LMFDB - Embedded newform 3850.2.c.o.1849.2

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:	$0$
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			1849.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3850.1849
Dual form			3850.2.c.o.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} -1.00000i q^{22} -6.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} +8.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} +12.0000 q^{41} -2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} +6.00000 q^{46} -12.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +12.0000 q^{51} -2.00000i q^{52} +4.00000 q^{54} -1.00000 q^{56} +4.00000i q^{57} +2.00000 q^{61} +8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} -12.0000 q^{69} +12.0000 q^{71} +1.00000i q^{72} +2.00000i q^{73} -4.00000 q^{74} +2.00000 q^{76} +1.00000i q^{77} +4.00000i q^{78} -14.0000 q^{79} -11.0000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +2.00000 q^{84} +4.00000 q^{86} +1.00000i q^{88} -6.00000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -16.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} -8.00000i q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{19} - 4 q^{21} - 4 q^{24} - 4 q^{26} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 24 q^{41} + 2 q^{44} + 12 q^{46} - 2 q^{49} + 24 q^{51} + 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 24 q^{69} + 24 q^{71} - 8 q^{74} + 4 q^{76} - 28 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 12 q^{89} + 4 q^{91} + 24 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 - 4 * q^19 - 4 * q^21 - 4 * q^24 - 4 * q^26 + 16 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 24 * q^41 + 2 * q^44 + 12 * q^46 - 2 * q^49 + 24 * q^51 + 8 * q^54 - 2 * q^56 + 4 * q^61 - 2 * q^64 - 4 * q^66 - 24 * q^69 + 24 * q^71 - 8 * q^74 + 4 * q^76 - 28 * q^79 - 22 * q^81 + 4 * q^84 + 8 * q^86 - 12 * q^89 + 4 * q^91 + 24 * 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	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	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
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	2.00000i	0.348155i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 2.00000i	− 0.324443i
$39$	4.00000	0.640513
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	− 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
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	12.0000	1.68034
$52$	− 2.00000i	− 0.277350i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	4.00000i	0.529813i
$58$	0	0
$59$	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	$0.459132\pi$
$62$	8.00000i	1.01600i
$63$	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	$-0.162521\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−12.0000	−1.44463
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000i	0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	2.00000	0.229416
$77$	1.00000i	0.113961i
$78$	4.00000i	0.452911i
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	12.0000i	1.32518i
$83$	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
$87$	0	0
$88$	1.00000i	0.106600i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	6.00000i	0.625543i
$93$	− 16.0000i	− 1.65912i
$94$	12.0000	1.23771
$95$	0	0
$96$	2.00000	0.204124
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	− 1.00000i	− 0.101015i
$99$	1.00000	0.100504
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	12.0000i	1.18818i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	4.00000i	0.384900i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	− 1.00000i	− 0.0944911i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000i	0.181071i
$123$	− 24.0000i	− 2.16401i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	− 2.00000i	− 0.174078i
$133$	2.00000i	0.173422i
$134$	8.00000	0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	− 12.0000i	− 1.02151i
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	12.0000i	1.00702i
$143$	− 2.00000i	− 0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	2.00000i	0.164957i
$148$	− 4.00000i	− 0.328798i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	2.00000i	0.162221i
$153$	− 6.00000i	− 0.485071i
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	− 14.0000i	− 1.11378i
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	− 11.0000i	− 0.864242i
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	−12.0000	−0.937043
$165$	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
$171$	2.00000	0.152944
$172$	4.00000i	0.304997i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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
$186$	16.0000	1.17318
$187$	− 6.00000i	− 0.438763i
$188$	12.0000i	0.875190i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	2.00000i	0.144338i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	1.00000i	0.0710669i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	16.0000	1.11477
$207$	6.00000i	0.417029i
$208$	2.00000i	0.138675i
$209$	2.00000	0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	12.0000	0.820303
$215$	0	0
$216$	−4.00000	−0.272166
$217$	− 8.00000i	− 0.543075i
$218$	16.0000i	1.08366i
$219$	4.00000	0.270295
$220$	0	0
$221$	−12.0000	−0.807207
$222$	8.00000i	0.536925i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	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	$-0.293301\pi$
$228$	− 4.00000i	− 0.264906i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	28.0000i	1.81880i
$238$	6.00000i	0.388922i
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	1.00000i	0.0642824i
$243$	10.0000i	0.641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	24.0000	1.53018
$247$	− 4.00000i	− 0.254514i
$248$	− 8.00000i	− 0.508001i
$249$	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
$257$	− 24.0000i	− 1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$	− 24.0000i	− 1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$	− 8.00000i	− 0.498058i
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	− 18.0000i	− 1.11204i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−2.00000	−0.122628
$267$	12.0000i	0.734388i
$268$	8.00000i	0.488678i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000i	0.363803i
$273$	− 4.00000i	− 0.242091i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	12.0000	0.722315
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 14.0000i	− 0.839664i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	− 24.0000i	− 1.42918i
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	2.00000	0.118262
$287$	− 12.0000i	− 0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−16.0000	−0.937937
$292$	− 2.00000i	− 0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	4.00000	0.232495
$297$	4.00000i	0.232104i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−4.00000	−0.230556
$302$	14.0000i	0.805609i
$303$	12.0000i	0.689382i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	6.00000	0.342997
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	−32.0000	−1.82042
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	− 4.00000i	− 0.226455i
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	14.0000	0.787562
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−24.0000	−1.33955
$322$	− 6.00000i	− 0.334367i
$323$	− 12.0000i	− 0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	−8.00000	−0.443079
$327$	− 32.0000i	− 1.76960i
$328$	− 12.0000i	− 0.662589i
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	9.00000i	0.489535i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−8.00000	−0.433224
$342$	2.00000i	0.108148i
$343$	1.00000i	0.0539949i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	18.0000	0.967686
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	− 1.00000i	− 0.0533002i
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	− 12.0000i	− 0.635107i
$358$	12.0000i	0.634220i
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000i	0.105118i
$363$	− 2.00000i	− 0.104973i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	4.00000	0.209083
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	− 6.00000i	− 0.312772i
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	16.0000i	0.829561i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	− 4.00000i	− 0.205738i
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 24.0000i	− 1.22795i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−2.00000	−0.101797
$387$	4.00000i	0.203331i
$388$	8.00000i	0.406138i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	1.00000i	0.0505076i
$393$	36.0000i	1.81596i
$394$	6.00000	0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	16.0000i	0.802008i
$399$	4.00000	0.200250
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	− 16.0000i	− 0.798007i
$403$	16.0000i	0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	− 4.00000i	− 0.198273i
$408$	− 12.0000i	− 0.594089i
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	16.0000i	0.788263i
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	28.0000i	1.37117i
$418$	2.00000i	0.0978232i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 4.00000i	− 0.194717i
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	24.0000	1.16280
$427$	− 2.00000i	− 0.0967868i
$428$	12.0000i	0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	12.0000i	0.574038i
$438$	4.00000i	0.191127i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 12.0000i	− 0.570782i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	6.00000i	0.282216i
$453$	− 28.0000i	− 1.31555i
$454$	24.0000	1.12638
$455$	0	0
$456$	4.00000	0.187317
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	10.0000i	0.467269i
$459$	24.0000	1.12022
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	2.00000i	0.0930484i
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	6.00000i	0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$	6.00000i	0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$	2.00000i	0.0924500i
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	4.00000i	0.183920i
$474$	−28.0000	−1.28608
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	− 18.0000i	− 0.823301i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	20.0000i	0.910975i
$483$	12.0000i	0.546019i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−10.0000	−0.453609
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	16.0000	0.723545
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	24.0000i	1.08200i
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	− 12.0000i	− 0.538274i
$498$	24.0000i	1.07547i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−6.00000	−0.266733
$507$	− 18.0000i	− 0.799408i
$508$	8.00000i	0.354943i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	1.00000i	0.0441942i
$513$	8.00000i	0.353209i
$514$	24.0000	1.05859
$515$	0	0
$516$	8.00000	0.352180
$517$	12.0000i	0.527759i
$518$	4.00000i	0.175750i
$519$	−36.0000	−1.58022
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	44.0000i	1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$	44.0000i	1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−24.0000	−1.04645
$527$	48.0000i	2.09091i
$528$	2.00000i	0.0870388i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 2.00000i	− 0.0867110i
$533$	24.0000i	1.03956i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−8.00000	−0.345547
$537$	− 24.0000i	− 1.03568i
$538$	18.0000i	0.776035i
$539$	1.00000	0.0430730
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	20.0000i	0.859074i
$543$	− 4.00000i	− 0.171656i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	4.00000	0.171184
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	12.0000i	0.510754i
$553$	14.0000i	0.595341i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	14.0000	0.593732
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	− 8.00000i	− 0.338667i
$559$	8.00000	0.338364
$560$	0	0
$561$	−12.0000	−0.506640
$562$	6.00000i	0.253095i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	16.0000	0.672530
$567$	11.0000i	0.461957i
$568$	− 12.0000i	− 0.503509i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	2.00000i	0.0836242i
$573$	48.0000i	2.00523i
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 8.00000i	− 0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$	− 8.00000i	− 0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$	− 19.0000i	− 0.790296i
$579$	4.00000	0.166234
$580$	0	0
$581$	12.0000	0.497844
$582$	− 16.0000i	− 0.663221i
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	30.0000i	1.23823i	0.785299	+	0.619116i	$0.212509\pi$
$587$	30.0000i	1.23823i	−0.785299	+	0.619116i	$0.787491\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−12.0000	−0.493614
$592$	4.00000i	0.164399i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	0	0
$597$	− 32.0000i	− 1.30967i
$598$	12.0000i	0.490716i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	− 4.00000i	− 0.163028i
$603$	8.00000i	0.325785i
$604$	−14.0000	−0.569652
$605$	0	0
$606$	−12.0000	−0.487467
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	6.00000i	0.242536i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	1.00000	0.0402911
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 32.0000i	− 1.28723i
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	24.0000i	0.962312i
$623$	6.00000i	0.240385i
$624$	4.00000	0.160128
$625$	0	0
$626$	−8.00000	−0.319744
$627$	− 4.00000i	− 0.159745i
$628$	14.0000i	0.558661i
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	14.0000i	0.556890i
$633$	8.00000i	0.317971i
$634$	−12.0000	−0.476581
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	−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	$0.701845\pi$
$642$	− 24.0000i	− 0.947204i
$643$	− 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
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	11.0000i	0.432121i
$649$	0	0
$650$	0	0
$651$	−16.0000	−0.627089
$652$	− 8.00000i	− 0.313304i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	32.0000	1.25130
$655$	0	0
$656$	12.0000	0.468521
$657$	− 2.00000i	− 0.0780274i
$658$	− 12.0000i	− 0.467809i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	− 28.0000i	− 1.08825i
$663$	24.0000i	0.932083i
$664$	12.0000	0.465690
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	− 2.00000i	− 0.0771517i
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	− 12.0000i	− 0.460857i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−48.0000	−1.83936
$682$	− 8.00000i	− 0.306336i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	− 20.0000i	− 0.763048i
$688$	− 4.00000i	− 0.152499i
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	18.0000i	0.684257i
$693$	− 1.00000i	− 0.0379869i
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	72.0000i	2.72719i
$698$	− 14.0000i	− 0.529908i
$699$	12.0000	0.453882
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	8.00000i	0.301941i
$703$	− 8.00000i	− 0.301726i
$704$	1.00000	0.0376889
$705$	0	0
$706$	24.0000	0.903252
$707$	6.00000i	0.225653i
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	6.00000i	0.224860i
$713$	− 48.0000i	− 1.79761i
$714$	12.0000	0.449089
$715$	0	0
$716$	−12.0000	−0.448461
$717$	36.0000i	1.34444i
$718$	− 30.0000i	− 1.11959i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	− 15.0000i	− 0.558242i
$723$	− 40.0000i	− 1.48762i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 20.0000i	− 0.741759i	−0.928681	−	0.370879i	$-0.879056\pi$
$727$	− 20.0000i	− 0.741759i	0.928681	−	0.370879i	$-0.120944\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	4.00000i	0.147844i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	6.00000	0.221163
$737$	8.00000i	0.294684i
$738$	− 12.0000i	− 0.441726i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−16.0000	−0.586588
$745$	0	0
$746$	−14.0000	−0.512576
$747$	− 12.0000i	− 0.439057i
$748$	6.00000i	0.219382i
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 12.0000i	− 0.437595i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	16.0000i	0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$	16.0000i	0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$	28.0000i	1.01701i
$759$	12.0000	0.435572
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	− 16.0000i	− 0.579619i
$763$	− 16.0000i	− 0.579239i
$764$	24.0000	0.868290
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−48.0000	−1.72868
$772$	− 2.00000i	− 0.0719816i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−8.00000	−0.287183
$777$	− 8.00000i	− 0.286998i
$778$	6.00000i	0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−12.0000	−0.429394
$782$	36.0000i	1.28736i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−36.0000	−1.28408
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	6.00000i	0.213741i
$789$	48.0000	1.70885
$790$	0	0
$791$	−6.00000	−0.213335
$792$	− 1.00000i	− 0.0355335i
$793$	4.00000i	0.142044i
$794$	38.0000	1.34857
$795$	0	0
$796$	−16.0000	−0.567105
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	4.00000i	0.141598i
$799$	72.0000	2.54718
$800$	0	0
$801$	6.00000	0.212000
$802$	30.0000i	1.05934i
$803$	− 2.00000i	− 0.0705785i
$804$	16.0000	0.564276
$805$	0	0
$806$	−16.0000	−0.563576
$807$	− 36.0000i	− 1.26726i
$808$	6.00000i	0.211079i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	− 40.0000i	− 1.40286i
$814$	4.00000	0.140200
$815$	0	0
$816$	12.0000	0.420084
$817$	8.00000i	0.279885i
$818$	4.00000i	0.139857i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	12.0000i	0.418548i
$823$	50.0000i	1.74289i	0.490493	+	0.871445i	$0.336817\pi$
$823$	50.0000i	1.74289i	−0.490493	+	0.871445i	$0.663183\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	− 2.00000i	− 0.0693375i
$833$	− 6.00000i	− 0.207888i
$834$	−28.0000	−0.969561
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	− 32.0000i	− 1.10608i
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 10.0000i	− 0.344623i
$843$	− 12.0000i	− 0.413302i
$844$	4.00000	0.137686
$845$	0	0
$846$	−12.0000	−0.412568
$847$	− 1.00000i	− 0.0343604i
$848$	0	0
$849$	−32.0000	−1.09824
$850$	0	0
$851$	24.0000	0.822709
$852$	24.0000i	0.822226i
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	− 4.00000i	− 0.136558i
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	30.0000i	1.02180i
$863$	18.0000i	0.612727i	0.951915	+	0.306364i	$0.0991123\pi$
$863$	18.0000i	0.612727i	−0.951915	+	0.306364i	$0.900888\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	4.00000	0.135926
$867$	38.0000i	1.29055i
$868$	8.00000i	0.271538i
$869$	14.0000	0.474917
$870$	0	0
$871$	16.0000	0.542139
$872$	− 16.0000i	− 0.541828i
$873$	8.00000i	0.270759i
$874$	−12.0000	−0.405906
$875$	0	0
$876$	−4.00000	−0.135147
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	28.0000i	0.944954i
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	1.00000i	0.0336718i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	− 8.00000i	− 0.268462i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	11.0000	0.368514
$892$	− 8.00000i	− 0.267860i
$893$	24.0000i	0.803129i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	− 24.0000i	− 0.801337i
$898$	18.0000i	0.600668i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 12.0000i	− 0.399556i
$903$	8.00000i	0.266223i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	28.0000	0.930238
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	24.0000i	0.796468i
$909$	6.00000	0.199007
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	4.00000i	0.132453i
$913$	− 12.0000i	− 0.397142i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−10.0000	−0.330409
$917$	18.0000i	0.594412i
$918$	24.0000i	0.792118i
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	− 18.0000i	− 0.592798i
$923$	24.0000i	0.789970i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	22.0000	0.722965
$927$	16.0000i	0.525509i
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	− 6.00000i	− 0.196537i
$933$	− 48.0000i	− 1.57145i
$934$	−6.00000	−0.196326
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	− 8.00000i	− 0.261209i
$939$	16.0000	0.522140
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	− 28.0000i	− 0.912289i
$943$	− 72.0000i	− 2.34464i
$944$	0	0
$945$	0	0
$946$	−4.00000	−0.130051
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	− 28.0000i	− 0.909398i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	24.0000	0.778253
$952$	− 6.00000i	− 0.194461i
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	0	0
$955$	0	0
$956$	18.0000	0.582162
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	6.00000	0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	− 8.00000i	− 0.257930i
$963$	12.0000i	0.386695i
$964$	−20.0000	−0.644157
$965$	0	0
$966$	−12.0000	−0.386094
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	− 10.0000i	− 0.320750i
$973$	14.0000i	0.448819i
$974$	2.00000	0.0640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	16.0000i	0.511624i
$979$	6.00000	0.191761
$980$	0	0
$981$	−16.0000	−0.510841
$982$	− 36.0000i	− 1.14881i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−24.0000	−0.765092
$985$	0	0
$986$	0	0
$987$	24.0000i	0.763928i
$988$	4.00000i	0.127257i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	8.00000i	0.254000i
$993$	56.0000i	1.77711i
$994$	12.0000	0.380617
$995$	0	0
$996$	−24.0000	−0.760469
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	4.00000i	0.126618i
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.o.1849.2		2
5.2	odd	4		770.2.a.a.1.1	✓	1
5.3	odd	4		3850.2.a.ba.1.1		1
5.4	even	2	inner	3850.2.c.o.1849.1		2
15.2	even	4		6930.2.a.bm.1.1		1
20.7	even	4		6160.2.a.k.1.1		1
35.27	even	4		5390.2.a.r.1.1		1
55.32	even	4		8470.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.a.1.1	✓	1	5.2	odd	4
3850.2.a.ba.1.1		1	5.3	odd	4
3850.2.c.o.1849.1		2	5.4	even	2	inner
3850.2.c.o.1849.2		2	1.1	even	1	trivial
5390.2.a.r.1.1		1	35.27	even	4
6160.2.a.k.1.1		1	20.7	even	4
6930.2.a.bm.1.1		1	15.2	even	4
8470.2.a.r.1.1		1	55.32	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} -1.00000i q^{22} -6.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} +8.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} +12.0000 q^{41} -2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} +6.00000 q^{46} -12.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +12.0000 q^{51} -2.00000i q^{52} +4.00000 q^{54} -1.00000 q^{56} +4.00000i q^{57} +2.00000 q^{61} +8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} -12.0000 q^{69} +12.0000 q^{71} +1.00000i q^{72} +2.00000i q^{73} -4.00000 q^{74} +2.00000 q^{76} +1.00000i q^{77} +4.00000i q^{78} -14.0000 q^{79} -11.0000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +2.00000 q^{84} +4.00000 q^{86} +1.00000i q^{88} -6.00000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -16.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} -8.00000i q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} - 4 q^{24} - 4 q^{26} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 24 q^{41} + 2 q^{44} + 12 q^{46} - 2 q^{49} + 24 q^{51} + 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 24 q^{69} + 24 q^{71} - 8 q^{74} + 4 q^{76} - 28 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 12 q^{89} + 4 q^{91} + 24 q^{94} + 4 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^19 - 4 * q^21 - 4 * q^24 - 4 * q^26 + 16 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 24 * q^41 + 2 * q^44 + 12 * q^46 - 2 * q^49 + 24 * q^51 + 8 * q^54 - 2 * q^56 + 4 * q^61 - 2 * q^64 - 4 * q^66 - 24 * q^69 + 24 * q^71 - 8 * q^74 + 4 * q^76 - 28 * q^79 - 22 * q^81 + 4 * q^84 + 8 * q^86 - 12 * q^89 + 4 * q^91 + 24 * q^94 + 4 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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