LMFDB - Embedded newform 2850.2.d.h.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:	yes
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.h.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -3.00000i q^{22} -3.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -3.00000 q^{29} -7.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} +10.0000i q^{43} -3.00000 q^{44} -3.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} +2.00000i q^{52} +9.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +3.00000i q^{58} -6.00000 q^{59} -1.00000 q^{61} +7.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +5.00000i q^{67} -6.00000i q^{68} -3.00000 q^{69} -1.00000i q^{72} +1.00000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{77} +2.00000i q^{78} +13.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -15.0000i q^{83} -2.00000 q^{84} +10.0000 q^{86} +3.00000i q^{87} +3.00000i q^{88} +15.0000 q^{89} +4.00000 q^{91} +3.00000i q^{92} +7.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} +8.00000i q^{97} -3.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{11} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} - 6 q^{29} - 14 q^{31} + 12 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 6 q^{44} - 6 q^{46} + 6 q^{49} + 12 q^{51} + 2 q^{54} - 4 q^{56} - 12 q^{59} - 2 q^{61} - 2 q^{64} - 6 q^{66} - 6 q^{69} + 4 q^{74} + 2 q^{76} + 26 q^{79} + 2 q^{81} - 4 q^{84} + 20 q^{86} + 30 q^{89} + 8 q^{91} + 24 q^{94} - 2 q^{96} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 6 * q^11 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 - 6 * q^29 - 14 * q^31 + 12 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 6 * q^44 - 6 * q^46 + 6 * q^49 + 12 * q^51 + 2 * q^54 - 4 * q^56 - 12 * q^59 - 2 * q^61 - 2 * q^64 - 6 * q^66 - 6 * q^69 + 4 * q^74 + 2 * q^76 + 26 * q^79 + 2 * q^81 - 4 * q^84 + 20 * q^86 + 30 * q^89 + 8 * q^91 + 24 * q^94 - 2 * q^96 - 6 * 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
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	1.00000i	0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	− 3.00000i	− 0.639602i
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	1.00000i	0.162221i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 2.00000i	− 0.308607i
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−3.00000	−0.442326
$47$	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
$51$	6.00000	0.840168
$52$	2.00000i	0.277350i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	1.00000i	0.132453i
$58$	3.00000i	0.393919i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	7.00000i	0.889001i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	1.00000i	0.117041i	0.998286	+	0.0585206i	$0.0186383\pi$
$73$	1.00000i	0.117041i	−0.998286	+	0.0585206i	$0.981362\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	6.00000i	0.683763i
$78$	2.00000i	0.226455i
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	− 15.0000i	− 1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	− 15.0000i	− 1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	3.00000i	0.321634i
$88$	3.00000i	0.319801i
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	3.00000i	0.312772i
$93$	7.00000i	0.725866i
$94$	12.0000	1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 6.00000i	− 0.594089i
$103$	7.00000i	0.689730i	0.938652	+	0.344865i	$0.112075\pi$
$103$	7.00000i	0.689730i	−0.938652	+	0.344865i	$0.887925\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	9.00000	0.874157
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−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
$111$	2.00000	0.189832
$112$	2.00000i	0.188982i
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	3.00000	0.278543
$117$	2.00000i	0.184900i
$118$	6.00000i	0.552345i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	1.00000i	0.0905357i
$123$	6.00000i	0.541002i
$124$	7.00000	0.628619
$125$	0	0
$126$	−2.00000	−0.178174
$127$	11.0000i	0.976092i	0.872818	+	0.488046i	$0.162290\pi$
$127$	11.0000i	0.976092i	−0.872818	+	0.488046i	$0.837710\pi$
$128$	1.00000i	0.0883883i
$129$	10.0000	0.880451
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	3.00000i	0.261116i
$133$	− 2.00000i	− 0.173422i
$134$	5.00000	0.431934
$135$	0	0
$136$	−6.00000	−0.514496
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	3.00000i	0.255377i
$139$	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
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	1.00000	0.0827606
$147$	− 3.00000i	− 0.247436i
$148$	− 2.00000i	− 0.164399i
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	− 6.00000i	− 0.485071i
$154$	6.00000	0.483494
$155$	0	0
$156$	2.00000	0.160128
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 13.0000i	− 1.03422i
$159$	9.00000	0.713746
$160$	0	0
$161$	6.00000	0.472866
$162$	− 1.00000i	− 0.0785674i
$163$	22.0000i	1.72317i	0.507611	+	0.861586i	$0.330529\pi$
$163$	22.0000i	1.72317i	−0.507611	+	0.861586i	$0.669471\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−15.0000	−1.16423
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	2.00000i	0.154303i
$169$	9.00000	0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 10.0000i	− 0.762493i
$173$	3.00000i	0.228086i	0.993476	+	0.114043i	$0.0363801\pi$
$173$	3.00000i	0.228086i	−0.993476	+	0.114043i	$0.963620\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	3.00000	0.226134
$177$	6.00000i	0.450988i
$178$	− 15.0000i	− 1.12430i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	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
$183$	1.00000i	0.0739221i
$184$	3.00000	0.221163
$185$	0	0
$186$	7.00000	0.513265
$187$	18.0000i	1.31629i
$188$	− 12.0000i	− 0.875190i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000i	0.0721688i
$193$	− 20.0000i	− 1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$	− 20.0000i	− 1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	3.00000i	0.213201i
$199$	−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
$201$	5.00000	0.352673
$202$	− 6.00000i	− 0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	7.00000	0.487713
$207$	3.00000i	0.208514i
$208$	− 2.00000i	− 0.138675i
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	− 9.00000i	− 0.618123i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 14.0000i	− 0.950382i
$218$	2.00000i	0.135457i
$219$	1.00000	0.0675737
$220$	0	0
$221$	12.0000	0.807207
$222$	− 2.00000i	− 0.134231i
$223$	− 5.00000i	− 0.334825i	−0.985887	−	0.167412i	$-0.946459\pi$
$223$	− 5.00000i	− 0.334825i	0.985887	−	0.167412i	$-0.0535411\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−9.00000	−0.598671
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	− 3.00000i	− 0.196960i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	− 13.0000i	− 0.844441i
$238$	12.0000i	0.777844i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	2.00000i	0.128565i
$243$	− 1.00000i	− 0.0641500i
$244$	1.00000	0.0640184
$245$	0	0
$246$	6.00000	0.382546
$247$	2.00000i	0.127257i
$248$	− 7.00000i	− 0.444500i
$249$	−15.0000	−0.950586
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	2.00000i	0.125988i
$253$	− 9.00000i	− 0.565825i
$254$	11.0000	0.690201
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 9.00000i	− 0.561405i	−0.959795	−	0.280702i	$-0.909433\pi$
$257$	− 9.00000i	− 0.561405i	0.959795	−	0.280702i	$-0.0905674\pi$
$258$	− 10.0000i	− 0.622573i
$259$	−4.00000	−0.248548
$260$	0	0
$261$	3.00000	0.185695
$262$	15.0000i	0.926703i
$263$	21.0000i	1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$	21.0000i	1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	−2.00000	−0.122628
$267$	− 15.0000i	− 0.917985i
$268$	− 5.00000i	− 0.305424i
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000i	0.363803i
$273$	− 4.00000i	− 0.242091i
$274$	12.0000	0.724947
$275$	0	0
$276$	3.00000	0.180579
$277$	11.0000i	0.660926i	0.943819	+	0.330463i	$0.107205\pi$
$277$	11.0000i	0.660926i	−0.943819	+	0.330463i	$0.892795\pi$
$278$	− 4.00000i	− 0.239904i
$279$	7.00000	0.419079
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	− 12.0000i	− 0.714590i
$283$	− 8.00000i	− 0.475551i	−0.971320	−	0.237775i	$-0.923582\pi$
$283$	− 8.00000i	− 0.475551i	0.971320	−	0.237775i	$-0.0764182\pi$
$284$	0	0
$285$	0	0
$286$	−6.00000	−0.354787
$287$	− 12.0000i	− 0.708338i
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	− 1.00000i	− 0.0585206i
$293$	15.0000i	0.876309i	0.898900	+	0.438155i	$0.144368\pi$
$293$	15.0000i	0.876309i	−0.898900	+	0.438155i	$0.855632\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	3.00000i	0.174078i
$298$	12.0000i	0.695141i
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−20.0000	−1.15278
$302$	− 20.0000i	− 1.15087i
$303$	− 6.00000i	− 0.344691i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 19.0000i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	− 19.0000i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$	− 6.00000i	− 0.341882i
$309$	7.00000	0.398216
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	− 2.00000i	− 0.113228i
$313$	13.0000i	0.734803i	0.930062	+	0.367402i	$0.119753\pi$
$313$	13.0000i	0.734803i	−0.930062	+	0.367402i	$0.880247\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−13.0000	−0.731307
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	− 9.00000i	− 0.504695i
$319$	−9.00000	−0.503903
$320$	0	0
$321$	0	0
$322$	− 6.00000i	− 0.334367i
$323$	− 6.00000i	− 0.333849i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	22.0000	1.21847
$327$	2.00000i	0.110600i
$328$	− 6.00000i	− 0.331295i
$329$	−24.0000	−1.32316
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	15.0000i	0.823232i
$333$	− 2.00000i	− 0.109599i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	2.00000	0.109109
$337$	32.0000i	1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$	32.0000i	1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−9.00000	−0.488813
$340$	0	0
$341$	−21.0000	−1.13721
$342$	− 1.00000i	− 0.0540738i
$343$	20.0000i	1.07990i
$344$	−10.0000	−0.539164
$345$	0	0
$346$	3.00000	0.161281
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	− 3.00000i	− 0.160817i
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 3.00000i	− 0.159901i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−15.0000	−0.794998
$357$	12.0000i	0.635107i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	10.0000i	0.525588i
$363$	2.00000i	0.104973i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	1.00000	0.0522708
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 3.00000i	− 0.156386i
$369$	6.00000	0.312348
$370$	0	0
$371$	−18.0000	−0.934513
$372$	− 7.00000i	− 0.362933i
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	−12.0000	−0.618853
$377$	6.00000i	0.309016i
$378$	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
$381$	11.0000	0.563547
$382$	− 3.00000i	− 0.153493i
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−20.0000	−1.01797
$387$	− 10.0000i	− 0.508329i
$388$	− 8.00000i	− 0.406138i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	3.00000i	0.151523i
$393$	15.0000i	0.756650i
$394$	6.00000	0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	17.0000i	0.853206i	0.904439	+	0.426603i	$0.140290\pi$
$397$	17.0000i	0.853206i	−0.904439	+	0.426603i	$0.859710\pi$
$398$	2.00000i	0.100251i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	− 5.00000i	− 0.249377i
$403$	14.0000i	0.697390i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	6.00000i	0.297409i
$408$	6.00000i	0.297044i
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	− 7.00000i	− 0.344865i
$413$	− 12.0000i	− 0.590481i
$414$	3.00000	0.147442
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 4.00000i	− 0.195881i
$418$	3.00000i	0.146735i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	13.0000i	0.632830i
$423$	− 12.0000i	− 0.583460i
$424$	−9.00000	−0.437079
$425$	0	0
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	1.00000i	0.0481125i
$433$	16.0000i	0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$	16.0000i	0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	−14.0000	−0.672022
$435$	0	0
$436$	2.00000	0.0957826
$437$	3.00000i	0.143509i
$438$	− 1.00000i	− 0.0477818i
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 12.0000i	− 0.570782i
$443$	− 27.0000i	− 1.28281i	−0.767203	−	0.641404i	$-0.778352\pi$
$443$	− 27.0000i	− 1.28281i	0.767203	−	0.641404i	$-0.221648\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−5.00000	−0.236757
$447$	12.0000i	0.567581i
$448$	− 2.00000i	− 0.0944911i
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	9.00000i	0.423324i
$453$	− 20.0000i	− 0.939682i
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	− 13.0000i	− 0.607450i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	− 6.00000i	− 0.279145i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 39.0000i	− 1.80470i	−0.430999	−	0.902352i	$-0.641839\pi$
$467$	− 39.0000i	− 1.80470i	0.430999	−	0.902352i	$-0.358161\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−10.0000	−0.461757
$470$	0	0
$471$	14.0000	0.645086
$472$	− 6.00000i	− 0.276172i
$473$	30.0000i	1.37940i
$474$	−13.0000	−0.597110
$475$	0	0
$476$	12.0000	0.550019
$477$	− 9.00000i	− 0.412082i
$478$	0	0
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	10.0000i	0.455488i
$483$	− 6.00000i	− 0.273009i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	− 1.00000i	− 0.0452679i
$489$	22.0000	0.994874
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 18.0000i	− 0.810679i
$494$	2.00000	0.0899843
$495$	0	0
$496$	−7.00000	−0.314309
$497$	0	0
$498$	15.0000i	0.672166i
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	− 24.0000i	− 1.07117i
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	−9.00000	−0.400099
$507$	− 9.00000i	− 0.399704i
$508$	− 11.0000i	− 0.488046i
$509$	−33.0000	−1.46270	−0.731350	−	0.682003i	$-0.761109\pi$
$509$	−33.0000	−1.46270	−0.731350	+	0.682003i	$0.761109\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	−9.00000	−0.396973
$515$	0	0
$516$	−10.0000	−0.440225
$517$	36.0000i	1.58328i
$518$	4.00000i	0.175750i
$519$	3.00000	0.131685
$520$	0	0
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	− 3.00000i	− 0.131306i
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	21.0000	0.915644
$527$	− 42.0000i	− 1.82955i
$528$	− 3.00000i	− 0.130558i
$529$	14.0000	0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	2.00000i	0.0867110i
$533$	12.0000i	0.519778i
$534$	−15.0000	−0.649113
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	30.0000i	1.29339i
$539$	9.00000	0.387657
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	− 20.0000i	− 0.859074i
$543$	10.0000i	0.429141i
$544$	6.00000	0.257248
$545$	0	0
$546$	−4.00000	−0.171184
$547$	11.0000i	0.470326i	0.971956	+	0.235163i	$0.0755624\pi$
$547$	11.0000i	0.470326i	−0.971956	+	0.235163i	$0.924438\pi$
$548$	− 12.0000i	− 0.512615i
$549$	1.00000	0.0426790
$550$	0	0
$551$	3.00000	0.127804
$552$	− 3.00000i	− 0.127688i
$553$	26.0000i	1.10563i
$554$	11.0000	0.467345
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 7.00000i	− 0.296334i
$559$	20.0000	0.845910
$560$	0	0
$561$	18.0000	0.759961
$562$	− 3.00000i	− 0.126547i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−8.00000	−0.336265
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	6.00000i	0.250873i
$573$	− 3.00000i	− 0.125327i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	23.0000i	0.957503i	0.877951	+	0.478751i	$0.158910\pi$
$577$	23.0000i	0.957503i	−0.877951	+	0.478751i	$0.841090\pi$
$578$	19.0000i	0.790296i
$579$	−20.0000	−0.831172
$580$	0	0
$581$	30.0000	1.24461
$582$	− 8.00000i	− 0.331611i
$583$	27.0000i	1.11823i
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	15.0000	0.619644
$587$	− 45.0000i	− 1.85735i	−0.370896	−	0.928674i	$-0.620949\pi$
$587$	− 45.0000i	− 1.85735i	0.370896	−	0.928674i	$-0.379051\pi$
$588$	3.00000i	0.123718i
$589$	7.00000	0.288430
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000i	0.0821995i
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	12.0000	0.491539
$597$	2.00000i	0.0818546i
$598$	6.00000i	0.245358i
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	20.0000i	0.815139i
$603$	− 5.00000i	− 0.203616i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−6.00000	−0.243733
$607$	23.0000i	0.933541i	0.884378	+	0.466771i	$0.154583\pi$
$607$	23.0000i	0.933541i	−0.884378	+	0.466771i	$0.845417\pi$
$608$	1.00000i	0.0405554i
$609$	−6.00000	−0.243132
$610$	0	0
$611$	24.0000	0.970936
$612$	6.00000i	0.242536i
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−19.0000	−0.766778
$615$	0	0
$616$	−6.00000	−0.241747
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 7.00000i	− 0.281581i
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	− 12.0000i	− 0.481156i
$623$	30.0000i	1.20192i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	13.0000	0.519584
$627$	3.00000i	0.119808i
$628$	− 14.0000i	− 0.558661i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	13.0000i	0.517112i
$633$	13.0000i	0.516704i
$634$	−3.00000	−0.119145
$635$	0	0
$636$	−9.00000	−0.356873
$637$	− 6.00000i	− 0.237729i
$638$	9.00000i	0.356313i
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	−6.00000	−0.236067
$647$	− 15.0000i	− 0.589711i	−0.955542	−	0.294855i	$-0.904729\pi$
$647$	− 15.0000i	− 0.589711i	0.955542	−	0.294855i	$-0.0952715\pi$
$648$	1.00000i	0.0392837i
$649$	−18.0000	−0.706562
$650$	0	0
$651$	−14.0000	−0.548703
$652$	− 22.0000i	− 0.861586i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 1.00000i	− 0.0390137i
$658$	24.0000i	0.935617i
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	− 5.00000i	− 0.194331i
$663$	− 12.0000i	− 0.466041i
$664$	15.0000	0.582113
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	9.00000i	0.348481i
$668$	18.0000i	0.696441i
$669$	−5.00000	−0.193311
$670$	0	0
$671$	−3.00000	−0.115814
$672$	− 2.00000i	− 0.0771517i
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	32.0000	1.23259
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 33.0000i	− 1.26829i	−0.773213	−	0.634147i	$-0.781352\pi$
$677$	− 33.0000i	− 1.26829i	0.773213	−	0.634147i	$-0.218648\pi$
$678$	9.00000i	0.345643i
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−18.0000	−0.689761
$682$	21.0000i	0.804132i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	− 13.0000i	− 0.495981i
$688$	10.0000i	0.381246i
$689$	18.0000	0.685745
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 3.00000i	− 0.114043i
$693$	− 6.00000i	− 0.227921i
$694$	12.0000	0.455514
$695$	0	0
$696$	−3.00000	−0.113715
$697$	− 36.0000i	− 1.36360i
$698$	− 19.0000i	− 0.719161i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 2.00000i	− 0.0754314i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	6.00000	0.225813
$707$	12.0000i	0.451306i
$708$	− 6.00000i	− 0.225494i
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	−13.0000	−0.487538
$712$	15.0000i	0.562149i
$713$	21.0000i	0.786456i
$714$	12.0000	0.449089
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	− 1.00000i	− 0.0372161i
$723$	10.0000i	0.371904i
$724$	10.0000	0.371647
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 34.0000i	− 1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$	− 34.0000i	− 1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$	4.00000i	0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−60.0000	−2.21918
$732$	− 1.00000i	− 0.0369611i
$733$	− 5.00000i	− 0.184679i	−0.995728	−	0.0923396i	$-0.970565\pi$
$733$	− 5.00000i	− 0.184679i	0.995728	−	0.0923396i	$-0.0294345\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−3.00000	−0.110581
$737$	15.0000i	0.552532i
$738$	− 6.00000i	− 0.220863i
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	18.0000i	0.660801i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	−7.00000	−0.256632
$745$	0	0
$746$	22.0000	0.805477
$747$	15.0000i	0.548821i
$748$	− 18.0000i	− 0.658145i
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	12.0000i	0.437595i
$753$	− 24.0000i	− 0.874609i
$754$	6.00000	0.218507
$755$	0	0
$756$	2.00000	0.0727393
$757$	29.0000i	1.05402i	0.849858	+	0.527011i	$0.176688\pi$
$757$	29.0000i	1.05402i	−0.849858	+	0.527011i	$0.823312\pi$
$758$	20.0000i	0.726433i
$759$	−9.00000	−0.326679
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	− 11.0000i	− 0.398488i
$763$	− 4.00000i	− 0.144810i
$764$	−3.00000	−0.108536
$765$	0	0
$766$	−12.0000	−0.433578
$767$	12.0000i	0.433295i
$768$	− 1.00000i	− 0.0360844i
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	−9.00000	−0.324127
$772$	20.0000i	0.719816i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−8.00000	−0.287183
$777$	4.00000i	0.143499i
$778$	− 6.00000i	− 0.215110i
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	− 18.0000i	− 0.643679i
$783$	− 3.00000i	− 0.107211i
$784$	3.00000	0.107143
$785$	0	0
$786$	15.0000	0.535032
$787$	− 49.0000i	− 1.74666i	−0.487128	−	0.873331i	$-0.661955\pi$
$787$	− 49.0000i	− 1.74666i	0.487128	−	0.873331i	$-0.338045\pi$
$788$	− 6.00000i	− 0.213741i
$789$	21.0000	0.747620
$790$	0	0
$791$	18.0000	0.640006
$792$	− 3.00000i	− 0.106600i
$793$	2.00000i	0.0710221i
$794$	17.0000	0.603307
$795$	0	0
$796$	2.00000	0.0708881
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	2.00000i	0.0707992i
$799$	−72.0000	−2.54718
$800$	0	0
$801$	−15.0000	−0.529999
$802$	− 9.00000i	− 0.317801i
$803$	3.00000i	0.105868i
$804$	−5.00000	−0.176336
$805$	0	0
$806$	14.0000	0.493129
$807$	30.0000i	1.05605i
$808$	6.00000i	0.211079i
$809$	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
$811$	41.0000	1.43970	0.719852	−	0.694127i	$-0.244209\pi$
$811$	41.0000	1.43970	0.719852	+	0.694127i	$0.244209\pi$
$812$	6.00000i	0.210559i
$813$	− 20.0000i	− 0.701431i
$814$	6.00000	0.210300
$815$	0	0
$816$	6.00000	0.210042
$817$	− 10.0000i	− 0.349856i
$818$	− 34.0000i	− 1.18878i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	− 12.0000i	− 0.418548i
$823$	− 26.0000i	− 0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$	− 26.0000i	− 0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$	−7.00000	−0.243857
$825$	0	0
$826$	−12.0000	−0.417533
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	− 3.00000i	− 0.104257i
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	11.0000	0.381586
$832$	2.00000i	0.0693375i
$833$	18.0000i	0.623663i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	3.00000	0.103757
$837$	− 7.00000i	− 0.241955i
$838$	12.0000i	0.414533i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	10.0000i	0.344623i
$843$	− 3.00000i	− 0.103325i
$844$	13.0000	0.447478
$845$	0	0
$846$	−12.0000	−0.412568
$847$	− 4.00000i	− 0.137442i
$848$	9.00000i	0.309061i
$849$	−8.00000	−0.274559
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	6.00000i	0.204837i
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	− 18.0000i	− 0.613082i
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	16.0000	0.543702
$867$	19.0000i	0.645274i
$868$	14.0000i	0.475191i
$869$	39.0000	1.32298
$870$	0	0
$871$	10.0000	0.338837
$872$	− 2.00000i	− 0.0677285i
$873$	− 8.00000i	− 0.270759i
$874$	3.00000	0.101477
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	35.0000i	1.18119i
$879$	15.0000	0.505937
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	3.00000i	0.101015i
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−27.0000	−0.907083
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	2.00000i	0.0671156i
$889$	−22.0000	−0.737856
$890$	0	0
$891$	3.00000	0.100504
$892$	5.00000i	0.167412i
$893$	− 12.0000i	− 0.401565i
$894$	12.0000	0.401340
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	6.00000i	0.200334i
$898$	− 33.0000i	− 1.10122i
$899$	21.0000	0.700389
$900$	0	0
$901$	−54.0000	−1.79900
$902$	18.0000i	0.599334i
$903$	20.0000i	0.665558i
$904$	9.00000	0.299336
$905$	0	0
$906$	−20.0000	−0.664455
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	18.0000i	0.597351i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	1.00000i	0.0331133i
$913$	− 45.0000i	− 1.48928i
$914$	2.00000	0.0661541
$915$	0	0
$916$	−13.0000	−0.429532
$917$	− 30.0000i	− 0.990687i
$918$	6.00000i	0.198030i
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−19.0000	−0.626071
$922$	12.0000i	0.395199i
$923$	0	0
$924$	−6.00000	−0.197386
$925$	0	0
$926$	16.0000	0.525793
$927$	− 7.00000i	− 0.229910i
$928$	3.00000i	0.0984798i
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	6.00000i	0.196537i
$933$	− 12.0000i	− 0.392862i
$934$	−39.0000	−1.27612
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	10.0000i	0.326512i
$939$	13.0000	0.424239
$940$	0	0
$941$	−51.0000	−1.66255	−0.831276	−	0.555860i	$-0.812389\pi$
$941$	−51.0000	−1.66255	−0.831276	+	0.555860i	$0.812389\pi$
$942$	− 14.0000i	− 0.456145i
$943$	18.0000i	0.586161i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	30.0000	0.975384
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	13.0000i	0.422220i
$949$	2.00000	0.0649227
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	− 12.0000i	− 0.388922i
$953$	33.0000i	1.06897i	0.845176	+	0.534487i	$0.179495\pi$
$953$	33.0000i	1.06897i	−0.845176	+	0.534487i	$0.820505\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	0	0
$957$	9.00000i	0.290929i
$958$	27.0000i	0.872330i
$959$	−24.0000	−0.775000
$960$	0	0
$961$	18.0000	0.580645
$962$	− 4.00000i	− 0.128965i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	−6.00000	−0.193047
$967$	38.0000i	1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$	38.0000i	1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000i	0.0320750i
$973$	8.00000i	0.256468i
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	− 22.0000i	− 0.703482i
$979$	45.0000	1.43821
$980$	0	0
$981$	2.00000	0.0638551
$982$	12.0000i	0.382935i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−18.0000	−0.573237
$987$	24.0000i	0.763928i
$988$	− 2.00000i	− 0.0636285i
$989$	30.0000	0.953945
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	7.00000i	0.222250i
$993$	− 5.00000i	− 0.158670i
$994$	0	0
$995$	0	0
$996$	15.0000	0.475293
$997$	− 19.0000i	− 0.601736i	−0.953666	−	0.300868i	$-0.902724\pi$
$997$	− 19.0000i	− 0.601736i	0.953666	−	0.300868i	$-0.0972764\pi$
$998$	14.0000i	0.443162i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.h.799.1		2
5.2	odd	4		2850.2.a.r.1.1	yes	1
5.3	odd	4		2850.2.a.o.1.1	✓	1
5.4	even	2	inner	2850.2.d.h.799.2		2
15.2	even	4		8550.2.a.d.1.1		1
15.8	even	4		8550.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.o.1.1	✓	1	5.3	odd	4
2850.2.a.r.1.1	yes	1	5.2	odd	4
2850.2.d.h.799.1		2	1.1	even	1	trivial
2850.2.d.h.799.2		2	5.4	even	2	inner
8550.2.a.d.1.1		1	15.2	even	4
8550.2.a.be.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -3.00000i q^{22} -3.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -3.00000 q^{29} -7.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} +10.0000i q^{43} -3.00000 q^{44} -3.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} +2.00000i q^{52} +9.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +3.00000i q^{58} -6.00000 q^{59} -1.00000 q^{61} +7.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +5.00000i q^{67} -6.00000i q^{68} -3.00000 q^{69} -1.00000i q^{72} +1.00000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{77} +2.00000i q^{78} +13.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -15.0000i q^{83} -2.00000 q^{84} +10.0000 q^{86} +3.00000i q^{87} +3.00000i q^{88} +15.0000 q^{89} +4.00000 q^{91} +3.00000i q^{92} +7.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} +8.00000i q^{97} -3.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 6 q^{11} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} - 6 q^{29} - 14 q^{31} + 12 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 6 q^{44} - 6 q^{46} + 6 q^{49} + 12 q^{51} + 2 q^{54} - 4 q^{56} - 12 q^{59} - 2 q^{61} - 2 q^{64} - 6 q^{66} - 6 q^{69} + 4 q^{74} + 2 q^{76} + 26 q^{79} + 2 q^{81} - 4 q^{84} + 20 q^{86} + 30 q^{89} + 8 q^{91} + 24 q^{94} - 2 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 6 * q^11 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 - 6 * q^29 - 14 * q^31 + 12 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 6 * q^44 - 6 * q^46 + 6 * q^49 + 12 * q^51 + 2 * q^54 - 4 * q^56 - 12 * q^59 - 2 * q^61 - 2 * q^64 - 6 * q^66 - 6 * q^69 + 4 * q^74 + 2 * q^76 + 26 * q^79 + 2 * q^81 - 4 * q^84 + 20 * q^86 + 30 * q^89 + 8 * q^91 + 24 * q^94 - 2 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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