LMFDB - Embedded newform 2850.2.d.q.799.1

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			799.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.q.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} -8.00000 q^{41} +2.00000i q^{42} -6.00000i q^{43} -2.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +2.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +14.0000i q^{73} -8.00000 q^{74} -1.00000 q^{76} +4.00000i q^{77} +4.00000i q^{78} +1.00000 q^{81} +8.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -6.00000 q^{86} +2.00000i q^{88} -8.00000 q^{91} -4.00000i q^{92} -8.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +12.0000i q^{97} -3.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 16 q^{31} + 4 q^{34} + 2 q^{36} - 8 q^{39} - 16 q^{41} - 4 q^{44} + 8 q^{46} + 6 q^{49} - 4 q^{51} - 2 q^{54} - 4 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 8 q^{69} - 16 q^{71} - 16 q^{74} - 2 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^21 - 2 * q^24 + 8 * q^26 - 16 * q^31 + 4 * q^34 + 2 * q^36 - 8 * q^39 - 16 * q^41 - 4 * q^44 + 8 * q^46 + 6 * q^49 - 4 * q^51 - 2 * q^54 - 4 * q^56 + 4 * q^61 - 2 * q^64 + 4 * q^66 - 8 * q^69 - 16 * q^71 - 16 * q^74 - 2 * q^76 + 2 * q^81 + 4 * q^84 - 12 * q^86 - 16 * q^91 + 24 * q^94 + 2 * q^96 - 4 * 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	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 1.00000i	− 0.288675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	− 2.00000i	− 0.426401i
$23$	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
$26$	4.00000	0.784465
$27$	− 1.00000i	− 0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	2.00000i	0.348155i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	− 1.00000i	− 0.162221i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	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	$-0.151253\pi$
$44$	−2.00000	−0.301511
$45$	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
$52$	− 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	$-0.135198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	1.00000i	0.132453i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$66$	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	$-0.162521\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 1.00000i	− 0.117851i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−1.00000	−0.114708
$77$	4.00000i	0.455842i
$78$	4.00000i	0.452911i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.00000i	0.883452i
$83$	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
$87$	0	0
$88$	2.00000i	0.213201i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	− 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	$0.208512\pi$
$97$	12.0000i	1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−2.00000	−0.201008
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	2.00000i	0.198030i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−4.00000	−0.392232
$105$	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
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	2.00000i	0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 2.00000i	− 0.181071i
$123$	− 8.00000i	− 0.721336i
$124$	8.00000	0.718421
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 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
$129$	6.00000	0.528271
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	− 2.00000i	− 0.174078i
$133$	2.00000i	0.173422i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	22.0000i	1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$	22.0000i	1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	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
$142$	8.00000i	0.671345i
$143$	8.00000i	0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	3.00000i	0.247436i
$148$	8.00000i	0.657596i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	1.00000i	0.0811107i
$153$	− 2.00000i	− 0.161690i
$154$	4.00000	0.322329
$155$	0	0
$156$	4.00000	0.320256
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	4.00000	0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	6.00000i	0.457496i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	8.00000i	0.592999i
$183$	2.00000i	0.147844i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−8.00000	−0.586588
$187$	4.00000i	0.292509i
$188$	− 12.0000i	− 0.875190i
$189$	2.00000	0.145479
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	24.0000i	1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$	24.0000i	1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	2.00000i	0.142134i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	4.00000	0.278693
$207$	− 4.00000i	− 0.278019i
$208$	4.00000i	0.277350i
$209$	2.00000	0.138343
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000i	0.412082i
$213$	− 8.00000i	− 0.548151i
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 16.0000i	− 1.08615i
$218$	10.0000i	0.677285i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−8.00000	−0.538138
$222$	− 8.00000i	− 0.536925i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	0	0
$237$	0	0
$238$	4.00000i	0.259281i
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	7.00000i	0.449977i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−8.00000	−0.510061
$247$	4.00000i	0.254514i
$248$	− 8.00000i	− 0.508001i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	2.00000i	0.125988i
$253$	8.00000i	0.502956i
$254$	−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	$-0.189739\pi$
$258$	− 6.00000i	− 0.373544i
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	18.0000i	1.11204i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	8.00000i	0.488678i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000i	0.121268i
$273$	− 8.00000i	− 0.484182i
$274$	22.0000	1.32907
$275$	0	0
$276$	4.00000	0.240772
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	12.0000i	0.714590i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	8.00000	0.473050
$287$	− 16.0000i	− 0.944450i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	− 14.0000i	− 0.819288i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	8.00000	0.464991
$297$	− 2.00000i	− 0.116052i
$298$	10.0000i	0.579284i
$299$	−16.0000	−0.925304
$300$	0	0
$301$	12.0000	0.691669
$302$	8.00000i	0.460348i
$303$	2.00000i	0.114897i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−2.00000	−0.114332
$307$	32.0000i	1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$	32.0000i	1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$	− 4.00000i	− 0.227921i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	− 4.00000i	− 0.226455i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	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
$326$	14.0000	0.775388
$327$	− 10.0000i	− 0.553001i
$328$	− 8.00000i	− 0.441726i
$329$	−24.0000	−1.32316
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 4.00000i	− 0.219529i
$333$	8.00000i	0.438397i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 28.0000i	− 1.52526i	−0.646837	−	0.762629i	$-0.723908\pi$
$337$	− 28.0000i	− 1.52526i	0.646837	−	0.762629i	$-0.276092\pi$
$338$	3.00000i	0.163178i
$339$	6.00000	0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	1.00000i	0.0540738i
$343$	20.0000i	1.07990i
$344$	6.00000	0.323498
$345$	0	0
$346$	14.0000	0.752645
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	− 2.00000i	− 0.106600i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 2.00000i	− 0.105118i
$363$	− 7.00000i	− 0.367405i
$364$	8.00000	0.419314
$365$	0	0
$366$	2.00000	0.104542
$367$	− 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
$369$	8.00000	0.416463
$370$	0	0
$371$	12.0000	0.623009
$372$	8.00000i	0.414781i
$373$	− 36.0000i	− 1.86401i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	− 36.0000i	− 1.86401i	0.362446	−	0.932005i	$-0.381942\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	− 2.00000i	− 0.102869i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	18.0000i	0.920960i
$383$	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
$386$	24.0000	1.22157
$387$	6.00000i	0.304997i
$388$	− 12.0000i	− 0.609208i
$389$	−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
$391$	−8.00000	−0.404577
$392$	3.00000i	0.151523i
$393$	− 18.0000i	− 0.907980i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	2.00000	0.100504
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 20.0000i	− 1.00251i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	− 8.00000i	− 0.399004i
$403$	− 32.0000i	− 1.59403i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	− 16.0000i	− 0.793091i
$408$	− 2.00000i	− 0.0990148i
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	− 4.00000i	− 0.197066i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	− 2.00000i	− 0.0978232i
$419$	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
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 12.0000i	− 0.584151i
$423$	− 12.0000i	− 0.583460i
$424$	6.00000	0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	4.00000i	0.193574i
$428$	− 12.0000i	− 0.580042i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	10.0000	0.478913
$437$	4.00000i	0.191346i
$438$	14.0000i	0.668946i
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	8.00000i	0.380521i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	4.00000	0.189405
$447$	− 10.0000i	− 0.472984i
$448$	− 2.00000i	− 0.0944911i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	6.00000i	0.282216i
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	10.0000i	0.467269i
$459$	2.00000	0.0933520
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	4.00000i	0.186097i
$463$	− 6.00000i	− 0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$	− 6.00000i	− 0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	4.00000i	0.184900i
$469$	16.0000	0.738811
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	6.00000i	0.274721i
$478$	10.0000i	0.457389i
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	− 2.00000i	− 0.0910975i
$483$	− 8.00000i	− 0.364013i
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	2.00000i	0.0905357i
$489$	−14.0000	−0.633102
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	8.00000i	0.360668i
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	−8.00000	−0.359211
$497$	− 16.0000i	− 0.717698i
$498$	4.00000i	0.179244i
$499$	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
$501$	8.00000	0.357414
$502$	− 2.00000i	− 0.0892644i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	8.00000	0.355643
$507$	− 3.00000i	− 0.133235i
$508$	8.00000i	0.354943i
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	− 1.00000i	− 0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−6.00000	−0.264135
$517$	24.0000i	1.05552i
$518$	− 16.0000i	− 0.703000i
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	24.0000	1.04645
$527$	− 16.0000i	− 0.696971i
$528$	2.00000i	0.0870388i
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	− 2.00000i	− 0.0867110i
$533$	− 32.0000i	− 1.38607i
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	− 12.0000i	− 0.515444i
$543$	2.00000i	0.0858282i
$544$	2.00000	0.0857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	− 22.0000i	− 0.939793i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	− 4.00000i	− 0.170251i
$553$	0	0
$554$	−18.0000	−0.764747
$555$	0	0
$556$	0	0
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	− 8.00000i	− 0.338667i
$559$	24.0000	1.01509
$560$	0	0
$561$	−4.00000	−0.168880
$562$	− 12.0000i	− 0.506189i
$563$	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
$566$	14.0000	0.588464
$567$	2.00000i	0.0839921i
$568$	− 8.00000i	− 0.335673i
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	− 8.00000i	− 0.334497i
$573$	− 18.0000i	− 0.751961i
$574$	−16.0000	−0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−8.00000	−0.331896
$582$	12.0000i	0.497416i
$583$	− 12.0000i	− 0.496989i
$584$	−14.0000	−0.579324
$585$	0	0
$586$	14.0000	0.578335
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	− 3.00000i	− 0.123718i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	18.0000	0.740421
$592$	− 8.00000i	− 0.328798i
$593$	− 46.0000i	− 1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	− 46.0000i	− 1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	10.0000	0.409616
$597$	20.0000i	0.818546i
$598$	16.0000i	0.654289i
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	− 12.0000i	− 0.489083i
$603$	8.00000i	0.325785i
$604$	8.00000	0.325515
$605$	0	0
$606$	2.00000	0.0812444
$607$	12.0000i	0.487065i	0.969893	+	0.243532i	$0.0783062\pi$
$607$	12.0000i	0.487065i	−0.969893	+	0.243532i	$0.921694\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	2.00000i	0.0808452i
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−4.00000	−0.161165
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	4.00000i	0.160904i
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	18.0000i	0.721734i
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−6.00000	−0.239808
$627$	2.00000i	0.0798723i
$628$	18.0000i	0.718278i
$629$	16.0000	0.637962
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	12.0000i	0.476957i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	12.0000i	0.475457i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	32.0000	1.26392	0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000	1.26392	0.631962	+	0.774999i	$0.282250\pi$
$642$	12.0000i	0.473602i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	2.00000	0.0786889
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	− 14.0000i	− 0.548282i
$653$	− 46.0000i	− 1.80012i	−0.435767	−	0.900060i	$-0.643523\pi$
$653$	− 46.0000i	− 1.80012i	0.435767	−	0.900060i	$-0.356477\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−8.00000	−0.312348
$657$	− 14.0000i	− 0.546192i
$658$	24.0000i	0.935617i
$659$	−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
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	− 12.0000i	− 0.466393i
$663$	− 8.00000i	− 0.310694i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	8.00000i	0.309529i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	4.00000	0.154418
$672$	2.00000i	0.0771517i
$673$	4.00000i	0.154189i	0.997024	+	0.0770943i	$0.0245643\pi$
$673$	4.00000i	0.154189i	−0.997024	+	0.0770943i	$0.975436\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	3.00000	0.115385
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	− 6.00000i	− 0.230429i
$679$	−24.0000	−0.921035
$680$	0	0
$681$	−12.0000	−0.459841
$682$	16.0000i	0.612672i
$683$	− 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
$687$	− 10.0000i	− 0.381524i
$688$	− 6.00000i	− 0.228748i
$689$	24.0000	0.914327
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	− 14.0000i	− 0.532200i
$693$	− 4.00000i	− 0.151947i
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	− 10.0000i	− 0.378506i
$699$	6.00000	0.226941
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	− 4.00000i	− 0.150970i
$703$	− 8.00000i	− 0.301726i
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−6.00000	−0.225813
$707$	4.00000i	0.150435i
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 32.0000i	− 1.19841i
$714$	−4.00000	−0.149696
$715$	0	0
$716$	0	0
$717$	− 10.0000i	− 0.373457i
$718$	− 10.0000i	− 0.373197i
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 1.00000i	− 0.0372161i
$723$	2.00000i	0.0743808i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−7.00000	−0.259794
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	− 2.00000i	− 0.0739221i
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	4.00000	0.147442
$737$	− 16.0000i	− 0.589368i
$738$	− 8.00000i	− 0.294484i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	− 12.0000i	− 0.440534i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−36.0000	−1.31805
$747$	− 4.00000i	− 0.146352i
$748$	− 4.00000i	− 0.146254i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	12.0000i	0.437595i
$753$	2.00000i	0.0728841i
$754$	0	0
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	− 20.0000i	− 0.726433i
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	− 8.00000i	− 0.289809i
$763$	− 20.0000i	− 0.724049i
$764$	18.0000	0.651217
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 24.0000i	− 0.863779i
$773$	34.0000i	1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$	34.0000i	1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$	6.00000	0.215666
$775$	0	0
$776$	−12.0000	−0.430775
$777$	16.0000i	0.573997i
$778$	30.0000i	1.07555i
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−16.0000	−0.572525
$782$	8.00000i	0.286079i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	−18.0000	−0.642039
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	18.0000i	0.641223i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	12.0000	0.426671
$792$	− 2.00000i	− 0.0710669i
$793$	8.00000i	0.284088i
$794$	22.0000	0.780751
$795$	0	0
$796$	−20.0000	−0.708881
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	2.00000i	0.0707992i
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	− 12.0000i	− 0.423735i
$803$	28.0000i	0.988099i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−32.0000	−1.12715
$807$	0	0
$808$	2.00000i	0.0703598i
$809$	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
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	12.0000i	0.420858i
$814$	−16.0000	−0.560800
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 6.00000i	− 0.209913i
$818$	30.0000i	1.04893i
$819$	8.00000	0.279543
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	22.0000i	0.767338i
$823$	34.0000i	1.18517i	0.805510	+	0.592583i	$0.201892\pi$
$823$	34.0000i	1.18517i	−0.805510	+	0.592583i	$0.798108\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	4.00000i	0.139010i
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	− 4.00000i	− 0.138675i
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	8.00000i	0.276520i
$838$	− 30.0000i	− 1.03633i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 22.0000i	− 0.758170i
$843$	12.0000i	0.413302i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−12.0000	−0.412568
$847$	− 14.0000i	− 0.481046i
$848$	− 6.00000i	− 0.206041i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	32.0000	1.09695
$852$	8.00000i	0.274075i
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	8.00000i	0.273115i
$859$	−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
$861$	16.0000	0.545279
$862$	− 12.0000i	− 0.408722i
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−16.0000	−0.543702
$867$	13.0000i	0.441503i
$868$	16.0000i	0.543075i
$869$	0	0
$870$	0	0
$871$	32.0000	1.08428
$872$	− 10.0000i	− 0.338643i
$873$	− 12.0000i	− 0.406138i
$874$	4.00000	0.135302
$875$	0	0
$876$	14.0000	0.473016
$877$	12.0000i	0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$	12.0000i	0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$	− 40.0000i	− 1.34993i
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	3.00000i	0.101015i
$883$	− 26.0000i	− 0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$	− 26.0000i	− 0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	24.0000	0.806296
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	8.00000i	0.268462i
$889$	16.0000	0.536623
$890$	0	0
$891$	2.00000	0.0670025
$892$	− 4.00000i	− 0.133930i
$893$	12.0000i	0.401565i
$894$	−10.0000	−0.334450
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	− 16.0000i	− 0.534224i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	16.0000i	0.532742i
$903$	12.0000i	0.399335i
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	1.00000i	0.0331133i
$913$	8.00000i	0.264761i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	10.0000	0.330409
$917$	− 36.0000i	− 1.18882i
$918$	− 2.00000i	− 0.0660098i
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	− 2.00000i	− 0.0658665i
$923$	− 32.0000i	− 1.05329i
$924$	4.00000	0.131590
$925$	0	0
$926$	−6.00000	−0.197172
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	6.00000i	0.196537i
$933$	− 18.0000i	− 0.589294i
$934$	−8.00000	−0.261768
$935$	0	0
$936$	4.00000	0.130744
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	− 16.0000i	− 0.522419i
$939$	6.00000	0.195803
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	− 18.0000i	− 0.586472i
$943$	− 32.0000i	− 1.04206i
$944$	0	0
$945$	0	0
$946$	−12.0000	−0.390154
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	−56.0000	−1.81784
$950$	0	0
$951$	18.0000	0.583690
$952$	− 4.00000i	− 0.129641i
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	10.0000	0.323423
$957$	0	0
$958$	− 10.0000i	− 0.323085i
$959$	−44.0000	−1.42083
$960$	0	0
$961$	33.0000	1.06452
$962$	− 32.0000i	− 1.03172i
$963$	− 12.0000i	− 0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−8.00000	−0.257396
$967$	− 58.0000i	− 1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	− 58.0000i	− 1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$	− 7.00000i	− 0.224989i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	−28.0000	−0.897178
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	14.0000i	0.447671i
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	− 22.0000i	− 0.702048i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	− 4.00000i	− 0.127257i
$989$	24.0000	0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	8.00000i	0.254000i
$993$	12.0000i	0.380808i
$994$	−16.0000	−0.507489
$995$	0	0
$996$	4.00000	0.126745
$997$	62.0000i	1.96356i	0.190022	+	0.981780i	$0.439144\pi$
$997$	62.0000i	1.96356i	−0.190022	+	0.981780i	$0.560856\pi$
$998$	− 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	2850.2.d.q.799.1		2
5.2	odd	4		570.2.a.l.1.1	✓	1
5.3	odd	4		2850.2.a.e.1.1		1
5.4	even	2	inner	2850.2.d.q.799.2		2
15.2	even	4		1710.2.a.b.1.1		1
15.8	even	4		8550.2.a.bf.1.1		1
20.7	even	4		4560.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.l.1.1	✓	1	5.2	odd	4
1710.2.a.b.1.1		1	15.2	even	4
2850.2.a.e.1.1		1	5.3	odd	4
2850.2.d.q.799.1		2	1.1	even	1	trivial
2850.2.d.q.799.2		2	5.4	even	2	inner
4560.2.a.o.1.1		1	20.7	even	4
8550.2.a.bf.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} -8.00000 q^{41} +2.00000i q^{42} -6.00000i q^{43} -2.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +2.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +14.0000i q^{73} -8.00000 q^{74} -1.00000 q^{76} +4.00000i q^{77} +4.00000i q^{78} +1.00000 q^{81} +8.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -6.00000 q^{86} +2.00000i q^{88} -8.00000 q^{91} -4.00000i q^{92} -8.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +12.0000i q^{97} -3.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 16 q^{31} + 4 q^{34} + 2 q^{36} - 8 q^{39} - 16 q^{41} - 4 q^{44} + 8 q^{46} + 6 q^{49} - 4 q^{51} - 2 q^{54} - 4 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 8 q^{69} - 16 q^{71} - 16 q^{74} - 2 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^21 - 2 * q^24 + 8 * q^26 - 16 * q^31 + 4 * q^34 + 2 * q^36 - 8 * q^39 - 16 * q^41 - 4 * q^44 + 8 * q^46 + 6 * q^49 - 4 * q^51 - 2 * q^54 - 4 * q^56 + 4 * q^61 - 2 * q^64 + 4 * q^66 - 8 * q^69 - 16 * q^71 - 16 * q^74 - 2 * q^76 + 2 * q^81 + 4 * q^84 - 12 * q^86 - 16 * q^91 + 24 * q^94 + 2 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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