LMFDB - Embedded newform 2550.2.d.q.2449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
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 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.q.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +4.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} -10.0000 q^{41} -2.00000i q^{42} +4.00000i q^{43} -6.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} -4.00000i q^{58} -12.0000 q^{59} -4.00000 q^{61} +6.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} +1.00000i q^{68} +6.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -4.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +4.00000i q^{87} +2.00000 q^{89} +12.0000 q^{91} +6.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{14} + 2 q^{16} - 8 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} + 8 q^{29} - 12 q^{31} - 2 q^{34} + 2 q^{36} - 12 q^{39} - 20 q^{41} - 12 q^{46} + 6 q^{49} + 2 q^{51} - 2 q^{54} + 4 q^{56} - 24 q^{59} - 8 q^{61} - 2 q^{64} + 12 q^{69} - 12 q^{71} - 8 q^{74} + 8 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 4 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 + 8 * q^29 - 12 * q^31 - 2 * q^34 + 2 * q^36 - 12 * q^39 - 20 * q^41 - 12 * q^46 + 6 * q^49 + 2 * q^51 - 2 * q^54 + 4 * q^56 - 24 * q^59 - 8 * q^61 - 2 * q^64 + 12 * q^69 - 12 * q^71 - 8 * q^74 + 8 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 4 * q^89 + 24 * q^91 + 8 * q^94 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	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
$21$	2.00000	0.436436
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	− 1.00000i	− 0.192450i
$28$	2.00000i	0.377964i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	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	$-0.106644\pi$
$38$	4.00000i	0.648886i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	− 2.00000i	− 0.308607i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	−6.00000	−0.884652
$47$	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
$51$	1.00000	0.140028
$52$	− 6.00000i	− 0.832050i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	− 4.00000i	− 0.529813i
$58$	− 4.00000i	− 0.525226i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	6.00000i	0.762001i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	1.00000i	0.121268i
$69$	6.00000	0.722315
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	6.00000i	0.679366i
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000i	1.10432i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	4.00000i	0.428845i
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	6.00000i	0.625543i
$93$	− 6.00000i	− 0.622171i
$94$	4.00000	0.412568
$95$	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	$0.901476\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.00000i	0.0962250i
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	− 2.00000i	− 0.188982i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−4.00000	−0.371391
$117$	− 6.00000i	− 0.554700i
$118$	12.0000i	1.10469i
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	4.00000i	0.362143i
$123$	− 10.0000i	− 0.901670i
$124$	6.00000	0.538816
$125$	0	0
$126$	2.00000	0.178174
$127$	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
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−12.0000	−1.03664
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 6.00000i	− 0.510754i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	6.00000i	0.503509i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	3.00000i	0.247436i
$148$	4.00000i	0.328798i
$149$	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
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	− 4.00000i	− 0.324443i
$153$	1.00000i	0.0808452i
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	10.0000i	0.795557i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−12.0000	−0.945732
$162$	− 1.00000i	− 0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	12.0000	0.931381
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	2.00000i	0.154303i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	− 4.00000i	− 0.304997i
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	− 2.00000i	− 0.149906i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	− 12.0000i	− 0.889499i
$183$	− 4.00000i	− 0.295689i
$184$	6.00000	0.442326
$185$	0	0
$186$	−6.00000	−0.439941
$187$	0	0
$188$	− 4.00000i	− 0.291730i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−3.00000	−0.214286
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	− 14.0000i	− 0.985037i
$203$	− 8.00000i	− 0.561490i
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−4.00000	−0.278693
$207$	6.00000i	0.417029i
$208$	6.00000i	0.416025i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	− 2.00000i	− 0.137361i
$213$	− 6.00000i	− 0.411113i
$214$	0	0
$215$	0	0
$216$	1.00000	0.0680414
$217$	12.0000i	0.814613i
$218$	16.0000i	1.08366i
$219$	2.00000	0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	− 4.00000i	− 0.268462i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−2.00000	−0.133038
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	4.00000i	0.264906i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	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	$0.937031\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	− 10.0000i	− 0.649570i
$238$	2.00000i	0.129641i
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	11.0000i	0.707107i
$243$	1.00000i	0.0641500i
$244$	4.00000	0.256074
$245$	0	0
$246$	−10.0000	−0.637577
$247$	− 24.0000i	− 1.52708i
$248$	− 6.00000i	− 0.381000i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	4.00000i	0.249029i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−4.00000	−0.247594
$262$	16.0000i	0.988483i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	2.00000i	0.122398i
$268$	12.0000i	0.733017i
$269$	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
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	12.0000i	0.726273i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	8.00000i	0.479808i
$279$	6.00000	0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	4.00000i	0.238197i
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	20.0000i	1.18056i
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	2.00000i	0.117041i
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	36.0000	2.08193
$300$	0	0
$301$	8.00000	0.461112
$302$	24.0000i	1.38104i
$303$	14.0000i	0.804279i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	− 6.00000i	− 0.339683i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	10.0000	0.562544
$317$	− 16.0000i	− 0.898650i	−0.893368	−	0.449325i	$-0.851665\pi$
$317$	− 16.0000i	− 0.898650i	0.893368	−	0.449325i	$-0.148335\pi$
$318$	2.00000i	0.112154i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	12.0000i	0.668734i
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	− 16.0000i	− 0.884802i
$328$	− 10.0000i	− 0.552158i
$329$	8.00000	0.441054
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 12.0000i	− 0.658586i
$333$	4.00000i	0.219199i
$334$	−2.00000	−0.109435
$335$	0	0
$336$	2.00000	0.109109
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	23.0000i	1.25104i
$339$	2.00000	0.108625
$340$	0	0
$341$	0	0
$342$	− 4.00000i	− 0.216295i
$343$	− 20.0000i	− 1.07990i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	4.00000	0.215041
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	− 4.00000i	− 0.214423i
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−2.00000	−0.106000
$357$	− 2.00000i	− 0.105851i
$358$	− 12.0000i	− 0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	20.0000i	1.05118i
$363$	− 11.0000i	− 0.577350i
$364$	−12.0000	−0.628971
$365$	0	0
$366$	−4.00000	−0.209083
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	− 6.00000i	− 0.312772i
$369$	10.0000	0.520579
$370$	0	0
$371$	4.00000	0.207670
$372$	6.00000i	0.311086i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	−4.00000	−0.206284
$377$	24.0000i	1.23606i
$378$	2.00000i	0.102869i
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	4.00000i	0.204658i
$383$	− 28.0000i	− 1.43073i	−0.698749	−	0.715367i	$-0.746260\pi$
$383$	− 28.0000i	− 1.43073i	0.698749	−	0.715367i	$-0.253740\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	− 4.00000i	− 0.203331i
$388$	− 6.00000i	− 0.304604i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	3.00000i	0.151523i
$393$	− 16.0000i	− 0.807093i
$394$	8.00000	0.403034
$395$	0	0
$396$	0	0
$397$	20.0000i	1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$	20.0000i	1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	14.0000i	0.701757i
$399$	−8.00000	−0.400501
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	− 12.0000i	− 0.598506i
$403$	− 36.0000i	− 1.79329i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	0	0
$408$	1.00000i	0.0495074i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	4.00000i	0.197066i
$413$	24.0000i	1.18096i
$414$	6.00000	0.294884
$415$	0	0
$416$	6.00000	0.294174
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	− 8.00000i	− 0.389434i
$423$	− 4.00000i	− 0.194487i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	−6.00000	−0.290701
$427$	8.00000i	0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	18.0000i	0.865025i	0.901628	+	0.432512i	$0.142373\pi$
$433$	18.0000i	0.865025i	−0.901628	+	0.432512i	$0.857627\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	16.0000	0.766261
$437$	24.0000i	1.14808i
$438$	− 2.00000i	− 0.0955637i
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 6.00000i	− 0.285391i
$443$	− 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
$446$	4.00000	0.189405
$447$	6.00000i	0.283790i
$448$	2.00000i	0.0944911i
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	0	0
$452$	2.00000i	0.0940721i
$453$	− 24.0000i	− 1.12762i
$454$	4.00000	0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	6.00000i	0.277350i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	−10.0000	−0.459315
$475$	0	0
$476$	2.00000	0.0916698
$477$	− 2.00000i	− 0.0915737i
$478$	− 20.0000i	− 0.914779i
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	18.0000i	0.819878i
$483$	− 12.0000i	− 0.546019i
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	− 4.00000i	− 0.181071i
$489$	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
$494$	−24.0000	−1.07981
$495$	0	0
$496$	−6.00000	−0.269408
$497$	12.0000i	0.538274i
$498$	12.0000i	0.537733i
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	− 12.0000i	− 0.535586i
$503$	26.0000i	1.15928i	0.814872	+	0.579641i	$0.196807\pi$
$503$	26.0000i	1.15928i	−0.814872	+	0.579641i	$0.803193\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	− 8.00000i	− 0.354943i
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	18.0000	0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	8.00000i	0.351500i
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	4.00000i	0.175075i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	12.0000	0.523225
$527$	6.00000i	0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	12.0000	0.520756
$532$	− 8.00000i	− 0.346844i
$533$	− 60.0000i	− 2.59889i
$534$	2.00000	0.0865485
$535$	0	0
$536$	12.0000	0.518321
$537$	12.0000i	0.517838i
$538$	− 12.0000i	− 0.517357i
$539$	0	0
$540$	0	0
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	16.0000i	0.687259i
$543$	− 20.0000i	− 0.858282i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	12.0000	0.513553
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	6.00000i	0.256307i
$549$	4.00000	0.170716
$550$	0	0
$551$	−16.0000	−0.681623
$552$	6.00000i	0.255377i
$553$	20.0000i	0.850487i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	8.00000	0.339276
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	− 6.00000i	− 0.254000i
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−32.0000	−1.34506
$567$	− 2.00000i	− 0.0839921i
$568$	− 6.00000i	− 0.251754i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	− 4.00000i	− 0.167102i
$574$	20.0000	0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	1.00000i	0.0415945i
$579$	6.00000	0.249351
$580$	0	0
$581$	24.0000	0.995688
$582$	6.00000i	0.248708i
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	− 3.00000i	− 0.123718i
$589$	24.0000	0.988903
$590$	0	0
$591$	−8.00000	−0.329076
$592$	− 4.00000i	− 0.164399i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 14.0000i	− 0.572982i
$598$	− 36.0000i	− 1.47215i
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	− 8.00000i	− 0.326056i
$603$	12.0000i	0.488678i
$604$	24.0000	0.976546
$605$	0	0
$606$	14.0000	0.568711
$607$	− 38.0000i	− 1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$	− 38.0000i	− 1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$	4.00000i	0.162221i
$609$	8.00000	0.324176
$610$	0	0
$611$	−24.0000	−0.970936
$612$	− 1.00000i	− 0.0404226i
$613$	− 30.0000i	− 1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$	− 30.0000i	− 1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 4.00000i	− 0.160904i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	− 30.0000i	− 1.20289i
$623$	− 4.00000i	− 0.160257i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	− 6.00000i	− 0.239426i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	− 10.0000i	− 0.397779i
$633$	8.00000i	0.317971i
$634$	−16.0000	−0.635441
$635$	0	0
$636$	2.00000	0.0793052
$637$	18.0000i	0.713186i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	4.00000	0.157378
$647$	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
$649$	0	0
$650$	0	0
$651$	−12.0000	−0.470317
$652$	12.0000i	0.469956i
$653$	− 20.0000i	− 0.782660i	−0.920250	−	0.391330i	$-0.872015\pi$
$653$	− 20.0000i	− 0.782660i	0.920250	−	0.391330i	$-0.127985\pi$
$654$	−16.0000	−0.625650
$655$	0	0
$656$	−10.0000	−0.390434
$657$	2.00000i	0.0780274i
$658$	− 8.00000i	− 0.311872i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	− 20.0000i	− 0.777322i
$663$	6.00000i	0.233021i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	4.00000	0.154997
$667$	− 24.0000i	− 0.929284i
$668$	2.00000i	0.0773823i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	0	0
$672$	− 2.00000i	− 0.0771517i
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	23.0000	0.884615
$677$	8.00000i	0.307465i	0.988113	+	0.153732i	$0.0491294\pi$
$677$	8.00000i	0.307465i	−0.988113	+	0.153732i	$0.950871\pi$
$678$	− 2.00000i	− 0.0768095i
$679$	12.0000	0.460518
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−20.0000	−0.763604
$687$	2.00000i	0.0763048i
$688$	4.00000i	0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	− 4.00000i	− 0.152057i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−4.00000	−0.151620
$697$	10.0000i	0.378777i
$698$	− 30.0000i	− 1.13552i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	− 6.00000i	− 0.226455i
$703$	16.0000i	0.603451i
$704$	0	0
$705$	0	0
$706$	−14.0000	−0.526897
$707$	− 28.0000i	− 1.05305i
$708$	12.0000i	0.450988i
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	2.00000i	0.0749532i
$713$	36.0000i	1.34821i
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	20.0000i	0.746914i
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000i	0.111648i
$723$	− 18.0000i	− 0.669427i
$724$	20.0000	0.743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	12.0000i	0.444750i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	4.00000i	0.147844i
$733$	− 18.0000i	− 0.664845i	−0.943131	−	0.332423i	$-0.892134\pi$
$733$	− 18.0000i	− 0.664845i	0.943131	−	0.332423i	$-0.107866\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	− 10.0000i	− 0.368105i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	− 4.00000i	− 0.146845i
$743$	− 38.0000i	− 1.39408i	−0.717030	−	0.697042i	$-0.754499\pi$
$743$	− 38.0000i	− 1.39408i	0.717030	−	0.697042i	$-0.245501\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	14.0000	0.512576
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	4.00000i	0.145865i
$753$	12.0000i	0.437304i
$754$	24.0000	0.874028
$755$	0	0
$756$	2.00000	0.0727393
$757$	14.0000i	0.508839i	0.967094	+	0.254419i	$0.0818843\pi$
$757$	14.0000i	0.508839i	−0.967094	+	0.254419i	$0.918116\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	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
$763$	32.0000i	1.15848i
$764$	4.00000	0.144715
$765$	0	0
$766$	−28.0000	−1.01168
$767$	− 72.0000i	− 2.59977i
$768$	1.00000i	0.0360844i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	6.00000i	0.215945i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−6.00000	−0.215387
$777$	− 8.00000i	− 0.286998i
$778$	− 14.0000i	− 0.501924i
$779$	40.0000	1.43315
$780$	0	0
$781$	0	0
$782$	6.00000i	0.214560i
$783$	− 4.00000i	− 0.142948i
$784$	3.00000	0.107143
$785$	0	0
$786$	−16.0000	−0.570701
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 8.00000i	− 0.284988i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	− 24.0000i	− 0.852265i
$794$	20.0000	0.709773
$795$	0	0
$796$	14.0000	0.496217
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	8.00000i	0.283197i
$799$	4.00000	0.141510
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	− 30.0000i	− 1.05934i
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−36.0000	−1.26805
$807$	12.0000i	0.422420i
$808$	14.0000i	0.492518i
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	8.00000i	0.280745i
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	1.00000	0.0350070
$817$	− 16.0000i	− 0.559769i
$818$	− 26.0000i	− 0.909069i
$819$	−12.0000	−0.419314
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 18.0000i	− 0.627441i	−0.949515	−	0.313720i	$-0.898425\pi$
$823$	− 18.0000i	− 0.627441i	0.949515	−	0.313720i	$-0.101575\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	− 6.00000i	− 0.208013i
$833$	− 3.00000i	− 0.103944i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	6.00000i	0.207390i
$838$	− 12.0000i	− 0.414533i
$839$	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
$841$	−13.0000	−0.448276
$842$	− 34.0000i	− 1.17172i
$843$	− 18.0000i	− 0.619953i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−4.00000	−0.137523
$847$	22.0000i	0.755929i
$848$	2.00000i	0.0686803i
$849$	32.0000	1.09824
$850$	0	0
$851$	−24.0000	−0.822709
$852$	6.00000i	0.205557i
$853$	16.0000i	0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$	16.0000i	0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	0	0
$857$	− 10.0000i	− 0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$	− 10.0000i	− 0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	− 14.0000i	− 0.476842i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	18.0000	0.611665
$867$	− 1.00000i	− 0.0339618i
$868$	− 12.0000i	− 0.407307i
$869$	0	0
$870$	0	0
$871$	72.0000	2.43963
$872$	− 16.0000i	− 0.541828i
$873$	− 6.00000i	− 0.203069i
$874$	24.0000	0.811812
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	− 10.0000i	− 0.337484i
$879$	2.00000	0.0674583
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	3.00000i	0.101015i
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−12.0000	−0.403148
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	4.00000i	0.134231i
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	− 16.0000i	− 0.535420i
$894$	6.00000	0.200670
$895$	0	0
$896$	2.00000	0.0668153
$897$	36.0000i	1.20201i
$898$	− 26.0000i	− 0.867631i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	2.00000	0.0666297
$902$	0	0
$903$	8.00000i	0.266223i
$904$	2.00000	0.0665190
$905$	0	0
$906$	−24.0000	−0.797347
$907$	24.0000i	0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$	24.0000i	0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$	− 4.00000i	− 0.132745i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	26.0000	0.861418	0.430709	−	0.902491i	$-0.358263\pi$
$911$	26.0000	0.861418	0.430709	+	0.902491i	$0.358263\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	32.0000i	1.05673i
$918$	1.00000i	0.0330049i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	10.0000i	0.329332i
$923$	− 36.0000i	− 1.18495i
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	4.00000i	0.131377i
$928$	− 4.00000i	− 0.131306i
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	− 6.00000i	− 0.196537i
$933$	30.0000i	0.982156i
$934$	−36.0000	−1.17796
$935$	0	0
$936$	6.00000	0.196116
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	24.0000i	0.783628i
$939$	−26.0000	−0.848478
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	6.00000i	0.195491i
$943$	60.0000i	1.95387i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	10.0000i	0.324785i
$949$	12.0000	0.389536
$950$	0	0
$951$	16.0000	0.518836
$952$	− 2.00000i	− 0.0648204i
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−20.0000	−0.646846
$957$	0	0
$958$	10.0000i	0.323085i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	5.00000	0.161290
$962$	− 24.0000i	− 0.773791i
$963$	0	0
$964$	18.0000	0.579741
$965$	0	0
$966$	−12.0000	−0.386094
$967$	− 44.0000i	− 1.41494i	−0.706741	−	0.707472i	$-0.749835\pi$
$967$	− 44.0000i	− 1.41494i	0.706741	−	0.707472i	$-0.250165\pi$
$968$	− 11.0000i	− 0.353553i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	16.0000i	0.512936i
$974$	−38.0000	−1.21760
$975$	0	0
$976$	−4.00000	−0.128037
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	− 12.0000i	− 0.383718i
$979$	0	0
$980$	0	0
$981$	16.0000	0.510841
$982$	20.0000i	0.638226i
$983$	38.0000i	1.21201i	0.795460	+	0.606006i	$0.207229\pi$
$983$	38.0000i	1.21201i	−0.795460	+	0.606006i	$0.792771\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	−4.00000	−0.127386
$987$	8.00000i	0.254643i
$988$	24.0000i	0.763542i
$989$	24.0000	0.763156
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	6.00000i	0.190500i
$993$	20.0000i	0.634681i
$994$	12.0000	0.380617
$995$	0	0
$996$	12.0000	0.380235
$997$	− 20.0000i	− 0.633406i	−0.948525	−	0.316703i	$-0.897424\pi$
$997$	− 20.0000i	− 0.633406i	0.948525	−	0.316703i	$-0.102576\pi$
$998$	32.0000i	1.01294i
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.q.2449.1		2
5.2	odd	4		2550.2.a.be.1.1		1
5.3	odd	4		102.2.a.a.1.1	✓	1
5.4	even	2	inner	2550.2.d.q.2449.2		2
15.2	even	4		7650.2.a.z.1.1		1
15.8	even	4		306.2.a.d.1.1		1
20.3	even	4		816.2.a.h.1.1		1
35.13	even	4		4998.2.a.x.1.1		1
40.3	even	4		3264.2.a.p.1.1		1
40.13	odd	4		3264.2.a.bf.1.1		1
60.23	odd	4		2448.2.a.t.1.1		1
85.8	odd	8		1734.2.f.g.829.2		4
85.13	odd	4		1734.2.b.d.577.1		2
85.33	odd	4		1734.2.a.h.1.1		1
85.38	odd	4		1734.2.b.d.577.2		2
85.43	odd	8		1734.2.f.g.829.1		4
85.53	odd	8		1734.2.f.g.1483.1		4
85.83	odd	8		1734.2.f.g.1483.2		4
120.53	even	4		9792.2.a.a.1.1		1
120.83	odd	4		9792.2.a.b.1.1		1
255.203	even	4		5202.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.a.1.1	✓	1	5.3	odd	4
306.2.a.d.1.1		1	15.8	even	4
816.2.a.h.1.1		1	20.3	even	4
1734.2.a.h.1.1		1	85.33	odd	4
1734.2.b.d.577.1		2	85.13	odd	4
1734.2.b.d.577.2		2	85.38	odd	4
1734.2.f.g.829.1		4	85.43	odd	8
1734.2.f.g.829.2		4	85.8	odd	8
1734.2.f.g.1483.1		4	85.53	odd	8
1734.2.f.g.1483.2		4	85.83	odd	8
2448.2.a.t.1.1		1	60.23	odd	4
2550.2.a.be.1.1		1	5.2	odd	4
2550.2.d.q.2449.1		2	1.1	even	1	trivial
2550.2.d.q.2449.2		2	5.4	even	2	inner
3264.2.a.p.1.1		1	40.3	even	4
3264.2.a.bf.1.1		1	40.13	odd	4
4998.2.a.x.1.1		1	35.13	even	4
5202.2.a.g.1.1		1	255.203	even	4
7650.2.a.z.1.1		1	15.2	even	4
9792.2.a.a.1.1		1	120.53	even	4
9792.2.a.b.1.1		1	120.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +4.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} -10.0000 q^{41} -2.00000i q^{42} +4.00000i q^{43} -6.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} -4.00000i q^{58} -12.0000 q^{59} -4.00000 q^{61} +6.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} +1.00000i q^{68} +6.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -4.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +4.00000i q^{87} +2.00000 q^{89} +12.0000 q^{91} +6.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 8 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} + 8 q^{29} - 12 q^{31} - 2 q^{34} + 2 q^{36} - 12 q^{39} - 20 q^{41} - 12 q^{46} + 6 q^{49} + 2 q^{51} - 2 q^{54} + 4 q^{56} - 24 q^{59} - 8 q^{61} - 2 q^{64} + 12 q^{69} - 12 q^{71} - 8 q^{74} + 8 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 4 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 + 8 * q^29 - 12 * q^31 - 2 * q^34 + 2 * q^36 - 12 * q^39 - 20 * q^41 - 12 * q^46 + 6 * q^49 + 2 * q^51 - 2 * q^54 + 4 * q^56 - 24 * q^59 - 8 * q^61 - 2 * q^64 + 12 * q^69 - 12 * q^71 - 8 * q^74 + 8 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 4 * q^89 + 24 * q^91 + 8 * q^94 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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