LMFDB - Embedded newform 7232.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7232,2,Mod(1,7232)] 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("7232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7232 = 2^{6} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.7478107418$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 226)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.17557$ of defining polynomial
Character	$\chi$	$=$	7232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.28408 q^{3} -0.273457 q^{5} -1.23607 q^{7} -1.35114 q^{9} -2.35114 q^{11} +5.04029 q^{13} -0.351141 q^{15} +4.47214 q^{17} +3.75621 q^{19} -1.58721 q^{21} +1.32739 q^{23} -4.92522 q^{25} -5.58721 q^{27} -5.31375 q^{29} +6.82328 q^{31} -3.01905 q^{33} +0.338012 q^{35} -1.86067 q^{37} +6.47214 q^{39} +1.10194 q^{41} +5.43945 q^{43} +0.369480 q^{45} +6.02967 q^{47} -5.47214 q^{49} +5.74258 q^{51} -8.49338 q^{53} +0.642937 q^{55} +4.82328 q^{57} +0.243785 q^{59} +6.85765 q^{61} +1.67010 q^{63} -1.37831 q^{65} -5.28408 q^{67} +1.70447 q^{69} -1.08540 q^{71} +13.7638 q^{73} -6.32437 q^{75} +2.90617 q^{77} +5.03186 q^{79} -3.12099 q^{81} -0.919299 q^{83} -1.22294 q^{85} -6.82328 q^{87} +14.0000 q^{89} -6.23015 q^{91} +8.76163 q^{93} -1.02717 q^{95} +13.2917 q^{97} +3.17672 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{13} + 8 q^{15} - 6 q^{19} + 12 q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - 4 q^{35} + 8 q^{37} + 8 q^{39} + 8 q^{41} - 6 q^{43} + 16 q^{45} + 6 q^{47} - 4 q^{49}+ \cdots + 40 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^13 + 8 * q^15 - 6 * q^19 + 12 * q^21 + 6 * q^23 + 4 * q^25 - 4 * q^27 - 4 * q^35 + 8 * q^37 + 8 * q^39 + 8 * q^41 - 6 * q^43 + 16 * q^45 + 6 * q^47 - 4 * q^49 - 20 * q^51 - 4 * q^53 + 20 * q^55 - 8 * q^57 + 22 * q^59 - 16 * q^61 + 4 * q^63 + 24 * q^65 - 18 * q^67 - 12 * q^69 - 22 * q^71 + 2 * q^75 + 2 * q^79 - 4 * q^81 + 12 * q^83 + 56 * q^89 - 24 * q^91 - 20 * q^93 + 16 * q^95 + 16 * q^97 + 40 * q^99

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
$3$	1.28408	0.741363	0.370682	−	0.928760i	$-0.379124\pi$
$3$	1.28408	0.741363	0.370682	+	0.928760i	$0.379124\pi$
$4$	0	0
$5$	−0.273457	−0.122294	−0.0611469	−	0.998129i	$-0.519476\pi$
$5$	−0.273457	−0.122294	−0.0611469	+	0.998129i	$0.519476\pi$
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	−1.35114	−0.450380
$10$	0	0
$11$	−2.35114	−0.708896	−0.354448	−	0.935076i	$-0.615331\pi$
$11$	−2.35114	−0.708896	−0.354448	+	0.935076i	$0.615331\pi$
$12$	0	0
$13$	5.04029	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$13$	5.04029	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$14$	0	0
$15$	−0.351141	−0.0906642
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	3.75621	0.861735	0.430867	−	0.902415i	$-0.358208\pi$
$19$	3.75621	0.861735	0.430867	+	0.902415i	$0.358208\pi$
$20$	0	0
$21$	−1.58721	−0.346357
$22$	0	0
$23$	1.32739	0.276780	0.138390	−	0.990378i	$-0.455807\pi$
$23$	1.32739	0.276780	0.138390	+	0.990378i	$0.455807\pi$
$24$	0	0
$25$	−4.92522	−0.985044
$26$	0	0
$27$	−5.58721	−1.07526
$28$	0	0
$29$	−5.31375	−0.986739	−0.493369	−	0.869820i	$-0.664235\pi$
$29$	−5.31375	−0.986739	−0.493369	+	0.869820i	$0.664235\pi$
$30$	0	0
$31$	6.82328	1.22550	0.612748	−	0.790278i	$-0.290064\pi$
$31$	6.82328	1.22550	0.612748	+	0.790278i	$0.290064\pi$
$32$	0	0
$33$	−3.01905	−0.525549
$34$	0	0
$35$	0.338012	0.0571345
$36$	0	0
$37$	−1.86067	−0.305892	−0.152946	−	0.988235i	$-0.548876\pi$
$37$	−1.86067	−0.305892	−0.152946	+	0.988235i	$0.548876\pi$
$38$	0	0
$39$	6.47214	1.03637
$40$	0	0
$41$	1.10194	0.172095	0.0860474	−	0.996291i	$-0.472576\pi$
$41$	1.10194	0.172095	0.0860474	+	0.996291i	$0.472576\pi$
$42$	0	0
$43$	5.43945	0.829508	0.414754	−	0.909934i	$-0.363868\pi$
$43$	5.43945	0.829508	0.414754	+	0.909934i	$0.363868\pi$
$44$	0	0
$45$	0.369480	0.0550788
$46$	0	0
$47$	6.02967	0.879518	0.439759	−	0.898116i	$-0.355064\pi$
$47$	6.02967	0.879518	0.439759	+	0.898116i	$0.355064\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	5.74258	0.804121
$52$	0	0
$53$	−8.49338	−1.16666	−0.583328	−	0.812237i	$-0.698250\pi$
$53$	−8.49338	−1.16666	−0.583328	+	0.812237i	$0.698250\pi$
$54$	0	0
$55$	0.642937	0.0866936
$56$	0	0
$57$	4.82328	0.638859
$58$	0	0
$59$	0.243785	0.0317381	0.0158691	−	0.999874i	$-0.494949\pi$
$59$	0.243785	0.0317381	0.0158691	+	0.999874i	$0.494949\pi$
$60$	0	0
$61$	6.85765	0.878032	0.439016	−	0.898479i	$-0.355327\pi$
$61$	6.85765	0.878032	0.439016	+	0.898479i	$0.355327\pi$
$62$	0	0
$63$	1.67010	0.210413
$64$	0	0
$65$	−1.37831	−0.170958
$66$	0	0
$67$	−5.28408	−0.645553	−0.322777	−	0.946475i	$-0.604616\pi$
$67$	−5.28408	−0.645553	−0.322777	+	0.946475i	$0.604616\pi$
$68$	0	0
$69$	1.70447	0.205195
$70$	0	0
$71$	−1.08540	−0.128813	−0.0644067	−	0.997924i	$-0.520515\pi$
$71$	−1.08540	−0.128813	−0.0644067	+	0.997924i	$0.520515\pi$
$72$	0	0
$73$	13.7638	1.61093	0.805467	−	0.592641i	$-0.201915\pi$
$73$	13.7638	1.61093	0.805467	+	0.592641i	$0.201915\pi$
$74$	0	0
$75$	−6.32437	−0.730276
$76$	0	0
$77$	2.90617	0.331189
$78$	0	0
$79$	5.03186	0.566129	0.283065	−	0.959101i	$-0.408649\pi$
$79$	5.03186	0.566129	0.283065	+	0.959101i	$0.408649\pi$
$80$	0	0
$81$	−3.12099	−0.346777
$82$	0	0
$83$	−0.919299	−0.100906	−0.0504531	−	0.998726i	$-0.516067\pi$
$83$	−0.919299	−0.100906	−0.0504531	+	0.998726i	$0.516067\pi$
$84$	0	0
$85$	−1.22294	−0.132646
$86$	0	0
$87$	−6.82328	−0.731532
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−6.23015	−0.653097
$92$	0	0
$93$	8.76163	0.908538
$94$	0	0
$95$	−1.02717	−0.105385
$96$	0	0
$97$	13.2917	1.34957	0.674783	−	0.738016i	$-0.264237\pi$
$97$	13.2917	1.34957	0.674783	+	0.738016i	$0.264237\pi$
$98$	0	0
$99$	3.17672	0.319273
$100$	0	0
$101$	−13.1311	−1.30659	−0.653297	−	0.757102i	$-0.726615\pi$
$101$	−13.1311	−1.30659	−0.653297	+	0.757102i	$0.726615\pi$
$102$	0	0
$103$	−10.3404	−1.01887	−0.509435	−	0.860509i	$-0.670146\pi$
$103$	−10.3404	−1.01887	−0.509435	+	0.860509i	$0.670146\pi$
$104$	0	0
$105$	0.434034	0.0423574
$106$	0	0
$107$	−5.43945	−0.525851	−0.262926	−	0.964816i	$-0.584687\pi$
$107$	−5.43945	−0.525851	−0.262926	+	0.964816i	$0.584687\pi$
$108$	0	0
$109$	−2.02124	−0.193600	−0.0968000	−	0.995304i	$-0.530861\pi$
$109$	−2.02124	−0.193600	−0.0968000	+	0.995304i	$0.530861\pi$
$110$	0	0
$111$	−2.38924	−0.226777
$112$	0	0
$113$	−1.00000	−0.0940721
$113$	−1.00000	−0.0940721
$114$	0	0
$115$	−0.362985	−0.0338485
$116$	0	0
$117$	−6.81015	−0.629598
$118$	0	0
$119$	−5.52786	−0.506738
$120$	0	0
$121$	−5.47214	−0.497467
$122$	0	0
$123$	1.41498	0.127585
$124$	0	0
$125$	2.71413	0.242759
$126$	0	0
$127$	21.5468	1.91197	0.955985	−	0.293416i	$-0.0947922\pi$
$127$	21.5468	1.91197	0.955985	+	0.293416i	$0.0947922\pi$
$128$	0	0
$129$	6.98468	0.614967
$130$	0	0
$131$	18.4449	1.61153	0.805767	−	0.592232i	$-0.201753\pi$
$131$	18.4449	1.61153	0.805767	+	0.592232i	$0.201753\pi$
$132$	0	0
$133$	−4.64294	−0.402594
$134$	0	0
$135$	1.52786	0.131498
$136$	0	0
$137$	13.2169	1.12920	0.564598	−	0.825366i	$-0.309031\pi$
$137$	13.2169	1.12920	0.564598	+	0.825366i	$0.309031\pi$
$138$	0	0
$139$	−0.313039	−0.0265516	−0.0132758	−	0.999912i	$-0.504226\pi$
$139$	−0.313039	−0.0265516	−0.0132758	+	0.999912i	$0.504226\pi$
$140$	0	0
$141$	7.74258	0.652043
$142$	0	0
$143$	−11.8504	−0.990984
$144$	0	0
$145$	1.45309	0.120672
$146$	0	0
$147$	−7.02666	−0.579549
$148$	0	0
$149$	−12.5850	−1.03100	−0.515502	−	0.856888i	$-0.672395\pi$
$149$	−12.5850	−1.03100	−0.515502	+	0.856888i	$0.672395\pi$
$150$	0	0
$151$	8.44246	0.687038	0.343519	−	0.939146i	$-0.388381\pi$
$151$	8.44246	0.687038	0.343519	+	0.939146i	$0.388381\pi$
$152$	0	0
$153$	−6.04249	−0.488506
$154$	0	0
$155$	−1.86588	−0.149871
$156$	0	0
$157$	0.997808	0.0796337	0.0398169	−	0.999207i	$-0.487323\pi$
$157$	0.997808	0.0796337	0.0398169	+	0.999207i	$0.487323\pi$
$158$	0	0
$159$	−10.9062	−0.864916
$160$	0	0
$161$	−1.64074	−0.129309
$162$	0	0
$163$	3.90398	0.305783	0.152892	−	0.988243i	$-0.451141\pi$
$163$	3.90398	0.305783	0.152892	+	0.988243i	$0.451141\pi$
$164$	0	0
$165$	0.825582	0.0642715
$166$	0	0
$167$	14.2275	1.10096	0.550480	−	0.834849i	$-0.314445\pi$
$167$	14.2275	1.10096	0.550480	+	0.834849i	$0.314445\pi$
$168$	0	0
$169$	12.4046	0.954197
$170$	0	0
$171$	−5.07518	−0.388108
$172$	0	0
$173$	−0.884927	−0.0672798	−0.0336399	−	0.999434i	$-0.510710\pi$
$173$	−0.884927	−0.0672798	−0.0336399	+	0.999434i	$0.510710\pi$
$174$	0	0
$175$	6.08791	0.460203
$176$	0	0
$177$	0.313039	0.0235295
$178$	0	0
$179$	15.7027	1.17367	0.586837	−	0.809705i	$-0.300373\pi$
$179$	15.7027	1.17367	0.586837	+	0.809705i	$0.300373\pi$
$180$	0	0
$181$	10.4501	0.776747	0.388374	−	0.921502i	$-0.373037\pi$
$181$	10.4501	0.776747	0.388374	+	0.921502i	$0.373037\pi$
$182$	0	0
$183$	8.80576	0.650941
$184$	0	0
$185$	0.508813	0.0374087
$186$	0	0
$187$	−10.5146	−0.768905
$188$	0	0
$189$	6.90617	0.502350
$190$	0	0
$191$	14.9833	1.08416	0.542078	−	0.840328i	$-0.317638\pi$
$191$	14.9833	1.08416	0.542078	+	0.840328i	$0.317638\pi$
$192$	0	0
$193$	3.49119	0.251301	0.125651	−	0.992075i	$-0.459898\pi$
$193$	3.49119	0.251301	0.125651	+	0.992075i	$0.459898\pi$
$194$	0	0
$195$	−1.76985	−0.126742
$196$	0	0
$197$	12.0279	0.856951	0.428475	−	0.903553i	$-0.359051\pi$
$197$	12.0279	0.856951	0.428475	+	0.903553i	$0.359051\pi$
$198$	0	0
$199$	5.44027	0.385651	0.192825	−	0.981233i	$-0.438235\pi$
$199$	5.44027	0.385651	0.192825	+	0.981233i	$0.438235\pi$
$200$	0	0
$201$	−6.78518	−0.478589
$202$	0	0
$203$	6.56816	0.460994
$204$	0	0
$205$	−0.301335	−0.0210461
$206$	0	0
$207$	−1.79349	−0.124656
$208$	0	0
$209$	−8.83139	−0.610880
$210$	0	0
$211$	1.83860	0.126574	0.0632872	−	0.997995i	$-0.479842\pi$
$211$	1.83860	0.126574	0.0632872	+	0.997995i	$0.479842\pi$
$212$	0	0
$213$	−1.39374	−0.0954975
$214$	0	0
$215$	−1.48746	−0.101444
$216$	0	0
$217$	−8.43403	−0.572540
$218$	0	0
$219$	17.6738	1.19429
$220$	0	0
$221$	22.5409	1.51626
$222$	0	0
$223$	20.1103	1.34668	0.673341	−	0.739332i	$-0.264858\pi$
$223$	20.1103	1.34668	0.673341	+	0.739332i	$0.264858\pi$
$224$	0	0
$225$	6.65467	0.443645
$226$	0	0
$227$	−15.1744	−1.00716	−0.503581	−	0.863948i	$-0.667984\pi$
$227$	−15.1744	−1.00716	−0.503581	+	0.863948i	$0.667984\pi$
$228$	0	0
$229$	7.08361	0.468098	0.234049	−	0.972225i	$-0.424802\pi$
$229$	7.08361	0.468098	0.234049	+	0.972225i	$0.424802\pi$
$230$	0	0
$231$	3.73175	0.245531
$232$	0	0
$233$	−19.8848	−1.30270	−0.651349	−	0.758779i	$-0.725796\pi$
$233$	−19.8848	−1.30270	−0.651349	+	0.758779i	$0.725796\pi$
$234$	0	0
$235$	−1.64886	−0.107560
$236$	0	0
$237$	6.46131	0.419707
$238$	0	0
$239$	5.09591	0.329627	0.164813	−	0.986325i	$-0.447298\pi$
$239$	5.09591	0.329627	0.164813	+	0.986325i	$0.447298\pi$
$240$	0	0
$241$	−3.01532	−0.194234	−0.0971170	−	0.995273i	$-0.530962\pi$
$241$	−3.01532	−0.194234	−0.0971170	+	0.995273i	$0.530962\pi$
$242$	0	0
$243$	12.7540	0.818171
$244$	0	0
$245$	1.49640	0.0956013
$246$	0	0
$247$	18.9324	1.20464
$248$	0	0
$249$	−1.18045	−0.0748082
$250$	0	0
$251$	26.9288	1.69973	0.849867	−	0.526998i	$-0.176682\pi$
$251$	26.9288	1.69973	0.849867	+	0.526998i	$0.176682\pi$
$252$	0	0
$253$	−3.12088	−0.196208
$254$	0	0
$255$	−1.57035	−0.0983392
$256$	0	0
$257$	20.2741	1.26466	0.632330	−	0.774699i	$-0.282099\pi$
$257$	20.2741	1.26466	0.632330	+	0.774699i	$0.282099\pi$
$258$	0	0
$259$	2.29991	0.142909
$260$	0	0
$261$	7.17963	0.444408
$262$	0	0
$263$	9.93365	0.612535	0.306268	−	0.951945i	$-0.400920\pi$
$263$	9.93365	0.612535	0.306268	+	0.951945i	$0.400920\pi$
$264$	0	0
$265$	2.32258	0.142675
$266$	0	0
$267$	17.9771	1.10018
$268$	0	0
$269$	−13.5924	−0.828744	−0.414372	−	0.910108i	$-0.635999\pi$
$269$	−13.5924	−0.828744	−0.414372	+	0.910108i	$0.635999\pi$
$270$	0	0
$271$	29.1506	1.77077	0.885385	−	0.464858i	$-0.153895\pi$
$271$	29.1506	1.77077	0.885385	+	0.464858i	$0.153895\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	11.5799	0.698294
$276$	0	0
$277$	29.8825	1.79547	0.897733	−	0.440540i	$-0.145213\pi$
$277$	29.8825	1.79547	0.897733	+	0.440540i	$0.145213\pi$
$278$	0	0
$279$	−9.21921	−0.551940
$280$	0	0
$281$	−21.2432	−1.26726	−0.633630	−	0.773636i	$-0.718436\pi$
$281$	−21.2432	−1.26726	−0.633630	+	0.773636i	$0.718436\pi$
$282$	0	0
$283$	−13.9061	−0.826629	−0.413315	−	0.910588i	$-0.635629\pi$
$283$	−13.9061	−0.826629	−0.413315	+	0.910588i	$0.635629\pi$
$284$	0	0
$285$	−1.31896	−0.0781285
$286$	0	0
$287$	−1.36208	−0.0804009
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	17.0676	1.00052
$292$	0	0
$293$	4.27346	0.249658	0.124829	−	0.992178i	$-0.460162\pi$
$293$	4.27346	0.249658	0.124829	+	0.992178i	$0.460162\pi$
$294$	0	0
$295$	−0.0666648	−0.00388138
$296$	0	0
$297$	13.1363	0.762246
$298$	0	0
$299$	6.69044	0.386918
$300$	0	0
$301$	−6.72353	−0.387538
$302$	0	0
$303$	−16.8614	−0.968661
$304$	0	0
$305$	−1.87528	−0.107378
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−13.2779	−0.755353
$310$	0	0
$311$	−9.69636	−0.549830	−0.274915	−	0.961469i	$-0.588650\pi$
$311$	−9.69636	−0.549830	−0.274915	+	0.961469i	$0.588650\pi$
$312$	0	0
$313$	18.5550	1.04879	0.524396	−	0.851474i	$-0.324291\pi$
$313$	18.5550	1.04879	0.524396	+	0.851474i	$0.324291\pi$
$314$	0	0
$315$	−0.456702	−0.0257322
$316$	0	0
$317$	10.8364	0.608633	0.304317	−	0.952571i	$-0.401572\pi$
$317$	10.8364	0.608633	0.304317	+	0.952571i	$0.401572\pi$
$318$	0	0
$319$	12.4934	0.699495
$320$	0	0
$321$	−6.98468	−0.389847
$322$	0	0
$323$	16.7983	0.934683
$324$	0	0
$325$	−24.8246	−1.37702
$326$	0	0
$327$	−2.59544	−0.143528
$328$	0	0
$329$	−7.45309	−0.410902
$330$	0	0
$331$	23.3914	1.28571	0.642855	−	0.765988i	$-0.277750\pi$
$331$	23.3914	1.28571	0.642855	+	0.765988i	$0.277750\pi$
$332$	0	0
$333$	2.51402	0.137768
$334$	0	0
$335$	1.44497	0.0789472
$336$	0	0
$337$	−30.6939	−1.67201	−0.836003	−	0.548725i	$-0.815113\pi$
$337$	−30.6939	−1.67201	−0.836003	+	0.548725i	$0.815113\pi$
$338$	0	0
$339$	−1.28408	−0.0697416
$340$	0	0
$341$	−16.0425	−0.868749
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	−0.466101	−0.0250940
$346$	0	0
$347$	3.41641	0.183402	0.0917012	−	0.995787i	$-0.470770\pi$
$347$	3.41641	0.183402	0.0917012	+	0.995787i	$0.470770\pi$
$348$	0	0
$349$	−36.2851	−1.94230	−0.971148	−	0.238478i	$-0.923351\pi$
$349$	−36.2851	−1.94230	−0.971148	+	0.238478i	$0.923351\pi$
$350$	0	0
$351$	−28.1612	−1.50313
$352$	0	0
$353$	−16.4684	−0.876525	−0.438262	−	0.898847i	$-0.644406\pi$
$353$	−16.4684	−0.876525	−0.438262	+	0.898847i	$0.644406\pi$
$354$	0	0
$355$	0.296811	0.0157531
$356$	0	0
$357$	−7.09821	−0.375677
$358$	0	0
$359$	−24.1197	−1.27299	−0.636493	−	0.771282i	$-0.719616\pi$
$359$	−24.1197	−1.27299	−0.636493	+	0.771282i	$0.719616\pi$
$360$	0	0
$361$	−4.89085	−0.257413
$362$	0	0
$363$	−7.02666	−0.368804
$364$	0	0
$365$	−3.76382	−0.197007
$366$	0	0
$367$	−34.4413	−1.79782	−0.898910	−	0.438134i	$-0.855640\pi$
$367$	−34.4413	−1.79782	−0.898910	+	0.438134i	$0.855640\pi$
$368$	0	0
$369$	−1.48888	−0.0775081
$370$	0	0
$371$	10.4984	0.545049
$372$	0	0
$373$	−26.3540	−1.36456	−0.682280	−	0.731091i	$-0.739012\pi$
$373$	−26.3540	−1.36456	−0.682280	+	0.731091i	$0.739012\pi$
$374$	0	0
$375$	3.48515	0.179972
$376$	0	0
$377$	−26.7829	−1.37939
$378$	0	0
$379$	28.5509	1.46656	0.733281	−	0.679925i	$-0.237988\pi$
$379$	28.5509	1.46656	0.733281	+	0.679925i	$0.237988\pi$
$380$	0	0
$381$	27.6678	1.41746
$382$	0	0
$383$	−22.7425	−1.16209	−0.581043	−	0.813873i	$-0.697355\pi$
$383$	−22.7425	−1.16209	−0.581043	+	0.813873i	$0.697355\pi$
$384$	0	0
$385$	−0.794714	−0.0405024
$386$	0	0
$387$	−7.34946	−0.373594
$388$	0	0
$389$	36.7249	1.86203	0.931014	−	0.364982i	$-0.118925\pi$
$389$	36.7249	1.86203	0.931014	+	0.364982i	$0.118925\pi$
$390$	0	0
$391$	5.93627	0.300210
$392$	0	0
$393$	23.6847	1.19473
$394$	0	0
$395$	−1.37600	−0.0692341
$396$	0	0
$397$	14.7183	0.738691	0.369346	−	0.929292i	$-0.379582\pi$
$397$	14.7183	0.738691	0.369346	+	0.929292i	$0.379582\pi$
$398$	0	0
$399$	−5.96190	−0.298468
$400$	0	0
$401$	−6.79702	−0.339427	−0.169713	−	0.985493i	$-0.554284\pi$
$401$	−6.79702	−0.339427	−0.169713	+	0.985493i	$0.554284\pi$
$402$	0	0
$403$	34.3913	1.71315
$404$	0	0
$405$	0.853459	0.0424087
$406$	0	0
$407$	4.37469	0.216845
$408$	0	0
$409$	9.91919	0.490472	0.245236	−	0.969463i	$-0.421135\pi$
$409$	9.91919	0.490472	0.245236	+	0.969463i	$0.421135\pi$
$410$	0	0
$411$	16.9715	0.837145
$412$	0	0
$413$	−0.301335	−0.0148277
$414$	0	0
$415$	0.251389	0.0123402
$416$	0	0
$417$	−0.401967	−0.0196844
$418$	0	0
$419$	−14.8498	−0.725461	−0.362731	−	0.931894i	$-0.618155\pi$
$419$	−14.8498	−0.725461	−0.362731	+	0.931894i	$0.618155\pi$
$420$	0	0
$421$	−3.49119	−0.170150	−0.0850750	−	0.996375i	$-0.527113\pi$
$421$	−3.49119	−0.170150	−0.0850750	+	0.996375i	$0.527113\pi$
$422$	0	0
$423$	−8.14694	−0.396118
$424$	0	0
$425$	−22.0263	−1.06843
$426$	0	0
$427$	−8.47652	−0.410208
$428$	0	0
$429$	−15.2169	−0.734679
$430$	0	0
$431$	38.2963	1.84467	0.922333	−	0.386395i	$-0.126280\pi$
$431$	38.2963	1.84467	0.922333	+	0.386395i	$0.126280\pi$
$432$	0	0
$433$	−39.4104	−1.89394	−0.946971	−	0.321320i	$-0.895874\pi$
$433$	−39.4104	−1.89394	−0.946971	+	0.321320i	$0.895874\pi$
$434$	0	0
$435$	1.86588	0.0894619
$436$	0	0
$437$	4.98596	0.238511
$438$	0	0
$439$	−32.1518	−1.53452	−0.767260	−	0.641336i	$-0.778381\pi$
$439$	−32.1518	−1.53452	−0.767260	+	0.641336i	$0.778381\pi$
$440$	0	0
$441$	7.39363	0.352077
$442$	0	0
$443$	−40.0402	−1.90237	−0.951183	−	0.308627i	$-0.900131\pi$
$443$	−40.0402	−1.90237	−0.951183	+	0.308627i	$0.900131\pi$
$444$	0	0
$445$	−3.82840	−0.181484
$446$	0	0
$447$	−16.1602	−0.764349
$448$	0	0
$449$	4.86368	0.229531	0.114766	−	0.993393i	$-0.463388\pi$
$449$	4.86368	0.229531	0.114766	+	0.993393i	$0.463388\pi$
$450$	0	0
$451$	−2.59083	−0.121997
$452$	0	0
$453$	10.8408	0.509345
$454$	0	0
$455$	1.70368	0.0798697
$456$	0	0
$457$	8.12472	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$457$	8.12472	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$458$	0	0
$459$	−24.9868	−1.16628
$460$	0	0
$461$	−5.57177	−0.259503	−0.129752	−	0.991547i	$-0.541418\pi$
$461$	−5.57177	−0.259503	−0.129752	+	0.991547i	$0.541418\pi$
$462$	0	0
$463$	26.7235	1.24195	0.620974	−	0.783831i	$-0.286737\pi$
$463$	26.7235	1.24195	0.620974	+	0.783831i	$0.286737\pi$
$464$	0	0
$465$	−2.39593	−0.111109
$466$	0	0
$467$	−28.4469	−1.31637	−0.658184	−	0.752857i	$-0.728675\pi$
$467$	−28.4469	−1.31637	−0.658184	+	0.752857i	$0.728675\pi$
$468$	0	0
$469$	6.53148	0.301596
$470$	0	0
$471$	1.28126	0.0590375
$472$	0	0
$473$	−12.7889	−0.588034
$474$	0	0
$475$	−18.5002	−0.848847
$476$	0	0
$477$	11.4758	0.525439
$478$	0	0
$479$	−5.05812	−0.231112	−0.115556	−	0.993301i	$-0.536865\pi$
$479$	−5.05812	−0.231112	−0.115556	+	0.993301i	$0.536865\pi$
$480$	0	0
$481$	−9.37831	−0.427614
$482$	0	0
$483$	−2.10685	−0.0958648
$484$	0	0
$485$	−3.63471	−0.165044
$486$	0	0
$487$	11.1660	0.505979	0.252990	−	0.967469i	$-0.418586\pi$
$487$	11.1660	0.505979	0.252990	+	0.967469i	$0.418586\pi$
$488$	0	0
$489$	5.01302	0.226696
$490$	0	0
$491$	−32.1169	−1.44942	−0.724708	−	0.689057i	$-0.758025\pi$
$491$	−32.1169	−1.44942	−0.724708	+	0.689057i	$0.758025\pi$
$492$	0	0
$493$	−23.7638	−1.07027
$494$	0	0
$495$	−0.868699	−0.0390451
$496$	0	0
$497$	1.34163	0.0601803
$498$	0	0
$499$	36.9671	1.65487	0.827437	−	0.561559i	$-0.189798\pi$
$499$	36.9671	1.65487	0.827437	+	0.561559i	$0.189798\pi$
$500$	0	0
$501$	18.2693	0.816211
$502$	0	0
$503$	13.7793	0.614387	0.307193	−	0.951647i	$-0.400610\pi$
$503$	13.7793	0.614387	0.307193	+	0.951647i	$0.400610\pi$
$504$	0	0
$505$	3.59080	0.159788
$506$	0	0
$507$	15.9284	0.707407
$508$	0	0
$509$	−13.6772	−0.606231	−0.303116	−	0.952954i	$-0.598027\pi$
$509$	−13.6772	−0.606231	−0.303116	+	0.952954i	$0.598027\pi$
$510$	0	0
$511$	−17.0130	−0.752612
$512$	0	0
$513$	−20.9868	−0.926588
$514$	0	0
$515$	2.82766	0.124602
$516$	0	0
$517$	−14.1766	−0.623487
$518$	0	0
$519$	−1.13632	−0.0498787
$520$	0	0
$521$	30.2360	1.32466	0.662331	−	0.749212i	$-0.269567\pi$
$521$	30.2360	1.32466	0.662331	+	0.749212i	$0.269567\pi$
$522$	0	0
$523$	25.1287	1.09880	0.549401	−	0.835559i	$-0.314856\pi$
$523$	25.1287	1.09880	0.549401	+	0.835559i	$0.314856\pi$
$524$	0	0
$525$	7.81736	0.341177
$526$	0	0
$527$	30.5146	1.32924
$528$	0	0
$529$	−21.2380	−0.923393
$530$	0	0
$531$	−0.329388	−0.0142942
$532$	0	0
$533$	5.55412	0.240576
$534$	0	0
$535$	1.48746	0.0643084
$536$	0	0
$537$	20.1635	0.870118
$538$	0	0
$539$	12.8658	0.554168
$540$	0	0
$541$	−43.1046	−1.85321	−0.926606	−	0.376033i	$-0.877288\pi$
$541$	−43.1046	−1.85321	−0.926606	+	0.376033i	$0.877288\pi$
$542$	0	0
$543$	13.4187	0.575852
$544$	0	0
$545$	0.552724	0.0236761
$546$	0	0
$547$	−19.3217	−0.826135	−0.413067	−	0.910700i	$-0.635543\pi$
$547$	−19.3217	−0.826135	−0.413067	+	0.910700i	$0.635543\pi$
$548$	0	0
$549$	−9.26565	−0.395448
$550$	0	0
$551$	−19.9596	−0.850307
$552$	0	0
$553$	−6.21973	−0.264490
$554$	0	0
$555$	0.653356	0.0277334
$556$	0	0
$557$	6.89677	0.292226	0.146113	−	0.989268i	$-0.453324\pi$
$557$	6.89677	0.292226	0.146113	+	0.989268i	$0.453324\pi$
$558$	0	0
$559$	27.4164	1.15959
$560$	0	0
$561$	−13.5016	−0.570038
$562$	0	0
$563$	1.36208	0.0574047	0.0287024	−	0.999588i	$-0.490862\pi$
$563$	1.36208	0.0574047	0.0287024	+	0.999588i	$0.490862\pi$
$564$	0	0
$565$	0.273457	0.0115044
$566$	0	0
$567$	3.85776	0.162011
$568$	0	0
$569$	23.9633	1.00459	0.502297	−	0.864695i	$-0.332488\pi$
$569$	23.9633	1.00459	0.502297	+	0.864695i	$0.332488\pi$
$570$	0	0
$571$	25.4028	1.06307	0.531536	−	0.847035i	$-0.321615\pi$
$571$	25.4028	1.06307	0.531536	+	0.847035i	$0.321615\pi$
$572$	0	0
$573$	19.2398	0.803754
$574$	0	0
$575$	−6.53769	−0.272641
$576$	0	0
$577$	32.6333	1.35854	0.679271	−	0.733887i	$-0.262296\pi$
$577$	32.6333	1.35854	0.679271	+	0.733887i	$0.262296\pi$
$578$	0	0
$579$	4.48296	0.186305
$580$	0	0
$581$	1.13632	0.0471423
$582$	0	0
$583$	19.9691	0.827037
$584$	0	0
$585$	1.86229	0.0769960
$586$	0	0
$587$	14.2953	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$587$	14.2953	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$588$	0	0
$589$	25.6297	1.05605
$590$	0	0
$591$	15.4447	0.635312
$592$	0	0
$593$	4.34741	0.178527	0.0892634	−	0.996008i	$-0.471549\pi$
$593$	4.34741	0.178527	0.0892634	+	0.996008i	$0.471549\pi$
$594$	0	0
$595$	1.51164	0.0619710
$596$	0	0
$597$	6.98574	0.285907
$598$	0	0
$599$	−18.1776	−0.742716	−0.371358	−	0.928490i	$-0.621108\pi$
$599$	−18.1776	−0.742716	−0.371358	+	0.928490i	$0.621108\pi$
$600$	0	0
$601$	25.6145	1.04484	0.522418	−	0.852689i	$-0.325030\pi$
$601$	25.6145	1.04484	0.522418	+	0.852689i	$0.325030\pi$
$602$	0	0
$603$	7.13954	0.290744
$604$	0	0
$605$	1.49640	0.0608372
$606$	0	0
$607$	13.2849	0.539218	0.269609	−	0.962970i	$-0.413106\pi$
$607$	13.2849	0.539218	0.269609	+	0.962970i	$0.413106\pi$
$608$	0	0
$609$	8.43403	0.341764
$610$	0	0
$611$	30.3913	1.22950
$612$	0	0
$613$	−22.3871	−0.904208	−0.452104	−	0.891965i	$-0.649326\pi$
$613$	−22.3871	−0.904208	−0.452104	+	0.891965i	$0.649326\pi$
$614$	0	0
$615$	−0.386938	−0.0156028
$616$	0	0
$617$	−21.4201	−0.862342	−0.431171	−	0.902270i	$-0.641899\pi$
$617$	−21.4201	−0.862342	−0.431171	+	0.902270i	$0.641899\pi$
$618$	0	0
$619$	6.03050	0.242386	0.121193	−	0.992629i	$-0.461328\pi$
$619$	6.03050	0.242386	0.121193	+	0.992629i	$0.461328\pi$
$620$	0	0
$621$	−7.41641	−0.297610
$622$	0	0
$623$	−17.3050	−0.693308
$624$	0	0
$625$	23.8839	0.955356
$626$	0	0
$627$	−11.3402	−0.452884
$628$	0	0
$629$	−8.32115	−0.331786
$630$	0	0
$631$	−6.17923	−0.245991	−0.122996	−	0.992407i	$-0.539250\pi$
$631$	−6.17923	−0.245991	−0.122996	+	0.992407i	$0.539250\pi$
$632$	0	0
$633$	2.36091	0.0938376
$634$	0	0
$635$	−5.89213	−0.233822
$636$	0	0
$637$	−27.5812	−1.09281
$638$	0	0
$639$	1.46653	0.0580150
$640$	0	0
$641$	−14.3549	−0.566983	−0.283492	−	0.958975i	$-0.591493\pi$
$641$	−14.3549	−0.566983	−0.283492	+	0.958975i	$0.591493\pi$
$642$	0	0
$643$	−38.5024	−1.51839	−0.759193	−	0.650865i	$-0.774406\pi$
$643$	−38.5024	−1.51839	−0.759193	+	0.650865i	$0.774406\pi$
$644$	0	0
$645$	−1.91001	−0.0752067
$646$	0	0
$647$	6.76601	0.265999	0.133000	−	0.991116i	$-0.457539\pi$
$647$	6.76601	0.265999	0.133000	+	0.991116i	$0.457539\pi$
$648$	0	0
$649$	−0.573173	−0.0224990
$650$	0	0
$651$	−10.8300	−0.424460
$652$	0	0
$653$	29.5718	1.15723	0.578616	−	0.815600i	$-0.303593\pi$
$653$	29.5718	1.15723	0.578616	+	0.815600i	$0.303593\pi$
$654$	0	0
$655$	−5.04388	−0.197081
$656$	0	0
$657$	−18.5969	−0.725533
$658$	0	0
$659$	−10.1953	−0.397151	−0.198576	−	0.980086i	$-0.563632\pi$
$659$	−10.1953	−0.397151	−0.198576	+	0.980086i	$0.563632\pi$
$660$	0	0
$661$	9.35624	0.363915	0.181958	−	0.983306i	$-0.441757\pi$
$661$	9.35624	0.363915	0.181958	+	0.983306i	$0.441757\pi$
$662$	0	0
$663$	28.9443	1.12410
$664$	0	0
$665$	1.26965	0.0492348
$666$	0	0
$667$	−7.05342	−0.273110
$668$	0	0
$669$	25.8232	0.998381
$670$	0	0
$671$	−16.1233	−0.622433
$672$	0	0
$673$	−28.4280	−1.09582	−0.547909	−	0.836538i	$-0.684576\pi$
$673$	−28.4280	−1.09582	−0.547909	+	0.836538i	$0.684576\pi$
$674$	0	0
$675$	27.5182	1.05918
$676$	0	0
$677$	−26.5453	−1.02022	−0.510109	−	0.860110i	$-0.670395\pi$
$677$	−26.5453	−1.02022	−0.510109	+	0.860110i	$0.670395\pi$
$678$	0	0
$679$	−16.4294	−0.630503
$680$	0	0
$681$	−19.4852	−0.746673
$682$	0	0
$683$	−45.0433	−1.72353	−0.861767	−	0.507305i	$-0.830642\pi$
$683$	−45.0433	−1.72353	−0.861767	+	0.507305i	$0.830642\pi$
$684$	0	0
$685$	−3.61426	−0.138094
$686$	0	0
$687$	9.09591	0.347031
$688$	0	0
$689$	−42.8091	−1.63090
$690$	0	0
$691$	18.9214	0.719803	0.359902	−	0.932990i	$-0.382810\pi$
$691$	18.9214	0.719803	0.359902	+	0.932990i	$0.382810\pi$
$692$	0	0
$693$	−3.92665	−0.149161
$694$	0	0
$695$	0.0856029	0.00324710
$696$	0	0
$697$	4.92804	0.186663
$698$	0	0
$699$	−25.5337	−0.965772
$700$	0	0
$701$	29.5285	1.11527	0.557637	−	0.830085i	$-0.311708\pi$
$701$	29.5285	1.11527	0.557637	+	0.830085i	$0.311708\pi$
$702$	0	0
$703$	−6.98906	−0.263598
$704$	0	0
$705$	−2.11727	−0.0797408
$706$	0	0
$707$	16.2309	0.610427
$708$	0	0
$709$	−37.3085	−1.40115	−0.700576	−	0.713578i	$-0.747073\pi$
$709$	−37.3085	−1.40115	−0.700576	+	0.713578i	$0.747073\pi$
$710$	0	0
$711$	−6.79876	−0.254973
$712$	0	0
$713$	9.05715	0.339193
$714$	0	0
$715$	3.24059	0.121191
$716$	0	0
$717$	6.54355	0.244373
$718$	0	0
$719$	−3.72967	−0.139093	−0.0695467	−	0.997579i	$-0.522155\pi$
$719$	−3.72967	−0.139093	−0.0695467	+	0.997579i	$0.522155\pi$
$720$	0	0
$721$	12.7814	0.476006
$722$	0	0
$723$	−3.87191	−0.143998
$724$	0	0
$725$	26.1714	0.971981
$726$	0	0
$727$	50.7068	1.88061	0.940306	−	0.340331i	$-0.110539\pi$
$727$	50.7068	1.88061	0.940306	+	0.340331i	$0.110539\pi$
$728$	0	0
$729$	25.7402	0.953339
$730$	0	0
$731$	24.3259	0.899727
$732$	0	0
$733$	−10.5036	−0.387959	−0.193980	−	0.981006i	$-0.562140\pi$
$733$	−10.5036	−0.387959	−0.193980	+	0.981006i	$0.562140\pi$
$734$	0	0
$735$	1.92149	0.0708753
$736$	0	0
$737$	12.4236	0.457630
$738$	0	0
$739$	36.9036	1.35752	0.678761	−	0.734359i	$-0.262517\pi$
$739$	36.9036	1.35752	0.678761	+	0.734359i	$0.262517\pi$
$740$	0	0
$741$	24.3107	0.893077
$742$	0	0
$743$	−27.4306	−1.00633	−0.503166	−	0.864190i	$-0.667832\pi$
$743$	−27.4306	−1.00633	−0.503166	+	0.864190i	$0.667832\pi$
$744$	0	0
$745$	3.44147	0.126086
$746$	0	0
$747$	1.24210	0.0454462
$748$	0	0
$749$	6.72353	0.245672
$750$	0	0
$751$	15.0962	0.550869	0.275435	−	0.961320i	$-0.411178\pi$
$751$	15.0962	0.550869	0.275435	+	0.961320i	$0.411178\pi$
$752$	0	0
$753$	34.5788	1.26012
$754$	0	0
$755$	−2.30865	−0.0840205
$756$	0	0
$757$	−0.208303	−0.00757092	−0.00378546	−	0.999993i	$-0.501205\pi$
$757$	−0.208303	−0.00757092	−0.00378546	+	0.999993i	$0.501205\pi$
$758$	0	0
$759$	−4.00746	−0.145462
$760$	0	0
$761$	−41.3758	−1.49987	−0.749935	−	0.661511i	$-0.769915\pi$
$761$	−41.3758	−1.49987	−0.749935	+	0.661511i	$0.769915\pi$
$762$	0	0
$763$	2.49839	0.0904479
$764$	0	0
$765$	1.65236	0.0597413
$766$	0	0
$767$	1.22875	0.0443675
$768$	0	0
$769$	−45.5051	−1.64096	−0.820478	−	0.571678i	$-0.806293\pi$
$769$	−45.5051	−1.64096	−0.820478	+	0.571678i	$0.806293\pi$
$770$	0	0
$771$	26.0335	0.937573
$772$	0	0
$773$	−46.8456	−1.68492	−0.842459	−	0.538760i	$-0.818893\pi$
$773$	−46.8456	−1.68492	−0.842459	+	0.538760i	$0.818893\pi$
$774$	0	0
$775$	−33.6061	−1.20717
$776$	0	0
$777$	2.95327	0.105948
$778$	0	0
$779$	4.13914	0.148300
$780$	0	0
$781$	2.55193	0.0913152
$782$	0	0
$783$	29.6890	1.06100
$784$	0	0
$785$	−0.272858	−0.00973872
$786$	0	0
$787$	−48.3311	−1.72282	−0.861409	−	0.507912i	$-0.830417\pi$
$787$	−48.3311	−1.72282	−0.861409	+	0.507912i	$0.830417\pi$
$788$	0	0
$789$	12.7556	0.454111
$790$	0	0
$791$	1.23607	0.0439495
$792$	0	0
$793$	34.5646	1.22742
$794$	0	0
$795$	2.98237	0.105774
$796$	0	0
$797$	−34.9772	−1.23895	−0.619477	−	0.785015i	$-0.712655\pi$
$797$	−34.9772	−1.23895	−0.619477	+	0.785015i	$0.712655\pi$
$798$	0	0
$799$	26.9655	0.953971
$800$	0	0
$801$	−18.9160	−0.668363
$802$	0	0
$803$	−32.3607	−1.14198
$804$	0	0
$805$	0.448674	0.0158137
$806$	0	0
$807$	−17.4537	−0.614401
$808$	0	0
$809$	47.8115	1.68096	0.840481	−	0.541841i	$-0.182273\pi$
$809$	47.8115	1.68096	0.840481	+	0.541841i	$0.182273\pi$
$810$	0	0
$811$	−12.7598	−0.448058	−0.224029	−	0.974582i	$-0.571921\pi$
$811$	−12.7598	−0.448058	−0.224029	+	0.974582i	$0.571921\pi$
$812$	0	0
$813$	37.4316	1.31278
$814$	0	0
$815$	−1.06757	−0.0373954
$816$	0	0
$817$	20.4317	0.714816
$818$	0	0
$819$	8.41781	0.294142
$820$	0	0
$821$	−43.5657	−1.52045	−0.760227	−	0.649657i	$-0.774912\pi$
$821$	−43.5657	−1.52045	−0.760227	+	0.649657i	$0.774912\pi$
$822$	0	0
$823$	1.44289	0.0502960	0.0251480	−	0.999684i	$-0.491994\pi$
$823$	1.44289	0.0502960	0.0251480	+	0.999684i	$0.491994\pi$
$824$	0	0
$825$	14.8695	0.517689
$826$	0	0
$827$	28.2397	0.981990	0.490995	−	0.871162i	$-0.336633\pi$
$827$	28.2397	0.981990	0.490995	+	0.871162i	$0.336633\pi$
$828$	0	0
$829$	6.33036	0.219862	0.109931	−	0.993939i	$-0.464937\pi$
$829$	6.33036	0.219862	0.109931	+	0.993939i	$0.464937\pi$
$830$	0	0
$831$	38.3715	1.33109
$832$	0	0
$833$	−24.4721	−0.847909
$834$	0	0
$835$	−3.89062	−0.134641
$836$	0	0
$837$	−38.1231	−1.31773
$838$	0	0
$839$	52.6519	1.81775	0.908873	−	0.417072i	$-0.136944\pi$
$839$	52.6519	1.81775	0.908873	+	0.417072i	$0.136944\pi$
$840$	0	0
$841$	−0.764045	−0.0263464
$842$	0	0
$843$	−27.2779	−0.939501
$844$	0	0
$845$	−3.39212	−0.116693
$846$	0	0
$847$	6.76393	0.232411
$848$	0	0
$849$	−17.8565	−0.612833
$850$	0	0
$851$	−2.46983	−0.0846647
$852$	0	0
$853$	−21.1090	−0.722760	−0.361380	−	0.932419i	$-0.617694\pi$
$853$	−21.1090	−0.722760	−0.361380	+	0.932419i	$0.617694\pi$
$854$	0	0
$855$	1.38784	0.0474633
$856$	0	0
$857$	7.29333	0.249136	0.124568	−	0.992211i	$-0.460246\pi$
$857$	7.29333	0.249136	0.124568	+	0.992211i	$0.460246\pi$
$858$	0	0
$859$	−1.13617	−0.0387657	−0.0193828	−	0.999812i	$-0.506170\pi$
$859$	−1.13617	−0.0387657	−0.0193828	+	0.999812i	$0.506170\pi$
$860$	0	0
$861$	−1.74902	−0.0596063
$862$	0	0
$863$	−1.73677	−0.0591202	−0.0295601	−	0.999563i	$-0.509411\pi$
$863$	−1.73677	−0.0591202	−0.0295601	+	0.999563i	$0.509411\pi$
$864$	0	0
$865$	0.241990	0.00822790
$866$	0	0
$867$	3.85224	0.130829
$868$	0	0
$869$	−11.8306	−0.401326
$870$	0	0
$871$	−26.6333	−0.902435
$872$	0	0
$873$	−17.9589	−0.607818
$874$	0	0
$875$	−3.35484	−0.113414
$876$	0	0
$877$	−2.27847	−0.0769385	−0.0384693	−	0.999260i	$-0.512248\pi$
$877$	−2.27847	−0.0769385	−0.0384693	+	0.999260i	$0.512248\pi$
$878$	0	0
$879$	5.48746	0.185087
$880$	0	0
$881$	28.9120	0.974069	0.487035	−	0.873383i	$-0.338079\pi$
$881$	28.9120	0.974069	0.487035	+	0.873383i	$0.338079\pi$
$882$	0	0
$883$	−41.0479	−1.38137	−0.690686	−	0.723155i	$-0.742691\pi$
$883$	−41.0479	−1.38137	−0.690686	+	0.723155i	$0.742691\pi$
$884$	0	0
$885$	−0.0856029	−0.00287751
$886$	0	0
$887$	−23.4411	−0.787074	−0.393537	−	0.919309i	$-0.628749\pi$
$887$	−23.4411	−0.787074	−0.393537	+	0.919309i	$0.628749\pi$
$888$	0	0
$889$	−26.6333	−0.893253
$890$	0	0
$891$	7.33790	0.245829
$892$	0	0
$893$	22.6487	0.757911
$894$	0	0
$895$	−4.29401	−0.143533
$896$	0	0
$897$	8.59105	0.286847
$898$	0	0
$899$	−36.2572	−1.20925
$900$	0	0
$901$	−37.9835	−1.26542
$902$	0	0
$903$	−8.63354	−0.287306
$904$	0	0
$905$	−2.85765	−0.0949915
$906$	0	0
$907$	−5.39798	−0.179237	−0.0896185	−	0.995976i	$-0.528565\pi$
$907$	−5.39798	−0.179237	−0.0896185	+	0.995976i	$0.528565\pi$
$908$	0	0
$909$	17.7420	0.588464
$910$	0	0
$911$	−5.06757	−0.167896	−0.0839481	−	0.996470i	$-0.526753\pi$
$911$	−5.06757	−0.167896	−0.0839481	+	0.996470i	$0.526753\pi$
$912$	0	0
$913$	2.16140	0.0715320
$914$	0	0
$915$	−2.40800	−0.0796061
$916$	0	0
$917$	−22.7991	−0.752893
$918$	0	0
$919$	45.2550	1.49282	0.746412	−	0.665484i	$-0.231775\pi$
$919$	45.2550	1.49282	0.746412	+	0.665484i	$0.231775\pi$
$920$	0	0
$921$	5.13632	0.169247
$922$	0	0
$923$	−5.47074	−0.180072
$924$	0	0
$925$	9.16419	0.301317
$926$	0	0
$927$	13.9713	0.458879
$928$	0	0
$929$	8.62169	0.282869	0.141434	−	0.989948i	$-0.454829\pi$
$929$	8.62169	0.282869	0.141434	+	0.989948i	$0.454829\pi$
$930$	0	0
$931$	−20.5545	−0.673647
$932$	0	0
$933$	−12.4509	−0.407624
$934$	0	0
$935$	2.87530	0.0940324
$936$	0	0
$937$	−17.5481	−0.573271	−0.286636	−	0.958040i	$-0.592537\pi$
$937$	−17.5481	−0.573271	−0.286636	+	0.958040i	$0.592537\pi$
$938$	0	0
$939$	23.8261	0.777536
$940$	0	0
$941$	32.7029	1.06608	0.533042	−	0.846089i	$-0.321049\pi$
$941$	32.7029	1.06608	0.533042	+	0.846089i	$0.321049\pi$
$942$	0	0
$943$	1.46271	0.0476324
$944$	0	0
$945$	−1.88854	−0.0614343
$946$	0	0
$947$	59.7085	1.94027	0.970133	−	0.242575i	$-0.0779921\pi$
$947$	59.7085	1.94027	0.970133	+	0.242575i	$0.0779921\pi$
$948$	0	0
$949$	69.3737	2.25197
$950$	0	0
$951$	13.9148	0.451218
$952$	0	0
$953$	−58.8800	−1.90731	−0.953654	−	0.300904i	$-0.902712\pi$
$953$	−58.8800	−1.90731	−0.953654	+	0.300904i	$0.902712\pi$
$954$	0	0
$955$	−4.09731	−0.132586
$956$	0	0
$957$	16.0425	0.518580
$958$	0	0
$959$	−16.3370	−0.527549
$960$	0	0
$961$	15.5571	0.501842
$962$	0	0
$963$	7.34946	0.236833
$964$	0	0
$965$	−0.954691	−0.0307326
$966$	0	0
$967$	−51.4799	−1.65548	−0.827741	−	0.561110i	$-0.810374\pi$
$967$	−51.4799	−1.65548	−0.827741	+	0.561110i	$0.810374\pi$
$968$	0	0
$969$	21.5704	0.692939
$970$	0	0
$971$	27.1359	0.870834	0.435417	−	0.900229i	$-0.356601\pi$
$971$	27.1359	0.870834	0.435417	+	0.900229i	$0.356601\pi$
$972$	0	0
$973$	0.386938	0.0124047
$974$	0	0
$975$	−31.8767	−1.02087
$976$	0	0
$977$	25.8667	0.827548	0.413774	−	0.910380i	$-0.364210\pi$
$977$	25.8667	0.827548	0.413774	+	0.910380i	$0.364210\pi$
$978$	0	0
$979$	−32.9160	−1.05200
$980$	0	0
$981$	2.73098	0.0871936
$982$	0	0
$983$	−49.7983	−1.58832	−0.794159	−	0.607710i	$-0.792088\pi$
$983$	−49.7983	−1.58832	−0.794159	+	0.607710i	$0.792088\pi$
$984$	0	0
$985$	−3.28911	−0.104800
$986$	0	0
$987$	−9.57035	−0.304628
$988$	0	0
$989$	7.22027	0.229591
$990$	0	0
$991$	−36.7879	−1.16861	−0.584303	−	0.811536i	$-0.698632\pi$
$991$	−36.7879	−1.16861	−0.584303	+	0.811536i	$0.698632\pi$
$992$	0	0
$993$	30.0365	0.953178
$994$	0	0
$995$	−1.48768	−0.0471627
$996$	0	0
$997$	−6.45172	−0.204328	−0.102164	−	0.994768i	$-0.532577\pi$
$997$	−6.45172	−0.204328	−0.102164	+	0.994768i	$0.532577\pi$
$998$	0	0
$999$	10.3959	0.328913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7232.2.a.v.1.3		4
4.3	odd	2		7232.2.a.u.1.2		4
8.3	odd	2		226.2.a.d.1.3	✓	4
8.5	even	2		1808.2.a.j.1.2		4
24.11	even	2		2034.2.a.r.1.3		4
40.19	odd	2		5650.2.a.o.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
226.2.a.d.1.3	✓	4	8.3	odd	2
1808.2.a.j.1.2		4	8.5	even	2
2034.2.a.r.1.3		4	24.11	even	2
5650.2.a.o.1.2		4	40.19	odd	2
7232.2.a.u.1.2		4	4.3	odd	2
7232.2.a.v.1.3		4	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:	\( N \)	\(=\)	\( 7232 = 2^{6} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28408 q^{3} -0.273457 q^{5} -1.23607 q^{7} -1.35114 q^{9} -2.35114 q^{11} +5.04029 q^{13} -0.351141 q^{15} +4.47214 q^{17} +3.75621 q^{19} -1.58721 q^{21} +1.32739 q^{23} -4.92522 q^{25} -5.58721 q^{27} -5.31375 q^{29} +6.82328 q^{31} -3.01905 q^{33} +0.338012 q^{35} -1.86067 q^{37} +6.47214 q^{39} +1.10194 q^{41} +5.43945 q^{43} +0.369480 q^{45} +6.02967 q^{47} -5.47214 q^{49} +5.74258 q^{51} -8.49338 q^{53} +0.642937 q^{55} +4.82328 q^{57} +0.243785 q^{59} +6.85765 q^{61} +1.67010 q^{63} -1.37831 q^{65} -5.28408 q^{67} +1.70447 q^{69} -1.08540 q^{71} +13.7638 q^{73} -6.32437 q^{75} +2.90617 q^{77} +5.03186 q^{79} -3.12099 q^{81} -0.919299 q^{83} -1.22294 q^{85} -6.82328 q^{87} +14.0000 q^{89} -6.23015 q^{91} +8.76163 q^{93} -1.02717 q^{95} +13.2917 q^{97} +3.17672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{13} + 8 q^{15} - 6 q^{19} + 12 q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - 4 q^{35} + 8 q^{37} + 8 q^{39} + 8 q^{41} - 6 q^{43} + 16 q^{45} + 6 q^{47} - 4 q^{49}+ \cdots + 40 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^13 + 8 * q^15 - 6 * q^19 + 12 * q^21 + 6 * q^23 + 4 * q^25 - 4 * q^27 - 4 * q^35 + 8 * q^37 + 8 * q^39 + 8 * q^41 - 6 * q^43 + 16 * q^45 + 6 * q^47 - 4 * q^49 - 20 * q^51 - 4 * q^53 + 20 * q^55 - 8 * q^57 + 22 * q^59 - 16 * q^61 + 4 * q^63 + 24 * q^65 - 18 * q^67 - 12 * q^69 - 22 * q^71 + 2 * q^75 + 2 * q^79 - 4 * q^81 + 12 * q^83 + 56 * q^89 - 24 * q^91 - 20 * q^93 + 16 * q^95 + 16 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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