LMFDB - Embedded newform 2550.2.d.k.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:	$0$
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 510)
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.k.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -1.00000i q^{48} +7.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} +4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +1.00000i q^{68} -1.00000i q^{72} -6.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} +4.00000i q^{88} +6.00000 q^{89} -8.00000i q^{93} -1.00000 q^{96} +14.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} - 8 q^{19} + 2 q^{24} - 4 q^{26} + 4 q^{29} + 16 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 8 q^{44} + 14 q^{49} - 2 q^{51} + 2 q^{54} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} - 12 q^{74} + 8 q^{76} - 16 q^{79} + 2 q^{81} - 8 q^{86} + 12 q^{89} - 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 - 8 * q^19 + 2 * q^24 - 4 * q^26 + 4 * q^29 + 16 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 8 * q^44 + 14 * q^49 - 2 * q^51 + 2 * q^54 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 - 12 * q^74 + 8 * q^76 - 16 * q^79 + 2 * q^81 - 8 * q^86 + 12 * q^89 - 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	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
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	0	0
$29$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	4.00000i	0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 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
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−1.00000	−0.140028
$52$	2.00000i	0.277350i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	1.00000i	0.121268i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	2.00000i	0.226455i
$79$	−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
$82$	6.00000i	0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	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
$91$	0	0
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	1.00000i	0.0990148i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−10.0000	−0.971286
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	2.00000i	0.184900i
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	6.00000i	0.541002i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	4.00000i	0.348155i
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	− 7.00000i	− 0.577350i
$148$	6.00000i	0.493197i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	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
$153$	1.00000i	0.0808452i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	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
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	− 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	$0.844270\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	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	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	− 4.00000i	− 0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	−8.00000	−0.586588
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	4.00000i	0.284268i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	10.0000i	0.703598i
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	10.0000i	0.686803i
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000i	0.948200i
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−2.00000	−0.134535
$222$	6.00000i	0.402694i
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	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	$-0.130377\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	6.00000	0.382546
$247$	8.00000i	0.509028i
$248$	8.00000i	0.508001i
$249$	−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	$0.376365\pi$
$252$	0	0
$253$	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	$-0.0198685\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	20.0000i	1.23560i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	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
$271$	−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
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	20.0000i	1.19952i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	14.0000	0.820695
$292$	6.00000i	0.351123i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	6.00000	0.348743
$297$	4.00000i	0.232104i
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	10.0000i	0.574485i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	− 2.00000i	− 0.113228i
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	8.00000	0.450035
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	10.0000i	0.560772i
$319$	8.00000	0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	14.0000i	0.774202i
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000i	0.658586i
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.0000i	1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$	30.0000i	1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	32.0000	1.73290
$342$	− 4.00000i	− 0.216295i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	2.00000i	0.107211i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 4.00000i	− 0.213201i
$353$	34.0000i	1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$	34.0000i	1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 22.0000i	− 1.15629i
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	−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
$381$	0	0
$382$	0	0
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	4.00000i	0.203331i
$388$	− 14.0000i	− 0.710742i
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	0	0
$392$	7.00000i	0.353553i
$393$	20.0000i	1.00887i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	4.00000	0.201008
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	4.00000i	0.199502i
$403$	− 16.0000i	− 0.797017i
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	− 24.0000i	− 1.18964i
$408$	− 1.00000i	− 0.0495074i
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	8.00000i	0.394132i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	20.0000i	0.979404i
$418$	16.0000i	0.782586i
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	− 12.0000i	− 0.584151i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	4.00000i	0.193347i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	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	$0.857627\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	6.00000i	0.286691i
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	2.00000i	0.0951303i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	14.0000i	0.658505i
$453$	− 8.00000i	− 0.375873i
$454$	−12.0000	−0.563188
$455$	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	$0.827956\pi$
$458$	6.00000i	0.280362i
$459$	1.00000	0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	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
$469$	0	0
$470$	0	0
$471$	18.0000	0.829396
$472$	4.00000i	0.184115i
$473$	− 16.0000i	− 0.735681i
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	16.0000i	0.731823i
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	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	$0.649037\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 2.00000i	− 0.0900755i
$494$	8.00000	0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	12.0000i	0.537733i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	2.00000i	0.0875376i
$523$	44.0000i	1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$	44.0000i	1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	− 8.00000i	− 0.348485i
$528$	− 4.00000i	− 0.174078i
$529$	23.0000	1.00000
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	12.0000i	0.519778i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	− 12.0000i	− 0.517838i
$538$	− 18.0000i	− 0.776035i
$539$	28.0000	1.20605
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	16.0000i	0.687259i
$543$	− 22.0000i	− 0.944110i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	10.0000i	0.427179i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	20.0000	0.848189
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	8.00000i	0.338667i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−4.00000	−0.168880
$562$	6.00000i	0.253095i
$563$	− 44.0000i	− 1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$	− 44.0000i	− 1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	0	0
$569$	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	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	8.00000i	0.334497i
$573$	0	0
$574$	0	0
$575$	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	$0.785320\pi$
$578$	1.00000i	0.0415945i
$579$	18.0000	0.748054
$580$	0	0
$581$	0	0
$582$	− 14.0000i	− 0.580319i
$583$	− 40.0000i	− 1.65663i
$584$	6.00000	0.248282
$585$	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	$-0.0796572\pi$
$588$	7.00000i	0.288675i
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 6.00000i	− 0.246598i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	10.0000	0.406222
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 1.00000i	− 0.0404226i
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	8.00000i	0.321807i
$619$	−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
$622$	− 16.0000i	− 0.641542i
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−22.0000	−0.879297
$627$	16.0000i	0.638978i
$628$	− 18.0000i	− 0.718278i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	− 8.00000i	− 0.318223i
$633$	− 12.0000i	− 0.476957i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	10.0000	0.396526
$637$	− 14.0000i	− 0.554700i
$638$	− 8.00000i	− 0.316723i
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000i	0.157867i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	1.00000i	0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 12.0000i	− 0.469956i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−6.00000	−0.234261
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000i	0.155464i
$663$	2.00000i	0.0776736i
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	50.0000i	1.92736i	0.267063	+	0.963679i	$0.413947\pi$
$673$	50.0000i	1.92736i	−0.267063	+	0.963679i	$0.586053\pi$
$674$	30.0000	1.15556
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	14.0000i	0.537667i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	− 32.0000i	− 1.22534i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	6.00000i	0.228914i
$688$	− 4.00000i	− 0.152499i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	2.00000	0.0758098
$697$	6.00000i	0.227266i
$698$	− 2.00000i	− 0.0757011i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	24.0000i	0.905177i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	34.0000	1.27961
$707$	0	0
$708$	4.00000i	0.150329i
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	16.0000i	0.597531i
$718$	24.0000i	0.895672i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	− 18.0000i	− 0.669427i
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	− 2.00000i	− 0.0739221i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	− 6.00000i	− 0.220863i
$739$	−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
$741$	8.00000	0.293887
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	6.00000	0.219676
$747$	12.0000i	0.439057i
$748$	4.00000i	0.146254i
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	4.00000i	0.145287i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−32.0000	−1.15621
$767$	− 8.00000i	− 0.288863i
$768$	− 1.00000i	− 0.0360844i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	− 18.0000i	− 0.647834i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	22.0000i	0.788738i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000i	0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	20.0000	0.713376
$787$	− 12.0000i	− 0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$	− 12.0000i	− 0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$	6.00000i	0.213741i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	− 4.00000i	− 0.142134i
$793$	4.00000i	0.142044i
$794$	18.0000	0.638796
$795$	0	0
$796$	−16.0000	−0.567105
$797$	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
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	30.0000i	1.05934i
$803$	− 24.0000i	− 0.846942i
$804$	4.00000	0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	− 18.0000i	− 0.633630i
$808$	− 10.0000i	− 0.351799i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	16.0000i	0.559769i
$818$	− 38.0000i	− 1.32864i
$819$	0	0
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	10.0000i	0.348790i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	2.00000i	0.0693375i
$833$	− 7.00000i	− 0.242536i
$834$	20.0000	0.692543
$835$	0	0
$836$	16.0000	0.553372
$837$	8.00000i	0.276520i
$838$	− 36.0000i	− 1.24360i
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 6.00000i	− 0.206774i
$843$	6.00000i	0.206651i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	4.00000	0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	8.00000i	0.273115i
$859$	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
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	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
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 14.0000i	− 0.474100i
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	6.00000	0.202721
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	− 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	$0.668641\pi$
$882$	7.00000i	0.235702i
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	12.0000	0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	0	0
$897$	0	0
$898$	18.0000i	0.600668i
$899$	16.0000	0.533630
$900$	0	0
$901$	−10.0000	−0.333148
$902$	24.0000i	0.799113i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	−8.00000	−0.265782
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	12.0000i	0.398234i
$909$	10.0000	0.331679
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	4.00000i	0.132453i
$913$	− 48.0000i	− 1.58857i
$914$	22.0000	0.727695
$915$	0	0
$916$	6.00000	0.198246
$917$	0	0
$918$	− 1.00000i	− 0.0330049i
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	− 2.00000i	− 0.0656532i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	22.0000i	0.720634i
$933$	− 16.0000i	− 0.523816i
$934$	12.0000	0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	− 18.0000i	− 0.586472i
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	20.0000i	0.649913i	0.945729	+	0.324956i	$0.105350\pi$
$947$	20.0000i	0.649913i	−0.945729	+	0.324956i	$0.894650\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	58.0000i	1.87880i	0.342817	+	0.939402i	$0.388619\pi$
$953$	58.0000i	1.87880i	−0.342817	+	0.939402i	$0.611381\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	16.0000	0.517477
$957$	− 8.00000i	− 0.258603i
$958$	− 40.0000i	− 1.29234i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	12.0000i	0.386896i
$963$	4.00000i	0.128898i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	5.00000i	0.160706i
$969$	4.00000	0.128499
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	46.0000i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$	46.0000i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	− 12.0000i	− 0.383718i
$979$	24.0000	0.767043
$980$	0	0
$981$	14.0000	0.446986
$982$	20.0000i	0.638226i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	− 8.00000i	− 0.254000i
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	− 20.0000i	− 0.633089i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.k.2449.1		2
5.2	odd	4		510.2.a.e.1.1	✓	1
5.3	odd	4		2550.2.a.l.1.1		1
5.4	even	2	inner	2550.2.d.k.2449.2		2
15.2	even	4		1530.2.a.b.1.1		1
15.8	even	4		7650.2.a.bx.1.1		1
20.7	even	4		4080.2.a.ba.1.1		1
85.67	odd	4		8670.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.e.1.1	✓	1	5.2	odd	4
1530.2.a.b.1.1		1	15.2	even	4
2550.2.a.l.1.1		1	5.3	odd	4
2550.2.d.k.2449.1		2	1.1	even	1	trivial
2550.2.d.k.2449.2		2	5.4	even	2	inner
4080.2.a.ba.1.1		1	20.7	even	4
7650.2.a.bx.1.1		1	15.8	even	4
8670.2.a.v.1.1		1	85.67	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} -1.00000i q^{48} +7.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} +4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +1.00000i q^{68} -1.00000i q^{72} -6.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} +4.00000i q^{88} +6.00000 q^{89} -8.00000i q^{93} -1.00000 q^{96} +14.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} - 8 q^{19} + 2 q^{24} - 4 q^{26} + 4 q^{29} + 16 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 8 q^{44} + 14 q^{49} - 2 q^{51} + 2 q^{54} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} - 12 q^{74} + 8 q^{76} - 16 q^{79} + 2 q^{81} - 8 q^{86} + 12 q^{89} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 - 8 * q^19 + 2 * q^24 - 4 * q^26 + 4 * q^29 + 16 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 8 * q^44 + 14 * q^49 - 2 * q^51 + 2 * q^54 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 - 12 * q^74 + 8 * q^76 - 16 * q^79 + 2 * q^81 - 8 * q^86 + 12 * q^89 - 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more