LMFDB - Embedded newform 6975.2.a.bc.1.2

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 = [3,0,0,2,0,0,-4,6,0,0,7,0,0,6,0,-4,-7] 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:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2325)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ 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)

$f(q)$	$=$	$q-0.539189 q^{2} -1.70928 q^{4} -3.70928 q^{7} +2.00000 q^{8} -1.87936 q^{11} +6.58864 q^{13} +2.00000 q^{14} +2.34017 q^{16} -3.09171 q^{17} -7.34017 q^{19} +1.01333 q^{22} +7.29791 q^{23} -3.55252 q^{26} +6.34017 q^{28} -0.170086 q^{29} -1.00000 q^{31} -5.26180 q^{32} +1.66701 q^{34} -6.82991 q^{37} +3.95774 q^{38} +7.95774 q^{41} -4.18342 q^{43} +3.21235 q^{44} -3.93495 q^{46} +11.7587 q^{47} +6.75872 q^{49} -11.2618 q^{52} -13.9288 q^{53} -7.41855 q^{56} +0.0917087 q^{58} +6.72261 q^{59} +1.84324 q^{61} +0.539189 q^{62} -1.84324 q^{64} +1.81432 q^{67} +5.28458 q^{68} +11.2618 q^{71} -2.92162 q^{73} +3.68261 q^{74} +12.5464 q^{76} +6.97107 q^{77} +6.24846 q^{79} -4.29072 q^{82} +6.21953 q^{83} +2.25565 q^{86} -3.75872 q^{88} +12.4030 q^{89} -24.4391 q^{91} -12.4741 q^{92} -6.34017 q^{94} -3.94441 q^{97} -3.64423 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} - 4 q^{7} + 6 q^{8} + 7 q^{11} + 6 q^{14} - 4 q^{16} - 7 q^{17} - 11 q^{19} + 4 q^{22} - 5 q^{23} - 10 q^{26} + 8 q^{28} + 5 q^{29} - 3 q^{31} - 8 q^{32} - 18 q^{34} - 26 q^{37} - 4 q^{38}+ \cdots - 8 q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 - 4 * q^7 + 6 * q^8 + 7 * q^11 + 6 * q^14 - 4 * q^16 - 7 * q^17 - 11 * q^19 + 4 * q^22 - 5 * q^23 - 10 * q^26 + 8 * q^28 + 5 * q^29 - 3 * q^31 - 8 * q^32 - 18 * q^34 - 26 * q^37 - 4 * q^38 + 8 * q^41 - 8 * q^43 + 20 * q^44 - 16 * q^46 + 10 * q^47 - 5 * q^49 - 26 * q^52 - 11 * q^53 - 8 * q^56 - 2 * q^58 + 14 * q^59 + 12 * q^61 - 12 * q^64 - 3 * q^67 - 12 * q^68 + 26 * q^71 - 12 * q^73 + 2 * q^74 + 2 * q^76 + 6 * q^77 + 10 * q^79 - 20 * q^82 - 5 * q^83 - 36 * q^86 + 14 * q^88 + 9 * q^89 - 26 * q^91 - 40 * q^92 - 8 * q^94 + 5 * q^97 - 8 * 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 $

$2$	−0.539189	−0.381264	−0.190632	−	0.981662i	$-0.561054\pi$
$2$	−0.539189	−0.381264	−0.190632	+	0.981662i	$0.561054\pi$
$3$	0	0
$3$	0	0
$4$	−1.70928	−0.854638
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.70928	−1.40197	−0.700987	−	0.713174i	$-0.747257\pi$
$7$	−3.70928	−1.40197	−0.700987	+	0.713174i	$0.747257\pi$
$8$	2.00000	0.707107
$9$	0	0
$10$	0	0
$11$	−1.87936	−0.566649	−0.283324	−	0.959024i	$-0.591437\pi$
$11$	−1.87936	−0.566649	−0.283324	+	0.959024i	$0.591437\pi$
$12$	0	0
$13$	6.58864	1.82736	0.913680	−	0.406435i	$-0.133228\pi$
$13$	6.58864	1.82736	0.913680	+	0.406435i	$0.133228\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	2.34017	0.585043
$17$	−3.09171	−0.749850	−0.374925	−	0.927055i	$-0.622331\pi$
$17$	−3.09171	−0.749850	−0.374925	+	0.927055i	$0.622331\pi$
$18$	0	0
$19$	−7.34017	−1.68395	−0.841976	−	0.539516i	$-0.818607\pi$
$19$	−7.34017	−1.68395	−0.841976	+	0.539516i	$0.818607\pi$
$20$	0	0
$21$	0	0
$22$	1.01333	0.216043
$23$	7.29791	1.52172	0.760860	−	0.648916i	$-0.224777\pi$
$23$	7.29791	1.52172	0.760860	+	0.648916i	$0.224777\pi$
$24$	0	0
$25$	0	0
$26$	−3.55252	−0.696706
$27$	0	0
$28$	6.34017	1.19818
$29$	−0.170086	−0.0315843	−0.0157921	−	0.999875i	$-0.505027\pi$
$29$	−0.170086	−0.0315843	−0.0157921	+	0.999875i	$0.505027\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.26180	−0.930163
$33$	0	0
$34$	1.66701	0.285891
$35$	0	0
$36$	0	0
$37$	−6.82991	−1.12283	−0.561415	−	0.827534i	$-0.689743\pi$
$37$	−6.82991	−1.12283	−0.561415	+	0.827534i	$0.689743\pi$
$38$	3.95774	0.642030
$39$	0	0
$40$	0	0
$41$	7.95774	1.24279	0.621395	−	0.783497i	$-0.286566\pi$
$41$	7.95774	1.24279	0.621395	+	0.783497i	$0.286566\pi$
$42$	0	0
$43$	−4.18342	−0.637965	−0.318983	−	0.947761i	$-0.603341\pi$
$43$	−4.18342	−0.637965	−0.318983	+	0.947761i	$0.603341\pi$
$44$	3.21235	0.484280
$45$	0	0
$46$	−3.93495	−0.580177
$47$	11.7587	1.71519	0.857593	−	0.514329i	$-0.171959\pi$
$47$	11.7587	1.71519	0.857593	+	0.514329i	$0.171959\pi$
$48$	0	0
$49$	6.75872	0.965532
$50$	0	0
$51$	0	0
$52$	−11.2618	−1.56173
$53$	−13.9288	−1.91327	−0.956635	−	0.291291i	$-0.905915\pi$
$53$	−13.9288	−1.91327	−0.956635	+	0.291291i	$0.905915\pi$
$54$	0	0
$55$	0	0
$56$	−7.41855	−0.991346
$57$	0	0
$58$	0.0917087	0.0120419
$59$	6.72261	0.875209	0.437604	−	0.899168i	$-0.355827\pi$
$59$	6.72261	0.875209	0.437604	+	0.899168i	$0.355827\pi$
$60$	0	0
$61$	1.84324	0.236003	0.118002	−	0.993013i	$-0.462351\pi$
$61$	1.84324	0.236003	0.118002	+	0.993013i	$0.462351\pi$
$62$	0.539189	0.0684771
$63$	0	0
$64$	−1.84324	−0.230406
$65$	0	0
$66$	0	0
$67$	1.81432	0.221654	0.110827	−	0.993840i	$-0.464650\pi$
$67$	1.81432	0.221654	0.110827	+	0.993840i	$0.464650\pi$
$68$	5.28458	0.640850
$69$	0	0
$70$	0	0
$71$	11.2618	1.33653	0.668265	−	0.743924i	$-0.267037\pi$
$71$	11.2618	1.33653	0.668265	+	0.743924i	$0.267037\pi$
$72$	0	0
$73$	−2.92162	−0.341950	−0.170975	−	0.985275i	$-0.554692\pi$
$73$	−2.92162	−0.341950	−0.170975	+	0.985275i	$0.554692\pi$
$74$	3.68261	0.428095
$75$	0	0
$76$	12.5464	1.43917
$77$	6.97107	0.794427
$78$	0	0
$79$	6.24846	0.703007	0.351504	−	0.936187i	$-0.385671\pi$
$79$	6.24846	0.703007	0.351504	+	0.936187i	$0.385671\pi$
$80$	0	0
$81$	0	0
$82$	−4.29072	−0.473831
$83$	6.21953	0.682683	0.341341	−	0.939939i	$-0.389119\pi$
$83$	6.21953	0.682683	0.341341	+	0.939939i	$0.389119\pi$
$84$	0	0
$85$	0	0
$86$	2.25565	0.243233
$87$	0	0
$88$	−3.75872	−0.400681
$89$	12.4030	1.31471	0.657355	−	0.753581i	$-0.271675\pi$
$89$	12.4030	1.31471	0.657355	+	0.753581i	$0.271675\pi$
$90$	0	0
$91$	−24.4391	−2.56191
$92$	−12.4741	−1.30052
$93$	0	0
$94$	−6.34017	−0.653939
$95$	0	0
$96$	0	0
$97$	−3.94441	−0.400494	−0.200247	−	0.979745i	$-0.564174\pi$
$97$	−3.94441	−0.400494	−0.200247	+	0.979745i	$0.564174\pi$
$98$	−3.64423	−0.368123
$99$	0	0
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−2.60197	−0.256380	−0.128190	−	0.991750i	$-0.540917\pi$
$103$	−2.60197	−0.256380	−0.128190	+	0.991750i	$0.540917\pi$
$104$	13.1773	1.29214
$105$	0	0
$106$	7.51026	0.729461
$107$	15.7431	1.52195	0.760973	−	0.648784i	$-0.224722\pi$
$107$	15.7431	1.52195	0.760973	+	0.648784i	$0.224722\pi$
$108$	0	0
$109$	−4.81432	−0.461128	−0.230564	−	0.973057i	$-0.574057\pi$
$109$	−4.81432	−0.461128	−0.230564	+	0.973057i	$0.574057\pi$
$110$	0	0
$111$	0	0
$112$	−8.68035	−0.820216
$113$	−9.91548	−0.932770	−0.466385	−	0.884582i	$-0.654444\pi$
$113$	−9.91548	−0.932770	−0.466385	+	0.884582i	$0.654444\pi$
$114$	0	0
$115$	0	0
$116$	0.290725	0.0269931
$117$	0	0
$118$	−3.62475	−0.333686
$119$	11.4680	1.05127
$120$	0	0
$121$	−7.46800	−0.678909
$122$	−0.993857	−0.0899796
$123$	0	0
$124$	1.70928	0.153497
$125$	0	0
$126$	0	0
$127$	6.68035	0.592785	0.296392	−	0.955066i	$-0.404216\pi$
$127$	6.68035	0.592785	0.296392	+	0.955066i	$0.404216\pi$
$128$	11.5174	1.01801
$129$	0	0
$130$	0	0
$131$	−3.98440	−0.348119	−0.174059	−	0.984735i	$-0.555688\pi$
$131$	−3.98440	−0.348119	−0.174059	+	0.984735i	$0.555688\pi$
$132$	0	0
$133$	27.2267	2.36086
$134$	−0.978259	−0.0845087
$135$	0	0
$136$	−6.18342	−0.530224
$137$	−23.1906	−1.98131	−0.990654	−	0.136402i	$-0.956446\pi$
$137$	−23.1906	−1.98131	−0.990654	+	0.136402i	$0.956446\pi$
$138$	0	0
$139$	−12.7454	−1.08105	−0.540525	−	0.841328i	$-0.681774\pi$
$139$	−12.7454	−1.08105	−0.540525	+	0.841328i	$0.681774\pi$
$140$	0	0
$141$	0	0
$142$	−6.07223	−0.509571
$143$	−12.3824	−1.03547
$144$	0	0
$145$	0	0
$146$	1.57531	0.130373
$147$	0	0
$148$	11.6742	0.959614
$149$	11.0361	0.904114	0.452057	−	0.891989i	$-0.350690\pi$
$149$	11.0361	0.904114	0.452057	+	0.891989i	$0.350690\pi$
$150$	0	0
$151$	−11.1050	−0.903715	−0.451858	−	0.892090i	$-0.649239\pi$
$151$	−11.1050	−0.903715	−0.451858	+	0.892090i	$0.649239\pi$
$152$	−14.6803	−1.19073
$153$	0	0
$154$	−3.75872	−0.302887
$155$	0	0
$156$	0	0
$157$	14.4186	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$157$	14.4186	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$158$	−3.36910	−0.268031
$159$	0	0
$160$	0	0
$161$	−27.0700	−2.13341
$162$	0	0
$163$	16.6742	1.30602	0.653012	−	0.757347i	$-0.273505\pi$
$163$	16.6742	1.30602	0.653012	+	0.757347i	$0.273505\pi$
$164$	−13.6020	−1.06214
$165$	0	0
$166$	−3.35350	−0.260282
$167$	−6.65368	−0.514878	−0.257439	−	0.966295i	$-0.582879\pi$
$167$	−6.65368	−0.514878	−0.257439	+	0.966295i	$0.582879\pi$
$168$	0	0
$169$	30.4101	2.33924
$170$	0	0
$171$	0	0
$172$	7.15061	0.545229
$173$	−19.7165	−1.49901	−0.749507	−	0.661996i	$-0.769710\pi$
$173$	−19.7165	−1.49901	−0.749507	+	0.661996i	$0.769710\pi$
$174$	0	0
$175$	0	0
$176$	−4.39803	−0.331514
$177$	0	0
$178$	−6.68753	−0.501252
$179$	3.68261	0.275251	0.137626	−	0.990484i	$-0.456053\pi$
$179$	3.68261	0.275251	0.137626	+	0.990484i	$0.456053\pi$
$180$	0	0
$181$	−1.51026	−0.112257	−0.0561284	−	0.998424i	$-0.517876\pi$
$181$	−1.51026	−0.112257	−0.0561284	+	0.998424i	$0.517876\pi$
$182$	13.1773	0.976765
$183$	0	0
$184$	14.5958	1.07602
$185$	0	0
$186$	0	0
$187$	5.81044	0.424901
$188$	−20.0989	−1.46586
$189$	0	0
$190$	0	0
$191$	7.43415	0.537916	0.268958	−	0.963152i	$-0.413321\pi$
$191$	7.43415	0.537916	0.268958	+	0.963152i	$0.413321\pi$
$192$	0	0
$193$	−22.9916	−1.65497	−0.827485	−	0.561487i	$-0.810229\pi$
$193$	−22.9916	−1.65497	−0.827485	+	0.561487i	$0.810229\pi$
$194$	2.12678	0.152694
$195$	0	0
$196$	−11.5525	−0.825180
$197$	−14.3135	−1.01980	−0.509898	−	0.860235i	$-0.670317\pi$
$197$	−14.3135	−1.01980	−0.509898	+	0.860235i	$0.670317\pi$
$198$	0	0
$199$	−4.64650	−0.329381	−0.164691	−	0.986345i	$-0.552663\pi$
$199$	−4.64650	−0.329381	−0.164691	+	0.986345i	$0.552663\pi$
$200$	0	0
$201$	0	0
$202$	−4.31351	−0.303498
$203$	0.630898	0.0442803
$204$	0	0
$205$	0	0
$206$	1.40295	0.0977483
$207$	0	0
$208$	15.4186	1.06908
$209$	13.7948	0.954209
$210$	0	0
$211$	26.6163	1.83234	0.916172	−	0.400785i	$-0.131262\pi$
$211$	26.6163	1.83234	0.916172	+	0.400785i	$0.131262\pi$
$212$	23.8082	1.63515
$213$	0	0
$214$	−8.48852	−0.580263
$215$	0	0
$216$	0	0
$217$	3.70928	0.251802
$218$	2.59583	0.175811
$219$	0	0
$220$	0	0
$221$	−20.3701	−1.37024
$222$	0	0
$223$	−17.0205	−1.13978	−0.569889	−	0.821722i	$-0.693014\pi$
$223$	−17.0205	−1.13978	−0.569889	+	0.821722i	$0.693014\pi$
$224$	19.5174	1.30406
$225$	0	0
$226$	5.34632	0.355632
$227$	−9.78539	−0.649479	−0.324739	−	0.945804i	$-0.605277\pi$
$227$	−9.78539	−0.649479	−0.324739	+	0.945804i	$0.605277\pi$
$228$	0	0
$229$	−22.3474	−1.47676	−0.738378	−	0.674388i	$-0.764408\pi$
$229$	−22.3474	−1.47676	−0.738378	+	0.674388i	$0.764408\pi$
$230$	0	0
$231$	0	0
$232$	−0.340173	−0.0223334
$233$	20.6803	1.35481	0.677407	−	0.735608i	$-0.263104\pi$
$233$	20.6803	1.35481	0.677407	+	0.735608i	$0.263104\pi$
$234$	0	0
$235$	0	0
$236$	−11.4908	−0.747986
$237$	0	0
$238$	−6.18342	−0.400811
$239$	3.43188	0.221990	0.110995	−	0.993821i	$-0.464596\pi$
$239$	3.43188	0.221990	0.110995	+	0.993821i	$0.464596\pi$
$240$	0	0
$241$	3.56812	0.229843	0.114921	−	0.993375i	$-0.463338\pi$
$241$	3.56812	0.229843	0.114921	+	0.993375i	$0.463338\pi$
$242$	4.02666	0.258844
$243$	0	0
$244$	−3.15061	−0.201697
$245$	0	0
$246$	0	0
$247$	−48.3617	−3.07718
$248$	−2.00000	−0.127000
$249$	0	0
$250$	0	0
$251$	−7.74539	−0.488885	−0.244442	−	0.969664i	$-0.578605\pi$
$251$	−7.74539	−0.488885	−0.244442	+	0.969664i	$0.578605\pi$
$252$	0	0
$253$	−13.7154	−0.862281
$254$	−3.60197	−0.226008
$255$	0	0
$256$	−2.52359	−0.157724
$257$	−20.4547	−1.27593	−0.637964	−	0.770067i	$-0.720223\pi$
$257$	−20.4547	−1.27593	−0.637964	+	0.770067i	$0.720223\pi$
$258$	0	0
$259$	25.3340	1.57418
$260$	0	0
$261$	0	0
$262$	2.14834	0.132725
$263$	−24.5113	−1.51143	−0.755716	−	0.654900i	$-0.772711\pi$
$263$	−24.5113	−1.51143	−0.755716	+	0.654900i	$0.772711\pi$
$264$	0	0
$265$	0	0
$266$	−14.6803	−0.900110
$267$	0	0
$268$	−3.10116	−0.189434
$269$	−27.2762	−1.66306	−0.831529	−	0.555482i	$-0.812534\pi$
$269$	−27.2762	−1.66306	−0.831529	+	0.555482i	$0.812534\pi$
$270$	0	0
$271$	−12.4897	−0.758698	−0.379349	−	0.925254i	$-0.623852\pi$
$271$	−12.4897	−0.758698	−0.379349	+	0.925254i	$0.623852\pi$
$272$	−7.23513	−0.438694
$273$	0	0
$274$	12.5041	0.755401
$275$	0	0
$276$	0	0
$277$	−14.8638	−0.893077	−0.446538	−	0.894764i	$-0.647343\pi$
$277$	−14.8638	−0.893077	−0.446538	+	0.894764i	$0.647343\pi$
$278$	6.87217	0.412166
$279$	0	0
$280$	0	0
$281$	8.83710	0.527177	0.263589	−	0.964635i	$-0.415094\pi$
$281$	8.83710	0.527177	0.263589	+	0.964635i	$0.415094\pi$
$282$	0	0
$283$	14.7009	0.873876	0.436938	−	0.899492i	$-0.356063\pi$
$283$	14.7009	0.873876	0.436938	+	0.899492i	$0.356063\pi$
$284$	−19.2495	−1.14225
$285$	0	0
$286$	6.67647	0.394788
$287$	−29.5174	−1.74236
$288$	0	0
$289$	−7.44134	−0.437726
$290$	0	0
$291$	0	0
$292$	4.99386	0.292243
$293$	−22.6381	−1.32253	−0.661266	−	0.750152i	$-0.729980\pi$
$293$	−22.6381	−1.32253	−0.661266	+	0.750152i	$0.729980\pi$
$294$	0	0
$295$	0	0
$296$	−13.6598	−0.793961
$297$	0	0
$298$	−5.95055	−0.344706
$299$	48.0833	2.78073
$300$	0	0
$301$	15.5174	0.894411
$302$	5.98771	0.344554
$303$	0	0
$304$	−17.1773	−0.985184
$305$	0	0
$306$	0	0
$307$	2.81432	0.160621	0.0803107	−	0.996770i	$-0.474409\pi$
$307$	2.81432	0.160621	0.0803107	+	0.996770i	$0.474409\pi$
$308$	−11.9155	−0.678947
$309$	0	0
$310$	0	0
$311$	−0.780465	−0.0442561	−0.0221281	−	0.999755i	$-0.507044\pi$
$311$	−0.780465	−0.0442561	−0.0221281	+	0.999755i	$0.507044\pi$
$312$	0	0
$313$	−33.3679	−1.88606	−0.943032	−	0.332702i	$-0.892040\pi$
$313$	−33.3679	−1.88606	−0.943032	+	0.332702i	$0.892040\pi$
$314$	−7.77432	−0.438730
$315$	0	0
$316$	−10.6803	−0.600816
$317$	−14.7805	−0.830154	−0.415077	−	0.909786i	$-0.636245\pi$
$317$	−14.7805	−0.830154	−0.415077	+	0.909786i	$0.636245\pi$
$318$	0	0
$319$	0.319654	0.0178972
$320$	0	0
$321$	0	0
$322$	14.5958	0.813394
$323$	22.6937	1.26271
$324$	0	0
$325$	0	0
$326$	−8.99054	−0.497940
$327$	0	0
$328$	15.9155	0.878785
$329$	−43.6163	−2.40465
$330$	0	0
$331$	2.16394	0.118941	0.0594705	−	0.998230i	$-0.481059\pi$
$331$	2.16394	0.118941	0.0594705	+	0.998230i	$0.481059\pi$
$332$	−10.6309	−0.583446
$333$	0	0
$334$	3.58759	0.196304
$335$	0	0
$336$	0	0
$337$	−33.3100	−1.81451	−0.907256	−	0.420578i	$-0.861827\pi$
$337$	−33.3100	−1.81451	−0.907256	+	0.420578i	$0.861827\pi$
$338$	−16.3968	−0.891869
$339$	0	0
$340$	0	0
$341$	1.87936	0.101773
$342$	0	0
$343$	0.894960	0.0483233
$344$	−8.36683	−0.451110
$345$	0	0
$346$	10.6309	0.571521
$347$	−2.88655	−0.154958	−0.0774791	−	0.996994i	$-0.524687\pi$
$347$	−2.88655	−0.154958	−0.0774791	+	0.996994i	$0.524687\pi$
$348$	0	0
$349$	3.34017	0.178795	0.0893977	−	0.995996i	$-0.471506\pi$
$349$	3.34017	0.178795	0.0893977	+	0.995996i	$0.471506\pi$
$350$	0	0
$351$	0	0
$352$	9.88882	0.527076
$353$	24.2111	1.28863	0.644314	−	0.764761i	$-0.277143\pi$
$353$	24.2111	1.28863	0.644314	+	0.764761i	$0.277143\pi$
$354$	0	0
$355$	0	0
$356$	−21.2001	−1.12360
$357$	0	0
$358$	−1.98562	−0.104943
$359$	−31.3751	−1.65591	−0.827956	−	0.560793i	$-0.810497\pi$
$359$	−31.3751	−1.65591	−0.827956	+	0.560793i	$0.810497\pi$
$360$	0	0
$361$	34.8781	1.83569
$362$	0.814315	0.0427995
$363$	0	0
$364$	41.7731	2.18951
$365$	0	0
$366$	0	0
$367$	13.6286	0.711409	0.355704	−	0.934599i	$-0.384241\pi$
$367$	13.6286	0.711409	0.355704	+	0.934599i	$0.384241\pi$
$368$	17.0784	0.890272
$369$	0	0
$370$	0	0
$371$	51.6658	2.68235
$372$	0	0
$373$	2.39576	0.124048	0.0620240	−	0.998075i	$-0.480244\pi$
$373$	2.39576	0.124048	0.0620240	+	0.998075i	$0.480244\pi$
$374$	−3.13292	−0.162000
$375$	0	0
$376$	23.5174	1.21282
$377$	−1.12064	−0.0577158
$378$	0	0
$379$	−11.3958	−0.585361	−0.292681	−	0.956210i	$-0.594547\pi$
$379$	−11.3958	−0.585361	−0.292681	+	0.956210i	$0.594547\pi$
$380$	0	0
$381$	0	0
$382$	−4.00841	−0.205088
$383$	−4.77820	−0.244155	−0.122077	−	0.992521i	$-0.538956\pi$
$383$	−4.77820	−0.244155	−0.122077	+	0.992521i	$0.538956\pi$
$384$	0	0
$385$	0	0
$386$	12.3968	0.630981
$387$	0	0
$388$	6.74208	0.342277
$389$	−3.84324	−0.194860	−0.0974301	−	0.995242i	$-0.531062\pi$
$389$	−3.84324	−0.194860	−0.0974301	+	0.995242i	$0.531062\pi$
$390$	0	0
$391$	−22.5630	−1.14106
$392$	13.5174	0.682734
$393$	0	0
$394$	7.71769	0.388811
$395$	0	0
$396$	0	0
$397$	12.7093	0.637860	0.318930	−	0.947778i	$-0.396676\pi$
$397$	12.7093	0.637860	0.318930	+	0.947778i	$0.396676\pi$
$398$	2.50534	0.125581
$399$	0	0
$400$	0	0
$401$	21.3968	1.06851	0.534253	−	0.845325i	$-0.320593\pi$
$401$	21.3968	1.06851	0.534253	+	0.845325i	$0.320593\pi$
$402$	0	0
$403$	−6.58864	−0.328203
$404$	−13.6742	−0.680317
$405$	0	0
$406$	−0.340173	−0.0168825
$407$	12.8359	0.636251
$408$	0	0
$409$	−36.1256	−1.78629	−0.893147	−	0.449765i	$-0.851508\pi$
$409$	−36.1256	−1.78629	−0.893147	+	0.449765i	$0.851508\pi$
$410$	0	0
$411$	0	0
$412$	4.44748	0.219112
$413$	−24.9360	−1.22702
$414$	0	0
$415$	0	0
$416$	−34.6681	−1.69974
$417$	0	0
$418$	−7.43802	−0.363806
$419$	2.92162	0.142731	0.0713653	−	0.997450i	$-0.477264\pi$
$419$	2.92162	0.142731	0.0713653	+	0.997450i	$0.477264\pi$
$420$	0	0
$421$	−28.6309	−1.39538	−0.697692	−	0.716398i	$-0.745790\pi$
$421$	−28.6309	−1.39538	−0.697692	+	0.716398i	$0.745790\pi$
$422$	−14.3512	−0.698607
$423$	0	0
$424$	−27.8576	−1.35289
$425$	0	0
$426$	0	0
$427$	−6.83710	−0.330871
$428$	−26.9093	−1.30071
$429$	0	0
$430$	0	0
$431$	2.41241	0.116202	0.0581008	−	0.998311i	$-0.481496\pi$
$431$	2.41241	0.116202	0.0581008	+	0.998311i	$0.481496\pi$
$432$	0	0
$433$	−26.2290	−1.26048	−0.630242	−	0.776398i	$-0.717044\pi$
$433$	−26.2290	−1.26048	−0.630242	+	0.776398i	$0.717044\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	8.22899	0.394097
$437$	−53.5679	−2.56250
$438$	0	0
$439$	28.5197	1.36117	0.680586	−	0.732668i	$-0.261725\pi$
$439$	28.5197	1.36117	0.680586	+	0.732668i	$0.261725\pi$
$440$	0	0
$441$	0	0
$442$	10.9834	0.522425
$443$	12.4235	0.590257	0.295128	−	0.955458i	$-0.404638\pi$
$443$	12.4235	0.590257	0.295128	+	0.955458i	$0.404638\pi$
$444$	0	0
$445$	0	0
$446$	9.17727	0.434557
$447$	0	0
$448$	6.83710	0.323023
$449$	17.0833	0.806211	0.403105	−	0.915154i	$-0.367931\pi$
$449$	17.0833	0.806211	0.403105	+	0.915154i	$0.367931\pi$
$450$	0	0
$451$	−14.9555	−0.704226
$452$	16.9483	0.797180
$453$	0	0
$454$	5.27617	0.247623
$455$	0	0
$456$	0	0
$457$	11.0277	0.515854	0.257927	−	0.966164i	$-0.416961\pi$
$457$	11.0277	0.515854	0.257927	+	0.966164i	$0.416961\pi$
$458$	12.0494	0.563034
$459$	0	0
$460$	0	0
$461$	−16.7165	−0.778563	−0.389282	−	0.921119i	$-0.627277\pi$
$461$	−16.7165	−0.778563	−0.389282	+	0.921119i	$0.627277\pi$
$462$	0	0
$463$	−12.4124	−0.576854	−0.288427	−	0.957502i	$-0.593132\pi$
$463$	−12.4124	−0.576854	−0.288427	+	0.957502i	$0.593132\pi$
$464$	−0.398032	−0.0184782
$465$	0	0
$466$	−11.1506	−0.516542
$467$	−14.7115	−0.680769	−0.340385	−	0.940286i	$-0.610557\pi$
$467$	−14.7115	−0.680769	−0.340385	+	0.940286i	$0.610557\pi$
$468$	0	0
$469$	−6.72979	−0.310753
$470$	0	0
$471$	0	0
$472$	13.4452	0.618866
$473$	7.86216	0.361502
$474$	0	0
$475$	0	0
$476$	−19.6020	−0.898455
$477$	0	0
$478$	−1.85043	−0.0846368
$479$	16.8371	0.769307	0.384653	−	0.923061i	$-0.374321\pi$
$479$	16.8371	0.769307	0.384653	+	0.923061i	$0.374321\pi$
$480$	0	0
$481$	−44.9998	−2.05182
$482$	−1.92389	−0.0876308
$483$	0	0
$484$	12.7649	0.580221
$485$	0	0
$486$	0	0
$487$	−2.39084	−0.108339	−0.0541697	−	0.998532i	$-0.517251\pi$
$487$	−2.39084	−0.108339	−0.0541697	+	0.998532i	$0.517251\pi$
$488$	3.68649	0.166880
$489$	0	0
$490$	0	0
$491$	9.24459	0.417202	0.208601	−	0.978001i	$-0.433109\pi$
$491$	9.24459	0.417202	0.208601	+	0.978001i	$0.433109\pi$
$492$	0	0
$493$	0.525858	0.0236834
$494$	26.0761	1.17322
$495$	0	0
$496$	−2.34017	−0.105077
$497$	−41.7731	−1.87378
$498$	0	0
$499$	7.72034	0.345610	0.172805	−	0.984956i	$-0.444717\pi$
$499$	7.72034	0.345610	0.172805	+	0.984956i	$0.444717\pi$
$500$	0	0
$501$	0	0
$502$	4.17623	0.186394
$503$	−20.3090	−0.905532	−0.452766	−	0.891629i	$-0.649563\pi$
$503$	−20.3090	−0.905532	−0.452766	+	0.891629i	$0.649563\pi$
$504$	0	0
$505$	0	0
$506$	7.39520	0.328757
$507$	0	0
$508$	−11.4186	−0.506616
$509$	31.1906	1.38250	0.691250	−	0.722616i	$-0.257060\pi$
$509$	31.1906	1.38250	0.691250	+	0.722616i	$0.257060\pi$
$510$	0	0
$511$	10.8371	0.479405
$512$	−21.6742	−0.957873
$513$	0	0
$514$	11.0289	0.486465
$515$	0	0
$516$	0	0
$517$	−22.0989	−0.971908
$518$	−13.6598	−0.600178
$519$	0	0
$520$	0	0
$521$	−14.6693	−0.642673	−0.321336	−	0.946965i	$-0.604132\pi$
$521$	−14.6693	−0.642673	−0.321336	+	0.946965i	$0.604132\pi$
$522$	0	0
$523$	2.52813	0.110547	0.0552736	−	0.998471i	$-0.482397\pi$
$523$	2.52813	0.110547	0.0552736	+	0.998471i	$0.482397\pi$
$524$	6.81044	0.297515
$525$	0	0
$526$	13.2162	0.576255
$527$	3.09171	0.134677
$528$	0	0
$529$	30.2595	1.31563
$530$	0	0
$531$	0	0
$532$	−46.5380	−2.01768
$533$	52.4307	2.27102
$534$	0	0
$535$	0	0
$536$	3.62863	0.156733
$537$	0	0
$538$	14.7070	0.634064
$539$	−12.7021	−0.547118
$540$	0	0
$541$	−5.08065	−0.218434	−0.109217	−	0.994018i	$-0.534834\pi$
$541$	−5.08065	−0.218434	−0.109217	+	0.994018i	$0.534834\pi$
$542$	6.73433	0.289264
$543$	0	0
$544$	16.2679	0.697482
$545$	0	0
$546$	0	0
$547$	40.2990	1.72306	0.861529	−	0.507708i	$-0.169507\pi$
$547$	40.2990	1.72306	0.861529	+	0.507708i	$0.169507\pi$
$548$	39.6391	1.69330
$549$	0	0
$550$	0	0
$551$	1.24846	0.0531864
$552$	0	0
$553$	−23.1773	−0.985598
$554$	8.01438	0.340498
$555$	0	0
$556$	21.7854	0.923906
$557$	11.5848	0.490862	0.245431	−	0.969414i	$-0.421071\pi$
$557$	11.5848	0.490862	0.245431	+	0.969414i	$0.421071\pi$
$558$	0	0
$559$	−27.5630	−1.16579
$560$	0	0
$561$	0	0
$562$	−4.76487	−0.200994
$563$	3.60319	0.151856	0.0759282	−	0.997113i	$-0.475808\pi$
$563$	3.60319	0.151856	0.0759282	+	0.997113i	$0.475808\pi$
$564$	0	0
$565$	0	0
$566$	−7.92654	−0.333177
$567$	0	0
$568$	22.5236	0.945069
$569$	−19.6959	−0.825697	−0.412848	−	0.910800i	$-0.635466\pi$
$569$	−19.6959	−0.825697	−0.412848	+	0.910800i	$0.635466\pi$
$570$	0	0
$571$	38.7214	1.62044	0.810220	−	0.586126i	$-0.199348\pi$
$571$	38.7214	1.62044	0.810220	+	0.586126i	$0.199348\pi$
$572$	21.1650	0.884953
$573$	0	0
$574$	15.9155	0.664299
$575$	0	0
$576$	0	0
$577$	8.23287	0.342739	0.171369	−	0.985207i	$-0.445181\pi$
$577$	8.23287	0.342739	0.171369	+	0.985207i	$0.445181\pi$
$578$	4.01229	0.166889
$579$	0	0
$580$	0	0
$581$	−23.0700	−0.957104
$582$	0	0
$583$	26.1773	1.08415
$584$	−5.84324	−0.241795
$585$	0	0
$586$	12.2062	0.504234
$587$	−18.6130	−0.768242	−0.384121	−	0.923283i	$-0.625495\pi$
$587$	−18.6130	−0.768242	−0.384121	+	0.923283i	$0.625495\pi$
$588$	0	0
$589$	7.34017	0.302447
$590$	0	0
$591$	0	0
$592$	−15.9832	−0.656905
$593$	9.03612	0.371069	0.185534	−	0.982638i	$-0.440598\pi$
$593$	9.03612	0.371069	0.185534	+	0.982638i	$0.440598\pi$
$594$	0	0
$595$	0	0
$596$	−18.8638	−0.772690
$597$	0	0
$598$	−25.9260	−1.06019
$599$	5.80098	0.237022	0.118511	−	0.992953i	$-0.462188\pi$
$599$	5.80098	0.237022	0.118511	+	0.992953i	$0.462188\pi$
$600$	0	0
$601$	−8.50412	−0.346890	−0.173445	−	0.984844i	$-0.555490\pi$
$601$	−8.50412	−0.346890	−0.173445	+	0.984844i	$0.555490\pi$
$602$	−8.36683	−0.341007
$603$	0	0
$604$	18.9816	0.772349
$605$	0	0
$606$	0	0
$607$	−6.90707	−0.280349	−0.140175	−	0.990127i	$-0.544766\pi$
$607$	−6.90707	−0.280349	−0.140175	+	0.990127i	$0.544766\pi$
$608$	38.6225	1.56635
$609$	0	0
$610$	0	0
$611$	77.4740	3.13426
$612$	0	0
$613$	−21.2618	−0.858756	−0.429378	−	0.903125i	$-0.641267\pi$
$613$	−21.2618	−0.858756	−0.429378	+	0.903125i	$0.641267\pi$
$614$	−1.51745	−0.0612392
$615$	0	0
$616$	13.9421	0.561745
$617$	−5.40295	−0.217515	−0.108757	−	0.994068i	$-0.534687\pi$
$617$	−5.40295	−0.217515	−0.108757	+	0.994068i	$0.534687\pi$
$618$	0	0
$619$	4.75258	0.191022	0.0955112	−	0.995428i	$-0.469551\pi$
$619$	4.75258	0.191022	0.0955112	+	0.995428i	$0.469551\pi$
$620$	0	0
$621$	0	0
$622$	0.420818	0.0168733
$623$	−46.0060	−1.84319
$624$	0	0
$625$	0	0
$626$	17.9916	0.719089
$627$	0	0
$628$	−24.6453	−0.983453
$629$	21.1161	0.841954
$630$	0	0
$631$	25.5369	1.01661	0.508304	−	0.861177i	$-0.330272\pi$
$631$	25.5369	1.01661	0.508304	+	0.861177i	$0.330272\pi$
$632$	12.4969	0.497101
$633$	0	0
$634$	7.96946	0.316508
$635$	0	0
$636$	0	0
$637$	44.5308	1.76437
$638$	−0.172354	−0.00682356
$639$	0	0
$640$	0	0
$641$	−8.08557	−0.319361	−0.159680	−	0.987169i	$-0.551046\pi$
$641$	−8.08557	−0.319361	−0.159680	+	0.987169i	$0.551046\pi$
$642$	0	0
$643$	−3.51479	−0.138610	−0.0693050	−	0.997596i	$-0.522078\pi$
$643$	−3.51479	−0.138610	−0.0693050	+	0.997596i	$0.522078\pi$
$644$	46.2700	1.82329
$645$	0	0
$646$	−12.2362	−0.481426
$647$	−21.0566	−0.827822	−0.413911	−	0.910317i	$-0.635837\pi$
$647$	−21.0566	−0.827822	−0.413911	+	0.910317i	$0.635837\pi$
$648$	0	0
$649$	−12.6342	−0.495936
$650$	0	0
$651$	0	0
$652$	−28.5008	−1.11618
$653$	19.2918	0.754945	0.377473	−	0.926021i	$-0.376793\pi$
$653$	19.2918	0.754945	0.377473	+	0.926021i	$0.376793\pi$
$654$	0	0
$655$	0	0
$656$	18.6225	0.727086
$657$	0	0
$658$	23.5174	0.916806
$659$	26.1399	1.01827	0.509134	−	0.860687i	$-0.329966\pi$
$659$	26.1399	1.01827	0.509134	+	0.860687i	$0.329966\pi$
$660$	0	0
$661$	−15.4475	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$661$	−15.4475	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$662$	−1.16677	−0.0453480
$663$	0	0
$664$	12.4391	0.482730
$665$	0	0
$666$	0	0
$667$	−1.24128	−0.0480624
$668$	11.3730	0.440034
$669$	0	0
$670$	0	0
$671$	−3.46412	−0.133731
$672$	0	0
$673$	−39.0277	−1.50441	−0.752204	−	0.658931i	$-0.771009\pi$
$673$	−39.0277	−1.50441	−0.752204	+	0.658931i	$0.771009\pi$
$674$	17.9604	0.691808
$675$	0	0
$676$	−51.9793	−1.99920
$677$	3.24354	0.124660	0.0623298	−	0.998056i	$-0.480147\pi$
$677$	3.24354	0.124660	0.0623298	+	0.998056i	$0.480147\pi$
$678$	0	0
$679$	14.6309	0.561482
$680$	0	0
$681$	0	0
$682$	−1.01333	−0.0388024
$683$	−47.4908	−1.81718	−0.908592	−	0.417684i	$-0.862842\pi$
$683$	−47.4908	−1.81718	−0.908592	+	0.417684i	$0.862842\pi$
$684$	0	0
$685$	0	0
$686$	−0.482553	−0.0184239
$687$	0	0
$688$	−9.78992	−0.373237
$689$	−91.7719	−3.49623
$690$	0	0
$691$	−9.71154	−0.369444	−0.184722	−	0.982791i	$-0.559139\pi$
$691$	−9.71154	−0.369444	−0.184722	+	0.982791i	$0.559139\pi$
$692$	33.7009	1.28111
$693$	0	0
$694$	1.55640	0.0590800
$695$	0	0
$696$	0	0
$697$	−24.6030	−0.931906
$698$	−1.80098	−0.0681683
$699$	0	0
$700$	0	0
$701$	−9.47519	−0.357873	−0.178936	−	0.983861i	$-0.557266\pi$
$701$	−9.47519	−0.357873	−0.178936	+	0.983861i	$0.557266\pi$
$702$	0	0
$703$	50.1327	1.89079
$704$	3.46412	0.130559
$705$	0	0
$706$	−13.0544	−0.491308
$707$	−29.6742	−1.11601
$708$	0	0
$709$	50.0482	1.87960	0.939800	−	0.341724i	$-0.111011\pi$
$709$	50.0482	1.87960	0.939800	+	0.341724i	$0.111011\pi$
$710$	0	0
$711$	0	0
$712$	24.8059	0.929641
$713$	−7.29791	−0.273309
$714$	0	0
$715$	0	0
$716$	−6.29460	−0.235240
$717$	0	0
$718$	16.9171	0.631340
$719$	−41.5753	−1.55050	−0.775249	−	0.631656i	$-0.782375\pi$
$719$	−41.5753	−1.55050	−0.775249	+	0.631656i	$0.782375\pi$
$720$	0	0
$721$	9.65142	0.359438
$722$	−18.8059	−0.699883
$723$	0	0
$724$	2.58145	0.0959388
$725$	0	0
$726$	0	0
$727$	2.99773	0.111180	0.0555899	−	0.998454i	$-0.482296\pi$
$727$	2.99773	0.111180	0.0555899	+	0.998454i	$0.482296\pi$
$728$	−48.8781	−1.81154
$729$	0	0
$730$	0	0
$731$	12.9339	0.478378
$732$	0	0
$733$	−36.6020	−1.35192	−0.675962	−	0.736936i	$-0.736272\pi$
$733$	−36.6020	−1.35192	−0.675962	+	0.736936i	$0.736272\pi$
$734$	−7.34841	−0.271235
$735$	0	0
$736$	−38.4001	−1.41545
$737$	−3.40975	−0.125600
$738$	0	0
$739$	4.16394	0.153173	0.0765866	−	0.997063i	$-0.475598\pi$
$739$	4.16394	0.153173	0.0765866	+	0.997063i	$0.475598\pi$
$740$	0	0
$741$	0	0
$742$	−27.8576	−1.02269
$743$	−17.3824	−0.637700	−0.318850	−	0.947805i	$-0.603297\pi$
$743$	−17.3824	−0.637700	−0.318850	+	0.947805i	$0.603297\pi$
$744$	0	0
$745$	0	0
$746$	−1.29177	−0.0472950
$747$	0	0
$748$	−9.93164	−0.363137
$749$	−58.3956	−2.13373
$750$	0	0
$751$	−33.4307	−1.21990	−0.609951	−	0.792439i	$-0.708811\pi$
$751$	−33.4307	−1.21990	−0.609951	+	0.792439i	$0.708811\pi$
$752$	27.5174	1.00346
$753$	0	0
$754$	0.604236	0.0220050
$755$	0	0
$756$	0	0
$757$	−5.02771	−0.182735	−0.0913676	−	0.995817i	$-0.529124\pi$
$757$	−5.02771	−0.182735	−0.0913676	+	0.995817i	$0.529124\pi$
$758$	6.14447	0.223177
$759$	0	0
$760$	0	0
$761$	−23.6560	−0.857528	−0.428764	−	0.903417i	$-0.641051\pi$
$761$	−23.6560	−0.857528	−0.428764	+	0.903417i	$0.641051\pi$
$762$	0	0
$763$	17.8576	0.646489
$764$	−12.7070	−0.459723
$765$	0	0
$766$	2.57635	0.0930873
$767$	44.2928	1.59932
$768$	0	0
$769$	−43.5197	−1.56936	−0.784681	−	0.619900i	$-0.787173\pi$
$769$	−43.5197	−1.56936	−0.784681	+	0.619900i	$0.787173\pi$
$770$	0	0
$771$	0	0
$772$	39.2990	1.41440
$773$	45.3267	1.63029	0.815143	−	0.579259i	$-0.196658\pi$
$773$	45.3267	1.63029	0.815143	+	0.579259i	$0.196658\pi$
$774$	0	0
$775$	0	0
$776$	−7.88882	−0.283192
$777$	0	0
$778$	2.07223	0.0742932
$779$	−58.4112	−2.09280
$780$	0	0
$781$	−21.1650	−0.757343
$782$	12.1657	0.435046
$783$	0	0
$784$	15.8166	0.564878
$785$	0	0
$786$	0	0
$787$	−11.2618	−0.401440	−0.200720	−	0.979649i	$-0.564328\pi$
$787$	−11.2618	−0.401440	−0.200720	+	0.979649i	$0.564328\pi$
$788$	24.4657	0.871556
$789$	0	0
$790$	0	0
$791$	36.7792	1.30772
$792$	0	0
$793$	12.1445	0.431263
$794$	−6.85270	−0.243193
$795$	0	0
$796$	7.94214	0.281502
$797$	8.00597	0.283586	0.141793	−	0.989896i	$-0.454713\pi$
$797$	8.00597	0.283586	0.141793	+	0.989896i	$0.454713\pi$
$798$	0	0
$799$	−36.3545	−1.28613
$800$	0	0
$801$	0	0
$802$	−11.5369	−0.407383
$803$	5.49079	0.193766
$804$	0	0
$805$	0	0
$806$	3.55252	0.125132
$807$	0	0
$808$	16.0000	0.562878
$809$	8.70805	0.306159	0.153079	−	0.988214i	$-0.451081\pi$
$809$	8.70805	0.306159	0.153079	+	0.988214i	$0.451081\pi$
$810$	0	0
$811$	21.5357	0.756221	0.378110	−	0.925761i	$-0.376574\pi$
$811$	21.5357	0.756221	0.378110	+	0.925761i	$0.376574\pi$
$812$	−1.07838	−0.0378436
$813$	0	0
$814$	−6.92096	−0.242580
$815$	0	0
$816$	0	0
$817$	30.7070	1.07430
$818$	19.4785	0.681050
$819$	0	0
$820$	0	0
$821$	−33.8092	−1.17995	−0.589975	−	0.807422i	$-0.700862\pi$
$821$	−33.8092	−1.17995	−0.589975	+	0.807422i	$0.700862\pi$
$822$	0	0
$823$	−50.2700	−1.75230	−0.876152	−	0.482036i	$-0.839898\pi$
$823$	−50.2700	−1.75230	−0.876152	+	0.482036i	$0.839898\pi$
$824$	−5.20394	−0.181288
$825$	0	0
$826$	13.4452	0.467819
$827$	−16.3268	−0.567740	−0.283870	−	0.958863i	$-0.591618\pi$
$827$	−16.3268	−0.567740	−0.283870	+	0.958863i	$0.591618\pi$
$828$	0	0
$829$	12.3207	0.427916	0.213958	−	0.976843i	$-0.431365\pi$
$829$	12.3207	0.427916	0.213958	+	0.976843i	$0.431365\pi$
$830$	0	0
$831$	0	0
$832$	−12.1445	−0.421034
$833$	−20.8960	−0.724004
$834$	0	0
$835$	0	0
$836$	−23.5792	−0.815503
$837$	0	0
$838$	−1.57531	−0.0544181
$839$	1.47641	0.0509713	0.0254857	−	0.999675i	$-0.491887\pi$
$839$	1.47641	0.0509713	0.0254857	+	0.999675i	$0.491887\pi$
$840$	0	0
$841$	−28.9711	−0.999002
$842$	15.4375	0.532010
$843$	0	0
$844$	−45.4947	−1.56599
$845$	0	0
$846$	0	0
$847$	27.7009	0.951813
$848$	−32.5958	−1.11935
$849$	0	0
$850$	0	0
$851$	−49.8441	−1.70863
$852$	0	0
$853$	22.2907	0.763220	0.381610	−	0.924323i	$-0.375370\pi$
$853$	22.2907	0.763220	0.381610	+	0.924323i	$0.375370\pi$
$854$	3.68649	0.126149
$855$	0	0
$856$	31.4863	1.07618
$857$	54.4235	1.85907	0.929535	−	0.368733i	$-0.120208\pi$
$857$	54.4235	1.85907	0.929535	+	0.368733i	$0.120208\pi$
$858$	0	0
$859$	−6.55479	−0.223646	−0.111823	−	0.993728i	$-0.535669\pi$
$859$	−6.55479	−0.223646	−0.111823	+	0.993728i	$0.535669\pi$
$860$	0	0
$861$	0	0
$862$	−1.30074	−0.0443035
$863$	38.1399	1.29830	0.649149	−	0.760661i	$-0.275125\pi$
$863$	38.1399	1.29830	0.649149	+	0.760661i	$0.275125\pi$
$864$	0	0
$865$	0	0
$866$	14.1424	0.480578
$867$	0	0
$868$	−6.34017	−0.215199
$869$	−11.7431	−0.398358
$870$	0	0
$871$	11.9539	0.405041
$872$	−9.62863	−0.326067
$873$	0	0
$874$	28.8832	0.976990
$875$	0	0
$876$	0	0
$877$	47.2762	1.59640	0.798201	−	0.602391i	$-0.205785\pi$
$877$	47.2762	1.59640	0.798201	+	0.602391i	$0.205785\pi$
$878$	−15.3775	−0.518966
$879$	0	0
$880$	0	0
$881$	−28.2557	−0.951957	−0.475979	−	0.879457i	$-0.657906\pi$
$881$	−28.2557	−0.951957	−0.475979	+	0.879457i	$0.657906\pi$
$882$	0	0
$883$	7.39908	0.248999	0.124499	−	0.992220i	$-0.460268\pi$
$883$	7.39908	0.248999	0.124499	+	0.992220i	$0.460268\pi$
$884$	34.8182	1.17106
$885$	0	0
$886$	−6.69860	−0.225044
$887$	−32.1822	−1.08057	−0.540286	−	0.841481i	$-0.681684\pi$
$887$	−32.1822	−1.08057	−0.540286	+	0.841481i	$0.681684\pi$
$888$	0	0
$889$	−24.7792	−0.831069
$890$	0	0
$891$	0	0
$892$	29.0928	0.974097
$893$	−86.3111	−2.88829
$894$	0	0
$895$	0	0
$896$	−42.7214	−1.42722
$897$	0	0
$898$	−9.21112	−0.307379
$899$	0.170086	0.00567270
$900$	0	0
$901$	43.0638	1.43466
$902$	8.06382	0.268496
$903$	0	0
$904$	−19.8310	−0.659568
$905$	0	0
$906$	0	0
$907$	−56.8453	−1.88752	−0.943759	−	0.330634i	$-0.892738\pi$
$907$	−56.8453	−1.88752	−0.943759	+	0.330634i	$0.892738\pi$
$908$	16.7259	0.555069
$909$	0	0
$910$	0	0
$911$	−17.6153	−0.583621	−0.291810	−	0.956476i	$-0.594258\pi$
$911$	−17.6153	−0.583621	−0.291810	+	0.956476i	$0.594258\pi$
$912$	0	0
$913$	−11.6888	−0.386841
$914$	−5.94602	−0.196677
$915$	0	0
$916$	38.1978	1.26209
$917$	14.7792	0.488054
$918$	0	0
$919$	−40.1012	−1.32282	−0.661408	−	0.750027i	$-0.730041\pi$
$919$	−40.1012	−1.32282	−0.661408	+	0.750027i	$0.730041\pi$
$920$	0	0
$921$	0	0
$922$	9.01333	0.296838
$923$	74.1999	2.44232
$924$	0	0
$925$	0	0
$926$	6.69263	0.219934
$927$	0	0
$928$	0.894960	0.0293785
$929$	0.326842	0.0107233	0.00536167	−	0.999986i	$-0.498293\pi$
$929$	0.326842	0.0107233	0.00536167	+	0.999986i	$0.498293\pi$
$930$	0	0
$931$	−49.6102	−1.62591
$932$	−35.3484	−1.15788
$933$	0	0
$934$	7.93230	0.259553
$935$	0	0
$936$	0	0
$937$	−35.1978	−1.14986	−0.574931	−	0.818202i	$-0.694971\pi$
$937$	−35.1978	−1.14986	−0.574931	+	0.818202i	$0.694971\pi$
$938$	3.62863	0.118479
$939$	0	0
$940$	0	0
$941$	−34.1038	−1.11175	−0.555876	−	0.831265i	$-0.687617\pi$
$941$	−34.1038	−1.11175	−0.555876	+	0.831265i	$0.687617\pi$
$942$	0	0
$943$	58.0749	1.89118
$944$	15.7321	0.512035
$945$	0	0
$946$	−4.23919	−0.137828
$947$	−14.6631	−0.476488	−0.238244	−	0.971205i	$-0.576572\pi$
$947$	−14.6631	−0.476488	−0.238244	+	0.971205i	$0.576572\pi$
$948$	0	0
$949$	−19.2495	−0.624866
$950$	0	0
$951$	0	0
$952$	22.9360	0.743360
$953$	−7.89657	−0.255795	−0.127897	−	0.991787i	$-0.540823\pi$
$953$	−7.89657	−0.255795	−0.127897	+	0.991787i	$0.540823\pi$
$954$	0	0
$955$	0	0
$956$	−5.86603	−0.189721
$957$	0	0
$958$	−9.07838	−0.293309
$959$	86.0203	2.77774
$960$	0	0
$961$	1.00000	0.0322581
$962$	24.2634	0.782284
$963$	0	0
$964$	−6.09890	−0.196432
$965$	0	0
$966$	0	0
$967$	11.7126	0.376651	0.188326	−	0.982107i	$-0.439694\pi$
$967$	11.7126	0.376651	0.188326	+	0.982107i	$0.439694\pi$
$968$	−14.9360	−0.480061
$969$	0	0
$970$	0	0
$971$	−2.24459	−0.0720323	−0.0360161	−	0.999351i	$-0.511467\pi$
$971$	−2.24459	−0.0720323	−0.0360161	+	0.999351i	$0.511467\pi$
$972$	0	0
$973$	47.2762	1.51560
$974$	1.28912	0.0413060
$975$	0	0
$976$	4.31351	0.138072
$977$	56.3701	1.80344	0.901720	−	0.432320i	$-0.142305\pi$
$977$	56.3701	1.80344	0.901720	+	0.432320i	$0.142305\pi$
$978$	0	0
$979$	−23.3096	−0.744979
$980$	0	0
$981$	0	0
$982$	−4.98458	−0.159064
$983$	−9.92881	−0.316680	−0.158340	−	0.987385i	$-0.550614\pi$
$983$	−9.92881	−0.316680	−0.158340	+	0.987385i	$0.550614\pi$
$984$	0	0
$985$	0	0
$986$	−0.283537	−0.00902965
$987$	0	0
$988$	82.6635	2.62988
$989$	−30.5302	−0.970804
$990$	0	0
$991$	−4.61530	−0.146610	−0.0733049	−	0.997310i	$-0.523355\pi$
$991$	−4.61530	−0.146610	−0.0733049	+	0.997310i	$0.523355\pi$
$992$	5.26180	0.167062
$993$	0	0
$994$	22.5236	0.714405
$995$	0	0
$996$	0	0
$997$	−8.72365	−0.276281	−0.138140	−	0.990413i	$-0.544113\pi$
$997$	−8.72365	−0.276281	−0.138140	+	0.990413i	$0.544113\pi$
$998$	−4.16272	−0.131769
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bc.1.2		3
3.2	odd	2		2325.2.a.u.1.2	yes	3
5.4	even	2		6975.2.a.bd.1.2		3
15.2	even	4		2325.2.c.m.1024.4		6
15.8	even	4		2325.2.c.m.1024.3		6
15.14	odd	2		2325.2.a.t.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.t.1.2	✓	3	15.14	odd	2
2325.2.a.u.1.2	yes	3	3.2	odd	2
2325.2.c.m.1024.3		6	15.8	even	4
2325.2.c.m.1024.4		6	15.2	even	4
6975.2.a.bc.1.2		3	1.1	even	1	trivial
6975.2.a.bd.1.2		3	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)

\(f(q)\)	\(=\)	\(q-0.539189 q^{2} -1.70928 q^{4} -3.70928 q^{7} +2.00000 q^{8} -1.87936 q^{11} +6.58864 q^{13} +2.00000 q^{14} +2.34017 q^{16} -3.09171 q^{17} -7.34017 q^{19} +1.01333 q^{22} +7.29791 q^{23} -3.55252 q^{26} +6.34017 q^{28} -0.170086 q^{29} -1.00000 q^{31} -5.26180 q^{32} +1.66701 q^{34} -6.82991 q^{37} +3.95774 q^{38} +7.95774 q^{41} -4.18342 q^{43} +3.21235 q^{44} -3.93495 q^{46} +11.7587 q^{47} +6.75872 q^{49} -11.2618 q^{52} -13.9288 q^{53} -7.41855 q^{56} +0.0917087 q^{58} +6.72261 q^{59} +1.84324 q^{61} +0.539189 q^{62} -1.84324 q^{64} +1.81432 q^{67} +5.28458 q^{68} +11.2618 q^{71} -2.92162 q^{73} +3.68261 q^{74} +12.5464 q^{76} +6.97107 q^{77} +6.24846 q^{79} -4.29072 q^{82} +6.21953 q^{83} +2.25565 q^{86} -3.75872 q^{88} +12.4030 q^{89} -24.4391 q^{91} -12.4741 q^{92} -6.34017 q^{94} -3.94441 q^{97} -3.64423 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} - 4 q^{7} + 6 q^{8} + 7 q^{11} + 6 q^{14} - 4 q^{16} - 7 q^{17} - 11 q^{19} + 4 q^{22} - 5 q^{23} - 10 q^{26} + 8 q^{28} + 5 q^{29} - 3 q^{31} - 8 q^{32} - 18 q^{34} - 26 q^{37} - 4 q^{38}+ \cdots - 8 q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 - 4 * q^7 + 6 * q^8 + 7 * q^11 + 6 * q^14 - 4 * q^16 - 7 * q^17 - 11 * q^19 + 4 * q^22 - 5 * q^23 - 10 * q^26 + 8 * q^28 + 5 * q^29 - 3 * q^31 - 8 * q^32 - 18 * q^34 - 26 * q^37 - 4 * q^38 + 8 * q^41 - 8 * q^43 + 20 * q^44 - 16 * q^46 + 10 * q^47 - 5 * q^49 - 26 * q^52 - 11 * q^53 - 8 * q^56 - 2 * q^58 + 14 * q^59 + 12 * q^61 - 12 * q^64 - 3 * q^67 - 12 * q^68 + 26 * q^71 - 12 * q^73 + 2 * q^74 + 2 * q^76 + 6 * q^77 + 10 * q^79 - 20 * q^82 - 5 * q^83 - 36 * q^86 + 14 * q^88 + 9 * q^89 - 26 * q^91 - 40 * q^92 - 8 * q^94 + 5 * q^97 - 8 * q^98

Introduction

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