LMFDB - Embedded newform 2550.2.d.e.2449.2

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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} -2.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +9.00000i q^{37} -4.00000i q^{38} -6.00000 q^{39} +11.0000 q^{41} +4.00000i q^{42} -10.0000i q^{43} +2.00000 q^{44} +5.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +1.00000 q^{51} -6.00000i q^{52} +11.0000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} +15.0000 q^{59} -1.00000 q^{61} +10.0000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -14.0000i q^{67} +1.00000i q^{68} +5.00000 q^{69} +11.0000 q^{71} +1.00000i q^{72} -8.00000i q^{73} -9.00000 q^{74} +4.00000 q^{76} +8.00000i q^{77} -6.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +11.0000i q^{82} +5.00000i q^{83} -4.00000 q^{84} +10.0000 q^{86} +2.00000i q^{88} +6.00000 q^{89} +24.0000 q^{91} +5.00000i q^{92} +10.0000i q^{93} -8.00000 q^{94} -1.00000 q^{96} +8.00000i q^{97} -9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} - 12 q^{26} + 20 q^{31} + 2 q^{34} + 2 q^{36} - 12 q^{39} + 22 q^{41} + 4 q^{44} + 10 q^{46} - 18 q^{49} + 2 q^{51} + 2 q^{54} - 8 q^{56} + 30 q^{59} - 2 q^{61} - 2 q^{64} + 4 q^{66} + 10 q^{69} + 22 q^{71} - 18 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{84} + 20 q^{86} + 12 q^{89} + 48 q^{91} - 16 q^{94} - 2 q^{96} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 8 * q^19 + 8 * q^21 + 2 * q^24 - 12 * q^26 + 20 * q^31 + 2 * q^34 + 2 * q^36 - 12 * q^39 + 22 * q^41 + 4 * q^44 + 10 * q^46 - 18 * q^49 + 2 * q^51 + 2 * q^54 - 8 * q^56 + 30 * q^59 - 2 * q^61 - 2 * q^64 + 4 * q^66 + 10 * q^69 + 22 * q^71 - 18 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^84 + 20 * q^86 + 12 * q^89 + 48 * q^91 - 16 * q^94 - 2 * q^96 + 4 * 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
$7$	− 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
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 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	$0.687167\pi$
$14$	4.00000	1.06904
$15$	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
$21$	4.00000	0.872872
$22$	− 2.00000i	− 0.426401i
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000i	0.176777i
$33$	− 2.00000i	− 0.348155i
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	9.00000i	1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$37$	9.00000i	1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	4.00000i	0.617213i
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	5.00000	0.737210
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	1.00000	0.140028
$52$	− 6.00000i	− 0.832050i
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	10.0000i	1.27000i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	1.00000i	0.121268i
$69$	5.00000	0.601929
$70$	0	0
$71$	11.0000	1.30546	0.652730	−	0.757591i	$-0.273624\pi$
$71$	11.0000	1.30546	0.652730	+	0.757591i	$0.273624\pi$
$72$	1.00000i	0.117851i
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	−9.00000	−1.04623
$75$	0	0
$76$	4.00000	0.458831
$77$	8.00000i	0.911685i
$78$	− 6.00000i	− 0.679366i
$79$	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
$82$	11.0000i	1.21475i
$83$	5.00000i	0.548821i	0.961613	+	0.274411i	$0.0884828\pi$
$83$	5.00000i	0.548821i	−0.961613	+	0.274411i	$0.911517\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	2.00000i	0.213201i
$89$	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
$91$	24.0000	2.51588
$92$	5.00000i	0.521286i
$93$	10.0000i	1.03695i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	− 9.00000i	− 0.909137i
$99$	2.00000	0.201008
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	1.00000i	0.0990148i
$103$	− 11.0000i	− 1.08386i	−0.840423	−	0.541931i	$-0.817693\pi$
$103$	− 11.0000i	− 1.08386i	0.840423	−	0.541931i	$-0.182307\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−11.0000	−1.06841
$107$	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
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	− 4.00000i	− 0.377964i
$113$	3.00000i	0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$	3.00000i	0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	15.0000i	1.38086i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 1.00000i	− 0.0905357i
$123$	11.0000i	0.991837i
$124$	−10.0000	−0.898027
$125$	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
$129$	10.0000	0.880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	2.00000i	0.174078i
$133$	16.0000i	1.38738i
$134$	14.0000	1.20942
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	5.00000i	0.425628i
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	11.0000i	0.923099i
$143$	− 12.0000i	− 1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	− 9.00000i	− 0.742307i
$148$	− 9.00000i	− 0.739795i
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	4.00000i	0.324443i
$153$	1.00000i	0.0808452i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	6.00000	0.480384
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	8.00000i	0.636446i
$159$	−11.0000	−0.872357
$160$	0	0
$161$	−20.0000	−1.57622
$162$	1.00000i	0.0785674i
$163$	5.00000i	0.391630i	0.980641	+	0.195815i	$0.0627352\pi$
$163$	5.00000i	0.391630i	−0.980641	+	0.195815i	$0.937265\pi$
$164$	−11.0000	−0.858956
$165$	0	0
$166$	−5.00000	−0.388075
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	− 4.00000i	− 0.308607i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	10.0000i	0.762493i
$173$	− 4.00000i	− 0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	− 4.00000i	− 0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	15.0000i	1.12747i
$178$	6.00000i	0.449719i
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	24.0000i	1.77900i
$183$	− 1.00000i	− 0.0739221i
$184$	−5.00000	−0.368605
$185$	0	0
$186$	−10.0000	−0.733236
$187$	2.00000i	0.146254i
$188$	− 8.00000i	− 0.583460i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	2.00000i	0.142134i
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	18.0000i	1.26648i
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	11.0000	0.766406
$207$	5.00000i	0.347524i
$208$	6.00000i	0.416025i
$209$	8.00000	0.553372
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 11.0000i	− 0.755483i
$213$	11.0000i	0.753708i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 40.0000i	− 2.71538i
$218$	14.0000i	0.948200i
$219$	8.00000	0.540590
$220$	0	0
$221$	6.00000	0.403604
$222$	− 9.00000i	− 0.604040i
$223$	7.00000i	0.468755i	0.972146	+	0.234377i	$0.0753051\pi$
$223$	7.00000i	0.468755i	−0.972146	+	0.234377i	$0.924695\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−3.00000	−0.199557
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	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
$231$	−8.00000	−0.526361
$232$	0	0
$233$	1.00000i	0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$	1.00000i	0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−15.0000	−0.976417
$237$	8.00000i	0.519656i
$238$	− 4.00000i	− 0.259281i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 7.00000i	− 0.449977i
$243$	1.00000i	0.0641500i
$244$	1.00000	0.0640184
$245$	0	0
$246$	−11.0000	−0.701334
$247$	− 24.0000i	− 1.52708i
$248$	− 10.0000i	− 0.635001i
$249$	−5.00000	−0.316862
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	− 4.00000i	− 0.251976i
$253$	10.0000i	0.628695i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	10.0000i	0.622573i
$259$	36.0000	2.23693
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	−16.0000	−0.981023
$267$	6.00000i	0.367194i
$268$	14.0000i	0.855186i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	24.0000i	1.45255i
$274$	12.0000	0.724947
$275$	0	0
$276$	−5.00000	−0.300965
$277$	− 3.00000i	− 0.180253i	−0.995930	−	0.0901263i	$-0.971273\pi$
$277$	− 3.00000i	− 0.180253i	0.995930	−	0.0901263i	$-0.0287271\pi$
$278$	− 11.0000i	− 0.659736i
$279$	−10.0000	−0.598684
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	− 8.00000i	− 0.476393i
$283$	− 3.00000i	− 0.178331i	−0.996017	−	0.0891657i	$-0.971580\pi$
$283$	− 3.00000i	− 0.178331i	0.996017	−	0.0891657i	$-0.0284201\pi$
$284$	−11.0000	−0.652730
$285$	0	0
$286$	12.0000	0.709575
$287$	− 44.0000i	− 2.59724i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	8.00000i	0.468165i
$293$	− 15.0000i	− 0.876309i	−0.898900	−	0.438155i	$-0.855632\pi$
$293$	− 15.0000i	− 0.876309i	0.898900	−	0.438155i	$-0.144368\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	9.00000	0.523114
$297$	2.00000i	0.116052i
$298$	1.00000i	0.0579284i
$299$	30.0000	1.73494
$300$	0	0
$301$	−40.0000	−2.30556
$302$	− 7.00000i	− 0.402805i
$303$	18.0000i	1.03407i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	− 8.00000i	− 0.455842i
$309$	11.0000	0.625768
$310$	0	0
$311$	1.00000	0.0567048	0.0283524	−	0.999598i	$-0.490974\pi$
$311$	1.00000	0.0567048	0.0283524	+	0.999598i	$0.490974\pi$
$312$	6.00000i	0.339683i
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 11.0000i	− 0.616849i
$319$	0	0
$320$	0	0
$321$	−4.00000	−0.223258
$322$	− 20.0000i	− 1.11456i
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−5.00000	−0.276924
$327$	14.0000i	0.774202i
$328$	− 11.0000i	− 0.607373i
$329$	32.0000	1.76422
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	− 5.00000i	− 0.274411i
$333$	− 9.00000i	− 0.493197i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	4.00000	0.218218
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	− 23.0000i	− 1.25104i
$339$	−3.00000	−0.162938
$340$	0	0
$341$	−20.0000	−1.08306
$342$	4.00000i	0.216295i
$343$	8.00000i	0.431959i
$344$	−10.0000	−0.539164
$345$	0	0
$346$	4.00000	0.215041
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	− 2.00000i	− 0.106600i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−15.0000	−0.797241
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 4.00000i	− 0.211702i
$358$	− 23.0000i	− 1.21559i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 7.00000i	− 0.367912i
$363$	− 7.00000i	− 0.367405i
$364$	−24.0000	−1.25794
$365$	0	0
$366$	1.00000	0.0522708
$367$	− 34.0000i	− 1.77479i	−0.461014	−	0.887393i	$-0.652514\pi$
$367$	− 34.0000i	− 1.77479i	0.461014	−	0.887393i	$-0.347486\pi$
$368$	− 5.00000i	− 0.260643i
$369$	−11.0000	−0.572637
$370$	0	0
$371$	44.0000	2.28437
$372$	− 10.0000i	− 0.518476i
$373$	24.0000i	1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$	24.0000i	1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	− 4.00000i	− 0.205738i
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	− 14.0000i	− 0.716302i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−4.00000	−0.203595
$387$	10.0000i	0.508329i
$388$	− 8.00000i	− 0.406138i
$389$	33.0000	1.67317	0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000	1.67317	0.836583	+	0.547840i	$0.184550\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	9.00000i	0.454569i
$393$	12.0000i	0.605320i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 7.00000i	− 0.351320i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$	− 7.00000i	− 0.351320i	0.984451	−	0.175660i	$-0.0562059\pi$
$398$	− 10.0000i	− 0.501255i
$399$	−16.0000	−0.801002
$400$	0	0
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	14.0000i	0.698257i
$403$	60.0000i	2.98881i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	− 18.0000i	− 0.892227i
$408$	− 1.00000i	− 0.0495074i
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	11.0000i	0.541931i
$413$	− 60.0000i	− 2.95241i
$414$	−5.00000	−0.245737
$415$	0	0
$416$	−6.00000	−0.294174
$417$	− 11.0000i	− 0.538672i
$418$	8.00000i	0.391293i
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	4.00000i	0.194717i
$423$	− 8.00000i	− 0.388973i
$424$	11.0000	0.534207
$425$	0	0
$426$	−11.0000	−0.532952
$427$	4.00000i	0.193574i
$428$	− 4.00000i	− 0.193347i
$429$	12.0000	0.579365
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 30.0000i	− 1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$	− 30.0000i	− 1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$	40.0000	1.92006
$435$	0	0
$436$	−14.0000	−0.670478
$437$	20.0000i	0.956730i
$438$	8.00000i	0.382255i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	6.00000i	0.285391i
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	9.00000	0.427121
$445$	0	0
$446$	−7.00000	−0.331460
$447$	1.00000i	0.0472984i
$448$	4.00000i	0.188982i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−22.0000	−1.03594
$452$	− 3.00000i	− 0.141108i
$453$	− 7.00000i	− 0.328889i
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 15.0000i	− 0.701670i	−0.936437	−	0.350835i	$-0.885898\pi$
$457$	− 15.0000i	− 0.701670i	0.936437	−	0.350835i	$-0.114102\pi$
$458$	2.00000i	0.0934539i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	− 8.00000i	− 0.372194i
$463$	− 19.0000i	− 0.883005i	−0.897260	−	0.441502i	$-0.854446\pi$
$463$	− 19.0000i	− 0.883005i	0.897260	−	0.441502i	$-0.145554\pi$
$464$	0	0
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	37.0000i	1.71216i	0.516847	+	0.856078i	$0.327106\pi$
$467$	37.0000i	1.71216i	−0.516847	+	0.856078i	$0.672894\pi$
$468$	6.00000i	0.277350i
$469$	−56.0000	−2.58584
$470$	0	0
$471$	18.0000	0.829396
$472$	− 15.0000i	− 0.690431i
$473$	20.0000i	0.919601i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	4.00000	0.183340
$477$	− 11.0000i	− 0.503655i
$478$	− 6.00000i	− 0.274434i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−54.0000	−2.46219
$482$	− 14.0000i	− 0.637683i
$483$	− 20.0000i	− 0.910032i
$484$	7.00000	0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	6.00000i	0.271886i	0.990717	+	0.135943i	$0.0434064\pi$
$487$	6.00000i	0.271886i	−0.990717	+	0.135943i	$0.956594\pi$
$488$	1.00000i	0.0452679i
$489$	−5.00000	−0.226108
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	− 11.0000i	− 0.495918i
$493$	0	0
$494$	24.0000	1.07981
$495$	0	0
$496$	10.0000	0.449013
$497$	− 44.0000i	− 1.97367i
$498$	− 5.00000i	− 0.224055i
$499$	27.0000	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$499$	27.0000	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	28.0000i	1.24970i
$503$	5.00000i	0.222939i	0.993768	+	0.111469i	$0.0355557\pi$
$503$	5.00000i	0.222939i	−0.993768	+	0.111469i	$0.964444\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	−10.0000	−0.444554
$507$	− 23.0000i	− 1.02147i
$508$	− 8.00000i	− 0.354943i
$509$	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
$511$	−32.0000	−1.41560
$512$	1.00000i	0.0441942i
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	−10.0000	−0.440225
$517$	− 16.0000i	− 0.703679i
$518$	36.0000i	1.58175i
$519$	4.00000	0.175581
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	6.00000i	0.262362i	0.991358	+	0.131181i	$0.0418769\pi$
$523$	6.00000i	0.262362i	−0.991358	+	0.131181i	$0.958123\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	8.00000	0.348817
$527$	− 10.0000i	− 0.435607i
$528$	− 2.00000i	− 0.0870388i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	−15.0000	−0.650945
$532$	− 16.0000i	− 0.693688i
$533$	66.0000i	2.85878i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−14.0000	−0.604708
$537$	− 23.0000i	− 0.992523i
$538$	− 10.0000i	− 0.431131i
$539$	18.0000	0.775315
$540$	0	0
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	19.0000i	0.816120i
$543$	− 7.00000i	− 0.300399i
$544$	1.00000	0.0428746
$545$	0	0
$546$	−24.0000	−1.02711
$547$	15.0000i	0.641354i	0.947189	+	0.320677i	$0.103910\pi$
$547$	15.0000i	0.641354i	−0.947189	+	0.320677i	$0.896090\pi$
$548$	12.0000i	0.512615i
$549$	1.00000	0.0426790
$550$	0	0
$551$	0	0
$552$	− 5.00000i	− 0.212814i
$553$	− 32.0000i	− 1.36078i
$554$	3.00000	0.127458
$555$	0	0
$556$	11.0000	0.466504
$557$	15.0000i	0.635570i	0.948163	+	0.317785i	$0.102939\pi$
$557$	15.0000i	0.635570i	−0.948163	+	0.317785i	$0.897061\pi$
$558$	− 10.0000i	− 0.423334i
$559$	60.0000	2.53773
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	14.0000i	0.590554i
$563$	− 41.0000i	− 1.72794i	−0.503540	−	0.863972i	$-0.667969\pi$
$563$	− 41.0000i	− 1.72794i	0.503540	−	0.863972i	$-0.332031\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	3.00000	0.126099
$567$	− 4.00000i	− 0.167984i
$568$	− 11.0000i	− 0.461550i
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	12.0000i	0.501745i
$573$	− 14.0000i	− 0.584858i
$574$	44.0000	1.83652
$575$	0	0
$576$	1.00000	0.0416667
$577$	31.0000i	1.29055i	0.763952	+	0.645273i	$0.223257\pi$
$577$	31.0000i	1.29055i	−0.763952	+	0.645273i	$0.776743\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−4.00000	−0.166234
$580$	0	0
$581$	20.0000	0.829740
$582$	− 8.00000i	− 0.331611i
$583$	− 22.0000i	− 0.911147i
$584$	−8.00000	−0.331042
$585$	0	0
$586$	15.0000	0.619644
$587$	23.0000i	0.949312i	0.880172	+	0.474656i	$0.157427\pi$
$587$	23.0000i	0.949312i	−0.880172	+	0.474656i	$0.842573\pi$
$588$	9.00000i	0.371154i
$589$	−40.0000	−1.64817
$590$	0	0
$591$	−6.00000	−0.246807
$592$	9.00000i	0.369898i
$593$	16.0000i	0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$	16.0000i	0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−1.00000	−0.0409616
$597$	− 10.0000i	− 0.409273i
$598$	30.0000i	1.22679i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	− 40.0000i	− 1.63028i
$603$	14.0000i	0.570124i
$604$	7.00000	0.284826
$605$	0	0
$606$	−18.0000	−0.731200
$607$	− 10.0000i	− 0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$	− 10.0000i	− 0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	− 1.00000i	− 0.0404226i
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	8.00000	0.322329
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	11.0000i	0.442485i
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	1.00000i	0.0400963i
$623$	− 24.0000i	− 0.961540i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	8.00000	0.319744
$627$	8.00000i	0.319489i
$628$	18.0000i	0.718278i
$629$	9.00000	0.358854
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	− 8.00000i	− 0.318223i
$633$	4.00000i	0.158986i
$634$	18.0000	0.714871
$635$	0	0
$636$	11.0000	0.436178
$637$	− 54.0000i	− 2.13956i
$638$	0	0
$639$	−11.0000	−0.435153
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 23.0000i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$643$	− 23.0000i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$644$	20.0000	0.788110
$645$	0	0
$646$	−4.00000	−0.157378
$647$	30.0000i	1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$	30.0000i	1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−30.0000	−1.17760
$650$	0	0
$651$	40.0000	1.56772
$652$	− 5.00000i	− 0.195815i
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	11.0000	0.429478
$657$	8.00000i	0.312110i
$658$	32.0000i	1.24749i
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	6.00000i	0.233197i
$663$	6.00000i	0.233021i
$664$	5.00000	0.194038
$665$	0	0
$666$	9.00000	0.348743
$667$	0	0
$668$	− 16.0000i	− 0.619059i
$669$	−7.00000	−0.270636
$670$	0	0
$671$	2.00000	0.0772091
$672$	4.00000i	0.154303i
$673$	− 36.0000i	− 1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$	− 36.0000i	− 1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	23.0000	0.884615
$677$	− 48.0000i	− 1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$	− 48.0000i	− 1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$	− 3.00000i	− 0.115214i
$679$	32.0000	1.22805
$680$	0	0
$681$	−18.0000	−0.689761
$682$	− 20.0000i	− 0.765840i
$683$	− 34.0000i	− 1.30097i	−0.759517	−	0.650487i	$-0.774565\pi$
$683$	− 34.0000i	− 1.30097i	0.759517	−	0.650487i	$-0.225435\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	2.00000i	0.0763048i
$688$	− 10.0000i	− 0.381246i
$689$	−66.0000	−2.51440
$690$	0	0
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	4.00000i	0.152057i
$693$	− 8.00000i	− 0.303895i
$694$	−8.00000	−0.303676
$695$	0	0
$696$	0	0
$697$	− 11.0000i	− 0.416655i
$698$	14.0000i	0.529908i
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	6.00000i	0.226455i
$703$	− 36.0000i	− 1.35777i
$704$	2.00000	0.0753778
$705$	0	0
$706$	0	0
$707$	− 72.0000i	− 2.70784i
$708$	− 15.0000i	− 0.563735i
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 6.00000i	− 0.224860i
$713$	− 50.0000i	− 1.87251i
$714$	4.00000	0.149696
$715$	0	0
$716$	23.0000	0.859550
$717$	− 6.00000i	− 0.224074i
$718$	− 8.00000i	− 0.298557i
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−44.0000	−1.63865
$722$	− 3.00000i	− 0.111648i
$723$	− 14.0000i	− 0.520666i
$724$	7.00000	0.260153
$725$	0	0
$726$	7.00000	0.259794
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	− 24.0000i	− 0.889499i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−10.0000	−0.369863
$732$	1.00000i	0.0369611i
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	5.00000	0.184302
$737$	28.0000i	1.03139i
$738$	− 11.0000i	− 0.404916i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	44.0000i	1.61529i
$743$	− 37.0000i	− 1.35740i	−0.734416	−	0.678699i	$-0.762544\pi$
$743$	− 37.0000i	− 1.35740i	0.734416	−	0.678699i	$-0.237456\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	−24.0000	−0.878702
$747$	− 5.00000i	− 0.182940i
$748$	− 2.00000i	− 0.0731272i
$749$	16.0000	0.584627
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	8.00000i	0.291730i
$753$	28.0000i	1.02038i
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	40.0000i	1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$	40.0000i	1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$	− 29.0000i	− 1.05333i
$759$	−10.0000	−0.362977
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	− 8.00000i	− 0.289809i
$763$	− 56.0000i	− 2.02734i
$764$	14.0000	0.506502
$765$	0	0
$766$	16.0000	0.578103
$767$	90.0000i	3.24971i
$768$	1.00000i	0.0360844i
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	0	0
$772$	− 4.00000i	− 0.143963i
$773$	17.0000i	0.611448i	0.952120	+	0.305724i	$0.0988984\pi$
$773$	17.0000i	0.611448i	−0.952120	+	0.305724i	$0.901102\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	8.00000	0.287183
$777$	36.0000i	1.29149i
$778$	33.0000i	1.18311i
$779$	−44.0000	−1.57646
$780$	0	0
$781$	−22.0000	−0.787222
$782$	− 5.00000i	− 0.178800i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−12.0000	−0.428026
$787$	1.00000i	0.0356462i	0.999841	+	0.0178231i	$0.00567356\pi$
$787$	1.00000i	0.0356462i	−0.999841	+	0.0178231i	$0.994326\pi$
$788$	− 6.00000i	− 0.213741i
$789$	8.00000	0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	− 2.00000i	− 0.0710669i
$793$	− 6.00000i	− 0.213066i
$794$	7.00000	0.248421
$795$	0	0
$796$	10.0000	0.354441
$797$	15.0000i	0.531327i	0.964066	+	0.265664i	$0.0855911\pi$
$797$	15.0000i	0.531327i	−0.964066	+	0.265664i	$0.914409\pi$
$798$	− 16.0000i	− 0.566394i
$799$	8.00000	0.283020
$800$	0	0
$801$	−6.00000	−0.212000
$802$	23.0000i	0.812158i
$803$	16.0000i	0.564628i
$804$	−14.0000	−0.493742
$805$	0	0
$806$	−60.0000	−2.11341
$807$	− 10.0000i	− 0.352017i
$808$	− 18.0000i	− 0.633238i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	19.0000i	0.666359i
$814$	18.0000	0.630900
$815$	0	0
$816$	1.00000	0.0350070
$817$	40.0000i	1.39942i
$818$	− 19.0000i	− 0.664319i
$819$	−24.0000	−0.838628
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	12.0000i	0.418548i
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	60.0000	2.08767
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	− 5.00000i	− 0.173762i
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	3.00000	0.104069
$832$	− 6.00000i	− 0.208013i
$833$	9.00000i	0.311832i
$834$	11.0000	0.380899
$835$	0	0
$836$	−8.00000	−0.276686
$837$	− 10.0000i	− 0.345651i
$838$	4.00000i	0.138178i
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	8.00000i	0.275698i
$843$	14.0000i	0.482186i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	28.0000i	0.962091i
$848$	11.0000i	0.377742i
$849$	3.00000	0.102960
$850$	0	0
$851$	45.0000	1.54258
$852$	− 11.0000i	− 0.376854i
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	4.00000	0.136717
$857$	19.0000i	0.649028i	0.945881	+	0.324514i	$0.105201\pi$
$857$	19.0000i	0.649028i	−0.945881	+	0.324514i	$0.894799\pi$
$858$	12.0000i	0.409673i
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	44.0000	1.49952
$862$	− 20.0000i	− 0.681203i
$863$	− 28.0000i	− 0.953131i	−0.879139	−	0.476566i	$-0.841881\pi$
$863$	− 28.0000i	− 0.953131i	0.879139	−	0.476566i	$-0.158119\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	30.0000	1.01944
$867$	− 1.00000i	− 0.0339618i
$868$	40.0000i	1.35769i
$869$	−16.0000	−0.542763
$870$	0	0
$871$	84.0000	2.84623
$872$	− 14.0000i	− 0.474100i
$873$	− 8.00000i	− 0.270759i
$874$	−20.0000	−0.676510
$875$	0	0
$876$	−8.00000	−0.270295
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	28.0000i	0.944954i
$879$	15.0000	0.505937
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	9.00000i	0.303046i
$883$	34.0000i	1.14419i	0.820187	+	0.572096i	$0.193869\pi$
$883$	34.0000i	1.14419i	−0.820187	+	0.572096i	$0.806131\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−9.00000	−0.302361
$887$	27.0000i	0.906571i	0.891365	+	0.453286i	$0.149748\pi$
$887$	27.0000i	0.906571i	−0.891365	+	0.453286i	$0.850252\pi$
$888$	9.00000i	0.302020i
$889$	32.0000	1.07325
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	− 7.00000i	− 0.234377i
$893$	− 32.0000i	− 1.07084i
$894$	−1.00000	−0.0334450
$895$	0	0
$896$	−4.00000	−0.133631
$897$	30.0000i	1.00167i
$898$	− 6.00000i	− 0.200223i
$899$	0	0
$900$	0	0
$901$	11.0000	0.366463
$902$	− 22.0000i	− 0.732520i
$903$	− 40.0000i	− 1.33112i
$904$	3.00000	0.0997785
$905$	0	0
$906$	7.00000	0.232559
$907$	13.0000i	0.431658i	0.976431	+	0.215829i	$0.0692454\pi$
$907$	13.0000i	0.431658i	−0.976431	+	0.215829i	$0.930755\pi$
$908$	− 18.0000i	− 0.597351i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 10.0000i	− 0.330952i
$914$	15.0000	0.496156
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	− 48.0000i	− 1.58510i
$918$	− 1.00000i	− 0.0330049i
$919$	−1.00000	−0.0329870	−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	−1.00000	−0.0329870	−0.0164935	+	0.999864i	$0.505250\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	− 15.0000i	− 0.493999i
$923$	66.0000i	2.17242i
$924$	8.00000	0.263181
$925$	0	0
$926$	19.0000	0.624379
$927$	11.0000i	0.361287i
$928$	0	0
$929$	−37.0000	−1.21393	−0.606965	−	0.794728i	$-0.707613\pi$
$929$	−37.0000	−1.21393	−0.606965	+	0.794728i	$0.707613\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	− 1.00000i	− 0.0327561i
$933$	1.00000i	0.0327385i
$934$	−37.0000	−1.21068
$935$	0	0
$936$	−6.00000	−0.196116
$937$	23.0000i	0.751377i	0.926746	+	0.375689i	$0.122594\pi$
$937$	23.0000i	0.751377i	−0.926746	+	0.375689i	$0.877406\pi$
$938$	− 56.0000i	− 1.82846i
$939$	8.00000	0.261070
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	18.0000i	0.586472i
$943$	− 55.0000i	− 1.79105i
$944$	15.0000	0.488208
$945$	0	0
$946$	−20.0000	−0.650256
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	− 8.00000i	− 0.259828i
$949$	48.0000	1.55815
$950$	0	0
$951$	18.0000	0.583690
$952$	4.00000i	0.129641i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	11.0000	0.356138
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	24.0000i	0.775405i
$959$	−48.0000	−1.55000
$960$	0	0
$961$	69.0000	2.22581
$962$	− 54.0000i	− 1.74103i
$963$	− 4.00000i	− 0.128898i
$964$	14.0000	0.450910
$965$	0	0
$966$	20.0000	0.643489
$967$	− 31.0000i	− 0.996893i	−0.866921	−	0.498446i	$-0.833904\pi$
$967$	− 31.0000i	− 0.996893i	0.866921	−	0.498446i	$-0.166096\pi$
$968$	7.00000i	0.224989i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	44.0000i	1.41058i
$974$	−6.00000	−0.192252
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	− 5.00000i	− 0.159882i
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−14.0000	−0.446986
$982$	33.0000i	1.05307i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	11.0000	0.350667
$985$	0	0
$986$	0	0
$987$	32.0000i	1.01857i
$988$	24.0000i	0.763542i
$989$	−50.0000	−1.58991
$990$	0	0
$991$	−60.0000	−1.90596	−0.952981	−	0.303029i	$-0.902002\pi$
$991$	−60.0000	−1.90596	−0.952981	+	0.303029i	$0.902002\pi$
$992$	10.0000i	0.317500i
$993$	6.00000i	0.190404i
$994$	44.0000	1.39560
$995$	0	0
$996$	5.00000	0.158431
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	27.0000i	0.854670i
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.e.2449.2		2
5.2	odd	4		2550.2.a.p.1.1	✓	1
5.3	odd	4		2550.2.a.r.1.1	yes	1
5.4	even	2	inner	2550.2.d.e.2449.1		2
15.2	even	4		7650.2.a.cm.1.1		1
15.8	even	4		7650.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.p.1.1	✓	1	5.2	odd	4
2550.2.a.r.1.1	yes	1	5.3	odd	4
2550.2.d.e.2449.1		2	5.4	even	2	inner
2550.2.d.e.2449.2		2	1.1	even	1	trivial
7650.2.a.d.1.1		1	15.8	even	4
7650.2.a.cm.1.1		1	15.2	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} -2.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +9.00000i q^{37} -4.00000i q^{38} -6.00000 q^{39} +11.0000 q^{41} +4.00000i q^{42} -10.0000i q^{43} +2.00000 q^{44} +5.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +1.00000 q^{51} -6.00000i q^{52} +11.0000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} +15.0000 q^{59} -1.00000 q^{61} +10.0000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -14.0000i q^{67} +1.00000i q^{68} +5.00000 q^{69} +11.0000 q^{71} +1.00000i q^{72} -8.00000i q^{73} -9.00000 q^{74} +4.00000 q^{76} +8.00000i q^{77} -6.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +11.0000i q^{82} +5.00000i q^{83} -4.00000 q^{84} +10.0000 q^{86} +2.00000i q^{88} +6.00000 q^{89} +24.0000 q^{91} +5.00000i q^{92} +10.0000i q^{93} -8.00000 q^{94} -1.00000 q^{96} +8.00000i q^{97} -9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} - 12 q^{26} + 20 q^{31} + 2 q^{34} + 2 q^{36} - 12 q^{39} + 22 q^{41} + 4 q^{44} + 10 q^{46} - 18 q^{49} + 2 q^{51} + 2 q^{54} - 8 q^{56} + 30 q^{59} - 2 q^{61} - 2 q^{64} + 4 q^{66} + 10 q^{69} + 22 q^{71} - 18 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{84} + 20 q^{86} + 12 q^{89} + 48 q^{91} - 16 q^{94} - 2 q^{96} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 8 * q^19 + 8 * q^21 + 2 * q^24 - 12 * q^26 + 20 * q^31 + 2 * q^34 + 2 * q^36 - 12 * q^39 + 22 * q^41 + 4 * q^44 + 10 * q^46 - 18 * q^49 + 2 * q^51 + 2 * q^54 - 8 * q^56 + 30 * q^59 - 2 * q^61 - 2 * q^64 + 4 * q^66 + 10 * q^69 + 22 * q^71 - 18 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^84 + 20 * q^86 + 12 * q^89 + 48 * q^91 - 16 * q^94 - 2 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more