LMFDB - Embedded newform 3850.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1,3850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.41421 q^{12} -3.24264 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.58579 q^{17} +1.00000 q^{18} +0.171573 q^{19} +1.41421 q^{21} +1.00000 q^{22} +1.58579 q^{23} -1.41421 q^{24} +3.24264 q^{26} -5.65685 q^{27} +1.00000 q^{28} +8.65685 q^{29} -0.171573 q^{31} -1.00000 q^{32} -1.41421 q^{33} -2.58579 q^{34} -1.00000 q^{36} +2.82843 q^{37} -0.171573 q^{38} -4.58579 q^{39} -6.82843 q^{41} -1.41421 q^{42} +10.8995 q^{43} -1.00000 q^{44} -1.58579 q^{46} -6.48528 q^{47} +1.41421 q^{48} +1.00000 q^{49} +3.65685 q^{51} -3.24264 q^{52} +12.2426 q^{53} +5.65685 q^{54} -1.00000 q^{56} +0.242641 q^{57} -8.65685 q^{58} +4.00000 q^{59} -12.8284 q^{61} +0.171573 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.41421 q^{66} +3.41421 q^{67} +2.58579 q^{68} +2.24264 q^{69} +10.0711 q^{71} +1.00000 q^{72} +15.3137 q^{73} -2.82843 q^{74} +0.171573 q^{76} -1.00000 q^{77} +4.58579 q^{78} +11.0711 q^{79} -5.00000 q^{81} +6.82843 q^{82} +13.8284 q^{83} +1.41421 q^{84} -10.8995 q^{86} +12.2426 q^{87} +1.00000 q^{88} -6.89949 q^{89} -3.24264 q^{91} +1.58579 q^{92} -0.242641 q^{93} +6.48528 q^{94} -1.41421 q^{96} +2.07107 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 6 q^{19} + 2 q^{22} + 6 q^{23} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 6 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{36} - 6 q^{38} - 12 q^{39} - 8 q^{41} + 2 q^{43} - 2 q^{44} - 6 q^{46} + 4 q^{47} + 2 q^{49} - 4 q^{51} + 2 q^{52} + 16 q^{53} - 2 q^{56} - 8 q^{57} - 6 q^{58} + 8 q^{59} - 20 q^{61} + 6 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 4 q^{69} + 6 q^{71} + 2 q^{72} + 8 q^{73} + 6 q^{76} - 2 q^{77} + 12 q^{78} + 8 q^{79} - 10 q^{81} + 8 q^{82} + 22 q^{83} - 2 q^{86} + 16 q^{87} + 2 q^{88} + 6 q^{89} + 2 q^{91} + 6 q^{92} + 8 q^{93} - 4 q^{94} - 10 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 8 * q^17 + 2 * q^18 + 6 * q^19 + 2 * q^22 + 6 * q^23 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 6 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^36 - 6 * q^38 - 12 * q^39 - 8 * q^41 + 2 * q^43 - 2 * q^44 - 6 * q^46 + 4 * q^47 + 2 * q^49 - 4 * q^51 + 2 * q^52 + 16 * q^53 - 2 * q^56 - 8 * q^57 - 6 * q^58 + 8 * q^59 - 20 * q^61 + 6 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^67 + 8 * q^68 - 4 * q^69 + 6 * q^71 + 2 * q^72 + 8 * q^73 + 6 * q^76 - 2 * q^77 + 12 * q^78 + 8 * q^79 - 10 * q^81 + 8 * q^82 + 22 * q^83 - 2 * q^86 + 16 * q^87 + 2 * q^88 + 6 * q^89 + 2 * q^91 + 6 * q^92 + 8 * q^93 - 4 * q^94 - 10 * q^97 - 2 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.41421	−0.577350
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.41421	0.408248
$13$	−3.24264	−0.899347	−0.449673	−	0.893193i	$-0.648460\pi$
$13$	−3.24264	−0.899347	−0.449673	+	0.893193i	$0.648460\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.58579	0.627145	0.313573	−	0.949564i	$-0.398474\pi$
$17$	2.58579	0.627145	0.313573	+	0.949564i	$0.398474\pi$
$18$	1.00000	0.235702
$19$	0.171573	0.0393615	0.0196808	−	0.999806i	$-0.493735\pi$
$19$	0.171573	0.0393615	0.0196808	+	0.999806i	$0.493735\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	1.00000	0.213201
$23$	1.58579	0.330659	0.165330	−	0.986238i	$-0.447131\pi$
$23$	1.58579	0.330659	0.165330	+	0.986238i	$0.447131\pi$
$24$	−1.41421	−0.288675
$25$	0	0
$26$	3.24264	0.635934
$27$	−5.65685	−1.08866
$28$	1.00000	0.188982
$29$	8.65685	1.60754	0.803769	−	0.594942i	$-0.202825\pi$
$29$	8.65685	1.60754	0.803769	+	0.594942i	$0.202825\pi$
$30$	0	0
$31$	−0.171573	−0.0308154	−0.0154077	−	0.999881i	$-0.504905\pi$
$31$	−0.171573	−0.0308154	−0.0154077	+	0.999881i	$0.504905\pi$
$32$	−1.00000	−0.176777
$33$	−1.41421	−0.246183
$34$	−2.58579	−0.443459
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.82843	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$37$	2.82843	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$38$	−0.171573	−0.0278328
$39$	−4.58579	−0.734314
$40$	0	0
$41$	−6.82843	−1.06642	−0.533211	−	0.845983i	$-0.679015\pi$
$41$	−6.82843	−1.06642	−0.533211	+	0.845983i	$0.679015\pi$
$42$	−1.41421	−0.218218
$43$	10.8995	1.66216	0.831079	−	0.556155i	$-0.187724\pi$
$43$	10.8995	1.66216	0.831079	+	0.556155i	$0.187724\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.58579	−0.233811
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	1.41421	0.204124
$49$	1.00000	0.142857
$50$	0	0
$51$	3.65685	0.512062
$52$	−3.24264	−0.449673
$53$	12.2426	1.68166	0.840828	−	0.541302i	$-0.182069\pi$
$53$	12.2426	1.68166	0.840828	+	0.541302i	$0.182069\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0.242641	0.0321385
$58$	−8.65685	−1.13670
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−12.8284	−1.64251	−0.821256	−	0.570560i	$-0.806726\pi$
$61$	−12.8284	−1.64251	−0.821256	+	0.570560i	$0.806726\pi$
$62$	0.171573	0.0217898
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	1.41421	0.174078
$67$	3.41421	0.417113	0.208556	−	0.978010i	$-0.433124\pi$
$67$	3.41421	0.417113	0.208556	+	0.978010i	$0.433124\pi$
$68$	2.58579	0.313573
$69$	2.24264	0.269982
$70$	0	0
$71$	10.0711	1.19522	0.597608	−	0.801788i	$-0.296118\pi$
$71$	10.0711	1.19522	0.597608	+	0.801788i	$0.296118\pi$
$72$	1.00000	0.117851
$73$	15.3137	1.79233	0.896167	−	0.443717i	$-0.146340\pi$
$73$	15.3137	1.79233	0.896167	+	0.443717i	$0.146340\pi$
$74$	−2.82843	−0.328798
$75$	0	0
$76$	0.171573	0.0196808
$77$	−1.00000	−0.113961
$78$	4.58579	0.519238
$79$	11.0711	1.24559	0.622796	−	0.782384i	$-0.285997\pi$
$79$	11.0711	1.24559	0.622796	+	0.782384i	$0.285997\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	6.82843	0.754074
$83$	13.8284	1.51787	0.758934	−	0.651168i	$-0.225721\pi$
$83$	13.8284	1.51787	0.758934	+	0.651168i	$0.225721\pi$
$84$	1.41421	0.154303
$85$	0	0
$86$	−10.8995	−1.17532
$87$	12.2426	1.31255
$88$	1.00000	0.106600
$89$	−6.89949	−0.731345	−0.365673	−	0.930744i	$-0.619161\pi$
$89$	−6.89949	−0.731345	−0.365673	+	0.930744i	$0.619161\pi$
$90$	0	0
$91$	−3.24264	−0.339921
$92$	1.58579	0.165330
$93$	−0.242641	−0.0251607
$94$	6.48528	0.668906
$95$	0	0
$96$	−1.41421	−0.144338
$97$	2.07107	0.210285	0.105143	−	0.994457i	$-0.466470\pi$
$97$	2.07107	0.210285	0.105143	+	0.994457i	$0.466470\pi$
$98$	−1.00000	−0.101015
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.24264	−0.919677	−0.459839	−	0.888003i	$-0.652093\pi$
$101$	−9.24264	−0.919677	−0.459839	+	0.888003i	$0.652093\pi$
$102$	−3.65685	−0.362083
$103$	6.17157	0.608103	0.304052	−	0.952656i	$-0.401660\pi$
$103$	6.17157	0.608103	0.304052	+	0.952656i	$0.401660\pi$
$104$	3.24264	0.317967
$105$	0	0
$106$	−12.2426	−1.18911
$107$	−2.07107	−0.200218	−0.100109	−	0.994976i	$-0.531919\pi$
$107$	−2.07107	−0.200218	−0.100109	+	0.994976i	$0.531919\pi$
$108$	−5.65685	−0.544331
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	1.00000	0.0944911
$113$	12.4853	1.17452	0.587258	−	0.809400i	$-0.300207\pi$
$113$	12.4853	1.17452	0.587258	+	0.809400i	$0.300207\pi$
$114$	−0.242641	−0.0227254
$115$	0	0
$116$	8.65685	0.803769
$117$	3.24264	0.299782
$118$	−4.00000	−0.368230
$119$	2.58579	0.237039
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.8284	1.16143
$123$	−9.65685	−0.870729
$124$	−0.171573	−0.0154077
$125$	0	0
$126$	1.00000	0.0890871
$127$	−10.8284	−0.960868	−0.480434	−	0.877031i	$-0.659521\pi$
$127$	−10.8284	−0.960868	−0.480434	+	0.877031i	$0.659521\pi$
$128$	−1.00000	−0.0883883
$129$	15.4142	1.35715
$130$	0	0
$131$	−2.17157	−0.189731	−0.0948656	−	0.995490i	$-0.530242\pi$
$131$	−2.17157	−0.189731	−0.0948656	+	0.995490i	$0.530242\pi$
$132$	−1.41421	−0.123091
$133$	0.171573	0.0148773
$134$	−3.41421	−0.294943
$135$	0	0
$136$	−2.58579	−0.221729
$137$	6.17157	0.527273	0.263637	−	0.964622i	$-0.415078\pi$
$137$	6.17157	0.527273	0.263637	+	0.964622i	$0.415078\pi$
$138$	−2.24264	−0.190906
$139$	−4.65685	−0.394989	−0.197495	−	0.980304i	$-0.563280\pi$
$139$	−4.65685	−0.394989	−0.197495	+	0.980304i	$0.563280\pi$
$140$	0	0
$141$	−9.17157	−0.772386
$142$	−10.0711	−0.845145
$143$	3.24264	0.271163
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−15.3137	−1.26737
$147$	1.41421	0.116642
$148$	2.82843	0.232495
$149$	16.4853	1.35053	0.675263	−	0.737577i	$-0.264030\pi$
$149$	16.4853	1.35053	0.675263	+	0.737577i	$0.264030\pi$
$150$	0	0
$151$	11.5563	0.940442	0.470221	−	0.882549i	$-0.344174\pi$
$151$	11.5563	0.940442	0.470221	+	0.882549i	$0.344174\pi$
$152$	−0.171573	−0.0139164
$153$	−2.58579	−0.209048
$154$	1.00000	0.0805823
$155$	0	0
$156$	−4.58579	−0.367157
$157$	9.07107	0.723950	0.361975	−	0.932188i	$-0.382103\pi$
$157$	9.07107	0.723950	0.361975	+	0.932188i	$0.382103\pi$
$158$	−11.0711	−0.880767
$159$	17.3137	1.37307
$160$	0	0
$161$	1.58579	0.124977
$162$	5.00000	0.392837
$163$	−20.4853	−1.60453	−0.802266	−	0.596967i	$-0.796372\pi$
$163$	−20.4853	−1.60453	−0.802266	+	0.596967i	$0.796372\pi$
$164$	−6.82843	−0.533211
$165$	0	0
$166$	−13.8284	−1.07329
$167$	22.7279	1.75874	0.879370	−	0.476140i	$-0.157964\pi$
$167$	22.7279	1.75874	0.879370	+	0.476140i	$0.157964\pi$
$168$	−1.41421	−0.109109
$169$	−2.48528	−0.191175
$170$	0	0
$171$	−0.171573	−0.0131205
$172$	10.8995	0.831079
$173$	18.0711	1.37392	0.686959	−	0.726696i	$-0.258945\pi$
$173$	18.0711	1.37392	0.686959	+	0.726696i	$0.258945\pi$
$174$	−12.2426	−0.928112
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	5.65685	0.425195
$178$	6.89949	0.517139
$179$	−18.3848	−1.37414	−0.687071	−	0.726590i	$-0.741104\pi$
$179$	−18.3848	−1.37414	−0.687071	+	0.726590i	$0.741104\pi$
$180$	0	0
$181$	0.485281	0.0360707	0.0180353	−	0.999837i	$-0.494259\pi$
$181$	0.485281	0.0360707	0.0180353	+	0.999837i	$0.494259\pi$
$182$	3.24264	0.240361
$183$	−18.1421	−1.34111
$184$	−1.58579	−0.116906
$185$	0	0
$186$	0.242641	0.0177913
$187$	−2.58579	−0.189091
$188$	−6.48528	−0.472988
$189$	−5.65685	−0.411476
$190$	0	0
$191$	−13.3848	−0.968488	−0.484244	−	0.874933i	$-0.660905\pi$
$191$	−13.3848	−0.968488	−0.484244	+	0.874933i	$0.660905\pi$
$192$	1.41421	0.102062
$193$	10.1421	0.730047	0.365023	−	0.930998i	$-0.381061\pi$
$193$	10.1421	0.730047	0.365023	+	0.930998i	$0.381061\pi$
$194$	−2.07107	−0.148694
$195$	0	0
$196$	1.00000	0.0714286
$197$	−20.6569	−1.47174	−0.735870	−	0.677123i	$-0.763227\pi$
$197$	−20.6569	−1.47174	−0.735870	+	0.677123i	$0.763227\pi$
$198$	−1.00000	−0.0710669
$199$	−22.7990	−1.61618	−0.808089	−	0.589061i	$-0.799498\pi$
$199$	−22.7990	−1.61618	−0.808089	+	0.589061i	$0.799498\pi$
$200$	0	0
$201$	4.82843	0.340571
$202$	9.24264	0.650310
$203$	8.65685	0.607592
$204$	3.65685	0.256031
$205$	0	0
$206$	−6.17157	−0.429994
$207$	−1.58579	−0.110220
$208$	−3.24264	−0.224837
$209$	−0.171573	−0.0118679
$210$	0	0
$211$	−13.6569	−0.940177	−0.470088	−	0.882619i	$-0.655778\pi$
$211$	−13.6569	−0.940177	−0.470088	+	0.882619i	$0.655778\pi$
$212$	12.2426	0.840828
$213$	14.2426	0.975890
$214$	2.07107	0.141575
$215$	0	0
$216$	5.65685	0.384900
$217$	−0.171573	−0.0116471
$218$	−3.00000	−0.203186
$219$	21.6569	1.46343
$220$	0	0
$221$	−8.38478	−0.564021
$222$	−4.00000	−0.268462
$223$	−20.1421	−1.34882	−0.674409	−	0.738358i	$-0.735601\pi$
$223$	−20.1421	−1.34882	−0.674409	+	0.738358i	$0.735601\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−12.4853	−0.830509
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	0.242641	0.0160693
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	0	0
$231$	−1.41421	−0.0930484
$232$	−8.65685	−0.568350
$233$	19.7574	1.29435	0.647174	−	0.762342i	$-0.275951\pi$
$233$	19.7574	1.29435	0.647174	+	0.762342i	$0.275951\pi$
$234$	−3.24264	−0.211978
$235$	0	0
$236$	4.00000	0.260378
$237$	15.6569	1.01702
$238$	−2.58579	−0.167612
$239$	22.4853	1.45445	0.727226	−	0.686398i	$-0.240809\pi$
$239$	22.4853	1.45445	0.727226	+	0.686398i	$0.240809\pi$
$240$	0	0
$241$	23.6569	1.52387	0.761936	−	0.647652i	$-0.224249\pi$
$241$	23.6569	1.52387	0.761936	+	0.647652i	$0.224249\pi$
$242$	−1.00000	−0.0642824
$243$	9.89949	0.635053
$244$	−12.8284	−0.821256
$245$	0	0
$246$	9.65685	0.615699
$247$	−0.556349	−0.0353997
$248$	0.171573	0.0108949
$249$	19.5563	1.23933
$250$	0	0
$251$	−16.5858	−1.04689	−0.523443	−	0.852061i	$-0.675353\pi$
$251$	−16.5858	−1.04689	−0.523443	+	0.852061i	$0.675353\pi$
$252$	−1.00000	−0.0629941
$253$	−1.58579	−0.0996975
$254$	10.8284	0.679436
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.414214	−0.0258379	−0.0129190	−	0.999917i	$-0.504112\pi$
$257$	−0.414214	−0.0258379	−0.0129190	+	0.999917i	$0.504112\pi$
$258$	−15.4142	−0.959647
$259$	2.82843	0.175750
$260$	0	0
$261$	−8.65685	−0.535846
$262$	2.17157	0.134160
$263$	10.3848	0.640353	0.320176	−	0.947358i	$-0.396258\pi$
$263$	10.3848	0.640353	0.320176	+	0.947358i	$0.396258\pi$
$264$	1.41421	0.0870388
$265$	0	0
$266$	−0.171573	−0.0105198
$267$	−9.75736	−0.597141
$268$	3.41421	0.208556
$269$	6.24264	0.380621	0.190310	−	0.981724i	$-0.439051\pi$
$269$	6.24264	0.380621	0.190310	+	0.981724i	$0.439051\pi$
$270$	0	0
$271$	−12.7279	−0.773166	−0.386583	−	0.922255i	$-0.626345\pi$
$271$	−12.7279	−0.773166	−0.386583	+	0.922255i	$0.626345\pi$
$272$	2.58579	0.156786
$273$	−4.58579	−0.277544
$274$	−6.17157	−0.372838
$275$	0	0
$276$	2.24264	0.134991
$277$	−14.6274	−0.878876	−0.439438	−	0.898273i	$-0.644822\pi$
$277$	−14.6274	−0.878876	−0.439438	+	0.898273i	$0.644822\pi$
$278$	4.65685	0.279300
$279$	0.171573	0.0102718
$280$	0	0
$281$	2.24264	0.133785	0.0668924	−	0.997760i	$-0.478692\pi$
$281$	2.24264	0.133785	0.0668924	+	0.997760i	$0.478692\pi$
$282$	9.17157	0.546159
$283$	−13.7990	−0.820265	−0.410132	−	0.912026i	$-0.634518\pi$
$283$	−13.7990	−0.820265	−0.410132	+	0.912026i	$0.634518\pi$
$284$	10.0711	0.597608
$285$	0	0
$286$	−3.24264	−0.191741
$287$	−6.82843	−0.403069
$288$	1.00000	0.0589256
$289$	−10.3137	−0.606689
$290$	0	0
$291$	2.92893	0.171697
$292$	15.3137	0.896167
$293$	−27.4558	−1.60399	−0.801994	−	0.597332i	$-0.796227\pi$
$293$	−27.4558	−1.60399	−0.801994	+	0.597332i	$0.796227\pi$
$294$	−1.41421	−0.0824786
$295$	0	0
$296$	−2.82843	−0.164399
$297$	5.65685	0.328244
$298$	−16.4853	−0.954967
$299$	−5.14214	−0.297377
$300$	0	0
$301$	10.8995	0.628236
$302$	−11.5563	−0.664993
$303$	−13.0711	−0.750913
$304$	0.171573	0.00984038
$305$	0	0
$306$	2.58579	0.147820
$307$	−0.970563	−0.0553929	−0.0276965	−	0.999616i	$-0.508817\pi$
$307$	−0.970563	−0.0553929	−0.0276965	+	0.999616i	$0.508817\pi$
$308$	−1.00000	−0.0569803
$309$	8.72792	0.496514
$310$	0	0
$311$	21.9706	1.24584	0.622918	−	0.782287i	$-0.285947\pi$
$311$	21.9706	1.24584	0.622918	+	0.782287i	$0.285947\pi$
$312$	4.58579	0.259619
$313$	−21.1716	−1.19669	−0.598344	−	0.801239i	$-0.704174\pi$
$313$	−21.1716	−1.19669	−0.598344	+	0.801239i	$0.704174\pi$
$314$	−9.07107	−0.511910
$315$	0	0
$316$	11.0711	0.622796
$317$	18.0416	1.01332	0.506659	−	0.862146i	$-0.330880\pi$
$317$	18.0416	1.01332	0.506659	+	0.862146i	$0.330880\pi$
$318$	−17.3137	−0.970905
$319$	−8.65685	−0.484691
$320$	0	0
$321$	−2.92893	−0.163477
$322$	−1.58579	−0.0883724
$323$	0.443651	0.0246854
$324$	−5.00000	−0.277778
$325$	0	0
$326$	20.4853	1.13457
$327$	4.24264	0.234619
$328$	6.82843	0.377037
$329$	−6.48528	−0.357545
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	13.8284	0.758934
$333$	−2.82843	−0.154997
$334$	−22.7279	−1.24362
$335$	0	0
$336$	1.41421	0.0771517
$337$	13.4142	0.730719	0.365359	−	0.930867i	$-0.380946\pi$
$337$	13.4142	0.730719	0.365359	+	0.930867i	$0.380946\pi$
$338$	2.48528	0.135181
$339$	17.6569	0.958989
$340$	0	0
$341$	0.171573	0.00929119
$342$	0.171573	0.00927760
$343$	1.00000	0.0539949
$344$	−10.8995	−0.587661
$345$	0	0
$346$	−18.0711	−0.971507
$347$	20.4853	1.09971	0.549854	−	0.835261i	$-0.314683\pi$
$347$	20.4853	1.09971	0.549854	+	0.835261i	$0.314683\pi$
$348$	12.2426	0.656274
$349$	10.2132	0.546700	0.273350	−	0.961915i	$-0.411868\pi$
$349$	10.2132	0.546700	0.273350	+	0.961915i	$0.411868\pi$
$350$	0	0
$351$	18.3431	0.979085
$352$	1.00000	0.0533002
$353$	5.58579	0.297301	0.148651	−	0.988890i	$-0.452507\pi$
$353$	5.58579	0.297301	0.148651	+	0.988890i	$0.452507\pi$
$354$	−5.65685	−0.300658
$355$	0	0
$356$	−6.89949	−0.365673
$357$	3.65685	0.193541
$358$	18.3848	0.971666
$359$	3.02944	0.159888	0.0799438	−	0.996799i	$-0.474526\pi$
$359$	3.02944	0.159888	0.0799438	+	0.996799i	$0.474526\pi$
$360$	0	0
$361$	−18.9706	−0.998451
$362$	−0.485281	−0.0255058
$363$	1.41421	0.0742270
$364$	−3.24264	−0.169961
$365$	0	0
$366$	18.1421	0.948305
$367$	−14.3137	−0.747170	−0.373585	−	0.927596i	$-0.621871\pi$
$367$	−14.3137	−0.747170	−0.373585	+	0.927596i	$0.621871\pi$
$368$	1.58579	0.0826648
$369$	6.82843	0.355474
$370$	0	0
$371$	12.2426	0.635606
$372$	−0.242641	−0.0125803
$373$	5.85786	0.303309	0.151654	−	0.988434i	$-0.451540\pi$
$373$	5.85786	0.303309	0.151654	+	0.988434i	$0.451540\pi$
$374$	2.58579	0.133708
$375$	0	0
$376$	6.48528	0.334453
$377$	−28.0711	−1.44573
$378$	5.65685	0.290957
$379$	9.27208	0.476275	0.238137	−	0.971231i	$-0.423463\pi$
$379$	9.27208	0.476275	0.238137	+	0.971231i	$0.423463\pi$
$380$	0	0
$381$	−15.3137	−0.784545
$382$	13.3848	0.684825
$383$	11.1421	0.569337	0.284668	−	0.958626i	$-0.408117\pi$
$383$	11.1421	0.569337	0.284668	+	0.958626i	$0.408117\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	−10.1421	−0.516221
$387$	−10.8995	−0.554052
$388$	2.07107	0.105143
$389$	−17.7574	−0.900334	−0.450167	−	0.892944i	$-0.648636\pi$
$389$	−17.7574	−0.900334	−0.450167	+	0.892944i	$0.648636\pi$
$390$	0	0
$391$	4.10051	0.207371
$392$	−1.00000	−0.0505076
$393$	−3.07107	−0.154915
$394$	20.6569	1.04068
$395$	0	0
$396$	1.00000	0.0502519
$397$	31.4558	1.57872	0.789362	−	0.613928i	$-0.210412\pi$
$397$	31.4558	1.57872	0.789362	+	0.613928i	$0.210412\pi$
$398$	22.7990	1.14281
$399$	0.242641	0.0121472
$400$	0	0
$401$	4.51472	0.225454	0.112727	−	0.993626i	$-0.464041\pi$
$401$	4.51472	0.225454	0.112727	+	0.993626i	$0.464041\pi$
$402$	−4.82843	−0.240820
$403$	0.556349	0.0277137
$404$	−9.24264	−0.459839
$405$	0	0
$406$	−8.65685	−0.429632
$407$	−2.82843	−0.140200
$408$	−3.65685	−0.181041
$409$	−5.41421	−0.267716	−0.133858	−	0.991001i	$-0.542737\pi$
$409$	−5.41421	−0.267716	−0.133858	+	0.991001i	$0.542737\pi$
$410$	0	0
$411$	8.72792	0.430517
$412$	6.17157	0.304052
$413$	4.00000	0.196827
$414$	1.58579	0.0779372
$415$	0	0
$416$	3.24264	0.158984
$417$	−6.58579	−0.322507
$418$	0.171573	0.00839190
$419$	−16.1421	−0.788595	−0.394297	−	0.918983i	$-0.629012\pi$
$419$	−16.1421	−0.788595	−0.394297	+	0.918983i	$0.629012\pi$
$420$	0	0
$421$	−7.07107	−0.344623	−0.172311	−	0.985043i	$-0.555124\pi$
$421$	−7.07107	−0.344623	−0.172311	+	0.985043i	$0.555124\pi$
$422$	13.6569	0.664805
$423$	6.48528	0.315325
$424$	−12.2426	−0.594555
$425$	0	0
$426$	−14.2426	−0.690058
$427$	−12.8284	−0.620811
$428$	−2.07107	−0.100109
$429$	4.58579	0.221404
$430$	0	0
$431$	9.17157	0.441779	0.220890	−	0.975299i	$-0.429104\pi$
$431$	9.17157	0.441779	0.220890	+	0.975299i	$0.429104\pi$
$432$	−5.65685	−0.272166
$433$	−7.58579	−0.364550	−0.182275	−	0.983248i	$-0.558346\pi$
$433$	−7.58579	−0.364550	−0.182275	+	0.983248i	$0.558346\pi$
$434$	0.171573	0.00823576
$435$	0	0
$436$	3.00000	0.143674
$437$	0.272078	0.0130153
$438$	−21.6569	−1.03480
$439$	4.48528	0.214071	0.107035	−	0.994255i	$-0.465864\pi$
$439$	4.48528	0.214071	0.107035	+	0.994255i	$0.465864\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	8.38478	0.398823
$443$	−3.31371	−0.157439	−0.0787195	−	0.996897i	$-0.525083\pi$
$443$	−3.31371	−0.157439	−0.0787195	+	0.996897i	$0.525083\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	20.1421	0.953758
$447$	23.3137	1.10270
$448$	1.00000	0.0472456
$449$	13.9706	0.659312	0.329656	−	0.944101i	$-0.393067\pi$
$449$	13.9706	0.659312	0.329656	+	0.944101i	$0.393067\pi$
$450$	0	0
$451$	6.82843	0.321538
$452$	12.4853	0.587258
$453$	16.3431	0.767868
$454$	−21.0000	−0.985579
$455$	0	0
$456$	−0.242641	−0.0113627
$457$	−19.6985	−0.921456	−0.460728	−	0.887541i	$-0.652412\pi$
$457$	−19.6985	−0.921456	−0.460728	+	0.887541i	$0.652412\pi$
$458$	1.31371	0.0613856
$459$	−14.6274	−0.682749
$460$	0	0
$461$	14.8284	0.690629	0.345314	−	0.938487i	$-0.387772\pi$
$461$	14.8284	0.690629	0.345314	+	0.938487i	$0.387772\pi$
$462$	1.41421	0.0657952
$463$	−30.6985	−1.42668	−0.713340	−	0.700818i	$-0.752818\pi$
$463$	−30.6985	−1.42668	−0.713340	+	0.700818i	$0.752818\pi$
$464$	8.65685	0.401884
$465$	0	0
$466$	−19.7574	−0.915242
$467$	−16.3431	−0.756271	−0.378135	−	0.925750i	$-0.623435\pi$
$467$	−16.3431	−0.756271	−0.378135	+	0.925750i	$0.623435\pi$
$468$	3.24264	0.149891
$469$	3.41421	0.157654
$470$	0	0
$471$	12.8284	0.591103
$472$	−4.00000	−0.184115
$473$	−10.8995	−0.501159
$474$	−15.6569	−0.719143
$475$	0	0
$476$	2.58579	0.118519
$477$	−12.2426	−0.560552
$478$	−22.4853	−1.02845
$479$	13.7990	0.630492	0.315246	−	0.949010i	$-0.397913\pi$
$479$	13.7990	0.630492	0.315246	+	0.949010i	$0.397913\pi$
$480$	0	0
$481$	−9.17157	−0.418188
$482$	−23.6569	−1.07754
$483$	2.24264	0.102044
$484$	1.00000	0.0454545
$485$	0	0
$486$	−9.89949	−0.449050
$487$	−34.4142	−1.55946	−0.779728	−	0.626118i	$-0.784643\pi$
$487$	−34.4142	−1.55946	−0.779728	+	0.626118i	$0.784643\pi$
$488$	12.8284	0.580716
$489$	−28.9706	−1.31009
$490$	0	0
$491$	40.5563	1.83028	0.915141	−	0.403133i	$-0.132079\pi$
$491$	40.5563	1.83028	0.915141	+	0.403133i	$0.132079\pi$
$492$	−9.65685	−0.435365
$493$	22.3848	1.00816
$494$	0.556349	0.0250313
$495$	0	0
$496$	−0.171573	−0.00770385
$497$	10.0711	0.451749
$498$	−19.5563	−0.876341
$499$	26.4853	1.18564	0.592822	−	0.805334i	$-0.298014\pi$
$499$	26.4853	1.18564	0.592822	+	0.805334i	$0.298014\pi$
$500$	0	0
$501$	32.1421	1.43600
$502$	16.5858	0.740260
$503$	−19.6569	−0.876456	−0.438228	−	0.898864i	$-0.644394\pi$
$503$	−19.6569	−0.876456	−0.438228	+	0.898864i	$0.644394\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	1.58579	0.0704968
$507$	−3.51472	−0.156094
$508$	−10.8284	−0.480434
$509$	4.10051	0.181752	0.0908758	−	0.995862i	$-0.471033\pi$
$509$	4.10051	0.181752	0.0908758	+	0.995862i	$0.471033\pi$
$510$	0	0
$511$	15.3137	0.677439
$512$	−1.00000	−0.0441942
$513$	−0.970563	−0.0428514
$514$	0.414214	0.0182702
$515$	0	0
$516$	15.4142	0.678573
$517$	6.48528	0.285222
$518$	−2.82843	−0.124274
$519$	25.5563	1.12180
$520$	0	0
$521$	−21.1005	−0.924430	−0.462215	−	0.886768i	$-0.652945\pi$
$521$	−21.1005	−0.924430	−0.462215	+	0.886768i	$0.652945\pi$
$522$	8.65685	0.378900
$523$	20.7990	0.909476	0.454738	−	0.890625i	$-0.349733\pi$
$523$	20.7990	0.909476	0.454738	+	0.890625i	$0.349733\pi$
$524$	−2.17157	−0.0948656
$525$	0	0
$526$	−10.3848	−0.452798
$527$	−0.443651	−0.0193257
$528$	−1.41421	−0.0615457
$529$	−20.4853	−0.890664
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0.171573	0.00743863
$533$	22.1421	0.959082
$534$	9.75736	0.422242
$535$	0	0
$536$	−3.41421	−0.147472
$537$	−26.0000	−1.12198
$538$	−6.24264	−0.269139
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−28.6569	−1.23205	−0.616027	−	0.787725i	$-0.711259\pi$
$541$	−28.6569	−1.23205	−0.616027	+	0.787725i	$0.711259\pi$
$542$	12.7279	0.546711
$543$	0.686292	0.0294516
$544$	−2.58579	−0.110865
$545$	0	0
$546$	4.58579	0.196254
$547$	1.92893	0.0824752	0.0412376	−	0.999149i	$-0.486870\pi$
$547$	1.92893	0.0824752	0.0412376	+	0.999149i	$0.486870\pi$
$548$	6.17157	0.263637
$549$	12.8284	0.547504
$550$	0	0
$551$	1.48528	0.0632751
$552$	−2.24264	−0.0954531
$553$	11.0711	0.470790
$554$	14.6274	0.621459
$555$	0	0
$556$	−4.65685	−0.197495
$557$	21.4853	0.910361	0.455180	−	0.890399i	$-0.349575\pi$
$557$	21.4853	0.910361	0.455180	+	0.890399i	$0.349575\pi$
$558$	−0.171573	−0.00726326
$559$	−35.3431	−1.49486
$560$	0	0
$561$	−3.65685	−0.154393
$562$	−2.24264	−0.0946001
$563$	20.1421	0.848890	0.424445	−	0.905454i	$-0.360469\pi$
$563$	20.1421	0.848890	0.424445	+	0.905454i	$0.360469\pi$
$564$	−9.17157	−0.386193
$565$	0	0
$566$	13.7990	0.580015
$567$	−5.00000	−0.209980
$568$	−10.0711	−0.422573
$569$	32.8701	1.37798	0.688992	−	0.724769i	$-0.258053\pi$
$569$	32.8701	1.37798	0.688992	+	0.724769i	$0.258053\pi$
$570$	0	0
$571$	9.10051	0.380844	0.190422	−	0.981702i	$-0.439014\pi$
$571$	9.10051	0.380844	0.190422	+	0.981702i	$0.439014\pi$
$572$	3.24264	0.135582
$573$	−18.9289	−0.790767
$574$	6.82843	0.285013
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	32.8284	1.36667	0.683333	−	0.730107i	$-0.260530\pi$
$577$	32.8284	1.36667	0.683333	+	0.730107i	$0.260530\pi$
$578$	10.3137	0.428994
$579$	14.3431	0.596081
$580$	0	0
$581$	13.8284	0.573700
$582$	−2.92893	−0.121408
$583$	−12.2426	−0.507038
$584$	−15.3137	−0.633686
$585$	0	0
$586$	27.4558	1.13419
$587$	9.17157	0.378551	0.189276	−	0.981924i	$-0.439386\pi$
$587$	9.17157	0.378551	0.189276	+	0.981924i	$0.439386\pi$
$588$	1.41421	0.0583212
$589$	−0.0294373	−0.00121294
$590$	0	0
$591$	−29.2132	−1.20167
$592$	2.82843	0.116248
$593$	23.4142	0.961507	0.480753	−	0.876856i	$-0.340363\pi$
$593$	23.4142	0.961507	0.480753	+	0.876856i	$0.340363\pi$
$594$	−5.65685	−0.232104
$595$	0	0
$596$	16.4853	0.675263
$597$	−32.2426	−1.31960
$598$	5.14214	0.210278
$599$	30.1421	1.23157	0.615787	−	0.787913i	$-0.288838\pi$
$599$	30.1421	1.23157	0.615787	+	0.787913i	$0.288838\pi$
$600$	0	0
$601$	−30.6274	−1.24932	−0.624659	−	0.780897i	$-0.714762\pi$
$601$	−30.6274	−1.24932	−0.624659	+	0.780897i	$0.714762\pi$
$602$	−10.8995	−0.444230
$603$	−3.41421	−0.139038
$604$	11.5563	0.470221
$605$	0	0
$606$	13.0711	0.530976
$607$	−16.6274	−0.674886	−0.337443	−	0.941346i	$-0.609562\pi$
$607$	−16.6274	−0.674886	−0.337443	+	0.941346i	$0.609562\pi$
$608$	−0.171573	−0.00695820
$609$	12.2426	0.496097
$610$	0	0
$611$	21.0294	0.850760
$612$	−2.58579	−0.104524
$613$	4.97056	0.200759	0.100380	−	0.994949i	$-0.467994\pi$
$613$	4.97056	0.200759	0.100380	+	0.994949i	$0.467994\pi$
$614$	0.970563	0.0391687
$615$	0	0
$616$	1.00000	0.0402911
$617$	−19.6863	−0.792540	−0.396270	−	0.918134i	$-0.629696\pi$
$617$	−19.6863	−0.792540	−0.396270	+	0.918134i	$0.629696\pi$
$618$	−8.72792	−0.351089
$619$	7.21320	0.289923	0.144962	−	0.989437i	$-0.453694\pi$
$619$	7.21320	0.289923	0.144962	+	0.989437i	$0.453694\pi$
$620$	0	0
$621$	−8.97056	−0.359976
$622$	−21.9706	−0.880939
$623$	−6.89949	−0.276422
$624$	−4.58579	−0.183578
$625$	0	0
$626$	21.1716	0.846186
$627$	−0.242641	−0.00969014
$628$	9.07107	0.361975
$629$	7.31371	0.291617
$630$	0	0
$631$	−4.62742	−0.184215	−0.0921073	−	0.995749i	$-0.529360\pi$
$631$	−4.62742	−0.184215	−0.0921073	+	0.995749i	$0.529360\pi$
$632$	−11.0711	−0.440383
$633$	−19.3137	−0.767651
$634$	−18.0416	−0.716525
$635$	0	0
$636$	17.3137	0.686533
$637$	−3.24264	−0.128478
$638$	8.65685	0.342728
$639$	−10.0711	−0.398405
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	2.92893	0.115596
$643$	6.04163	0.238259	0.119129	−	0.992879i	$-0.461990\pi$
$643$	6.04163	0.238259	0.119129	+	0.992879i	$0.461990\pi$
$644$	1.58579	0.0624887
$645$	0	0
$646$	−0.443651	−0.0174552
$647$	2.20101	0.0865306	0.0432653	−	0.999064i	$-0.486224\pi$
$647$	2.20101	0.0865306	0.0432653	+	0.999064i	$0.486224\pi$
$648$	5.00000	0.196419
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−0.242641	−0.00950984
$652$	−20.4853	−0.802266
$653$	16.2426	0.635624	0.317812	−	0.948154i	$-0.397052\pi$
$653$	16.2426	0.635624	0.317812	+	0.948154i	$0.397052\pi$
$654$	−4.24264	−0.165900
$655$	0	0
$656$	−6.82843	−0.266605
$657$	−15.3137	−0.597445
$658$	6.48528	0.252823
$659$	−32.4142	−1.26268	−0.631339	−	0.775507i	$-0.717494\pi$
$659$	−32.4142	−1.26268	−0.631339	+	0.775507i	$0.717494\pi$
$660$	0	0
$661$	12.7279	0.495059	0.247529	−	0.968880i	$-0.420381\pi$
$661$	12.7279	0.495059	0.247529	+	0.968880i	$0.420381\pi$
$662$	−8.48528	−0.329790
$663$	−11.8579	−0.460521
$664$	−13.8284	−0.536647
$665$	0	0
$666$	2.82843	0.109599
$667$	13.7279	0.531547
$668$	22.7279	0.879370
$669$	−28.4853	−1.10130
$670$	0	0
$671$	12.8284	0.495236
$672$	−1.41421	−0.0545545
$673$	1.65685	0.0638670	0.0319335	−	0.999490i	$-0.489834\pi$
$673$	1.65685	0.0638670	0.0319335	+	0.999490i	$0.489834\pi$
$674$	−13.4142	−0.516696
$675$	0	0
$676$	−2.48528	−0.0955877
$677$	−40.2132	−1.54552	−0.772759	−	0.634699i	$-0.781124\pi$
$677$	−40.2132	−1.54552	−0.772759	+	0.634699i	$0.781124\pi$
$678$	−17.6569	−0.678107
$679$	2.07107	0.0794803
$680$	0	0
$681$	29.6985	1.13805
$682$	−0.171573	−0.00656986
$683$	29.8995	1.14407	0.572036	−	0.820228i	$-0.306154\pi$
$683$	29.8995	1.14407	0.572036	+	0.820228i	$0.306154\pi$
$684$	−0.171573	−0.00656025
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−1.85786	−0.0708819
$688$	10.8995	0.415539
$689$	−39.6985	−1.51239
$690$	0	0
$691$	4.48528	0.170628	0.0853141	−	0.996354i	$-0.472811\pi$
$691$	4.48528	0.170628	0.0853141	+	0.996354i	$0.472811\pi$
$692$	18.0711	0.686959
$693$	1.00000	0.0379869
$694$	−20.4853	−0.777611
$695$	0	0
$696$	−12.2426	−0.464056
$697$	−17.6569	−0.668801
$698$	−10.2132	−0.386575
$699$	27.9411	1.05683
$700$	0	0
$701$	−18.6569	−0.704660	−0.352330	−	0.935876i	$-0.614610\pi$
$701$	−18.6569	−0.704660	−0.352330	+	0.935876i	$0.614610\pi$
$702$	−18.3431	−0.692317
$703$	0.485281	0.0183027
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−5.58579	−0.210224
$707$	−9.24264	−0.347605
$708$	5.65685	0.212598
$709$	49.5563	1.86113	0.930564	−	0.366130i	$-0.119317\pi$
$709$	49.5563	1.86113	0.930564	+	0.366130i	$0.119317\pi$
$710$	0	0
$711$	−11.0711	−0.415197
$712$	6.89949	0.258570
$713$	−0.272078	−0.0101894
$714$	−3.65685	−0.136854
$715$	0	0
$716$	−18.3848	−0.687071
$717$	31.7990	1.18756
$718$	−3.02944	−0.113058
$719$	35.4558	1.32228	0.661140	−	0.750263i	$-0.270073\pi$
$719$	35.4558	1.32228	0.661140	+	0.750263i	$0.270073\pi$
$720$	0	0
$721$	6.17157	0.229841
$722$	18.9706	0.706011
$723$	33.4558	1.24424
$724$	0.485281	0.0180353
$725$	0	0
$726$	−1.41421	−0.0524864
$727$	2.85786	0.105992	0.0529962	−	0.998595i	$-0.483123\pi$
$727$	2.85786	0.105992	0.0529962	+	0.998595i	$0.483123\pi$
$728$	3.24264	0.120180
$729$	29.0000	1.07407
$730$	0	0
$731$	28.1838	1.04241
$732$	−18.1421	−0.670553
$733$	−21.2426	−0.784615	−0.392307	−	0.919834i	$-0.628323\pi$
$733$	−21.2426	−0.784615	−0.392307	+	0.919834i	$0.628323\pi$
$734$	14.3137	0.528329
$735$	0	0
$736$	−1.58579	−0.0584529
$737$	−3.41421	−0.125764
$738$	−6.82843	−0.251358
$739$	32.7696	1.20545	0.602724	−	0.797950i	$-0.294082\pi$
$739$	32.7696	1.20545	0.602724	+	0.797950i	$0.294082\pi$
$740$	0	0
$741$	−0.786797	−0.0289037
$742$	−12.2426	−0.449441
$743$	−6.24264	−0.229020	−0.114510	−	0.993422i	$-0.536530\pi$
$743$	−6.24264	−0.229020	−0.114510	+	0.993422i	$0.536530\pi$
$744$	0.242641	0.00889564
$745$	0	0
$746$	−5.85786	−0.214472
$747$	−13.8284	−0.505956
$748$	−2.58579	−0.0945457
$749$	−2.07107	−0.0756752
$750$	0	0
$751$	11.4437	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$751$	11.4437	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$752$	−6.48528	−0.236494
$753$	−23.4558	−0.854778
$754$	28.0711	1.02229
$755$	0	0
$756$	−5.65685	−0.205738
$757$	−45.2548	−1.64481	−0.822407	−	0.568899i	$-0.807370\pi$
$757$	−45.2548	−1.64481	−0.822407	+	0.568899i	$0.807370\pi$
$758$	−9.27208	−0.336777
$759$	−2.24264	−0.0814027
$760$	0	0
$761$	−33.2132	−1.20398	−0.601989	−	0.798504i	$-0.705625\pi$
$761$	−33.2132	−1.20398	−0.601989	+	0.798504i	$0.705625\pi$
$762$	15.3137	0.554757
$763$	3.00000	0.108607
$764$	−13.3848	−0.484244
$765$	0	0
$766$	−11.1421	−0.402582
$767$	−12.9706	−0.468340
$768$	1.41421	0.0510310
$769$	−46.2426	−1.66755	−0.833776	−	0.552103i	$-0.813826\pi$
$769$	−46.2426	−1.66755	−0.833776	+	0.552103i	$0.813826\pi$
$770$	0	0
$771$	−0.585786	−0.0210966
$772$	10.1421	0.365023
$773$	−41.2548	−1.48383	−0.741917	−	0.670492i	$-0.766083\pi$
$773$	−41.2548	−1.48383	−0.741917	+	0.670492i	$0.766083\pi$
$774$	10.8995	0.391774
$775$	0	0
$776$	−2.07107	−0.0743470
$777$	4.00000	0.143499
$778$	17.7574	0.636632
$779$	−1.17157	−0.0419760
$780$	0	0
$781$	−10.0711	−0.360371
$782$	−4.10051	−0.146634
$783$	−48.9706	−1.75007
$784$	1.00000	0.0357143
$785$	0	0
$786$	3.07107	0.109541
$787$	46.7696	1.66715	0.833577	−	0.552403i	$-0.186289\pi$
$787$	46.7696	1.66715	0.833577	+	0.552403i	$0.186289\pi$
$788$	−20.6569	−0.735870
$789$	14.6863	0.522846
$790$	0	0
$791$	12.4853	0.443925
$792$	−1.00000	−0.0355335
$793$	41.5980	1.47719
$794$	−31.4558	−1.11633
$795$	0	0
$796$	−22.7990	−0.808089
$797$	−27.0711	−0.958906	−0.479453	−	0.877567i	$-0.659165\pi$
$797$	−27.0711	−0.958906	−0.479453	+	0.877567i	$0.659165\pi$
$798$	−0.242641	−0.00858939
$799$	−16.7696	−0.593264
$800$	0	0
$801$	6.89949	0.243782
$802$	−4.51472	−0.159420
$803$	−15.3137	−0.540409
$804$	4.82843	0.170285
$805$	0	0
$806$	−0.556349	−0.0195966
$807$	8.82843	0.310775
$808$	9.24264	0.325155
$809$	−13.4142	−0.471619	−0.235809	−	0.971799i	$-0.575774\pi$
$809$	−13.4142	−0.471619	−0.235809	+	0.971799i	$0.575774\pi$
$810$	0	0
$811$	40.1421	1.40958	0.704791	−	0.709415i	$-0.251041\pi$
$811$	40.1421	1.40958	0.704791	+	0.709415i	$0.251041\pi$
$812$	8.65685	0.303796
$813$	−18.0000	−0.631288
$814$	2.82843	0.0991363
$815$	0	0
$816$	3.65685	0.128016
$817$	1.87006	0.0654250
$818$	5.41421	0.189304
$819$	3.24264	0.113307
$820$	0	0
$821$	−31.7696	−1.10877	−0.554383	−	0.832262i	$-0.687046\pi$
$821$	−31.7696	−1.10877	−0.554383	+	0.832262i	$0.687046\pi$
$822$	−8.72792	−0.304421
$823$	−17.6569	−0.615479	−0.307740	−	0.951471i	$-0.599573\pi$
$823$	−17.6569	−0.615479	−0.307740	+	0.951471i	$0.599573\pi$
$824$	−6.17157	−0.214997
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−50.8995	−1.76995	−0.884974	−	0.465640i	$-0.845824\pi$
$827$	−50.8995	−1.76995	−0.884974	+	0.465640i	$0.845824\pi$
$828$	−1.58579	−0.0551099
$829$	7.37258	0.256060	0.128030	−	0.991770i	$-0.459135\pi$
$829$	7.37258	0.256060	0.128030	+	0.991770i	$0.459135\pi$
$830$	0	0
$831$	−20.6863	−0.717600
$832$	−3.24264	−0.112418
$833$	2.58579	0.0895922
$834$	6.58579	0.228047
$835$	0	0
$836$	−0.171573	−0.00593397
$837$	0.970563	0.0335476
$838$	16.1421	0.557621
$839$	−36.8284	−1.27146	−0.635729	−	0.771912i	$-0.719301\pi$
$839$	−36.8284	−1.27146	−0.635729	+	0.771912i	$0.719301\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	7.07107	0.243685
$843$	3.17157	0.109235
$844$	−13.6569	−0.470088
$845$	0	0
$846$	−6.48528	−0.222969
$847$	1.00000	0.0343604
$848$	12.2426	0.420414
$849$	−19.5147	−0.669743
$850$	0	0
$851$	4.48528	0.153753
$852$	14.2426	0.487945
$853$	−48.1421	−1.64836	−0.824178	−	0.566331i	$-0.808362\pi$
$853$	−48.1421	−1.64836	−0.824178	+	0.566331i	$0.808362\pi$
$854$	12.8284	0.438980
$855$	0	0
$856$	2.07107	0.0707876
$857$	−49.2132	−1.68109	−0.840546	−	0.541741i	$-0.817765\pi$
$857$	−49.2132	−1.68109	−0.840546	+	0.541741i	$0.817765\pi$
$858$	−4.58579	−0.156556
$859$	17.4142	0.594165	0.297083	−	0.954852i	$-0.403986\pi$
$859$	17.4142	0.594165	0.297083	+	0.954852i	$0.403986\pi$
$860$	0	0
$861$	−9.65685	−0.329105
$862$	−9.17157	−0.312385
$863$	2.61522	0.0890232	0.0445116	−	0.999009i	$-0.485827\pi$
$863$	2.61522	0.0890232	0.0445116	+	0.999009i	$0.485827\pi$
$864$	5.65685	0.192450
$865$	0	0
$866$	7.58579	0.257776
$867$	−14.5858	−0.495359
$868$	−0.171573	−0.00582356
$869$	−11.0711	−0.375560
$870$	0	0
$871$	−11.0711	−0.375129
$872$	−3.00000	−0.101593
$873$	−2.07107	−0.0700950
$874$	−0.272078	−0.00920317
$875$	0	0
$876$	21.6569	0.731717
$877$	−43.4853	−1.46839	−0.734197	−	0.678937i	$-0.762441\pi$
$877$	−43.4853	−1.46839	−0.734197	+	0.678937i	$0.762441\pi$
$878$	−4.48528	−0.151371
$879$	−38.8284	−1.30965
$880$	0	0
$881$	−23.3848	−0.787853	−0.393927	−	0.919142i	$-0.628884\pi$
$881$	−23.3848	−0.787853	−0.393927	+	0.919142i	$0.628884\pi$
$882$	1.00000	0.0336718
$883$	−43.1127	−1.45086	−0.725429	−	0.688297i	$-0.758359\pi$
$883$	−43.1127	−1.45086	−0.725429	+	0.688297i	$0.758359\pi$
$884$	−8.38478	−0.282011
$885$	0	0
$886$	3.31371	0.111326
$887$	18.6863	0.627424	0.313712	−	0.949518i	$-0.398427\pi$
$887$	18.6863	0.627424	0.313712	+	0.949518i	$0.398427\pi$
$888$	−4.00000	−0.134231
$889$	−10.8284	−0.363174
$890$	0	0
$891$	5.00000	0.167506
$892$	−20.1421	−0.674409
$893$	−1.11270	−0.0372350
$894$	−23.3137	−0.779727
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−7.27208	−0.242808
$898$	−13.9706	−0.466204
$899$	−1.48528	−0.0495369
$900$	0	0
$901$	31.6569	1.05464
$902$	−6.82843	−0.227362
$903$	15.4142	0.512953
$904$	−12.4853	−0.415254
$905$	0	0
$906$	−16.3431	−0.542965
$907$	21.5563	0.715767	0.357883	−	0.933766i	$-0.383499\pi$
$907$	21.5563	0.715767	0.357883	+	0.933766i	$0.383499\pi$
$908$	21.0000	0.696909
$909$	9.24264	0.306559
$910$	0	0
$911$	−10.6863	−0.354053	−0.177026	−	0.984206i	$-0.556648\pi$
$911$	−10.6863	−0.354053	−0.177026	+	0.984206i	$0.556648\pi$
$912$	0.242641	0.00803464
$913$	−13.8284	−0.457654
$914$	19.6985	0.651568
$915$	0	0
$916$	−1.31371	−0.0434062
$917$	−2.17157	−0.0717117
$918$	14.6274	0.482777
$919$	−50.6274	−1.67004	−0.835022	−	0.550216i	$-0.814545\pi$
$919$	−50.6274	−1.67004	−0.835022	+	0.550216i	$0.814545\pi$
$920$	0	0
$921$	−1.37258	−0.0452281
$922$	−14.8284	−0.488348
$923$	−32.6569	−1.07491
$924$	−1.41421	−0.0465242
$925$	0	0
$926$	30.6985	1.00881
$927$	−6.17157	−0.202701
$928$	−8.65685	−0.284175
$929$	−14.7574	−0.484173	−0.242087	−	0.970255i	$-0.577832\pi$
$929$	−14.7574	−0.484173	−0.242087	+	0.970255i	$0.577832\pi$
$930$	0	0
$931$	0.171573	0.00562307
$932$	19.7574	0.647174
$933$	31.0711	1.01722
$934$	16.3431	0.534764
$935$	0	0
$936$	−3.24264	−0.105989
$937$	22.0416	0.720069	0.360034	−	0.932939i	$-0.382765\pi$
$937$	22.0416	0.720069	0.360034	+	0.932939i	$0.382765\pi$
$938$	−3.41421	−0.111478
$939$	−29.9411	−0.977092
$940$	0	0
$941$	−31.1716	−1.01616	−0.508082	−	0.861309i	$-0.669645\pi$
$941$	−31.1716	−1.01616	−0.508082	+	0.861309i	$0.669645\pi$
$942$	−12.8284	−0.417973
$943$	−10.8284	−0.352622
$944$	4.00000	0.130189
$945$	0	0
$946$	10.8995	0.354373
$947$	−28.8701	−0.938151	−0.469075	−	0.883158i	$-0.655413\pi$
$947$	−28.8701	−0.938151	−0.469075	+	0.883158i	$0.655413\pi$
$948$	15.6569	0.508511
$949$	−49.6569	−1.61193
$950$	0	0
$951$	25.5147	0.827371
$952$	−2.58579	−0.0838058
$953$	57.0122	1.84681	0.923403	−	0.383832i	$-0.125396\pi$
$953$	57.0122	1.84681	0.923403	+	0.383832i	$0.125396\pi$
$954$	12.2426	0.396370
$955$	0	0
$956$	22.4853	0.727226
$957$	−12.2426	−0.395748
$958$	−13.7990	−0.445825
$959$	6.17157	0.199290
$960$	0	0
$961$	−30.9706	−0.999050
$962$	9.17157	0.295703
$963$	2.07107	0.0667392
$964$	23.6569	0.761936
$965$	0	0
$966$	−2.24264	−0.0721558
$967$	−11.6152	−0.373520	−0.186760	−	0.982406i	$-0.559799\pi$
$967$	−11.6152	−0.373520	−0.186760	+	0.982406i	$0.559799\pi$
$968$	−1.00000	−0.0321412
$969$	0.627417	0.0201555
$970$	0	0
$971$	17.0294	0.546501	0.273250	−	0.961943i	$-0.411901\pi$
$971$	17.0294	0.546501	0.273250	+	0.961943i	$0.411901\pi$
$972$	9.89949	0.317526
$973$	−4.65685	−0.149292
$974$	34.4142	1.10270
$975$	0	0
$976$	−12.8284	−0.410628
$977$	2.62742	0.0840585	0.0420293	−	0.999116i	$-0.486618\pi$
$977$	2.62742	0.0840585	0.0420293	+	0.999116i	$0.486618\pi$
$978$	28.9706	0.926376
$979$	6.89949	0.220509
$980$	0	0
$981$	−3.00000	−0.0957826
$982$	−40.5563	−1.29421
$983$	3.34315	0.106630	0.0533149	−	0.998578i	$-0.483021\pi$
$983$	3.34315	0.106630	0.0533149	+	0.998578i	$0.483021\pi$
$984$	9.65685	0.307849
$985$	0	0
$986$	−22.3848	−0.712877
$987$	−9.17157	−0.291934
$988$	−0.556349	−0.0176998
$989$	17.2843	0.549608
$990$	0	0
$991$	12.9706	0.412024	0.206012	−	0.978550i	$-0.433951\pi$
$991$	12.9706	0.412024	0.206012	+	0.978550i	$0.433951\pi$
$992$	0.171573	0.00544744
$993$	12.0000	0.380808
$994$	−10.0711	−0.319435
$995$	0	0
$996$	19.5563	0.619667
$997$	30.0000	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$998$	−26.4853	−0.838377
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bg.1.2	✓	2
5.2	odd	4		3850.2.c.w.1849.1		4
5.3	odd	4		3850.2.c.w.1849.4		4
5.4	even	2		3850.2.a.bn.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3850.2.a.bg.1.2	✓	2	1.1	even	1	trivial
3850.2.a.bn.1.1	yes	2	5.4	even	2
3850.2.c.w.1849.1		4	5.2	odd	4
3850.2.c.w.1849.4		4	5.3	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.41421 q^{12} -3.24264 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.58579 q^{17} +1.00000 q^{18} +0.171573 q^{19} +1.41421 q^{21} +1.00000 q^{22} +1.58579 q^{23} -1.41421 q^{24} +3.24264 q^{26} -5.65685 q^{27} +1.00000 q^{28} +8.65685 q^{29} -0.171573 q^{31} -1.00000 q^{32} -1.41421 q^{33} -2.58579 q^{34} -1.00000 q^{36} +2.82843 q^{37} -0.171573 q^{38} -4.58579 q^{39} -6.82843 q^{41} -1.41421 q^{42} +10.8995 q^{43} -1.00000 q^{44} -1.58579 q^{46} -6.48528 q^{47} +1.41421 q^{48} +1.00000 q^{49} +3.65685 q^{51} -3.24264 q^{52} +12.2426 q^{53} +5.65685 q^{54} -1.00000 q^{56} +0.242641 q^{57} -8.65685 q^{58} +4.00000 q^{59} -12.8284 q^{61} +0.171573 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.41421 q^{66} +3.41421 q^{67} +2.58579 q^{68} +2.24264 q^{69} +10.0711 q^{71} +1.00000 q^{72} +15.3137 q^{73} -2.82843 q^{74} +0.171573 q^{76} -1.00000 q^{77} +4.58579 q^{78} +11.0711 q^{79} -5.00000 q^{81} +6.82843 q^{82} +13.8284 q^{83} +1.41421 q^{84} -10.8995 q^{86} +12.2426 q^{87} +1.00000 q^{88} -6.89949 q^{89} -3.24264 q^{91} +1.58579 q^{92} -0.242641 q^{93} +6.48528 q^{94} -1.41421 q^{96} +2.07107 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 6 q^{19} + 2 q^{22} + 6 q^{23} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 6 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{36} - 6 q^{38} - 12 q^{39} - 8 q^{41} + 2 q^{43} - 2 q^{44} - 6 q^{46} + 4 q^{47} + 2 q^{49} - 4 q^{51} + 2 q^{52} + 16 q^{53} - 2 q^{56} - 8 q^{57} - 6 q^{58} + 8 q^{59} - 20 q^{61} + 6 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 4 q^{69} + 6 q^{71} + 2 q^{72} + 8 q^{73} + 6 q^{76} - 2 q^{77} + 12 q^{78} + 8 q^{79} - 10 q^{81} + 8 q^{82} + 22 q^{83} - 2 q^{86} + 16 q^{87} + 2 q^{88} + 6 q^{89} + 2 q^{91} + 6 q^{92} + 8 q^{93} - 4 q^{94} - 10 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 8 * q^17 + 2 * q^18 + 6 * q^19 + 2 * q^22 + 6 * q^23 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 6 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^36 - 6 * q^38 - 12 * q^39 - 8 * q^41 + 2 * q^43 - 2 * q^44 - 6 * q^46 + 4 * q^47 + 2 * q^49 - 4 * q^51 + 2 * q^52 + 16 * q^53 - 2 * q^56 - 8 * q^57 - 6 * q^58 + 8 * q^59 - 20 * q^61 + 6 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^67 + 8 * q^68 - 4 * q^69 + 6 * q^71 + 2 * q^72 + 8 * q^73 + 6 * q^76 - 2 * q^77 + 12 * q^78 + 8 * q^79 - 10 * q^81 + 8 * q^82 + 22 * q^83 - 2 * q^86 + 16 * q^87 + 2 * q^88 + 6 * q^89 + 2 * q^91 + 6 * q^92 + 8 * q^93 - 4 * q^94 - 10 * q^97 - 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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