LMFDB - Embedded newform 6975.2.a.bd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [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:	$0$
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			1.3
Root			$-1.48119$ 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)

$f(q)$	$=$	$q+1.67513 q^{2} +0.806063 q^{4} +1.19394 q^{7} -2.00000 q^{8} +4.28726 q^{11} +2.09332 q^{13} +2.00000 q^{14} -4.96239 q^{16} -2.83146 q^{17} -0.0376114 q^{19} +7.18172 q^{22} +3.89938 q^{23} +3.50659 q^{26} +0.962389 q^{28} +3.48119 q^{29} -1.00000 q^{31} -4.31265 q^{32} -4.74306 q^{34} +10.4812 q^{37} -0.0630040 q^{38} +4.06300 q^{41} -7.66291 q^{43} +3.45580 q^{44} +6.53198 q^{46} +0.574515 q^{47} -5.57452 q^{49} +1.68735 q^{52} -2.05571 q^{53} -2.38787 q^{56} +5.83146 q^{58} -3.98778 q^{59} -2.70052 q^{61} -1.67513 q^{62} +2.70052 q^{64} +14.8192 q^{67} -2.28233 q^{68} +1.68735 q^{71} +0.649738 q^{73} +17.5574 q^{74} -0.0303172 q^{76} +5.11871 q^{77} +4.86907 q^{79} +6.80606 q^{82} +7.24965 q^{83} -12.8364 q^{86} -8.57452 q^{88} -12.9126 q^{89} +2.49929 q^{91} +3.14315 q^{92} +0.962389 q^{94} +8.24472 q^{97} -9.33804 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.67513	1.18450	0.592248	−	0.805756i	$-0.298240\pi$
$2$	1.67513	1.18450	0.592248	+	0.805756i	$0.298240\pi$
$3$	0	0
$3$	0	0
$4$	0.806063	0.403032
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.19394	0.451266	0.225633	−	0.974212i	$-0.427555\pi$
$7$	1.19394	0.451266	0.225633	+	0.974212i	$0.427555\pi$
$8$	−2.00000	−0.707107
$9$	0	0
$10$	0	0
$11$	4.28726	1.29266	0.646328	−	0.763059i	$-0.276304\pi$
$11$	4.28726	1.29266	0.646328	+	0.763059i	$0.276304\pi$
$12$	0	0
$13$	2.09332	0.580583	0.290291	−	0.956938i	$-0.406248\pi$
$13$	2.09332	0.580583	0.290291	+	0.956938i	$0.406248\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.96239	−1.24060
$17$	−2.83146	−0.686729	−0.343364	−	0.939202i	$-0.611567\pi$
$17$	−2.83146	−0.686729	−0.343364	+	0.939202i	$0.611567\pi$
$18$	0	0
$19$	−0.0376114	−0.00862865	−0.00431432	−	0.999991i	$-0.501373\pi$
$19$	−0.0376114	−0.00862865	−0.00431432	+	0.999991i	$0.501373\pi$
$20$	0	0
$21$	0	0
$22$	7.18172	1.53115
$23$	3.89938	0.813078	0.406539	−	0.913633i	$-0.366736\pi$
$23$	3.89938	0.813078	0.406539	+	0.913633i	$0.366736\pi$
$24$	0	0
$25$	0	0
$26$	3.50659	0.687698
$27$	0	0
$28$	0.962389	0.181874
$29$	3.48119	0.646442	0.323221	−	0.946324i	$-0.395234\pi$
$29$	3.48119	0.646442	0.323221	+	0.946324i	$0.395234\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−4.31265	−0.762376
$33$	0	0
$34$	−4.74306	−0.813428
$35$	0	0
$36$	0	0
$37$	10.4812	1.72310	0.861549	−	0.507675i	$-0.169495\pi$
$37$	10.4812	1.72310	0.861549	+	0.507675i	$0.169495\pi$
$38$	−0.0630040	−0.0102206
$39$	0	0
$40$	0	0
$41$	4.06300	0.634535	0.317267	−	0.948336i	$-0.397235\pi$
$41$	4.06300	0.634535	0.317267	+	0.948336i	$0.397235\pi$
$42$	0	0
$43$	−7.66291	−1.16858	−0.584292	−	0.811544i	$-0.698628\pi$
$43$	−7.66291	−1.16858	−0.584292	+	0.811544i	$0.698628\pi$
$44$	3.45580	0.520982
$45$	0	0
$46$	6.53198	0.963088
$47$	0.574515	0.0838017	0.0419008	−	0.999122i	$-0.486659\pi$
$47$	0.574515	0.0838017	0.0419008	+	0.999122i	$0.486659\pi$
$48$	0	0
$49$	−5.57452	−0.796359
$50$	0	0
$51$	0	0
$52$	1.68735	0.233993
$53$	−2.05571	−0.282373	−0.141187	−	0.989983i	$-0.545092\pi$
$53$	−2.05571	−0.282373	−0.141187	+	0.989983i	$0.545092\pi$
$54$	0	0
$55$	0	0
$56$	−2.38787	−0.319093
$57$	0	0
$58$	5.83146	0.765708
$59$	−3.98778	−0.519165	−0.259582	−	0.965721i	$-0.583585\pi$
$59$	−3.98778	−0.519165	−0.259582	+	0.965721i	$0.583585\pi$
$60$	0	0
$61$	−2.70052	−0.345767	−0.172883	−	0.984942i	$-0.555308\pi$
$61$	−2.70052	−0.345767	−0.172883	+	0.984942i	$0.555308\pi$
$62$	−1.67513	−0.212742
$63$	0	0
$64$	2.70052	0.337565
$65$	0	0
$66$	0	0
$67$	14.8192	1.81046	0.905229	−	0.424924i	$-0.139699\pi$
$67$	14.8192	1.81046	0.905229	+	0.424924i	$0.139699\pi$
$68$	−2.28233	−0.276774
$69$	0	0
$70$	0	0
$71$	1.68735	0.200252	0.100126	−	0.994975i	$-0.468075\pi$
$71$	1.68735	0.200252	0.100126	+	0.994975i	$0.468075\pi$
$72$	0	0
$73$	0.649738	0.0760461	0.0380231	−	0.999277i	$-0.487894\pi$
$73$	0.649738	0.0760461	0.0380231	+	0.999277i	$0.487894\pi$
$74$	17.5574	2.04100
$75$	0	0
$76$	−0.0303172	−0.00347762
$77$	5.11871	0.583332
$78$	0	0
$79$	4.86907	0.547813	0.273906	−	0.961756i	$-0.411684\pi$
$79$	4.86907	0.547813	0.273906	+	0.961756i	$0.411684\pi$
$80$	0	0
$81$	0	0
$82$	6.80606	0.751604
$83$	7.24965	0.795752	0.397876	−	0.917439i	$-0.369747\pi$
$83$	7.24965	0.795752	0.397876	+	0.917439i	$0.369747\pi$
$84$	0	0
$85$	0	0
$86$	−12.8364	−1.38418
$87$	0	0
$88$	−8.57452	−0.914046
$89$	−12.9126	−1.36873	−0.684364	−	0.729140i	$-0.739920\pi$
$89$	−12.9126	−1.36873	−0.684364	+	0.729140i	$0.739920\pi$
$90$	0	0
$91$	2.49929	0.261997
$92$	3.14315	0.327696
$93$	0	0
$94$	0.962389	0.0992628
$95$	0	0
$96$	0	0
$97$	8.24472	0.837125	0.418562	−	0.908188i	$-0.362534\pi$
$97$	8.24472	0.837125	0.418562	+	0.908188i	$0.362534\pi$
$98$	−9.33804	−0.943285
$99$	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
$103$	−14.2750	−1.40656	−0.703281	−	0.710912i	$-0.748282\pi$
$103$	−14.2750	−1.40656	−0.703281	+	0.710912i	$0.748282\pi$
$104$	−4.18664	−0.410534
$105$	0	0
$106$	−3.44358	−0.334470
$107$	16.8749	1.63136	0.815681	−	0.578501i	$-0.196362\pi$
$107$	16.8749	1.63136	0.815681	+	0.578501i	$0.196362\pi$
$108$	0	0
$109$	11.8192	1.13208	0.566039	−	0.824379i	$-0.308475\pi$
$109$	11.8192	1.13208	0.566039	+	0.824379i	$0.308475\pi$
$110$	0	0
$111$	0	0
$112$	−5.92478	−0.559839
$113$	2.12601	0.199998	0.0999990	−	0.994988i	$-0.468116\pi$
$113$	2.12601	0.199998	0.0999990	+	0.994988i	$0.468116\pi$
$114$	0	0
$115$	0	0
$116$	2.80606	0.260536
$117$	0	0
$118$	−6.68006	−0.614949
$119$	−3.38058	−0.309897
$120$	0	0
$121$	7.38058	0.670962
$122$	−4.52373	−0.409559
$123$	0	0
$124$	−0.806063	−0.0723866
$125$	0	0
$126$	0	0
$127$	7.92478	0.703210	0.351605	−	0.936148i	$-0.385636\pi$
$127$	7.92478	0.703210	0.351605	+	0.936148i	$0.385636\pi$
$128$	13.1490	1.16222
$129$	0	0
$130$	0	0
$131$	16.3004	1.42418	0.712088	−	0.702091i	$-0.247750\pi$
$131$	16.3004	1.42418	0.712088	+	0.702091i	$0.247750\pi$
$132$	0	0
$133$	−0.0449056	−0.00389381
$134$	24.8242	2.14448
$135$	0	0
$136$	5.66291	0.485591
$137$	−2.36836	−0.202343	−0.101171	−	0.994869i	$-0.532259\pi$
$137$	−2.36836	−0.202343	−0.101171	+	0.994869i	$0.532259\pi$
$138$	0	0
$139$	−8.60720	−0.730053	−0.365027	−	0.930997i	$-0.618940\pi$
$139$	−8.60720	−0.730053	−0.365027	+	0.930997i	$0.618940\pi$
$140$	0	0
$141$	0	0
$142$	2.82653	0.237197
$143$	8.97461	0.750494
$144$	0	0
$145$	0	0
$146$	1.08840	0.0900763
$147$	0	0
$148$	8.44851	0.694463
$149$	9.41327	0.771165	0.385582	−	0.922673i	$-0.374001\pi$
$149$	9.41327	0.771165	0.385582	+	0.922673i	$0.374001\pi$
$150$	0	0
$151$	3.01317	0.245209	0.122604	−	0.992456i	$-0.460875\pi$
$151$	3.01317	0.245209	0.122604	+	0.992456i	$0.460875\pi$
$152$	0.0752228	0.00610137
$153$	0	0
$154$	8.57452	0.690954
$155$	0	0
$156$	0	0
$157$	−9.38787	−0.749234	−0.374617	−	0.927180i	$-0.622226\pi$
$157$	−9.38787	−0.749234	−0.374617	+	0.927180i	$0.622226\pi$
$158$	8.15633	0.648882
$159$	0	0
$160$	0	0
$161$	4.65562	0.366914
$162$	0	0
$163$	3.44851	0.270108	0.135054	−	0.990838i	$-0.456879\pi$
$163$	3.44851	0.270108	0.135054	+	0.990838i	$0.456879\pi$
$164$	3.27504	0.255738
$165$	0	0
$166$	12.1441	0.942565
$167$	8.43866	0.653003	0.326501	−	0.945197i	$-0.394130\pi$
$167$	8.43866	0.653003	0.326501	+	0.945197i	$0.394130\pi$
$168$	0	0
$169$	−8.61801	−0.662924
$170$	0	0
$171$	0	0
$172$	−6.17679	−0.470976
$173$	3.48849	0.265225	0.132612	−	0.991168i	$-0.457663\pi$
$173$	3.48849	0.265225	0.132612	+	0.991168i	$0.457663\pi$
$174$	0	0
$175$	0	0
$176$	−21.2750	−1.60367
$177$	0	0
$178$	−21.6302	−1.62125
$179$	17.5574	1.31230	0.656150	−	0.754631i	$-0.272184\pi$
$179$	17.5574	1.31230	0.656150	+	0.754631i	$0.272184\pi$
$180$	0	0
$181$	9.44358	0.701936	0.350968	−	0.936387i	$-0.385853\pi$
$181$	9.44358	0.701936	0.350968	+	0.936387i	$0.385853\pi$
$182$	4.18664	0.310335
$183$	0	0
$184$	−7.79877	−0.574933
$185$	0	0
$186$	0	0
$187$	−12.1392	−0.887705
$188$	0.463096	0.0337747
$189$	0	0
$190$	0	0
$191$	22.6883	1.64167	0.820834	−	0.571167i	$-0.193509\pi$
$191$	22.6883	1.64167	0.820834	+	0.571167i	$0.193509\pi$
$192$	0	0
$193$	−11.0059	−0.792221	−0.396110	−	0.918203i	$-0.629640\pi$
$193$	−11.0059	−0.792221	−0.396110	+	0.918203i	$0.629640\pi$
$194$	13.8110	0.991571
$195$	0	0
$196$	−4.49341	−0.320958
$197$	23.4010	1.66726	0.833628	−	0.552327i	$-0.186260\pi$
$197$	23.4010	1.66726	0.833628	+	0.552327i	$0.186260\pi$
$198$	0	0
$199$	−20.1441	−1.42798	−0.713989	−	0.700157i	$-0.753113\pi$
$199$	−20.1441	−1.42798	−0.713989	+	0.700157i	$0.753113\pi$
$200$	0	0
$201$	0	0
$202$	13.4010	0.942894
$203$	4.15633	0.291717
$204$	0	0
$205$	0	0
$206$	−23.9126	−1.66607
$207$	0	0
$208$	−10.3879	−0.720269
$209$	−0.161250	−0.0111539
$210$	0	0
$211$	−17.6859	−1.21755	−0.608775	−	0.793343i	$-0.708339\pi$
$211$	−17.6859	−1.21755	−0.608775	+	0.793343i	$0.708339\pi$
$212$	−1.65703	−0.113805
$213$	0	0
$214$	28.2677	1.93234
$215$	0	0
$216$	0	0
$217$	−1.19394	−0.0810497
$218$	19.7988	1.34094
$219$	0	0
$220$	0	0
$221$	−5.92715	−0.398703
$222$	0	0
$223$	−4.88717	−0.327269	−0.163634	−	0.986521i	$-0.552322\pi$
$223$	−4.88717	−0.327269	−0.163634	+	0.986521i	$0.552322\pi$
$224$	−5.14903	−0.344034
$225$	0	0
$226$	3.56134	0.236897
$227$	−18.9380	−1.25696	−0.628478	−	0.777827i	$-0.716322\pi$
$227$	−18.9380	−1.25696	−0.628478	+	0.777827i	$0.716322\pi$
$228$	0	0
$229$	−1.33216	−0.0880318	−0.0440159	−	0.999031i	$-0.514015\pi$
$229$	−1.33216	−0.0880318	−0.0440159	+	0.999031i	$0.514015\pi$
$230$	0	0
$231$	0	0
$232$	−6.96239	−0.457103
$233$	−6.07522	−0.398001	−0.199001	−	0.979999i	$-0.563770\pi$
$233$	−6.07522	−0.398001	−0.199001	+	0.979999i	$0.563770\pi$
$234$	0	0
$235$	0	0
$236$	−3.21440	−0.209240
$237$	0	0
$238$	−5.66291	−0.367072
$239$	−9.79384	−0.633511	−0.316756	−	0.948507i	$-0.602593\pi$
$239$	−9.79384	−0.633511	−0.316756	+	0.948507i	$0.602593\pi$
$240$	0	0
$241$	16.7938	1.08179	0.540893	−	0.841091i	$-0.318086\pi$
$241$	16.7938	1.08179	0.540893	+	0.841091i	$0.318086\pi$
$242$	12.3634	0.794752
$243$	0	0
$244$	−2.17679	−0.139355
$245$	0	0
$246$	0	0
$247$	−0.0787327	−0.00500964
$248$	2.00000	0.127000
$249$	0	0
$250$	0	0
$251$	−3.60720	−0.227685	−0.113842	−	0.993499i	$-0.536316\pi$
$251$	−3.60720	−0.227685	−0.113842	+	0.993499i	$0.536316\pi$
$252$	0	0
$253$	16.7177	1.05103
$254$	13.2750	0.832950
$255$	0	0
$256$	16.6253	1.03908
$257$	13.8011	0.860891	0.430446	−	0.902616i	$-0.358356\pi$
$257$	13.8011	0.860891	0.430446	+	0.902616i	$0.358356\pi$
$258$	0	0
$259$	12.5139	0.777575
$260$	0	0
$261$	0	0
$262$	27.3054	1.68693
$263$	−5.67276	−0.349797	−0.174899	−	0.984586i	$-0.555960\pi$
$263$	−5.67276	−0.349797	−0.174899	+	0.984586i	$0.555960\pi$
$264$	0	0
$265$	0	0
$266$	−0.0752228	−0.00461220
$267$	0	0
$268$	11.9452	0.729672
$269$	9.72355	0.592855	0.296428	−	0.955055i	$-0.404205\pi$
$269$	9.72355	0.592855	0.296428	+	0.955055i	$0.404205\pi$
$270$	0	0
$271$	−23.4436	−1.42410	−0.712048	−	0.702131i	$-0.752232\pi$
$271$	−23.4436	−1.42410	−0.712048	+	0.702131i	$0.752232\pi$
$272$	14.0508	0.851954
$273$	0	0
$274$	−3.96731	−0.239674
$275$	0	0
$276$	0	0
$277$	−11.5877	−0.696237	−0.348118	−	0.937451i	$-0.613179\pi$
$277$	−11.5877	−0.696237	−0.348118	+	0.937451i	$0.613179\pi$
$278$	−14.4182	−0.864746
$279$	0	0
$280$	0	0
$281$	−1.22425	−0.0730329	−0.0365164	−	0.999333i	$-0.511626\pi$
$281$	−1.22425	−0.0730329	−0.0365164	+	0.999333i	$0.511626\pi$
$282$	0	0
$283$	21.8119	1.29659	0.648293	−	0.761391i	$-0.275483\pi$
$283$	21.8119	1.29659	0.648293	+	0.761391i	$0.275483\pi$
$284$	1.36011	0.0807077
$285$	0	0
$286$	15.0336	0.888958
$287$	4.85097	0.286344
$288$	0	0
$289$	−8.98286	−0.528403
$290$	0	0
$291$	0	0
$292$	0.523730	0.0306490
$293$	4.13823	0.241758	0.120879	−	0.992667i	$-0.461429\pi$
$293$	4.13823	0.241758	0.120879	+	0.992667i	$0.461429\pi$
$294$	0	0
$295$	0	0
$296$	−20.9624	−1.21841
$297$	0	0
$298$	15.7685	0.913442
$299$	8.16266	0.472059
$300$	0	0
$301$	−9.14903	−0.527341
$302$	5.04746	0.290449
$303$	0	0
$304$	0.186642	0.0107047
$305$	0	0
$306$	0	0
$307$	13.8192	0.788706	0.394353	−	0.918959i	$-0.370969\pi$
$307$	13.8192	0.788706	0.394353	+	0.918959i	$0.370969\pi$
$308$	4.12601	0.235101
$309$	0	0
$310$	0	0
$311$	−14.2496	−0.808023	−0.404012	−	0.914754i	$-0.632384\pi$
$311$	−14.2496	−0.808023	−0.404012	+	0.914754i	$0.632384\pi$
$312$	0	0
$313$	−9.55500	−0.540081	−0.270040	−	0.962849i	$-0.587037\pi$
$313$	−9.55500	−0.540081	−0.270040	+	0.962849i	$0.587037\pi$
$314$	−15.7259	−0.887465
$315$	0	0
$316$	3.92478	0.220786
$317$	28.2496	1.58666	0.793329	−	0.608793i	$-0.208346\pi$
$317$	28.2496	1.58666	0.793329	+	0.608793i	$0.208346\pi$
$318$	0	0
$319$	14.9248	0.835627
$320$	0	0
$321$	0	0
$322$	7.79877	0.434608
$323$	0.106495	0.00592554
$324$	0	0
$325$	0	0
$326$	5.77670	0.319942
$327$	0	0
$328$	−8.12601	−0.448684
$329$	0.685935	0.0378168
$330$	0	0
$331$	−7.00492	−0.385025	−0.192513	−	0.981294i	$-0.561664\pi$
$331$	−7.00492	−0.385025	−0.192513	+	0.981294i	$0.561664\pi$
$332$	5.84367	0.320713
$333$	0	0
$334$	14.1359	0.773480
$335$	0	0
$336$	0	0
$337$	−33.7924	−1.84079	−0.920395	−	0.390989i	$-0.872133\pi$
$337$	−33.7924	−1.84079	−0.920395	+	0.390989i	$0.872133\pi$
$338$	−14.4363	−0.785231
$339$	0	0
$340$	0	0
$341$	−4.28726	−0.232168
$342$	0	0
$343$	−15.0132	−0.810635
$344$	15.3258	0.826313
$345$	0	0
$346$	5.84367	0.314158
$347$	−16.9927	−0.912216	−0.456108	−	0.889924i	$-0.650757\pi$
$347$	−16.9927	−0.912216	−0.456108	+	0.889924i	$0.650757\pi$
$348$	0	0
$349$	−3.96239	−0.212102	−0.106051	−	0.994361i	$-0.533821\pi$
$349$	−3.96239	−0.212102	−0.106051	+	0.994361i	$0.533821\pi$
$350$	0	0
$351$	0	0
$352$	−18.4894	−0.985491
$353$	23.2555	1.23777	0.618883	−	0.785483i	$-0.287585\pi$
$353$	23.2555	1.23777	0.618883	+	0.785483i	$0.287585\pi$
$354$	0	0
$355$	0	0
$356$	−10.4083	−0.551641
$357$	0	0
$358$	29.4109	1.55441
$359$	25.2605	1.33320	0.666598	−	0.745418i	$-0.267750\pi$
$359$	25.2605	1.33320	0.666598	+	0.745418i	$0.267750\pi$
$360$	0	0
$361$	−18.9986	−0.999926
$362$	15.8192	0.831441
$363$	0	0
$364$	2.01459	0.105593
$365$	0	0
$366$	0	0
$367$	19.6385	1.02512	0.512560	−	0.858651i	$-0.328697\pi$
$367$	19.6385	1.02512	0.512560	+	0.858651i	$0.328697\pi$
$368$	−19.3503	−1.00870
$369$	0	0
$370$	0	0
$371$	−2.45439	−0.127425
$372$	0	0
$373$	9.20711	0.476726	0.238363	−	0.971176i	$-0.423389\pi$
$373$	9.20711	0.476726	0.238363	+	0.971176i	$0.423389\pi$
$374$	−20.3347	−1.05148
$375$	0	0
$376$	−1.14903	−0.0592567
$377$	7.28726	0.375313
$378$	0	0
$379$	0.207110	0.0106385	0.00531927	−	0.999986i	$-0.498307\pi$
$379$	0.207110	0.0106385	0.00531927	+	0.999986i	$0.498307\pi$
$380$	0	0
$381$	0	0
$382$	38.0059	1.94455
$383$	−10.2325	−0.522856	−0.261428	−	0.965223i	$-0.584193\pi$
$383$	−10.2325	−0.522856	−0.261428	+	0.965223i	$0.584193\pi$
$384$	0	0
$385$	0	0
$386$	−18.4363	−0.938382
$387$	0	0
$388$	6.64577	0.337388
$389$	0.700523	0.0355180	0.0177590	−	0.999842i	$-0.494347\pi$
$389$	0.700523	0.0355180	0.0177590	+	0.999842i	$0.494347\pi$
$390$	0	0
$391$	−11.0409	−0.558364
$392$	11.1490	0.563111
$393$	0	0
$394$	39.1998	1.97486
$395$	0	0
$396$	0	0
$397$	−10.1939	−0.511619	−0.255810	−	0.966727i	$-0.582342\pi$
$397$	−10.1939	−0.511619	−0.255810	+	0.966727i	$0.582342\pi$
$398$	−33.7440	−1.69143
$399$	0	0
$400$	0	0
$401$	−9.43629	−0.471226	−0.235613	−	0.971847i	$-0.575710\pi$
$401$	−9.43629	−0.471226	−0.235613	+	0.971847i	$0.575710\pi$
$402$	0	0
$403$	−2.09332	−0.104276
$404$	6.44851	0.320825
$405$	0	0
$406$	6.96239	0.345538
$407$	44.9356	2.22737
$408$	0	0
$409$	−0.0996603	−0.00492789	−0.00246394	−	0.999997i	$-0.500784\pi$
$409$	−0.0996603	−0.00492789	−0.00246394	+	0.999997i	$0.500784\pi$
$410$	0	0
$411$	0	0
$412$	−11.5066	−0.566889
$413$	−4.76116	−0.234281
$414$	0	0
$415$	0	0
$416$	−9.02776	−0.442622
$417$	0	0
$418$	−0.270114	−0.0132117
$419$	0.649738	0.0317418	0.0158709	−	0.999874i	$-0.494948\pi$
$419$	0.649738	0.0317418	0.0158709	+	0.999874i	$0.494948\pi$
$420$	0	0
$421$	−23.8437	−1.16207	−0.581035	−	0.813879i	$-0.697352\pi$
$421$	−23.8437	−1.16207	−0.581035	+	0.813879i	$0.697352\pi$
$422$	−29.6263	−1.44218
$423$	0	0
$424$	4.11142	0.199668
$425$	0	0
$426$	0	0
$427$	−3.22425	−0.156033
$428$	13.6023	0.657491
$429$	0	0
$430$	0	0
$431$	−8.13586	−0.391890	−0.195945	−	0.980615i	$-0.562777\pi$
$431$	−8.13586	−0.391890	−0.195945	+	0.980615i	$0.562777\pi$
$432$	0	0
$433$	27.5271	1.32287	0.661433	−	0.750004i	$-0.269949\pi$
$433$	27.5271	1.32287	0.661433	+	0.750004i	$0.269949\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	9.52705	0.456263
$437$	−0.146661	−0.00701576
$438$	0	0
$439$	32.3331	1.54318	0.771588	−	0.636123i	$-0.219463\pi$
$439$	32.3331	1.54318	0.771588	+	0.636123i	$0.219463\pi$
$440$	0	0
$441$	0	0
$442$	−9.92875	−0.472262
$443$	34.7997	1.65338	0.826692	−	0.562654i	$-0.190220\pi$
$443$	34.7997	1.65338	0.826692	+	0.562654i	$0.190220\pi$
$444$	0	0
$445$	0	0
$446$	−8.18664	−0.387649
$447$	0	0
$448$	3.22425	0.152332
$449$	−22.8373	−1.07776	−0.538880	−	0.842382i	$-0.681152\pi$
$449$	−22.8373	−1.07776	−0.538880	+	0.842382i	$0.681152\pi$
$450$	0	0
$451$	17.4191	0.820236
$452$	1.71370	0.0806055
$453$	0	0
$454$	−31.7235	−1.48886
$455$	0	0
$456$	0	0
$457$	24.5926	1.15039	0.575197	−	0.818015i	$-0.304925\pi$
$457$	24.5926	1.15039	0.575197	+	0.818015i	$0.304925\pi$
$458$	−2.23155	−0.104273
$459$	0	0
$460$	0	0
$461$	−0.488489	−0.0227512	−0.0113756	−	0.999935i	$-0.503621\pi$
$461$	−0.488489	−0.0227512	−0.0113756	+	0.999935i	$0.503621\pi$
$462$	0	0
$463$	1.86414	0.0866341	0.0433170	−	0.999061i	$-0.486207\pi$
$463$	1.86414	0.0866341	0.0433170	+	0.999061i	$0.486207\pi$
$464$	−17.2750	−0.801974
$465$	0	0
$466$	−10.1768	−0.471431
$467$	40.6761	1.88226	0.941132	−	0.338038i	$-0.109763\pi$
$467$	40.6761	1.88226	0.941132	+	0.338038i	$0.109763\pi$
$468$	0	0
$469$	17.6932	0.816997
$470$	0	0
$471$	0	0
$472$	7.97556	0.367105
$473$	−32.8529	−1.51058
$474$	0	0
$475$	0	0
$476$	−2.72496	−0.124898
$477$	0	0
$478$	−16.4060	−0.750392
$479$	6.77575	0.309592	0.154796	−	0.987946i	$-0.450528\pi$
$479$	6.77575	0.309592	0.154796	+	0.987946i	$0.450528\pi$
$480$	0	0
$481$	21.9405	1.00040
$482$	28.1319	1.28137
$483$	0	0
$484$	5.94921	0.270419
$485$	0	0
$486$	0	0
$487$	32.9805	1.49449	0.747244	−	0.664549i	$-0.231377\pi$
$487$	32.9805	1.49449	0.747244	+	0.664549i	$0.231377\pi$
$488$	5.40105	0.244494
$489$	0	0
$490$	0	0
$491$	30.8275	1.39122	0.695612	−	0.718417i	$-0.255133\pi$
$491$	30.8275	1.39122	0.695612	+	0.718417i	$0.255133\pi$
$492$	0	0
$493$	−9.85685	−0.443930
$494$	−0.131888	−0.00593390
$495$	0	0
$496$	4.96239	0.222818
$497$	2.01459	0.0903666
$498$	0	0
$499$	−31.4699	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$499$	−31.4699	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$500$	0	0
$501$	0	0
$502$	−6.04254	−0.269692
$503$	−27.5633	−1.22898	−0.614492	−	0.788923i	$-0.710639\pi$
$503$	−27.5633	−1.22898	−0.614492	+	0.788923i	$0.710639\pi$
$504$	0	0
$505$	0	0
$506$	28.0043	1.24494
$507$	0	0
$508$	6.38787	0.283416
$509$	5.63164	0.249618	0.124809	−	0.992181i	$-0.460168\pi$
$509$	5.63164	0.249618	0.124809	+	0.992181i	$0.460168\pi$
$510$	0	0
$511$	0.775746	0.0343170
$512$	1.55149	0.0685669
$513$	0	0
$514$	23.1187	1.01972
$515$	0	0
$516$	0	0
$517$	2.46310	0.108327
$518$	20.9624	0.921034
$519$	0	0
$520$	0	0
$521$	−36.7391	−1.60957	−0.804784	−	0.593567i	$-0.797719\pi$
$521$	−36.7391	−1.60957	−0.804784	+	0.593567i	$0.797719\pi$
$522$	0	0
$523$	−40.3390	−1.76390	−0.881951	−	0.471342i	$-0.843770\pi$
$523$	−40.3390	−1.76390	−0.881951	+	0.471342i	$0.843770\pi$
$524$	13.1392	0.573988
$525$	0	0
$526$	−9.50262	−0.414334
$527$	2.83146	0.123340
$528$	0	0
$529$	−7.79480	−0.338904
$530$	0	0
$531$	0	0
$532$	−0.0361968	−0.00156933
$533$	8.50517	0.368400
$534$	0	0
$535$	0	0
$536$	−29.6385	−1.28019
$537$	0	0
$538$	16.2882	0.702235
$539$	−23.8994	−1.02942
$540$	0	0
$541$	−35.8324	−1.54056	−0.770278	−	0.637708i	$-0.779882\pi$
$541$	−35.8324	−1.54056	−0.770278	+	0.637708i	$0.779882\pi$
$542$	−39.2711	−1.68684
$543$	0	0
$544$	12.2111	0.523546
$545$	0	0
$546$	0	0
$547$	−9.87144	−0.422072	−0.211036	−	0.977478i	$-0.567684\pi$
$547$	−9.87144	−0.422072	−0.211036	+	0.977478i	$0.567684\pi$
$548$	−1.90905	−0.0815505
$549$	0	0
$550$	0	0
$551$	−0.130933	−0.00557791
$552$	0	0
$553$	5.81336	0.247209
$554$	−19.4109	−0.824690
$555$	0	0
$556$	−6.93795	−0.294235
$557$	−25.8651	−1.09594	−0.547970	−	0.836498i	$-0.684599\pi$
$557$	−25.8651	−1.09594	−0.547970	+	0.836498i	$0.684599\pi$
$558$	0	0
$559$	−16.0409	−0.678459
$560$	0	0
$561$	0	0
$562$	−2.05079	−0.0865072
$563$	−34.4363	−1.45132	−0.725658	−	0.688055i	$-0.758465\pi$
$563$	−34.4363	−1.45132	−0.725658	+	0.688055i	$0.758465\pi$
$564$	0	0
$565$	0	0
$566$	36.5379	1.53580
$567$	0	0
$568$	−3.37470	−0.141599
$569$	−25.3757	−1.06380	−0.531901	−	0.846806i	$-0.678522\pi$
$569$	−25.3757	−1.06380	−0.531901	+	0.846806i	$0.678522\pi$
$570$	0	0
$571$	−19.6991	−0.824382	−0.412191	−	0.911097i	$-0.635236\pi$
$571$	−19.6991	−0.824382	−0.412191	+	0.911097i	$0.635236\pi$
$572$	7.23410	0.302473
$573$	0	0
$574$	8.12601	0.339173
$575$	0	0
$576$	0	0
$577$	13.4314	0.559155	0.279578	−	0.960123i	$-0.409806\pi$
$577$	13.4314	0.559155	0.279578	+	0.960123i	$0.409806\pi$
$578$	−15.0475	−0.625892
$579$	0	0
$580$	0	0
$581$	8.65562	0.359096
$582$	0	0
$583$	−8.81336	−0.365012
$584$	−1.29948	−0.0537727
$585$	0	0
$586$	6.93207	0.286361
$587$	−34.9389	−1.44208	−0.721041	−	0.692892i	$-0.756336\pi$
$587$	−34.9389	−1.44208	−0.721041	+	0.692892i	$0.756336\pi$
$588$	0	0
$589$	0.0376114	0.00154975
$590$	0	0
$591$	0	0
$592$	−52.0118	−2.13767
$593$	−7.41327	−0.304426	−0.152213	−	0.988348i	$-0.548640\pi$
$593$	−7.41327	−0.304426	−0.152213	+	0.988348i	$0.548640\pi$
$594$	0	0
$595$	0	0
$596$	7.58769	0.310804
$597$	0	0
$598$	13.6735	0.559152
$599$	−2.63752	−0.107766	−0.0538831	−	0.998547i	$-0.517160\pi$
$599$	−2.63752	−0.107766	−0.0538831	+	0.998547i	$0.517160\pi$
$600$	0	0
$601$	7.96731	0.324993	0.162497	−	0.986709i	$-0.448045\pi$
$601$	7.96731	0.324993	0.162497	+	0.986709i	$0.448045\pi$
$602$	−15.3258	−0.624634
$603$	0	0
$604$	2.42881	0.0988268
$605$	0	0
$606$	0	0
$607$	−34.8799	−1.41573	−0.707865	−	0.706348i	$-0.750342\pi$
$607$	−34.8799	−1.41573	−0.707865	+	0.706348i	$0.750342\pi$
$608$	0.162205	0.00657827
$609$	0	0
$610$	0	0
$611$	1.20265	0.0486538
$612$	0	0
$613$	11.6873	0.472048	0.236024	−	0.971747i	$-0.424156\pi$
$613$	11.6873	0.472048	0.236024	+	0.971747i	$0.424156\pi$
$614$	23.1490	0.934219
$615$	0	0
$616$	−10.2374	−0.412478
$617$	−19.9126	−0.801649	−0.400825	−	0.916155i	$-0.631276\pi$
$617$	−19.9126	−0.801649	−0.400825	+	0.916155i	$0.631276\pi$
$618$	0	0
$619$	−13.0982	−0.526463	−0.263231	−	0.964733i	$-0.584788\pi$
$619$	−13.0982	−0.526463	−0.263231	+	0.964733i	$0.584788\pi$
$620$	0	0
$621$	0	0
$622$	−23.8700	−0.957101
$623$	−15.4168	−0.617660
$624$	0	0
$625$	0	0
$626$	−16.0059	−0.639724
$627$	0	0
$628$	−7.56722	−0.301965
$629$	−29.6770	−1.18330
$630$	0	0
$631$	−1.80702	−0.0719363	−0.0359681	−	0.999353i	$-0.511451\pi$
$631$	−1.80702	−0.0719363	−0.0359681	+	0.999353i	$0.511451\pi$
$632$	−9.73813	−0.387362
$633$	0	0
$634$	47.3219	1.87939
$635$	0	0
$636$	0	0
$637$	−11.6693	−0.462353
$638$	25.0010	0.989797
$639$	0	0
$640$	0	0
$641$	3.35519	0.132522	0.0662609	−	0.997802i	$-0.478893\pi$
$641$	3.35519	0.132522	0.0662609	+	0.997802i	$0.478893\pi$
$642$	0	0
$643$	49.5207	1.95291	0.976453	−	0.215729i	$-0.0692129\pi$
$643$	49.5207	1.95291	0.976453	+	0.215729i	$0.0692129\pi$
$644$	3.75272	0.147878
$645$	0	0
$646$	0.178393	0.00701878
$647$	−2.47390	−0.0972590	−0.0486295	−	0.998817i	$-0.515485\pi$
$647$	−2.47390	−0.0972590	−0.0486295	+	0.998817i	$0.515485\pi$
$648$	0	0
$649$	−17.0966	−0.671102
$650$	0	0
$651$	0	0
$652$	2.77972	0.108862
$653$	−2.57689	−0.100841	−0.0504207	−	0.998728i	$-0.516056\pi$
$653$	−2.57689	−0.100841	−0.0504207	+	0.998728i	$0.516056\pi$
$654$	0	0
$655$	0	0
$656$	−20.1622	−0.787202
$657$	0	0
$658$	1.14903	0.0447939
$659$	−37.3112	−1.45344	−0.726720	−	0.686934i	$-0.758956\pi$
$659$	−37.3112	−1.45344	−0.726720	+	0.686934i	$0.758956\pi$
$660$	0	0
$661$	−22.5066	−0.875405	−0.437702	−	0.899120i	$-0.644208\pi$
$661$	−22.5066	−0.875405	−0.437702	+	0.899120i	$0.644208\pi$
$662$	−11.7342	−0.456061
$663$	0	0
$664$	−14.4993	−0.562682
$665$	0	0
$666$	0	0
$667$	13.5745	0.525607
$668$	6.80209	0.263181
$669$	0	0
$670$	0	0
$671$	−11.5778	−0.446958
$672$	0	0
$673$	3.40739	0.131345	0.0656725	−	0.997841i	$-0.479081\pi$
$673$	3.40739	0.131345	0.0656725	+	0.997841i	$0.479081\pi$
$674$	−56.6067	−2.18041
$675$	0	0
$676$	−6.94666	−0.267179
$677$	−44.0567	−1.69324	−0.846618	−	0.532202i	$-0.821365\pi$
$677$	−44.0567	−1.69324	−0.846618	+	0.532202i	$0.821365\pi$
$678$	0	0
$679$	9.84367	0.377766
$680$	0	0
$681$	0	0
$682$	−7.18172	−0.275002
$683$	39.2144	1.50050	0.750249	−	0.661156i	$-0.229934\pi$
$683$	39.2144	1.50050	0.750249	+	0.661156i	$0.229934\pi$
$684$	0	0
$685$	0	0
$686$	−25.1490	−0.960194
$687$	0	0
$688$	38.0263	1.44974
$689$	−4.30326	−0.163941
$690$	0	0
$691$	−35.6761	−1.35718	−0.678591	−	0.734516i	$-0.737409\pi$
$691$	−35.6761	−1.35718	−0.678591	+	0.734516i	$0.737409\pi$
$692$	2.81194	0.106894
$693$	0	0
$694$	−28.4650	−1.08052
$695$	0	0
$696$	0	0
$697$	−11.5042	−0.435753
$698$	−6.63752	−0.251234
$699$	0	0
$700$	0	0
$701$	19.0860	0.720869	0.360435	−	0.932784i	$-0.382628\pi$
$701$	19.0860	0.720869	0.360435	+	0.932784i	$0.382628\pi$
$702$	0	0
$703$	−0.394212	−0.0148680
$704$	11.5778	0.436356
$705$	0	0
$706$	38.9560	1.46613
$707$	9.55149	0.359221
$708$	0	0
$709$	−7.47978	−0.280909	−0.140455	−	0.990087i	$-0.544856\pi$
$709$	−7.47978	−0.280909	−0.140455	+	0.990087i	$0.544856\pi$
$710$	0	0
$711$	0	0
$712$	25.8251	0.967837
$713$	−3.89938	−0.146033
$714$	0	0
$715$	0	0
$716$	14.1524	0.528898
$717$	0	0
$718$	42.3146	1.57917
$719$	−41.0884	−1.53234	−0.766169	−	0.642639i	$-0.777840\pi$
$719$	−41.0884	−1.53234	−0.766169	+	0.642639i	$0.777840\pi$
$720$	0	0
$721$	−17.0435	−0.634733
$722$	−31.8251	−1.18441
$723$	0	0
$724$	7.61213	0.282902
$725$	0	0
$726$	0	0
$727$	25.4821	0.945081	0.472540	−	0.881309i	$-0.343337\pi$
$727$	25.4821	0.945081	0.472540	+	0.881309i	$0.343337\pi$
$728$	−4.99859	−0.185260
$729$	0	0
$730$	0	0
$731$	21.6972	0.802500
$732$	0	0
$733$	19.7250	0.728558	0.364279	−	0.931290i	$-0.381315\pi$
$733$	19.7250	0.728558	0.364279	+	0.931290i	$0.381315\pi$
$734$	32.8970	1.21425
$735$	0	0
$736$	−16.8167	−0.619871
$737$	63.5339	2.34030
$738$	0	0
$739$	−5.00492	−0.184109	−0.0920546	−	0.995754i	$-0.529343\pi$
$739$	−5.00492	−0.184109	−0.0920546	+	0.995754i	$0.529343\pi$
$740$	0	0
$741$	0	0
$742$	−4.11142	−0.150935
$743$	13.9746	0.512679	0.256339	−	0.966587i	$-0.417484\pi$
$743$	13.9746	0.512679	0.256339	+	0.966587i	$0.417484\pi$
$744$	0	0
$745$	0	0
$746$	15.4231	0.564680
$747$	0	0
$748$	−9.78495	−0.357773
$749$	20.1476	0.736178
$750$	0	0
$751$	27.5052	1.00368	0.501839	−	0.864961i	$-0.332657\pi$
$751$	27.5052	1.00368	0.501839	+	0.864961i	$0.332657\pi$
$752$	−2.85097	−0.103964
$753$	0	0
$754$	12.2071	0.444557
$755$	0	0
$756$	0	0
$757$	−30.5926	−1.11191	−0.555954	−	0.831213i	$-0.687647\pi$
$757$	−30.5926	−1.11191	−0.555954	+	0.831213i	$0.687647\pi$
$758$	0.346937	0.0126013
$759$	0	0
$760$	0	0
$761$	−53.9208	−1.95463	−0.977314	−	0.211796i	$-0.932069\pi$
$761$	−53.9208	−1.95463	−0.977314	+	0.211796i	$0.932069\pi$
$762$	0	0
$763$	14.1114	0.510868
$764$	18.2882	0.661644
$765$	0	0
$766$	−17.1408	−0.619322
$767$	−8.34771	−0.301418
$768$	0	0
$769$	−47.3331	−1.70688	−0.853438	−	0.521194i	$-0.825487\pi$
$769$	−47.3331	−1.70688	−0.853438	+	0.521194i	$0.825487\pi$
$770$	0	0
$771$	0	0
$772$	−8.87144	−0.319290
$773$	20.7212	0.745289	0.372644	−	0.927974i	$-0.378451\pi$
$773$	20.7212	0.745289	0.372644	+	0.927974i	$0.378451\pi$
$774$	0	0
$775$	0	0
$776$	−16.4894	−0.591937
$777$	0	0
$778$	1.17347	0.0420709
$779$	−0.152815	−0.00547517
$780$	0	0
$781$	7.23410	0.258857
$782$	−18.4950	−0.661380
$783$	0	0
$784$	27.6629	0.987961
$785$	0	0
$786$	0	0
$787$	1.68735	0.0601475	0.0300738	−	0.999548i	$-0.490426\pi$
$787$	1.68735	0.0601475	0.0300738	+	0.999548i	$0.490426\pi$
$788$	18.8627	0.671957
$789$	0	0
$790$	0	0
$791$	2.53832	0.0902522
$792$	0	0
$793$	−5.65306	−0.200746
$794$	−17.0762	−0.606011
$795$	0	0
$796$	−16.2374	−0.575520
$797$	53.4168	1.89212	0.946060	−	0.323993i	$-0.105025\pi$
$797$	53.4168	1.89212	0.946060	+	0.323993i	$0.105025\pi$
$798$	0	0
$799$	−1.62672	−0.0575491
$800$	0	0
$801$	0	0
$802$	−15.8070	−0.558165
$803$	2.78560	0.0983015
$804$	0	0
$805$	0	0
$806$	−3.50659	−0.123514
$807$	0	0
$808$	−16.0000	−0.562878
$809$	−41.5174	−1.45967	−0.729837	−	0.683621i	$-0.760404\pi$
$809$	−41.5174	−1.45967	−0.729837	+	0.683621i	$0.760404\pi$
$810$	0	0
$811$	−53.5183	−1.87928	−0.939642	−	0.342160i	$-0.888841\pi$
$811$	−53.5183	−1.87928	−0.939642	+	0.342160i	$0.888841\pi$
$812$	3.35026	0.117571
$813$	0	0
$814$	75.2730	2.63832
$815$	0	0
$816$	0	0
$817$	0.288213	0.0100833
$818$	−0.166944	−0.00583706
$819$	0	0
$820$	0	0
$821$	7.57215	0.264270	0.132135	−	0.991232i	$-0.457817\pi$
$821$	7.57215	0.264270	0.132135	+	0.991232i	$0.457817\pi$
$822$	0	0
$823$	7.75272	0.270243	0.135121	−	0.990829i	$-0.456858\pi$
$823$	7.75272	0.270243	0.135121	+	0.990829i	$0.456858\pi$
$824$	28.5501	0.994589
$825$	0	0
$826$	−7.97556	−0.277505
$827$	17.2193	0.598775	0.299387	−	0.954132i	$-0.403218\pi$
$827$	17.2193	0.598775	0.299387	+	0.954132i	$0.403218\pi$
$828$	0	0
$829$	7.69560	0.267279	0.133640	−	0.991030i	$-0.457334\pi$
$829$	7.69560	0.267279	0.133640	+	0.991030i	$0.457334\pi$
$830$	0	0
$831$	0	0
$832$	5.65306	0.195985
$833$	15.7840	0.546883
$834$	0	0
$835$	0	0
$836$	−0.129978	−0.00449537
$837$	0	0
$838$	1.08840	0.0375980
$839$	20.6253	0.712064	0.356032	−	0.934474i	$-0.384129\pi$
$839$	20.6253	0.712064	0.356032	+	0.934474i	$0.384129\pi$
$840$	0	0
$841$	−16.8813	−0.582113
$842$	−39.9413	−1.37647
$843$	0	0
$844$	−14.2560	−0.490711
$845$	0	0
$846$	0	0
$847$	8.81194	0.302782
$848$	10.2012	0.350312
$849$	0	0
$850$	0	0
$851$	40.8702	1.40101
$852$	0	0
$853$	−24.8061	−0.849343	−0.424672	−	0.905347i	$-0.639610\pi$
$853$	−24.8061	−0.849343	−0.424672	+	0.905347i	$0.639610\pi$
$854$	−5.40105	−0.184820
$855$	0	0
$856$	−33.7499	−1.15355
$857$	−7.20028	−0.245957	−0.122978	−	0.992409i	$-0.539245\pi$
$857$	−7.20028	−0.245957	−0.122978	+	0.992409i	$0.539245\pi$
$858$	0	0
$859$	−27.9756	−0.954514	−0.477257	−	0.878764i	$-0.658369\pi$
$859$	−27.9756	−0.954514	−0.477257	+	0.878764i	$0.658369\pi$
$860$	0	0
$861$	0	0
$862$	−13.6286	−0.464193
$863$	25.3112	0.861604	0.430802	−	0.902446i	$-0.358231\pi$
$863$	25.3112	0.861604	0.430802	+	0.902446i	$0.358231\pi$
$864$	0	0
$865$	0	0
$866$	46.1114	1.56693
$867$	0	0
$868$	−0.962389	−0.0326656
$869$	20.8749	0.708134
$870$	0	0
$871$	31.0214	1.05112
$872$	−23.6385	−0.800500
$873$	0	0
$874$	−0.245677	−0.00831014
$875$	0	0
$876$	0	0
$877$	−10.2765	−0.347011	−0.173506	−	0.984833i	$-0.555509\pi$
$877$	−10.2765	−0.347011	−0.173506	+	0.984833i	$0.555509\pi$
$878$	54.1622	1.82789
$879$	0	0
$880$	0	0
$881$	−13.1636	−0.443494	−0.221747	−	0.975104i	$-0.571176\pi$
$881$	−13.1636	−0.443494	−0.221747	+	0.975104i	$0.571176\pi$
$882$	0	0
$883$	−5.04586	−0.169807	−0.0849034	−	0.996389i	$-0.527058\pi$
$883$	−5.04586	−0.169807	−0.0849034	+	0.996389i	$0.527058\pi$
$884$	−4.77766	−0.160690
$885$	0	0
$886$	58.2941	1.95843
$887$	−27.3742	−0.919137	−0.459569	−	0.888142i	$-0.651996\pi$
$887$	−27.3742	−0.919137	−0.459569	+	0.888142i	$0.651996\pi$
$888$	0	0
$889$	9.46168	0.317335
$890$	0	0
$891$	0	0
$892$	−3.93937	−0.131900
$893$	−0.0216083	−0.000723095 0
$894$	0	0
$895$	0	0
$896$	15.6991	0.524470
$897$	0	0
$898$	−38.2555	−1.27660
$899$	−3.48119	−0.116104
$900$	0	0
$901$	5.82065	0.193914
$902$	29.1793	0.971566
$903$	0	0
$904$	−4.25202	−0.141420
$905$	0	0
$906$	0	0
$907$	13.8411	0.459587	0.229793	−	0.973239i	$-0.426195\pi$
$907$	13.8411	0.459587	0.229793	+	0.973239i	$0.426195\pi$
$908$	−15.2652	−0.506593
$909$	0	0
$910$	0	0
$911$	7.45676	0.247053	0.123527	−	0.992341i	$-0.460580\pi$
$911$	7.45676	0.247053	0.123527	+	0.992341i	$0.460580\pi$
$912$	0	0
$913$	31.0811	1.02863
$914$	41.1958	1.36264
$915$	0	0
$916$	−1.07381	−0.0354796
$917$	19.4617	0.642681
$918$	0	0
$919$	−48.9452	−1.61455	−0.807277	−	0.590172i	$-0.799060\pi$
$919$	−48.9452	−1.61455	−0.807277	+	0.590172i	$0.799060\pi$
$920$	0	0
$921$	0	0
$922$	−0.818282	−0.0269487
$923$	3.53216	0.116263
$924$	0	0
$925$	0	0
$926$	3.12268	0.102618
$927$	0	0
$928$	−15.0132	−0.492832
$929$	1.21933	0.0400049	0.0200024	−	0.999800i	$-0.493633\pi$
$929$	1.21933	0.0400049	0.0200024	+	0.999800i	$0.493633\pi$
$930$	0	0
$931$	0.209665	0.00687150
$932$	−4.89701	−0.160407
$933$	0	0
$934$	68.1378	2.22954
$935$	0	0
$936$	0	0
$937$	−4.07381	−0.133086	−0.0665428	−	0.997784i	$-0.521197\pi$
$937$	−4.07381	−0.133086	−0.0665428	+	0.997784i	$0.521197\pi$
$938$	29.6385	0.967730
$939$	0	0
$940$	0	0
$941$	27.7245	0.903793	0.451896	−	0.892070i	$-0.350748\pi$
$941$	27.7245	0.903793	0.451896	+	0.892070i	$0.350748\pi$
$942$	0	0
$943$	15.8432	0.515926
$944$	19.7889	0.644074
$945$	0	0
$946$	−55.0329	−1.78927
$947$	31.2154	1.01436	0.507181	−	0.861839i	$-0.330687\pi$
$947$	31.2154	1.01436	0.507181	+	0.861839i	$0.330687\pi$
$948$	0	0
$949$	1.36011	0.0441511
$950$	0	0
$951$	0	0
$952$	6.76116	0.219130
$953$	−29.4274	−0.953247	−0.476623	−	0.879108i	$-0.658139\pi$
$953$	−29.4274	−0.953247	−0.476623	+	0.879108i	$0.658139\pi$
$954$	0	0
$955$	0	0
$956$	−7.89446	−0.255325
$957$	0	0
$958$	11.3503	0.366710
$959$	−2.82767	−0.0913103
$960$	0	0
$961$	1.00000	0.0322581
$962$	36.7532	1.18497
$963$	0	0
$964$	13.5369	0.435994
$965$	0	0
$966$	0	0
$967$	−18.4469	−0.593213	−0.296606	−	0.955000i	$-0.595855\pi$
$967$	−18.4469	−0.593213	−0.296606	+	0.955000i	$0.595855\pi$
$968$	−14.7612	−0.474442
$969$	0	0
$970$	0	0
$971$	−23.8275	−0.764660	−0.382330	−	0.924026i	$-0.624878\pi$
$971$	−23.8275	−0.764660	−0.382330	+	0.924026i	$0.624878\pi$
$972$	0	0
$973$	−10.2765	−0.329448
$974$	55.2466	1.77022
$975$	0	0
$976$	13.4010	0.428957
$977$	−41.9271	−1.34137	−0.670684	−	0.741743i	$-0.733999\pi$
$977$	−41.9271	−1.34137	−0.670684	+	0.741743i	$0.733999\pi$
$978$	0	0
$979$	−55.3595	−1.76930
$980$	0	0
$981$	0	0
$982$	51.6401	1.64790
$983$	−6.05571	−0.193147	−0.0965736	−	0.995326i	$-0.530788\pi$
$983$	−6.05571	−0.193147	−0.0965736	+	0.995326i	$0.530788\pi$
$984$	0	0
$985$	0	0
$986$	−16.5115	−0.525834
$987$	0	0
$988$	−0.0634636	−0.00201905
$989$	−29.8806	−0.950149
$990$	0	0
$991$	20.4568	0.649830	0.324915	−	0.945743i	$-0.394664\pi$
$991$	20.4568	0.649830	0.324915	+	0.945743i	$0.394664\pi$
$992$	4.31265	0.136927
$993$	0	0
$994$	3.37470	0.107039
$995$	0	0
$996$	0	0
$997$	−21.2170	−0.671948	−0.335974	−	0.941871i	$-0.609065\pi$
$997$	−21.2170	−0.671948	−0.335974	+	0.941871i	$0.609065\pi$
$998$	−52.7163	−1.66870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bd.1.3		3
3.2	odd	2		2325.2.a.t.1.1	✓	3
5.4	even	2		6975.2.a.bc.1.1		3
15.2	even	4		2325.2.c.m.1024.2		6
15.8	even	4		2325.2.c.m.1024.5		6
15.14	odd	2		2325.2.a.u.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.t.1.1	✓	3	3.2	odd	2
2325.2.a.u.1.3	yes	3	15.14	odd	2
2325.2.c.m.1024.2		6	15.2	even	4
2325.2.c.m.1024.5		6	15.8	even	4
6975.2.a.bc.1.1		3	5.4	even	2
6975.2.a.bd.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.67513 q^{2} +0.806063 q^{4} +1.19394 q^{7} -2.00000 q^{8} +4.28726 q^{11} +2.09332 q^{13} +2.00000 q^{14} -4.96239 q^{16} -2.83146 q^{17} -0.0376114 q^{19} +7.18172 q^{22} +3.89938 q^{23} +3.50659 q^{26} +0.962389 q^{28} +3.48119 q^{29} -1.00000 q^{31} -4.31265 q^{32} -4.74306 q^{34} +10.4812 q^{37} -0.0630040 q^{38} +4.06300 q^{41} -7.66291 q^{43} +3.45580 q^{44} +6.53198 q^{46} +0.574515 q^{47} -5.57452 q^{49} +1.68735 q^{52} -2.05571 q^{53} -2.38787 q^{56} +5.83146 q^{58} -3.98778 q^{59} -2.70052 q^{61} -1.67513 q^{62} +2.70052 q^{64} +14.8192 q^{67} -2.28233 q^{68} +1.68735 q^{71} +0.649738 q^{73} +17.5574 q^{74} -0.0303172 q^{76} +5.11871 q^{77} +4.86907 q^{79} +6.80606 q^{82} +7.24965 q^{83} -12.8364 q^{86} -8.57452 q^{88} -12.9126 q^{89} +2.49929 q^{91} +3.14315 q^{92} +0.962389 q^{94} +8.24472 q^{97} -9.33804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} + 4 q^{7} - 6 q^{8} + 7 q^{11} + 6 q^{14} - 4 q^{16} + 7 q^{17} - 11 q^{19} - 4 q^{22} + 5 q^{23} - 10 q^{26} - 8 q^{28} + 5 q^{29} - 3 q^{31} + 8 q^{32} - 18 q^{34} + 26 q^{37} + 4 q^{38}+ \cdots + 8 q^{98}+O(q^{100}) \) 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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