LMFDB - Embedded newform 6975.2.a.bt.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 = [5,-1,0,3,0,0,-2,-3,0,0,6,0,-4,8,0,-5,-12] 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:	$5$
Coefficient field:	5.5.582992.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x^{2} + 7x - 3 $ x^5 - x^4 - 6x^3 + 4x^2 + 7*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.38937$ 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)

$f(q)$	$=$	$q-2.38937 q^{2} +3.70910 q^{4} -1.81408 q^{7} -4.08369 q^{8} +0.424709 q^{11} -1.94988 q^{13} +4.33452 q^{14} +2.33925 q^{16} -2.94515 q^{17} -5.97871 q^{19} -1.01479 q^{22} +2.28440 q^{23} +4.65898 q^{26} -6.72862 q^{28} +4.44423 q^{29} -1.00000 q^{31} +2.57804 q^{32} +7.03706 q^{34} -4.85496 q^{37} +14.2854 q^{38} +8.11450 q^{41} +9.93134 q^{43} +1.57529 q^{44} -5.45827 q^{46} +7.11253 q^{47} -3.70910 q^{49} -7.23229 q^{52} +3.16339 q^{53} +7.40815 q^{56} -10.6189 q^{58} +1.05559 q^{59} +6.75746 q^{61} +2.38937 q^{62} -10.8384 q^{64} -3.33851 q^{67} -10.9239 q^{68} -0.768686 q^{71} +9.02234 q^{73} +11.6003 q^{74} -22.1757 q^{76} -0.770457 q^{77} +5.54669 q^{79} -19.3886 q^{82} -0.355067 q^{83} -23.7297 q^{86} -1.73438 q^{88} +9.93939 q^{89} +3.53723 q^{91} +8.47306 q^{92} -16.9945 q^{94} -0.875505 q^{97} +8.86244 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 6 q^{11} - 4 q^{13} + 8 q^{14} - 5 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{22} - 8 q^{23} + 2 q^{26} - 6 q^{28} + 14 q^{29} - 5 q^{31} - 7 q^{32} - 2 q^{34} - 4 q^{37}+ \cdots + 5 q^{98}+O(q^{100}) $ 5 * q - q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8 + 6 * q^11 - 4 * q^13 + 8 * q^14 - 5 * q^16 - 12 * q^17 - 4 * q^19 + 4 * q^22 - 8 * q^23 + 2 * q^26 - 6 * q^28 + 14 * q^29 - 5 * q^31 - 7 * q^32 - 2 * q^34 - 4 * q^37 - 2 * q^38 + 16 * q^41 + 2 * q^43 + 4 * q^44 - 8 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^52 - 14 * q^53 + 12 * q^56 - 20 * q^58 + 10 * q^59 - 14 * q^61 + q^62 - 5 * q^64 + 2 * q^67 - 26 * q^68 + 12 * q^71 - 10 * q^73 + 10 * q^74 - 32 * q^76 - 26 * q^77 - 12 * q^79 + 6 * q^82 - 12 * q^83 - 10 * q^86 - 14 * q^88 + 14 * q^89 - 16 * q^91 + 14 * q^92 + 4 * q^94 - 8 * q^97 + 5 * 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 $

$2$	−2.38937	−1.68954	−0.844771	−	0.535128i	$-0.820263\pi$
$2$	−2.38937	−1.68954	−0.844771	+	0.535128i	$0.820263\pi$
$3$	0	0
$3$	0	0
$4$	3.70910	1.85455
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.81408	−0.685659	−0.342829	−	0.939398i	$-0.611385\pi$
$7$	−1.81408	−0.685659	−0.342829	+	0.939398i	$0.611385\pi$
$8$	−4.08369	−1.44380
$9$	0	0
$10$	0	0
$11$	0.424709	0.128055	0.0640273	−	0.997948i	$-0.479606\pi$
$11$	0.424709	0.128055	0.0640273	+	0.997948i	$0.479606\pi$
$12$	0	0
$13$	−1.94988	−0.540798	−0.270399	−	0.962748i	$-0.587156\pi$
$13$	−1.94988	−0.540798	−0.270399	+	0.962748i	$0.587156\pi$
$14$	4.33452	1.15845
$15$	0	0
$16$	2.33925	0.584812
$17$	−2.94515	−0.714303	−0.357152	−	0.934046i	$-0.616252\pi$
$17$	−2.94515	−0.714303	−0.357152	+	0.934046i	$0.616252\pi$
$18$	0	0
$19$	−5.97871	−1.37161	−0.685805	−	0.727785i	$-0.740550\pi$
$19$	−5.97871	−1.37161	−0.685805	+	0.727785i	$0.740550\pi$
$20$	0	0
$21$	0	0
$22$	−1.01479	−0.216354
$23$	2.28440	0.476329	0.238165	−	0.971225i	$-0.423454\pi$
$23$	2.28440	0.476329	0.238165	+	0.971225i	$0.423454\pi$
$24$	0	0
$25$	0	0
$26$	4.65898	0.913701
$27$	0	0
$28$	−6.72862	−1.27159
$29$	4.44423	0.825272	0.412636	−	0.910896i	$-0.364608\pi$
$29$	4.44423	0.825272	0.412636	+	0.910896i	$0.364608\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	2.57804	0.455737
$33$	0	0
$34$	7.03706	1.20685
$35$	0	0
$36$	0	0
$37$	−4.85496	−0.798150	−0.399075	−	0.916918i	$-0.630669\pi$
$37$	−4.85496	−0.798150	−0.399075	+	0.916918i	$0.630669\pi$
$38$	14.2854	2.31739
$39$	0	0
$40$	0	0
$41$	8.11450	1.26727	0.633636	−	0.773631i	$-0.281562\pi$
$41$	8.11450	1.26727	0.633636	+	0.773631i	$0.281562\pi$
$42$	0	0
$43$	9.93134	1.51451	0.757257	−	0.653117i	$-0.226539\pi$
$43$	9.93134	1.51451	0.757257	+	0.653117i	$0.226539\pi$
$44$	1.57529	0.237484
$45$	0	0
$46$	−5.45827	−0.804779
$47$	7.11253	1.03747	0.518734	−	0.854936i	$-0.326403\pi$
$47$	7.11253	1.03747	0.518734	+	0.854936i	$0.326403\pi$
$48$	0	0
$49$	−3.70910	−0.529872
$50$	0	0
$51$	0	0
$52$	−7.23229	−1.00294
$53$	3.16339	0.434525	0.217263	−	0.976113i	$-0.430287\pi$
$53$	3.16339	0.434525	0.217263	+	0.976113i	$0.430287\pi$
$54$	0	0
$55$	0	0
$56$	7.40815	0.989956
$57$	0	0
$58$	−10.6189	−1.39433
$59$	1.05559	0.137427	0.0687133	−	0.997636i	$-0.478111\pi$
$59$	1.05559	0.137427	0.0687133	+	0.997636i	$0.478111\pi$
$60$	0	0
$61$	6.75746	0.865204	0.432602	−	0.901585i	$-0.357596\pi$
$61$	6.75746	0.865204	0.432602	+	0.901585i	$0.357596\pi$
$62$	2.38937	0.303451
$63$	0	0
$64$	−10.8384	−1.35480
$65$	0	0
$66$	0	0
$67$	−3.33851	−0.407864	−0.203932	−	0.978985i	$-0.565372\pi$
$67$	−3.33851	−0.407864	−0.203932	+	0.978985i	$0.565372\pi$
$68$	−10.9239	−1.32471
$69$	0	0
$70$	0	0
$71$	−0.768686	−0.0912263	−0.0456131	−	0.998959i	$-0.514524\pi$
$71$	−0.768686	−0.0912263	−0.0456131	+	0.998959i	$0.514524\pi$
$72$	0	0
$73$	9.02234	1.05598	0.527992	−	0.849249i	$-0.322945\pi$
$73$	9.02234	1.05598	0.527992	+	0.849249i	$0.322945\pi$
$74$	11.6003	1.34851
$75$	0	0
$76$	−22.1757	−2.54372
$77$	−0.770457	−0.0878018
$78$	0	0
$79$	5.54669	0.624052	0.312026	−	0.950074i	$-0.398992\pi$
$79$	5.54669	0.624052	0.312026	+	0.950074i	$0.398992\pi$
$80$	0	0
$81$	0	0
$82$	−19.3886	−2.14111
$83$	−0.355067	−0.0389737	−0.0194869	−	0.999810i	$-0.506203\pi$
$83$	−0.355067	−0.0389737	−0.0194869	+	0.999810i	$0.506203\pi$
$84$	0	0
$85$	0	0
$86$	−23.7297	−2.55884
$87$	0	0
$88$	−1.73438	−0.184886
$89$	9.93939	1.05357	0.526787	−	0.849998i	$-0.323397\pi$
$89$	9.93939	1.05357	0.526787	+	0.849998i	$0.323397\pi$
$90$	0	0
$91$	3.53723	0.370803
$92$	8.47306	0.883378
$93$	0	0
$94$	−16.9945	−1.75285
$95$	0	0
$96$	0	0
$97$	−0.875505	−0.0888941	−0.0444470	−	0.999012i	$-0.514153\pi$
$97$	−0.875505	−0.0888941	−0.0444470	+	0.999012i	$0.514153\pi$
$98$	8.86244	0.895241
$99$	0	0
$100$	0	0
$101$	−15.1999	−1.51245	−0.756224	−	0.654313i	$-0.772958\pi$
$101$	−15.1999	−1.51245	−0.756224	+	0.654313i	$0.772958\pi$
$102$	0	0
$103$	6.75820	0.665905	0.332953	−	0.942944i	$-0.391955\pi$
$103$	6.75820	0.665905	0.332953	+	0.942944i	$0.391955\pi$
$104$	7.96269	0.780806
$105$	0	0
$106$	−7.55852	−0.734149
$107$	−7.51115	−0.726130	−0.363065	−	0.931764i	$-0.618270\pi$
$107$	−7.51115	−0.726130	−0.363065	+	0.931764i	$0.618270\pi$
$108$	0	0
$109$	−8.63564	−0.827145	−0.413572	−	0.910471i	$-0.635719\pi$
$109$	−8.63564	−0.827145	−0.413572	+	0.910471i	$0.635719\pi$
$110$	0	0
$111$	0	0
$112$	−4.24359	−0.400982
$113$	3.06766	0.288581	0.144291	−	0.989535i	$-0.453910\pi$
$113$	3.06766	0.288581	0.144291	+	0.989535i	$0.453910\pi$
$114$	0	0
$115$	0	0
$116$	16.4841	1.53051
$117$	0	0
$118$	−2.52221	−0.232188
$119$	5.34274	0.489768
$120$	0	0
$121$	−10.8196	−0.983602
$122$	−16.1461	−1.46180
$123$	0	0
$124$	−3.70910	−0.333087
$125$	0	0
$126$	0	0
$127$	2.37733	0.210954	0.105477	−	0.994422i	$-0.466363\pi$
$127$	2.37733	0.210954	0.105477	+	0.994422i	$0.466363\pi$
$128$	20.7409	1.83325
$129$	0	0
$130$	0	0
$131$	12.3721	1.08095	0.540476	−	0.841359i	$-0.318244\pi$
$131$	12.3721	1.08095	0.540476	+	0.841359i	$0.318244\pi$
$132$	0	0
$133$	10.8459	0.940457
$134$	7.97694	0.689103
$135$	0	0
$136$	12.0271	1.03131
$137$	−4.50381	−0.384786	−0.192393	−	0.981318i	$-0.561625\pi$
$137$	−4.50381	−0.384786	−0.192393	+	0.981318i	$0.561625\pi$
$138$	0	0
$139$	−11.8760	−1.00731	−0.503654	−	0.863905i	$-0.668011\pi$
$139$	−11.8760	−1.00731	−0.503654	+	0.863905i	$0.668011\pi$
$140$	0	0
$141$	0	0
$142$	1.83668	0.154131
$143$	−0.828130	−0.0692517
$144$	0	0
$145$	0	0
$146$	−21.5577	−1.78413
$147$	0	0
$148$	−18.0075	−1.48021
$149$	5.19796	0.425833	0.212917	−	0.977070i	$-0.431704\pi$
$149$	5.19796	0.425833	0.212917	+	0.977070i	$0.431704\pi$
$150$	0	0
$151$	−20.4139	−1.66126	−0.830628	−	0.556828i	$-0.812018\pi$
$151$	−20.4139	−1.66126	−0.830628	+	0.556828i	$0.812018\pi$
$152$	24.4152	1.98033
$153$	0	0
$154$	1.84091	0.148345
$155$	0	0
$156$	0	0
$157$	14.7116	1.17411	0.587055	−	0.809547i	$-0.300287\pi$
$157$	14.7116	1.17411	0.587055	+	0.809547i	$0.300287\pi$
$158$	−13.2531	−1.05436
$159$	0	0
$160$	0	0
$161$	−4.14408	−0.326599
$162$	0	0
$163$	4.34733	0.340509	0.170255	−	0.985400i	$-0.445541\pi$
$163$	4.34733	0.340509	0.170255	+	0.985400i	$0.445541\pi$
$164$	30.0975	2.35022
$165$	0	0
$166$	0.848388	0.0658477
$167$	−9.11984	−0.705714	−0.352857	−	0.935677i	$-0.614790\pi$
$167$	−9.11984	−0.705714	−0.352857	+	0.935677i	$0.614790\pi$
$168$	0	0
$169$	−9.19799	−0.707537
$170$	0	0
$171$	0	0
$172$	36.8364	2.80875
$173$	0.0173305	0.00131761	0.000658807	−	1.00000i	$-0.499790\pi$
$173$	0.0173305	0.00131761	0.000658807		1.00000i	$0.499790\pi$
$174$	0	0
$175$	0	0
$176$	0.993500	0.0748879
$177$	0	0
$178$	−23.7489	−1.78006
$179$	18.3939	1.37483	0.687413	−	0.726267i	$-0.258746\pi$
$179$	18.3939	1.37483	0.687413	+	0.726267i	$0.258746\pi$
$180$	0	0
$181$	−7.13078	−0.530026	−0.265013	−	0.964245i	$-0.585376\pi$
$181$	−7.13078	−0.530026	−0.265013	+	0.964245i	$0.585376\pi$
$182$	−8.45177	−0.626487
$183$	0	0
$184$	−9.32876	−0.687725
$185$	0	0
$186$	0	0
$187$	−1.25083	−0.0914698
$188$	26.3811	1.92404
$189$	0	0
$190$	0	0
$191$	−16.5370	−1.19657	−0.598287	−	0.801282i	$-0.704152\pi$
$191$	−16.5370	−1.19657	−0.598287	+	0.801282i	$0.704152\pi$
$192$	0	0
$193$	−8.00201	−0.575997	−0.287999	−	0.957631i	$-0.592990\pi$
$193$	−8.00201	−0.575997	−0.287999	+	0.957631i	$0.592990\pi$
$194$	2.09191	0.150190
$195$	0	0
$196$	−13.7575	−0.982676
$197$	−8.78210	−0.625699	−0.312849	−	0.949803i	$-0.601283\pi$
$197$	−8.78210	−0.625699	−0.312849	+	0.949803i	$0.601283\pi$
$198$	0	0
$199$	3.58662	0.254249	0.127124	−	0.991887i	$-0.459425\pi$
$199$	3.58662	0.254249	0.127124	+	0.991887i	$0.459425\pi$
$200$	0	0
$201$	0	0
$202$	36.3183	2.55534
$203$	−8.06219	−0.565855
$204$	0	0
$205$	0	0
$206$	−16.1479	−1.12507
$207$	0	0
$208$	−4.56124	−0.316265
$209$	−2.53921	−0.175641
$210$	0	0
$211$	−12.0390	−0.828801	−0.414401	−	0.910095i	$-0.636009\pi$
$211$	−12.0390	−0.828801	−0.414401	+	0.910095i	$0.636009\pi$
$212$	11.7334	0.805850
$213$	0	0
$214$	17.9469	1.22683
$215$	0	0
$216$	0	0
$217$	1.81408	0.123148
$218$	20.6338	1.39750
$219$	0	0
$220$	0	0
$221$	5.74267	0.386294
$222$	0	0
$223$	−22.6690	−1.51803	−0.759014	−	0.651074i	$-0.774319\pi$
$223$	−22.6690	−1.51803	−0.759014	+	0.651074i	$0.774319\pi$
$224$	−4.67678	−0.312480
$225$	0	0
$226$	−7.32979	−0.487570
$227$	−21.5104	−1.42770	−0.713848	−	0.700301i	$-0.753049\pi$
$227$	−21.5104	−1.42770	−0.713848	+	0.700301i	$0.753049\pi$
$228$	0	0
$229$	−10.8577	−0.717498	−0.358749	−	0.933434i	$-0.616797\pi$
$229$	−10.8577	−0.717498	−0.358749	+	0.933434i	$0.616797\pi$
$230$	0	0
$231$	0	0
$232$	−18.1488	−1.19153
$233$	9.37335	0.614068	0.307034	−	0.951698i	$-0.400663\pi$
$233$	9.37335	0.614068	0.307034	+	0.951698i	$0.400663\pi$
$234$	0	0
$235$	0	0
$236$	3.91531	0.254865
$237$	0	0
$238$	−12.7658	−0.827484
$239$	−3.69418	−0.238957	−0.119478	−	0.992837i	$-0.538122\pi$
$239$	−3.69418	−0.238957	−0.119478	+	0.992837i	$0.538122\pi$
$240$	0	0
$241$	15.6033	1.00510	0.502550	−	0.864548i	$-0.332395\pi$
$241$	15.6033	1.00510	0.502550	+	0.864548i	$0.332395\pi$
$242$	25.8521	1.66184
$243$	0	0
$244$	25.0641	1.60457
$245$	0	0
$246$	0	0
$247$	11.6577	0.741764
$248$	4.08369	0.259315
$249$	0	0
$250$	0	0
$251$	−7.36999	−0.465190	−0.232595	−	0.972574i	$-0.574722\pi$
$251$	−7.36999	−0.465190	−0.232595	+	0.972574i	$0.574722\pi$
$252$	0	0
$253$	0.970204	0.0609962
$254$	−5.68034	−0.356416
$255$	0	0
$256$	−27.8810	−1.74256
$257$	−26.7289	−1.66730	−0.833650	−	0.552293i	$-0.813753\pi$
$257$	−26.7289	−1.66730	−0.833650	+	0.552293i	$0.813753\pi$
$258$	0	0
$259$	8.80729	0.547259
$260$	0	0
$261$	0	0
$262$	−29.5615	−1.82632
$263$	11.7852	0.726706	0.363353	−	0.931651i	$-0.381632\pi$
$263$	11.7852	0.726706	0.363353	+	0.931651i	$0.381632\pi$
$264$	0	0
$265$	0	0
$266$	−25.9148	−1.58894
$267$	0	0
$268$	−12.3829	−0.756404
$269$	28.1171	1.71433	0.857164	−	0.515044i	$-0.172224\pi$
$269$	28.1171	1.71433	0.857164	+	0.515044i	$0.172224\pi$
$270$	0	0
$271$	12.2533	0.744336	0.372168	−	0.928165i	$-0.378614\pi$
$271$	12.2533	0.744336	0.372168	+	0.928165i	$0.378614\pi$
$272$	−6.88943	−0.417733
$273$	0	0
$274$	10.7613	0.650113
$275$	0	0
$276$	0	0
$277$	−21.3064	−1.28018	−0.640089	−	0.768301i	$-0.721102\pi$
$277$	−21.3064	−1.28018	−0.640089	+	0.768301i	$0.721102\pi$
$278$	28.3762	1.70189
$279$	0	0
$280$	0	0
$281$	−31.0108	−1.84995	−0.924974	−	0.380030i	$-0.875914\pi$
$281$	−31.0108	−1.84995	−0.924974	+	0.380030i	$0.875914\pi$
$282$	0	0
$283$	5.64316	0.335451	0.167726	−	0.985834i	$-0.446358\pi$
$283$	5.64316	0.335451	0.167726	+	0.985834i	$0.446358\pi$
$284$	−2.85114	−0.169184
$285$	0	0
$286$	1.97871	0.117004
$287$	−14.7204	−0.868917
$288$	0	0
$289$	−8.32611	−0.489771
$290$	0	0
$291$	0	0
$292$	33.4648	1.95838
$293$	−21.4365	−1.25233	−0.626167	−	0.779689i	$-0.715377\pi$
$293$	−21.4365	−1.25233	−0.626167	+	0.779689i	$0.715377\pi$
$294$	0	0
$295$	0	0
$296$	19.8261	1.15237
$297$	0	0
$298$	−12.4199	−0.719463
$299$	−4.45429	−0.257598
$300$	0	0
$301$	−18.0163	−1.03844
$302$	48.7763	2.80676
$303$	0	0
$304$	−13.9857	−0.802135
$305$	0	0
$306$	0	0
$307$	2.58585	0.147582	0.0737911	−	0.997274i	$-0.476490\pi$
$307$	2.58585	0.147582	0.0737911	+	0.997274i	$0.476490\pi$
$308$	−2.85771	−0.162833
$309$	0	0
$310$	0	0
$311$	−10.5398	−0.597658	−0.298829	−	0.954307i	$-0.596596\pi$
$311$	−10.5398	−0.597658	−0.298829	+	0.954307i	$0.596596\pi$
$312$	0	0
$313$	−26.8268	−1.51634	−0.758171	−	0.652056i	$-0.773907\pi$
$313$	−26.8268	−1.51634	−0.758171	+	0.652056i	$0.773907\pi$
$314$	−35.1514	−1.98371
$315$	0	0
$316$	20.5733	1.15734
$317$	18.0374	1.01308	0.506540	−	0.862217i	$-0.330924\pi$
$317$	18.0374	1.01308	0.506540	+	0.862217i	$0.330924\pi$
$318$	0	0
$319$	1.88750	0.105680
$320$	0	0
$321$	0	0
$322$	9.90176	0.551803
$323$	17.6082	0.979746
$324$	0	0
$325$	0	0
$326$	−10.3874	−0.575305
$327$	0	0
$328$	−33.1371	−1.82969
$329$	−12.9027	−0.711349
$330$	0	0
$331$	27.9836	1.53812	0.769059	−	0.639178i	$-0.220725\pi$
$331$	27.9836	1.53812	0.769059	+	0.639178i	$0.220725\pi$
$332$	−1.31698	−0.0722788
$333$	0	0
$334$	21.7907	1.19233
$335$	0	0
$336$	0	0
$337$	17.9294	0.976678	0.488339	−	0.872654i	$-0.337603\pi$
$337$	17.9294	0.976678	0.488339	+	0.872654i	$0.337603\pi$
$338$	21.9774	1.19541
$339$	0	0
$340$	0	0
$341$	−0.424709	−0.0229993
$342$	0	0
$343$	19.4272	1.04897
$344$	−40.5565	−2.18666
$345$	0	0
$346$	−0.0414091	−0.00222616
$347$	11.6444	0.625106	0.312553	−	0.949900i	$-0.398816\pi$
$347$	11.6444	0.625106	0.312553	+	0.949900i	$0.398816\pi$
$348$	0	0
$349$	−9.70857	−0.519688	−0.259844	−	0.965651i	$-0.583671\pi$
$349$	−9.70857	−0.519688	−0.259844	+	0.965651i	$0.583671\pi$
$350$	0	0
$351$	0	0
$352$	1.09492	0.0583593
$353$	0.162858	0.00866805	0.00433403	−	0.999991i	$-0.498620\pi$
$353$	0.162858	0.00866805	0.00433403	+	0.999991i	$0.498620\pi$
$354$	0	0
$355$	0	0
$356$	36.8662	1.95391
$357$	0	0
$358$	−43.9499	−2.32283
$359$	0.721311	0.0380693	0.0190347	−	0.999819i	$-0.493941\pi$
$359$	0.721311	0.0380693	0.0190347	+	0.999819i	$0.493941\pi$
$360$	0	0
$361$	16.7450	0.881315
$362$	17.0381	0.895502
$363$	0	0
$364$	13.1200	0.687674
$365$	0	0
$366$	0	0
$367$	19.0724	0.995572	0.497786	−	0.867300i	$-0.334147\pi$
$367$	19.0724	0.995572	0.497786	+	0.867300i	$0.334147\pi$
$368$	5.34377	0.278563
$369$	0	0
$370$	0	0
$371$	−5.73865	−0.297936
$372$	0	0
$373$	−5.79359	−0.299981	−0.149990	−	0.988687i	$-0.547924\pi$
$373$	−5.79359	−0.299981	−0.149990	+	0.988687i	$0.547924\pi$
$374$	2.98870	0.154542
$375$	0	0
$376$	−29.0453	−1.49790
$377$	−8.66569	−0.446306
$378$	0	0
$379$	4.91686	0.252562	0.126281	−	0.991994i	$-0.459696\pi$
$379$	4.91686	0.252562	0.126281	+	0.991994i	$0.459696\pi$
$380$	0	0
$381$	0	0
$382$	39.5130	2.02166
$383$	−8.16007	−0.416960	−0.208480	−	0.978027i	$-0.566852\pi$
$383$	−8.16007	−0.416960	−0.208480	+	0.978027i	$0.566852\pi$
$384$	0	0
$385$	0	0
$386$	19.1198	0.973171
$387$	0	0
$388$	−3.24734	−0.164859
$389$	33.4247	1.69470	0.847350	−	0.531034i	$-0.178196\pi$
$389$	33.4247	1.69470	0.847350	+	0.531034i	$0.178196\pi$
$390$	0	0
$391$	−6.72788	−0.340244
$392$	15.1468	0.765031
$393$	0	0
$394$	20.9837	1.05714
$395$	0	0
$396$	0	0
$397$	−8.85533	−0.444436	−0.222218	−	0.974997i	$-0.571330\pi$
$397$	−8.85533	−0.444436	−0.222218	+	0.974997i	$0.571330\pi$
$398$	−8.56977	−0.429564
$399$	0	0
$400$	0	0
$401$	−11.1275	−0.555682	−0.277841	−	0.960627i	$-0.589619\pi$
$401$	−11.1275	−0.555682	−0.277841	+	0.960627i	$0.589619\pi$
$402$	0	0
$403$	1.94988	0.0971302
$404$	−56.3781	−2.80491
$405$	0	0
$406$	19.2636	0.956036
$407$	−2.06195	−0.102207
$408$	0	0
$409$	7.88544	0.389910	0.194955	−	0.980812i	$-0.437544\pi$
$409$	7.88544	0.389910	0.194955	+	0.980812i	$0.437544\pi$
$410$	0	0
$411$	0	0
$412$	25.0669	1.23496
$413$	−1.91494	−0.0942278
$414$	0	0
$415$	0	0
$416$	−5.02686	−0.246462
$417$	0	0
$418$	6.06713	0.296753
$419$	−6.33770	−0.309617	−0.154808	−	0.987945i	$-0.549476\pi$
$419$	−6.33770	−0.309617	−0.154808	+	0.987945i	$0.549476\pi$
$420$	0	0
$421$	24.7757	1.20749	0.603746	−	0.797177i	$-0.293674\pi$
$421$	24.7757	1.20749	0.603746	+	0.797177i	$0.293674\pi$
$422$	28.7657	1.40029
$423$	0	0
$424$	−12.9183	−0.627368
$425$	0	0
$426$	0	0
$427$	−12.2586	−0.593235
$428$	−27.8596	−1.34665
$429$	0	0
$430$	0	0
$431$	−28.8224	−1.38833	−0.694163	−	0.719818i	$-0.744225\pi$
$431$	−28.8224	−1.38833	−0.694163	+	0.719818i	$0.744225\pi$
$432$	0	0
$433$	1.93339	0.0929126	0.0464563	−	0.998920i	$-0.485207\pi$
$433$	1.93339	0.0929126	0.0464563	+	0.998920i	$0.485207\pi$
$434$	−4.33452	−0.208064
$435$	0	0
$436$	−32.0305	−1.53398
$437$	−13.6577	−0.653338
$438$	0	0
$439$	34.5814	1.65048	0.825240	−	0.564782i	$-0.191040\pi$
$439$	34.5814	1.65048	0.825240	+	0.564782i	$0.191040\pi$
$440$	0	0
$441$	0	0
$442$	−13.7214	−0.652660
$443$	−37.3300	−1.77360	−0.886801	−	0.462151i	$-0.847078\pi$
$443$	−37.3300	−1.77360	−0.886801	+	0.462151i	$0.847078\pi$
$444$	0	0
$445$	0	0
$446$	54.1647	2.56477
$447$	0	0
$448$	19.6617	0.928930
$449$	12.1643	0.574069	0.287034	−	0.957920i	$-0.407331\pi$
$449$	12.1643	0.574069	0.287034	+	0.957920i	$0.407331\pi$
$450$	0	0
$451$	3.44630	0.162280
$452$	11.3783	0.535189
$453$	0	0
$454$	51.3964	2.41215
$455$	0	0
$456$	0	0
$457$	−32.5798	−1.52402	−0.762010	−	0.647565i	$-0.775787\pi$
$457$	−32.5798	−1.52402	−0.762010	+	0.647565i	$0.775787\pi$
$458$	25.9431	1.21224
$459$	0	0
$460$	0	0
$461$	34.9354	1.62710	0.813552	−	0.581492i	$-0.197531\pi$
$461$	34.9354	1.62710	0.813552	+	0.581492i	$0.197531\pi$
$462$	0	0
$463$	−20.6955	−0.961803	−0.480902	−	0.876775i	$-0.659691\pi$
$463$	−20.6955	−0.961803	−0.480902	+	0.876775i	$0.659691\pi$
$464$	10.3962	0.482629
$465$	0	0
$466$	−22.3964	−1.03749
$467$	29.0620	1.34483	0.672414	−	0.740175i	$-0.265258\pi$
$467$	29.0620	1.34483	0.672414	+	0.740175i	$0.265258\pi$
$468$	0	0
$469$	6.05633	0.279655
$470$	0	0
$471$	0	0
$472$	−4.31072	−0.198417
$473$	4.21793	0.193941
$474$	0	0
$475$	0	0
$476$	19.8168	0.908301
$477$	0	0
$478$	8.82677	0.403727
$479$	19.1848	0.876575	0.438287	−	0.898835i	$-0.355585\pi$
$479$	19.1848	0.876575	0.438287	+	0.898835i	$0.355585\pi$
$480$	0	0
$481$	9.46656	0.431638
$482$	−37.2822	−1.69816
$483$	0	0
$484$	−40.1311	−1.82414
$485$	0	0
$486$	0	0
$487$	−1.00212	−0.0454102	−0.0227051	−	0.999742i	$-0.507228\pi$
$487$	−1.00212	−0.0454102	−0.0227051	+	0.999742i	$0.507228\pi$
$488$	−27.5954	−1.24918
$489$	0	0
$490$	0	0
$491$	−10.0390	−0.453055	−0.226528	−	0.974005i	$-0.572737\pi$
$491$	−10.0390	−0.453055	−0.226528	+	0.974005i	$0.572737\pi$
$492$	0	0
$493$	−13.0889	−0.589494
$494$	−27.8547	−1.25324
$495$	0	0
$496$	−2.33925	−0.105035
$497$	1.39446	0.0625501
$498$	0	0
$499$	32.6536	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$499$	32.6536	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$500$	0	0
$501$	0	0
$502$	17.6097	0.785958
$503$	−18.7838	−0.837529	−0.418765	−	0.908095i	$-0.637537\pi$
$503$	−18.7838	−0.837529	−0.418765	+	0.908095i	$0.637537\pi$
$504$	0	0
$505$	0	0
$506$	−2.31818	−0.103056
$507$	0	0
$508$	8.81778	0.391226
$509$	−38.4971	−1.70636	−0.853178	−	0.521620i	$-0.825328\pi$
$509$	−38.4971	−1.70636	−0.853178	+	0.521620i	$0.825328\pi$
$510$	0	0
$511$	−16.3673	−0.724045
$512$	25.1362	1.11087
$513$	0	0
$514$	63.8652	2.81697
$515$	0	0
$516$	0	0
$517$	3.02075	0.132853
$518$	−21.0439	−0.924616
$519$	0	0
$520$	0	0
$521$	37.0045	1.62120	0.810598	−	0.585603i	$-0.199142\pi$
$521$	37.0045	1.62120	0.810598	+	0.585603i	$0.199142\pi$
$522$	0	0
$523$	−18.4745	−0.807832	−0.403916	−	0.914796i	$-0.632351\pi$
$523$	−18.4745	−0.807832	−0.403916	+	0.914796i	$0.632351\pi$
$524$	45.8893	2.00468
$525$	0	0
$526$	−28.1592	−1.22780
$527$	2.94515	0.128293
$528$	0	0
$529$	−17.7815	−0.773110
$530$	0	0
$531$	0	0
$532$	40.2285	1.74413
$533$	−15.8223	−0.685339
$534$	0	0
$535$	0	0
$536$	13.6334	0.588874
$537$	0	0
$538$	−67.1822	−2.89643
$539$	−1.57529	−0.0678526
$540$	0	0
$541$	−7.62001	−0.327610	−0.163805	−	0.986493i	$-0.552377\pi$
$541$	−7.62001	−0.327610	−0.163805	+	0.986493i	$0.552377\pi$
$542$	−29.2778	−1.25759
$543$	0	0
$544$	−7.59271	−0.325535
$545$	0	0
$546$	0	0
$547$	−10.7821	−0.461011	−0.230505	−	0.973071i	$-0.574038\pi$
$547$	−10.7821	−0.461011	−0.230505	+	0.973071i	$0.574038\pi$
$548$	−16.7051	−0.713606
$549$	0	0
$550$	0	0
$551$	−26.5707	−1.13195
$552$	0	0
$553$	−10.0622	−0.427886
$554$	50.9090	2.16291
$555$	0	0
$556$	−44.0493	−1.86811
$557$	−26.6085	−1.12744	−0.563720	−	0.825966i	$-0.690630\pi$
$557$	−26.6085	−1.12744	−0.563720	+	0.825966i	$0.690630\pi$
$558$	0	0
$559$	−19.3649	−0.819047
$560$	0	0
$561$	0	0
$562$	74.0963	3.12556
$563$	6.08008	0.256245	0.128122	−	0.991758i	$-0.459105\pi$
$563$	6.08008	0.256245	0.128122	+	0.991758i	$0.459105\pi$
$564$	0	0
$565$	0	0
$566$	−13.4836	−0.566759
$567$	0	0
$568$	3.13908	0.131713
$569$	10.8240	0.453767	0.226884	−	0.973922i	$-0.427146\pi$
$569$	10.8240	0.453767	0.226884	+	0.973922i	$0.427146\pi$
$570$	0	0
$571$	−26.0503	−1.09017	−0.545087	−	0.838380i	$-0.683503\pi$
$571$	−26.0503	−1.09017	−0.545087	+	0.838380i	$0.683503\pi$
$572$	−3.07162	−0.128431
$573$	0	0
$574$	35.1725	1.46807
$575$	0	0
$576$	0	0
$577$	−21.6689	−0.902089	−0.451045	−	0.892501i	$-0.648949\pi$
$577$	−21.6689	−0.902089	−0.451045	+	0.892501i	$0.648949\pi$
$578$	19.8942	0.827489
$579$	0	0
$580$	0	0
$581$	0.644121	0.0267227
$582$	0	0
$583$	1.34352	0.0556430
$584$	−36.8444	−1.52463
$585$	0	0
$586$	51.2198	2.11587
$587$	−40.8572	−1.68636	−0.843178	−	0.537634i	$-0.819318\pi$
$587$	−40.8572	−1.68636	−0.843178	+	0.537634i	$0.819318\pi$
$588$	0	0
$589$	5.97871	0.246349
$590$	0	0
$591$	0	0
$592$	−11.3570	−0.466768
$593$	−14.9102	−0.612289	−0.306145	−	0.951985i	$-0.599039\pi$
$593$	−14.9102	−0.612289	−0.306145	+	0.951985i	$0.599039\pi$
$594$	0	0
$595$	0	0
$596$	19.2798	0.789730
$597$	0	0
$598$	10.6430	0.435223
$599$	−34.9636	−1.42857	−0.714287	−	0.699853i	$-0.753249\pi$
$599$	−34.9636	−1.42857	−0.714287	+	0.699853i	$0.753249\pi$
$600$	0	0
$601$	−38.2392	−1.55981	−0.779904	−	0.625899i	$-0.784732\pi$
$601$	−38.2392	−1.55981	−0.779904	+	0.625899i	$0.784732\pi$
$602$	43.0476	1.75449
$603$	0	0
$604$	−75.7171	−3.08089
$605$	0	0
$606$	0	0
$607$	−9.16659	−0.372061	−0.186030	−	0.982544i	$-0.559562\pi$
$607$	−9.16659	−0.372061	−0.186030	+	0.982544i	$0.559562\pi$
$608$	−15.4134	−0.625094
$609$	0	0
$610$	0	0
$611$	−13.8685	−0.561061
$612$	0	0
$613$	42.8459	1.73053	0.865265	−	0.501314i	$-0.167150\pi$
$613$	42.8459	1.73053	0.865265	+	0.501314i	$0.167150\pi$
$614$	−6.17856	−0.249346
$615$	0	0
$616$	3.14631	0.126768
$617$	−12.8553	−0.517536	−0.258768	−	0.965939i	$-0.583317\pi$
$617$	−12.8553	−0.517536	−0.258768	+	0.965939i	$0.583317\pi$
$618$	0	0
$619$	−24.5706	−0.987574	−0.493787	−	0.869583i	$-0.664388\pi$
$619$	−24.5706	−0.987574	−0.493787	+	0.869583i	$0.664388\pi$
$620$	0	0
$621$	0	0
$622$	25.1836	1.00977
$623$	−18.0309	−0.722392
$624$	0	0
$625$	0	0
$626$	64.0993	2.56192
$627$	0	0
$628$	54.5667	2.17745
$629$	14.2986	0.570121
$630$	0	0
$631$	20.2039	0.804306	0.402153	−	0.915572i	$-0.368262\pi$
$631$	20.2039	0.804306	0.402153	+	0.915572i	$0.368262\pi$
$632$	−22.6510	−0.901007
$633$	0	0
$634$	−43.0980	−1.71164
$635$	0	0
$636$	0	0
$637$	7.23229	0.286554
$638$	−4.50995	−0.178551
$639$	0	0
$640$	0	0
$641$	37.2532	1.47141	0.735706	−	0.677301i	$-0.236851\pi$
$641$	37.2532	1.47141	0.735706	+	0.677301i	$0.236851\pi$
$642$	0	0
$643$	29.8201	1.17599	0.587995	−	0.808865i	$-0.299918\pi$
$643$	29.8201	1.17599	0.587995	+	0.808865i	$0.299918\pi$
$644$	−15.3708	−0.605696
$645$	0	0
$646$	−42.0725	−1.65532
$647$	35.1376	1.38140	0.690702	−	0.723140i	$-0.257302\pi$
$647$	35.1376	1.38140	0.690702	+	0.723140i	$0.257302\pi$
$648$	0	0
$649$	0.448321	0.0175981
$650$	0	0
$651$	0	0
$652$	16.1247	0.631492
$653$	−23.8904	−0.934903	−0.467451	−	0.884019i	$-0.654828\pi$
$653$	−23.8904	−0.934903	−0.467451	+	0.884019i	$0.654828\pi$
$654$	0	0
$655$	0	0
$656$	18.9818	0.741116
$657$	0	0
$658$	30.8294	1.20185
$659$	24.6184	0.958995	0.479498	−	0.877543i	$-0.340819\pi$
$659$	24.6184	0.958995	0.479498	+	0.877543i	$0.340819\pi$
$660$	0	0
$661$	10.2449	0.398479	0.199239	−	0.979951i	$-0.436153\pi$
$661$	10.2449	0.398479	0.199239	+	0.979951i	$0.436153\pi$
$662$	−66.8633	−2.59871
$663$	0	0
$664$	1.44998	0.0562703
$665$	0	0
$666$	0	0
$667$	10.1524	0.393101
$668$	−33.8264	−1.30878
$669$	0	0
$670$	0	0
$671$	2.86995	0.110793
$672$	0	0
$673$	−13.8607	−0.534291	−0.267145	−	0.963656i	$-0.586080\pi$
$673$	−13.8607	−0.534291	−0.267145	+	0.963656i	$0.586080\pi$
$674$	−42.8401	−1.65014
$675$	0	0
$676$	−34.1163	−1.31217
$677$	−31.4606	−1.20913	−0.604564	−	0.796556i	$-0.706653\pi$
$677$	−31.4606	−1.20913	−0.604564	+	0.796556i	$0.706653\pi$
$678$	0	0
$679$	1.58824	0.0609510
$680$	0	0
$681$	0	0
$682$	1.01479	0.0388583
$683$	31.8992	1.22059	0.610295	−	0.792174i	$-0.291051\pi$
$683$	31.8992	1.22059	0.610295	+	0.792174i	$0.291051\pi$
$684$	0	0
$685$	0	0
$686$	−46.4188	−1.77228
$687$	0	0
$688$	23.2319	0.885707
$689$	−6.16822	−0.234990
$690$	0	0
$691$	45.6534	1.73674	0.868370	−	0.495918i	$-0.165168\pi$
$691$	45.6534	1.73674	0.868370	+	0.495918i	$0.165168\pi$
$692$	0.0642807	0.00244358
$693$	0	0
$694$	−27.8229	−1.05614
$695$	0	0
$696$	0	0
$697$	−23.8984	−0.905217
$698$	23.1974	0.878035
$699$	0	0
$700$	0	0
$701$	−38.5030	−1.45424	−0.727119	−	0.686512i	$-0.759141\pi$
$701$	−38.5030	−1.45424	−0.727119	+	0.686512i	$0.759141\pi$
$702$	0	0
$703$	29.0264	1.09475
$704$	−4.60317	−0.173488
$705$	0	0
$706$	−0.389128	−0.0146450
$707$	27.5739	1.03702
$708$	0	0
$709$	3.52248	0.132289	0.0661447	−	0.997810i	$-0.478930\pi$
$709$	3.52248	0.132289	0.0661447	+	0.997810i	$0.478930\pi$
$710$	0	0
$711$	0	0
$712$	−40.5894	−1.52115
$713$	−2.28440	−0.0855513
$714$	0	0
$715$	0	0
$716$	68.2249	2.54969
$717$	0	0
$718$	−1.72348	−0.0643198
$719$	−48.0264	−1.79108	−0.895540	−	0.444980i	$-0.853211\pi$
$719$	−48.0264	−1.79108	−0.895540	+	0.444980i	$0.853211\pi$
$720$	0	0
$721$	−12.2599	−0.456584
$722$	−40.0100	−1.48902
$723$	0	0
$724$	−26.4488	−0.982962
$725$	0	0
$726$	0	0
$727$	−41.8513	−1.55218	−0.776090	−	0.630623i	$-0.782800\pi$
$727$	−41.8513	−1.55218	−0.776090	+	0.630623i	$0.782800\pi$
$728$	−14.4450	−0.535366
$729$	0	0
$730$	0	0
$731$	−29.2492	−1.08182
$732$	0	0
$733$	−48.6851	−1.79823	−0.899113	−	0.437717i	$-0.855787\pi$
$733$	−48.6851	−1.79823	−0.899113	+	0.437717i	$0.855787\pi$
$734$	−45.5711	−1.68206
$735$	0	0
$736$	5.88926	0.217081
$737$	−1.41789	−0.0522288
$738$	0	0
$739$	−36.1228	−1.32880	−0.664399	−	0.747378i	$-0.731312\pi$
$739$	−36.1228	−1.32880	−0.664399	+	0.747378i	$0.731312\pi$
$740$	0	0
$741$	0	0
$742$	13.7118	0.503375
$743$	−6.18512	−0.226910	−0.113455	−	0.993543i	$-0.536192\pi$
$743$	−6.18512	−0.226910	−0.113455	+	0.993543i	$0.536192\pi$
$744$	0	0
$745$	0	0
$746$	13.8430	0.506830
$747$	0	0
$748$	−4.63946	−0.169636
$749$	13.6258	0.497877
$750$	0	0
$751$	−8.75867	−0.319608	−0.159804	−	0.987149i	$-0.551086\pi$
$751$	−8.75867	−0.319608	−0.159804	+	0.987149i	$0.551086\pi$
$752$	16.6380	0.606724
$753$	0	0
$754$	20.7056	0.754052
$755$	0	0
$756$	0	0
$757$	29.2031	1.06140	0.530702	−	0.847559i	$-0.321928\pi$
$757$	29.2031	1.06140	0.530702	+	0.847559i	$0.321928\pi$
$758$	−11.7482	−0.426715
$759$	0	0
$760$	0	0
$761$	−13.0911	−0.474554	−0.237277	−	0.971442i	$-0.576255\pi$
$761$	−13.0911	−0.474554	−0.237277	+	0.971442i	$0.576255\pi$
$762$	0	0
$763$	15.6658	0.567139
$764$	−61.3374	−2.21911
$765$	0	0
$766$	19.4974	0.704471
$767$	−2.05828	−0.0743201
$768$	0	0
$769$	−34.5946	−1.24751	−0.623757	−	0.781619i	$-0.714394\pi$
$769$	−34.5946	−1.24751	−0.623757	+	0.781619i	$0.714394\pi$
$770$	0	0
$771$	0	0
$772$	−29.6803	−1.06822
$773$	−33.1294	−1.19158	−0.595790	−	0.803140i	$-0.703161\pi$
$773$	−33.1294	−1.19158	−0.595790	+	0.803140i	$0.703161\pi$
$774$	0	0
$775$	0	0
$776$	3.57529	0.128345
$777$	0	0
$778$	−79.8642	−2.86327
$779$	−48.5143	−1.73820
$780$	0	0
$781$	−0.326468	−0.0116819
$782$	16.0754	0.574856
$783$	0	0
$784$	−8.67652	−0.309876
$785$	0	0
$786$	0	0
$787$	−49.5637	−1.76675	−0.883377	−	0.468663i	$-0.844736\pi$
$787$	−49.5637	−1.76675	−0.883377	+	0.468663i	$0.844736\pi$
$788$	−32.5737	−1.16039
$789$	0	0
$790$	0	0
$791$	−5.56499	−0.197868
$792$	0	0
$793$	−13.1762	−0.467901
$794$	21.1587	0.750894
$795$	0	0
$796$	13.3031	0.471517
$797$	−1.17237	−0.0415276	−0.0207638	−	0.999784i	$-0.506610\pi$
$797$	−1.17237	−0.0415276	−0.0207638	+	0.999784i	$0.506610\pi$
$798$	0	0
$799$	−20.9474	−0.741067
$800$	0	0
$801$	0	0
$802$	26.5878	0.938848
$803$	3.83187	0.135224
$804$	0	0
$805$	0	0
$806$	−4.65898	−0.164106
$807$	0	0
$808$	62.0717	2.18368
$809$	1.01075	0.0355362	0.0177681	−	0.999842i	$-0.494344\pi$
$809$	1.01075	0.0355362	0.0177681	+	0.999842i	$0.494344\pi$
$810$	0	0
$811$	−9.60203	−0.337173	−0.168586	−	0.985687i	$-0.553920\pi$
$811$	−9.60203	−0.337173	−0.168586	+	0.985687i	$0.553920\pi$
$812$	−29.9035	−1.04941
$813$	0	0
$814$	4.92676	0.172683
$815$	0	0
$816$	0	0
$817$	−59.3766	−2.07732
$818$	−18.8413	−0.658770
$819$	0	0
$820$	0	0
$821$	−46.5993	−1.62633	−0.813164	−	0.582034i	$-0.802257\pi$
$821$	−46.5993	−1.62633	−0.813164	+	0.582034i	$0.802257\pi$
$822$	0	0
$823$	30.7737	1.07270	0.536351	−	0.843995i	$-0.319802\pi$
$823$	30.7737	1.07270	0.536351	+	0.843995i	$0.319802\pi$
$824$	−27.5984	−0.961435
$825$	0	0
$826$	4.57550	0.159202
$827$	37.8513	1.31622	0.658110	−	0.752922i	$-0.271356\pi$
$827$	37.8513	1.31622	0.658110	+	0.752922i	$0.271356\pi$
$828$	0	0
$829$	−9.71706	−0.337487	−0.168744	−	0.985660i	$-0.553971\pi$
$829$	−9.71706	−0.337487	−0.168744	+	0.985660i	$0.553971\pi$
$830$	0	0
$831$	0	0
$832$	21.1335	0.732673
$833$	10.9239	0.378489
$834$	0	0
$835$	0	0
$836$	−9.41821	−0.325736
$837$	0	0
$838$	15.1431	0.523110
$839$	−14.5204	−0.501300	−0.250650	−	0.968078i	$-0.580644\pi$
$839$	−14.5204	−0.501300	−0.250650	+	0.968078i	$0.580644\pi$
$840$	0	0
$841$	−9.24885	−0.318926
$842$	−59.1983	−2.04011
$843$	0	0
$844$	−44.6540	−1.53706
$845$	0	0
$846$	0	0
$847$	19.6277	0.674415
$848$	7.39996	0.254116
$849$	0	0
$850$	0	0
$851$	−11.0906	−0.380182
$852$	0	0
$853$	−14.2821	−0.489010	−0.244505	−	0.969648i	$-0.578626\pi$
$853$	−14.2821	−0.489010	−0.244505	+	0.969648i	$0.578626\pi$
$854$	29.2903	1.00229
$855$	0	0
$856$	30.6732	1.04839
$857$	−18.7661	−0.641036	−0.320518	−	0.947242i	$-0.603857\pi$
$857$	−18.7661	−0.641036	−0.320518	+	0.947242i	$0.603857\pi$
$858$	0	0
$859$	31.5981	1.07811	0.539056	−	0.842270i	$-0.318781\pi$
$859$	31.5981	1.07811	0.539056	+	0.842270i	$0.318781\pi$
$860$	0	0
$861$	0	0
$862$	68.8675	2.34564
$863$	48.5052	1.65113	0.825567	−	0.564304i	$-0.190855\pi$
$863$	48.5052	1.65113	0.825567	+	0.564304i	$0.190855\pi$
$864$	0	0
$865$	0	0
$866$	−4.61958	−0.156980
$867$	0	0
$868$	6.72862	0.228384
$869$	2.35573	0.0799127
$870$	0	0
$871$	6.50967	0.220572
$872$	35.2653	1.19423
$873$	0	0
$874$	32.6334	1.10384
$875$	0	0
$876$	0	0
$877$	−35.1135	−1.18570	−0.592849	−	0.805314i	$-0.701997\pi$
$877$	−35.1135	−1.18570	−0.592849	+	0.805314i	$0.701997\pi$
$878$	−82.6279	−2.78856
$879$	0	0
$880$	0	0
$881$	20.8946	0.703955	0.351978	−	0.936008i	$-0.385509\pi$
$881$	20.8946	0.703955	0.351978	+	0.936008i	$0.385509\pi$
$882$	0	0
$883$	18.3419	0.617253	0.308627	−	0.951183i	$-0.400131\pi$
$883$	18.3419	0.617253	0.308627	+	0.951183i	$0.400131\pi$
$884$	21.3002	0.716402
$885$	0	0
$886$	89.1953	2.99658
$887$	−49.8273	−1.67304	−0.836519	−	0.547939i	$-0.815413\pi$
$887$	−49.8273	−1.67304	−0.836519	+	0.547939i	$0.815413\pi$
$888$	0	0
$889$	−4.31268	−0.144643
$890$	0	0
$891$	0	0
$892$	−84.0817	−2.81526
$893$	−42.5237	−1.42300
$894$	0	0
$895$	0	0
$896$	−37.6257	−1.25699
$897$	0	0
$898$	−29.0651	−0.969913
$899$	−4.44423	−0.148223
$900$	0	0
$901$	−9.31665	−0.310383
$902$	−8.23451	−0.274179
$903$	0	0
$904$	−12.5274	−0.416655
$905$	0	0
$906$	0	0
$907$	−57.8644	−1.92136	−0.960678	−	0.277665i	$-0.910439\pi$
$907$	−57.8644	−1.92136	−0.960678	+	0.277665i	$0.910439\pi$
$908$	−79.7843	−2.64774
$909$	0	0
$910$	0	0
$911$	−6.48386	−0.214820	−0.107410	−	0.994215i	$-0.534256\pi$
$911$	−6.48386	−0.214820	−0.107410	+	0.994215i	$0.534256\pi$
$912$	0	0
$913$	−0.150800	−0.00499076
$914$	77.8454	2.57490
$915$	0	0
$916$	−40.2724	−1.33064
$917$	−22.4440	−0.741165
$918$	0	0
$919$	6.01521	0.198424	0.0992118	−	0.995066i	$-0.468368\pi$
$919$	6.01521	0.198424	0.0992118	+	0.995066i	$0.468368\pi$
$920$	0	0
$921$	0	0
$922$	−83.4738	−2.74906
$923$	1.49884	0.0493350
$924$	0	0
$925$	0	0
$926$	49.4494	1.62501
$927$	0	0
$928$	11.4574	0.376107
$929$	−32.4939	−1.06609	−0.533044	−	0.846087i	$-0.678952\pi$
$929$	−32.4939	−1.06609	−0.533044	+	0.846087i	$0.678952\pi$
$930$	0	0
$931$	22.1757	0.726778
$932$	34.7667	1.13882
$933$	0	0
$934$	−69.4399	−2.27214
$935$	0	0
$936$	0	0
$937$	−27.9624	−0.913492	−0.456746	−	0.889597i	$-0.650985\pi$
$937$	−27.9624	−0.913492	−0.456746	+	0.889597i	$0.650985\pi$
$938$	−14.4708	−0.472489
$939$	0	0
$940$	0	0
$941$	−41.0719	−1.33890	−0.669452	−	0.742855i	$-0.733471\pi$
$941$	−41.0719	−1.33890	−0.669452	+	0.742855i	$0.733471\pi$
$942$	0	0
$943$	18.5367	0.603639
$944$	2.46930	0.0803688
$945$	0	0
$946$	−10.0782	−0.327671
$947$	−51.6780	−1.67931	−0.839655	−	0.543120i	$-0.817243\pi$
$947$	−51.6780	−1.67931	−0.839655	+	0.543120i	$0.817243\pi$
$948$	0	0
$949$	−17.5924	−0.571075
$950$	0	0
$951$	0	0
$952$	−21.8181	−0.707128
$953$	25.6976	0.832426	0.416213	−	0.909267i	$-0.363357\pi$
$953$	25.6976	0.832426	0.416213	+	0.909267i	$0.363357\pi$
$954$	0	0
$955$	0	0
$956$	−13.7021	−0.443157
$957$	0	0
$958$	−45.8396	−1.48101
$959$	8.17028	0.263832
$960$	0	0
$961$	1.00000	0.0322581
$962$	−22.6192	−0.729271
$963$	0	0
$964$	57.8744	1.86401
$965$	0	0
$966$	0	0
$967$	−58.3618	−1.87679	−0.938395	−	0.345566i	$-0.887687\pi$
$967$	−58.3618	−1.87679	−0.938395	+	0.345566i	$0.887687\pi$
$968$	44.1840	1.42013
$969$	0	0
$970$	0	0
$971$	−9.95338	−0.319419	−0.159710	−	0.987164i	$-0.551056\pi$
$971$	−9.95338	−0.319419	−0.159710	+	0.987164i	$0.551056\pi$
$972$	0	0
$973$	21.5440	0.690670
$974$	2.39443	0.0767225
$975$	0	0
$976$	15.8074	0.505982
$977$	20.0430	0.641232	0.320616	−	0.947209i	$-0.396110\pi$
$977$	20.0430	0.641232	0.320616	+	0.947209i	$0.396110\pi$
$978$	0	0
$979$	4.22135	0.134915
$980$	0	0
$981$	0	0
$982$	23.9870	0.765456
$983$	49.5990	1.58196	0.790982	−	0.611840i	$-0.209570\pi$
$983$	49.5990	1.58196	0.790982	+	0.611840i	$0.209570\pi$
$984$	0	0
$985$	0	0
$986$	31.2743	0.995976
$987$	0	0
$988$	43.2398	1.37564
$989$	22.6871	0.721408
$990$	0	0
$991$	−1.81669	−0.0577091	−0.0288546	−	0.999584i	$-0.509186\pi$
$991$	−1.81669	−0.0577091	−0.0288546	+	0.999584i	$0.509186\pi$
$992$	−2.57804	−0.0818529
$993$	0	0
$994$	−3.33189	−0.105681
$995$	0	0
$996$	0	0
$997$	57.8849	1.83323	0.916616	−	0.399768i	$-0.130909\pi$
$997$	57.8849	1.83323	0.916616	+	0.399768i	$0.130909\pi$
$998$	−78.0216	−2.46973
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bt.1.1		5
3.2	odd	2		6975.2.a.bu.1.5		5
5.4	even	2		1395.2.a.p.1.5	yes	5
15.14	odd	2		1395.2.a.o.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.a.o.1.1	✓	5	15.14	odd	2
1395.2.a.p.1.5	yes	5	5.4	even	2
6975.2.a.bt.1.1		5	1.1	even	1	trivial
6975.2.a.bu.1.5		5	3.2	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)

\(f(q)\)	\(=\)	\(q-2.38937 q^{2} +3.70910 q^{4} -1.81408 q^{7} -4.08369 q^{8} +0.424709 q^{11} -1.94988 q^{13} +4.33452 q^{14} +2.33925 q^{16} -2.94515 q^{17} -5.97871 q^{19} -1.01479 q^{22} +2.28440 q^{23} +4.65898 q^{26} -6.72862 q^{28} +4.44423 q^{29} -1.00000 q^{31} +2.57804 q^{32} +7.03706 q^{34} -4.85496 q^{37} +14.2854 q^{38} +8.11450 q^{41} +9.93134 q^{43} +1.57529 q^{44} -5.45827 q^{46} +7.11253 q^{47} -3.70910 q^{49} -7.23229 q^{52} +3.16339 q^{53} +7.40815 q^{56} -10.6189 q^{58} +1.05559 q^{59} +6.75746 q^{61} +2.38937 q^{62} -10.8384 q^{64} -3.33851 q^{67} -10.9239 q^{68} -0.768686 q^{71} +9.02234 q^{73} +11.6003 q^{74} -22.1757 q^{76} -0.770457 q^{77} +5.54669 q^{79} -19.3886 q^{82} -0.355067 q^{83} -23.7297 q^{86} -1.73438 q^{88} +9.93939 q^{89} +3.53723 q^{91} +8.47306 q^{92} -16.9945 q^{94} -0.875505 q^{97} +8.86244 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 6 q^{11} - 4 q^{13} + 8 q^{14} - 5 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{22} - 8 q^{23} + 2 q^{26} - 6 q^{28} + 14 q^{29} - 5 q^{31} - 7 q^{32} - 2 q^{34} - 4 q^{37}+ \cdots + 5 q^{98}+O(q^{100}) \) 5 * q - q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8 + 6 * q^11 - 4 * q^13 + 8 * q^14 - 5 * q^16 - 12 * q^17 - 4 * q^19 + 4 * q^22 - 8 * q^23 + 2 * q^26 - 6 * q^28 + 14 * q^29 - 5 * q^31 - 7 * q^32 - 2 * q^34 - 4 * q^37 - 2 * q^38 + 16 * q^41 + 2 * q^43 + 4 * q^44 - 8 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^52 - 14 * q^53 + 12 * q^56 - 20 * q^58 + 10 * q^59 - 14 * q^61 + q^62 - 5 * q^64 + 2 * q^67 - 26 * q^68 + 12 * q^71 - 10 * q^73 + 10 * q^74 - 32 * q^76 - 26 * q^77 - 12 * q^79 + 6 * q^82 - 12 * q^83 - 10 * q^86 - 14 * q^88 + 14 * q^89 - 16 * q^91 + 14 * q^92 + 4 * q^94 - 8 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

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Database

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