LMFDB - Embedded newform 950.2.b.a.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.b (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:	$7.58578819202$
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 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.2.b.a.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} -4.00000 q^{11} +3.00000i q^{12} -1.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +6.00000i q^{18} -1.00000 q^{19} +15.0000 q^{21} +4.00000i q^{22} +7.00000i q^{23} +3.00000 q^{24} -1.00000 q^{26} +9.00000i q^{27} -5.00000i q^{28} +3.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} +12.0000i q^{33} +3.00000 q^{34} +6.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -3.00000 q^{39} -6.00000 q^{41} -15.0000i q^{42} +6.00000i q^{43} +4.00000 q^{44} +7.00000 q^{46} -3.00000i q^{48} -18.0000 q^{49} +9.00000 q^{51} +1.00000i q^{52} -13.0000i q^{53} +9.00000 q^{54} -5.00000 q^{56} +3.00000i q^{57} -3.00000i q^{58} +9.00000 q^{59} -12.0000 q^{61} +2.00000i q^{62} -30.0000i q^{63} -1.00000 q^{64} +12.0000 q^{66} +3.00000i q^{67} -3.00000i q^{68} +21.0000 q^{69} -6.00000i q^{72} +11.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} -20.0000i q^{77} +3.00000i q^{78} +2.00000 q^{79} +9.00000 q^{81} +6.00000i q^{82} -10.0000i q^{83} -15.0000 q^{84} +6.00000 q^{86} -9.00000i q^{87} -4.00000i q^{88} -2.00000 q^{89} +5.00000 q^{91} -7.00000i q^{92} +6.00000i q^{93} -3.00000 q^{96} +2.00000i q^{97} +18.0000i q^{98} +24.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9	$ 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 8 q^{11} + 10 q^{14} + 2 q^{16} - 2 q^{19} + 30 q^{21} + 6 q^{24} - 2 q^{26} + 6 q^{29} - 4 q^{31} + 6 q^{34} + 12 q^{36} - 6 q^{39} - 12 q^{41} + 8 q^{44} + 14 q^{46} - 36 q^{49} + 18 q^{51} + 18 q^{54} - 10 q^{56} + 18 q^{59} - 24 q^{61} - 2 q^{64} + 24 q^{66} + 42 q^{69} + 4 q^{74} + 2 q^{76} + 4 q^{79} + 18 q^{81} - 30 q^{84} + 12 q^{86} - 4 q^{89} + 10 q^{91} - 6 q^{96} + 48 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 - 8 * q^11 + 10 * q^14 + 2 * q^16 - 2 * q^19 + 30 * q^21 + 6 * q^24 - 2 * q^26 + 6 * q^29 - 4 * q^31 + 6 * q^34 + 12 * q^36 - 6 * q^39 - 12 * q^41 + 8 * q^44 + 14 * q^46 - 36 * q^49 + 18 * q^51 + 18 * q^54 - 10 * q^56 + 18 * q^59 - 24 * q^61 - 2 * q^64 + 24 * q^66 + 42 * q^69 + 4 * q^74 + 2 * q^76 + 4 * q^79 + 18 * q^81 - 30 * q^84 + 12 * q^86 - 4 * q^89 + 10 * q^91 - 6 * q^96 + 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	5.00000i	1.88982i	0.327327	+	0.944911i	$0.393852\pi$
$7$	5.00000i	1.88982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$	1.00000i	0.353553i
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	3.00000i	0.866025i
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	6.00000i	1.41421i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	15.0000	3.27327
$22$	4.00000i	0.852803i
$23$	7.00000i	1.45960i	0.683660	+	0.729800i	$0.260387\pi$
$23$	7.00000i	1.45960i	−0.683660	+	0.729800i	$0.739613\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−1.00000	−0.196116
$27$	9.00000i	1.73205i
$28$	− 5.00000i	− 0.944911i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	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
$33$	12.0000i	2.08893i
$34$	3.00000	0.514496
$35$	0	0
$36$	6.00000	1.00000
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	1.00000i	0.162221i
$39$	−3.00000	−0.480384
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 15.0000i	− 2.31455i
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	7.00000	1.03209
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 3.00000i	− 0.433013i
$49$	−18.0000	−2.57143
$50$	0	0
$51$	9.00000	1.26025
$52$	1.00000i	0.138675i
$53$	− 13.0000i	− 1.78569i	−0.450367	−	0.892844i	$-0.648707\pi$
$53$	− 13.0000i	− 1.78569i	0.450367	−	0.892844i	$-0.351293\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	−5.00000	−0.668153
$57$	3.00000i	0.397360i
$58$	− 3.00000i	− 0.393919i
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	2.00000i	0.254000i
$63$	− 30.0000i	− 3.77964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	12.0000	1.47710
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	− 3.00000i	− 0.363803i
$69$	21.0000	2.52810
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 6.00000i	− 0.707107i
$73$	11.0000i	1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$	11.0000i	1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	− 20.0000i	− 2.27921i
$78$	3.00000i	0.339683i
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	6.00000i	0.662589i
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	−15.0000	−1.63663
$85$	0	0
$86$	6.00000	0.646997
$87$	− 9.00000i	− 0.964901i
$88$	− 4.00000i	− 0.426401i
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	− 7.00000i	− 0.729800i
$93$	6.00000i	0.622171i
$94$	0	0
$95$	0	0
$96$	−3.00000	−0.306186
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	18.0000i	1.81827i
$99$	24.0000	2.41209
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	− 9.00000i	− 0.891133i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−13.0000	−1.26267
$107$	13.0000i	1.25676i	0.777908	+	0.628379i	$0.216281\pi$
$107$	13.0000i	1.25676i	−0.777908	+	0.628379i	$0.783719\pi$
$108$	− 9.00000i	− 0.866025i
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	5.00000i	0.472456i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	−3.00000	−0.278543
$117$	6.00000i	0.554700i
$118$	− 9.00000i	− 0.828517i
$119$	−15.0000	−1.37505
$120$	0	0
$121$	5.00000	0.454545
$122$	12.0000i	1.08643i
$123$	18.0000i	1.62301i
$124$	2.00000	0.179605
$125$	0	0
$126$	−30.0000	−2.67261
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	1.00000i	0.0883883i
$129$	18.0000	1.58481
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	− 12.0000i	− 1.04447i
$133$	− 5.00000i	− 0.433555i
$134$	3.00000	0.259161
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	− 21.0000i	− 1.78764i
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000i	0.334497i
$144$	−6.00000	−0.500000
$145$	0	0
$146$	11.0000	0.910366
$147$	54.0000i	4.45384i
$148$	− 2.00000i	− 0.164399i
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	− 18.0000i	− 1.45521i
$154$	−20.0000	−1.61165
$155$	0	0
$156$	3.00000	0.240192
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 2.00000i	− 0.159111i
$159$	−39.0000	−3.09290
$160$	0	0
$161$	−35.0000	−2.75839
$162$	− 9.00000i	− 0.707107i
$163$	22.0000i	1.72317i	0.507611	+	0.861586i	$0.330529\pi$
$163$	22.0000i	1.72317i	−0.507611	+	0.861586i	$0.669471\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−10.0000	−0.776151
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	15.0000i	1.15728i
$169$	12.0000	0.923077
$170$	0	0
$171$	6.00000	0.458831
$172$	− 6.00000i	− 0.457496i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 27.0000i	− 2.02944i
$178$	2.00000i	0.149906i
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	− 5.00000i	− 0.370625i
$183$	36.0000i	2.66120i
$184$	−7.00000	−0.516047
$185$	0	0
$186$	6.00000	0.439941
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	−45.0000	−3.27327
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	3.00000i	0.216506i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	18.0000	1.28571
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	− 24.0000i	− 1.70561i
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	8.00000i	0.562878i
$203$	15.0000i	1.05279i
$204$	−9.00000	−0.630126
$205$	0	0
$206$	4.00000	0.278693
$207$	− 42.0000i	− 2.91920i
$208$	− 1.00000i	− 0.0693375i
$209$	4.00000	0.276686
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	13.0000i	0.892844i
$213$	0	0
$214$	13.0000	0.888662
$215$	0	0
$216$	−9.00000	−0.612372
$217$	− 10.0000i	− 0.678844i
$218$	19.0000i	1.28684i
$219$	33.0000	2.22993
$220$	0	0
$221$	3.00000	0.201802
$222$	− 6.00000i	− 0.402694i
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	0	0
$227$	− 5.00000i	− 0.331862i	−0.986137	−	0.165931i	$-0.946937\pi$
$227$	− 5.00000i	− 0.331862i	0.986137	−	0.165931i	$-0.0530628\pi$
$228$	− 3.00000i	− 0.198680i
$229$	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
$231$	−60.0000	−3.94771
$232$	3.00000i	0.196960i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−9.00000	−0.585850
$237$	− 6.00000i	− 0.389742i
$238$	15.0000i	0.972306i
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	12.0000	0.768221
$245$	0	0
$246$	18.0000	1.14764
$247$	1.00000i	0.0636285i
$248$	− 2.00000i	− 0.127000i
$249$	−30.0000	−1.90117
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	30.0000i	1.88982i
$253$	− 28.0000i	− 1.76034i
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	− 18.0000i	− 1.12063i
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−18.0000	−1.11417
$262$	− 16.0000i	− 0.988483i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−5.00000	−0.306570
$267$	6.00000i	0.367194i
$268$	− 3.00000i	− 0.183254i
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−27.0000	−1.64013	−0.820067	−	0.572268i	$-0.806064\pi$
$271$	−27.0000	−1.64013	−0.820067	+	0.572268i	$0.806064\pi$
$272$	3.00000i	0.181902i
$273$	− 15.0000i	− 0.907841i
$274$	−9.00000	−0.543710
$275$	0	0
$276$	−21.0000	−1.26405
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	16.0000i	0.959616i
$279$	12.0000	0.718421
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	− 30.0000i	− 1.77084i
$288$	6.00000i	0.353553i
$289$	8.00000	0.470588
$290$	0	0
$291$	6.00000	0.351726
$292$	− 11.0000i	− 0.643726i
$293$	− 27.0000i	− 1.57736i	−0.614806	−	0.788678i	$-0.710766\pi$
$293$	− 27.0000i	− 1.57736i	0.614806	−	0.788678i	$-0.289234\pi$
$294$	54.0000	3.14934
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 36.0000i	− 2.08893i
$298$	− 4.00000i	− 0.231714i
$299$	7.00000	0.404820
$300$	0	0
$301$	−30.0000	−1.72917
$302$	10.0000i	0.575435i
$303$	24.0000i	1.37876i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−18.0000	−1.02899
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	20.0000i	1.13961i
$309$	12.0000	0.682656
$310$	0	0
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	− 3.00000i	− 0.169842i
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−2.00000	−0.112509
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	39.0000i	2.18701i
$319$	−12.0000	−0.671871
$320$	0	0
$321$	39.0000	2.17677
$322$	35.0000i	1.95047i
$323$	− 3.00000i	− 0.166924i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	22.0000	1.21847
$327$	57.0000i	3.15211i
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	10.0000i	0.548821i
$333$	− 12.0000i	− 0.657596i
$334$	2.00000	0.109435
$335$	0	0
$336$	15.0000	0.818317
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	− 12.0000i	− 0.652714i
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	− 6.00000i	− 0.324443i
$343$	− 55.0000i	− 2.96972i
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−14.0000	−0.752645
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	9.00000i	0.482451i
$349$	−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
$351$	9.00000	0.480384
$352$	4.00000i	0.213201i
$353$	7.00000i	0.372572i	0.982496	+	0.186286i	$0.0596452\pi$
$353$	7.00000i	0.372572i	−0.982496	+	0.186286i	$0.940355\pi$
$354$	−27.0000	−1.43503
$355$	0	0
$356$	2.00000	0.106000
$357$	45.0000i	2.38165i
$358$	− 8.00000i	− 0.422813i
$359$	5.00000	0.263890	0.131945	−	0.991257i	$-0.457878\pi$
$359$	5.00000	0.263890	0.131945	+	0.991257i	$0.457878\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 26.0000i	− 1.36653i
$363$	− 15.0000i	− 0.787296i
$364$	−5.00000	−0.262071
$365$	0	0
$366$	36.0000	1.88175
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	7.00000i	0.364900i
$369$	36.0000	1.87409
$370$	0	0
$371$	65.0000	3.37463
$372$	− 6.00000i	− 0.311086i
$373$	− 23.0000i	− 1.19089i	−0.803394	−	0.595447i	$-0.796975\pi$
$373$	− 23.0000i	− 1.19089i	0.803394	−	0.595447i	$-0.203025\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	0	0
$377$	− 3.00000i	− 0.154508i
$378$	45.0000i	2.31455i
$379$	33.0000	1.69510	0.847548	−	0.530719i	$-0.178078\pi$
$379$	33.0000	1.69510	0.847548	+	0.530719i	$0.178078\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	− 9.00000i	− 0.460480i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	10.0000	0.508987
$387$	− 36.0000i	− 1.82998i
$388$	− 2.00000i	− 0.101535i
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	−21.0000	−1.06202
$392$	− 18.0000i	− 0.909137i
$393$	− 48.0000i	− 2.42128i
$394$	22.0000	1.10834
$395$	0	0
$396$	−24.0000	−1.20605
$397$	16.0000i	0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$	16.0000i	0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$	− 15.0000i	− 0.751882i
$399$	−15.0000	−0.750939
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	− 9.00000i	− 0.448879i
$403$	2.00000i	0.0996271i
$404$	8.00000	0.398015
$405$	0	0
$406$	15.0000	0.744438
$407$	− 8.00000i	− 0.396545i
$408$	9.00000i	0.445566i
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−27.0000	−1.33181
$412$	− 4.00000i	− 0.197066i
$413$	45.0000i	2.21431i
$414$	−42.0000	−2.06419
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	48.0000i	2.35057i
$418$	− 4.00000i	− 0.195646i
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	5.00000i	0.243396i
$423$	0	0
$424$	13.0000	0.631336
$425$	0	0
$426$	0	0
$427$	− 60.0000i	− 2.90360i
$428$	− 13.0000i	− 0.628379i
$429$	12.0000	0.579365
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	9.00000i	0.433013i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	19.0000	0.909935
$437$	− 7.00000i	− 0.334855i
$438$	− 33.0000i	− 1.57680i
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	108.000	5.14286
$442$	− 3.00000i	− 0.142695i
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	− 12.0000i	− 0.567581i
$448$	− 5.00000i	− 0.236228i
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	30.0000i	1.40952i
$454$	−5.00000	−0.234662
$455$	0	0
$456$	−3.00000	−0.140488
$457$	29.0000i	1.35656i	0.734802	+	0.678281i	$0.237275\pi$
$457$	29.0000i	1.35656i	−0.734802	+	0.678281i	$0.762725\pi$
$458$	− 6.00000i	− 0.280362i
$459$	−27.0000	−1.26025
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	60.0000i	2.79145i
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	10.0000	0.463241
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	− 6.00000i	− 0.277350i
$469$	−15.0000	−0.692636
$470$	0	0
$471$	−18.0000	−0.829396
$472$	9.00000i	0.414259i
$473$	− 24.0000i	− 1.10352i
$474$	−6.00000	−0.275589
$475$	0	0
$476$	15.0000	0.687524
$477$	78.0000i	3.57137i
$478$	− 11.0000i	− 0.503128i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	12.0000i	0.546585i
$483$	105.000i	4.77767i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	− 12.0000i	− 0.543214i
$489$	66.0000	2.98462
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	− 18.0000i	− 0.811503i
$493$	9.00000i	0.405340i
$494$	1.00000	0.0449921
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	30.0000i	1.34433i
$499$	−42.0000	−1.88018	−0.940089	−	0.340929i	$-0.889258\pi$
$499$	−42.0000	−1.88018	−0.940089	+	0.340929i	$0.889258\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	12.0000i	0.535586i
$503$	− 21.0000i	− 0.936344i	−0.883637	−	0.468172i	$-0.844913\pi$
$503$	− 21.0000i	− 0.936344i	0.883637	−	0.468172i	$-0.155087\pi$
$504$	30.0000	1.33631
$505$	0	0
$506$	−28.0000	−1.24475
$507$	− 36.0000i	− 1.59882i
$508$	− 6.00000i	− 0.266207i
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−55.0000	−2.43306
$512$	− 1.00000i	− 0.0441942i
$513$	− 9.00000i	− 0.397360i
$514$	−22.0000	−0.970378
$515$	0	0
$516$	−18.0000	−0.792406
$517$	0	0
$518$	10.0000i	0.439375i
$519$	−42.0000	−1.84360
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	18.0000i	0.787839i
$523$	− 9.00000i	− 0.393543i	−0.980449	−	0.196771i	$-0.936954\pi$
$523$	− 9.00000i	− 0.393543i	0.980449	−	0.196771i	$-0.0630456\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	8.00000	0.348817
$527$	− 6.00000i	− 0.261364i
$528$	12.0000i	0.522233i
$529$	−26.0000	−1.13043
$530$	0	0
$531$	−54.0000	−2.34340
$532$	5.00000i	0.216777i
$533$	6.00000i	0.259889i
$534$	6.00000	0.259645
$535$	0	0
$536$	−3.00000	−0.129580
$537$	− 24.0000i	− 1.03568i
$538$	2.00000i	0.0862261i
$539$	72.0000	3.10126
$540$	0	0
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	27.0000i	1.15975i
$543$	− 78.0000i	− 3.34730i
$544$	3.00000	0.128624
$545$	0	0
$546$	−15.0000	−0.641941
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	9.00000i	0.384461i
$549$	72.0000	3.07289
$550$	0	0
$551$	−3.00000	−0.127804
$552$	21.0000i	0.893819i
$553$	10.0000i	0.425243i
$554$	8.00000	0.339887
$555$	0	0
$556$	16.0000	0.678551
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	− 12.0000i	− 0.508001i
$559$	6.00000	0.253773
$560$	0	0
$561$	−36.0000	−1.51992
$562$	18.0000i	0.759284i
$563$	20.0000i	0.842900i	0.906852	+	0.421450i	$0.138479\pi$
$563$	20.0000i	0.842900i	−0.906852	+	0.421450i	$0.861521\pi$
$564$	0	0
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	45.0000i	1.88982i
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	6.00000	0.251092	0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000	0.251092	0.125546	+	0.992088i	$0.459932\pi$
$572$	− 4.00000i	− 0.167248i
$573$	− 27.0000i	− 1.12794i
$574$	−30.0000	−1.25218
$575$	0	0
$576$	6.00000	0.250000
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	− 8.00000i	− 0.332756i
$579$	30.0000	1.24676
$580$	0	0
$581$	50.0000	2.07435
$582$	− 6.00000i	− 0.248708i
$583$	52.0000i	2.15362i
$584$	−11.0000	−0.455183
$585$	0	0
$586$	−27.0000	−1.11536
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	− 54.0000i	− 2.22692i
$589$	2.00000	0.0824086
$590$	0	0
$591$	66.0000	2.71488
$592$	2.00000i	0.0821995i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	−36.0000	−1.47710
$595$	0	0
$596$	−4.00000	−0.163846
$597$	− 45.0000i	− 1.84173i
$598$	− 7.00000i	− 0.286251i
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	30.0000i	1.22271i
$603$	− 18.0000i	− 0.733017i
$604$	10.0000	0.406894
$605$	0	0
$606$	24.0000	0.974933
$607$	26.0000i	1.05531i	0.849460	+	0.527654i	$0.176928\pi$
$607$	26.0000i	1.05531i	−0.849460	+	0.527654i	$0.823072\pi$
$608$	1.00000i	0.0405554i
$609$	45.0000	1.82349
$610$	0	0
$611$	0	0
$612$	18.0000i	0.727607i
$613$	− 20.0000i	− 0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$	− 20.0000i	− 0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	20.0000	0.805823
$617$	− 14.0000i	− 0.563619i	−0.959470	−	0.281809i	$-0.909065\pi$
$617$	− 14.0000i	− 0.563619i	0.959470	−	0.281809i	$-0.0909346\pi$
$618$	− 12.0000i	− 0.482711i
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	−63.0000	−2.52810
$622$	− 25.0000i	− 1.00241i
$623$	− 10.0000i	− 0.400642i
$624$	−3.00000	−0.120096
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	− 12.0000i	− 0.479234i
$628$	6.00000i	0.239426i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	2.00000i	0.0795557i
$633$	15.0000i	0.596196i
$634$	−9.00000	−0.357436
$635$	0	0
$636$	39.0000	1.54645
$637$	18.0000i	0.713186i
$638$	12.0000i	0.475085i
$639$	0	0
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	− 39.0000i	− 1.53921i
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	35.0000	1.37919
$645$	0	0
$646$	−3.00000	−0.118033
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	9.00000i	0.353553i
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−30.0000	−1.17579
$652$	− 22.0000i	− 0.861586i
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	57.0000	2.22888
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 66.0000i	− 2.57491i
$658$	0	0
$659$	−33.0000	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$660$	0	0
$661$	15.0000	0.583432	0.291716	−	0.956505i	$-0.405774\pi$
$661$	15.0000	0.583432	0.291716	+	0.956505i	$0.405774\pi$
$662$	7.00000i	0.272063i
$663$	− 9.00000i	− 0.349531i
$664$	10.0000	0.388075
$665$	0	0
$666$	−12.0000	−0.464991
$667$	21.0000i	0.813123i
$668$	− 2.00000i	− 0.0773823i
$669$	−6.00000	−0.231973
$670$	0	0
$671$	48.0000	1.85302
$672$	− 15.0000i	− 0.578638i
$673$	− 44.0000i	− 1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$	− 44.0000i	− 1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	−12.0000	−0.461538
$677$	39.0000i	1.49889i	0.662066	+	0.749446i	$0.269680\pi$
$677$	39.0000i	1.49889i	−0.662066	+	0.749446i	$0.730320\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−15.0000	−0.574801
$682$	− 8.00000i	− 0.306336i
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−55.0000	−2.09991
$687$	− 18.0000i	− 0.686743i
$688$	6.00000i	0.228748i
$689$	−13.0000	−0.495261
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	14.0000i	0.532200i
$693$	120.000i	4.55842i
$694$	6.00000	0.227757
$695$	0	0
$696$	9.00000	0.341144
$697$	− 18.0000i	− 0.681799i
$698$	14.0000i	0.529908i
$699$	30.0000	1.13470
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	− 9.00000i	− 0.339683i
$703$	− 2.00000i	− 0.0754314i
$704$	4.00000	0.150756
$705$	0	0
$706$	7.00000	0.263448
$707$	− 40.0000i	− 1.50435i
$708$	27.0000i	1.01472i
$709$	−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
$711$	−12.0000	−0.450035
$712$	− 2.00000i	− 0.0749532i
$713$	− 14.0000i	− 0.524304i
$714$	45.0000	1.68408
$715$	0	0
$716$	−8.00000	−0.298974
$717$	− 33.0000i	− 1.23241i
$718$	− 5.00000i	− 0.186598i
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	− 1.00000i	− 0.0372161i
$723$	36.0000i	1.33885i
$724$	−26.0000	−0.966282
$725$	0	0
$726$	−15.0000	−0.556702
$727$	− 23.0000i	− 0.853023i	−0.904482	−	0.426511i	$-0.859742\pi$
$727$	− 23.0000i	− 0.853023i	0.904482	−	0.426511i	$-0.140258\pi$
$728$	5.00000i	0.185312i
$729$	27.0000	1.00000
$730$	0	0
$731$	−18.0000	−0.665754
$732$	− 36.0000i	− 1.33060i
$733$	36.0000i	1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$	36.0000i	1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	7.00000	0.258023
$737$	− 12.0000i	− 0.442026i
$738$	− 36.0000i	− 1.32518i
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	− 65.0000i	− 2.38623i
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−23.0000	−0.842090
$747$	60.0000i	2.19529i
$748$	12.0000i	0.438763i
$749$	−65.0000	−2.37505
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	36.0000i	1.31191i
$754$	−3.00000	−0.109254
$755$	0	0
$756$	45.0000	1.63663
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	− 33.0000i	− 1.19861i
$759$	−84.0000	−3.04901
$760$	0	0
$761$	11.0000	0.398750	0.199375	−	0.979923i	$-0.436109\pi$
$761$	11.0000	0.398750	0.199375	+	0.979923i	$0.436109\pi$
$762$	− 18.0000i	− 0.652071i
$763$	− 95.0000i	− 3.43923i
$764$	−9.00000	−0.325609
$765$	0	0
$766$	−4.00000	−0.144526
$767$	− 9.00000i	− 0.324971i
$768$	− 3.00000i	− 0.108253i
$769$	47.0000	1.69486	0.847432	−	0.530904i	$-0.178148\pi$
$769$	47.0000	1.69486	0.847432	+	0.530904i	$0.178148\pi$
$770$	0	0
$771$	−66.0000	−2.37693
$772$	− 10.0000i	− 0.359908i
$773$	− 51.0000i	− 1.83434i	−0.398493	−	0.917171i	$-0.630467\pi$
$773$	− 51.0000i	− 1.83434i	0.398493	−	0.917171i	$-0.369533\pi$
$774$	−36.0000	−1.29399
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	30.0000i	1.07624i
$778$	− 4.00000i	− 0.143407i
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	21.0000i	0.750958i
$783$	27.0000i	0.964901i
$784$	−18.0000	−0.642857
$785$	0	0
$786$	−48.0000	−1.71210
$787$	39.0000i	1.39020i	0.718913	+	0.695100i	$0.244640\pi$
$787$	39.0000i	1.39020i	−0.718913	+	0.695100i	$0.755360\pi$
$788$	− 22.0000i	− 0.783718i
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	24.0000i	0.852803i
$793$	12.0000i	0.426132i
$794$	16.0000	0.567819
$795$	0	0
$796$	−15.0000	−0.531661
$797$	− 31.0000i	− 1.09808i	−0.835797	−	0.549038i	$-0.814994\pi$
$797$	− 31.0000i	− 1.09808i	0.835797	−	0.549038i	$-0.185006\pi$
$798$	15.0000i	0.530994i
$799$	0	0
$800$	0	0
$801$	12.0000	0.423999
$802$	6.00000i	0.211867i
$803$	− 44.0000i	− 1.55273i
$804$	−9.00000	−0.317406
$805$	0	0
$806$	2.00000	0.0704470
$807$	6.00000i	0.211210i
$808$	− 8.00000i	− 0.281439i
$809$	−25.0000	−0.878953	−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000	−0.878953	−0.439477	+	0.898254i	$0.644836\pi$
$810$	0	0
$811$	37.0000	1.29925	0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000	1.29925	0.649623	+	0.760257i	$0.274927\pi$
$812$	− 15.0000i	− 0.526397i
$813$	81.0000i	2.84079i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	9.00000	0.315063
$817$	− 6.00000i	− 0.209913i
$818$	22.0000i	0.769212i
$819$	−30.0000	−1.04828
$820$	0	0
$821$	−52.0000	−1.81481	−0.907406	−	0.420255i	$-0.861941\pi$
$821$	−52.0000	−1.81481	−0.907406	+	0.420255i	$0.861941\pi$
$822$	27.0000i	0.941733i
$823$	− 43.0000i	− 1.49889i	−0.662069	−	0.749443i	$-0.730321\pi$
$823$	− 43.0000i	− 1.49889i	0.662069	−	0.749443i	$-0.269679\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	45.0000	1.56575
$827$	3.00000i	0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$	3.00000i	0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$	42.0000i	1.45960i
$829$	−35.0000	−1.21560	−0.607800	−	0.794090i	$-0.707948\pi$
$829$	−35.0000	−1.21560	−0.607800	+	0.794090i	$0.707948\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	1.00000i	0.0346688i
$833$	− 54.0000i	− 1.87099i
$834$	48.0000	1.66210
$835$	0	0
$836$	−4.00000	−0.138343
$837$	− 18.0000i	− 0.622171i
$838$	14.0000i	0.483622i
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 1.00000i	− 0.0344623i
$843$	54.0000i	1.85986i
$844$	5.00000	0.172107
$845$	0	0
$846$	0	0
$847$	25.0000i	0.859010i
$848$	− 13.0000i	− 0.446422i
$849$	−6.00000	−0.205919
$850$	0	0
$851$	−14.0000	−0.479914
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	−60.0000	−2.05316
$855$	0	0
$856$	−13.0000	−0.444331
$857$	40.0000i	1.36637i	0.730243	+	0.683187i	$0.239407\pi$
$857$	40.0000i	1.36637i	−0.730243	+	0.683187i	$0.760593\pi$
$858$	− 12.0000i	− 0.409673i
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	−90.0000	−3.06719
$862$	36.0000i	1.22616i
$863$	56.0000i	1.90626i	0.302558	+	0.953131i	$0.402160\pi$
$863$	56.0000i	1.90626i	−0.302558	+	0.953131i	$0.597840\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	−16.0000	−0.543702
$867$	− 24.0000i	− 0.815083i
$868$	10.0000i	0.339422i
$869$	−8.00000	−0.271381
$870$	0	0
$871$	3.00000	0.101651
$872$	− 19.0000i	− 0.643421i
$873$	− 12.0000i	− 0.406138i
$874$	−7.00000	−0.236779
$875$	0	0
$876$	−33.0000	−1.11497
$877$	− 33.0000i	− 1.11433i	−0.830402	−	0.557165i	$-0.811889\pi$
$877$	− 33.0000i	− 1.11433i	0.830402	−	0.557165i	$-0.188111\pi$
$878$	26.0000i	0.877457i
$879$	−81.0000	−2.73206
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	− 108.000i	− 3.63655i
$883$	30.0000i	1.00958i	0.863242	+	0.504790i	$0.168430\pi$
$883$	30.0000i	1.00958i	−0.863242	+	0.504790i	$0.831570\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	36.0000	1.20944
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	6.00000i	0.201347i
$889$	−30.0000	−1.00617
$890$	0	0
$891$	−36.0000	−1.20605
$892$	2.00000i	0.0669650i
$893$	0	0
$894$	−12.0000	−0.401340
$895$	0	0
$896$	−5.00000	−0.167038
$897$	− 21.0000i	− 0.701170i
$898$	− 22.0000i	− 0.734150i
$899$	−6.00000	−0.200111
$900$	0	0
$901$	39.0000	1.29928
$902$	− 24.0000i	− 0.799113i
$903$	90.0000i	2.99501i
$904$	0	0
$905$	0	0
$906$	30.0000	0.996683
$907$	1.00000i	0.0332045i	0.999862	+	0.0166022i	$0.00528490\pi$
$907$	1.00000i	0.0332045i	−0.999862	+	0.0166022i	$0.994715\pi$
$908$	5.00000i	0.165931i
$909$	48.0000	1.59206
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	3.00000i	0.0993399i
$913$	40.0000i	1.32381i
$914$	29.0000	0.959235
$915$	0	0
$916$	−6.00000	−0.198246
$917$	80.0000i	2.64183i
$918$	27.0000i	0.891133i
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	− 18.0000i	− 0.592798i
$923$	0	0
$924$	60.0000	1.97386
$925$	0	0
$926$	−8.00000	−0.262896
$927$	− 24.0000i	− 0.788263i
$928$	− 3.00000i	− 0.0984798i
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	− 10.0000i	− 0.327561i
$933$	− 75.0000i	− 2.45539i
$934$	8.00000	0.261768
$935$	0	0
$936$	−6.00000	−0.196116
$937$	− 47.0000i	− 1.53542i	−0.640796	−	0.767712i	$-0.721395\pi$
$937$	− 47.0000i	− 1.53542i	0.640796	−	0.767712i	$-0.278605\pi$
$938$	15.0000i	0.489767i
$939$	−3.00000	−0.0979013
$940$	0	0
$941$	−51.0000	−1.66255	−0.831276	−	0.555860i	$-0.812389\pi$
$941$	−51.0000	−1.66255	−0.831276	+	0.555860i	$0.812389\pi$
$942$	18.0000i	0.586472i
$943$	− 42.0000i	− 1.36771i
$944$	9.00000	0.292925
$945$	0	0
$946$	−24.0000	−0.780307
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	6.00000i	0.194871i
$949$	11.0000	0.357075
$950$	0	0
$951$	−27.0000	−0.875535
$952$	− 15.0000i	− 0.486153i
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	78.0000	2.52534
$955$	0	0
$956$	−11.0000	−0.355765
$957$	36.0000i	1.16371i
$958$	0	0
$959$	45.0000	1.45313
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 2.00000i	− 0.0644826i
$963$	− 78.0000i	− 2.51351i
$964$	12.0000	0.386494
$965$	0	0
$966$	105.000	3.37832
$967$	44.0000i	1.41494i	0.706741	+	0.707472i	$0.250165\pi$
$967$	44.0000i	1.41494i	−0.706741	+	0.707472i	$0.749835\pi$
$968$	5.00000i	0.160706i
$969$	−9.00000	−0.289122
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	− 80.0000i	− 2.56468i
$974$	38.0000	1.21760
$975$	0	0
$976$	−12.0000	−0.384111
$977$	62.0000i	1.98356i	0.127971	+	0.991778i	$0.459153\pi$
$977$	62.0000i	1.98356i	−0.127971	+	0.991778i	$0.540847\pi$
$978$	− 66.0000i	− 2.11045i
$979$	8.00000	0.255681
$980$	0	0
$981$	114.000	3.63974
$982$	− 18.0000i	− 0.574403i
$983$	− 42.0000i	− 1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$	− 42.0000i	− 1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$	−18.0000	−0.573819
$985$	0	0
$986$	9.00000	0.286618
$987$	0	0
$988$	− 1.00000i	− 0.0318142i
$989$	−42.0000	−1.33552
$990$	0	0
$991$	30.0000	0.952981	0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000	0.952981	0.476491	+	0.879180i	$0.341909\pi$
$992$	2.00000i	0.0635001i
$993$	21.0000i	0.666415i
$994$	0	0
$995$	0	0
$996$	30.0000	0.950586
$997$	50.0000i	1.58352i	0.610835	+	0.791758i	$0.290834\pi$
$997$	50.0000i	1.58352i	−0.610835	+	0.791758i	$0.709166\pi$
$998$	42.0000i	1.32949i
$999$	−18.0000	−0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.b.a.799.1		2
5.2	odd	4		190.2.a.b.1.1	✓	1
5.3	odd	4		950.2.a.c.1.1		1
5.4	even	2	inner	950.2.b.a.799.2		2
15.2	even	4		1710.2.a.g.1.1		1
15.8	even	4		8550.2.a.bm.1.1		1
20.3	even	4		7600.2.a.a.1.1		1
20.7	even	4		1520.2.a.j.1.1		1
35.27	even	4		9310.2.a.u.1.1		1
40.27	even	4		6080.2.a.b.1.1		1
40.37	odd	4		6080.2.a.x.1.1		1
95.37	even	4		3610.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.b.1.1	✓	1	5.2	odd	4
950.2.a.c.1.1		1	5.3	odd	4
950.2.b.a.799.1		2	1.1	even	1	trivial
950.2.b.a.799.2		2	5.4	even	2	inner
1520.2.a.j.1.1		1	20.7	even	4
1710.2.a.g.1.1		1	15.2	even	4
3610.2.a.e.1.1		1	95.37	even	4
6080.2.a.b.1.1		1	40.27	even	4
6080.2.a.x.1.1		1	40.37	odd	4
7600.2.a.a.1.1		1	20.3	even	4
8550.2.a.bm.1.1		1	15.8	even	4
9310.2.a.u.1.1		1	35.27	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} -4.00000 q^{11} +3.00000i q^{12} -1.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +6.00000i q^{18} -1.00000 q^{19} +15.0000 q^{21} +4.00000i q^{22} +7.00000i q^{23} +3.00000 q^{24} -1.00000 q^{26} +9.00000i q^{27} -5.00000i q^{28} +3.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} +12.0000i q^{33} +3.00000 q^{34} +6.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -3.00000 q^{39} -6.00000 q^{41} -15.0000i q^{42} +6.00000i q^{43} +4.00000 q^{44} +7.00000 q^{46} -3.00000i q^{48} -18.0000 q^{49} +9.00000 q^{51} +1.00000i q^{52} -13.0000i q^{53} +9.00000 q^{54} -5.00000 q^{56} +3.00000i q^{57} -3.00000i q^{58} +9.00000 q^{59} -12.0000 q^{61} +2.00000i q^{62} -30.0000i q^{63} -1.00000 q^{64} +12.0000 q^{66} +3.00000i q^{67} -3.00000i q^{68} +21.0000 q^{69} -6.00000i q^{72} +11.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} -20.0000i q^{77} +3.00000i q^{78} +2.00000 q^{79} +9.00000 q^{81} +6.00000i q^{82} -10.0000i q^{83} -15.0000 q^{84} +6.00000 q^{86} -9.00000i q^{87} -4.00000i q^{88} -2.00000 q^{89} +5.00000 q^{91} -7.00000i q^{92} +6.00000i q^{93} -3.00000 q^{96} +2.00000i q^{97} +18.0000i q^{98} +24.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9	\( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 8 q^{11} + 10 q^{14} + 2 q^{16} - 2 q^{19} + 30 q^{21} + 6 q^{24} - 2 q^{26} + 6 q^{29} - 4 q^{31} + 6 q^{34} + 12 q^{36} - 6 q^{39} - 12 q^{41} + 8 q^{44} + 14 q^{46} - 36 q^{49} + 18 q^{51} + 18 q^{54} - 10 q^{56} + 18 q^{59} - 24 q^{61} - 2 q^{64} + 24 q^{66} + 42 q^{69} + 4 q^{74} + 2 q^{76} + 4 q^{79} + 18 q^{81} - 30 q^{84} + 12 q^{86} - 4 q^{89} + 10 q^{91} - 6 q^{96} + 48 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 - 8 * q^11 + 10 * q^14 + 2 * q^16 - 2 * q^19 + 30 * q^21 + 6 * q^24 - 2 * q^26 + 6 * q^29 - 4 * q^31 + 6 * q^34 + 12 * q^36 - 6 * q^39 - 12 * q^41 + 8 * q^44 + 14 * q^46 - 36 * q^49 + 18 * q^51 + 18 * q^54 - 10 * q^56 + 18 * q^59 - 24 * q^61 - 2 * q^64 + 24 * q^66 + 42 * q^69 + 4 * q^74 + 2 * q^76 + 4 * q^79 + 18 * q^81 - 30 * q^84 + 12 * q^86 - 4 * q^89 + 10 * q^91 - 6 * q^96 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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