LMFDB - Embedded newform 4650.2.d.g.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:	$1$
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.g.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +2.00000 q^{21} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -8.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +4.00000 q^{39} +10.0000 q^{41} -2.00000i q^{42} +8.00000i q^{43} +4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -4.00000i q^{52} -14.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +8.00000i q^{58} -14.0000 q^{59} -6.00000 q^{61} +1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +10.0000i q^{67} +6.00000i q^{68} +6.00000 q^{71} -1.00000i q^{72} -8.00000i q^{73} -4.00000 q^{74} -4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +8.00000 q^{86} +8.00000i q^{87} -16.0000 q^{89} -8.00000 q^{91} +1.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} + 8 q^{26} - 16 q^{29} - 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 20 q^{41} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 4 q^{56}+ \cdots - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 + 4 * q^21 + 2 * q^24 + 8 * q^26 - 16 * q^29 - 2 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 20 * q^41 + 6 * q^49 - 12 * q^51 + 2 * q^54 - 4 * q^56 - 28 * q^59 - 12 * q^61 - 2 * q^64 + 12 * q^71 - 8 * q^74 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 16 * q^86 - 32 * q^89 - 16 * q^91 + 8 * q^94 - 2 * q^96

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	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	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
$22$	0	0
$23$	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
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−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
$41$	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	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	− 4.00000i	− 0.554700i
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	8.00000i	1.05045i
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	1.00000i	0.127000i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	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
$71$	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
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	0	0
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	8.00000	0.862662
$87$	8.00000i	0.857690i
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	1.00000i	0.103695i
$94$	4.00000	0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000i	0.594089i
$103$	− 6.00000i	− 0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$	− 6.00000i	− 0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−14.0000	−1.35980
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	− 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
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	8.00000	0.742781
$117$	− 4.00000i	− 0.369800i
$118$	14.0000i	1.28880i
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000i	0.543214i
$123$	− 10.0000i	− 0.901670i
$124$	1.00000	0.0898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	8.00000	0.704361
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	10.0000	0.863868
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−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
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−8.00000	−0.662085
$147$	− 3.00000i	− 0.247436i
$148$	4.00000i	0.328798i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	8.00000i	0.636446i
$159$	−14.0000	−1.11027
$160$	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	$0.924498\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	2.00000i	0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	− 8.00000i	− 0.609994i
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	0	0
$177$	14.0000i	1.05230i
$178$	16.0000i	1.19925i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	8.00000i	0.592999i
$183$	6.00000i	0.443533i
$184$	0	0
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	− 4.00000i	− 0.291730i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	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	$-0.0229323\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	6.00000i	0.422159i
$203$	− 16.0000i	− 1.12298i
$204$	6.00000	0.420084
$205$	0	0
$206$	−6.00000	−0.418040
$207$	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	$0.314345\pi$
$212$	14.0000i	0.961524i
$213$	− 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
$219$	−8.00000	−0.540590
$220$	0	0
$221$	24.0000	1.61441
$222$	4.00000i	0.268462i
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	14.0000	0.931266
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	− 8.00000i	− 0.525226i
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	14.0000	0.911322
$237$	8.00000i	0.519656i
$238$	− 12.0000i	− 0.777844i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	− 1.00000i	− 0.0635001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	− 8.00000i	− 0.498058i
$259$	8.00000	0.497096
$260$	0	0
$261$	8.00000	0.495188
$262$	18.0000i	1.11204i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.0000i	0.979184i
$268$	− 10.0000i	− 0.610847i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 6.00000i	− 0.363803i
$273$	8.00000i	0.484182i
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 16.0000i	− 0.961347i	−0.876900	−	0.480673i	$-0.840392\pi$
$277$	− 16.0000i	− 0.961347i	0.876900	−	0.480673i	$-0.159608\pi$
$278$	4.00000i	0.239904i
$279$	1.00000	0.0598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	− 4.00000i	− 0.238197i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	20.0000i	1.18056i
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	10.0000	0.586210
$292$	8.00000i	0.468165i
$293$	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
$296$	4.00000	0.232495
$297$	0	0
$298$	18.0000i	1.04271i
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	8.00000i	0.460348i
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	6.00000	0.342997
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	4.00000i	0.226455i
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	8.00000	0.450035
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	14.0000i	0.785081i
$319$	0	0
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	6.00000	0.332309
$327$	18.0000i	0.995402i
$328$	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	$0.606978\pi$
$332$	12.0000i	0.658586i
$333$	4.00000i	0.219199i
$334$	8.00000	0.437741
$335$	0	0
$336$	2.00000	0.109109
$337$	− 4.00000i	− 0.217894i	−0.994048	−	0.108947i	$-0.965252\pi$
$337$	− 4.00000i	− 0.217894i	0.994048	−	0.108947i	$-0.0347479\pi$
$338$	3.00000i	0.163178i
$339$	14.0000	0.760376
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	10.0000	0.537603
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	− 8.00000i	− 0.428845i
$349$	−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
$352$	0	0
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	14.0000	0.744092
$355$	0	0
$356$	16.0000	0.847998
$357$	− 12.0000i	− 0.635107i
$358$	0	0
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.00000i	0.315353i
$363$	11.0000i	0.577350i
$364$	8.00000	0.419314
$365$	0	0
$366$	6.00000	0.313625
$367$	36.0000i	1.87918i	0.342296	+	0.939592i	$0.388796\pi$
$367$	36.0000i	1.87918i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	28.0000	1.45369
$372$	− 1.00000i	− 0.0518476i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	0	0
$376$	−4.00000	−0.206284
$377$	− 32.0000i	− 1.64808i
$378$	2.00000i	0.102869i
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	0	0
$382$	14.0000i	0.716302i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	− 8.00000i	− 0.406663i
$388$	− 10.0000i	− 0.507673i
$389$	32.0000	1.62246	0.811232	−	0.584724i	$-0.198797\pi$
$389$	32.0000	1.62246	0.811232	+	0.584724i	$0.198797\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	18.0000i	0.907980i
$394$	2.00000	0.100759
$395$	0	0
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	− 10.0000i	− 0.498755i
$403$	− 4.00000i	− 0.199254i
$404$	6.00000	0.298511
$405$	0	0
$406$	−16.0000	−0.794067
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	6.00000i	0.295599i
$413$	− 28.0000i	− 1.37779i
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	4.00000i	0.195881i
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	− 16.0000i	− 0.778868i
$423$	− 4.00000i	− 0.194487i
$424$	14.0000	0.679900
$425$	0	0
$426$	−6.00000	−0.290701
$427$	− 12.0000i	− 0.580721i
$428$	8.00000i	0.386695i
$429$	0	0
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	1.00000i	0.0481125i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	18.0000	0.862044
$437$	0	0
$438$	8.00000i	0.382255i
$439$	−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
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−20.0000	−0.947027
$447$	18.0000i	0.851371i
$448$	− 2.00000i	− 0.0944911i
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	0	0
$452$	− 14.0000i	− 0.658505i
$453$	8.00000i	0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	− 10.0000i	− 0.467269i
$459$	6.00000	0.280056
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−14.0000	−0.648537
$467$	32.0000i	1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$	32.0000i	1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$	4.00000i	0.184900i
$469$	−20.0000	−0.923514
$470$	0	0
$471$	18.0000	0.829396
$472$	− 14.0000i	− 0.644402i
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	14.0000i	0.641016i
$478$	− 12.0000i	− 0.548867i
$479$	42.0000	1.91903	0.959514	−	0.281659i	$-0.0908848\pi$
$479$	42.0000	1.91903	0.959514	+	0.281659i	$0.0908848\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	14.0000i	0.637683i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	− 6.00000i	− 0.271607i
$489$	6.00000	0.271329
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	10.0000i	0.450835i
$493$	48.0000i	2.16181i
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	12.0000i	0.538274i
$498$	12.0000i	0.537733i
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	− 20.0000i	− 0.891756i	−0.895094	−	0.445878i	$-0.852892\pi$
$503$	− 20.0000i	− 0.891756i	0.895094	−	0.445878i	$-0.147108\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	10.0000	0.438951
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	− 8.00000i	− 0.350150i
$523$	24.0000i	1.04945i	0.851273	+	0.524723i	$0.175831\pi$
$523$	24.0000i	1.04945i	−0.851273	+	0.524723i	$0.824169\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	16.0000	0.697633
$527$	6.00000i	0.261364i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	14.0000	0.607548
$532$	0	0
$533$	40.0000i	1.73259i
$534$	16.0000	0.692388
$535$	0	0
$536$	−10.0000	−0.431934
$537$	0	0
$538$	24.0000i	1.03471i
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	16.0000i	0.687259i
$543$	6.00000i	0.257485i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	8.00000	0.342368
$547$	− 6.00000i	− 0.256541i	−0.991739	−	0.128271i	$-0.959057\pi$
$547$	− 6.00000i	− 0.256541i	0.991739	−	0.128271i	$-0.0409426\pi$
$548$	− 6.00000i	− 0.256307i
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−16.0000	−0.679775
$555$	0	0
$556$	4.00000	0.169638
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	−32.0000	−1.35346
$560$	0	0
$561$	0	0
$562$	− 30.0000i	− 1.26547i
$563$	− 32.0000i	− 1.34864i	−0.738440	−	0.674320i	$-0.764437\pi$
$563$	− 32.0000i	− 1.34864i	0.738440	−	0.674320i	$-0.235563\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	14.0000	0.588464
$567$	2.00000i	0.0839921i
$568$	6.00000i	0.251754i
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	14.0000i	0.584858i
$574$	20.0000	0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	19.0000i	0.790296i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	24.0000	0.995688
$582$	− 10.0000i	− 0.414513i
$583$	0	0
$584$	8.00000	0.331042
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	3.00000i	0.123718i
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	− 4.00000i	− 0.164399i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	− 24.0000i	− 0.982255i
$598$	0	0
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	16.0000i	0.652111i
$603$	− 10.0000i	− 0.407231i
$604$	8.00000	0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	− 2.00000i	− 0.0811775i	−0.999176	−	0.0405887i	$-0.987077\pi$
$607$	− 2.00000i	− 0.0811775i	0.999176	−	0.0405887i	$-0.0129233\pi$
$608$	0	0
$609$	−16.0000	−0.648353
$610$	0	0
$611$	−16.0000	−0.647291
$612$	− 6.00000i	− 0.242536i
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	0	0
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	6.00000i	0.241355i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	− 32.0000i	− 1.28205i
$624$	4.00000	0.160128
$625$	0	0
$626$	−16.0000	−0.639489
$627$	0	0
$628$	− 18.0000i	− 0.718278i
$629$	−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
$634$	10.0000	0.397151
$635$	0	0
$636$	14.0000	0.555136
$637$	12.0000i	0.475457i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−44.0000	−1.73790	−0.868948	−	0.494904i	$-0.835203\pi$
$641$	−44.0000	−1.73790	−0.868948	+	0.494904i	$0.835203\pi$
$642$	8.00000i	0.315735i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.0000i	0.629025i	0.949253	+	0.314512i	$0.101841\pi$
$647$	16.0000i	0.629025i	−0.949253	+	0.314512i	$0.898159\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	− 6.00000i	− 0.234978i
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	10.0000	0.390434
$657$	8.00000i	0.312110i
$658$	8.00000i	0.311872i
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	12.0000i	0.466393i
$663$	− 24.0000i	− 0.932083i
$664$	12.0000	0.465690
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	− 8.00000i	− 0.309529i
$669$	−20.0000	−0.773245
$670$	0	0
$671$	0	0
$672$	− 2.00000i	− 0.0771517i
$673$	− 20.0000i	− 0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$	− 20.0000i	− 0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	3.00000	0.115385
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	− 14.0000i	− 0.537667i
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	− 10.0000i	− 0.381524i
$688$	8.00000i	0.304997i
$689$	56.0000	2.13343
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	− 10.0000i	− 0.380143i
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	−8.00000	−0.303239
$697$	− 60.0000i	− 2.27266i
$698$	2.00000i	0.0757011i
$699$	−14.0000	−0.529529
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	4.00000i	0.150970i
$703$	0	0
$704$	0	0
$705$	0	0
$706$	14.0000	0.526897
$707$	− 12.0000i	− 0.451306i
$708$	− 14.0000i	− 0.526152i
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 16.0000i	− 0.599625i
$713$	0	0
$714$	−12.0000	−0.449089
$715$	0	0
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	18.0000i	0.671754i
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	19.0000i	0.707107i
$723$	14.0000i	0.520666i
$724$	6.00000	0.222988
$725$	0	0
$726$	11.0000	0.408248
$727$	10.0000i	0.370879i	0.982656	+	0.185440i	$0.0593710\pi$
$727$	10.0000i	0.370879i	−0.982656	+	0.185440i	$0.940629\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	− 6.00000i	− 0.221766i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	0	0
$737$	0	0
$738$	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
$741$	0	0
$742$	− 28.0000i	− 1.02791i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	6.00000	0.219676
$747$	12.0000i	0.439057i
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	4.00000i	0.145865i
$753$	0	0
$754$	−32.0000	−1.16537
$755$	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
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	− 36.0000i	− 1.30329i
$764$	14.0000	0.506502
$765$	0	0
$766$	0	0
$767$	− 56.0000i	− 2.02204i
$768$	− 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
$773$	− 46.0000i	− 1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$	− 46.0000i	− 1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−10.0000	−0.358979
$777$	− 8.00000i	− 0.286998i
$778$	− 32.0000i	− 1.14726i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 8.00000i	− 0.285897i
$784$	3.00000	0.107143
$785$	0	0
$786$	18.0000	0.642039
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	16.0000	0.569615
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	− 24.0000i	− 0.852265i
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−24.0000	−0.850657
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	16.0000	0.565332
$802$	− 4.00000i	− 0.141245i
$803$	0	0
$804$	−10.0000	−0.352673
$805$	0	0
$806$	−4.00000	−0.140894
$807$	24.0000i	0.844840i
$808$	− 6.00000i	− 0.211079i
$809$	−4.00000	−0.140633	−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000	−0.140633	−0.0703163	+	0.997525i	$0.522401\pi$
$810$	0	0
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	16.0000i	0.561490i
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	0	0
$818$	30.0000i	1.04893i
$819$	8.00000	0.279543
$820$	0	0
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	− 6.00000i	− 0.209274i
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−28.0000	−0.974245
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	− 4.00000i	− 0.138675i
$833$	− 18.0000i	− 0.623663i
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	− 6.00000i	− 0.207267i
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	6.00000i	0.206774i
$843$	− 30.0000i	− 1.03325i
$844$	−16.0000	−0.550743
$845$	0	0
$846$	−4.00000	−0.137523
$847$	− 22.0000i	− 0.755929i
$848$	− 14.0000i	− 0.480762i
$849$	14.0000	0.480479
$850$	0	0
$851$	0	0
$852$	6.00000i	0.205557i
$853$	18.0000i	0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$	18.0000i	0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$	−12.0000	−0.410632
$855$	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	$0.945366\pi$
$858$	0	0
$859$	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
$862$	6.00000i	0.204361i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−16.0000	−0.543702
$867$	19.0000i	0.645274i
$868$	2.00000i	0.0678844i
$869$	0	0
$870$	0	0
$871$	−40.0000	−1.35535
$872$	− 18.0000i	− 0.609557i
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	24.0000i	0.809961i
$879$	6.00000	0.202375
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	3.00000i	0.101015i
$883$	− 12.0000i	− 0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	− 12.0000i	− 0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−36.0000	−1.20944
$887$	56.0000i	1.88030i	0.340766	+	0.940148i	$0.389313\pi$
$887$	56.0000i	1.88030i	−0.340766	+	0.940148i	$0.610687\pi$
$888$	− 4.00000i	− 0.134231i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	20.0000i	0.669650i
$893$	0	0
$894$	18.0000	0.602010
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	− 20.0000i	− 0.667409i
$899$	8.00000	0.266815
$900$	0	0
$901$	−84.0000	−2.79845
$902$	0	0
$903$	16.0000i	0.532447i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	8.00000	0.265782
$907$	− 2.00000i	− 0.0664089i	−0.999449	−	0.0332045i	$-0.989429\pi$
$907$	− 2.00000i	− 0.0664089i	0.999449	−	0.0332045i	$-0.0105712\pi$
$908$	12.0000i	0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	0	0
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 36.0000i	− 1.18882i
$918$	− 6.00000i	− 0.198030i
$919$	60.0000	1.97922	0.989609	−	0.143787i	$-0.0459280\pi$
$919$	60.0000	1.97922	0.989609	+	0.143787i	$0.0459280\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	12.0000i	0.395199i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.00000i	0.197066i
$928$	8.00000i	0.262613i
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	14.0000i	0.458585i
$933$	− 18.0000i	− 0.589294i
$934$	32.0000	1.04707
$935$	0	0
$936$	4.00000	0.130744
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	20.0000i	0.653023i
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	− 18.0000i	− 0.586472i
$943$	0	0
$944$	−14.0000	−0.455661
$945$	0	0
$946$	0	0
$947$	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
$949$	32.0000	1.03876
$950$	0	0
$951$	10.0000	0.324272
$952$	12.0000i	0.388922i
$953$	− 46.0000i	− 1.49009i	−0.667016	−	0.745043i	$-0.732429\pi$
$953$	− 46.0000i	− 1.49009i	0.667016	−	0.745043i	$-0.267571\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	− 42.0000i	− 1.35696i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 16.0000i	− 0.515861i
$963$	8.00000i	0.257796i
$964$	14.0000	0.450910
$965$	0	0
$966$	0	0
$967$	− 52.0000i	− 1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$	− 52.0000i	− 1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	1.00000i	0.0320750i
$973$	− 8.00000i	− 0.256468i
$974$	8.00000	0.256337
$975$	0	0
$976$	−6.00000	−0.192055
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	− 6.00000i	− 0.191859i
$979$	0	0
$980$	0	0
$981$	18.0000	0.574696
$982$	− 36.0000i	− 1.14881i
$983$	− 40.0000i	− 1.27580i	−0.770118	−	0.637901i	$-0.779803\pi$
$983$	− 40.0000i	− 1.27580i	0.770118	−	0.637901i	$-0.220197\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	48.0000	1.52863
$987$	8.00000i	0.254643i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	1.00000i	0.0317500i
$993$	12.0000i	0.380808i
$994$	12.0000	0.380617
$995$	0	0
$996$	12.0000	0.380235
$997$	− 6.00000i	− 0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$	− 6.00000i	− 0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$	20.0000i	0.633089i
$999$	4.00000	0.126554

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +2.00000 q^{21} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -8.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +4.00000 q^{39} +10.0000 q^{41} -2.00000i q^{42} +8.00000i q^{43} +4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -4.00000i q^{52} -14.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +8.00000i q^{58} -14.0000 q^{59} -6.00000 q^{61} +1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +10.0000i q^{67} +6.00000i q^{68} +6.00000 q^{71} -1.00000i q^{72} -8.00000i q^{73} -4.00000 q^{74} -4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +8.00000 q^{86} +8.00000i q^{87} -16.0000 q^{89} -8.00000 q^{91} +1.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} + 8 q^{26} - 16 q^{29} - 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 20 q^{41} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 4 q^{56}+ \cdots - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 + 4 * q^21 + 2 * q^24 + 8 * q^26 - 16 * q^29 - 2 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 20 * q^41 + 6 * q^49 - 12 * q^51 + 2 * q^54 - 4 * q^56 - 28 * q^59 - 12 * q^61 - 2 * q^64 + 12 * q^71 - 8 * q^74 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 16 * q^86 - 32 * q^89 - 16 * q^91 + 8 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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