LMFDB - Embedded newform 4650.2.d.p.3349.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{21} -4.00000i q^{22} -1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} +4.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{58} -6.00000 q^{59} +10.0000 q^{61} +1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +10.0000i q^{67} -6.00000i q^{68} -14.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +4.00000 q^{74} +8.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} +8.00000 q^{86} +4.00000i q^{87} +4.00000i q^{88} +8.00000 q^{89} -8.00000 q^{91} -1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 8 q^{29} + 2 q^{31} - 12 q^{34} + 2 q^{36} - 8 q^{39} - 12 q^{41} + 8 q^{44} + 6 q^{49} + 12 q^{51} - 2 q^{54} - 4 q^{56} - 12 q^{59} + 20 q^{61} - 2 q^{64} - 8 q^{66} - 28 q^{71} + 8 q^{74} + 16 q^{79} + 2 q^{81} + 4 q^{84} + 16 q^{86} + 16 q^{89} - 16 q^{91} - 24 q^{94} + 2 q^{96} + 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 - 4 * q^21 - 2 * q^24 + 8 * q^26 - 8 * q^29 + 2 * q^31 - 12 * q^34 + 2 * q^36 - 8 * q^39 - 12 * q^41 + 8 * q^44 + 6 * q^49 + 12 * q^51 - 2 * q^54 - 4 * q^56 - 12 * q^59 + 20 * q^61 - 2 * q^64 - 8 * q^66 - 28 * q^71 + 8 * q^74 + 16 * q^79 + 2 * q^81 + 4 * q^84 + 16 * q^86 + 16 * q^89 - 16 * q^91 - 24 * q^94 + 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

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	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	− 4.00000i	− 0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	1.00000i	0.192450i
$28$	2.00000i	0.377964i
$29$	−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
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 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
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	6.00000	0.840168
$52$	4.00000i	0.554700i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	− 4.00000i	− 0.525226i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	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
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	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	$0.924791\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	0	0
$77$	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
$82$	− 6.00000i	− 0.662589i
$83$	− 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
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	3.00000i	0.303046i
$99$	4.00000	0.402015
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	6.00000i	0.594089i
$103$	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
$106$	2.00000	0.194257
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	− 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
$111$	−4.00000	−0.379663
$112$	− 2.00000i	− 0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	4.00000i	0.369800i
$118$	− 6.00000i	− 0.552345i
$119$	12.0000	1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	6.00000i	0.541002i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	−10.0000	−0.863868
$135$	0	0
$136$	6.00000	0.514496
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	− 14.0000i	− 1.17485i
$143$	16.0000i	1.33799i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 3.00000i	− 0.247436i
$148$	4.00000i	0.328798i
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	4.00000	0.320256
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	8.00000i	0.636446i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	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	$0.924498\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	2.00000i	0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−4.00000	−0.301511
$177$	6.00000i	0.450988i
$178$	8.00000i	0.599625i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	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
$184$	0	0
$185$	0	0
$186$	1.00000	0.0733236
$187$	− 24.0000i	− 1.75505i
$188$	− 12.0000i	− 0.875190i
$189$	2.00000	0.145479
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	1.00000i	0.0721688i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	4.00000i	0.284268i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	14.0000i	0.985037i
$203$	8.00000i	0.561490i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	− 4.00000i	− 0.277350i
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	2.00000i	0.137361i
$213$	14.0000i	0.959264i
$214$	−16.0000	−1.09374
$215$	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
$221$	24.0000	1.61441
$222$	− 4.00000i	− 0.268462i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	4.00000i	0.262613i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	− 8.00000i	− 0.519656i
$238$	12.0000i	0.777844i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	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
$246$	−6.00000	−0.382546
$247$	0	0
$248$	− 1.00000i	− 0.0635001i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	− 8.00000i	− 0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	4.00000	0.247594
$262$	14.0000i	0.864923i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	− 8.00000i	− 0.489592i
$268$	− 10.0000i	− 0.610847i
$269$	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
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	6.00000i	0.363803i
$273$	8.00000i	0.484182i
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	12.0000i	0.719712i
$279$	−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	$0.442725\pi$
$282$	12.0000i	0.714590i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	14.0000	0.830747
$285$	0	0
$286$	−16.0000	−0.946100
$287$	12.0000i	0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	− 4.00000i	− 0.234082i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−4.00000	−0.232495
$297$	− 4.00000i	− 0.232104i
$298$	− 14.0000i	− 0.810998i
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	− 16.0000i	− 0.920697i
$303$	− 14.0000i	− 0.804279i
$304$	0	0
$305$	0	0
$306$	6.00000	0.342997
$307$	− 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
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	4.00000i	0.226455i
$313$	− 4.00000i	− 0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$	− 4.00000i	− 0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	− 2.00000i	− 0.112154i
$319$	16.0000	0.895828
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	18.0000i	0.995402i
$328$	6.00000i	0.331295i
$329$	24.0000	1.32316
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	4.00000i	0.219529i
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	−2.00000	−0.109109
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	− 3.00000i	− 0.163178i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 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
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	− 4.00000i	− 0.213201i
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−8.00000	−0.423999
$357$	− 12.0000i	− 0.635107i
$358$	4.00000i	0.211407i
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	18.0000i	0.946059i
$363$	− 5.00000i	− 0.262432i
$364$	8.00000	0.419314
$365$	0	0
$366$	10.0000	0.522708
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−4.00000	−0.207670
$372$	1.00000i	0.0518476i
$373$	18.0000i	0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$	18.0000i	0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	12.0000	0.618853
$377$	16.0000i	0.824042i
$378$	2.00000i	0.102869i
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	− 10.0000i	− 0.511645i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	8.00000i	0.406663i
$388$	− 2.00000i	− 0.101535i
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	− 14.0000i	− 0.706207i
$394$	2.00000	0.100759
$395$	0	0
$396$	−4.00000	−0.201008
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	10.0000i	0.498755i
$403$	− 4.00000i	− 0.199254i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	16.0000i	0.793091i
$408$	− 6.00000i	− 0.297044i
$409$	−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
$411$	18.0000	0.887875
$412$	− 14.0000i	− 0.689730i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	16.0000i	0.778868i
$423$	− 12.0000i	− 0.583460i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	−14.0000	−0.678302
$427$	− 20.0000i	− 0.967868i
$428$	− 16.0000i	− 0.773389i
$429$	16.0000	0.772487
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	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
$437$	0	0
$438$	4.00000i	0.191127i
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	24.0000i	1.14156i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−16.0000	−0.757622
$447$	14.0000i	0.662177i
$448$	2.00000i	0.0944911i
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	6.00000i	0.282216i
$453$	16.0000i	0.751746i
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	4.00000i	0.187112i	0.995614	+	0.0935561i	$0.0298234\pi$
$457$	4.00000i	0.187112i	−0.995614	+	0.0935561i	$0.970177\pi$
$458$	26.0000i	1.21490i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	8.00000i	0.372194i
$463$	12.0000i	0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$	12.0000i	0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−6.00000	−0.277945
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	− 4.00000i	− 0.184900i
$469$	20.0000	0.923514
$470$	0	0
$471$	6.00000	0.276465
$472$	6.00000i	0.276172i
$473$	32.0000i	1.47136i
$474$	8.00000	0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	2.00000i	0.0915737i
$478$	− 20.0000i	− 0.914779i
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	28.0000i	1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$	28.0000i	1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	− 10.0000i	− 0.452679i
$489$	6.00000	0.271329
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 24.0000i	− 1.08091i
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	28.0000i	1.25597i
$498$	− 4.00000i	− 0.179244i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 28.0000i	− 1.24970i
$503$	− 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
$506$	0	0
$507$	3.00000i	0.133235i
$508$	− 12.0000i	− 0.532414i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	8.00000	0.352180
$517$	− 48.0000i	− 2.11104i
$518$	− 8.00000i	− 0.351500i
$519$	6.00000	0.263371
$520$	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			$\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−24.0000	−1.04645
$527$	6.00000i	0.261364i
$528$	4.00000i	0.174078i
$529$	23.0000	1.00000
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	24.0000i	1.03956i
$534$	8.00000	0.346194
$535$	0	0
$536$	10.0000	0.431934
$537$	− 4.00000i	− 0.172613i
$538$	12.0000i	0.517357i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	− 24.0000i	− 1.03089i
$543$	− 18.0000i	− 0.772454i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 46.0000i	− 1.96682i	−0.181402	−	0.983409i	$-0.558064\pi$
$547$	− 46.0000i	− 1.96682i	0.181402	−	0.983409i	$-0.441936\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−12.0000	−0.508913
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	−32.0000	−1.35346
$560$	0	0
$561$	−24.0000	−1.01328
$562$	6.00000i	0.253095i
$563$	8.00000i	0.337160i	0.985688	+	0.168580i	$0.0539181\pi$
$563$	8.00000i	0.337160i	−0.985688	+	0.168580i	$0.946082\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−14.0000	−0.588464
$567$	− 2.00000i	− 0.0839921i
$568$	14.0000i	0.587427i
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	− 16.0000i	− 0.668994i
$573$	10.0000i	0.417756i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 42.0000i	− 1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$	− 42.0000i	− 1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$	− 19.0000i	− 0.790296i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	−8.00000	−0.331896
$582$	2.00000i	0.0829027i
$583$	8.00000i	0.331326i
$584$	4.00000	0.165521
$585$	0	0
$586$	−26.0000	−1.07405
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	− 4.00000i	− 0.164399i
$593$	− 46.0000i	− 1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	− 46.0000i	− 1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	14.0000	0.573462
$597$	0	0
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	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
$603$	− 10.0000i	− 0.407231i
$604$	16.0000	0.651031
$605$	0	0
$606$	14.0000	0.568711
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	48.0000	1.94187
$612$	6.00000i	0.242536i
$613$	− 24.0000i	− 0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$	− 24.0000i	− 0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	8.00000	0.322329
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	14.0000i	0.563163i
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	6.00000i	0.240578i
$623$	− 16.0000i	− 0.641026i
$624$	−4.00000	−0.160128
$625$	0	0
$626$	4.00000	0.159872
$627$	0	0
$628$	− 6.00000i	− 0.239426i
$629$	24.0000	0.956943
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	− 8.00000i	− 0.318223i
$633$	− 16.0000i	− 0.635943i
$634$	26.0000	1.03259
$635$	0	0
$636$	2.00000	0.0793052
$637$	− 12.0000i	− 0.475457i
$638$	16.0000i	0.633446i
$639$	14.0000	0.553831
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	16.0000i	0.631470i
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 32.0000i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$	− 32.0000i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	24.0000	0.942082
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	− 6.00000i	− 0.234978i
$653$	− 22.0000i	− 0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$	− 22.0000i	− 0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 4.00000i	− 0.156055i
$658$	24.0000i	0.935617i
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	− 28.0000i	− 1.08825i
$663$	− 24.0000i	− 0.932083i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−40.0000	−1.54418
$672$	− 2.00000i	− 0.0771517i
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	3.00000	0.115385
$677$	34.0000i	1.30673i	0.757045	+	0.653363i	$0.226642\pi$
$677$	34.0000i	1.30673i	−0.757045	+	0.653363i	$0.773358\pi$
$678$	− 6.00000i	− 0.230429i
$679$	4.00000	0.153506
$680$	0	0
$681$	−20.0000	−0.766402
$682$	− 4.00000i	− 0.153168i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	− 26.0000i	− 0.991962i
$688$	− 8.00000i	− 0.304997i
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	− 6.00000i	− 0.228086i
$693$	− 8.00000i	− 0.303895i
$694$	4.00000	0.151838
$695$	0	0
$696$	4.00000	0.151620
$697$	− 36.0000i	− 1.36360i
$698$	− 2.00000i	− 0.0757011i
$699$	6.00000	0.226941
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	4.00000i	0.150970i
$703$	0	0
$704$	4.00000	0.150756
$705$	0	0
$706$	−10.0000	−0.376355
$707$	− 28.0000i	− 1.05305i
$708$	− 6.00000i	− 0.225494i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 8.00000i	− 0.299813i
$713$	0	0
$714$	12.0000	0.449089
$715$	0	0
$716$	−4.00000	−0.149487
$717$	20.0000i	0.746914i
$718$	− 30.0000i	− 1.11959i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	− 19.0000i	− 0.707107i
$723$	− 10.0000i	− 0.371904i
$724$	−18.0000	−0.668965
$725$	0	0
$726$	5.00000	0.185567
$727$	30.0000i	1.11264i	0.830969	+	0.556319i	$0.187787\pi$
$727$	30.0000i	1.11264i	−0.830969	+	0.556319i	$0.812213\pi$
$728$	8.00000i	0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	10.0000i	0.369611i
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 40.0000i	− 1.47342i
$738$	6.00000i	0.220863i
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	0	0
$742$	− 4.00000i	− 0.146845i
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−18.0000	−0.659027
$747$	4.00000i	0.146352i
$748$	24.0000i	0.877527i
$749$	32.0000	1.16925
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	12.0000i	0.437595i
$753$	28.0000i	1.02038i
$754$	−16.0000	−0.582686
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	12.0000i	0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$	12.0000i	0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$	8.00000i	0.290573i
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	12.0000i	0.434714i
$763$	36.0000i	1.30329i
$764$	10.0000	0.361787
$765$	0	0
$766$	−24.0000	−0.867155
$767$	24.0000i	0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	2.00000i	0.0719816i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	2.00000	0.0717958
$777$	8.00000i	0.286998i
$778$	− 36.0000i	− 1.29066i
$779$	0	0
$780$	0	0
$781$	56.0000	2.00384
$782$	0	0
$783$	− 4.00000i	− 0.142948i
$784$	3.00000	0.107143
$785$	0	0
$786$	14.0000	0.499363
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	2.00000i	0.0712470i
$789$	24.0000	0.854423
$790$	0	0
$791$	−12.0000	−0.426671
$792$	− 4.00000i	− 0.142134i
$793$	− 40.0000i	− 1.42044i
$794$	−18.0000	−0.638796
$795$	0	0
$796$	0	0
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	−8.00000	−0.282666
$802$	− 4.00000i	− 0.141245i
$803$	− 16.0000i	− 0.564628i
$804$	−10.0000	−0.352673
$805$	0	0
$806$	4.00000	0.140894
$807$	− 12.0000i	− 0.422420i
$808$	− 14.0000i	− 0.492518i
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	− 8.00000i	− 0.280745i
$813$	24.0000i	0.841717i
$814$	−16.0000	−0.560800
$815$	0	0
$816$	6.00000	0.210042
$817$	0	0
$818$	− 6.00000i	− 0.209785i
$819$	8.00000	0.279543
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	18.0000i	0.627822i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	−12.0000	−0.417533
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	4.00000i	0.138675i
$833$	18.0000i	0.623663i
$834$	12.0000	0.415526
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	30.0000i	1.03633i
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 30.0000i	− 1.03387i
$843$	− 6.00000i	− 0.206651i
$844$	−16.0000	−0.550743
$845$	0	0
$846$	12.0000	0.412568
$847$	− 10.0000i	− 0.343604i
$848$	− 2.00000i	− 0.0686803i
$849$	14.0000	0.480479
$850$	0	0
$851$	0	0
$852$	− 14.0000i	− 0.479632i
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	16.0000	0.546869
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	16.0000i	0.546231i
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	− 10.0000i	− 0.340601i
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	19.0000i	0.645274i
$868$	2.00000i	0.0678844i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	40.0000	1.35535
$872$	18.0000i	0.609557i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	− 24.0000i	− 0.809961i
$879$	26.0000	0.876958
$880$	0	0
$881$	−56.0000	−1.88669	−0.943344	−	0.331816i	$-0.892339\pi$
$881$	−56.0000	−1.88669	−0.943344	+	0.331816i	$0.892339\pi$
$882$	− 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	$-0.109252\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	12.0000	0.403148
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	4.00000i	0.134231i
$889$	24.0000	0.804934
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	−14.0000	−0.468230
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	− 20.0000i	− 0.667409i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	12.0000	0.399778
$902$	24.0000i	0.799113i
$903$	16.0000i	0.532447i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−16.0000	−0.531564
$907$	− 58.0000i	− 1.92586i	−0.269754	−	0.962929i	$-0.586942\pi$
$907$	− 58.0000i	− 1.92586i	0.269754	−	0.962929i	$-0.413058\pi$
$908$	20.0000i	0.663723i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	−4.00000	−0.132308
$915$	0	0
$916$	−26.0000	−0.859064
$917$	− 28.0000i	− 0.924641i
$918$	− 6.00000i	− 0.198030i
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	24.0000i	0.790398i
$923$	56.0000i	1.84326i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−12.0000	−0.394344
$927$	− 14.0000i	− 0.459820i
$928$	− 4.00000i	− 0.131306i
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	0	0
$932$	− 6.00000i	− 0.196537i
$933$	− 6.00000i	− 0.196431i
$934$	−24.0000	−0.785304
$935$	0	0
$936$	4.00000	0.130744
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	20.0000i	0.653023i
$939$	−4.00000	−0.130535
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	6.00000i	0.195491i
$943$	0	0
$944$	−6.00000	−0.195283
$945$	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
$951$	−26.0000	−0.843108
$952$	− 12.0000i	− 0.388922i
$953$	− 50.0000i	− 1.61966i	−0.586665	−	0.809829i	$-0.699560\pi$
$953$	− 50.0000i	− 1.61966i	0.586665	−	0.809829i	$-0.300440\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	20.0000	0.646846
$957$	− 16.0000i	− 0.517207i
$958$	14.0000i	0.452319i
$959$	36.0000	1.16250
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 16.0000i	− 0.515861i
$963$	− 16.0000i	− 0.515593i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000i	0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	−28.0000	−0.897178
$975$	0	0
$976$	10.0000	0.320092
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	6.00000i	0.191859i
$979$	−32.0000	−1.02272
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	24.0000	0.764316
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	1.00000i	0.0317500i
$993$	28.0000i	0.888553i
$994$	−28.0000	−0.888106
$995$	0	0
$996$	4.00000	0.126745
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	4.00000i	0.126618i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.p.3349.2		2
5.2	odd	4		930.2.a.d.1.1	✓	1
5.3	odd	4		4650.2.a.bl.1.1		1
5.4	even	2	inner	4650.2.d.p.3349.1		2
15.2	even	4		2790.2.a.t.1.1		1
20.7	even	4		7440.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.d.1.1	✓	1	5.2	odd	4
2790.2.a.t.1.1		1	15.2	even	4
4650.2.a.bl.1.1		1	5.3	odd	4
4650.2.d.p.3349.1		2	5.4	even	2	inner
4650.2.d.p.3349.2		2	1.1	even	1	trivial
7440.2.a.y.1.1		1	20.7	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{21} -4.00000i q^{22} -1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} +4.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{58} -6.00000 q^{59} +10.0000 q^{61} +1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +10.0000i q^{67} -6.00000i q^{68} -14.0000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +4.00000 q^{74} +8.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} +8.00000 q^{86} +4.00000i q^{87} +4.00000i q^{88} +8.00000 q^{89} -8.00000 q^{91} -1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 8 q^{29} + 2 q^{31} - 12 q^{34} + 2 q^{36} - 8 q^{39} - 12 q^{41} + 8 q^{44} + 6 q^{49} + 12 q^{51} - 2 q^{54} - 4 q^{56} - 12 q^{59} + 20 q^{61} - 2 q^{64} - 8 q^{66} - 28 q^{71} + 8 q^{74} + 16 q^{79} + 2 q^{81} + 4 q^{84} + 16 q^{86} + 16 q^{89} - 16 q^{91} - 24 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 4 * q^14 + 2 * q^16 - 4 * q^21 - 2 * q^24 + 8 * q^26 - 8 * q^29 + 2 * q^31 - 12 * q^34 + 2 * q^36 - 8 * q^39 - 12 * q^41 + 8 * q^44 + 6 * q^49 + 12 * q^51 - 2 * q^54 - 4 * q^56 - 12 * q^59 + 20 * q^61 - 2 * q^64 - 8 * q^66 - 28 * q^71 + 8 * q^74 + 16 * q^79 + 2 * q^81 + 4 * q^84 + 16 * q^86 + 16 * q^89 - 16 * q^91 - 24 * q^94 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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