LMFDB - Embedded newform 1950.2.f.i.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(649,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1950.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.f (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:	$15.5708283941$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.649
Dual form			1950.2.f.i.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +1.00000 q^{8} -1.00000 q^{9} +4.00000i q^{11} +1.00000i q^{12} +(2.00000 + 3.00000i) q^{13} +1.00000 q^{16} +4.00000i q^{17} -1.00000 q^{18} -4.00000i q^{19} +4.00000i q^{22} -6.00000i q^{23} +1.00000i q^{24} +(2.00000 + 3.00000i) q^{26} -1.00000i q^{27} -4.00000 q^{29} +10.0000i q^{31} +1.00000 q^{32} -4.00000 q^{33} +4.00000i q^{34} -1.00000 q^{36} +4.00000 q^{37} -4.00000i q^{38} +(-3.00000 + 2.00000i) q^{39} -2.00000i q^{41} +12.0000i q^{43} +4.00000i q^{44} -6.00000i q^{46} +1.00000i q^{48} -7.00000 q^{49} -4.00000 q^{51} +(2.00000 + 3.00000i) q^{52} +2.00000i q^{53} -1.00000i q^{54} +4.00000 q^{57} -4.00000 q^{58} -4.00000i q^{59} -10.0000 q^{61} +10.0000i q^{62} +1.00000 q^{64} -4.00000 q^{66} +10.0000 q^{67} +4.00000i q^{68} +6.00000 q^{69} +12.0000i q^{71} -1.00000 q^{72} +6.00000 q^{73} +4.00000 q^{74} -4.00000i q^{76} +(-3.00000 + 2.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +12.0000i q^{86} -4.00000i q^{87} +4.00000i q^{88} +10.0000i q^{89} -6.00000i q^{92} -10.0000 q^{93} +1.00000i q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 4 q^{13} + 2 q^{16} - 2 q^{18} + 4 q^{26} - 8 q^{29} + 2 q^{32} - 8 q^{33} - 2 q^{36} + 8 q^{37} - 6 q^{39} - 14 q^{49} - 8 q^{51} + 4 q^{52} + 8 q^{57} - 8 q^{58} - 20 q^{61} + 2 q^{64} - 8 q^{66} + 20 q^{67} + 12 q^{69} - 2 q^{72} + 12 q^{73} + 8 q^{74} - 6 q^{78} + 16 q^{79} + 2 q^{81} - 20 q^{93} + 28 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 4 * q^13 + 2 * q^16 - 2 * q^18 + 4 * q^26 - 8 * q^29 + 2 * q^32 - 8 * q^33 - 2 * q^36 + 8 * q^37 - 6 * q^39 - 14 * q^49 - 8 * q^51 + 4 * q^52 + 8 * q^57 - 8 * q^58 - 20 * q^61 + 2 * q^64 - 8 * q^66 + 20 * q^67 + 12 * q^69 - 2 * q^72 + 12 * q^73 + 8 * q^74 - 6 * q^78 + 16 * q^79 + 2 * q^81 - 20 * q^93 + 28 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			1.00000i			0.408248i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	1.00000			0.353553
$9$	−1.00000			−0.333333
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$			1.00000i			0.288675i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	−1.00000			−0.235702
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		0			0
$22$			4.00000i			0.852803i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$			1.00000i			0.204124i
$25$		0			0
$26$	2.00000	+	3.00000i	0.392232	+	0.588348i
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$	1.00000			0.176777
$33$	−4.00000			−0.696311
$34$			4.00000i			0.685994i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		−	4.00000i		−	0.648886i
$39$	−3.00000	+	2.00000i	−0.480384	+	0.320256i
$40$		0			0
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$		0			0
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$			4.00000i			0.603023i
$45$		0			0
$46$		−	6.00000i		−	0.884652i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$			1.00000i			0.144338i
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−4.00000			−0.560112
$52$	2.00000	+	3.00000i	0.277350	+	0.416025i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$		0			0
$57$	4.00000			0.529813
$58$	−4.00000			−0.525226
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$			10.0000i			1.27000i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$	−4.00000			−0.492366
$67$	10.0000			1.22169			0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000			1.22169			0.610847	+	0.791748i	$0.290829\pi$
$68$			4.00000i			0.485071i
$69$	6.00000			0.722315
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	−1.00000			−0.117851
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$	4.00000			0.464991
$75$		0			0
$76$		−	4.00000i		−	0.458831i
$77$		0			0
$78$	−3.00000	+	2.00000i	−0.339683	+	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
$82$		−	2.00000i		−	0.220863i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$			12.0000i			1.29399i
$87$		−	4.00000i		−	0.428845i
$88$			4.00000i			0.426401i
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$		0			0
$92$		−	6.00000i		−	0.625543i
$93$	−10.0000			−1.03695
$94$		0			0
$95$		0			0
$96$			1.00000i			0.102062i
$97$	14.0000			1.42148			0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000			1.42148			0.710742	+	0.703452i	$0.248359\pi$
$98$	−7.00000			−0.707107
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$	−4.00000			−0.396059
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	2.00000	+	3.00000i	0.196116	+	0.294174i
$105$		0			0
$106$			2.00000i			0.194257i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		−	1.00000i		−	0.0962250i
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$			4.00000i			0.379663i
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	4.00000			0.374634
$115$		0			0
$116$	−4.00000			−0.371391
$117$	−2.00000	−	3.00000i	−0.184900	−	0.277350i
$118$		−	4.00000i		−	0.368230i
$119$		0			0
$120$		0			0
$121$	−5.00000			−0.454545
$122$	−10.0000			−0.905357
$123$	2.00000			0.180334
$124$			10.0000i			0.898027i
$125$		0			0
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000			0.0883883
$129$	−12.0000			−1.05654
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$	−4.00000			−0.348155
$133$		0			0
$134$	10.0000			0.863868
$135$		0			0
$136$			4.00000i			0.342997i
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$	6.00000			0.510754
$139$	−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
$142$			12.0000i			1.00702i
$143$	−12.0000	+	8.00000i	−1.00349	+	0.668994i
$144$	−1.00000			−0.0833333
$145$		0			0
$146$	6.00000			0.496564
$147$		−	7.00000i		−	0.577350i
$148$	4.00000			0.328798
$149$			2.00000i			0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$			2.00000i			0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$		0			0
$151$		−	22.0000i		−	1.79033i	−0.445730	−	0.895167i	$-0.647056\pi$
$151$		−	22.0000i		−	1.79033i	0.445730	−	0.895167i	$-0.352944\pi$
$152$		−	4.00000i		−	0.324443i
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$		0			0
$156$	−3.00000	+	2.00000i	−0.240192	+	0.160128i
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	8.00000			0.636446
$159$	−2.00000			−0.158610
$160$		0			0
$161$		0			0
$162$	1.00000			0.0785674
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		−	2.00000i		−	0.156174i
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$			4.00000i			0.305888i
$172$			12.0000i			0.914991i
$173$		−	22.0000i		−	1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$		−	22.0000i		−	1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$		−	4.00000i		−	0.303239i
$175$		0			0
$176$			4.00000i			0.301511i
$177$	4.00000			0.300658
$178$			10.0000i			0.749532i
$179$	−22.0000			−1.64436			−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000			−1.64436			−0.822179	+	0.569230i	$0.807242\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		−	6.00000i		−	0.442326i
$185$		0			0
$186$	−10.0000			−0.733236
$187$	−16.0000			−1.17004
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$			1.00000i			0.0721688i
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$	14.0000			1.00514
$195$		0			0
$196$	−7.00000			−0.500000
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		−	4.00000i		−	0.284268i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			10.0000i			0.705346i
$202$	8.00000			0.562878
$203$		0			0
$204$	−4.00000			−0.280056
$205$		0			0
$206$		−	4.00000i		−	0.278693i
$207$			6.00000i			0.417029i
$208$	2.00000	+	3.00000i	0.138675	+	0.208013i
$209$	16.0000			1.10674
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$			2.00000i			0.137361i
$213$	−12.0000			−0.822226
$214$		0			0
$215$		0			0
$216$		−	1.00000i		−	0.0680414i
$217$		0			0
$218$		−	6.00000i		−	0.406371i
$219$			6.00000i			0.405442i
$220$		0			0
$221$	−12.0000	+	8.00000i	−0.807207	+	0.538138i
$222$			4.00000i			0.268462i
$223$	20.0000			1.33930			0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000			1.33930			0.669650	+	0.742677i	$0.266444\pi$
$224$		0			0
$225$		0			0
$226$		−	12.0000i		−	0.798228i
$227$	−16.0000			−1.06196			−0.530979	−	0.847385i	$-0.678176\pi$
$227$	−16.0000			−1.06196			−0.530979	+	0.847385i	$0.678176\pi$
$228$	4.00000			0.264906
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$		0			0
$231$		0			0
$232$	−4.00000			−0.262613
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	−2.00000	−	3.00000i	−0.130744	−	0.196116i
$235$		0			0
$236$		−	4.00000i		−	0.260378i
$237$			8.00000i			0.519656i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			4.00000i			0.257663i	0.991667	+	0.128831i	$0.0411226\pi$
$241$			4.00000i			0.257663i	−0.991667	+	0.128831i	$0.958877\pi$
$242$	−5.00000			−0.321412
$243$			1.00000i			0.0641500i
$244$	−10.0000			−0.640184
$245$		0			0
$246$	2.00000			0.127515
$247$	12.0000	−	8.00000i	0.763542	−	0.509028i
$248$			10.0000i			0.635001i
$249$		0			0
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$			8.00000i			0.501965i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	12.0000i		−	0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$		−	12.0000i		−	0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	−12.0000			−0.747087
$259$		0			0
$260$		0			0
$261$	4.00000			0.247594
$262$	−18.0000			−1.11204
$263$			14.0000i			0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$			14.0000i			0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	−4.00000			−0.246183
$265$		0			0
$266$		0			0
$267$	−10.0000			−0.611990
$268$	10.0000			0.610847
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		−	2.00000i		−	0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$		−	2.00000i		−	0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$			4.00000i			0.242536i
$273$		0			0
$274$	2.00000			0.120824
$275$		0			0
$276$	6.00000			0.361158
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	−16.0000			−0.959616
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$		−	30.0000i		−	1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$		−	30.0000i		−	1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$			12.0000i			0.712069i
$285$		0			0
$286$	−12.0000	+	8.00000i	−0.709575	+	0.473050i
$287$		0			0
$288$	−1.00000			−0.0589256
$289$	1.00000			0.0588235
$290$		0			0
$291$			14.0000i			0.820695i
$292$	6.00000			0.351123
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		−	7.00000i		−	0.408248i
$295$		0			0
$296$	4.00000			0.232495
$297$	4.00000			0.232104
$298$			2.00000i			0.115857i
$299$	18.0000	−	12.0000i	1.04097	−	0.693978i
$300$		0			0
$301$		0			0
$302$		−	22.0000i		−	1.26596i
$303$			8.00000i			0.459588i
$304$		−	4.00000i		−	0.229416i
$305$		0			0
$306$		−	4.00000i		−	0.228665i
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$	−3.00000	+	2.00000i	−0.169842	+	0.113228i
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		−	14.0000i		−	0.790066i
$315$		0			0
$316$	8.00000			0.450035
$317$	−6.00000			−0.336994			−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000			−0.336994			−0.168497	+	0.985702i	$0.553891\pi$
$318$	−2.00000			−0.112154
$319$		−	16.0000i		−	0.895828i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000			0.890264
$324$	1.00000			0.0555556
$325$		0			0
$326$	−2.00000			−0.110770
$327$	6.00000			0.331801
$328$		−	2.00000i		−	0.110432i
$329$		0			0
$330$		0			0
$331$		−	32.0000i		−	1.75888i	−0.476011	−	0.879440i	$-0.657918\pi$
$331$		−	32.0000i		−	1.75888i	0.476011	−	0.879440i	$-0.342082\pi$
$332$		0			0
$333$	−4.00000			−0.219199
$334$		0			0
$335$		0			0
$336$		0			0
$337$			2.00000i			0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$			2.00000i			0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	−5.00000	+	12.0000i	−0.271964	+	0.652714i
$339$	12.0000			0.651751
$340$		0			0
$341$	−40.0000			−2.16612
$342$			4.00000i			0.216295i
$343$		0			0
$344$			12.0000i			0.646997i
$345$		0			0
$346$		−	22.0000i		−	1.18273i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		−	4.00000i		−	0.214423i
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	3.00000	−	2.00000i	0.160128	−	0.106752i
$352$			4.00000i			0.213201i
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$	4.00000			0.212598
$355$		0			0
$356$			10.0000i			0.529999i
$357$		0			0
$358$	−22.0000			−1.16274
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	22.0000			1.15629
$363$		−	5.00000i		−	0.262432i
$364$		0			0
$365$		0			0
$366$		−	10.0000i		−	0.522708i
$367$			16.0000i			0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$			16.0000i			0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$		−	6.00000i		−	0.312772i
$369$			2.00000i			0.104116i
$370$		0			0
$371$		0			0
$372$	−10.0000			−0.518476
$373$			26.0000i			1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$			26.0000i			1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	−16.0000			−0.827340
$375$		0			0
$376$		0			0
$377$	−8.00000	−	12.0000i	−0.412021	−	0.618031i
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$	−12.0000			−0.613973
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$	14.0000			0.712581
$387$		−	12.0000i		−	0.609994i
$388$	14.0000			0.710742
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$	−7.00000			−0.353553
$393$		−	18.0000i		−	0.907980i
$394$	26.0000			1.30986
$395$		0			0
$396$		−	4.00000i		−	0.201008i
$397$	−28.0000			−1.40528			−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000			−1.40528			−0.702640	+	0.711546i	$0.747995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	26.0000i		−	1.29838i	−0.760627	−	0.649189i	$-0.775108\pi$
$401$		−	26.0000i		−	1.29838i	0.760627	−	0.649189i	$-0.224892\pi$
$402$			10.0000i			0.498755i
$403$	−30.0000	+	20.0000i	−1.49441	+	0.996271i
$404$	8.00000			0.398015
$405$		0			0
$406$		0			0
$407$			16.0000i			0.793091i
$408$	−4.00000			−0.198030
$409$			8.00000i			0.395575i	0.980245	+	0.197787i	$0.0633755\pi$
$409$			8.00000i			0.395575i	−0.980245	+	0.197787i	$0.936624\pi$
$410$		0			0
$411$			2.00000i			0.0986527i
$412$		−	4.00000i		−	0.197066i
$413$		0			0
$414$			6.00000i			0.294884i
$415$		0			0
$416$	2.00000	+	3.00000i	0.0980581	+	0.147087i
$417$		−	16.0000i		−	0.783523i
$418$	16.0000			0.782586
$419$	−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
$421$		−	14.0000i		−	0.682318i	−0.940006	−	0.341159i	$-0.889181\pi$
$421$		−	14.0000i		−	0.682318i	0.940006	−	0.341159i	$-0.110819\pi$
$422$	24.0000			1.16830
$423$		0			0
$424$			2.00000i			0.0971286i
$425$		0			0
$426$	−12.0000			−0.581402
$427$		0			0
$428$		0			0
$429$	−8.00000	−	12.0000i	−0.386244	−	0.579365i
$430$		0			0
$431$		−	12.0000i		−	0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$		−	12.0000i		−	0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$		−	1.00000i		−	0.0481125i
$433$		−	18.0000i		−	0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$		−	18.0000i		−	0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$		0			0
$435$		0			0
$436$		−	6.00000i		−	0.287348i
$437$	−24.0000			−1.14808
$438$			6.00000i			0.286691i
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$	−12.0000	+	8.00000i	−0.570782	+	0.380521i
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$			4.00000i			0.189832i
$445$		0			0
$446$	20.0000			0.947027
$447$	−2.00000			−0.0945968
$448$		0			0
$449$		−	30.0000i		−	1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$		−	30.0000i		−	1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		−	12.0000i		−	0.564433i
$453$	22.0000			1.03365
$454$	−16.0000			−0.750917
$455$		0			0
$456$	4.00000			0.187317
$457$	14.0000			0.654892			0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000			0.654892			0.327446	+	0.944870i	$0.393812\pi$
$458$		−	2.00000i		−	0.0934539i
$459$	4.00000			0.186704
$460$		0			0
$461$			26.0000i			1.21094i	0.795868	+	0.605470i	$0.207015\pi$
$461$			26.0000i			1.21094i	−0.795868	+	0.605470i	$0.792985\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$	−4.00000			−0.185695
$465$		0			0
$466$		−	24.0000i		−	1.11178i
$467$		−	4.00000i		−	0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$		−	4.00000i		−	0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	−2.00000	−	3.00000i	−0.0924500	−	0.138675i
$469$		0			0
$470$		0			0
$471$	14.0000			0.645086
$472$		−	4.00000i		−	0.184115i
$473$	−48.0000			−2.20704
$474$			8.00000i			0.367452i
$475$		0			0
$476$		0			0
$477$		−	2.00000i		−	0.0915737i
$478$		0			0
$479$		−	16.0000i		−	0.731059i	−0.930800	−	0.365529i	$-0.880888\pi$
$479$		−	16.0000i		−	0.731059i	0.930800	−	0.365529i	$-0.119112\pi$
$480$		0			0
$481$	8.00000	+	12.0000i	0.364769	+	0.547153i
$482$			4.00000i			0.182195i
$483$		0			0
$484$	−5.00000			−0.227273
$485$		0			0
$486$			1.00000i			0.0453609i
$487$	24.0000			1.08754			0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000			1.08754			0.543772	+	0.839233i	$0.316996\pi$
$488$	−10.0000			−0.452679
$489$		−	2.00000i		−	0.0904431i
$490$		0			0
$491$	22.0000			0.992846			0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000			0.992846			0.496423	+	0.868081i	$0.334646\pi$
$492$	2.00000			0.0901670
$493$		−	16.0000i		−	0.720604i
$494$	12.0000	−	8.00000i	0.539906	−	0.359937i
$495$		0			0
$496$			10.0000i			0.449013i
$497$		0			0
$498$		0			0
$499$			4.00000i			0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$			4.00000i			0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$		0			0
$501$		0			0
$502$	2.00000			0.0892644
$503$			22.0000i			0.980932i	0.871460	+	0.490466i	$0.163173\pi$
$503$			22.0000i			0.980932i	−0.871460	+	0.490466i	$0.836827\pi$
$504$		0			0
$505$		0			0
$506$	24.0000			1.06693
$507$	−12.0000	−	5.00000i	−0.532939	−	0.222058i
$508$			8.00000i			0.354943i
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$	−4.00000			−0.176604
$514$		−	12.0000i		−	0.529297i
$515$		0			0
$516$	−12.0000			−0.528271
$517$		0			0
$518$		0			0
$519$	22.0000			0.965693
$520$		0			0
$521$	34.0000			1.48957			0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000			1.48957			0.744784	+	0.667306i	$0.232553\pi$
$522$	4.00000			0.175075
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−18.0000			−0.786334
$525$		0			0
$526$			14.0000i			0.610429i
$527$	−40.0000			−1.74243
$528$	−4.00000			−0.174078
$529$	−13.0000			−0.565217
$530$		0			0
$531$			4.00000i			0.173585i
$532$		0			0
$533$	6.00000	−	4.00000i	0.259889	−	0.173259i
$534$	−10.0000			−0.432742
$535$		0			0
$536$	10.0000			0.431934
$537$		−	22.0000i		−	0.949370i
$538$		0			0
$539$		−	28.0000i		−	1.20605i
$540$		0			0
$541$		−	18.0000i		−	0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$		−	18.0000i		−	0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$		−	2.00000i		−	0.0859074i
$543$			22.0000i			0.944110i
$544$			4.00000i			0.171499i
$545$		0			0
$546$		0			0
$547$			12.0000i			0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$			12.0000i			0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	2.00000			0.0854358
$549$	10.0000			0.426790
$550$		0			0
$551$			16.0000i			0.681623i
$552$	6.00000			0.255377
$553$		0			0
$554$		−	22.0000i		−	0.934690i
$555$		0			0
$556$	−16.0000			−0.678551
$557$	−30.0000			−1.27114			−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000			−1.27114			−0.635570	+	0.772043i	$0.719235\pi$
$558$		−	10.0000i		−	0.423334i
$559$	−36.0000	+	24.0000i	−1.52264	+	1.01509i
$560$		0			0
$561$		−	16.0000i		−	0.675521i
$562$		−	30.0000i		−	1.26547i
$563$		−	20.0000i		−	0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$		−	20.0000i		−	0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$		0			0
$565$		0			0
$566$		−	4.00000i		−	0.168133i
$567$		0			0
$568$			12.0000i			0.503509i
$569$	−46.0000			−1.92842			−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000			−1.92842			−0.964210	+	0.265139i	$0.914582\pi$
$570$		0			0
$571$	24.0000			1.00437			0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000			1.00437			0.502184	+	0.864761i	$0.332530\pi$
$572$	−12.0000	+	8.00000i	−0.501745	+	0.334497i
$573$		−	12.0000i		−	0.501307i
$574$		0			0
$575$		0			0
$576$	−1.00000			−0.0416667
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$	1.00000			0.0415945
$579$			14.0000i			0.581820i
$580$		0			0
$581$		0			0
$582$			14.0000i			0.580319i
$583$	−8.00000			−0.331326
$584$	6.00000			0.248282
$585$		0			0
$586$	14.0000			0.578335
$587$	−44.0000			−1.81607			−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000			−1.81607			−0.908037	+	0.418890i	$0.862419\pi$
$588$		−	7.00000i		−	0.288675i
$589$	40.0000			1.64817
$590$		0			0
$591$			26.0000i			1.06950i
$592$	4.00000			0.164399
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$	4.00000			0.164122
$595$		0			0
$596$			2.00000i			0.0819232i
$597$		0			0
$598$	18.0000	−	12.0000i	0.736075	−	0.490716i
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$	−10.0000			−0.407231
$604$		−	22.0000i		−	0.895167i
$605$		0			0
$606$			8.00000i			0.324978i
$607$		−	16.0000i		−	0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$		−	16.0000i		−	0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$		−	4.00000i		−	0.162221i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		−	4.00000i		−	0.161690i
$613$	36.0000			1.45403			0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000			1.45403			0.727013	+	0.686624i	$0.240908\pi$
$614$	2.00000			0.0807134
$615$		0			0
$616$		0			0
$617$	26.0000			1.04672			0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000			1.04672			0.523360	+	0.852111i	$0.324678\pi$
$618$	4.00000			0.160904
$619$			40.0000i			1.60774i	0.594808	+	0.803868i	$0.297228\pi$
$619$			40.0000i			1.60774i	−0.594808	+	0.803868i	$0.702772\pi$
$620$		0			0
$621$	−6.00000			−0.240772
$622$	12.0000			0.481156
$623$		0			0
$624$	−3.00000	+	2.00000i	−0.120096	+	0.0800641i
$625$		0			0
$626$			10.0000i			0.399680i
$627$			16.0000i			0.638978i
$628$		−	14.0000i		−	0.558661i
$629$			16.0000i			0.637962i
$630$		0			0
$631$			22.0000i			0.875806i	0.899022	+	0.437903i	$0.144279\pi$
$631$			22.0000i			0.875806i	−0.899022	+	0.437903i	$0.855721\pi$
$632$	8.00000			0.318223
$633$			24.0000i			0.953914i
$634$	−6.00000			−0.238290
$635$		0			0
$636$	−2.00000			−0.0793052
$637$	−14.0000	−	21.0000i	−0.554700	−	0.832050i
$638$		−	16.0000i		−	0.633446i
$639$		−	12.0000i		−	0.474713i
$640$		0			0
$641$	−10.0000			−0.394976			−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000			−0.394976			−0.197488	+	0.980305i	$0.563278\pi$
$642$		0			0
$643$	−50.0000			−1.97181			−0.985904	−	0.167313i	$-0.946491\pi$
$643$	−50.0000			−1.97181			−0.985904	+	0.167313i	$0.946491\pi$
$644$		0			0
$645$		0			0
$646$	16.0000			0.629512
$647$			38.0000i			1.49393i	0.664861	+	0.746967i	$0.268491\pi$
$647$			38.0000i			1.49393i	−0.664861	+	0.746967i	$0.731509\pi$
$648$	1.00000			0.0392837
$649$	16.0000			0.628055
$650$		0			0
$651$		0			0
$652$	−2.00000			−0.0783260
$653$			10.0000i			0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$			10.0000i			0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	6.00000			0.234619
$655$		0			0
$656$		−	2.00000i		−	0.0780869i
$657$	−6.00000			−0.234082
$658$		0			0
$659$	−42.0000			−1.63609			−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000			−1.63609			−0.818044	+	0.575156i	$0.804941\pi$
$660$		0			0
$661$		−	22.0000i		−	0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$		−	22.0000i		−	0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$		−	32.0000i		−	1.24372i
$663$	−8.00000	−	12.0000i	−0.310694	−	0.466041i
$664$		0			0
$665$		0			0
$666$	−4.00000			−0.154997
$667$			24.0000i			0.929284i
$668$		0			0
$669$			20.0000i			0.773245i
$670$		0			0
$671$		−	40.0000i		−	1.54418i
$672$		0			0
$673$			26.0000i			1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$			26.0000i			1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$			2.00000i			0.0770371i
$675$		0			0
$676$	−5.00000	+	12.0000i	−0.192308	+	0.461538i
$677$			34.0000i			1.30673i	0.757045	+	0.653363i	$0.226642\pi$
$677$			34.0000i			1.30673i	−0.757045	+	0.653363i	$0.773358\pi$
$678$	12.0000			0.460857
$679$		0			0
$680$		0			0
$681$		−	16.0000i		−	0.613121i
$682$	−40.0000			−1.53168
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$			4.00000i			0.152944i
$685$		0			0
$686$		0			0
$687$	2.00000			0.0763048
$688$			12.0000i			0.457496i
$689$	−6.00000	+	4.00000i	−0.228582	+	0.152388i
$690$		0			0
$691$		−	8.00000i		−	0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		−	22.0000i		−	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		−	4.00000i		−	0.151620i
$697$	8.00000			0.303022
$698$		−	14.0000i		−	0.529908i
$699$	24.0000			0.907763
$700$		0			0
$701$	−36.0000			−1.35970			−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000			−1.35970			−0.679851	+	0.733351i	$0.737955\pi$
$702$	3.00000	−	2.00000i	0.113228	−	0.0754851i
$703$		−	16.0000i		−	0.603451i
$704$			4.00000i			0.150756i
$705$		0			0
$706$	−6.00000			−0.225813
$707$		0			0
$708$	4.00000			0.150329
$709$		−	50.0000i		−	1.87779i	−0.344204	−	0.938895i	$-0.611851\pi$
$709$		−	50.0000i		−	1.87779i	0.344204	−	0.938895i	$-0.388149\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$			10.0000i			0.374766i
$713$	60.0000			2.24702
$714$		0			0
$715$		0			0
$716$	−22.0000			−0.822179
$717$		0			0
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$		0			0
$722$	3.00000			0.111648
$723$	−4.00000			−0.148762
$724$	22.0000			0.817624
$725$		0			0
$726$		−	5.00000i		−	0.185567i
$727$			4.00000i			0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$			4.00000i			0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−48.0000			−1.77534
$732$		−	10.0000i		−	0.369611i
$733$	−40.0000			−1.47743			−0.738717	−	0.674016i	$-0.764568\pi$
$733$	−40.0000			−1.47743			−0.738717	+	0.674016i	$0.764568\pi$
$734$			16.0000i			0.590571i
$735$		0			0
$736$		−	6.00000i		−	0.221163i
$737$			40.0000i			1.47342i
$738$			2.00000i			0.0736210i
$739$			8.00000i			0.294285i	0.989115	+	0.147142i	$0.0470076\pi$
$739$			8.00000i			0.294285i	−0.989115	+	0.147142i	$0.952992\pi$
$740$		0			0
$741$	8.00000	+	12.0000i	0.293887	+	0.440831i
$742$		0			0
$743$	−32.0000			−1.17397			−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000			−1.17397			−0.586983	+	0.809599i	$0.699684\pi$
$744$	−10.0000			−0.366618
$745$		0			0
$746$			26.0000i			0.951928i
$747$		0			0
$748$	−16.0000			−0.585018
$749$		0			0
$750$		0			0
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$		0			0
$753$			2.00000i			0.0728841i
$754$	−8.00000	−	12.0000i	−0.291343	−	0.437014i
$755$		0			0
$756$		0			0
$757$			26.0000i			0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$			26.0000i			0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$			16.0000i			0.581146i
$759$			24.0000i			0.871145i
$760$		0			0
$761$			18.0000i			0.652499i	0.945284	+	0.326250i	$0.105785\pi$
$761$			18.0000i			0.652499i	−0.945284	+	0.326250i	$0.894215\pi$
$762$	−8.00000			−0.289809
$763$		0			0
$764$	−12.0000			−0.434145
$765$		0			0
$766$	−16.0000			−0.578103
$767$	12.0000	−	8.00000i	0.433295	−	0.288863i
$768$			1.00000i			0.0360844i
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$	14.0000			0.503871
$773$	26.0000			0.935155			0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000			0.935155			0.467578	+	0.883952i	$0.345127\pi$
$774$		−	12.0000i		−	0.431331i
$775$		0			0
$776$	14.0000			0.502571
$777$		0			0
$778$	20.0000			0.717035
$779$	−8.00000			−0.286630
$780$		0			0
$781$	−48.0000			−1.71758
$782$	24.0000			0.858238
$783$			4.00000i			0.142948i
$784$	−7.00000			−0.250000
$785$		0			0
$786$		−	18.0000i		−	0.642039i
$787$	38.0000			1.35455			0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000			1.35455			0.677277	+	0.735728i	$0.263160\pi$
$788$	26.0000			0.926212
$789$	−14.0000			−0.498413
$790$		0			0
$791$		0			0
$792$		−	4.00000i		−	0.142134i
$793$	−20.0000	−	30.0000i	−0.710221	−	1.06533i
$794$	−28.0000			−0.993683
$795$		0			0
$796$		0			0
$797$			42.0000i			1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$			42.0000i			1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	10.0000i		−	0.353333i
$802$		−	26.0000i		−	0.918092i
$803$			24.0000i			0.846942i
$804$			10.0000i			0.352673i
$805$		0			0
$806$	−30.0000	+	20.0000i	−1.05670	+	0.704470i
$807$		0			0
$808$	8.00000			0.281439
$809$	34.0000			1.19538			0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000			1.19538			0.597688	+	0.801729i	$0.296086\pi$
$810$		0			0
$811$			52.0000i			1.82597i	0.407997	+	0.912983i	$0.366228\pi$
$811$			52.0000i			1.82597i	−0.407997	+	0.912983i	$0.633772\pi$
$812$		0			0
$813$	2.00000			0.0701431
$814$			16.0000i			0.560800i
$815$		0			0
$816$	−4.00000			−0.140028
$817$	48.0000			1.67931
$818$			8.00000i			0.279713i
$819$		0			0
$820$		0			0
$821$		−	34.0000i		−	1.18661i	−0.804978	−	0.593304i	$-0.797823\pi$
$821$		−	34.0000i		−	1.18661i	0.804978	−	0.593304i	$-0.202177\pi$
$822$			2.00000i			0.0697580i
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$		−	4.00000i		−	0.139347i
$825$		0			0
$826$		0			0
$827$	52.0000			1.80822			0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000			1.80822			0.904109	+	0.427303i	$0.140536\pi$
$828$			6.00000i			0.208514i
$829$	22.0000			0.764092			0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000			0.764092			0.382046	+	0.924143i	$0.375220\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$	2.00000	+	3.00000i	0.0693375	+	0.104006i
$833$		−	28.0000i		−	0.970143i
$834$		−	16.0000i		−	0.554035i
$835$		0			0
$836$	16.0000			0.553372
$837$	10.0000			0.345651
$838$	−14.0000			−0.483622
$839$			48.0000i			1.65714i	0.559883	+	0.828572i	$0.310846\pi$
$839$			48.0000i			1.65714i	−0.559883	+	0.828572i	$0.689154\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		−	14.0000i		−	0.482472i
$843$	30.0000			1.03325
$844$	24.0000			0.826114
$845$		0			0
$846$		0			0
$847$		0			0
$848$			2.00000i			0.0686803i
$849$	4.00000			0.137280
$850$		0			0
$851$		−	24.0000i		−	0.822709i
$852$	−12.0000			−0.411113
$853$	44.0000			1.50653			0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000			1.50653			0.753266	+	0.657716i	$0.228477\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	52.0000i		−	1.77629i	−0.459567	−	0.888143i	$-0.651995\pi$
$857$		−	52.0000i		−	1.77629i	0.459567	−	0.888143i	$-0.348005\pi$
$858$	−8.00000	−	12.0000i	−0.273115	−	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
$862$		−	12.0000i		−	0.408722i
$863$	−8.00000			−0.272323			−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000			−0.272323			−0.136162	+	0.990687i	$0.543477\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		0			0
$866$		−	18.0000i		−	0.611665i
$867$			1.00000i			0.0339618i
$868$		0			0
$869$			32.0000i			1.08553i
$870$		0			0
$871$	20.0000	+	30.0000i	0.677674	+	1.01651i
$872$		−	6.00000i		−	0.203186i
$873$	−14.0000			−0.473828
$874$	−24.0000			−0.811812
$875$		0			0
$876$			6.00000i			0.202721i
$877$	−48.0000			−1.62084			−0.810422	−	0.585846i	$-0.800762\pi$
$877$	−48.0000			−1.62084			−0.810422	+	0.585846i	$0.800762\pi$
$878$	32.0000			1.07995
$879$			14.0000i			0.472208i
$880$		0			0
$881$	−26.0000			−0.875962			−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000			−0.875962			−0.437981	+	0.898984i	$0.644306\pi$
$882$	7.00000			0.235702
$883$		−	4.00000i		−	0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$		−	4.00000i		−	0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−12.0000	+	8.00000i	−0.403604	+	0.269069i
$885$		0			0
$886$			36.0000i			1.20944i
$887$		−	30.0000i		−	1.00730i	−0.863907	−	0.503651i	$-0.831990\pi$
$887$		−	30.0000i		−	1.00730i	0.863907	−	0.503651i	$-0.168010\pi$
$888$			4.00000i			0.134231i
$889$		0			0
$890$		0			0
$891$			4.00000i			0.134005i
$892$	20.0000			0.669650
$893$		0			0
$894$	−2.00000			−0.0668900
$895$		0			0
$896$		0			0
$897$	12.0000	+	18.0000i	0.400668	+	0.601003i
$898$		−	30.0000i		−	1.00111i
$899$		−	40.0000i		−	1.33407i
$900$		0			0
$901$	−8.00000			−0.266519
$902$	8.00000			0.266371
$903$		0			0
$904$		−	12.0000i		−	0.399114i
$905$		0			0
$906$	22.0000			0.730901
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	−16.0000			−0.530979
$909$	−8.00000			−0.265343
$910$		0			0
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$	4.00000			0.132453
$913$		0			0
$914$	14.0000			0.463079
$915$		0			0
$916$		−	2.00000i		−	0.0660819i
$917$		0			0
$918$	4.00000			0.132020
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$			2.00000i			0.0659022i
$922$			26.0000i			0.856264i
$923$	−36.0000	+	24.0000i	−1.18495	+	0.789970i
$924$		0			0
$925$		0			0
$926$	20.0000			0.657241
$927$			4.00000i			0.131377i
$928$	−4.00000			−0.131306
$929$			10.0000i			0.328089i	0.986453	+	0.164045i	$0.0524541\pi$
$929$			10.0000i			0.328089i	−0.986453	+	0.164045i	$0.947546\pi$
$930$		0			0
$931$			28.0000i			0.917663i
$932$		−	24.0000i		−	0.786146i
$933$			12.0000i			0.392862i
$934$		−	4.00000i		−	0.130884i
$935$		0			0
$936$	−2.00000	−	3.00000i	−0.0653720	−	0.0980581i
$937$		−	6.00000i		−	0.196011i	−0.995186	−	0.0980057i	$-0.968754\pi$
$937$		−	6.00000i		−	0.196011i	0.995186	−	0.0980057i	$-0.0312463\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		0			0
$941$			18.0000i			0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$			18.0000i			0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	14.0000			0.456145
$943$	−12.0000			−0.390774
$944$		−	4.00000i		−	0.130189i
$945$		0			0
$946$	−48.0000			−1.56061
$947$	48.0000			1.55979			0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000			1.55979			0.779895	+	0.625910i	$0.215272\pi$
$948$			8.00000i			0.259828i
$949$	12.0000	+	18.0000i	0.389536	+	0.584305i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		−	2.00000i		−	0.0647524i
$955$		0			0
$956$		0			0
$957$	16.0000			0.517207
$958$		−	16.0000i		−	0.516937i
$959$		0			0
$960$		0			0
$961$	−69.0000			−2.22581
$962$	8.00000	+	12.0000i	0.257930	+	0.386896i
$963$		0			0
$964$			4.00000i			0.128831i
$965$		0			0
$966$		0			0
$967$	−48.0000			−1.54358			−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000			−1.54358			−0.771788	+	0.635880i	$0.780637\pi$
$968$	−5.00000			−0.160706
$969$			16.0000i			0.513994i
$970$		0			0
$971$	−34.0000			−1.09111			−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000			−1.09111			−0.545556	+	0.838074i	$0.683681\pi$
$972$			1.00000i			0.0320750i
$973$		0			0
$974$	24.0000			0.769010
$975$		0			0
$976$	−10.0000			−0.320092
$977$	−2.00000			−0.0639857			−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000			−0.0639857			−0.0319928	+	0.999488i	$0.510185\pi$
$978$		−	2.00000i		−	0.0639529i
$979$	−40.0000			−1.27841
$980$		0			0
$981$			6.00000i			0.191565i
$982$	22.0000			0.702048
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	2.00000			0.0637577
$985$		0			0
$986$		−	16.0000i		−	0.509544i
$987$		0			0
$988$	12.0000	−	8.00000i	0.381771	−	0.254514i
$989$	72.0000			2.28947
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$			10.0000i			0.317500i
$993$	32.0000			1.01549
$994$		0			0
$995$		0			0
$996$		0			0
$997$			42.0000i			1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$			42.0000i			1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$			4.00000i			0.126618i
$999$		−	4.00000i		−	0.126554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.f.i.649.2		2
5.2	odd	4		1950.2.b.g.1351.2		2
5.3	odd	4		390.2.b.a.181.1	✓	2
5.4	even	2		1950.2.f.b.649.1		2
13.12	even	2		1950.2.f.b.649.2		2
15.8	even	4		1170.2.b.b.181.2		2
20.3	even	4		3120.2.g.f.961.2		2
65.8	even	4		5070.2.a.m.1.1		1
65.12	odd	4		1950.2.b.g.1351.1		2
65.18	even	4		5070.2.a.g.1.1		1
65.38	odd	4		390.2.b.a.181.2	yes	2
65.64	even	2	inner	1950.2.f.i.649.1		2
195.38	even	4		1170.2.b.b.181.1		2
260.103	even	4		3120.2.g.f.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.a.181.1	✓	2	5.3	odd	4
390.2.b.a.181.2	yes	2	65.38	odd	4
1170.2.b.b.181.1		2	195.38	even	4
1170.2.b.b.181.2		2	15.8	even	4
1950.2.b.g.1351.1		2	65.12	odd	4
1950.2.b.g.1351.2		2	5.2	odd	4
1950.2.f.b.649.1		2	5.4	even	2
1950.2.f.b.649.2		2	13.12	even	2
1950.2.f.i.649.1		2	65.64	even	2	inner
1950.2.f.i.649.2		2	1.1	even	1	trivial
3120.2.g.f.961.1		2	260.103	even	4
3120.2.g.f.961.2		2	20.3	even	4
5070.2.a.g.1.1		1	65.18	even	4
5070.2.a.m.1.1		1	65.8	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +1.00000 q^{8} -1.00000 q^{9} +4.00000i q^{11} +1.00000i q^{12} +(2.00000 + 3.00000i) q^{13} +1.00000 q^{16} +4.00000i q^{17} -1.00000 q^{18} -4.00000i q^{19} +4.00000i q^{22} -6.00000i q^{23} +1.00000i q^{24} +(2.00000 + 3.00000i) q^{26} -1.00000i q^{27} -4.00000 q^{29} +10.0000i q^{31} +1.00000 q^{32} -4.00000 q^{33} +4.00000i q^{34} -1.00000 q^{36} +4.00000 q^{37} -4.00000i q^{38} +(-3.00000 + 2.00000i) q^{39} -2.00000i q^{41} +12.0000i q^{43} +4.00000i q^{44} -6.00000i q^{46} +1.00000i q^{48} -7.00000 q^{49} -4.00000 q^{51} +(2.00000 + 3.00000i) q^{52} +2.00000i q^{53} -1.00000i q^{54} +4.00000 q^{57} -4.00000 q^{58} -4.00000i q^{59} -10.0000 q^{61} +10.0000i q^{62} +1.00000 q^{64} -4.00000 q^{66} +10.0000 q^{67} +4.00000i q^{68} +6.00000 q^{69} +12.0000i q^{71} -1.00000 q^{72} +6.00000 q^{73} +4.00000 q^{74} -4.00000i q^{76} +(-3.00000 + 2.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +12.0000i q^{86} -4.00000i q^{87} +4.00000i q^{88} +10.0000i q^{89} -6.00000i q^{92} -10.0000 q^{93} +1.00000i q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 4 q^{13} + 2 q^{16} - 2 q^{18} + 4 q^{26} - 8 q^{29} + 2 q^{32} - 8 q^{33} - 2 q^{36} + 8 q^{37} - 6 q^{39} - 14 q^{49} - 8 q^{51} + 4 q^{52} + 8 q^{57} - 8 q^{58} - 20 q^{61} + 2 q^{64} - 8 q^{66} + 20 q^{67} + 12 q^{69} - 2 q^{72} + 12 q^{73} + 8 q^{74} - 6 q^{78} + 16 q^{79} + 2 q^{81} - 20 q^{93} + 28 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 4 * q^13 + 2 * q^16 - 2 * q^18 + 4 * q^26 - 8 * q^29 + 2 * q^32 - 8 * q^33 - 2 * q^36 + 8 * q^37 - 6 * q^39 - 14 * q^49 - 8 * q^51 + 4 * q^52 + 8 * q^57 - 8 * q^58 - 20 * q^61 + 2 * q^64 - 8 * q^66 + 20 * q^67 + 12 * q^69 - 2 * q^72 + 12 * q^73 + 8 * q^74 - 6 * q^78 + 16 * q^79 + 2 * q^81 - 20 * q^93 + 28 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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