LMFDB - Embedded newform 3850.2.a.bg.1.1

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			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.41421 q^{12} +5.24264 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.41421 q^{17} +1.00000 q^{18} +5.82843 q^{19} -1.41421 q^{21} +1.00000 q^{22} +4.41421 q^{23} +1.41421 q^{24} -5.24264 q^{26} +5.65685 q^{27} +1.00000 q^{28} -2.65685 q^{29} -5.82843 q^{31} -1.00000 q^{32} +1.41421 q^{33} -5.41421 q^{34} -1.00000 q^{36} -2.82843 q^{37} -5.82843 q^{38} -7.41421 q^{39} -1.17157 q^{41} +1.41421 q^{42} -8.89949 q^{43} -1.00000 q^{44} -4.41421 q^{46} +10.4853 q^{47} -1.41421 q^{48} +1.00000 q^{49} -7.65685 q^{51} +5.24264 q^{52} +3.75736 q^{53} -5.65685 q^{54} -1.00000 q^{56} -8.24264 q^{57} +2.65685 q^{58} +4.00000 q^{59} -7.17157 q^{61} +5.82843 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.41421 q^{66} +0.585786 q^{67} +5.41421 q^{68} -6.24264 q^{69} -4.07107 q^{71} +1.00000 q^{72} -7.31371 q^{73} +2.82843 q^{74} +5.82843 q^{76} -1.00000 q^{77} +7.41421 q^{78} -3.07107 q^{79} -5.00000 q^{81} +1.17157 q^{82} +8.17157 q^{83} -1.41421 q^{84} +8.89949 q^{86} +3.75736 q^{87} +1.00000 q^{88} +12.8995 q^{89} +5.24264 q^{91} +4.41421 q^{92} +8.24264 q^{93} -10.4853 q^{94} +1.41421 q^{96} -12.0711 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	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
$13$	5.24264	1.45405	0.727023	−	0.686613i	$-0.240903\pi$
$13$	5.24264	1.45405	0.727023	+	0.686613i	$0.240903\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.41421	1.31314	0.656570	−	0.754265i	$-0.272007\pi$
$17$	5.41421	1.31314	0.656570	+	0.754265i	$0.272007\pi$
$18$	1.00000	0.235702
$19$	5.82843	1.33713	0.668566	−	0.743652i	$-0.266908\pi$
$19$	5.82843	1.33713	0.668566	+	0.743652i	$0.266908\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	1.00000	0.213201
$23$	4.41421	0.920427	0.460214	−	0.887808i	$-0.347773\pi$
$23$	4.41421	0.920427	0.460214	+	0.887808i	$0.347773\pi$
$24$	1.41421	0.288675
$25$	0	0
$26$	−5.24264	−1.02817
$27$	5.65685	1.08866
$28$	1.00000	0.188982
$29$	−2.65685	−0.493365	−0.246683	−	0.969096i	$-0.579341\pi$
$29$	−2.65685	−0.493365	−0.246683	+	0.969096i	$0.579341\pi$
$30$	0	0
$31$	−5.82843	−1.04682	−0.523408	−	0.852082i	$-0.675340\pi$
$31$	−5.82843	−1.04682	−0.523408	+	0.852082i	$0.675340\pi$
$32$	−1.00000	−0.176777
$33$	1.41421	0.246183
$34$	−5.41421	−0.928530
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.82843	−0.464991	−0.232495	−	0.972598i	$-0.574689\pi$
$37$	−2.82843	−0.464991	−0.232495	+	0.972598i	$0.574689\pi$
$38$	−5.82843	−0.945496
$39$	−7.41421	−1.18722
$40$	0	0
$41$	−1.17157	−0.182969	−0.0914845	−	0.995807i	$-0.529161\pi$
$41$	−1.17157	−0.182969	−0.0914845	+	0.995807i	$0.529161\pi$
$42$	1.41421	0.218218
$43$	−8.89949	−1.35716	−0.678580	−	0.734526i	$-0.737404\pi$
$43$	−8.89949	−1.35716	−0.678580	+	0.734526i	$0.737404\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−4.41421	−0.650840
$47$	10.4853	1.52944	0.764718	−	0.644365i	$-0.222878\pi$
$47$	10.4853	1.52944	0.764718	+	0.644365i	$0.222878\pi$
$48$	−1.41421	−0.204124
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.65685	−1.07217
$52$	5.24264	0.727023
$53$	3.75736	0.516113	0.258056	−	0.966130i	$-0.416918\pi$
$53$	3.75736	0.516113	0.258056	+	0.966130i	$0.416918\pi$
$54$	−5.65685	−0.769800
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−8.24264	−1.09176
$58$	2.65685	0.348862
$59$	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
$61$	−7.17157	−0.918226	−0.459113	−	0.888378i	$-0.651833\pi$
$61$	−7.17157	−0.918226	−0.459113	+	0.888378i	$0.651833\pi$
$62$	5.82843	0.740211
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.41421	−0.174078
$67$	0.585786	0.0715652	0.0357826	−	0.999360i	$-0.488608\pi$
$67$	0.585786	0.0715652	0.0357826	+	0.999360i	$0.488608\pi$
$68$	5.41421	0.656570
$69$	−6.24264	−0.751526
$70$	0	0
$71$	−4.07107	−0.483147	−0.241573	−	0.970383i	$-0.577663\pi$
$71$	−4.07107	−0.483147	−0.241573	+	0.970383i	$0.577663\pi$
$72$	1.00000	0.117851
$73$	−7.31371	−0.856005	−0.428002	−	0.903778i	$-0.640783\pi$
$73$	−7.31371	−0.856005	−0.428002	+	0.903778i	$0.640783\pi$
$74$	2.82843	0.328798
$75$	0	0
$76$	5.82843	0.668566
$77$	−1.00000	−0.113961
$78$	7.41421	0.839494
$79$	−3.07107	−0.345522	−0.172761	−	0.984964i	$-0.555269\pi$
$79$	−3.07107	−0.345522	−0.172761	+	0.984964i	$0.555269\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	1.17157	0.129379
$83$	8.17157	0.896947	0.448473	−	0.893796i	$-0.351968\pi$
$83$	8.17157	0.896947	0.448473	+	0.893796i	$0.351968\pi$
$84$	−1.41421	−0.154303
$85$	0	0
$86$	8.89949	0.959657
$87$	3.75736	0.402831
$88$	1.00000	0.106600
$89$	12.8995	1.36734	0.683672	−	0.729790i	$-0.260382\pi$
$89$	12.8995	1.36734	0.683672	+	0.729790i	$0.260382\pi$
$90$	0	0
$91$	5.24264	0.549578
$92$	4.41421	0.460214
$93$	8.24264	0.854722
$94$	−10.4853	−1.08147
$95$	0	0
$96$	1.41421	0.144338
$97$	−12.0711	−1.22563	−0.612816	−	0.790226i	$-0.709963\pi$
$97$	−12.0711	−1.22563	−0.612816	+	0.790226i	$0.709963\pi$
$98$	−1.00000	−0.101015
$99$	1.00000	0.100504
$100$	0	0
$101$	−0.757359	−0.0753601	−0.0376800	−	0.999290i	$-0.511997\pi$
$101$	−0.757359	−0.0753601	−0.0376800	+	0.999290i	$0.511997\pi$
$102$	7.65685	0.758142
$103$	11.8284	1.16549	0.582745	−	0.812655i	$-0.301979\pi$
$103$	11.8284	1.16549	0.582745	+	0.812655i	$0.301979\pi$
$104$	−5.24264	−0.514083
$105$	0	0
$106$	−3.75736	−0.364947
$107$	12.0711	1.16695	0.583477	−	0.812130i	$-0.301692\pi$
$107$	12.0711	1.16695	0.583477	+	0.812130i	$0.301692\pi$
$108$	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
$113$	−4.48528	−0.421940	−0.210970	−	0.977493i	$-0.567662\pi$
$113$	−4.48528	−0.421940	−0.210970	+	0.977493i	$0.567662\pi$
$114$	8.24264	0.771994
$115$	0	0
$116$	−2.65685	−0.246683
$117$	−5.24264	−0.484682
$118$	−4.00000	−0.368230
$119$	5.41421	0.496320
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.17157	0.649284
$123$	1.65685	0.149394
$124$	−5.82843	−0.523408
$125$	0	0
$126$	1.00000	0.0890871
$127$	−5.17157	−0.458903	−0.229451	−	0.973320i	$-0.573693\pi$
$127$	−5.17157	−0.458903	−0.229451	+	0.973320i	$0.573693\pi$
$128$	−1.00000	−0.0883883
$129$	12.5858	1.10812
$130$	0	0
$131$	−7.82843	−0.683973	−0.341986	−	0.939705i	$-0.611100\pi$
$131$	−7.82843	−0.683973	−0.341986	+	0.939705i	$0.611100\pi$
$132$	1.41421	0.123091
$133$	5.82843	0.505389
$134$	−0.585786	−0.0506042
$135$	0	0
$136$	−5.41421	−0.464265
$137$	11.8284	1.01057	0.505285	−	0.862952i	$-0.331387\pi$
$137$	11.8284	1.01057	0.505285	+	0.862952i	$0.331387\pi$
$138$	6.24264	0.531409
$139$	6.65685	0.564627	0.282314	−	0.959322i	$-0.408898\pi$
$139$	6.65685	0.564627	0.282314	+	0.959322i	$0.408898\pi$
$140$	0	0
$141$	−14.8284	−1.24878
$142$	4.07107	0.341636
$143$	−5.24264	−0.438412
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	7.31371	0.605287
$147$	−1.41421	−0.116642
$148$	−2.82843	−0.232495
$149$	−0.485281	−0.0397558	−0.0198779	−	0.999802i	$-0.506328\pi$
$149$	−0.485281	−0.0397558	−0.0198779	+	0.999802i	$0.506328\pi$
$150$	0	0
$151$	−19.5563	−1.59147	−0.795737	−	0.605643i	$-0.792916\pi$
$151$	−19.5563	−1.59147	−0.795737	+	0.605643i	$0.792916\pi$
$152$	−5.82843	−0.472748
$153$	−5.41421	−0.437713
$154$	1.00000	0.0805823
$155$	0	0
$156$	−7.41421	−0.593612
$157$	−5.07107	−0.404715	−0.202358	−	0.979312i	$-0.564860\pi$
$157$	−5.07107	−0.404715	−0.202358	+	0.979312i	$0.564860\pi$
$158$	3.07107	0.244321
$159$	−5.31371	−0.421404
$160$	0	0
$161$	4.41421	0.347889
$162$	5.00000	0.392837
$163$	−3.51472	−0.275294	−0.137647	−	0.990481i	$-0.543954\pi$
$163$	−3.51472	−0.275294	−0.137647	+	0.990481i	$0.543954\pi$
$164$	−1.17157	−0.0914845
$165$	0	0
$166$	−8.17157	−0.634237
$167$	−2.72792	−0.211093	−0.105546	−	0.994414i	$-0.533659\pi$
$167$	−2.72792	−0.211093	−0.105546	+	0.994414i	$0.533659\pi$
$168$	1.41421	0.109109
$169$	14.4853	1.11425
$170$	0	0
$171$	−5.82843	−0.445711
$172$	−8.89949	−0.678580
$173$	3.92893	0.298711	0.149356	−	0.988784i	$-0.452280\pi$
$173$	3.92893	0.298711	0.149356	+	0.988784i	$0.452280\pi$
$174$	−3.75736	−0.284845
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−5.65685	−0.425195
$178$	−12.8995	−0.966858
$179$	18.3848	1.37414	0.687071	−	0.726590i	$-0.258896\pi$
$179$	18.3848	1.37414	0.687071	+	0.726590i	$0.258896\pi$
$180$	0	0
$181$	−16.4853	−1.22534	−0.612671	−	0.790338i	$-0.709905\pi$
$181$	−16.4853	−1.22534	−0.612671	+	0.790338i	$0.709905\pi$
$182$	−5.24264	−0.388610
$183$	10.1421	0.749728
$184$	−4.41421	−0.325420
$185$	0	0
$186$	−8.24264	−0.604380
$187$	−5.41421	−0.395927
$188$	10.4853	0.764718
$189$	5.65685	0.411476
$190$	0	0
$191$	23.3848	1.69206	0.846032	−	0.533133i	$-0.178985\pi$
$191$	23.3848	1.69206	0.846032	+	0.533133i	$0.178985\pi$
$192$	−1.41421	−0.102062
$193$	−18.1421	−1.30590	−0.652950	−	0.757401i	$-0.726469\pi$
$193$	−18.1421	−1.30590	−0.652950	+	0.757401i	$0.726469\pi$
$194$	12.0711	0.866652
$195$	0	0
$196$	1.00000	0.0714286
$197$	−9.34315	−0.665672	−0.332836	−	0.942985i	$-0.608005\pi$
$197$	−9.34315	−0.665672	−0.332836	+	0.942985i	$0.608005\pi$
$198$	−1.00000	−0.0710669
$199$	16.7990	1.19085	0.595424	−	0.803411i	$-0.296984\pi$
$199$	16.7990	1.19085	0.595424	+	0.803411i	$0.296984\pi$
$200$	0	0
$201$	−0.828427	−0.0584327
$202$	0.757359	0.0532876
$203$	−2.65685	−0.186475
$204$	−7.65685	−0.536087
$205$	0	0
$206$	−11.8284	−0.824126
$207$	−4.41421	−0.306809
$208$	5.24264	0.363512
$209$	−5.82843	−0.403161
$210$	0	0
$211$	−2.34315	−0.161309	−0.0806544	−	0.996742i	$-0.525701\pi$
$211$	−2.34315	−0.161309	−0.0806544	+	0.996742i	$0.525701\pi$
$212$	3.75736	0.258056
$213$	5.75736	0.394488
$214$	−12.0711	−0.825161
$215$	0	0
$216$	−5.65685	−0.384900
$217$	−5.82843	−0.395659
$218$	−3.00000	−0.203186
$219$	10.3431	0.698925
$220$	0	0
$221$	28.3848	1.90937
$222$	−4.00000	−0.268462
$223$	8.14214	0.545238	0.272619	−	0.962122i	$-0.412110\pi$
$223$	8.14214	0.545238	0.272619	+	0.962122i	$0.412110\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	4.48528	0.298356
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	−8.24264	−0.545882
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	0	0
$231$	1.41421	0.0930484
$232$	2.65685	0.174431
$233$	28.2426	1.85024	0.925118	−	0.379679i	$-0.123965\pi$
$233$	28.2426	1.85024	0.925118	+	0.379679i	$0.123965\pi$
$234$	5.24264	0.342722
$235$	0	0
$236$	4.00000	0.260378
$237$	4.34315	0.282118
$238$	−5.41421	−0.350951
$239$	5.51472	0.356717	0.178359	−	0.983966i	$-0.442921\pi$
$239$	5.51472	0.356717	0.178359	+	0.983966i	$0.442921\pi$
$240$	0	0
$241$	12.3431	0.795092	0.397546	−	0.917582i	$-0.369862\pi$
$241$	12.3431	0.795092	0.397546	+	0.917582i	$0.369862\pi$
$242$	−1.00000	−0.0642824
$243$	−9.89949	−0.635053
$244$	−7.17157	−0.459113
$245$	0	0
$246$	−1.65685	−0.105637
$247$	30.5563	1.94425
$248$	5.82843	0.370105
$249$	−11.5563	−0.732354
$250$	0	0
$251$	−19.4142	−1.22541	−0.612707	−	0.790310i	$-0.709919\pi$
$251$	−19.4142	−1.22541	−0.612707	+	0.790310i	$0.709919\pi$
$252$	−1.00000	−0.0629941
$253$	−4.41421	−0.277519
$254$	5.17157	0.324493
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.41421	0.150595	0.0752973	−	0.997161i	$-0.476009\pi$
$257$	2.41421	0.150595	0.0752973	+	0.997161i	$0.476009\pi$
$258$	−12.5858	−0.783557
$259$	−2.82843	−0.175750
$260$	0	0
$261$	2.65685	0.164455
$262$	7.82843	0.483642
$263$	−26.3848	−1.62696	−0.813478	−	0.581596i	$-0.802428\pi$
$263$	−26.3848	−1.62696	−0.813478	+	0.581596i	$0.802428\pi$
$264$	−1.41421	−0.0870388
$265$	0	0
$266$	−5.82843	−0.357364
$267$	−18.2426	−1.11643
$268$	0.585786	0.0357826
$269$	−2.24264	−0.136736	−0.0683681	−	0.997660i	$-0.521779\pi$
$269$	−2.24264	−0.136736	−0.0683681	+	0.997660i	$0.521779\pi$
$270$	0	0
$271$	12.7279	0.773166	0.386583	−	0.922255i	$-0.373655\pi$
$271$	12.7279	0.773166	0.386583	+	0.922255i	$0.373655\pi$
$272$	5.41421	0.328285
$273$	−7.41421	−0.448729
$274$	−11.8284	−0.714581
$275$	0	0
$276$	−6.24264	−0.375763
$277$	30.6274	1.84022	0.920112	−	0.391656i	$-0.128098\pi$
$277$	30.6274	1.84022	0.920112	+	0.391656i	$0.128098\pi$
$278$	−6.65685	−0.399252
$279$	5.82843	0.348939
$280$	0	0
$281$	−6.24264	−0.372405	−0.186202	−	0.982511i	$-0.559618\pi$
$281$	−6.24264	−0.372405	−0.186202	+	0.982511i	$0.559618\pi$
$282$	14.8284	0.883020
$283$	25.7990	1.53359	0.766795	−	0.641892i	$-0.221850\pi$
$283$	25.7990	1.53359	0.766795	+	0.641892i	$0.221850\pi$
$284$	−4.07107	−0.241573
$285$	0	0
$286$	5.24264	0.310004
$287$	−1.17157	−0.0691558
$288$	1.00000	0.0589256
$289$	12.3137	0.724336
$290$	0	0
$291$	17.0711	1.00072
$292$	−7.31371	−0.428002
$293$	23.4558	1.37031	0.685153	−	0.728400i	$-0.259736\pi$
$293$	23.4558	1.37031	0.685153	+	0.728400i	$0.259736\pi$
$294$	1.41421	0.0824786
$295$	0	0
$296$	2.82843	0.164399
$297$	−5.65685	−0.328244
$298$	0.485281	0.0281116
$299$	23.1421	1.33834
$300$	0	0
$301$	−8.89949	−0.512958
$302$	19.5563	1.12534
$303$	1.07107	0.0615312
$304$	5.82843	0.334283
$305$	0	0
$306$	5.41421	0.309510
$307$	32.9706	1.88173	0.940865	−	0.338783i	$-0.110015\pi$
$307$	32.9706	1.88173	0.940865	+	0.338783i	$0.110015\pi$
$308$	−1.00000	−0.0569803
$309$	−16.7279	−0.951618
$310$	0	0
$311$	−11.9706	−0.678788	−0.339394	−	0.940644i	$-0.610222\pi$
$311$	−11.9706	−0.678788	−0.339394	+	0.940644i	$0.610222\pi$
$312$	7.41421	0.419747
$313$	−26.8284	−1.51643	−0.758216	−	0.652003i	$-0.773929\pi$
$313$	−26.8284	−1.51643	−0.758216	+	0.652003i	$0.773929\pi$
$314$	5.07107	0.286177
$315$	0	0
$316$	−3.07107	−0.172761
$317$	−30.0416	−1.68731	−0.843653	−	0.536889i	$-0.819599\pi$
$317$	−30.0416	−1.68731	−0.843653	+	0.536889i	$0.819599\pi$
$318$	5.31371	0.297978
$319$	2.65685	0.148755
$320$	0	0
$321$	−17.0711	−0.952814
$322$	−4.41421	−0.245995
$323$	31.5563	1.75584
$324$	−5.00000	−0.277778
$325$	0	0
$326$	3.51472	0.194662
$327$	−4.24264	−0.234619
$328$	1.17157	0.0646893
$329$	10.4853	0.578072
$330$	0	0
$331$	−8.48528	−0.466393	−0.233197	−	0.972430i	$-0.574919\pi$
$331$	−8.48528	−0.466393	−0.233197	+	0.972430i	$0.574919\pi$
$332$	8.17157	0.448473
$333$	2.82843	0.154997
$334$	2.72792	0.149265
$335$	0	0
$336$	−1.41421	−0.0771517
$337$	10.5858	0.576645	0.288322	−	0.957533i	$-0.406903\pi$
$337$	10.5858	0.576645	0.288322	+	0.957533i	$0.406903\pi$
$338$	−14.4853	−0.787895
$339$	6.34315	0.344512
$340$	0	0
$341$	5.82843	0.315627
$342$	5.82843	0.315165
$343$	1.00000	0.0539949
$344$	8.89949	0.479829
$345$	0	0
$346$	−3.92893	−0.211221
$347$	3.51472	0.188680	0.0943400	−	0.995540i	$-0.469926\pi$
$347$	3.51472	0.188680	0.0943400	+	0.995540i	$0.469926\pi$
$348$	3.75736	0.201416
$349$	−32.2132	−1.72433	−0.862167	−	0.506625i	$-0.830893\pi$
$349$	−32.2132	−1.72433	−0.862167	+	0.506625i	$0.830893\pi$
$350$	0	0
$351$	29.6569	1.58297
$352$	1.00000	0.0533002
$353$	8.41421	0.447843	0.223922	−	0.974607i	$-0.428114\pi$
$353$	8.41421	0.447843	0.223922	+	0.974607i	$0.428114\pi$
$354$	5.65685	0.300658
$355$	0	0
$356$	12.8995	0.683672
$357$	−7.65685	−0.405244
$358$	−18.3848	−0.971666
$359$	36.9706	1.95123	0.975616	−	0.219485i	$-0.0704378\pi$
$359$	36.9706	1.95123	0.975616	+	0.219485i	$0.0704378\pi$
$360$	0	0
$361$	14.9706	0.787924
$362$	16.4853	0.866447
$363$	−1.41421	−0.0742270
$364$	5.24264	0.274789
$365$	0	0
$366$	−10.1421	−0.530138
$367$	8.31371	0.433972	0.216986	−	0.976175i	$-0.430377\pi$
$367$	8.31371	0.433972	0.216986	+	0.976175i	$0.430377\pi$
$368$	4.41421	0.230107
$369$	1.17157	0.0609896
$370$	0	0
$371$	3.75736	0.195072
$372$	8.24264	0.427361
$373$	34.1421	1.76781	0.883906	−	0.467664i	$-0.154904\pi$
$373$	34.1421	1.76781	0.883906	+	0.467664i	$0.154904\pi$
$374$	5.41421	0.279962
$375$	0	0
$376$	−10.4853	−0.540737
$377$	−13.9289	−0.717377
$378$	−5.65685	−0.290957
$379$	34.7279	1.78385	0.891927	−	0.452180i	$-0.149353\pi$
$379$	34.7279	1.78385	0.891927	+	0.452180i	$0.149353\pi$
$380$	0	0
$381$	7.31371	0.374693
$382$	−23.3848	−1.19647
$383$	−17.1421	−0.875922	−0.437961	−	0.898994i	$-0.644299\pi$
$383$	−17.1421	−0.875922	−0.437961	+	0.898994i	$0.644299\pi$
$384$	1.41421	0.0721688
$385$	0	0
$386$	18.1421	0.923410
$387$	8.89949	0.452387
$388$	−12.0711	−0.612816
$389$	−26.2426	−1.33055	−0.665277	−	0.746596i	$-0.731687\pi$
$389$	−26.2426	−1.33055	−0.665277	+	0.746596i	$0.731687\pi$
$390$	0	0
$391$	23.8995	1.20865
$392$	−1.00000	−0.0505076
$393$	11.0711	0.558461
$394$	9.34315	0.470701
$395$	0	0
$396$	1.00000	0.0502519
$397$	−19.4558	−0.976461	−0.488230	−	0.872715i	$-0.662357\pi$
$397$	−19.4558	−0.976461	−0.488230	+	0.872715i	$0.662357\pi$
$398$	−16.7990	−0.842057
$399$	−8.24264	−0.412648
$400$	0	0
$401$	21.4853	1.07292	0.536462	−	0.843925i	$-0.319761\pi$
$401$	21.4853	1.07292	0.536462	+	0.843925i	$0.319761\pi$
$402$	0.828427	0.0413182
$403$	−30.5563	−1.52212
$404$	−0.757359	−0.0376800
$405$	0	0
$406$	2.65685	0.131857
$407$	2.82843	0.140200
$408$	7.65685	0.379071
$409$	−2.58579	−0.127859	−0.0639295	−	0.997954i	$-0.520363\pi$
$409$	−2.58579	−0.127859	−0.0639295	+	0.997954i	$0.520363\pi$
$410$	0	0
$411$	−16.7279	−0.825128
$412$	11.8284	0.582745
$413$	4.00000	0.196827
$414$	4.41421	0.216947
$415$	0	0
$416$	−5.24264	−0.257042
$417$	−9.41421	−0.461016
$418$	5.82843	0.285078
$419$	12.1421	0.593182	0.296591	−	0.955005i	$-0.404150\pi$
$419$	12.1421	0.593182	0.296591	+	0.955005i	$0.404150\pi$
$420$	0	0
$421$	7.07107	0.344623	0.172311	−	0.985043i	$-0.444876\pi$
$421$	7.07107	0.344623	0.172311	+	0.985043i	$0.444876\pi$
$422$	2.34315	0.114063
$423$	−10.4853	−0.509812
$424$	−3.75736	−0.182473
$425$	0	0
$426$	−5.75736	−0.278945
$427$	−7.17157	−0.347057
$428$	12.0711	0.583477
$429$	7.41421	0.357962
$430$	0	0
$431$	14.8284	0.714260	0.357130	−	0.934055i	$-0.383755\pi$
$431$	14.8284	0.714260	0.357130	+	0.934055i	$0.383755\pi$
$432$	5.65685	0.272166
$433$	−10.4142	−0.500475	−0.250238	−	0.968184i	$-0.580509\pi$
$433$	−10.4142	−0.500475	−0.250238	+	0.968184i	$0.580509\pi$
$434$	5.82843	0.279773
$435$	0	0
$436$	3.00000	0.143674
$437$	25.7279	1.23073
$438$	−10.3431	−0.494215
$439$	−12.4853	−0.595890	−0.297945	−	0.954583i	$-0.596301\pi$
$439$	−12.4853	−0.595890	−0.297945	+	0.954583i	$0.596301\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	−28.3848	−1.35013
$443$	19.3137	0.917622	0.458811	−	0.888534i	$-0.348276\pi$
$443$	19.3137	0.917622	0.458811	+	0.888534i	$0.348276\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−8.14214	−0.385541
$447$	0.686292	0.0324605
$448$	1.00000	0.0472456
$449$	−19.9706	−0.942469	−0.471235	−	0.882008i	$-0.656191\pi$
$449$	−19.9706	−0.942469	−0.471235	+	0.882008i	$0.656191\pi$
$450$	0	0
$451$	1.17157	0.0551672
$452$	−4.48528	−0.210970
$453$	27.6569	1.29943
$454$	−21.0000	−0.985579
$455$	0	0
$456$	8.24264	0.385997
$457$	39.6985	1.85702	0.928508	−	0.371311i	$-0.121092\pi$
$457$	39.6985	1.85702	0.928508	+	0.371311i	$0.121092\pi$
$458$	−21.3137	−0.995924
$459$	30.6274	1.42957
$460$	0	0
$461$	9.17157	0.427163	0.213581	−	0.976925i	$-0.431487\pi$
$461$	9.17157	0.427163	0.213581	+	0.976925i	$0.431487\pi$
$462$	−1.41421	−0.0657952
$463$	28.6985	1.33373	0.666866	−	0.745178i	$-0.267635\pi$
$463$	28.6985	1.33373	0.666866	+	0.745178i	$0.267635\pi$
$464$	−2.65685	−0.123341
$465$	0	0
$466$	−28.2426	−1.30832
$467$	−27.6569	−1.27981	−0.639903	−	0.768455i	$-0.721026\pi$
$467$	−27.6569	−1.27981	−0.639903	+	0.768455i	$0.721026\pi$
$468$	−5.24264	−0.242341
$469$	0.585786	0.0270491
$470$	0	0
$471$	7.17157	0.330449
$472$	−4.00000	−0.184115
$473$	8.89949	0.409199
$474$	−4.34315	−0.199487
$475$	0	0
$476$	5.41421	0.248160
$477$	−3.75736	−0.172038
$478$	−5.51472	−0.252237
$479$	−25.7990	−1.17879	−0.589393	−	0.807846i	$-0.700633\pi$
$479$	−25.7990	−1.17879	−0.589393	+	0.807846i	$0.700633\pi$
$480$	0	0
$481$	−14.8284	−0.676118
$482$	−12.3431	−0.562215
$483$	−6.24264	−0.284050
$484$	1.00000	0.0454545
$485$	0	0
$486$	9.89949	0.449050
$487$	−31.5858	−1.43129	−0.715644	−	0.698465i	$-0.753867\pi$
$487$	−31.5858	−1.43129	−0.715644	+	0.698465i	$0.753867\pi$
$488$	7.17157	0.324642
$489$	4.97056	0.224777
$490$	0	0
$491$	9.44365	0.426186	0.213093	−	0.977032i	$-0.431646\pi$
$491$	9.44365	0.426186	0.213093	+	0.977032i	$0.431646\pi$
$492$	1.65685	0.0746968
$493$	−14.3848	−0.647858
$494$	−30.5563	−1.37480
$495$	0	0
$496$	−5.82843	−0.261704
$497$	−4.07107	−0.182612
$498$	11.5563	0.517852
$499$	9.51472	0.425937	0.212969	−	0.977059i	$-0.431687\pi$
$499$	9.51472	0.425937	0.212969	+	0.977059i	$0.431687\pi$
$500$	0	0
$501$	3.85786	0.172357
$502$	19.4142	0.866499
$503$	−8.34315	−0.372002	−0.186001	−	0.982550i	$-0.559553\pi$
$503$	−8.34315	−0.372002	−0.186001	+	0.982550i	$0.559553\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	4.41421	0.196236
$507$	−20.4853	−0.909783
$508$	−5.17157	−0.229451
$509$	23.8995	1.05933	0.529663	−	0.848208i	$-0.322318\pi$
$509$	23.8995	1.05933	0.529663	+	0.848208i	$0.322318\pi$
$510$	0	0
$511$	−7.31371	−0.323539
$512$	−1.00000	−0.0441942
$513$	32.9706	1.45569
$514$	−2.41421	−0.106486
$515$	0	0
$516$	12.5858	0.554058
$517$	−10.4853	−0.461142
$518$	2.82843	0.124274
$519$	−5.55635	−0.243897
$520$	0	0
$521$	−40.8995	−1.79184	−0.895920	−	0.444216i	$-0.853482\pi$
$521$	−40.8995	−1.79184	−0.895920	+	0.444216i	$0.853482\pi$
$522$	−2.65685	−0.116287
$523$	−18.7990	−0.822022	−0.411011	−	0.911630i	$-0.634824\pi$
$523$	−18.7990	−0.822022	−0.411011	+	0.911630i	$0.634824\pi$
$524$	−7.82843	−0.341986
$525$	0	0
$526$	26.3848	1.15043
$527$	−31.5563	−1.37462
$528$	1.41421	0.0615457
$529$	−3.51472	−0.152814
$530$	0	0
$531$	−4.00000	−0.173585
$532$	5.82843	0.252694
$533$	−6.14214	−0.266045
$534$	18.2426	0.789436
$535$	0	0
$536$	−0.585786	−0.0253021
$537$	−26.0000	−1.12198
$538$	2.24264	0.0966871
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−17.3431	−0.745640	−0.372820	−	0.927904i	$-0.621609\pi$
$541$	−17.3431	−0.745640	−0.372820	+	0.927904i	$0.621609\pi$
$542$	−12.7279	−0.546711
$543$	23.3137	1.00049
$544$	−5.41421	−0.232132
$545$	0	0
$546$	7.41421	0.317299
$547$	16.0711	0.687149	0.343575	−	0.939125i	$-0.388362\pi$
$547$	16.0711	0.687149	0.343575	+	0.939125i	$0.388362\pi$
$548$	11.8284	0.505285
$549$	7.17157	0.306075
$550$	0	0
$551$	−15.4853	−0.659695
$552$	6.24264	0.265704
$553$	−3.07107	−0.130595
$554$	−30.6274	−1.30123
$555$	0	0
$556$	6.65685	0.282314
$557$	4.51472	0.191295	0.0956474	−	0.995415i	$-0.469508\pi$
$557$	4.51472	0.191295	0.0956474	+	0.995415i	$0.469508\pi$
$558$	−5.82843	−0.246737
$559$	−46.6569	−1.97337
$560$	0	0
$561$	7.65685	0.323273
$562$	6.24264	0.263330
$563$	−8.14214	−0.343150	−0.171575	−	0.985171i	$-0.554886\pi$
$563$	−8.14214	−0.343150	−0.171575	+	0.985171i	$0.554886\pi$
$564$	−14.8284	−0.624389
$565$	0	0
$566$	−25.7990	−1.08441
$567$	−5.00000	−0.209980
$568$	4.07107	0.170818
$569$	−20.8701	−0.874918	−0.437459	−	0.899238i	$-0.644122\pi$
$569$	−20.8701	−0.874918	−0.437459	+	0.899238i	$0.644122\pi$
$570$	0	0
$571$	28.8995	1.20941	0.604703	−	0.796451i	$-0.293292\pi$
$571$	28.8995	1.20941	0.604703	+	0.796451i	$0.293292\pi$
$572$	−5.24264	−0.219206
$573$	−33.0711	−1.38156
$574$	1.17157	0.0489005
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	27.1716	1.13117	0.565584	−	0.824691i	$-0.308651\pi$
$577$	27.1716	1.13117	0.565584	+	0.824691i	$0.308651\pi$
$578$	−12.3137	−0.512183
$579$	25.6569	1.06626
$580$	0	0
$581$	8.17157	0.339014
$582$	−17.0711	−0.707619
$583$	−3.75736	−0.155614
$584$	7.31371	0.302643
$585$	0	0
$586$	−23.4558	−0.968952
$587$	14.8284	0.612035	0.306017	−	0.952026i	$-0.401003\pi$
$587$	14.8284	0.612035	0.306017	+	0.952026i	$0.401003\pi$
$588$	−1.41421	−0.0583212
$589$	−33.9706	−1.39973
$590$	0	0
$591$	13.2132	0.543519
$592$	−2.82843	−0.116248
$593$	20.5858	0.845357	0.422678	−	0.906280i	$-0.361090\pi$
$593$	20.5858	0.845357	0.422678	+	0.906280i	$0.361090\pi$
$594$	5.65685	0.232104
$595$	0	0
$596$	−0.485281	−0.0198779
$597$	−23.7574	−0.972324
$598$	−23.1421	−0.946352
$599$	1.85786	0.0759103	0.0379551	−	0.999279i	$-0.487916\pi$
$599$	1.85786	0.0759103	0.0379551	+	0.999279i	$0.487916\pi$
$600$	0	0
$601$	14.6274	0.596665	0.298332	−	0.954462i	$-0.403570\pi$
$601$	14.6274	0.596665	0.298332	+	0.954462i	$0.403570\pi$
$602$	8.89949	0.362716
$603$	−0.585786	−0.0238551
$604$	−19.5563	−0.795737
$605$	0	0
$606$	−1.07107	−0.0435092
$607$	28.6274	1.16195	0.580976	−	0.813921i	$-0.302671\pi$
$607$	28.6274	1.16195	0.580976	+	0.813921i	$0.302671\pi$
$608$	−5.82843	−0.236374
$609$	3.75736	0.152256
$610$	0	0
$611$	54.9706	2.22387
$612$	−5.41421	−0.218857
$613$	−28.9706	−1.17011	−0.585055	−	0.810994i	$-0.698927\pi$
$613$	−28.9706	−1.17011	−0.585055	+	0.810994i	$0.698927\pi$
$614$	−32.9706	−1.33058
$615$	0	0
$616$	1.00000	0.0402911
$617$	−42.3137	−1.70349	−0.851743	−	0.523960i	$-0.824454\pi$
$617$	−42.3137	−1.70349	−0.851743	+	0.523960i	$0.824454\pi$
$618$	16.7279	0.672896
$619$	−35.2132	−1.41534	−0.707669	−	0.706544i	$-0.750253\pi$
$619$	−35.2132	−1.41534	−0.707669	+	0.706544i	$0.750253\pi$
$620$	0	0
$621$	24.9706	1.00203
$622$	11.9706	0.479976
$623$	12.8995	0.516807
$624$	−7.41421	−0.296806
$625$	0	0
$626$	26.8284	1.07228
$627$	8.24264	0.329179
$628$	−5.07107	−0.202358
$629$	−15.3137	−0.610598
$630$	0	0
$631$	40.6274	1.61735	0.808676	−	0.588254i	$-0.200185\pi$
$631$	40.6274	1.61735	0.808676	+	0.588254i	$0.200185\pi$
$632$	3.07107	0.122161
$633$	3.31371	0.131708
$634$	30.0416	1.19311
$635$	0	0
$636$	−5.31371	−0.210702
$637$	5.24264	0.207721
$638$	−2.65685	−0.105186
$639$	4.07107	0.161049
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	17.0711	0.673741
$643$	−42.0416	−1.65796	−0.828980	−	0.559278i	$-0.811078\pi$
$643$	−42.0416	−1.65796	−0.828980	+	0.559278i	$0.811078\pi$
$644$	4.41421	0.173944
$645$	0	0
$646$	−31.5563	−1.24157
$647$	41.7990	1.64329	0.821644	−	0.570001i	$-0.193057\pi$
$647$	41.7990	1.64329	0.821644	+	0.570001i	$0.193057\pi$
$648$	5.00000	0.196419
$649$	−4.00000	−0.157014
$650$	0	0
$651$	8.24264	0.323055
$652$	−3.51472	−0.137647
$653$	7.75736	0.303569	0.151784	−	0.988414i	$-0.451498\pi$
$653$	7.75736	0.303569	0.151784	+	0.988414i	$0.451498\pi$
$654$	4.24264	0.165900
$655$	0	0
$656$	−1.17157	−0.0457422
$657$	7.31371	0.285335
$658$	−10.4853	−0.408759
$659$	−29.5858	−1.15250	−0.576249	−	0.817274i	$-0.695484\pi$
$659$	−29.5858	−1.15250	−0.576249	+	0.817274i	$0.695484\pi$
$660$	0	0
$661$	−12.7279	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$661$	−12.7279	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$662$	8.48528	0.329790
$663$	−40.1421	−1.55899
$664$	−8.17157	−0.317119
$665$	0	0
$666$	−2.82843	−0.109599
$667$	−11.7279	−0.454107
$668$	−2.72792	−0.105546
$669$	−11.5147	−0.445185
$670$	0	0
$671$	7.17157	0.276856
$672$	1.41421	0.0545545
$673$	−9.65685	−0.372244	−0.186122	−	0.982527i	$-0.559592\pi$
$673$	−9.65685	−0.372244	−0.186122	+	0.982527i	$0.559592\pi$
$674$	−10.5858	−0.407749
$675$	0	0
$676$	14.4853	0.557126
$677$	2.21320	0.0850603	0.0425302	−	0.999095i	$-0.486458\pi$
$677$	2.21320	0.0850603	0.0425302	+	0.999095i	$0.486458\pi$
$678$	−6.34315	−0.243607
$679$	−12.0711	−0.463245
$680$	0	0
$681$	−29.6985	−1.13805
$682$	−5.82843	−0.223182
$683$	10.1005	0.386485	0.193243	−	0.981151i	$-0.438100\pi$
$683$	10.1005	0.386485	0.193243	+	0.981151i	$0.438100\pi$
$684$	−5.82843	−0.222855
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−30.1421	−1.14999
$688$	−8.89949	−0.339290
$689$	19.6985	0.750453
$690$	0	0
$691$	−12.4853	−0.474962	−0.237481	−	0.971392i	$-0.576322\pi$
$691$	−12.4853	−0.474962	−0.237481	+	0.971392i	$0.576322\pi$
$692$	3.92893	0.149356
$693$	1.00000	0.0379869
$694$	−3.51472	−0.133417
$695$	0	0
$696$	−3.75736	−0.142422
$697$	−6.34315	−0.240264
$698$	32.2132	1.21929
$699$	−39.9411	−1.51071
$700$	0	0
$701$	−7.34315	−0.277347	−0.138673	−	0.990338i	$-0.544284\pi$
$701$	−7.34315	−0.277347	−0.138673	+	0.990338i	$0.544284\pi$
$702$	−29.6569	−1.11933
$703$	−16.4853	−0.621754
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−8.41421	−0.316673
$707$	−0.757359	−0.0284834
$708$	−5.65685	−0.212598
$709$	18.4437	0.692666	0.346333	−	0.938112i	$-0.387427\pi$
$709$	18.4437	0.692666	0.346333	+	0.938112i	$0.387427\pi$
$710$	0	0
$711$	3.07107	0.115174
$712$	−12.8995	−0.483429
$713$	−25.7279	−0.963518
$714$	7.65685	0.286551
$715$	0	0
$716$	18.3848	0.687071
$717$	−7.79899	−0.291259
$718$	−36.9706	−1.37973
$719$	−15.4558	−0.576406	−0.288203	−	0.957569i	$-0.593058\pi$
$719$	−15.4558	−0.576406	−0.288203	+	0.957569i	$0.593058\pi$
$720$	0	0
$721$	11.8284	0.440514
$722$	−14.9706	−0.557147
$723$	−17.4558	−0.649190
$724$	−16.4853	−0.612671
$725$	0	0
$726$	1.41421	0.0524864
$727$	31.1421	1.15500	0.577499	−	0.816391i	$-0.304029\pi$
$727$	31.1421	1.15500	0.577499	+	0.816391i	$0.304029\pi$
$728$	−5.24264	−0.194305
$729$	29.0000	1.07407
$730$	0	0
$731$	−48.1838	−1.78214
$732$	10.1421	0.374864
$733$	−12.7574	−0.471204	−0.235602	−	0.971850i	$-0.575706\pi$
$733$	−12.7574	−0.471204	−0.235602	+	0.971850i	$0.575706\pi$
$734$	−8.31371	−0.306865
$735$	0	0
$736$	−4.41421	−0.162710
$737$	−0.585786	−0.0215777
$738$	−1.17157	−0.0431262
$739$	−40.7696	−1.49973	−0.749866	−	0.661590i	$-0.769882\pi$
$739$	−40.7696	−1.49973	−0.749866	+	0.661590i	$0.769882\pi$
$740$	0	0
$741$	−43.2132	−1.58748
$742$	−3.75736	−0.137937
$743$	2.24264	0.0822745	0.0411373	−	0.999154i	$-0.486902\pi$
$743$	2.24264	0.0822745	0.0411373	+	0.999154i	$0.486902\pi$
$744$	−8.24264	−0.302190
$745$	0	0
$746$	−34.1421	−1.25003
$747$	−8.17157	−0.298982
$748$	−5.41421	−0.197963
$749$	12.0711	0.441067
$750$	0	0
$751$	42.5563	1.55290	0.776452	−	0.630177i	$-0.217018\pi$
$751$	42.5563	1.55290	0.776452	+	0.630177i	$0.217018\pi$
$752$	10.4853	0.382359
$753$	27.4558	1.00055
$754$	13.9289	0.507262
$755$	0	0
$756$	5.65685	0.205738
$757$	45.2548	1.64481	0.822407	−	0.568899i	$-0.192630\pi$
$757$	45.2548	1.64481	0.822407	+	0.568899i	$0.192630\pi$
$758$	−34.7279	−1.26137
$759$	6.24264	0.226594
$760$	0	0
$761$	9.21320	0.333978	0.166989	−	0.985959i	$-0.446596\pi$
$761$	9.21320	0.333978	0.166989	+	0.985959i	$0.446596\pi$
$762$	−7.31371	−0.264948
$763$	3.00000	0.108607
$764$	23.3848	0.846032
$765$	0	0
$766$	17.1421	0.619371
$767$	20.9706	0.757203
$768$	−1.41421	−0.0510310
$769$	−37.7574	−1.36157	−0.680783	−	0.732486i	$-0.738360\pi$
$769$	−37.7574	−1.36157	−0.680783	+	0.732486i	$0.738360\pi$
$770$	0	0
$771$	−3.41421	−0.122960
$772$	−18.1421	−0.652950
$773$	49.2548	1.77157	0.885787	−	0.464093i	$-0.153620\pi$
$773$	49.2548	1.77157	0.885787	+	0.464093i	$0.153620\pi$
$774$	−8.89949	−0.319886
$775$	0	0
$776$	12.0711	0.433326
$777$	4.00000	0.143499
$778$	26.2426	0.940844
$779$	−6.82843	−0.244654
$780$	0	0
$781$	4.07107	0.145674
$782$	−23.8995	−0.854644
$783$	−15.0294	−0.537108
$784$	1.00000	0.0357143
$785$	0	0
$786$	−11.0711	−0.394892
$787$	−26.7696	−0.954231	−0.477116	−	0.878840i	$-0.658318\pi$
$787$	−26.7696	−0.954231	−0.477116	+	0.878840i	$0.658318\pi$
$788$	−9.34315	−0.332836
$789$	37.3137	1.32840
$790$	0	0
$791$	−4.48528	−0.159478
$792$	−1.00000	−0.0355335
$793$	−37.5980	−1.33514
$794$	19.4558	0.690462
$795$	0	0
$796$	16.7990	0.595424
$797$	−12.9289	−0.457966	−0.228983	−	0.973430i	$-0.573540\pi$
$797$	−12.9289	−0.457966	−0.228983	+	0.973430i	$0.573540\pi$
$798$	8.24264	0.291786
$799$	56.7696	2.00836
$800$	0	0
$801$	−12.8995	−0.455781
$802$	−21.4853	−0.758672
$803$	7.31371	0.258095
$804$	−0.828427	−0.0292164
$805$	0	0
$806$	30.5563	1.07630
$807$	3.17157	0.111645
$808$	0.757359	0.0266438
$809$	−10.5858	−0.372176	−0.186088	−	0.982533i	$-0.559581\pi$
$809$	−10.5858	−0.372176	−0.186088	+	0.982533i	$0.559581\pi$
$810$	0	0
$811$	11.8579	0.416386	0.208193	−	0.978088i	$-0.433242\pi$
$811$	11.8579	0.416386	0.208193	+	0.978088i	$0.433242\pi$
$812$	−2.65685	−0.0932373
$813$	−18.0000	−0.631288
$814$	−2.82843	−0.0991363
$815$	0	0
$816$	−7.65685	−0.268044
$817$	−51.8701	−1.81470
$818$	2.58579	0.0904099
$819$	−5.24264	−0.183193
$820$	0	0
$821$	41.7696	1.45777	0.728884	−	0.684638i	$-0.240039\pi$
$821$	41.7696	1.45777	0.728884	+	0.684638i	$0.240039\pi$
$822$	16.7279	0.583453
$823$	−6.34315	−0.221108	−0.110554	−	0.993870i	$-0.535263\pi$
$823$	−6.34315	−0.221108	−0.110554	+	0.993870i	$0.535263\pi$
$824$	−11.8284	−0.412063
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−31.1005	−1.08147	−0.540735	−	0.841193i	$-0.681854\pi$
$827$	−31.1005	−1.08147	−0.540735	+	0.841193i	$0.681854\pi$
$828$	−4.41421	−0.153405
$829$	52.6274	1.82783	0.913913	−	0.405910i	$-0.133046\pi$
$829$	52.6274	1.82783	0.913913	+	0.405910i	$0.133046\pi$
$830$	0	0
$831$	−43.3137	−1.50254
$832$	5.24264	0.181756
$833$	5.41421	0.187591
$834$	9.41421	0.325988
$835$	0	0
$836$	−5.82843	−0.201580
$837$	−32.9706	−1.13963
$838$	−12.1421	−0.419443
$839$	−31.1716	−1.07616	−0.538081	−	0.842893i	$-0.680851\pi$
$839$	−31.1716	−1.07616	−0.538081	+	0.842893i	$0.680851\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	−7.07107	−0.243685
$843$	8.82843	0.304067
$844$	−2.34315	−0.0806544
$845$	0	0
$846$	10.4853	0.360491
$847$	1.00000	0.0343604
$848$	3.75736	0.129028
$849$	−36.4853	−1.25217
$850$	0	0
$851$	−12.4853	−0.427990
$852$	5.75736	0.197244
$853$	−19.8579	−0.679920	−0.339960	−	0.940440i	$-0.610414\pi$
$853$	−19.8579	−0.679920	−0.339960	+	0.940440i	$0.610414\pi$
$854$	7.17157	0.245406
$855$	0	0
$856$	−12.0711	−0.412581
$857$	−6.78680	−0.231833	−0.115916	−	0.993259i	$-0.536980\pi$
$857$	−6.78680	−0.231833	−0.115916	+	0.993259i	$0.536980\pi$
$858$	−7.41421	−0.253117
$859$	14.5858	0.497661	0.248830	−	0.968547i	$-0.419954\pi$
$859$	14.5858	0.497661	0.248830	+	0.968547i	$0.419954\pi$
$860$	0	0
$861$	1.65685	0.0564654
$862$	−14.8284	−0.505058
$863$	39.3848	1.34067	0.670337	−	0.742057i	$-0.266150\pi$
$863$	39.3848	1.34067	0.670337	+	0.742057i	$0.266150\pi$
$864$	−5.65685	−0.192450
$865$	0	0
$866$	10.4142	0.353889
$867$	−17.4142	−0.591418
$868$	−5.82843	−0.197830
$869$	3.07107	0.104179
$870$	0	0
$871$	3.07107	0.104059
$872$	−3.00000	−0.101593
$873$	12.0711	0.408544
$874$	−25.7279	−0.870260
$875$	0	0
$876$	10.3431	0.349463
$877$	−26.5147	−0.895338	−0.447669	−	0.894199i	$-0.647746\pi$
$877$	−26.5147	−0.895338	−0.447669	+	0.894199i	$0.647746\pi$
$878$	12.4853	0.421358
$879$	−33.1716	−1.11885
$880$	0	0
$881$	13.3848	0.450945	0.225472	−	0.974250i	$-0.427607\pi$
$881$	13.3848	0.450945	0.225472	+	0.974250i	$0.427607\pi$
$882$	1.00000	0.0336718
$883$	19.1127	0.643194	0.321597	−	0.946877i	$-0.395780\pi$
$883$	19.1127	0.643194	0.321597	+	0.946877i	$0.395780\pi$
$884$	28.3848	0.954683
$885$	0	0
$886$	−19.3137	−0.648857
$887$	41.3137	1.38718	0.693589	−	0.720371i	$-0.256028\pi$
$887$	41.3137	1.38718	0.693589	+	0.720371i	$0.256028\pi$
$888$	−4.00000	−0.134231
$889$	−5.17157	−0.173449
$890$	0	0
$891$	5.00000	0.167506
$892$	8.14214	0.272619
$893$	61.1127	2.04506
$894$	−0.686292	−0.0229530
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−32.7279	−1.09275
$898$	19.9706	0.666426
$899$	15.4853	0.516463
$900$	0	0
$901$	20.3431	0.677728
$902$	−1.17157	−0.0390091
$903$	12.5858	0.418829
$904$	4.48528	0.149178
$905$	0	0
$906$	−27.6569	−0.918837
$907$	−9.55635	−0.317313	−0.158657	−	0.987334i	$-0.550716\pi$
$907$	−9.55635	−0.317313	−0.158657	+	0.987334i	$0.550716\pi$
$908$	21.0000	0.696909
$909$	0.757359	0.0251200
$910$	0	0
$911$	−33.3137	−1.10373	−0.551866	−	0.833933i	$-0.686084\pi$
$911$	−33.3137	−1.10373	−0.551866	+	0.833933i	$0.686084\pi$
$912$	−8.24264	−0.272941
$913$	−8.17157	−0.270440
$914$	−39.6985	−1.31311
$915$	0	0
$916$	21.3137	0.704225
$917$	−7.82843	−0.258517
$918$	−30.6274	−1.01086
$919$	−5.37258	−0.177225	−0.0886126	−	0.996066i	$-0.528243\pi$
$919$	−5.37258	−0.177225	−0.0886126	+	0.996066i	$0.528243\pi$
$920$	0	0
$921$	−46.6274	−1.53643
$922$	−9.17157	−0.302050
$923$	−21.3431	−0.702518
$924$	1.41421	0.0465242
$925$	0	0
$926$	−28.6985	−0.943091
$927$	−11.8284	−0.388497
$928$	2.65685	0.0872155
$929$	−23.2426	−0.762566	−0.381283	−	0.924458i	$-0.624518\pi$
$929$	−23.2426	−0.762566	−0.381283	+	0.924458i	$0.624518\pi$
$930$	0	0
$931$	5.82843	0.191019
$932$	28.2426	0.925118
$933$	16.9289	0.554228
$934$	27.6569	0.904960
$935$	0	0
$936$	5.24264	0.171361
$937$	−26.0416	−0.850743	−0.425371	−	0.905019i	$-0.639857\pi$
$937$	−26.0416	−0.850743	−0.425371	+	0.905019i	$0.639857\pi$
$938$	−0.585786	−0.0191266
$939$	37.9411	1.23816
$940$	0	0
$941$	−36.8284	−1.20057	−0.600286	−	0.799785i	$-0.704947\pi$
$941$	−36.8284	−1.20057	−0.600286	+	0.799785i	$0.704947\pi$
$942$	−7.17157	−0.233662
$943$	−5.17157	−0.168410
$944$	4.00000	0.130189
$945$	0	0
$946$	−8.89949	−0.289348
$947$	24.8701	0.808168	0.404084	−	0.914722i	$-0.367590\pi$
$947$	24.8701	0.808168	0.404084	+	0.914722i	$0.367590\pi$
$948$	4.34315	0.141059
$949$	−38.3431	−1.24467
$950$	0	0
$951$	42.4853	1.37768
$952$	−5.41421	−0.175476
$953$	−25.0122	−0.810224	−0.405112	−	0.914267i	$-0.632768\pi$
$953$	−25.0122	−0.810224	−0.405112	+	0.914267i	$0.632768\pi$
$954$	3.75736	0.121649
$955$	0	0
$956$	5.51472	0.178359
$957$	−3.75736	−0.121458
$958$	25.7990	0.833528
$959$	11.8284	0.381960
$960$	0	0
$961$	2.97056	0.0958246
$962$	14.8284	0.478088
$963$	−12.0711	−0.388985
$964$	12.3431	0.397546
$965$	0	0
$966$	6.24264	0.200854
$967$	−48.3848	−1.55595	−0.777975	−	0.628296i	$-0.783753\pi$
$967$	−48.3848	−1.55595	−0.777975	+	0.628296i	$0.783753\pi$
$968$	−1.00000	−0.0321412
$969$	−44.6274	−1.43364
$970$	0	0
$971$	50.9706	1.63572	0.817862	−	0.575415i	$-0.195159\pi$
$971$	50.9706	1.63572	0.817862	+	0.575415i	$0.195159\pi$
$972$	−9.89949	−0.317526
$973$	6.65685	0.213409
$974$	31.5858	1.01207
$975$	0	0
$976$	−7.17157	−0.229556
$977$	−42.6274	−1.36377	−0.681886	−	0.731459i	$-0.738840\pi$
$977$	−42.6274	−1.36377	−0.681886	+	0.731459i	$0.738840\pi$
$978$	−4.97056	−0.158941
$979$	−12.8995	−0.412270
$980$	0	0
$981$	−3.00000	−0.0957826
$982$	−9.44365	−0.301359
$983$	14.6569	0.467481	0.233741	−	0.972299i	$-0.424903\pi$
$983$	14.6569	0.467481	0.233741	+	0.972299i	$0.424903\pi$
$984$	−1.65685	−0.0528186
$985$	0	0
$986$	14.3848	0.458105
$987$	−14.8284	−0.471994
$988$	30.5563	0.972127
$989$	−39.2843	−1.24917
$990$	0	0
$991$	−20.9706	−0.666152	−0.333076	−	0.942900i	$-0.608087\pi$
$991$	−20.9706	−0.666152	−0.333076	+	0.942900i	$0.608087\pi$
$992$	5.82843	0.185053
$993$	12.0000	0.380808
$994$	4.07107	0.129126
$995$	0	0
$996$	−11.5563	−0.366177
$997$	30.0000	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$998$	−9.51472	−0.301183
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bg.1.1	✓	2
5.2	odd	4		3850.2.c.w.1849.2		4
5.3	odd	4		3850.2.c.w.1849.3		4
5.4	even	2		3850.2.a.bn.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3850.2.a.bg.1.1	✓	2	1.1	even	1	trivial
3850.2.a.bn.1.2	yes	2	5.4	even	2
3850.2.c.w.1849.2		4	5.2	odd	4
3850.2.c.w.1849.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.41421 q^{12} +5.24264 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.41421 q^{17} +1.00000 q^{18} +5.82843 q^{19} -1.41421 q^{21} +1.00000 q^{22} +4.41421 q^{23} +1.41421 q^{24} -5.24264 q^{26} +5.65685 q^{27} +1.00000 q^{28} -2.65685 q^{29} -5.82843 q^{31} -1.00000 q^{32} +1.41421 q^{33} -5.41421 q^{34} -1.00000 q^{36} -2.82843 q^{37} -5.82843 q^{38} -7.41421 q^{39} -1.17157 q^{41} +1.41421 q^{42} -8.89949 q^{43} -1.00000 q^{44} -4.41421 q^{46} +10.4853 q^{47} -1.41421 q^{48} +1.00000 q^{49} -7.65685 q^{51} +5.24264 q^{52} +3.75736 q^{53} -5.65685 q^{54} -1.00000 q^{56} -8.24264 q^{57} +2.65685 q^{58} +4.00000 q^{59} -7.17157 q^{61} +5.82843 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.41421 q^{66} +0.585786 q^{67} +5.41421 q^{68} -6.24264 q^{69} -4.07107 q^{71} +1.00000 q^{72} -7.31371 q^{73} +2.82843 q^{74} +5.82843 q^{76} -1.00000 q^{77} +7.41421 q^{78} -3.07107 q^{79} -5.00000 q^{81} +1.17157 q^{82} +8.17157 q^{83} -1.41421 q^{84} +8.89949 q^{86} +3.75736 q^{87} +1.00000 q^{88} +12.8995 q^{89} +5.24264 q^{91} +4.41421 q^{92} +8.24264 q^{93} -10.4853 q^{94} +1.41421 q^{96} -12.0711 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

Properties

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