LMFDB - Embedded newform 1875.2.a.m.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1875,2,Mod(1,1875)] 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("1875.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.9719503790$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.53767$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.53767 q^{2} +1.00000 q^{3} +0.364440 q^{4} +1.53767 q^{6} -1.68601 q^{7} -2.51496 q^{8} +1.00000 q^{9} -2.97757 q^{11} +0.364440 q^{12} -0.232892 q^{13} -2.59253 q^{14} -4.59606 q^{16} -7.45901 q^{17} +1.53767 q^{18} +0.753527 q^{19} -1.68601 q^{21} -4.57853 q^{22} -0.872721 q^{23} -2.51496 q^{24} -0.358112 q^{26} +1.00000 q^{27} -0.614448 q^{28} +6.87482 q^{29} -9.81929 q^{31} -2.03733 q^{32} -2.97757 q^{33} -11.4695 q^{34} +0.364440 q^{36} -10.1272 q^{37} +1.15868 q^{38} -0.232892 q^{39} +3.79732 q^{41} -2.59253 q^{42} +5.27322 q^{43} -1.08514 q^{44} -1.34196 q^{46} +8.56747 q^{47} -4.59606 q^{48} -4.15738 q^{49} -7.45901 q^{51} -0.0848751 q^{52} -5.97876 q^{53} +1.53767 q^{54} +4.24024 q^{56} +0.753527 q^{57} +10.5712 q^{58} -3.85114 q^{59} -4.39643 q^{61} -15.0989 q^{62} -1.68601 q^{63} +6.05938 q^{64} -4.57853 q^{66} -1.79282 q^{67} -2.71836 q^{68} -0.872721 q^{69} -4.37450 q^{71} -2.51496 q^{72} +15.0528 q^{73} -15.5723 q^{74} +0.274615 q^{76} +5.02021 q^{77} -0.358112 q^{78} +7.37584 q^{79} +1.00000 q^{81} +5.83904 q^{82} -4.34451 q^{83} -0.614448 q^{84} +8.10849 q^{86} +6.87482 q^{87} +7.48847 q^{88} +12.1032 q^{89} +0.392657 q^{91} -0.318054 q^{92} -9.81929 q^{93} +13.1740 q^{94} -2.03733 q^{96} -9.47426 q^{97} -6.39269 q^{98} -2.97757 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^11 + 4 * q^12 - 16 * q^13 + 6 * q^14 - 16 * q^17 - 4 * q^18 - 14 * q^19 - 8 * q^21 - 12 * q^22 - 14 * q^23 - 12 * q^24 + 6 * q^26 + 8 * q^27 - 16 * q^28 + 2 * q^29 - 22 * q^31 + 2 * q^32 + 2 * q^33 - 12 * q^34 + 4 * q^36 - 28 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 + 6 * q^42 - 20 * q^43 + 22 * q^44 - 2 * q^46 - 10 * q^47 - 16 * q^51 - 16 * q^52 - 44 * q^53 - 4 * q^54 + 30 * q^56 - 14 * q^57 - 8 * q^58 + 14 * q^59 - 20 * q^61 - 16 * q^62 - 8 * q^63 + 6 * q^64 - 12 * q^66 - 16 * q^67 + 2 * q^68 - 14 * q^69 + 16 * q^71 - 12 * q^72 - 24 * q^73 + 26 * q^74 - 16 * q^76 - 46 * q^77 + 6 * q^78 - 30 * q^79 + 8 * q^81 - 16 * q^82 - 12 * q^83 - 16 * q^84 + 32 * q^86 + 2 * q^87 - 32 * q^88 + 16 * q^89 - 12 * q^91 + 2 * q^92 - 22 * q^93 + 14 * q^94 + 2 * q^96 - 16 * q^97 - 4 * q^98 + 2 * 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 $

$2$	1.53767	1.08730	0.543650	−	0.839312i	$-0.317042\pi$
$2$	1.53767	1.08730	0.543650	+	0.839312i	$0.317042\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.364440	0.182220
$5$	0	0
$5$	0	0
$6$	1.53767	0.627753
$7$	−1.68601	−0.637251	−0.318625	−	0.947881i	$-0.603221\pi$
$7$	−1.68601	−0.637251	−0.318625	+	0.947881i	$0.603221\pi$
$8$	−2.51496	−0.889172
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.97757	−0.897772	−0.448886	−	0.893589i	$-0.648179\pi$
$11$	−2.97757	−0.897772	−0.448886	+	0.893589i	$0.648179\pi$
$12$	0.364440	0.105205
$13$	−0.232892	−0.0645926	−0.0322963	−	0.999478i	$-0.510282\pi$
$13$	−0.232892	−0.0645926	−0.0322963	+	0.999478i	$0.510282\pi$
$14$	−2.59253	−0.692882
$15$	0	0
$16$	−4.59606	−1.14902
$17$	−7.45901	−1.80908	−0.904538	−	0.426392i	$-0.859784\pi$
$17$	−7.45901	−1.80908	−0.904538	+	0.426392i	$0.859784\pi$
$18$	1.53767	0.362433
$19$	0.753527	0.172871	0.0864354	−	0.996257i	$-0.472452\pi$
$19$	0.753527	0.172871	0.0864354	+	0.996257i	$0.472452\pi$
$20$	0	0
$21$	−1.68601	−0.367917
$22$	−4.57853	−0.976146
$23$	−0.872721	−0.181975	−0.0909874	−	0.995852i	$-0.529002\pi$
$23$	−0.872721	−0.181975	−0.0909874	+	0.995852i	$0.529002\pi$
$24$	−2.51496	−0.513364
$25$	0	0
$26$	−0.358112	−0.0702315
$27$	1.00000	0.192450
$28$	−0.614448	−0.116120
$29$	6.87482	1.27662	0.638311	−	0.769778i	$-0.279633\pi$
$29$	6.87482	1.27662	0.638311	+	0.769778i	$0.279633\pi$
$30$	0	0
$31$	−9.81929	−1.76360	−0.881798	−	0.471627i	$-0.843667\pi$
$31$	−9.81929	−1.76360	−0.881798	+	0.471627i	$0.843667\pi$
$32$	−2.03733	−0.360152
$33$	−2.97757	−0.518329
$34$	−11.4695	−1.96701
$35$	0	0
$36$	0.364440	0.0607399
$37$	−10.1272	−1.66489	−0.832447	−	0.554105i	$-0.813061\pi$
$37$	−10.1272	−1.66489	−0.832447	+	0.554105i	$0.813061\pi$
$38$	1.15868	0.187962
$39$	−0.232892	−0.0372926
$40$	0	0
$41$	3.79732	0.593042	0.296521	−	0.955026i	$-0.404174\pi$
$41$	3.79732	0.593042	0.296521	+	0.955026i	$0.404174\pi$
$42$	−2.59253	−0.400036
$43$	5.27322	0.804159	0.402079	−	0.915605i	$-0.368288\pi$
$43$	5.27322	0.804159	0.402079	+	0.915605i	$0.368288\pi$
$44$	−1.08514	−0.163592
$45$	0	0
$46$	−1.34196	−0.197861
$47$	8.56747	1.24969	0.624847	−	0.780747i	$-0.285161\pi$
$47$	8.56747	1.24969	0.624847	+	0.780747i	$0.285161\pi$
$48$	−4.59606	−0.663385
$49$	−4.15738	−0.593911
$50$	0	0
$51$	−7.45901	−1.04447
$52$	−0.0848751	−0.0117701
$53$	−5.97876	−0.821246	−0.410623	−	0.911805i	$-0.634689\pi$
$53$	−5.97876	−0.821246	−0.410623	+	0.911805i	$0.634689\pi$
$54$	1.53767	0.209251
$55$	0	0
$56$	4.24024	0.566625
$57$	0.753527	0.0998070
$58$	10.5712	1.38807
$59$	−3.85114	−0.501376	−0.250688	−	0.968068i	$-0.580657\pi$
$59$	−3.85114	−0.501376	−0.250688	+	0.968068i	$0.580657\pi$
$60$	0	0
$61$	−4.39643	−0.562905	−0.281452	−	0.959575i	$-0.590816\pi$
$61$	−4.39643	−0.562905	−0.281452	+	0.959575i	$0.590816\pi$
$62$	−15.0989	−1.91756
$63$	−1.68601	−0.212417
$64$	6.05938	0.757423
$65$	0	0
$66$	−4.57853	−0.563578
$67$	−1.79282	−0.219028	−0.109514	−	0.993985i	$-0.534930\pi$
$67$	−1.79282	−0.219028	−0.109514	+	0.993985i	$0.534930\pi$
$68$	−2.71836	−0.329650
$69$	−0.872721	−0.105063
$70$	0	0
$71$	−4.37450	−0.519157	−0.259579	−	0.965722i	$-0.583584\pi$
$71$	−4.37450	−0.519157	−0.259579	+	0.965722i	$0.583584\pi$
$72$	−2.51496	−0.296391
$73$	15.0528	1.76180	0.880900	−	0.473303i	$-0.156938\pi$
$73$	15.0528	1.76180	0.880900	+	0.473303i	$0.156938\pi$
$74$	−15.5723	−1.81024
$75$	0	0
$76$	0.274615	0.0315005
$77$	5.02021	0.572106
$78$	−0.358112	−0.0405482
$79$	7.37584	0.829847	0.414924	−	0.909856i	$-0.363808\pi$
$79$	7.37584	0.829847	0.414924	+	0.909856i	$0.363808\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.83904	0.644814
$83$	−4.34451	−0.476872	−0.238436	−	0.971158i	$-0.576635\pi$
$83$	−4.34451	−0.476872	−0.238436	+	0.971158i	$0.576635\pi$
$84$	−0.614448	−0.0670417
$85$	0	0
$86$	8.10849	0.874361
$87$	6.87482	0.737058
$88$	7.48847	0.798273
$89$	12.1032	1.28294	0.641469	−	0.767149i	$-0.278325\pi$
$89$	12.1032	1.28294	0.641469	+	0.767149i	$0.278325\pi$
$90$	0	0
$91$	0.392657	0.0411617
$92$	−0.318054	−0.0331594
$93$	−9.81929	−1.01821
$94$	13.1740	1.35879
$95$	0	0
$96$	−2.03733	−0.207934
$97$	−9.47426	−0.961965	−0.480982	−	0.876730i	$-0.659720\pi$
$97$	−9.47426	−0.961965	−0.480982	+	0.876730i	$0.659720\pi$
$98$	−6.39269	−0.645760
$99$	−2.97757	−0.299257
$100$	0	0
$101$	−6.54468	−0.651220	−0.325610	−	0.945504i	$-0.605570\pi$
$101$	−6.54468	−0.651220	−0.325610	+	0.945504i	$0.605570\pi$
$102$	−11.4695	−1.13565
$103$	0.700804	0.0690522	0.0345261	−	0.999404i	$-0.489008\pi$
$103$	0.700804	0.0690522	0.0345261	+	0.999404i	$0.489008\pi$
$104$	0.585713	0.0574339
$105$	0	0
$106$	−9.19338	−0.892940
$107$	−12.5288	−1.21120	−0.605602	−	0.795768i	$-0.707067\pi$
$107$	−12.5288	−1.21120	−0.605602	+	0.795768i	$0.707067\pi$
$108$	0.364440	0.0350682
$109$	4.28348	0.410283	0.205141	−	0.978732i	$-0.434235\pi$
$109$	4.28348	0.410283	0.205141	+	0.978732i	$0.434235\pi$
$110$	0	0
$111$	−10.1272	−0.961227
$112$	7.74899	0.732211
$113$	−8.22761	−0.773988	−0.386994	−	0.922082i	$-0.626487\pi$
$113$	−8.22761	−0.773988	−0.386994	+	0.922082i	$0.626487\pi$
$114$	1.15868	0.108520
$115$	0	0
$116$	2.50546	0.232626
$117$	−0.232892	−0.0215309
$118$	−5.92180	−0.545146
$119$	12.5760	1.15284
$120$	0	0
$121$	−2.13407	−0.194006
$122$	−6.76027	−0.612046
$123$	3.79732	0.342393
$124$	−3.57854	−0.321362
$125$	0	0
$126$	−2.59253	−0.230961
$127$	−11.9482	−1.06023	−0.530114	−	0.847926i	$-0.677851\pi$
$127$	−11.9482	−1.06023	−0.530114	+	0.847926i	$0.677851\pi$
$128$	13.3920	1.18370
$129$	5.27322	0.464281
$130$	0	0
$131$	21.4573	1.87474	0.937369	−	0.348339i	$-0.113254\pi$
$131$	21.4573	1.87474	0.937369	+	0.348339i	$0.113254\pi$
$132$	−1.08514	−0.0944497
$133$	−1.27045	−0.110162
$134$	−2.75678	−0.238149
$135$	0	0
$136$	18.7591	1.60858
$137$	−10.0192	−0.855998	−0.427999	−	0.903779i	$-0.640781\pi$
$137$	−10.0192	−0.855998	−0.427999	+	0.903779i	$0.640781\pi$
$138$	−1.34196	−0.114235
$139$	−6.57467	−0.557656	−0.278828	−	0.960341i	$-0.589946\pi$
$139$	−6.57467	−0.557656	−0.278828	+	0.960341i	$0.589946\pi$
$140$	0	0
$141$	8.56747	0.721511
$142$	−6.72655	−0.564479
$143$	0.693452	0.0579894
$144$	−4.59606	−0.383005
$145$	0	0
$146$	23.1463	1.91560
$147$	−4.15738	−0.342895
$148$	−3.69074	−0.303377
$149$	10.9143	0.894132	0.447066	−	0.894501i	$-0.352469\pi$
$149$	10.9143	0.894132	0.447066	+	0.894501i	$0.352469\pi$
$150$	0	0
$151$	20.4128	1.66117	0.830584	−	0.556894i	$-0.188007\pi$
$151$	20.4128	1.66117	0.830584	+	0.556894i	$0.188007\pi$
$152$	−1.89509	−0.153712
$153$	−7.45901	−0.603026
$154$	7.71944	0.622050
$155$	0	0
$156$	−0.0848751	−0.00679544
$157$	−3.49944	−0.279286	−0.139643	−	0.990202i	$-0.544595\pi$
$157$	−3.49944	−0.279286	−0.139643	+	0.990202i	$0.544595\pi$
$158$	11.3416	0.902292
$159$	−5.97876	−0.474146
$160$	0	0
$161$	1.47141	0.115964
$162$	1.53767	0.120811
$163$	−5.97357	−0.467886	−0.233943	−	0.972250i	$-0.575163\pi$
$163$	−5.97357	−0.467886	−0.233943	+	0.972250i	$0.575163\pi$
$164$	1.38389	0.108064
$165$	0	0
$166$	−6.68044	−0.518503
$167$	2.33767	0.180895	0.0904473	−	0.995901i	$-0.471170\pi$
$167$	2.33767	0.180895	0.0904473	+	0.995901i	$0.471170\pi$
$168$	4.24024	0.327141
$169$	−12.9458	−0.995828
$170$	0	0
$171$	0.753527	0.0576236
$172$	1.92177	0.146534
$173$	−6.07099	−0.461569	−0.230784	−	0.973005i	$-0.574129\pi$
$173$	−6.07099	−0.461569	−0.230784	+	0.973005i	$0.574129\pi$
$174$	10.5712	0.801403
$175$	0	0
$176$	13.6851	1.03155
$177$	−3.85114	−0.289470
$178$	18.6108	1.39494
$179$	11.4235	0.853835	0.426917	−	0.904291i	$-0.359599\pi$
$179$	11.4235	0.853835	0.426917	+	0.904291i	$0.359599\pi$
$180$	0	0
$181$	−7.85007	−0.583491	−0.291746	−	0.956496i	$-0.594236\pi$
$181$	−7.85007	−0.583491	−0.291746	+	0.956496i	$0.594236\pi$
$182$	0.603779	0.0447551
$183$	−4.39643	−0.324993
$184$	2.19486	0.161807
$185$	0	0
$186$	−15.0989	−1.10710
$187$	22.2097	1.62414
$188$	3.12232	0.227719
$189$	−1.68601	−0.122639
$190$	0	0
$191$	12.5641	0.909110	0.454555	−	0.890719i	$-0.349798\pi$
$191$	12.5641	0.909110	0.454555	+	0.890719i	$0.349798\pi$
$192$	6.05938	0.437298
$193$	10.1437	0.730161	0.365081	−	0.930976i	$-0.381041\pi$
$193$	10.1437	0.730161	0.365081	+	0.930976i	$0.381041\pi$
$194$	−14.5683	−1.04594
$195$	0	0
$196$	−1.51511	−0.108222
$197$	2.04703	0.145845	0.0729223	−	0.997338i	$-0.476767\pi$
$197$	2.04703	0.145845	0.0729223	+	0.997338i	$0.476767\pi$
$198$	−4.57853	−0.325382
$199$	3.57125	0.253159	0.126580	−	0.991956i	$-0.459600\pi$
$199$	3.57125	0.253159	0.126580	+	0.991956i	$0.459600\pi$
$200$	0	0
$201$	−1.79282	−0.126456
$202$	−10.0636	−0.708071
$203$	−11.5910	−0.813529
$204$	−2.71836	−0.190323
$205$	0	0
$206$	1.07761	0.0750805
$207$	−0.872721	−0.0606583
$208$	1.07039	0.0742179
$209$	−2.24368	−0.155199
$210$	0	0
$211$	3.72643	0.256538	0.128269	−	0.991739i	$-0.459058\pi$
$211$	3.72643	0.256538	0.128269	+	0.991739i	$0.459058\pi$
$212$	−2.17890	−0.149647
$213$	−4.37450	−0.299736
$214$	−19.2652	−1.31694
$215$	0	0
$216$	−2.51496	−0.171121
$217$	16.5554	1.12385
$218$	6.58659	0.446100
$219$	15.0528	1.01718
$220$	0	0
$221$	1.73714	0.116853
$222$	−15.5723	−1.04514
$223$	−26.2488	−1.75775	−0.878875	−	0.477051i	$-0.841706\pi$
$223$	−26.2488	−1.75775	−0.878875	+	0.477051i	$0.841706\pi$
$224$	3.43495	0.229507
$225$	0	0
$226$	−12.6514	−0.841557
$227$	4.49111	0.298085	0.149043	−	0.988831i	$-0.452381\pi$
$227$	4.49111	0.298085	0.149043	+	0.988831i	$0.452381\pi$
$228$	0.274615	0.0181868
$229$	−7.53935	−0.498214	−0.249107	−	0.968476i	$-0.580137\pi$
$229$	−7.53935	−0.498214	−0.249107	+	0.968476i	$0.580137\pi$
$230$	0	0
$231$	5.02021	0.330305
$232$	−17.2899	−1.13514
$233$	−10.3640	−0.678971	−0.339485	−	0.940611i	$-0.610253\pi$
$233$	−10.3640	−0.678971	−0.339485	+	0.940611i	$0.610253\pi$
$234$	−0.358112	−0.0234105
$235$	0	0
$236$	−1.40351	−0.0913607
$237$	7.37584	0.479112
$238$	19.3377	1.25348
$239$	−18.5738	−1.20144	−0.600719	−	0.799460i	$-0.705119\pi$
$239$	−18.5738	−1.20144	−0.600719	+	0.799460i	$0.705119\pi$
$240$	0	0
$241$	−5.43543	−0.350127	−0.175063	−	0.984557i	$-0.556013\pi$
$241$	−5.43543	−0.350127	−0.175063	+	0.984557i	$0.556013\pi$
$242$	−3.28150	−0.210943
$243$	1.00000	0.0641500
$244$	−1.60223	−0.102572
$245$	0	0
$246$	5.83904	0.372283
$247$	−0.175490	−0.0111662
$248$	24.6951	1.56814
$249$	−4.34451	−0.275322
$250$	0	0
$251$	23.3577	1.47432	0.737162	−	0.675716i	$-0.236166\pi$
$251$	23.3577	1.47432	0.737162	+	0.675716i	$0.236166\pi$
$252$	−0.614448	−0.0387066
$253$	2.59859	0.163372
$254$	−18.3724	−1.15279
$255$	0	0
$256$	8.47377	0.529611
$257$	−4.48380	−0.279692	−0.139846	−	0.990173i	$-0.544661\pi$
$257$	−4.48380	−0.279692	−0.139846	+	0.990173i	$0.544661\pi$
$258$	8.10849	0.504813
$259$	17.0745	1.06095
$260$	0	0
$261$	6.87482	0.425541
$262$	32.9944	2.03840
$263$	7.72550	0.476375	0.238187	−	0.971219i	$-0.423447\pi$
$263$	7.72550	0.476375	0.238187	+	0.971219i	$0.423447\pi$
$264$	7.48847	0.460883
$265$	0	0
$266$	−1.95354	−0.119779
$267$	12.1032	0.740704
$268$	−0.653376	−0.0399113
$269$	10.7394	0.654791	0.327396	−	0.944887i	$-0.393829\pi$
$269$	10.7394	0.654791	0.327396	+	0.944887i	$0.393829\pi$
$270$	0	0
$271$	−25.7126	−1.56193	−0.780964	−	0.624576i	$-0.785272\pi$
$271$	−25.7126	−1.56193	−0.780964	+	0.624576i	$0.785272\pi$
$272$	34.2821	2.07866
$273$	0.392657	0.0237647
$274$	−15.4063	−0.930726
$275$	0	0
$276$	−0.318054	−0.0191446
$277$	−9.62446	−0.578278	−0.289139	−	0.957287i	$-0.593369\pi$
$277$	−9.62446	−0.578278	−0.289139	+	0.957287i	$0.593369\pi$
$278$	−10.1097	−0.606340
$279$	−9.81929	−0.587865
$280$	0	0
$281$	−2.83275	−0.168988	−0.0844939	−	0.996424i	$-0.526927\pi$
$281$	−2.83275	−0.168988	−0.0844939	+	0.996424i	$0.526927\pi$
$282$	13.1740	0.784498
$283$	−23.6634	−1.40664	−0.703321	−	0.710873i	$-0.748300\pi$
$283$	−23.6634	−1.40664	−0.703321	+	0.710873i	$0.748300\pi$
$284$	−1.59424	−0.0946007
$285$	0	0
$286$	1.06630	0.0630518
$287$	−6.40231	−0.377916
$288$	−2.03733	−0.120051
$289$	38.6369	2.27276
$290$	0	0
$291$	−9.47426	−0.555391
$292$	5.48584	0.321035
$293$	−23.2376	−1.35755	−0.678777	−	0.734345i	$-0.737490\pi$
$293$	−23.2376	−1.35755	−0.678777	+	0.734345i	$0.737490\pi$
$294$	−6.39269	−0.372829
$295$	0	0
$296$	25.4694	1.48038
$297$	−2.97757	−0.172776
$298$	16.7826	0.972189
$299$	0.203250	0.0117542
$300$	0	0
$301$	−8.89069	−0.512451
$302$	31.3882	1.80619
$303$	−6.54468	−0.375982
$304$	−3.46326	−0.198631
$305$	0	0
$306$	−11.4695	−0.655669
$307$	21.8131	1.24494	0.622470	−	0.782643i	$-0.286129\pi$
$307$	21.8131	1.24494	0.622470	+	0.782643i	$0.286129\pi$
$308$	1.82956	0.104249
$309$	0.700804	0.0398673
$310$	0	0
$311$	10.0414	0.569394	0.284697	−	0.958618i	$-0.408107\pi$
$311$	10.0414	0.569394	0.284697	+	0.958618i	$0.408107\pi$
$312$	0.585713	0.0331595
$313$	−6.05366	−0.342173	−0.171087	−	0.985256i	$-0.554728\pi$
$313$	−6.05366	−0.342173	−0.171087	+	0.985256i	$0.554728\pi$
$314$	−5.38099	−0.303667
$315$	0	0
$316$	2.68805	0.151215
$317$	8.49907	0.477355	0.238678	−	0.971099i	$-0.423286\pi$
$317$	8.49907	0.477355	0.238678	+	0.971099i	$0.423286\pi$
$318$	−9.19338	−0.515539
$319$	−20.4703	−1.14612
$320$	0	0
$321$	−12.5288	−0.699289
$322$	2.26255	0.126087
$323$	−5.62057	−0.312737
$324$	0.364440	0.0202466
$325$	0	0
$326$	−9.18540	−0.508733
$327$	4.28348	0.236877
$328$	−9.55010	−0.527316
$329$	−14.4448	−0.796368
$330$	0	0
$331$	16.2945	0.895624	0.447812	−	0.894128i	$-0.352203\pi$
$331$	16.2945	0.895624	0.447812	+	0.894128i	$0.352203\pi$
$332$	−1.58331	−0.0868955
$333$	−10.1272	−0.554965
$334$	3.59458	0.196687
$335$	0	0
$336$	7.74899	0.422742
$337$	−24.6962	−1.34529	−0.672644	−	0.739966i	$-0.734841\pi$
$337$	−24.6962	−1.34529	−0.672644	+	0.739966i	$0.734841\pi$
$338$	−19.9064	−1.08276
$339$	−8.22761	−0.446862
$340$	0	0
$341$	29.2376	1.58331
$342$	1.15868	0.0626541
$343$	18.8114	1.01572
$344$	−13.2619	−0.715035
$345$	0	0
$346$	−9.33520	−0.501864
$347$	13.9466	0.748695	0.374347	−	0.927289i	$-0.377867\pi$
$347$	13.9466	0.748695	0.374347	+	0.927289i	$0.377867\pi$
$348$	2.50546	0.134307
$349$	18.4966	0.990099	0.495049	−	0.868865i	$-0.335150\pi$
$349$	18.4966	0.990099	0.495049	+	0.868865i	$0.335150\pi$
$350$	0	0
$351$	−0.232892	−0.0124309
$352$	6.06629	0.323334
$353$	0.441733	0.0235111	0.0117555	−	0.999931i	$-0.496258\pi$
$353$	0.441733	0.0235111	0.0117555	+	0.999931i	$0.496258\pi$
$354$	−5.92180	−0.314740
$355$	0	0
$356$	4.41089	0.233777
$357$	12.5760	0.665590
$358$	17.5657	0.928374
$359$	4.62127	0.243901	0.121951	−	0.992536i	$-0.461085\pi$
$359$	4.62127	0.243901	0.121951	+	0.992536i	$0.461085\pi$
$360$	0	0
$361$	−18.4322	−0.970116
$362$	−12.0708	−0.634429
$363$	−2.13407	−0.112010
$364$	0.143100	0.00750047
$365$	0	0
$366$	−6.76027	−0.353365
$367$	−8.77696	−0.458153	−0.229077	−	0.973408i	$-0.573571\pi$
$367$	−8.77696	−0.458153	−0.229077	+	0.973408i	$0.573571\pi$
$368$	4.01108	0.209092
$369$	3.79732	0.197681
$370$	0	0
$371$	10.0802	0.523339
$372$	−3.57854	−0.185538
$373$	15.4955	0.802328	0.401164	−	0.916006i	$-0.368606\pi$
$373$	15.4955	0.802328	0.401164	+	0.916006i	$0.368606\pi$
$374$	34.1513	1.76592
$375$	0	0
$376$	−21.5468	−1.11119
$377$	−1.60109	−0.0824604
$378$	−2.59253	−0.133345
$379$	−27.2931	−1.40195	−0.700977	−	0.713184i	$-0.747252\pi$
$379$	−27.2931	−1.40195	−0.700977	+	0.713184i	$0.747252\pi$
$380$	0	0
$381$	−11.9482	−0.612123
$382$	19.3196	0.988474
$383$	13.1042	0.669592	0.334796	−	0.942291i	$-0.391333\pi$
$383$	13.1042	0.669592	0.334796	+	0.942291i	$0.391333\pi$
$384$	13.3920	0.683408
$385$	0	0
$386$	15.5977	0.793904
$387$	5.27322	0.268053
$388$	−3.45279	−0.175289
$389$	16.9056	0.857147	0.428574	−	0.903507i	$-0.359016\pi$
$389$	16.9056	0.857147	0.428574	+	0.903507i	$0.359016\pi$
$390$	0	0
$391$	6.50964	0.329207
$392$	10.4556	0.528089
$393$	21.4573	1.08238
$394$	3.14766	0.158577
$395$	0	0
$396$	−1.08514	−0.0545306
$397$	−10.4078	−0.522353	−0.261177	−	0.965291i	$-0.584110\pi$
$397$	−10.4078	−0.522353	−0.261177	+	0.965291i	$0.584110\pi$
$398$	5.49142	0.275260
$399$	−1.27045	−0.0636021
$400$	0	0
$401$	−0.694800	−0.0346967	−0.0173483	−	0.999850i	$-0.505522\pi$
$401$	−0.694800	−0.0346967	−0.0173483	+	0.999850i	$0.505522\pi$
$402$	−2.75678	−0.137496
$403$	2.28683	0.113915
$404$	−2.38514	−0.118665
$405$	0	0
$406$	−17.8232	−0.884549
$407$	30.1543	1.49469
$408$	18.7591	0.928714
$409$	−1.18910	−0.0587972	−0.0293986	−	0.999568i	$-0.509359\pi$
$409$	−1.18910	−0.0587972	−0.0293986	+	0.999568i	$0.509359\pi$
$410$	0	0
$411$	−10.0192	−0.494211
$412$	0.255401	0.0125827
$413$	6.49306	0.319502
$414$	−1.34196	−0.0659537
$415$	0	0
$416$	0.474477	0.0232632
$417$	−6.57467	−0.321963
$418$	−3.45005	−0.168747
$419$	1.97841	0.0966515	0.0483258	−	0.998832i	$-0.484611\pi$
$419$	1.97841	0.0966515	0.0483258	+	0.998832i	$0.484611\pi$
$420$	0	0
$421$	11.0825	0.540126	0.270063	−	0.962843i	$-0.412955\pi$
$421$	11.0825	0.540126	0.270063	+	0.962843i	$0.412955\pi$
$422$	5.73003	0.278933
$423$	8.56747	0.416564
$424$	15.0363	0.730229
$425$	0	0
$426$	−6.72655	−0.325902
$427$	7.41241	0.358712
$428$	−4.56599	−0.220705
$429$	0.693452	0.0334802
$430$	0	0
$431$	−10.4137	−0.501608	−0.250804	−	0.968038i	$-0.580695\pi$
$431$	−10.4137	−0.501608	−0.250804	+	0.968038i	$0.580695\pi$
$432$	−4.59606	−0.221128
$433$	−10.2643	−0.493272	−0.246636	−	0.969108i	$-0.579325\pi$
$433$	−10.2643	−0.493272	−0.246636	+	0.969108i	$0.579325\pi$
$434$	25.4568	1.22196
$435$	0	0
$436$	1.56107	0.0747616
$437$	−0.657618	−0.0314582
$438$	23.1463	1.10597
$439$	−24.7115	−1.17942	−0.589708	−	0.807616i	$-0.700757\pi$
$439$	−24.7115	−1.17942	−0.589708	+	0.807616i	$0.700757\pi$
$440$	0	0
$441$	−4.15738	−0.197970
$442$	2.67116	0.127054
$443$	7.52935	0.357730	0.178865	−	0.983874i	$-0.442757\pi$
$443$	7.52935	0.357730	0.178865	+	0.983874i	$0.442757\pi$
$444$	−3.69074	−0.175155
$445$	0	0
$446$	−40.3621	−1.91120
$447$	10.9143	0.516227
$448$	−10.2162	−0.482668
$449$	−31.6627	−1.49426	−0.747128	−	0.664681i	$-0.768568\pi$
$449$	−31.6627	−1.49426	−0.747128	+	0.664681i	$0.768568\pi$
$450$	0	0
$451$	−11.3068	−0.532416
$452$	−2.99847	−0.141036
$453$	20.4128	0.959075
$454$	6.90585	0.324108
$455$	0	0
$456$	−1.89509	−0.0887456
$457$	2.95742	0.138342	0.0691712	−	0.997605i	$-0.477965\pi$
$457$	2.95742	0.138342	0.0691712	+	0.997605i	$0.477965\pi$
$458$	−11.5931	−0.541708
$459$	−7.45901	−0.348157
$460$	0	0
$461$	−17.6011	−0.819765	−0.409883	−	0.912138i	$-0.634430\pi$
$461$	−17.6011	−0.819765	−0.409883	+	0.912138i	$0.634430\pi$
$462$	7.71944	0.359141
$463$	−26.3421	−1.22422	−0.612110	−	0.790773i	$-0.709679\pi$
$463$	−26.3421	−1.22422	−0.612110	+	0.790773i	$0.709679\pi$
$464$	−31.5971	−1.46686
$465$	0	0
$466$	−15.9365	−0.738244
$467$	−7.35906	−0.340537	−0.170268	−	0.985398i	$-0.554463\pi$
$467$	−7.35906	−0.340537	−0.170268	+	0.985398i	$0.554463\pi$
$468$	−0.0848751	−0.00392335
$469$	3.02271	0.139576
$470$	0	0
$471$	−3.49944	−0.161246
$472$	9.68547	0.445810
$473$	−15.7014	−0.721951
$474$	11.3416	0.520939
$475$	0	0
$476$	4.58317	0.210069
$477$	−5.97876	−0.273749
$478$	−28.5604	−1.30632
$479$	−28.4670	−1.30069	−0.650346	−	0.759638i	$-0.725376\pi$
$479$	−28.4670	−1.30069	−0.650346	+	0.759638i	$0.725376\pi$
$480$	0	0
$481$	2.35853	0.107540
$482$	−8.35791	−0.380692
$483$	1.47141	0.0669516
$484$	−0.777739	−0.0353518
$485$	0	0
$486$	1.53767	0.0697503
$487$	−2.38406	−0.108032	−0.0540161	−	0.998540i	$-0.517202\pi$
$487$	−2.38406	−0.108032	−0.0540161	+	0.998540i	$0.517202\pi$
$488$	11.0568	0.500519
$489$	−5.97357	−0.270134
$490$	0	0
$491$	−25.7231	−1.16087	−0.580434	−	0.814307i	$-0.697117\pi$
$491$	−25.7231	−1.16087	−0.580434	+	0.814307i	$0.697117\pi$
$492$	1.38389	0.0623907
$493$	−51.2794	−2.30951
$494$	−0.269847	−0.0121410
$495$	0	0
$496$	45.1301	2.02640
$497$	7.37543	0.330833
$498$	−6.68044	−0.299358
$499$	29.9989	1.34293	0.671467	−	0.741035i	$-0.265665\pi$
$499$	29.9989	1.34293	0.671467	+	0.741035i	$0.265665\pi$
$500$	0	0
$501$	2.33767	0.104440
$502$	35.9165	1.60303
$503$	13.8250	0.616426	0.308213	−	0.951317i	$-0.400269\pi$
$503$	13.8250	0.616426	0.308213	+	0.951317i	$0.400269\pi$
$504$	4.24024	0.188875
$505$	0	0
$506$	3.99578	0.177634
$507$	−12.9458	−0.574941
$508$	−4.35439	−0.193195
$509$	31.9537	1.41632	0.708161	−	0.706051i	$-0.249525\pi$
$509$	31.9537	1.41632	0.708161	+	0.706051i	$0.249525\pi$
$510$	0	0
$511$	−25.3792	−1.12271
$512$	−13.7541	−0.607852
$513$	0.753527	0.0332690
$514$	−6.89462	−0.304109
$515$	0	0
$516$	1.92177	0.0846012
$517$	−25.5102	−1.12194
$518$	26.2549	1.15358
$519$	−6.07099	−0.266487
$520$	0	0
$521$	−38.7968	−1.69972	−0.849859	−	0.527010i	$-0.823313\pi$
$521$	−38.7968	−1.69972	−0.849859	+	0.527010i	$0.823313\pi$
$522$	10.5712	0.462690
$523$	−23.7143	−1.03695	−0.518477	−	0.855091i	$-0.673501\pi$
$523$	−23.7143	−1.03695	−0.518477	+	0.855091i	$0.673501\pi$
$524$	7.81991	0.341614
$525$	0	0
$526$	11.8793	0.517962
$527$	73.2422	3.19048
$528$	13.6851	0.595568
$529$	−22.2384	−0.966885
$530$	0	0
$531$	−3.85114	−0.167125
$532$	−0.463003	−0.0200737
$533$	−0.884365	−0.0383061
$534$	18.6108	0.805367
$535$	0	0
$536$	4.50888	0.194754
$537$	11.4235	0.492962
$538$	16.5137	0.711954
$539$	12.3789	0.533197
$540$	0	0
$541$	2.81765	0.121140	0.0605702	−	0.998164i	$-0.480708\pi$
$541$	2.81765	0.121140	0.0605702	+	0.998164i	$0.480708\pi$
$542$	−39.5376	−1.69828
$543$	−7.85007	−0.336879
$544$	15.1965	0.651543
$545$	0	0
$546$	0.603779	0.0258394
$547$	4.42379	0.189148	0.0945739	−	0.995518i	$-0.469851\pi$
$547$	4.42379	0.189148	0.0945739	+	0.995518i	$0.469851\pi$
$548$	−3.65139	−0.155980
$549$	−4.39643	−0.187635
$550$	0	0
$551$	5.18036	0.220691
$552$	2.19486	0.0934193
$553$	−12.4357	−0.528821
$554$	−14.7993	−0.628761
$555$	0	0
$556$	−2.39607	−0.101616
$557$	−7.20182	−0.305151	−0.152575	−	0.988292i	$-0.548757\pi$
$557$	−7.20182	−0.305151	−0.152575	+	0.988292i	$0.548757\pi$
$558$	−15.0989	−0.639185
$559$	−1.22809	−0.0519427
$560$	0	0
$561$	22.2097	0.937696
$562$	−4.35585	−0.183740
$563$	23.2509	0.979907	0.489954	−	0.871749i	$-0.337014\pi$
$563$	23.2509	0.979907	0.489954	+	0.871749i	$0.337014\pi$
$564$	3.12232	0.131474
$565$	0	0
$566$	−36.3865	−1.52944
$567$	−1.68601	−0.0708056
$568$	11.0017	0.461620
$569$	−20.6237	−0.864589	−0.432295	−	0.901732i	$-0.642296\pi$
$569$	−20.6237	−0.864589	−0.432295	+	0.901732i	$0.642296\pi$
$570$	0	0
$571$	1.80372	0.0754834	0.0377417	−	0.999288i	$-0.487984\pi$
$571$	1.80372	0.0754834	0.0377417	+	0.999288i	$0.487984\pi$
$572$	0.252722	0.0105668
$573$	12.5641	0.524875
$574$	−9.84466	−0.410908
$575$	0	0
$576$	6.05938	0.252474
$577$	16.4863	0.686335	0.343168	−	0.939274i	$-0.388500\pi$
$577$	16.4863	0.686335	0.343168	+	0.939274i	$0.388500\pi$
$578$	59.4109	2.47117
$579$	10.1437	0.421559
$580$	0	0
$581$	7.32488	0.303887
$582$	−14.5683	−0.603876
$583$	17.8022	0.737291
$584$	−37.8572	−1.56654
$585$	0	0
$586$	−35.7318	−1.47607
$587$	7.27161	0.300131	0.150066	−	0.988676i	$-0.452051\pi$
$587$	7.27161	0.300131	0.150066	+	0.988676i	$0.452051\pi$
$588$	−1.51511	−0.0624822
$589$	−7.39909	−0.304874
$590$	0	0
$591$	2.04703	0.0842034
$592$	46.5450	1.91299
$593$	−2.47898	−0.101800	−0.0508998	−	0.998704i	$-0.516209\pi$
$593$	−2.47898	−0.101800	−0.0508998	+	0.998704i	$0.516209\pi$
$594$	−4.57853	−0.187859
$595$	0	0
$596$	3.97759	0.162928
$597$	3.57125	0.146162
$598$	0.312532	0.0127804
$599$	30.2951	1.23782	0.618912	−	0.785460i	$-0.287574\pi$
$599$	30.2951	1.23782	0.618912	+	0.785460i	$0.287574\pi$
$600$	0	0
$601$	4.46130	0.181980	0.0909900	−	0.995852i	$-0.470997\pi$
$601$	4.46130	0.181980	0.0909900	+	0.995852i	$0.470997\pi$
$602$	−13.6710	−0.557187
$603$	−1.79282	−0.0730094
$604$	7.43922	0.302698
$605$	0	0
$606$	−10.0636	−0.408805
$607$	−17.2931	−0.701906	−0.350953	−	0.936393i	$-0.614142\pi$
$607$	−17.2931	−0.701906	−0.350953	+	0.936393i	$0.614142\pi$
$608$	−1.53518	−0.0622598
$609$	−11.5910	−0.469691
$610$	0	0
$611$	−1.99529	−0.0807210
$612$	−2.71836	−0.109883
$613$	−14.3129	−0.578094	−0.289047	−	0.957315i	$-0.593338\pi$
$613$	−14.3129	−0.578094	−0.289047	+	0.957315i	$0.593338\pi$
$614$	33.5415	1.35362
$615$	0	0
$616$	−12.6256	−0.508700
$617$	26.3569	1.06109	0.530544	−	0.847658i	$-0.321988\pi$
$617$	26.3569	1.06109	0.530544	+	0.847658i	$0.321988\pi$
$618$	1.07761	0.0433477
$619$	−2.79825	−0.112471	−0.0562356	−	0.998418i	$-0.517910\pi$
$619$	−2.79825	−0.112471	−0.0562356	+	0.998418i	$0.517910\pi$
$620$	0	0
$621$	−0.872721	−0.0350211
$622$	15.4403	0.619101
$623$	−20.4061	−0.817553
$624$	1.07039	0.0428497
$625$	0	0
$626$	−9.30856	−0.372045
$627$	−2.24368	−0.0896039
$628$	−1.27533	−0.0508914
$629$	75.5386	3.01192
$630$	0	0
$631$	35.4035	1.40939	0.704696	−	0.709510i	$-0.251084\pi$
$631$	35.4035	1.40939	0.704696	+	0.709510i	$0.251084\pi$
$632$	−18.5499	−0.737877
$633$	3.72643	0.148112
$634$	13.0688	0.519028
$635$	0	0
$636$	−2.17890	−0.0863989
$637$	0.968220	0.0383623
$638$	−31.4766	−1.24617
$639$	−4.37450	−0.173052
$640$	0	0
$641$	16.9334	0.668829	0.334415	−	0.942426i	$-0.391461\pi$
$641$	16.9334	0.668829	0.334415	+	0.942426i	$0.391461\pi$
$642$	−19.2652	−0.760336
$643$	−25.9118	−1.02186	−0.510931	−	0.859622i	$-0.670699\pi$
$643$	−25.9118	−1.02186	−0.510931	+	0.859622i	$0.670699\pi$
$644$	0.536241	0.0211309
$645$	0	0
$646$	−8.64260	−0.340038
$647$	−10.9259	−0.429542	−0.214771	−	0.976664i	$-0.568900\pi$
$647$	−10.9259	−0.429542	−0.214771	+	0.976664i	$0.568900\pi$
$648$	−2.51496	−0.0987969
$649$	11.4671	0.450121
$650$	0	0
$651$	16.5554	0.648857
$652$	−2.17701	−0.0852582
$653$	−14.2580	−0.557958	−0.278979	−	0.960297i	$-0.589996\pi$
$653$	−14.2580	−0.557958	−0.278979	+	0.960297i	$0.589996\pi$
$654$	6.58659	0.257556
$655$	0	0
$656$	−17.4527	−0.681414
$657$	15.0528	0.587267
$658$	−22.2114	−0.865890
$659$	22.8894	0.891644	0.445822	−	0.895122i	$-0.352912\pi$
$659$	22.8894	0.891644	0.445822	+	0.895122i	$0.352912\pi$
$660$	0	0
$661$	−39.9050	−1.55213	−0.776063	−	0.630656i	$-0.782786\pi$
$661$	−39.9050	−1.55213	−0.776063	+	0.630656i	$0.782786\pi$
$662$	25.0555	0.973811
$663$	1.73714	0.0674651
$664$	10.9263	0.424021
$665$	0	0
$666$	−15.5723	−0.603413
$667$	−5.99980	−0.232313
$668$	0.851941	0.0329626
$669$	−26.2488	−1.01484
$670$	0	0
$671$	13.0907	0.505360
$672$	3.43495	0.132506
$673$	2.15847	0.0832029	0.0416015	−	0.999134i	$-0.486754\pi$
$673$	2.15847	0.0832029	0.0416015	+	0.999134i	$0.486754\pi$
$674$	−37.9747	−1.46273
$675$	0	0
$676$	−4.71795	−0.181460
$677$	−41.9740	−1.61319	−0.806596	−	0.591103i	$-0.798693\pi$
$677$	−41.9740	−1.61319	−0.806596	+	0.591103i	$0.798693\pi$
$678$	−12.6514	−0.485873
$679$	15.9737	0.613013
$680$	0	0
$681$	4.49111	0.172100
$682$	44.9579	1.72153
$683$	29.2062	1.11754	0.558772	−	0.829321i	$-0.311273\pi$
$683$	29.2062	1.11754	0.558772	+	0.829321i	$0.311273\pi$
$684$	0.274615	0.0105002
$685$	0	0
$686$	28.9258	1.10439
$687$	−7.53935	−0.287644
$688$	−24.2361	−0.923991
$689$	1.39240	0.0530464
$690$	0	0
$691$	44.5257	1.69384	0.846919	−	0.531721i	$-0.178455\pi$
$691$	44.5257	1.69384	0.846919	+	0.531721i	$0.178455\pi$
$692$	−2.21251	−0.0841070
$693$	5.02021	0.190702
$694$	21.4454	0.814055
$695$	0	0
$696$	−17.2899	−0.655372
$697$	−28.3243	−1.07286
$698$	28.4417	1.07653
$699$	−10.3640	−0.392004
$700$	0	0
$701$	−4.50567	−0.170177	−0.0850884	−	0.996373i	$-0.527117\pi$
$701$	−4.50567	−0.170177	−0.0850884	+	0.996373i	$0.527117\pi$
$702$	−0.358112	−0.0135161
$703$	−7.63108	−0.287812
$704$	−18.0422	−0.679992
$705$	0	0
$706$	0.679241	0.0255636
$707$	11.0344	0.414990
$708$	−1.40351	−0.0527471
$709$	−51.2706	−1.92551	−0.962754	−	0.270380i	$-0.912851\pi$
$709$	−51.2706	−1.92551	−0.962754	+	0.270380i	$0.912851\pi$
$710$	0	0
$711$	7.37584	0.276616
$712$	−30.4390	−1.14075
$713$	8.56950	0.320930
$714$	19.3377	0.723695
$715$	0	0
$716$	4.16319	0.155586
$717$	−18.5738	−0.693651
$718$	7.10600	0.265194
$719$	35.2653	1.31517	0.657586	−	0.753379i	$-0.271578\pi$
$719$	35.2653	1.31517	0.657586	+	0.753379i	$0.271578\pi$
$720$	0	0
$721$	−1.18156	−0.0440036
$722$	−28.3427	−1.05481
$723$	−5.43543	−0.202146
$724$	−2.86088	−0.106324
$725$	0	0
$726$	−3.28150	−0.121788
$727$	−44.0038	−1.63201	−0.816005	−	0.578045i	$-0.803816\pi$
$727$	−44.0038	−1.63201	−0.816005	+	0.578045i	$0.803816\pi$
$728$	−0.987517	−0.0365998
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.3330	−1.45478
$732$	−1.60223	−0.0592202
$733$	27.8741	1.02955	0.514776	−	0.857325i	$-0.327875\pi$
$733$	27.8741	1.02955	0.514776	+	0.857325i	$0.327875\pi$
$734$	−13.4961	−0.498150
$735$	0	0
$736$	1.77802	0.0655386
$737$	5.33826	0.196637
$738$	5.83904	0.214938
$739$	−32.4413	−1.19337	−0.596687	−	0.802474i	$-0.703516\pi$
$739$	−32.4413	−1.19337	−0.596687	+	0.802474i	$0.703516\pi$
$740$	0	0
$741$	−0.175490	−0.00644680
$742$	15.5001	0.569027
$743$	−9.09256	−0.333574	−0.166787	−	0.985993i	$-0.553339\pi$
$743$	−9.09256	−0.333574	−0.166787	+	0.985993i	$0.553339\pi$
$744$	24.6951	0.905366
$745$	0	0
$746$	23.8271	0.872370
$747$	−4.34451	−0.158957
$748$	8.09411	0.295950
$749$	21.1236	0.771840
$750$	0	0
$751$	−49.8861	−1.82037	−0.910185	−	0.414202i	$-0.864061\pi$
$751$	−49.8861	−1.82037	−0.910185	+	0.414202i	$0.864061\pi$
$752$	−39.3766	−1.43592
$753$	23.3577	0.851201
$754$	−2.46196	−0.0896591
$755$	0	0
$756$	−0.614448	−0.0223472
$757$	−31.1239	−1.13122	−0.565608	−	0.824674i	$-0.691358\pi$
$757$	−31.1239	−1.13122	−0.565608	+	0.824674i	$0.691358\pi$
$758$	−41.9679	−1.52434
$759$	2.59859	0.0943228
$760$	0	0
$761$	39.1269	1.41835	0.709175	−	0.705032i	$-0.249067\pi$
$761$	39.1269	1.41835	0.709175	+	0.705032i	$0.249067\pi$
$762$	−18.3724	−0.665561
$763$	−7.22197	−0.261453
$764$	4.57887	0.165658
$765$	0	0
$766$	20.1499	0.728047
$767$	0.896901	0.0323852
$768$	8.47377	0.305771
$769$	14.3382	0.517050	0.258525	−	0.966005i	$-0.416764\pi$
$769$	14.3382	0.517050	0.258525	+	0.966005i	$0.416764\pi$
$770$	0	0
$771$	−4.48380	−0.161480
$772$	3.69677	0.133050
$773$	4.24997	0.152861	0.0764304	−	0.997075i	$-0.475648\pi$
$773$	4.24997	0.152861	0.0764304	+	0.997075i	$0.475648\pi$
$774$	8.10849	0.291454
$775$	0	0
$776$	23.8274	0.855352
$777$	17.0745	0.612543
$778$	25.9953	0.931975
$779$	2.86138	0.102520
$780$	0	0
$781$	13.0254	0.466085
$782$	10.0097	0.357946
$783$	6.87482	0.245686
$784$	19.1076	0.682414
$785$	0	0
$786$	32.9944	1.17687
$787$	27.1296	0.967067	0.483534	−	0.875326i	$-0.339353\pi$
$787$	27.1296	0.967067	0.483534	+	0.875326i	$0.339353\pi$
$788$	0.746017	0.0265758
$789$	7.72550	0.275035
$790$	0	0
$791$	13.8718	0.493225
$792$	7.48847	0.266091
$793$	1.02389	0.0363595
$794$	−16.0038	−0.567954
$795$	0	0
$796$	1.30151	0.0461306
$797$	12.6175	0.446935	0.223468	−	0.974711i	$-0.428262\pi$
$797$	12.6175	0.446935	0.223468	+	0.974711i	$0.428262\pi$
$798$	−1.95354	−0.0691545
$799$	−63.9049	−2.26079
$800$	0	0
$801$	12.1032	0.427646
$802$	−1.06838	−0.0377257
$803$	−44.8209	−1.58169
$804$	−0.653376	−0.0230428
$805$	0	0
$806$	3.51640	0.123860
$807$	10.7394	0.378044
$808$	16.4596	0.579047
$809$	−37.4138	−1.31540	−0.657699	−	0.753280i	$-0.728470\pi$
$809$	−37.4138	−1.31540	−0.657699	+	0.753280i	$0.728470\pi$
$810$	0	0
$811$	37.1992	1.30624	0.653120	−	0.757254i	$-0.273460\pi$
$811$	37.1992	1.30624	0.653120	+	0.757254i	$0.273460\pi$
$812$	−4.22422	−0.148241
$813$	−25.7126	−0.901780
$814$	46.3675	1.62518
$815$	0	0
$816$	34.2821	1.20011
$817$	3.97351	0.139016
$818$	−1.82845	−0.0639302
$819$	0.392657	0.0137206
$820$	0	0
$821$	27.4740	0.958850	0.479425	−	0.877583i	$-0.340845\pi$
$821$	27.4740	0.958850	0.479425	+	0.877583i	$0.340845\pi$
$822$	−15.4063	−0.537355
$823$	23.4267	0.816602	0.408301	−	0.912847i	$-0.366121\pi$
$823$	23.4267	0.816602	0.408301	+	0.912847i	$0.366121\pi$
$824$	−1.76249	−0.0613993
$825$	0	0
$826$	9.98420	0.347395
$827$	−32.4570	−1.12864	−0.564320	−	0.825556i	$-0.690862\pi$
$827$	−32.4570	−1.12864	−0.564320	+	0.825556i	$0.690862\pi$
$828$	−0.318054	−0.0110531
$829$	−0.154106	−0.00535231	−0.00267615	−	0.999996i	$-0.500852\pi$
$829$	−0.154106	−0.00535231	−0.00267615	+	0.999996i	$0.500852\pi$
$830$	0	0
$831$	−9.62446	−0.333869
$832$	−1.41118	−0.0489239
$833$	31.0100	1.07443
$834$	−10.1097	−0.350070
$835$	0	0
$836$	−0.817686	−0.0282802
$837$	−9.81929	−0.339404
$838$	3.04214	0.105089
$839$	−35.4217	−1.22289	−0.611447	−	0.791285i	$-0.709412\pi$
$839$	−35.4217	−1.22289	−0.611447	+	0.791285i	$0.709412\pi$
$840$	0	0
$841$	18.2632	0.629765
$842$	17.0412	0.587279
$843$	−2.83275	−0.0975651
$844$	1.35806	0.0467462
$845$	0	0
$846$	13.1740	0.452930
$847$	3.59806	0.123631
$848$	27.4788	0.943624
$849$	−23.6634	−0.812125
$850$	0	0
$851$	8.83818	0.302969
$852$	−1.59424	−0.0546178
$853$	−36.6066	−1.25339	−0.626694	−	0.779266i	$-0.715592\pi$
$853$	−36.6066	−1.25339	−0.626694	+	0.779266i	$0.715592\pi$
$854$	11.3979	0.390027
$855$	0	0
$856$	31.5094	1.07697
$857$	−2.04867	−0.0699813	−0.0349907	−	0.999388i	$-0.511140\pi$
$857$	−2.04867	−0.0699813	−0.0349907	+	0.999388i	$0.511140\pi$
$858$	1.06630	0.0364030
$859$	14.1821	0.483888	0.241944	−	0.970290i	$-0.422215\pi$
$859$	14.1821	0.483888	0.241944	+	0.970290i	$0.422215\pi$
$860$	0	0
$861$	−6.40231	−0.218190
$862$	−16.0128	−0.545398
$863$	43.1358	1.46836	0.734180	−	0.678955i	$-0.237567\pi$
$863$	43.1358	1.46836	0.734180	+	0.678955i	$0.237567\pi$
$864$	−2.03733	−0.0693113
$865$	0	0
$866$	−15.7832	−0.536334
$867$	38.6369	1.31218
$868$	6.03344	0.204788
$869$	−21.9621	−0.745013
$870$	0	0
$871$	0.417534	0.0141476
$872$	−10.7728	−0.364812
$873$	−9.47426	−0.320655
$874$	−1.01120	−0.0342044
$875$	0	0
$876$	5.48584	0.185350
$877$	−6.74680	−0.227823	−0.113912	−	0.993491i	$-0.536338\pi$
$877$	−6.74680	−0.227823	−0.113912	+	0.993491i	$0.536338\pi$
$878$	−37.9983	−1.28238
$879$	−23.2376	−0.783784
$880$	0	0
$881$	17.1783	0.578752	0.289376	−	0.957216i	$-0.406552\pi$
$881$	17.1783	0.578752	0.289376	+	0.957216i	$0.406552\pi$
$882$	−6.39269	−0.215253
$883$	−56.8617	−1.91355	−0.956774	−	0.290831i	$-0.906068\pi$
$883$	−56.8617	−1.91355	−0.956774	+	0.290831i	$0.906068\pi$
$884$	0.633084	0.0212929
$885$	0	0
$886$	11.5777	0.388960
$887$	36.9297	1.23998	0.619989	−	0.784610i	$-0.287137\pi$
$887$	36.9297	1.23998	0.619989	+	0.784610i	$0.287137\pi$
$888$	25.4694	0.854696
$889$	20.1447	0.675632
$890$	0	0
$891$	−2.97757	−0.0997524
$892$	−9.56611	−0.320297
$893$	6.45581	0.216036
$894$	16.7826	0.561293
$895$	0	0
$896$	−22.5790	−0.754312
$897$	0.203250	0.00678631
$898$	−48.6869	−1.62470
$899$	−67.5058	−2.25145
$900$	0	0
$901$	44.5956	1.48570
$902$	−17.3862	−0.578895
$903$	−8.89069	−0.295864
$904$	20.6921	0.688209
$905$	0	0
$906$	31.3882	1.04280
$907$	−25.7833	−0.856120	−0.428060	−	0.903750i	$-0.640803\pi$
$907$	−25.7833	−0.856120	−0.428060	+	0.903750i	$0.640803\pi$
$908$	1.63674	0.0543170
$909$	−6.54468	−0.217073
$910$	0	0
$911$	−27.0066	−0.894768	−0.447384	−	0.894342i	$-0.647644\pi$
$911$	−27.0066	−0.894768	−0.447384	+	0.894342i	$0.647644\pi$
$912$	−3.46326	−0.114680
$913$	12.9361	0.428122
$914$	4.54755	0.150420
$915$	0	0
$916$	−2.74764	−0.0907845
$917$	−36.1772	−1.19468
$918$	−11.4695	−0.378551
$919$	−25.9342	−0.855492	−0.427746	−	0.903899i	$-0.640692\pi$
$919$	−25.9342	−0.855492	−0.427746	+	0.903899i	$0.640692\pi$
$920$	0	0
$921$	21.8131	0.718767
$922$	−27.0647	−0.891330
$923$	1.01879	0.0335337
$924$	1.82956	0.0601882
$925$	0	0
$926$	−40.5055	−1.33109
$927$	0.700804	0.0230174
$928$	−14.0063	−0.459778
$929$	15.0866	0.494974	0.247487	−	0.968891i	$-0.420395\pi$
$929$	15.0866	0.494974	0.247487	+	0.968891i	$0.420395\pi$
$930$	0	0
$931$	−3.13270	−0.102670
$932$	−3.77707	−0.123722
$933$	10.0414	0.328740
$934$	−11.3158	−0.370265
$935$	0	0
$936$	0.585713	0.0191446
$937$	0.655563	0.0214163	0.0107082	−	0.999943i	$-0.496591\pi$
$937$	0.655563	0.0214163	0.0107082	+	0.999943i	$0.496591\pi$
$938$	4.64795	0.151761
$939$	−6.05366	−0.197554
$940$	0	0
$941$	−36.4140	−1.18706	−0.593531	−	0.804811i	$-0.702267\pi$
$941$	−36.4140	−1.18706	−0.593531	+	0.804811i	$0.702267\pi$
$942$	−5.38099	−0.175322
$943$	−3.31400	−0.107919
$944$	17.7001	0.576089
$945$	0	0
$946$	−24.1436	−0.784976
$947$	−45.4127	−1.47571	−0.737857	−	0.674957i	$-0.764162\pi$
$947$	−45.4127	−1.47571	−0.737857	+	0.674957i	$0.764162\pi$
$948$	2.68805	0.0873038
$949$	−3.50568	−0.113799
$950$	0	0
$951$	8.49907	0.275601
$952$	−31.6280	−1.02507
$953$	46.2536	1.49830	0.749151	−	0.662399i	$-0.230462\pi$
$953$	46.2536	1.49830	0.749151	+	0.662399i	$0.230462\pi$
$954$	−9.19338	−0.297647
$955$	0	0
$956$	−6.76903	−0.218926
$957$	−20.4703	−0.661710
$958$	−43.7730	−1.41424
$959$	16.8924	0.545485
$960$	0	0
$961$	65.4184	2.11027
$962$	3.62665	0.116928
$963$	−12.5288	−0.403734
$964$	−1.98088	−0.0638000
$965$	0	0
$966$	2.26255	0.0727965
$967$	2.91740	0.0938172	0.0469086	−	0.998899i	$-0.485063\pi$
$967$	2.91740	0.0938172	0.0469086	+	0.998899i	$0.485063\pi$
$968$	5.36709	0.172505
$969$	−5.62057	−0.180559
$970$	0	0
$971$	−1.21820	−0.0390940	−0.0195470	−	0.999809i	$-0.506222\pi$
$971$	−1.21820	−0.0390940	−0.0195470	+	0.999809i	$0.506222\pi$
$972$	0.364440	0.0116894
$973$	11.0849	0.355367
$974$	−3.66591	−0.117463
$975$	0	0
$976$	20.2063	0.646787
$977$	−44.6720	−1.42918	−0.714591	−	0.699542i	$-0.753387\pi$
$977$	−44.6720	−1.42918	−0.714591	+	0.699542i	$0.753387\pi$
$978$	−9.18540	−0.293717
$979$	−36.0382	−1.15178
$980$	0	0
$981$	4.28348	0.136761
$982$	−39.5538	−1.26221
$983$	36.1310	1.15240	0.576201	−	0.817308i	$-0.304535\pi$
$983$	36.1310	1.15240	0.576201	+	0.817308i	$0.304535\pi$
$984$	−9.55010	−0.304446
$985$	0	0
$986$	−78.8510	−2.51113
$987$	−14.4448	−0.459783
$988$	−0.0639556	−0.00203470
$989$	−4.60205	−0.146337
$990$	0	0
$991$	−40.0195	−1.27126	−0.635631	−	0.771993i	$-0.719260\pi$
$991$	−40.0195	−1.27126	−0.635631	+	0.771993i	$0.719260\pi$
$992$	20.0051	0.635163
$993$	16.2945	0.517089
$994$	11.3410	0.359715
$995$	0	0
$996$	−1.58331	−0.0501692
$997$	−3.24338	−0.102719	−0.0513595	−	0.998680i	$-0.516355\pi$
$997$	−3.24338	−0.102719	−0.0513595	+	0.998680i	$0.516355\pi$
$998$	46.1285	1.46017
$999$	−10.1272	−0.320409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.m.1.8		8
3.2	odd	2		5625.2.a.bd.1.1		8
5.2	odd	4		1875.2.b.h.1249.14		16
5.3	odd	4		1875.2.b.h.1249.3		16
5.4	even	2		1875.2.a.p.1.1		8
15.14	odd	2		5625.2.a.t.1.8		8
25.3	odd	20		75.2.i.a.34.1	✓	16
25.4	even	10		375.2.g.d.76.4		16
25.6	even	5		375.2.g.e.301.1		16
25.8	odd	20		375.2.i.c.199.4		16
25.17	odd	20		75.2.i.a.64.1	yes	16
25.19	even	10		375.2.g.d.301.4		16
25.21	even	5		375.2.g.e.76.1		16
25.22	odd	20		375.2.i.c.49.4		16
75.17	even	20		225.2.m.b.64.4		16
75.53	even	20		225.2.m.b.109.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.1	✓	16	25.3	odd	20
75.2.i.a.64.1	yes	16	25.17	odd	20
225.2.m.b.64.4		16	75.17	even	20
225.2.m.b.109.4		16	75.53	even	20
375.2.g.d.76.4		16	25.4	even	10
375.2.g.d.301.4		16	25.19	even	10
375.2.g.e.76.1		16	25.21	even	5
375.2.g.e.301.1		16	25.6	even	5
375.2.i.c.49.4		16	25.22	odd	20
375.2.i.c.199.4		16	25.8	odd	20
1875.2.a.m.1.8		8	1.1	even	1	trivial
1875.2.a.p.1.1		8	5.4	even	2
1875.2.b.h.1249.3		16	5.3	odd	4
1875.2.b.h.1249.14		16	5.2	odd	4
5625.2.a.t.1.8		8	15.14	odd	2
5625.2.a.bd.1.1		8	3.2	odd	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53767 q^{2} +1.00000 q^{3} +0.364440 q^{4} +1.53767 q^{6} -1.68601 q^{7} -2.51496 q^{8} +1.00000 q^{9} -2.97757 q^{11} +0.364440 q^{12} -0.232892 q^{13} -2.59253 q^{14} -4.59606 q^{16} -7.45901 q^{17} +1.53767 q^{18} +0.753527 q^{19} -1.68601 q^{21} -4.57853 q^{22} -0.872721 q^{23} -2.51496 q^{24} -0.358112 q^{26} +1.00000 q^{27} -0.614448 q^{28} +6.87482 q^{29} -9.81929 q^{31} -2.03733 q^{32} -2.97757 q^{33} -11.4695 q^{34} +0.364440 q^{36} -10.1272 q^{37} +1.15868 q^{38} -0.232892 q^{39} +3.79732 q^{41} -2.59253 q^{42} +5.27322 q^{43} -1.08514 q^{44} -1.34196 q^{46} +8.56747 q^{47} -4.59606 q^{48} -4.15738 q^{49} -7.45901 q^{51} -0.0848751 q^{52} -5.97876 q^{53} +1.53767 q^{54} +4.24024 q^{56} +0.753527 q^{57} +10.5712 q^{58} -3.85114 q^{59} -4.39643 q^{61} -15.0989 q^{62} -1.68601 q^{63} +6.05938 q^{64} -4.57853 q^{66} -1.79282 q^{67} -2.71836 q^{68} -0.872721 q^{69} -4.37450 q^{71} -2.51496 q^{72} +15.0528 q^{73} -15.5723 q^{74} +0.274615 q^{76} +5.02021 q^{77} -0.358112 q^{78} +7.37584 q^{79} +1.00000 q^{81} +5.83904 q^{82} -4.34451 q^{83} -0.614448 q^{84} +8.10849 q^{86} +6.87482 q^{87} +7.48847 q^{88} +12.1032 q^{89} +0.392657 q^{91} -0.318054 q^{92} -9.81929 q^{93} +13.1740 q^{94} -2.03733 q^{96} -9.47426 q^{97} -6.39269 q^{98} -2.97757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^11 + 4 * q^12 - 16 * q^13 + 6 * q^14 - 16 * q^17 - 4 * q^18 - 14 * q^19 - 8 * q^21 - 12 * q^22 - 14 * q^23 - 12 * q^24 + 6 * q^26 + 8 * q^27 - 16 * q^28 + 2 * q^29 - 22 * q^31 + 2 * q^32 + 2 * q^33 - 12 * q^34 + 4 * q^36 - 28 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 + 6 * q^42 - 20 * q^43 + 22 * q^44 - 2 * q^46 - 10 * q^47 - 16 * q^51 - 16 * q^52 - 44 * q^53 - 4 * q^54 + 30 * q^56 - 14 * q^57 - 8 * q^58 + 14 * q^59 - 20 * q^61 - 16 * q^62 - 8 * q^63 + 6 * q^64 - 12 * q^66 - 16 * q^67 + 2 * q^68 - 14 * q^69 + 16 * q^71 - 12 * q^72 - 24 * q^73 + 26 * q^74 - 16 * q^76 - 46 * q^77 + 6 * q^78 - 30 * q^79 + 8 * q^81 - 16 * q^82 - 12 * q^83 - 16 * q^84 + 32 * q^86 + 2 * q^87 - 32 * q^88 + 16 * q^89 - 12 * q^91 + 2 * q^92 - 22 * q^93 + 14 * q^94 + 2 * q^96 - 16 * q^97 - 4 * q^98 + 2 * q^99

Introduction

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