LMFDB - Embedded newform 3850.2.c.d.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
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.d.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} +4.00000i q^{23} +2.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} -1.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} +8.00000 q^{39} -2.00000i q^{42} -4.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} -10.0000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +4.00000i q^{52} -14.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} -2.00000i q^{58} -10.0000 q^{59} -8.00000 q^{61} -10.0000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -8.00000i q^{67} -8.00000 q^{69} -4.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +1.00000i q^{77} +8.00000i q^{78} -16.0000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000i q^{87} -1.00000i q^{88} -10.0000 q^{89} +4.00000 q^{91} -4.00000i q^{92} -20.0000i q^{93} +10.0000 q^{94} -2.00000 q^{96} -6.00000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{19} - 4 q^{21} + 4 q^{24} + 8 q^{26} - 4 q^{29} - 20 q^{31} + 2 q^{36} + 16 q^{39} - 2 q^{44} - 8 q^{46} - 2 q^{49} - 8 q^{54} + 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} - 8 q^{71} - 12 q^{74} + 8 q^{76} - 32 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 20 q^{89} + 8 q^{91} + 20 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 - 8 * q^19 - 4 * q^21 + 4 * q^24 + 8 * q^26 - 4 * q^29 - 20 * q^31 + 2 * q^36 + 16 * q^39 - 2 * q^44 - 8 * q^46 - 2 * q^49 - 8 * q^54 + 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 - 4 * q^66 - 16 * q^69 - 8 * q^71 - 12 * q^74 + 8 * q^76 - 32 * q^79 - 22 * q^81 + 4 * q^84 + 8 * q^86 - 20 * q^89 + 8 * q^91 + 20 * 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	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−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
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	1.00000i	0.213201i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	4.00000	0.784465
$27$	4.00000i	0.769800i
$28$	− 1.00000i	− 0.188982i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000i	0.176777i
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 4.00000i	− 0.648886i
$39$	8.00000	1.28103
$40$	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
$46$	−4.00000	−0.589768
$47$	− 10.0000i	− 1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$	− 10.0000i	− 1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$	2.00000i	0.288675i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	4.00000i	0.554700i
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	− 8.00000i	− 1.05963i
$58$	− 2.00000i	− 0.262613i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	− 10.0000i	− 1.27000i
$63$	− 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	$-0.162521\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	1.00000i	0.117851i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	1.00000i	0.113961i
$78$	8.00000i	0.905822i
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	− 4.00000i	− 0.428845i
$88$	− 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
$91$	4.00000	0.419314
$92$	− 4.00000i	− 0.417029i
$93$	− 20.0000i	− 2.07390i
$94$	10.0000	1.03142
$95$	0	0
$96$	−2.00000	−0.204124
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	− 1.00000i	− 0.101015i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	14.0000	1.35980
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 4.00000i	− 0.384900i
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	1.00000i	0.0944911i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	2.00000	0.185695
$117$	4.00000i	0.369800i
$118$	− 10.0000i	− 0.920575i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	10.0000	0.898027
$125$	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
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	− 2.00000i	− 0.174078i
$133$	− 4.00000i	− 0.346844i
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 8.00000i	− 0.681005i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	20.0000	1.68430
$142$	− 4.00000i	− 0.335673i
$143$	− 4.00000i	− 0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 2.00000i	− 0.164957i
$148$	− 6.00000i	− 0.493197i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−8.00000	−0.640513
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 16.0000i	− 1.27289i
$159$	28.0000	2.22054
$160$	0	0
$161$	−4.00000	−0.315244
$162$	− 11.0000i	− 0.864242i
$163$	24.0000i	1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$	24.0000i	1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	0	0
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	2.00000i	0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000i	0.304997i
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	1.00000	0.0753778
$177$	− 20.0000i	− 1.50329i
$178$	− 10.0000i	− 0.749532i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	4.00000i	0.296500i
$183$	− 16.0000i	− 1.18275i
$184$	4.00000	0.294884
$185$	0	0
$186$	20.0000	1.46647
$187$	0	0
$188$	10.0000i	0.729325i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 2.00000i	− 0.144338i
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	12.0000i	0.844317i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	−2.00000	−0.139347
$207$	− 4.00000i	− 0.278019i
$208$	− 4.00000i	− 0.277350i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	14.0000i	0.961524i
$213$	− 8.00000i	− 0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	4.00000	0.272166
$217$	− 10.0000i	− 0.678844i
$218$	14.0000i	0.948200i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	0	0
$222$	− 12.0000i	− 0.805387i
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	8.00000i	0.529813i
$229$	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
$232$	2.00000i	0.131306i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	10.0000	0.650945
$237$	− 32.0000i	− 2.07862i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	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
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	10.0000i	0.635001i
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	1.00000i	0.0629941i
$253$	4.00000i	0.251478i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	8.00000i	0.498058i
$259$	−6.00000	−0.372822
$260$	0	0
$261$	2.00000	0.123797
$262$	8.00000i	0.494242i
$263$	− 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
$266$	4.00000	0.245256
$267$	− 20.0000i	− 1.22398i
$268$	8.00000i	0.488678i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	6.00000	0.362473
$275$	0	0
$276$	8.00000	0.481543
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	− 20.0000i	− 1.19952i
$279$	10.0000	0.598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	20.0000i	1.19098i
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	12.0000	0.703452
$292$	− 4.00000i	− 0.234082i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	6.00000	0.348743
$297$	4.00000i	0.232104i
$298$	− 22.0000i	− 1.27443i
$299$	16.0000	0.925304
$300$	0	0
$301$	4.00000	0.230556
$302$	16.0000i	0.920697i
$303$	24.0000i	1.37876i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	− 8.00000i	− 0.452911i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	16.0000	0.900070
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	28.0000i	1.57016i
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−24.0000	−1.33955
$322$	− 4.00000i	− 0.222911i
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	−24.0000	−1.32924
$327$	28.0000i	1.54840i
$328$	0	0
$329$	10.0000	0.551318
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−2.00000	−0.109109
$337$	34.0000i	1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$	34.0000i	1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$	− 3.00000i	− 0.163178i
$339$	28.0000	1.52075
$340$	0	0
$341$	−10.0000	−0.541530
$342$	4.00000i	0.216295i
$343$	− 1.00000i	− 0.0539949i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−4.00000	−0.215041
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	4.00000i	0.214423i
$349$	−32.0000	−1.71292	−0.856460	−	0.516213i	$-0.827341\pi$
$349$	−32.0000	−1.71292	−0.856460	+	0.516213i	$0.827341\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	1.00000i	0.0533002i
$353$	2.00000i	0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$	2.00000i	0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$	20.0000	1.06299
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	14.0000i	0.735824i
$363$	2.00000i	0.104973i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	16.0000	0.836333
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	14.0000	0.726844
$372$	20.0000i	1.03695i
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	0	0
$375$	0	0
$376$	−10.0000	−0.515711
$377$	8.00000i	0.412021i
$378$	− 4.00000i	− 0.205738i
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−32.0000	−1.63941
$382$	8.00000i	0.409316i
$383$	14.0000i	0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$	14.0000i	0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	6.00000	0.305392
$387$	4.00000i	0.203331i
$388$	6.00000i	0.304604i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	0	0
$392$	1.00000i	0.0505076i
$393$	16.0000i	0.807093i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	1.00000	0.0502519
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	14.0000i	0.701757i
$399$	8.00000	0.400501
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	16.0000i	0.798007i
$403$	40.0000i	1.99254i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	2.00000	0.0992583
$407$	6.00000i	0.297409i
$408$	0	0
$409$	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
$412$	− 2.00000i	− 0.0985329i
$413$	− 10.0000i	− 0.492068i
$414$	4.00000	0.196589
$415$	0	0
$416$	4.00000	0.196116
$417$	− 40.0000i	− 1.95881i
$418$	− 4.00000i	− 0.195646i
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	− 4.00000i	− 0.194717i
$423$	10.0000i	0.486217i
$424$	−14.0000	−0.679900
$425$	0	0
$426$	8.00000	0.387601
$427$	− 8.00000i	− 0.387147i
$428$	− 12.0000i	− 0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	4.00000i	0.192450i
$433$	− 10.0000i	− 0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$	− 10.0000i	− 0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	−14.0000	−0.670478
$437$	− 16.0000i	− 0.765384i
$438$	− 8.00000i	− 0.382255i
$439$	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
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	12.0000	0.569495
$445$	0	0
$446$	14.0000	0.662919
$447$	− 44.0000i	− 2.08113i
$448$	− 1.00000i	− 0.0472456i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	14.0000i	0.658505i
$453$	32.0000i	1.50349i
$454$	8.00000	0.375459
$455$	0	0
$456$	−8.00000	−0.374634
$457$	38.0000i	1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$	38.0000i	1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$	10.0000i	0.467269i
$459$	0	0
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	− 2.00000i	− 0.0930484i
$463$	12.0000i	0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$	12.0000i	0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 14.0000i	− 0.647843i	−0.946084	−	0.323921i	$-0.894999\pi$
$467$	− 14.0000i	− 0.647843i	0.946084	−	0.323921i	$-0.105001\pi$
$468$	− 4.00000i	− 0.184900i
$469$	8.00000	0.369406
$470$	0	0
$471$	20.0000	0.921551
$472$	10.0000i	0.460287i
$473$	− 4.00000i	− 0.183920i
$474$	32.0000	1.46981
$475$	0	0
$476$	0	0
$477$	14.0000i	0.641016i
$478$	8.00000i	0.365911i
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	8.00000i	0.364390i
$483$	− 8.00000i	− 0.364013i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	10.0000	0.453609
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	8.00000i	0.362143i
$489$	−48.0000	−2.17064
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	0	0
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 4.00000i	− 0.179425i
$498$	− 8.00000i	− 0.358489i
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	− 26.0000i	− 1.16044i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−4.00000	−0.177822
$507$	− 6.00000i	− 0.266469i
$508$	− 16.0000i	− 0.709885i
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000i	0.0441942i
$513$	− 16.0000i	− 0.706417i
$514$	2.00000	0.0882162
$515$	0	0
$516$	−8.00000	−0.352180
$517$	− 10.0000i	− 0.439799i
$518$	− 6.00000i	− 0.263625i
$519$	−8.00000	−0.351161
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	2.00000i	0.0875376i
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	7.00000	0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	4.00000i	0.173422i
$533$	0	0
$534$	20.0000	0.865485
$535$	0	0
$536$	−8.00000	−0.345547
$537$	− 24.0000i	− 1.03568i
$538$	− 14.0000i	− 0.603583i
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	− 28.0000i	− 1.20270i
$543$	28.0000i	1.20160i
$544$	0	0
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	6.00000i	0.256307i
$549$	8.00000	0.341432
$550$	0	0
$551$	8.00000	0.340811
$552$	8.00000i	0.340503i
$553$	− 16.0000i	− 0.680389i
$554$	6.00000	0.254916
$555$	0	0
$556$	20.0000	0.848189
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	10.0000i	0.423334i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	30.0000i	1.26547i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	−20.0000	−0.842152
$565$	0	0
$566$	0	0
$567$	− 11.0000i	− 0.461957i
$568$	4.00000i	0.167836i
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	4.00000i	0.167248i
$573$	16.0000i	0.668410i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	42.0000i	1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$	42.0000i	1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$	17.0000i	0.707107i
$579$	12.0000	0.498703
$580$	0	0
$581$	−4.00000	−0.165948
$582$	12.0000i	0.497416i
$583$	− 14.0000i	− 0.579821i
$584$	4.00000	0.165521
$585$	0	0
$586$	0	0
$587$	42.0000i	1.73353i	0.498721	+	0.866763i	$0.333803\pi$
$587$	42.0000i	1.73353i	−0.498721	+	0.866763i	$0.666197\pi$
$588$	2.00000i	0.0824786i
$589$	40.0000	1.64817
$590$	0	0
$591$	−36.0000	−1.48084
$592$	6.00000i	0.246598i
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	22.0000	0.901155
$597$	28.0000i	1.14596i
$598$	16.0000i	0.654289i
$599$	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
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	4.00000i	0.163028i
$603$	8.00000i	0.325785i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−24.0000	−0.974933
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	− 4.00000i	− 0.162221i
$609$	4.00000	0.162088
$610$	0	0
$611$	−40.0000	−1.61823
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	1.00000	0.0402911
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	− 4.00000i	− 0.160904i
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	− 6.00000i	− 0.240578i
$623$	− 10.0000i	− 0.400642i
$624$	8.00000	0.320256
$625$	0	0
$626$	6.00000	0.239808
$627$	− 8.00000i	− 0.319489i
$628$	10.0000i	0.399043i
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	16.0000i	0.636446i
$633$	− 8.00000i	− 0.317971i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−28.0000	−1.11027
$637$	4.00000i	0.158486i
$638$	− 2.00000i	− 0.0791808i
$639$	4.00000	0.158238
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	− 24.0000i	− 0.947204i
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	11.0000i	0.432121i
$649$	−10.0000	−0.392534
$650$	0	0
$651$	20.0000	0.783862
$652$	− 24.0000i	− 0.939913i
$653$	46.0000i	1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$	46.0000i	1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$	−28.0000	−1.09489
$655$	0	0
$656$	0	0
$657$	− 4.00000i	− 0.156055i
$658$	10.0000i	0.389841i
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	− 8.00000i	− 0.309761i
$668$	− 8.00000i	− 0.309529i
$669$	28.0000	1.08254
$670$	0	0
$671$	−8.00000	−0.308837
$672$	− 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	$-0.0617360\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	3.00000	0.115385
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	28.0000i	1.07533i
$679$	6.00000	0.230259
$680$	0	0
$681$	16.0000	0.613121
$682$	− 10.0000i	− 0.382920i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	1.00000	0.0381802
$687$	20.0000i	0.763048i
$688$	− 4.00000i	− 0.152499i
$689$	−56.0000	−2.13343
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	− 4.00000i	− 0.152057i
$693$	− 1.00000i	− 0.0379869i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−4.00000	−0.151620
$697$	0	0
$698$	− 32.0000i	− 1.21122i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	16.0000i	0.603881i
$703$	− 24.0000i	− 0.905177i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	12.0000i	0.451306i
$708$	20.0000i	0.751646i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	10.0000i	0.374766i
$713$	− 40.0000i	− 1.49801i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	16.0000i	0.597531i
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	− 3.00000i	− 0.111648i
$723$	16.0000i	0.595046i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	− 46.0000i	− 1.70605i	−0.521874	−	0.853023i	$-0.674767\pi$
$727$	− 46.0000i	− 1.70605i	0.521874	−	0.853023i	$-0.325233\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	16.0000i	0.591377i
$733$	− 8.00000i	− 0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$	− 8.00000i	− 0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	14.0000i	0.513956i
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	−20.0000	−0.733236
$745$	0	0
$746$	34.0000	1.24483
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	− 10.0000i	− 0.364662i
$753$	− 52.0000i	− 1.89499i
$754$	−8.00000	−0.291343
$755$	0	0
$756$	4.00000	0.145479
$757$	− 30.0000i	− 1.09037i	−0.838316	−	0.545184i	$-0.816460\pi$
$757$	− 30.0000i	− 1.09037i	0.838316	−	0.545184i	$-0.183540\pi$
$758$	− 8.00000i	− 0.290573i
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	− 32.0000i	− 1.15924i
$763$	14.0000i	0.506834i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−14.0000	−0.505841
$767$	40.0000i	1.44432i
$768$	2.00000i	0.0721688i
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	6.00000i	0.215945i
$773$	− 34.0000i	− 1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	− 34.0000i	− 1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−6.00000	−0.215387
$777$	− 12.0000i	− 0.430498i
$778$	18.0000i	0.645331i
$779$	0	0
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	− 8.00000i	− 0.285897i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−16.0000	−0.570701
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	− 18.0000i	− 0.641223i
$789$	48.0000	1.70885
$790$	0	0
$791$	14.0000	0.497783
$792$	1.00000i	0.0355335i
$793$	32.0000i	1.13635i
$794$	18.0000	0.638796
$795$	0	0
$796$	−14.0000	−0.496217
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	8.00000i	0.283197i
$799$	0	0
$800$	0	0
$801$	10.0000	0.353333
$802$	10.0000i	0.353112i
$803$	4.00000i	0.141157i
$804$	−16.0000	−0.564276
$805$	0	0
$806$	−40.0000	−1.40894
$807$	− 28.0000i	− 0.985647i
$808$	− 12.0000i	− 0.422159i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	2.00000i	0.0701862i
$813$	− 56.0000i	− 1.96401i
$814$	−6.00000	−0.210300
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	4.00000i	0.139857i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	12.0000i	0.418548i
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	2.00000	0.0696733
$825$	0	0
$826$	10.0000	0.347945
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	4.00000i	0.139010i
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	4.00000i	0.138675i
$833$	0	0
$834$	40.0000	1.38509
$835$	0	0
$836$	4.00000	0.138343
$837$	− 40.0000i	− 1.38260i
$838$	30.0000i	1.03633i
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	10.0000i	0.344623i
$843$	60.0000i	2.06651i
$844$	4.00000	0.137686
$845$	0	0
$846$	−10.0000	−0.343807
$847$	1.00000i	0.0343604i
$848$	− 14.0000i	− 0.480762i
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	8.00000i	0.274075i
$853$	4.00000i	0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$	4.00000i	0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	12.0000	0.410152
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	8.00000i	0.273115i
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	− 16.0000i	− 0.544962i
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	10.0000	0.339814
$867$	34.0000i	1.15470i
$868$	10.0000i	0.339422i
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−32.0000	−1.08428
$872$	− 14.0000i	− 0.474100i
$873$	6.00000i	0.203069i
$874$	16.0000	0.541208
$875$	0	0
$876$	8.00000	0.270295
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	28.0000i	0.944954i
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	1.00000i	0.0336718i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	−4.00000	−0.134383
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	12.0000i	0.402694i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−11.0000	−0.368514
$892$	14.0000i	0.468755i
$893$	40.0000i	1.33855i
$894$	44.0000	1.47158
$895$	0	0
$896$	1.00000	0.0334077
$897$	32.0000i	1.06845i
$898$	− 6.00000i	− 0.200223i
$899$	20.0000	0.667037
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.00000i	0.266223i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−32.0000	−1.06313
$907$	− 48.0000i	− 1.59381i	−0.604102	−	0.796907i	$-0.706468\pi$
$907$	− 48.0000i	− 1.59381i	0.604102	−	0.796907i	$-0.293532\pi$
$908$	8.00000i	0.265489i
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	− 8.00000i	− 0.264906i
$913$	4.00000i	0.132381i
$914$	−38.0000	−1.25693
$915$	0	0
$916$	−10.0000	−0.330409
$917$	8.00000i	0.264183i
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	32.0000i	1.05386i
$923$	16.0000i	0.526646i
$924$	2.00000	0.0657952
$925$	0	0
$926$	−12.0000	−0.394344
$927$	− 2.00000i	− 0.0656886i
$928$	− 2.00000i	− 0.0656532i
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	− 6.00000i	− 0.196537i
$933$	− 12.0000i	− 0.392862i
$934$	14.0000	0.458094
$935$	0	0
$936$	4.00000	0.130744
$937$	52.0000i	1.69877i	0.527777	+	0.849383i	$0.323026\pi$
$937$	52.0000i	1.69877i	−0.527777	+	0.849383i	$0.676974\pi$
$938$	8.00000i	0.261209i
$939$	12.0000	0.391605
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	20.0000i	0.651635i
$943$	0	0
$944$	−10.0000	−0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	32.0000i	1.03931i
$949$	16.0000	0.519382
$950$	0	0
$951$	−36.0000	−1.16738
$952$	0	0
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	−14.0000	−0.453267
$955$	0	0
$956$	−8.00000	−0.258738
$957$	− 4.00000i	− 0.129302i
$958$	− 12.0000i	− 0.387702i
$959$	6.00000	0.193750
$960$	0	0
$961$	69.0000	2.22581
$962$	24.0000i	0.773791i
$963$	− 12.0000i	− 0.386695i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	8.00000	0.257396
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	10.0000i	0.320750i
$973$	− 20.0000i	− 0.641171i
$974$	12.0000	0.384505
$975$	0	0
$976$	−8.00000	−0.256074
$977$	22.0000i	0.703842i	0.936030	+	0.351921i	$0.114471\pi$
$977$	22.0000i	0.703842i	−0.936030	+	0.351921i	$0.885529\pi$
$978$	− 48.0000i	− 1.53487i
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−14.0000	−0.446986
$982$	− 28.0000i	− 0.893516i
$983$	26.0000i	0.829271i	0.909988	+	0.414636i	$0.136091\pi$
$983$	26.0000i	0.829271i	−0.909988	+	0.414636i	$0.863909\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	20.0000i	0.636607i
$988$	− 16.0000i	− 0.509028i
$989$	16.0000	0.508770
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	− 10.0000i	− 0.317500i
$993$	− 40.0000i	− 1.26936i
$994$	4.00000	0.126872
$995$	0	0
$996$	8.00000	0.253490
$997$	− 36.0000i	− 1.14013i	−0.821599	−	0.570066i	$-0.806918\pi$
$997$	− 36.0000i	− 1.14013i	0.821599	−	0.570066i	$-0.193082\pi$
$998$	16.0000i	0.506471i
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.d.1849.2		2
5.2	odd	4		154.2.a.b.1.1	✓	1
5.3	odd	4		3850.2.a.o.1.1		1
5.4	even	2	inner	3850.2.c.d.1849.1		2
15.2	even	4		1386.2.a.f.1.1		1
20.7	even	4		1232.2.a.c.1.1		1
35.2	odd	12		1078.2.e.h.67.1		2
35.12	even	12		1078.2.e.l.67.1		2
35.17	even	12		1078.2.e.l.177.1		2
35.27	even	4		1078.2.a.b.1.1		1
35.32	odd	12		1078.2.e.h.177.1		2
40.27	even	4		4928.2.a.bf.1.1		1
40.37	odd	4		4928.2.a.d.1.1		1
55.32	even	4		1694.2.a.i.1.1		1
105.62	odd	4		9702.2.a.bz.1.1		1
140.27	odd	4		8624.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.b.1.1	✓	1	5.2	odd	4
1078.2.a.b.1.1		1	35.27	even	4
1078.2.e.h.67.1		2	35.2	odd	12
1078.2.e.h.177.1		2	35.32	odd	12
1078.2.e.l.67.1		2	35.12	even	12
1078.2.e.l.177.1		2	35.17	even	12
1232.2.a.c.1.1		1	20.7	even	4
1386.2.a.f.1.1		1	15.2	even	4
1694.2.a.i.1.1		1	55.32	even	4
3850.2.a.o.1.1		1	5.3	odd	4
3850.2.c.d.1849.1		2	5.4	even	2	inner
3850.2.c.d.1849.2		2	1.1	even	1	trivial
4928.2.a.d.1.1		1	40.37	odd	4
4928.2.a.bf.1.1		1	40.27	even	4
8624.2.a.z.1.1		1	140.27	odd	4
9702.2.a.bz.1.1		1	105.62	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} +4.00000i q^{23} +2.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} -1.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} +8.00000 q^{39} -2.00000i q^{42} -4.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} -10.0000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +4.00000i q^{52} -14.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} -2.00000i q^{58} -10.0000 q^{59} -8.00000 q^{61} -10.0000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -8.00000i q^{67} -8.00000 q^{69} -4.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +1.00000i q^{77} +8.00000i q^{78} -16.0000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000i q^{87} -1.00000i q^{88} -10.0000 q^{89} +4.00000 q^{91} -4.00000i q^{92} -20.0000i q^{93} +10.0000 q^{94} -2.00000 q^{96} -6.00000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 4 q^{24} + 8 q^{26} - 4 q^{29} - 20 q^{31} + 2 q^{36} + 16 q^{39} - 2 q^{44} - 8 q^{46} - 2 q^{49} - 8 q^{54} + 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} - 8 q^{71} - 12 q^{74} + 8 q^{76} - 32 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 20 q^{89} + 8 q^{91} + 20 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 2 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 4 * q^24 + 8 * q^26 - 4 * q^29 - 20 * q^31 + 2 * q^36 + 16 * q^39 - 2 * q^44 - 8 * q^46 - 2 * q^49 - 8 * q^54 + 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 - 4 * q^66 - 16 * q^69 - 8 * q^71 - 12 * q^74 + 8 * q^76 - 32 * q^79 - 22 * q^81 + 4 * q^84 + 8 * q^86 - 20 * q^89 + 8 * q^91 + 20 * q^94 - 4 * q^96 - 2 * q^99

Introduction

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