LMFDB - Embedded newform 4650.2.d.c.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.c.3349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} -4.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} -8.00000 q^{59} +10.0000 q^{61} -1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} +1.00000i q^{72} +14.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +16.0000i q^{77} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +6.00000 q^{89} +8.00000 q^{91} -4.00000i q^{92} -1.00000i q^{93} +8.00000 q^{94} -1.00000 q^{96} +6.00000i q^{97} -9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} + 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} - 4 q^{39} + 20 q^{41} + 8 q^{44} - 8 q^{46} - 18 q^{49}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 8 * q^14 + 2 * q^16 - 8 * q^19 + 8 * q^21 + 2 * q^24 - 4 * q^26 + 4 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 - 4 * q^39 + 20 * q^41 + 8 * q^44 - 8 * q^46 - 18 * q^49 + 4 * q^51 + 2 * q^54 - 8 * q^56 - 16 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 - 12 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^84 + 8 * q^86 + 12 * q^89 + 16 * q^91 + 16 * 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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 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
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	− 4.00000i	− 0.852803i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	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
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	4.00000i	0.617213i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	− 2.00000i	− 0.277350i
$53$	− 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
$56$	−4.00000	−0.534522
$57$	− 4.00000i	− 0.529813i
$58$	2.00000i	0.262613i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	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
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	2.00000i	0.242536i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	16.0000i	1.82337i
$78$	− 2.00000i	− 0.226455i
$79$	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	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	2.00000i	0.214423i
$88$	4.00000i	0.426401i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	− 4.00000i	− 0.417029i
$93$	− 1.00000i	− 0.103695i
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 9.00000i	− 0.909137i
$99$	4.00000	0.402015
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	2.00000i	0.198030i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	− 4.00000i	− 0.377964i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	− 2.00000i	− 0.184900i
$118$	− 8.00000i	− 0.736460i
$119$	−8.00000	−0.733359
$120$	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
$126$	−4.00000	−0.356348
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	4.00000i	0.348155i
$133$	16.0000i	1.38738i
$134$	−12.0000	−1.03664
$135$	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	$0.917494\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 9.00000i	− 0.742307i
$148$	− 6.00000i	− 0.493197i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	4.00000i	0.324443i
$153$	2.00000i	0.161690i
$154$	−16.0000	−1.28932
$155$	0	0
$156$	2.00000	0.160128
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	8.00000i	0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	16.0000	1.26098
$162$	1.00000i	0.0785674i
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−4.00000	−0.310460
$167$	20.0000i	1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$	20.0000i	1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$	− 4.00000i	− 0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000i	0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 8.00000i	− 0.601317i
$178$	6.00000i	0.449719i
$179$	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
$181$	−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
$183$	10.0000i	0.739221i
$184$	4.00000	0.294884
$185$	0	0
$186$	1.00000	0.0733236
$187$	8.00000i	0.585018i
$188$	8.00000i	0.583460i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	9.00000	0.642857
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	4.00000i	0.284268i
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	18.0000i	1.26648i
$203$	− 8.00000i	− 0.561490i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−4.00000	−0.278693
$207$	− 4.00000i	− 0.278019i
$208$	2.00000i	0.138675i
$209$	16.0000	1.10674
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	4.00000i	0.271538i
$218$	− 6.00000i	− 0.406371i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	4.00000	0.269069
$222$	− 6.00000i	− 0.402694i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	4.00000i	0.264906i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	− 2.00000i	− 0.131306i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	8.00000	0.520756
$237$	8.00000i	0.519656i
$238$	− 8.00000i	− 0.518563i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000i	0.321412i
$243$	1.00000i	0.0641500i
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−10.0000	−0.637577
$247$	− 8.00000i	− 0.509028i
$248$	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	$0.844936\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 16.0000i	− 1.00591i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	4.00000i	0.249029i
$259$	24.0000	1.49129
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 8.00000i	− 0.494242i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	−16.0000	−0.981023
$267$	6.00000i	0.367194i
$268$	− 12.0000i	− 0.733017i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	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
$276$	4.00000	0.240772
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 16.0000i	− 0.959616i
$279$	1.00000	0.0598684
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	8.00000i	0.476393i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	− 40.0000i	− 2.36113i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−6.00000	−0.351726
$292$	− 14.0000i	− 0.819288i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	6.00000	0.348743
$297$	4.00000i	0.232104i
$298$	6.00000i	0.347571i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−16.0000	−0.922225
$302$	8.00000i	0.460348i
$303$	18.0000i	1.03407i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−2.00000	−0.114332
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	− 16.0000i	− 0.911685i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	2.00000i	0.113228i
$313$	− 34.0000i	− 1.92179i	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	− 34.0000i	− 1.92179i	0.276907	−	0.960897i	$-0.410691\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	6.00000i	0.336463i
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	16.0000i	0.891645i
$323$	8.00000i	0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	− 6.00000i	− 0.331801i
$328$	− 10.0000i	− 0.552158i
$329$	−32.0000	−1.76422
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	−20.0000	−1.09435
$335$	0	0
$336$	4.00000	0.218218
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	9.00000i	0.489535i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	4.00000	0.216612
$342$	4.00000i	0.216295i
$343$	8.00000i	0.431959i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	− 2.00000i	− 0.107211i
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 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	$0.914257\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 8.00000i	− 0.423405i
$358$	20.0000i	1.05703i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 6.00000i	− 0.315353i
$363$	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	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	4.00000i	0.208514i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−24.0000	−1.24602
$372$	1.00000i	0.0518476i
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−8.00000	−0.412568
$377$	4.00000i	0.206010i
$378$	− 4.00000i	− 0.205738i
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	24.0000i	1.22795i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	4.00000i	0.203331i
$388$	− 6.00000i	− 0.304604i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	9.00000i	0.454569i
$393$	− 8.00000i	− 0.403547i
$394$	−14.0000	−0.705310
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	8.00000i	0.401004i
$399$	−16.0000	−0.801002
$400$	0	0
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	− 12.0000i	− 0.598506i
$403$	− 2.00000i	− 0.0996271i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	− 24.0000i	− 1.18964i
$408$	− 2.00000i	− 0.0990148i
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 4.00000i	− 0.197066i
$413$	32.0000i	1.57462i
$414$	4.00000	0.196589
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 16.0000i	− 0.783523i
$418$	16.0000i	0.782586i
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 12.0000i	− 0.584151i
$423$	8.00000i	0.388973i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	− 4.00000i	− 0.193347i
$429$	8.00000	0.386244
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	6.00000	0.287348
$437$	− 16.0000i	− 0.765384i
$438$	− 14.0000i	− 0.668946i
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	4.00000i	0.190261i
$443$	28.0000i	1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$	28.0000i	1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−16.0000	−0.757622
$447$	6.00000i	0.283790i
$448$	4.00000i	0.188982i
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	− 6.00000i	− 0.282216i
$453$	8.00000i	0.375873i
$454$	4.00000	0.187729
$455$	0	0
$456$	−4.00000	−0.187317
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	14.0000i	0.654177i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	− 16.0000i	− 0.744387i
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	2.00000i	0.0924500i
$469$	48.0000	2.21643
$470$	0	0
$471$	−22.0000	−1.01371
$472$	8.00000i	0.368230i
$473$	16.0000i	0.735681i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	6.00000i	0.274721i
$478$	− 8.00000i	− 0.365911i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 14.0000i	− 0.637683i
$483$	16.0000i	0.728025i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 24.0000i	− 1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$	− 24.0000i	− 1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 4.00000i	− 0.180151i
$494$	8.00000	0.359937
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	− 4.00000i	− 0.179244i
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	− 28.0000i	− 1.24970i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	16.0000	0.711287
$507$	9.00000i	0.399704i
$508$	− 8.00000i	− 0.354943i
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	1.00000i	0.0441942i
$513$	4.00000i	0.176604i
$514$	6.00000	0.264649
$515$	0	0
$516$	−4.00000	−0.176090
$517$	32.0000i	1.40736i
$518$	24.0000i	1.05450i
$519$	14.0000	0.614532
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	12.0000	0.523225
$527$	2.00000i	0.0871214i
$528$	− 4.00000i	− 0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	8.00000	0.347170
$532$	− 16.0000i	− 0.693688i
$533$	20.0000i	0.866296i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	12.0000	0.518321
$537$	20.0000i	0.863064i
$538$	18.0000i	0.776035i
$539$	36.0000	1.55063
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	8.00000i	0.343629i
$543$	− 6.00000i	− 0.257485i
$544$	2.00000	0.0857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	4.00000i	0.170251i
$553$	− 32.0000i	− 1.36078i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	16.0000	0.678551
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	1.00000i	0.0423334i
$559$	8.00000	0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	− 30.0000i	− 1.26547i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−20.0000	−0.840663
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	8.00000i	0.334497i
$573$	24.0000i	1.00261i
$574$	40.0000	1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	13.0000i	0.540729i
$579$	14.0000	0.581820
$580$	0	0
$581$	16.0000	0.663792
$582$	− 6.00000i	− 0.248708i
$583$	24.0000i	0.993978i
$584$	14.0000	0.579324
$585$	0	0
$586$	6.00000	0.247858
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	9.00000i	0.371154i
$589$	4.00000	0.164817
$590$	0	0
$591$	−14.0000	−0.575883
$592$	6.00000i	0.246598i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	8.00000i	0.327418i
$598$	− 8.00000i	− 0.327144i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	− 16.0000i	− 0.652111i
$603$	− 12.0000i	− 0.488678i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−18.0000	−0.731200
$607$	− 28.0000i	− 1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$	− 28.0000i	− 1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$	− 4.00000i	− 0.162221i
$609$	8.00000	0.324176
$610$	0	0
$611$	16.0000	0.647291
$612$	− 2.00000i	− 0.0808452i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	16.0000	0.644658
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 4.00000i	− 0.160904i
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	24.0000i	0.962312i
$623$	− 24.0000i	− 0.961540i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	34.0000	1.35891
$627$	16.0000i	0.638978i
$628$	− 22.0000i	− 0.877896i
$629$	12.0000	0.478471
$630$	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
$633$	− 12.0000i	− 0.476957i
$634$	2.00000	0.0794301
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 18.0000i	− 0.713186i
$638$	− 8.00000i	− 0.316723i
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−8.00000	−0.314756
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	32.0000	1.25611
$650$	0	0
$651$	−4.00000	−0.156772
$652$	− 12.0000i	− 0.469956i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	10.0000	0.390434
$657$	− 14.0000i	− 0.546192i
$658$	− 32.0000i	− 1.24749i
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	24.0000i	0.932786i
$663$	4.00000i	0.155347i
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	8.00000i	0.309761i
$668$	− 20.0000i	− 0.773823i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−40.0000	−1.54418
$672$	4.00000i	0.154303i
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	− 6.00000i	− 0.230429i
$679$	24.0000	0.921035
$680$	0	0
$681$	4.00000	0.153280
$682$	4.00000i	0.153168i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	14.0000i	0.534133i
$688$	− 4.00000i	− 0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	14.0000i	0.532200i
$693$	− 16.0000i	− 0.607790i
$694$	20.0000	0.759190
$695$	0	0
$696$	2.00000	0.0758098
$697$	− 20.0000i	− 0.757554i
$698$	− 30.0000i	− 1.13552i
$699$	10.0000	0.378235
$700$	0	0
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	2.00000i	0.0754851i
$703$	− 24.0000i	− 0.905177i
$704$	4.00000	0.150756
$705$	0	0
$706$	−10.0000	−0.376355
$707$	− 72.0000i	− 2.70784i
$708$	8.00000i	0.300658i
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 6.00000i	− 0.224860i
$713$	− 4.00000i	− 0.149801i
$714$	8.00000	0.299392
$715$	0	0
$716$	−20.0000	−0.747435
$717$	− 8.00000i	− 0.298765i
$718$	− 16.0000i	− 0.597115i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 3.00000i	− 0.111648i
$723$	− 14.0000i	− 0.520666i
$724$	6.00000	0.222988
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	− 10.0000i	− 0.369611i
$733$	18.0000i	0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$	18.0000i	0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 48.0000i	− 1.76810i
$738$	− 10.0000i	− 0.368105i
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	− 24.0000i	− 0.881068i
$743$	− 4.00000i	− 0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$	− 4.00000i	− 0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	14.0000	0.512576
$747$	− 4.00000i	− 0.146352i
$748$	− 8.00000i	− 0.292509i
$749$	16.0000	0.584627
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 8.00000i	− 0.291730i
$753$	− 28.0000i	− 1.02038i
$754$	−4.00000	−0.145671
$755$	0	0
$756$	4.00000	0.145479
$757$	54.0000i	1.96266i	0.192323	+	0.981332i	$0.438398\pi$
$757$	54.0000i	1.96266i	−0.192323	+	0.981332i	$0.561602\pi$
$758$	− 4.00000i	− 0.145287i
$759$	16.0000	0.580763
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	− 8.00000i	− 0.289809i
$763$	24.0000i	0.868858i
$764$	−24.0000	−0.868290
$765$	0	0
$766$	4.00000	0.144526
$767$	− 16.0000i	− 0.577727i
$768$	1.00000i	0.0360844i
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	14.0000i	0.503871i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	6.00000	0.215387
$777$	24.0000i	0.860995i
$778$	10.0000i	0.358517i
$779$	−40.0000	−1.43315
$780$	0	0
$781$	0	0
$782$	8.00000i	0.286079i
$783$	− 2.00000i	− 0.0714742i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	8.00000	0.285351
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	− 14.0000i	− 0.498729i
$789$	12.0000	0.427211
$790$	0	0
$791$	24.0000	0.853342
$792$	− 4.00000i	− 0.142134i
$793$	20.0000i	0.710221i
$794$	2.00000	0.0709773
$795$	0	0
$796$	−8.00000	−0.283552
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	− 16.0000i	− 0.566394i
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	34.0000i	1.20058i
$803$	− 56.0000i	− 1.97620i
$804$	12.0000	0.423207
$805$	0	0
$806$	2.00000	0.0704470
$807$	18.0000i	0.633630i
$808$	− 18.0000i	− 0.633238i
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	8.00000i	0.280745i
$813$	8.00000i	0.280572i
$814$	24.0000	0.841200
$815$	0	0
$816$	2.00000	0.0700140
$817$	16.0000i	0.559769i
$818$	− 18.0000i	− 0.629355i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	− 6.00000i	− 0.209274i
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−32.0000	−1.11342
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	4.00000i	0.139010i
$829$	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
$831$	−22.0000	−0.763172
$832$	− 2.00000i	− 0.0693375i
$833$	18.0000i	0.623663i
$834$	16.0000	0.554035
$835$	0	0
$836$	−16.0000	−0.553372
$837$	1.00000i	0.0345651i
$838$	− 8.00000i	− 0.276355i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	14.0000i	0.482472i
$843$	− 30.0000i	− 1.03325i
$844$	12.0000	0.413057
$845$	0	0
$846$	−8.00000	−0.275046
$847$	− 20.0000i	− 0.687208i
$848$	− 6.00000i	− 0.206041i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	4.00000	0.136717
$857$	− 22.0000i	− 0.751506i	−0.926720	−	0.375753i	$-0.877384\pi$
$857$	− 22.0000i	− 0.751506i	0.926720	−	0.375753i	$-0.122616\pi$
$858$	8.00000i	0.273115i
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	40.0000	1.36320
$862$	− 24.0000i	− 0.817443i
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	26.0000	0.883516
$867$	13.0000i	0.441503i
$868$	− 4.00000i	− 0.135769i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−24.0000	−0.813209
$872$	6.00000i	0.203186i
$873$	− 6.00000i	− 0.203069i
$874$	16.0000	0.541208
$875$	0	0
$876$	14.0000	0.473016
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	− 32.0000i	− 1.07995i
$879$	6.00000	0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000i	0.303046i
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−28.0000	−0.940678
$887$	− 40.0000i	− 1.34307i	−0.740973	−	0.671534i	$-0.765636\pi$
$887$	− 40.0000i	− 1.34307i	0.740973	−	0.671534i	$-0.234364\pi$
$888$	6.00000i	0.201347i
$889$	32.0000	1.07325
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 16.0000i	− 0.535720i
$893$	32.0000i	1.07084i
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−4.00000	−0.133631
$897$	− 8.00000i	− 0.267112i
$898$	14.0000i	0.467186i
$899$	−2.00000	−0.0667037
$900$	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
$906$	−8.00000	−0.265782
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	4.00000i	0.132745i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 16.0000i	− 0.529523i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−14.0000	−0.462573
$917$	32.0000i	1.05673i
$918$	− 2.00000i	− 0.0660098i
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	38.0000i	1.25146i
$923$	0	0
$924$	16.0000	0.526361
$925$	0	0
$926$	24.0000	0.788689
$927$	− 4.00000i	− 0.131377i
$928$	2.00000i	0.0656532i
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	10.0000i	0.327561i
$933$	24.0000i	0.785725i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	14.0000i	0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$	14.0000i	0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	48.0000i	1.56726i
$939$	34.0000	1.10955
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	− 22.0000i	− 0.716799i
$943$	40.0000i	1.30258i
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−16.0000	−0.520205
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−28.0000	−0.908918
$950$	0	0
$951$	2.00000	0.0648544
$952$	8.00000i	0.259281i
$953$	− 38.0000i	− 1.23094i	−0.788160	−	0.615470i	$-0.788966\pi$
$953$	− 38.0000i	− 1.23094i	0.788160	−	0.615470i	$-0.211034\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	− 8.00000i	− 0.258603i
$958$	16.0000i	0.516937i
$959$	24.0000	0.775000
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 12.0000i	− 0.386896i
$963$	− 4.00000i	− 0.128898i
$964$	14.0000	0.450910
$965$	0	0
$966$	−16.0000	−0.514792
$967$	− 48.0000i	− 1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$	− 48.0000i	− 1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$	− 5.00000i	− 0.160706i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	64.0000i	2.05175i
$974$	24.0000	0.769010
$975$	0	0
$976$	10.0000	0.320092
$977$	10.0000i	0.319928i	0.987123	+	0.159964i	$0.0511379\pi$
$977$	10.0000i	0.319928i	−0.987123	+	0.159964i	$0.948862\pi$
$978$	− 12.0000i	− 0.383718i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	6.00000	0.191565
$982$	− 20.0000i	− 0.638226i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	4.00000	0.127386
$987$	− 32.0000i	− 1.01857i
$988$	8.00000i	0.254514i
$989$	16.0000	0.508770
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	24.0000i	0.761617i
$994$	0	0
$995$	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	$-0.0920045\pi$
$998$	24.0000i	0.759707i
$999$	6.00000	0.189832

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} -4.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} -8.00000 q^{59} +10.0000 q^{61} -1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} +1.00000i q^{72} +14.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +16.0000i q^{77} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +6.00000 q^{89} +8.00000 q^{91} -4.00000i q^{92} -1.00000i q^{93} +8.00000 q^{94} -1.00000 q^{96} +6.00000i q^{97} -9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} + 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} - 4 q^{39} + 20 q^{41} + 8 q^{44} - 8 q^{46} - 18 q^{49}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 8 * q^14 + 2 * q^16 - 8 * q^19 + 8 * q^21 + 2 * q^24 - 4 * q^26 + 4 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 - 4 * q^39 + 20 * q^41 + 8 * q^44 - 8 * q^46 - 18 * q^49 + 4 * q^51 + 2 * q^54 - 8 * q^56 - 16 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 - 12 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^84 + 8 * q^86 + 12 * q^89 + 16 * q^91 + 16 * q^94 - 2 * q^96 + 8 * q^99

Introduction

L-functions

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