LMFDB - Embedded newform 6975.2.a.ck.1.7

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 = [16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,44,0] 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 152x^{12} - 571x^{10} + 1130x^{8} - 1138x^{6} + 492x^{4} - 43x^{2} + 1 $ x^16 - 20x^14 + 152x^12 - 571x^10 + 1130x^8 - 1138x^6 + 492x^4 - 43*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.13362$ 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)

$f(q)$	$=$	$q-0.496936 q^{2} -1.75305 q^{4} -1.76854 q^{7} +1.86503 q^{8} -3.54700 q^{11} -4.37401 q^{13} +0.878853 q^{14} +2.57931 q^{16} -5.52017 q^{17} +6.04176 q^{19} +1.76263 q^{22} +6.47065 q^{23} +2.17361 q^{26} +3.10035 q^{28} +2.89707 q^{29} -1.00000 q^{31} -5.01181 q^{32} +2.74317 q^{34} -10.0608 q^{37} -3.00237 q^{38} -10.8874 q^{41} -2.90684 q^{43} +6.21809 q^{44} -3.21550 q^{46} +7.89146 q^{47} -3.87225 q^{49} +7.66788 q^{52} -3.26414 q^{53} -3.29838 q^{56} -1.43966 q^{58} -9.66892 q^{59} -10.6234 q^{61} +0.496936 q^{62} -2.66807 q^{64} -14.1118 q^{67} +9.67717 q^{68} -2.72979 q^{71} +7.66788 q^{73} +4.99956 q^{74} -10.5915 q^{76} +6.27303 q^{77} +3.25683 q^{79} +5.41037 q^{82} -5.94711 q^{83} +1.44451 q^{86} -6.61526 q^{88} +4.92040 q^{89} +7.73563 q^{91} -11.3434 q^{92} -3.92155 q^{94} -1.46127 q^{97} +1.92426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) $ 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

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 $

$2$	−0.496936	−0.351387	−0.175693	−	0.984445i	$-0.556217\pi$
$2$	−0.496936	−0.351387	−0.175693	+	0.984445i	$0.556217\pi$
$3$	0	0
$3$	0	0
$4$	−1.75305	−0.876527
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.76854	−0.668447	−0.334223	−	0.942494i	$-0.608474\pi$
$7$	−1.76854	−0.668447	−0.334223	+	0.942494i	$0.608474\pi$
$8$	1.86503	0.659387
$9$	0	0
$10$	0	0
$11$	−3.54700	−1.06946	−0.534730	−	0.845023i	$-0.679587\pi$
$11$	−3.54700	−1.06946	−0.534730	+	0.845023i	$0.679587\pi$
$12$	0	0
$13$	−4.37401	−1.21313	−0.606566	−	0.795033i	$-0.707454\pi$
$13$	−4.37401	−1.21313	−0.606566	+	0.795033i	$0.707454\pi$
$14$	0.878853	0.234883
$15$	0	0
$16$	2.57931	0.644827
$17$	−5.52017	−1.33884	−0.669419	−	0.742885i	$-0.733457\pi$
$17$	−5.52017	−1.33884	−0.669419	+	0.742885i	$0.733457\pi$
$18$	0	0
$19$	6.04176	1.38607	0.693037	−	0.720902i	$-0.256272\pi$
$19$	6.04176	1.38607	0.693037	+	0.720902i	$0.256272\pi$
$20$	0	0
$21$	0	0
$22$	1.76263	0.375795
$23$	6.47065	1.34922	0.674612	−	0.738172i	$-0.264311\pi$
$23$	6.47065	1.34922	0.674612	+	0.738172i	$0.264311\pi$
$24$	0	0
$25$	0	0
$26$	2.17361	0.426279
$27$	0	0
$28$	3.10035	0.585912
$29$	2.89707	0.537972	0.268986	−	0.963144i	$-0.413311\pi$
$29$	2.89707	0.537972	0.268986	+	0.963144i	$0.413311\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.01181	−0.885971
$33$	0	0
$34$	2.74317	0.470451
$35$	0	0
$36$	0	0
$37$	−10.0608	−1.65398	−0.826990	−	0.562217i	$-0.809949\pi$
$37$	−10.0608	−1.65398	−0.826990	+	0.562217i	$0.809949\pi$
$38$	−3.00237	−0.487048
$39$	0	0
$40$	0	0
$41$	−10.8874	−1.70033	−0.850167	−	0.526513i	$-0.823499\pi$
$41$	−10.8874	−1.70033	−0.850167	+	0.526513i	$0.823499\pi$
$42$	0	0
$43$	−2.90684	−0.443288	−0.221644	−	0.975128i	$-0.571142\pi$
$43$	−2.90684	−0.443288	−0.221644	+	0.975128i	$0.571142\pi$
$44$	6.21809	0.937412
$45$	0	0
$46$	−3.21550	−0.474100
$47$	7.89146	1.15109	0.575544	−	0.817771i	$-0.304790\pi$
$47$	7.89146	1.15109	0.575544	+	0.817771i	$0.304790\pi$
$48$	0	0
$49$	−3.87225	−0.553179
$50$	0	0
$51$	0	0
$52$	7.66788	1.06334
$53$	−3.26414	−0.448364	−0.224182	−	0.974547i	$-0.571971\pi$
$53$	−3.26414	−0.448364	−0.224182	+	0.974547i	$0.571971\pi$
$54$	0	0
$55$	0	0
$56$	−3.29838	−0.440765
$57$	0	0
$58$	−1.43966	−0.189036
$59$	−9.66892	−1.25879	−0.629393	−	0.777087i	$-0.716696\pi$
$59$	−9.66892	−1.25879	−0.629393	+	0.777087i	$0.716696\pi$
$60$	0	0
$61$	−10.6234	−1.36019	−0.680094	−	0.733125i	$-0.738061\pi$
$61$	−10.6234	−1.36019	−0.680094	+	0.733125i	$0.738061\pi$
$62$	0.496936	0.0631110
$63$	0	0
$64$	−2.66807	−0.333509
$65$	0	0
$66$	0	0
$67$	−14.1118	−1.72403	−0.862016	−	0.506882i	$-0.830798\pi$
$67$	−14.1118	−1.72403	−0.862016	+	0.506882i	$0.830798\pi$
$68$	9.67717	1.17353
$69$	0	0
$70$	0	0
$71$	−2.72979	−0.323966	−0.161983	−	0.986794i	$-0.551789\pi$
$71$	−2.72979	−0.323966	−0.161983	+	0.986794i	$0.551789\pi$
$72$	0	0
$73$	7.66788	0.897458	0.448729	−	0.893668i	$-0.351877\pi$
$73$	7.66788	0.897458	0.448729	+	0.893668i	$0.351877\pi$
$74$	4.99956	0.581187
$75$	0	0
$76$	−10.5915	−1.21493
$77$	6.27303	0.714878
$78$	0	0
$79$	3.25683	0.366422	0.183211	−	0.983074i	$-0.441351\pi$
$79$	3.25683	0.366422	0.183211	+	0.983074i	$0.441351\pi$
$80$	0	0
$81$	0	0
$82$	5.41037	0.597475
$83$	−5.94711	−0.652780	−0.326390	−	0.945235i	$-0.605832\pi$
$83$	−5.94711	−0.652780	−0.326390	+	0.945235i	$0.605832\pi$
$84$	0	0
$85$	0	0
$86$	1.44451	0.155766
$87$	0	0
$88$	−6.61526	−0.705189
$89$	4.92040	0.521561	0.260780	−	0.965398i	$-0.416020\pi$
$89$	4.92040	0.521561	0.260780	+	0.965398i	$0.416020\pi$
$90$	0	0
$91$	7.73563	0.810915
$92$	−11.3434	−1.18263
$93$	0	0
$94$	−3.92155	−0.404477
$95$	0	0
$96$	0	0
$97$	−1.46127	−0.148369	−0.0741846	−	0.997245i	$-0.523635\pi$
$97$	−1.46127	−0.148369	−0.0741846	+	0.997245i	$0.523635\pi$
$98$	1.92426	0.194380
$99$	0	0
$100$	0	0
$101$	9.36377	0.931730	0.465865	−	0.884856i	$-0.345743\pi$
$101$	9.36377	0.931730	0.465865	+	0.884856i	$0.345743\pi$
$102$	0	0
$103$	10.2734	1.01226	0.506132	−	0.862456i	$-0.331075\pi$
$103$	10.2734	1.01226	0.506132	+	0.862456i	$0.331075\pi$
$104$	−8.15766	−0.799924
$105$	0	0
$106$	1.62207	0.157549
$107$	−14.9176	−1.44214	−0.721070	−	0.692862i	$-0.756350\pi$
$107$	−14.9176	−1.44214	−0.721070	+	0.692862i	$0.756350\pi$
$108$	0	0
$109$	18.8309	1.80368	0.901838	−	0.432074i	$-0.142218\pi$
$109$	18.8309	1.80368	0.901838	+	0.432074i	$0.142218\pi$
$110$	0	0
$111$	0	0
$112$	−4.56162	−0.431033
$113$	−13.0971	−1.23208	−0.616038	−	0.787716i	$-0.711263\pi$
$113$	−13.0971	−1.23208	−0.616038	+	0.787716i	$0.711263\pi$
$114$	0	0
$115$	0	0
$116$	−5.07872	−0.471547
$117$	0	0
$118$	4.80484	0.442321
$119$	9.76267	0.894942
$120$	0	0
$121$	1.58122	0.143747
$122$	5.27915	0.477952
$123$	0	0
$124$	1.75305	0.157429
$125$	0	0
$126$	0	0
$127$	13.8045	1.22495	0.612477	−	0.790488i	$-0.290173\pi$
$127$	13.8045	1.22495	0.612477	+	0.790488i	$0.290173\pi$
$128$	11.3495	1.00316
$129$	0	0
$130$	0	0
$131$	12.6622	1.10630	0.553149	−	0.833082i	$-0.313426\pi$
$131$	12.6622	1.10630	0.553149	+	0.833082i	$0.313426\pi$
$132$	0	0
$133$	−10.6851	−0.926516
$134$	7.01267	0.605802
$135$	0	0
$136$	−10.2953	−0.882813
$137$	−6.94098	−0.593008	−0.296504	−	0.955032i	$-0.595821\pi$
$137$	−6.94098	−0.593008	−0.296504	+	0.955032i	$0.595821\pi$
$138$	0	0
$139$	3.03378	0.257322	0.128661	−	0.991689i	$-0.458932\pi$
$139$	3.03378	0.257322	0.128661	+	0.991689i	$0.458932\pi$
$140$	0	0
$141$	0	0
$142$	1.35653	0.113838
$143$	15.5146	1.29740
$144$	0	0
$145$	0	0
$146$	−3.81045	−0.315355
$147$	0	0
$148$	17.6371	1.44976
$149$	15.5568	1.27446	0.637232	−	0.770672i	$-0.280079\pi$
$149$	15.5568	1.27446	0.637232	+	0.770672i	$0.280079\pi$
$150$	0	0
$151$	12.8878	1.04879	0.524396	−	0.851475i	$-0.324291\pi$
$151$	12.8878	1.04879	0.524396	+	0.851475i	$0.324291\pi$
$152$	11.2680	0.913959
$153$	0	0
$154$	−3.11729	−0.251199
$155$	0	0
$156$	0	0
$157$	18.6484	1.48831	0.744154	−	0.668008i	$-0.232853\pi$
$157$	18.6484	1.48831	0.744154	+	0.668008i	$0.232853\pi$
$158$	−1.61843	−0.128756
$159$	0	0
$160$	0	0
$161$	−11.4436	−0.901884
$162$	0	0
$163$	−13.2690	−1.03931	−0.519653	−	0.854377i	$-0.673939\pi$
$163$	−13.2690	−1.03931	−0.519653	+	0.854377i	$0.673939\pi$
$164$	19.0863	1.49039
$165$	0	0
$166$	2.95533	0.229378
$167$	−2.46789	−0.190971	−0.0954857	−	0.995431i	$-0.530440\pi$
$167$	−2.46789	−0.190971	−0.0954857	+	0.995431i	$0.530440\pi$
$168$	0	0
$169$	6.13199	0.471692
$170$	0	0
$171$	0	0
$172$	5.09584	0.388554
$173$	−9.87479	−0.750767	−0.375383	−	0.926870i	$-0.622489\pi$
$173$	−9.87479	−0.750767	−0.375383	+	0.926870i	$0.622489\pi$
$174$	0	0
$175$	0	0
$176$	−9.14881	−0.689618
$177$	0	0
$178$	−2.44512	−0.183270
$179$	−16.5143	−1.23434	−0.617169	−	0.786831i	$-0.711720\pi$
$179$	−16.5143	−1.23434	−0.617169	+	0.786831i	$0.711720\pi$
$180$	0	0
$181$	13.5436	1.00669	0.503345	−	0.864086i	$-0.332103\pi$
$181$	13.5436	1.00669	0.503345	+	0.864086i	$0.332103\pi$
$182$	−3.84412	−0.284945
$183$	0	0
$184$	12.0680	0.889661
$185$	0	0
$186$	0	0
$187$	19.5801	1.43184
$188$	−13.8342	−1.00896
$189$	0	0
$190$	0	0
$191$	1.05124	0.0760654	0.0380327	−	0.999276i	$-0.487891\pi$
$191$	1.05124	0.0760654	0.0380327	+	0.999276i	$0.487891\pi$
$192$	0	0
$193$	−1.35679	−0.0976638	−0.0488319	−	0.998807i	$-0.515550\pi$
$193$	−1.35679	−0.0976638	−0.0488319	+	0.998807i	$0.515550\pi$
$194$	0.726156	0.0521350
$195$	0	0
$196$	6.78827	0.484877
$197$	−19.0284	−1.35572	−0.677859	−	0.735192i	$-0.737092\pi$
$197$	−19.0284	−1.35572	−0.677859	+	0.735192i	$0.737092\pi$
$198$	0	0
$199$	25.3863	1.79959	0.899795	−	0.436313i	$-0.143716\pi$
$199$	25.3863	1.79959	0.899795	+	0.436313i	$0.143716\pi$
$200$	0	0
$201$	0	0
$202$	−4.65320	−0.327398
$203$	−5.12359	−0.359606
$204$	0	0
$205$	0	0
$206$	−5.10520	−0.355696
$207$	0	0
$208$	−11.2819	−0.782261
$209$	−21.4301	−1.48235
$210$	0	0
$211$	5.91692	0.407338	0.203669	−	0.979040i	$-0.434713\pi$
$211$	5.91692	0.407338	0.203669	+	0.979040i	$0.434713\pi$
$212$	5.72221	0.393003
$213$	0	0
$214$	7.41311	0.506749
$215$	0	0
$216$	0	0
$217$	1.76854	0.120057
$218$	−9.35777	−0.633788
$219$	0	0
$220$	0	0
$221$	24.1453	1.62419
$222$	0	0
$223$	18.9498	1.26897	0.634487	−	0.772934i	$-0.281212\pi$
$223$	18.9498	1.26897	0.634487	+	0.772934i	$0.281212\pi$
$224$	8.86360	0.592224
$225$	0	0
$226$	6.50845	0.432935
$227$	18.2800	1.21328	0.606642	−	0.794975i	$-0.292516\pi$
$227$	18.2800	1.21328	0.606642	+	0.794975i	$0.292516\pi$
$228$	0	0
$229$	−3.63094	−0.239940	−0.119970	−	0.992778i	$-0.538280\pi$
$229$	−3.63094	−0.239940	−0.119970	+	0.992778i	$0.538280\pi$
$230$	0	0
$231$	0	0
$232$	5.40312	0.354732
$233$	−13.9237	−0.912175	−0.456088	−	0.889935i	$-0.650750\pi$
$233$	−13.9237	−0.912175	−0.456088	+	0.889935i	$0.650750\pi$
$234$	0	0
$235$	0	0
$236$	16.9501	1.10336
$237$	0	0
$238$	−4.85142	−0.314471
$239$	−16.7799	−1.08540	−0.542702	−	0.839926i	$-0.682599\pi$
$239$	−16.7799	−1.08540	−0.542702	+	0.839926i	$0.682599\pi$
$240$	0	0
$241$	−1.11729	−0.0719712	−0.0359856	−	0.999352i	$-0.511457\pi$
$241$	−1.11729	−0.0719712	−0.0359856	+	0.999352i	$0.511457\pi$
$242$	−0.785763	−0.0505108
$243$	0	0
$244$	18.6234	1.19224
$245$	0	0
$246$	0	0
$247$	−26.4267	−1.68149
$248$	−1.86503	−0.118429
$249$	0	0
$250$	0	0
$251$	−2.58950	−0.163448	−0.0817240	−	0.996655i	$-0.526043\pi$
$251$	−2.58950	−0.163448	−0.0817240	+	0.996655i	$0.526043\pi$
$252$	0	0
$253$	−22.9514	−1.44294
$254$	−6.85997	−0.430433
$255$	0	0
$256$	−0.303829	−0.0189893
$257$	−24.9870	−1.55864	−0.779322	−	0.626624i	$-0.784436\pi$
$257$	−24.9870	−1.55864	−0.779322	+	0.626624i	$0.784436\pi$
$258$	0	0
$259$	17.7929	1.10560
$260$	0	0
$261$	0	0
$262$	−6.29228	−0.388739
$263$	17.5024	1.07924	0.539621	−	0.841908i	$-0.318567\pi$
$263$	17.5024	1.07924	0.539621	+	0.841908i	$0.318567\pi$
$264$	0	0
$265$	0	0
$266$	5.30982	0.325566
$267$	0	0
$268$	24.7388	1.51116
$269$	−21.5139	−1.31172	−0.655862	−	0.754881i	$-0.727695\pi$
$269$	−21.5139	−1.31172	−0.655862	+	0.754881i	$0.727695\pi$
$270$	0	0
$271$	19.1371	1.16249	0.581247	−	0.813727i	$-0.302565\pi$
$271$	19.1371	1.16249	0.581247	+	0.813727i	$0.302565\pi$
$272$	−14.2382	−0.863320
$273$	0	0
$274$	3.44922	0.208375
$275$	0	0
$276$	0	0
$277$	−24.4680	−1.47014	−0.735071	−	0.677990i	$-0.762851\pi$
$277$	−24.4680	−1.47014	−0.735071	+	0.677990i	$0.762851\pi$
$278$	−1.50760	−0.0904196
$279$	0	0
$280$	0	0
$281$	8.93983	0.533305	0.266653	−	0.963793i	$-0.414082\pi$
$281$	8.93983	0.533305	0.266653	+	0.963793i	$0.414082\pi$
$282$	0	0
$283$	1.46718	0.0872146	0.0436073	−	0.999049i	$-0.486115\pi$
$283$	1.46718	0.0872146	0.0436073	+	0.999049i	$0.486115\pi$
$284$	4.78547	0.283965
$285$	0	0
$286$	−7.70978	−0.455889
$287$	19.2549	1.13658
$288$	0	0
$289$	13.4723	0.792490
$290$	0	0
$291$	0	0
$292$	−13.4422	−0.786646
$293$	−14.5294	−0.848816	−0.424408	−	0.905471i	$-0.639518\pi$
$293$	−14.5294	−0.848816	−0.424408	+	0.905471i	$0.639518\pi$
$294$	0	0
$295$	0	0
$296$	−18.7636	−1.09061
$297$	0	0
$298$	−7.73074	−0.447830
$299$	−28.3027	−1.63679
$300$	0	0
$301$	5.14087	0.296315
$302$	−6.40440	−0.368532
$303$	0	0
$304$	15.5836	0.893778
$305$	0	0
$306$	0	0
$307$	−17.2088	−0.982157	−0.491079	−	0.871115i	$-0.663397\pi$
$307$	−17.2088	−0.982157	−0.491079	+	0.871115i	$0.663397\pi$
$308$	−10.9970	−0.626610
$309$	0	0
$310$	0	0
$311$	28.6644	1.62541	0.812704	−	0.582676i	$-0.197994\pi$
$311$	28.6644	1.62541	0.812704	+	0.582676i	$0.197994\pi$
$312$	0	0
$313$	0.694311	0.0392448	0.0196224	−	0.999807i	$-0.493754\pi$
$313$	0.694311	0.0392448	0.0196224	+	0.999807i	$0.493754\pi$
$314$	−9.26709	−0.522972
$315$	0	0
$316$	−5.70939	−0.321178
$317$	9.40607	0.528298	0.264149	−	0.964482i	$-0.414909\pi$
$317$	9.40607	0.528298	0.264149	+	0.964482i	$0.414909\pi$
$318$	0	0
$319$	−10.2759	−0.575341
$320$	0	0
$321$	0	0
$322$	5.68675	0.316910
$323$	−33.3515	−1.85573
$324$	0	0
$325$	0	0
$326$	6.59383	0.365199
$327$	0	0
$328$	−20.3054	−1.12118
$329$	−13.9564	−0.769441
$330$	0	0
$331$	16.3366	0.897941	0.448971	−	0.893547i	$-0.351791\pi$
$331$	16.3366	0.897941	0.448971	+	0.893547i	$0.351791\pi$
$332$	10.4256	0.572179
$333$	0	0
$334$	1.22639	0.0671048
$335$	0	0
$336$	0	0
$337$	4.99244	0.271956	0.135978	−	0.990712i	$-0.456582\pi$
$337$	4.99244	0.271956	0.135978	+	0.990712i	$0.456582\pi$
$338$	−3.04721	−0.165746
$339$	0	0
$340$	0	0
$341$	3.54700	0.192081
$342$	0	0
$343$	19.2281	1.03822
$344$	−5.42133	−0.292299
$345$	0	0
$346$	4.90714	0.263810
$347$	29.3780	1.57709	0.788546	−	0.614976i	$-0.210834\pi$
$347$	29.3780	1.57709	0.788546	+	0.614976i	$0.210834\pi$
$348$	0	0
$349$	7.15823	0.383171	0.191586	−	0.981476i	$-0.438637\pi$
$349$	7.15823	0.383171	0.191586	+	0.981476i	$0.438637\pi$
$350$	0	0
$351$	0	0
$352$	17.7769	0.947511
$353$	24.5213	1.30514	0.652569	−	0.757729i	$-0.273691\pi$
$353$	24.5213	1.30514	0.652569	+	0.757729i	$0.273691\pi$
$354$	0	0
$355$	0	0
$356$	−8.62572	−0.457162
$357$	0	0
$358$	8.20656	0.433730
$359$	−1.27506	−0.0672950	−0.0336475	−	0.999434i	$-0.510712\pi$
$359$	−1.27506	−0.0672950	−0.0336475	+	0.999434i	$0.510712\pi$
$360$	0	0
$361$	17.5028	0.921201
$362$	−6.73031	−0.353738
$363$	0	0
$364$	−13.5610	−0.710789
$365$	0	0
$366$	0	0
$367$	16.5844	0.865701	0.432851	−	0.901466i	$-0.357508\pi$
$367$	16.5844	0.865701	0.432851	+	0.901466i	$0.357508\pi$
$368$	16.6898	0.870017
$369$	0	0
$370$	0	0
$371$	5.77277	0.299707
$372$	0	0
$373$	−36.0176	−1.86492	−0.932460	−	0.361274i	$-0.882342\pi$
$373$	−36.0176	−1.86492	−0.932460	+	0.361274i	$0.882342\pi$
$374$	−9.73004	−0.503129
$375$	0	0
$376$	14.7178	0.759012
$377$	−12.6718	−0.652632
$378$	0	0
$379$	1.56719	0.0805013	0.0402506	−	0.999190i	$-0.487184\pi$
$379$	1.56719	0.0805013	0.0402506	+	0.999190i	$0.487184\pi$
$380$	0	0
$381$	0	0
$382$	−0.522401	−0.0267284
$383$	−0.465919	−0.0238073	−0.0119037	−	0.999929i	$-0.503789\pi$
$383$	−0.465919	−0.0238073	−0.0119037	+	0.999929i	$0.503789\pi$
$384$	0	0
$385$	0	0
$386$	0.674237	0.0343178
$387$	0	0
$388$	2.56168	0.130050
$389$	26.3505	1.33602	0.668012	−	0.744151i	$-0.267146\pi$
$389$	26.3505	1.33602	0.668012	+	0.744151i	$0.267146\pi$
$390$	0	0
$391$	−35.7191	−1.80639
$392$	−7.22186	−0.364759
$393$	0	0
$394$	9.45591	0.476382
$395$	0	0
$396$	0	0
$397$	39.0250	1.95861	0.979305	−	0.202391i	$-0.0648712\pi$
$397$	39.0250	1.95861	0.979305	+	0.202391i	$0.0648712\pi$
$398$	−12.6154	−0.632352
$399$	0	0
$400$	0	0
$401$	8.20178	0.409577	0.204789	−	0.978806i	$-0.434349\pi$
$401$	8.20178	0.409577	0.204789	+	0.978806i	$0.434349\pi$
$402$	0	0
$403$	4.37401	0.217885
$404$	−16.4152	−0.816687
$405$	0	0
$406$	2.54610	0.126361
$407$	35.6855	1.76887
$408$	0	0
$409$	16.7022	0.825873	0.412936	−	0.910760i	$-0.364503\pi$
$409$	16.7022	0.825873	0.412936	+	0.910760i	$0.364503\pi$
$410$	0	0
$411$	0	0
$412$	−18.0097	−0.887277
$413$	17.0999	0.841431
$414$	0	0
$415$	0	0
$416$	21.9217	1.07480
$417$	0	0
$418$	10.6494	0.520879
$419$	0.428312	0.0209244	0.0104622	−	0.999945i	$-0.496670\pi$
$419$	0.428312	0.0209244	0.0104622	+	0.999945i	$0.496670\pi$
$420$	0	0
$421$	32.8967	1.60329	0.801643	−	0.597803i	$-0.203959\pi$
$421$	32.8967	1.60329	0.801643	+	0.597803i	$0.203959\pi$
$422$	−2.94033	−0.143133
$423$	0	0
$424$	−6.08771	−0.295645
$425$	0	0
$426$	0	0
$427$	18.7879	0.909213
$428$	26.1514	1.26408
$429$	0	0
$430$	0	0
$431$	−4.38916	−0.211418	−0.105709	−	0.994397i	$-0.533711\pi$
$431$	−4.38916	−0.211418	−0.105709	+	0.994397i	$0.533711\pi$
$432$	0	0
$433$	25.0095	1.20188	0.600940	−	0.799294i	$-0.294793\pi$
$433$	25.0095	1.20188	0.600940	+	0.799294i	$0.294793\pi$
$434$	−0.878853	−0.0421863
$435$	0	0
$436$	−33.0116	−1.58097
$437$	39.0941	1.87012
$438$	0	0
$439$	−2.57316	−0.122810	−0.0614051	−	0.998113i	$-0.519558\pi$
$439$	−2.57316	−0.122810	−0.0614051	+	0.998113i	$0.519558\pi$
$440$	0	0
$441$	0	0
$442$	−11.9987	−0.570719
$443$	16.9716	0.806346	0.403173	−	0.915124i	$-0.367907\pi$
$443$	16.9716	0.806346	0.403173	+	0.915124i	$0.367907\pi$
$444$	0	0
$445$	0	0
$446$	−9.41685	−0.445901
$447$	0	0
$448$	4.71859	0.222933
$449$	38.1830	1.80197	0.900985	−	0.433851i	$-0.142846\pi$
$449$	38.1830	1.80197	0.900985	+	0.433851i	$0.142846\pi$
$450$	0	0
$451$	38.6178	1.81844
$452$	22.9600	1.07995
$453$	0	0
$454$	−9.08398	−0.426332
$455$	0	0
$456$	0	0
$457$	10.9724	0.513266	0.256633	−	0.966509i	$-0.417387\pi$
$457$	10.9724	0.513266	0.256633	+	0.966509i	$0.417387\pi$
$458$	1.80435	0.0843116
$459$	0	0
$460$	0	0
$461$	−26.3189	−1.22579	−0.612897	−	0.790163i	$-0.709996\pi$
$461$	−26.3189	−1.22579	−0.612897	+	0.790163i	$0.709996\pi$
$462$	0	0
$463$	20.3529	0.945881	0.472940	−	0.881094i	$-0.343193\pi$
$463$	20.3529	0.945881	0.472940	+	0.881094i	$0.343193\pi$
$464$	7.47244	0.346899
$465$	0	0
$466$	6.91921	0.320526
$467$	−6.80800	−0.315037	−0.157518	−	0.987516i	$-0.550349\pi$
$467$	−6.80800	−0.315037	−0.157518	+	0.987516i	$0.550349\pi$
$468$	0	0
$469$	24.9573	1.15242
$470$	0	0
$471$	0	0
$472$	−18.0328	−0.830028
$473$	10.3105	0.474080
$474$	0	0
$475$	0	0
$476$	−17.1145	−0.784441
$477$	0	0
$478$	8.33855	0.381396
$479$	−12.8397	−0.586659	−0.293329	−	0.956011i	$-0.594763\pi$
$479$	−12.8397	−0.586659	−0.293329	+	0.956011i	$0.594763\pi$
$480$	0	0
$481$	44.0059	2.00650
$482$	0.555223	0.0252897
$483$	0	0
$484$	−2.77196	−0.125998
$485$	0	0
$486$	0	0
$487$	−18.0606	−0.818406	−0.409203	−	0.912443i	$-0.634193\pi$
$487$	−18.0606	−0.818406	−0.409203	+	0.912443i	$0.634193\pi$
$488$	−19.8129	−0.896890
$489$	0	0
$490$	0	0
$491$	16.0273	0.723301	0.361651	−	0.932314i	$-0.382213\pi$
$491$	16.0273	0.723301	0.361651	+	0.932314i	$0.382213\pi$
$492$	0	0
$493$	−15.9923	−0.720258
$494$	13.1324	0.590854
$495$	0	0
$496$	−2.57931	−0.115814
$497$	4.82775	0.216554
$498$	0	0
$499$	3.99151	0.178685	0.0893423	−	0.996001i	$-0.471523\pi$
$499$	3.99151	0.178685	0.0893423	+	0.996001i	$0.471523\pi$
$500$	0	0
$501$	0	0
$502$	1.28682	0.0574335
$503$	22.6460	1.00974	0.504868	−	0.863196i	$-0.331541\pi$
$503$	22.6460	1.00974	0.504868	+	0.863196i	$0.331541\pi$
$504$	0	0
$505$	0	0
$506$	11.4054	0.507031
$507$	0	0
$508$	−24.2001	−1.07371
$509$	−5.45499	−0.241788	−0.120894	−	0.992665i	$-0.538576\pi$
$509$	−5.45499	−0.241788	−0.120894	+	0.992665i	$0.538576\pi$
$510$	0	0
$511$	−13.5610	−0.599903
$512$	−22.5480	−0.996489
$513$	0	0
$514$	12.4169	0.547687
$515$	0	0
$516$	0	0
$517$	−27.9910	−1.23104
$518$	−8.84194	−0.388492
$519$	0	0
$520$	0	0
$521$	−26.8921	−1.17816	−0.589082	−	0.808073i	$-0.700511\pi$
$521$	−26.8921	−1.17816	−0.589082	+	0.808073i	$0.700511\pi$
$522$	0	0
$523$	−11.7605	−0.514253	−0.257126	−	0.966378i	$-0.582776\pi$
$523$	−11.7605	−0.514253	−0.257126	+	0.966378i	$0.582776\pi$
$524$	−22.1974	−0.969700
$525$	0	0
$526$	−8.69756	−0.379232
$527$	5.52017	0.240463
$528$	0	0
$529$	18.8693	0.820406
$530$	0	0
$531$	0	0
$532$	18.7316	0.812117
$533$	47.6218	2.06273
$534$	0	0
$535$	0	0
$536$	−26.3189	−1.13680
$537$	0	0
$538$	10.6910	0.460923
$539$	13.7349	0.591604
$540$	0	0
$541$	−19.9004	−0.855586	−0.427793	−	0.903877i	$-0.640709\pi$
$541$	−19.9004	−0.855586	−0.427793	+	0.903877i	$0.640709\pi$
$542$	−9.50989	−0.408485
$543$	0	0
$544$	27.6661	1.18617
$545$	0	0
$546$	0	0
$547$	−25.7749	−1.10205	−0.551027	−	0.834488i	$-0.685764\pi$
$547$	−25.7749	−1.10205	−0.551027	+	0.834488i	$0.685764\pi$
$548$	12.1679	0.519788
$549$	0	0
$550$	0	0
$551$	17.5034	0.745669
$552$	0	0
$553$	−5.75984	−0.244933
$554$	12.1590	0.516588
$555$	0	0
$556$	−5.31838	−0.225550
$557$	−8.99498	−0.381130	−0.190565	−	0.981675i	$-0.561032\pi$
$557$	−8.99498	−0.381130	−0.190565	+	0.981675i	$0.561032\pi$
$558$	0	0
$559$	12.7145	0.537768
$560$	0	0
$561$	0	0
$562$	−4.44252	−0.187397
$563$	−35.1487	−1.48134	−0.740670	−	0.671869i	$-0.765492\pi$
$563$	−35.1487	−1.48134	−0.740670	+	0.671869i	$0.765492\pi$
$564$	0	0
$565$	0	0
$566$	−0.729093	−0.0306461
$567$	0	0
$568$	−5.09113	−0.213619
$569$	31.2790	1.31128	0.655641	−	0.755073i	$-0.272399\pi$
$569$	31.2790	1.31128	0.655641	+	0.755073i	$0.272399\pi$
$570$	0	0
$571$	35.8655	1.50093	0.750463	−	0.660913i	$-0.229831\pi$
$571$	35.8655	1.50093	0.750463	+	0.660913i	$0.229831\pi$
$572$	−27.1980	−1.13720
$573$	0	0
$574$	−9.56847	−0.399380
$575$	0	0
$576$	0	0
$577$	−12.0301	−0.500819	−0.250409	−	0.968140i	$-0.580565\pi$
$577$	−12.0301	−0.500819	−0.250409	+	0.968140i	$0.580565\pi$
$578$	−6.69489	−0.278471
$579$	0	0
$580$	0	0
$581$	10.5177	0.436349
$582$	0	0
$583$	11.5779	0.479508
$584$	14.3008	0.591772
$585$	0	0
$586$	7.22018	0.298263
$587$	−19.5447	−0.806694	−0.403347	−	0.915047i	$-0.632153\pi$
$587$	−19.5447	−0.806694	−0.403347	+	0.915047i	$0.632153\pi$
$588$	0	0
$589$	−6.04176	−0.248946
$590$	0	0
$591$	0	0
$592$	−25.9498	−1.06653
$593$	5.29367	0.217385	0.108692	−	0.994075i	$-0.465334\pi$
$593$	5.29367	0.217385	0.108692	+	0.994075i	$0.465334\pi$
$594$	0	0
$595$	0	0
$596$	−27.2719	−1.11710
$597$	0	0
$598$	14.0646	0.575146
$599$	−43.4560	−1.77557	−0.887783	−	0.460263i	$-0.847755\pi$
$599$	−43.4560	−1.77557	−0.887783	+	0.460263i	$0.847755\pi$
$600$	0	0
$601$	1.81408	0.0739979	0.0369989	−	0.999315i	$-0.488220\pi$
$601$	1.81408	0.0739979	0.0369989	+	0.999315i	$0.488220\pi$
$602$	−2.55468	−0.104121
$603$	0	0
$604$	−22.5930	−0.919295
$605$	0	0
$606$	0	0
$607$	3.87068	0.157106	0.0785530	−	0.996910i	$-0.474970\pi$
$607$	3.87068	0.157106	0.0785530	+	0.996910i	$0.474970\pi$
$608$	−30.2801	−1.22802
$609$	0	0
$610$	0	0
$611$	−34.5173	−1.39642
$612$	0	0
$613$	−6.49616	−0.262378	−0.131189	−	0.991357i	$-0.541879\pi$
$613$	−6.49616	−0.262378	−0.131189	+	0.991357i	$0.541879\pi$
$614$	8.55167	0.345117
$615$	0	0
$616$	11.6994	0.471381
$617$	−34.4420	−1.38658	−0.693292	−	0.720657i	$-0.743840\pi$
$617$	−34.4420	−1.38658	−0.693292	+	0.720657i	$0.743840\pi$
$618$	0	0
$619$	34.1864	1.37407	0.687034	−	0.726626i	$-0.258913\pi$
$619$	34.1864	1.37407	0.687034	+	0.726626i	$0.258913\pi$
$620$	0	0
$621$	0	0
$622$	−14.2444	−0.571147
$623$	−8.70193	−0.348636
$624$	0	0
$625$	0	0
$626$	−0.345028	−0.0137901
$627$	0	0
$628$	−32.6917	−1.30454
$629$	55.5372	2.21441
$630$	0	0
$631$	−3.67694	−0.146377	−0.0731884	−	0.997318i	$-0.523317\pi$
$631$	−3.67694	−0.146377	−0.0731884	+	0.997318i	$0.523317\pi$
$632$	6.07407	0.241614
$633$	0	0
$634$	−4.67422	−0.185637
$635$	0	0
$636$	0	0
$637$	16.9373	0.671080
$638$	5.10647	0.202167
$639$	0	0
$640$	0	0
$641$	17.6537	0.697278	0.348639	−	0.937257i	$-0.386644\pi$
$641$	17.6537	0.697278	0.348639	+	0.937257i	$0.386644\pi$
$642$	0	0
$643$	41.6411	1.64216	0.821082	−	0.570810i	$-0.193371\pi$
$643$	41.6411	1.64216	0.821082	+	0.570810i	$0.193371\pi$
$644$	20.0613	0.790526
$645$	0	0
$646$	16.5736	0.652079
$647$	−31.6340	−1.24366	−0.621830	−	0.783152i	$-0.713611\pi$
$647$	−31.6340	−1.24366	−0.621830	+	0.783152i	$0.713611\pi$
$648$	0	0
$649$	34.2957	1.34622
$650$	0	0
$651$	0	0
$652$	23.2612	0.910980
$653$	−16.5171	−0.646365	−0.323183	−	0.946337i	$-0.604753\pi$
$653$	−16.5171	−0.646365	−0.323183	+	0.946337i	$0.604753\pi$
$654$	0	0
$655$	0	0
$656$	−28.0821	−1.09642
$657$	0	0
$658$	6.93543	0.270371
$659$	−2.61904	−0.102023	−0.0510116	−	0.998698i	$-0.516245\pi$
$659$	−2.61904	−0.102023	−0.0510116	+	0.998698i	$0.516245\pi$
$660$	0	0
$661$	−2.36949	−0.0921624	−0.0460812	−	0.998938i	$-0.514673\pi$
$661$	−2.36949	−0.0921624	−0.0460812	+	0.998938i	$0.514673\pi$
$662$	−8.11825	−0.315525
$663$	0	0
$664$	−11.0915	−0.430435
$665$	0	0
$666$	0	0
$667$	18.7459	0.725845
$668$	4.32635	0.167392
$669$	0	0
$670$	0	0
$671$	37.6812	1.45467
$672$	0	0
$673$	41.0800	1.58352	0.791759	−	0.610834i	$-0.209166\pi$
$673$	41.0800	1.58352	0.791759	+	0.610834i	$0.209166\pi$
$674$	−2.48093	−0.0955617
$675$	0	0
$676$	−10.7497	−0.413450
$677$	−0.0142301	−0.000546908 0	−0.000273454	−	1.00000i	$-0.500087\pi$
$677$	−0.0142301	−0.000546908 0	−0.000273454		1.00000i	$0.500087\pi$
$678$	0	0
$679$	2.58431	0.0991769
$680$	0	0
$681$	0	0
$682$	−1.76263	−0.0674947
$683$	24.2385	0.927461	0.463731	−	0.885976i	$-0.346511\pi$
$683$	24.2385	0.927461	0.463731	+	0.885976i	$0.346511\pi$
$684$	0	0
$685$	0	0
$686$	−9.55512	−0.364816
$687$	0	0
$688$	−7.49763	−0.285844
$689$	14.2774	0.543925
$690$	0	0
$691$	−31.2970	−1.19060	−0.595298	−	0.803505i	$-0.702966\pi$
$691$	−31.2970	−1.19060	−0.595298	+	0.803505i	$0.702966\pi$
$692$	17.3111	0.658067
$693$	0	0
$694$	−14.5990	−0.554169
$695$	0	0
$696$	0	0
$697$	60.1006	2.27647
$698$	−3.55718	−0.134641
$699$	0	0
$700$	0	0
$701$	−13.6342	−0.514958	−0.257479	−	0.966284i	$-0.582892\pi$
$701$	−13.6342	−0.514958	−0.257479	+	0.966284i	$0.582892\pi$
$702$	0	0
$703$	−60.7847	−2.29254
$704$	9.46364	0.356674
$705$	0	0
$706$	−12.1855	−0.458609
$707$	−16.5602	−0.622812
$708$	0	0
$709$	−5.42727	−0.203826	−0.101913	−	0.994793i	$-0.532496\pi$
$709$	−5.42727	−0.203826	−0.101913	+	0.994793i	$0.532496\pi$
$710$	0	0
$711$	0	0
$712$	9.17668	0.343911
$713$	−6.47065	−0.242328
$714$	0	0
$715$	0	0
$716$	28.9505	1.08193
$717$	0	0
$718$	0.633622	0.0236466
$719$	39.2467	1.46365	0.731827	−	0.681490i	$-0.238668\pi$
$719$	39.2467	1.46365	0.731827	+	0.681490i	$0.238668\pi$
$720$	0	0
$721$	−18.1689	−0.676644
$722$	−8.69778	−0.323698
$723$	0	0
$724$	−23.7427	−0.882391
$725$	0	0
$726$	0	0
$727$	−0.160347	−0.00594693	−0.00297346	−	0.999996i	$-0.500946\pi$
$727$	−0.160347	−0.00594693	−0.00297346	+	0.999996i	$0.500946\pi$
$728$	14.4272	0.534707
$729$	0	0
$730$	0	0
$731$	16.0462	0.593492
$732$	0	0
$733$	−4.92088	−0.181757	−0.0908784	−	0.995862i	$-0.528967\pi$
$733$	−4.92088	−0.181757	−0.0908784	+	0.995862i	$0.528967\pi$
$734$	−8.24141	−0.304196
$735$	0	0
$736$	−32.4297	−1.19537
$737$	50.0546	1.84378
$738$	0	0
$739$	9.52146	0.350253	0.175126	−	0.984546i	$-0.443967\pi$
$739$	9.52146	0.350253	0.175126	+	0.984546i	$0.443967\pi$
$740$	0	0
$741$	0	0
$742$	−2.86870	−0.105313
$743$	−36.3332	−1.33293	−0.666467	−	0.745534i	$-0.732194\pi$
$743$	−36.3332	−1.33293	−0.666467	+	0.745534i	$0.732194\pi$
$744$	0	0
$745$	0	0
$746$	17.8984	0.655308
$747$	0	0
$748$	−34.3249	−1.25504
$749$	26.3825	0.963994
$750$	0	0
$751$	−27.7863	−1.01394	−0.506968	−	0.861965i	$-0.669234\pi$
$751$	−27.7863	−1.01394	−0.506968	+	0.861965i	$0.669234\pi$
$752$	20.3545	0.742253
$753$	0	0
$754$	6.29709	0.229326
$755$	0	0
$756$	0	0
$757$	−24.3792	−0.886079	−0.443039	−	0.896502i	$-0.646100\pi$
$757$	−24.3792	−0.886079	−0.443039	+	0.896502i	$0.646100\pi$
$758$	−0.778795	−0.0282871
$759$	0	0
$760$	0	0
$761$	−34.7366	−1.25920	−0.629600	−	0.776920i	$-0.716781\pi$
$761$	−34.7366	−1.25920	−0.629600	+	0.776920i	$0.716781\pi$
$762$	0	0
$763$	−33.3033	−1.20566
$764$	−1.84289	−0.0666734
$765$	0	0
$766$	0.231532	0.00836558
$767$	42.2920	1.52708
$768$	0	0
$769$	3.02648	0.109138	0.0545689	−	0.998510i	$-0.482622\pi$
$769$	3.02648	0.109138	0.0545689	+	0.998510i	$0.482622\pi$
$770$	0	0
$771$	0	0
$772$	2.37852	0.0856050
$773$	24.1577	0.868894	0.434447	−	0.900697i	$-0.356944\pi$
$773$	24.1577	0.868894	0.434447	+	0.900697i	$0.356944\pi$
$774$	0	0
$775$	0	0
$776$	−2.72530	−0.0978327
$777$	0	0
$778$	−13.0945	−0.469461
$779$	−65.7793	−2.35679
$780$	0	0
$781$	9.68256	0.346469
$782$	17.7501	0.634743
$783$	0	0
$784$	−9.98774	−0.356705
$785$	0	0
$786$	0	0
$787$	−12.5834	−0.448549	−0.224274	−	0.974526i	$-0.572001\pi$
$787$	−12.5834	−0.448549	−0.224274	+	0.974526i	$0.572001\pi$
$788$	33.3579	1.18832
$789$	0	0
$790$	0	0
$791$	23.1629	0.823577
$792$	0	0
$793$	46.4669	1.65009
$794$	−19.3929	−0.688230
$795$	0	0
$796$	−44.5036	−1.57739
$797$	−18.4732	−0.654354	−0.327177	−	0.944963i	$-0.606097\pi$
$797$	−18.4732	−0.654354	−0.327177	+	0.944963i	$0.606097\pi$
$798$	0	0
$799$	−43.5622	−1.54112
$800$	0	0
$801$	0	0
$802$	−4.07576	−0.143920
$803$	−27.1980	−0.959796
$804$	0	0
$805$	0	0
$806$	−2.17361	−0.0765620
$807$	0	0
$808$	17.4637	0.614371
$809$	45.3019	1.59273	0.796364	−	0.604817i	$-0.206754\pi$
$809$	45.3019	1.59273	0.796364	+	0.604817i	$0.206754\pi$
$810$	0	0
$811$	2.51365	0.0882662	0.0441331	−	0.999026i	$-0.485947\pi$
$811$	2.51365	0.0882662	0.0441331	+	0.999026i	$0.485947\pi$
$812$	8.98194	0.315204
$813$	0	0
$814$	−17.7334	−0.621557
$815$	0	0
$816$	0	0
$817$	−17.5624	−0.614430
$818$	−8.29995	−0.290201
$819$	0	0
$820$	0	0
$821$	0.147861	0.00516040	0.00258020	−	0.999997i	$-0.499179\pi$
$821$	0.147861	0.00516040	0.00258020	+	0.999997i	$0.499179\pi$
$822$	0	0
$823$	−28.6263	−0.997852	−0.498926	−	0.866645i	$-0.666272\pi$
$823$	−28.6263	−0.997852	−0.498926	+	0.866645i	$0.666272\pi$
$824$	19.1601	0.667474
$825$	0	0
$826$	−8.49756	−0.295668
$827$	19.4442	0.676141	0.338071	−	0.941121i	$-0.390226\pi$
$827$	19.4442	0.676141	0.338071	+	0.941121i	$0.390226\pi$
$828$	0	0
$829$	13.3361	0.463183	0.231592	−	0.972813i	$-0.425607\pi$
$829$	13.3361	0.463183	0.231592	+	0.972813i	$0.425607\pi$
$830$	0	0
$831$	0	0
$832$	11.6702	0.404590
$833$	21.3755	0.740618
$834$	0	0
$835$	0	0
$836$	37.5682	1.29932
$837$	0	0
$838$	−0.212844	−0.00735256
$839$	−23.5246	−0.812158	−0.406079	−	0.913838i	$-0.633104\pi$
$839$	−23.5246	−0.812158	−0.406079	+	0.913838i	$0.633104\pi$
$840$	0	0
$841$	−20.6070	−0.710586
$842$	−16.3476	−0.563374
$843$	0	0
$844$	−10.3727	−0.357042
$845$	0	0
$846$	0	0
$847$	−2.79645	−0.0960871
$848$	−8.41922	−0.289117
$849$	0	0
$850$	0	0
$851$	−65.0997	−2.23159
$852$	0	0
$853$	5.51542	0.188844	0.0944222	−	0.995532i	$-0.469900\pi$
$853$	5.51542	0.188844	0.0944222	+	0.995532i	$0.469900\pi$
$854$	−9.33641	−0.319485
$855$	0	0
$856$	−27.8218	−0.950929
$857$	−7.80908	−0.266753	−0.133377	−	0.991065i	$-0.542582\pi$
$857$	−7.80908	−0.266753	−0.133377	+	0.991065i	$0.542582\pi$
$858$	0	0
$859$	−9.74652	−0.332547	−0.166273	−	0.986080i	$-0.553173\pi$
$859$	−9.74652	−0.332547	−0.166273	+	0.986080i	$0.553173\pi$
$860$	0	0
$861$	0	0
$862$	2.18113	0.0742896
$863$	−13.7896	−0.469403	−0.234701	−	0.972068i	$-0.575411\pi$
$863$	−13.7896	−0.469403	−0.234701	+	0.972068i	$0.575411\pi$
$864$	0	0
$865$	0	0
$866$	−12.4281	−0.422325
$867$	0	0
$868$	−3.10035	−0.105233
$869$	−11.5520	−0.391874
$870$	0	0
$871$	61.7252	2.09148
$872$	35.1202	1.18932
$873$	0	0
$874$	−19.4273	−0.657137
$875$	0	0
$876$	0	0
$877$	4.29114	0.144902	0.0724508	−	0.997372i	$-0.476918\pi$
$877$	4.29114	0.144902	0.0724508	+	0.997372i	$0.476918\pi$
$878$	1.27870	0.0431539
$879$	0	0
$880$	0	0
$881$	−36.4210	−1.22705	−0.613527	−	0.789674i	$-0.710250\pi$
$881$	−36.4210	−1.22705	−0.613527	+	0.789674i	$0.710250\pi$
$882$	0	0
$883$	18.2807	0.615196	0.307598	−	0.951516i	$-0.400475\pi$
$883$	18.2807	0.615196	0.307598	+	0.951516i	$0.400475\pi$
$884$	−42.3281	−1.42365
$885$	0	0
$886$	−8.43381	−0.283339
$887$	−2.37100	−0.0796104	−0.0398052	−	0.999207i	$-0.512674\pi$
$887$	−2.37100	−0.0796104	−0.0398052	+	0.999207i	$0.512674\pi$
$888$	0	0
$889$	−24.4139	−0.818816
$890$	0	0
$891$	0	0
$892$	−33.2201	−1.11229
$893$	47.6783	1.59549
$894$	0	0
$895$	0	0
$896$	−20.0720	−0.670560
$897$	0	0
$898$	−18.9745	−0.633188
$899$	−2.89707	−0.0966227
$900$	0	0
$901$	18.0186	0.600287
$902$	−19.1906	−0.638976
$903$	0	0
$904$	−24.4266	−0.812415
$905$	0	0
$906$	0	0
$907$	−1.60903	−0.0534269	−0.0267135	−	0.999643i	$-0.508504\pi$
$907$	−1.60903	−0.0534269	−0.0267135	+	0.999643i	$0.508504\pi$
$908$	−32.0458	−1.06348
$909$	0	0
$910$	0	0
$911$	−53.4697	−1.77153	−0.885766	−	0.464132i	$-0.846366\pi$
$911$	−53.4697	−1.77153	−0.885766	+	0.464132i	$0.846366\pi$
$912$	0	0
$913$	21.0944	0.698123
$914$	−5.45257	−0.180355
$915$	0	0
$916$	6.36524	0.210314
$917$	−22.3936	−0.739501
$918$	0	0
$919$	−29.0385	−0.957890	−0.478945	−	0.877845i	$-0.658981\pi$
$919$	−29.0385	−0.957890	−0.478945	+	0.877845i	$0.658981\pi$
$920$	0	0
$921$	0	0
$922$	13.0788	0.430728
$923$	11.9401	0.393014
$924$	0	0
$925$	0	0
$926$	−10.1141	−0.332370
$927$	0	0
$928$	−14.5196	−0.476628
$929$	26.5506	0.871098	0.435549	−	0.900165i	$-0.356554\pi$
$929$	26.5506	0.871098	0.435549	+	0.900165i	$0.356554\pi$
$930$	0	0
$931$	−23.3952	−0.766747
$932$	24.4091	0.799546
$933$	0	0
$934$	3.38314	0.110700
$935$	0	0
$936$	0	0
$937$	16.2980	0.532433	0.266216	−	0.963913i	$-0.414226\pi$
$937$	16.2980	0.532433	0.266216	+	0.963913i	$0.414226\pi$
$938$	−12.4022	−0.404946
$939$	0	0
$940$	0	0
$941$	21.5930	0.703913	0.351956	−	0.936016i	$-0.385517\pi$
$941$	21.5930	0.703913	0.351956	+	0.936016i	$0.385517\pi$
$942$	0	0
$943$	−70.4489	−2.29413
$944$	−24.9391	−0.811700
$945$	0	0
$946$	−5.12369	−0.166585
$947$	−14.9550	−0.485972	−0.242986	−	0.970030i	$-0.578127\pi$
$947$	−14.9550	−0.485972	−0.242986	+	0.970030i	$0.578127\pi$
$948$	0	0
$949$	−33.5394	−1.08874
$950$	0	0
$951$	0	0
$952$	18.2077	0.590114
$953$	35.5313	1.15097	0.575486	−	0.817812i	$-0.304813\pi$
$953$	35.5313	1.15097	0.575486	+	0.817812i	$0.304813\pi$
$954$	0	0
$955$	0	0
$956$	29.4161	0.951385
$957$	0	0
$958$	6.38049	0.206144
$959$	12.2754	0.396394
$960$	0	0
$961$	1.00000	0.0322581
$962$	−21.8681	−0.705057
$963$	0	0
$964$	1.95868	0.0630847
$965$	0	0
$966$	0	0
$967$	18.9498	0.609385	0.304692	−	0.952451i	$-0.401446\pi$
$967$	18.9498	0.609385	0.304692	+	0.952451i	$0.401446\pi$
$968$	2.94901	0.0947848
$969$	0	0
$970$	0	0
$971$	−3.16373	−0.101529	−0.0507644	−	0.998711i	$-0.516166\pi$
$971$	−3.16373	−0.101529	−0.0507644	+	0.998711i	$0.516166\pi$
$972$	0	0
$973$	−5.36538	−0.172006
$974$	8.97499	0.287577
$975$	0	0
$976$	−27.4010	−0.877086
$977$	12.4565	0.398517	0.199259	−	0.979947i	$-0.436147\pi$
$977$	12.4565	0.398517	0.199259	+	0.979947i	$0.436147\pi$
$978$	0	0
$979$	−17.4527	−0.557789
$980$	0	0
$981$	0	0
$982$	−7.96454	−0.254159
$983$	40.0576	1.27764	0.638820	−	0.769356i	$-0.279423\pi$
$983$	40.0576	1.27764	0.638820	+	0.769356i	$0.279423\pi$
$984$	0	0
$985$	0	0
$986$	7.94717	0.253089
$987$	0	0
$988$	46.3275	1.47387
$989$	−18.8091	−0.598095
$990$	0	0
$991$	17.6197	0.559707	0.279854	−	0.960043i	$-0.409714\pi$
$991$	17.6197	0.559707	0.279854	+	0.960043i	$0.409714\pi$
$992$	5.01181	0.159125
$993$	0	0
$994$	−2.39908	−0.0760943
$995$	0	0
$996$	0	0
$997$	41.5224	1.31503	0.657513	−	0.753443i	$-0.271608\pi$
$997$	41.5224	1.31503	0.657513	+	0.753443i	$0.271608\pi$
$998$	−1.98353	−0.0627874
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ck.1.7		16
3.2	odd	2	inner	6975.2.a.ck.1.9		16
5.2	odd	4		1395.2.c.g.559.7	✓	16
5.3	odd	4		1395.2.c.g.559.9	yes	16
5.4	even	2	inner	6975.2.a.ck.1.10		16
15.2	even	4		1395.2.c.g.559.10	yes	16
15.8	even	4		1395.2.c.g.559.8	yes	16
15.14	odd	2	inner	6975.2.a.ck.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.c.g.559.7	✓	16	5.2	odd	4
1395.2.c.g.559.8	yes	16	15.8	even	4
1395.2.c.g.559.9	yes	16	5.3	odd	4
1395.2.c.g.559.10	yes	16	15.2	even	4
6975.2.a.ck.1.7		16	1.1	even	1	trivial
6975.2.a.ck.1.8		16	15.14	odd	2	inner
6975.2.a.ck.1.9		16	3.2	odd	2	inner
6975.2.a.ck.1.10		16	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.496936 q^{2} -1.75305 q^{4} -1.76854 q^{7} +1.86503 q^{8} -3.54700 q^{11} -4.37401 q^{13} +0.878853 q^{14} +2.57931 q^{16} -5.52017 q^{17} +6.04176 q^{19} +1.76263 q^{22} +6.47065 q^{23} +2.17361 q^{26} +3.10035 q^{28} +2.89707 q^{29} -1.00000 q^{31} -5.01181 q^{32} +2.74317 q^{34} -10.0608 q^{37} -3.00237 q^{38} -10.8874 q^{41} -2.90684 q^{43} +6.21809 q^{44} -3.21550 q^{46} +7.89146 q^{47} -3.87225 q^{49} +7.66788 q^{52} -3.26414 q^{53} -3.29838 q^{56} -1.43966 q^{58} -9.66892 q^{59} -10.6234 q^{61} +0.496936 q^{62} -2.66807 q^{64} -14.1118 q^{67} +9.67717 q^{68} -2.72979 q^{71} +7.66788 q^{73} +4.99956 q^{74} -10.5915 q^{76} +6.27303 q^{77} +3.25683 q^{79} +5.41037 q^{82} -5.94711 q^{83} +1.44451 q^{86} -6.61526 q^{88} +4.92040 q^{89} +7.73563 q^{91} -11.3434 q^{92} -3.92155 q^{94} -1.46127 q^{97} +1.92426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) \) 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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