LMFDB - Embedded newform 3850.2.c.c.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1849,3850)]

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([1, 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("3850.1849");

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.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.7424047782$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3850.1849
Dual form			3850.2.c.c.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} -6.00000 q^{19} +2.00000 q^{21} +1.00000i q^{22} -6.00000i q^{23} +2.00000 q^{24} -2.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} -4.00000 q^{29} -1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} +6.00000i q^{38} -4.00000 q^{39} -2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -4.00000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} +12.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} +12.0000i q^{57} +4.00000i q^{58} +2.00000 q^{61} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} -1.00000i q^{72} +6.00000i q^{73} +8.00000 q^{74} +6.00000 q^{76} -1.00000i q^{77} +4.00000i q^{78} -10.0000 q^{79} -11.0000 q^{81} +12.0000i q^{83} -2.00000 q^{84} -4.00000 q^{86} +8.00000i q^{87} -1.00000i q^{88} -14.0000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -4.00000 q^{94} -2.00000 q^{96} +4.00000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 4 q^{21} + 4 q^{24} - 4 q^{26} - 8 q^{29} + 4 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} + 8 q^{51} - 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} + 16 q^{74} + 12 q^{76} - 20 q^{79} - 22 q^{81} - 4 q^{84} - 8 q^{86} - 28 q^{89} + 4 q^{91} - 8 q^{94} - 4 q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 4 * q^21 + 4 * q^24 - 4 * q^26 - 8 * q^29 + 4 * q^34 + 2 * q^36 - 8 * q^39 + 2 * q^44 - 12 * q^46 - 2 * q^49 + 8 * q^51 - 8 * q^54 - 2 * q^56 + 4 * q^61 - 2 * q^64 + 4 * q^66 - 24 * q^69 - 24 * q^71 + 16 * q^74 + 12 * q^76 - 20 * q^79 - 22 * q^81 - 4 * q^84 - 8 * q^86 - 28 * q^89 + 4 * q^91 - 8 * q^94 - 4 * q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$.

$n$	$1751$	$2201$	$2927$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.00000i	0.577350i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	1.00000i	0.235702i
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	1.00000i	0.213201i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 4.00000i	− 0.769800i
$28$	− 1.00000i	− 0.188982i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	2.00000i	0.348155i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	6.00000i	0.973329i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	2.00000i	0.277350i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	12.0000i	1.58944i
$58$	4.00000i	0.525226i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−12.0000	−1.44463
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	− 1.00000i	− 0.117851i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	6.00000	0.688247
$77$	− 1.00000i	− 0.113961i
$78$	4.00000i	0.452911i
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	8.00000i	0.857690i
$88$	− 1.00000i	− 0.106600i
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	6.00000i	0.625543i
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−2.00000	−0.204124
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	1.00000i	0.101015i
$99$	1.00000	0.100504
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	− 4.00000i	− 0.396059i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	4.00000i	0.384900i
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	16.0000	1.51865
$112$	1.00000i	0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	12.0000	1.12390
$115$	0	0
$116$	4.00000	0.371391
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	− 2.00000i	− 0.174078i
$133$	− 6.00000i	− 0.520266i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	− 22.0000i	− 1.87959i	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	− 22.0000i	− 1.87959i	0.341743	−	0.939793i	$-0.388983\pi$
$138$	12.0000i	1.02151i
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	12.0000i	1.00702i
$143$	2.00000i	0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	2.00000i	0.164957i
$148$	− 8.00000i	− 0.657596i
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	− 6.00000i	− 0.486664i
$153$	− 2.00000i	− 0.161690i
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	4.00000	0.320256
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	10.0000i	0.795557i
$159$	24.0000	1.90332
$160$	0	0
$161$	6.00000	0.472866
$162$	11.0000i	0.864242i
$163$	24.0000i	1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$	24.0000i	1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	2.00000i	0.154303i
$169$	9.00000	0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	4.00000i	0.304997i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	14.0000i	1.04934i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 2.00000i	− 0.148250i
$183$	− 4.00000i	− 0.295689i
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	4.00000i	0.291730i
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	2.00000i	0.144338i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	− 10.0000i	− 0.703598i
$203$	− 4.00000i	− 0.280745i
$204$	−4.00000	−0.280056
$205$	0	0
$206$	8.00000	0.557386
$207$	6.00000i	0.417029i
$208$	− 2.00000i	− 0.138675i
$209$	6.00000	0.415029
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	− 12.0000i	− 0.824163i
$213$	24.0000i	1.64445i
$214$	20.0000	1.36717
$215$	0	0
$216$	4.00000	0.272166
$217$	0	0
$218$	4.00000i	0.270914i
$219$	12.0000	0.810885
$220$	0	0
$221$	4.00000	0.269069
$222$	− 16.0000i	− 1.07385i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	− 12.0000i	− 0.794719i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	− 4.00000i	− 0.262613i
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	20.0000i	1.29914i
$238$	2.00000i	0.129641i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	10.0000i	0.641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	24.0000	1.52094
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000i	0.0629941i
$253$	6.00000i	0.377217i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 28.0000i	− 1.74659i	−0.487190	−	0.873296i	$-0.661978\pi$
$257$	− 28.0000i	− 1.74659i	0.487190	−	0.873296i	$-0.338022\pi$
$258$	8.00000i	0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	4.00000	0.247594
$262$	6.00000i	0.370681i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	−6.00000	−0.367884
$267$	28.0000i	1.71357i
$268$	8.00000i	0.488678i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	2.00000i	0.121268i
$273$	− 4.00000i	− 0.242091i
$274$	−22.0000	−1.32907
$275$	0	0
$276$	12.0000	0.722315
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	− 6.00000i	− 0.359856i
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	8.00000i	0.476393i
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	8.00000	0.468968
$292$	− 6.00000i	− 0.351123i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	−8.00000	−0.464991
$297$	4.00000i	0.232104i
$298$	4.00000i	0.231714i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	4.00000	0.230556
$302$	6.00000i	0.345261i
$303$	− 20.0000i	− 1.14897i
$304$	−6.00000	−0.344124
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	1.00000i	0.0569803i
$309$	16.0000	0.910208
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	− 4.00000i	− 0.226455i
$313$	12.0000i	0.678280i	0.940736	+	0.339140i	$0.110136\pi$
$313$	12.0000i	0.678280i	−0.940736	+	0.339140i	$0.889864\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	10.0000	0.562544
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	− 24.0000i	− 1.34585i
$319$	4.00000	0.223957
$320$	0	0
$321$	40.0000	2.23258
$322$	− 6.00000i	− 0.334367i
$323$	− 12.0000i	− 0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	24.0000	1.32924
$327$	8.00000i	0.442401i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 8.00000i	− 0.438397i
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	− 9.00000i	− 0.489535i
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	− 6.00000i	− 0.324443i
$343$	− 1.00000i	− 0.0539949i
$344$	4.00000	0.215666
$345$	0	0
$346$	18.0000	0.967686
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	− 8.00000i	− 0.428845i
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	1.00000i	0.0533002i
$353$	− 20.0000i	− 1.06449i	−0.846590	−	0.532246i	$-0.821348\pi$
$353$	− 20.0000i	− 1.06449i	0.846590	−	0.532246i	$-0.178652\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	4.00000i	0.211702i
$358$	− 4.00000i	− 0.211407i
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 2.00000i	− 0.105118i
$363$	− 2.00000i	− 0.104973i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	−4.00000	−0.209083
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	4.00000	0.206284
$377$	8.00000i	0.412021i
$378$	− 4.00000i	− 0.205738i
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	− 8.00000i	− 0.409316i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−2.00000	−0.101797
$387$	4.00000i	0.203331i
$388$	− 4.00000i	− 0.203069i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	− 1.00000i	− 0.0505076i
$393$	12.0000i	0.605320i
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	16.0000i	0.802008i
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	16.0000i	0.798007i
$403$	0	0
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−4.00000	−0.198517
$407$	− 8.00000i	− 0.396545i
$408$	4.00000i	0.198030i
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	−44.0000	−2.17036
$412$	− 8.00000i	− 0.394132i
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 12.0000i	− 0.587643i
$418$	− 6.00000i	− 0.293470i
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	20.0000i	0.973585i
$423$	4.00000i	0.194487i
$424$	−12.0000	−0.582772
$425$	0	0
$426$	24.0000	1.16280
$427$	2.00000i	0.0967868i
$428$	− 20.0000i	− 0.966736i
$429$	4.00000	0.193122
$430$	0	0
$431$	−22.0000	−1.05970	−0.529851	−	0.848091i	$-0.677752\pi$
$431$	−22.0000	−1.05970	−0.529851	+	0.848091i	$0.677752\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	0	0
$435$	0	0
$436$	4.00000	0.191565
$437$	36.0000i	1.72211i
$438$	− 12.0000i	− 0.573382i
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 4.00000i	− 0.190261i
$443$	− 40.0000i	− 1.90046i	−0.311553	−	0.950229i	$-0.600849\pi$
$443$	− 40.0000i	− 1.90046i	0.311553	−	0.950229i	$-0.399151\pi$
$444$	−16.0000	−0.759326
$445$	0	0
$446$	−16.0000	−0.757622
$447$	8.00000i	0.378387i
$448$	− 1.00000i	− 0.0472456i
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	0	0
$452$	− 6.00000i	− 0.282216i
$453$	12.0000i	0.563809i
$454$	−24.0000	−1.12638
$455$	0	0
$456$	−12.0000	−0.561951
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	6.00000i	0.280362i
$459$	8.00000	0.373408
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	2.00000i	0.0930484i
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	26.0000	1.20443
$467$	38.0000i	1.75843i	0.476425	+	0.879215i	$0.341932\pi$
$467$	38.0000i	1.75843i	−0.476425	+	0.879215i	$0.658068\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	8.00000	0.369406
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	4.00000i	0.183920i
$474$	20.0000	0.918630
$475$	0	0
$476$	2.00000	0.0916698
$477$	− 12.0000i	− 0.549442i
$478$	6.00000i	0.274434i
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	8.00000i	0.364390i
$483$	− 12.0000i	− 0.546019i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	10.0000	0.453609
$487$	− 18.0000i	− 0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$	− 18.0000i	− 0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$	2.00000i	0.0905357i
$489$	48.0000	2.17064
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	12.0000	0.539906
$495$	0	0
$496$	0	0
$497$	− 12.0000i	− 0.538274i
$498$	− 24.0000i	− 1.07547i
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	6.00000	0.266733
$507$	− 18.0000i	− 0.799408i
$508$	− 8.00000i	− 0.354943i
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	− 1.00000i	− 0.0441942i
$513$	24.0000i	1.05963i
$514$	−28.0000	−1.23503
$515$	0	0
$516$	8.00000	0.352180
$517$	4.00000i	0.175920i
$518$	8.00000i	0.351500i
$519$	36.0000	1.58022
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	− 4.00000i	− 0.175075i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−8.00000	−0.348817
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	6.00000i	0.260133i
$533$	0	0
$534$	28.0000	1.21168
$535$	0	0
$536$	8.00000	0.345547
$537$	− 8.00000i	− 0.345225i
$538$	14.0000i	0.603583i
$539$	1.00000	0.0430730
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	− 28.0000i	− 1.20270i
$543$	− 4.00000i	− 0.171656i
$544$	2.00000	0.0857493
$545$	0	0
$546$	−4.00000	−0.171184
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	22.0000i	0.939793i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	− 12.0000i	− 0.510754i
$553$	− 10.0000i	− 0.425243i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−6.00000	−0.254457
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−4.00000	−0.168880
$562$	2.00000i	0.0843649i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	−32.0000	−1.34506
$567$	− 11.0000i	− 0.461957i
$568$	− 12.0000i	− 0.503509i
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.00000i	0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$	4.00000i	0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−12.0000	−0.497844
$582$	− 8.00000i	− 0.331611i
$583$	− 12.0000i	− 0.496989i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	6.00000	0.247858
$587$	− 2.00000i	− 0.0825488i	−0.999148	−	0.0412744i	$-0.986858\pi$
$587$	− 2.00000i	− 0.0825488i	0.999148	−	0.0412744i	$-0.0131418\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	0	0
$590$	0	0
$591$	−4.00000	−0.164538
$592$	8.00000i	0.328798i
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	4.00000	0.163846
$597$	32.0000i	1.30967i
$598$	12.0000i	0.490716i
$599$	−44.0000	−1.79779	−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000	−1.79779	−0.898896	+	0.438163i	$0.855629\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	− 4.00000i	− 0.163028i
$603$	8.00000i	0.325785i
$604$	6.00000	0.244137
$605$	0	0
$606$	−20.0000	−0.812444
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	6.00000i	0.243332i
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−8.00000	−0.323645
$612$	2.00000i	0.0808452i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	1.00000	0.0402911
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	− 16.0000i	− 0.643614i
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	− 16.0000i	− 0.641542i
$623$	− 14.0000i	− 0.560898i
$624$	−4.00000	−0.160128
$625$	0	0
$626$	12.0000	0.479616
$627$	− 12.0000i	− 0.479234i
$628$	10.0000i	0.399043i
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	− 10.0000i	− 0.397779i
$633$	40.0000i	1.58986i
$634$	0	0
$635$	0	0
$636$	−24.0000	−0.951662
$637$	2.00000i	0.0792429i
$638$	− 4.00000i	− 0.158362i
$639$	12.0000	0.474713
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 40.0000i	− 1.57867i
$643$	− 34.0000i	− 1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$	− 34.0000i	− 1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	−12.0000	−0.472134
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 11.0000i	− 0.432121i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 24.0000i	− 0.939913i
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	− 4.00000i	− 0.155936i
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	− 12.0000i	− 0.466393i
$663$	− 8.00000i	− 0.310694i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−8.00000	−0.309994
$667$	24.0000i	0.929284i
$668$	0	0
$669$	−32.0000	−1.23719
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	− 2.00000i	− 0.0771517i
$673$	42.0000i	1.61898i	0.587133	+	0.809491i	$0.300257\pi$
$673$	42.0000i	1.61898i	−0.587133	+	0.809491i	$0.699743\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	− 12.0000i	− 0.460857i
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−48.0000	−1.83936
$682$	0	0
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	12.0000i	0.457829i
$688$	− 4.00000i	− 0.152499i
$689$	24.0000	0.914327
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	− 18.0000i	− 0.684257i
$693$	1.00000i	0.0379869i
$694$	20.0000	0.759190
$695$	0	0
$696$	−8.00000	−0.303239
$697$	0	0
$698$	6.00000i	0.227103i
$699$	52.0000	1.96682
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	8.00000i	0.301941i
$703$	− 48.0000i	− 1.81035i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−20.0000	−0.752710
$707$	10.0000i	0.376089i
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	− 14.0000i	− 0.524672i
$713$	0	0
$714$	4.00000	0.149696
$715$	0	0
$716$	−4.00000	−0.149487
$717$	12.0000i	0.448148i
$718$	− 6.00000i	− 0.223918i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 17.0000i	− 0.632674i
$723$	16.0000i	0.595046i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	− 44.0000i	− 1.63187i	−0.578144	−	0.815935i	$-0.696223\pi$
$727$	− 44.0000i	− 1.63187i	0.578144	−	0.815935i	$-0.303777\pi$
$728$	2.00000i	0.0741249i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	8.00000	0.295891
$732$	4.00000i	0.147844i
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	0	0
$735$	0	0
$736$	−6.00000	−0.221163
$737$	8.00000i	0.294684i
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	12.0000i	0.440534i
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	26.0000	0.951928
$747$	− 12.0000i	− 0.439057i
$748$	2.00000i	0.0731272i
$749$	−20.0000	−0.730784
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	− 4.00000i	− 0.145865i
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	−4.00000	−0.145479
$757$	52.0000i	1.88997i	0.327111	+	0.944986i	$0.393925\pi$
$757$	52.0000i	1.88997i	−0.327111	+	0.944986i	$0.606075\pi$
$758$	4.00000i	0.145287i
$759$	12.0000	0.435572
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	− 16.0000i	− 0.579619i
$763$	− 4.00000i	− 0.144810i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	−56.0000	−2.01679
$772$	2.00000i	0.0719816i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−4.00000	−0.143592
$777$	16.0000i	0.573997i
$778$	18.0000i	0.645331i
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	− 12.0000i	− 0.429119i
$783$	16.0000i	0.571793i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	12.0000	0.428026
$787$	− 48.0000i	− 1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$	− 48.0000i	− 1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$	2.00000i	0.0712470i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−6.00000	−0.213335
$792$	1.00000i	0.0355335i
$793$	− 4.00000i	− 0.142044i
$794$	14.0000	0.496841
$795$	0	0
$796$	16.0000	0.567105
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	12.0000i	0.424795i
$799$	8.00000	0.283020
$800$	0	0
$801$	14.0000	0.494666
$802$	34.0000i	1.20058i
$803$	− 6.00000i	− 0.211735i
$804$	16.0000	0.564276
$805$	0	0
$806$	0	0
$807$	28.0000i	0.985647i
$808$	10.0000i	0.351799i
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	4.00000i	0.140372i
$813$	− 56.0000i	− 1.96401i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	4.00000	0.140028
$817$	24.0000i	0.839654i
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	44.0000i	1.53468i
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 6.00000i	− 0.208514i
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−44.0000	−1.52634
$832$	2.00000i	0.0693375i
$833$	− 2.00000i	− 0.0692959i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	−6.00000	−0.207514
$837$	0	0
$838$	− 16.0000i	− 0.552711i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	26.0000i	0.896019i
$843$	4.00000i	0.137767i
$844$	20.0000	0.688428
$845$	0	0
$846$	4.00000	0.137523
$847$	1.00000i	0.0343604i
$848$	12.0000i	0.412082i
$849$	−64.0000	−2.19647
$850$	0	0
$851$	48.0000	1.64542
$852$	− 24.0000i	− 0.822226i
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−20.0000	−0.683586
$857$	− 30.0000i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$	− 30.0000i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$	− 4.00000i	− 0.136558i
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	0	0
$861$	0	0
$862$	22.0000i	0.749323i
$863$	− 30.0000i	− 1.02121i	−0.859815	−	0.510606i	$-0.829421\pi$
$863$	− 30.0000i	− 1.02121i	0.859815	−	0.510606i	$-0.170579\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−16.0000	−0.543702
$867$	− 26.0000i	− 0.883006i
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	−16.0000	−0.542139
$872$	− 4.00000i	− 0.135457i
$873$	− 4.00000i	− 0.135379i
$874$	36.0000	1.21772
$875$	0	0
$876$	−12.0000	−0.405442
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	12.0000i	0.404980i
$879$	12.0000	0.404750
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−40.0000	−1.34383
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	16.0000i	0.536925i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	11.0000	0.368514
$892$	16.0000i	0.535720i
$893$	24.0000i	0.803129i
$894$	8.00000	0.267560
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	24.0000i	0.801337i
$898$	14.0000i	0.467186i
$899$	0	0
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	12.0000	0.398673
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	24.0000i	0.796468i
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	12.0000i	0.397360i
$913$	− 12.0000i	− 0.397142i
$914$	22.0000	0.727695
$915$	0	0
$916$	6.00000	0.198246
$917$	− 6.00000i	− 0.198137i
$918$	− 8.00000i	− 0.264039i
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	26.0000i	0.856264i
$923$	24.0000i	0.789970i
$924$	2.00000	0.0657952
$925$	0	0
$926$	10.0000	0.328620
$927$	− 8.00000i	− 0.262754i
$928$	4.00000i	0.131306i
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 26.0000i	− 0.851658i
$933$	− 32.0000i	− 1.04763i
$934$	38.0000	1.24340
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	6.00000i	0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$	6.00000i	0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$	− 8.00000i	− 0.261209i
$939$	24.0000	0.783210
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	20.0000i	0.651635i
$943$	0	0
$944$	0	0
$945$	0	0
$946$	4.00000	0.130051
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	− 20.0000i	− 0.649570i
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	− 2.00000i	− 0.0648204i
$953$	− 18.0000i	− 0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$	− 18.0000i	− 0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	6.00000	0.194054
$957$	− 8.00000i	− 0.258603i
$958$	8.00000i	0.258468i
$959$	22.0000	0.710417
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 16.0000i	− 0.515861i
$963$	− 20.0000i	− 0.644491i
$964$	8.00000	0.257663
$965$	0	0
$966$	−12.0000	−0.386094
$967$	− 56.0000i	− 1.80084i	−0.435023	−	0.900419i	$-0.643260\pi$
$967$	− 56.0000i	− 1.80084i	0.435023	−	0.900419i	$-0.356740\pi$
$968$	1.00000i	0.0321412i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	− 10.0000i	− 0.320750i
$973$	6.00000i	0.192351i
$974$	−18.0000	−0.576757
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	− 48.0000i	− 1.53487i
$979$	14.0000	0.447442
$980$	0	0
$981$	4.00000	0.127710
$982$	− 12.0000i	− 0.382935i
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	0	0
$985$	0	0
$986$	−8.00000	−0.254772
$987$	− 8.00000i	− 0.254643i
$988$	− 12.0000i	− 0.381771i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	− 24.0000i	− 0.761617i
$994$	−12.0000	−0.380617
$995$	0	0
$996$	−24.0000	−0.760469
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	− 28.0000i	− 0.886325i
$999$	32.0000	1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.c.1849.1		2
5.2	odd	4		3850.2.a.m.1.1		1
5.3	odd	4		770.2.a.e.1.1	✓	1
5.4	even	2	inner	3850.2.c.c.1849.2		2
15.8	even	4		6930.2.a.bk.1.1		1
20.3	even	4		6160.2.a.a.1.1		1
35.13	even	4		5390.2.a.c.1.1		1
55.43	even	4		8470.2.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.e.1.1	✓	1	5.3	odd	4
3850.2.a.m.1.1		1	5.2	odd	4
3850.2.c.c.1849.1		2	1.1	even	1	trivial
3850.2.c.c.1849.2		2	5.4	even	2	inner
5390.2.a.c.1.1		1	35.13	even	4
6160.2.a.a.1.1		1	20.3	even	4
6930.2.a.bk.1.1		1	15.8	even	4
8470.2.a.bg.1.1		1	55.43	even	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} -6.00000 q^{19} +2.00000 q^{21} +1.00000i q^{22} -6.00000i q^{23} +2.00000 q^{24} -2.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} -4.00000 q^{29} -1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} +6.00000i q^{38} -4.00000 q^{39} -2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -4.00000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} +12.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} +12.0000i q^{57} +4.00000i q^{58} +2.00000 q^{61} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} -1.00000i q^{72} +6.00000i q^{73} +8.00000 q^{74} +6.00000 q^{76} -1.00000i q^{77} +4.00000i q^{78} -10.0000 q^{79} -11.0000 q^{81} +12.0000i q^{83} -2.00000 q^{84} -4.00000 q^{86} +8.00000i q^{87} -1.00000i q^{88} -14.0000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -4.00000 q^{94} -2.00000 q^{96} +4.00000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 4 q^{21} + 4 q^{24} - 4 q^{26} - 8 q^{29} + 4 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} + 8 q^{51} - 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} + 16 q^{74} + 12 q^{76} - 20 q^{79} - 22 q^{81} - 4 q^{84} - 8 q^{86} - 28 q^{89} + 4 q^{91} - 8 q^{94} - 4 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 4 * q^21 + 4 * q^24 - 4 * q^26 - 8 * q^29 + 4 * q^34 + 2 * q^36 - 8 * q^39 + 2 * q^44 - 12 * q^46 - 2 * q^49 + 8 * q^51 - 8 * q^54 - 2 * q^56 + 4 * q^61 - 2 * q^64 + 4 * q^66 - 24 * q^69 - 24 * q^71 + 16 * q^74 + 12 * q^76 - 20 * q^79 - 22 * q^81 - 4 * q^84 - 8 * q^86 - 28 * q^89 + 4 * q^91 - 8 * q^94 - 4 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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