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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} -6.00000i q^{23} +2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +2.00000i q^{38} +4.00000 q^{39} +2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -2.00000i q^{48} -1.00000 q^{49} -12.0000 q^{51} -2.00000i q^{52} +12.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} +4.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} -8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +4.00000i q^{67} +6.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} -1.00000i q^{72} +14.0000i q^{73} -8.00000 q^{74} +2.00000 q^{76} +1.00000i q^{77} -4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +2.00000 q^{84} -4.00000 q^{86} -1.00000i q^{88} +18.0000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -16.0000i q^{93} -2.00000 q^{96} -8.00000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{44} - 12 q^{46} - 2 q^{49} - 24 q^{51} - 8 q^{54} + 2 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} - 16 q^{74} + 4 q^{76} + 20 q^{79} - 22 q^{81} + 4 q^{84} - 8 q^{86} + 36 q^{89} + 4 q^{91} - 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 + 2 * q^44 - 12 * q^46 - 2 * q^49 - 24 * q^51 - 8 * q^54 + 2 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 + 4 * q^66 - 24 * q^69 - 24 * q^71 - 16 * q^74 + 4 * q^76 + 20 * q^79 - 22 * q^81 + 4 * q^84 - 8 * q^86 + 36 * q^89 + 4 * q^91 - 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	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	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
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	2.00000i	0.324443i
$39$	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
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−12.0000	−1.68034
$52$	− 2.00000i	− 0.277350i
$53$	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
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 8.00000i	− 1.01600i
$63$	1.00000i	0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	6.00000i	0.727607i
$69$	−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
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	2.00000	0.229416
$77$	1.00000i	0.113961i
$78$	− 4.00000i	− 0.452911i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	− 1.00000i	− 0.106600i
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	6.00000i	0.625543i
$93$	− 16.0000i	− 1.65912i
$94$	0	0
$95$	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
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	12.0000i	1.18818i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	− 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
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	−16.0000	−1.51865
$112$	− 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
$118$	12.0000i	1.10469i
$119$	−6.00000	−0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	10.0000i	0.905357i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	1.00000	0.0890871
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	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
$134$	4.00000	0.345547
$135$	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
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	− 2.00000i	− 0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	2.00000i	0.164957i
$148$	8.00000i	0.657596i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 2.00000i	− 0.162221i
$153$	6.00000i	0.485071i
$154$	1.00000	0.0805823
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 10.0000i	− 0.795557i
$159$	24.0000	1.90332
$160$	0	0
$161$	−6.00000	−0.472866
$162$	11.0000i	0.864242i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	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
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	24.0000i	1.80395i
$178$	− 18.0000i	− 1.34916i
$179$	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
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	− 2.00000i	− 0.148250i
$183$	20.0000i	1.47844i
$184$	6.00000	0.442326
$185$	0	0
$186$	−16.0000	−1.17318
$187$	6.00000i	0.438763i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000i	0.144338i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	18.0000i	1.26648i
$203$	0	0
$204$	12.0000	0.840168
$205$	0	0
$206$	−4.00000	−0.278693
$207$	6.00000i	0.417029i
$208$	2.00000i	0.138675i
$209$	2.00000	0.138343
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 12.0000i	− 0.824163i
$213$	24.0000i	1.64445i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	4.00000	0.272166
$217$	− 8.00000i	− 0.543075i
$218$	8.00000i	0.541828i
$219$	28.0000	1.89206
$220$	0	0
$221$	12.0000	0.807207
$222$	16.0000i	1.07385i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	− 20.0000i	− 1.29914i
$238$	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
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	10.0000i	0.641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	8.00000i	0.508001i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	6.00000i	0.377217i
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0000i	1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$	24.0000i	1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$	8.00000i	0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	18.0000i	1.11204i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	2.00000	0.122628
$267$	− 36.0000i	− 2.20316i
$268$	− 4.00000i	− 0.244339i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	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
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	− 10.0000i	− 0.599760i
$279$	−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	$0.442725\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−16.0000	−0.937937
$292$	− 14.0000i	− 0.819288i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	8.00000	0.464991
$297$	4.00000i	0.232104i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−4.00000	−0.230556
$302$	10.0000i	0.575435i
$303$	36.0000i	2.06815i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	6.00000	0.342997
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	−8.00000	−0.455104
$310$	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
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	− 24.0000i	− 1.34585i
$319$	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
$326$	−4.00000	−0.221540
$327$	16.0000i	0.884802i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	0	0
$336$	−2.00000	−0.109109
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 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
$346$	6.00000	0.322562
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	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	$-0.220526\pi$
$354$	24.0000	1.27559
$355$	0	0
$356$	−18.0000	−0.953998
$357$	12.0000i	0.635107i
$358$	− 12.0000i	− 0.634220i
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	10.0000i	0.525588i
$363$	− 2.00000i	− 0.104973i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	20.0000	1.04542
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	16.0000i	0.829561i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	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
$381$	32.0000	1.63941
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	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
$394$	−18.0000	−0.906827
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	8.00000i	0.401004i
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	− 8.00000i	− 0.399004i
$403$	16.0000i	0.797017i
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	− 12.0000i	− 0.594089i
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	4.00000i	0.197066i
$413$	12.0000i	0.590481i
$414$	6.00000	0.294884
$415$	0	0
$416$	2.00000	0.0980581
$417$	− 20.0000i	− 0.979404i
$418$	− 2.00000i	− 0.0978232i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 20.0000i	− 0.973585i
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	24.0000	1.16280
$427$	10.0000i	0.483934i
$428$	12.0000i	0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	− 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
$436$	8.00000	0.383131
$437$	12.0000i	0.574038i
$438$	− 28.0000i	− 1.33789i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 12.0000i	− 0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	16.0000	0.759326
$445$	0	0
$446$	−4.00000	−0.189405
$447$	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
$451$	0	0
$452$	6.00000i	0.282216i
$453$	20.0000i	0.939682i
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	26.0000i	1.21490i
$459$	−24.0000	−1.12022
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	− 2.00000i	− 0.0930484i
$463$	2.00000i	0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$	2.00000i	0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	2.00000i	0.0924500i
$469$	4.00000	0.184703
$470$	0	0
$471$	−4.00000	−0.184310
$472$	− 12.0000i	− 0.552345i
$473$	4.00000i	0.183920i
$474$	−20.0000	−0.918630
$475$	0	0
$476$	6.00000	0.275010
$477$	− 12.0000i	− 0.549442i
$478$	18.0000i	0.823301i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	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
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−8.00000	−0.361773
$490$	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
$493$	0	0
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	12.0000i	0.538274i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	12.0000i	0.535586i
$503$	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
$508$	− 16.0000i	− 0.709885i
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	− 1.00000i	− 0.0441942i
$513$	8.00000i	0.353209i
$514$	24.0000	1.05859
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	8.00000i	0.351500i
$519$	12.0000	0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	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
$531$	12.0000	0.520756
$532$	− 2.00000i	− 0.0867110i
$533$	0	0
$534$	−36.0000	−1.55787
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 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
$543$	20.0000i	0.858282i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−4.00000	−0.171184
$547$	− 44.0000i	− 1.88130i	−0.339372	−	0.940652i	$-0.610215\pi$
$547$	− 44.0000i	− 1.88130i	0.339372	−	0.940652i	$-0.389785\pi$
$548$	− 6.00000i	− 0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	0	0
$552$	− 12.0000i	− 0.510754i
$553$	− 10.0000i	− 0.425243i
$554$	10.0000	0.424859
$555$	0	0
$556$	−10.0000	−0.424094
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	8.00000i	0.338667i
$559$	8.00000	0.338364
$560$	0	0
$561$	12.0000	0.506640
$562$	− 6.00000i	− 0.253095i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	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
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	16.0000i	0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$	16.0000i	0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$	19.0000i	0.790296i
$579$	28.0000	1.16364
$580$	0	0
$581$	0	0
$582$	16.0000i	0.663221i
$583$	− 12.0000i	− 0.496989i
$584$	−14.0000	−0.579324
$585$	0	0
$586$	18.0000	0.743573
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−36.0000	−1.48084
$592$	− 8.00000i	− 0.328798i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	0	0
$597$	16.0000i	0.654836i
$598$	− 12.0000i	− 0.490716i
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	4.00000i	0.163028i
$603$	− 4.00000i	− 0.162893i
$604$	10.0000	0.406894
$605$	0	0
$606$	36.0000	1.46240
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 6.00000i	− 0.242536i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	8.00000i	0.321807i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	− 24.0000i	− 0.962312i
$623$	− 18.0000i	− 0.721155i
$624$	4.00000	0.160128
$625$	0	0
$626$	−16.0000	−0.639489
$627$	− 4.00000i	− 0.159745i
$628$	2.00000i	0.0798087i
$629$	−48.0000	−1.91389
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	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
$638$	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
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	12.0000	0.472134
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	− 11.0000i	− 0.432121i
$649$	12.0000	0.471041
$650$	0	0
$651$	−16.0000	−0.627089
$652$	4.00000i	0.156652i
$653$	− 36.0000i	− 1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$	− 36.0000i	− 1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$	16.0000	0.625650
$655$	0	0
$656$	0	0
$657$	− 14.0000i	− 0.546192i
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 20.0000i	− 0.777322i
$663$	− 24.0000i	− 0.932083i
$664$	0	0
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	10.0000	0.386046
$672$	2.00000i	0.0771517i
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	12.0000i	0.460857i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	24.0000	0.919682
$682$	8.00000i	0.306336i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	1.00000	0.0381802
$687$	52.0000i	1.98392i
$688$	− 4.00000i	− 0.152499i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 6.00000i	− 0.228086i
$693$	− 1.00000i	− 0.0379869i
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.00000i	0.0757011i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	− 8.00000i	− 0.301941i
$703$	16.0000i	0.603451i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−24.0000	−0.903252
$707$	18.0000i	0.676960i
$708$	− 24.0000i	− 0.901975i
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	18.0000i	0.674579i
$713$	− 48.0000i	− 1.79761i
$714$	12.0000	0.449089
$715$	0	0
$716$	−12.0000	−0.448461
$717$	36.0000i	1.34444i
$718$	− 18.0000i	− 0.671754i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	15.0000i	0.558242i
$723$	− 16.0000i	− 0.595046i
$724$	10.0000	0.371647
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	2.00000i	0.0741249i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	− 20.0000i	− 0.739221i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	−6.00000	−0.221163
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	− 12.0000i	− 0.440534i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	16.0000	0.586588
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	− 6.00000i	− 0.219382i
$749$	−12.0000	−0.438470
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	0	0
$753$	24.0000i	0.874609i
$754$	0	0
$755$	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	$0.606075\pi$
$758$	− 28.0000i	− 1.01701i
$759$	12.0000	0.435572
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	− 32.0000i	− 1.15924i
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	− 24.0000i	− 0.866590i
$768$	− 2.00000i	− 0.0721688i
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	48.0000	1.72868
$772$	− 14.0000i	− 0.503871i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	8.00000	0.287183
$777$	16.0000i	0.573997i
$778$	− 6.00000i	− 0.215110i
$779$	0	0
$780$	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
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	18.0000i	0.641223i
$789$	−48.0000	−1.70885
$790$	0	0
$791$	−6.00000	−0.213335
$792$	1.00000i	0.0355335i
$793$	− 20.0000i	− 0.710221i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	− 4.00000i	− 0.141598i
$799$	0	0
$800$	0	0
$801$	−18.0000	−0.635999
$802$	18.0000i	0.635602i
$803$	− 14.0000i	− 0.494049i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	16.0000	0.563576
$807$	36.0000i	1.26726i
$808$	− 18.0000i	− 0.633238i
$809$	−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
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	0	0
$813$	− 40.0000i	− 1.40286i
$814$	8.00000	0.280400
$815$	0	0
$816$	−12.0000	−0.420084
$817$	8.00000i	0.279885i
$818$	8.00000i	0.279713i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	− 12.0000i	− 0.418548i
$823$	2.00000i	0.0697156i	0.999392	+	0.0348578i	$0.0110978\pi$
$823$	2.00000i	0.0697156i	−0.999392	+	0.0348578i	$0.988902\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	12.0000	0.417533
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	− 2.00000i	− 0.0693375i
$833$	6.00000i	0.207888i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	− 32.0000i	− 1.10608i
$838$	− 12.0000i	− 0.414533i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 14.0000i	− 0.482472i
$843$	− 12.0000i	− 0.413302i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	− 1.00000i	− 0.0343604i
$848$	12.0000i	0.412082i
$849$	−8.00000	−0.274559
$850$	0	0
$851$	−48.0000	−1.64542
$852$	− 24.0000i	− 0.822226i
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	12.0000	0.410152
$857$	− 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
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	18.0000i	0.613082i
$863$	− 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
$866$	−4.00000	−0.135926
$867$	38.0000i	1.29055i
$868$	8.00000i	0.271538i
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 8.00000i	− 0.270914i
$873$	8.00000i	0.270759i
$874$	12.0000	0.405906
$875$	0	0
$876$	−28.0000	−0.946032
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	20.0000i	0.674967i
$879$	36.0000	1.21425
$880$	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
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 16.0000i	− 0.536925i
$889$	16.0000	0.536623
$890$	0	0
$891$	11.0000	0.368514
$892$	4.00000i	0.133930i
$893$	0	0
$894$	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
$901$	72.0000	2.39867
$902$	0	0
$903$	8.00000i	0.266223i
$904$	6.00000	0.199557
$905$	0	0
$906$	20.0000	0.664455
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	− 12.0000i	− 0.398234i
$909$	18.0000	0.597022
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	−26.0000	−0.860004
$915$	0	0
$916$	26.0000	0.859064
$917$	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
$921$	−64.0000	−2.10887
$922$	30.0000i	0.987997i
$923$	− 24.0000i	− 0.789970i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	2.00000	0.0657241
$927$	4.00000i	0.131377i
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	6.00000i	0.196537i
$933$	− 48.0000i	− 1.57145i
$934$	−18.0000	−0.588978
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	− 4.00000i	− 0.130605i
$939$	−32.0000	−1.04428
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	4.00000i	0.130327i
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	4.00000	0.130051
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	20.0000i	0.649570i
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	− 6.00000i	− 0.194461i
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	18.0000	0.582162
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	− 16.0000i	− 0.515861i
$963$	12.0000i	0.386695i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	12.0000	0.386094
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	1.00000i	0.0321412i
$969$	24.0000	0.770991
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	− 10.0000i	− 0.320750i
$973$	− 10.0000i	− 0.320585i
$974$	22.0000	0.704925
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	8.00000i	0.255812i
$979$	−18.0000	−0.575282
$980$	0	0
$981$	8.00000	0.255420
$982$	− 12.0000i	− 0.382935i
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	4.00000i	0.127257i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 40.0000i	− 1.26936i
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	20.0000i	0.633089i
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.a.1849.1		2
5.2	odd	4		770.2.a.g.1.1	✓	1
5.3	odd	4		3850.2.a.j.1.1		1
5.4	even	2	inner	3850.2.c.a.1849.2		2
15.2	even	4		6930.2.a.f.1.1		1
20.7	even	4		6160.2.a.n.1.1		1
35.27	even	4		5390.2.a.bh.1.1		1
55.32	even	4		8470.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.g.1.1	✓	1	5.2	odd	4
3850.2.a.j.1.1		1	5.3	odd	4
3850.2.c.a.1849.1		2	1.1	even	1	trivial
3850.2.c.a.1849.2		2	5.4	even	2	inner
5390.2.a.bh.1.1		1	35.27	even	4
6160.2.a.n.1.1		1	20.7	even	4
6930.2.a.f.1.1		1	15.2	even	4
8470.2.a.e.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} -6.00000i q^{23} +2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +2.00000i q^{38} +4.00000 q^{39} +2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -2.00000i q^{48} -1.00000 q^{49} -12.0000 q^{51} -2.00000i q^{52} +12.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} +4.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} -8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +4.00000i q^{67} +6.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} -1.00000i q^{72} +14.0000i q^{73} -8.00000 q^{74} +2.00000 q^{76} +1.00000i q^{77} -4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +2.00000 q^{84} -4.00000 q^{86} -1.00000i q^{88} +18.0000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -16.0000i q^{93} -2.00000 q^{96} -8.00000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} + 4 q^{24} + 4 q^{26} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} - 24 q^{51} - 8 q^{54} + 2 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} - 16 q^{74} + 4 q^{76} + 20 q^{79} - 22 q^{81} + 4 q^{84} - 8 q^{86} + 36 q^{89} + 4 q^{91} - 4 q^{96} + 2 q^{99}+O(q^{100}) \) 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 + 2 * q^44 - 12 * q^46 - 2 * q^49 - 24 * q^51 - 8 * q^54 + 2 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 + 4 * q^66 - 24 * q^69 - 24 * q^71 - 16 * q^74 + 4 * q^76 + 20 * q^79 - 22 * q^81 + 4 * q^84 - 8 * q^86 + 36 * q^89 + 4 * q^91 - 4 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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