LMFDB - Embedded newform 415.2.a.e.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [415,2,Mod(1,415)] 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("415.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 415 = 5 \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	415.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.31379168388$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 20x^{9} - x^{8} + 146x^{7} + 15x^{6} - 464x^{5} - 76x^{4} + 567x^{3} + 136x^{2} - 100x - 8 $ x^11 - 20x^9 - x^8 + 146x^7 + 15x^6 - 464x^5 - 76x^4 + 567x^3 + 136x^2 - 100x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.381602$ of defining polynomial
Character	$\chi$	$=$	415.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.381602 q^{2} +1.63018 q^{3} -1.85438 q^{4} +1.00000 q^{5} +0.622079 q^{6} -0.0761781 q^{7} -1.47084 q^{8} -0.342523 q^{9} +0.381602 q^{10} +4.79348 q^{11} -3.02297 q^{12} +6.08842 q^{13} -0.0290697 q^{14} +1.63018 q^{15} +3.14748 q^{16} +7.73742 q^{17} -0.130708 q^{18} -0.833555 q^{19} -1.85438 q^{20} -0.124184 q^{21} +1.82920 q^{22} -8.25464 q^{23} -2.39773 q^{24} +1.00000 q^{25} +2.32336 q^{26} -5.44890 q^{27} +0.141263 q^{28} -6.42032 q^{29} +0.622079 q^{30} -4.72443 q^{31} +4.14277 q^{32} +7.81422 q^{33} +2.95262 q^{34} -0.0761781 q^{35} +0.635168 q^{36} +0.579319 q^{37} -0.318086 q^{38} +9.92521 q^{39} -1.47084 q^{40} -4.08551 q^{41} -0.0473888 q^{42} +3.19950 q^{43} -8.88893 q^{44} -0.342523 q^{45} -3.14999 q^{46} -5.36358 q^{47} +5.13095 q^{48} -6.99420 q^{49} +0.381602 q^{50} +12.6134 q^{51} -11.2903 q^{52} -0.465047 q^{53} -2.07931 q^{54} +4.79348 q^{55} +0.112046 q^{56} -1.35884 q^{57} -2.45001 q^{58} +7.57236 q^{59} -3.02297 q^{60} -5.94859 q^{61} -1.80285 q^{62} +0.0260928 q^{63} -4.71408 q^{64} +6.08842 q^{65} +2.98192 q^{66} +6.35636 q^{67} -14.3481 q^{68} -13.4565 q^{69} -0.0290697 q^{70} +2.40089 q^{71} +0.503797 q^{72} +1.19045 q^{73} +0.221069 q^{74} +1.63018 q^{75} +1.54573 q^{76} -0.365158 q^{77} +3.78748 q^{78} +2.25348 q^{79} +3.14748 q^{80} -7.85511 q^{81} -1.55904 q^{82} -1.00000 q^{83} +0.230284 q^{84} +7.73742 q^{85} +1.22094 q^{86} -10.4663 q^{87} -7.05044 q^{88} +4.10871 q^{89} -0.130708 q^{90} -0.463805 q^{91} +15.3072 q^{92} -7.70166 q^{93} -2.04676 q^{94} -0.833555 q^{95} +6.75344 q^{96} -12.1829 q^{97} -2.66900 q^{98} -1.64188 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 18 q^{4} + 11 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 21 q^{9} - q^{11} + 2 q^{12} + q^{13} + 12 q^{14} + 20 q^{16} + 28 q^{17} - 26 q^{18} - 2 q^{19} + 18 q^{20} + 3 q^{21} - 20 q^{22} + q^{23}+ \cdots - 36 q^{99}+O(q^{100}) $ 11 * q + 18 * q^4 + 11 * q^5 - 5 * q^6 - 3 * q^7 + 3 * q^8 + 21 * q^9 - q^11 + 2 * q^12 + q^13 + 12 * q^14 + 20 * q^16 + 28 * q^17 - 26 * q^18 - 2 * q^19 + 18 * q^20 + 3 * q^21 - 20 * q^22 + q^23 - 10 * q^24 + 11 * q^25 + 21 * q^26 - 12 * q^27 - 22 * q^28 + 21 * q^29 - 5 * q^30 + 4 * q^31 + q^32 + 10 * q^33 - 4 * q^34 - 3 * q^35 + 45 * q^36 - 10 * q^37 + 17 * q^38 - 2 * q^39 + 3 * q^40 + 39 * q^41 - 29 * q^42 - 13 * q^43 - 25 * q^44 + 21 * q^45 - 13 * q^46 + 8 * q^47 + 17 * q^48 + 32 * q^49 + 32 * q^51 - 65 * q^52 + 21 * q^53 - 72 * q^54 - q^55 + 57 * q^56 - 19 * q^57 - 6 * q^58 + 5 * q^59 + 2 * q^60 + 14 * q^61 - q^62 - 30 * q^63 - 21 * q^64 + q^65 - 11 * q^66 - 42 * q^67 + 61 * q^68 - 7 * q^69 + 12 * q^70 + 3 * q^71 - 69 * q^72 + 3 * q^73 - 54 * q^74 + 13 * q^76 - 42 * q^77 - 7 * q^78 - 20 * q^79 + 20 * q^80 + 39 * q^81 - 43 * q^82 - 11 * q^83 - 116 * q^84 + 28 * q^85 + 22 * q^86 - 16 * q^87 - 75 * q^88 + 37 * q^89 - 26 * q^90 - 17 * q^91 + 7 * q^92 - 55 * q^93 - 54 * q^94 - 2 * q^95 - 58 * q^96 - 2 * q^97 - 26 * q^98 - 36 * 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 $

$2$	0.381602	0.269834	0.134917	−	0.990857i	$-0.456923\pi$
$2$	0.381602	0.269834	0.134917	+	0.990857i	$0.456923\pi$
$3$	1.63018	0.941183	0.470592	−	0.882351i	$-0.344041\pi$
$3$	1.63018	0.941183	0.470592	+	0.882351i	$0.344041\pi$
$4$	−1.85438	−0.927190
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.622079	0.253963
$7$	−0.0761781	−0.0287926	−0.0143963	−	0.999896i	$-0.504583\pi$
$7$	−0.0761781	−0.0287926	−0.0143963	+	0.999896i	$0.504583\pi$
$8$	−1.47084	−0.520020
$9$	−0.342523	−0.114174
$10$	0.381602	0.120673
$11$	4.79348	1.44529	0.722644	−	0.691220i	$-0.242926\pi$
$11$	4.79348	1.44529	0.722644	+	0.691220i	$0.242926\pi$
$12$	−3.02297	−0.872655
$13$	6.08842	1.68863	0.844313	−	0.535851i	$-0.180009\pi$
$13$	6.08842	1.68863	0.844313	+	0.535851i	$0.180009\pi$
$14$	−0.0290697	−0.00776921
$15$	1.63018	0.420910
$16$	3.14748	0.786871
$17$	7.73742	1.87660	0.938300	−	0.345823i	$-0.112400\pi$
$17$	7.73742	1.87660	0.938300	+	0.345823i	$0.112400\pi$
$18$	−0.130708	−0.0308081
$19$	−0.833555	−0.191231	−0.0956153	−	0.995418i	$-0.530482\pi$
$19$	−0.833555	−0.191231	−0.0956153	+	0.995418i	$0.530482\pi$
$20$	−1.85438	−0.414652
$21$	−0.124184	−0.0270991
$22$	1.82920	0.389987
$23$	−8.25464	−1.72121	−0.860606	−	0.509271i	$-0.829915\pi$
$23$	−8.25464	−1.72121	−0.860606	+	0.509271i	$0.829915\pi$
$24$	−2.39773	−0.489434
$25$	1.00000	0.200000
$26$	2.32336	0.455648
$27$	−5.44890	−1.04864
$28$	0.141263	0.0266962
$29$	−6.42032	−1.19222	−0.596112	−	0.802901i	$-0.703289\pi$
$29$	−6.42032	−1.19222	−0.596112	+	0.802901i	$0.703289\pi$
$30$	0.622079	0.113576
$31$	−4.72443	−0.848533	−0.424267	−	0.905537i	$-0.639468\pi$
$31$	−4.72443	−0.848533	−0.424267	+	0.905537i	$0.639468\pi$
$32$	4.14277	0.732345
$33$	7.81422	1.36028
$34$	2.95262	0.506370
$35$	−0.0761781	−0.0128764
$36$	0.635168	0.105861
$37$	0.579319	0.0952394	0.0476197	−	0.998866i	$-0.484836\pi$
$37$	0.579319	0.0952394	0.0476197	+	0.998866i	$0.484836\pi$
$38$	−0.318086	−0.0516004
$39$	9.92521	1.58931
$40$	−1.47084	−0.232560
$41$	−4.08551	−0.638049	−0.319025	−	0.947746i	$-0.603355\pi$
$41$	−4.08551	−0.638049	−0.319025	+	0.947746i	$0.603355\pi$
$42$	−0.0473888	−0.00731225
$43$	3.19950	0.487920	0.243960	−	0.969785i	$-0.421553\pi$
$43$	3.19950	0.487920	0.243960	+	0.969785i	$0.421553\pi$
$44$	−8.88893	−1.34006
$45$	−0.342523	−0.0510604
$46$	−3.14999	−0.464441
$47$	−5.36358	−0.782359	−0.391179	−	0.920314i	$-0.627933\pi$
$47$	−5.36358	−0.782359	−0.391179	+	0.920314i	$0.627933\pi$
$48$	5.13095	0.740590
$49$	−6.99420	−0.999171
$50$	0.381602	0.0539667
$51$	12.6134	1.76622
$52$	−11.2903	−1.56568
$53$	−0.465047	−0.0638791	−0.0319395	−	0.999490i	$-0.510168\pi$
$53$	−0.465047	−0.0638791	−0.0319395	+	0.999490i	$0.510168\pi$
$54$	−2.07931	−0.282959
$55$	4.79348	0.646353
$56$	0.112046	0.0149727
$57$	−1.35884	−0.179983
$58$	−2.45001	−0.321702
$59$	7.57236	0.985838	0.492919	−	0.870075i	$-0.335930\pi$
$59$	7.57236	0.985838	0.492919	+	0.870075i	$0.335930\pi$
$60$	−3.02297	−0.390263
$61$	−5.94859	−0.761638	−0.380819	−	0.924650i	$-0.624358\pi$
$61$	−5.94859	−0.761638	−0.380819	+	0.924650i	$0.624358\pi$
$62$	−1.80285	−0.228963
$63$	0.0260928	0.00328738
$64$	−4.71408	−0.589260
$65$	6.08842	0.755176
$66$	2.98192	0.367050
$67$	6.35636	0.776553	0.388276	−	0.921543i	$-0.373071\pi$
$67$	6.35636	0.776553	0.388276	+	0.921543i	$0.373071\pi$
$68$	−14.3481	−1.73996
$69$	−13.4565	−1.61998
$70$	−0.0290697	−0.00347450
$71$	2.40089	0.284933	0.142466	−	0.989800i	$-0.454497\pi$
$71$	2.40089	0.284933	0.142466	+	0.989800i	$0.454497\pi$
$72$	0.503797	0.0593731
$73$	1.19045	0.139332	0.0696658	−	0.997570i	$-0.477807\pi$
$73$	1.19045	0.139332	0.0696658	+	0.997570i	$0.477807\pi$
$74$	0.221069	0.0256988
$75$	1.63018	0.188237
$76$	1.54573	0.177307
$77$	−0.365158	−0.0416136
$78$	3.78748	0.428848
$79$	2.25348	0.253537	0.126768	−	0.991932i	$-0.459540\pi$
$79$	2.25348	0.253537	0.126768	+	0.991932i	$0.459540\pi$
$80$	3.14748	0.351899
$81$	−7.85511	−0.872790
$82$	−1.55904	−0.172167
$83$	−1.00000	−0.109764
$83$	−1.00000	−0.109764
$84$	0.230284	0.0251260
$85$	7.73742	0.839241
$86$	1.22094	0.131657
$87$	−10.4663	−1.12210
$88$	−7.05044	−0.751580
$89$	4.10871	0.435523	0.217761	−	0.976002i	$-0.430125\pi$
$89$	4.10871	0.435523	0.217761	+	0.976002i	$0.430125\pi$
$90$	−0.130708	−0.0137778
$91$	−0.463805	−0.0486199
$92$	15.3072	1.59589
$93$	−7.70166	−0.798625
$94$	−2.04676	−0.211107
$95$	−0.833555	−0.0855209
$96$	6.75344	0.689270
$97$	−12.1829	−1.23698	−0.618492	−	0.785791i	$-0.712256\pi$
$97$	−12.1829	−1.23698	−0.618492	+	0.785791i	$0.712256\pi$
$98$	−2.66900	−0.269610
$99$	−1.64188	−0.165015
$100$	−1.85438	−0.185438
$101$	13.3255	1.32593	0.662966	−	0.748649i	$-0.269297\pi$
$101$	13.3255	1.32593	0.662966	+	0.748649i	$0.269297\pi$
$102$	4.81329	0.476586
$103$	−10.3909	−1.02385	−0.511923	−	0.859031i	$-0.671067\pi$
$103$	−10.3909	−1.02385	−0.511923	+	0.859031i	$0.671067\pi$
$104$	−8.95510	−0.878120
$105$	−0.124184	−0.0121191
$106$	−0.177463	−0.0172367
$107$	−15.6812	−1.51596	−0.757979	−	0.652279i	$-0.773813\pi$
$107$	−15.6812	−1.51596	−0.757979	+	0.652279i	$0.773813\pi$
$108$	10.1043	0.972290
$109$	17.0197	1.63019	0.815095	−	0.579328i	$-0.196685\pi$
$109$	17.0197	1.63019	0.815095	+	0.579328i	$0.196685\pi$
$110$	1.82920	0.174408
$111$	0.944392	0.0896377
$112$	−0.239769	−0.0226561
$113$	−2.67610	−0.251746	−0.125873	−	0.992046i	$-0.540173\pi$
$113$	−2.67610	−0.251746	−0.125873	+	0.992046i	$0.540173\pi$
$114$	−0.518537	−0.0485654
$115$	−8.25464	−0.769749
$116$	11.9057	1.10542
$117$	−2.08543	−0.192798
$118$	2.88963	0.266012
$119$	−0.589422	−0.0540322
$120$	−2.39773	−0.218882
$121$	11.9775	1.08886
$122$	−2.26999	−0.205516
$123$	−6.66010	−0.600521
$124$	8.76089	0.786751
$125$	1.00000	0.0894427
$126$	0.00995706	0.000887046 0
$127$	−2.16302	−0.191937	−0.0959684	−	0.995384i	$-0.530595\pi$
$127$	−2.16302	−0.191937	−0.0959684	+	0.995384i	$0.530595\pi$
$128$	−10.0844	−0.891347
$129$	5.21576	0.459222
$130$	2.32336	0.203772
$131$	18.3932	1.60702	0.803509	−	0.595293i	$-0.202964\pi$
$131$	18.3932	1.60702	0.803509	+	0.595293i	$0.202964\pi$
$132$	−14.4905	−1.26124
$133$	0.0634986	0.00550603
$134$	2.42560	0.209540
$135$	−5.44890	−0.468967
$136$	−11.3805	−0.975870
$137$	−17.4166	−1.48800	−0.744001	−	0.668178i	$-0.767074\pi$
$137$	−17.4166	−1.48800	−0.744001	+	0.668178i	$0.767074\pi$
$138$	−5.13504	−0.437124
$139$	−13.2880	−1.12707	−0.563535	−	0.826092i	$-0.690559\pi$
$139$	−13.2880	−1.12707	−0.563535	+	0.826092i	$0.690559\pi$
$140$	0.141263	0.0119389
$141$	−8.74359	−0.736343
$142$	0.916184	0.0768845
$143$	29.1847	2.44055
$144$	−1.07809	−0.0898406
$145$	−6.42032	−0.533179
$146$	0.454278	0.0375963
$147$	−11.4018	−0.940403
$148$	−1.07428	−0.0883050
$149$	17.6410	1.44521	0.722605	−	0.691261i	$-0.242944\pi$
$149$	17.6410	1.44521	0.722605	+	0.691261i	$0.242944\pi$
$150$	0.622079	0.0507926
$151$	−16.8357	−1.37007	−0.685033	−	0.728512i	$-0.740212\pi$
$151$	−16.8357	−1.37007	−0.685033	+	0.728512i	$0.740212\pi$
$152$	1.22603	0.0994438
$153$	−2.65025	−0.214260
$154$	−0.139345	−0.0112288
$155$	−4.72443	−0.379476
$156$	−18.4051	−1.47359
$157$	−13.6620	−1.09035	−0.545175	−	0.838322i	$-0.683537\pi$
$157$	−13.6620	−1.09035	−0.545175	+	0.838322i	$0.683537\pi$
$158$	0.859934	0.0684127
$159$	−0.758108	−0.0601219
$160$	4.14277	0.327514
$161$	0.628823	0.0495582
$162$	−2.99753	−0.235508
$163$	1.28882	0.100948	0.0504741	−	0.998725i	$-0.483927\pi$
$163$	1.28882	0.100948	0.0504741	+	0.998725i	$0.483927\pi$
$164$	7.57608	0.591593
$165$	7.81422	0.608336
$166$	−0.381602	−0.0296181
$167$	−17.5230	−1.35597	−0.677985	−	0.735076i	$-0.737146\pi$
$167$	−17.5230	−1.35597	−0.677985	+	0.735076i	$0.737146\pi$
$168$	0.182654	0.0140921
$169$	24.0689	1.85145
$170$	2.95262	0.226455
$171$	0.285512	0.0218336
$172$	−5.93309	−0.452394
$173$	16.7053	1.27008	0.635040	−	0.772479i	$-0.280983\pi$
$173$	16.7053	1.27008	0.635040	+	0.772479i	$0.280983\pi$
$174$	−3.99395	−0.302781
$175$	−0.0761781	−0.00575852
$176$	15.0874	1.13726
$177$	12.3443	0.927854
$178$	1.56789	0.117519
$179$	1.30846	0.0977988	0.0488994	−	0.998804i	$-0.484429\pi$
$179$	1.30846	0.0977988	0.0488994	+	0.998804i	$0.484429\pi$
$180$	0.635168	0.0473427
$181$	−10.8110	−0.803578	−0.401789	−	0.915732i	$-0.631611\pi$
$181$	−10.8110	−0.803578	−0.401789	+	0.915732i	$0.631611\pi$
$182$	−0.176989	−0.0131193
$183$	−9.69725	−0.716841
$184$	12.1413	0.895065
$185$	0.579319	0.0425924
$186$	−2.93897	−0.215496
$187$	37.0892	2.71223
$188$	9.94612	0.725395
$189$	0.415087	0.0301931
$190$	−0.318086	−0.0230764
$191$	−13.5911	−0.983415	−0.491707	−	0.870760i	$-0.663627\pi$
$191$	−13.5911	−0.983415	−0.491707	+	0.870760i	$0.663627\pi$
$192$	−7.68478	−0.554601
$193$	−9.19181	−0.661641	−0.330820	−	0.943694i	$-0.607325\pi$
$193$	−9.19181	−0.661641	−0.330820	+	0.943694i	$0.607325\pi$
$194$	−4.64901	−0.333780
$195$	9.92521	0.710759
$196$	12.9699	0.926421
$197$	24.8615	1.77131	0.885655	−	0.464344i	$-0.153710\pi$
$197$	24.8615	1.77131	0.885655	+	0.464344i	$0.153710\pi$
$198$	−0.626545	−0.0445266
$199$	−11.0180	−0.781048	−0.390524	−	0.920593i	$-0.627706\pi$
$199$	−11.0180	−0.781048	−0.390524	+	0.920593i	$0.627706\pi$
$200$	−1.47084	−0.104004
$201$	10.3620	0.730878
$202$	5.08502	0.357781
$203$	0.489088	0.0343272
$204$	−23.3900	−1.63762
$205$	−4.08551	−0.285344
$206$	−3.96519	−0.276268
$207$	2.82741	0.196518
$208$	19.1632	1.32873
$209$	−3.99563	−0.276383
$210$	−0.0473888	−0.00327014
$211$	22.5868	1.55494	0.777471	−	0.628918i	$-0.216502\pi$
$211$	22.5868	1.55494	0.777471	+	0.628918i	$0.216502\pi$
$212$	0.862373	0.0592280
$213$	3.91387	0.268174
$214$	−5.98398	−0.409056
$215$	3.19950	0.218204
$216$	8.01447	0.545315
$217$	0.359898	0.0244315
$218$	6.49475	0.439880
$219$	1.94064	0.131136
$220$	−8.88893	−0.599292
$221$	47.1087	3.16887
$222$	0.360382	0.0241873
$223$	−14.8867	−0.996887	−0.498444	−	0.866922i	$-0.666095\pi$
$223$	−14.8867	−0.996887	−0.498444	+	0.866922i	$0.666095\pi$
$224$	−0.315588	−0.0210861
$225$	−0.342523	−0.0228349
$226$	−1.02121	−0.0679296
$227$	9.68451	0.642784	0.321392	−	0.946946i	$-0.395849\pi$
$227$	9.68451	0.642784	0.321392	+	0.946946i	$0.395849\pi$
$228$	2.51981	0.166878
$229$	−20.2691	−1.33942	−0.669711	−	0.742621i	$-0.733582\pi$
$229$	−20.2691	−1.33942	−0.669711	+	0.742621i	$0.733582\pi$
$230$	−3.14999	−0.207704
$231$	−0.595272	−0.0391661
$232$	9.44327	0.619981
$233$	19.7228	1.29208	0.646041	−	0.763303i	$-0.276423\pi$
$233$	19.7228	1.29208	0.646041	+	0.763303i	$0.276423\pi$
$234$	−0.795804	−0.0520233
$235$	−5.36358	−0.349882
$236$	−14.0420	−0.914059
$237$	3.67357	0.238624
$238$	−0.224925	−0.0145797
$239$	23.2525	1.50408	0.752039	−	0.659119i	$-0.229070\pi$
$239$	23.2525	1.50408	0.752039	+	0.659119i	$0.229070\pi$
$240$	5.13095	0.331202
$241$	−10.4213	−0.671298	−0.335649	−	0.941987i	$-0.608956\pi$
$241$	−10.4213	−0.671298	−0.335649	+	0.941987i	$0.608956\pi$
$242$	4.57062	0.293811
$243$	3.54150	0.227187
$244$	11.0309	0.706183
$245$	−6.99420	−0.446843
$246$	−2.54151	−0.162041
$247$	−5.07503	−0.322917
$248$	6.94888	0.441255
$249$	−1.63018	−0.103308
$250$	0.381602	0.0241346
$251$	−3.62296	−0.228679	−0.114340	−	0.993442i	$-0.536475\pi$
$251$	−3.62296	−0.228679	−0.114340	+	0.993442i	$0.536475\pi$
$252$	−0.0483859	−0.00304803
$253$	−39.5685	−2.48765
$254$	−0.825413	−0.0517910
$255$	12.6134	0.789879
$256$	5.57991	0.348744
$257$	6.32617	0.394616	0.197308	−	0.980342i	$-0.436780\pi$
$257$	6.32617	0.394616	0.197308	+	0.980342i	$0.436780\pi$
$258$	1.99034	0.123913
$259$	−0.0441314	−0.00274219
$260$	−11.2903	−0.700192
$261$	2.19911	0.136122
$262$	7.01887	0.433627
$263$	−15.4402	−0.952082	−0.476041	−	0.879423i	$-0.657929\pi$
$263$	−15.4402	−0.952082	−0.476041	+	0.879423i	$0.657929\pi$
$264$	−11.4935	−0.707374
$265$	−0.465047	−0.0285676
$266$	0.0242312	0.00148571
$267$	6.69793	0.409907
$268$	−11.7871	−0.720012
$269$	12.1989	0.743777	0.371889	−	0.928277i	$-0.378710\pi$
$269$	12.1989	0.743777	0.371889	+	0.928277i	$0.378710\pi$
$270$	−2.07931	−0.126543
$271$	−3.43954	−0.208937	−0.104469	−	0.994528i	$-0.533314\pi$
$271$	−3.43954	−0.208937	−0.104469	+	0.994528i	$0.533314\pi$
$272$	24.3534	1.47664
$273$	−0.756083	−0.0457603
$274$	−6.64622	−0.401513
$275$	4.79348	0.289058
$276$	24.9535	1.50202
$277$	−9.74712	−0.585648	−0.292824	−	0.956166i	$-0.594595\pi$
$277$	−9.74712	−0.585648	−0.292824	+	0.956166i	$0.594595\pi$
$278$	−5.07072	−0.304122
$279$	1.61823	0.0968808
$280$	0.112046	0.00669602
$281$	−6.98593	−0.416746	−0.208373	−	0.978049i	$-0.566817\pi$
$281$	−6.98593	−0.416746	−0.208373	+	0.978049i	$0.566817\pi$
$282$	−3.33657	−0.198690
$283$	−19.2612	−1.14496	−0.572478	−	0.819920i	$-0.694018\pi$
$283$	−19.2612	−1.14496	−0.572478	+	0.819920i	$0.694018\pi$
$284$	−4.45216	−0.264187
$285$	−1.35884	−0.0804908
$286$	11.1370	0.658543
$287$	0.311226	0.0183711
$288$	−1.41899	−0.0836151
$289$	42.8677	2.52163
$290$	−2.45001	−0.143870
$291$	−19.8602	−1.16423
$292$	−2.20754	−0.129187
$293$	15.3470	0.896579	0.448289	−	0.893888i	$-0.352033\pi$
$293$	15.3470	0.896579	0.448289	+	0.893888i	$0.352033\pi$
$294$	−4.35094	−0.253752
$295$	7.57236	0.440880
$296$	−0.852085	−0.0495265
$297$	−26.1192	−1.51559
$298$	6.73186	0.389966
$299$	−50.2578	−2.90648
$300$	−3.02297	−0.174531
$301$	−0.243732	−0.0140485
$302$	−6.42453	−0.369690
$303$	21.7228	1.24794
$304$	−2.62360	−0.150474
$305$	−5.94859	−0.340615
$306$	−1.01134	−0.0578145
$307$	−6.92676	−0.395331	−0.197665	−	0.980270i	$-0.563336\pi$
$307$	−6.92676	−0.395331	−0.197665	+	0.980270i	$0.563336\pi$
$308$	0.677142	0.0385837
$309$	−16.9390	−0.963627
$310$	−1.80285	−0.102395
$311$	−19.6813	−1.11602	−0.558011	−	0.829833i	$-0.688435\pi$
$311$	−19.6813	−1.11602	−0.558011	+	0.829833i	$0.688435\pi$
$312$	−14.5984	−0.826471
$313$	−3.87220	−0.218869	−0.109435	−	0.993994i	$-0.534904\pi$
$313$	−3.87220	−0.218869	−0.109435	+	0.993994i	$0.534904\pi$
$314$	−5.21347	−0.294213
$315$	0.0260928	0.00147016
$316$	−4.17881	−0.235076
$317$	17.7095	0.994666	0.497333	−	0.867560i	$-0.334313\pi$
$317$	17.7095	0.994666	0.497333	+	0.867560i	$0.334313\pi$
$318$	−0.289296	−0.0162229
$319$	−30.7757	−1.72311
$320$	−4.71408	−0.263525
$321$	−25.5631	−1.42679
$322$	0.239960	0.0133725
$323$	−6.44956	−0.358863
$324$	14.5664	0.809242
$325$	6.08842	0.337725
$326$	0.491816	0.0272392
$327$	27.7451	1.53431
$328$	6.00913	0.331799
$329$	0.408587	0.0225262
$330$	2.98192	0.164150
$331$	10.7130	0.588839	0.294419	−	0.955676i	$-0.404874\pi$
$331$	10.7130	0.588839	0.294419	+	0.955676i	$0.404874\pi$
$332$	1.85438	0.101772
$333$	−0.198430	−0.0108739
$334$	−6.68681	−0.365886
$335$	6.35636	0.347285
$336$	−0.390866	−0.0213235
$337$	16.0911	0.876540	0.438270	−	0.898843i	$-0.355591\pi$
$337$	16.0911	0.876540	0.438270	+	0.898843i	$0.355591\pi$
$338$	9.18475	0.499585
$339$	−4.36251	−0.236939
$340$	−14.3481	−0.778136
$341$	−22.6465	−1.22638
$342$	0.108952	0.00589145
$343$	1.06605	0.0575614
$344$	−4.70596	−0.253728
$345$	−13.4565	−0.724475
$346$	6.37478	0.342710
$347$	3.82853	0.205526	0.102763	−	0.994706i	$-0.467232\pi$
$347$	3.82853	0.205526	0.102763	+	0.994706i	$0.467232\pi$
$348$	19.4084	1.04040
$349$	15.5449	0.832101	0.416050	−	0.909342i	$-0.363414\pi$
$349$	15.5449	0.832101	0.416050	+	0.909342i	$0.363414\pi$
$350$	−0.0290697	−0.00155384
$351$	−33.1752	−1.77076
$352$	19.8583	1.05845
$353$	9.50263	0.505774	0.252887	−	0.967496i	$-0.418620\pi$
$353$	9.50263	0.505774	0.252887	+	0.967496i	$0.418620\pi$
$354$	4.71061	0.250366
$355$	2.40089	0.127426
$356$	−7.61911	−0.403812
$357$	−0.960862	−0.0508542
$358$	0.499311	0.0263894
$359$	−3.38446	−0.178625	−0.0893125	−	0.996004i	$-0.528467\pi$
$359$	−3.38446	−0.178625	−0.0893125	+	0.996004i	$0.528467\pi$
$360$	0.503797	0.0265524
$361$	−18.3052	−0.963431
$362$	−4.12551	−0.216832
$363$	19.5254	1.02482
$364$	0.860070	0.0450799
$365$	1.19045	0.0623110
$366$	−3.70049	−0.193428
$367$	2.22778	0.116289	0.0581446	−	0.998308i	$-0.481482\pi$
$367$	2.22778	0.116289	0.0581446	+	0.998308i	$0.481482\pi$
$368$	−25.9814	−1.35437
$369$	1.39938	0.0728489
$370$	0.221069	0.0114929
$371$	0.0354264	0.00183924
$372$	14.2818	0.740477
$373$	−33.4446	−1.73170	−0.865849	−	0.500306i	$-0.833221\pi$
$373$	−33.4446	−1.73170	−0.865849	+	0.500306i	$0.833221\pi$
$374$	14.1533	0.731850
$375$	1.63018	0.0841820
$376$	7.88897	0.406843
$377$	−39.0897	−2.01322
$378$	0.158398	0.00814712
$379$	29.3505	1.50763	0.753817	−	0.657084i	$-0.228210\pi$
$379$	29.3505	1.50763	0.753817	+	0.657084i	$0.228210\pi$
$380$	1.54573	0.0792941
$381$	−3.52610	−0.180648
$382$	−5.18638	−0.265358
$383$	−4.51068	−0.230485	−0.115243	−	0.993337i	$-0.536765\pi$
$383$	−4.51068	−0.230485	−0.115243	+	0.993337i	$0.536765\pi$
$384$	−16.4394	−0.838920
$385$	−0.365158	−0.0186102
$386$	−3.50762	−0.178533
$387$	−1.09590	−0.0557080
$388$	22.5917	1.14692
$389$	−7.92259	−0.401691	−0.200846	−	0.979623i	$-0.564369\pi$
$389$	−7.92259	−0.401691	−0.200846	+	0.979623i	$0.564369\pi$
$390$	3.78748	0.191787
$391$	−63.8696	−3.23003
$392$	10.2873	0.519589
$393$	29.9841	1.51250
$394$	9.48721	0.477959
$395$	2.25348	0.113385
$396$	3.04467	0.153000
$397$	−0.720345	−0.0361531	−0.0180765	−	0.999837i	$-0.505754\pi$
$397$	−0.720345	−0.0361531	−0.0180765	+	0.999837i	$0.505754\pi$
$398$	−4.20451	−0.210753
$399$	0.103514	0.00518218
$400$	3.14748	0.157374
$401$	39.7909	1.98706	0.993531	−	0.113563i	$-0.0362264\pi$
$401$	39.7909	1.98706	0.993531	+	0.113563i	$0.0362264\pi$
$402$	3.95416	0.197215
$403$	−28.7643	−1.43285
$404$	−24.7104	−1.22939
$405$	−7.85511	−0.390323
$406$	0.186637	0.00926264
$407$	2.77695	0.137648
$408$	−18.5522	−0.918473
$409$	4.80091	0.237390	0.118695	−	0.992931i	$-0.462129\pi$
$409$	4.80091	0.237390	0.118695	+	0.992931i	$0.462129\pi$
$410$	−1.55904	−0.0769954
$411$	−28.3922	−1.40048
$412$	19.2687	0.949300
$413$	−0.576848	−0.0283848
$414$	1.07895	0.0530273
$415$	−1.00000	−0.0490881
$416$	25.2229	1.23666
$417$	−21.6617	−1.06078
$418$	−1.52474	−0.0745775
$419$	32.3407	1.57995	0.789973	−	0.613141i	$-0.210094\pi$
$419$	32.3407	1.57995	0.789973	+	0.613141i	$0.210094\pi$
$420$	0.230284	0.0112367
$421$	26.2043	1.27712	0.638560	−	0.769572i	$-0.279530\pi$
$421$	26.2043	1.27712	0.638560	+	0.769572i	$0.279530\pi$
$422$	8.61919	0.419576
$423$	1.83715	0.0893254
$424$	0.684009	0.0332184
$425$	7.73742	0.375320
$426$	1.49354	0.0723624
$427$	0.453152	0.0219296
$428$	29.0789	1.40558
$429$	47.5763	2.29701
$430$	1.22094	0.0588788
$431$	14.2425	0.686036	0.343018	−	0.939329i	$-0.388551\pi$
$431$	14.2425	0.686036	0.343018	+	0.939329i	$0.388551\pi$
$432$	−17.1503	−0.825146
$433$	18.8710	0.906882	0.453441	−	0.891286i	$-0.350196\pi$
$433$	18.8710	0.906882	0.453441	+	0.891286i	$0.350196\pi$
$434$	0.137338	0.00659243
$435$	−10.4663	−0.501819
$436$	−31.5609	−1.51150
$437$	6.88069	0.329148
$438$	0.740554	0.0353850
$439$	25.5166	1.21784	0.608920	−	0.793232i	$-0.291603\pi$
$439$	25.5166	1.21784	0.608920	+	0.793232i	$0.291603\pi$
$440$	−7.05044	−0.336117
$441$	2.39568	0.114080
$442$	17.9768	0.855068
$443$	−27.9646	−1.32864	−0.664318	−	0.747450i	$-0.731278\pi$
$443$	−27.9646	−1.32864	−0.664318	+	0.747450i	$0.731278\pi$
$444$	−1.75126	−0.0831112
$445$	4.10871	0.194772
$446$	−5.68080	−0.268994
$447$	28.7580	1.36021
$448$	0.359109	0.0169663
$449$	−20.4268	−0.964002	−0.482001	−	0.876171i	$-0.660090\pi$
$449$	−20.4268	−0.964002	−0.482001	+	0.876171i	$0.660090\pi$
$450$	−0.130708	−0.00616162
$451$	−19.5838	−0.922165
$452$	4.96250	0.233416
$453$	−27.4451	−1.28948
$454$	3.69563	0.173445
$455$	−0.463805	−0.0217435
$456$	1.99864	0.0935948
$457$	10.9786	0.513559	0.256780	−	0.966470i	$-0.417339\pi$
$457$	10.9786	0.513559	0.256780	+	0.966470i	$0.417339\pi$
$458$	−7.73475	−0.361421
$459$	−42.1605	−1.96788
$460$	15.3072	0.713704
$461$	−38.1410	−1.77640	−0.888201	−	0.459455i	$-0.848045\pi$
$461$	−38.1410	−1.77640	−0.888201	+	0.459455i	$0.848045\pi$
$462$	−0.227157	−0.0105683
$463$	−20.1515	−0.936519	−0.468259	−	0.883591i	$-0.655119\pi$
$463$	−20.1515	−0.936519	−0.468259	+	0.883591i	$0.655119\pi$
$464$	−20.2079	−0.938126
$465$	−7.70166	−0.357156
$466$	7.52626	0.348647
$467$	31.6392	1.46409	0.732045	−	0.681257i	$-0.238566\pi$
$467$	31.6392	1.46409	0.732045	+	0.681257i	$0.238566\pi$
$468$	3.86718	0.178760
$469$	−0.484215	−0.0223590
$470$	−2.04676	−0.0944098
$471$	−22.2715	−1.02622
$472$	−11.1377	−0.512656
$473$	15.3368	0.705185
$474$	1.40184	0.0643888
$475$	−0.833555	−0.0382461
$476$	1.09301	0.0500981
$477$	0.159289	0.00729336
$478$	8.87320	0.405851
$479$	23.8341	1.08901	0.544505	−	0.838758i	$-0.316718\pi$
$479$	23.8341	1.08901	0.544505	+	0.838758i	$0.316718\pi$
$480$	6.75344	0.308251
$481$	3.52714	0.160824
$482$	−3.97681	−0.181139
$483$	1.02509	0.0466433
$484$	−22.2108	−1.00958
$485$	−12.1829	−0.553196
$486$	1.35144	0.0613027
$487$	−11.4881	−0.520575	−0.260287	−	0.965531i	$-0.583817\pi$
$487$	−11.4881	−0.520575	−0.260287	+	0.965531i	$0.583817\pi$
$488$	8.74942	0.396068
$489$	2.10100	0.0950107
$490$	−2.66900	−0.120573
$491$	−27.1086	−1.22339	−0.611696	−	0.791093i	$-0.709513\pi$
$491$	−27.1086	−1.22339	−0.611696	+	0.791093i	$0.709513\pi$
$492$	12.3504	0.556797
$493$	−49.6767	−2.23733
$494$	−1.93664	−0.0871338
$495$	−1.64188	−0.0737970
$496$	−14.8701	−0.667686
$497$	−0.182895	−0.00820396
$498$	−0.622079	−0.0278760
$499$	0.465902	0.0208566	0.0104283	−	0.999946i	$-0.496681\pi$
$499$	0.465902	0.0208566	0.0104283	+	0.999946i	$0.496681\pi$
$500$	−1.85438	−0.0829304
$501$	−28.5656	−1.27622
$502$	−1.38253	−0.0617053
$503$	−40.9594	−1.82629	−0.913144	−	0.407638i	$-0.866353\pi$
$503$	−40.9594	−1.82629	−0.913144	+	0.407638i	$0.866353\pi$
$504$	−0.0383783	−0.00170951
$505$	13.3255	0.592975
$506$	−15.0994	−0.671251
$507$	39.2366	1.74256
$508$	4.01106	0.177962
$509$	−5.75767	−0.255204	−0.127602	−	0.991825i	$-0.540728\pi$
$509$	−5.75767	−0.255204	−0.127602	+	0.991825i	$0.540728\pi$
$510$	4.81329	0.213136
$511$	−0.0906862	−0.00401172
$512$	22.2982	0.985450
$513$	4.54196	0.200532
$514$	2.41408	0.106481
$515$	−10.3909	−0.457878
$516$	−9.67199	−0.425786
$517$	−25.7102	−1.13073
$518$	−0.0168406	−0.000739935 0
$519$	27.2326	1.19538
$520$	−8.95510	−0.392707
$521$	39.7262	1.74044	0.870218	−	0.492666i	$-0.163978\pi$
$521$	39.7262	1.74044	0.870218	+	0.492666i	$0.163978\pi$
$522$	0.839186	0.0367302
$523$	16.0767	0.702985	0.351493	−	0.936191i	$-0.385674\pi$
$523$	16.0767	0.702985	0.351493	+	0.936191i	$0.385674\pi$
$524$	−34.1079	−1.49001
$525$	−0.124184	−0.00541982
$526$	−5.89201	−0.256904
$527$	−36.5549	−1.59236
$528$	24.5951	1.07037
$529$	45.1391	1.96257
$530$	−0.177463	−0.00770849
$531$	−2.59371	−0.112557
$532$	−0.117751	−0.00510513
$533$	−24.8743	−1.07743
$534$	2.55594	0.110607
$535$	−15.6812	−0.677957
$536$	−9.34918	−0.403823
$537$	2.13302	0.0920466
$538$	4.65511	0.200696
$539$	−33.5265	−1.44409
$540$	10.1043	0.434821
$541$	37.9691	1.63242	0.816210	−	0.577755i	$-0.196071\pi$
$541$	37.9691	1.63242	0.816210	+	0.577755i	$0.196071\pi$
$542$	−1.31254	−0.0563782
$543$	−17.6239	−0.756314
$544$	32.0543	1.37432
$545$	17.0197	0.729043
$546$	−0.288523	−0.0123477
$547$	−2.29088	−0.0979509	−0.0489754	−	0.998800i	$-0.515596\pi$
$547$	−2.29088	−0.0979509	−0.0489754	+	0.998800i	$0.515596\pi$
$548$	32.2970	1.37966
$549$	2.03753	0.0869597
$550$	1.82920	0.0779975
$551$	5.35169	0.227990
$552$	19.7924	0.842420
$553$	−0.171666	−0.00729998
$554$	−3.71952	−0.158027
$555$	0.944392	0.0400872
$556$	24.6409	1.04501
$557$	−4.96293	−0.210286	−0.105143	−	0.994457i	$-0.533530\pi$
$557$	−4.96293	−0.210286	−0.105143	+	0.994457i	$0.533530\pi$
$558$	0.617520	0.0261417
$559$	19.4799	0.823913
$560$	−0.239769	−0.0101321
$561$	60.4619	2.55270
$562$	−2.66585	−0.112452
$563$	8.06701	0.339984	0.169992	−	0.985445i	$-0.445626\pi$
$563$	8.06701	0.339984	0.169992	+	0.985445i	$0.445626\pi$
$564$	16.2139	0.682730
$565$	−2.67610	−0.112584
$566$	−7.35010	−0.308948
$567$	0.598387	0.0251299
$568$	−3.53132	−0.148171
$569$	−3.30222	−0.138436	−0.0692182	−	0.997602i	$-0.522050\pi$
$569$	−3.30222	−0.138436	−0.0692182	+	0.997602i	$0.522050\pi$
$570$	−0.518537	−0.0217191
$571$	−17.3300	−0.725238	−0.362619	−	0.931937i	$-0.618117\pi$
$571$	−17.3300	−0.725238	−0.362619	+	0.931937i	$0.618117\pi$
$572$	−54.1196	−2.26285
$573$	−22.1558	−0.925573
$574$	0.118765	0.00495714
$575$	−8.25464	−0.344242
$576$	1.61468	0.0672784
$577$	9.99784	0.416216	0.208108	−	0.978106i	$-0.433270\pi$
$577$	9.99784	0.416216	0.208108	+	0.978106i	$0.433270\pi$
$578$	16.3584	0.680420
$579$	−14.9843	−0.622725
$580$	11.9057	0.494358
$581$	0.0761781	0.00316040
$582$	−7.57871	−0.314148
$583$	−2.22919	−0.0923237
$584$	−1.75096	−0.0724553
$585$	−2.08543	−0.0862218
$586$	5.85643	0.241927
$587$	17.9292	0.740018	0.370009	−	0.929028i	$-0.379355\pi$
$587$	17.9292	0.740018	0.370009	+	0.929028i	$0.379355\pi$
$588$	21.1432	0.871932
$589$	3.93807	0.162265
$590$	2.88963	0.118964
$591$	40.5287	1.66713
$592$	1.82340	0.0749411
$593$	7.04507	0.289306	0.144653	−	0.989482i	$-0.453793\pi$
$593$	7.04507	0.289306	0.144653	+	0.989482i	$0.453793\pi$
$594$	−9.96715	−0.408957
$595$	−0.589422	−0.0241639
$596$	−32.7132	−1.33998
$597$	−17.9613	−0.735109
$598$	−19.1785	−0.784266
$599$	−16.1538	−0.660028	−0.330014	−	0.943976i	$-0.607053\pi$
$599$	−16.1538	−0.660028	−0.330014	+	0.943976i	$0.607053\pi$
$600$	−2.39773	−0.0978869
$601$	−41.6254	−1.69794	−0.848968	−	0.528444i	$-0.822776\pi$
$601$	−41.6254	−1.69794	−0.848968	+	0.528444i	$0.822776\pi$
$602$	−0.0930087	−0.00379075
$603$	−2.17720	−0.0886625
$604$	31.2197	1.27031
$605$	11.9775	0.486953
$606$	8.28949	0.336737
$607$	−2.47890	−0.100616	−0.0503078	−	0.998734i	$-0.516020\pi$
$607$	−2.47890	−0.100616	−0.0503078	+	0.998734i	$0.516020\pi$
$608$	−3.45322	−0.140047
$609$	0.797300	0.0323082
$610$	−2.26999	−0.0919094
$611$	−32.6558	−1.32111
$612$	4.91457	0.198660
$613$	46.9158	1.89491	0.947455	−	0.319889i	$-0.103646\pi$
$613$	46.9158	1.89491	0.947455	+	0.319889i	$0.103646\pi$
$614$	−2.64327	−0.106674
$615$	−6.66010	−0.268561
$616$	0.537089	0.0216399
$617$	3.90275	0.157119	0.0785594	−	0.996909i	$-0.474968\pi$
$617$	3.90275	0.157119	0.0785594	+	0.996909i	$0.474968\pi$
$618$	−6.46397	−0.260019
$619$	26.5939	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$619$	26.5939	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$620$	8.76089	0.351846
$621$	44.9788	1.80494
$622$	−7.51042	−0.301140
$623$	−0.312994	−0.0125398
$624$	31.2394	1.25058
$625$	1.00000	0.0400000
$626$	−1.47764	−0.0590583
$627$	−6.51358	−0.260127
$628$	25.3346	1.01096
$629$	4.48243	0.178726
$630$	0.00995706	0.000396699 0
$631$	−28.3391	−1.12816	−0.564082	−	0.825719i	$-0.690770\pi$
$631$	−28.3391	−1.12816	−0.564082	+	0.825719i	$0.690770\pi$
$632$	−3.31451	−0.131844
$633$	36.8206	1.46349
$634$	6.75799	0.268394
$635$	−2.16302	−0.0858368
$636$	1.40582	0.0557444
$637$	−42.5836	−1.68723
$638$	−11.7441	−0.464952
$639$	−0.822360	−0.0325321
$640$	−10.0844	−0.398622
$641$	37.6459	1.48692	0.743462	−	0.668778i	$-0.233182\pi$
$641$	37.6459	1.48692	0.743462	+	0.668778i	$0.233182\pi$
$642$	−9.75494	−0.384997
$643$	−10.5720	−0.416920	−0.208460	−	0.978031i	$-0.566845\pi$
$643$	−10.5720	−0.416920	−0.208460	+	0.978031i	$0.566845\pi$
$644$	−1.16608	−0.0459498
$645$	5.21576	0.205370
$646$	−2.46117	−0.0968333
$647$	28.6602	1.12675	0.563374	−	0.826202i	$-0.309503\pi$
$647$	28.6602	1.12675	0.563374	+	0.826202i	$0.309503\pi$
$648$	11.5536	0.453869
$649$	36.2980	1.42482
$650$	2.32336	0.0911295
$651$	0.586698	0.0229945
$652$	−2.38996	−0.0935981
$653$	8.23646	0.322318	0.161159	−	0.986928i	$-0.448477\pi$
$653$	8.23646	0.322318	0.161159	+	0.986928i	$0.448477\pi$
$654$	10.5876	0.414007
$655$	18.3932	0.718680
$656$	−12.8591	−0.502062
$657$	−0.407757	−0.0159081
$658$	0.155918	0.00607831
$659$	6.51555	0.253810	0.126905	−	0.991915i	$-0.459496\pi$
$659$	6.51555	0.253810	0.126905	+	0.991915i	$0.459496\pi$
$660$	−14.4905	−0.564043
$661$	30.5093	1.18668	0.593338	−	0.804954i	$-0.297810\pi$
$661$	30.5093	1.18668	0.593338	+	0.804954i	$0.297810\pi$
$662$	4.08810	0.158888
$663$	76.7955	2.98249
$664$	1.47084	0.0570797
$665$	0.0634986	0.00246237
$666$	−0.0757214	−0.00293415
$667$	52.9975	2.05207
$668$	32.4943	1.25724
$669$	−24.2680	−0.938253
$670$	2.42560	0.0937091
$671$	−28.5144	−1.10079
$672$	−0.514464	−0.0198459
$673$	30.8237	1.18817	0.594084	−	0.804403i	$-0.297515\pi$
$673$	30.8237	1.18817	0.594084	+	0.804403i	$0.297515\pi$
$674$	6.14041	0.236520
$675$	−5.44890	−0.209728
$676$	−44.6329	−1.71665
$677$	30.9153	1.18817	0.594086	−	0.804402i	$-0.297514\pi$
$677$	30.9153	1.18817	0.594086	+	0.804402i	$0.297514\pi$
$678$	−1.66474	−0.0639341
$679$	0.928068	0.0356160
$680$	−11.3805	−0.436423
$681$	15.7875	0.604977
$682$	−8.64195	−0.330917
$683$	−0.759911	−0.0290772	−0.0145386	−	0.999894i	$-0.504628\pi$
$683$	−0.759911	−0.0290772	−0.0145386	+	0.999894i	$0.504628\pi$
$684$	−0.529448	−0.0202439
$685$	−17.4166	−0.665455
$686$	0.406808	0.0155320
$687$	−33.0423	−1.26064
$688$	10.0704	0.383930
$689$	−2.83140	−0.107868
$690$	−5.13504	−0.195488
$691$	35.9938	1.36927	0.684634	−	0.728887i	$-0.259962\pi$
$691$	35.9938	1.36927	0.684634	+	0.728887i	$0.259962\pi$
$692$	−30.9780	−1.17761
$693$	0.125075	0.00475122
$694$	1.46098	0.0554579
$695$	−13.2880	−0.504041
$696$	15.3942	0.583516
$697$	−31.6113	−1.19736
$698$	5.93198	0.224529
$699$	32.1516	1.21609
$700$	0.141263	0.00533924
$701$	−8.20111	−0.309752	−0.154876	−	0.987934i	$-0.549498\pi$
$701$	−8.20111	−0.309752	−0.154876	+	0.987934i	$0.549498\pi$
$702$	−12.6597	−0.477811
$703$	−0.482894	−0.0182127
$704$	−22.5968	−0.851650
$705$	−8.74359	−0.329303
$706$	3.62623	0.136475
$707$	−1.01511	−0.0381770
$708$	−22.8910	−0.860296
$709$	−22.3663	−0.839984	−0.419992	−	0.907528i	$-0.637967\pi$
$709$	−22.3663	−0.839984	−0.419992	+	0.907528i	$0.637967\pi$
$710$	0.916184	0.0343838
$711$	−0.771870	−0.0289474
$712$	−6.04326	−0.226481
$713$	38.9985	1.46051
$714$	−0.366667	−0.0137222
$715$	29.1847	1.09145
$716$	−2.42638	−0.0906781
$717$	37.9057	1.41561
$718$	−1.29152	−0.0481990
$719$	−8.41809	−0.313942	−0.156971	−	0.987603i	$-0.550173\pi$
$719$	−8.41809	−0.313942	−0.156971	+	0.987603i	$0.550173\pi$
$720$	−1.07809	−0.0401779
$721$	0.791560	0.0294792
$722$	−6.98530	−0.259966
$723$	−16.9886	−0.631814
$724$	20.0478	0.745069
$725$	−6.42032	−0.238445
$726$	7.45093	0.276530
$727$	32.7173	1.21342	0.606709	−	0.794924i	$-0.292489\pi$
$727$	32.7173	1.21342	0.606709	+	0.794924i	$0.292489\pi$
$728$	0.682182	0.0252834
$729$	29.3386	1.08661
$730$	0.454278	0.0168136
$731$	24.7559	0.915630
$732$	17.9824	0.664648
$733$	−17.7563	−0.655844	−0.327922	−	0.944705i	$-0.606348\pi$
$733$	−17.7563	−0.655844	−0.327922	+	0.944705i	$0.606348\pi$
$734$	0.850126	0.0313787
$735$	−11.4018	−0.420561
$736$	−34.1971	−1.26052
$737$	30.4691	1.12234
$738$	0.534007	0.0196571
$739$	16.2783	0.598805	0.299403	−	0.954127i	$-0.403213\pi$
$739$	16.2783	0.598805	0.299403	+	0.954127i	$0.403213\pi$
$740$	−1.07428	−0.0394912
$741$	−8.27320	−0.303924
$742$	0.0135188	0.000496290 0
$743$	−39.4194	−1.44616	−0.723079	−	0.690765i	$-0.757274\pi$
$743$	−39.4194	−1.44616	−0.723079	+	0.690765i	$0.757274\pi$
$744$	11.3279	0.415301
$745$	17.6410	0.646318
$746$	−12.7626	−0.467270
$747$	0.342523	0.0125323
$748$	−68.7774	−2.51475
$749$	1.19456	0.0436484
$750$	0.622079	0.0227151
$751$	−32.5955	−1.18943	−0.594714	−	0.803937i	$-0.702735\pi$
$751$	−32.5955	−1.18943	−0.594714	+	0.803937i	$0.702735\pi$
$752$	−16.8818	−0.615615
$753$	−5.90607	−0.215229
$754$	−14.9167	−0.543234
$755$	−16.8357	−0.612712
$756$	−0.769729	−0.0279948
$757$	−38.1681	−1.38724	−0.693621	−	0.720340i	$-0.743986\pi$
$757$	−38.1681	−1.38724	−0.693621	+	0.720340i	$0.743986\pi$
$758$	11.2002	0.406811
$759$	−64.5036	−2.34133
$760$	1.22603	0.0444726
$761$	−14.3504	−0.520201	−0.260100	−	0.965582i	$-0.583756\pi$
$761$	−14.3504	−0.520201	−0.260100	+	0.965582i	$0.583756\pi$
$762$	−1.34557	−0.0487448
$763$	−1.29653	−0.0469374
$764$	25.2030	0.911812
$765$	−2.65025	−0.0958199
$766$	−1.72129	−0.0621926
$767$	46.1038	1.66471
$768$	9.09624	0.328232
$769$	−50.4182	−1.81813	−0.909064	−	0.416657i	$-0.863201\pi$
$769$	−50.4182	−1.81813	−0.909064	+	0.416657i	$0.863201\pi$
$770$	−0.139345	−0.00502165
$771$	10.3128	0.371405
$772$	17.0451	0.613467
$773$	−12.5954	−0.453025	−0.226512	−	0.974008i	$-0.572732\pi$
$773$	−12.5954	−0.453025	−0.226512	+	0.974008i	$0.572732\pi$
$774$	−0.418200	−0.0150319
$775$	−4.72443	−0.169707
$776$	17.9191	0.643257
$777$	−0.0719420	−0.00258090
$778$	−3.02328	−0.108390
$779$	3.40549	0.122014
$780$	−18.4051	−0.659008
$781$	11.5086	0.411810
$782$	−24.3728	−0.871569
$783$	34.9837	1.25022
$784$	−22.0141	−0.786219
$785$	−13.6620	−0.487619
$786$	11.4420	0.408123
$787$	−44.8914	−1.60021	−0.800103	−	0.599862i	$-0.795222\pi$
$787$	−44.8914	−1.60021	−0.800103	+	0.599862i	$0.795222\pi$
$788$	−46.1027	−1.64234
$789$	−25.1702	−0.896084
$790$	0.859934	0.0305951
$791$	0.203860	0.00724843
$792$	2.41494	0.0858112
$793$	−36.2175	−1.28612
$794$	−0.274885	−0.00975531
$795$	−0.758108	−0.0268873
$796$	20.4316	0.724179
$797$	2.08094	0.0737105	0.0368553	−	0.999321i	$-0.488266\pi$
$797$	2.08094	0.0737105	0.0368553	+	0.999321i	$0.488266\pi$
$798$	0.0395012	0.00139833
$799$	−41.5003	−1.46817
$800$	4.14277	0.146469
$801$	−1.40733	−0.0497256
$802$	15.1843	0.536176
$803$	5.70640	0.201374
$804$	−19.2151	−0.677663
$805$	0.628823	0.0221631
$806$	−10.9765	−0.386632
$807$	19.8863	0.700031
$808$	−19.5996	−0.689512
$809$	−13.3820	−0.470485	−0.235243	−	0.971937i	$-0.575588\pi$
$809$	−13.3820	−0.470485	−0.235243	+	0.971937i	$0.575588\pi$
$810$	−2.99753	−0.105322
$811$	−1.57763	−0.0553981	−0.0276990	−	0.999616i	$-0.508818\pi$
$811$	−1.57763	−0.0553981	−0.0276990	+	0.999616i	$0.508818\pi$
$812$	−0.906955	−0.0318279
$813$	−5.60706	−0.196648
$814$	1.05969	0.0371422
$815$	1.28882	0.0451454
$816$	39.7003	1.38979
$817$	−2.66696	−0.0933051
$818$	1.83204	0.0640557
$819$	0.158864	0.00555115
$820$	7.57608	0.264568
$821$	−20.3887	−0.711570	−0.355785	−	0.934568i	$-0.615786\pi$
$821$	−20.3887	−0.711570	−0.355785	+	0.934568i	$0.615786\pi$
$822$	−10.8345	−0.377897
$823$	11.1627	0.389106	0.194553	−	0.980892i	$-0.437674\pi$
$823$	11.1627	0.389106	0.194553	+	0.980892i	$0.437674\pi$
$824$	15.2834	0.532421
$825$	7.81422	0.272056
$826$	−0.220127	−0.00765918
$827$	−14.2386	−0.495124	−0.247562	−	0.968872i	$-0.579629\pi$
$827$	−14.2386	−0.495124	−0.247562	+	0.968872i	$0.579629\pi$
$828$	−5.24309	−0.182210
$829$	15.9964	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$829$	15.9964	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$830$	−0.381602	−0.0132456
$831$	−15.8895	−0.551202
$832$	−28.7013	−0.995039
$833$	−54.1170	−1.87504
$834$	−8.26617	−0.286234
$835$	−17.5230	−0.606408
$836$	7.40941	0.256260
$837$	25.7430	0.889808
$838$	12.3413	0.426323
$839$	44.3604	1.53149	0.765746	−	0.643143i	$-0.222370\pi$
$839$	44.3604	1.53149	0.765746	+	0.643143i	$0.222370\pi$
$840$	0.182654	0.00630218
$841$	12.2206	0.421398
$842$	9.99963	0.344610
$843$	−11.3883	−0.392234
$844$	−41.8846	−1.44173
$845$	24.0689	0.827996
$846$	0.701062	0.0241030
$847$	−0.912420	−0.0313511
$848$	−1.46373	−0.0502646
$849$	−31.3991	−1.07761
$850$	2.95262	0.101274
$851$	−4.78207	−0.163927
$852$	−7.25780	−0.248648
$853$	42.9204	1.46957	0.734783	−	0.678302i	$-0.237284\pi$
$853$	42.9204	1.46957	0.734783	+	0.678302i	$0.237284\pi$
$854$	0.172924	0.00591733
$855$	0.285512	0.00976430
$856$	23.0645	0.788329
$857$	18.2282	0.622662	0.311331	−	0.950302i	$-0.399225\pi$
$857$	18.2282	0.622662	0.311331	+	0.950302i	$0.399225\pi$
$858$	18.1552	0.619809
$859$	−8.29963	−0.283180	−0.141590	−	0.989925i	$-0.545221\pi$
$859$	−8.29963	−0.283180	−0.141590	+	0.989925i	$0.545221\pi$
$860$	−5.93309	−0.202317
$861$	0.507354	0.0172906
$862$	5.43496	0.185116
$863$	23.6589	0.805360	0.402680	−	0.915341i	$-0.368079\pi$
$863$	23.6589	0.805360	0.402680	+	0.915341i	$0.368079\pi$
$864$	−22.5735	−0.767967
$865$	16.7053	0.567997
$866$	7.20121	0.244707
$867$	69.8819	2.37331
$868$	−0.667388	−0.0226526
$869$	10.8020	0.366433
$870$	−3.99395	−0.135408
$871$	38.7002	1.31131
$872$	−25.0332	−0.847732
$873$	4.17292	0.141232
$874$	2.62569	0.0888152
$875$	−0.0761781	−0.00257529
$876$	−3.59869	−0.121588
$877$	−53.7913	−1.81640	−0.908202	−	0.418532i	$-0.862545\pi$
$877$	−53.7913	−1.81640	−0.908202	+	0.418532i	$0.862545\pi$
$878$	9.73719	0.328614
$879$	25.0183	0.843845
$880$	15.0874	0.508596
$881$	12.8646	0.433419	0.216710	−	0.976236i	$-0.430468\pi$
$881$	12.8646	0.433419	0.216710	+	0.976236i	$0.430468\pi$
$882$	0.914196	0.0307826
$883$	−16.6092	−0.558945	−0.279472	−	0.960154i	$-0.590160\pi$
$883$	−16.6092	−0.558945	−0.279472	+	0.960154i	$0.590160\pi$
$884$	−87.3574	−2.93815
$885$	12.3443	0.414949
$886$	−10.6713	−0.358511
$887$	−19.0785	−0.640593	−0.320296	−	0.947317i	$-0.603782\pi$
$887$	−19.0785	−0.640593	−0.320296	+	0.947317i	$0.603782\pi$
$888$	−1.38905	−0.0466135
$889$	0.164775	0.00552636
$890$	1.56789	0.0525559
$891$	−37.6533	−1.26143
$892$	27.6056	0.924304
$893$	4.47084	0.149611
$894$	10.9741	0.367030
$895$	1.30846	0.0437370
$896$	0.768213	0.0256642
$897$	−81.9290	−2.73553
$898$	−7.79492	−0.260120
$899$	30.3324	1.01164
$900$	0.635168	0.0211723
$901$	−3.59826	−0.119875
$902$	−7.47322	−0.248831
$903$	−0.397326	−0.0132222
$904$	3.93611	0.130913
$905$	−10.8110	−0.359371
$906$	−10.4731	−0.347946
$907$	−11.0744	−0.367720	−0.183860	−	0.982952i	$-0.558859\pi$
$907$	−11.0744	−0.367720	−0.183860	+	0.982952i	$0.558859\pi$
$908$	−17.9588	−0.595982
$909$	−4.56428	−0.151388
$910$	−0.176989	−0.00586712
$911$	20.8399	0.690456	0.345228	−	0.938519i	$-0.387802\pi$
$911$	20.8399	0.690456	0.345228	+	0.938519i	$0.387802\pi$
$912$	−4.27693	−0.141623
$913$	−4.79348	−0.158641
$914$	4.18947	0.138575
$915$	−9.69725	−0.320581
$916$	37.5867	1.24190
$917$	−1.40116	−0.0462702
$918$	−16.0885	−0.531001
$919$	−43.4535	−1.43340	−0.716700	−	0.697382i	$-0.754348\pi$
$919$	−43.4535	−1.43340	−0.716700	+	0.697382i	$0.754348\pi$
$920$	12.1413	0.400285
$921$	−11.2918	−0.372079
$922$	−14.5547	−0.479333
$923$	14.6176	0.481145
$924$	1.10386	0.0363144
$925$	0.579319	0.0190479
$926$	−7.68985	−0.252704
$927$	3.55913	0.116897
$928$	−26.5979	−0.873119
$929$	−13.8883	−0.455660	−0.227830	−	0.973701i	$-0.573163\pi$
$929$	−13.8883	−0.455660	−0.227830	+	0.973701i	$0.573163\pi$
$930$	−2.93897	−0.0963727
$931$	5.83004	0.191072
$932$	−36.5735	−1.19801
$933$	−32.0840	−1.05038
$934$	12.0736	0.395060
$935$	37.0892	1.21295
$936$	3.06733	0.100259
$937$	−0.990239	−0.0323497	−0.0161748	−	0.999869i	$-0.505149\pi$
$937$	−0.990239	−0.0323497	−0.0161748	+	0.999869i	$0.505149\pi$
$938$	−0.184778	−0.00603320
$939$	−6.31236	−0.205996
$940$	9.94612	0.324407
$941$	21.6651	0.706263	0.353131	−	0.935574i	$-0.385117\pi$
$941$	21.6651	0.706263	0.353131	+	0.935574i	$0.385117\pi$
$942$	−8.49887	−0.276908
$943$	33.7244	1.09822
$944$	23.8339	0.775727
$945$	0.415087	0.0135028
$946$	5.85254	0.190283
$947$	7.53272	0.244780	0.122390	−	0.992482i	$-0.460944\pi$
$947$	7.53272	0.244780	0.122390	+	0.992482i	$0.460944\pi$
$948$	−6.81220	−0.221250
$949$	7.24796	0.235279
$950$	−0.318086	−0.0103201
$951$	28.8696	0.936162
$952$	0.866945	0.0280979
$953$	6.12024	0.198254	0.0991269	−	0.995075i	$-0.468395\pi$
$953$	6.12024	0.198254	0.0991269	+	0.995075i	$0.468395\pi$
$954$	0.0607852	0.00196799
$955$	−13.5911	−0.439797
$956$	−43.1189	−1.39457
$957$	−50.1698	−1.62176
$958$	9.09516	0.293851
$959$	1.32677	0.0428435
$960$	−7.68478	−0.248025
$961$	−8.67974	−0.279992
$962$	1.34596	0.0433956
$963$	5.37118	0.173084
$964$	19.3251	0.622421
$965$	−9.19181	−0.295895
$966$	0.391178	0.0125859
$967$	−37.9234	−1.21953	−0.609766	−	0.792581i	$-0.708737\pi$
$967$	−37.9234	−1.21953	−0.609766	+	0.792581i	$0.708737\pi$
$968$	−17.6169	−0.566229
$969$	−10.5139	−0.337756
$970$	−4.64901	−0.149271
$971$	40.4241	1.29727	0.648636	−	0.761099i	$-0.275340\pi$
$971$	40.4241	1.29727	0.648636	+	0.761099i	$0.275340\pi$
$972$	−6.56728	−0.210646
$973$	1.01225	0.0324513
$974$	−4.38388	−0.140469
$975$	9.92521	0.317861
$976$	−18.7231	−0.599311
$977$	1.51918	0.0486030	0.0243015	−	0.999705i	$-0.492264\pi$
$977$	1.51918	0.0486030	0.0243015	+	0.999705i	$0.492264\pi$
$978$	0.801748	0.0256371
$979$	19.6950	0.629456
$980$	12.9699	0.414308
$981$	−5.82964	−0.186126
$982$	−10.3447	−0.330112
$983$	−43.5665	−1.38956	−0.694778	−	0.719224i	$-0.744497\pi$
$983$	−43.5665	−1.38956	−0.694778	+	0.719224i	$0.744497\pi$
$984$	9.79594	0.312283
$985$	24.8615	0.792154
$986$	−18.9568	−0.603706
$987$	0.666070	0.0212012
$988$	9.41104	0.299405
$989$	−26.4108	−0.839813
$990$	−0.626545	−0.0199129
$991$	60.2012	1.91235	0.956177	−	0.292789i	$-0.0945832\pi$
$991$	60.2012	1.91235	0.956177	+	0.292789i	$0.0945832\pi$
$992$	−19.5722	−0.621419
$993$	17.4641	0.554205
$994$	−0.0697932	−0.00221370
$995$	−11.0180	−0.349295
$996$	3.02297	0.0957864
$997$	22.9234	0.725990	0.362995	−	0.931791i	$-0.381754\pi$
$997$	22.9234	0.725990	0.362995	+	0.931791i	$0.381754\pi$
$998$	0.177789	0.00562782
$999$	−3.15665	−0.0998721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	415.2.a.e.1.7	✓	11
3.2	odd	2		3735.2.a.s.1.5		11
4.3	odd	2		6640.2.a.bi.1.3		11
5.4	even	2		2075.2.a.k.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.2.a.e.1.7	✓	11	1.1	even	1	trivial
2075.2.a.k.1.5		11	5.4	even	2
3735.2.a.s.1.5		11	3.2	odd	2
6640.2.a.bi.1.3		11	4.3	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:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	415.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381602 q^{2} +1.63018 q^{3} -1.85438 q^{4} +1.00000 q^{5} +0.622079 q^{6} -0.0761781 q^{7} -1.47084 q^{8} -0.342523 q^{9} +0.381602 q^{10} +4.79348 q^{11} -3.02297 q^{12} +6.08842 q^{13} -0.0290697 q^{14} +1.63018 q^{15} +3.14748 q^{16} +7.73742 q^{17} -0.130708 q^{18} -0.833555 q^{19} -1.85438 q^{20} -0.124184 q^{21} +1.82920 q^{22} -8.25464 q^{23} -2.39773 q^{24} +1.00000 q^{25} +2.32336 q^{26} -5.44890 q^{27} +0.141263 q^{28} -6.42032 q^{29} +0.622079 q^{30} -4.72443 q^{31} +4.14277 q^{32} +7.81422 q^{33} +2.95262 q^{34} -0.0761781 q^{35} +0.635168 q^{36} +0.579319 q^{37} -0.318086 q^{38} +9.92521 q^{39} -1.47084 q^{40} -4.08551 q^{41} -0.0473888 q^{42} +3.19950 q^{43} -8.88893 q^{44} -0.342523 q^{45} -3.14999 q^{46} -5.36358 q^{47} +5.13095 q^{48} -6.99420 q^{49} +0.381602 q^{50} +12.6134 q^{51} -11.2903 q^{52} -0.465047 q^{53} -2.07931 q^{54} +4.79348 q^{55} +0.112046 q^{56} -1.35884 q^{57} -2.45001 q^{58} +7.57236 q^{59} -3.02297 q^{60} -5.94859 q^{61} -1.80285 q^{62} +0.0260928 q^{63} -4.71408 q^{64} +6.08842 q^{65} +2.98192 q^{66} +6.35636 q^{67} -14.3481 q^{68} -13.4565 q^{69} -0.0290697 q^{70} +2.40089 q^{71} +0.503797 q^{72} +1.19045 q^{73} +0.221069 q^{74} +1.63018 q^{75} +1.54573 q^{76} -0.365158 q^{77} +3.78748 q^{78} +2.25348 q^{79} +3.14748 q^{80} -7.85511 q^{81} -1.55904 q^{82} -1.00000 q^{83} +0.230284 q^{84} +7.73742 q^{85} +1.22094 q^{86} -10.4663 q^{87} -7.05044 q^{88} +4.10871 q^{89} -0.130708 q^{90} -0.463805 q^{91} +15.3072 q^{92} -7.70166 q^{93} -2.04676 q^{94} -0.833555 q^{95} +6.75344 q^{96} -12.1829 q^{97} -2.66900 q^{98} -1.64188 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 18 q^{4} + 11 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 21 q^{9} - q^{11} + 2 q^{12} + q^{13} + 12 q^{14} + 20 q^{16} + 28 q^{17} - 26 q^{18} - 2 q^{19} + 18 q^{20} + 3 q^{21} - 20 q^{22} + q^{23}+ \cdots - 36 q^{99}+O(q^{100}) \) 11 * q + 18 * q^4 + 11 * q^5 - 5 * q^6 - 3 * q^7 + 3 * q^8 + 21 * q^9 - q^11 + 2 * q^12 + q^13 + 12 * q^14 + 20 * q^16 + 28 * q^17 - 26 * q^18 - 2 * q^19 + 18 * q^20 + 3 * q^21 - 20 * q^22 + q^23 - 10 * q^24 + 11 * q^25 + 21 * q^26 - 12 * q^27 - 22 * q^28 + 21 * q^29 - 5 * q^30 + 4 * q^31 + q^32 + 10 * q^33 - 4 * q^34 - 3 * q^35 + 45 * q^36 - 10 * q^37 + 17 * q^38 - 2 * q^39 + 3 * q^40 + 39 * q^41 - 29 * q^42 - 13 * q^43 - 25 * q^44 + 21 * q^45 - 13 * q^46 + 8 * q^47 + 17 * q^48 + 32 * q^49 + 32 * q^51 - 65 * q^52 + 21 * q^53 - 72 * q^54 - q^55 + 57 * q^56 - 19 * q^57 - 6 * q^58 + 5 * q^59 + 2 * q^60 + 14 * q^61 - q^62 - 30 * q^63 - 21 * q^64 + q^65 - 11 * q^66 - 42 * q^67 + 61 * q^68 - 7 * q^69 + 12 * q^70 + 3 * q^71 - 69 * q^72 + 3 * q^73 - 54 * q^74 + 13 * q^76 - 42 * q^77 - 7 * q^78 - 20 * q^79 + 20 * q^80 + 39 * q^81 - 43 * q^82 - 11 * q^83 - 116 * q^84 + 28 * q^85 + 22 * q^86 - 16 * q^87 - 75 * q^88 + 37 * q^89 - 26 * q^90 - 17 * q^91 + 7 * q^92 - 55 * q^93 - 54 * q^94 - 2 * q^95 - 58 * q^96 - 2 * q^97 - 26 * q^98 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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