LMFDB - Embedded newform 2550.2.d.t.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.t.2449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +6.00000i q^{22} +5.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +5.00000 q^{41} -2.00000i q^{43} -6.00000 q^{44} -5.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -1.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +3.00000 q^{59} +5.00000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} -2.00000i q^{67} -1.00000i q^{68} +5.00000 q^{69} +5.00000 q^{71} +1.00000i q^{72} -3.00000 q^{74} +4.00000 q^{76} +2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +5.00000i q^{82} +1.00000i q^{83} +2.00000 q^{86} -6.00000i q^{88} -14.0000 q^{89} -5.00000i q^{92} +2.00000i q^{93} +1.00000 q^{96} +16.0000i q^{97} +7.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} - 4 q^{31} - 2 q^{34} + 2 q^{36} + 4 q^{39} + 10 q^{41} - 12 q^{44} - 10 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 6 q^{59} + 10 q^{61} - 2 q^{64} + 12 q^{66} + 10 q^{69} + 10 q^{71} - 6 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{86} - 28 q^{89} + 2 q^{96} - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 12 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 - 4 * q^31 - 2 * q^34 + 2 * q^36 + 4 * q^39 + 10 * q^41 - 12 * q^44 - 10 * q^46 + 14 * q^49 + 2 * q^51 - 2 * q^54 + 6 * q^59 + 10 * q^61 - 2 * q^64 + 12 * q^66 + 10 * q^69 + 10 * q^71 - 6 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^86 - 28 * q^89 + 2 * q^96 - 12 * 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
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000i	0.288675i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	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
$22$	6.00000i	1.27920i
$23$	5.00000i	1.04257i	0.853382	+	0.521286i	$0.174548\pi$
$23$	5.00000i	1.04257i	−0.853382	+	0.521286i	$0.825452\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000i	0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	− 4.00000i	− 0.648886i
$39$	2.00000	0.320256
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	−5.00000	−0.737210
$47$	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
$53$	− 1.00000i	− 0.137361i	−0.997639	−	0.0686803i	$-0.978121\pi$
$53$	− 1.00000i	− 0.137361i	0.997639	−	0.0686803i	$-0.0218788\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	− 2.00000i	− 0.254000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	− 1.00000i	− 0.121268i
$69$	5.00000	0.601929
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	1.00000i	0.117851i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	2.00000i	0.226455i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.00000i	0.552158i
$83$	1.00000i	0.109764i	0.998493	+	0.0548821i	$0.0174783\pi$
$83$	1.00000i	0.109764i	−0.998493	+	0.0548821i	$0.982522\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	− 5.00000i	− 0.521286i
$93$	2.00000i	0.207390i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	7.00000i	0.707107i
$99$	−6.00000	−0.603023
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	1.00000i	0.0990148i
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	1.00000	0.0971286
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	13.0000i	1.22294i	0.791269	+	0.611469i	$0.209421\pi$
$113$	13.0000i	1.22294i	−0.791269	+	0.611469i	$0.790579\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	3.00000i	0.276172i
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	5.00000i	0.452679i
$123$	− 5.00000i	− 0.450835i
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	6.00000i	0.522233i
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 20.0000i	− 1.70872i	−0.519685	−	0.854358i	$-0.673951\pi$
$137$	− 20.0000i	− 1.70872i	0.519685	−	0.854358i	$-0.326049\pi$
$138$	5.00000i	0.425628i
$139$	−1.00000	−0.0848189	−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000	−0.0848189	−0.0424094	+	0.999100i	$0.513503\pi$
$140$	0	0
$141$	0	0
$142$	5.00000i	0.419591i
$143$	12.0000i	1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	0	0
$147$	− 7.00000i	− 0.577350i
$148$	− 3.00000i	− 0.246598i
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	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
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 9.00000i	− 0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$	− 9.00000i	− 0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	2.00000i	0.152499i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	− 3.00000i	− 0.225494i
$178$	− 14.0000i	− 1.04934i
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	0	0
$183$	− 5.00000i	− 0.369611i
$184$	5.00000	0.368605
$185$	0	0
$186$	−2.00000	−0.146647
$187$	6.00000i	0.438763i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	1.00000i	0.0721688i
$193$	12.0000i	0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$	12.0000i	0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	−16.0000	−1.14873
$195$	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	$-0.0685649\pi$
$198$	− 6.00000i	− 0.426401i
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	10.0000i	0.703598i
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−13.0000	−0.905753
$207$	− 5.00000i	− 0.347524i
$208$	2.00000i	0.138675i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	1.00000i	0.0686803i
$213$	− 5.00000i	− 0.342594i
$214$	−16.0000	−1.09374
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000i	0.677285i
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	3.00000i	0.201347i
$223$	− 1.00000i	− 0.0669650i	−0.999439	−	0.0334825i	$-0.989340\pi$
$223$	− 1.00000i	− 0.0669650i	0.999439	−	0.0334825i	$-0.0106598\pi$
$224$	0	0
$225$	0	0
$226$	−13.0000	−0.864747
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	− 4.00000i	− 0.264906i
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.00000i	− 0.0655122i	−0.999463	−	0.0327561i	$-0.989572\pi$
$233$	− 1.00000i	− 0.0655122i	0.999463	−	0.0327561i	$-0.0104285\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−3.00000	−0.195283
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	25.0000i	1.60706i
$243$	− 1.00000i	− 0.0641500i
$244$	−5.00000	−0.320092
$245$	0	0
$246$	5.00000	0.318788
$247$	− 8.00000i	− 0.509028i
$248$	2.00000i	0.127000i
$249$	1.00000	0.0633724
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	30.0000i	1.88608i
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 28.0000i	− 1.74659i	−0.487190	−	0.873296i	$-0.661978\pi$
$257$	− 28.0000i	− 1.74659i	0.487190	−	0.873296i	$-0.338022\pi$
$258$	− 2.00000i	− 0.124515i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	14.0000i	0.856786i
$268$	2.00000i	0.122169i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	1.00000i	0.0606339i
$273$	0	0
$274$	20.0000	1.20824
$275$	0	0
$276$	−5.00000	−0.300965
$277$	− 25.0000i	− 1.50210i	−0.660243	−	0.751052i	$-0.729547\pi$
$277$	− 25.0000i	− 1.50210i	0.660243	−	0.751052i	$-0.270453\pi$
$278$	− 1.00000i	− 0.0599760i
$279$	2.00000	0.119737
$280$	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
$283$	− 17.0000i	− 1.01055i	−0.862960	−	0.505273i	$-0.831392\pi$
$283$	− 17.0000i	− 1.01055i	0.862960	−	0.505273i	$-0.168608\pi$
$284$	−5.00000	−0.296695
$285$	0	0
$286$	−12.0000	−0.709575
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.0000	0.937937
$292$	0	0
$293$	21.0000i	1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$	21.0000i	1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	3.00000	0.174371
$297$	6.00000i	0.348155i
$298$	5.00000i	0.289642i
$299$	−10.0000	−0.578315
$300$	0	0
$301$	0	0
$302$	9.00000i	0.517892i
$303$	− 10.0000i	− 0.574485i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	13.0000	0.739544
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	− 2.00000i	− 0.113228i
$313$	20.0000i	1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$	20.0000i	1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	− 1.00000i	− 0.0560772i
$319$	0	0
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	9.00000	0.498464
$327$	− 10.0000i	− 0.553001i
$328$	− 5.00000i	− 0.276079i
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	− 1.00000i	− 0.0548821i
$333$	− 3.00000i	− 0.164399i
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	6.00000i	0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$	6.00000i	0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	9.00000i	0.489535i
$339$	13.0000	0.706063
$340$	0	0
$341$	−12.0000	−0.649836
$342$	4.00000i	0.216295i
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−24.0000	−1.29025
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	6.00000i	0.319801i
$353$	− 20.0000i	− 1.06449i	−0.846590	−	0.532246i	$-0.821348\pi$
$353$	− 20.0000i	− 1.06449i	0.846590	−	0.532246i	$-0.178652\pi$
$354$	3.00000	0.159448
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	− 3.00000i	− 0.158555i
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	19.0000i	0.998618i
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	5.00000	0.261354
$367$	14.0000i	0.730794i	0.930852	+	0.365397i	$0.119067\pi$
$367$	14.0000i	0.730794i	−0.930852	+	0.365397i	$0.880933\pi$
$368$	5.00000i	0.260643i
$369$	−5.00000	−0.260290
$370$	0	0
$371$	0	0
$372$	− 2.00000i	− 0.103695i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−31.0000	−1.59236	−0.796182	−	0.605058i	$-0.793150\pi$
$379$	−31.0000	−1.59236	−0.796182	+	0.605058i	$0.793150\pi$
$380$	0	0
$381$	0	0
$382$	− 6.00000i	− 0.306987i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−12.0000	−0.610784
$387$	2.00000i	0.101666i
$388$	− 16.0000i	− 0.812277i
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	− 7.00000i	− 0.353553i
$393$	12.0000i	0.605320i
$394$	6.00000	0.302276
$395$	0	0
$396$	6.00000	0.301511
$397$	− 21.0000i	− 1.05396i	−0.849878	−	0.526980i	$-0.823324\pi$
$397$	− 21.0000i	− 1.05396i	0.849878	−	0.526980i	$-0.176676\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	0	0
$401$	−39.0000	−1.94757	−0.973784	−	0.227477i	$-0.926952\pi$
$401$	−39.0000	−1.94757	−0.973784	+	0.227477i	$0.926952\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	− 4.00000i	− 0.199254i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	0	0
$407$	18.0000i	0.892227i
$408$	− 1.00000i	− 0.0495074i
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	− 13.0000i	− 0.640464i
$413$	0	0
$414$	5.00000	0.245737
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	1.00000i	0.0489702i
$418$	− 24.0000i	− 1.17388i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	0	0
$426$	5.00000	0.242251
$427$	0	0
$428$	− 16.0000i	− 0.773389i
$429$	12.0000	0.579365
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	1.00000i	0.0481125i
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 20.0000i	− 0.956730i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 2.00000i	− 0.0951303i
$443$	− 11.0000i	− 0.522626i	−0.965254	−	0.261313i	$-0.915845\pi$
$443$	− 11.0000i	− 0.522626i	0.965254	−	0.261313i	$-0.0841554\pi$
$444$	−3.00000	−0.142374
$445$	0	0
$446$	1.00000	0.0473514
$447$	− 5.00000i	− 0.236492i
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	30.0000	1.41264
$452$	− 13.0000i	− 0.611469i
$453$	− 9.00000i	− 0.422857i
$454$	6.00000	0.281594
$455$	0	0
$456$	4.00000	0.187317
$457$	25.0000i	1.16945i	0.811231	+	0.584725i	$0.198798\pi$
$457$	25.0000i	1.16945i	−0.811231	+	0.584725i	$0.801202\pi$
$458$	6.00000i	0.280362i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−35.0000	−1.63011	−0.815056	−	0.579382i	$-0.803294\pi$
$461$	−35.0000	−1.63011	−0.815056	+	0.579382i	$0.803294\pi$
$462$	0	0
$463$	− 27.0000i	− 1.25480i	−0.778699	−	0.627398i	$-0.784120\pi$
$463$	− 27.0000i	− 1.25480i	0.778699	−	0.627398i	$-0.215880\pi$
$464$	0	0
$465$	0	0
$466$	1.00000	0.0463241
$467$	33.0000i	1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$	33.0000i	1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$	2.00000i	0.0924500i
$469$	0	0
$470$	0	0
$471$	18.0000	0.829396
$472$	− 3.00000i	− 0.138086i
$473$	− 12.0000i	− 0.551761i
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	1.00000i	0.0457869i
$478$	30.0000i	1.37217i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	− 5.00000i	− 0.226339i
$489$	−9.00000	−0.406994
$490$	0	0
$491$	−43.0000	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$491$	−43.0000	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$492$	5.00000i	0.225417i
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	1.00000i	0.0448111i
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	− 12.0000i	− 0.535586i
$503$	− 5.00000i	− 0.222939i	−0.993768	−	0.111469i	$-0.964444\pi$
$503$	− 5.00000i	− 0.222939i	0.993768	−	0.111469i	$-0.0355557\pi$
$504$	0	0
$505$	0	0
$506$	−30.0000	−1.33366
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	28.0000	1.23503
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	− 26.0000i	− 1.13690i	−0.822718	−	0.568450i	$-0.807543\pi$
$523$	− 26.0000i	− 1.13690i	0.822718	−	0.568450i	$-0.192457\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−8.00000	−0.348817
$527$	− 2.00000i	− 0.0871214i
$528$	− 6.00000i	− 0.261116i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	−3.00000	−0.130189
$532$	0	0
$533$	10.0000i	0.433148i
$534$	−14.0000	−0.605839
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	3.00000i	0.129460i
$538$	− 26.0000i	− 1.12094i
$539$	42.0000	1.80907
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	11.0000i	0.472490i
$543$	− 19.0000i	− 0.815368i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	13.0000i	0.555840i	0.960604	+	0.277920i	$0.0896450\pi$
$547$	13.0000i	0.555840i	−0.960604	+	0.277920i	$0.910355\pi$
$548$	20.0000i	0.854358i
$549$	−5.00000	−0.213395
$550$	0	0
$551$	0	0
$552$	− 5.00000i	− 0.212814i
$553$	0	0
$554$	25.0000	1.06215
$555$	0	0
$556$	1.00000	0.0424094
$557$	11.0000i	0.466085i	0.972467	+	0.233042i	$0.0748681\pi$
$557$	11.0000i	0.466085i	−0.972467	+	0.233042i	$0.925132\pi$
$558$	2.00000i	0.0846668i
$559$	4.00000	0.169182
$560$	0	0
$561$	6.00000	0.253320
$562$	− 6.00000i	− 0.253095i
$563$	− 21.0000i	− 0.885044i	−0.896758	−	0.442522i	$-0.854084\pi$
$563$	− 21.0000i	− 0.885044i	0.896758	−	0.442522i	$-0.145916\pi$
$564$	0	0
$565$	0	0
$566$	17.0000	0.714563
$567$	0	0
$568$	− 5.00000i	− 0.209795i
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	− 12.0000i	− 0.501745i
$573$	6.00000i	0.250654i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	12.0000	0.498703
$580$	0	0
$581$	0	0
$582$	16.0000i	0.663221i
$583$	− 6.00000i	− 0.248495i
$584$	0	0
$585$	0	0
$586$	−21.0000	−0.867502
$587$	− 45.0000i	− 1.85735i	−0.370896	−	0.928674i	$-0.620949\pi$
$587$	− 45.0000i	− 1.85735i	0.370896	−	0.928674i	$-0.379051\pi$
$588$	7.00000i	0.288675i
$589$	8.00000	0.329634
$590$	0	0
$591$	−6.00000	−0.246807
$592$	3.00000i	0.123299i
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−5.00000	−0.204808
$597$	− 10.0000i	− 0.409273i
$598$	− 10.0000i	− 0.408930i
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	−9.00000	−0.366205
$605$	0	0
$606$	10.0000	0.406222
$607$	− 34.0000i	− 1.38002i	−0.723801	−	0.690009i	$-0.757607\pi$
$607$	− 34.0000i	− 1.38002i	0.723801	−	0.690009i	$-0.242393\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.00000i	0.0404226i
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	2.00000	0.0807134
$615$	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	$-0.0385381\pi$
$618$	13.0000i	0.522937i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	15.0000i	0.601445i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	−20.0000	−0.799361
$627$	24.0000i	0.958468i
$628$	− 18.0000i	− 0.718278i
$629$	−3.00000	−0.119618
$630$	0	0
$631$	41.0000	1.63218	0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000	1.63218	0.816092	+	0.577922i	$0.196136\pi$
$632$	− 8.00000i	− 0.318223i
$633$	− 12.0000i	− 0.476957i
$634$	14.0000	0.556011
$635$	0	0
$636$	1.00000	0.0396526
$637$	14.0000i	0.554700i
$638$	0	0
$639$	−5.00000	−0.197797
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	16.0000i	0.631470i
$643$	− 37.0000i	− 1.45914i	−0.683907	−	0.729569i	$-0.739721\pi$
$643$	− 37.0000i	− 1.45914i	0.683907	−	0.729569i	$-0.260279\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	− 10.0000i	− 0.393141i	−0.980490	−	0.196570i	$-0.937020\pi$
$647$	− 10.0000i	− 0.393141i	0.980490	−	0.196570i	$-0.0629804\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	18.0000	0.706562
$650$	0	0
$651$	0	0
$652$	9.00000i	0.352467i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	5.00000	0.195217
$657$	0	0
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	10.0000i	0.388661i
$663$	2.00000i	0.0776736i
$664$	1.00000	0.0388075
$665$	0	0
$666$	3.00000	0.116248
$667$	0	0
$668$	24.0000i	0.928588i
$669$	−1.00000	−0.0386622
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	16.0000i	0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$	16.0000i	0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	−9.00000	−0.346154
$677$	36.0000i	1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$	36.0000i	1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$	13.0000i	0.499262i
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	− 12.0000i	− 0.459504i
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	− 6.00000i	− 0.228914i
$688$	− 2.00000i	− 0.0762493i
$689$	2.00000	0.0761939
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	− 24.0000i	− 0.912343i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	5.00000i	0.189389i
$698$	− 14.0000i	− 0.529908i
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 12.0000i	− 0.452589i
$704$	−6.00000	−0.226134
$705$	0	0
$706$	20.0000	0.752710
$707$	0	0
$708$	3.00000i	0.112747i
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	14.0000i	0.524672i
$713$	− 10.0000i	− 0.374503i
$714$	0	0
$715$	0	0
$716$	3.00000	0.112115
$717$	− 30.0000i	− 1.12037i
$718$	4.00000i	0.149279i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 3.00000i	− 0.111648i
$723$	18.0000i	0.669427i
$724$	−19.0000	−0.706129
$725$	0	0
$726$	25.0000	0.927837
$727$	52.0000i	1.92857i	0.264861	+	0.964287i	$0.414674\pi$
$727$	52.0000i	1.92857i	−0.264861	+	0.964287i	$0.585326\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	5.00000i	0.184805i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	−5.00000	−0.184302
$737$	− 12.0000i	− 0.442026i
$738$	− 5.00000i	− 0.184053i
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	− 19.0000i	− 0.697042i	−0.937301	−	0.348521i	$-0.886684\pi$
$743$	− 19.0000i	− 0.697042i	0.937301	−	0.348521i	$-0.113316\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−4.00000	−0.146450
$747$	− 1.00000i	− 0.0365881i
$748$	− 6.00000i	− 0.219382i
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 40.0000i	− 1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$	− 40.0000i	− 1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$	− 31.0000i	− 1.12597i
$759$	30.0000	1.08893
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	0	0
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	4.00000	0.144526
$767$	6.00000i	0.216647i
$768$	− 1.00000i	− 0.0360844i
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	− 12.0000i	− 0.431889i
$773$	− 3.00000i	− 0.107903i	−0.998544	−	0.0539513i	$-0.982818\pi$
$773$	− 3.00000i	− 0.107903i	0.998544	−	0.0539513i	$-0.0171816\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	16.0000	0.574367
$777$	0	0
$778$	− 19.0000i	− 0.681183i
$779$	−20.0000	−0.716574
$780$	0	0
$781$	30.0000	1.07348
$782$	− 5.00000i	− 0.178800i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	− 21.0000i	− 0.748569i	−0.927314	−	0.374285i	$-0.877888\pi$
$787$	− 21.0000i	− 0.748569i	0.927314	−	0.374285i	$-0.122112\pi$
$788$	6.00000i	0.213741i
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	6.00000i	0.213201i
$793$	10.0000i	0.355110i
$794$	21.0000	0.745262
$795$	0	0
$796$	−10.0000	−0.354441
$797$	19.0000i	0.673015i	0.941681	+	0.336507i	$0.109246\pi$
$797$	19.0000i	0.673015i	−0.941681	+	0.336507i	$0.890754\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	14.0000	0.494666
$802$	− 39.0000i	− 1.37714i
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	4.00000	0.140894
$807$	26.0000i	0.915243i
$808$	− 10.0000i	− 0.351799i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	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
$813$	− 11.0000i	− 0.385787i
$814$	−18.0000	−0.630900
$815$	0	0
$816$	1.00000	0.0350070
$817$	8.00000i	0.279885i
$818$	− 11.0000i	− 0.384606i
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	− 20.0000i	− 0.697580i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	0	0
$827$	10.0000i	0.347734i	0.984769	+	0.173867i	$0.0556263\pi$
$827$	10.0000i	0.347734i	−0.984769	+	0.173867i	$0.944374\pi$
$828$	5.00000i	0.173762i
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	−25.0000	−0.867240
$832$	− 2.00000i	− 0.0693375i
$833$	7.00000i	0.242536i
$834$	−1.00000	−0.0346272
$835$	0	0
$836$	24.0000	0.830057
$837$	− 2.00000i	− 0.0691301i
$838$	− 4.00000i	− 0.138178i
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 28.0000i	− 0.964944i
$843$	6.00000i	0.206651i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 1.00000i	− 0.0343401i
$849$	−17.0000	−0.583438
$850$	0	0
$851$	−15.0000	−0.514193
$852$	5.00000i	0.171297i
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	0	0
$856$	16.0000	0.546869
$857$	− 11.0000i	− 0.375753i	−0.982193	−	0.187876i	$-0.939840\pi$
$857$	− 11.0000i	− 0.375753i	0.982193	−	0.187876i	$-0.0601604\pi$
$858$	12.0000i	0.409673i
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	0	0
$862$	− 28.0000i	− 0.953684i
$863$	− 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
$866$	6.00000	0.203888
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	4.00000	0.135535
$872$	− 10.0000i	− 0.338643i
$873$	− 16.0000i	− 0.541518i
$874$	20.0000	0.676510
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	4.00000i	0.134993i
$879$	21.0000	0.708312
$880$	0	0
$881$	−53.0000	−1.78562	−0.892808	−	0.450438i	$-0.851268\pi$
$881$	−53.0000	−1.78562	−0.892808	+	0.450438i	$0.851268\pi$
$882$	− 7.00000i	− 0.235702i
$883$	− 6.00000i	− 0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$	− 6.00000i	− 0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	11.0000	0.369552
$887$	− 3.00000i	− 0.100730i	−0.998731	−	0.0503651i	$-0.983962\pi$
$887$	− 3.00000i	− 0.100730i	0.998731	−	0.0503651i	$-0.0160385\pi$
$888$	− 3.00000i	− 0.100673i
$889$	0	0
$890$	0	0
$891$	6.00000	0.201008
$892$	1.00000i	0.0334825i
$893$	0	0
$894$	5.00000	0.167225
$895$	0	0
$896$	0	0
$897$	10.0000i	0.333890i
$898$	− 2.00000i	− 0.0667409i
$899$	0	0
$900$	0	0
$901$	1.00000	0.0333148
$902$	30.0000i	0.998891i
$903$	0	0
$904$	13.0000	0.432374
$905$	0	0
$906$	9.00000	0.299005
$907$	− 17.0000i	− 0.564476i	−0.959344	−	0.282238i	$-0.908923\pi$
$907$	− 17.0000i	− 0.564476i	0.959344	−	0.282238i	$-0.0910767\pi$
$908$	6.00000i	0.199117i
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	4.00000i	0.132453i
$913$	6.00000i	0.198571i
$914$	−25.0000	−0.826927
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	− 1.00000i	− 0.0330049i
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	− 35.0000i	− 1.15266i
$923$	10.0000i	0.329154i
$924$	0	0
$925$	0	0
$926$	27.0000	0.887275
$927$	− 13.0000i	− 0.426976i
$928$	0	0
$929$	13.0000	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$929$	13.0000	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	1.00000i	0.0327561i
$933$	− 15.0000i	− 0.491078i
$934$	−33.0000	−1.07979
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	23.0000i	0.751377i	0.926746	+	0.375689i	$0.122594\pi$
$937$	23.0000i	0.751377i	−0.926746	+	0.375689i	$0.877406\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	18.0000i	0.586472i
$943$	25.0000i	0.814112i
$944$	3.00000	0.0976417
$945$	0	0
$946$	12.0000	0.390154
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	8.00000i	0.259828i
$949$	0	0
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	− 14.0000i	− 0.453504i	−0.973952	−	0.226752i	$-0.927189\pi$
$953$	− 14.0000i	− 0.453504i	0.973952	−	0.226752i	$-0.0728108\pi$
$954$	−1.00000	−0.0323762
$955$	0	0
$956$	−30.0000	−0.970269
$957$	0	0
$958$	16.0000i	0.516937i
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 6.00000i	− 0.193448i
$963$	− 16.0000i	− 0.515593i
$964$	18.0000	0.579741
$965$	0	0
$966$	0	0
$967$	57.0000i	1.83300i	0.400039	+	0.916498i	$0.368997\pi$
$967$	57.0000i	1.83300i	−0.400039	+	0.916498i	$0.631003\pi$
$968$	− 25.0000i	− 0.803530i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	26.0000	0.833094
$975$	0	0
$976$	5.00000	0.160046
$977$	34.0000i	1.08776i	0.839164	+	0.543878i	$0.183045\pi$
$977$	34.0000i	1.08776i	−0.839164	+	0.543878i	$0.816955\pi$
$978$	− 9.00000i	− 0.287788i
$979$	−84.0000	−2.68465
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 43.0000i	− 1.37219i
$983$	− 8.00000i	− 0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$	− 8.00000i	− 0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$	−5.00000	−0.159394
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000i	0.254514i
$989$	10.0000	0.317982
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	− 10.0000i	− 0.317340i
$994$	0	0
$995$	0	0
$996$	−1.00000	−0.0316862
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	41.0000i	1.29783i
$999$	−3.00000	−0.0949158

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.e.1.1	✓	1	5.2	odd	4
2550.2.a.bd.1.1	yes	1	5.3	odd	4
2550.2.d.t.2449.1		2	5.4	even	2	inner
2550.2.d.t.2449.2		2	1.1	even	1	trivial
7650.2.a.p.1.1		1	15.8	even	4
7650.2.a.bv.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +6.00000i q^{22} +5.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +5.00000 q^{41} -2.00000i q^{43} -6.00000 q^{44} -5.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -1.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +3.00000 q^{59} +5.00000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} -2.00000i q^{67} -1.00000i q^{68} +5.00000 q^{69} +5.00000 q^{71} +1.00000i q^{72} -3.00000 q^{74} +4.00000 q^{76} +2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +5.00000i q^{82} +1.00000i q^{83} +2.00000 q^{86} -6.00000i q^{88} -14.0000 q^{89} -5.00000i q^{92} +2.00000i q^{93} +1.00000 q^{96} +16.0000i q^{97} +7.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 12 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} - 4 q^{31} - 2 q^{34} + 2 q^{36} + 4 q^{39} + 10 q^{41} - 12 q^{44} - 10 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 6 q^{59} + 10 q^{61} - 2 q^{64} + 12 q^{66} + 10 q^{69} + 10 q^{71} - 6 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{86} - 28 q^{89} + 2 q^{96} - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 12 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 - 4 * q^31 - 2 * q^34 + 2 * q^36 + 4 * q^39 + 10 * q^41 - 12 * q^44 - 10 * q^46 + 14 * q^49 + 2 * q^51 - 2 * q^54 + 6 * q^59 + 10 * q^61 - 2 * q^64 + 12 * q^66 + 10 * q^69 + 10 * q^71 - 6 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^86 - 28 * q^89 + 2 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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