LMFDB - Embedded newform 7168.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.2367681689$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 $ x^12 - 24x^10 + 221x^8 - 968x^6 + 2008x^4 - 1640*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.377920$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.589521 q^{3} -1.60045 q^{5} +1.00000 q^{7} -2.65247 q^{9} +O(q^{10})$	$q-0.589521 q^{3} -1.60045 q^{5} +1.00000 q^{7} -2.65247 q^{9} -5.45487 q^{11} +6.59464 q^{13} +0.943500 q^{15} +5.33230 q^{17} -3.61925 q^{19} -0.589521 q^{21} -2.60484 q^{23} -2.43855 q^{25} +3.33225 q^{27} +1.72929 q^{29} -0.833708 q^{31} +3.21576 q^{33} -1.60045 q^{35} -6.26450 q^{37} -3.88768 q^{39} -0.263382 q^{41} -1.77107 q^{43} +4.24514 q^{45} +10.7559 q^{47} +1.00000 q^{49} -3.14350 q^{51} -0.0673478 q^{53} +8.73026 q^{55} +2.13362 q^{57} -5.10081 q^{59} -6.31304 q^{61} -2.65247 q^{63} -10.5544 q^{65} -13.4487 q^{67} +1.53561 q^{69} -2.05301 q^{71} -5.48268 q^{73} +1.43758 q^{75} -5.45487 q^{77} -5.21576 q^{79} +5.99297 q^{81} +8.25965 q^{83} -8.53410 q^{85} -1.01945 q^{87} +6.32651 q^{89} +6.59464 q^{91} +0.491488 q^{93} +5.79243 q^{95} +18.8089 q^{97} +14.4689 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 + 12 * q^9	$ 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

Display 100 coefficients 10 coefficients

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	0
$2$	0	0
$3$	−0.589521	−0.340360	−0.170180	−	0.985413i	$-0.554435\pi$
$3$	−0.589521	−0.340360	−0.170180	+	0.985413i	$0.554435\pi$
$4$	0	0
$5$	−1.60045	−0.715744	−0.357872	−	0.933771i	$-0.616498\pi$
$5$	−1.60045	−0.715744	−0.357872	+	0.933771i	$0.616498\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.65247	−0.884155
$10$	0	0
$11$	−5.45487	−1.64471	−0.822353	−	0.568978i	$-0.807339\pi$
$11$	−5.45487	−1.64471	−0.822353	+	0.568978i	$0.807339\pi$
$12$	0	0
$13$	6.59464	1.82902	0.914512	−	0.404559i	$-0.132575\pi$
$13$	6.59464	1.82902	0.914512	+	0.404559i	$0.132575\pi$
$14$	0	0
$15$	0.943500	0.243611
$16$	0	0
$17$	5.33230	1.29327	0.646637	−	0.762798i	$-0.276175\pi$
$17$	5.33230	1.29327	0.646637	+	0.762798i	$0.276175\pi$
$18$	0	0
$19$	−3.61925	−0.830312	−0.415156	−	0.909750i	$-0.636273\pi$
$19$	−3.61925	−0.830312	−0.415156	+	0.909750i	$0.636273\pi$
$20$	0	0
$21$	−0.589521	−0.128644
$22$	0	0
$23$	−2.60484	−0.543147	−0.271574	−	0.962418i	$-0.587544\pi$
$23$	−2.60484	−0.543147	−0.271574	+	0.962418i	$0.587544\pi$
$24$	0	0
$25$	−2.43855	−0.487710
$26$	0	0
$27$	3.33225	0.641291
$28$	0	0
$29$	1.72929	0.321121	0.160560	−	0.987026i	$-0.448670\pi$
$29$	1.72929	0.321121	0.160560	+	0.987026i	$0.448670\pi$
$30$	0	0
$31$	−0.833708	−0.149738	−0.0748692	−	0.997193i	$-0.523854\pi$
$31$	−0.833708	−0.149738	−0.0748692	+	0.997193i	$0.523854\pi$
$32$	0	0
$33$	3.21576	0.559792
$34$	0	0
$35$	−1.60045	−0.270526
$36$	0	0
$37$	−6.26450	−1.02988	−0.514939	−	0.857227i	$-0.672185\pi$
$37$	−6.26450	−1.02988	−0.514939	+	0.857227i	$0.672185\pi$
$38$	0	0
$39$	−3.88768	−0.622526
$40$	0	0
$41$	−0.263382	−0.0411333	−0.0205667	−	0.999788i	$-0.506547\pi$
$41$	−0.263382	−0.0411333	−0.0205667	+	0.999788i	$0.506547\pi$
$42$	0	0
$43$	−1.77107	−0.270085	−0.135043	−	0.990840i	$-0.543117\pi$
$43$	−1.77107	−0.270085	−0.135043	+	0.990840i	$0.543117\pi$
$44$	0	0
$45$	4.24514	0.632829
$46$	0	0
$47$	10.7559	1.56891	0.784455	−	0.620186i	$-0.212943\pi$
$47$	10.7559	1.56891	0.784455	+	0.620186i	$0.212943\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.14350	−0.440178
$52$	0	0
$53$	−0.0673478	−0.00925093	−0.00462546	−	0.999989i	$-0.501472\pi$
$53$	−0.0673478	−0.00925093	−0.00462546	+	0.999989i	$0.501472\pi$
$54$	0	0
$55$	8.73026	1.17719
$56$	0	0
$57$	2.13362	0.282605
$58$	0	0
$59$	−5.10081	−0.664069	−0.332034	−	0.943267i	$-0.607735\pi$
$59$	−5.10081	−0.664069	−0.332034	+	0.943267i	$0.607735\pi$
$60$	0	0
$61$	−6.31304	−0.808302	−0.404151	−	0.914692i	$-0.632433\pi$
$61$	−6.31304	−0.808302	−0.404151	+	0.914692i	$0.632433\pi$
$62$	0	0
$63$	−2.65247	−0.334179
$64$	0	0
$65$	−10.5544	−1.30911
$66$	0	0
$67$	−13.4487	−1.64302	−0.821508	−	0.570197i	$-0.806867\pi$
$67$	−13.4487	−1.64302	−0.821508	+	0.570197i	$0.806867\pi$
$68$	0	0
$69$	1.53561	0.184866
$70$	0	0
$71$	−2.05301	−0.243647	−0.121824	−	0.992552i	$-0.538874\pi$
$71$	−2.05301	−0.243647	−0.121824	+	0.992552i	$0.538874\pi$
$72$	0	0
$73$	−5.48268	−0.641700	−0.320850	−	0.947130i	$-0.603968\pi$
$73$	−5.48268	−0.641700	−0.320850	+	0.947130i	$0.603968\pi$
$74$	0	0
$75$	1.43758	0.165997
$76$	0	0
$77$	−5.45487	−0.621640
$78$	0	0
$79$	−5.21576	−0.586819	−0.293409	−	0.955987i	$-0.594790\pi$
$79$	−5.21576	−0.586819	−0.293409	+	0.955987i	$0.594790\pi$
$80$	0	0
$81$	5.99297	0.665885
$82$	0	0
$83$	8.25965	0.906614	0.453307	−	0.891354i	$-0.350244\pi$
$83$	8.25965	0.906614	0.453307	+	0.891354i	$0.350244\pi$
$84$	0	0
$85$	−8.53410	−0.925653
$86$	0	0
$87$	−1.01945	−0.109297
$88$	0	0
$89$	6.32651	0.670609	0.335304	−	0.942110i	$-0.391161\pi$
$89$	6.32651	0.670609	0.335304	+	0.942110i	$0.391161\pi$
$90$	0	0
$91$	6.59464	0.691306
$92$	0	0
$93$	0.491488	0.0509650
$94$	0	0
$95$	5.79243	0.594291
$96$	0	0
$97$	18.8089	1.90976	0.954878	−	0.296999i	$-0.0959858\pi$
$97$	18.8089	1.90976	0.954878	+	0.296999i	$0.0959858\pi$
$98$	0	0
$99$	14.4689	1.45417
$100$	0	0
$101$	−4.08033	−0.406008	−0.203004	−	0.979178i	$-0.565070\pi$
$101$	−4.08033	−0.406008	−0.203004	+	0.979178i	$0.565070\pi$
$102$	0	0
$103$	7.74040	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$103$	7.74040	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$104$	0	0
$105$	0.943500	0.0920762
$106$	0	0
$107$	3.30144	0.319162	0.159581	−	0.987185i	$-0.448986\pi$
$107$	3.30144	0.319162	0.159581	+	0.987185i	$0.448986\pi$
$108$	0	0
$109$	−6.96962	−0.667569	−0.333784	−	0.942649i	$-0.608326\pi$
$109$	−6.96962	−0.667569	−0.333784	+	0.942649i	$0.608326\pi$
$110$	0	0
$111$	3.69306	0.350529
$112$	0	0
$113$	−5.24381	−0.493296	−0.246648	−	0.969105i	$-0.579329\pi$
$113$	−5.24381	−0.493296	−0.246648	+	0.969105i	$0.579329\pi$
$114$	0	0
$115$	4.16893	0.388755
$116$	0	0
$117$	−17.4921	−1.61714
$118$	0	0
$119$	5.33230	0.488811
$120$	0	0
$121$	18.7556	1.70506
$122$	0	0
$123$	0.155269	0.0140001
$124$	0	0
$125$	11.9050	1.06482
$126$	0	0
$127$	−17.6789	−1.56875	−0.784373	−	0.620290i	$-0.787015\pi$
$127$	−17.6789	−1.56875	−0.784373	+	0.620290i	$0.787015\pi$
$128$	0	0
$129$	1.04408	0.0919262
$130$	0	0
$131$	3.08173	0.269252	0.134626	−	0.990896i	$-0.457017\pi$
$131$	3.08173	0.269252	0.134626	+	0.990896i	$0.457017\pi$
$132$	0	0
$133$	−3.61925	−0.313828
$134$	0	0
$135$	−5.33310	−0.459000
$136$	0	0
$137$	8.54650	0.730177	0.365088	−	0.930973i	$-0.381039\pi$
$137$	8.54650	0.730177	0.365088	+	0.930973i	$0.381039\pi$
$138$	0	0
$139$	8.09706	0.686784	0.343392	−	0.939192i	$-0.388424\pi$
$139$	8.09706	0.686784	0.343392	+	0.939192i	$0.388424\pi$
$140$	0	0
$141$	−6.34083	−0.533994
$142$	0	0
$143$	−35.9729	−3.00821
$144$	0	0
$145$	−2.76764	−0.229840
$146$	0	0
$147$	−0.589521	−0.0486229
$148$	0	0
$149$	−0.558836	−0.0457816	−0.0228908	−	0.999738i	$-0.507287\pi$
$149$	−0.558836	−0.0457816	−0.0228908	+	0.999738i	$0.507287\pi$
$150$	0	0
$151$	−3.71559	−0.302371	−0.151185	−	0.988505i	$-0.548309\pi$
$151$	−3.71559	−0.302371	−0.151185	+	0.988505i	$0.548309\pi$
$152$	0	0
$153$	−14.1437	−1.14345
$154$	0	0
$155$	1.33431	0.107174
$156$	0	0
$157$	−5.65037	−0.450949	−0.225474	−	0.974249i	$-0.572393\pi$
$157$	−5.65037	−0.450949	−0.225474	+	0.974249i	$0.572393\pi$
$158$	0	0
$159$	0.0397029	0.00314864
$160$	0	0
$161$	−2.60484	−0.205290
$162$	0	0
$163$	−4.76827	−0.373480	−0.186740	−	0.982409i	$-0.559792\pi$
$163$	−4.76827	−0.373480	−0.186740	+	0.982409i	$0.559792\pi$
$164$	0	0
$165$	−5.14667	−0.400668
$166$	0	0
$167$	12.4233	0.961345	0.480673	−	0.876900i	$-0.340393\pi$
$167$	12.4233	0.961345	0.480673	+	0.876900i	$0.340393\pi$
$168$	0	0
$169$	30.4893	2.34533
$170$	0	0
$171$	9.59993	0.734125
$172$	0	0
$173$	13.0943	0.995542	0.497771	−	0.867308i	$-0.334152\pi$
$173$	13.0943	0.995542	0.497771	+	0.867308i	$0.334152\pi$
$174$	0	0
$175$	−2.43855	−0.184337
$176$	0	0
$177$	3.00703	0.226022
$178$	0	0
$179$	−14.0788	−1.05230	−0.526150	−	0.850392i	$-0.676365\pi$
$179$	−14.0788	−1.05230	−0.526150	+	0.850392i	$0.676365\pi$
$180$	0	0
$181$	7.18598	0.534130	0.267065	−	0.963679i	$-0.413946\pi$
$181$	7.18598	0.534130	0.267065	+	0.963679i	$0.413946\pi$
$182$	0	0
$183$	3.72167	0.275114
$184$	0	0
$185$	10.0260	0.737129
$186$	0	0
$187$	−29.0870	−2.12705
$188$	0	0
$189$	3.33225	0.242385
$190$	0	0
$191$	20.7927	1.50451	0.752254	−	0.658873i	$-0.228967\pi$
$191$	20.7927	1.50451	0.752254	+	0.658873i	$0.228967\pi$
$192$	0	0
$193$	13.3447	0.960574	0.480287	−	0.877111i	$-0.340533\pi$
$193$	13.3447	0.960574	0.480287	+	0.877111i	$0.340533\pi$
$194$	0	0
$195$	6.22204	0.445570
$196$	0	0
$197$	0.275011	0.0195937	0.00979687	−	0.999952i	$-0.496882\pi$
$197$	0.275011	0.0195937	0.00979687	+	0.999952i	$0.496882\pi$
$198$	0	0
$199$	14.4003	1.02081	0.510403	−	0.859935i	$-0.329496\pi$
$199$	14.4003	1.02081	0.510403	+	0.859935i	$0.329496\pi$
$200$	0	0
$201$	7.92827	0.559217
$202$	0	0
$203$	1.72929	0.121372
$204$	0	0
$205$	0.421530	0.0294409
$206$	0	0
$207$	6.90926	0.480227
$208$	0	0
$209$	19.7425	1.36562
$210$	0	0
$211$	10.9228	0.751954	0.375977	−	0.926629i	$-0.377307\pi$
$211$	10.9228	0.751954	0.375977	+	0.926629i	$0.377307\pi$
$212$	0	0
$213$	1.21029	0.0829278
$214$	0	0
$215$	2.83451	0.193312
$216$	0	0
$217$	−0.833708	−0.0565958
$218$	0	0
$219$	3.23216	0.218409
$220$	0	0
$221$	35.1646	2.36543
$222$	0	0
$223$	−7.06285	−0.472963	−0.236482	−	0.971636i	$-0.575994\pi$
$223$	−7.06285	−0.472963	−0.236482	+	0.971636i	$0.575994\pi$
$224$	0	0
$225$	6.46817	0.431212
$226$	0	0
$227$	16.7200	1.10975	0.554874	−	0.831934i	$-0.312766\pi$
$227$	16.7200	1.10975	0.554874	+	0.831934i	$0.312766\pi$
$228$	0	0
$229$	−13.2627	−0.876425	−0.438212	−	0.898872i	$-0.644388\pi$
$229$	−13.2627	−0.876425	−0.438212	+	0.898872i	$0.644388\pi$
$230$	0	0
$231$	3.21576	0.211581
$232$	0	0
$233$	24.4385	1.60102	0.800511	−	0.599318i	$-0.204561\pi$
$233$	24.4385	1.60102	0.800511	+	0.599318i	$0.204561\pi$
$234$	0	0
$235$	−17.2143	−1.12294
$236$	0	0
$237$	3.07480	0.199730
$238$	0	0
$239$	6.27660	0.406000	0.203000	−	0.979179i	$-0.434931\pi$
$239$	6.27660	0.406000	0.203000	+	0.979179i	$0.434931\pi$
$240$	0	0
$241$	15.0124	0.967034	0.483517	−	0.875335i	$-0.339359\pi$
$241$	15.0124	0.967034	0.483517	+	0.875335i	$0.339359\pi$
$242$	0	0
$243$	−13.5297	−0.867932
$244$	0	0
$245$	−1.60045	−0.102249
$246$	0	0
$247$	−23.8676	−1.51866
$248$	0	0
$249$	−4.86923	−0.308575
$250$	0	0
$251$	16.6772	1.05265	0.526327	−	0.850282i	$-0.323569\pi$
$251$	16.6772	1.05265	0.526327	+	0.850282i	$0.323569\pi$
$252$	0	0
$253$	14.2091	0.893317
$254$	0	0
$255$	5.03103	0.315055
$256$	0	0
$257$	20.3977	1.27237	0.636186	−	0.771536i	$-0.280511\pi$
$257$	20.3977	1.27237	0.636186	+	0.771536i	$0.280511\pi$
$258$	0	0
$259$	−6.26450	−0.389257
$260$	0	0
$261$	−4.58688	−0.283921
$262$	0	0
$263$	−23.3452	−1.43953	−0.719764	−	0.694219i	$-0.755750\pi$
$263$	−23.3452	−1.43953	−0.719764	+	0.694219i	$0.755750\pi$
$264$	0	0
$265$	0.107787	0.00662130
$266$	0	0
$267$	−3.72961	−0.228248
$268$	0	0
$269$	31.2124	1.90306	0.951528	−	0.307563i	$-0.0995134\pi$
$269$	31.2124	1.90306	0.951528	+	0.307563i	$0.0995134\pi$
$270$	0	0
$271$	−1.46392	−0.0889266	−0.0444633	−	0.999011i	$-0.514158\pi$
$271$	−1.46392	−0.0889266	−0.0444633	+	0.999011i	$0.514158\pi$
$272$	0	0
$273$	−3.88768	−0.235293
$274$	0	0
$275$	13.3020	0.802140
$276$	0	0
$277$	1.96907	0.118310	0.0591550	−	0.998249i	$-0.481159\pi$
$277$	1.96907	0.118310	0.0591550	+	0.998249i	$0.481159\pi$
$278$	0	0
$279$	2.21138	0.132392
$280$	0	0
$281$	5.66742	0.338090	0.169045	−	0.985608i	$-0.445932\pi$
$281$	5.66742	0.338090	0.169045	+	0.985608i	$0.445932\pi$
$282$	0	0
$283$	−23.5629	−1.40067	−0.700336	−	0.713814i	$-0.746966\pi$
$283$	−23.5629	−1.40067	−0.700336	+	0.713814i	$0.746966\pi$
$284$	0	0
$285$	−3.41476	−0.202273
$286$	0	0
$287$	−0.263382	−0.0155469
$288$	0	0
$289$	11.4334	0.672555
$290$	0	0
$291$	−11.0882	−0.650004
$292$	0	0
$293$	24.2001	1.41379	0.706893	−	0.707320i	$-0.250096\pi$
$293$	24.2001	1.41379	0.706893	+	0.707320i	$0.250096\pi$
$294$	0	0
$295$	8.16360	0.475303
$296$	0	0
$297$	−18.1770	−1.05473
$298$	0	0
$299$	−17.1780	−0.993429
$300$	0	0
$301$	−1.77107	−0.102083
$302$	0	0
$303$	2.40544	0.138189
$304$	0	0
$305$	10.1037	0.578537
$306$	0	0
$307$	−13.5741	−0.774717	−0.387359	−	0.921929i	$-0.626612\pi$
$307$	−13.5741	−0.774717	−0.387359	+	0.921929i	$0.626612\pi$
$308$	0	0
$309$	−4.56312	−0.259587
$310$	0	0
$311$	−19.1866	−1.08797	−0.543987	−	0.839094i	$-0.683086\pi$
$311$	−19.1866	−1.08797	−0.543987	+	0.839094i	$0.683086\pi$
$312$	0	0
$313$	5.03963	0.284857	0.142428	−	0.989805i	$-0.454509\pi$
$313$	5.03963	0.284857	0.142428	+	0.989805i	$0.454509\pi$
$314$	0	0
$315$	4.24514	0.239187
$316$	0	0
$317$	−7.81489	−0.438928	−0.219464	−	0.975621i	$-0.570431\pi$
$317$	−7.81489	−0.438928	−0.219464	+	0.975621i	$0.570431\pi$
$318$	0	0
$319$	−9.43305	−0.528149
$320$	0	0
$321$	−1.94627	−0.108630
$322$	0	0
$323$	−19.2989	−1.07382
$324$	0	0
$325$	−16.0814	−0.892034
$326$	0	0
$327$	4.10874	0.227214
$328$	0	0
$329$	10.7559	0.592992
$330$	0	0
$331$	20.8617	1.14666	0.573331	−	0.819324i	$-0.305651\pi$
$331$	20.8617	1.14666	0.573331	+	0.819324i	$0.305651\pi$
$332$	0	0
$333$	16.6164	0.910572
$334$	0	0
$335$	21.5239	1.17598
$336$	0	0
$337$	−16.7111	−0.910311	−0.455156	−	0.890412i	$-0.650416\pi$
$337$	−16.7111	−0.910311	−0.455156	+	0.890412i	$0.650416\pi$
$338$	0	0
$339$	3.09133	0.167898
$340$	0	0
$341$	4.54777	0.246276
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.45767	−0.132316
$346$	0	0
$347$	7.89025	0.423571	0.211785	−	0.977316i	$-0.432072\pi$
$347$	7.89025	0.423571	0.211785	+	0.977316i	$0.432072\pi$
$348$	0	0
$349$	25.4260	1.36102	0.680511	−	0.732738i	$-0.261758\pi$
$349$	25.4260	1.36102	0.680511	+	0.732738i	$0.261758\pi$
$350$	0	0
$351$	21.9750	1.17294
$352$	0	0
$353$	−25.0318	−1.33231	−0.666155	−	0.745814i	$-0.732061\pi$
$353$	−25.0318	−1.33231	−0.666155	+	0.745814i	$0.732061\pi$
$354$	0	0
$355$	3.28574	0.174389
$356$	0	0
$357$	−3.14350	−0.166372
$358$	0	0
$359$	−17.9910	−0.949526	−0.474763	−	0.880114i	$-0.657466\pi$
$359$	−17.9910	−0.949526	−0.474763	+	0.880114i	$0.657466\pi$
$360$	0	0
$361$	−5.90105	−0.310582
$362$	0	0
$363$	−11.0568	−0.580333
$364$	0	0
$365$	8.77477	0.459293
$366$	0	0
$367$	−10.3077	−0.538060	−0.269030	−	0.963132i	$-0.586703\pi$
$367$	−10.3077	−0.538060	−0.269030	+	0.963132i	$0.586703\pi$
$368$	0	0
$369$	0.698611	0.0363682
$370$	0	0
$371$	−0.0673478	−0.00349652
$372$	0	0
$373$	−22.7060	−1.17567	−0.587835	−	0.808981i	$-0.700019\pi$
$373$	−22.7060	−1.17567	−0.587835	+	0.808981i	$0.700019\pi$
$374$	0	0
$375$	−7.01827	−0.362422
$376$	0	0
$377$	11.4040	0.587338
$378$	0	0
$379$	22.3335	1.14720	0.573598	−	0.819137i	$-0.305547\pi$
$379$	22.3335	1.14720	0.573598	+	0.819137i	$0.305547\pi$
$380$	0	0
$381$	10.4221	0.533938
$382$	0	0
$383$	18.3633	0.938319	0.469160	−	0.883113i	$-0.344557\pi$
$383$	18.3633	0.938319	0.469160	+	0.883113i	$0.344557\pi$
$384$	0	0
$385$	8.73026	0.444935
$386$	0	0
$387$	4.69769	0.238797
$388$	0	0
$389$	19.7359	1.00065	0.500324	−	0.865838i	$-0.333214\pi$
$389$	19.7359	1.00065	0.500324	+	0.865838i	$0.333214\pi$
$390$	0	0
$391$	−13.8898	−0.702438
$392$	0	0
$393$	−1.81674	−0.0916426
$394$	0	0
$395$	8.34758	0.420012
$396$	0	0
$397$	−1.59544	−0.0800730	−0.0400365	−	0.999198i	$-0.512747\pi$
$397$	−1.59544	−0.0800730	−0.0400365	+	0.999198i	$0.512747\pi$
$398$	0	0
$399$	2.13362	0.106815
$400$	0	0
$401$	14.3470	0.716454	0.358227	−	0.933635i	$-0.383381\pi$
$401$	14.3470	0.716454	0.358227	+	0.933635i	$0.383381\pi$
$402$	0	0
$403$	−5.49800	−0.273875
$404$	0	0
$405$	−9.59146	−0.476604
$406$	0	0
$407$	34.1721	1.69385
$408$	0	0
$409$	39.2518	1.94088	0.970438	−	0.241350i	$-0.0775901\pi$
$409$	39.2518	1.94088	0.970438	+	0.241350i	$0.0775901\pi$
$410$	0	0
$411$	−5.03834	−0.248523
$412$	0	0
$413$	−5.10081	−0.250994
$414$	0	0
$415$	−13.2192	−0.648904
$416$	0	0
$417$	−4.77338	−0.233754
$418$	0	0
$419$	6.24337	0.305008	0.152504	−	0.988303i	$-0.451266\pi$
$419$	6.24337	0.305008	0.152504	+	0.988303i	$0.451266\pi$
$420$	0	0
$421$	−10.7126	−0.522099	−0.261050	−	0.965325i	$-0.584069\pi$
$421$	−10.7126	−0.522099	−0.261050	+	0.965325i	$0.584069\pi$
$422$	0	0
$423$	−28.5297	−1.38716
$424$	0	0
$425$	−13.0031	−0.630743
$426$	0	0
$427$	−6.31304	−0.305509
$428$	0	0
$429$	21.2068	1.02387
$430$	0	0
$431$	18.5396	0.893020	0.446510	−	0.894779i	$-0.352667\pi$
$431$	18.5396	0.893020	0.446510	+	0.894779i	$0.352667\pi$
$432$	0	0
$433$	−7.21190	−0.346582	−0.173291	−	0.984871i	$-0.555440\pi$
$433$	−7.21190	−0.346582	−0.173291	+	0.984871i	$0.555440\pi$
$434$	0	0
$435$	1.63158	0.0782285
$436$	0	0
$437$	9.42757	0.450982
$438$	0	0
$439$	15.1615	0.723617	0.361809	−	0.932252i	$-0.382159\pi$
$439$	15.1615	0.723617	0.361809	+	0.932252i	$0.382159\pi$
$440$	0	0
$441$	−2.65247	−0.126308
$442$	0	0
$443$	−2.99454	−0.142275	−0.0711374	−	0.997467i	$-0.522663\pi$
$443$	−2.99454	−0.142275	−0.0711374	+	0.997467i	$0.522663\pi$
$444$	0	0
$445$	−10.1253	−0.479984
$446$	0	0
$447$	0.329445	0.0155822
$448$	0	0
$449$	4.29509	0.202698	0.101349	−	0.994851i	$-0.467684\pi$
$449$	4.29509	0.202698	0.101349	+	0.994851i	$0.467684\pi$
$450$	0	0
$451$	1.43671	0.0676522
$452$	0	0
$453$	2.19042	0.102915
$454$	0	0
$455$	−10.5544	−0.494798
$456$	0	0
$457$	−27.7833	−1.29965	−0.649823	−	0.760085i	$-0.725157\pi$
$457$	−27.7833	−1.29965	−0.649823	+	0.760085i	$0.725157\pi$
$458$	0	0
$459$	17.7685	0.829364
$460$	0	0
$461$	−9.20528	−0.428733	−0.214366	−	0.976753i	$-0.568769\pi$
$461$	−9.20528	−0.428733	−0.214366	+	0.976753i	$0.568769\pi$
$462$	0	0
$463$	39.1018	1.81722	0.908608	−	0.417650i	$-0.137146\pi$
$463$	39.1018	1.81722	0.908608	+	0.417650i	$0.137146\pi$
$464$	0	0
$465$	−0.786604	−0.0364779
$466$	0	0
$467$	23.2805	1.07729	0.538647	−	0.842531i	$-0.318936\pi$
$467$	23.2805	1.07729	0.538647	+	0.842531i	$0.318936\pi$
$468$	0	0
$469$	−13.4487	−0.621002
$470$	0	0
$471$	3.33101	0.153485
$472$	0	0
$473$	9.66094	0.444210
$474$	0	0
$475$	8.82572	0.404952
$476$	0	0
$477$	0.178638	0.00817925
$478$	0	0
$479$	−37.2565	−1.70229	−0.851145	−	0.524930i	$-0.824092\pi$
$479$	−37.2565	−1.70229	−0.851145	+	0.524930i	$0.824092\pi$
$480$	0	0
$481$	−41.3121	−1.88367
$482$	0	0
$483$	1.53561	0.0698726
$484$	0	0
$485$	−30.1028	−1.36690
$486$	0	0
$487$	38.4219	1.74106	0.870531	−	0.492113i	$-0.163775\pi$
$487$	38.4219	1.74106	0.870531	+	0.492113i	$0.163775\pi$
$488$	0	0
$489$	2.81100	0.127118
$490$	0	0
$491$	−13.4680	−0.607802	−0.303901	−	0.952704i	$-0.598289\pi$
$491$	−13.4680	−0.607802	−0.303901	+	0.952704i	$0.598289\pi$
$492$	0	0
$493$	9.22109	0.415297
$494$	0	0
$495$	−23.1567	−1.04082
$496$	0	0
$497$	−2.05301	−0.0920901
$498$	0	0
$499$	13.8337	0.619281	0.309641	−	0.950854i	$-0.399791\pi$
$499$	13.8337	0.619281	0.309641	+	0.950854i	$0.399791\pi$
$500$	0	0
$501$	−7.32380	−0.327203
$502$	0	0
$503$	−11.0554	−0.492938	−0.246469	−	0.969151i	$-0.579270\pi$
$503$	−11.0554	−0.492938	−0.246469	+	0.969151i	$0.579270\pi$
$504$	0	0
$505$	6.53038	0.290598
$506$	0	0
$507$	−17.9741	−0.798256
$508$	0	0
$509$	0.244998	0.0108593	0.00542967	−	0.999985i	$-0.498272\pi$
$509$	0.244998	0.0108593	0.00542967	+	0.999985i	$0.498272\pi$
$510$	0	0
$511$	−5.48268	−0.242540
$512$	0	0
$513$	−12.0602	−0.532472
$514$	0	0
$515$	−12.3881	−0.545886
$516$	0	0
$517$	−58.6721	−2.58039
$518$	0	0
$519$	−7.71937	−0.338843
$520$	0	0
$521$	29.9861	1.31371	0.656857	−	0.754015i	$-0.271885\pi$
$521$	29.9861	1.31371	0.656857	+	0.754015i	$0.271885\pi$
$522$	0	0
$523$	−2.10620	−0.0920976	−0.0460488	−	0.998939i	$-0.514663\pi$
$523$	−2.10620	−0.0920976	−0.0460488	+	0.998939i	$0.514663\pi$
$524$	0	0
$525$	1.43758	0.0627410
$526$	0	0
$527$	−4.44558	−0.193653
$528$	0	0
$529$	−16.2148	−0.704991
$530$	0	0
$531$	13.5297	0.587140
$532$	0	0
$533$	−1.73691	−0.0752338
$534$	0	0
$535$	−5.28380	−0.228439
$536$	0	0
$537$	8.29975	0.358161
$538$	0	0
$539$	−5.45487	−0.234958
$540$	0	0
$541$	−17.0889	−0.734711	−0.367355	−	0.930081i	$-0.619737\pi$
$541$	−17.0889	−0.734711	−0.367355	+	0.930081i	$0.619737\pi$
$542$	0	0
$543$	−4.23628	−0.181796
$544$	0	0
$545$	11.1545	0.477808
$546$	0	0
$547$	−17.3471	−0.741710	−0.370855	−	0.928691i	$-0.620935\pi$
$547$	−17.3471	−0.741710	−0.370855	+	0.928691i	$0.620935\pi$
$548$	0	0
$549$	16.7451	0.714664
$550$	0	0
$551$	−6.25872	−0.266631
$552$	0	0
$553$	−5.21576	−0.221797
$554$	0	0
$555$	−5.91056	−0.250889
$556$	0	0
$557$	−3.29115	−0.139451	−0.0697253	−	0.997566i	$-0.522212\pi$
$557$	−3.29115	−0.139451	−0.0697253	+	0.997566i	$0.522212\pi$
$558$	0	0
$559$	−11.6795	−0.493992
$560$	0	0
$561$	17.1474	0.723964
$562$	0	0
$563$	−17.0513	−0.718627	−0.359314	−	0.933217i	$-0.616989\pi$
$563$	−17.0513	−0.718627	−0.359314	+	0.933217i	$0.616989\pi$
$564$	0	0
$565$	8.39246	0.353074
$566$	0	0
$567$	5.99297	0.251681
$568$	0	0
$569$	7.99770	0.335281	0.167641	−	0.985848i	$-0.446385\pi$
$569$	7.99770	0.335281	0.167641	+	0.985848i	$0.446385\pi$
$570$	0	0
$571$	31.1857	1.30508	0.652541	−	0.757754i	$-0.273703\pi$
$571$	31.1857	1.30508	0.652541	+	0.757754i	$0.273703\pi$
$572$	0	0
$573$	−12.2577	−0.512074
$574$	0	0
$575$	6.35204	0.264899
$576$	0	0
$577$	43.4199	1.80760	0.903798	−	0.427960i	$-0.140768\pi$
$577$	43.4199	1.80760	0.903798	+	0.427960i	$0.140768\pi$
$578$	0	0
$579$	−7.86699	−0.326941
$580$	0	0
$581$	8.25965	0.342668
$582$	0	0
$583$	0.367373	0.0152150
$584$	0	0
$585$	27.9952	1.15746
$586$	0	0
$587$	25.7775	1.06395	0.531975	−	0.846760i	$-0.321450\pi$
$587$	25.7775	1.06395	0.531975	+	0.846760i	$0.321450\pi$
$588$	0	0
$589$	3.01740	0.124330
$590$	0	0
$591$	−0.162125	−0.00666892
$592$	0	0
$593$	39.7514	1.63239	0.816197	−	0.577773i	$-0.196078\pi$
$593$	39.7514	1.63239	0.816197	+	0.577773i	$0.196078\pi$
$594$	0	0
$595$	−8.53410	−0.349864
$596$	0	0
$597$	−8.48925	−0.347442
$598$	0	0
$599$	37.5296	1.53342	0.766708	−	0.641996i	$-0.221893\pi$
$599$	37.5296	1.53342	0.766708	+	0.641996i	$0.221893\pi$
$600$	0	0
$601$	−3.99899	−0.163122	−0.0815611	−	0.996668i	$-0.525991\pi$
$601$	−3.99899	−0.163122	−0.0815611	+	0.996668i	$0.525991\pi$
$602$	0	0
$603$	35.6721	1.45268
$604$	0	0
$605$	−30.0175	−1.22038
$606$	0	0
$607$	−24.3672	−0.989035	−0.494517	−	0.869168i	$-0.664655\pi$
$607$	−24.3672	−0.989035	−0.494517	+	0.869168i	$0.664655\pi$
$608$	0	0
$609$	−1.01945	−0.0413103
$610$	0	0
$611$	70.9313	2.86957
$612$	0	0
$613$	−23.6409	−0.954848	−0.477424	−	0.878673i	$-0.658429\pi$
$613$	−23.6409	−0.954848	−0.477424	+	0.878673i	$0.658429\pi$
$614$	0	0
$615$	−0.248501	−0.0100205
$616$	0	0
$617$	2.64202	0.106364	0.0531819	−	0.998585i	$-0.483064\pi$
$617$	2.64202	0.106364	0.0531819	+	0.998585i	$0.483064\pi$
$618$	0	0
$619$	−38.1516	−1.53344	−0.766721	−	0.641981i	$-0.778113\pi$
$619$	−38.1516	−1.53344	−0.766721	+	0.641981i	$0.778113\pi$
$620$	0	0
$621$	−8.67998	−0.348315
$622$	0	0
$623$	6.32651	0.253466
$624$	0	0
$625$	−6.86071	−0.274428
$626$	0	0
$627$	−11.6386	−0.464802
$628$	0	0
$629$	−33.4042	−1.33191
$630$	0	0
$631$	−18.9710	−0.755223	−0.377611	−	0.925964i	$-0.623254\pi$
$631$	−18.9710	−0.755223	−0.377611	+	0.925964i	$0.623254\pi$
$632$	0	0
$633$	−6.43919	−0.255935
$634$	0	0
$635$	28.2942	1.12282
$636$	0	0
$637$	6.59464	0.261289
$638$	0	0
$639$	5.44554	0.215422
$640$	0	0
$641$	31.3762	1.23929	0.619643	−	0.784884i	$-0.287277\pi$
$641$	31.3762	1.23929	0.619643	+	0.784884i	$0.287277\pi$
$642$	0	0
$643$	−19.1895	−0.756760	−0.378380	−	0.925650i	$-0.623519\pi$
$643$	−19.1895	−0.756760	−0.378380	+	0.925650i	$0.623519\pi$
$644$	0	0
$645$	−1.67100	−0.0657956
$646$	0	0
$647$	39.6587	1.55915	0.779573	−	0.626312i	$-0.215436\pi$
$647$	39.6587	1.55915	0.779573	+	0.626312i	$0.215436\pi$
$648$	0	0
$649$	27.8242	1.09220
$650$	0	0
$651$	0.491488	0.0192629
$652$	0	0
$653$	4.19547	0.164181	0.0820907	−	0.996625i	$-0.473840\pi$
$653$	4.19547	0.164181	0.0820907	+	0.996625i	$0.473840\pi$
$654$	0	0
$655$	−4.93216	−0.192715
$656$	0	0
$657$	14.5426	0.567362
$658$	0	0
$659$	−27.5882	−1.07468	−0.537341	−	0.843365i	$-0.680571\pi$
$659$	−27.5882	−1.07468	−0.537341	+	0.843365i	$0.680571\pi$
$660$	0	0
$661$	−16.9340	−0.658655	−0.329327	−	0.944216i	$-0.606822\pi$
$661$	−16.9340	−0.658655	−0.329327	+	0.944216i	$0.606822\pi$
$662$	0	0
$663$	−20.7303	−0.805097
$664$	0	0
$665$	5.79243	0.224621
$666$	0	0
$667$	−4.50453	−0.174416
$668$	0	0
$669$	4.16369	0.160978
$670$	0	0
$671$	34.4368	1.32942
$672$	0	0
$673$	−30.9400	−1.19265	−0.596324	−	0.802744i	$-0.703373\pi$
$673$	−30.9400	−1.19265	−0.596324	+	0.802744i	$0.703373\pi$
$674$	0	0
$675$	−8.12585	−0.312764
$676$	0	0
$677$	−0.0174449	−0.000670463 0	−0.000335231	−	1.00000i	$-0.500107\pi$
$677$	−0.0174449	−0.000670463 0	−0.000335231		1.00000i	$0.500107\pi$
$678$	0	0
$679$	18.8089	0.721820
$680$	0	0
$681$	−9.85681	−0.377714
$682$	0	0
$683$	19.9899	0.764892	0.382446	−	0.923978i	$-0.375082\pi$
$683$	19.9899	0.764892	0.382446	+	0.923978i	$0.375082\pi$
$684$	0	0
$685$	−13.6783	−0.522620
$686$	0	0
$687$	7.81864	0.298300
$688$	0	0
$689$	−0.444134	−0.0169202
$690$	0	0
$691$	44.5980	1.69659	0.848293	−	0.529527i	$-0.177631\pi$
$691$	44.5980	1.69659	0.848293	+	0.529527i	$0.177631\pi$
$692$	0	0
$693$	14.4689	0.549626
$694$	0	0
$695$	−12.9590	−0.491561
$696$	0	0
$697$	−1.40443	−0.0531966
$698$	0	0
$699$	−14.4070	−0.544924
$700$	0	0
$701$	23.8513	0.900851	0.450425	−	0.892814i	$-0.351272\pi$
$701$	23.8513	0.900851	0.450425	+	0.892814i	$0.351272\pi$
$702$	0	0
$703$	22.6728	0.855120
$704$	0	0
$705$	10.1482	0.382203
$706$	0	0
$707$	−4.08033	−0.153457
$708$	0	0
$709$	30.5955	1.14904	0.574519	−	0.818491i	$-0.305189\pi$
$709$	30.5955	1.14904	0.574519	+	0.818491i	$0.305189\pi$
$710$	0	0
$711$	13.8346	0.518839
$712$	0	0
$713$	2.17168	0.0813300
$714$	0	0
$715$	57.5729	2.15311
$716$	0	0
$717$	−3.70019	−0.138186
$718$	0	0
$719$	−40.3698	−1.50554	−0.752769	−	0.658284i	$-0.771283\pi$
$719$	−40.3698	−1.50554	−0.752769	+	0.658284i	$0.771283\pi$
$720$	0	0
$721$	7.74040	0.288267
$722$	0	0
$723$	−8.85013	−0.329140
$724$	0	0
$725$	−4.21696	−0.156614
$726$	0	0
$727$	−3.43634	−0.127447	−0.0637234	−	0.997968i	$-0.520298\pi$
$727$	−3.43634	−0.127447	−0.0637234	+	0.997968i	$0.520298\pi$
$728$	0	0
$729$	−10.0029	−0.370476
$730$	0	0
$731$	−9.44386	−0.349294
$732$	0	0
$733$	38.6636	1.42807	0.714037	−	0.700108i	$-0.246865\pi$
$733$	38.6636	1.42807	0.714037	+	0.700108i	$0.246865\pi$
$734$	0	0
$735$	0.943500	0.0348015
$736$	0	0
$737$	73.3607	2.70228
$738$	0	0
$739$	40.1059	1.47532	0.737659	−	0.675173i	$-0.235931\pi$
$739$	40.1059	1.47532	0.737659	+	0.675173i	$0.235931\pi$
$740$	0	0
$741$	14.0705	0.516891
$742$	0	0
$743$	34.1733	1.25370	0.626848	−	0.779141i	$-0.284344\pi$
$743$	34.1733	1.25370	0.626848	+	0.779141i	$0.284344\pi$
$744$	0	0
$745$	0.894391	0.0327679
$746$	0	0
$747$	−21.9084	−0.801588
$748$	0	0
$749$	3.30144	0.120632
$750$	0	0
$751$	−8.55791	−0.312282	−0.156141	−	0.987735i	$-0.549905\pi$
$751$	−8.55791	−0.312282	−0.156141	+	0.987735i	$0.549905\pi$
$752$	0	0
$753$	−9.83155	−0.358281
$754$	0	0
$755$	5.94663	0.216420
$756$	0	0
$757$	35.4249	1.28754	0.643770	−	0.765219i	$-0.277369\pi$
$757$	35.4249	1.28754	0.643770	+	0.765219i	$0.277369\pi$
$758$	0	0
$759$	−8.37655	−0.304050
$760$	0	0
$761$	−28.4224	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$761$	−28.4224	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$762$	0	0
$763$	−6.96962	−0.252317
$764$	0	0
$765$	22.6364	0.818420
$766$	0	0
$767$	−33.6380	−1.21460
$768$	0	0
$769$	−12.0189	−0.433413	−0.216707	−	0.976237i	$-0.569531\pi$
$769$	−12.0189	−0.433413	−0.216707	+	0.976237i	$0.569531\pi$
$770$	0	0
$771$	−12.0249	−0.433065
$772$	0	0
$773$	7.16883	0.257845	0.128922	−	0.991655i	$-0.458848\pi$
$773$	7.16883	0.257845	0.128922	+	0.991655i	$0.458848\pi$
$774$	0	0
$775$	2.03304	0.0730290
$776$	0	0
$777$	3.69306	0.132488
$778$	0	0
$779$	0.953244	0.0341535
$780$	0	0
$781$	11.1989	0.400728
$782$	0	0
$783$	5.76241	0.205932
$784$	0	0
$785$	9.04315	0.322764
$786$	0	0
$787$	23.9900	0.855152	0.427576	−	0.903979i	$-0.359368\pi$
$787$	23.9900	0.855152	0.427576	+	0.903979i	$0.359368\pi$
$788$	0	0
$789$	13.7625	0.489958
$790$	0	0
$791$	−5.24381	−0.186448
$792$	0	0
$793$	−41.6322	−1.47840
$794$	0	0
$795$	−0.0635426	−0.00225362
$796$	0	0
$797$	−50.2908	−1.78139	−0.890695	−	0.454601i	$-0.849782\pi$
$797$	−50.2908	−1.78139	−0.890695	+	0.454601i	$0.849782\pi$
$798$	0	0
$799$	57.3537	2.02903
$800$	0	0
$801$	−16.7808	−0.592922
$802$	0	0
$803$	29.9073	1.05541
$804$	0	0
$805$	4.16893	0.146935
$806$	0	0
$807$	−18.4004	−0.647724
$808$	0	0
$809$	11.8621	0.417050	0.208525	−	0.978017i	$-0.433134\pi$
$809$	11.8621	0.417050	0.208525	+	0.978017i	$0.433134\pi$
$810$	0	0
$811$	−28.9714	−1.01732	−0.508662	−	0.860966i	$-0.669860\pi$
$811$	−28.9714	−1.01732	−0.508662	+	0.860966i	$0.669860\pi$
$812$	0	0
$813$	0.863010	0.0302671
$814$	0	0
$815$	7.63140	0.267316
$816$	0	0
$817$	6.40993	0.224255
$818$	0	0
$819$	−17.4921	−0.611222
$820$	0	0
$821$	41.1122	1.43483	0.717413	−	0.696648i	$-0.245326\pi$
$821$	41.1122	1.43483	0.717413	+	0.696648i	$0.245326\pi$
$822$	0	0
$823$	51.3595	1.79028	0.895140	−	0.445785i	$-0.147075\pi$
$823$	51.3595	1.79028	0.895140	+	0.445785i	$0.147075\pi$
$824$	0	0
$825$	−7.84180	−0.273016
$826$	0	0
$827$	−34.5654	−1.20196	−0.600979	−	0.799265i	$-0.705222\pi$
$827$	−34.5654	−1.20196	−0.600979	+	0.799265i	$0.705222\pi$
$828$	0	0
$829$	16.1023	0.559257	0.279629	−	0.960108i	$-0.409789\pi$
$829$	16.1023	0.559257	0.279629	+	0.960108i	$0.409789\pi$
$830$	0	0
$831$	−1.16081	−0.0402680
$832$	0	0
$833$	5.33230	0.184753
$834$	0	0
$835$	−19.8829	−0.688077
$836$	0	0
$837$	−2.77812	−0.0960259
$838$	0	0
$839$	−35.2906	−1.21837	−0.609184	−	0.793029i	$-0.708503\pi$
$839$	−35.2906	−1.21837	−0.609184	+	0.793029i	$0.708503\pi$
$840$	0	0
$841$	−26.0096	−0.896881
$842$	0	0
$843$	−3.34106	−0.115072
$844$	0	0
$845$	−48.7966	−1.67865
$846$	0	0
$847$	18.7556	0.644451
$848$	0	0
$849$	13.8908	0.476732
$850$	0	0
$851$	16.3181	0.559376
$852$	0	0
$853$	9.80684	0.335780	0.167890	−	0.985806i	$-0.446305\pi$
$853$	9.80684	0.335780	0.167890	+	0.985806i	$0.446305\pi$
$854$	0	0
$855$	−15.3642	−0.525445
$856$	0	0
$857$	−29.1791	−0.996737	−0.498369	−	0.866965i	$-0.666067\pi$
$857$	−29.1791	−0.996737	−0.498369	+	0.866965i	$0.666067\pi$
$858$	0	0
$859$	−2.29972	−0.0784653	−0.0392326	−	0.999230i	$-0.512491\pi$
$859$	−2.29972	−0.0784653	−0.0392326	+	0.999230i	$0.512491\pi$
$860$	0	0
$861$	0.155269	0.00529155
$862$	0	0
$863$	−33.7059	−1.14736	−0.573681	−	0.819079i	$-0.694485\pi$
$863$	−33.7059	−1.14736	−0.573681	+	0.819079i	$0.694485\pi$
$864$	0	0
$865$	−20.9568	−0.712554
$866$	0	0
$867$	−6.74025	−0.228911
$868$	0	0
$869$	28.4513	0.965144
$870$	0	0
$871$	−88.6891	−3.00512
$872$	0	0
$873$	−49.8900	−1.68852
$874$	0	0
$875$	11.9050	0.402464
$876$	0	0
$877$	−35.8993	−1.21223	−0.606116	−	0.795376i	$-0.707273\pi$
$877$	−35.8993	−1.21223	−0.606116	+	0.795376i	$0.707273\pi$
$878$	0	0
$879$	−14.2665	−0.481196
$880$	0	0
$881$	−13.0482	−0.439606	−0.219803	−	0.975544i	$-0.570541\pi$
$881$	−13.0482	−0.439606	−0.219803	+	0.975544i	$0.570541\pi$
$882$	0	0
$883$	−34.3505	−1.15599	−0.577994	−	0.816041i	$-0.696164\pi$
$883$	−34.3505	−1.15599	−0.577994	+	0.816041i	$0.696164\pi$
$884$	0	0
$885$	−4.81261	−0.161774
$886$	0	0
$887$	−19.8489	−0.666461	−0.333230	−	0.942845i	$-0.608139\pi$
$887$	−19.8489	−0.666461	−0.333230	+	0.942845i	$0.608139\pi$
$888$	0	0
$889$	−17.6789	−0.592930
$890$	0	0
$891$	−32.6909	−1.09519
$892$	0	0
$893$	−38.9283	−1.30268
$894$	0	0
$895$	22.5325	0.753178
$896$	0	0
$897$	10.1268	0.338124
$898$	0	0
$899$	−1.44172	−0.0480841
$900$	0	0
$901$	−0.359119	−0.0119640
$902$	0	0
$903$	1.04408	0.0347448
$904$	0	0
$905$	−11.5008	−0.382300
$906$	0	0
$907$	56.0335	1.86056	0.930281	−	0.366848i	$-0.119563\pi$
$907$	56.0335	1.86056	0.930281	+	0.366848i	$0.119563\pi$
$908$	0	0
$909$	10.8229	0.358974
$910$	0	0
$911$	−11.6260	−0.385188	−0.192594	−	0.981279i	$-0.561690\pi$
$911$	−11.6260	−0.385188	−0.192594	+	0.981279i	$0.561690\pi$
$912$	0	0
$913$	−45.0553	−1.49111
$914$	0	0
$915$	−5.95635	−0.196911
$916$	0	0
$917$	3.08173	0.101768
$918$	0	0
$919$	−42.1056	−1.38894	−0.694468	−	0.719523i	$-0.744360\pi$
$919$	−42.1056	−1.38894	−0.694468	+	0.719523i	$0.744360\pi$
$920$	0	0
$921$	8.00224	0.263683
$922$	0	0
$923$	−13.5389	−0.445637
$924$	0	0
$925$	15.2763	0.502282
$926$	0	0
$927$	−20.5311	−0.674331
$928$	0	0
$929$	30.3239	0.994894	0.497447	−	0.867494i	$-0.334271\pi$
$929$	30.3239	0.994894	0.497447	+	0.867494i	$0.334271\pi$
$930$	0	0
$931$	−3.61925	−0.118616
$932$	0	0
$933$	11.3109	0.370303
$934$	0	0
$935$	46.5524	1.52243
$936$	0	0
$937$	17.7772	0.580757	0.290378	−	0.956912i	$-0.406219\pi$
$937$	17.7772	0.580757	0.290378	+	0.956912i	$0.406219\pi$
$938$	0	0
$939$	−2.97097	−0.0969538
$940$	0	0
$941$	−29.9242	−0.975500	−0.487750	−	0.872983i	$-0.662182\pi$
$941$	−29.9242	−0.975500	−0.487750	+	0.872983i	$0.662182\pi$
$942$	0	0
$943$	0.686068	0.0223415
$944$	0	0
$945$	−5.33310	−0.173486
$946$	0	0
$947$	16.7581	0.544565	0.272283	−	0.962217i	$-0.412221\pi$
$947$	16.7581	0.544565	0.272283	+	0.962217i	$0.412221\pi$
$948$	0	0
$949$	−36.1563	−1.17368
$950$	0	0
$951$	4.60704	0.149394
$952$	0	0
$953$	14.1855	0.459513	0.229757	−	0.973248i	$-0.426207\pi$
$953$	14.1855	0.459513	0.229757	+	0.973248i	$0.426207\pi$
$954$	0	0
$955$	−33.2778	−1.07684
$956$	0	0
$957$	5.56098	0.179761
$958$	0	0
$959$	8.54650	0.275981
$960$	0	0
$961$	−30.3049	−0.977578
$962$	0	0
$963$	−8.75696	−0.282189
$964$	0	0
$965$	−21.3576	−0.687525
$966$	0	0
$967$	−8.54873	−0.274909	−0.137454	−	0.990508i	$-0.543892\pi$
$967$	−8.54873	−0.274909	−0.137454	+	0.990508i	$0.543892\pi$
$968$	0	0
$969$	11.3771	0.365485
$970$	0	0
$971$	36.0750	1.15770	0.578851	−	0.815433i	$-0.303501\pi$
$971$	36.0750	1.15770	0.578851	+	0.815433i	$0.303501\pi$
$972$	0	0
$973$	8.09706	0.259580
$974$	0	0
$975$	9.48030	0.303613
$976$	0	0
$977$	−13.2375	−0.423506	−0.211753	−	0.977323i	$-0.567917\pi$
$977$	−13.2375	−0.423506	−0.211753	+	0.977323i	$0.567917\pi$
$978$	0	0
$979$	−34.5103	−1.10295
$980$	0	0
$981$	18.4867	0.590234
$982$	0	0
$983$	−19.9233	−0.635455	−0.317727	−	0.948182i	$-0.602920\pi$
$983$	−19.9233	−0.635455	−0.317727	+	0.948182i	$0.602920\pi$
$984$	0	0
$985$	−0.440142	−0.0140241
$986$	0	0
$987$	−6.34083	−0.201831
$988$	0	0
$989$	4.61335	0.146696
$990$	0	0
$991$	54.3594	1.72678	0.863392	−	0.504533i	$-0.168335\pi$
$991$	54.3594	1.72678	0.863392	+	0.504533i	$0.168335\pi$
$992$	0	0
$993$	−12.2984	−0.390278
$994$	0	0
$995$	−23.0469	−0.730637
$996$	0	0
$997$	20.3692	0.645098	0.322549	−	0.946553i	$-0.395460\pi$
$997$	20.3692	0.645098	0.322549	+	0.946553i	$0.395460\pi$
$998$	0	0
$999$	−20.8749	−0.660452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bj.1.6		12
4.3	odd	2		7168.2.a.bi.1.7		12
8.3	odd	2		7168.2.a.bi.1.6		12
8.5	even	2	inner	7168.2.a.bj.1.7		12
32.3	odd	8		896.2.m.h.673.3		12
32.5	even	8		112.2.m.d.85.5	yes	12
32.11	odd	8		896.2.m.h.225.3		12
32.13	even	8		112.2.m.d.29.5	✓	12
32.19	odd	8		448.2.m.d.337.4		12
32.21	even	8		896.2.m.g.225.4		12
32.27	odd	8		448.2.m.d.113.4		12
32.29	even	8		896.2.m.g.673.4		12
224.5	odd	24		784.2.x.m.165.1		24
224.13	odd	8		784.2.m.h.589.5		12
224.37	even	24		784.2.x.l.165.1		24
224.45	odd	24		784.2.x.m.765.1		24
224.69	odd	8		784.2.m.h.197.5		12
224.101	odd	24		784.2.x.m.373.4		24
224.109	even	24		784.2.x.l.765.1		24
224.165	even	24		784.2.x.l.373.4		24
224.173	odd	24		784.2.x.m.557.4		24
224.205	even	24		784.2.x.l.557.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.5	✓	12	32.13	even	8
112.2.m.d.85.5	yes	12	32.5	even	8
448.2.m.d.113.4		12	32.27	odd	8
448.2.m.d.337.4		12	32.19	odd	8
784.2.m.h.197.5		12	224.69	odd	8
784.2.m.h.589.5		12	224.13	odd	8
784.2.x.l.165.1		24	224.37	even	24
784.2.x.l.373.4		24	224.165	even	24
784.2.x.l.557.4		24	224.205	even	24
784.2.x.l.765.1		24	224.109	even	24
784.2.x.m.165.1		24	224.5	odd	24
784.2.x.m.373.4		24	224.101	odd	24
784.2.x.m.557.4		24	224.173	odd	24
784.2.x.m.765.1		24	224.45	odd	24
896.2.m.g.225.4		12	32.21	even	8
896.2.m.g.673.4		12	32.29	even	8
896.2.m.h.225.3		12	32.11	odd	8
896.2.m.h.673.3		12	32.3	odd	8
7168.2.a.bi.1.6		12	8.3	odd	2
7168.2.a.bi.1.7		12	4.3	odd	2
7168.2.a.bj.1.6		12	1.1	even	1	trivial
7168.2.a.bj.1.7		12	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-0.589521 q^{3} -1.60045 q^{5} +1.00000 q^{7} -2.65247 q^{9} +O(q^{10})\)	\(q-0.589521 q^{3} -1.60045 q^{5} +1.00000 q^{7} -2.65247 q^{9} -5.45487 q^{11} +6.59464 q^{13} +0.943500 q^{15} +5.33230 q^{17} -3.61925 q^{19} -0.589521 q^{21} -2.60484 q^{23} -2.43855 q^{25} +3.33225 q^{27} +1.72929 q^{29} -0.833708 q^{31} +3.21576 q^{33} -1.60045 q^{35} -6.26450 q^{37} -3.88768 q^{39} -0.263382 q^{41} -1.77107 q^{43} +4.24514 q^{45} +10.7559 q^{47} +1.00000 q^{49} -3.14350 q^{51} -0.0673478 q^{53} +8.73026 q^{55} +2.13362 q^{57} -5.10081 q^{59} -6.31304 q^{61} -2.65247 q^{63} -10.5544 q^{65} -13.4487 q^{67} +1.53561 q^{69} -2.05301 q^{71} -5.48268 q^{73} +1.43758 q^{75} -5.45487 q^{77} -5.21576 q^{79} +5.99297 q^{81} +8.25965 q^{83} -8.53410 q^{85} -1.01945 q^{87} +6.32651 q^{89} +6.59464 q^{91} +0.491488 q^{93} +5.79243 q^{95} +18.8089 q^{97} +14.4689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 + 12 * q^9	\( 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

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