LMFDB - Embedded newform 2850.2.d.t.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.t.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -6.00000i q^{22} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +1.00000i q^{38} +4.00000 q^{39} -12.0000 q^{41} -2.00000i q^{42} +2.00000i q^{43} -6.00000 q^{44} +1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +12.0000 q^{59} +2.00000 q^{61} -8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +16.0000i q^{67} -6.00000i q^{68} -1.00000i q^{72} -10.0000i q^{73} -8.00000 q^{74} +1.00000 q^{76} -12.0000i q^{77} -4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{82} -2.00000 q^{84} +2.00000 q^{86} +6.00000i q^{88} +12.0000 q^{89} -8.00000 q^{91} +8.00000i q^{93} +1.00000 q^{96} -8.00000i q^{97} -3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{26} + 16 q^{31} + 12 q^{34} + 2 q^{36} + 8 q^{39} - 24 q^{41} - 12 q^{44} + 6 q^{49} - 12 q^{51} - 2 q^{54} + 4 q^{56} + 24 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{66} - 16 q^{74} + 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{86} + 24 q^{89} - 16 q^{91} + 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 - 8 * q^26 + 16 * q^31 + 12 * q^34 + 2 * q^36 + 8 * q^39 - 24 * q^41 - 12 * q^44 + 6 * q^49 - 12 * q^51 - 2 * q^54 + 4 * q^56 + 24 * q^59 + 4 * q^61 - 2 * q^64 + 12 * q^66 - 16 * q^74 + 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 4 * q^86 + 24 * q^89 - 16 * q^91 + 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	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−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
$22$	− 6.00000i	− 1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−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	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	6.00000i	1.04447i
$34$	6.00000	1.02899
$35$	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
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	− 2.00000i	− 0.308607i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	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
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	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
$67$	16.0000i	1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$	16.0000i	1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	1.00000	0.114708
$77$	− 12.0000i	− 1.36753i
$78$	− 4.00000i	− 0.452911i
$79$	−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
$82$	12.0000i	1.32518i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	6.00000i	0.639602i
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−6.00000	−0.603023
$100$	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
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	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
$109$	−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
$111$	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
$118$	− 12.0000i	− 1.10469i
$119$	12.0000	1.10004
$120$	0	0
$121$	25.0000	2.27273
$122$	− 2.00000i	− 0.181071i
$123$	− 12.0000i	− 1.08200i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	2.00000	0.178174
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	− 6.00000i	− 0.522233i
$133$	2.00000i	0.173422i
$134$	16.0000	1.38219
$135$	0	0
$136$	−6.00000	−0.514496
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 24.0000i	− 2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	3.00000i	0.247436i
$148$	8.00000i	0.657596i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	− 6.00000i	− 0.485071i
$154$	−12.0000	−0.966988
$155$	0	0
$156$	−4.00000	−0.320256
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	8.00000i	0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	2.00000i	0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 2.00000i	− 0.152499i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	12.0000i	0.901975i
$178$	− 12.0000i	− 0.899438i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	36.0000i	2.63258i
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	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
$201$	−16.0000	−1.12855
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	− 4.00000i	− 0.277350i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 16.0000i	− 1.08615i
$218$	2.00000i	0.135457i
$219$	10.0000	0.675737
$220$	0	0
$221$	24.0000	1.61441
$222$	− 8.00000i	− 0.536925i
$223$	− 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
$226$	−6.00000	−0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	1.00000i	0.0662266i
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−12.0000	−0.781133
$237$	− 8.00000i	− 0.519656i
$238$	− 12.0000i	− 0.777844i
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 25.0000i	− 1.60706i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−12.0000	−0.765092
$247$	4.00000i	0.254514i
$248$	8.00000i	0.508001i
$249$	0	0
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	2.00000i	0.124515i
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	− 18.0000i	− 1.11204i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	2.00000	0.122628
$267$	12.0000i	0.734388i
$268$	− 16.0000i	− 0.977356i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	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
$273$	− 8.00000i	− 0.484182i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	8.00000i	0.479808i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0	0
$285$	0	0
$286$	−24.0000	−1.41915
$287$	24.0000i	1.41668i
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	10.0000i	0.585206i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	8.00000	0.464991
$297$	− 6.00000i	− 0.348155i
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	− 8.00000i	− 0.460348i
$303$	6.00000i	0.344691i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	12.0000i	0.683763i
$309$	16.0000	0.910208
$310$	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
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	8.00000	0.450035
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−10.0000	−0.553849
$327$	− 2.00000i	− 0.110600i
$328$	− 12.0000i	− 0.662589i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	2.00000	0.109109
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	3.00000i	0.163178i
$339$	6.00000	0.325875
$340$	0	0
$341$	48.0000	2.59935
$342$	− 1.00000i	− 0.0540738i
$343$	− 20.0000i	− 1.07990i
$344$	−2.00000	−0.107833
$345$	0	0
$346$	6.00000	0.322562
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	− 6.00000i	− 0.319801i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−12.0000	−0.635999
$357$	12.0000i	0.635107i
$358$	− 12.0000i	− 0.634220i
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 2.00000i	− 0.105118i
$363$	25.0000i	1.31216i
$364$	8.00000	0.419314
$365$	0	0
$366$	2.00000	0.104542
$367$	34.0000i	1.77479i	0.461014	+	0.887393i	$0.347486\pi$
$367$	34.0000i	1.77479i	−0.461014	+	0.887393i	$0.652514\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	−12.0000	−0.623009
$372$	− 8.00000i	− 0.414781i
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	36.0000	1.86152
$375$	0	0
$376$	0	0
$377$	0	0
$378$	2.00000i	0.102869i
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	− 6.00000i	− 0.306987i
$383$	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
$386$	20.0000	1.01797
$387$	− 2.00000i	− 0.101666i
$388$	8.00000i	0.406138i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	18.0000i	0.907980i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	6.00000	0.301511
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	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	$0.5\pi$
$402$	16.0000i	0.798007i
$403$	− 32.0000i	− 1.59403i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	− 48.0000i	− 2.37927i
$408$	− 6.00000i	− 0.297044i
$409$	−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
$412$	16.0000i	0.788263i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	− 8.00000i	− 0.391762i
$418$	6.00000i	0.293470i
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	− 12.0000i	− 0.580042i
$429$	24.0000	1.15873
$430$	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
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	− 10.0000i	− 0.477818i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 24.0000i	− 1.14156i
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−16.0000	−0.757622
$447$	18.0000i	0.851371i
$448$	2.00000i	0.0944911i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−72.0000	−3.39035
$452$	6.00000i	0.282216i
$453$	8.00000i	0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	1.00000	0.0468293
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	2.00000i	0.0934539i
$459$	6.00000	0.280056
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	− 12.0000i	− 0.558291i
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	− 4.00000i	− 0.184900i
$469$	32.0000	1.47762
$470$	0	0
$471$	−10.0000	−0.460776
$472$	12.0000i	0.552345i
$473$	12.0000i	0.551761i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	6.00000i	0.274721i
$478$	− 6.00000i	− 0.274434i
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−32.0000	−1.45907
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	2.00000i	0.0905357i
$489$	10.0000	0.452216
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	12.0000i	0.541002i
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	− 6.00000i	− 0.267793i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	20.0000i	0.887357i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	6.00000	0.264649
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	16.0000i	0.703000i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	−12.0000	−0.523225
$527$	48.0000i	2.09091i
$528$	6.00000i	0.261116i
$529$	23.0000	1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	− 2.00000i	− 0.0867110i
$533$	48.0000i	2.07911i
$534$	12.0000	0.519291
$535$	0	0
$536$	−16.0000	−0.691095
$537$	12.0000i	0.517838i
$538$	24.0000i	1.03471i
$539$	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
$544$	6.00000	0.257248
$545$	0	0
$546$	−8.00000	−0.342368
$547$	16.0000i	0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$	16.0000i	0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$	6.00000i	0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000i	0.680389i
$554$	10.0000	0.424859
$555$	0	0
$556$	8.00000	0.339276
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	8.00000i	0.338667i
$559$	8.00000	0.338364
$560$	0	0
$561$	−36.0000	−1.51992
$562$	− 24.0000i	− 1.01238i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	14.0000	0.588464
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	24.0000i	1.00349i
$573$	6.00000i	0.250654i
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	19.0000i	0.790296i
$579$	−20.0000	−0.831172
$580$	0	0
$581$	0	0
$582$	− 8.00000i	− 0.331611i
$583$	− 36.0000i	− 1.49097i
$584$	10.0000	0.413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	− 3.00000i	− 0.123718i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	− 8.00000i	− 0.328798i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−18.0000	−0.737309
$597$	− 20.0000i	− 0.818546i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	− 4.00000i	− 0.163028i
$603$	− 16.0000i	− 0.651570i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	1.00000i	0.0405554i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	6.00000i	0.242536i
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	12.0000	0.483494
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	− 16.0000i	− 0.643614i
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	0	0
$622$	18.0000i	0.721734i
$623$	− 24.0000i	− 0.961540i
$624$	4.00000	0.160128
$625$	0	0
$626$	26.0000	1.03917
$627$	− 6.00000i	− 0.239617i
$628$	− 10.0000i	− 0.399043i
$629$	48.0000	1.91389
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 8.00000i	− 0.318223i
$633$	− 4.00000i	− 0.158986i
$634$	30.0000	1.19145
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	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	$0.910973\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	1.00000i	0.0392837i
$649$	72.0000	2.82625
$650$	0	0
$651$	16.0000	0.627089
$652$	10.0000i	0.391630i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−12.0000	−0.468521
$657$	10.0000i	0.390137i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	28.0000i	1.08825i
$663$	24.0000i	0.932083i
$664$	0	0
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	24.0000i	0.928588i
$669$	16.0000	0.618596
$670$	0	0
$671$	12.0000	0.463255
$672$	− 2.00000i	− 0.0771517i
$673$	8.00000i	0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$	8.00000i	0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	3.00000	0.115385
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	− 6.00000i	− 0.230429i
$679$	−16.0000	−0.614024
$680$	0	0
$681$	12.0000	0.459841
$682$	− 48.0000i	− 1.83801i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 2.00000i	− 0.0763048i
$688$	2.00000i	0.0762493i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 6.00000i	− 0.228086i
$693$	12.0000i	0.455842i
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	− 72.0000i	− 2.72719i
$698$	14.0000i	0.529908i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	4.00000i	0.150970i
$703$	8.00000i	0.301726i
$704$	−6.00000	−0.226134
$705$	0	0
$706$	30.0000	1.12906
$707$	− 12.0000i	− 0.451306i
$708$	− 12.0000i	− 0.450988i
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	12.0000i	0.449719i
$713$	0	0
$714$	12.0000	0.449089
$715$	0	0
$716$	−12.0000	−0.448461
$717$	6.00000i	0.224074i
$718$	6.00000i	0.223918i
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	− 1.00000i	− 0.0372161i
$723$	2.00000i	0.0743808i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	25.0000	0.927837
$727$	10.0000i	0.370879i	0.982656	+	0.185440i	$0.0593710\pi$
$727$	10.0000i	0.370879i	−0.982656	+	0.185440i	$0.940629\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	− 2.00000i	− 0.0739221i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	0	0
$737$	96.0000i	3.53621i
$738$	− 12.0000i	− 0.441726i
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	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
$746$	−4.00000	−0.146450
$747$	0	0
$748$	− 36.0000i	− 1.31629i
$749$	24.0000	0.876941
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	6.00000i	0.218652i
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	− 20.0000i	− 0.724524i
$763$	4.00000i	0.144810i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	24.0000	0.867155
$767$	− 48.0000i	− 1.73318i
$768$	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
$771$	−6.00000	−0.216085
$772$	− 20.0000i	− 0.719816i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	8.00000	0.287183
$777$	− 16.0000i	− 0.573997i
$778$	18.0000i	0.645331i
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	18.0000	0.642039
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	6.00000i	0.213741i
$789$	12.0000	0.427211
$790$	0	0
$791$	−12.0000	−0.426671
$792$	− 6.00000i	− 0.213201i
$793$	− 8.00000i	− 0.284088i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	20.0000	0.708881
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	2.00000i	0.0707992i
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	− 60.0000i	− 2.11735i
$804$	16.0000	0.564276
$805$	0	0
$806$	−32.0000	−1.12715
$807$	− 24.0000i	− 0.844840i
$808$	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
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	20.0000i	0.701431i
$814$	−48.0000	−1.68240
$815$	0	0
$816$	−6.00000	−0.210042
$817$	− 2.00000i	− 0.0699711i
$818$	14.0000i	0.489499i
$819$	8.00000	0.279543
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	− 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	$0.921539\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−24.0000	−0.835067
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	4.00000i	0.138675i
$833$	18.0000i	0.623663i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	6.00000	0.207514
$837$	− 8.00000i	− 0.276520i
$838$	30.0000i	1.03633i
$839$	−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
$841$	−29.0000	−1.00000
$842$	10.0000i	0.344623i
$843$	24.0000i	0.826604i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 50.0000i	− 1.71802i
$848$	− 6.00000i	− 0.206041i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	− 24.0000i	− 0.819346i
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	− 12.0000i	− 0.408722i
$863$	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
$866$	−4.00000	−0.135926
$867$	− 19.0000i	− 0.645274i
$868$	16.0000i	0.543075i
$869$	−48.0000	−1.62829
$870$	0	0
$871$	64.0000	2.16856
$872$	− 2.00000i	− 0.0677285i
$873$	8.00000i	0.270759i
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	− 20.0000i	− 0.675352i	−0.941262	−	0.337676i	$-0.890359\pi$
$877$	− 20.0000i	− 0.675352i	0.941262	−	0.337676i	$-0.109641\pi$
$878$	8.00000i	0.269987i
$879$	18.0000	0.607125
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	3.00000i	0.101015i
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−36.0000	−1.20944
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	8.00000i	0.268462i
$889$	−40.0000	−1.34156
$890$	0	0
$891$	6.00000	0.201008
$892$	16.0000i	0.535720i
$893$	0	0
$894$	18.0000	0.602010
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	12.0000i	0.400445i
$899$	0	0
$900$	0	0
$901$	36.0000	1.19933
$902$	72.0000i	2.39734i
$903$	4.00000i	0.133112i
$904$	6.00000	0.199557
$905$	0	0
$906$	8.00000	0.265782
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	12.0000i	0.398234i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	2.00000	0.0660819
$917$	− 36.0000i	− 1.18882i
$918$	− 6.00000i	− 0.198030i
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	42.0000i	1.38320i
$923$	0	0
$924$	−12.0000	−0.394771
$925$	0	0
$926$	14.0000	0.460069
$927$	16.0000i	0.525509i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	− 6.00000i	− 0.196537i
$933$	− 18.0000i	− 0.589294i
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−4.00000	−0.130744
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	− 32.0000i	− 1.04484i
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	10.0000i	0.325818i
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	12.0000	0.390154
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	8.00000i	0.259828i
$949$	−40.0000	−1.29845
$950$	0	0
$951$	−30.0000	−0.972817
$952$	12.0000i	0.388922i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	30.0000i	0.969256i
$959$	−12.0000	−0.387500
$960$	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
$966$	0	0
$967$	34.0000i	1.09337i	0.837340	+	0.546683i	$0.184110\pi$
$967$	34.0000i	1.09337i	−0.837340	+	0.546683i	$0.815890\pi$
$968$	25.0000i	0.803530i
$969$	6.00000	0.192748
$970$	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
$973$	16.0000i	0.512936i
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	− 10.0000i	− 0.319765i
$979$	72.0000	2.30113
$980$	0	0
$981$	2.00000	0.0638551
$982$	6.00000i	0.191468i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	12.0000	0.382546
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	32.0000i	1.01294i
$999$	−8.00000	−0.253109

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.k.1.1	✓	1	5.2	odd	4
1710.2.a.j.1.1		1	15.2	even	4
2850.2.a.c.1.1		1	5.3	odd	4
2850.2.d.t.799.1		2	1.1	even	1	trivial
2850.2.d.t.799.2		2	5.4	even	2	inner
4560.2.a.b.1.1		1	20.7	even	4
8550.2.a.v.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -6.00000i q^{22} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +1.00000i q^{38} +4.00000 q^{39} -12.0000 q^{41} -2.00000i q^{42} +2.00000i q^{43} -6.00000 q^{44} +1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +12.0000 q^{59} +2.00000 q^{61} -8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +16.0000i q^{67} -6.00000i q^{68} -1.00000i q^{72} -10.0000i q^{73} -8.00000 q^{74} +1.00000 q^{76} -12.0000i q^{77} -4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{82} -2.00000 q^{84} +2.00000 q^{86} +6.00000i q^{88} +12.0000 q^{89} -8.00000 q^{91} +8.00000i q^{93} +1.00000 q^{96} -8.00000i q^{97} -3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 12 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} - 8 q^{26} + 16 q^{31} + 12 q^{34} + 2 q^{36} + 8 q^{39} - 24 q^{41} - 12 q^{44} + 6 q^{49} - 12 q^{51} - 2 q^{54} + 4 q^{56} + 24 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{66} - 16 q^{74} + 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{86} + 24 q^{89} - 16 q^{91} + 2 q^{96} - 12 q^{99}+O(q^{100}) \) 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 - 8 * q^26 + 16 * q^31 + 12 * q^34 + 2 * q^36 + 8 * q^39 - 24 * q^41 - 12 * q^44 + 6 * q^49 - 12 * q^51 - 2 * q^54 + 4 * q^56 + 24 * q^59 + 4 * q^61 - 2 * q^64 + 12 * q^66 - 16 * q^74 + 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 4 * q^86 + 24 * q^89 - 16 * q^91 + 2 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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