LMFDB - Embedded newform 6975.2.a.bw.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 = [5,3,0,3,0,0,6,9,0,0,-2,0,4,0,0,11,12] 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:	$0$
Dimension:	$5$
Coefficient field:	5.5.223824.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 2x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 2x - 2
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			$2.57723$ 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)

$f(q)$	$=$	$q-1.57723 q^{2} +0.487640 q^{4} +5.00807 q^{7} +2.38533 q^{8} +5.58529 q^{11} +3.35325 q^{13} -7.89885 q^{14} -4.73749 q^{16} +0.280771 q^{17} -5.64647 q^{19} -8.80926 q^{22} +1.14781 q^{23} -5.28884 q^{26} +2.44213 q^{28} -4.93739 q^{29} +1.00000 q^{31} +2.70142 q^{32} -0.442839 q^{34} +11.1378 q^{37} +8.90576 q^{38} -3.66017 q^{41} +8.88866 q^{43} +2.72361 q^{44} -1.81036 q^{46} -5.65597 q^{47} +18.0807 q^{49} +1.63518 q^{52} +4.13186 q^{53} +11.9459 q^{56} +7.78738 q^{58} -12.4349 q^{59} +1.50798 q^{61} -1.57723 q^{62} +5.21423 q^{64} -2.37797 q^{67} +0.136915 q^{68} +15.6481 q^{71} +1.81490 q^{73} -17.5668 q^{74} -2.75344 q^{76} +27.9715 q^{77} +8.69779 q^{79} +5.77292 q^{82} -7.65060 q^{83} -14.0194 q^{86} +13.3228 q^{88} +9.58898 q^{89} +16.7933 q^{91} +0.559720 q^{92} +8.92073 q^{94} +9.26597 q^{97} -28.5174 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 3 q^{4} + 6 q^{7} + 9 q^{8} - 2 q^{11} + 4 q^{13} + 11 q^{16} + 12 q^{17} - 4 q^{19} - 16 q^{22} + 4 q^{23} - 18 q^{26} + 18 q^{28} - 18 q^{29} + 5 q^{31} + 21 q^{32} + 22 q^{34} + 16 q^{37}+ \cdots - 47 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 3 * q^4 + 6 * q^7 + 9 * q^8 - 2 * q^11 + 4 * q^13 + 11 * q^16 + 12 * q^17 - 4 * q^19 - 16 * q^22 + 4 * q^23 - 18 * q^26 + 18 * q^28 - 18 * q^29 + 5 * q^31 + 21 * q^32 + 22 * q^34 + 16 * q^37 + 22 * q^38 - 16 * q^41 + 10 * q^43 + 14 * q^47 + 5 * q^49 - 30 * q^52 + 14 * q^53 + 28 * q^56 - 4 * q^58 - 14 * q^59 + 10 * q^61 + 3 * q^62 + 27 * q^64 + 2 * q^67 + 34 * q^68 + 12 * q^71 + 22 * q^73 + 10 * q^74 + 16 * q^76 + 34 * q^77 + 12 * q^79 + 6 * q^82 + 8 * q^83 - 10 * q^86 + 2 * q^88 + 2 * q^89 + 8 * q^91 - 26 * q^92 + 20 * q^94 + 16 * q^97 - 47 * 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 $

$2$	−1.57723	−1.11527	−0.557633	−	0.830087i	$-0.688290\pi$
$2$	−1.57723	−1.11527	−0.557633	+	0.830087i	$0.688290\pi$
$3$	0	0
$3$	0	0
$4$	0.487640	0.243820
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00807	1.89287	0.946435	−	0.322893i	$-0.104655\pi$
$7$	5.00807	1.89287	0.946435	+	0.322893i	$0.104655\pi$
$8$	2.38533	0.843343
$9$	0	0
$10$	0	0
$11$	5.58529	1.68403	0.842014	−	0.539455i	$-0.181370\pi$
$11$	5.58529	1.68403	0.842014	+	0.539455i	$0.181370\pi$
$12$	0	0
$13$	3.35325	0.930025	0.465013	−	0.885304i	$-0.346050\pi$
$13$	3.35325	0.930025	0.465013	+	0.885304i	$0.346050\pi$
$14$	−7.89885	−2.11106
$15$	0	0
$16$	−4.73749	−1.18437
$17$	0.280771	0.0680970	0.0340485	−	0.999420i	$-0.489160\pi$
$17$	0.280771	0.0680970	0.0340485	+	0.999420i	$0.489160\pi$
$18$	0	0
$19$	−5.64647	−1.29539	−0.647695	−	0.761900i	$-0.724267\pi$
$19$	−5.64647	−1.29539	−0.647695	+	0.761900i	$0.724267\pi$
$20$	0	0
$21$	0	0
$22$	−8.80926	−1.87814
$23$	1.14781	0.239336	0.119668	−	0.992814i	$-0.461817\pi$
$23$	1.14781	0.239336	0.119668	+	0.992814i	$0.461817\pi$
$24$	0	0
$25$	0	0
$26$	−5.28884	−1.03723
$27$	0	0
$28$	2.44213	0.461520
$29$	−4.93739	−0.916850	−0.458425	−	0.888733i	$-0.651586\pi$
$29$	−4.93739	−0.916850	−0.458425	+	0.888733i	$0.651586\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	2.70142	0.477548
$33$	0	0
$34$	−0.442839	−0.0759463
$35$	0	0
$36$	0	0
$37$	11.1378	1.83104	0.915521	−	0.402270i	$-0.131779\pi$
$37$	11.1378	1.83104	0.915521	+	0.402270i	$0.131779\pi$
$38$	8.90576	1.44470
$39$	0	0
$40$	0	0
$41$	−3.66017	−0.571623	−0.285812	−	0.958286i	$-0.592263\pi$
$41$	−3.66017	−0.571623	−0.285812	+	0.958286i	$0.592263\pi$
$42$	0	0
$43$	8.88866	1.35551	0.677754	−	0.735289i	$-0.262954\pi$
$43$	8.88866	1.35551	0.677754	+	0.735289i	$0.262954\pi$
$44$	2.72361	0.410600
$45$	0	0
$46$	−1.81036	−0.266923
$47$	−5.65597	−0.825008	−0.412504	−	0.910956i	$-0.635346\pi$
$47$	−5.65597	−0.825008	−0.412504	+	0.910956i	$0.635346\pi$
$48$	0	0
$49$	18.0807	2.58296
$50$	0	0
$51$	0	0
$52$	1.63518	0.226759
$53$	4.13186	0.567554	0.283777	−	0.958890i	$-0.408412\pi$
$53$	4.13186	0.567554	0.283777	+	0.958890i	$0.408412\pi$
$54$	0	0
$55$	0	0
$56$	11.9459	1.59634
$57$	0	0
$58$	7.78738	1.02253
$59$	−12.4349	−1.61889	−0.809446	−	0.587194i	$-0.800233\pi$
$59$	−12.4349	−1.61889	−0.809446	+	0.587194i	$0.800233\pi$
$60$	0	0
$61$	1.50798	0.193077	0.0965384	−	0.995329i	$-0.469223\pi$
$61$	1.50798	0.193077	0.0965384	+	0.995329i	$0.469223\pi$
$62$	−1.57723	−0.200308
$63$	0	0
$64$	5.21423	0.651778
$65$	0	0
$66$	0	0
$67$	−2.37797	−0.290516	−0.145258	−	0.989394i	$-0.546401\pi$
$67$	−2.37797	−0.290516	−0.145258	+	0.989394i	$0.546401\pi$
$68$	0.136915	0.0166034
$69$	0	0
$70$	0	0
$71$	15.6481	1.85708	0.928542	−	0.371226i	$-0.121063\pi$
$71$	15.6481	1.85708	0.928542	+	0.371226i	$0.121063\pi$
$72$	0	0
$73$	1.81490	0.212418	0.106209	−	0.994344i	$-0.466129\pi$
$73$	1.81490	0.212418	0.106209	+	0.994344i	$0.466129\pi$
$74$	−17.5668	−2.04210
$75$	0	0
$76$	−2.75344	−0.315842
$77$	27.9715	3.18765
$78$	0	0
$79$	8.69779	0.978578	0.489289	−	0.872122i	$-0.337256\pi$
$79$	8.69779	0.978578	0.489289	+	0.872122i	$0.337256\pi$
$80$	0	0
$81$	0	0
$82$	5.77292	0.637512
$83$	−7.65060	−0.839763	−0.419881	−	0.907579i	$-0.637928\pi$
$83$	−7.65060	−0.839763	−0.419881	+	0.907579i	$0.637928\pi$
$84$	0	0
$85$	0	0
$86$	−14.0194	−1.51175
$87$	0	0
$88$	13.3228	1.42021
$89$	9.58898	1.01643	0.508215	−	0.861230i	$-0.330306\pi$
$89$	9.58898	1.01643	0.508215	+	0.861230i	$0.330306\pi$
$90$	0	0
$91$	16.7933	1.76042
$92$	0.559720	0.0583549
$93$	0	0
$94$	8.92073	0.920104
$95$	0	0
$96$	0	0
$97$	9.26597	0.940817	0.470408	−	0.882449i	$-0.344107\pi$
$97$	9.26597	0.940817	0.470408	+	0.882449i	$0.344107\pi$
$98$	−28.5174	−2.88069
$99$	0	0
$100$	0	0
$101$	−13.1628	−1.30975	−0.654873	−	0.755739i	$-0.727278\pi$
$101$	−13.1628	−1.30975	−0.654873	+	0.755739i	$0.727278\pi$
$102$	0	0
$103$	1.14413	0.112735	0.0563673	−	0.998410i	$-0.482048\pi$
$103$	1.14413	0.112735	0.0563673	+	0.998410i	$0.482048\pi$
$104$	7.99863	0.784330
$105$	0	0
$106$	−6.51687	−0.632974
$107$	5.11361	0.494351	0.247176	−	0.968971i	$-0.420498\pi$
$107$	5.11361	0.494351	0.247176	+	0.968971i	$0.420498\pi$
$108$	0	0
$109$	9.55661	0.915357	0.457679	−	0.889118i	$-0.348681\pi$
$109$	9.55661	0.915357	0.457679	+	0.889118i	$0.348681\pi$
$110$	0	0
$111$	0	0
$112$	−23.7256	−2.24186
$113$	−15.8921	−1.49500	−0.747502	−	0.664259i	$-0.768747\pi$
$113$	−15.8921	−1.49500	−0.747502	+	0.664259i	$0.768747\pi$
$114$	0	0
$115$	0	0
$116$	−2.40767	−0.223546
$117$	0	0
$118$	19.6127	1.80550
$119$	1.40612	0.128899
$120$	0	0
$121$	20.1955	1.83595
$122$	−2.37842	−0.215332
$123$	0	0
$124$	0.487640	0.0437913
$125$	0	0
$126$	0	0
$127$	−8.39706	−0.745118	−0.372559	−	0.928008i	$-0.621520\pi$
$127$	−8.39706	−0.745118	−0.372559	+	0.928008i	$0.621520\pi$
$128$	−13.6268	−1.20445
$129$	0	0
$130$	0	0
$131$	3.11942	0.272545	0.136272	−	0.990671i	$-0.456488\pi$
$131$	3.11942	0.272545	0.136272	+	0.990671i	$0.456488\pi$
$132$	0	0
$133$	−28.2779	−2.45201
$134$	3.75060	0.324003
$135$	0	0
$136$	0.669732	0.0574291
$137$	1.93010	0.164900	0.0824499	−	0.996595i	$-0.473726\pi$
$137$	1.93010	0.164900	0.0824499	+	0.996595i	$0.473726\pi$
$138$	0	0
$139$	−7.06969	−0.599644	−0.299822	−	0.953995i	$-0.596927\pi$
$139$	−7.06969	−0.599644	−0.299822	+	0.953995i	$0.596927\pi$
$140$	0	0
$141$	0	0
$142$	−24.6805	−2.07115
$143$	18.7289	1.56619
$144$	0	0
$145$	0	0
$146$	−2.86251	−0.236903
$147$	0	0
$148$	5.43123	0.446445
$149$	−17.5898	−1.44101	−0.720506	−	0.693448i	$-0.756091\pi$
$149$	−17.5898	−1.44101	−0.720506	+	0.693448i	$0.756091\pi$
$150$	0	0
$151$	3.07390	0.250151	0.125075	−	0.992147i	$-0.460083\pi$
$151$	3.07390	0.250151	0.125075	+	0.992147i	$0.460083\pi$
$152$	−13.4687	−1.09246
$153$	0	0
$154$	−44.1174	−3.55508
$155$	0	0
$156$	0	0
$157$	−10.3206	−0.823677	−0.411838	−	0.911257i	$-0.635113\pi$
$157$	−10.3206	−0.823677	−0.411838	+	0.911257i	$0.635113\pi$
$158$	−13.7184	−1.09138
$159$	0	0
$160$	0	0
$161$	5.74833	0.453032
$162$	0	0
$163$	−15.4991	−1.21398	−0.606992	−	0.794708i	$-0.707624\pi$
$163$	−15.4991	−1.21398	−0.606992	+	0.794708i	$0.707624\pi$
$164$	−1.78485	−0.139373
$165$	0	0
$166$	12.0667	0.936560
$167$	1.64933	0.127629	0.0638145	−	0.997962i	$-0.479673\pi$
$167$	1.64933	0.127629	0.0638145	+	0.997962i	$0.479673\pi$
$168$	0	0
$169$	−1.75569	−0.135053
$170$	0	0
$171$	0	0
$172$	4.33446	0.330500
$173$	14.0857	1.07091	0.535456	−	0.844563i	$-0.320140\pi$
$173$	14.0857	1.07091	0.535456	+	0.844563i	$0.320140\pi$
$174$	0	0
$175$	0	0
$176$	−26.4602	−1.99452
$177$	0	0
$178$	−15.1240	−1.13359
$179$	−10.9281	−0.816808	−0.408404	−	0.912801i	$-0.633915\pi$
$179$	−10.9281	−0.816808	−0.408404	+	0.912801i	$0.633915\pi$
$180$	0	0
$181$	9.81237	0.729348	0.364674	−	0.931135i	$-0.381180\pi$
$181$	9.81237	0.729348	0.364674	+	0.931135i	$0.381180\pi$
$182$	−26.4868	−1.96334
$183$	0	0
$184$	2.73792	0.201842
$185$	0	0
$186$	0	0
$187$	1.56819	0.114677
$188$	−2.75807	−0.201153
$189$	0	0
$190$	0	0
$191$	6.66862	0.482524	0.241262	−	0.970460i	$-0.422439\pi$
$191$	6.66862	0.482524	0.241262	+	0.970460i	$0.422439\pi$
$192$	0	0
$193$	15.7684	1.13504	0.567518	−	0.823361i	$-0.307904\pi$
$193$	15.7684	1.13504	0.567518	+	0.823361i	$0.307904\pi$
$194$	−14.6145	−1.04926
$195$	0	0
$196$	8.81688	0.629777
$197$	22.6660	1.61489	0.807443	−	0.589945i	$-0.200851\pi$
$197$	22.6660	1.61489	0.807443	+	0.589945i	$0.200851\pi$
$198$	0	0
$199$	−21.3925	−1.51647	−0.758236	−	0.651980i	$-0.773939\pi$
$199$	−21.3925	−1.51647	−0.758236	+	0.651980i	$0.773939\pi$
$200$	0	0
$201$	0	0
$202$	20.7607	1.46072
$203$	−24.7268	−1.73548
$204$	0	0
$205$	0	0
$206$	−1.80455	−0.125729
$207$	0	0
$208$	−15.8860	−1.10150
$209$	−31.5372	−2.18147
$210$	0	0
$211$	−15.6047	−1.07427	−0.537136	−	0.843496i	$-0.680494\pi$
$211$	−15.6047	−1.07427	−0.537136	+	0.843496i	$0.680494\pi$
$212$	2.01486	0.138381
$213$	0	0
$214$	−8.06531	−0.551333
$215$	0	0
$216$	0	0
$217$	5.00807	0.339970
$218$	−15.0729	−1.02087
$219$	0	0
$220$	0	0
$221$	0.941497	0.0633319
$222$	0	0
$223$	19.9499	1.33594	0.667972	−	0.744187i	$-0.267163\pi$
$223$	19.9499	1.33594	0.667972	+	0.744187i	$0.267163\pi$
$224$	13.5289	0.903937
$225$	0	0
$226$	25.0654	1.66733
$227$	6.10015	0.404881	0.202441	−	0.979295i	$-0.435113\pi$
$227$	6.10015	0.404881	0.202441	+	0.979295i	$0.435113\pi$
$228$	0	0
$229$	−15.6787	−1.03608	−0.518040	−	0.855356i	$-0.673338\pi$
$229$	−15.6787	−1.03608	−0.518040	+	0.855356i	$0.673338\pi$
$230$	0	0
$231$	0	0
$232$	−11.7773	−0.773219
$233$	2.48002	0.162472	0.0812358	−	0.996695i	$-0.474113\pi$
$233$	2.48002	0.162472	0.0812358	+	0.996695i	$0.474113\pi$
$234$	0	0
$235$	0	0
$236$	−6.06378	−0.394718
$237$	0	0
$238$	−2.21777	−0.143757
$239$	26.9693	1.74450	0.872248	−	0.489064i	$-0.162662\pi$
$239$	26.9693	1.74450	0.872248	+	0.489064i	$0.162662\pi$
$240$	0	0
$241$	−4.37932	−0.282097	−0.141048	−	0.990003i	$-0.545047\pi$
$241$	−4.37932	−0.282097	−0.141048	+	0.990003i	$0.545047\pi$
$242$	−31.8528	−2.04758
$243$	0	0
$244$	0.735350	0.0470760
$245$	0	0
$246$	0	0
$247$	−18.9341	−1.20474
$248$	2.38533	0.151469
$249$	0	0
$250$	0	0
$251$	20.4073	1.28810	0.644048	−	0.764985i	$-0.277254\pi$
$251$	20.4073	1.28810	0.644048	+	0.764985i	$0.277254\pi$
$252$	0	0
$253$	6.41088	0.403048
$254$	13.2441	0.831006
$255$	0	0
$256$	11.0642	0.691510
$257$	−26.3849	−1.64584	−0.822921	−	0.568155i	$-0.807657\pi$
$257$	−26.3849	−1.64584	−0.822921	+	0.568155i	$0.807657\pi$
$258$	0	0
$259$	55.7788	3.46593
$260$	0	0
$261$	0	0
$262$	−4.92003	−0.303960
$263$	1.56283	0.0963679	0.0481840	−	0.998838i	$-0.484657\pi$
$263$	1.56283	0.0963679	0.0481840	+	0.998838i	$0.484657\pi$
$264$	0	0
$265$	0	0
$266$	44.6006	2.73464
$267$	0	0
$268$	−1.15959	−0.0708335
$269$	−23.0685	−1.40651	−0.703257	−	0.710936i	$-0.748272\pi$
$269$	−23.0685	−1.40651	−0.703257	+	0.710936i	$0.748272\pi$
$270$	0	0
$271$	27.0894	1.64557	0.822783	−	0.568355i	$-0.192420\pi$
$271$	27.0894	1.64557	0.822783	+	0.568355i	$0.192420\pi$
$272$	−1.33015	−0.0806522
$273$	0	0
$274$	−3.04421	−0.183907
$275$	0	0
$276$	0	0
$277$	13.0846	0.786177	0.393089	−	0.919501i	$-0.371407\pi$
$277$	13.0846	0.786177	0.393089	+	0.919501i	$0.371407\pi$
$278$	11.1505	0.668763
$279$	0	0
$280$	0	0
$281$	−10.1297	−0.604289	−0.302145	−	0.953262i	$-0.597703\pi$
$281$	−10.1297	−0.604289	−0.302145	+	0.953262i	$0.597703\pi$
$282$	0	0
$283$	−10.5172	−0.625185	−0.312593	−	0.949887i	$-0.601197\pi$
$283$	−10.5172	−0.625185	−0.312593	+	0.949887i	$0.601197\pi$
$284$	7.63062	0.452794
$285$	0	0
$286$	−29.5397	−1.74672
$287$	−18.3304	−1.08201
$288$	0	0
$289$	−16.9212	−0.995363
$290$	0	0
$291$	0	0
$292$	0.885017	0.0517917
$293$	4.00213	0.233807	0.116903	−	0.993143i	$-0.462703\pi$
$293$	4.00213	0.233807	0.116903	+	0.993143i	$0.462703\pi$
$294$	0	0
$295$	0	0
$296$	26.5674	1.54420
$297$	0	0
$298$	27.7431	1.60711
$299$	3.84891	0.222588
$300$	0	0
$301$	44.5150	2.56580
$302$	−4.84824	−0.278985
$303$	0	0
$304$	26.7501	1.53422
$305$	0	0
$306$	0	0
$307$	−15.8305	−0.903493	−0.451747	−	0.892146i	$-0.649199\pi$
$307$	−15.8305	−0.903493	−0.451747	+	0.892146i	$0.649199\pi$
$308$	13.6400	0.777212
$309$	0	0
$310$	0	0
$311$	−13.2168	−0.749457	−0.374728	−	0.927135i	$-0.622264\pi$
$311$	−13.2168	−0.749457	−0.374728	+	0.927135i	$0.622264\pi$
$312$	0	0
$313$	15.8035	0.893267	0.446634	−	0.894717i	$-0.352623\pi$
$313$	15.8035	0.893267	0.446634	+	0.894717i	$0.352623\pi$
$314$	16.2780	0.918619
$315$	0	0
$316$	4.24139	0.238597
$317$	−13.9027	−0.780853	−0.390427	−	0.920634i	$-0.627672\pi$
$317$	−13.9027	−0.780853	−0.390427	+	0.920634i	$0.627672\pi$
$318$	0	0
$319$	−27.5768	−1.54400
$320$	0	0
$321$	0	0
$322$	−9.06641	−0.505251
$323$	−1.58537	−0.0882121
$324$	0	0
$325$	0	0
$326$	24.4456	1.35392
$327$	0	0
$328$	−8.73073	−0.482074
$329$	−28.3255	−1.56163
$330$	0	0
$331$	−19.0250	−1.04571	−0.522855	−	0.852422i	$-0.675133\pi$
$331$	−19.0250	−1.04571	−0.522855	+	0.852422i	$0.675133\pi$
$332$	−3.73074	−0.204751
$333$	0	0
$334$	−2.60137	−0.142340
$335$	0	0
$336$	0	0
$337$	9.61709	0.523876	0.261938	−	0.965085i	$-0.415638\pi$
$337$	9.61709	0.523876	0.261938	+	0.965085i	$0.415638\pi$
$338$	2.76912	0.150620
$339$	0	0
$340$	0	0
$341$	5.58529	0.302460
$342$	0	0
$343$	55.4930	2.99634
$344$	21.2024	1.14316
$345$	0	0
$346$	−22.2162	−1.19435
$347$	−23.3944	−1.25588	−0.627938	−	0.778263i	$-0.716101\pi$
$347$	−23.3944	−1.25588	−0.627938	+	0.778263i	$0.716101\pi$
$348$	0	0
$349$	−21.7353	−1.16346	−0.581732	−	0.813381i	$-0.697625\pi$
$349$	−21.7353	−1.16346	−0.581732	+	0.813381i	$0.697625\pi$
$350$	0	0
$351$	0	0
$352$	15.0882	0.804204
$353$	−11.2284	−0.597628	−0.298814	−	0.954311i	$-0.596591\pi$
$353$	−11.2284	−0.597628	−0.298814	+	0.954311i	$0.596591\pi$
$354$	0	0
$355$	0	0
$356$	4.67597	0.247826
$357$	0	0
$358$	17.2362	0.910959
$359$	−4.00083	−0.211155	−0.105578	−	0.994411i	$-0.533669\pi$
$359$	−4.00083	−0.211155	−0.105578	+	0.994411i	$0.533669\pi$
$360$	0	0
$361$	12.8826	0.678034
$362$	−15.4763	−0.813417
$363$	0	0
$364$	8.18909	0.429225
$365$	0	0
$366$	0	0
$367$	24.7834	1.29368	0.646841	−	0.762625i	$-0.276090\pi$
$367$	24.7834	1.29368	0.646841	+	0.762625i	$0.276090\pi$
$368$	−5.43776	−0.283463
$369$	0	0
$370$	0	0
$371$	20.6926	1.07431
$372$	0	0
$373$	25.0935	1.29929	0.649646	−	0.760237i	$-0.274917\pi$
$373$	25.0935	1.29929	0.649646	+	0.760237i	$0.274917\pi$
$374$	−2.47339	−0.127896
$375$	0	0
$376$	−13.4914	−0.695764
$377$	−16.5563	−0.852694
$378$	0	0
$379$	35.1856	1.80736	0.903682	−	0.428204i	$-0.140854\pi$
$379$	35.1856	1.80736	0.903682	+	0.428204i	$0.140854\pi$
$380$	0	0
$381$	0	0
$382$	−10.5179	−0.538143
$383$	1.15429	0.0589813	0.0294906	−	0.999565i	$-0.490611\pi$
$383$	1.15429	0.0589813	0.0294906	+	0.999565i	$0.490611\pi$
$384$	0	0
$385$	0	0
$386$	−24.8703	−1.26587
$387$	0	0
$388$	4.51846	0.229390
$389$	18.0822	0.916804	0.458402	−	0.888745i	$-0.348422\pi$
$389$	18.0822	0.916804	0.458402	+	0.888745i	$0.348422\pi$
$390$	0	0
$391$	0.322273	0.0162981
$392$	43.1285	2.17832
$393$	0	0
$394$	−35.7494	−1.80103
$395$	0	0
$396$	0	0
$397$	23.4453	1.17668	0.588342	−	0.808612i	$-0.299781\pi$
$397$	23.4453	1.17668	0.588342	+	0.808612i	$0.299781\pi$
$398$	33.7408	1.69127
$399$	0	0
$400$	0	0
$401$	7.11418	0.355265	0.177632	−	0.984097i	$-0.443156\pi$
$401$	7.11418	0.355265	0.177632	+	0.984097i	$0.443156\pi$
$402$	0	0
$403$	3.35325	0.167037
$404$	−6.41870	−0.319342
$405$	0	0
$406$	38.9997	1.93552
$407$	62.2078	3.08353
$408$	0	0
$409$	8.69609	0.429994	0.214997	−	0.976615i	$-0.431026\pi$
$409$	8.69609	0.429994	0.214997	+	0.976615i	$0.431026\pi$
$410$	0	0
$411$	0	0
$412$	0.557923	0.0274869
$413$	−62.2750	−3.06435
$414$	0	0
$415$	0	0
$416$	9.05854	0.444132
$417$	0	0
$418$	49.7413	2.43292
$419$	4.17671	0.204046	0.102023	−	0.994782i	$-0.467468\pi$
$419$	4.17671	0.204046	0.102023	+	0.994782i	$0.467468\pi$
$420$	0	0
$421$	−9.81464	−0.478336	−0.239168	−	0.970978i	$-0.576875\pi$
$421$	−9.81464	−0.478336	−0.239168	+	0.970978i	$0.576875\pi$
$422$	24.6121	1.19810
$423$	0	0
$424$	9.85585	0.478643
$425$	0	0
$426$	0	0
$427$	7.55206	0.365470
$428$	2.49360	0.120533
$429$	0	0
$430$	0	0
$431$	−6.88478	−0.331628	−0.165814	−	0.986157i	$-0.553025\pi$
$431$	−6.88478	−0.331628	−0.165814	+	0.986157i	$0.553025\pi$
$432$	0	0
$433$	−32.2480	−1.54974	−0.774869	−	0.632121i	$-0.782184\pi$
$433$	−32.2480	−1.54974	−0.774869	+	0.632121i	$0.782184\pi$
$434$	−7.89885	−0.379157
$435$	0	0
$436$	4.66018	0.223182
$437$	−6.48110	−0.310033
$438$	0	0
$439$	0.425908	0.0203275	0.0101637	−	0.999948i	$-0.496765\pi$
$439$	0.425908	0.0203275	0.0101637	+	0.999948i	$0.496765\pi$
$440$	0	0
$441$	0	0
$442$	−1.48495	−0.0706320
$443$	17.2781	0.820908	0.410454	−	0.911881i	$-0.365370\pi$
$443$	17.2781	0.820908	0.410454	+	0.911881i	$0.365370\pi$
$444$	0	0
$445$	0	0
$446$	−31.4655	−1.48993
$447$	0	0
$448$	26.1132	1.23373
$449$	32.4566	1.53172	0.765860	−	0.643007i	$-0.222314\pi$
$449$	32.4566	1.53172	0.765860	+	0.643007i	$0.222314\pi$
$450$	0	0
$451$	−20.4431	−0.962630
$452$	−7.74963	−0.364512
$453$	0	0
$454$	−9.62132	−0.451551
$455$	0	0
$456$	0	0
$457$	−6.28438	−0.293971	−0.146985	−	0.989139i	$-0.546957\pi$
$457$	−6.28438	−0.293971	−0.146985	+	0.989139i	$0.546957\pi$
$458$	24.7289	1.15551
$459$	0	0
$460$	0	0
$461$	−24.4903	−1.14063	−0.570314	−	0.821426i	$-0.693179\pi$
$461$	−24.4903	−1.14063	−0.570314	+	0.821426i	$0.693179\pi$
$462$	0	0
$463$	−2.29936	−0.106860	−0.0534302	−	0.998572i	$-0.517015\pi$
$463$	−2.29936	−0.106860	−0.0534302	+	0.998572i	$0.517015\pi$
$464$	23.3908	1.08589
$465$	0	0
$466$	−3.91155	−0.181199
$467$	6.06520	0.280664	0.140332	−	0.990105i	$-0.455183\pi$
$467$	6.06520	0.280664	0.140332	+	0.990105i	$0.455183\pi$
$468$	0	0
$469$	−11.9091	−0.549909
$470$	0	0
$471$	0	0
$472$	−29.6615	−1.36528
$473$	49.6457	2.28271
$474$	0	0
$475$	0	0
$476$	0.685680	0.0314281
$477$	0	0
$478$	−42.5366	−1.94558
$479$	−40.3693	−1.84452	−0.922261	−	0.386569i	$-0.873660\pi$
$479$	−40.3693	−1.84452	−0.922261	+	0.386569i	$0.873660\pi$
$480$	0	0
$481$	37.3479	1.70292
$482$	6.90717	0.314613
$483$	0	0
$484$	9.84812	0.447642
$485$	0	0
$486$	0	0
$487$	−12.0082	−0.544144	−0.272072	−	0.962277i	$-0.587709\pi$
$487$	−12.0082	−0.544144	−0.272072	+	0.962277i	$0.587709\pi$
$488$	3.59703	0.162830
$489$	0	0
$490$	0	0
$491$	36.4834	1.64647	0.823237	−	0.567698i	$-0.192166\pi$
$491$	36.4834	1.64647	0.823237	+	0.567698i	$0.192166\pi$
$492$	0	0
$493$	−1.38628	−0.0624347
$494$	29.8633	1.34361
$495$	0	0
$496$	−4.73749	−0.212719
$497$	78.3666	3.51522
$498$	0	0
$499$	28.4505	1.27362	0.636810	−	0.771021i	$-0.280254\pi$
$499$	28.4505	1.27362	0.636810	+	0.771021i	$0.280254\pi$
$500$	0	0
$501$	0	0
$502$	−32.1869	−1.43657
$503$	−11.0760	−0.493854	−0.246927	−	0.969034i	$-0.579421\pi$
$503$	−11.0760	−0.493854	−0.246927	+	0.969034i	$0.579421\pi$
$504$	0	0
$505$	0	0
$506$	−10.1114	−0.449507
$507$	0	0
$508$	−4.09474	−0.181675
$509$	−36.0031	−1.59581	−0.797904	−	0.602785i	$-0.794058\pi$
$509$	−36.0031	−1.59581	−0.797904	+	0.602785i	$0.794058\pi$
$510$	0	0
$511$	9.08914	0.402080
$512$	9.80303	0.433237
$513$	0	0
$514$	41.6149	1.83555
$515$	0	0
$516$	0	0
$517$	−31.5902	−1.38934
$518$	−87.9758	−3.86543
$519$	0	0
$520$	0	0
$521$	1.94132	0.0850509	0.0425254	−	0.999095i	$-0.486460\pi$
$521$	1.94132	0.0850509	0.0425254	+	0.999095i	$0.486460\pi$
$522$	0	0
$523$	−19.9093	−0.870574	−0.435287	−	0.900292i	$-0.643353\pi$
$523$	−19.9093	−0.870574	−0.435287	+	0.900292i	$0.643353\pi$
$524$	1.52115	0.0664519
$525$	0	0
$526$	−2.46493	−0.107476
$527$	0.280771	0.0122306
$528$	0	0
$529$	−21.6825	−0.942718
$530$	0	0
$531$	0	0
$532$	−13.7894	−0.597848
$533$	−12.2735	−0.531624
$534$	0	0
$535$	0	0
$536$	−5.67226	−0.245004
$537$	0	0
$538$	36.3843	1.56864
$539$	100.986	4.34978
$540$	0	0
$541$	−23.0466	−0.990851	−0.495426	−	0.868650i	$-0.664988\pi$
$541$	−23.0466	−0.990851	−0.495426	+	0.868650i	$0.664988\pi$
$542$	−42.7262	−1.83525
$543$	0	0
$544$	0.758480	0.0325196
$545$	0	0
$546$	0	0
$547$	−6.07899	−0.259919	−0.129959	−	0.991519i	$-0.541485\pi$
$547$	−6.07899	−0.259919	−0.129959	+	0.991519i	$0.541485\pi$
$548$	0.941195	0.0402058
$549$	0	0
$550$	0	0
$551$	27.8788	1.18768
$552$	0	0
$553$	43.5591	1.85232
$554$	−20.6374	−0.876798
$555$	0	0
$556$	−3.44746	−0.146205
$557$	24.5583	1.04057	0.520285	−	0.853993i	$-0.325826\pi$
$557$	24.5583	1.04057	0.520285	+	0.853993i	$0.325826\pi$
$558$	0	0
$559$	29.8059	1.26066
$560$	0	0
$561$	0	0
$562$	15.9769	0.673943
$563$	8.93479	0.376556	0.188278	−	0.982116i	$-0.439709\pi$
$563$	8.93479	0.376556	0.188278	+	0.982116i	$0.439709\pi$
$564$	0	0
$565$	0	0
$566$	16.5881	0.697248
$567$	0	0
$568$	37.3259	1.56616
$569$	−3.41217	−0.143046	−0.0715228	−	0.997439i	$-0.522786\pi$
$569$	−3.41217	−0.143046	−0.0715228	+	0.997439i	$0.522786\pi$
$570$	0	0
$571$	−16.0935	−0.673493	−0.336746	−	0.941595i	$-0.609327\pi$
$571$	−16.0935	−0.673493	−0.336746	+	0.941595i	$0.609327\pi$
$572$	9.13296	0.381868
$573$	0	0
$574$	28.9112	1.20673
$575$	0	0
$576$	0	0
$577$	−25.9663	−1.08099	−0.540495	−	0.841347i	$-0.681763\pi$
$577$	−25.9663	−1.08099	−0.540495	+	0.841347i	$0.681763\pi$
$578$	26.6885	1.11010
$579$	0	0
$580$	0	0
$581$	−38.3147	−1.58956
$582$	0	0
$583$	23.0776	0.955778
$584$	4.32914	0.179141
$585$	0	0
$586$	−6.31226	−0.260757
$587$	44.1926	1.82402	0.912012	−	0.410164i	$-0.134528\pi$
$587$	44.1926	1.82402	0.912012	+	0.410164i	$0.134528\pi$
$588$	0	0
$589$	−5.64647	−0.232659
$590$	0	0
$591$	0	0
$592$	−52.7652	−2.16863
$593$	−13.8500	−0.568753	−0.284376	−	0.958713i	$-0.591787\pi$
$593$	−13.8500	−0.568753	−0.284376	+	0.958713i	$0.591787\pi$
$594$	0	0
$595$	0	0
$596$	−8.57749	−0.351348
$597$	0	0
$598$	−6.07060	−0.248245
$599$	−22.7377	−0.929038	−0.464519	−	0.885563i	$-0.653773\pi$
$599$	−22.7377	−0.929038	−0.464519	+	0.885563i	$0.653773\pi$
$600$	0	0
$601$	−17.9588	−0.732554	−0.366277	−	0.930506i	$-0.619368\pi$
$601$	−17.9588	−0.732554	−0.366277	+	0.930506i	$0.619368\pi$
$602$	−70.2101	−2.86155
$603$	0	0
$604$	1.49896	0.0609917
$605$	0	0
$606$	0	0
$607$	29.4317	1.19460	0.597298	−	0.802020i	$-0.296241\pi$
$607$	29.4317	1.19460	0.597298	+	0.802020i	$0.296241\pi$
$608$	−15.2535	−0.618611
$609$	0	0
$610$	0	0
$611$	−18.9659	−0.767278
$612$	0	0
$613$	−10.4561	−0.422316	−0.211158	−	0.977452i	$-0.567724\pi$
$613$	−10.4561	−0.422316	−0.211158	+	0.977452i	$0.567724\pi$
$614$	24.9682	1.00764
$615$	0	0
$616$	66.7213	2.68828
$617$	−14.7998	−0.595819	−0.297909	−	0.954594i	$-0.596289\pi$
$617$	−14.7998	−0.595819	−0.297909	+	0.954594i	$0.596289\pi$
$618$	0	0
$619$	9.82309	0.394823	0.197412	−	0.980321i	$-0.436746\pi$
$619$	9.82309	0.394823	0.197412	+	0.980321i	$0.436746\pi$
$620$	0	0
$621$	0	0
$622$	20.8459	0.835844
$623$	48.0222	1.92397
$624$	0	0
$625$	0	0
$626$	−24.9257	−0.996231
$627$	0	0
$628$	−5.03276	−0.200829
$629$	3.12717	0.124688
$630$	0	0
$631$	−29.3447	−1.16820	−0.584098	−	0.811683i	$-0.698552\pi$
$631$	−29.3447	−1.16820	−0.584098	+	0.811683i	$0.698552\pi$
$632$	20.7471	0.825276
$633$	0	0
$634$	21.9277	0.870860
$635$	0	0
$636$	0	0
$637$	60.6292	2.40222
$638$	43.4948	1.72197
$639$	0	0
$640$	0	0
$641$	−20.0918	−0.793580	−0.396790	−	0.917909i	$-0.629876\pi$
$641$	−20.0918	−0.793580	−0.396790	+	0.917909i	$0.629876\pi$
$642$	0	0
$643$	1.43114	0.0564387	0.0282193	−	0.999602i	$-0.491016\pi$
$643$	1.43114	0.0564387	0.0282193	+	0.999602i	$0.491016\pi$
$644$	2.80311	0.110458
$645$	0	0
$646$	2.50048	0.0983800
$647$	−31.0740	−1.22165	−0.610823	−	0.791767i	$-0.709161\pi$
$647$	−31.0740	−1.22165	−0.610823	+	0.791767i	$0.709161\pi$
$648$	0	0
$649$	−69.4528	−2.72626
$650$	0	0
$651$	0	0
$652$	−7.55799	−0.295994
$653$	50.2166	1.96513	0.982564	−	0.185924i	$-0.0595277\pi$
$653$	50.2166	1.96513	0.982564	+	0.185924i	$0.0595277\pi$
$654$	0	0
$655$	0	0
$656$	17.3400	0.677014
$657$	0	0
$658$	44.6756	1.74164
$659$	−26.4863	−1.03176	−0.515879	−	0.856661i	$-0.672535\pi$
$659$	−26.4863	−1.03176	−0.515879	+	0.856661i	$0.672535\pi$
$660$	0	0
$661$	−22.0125	−0.856188	−0.428094	−	0.903734i	$-0.640815\pi$
$661$	−22.0125	−0.856188	−0.428094	+	0.903734i	$0.640815\pi$
$662$	30.0067	1.16625
$663$	0	0
$664$	−18.2492	−0.708208
$665$	0	0
$666$	0	0
$667$	−5.66721	−0.219435
$668$	0.804280	0.0311185
$669$	0	0
$670$	0	0
$671$	8.42250	0.325147
$672$	0	0
$673$	−49.7148	−1.91637	−0.958183	−	0.286157i	$-0.907622\pi$
$673$	−49.7148	−1.91637	−0.958183	+	0.286157i	$0.907622\pi$
$674$	−15.1683	−0.584262
$675$	0	0
$676$	−0.856144	−0.0329286
$677$	22.0765	0.848467	0.424234	−	0.905553i	$-0.360544\pi$
$677$	22.0765	0.848467	0.424234	+	0.905553i	$0.360544\pi$
$678$	0	0
$679$	46.4046	1.78084
$680$	0	0
$681$	0	0
$682$	−8.80926	−0.337324
$683$	−30.1274	−1.15279	−0.576396	−	0.817171i	$-0.695541\pi$
$683$	−30.1274	−1.15279	−0.576396	+	0.817171i	$0.695541\pi$
$684$	0	0
$685$	0	0
$686$	−87.5249	−3.34172
$687$	0	0
$688$	−42.1099	−1.60542
$689$	13.8552	0.527840
$690$	0	0
$691$	−45.2741	−1.72231	−0.861154	−	0.508343i	$-0.830258\pi$
$691$	−45.2741	−1.72231	−0.861154	+	0.508343i	$0.830258\pi$
$692$	6.86872	0.261110
$693$	0	0
$694$	36.8982	1.40064
$695$	0	0
$696$	0	0
$697$	−1.02767	−0.0389258
$698$	34.2814	1.29757
$699$	0	0
$700$	0	0
$701$	−2.78456	−0.105171	−0.0525857	−	0.998616i	$-0.516746\pi$
$701$	−2.78456	−0.105171	−0.0525857	+	0.998616i	$0.516746\pi$
$702$	0	0
$703$	−62.8893	−2.37191
$704$	29.1230	1.09761
$705$	0	0
$706$	17.7097	0.666515
$707$	−65.9201	−2.47918
$708$	0	0
$709$	23.6776	0.889232	0.444616	−	0.895721i	$-0.353340\pi$
$709$	23.6776	0.889232	0.444616	+	0.895721i	$0.353340\pi$
$710$	0	0
$711$	0	0
$712$	22.8729	0.857198
$713$	1.14781	0.0429860
$714$	0	0
$715$	0	0
$716$	−5.32900	−0.199154
$717$	0	0
$718$	6.31020	0.235495
$719$	−25.7767	−0.961308	−0.480654	−	0.876910i	$-0.659601\pi$
$719$	−25.7767	−0.961308	−0.480654	+	0.876910i	$0.659601\pi$
$720$	0	0
$721$	5.72988	0.213392
$722$	−20.3188	−0.756189
$723$	0	0
$724$	4.78490	0.177830
$725$	0	0
$726$	0	0
$727$	32.0094	1.18716	0.593582	−	0.804774i	$-0.297713\pi$
$727$	32.0094	1.18716	0.593582	+	0.804774i	$0.297713\pi$
$728$	40.0576	1.48464
$729$	0	0
$730$	0	0
$731$	2.49568	0.0923060
$732$	0	0
$733$	−38.7807	−1.43240	−0.716199	−	0.697896i	$-0.754120\pi$
$733$	−38.7807	−1.43240	−0.716199	+	0.697896i	$0.754120\pi$
$734$	−39.0890	−1.44280
$735$	0	0
$736$	3.10073	0.114294
$737$	−13.2817	−0.489237
$738$	0	0
$739$	49.2404	1.81134	0.905669	−	0.423986i	$-0.139369\pi$
$739$	49.2404	1.81134	0.905669	+	0.423986i	$0.139369\pi$
$740$	0	0
$741$	0	0
$742$	−32.6369	−1.19814
$743$	50.7533	1.86196	0.930978	−	0.365074i	$-0.118956\pi$
$743$	50.7533	1.86196	0.930978	+	0.365074i	$0.118956\pi$
$744$	0	0
$745$	0	0
$746$	−39.5781	−1.44906
$747$	0	0
$748$	0.764711	0.0279606
$749$	25.6093	0.935743
$750$	0	0
$751$	20.2448	0.738743	0.369371	−	0.929282i	$-0.379573\pi$
$751$	20.2448	0.738743	0.369371	+	0.929282i	$0.379573\pi$
$752$	26.7951	0.977116
$753$	0	0
$754$	26.1130	0.950981
$755$	0	0
$756$	0	0
$757$	24.4549	0.888827	0.444413	−	0.895822i	$-0.353412\pi$
$757$	24.4549	0.888827	0.444413	+	0.895822i	$0.353412\pi$
$758$	−55.4957	−2.01569
$759$	0	0
$760$	0	0
$761$	−31.6937	−1.14890	−0.574448	−	0.818541i	$-0.694783\pi$
$761$	−31.6937	−1.14890	−0.574448	+	0.818541i	$0.694783\pi$
$762$	0	0
$763$	47.8601	1.73265
$764$	3.25188	0.117649
$765$	0	0
$766$	−1.82057	−0.0657799
$767$	−41.6975	−1.50561
$768$	0	0
$769$	7.17944	0.258897	0.129449	−	0.991586i	$-0.458679\pi$
$769$	7.17944	0.258897	0.129449	+	0.991586i	$0.458679\pi$
$770$	0	0
$771$	0	0
$772$	7.68931	0.276744
$773$	−11.6966	−0.420699	−0.210350	−	0.977626i	$-0.567460\pi$
$773$	−11.6966	−0.420699	−0.210350	+	0.977626i	$0.567460\pi$
$774$	0	0
$775$	0	0
$776$	22.1024	0.793431
$777$	0	0
$778$	−28.5197	−1.02248
$779$	20.6671	0.740475
$780$	0	0
$781$	87.3991	3.12738
$782$	−0.508297	−0.0181767
$783$	0	0
$784$	−85.6572	−3.05919
$785$	0	0
$786$	0	0
$787$	4.22457	0.150590	0.0752948	−	0.997161i	$-0.476010\pi$
$787$	4.22457	0.150590	0.0752948	+	0.997161i	$0.476010\pi$
$788$	11.0529	0.393742
$789$	0	0
$790$	0	0
$791$	−79.5887	−2.82985
$792$	0	0
$793$	5.05664	0.179566
$794$	−36.9785	−1.31232
$795$	0	0
$796$	−10.4318	−0.369746
$797$	46.6401	1.65208	0.826038	−	0.563614i	$-0.190590\pi$
$797$	46.6401	1.65208	0.826038	+	0.563614i	$0.190590\pi$
$798$	0	0
$799$	−1.58803	−0.0561805
$800$	0	0
$801$	0	0
$802$	−11.2207	−0.396215
$803$	10.1367	0.357718
$804$	0	0
$805$	0	0
$806$	−5.28884	−0.186291
$807$	0	0
$808$	−31.3976	−1.10457
$809$	−17.0229	−0.598494	−0.299247	−	0.954176i	$-0.596735\pi$
$809$	−17.0229	−0.598494	−0.299247	+	0.954176i	$0.596735\pi$
$810$	0	0
$811$	−3.57750	−0.125623	−0.0628115	−	0.998025i	$-0.520007\pi$
$811$	−3.57750	−0.125623	−0.0628115	+	0.998025i	$0.520007\pi$
$812$	−12.0578	−0.423144
$813$	0	0
$814$	−98.1158	−3.43896
$815$	0	0
$816$	0	0
$817$	−50.1895	−1.75591
$818$	−13.7157	−0.479558
$819$	0	0
$820$	0	0
$821$	43.7349	1.52636	0.763179	−	0.646188i	$-0.223638\pi$
$821$	43.7349	1.52636	0.763179	+	0.646188i	$0.223638\pi$
$822$	0	0
$823$	3.83204	0.133577	0.0667883	−	0.997767i	$-0.478725\pi$
$823$	3.83204	0.133577	0.0667883	+	0.997767i	$0.478725\pi$
$824$	2.72913	0.0950738
$825$	0	0
$826$	98.2218	3.41757
$827$	−5.45334	−0.189631	−0.0948156	−	0.995495i	$-0.530226\pi$
$827$	−5.45334	−0.189631	−0.0948156	+	0.995495i	$0.530226\pi$
$828$	0	0
$829$	31.6739	1.10008	0.550040	−	0.835138i	$-0.314613\pi$
$829$	31.6739	1.10008	0.550040	+	0.835138i	$0.314613\pi$
$830$	0	0
$831$	0	0
$832$	17.4846	0.606170
$833$	5.07654	0.175892
$834$	0	0
$835$	0	0
$836$	−15.3788	−0.531887
$837$	0	0
$838$	−6.58762	−0.227566
$839$	−20.2645	−0.699610	−0.349805	−	0.936823i	$-0.613752\pi$
$839$	−20.2645	−0.699610	−0.349805	+	0.936823i	$0.613752\pi$
$840$	0	0
$841$	−4.62218	−0.159386
$842$	15.4799	0.533472
$843$	0	0
$844$	−7.60948	−0.261929
$845$	0	0
$846$	0	0
$847$	101.140	3.47522
$848$	−19.5746	−0.672195
$849$	0	0
$850$	0	0
$851$	12.7841	0.438234
$852$	0	0
$853$	3.62200	0.124015	0.0620075	−	0.998076i	$-0.480250\pi$
$853$	3.62200	0.124015	0.0620075	+	0.998076i	$0.480250\pi$
$854$	−11.9113	−0.407596
$855$	0	0
$856$	12.1977	0.416907
$857$	−1.05356	−0.0359890	−0.0179945	−	0.999838i	$-0.505728\pi$
$857$	−1.05356	−0.0359890	−0.0179945	+	0.999838i	$0.505728\pi$
$858$	0	0
$859$	−50.9355	−1.73790	−0.868948	−	0.494903i	$-0.835204\pi$
$859$	−50.9355	−1.73790	−0.868948	+	0.494903i	$0.835204\pi$
$860$	0	0
$861$	0	0
$862$	10.8588	0.369854
$863$	27.3752	0.931863	0.465931	−	0.884821i	$-0.345719\pi$
$863$	27.3752	0.931863	0.465931	+	0.884821i	$0.345719\pi$
$864$	0	0
$865$	0	0
$866$	50.8623	1.72837
$867$	0	0
$868$	2.44213	0.0828914
$869$	48.5797	1.64795
$870$	0	0
$871$	−7.97395	−0.270187
$872$	22.7957	0.771960
$873$	0	0
$874$	10.2222	0.345770
$875$	0	0
$876$	0	0
$877$	−1.97499	−0.0666906	−0.0333453	−	0.999444i	$-0.510616\pi$
$877$	−1.97499	−0.0666906	−0.0333453	+	0.999444i	$0.510616\pi$
$878$	−0.671753	−0.0226706
$879$	0	0
$880$	0	0
$881$	3.16894	0.106764	0.0533821	−	0.998574i	$-0.483000\pi$
$881$	3.16894	0.106764	0.0533821	+	0.998574i	$0.483000\pi$
$882$	0	0
$883$	22.6813	0.763286	0.381643	−	0.924310i	$-0.375358\pi$
$883$	22.6813	0.763286	0.381643	+	0.924310i	$0.375358\pi$
$884$	0.459111	0.0154416
$885$	0	0
$886$	−27.2515	−0.915531
$887$	−39.9446	−1.34121	−0.670605	−	0.741815i	$-0.733965\pi$
$887$	−39.9446	−1.34121	−0.670605	+	0.741815i	$0.733965\pi$
$888$	0	0
$889$	−42.0530	−1.41041
$890$	0	0
$891$	0	0
$892$	9.72836	0.325730
$893$	31.9363	1.06871
$894$	0	0
$895$	0	0
$896$	−68.2442	−2.27988
$897$	0	0
$898$	−51.1913	−1.70828
$899$	−4.93739	−0.164671
$900$	0	0
$901$	1.16011	0.0386487
$902$	32.2434	1.07359
$903$	0	0
$904$	−37.9080	−1.26080
$905$	0	0
$906$	0	0
$907$	−43.5661	−1.44659	−0.723294	−	0.690540i	$-0.757373\pi$
$907$	−43.5661	−1.44659	−0.723294	+	0.690540i	$0.757373\pi$
$908$	2.97468	0.0987182
$909$	0	0
$910$	0	0
$911$	52.3509	1.73446	0.867232	−	0.497905i	$-0.165897\pi$
$911$	52.3509	1.73446	0.867232	+	0.497905i	$0.165897\pi$
$912$	0	0
$913$	−42.7309	−1.41418
$914$	9.91188	0.327856
$915$	0	0
$916$	−7.64558	−0.252617
$917$	15.6223	0.515892
$918$	0	0
$919$	16.1202	0.531757	0.265878	−	0.964007i	$-0.414338\pi$
$919$	16.1202	0.531757	0.265878	+	0.964007i	$0.414338\pi$
$920$	0	0
$921$	0	0
$922$	38.6268	1.27211
$923$	52.4720	1.72714
$924$	0	0
$925$	0	0
$926$	3.62661	0.119178
$927$	0	0
$928$	−13.3380	−0.437840
$929$	45.4771	1.49205	0.746027	−	0.665916i	$-0.231959\pi$
$929$	45.4771	1.49205	0.746027	+	0.665916i	$0.231959\pi$
$930$	0	0
$931$	−102.092	−3.34594
$932$	1.20936	0.0396138
$933$	0	0
$934$	−9.56618	−0.313015
$935$	0	0
$936$	0	0
$937$	0.940202	0.0307151	0.0153575	−	0.999882i	$-0.495111\pi$
$937$	0.940202	0.0307151	0.0153575	+	0.999882i	$0.495111\pi$
$938$	18.7833	0.613295
$939$	0	0
$940$	0	0
$941$	12.0715	0.393519	0.196759	−	0.980452i	$-0.436958\pi$
$941$	12.0715	0.393519	0.196759	+	0.980452i	$0.436958\pi$
$942$	0	0
$943$	−4.20120	−0.136810
$944$	58.9104	1.91737
$945$	0	0
$946$	−78.3025	−2.54583
$947$	43.0074	1.39755	0.698777	−	0.715340i	$-0.253728\pi$
$947$	43.0074	1.39755	0.698777	+	0.715340i	$0.253728\pi$
$948$	0	0
$949$	6.08582	0.197554
$950$	0	0
$951$	0	0
$952$	3.35406	0.108706
$953$	26.7708	0.867193	0.433596	−	0.901107i	$-0.357244\pi$
$953$	26.7708	0.867193	0.433596	+	0.901107i	$0.357244\pi$
$954$	0	0
$955$	0	0
$956$	13.1513	0.425343
$957$	0	0
$958$	63.6715	2.05713
$959$	9.66608	0.312134
$960$	0	0
$961$	1.00000	0.0322581
$962$	−58.9060	−1.89921
$963$	0	0
$964$	−2.13553	−0.0687808
$965$	0	0
$966$	0	0
$967$	−58.3066	−1.87502	−0.937508	−	0.347965i	$-0.886873\pi$
$967$	−58.3066	−1.87502	−0.937508	+	0.347965i	$0.886873\pi$
$968$	48.1729	1.54834
$969$	0	0
$970$	0	0
$971$	−25.3931	−0.814905	−0.407452	−	0.913226i	$-0.633583\pi$
$971$	−25.3931	−0.814905	−0.407452	+	0.913226i	$0.633583\pi$
$972$	0	0
$973$	−35.4055	−1.13505
$974$	18.9397	0.606866
$975$	0	0
$976$	−7.14403	−0.228675
$977$	−37.6544	−1.20467	−0.602336	−	0.798243i	$-0.705763\pi$
$977$	−37.6544	−1.20467	−0.602336	+	0.798243i	$0.705763\pi$
$978$	0	0
$979$	53.5572	1.71170
$980$	0	0
$981$	0	0
$982$	−57.5426	−1.83626
$983$	18.6108	0.593592	0.296796	−	0.954941i	$-0.404082\pi$
$983$	18.6108	0.593592	0.296796	+	0.954941i	$0.404082\pi$
$984$	0	0
$985$	0	0
$986$	2.18647	0.0696314
$987$	0	0
$988$	−9.23300	−0.293741
$989$	10.2025	0.324422
$990$	0	0
$991$	14.8332	0.471192	0.235596	−	0.971851i	$-0.424296\pi$
$991$	14.8332	0.471192	0.235596	+	0.971851i	$0.424296\pi$
$992$	2.70142	0.0857701
$993$	0	0
$994$	−123.602	−3.92041
$995$	0	0
$996$	0	0
$997$	31.7876	1.00672	0.503362	−	0.864076i	$-0.332096\pi$
$997$	31.7876	1.00672	0.503362	+	0.864076i	$0.332096\pi$
$998$	−44.8729	−1.42043
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bw.1.1		5
3.2	odd	2		6975.2.a.br.1.5		5
5.4	even	2		1395.2.a.n.1.5	✓	5
15.14	odd	2		1395.2.a.q.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.a.n.1.5	✓	5	5.4	even	2
1395.2.a.q.1.1	yes	5	15.14	odd	2
6975.2.a.br.1.5		5	3.2	odd	2
6975.2.a.bw.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.57723 q^{2} +0.487640 q^{4} +5.00807 q^{7} +2.38533 q^{8} +5.58529 q^{11} +3.35325 q^{13} -7.89885 q^{14} -4.73749 q^{16} +0.280771 q^{17} -5.64647 q^{19} -8.80926 q^{22} +1.14781 q^{23} -5.28884 q^{26} +2.44213 q^{28} -4.93739 q^{29} +1.00000 q^{31} +2.70142 q^{32} -0.442839 q^{34} +11.1378 q^{37} +8.90576 q^{38} -3.66017 q^{41} +8.88866 q^{43} +2.72361 q^{44} -1.81036 q^{46} -5.65597 q^{47} +18.0807 q^{49} +1.63518 q^{52} +4.13186 q^{53} +11.9459 q^{56} +7.78738 q^{58} -12.4349 q^{59} +1.50798 q^{61} -1.57723 q^{62} +5.21423 q^{64} -2.37797 q^{67} +0.136915 q^{68} +15.6481 q^{71} +1.81490 q^{73} -17.5668 q^{74} -2.75344 q^{76} +27.9715 q^{77} +8.69779 q^{79} +5.77292 q^{82} -7.65060 q^{83} -14.0194 q^{86} +13.3228 q^{88} +9.58898 q^{89} +16.7933 q^{91} +0.559720 q^{92} +8.92073 q^{94} +9.26597 q^{97} -28.5174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 3 q^{4} + 6 q^{7} + 9 q^{8} - 2 q^{11} + 4 q^{13} + 11 q^{16} + 12 q^{17} - 4 q^{19} - 16 q^{22} + 4 q^{23} - 18 q^{26} + 18 q^{28} - 18 q^{29} + 5 q^{31} + 21 q^{32} + 22 q^{34} + 16 q^{37}+ \cdots - 47 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 3 * q^4 + 6 * q^7 + 9 * q^8 - 2 * q^11 + 4 * q^13 + 11 * q^16 + 12 * q^17 - 4 * q^19 - 16 * q^22 + 4 * q^23 - 18 * q^26 + 18 * q^28 - 18 * q^29 + 5 * q^31 + 21 * q^32 + 22 * q^34 + 16 * q^37 + 22 * q^38 - 16 * q^41 + 10 * q^43 + 14 * q^47 + 5 * q^49 - 30 * q^52 + 14 * q^53 + 28 * q^56 - 4 * q^58 - 14 * q^59 + 10 * q^61 + 3 * q^62 + 27 * q^64 + 2 * q^67 + 34 * q^68 + 12 * q^71 + 22 * q^73 + 10 * q^74 + 16 * q^76 + 34 * q^77 + 12 * q^79 + 6 * q^82 + 8 * q^83 - 10 * q^86 + 2 * q^88 + 2 * q^89 + 8 * q^91 - 26 * q^92 + 20 * q^94 + 16 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more