LMFDB - Embedded newform 1450.2.b.b.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(349,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.b (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:	$11.5783082931$
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 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1450.349
Dual form			1450.2.b.b.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +2.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} -8.00000 q^{34} -2.00000 q^{36} -8.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -2.00000i q^{42} -11.0000i q^{43} +3.00000 q^{44} +4.00000 q^{46} -13.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -11.0000i q^{53} -5.00000 q^{54} -2.00000 q^{56} -1.00000i q^{58} -8.00000 q^{61} +3.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +12.0000i q^{67} +8.00000i q^{68} +4.00000 q^{69} +2.00000 q^{71} +2.00000i q^{72} +4.00000i q^{73} -8.00000 q^{74} -6.00000i q^{77} +1.00000i q^{78} -15.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} -11.0000 q^{86} -1.00000i q^{87} -3.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000i q^{92} +3.00000i q^{93} -13.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} - 6 q^{31} - 16 q^{34} - 4 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 6 q^{49} - 16 q^{51} - 10 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 4 q^{71} - 16 q^{74} - 30 q^{79} + 2 q^{81} - 4 q^{84} - 22 q^{86} + 20 q^{89} + 4 q^{91} - 26 q^{94} - 2 q^{96} - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 6 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^21 + 2 * q^24 - 2 * q^26 + 2 * q^29 - 6 * q^31 - 16 * q^34 - 4 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 + 6 * q^49 - 16 * q^51 - 10 * q^54 - 4 * q^56 - 16 * q^61 - 2 * q^64 + 6 * q^66 + 8 * q^69 + 4 * q^71 - 16 * q^74 - 30 * q^79 + 2 * q^81 - 4 * q^84 - 22 * q^86 + 20 * q^89 + 4 * q^91 - 26 * q^94 - 2 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$
$\chi(n)$	$1$	$-1$

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	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	−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
$9$	2.00000	0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000i	0.288675i
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 8.00000i	− 1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	− 8.00000i	− 1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$	− 2.00000i	− 0.471405i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	3.00000i	0.639602i
$23$	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
$26$	−1.00000	−0.196116
$27$	− 5.00000i	− 0.962250i
$28$	− 2.00000i	− 0.377964i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 1.00000i	− 0.176777i
$33$	3.00000i	0.522233i
$34$	−8.00000	−1.37199
$35$	0	0
$36$	−2.00000	−0.333333
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 11.0000i	− 1.67748i	−0.544529	−	0.838742i	$-0.683292\pi$
$43$	− 11.0000i	− 1.67748i	0.544529	−	0.838742i	$-0.316708\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	− 13.0000i	− 1.89624i	−0.317905	−	0.948122i	$-0.602979\pi$
$47$	− 13.0000i	− 1.89624i	0.317905	−	0.948122i	$-0.397021\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−8.00000	−1.12022
$52$	1.00000i	0.138675i
$53$	− 11.0000i	− 1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$	− 11.0000i	− 1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	− 1.00000i	− 0.131306i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	3.00000i	0.381000i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	8.00000i	0.970143i
$69$	4.00000	0.481543
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	2.00000i	0.235702i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	1.00000i	0.113228i
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	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
$86$	−11.0000	−1.18616
$87$	− 1.00000i	− 0.107211i
$88$	− 3.00000i	− 0.319801i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	− 4.00000i	− 0.417029i
$93$	3.00000i	0.311086i
$94$	−13.0000	−1.34085
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	8.00000i	0.792118i
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−11.0000	−1.06841
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	5.00000i	0.481125i
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	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	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000i	0.724286i
$123$	− 2.00000i	− 0.180334i
$124$	3.00000	0.269408
$125$	0	0
$126$	4.00000	0.356348
$127$	− 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
$129$	−11.0000	−0.968496
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 3.00000i	− 0.261116i
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	8.00000	0.685994
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 4.00000i	− 0.340503i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−13.0000	−1.09480
$142$	− 2.00000i	− 0.167836i
$143$	3.00000i	0.250873i
$144$	2.00000	0.166667
$145$	0	0
$146$	4.00000	0.331042
$147$	− 3.00000i	− 0.247436i
$148$	8.00000i	0.657596i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	− 16.0000i	− 1.29352i
$154$	−6.00000	−0.483494
$155$	0	0
$156$	1.00000	0.0800641
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	15.0000i	1.19334i
$159$	−11.0000	−0.872357
$160$	0	0
$161$	−8.00000	−0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	9.00000i	0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$	9.00000i	0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	2.00000i	0.154303i
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	11.0000i	0.838742i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	− 2.00000i	− 0.148250i
$183$	8.00000i	0.591377i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	3.00000	0.219971
$187$	24.0000i	1.75505i
$188$	13.0000i	0.948122i
$189$	10.0000	0.727393
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000i	0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	2.00000	0.143592
$195$	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	$-0.221573\pi$
$198$	6.00000i	0.426401i
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	8.00000i	0.562878i
$203$	2.00000i	0.140372i
$204$	8.00000	0.560112
$205$	0	0
$206$	14.0000	0.975426
$207$	8.00000i	0.556038i
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	11.0000i	0.755483i
$213$	− 2.00000i	− 0.137038i
$214$	2.00000	0.136717
$215$	0	0
$216$	5.00000	0.340207
$217$	− 6.00000i	− 0.407307i
$218$	5.00000i	0.338643i
$219$	4.00000	0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	8.00000i	0.536925i
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	−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
$231$	−6.00000	−0.394771
$232$	1.00000i	0.0656532i
$233$	− 1.00000i	− 0.0655122i	−0.999463	−	0.0327561i	$-0.989572\pi$
$233$	− 1.00000i	− 0.0655122i	0.999463	−	0.0327561i	$-0.0104285\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	15.0000i	0.974355i
$238$	− 16.0000i	− 1.03713i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	2.00000i	0.128565i
$243$	− 16.0000i	− 1.02640i
$244$	8.00000	0.512148
$245$	0	0
$246$	−2.00000	−0.127515
$247$	0	0
$248$	− 3.00000i	− 0.190500i
$249$	4.00000	0.253490
$250$	0	0
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 12.0000i	− 0.754434i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 13.0000i	− 0.810918i	−0.914113	−	0.405459i	$-0.867112\pi$
$257$	− 13.0000i	− 0.810918i	0.914113	−	0.405459i	$-0.132888\pi$
$258$	11.0000i	0.684830i
$259$	16.0000	0.994192
$260$	0	0
$261$	2.00000	0.123797
$262$	− 12.0000i	− 0.741362i
$263$	9.00000i	0.554964i	0.960731	+	0.277482i	$0.0894999\pi$
$263$	9.00000i	0.554964i	−0.960731	+	0.277482i	$0.910500\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	− 12.0000i	− 0.733017i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	− 8.00000i	− 0.485071i
$273$	− 2.00000i	− 0.121046i
$274$	12.0000	0.724947
$275$	0	0
$276$	−4.00000	−0.240772
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 20.0000i	− 1.19952i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	13.0000i	0.774139i
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	3.00000	0.177394
$287$	4.00000i	0.236113i
$288$	− 2.00000i	− 0.117851i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	2.00000	0.117242
$292$	− 4.00000i	− 0.234082i
$293$	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
$297$	15.0000i	0.870388i
$298$	15.0000i	0.868927i
$299$	4.00000	0.231326
$300$	0	0
$301$	22.0000	1.26806
$302$	− 2.00000i	− 0.115087i
$303$	8.00000i	0.459588i
$304$	0	0
$305$	0	0
$306$	−16.0000	−0.914659
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	6.00000i	0.341882i
$309$	14.0000	0.796432
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	9.00000i	0.508710i	0.967111	+	0.254355i	$0.0818632\pi$
$313$	9.00000i	0.508710i	−0.967111	+	0.254355i	$0.918137\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	15.0000	0.843816
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	11.0000i	0.616849i
$319$	−3.00000	−0.167968
$320$	0	0
$321$	2.00000	0.111629
$322$	8.00000i	0.445823i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	9.00000	0.498464
$327$	5.00000i	0.276501i
$328$	2.00000i	0.110432i
$329$	26.0000	1.43343
$330$	0	0
$331$	−23.0000	−1.26419	−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000	−1.26419	−0.632097	+	0.774889i	$0.717806\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 16.0000i	− 0.876795i
$334$	2.00000	0.109435
$335$	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	$0.663099\pi$
$338$	− 12.0000i	− 0.652714i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	9.00000	0.487377
$342$	0	0
$343$	20.0000i	1.07990i
$344$	11.0000	0.593080
$345$	0	0
$346$	−6.00000	−0.322562
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	1.00000i	0.0536056i
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	3.00000i	0.159901i
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	− 16.0000i	− 0.846810i
$358$	− 10.0000i	− 0.528516i
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 7.00000i	− 0.367912i
$363$	2.00000i	0.104973i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	8.00000	0.418167
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	4.00000i	0.208514i
$369$	4.00000	0.208232
$370$	0	0
$371$	22.0000	1.14218
$372$	− 3.00000i	− 0.155543i
$373$	− 21.0000i	− 1.08734i	−0.839299	−	0.543669i	$-0.817035\pi$
$373$	− 21.0000i	− 1.08734i	0.839299	−	0.543669i	$-0.182965\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	13.0000	0.670424
$377$	− 1.00000i	− 0.0515026i
$378$	− 10.0000i	− 0.514344i
$379$	−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
$381$	−8.00000	−0.409852
$382$	8.00000i	0.409316i
$383$	14.0000i	0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$	14.0000i	0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	− 22.0000i	− 1.11832i
$388$	− 2.00000i	− 0.101535i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	3.00000i	0.151523i
$393$	− 12.0000i	− 0.605320i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	6.00000	0.301511
$397$	17.0000i	0.853206i	0.904439	+	0.426603i	$0.140290\pi$
$397$	17.0000i	0.853206i	−0.904439	+	0.426603i	$0.859710\pi$
$398$	− 10.0000i	− 0.501255i
$399$	0	0
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	− 12.0000i	− 0.598506i
$403$	3.00000i	0.149441i
$404$	8.00000	0.398015
$405$	0	0
$406$	2.00000	0.0992583
$407$	24.0000i	1.18964i
$408$	− 8.00000i	− 0.396059i
$409$	−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
$411$	12.0000	0.591916
$412$	− 14.0000i	− 0.689730i
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	3.00000i	0.146038i
$423$	− 26.0000i	− 1.26416i
$424$	11.0000	0.534207
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	− 16.0000i	− 0.774294i
$428$	− 2.00000i	− 0.0966736i
$429$	3.00000	0.144841
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	− 5.00000i	− 0.240563i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	5.00000	0.239457
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	8.00000i	0.380521i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−26.0000	−1.23114
$447$	15.0000i	0.709476i
$448$	− 2.00000i	− 0.0944911i
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	6.00000i	0.282216i
$453$	− 2.00000i	− 0.0939682i
$454$	−18.0000	−0.844782
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	10.0000i	0.467269i
$459$	−40.0000	−1.86704
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	6.00000i	0.279145i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	27.0000i	1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$	27.0000i	1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$	2.00000i	0.0924500i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	33.0000i	1.51734i
$474$	15.0000	0.688973
$475$	0	0
$476$	−16.0000	−0.733359
$477$	− 22.0000i	− 1.00731i
$478$	0	0
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	− 17.0000i	− 0.774329i
$483$	8.00000i	0.364013i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−16.0000	−0.725775
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	− 8.00000i	− 0.362143i
$489$	9.00000	0.406994
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	2.00000i	0.0901670i
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	0	0
$496$	−3.00000	−0.134704
$497$	4.00000i	0.179425i
$498$	− 4.00000i	− 0.179244i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	− 27.0000i	− 1.20507i
$503$	19.0000i	0.847168i	0.905857	+	0.423584i	$0.139228\pi$
$503$	19.0000i	0.847168i	−0.905857	+	0.423584i	$0.860772\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	−12.0000	−0.533465
$507$	− 12.0000i	− 0.532939i
$508$	8.00000i	0.354943i
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−13.0000	−0.573405
$515$	0	0
$516$	11.0000	0.484248
$517$	39.0000i	1.71522i
$518$	− 16.0000i	− 0.703000i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−13.0000	−0.569540	−0.284770	−	0.958596i	$-0.591917\pi$
$521$	−13.0000	−0.569540	−0.284770	+	0.958596i	$0.591917\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	24.0000i	1.04945i	0.851273	+	0.524723i	$0.175831\pi$
$523$	24.0000i	1.04945i	−0.851273	+	0.524723i	$0.824169\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	9.00000	0.392419
$527$	24.0000i	1.04546i
$528$	3.00000i	0.130558i
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.00000i	− 0.0866296i
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 10.0000i	− 0.431532i
$538$	0	0
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	13.0000i	0.558398i
$543$	− 7.00000i	− 0.300399i
$544$	−8.00000	−0.342997
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	− 38.0000i	− 1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$547$	− 38.0000i	− 1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$548$	− 12.0000i	− 0.512615i
$549$	−16.0000	−0.682863
$550$	0	0
$551$	0	0
$552$	4.00000i	0.170251i
$553$	− 30.0000i	− 1.27573i
$554$	2.00000	0.0849719
$555$	0	0
$556$	−20.0000	−0.848189
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	6.00000i	0.254000i
$559$	−11.0000	−0.465250
$560$	0	0
$561$	24.0000	1.01328
$562$	− 27.0000i	− 1.13893i
$563$	− 11.0000i	− 0.463595i	−0.972764	−	0.231797i	$-0.925539\pi$
$563$	− 11.0000i	− 0.463595i	0.972764	−	0.231797i	$-0.0744606\pi$
$564$	13.0000	0.547399
$565$	0	0
$566$	4.00000	0.168133
$567$	2.00000i	0.0839921i
$568$	2.00000i	0.0839181i
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	− 3.00000i	− 0.125436i
$573$	8.00000i	0.334205i
$574$	4.00000	0.166957
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	− 8.00000i	− 0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$	− 8.00000i	− 0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$	47.0000i	1.95494i
$579$	14.0000	0.581820
$580$	0	0
$581$	−8.00000	−0.331896
$582$	− 2.00000i	− 0.0829027i
$583$	33.0000i	1.36672i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	14.0000	0.578335
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	− 8.00000i	− 0.328798i
$593$	39.0000i	1.60154i	0.598973	+	0.800769i	$0.295576\pi$
$593$	39.0000i	1.60154i	−0.598973	+	0.800769i	$0.704424\pi$
$594$	15.0000	0.615457
$595$	0	0
$596$	15.0000	0.614424
$597$	− 10.0000i	− 0.409273i
$598$	− 4.00000i	− 0.163572i
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	− 22.0000i	− 0.896653i
$603$	24.0000i	0.977356i
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	8.00000	0.324978
$607$	− 3.00000i	− 0.121766i	−0.998145	−	0.0608831i	$-0.980608\pi$
$607$	− 3.00000i	− 0.121766i	0.998145	−	0.0608831i	$-0.0193917\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	−13.0000	−0.525924
$612$	16.0000i	0.646762i
$613$	− 31.0000i	− 1.25208i	−0.779792	−	0.626039i	$-0.784675\pi$
$613$	− 31.0000i	− 1.25208i	0.779792	−	0.626039i	$-0.215325\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	6.00000	0.241747
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	− 14.0000i	− 0.563163i
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	8.00000i	0.320771i
$623$	20.0000i	0.801283i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	9.00000	0.359712
$627$	0	0
$628$	18.0000i	0.718278i
$629$	−64.0000	−2.55185
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	− 15.0000i	− 0.596668i
$633$	3.00000i	0.119239i
$634$	12.0000	0.476581
$635$	0	0
$636$	11.0000	0.436178
$637$	− 3.00000i	− 0.118864i
$638$	3.00000i	0.118771i
$639$	4.00000	0.158238
$640$	0	0
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	− 2.00000i	− 0.0789337i
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	0	0
$647$	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
$651$	−6.00000	−0.235159
$652$	− 9.00000i	− 0.352467i
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	5.00000	0.195515
$655$	0	0
$656$	2.00000	0.0780869
$657$	8.00000i	0.312110i
$658$	− 26.0000i	− 1.01359i
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	23.0000i	0.893920i
$663$	8.00000i	0.310694i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−16.0000	−0.619987
$667$	4.00000i	0.154881i
$668$	− 2.00000i	− 0.0773823i
$669$	−26.0000	−1.00522
$670$	0	0
$671$	24.0000	0.926510
$672$	− 2.00000i	− 0.0771517i
$673$	9.00000i	0.346925i	0.984841	+	0.173462i	$0.0554955\pi$
$673$	9.00000i	0.346925i	−0.984841	+	0.173462i	$0.944505\pi$
$674$	32.0000	1.23259
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	6.00000i	0.230429i
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−18.0000	−0.689761
$682$	− 9.00000i	− 0.344628i
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	10.0000i	0.381524i
$688$	− 11.0000i	− 0.419371i
$689$	−11.0000	−0.419067
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000i	0.228086i
$693$	− 12.0000i	− 0.455842i
$694$	2.00000	0.0759190
$695$	0	0
$696$	1.00000	0.0379049
$697$	− 16.0000i	− 0.606043i
$698$	− 15.0000i	− 0.567758i
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	5.00000i	0.188713i
$703$	0	0
$704$	3.00000	0.113067
$705$	0	0
$706$	−26.0000	−0.978523
$707$	− 16.0000i	− 0.601742i
$708$	0	0
$709$	−15.0000	−0.563337	−0.281668	−	0.959512i	$-0.590888\pi$
$709$	−15.0000	−0.563337	−0.281668	+	0.959512i	$0.590888\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	10.0000i	0.374766i
$713$	− 12.0000i	− 0.449404i
$714$	−16.0000	−0.598785
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	− 25.0000i	− 0.932992i
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	19.0000i	0.707107i
$723$	− 17.0000i	− 0.632237i
$724$	−7.00000	−0.260153
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	2.00000i	0.0741249i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−88.0000	−3.25480
$732$	− 8.00000i	− 0.295689i
$733$	24.0000i	0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$	24.0000i	0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	4.00000	0.147442
$737$	− 36.0000i	− 1.32608i
$738$	− 4.00000i	− 0.147242i
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	0	0
$742$	− 22.0000i	− 0.807645i
$743$	44.0000i	1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$	44.0000i	1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	−21.0000	−0.768865
$747$	8.00000i	0.292705i
$748$	− 24.0000i	− 0.877527i
$749$	−4.00000	−0.146157
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 13.0000i	− 0.474061i
$753$	− 27.0000i	− 0.983935i
$754$	−1.00000	−0.0364179
$755$	0	0
$756$	−10.0000	−0.363696
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	20.0000i	0.726433i
$759$	−12.0000	−0.435572
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	8.00000i	0.289809i
$763$	− 10.0000i	− 0.362024i
$764$	8.00000	0.289430
$765$	0	0
$766$	14.0000	0.505841
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	−13.0000	−0.468184
$772$	− 14.0000i	− 0.503871i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	−22.0000	−0.790774
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	− 16.0000i	− 0.573997i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−6.00000	−0.214697
$782$	− 32.0000i	− 1.14432i
$783$	− 5.00000i	− 0.178685i
$784$	3.00000	0.107143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	18.0000i	0.641223i
$789$	9.00000	0.320408
$790$	0	0
$791$	12.0000	0.426671
$792$	− 6.00000i	− 0.213201i
$793$	8.00000i	0.284088i
$794$	17.0000	0.603307
$795$	0	0
$796$	−10.0000	−0.354441
$797$	32.0000i	1.13350i	0.823890	+	0.566749i	$0.191799\pi$
$797$	32.0000i	1.13350i	−0.823890	+	0.566749i	$0.808201\pi$
$798$	0	0
$799$	−104.000	−3.67926
$800$	0	0
$801$	20.0000	0.706665
$802$	− 27.0000i	− 0.953403i
$803$	− 12.0000i	− 0.423471i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	3.00000	0.105670
$807$	0	0
$808$	− 8.00000i	− 0.281439i
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	13.0000i	0.455930i
$814$	24.0000	0.841200
$815$	0	0
$816$	−8.00000	−0.280056
$817$	0	0
$818$	30.0000i	1.04893i
$819$	4.00000	0.139771
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	− 12.0000i	− 0.418548i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	0	0
$827$	− 13.0000i	− 0.452054i	−0.974121	−	0.226027i	$-0.927426\pi$
$827$	− 13.0000i	− 0.452054i	0.974121	−	0.226027i	$-0.0725738\pi$
$828$	− 8.00000i	− 0.278019i
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	1.00000i	0.0346688i
$833$	− 24.0000i	− 0.831551i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	15.0000i	0.518476i
$838$	− 10.0000i	− 0.345444i
$839$	−45.0000	−1.55357	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000	−1.55357	−0.776786	+	0.629764i	$0.783151\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 32.0000i	− 1.10279i
$843$	− 27.0000i	− 0.929929i
$844$	3.00000	0.103264
$845$	0	0
$846$	−26.0000	−0.893898
$847$	− 4.00000i	− 0.137442i
$848$	− 11.0000i	− 0.377742i
$849$	4.00000	0.137280
$850$	0	0
$851$	32.0000	1.09695
$852$	2.00000i	0.0685189i
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	27.0000i	0.922302i	0.887322	+	0.461151i	$0.152563\pi$
$857$	27.0000i	0.922302i	−0.887322	+	0.461151i	$0.847437\pi$
$858$	− 3.00000i	− 0.102418i
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	− 32.0000i	− 1.08992i
$863$	− 46.0000i	− 1.56586i	−0.622111	−	0.782929i	$-0.713725\pi$
$863$	− 46.0000i	− 1.56586i	0.622111	−	0.782929i	$-0.286275\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−16.0000	−0.543702
$867$	47.0000i	1.59620i
$868$	6.00000i	0.203653i
$869$	45.0000	1.52652
$870$	0	0
$871$	12.0000	0.406604
$872$	− 5.00000i	− 0.169321i
$873$	4.00000i	0.135379i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 13.0000i	− 0.438979i	−0.975615	−	0.219489i	$-0.929561\pi$
$877$	− 13.0000i	− 0.438979i	0.975615	−	0.219489i	$-0.0704391\pi$
$878$	20.0000i	0.674967i
$879$	14.0000	0.472208
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 6.00000i	− 0.202031i
$883$	− 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
$886$	4.00000	0.134383
$887$	− 33.0000i	− 1.10803i	−0.832506	−	0.554016i	$-0.813095\pi$
$887$	− 33.0000i	− 1.10803i	0.832506	−	0.554016i	$-0.186905\pi$
$888$	− 8.00000i	− 0.268462i
$889$	16.0000	0.536623
$890$	0	0
$891$	−3.00000	−0.100504
$892$	26.0000i	0.870544i
$893$	0	0
$894$	15.0000	0.501675
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	− 4.00000i	− 0.133556i
$898$	− 10.0000i	− 0.333704i
$899$	−3.00000	−0.100056
$900$	0	0
$901$	−88.0000	−2.93171
$902$	6.00000i	0.199778i
$903$	− 22.0000i	− 0.732114i
$904$	6.00000	0.199557
$905$	0	0
$906$	−2.00000	−0.0664455
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	18.0000i	0.597351i
$909$	−16.0000	−0.530687
$910$	0	0
$911$	−13.0000	−0.430709	−0.215355	−	0.976536i	$-0.569091\pi$
$911$	−13.0000	−0.430709	−0.215355	+	0.976536i	$0.569091\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	2.00000	0.0661541
$915$	0	0
$916$	10.0000	0.330409
$917$	24.0000i	0.792550i
$918$	40.0000i	1.32020i
$919$	−30.0000	−0.989609	−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000	−0.989609	−0.494804	+	0.869004i	$0.664760\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	− 2.00000i	− 0.0658665i
$923$	− 2.00000i	− 0.0658308i
$924$	6.00000	0.197386
$925$	0	0
$926$	4.00000	0.131448
$927$	28.0000i	0.919641i
$928$	− 1.00000i	− 0.0328266i
$929$	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
$931$	0	0
$932$	1.00000i	0.0327561i
$933$	8.00000i	0.261908i
$934$	27.0000	0.883467
$935$	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	$0.989599\pi$
$938$	24.0000i	0.783628i
$939$	9.00000	0.293704
$940$	0	0
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	18.0000i	0.586472i
$943$	8.00000i	0.260516i
$944$	0	0
$945$	0	0
$946$	33.0000	1.07292
$947$	− 33.0000i	− 1.07236i	−0.844105	−	0.536178i	$-0.819868\pi$
$947$	− 33.0000i	− 1.07236i	0.844105	−	0.536178i	$-0.180132\pi$
$948$	− 15.0000i	− 0.487177i
$949$	4.00000	0.129845
$950$	0	0
$951$	12.0000	0.389127
$952$	16.0000i	0.518563i
$953$	− 1.00000i	− 0.0323932i	−0.999869	−	0.0161966i	$-0.994844\pi$
$953$	− 1.00000i	− 0.0323932i	0.999869	−	0.0161966i	$-0.00515576\pi$
$954$	−22.0000	−0.712276
$955$	0	0
$956$	0	0
$957$	3.00000i	0.0969762i
$958$	− 5.00000i	− 0.161543i
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−22.0000	−0.709677
$962$	8.00000i	0.257930i
$963$	4.00000i	0.128898i
$964$	−17.0000	−0.547533
$965$	0	0
$966$	8.00000	0.257396
$967$	− 13.0000i	− 0.418052i	−0.977910	−	0.209026i	$-0.932971\pi$
$967$	− 13.0000i	− 0.418052i	0.977910	−	0.209026i	$-0.0670293\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	16.0000i	0.513200i
$973$	40.0000i	1.28234i
$974$	22.0000	0.704925
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 13.0000i	− 0.415907i	−0.978139	−	0.207953i	$-0.933320\pi$
$977$	− 13.0000i	− 0.415907i	0.978139	−	0.207953i	$-0.0666802\pi$
$978$	− 9.00000i	− 0.287788i
$979$	−30.0000	−0.958804
$980$	0	0
$981$	−10.0000	−0.319275
$982$	33.0000i	1.05307i
$983$	49.0000i	1.56286i	0.623995	+	0.781429i	$0.285509\pi$
$983$	49.0000i	1.56286i	−0.623995	+	0.781429i	$0.714491\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−8.00000	−0.254772
$987$	− 26.0000i	− 0.827589i
$988$	0	0
$989$	44.0000	1.39912
$990$	0	0
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	3.00000i	0.0952501i
$993$	23.0000i	0.729883i
$994$	4.00000	0.126872
$995$	0	0
$996$	−4.00000	−0.126745
$997$	− 8.00000i	− 0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$	− 8.00000i	− 0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$	− 20.0000i	− 0.633089i
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.b.b.349.1		2
5.2	odd	4		58.2.a.b.1.1	✓	1
5.3	odd	4		1450.2.a.c.1.1		1
5.4	even	2	inner	1450.2.b.b.349.2		2
15.2	even	4		522.2.a.b.1.1		1
20.7	even	4		464.2.a.e.1.1		1
35.27	even	4		2842.2.a.e.1.1		1
40.27	even	4		1856.2.a.f.1.1		1
40.37	odd	4		1856.2.a.k.1.1		1
55.32	even	4		7018.2.a.a.1.1		1
60.47	odd	4		4176.2.a.n.1.1		1
65.12	odd	4		9802.2.a.a.1.1		1
145.12	even	4		1682.2.b.a.1681.2		2
145.17	even	4		1682.2.b.a.1681.1		2
145.57	odd	4		1682.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.b.1.1	✓	1	5.2	odd	4
464.2.a.e.1.1		1	20.7	even	4
522.2.a.b.1.1		1	15.2	even	4
1450.2.a.c.1.1		1	5.3	odd	4
1450.2.b.b.349.1		2	1.1	even	1	trivial
1450.2.b.b.349.2		2	5.4	even	2	inner
1682.2.a.d.1.1		1	145.57	odd	4
1682.2.b.a.1681.1		2	145.17	even	4
1682.2.b.a.1681.2		2	145.12	even	4
1856.2.a.f.1.1		1	40.27	even	4
1856.2.a.k.1.1		1	40.37	odd	4
2842.2.a.e.1.1		1	35.27	even	4
4176.2.a.n.1.1		1	60.47	odd	4
7018.2.a.a.1.1		1	55.32	even	4
9802.2.a.a.1.1		1	65.12	odd	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:	\( N \)	\(=\)	\( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1450.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +2.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} -8.00000 q^{34} -2.00000 q^{36} -8.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -2.00000i q^{42} -11.0000i q^{43} +3.00000 q^{44} +4.00000 q^{46} -13.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -11.0000i q^{53} -5.00000 q^{54} -2.00000 q^{56} -1.00000i q^{58} -8.00000 q^{61} +3.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +12.0000i q^{67} +8.00000i q^{68} +4.00000 q^{69} +2.00000 q^{71} +2.00000i q^{72} +4.00000i q^{73} -8.00000 q^{74} -6.00000i q^{77} +1.00000i q^{78} -15.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} -11.0000 q^{86} -1.00000i q^{87} -3.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000i q^{92} +3.00000i q^{93} -13.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} - 6 q^{31} - 16 q^{34} - 4 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 6 q^{49} - 16 q^{51} - 10 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 4 q^{71} - 16 q^{74} - 30 q^{79} + 2 q^{81} - 4 q^{84} - 22 q^{86} + 20 q^{89} + 4 q^{91} - 26 q^{94} - 2 q^{96} - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 6 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^21 + 2 * q^24 - 2 * q^26 + 2 * q^29 - 6 * q^31 - 16 * q^34 - 4 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 + 6 * q^49 - 16 * q^51 - 10 * q^54 - 4 * q^56 - 16 * q^61 - 2 * q^64 + 6 * q^66 + 8 * q^69 + 4 * q^71 - 16 * q^74 - 30 * q^79 + 2 * q^81 - 4 * q^84 - 22 * q^86 + 20 * q^89 + 4 * q^91 - 26 * q^94 - 2 * q^96 - 12 * q^99

Introduction

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