LMFDB - Embedded newform 1682.2.b.a.1681.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1681,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1682.1681");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
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			1681.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1682.1681
Dual form			1682.2.b.a.1681.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000i q^{8} +2.00000 q^{9} +1.00000i q^{10} +3.00000i q^{11} -1.00000i q^{12} +1.00000 q^{13} +2.00000i q^{14} -1.00000i q^{15} +1.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +1.00000 q^{20} -2.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000i q^{26} +5.00000i q^{27} +2.00000 q^{28} -1.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} -3.00000 q^{33} -8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{36} +8.00000i q^{37} +1.00000i q^{39} -1.00000i q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000i q^{43} -3.00000i q^{44} -2.00000 q^{45} -4.00000i q^{46} +13.0000i q^{47} +1.00000i q^{48} -3.00000 q^{49} +4.00000i q^{50} +8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} -3.00000i q^{55} -2.00000i q^{56} +1.00000i q^{60} +8.00000i q^{61} +3.00000 q^{62} -4.00000 q^{63} -1.00000 q^{64} -1.00000 q^{65} +3.00000i q^{66} +12.0000 q^{67} +8.00000i q^{68} +4.00000i q^{69} -2.00000i q^{70} -2.00000 q^{71} +2.00000i q^{72} +4.00000i q^{73} +8.00000 q^{74} -4.00000i q^{75} -6.00000i q^{77} +1.00000 q^{78} -15.0000i q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +2.00000i q^{84} +8.00000i q^{85} +11.0000 q^{86} -3.00000 q^{88} +10.0000i q^{89} +2.00000i q^{90} -2.00000 q^{91} -4.00000 q^{92} -3.00000 q^{93} +13.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} +6.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9	$ 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} + 2 q^{20} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{25} + 4 q^{28} - 2 q^{30} - 6 q^{33} - 16 q^{34} + 4 q^{35} - 4 q^{36} - 4 q^{42} - 4 q^{45} - 6 q^{49} + 16 q^{51} - 2 q^{52} - 22 q^{53} + 10 q^{54} + 6 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 24 q^{67} - 4 q^{71} + 16 q^{74} + 2 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 22 q^{86} - 6 q^{88} - 4 q^{91} - 8 q^{92} - 6 q^{93} + 26 q^{94} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 + 2 * q^13 + 2 * q^16 + 2 * q^20 + 6 * q^22 + 8 * q^23 - 2 * q^24 - 8 * q^25 + 4 * q^28 - 2 * q^30 - 6 * q^33 - 16 * q^34 + 4 * q^35 - 4 * q^36 - 4 * q^42 - 4 * q^45 - 6 * q^49 + 16 * q^51 - 2 * q^52 - 22 * q^53 + 10 * q^54 + 6 * q^62 - 8 * q^63 - 2 * q^64 - 2 * q^65 + 24 * q^67 - 4 * q^71 + 16 * q^74 + 2 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 + 22 * q^86 - 6 * q^88 - 4 * q^91 - 8 * q^92 - 6 * q^93 + 26 * q^94 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times$.

$n$	$843$
$\chi(n)$	$-1$

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	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000i	0.353553i
$9$	2.00000	0.666667
$10$	1.00000i	0.316228i
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	− 1.00000i	− 0.288675i
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	2.00000i	0.534522i
$15$	− 1.00000i	− 0.258199i
$16$	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	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.00000	0.223607
$21$	− 2.00000i	− 0.436436i
$22$	3.00000	0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	− 1.00000i	− 0.196116i
$27$	5.00000i	0.962250i
$28$	2.00000	0.377964
$29$	0	0
$29$	0	0
$30$	−1.00000	−0.182574
$31$	3.00000i	0.538816i	0.963026	+	0.269408i	$0.0868280\pi$
$31$	3.00000i	0.538816i	−0.963026	+	0.269408i	$0.913172\pi$
$32$	− 1.00000i	− 0.176777i
$33$	−3.00000	−0.522233
$34$	−8.00000	−1.37199
$35$	2.00000	0.338062
$36$	−2.00000	−0.333333
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	1.00000i	0.160128i
$40$	− 1.00000i	− 0.158114i
$41$	2.00000i	0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$	2.00000i	0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$	−2.00000	−0.308607
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	− 3.00000i	− 0.452267i
$45$	−2.00000	−0.298142
$46$	− 4.00000i	− 0.589768i
$47$	13.0000i	1.89624i	0.317905	+	0.948122i	$0.397021\pi$
$47$	13.0000i	1.89624i	−0.317905	+	0.948122i	$0.602979\pi$
$48$	1.00000i	0.144338i
$49$	−3.00000	−0.428571
$50$	4.00000i	0.565685i
$51$	8.00000	1.12022
$52$	−1.00000	−0.138675
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	5.00000	0.680414
$55$	− 3.00000i	− 0.404520i
$56$	− 2.00000i	− 0.267261i
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000i	0.129099i
$61$	8.00000i	1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$	8.00000i	1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$	3.00000	0.381000
$63$	−4.00000	−0.503953
$64$	−1.00000	−0.125000
$65$	−1.00000	−0.124035
$66$	3.00000i	0.369274i
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	8.00000i	0.970143i
$69$	4.00000i	0.481543i
$70$	− 2.00000i	− 0.239046i
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	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
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	1.00000	0.113228
$79$	− 15.0000i	− 1.68763i	−0.536633	−	0.843816i	$-0.680304\pi$
$79$	− 15.0000i	− 1.68763i	0.536633	−	0.843816i	$-0.319696\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	2.00000i	0.218218i
$85$	8.00000i	0.867722i
$86$	11.0000	1.18616
$87$	0	0
$88$	−3.00000	−0.319801
$89$	10.0000i	1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$	10.0000i	1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	2.00000i	0.210819i
$91$	−2.00000	−0.209657
$92$	−4.00000	−0.417029
$93$	−3.00000	−0.311086
$94$	13.0000	1.34085
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	3.00000i	0.303046i
$99$	6.00000i	0.603023i
$100$	4.00000	0.400000
$101$	8.00000i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$101$	8.00000i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$102$	− 8.00000i	− 0.792118i
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	1.00000i	0.0980581i
$105$	2.00000i	0.195180i
$106$	11.0000i	1.06841i
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	− 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	$0.576969\pi$
$110$	−3.00000	−0.286039
$111$	−8.00000	−0.759326
$112$	−2.00000	−0.188982
$113$	− 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
$115$	−4.00000	−0.373002
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	16.0000i	1.46672i
$120$	1.00000	0.0912871
$121$	2.00000	0.181818
$122$	8.00000	0.724286
$123$	−2.00000	−0.180334
$124$	− 3.00000i	− 0.269408i
$125$	9.00000	0.804984
$126$	4.00000i	0.356348i
$127$	− 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
$130$	1.00000i	0.0877058i
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	3.00000	0.261116
$133$	0	0
$134$	− 12.0000i	− 1.03664i
$135$	− 5.00000i	− 0.430331i
$136$	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	$0.828677\pi$
$138$	4.00000	0.340503
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−2.00000	−0.169031
$141$	−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	$0.710612\pi$
$150$	−4.00000	−0.326599
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	− 16.0000i	− 1.29352i
$154$	−6.00000	−0.483494
$155$	− 3.00000i	− 0.240966i
$156$	− 1.00000i	− 0.0800641i
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	−15.0000	−1.19334
$159$	− 11.0000i	− 0.872357i
$160$	1.00000i	0.0790569i
$161$	−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	$0.885343\pi$
$164$	− 2.00000i	− 0.156174i
$165$	3.00000	0.233550
$166$	− 4.00000i	− 0.310460i
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	2.00000	0.154303
$169$	−12.0000	−0.923077
$170$	8.00000	0.613572
$171$	0	0
$172$	− 11.0000i	− 0.838742i
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	3.00000i	0.226134i
$177$	0	0
$178$	10.0000	0.749532
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	2.00000	0.149071
$181$	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
$183$	−8.00000	−0.591377
$184$	4.00000i	0.294884i
$185$	− 8.00000i	− 0.588172i
$186$	3.00000i	0.219971i
$187$	24.0000	1.75505
$188$	− 13.0000i	− 0.948122i
$189$	− 10.0000i	− 0.727393i
$190$	0	0
$191$	8.00000i	0.578860i	0.957199	+	0.289430i	$0.0934657\pi$
$191$	8.00000i	0.578860i	−0.957199	+	0.289430i	$0.906534\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−2.00000	−0.143592
$195$	− 1.00000i	− 0.0716115i
$196$	3.00000	0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	6.00000	0.426401
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	− 4.00000i	− 0.282843i
$201$	12.0000i	0.846415i
$202$	8.00000	0.562878
$203$	0	0
$204$	−8.00000	−0.560112
$205$	− 2.00000i	− 0.139686i
$206$	− 14.0000i	− 0.975426i
$207$	8.00000	0.556038
$208$	1.00000	0.0693375
$209$	0	0
$210$	2.00000	0.138013
$211$	− 3.00000i	− 0.206529i	−0.994654	−	0.103264i	$-0.967071\pi$
$211$	− 3.00000i	− 0.206529i	0.994654	−	0.103264i	$-0.0329287\pi$
$212$	11.0000	0.755483
$213$	− 2.00000i	− 0.137038i
$214$	2.00000i	0.136717i
$215$	− 11.0000i	− 0.750194i
$216$	−5.00000	−0.340207
$217$	− 6.00000i	− 0.407307i
$218$	5.00000i	0.338643i
$219$	−4.00000	−0.270295
$220$	3.00000i	0.202260i
$221$	− 8.00000i	− 0.538138i
$222$	8.00000i	0.536925i
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	2.00000i	0.133631i
$225$	−8.00000	−0.533333
$226$	−6.00000	−0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	4.00000i	0.263752i
$231$	6.00000	0.394771
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	− 2.00000i	− 0.130744i
$235$	− 13.0000i	− 0.848026i
$236$	0	0
$237$	15.0000	0.974355
$238$	16.0000	1.03713
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	− 2.00000i	− 0.128565i
$243$	16.0000i	1.02640i
$244$	− 8.00000i	− 0.512148i
$245$	3.00000	0.191663
$246$	2.00000i	0.127515i
$247$	0	0
$248$	−3.00000	−0.190500
$249$	4.00000i	0.253490i
$250$	− 9.00000i	− 0.569210i
$251$	− 27.0000i	− 1.70422i	−0.523359	−	0.852112i	$-0.675321\pi$
$251$	− 27.0000i	− 1.70422i	0.523359	−	0.852112i	$-0.324679\pi$
$252$	4.00000	0.251976
$253$	12.0000i	0.754434i
$254$	−8.00000	−0.501965
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	11.0000i	0.684830i
$259$	− 16.0000i	− 0.994192i
$260$	1.00000	0.0620174
$261$	0	0
$262$	12.0000	0.741362
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	− 3.00000i	− 0.184637i
$265$	11.0000	0.675725
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−12.0000	−0.733017
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−5.00000	−0.304290
$271$	− 13.0000i	− 0.789694i	−0.918747	−	0.394847i	$-0.870798\pi$
$271$	− 13.0000i	− 0.789694i	0.918747	−	0.394847i	$-0.129202\pi$
$272$	− 8.00000i	− 0.485071i
$273$	− 2.00000i	− 0.121046i
$274$	12.0000	0.724947
$275$	− 12.0000i	− 0.723627i
$276$	− 4.00000i	− 0.240772i
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	20.0000i	1.19952i
$279$	6.00000i	0.359211i
$280$	2.00000i	0.119523i
$281$	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
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	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	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−8.00000	−0.464991
$297$	−15.0000	−0.870388
$298$	15.0000i	0.868927i
$299$	4.00000	0.231326
$300$	4.00000i	0.230940i
$301$	− 22.0000i	− 1.26806i
$302$	2.00000i	0.115087i
$303$	−8.00000	−0.459588
$304$	0	0
$305$	− 8.00000i	− 0.458079i
$306$	−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
$309$	14.0000i	0.796432i
$310$	−3.00000	−0.170389
$311$	8.00000i	0.453638i	0.973937	+	0.226819i	$0.0728326\pi$
$311$	8.00000i	0.453638i	−0.973937	+	0.226819i	$0.927167\pi$
$312$	−1.00000	−0.0566139
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	18.0000	1.01580
$315$	4.00000	0.225374
$316$	15.0000i	0.843816i
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	−11.0000	−0.616849
$319$	0	0
$320$	1.00000	0.0559017
$321$	− 2.00000i	− 0.111629i
$322$	8.00000i	0.445823i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	−4.00000	−0.221880
$326$	9.00000	0.498464
$327$	− 5.00000i	− 0.276501i
$328$	−2.00000	−0.110432
$329$	− 26.0000i	− 1.43343i
$330$	− 3.00000i	− 0.165145i
$331$	− 23.0000i	− 1.26419i	−0.774889	−	0.632097i	$-0.782194\pi$
$331$	− 23.0000i	− 1.26419i	0.774889	−	0.632097i	$-0.217806\pi$
$332$	−4.00000	−0.219529
$333$	16.0000i	0.876795i
$334$	− 2.00000i	− 0.109435i
$335$	−12.0000	−0.655630
$336$	− 2.00000i	− 0.109109i
$337$	− 32.0000i	− 1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	− 32.0000i	− 1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$	12.0000i	0.652714i
$339$	6.00000	0.325875
$340$	− 8.00000i	− 0.433861i
$341$	−9.00000	−0.487377
$342$	0	0
$343$	20.0000	1.07990
$344$	−11.0000	−0.593080
$345$	− 4.00000i	− 0.215353i
$346$	− 6.00000i	− 0.322562i
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	− 8.00000i	− 0.427618i
$351$	5.00000i	0.266880i
$352$	3.00000	0.159901
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	− 10.0000i	− 0.529999i
$357$	−16.0000	−0.846810
$358$	− 10.0000i	− 0.528516i
$359$	25.0000i	1.31945i	0.751507	+	0.659725i	$0.229327\pi$
$359$	25.0000i	1.31945i	−0.751507	+	0.659725i	$0.770673\pi$
$360$	− 2.00000i	− 0.105409i
$361$	19.0000	1.00000
$362$	− 7.00000i	− 0.367912i
$363$	2.00000i	0.104973i
$364$	2.00000	0.104828
$365$	− 4.00000i	− 0.209370i
$366$	8.00000i	0.418167i
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	4.00000	0.208514
$369$	4.00000i	0.208232i
$370$	−8.00000	−0.415900
$371$	22.0000	1.14218
$372$	3.00000	0.155543
$373$	−21.0000	−1.08734	−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000	−1.08734	−0.543669	+	0.839299i	$0.682965\pi$
$374$	− 24.0000i	− 1.24101i
$375$	9.00000i	0.464758i
$376$	−13.0000	−0.670424
$377$	0	0
$378$	−10.0000	−0.514344
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	8.00000	0.409316
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	−1.00000	−0.0510310
$385$	6.00000i	0.305788i
$386$	−14.0000	−0.712581
$387$	22.0000i	1.11832i
$388$	2.00000i	0.101535i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	−1.00000	−0.0506370
$391$	− 32.0000i	− 1.61831i
$392$	− 3.00000i	− 0.151523i
$393$	−12.0000	−0.605320
$394$	− 18.0000i	− 0.906827i
$395$	15.0000i	0.754732i
$396$	− 6.00000i	− 0.301511i
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	−4.00000	−0.200000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	12.0000	0.598506
$403$	3.00000i	0.149441i
$404$	− 8.00000i	− 0.398015i
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	8.00000i	0.396059i
$409$	− 30.0000i	− 1.48340i	−0.670729	−	0.741702i	$-0.734019\pi$
$409$	− 30.0000i	− 1.48340i	0.670729	−	0.741702i	$-0.265981\pi$
$410$	−2.00000	−0.0987730
$411$	−12.0000	−0.591916
$412$	−14.0000	−0.689730
$413$	0	0
$414$	− 8.00000i	− 0.393179i
$415$	−4.00000	−0.196352
$416$	− 1.00000i	− 0.0490290i
$417$	− 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	$0.421453\pi$
$420$	− 2.00000i	− 0.0975900i
$421$	32.0000i	1.55958i	0.626038	+	0.779792i	$0.284675\pi$
$421$	32.0000i	1.55958i	−0.626038	+	0.779792i	$0.715325\pi$
$422$	−3.00000	−0.146038
$423$	26.0000i	1.26416i
$424$	− 11.0000i	− 0.534207i
$425$	32.0000i	1.55223i
$426$	−2.00000	−0.0969003
$427$	− 16.0000i	− 0.774294i
$428$	2.00000	0.0966736
$429$	−3.00000	−0.144841
$430$	−11.0000	−0.530467
$431$	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	$0.658375\pi$
$440$	3.00000	0.143019
$441$	−6.00000	−0.285714
$442$	−8.00000	−0.380521
$443$	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
$445$	− 10.0000i	− 0.474045i
$446$	26.0000i	1.23114i
$447$	− 15.0000i	− 0.709476i
$448$	2.00000	0.0944911
$449$	10.0000i	0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$449$	10.0000i	0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$450$	8.00000i	0.377124i
$451$	−6.00000	−0.282529
$452$	6.00000i	0.282216i
$453$	− 2.00000i	− 0.0939682i
$454$	− 18.0000i	− 0.844782i
$455$	2.00000	0.0937614
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	10.0000	0.467269
$459$	40.0000	1.86704
$460$	4.00000	0.186501
$461$	2.00000i	0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$	2.00000i	0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$	− 6.00000i	− 0.279145i
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	3.00000	0.139122
$466$	1.00000i	0.0463241i
$467$	27.0000i	1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$	27.0000i	1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$	−2.00000	−0.0924500
$469$	−24.0000	−1.10822
$470$	−13.0000	−0.599645
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−33.0000	−1.51734
$474$	− 15.0000i	− 0.688973i
$475$	0	0
$476$	− 16.0000i	− 0.733359i
$477$	−22.0000	−1.00731
$478$	0	0
$479$	− 5.00000i	− 0.228456i	−0.993455	−	0.114228i	$-0.963561\pi$
$479$	− 5.00000i	− 0.228456i	0.993455	−	0.114228i	$-0.0364394\pi$
$480$	−1.00000	−0.0456435
$481$	8.00000i	0.364769i
$482$	17.0000i	0.774329i
$483$	− 8.00000i	− 0.364013i
$484$	−2.00000	−0.0909091
$485$	2.00000i	0.0908153i
$486$	16.0000	0.725775
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−8.00000	−0.362143
$489$	−9.00000	−0.406994
$490$	− 3.00000i	− 0.135526i
$491$	− 33.0000i	− 1.48927i	−0.667472	−	0.744635i	$-0.732624\pi$
$491$	− 33.0000i	− 1.48927i	0.667472	−	0.744635i	$-0.267376\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	0	0
$495$	− 6.00000i	− 0.269680i
$496$	3.00000i	0.134704i
$497$	4.00000	0.179425
$498$	4.00000	0.179244
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−9.00000	−0.402492
$501$	2.00000i	0.0893534i
$502$	−27.0000	−1.20507
$503$	19.0000i	0.847168i	0.905857	+	0.423584i	$0.139228\pi$
$503$	19.0000i	0.847168i	−0.905857	+	0.423584i	$0.860772\pi$
$504$	− 4.00000i	− 0.178174i
$505$	− 8.00000i	− 0.355995i
$506$	12.0000	0.533465
$507$	− 12.0000i	− 0.532939i
$508$	8.00000i	0.354943i
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	8.00000i	0.354246i
$511$	− 8.00000i	− 0.353899i
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 13.0000i	− 0.573405i
$515$	−14.0000	−0.616914
$516$	11.0000	0.484248
$517$	−39.0000	−1.71522
$518$	−16.0000	−0.703000
$519$	6.00000i	0.263371i
$520$	− 1.00000i	− 0.0438529i
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	− 12.0000i	− 0.524222i
$525$	8.00000i	0.349149i
$526$	−9.00000	−0.392419
$527$	24.0000	1.04546
$528$	−3.00000	−0.130558
$529$	−7.00000	−0.304348
$530$	− 11.0000i	− 0.477809i
$531$	0	0
$532$	0	0
$533$	2.00000i	0.0866296i
$534$	10.0000i	0.432742i
$535$	2.00000	0.0864675
$536$	12.0000i	0.518321i
$537$	10.0000i	0.431532i
$538$	0	0
$539$	− 9.00000i	− 0.387657i
$540$	5.00000i	0.215166i
$541$	8.00000i	0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$	8.00000i	0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$	−13.0000	−0.558398
$543$	7.00000i	0.300399i
$544$	−8.00000	−0.342997
$545$	5.00000	0.214176
$546$	−2.00000	−0.0855921
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	− 12.0000i	− 0.512615i
$549$	16.0000i	0.682863i
$550$	−12.0000	−0.511682
$551$	0	0
$552$	−4.00000	−0.170251
$553$	30.0000i	1.27573i
$554$	2.00000i	0.0849719i
$555$	8.00000	0.339581
$556$	20.0000	0.848189
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	6.00000	0.254000
$559$	11.0000i	0.465250i
$560$	2.00000	0.0845154
$561$	24.0000i	1.01328i
$562$	− 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
$565$	6.00000i	0.252422i
$566$	4.00000i	0.168133i
$567$	−2.00000	−0.0839921
$568$	− 2.00000i	− 0.0839181i
$569$	− 30.0000i	− 1.25767i	−0.777541	−	0.628833i	$-0.783533\pi$
$569$	− 30.0000i	− 1.25767i	0.777541	−	0.628833i	$-0.216467\pi$
$570$	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
$573$	−8.00000	−0.334205
$574$	−4.00000	−0.166957
$575$	−16.0000	−0.667246
$576$	−2.00000	−0.0833333
$577$	8.00000i	0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$	8.00000i	0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$	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
$585$	−2.00000	−0.0826898
$586$	−14.0000	−0.578335
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	18.0000i	0.740421i
$592$	8.00000i	0.328798i
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	15.0000i	0.615457i
$595$	− 16.0000i	− 0.655936i
$596$	15.0000	0.614424
$597$	− 10.0000i	− 0.409273i
$598$	− 4.00000i	− 0.163572i
$599$	5.00000i	0.204294i	0.994769	+	0.102147i	$0.0325713\pi$
$599$	5.00000i	0.204294i	−0.994769	+	0.102147i	$0.967429\pi$
$600$	4.00000	0.163299
$601$	− 2.00000i	− 0.0815817i	−0.999168	−	0.0407909i	$-0.987012\pi$
$601$	− 2.00000i	− 0.0815817i	0.999168	−	0.0407909i	$-0.0129877\pi$
$602$	−22.0000	−0.896653
$603$	24.0000	0.977356
$604$	2.00000	0.0813788
$605$	−2.00000	−0.0813116
$606$	8.00000i	0.324978i
$607$	3.00000i	0.121766i	0.998145	+	0.0608831i	$0.0193917\pi$
$607$	3.00000i	0.121766i	−0.998145	+	0.0608831i	$0.980608\pi$
$608$	0	0
$609$	0	0
$610$	−8.00000	−0.323911
$611$	13.0000i	0.525924i
$612$	16.0000i	0.646762i
$613$	31.0000	1.25208	0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	7.00000	0.282497
$615$	2.00000	0.0806478
$616$	6.00000	0.241747
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	14.0000	0.563163
$619$	− 35.0000i	− 1.40677i	−0.710810	−	0.703384i	$-0.751671\pi$
$619$	− 35.0000i	− 1.40677i	0.710810	−	0.703384i	$-0.248329\pi$
$620$	3.00000i	0.120483i
$621$	20.0000i	0.802572i
$622$	8.00000	0.320771
$623$	− 20.0000i	− 0.801283i
$624$	1.00000i	0.0400320i
$625$	11.0000	0.440000
$626$	− 9.00000i	− 0.359712i
$627$	0	0
$628$	− 18.0000i	− 0.718278i
$629$	64.0000	2.55185
$630$	− 4.00000i	− 0.159364i
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	15.0000	0.596668
$633$	3.00000	0.119239
$634$	−12.0000	−0.476581
$635$	8.00000i	0.317470i
$636$	11.0000i	0.436178i
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−4.00000	−0.158238
$640$	− 1.00000i	− 0.0395285i
$641$	8.00000i	0.315981i	0.987441	+	0.157991i	$0.0505015\pi$
$641$	8.00000i	0.315981i	−0.987441	+	0.157991i	$0.949498\pi$
$642$	−2.00000	−0.0789337
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	8.00000	0.315244
$645$	11.0000	0.433125
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	4.00000i	0.156893i
$651$	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
$655$	− 12.0000i	− 0.468879i
$656$	2.00000i	0.0780869i
$657$	8.00000i	0.312110i
$658$	−26.0000	−1.01359
$659$	− 15.0000i	− 0.584317i	−0.956370	−	0.292159i	$-0.905627\pi$
$659$	− 15.0000i	− 0.584317i	0.956370	−	0.292159i	$-0.0943735\pi$
$660$	−3.00000	−0.116775
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−23.0000	−0.893920
$663$	8.00000	0.310694
$664$	4.00000i	0.155230i
$665$	0	0
$666$	16.0000	0.619987
$667$	0	0
$668$	−2.00000	−0.0773823
$669$	− 26.0000i	− 1.00522i
$670$	12.0000i	0.463600i
$671$	−24.0000	−0.926510
$672$	−2.00000	−0.0771517
$673$	−9.00000	−0.346925	−0.173462	−	0.984841i	$-0.555495\pi$
$673$	−9.00000	−0.346925	−0.173462	+	0.984841i	$0.555495\pi$
$674$	−32.0000	−1.23259
$675$	− 20.0000i	− 0.769800i
$676$	12.0000	0.461538
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	− 6.00000i	− 0.230429i
$679$	4.00000i	0.153506i
$680$	−8.00000	−0.306786
$681$	18.0000i	0.689761i
$682$	9.00000i	0.344628i
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	− 12.0000i	− 0.458496i
$686$	− 20.0000i	− 0.763604i
$687$	−10.0000	−0.381524
$688$	11.0000i	0.419371i
$689$	−11.0000	−0.419067
$690$	−4.00000	−0.152277
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000	−0.228086
$693$	− 12.0000i	− 0.455842i
$694$	− 2.00000i	− 0.0759190i
$695$	20.0000	0.758643
$696$	0	0
$697$	16.0000	0.606043
$698$	15.0000i	0.567758i
$699$	− 1.00000i	− 0.0378235i
$700$	−8.00000	−0.302372
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	5.00000	0.188713
$703$	0	0
$704$	− 3.00000i	− 0.113067i
$705$	13.0000	0.489608
$706$	− 26.0000i	− 0.978523i
$707$	− 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	$0.590888\pi$
$710$	− 2.00000i	− 0.0750587i
$711$	− 30.0000i	− 1.12509i
$712$	−10.0000	−0.374766
$713$	12.0000i	0.449404i
$714$	16.0000i	0.598785i
$715$	− 3.00000i	− 0.112194i
$716$	−10.0000	−0.373718
$717$	0	0
$718$	25.0000	0.932992
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	−2.00000	−0.0745356
$721$	−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
$730$	−4.00000	−0.148047
$731$	88.0000	3.25480
$732$	8.00000	0.295689
$733$	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
$735$	3.00000i	0.110657i
$736$	− 4.00000i	− 0.147442i
$737$	36.0000i	1.32608i
$738$	4.00000	0.147242
$739$	5.00000i	0.183928i	0.995762	+	0.0919640i	$0.0293145\pi$
$739$	5.00000i	0.183928i	−0.995762	+	0.0919640i	$0.970686\pi$
$740$	8.00000i	0.294086i
$741$	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	$0.701035\pi$
$744$	− 3.00000i	− 0.109985i
$745$	15.0000	0.549557
$746$	21.0000i	0.768865i
$747$	8.00000	0.292705
$748$	−24.0000	−0.877527
$749$	4.00000	0.146157
$750$	9.00000	0.328634
$751$	32.0000i	1.16770i	0.811863	+	0.583848i	$0.198454\pi$
$751$	32.0000i	1.16770i	−0.811863	+	0.583848i	$0.801546\pi$
$752$	13.0000i	0.474061i
$753$	27.0000	0.983935
$754$	0	0
$755$	2.00000	0.0727875
$756$	10.0000i	0.363696i
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	−20.0000	−0.726433
$759$	−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
$763$	10.0000	0.362024
$764$	− 8.00000i	− 0.289430i
$765$	16.0000i	0.578481i
$766$	14.0000i	0.505841i
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	− 20.0000i	− 0.721218i	−0.932717	−	0.360609i	$-0.882569\pi$
$769$	− 20.0000i	− 0.721218i	0.932717	−	0.360609i	$-0.117431\pi$
$770$	6.00000	0.216225
$771$	13.0000i	0.468184i
$772$	14.0000i	0.503871i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	22.0000	0.790774
$775$	− 12.0000i	− 0.431053i
$776$	2.00000	0.0717958
$777$	16.0000	0.573997
$778$	0	0
$779$	0	0
$780$	1.00000i	0.0358057i
$781$	− 6.00000i	− 0.214697i
$782$	−32.0000	−1.14432
$783$	0	0
$784$	−3.00000	−0.107143
$785$	− 18.0000i	− 0.642448i
$786$	12.0000i	0.428026i
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−18.0000	−0.641223
$789$	9.00000	0.320408
$790$	15.0000	0.533676
$791$	12.0000i	0.426671i
$792$	−6.00000	−0.213201
$793$	8.00000i	0.284088i
$794$	17.0000i	0.603307i
$795$	11.0000i	0.390130i
$796$	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
$800$	4.00000i	0.141421i
$801$	20.0000i	0.706665i
$802$	− 27.0000i	− 0.953403i
$803$	−12.0000	−0.423471
$804$	− 12.0000i	− 0.423207i
$805$	8.00000	0.281963
$806$	3.00000	0.105670
$807$	0	0
$808$	−8.00000	−0.281439
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.00000i	0.0351364i
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	0	0
$813$	13.0000	0.455930
$814$	24.0000i	0.841200i
$815$	− 9.00000i	− 0.315256i
$816$	8.00000	0.280056
$817$	0	0
$818$	−30.0000	−1.04893
$819$	−4.00000	−0.139771
$820$	2.00000i	0.0698430i
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	12.0000i	0.418548i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	14.0000i	0.487713i
$825$	12.0000	0.417786
$826$	0	0
$827$	13.0000i	0.452054i	0.974121	+	0.226027i	$0.0725738\pi$
$827$	13.0000i	0.452054i	−0.974121	+	0.226027i	$0.927426\pi$
$828$	−8.00000	−0.278019
$829$	− 40.0000i	− 1.38926i	−0.719368	−	0.694629i	$-0.755569\pi$
$829$	− 40.0000i	− 1.38926i	0.719368	−	0.694629i	$-0.244431\pi$
$830$	4.00000i	0.138842i
$831$	− 2.00000i	− 0.0693792i
$832$	−1.00000	−0.0346688
$833$	24.0000i	0.831551i
$834$	−20.0000	−0.692543
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	−15.0000	−0.518476
$838$	− 10.0000i	− 0.345444i
$839$	45.0000i	1.55357i	0.629764	+	0.776786i	$0.283151\pi$
$839$	45.0000i	1.55357i	−0.629764	+	0.776786i	$0.716849\pi$
$840$	−2.00000	−0.0690066
$841$	0	0
$842$	32.0000	1.10279
$843$	27.0000i	0.929929i
$844$	3.00000i	0.103264i
$845$	12.0000	0.412813
$846$	26.0000	0.893898
$847$	−4.00000	−0.137442
$848$	−11.0000	−0.377742
$849$	− 4.00000i	− 0.137280i
$850$	32.0000	1.09759
$851$	32.0000i	1.09695i
$852$	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
$856$	− 2.00000i	− 0.0683586i
$857$	−27.0000	−0.922302	−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000	−0.922302	−0.461151	+	0.887322i	$0.652563\pi$
$858$	3.00000i	0.102418i
$859$	− 25.0000i	− 0.852989i	−0.904490	−	0.426494i	$-0.859748\pi$
$859$	− 25.0000i	− 0.852989i	0.904490	−	0.426494i	$-0.140252\pi$
$860$	11.0000i	0.375097i
$861$	4.00000	0.136320
$862$	− 32.0000i	− 1.08992i
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	5.00000	0.170103
$865$	−6.00000	−0.204006
$866$	−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
$875$	−18.0000	−0.608511
$876$	4.00000	0.135147
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	20.0000i	0.674967i
$879$	14.0000	0.472208
$880$	− 3.00000i	− 0.101130i
$881$	− 42.0000i	− 1.41502i	−0.706705	−	0.707508i	$-0.749819\pi$
$881$	− 42.0000i	− 1.41502i	0.706705	−	0.707508i	$-0.250181\pi$
$882$	6.00000i	0.202031i
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	8.00000i	0.269069i
$885$	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
$889$	16.0000i	0.536623i
$890$	−10.0000	−0.335201
$891$	3.00000i	0.100504i
$892$	26.0000	0.870544
$893$	0	0
$894$	−15.0000	−0.501675
$895$	−10.0000	−0.334263
$896$	− 2.00000i	− 0.0668153i
$897$	4.00000i	0.133556i
$898$	10.0000	0.333704
$899$	0	0
$900$	8.00000	0.266667
$901$	88.0000i	2.93171i
$902$	6.00000i	0.199778i
$903$	22.0000	0.732114
$904$	6.00000	0.199557
$905$	−7.00000	−0.232688
$906$	−2.00000	−0.0664455
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	−18.0000	−0.597351
$909$	16.0000i	0.530687i
$910$	− 2.00000i	− 0.0662994i
$911$	− 13.0000i	− 0.430709i	−0.976536	−	0.215355i	$-0.930909\pi$
$911$	− 13.0000i	− 0.430709i	0.976536	−	0.215355i	$-0.0690907\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	− 2.00000i	− 0.0661541i
$915$	8.00000	0.264472
$916$	− 10.0000i	− 0.330409i
$917$	− 24.0000i	− 0.792550i
$918$	− 40.0000i	− 1.32020i
$919$	30.0000	0.989609	0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000	0.989609	0.494804	+	0.869004i	$0.335240\pi$
$920$	− 4.00000i	− 0.131876i
$921$	−7.00000	−0.230658
$922$	2.00000	0.0658665
$923$	−2.00000	−0.0658308
$924$	−6.00000	−0.197386
$925$	− 32.0000i	− 1.05215i
$926$	4.00000i	0.131448i
$927$	28.0000	0.919641
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	− 3.00000i	− 0.0983739i
$931$	0	0
$932$	1.00000	0.0327561
$933$	−8.00000	−0.261908
$934$	27.0000	0.883467
$935$	−24.0000	−0.784884
$936$	2.00000i	0.0653720i
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	24.0000i	0.783628i
$939$	9.00000i	0.293704i
$940$	13.0000i	0.424013i
$941$	−37.0000	−1.20617	−0.603083	−	0.797679i	$-0.706061\pi$
$941$	−37.0000	−1.20617	−0.603083	+	0.797679i	$0.706061\pi$
$942$	18.0000i	0.586472i
$943$	8.00000i	0.260516i
$944$	0	0
$945$	10.0000i	0.325300i
$946$	33.0000i	1.07292i
$947$	− 33.0000i	− 1.07236i	−0.844105	−	0.536178i	$-0.819868\pi$
$947$	− 33.0000i	− 1.07236i	0.844105	−	0.536178i	$-0.180132\pi$
$948$	−15.0000	−0.487177
$949$	4.00000i	0.129845i
$950$	0	0
$951$	12.0000	0.389127
$952$	−16.0000	−0.518563
$953$	−1.00000	−0.0323932	−0.0161966	−	0.999869i	$-0.505156\pi$
$953$	−1.00000	−0.0323932	−0.0161966	+	0.999869i	$0.505156\pi$
$954$	22.0000i	0.712276i
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	0	0
$958$	−5.00000	−0.161543
$959$	− 24.0000i	− 0.775000i
$960$	1.00000i	0.0322749i
$961$	22.0000	0.709677
$962$	8.00000	0.257930
$963$	−4.00000	−0.128898
$964$	17.0000	0.547533
$965$	14.0000i	0.450676i
$966$	−8.00000	−0.257396
$967$	13.0000i	0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$	13.0000i	0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	2.00000	0.0642161
$971$	28.0000i	0.898563i	0.893390	+	0.449281i	$0.148320\pi$
$971$	28.0000i	0.898563i	−0.893390	+	0.449281i	$0.851680\pi$
$972$	− 16.0000i	− 0.513200i
$973$	40.0000	1.28234
$974$	22.0000i	0.704925i
$975$	− 4.00000i	− 0.128103i
$976$	8.00000i	0.256074i
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	9.00000i	0.287788i
$979$	−30.0000	−0.958804
$980$	−3.00000	−0.0958315
$981$	−10.0000	−0.319275
$982$	−33.0000	−1.05307
$983$	49.0000i	1.56286i	0.623995	+	0.781429i	$0.285509\pi$
$983$	49.0000i	1.56286i	−0.623995	+	0.781429i	$0.714491\pi$
$984$	− 2.00000i	− 0.0637577i
$985$	−18.0000	−0.573528
$986$	0	0
$987$	26.0000	0.827589
$988$	0	0
$989$	44.0000i	1.39912i
$990$	−6.00000	−0.190693
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	3.00000	0.0952501
$993$	23.0000	0.729883
$994$	− 4.00000i	− 0.126872i
$995$	10.0000	0.317021
$996$	− 4.00000i	− 0.126745i
$997$	− 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	1682.2.b.a.1681.1		2
29.12	odd	4		58.2.a.b.1.1	✓	1
29.17	odd	4		1682.2.a.d.1.1		1
29.28	even	2	inner	1682.2.b.a.1681.2		2
87.41	even	4		522.2.a.b.1.1		1
116.99	even	4		464.2.a.e.1.1		1
145.12	even	4		1450.2.b.b.349.2		2
145.99	odd	4		1450.2.a.c.1.1		1
145.128	even	4		1450.2.b.b.349.1		2
203.41	even	4		2842.2.a.e.1.1		1
232.99	even	4		1856.2.a.f.1.1		1
232.157	odd	4		1856.2.a.k.1.1		1
319.186	even	4		7018.2.a.a.1.1		1
348.215	odd	4		4176.2.a.n.1.1		1
377.12	odd	4		9802.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.b.1.1	✓	1	29.12	odd	4
464.2.a.e.1.1		1	116.99	even	4
522.2.a.b.1.1		1	87.41	even	4
1450.2.a.c.1.1		1	145.99	odd	4
1450.2.b.b.349.1		2	145.128	even	4
1450.2.b.b.349.2		2	145.12	even	4
1682.2.a.d.1.1		1	29.17	odd	4
1682.2.b.a.1681.1		2	1.1	even	1	trivial
1682.2.b.a.1681.2		2	29.28	even	2	inner
1856.2.a.f.1.1		1	232.99	even	4
1856.2.a.k.1.1		1	232.157	odd	4
2842.2.a.e.1.1		1	203.41	even	4
4176.2.a.n.1.1		1	348.215	odd	4
7018.2.a.a.1.1		1	319.186	even	4
9802.2.a.a.1.1		1	377.12	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:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000i q^{8} +2.00000 q^{9} +1.00000i q^{10} +3.00000i q^{11} -1.00000i q^{12} +1.00000 q^{13} +2.00000i q^{14} -1.00000i q^{15} +1.00000 q^{16} -8.00000i q^{17} -2.00000i q^{18} +1.00000 q^{20} -2.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000i q^{26} +5.00000i q^{27} +2.00000 q^{28} -1.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} -3.00000 q^{33} -8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{36} +8.00000i q^{37} +1.00000i q^{39} -1.00000i q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000i q^{43} -3.00000i q^{44} -2.00000 q^{45} -4.00000i q^{46} +13.0000i q^{47} +1.00000i q^{48} -3.00000 q^{49} +4.00000i q^{50} +8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} -3.00000i q^{55} -2.00000i q^{56} +1.00000i q^{60} +8.00000i q^{61} +3.00000 q^{62} -4.00000 q^{63} -1.00000 q^{64} -1.00000 q^{65} +3.00000i q^{66} +12.0000 q^{67} +8.00000i q^{68} +4.00000i q^{69} -2.00000i q^{70} -2.00000 q^{71} +2.00000i q^{72} +4.00000i q^{73} +8.00000 q^{74} -4.00000i q^{75} -6.00000i q^{77} +1.00000 q^{78} -15.0000i q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +2.00000i q^{84} +8.00000i q^{85} +11.0000 q^{86} -3.00000 q^{88} +10.0000i q^{89} +2.00000i q^{90} -2.00000 q^{91} -4.00000 q^{92} -3.00000 q^{93} +13.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9	\( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} + 2 q^{20} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{25} + 4 q^{28} - 2 q^{30} - 6 q^{33} - 16 q^{34} + 4 q^{35} - 4 q^{36} - 4 q^{42} - 4 q^{45} - 6 q^{49} + 16 q^{51} - 2 q^{52} - 22 q^{53} + 10 q^{54} + 6 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 24 q^{67} - 4 q^{71} + 16 q^{74} + 2 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 22 q^{86} - 6 q^{88} - 4 q^{91} - 8 q^{92} - 6 q^{93} + 26 q^{94} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 + 2 * q^13 + 2 * q^16 + 2 * q^20 + 6 * q^22 + 8 * q^23 - 2 * q^24 - 8 * q^25 + 4 * q^28 - 2 * q^30 - 6 * q^33 - 16 * q^34 + 4 * q^35 - 4 * q^36 - 4 * q^42 - 4 * q^45 - 6 * q^49 + 16 * q^51 - 2 * q^52 - 22 * q^53 + 10 * q^54 + 6 * q^62 - 8 * q^63 - 2 * q^64 - 2 * q^65 + 24 * q^67 - 4 * q^71 + 16 * q^74 + 2 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 + 22 * q^86 - 6 * q^88 - 4 * q^91 - 8 * q^92 - 6 * q^93 + 26 * q^94 + 2 * q^96

Introduction

L-functions

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Varieties

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Database

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