.3946	−1.59024	−0.795122	+	0.606449i	$0.792593\pi$
$182$	−4.96252	−0.367846
$183$	0	0
$184$	−6.12395	−0.451463
$185$	0	0
$186$	0	0
$187$	−8.71416	−0.637243
$188$	10.9609	0.799403
$189$	0	0
$190$	0	0
$191$	−0.201050	−0.0145475	−0.00727373	−	0.999974i	$-0.502315\pi$
$191$	−0.201050	−0.0145475	−0.00727373	+	0.999974i	$0.502315\pi$
$192$	0	0
$193$	14.5822	1.04965	0.524825	−	0.851210i	$-0.324131\pi$
$193$	14.5822	1.04965	0.524825	+	0.851210i	$0.324131\pi$
$194$	25.7801	1.85090
$195$	0	0
$196$	−0.801656	−0.0572611
$197$	1.78006	0.126824	0.0634121	−	0.997987i	$-0.479802\pi$
$197$	1.78006	0.126824	0.0634121	+	0.997987i	$0.479802\pi$
$198$	0	0
$199$	21.4144	1.51803	0.759013	−	0.651075i	$-0.225682\pi$
$199$	21.4144	1.51803	0.759013	+	0.651075i	$0.225682\pi$
$200$	0	0
$201$	0	0
$202$	−9.09931	−0.640225
$203$	−4.75831	−0.333968
$204$	0	0
$205$	0	0
$206$	−4.46356	−0.310991
$207$	0	0
$208$	5.54046	0.384162
$209$	−23.0959	−1.59757
$210$	0	0
$211$	1.67572	0.115361	0.0576806	−	0.998335i	$-0.481629\pi$
$211$	1.67572	0.115361	0.0576806	+	0.998335i	$0.481629\pi$
$212$	−1.98695	−0.136464
$213$	0	0
$214$	−13.8615	−0.947553
$215$	0	0
$216$	0	0
$217$	2.51081	0.170445
$218$	−0.848424	−0.0574625
$219$	0	0
$220$	0	0
$221$	−2.85278	−0.191899
$222$	0	0
$223$	−13.6166	−0.911837	−0.455919	−	0.890021i	$-0.650689\pi$
$223$	−13.6166	−0.911837	−0.455919	+	0.890021i	$0.650689\pi$
$224$	−14.6260	−0.977238
$225$	0	0
$226$	28.8434	1.91863
$227$	12.1319	0.805225	0.402613	−	0.915371i	$-0.368102\pi$
$227$	12.1319	0.805225	0.402613	+	0.915371i	$0.368102\pi$
$228$	0	0
$229$	−14.6917	−0.970855	−0.485428	−	0.874277i	$-0.661336\pi$
$229$	−14.6917	−0.970855	−0.485428	+	0.874277i	$0.661336\pi$
$230$	0	0
$231$	0	0
$232$	2.85292	0.187304
$233$	2.48126	0.162553	0.0812764	−	0.996692i	$-0.474100\pi$
$233$	2.48126	0.162553	0.0812764	+	0.996692i	$0.474100\pi$
$234$	0	0
$235$	0	0
$236$	−8.14647	−0.530290
$237$	0	0
$238$	11.4233	0.740464
$239$	29.3310	1.89726	0.948632	−	0.316383i	$-0.102468\pi$
$239$	29.3310	1.89726	0.948632	+	0.316383i	$0.102468\pi$
$240$	0	0
$241$	−11.6673	−0.751556	−0.375778	−	0.926710i	$-0.622625\pi$
$241$	−11.6673	−0.751556	−0.375778	+	0.926710i	$0.622625\pi$
$242$	−1.00060	−0.0643210
$243$	0	0
$244$	15.8876	1.01710
$245$	0	0
$246$	0	0
$247$	−7.56097	−0.481093
$248$	−1.50540	−0.0955929
$249$	0	0
$250$	0	0
$251$	−12.0401	−0.759966	−0.379983	−	0.924994i	$-0.624070\pi$
$251$	−12.0401	−0.759966	−0.379983	+	0.924994i	$0.624070\pi$
$252$	0	0
$253$	13.8333	0.869693
$254$	1.64332	0.103111
$255$	0	0
$256$	20.2368	1.26480
$257$	−4.58981	−0.286304	−0.143152	−	0.989701i	$-0.545724\pi$
$257$	−4.58981	−0.286304	−0.143152	+	0.989701i	$0.545724\pi$
$258$	0	0
$259$	9.09931	0.565403
$260$	0	0
$261$	0	0
$262$	−31.5704	−1.95042
$263$	−27.8054	−1.71455	−0.857277	−	0.514855i	$-0.827846\pi$
$263$	−27.8054	−1.71455	−0.857277	+	0.514855i	$0.827846\pi$
$264$	0	0
$265$	0	0
$266$	30.2762	1.85635
$267$	0	0
$268$	−6.38686	−0.390140
$269$	−2.67766	−0.163260	−0.0816299	−	0.996663i	$-0.526013\pi$
$269$	−2.67766	−0.163260	−0.0816299	+	0.996663i	$0.526013\pi$
$270$	0	0
$271$	0.147957	0.00898772	0.00449386	−	0.999990i	$-0.498570\pi$
$271$	0.147957	0.00898772	0.00449386	+	0.999990i	$0.498570\pi$
$272$	−12.7537	−0.773306
$273$	0	0
$274$	−3.18441	−0.192377
$275$	0	0
$276$	0	0
$277$	21.7361	1.30599	0.652996	−	0.757361i	$-0.273512\pi$
$277$	21.7361	1.30599	0.652996	+	0.757361i	$0.273512\pi$
$278$	−11.2262	−0.673304
$279$	0	0
$280$	0	0
$281$	4.96133	0.295968	0.147984	−	0.988990i	$-0.452722\pi$
$281$	4.96133	0.295968	0.147984	+	0.988990i	$0.452722\pi$
$282$	0	0
$283$	−32.3990	−1.92592	−0.962959	−	0.269648i	$-0.913093\pi$
$283$	−32.3990	−1.92592	−0.962959	+	0.269648i	$0.913093\pi$
$284$	15.9376	0.945725
$285$	0	0
$286$	−6.72101	−0.397422
$287$	−8.23749	−0.486244
$288$	0	0
$289$	−10.4331	−0.613713
$290$	0	0
$291$	0	0
$292$	−1.47760	−0.0864702
$293$	−1.84437	−0.107749	−0.0538747	−	0.998548i	$-0.517157\pi$
$293$	−1.84437	−0.107749	−0.0538747	+	0.998548i	$0.517157\pi$
$294$	0	0
$295$	0	0
$296$	−5.45564	−0.317103
$297$	0	0
$298$	27.8929	1.61579
$299$	4.52865	0.261899
$300$	0	0
$301$	23.4980	1.35440
$302$	17.3846	1.00037
$303$	0	0
$304$	−33.8022	−1.93869
$305$	0	0
$306$	0	0
$307$	10.9439	0.624603	0.312302	−	0.949983i	$-0.398900\pi$
$307$	10.9439	0.624603	0.312302	+	0.949983i	$0.398900\pi$
$308$	9.83659	0.560492
$309$	0	0
$310$	0	0
$311$	−34.6696	−1.96593	−0.982966	−	0.183789i	$-0.941164\pi$
$311$	−34.6696	−1.96593	−0.982966	+	0.183789i	$0.941164\pi$
$312$	0	0
$313$	17.6409	0.997121	0.498561	−	0.866855i	$-0.333862\pi$
$313$	17.6409	0.997121	0.498561	+	0.866855i	$0.333862\pi$
$314$	27.2127	1.53570
$315$	0	0
$316$	0.633248	0.0356230
$317$	16.7046	0.938226	0.469113	−	0.883138i	$-0.344574\pi$
$317$	16.7046	0.938226	0.469113	+	0.883138i	$0.344574\pi$
$318$	0	0
$319$	−6.44444	−0.360819
$320$	0	0
$321$	0	0
$322$	−18.1340	−1.01057
$323$	17.4047	0.968426
$324$	0	0
$325$	0	0
$326$	3.45396	0.191297
$327$	0	0
$328$	4.93893	0.272707
$329$	−23.8877	−1.31697
$330$	0	0
$331$	26.5137	1.45732	0.728662	−	0.684873i	$-0.240142\pi$
$331$	26.5137	1.45732	0.728662	+	0.684873i	$0.240142\pi$
$332$	10.9841	0.602832
$333$	0	0
$334$	−25.6232	−1.40204
$335$	0	0
$336$	0	0
$337$	8.06876	0.439533	0.219767	−	0.975552i	$-0.429470\pi$
$337$	8.06876	0.439533	0.219767	+	0.975552i	$0.429470\pi$
$338$	20.8801	1.13573
$339$	0	0
$340$	0	0
$341$	3.40053	0.184149
$342$	0	0
$343$	19.3228	1.04333
$344$	−14.0886	−0.759607
$345$	0	0
$346$	20.7571	1.11591
$347$	−8.17121	−0.438653	−0.219327	−	0.975651i	$-0.570386\pi$
$347$	−8.17121	−0.438653	−0.219327	+	0.975651i	$0.570386\pi$
$348$	0	0
$349$	12.0729	0.646247	0.323123	−	0.946357i	$-0.395267\pi$
$349$	12.0729	0.646247	0.323123	+	0.946357i	$0.395267\pi$
$350$	0	0
$351$	0	0
$352$	−19.8087	−1.05581
$353$	−28.9900	−1.54298	−0.771492	−	0.636239i	$-0.780489\pi$
$353$	−28.9900	−1.54298	−0.771492	+	0.636239i	$0.780489\pi$
$354$	0	0
$355$	0	0
$356$	11.2510	0.596300
$357$	0	0
$358$	−34.0086	−1.79741
$359$	−10.3260	−0.544983	−0.272491	−	0.962158i	$-0.587848\pi$
$359$	−10.3260	−0.544983	−0.272491	+	0.962158i	$0.587848\pi$
$360$	0	0
$361$	27.1292	1.42785
$362$	37.9841	1.99640
$363$	0	0
$364$	3.22024	0.168786
$365$	0	0
$366$	0	0
$367$	6.46606	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$367$	6.46606	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$368$	20.2459	1.05539
$369$	0	0
$370$	0	0
$371$	4.33028	0.224817
$372$	0	0
$373$	8.21366	0.425287	0.212644	−	0.977130i	$-0.431793\pi$
$373$	8.21366	0.425287	0.212644	+	0.977130i	$0.431793\pi$
$374$	15.4712	0.799998
$375$	0	0
$376$	14.3223	0.738615
$377$	−2.10974	−0.108657
$378$	0	0
$379$	−28.1179	−1.44432	−0.722160	−	0.691726i	$-0.756851\pi$
$379$	−28.1179	−1.44432	−0.722160	+	0.691726i	$0.756851\pi$
$380$	0	0
$381$	0	0
$382$	0.356946	0.0182630
$383$	−23.5747	−1.20461	−0.602307	−	0.798265i	$-0.705752\pi$
$383$	−23.5747	−1.20461	−0.602307	+	0.798265i	$0.705752\pi$
$384$	0	0
$385$	0	0
$386$	−25.8894	−1.31773
$387$	0	0
$388$	−16.7290	−0.849285
$389$	−6.83628	−0.346613	−0.173307	−	0.984868i	$-0.555445\pi$
$389$	−6.83628	−0.346613	−0.173307	+	0.984868i	$0.555445\pi$
$390$	0	0
$391$	−10.4246	−0.527195
$392$	−1.04750	−0.0529069
$393$	0	0
$394$	−3.16034	−0.159216
$395$	0	0
$396$	0	0
$397$	−27.5209	−1.38123	−0.690617	−	0.723221i	$-0.742661\pi$
$397$	−27.5209	−1.38123	−0.690617	+	0.723221i	$0.742661\pi$
$398$	−38.0194	−1.90574
$399$	0	0
$400$	0	0
$401$	−13.0187	−0.650121	−0.325060	−	0.945693i	$-0.605385\pi$
$401$	−13.0187	−0.650121	−0.325060	+	0.945693i	$0.605385\pi$
$402$	0	0
$403$	1.11324	0.0554545
$404$	5.90464	0.293767
$405$	0	0
$406$	8.44796	0.419265
$407$	12.3237	0.610863
$408$	0	0
$409$	4.28433	0.211846	0.105923	−	0.994374i	$-0.466220\pi$
$409$	4.28433	0.211846	0.105923	+	0.994374i	$0.466220\pi$
$410$	0	0
$411$	0	0
$412$	2.89645	0.142698
$413$	17.7541	0.873623
$414$	0	0
$415$	0	0
$416$	−6.48485	−0.317946
$417$	0	0
$418$	41.0047	2.00560
$419$	−20.3954	−0.996378	−0.498189	−	0.867068i	$-0.666001\pi$
$419$	−20.3954	−0.996378	−0.498189	+	0.867068i	$0.666001\pi$
$420$	0	0
$421$	−0.314475	−0.0153266	−0.00766329	−	0.999971i	$-0.502439\pi$
$421$	−0.314475	−0.0153266	−0.00766329	+	0.999971i	$0.502439\pi$
$422$	−2.97509	−0.144825
$423$	0	0
$424$	−2.59629	−0.126087
$425$	0	0
$426$	0	0
$427$	−34.6249	−1.67562
$428$	8.99489	0.434785
$429$	0	0
$430$	0	0
$431$	23.2462	1.11973	0.559864	−	0.828584i	$-0.310853\pi$
$431$	23.2462	1.11973	0.559864	+	0.828584i	$0.310853\pi$
$432$	0	0
$433$	−3.98016	−0.191274	−0.0956371	−	0.995416i	$-0.530489\pi$
$433$	−3.98016	−0.191274	−0.0956371	+	0.995416i	$0.530489\pi$
$434$	−4.45772	−0.213977
$435$	0	0
$436$	0.550552	0.0263667
$437$	−27.6292	−1.32168
$438$	0	0
$439$	−16.8778	−0.805535	−0.402768	−	0.915302i	$-0.631952\pi$
$439$	−16.8778	−0.805535	−0.402768	+	0.915302i	$0.631952\pi$
$440$	0	0
$441$	0	0
$442$	5.06486	0.240911
$443$	15.5414	0.738393	0.369196	−	0.929351i	$-0.379633\pi$
$443$	15.5414	0.738393	0.369196	+	0.929351i	$0.379633\pi$
$444$	0	0
$445$	0	0
$446$	24.1751	1.14473
$447$	0	0
$448$	0.975132	0.0460707
$449$	−9.23621	−0.435884	−0.217942	−	0.975962i	$-0.569934\pi$
$449$	−9.23621	−0.435884	−0.217942	+	0.975962i	$0.569934\pi$
$450$	0	0
$451$	−11.1565	−0.525338
$452$	−18.7168	−0.880364
$453$	0	0
$454$	−21.5392	−1.01088
$455$	0	0
$456$	0	0
$457$	−36.7035	−1.71692	−0.858459	−	0.512882i	$-0.828578\pi$
$457$	−36.7035	−1.71692	−0.858459	+	0.512882i	$0.828578\pi$
$458$	26.0838	1.21882
$459$	0	0
$460$	0	0
$461$	−3.05958	−0.142499	−0.0712494	−	0.997459i	$-0.522699\pi$
$461$	−3.05958	−0.142499	−0.0712494	+	0.997459i	$0.522699\pi$
$462$	0	0
$463$	−25.7494	−1.19668	−0.598338	−	0.801244i	$-0.704172\pi$
$463$	−25.7494	−1.19668	−0.598338	+	0.801244i	$0.704172\pi$
$464$	−9.43181	−0.437861
$465$	0	0
$466$	−4.40526	−0.204070
$467$	10.0435	0.464757	0.232378	−	0.972625i	$-0.425349\pi$
$467$	10.0435	0.464757	0.232378	+	0.972625i	$0.425349\pi$
$468$	0	0
$469$	13.9193	0.642733
$470$	0	0
$471$	0	0
$472$	−10.6448	−0.489965
$473$	31.8246	1.46330
$474$	0	0
$475$	0	0
$476$	−7.41272	−0.339762
$477$	0	0
$478$	−52.0745	−2.38183
$479$	−28.8424	−1.31784	−0.658922	−	0.752211i	$-0.728987\pi$
$479$	−28.8424	−1.31784	−0.658922	+	0.752211i	$0.728987\pi$
$480$	0	0
$481$	4.03444	0.183955
$482$	20.7142	0.943508
$483$	0	0
$484$	0.649301	0.0295137
$485$	0	0
$486$	0	0
$487$	30.4543	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$487$	30.4543	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$488$	20.7600	0.939759
$489$	0	0
$490$	0	0
$491$	−9.99856	−0.451229	−0.225614	−	0.974217i	$-0.572439\pi$
$491$	−9.99856	−0.451229	−0.225614	+	0.974217i	$0.572439\pi$
$492$	0	0
$493$	4.85644	0.218723
$494$	13.4238	0.603967
$495$	0	0
$496$	4.97687	0.223468
$497$	−34.7339	−1.55803
$498$	0	0
$499$	−0.305009	−0.0136541	−0.00682704	−	0.999977i	$-0.502173\pi$
$499$	−0.305009	−0.0136541	−0.00682704	+	0.999977i	$0.502173\pi$
$500$	0	0
$501$	0	0
$502$	21.3762	0.954065
$503$	−19.3517	−0.862851	−0.431425	−	0.902149i	$-0.641989\pi$
$503$	−19.3517	−0.862851	−0.431425	+	0.902149i	$0.641989\pi$
$504$	0	0
$505$	0	0
$506$	−24.5598	−1.09182
$507$	0	0
$508$	−1.06637	−0.0473126
$509$	−9.75913	−0.432566	−0.216283	−	0.976331i	$-0.569393\pi$
$509$	−9.75913	−0.432566	−0.216283	+	0.976331i	$0.569393\pi$
$510$	0	0
$511$	3.22024	0.142455
$512$	−14.0069	−0.619023
$513$	0	0
$514$	8.14880	0.359428
$515$	0	0
$516$	0	0
$517$	−32.3524	−1.42286
$518$	−16.1550	−0.709810
$519$	0	0
$520$	0	0
$521$	4.53914	0.198863	0.0994317	−	0.995044i	$-0.468298\pi$
$521$	4.53914	0.198863	0.0994317	+	0.995044i	$0.468298\pi$
$522$	0	0
$523$	−19.8665	−0.868702	−0.434351	−	0.900744i	$-0.643022\pi$
$523$	−19.8665	−0.868702	−0.434351	+	0.900744i	$0.643022\pi$
$524$	20.4864	0.894952
$525$	0	0
$526$	49.3660	2.15246
$527$	−2.56259	−0.111628
$528$	0	0
$529$	−6.45146	−0.280498
$530$	0	0
$531$	0	0
$532$	−19.6466	−0.851786
$533$	−3.65233	−0.158200
$534$	0	0
$535$	0	0
$536$	−8.34555	−0.360473
$537$	0	0
$538$	4.75395	0.204957
$539$	2.36619	0.101919
$540$	0	0
$541$	−30.5214	−1.31222	−0.656108	−	0.754667i	$-0.727799\pi$
$541$	−30.5214	−1.31222	−0.656108	+	0.754667i	$0.727799\pi$
$542$	−0.262684	−0.0112832
$543$	0	0
$544$	14.9276	0.640015
$545$	0	0
$546$	0	0
$547$	27.5724	1.17891	0.589455	−	0.807801i	$-0.299343\pi$
$547$	27.5724	1.17891	0.589455	+	0.807801i	$0.299343\pi$
$548$	2.06640	0.0882722
$549$	0	0
$550$	0	0
$551$	12.8714	0.548341
$552$	0	0
$553$	−1.38008	−0.0586868
$554$	−38.5904	−1.63955
$555$	0	0
$556$	7.28483	0.308946
$557$	−35.9720	−1.52418	−0.762092	−	0.647468i	$-0.775828\pi$
$557$	−35.9720	−1.52418	−0.762092	+	0.647468i	$0.775828\pi$
$558$	0	0
$559$	10.4185	0.440656
$560$	0	0
$561$	0	0
$562$	−8.80840	−0.371560
$563$	26.0911	1.09961	0.549805	−	0.835293i	$-0.314702\pi$
$563$	26.0911	1.09961	0.549805	+	0.835293i	$0.314702\pi$
$564$	0	0
$565$	0	0
$566$	57.5215	2.41781
$567$	0	0
$568$	20.8253	0.873810
$569$	6.35017	0.266213	0.133107	−	0.991102i	$-0.457505\pi$
$569$	6.35017	0.266213	0.133107	+	0.991102i	$0.457505\pi$
$570$	0	0
$571$	22.0717	0.923670	0.461835	−	0.886966i	$-0.347191\pi$
$571$	22.0717	0.923670	0.461835	+	0.886966i	$0.347191\pi$
$572$	4.36134	0.182357
$573$	0	0
$574$	14.6249	0.610433
$575$	0	0
$576$	0	0
$577$	14.9885	0.623978	0.311989	−	0.950086i	$-0.399005\pi$
$577$	14.9885	0.623978	0.311989	+	0.950086i	$0.399005\pi$
$578$	18.5231	0.770459
$579$	0	0
$580$	0	0
$581$	−23.9384	−0.993131
$582$	0	0
$583$	5.86474	0.242893
$584$	−1.93075	−0.0798948
$585$	0	0
$586$	3.27452	0.135269
$587$	15.2241	0.628365	0.314182	−	0.949363i	$-0.398270\pi$
$587$	15.2241	0.628365	0.314182	+	0.949363i	$0.398270\pi$
$588$	0	0
$589$	−6.79185	−0.279853
$590$	0	0
$591$	0	0
$592$	18.0364	0.741293
$593$	−19.9042	−0.817368	−0.408684	−	0.912676i	$-0.634012\pi$
$593$	−19.9042	−0.817368	−0.408684	+	0.912676i	$0.634012\pi$
$594$	0	0
$595$	0	0
$596$	−18.1000	−0.741405
$597$	0	0
$598$	−8.04022	−0.328789
$599$	26.0963	1.06626	0.533132	−	0.846032i	$-0.321015\pi$
$599$	26.0963	1.06626	0.533132	+	0.846032i	$0.321015\pi$
$600$	0	0
$601$	−13.7362	−0.560311	−0.280155	−	0.959955i	$-0.590386\pi$
$601$	−13.7362	−0.560311	−0.280155	+	0.959955i	$0.590386\pi$
$602$	−41.7186	−1.70032
$603$	0	0
$604$	−11.2810	−0.459019
$605$	0	0
$606$	0	0
$607$	−9.94243	−0.403551	−0.201775	−	0.979432i	$-0.564671\pi$
$607$	−9.94243	−0.403551	−0.201775	+	0.979432i	$0.564671\pi$
$608$	39.5638	1.60453
$609$	0	0
$610$	0	0
$611$	−10.5913	−0.428479
$612$	0	0
$613$	−36.4235	−1.47113	−0.735566	−	0.677453i	$-0.763084\pi$
$613$	−36.4235	−1.47113	−0.735566	+	0.677453i	$0.763084\pi$
$614$	−19.4300	−0.784130
$615$	0	0
$616$	12.8532	0.517871
$617$	27.5706	1.10995	0.554976	−	0.831867i	$-0.312728\pi$
$617$	27.5706	1.10995	0.554976	+	0.831867i	$0.312728\pi$
$618$	0	0
$619$	−11.1689	−0.448916	−0.224458	−	0.974484i	$-0.572061\pi$
$619$	−11.1689	−0.448916	−0.224458	+	0.974484i	$0.572061\pi$
$620$	0	0
$621$	0	0
$622$	61.5528	2.46804
$623$	−24.5199	−0.982371
$624$	0	0
$625$	0	0
$626$	−31.3198	−1.25179
$627$	0	0
$628$	−17.6587	−0.704657
$629$	−9.28697	−0.370296
$630$	0	0
$631$	−44.1384	−1.75712	−0.878560	−	0.477632i	$-0.841495\pi$
$631$	−44.1384	−1.75712	−0.878560	+	0.477632i	$0.841495\pi$
$632$	0.827448	0.0329141
$633$	0	0
$634$	−29.6576	−1.17785
$635$	0	0
$636$	0	0
$637$	0.774628	0.0306919
$638$	11.4415	0.452974
$639$	0	0
$640$	0	0
$641$	−48.1194	−1.90060	−0.950301	−	0.311331i	$-0.899225\pi$
$641$	−48.1194	−1.90060	−0.950301	+	0.311331i	$0.899225\pi$
$642$	0	0
$643$	−36.0096	−1.42008	−0.710041	−	0.704161i	$-0.751323\pi$
$643$	−36.0096	−1.42008	−0.710041	+	0.704161i	$0.751323\pi$
$644$	11.7673	0.463698
$645$	0	0
$646$	−30.9006	−1.21577
$647$	4.17999	0.164332	0.0821662	−	0.996619i	$-0.473816\pi$
$647$	4.17999	0.164332	0.0821662	+	0.996619i	$0.473816\pi$
$648$	0	0
$649$	24.0454	0.943863
$650$	0	0
$651$	0	0
$652$	−2.24132	−0.0877767
$653$	−8.94602	−0.350085	−0.175042	−	0.984561i	$-0.556006\pi$
$653$	−8.94602	−0.350085	−0.175042	+	0.984561i	$0.556006\pi$
$654$	0	0
$655$	0	0
$656$	−16.3282	−0.637508
$657$	0	0
$658$	42.4105	1.65333
$659$	−43.2813	−1.68600	−0.842999	−	0.537914i	$-0.819212\pi$
$659$	−43.2813	−1.68600	−0.842999	+	0.537914i	$0.819212\pi$
$660$	0	0
$661$	10.2049	0.396923	0.198462	−	0.980109i	$-0.436406\pi$
$661$	10.2049	0.396923	0.198462	+	0.980109i	$0.436406\pi$
$662$	−47.0727	−1.82953
$663$	0	0
$664$	14.3526	0.556991
$665$	0	0
$666$	0	0
$667$	−7.70937	−0.298508
$668$	16.6272	0.643324
$669$	0	0
$670$	0	0
$671$	−46.8944	−1.81034
$672$	0	0
$673$	−11.5778	−0.446291	−0.223146	−	0.974785i	$-0.571633\pi$
$673$	−11.5778	−0.446291	−0.223146	+	0.974785i	$0.571633\pi$
$674$	−14.3254	−0.551793
$675$	0	0
$676$	−13.5493	−0.521127
$677$	−3.44545	−0.132419	−0.0662096	−	0.997806i	$-0.521091\pi$
$677$	−3.44545	−0.132419	−0.0662096	+	0.997806i	$0.521091\pi$
$678$	0	0
$679$	36.4585	1.39915
$680$	0	0
$681$	0	0
$682$	−6.03733	−0.231181
$683$	−1.45473	−0.0556638	−0.0278319	−	0.999613i	$-0.508860\pi$
$683$	−1.45473	−0.0556638	−0.0278319	+	0.999613i	$0.508860\pi$
$684$	0	0
$685$	0	0
$686$	−34.3059	−1.30980
$687$	0	0
$688$	46.5772	1.77574
$689$	1.91996	0.0731446
$690$	0	0
$691$	26.3645	1.00295	0.501477	−	0.865171i	$-0.332790\pi$
$691$	26.3645	1.00295	0.501477	+	0.865171i	$0.332790\pi$
$692$	−13.4695	−0.512035
$693$	0	0
$694$	14.5073	0.550688
$695$	0	0
$696$	0	0
$697$	8.40738	0.318452
$698$	−21.4343	−0.811302
$699$	0	0
$700$	0	0
$701$	−12.7869	−0.482956	−0.241478	−	0.970406i	$-0.577632\pi$
$701$	−12.7869	−0.482956	−0.241478	+	0.970406i	$0.577632\pi$
$702$	0	0
$703$	−24.6140	−0.928335
$704$	1.32067	0.0497748
$705$	0	0
$706$	51.4692	1.93707
$707$	−12.8684	−0.483965
$708$	0	0
$709$	5.39099	0.202463	0.101231	−	0.994863i	$-0.467722\pi$
$709$	5.39099	0.202463	0.101231	+	0.994863i	$0.467722\pi$
$710$	0	0
$711$	0	0
$712$	14.7013	0.550956
$713$	4.06799	0.152347
$714$	0	0
$715$	0	0
$716$	22.0685	0.824740
$717$	0	0
$718$	18.3328	0.684174
$719$	−4.88575	−0.182208	−0.0911038	−	0.995841i	$-0.529040\pi$
$719$	−4.88575	−0.182208	−0.0911038	+	0.995841i	$0.529040\pi$
$720$	0	0
$721$	−6.31243	−0.235087
$722$	−48.1655	−1.79254
$723$	0	0
$724$	−24.6483	−0.916048
$725$	0	0
$726$	0	0
$727$	4.49621	0.166755	0.0833776	−	0.996518i	$-0.473429\pi$
$727$	4.49621	0.166755	0.0833776	+	0.996518i	$0.473429\pi$
$728$	4.20780	0.155951
$729$	0	0
$730$	0	0
$731$	−23.9826	−0.887028
$732$	0	0
$733$	15.3670	0.567594	0.283797	−	0.958884i	$-0.408406\pi$
$733$	15.3670	0.567594	0.283797	+	0.958884i	$0.408406\pi$
$734$	−11.4799	−0.423731
$735$	0	0
$736$	−23.6968	−0.873477
$737$	18.8517	0.694410
$738$	0	0
$739$	28.9710	1.06571	0.532857	−	0.846205i	$-0.321118\pi$
$739$	28.9710	1.06571	0.532857	+	0.846205i	$0.321118\pi$
$740$	0	0
$741$	0	0
$742$	−7.68803	−0.282237
$743$	−33.5233	−1.22985	−0.614926	−	0.788585i	$-0.710814\pi$
$743$	−33.5233	−1.22985	−0.614926	+	0.788585i	$0.710814\pi$
$744$	0	0
$745$	0	0
$746$	−14.5826	−0.533908
$747$	0	0
$748$	−10.0395	−0.367079
$749$	−19.6031	−0.716283
$750$	0	0
$751$	−3.70436	−0.135174	−0.0675870	−	0.997713i	$-0.521530\pi$
$751$	−3.70436	−0.135174	−0.0675870	+	0.997713i	$0.521530\pi$
$752$	−47.3497	−1.72666
$753$	0	0
$754$	3.74565	0.136408
$755$	0	0
$756$	0	0
$757$	−31.1959	−1.13384	−0.566918	−	0.823774i	$-0.691864\pi$
$757$	−31.1959	−1.13384	−0.566918	+	0.823774i	$0.691864\pi$
$758$	49.9209	1.81321
$759$	0	0
$760$	0	0
$761$	−7.38412	−0.267674	−0.133837	−	0.991003i	$-0.542730\pi$
$761$	−7.38412	−0.267674	−0.133837	+	0.991003i	$0.542730\pi$
$762$	0	0
$763$	−1.19985	−0.0434376
$764$	−0.231627	−0.00837996
$765$	0	0
$766$	41.8549	1.51228
$767$	7.87181	0.284234
$768$	0	0
$769$	−21.5475	−0.777022	−0.388511	−	0.921444i	$-0.627010\pi$
$769$	−21.5475	−0.777022	−0.388511	+	0.921444i	$0.627010\pi$
$770$	0	0
$771$	0	0
$772$	16.7999	0.604642
$773$	−42.3698	−1.52394	−0.761968	−	0.647615i	$-0.775767\pi$
$773$	−42.3698	−1.52394	−0.761968	+	0.647615i	$0.775767\pi$
$774$	0	0
$775$	0	0
$776$	−21.8593	−0.784704
$777$	0	0
$778$	12.1372	0.435140
$779$	22.2828	0.798363
$780$	0	0
$781$	−47.0420	−1.68330
$782$	18.5079	0.661843
$783$	0	0
$784$	3.46306	0.123681
$785$	0	0
$786$	0	0
$787$	12.6001	0.449144	0.224572	−	0.974457i	$-0.427902\pi$
$787$	12.6001	0.449144	0.224572	+	0.974457i	$0.427902\pi$
$788$	2.05078	0.0730561
$789$	0	0
$790$	0	0
$791$	40.7907	1.45035
$792$	0	0
$793$	−15.3520	−0.545165
$794$	48.8609	1.73401
$795$	0	0
$796$	24.6712	0.874447
$797$	20.4137	0.723089	0.361544	−	0.932355i	$-0.382250\pi$
$797$	20.4137	0.723089	0.361544	+	0.932355i	$0.382250\pi$
$798$	0	0
$799$	24.3803	0.862515
$800$	0	0
$801$	0	0
$802$	23.1135	0.816165
$803$	4.36134	0.153908
$804$	0	0
$805$	0	0
$806$	−1.97646	−0.0696179
$807$	0	0
$808$	7.71544	0.271428
$809$	−4.41162	−0.155104	−0.0775521	−	0.996988i	$-0.524710\pi$
$809$	−4.41162	−0.155104	−0.0775521	+	0.996988i	$0.524710\pi$
$810$	0	0
$811$	−4.32233	−0.151778	−0.0758888	−	0.997116i	$-0.524179\pi$
$811$	−4.32233	−0.151778	−0.0758888	+	0.997116i	$0.524179\pi$
$812$	−5.48198	−0.192380
$813$	0	0
$814$	−21.8796	−0.766880
$815$	0	0
$816$	0	0
$817$	−63.5631	−2.22379
$818$	−7.60644	−0.265953
$819$	0	0
$820$	0	0
$821$	43.2855	1.51068	0.755338	−	0.655336i	$-0.227473\pi$
$821$	43.2855	1.51068	0.755338	+	0.655336i	$0.227473\pi$
$822$	0	0
$823$	45.5754	1.58866	0.794330	−	0.607486i	$-0.207822\pi$
$823$	45.5754	1.58866	0.794330	+	0.607486i	$0.207822\pi$
$824$	3.78472	0.131847
$825$	0	0
$826$	−31.5208	−1.09675
$827$	−6.17189	−0.214618	−0.107309	−	0.994226i	$-0.534223\pi$
$827$	−6.17189	−0.214618	−0.107309	+	0.994226i	$0.534223\pi$
$828$	0	0
$829$	−8.46744	−0.294086	−0.147043	−	0.989130i	$-0.546976\pi$
$829$	−8.46744	−0.294086	−0.147043	+	0.989130i	$0.546976\pi$
$830$	0	0
$831$	0	0
$832$	0.432353	0.0149892
$833$	−1.78313	−0.0617818
$834$	0	0
$835$	0	0
$836$	−26.6084	−0.920271
$837$	0	0
$838$	36.2101	1.25086
$839$	−46.1580	−1.59355	−0.796776	−	0.604275i	$-0.793463\pi$
$839$	−46.1580	−1.59355	−0.796776	+	0.604275i	$0.793463\pi$
$840$	0	0
$841$	−25.4085	−0.876155
$842$	0.558323	0.0192411
$843$	0	0
$844$	1.93057	0.0664529
$845$	0	0
$846$	0	0
$847$	−1.41506	−0.0486221
$848$	8.58339	0.294755
$849$	0	0
$850$	0	0
$851$	14.7426	0.505370
$852$	0	0
$853$	−10.7820	−0.369170	−0.184585	−	0.982817i	$-0.559094\pi$
$853$	−10.7820	−0.369170	−0.184585	+	0.982817i	$0.559094\pi$
$854$	61.4735	2.10358
$855$	0	0
$856$	11.7534	0.401723
$857$	0.208859	0.00713450	0.00356725	−	0.999994i	$-0.498865\pi$
$857$	0.208859	0.00713450	0.00356725	+	0.999994i	$0.498865\pi$
$858$	0	0
$859$	−28.8449	−0.984175	−0.492088	−	0.870546i	$-0.663766\pi$
$859$	−28.8449	−0.984175	−0.492088	+	0.870546i	$0.663766\pi$
$860$	0	0
$861$	0	0
$862$	−41.2715	−1.40571
$863$	37.7540	1.28516	0.642580	−	0.766218i	$-0.277864\pi$
$863$	37.7540	1.28516	0.642580	+	0.766218i	$0.277864\pi$
$864$	0	0
$865$	0	0
$866$	7.06642	0.240127
$867$	0	0
$868$	2.89267	0.0981835
$869$	−1.86911	−0.0634053
$870$	0	0
$871$	6.17153	0.209114
$872$	0.719392	0.0243617
$873$	0	0
$874$	49.0531	1.65925
$875$	0	0
$876$	0	0
$877$	−25.0400	−0.845539	−0.422770	−	0.906237i	$-0.638942\pi$
$877$	−25.0400	−0.845539	−0.422770	+	0.906237i	$0.638942\pi$
$878$	29.9651	1.01127
$879$	0	0
$880$	0	0
$881$	−34.9760	−1.17837	−0.589186	−	0.807998i	$-0.700551\pi$
$881$	−34.9760	−1.17837	−0.589186	+	0.807998i	$0.700551\pi$
$882$	0	0
$883$	−47.2590	−1.59039	−0.795196	−	0.606352i	$-0.792632\pi$
$883$	−47.2590	−1.59039	−0.795196	+	0.606352i	$0.792632\pi$
$884$	−3.28665	−0.110542
$885$	0	0
$886$	−27.5923	−0.926982
$887$	−1.81183	−0.0608352	−0.0304176	−	0.999537i	$-0.509684\pi$
$887$	−1.81183	−0.0608352	−0.0304176	+	0.999537i	$0.509684\pi$
$888$	0	0
$889$	2.32401	0.0779448
$890$	0	0
$891$	0	0
$892$	−15.6875	−0.525257
$893$	64.6173	2.16233
$894$	0	0
$895$	0	0
$896$	27.5207	0.919401
$897$	0	0
$898$	16.3981	0.547211
$899$	−1.89513	−0.0632061
$900$	0	0
$901$	−4.41959	−0.147238
$902$	19.8073	0.659512
$903$	0	0
$904$	−24.4568	−0.813420
$905$	0	0
$906$	0	0
$907$	33.8683	1.12458	0.562289	−	0.826941i	$-0.309921\pi$
$907$	33.8683	1.12458	0.562289	+	0.826941i	$0.309921\pi$
$908$	13.9770	0.463844
$909$	0	0
$910$	0	0
$911$	−47.9463	−1.58853	−0.794265	−	0.607571i	$-0.792144\pi$
$911$	−47.9463	−1.58853	−0.794265	+	0.607571i	$0.792144\pi$
$912$	0	0
$913$	−32.4210	−1.07298
$914$	65.1638	2.15543
$915$	0	0
$916$	−16.9261	−0.559254
$917$	−44.6473	−1.47438
$918$	0	0
$919$	30.1556	0.994743	0.497371	−	0.867538i	$-0.334299\pi$
$919$	30.1556	0.994743	0.497371	+	0.867538i	$0.334299\pi$
$920$	0	0
$921$	0	0
$922$	5.43201	0.178894
$923$	−15.4003	−0.506907
$924$	0	0
$925$	0	0
$926$	45.7157	1.50231
$927$	0	0
$928$	11.0395	0.362389
$929$	7.35345	0.241259	0.120629	−	0.992698i	$-0.461509\pi$
$929$	7.35345	0.241259	0.120629	+	0.992698i	$0.461509\pi$
$930$	0	0
$931$	−4.72598	−0.154888
$932$	2.85862	0.0936372
$933$	0	0
$934$	−17.8313	−0.583458
$935$	0	0
$936$	0	0
$937$	−16.7720	−0.547918	−0.273959	−	0.961741i	$-0.588333\pi$
$937$	−16.7720	−0.547918	−0.273959	+	0.961741i	$0.588333\pi$
$938$	−24.7125	−0.806891
$939$	0	0
$940$	0	0
$941$	−38.3413	−1.24989	−0.624945	−	0.780668i	$-0.714879\pi$
$941$	−38.3413	−1.24989	−0.624945	+	0.780668i	$0.714879\pi$
$942$	0	0
$943$	−13.3463	−0.434615
$944$	35.1918	1.14540
$945$	0	0
$946$	−56.5018	−1.83703
$947$	9.03981	0.293754	0.146877	−	0.989155i	$-0.453078\pi$
$947$	9.03981	0.293754	0.146877	+	0.989155i	$0.453078\pi$
$948$	0	0
$949$	1.42779	0.0463479
$950$	0	0
$951$	0	0
$952$	−9.68601	−0.313925
$953$	40.5370	1.31312	0.656562	−	0.754272i	$-0.272010\pi$
$953$	40.5370	1.31312	0.656562	+	0.754272i	$0.272010\pi$
$954$	0	0
$955$	0	0
$956$	33.7918	1.09290
$957$	0	0
$958$	51.2071	1.65443
$959$	−4.50344	−0.145424
$960$	0	0
$961$	1.00000	0.0322581
$962$	−7.16280	−0.230938
$963$	0	0
$964$	−13.4417	−0.432928
$965$	0	0
$966$	0	0
$967$	17.2224	0.553834	0.276917	−	0.960894i	$-0.410687\pi$
$967$	17.2224	0.553834	0.276917	+	0.960894i	$0.410687\pi$
$968$	0.848424	0.0272694
$969$	0	0
$970$	0	0
$971$	−42.0718	−1.35015	−0.675074	−	0.737750i	$-0.735888\pi$
$971$	−42.0718	−1.35015	−0.675074	+	0.737750i	$0.735888\pi$
$972$	0	0
$973$	−15.8763	−0.508970
$974$	−54.0690	−1.73248
$975$	0	0
$976$	−68.6327	−2.19688
$977$	39.7622	1.27211	0.636053	−	0.771645i	$-0.280566\pi$
$977$	39.7622	1.27211	0.636053	+	0.771645i	$0.280566\pi$
$978$	0	0
$979$	−33.2087	−1.06135
$980$	0	0
$981$	0	0
$982$	17.7516	0.566475
$983$	−22.7537	−0.725732	−0.362866	−	0.931841i	$-0.618202\pi$
$983$	−22.7537	−0.725732	−0.362866	+	0.931841i	$0.618202\pi$
$984$	0	0
$985$	0	0
$986$	−8.62218	−0.274586
$987$	0	0
$988$	−8.71088	−0.277130
$989$	38.0712	1.21059
$990$	0	0
$991$	7.11452	0.226000	0.113000	−	0.993595i	$-0.463954\pi$
$991$	7.11452	0.226000	0.113000	+	0.993595i	$0.463954\pi$
$992$	−5.82519	−0.184950
$993$	0	0
$994$	61.6669	1.95596
$995$	0	0
$996$	0	0
$997$	27.7159	0.877773	0.438886	−	0.898543i	$-0.355373\pi$
$997$	27.7159	0.877773	0.438886	+	0.898543i	$0.355373\pi$
$998$	0.541516	0.0171414
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cf.1.3	✓	8
3.2	odd	2	inner	6975.2.a.cf.1.6	yes	8
5.4	even	2		6975.2.a.cg.1.6	yes	8
15.14	odd	2		6975.2.a.cg.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.cf.1.3	✓	8	1.1	even	1	trivial
6975.2.a.cf.1.6	yes	8	3.2	odd	2	inner
6975.2.a.cg.1.3	yes	8	15.14	odd	2
6975.2.a.cg.1.6	yes	8	5.4	even	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)

\(f(q)\)	\(=\)	\(q-1.77541 q^{2} +1.15208 q^{4} -2.51081 q^{7} +1.50540 q^{8} -3.40053 q^{11} -1.11324 q^{13} +4.45772 q^{14} -4.97687 q^{16} +2.56259 q^{17} +6.79185 q^{19} +6.03733 q^{22} -4.06799 q^{23} +1.97646 q^{26} -2.89267 q^{28} +1.89513 q^{29} -1.00000 q^{31} +5.82519 q^{32} -4.54965 q^{34} -3.62405 q^{37} -12.0583 q^{38} +3.28081 q^{41} -9.35873 q^{43} -3.91770 q^{44} +7.22235 q^{46} +9.51394 q^{47} -0.695831 q^{49} -1.28255 q^{52} -1.72466 q^{53} -3.77977 q^{56} -3.36463 q^{58} -7.07107 q^{59} +13.7903 q^{61} +1.77541 q^{62} -0.388373 q^{64} -5.54375 q^{67} +2.95232 q^{68} +13.8337 q^{71} -1.28255 q^{73} +6.43418 q^{74} +7.82479 q^{76} +8.53808 q^{77} +0.549654 q^{79} -5.82479 q^{82} +9.53412 q^{83} +16.6156 q^{86} -5.11915 q^{88} +9.76575 q^{89} +2.79514 q^{91} -4.68667 q^{92} -16.8912 q^{94} -14.5206 q^{97} +1.23539 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 4 q^{7} - 8 q^{13} + 12 q^{16} - 28 q^{22} - 36 q^{28} - 8 q^{31} - 28 q^{34} - 12 q^{37} - 52 q^{43} - 16 q^{46} + 8 q^{49} - 56 q^{52} - 16 q^{58} - 28 q^{61} + 48 q^{64} - 24 q^{67}+ \cdots - 44 q^{97}+O(q^{100}) \) 8 * q + 16 * q^4 - 4 * q^7 - 8 * q^13 + 12 * q^16 - 28 * q^22 - 36 * q^28 - 8 * q^31 - 28 * q^34 - 12 * q^37 - 52 * q^43 - 16 * q^46 + 8 * q^49 - 56 * q^52 - 16 * q^58 - 28 * q^61 + 48 * q^64 - 24 * q^67 - 56 * q^73 + 4 * q^76 - 4 * q^79 + 12 * q^82 - 52 * q^88 - 8 * q^91 - 64 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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