LMFDB - Embedded newform 2925.2.a.bf.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2925,2,Mod(1,2925)] 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("2925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,5,0,0,2,-9,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.3562425912$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21432 q^{2} -0.525428 q^{4} +2.90321 q^{7} -3.06668 q^{8} -0.214320 q^{11} -1.00000 q^{13} +3.52543 q^{14} -2.67307 q^{16} -6.42864 q^{17} +2.21432 q^{19} -0.260253 q^{22} -4.68889 q^{23} -1.21432 q^{26} -1.52543 q^{28} -8.70964 q^{29} -5.59210 q^{31} +2.88739 q^{32} -7.80642 q^{34} +2.28100 q^{37} +2.68889 q^{38} -3.05086 q^{41} +6.36196 q^{43} +0.112610 q^{44} -5.69381 q^{46} -1.09679 q^{47} +1.42864 q^{49} +0.525428 q^{52} +6.23506 q^{53} -8.90321 q^{56} -10.5763 q^{58} +9.26517 q^{59} -0.280996 q^{61} -6.79060 q^{62} +8.85236 q^{64} -7.76049 q^{67} +3.37778 q^{68} +6.08097 q^{71} -10.2810 q^{73} +2.76986 q^{74} -1.16346 q^{76} -0.622216 q^{77} -14.2351 q^{79} -3.70471 q^{82} -9.52543 q^{83} +7.72546 q^{86} +0.657249 q^{88} +5.61285 q^{89} -2.90321 q^{91} +2.46367 q^{92} -1.33185 q^{94} -18.0415 q^{97} +1.73483 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} + 6 q^{11} - 3 q^{13} + 4 q^{14} + 5 q^{16} - 6 q^{17} - 14 q^{22} - 14 q^{23} + 3 q^{26} + 2 q^{28} - 6 q^{29} - 10 q^{31} - 11 q^{32} - 10 q^{34} + 8 q^{38}+ \cdots + 25 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8 + 6 * q^11 - 3 * q^13 + 4 * q^14 + 5 * q^16 - 6 * q^17 - 14 * q^22 - 14 * q^23 + 3 * q^26 + 2 * q^28 - 6 * q^29 - 10 * q^31 - 11 * q^32 - 10 * q^34 + 8 * q^38 + 4 * q^41 + 6 * q^43 + 20 * q^44 + 16 * q^46 - 10 * q^47 - 9 * q^49 - 5 * q^52 - 8 * q^53 - 20 * q^56 - 12 * q^58 + 8 * q^59 + 6 * q^61 + 6 * q^62 + 33 * q^64 + 10 * q^67 + 10 * q^68 + 12 * q^71 - 24 * q^73 + 2 * q^74 - 10 * q^76 - 2 * q^77 - 16 * q^79 - 24 * q^82 - 22 * q^83 + 16 * q^86 - 24 * q^88 - 10 * q^89 - 2 * q^91 - 32 * q^92 + 16 * q^94 - 14 * q^97 + 25 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.21432	0.858654	0.429327	−	0.903149i	$-0.358751\pi$
$2$	1.21432	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$3$	0	0
$3$	0	0
$4$	−0.525428	−0.262714
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.90321	1.09731	0.548655	−	0.836049i	$-0.315140\pi$
$7$	2.90321	1.09731	0.548655	+	0.836049i	$0.315140\pi$
$8$	−3.06668	−1.08423
$9$	0	0
$10$	0	0
$11$	−0.214320	−0.0646198	−0.0323099	−	0.999478i	$-0.510286\pi$
$11$	−0.214320	−0.0646198	−0.0323099	+	0.999478i	$0.510286\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	3.52543	0.942210
$15$	0	0
$16$	−2.67307	−0.668268
$17$	−6.42864	−1.55917	−0.779587	−	0.626294i	$-0.784571\pi$
$17$	−6.42864	−1.55917	−0.779587	+	0.626294i	$0.784571\pi$
$18$	0	0
$19$	2.21432	0.508000	0.254000	−	0.967204i	$-0.418254\pi$
$19$	2.21432	0.508000	0.254000	+	0.967204i	$0.418254\pi$
$20$	0	0
$21$	0	0
$22$	−0.260253	−0.0554861
$23$	−4.68889	−0.977702	−0.488851	−	0.872367i	$-0.662584\pi$
$23$	−4.68889	−0.977702	−0.488851	+	0.872367i	$0.662584\pi$
$24$	0	0
$25$	0	0
$26$	−1.21432	−0.238148
$27$	0	0
$28$	−1.52543	−0.288279
$29$	−8.70964	−1.61734	−0.808669	−	0.588263i	$-0.799812\pi$
$29$	−8.70964	−1.61734	−0.808669	+	0.588263i	$0.799812\pi$
$30$	0	0
$31$	−5.59210	−1.00437	−0.502186	−	0.864760i	$-0.667471\pi$
$31$	−5.59210	−1.00437	−0.502186	+	0.864760i	$0.667471\pi$
$32$	2.88739	0.510423
$33$	0	0
$34$	−7.80642	−1.33879
$35$	0	0
$36$	0	0
$37$	2.28100	0.374993	0.187497	−	0.982265i	$-0.439963\pi$
$37$	2.28100	0.374993	0.187497	+	0.982265i	$0.439963\pi$
$38$	2.68889	0.436196
$39$	0	0
$40$	0	0
$41$	−3.05086	−0.476464	−0.238232	−	0.971208i	$-0.576568\pi$
$41$	−3.05086	−0.476464	−0.238232	+	0.971208i	$0.576568\pi$
$42$	0	0
$43$	6.36196	0.970190	0.485095	−	0.874461i	$-0.338785\pi$
$43$	6.36196	0.970190	0.485095	+	0.874461i	$0.338785\pi$
$44$	0.112610	0.0169765
$45$	0	0
$46$	−5.69381	−0.839507
$47$	−1.09679	−0.159983	−0.0799915	−	0.996796i	$-0.525489\pi$
$47$	−1.09679	−0.159983	−0.0799915	+	0.996796i	$0.525489\pi$
$48$	0	0
$49$	1.42864	0.204091
$50$	0	0
$51$	0	0
$52$	0.525428	0.0728637
$53$	6.23506	0.856452	0.428226	−	0.903672i	$-0.359139\pi$
$53$	6.23506	0.856452	0.428226	+	0.903672i	$0.359139\pi$
$54$	0	0
$55$	0	0
$56$	−8.90321	−1.18974
$57$	0	0
$58$	−10.5763	−1.38873
$59$	9.26517	1.20622	0.603112	−	0.797657i	$-0.293927\pi$
$59$	9.26517	1.20622	0.603112	+	0.797657i	$0.293927\pi$
$60$	0	0
$61$	−0.280996	−0.0359779	−0.0179889	−	0.999838i	$-0.505726\pi$
$61$	−0.280996	−0.0359779	−0.0179889	+	0.999838i	$0.505726\pi$
$62$	−6.79060	−0.862407
$63$	0	0
$64$	8.85236	1.10654
$65$	0	0
$66$	0	0
$67$	−7.76049	−0.948095	−0.474047	−	0.880499i	$-0.657207\pi$
$67$	−7.76049	−0.948095	−0.474047	+	0.880499i	$0.657207\pi$
$68$	3.37778	0.409617
$69$	0	0
$70$	0	0
$71$	6.08097	0.721678	0.360839	−	0.932628i	$-0.382490\pi$
$71$	6.08097	0.721678	0.360839	+	0.932628i	$0.382490\pi$
$72$	0	0
$73$	−10.2810	−1.20330	−0.601650	−	0.798760i	$-0.705490\pi$
$73$	−10.2810	−1.20330	−0.601650	+	0.798760i	$0.705490\pi$
$74$	2.76986	0.321990
$75$	0	0
$76$	−1.16346	−0.133459
$77$	−0.622216	−0.0709081
$78$	0	0
$79$	−14.2351	−1.60157	−0.800785	−	0.598952i	$-0.795584\pi$
$79$	−14.2351	−1.60157	−0.800785	+	0.598952i	$0.795584\pi$
$80$	0	0
$81$	0	0
$82$	−3.70471	−0.409117
$83$	−9.52543	−1.04555	−0.522776	−	0.852470i	$-0.675103\pi$
$83$	−9.52543	−1.04555	−0.522776	+	0.852470i	$0.675103\pi$
$84$	0	0
$85$	0	0
$86$	7.72546	0.833057
$87$	0	0
$88$	0.657249	0.0700630
$89$	5.61285	0.594961	0.297480	−	0.954728i	$-0.403854\pi$
$89$	5.61285	0.594961	0.297480	+	0.954728i	$0.403854\pi$
$90$	0	0
$91$	−2.90321	−0.304339
$92$	2.46367	0.256856
$93$	0	0
$94$	−1.33185	−0.137370
$95$	0	0
$96$	0	0
$97$	−18.0415	−1.83184	−0.915918	−	0.401366i	$-0.868536\pi$
$97$	−18.0415	−1.83184	−0.915918	+	0.401366i	$0.868536\pi$
$98$	1.73483	0.175244
$99$	0	0
$100$	0	0
$101$	−3.93978	−0.392022	−0.196011	−	0.980602i	$-0.562799\pi$
$101$	−3.93978	−0.392022	−0.196011	+	0.980602i	$0.562799\pi$
$102$	0	0
$103$	2.82225	0.278084	0.139042	−	0.990286i	$-0.455598\pi$
$103$	2.82225	0.278084	0.139042	+	0.990286i	$0.455598\pi$
$104$	3.06668	0.300712
$105$	0	0
$106$	7.57136	0.735396
$107$	−17.1175	−1.65481	−0.827407	−	0.561603i	$-0.810185\pi$
$107$	−17.1175	−1.65481	−0.827407	+	0.561603i	$0.810185\pi$
$108$	0	0
$109$	−16.7239	−1.60186	−0.800931	−	0.598757i	$-0.795662\pi$
$109$	−16.7239	−1.60186	−0.800931	+	0.598757i	$0.795662\pi$
$110$	0	0
$111$	0	0
$112$	−7.76049	−0.733297
$113$	−1.18421	−0.111401	−0.0557005	−	0.998448i	$-0.517739\pi$
$113$	−1.18421	−0.111401	−0.0557005	+	0.998448i	$0.517739\pi$
$114$	0	0
$115$	0	0
$116$	4.57628	0.424897
$117$	0	0
$118$	11.2509	1.03573
$119$	−18.6637	−1.71090
$120$	0	0
$121$	−10.9541	−0.995824
$122$	−0.341219	−0.0308925
$123$	0	0
$124$	2.93825	0.263862
$125$	0	0
$126$	0	0
$127$	2.30174	0.204246	0.102123	−	0.994772i	$-0.467436\pi$
$127$	2.30174	0.204246	0.102123	+	0.994772i	$0.467436\pi$
$128$	4.97481	0.439715
$129$	0	0
$130$	0	0
$131$	13.4193	1.17245	0.586224	−	0.810149i	$-0.300614\pi$
$131$	13.4193	1.17245	0.586224	+	0.810149i	$0.300614\pi$
$132$	0	0
$133$	6.42864	0.557434
$134$	−9.42372	−0.814085
$135$	0	0
$136$	19.7146	1.69051
$137$	−19.1526	−1.63631	−0.818157	−	0.574995i	$-0.805004\pi$
$137$	−19.1526	−1.63631	−0.818157	+	0.574995i	$0.805004\pi$
$138$	0	0
$139$	19.0923	1.61939	0.809696	−	0.586850i	$-0.199632\pi$
$139$	19.0923	1.61939	0.809696	+	0.586850i	$0.199632\pi$
$140$	0	0
$141$	0	0
$142$	7.38424	0.619671
$143$	0.214320	0.0179223
$144$	0	0
$145$	0	0
$146$	−12.4844	−1.03322
$147$	0	0
$148$	−1.19850	−0.0985160
$149$	3.57136	0.292577	0.146289	−	0.989242i	$-0.453267\pi$
$149$	3.57136	0.292577	0.146289	+	0.989242i	$0.453267\pi$
$150$	0	0
$151$	−1.26517	−0.102958	−0.0514792	−	0.998674i	$-0.516394\pi$
$151$	−1.26517	−0.102958	−0.0514792	+	0.998674i	$0.516394\pi$
$152$	−6.79060	−0.550791
$153$	0	0
$154$	−0.755569	−0.0608855
$155$	0	0
$156$	0	0
$157$	−5.61285	−0.447954	−0.223977	−	0.974594i	$-0.571904\pi$
$157$	−5.61285	−0.447954	−0.223977	+	0.974594i	$0.571904\pi$
$158$	−17.2859	−1.37519
$159$	0	0
$160$	0	0
$161$	−13.6128	−1.07284
$162$	0	0
$163$	3.71900	0.291295	0.145647	−	0.989337i	$-0.453473\pi$
$163$	3.71900	0.291295	0.145647	+	0.989337i	$0.453473\pi$
$164$	1.60300	0.125174
$165$	0	0
$166$	−11.5669	−0.897767
$167$	7.03657	0.544506	0.272253	−	0.962226i	$-0.412231\pi$
$167$	7.03657	0.544506	0.272253	+	0.962226i	$0.412231\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−3.34275	−0.254882
$173$	−0.723926	−0.0550391	−0.0275195	−	0.999621i	$-0.508761\pi$
$173$	−0.723926	−0.0550391	−0.0275195	+	0.999621i	$0.508761\pi$
$174$	0	0
$175$	0	0
$176$	0.572892	0.0431833
$177$	0	0
$178$	6.81579	0.510865
$179$	4.04149	0.302075	0.151037	−	0.988528i	$-0.451739\pi$
$179$	4.04149	0.302075	0.151037	+	0.988528i	$0.451739\pi$
$180$	0	0
$181$	2.34122	0.174021	0.0870107	−	0.996207i	$-0.472269\pi$
$181$	2.34122	0.174021	0.0870107	+	0.996207i	$0.472269\pi$
$182$	−3.52543	−0.261322
$183$	0	0
$184$	14.3793	1.06006
$185$	0	0
$186$	0	0
$187$	1.37778	0.100754
$188$	0.576283	0.0420297
$189$	0	0
$190$	0	0
$191$	2.10171	0.152074	0.0760372	−	0.997105i	$-0.475773\pi$
$191$	2.10171	0.152074	0.0760372	+	0.997105i	$0.475773\pi$
$192$	0	0
$193$	13.5210	0.973262	0.486631	−	0.873608i	$-0.338226\pi$
$193$	13.5210	0.973262	0.486631	+	0.873608i	$0.338226\pi$
$194$	−21.9081	−1.57291
$195$	0	0
$196$	−0.750647	−0.0536176
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	22.1432	1.56969	0.784845	−	0.619692i	$-0.212743\pi$
$199$	22.1432	1.56969	0.784845	+	0.619692i	$0.212743\pi$
$200$	0	0
$201$	0	0
$202$	−4.78415	−0.336612
$203$	−25.2859	−1.77472
$204$	0	0
$205$	0	0
$206$	3.42711	0.238778
$207$	0	0
$208$	2.67307	0.185344
$209$	−0.474572	−0.0328269
$210$	0	0
$211$	19.6543	1.35306	0.676530	−	0.736415i	$-0.263483\pi$
$211$	19.6543	1.35306	0.676530	+	0.736415i	$0.263483\pi$
$212$	−3.27607	−0.225002
$213$	0	0
$214$	−20.7862	−1.42091
$215$	0	0
$216$	0	0
$217$	−16.2351	−1.10211
$218$	−20.3082	−1.37544
$219$	0	0
$220$	0	0
$221$	6.42864	0.432437
$222$	0	0
$223$	−19.6686	−1.31711	−0.658554	−	0.752533i	$-0.728832\pi$
$223$	−19.6686	−1.31711	−0.658554	+	0.752533i	$0.728832\pi$
$224$	8.38271	0.560093
$225$	0	0
$226$	−1.43801	−0.0956548
$227$	−13.2716	−0.880869	−0.440434	−	0.897785i	$-0.645176\pi$
$227$	−13.2716	−0.880869	−0.440434	+	0.897785i	$0.645176\pi$
$228$	0	0
$229$	−2.42864	−0.160489	−0.0802445	−	0.996775i	$-0.525570\pi$
$229$	−2.42864	−0.160489	−0.0802445	+	0.996775i	$0.525570\pi$
$230$	0	0
$231$	0	0
$232$	26.7096	1.75357
$233$	16.1748	1.05965	0.529825	−	0.848107i	$-0.322258\pi$
$233$	16.1748	1.05965	0.529825	+	0.848107i	$0.322258\pi$
$234$	0	0
$235$	0	0
$236$	−4.86818	−0.316891
$237$	0	0
$238$	−22.6637	−1.46907
$239$	12.7763	0.826431	0.413215	−	0.910633i	$-0.364406\pi$
$239$	12.7763	0.826431	0.413215	+	0.910633i	$0.364406\pi$
$240$	0	0
$241$	−5.89829	−0.379942	−0.189971	−	0.981790i	$-0.560839\pi$
$241$	−5.89829	−0.379942	−0.189971	+	0.981790i	$0.560839\pi$
$242$	−13.3017	−0.855068
$243$	0	0
$244$	0.147643	0.00945189
$245$	0	0
$246$	0	0
$247$	−2.21432	−0.140894
$248$	17.1492	1.08897
$249$	0	0
$250$	0	0
$251$	2.07313	0.130855	0.0654274	−	0.997857i	$-0.479159\pi$
$251$	2.07313	0.130855	0.0654274	+	0.997857i	$0.479159\pi$
$252$	0	0
$253$	1.00492	0.0631789
$254$	2.79505	0.175377
$255$	0	0
$256$	−11.6637	−0.728981
$257$	18.3970	1.14757	0.573787	−	0.819005i	$-0.305474\pi$
$257$	18.3970	1.14757	0.573787	+	0.819005i	$0.305474\pi$
$258$	0	0
$259$	6.62222	0.411484
$260$	0	0
$261$	0	0
$262$	16.2953	1.00673
$263$	11.0257	0.679872	0.339936	−	0.940449i	$-0.389595\pi$
$263$	11.0257	0.679872	0.339936	+	0.940449i	$0.389595\pi$
$264$	0	0
$265$	0	0
$266$	7.80642	0.478643
$267$	0	0
$268$	4.07758	0.249078
$269$	16.1432	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$269$	16.1432	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$270$	0	0
$271$	13.0114	0.790385	0.395192	−	0.918598i	$-0.370678\pi$
$271$	13.0114	0.790385	0.395192	+	0.918598i	$0.370678\pi$
$272$	17.1842	1.04195
$273$	0	0
$274$	−23.2573	−1.40503
$275$	0	0
$276$	0	0
$277$	−7.57136	−0.454919	−0.227459	−	0.973788i	$-0.573042\pi$
$277$	−7.57136	−0.454919	−0.227459	+	0.973788i	$0.573042\pi$
$278$	23.1842	1.39050
$279$	0	0
$280$	0	0
$281$	−6.75557	−0.403003	−0.201502	−	0.979488i	$-0.564582\pi$
$281$	−6.75557	−0.403003	−0.201502	+	0.979488i	$0.564582\pi$
$282$	0	0
$283$	−19.0859	−1.13454	−0.567269	−	0.823532i	$-0.692000\pi$
$283$	−19.0859	−1.13454	−0.567269	+	0.823532i	$0.692000\pi$
$284$	−3.19511	−0.189595
$285$	0	0
$286$	0.260253	0.0153891
$287$	−8.85728	−0.522829
$288$	0	0
$289$	24.3274	1.43102
$290$	0	0
$291$	0	0
$292$	5.40192	0.316123
$293$	−8.08742	−0.472472	−0.236236	−	0.971696i	$-0.575914\pi$
$293$	−8.08742	−0.472472	−0.236236	+	0.971696i	$0.575914\pi$
$294$	0	0
$295$	0	0
$296$	−6.99508	−0.406581
$297$	0	0
$298$	4.33677	0.251223
$299$	4.68889	0.271166
$300$	0	0
$301$	18.4701	1.06460
$302$	−1.53633	−0.0884057
$303$	0	0
$304$	−5.91903	−0.339480
$305$	0	0
$306$	0	0
$307$	13.4336	0.766694	0.383347	−	0.923604i	$-0.374771\pi$
$307$	13.4336	0.766694	0.383347	+	0.923604i	$0.374771\pi$
$308$	0.326929	0.0186285
$309$	0	0
$310$	0	0
$311$	−20.2034	−1.14563	−0.572815	−	0.819684i	$-0.694149\pi$
$311$	−20.2034	−1.14563	−0.572815	+	0.819684i	$0.694149\pi$
$312$	0	0
$313$	−15.1111	−0.854129	−0.427064	−	0.904221i	$-0.640452\pi$
$313$	−15.1111	−0.854129	−0.427064	+	0.904221i	$0.640452\pi$
$314$	−6.81579	−0.384637
$315$	0	0
$316$	7.47949	0.420754
$317$	22.2810	1.25143	0.625713	−	0.780054i	$-0.284808\pi$
$317$	22.2810	1.25143	0.625713	+	0.780054i	$0.284808\pi$
$318$	0	0
$319$	1.86665	0.104512
$320$	0	0
$321$	0	0
$322$	−16.5303	−0.921200
$323$	−14.2351	−0.792060
$324$	0	0
$325$	0	0
$326$	4.51606	0.250121
$327$	0	0
$328$	9.35599	0.516598
$329$	−3.18421	−0.175551
$330$	0	0
$331$	8.25581	0.453780	0.226890	−	0.973920i	$-0.427144\pi$
$331$	8.25581	0.453780	0.226890	+	0.973920i	$0.427144\pi$
$332$	5.00492	0.274681
$333$	0	0
$334$	8.54464	0.467542
$335$	0	0
$336$	0	0
$337$	13.7462	0.748803	0.374402	−	0.927267i	$-0.377848\pi$
$337$	13.7462	0.748803	0.374402	+	0.927267i	$0.377848\pi$
$338$	1.21432	0.0660503
$339$	0	0
$340$	0	0
$341$	1.19850	0.0649023
$342$	0	0
$343$	−16.1748	−0.873359
$344$	−19.5101	−1.05191
$345$	0	0
$346$	−0.879077	−0.0472595
$347$	1.21924	0.0654523	0.0327262	−	0.999464i	$-0.489581\pi$
$347$	1.21924	0.0654523	0.0327262	+	0.999464i	$0.489581\pi$
$348$	0	0
$349$	22.5116	1.20502	0.602510	−	0.798112i	$-0.294168\pi$
$349$	22.5116	1.20502	0.602510	+	0.798112i	$0.294168\pi$
$350$	0	0
$351$	0	0
$352$	−0.618825	−0.0329835
$353$	−14.2810	−0.760101	−0.380050	−	0.924966i	$-0.624093\pi$
$353$	−14.2810	−0.760101	−0.380050	+	0.924966i	$0.624093\pi$
$354$	0	0
$355$	0	0
$356$	−2.94914	−0.156304
$357$	0	0
$358$	4.90766	0.259378
$359$	12.1541	0.641469	0.320734	−	0.947169i	$-0.396070\pi$
$359$	12.1541	0.641469	0.320734	+	0.947169i	$0.396070\pi$
$360$	0	0
$361$	−14.0968	−0.741936
$362$	2.84299	0.149424
$363$	0	0
$364$	1.52543	0.0799541
$365$	0	0
$366$	0	0
$367$	4.65725	0.243106	0.121553	−	0.992585i	$-0.461212\pi$
$367$	4.65725	0.243106	0.121553	+	0.992585i	$0.461212\pi$
$368$	12.5337	0.653366
$369$	0	0
$370$	0	0
$371$	18.1017	0.939794
$372$	0	0
$373$	−34.9403	−1.80914	−0.904569	−	0.426328i	$-0.859807\pi$
$373$	−34.9403	−1.80914	−0.904569	+	0.426328i	$0.859807\pi$
$374$	1.67307	0.0865124
$375$	0	0
$376$	3.36349	0.173459
$377$	8.70964	0.448569
$378$	0	0
$379$	−17.4717	−0.897459	−0.448729	−	0.893668i	$-0.648123\pi$
$379$	−17.4717	−0.897459	−0.448729	+	0.893668i	$0.648123\pi$
$380$	0	0
$381$	0	0
$382$	2.55215	0.130579
$383$	−18.6780	−0.954401	−0.477200	−	0.878794i	$-0.658348\pi$
$383$	−18.6780	−0.954401	−0.477200	+	0.878794i	$0.658348\pi$
$384$	0	0
$385$	0	0
$386$	16.4188	0.835695
$387$	0	0
$388$	9.47949	0.481248
$389$	−1.61285	−0.0817746	−0.0408873	−	0.999164i	$-0.513018\pi$
$389$	−1.61285	−0.0817746	−0.0408873	+	0.999164i	$0.513018\pi$
$390$	0	0
$391$	30.1432	1.52441
$392$	−4.38118	−0.221283
$393$	0	0
$394$	−2.42864	−0.122353
$395$	0	0
$396$	0	0
$397$	−6.57628	−0.330054	−0.165027	−	0.986289i	$-0.552771\pi$
$397$	−6.57628	−0.330054	−0.165027	+	0.986289i	$0.552771\pi$
$398$	26.8889	1.34782
$399$	0	0
$400$	0	0
$401$	21.9081	1.09404	0.547020	−	0.837120i	$-0.315762\pi$
$401$	21.9081	1.09404	0.547020	+	0.837120i	$0.315762\pi$
$402$	0	0
$403$	5.59210	0.278563
$404$	2.07007	0.102990
$405$	0	0
$406$	−30.7052	−1.52387
$407$	−0.488863	−0.0242320
$408$	0	0
$409$	−10.1936	−0.504040	−0.252020	−	0.967722i	$-0.581095\pi$
$409$	−10.1936	−0.504040	−0.252020	+	0.967722i	$0.581095\pi$
$410$	0	0
$411$	0	0
$412$	−1.48289	−0.0730565
$413$	26.8988	1.32360
$414$	0	0
$415$	0	0
$416$	−2.88739	−0.141566
$417$	0	0
$418$	−0.576283	−0.0281869
$419$	−7.31756	−0.357486	−0.178743	−	0.983896i	$-0.557203\pi$
$419$	−7.31756	−0.357486	−0.178743	+	0.983896i	$0.557203\pi$
$420$	0	0
$421$	−7.86665	−0.383397	−0.191698	−	0.981454i	$-0.561400\pi$
$421$	−7.86665	−0.383397	−0.191698	+	0.981454i	$0.561400\pi$
$422$	23.8666	1.16181
$423$	0	0
$424$	−19.1209	−0.928594
$425$	0	0
$426$	0	0
$427$	−0.815792	−0.0394789
$428$	8.99402	0.434743
$429$	0	0
$430$	0	0
$431$	38.9195	1.87469	0.937343	−	0.348407i	$-0.113277\pi$
$431$	38.9195	1.87469	0.937343	+	0.348407i	$0.113277\pi$
$432$	0	0
$433$	20.2034	0.970914	0.485457	−	0.874260i	$-0.338653\pi$
$433$	20.2034	0.970914	0.485457	+	0.874260i	$0.338653\pi$
$434$	−19.7146	−0.946329
$435$	0	0
$436$	8.78721	0.420831
$437$	−10.3827	−0.496672
$438$	0	0
$439$	10.8889	0.519700	0.259850	−	0.965649i	$-0.416327\pi$
$439$	10.8889	0.519700	0.259850	+	0.965649i	$0.416327\pi$
$440$	0	0
$441$	0	0
$442$	7.80642	0.371314
$443$	−28.6287	−1.36019	−0.680095	−	0.733124i	$-0.738061\pi$
$443$	−28.6287	−1.36019	−0.680095	+	0.733124i	$0.738061\pi$
$444$	0	0
$445$	0	0
$446$	−23.8840	−1.13094
$447$	0	0
$448$	25.7003	1.21422
$449$	10.9304	0.515838	0.257919	−	0.966167i	$-0.416963\pi$
$449$	10.9304	0.515838	0.257919	+	0.966167i	$0.416963\pi$
$450$	0	0
$451$	0.653858	0.0307890
$452$	0.622216	0.0292666
$453$	0	0
$454$	−16.1160	−0.756361
$455$	0	0
$456$	0	0
$457$	11.4064	0.533567	0.266784	−	0.963756i	$-0.414039\pi$
$457$	11.4064	0.533567	0.266784	+	0.963756i	$0.414039\pi$
$458$	−2.94914	−0.137804
$459$	0	0
$460$	0	0
$461$	26.1334	1.21715	0.608576	−	0.793496i	$-0.291741\pi$
$461$	26.1334	1.21715	0.608576	+	0.793496i	$0.291741\pi$
$462$	0	0
$463$	7.92242	0.368186	0.184093	−	0.982909i	$-0.441065\pi$
$463$	7.92242	0.368186	0.184093	+	0.982909i	$0.441065\pi$
$464$	23.2815	1.08082
$465$	0	0
$466$	19.6414	0.909872
$467$	10.8923	0.504036	0.252018	−	0.967723i	$-0.418906\pi$
$467$	10.8923	0.504036	0.252018	+	0.967723i	$0.418906\pi$
$468$	0	0
$469$	−22.5303	−1.04035
$470$	0	0
$471$	0	0
$472$	−28.4133	−1.30783
$473$	−1.36349	−0.0626935
$474$	0	0
$475$	0	0
$476$	9.80642	0.449477
$477$	0	0
$478$	15.5145	0.709618
$479$	9.13182	0.417244	0.208622	−	0.977996i	$-0.433102\pi$
$479$	9.13182	0.417244	0.208622	+	0.977996i	$0.433102\pi$
$480$	0	0
$481$	−2.28100	−0.104004
$482$	−7.16241	−0.326239
$483$	0	0
$484$	5.75557	0.261617
$485$	0	0
$486$	0	0
$487$	16.1891	0.733600	0.366800	−	0.930300i	$-0.380453\pi$
$487$	16.1891	0.733600	0.366800	+	0.930300i	$0.380453\pi$
$488$	0.861725	0.0390084
$489$	0	0
$490$	0	0
$491$	−26.2636	−1.18526	−0.592631	−	0.805474i	$-0.701911\pi$
$491$	−26.2636	−1.18526	−0.592631	+	0.805474i	$0.701911\pi$
$492$	0	0
$493$	55.9911	2.52171
$494$	−2.68889	−0.120979
$495$	0	0
$496$	14.9481	0.671189
$497$	17.6543	0.791905
$498$	0	0
$499$	30.0306	1.34435	0.672177	−	0.740391i	$-0.265359\pi$
$499$	30.0306	1.34435	0.672177	+	0.740391i	$0.265359\pi$
$500$	0	0
$501$	0	0
$502$	2.51744	0.112359
$503$	−16.7304	−0.745971	−0.372985	−	0.927837i	$-0.621666\pi$
$503$	−16.7304	−0.745971	−0.372985	+	0.927837i	$0.621666\pi$
$504$	0	0
$505$	0	0
$506$	1.22030	0.0542488
$507$	0	0
$508$	−1.20940	−0.0536583
$509$	11.9684	0.530488	0.265244	−	0.964181i	$-0.414547\pi$
$509$	11.9684	0.530488	0.265244	+	0.964181i	$0.414547\pi$
$510$	0	0
$511$	−29.8479	−1.32039
$512$	−24.1131	−1.06566
$513$	0	0
$514$	22.3398	0.985368
$515$	0	0
$516$	0	0
$517$	0.235063	0.0103381
$518$	8.04149	0.353323
$519$	0	0
$520$	0	0
$521$	−5.75065	−0.251940	−0.125970	−	0.992034i	$-0.540204\pi$
$521$	−5.75065	−0.251940	−0.125970	+	0.992034i	$0.540204\pi$
$522$	0	0
$523$	20.8035	0.909674	0.454837	−	0.890575i	$-0.349698\pi$
$523$	20.8035	0.909674	0.454837	+	0.890575i	$0.349698\pi$
$524$	−7.05086	−0.308018
$525$	0	0
$526$	13.3887	0.583774
$527$	35.9496	1.56599
$528$	0	0
$529$	−1.01429	−0.0440996
$530$	0	0
$531$	0	0
$532$	−3.37778	−0.146446
$533$	3.05086	0.132147
$534$	0	0
$535$	0	0
$536$	23.7989	1.02796
$537$	0	0
$538$	19.6030	0.845145
$539$	−0.306186	−0.0131883
$540$	0	0
$541$	16.6222	0.714645	0.357322	−	0.933981i	$-0.383690\pi$
$541$	16.6222	0.714645	0.357322	+	0.933981i	$0.383690\pi$
$542$	15.8000	0.678667
$543$	0	0
$544$	−18.5620	−0.795839
$545$	0	0
$546$	0	0
$547$	29.9748	1.28163	0.640815	−	0.767695i	$-0.278596\pi$
$547$	29.9748	1.28163	0.640815	+	0.767695i	$0.278596\pi$
$548$	10.0633	0.429882
$549$	0	0
$550$	0	0
$551$	−19.2859	−0.821608
$552$	0	0
$553$	−41.3274	−1.75742
$554$	−9.19405	−0.390618
$555$	0	0
$556$	−10.0316	−0.425436
$557$	−5.03657	−0.213406	−0.106703	−	0.994291i	$-0.534029\pi$
$557$	−5.03657	−0.213406	−0.106703	+	0.994291i	$0.534029\pi$
$558$	0	0
$559$	−6.36196	−0.269082
$560$	0	0
$561$	0	0
$562$	−8.20342	−0.346040
$563$	2.88247	0.121482	0.0607408	−	0.998154i	$-0.480654\pi$
$563$	2.88247	0.121482	0.0607408	+	0.998154i	$0.480654\pi$
$564$	0	0
$565$	0	0
$566$	−23.1764	−0.974176
$567$	0	0
$568$	−18.6484	−0.782468
$569$	4.37286	0.183320	0.0916600	−	0.995790i	$-0.470783\pi$
$569$	4.37286	0.183320	0.0916600	+	0.995790i	$0.470783\pi$
$570$	0	0
$571$	−1.58120	−0.0661714	−0.0330857	−	0.999453i	$-0.510533\pi$
$571$	−1.58120	−0.0661714	−0.0330857	+	0.999453i	$0.510533\pi$
$572$	−0.112610	−0.00470844
$573$	0	0
$574$	−10.7556	−0.448929
$575$	0	0
$576$	0	0
$577$	7.61729	0.317112	0.158556	−	0.987350i	$-0.449316\pi$
$577$	7.61729	0.317112	0.158556	+	0.987350i	$0.449316\pi$
$578$	29.5412	1.22875
$579$	0	0
$580$	0	0
$581$	−27.6543	−1.14730
$582$	0	0
$583$	−1.33630	−0.0553438
$584$	31.5285	1.30466
$585$	0	0
$586$	−9.82071	−0.405690
$587$	−46.8243	−1.93264	−0.966322	−	0.257336i	$-0.917155\pi$
$587$	−46.8243	−1.93264	−0.966322	+	0.257336i	$0.917155\pi$
$588$	0	0
$589$	−12.3827	−0.510221
$590$	0	0
$591$	0	0
$592$	−6.09726	−0.250596
$593$	−15.9398	−0.654568	−0.327284	−	0.944926i	$-0.606133\pi$
$593$	−15.9398	−0.654568	−0.327284	+	0.944926i	$0.606133\pi$
$594$	0	0
$595$	0	0
$596$	−1.87649	−0.0768641
$597$	0	0
$598$	5.69381	0.232837
$599$	−18.4889	−0.755434	−0.377717	−	0.925921i	$-0.623291\pi$
$599$	−18.4889	−0.755434	−0.377717	+	0.925921i	$0.623291\pi$
$600$	0	0
$601$	20.7556	0.846637	0.423319	−	0.905981i	$-0.360865\pi$
$601$	20.7556	0.846637	0.423319	+	0.905981i	$0.360865\pi$
$602$	22.4286	0.914123
$603$	0	0
$604$	0.664758	0.0270486
$605$	0	0
$606$	0	0
$607$	−36.0765	−1.46430	−0.732150	−	0.681143i	$-0.761483\pi$
$607$	−36.0765	−1.46430	−0.732150	+	0.681143i	$0.761483\pi$
$608$	6.39361	0.259295
$609$	0	0
$610$	0	0
$611$	1.09679	0.0443713
$612$	0	0
$613$	−9.94962	−0.401861	−0.200931	−	0.979605i	$-0.564397\pi$
$613$	−9.94962	−0.401861	−0.200931	+	0.979605i	$0.564397\pi$
$614$	16.3126	0.658325
$615$	0	0
$616$	1.90813	0.0768809
$617$	2.09187	0.0842154	0.0421077	−	0.999113i	$-0.486593\pi$
$617$	2.09187	0.0842154	0.0421077	+	0.999113i	$0.486593\pi$
$618$	0	0
$619$	18.4681	0.742296	0.371148	−	0.928574i	$-0.378964\pi$
$619$	18.4681	0.742296	0.371148	+	0.928574i	$0.378964\pi$
$620$	0	0
$621$	0	0
$622$	−24.5334	−0.983700
$623$	16.2953	0.652857
$624$	0	0
$625$	0	0
$626$	−18.3497	−0.733401
$627$	0	0
$628$	2.94914	0.117684
$629$	−14.6637	−0.584680
$630$	0	0
$631$	−38.6657	−1.53926	−0.769629	−	0.638492i	$-0.779559\pi$
$631$	−38.6657	−1.53926	−0.769629	+	0.638492i	$0.779559\pi$
$632$	43.6543	1.73648
$633$	0	0
$634$	27.0563	1.07454
$635$	0	0
$636$	0	0
$637$	−1.42864	−0.0566048
$638$	2.26671	0.0897398
$639$	0	0
$640$	0	0
$641$	−24.5718	−0.970529	−0.485265	−	0.874367i	$-0.661277\pi$
$641$	−24.5718	−0.970529	−0.485265	+	0.874367i	$0.661277\pi$
$642$	0	0
$643$	−27.4938	−1.08425	−0.542125	−	0.840298i	$-0.682380\pi$
$643$	−27.4938	−1.08425	−0.542125	+	0.840298i	$0.682380\pi$
$644$	7.15257	0.281851
$645$	0	0
$646$	−17.2859	−0.680105
$647$	13.7812	0.541796	0.270898	−	0.962608i	$-0.412679\pi$
$647$	13.7812	0.541796	0.270898	+	0.962608i	$0.412679\pi$
$648$	0	0
$649$	−1.98571	−0.0779459
$650$	0	0
$651$	0	0
$652$	−1.95407	−0.0765272
$653$	−2.12045	−0.0829795	−0.0414897	−	0.999139i	$-0.513210\pi$
$653$	−2.12045	−0.0829795	−0.0414897	+	0.999139i	$0.513210\pi$
$654$	0	0
$655$	0	0
$656$	8.15515	0.318405
$657$	0	0
$658$	−3.86665	−0.150738
$659$	−33.8894	−1.32014	−0.660072	−	0.751203i	$-0.729474\pi$
$659$	−33.8894	−1.32014	−0.660072	+	0.751203i	$0.729474\pi$
$660$	0	0
$661$	−37.3689	−1.45348	−0.726741	−	0.686912i	$-0.758966\pi$
$661$	−37.3689	−1.45348	−0.726741	+	0.686912i	$0.758966\pi$
$662$	10.0252	0.389640
$663$	0	0
$664$	29.2114	1.13362
$665$	0	0
$666$	0	0
$667$	40.8385	1.58127
$668$	−3.69721	−0.143049
$669$	0	0
$670$	0	0
$671$	0.0602231	0.00232489
$672$	0	0
$673$	35.4608	1.36691	0.683456	−	0.729992i	$-0.260476\pi$
$673$	35.4608	1.36691	0.683456	+	0.729992i	$0.260476\pi$
$674$	16.6923	0.642963
$675$	0	0
$676$	−0.525428	−0.0202088
$677$	15.3047	0.588206	0.294103	−	0.955774i	$-0.404979\pi$
$677$	15.3047	0.588206	0.294103	+	0.955774i	$0.404979\pi$
$678$	0	0
$679$	−52.3783	−2.01009
$680$	0	0
$681$	0	0
$682$	1.45536	0.0557286
$683$	−13.0968	−0.501135	−0.250567	−	0.968099i	$-0.580617\pi$
$683$	−13.0968	−0.501135	−0.250567	+	0.968099i	$0.580617\pi$
$684$	0	0
$685$	0	0
$686$	−19.6414	−0.749913
$687$	0	0
$688$	−17.0060	−0.648347
$689$	−6.23506	−0.237537
$690$	0	0
$691$	−18.4079	−0.700269	−0.350135	−	0.936699i	$-0.613864\pi$
$691$	−18.4079	−0.700269	−0.350135	+	0.936699i	$0.613864\pi$
$692$	0.380371	0.0144595
$693$	0	0
$694$	1.48055	0.0562009
$695$	0	0
$696$	0	0
$697$	19.6128	0.742890
$698$	27.3363	1.03469
$699$	0	0
$700$	0	0
$701$	31.3689	1.18479	0.592393	−	0.805649i	$-0.298183\pi$
$701$	31.3689	1.18479	0.592393	+	0.805649i	$0.298183\pi$
$702$	0	0
$703$	5.05086	0.190497
$704$	−1.89723	−0.0715047
$705$	0	0
$706$	−17.3417	−0.652663
$707$	−11.4380	−0.430171
$708$	0	0
$709$	9.47949	0.356010	0.178005	−	0.984030i	$-0.443036\pi$
$709$	9.47949	0.356010	0.178005	+	0.984030i	$0.443036\pi$
$710$	0	0
$711$	0	0
$712$	−17.2128	−0.645077
$713$	26.2208	0.981976
$714$	0	0
$715$	0	0
$716$	−2.12351	−0.0793592
$717$	0	0
$718$	14.7590	0.550799
$719$	29.6227	1.10474	0.552370	−	0.833599i	$-0.313724\pi$
$719$	29.6227	1.10474	0.552370	+	0.833599i	$0.313724\pi$
$720$	0	0
$721$	8.19358	0.305145
$722$	−17.1180	−0.637066
$723$	0	0
$724$	−1.23014	−0.0457178
$725$	0	0
$726$	0	0
$727$	−42.6702	−1.58255	−0.791274	−	0.611461i	$-0.790582\pi$
$727$	−42.6702	−1.58255	−0.791274	+	0.611461i	$0.790582\pi$
$728$	8.90321	0.329975
$729$	0	0
$730$	0	0
$731$	−40.8988	−1.51270
$732$	0	0
$733$	−26.0830	−0.963397	−0.481698	−	0.876337i	$-0.659980\pi$
$733$	−26.0830	−0.963397	−0.481698	+	0.876337i	$0.659980\pi$
$734$	5.65539	0.208744
$735$	0	0
$736$	−13.5387	−0.499042
$737$	1.66323	0.0612657
$738$	0	0
$739$	−28.2687	−1.03988	−0.519941	−	0.854202i	$-0.674046\pi$
$739$	−28.2687	−1.03988	−0.519941	+	0.854202i	$0.674046\pi$
$740$	0	0
$741$	0	0
$742$	21.9813	0.806958
$743$	20.6681	0.758241	0.379120	−	0.925347i	$-0.376227\pi$
$743$	20.6681	0.758241	0.379120	+	0.925347i	$0.376227\pi$
$744$	0	0
$745$	0	0
$746$	−42.4286	−1.55342
$747$	0	0
$748$	−0.723926	−0.0264694
$749$	−49.6958	−1.81585
$750$	0	0
$751$	2.46028	0.0897770	0.0448885	−	0.998992i	$-0.485707\pi$
$751$	2.46028	0.0897770	0.0448885	+	0.998992i	$0.485707\pi$
$752$	2.93179	0.106911
$753$	0	0
$754$	10.5763	0.385165
$755$	0	0
$756$	0	0
$757$	48.6035	1.76652	0.883262	−	0.468880i	$-0.155342\pi$
$757$	48.6035	1.76652	0.883262	+	0.468880i	$0.155342\pi$
$758$	−21.2162	−0.770606
$759$	0	0
$760$	0	0
$761$	13.8252	0.501162	0.250581	−	0.968096i	$-0.419378\pi$
$761$	13.8252	0.501162	0.250581	+	0.968096i	$0.419378\pi$
$762$	0	0
$763$	−48.5531	−1.75774
$764$	−1.10430	−0.0399520
$765$	0	0
$766$	−22.6811	−0.819500
$767$	−9.26517	−0.334546
$768$	0	0
$769$	−38.9688	−1.40525	−0.702626	−	0.711559i	$-0.747989\pi$
$769$	−38.9688	−1.40525	−0.702626	+	0.711559i	$0.747989\pi$
$770$	0	0
$771$	0	0
$772$	−7.10430	−0.255689
$773$	−0.445992	−0.0160412	−0.00802061	−	0.999968i	$-0.502553\pi$
$773$	−0.445992	−0.0160412	−0.00802061	+	0.999968i	$0.502553\pi$
$774$	0	0
$775$	0	0
$776$	55.3274	1.98614
$777$	0	0
$778$	−1.95851	−0.0702161
$779$	−6.75557	−0.242043
$780$	0	0
$781$	−1.30327	−0.0466347
$782$	36.6035	1.30894
$783$	0	0
$784$	−3.81885	−0.136388
$785$	0	0
$786$	0	0
$787$	33.9037	1.20854	0.604268	−	0.796781i	$-0.293466\pi$
$787$	33.9037	1.20854	0.604268	+	0.796781i	$0.293466\pi$
$788$	1.05086	0.0374352
$789$	0	0
$790$	0	0
$791$	−3.43801	−0.122241
$792$	0	0
$793$	0.280996	0.00997847
$794$	−7.98571	−0.283402
$795$	0	0
$796$	−11.6346	−0.412379
$797$	10.2953	0.364678	0.182339	−	0.983236i	$-0.441633\pi$
$797$	10.2953	0.364678	0.182339	+	0.983236i	$0.441633\pi$
$798$	0	0
$799$	7.05086	0.249441
$800$	0	0
$801$	0	0
$802$	26.6035	0.939402
$803$	2.20342	0.0777570
$804$	0	0
$805$	0	0
$806$	6.79060	0.239189
$807$	0	0
$808$	12.0820	0.425044
$809$	−7.94422	−0.279304	−0.139652	−	0.990201i	$-0.544598\pi$
$809$	−7.94422	−0.279304	−0.139652	+	0.990201i	$0.544598\pi$
$810$	0	0
$811$	−8.12245	−0.285218	−0.142609	−	0.989779i	$-0.545549\pi$
$811$	−8.12245	−0.285218	−0.142609	+	0.989779i	$0.545549\pi$
$812$	13.2859	0.466244
$813$	0	0
$814$	−0.593635	−0.0208069
$815$	0	0
$816$	0	0
$817$	14.0874	0.492856
$818$	−12.3783	−0.432796
$819$	0	0
$820$	0	0
$821$	22.2065	0.775012	0.387506	−	0.921867i	$-0.373337\pi$
$821$	22.2065	0.775012	0.387506	+	0.921867i	$0.373337\pi$
$822$	0	0
$823$	11.1175	0.387533	0.193766	−	0.981048i	$-0.437930\pi$
$823$	11.1175	0.387533	0.193766	+	0.981048i	$0.437930\pi$
$824$	−8.65491	−0.301508
$825$	0	0
$826$	32.6637	1.13652
$827$	−23.1570	−0.805248	−0.402624	−	0.915365i	$-0.631902\pi$
$827$	−23.1570	−0.805248	−0.402624	+	0.915365i	$0.631902\pi$
$828$	0	0
$829$	−27.1195	−0.941901	−0.470950	−	0.882160i	$-0.656089\pi$
$829$	−27.1195	−0.941901	−0.470950	+	0.882160i	$0.656089\pi$
$830$	0	0
$831$	0	0
$832$	−8.85236	−0.306900
$833$	−9.18421	−0.318214
$834$	0	0
$835$	0	0
$836$	0.249353	0.00862407
$837$	0	0
$838$	−8.88586	−0.306957
$839$	−25.3955	−0.876749	−0.438374	−	0.898792i	$-0.644446\pi$
$839$	−25.3955	−0.876749	−0.438374	+	0.898792i	$0.644446\pi$
$840$	0	0
$841$	46.8578	1.61578
$842$	−9.55262	−0.329205
$843$	0	0
$844$	−10.3269	−0.355468
$845$	0	0
$846$	0	0
$847$	−31.8020	−1.09273
$848$	−16.6668	−0.572339
$849$	0	0
$850$	0	0
$851$	−10.6953	−0.366632
$852$	0	0
$853$	25.0651	0.858214	0.429107	−	0.903254i	$-0.358828\pi$
$853$	25.0651	0.858214	0.429107	+	0.903254i	$0.358828\pi$
$854$	−0.990632	−0.0338987
$855$	0	0
$856$	52.4939	1.79421
$857$	−7.61285	−0.260050	−0.130025	−	0.991511i	$-0.541506\pi$
$857$	−7.61285	−0.260050	−0.130025	+	0.991511i	$0.541506\pi$
$858$	0	0
$859$	42.1432	1.43791	0.718954	−	0.695058i	$-0.244621\pi$
$859$	42.1432	1.43791	0.718954	+	0.695058i	$0.244621\pi$
$860$	0	0
$861$	0	0
$862$	47.2607	1.60971
$863$	−51.5768	−1.75569	−0.877847	−	0.478942i	$-0.841020\pi$
$863$	−51.5768	−1.75569	−0.877847	+	0.478942i	$0.841020\pi$
$864$	0	0
$865$	0	0
$866$	24.5334	0.833679
$867$	0	0
$868$	8.53035	0.289539
$869$	3.05086	0.103493
$870$	0	0
$871$	7.76049	0.262954
$872$	51.2869	1.73679
$873$	0	0
$874$	−12.6079	−0.426469
$875$	0	0
$876$	0	0
$877$	−34.0701	−1.15046	−0.575232	−	0.817990i	$-0.695088\pi$
$877$	−34.0701	−1.15046	−0.575232	+	0.817990i	$0.695088\pi$
$878$	13.2226	0.446242
$879$	0	0
$880$	0	0
$881$	−3.71900	−0.125296	−0.0626482	−	0.998036i	$-0.519955\pi$
$881$	−3.71900	−0.125296	−0.0626482	+	0.998036i	$0.519955\pi$
$882$	0	0
$883$	−42.0163	−1.41396	−0.706981	−	0.707233i	$-0.749943\pi$
$883$	−42.0163	−1.41396	−0.706981	+	0.707233i	$0.749943\pi$
$884$	−3.37778	−0.113607
$885$	0	0
$886$	−34.7644	−1.16793
$887$	40.3116	1.35353	0.676765	−	0.736199i	$-0.263381\pi$
$887$	40.3116	1.35353	0.676765	+	0.736199i	$0.263381\pi$
$888$	0	0
$889$	6.68244	0.224122
$890$	0	0
$891$	0	0
$892$	10.3344	0.346023
$893$	−2.42864	−0.0812713
$894$	0	0
$895$	0	0
$896$	14.4429	0.482504
$897$	0	0
$898$	13.2730	0.442926
$899$	48.7052	1.62441
$900$	0	0
$901$	−40.0830	−1.33536
$902$	0.793993	0.0264371
$903$	0	0
$904$	3.63158	0.120785
$905$	0	0
$906$	0	0
$907$	34.8419	1.15691	0.578454	−	0.815715i	$-0.303656\pi$
$907$	34.8419	1.15691	0.578454	+	0.815715i	$0.303656\pi$
$908$	6.97328	0.231416
$909$	0	0
$910$	0	0
$911$	−23.2672	−0.770876	−0.385438	−	0.922734i	$-0.625950\pi$
$911$	−23.2672	−0.770876	−0.385438	+	0.922734i	$0.625950\pi$
$912$	0	0
$913$	2.04149	0.0675634
$914$	13.8510	0.458149
$915$	0	0
$916$	1.27607	0.0421627
$917$	38.9590	1.28654
$918$	0	0
$919$	−3.22570	−0.106406	−0.0532029	−	0.998584i	$-0.516943\pi$
$919$	−3.22570	−0.106406	−0.0532029	+	0.998584i	$0.516943\pi$
$920$	0	0
$921$	0	0
$922$	31.7342	1.04511
$923$	−6.08097	−0.200157
$924$	0	0
$925$	0	0
$926$	9.62036	0.316145
$927$	0	0
$928$	−25.1481	−0.825527
$929$	39.3461	1.29091	0.645453	−	0.763800i	$-0.276669\pi$
$929$	39.3461	1.29091	0.645453	+	0.763800i	$0.276669\pi$
$930$	0	0
$931$	3.16346	0.103678
$932$	−8.49871	−0.278384
$933$	0	0
$934$	13.2268	0.432792
$935$	0	0
$936$	0	0
$937$	−51.6040	−1.68583	−0.842914	−	0.538048i	$-0.819162\pi$
$937$	−51.6040	−1.68583	−0.842914	+	0.538048i	$0.819162\pi$
$938$	−27.3590	−0.893305
$939$	0	0
$940$	0	0
$941$	−37.5081	−1.22273	−0.611364	−	0.791349i	$-0.709379\pi$
$941$	−37.5081	−1.22273	−0.611364	+	0.791349i	$0.709379\pi$
$942$	0	0
$943$	14.3051	0.465839
$944$	−24.7665	−0.806080
$945$	0	0
$946$	−1.65572	−0.0538320
$947$	−38.1160	−1.23860	−0.619302	−	0.785153i	$-0.712584\pi$
$947$	−38.1160	−1.23860	−0.619302	+	0.785153i	$0.712584\pi$
$948$	0	0
$949$	10.2810	0.333735
$950$	0	0
$951$	0	0
$952$	57.2355	1.85501
$953$	−28.7368	−0.930877	−0.465439	−	0.885080i	$-0.654103\pi$
$953$	−28.7368	−0.930877	−0.465439	+	0.885080i	$0.654103\pi$
$954$	0	0
$955$	0	0
$956$	−6.71303	−0.217115
$957$	0	0
$958$	11.0890	0.358268
$959$	−55.6040	−1.79555
$960$	0	0
$961$	0.271628	0.00876221
$962$	−2.76986	−0.0893038
$963$	0	0
$964$	3.09912	0.0998161
$965$	0	0
$966$	0	0
$967$	−29.0593	−0.934485	−0.467242	−	0.884129i	$-0.654752\pi$
$967$	−29.0593	−0.934485	−0.467242	+	0.884129i	$0.654752\pi$
$968$	33.5926	1.07971
$969$	0	0
$970$	0	0
$971$	39.8578	1.27910	0.639548	−	0.768751i	$-0.279121\pi$
$971$	39.8578	1.27910	0.639548	+	0.768751i	$0.279121\pi$
$972$	0	0
$973$	55.4291	1.77698
$974$	19.6588	0.629908
$975$	0	0
$976$	0.751123	0.0240429
$977$	−12.8617	−0.411483	−0.205742	−	0.978606i	$-0.565961\pi$
$977$	−12.8617	−0.411483	−0.205742	+	0.978606i	$0.565961\pi$
$978$	0	0
$979$	−1.20294	−0.0384463
$980$	0	0
$981$	0	0
$982$	−31.8925	−1.01773
$983$	45.4880	1.45084	0.725420	−	0.688306i	$-0.241645\pi$
$983$	45.4880	1.45084	0.725420	+	0.688306i	$0.241645\pi$
$984$	0	0
$985$	0	0
$986$	67.9911	2.16528
$987$	0	0
$988$	1.16346	0.0370147
$989$	−29.8306	−0.948557
$990$	0	0
$991$	8.07007	0.256354	0.128177	−	0.991751i	$-0.459087\pi$
$991$	8.07007	0.256354	0.128177	+	0.991751i	$0.459087\pi$
$992$	−16.1466	−0.512655
$993$	0	0
$994$	21.4380	0.679972
$995$	0	0
$996$	0	0
$997$	32.8158	1.03929	0.519643	−	0.854383i	$-0.326065\pi$
$997$	32.8158	1.03929	0.519643	+	0.854383i	$0.326065\pi$
$998$	36.4667	1.15433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bf.1.3		3
3.2	odd	2		325.2.a.k.1.1		3
5.2	odd	4		585.2.c.b.469.4		6
5.3	odd	4		585.2.c.b.469.3		6
5.4	even	2		2925.2.a.bj.1.1		3
12.11	even	2		5200.2.a.cb.1.2		3
15.2	even	4		65.2.b.a.14.3	✓	6
15.8	even	4		65.2.b.a.14.4	yes	6
15.14	odd	2		325.2.a.j.1.3		3
39.38	odd	2		4225.2.a.ba.1.3		3
60.23	odd	4		1040.2.d.c.209.2		6
60.47	odd	4		1040.2.d.c.209.5		6
60.59	even	2		5200.2.a.cj.1.2		3
195.2	odd	12		845.2.l.d.654.6		12
195.8	odd	4		845.2.d.b.844.2		6
195.17	even	12		845.2.n.g.484.4		12
195.23	even	12		845.2.n.g.529.4		12
195.32	odd	12		845.2.l.d.699.5		12
195.38	even	4		845.2.b.c.339.3		6
195.47	odd	4		845.2.d.a.844.5		6
195.62	even	12		845.2.n.g.529.3		12
195.68	even	12		845.2.n.f.529.3		12
195.77	even	4		845.2.b.c.339.4		6
195.83	odd	4		845.2.d.a.844.6		6
195.98	odd	12		845.2.l.d.699.6		12
195.107	even	12		845.2.n.f.529.4		12
195.113	even	12		845.2.n.f.484.4		12
195.122	odd	4		845.2.d.b.844.1		6
195.128	odd	12		845.2.l.d.654.5		12
195.137	odd	12		845.2.l.e.699.1		12
195.152	even	12		845.2.n.f.484.3		12
195.158	odd	12		845.2.l.e.654.1		12
195.167	odd	12		845.2.l.e.654.2		12
195.173	even	12		845.2.n.g.484.3		12
195.188	odd	12		845.2.l.e.699.2		12
195.194	odd	2		4225.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.3	✓	6	15.2	even	4
65.2.b.a.14.4	yes	6	15.8	even	4
325.2.a.j.1.3		3	15.14	odd	2
325.2.a.k.1.1		3	3.2	odd	2
585.2.c.b.469.3		6	5.3	odd	4
585.2.c.b.469.4		6	5.2	odd	4
845.2.b.c.339.3		6	195.38	even	4
845.2.b.c.339.4		6	195.77	even	4
845.2.d.a.844.5		6	195.47	odd	4
845.2.d.a.844.6		6	195.83	odd	4
845.2.d.b.844.1		6	195.122	odd	4
845.2.d.b.844.2		6	195.8	odd	4
845.2.l.d.654.5		12	195.128	odd	12
845.2.l.d.654.6		12	195.2	odd	12
845.2.l.d.699.5		12	195.32	odd	12
845.2.l.d.699.6		12	195.98	odd	12
845.2.l.e.654.1		12	195.158	odd	12
845.2.l.e.654.2		12	195.167	odd	12
845.2.l.e.699.1		12	195.137	odd	12
845.2.l.e.699.2		12	195.188	odd	12
845.2.n.f.484.3		12	195.152	even	12
845.2.n.f.484.4		12	195.113	even	12
845.2.n.f.529.3		12	195.68	even	12
845.2.n.f.529.4		12	195.107	even	12
845.2.n.g.484.3		12	195.173	even	12
845.2.n.g.484.4		12	195.17	even	12
845.2.n.g.529.3		12	195.62	even	12
845.2.n.g.529.4		12	195.23	even	12
1040.2.d.c.209.2		6	60.23	odd	4
1040.2.d.c.209.5		6	60.47	odd	4
2925.2.a.bf.1.3		3	1.1	even	1	trivial
2925.2.a.bj.1.1		3	5.4	even	2
4225.2.a.ba.1.3		3	39.38	odd	2
4225.2.a.bh.1.1		3	195.194	odd	2
5200.2.a.cb.1.2		3	12.11	even	2
5200.2.a.cj.1.2		3	60.59	even	2

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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21432 q^{2} -0.525428 q^{4} +2.90321 q^{7} -3.06668 q^{8} -0.214320 q^{11} -1.00000 q^{13} +3.52543 q^{14} -2.67307 q^{16} -6.42864 q^{17} +2.21432 q^{19} -0.260253 q^{22} -4.68889 q^{23} -1.21432 q^{26} -1.52543 q^{28} -8.70964 q^{29} -5.59210 q^{31} +2.88739 q^{32} -7.80642 q^{34} +2.28100 q^{37} +2.68889 q^{38} -3.05086 q^{41} +6.36196 q^{43} +0.112610 q^{44} -5.69381 q^{46} -1.09679 q^{47} +1.42864 q^{49} +0.525428 q^{52} +6.23506 q^{53} -8.90321 q^{56} -10.5763 q^{58} +9.26517 q^{59} -0.280996 q^{61} -6.79060 q^{62} +8.85236 q^{64} -7.76049 q^{67} +3.37778 q^{68} +6.08097 q^{71} -10.2810 q^{73} +2.76986 q^{74} -1.16346 q^{76} -0.622216 q^{77} -14.2351 q^{79} -3.70471 q^{82} -9.52543 q^{83} +7.72546 q^{86} +0.657249 q^{88} +5.61285 q^{89} -2.90321 q^{91} +2.46367 q^{92} -1.33185 q^{94} -18.0415 q^{97} +1.73483 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} + 6 q^{11} - 3 q^{13} + 4 q^{14} + 5 q^{16} - 6 q^{17} - 14 q^{22} - 14 q^{23} + 3 q^{26} + 2 q^{28} - 6 q^{29} - 10 q^{31} - 11 q^{32} - 10 q^{34} + 8 q^{38}+ \cdots + 25 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8 + 6 * q^11 - 3 * q^13 + 4 * q^14 + 5 * q^16 - 6 * q^17 - 14 * q^22 - 14 * q^23 + 3 * q^26 + 2 * q^28 - 6 * q^29 - 10 * q^31 - 11 * q^32 - 10 * q^34 + 8 * q^38 + 4 * q^41 + 6 * q^43 + 20 * q^44 + 16 * q^46 - 10 * q^47 - 9 * q^49 - 5 * q^52 - 8 * q^53 - 20 * q^56 - 12 * q^58 + 8 * q^59 + 6 * q^61 + 6 * q^62 + 33 * q^64 + 10 * q^67 + 10 * q^68 + 12 * q^71 - 24 * q^73 + 2 * q^74 - 10 * q^76 - 2 * q^77 - 16 * q^79 - 24 * q^82 - 22 * q^83 + 16 * q^86 - 24 * q^88 - 10 * q^89 - 2 * q^91 - 32 * q^92 + 16 * q^94 - 14 * q^97 + 25 * q^98

Introduction

L-functions

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