LMFDB - Embedded newform 3850.2.c.b.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.b.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} +2.00000 q^{24} -4.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -8.00000 q^{39} -12.0000 q^{41} +2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} -6.00000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +4.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} -4.00000 q^{61} +10.0000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +4.00000i q^{67} +12.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +1.00000i q^{77} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +2.00000 q^{84} -4.00000 q^{86} -12.0000i q^{87} -1.00000i q^{88} -18.0000 q^{89} -4.00000 q^{91} +20.0000i q^{93} -6.00000 q^{94} -2.00000 q^{96} +10.0000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{29} - 20 q^{31} + 2 q^{36} - 16 q^{39} - 24 q^{41} + 2 q^{44} - 2 q^{49} - 8 q^{54} + 2 q^{56} + 12 q^{59} - 8 q^{61} - 2 q^{64} + 4 q^{66} + 24 q^{71} - 4 q^{74} - 8 q^{76} - 16 q^{79} - 22 q^{81} + 4 q^{84} - 8 q^{86} - 36 q^{89} - 8 q^{91} - 12 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 + 12 * q^29 - 20 * q^31 + 2 * q^36 - 16 * q^39 - 24 * q^41 + 2 * q^44 - 2 * q^49 - 8 * q^54 + 2 * q^56 + 12 * q^59 - 8 * q^61 - 2 * q^64 + 4 * q^66 + 24 * q^71 - 4 * q^74 - 8 * q^76 - 16 * q^79 - 22 * q^81 + 4 * q^84 - 8 * q^86 - 36 * q^89 - 8 * q^91 - 12 * q^94 - 4 * q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.00000i	0.577350i
$13$	− 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	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	1.00000i	0.213201i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	−4.00000	−0.784465
$27$	− 4.00000i	− 0.769800i
$28$	1.00000i	0.188982i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	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
$46$	0	0
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	4.00000i	0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	− 8.00000i	− 1.05963i
$58$	− 6.00000i	− 0.787839i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	10.0000i	1.27000i
$63$	1.00000i	0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	1.00000i	0.113961i
$78$	8.00000i	0.905822i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	12.0000i	1.32518i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 12.0000i	− 1.28654i
$88$	− 1.00000i	− 0.106600i
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	20.0000i	2.07390i
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−2.00000	−0.204124
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	1.00000i	0.101015i
$99$	1.00000	0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 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	$-0.0910696\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	−6.00000	−0.557086
$117$	4.00000i	0.369800i
$118$	− 6.00000i	− 0.552345i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.00000i	0.362143i
$123$	24.0000i	2.16401i
$124$	10.0000	0.898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 2.00000i	− 0.174078i
$133$	− 4.00000i	− 0.346844i
$134$	4.00000	0.345547
$135$	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	$-0.0825059\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	− 12.0000i	− 1.00702i
$143$	4.00000i	0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	2.00000i	0.164957i
$148$	2.00000i	0.164399i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	8.00000	0.640513
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	8.00000i	0.636446i
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	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	$-0.0500701\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	12.0000	0.931381
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000i	0.304997i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 12.0000i	− 0.901975i
$178$	18.0000i	1.34916i
$179$	−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
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.00000i	0.296500i
$183$	8.00000i	0.591377i
$184$	0	0
$185$	0	0
$186$	20.0000	1.46647
$187$	0	0
$188$	6.00000i	0.437595i
$189$	−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	$0.831909\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	14.0000	0.975426
$207$	0	0
$208$	− 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	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	− 24.0000i	− 1.64445i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	4.00000	0.272166
$217$	10.0000i	0.678844i
$218$	2.00000i	0.135457i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	0	0
$222$	4.00000i	0.268462i
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	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
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−6.00000	−0.390567
$237$	16.0000i	1.03931i
$238$	0	0
$239$	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
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	10.0000i	0.641500i
$244$	4.00000	0.256074
$245$	0	0
$246$	24.0000	1.53018
$247$	− 16.0000i	− 1.01806i
$248$	− 10.0000i	− 0.635001i
$249$	24.0000	1.52094
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	0	0
$254$	−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	$-0.189739\pi$
$258$	8.00000i	0.498058i
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	−4.00000	−0.245256
$267$	36.0000i	2.20316i
$268$	− 4.00000i	− 0.244339i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 4.00000i	− 0.239904i
$279$	10.0000	0.598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	12.0000i	0.714590i
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	4.00000	0.236525
$287$	12.0000i	0.708338i
$288$	1.00000i	0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	20.0000	1.17242
$292$	4.00000i	0.234082i
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	2.00000	0.116248
$297$	4.00000i	0.232104i
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	16.0000i	0.920697i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 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
$309$	28.0000	1.59286
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	− 8.00000i	− 0.452911i
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	12.0000i	0.672927i
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	4.00000i	0.221201i
$328$	− 12.0000i	− 0.662589i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 12.0000i	− 0.658586i
$333$	2.00000i	0.109599i
$334$	−24.0000	−1.31322
$335$	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	$0.795482\pi$
$338$	3.00000i	0.163178i
$339$	−12.0000	−0.651751
$340$	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
$346$	12.0000	0.645124
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	12.0000i	0.643268i
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	1.00000i	0.0533002i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	12.0000i	0.634220i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000i	0.525588i
$363$	− 2.00000i	− 0.104973i
$364$	4.00000	0.209657
$365$	0	0
$366$	8.00000	0.418167
$367$	34.0000i	1.77479i	0.461014	+	0.887393i	$0.347486\pi$
$367$	34.0000i	1.77479i	−0.461014	+	0.887393i	$0.652514\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	−6.00000	−0.311504
$372$	− 20.0000i	− 1.03695i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	− 24.0000i	− 1.23606i
$378$	4.00000i	0.205738i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	4.00000i	0.203331i
$388$	− 10.0000i	− 0.507673i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	6.00000	0.302276
$395$	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	$-0.0159822\pi$
$398$	2.00000i	0.100251i
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	− 8.00000i	− 0.399004i
$403$	40.0000i	1.99254i
$404$	0	0
$405$	0	0
$406$	−6.00000	−0.297775
$407$	2.00000i	0.0991363i
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	− 14.0000i	− 0.689730i
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	− 8.00000i	− 0.391762i
$418$	4.00000i	0.195646i
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	4.00000i	0.194717i
$423$	6.00000i	0.291730i
$424$	6.00000	0.291386
$425$	0	0
$426$	−24.0000	−1.16280
$427$	4.00000i	0.193574i
$428$	12.0000i	0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	− 4.00000i	− 0.192450i
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	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
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−10.0000	−0.473514
$447$	− 36.0000i	− 1.70274i
$448$	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
$451$	12.0000	0.565058
$452$	6.00000i	0.282216i
$453$	32.0000i	1.50349i
$454$	24.0000	1.12638
$455$	0	0
$456$	8.00000	0.374634
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	− 2.00000i	− 0.0930484i
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 42.0000i	− 1.94353i	−0.235954	−	0.971764i	$-0.575822\pi$
$467$	− 42.0000i	− 1.94353i	0.235954	−	0.971764i	$-0.424178\pi$
$468$	− 4.00000i	− 0.184900i
$469$	4.00000	0.184703
$470$	0	0
$471$	44.0000	2.02741
$472$	6.00000i	0.276172i
$473$	4.00000i	0.183920i
$474$	16.0000	0.734904
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	− 24.0000i	− 1.09773i
$479$	−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
$481$	−8.00000	−0.364769
$482$	4.00000i	0.182195i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	10.0000	0.453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	− 4.00000i	− 0.181071i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	− 24.0000i	− 1.08200i
$493$	0	0
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 12.0000i	− 0.538274i
$498$	− 24.0000i	− 1.07547i
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	18.0000i	0.803379i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	0	0
$507$	6.00000i	0.266469i
$508$	8.00000i	0.354943i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	− 16.0000i	− 0.706417i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	8.00000	0.352180
$517$	6.00000i	0.263880i
$518$	2.00000i	0.0878750i
$519$	24.0000	1.05348
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	6.00000i	0.262613i
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	23.0000	1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	4.00000i	0.173422i
$533$	48.0000i	2.07911i
$534$	36.0000	1.55787
$535$	0	0
$536$	−4.00000	−0.172774
$537$	24.0000i	1.03568i
$538$	6.00000i	0.258678i
$539$	1.00000	0.0430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 20.0000i	− 0.859074i
$543$	20.0000i	0.858282i
$544$	0	0
$545$	0	0
$546$	8.00000	0.342368
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	6.00000i	0.256307i
$549$	4.00000	0.170716
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	8.00000i	0.340195i
$554$	22.0000	0.934690
$555$	0	0
$556$	−4.00000	−0.169638
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	− 10.0000i	− 0.423334i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−16.0000	−0.672530
$567$	11.0000i	0.461957i
$568$	12.0000i	0.503509i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	− 17.0000i	− 0.707107i
$579$	28.0000	1.16364
$580$	0	0
$581$	12.0000	0.497844
$582$	− 20.0000i	− 0.829027i
$583$	6.00000i	0.248495i
$584$	4.00000	0.165521
$585$	0	0
$586$	24.0000	0.991431
$587$	− 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
$589$	−40.0000	−1.64817
$590$	0	0
$591$	12.0000	0.493614
$592$	− 2.00000i	− 0.0821995i
$593$	− 12.0000i	− 0.492781i	−0.969171	−	0.246390i	$-0.920755\pi$
$593$	− 12.0000i	− 0.492781i	0.969171	−	0.246390i	$-0.0792446\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−18.0000	−0.737309
$597$	4.00000i	0.163709i
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	4.00000i	0.163028i
$603$	− 4.00000i	− 0.162893i
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	− 4.00000i	− 0.162221i
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	− 10.0000i	− 0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	− 10.0000i	− 0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 28.0000i	− 1.12633i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	18.0000i	0.721155i
$624$	−8.00000	−0.320256
$625$	0	0
$626$	2.00000	0.0799361
$627$	8.00000i	0.319489i
$628$	− 22.0000i	− 0.877896i
$629$	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	$0.449096\pi$
$632$	− 8.00000i	− 0.318223i
$633$	8.00000i	0.317971i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	12.0000	0.475831
$637$	4.00000i	0.158486i
$638$	6.00000i	0.237542i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	24.0000i	0.947204i
$643$	2.00000i	0.0788723i	0.999222	+	0.0394362i	$0.0125562\pi$
$643$	2.00000i	0.0788723i	−0.999222	+	0.0394362i	$0.987444\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	− 11.0000i	− 0.432121i
$649$	−6.00000	−0.235521
$650$	0	0
$651$	20.0000	0.783862
$652$	4.00000i	0.156652i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	−12.0000	−0.468521
$657$	4.00000i	0.156055i
$658$	6.00000i	0.233904i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	24.0000i	0.928588i
$669$	−20.0000	−0.773245
$670$	0	0
$671$	4.00000	0.154418
$672$	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
$676$	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	$-0.0740685\pi$
$678$	12.0000i	0.460857i
$679$	10.0000	0.383765
$680$	0	0
$681$	48.0000	1.83936
$682$	− 10.0000i	− 0.382920i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	1.00000	0.0381802
$687$	− 20.0000i	− 0.763048i
$688$	− 4.00000i	− 0.152499i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	− 12.0000i	− 0.456172i
$693$	− 1.00000i	− 0.0379869i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	12.0000	0.454859
$697$	0	0
$698$	20.0000i	0.757011i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	16.0000i	0.603881i
$703$	− 8.00000i	− 0.301726i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	12.0000i	0.450988i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 18.0000i	− 0.674579i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 48.0000i	− 1.79259i
$718$	− 24.0000i	− 0.895672i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	3.00000i	0.111648i
$723$	8.00000i	0.297523i
$724$	10.0000	0.371647
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	− 2.00000i	− 0.0741759i	−0.999312	−	0.0370879i	$-0.988192\pi$
$727$	− 2.00000i	− 0.0741759i	0.999312	−	0.0370879i	$-0.0118082\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	− 8.00000i	− 0.295689i
$733$	− 40.0000i	− 1.47743i	−0.674016	−	0.738717i	$-0.735432\pi$
$733$	− 40.0000i	− 1.47743i	0.674016	−	0.738717i	$-0.264568\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	0	0
$737$	− 4.00000i	− 0.147342i
$738$	− 12.0000i	− 0.441726i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	6.00000i	0.220267i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	−20.0000	−0.733236
$745$	0	0
$746$	−22.0000	−0.805477
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	− 6.00000i	− 0.218797i
$753$	36.0000i	1.31191i
$754$	−24.0000	−0.874028
$755$	0	0
$756$	4.00000	0.145479
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	− 16.0000i	− 0.581146i
$759$	0	0
$760$	0	0
$761$	24.0000	0.869999	0.435000	−	0.900431i	$-0.356748\pi$
$761$	24.0000	0.869999	0.435000	+	0.900431i	$0.356748\pi$
$762$	16.0000i	0.579619i
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	0	0
$766$	−6.00000	−0.216789
$767$	− 24.0000i	− 0.866590i
$768$	− 2.00000i	− 0.0721688i
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	− 14.0000i	− 0.503871i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	4.00000i	0.143499i
$778$	30.0000i	1.07555i
$779$	−48.0000	−1.71978
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	− 24.0000i	− 0.857690i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	1.00000i	0.0355335i
$793$	16.0000i	0.568177i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	2.00000	0.0708881
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	8.00000i	0.283197i
$799$	0	0
$800$	0	0
$801$	18.0000	0.635999
$802$	6.00000i	0.211867i
$803$	4.00000i	0.141157i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	40.0000	1.40894
$807$	12.0000i	0.422420i
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	6.00000i	0.210559i
$813$	− 40.0000i	− 1.40286i
$814$	2.00000	0.0701000
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	32.0000i	1.11885i
$819$	4.00000	0.139771
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	12.0000i	0.418548i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−6.00000	−0.208767
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	4.00000i	0.138675i
$833$	0	0
$834$	−8.00000	−0.277017
$835$	0	0
$836$	4.00000	0.138343
$837$	40.0000i	1.38260i
$838$	− 6.00000i	− 0.207267i
$839$	−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
$841$	7.00000	0.241379
$842$	− 26.0000i	− 0.896019i
$843$	36.0000i	1.23991i
$844$	4.00000	0.137686
$845$	0	0
$846$	6.00000	0.206284
$847$	− 1.00000i	− 0.0343604i
$848$	− 6.00000i	− 0.206041i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	0	0
$852$	24.0000i	0.822226i
$853$	− 4.00000i	− 0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$	− 4.00000i	− 0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	12.0000	0.410152
$857$	12.0000i	0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$	12.0000i	0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$	− 8.00000i	− 0.273115i
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	− 24.0000i	− 0.817443i
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	14.0000	0.475739
$867$	− 34.0000i	− 1.15470i
$868$	− 10.0000i	− 0.339422i
$869$	8.00000	0.271381
$870$	0	0
$871$	16.0000	0.542139
$872$	− 2.00000i	− 0.0677285i
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	− 28.0000i	− 0.944954i
$879$	48.0000	1.61900
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	− 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	$-0.0867663\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	− 4.00000i	− 0.134231i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	11.0000	0.368514
$892$	10.0000i	0.334825i
$893$	− 24.0000i	− 0.803129i
$894$	−36.0000	−1.20402
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	6.00000i	0.200223i
$899$	−60.0000	−2.00111
$900$	0	0
$901$	0	0
$902$	− 12.0000i	− 0.399556i
$903$	8.00000i	0.266223i
$904$	6.00000	0.199557
$905$	0	0
$906$	32.0000	1.06313
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	− 24.0000i	− 0.796468i
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	− 8.00000i	− 0.264906i
$913$	− 12.0000i	− 0.397142i
$914$	10.0000	0.330771
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−64.0000	−2.10887
$922$	36.0000i	1.18560i
$923$	− 48.0000i	− 1.57994i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	−40.0000	−1.31448
$927$	− 14.0000i	− 0.459820i
$928$	− 6.00000i	− 0.196960i
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	6.00000i	0.196537i
$933$	− 36.0000i	− 1.17859i
$934$	−42.0000	−1.37428
$935$	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	$0.676974\pi$
$938$	− 4.00000i	− 0.130605i
$939$	4.00000	0.130535
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	− 44.0000i	− 1.43360i
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	4.00000	0.130051
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	− 16.0000i	− 0.519656i
$949$	−16.0000	−0.519382
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	− 18.0000i	− 0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$	− 18.0000i	− 0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	12.0000i	0.387905i
$958$	12.0000i	0.387702i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	69.0000	2.22581
$962$	8.00000i	0.257930i
$963$	12.0000i	0.386695i
$964$	4.00000	0.128831
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	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
$973$	− 4.00000i	− 0.128234i
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−4.00000	−0.128037
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	8.00000i	0.255812i
$979$	18.0000	0.575282
$980$	0	0
$981$	2.00000	0.0638551
$982$	12.0000i	0.382935i
$983$	− 42.0000i	− 1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$	− 42.0000i	− 1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$	−24.0000	−0.765092
$985$	0	0
$986$	0	0
$987$	12.0000i	0.381964i
$988$	16.0000i	0.509028i
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	10.0000i	0.317500i
$993$	− 40.0000i	− 1.26936i
$994$	−12.0000	−0.380617
$995$	0	0
$996$	−24.0000	−0.760469
$997$	52.0000i	1.64686i	0.567420	+	0.823428i	$0.307941\pi$
$997$	52.0000i	1.64686i	−0.567420	+	0.823428i	$0.692059\pi$
$998$	− 40.0000i	− 1.26618i
$999$	−8.00000	−0.253109

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} +2.00000 q^{24} -4.00000 q^{26} -4.00000i q^{27} +1.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -8.00000 q^{39} -12.0000 q^{41} +2.00000i q^{42} -4.00000i q^{43} +1.00000 q^{44} -6.00000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +4.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} -4.00000 q^{61} +10.0000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +4.00000i q^{67} +12.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +1.00000i q^{77} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +2.00000 q^{84} -4.00000 q^{86} -12.0000i q^{87} -1.00000i q^{88} -18.0000 q^{89} -4.00000 q^{91} +20.0000i q^{93} -6.00000 q^{94} -2.00000 q^{96} +10.0000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} - 4 q^{21} + 4 q^{24} - 8 q^{26} + 12 q^{29} - 20 q^{31} + 2 q^{36} - 16 q^{39} - 24 q^{41} + 2 q^{44} - 2 q^{49} - 8 q^{54} + 2 q^{56} + 12 q^{59} - 8 q^{61} - 2 q^{64} + 4 q^{66} + 24 q^{71} - 4 q^{74} - 8 q^{76} - 16 q^{79} - 22 q^{81} + 4 q^{84} - 8 q^{86} - 36 q^{89} - 8 q^{91} - 12 q^{94} - 4 q^{96} + 2 q^{99}+O(q^{100}) \) 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 + 12 * q^29 - 20 * q^31 + 2 * q^36 - 16 * q^39 - 24 * q^41 + 2 * q^44 - 2 * q^49 - 8 * q^54 + 2 * q^56 + 12 * q^59 - 8 * q^61 - 2 * q^64 + 4 * q^66 + 24 * q^71 - 4 * q^74 - 8 * q^76 - 16 * q^79 - 22 * q^81 + 4 * q^84 - 8 * q^86 - 36 * q^89 - 8 * q^91 - 12 * q^94 - 4 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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