LMFDB - Embedded newform 2850.2.d.j.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.d (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:	$22.7573645761$
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 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.j.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} +2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +2.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -4.00000 q^{41} -2.00000i q^{42} -6.00000i q^{43} +6.00000 q^{44} -4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -1.00000i q^{57} +8.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} -8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} +4.00000 q^{69} +1.00000i q^{72} +6.00000i q^{73} -4.00000 q^{74} -1.00000 q^{76} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} +6.00000 q^{86} -8.00000i q^{87} +6.00000i q^{88} +4.00000 q^{89} -4.00000i q^{92} +8.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -12.0000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 12 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{21} - 2 q^{24} + 16 q^{29} - 16 q^{31} + 4 q^{34} + 2 q^{36} - 8 q^{41} + 12 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} - 2 q^{54} - 4 q^{56} + 8 q^{59} + 4 q^{61} - 2 q^{64} - 12 q^{66} + 8 q^{69} - 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} + 4 q^{84} + 12 q^{86} + 8 q^{89} - 24 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 12 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^21 - 2 * q^24 + 16 * q^29 - 16 * q^31 + 4 * q^34 + 2 * q^36 - 8 * q^41 + 12 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^51 - 2 * q^54 - 4 * q^56 + 8 * q^59 + 4 * q^61 - 2 * q^64 - 12 * q^66 + 8 * q^69 - 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 + 4 * q^84 + 12 * q^86 + 8 * q^89 - 24 * q^94 + 2 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000i	0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	− 6.00000i	− 1.27920i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000i	0.176777i
$33$	6.00000i	1.04447i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−4.00000	−0.589768
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	− 1.00000i	− 0.132453i
$58$	8.00000i	1.05045i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 8.00000i	− 1.01600i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	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
$69$	4.00000	0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−1.00000	−0.114708
$77$	12.0000i	1.36753i
$78$	0	0
$79$	−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
$81$	1.00000	0.111111
$82$	− 4.00000i	− 0.441726i
$83$	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
$86$	6.00000	0.646997
$87$	− 8.00000i	− 0.857690i
$88$	6.00000i	0.639602i
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	− 4.00000i	− 0.417029i
$93$	8.00000i	0.829561i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	− 12.0000i	− 1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$	− 12.0000i	− 1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$	3.00000i	0.303046i
$99$	6.00000	0.603023
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 2.00000i	− 0.198030i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	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
$111$	4.00000	0.379663
$112$	− 2.00000i	− 0.188982i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−8.00000	−0.742781
$117$	0	0
$118$	4.00000i	0.368230i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	2.00000i	0.181071i
$123$	4.00000i	0.360668i
$124$	8.00000	0.718421
$125$	0	0
$126$	−2.00000	−0.178174
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−6.00000	−0.528271
$130$	0	0
$131$	22.0000	1.92215	0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000	1.92215	0.961074	+	0.276289i	$0.0891049\pi$
$132$	− 6.00000i	− 0.522233i
$133$	− 2.00000i	− 0.173422i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	4.00000i	0.340503i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	− 3.00000i	− 0.247436i
$148$	− 4.00000i	− 0.328798i
$149$	−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
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	2.00000i	0.161690i
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 8.00000i	− 0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	8.00000	0.630488
$162$	1.00000i	0.0785674i
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−4.00000	−0.310460
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	2.00000i	0.154303i
$169$	13.0000	1.00000
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	6.00000i	0.457496i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	−6.00000	−0.452267
$177$	− 4.00000i	− 0.300658i
$178$	4.00000i	0.299813i
$179$	−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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	4.00000	0.294884
$185$	0	0
$186$	−8.00000	−0.586588
$187$	12.0000i	0.877527i
$188$	− 12.0000i	− 0.875190i
$189$	2.00000	0.145479
$190$	0	0
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	1.00000i	0.0721688i
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	6.00000i	0.426401i
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 10.0000i	− 0.703598i
$203$	− 16.0000i	− 1.12298i
$204$	2.00000	0.140028
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	16.0000i	1.08615i
$218$	14.0000i	0.948200i
$219$	6.00000	0.405442
$220$	0	0
$221$	0	0
$222$	4.00000i	0.268462i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−14.0000	−0.931266
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	1.00000i	0.0662266i
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	− 8.00000i	− 0.525226i
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	8.00000i	0.519656i
$238$	− 4.00000i	− 0.259281i
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	25.0000i	1.60706i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−4.00000	−0.255031
$247$	0	0
$248$	8.00000i	0.508001i
$249$	4.00000	0.253490
$250$	0	0
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	− 2.00000i	− 0.125988i
$253$	− 24.0000i	− 1.50887i
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	− 6.00000i	− 0.373544i
$259$	8.00000	0.497096
$260$	0	0
$261$	−8.00000	−0.495188
$262$	22.0000i	1.35916i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	2.00000	0.122628
$267$	− 4.00000i	− 0.244796i
$268$	− 8.00000i	− 0.488678i
$269$	−8.00000	−0.487769	−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000	−0.487769	−0.243884	+	0.969804i	$0.578422\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	−4.00000	−0.240772
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	12.0000i	0.714590i
$283$	22.0000i	1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$	22.0000i	1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000i	0.472225i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	− 6.00000i	− 0.351123i
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	4.00000	0.232495
$297$	− 6.00000i	− 0.348155i
$298$	− 22.0000i	− 1.27443i
$299$	0	0
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	10.0000i	0.574485i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−2.00000	−0.114332
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	− 12.0000i	− 0.683763i
$309$	0	0
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	6.00000	0.338600
$315$	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	$-0.0538913\pi$
$318$	6.00000i	0.336463i
$319$	−48.0000	−2.68748
$320$	0	0
$321$	12.0000	0.669775
$322$	8.00000i	0.445823i
$323$	− 2.00000i	− 0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	− 14.0000i	− 0.774202i
$328$	4.00000i	0.220863i
$329$	24.0000	1.32316
$330$	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
$333$	− 4.00000i	− 0.219199i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−2.00000	−0.109109
$337$	20.0000i	1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$	20.0000i	1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	13.0000i	0.707107i
$339$	14.0000	0.760376
$340$	0	0
$341$	48.0000	2.59935
$342$	− 1.00000i	− 0.0540738i
$343$	− 20.0000i	− 1.07990i
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 36.0000i	− 1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$	− 36.0000i	− 1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$	8.00000i	0.428845i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	− 6.00000i	− 0.319801i
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	−4.00000	−0.212000
$357$	4.00000i	0.211702i
$358$	− 12.0000i	− 0.634220i
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.00000i	0.105118i
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	4.00000i	0.208514i
$369$	4.00000	0.208232
$370$	0	0
$371$	12.0000	0.623009
$372$	− 8.00000i	− 0.414781i
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	12.0000	0.618853
$377$	0	0
$378$	2.00000i	0.102869i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	− 22.0000i	− 1.12562i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	8.00000	0.407189
$387$	6.00000i	0.304997i
$388$	12.0000i	0.609208i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	− 3.00000i	− 0.151523i
$393$	− 22.0000i	− 1.10975i
$394$	14.0000	0.705310
$395$	0	0
$396$	−6.00000	−0.301511
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	− 20.0000i	− 1.00251i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	8.00000i	0.399004i
$403$	0	0
$404$	10.0000	0.497519
$405$	0	0
$406$	16.0000	0.794067
$407$	− 24.0000i	− 1.18964i
$408$	2.00000i	0.0990148i
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	4.00000	0.196589
$415$	0	0
$416$	0	0
$417$	0	0
$418$	− 6.00000i	− 0.293470i
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	− 28.0000i	− 1.36302i
$423$	− 12.0000i	− 0.583460i
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	1.00000i	0.0481125i
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−14.0000	−0.670478
$437$	4.00000i	0.191346i
$438$	6.00000i	0.286691i
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	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
$446$	16.0000	0.757622
$447$	22.0000i	1.04056i
$448$	2.00000i	0.0944911i
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	14.0000i	0.654892i	0.944870	+	0.327446i	$0.106188\pi$
$457$	14.0000i	0.654892i	−0.944870	+	0.327446i	$0.893812\pi$
$458$	− 18.0000i	− 0.841085i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	12.0000i	0.558291i
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	−14.0000	−0.648537
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−6.00000	−0.276465
$472$	− 4.00000i	− 0.184115i
$473$	36.0000i	1.65528i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	− 6.00000i	− 0.274721i
$478$	18.0000i	0.823301i
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	0	0
$482$	− 22.0000i	− 1.00207i
$483$	− 8.00000i	− 0.364013i
$484$	−25.0000	−1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	6.00000	0.271329
$490$	0	0
$491$	−34.0000	−1.53440	−0.767199	−	0.641409i	$-0.778350\pi$
$491$	−34.0000	−1.53440	−0.767199	+	0.641409i	$0.778350\pi$
$492$	− 4.00000i	− 0.180334i
$493$	− 16.0000i	− 0.720604i
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	− 14.0000i	− 0.624851i
$503$	40.0000i	1.78351i	0.452517	+	0.891756i	$0.350526\pi$
$503$	40.0000i	1.78351i	−0.452517	+	0.891756i	$0.649474\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	24.0000	1.06693
$507$	− 13.0000i	− 0.577350i
$508$	− 12.0000i	− 0.532414i
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	1.00000i	0.0441942i
$513$	1.00000i	0.0441511i
$514$	6.00000	0.264649
$515$	0	0
$516$	6.00000	0.264135
$517$	− 72.0000i	− 3.16656i
$518$	8.00000i	0.351500i
$519$	18.0000	0.790112
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	− 8.00000i	− 0.350150i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−22.0000	−0.961074
$525$	0	0
$526$	−16.0000	−0.697633
$527$	16.0000i	0.696971i
$528$	6.00000i	0.261116i
$529$	7.00000	0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	2.00000i	0.0867110i
$533$	0	0
$534$	4.00000	0.173097
$535$	0	0
$536$	8.00000	0.345547
$537$	12.0000i	0.517838i
$538$	− 8.00000i	− 0.344904i
$539$	−18.0000	−0.775315
$540$	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
$543$	− 2.00000i	− 0.0858282i
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	− 40.0000i	− 1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$	− 40.0000i	− 1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$	6.00000i	0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	8.00000	0.340811
$552$	− 4.00000i	− 0.170251i
$553$	16.0000i	0.680389i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	8.00000i	0.338667i
$559$	0	0
$560$	0	0
$561$	12.0000	0.506640
$562$	16.0000i	0.674919i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−22.0000	−0.924729
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	22.0000i	0.919063i
$574$	−8.00000	−0.333914
$575$	0	0
$576$	1.00000	0.0416667
$577$	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
$579$	−8.00000	−0.332469
$580$	0	0
$581$	8.00000	0.331896
$582$	− 12.0000i	− 0.497416i
$583$	− 36.0000i	− 1.49097i
$584$	6.00000	0.248282
$585$	0	0
$586$	−10.0000	−0.413096
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	3.00000i	0.123718i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−14.0000	−0.575883
$592$	4.00000i	0.164399i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	22.0000	0.901155
$597$	20.0000i	0.818546i
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 12.0000i	− 0.489083i
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	0	0
$606$	−10.0000	−0.406222
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	1.00000i	0.0405554i
$609$	−16.0000	−0.648353
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	− 10.0000i	− 0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	− 10.0000i	− 0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	12.0000	0.483494
$617$	38.0000i	1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$	38.0000i	1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 6.00000i	− 0.240578i
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	14.0000	0.559553
$627$	6.00000i	0.239617i
$628$	6.00000i	0.239426i
$629$	8.00000	0.318981
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	8.00000i	0.318223i
$633$	28.0000i	1.11290i
$634$	6.00000	0.238290
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	− 48.0000i	− 1.90034i
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	12.0000i	0.473602i
$643$	− 26.0000i	− 1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$	− 26.0000i	− 1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	2.00000	0.0786889
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−24.0000	−0.942082
$650$	0	0
$651$	16.0000	0.627089
$652$	− 6.00000i	− 0.234978i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−4.00000	−0.156174
$657$	− 6.00000i	− 0.234082i
$658$	24.0000i	0.935617i
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	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
$664$	4.00000	0.155230
$665$	0	0
$666$	4.00000	0.154997
$667$	32.0000i	1.23904i
$668$	− 16.0000i	− 0.619059i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−12.0000	−0.463255
$672$	− 2.00000i	− 0.0771517i
$673$	− 28.0000i	− 1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$	− 28.0000i	− 1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	14.0000i	0.537667i
$679$	−24.0000	−0.921035
$680$	0	0
$681$	28.0000	1.07296
$682$	48.0000i	1.83801i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	18.0000i	0.686743i
$688$	− 6.00000i	− 0.228748i
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	− 18.0000i	− 0.684257i
$693$	− 12.0000i	− 0.455842i
$694$	36.0000	1.36654
$695$	0	0
$696$	−8.00000	−0.303239
$697$	8.00000i	0.303022i
$698$	2.00000i	0.0757011i
$699$	14.0000	0.529529
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	4.00000i	0.150863i
$704$	6.00000	0.226134
$705$	0	0
$706$	18.0000	0.677439
$707$	20.0000i	0.752177i
$708$	4.00000i	0.150329i
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 4.00000i	− 0.149906i
$713$	− 32.0000i	− 1.19841i
$714$	−4.00000	−0.149696
$715$	0	0
$716$	12.0000	0.448461
$717$	− 18.0000i	− 0.672222i
$718$	6.00000i	0.223918i
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	0	0
$721$	0	0
$722$	1.00000i	0.0372161i
$723$	22.0000i	0.818189i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	25.0000	0.927837
$727$	− 14.0000i	− 0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$	− 14.0000i	− 0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	2.00000i	0.0739221i
$733$	18.0000i	0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$	18.0000i	0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 48.0000i	− 1.76810i
$738$	4.00000i	0.147242i
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	12.0000i	0.440534i
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	24.0000	0.878702
$747$	− 4.00000i	− 0.146352i
$748$	− 12.0000i	− 0.438763i
$749$	24.0000	0.876941
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	12.0000i	0.437595i
$753$	14.0000i	0.510188i
$754$	0	0
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	− 20.0000i	− 0.726433i
$759$	−24.0000	−0.871145
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	12.0000i	0.434714i
$763$	− 28.0000i	− 1.01367i
$764$	22.0000	0.795932
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	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
$771$	−6.00000	−0.216085
$772$	8.00000i	0.287926i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	−12.0000	−0.430775
$777$	− 8.00000i	− 0.286998i
$778$	30.0000i	1.07555i
$779$	−4.00000	−0.143315
$780$	0	0
$781$	0	0
$782$	8.00000i	0.286079i
$783$	8.00000i	0.285897i
$784$	3.00000	0.107143
$785$	0	0
$786$	22.0000	0.784714
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	14.0000i	0.498729i
$789$	16.0000	0.569615
$790$	0	0
$791$	28.0000	0.995565
$792$	− 6.00000i	− 0.213201i
$793$	0	0
$794$	22.0000	0.780751
$795$	0	0
$796$	20.0000	0.708881
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	− 2.00000i	− 0.0707992i
$799$	24.0000	0.849059
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	− 36.0000i	− 1.27041i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	0	0
$807$	8.00000i	0.281613i
$808$	10.0000i	0.351799i
$809$	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
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	16.0000i	0.561490i
$813$	− 20.0000i	− 0.701431i
$814$	24.0000	0.841200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 6.00000i	− 0.209913i
$818$	− 14.0000i	− 0.489499i
$819$	0	0
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	− 6.00000i	− 0.209274i
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	0	0
$825$	0	0
$826$	8.00000	0.278356
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	4.00000i	0.139010i
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	− 8.00000i	− 0.276520i
$838$	14.0000i	0.483622i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	− 2.00000i	− 0.0689246i
$843$	− 16.0000i	− 0.551069i
$844$	28.0000	0.963800
$845$	0	0
$846$	12.0000	0.412568
$847$	− 50.0000i	− 1.71802i
$848$	6.00000i	0.206041i
$849$	22.0000	0.755038
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	12.0000	0.410152
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	36.0000i	1.22616i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	− 16.0000i	− 0.543075i
$869$	48.0000	1.62829
$870$	0	0
$871$	0	0
$872$	− 14.0000i	− 0.474100i
$873$	12.0000i	0.406138i
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−6.00000	−0.202721
$877$	32.0000i	1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$	32.0000i	1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$	8.00000i	0.269987i
$879$	10.0000	0.337292
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	− 3.00000i	− 0.101015i
$883$	14.0000i	0.471138i	0.971858	+	0.235569i	$0.0756953\pi$
$883$	14.0000i	0.471138i	−0.971858	+	0.235569i	$0.924305\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	− 4.00000i	− 0.134231i
$889$	24.0000	0.804934
$890$	0	0
$891$	−6.00000	−0.201008
$892$	16.0000i	0.535720i
$893$	12.0000i	0.401565i
$894$	−22.0000	−0.735790
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	4.00000i	0.133482i
$899$	−64.0000	−2.13452
$900$	0	0
$901$	12.0000	0.399778
$902$	24.0000i	0.799113i
$903$	12.0000i	0.399335i
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	− 28.0000i	− 0.929213i
$909$	10.0000	0.331679
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	− 24.0000i	− 0.794284i
$914$	−14.0000	−0.463079
$915$	0	0
$916$	18.0000	0.594737
$917$	− 44.0000i	− 1.45301i
$918$	2.00000i	0.0660098i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	− 34.0000i	− 1.11973i
$923$	0	0
$924$	−12.0000	−0.394771
$925$	0	0
$926$	26.0000	0.854413
$927$	0	0
$928$	8.00000i	0.262613i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	− 14.0000i	− 0.458585i
$933$	6.00000i	0.196431i
$934$	−8.00000	−0.261768
$935$	0	0
$936$	0	0
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	16.0000i	0.522419i
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−40.0000	−1.30396	−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000	−1.30396	−0.651981	+	0.758235i	$0.726062\pi$
$942$	− 6.00000i	− 0.195491i
$943$	− 16.0000i	− 0.521032i
$944$	4.00000	0.130189
$945$	0	0
$946$	−36.0000	−1.17046
$947$	− 44.0000i	− 1.42981i	−0.699223	−	0.714904i	$-0.746470\pi$
$947$	− 44.0000i	− 1.42981i	0.699223	−	0.714904i	$-0.253530\pi$
$948$	− 8.00000i	− 0.259828i
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	4.00000i	0.129641i
$953$	− 34.0000i	− 1.10137i	−0.834714	−	0.550684i	$-0.814367\pi$
$953$	− 34.0000i	− 1.10137i	0.834714	−	0.550684i	$-0.185633\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−18.0000	−0.582162
$957$	48.0000i	1.55162i
$958$	− 18.0000i	− 0.581554i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	22.0000	0.708572
$965$	0	0
$966$	8.00000	0.257396
$967$	18.0000i	0.578841i	0.957202	+	0.289420i	$0.0934626\pi$
$967$	18.0000i	0.578841i	−0.957202	+	0.289420i	$0.906537\pi$
$968$	− 25.0000i	− 0.803530i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	6.00000i	0.191859i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−14.0000	−0.446986
$982$	− 34.0000i	− 1.08498i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	4.00000	0.127515
$985$	0	0
$986$	16.0000	0.509544
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	− 8.00000i	− 0.254000i
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	58.0000i	1.83688i	0.395562	+	0.918439i	$0.370550\pi$
$997$	58.0000i	1.83688i	−0.395562	+	0.918439i	$0.629450\pi$
$998$	− 24.0000i	− 0.759707i
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.j.799.2		2
5.2	odd	4		570.2.a.b.1.1	✓	1
5.3	odd	4		2850.2.a.y.1.1		1
5.4	even	2	inner	2850.2.d.j.799.1		2
15.2	even	4		1710.2.a.s.1.1		1
15.8	even	4		8550.2.a.g.1.1		1
20.7	even	4		4560.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.b.1.1	✓	1	5.2	odd	4
1710.2.a.s.1.1		1	15.2	even	4
2850.2.a.y.1.1		1	5.3	odd	4
2850.2.d.j.799.1		2	5.4	even	2	inner
2850.2.d.j.799.2		2	1.1	even	1	trivial
4560.2.a.r.1.1		1	20.7	even	4
8550.2.a.g.1.1		1	15.8	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} +2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +2.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -4.00000 q^{41} -2.00000i q^{42} -6.00000i q^{43} +6.00000 q^{44} -4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -1.00000i q^{57} +8.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} -8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} +4.00000 q^{69} +1.00000i q^{72} +6.00000i q^{73} -4.00000 q^{74} -1.00000 q^{76} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} +6.00000 q^{86} -8.00000i q^{87} +6.00000i q^{88} +4.00000 q^{89} -4.00000i q^{92} +8.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -12.0000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 12 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{21} - 2 q^{24} + 16 q^{29} - 16 q^{31} + 4 q^{34} + 2 q^{36} - 8 q^{41} + 12 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} - 2 q^{54} - 4 q^{56} + 8 q^{59} + 4 q^{61} - 2 q^{64} - 12 q^{66} + 8 q^{69} - 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} + 4 q^{84} + 12 q^{86} + 8 q^{89} - 24 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 12 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^21 - 2 * q^24 + 16 * q^29 - 16 * q^31 + 4 * q^34 + 2 * q^36 - 8 * q^41 + 12 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^51 - 2 * q^54 - 4 * q^56 + 8 * q^59 + 4 * q^61 - 2 * q^64 - 12 * q^66 + 8 * q^69 - 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 + 4 * q^84 + 12 * q^86 + 8 * q^89 - 24 * q^94 + 2 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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