LMFDB - Embedded newform 4650.2.d.b.3349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.1304369399$
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 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.b.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} -2.00000 q^{21} +4.00000i q^{22} +8.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -12.0000i q^{37} -8.00000i q^{38} +4.00000 q^{39} +10.0000 q^{41} +2.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -8.00000i q^{57} +4.00000i q^{58} -2.00000 q^{59} +10.0000 q^{61} +1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -6.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} +6.00000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -12.0000 q^{74} -8.00000 q^{76} +8.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -8.00000 q^{86} +4.00000i q^{87} -4.00000i q^{88} +8.00000 q^{91} -8.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 8 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 8 q^{39} + 20 q^{41} + 8 q^{44} + 16 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 - 4 * q^14 + 2 * q^16 + 16 * q^19 - 4 * q^21 + 2 * q^24 + 8 * q^26 - 8 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 + 8 * q^39 + 20 * q^41 + 8 * q^44 + 16 * q^46 + 6 * q^49 + 4 * q^51 + 2 * q^54 + 4 * q^56 - 4 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 + 16 * q^69 + 12 * q^71 - 24 * q^74 - 16 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^84 - 16 * q^86 + 16 * q^91 - 8 * q^94 - 2 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000i	0.288675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	1.00000i	0.235702i
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.00000i	0.852803i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 1.00000i	− 0.176777i
$33$	4.00000i	0.696311i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 12.0000i	− 1.97279i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	− 12.0000i	− 1.97279i	0.164399	−	0.986394i	$-0.447432\pi$
$38$	− 8.00000i	− 1.29777i
$39$	4.00000	0.640513
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000i	0.308607i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	− 4.00000i	− 0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	− 8.00000i	− 1.05963i
$58$	4.00000i	0.525226i
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	1.00000i	0.127000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	− 2.00000i	− 0.242536i
$69$	8.00000	0.963087
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	− 1.00000i	− 0.117851i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−12.0000	−1.39497
$75$	0	0
$76$	−8.00000	−0.917663
$77$	8.00000i	0.911685i
$78$	− 4.00000i	− 0.452911i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−8.00000	−0.862662
$87$	4.00000i	0.428845i
$88$	− 4.00000i	− 0.426401i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	− 8.00000i	− 0.834058i
$93$	1.00000i	0.103695i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 18.0000i	− 1.82762i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	− 18.0000i	− 1.82762i	0.406138	−	0.913812i	$-0.366875\pi$
$98$	− 3.00000i	− 0.303046i
$99$	4.00000	0.402015
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	− 2.00000i	− 0.198030i
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	− 2.00000i	− 0.188982i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	4.00000	0.371391
$117$	− 4.00000i	− 0.369800i
$118$	2.00000i	0.184115i
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	− 10.0000i	− 0.905357i
$123$	− 10.0000i	− 0.901670i
$124$	1.00000	0.0898027
$125$	0	0
$126$	2.00000	0.178174
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	− 4.00000i	− 0.348155i
$133$	− 16.0000i	− 1.38738i
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−2.00000	−0.171499
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	− 8.00000i	− 0.681005i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	− 6.00000i	− 0.503509i
$143$	− 16.0000i	− 1.33799i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	− 3.00000i	− 0.247436i
$148$	12.0000i	0.986394i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	8.00000i	0.648886i
$153$	− 2.00000i	− 0.161690i
$154$	8.00000	0.644658
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	16.0000	1.26098
$162$	− 1.00000i	− 0.0785674i
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−8.00000	−0.611775
$172$	8.00000i	0.609994i
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	−4.00000	−0.301511
$177$	2.00000i	0.150329i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	− 8.00000i	− 0.592999i
$183$	− 10.0000i	− 0.739221i
$184$	−8.00000	−0.589768
$185$	0	0
$186$	1.00000	0.0733236
$187$	− 8.00000i	− 0.585018i
$188$	4.00000i	0.291730i
$189$	2.00000	0.145479
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	1.00000i	0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	− 4.00000i	− 0.284268i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	18.0000i	1.26648i
$203$	8.00000i	0.561490i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	14.0000	0.975426
$207$	− 8.00000i	− 0.556038i
$208$	4.00000i	0.277350i
$209$	−32.0000	−2.21349
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	6.00000i	0.412082i
$213$	− 6.00000i	− 0.411113i
$214$	8.00000	0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	2.00000i	0.135769i
$218$	18.0000i	1.21911i
$219$	4.00000	0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	12.0000i	0.805387i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	8.00000i	0.529813i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	− 4.00000i	− 0.262613i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	2.00000	0.130189
$237$	− 8.00000i	− 0.519656i
$238$	− 4.00000i	− 0.259281i
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−10.0000	−0.637577
$247$	32.0000i	2.03611i
$248$	− 1.00000i	− 0.0635001i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	− 2.00000i	− 0.125988i
$253$	− 32.0000i	− 2.01182i
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	8.00000i	0.498058i
$259$	−24.0000	−1.49129
$260$	0	0
$261$	4.00000	0.247594
$262$	− 10.0000i	− 0.617802i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	−16.0000	−0.981023
$267$	0	0
$268$	6.00000i	0.366508i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000i	0.121268i
$273$	− 8.00000i	− 0.484182i
$274$	6.00000	0.362473
$275$	0	0
$276$	−8.00000	−0.481543
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	4.00000i	0.239904i
$279$	1.00000	0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	4.00000i	0.238197i
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−16.0000	−0.946100
$287$	− 20.0000i	− 1.18056i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	− 4.00000i	− 0.234082i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	12.0000	0.697486
$297$	− 4.00000i	− 0.232104i
$298$	− 18.0000i	− 1.04271i
$299$	−32.0000	−1.85061
$300$	0	0
$301$	−16.0000	−0.922225
$302$	16.0000i	0.920697i
$303$	18.0000i	1.03407i
$304$	8.00000	0.458831
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	− 8.00000i	− 0.455842i
$309$	14.0000	0.796432
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	4.00000i	0.226455i
$313$	− 20.0000i	− 1.13047i	−0.824931	−	0.565233i	$-0.808786\pi$
$313$	− 20.0000i	− 1.13047i	0.824931	−	0.565233i	$-0.191214\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	6.00000i	0.336463i
$319$	16.0000	0.895828
$320$	0	0
$321$	8.00000	0.446516
$322$	− 16.0000i	− 0.891645i
$323$	16.0000i	0.890264i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	6.00000	0.332309
$327$	18.0000i	0.995402i
$328$	10.0000i	0.552158i
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	4.00000i	0.219529i
$333$	12.0000i	0.657596i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 32.0000i	− 1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	− 32.0000i	− 1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$	3.00000i	0.163178i
$339$	6.00000	0.325875
$340$	0	0
$341$	4.00000	0.216612
$342$	8.00000i	0.432590i
$343$	− 20.0000i	− 1.07990i
$344$	8.00000	0.431331
$345$	0	0
$346$	−22.0000	−1.18273
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	− 4.00000i	− 0.214423i
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	4.00000i	0.213201i
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	2.00000	0.106299
$355$	0	0
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	− 20.0000i	− 1.05703i
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 18.0000i	− 0.946059i
$363$	− 5.00000i	− 0.262432i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−10.0000	−0.522708
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	8.00000i	0.417029i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−12.0000	−0.623009
$372$	− 1.00000i	− 0.0518476i
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	4.00000	0.206284
$377$	− 16.0000i	− 0.824042i
$378$	− 2.00000i	− 0.102869i
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	− 18.0000i	− 0.920960i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	8.00000i	0.406663i
$388$	18.0000i	0.913812i
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	3.00000i	0.151523i
$393$	− 10.0000i	− 0.504433i
$394$	10.0000	0.503793
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	16.0000i	0.802008i
$399$	−16.0000	−0.801002
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	6.00000i	0.299253i
$403$	− 4.00000i	− 0.199254i
$404$	18.0000	0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	48.0000i	2.37927i
$408$	2.00000i	0.0990148i
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	− 14.0000i	− 0.689730i
$413$	4.00000i	0.196827i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	4.00000	0.196116
$417$	4.00000i	0.195881i
$418$	32.0000i	1.56517i
$419$	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
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 24.0000i	− 1.16830i
$423$	4.00000i	0.194487i
$424$	6.00000	0.291386
$425$	0	0
$426$	−6.00000	−0.290701
$427$	− 20.0000i	− 0.967868i
$428$	− 8.00000i	− 0.386695i
$429$	−16.0000	−0.772487
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	1.00000i	0.0481125i
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	18.0000	0.862044
$437$	64.0000i	3.06154i
$438$	− 4.00000i	− 0.191127i
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	8.00000i	0.380521i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	12.0000	0.569495
$445$	0	0
$446$	8.00000	0.378811
$447$	− 18.0000i	− 0.851371i
$448$	2.00000i	0.0944911i
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	− 6.00000i	− 0.282216i
$453$	16.0000i	0.751746i
$454$	4.00000	0.187729
$455$	0	0
$456$	8.00000	0.374634
$457$	36.0000i	1.68401i	0.539471	+	0.842004i	$0.318624\pi$
$457$	36.0000i	1.68401i	−0.539471	+	0.842004i	$0.681376\pi$
$458$	22.0000i	1.02799i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−40.0000	−1.86299	−0.931493	−	0.363760i	$-0.881493\pi$
$461$	−40.0000	−1.86299	−0.931493	+	0.363760i	$0.881493\pi$
$462$	− 8.00000i	− 0.372194i
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	10.0000	0.463241
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	4.00000i	0.184900i
$469$	−12.0000	−0.554109
$470$	0	0
$471$	−22.0000	−1.01371
$472$	− 2.00000i	− 0.0920575i
$473$	32.0000i	1.47136i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−4.00000	−0.183340
$477$	6.00000i	0.274721i
$478$	− 4.00000i	− 0.182956i
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	48.0000	2.18861
$482$	− 10.0000i	− 0.455488i
$483$	− 16.0000i	− 0.728025i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	10.0000i	0.452679i
$489$	6.00000	0.271329
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	10.0000i	0.450835i
$493$	− 8.00000i	− 0.360302i
$494$	32.0000	1.43975
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	− 12.0000i	− 0.538274i
$498$	4.00000i	0.179244i
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	− 20.0000i	− 0.892644i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−32.0000	−1.42257
$507$	3.00000i	0.133235i
$508$	− 4.00000i	− 0.177471i
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	8.00000i	0.353209i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	8.00000	0.352180
$517$	16.0000i	0.703679i
$518$	24.0000i	1.05450i
$519$	−22.0000	−0.965693
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	− 4.00000i	− 0.175075i
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	0	0
$527$	− 2.00000i	− 0.0871214i
$528$	4.00000i	0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	2.00000	0.0867926
$532$	16.0000i	0.693688i
$533$	40.0000i	1.73259i
$534$	0	0
$535$	0	0
$536$	6.00000	0.259161
$537$	− 20.0000i	− 0.863064i
$538$	− 12.0000i	− 0.517357i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 8.00000i	− 0.343629i
$543$	− 18.0000i	− 0.772454i
$544$	2.00000	0.0857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−32.0000	−1.36325
$552$	8.00000i	0.340503i
$553$	− 16.0000i	− 0.680389i
$554$	8.00000	0.339887
$555$	0	0
$556$	4.00000	0.169638
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	32.0000	1.35346
$560$	0	0
$561$	−8.00000	−0.337760
$562$	− 6.00000i	− 0.253095i
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	− 2.00000i	− 0.0839921i
$568$	6.00000i	0.251754i
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	16.0000i	0.668994i
$573$	− 18.0000i	− 0.751961i
$574$	−20.0000	−0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 6.00000i	− 0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$	− 6.00000i	− 0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$	− 13.0000i	− 0.540729i
$579$	2.00000	0.0831172
$580$	0	0
$581$	−8.00000	−0.331896
$582$	18.0000i	0.746124i
$583$	24.0000i	0.993978i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	3.00000i	0.123718i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	10.0000	0.411345
$592$	− 12.0000i	− 0.493197i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−18.0000	−0.737309
$597$	16.0000i	0.654836i
$598$	32.0000i	1.30858i
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	16.0000i	0.652111i
$603$	6.00000i	0.244339i
$604$	16.0000	0.651031
$605$	0	0
$606$	18.0000	0.731200
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	− 8.00000i	− 0.324443i
$609$	8.00000	0.324176
$610$	0	0
$611$	16.0000	0.647291
$612$	2.00000i	0.0808452i
$613$	− 8.00000i	− 0.323117i	−0.986863	−	0.161558i	$-0.948348\pi$
$613$	− 8.00000i	− 0.323117i	0.986863	−	0.161558i	$-0.0516520\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−8.00000	−0.322329
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	− 14.0000i	− 0.563163i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	30.0000i	1.20289i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−20.0000	−0.799361
$627$	32.0000i	1.27796i
$628$	22.0000i	0.877896i
$629$	24.0000	0.956943
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000i	0.318223i
$633$	− 24.0000i	− 0.953914i
$634$	−22.0000	−0.873732
$635$	0	0
$636$	6.00000	0.237915
$637$	12.0000i	0.475457i
$638$	− 16.0000i	− 0.633446i
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	− 8.00000i	− 0.315735i
$643$	8.00000i	0.315489i	0.987480	+	0.157745i	$0.0504223\pi$
$643$	8.00000i	0.315489i	−0.987480	+	0.157745i	$0.949578\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	16.0000	0.629512
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	1.00000i	0.0392837i
$649$	8.00000	0.314027
$650$	0	0
$651$	2.00000	0.0783862
$652$	− 6.00000i	− 0.234978i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	10.0000	0.390434
$657$	− 4.00000i	− 0.156055i
$658$	8.00000i	0.311872i
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	12.0000i	0.466393i
$663$	8.00000i	0.310694i
$664$	4.00000	0.155230
$665$	0	0
$666$	12.0000	0.464991
$667$	− 32.0000i	− 1.23904i
$668$	8.00000i	0.309529i
$669$	8.00000	0.309298
$670$	0	0
$671$	−40.0000	−1.54418
$672$	2.00000i	0.0771517i
$673$	− 8.00000i	− 0.308377i	−0.988041	−	0.154189i	$-0.950724\pi$
$673$	− 8.00000i	− 0.308377i	0.988041	−	0.154189i	$-0.0492764\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	3.00000	0.115385
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	− 6.00000i	− 0.230429i
$679$	−36.0000	−1.38155
$680$	0	0
$681$	4.00000	0.153280
$682$	− 4.00000i	− 0.153168i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	−20.0000	−0.763604
$687$	22.0000i	0.839352i
$688$	− 8.00000i	− 0.304997i
$689$	24.0000	0.914327
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	22.0000i	0.836315i
$693$	− 8.00000i	− 0.303895i
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−4.00000	−0.151620
$697$	20.0000i	0.757554i
$698$	− 30.0000i	− 1.13552i
$699$	10.0000	0.378235
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	4.00000i	0.150970i
$703$	− 96.0000i	− 3.62071i
$704$	4.00000	0.150756
$705$	0	0
$706$	−34.0000	−1.27961
$707$	36.0000i	1.35392i
$708$	− 2.00000i	− 0.0751646i
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−20.0000	−0.747435
$717$	− 4.00000i	− 0.149383i
$718$	10.0000i	0.373197i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	− 45.0000i	− 1.67473i
$723$	− 10.0000i	− 0.371904i
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 2.00000i	− 0.0741759i	−0.999312	−	0.0370879i	$-0.988192\pi$
$727$	− 2.00000i	− 0.0741759i	0.999312	−	0.0370879i	$-0.0118082\pi$
$728$	8.00000i	0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	10.0000i	0.369611i
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	8.00000	0.294884
$737$	24.0000i	0.884051i
$738$	10.0000i	0.368105i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	32.0000	1.17555
$742$	12.0000i	0.440534i
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−34.0000	−1.24483
$747$	4.00000i	0.146352i
$748$	8.00000i	0.292509i
$749$	16.0000	0.584627
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	− 4.00000i	− 0.145865i
$753$	− 20.0000i	− 0.728841i
$754$	−16.0000	−0.582686
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	36.0000i	1.30844i	0.756303	+	0.654221i	$0.227003\pi$
$757$	36.0000i	1.30844i	−0.756303	+	0.654221i	$0.772997\pi$
$758$	16.0000i	0.581146i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	− 4.00000i	− 0.144905i
$763$	36.0000i	1.30329i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	16.0000	0.578103
$767$	− 8.00000i	− 0.288863i
$768$	− 1.00000i	− 0.0360844i
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	− 2.00000i	− 0.0719816i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	18.0000	0.646162
$777$	24.0000i	0.860995i
$778$	− 28.0000i	− 1.00385i
$779$	80.0000	2.86630
$780$	0	0
$781$	−24.0000	−0.858788
$782$	16.0000i	0.572159i
$783$	− 4.00000i	− 0.142948i
$784$	3.00000	0.107143
$785$	0	0
$786$	−10.0000	−0.356688
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 10.0000i	− 0.356235i
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	4.00000i	0.142134i
$793$	40.0000i	1.42044i
$794$	−34.0000	−1.20661
$795$	0	0
$796$	16.0000	0.567105
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	16.0000i	0.566394i
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	− 4.00000i	− 0.141245i
$803$	− 16.0000i	− 0.564628i
$804$	6.00000	0.211604
$805$	0	0
$806$	−4.00000	−0.140894
$807$	− 12.0000i	− 0.422420i
$808$	− 18.0000i	− 0.633238i
$809$	−28.0000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$809$	−28.0000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	− 8.00000i	− 0.280745i
$813$	− 8.00000i	− 0.280572i
$814$	48.0000	1.68240
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 64.0000i	− 2.23908i
$818$	6.00000i	0.209785i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	4.00000	0.139178
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	8.00000i	0.278019i
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	− 4.00000i	− 0.138675i
$833$	6.00000i	0.207888i
$834$	4.00000	0.138509
$835$	0	0
$836$	32.0000	1.10674
$837$	− 1.00000i	− 0.0345651i
$838$	− 10.0000i	− 0.345444i
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 2.00000i	− 0.0689246i
$843$	− 6.00000i	− 0.206651i
$844$	−24.0000	−0.826114
$845$	0	0
$846$	4.00000	0.137523
$847$	− 10.0000i	− 0.343604i
$848$	− 6.00000i	− 0.206041i
$849$	−2.00000	−0.0686398
$850$	0	0
$851$	96.0000	3.29084
$852$	6.00000i	0.205557i
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	−8.00000	−0.273434
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	16.0000i	0.546231i
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	30.0000i	1.02180i
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−4.00000	−0.135926
$867$	− 13.0000i	− 0.441503i
$868$	− 2.00000i	− 0.0678844i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	24.0000	0.813209
$872$	− 18.0000i	− 0.609557i
$873$	18.0000i	0.609208i
$874$	64.0000	2.16483
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	− 40.0000i	− 1.34993i
$879$	6.00000	0.202375
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	3.00000i	0.101015i
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 56.0000i	− 1.88030i	−0.340766	−	0.940148i	$-0.610687\pi$
$887$	− 56.0000i	− 1.88030i	0.340766	−	0.940148i	$-0.389313\pi$
$888$	− 12.0000i	− 0.402694i
$889$	8.00000	0.268311
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 8.00000i	− 0.267860i
$893$	− 32.0000i	− 1.07084i
$894$	−18.0000	−0.602010
$895$	0	0
$896$	2.00000	0.0668153
$897$	32.0000i	1.06845i
$898$	28.0000i	0.934372i
$899$	4.00000	0.133407
$900$	0	0
$901$	12.0000	0.399778
$902$	40.0000i	1.33185i
$903$	16.0000i	0.532447i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	16.0000	0.531564
$907$	− 10.0000i	− 0.332045i	−0.986122	−	0.166022i	$-0.946908\pi$
$907$	− 10.0000i	− 0.332045i	0.986122	−	0.166022i	$-0.0530924\pi$
$908$	− 4.00000i	− 0.132745i
$909$	18.0000	0.597022
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	− 8.00000i	− 0.264906i
$913$	16.0000i	0.529523i
$914$	36.0000	1.19077
$915$	0	0
$916$	22.0000	0.726900
$917$	− 20.0000i	− 0.660458i
$918$	2.00000i	0.0660098i
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	40.0000i	1.31733i
$923$	24.0000i	0.789970i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	36.0000	1.18303
$927$	− 14.0000i	− 0.459820i
$928$	4.00000i	0.131306i
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	− 10.0000i	− 0.327561i
$933$	30.0000i	0.982156i
$934$	0	0
$935$	0	0
$936$	4.00000	0.130744
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	12.0000i	0.391814i
$939$	−20.0000	−0.652675
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	22.0000i	0.716799i
$943$	80.0000i	2.60516i
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	32.0000	1.04041
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	8.00000i	0.259828i
$949$	−16.0000	−0.519382
$950$	0	0
$951$	−22.0000	−0.713399
$952$	4.00000i	0.129641i
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−4.00000	−0.129369
$957$	− 16.0000i	− 0.517207i
$958$	− 10.0000i	− 0.323085i
$959$	12.0000	0.387500
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 48.0000i	− 1.54758i
$963$	− 8.00000i	− 0.257796i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−16.0000	−0.514792
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	5.00000i	0.160706i
$969$	16.0000	0.513994
$970$	0	0
$971$	−2.00000	−0.0641831	−0.0320915	−	0.999485i	$-0.510217\pi$
$971$	−2.00000	−0.0641831	−0.0320915	+	0.999485i	$0.510217\pi$
$972$	1.00000i	0.0320750i
$973$	8.00000i	0.256468i
$974$	−12.0000	−0.384505
$975$	0	0
$976$	10.0000	0.320092
$977$	− 46.0000i	− 1.47167i	−0.677161	−	0.735835i	$-0.736790\pi$
$977$	− 46.0000i	− 1.47167i	0.677161	−	0.735835i	$-0.263210\pi$
$978$	− 6.00000i	− 0.191859i
$979$	0	0
$980$	0	0
$981$	18.0000	0.574696
$982$	8.00000i	0.255290i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	−8.00000	−0.254772
$987$	8.00000i	0.254643i
$988$	− 32.0000i	− 1.01806i
$989$	64.0000	2.03508
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	1.00000i	0.0317500i
$993$	12.0000i	0.380808i
$994$	−12.0000	−0.380617
$995$	0	0
$996$	4.00000	0.126745
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	− 36.0000i	− 1.13956i
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.b.3349.1		2
5.2	odd	4		4650.2.a.bg.1.1		1
5.3	odd	4		930.2.a.h.1.1	✓	1
5.4	even	2	inner	4650.2.d.b.3349.2		2
15.8	even	4		2790.2.a.p.1.1		1
20.3	even	4		7440.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.h.1.1	✓	1	5.3	odd	4
2790.2.a.p.1.1		1	15.8	even	4
4650.2.a.bg.1.1		1	5.2	odd	4
4650.2.d.b.3349.1		2	1.1	even	1	trivial
4650.2.d.b.3349.2		2	5.4	even	2	inner
7440.2.a.m.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} -2.00000 q^{21} +4.00000i q^{22} +8.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -12.0000i q^{37} -8.00000i q^{38} +4.00000 q^{39} +10.0000 q^{41} +2.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -8.00000i q^{57} +4.00000i q^{58} -2.00000 q^{59} +10.0000 q^{61} +1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -6.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} +6.00000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -12.0000 q^{74} -8.00000 q^{76} +8.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -8.00000 q^{86} +4.00000i q^{87} -4.00000i q^{88} +8.00000 q^{91} -8.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 8 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 8 q^{39} + 20 q^{41} + 8 q^{44} + 16 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 - 4 * q^14 + 2 * q^16 + 16 * q^19 - 4 * q^21 + 2 * q^24 + 8 * q^26 - 8 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 + 8 * q^39 + 20 * q^41 + 8 * q^44 + 16 * q^46 + 6 * q^49 + 4 * q^51 + 2 * q^54 + 4 * q^56 - 4 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 + 16 * q^69 + 12 * q^71 - 24 * q^74 - 16 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^84 - 16 * q^86 + 16 * q^91 - 8 * q^94 - 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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