LMFDB - Embedded newform 3850.2.c.m.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.m.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -2.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -3.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} -10.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +1.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +1.00000i q^{88} -10.0000 q^{89} -6.00000 q^{91} +4.00000i q^{92} +4.00000 q^{94} +6.00000i q^{97} -1.00000i q^{98} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^9	$ 2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 12 q^{26} - 12 q^{29} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 2 q^{44} + 8 q^{46} - 2 q^{49} - 2 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 16 q^{71} - 4 q^{74} - 8 q^{76} + 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} - 12 q^{91} + 8 q^{94} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 12 * q^26 - 12 * q^29 - 4 * q^34 - 6 * q^36 - 12 * q^41 + 2 * q^44 + 8 * q^46 - 2 * q^49 - 2 * q^56 - 24 * q^59 - 4 * q^61 - 2 * q^64 - 16 * q^71 - 4 * q^74 - 8 * q^76 + 16 * q^79 + 18 * q^81 + 8 * q^86 - 20 * q^89 - 12 * q^91 + 8 * q^94 - 6 * 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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	3.00000	1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	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	$0.922021\pi$
$18$	3.00000i	0.707107i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	1.00000i	0.188982i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−3.00000	−0.500000
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 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
$46$	4.00000	0.589768
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	6.00000i	0.832050i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 3.00000i	− 0.353553i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	1.00000i	0.113961i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 6.00000i	− 0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	1.00000i	0.106600i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	4.00000i	0.417029i
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 1.00000i	− 0.101015i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	− 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	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	− 18.0000i	− 1.66410i
$118$	− 12.0000i	− 1.10469i
$119$	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
$126$	3.00000	0.267261
$127$	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
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	6.00000i	0.501745i
$144$	3.00000	0.250000
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 4.00000i	− 0.324443i
$153$	6.00000i	0.485071i
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	9.00000i	0.707107i
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	12.0000	0.917663
$172$	4.00000i	0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	− 6.00000i	− 0.444750i
$183$	0	0
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	−6.00000	−0.430775
$195$	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	$-0.221573\pi$
$198$	− 3.00000i	− 0.213201i
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	0	0
$202$	14.0000i	0.985037i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	− 12.0000i	− 0.834058i
$208$	− 6.00000i	− 0.416025i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000i	0.137361i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 14.0000i	− 0.948200i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	− 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
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	18.0000	1.17670
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	2.00000i	0.129641i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	− 24.0000i	− 1.52708i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	3.00000i	0.188982i
$253$	4.00000i	0.251478i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−18.0000	−1.11417
$262$	− 12.0000i	− 0.741362i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	20.0000i	1.19952i
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−6.00000	−0.354787
$287$	6.00000i	0.354169i
$288$	3.00000i	0.176777i
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	10.0000i	0.585206i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−24.0000	−1.38796
$300$	0	0
$301$	−4.00000	−0.230556
$302$	8.00000i	0.460348i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	−6.00000	−0.342997
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	− 4.00000i	− 0.222911i
$323$	8.00000i	0.445132i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−8.00000	−0.443079
$327$	0	0
$328$	6.00000i	0.331295i
$329$	−4.00000	−0.220527
$330$	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
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	− 23.0000i	− 1.25104i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	12.0000i	0.648886i
$343$	1.00000i	0.0539949i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	− 20.0000i	− 1.05703i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	6.00000i	0.315353i
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−18.0000	−0.937043
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	34.0000i	1.76045i	0.474554	+	0.880227i	$0.342610\pi$
$373$	34.0000i	1.76045i	−0.474554	+	0.880227i	$0.657390\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−4.00000	−0.206284
$377$	36.0000i	1.85409i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	− 28.0000i	− 1.43073i	−0.698749	−	0.715367i	$-0.746260\pi$
$383$	− 28.0000i	− 1.43073i	0.698749	−	0.715367i	$-0.253740\pi$
$384$	0	0
$385$	0	0
$386$	−6.00000	−0.305392
$387$	− 12.0000i	− 0.609994i
$388$	− 6.00000i	− 0.304604i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	3.00000	0.150756
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	24.0000i	1.20301i
$399$	0	0
$400$	0	0
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	0	0
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−6.00000	−0.297775
$407$	− 2.00000i	− 0.0991363i
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	12.0000i	0.590481i
$414$	12.0000	0.589768
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	− 4.00000i	− 0.195646i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	4.00000i	0.194717i
$423$	− 12.0000i	− 0.583460i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	12.0000i	0.570782i
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	0	0
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	6.00000i	0.282216i
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	− 6.00000i	− 0.280362i
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	18.0000i	0.832050i
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	4.00000i	0.183920i
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 6.00000i	− 0.274721i
$478$	24.0000i	1.09773i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	28.0000i	1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$	28.0000i	1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	2.00000i	0.0905357i
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	24.0000	1.07981
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	0	0
$502$	20.0000i	0.892644i
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	4.00000i	0.175920i
$518$	2.00000i	0.0878750i
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	− 18.0000i	− 0.787839i
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−36.0000	−1.56227
$532$	4.00000i	0.173422i
$533$	36.0000i	1.55933i
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	2.00000i	0.0862261i
$539$	1.00000	0.0430730
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	18.0000i	0.768922i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	26.0000	1.10463
$555$	0	0
$556$	−20.0000	−0.848189
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	− 30.0000i	− 1.26547i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	20.0000	0.840663
$567$	− 9.00000i	− 0.377964i
$568$	8.00000i	0.335673i
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	−3.00000	−0.125000
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	2.00000i	0.0828315i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	14.0000	0.578335
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	− 24.0000i	− 0.981433i
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	− 4.00000i	− 0.163028i
$603$	24.0000i	0.977356i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	− 6.00000i	− 0.242536i
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	1.00000	0.0402911
$617$	− 34.0000i	− 1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$	− 34.0000i	− 1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.0000i	0.400642i
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	14.0000	0.556011
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	6.00000i	0.237542i
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−8.00000	−0.314756
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	− 9.00000i	− 0.353553i
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	− 8.00000i	− 0.313304i
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 30.0000i	− 1.17041i
$658$	− 4.00000i	− 0.155936i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	− 28.0000i	− 1.08825i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000i	0.929284i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	23.0000	0.884615
$677$	14.0000i	0.538064i	0.963131	+	0.269032i	$0.0867037\pi$
$677$	14.0000i	0.538064i	−0.963131	+	0.269032i	$0.913296\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	−12.0000	−0.458831
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	14.0000i	0.532200i
$693$	3.00000i	0.113961i
$694$	−28.0000	−1.06287
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	34.0000i	1.28692i
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	1.00000	0.0376889
$705$	0	0
$706$	14.0000	0.526897
$707$	− 14.0000i	− 0.526524i
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	10.0000i	0.374766i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	16.0000i	0.597115i
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	−6.00000	−0.222988
$725$	0	0
$726$	0	0
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	6.00000i	0.222375i
$729$	27.0000	1.00000
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	4.00000	0.147442
$737$	− 8.00000i	− 0.294684i
$738$	− 18.0000i	− 0.662589i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	− 2.00000i	− 0.0734223i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−34.0000	−1.24483
$747$	− 36.0000i	− 1.31717i
$748$	2.00000i	0.0731272i
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	− 4.00000i	− 0.145865i
$753$	0	0
$754$	−36.0000	−1.31104
$755$	0	0
$756$	0	0
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	12.0000i	0.435860i
$759$	0	0
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	14.0000i	0.506834i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	28.0000	1.01168
$767$	72.0000i	2.59977i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 6.00000i	− 0.215945i
$773$	30.0000i	1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$	30.0000i	1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	8.00000	0.286263
$782$	8.00000i	0.286079i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	3.00000i	0.106600i
$793$	12.0000i	0.426132i
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−24.0000	−0.850657
$797$	34.0000i	1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$	34.0000i	1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	−30.0000	−1.06000
$802$	34.0000i	1.20058i
$803$	10.0000i	0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 14.0000i	− 0.492518i
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	2.00000	0.0701000
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	− 34.0000i	− 1.18878i
$819$	−18.0000	−0.628971
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	12.0000i	0.417029i
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	0	0
$832$	6.00000i	0.208013i
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	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
$841$	7.00000	0.241379
$842$	− 26.0000i	− 0.896019i
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	12.0000	0.412568
$847$	− 1.00000i	− 0.0343604i
$848$	− 2.00000i	− 0.0686803i
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−4.00000	−0.136717
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	8.00000i	0.272481i
$863$	− 52.0000i	− 1.77010i	−0.465495	−	0.885050i	$-0.654124\pi$
$863$	− 52.0000i	− 1.77010i	0.465495	−	0.885050i	$-0.345876\pi$
$864$	0	0
$865$	0	0
$866$	6.00000	0.203888
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	48.0000	1.62642
$872$	14.0000i	0.474100i
$873$	18.0000i	0.609208i
$874$	16.0000	0.541208
$875$	0	0
$876$	0	0
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	24.0000i	0.809961i
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	− 3.00000i	− 0.101015i
$883$	56.0000i	1.88455i	0.334840	+	0.942275i	$0.391318\pi$
$883$	56.0000i	1.88455i	−0.334840	+	0.942275i	$0.608682\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−16.0000	−0.537531
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−9.00000	−0.301511
$892$	4.00000i	0.133930i
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	− 2.00000i	− 0.0667409i
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	6.00000i	0.199778i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	4.00000i	0.132745i
$909$	42.0000	1.39305
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	−26.0000	−0.860004
$915$	0	0
$916$	6.00000	0.198246
$917$	12.0000i	0.396275i
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	0	0
$922$	6.00000i	0.197599i
$923$	48.0000i	1.57994i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	− 12.0000i	− 0.394132i
$928$	− 6.00000i	− 0.196960i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	10.0000i	0.327561i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−18.0000	−0.588348
$937$	50.0000i	1.63343i	0.577042	+	0.816714i	$0.304207\pi$
$937$	50.0000i	1.63343i	−0.577042	+	0.816714i	$0.695793\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	0	0
$952$	− 2.00000i	− 0.0648204i
$953$	22.0000i	0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$	22.0000i	0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	24.0000i	0.775405i
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	12.0000i	0.386896i
$963$	− 12.0000i	− 0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	−28.0000	−0.897178
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	−42.0000	−1.34096
$982$	20.0000i	0.638226i
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	24.0000i	0.763542i
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	0	0
$994$	−8.00000	−0.253745
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 28.0000i	− 0.886325i
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} -4.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -2.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -3.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} -10.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +1.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +1.00000i q^{88} -10.0000 q^{89} -6.00000 q^{91} +4.00000i q^{92} +4.00000 q^{94} +6.00000i q^{97} -1.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^9	\( 2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 12 q^{26} - 12 q^{29} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 2 q^{44} + 8 q^{46} - 2 q^{49} - 2 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 16 q^{71} - 4 q^{74} - 8 q^{76} + 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} - 12 q^{91} + 8 q^{94} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 12 * q^26 - 12 * q^29 - 4 * q^34 - 6 * q^36 - 12 * q^41 + 2 * q^44 + 8 * q^46 - 2 * q^49 - 2 * q^56 - 24 * q^59 - 4 * q^61 - 2 * q^64 - 16 * q^71 - 4 * q^74 - 8 * q^76 + 16 * q^79 + 18 * q^81 + 8 * q^86 - 20 * q^89 - 12 * q^91 + 8 * q^94 - 6 * q^99

Introduction

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