LMFDB - Embedded newform 5225.2.a.u.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-1,4,13,0,-1,17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 $ x^15 - x^14 - 21x^13 + 21x^12 + 168x^11 - 165x^10 - 645x^9 + 606x^8 + 1239x^7 - 1055x^6 - 1160x^5 + 793x^4 + 485x^3 - 171x^2 - 93*x - 5
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			1.11
Root			$-1.26824$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.26824 q^{2} +1.37964 q^{3} -0.391576 q^{4} +1.74971 q^{6} +2.74560 q^{7} -3.03308 q^{8} -1.09659 q^{9} -1.00000 q^{11} -0.540234 q^{12} +2.90239 q^{13} +3.48208 q^{14} -3.06352 q^{16} +5.83714 q^{17} -1.39074 q^{18} +1.00000 q^{19} +3.78795 q^{21} -1.26824 q^{22} -1.00708 q^{23} -4.18457 q^{24} +3.68092 q^{26} -5.65182 q^{27} -1.07511 q^{28} +8.92770 q^{29} +5.91308 q^{31} +2.18091 q^{32} -1.37964 q^{33} +7.40287 q^{34} +0.429399 q^{36} +8.15096 q^{37} +1.26824 q^{38} +4.00426 q^{39} -7.98447 q^{41} +4.80401 q^{42} -6.88762 q^{43} +0.391576 q^{44} -1.27722 q^{46} -10.9565 q^{47} -4.22655 q^{48} +0.538344 q^{49} +8.05315 q^{51} -1.13651 q^{52} -5.29137 q^{53} -7.16785 q^{54} -8.32765 q^{56} +1.37964 q^{57} +11.3224 q^{58} -5.09173 q^{59} +7.63872 q^{61} +7.49919 q^{62} -3.01081 q^{63} +8.89294 q^{64} -1.74971 q^{66} +15.6685 q^{67} -2.28568 q^{68} -1.38941 q^{69} -0.653706 q^{71} +3.32605 q^{72} -4.93939 q^{73} +10.3373 q^{74} -0.391576 q^{76} -2.74560 q^{77} +5.07835 q^{78} +12.8034 q^{79} -4.50771 q^{81} -10.1262 q^{82} +12.7259 q^{83} -1.48327 q^{84} -8.73513 q^{86} +12.3170 q^{87} +3.03308 q^{88} +11.9827 q^{89} +7.96882 q^{91} +0.394349 q^{92} +8.15793 q^{93} -13.8954 q^{94} +3.00887 q^{96} -0.982691 q^{97} +0.682748 q^{98} +1.09659 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23}+ \cdots - 15 q^{99}+O(q^{100}) $ 15 * q - q^2 + 4 * q^3 + 13 * q^4 - q^6 + 17 * q^7 + 3 * q^8 + 15 * q^9 - 15 * q^11 + 9 * q^12 + 3 * q^13 - 11 * q^14 + 13 * q^16 + q^17 - 10 * q^18 + 15 * q^19 + 16 * q^21 + q^22 + 18 * q^23 - 27 * q^24 + 15 * q^26 + 31 * q^27 + 34 * q^28 + 11 * q^29 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 13 * q^34 + 8 * q^36 + 27 * q^37 - q^38 + 4 * q^39 + 6 * q^41 + 2 * q^42 + 38 * q^43 - 13 * q^44 - 9 * q^46 + 14 * q^47 + 32 * q^48 + 28 * q^49 - 32 * q^51 + 16 * q^52 + 11 * q^53 - 11 * q^54 + 2 * q^56 + 4 * q^57 - 6 * q^58 + 13 * q^59 + 8 * q^61 + 7 * q^62 + 49 * q^63 + 9 * q^64 + q^66 + 31 * q^67 - 26 * q^68 + 39 * q^69 - 3 * q^71 - 18 * q^72 + 18 * q^73 + 7 * q^74 + 13 * q^76 - 17 * q^77 + 18 * q^78 + 10 * q^79 + 31 * q^81 + 58 * q^82 + 16 * q^83 + 112 * q^84 - 63 * q^86 + 67 * q^87 - 3 * q^88 - 7 * q^89 - 50 * q^91 + 98 * q^92 + 26 * q^93 + 22 * q^94 - 37 * q^96 + 24 * q^97 - 46 * q^98 - 15 * 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 $

$2$	1.26824	0.896779	0.448389	−	0.893838i	$-0.351998\pi$
$2$	1.26824	0.896779	0.448389	+	0.893838i	$0.351998\pi$
$3$	1.37964	0.796536	0.398268	−	0.917269i	$-0.369611\pi$
$3$	1.37964	0.796536	0.398268	+	0.917269i	$0.369611\pi$
$4$	−0.391576	−0.195788
$5$	0	0
$5$	0	0
$6$	1.74971	0.714316
$7$	2.74560	1.03774	0.518870	−	0.854853i	$-0.326353\pi$
$7$	2.74560	1.03774	0.518870	+	0.854853i	$0.326353\pi$
$8$	−3.03308	−1.07236
$9$	−1.09659	−0.365531
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.540234	−0.155952
$13$	2.90239	0.804979	0.402489	−	0.915425i	$-0.368145\pi$
$13$	2.90239	0.804979	0.402489	+	0.915425i	$0.368145\pi$
$14$	3.48208	0.930624
$15$	0	0
$16$	−3.06352	−0.765879
$17$	5.83714	1.41571	0.707857	−	0.706356i	$-0.249662\pi$
$17$	5.83714	1.41571	0.707857	+	0.706356i	$0.249662\pi$
$18$	−1.39074	−0.327800
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.78795	0.826598
$22$	−1.26824	−0.270389
$23$	−1.00708	−0.209991	−0.104996	−	0.994473i	$-0.533483\pi$
$23$	−1.00708	−0.209991	−0.104996	+	0.994473i	$0.533483\pi$
$24$	−4.18457	−0.854171
$25$	0	0
$26$	3.68092	0.721888
$27$	−5.65182	−1.08769
$28$	−1.07511	−0.203177
$29$	8.92770	1.65783	0.828916	−	0.559373i	$-0.188958\pi$
$29$	8.92770	1.65783	0.828916	+	0.559373i	$0.188958\pi$
$30$	0	0
$31$	5.91308	1.06202	0.531011	−	0.847365i	$-0.321812\pi$
$31$	5.91308	1.06202	0.531011	+	0.847365i	$0.321812\pi$
$32$	2.18091	0.385533
$33$	−1.37964	−0.240165
$34$	7.40287	1.26958
$35$	0	0
$36$	0.429399	0.0715665
$37$	8.15096	1.34001	0.670004	−	0.742357i	$-0.266292\pi$
$37$	8.15096	1.34001	0.670004	+	0.742357i	$0.266292\pi$
$38$	1.26824	0.205735
$39$	4.00426	0.641195
$40$	0	0
$41$	−7.98447	−1.24696	−0.623482	−	0.781837i	$-0.714283\pi$
$41$	−7.98447	−1.24696	−0.623482	+	0.781837i	$0.714283\pi$
$42$	4.80401	0.741275
$43$	−6.88762	−1.05035	−0.525176	−	0.850994i	$-0.676001\pi$
$43$	−6.88762	−1.05035	−0.525176	+	0.850994i	$0.676001\pi$
$44$	0.391576	0.0590323
$45$	0	0
$46$	−1.27722	−0.188315
$47$	−10.9565	−1.59816	−0.799082	−	0.601222i	$-0.794681\pi$
$47$	−10.9565	−1.59816	−0.799082	+	0.601222i	$0.794681\pi$
$48$	−4.22655	−0.610050
$49$	0.538344	0.0769063
$50$	0	0
$51$	8.05315	1.12767
$52$	−1.13651	−0.157605
$53$	−5.29137	−0.726825	−0.363413	−	0.931628i	$-0.618388\pi$
$53$	−5.29137	−0.726825	−0.363413	+	0.931628i	$0.618388\pi$
$54$	−7.16785	−0.975421
$55$	0	0
$56$	−8.32765	−1.11283
$57$	1.37964	0.182738
$58$	11.3224	1.48671
$59$	−5.09173	−0.662886	−0.331443	−	0.943475i	$-0.607536\pi$
$59$	−5.09173	−0.662886	−0.331443	+	0.943475i	$0.607536\pi$
$60$	0	0
$61$	7.63872	0.978038	0.489019	−	0.872273i	$-0.337355\pi$
$61$	7.63872	0.978038	0.489019	+	0.872273i	$0.337355\pi$
$62$	7.49919	0.952398
$63$	−3.01081	−0.379326
$64$	8.89294	1.11162
$65$	0	0
$66$	−1.74971	−0.215375
$67$	15.6685	1.91421	0.957104	−	0.289743i	$-0.0935699\pi$
$67$	15.6685	1.91421	0.957104	+	0.289743i	$0.0935699\pi$
$68$	−2.28568	−0.277180
$69$	−1.38941	−0.167265
$70$	0	0
$71$	−0.653706	−0.0775806	−0.0387903	−	0.999247i	$-0.512350\pi$
$71$	−0.653706	−0.0775806	−0.0387903	+	0.999247i	$0.512350\pi$
$72$	3.32605	0.391979
$73$	−4.93939	−0.578112	−0.289056	−	0.957312i	$-0.593341\pi$
$73$	−4.93939	−0.578112	−0.289056	+	0.957312i	$0.593341\pi$
$74$	10.3373	1.20169
$75$	0	0
$76$	−0.391576	−0.0449168
$77$	−2.74560	−0.312891
$78$	5.07835	0.575010
$79$	12.8034	1.44049	0.720245	−	0.693720i	$-0.244029\pi$
$79$	12.8034	1.44049	0.720245	+	0.693720i	$0.244029\pi$
$80$	0	0
$81$	−4.50771	−0.500857
$82$	−10.1262	−1.11825
$83$	12.7259	1.39685	0.698424	−	0.715685i	$-0.253885\pi$
$83$	12.7259	1.39685	0.698424	+	0.715685i	$0.253885\pi$
$84$	−1.48327	−0.161838
$85$	0	0
$86$	−8.73513	−0.941933
$87$	12.3170	1.32052
$88$	3.03308	0.323328
$89$	11.9827	1.27016	0.635081	−	0.772445i	$-0.280967\pi$
$89$	11.9827	1.27016	0.635081	+	0.772445i	$0.280967\pi$
$90$	0	0
$91$	7.96882	0.835359
$92$	0.394349	0.0411137
$93$	8.15793	0.845938
$94$	−13.8954	−1.43320
$95$	0	0
$96$	3.00887	0.307091
$97$	−0.982691	−0.0997771	−0.0498886	−	0.998755i	$-0.515887\pi$
$97$	−0.982691	−0.0997771	−0.0498886	+	0.998755i	$0.515887\pi$
$98$	0.682748	0.0689680
$99$	1.09659	0.110212
$100$	0	0
$101$	0.585289	0.0582384	0.0291192	−	0.999576i	$-0.490730\pi$
$101$	0.585289	0.0582384	0.0291192	+	0.999576i	$0.490730\pi$
$102$	10.2133	1.01127
$103$	0.163288	0.0160893	0.00804464	−	0.999968i	$-0.497439\pi$
$103$	0.163288	0.0160893	0.00804464	+	0.999968i	$0.497439\pi$
$104$	−8.80320	−0.863225
$105$	0	0
$106$	−6.71071	−0.651801
$107$	8.57415	0.828894	0.414447	−	0.910073i	$-0.363975\pi$
$107$	8.57415	0.828894	0.414447	+	0.910073i	$0.363975\pi$
$108$	2.21312	0.212957
$109$	16.1503	1.54692	0.773458	−	0.633848i	$-0.218525\pi$
$109$	16.1503	1.54692	0.773458	+	0.633848i	$0.218525\pi$
$110$	0	0
$111$	11.2454	1.06737
$112$	−8.41120	−0.794784
$113$	9.09077	0.855188	0.427594	−	0.903971i	$-0.359361\pi$
$113$	9.09077	0.855188	0.427594	+	0.903971i	$0.359361\pi$
$114$	1.74971	0.163875
$115$	0	0
$116$	−3.49587	−0.324584
$117$	−3.18274	−0.294244
$118$	−6.45751	−0.594462
$119$	16.0265	1.46914
$120$	0	0
$121$	1.00000	0.0909091
$122$	9.68770	0.877083
$123$	−11.0157	−0.993252
$124$	−2.31542	−0.207931
$125$	0	0
$126$	−3.81842	−0.340171
$127$	13.6931	1.21506	0.607531	−	0.794296i	$-0.292160\pi$
$127$	13.6931	1.21506	0.607531	+	0.794296i	$0.292160\pi$
$128$	6.91654	0.611341
$129$	−9.50244	−0.836643
$130$	0	0
$131$	14.7215	1.28623	0.643114	−	0.765771i	$-0.277642\pi$
$131$	14.7215	1.28623	0.643114	+	0.765771i	$0.277642\pi$
$132$	0.540234	0.0470214
$133$	2.74560	0.238074
$134$	19.8713	1.71662
$135$	0	0
$136$	−17.7045	−1.51815
$137$	−7.29444	−0.623206	−0.311603	−	0.950212i	$-0.600866\pi$
$137$	−7.29444	−0.623206	−0.311603	+	0.950212i	$0.600866\pi$
$138$	−1.76210	−0.150000
$139$	16.5868	1.40687	0.703435	−	0.710760i	$-0.251649\pi$
$139$	16.5868	1.40687	0.703435	+	0.710760i	$0.251649\pi$
$140$	0	0
$141$	−15.1160	−1.27299
$142$	−0.829054	−0.0695727
$143$	−2.90239	−0.242710
$144$	3.35943	0.279952
$145$	0	0
$146$	−6.26432	−0.518439
$147$	0.742722	0.0612587
$148$	−3.19172	−0.262358
$149$	−6.10805	−0.500391	−0.250195	−	0.968195i	$-0.580495\pi$
$149$	−6.10805	−0.500391	−0.250195	+	0.968195i	$0.580495\pi$
$150$	0	0
$151$	7.21136	0.586852	0.293426	−	0.955982i	$-0.405205\pi$
$151$	7.21136	0.586852	0.293426	+	0.955982i	$0.405205\pi$
$152$	−3.03308	−0.246016
$153$	−6.40095	−0.517487
$154$	−3.48208	−0.280594
$155$	0	0
$156$	−1.56797	−0.125538
$157$	−8.21699	−0.655787	−0.327894	−	0.944715i	$-0.606339\pi$
$157$	−8.21699	−0.655787	−0.327894	+	0.944715i	$0.606339\pi$
$158$	16.2377	1.29180
$159$	−7.30019	−0.578942
$160$	0	0
$161$	−2.76505	−0.217916
$162$	−5.71685	−0.449158
$163$	−15.5417	−1.21732	−0.608660	−	0.793431i	$-0.708292\pi$
$163$	−15.5417	−1.21732	−0.608660	+	0.793431i	$0.708292\pi$
$164$	3.12653	0.244141
$165$	0	0
$166$	16.1394	1.25266
$167$	−9.04791	−0.700148	−0.350074	−	0.936722i	$-0.613844\pi$
$167$	−9.04791	−0.700148	−0.350074	+	0.936722i	$0.613844\pi$
$168$	−11.4892	−0.886408
$169$	−4.57612	−0.352009
$170$	0	0
$171$	−1.09659	−0.0838585
$172$	2.69703	0.205646
$173$	−4.77616	−0.363124	−0.181562	−	0.983379i	$-0.558115\pi$
$173$	−4.77616	−0.363124	−0.181562	+	0.983379i	$0.558115\pi$
$174$	15.6209	1.18422
$175$	0	0
$176$	3.06352	0.230921
$177$	−7.02475	−0.528013
$178$	15.1969	1.13905
$179$	2.94093	0.219816	0.109908	−	0.993942i	$-0.464944\pi$
$179$	2.94093	0.219816	0.109908	+	0.993942i	$0.464944\pi$
$180$	0	0
$181$	1.63416	0.121466	0.0607330	−	0.998154i	$-0.480656\pi$
$181$	1.63416	0.121466	0.0607330	+	0.998154i	$0.480656\pi$
$182$	10.1064	0.749133
$183$	10.5387	0.779042
$184$	3.05456	0.225185
$185$	0	0
$186$	10.3462	0.758619
$187$	−5.83714	−0.426854
$188$	4.29029	0.312901
$189$	−15.5177	−1.12874
$190$	0	0
$191$	−6.07300	−0.439427	−0.219714	−	0.975564i	$-0.570512\pi$
$191$	−6.07300	−0.439427	−0.219714	+	0.975564i	$0.570512\pi$
$192$	12.2691	0.885443
$193$	0.167517	0.0120582	0.00602908	−	0.999982i	$-0.498081\pi$
$193$	0.167517	0.0120582	0.00602908	+	0.999982i	$0.498081\pi$
$194$	−1.24628	−0.0894780
$195$	0	0
$196$	−0.210803	−0.0150573
$197$	−17.0070	−1.21170	−0.605850	−	0.795579i	$-0.707167\pi$
$197$	−17.0070	−1.21170	−0.605850	+	0.795579i	$0.707167\pi$
$198$	1.39074	0.0988354
$199$	−19.1932	−1.36057	−0.680286	−	0.732947i	$-0.738144\pi$
$199$	−19.1932	−1.36057	−0.680286	+	0.732947i	$0.738144\pi$
$200$	0	0
$201$	21.6169	1.52474
$202$	0.742285	0.0522270
$203$	24.5119	1.72040
$204$	−3.15342	−0.220784
$205$	0	0
$206$	0.207088	0.0144285
$207$	1.10436	0.0767581
$208$	−8.89153	−0.616516
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	16.0927	1.10786	0.553932	−	0.832562i	$-0.313127\pi$
$211$	16.0927	1.10786	0.553932	+	0.832562i	$0.313127\pi$
$212$	2.07197	0.142304
$213$	−0.901880	−0.0617958
$214$	10.8740	0.743334
$215$	0	0
$216$	17.1425	1.16640
$217$	16.2350	1.10210
$218$	20.4824	1.38724
$219$	−6.81459	−0.460487
$220$	0	0
$221$	16.9417	1.13962
$222$	14.2618	0.957190
$223$	−24.8789	−1.66601	−0.833006	−	0.553263i	$-0.813382\pi$
$223$	−24.8789	−1.66601	−0.833006	+	0.553263i	$0.813382\pi$
$224$	5.98790	0.400084
$225$	0	0
$226$	11.5292	0.766914
$227$	27.5594	1.82918	0.914591	−	0.404381i	$-0.132513\pi$
$227$	27.5594	1.82918	0.914591	+	0.404381i	$0.132513\pi$
$228$	−0.540234	−0.0357779
$229$	−17.5798	−1.16171	−0.580854	−	0.814008i	$-0.697281\pi$
$229$	−17.5798	−1.16171	−0.580854	+	0.814008i	$0.697281\pi$
$230$	0	0
$231$	−3.78795	−0.249229
$232$	−27.0785	−1.77779
$233$	4.90663	0.321444	0.160722	−	0.987000i	$-0.448618\pi$
$233$	4.90663	0.321444	0.160722	+	0.987000i	$0.448618\pi$
$234$	−4.03647	−0.263872
$235$	0	0
$236$	1.99380	0.129785
$237$	17.6640	1.14740
$238$	20.3254	1.31750
$239$	4.67250	0.302239	0.151119	−	0.988516i	$-0.451712\pi$
$239$	4.67250	0.302239	0.151119	+	0.988516i	$0.451712\pi$
$240$	0	0
$241$	−24.3107	−1.56599	−0.782994	−	0.622029i	$-0.786308\pi$
$241$	−24.3107	−1.56599	−0.782994	+	0.622029i	$0.786308\pi$
$242$	1.26824	0.0815253
$243$	10.7365	0.688744
$244$	−2.99114	−0.191488
$245$	0	0
$246$	−13.9705	−0.890727
$247$	2.90239	0.184675
$248$	−17.9349	−1.13887
$249$	17.5571	1.11264
$250$	0	0
$251$	−17.6394	−1.11339	−0.556694	−	0.830718i	$-0.687930\pi$
$251$	−17.6394	−1.11339	−0.556694	+	0.830718i	$0.687930\pi$
$252$	1.17896	0.0742675
$253$	1.00708	0.0633147
$254$	17.3660	1.08964
$255$	0	0
$256$	−9.01407	−0.563379
$257$	7.03977	0.439129	0.219564	−	0.975598i	$-0.429536\pi$
$257$	7.03977	0.439129	0.219564	+	0.975598i	$0.429536\pi$
$258$	−12.0513	−0.750284
$259$	22.3793	1.39058
$260$	0	0
$261$	−9.79004	−0.605988
$262$	18.6704	1.15346
$263$	−9.19090	−0.566735	−0.283368	−	0.959011i	$-0.591452\pi$
$263$	−9.19090	−0.566735	−0.283368	+	0.959011i	$0.591452\pi$
$264$	4.18457	0.257542
$265$	0	0
$266$	3.48208	0.213500
$267$	16.5318	1.01173
$268$	−6.13540	−0.374779
$269$	−16.7645	−1.02215	−0.511075	−	0.859536i	$-0.670753\pi$
$269$	−16.7645	−1.02215	−0.511075	+	0.859536i	$0.670753\pi$
$270$	0	0
$271$	−0.336276	−0.0204273	−0.0102137	−	0.999948i	$-0.503251\pi$
$271$	−0.336276	−0.0204273	−0.0102137	+	0.999948i	$0.503251\pi$
$272$	−17.8822	−1.08427
$273$	10.9941	0.665394
$274$	−9.25108	−0.558878
$275$	0	0
$276$	0.544060	0.0327486
$277$	0.773144	0.0464537	0.0232269	−	0.999730i	$-0.492606\pi$
$277$	0.773144	0.0464537	0.0232269	+	0.999730i	$0.492606\pi$
$278$	21.0359	1.26165
$279$	−6.48424	−0.388201
$280$	0	0
$281$	−13.2396	−0.789806	−0.394903	−	0.918723i	$-0.629222\pi$
$281$	−13.2396	−0.789806	−0.394903	+	0.918723i	$0.629222\pi$
$282$	−19.1706	−1.14159
$283$	3.64509	0.216678	0.108339	−	0.994114i	$-0.465447\pi$
$283$	3.64509	0.216678	0.108339	+	0.994114i	$0.465447\pi$
$284$	0.255976	0.0151894
$285$	0	0
$286$	−3.68092	−0.217657
$287$	−21.9222	−1.29403
$288$	−2.39156	−0.140924
$289$	17.0722	1.00424
$290$	0	0
$291$	−1.35576	−0.0794761
$292$	1.93415	0.113187
$293$	−9.46530	−0.552969	−0.276484	−	0.961018i	$-0.589169\pi$
$293$	−9.46530	−0.552969	−0.276484	+	0.961018i	$0.589169\pi$
$294$	0.941947	0.0549355
$295$	0	0
$296$	−24.7225	−1.43697
$297$	5.65182	0.327952
$298$	−7.74645	−0.448740
$299$	−2.92295	−0.169038
$300$	0	0
$301$	−18.9107	−1.08999
$302$	9.14571	0.526276
$303$	0.807489	0.0463890
$304$	−3.06352	−0.175705
$305$	0	0
$306$	−8.11793	−0.464071
$307$	34.1985	1.95181	0.975906	−	0.218190i	$-0.0700154\pi$
$307$	34.1985	1.95181	0.975906	+	0.218190i	$0.0700154\pi$
$308$	1.07511	0.0612602
$309$	0.225279	0.0128157
$310$	0	0
$311$	−31.2004	−1.76921	−0.884607	−	0.466337i	$-0.845573\pi$
$311$	−31.2004	−1.76921	−0.884607	+	0.466337i	$0.845573\pi$
$312$	−12.1453	−0.687590
$313$	8.52890	0.482082	0.241041	−	0.970515i	$-0.422511\pi$
$313$	8.52890	0.482082	0.241041	+	0.970515i	$0.422511\pi$
$314$	−10.4211	−0.588096
$315$	0	0
$316$	−5.01349	−0.282031
$317$	−13.9541	−0.783741	−0.391870	−	0.920020i	$-0.628172\pi$
$317$	−13.9541	−0.783741	−0.391870	+	0.920020i	$0.628172\pi$
$318$	−9.25836	−0.519183
$319$	−8.92770	−0.499855
$320$	0	0
$321$	11.8292	0.660244
$322$	−3.50673	−0.195423
$323$	5.83714	0.324787
$324$	1.76511	0.0980618
$325$	0	0
$326$	−19.7105	−1.09167
$327$	22.2816	1.23217
$328$	24.2176	1.33719
$329$	−30.0821	−1.65848
$330$	0	0
$331$	−20.4997	−1.12676	−0.563382	−	0.826196i	$-0.690500\pi$
$331$	−20.4997	−1.12676	−0.563382	+	0.826196i	$0.690500\pi$
$332$	−4.98315	−0.273486
$333$	−8.93827	−0.489814
$334$	−11.4749	−0.627878
$335$	0	0
$336$	−11.6044	−0.633074
$337$	31.9046	1.73795	0.868977	−	0.494853i	$-0.164778\pi$
$337$	31.9046	1.73795	0.868977	+	0.494853i	$0.164778\pi$
$338$	−5.80360	−0.315674
$339$	12.5420	0.681188
$340$	0	0
$341$	−5.91308	−0.320211
$342$	−1.39074	−0.0752025
$343$	−17.7412	−0.957932
$344$	20.8907	1.12635
$345$	0	0
$346$	−6.05730	−0.325642
$347$	14.1093	0.757426	0.378713	−	0.925514i	$-0.376367\pi$
$347$	14.1093	0.757426	0.378713	+	0.925514i	$0.376367\pi$
$348$	−4.82305	−0.258542
$349$	−28.3982	−1.52012	−0.760060	−	0.649853i	$-0.774831\pi$
$349$	−28.3982	−1.52012	−0.760060	+	0.649853i	$0.774831\pi$
$350$	0	0
$351$	−16.4038	−0.875571
$352$	−2.18091	−0.116243
$353$	−23.4620	−1.24875	−0.624377	−	0.781123i	$-0.714647\pi$
$353$	−23.4620	−1.24875	−0.624377	+	0.781123i	$0.714647\pi$
$354$	−8.90905	−0.473511
$355$	0	0
$356$	−4.69213	−0.248683
$357$	22.1108	1.17023
$358$	3.72980	0.197126
$359$	11.4832	0.606061	0.303031	−	0.952981i	$-0.402002\pi$
$359$	11.4832	0.606061	0.303031	+	0.952981i	$0.402002\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.07250	0.108928
$363$	1.37964	0.0724124
$364$	−3.12040	−0.163553
$365$	0	0
$366$	13.3655	0.698628
$367$	−25.0513	−1.30767	−0.653833	−	0.756639i	$-0.726840\pi$
$367$	−25.0513	−1.30767	−0.653833	+	0.756639i	$0.726840\pi$
$368$	3.08521	0.160828
$369$	8.75570	0.455804
$370$	0	0
$371$	−14.5280	−0.754256
$372$	−3.19445	−0.165625
$373$	−23.5796	−1.22090	−0.610451	−	0.792054i	$-0.709012\pi$
$373$	−23.5796	−1.22090	−0.610451	+	0.792054i	$0.709012\pi$
$374$	−7.40287	−0.382793
$375$	0	0
$376$	33.2319	1.71380
$377$	25.9117	1.33452
$378$	−19.6801	−1.01223
$379$	−26.7205	−1.37254	−0.686269	−	0.727348i	$-0.740753\pi$
$379$	−26.7205	−1.37254	−0.686269	+	0.727348i	$0.740753\pi$
$380$	0	0
$381$	18.8915	0.967841
$382$	−7.70200	−0.394069
$383$	−24.2328	−1.23824	−0.619119	−	0.785297i	$-0.712510\pi$
$383$	−24.2328	−1.23824	−0.619119	+	0.785297i	$0.712510\pi$
$384$	9.54234	0.486955
$385$	0	0
$386$	0.212452	0.0108135
$387$	7.55290	0.383936
$388$	0.384798	0.0195352
$389$	5.28290	0.267854	0.133927	−	0.990991i	$-0.457241\pi$
$389$	5.28290	0.267854	0.133927	+	0.990991i	$0.457241\pi$
$390$	0	0
$391$	−5.87847	−0.297287
$392$	−1.63284	−0.0824711
$393$	20.3104	1.02453
$394$	−21.5689	−1.08663
$395$	0	0
$396$	−0.429399	−0.0215781
$397$	2.09703	0.105247	0.0526235	−	0.998614i	$-0.483242\pi$
$397$	2.09703	0.105247	0.0526235	+	0.998614i	$0.483242\pi$
$398$	−24.3415	−1.22013
$399$	3.78795	0.189635
$400$	0	0
$401$	20.2889	1.01318	0.506590	−	0.862187i	$-0.330906\pi$
$401$	20.2889	1.01318	0.506590	+	0.862187i	$0.330906\pi$
$402$	27.4153	1.36735
$403$	17.1621	0.854905
$404$	−0.229185	−0.0114024
$405$	0	0
$406$	31.0869	1.54282
$407$	−8.15096	−0.404028
$408$	−24.4259	−1.20926
$409$	33.3628	1.64969	0.824843	−	0.565362i	$-0.191263\pi$
$409$	33.3628	1.64969	0.824843	+	0.565362i	$0.191263\pi$
$410$	0	0
$411$	−10.0637	−0.496406
$412$	−0.0639398	−0.00315009
$413$	−13.9799	−0.687904
$414$	1.40059	0.0688350
$415$	0	0
$416$	6.32984	0.310346
$417$	22.8838	1.12062
$418$	−1.26824	−0.0620315
$419$	11.4626	0.559983	0.279992	−	0.960002i	$-0.409668\pi$
$419$	11.4626	0.559983	0.279992	+	0.960002i	$0.409668\pi$
$420$	0	0
$421$	21.4615	1.04597	0.522986	−	0.852341i	$-0.324818\pi$
$421$	21.4615	1.04597	0.522986	+	0.852341i	$0.324818\pi$
$422$	20.4093	0.993509
$423$	12.0148	0.584177
$424$	16.0492	0.779416
$425$	0	0
$426$	−1.14380	−0.0554171
$427$	20.9729	1.01495
$428$	−3.35743	−0.162287
$429$	−4.00426	−0.193327
$430$	0	0
$431$	−1.86277	−0.0897266	−0.0448633	−	0.998993i	$-0.514285\pi$
$431$	−1.86277	−0.0897266	−0.0448633	+	0.998993i	$0.514285\pi$
$432$	17.3145	0.833042
$433$	−16.5406	−0.794890	−0.397445	−	0.917626i	$-0.630103\pi$
$433$	−16.5406	−0.794890	−0.397445	+	0.917626i	$0.630103\pi$
$434$	20.5898	0.988342
$435$	0	0
$436$	−6.32406	−0.302868
$437$	−1.00708	−0.0481752
$438$	−8.64251	−0.412955
$439$	13.4407	0.641488	0.320744	−	0.947166i	$-0.396067\pi$
$439$	13.4407	0.641488	0.320744	+	0.947166i	$0.396067\pi$
$440$	0	0
$441$	−0.590344	−0.0281116
$442$	21.4860	1.02199
$443$	15.6710	0.744551	0.372276	−	0.928122i	$-0.378578\pi$
$443$	15.6710	0.744551	0.372276	+	0.928122i	$0.378578\pi$
$444$	−4.40342	−0.208977
$445$	0	0
$446$	−31.5523	−1.49404
$447$	−8.42691	−0.398579
$448$	24.4165	1.15357
$449$	−39.1867	−1.84933	−0.924667	−	0.380778i	$-0.875656\pi$
$449$	−39.1867	−1.84933	−0.924667	+	0.380778i	$0.875656\pi$
$450$	0	0
$451$	7.98447	0.375974
$452$	−3.55973	−0.167435
$453$	9.94908	0.467449
$454$	34.9518	1.64037
$455$	0	0
$456$	−4.18457	−0.195960
$457$	−16.2611	−0.760662	−0.380331	−	0.924850i	$-0.624190\pi$
$457$	−16.2611	−0.760662	−0.380331	+	0.924850i	$0.624190\pi$
$458$	−22.2954	−1.04179
$459$	−32.9905	−1.53986
$460$	0	0
$461$	30.0292	1.39860	0.699299	−	0.714829i	$-0.253496\pi$
$461$	30.0292	1.39860	0.699299	+	0.714829i	$0.253496\pi$
$462$	−4.80401	−0.223503
$463$	34.3384	1.59584	0.797921	−	0.602762i	$-0.205933\pi$
$463$	34.3384	1.59584	0.797921	+	0.602762i	$0.205933\pi$
$464$	−27.3501	−1.26970
$465$	0	0
$466$	6.22276	0.288264
$467$	−32.9295	−1.52380	−0.761898	−	0.647697i	$-0.775732\pi$
$467$	−32.9295	−1.52380	−0.761898	+	0.647697i	$0.775732\pi$
$468$	1.24628	0.0576095
$469$	43.0194	1.98645
$470$	0	0
$471$	−11.3365	−0.522358
$472$	15.4436	0.710851
$473$	6.88762	0.316693
$474$	22.4022	1.02897
$475$	0	0
$476$	−6.27558	−0.287641
$477$	5.80247	0.265677
$478$	5.92583	0.271041
$479$	−33.5237	−1.53174	−0.765869	−	0.642997i	$-0.777691\pi$
$479$	−33.5237	−1.53174	−0.765869	+	0.642997i	$0.777691\pi$
$480$	0	0
$481$	23.6573	1.07868
$482$	−30.8317	−1.40434
$483$	−3.81477	−0.173578
$484$	−0.391576	−0.0177989
$485$	0	0
$486$	13.6164	0.617651
$487$	20.6826	0.937216	0.468608	−	0.883406i	$-0.344756\pi$
$487$	20.6826	0.937216	0.468608	+	0.883406i	$0.344756\pi$
$488$	−23.1689	−1.04881
$489$	−21.4419	−0.969638
$490$	0	0
$491$	−22.7879	−1.02840	−0.514202	−	0.857669i	$-0.671912\pi$
$491$	−22.7879	−1.02840	−0.514202	+	0.857669i	$0.671912\pi$
$492$	4.31348	0.194467
$493$	52.1122	2.34702
$494$	3.68092	0.165612
$495$	0	0
$496$	−18.1148	−0.813380
$497$	−1.79482	−0.0805086
$498$	22.2666	0.997791
$499$	−18.7580	−0.839723	−0.419861	−	0.907588i	$-0.637921\pi$
$499$	−18.7580	−0.839723	−0.419861	+	0.907588i	$0.637921\pi$
$500$	0	0
$501$	−12.4829	−0.557693
$502$	−22.3709	−0.998463
$503$	24.1464	1.07663	0.538317	−	0.842742i	$-0.319060\pi$
$503$	24.1464	1.07663	0.538317	+	0.842742i	$0.319060\pi$
$504$	9.13203	0.406773
$505$	0	0
$506$	1.27722	0.0567792
$507$	−6.31340	−0.280388
$508$	−5.36187	−0.237895
$509$	−10.5884	−0.469322	−0.234661	−	0.972077i	$-0.575398\pi$
$509$	−10.5884	−0.469322	−0.234661	+	0.972077i	$0.575398\pi$
$510$	0	0
$511$	−13.5616	−0.599931
$512$	−25.2650	−1.11657
$513$	−5.65182	−0.249534
$514$	8.92809	0.393801
$515$	0	0
$516$	3.72093	0.163805
$517$	10.9565	0.481864
$518$	28.3822	1.24704
$519$	−6.58938	−0.289242
$520$	0	0
$521$	4.39548	0.192570	0.0962848	−	0.995354i	$-0.469304\pi$
$521$	4.39548	0.192570	0.0962848	+	0.995354i	$0.469304\pi$
$522$	−12.4161	−0.543437
$523$	31.2908	1.36825	0.684125	−	0.729364i	$-0.260184\pi$
$523$	31.2908	1.36825	0.684125	+	0.729364i	$0.260184\pi$
$524$	−5.76461	−0.251828
$525$	0	0
$526$	−11.6562	−0.508236
$527$	34.5155	1.50352
$528$	4.22655	0.183937
$529$	−21.9858	−0.955904
$530$	0	0
$531$	5.58355	0.242305
$532$	−1.07511	−0.0466121
$533$	−23.1741	−1.00378
$534$	20.9662	0.907298
$535$	0	0
$536$	−47.5238	−2.05272
$537$	4.05743	0.175091
$538$	−21.2614	−0.916643
$539$	−0.538344	−0.0231881
$540$	0	0
$541$	12.9098	0.555035	0.277518	−	0.960721i	$-0.410488\pi$
$541$	12.9098	0.555035	0.277518	+	0.960721i	$0.410488\pi$
$542$	−0.426477	−0.0183188
$543$	2.25455	0.0967521
$544$	12.7302	0.545805
$545$	0	0
$546$	13.9431	0.596711
$547$	17.1687	0.734081	0.367041	−	0.930205i	$-0.380371\pi$
$547$	17.1687	0.734081	0.367041	+	0.930205i	$0.380371\pi$
$548$	2.85633	0.122016
$549$	−8.37655	−0.357503
$550$	0	0
$551$	8.92770	0.380333
$552$	4.21420	0.179368
$553$	35.1530	1.49486
$554$	0.980529	0.0416587
$555$	0	0
$556$	−6.49497	−0.275448
$557$	19.6073	0.830788	0.415394	−	0.909642i	$-0.363644\pi$
$557$	19.6073	0.830788	0.415394	+	0.909642i	$0.363644\pi$
$558$	−8.22355	−0.348131
$559$	−19.9906	−0.845511
$560$	0	0
$561$	−8.05315	−0.340004
$562$	−16.7909	−0.708281
$563$	−3.06142	−0.129023	−0.0645117	−	0.997917i	$-0.520549\pi$
$563$	−3.06142	−0.129023	−0.0645117	+	0.997917i	$0.520549\pi$
$564$	5.91905	0.249237
$565$	0	0
$566$	4.62284	0.194312
$567$	−12.3764	−0.519760
$568$	1.98275	0.0831942
$569$	−35.8142	−1.50141	−0.750704	−	0.660639i	$-0.770285\pi$
$569$	−35.8142	−1.50141	−0.750704	+	0.660639i	$0.770285\pi$
$570$	0	0
$571$	−33.6489	−1.40816	−0.704081	−	0.710120i	$-0.748641\pi$
$571$	−33.6489	−1.40816	−0.704081	+	0.710120i	$0.748641\pi$
$572$	1.13651	0.0475198
$573$	−8.37856	−0.350019
$574$	−27.8025	−1.16046
$575$	0	0
$576$	−9.75192	−0.406330
$577$	−29.8985	−1.24469	−0.622345	−	0.782743i	$-0.713820\pi$
$577$	−29.8985	−1.24469	−0.622345	+	0.782743i	$0.713820\pi$
$578$	21.6515	0.900586
$579$	0.231114	0.00960475
$580$	0	0
$581$	34.9402	1.44957
$582$	−1.71942	−0.0712724
$583$	5.29137	0.219146
$584$	14.9816	0.619943
$585$	0	0
$586$	−12.0042	−0.495891
$587$	33.6644	1.38948	0.694740	−	0.719261i	$-0.255520\pi$
$587$	33.6644	1.38948	0.694740	+	0.719261i	$0.255520\pi$
$588$	−0.290832	−0.0119937
$589$	5.91308	0.243644
$590$	0	0
$591$	−23.4636	−0.965163
$592$	−24.9706	−1.02628
$593$	2.13964	0.0878646	0.0439323	−	0.999035i	$-0.486011\pi$
$593$	2.13964	0.0878646	0.0439323	+	0.999035i	$0.486011\pi$
$594$	7.16785	0.294100
$595$	0	0
$596$	2.39176	0.0979705
$597$	−26.4798	−1.08374
$598$	−3.70699	−0.151590
$599$	25.3896	1.03739	0.518696	−	0.854959i	$-0.326418\pi$
$599$	25.3896	1.03739	0.518696	+	0.854959i	$0.326418\pi$
$600$	0	0
$601$	−16.6132	−0.677666	−0.338833	−	0.940846i	$-0.610032\pi$
$601$	−16.6132	−0.677666	−0.338833	+	0.940846i	$0.610032\pi$
$602$	−23.9832	−0.977483
$603$	−17.1819	−0.699702
$604$	−2.82379	−0.114899
$605$	0	0
$606$	1.02409	0.0416007
$607$	−15.5197	−0.629927	−0.314963	−	0.949104i	$-0.601992\pi$
$607$	−15.5197	−0.629927	−0.314963	+	0.949104i	$0.601992\pi$
$608$	2.18091	0.0884474
$609$	33.8177	1.37036
$610$	0	0
$611$	−31.7999	−1.28649
$612$	2.50646	0.101318
$613$	−28.8792	−1.16642	−0.583210	−	0.812322i	$-0.698203\pi$
$613$	−28.8792	−1.16642	−0.583210	+	0.812322i	$0.698203\pi$
$614$	43.3718	1.75034
$615$	0	0
$616$	8.32765	0.335531
$617$	−44.7531	−1.80169	−0.900846	−	0.434138i	$-0.857053\pi$
$617$	−44.7531	−1.80169	−0.900846	+	0.434138i	$0.857053\pi$
$618$	0.285707	0.0114928
$619$	−31.6774	−1.27322	−0.636610	−	0.771186i	$-0.719664\pi$
$619$	−31.6774	−1.27322	−0.636610	+	0.771186i	$0.719664\pi$
$620$	0	0
$621$	5.69185	0.228406
$622$	−39.5695	−1.58659
$623$	32.8997	1.31810
$624$	−12.2671	−0.491077
$625$	0	0
$626$	10.8167	0.432321
$627$	−1.37964	−0.0550975
$628$	3.21758	0.128395
$629$	47.5782	1.89707
$630$	0	0
$631$	−22.3735	−0.890677	−0.445338	−	0.895362i	$-0.646917\pi$
$631$	−22.3735	−0.890677	−0.445338	+	0.895362i	$0.646917\pi$
$632$	−38.8337	−1.54472
$633$	22.2021	0.882454
$634$	−17.6971	−0.702842
$635$	0	0
$636$	2.85858	0.113350
$637$	1.56249	0.0619080
$638$	−11.3224	−0.448259
$639$	0.716849	0.0283581
$640$	0	0
$641$	5.11199	0.201911	0.100956	−	0.994891i	$-0.467810\pi$
$641$	5.11199	0.201911	0.100956	+	0.994891i	$0.467810\pi$
$642$	15.0023	0.592093
$643$	19.5913	0.772606	0.386303	−	0.922372i	$-0.373752\pi$
$643$	19.5913	0.772606	0.386303	+	0.922372i	$0.373752\pi$
$644$	1.08273	0.0426654
$645$	0	0
$646$	7.40287	0.291262
$647$	22.5387	0.886089	0.443045	−	0.896500i	$-0.353898\pi$
$647$	22.5387	0.886089	0.443045	+	0.896500i	$0.353898\pi$
$648$	13.6723	0.537098
$649$	5.09173	0.199868
$650$	0	0
$651$	22.3985	0.877865
$652$	6.08575	0.238337
$653$	19.5580	0.765364	0.382682	−	0.923880i	$-0.375001\pi$
$653$	19.5580	0.765364	0.382682	+	0.923880i	$0.375001\pi$
$654$	28.2583	1.10499
$655$	0	0
$656$	24.4606	0.955024
$657$	5.41649	0.211318
$658$	−38.1512	−1.48729
$659$	−32.6875	−1.27332	−0.636662	−	0.771143i	$-0.719685\pi$
$659$	−32.6875	−1.27332	−0.636662	+	0.771143i	$0.719685\pi$
$660$	0	0
$661$	15.4629	0.601437	0.300718	−	0.953713i	$-0.402774\pi$
$661$	15.4629	0.601437	0.300718	+	0.953713i	$0.402774\pi$
$662$	−25.9984	−1.01046
$663$	23.3734	0.907748
$664$	−38.5987	−1.49792
$665$	0	0
$666$	−11.3358	−0.439255
$667$	−8.99092	−0.348130
$668$	3.54294	0.137081
$669$	−34.3239	−1.32704
$670$	0	0
$671$	−7.63872	−0.294889
$672$	8.26115	0.318681
$673$	24.8554	0.958105	0.479053	−	0.877786i	$-0.340980\pi$
$673$	24.8554	0.958105	0.479053	+	0.877786i	$0.340980\pi$
$674$	40.4626	1.55856
$675$	0	0
$676$	1.79190	0.0689192
$677$	−48.0537	−1.84685	−0.923427	−	0.383773i	$-0.874624\pi$
$677$	−48.0537	−1.84685	−0.923427	+	0.383773i	$0.874624\pi$
$678$	15.9062	0.610875
$679$	−2.69808	−0.103543
$680$	0	0
$681$	38.0221	1.45701
$682$	−7.49919	−0.287159
$683$	−10.6837	−0.408799	−0.204400	−	0.978888i	$-0.565524\pi$
$683$	−10.6837	−0.408799	−0.204400	+	0.978888i	$0.565524\pi$
$684$	0.429399	0.0164185
$685$	0	0
$686$	−22.5000	−0.859053
$687$	−24.2538	−0.925342
$688$	21.1003	0.804443
$689$	−15.3576	−0.585079
$690$	0	0
$691$	15.2228	0.579101	0.289551	−	0.957163i	$-0.406494\pi$
$691$	15.2228	0.579101	0.289551	+	0.957163i	$0.406494\pi$
$692$	1.87023	0.0710954
$693$	3.01081	0.114371
$694$	17.8939	0.679243
$695$	0	0
$696$	−37.3585	−1.41607
$697$	−46.6064	−1.76534
$698$	−36.0156	−1.36321
$699$	6.76938	0.256042
$700$	0	0
$701$	2.56955	0.0970504	0.0485252	−	0.998822i	$-0.484548\pi$
$701$	2.56955	0.0970504	0.0485252	+	0.998822i	$0.484548\pi$
$702$	−20.8039	−0.785193
$703$	8.15096	0.307419
$704$	−8.89294	−0.335165
$705$	0	0
$706$	−29.7553	−1.11986
$707$	1.60697	0.0604364
$708$	2.75073	0.103379
$709$	−0.432067	−0.0162266	−0.00811331	−	0.999967i	$-0.502583\pi$
$709$	−0.432067	−0.0162266	−0.00811331	+	0.999967i	$0.502583\pi$
$710$	0	0
$711$	−14.0401	−0.526543
$712$	−36.3445	−1.36207
$713$	−5.95496	−0.223015
$714$	28.0417	1.04943
$715$	0	0
$716$	−1.15160	−0.0430373
$717$	6.44637	0.240744
$718$	14.5634	0.543503
$719$	32.4602	1.21056	0.605281	−	0.796012i	$-0.293061\pi$
$719$	32.4602	1.21056	0.605281	+	0.796012i	$0.293061\pi$
$720$	0	0
$721$	0.448325	0.0166965
$722$	1.26824	0.0471989
$723$	−33.5400	−1.24737
$724$	−0.639897	−0.0237816
$725$	0	0
$726$	1.74971	0.0649379
$727$	17.6543	0.654763	0.327381	−	0.944892i	$-0.393834\pi$
$727$	17.6543	0.654763	0.327381	+	0.944892i	$0.393834\pi$
$728$	−24.1701	−0.895804
$729$	28.3356	1.04947
$730$	0	0
$731$	−40.2040	−1.48700
$732$	−4.12670	−0.152527
$733$	12.7227	0.469925	0.234962	−	0.972004i	$-0.424503\pi$
$733$	12.7227	0.469925	0.234962	+	0.972004i	$0.424503\pi$
$734$	−31.7710	−1.17269
$735$	0	0
$736$	−2.19635	−0.0809585
$737$	−15.6685	−0.577156
$738$	11.1043	0.408755
$739$	5.08759	0.187150	0.0935750	−	0.995612i	$-0.470171\pi$
$739$	5.08759	0.187150	0.0935750	+	0.995612i	$0.470171\pi$
$740$	0	0
$741$	4.00426	0.147100
$742$	−18.4249	−0.676401
$743$	−38.3289	−1.40615	−0.703075	−	0.711116i	$-0.748190\pi$
$743$	−38.3289	−1.40615	−0.703075	+	0.711116i	$0.748190\pi$
$744$	−24.7437	−0.907148
$745$	0	0
$746$	−29.9044	−1.09488
$747$	−13.9551	−0.510590
$748$	2.28568	0.0835728
$749$	23.5412	0.860177
$750$	0	0
$751$	−39.9351	−1.45725	−0.728626	−	0.684911i	$-0.759841\pi$
$751$	−39.9351	−1.45725	−0.728626	+	0.684911i	$0.759841\pi$
$752$	33.5653	1.22400
$753$	−24.3360	−0.886854
$754$	32.8621	1.19677
$755$	0	0
$756$	6.07635	0.220995
$757$	44.8404	1.62975	0.814876	−	0.579636i	$-0.196805\pi$
$757$	44.8404	1.62975	0.814876	+	0.579636i	$0.196805\pi$
$758$	−33.8879	−1.23086
$759$	1.38941	0.0504324
$760$	0	0
$761$	43.9069	1.59163	0.795813	−	0.605543i	$-0.207044\pi$
$761$	43.9069	1.59163	0.795813	+	0.605543i	$0.207044\pi$
$762$	23.9589	0.867939
$763$	44.3423	1.60530
$764$	2.37804	0.0860346
$765$	0	0
$766$	−30.7329	−1.11043
$767$	−14.7782	−0.533609
$768$	−12.4362	−0.448752
$769$	−11.9549	−0.431106	−0.215553	−	0.976492i	$-0.569155\pi$
$769$	−11.9549	−0.431106	−0.215553	+	0.976492i	$0.569155\pi$
$770$	0	0
$771$	9.71235	0.349782
$772$	−0.0655957	−0.00236084
$773$	−4.78825	−0.172221	−0.0861107	−	0.996286i	$-0.527444\pi$
$773$	−4.78825	−0.172221	−0.0861107	+	0.996286i	$0.527444\pi$
$774$	9.57887	0.344305
$775$	0	0
$776$	2.98058	0.106997
$777$	30.8754	1.10765
$778$	6.69997	0.240206
$779$	−7.98447	−0.286073
$780$	0	0
$781$	0.653706	0.0233914
$782$	−7.45529	−0.266601
$783$	−50.4578	−1.80321
$784$	−1.64923	−0.0589010
$785$	0	0
$786$	25.7585	0.918774
$787$	−35.2409	−1.25620	−0.628100	−	0.778132i	$-0.716167\pi$
$787$	−35.2409	−1.25620	−0.628100	+	0.778132i	$0.716167\pi$
$788$	6.65955	0.237236
$789$	−12.6801	−0.451425
$790$	0	0
$791$	24.9597	0.887463
$792$	−3.32605	−0.118186
$793$	22.1706	0.787300
$794$	2.65953	0.0943833
$795$	0	0
$796$	7.51561	0.266384
$797$	45.1345	1.59875	0.799373	−	0.600835i	$-0.205165\pi$
$797$	45.1345	1.59875	0.799373	+	0.600835i	$0.205165\pi$
$798$	4.80401	0.170060
$799$	−63.9543	−2.26254
$800$	0	0
$801$	−13.1401	−0.464283
$802$	25.7312	0.908599
$803$	4.93939	0.174307
$804$	−8.46465	−0.298525
$805$	0	0
$806$	21.7656	0.766660
$807$	−23.1290	−0.814180
$808$	−1.77523	−0.0624524
$809$	−42.3579	−1.48922	−0.744612	−	0.667498i	$-0.767365\pi$
$809$	−42.3579	−1.48922	−0.744612	+	0.667498i	$0.767365\pi$
$810$	0	0
$811$	32.2926	1.13395	0.566973	−	0.823736i	$-0.308114\pi$
$811$	32.2926	1.13395	0.566973	+	0.823736i	$0.308114\pi$
$812$	−9.59828	−0.336834
$813$	−0.463940	−0.0162711
$814$	−10.3373	−0.362324
$815$	0	0
$816$	−24.6710	−0.863656
$817$	−6.88762	−0.240967
$818$	42.3120	1.47940
$819$	−8.73854	−0.305349
$820$	0	0
$821$	1.70149	0.0593824	0.0296912	−	0.999559i	$-0.490548\pi$
$821$	1.70149	0.0593824	0.0296912	+	0.999559i	$0.490548\pi$
$822$	−12.7632	−0.445167
$823$	2.68188	0.0934844	0.0467422	−	0.998907i	$-0.485116\pi$
$823$	2.68188	0.0934844	0.0467422	+	0.998907i	$0.485116\pi$
$824$	−0.495267	−0.0172535
$825$	0	0
$826$	−17.7298	−0.616898
$827$	−11.9898	−0.416928	−0.208464	−	0.978030i	$-0.566846\pi$
$827$	−11.9898	−0.416928	−0.208464	+	0.978030i	$0.566846\pi$
$828$	−0.432440	−0.0150283
$829$	−40.3322	−1.40079	−0.700397	−	0.713753i	$-0.746994\pi$
$829$	−40.3322	−1.40079	−0.700397	+	0.713753i	$0.746994\pi$
$830$	0	0
$831$	1.06666	0.0370021
$832$	25.8108	0.894828
$833$	3.14239	0.108877
$834$	29.0220	1.00495
$835$	0	0
$836$	0.391576	0.0135429
$837$	−33.4197	−1.15515
$838$	14.5372	0.502181
$839$	19.1737	0.661950	0.330975	−	0.943639i	$-0.392622\pi$
$839$	19.1737	0.661950	0.330975	+	0.943639i	$0.392622\pi$
$840$	0	0
$841$	50.7038	1.74841
$842$	27.2183	0.938005
$843$	−18.2658	−0.629109
$844$	−6.30150	−0.216907
$845$	0	0
$846$	15.2376	0.523878
$847$	2.74560	0.0943401
$848$	16.2102	0.556660
$849$	5.02892	0.172592
$850$	0	0
$851$	−8.20868	−0.281390
$852$	0.353154	0.0120989
$853$	22.8894	0.783717	0.391858	−	0.920026i	$-0.371832\pi$
$853$	22.8894	0.783717	0.391858	+	0.920026i	$0.371832\pi$
$854$	26.5986	0.910185
$855$	0	0
$856$	−26.0061	−0.888870
$857$	−39.5732	−1.35180	−0.675898	−	0.736995i	$-0.736244\pi$
$857$	−39.5732	−1.35180	−0.675898	+	0.736995i	$0.736244\pi$
$858$	−5.07835	−0.173372
$859$	−26.7629	−0.913138	−0.456569	−	0.889688i	$-0.650922\pi$
$859$	−26.7629	−0.913138	−0.456569	+	0.889688i	$0.650922\pi$
$860$	0	0
$861$	−30.2448	−1.03074
$862$	−2.36244	−0.0804649
$863$	6.34654	0.216039	0.108019	−	0.994149i	$-0.465549\pi$
$863$	6.34654	0.216039	0.108019	+	0.994149i	$0.465549\pi$
$864$	−12.3261	−0.419342
$865$	0	0
$866$	−20.9774	−0.712840
$867$	23.5535	0.799917
$868$	−6.35723	−0.215779
$869$	−12.8034	−0.434324
$870$	0	0
$871$	45.4761	1.54090
$872$	−48.9851	−1.65885
$873$	1.07761	0.0364716
$874$	−1.27722	−0.0432025
$875$	0	0
$876$	2.66843	0.0901578
$877$	−47.2905	−1.59689	−0.798444	−	0.602070i	$-0.794343\pi$
$877$	−47.2905	−1.59689	−0.798444	+	0.602070i	$0.794343\pi$
$878$	17.0460	0.575273
$879$	−13.0587	−0.440460
$880$	0	0
$881$	−4.11832	−0.138750	−0.0693748	−	0.997591i	$-0.522100\pi$
$881$	−4.11832	−0.138750	−0.0693748	+	0.997591i	$0.522100\pi$
$882$	−0.748696	−0.0252099
$883$	−9.63872	−0.324369	−0.162184	−	0.986760i	$-0.551854\pi$
$883$	−9.63872	−0.324369	−0.162184	+	0.986760i	$0.551854\pi$
$884$	−6.63395	−0.223124
$885$	0	0
$886$	19.8745	0.667698
$887$	−46.9180	−1.57535	−0.787677	−	0.616089i	$-0.788716\pi$
$887$	−46.9180	−1.57535	−0.787677	+	0.616089i	$0.788716\pi$
$888$	−34.1082	−1.14460
$889$	37.5957	1.26092
$890$	0	0
$891$	4.50771	0.151014
$892$	9.74197	0.326185
$893$	−10.9565	−0.366644
$894$	−10.6873	−0.357437
$895$	0	0
$896$	18.9901	0.634414
$897$	−4.03261	−0.134645
$898$	−49.6980	−1.65844
$899$	52.7902	1.76065
$900$	0	0
$901$	−30.8864	−1.02898
$902$	10.1262	0.337165
$903$	−26.0899	−0.868219
$904$	−27.5731	−0.917067
$905$	0	0
$906$	12.6178	0.419198
$907$	24.3781	0.809462	0.404731	−	0.914436i	$-0.367365\pi$
$907$	24.3781	0.809462	0.404731	+	0.914436i	$0.367365\pi$
$908$	−10.7916	−0.358132
$909$	−0.641823	−0.0212879
$910$	0	0
$911$	55.1931	1.82863	0.914315	−	0.405005i	$-0.132730\pi$
$911$	55.1931	1.82863	0.914315	+	0.405005i	$0.132730\pi$
$912$	−4.22655	−0.139955
$913$	−12.7259	−0.421165
$914$	−20.6229	−0.682145
$915$	0	0
$916$	6.88384	0.227448
$917$	40.4196	1.33477
$918$	−41.8397	−1.38092
$919$	−36.4646	−1.20286	−0.601428	−	0.798927i	$-0.705401\pi$
$919$	−36.4646	−1.20286	−0.601428	+	0.798927i	$0.705401\pi$
$920$	0	0
$921$	47.1817	1.55469
$922$	38.0841	1.25423
$923$	−1.89731	−0.0624508
$924$	1.48327	0.0487960
$925$	0	0
$926$	43.5493	1.43112
$927$	−0.179061	−0.00588112
$928$	19.4705	0.639149
$929$	13.8456	0.454259	0.227130	−	0.973865i	$-0.427066\pi$
$929$	13.8456	0.454259	0.227130	+	0.973865i	$0.427066\pi$
$930$	0	0
$931$	0.538344	0.0176435
$932$	−1.92132	−0.0629348
$933$	−43.0454	−1.40924
$934$	−41.7624	−1.36651
$935$	0	0
$936$	9.65351	0.315535
$937$	6.21716	0.203106	0.101553	−	0.994830i	$-0.467619\pi$
$937$	6.21716	0.203106	0.101553	+	0.994830i	$0.467619\pi$
$938$	54.5588	1.78141
$939$	11.7668	0.383995
$940$	0	0
$941$	8.07984	0.263395	0.131698	−	0.991290i	$-0.457957\pi$
$941$	8.07984	0.263395	0.131698	+	0.991290i	$0.457957\pi$
$942$	−14.3774	−0.468439
$943$	8.04101	0.261851
$944$	15.5986	0.507691
$945$	0	0
$946$	8.73513	0.284004
$947$	−33.0342	−1.07347	−0.536734	−	0.843752i	$-0.680342\pi$
$947$	−33.0342	−1.07347	−0.536734	+	0.843752i	$0.680342\pi$
$948$	−6.91681	−0.224648
$949$	−14.3361	−0.465368
$950$	0	0
$951$	−19.2517	−0.624278
$952$	−48.6096	−1.57545
$953$	1.71006	0.0553943	0.0276972	−	0.999616i	$-0.491183\pi$
$953$	1.71006	0.0553943	0.0276972	+	0.999616i	$0.491183\pi$
$954$	7.35890	0.238253
$955$	0	0
$956$	−1.82964	−0.0591747
$957$	−12.3170	−0.398153
$958$	−42.5160	−1.37363
$959$	−20.0277	−0.646727
$960$	0	0
$961$	3.96456	0.127889
$962$	30.0030	0.967336
$963$	−9.40234	−0.302986
$964$	9.51947	0.306602
$965$	0	0
$966$	−4.83803	−0.155661
$967$	−43.3980	−1.39558	−0.697792	−	0.716300i	$-0.745834\pi$
$967$	−43.3980	−1.39558	−0.697792	+	0.716300i	$0.745834\pi$
$968$	−3.03308	−0.0974870
$969$	8.05315	0.258704
$970$	0	0
$971$	−45.2884	−1.45337	−0.726687	−	0.686969i	$-0.758941\pi$
$971$	−45.2884	−1.45337	−0.726687	+	0.686969i	$0.758941\pi$
$972$	−4.20414	−0.134848
$973$	45.5407	1.45997
$974$	26.2304	0.840476
$975$	0	0
$976$	−23.4013	−0.749059
$977$	−13.2323	−0.423340	−0.211670	−	0.977341i	$-0.567890\pi$
$977$	−13.2323	−0.423340	−0.211670	+	0.977341i	$0.567890\pi$
$978$	−27.1935	−0.869551
$979$	−11.9827	−0.382968
$980$	0	0
$981$	−17.7103	−0.565445
$982$	−28.9004	−0.922250
$983$	49.0140	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$983$	49.0140	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$984$	33.4115	1.06512
$985$	0	0
$986$	66.0906	2.10475
$987$	−41.5025	−1.32104
$988$	−1.13651	−0.0361571
$989$	6.93639	0.220564
$990$	0	0
$991$	24.0457	0.763836	0.381918	−	0.924196i	$-0.375264\pi$
$991$	24.0457	0.763836	0.381918	+	0.924196i	$0.375264\pi$
$992$	12.8959	0.409444
$993$	−28.2822	−0.897508
$994$	−2.27625	−0.0721984
$995$	0	0
$996$	−6.87496	−0.217841
$997$	36.6608	1.16106	0.580530	−	0.814239i	$-0.302845\pi$
$997$	36.6608	1.16106	0.580530	+	0.814239i	$0.302845\pi$
$998$	−23.7896	−0.753045
$999$	−46.0678	−1.45752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.u.1.11	✓	15
5.4	even	2		5225.2.a.v.1.5	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.u.1.11	✓	15	1.1	even	1	trivial
5225.2.a.v.1.5	yes	15	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26824 q^{2} +1.37964 q^{3} -0.391576 q^{4} +1.74971 q^{6} +2.74560 q^{7} -3.03308 q^{8} -1.09659 q^{9} -1.00000 q^{11} -0.540234 q^{12} +2.90239 q^{13} +3.48208 q^{14} -3.06352 q^{16} +5.83714 q^{17} -1.39074 q^{18} +1.00000 q^{19} +3.78795 q^{21} -1.26824 q^{22} -1.00708 q^{23} -4.18457 q^{24} +3.68092 q^{26} -5.65182 q^{27} -1.07511 q^{28} +8.92770 q^{29} +5.91308 q^{31} +2.18091 q^{32} -1.37964 q^{33} +7.40287 q^{34} +0.429399 q^{36} +8.15096 q^{37} +1.26824 q^{38} +4.00426 q^{39} -7.98447 q^{41} +4.80401 q^{42} -6.88762 q^{43} +0.391576 q^{44} -1.27722 q^{46} -10.9565 q^{47} -4.22655 q^{48} +0.538344 q^{49} +8.05315 q^{51} -1.13651 q^{52} -5.29137 q^{53} -7.16785 q^{54} -8.32765 q^{56} +1.37964 q^{57} +11.3224 q^{58} -5.09173 q^{59} +7.63872 q^{61} +7.49919 q^{62} -3.01081 q^{63} +8.89294 q^{64} -1.74971 q^{66} +15.6685 q^{67} -2.28568 q^{68} -1.38941 q^{69} -0.653706 q^{71} +3.32605 q^{72} -4.93939 q^{73} +10.3373 q^{74} -0.391576 q^{76} -2.74560 q^{77} +5.07835 q^{78} +12.8034 q^{79} -4.50771 q^{81} -10.1262 q^{82} +12.7259 q^{83} -1.48327 q^{84} -8.73513 q^{86} +12.3170 q^{87} +3.03308 q^{88} +11.9827 q^{89} +7.96882 q^{91} +0.394349 q^{92} +8.15793 q^{93} -13.8954 q^{94} +3.00887 q^{96} -0.982691 q^{97} +0.682748 q^{98} +1.09659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23}+ \cdots - 15 q^{99}+O(q^{100}) \) 15 * q - q^2 + 4 * q^3 + 13 * q^4 - q^6 + 17 * q^7 + 3 * q^8 + 15 * q^9 - 15 * q^11 + 9 * q^12 + 3 * q^13 - 11 * q^14 + 13 * q^16 + q^17 - 10 * q^18 + 15 * q^19 + 16 * q^21 + q^22 + 18 * q^23 - 27 * q^24 + 15 * q^26 + 31 * q^27 + 34 * q^28 + 11 * q^29 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 13 * q^34 + 8 * q^36 + 27 * q^37 - q^38 + 4 * q^39 + 6 * q^41 + 2 * q^42 + 38 * q^43 - 13 * q^44 - 9 * q^46 + 14 * q^47 + 32 * q^48 + 28 * q^49 - 32 * q^51 + 16 * q^52 + 11 * q^53 - 11 * q^54 + 2 * q^56 + 4 * q^57 - 6 * q^58 + 13 * q^59 + 8 * q^61 + 7 * q^62 + 49 * q^63 + 9 * q^64 + q^66 + 31 * q^67 - 26 * q^68 + 39 * q^69 - 3 * q^71 - 18 * q^72 + 18 * q^73 + 7 * q^74 + 13 * q^76 - 17 * q^77 + 18 * q^78 + 10 * q^79 + 31 * q^81 + 58 * q^82 + 16 * q^83 + 112 * q^84 - 63 * q^86 + 67 * q^87 - 3 * q^88 - 7 * q^89 - 50 * q^91 + 98 * q^92 + 26 * q^93 + 22 * q^94 - 37 * q^96 + 24 * q^97 - 46 * q^98 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more