LMFDB - Embedded newform 6975.2.a.cj.1.5

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 = [11,3,0,15,0,0,-8,9,0,0,0,0,-14,14,0,27,12] 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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 $ x^11 - 3x^10 - 14x^9 + 44x^8 + 61x^7 - 211x^6 - 83x^5 + 369x^4 + 10x^3 - 168x^2 - 31x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.364910$ 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)

$f(q)$	$=$	$q-0.364910 q^{2} -1.86684 q^{4} -0.715024 q^{7} +1.41105 q^{8} -3.71533 q^{11} -5.90954 q^{13} +0.260920 q^{14} +3.21877 q^{16} -6.46804 q^{17} -4.69100 q^{19} +1.35576 q^{22} -2.26578 q^{23} +2.15645 q^{26} +1.33484 q^{28} -9.36642 q^{29} +1.00000 q^{31} -3.99667 q^{32} +2.36026 q^{34} -8.82327 q^{37} +1.71179 q^{38} +2.31308 q^{41} -0.929609 q^{43} +6.93593 q^{44} +0.826808 q^{46} +8.18267 q^{47} -6.48874 q^{49} +11.0322 q^{52} -2.25353 q^{53} -1.00894 q^{56} +3.41790 q^{58} +5.09443 q^{59} -8.02281 q^{61} -0.364910 q^{62} -4.97912 q^{64} +2.00786 q^{67} +12.0748 q^{68} -3.94713 q^{71} +0.757173 q^{73} +3.21970 q^{74} +8.75735 q^{76} +2.65655 q^{77} +12.1793 q^{79} -0.844067 q^{82} +5.28381 q^{83} +0.339224 q^{86} -5.24252 q^{88} +5.44397 q^{89} +4.22547 q^{91} +4.22986 q^{92} -2.98594 q^{94} -17.7057 q^{97} +2.36781 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{2} + 15 q^{4} - 8 q^{7} + 9 q^{8} - 14 q^{13} + 14 q^{14} + 27 q^{16} + 12 q^{17} + 12 q^{19} - 10 q^{22} + 12 q^{23} + 6 q^{26} - 22 q^{28} + 8 q^{29} + 11 q^{31} + 21 q^{32} + 2 q^{34} - 16 q^{37}+ \cdots + 17 q^{98}+O(q^{100}) $ 11 * q + 3 * q^2 + 15 * q^4 - 8 * q^7 + 9 * q^8 - 14 * q^13 + 14 * q^14 + 27 * q^16 + 12 * q^17 + 12 * q^19 - 10 * q^22 + 12 * q^23 + 6 * q^26 - 22 * q^28 + 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 - 16 * q^37 + 26 * q^38 + 8 * q^41 - 28 * q^43 + 26 * q^44 + 8 * q^46 + 14 * q^47 + 23 * q^49 - 18 * q^52 + 26 * q^53 + 42 * q^56 - 6 * q^58 + 6 * q^59 + 14 * q^61 + 3 * q^62 + 29 * q^64 - 6 * q^67 + 54 * q^68 + 18 * q^71 - 26 * q^73 + 40 * q^74 + 10 * q^76 + 2 * q^77 + 8 * q^79 + 30 * q^82 + 48 * q^83 + 12 * q^86 - 40 * q^88 + 14 * q^89 - 4 * q^91 - 30 * q^92 - 4 * q^94 - 16 * q^97 + 17 * 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 $

$2$	−0.364910	−0.258031	−0.129015	−	0.991643i	$-0.541182\pi$
$2$	−0.364910	−0.258031	−0.129015	+	0.991643i	$0.541182\pi$
$3$	0	0
$3$	0	0
$4$	−1.86684	−0.933420
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.715024	−0.270254	−0.135127	−	0.990828i	$-0.543144\pi$
$7$	−0.715024	−0.270254	−0.135127	+	0.990828i	$0.543144\pi$
$8$	1.41105	0.498882
$9$	0	0
$10$	0	0
$11$	−3.71533	−1.12021	−0.560107	−	0.828420i	$-0.689240\pi$
$11$	−3.71533	−1.12021	−0.560107	+	0.828420i	$0.689240\pi$
$12$	0	0
$13$	−5.90954	−1.63901	−0.819506	−	0.573071i	$-0.805752\pi$
$13$	−5.90954	−1.63901	−0.819506	+	0.573071i	$0.805752\pi$
$14$	0.260920	0.0697337
$15$	0	0
$16$	3.21877	0.804693
$17$	−6.46804	−1.56873	−0.784365	−	0.620300i	$-0.787011\pi$
$17$	−6.46804	−1.56873	−0.784365	+	0.620300i	$0.787011\pi$
$18$	0	0
$19$	−4.69100	−1.07619	−0.538095	−	0.842884i	$-0.680856\pi$
$19$	−4.69100	−1.07619	−0.538095	+	0.842884i	$0.680856\pi$
$20$	0	0
$21$	0	0
$22$	1.35576	0.289049
$23$	−2.26578	−0.472449	−0.236224	−	0.971699i	$-0.575910\pi$
$23$	−2.26578	−0.472449	−0.236224	+	0.971699i	$0.575910\pi$
$24$	0	0
$25$	0	0
$26$	2.15645	0.422915
$27$	0	0
$28$	1.33484	0.252260
$29$	−9.36642	−1.73930	−0.869650	−	0.493669i	$-0.835656\pi$
$29$	−9.36642	−1.73930	−0.869650	+	0.493669i	$0.835656\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.99667	−0.706517
$33$	0	0
$34$	2.36026	0.404780
$35$	0	0
$36$	0	0
$37$	−8.82327	−1.45054	−0.725268	−	0.688467i	$-0.758284\pi$
$37$	−8.82327	−1.45054	−0.725268	+	0.688467i	$0.758284\pi$
$38$	1.71179	0.277690
$39$	0	0
$40$	0	0
$41$	2.31308	0.361243	0.180621	−	0.983553i	$-0.442189\pi$
$41$	2.31308	0.361243	0.180621	+	0.983553i	$0.442189\pi$
$42$	0	0
$43$	−0.929609	−0.141764	−0.0708820	−	0.997485i	$-0.522581\pi$
$43$	−0.929609	−0.141764	−0.0708820	+	0.997485i	$0.522581\pi$
$44$	6.93593	1.04563
$45$	0	0
$46$	0.826808	0.121906
$47$	8.18267	1.19356	0.596782	−	0.802403i	$-0.296446\pi$
$47$	8.18267	1.19356	0.596782	+	0.802403i	$0.296446\pi$
$48$	0	0
$49$	−6.48874	−0.926963
$50$	0	0
$51$	0	0
$52$	11.0322	1.52989
$53$	−2.25353	−0.309546	−0.154773	−	0.987950i	$-0.549465\pi$
$53$	−2.25353	−0.309546	−0.154773	+	0.987950i	$0.549465\pi$
$54$	0	0
$55$	0	0
$56$	−1.00894	−0.134825
$57$	0	0
$58$	3.41790	0.448793
$59$	5.09443	0.663238	0.331619	−	0.943413i	$-0.392405\pi$
$59$	5.09443	0.663238	0.331619	+	0.943413i	$0.392405\pi$
$60$	0	0
$61$	−8.02281	−1.02722	−0.513608	−	0.858025i	$-0.671691\pi$
$61$	−8.02281	−1.02722	−0.513608	+	0.858025i	$0.671691\pi$
$62$	−0.364910	−0.0463437
$63$	0	0
$64$	−4.97912	−0.622390
$65$	0	0
$66$	0	0
$67$	2.00786	0.245299	0.122650	−	0.992450i	$-0.460861\pi$
$67$	2.00786	0.245299	0.122650	+	0.992450i	$0.460861\pi$
$68$	12.0748	1.46428
$69$	0	0
$70$	0	0
$71$	−3.94713	−0.468438	−0.234219	−	0.972184i	$-0.575253\pi$
$71$	−3.94713	−0.468438	−0.234219	+	0.972184i	$0.575253\pi$
$72$	0	0
$73$	0.757173	0.0886204	0.0443102	−	0.999018i	$-0.485891\pi$
$73$	0.757173	0.0886204	0.0443102	+	0.999018i	$0.485891\pi$
$74$	3.21970	0.374283
$75$	0	0
$76$	8.75735	1.00454
$77$	2.65655	0.302742
$78$	0	0
$79$	12.1793	1.37028	0.685141	−	0.728410i	$-0.259740\pi$
$79$	12.1793	1.37028	0.685141	+	0.728410i	$0.259740\pi$
$80$	0	0
$81$	0	0
$82$	−0.844067	−0.0932117
$83$	5.28381	0.579974	0.289987	−	0.957031i	$-0.406349\pi$
$83$	5.28381	0.579974	0.289987	+	0.957031i	$0.406349\pi$
$84$	0	0
$85$	0	0
$86$	0.339224	0.0365795
$87$	0	0
$88$	−5.24252	−0.558854
$89$	5.44397	0.577060	0.288530	−	0.957471i	$-0.406834\pi$
$89$	5.44397	0.577060	0.288530	+	0.957471i	$0.406834\pi$
$90$	0	0
$91$	4.22547	0.442949
$92$	4.22986	0.440993
$93$	0	0
$94$	−2.98594	−0.307976
$95$	0	0
$96$	0	0
$97$	−17.7057	−1.79774	−0.898869	−	0.438217i	$-0.855610\pi$
$97$	−17.7057	−1.79774	−0.898869	+	0.438217i	$0.855610\pi$
$98$	2.36781	0.239185
$99$	0	0
$100$	0	0
$101$	−11.7545	−1.16962	−0.584809	−	0.811171i	$-0.698831\pi$
$101$	−11.7545	−1.16962	−0.584809	+	0.811171i	$0.698831\pi$
$102$	0	0
$103$	−11.0229	−1.08612	−0.543060	−	0.839694i	$-0.682734\pi$
$103$	−11.0229	−1.08612	−0.543060	+	0.839694i	$0.682734\pi$
$104$	−8.33866	−0.817673
$105$	0	0
$106$	0.822336	0.0798723
$107$	18.2442	1.76373	0.881867	−	0.471499i	$-0.156287\pi$
$107$	18.2442	1.76373	0.881867	+	0.471499i	$0.156287\pi$
$108$	0	0
$109$	−2.25344	−0.215841	−0.107920	−	0.994160i	$-0.534419\pi$
$109$	−2.25344	−0.215841	−0.107920	+	0.994160i	$0.534419\pi$
$110$	0	0
$111$	0	0
$112$	−2.30150	−0.217471
$113$	−8.56917	−0.806120	−0.403060	−	0.915174i	$-0.632053\pi$
$113$	−8.56917	−0.806120	−0.403060	+	0.915174i	$0.632053\pi$
$114$	0	0
$115$	0	0
$116$	17.4856	1.62350
$117$	0	0
$118$	−1.85901	−0.171136
$119$	4.62480	0.423955
$120$	0	0
$121$	2.80367	0.254879
$122$	2.92761	0.265053
$123$	0	0
$124$	−1.86684	−0.167647
$125$	0	0
$126$	0	0
$127$	−17.8348	−1.58258	−0.791292	−	0.611438i	$-0.790591\pi$
$127$	−17.8348	−1.58258	−0.791292	+	0.611438i	$0.790591\pi$
$128$	9.81026	0.867113
$129$	0	0
$130$	0	0
$131$	15.7384	1.37507	0.687534	−	0.726152i	$-0.258693\pi$
$131$	15.7384	1.37507	0.687534	+	0.726152i	$0.258693\pi$
$132$	0	0
$133$	3.35418	0.290844
$134$	−0.732690	−0.0632948
$135$	0	0
$136$	−9.12673	−0.782611
$137$	0.562050	0.0480191	0.0240096	−	0.999712i	$-0.492357\pi$
$137$	0.562050	0.0480191	0.0240096	+	0.999712i	$0.492357\pi$
$138$	0	0
$139$	11.8044	1.00124	0.500618	−	0.865668i	$-0.333106\pi$
$139$	11.8044	1.00124	0.500618	+	0.865668i	$0.333106\pi$
$140$	0	0
$141$	0	0
$142$	1.44035	0.120871
$143$	21.9559	1.83604
$144$	0	0
$145$	0	0
$146$	−0.276300	−0.0228668
$147$	0	0
$148$	16.4716	1.35396
$149$	0.0410925	0.00336643	0.00168321	−	0.999999i	$-0.499464\pi$
$149$	0.0410925	0.00336643	0.00168321	+	0.999999i	$0.499464\pi$
$150$	0	0
$151$	−2.77322	−0.225681	−0.112841	−	0.993613i	$-0.535995\pi$
$151$	−2.77322	−0.225681	−0.112841	+	0.993613i	$0.535995\pi$
$152$	−6.61924	−0.536891
$153$	0	0
$154$	−0.969403	−0.0781167
$155$	0	0
$156$	0	0
$157$	−16.1973	−1.29269	−0.646343	−	0.763047i	$-0.723702\pi$
$157$	−16.1973	−1.29269	−0.646343	+	0.763047i	$0.723702\pi$
$158$	−4.44437	−0.353575
$159$	0	0
$160$	0	0
$161$	1.62009	0.127681
$162$	0	0
$163$	−10.9559	−0.858130	−0.429065	−	0.903274i	$-0.641157\pi$
$163$	−10.9559	−0.858130	−0.429065	+	0.903274i	$0.641157\pi$
$164$	−4.31815	−0.337191
$165$	0	0
$166$	−1.92812	−0.149651
$167$	−4.09325	−0.316745	−0.158372	−	0.987379i	$-0.550625\pi$
$167$	−4.09325	−0.316745	−0.158372	+	0.987379i	$0.550625\pi$
$168$	0	0
$169$	21.9227	1.68636
$170$	0	0
$171$	0	0
$172$	1.73543	0.132325
$173$	17.8985	1.36080	0.680398	−	0.732843i	$-0.261807\pi$
$173$	17.8985	1.36080	0.680398	+	0.732843i	$0.261807\pi$
$174$	0	0
$175$	0	0
$176$	−11.9588	−0.901429
$177$	0	0
$178$	−1.98656	−0.148899
$179$	19.1255	1.42951	0.714754	−	0.699376i	$-0.246539\pi$
$179$	19.1255	1.42951	0.714754	+	0.699376i	$0.246539\pi$
$180$	0	0
$181$	−4.59215	−0.341332	−0.170666	−	0.985329i	$-0.554592\pi$
$181$	−4.59215	−0.341332	−0.170666	+	0.985329i	$0.554592\pi$
$182$	−1.54192	−0.114294
$183$	0	0
$184$	−3.19714	−0.235696
$185$	0	0
$186$	0	0
$187$	24.0309	1.75731
$188$	−15.2757	−1.11410
$189$	0	0
$190$	0	0
$191$	−19.1508	−1.38571	−0.692853	−	0.721079i	$-0.743646\pi$
$191$	−19.1508	−1.38571	−0.692853	+	0.721079i	$0.743646\pi$
$192$	0	0
$193$	−5.95401	−0.428579	−0.214289	−	0.976770i	$-0.568744\pi$
$193$	−5.95401	−0.428579	−0.214289	+	0.976770i	$0.568744\pi$
$194$	6.46098	0.463872
$195$	0	0
$196$	12.1134	0.865246
$197$	−12.5481	−0.894018	−0.447009	−	0.894530i	$-0.647511\pi$
$197$	−12.5481	−0.894018	−0.447009	+	0.894530i	$0.647511\pi$
$198$	0	0
$199$	−15.6893	−1.11219	−0.556093	−	0.831120i	$-0.687700\pi$
$199$	−15.6893	−1.11219	−0.556093	+	0.831120i	$0.687700\pi$
$200$	0	0
$201$	0	0
$202$	4.28935	0.301798
$203$	6.69721	0.470052
$204$	0	0
$205$	0	0
$206$	4.02237	0.280252
$207$	0	0
$208$	−19.0215	−1.31890
$209$	17.4286	1.20556
$210$	0	0
$211$	−17.1212	−1.17867	−0.589334	−	0.807889i	$-0.700610\pi$
$211$	−17.1212	−1.17867	−0.589334	+	0.807889i	$0.700610\pi$
$212$	4.20698	0.288936
$213$	0	0
$214$	−6.65750	−0.455097
$215$	0	0
$216$	0	0
$217$	−0.715024	−0.0485390
$218$	0.822305	0.0556936
$219$	0	0
$220$	0	0
$221$	38.2232	2.57117
$222$	0	0
$223$	−17.7779	−1.19049	−0.595247	−	0.803543i	$-0.702946\pi$
$223$	−17.7779	−1.19049	−0.595247	+	0.803543i	$0.702946\pi$
$224$	2.85771	0.190939
$225$	0	0
$226$	3.12698	0.208004
$227$	−7.64338	−0.507309	−0.253654	−	0.967295i	$-0.581633\pi$
$227$	−7.64338	−0.507309	−0.253654	+	0.967295i	$0.581633\pi$
$228$	0	0
$229$	23.9756	1.58435	0.792177	−	0.610292i	$-0.208948\pi$
$229$	23.9756	1.58435	0.792177	+	0.610292i	$0.208948\pi$
$230$	0	0
$231$	0	0
$232$	−13.2165	−0.867705
$233$	−24.6240	−1.61317	−0.806585	−	0.591119i	$-0.798686\pi$
$233$	−24.6240	−1.61317	−0.806585	+	0.591119i	$0.798686\pi$
$234$	0	0
$235$	0	0
$236$	−9.51049	−0.619080
$237$	0	0
$238$	−1.68764	−0.109393
$239$	1.69260	0.109485	0.0547426	−	0.998500i	$-0.482566\pi$
$239$	1.69260	0.109485	0.0547426	+	0.998500i	$0.482566\pi$
$240$	0	0
$241$	24.3340	1.56749	0.783745	−	0.621083i	$-0.213307\pi$
$241$	24.3340	1.56749	0.783745	+	0.621083i	$0.213307\pi$
$242$	−1.02309	−0.0657666
$243$	0	0
$244$	14.9773	0.958824
$245$	0	0
$246$	0	0
$247$	27.7217	1.76389
$248$	1.41105	0.0896018
$249$	0	0
$250$	0	0
$251$	−6.48374	−0.409250	−0.204625	−	0.978840i	$-0.565597\pi$
$251$	−6.48374	−0.409250	−0.204625	+	0.978840i	$0.565597\pi$
$252$	0	0
$253$	8.41813	0.529243
$254$	6.50811	0.408355
$255$	0	0
$256$	6.37838	0.398649
$257$	16.8234	1.04941	0.524707	−	0.851283i	$-0.324175\pi$
$257$	16.8234	1.04941	0.524707	+	0.851283i	$0.324175\pi$
$258$	0	0
$259$	6.30885	0.392013
$260$	0	0
$261$	0	0
$262$	−5.74310	−0.354810
$263$	−28.6460	−1.76639	−0.883195	−	0.469005i	$-0.844612\pi$
$263$	−28.6460	−1.76639	−0.883195	+	0.469005i	$0.844612\pi$
$264$	0	0
$265$	0	0
$266$	−1.22397	−0.0750467
$267$	0	0
$268$	−3.74836	−0.228967
$269$	19.1915	1.17013	0.585064	−	0.810987i	$-0.301069\pi$
$269$	19.1915	1.17013	0.585064	+	0.810987i	$0.301069\pi$
$270$	0	0
$271$	1.69900	0.103207	0.0516035	−	0.998668i	$-0.483567\pi$
$271$	1.69900	0.103207	0.0516035	+	0.998668i	$0.483567\pi$
$272$	−20.8192	−1.26235
$273$	0	0
$274$	−0.205098	−0.0123904
$275$	0	0
$276$	0	0
$277$	−3.01799	−0.181334	−0.0906668	−	0.995881i	$-0.528900\pi$
$277$	−3.01799	−0.181334	−0.0906668	+	0.995881i	$0.528900\pi$
$278$	−4.30755	−0.258349
$279$	0	0
$280$	0	0
$281$	18.3874	1.09690	0.548452	−	0.836182i	$-0.315218\pi$
$281$	18.3874	1.09690	0.548452	+	0.836182i	$0.315218\pi$
$282$	0	0
$283$	12.0881	0.718565	0.359283	−	0.933229i	$-0.383021\pi$
$283$	12.0881	0.718565	0.359283	+	0.933229i	$0.383021\pi$
$284$	7.36866	0.437249
$285$	0	0
$286$	−8.01193	−0.473756
$287$	−1.65391	−0.0976271
$288$	0	0
$289$	24.8355	1.46091
$290$	0	0
$291$	0	0
$292$	−1.41352	−0.0827201
$293$	−1.48343	−0.0866630	−0.0433315	−	0.999061i	$-0.513797\pi$
$293$	−1.48343	−0.0866630	−0.0433315	+	0.999061i	$0.513797\pi$
$294$	0	0
$295$	0	0
$296$	−12.4501	−0.723646
$297$	0	0
$298$	−0.0149951	−0.000868641 0
$299$	13.3897	0.774349
$300$	0	0
$301$	0.664693	0.0383122
$302$	1.01198	0.0582326
$303$	0	0
$304$	−15.0993	−0.866002
$305$	0	0
$306$	0	0
$307$	−2.99900	−0.171162	−0.0855809	−	0.996331i	$-0.527275\pi$
$307$	−2.99900	−0.171162	−0.0855809	+	0.996331i	$0.527275\pi$
$308$	−4.95935	−0.282585
$309$	0	0
$310$	0	0
$311$	2.98271	0.169134	0.0845670	−	0.996418i	$-0.473049\pi$
$311$	2.98271	0.169134	0.0845670	+	0.996418i	$0.473049\pi$
$312$	0	0
$313$	−14.6078	−0.825684	−0.412842	−	0.910803i	$-0.635464\pi$
$313$	−14.6078	−0.825684	−0.412842	+	0.910803i	$0.635464\pi$
$314$	5.91057	0.333553
$315$	0	0
$316$	−22.7369	−1.27905
$317$	−30.4040	−1.70766	−0.853830	−	0.520552i	$-0.825726\pi$
$317$	−30.4040	−1.70766	−0.853830	+	0.520552i	$0.825726\pi$
$318$	0	0
$319$	34.7993	1.94839
$320$	0	0
$321$	0	0
$322$	−0.591188	−0.0329456
$323$	30.3416	1.68825
$324$	0	0
$325$	0	0
$326$	3.99791	0.221424
$327$	0	0
$328$	3.26387	0.180217
$329$	−5.85081	−0.322565
$330$	0	0
$331$	−31.2630	−1.71837	−0.859186	−	0.511664i	$-0.829029\pi$
$331$	−31.2630	−1.71837	−0.859186	+	0.511664i	$0.829029\pi$
$332$	−9.86404	−0.541359
$333$	0	0
$334$	1.49367	0.0817299
$335$	0	0
$336$	0	0
$337$	21.1684	1.15312	0.576559	−	0.817056i	$-0.304395\pi$
$337$	21.1684	1.15312	0.576559	+	0.817056i	$0.304395\pi$
$338$	−7.99982	−0.435133
$339$	0	0
$340$	0	0
$341$	−3.71533	−0.201196
$342$	0	0
$343$	9.64477	0.520769
$344$	−1.31172	−0.0707235
$345$	0	0
$346$	−6.53134	−0.351127
$347$	−16.2791	−0.873907	−0.436953	−	0.899484i	$-0.643942\pi$
$347$	−16.2791	−0.873907	−0.436953	+	0.899484i	$0.643942\pi$
$348$	0	0
$349$	−11.3671	−0.608465	−0.304233	−	0.952598i	$-0.598400\pi$
$349$	−11.3671	−0.608465	−0.304233	+	0.952598i	$0.598400\pi$
$350$	0	0
$351$	0	0
$352$	14.8489	0.791450
$353$	−19.1430	−1.01888	−0.509439	−	0.860507i	$-0.670147\pi$
$353$	−19.1430	−1.01888	−0.509439	+	0.860507i	$0.670147\pi$
$354$	0	0
$355$	0	0
$356$	−10.1630	−0.538639
$357$	0	0
$358$	−6.97910	−0.368857
$359$	−2.30596	−0.121704	−0.0608518	−	0.998147i	$-0.519382\pi$
$359$	−2.30596	−0.121704	−0.0608518	+	0.998147i	$0.519382\pi$
$360$	0	0
$361$	3.00548	0.158183
$362$	1.67572	0.0880741
$363$	0	0
$364$	−7.88827	−0.413458
$365$	0	0
$366$	0	0
$367$	0.203852	0.0106410	0.00532049	−	0.999986i	$-0.498306\pi$
$367$	0.203852	0.0106410	0.00532049	+	0.999986i	$0.498306\pi$
$368$	−7.29305	−0.380176
$369$	0	0
$370$	0	0
$371$	1.61133	0.0836559
$372$	0	0
$373$	14.6680	0.759480	0.379740	−	0.925093i	$-0.376013\pi$
$373$	14.6680	0.759480	0.379740	+	0.925093i	$0.376013\pi$
$374$	−8.76912	−0.453441
$375$	0	0
$376$	11.5462	0.595448
$377$	55.3512	2.85073
$378$	0	0
$379$	−25.3827	−1.30382	−0.651910	−	0.758297i	$-0.726032\pi$
$379$	−25.3827	−1.30382	−0.651910	+	0.758297i	$0.726032\pi$
$380$	0	0
$381$	0	0
$382$	6.98834	0.357555
$383$	7.50401	0.383437	0.191718	−	0.981450i	$-0.438594\pi$
$383$	7.50401	0.383437	0.191718	+	0.981450i	$0.438594\pi$
$384$	0	0
$385$	0	0
$386$	2.17268	0.110586
$387$	0	0
$388$	33.0537	1.67805
$389$	11.9675	0.606777	0.303389	−	0.952867i	$-0.401882\pi$
$389$	11.9675	0.606777	0.303389	+	0.952867i	$0.401882\pi$
$390$	0	0
$391$	14.6552	0.741144
$392$	−9.15594	−0.462445
$393$	0	0
$394$	4.57895	0.230684
$395$	0	0
$396$	0	0
$397$	−4.58876	−0.230303	−0.115152	−	0.993348i	$-0.536735\pi$
$397$	−4.58876	−0.230303	−0.115152	+	0.993348i	$0.536735\pi$
$398$	5.72520	0.286978
$399$	0	0
$400$	0	0
$401$	18.0963	0.903686	0.451843	−	0.892098i	$-0.350767\pi$
$401$	18.0963	0.903686	0.451843	+	0.892098i	$0.350767\pi$
$402$	0	0
$403$	−5.90954	−0.294375
$404$	21.9438	1.09175
$405$	0	0
$406$	−2.44388	−0.121288
$407$	32.7813	1.62491
$408$	0	0
$409$	16.2103	0.801548	0.400774	−	0.916177i	$-0.368741\pi$
$409$	16.2103	0.801548	0.400774	+	0.916177i	$0.368741\pi$
$410$	0	0
$411$	0	0
$412$	20.5780	1.01381
$413$	−3.64264	−0.179243
$414$	0	0
$415$	0	0
$416$	23.6185	1.15799
$417$	0	0
$418$	−6.35988	−0.311072
$419$	10.3314	0.504723	0.252361	−	0.967633i	$-0.418793\pi$
$419$	10.3314	0.504723	0.252361	+	0.967633i	$0.418793\pi$
$420$	0	0
$421$	2.33725	0.113911	0.0569554	−	0.998377i	$-0.481861\pi$
$421$	2.33725	0.113911	0.0569554	+	0.998377i	$0.481861\pi$
$422$	6.24769	0.304133
$423$	0	0
$424$	−3.17984	−0.154427
$425$	0	0
$426$	0	0
$427$	5.73650	0.277609
$428$	−34.0590	−1.64630
$429$	0	0
$430$	0	0
$431$	0.916475	0.0441450	0.0220725	−	0.999756i	$-0.492974\pi$
$431$	0.916475	0.0441450	0.0220725	+	0.999756i	$0.492974\pi$
$432$	0	0
$433$	15.6700	0.753052	0.376526	−	0.926406i	$-0.377118\pi$
$433$	15.6700	0.753052	0.376526	+	0.926406i	$0.377118\pi$
$434$	0.260920	0.0125245
$435$	0	0
$436$	4.20682	0.201470
$437$	10.6288	0.508444
$438$	0	0
$439$	41.6107	1.98597	0.992986	−	0.118231i	$-0.0377225\pi$
$439$	41.6107	1.98597	0.992986	+	0.118231i	$0.0377225\pi$
$440$	0	0
$441$	0	0
$442$	−13.9480	−0.663440
$443$	0.792406	0.0376484	0.0188242	−	0.999823i	$-0.494008\pi$
$443$	0.792406	0.0376484	0.0188242	+	0.999823i	$0.494008\pi$
$444$	0	0
$445$	0	0
$446$	6.48733	0.307184
$447$	0	0
$448$	3.56019	0.168203
$449$	12.4223	0.586245	0.293122	−	0.956075i	$-0.405306\pi$
$449$	12.4223	0.586245	0.293122	+	0.956075i	$0.405306\pi$
$450$	0	0
$451$	−8.59386	−0.404669
$452$	15.9973	0.752448
$453$	0	0
$454$	2.78915	0.130901
$455$	0	0
$456$	0	0
$457$	−24.0840	−1.12660	−0.563301	−	0.826252i	$-0.690469\pi$
$457$	−24.0840	−1.12660	−0.563301	+	0.826252i	$0.690469\pi$
$458$	−8.74896	−0.408812
$459$	0	0
$460$	0	0
$461$	−0.553124	−0.0257616	−0.0128808	−	0.999917i	$-0.504100\pi$
$461$	−0.553124	−0.0257616	−0.0128808	+	0.999917i	$0.504100\pi$
$462$	0	0
$463$	37.1991	1.72879	0.864395	−	0.502814i	$-0.167702\pi$
$463$	37.1991	1.72879	0.864395	+	0.502814i	$0.167702\pi$
$464$	−30.1484	−1.39960
$465$	0	0
$466$	8.98554	0.416247
$467$	4.40335	0.203763	0.101881	−	0.994797i	$-0.467514\pi$
$467$	4.40335	0.203763	0.101881	+	0.994797i	$0.467514\pi$
$468$	0	0
$469$	−1.43567	−0.0662931
$470$	0	0
$471$	0	0
$472$	7.18850	0.330877
$473$	3.45380	0.158806
$474$	0	0
$475$	0	0
$476$	−8.63377	−0.395728
$477$	0	0
$478$	−0.617648	−0.0282506
$479$	1.51616	0.0692752	0.0346376	−	0.999400i	$-0.488972\pi$
$479$	1.51616	0.0692752	0.0346376	+	0.999400i	$0.488972\pi$
$480$	0	0
$481$	52.1415	2.37745
$482$	−8.87973	−0.404461
$483$	0	0
$484$	−5.23400	−0.237909
$485$	0	0
$486$	0	0
$487$	12.7509	0.577800	0.288900	−	0.957359i	$-0.406710\pi$
$487$	12.7509	0.577800	0.288900	+	0.957359i	$0.406710\pi$
$488$	−11.3206	−0.512459
$489$	0	0
$490$	0	0
$491$	5.40923	0.244115	0.122058	−	0.992523i	$-0.461051\pi$
$491$	5.40923	0.244115	0.122058	+	0.992523i	$0.461051\pi$
$492$	0	0
$493$	60.5824	2.72849
$494$	−10.1159	−0.455137
$495$	0	0
$496$	3.21877	0.144527
$497$	2.82229	0.126597
$498$	0	0
$499$	−5.36026	−0.239958	−0.119979	−	0.992776i	$-0.538283\pi$
$499$	−5.36026	−0.239958	−0.119979	+	0.992776i	$0.538283\pi$
$500$	0	0
$501$	0	0
$502$	2.36598	0.105599
$503$	29.5491	1.31753	0.658764	−	0.752349i	$-0.271079\pi$
$503$	29.5491	1.31753	0.658764	+	0.752349i	$0.271079\pi$
$504$	0	0
$505$	0	0
$506$	−3.07186	−0.136561
$507$	0	0
$508$	33.2948	1.47722
$509$	−14.3374	−0.635495	−0.317747	−	0.948175i	$-0.602926\pi$
$509$	−14.3374	−0.635495	−0.317747	+	0.948175i	$0.602926\pi$
$510$	0	0
$511$	−0.541397	−0.0239500
$512$	−21.9481	−0.969977
$513$	0	0
$514$	−6.13902	−0.270781
$515$	0	0
$516$	0	0
$517$	−30.4013	−1.33705
$518$	−2.30216	−0.101151
$519$	0	0
$520$	0	0
$521$	−21.4779	−0.940966	−0.470483	−	0.882409i	$-0.655920\pi$
$521$	−21.4779	−0.940966	−0.470483	+	0.882409i	$0.655920\pi$
$522$	0	0
$523$	−30.7154	−1.34309	−0.671546	−	0.740963i	$-0.734369\pi$
$523$	−30.7154	−1.34309	−0.671546	+	0.740963i	$0.734369\pi$
$524$	−29.3810	−1.28352
$525$	0	0
$526$	10.4532	0.455783
$527$	−6.46804	−0.281752
$528$	0	0
$529$	−17.8662	−0.776792
$530$	0	0
$531$	0	0
$532$	−6.26171	−0.271480
$533$	−13.6693	−0.592081
$534$	0	0
$535$	0	0
$536$	2.83320	0.122375
$537$	0	0
$538$	−7.00319	−0.301929
$539$	24.1078	1.03840
$540$	0	0
$541$	15.3718	0.660886	0.330443	−	0.943826i	$-0.392802\pi$
$541$	15.3718	0.660886	0.330443	+	0.943826i	$0.392802\pi$
$542$	−0.619983	−0.0266306
$543$	0	0
$544$	25.8506	1.10833
$545$	0	0
$546$	0	0
$547$	7.00541	0.299530	0.149765	−	0.988722i	$-0.452148\pi$
$547$	7.00541	0.299530	0.149765	+	0.988722i	$0.452148\pi$
$548$	−1.04926	−0.0448220
$549$	0	0
$550$	0	0
$551$	43.9379	1.87182
$552$	0	0
$553$	−8.70852	−0.370324
$554$	1.10130	0.0467896
$555$	0	0
$556$	−22.0369	−0.934574
$557$	−5.32428	−0.225597	−0.112798	−	0.993618i	$-0.535981\pi$
$557$	−5.32428	−0.225597	−0.112798	+	0.993618i	$0.535981\pi$
$558$	0	0
$559$	5.49356	0.232353
$560$	0	0
$561$	0	0
$562$	−6.70977	−0.283035
$563$	22.8713	0.963909	0.481954	−	0.876196i	$-0.339927\pi$
$563$	22.8713	0.963909	0.481954	+	0.876196i	$0.339927\pi$
$564$	0	0
$565$	0	0
$566$	−4.41109	−0.185412
$567$	0	0
$568$	−5.56960	−0.233695
$569$	33.6790	1.41190	0.705949	−	0.708263i	$-0.250521\pi$
$569$	33.6790	1.41190	0.705949	+	0.708263i	$0.250521\pi$
$570$	0	0
$571$	20.8102	0.870881	0.435440	−	0.900218i	$-0.356593\pi$
$571$	20.8102	0.870881	0.435440	+	0.900218i	$0.356593\pi$
$572$	−40.9881	−1.71380
$573$	0	0
$574$	0.603529	0.0251908
$575$	0	0
$576$	0	0
$577$	30.2603	1.25975	0.629877	−	0.776695i	$-0.283105\pi$
$577$	30.2603	1.25975	0.629877	+	0.776695i	$0.283105\pi$
$578$	−9.06275	−0.376961
$579$	0	0
$580$	0	0
$581$	−3.77805	−0.156740
$582$	0	0
$583$	8.37260	0.346758
$584$	1.06841	0.0442111
$585$	0	0
$586$	0.541320	0.0223617
$587$	−38.1800	−1.57586	−0.787928	−	0.615768i	$-0.788846\pi$
$587$	−38.1800	−1.57586	−0.787928	+	0.615768i	$0.788846\pi$
$588$	0	0
$589$	−4.69100	−0.193289
$590$	0	0
$591$	0	0
$592$	−28.4001	−1.16724
$593$	−12.2732	−0.504000	−0.252000	−	0.967727i	$-0.581088\pi$
$593$	−12.2732	−0.504000	−0.252000	+	0.967727i	$0.581088\pi$
$594$	0	0
$595$	0	0
$596$	−0.0767131	−0.00314229
$597$	0	0
$598$	−4.88606	−0.199806
$599$	16.8528	0.688585	0.344293	−	0.938862i	$-0.388119\pi$
$599$	16.8528	0.688585	0.344293	+	0.938862i	$0.388119\pi$
$600$	0	0
$601$	−4.89166	−0.199535	−0.0997674	−	0.995011i	$-0.531810\pi$
$601$	−4.89166	−0.199535	−0.0997674	+	0.995011i	$0.531810\pi$
$602$	−0.242553	−0.00988573
$603$	0	0
$604$	5.17715	0.210655
$605$	0	0
$606$	0	0
$607$	−45.7290	−1.85608	−0.928042	−	0.372476i	$-0.878509\pi$
$607$	−45.7290	−1.85608	−0.928042	+	0.372476i	$0.878509\pi$
$608$	18.7484	0.760346
$609$	0	0
$610$	0	0
$611$	−48.3558	−1.95627
$612$	0	0
$613$	−42.2505	−1.70648	−0.853240	−	0.521518i	$-0.825366\pi$
$613$	−42.2505	−1.70648	−0.853240	+	0.521518i	$0.825366\pi$
$614$	1.09436	0.0441650
$615$	0	0
$616$	3.74853	0.151032
$617$	35.0465	1.41092	0.705459	−	0.708751i	$-0.250741\pi$
$617$	35.0465	1.41092	0.705459	+	0.708751i	$0.250741\pi$
$618$	0	0
$619$	−43.9185	−1.76523	−0.882617	−	0.470093i	$-0.844220\pi$
$619$	−43.9185	−1.76523	−0.882617	+	0.470093i	$0.844220\pi$
$620$	0	0
$621$	0	0
$622$	−1.08842	−0.0436417
$623$	−3.89257	−0.155953
$624$	0	0
$625$	0	0
$626$	5.33056	0.213052
$627$	0	0
$628$	30.2378	1.20662
$629$	57.0692	2.27550
$630$	0	0
$631$	8.32224	0.331303	0.165651	−	0.986184i	$-0.447027\pi$
$631$	8.32224	0.331303	0.165651	+	0.986184i	$0.447027\pi$
$632$	17.1857	0.683609
$633$	0	0
$634$	11.0947	0.440629
$635$	0	0
$636$	0	0
$637$	38.3455	1.51930
$638$	−12.6986	−0.502744
$639$	0	0
$640$	0	0
$641$	31.5971	1.24801	0.624005	−	0.781421i	$-0.285505\pi$
$641$	31.5971	1.24801	0.624005	+	0.781421i	$0.285505\pi$
$642$	0	0
$643$	−48.3376	−1.90625	−0.953124	−	0.302579i	$-0.902152\pi$
$643$	−48.3376	−1.90625	−0.953124	+	0.302579i	$0.902152\pi$
$644$	−3.02445	−0.119180
$645$	0	0
$646$	−11.0720	−0.435620
$647$	25.7353	1.01176	0.505880	−	0.862604i	$-0.331168\pi$
$647$	25.7353	1.01176	0.505880	+	0.862604i	$0.331168\pi$
$648$	0	0
$649$	−18.9275	−0.742969
$650$	0	0
$651$	0	0
$652$	20.4529	0.800995
$653$	−18.7287	−0.732911	−0.366455	−	0.930436i	$-0.619429\pi$
$653$	−18.7287	−0.732911	−0.366455	+	0.930436i	$0.619429\pi$
$654$	0	0
$655$	0	0
$656$	7.44528	0.290690
$657$	0	0
$658$	2.13502	0.0832317
$659$	−20.9929	−0.817767	−0.408883	−	0.912587i	$-0.634082\pi$
$659$	−20.9929	−0.817767	−0.408883	+	0.912587i	$0.634082\pi$
$660$	0	0
$661$	−1.44236	−0.0561014	−0.0280507	−	0.999607i	$-0.508930\pi$
$661$	−1.44236	−0.0561014	−0.0280507	+	0.999607i	$0.508930\pi$
$662$	11.4082	0.443393
$663$	0	0
$664$	7.45573	0.289338
$665$	0	0
$666$	0	0
$667$	21.2223	0.821730
$668$	7.64144	0.295656
$669$	0	0
$670$	0	0
$671$	29.8074	1.15070
$672$	0	0
$673$	−2.48056	−0.0956187	−0.0478094	−	0.998856i	$-0.515224\pi$
$673$	−2.48056	−0.0956187	−0.0478094	+	0.998856i	$0.515224\pi$
$674$	−7.72458	−0.297540
$675$	0	0
$676$	−40.9262	−1.57408
$677$	−29.2749	−1.12513	−0.562564	−	0.826754i	$-0.690185\pi$
$677$	−29.2749	−1.12513	−0.562564	+	0.826754i	$0.690185\pi$
$678$	0	0
$679$	12.6600	0.485845
$680$	0	0
$681$	0	0
$682$	1.35576	0.0519148
$683$	22.7626	0.870987	0.435494	−	0.900192i	$-0.356574\pi$
$683$	22.7626	0.870987	0.435494	+	0.900192i	$0.356574\pi$
$684$	0	0
$685$	0	0
$686$	−3.51948	−0.134374
$687$	0	0
$688$	−2.99220	−0.114077
$689$	13.3173	0.507349
$690$	0	0
$691$	−38.7666	−1.47475	−0.737376	−	0.675483i	$-0.763935\pi$
$691$	−38.7666	−1.47475	−0.737376	+	0.675483i	$0.763935\pi$
$692$	−33.4136	−1.27019
$693$	0	0
$694$	5.94041	0.225495
$695$	0	0
$696$	0	0
$697$	−14.9611	−0.566692
$698$	4.14796	0.157003
$699$	0	0
$700$	0	0
$701$	30.6440	1.15741	0.578704	−	0.815537i	$-0.303559\pi$
$701$	30.6440	1.15741	0.578704	+	0.815537i	$0.303559\pi$
$702$	0	0
$703$	41.3899	1.56105
$704$	18.4991	0.697210
$705$	0	0
$706$	6.98547	0.262902
$707$	8.40477	0.316094
$708$	0	0
$709$	−28.2931	−1.06257	−0.531286	−	0.847193i	$-0.678291\pi$
$709$	−28.2931	−1.06257	−0.531286	+	0.847193i	$0.678291\pi$
$710$	0	0
$711$	0	0
$712$	7.68172	0.287885
$713$	−2.26578	−0.0848543
$714$	0	0
$715$	0	0
$716$	−35.7043	−1.33433
$717$	0	0
$718$	0.841467	0.0314033
$719$	−23.2579	−0.867373	−0.433687	−	0.901064i	$-0.642788\pi$
$719$	−23.2579	−0.867373	−0.433687	+	0.901064i	$0.642788\pi$
$720$	0	0
$721$	7.88164	0.293528
$722$	−1.09673	−0.0408161
$723$	0	0
$724$	8.57282	0.318606
$725$	0	0
$726$	0	0
$727$	−14.0608	−0.521485	−0.260742	−	0.965408i	$-0.583967\pi$
$727$	−14.0608	−0.521485	−0.260742	+	0.965408i	$0.583967\pi$
$728$	5.96234	0.220979
$729$	0	0
$730$	0	0
$731$	6.01275	0.222389
$732$	0	0
$733$	25.2227	0.931623	0.465811	−	0.884884i	$-0.345763\pi$
$733$	25.2227	0.931623	0.465811	+	0.884884i	$0.345763\pi$
$734$	−0.0743877	−0.00274570
$735$	0	0
$736$	9.05558	0.333793
$737$	−7.45987	−0.274788
$738$	0	0
$739$	−45.4258	−1.67101	−0.835507	−	0.549479i	$-0.814826\pi$
$739$	−45.4258	−1.67101	−0.835507	+	0.549479i	$0.814826\pi$
$740$	0	0
$741$	0	0
$742$	−0.587990	−0.0215858
$743$	10.7035	0.392672	0.196336	−	0.980537i	$-0.437096\pi$
$743$	10.7035	0.392672	0.196336	+	0.980537i	$0.437096\pi$
$744$	0	0
$745$	0	0
$746$	−5.35251	−0.195969
$747$	0	0
$748$	−44.8618	−1.64031
$749$	−13.0450	−0.476656
$750$	0	0
$751$	−8.19053	−0.298877	−0.149438	−	0.988771i	$-0.547747\pi$
$751$	−8.19053	−0.298877	−0.149438	+	0.988771i	$0.547747\pi$
$752$	26.3382	0.960454
$753$	0	0
$754$	−20.1982	−0.735577
$755$	0	0
$756$	0	0
$757$	11.5266	0.418940	0.209470	−	0.977815i	$-0.432826\pi$
$757$	11.5266	0.418940	0.209470	+	0.977815i	$0.432826\pi$
$758$	9.26240	0.336425
$759$	0	0
$760$	0	0
$761$	8.45784	0.306596	0.153298	−	0.988180i	$-0.451011\pi$
$761$	8.45784	0.306596	0.153298	+	0.988180i	$0.451011\pi$
$762$	0	0
$763$	1.61127	0.0583318
$764$	35.7516	1.29345
$765$	0	0
$766$	−2.73829	−0.0989385
$767$	−30.1057	−1.08706
$768$	0	0
$769$	23.8489	0.860013	0.430007	−	0.902826i	$-0.358511\pi$
$769$	23.8489	0.860013	0.430007	+	0.902826i	$0.358511\pi$
$770$	0	0
$771$	0	0
$772$	11.1152	0.400044
$773$	−51.8506	−1.86494	−0.932468	−	0.361252i	$-0.882349\pi$
$773$	−51.8506	−1.86494	−0.932468	+	0.361252i	$0.882349\pi$
$774$	0	0
$775$	0	0
$776$	−24.9836	−0.896859
$777$	0	0
$778$	−4.36707	−0.156567
$779$	−10.8507	−0.388765
$780$	0	0
$781$	14.6649	0.524751
$782$	−5.34783	−0.191238
$783$	0	0
$784$	−20.8858	−0.745921
$785$	0	0
$786$	0	0
$787$	−31.0467	−1.10670	−0.553348	−	0.832950i	$-0.686650\pi$
$787$	−31.0467	−1.10670	−0.553348	+	0.832950i	$0.686650\pi$
$788$	23.4254	0.834494
$789$	0	0
$790$	0	0
$791$	6.12716	0.217857
$792$	0	0
$793$	47.4111	1.68362
$794$	1.67449	0.0594253
$795$	0	0
$796$	29.2895	1.03814
$797$	−38.0709	−1.34854	−0.674271	−	0.738484i	$-0.735542\pi$
$797$	−38.0709	−1.34854	−0.674271	+	0.738484i	$0.735542\pi$
$798$	0	0
$799$	−52.9258	−1.87238
$800$	0	0
$801$	0	0
$802$	−6.60353	−0.233179
$803$	−2.81315	−0.0992738
$804$	0	0
$805$	0	0
$806$	2.15645	0.0759578
$807$	0	0
$808$	−16.5862	−0.583501
$809$	−31.4393	−1.10535	−0.552674	−	0.833397i	$-0.686393\pi$
$809$	−31.4393	−1.10535	−0.552674	+	0.833397i	$0.686393\pi$
$810$	0	0
$811$	−30.7935	−1.08130	−0.540652	−	0.841246i	$-0.681822\pi$
$811$	−30.7935	−1.08130	−0.540652	+	0.841246i	$0.681822\pi$
$812$	−12.5026	−0.438756
$813$	0	0
$814$	−11.9623	−0.419277
$815$	0	0
$816$	0	0
$817$	4.36079	0.152565
$818$	−5.91531	−0.206824
$819$	0	0
$820$	0	0
$821$	−3.43296	−0.119811	−0.0599056	−	0.998204i	$-0.519080\pi$
$821$	−3.43296	−0.119811	−0.0599056	+	0.998204i	$0.519080\pi$
$822$	0	0
$823$	−44.1890	−1.54033	−0.770166	−	0.637843i	$-0.779827\pi$
$823$	−44.1890	−1.54033	−0.770166	+	0.637843i	$0.779827\pi$
$824$	−15.5539	−0.541845
$825$	0	0
$826$	1.32924	0.0462501
$827$	36.1407	1.25674	0.628368	−	0.777916i	$-0.283723\pi$
$827$	36.1407	1.25674	0.628368	+	0.777916i	$0.283723\pi$
$828$	0	0
$829$	41.9005	1.45526	0.727632	−	0.685967i	$-0.240621\pi$
$829$	41.9005	1.45526	0.727632	+	0.685967i	$0.240621\pi$
$830$	0	0
$831$	0	0
$832$	29.4243	1.02011
$833$	41.9694	1.45415
$834$	0	0
$835$	0	0
$836$	−32.5364	−1.12530
$837$	0	0
$838$	−3.77004	−0.130234
$839$	−12.5707	−0.433989	−0.216995	−	0.976173i	$-0.569625\pi$
$839$	−12.5707	−0.433989	−0.216995	+	0.976173i	$0.569625\pi$
$840$	0	0
$841$	58.7298	2.02517
$842$	−0.852888	−0.0293925
$843$	0	0
$844$	31.9625	1.10019
$845$	0	0
$846$	0	0
$847$	−2.00469	−0.0688819
$848$	−7.25360	−0.249090
$849$	0	0
$850$	0	0
$851$	19.9916	0.685304
$852$	0	0
$853$	11.8787	0.406719	0.203359	−	0.979104i	$-0.434814\pi$
$853$	11.8787	0.406719	0.203359	+	0.979104i	$0.434814\pi$
$854$	−2.09331	−0.0716316
$855$	0	0
$856$	25.7435	0.879894
$857$	10.2033	0.348539	0.174269	−	0.984698i	$-0.444244\pi$
$857$	10.2033	0.348539	0.174269	+	0.984698i	$0.444244\pi$
$858$	0	0
$859$	38.8826	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$859$	38.8826	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$860$	0	0
$861$	0	0
$862$	−0.334431	−0.0113908
$863$	−46.7216	−1.59042	−0.795211	−	0.606332i	$-0.792640\pi$
$863$	−46.7216	−1.59042	−0.795211	+	0.606332i	$0.792640\pi$
$864$	0	0
$865$	0	0
$866$	−5.71815	−0.194311
$867$	0	0
$868$	1.33484	0.0453073
$869$	−45.2503	−1.53501
$870$	0	0
$871$	−11.8655	−0.402049
$872$	−3.17972	−0.107679
$873$	0	0
$874$	−3.87856	−0.131194
$875$	0	0
$876$	0	0
$877$	41.7255	1.40897	0.704485	−	0.709719i	$-0.251178\pi$
$877$	41.7255	1.40897	0.704485	+	0.709719i	$0.251178\pi$
$878$	−15.1842	−0.512442
$879$	0	0
$880$	0	0
$881$	−16.8496	−0.567677	−0.283839	−	0.958872i	$-0.591608\pi$
$881$	−16.8496	−0.567677	−0.283839	+	0.958872i	$0.591608\pi$
$882$	0	0
$883$	13.6867	0.460595	0.230297	−	0.973120i	$-0.426030\pi$
$883$	13.6867	0.460595	0.230297	+	0.973120i	$0.426030\pi$
$884$	−71.3565	−2.39998
$885$	0	0
$886$	−0.289157	−0.00971443
$887$	−9.91872	−0.333038	−0.166519	−	0.986038i	$-0.553253\pi$
$887$	−9.91872	−0.333038	−0.166519	+	0.986038i	$0.553253\pi$
$888$	0	0
$889$	12.7523	0.427699
$890$	0	0
$891$	0	0
$892$	33.1885	1.11123
$893$	−38.3849	−1.28450
$894$	0	0
$895$	0	0
$896$	−7.01458	−0.234341
$897$	0	0
$898$	−4.53303	−0.151269
$899$	−9.36642	−0.312388
$900$	0	0
$901$	14.5759	0.485594
$902$	3.13599	0.104417
$903$	0	0
$904$	−12.0915	−0.402158
$905$	0	0
$906$	0	0
$907$	2.68537	0.0891664	0.0445832	−	0.999006i	$-0.485804\pi$
$907$	2.68537	0.0891664	0.0445832	+	0.999006i	$0.485804\pi$
$908$	14.2690	0.473532
$909$	0	0
$910$	0	0
$911$	3.11531	0.103215	0.0516075	−	0.998667i	$-0.483566\pi$
$911$	3.11531	0.103215	0.0516075	+	0.998667i	$0.483566\pi$
$912$	0	0
$913$	−19.6311	−0.649695
$914$	8.78851	0.290698
$915$	0	0
$916$	−44.7587	−1.47887
$917$	−11.2533	−0.371617
$918$	0	0
$919$	−25.1517	−0.829678	−0.414839	−	0.909895i	$-0.636162\pi$
$919$	−25.1517	−0.829678	−0.414839	+	0.909895i	$0.636162\pi$
$920$	0	0
$921$	0	0
$922$	0.201841	0.00664727
$923$	23.3257	0.767776
$924$	0	0
$925$	0	0
$926$	−13.5743	−0.446081
$927$	0	0
$928$	37.4344	1.22885
$929$	−34.7201	−1.13913	−0.569565	−	0.821946i	$-0.692888\pi$
$929$	−34.7201	−1.13913	−0.569565	+	0.821946i	$0.692888\pi$
$930$	0	0
$931$	30.4387	0.997587
$932$	45.9690	1.50576
$933$	0	0
$934$	−1.60683	−0.0525770
$935$	0	0
$936$	0	0
$937$	5.96639	0.194913	0.0974567	−	0.995240i	$-0.468929\pi$
$937$	5.96639	0.194913	0.0974567	+	0.995240i	$0.468929\pi$
$938$	0.523891	0.0171056
$939$	0	0
$940$	0	0
$941$	37.4254	1.22003	0.610017	−	0.792389i	$-0.291163\pi$
$941$	37.4254	1.22003	0.610017	+	0.792389i	$0.291163\pi$
$942$	0	0
$943$	−5.24094	−0.170669
$944$	16.3978	0.533703
$945$	0	0
$946$	−1.26033	−0.0409768
$947$	7.68430	0.249706	0.124853	−	0.992175i	$-0.460154\pi$
$947$	7.68430	0.249706	0.124853	+	0.992175i	$0.460154\pi$
$948$	0	0
$949$	−4.47455	−0.145250
$950$	0	0
$951$	0	0
$952$	6.52583	0.211503
$953$	−27.1134	−0.878289	−0.439144	−	0.898416i	$-0.644718\pi$
$953$	−27.1134	−0.878289	−0.439144	+	0.898416i	$0.644718\pi$
$954$	0	0
$955$	0	0
$956$	−3.15982	−0.102196
$957$	0	0
$958$	−0.553263	−0.0178751
$959$	−0.401879	−0.0129773
$960$	0	0
$961$	1.00000	0.0322581
$962$	−19.0270	−0.613454
$963$	0	0
$964$	−45.4277	−1.46313
$965$	0	0
$966$	0	0
$967$	−6.16746	−0.198332	−0.0991661	−	0.995071i	$-0.531618\pi$
$967$	−6.16746	−0.198332	−0.0991661	+	0.995071i	$0.531618\pi$
$968$	3.95612	0.127154
$969$	0	0
$970$	0	0
$971$	−14.7787	−0.474272	−0.237136	−	0.971476i	$-0.576209\pi$
$971$	−14.7787	−0.474272	−0.237136	+	0.971476i	$0.576209\pi$
$972$	0	0
$973$	−8.44043	−0.270588
$974$	−4.65295	−0.149090
$975$	0	0
$976$	−25.8236	−0.826594
$977$	−29.5389	−0.945033	−0.472517	−	0.881322i	$-0.656654\pi$
$977$	−29.5389	−0.945033	−0.472517	+	0.881322i	$0.656654\pi$
$978$	0	0
$979$	−20.2261	−0.646430
$980$	0	0
$981$	0	0
$982$	−1.97389	−0.0629892
$983$	6.85969	0.218790	0.109395	−	0.993998i	$-0.465109\pi$
$983$	6.85969	0.218790	0.109395	+	0.993998i	$0.465109\pi$
$984$	0	0
$985$	0	0
$986$	−22.1071	−0.704035
$987$	0	0
$988$	−51.7519	−1.64645
$989$	2.10629	0.0669762
$990$	0	0
$991$	−9.32528	−0.296227	−0.148114	−	0.988970i	$-0.547320\pi$
$991$	−9.32528	−0.296227	−0.148114	+	0.988970i	$0.547320\pi$
$992$	−3.99667	−0.126894
$993$	0	0
$994$	−1.02988	−0.0326659
$995$	0	0
$996$	0	0
$997$	−32.1879	−1.01940	−0.509700	−	0.860352i	$-0.670244\pi$
$997$	−32.1879	−1.01940	−0.509700	+	0.860352i	$0.670244\pi$
$998$	1.95602	0.0619166
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cj.1.5		11
3.2	odd	2		2325.2.a.bc.1.7		11
5.2	odd	4		1395.2.c.h.559.10		22
5.3	odd	4		1395.2.c.h.559.13		22
5.4	even	2		6975.2.a.ci.1.7		11
15.2	even	4		465.2.c.b.94.13	yes	22
15.8	even	4		465.2.c.b.94.10	✓	22
15.14	odd	2		2325.2.a.bd.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.b.94.10	✓	22	15.8	even	4
465.2.c.b.94.13	yes	22	15.2	even	4
1395.2.c.h.559.10		22	5.2	odd	4
1395.2.c.h.559.13		22	5.3	odd	4
2325.2.a.bc.1.7		11	3.2	odd	2
2325.2.a.bd.1.5		11	15.14	odd	2
6975.2.a.ci.1.7		11	5.4	even	2
6975.2.a.cj.1.5		11	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)

\(f(q)\)	\(=\)	\(q-0.364910 q^{2} -1.86684 q^{4} -0.715024 q^{7} +1.41105 q^{8} -3.71533 q^{11} -5.90954 q^{13} +0.260920 q^{14} +3.21877 q^{16} -6.46804 q^{17} -4.69100 q^{19} +1.35576 q^{22} -2.26578 q^{23} +2.15645 q^{26} +1.33484 q^{28} -9.36642 q^{29} +1.00000 q^{31} -3.99667 q^{32} +2.36026 q^{34} -8.82327 q^{37} +1.71179 q^{38} +2.31308 q^{41} -0.929609 q^{43} +6.93593 q^{44} +0.826808 q^{46} +8.18267 q^{47} -6.48874 q^{49} +11.0322 q^{52} -2.25353 q^{53} -1.00894 q^{56} +3.41790 q^{58} +5.09443 q^{59} -8.02281 q^{61} -0.364910 q^{62} -4.97912 q^{64} +2.00786 q^{67} +12.0748 q^{68} -3.94713 q^{71} +0.757173 q^{73} +3.21970 q^{74} +8.75735 q^{76} +2.65655 q^{77} +12.1793 q^{79} -0.844067 q^{82} +5.28381 q^{83} +0.339224 q^{86} -5.24252 q^{88} +5.44397 q^{89} +4.22547 q^{91} +4.22986 q^{92} -2.98594 q^{94} -17.7057 q^{97} +2.36781 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{2} + 15 q^{4} - 8 q^{7} + 9 q^{8} - 14 q^{13} + 14 q^{14} + 27 q^{16} + 12 q^{17} + 12 q^{19} - 10 q^{22} + 12 q^{23} + 6 q^{26} - 22 q^{28} + 8 q^{29} + 11 q^{31} + 21 q^{32} + 2 q^{34} - 16 q^{37}+ \cdots + 17 q^{98}+O(q^{100}) \) 11 * q + 3 * q^2 + 15 * q^4 - 8 * q^7 + 9 * q^8 - 14 * q^13 + 14 * q^14 + 27 * q^16 + 12 * q^17 + 12 * q^19 - 10 * q^22 + 12 * q^23 + 6 * q^26 - 22 * q^28 + 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 - 16 * q^37 + 26 * q^38 + 8 * q^41 - 28 * q^43 + 26 * q^44 + 8 * q^46 + 14 * q^47 + 23 * q^49 - 18 * q^52 + 26 * q^53 + 42 * q^56 - 6 * q^58 + 6 * q^59 + 14 * q^61 + 3 * q^62 + 29 * q^64 - 6 * q^67 + 54 * q^68 + 18 * q^71 - 26 * q^73 + 40 * q^74 + 10 * q^76 + 2 * q^77 + 8 * q^79 + 30 * q^82 + 48 * q^83 + 12 * q^86 - 40 * q^88 + 14 * q^89 - 4 * q^91 - 30 * q^92 - 4 * q^94 - 16 * q^97 + 17 * q^98

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