LMFDB - Embedded newform 1950.2.f.m.649.3

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:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	1950.649
Dual form			1950.2.f.m.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +1.00000i q^{12} +3.60555i q^{13} +2.60555 q^{14} +1.00000 q^{16} -2.60555i q^{17} +1.00000 q^{18} -2.60555i q^{19} -2.60555i q^{21} +8.60555i q^{23} -1.00000i q^{24} -3.60555i q^{26} -1.00000i q^{27} -2.60555 q^{28} +2.60555 q^{29} +6.00000i q^{31} -1.00000 q^{32} +2.60555i q^{34} -1.00000 q^{36} -5.21110 q^{37} +2.60555i q^{38} -3.60555 q^{39} -11.2111i q^{41} +2.60555i q^{42} -8.00000i q^{43} -8.60555i q^{46} -5.21110 q^{47} +1.00000i q^{48} -0.211103 q^{49} +2.60555 q^{51} +3.60555i q^{52} -6.00000i q^{53} +1.00000i q^{54} +2.60555 q^{56} +2.60555 q^{57} -2.60555 q^{58} -5.21110i q^{59} +3.21110 q^{61} -6.00000i q^{62} +2.60555 q^{63} +1.00000 q^{64} -11.2111 q^{67} -2.60555i q^{68} -8.60555 q^{69} -5.21110i q^{71} +1.00000 q^{72} -8.60555 q^{73} +5.21110 q^{74} -2.60555i q^{76} +3.60555 q^{78} -14.4222 q^{79} +1.00000 q^{81} +11.2111i q^{82} +17.2111 q^{83} -2.60555i q^{84} +8.00000i q^{86} +2.60555i q^{87} -0.788897i q^{89} -9.39445i q^{91} +8.60555i q^{92} -6.00000 q^{93} +5.21110 q^{94} -1.00000i q^{96} -8.60555 q^{97} +0.211103 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9} - 4 q^{14} + 4 q^{16} + 4 q^{18} + 4 q^{28} - 4 q^{29} - 4 q^{32} - 4 q^{36} + 8 q^{37} + 8 q^{47} + 28 q^{49} - 4 q^{51} - 4 q^{56} - 4 q^{57} + 4 q^{58} - 16 q^{61} - 4 q^{63} + 4 q^{64} - 16 q^{67} - 20 q^{69} + 4 q^{72} - 20 q^{73} - 8 q^{74} + 4 q^{81} + 40 q^{83} - 24 q^{93} - 8 q^{94} - 20 q^{97} - 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 4 * q^9 - 4 * q^14 + 4 * q^16 + 4 * q^18 + 4 * q^28 - 4 * q^29 - 4 * q^32 - 4 * q^36 + 8 * q^37 + 8 * q^47 + 28 * q^49 - 4 * q^51 - 4 * q^56 - 4 * q^57 + 4 * q^58 - 16 * q^61 - 4 * q^63 + 4 * q^64 - 16 * q^67 - 20 * q^69 + 4 * q^72 - 20 * q^73 - 8 * q^74 + 4 * q^81 + 40 * q^83 - 24 * q^93 - 8 * q^94 - 20 * q^97 - 28 * 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
$7$	−2.60555	−0.984806	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	−2.60555	−0.984806	−0.492403	+	0.870367i	$0.663881\pi$
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.00000i	0.288675i
$13$	3.60555i	1.00000i
$13$	3.60555i	1.00000i
$14$	2.60555	0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.60555i	− 0.631939i	−0.948769	−	0.315970i	$-0.897670\pi$
$17$	− 2.60555i	− 0.631939i	0.948769	−	0.315970i	$-0.102330\pi$
$18$	1.00000	0.235702
$19$	− 2.60555i	− 0.597754i	−0.954292	−	0.298877i	$-0.903388\pi$
$19$	− 2.60555i	− 0.597754i	0.954292	−	0.298877i	$-0.0966121\pi$
$20$	0	0
$21$	− 2.60555i	− 0.568578i
$22$	0	0
$23$	8.60555i	1.79438i	0.441643	+	0.897191i	$0.354396\pi$
$23$	8.60555i	1.79438i	−0.441643	+	0.897191i	$0.645604\pi$
$24$	− 1.00000i	− 0.204124i
$25$	0	0
$26$	− 3.60555i	− 0.707107i
$27$	− 1.00000i	− 0.192450i
$28$	−2.60555	−0.492403
$29$	2.60555	0.483839	0.241919	−	0.970296i	$-0.422223\pi$
$29$	2.60555	0.483839	0.241919	+	0.970296i	$0.422223\pi$
$30$	0	0
$31$	6.00000i	1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$	6.00000i	1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.60555i	0.446848i
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−5.21110	−0.856700	−0.428350	−	0.903613i	$-0.640905\pi$
$37$	−5.21110	−0.856700	−0.428350	+	0.903613i	$0.640905\pi$
$38$	2.60555i	0.422676i
$39$	−3.60555	−0.577350
$40$	0	0
$41$	− 11.2111i	− 1.75088i	−0.483327	−	0.875440i	$-0.660572\pi$
$41$	− 11.2111i	− 1.75088i	0.483327	−	0.875440i	$-0.339428\pi$
$42$	2.60555i	0.402045i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	− 8.60555i	− 1.26882i
$47$	−5.21110	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	1.00000i	0.144338i
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	2.60555	0.364850
$52$	3.60555i	0.500000i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000i	0.136083i
$55$	0	0
$56$	2.60555	0.348181
$57$	2.60555	0.345114
$58$	−2.60555	−0.342126
$59$	− 5.21110i	− 0.678428i	−0.940709	−	0.339214i	$-0.889839\pi$
$59$	− 5.21110i	− 0.678428i	0.940709	−	0.339214i	$-0.110161\pi$
$60$	0	0
$61$	3.21110	0.411140	0.205570	−	0.978642i	$-0.434095\pi$
$61$	3.21110	0.411140	0.205570	+	0.978642i	$0.434095\pi$
$62$	− 6.00000i	− 0.762001i
$63$	2.60555	0.328269
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−11.2111	−1.36965	−0.684827	−	0.728706i	$-0.740122\pi$
$67$	−11.2111	−1.36965	−0.684827	+	0.728706i	$0.740122\pi$
$68$	− 2.60555i	− 0.315970i
$69$	−8.60555	−1.03599
$70$	0	0
$71$	− 5.21110i	− 0.618444i	−0.950990	−	0.309222i	$-0.899931\pi$
$71$	− 5.21110i	− 0.618444i	0.950990	−	0.309222i	$-0.100069\pi$
$72$	1.00000	0.117851
$73$	−8.60555	−1.00720	−0.503602	−	0.863936i	$-0.667992\pi$
$73$	−8.60555	−1.00720	−0.503602	+	0.863936i	$0.667992\pi$
$74$	5.21110	0.605778
$75$	0	0
$76$	− 2.60555i	− 0.298877i
$77$	0	0
$78$	3.60555	0.408248
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.2111i	1.23806i
$83$	17.2111	1.88916	0.944582	−	0.328276i	$-0.106467\pi$
$83$	17.2111	1.88916	0.944582	+	0.328276i	$0.106467\pi$
$84$	− 2.60555i	− 0.284289i
$85$	0	0
$86$	8.00000i	0.862662i
$87$	2.60555i	0.279344i
$88$	0	0
$89$	− 0.788897i	− 0.0836230i	−0.999126	−	0.0418115i	$-0.986687\pi$
$89$	− 0.788897i	− 0.0836230i	0.999126	−	0.0418115i	$-0.0133129\pi$
$90$	0	0
$91$	− 9.39445i	− 0.984806i
$92$	8.60555i	0.897191i
$93$	−6.00000	−0.622171
$94$	5.21110	0.537484
$95$	0	0
$96$	− 1.00000i	− 0.102062i
$97$	−8.60555	−0.873761	−0.436881	−	0.899519i	$-0.643917\pi$
$97$	−8.60555	−0.873761	−0.436881	+	0.899519i	$0.643917\pi$
$98$	0.211103	0.0213246
$99$	0	0
$100$	0	0
$101$	−14.6056	−1.45331	−0.726653	−	0.687004i	$-0.758925\pi$
$101$	−14.6056	−1.45331	−0.726653	+	0.687004i	$0.758925\pi$
$102$	−2.60555	−0.257988
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	− 3.60555i	− 0.353553i
$105$	0	0
$106$	6.00000i	0.582772i
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	− 8.60555i	− 0.824262i	−0.911125	−	0.412131i	$-0.864785\pi$
$109$	− 8.60555i	− 0.824262i	0.911125	−	0.412131i	$-0.135215\pi$
$110$	0	0
$111$	− 5.21110i	− 0.494616i
$112$	−2.60555	−0.246201
$113$	− 7.81665i	− 0.735329i	−0.929959	−	0.367664i	$-0.880157\pi$
$113$	− 7.81665i	− 0.735329i	0.929959	−	0.367664i	$-0.119843\pi$
$114$	−2.60555	−0.244032
$115$	0	0
$116$	2.60555	0.241919
$117$	− 3.60555i	− 0.333333i
$118$	5.21110i	0.479721i
$119$	6.78890i	0.622337i
$120$	0	0
$121$	11.0000	1.00000
$122$	−3.21110	−0.290720
$123$	11.2111	1.01087
$124$	6.00000i	0.538816i
$125$	0	0
$126$	−2.60555	−0.232121
$127$	− 13.2111i	− 1.17230i	−0.810204	−	0.586148i	$-0.800644\pi$
$127$	− 13.2111i	− 1.17230i	0.810204	−	0.586148i	$-0.199356\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	15.3944	1.34502	0.672510	−	0.740088i	$-0.265216\pi$
$131$	15.3944	1.34502	0.672510	+	0.740088i	$0.265216\pi$
$132$	0	0
$133$	6.78890i	0.588672i
$134$	11.2111	0.968492
$135$	0	0
$136$	2.60555i	0.223424i
$137$	11.2111	0.957829	0.478915	−	0.877862i	$-0.341030\pi$
$137$	11.2111	0.957829	0.478915	+	0.877862i	$0.341030\pi$
$138$	8.60555	0.732553
$139$	−2.78890	−0.236551	−0.118276	−	0.992981i	$-0.537737\pi$
$139$	−2.78890	−0.236551	−0.118276	+	0.992981i	$0.537737\pi$
$140$	0	0
$141$	− 5.21110i	− 0.438854i
$142$	5.21110i	0.437306i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.60555	0.712200
$147$	− 0.211103i	− 0.0174114i
$148$	−5.21110	−0.428350
$149$	− 0.788897i	− 0.0646290i	−0.999478	−	0.0323145i	$-0.989712\pi$
$149$	− 0.788897i	− 0.0646290i	0.999478	−	0.0323145i	$-0.0102878\pi$
$150$	0	0
$151$	6.00000i	0.488273i	0.969741	+	0.244137i	$0.0785045\pi$
$151$	6.00000i	0.488273i	−0.969741	+	0.244137i	$0.921495\pi$
$152$	2.60555i	0.211338i
$153$	2.60555i	0.210646i
$154$	0	0
$155$	0	0
$156$	−3.60555	−0.288675
$157$	− 8.42221i	− 0.672165i	−0.941833	−	0.336083i	$-0.890898\pi$
$157$	− 8.42221i	− 0.672165i	0.941833	−	0.336083i	$-0.109102\pi$
$158$	14.4222	1.14737
$159$	6.00000	0.475831
$160$	0	0
$161$	− 22.4222i	− 1.76712i
$162$	−1.00000	−0.0785674
$163$	−4.42221	−0.346374	−0.173187	−	0.984889i	$-0.555406\pi$
$163$	−4.42221	−0.346374	−0.173187	+	0.984889i	$0.555406\pi$
$164$	− 11.2111i	− 0.875440i
$165$	0	0
$166$	−17.2111	−1.33584
$167$	5.21110	0.403247	0.201624	−	0.979463i	$-0.435378\pi$
$167$	5.21110	0.403247	0.201624	+	0.979463i	$0.435378\pi$
$168$	2.60555i	0.201023i
$169$	−13.0000	−1.00000
$170$	0	0
$171$	2.60555i	0.199251i
$172$	− 8.00000i	− 0.609994i
$173$	− 16.4222i	− 1.24856i	−0.781202	−	0.624279i	$-0.785393\pi$
$173$	− 16.4222i	− 1.24856i	0.781202	−	0.624279i	$-0.214607\pi$
$174$	− 2.60555i	− 0.197526i
$175$	0	0
$176$	0	0
$177$	5.21110	0.391690
$178$	0.788897i	0.0591304i
$179$	−1.81665	−0.135783	−0.0678915	−	0.997693i	$-0.521627\pi$
$179$	−1.81665	−0.135783	−0.0678915	+	0.997693i	$0.521627\pi$
$180$	0	0
$181$	−20.4222	−1.51797	−0.758985	−	0.651108i	$-0.774305\pi$
$181$	−20.4222	−1.51797	−0.758985	+	0.651108i	$0.774305\pi$
$182$	9.39445i	0.696363i
$183$	3.21110i	0.237372i
$184$	− 8.60555i	− 0.634410i
$185$	0	0
$186$	6.00000	0.439941
$187$	0	0
$188$	−5.21110	−0.380059
$189$	2.60555i	0.189526i
$190$	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
$193$	−13.8167	−0.994545	−0.497272	−	0.867595i	$-0.665665\pi$
$193$	−13.8167	−0.994545	−0.497272	+	0.867595i	$0.665665\pi$
$194$	8.60555	0.617843
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	6.42221	0.455258	0.227629	−	0.973748i	$-0.426903\pi$
$199$	6.42221	0.455258	0.227629	+	0.973748i	$0.426903\pi$
$200$	0	0
$201$	− 11.2111i	− 0.790770i
$202$	14.6056	1.02764
$203$	−6.78890	−0.476487
$204$	2.60555	0.182425
$205$	0	0
$206$	− 4.00000i	− 0.278693i
$207$	− 8.60555i	− 0.598127i
$208$	3.60555i	0.250000i
$209$	0	0
$210$	0	0
$211$	−2.78890	−0.191996	−0.0959978	−	0.995382i	$-0.530604\pi$
$211$	−2.78890	−0.191996	−0.0959978	+	0.995382i	$0.530604\pi$
$212$	− 6.00000i	− 0.412082i
$213$	5.21110	0.357059
$214$	0	0
$215$	0	0
$216$	1.00000i	0.0680414i
$217$	− 15.6333i	− 1.06126i
$218$	8.60555i	0.582841i
$219$	− 8.60555i	− 0.581509i
$220$	0	0
$221$	9.39445	0.631939
$222$	5.21110i	0.349746i
$223$	−19.8167	−1.32702	−0.663511	−	0.748167i	$-0.730934\pi$
$223$	−19.8167	−1.32702	−0.663511	+	0.748167i	$0.730934\pi$
$224$	2.60555	0.174091
$225$	0	0
$226$	7.81665i	0.519956i
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	2.60555	0.172557
$229$	− 1.81665i	− 0.120048i	−0.998197	−	0.0600239i	$-0.980882\pi$
$229$	− 1.81665i	− 0.120048i	0.998197	−	0.0600239i	$-0.0191177\pi$
$230$	0	0
$231$	0	0
$232$	−2.60555	−0.171063
$233$	− 19.8167i	− 1.29823i	−0.760689	−	0.649116i	$-0.775139\pi$
$233$	− 19.8167i	− 1.29823i	0.760689	−	0.649116i	$-0.224861\pi$
$234$	3.60555i	0.235702i
$235$	0	0
$236$	− 5.21110i	− 0.339214i
$237$	− 14.4222i	− 0.936823i
$238$	− 6.78890i	− 0.440059i
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	22.4222i	1.44434i	0.691715	+	0.722171i	$0.256855\pi$
$241$	22.4222i	1.44434i	−0.691715	+	0.722171i	$0.743145\pi$
$242$	−11.0000	−0.707107
$243$	1.00000i	0.0641500i
$244$	3.21110	0.205570
$245$	0	0
$246$	−11.2111	−0.714794
$247$	9.39445	0.597754
$248$	− 6.00000i	− 0.381000i
$249$	17.2111i	1.09071i
$250$	0	0
$251$	−20.6056	−1.30061	−0.650305	−	0.759673i	$-0.725359\pi$
$251$	−20.6056	−1.30061	−0.650305	+	0.759673i	$0.725359\pi$
$252$	2.60555	0.164134
$253$	0	0
$254$	13.2111i	0.828938i
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 21.3944i	− 1.33455i	−0.744812	−	0.667275i	$-0.767461\pi$
$257$	− 21.3944i	− 1.33455i	0.744812	−	0.667275i	$-0.232539\pi$
$258$	−8.00000	−0.498058
$259$	13.5778	0.843683
$260$	0	0
$261$	−2.60555	−0.161280
$262$	−15.3944	−0.951072
$263$	− 13.8167i	− 0.851971i	−0.904730	−	0.425986i	$-0.859927\pi$
$263$	− 13.8167i	− 0.851971i	0.904730	−	0.425986i	$-0.140073\pi$
$264$	0	0
$265$	0	0
$266$	− 6.78890i	− 0.416254i
$267$	0.788897	0.0482797
$268$	−11.2111	−0.684827
$269$	−4.18335	−0.255063	−0.127532	−	0.991835i	$-0.540705\pi$
$269$	−4.18335	−0.255063	−0.127532	+	0.991835i	$0.540705\pi$
$270$	0	0
$271$	28.4222i	1.72653i	0.504754	+	0.863263i	$0.331583\pi$
$271$	28.4222i	1.72653i	−0.504754	+	0.863263i	$0.668417\pi$
$272$	− 2.60555i	− 0.157985i
$273$	9.39445	0.568578
$274$	−11.2111	−0.677287
$275$	0	0
$276$	−8.60555	−0.517993
$277$	12.4222i	0.746378i	0.927755	+	0.373189i	$0.121736\pi$
$277$	12.4222i	0.746378i	−0.927755	+	0.373189i	$0.878264\pi$
$278$	2.78890	0.167267
$279$	− 6.00000i	− 0.359211i
$280$	0	0
$281$	12.7889i	0.762922i	0.924385	+	0.381461i	$0.124579\pi$
$281$	12.7889i	0.762922i	−0.924385	+	0.381461i	$0.875421\pi$
$282$	5.21110i	0.310317i
$283$	18.4222i	1.09509i	0.836777	+	0.547543i	$0.184437\pi$
$283$	18.4222i	1.09509i	−0.836777	+	0.547543i	$0.815563\pi$
$284$	− 5.21110i	− 0.309222i
$285$	0	0
$286$	0	0
$287$	29.2111i	1.72428i
$288$	1.00000	0.0589256
$289$	10.2111	0.600653
$290$	0	0
$291$	− 8.60555i	− 0.504466i
$292$	−8.60555	−0.503602
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0.211103i	0.0123118i
$295$	0	0
$296$	5.21110	0.302889
$297$	0	0
$298$	0.788897i	0.0456996i
$299$	−31.0278	−1.79438
$300$	0	0
$301$	20.8444i	1.20145i
$302$	− 6.00000i	− 0.345261i
$303$	− 14.6056i	− 0.839067i
$304$	− 2.60555i	− 0.149439i
$305$	0	0
$306$	− 2.60555i	− 0.148949i
$307$	23.2111	1.32473	0.662364	−	0.749182i	$-0.269553\pi$
$307$	23.2111	1.32473	0.662364	+	0.749182i	$0.269553\pi$
$308$	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	$0.389495\pi$
$312$	3.60555	0.204124
$313$	− 32.4222i	− 1.83261i	−0.400480	−	0.916306i	$-0.631156\pi$
$313$	− 32.4222i	− 1.83261i	0.400480	−	0.916306i	$-0.368844\pi$
$314$	8.42221i	0.475293i
$315$	0	0
$316$	−14.4222	−0.811312
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	0	0
$322$	22.4222i	1.24954i
$323$	−6.78890	−0.377744
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.42221	0.244923
$327$	8.60555	0.475888
$328$	11.2111i	0.619030i
$329$	13.5778	0.748568
$330$	0	0
$331$	− 9.39445i	− 0.516366i	−0.966096	−	0.258183i	$-0.916876\pi$
$331$	− 9.39445i	− 0.516366i	0.966096	−	0.258183i	$-0.0831237\pi$
$332$	17.2111	0.944582
$333$	5.21110	0.285567
$334$	−5.21110	−0.285139
$335$	0	0
$336$	− 2.60555i	− 0.142144i
$337$	− 29.6333i	− 1.61423i	−0.590395	−	0.807115i	$-0.701028\pi$
$337$	− 29.6333i	− 1.61423i	0.590395	−	0.807115i	$-0.298972\pi$
$338$	13.0000	0.707107
$339$	7.81665	0.424542
$340$	0	0
$341$	0	0
$342$	− 2.60555i	− 0.140892i
$343$	18.7889	1.01451
$344$	8.00000i	0.431331i
$345$	0	0
$346$	16.4222i	0.882863i
$347$	15.6333i	0.839240i	0.907700	+	0.419620i	$0.137837\pi$
$347$	15.6333i	0.839240i	−0.907700	+	0.419620i	$0.862163\pi$
$348$	2.60555i	0.139672i
$349$	− 13.8167i	− 0.739589i	−0.929114	−	0.369794i	$-0.879428\pi$
$349$	− 13.8167i	− 0.739589i	0.929114	−	0.369794i	$-0.120572\pi$
$350$	0	0
$351$	3.60555	0.192450
$352$	0	0
$353$	−23.2111	−1.23540	−0.617701	−	0.786413i	$-0.711936\pi$
$353$	−23.2111	−1.23540	−0.617701	+	0.786413i	$0.711936\pi$
$354$	−5.21110	−0.276967
$355$	0	0
$356$	− 0.788897i	− 0.0418115i
$357$	−6.78890	−0.359307
$358$	1.81665	0.0960131
$359$	27.6333i	1.45843i	0.684285	+	0.729215i	$0.260115\pi$
$359$	27.6333i	1.45843i	−0.684285	+	0.729215i	$0.739885\pi$
$360$	0	0
$361$	12.2111	0.642690
$362$	20.4222	1.07337
$363$	11.0000i	0.577350i
$364$	− 9.39445i	− 0.492403i
$365$	0	0
$366$	− 3.21110i	− 0.167847i
$367$	23.6333i	1.23365i	0.787101	+	0.616824i	$0.211581\pi$
$367$	23.6333i	1.23365i	−0.787101	+	0.616824i	$0.788419\pi$
$368$	8.60555i	0.448595i
$369$	11.2111i	0.583627i
$370$	0	0
$371$	15.6333i	0.811641i
$372$	−6.00000	−0.311086
$373$	− 8.42221i	− 0.436085i	−0.975939	−	0.218043i	$-0.930033\pi$
$373$	− 8.42221i	− 0.436085i	0.975939	−	0.218043i	$-0.0699672\pi$
$374$	0	0
$375$	0	0
$376$	5.21110	0.268742
$377$	9.39445i	0.483839i
$378$	− 2.60555i	− 0.134015i
$379$	1.02776i	0.0527923i	0.999652	+	0.0263961i	$0.00840313\pi$
$379$	1.02776i	0.0527923i	−0.999652	+	0.0263961i	$0.991597\pi$
$380$	0	0
$381$	13.2111	0.676825
$382$	12.0000	0.613973
$383$	15.6333	0.798825	0.399412	−	0.916771i	$-0.369214\pi$
$383$	15.6333	0.798825	0.399412	+	0.916771i	$0.369214\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	0	0
$386$	13.8167	0.703249
$387$	8.00000i	0.406663i
$388$	−8.60555	−0.436881
$389$	2.60555	0.132107	0.0660533	−	0.997816i	$-0.478959\pi$
$389$	2.60555	0.132107	0.0660533	+	0.997816i	$0.478959\pi$
$390$	0	0
$391$	22.4222	1.13394
$392$	0.211103	0.0106623
$393$	15.3944i	0.776547i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	−39.6333	−1.98914	−0.994569	−	0.104076i	$-0.966811\pi$
$397$	−39.6333	−1.98914	−0.994569	+	0.104076i	$0.966811\pi$
$398$	−6.42221	−0.321916
$399$	−6.78890	−0.339870
$400$	0	0
$401$	23.2111i	1.15911i	0.814934	+	0.579554i	$0.196773\pi$
$401$	23.2111i	1.15911i	−0.814934	+	0.579554i	$0.803227\pi$
$402$	11.2111i	0.559159i
$403$	−21.6333	−1.07763
$404$	−14.6056	−0.726653
$405$	0	0
$406$	6.78890	0.336927
$407$	0	0
$408$	−2.60555	−0.128994
$409$	29.2111i	1.44440i	0.691686	+	0.722198i	$0.256868\pi$
$409$	29.2111i	1.44440i	−0.691686	+	0.722198i	$0.743132\pi$
$410$	0	0
$411$	11.2111i	0.553003i
$412$	4.00000i	0.197066i
$413$	13.5778i	0.668120i
$414$	8.60555i	0.422940i
$415$	0	0
$416$	− 3.60555i	− 0.176777i
$417$	− 2.78890i	− 0.136573i
$418$	0	0
$419$	−25.8167	−1.26123	−0.630613	−	0.776097i	$-0.717196\pi$
$419$	−25.8167	−1.26123	−0.630613	+	0.776097i	$0.717196\pi$
$420$	0	0
$421$	1.81665i	0.0885383i	0.999020	+	0.0442691i	$0.0140959\pi$
$421$	1.81665i	0.0885383i	−0.999020	+	0.0442691i	$0.985904\pi$
$422$	2.78890	0.135761
$423$	5.21110	0.253372
$424$	6.00000i	0.291386i
$425$	0	0
$426$	−5.21110	−0.252479
$427$	−8.36669	−0.404893
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000i	0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$	12.0000i	0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 4.78890i	− 0.230140i	−0.993357	−	0.115070i	$-0.963291\pi$
$433$	− 4.78890i	− 0.230140i	0.993357	−	0.115070i	$-0.0367092\pi$
$434$	15.6333i	0.750423i
$435$	0	0
$436$	− 8.60555i	− 0.412131i
$437$	22.4222	1.07260
$438$	8.60555i	0.411189i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0.211103	0.0100525
$442$	−9.39445	−0.446848
$443$	27.6333i	1.31290i	0.754370	+	0.656449i	$0.227942\pi$
$443$	27.6333i	1.31290i	−0.754370	+	0.656449i	$0.772058\pi$
$444$	− 5.21110i	− 0.247308i
$445$	0	0
$446$	19.8167	0.938346
$447$	0.788897	0.0373136
$448$	−2.60555	−0.123101
$449$	9.63331i	0.454624i	0.973822	+	0.227312i	$0.0729937\pi$
$449$	9.63331i	0.454624i	−0.973822	+	0.227312i	$0.927006\pi$
$450$	0	0
$451$	0	0
$452$	− 7.81665i	− 0.367664i
$453$	−6.00000	−0.281905
$454$	24.0000	1.12638
$455$	0	0
$456$	−2.60555	−0.122016
$457$	12.2389	0.572510	0.286255	−	0.958154i	$-0.407590\pi$
$457$	12.2389	0.572510	0.286255	+	0.958154i	$0.407590\pi$
$458$	1.81665i	0.0848867i
$459$	−2.60555	−0.121617
$460$	0	0
$461$	9.63331i	0.448668i	0.974512	+	0.224334i	$0.0720206\pi$
$461$	9.63331i	0.448668i	−0.974512	+	0.224334i	$0.927979\pi$
$462$	0	0
$463$	−38.6056	−1.79415	−0.897076	−	0.441876i	$-0.854313\pi$
$463$	−38.6056	−1.79415	−0.897076	+	0.441876i	$0.854313\pi$
$464$	2.60555	0.120960
$465$	0	0
$466$	19.8167i	0.917989i
$467$	− 1.57779i	− 0.0730116i	−0.999333	−	0.0365058i	$-0.988377\pi$
$467$	− 1.57779i	− 0.0730116i	0.999333	−	0.0365058i	$-0.0116227\pi$
$468$	− 3.60555i	− 0.166667i
$469$	29.2111	1.34884
$470$	0	0
$471$	8.42221	0.388075
$472$	5.21110i	0.239860i
$473$	0	0
$474$	14.4222i	0.662434i
$475$	0	0
$476$	6.78890i	0.311169i
$477$	6.00000i	0.274721i
$478$	0	0
$479$	− 34.4222i	− 1.57279i	−0.617724	−	0.786395i	$-0.711945\pi$
$479$	− 34.4222i	− 1.57279i	0.617724	−	0.786395i	$-0.288055\pi$
$480$	0	0
$481$	− 18.7889i	− 0.856700i
$482$	− 22.4222i	− 1.02130i
$483$	22.4222	1.02025
$484$	11.0000	0.500000
$485$	0	0
$486$	− 1.00000i	− 0.0453609i
$487$	37.0278	1.67789	0.838944	−	0.544218i	$-0.183174\pi$
$487$	37.0278	1.67789	0.838944	+	0.544218i	$0.183174\pi$
$488$	−3.21110	−0.145360
$489$	− 4.42221i	− 0.199979i
$490$	0	0
$491$	−13.8167	−0.623537	−0.311768	−	0.950158i	$-0.600921\pi$
$491$	−13.8167	−0.623537	−0.311768	+	0.950158i	$0.600921\pi$
$492$	11.2111	0.505436
$493$	− 6.78890i	− 0.305757i
$494$	−9.39445	−0.422676
$495$	0	0
$496$	6.00000i	0.269408i
$497$	13.5778i	0.609047i
$498$	− 17.2111i	− 0.771248i
$499$	13.0278i	0.583202i	0.956540	+	0.291601i	$0.0941880\pi$
$499$	13.0278i	0.583202i	−0.956540	+	0.291601i	$0.905812\pi$
$500$	0	0
$501$	5.21110i	0.232815i
$502$	20.6056	0.919671
$503$	41.4500i	1.84816i	0.382196	+	0.924081i	$0.375168\pi$
$503$	41.4500i	1.84816i	−0.382196	+	0.924081i	$0.624832\pi$
$504$	−2.60555	−0.116060
$505$	0	0
$506$	0	0
$507$	− 13.0000i	− 0.577350i
$508$	− 13.2111i	− 0.586148i
$509$	− 9.63331i	− 0.426989i	−0.976944	−	0.213494i	$-0.931515\pi$
$509$	− 9.63331i	− 0.426989i	0.976944	−	0.213494i	$-0.0684845\pi$
$510$	0	0
$511$	22.4222	0.991900
$512$	−1.00000	−0.0441942
$513$	−2.60555	−0.115038
$514$	21.3944i	0.943669i
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	−13.5778	−0.596574
$519$	16.4222	0.720855
$520$	0	0
$521$	−21.6333	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$521$	−21.6333	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$522$	2.60555	0.114042
$523$	24.8444i	1.08637i	0.839613	+	0.543185i	$0.182782\pi$
$523$	24.8444i	1.08637i	−0.839613	+	0.543185i	$0.817218\pi$
$524$	15.3944	0.672510
$525$	0	0
$526$	13.8167i	0.602435i
$527$	15.6333	0.680998
$528$	0	0
$529$	−51.0555	−2.21980
$530$	0	0
$531$	5.21110i	0.226143i
$532$	6.78890i	0.294336i
$533$	40.4222	1.75088
$534$	−0.788897	−0.0341389
$535$	0	0
$536$	11.2111	0.484246
$537$	− 1.81665i	− 0.0783944i
$538$	4.18335	0.180357
$539$	0	0
$540$	0	0
$541$	43.0278i	1.84991i	0.380079	+	0.924954i	$0.375897\pi$
$541$	43.0278i	1.84991i	−0.380079	+	0.924954i	$0.624103\pi$
$542$	− 28.4222i	− 1.22084i
$543$	− 20.4222i	− 0.876401i
$544$	2.60555i	0.111712i
$545$	0	0
$546$	−9.39445	−0.402045
$547$	− 14.4222i	− 0.616649i	−0.951281	−	0.308324i	$-0.900232\pi$
$547$	− 14.4222i	− 0.616649i	0.951281	−	0.308324i	$-0.0997682\pi$
$548$	11.2111	0.478915
$549$	−3.21110	−0.137047
$550$	0	0
$551$	− 6.78890i	− 0.289217i
$552$	8.60555	0.366277
$553$	37.5778	1.59797
$554$	− 12.4222i	− 0.527769i
$555$	0	0
$556$	−2.78890	−0.118276
$557$	−40.4222	−1.71274	−0.856372	−	0.516360i	$-0.827287\pi$
$557$	−40.4222	−1.71274	−0.856372	+	0.516360i	$0.827287\pi$
$558$	6.00000i	0.254000i
$559$	28.8444	1.21999
$560$	0	0
$561$	0	0
$562$	− 12.7889i	− 0.539467i
$563$	38.0555i	1.60385i	0.597426	+	0.801924i	$0.296190\pi$
$563$	38.0555i	1.60385i	−0.597426	+	0.801924i	$0.703810\pi$
$564$	− 5.21110i	− 0.219427i
$565$	0	0
$566$	− 18.4222i	− 0.774343i
$567$	−2.60555	−0.109423
$568$	5.21110i	0.218653i
$569$	9.63331	0.403849	0.201925	−	0.979401i	$-0.435280\pi$
$569$	9.63331	0.403849	0.201925	+	0.979401i	$0.435280\pi$
$570$	0	0
$571$	42.0555	1.75997	0.879984	−	0.475003i	$-0.157553\pi$
$571$	42.0555	1.75997	0.879984	+	0.475003i	$0.157553\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	− 29.2111i	− 1.21925i
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	44.6056	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$577$	44.6056	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$578$	−10.2111	−0.424726
$579$	− 13.8167i	− 0.574201i
$580$	0	0
$581$	−44.8444	−1.86046
$582$	8.60555i	0.356712i
$583$	0	0
$584$	8.60555	0.356100
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−22.4222	−0.925463	−0.462732	−	0.886498i	$-0.653131\pi$
$587$	−22.4222	−0.925463	−0.462732	+	0.886498i	$0.653131\pi$
$588$	− 0.211103i	− 0.00870572i
$589$	15.6333	0.644159
$590$	0	0
$591$	6.00000i	0.246807i
$592$	−5.21110	−0.214175
$593$	4.42221	0.181598	0.0907991	−	0.995869i	$-0.471058\pi$
$593$	4.42221	0.181598	0.0907991	+	0.995869i	$0.471058\pi$
$594$	0	0
$595$	0	0
$596$	− 0.788897i	− 0.0323145i
$597$	6.42221i	0.262843i
$598$	31.0278	1.26882
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−41.6333	−1.69826	−0.849129	−	0.528185i	$-0.822873\pi$
$601$	−41.6333	−1.69826	−0.849129	+	0.528185i	$0.822873\pi$
$602$	− 20.8444i	− 0.849555i
$603$	11.2111	0.456551
$604$	6.00000i	0.244137i
$605$	0	0
$606$	14.6056i	0.593310i
$607$	2.78890i	0.113198i	0.998397	+	0.0565989i	$0.0180256\pi$
$607$	2.78890i	0.113198i	−0.998397	+	0.0565989i	$0.981974\pi$
$608$	2.60555i	0.105669i
$609$	− 6.78890i	− 0.275100i
$610$	0	0
$611$	− 18.7889i	− 0.760117i
$612$	2.60555i	0.105323i
$613$	−18.7889	−0.758876	−0.379438	−	0.925217i	$-0.623883\pi$
$613$	−18.7889	−0.758876	−0.379438	+	0.925217i	$0.623883\pi$
$614$	−23.2111	−0.936724
$615$	0	0
$616$	0	0
$617$	16.4222	0.661133	0.330567	−	0.943783i	$-0.392760\pi$
$617$	16.4222	0.661133	0.330567	+	0.943783i	$0.392760\pi$
$618$	4.00000	0.160904
$619$	− 4.18335i	− 0.168143i	−0.996460	−	0.0840714i	$-0.973208\pi$
$619$	− 4.18335i	− 0.168143i	0.996460	−	0.0840714i	$-0.0267924\pi$
$620$	0	0
$621$	8.60555	0.345329
$622$	−12.0000	−0.481156
$623$	2.05551i	0.0823524i
$624$	−3.60555	−0.144338
$625$	0	0
$626$	32.4222i	1.29585i
$627$	0	0
$628$	− 8.42221i	− 0.336083i
$629$	13.5778i	0.541382i
$630$	0	0
$631$	− 11.2111i	− 0.446307i	−0.974783	−	0.223153i	$-0.928365\pi$
$631$	− 11.2111i	− 0.446307i	0.974783	−	0.223153i	$-0.0716351\pi$
$632$	14.4222	0.573685
$633$	− 2.78890i	− 0.110849i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	− 0.761141i	− 0.0301575i
$638$	0	0
$639$	5.21110i	0.206148i
$640$	0	0
$641$	−28.4222	−1.12261	−0.561305	−	0.827609i	$-0.689700\pi$
$641$	−28.4222	−1.12261	−0.561305	+	0.827609i	$0.689700\pi$
$642$	0	0
$643$	−33.6333	−1.32637	−0.663184	−	0.748456i	$-0.730795\pi$
$643$	−33.6333	−1.32637	−0.663184	+	0.748456i	$0.730795\pi$
$644$	− 22.4222i	− 0.883559i
$645$	0	0
$646$	6.78890	0.267106
$647$	− 27.3944i	− 1.07699i	−0.842630	−	0.538493i	$-0.818994\pi$
$647$	− 27.3944i	− 1.07699i	0.842630	−	0.538493i	$-0.181006\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	0	0
$651$	15.6333	0.612718
$652$	−4.42221	−0.173187
$653$	− 24.7889i	− 0.970065i	−0.874496	−	0.485032i	$-0.838808\pi$
$653$	− 24.7889i	− 0.970065i	0.874496	−	0.485032i	$-0.161192\pi$
$654$	−8.60555	−0.336504
$655$	0	0
$656$	− 11.2111i	− 0.437720i
$657$	8.60555	0.335735
$658$	−13.5778	−0.529318
$659$	−24.2389	−0.944212	−0.472106	−	0.881542i	$-0.656506\pi$
$659$	−24.2389	−0.944212	−0.472106	+	0.881542i	$0.656506\pi$
$660$	0	0
$661$	− 0.238859i	− 0.00929054i	−0.999989	−	0.00464527i	$-0.998521\pi$
$661$	− 0.238859i	− 0.00929054i	0.999989	−	0.00464527i	$-0.00147864\pi$
$662$	9.39445i	0.365126i
$663$	9.39445i	0.364850i
$664$	−17.2111	−0.667920
$665$	0	0
$666$	−5.21110	−0.201926
$667$	22.4222i	0.868191i
$668$	5.21110	0.201624
$669$	− 19.8167i	− 0.766156i
$670$	0	0
$671$	0	0
$672$	2.60555i	0.100511i
$673$	− 5.63331i	− 0.217148i	−0.994088	−	0.108574i	$-0.965372\pi$
$673$	− 5.63331i	− 0.217148i	0.994088	−	0.108574i	$-0.0346284\pi$
$674$	29.6333i	1.14143i
$675$	0	0
$676$	−13.0000	−0.500000
$677$	44.0555i	1.69319i	0.532237	+	0.846595i	$0.321352\pi$
$677$	44.0555i	1.69319i	−0.532237	+	0.846595i	$0.678648\pi$
$678$	−7.81665	−0.300197
$679$	22.4222	0.860485
$680$	0	0
$681$	− 24.0000i	− 0.919682i
$682$	0	0
$683$	−5.21110	−0.199397	−0.0996986	−	0.995018i	$-0.531788\pi$
$683$	−5.21110	−0.199397	−0.0996986	+	0.995018i	$0.531788\pi$
$684$	2.60555i	0.0996257i
$685$	0	0
$686$	−18.7889	−0.717363
$687$	1.81665	0.0693097
$688$	− 8.00000i	− 0.304997i
$689$	21.6333	0.824163
$690$	0	0
$691$	− 30.2389i	− 1.15034i	−0.818034	−	0.575170i	$-0.804936\pi$
$691$	− 30.2389i	− 1.15034i	0.818034	−	0.575170i	$-0.195064\pi$
$692$	− 16.4222i	− 0.624279i
$693$	0	0
$694$	− 15.6333i	− 0.593432i
$695$	0	0
$696$	− 2.60555i	− 0.0987632i
$697$	−29.2111	−1.10645
$698$	13.8167i	0.522968i
$699$	19.8167	0.749535
$700$	0	0
$701$	10.9722	0.414416	0.207208	−	0.978297i	$-0.433562\pi$
$701$	10.9722	0.414416	0.207208	+	0.978297i	$0.433562\pi$
$702$	−3.60555	−0.136083
$703$	13.5778i	0.512096i
$704$	0	0
$705$	0	0
$706$	23.2111	0.873561
$707$	38.0555	1.43122
$708$	5.21110	0.195845
$709$	8.60555i	0.323188i	0.986857	+	0.161594i	$0.0516635\pi$
$709$	8.60555i	0.323188i	−0.986857	+	0.161594i	$0.948336\pi$
$710$	0	0
$711$	14.4222	0.540875
$712$	0.788897i	0.0295652i
$713$	−51.6333	−1.93368
$714$	6.78890	0.254068
$715$	0	0
$716$	−1.81665	−0.0678915
$717$	0	0
$718$	− 27.6333i	− 1.03127i
$719$	8.36669	0.312025	0.156012	−	0.987755i	$-0.450136\pi$
$719$	8.36669	0.312025	0.156012	+	0.987755i	$0.450136\pi$
$720$	0	0
$721$	− 10.4222i	− 0.388143i
$722$	−12.2111	−0.454450
$723$	−22.4222	−0.833891
$724$	−20.4222	−0.758985
$725$	0	0
$726$	− 11.0000i	− 0.408248i
$727$	14.4222i	0.534890i	0.963573	+	0.267445i	$0.0861794\pi$
$727$	14.4222i	0.534890i	−0.963573	+	0.267445i	$0.913821\pi$
$728$	9.39445i	0.348181i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−20.8444	−0.770958
$732$	3.21110i	0.118686i
$733$	38.0555	1.40561	0.702806	−	0.711381i	$-0.251930\pi$
$733$	38.0555	1.40561	0.702806	+	0.711381i	$0.251930\pi$
$734$	− 23.6333i	− 0.872321i
$735$	0	0
$736$	− 8.60555i	− 0.317205i
$737$	0	0
$738$	− 11.2111i	− 0.412686i
$739$	30.2389i	1.11235i	0.831064	+	0.556177i	$0.187732\pi$
$739$	30.2389i	1.11235i	−0.831064	+	0.556177i	$0.812268\pi$
$740$	0	0
$741$	9.39445i	0.345114i
$742$	− 15.6333i	− 0.573917i
$743$	−20.8444	−0.764707	−0.382354	−	0.924016i	$-0.624886\pi$
$743$	−20.8444	−0.764707	−0.382354	+	0.924016i	$0.624886\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	8.42221i	0.308359i
$747$	−17.2111	−0.629721
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.4222	0.672236	0.336118	−	0.941820i	$-0.390886\pi$
$751$	18.4222	0.672236	0.336118	+	0.941820i	$0.390886\pi$
$752$	−5.21110	−0.190029
$753$	− 20.6056i	− 0.750908i
$754$	− 9.39445i	− 0.342126i
$755$	0	0
$756$	2.60555i	0.0947630i
$757$	27.2111i	0.989004i	0.869176	+	0.494502i	$0.164650\pi$
$757$	27.2111i	0.989004i	−0.869176	+	0.494502i	$0.835350\pi$
$758$	− 1.02776i	− 0.0373298i
$759$	0	0
$760$	0	0
$761$	− 9.63331i	− 0.349207i	−0.984639	−	0.174604i	$-0.944136\pi$
$761$	− 9.63331i	− 0.349207i	0.984639	−	0.174604i	$-0.0558644\pi$
$762$	−13.2111	−0.478588
$763$	22.4222i	0.811738i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−15.6333	−0.564854
$767$	18.7889	0.678428
$768$	1.00000i	0.0360844i
$769$	− 44.8444i	− 1.61713i	−0.588406	−	0.808565i	$-0.700244\pi$
$769$	− 44.8444i	− 1.61713i	0.588406	−	0.808565i	$-0.299756\pi$
$770$	0	0
$771$	21.3944	0.770502
$772$	−13.8167	−0.497272
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	− 8.00000i	− 0.287554i
$775$	0	0
$776$	8.60555	0.308921
$777$	13.5778i	0.487101i
$778$	−2.60555	−0.0934135
$779$	−29.2111	−1.04660
$780$	0	0
$781$	0	0
$782$	−22.4222	−0.801816
$783$	− 2.60555i	− 0.0931148i
$784$	−0.211103	−0.00753938
$785$	0	0
$786$	− 15.3944i	− 0.549102i
$787$	37.2666	1.32841	0.664206	−	0.747550i	$-0.268770\pi$
$787$	37.2666	1.32841	0.664206	+	0.747550i	$0.268770\pi$
$788$	6.00000	0.213741
$789$	13.8167	0.491886
$790$	0	0
$791$	20.3667i	0.724156i
$792$	0	0
$793$	11.5778i	0.411140i
$794$	39.6333	1.40653
$795$	0	0
$796$	6.42221	0.227629
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	6.78890	0.240324
$799$	13.5778i	0.480348i
$800$	0	0
$801$	0.788897i	0.0278743i
$802$	− 23.2111i	− 0.819613i
$803$	0	0
$804$	− 11.2111i	− 0.395385i
$805$	0	0
$806$	21.6333	0.762001
$807$	− 4.18335i	− 0.147261i
$808$	14.6056	0.513822
$809$	−50.8444	−1.78759	−0.893797	−	0.448471i	$-0.851969\pi$
$809$	−50.8444	−1.78759	−0.893797	+	0.448471i	$0.851969\pi$
$810$	0	0
$811$	− 18.2389i	− 0.640453i	−0.947341	−	0.320226i	$-0.896241\pi$
$811$	− 18.2389i	− 0.640453i	0.947341	−	0.320226i	$-0.103759\pi$
$812$	−6.78890	−0.238244
$813$	−28.4222	−0.996810
$814$	0	0
$815$	0	0
$816$	2.60555	0.0912125
$817$	−20.8444	−0.729254
$818$	− 29.2111i	− 1.02134i
$819$	9.39445i	0.328269i
$820$	0	0
$821$	11.2111i	0.391270i	0.980677	+	0.195635i	$0.0626768\pi$
$821$	11.2111i	0.391270i	−0.980677	+	0.195635i	$0.937323\pi$
$822$	− 11.2111i	− 0.391032i
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	− 4.00000i	− 0.139347i
$825$	0	0
$826$	− 13.5778i	− 0.472432i
$827$	−15.6333	−0.543623	−0.271812	−	0.962350i	$-0.587623\pi$
$827$	−15.6333	−0.543623	−0.271812	+	0.962350i	$0.587623\pi$
$828$	− 8.60555i	− 0.299064i
$829$	10.8444	0.376642	0.188321	−	0.982108i	$-0.439695\pi$
$829$	10.8444	0.376642	0.188321	+	0.982108i	$0.439695\pi$
$830$	0	0
$831$	−12.4222	−0.430922
$832$	3.60555i	0.125000i
$833$	0.550039i	0.0190577i
$834$	2.78890i	0.0965716i
$835$	0	0
$836$	0	0
$837$	6.00000	0.207390
$838$	25.8167	0.891822
$839$	− 10.4222i	− 0.359814i	−0.983684	−	0.179907i	$-0.942420\pi$
$839$	− 10.4222i	− 0.359814i	0.983684	−	0.179907i	$-0.0575798\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	− 1.81665i	− 0.0626060i
$843$	−12.7889	−0.440473
$844$	−2.78890	−0.0959978
$845$	0	0
$846$	−5.21110	−0.179161
$847$	−28.6611	−0.984806
$848$	− 6.00000i	− 0.206041i
$849$	−18.4222	−0.632248
$850$	0	0
$851$	− 44.8444i	− 1.53725i
$852$	5.21110	0.178529
$853$	−29.2111	−1.00017	−0.500085	−	0.865977i	$-0.666698\pi$
$853$	−29.2111	−1.00017	−0.500085	+	0.865977i	$0.666698\pi$
$854$	8.36669	0.286302
$855$	0	0
$856$	0	0
$857$	13.0278i	0.445020i	0.974930	+	0.222510i	$0.0714249\pi$
$857$	13.0278i	0.445020i	−0.974930	+	0.222510i	$0.928575\pi$
$858$	0	0
$859$	−10.7889	−0.368112	−0.184056	−	0.982916i	$-0.558923\pi$
$859$	−10.7889	−0.368112	−0.184056	+	0.982916i	$0.558923\pi$
$860$	0	0
$861$	−29.2111	−0.995512
$862$	− 12.0000i	− 0.408722i
$863$	8.36669	0.284806	0.142403	−	0.989809i	$-0.454517\pi$
$863$	8.36669	0.284806	0.142403	+	0.989809i	$0.454517\pi$
$864$	1.00000i	0.0340207i
$865$	0	0
$866$	4.78890i	0.162733i
$867$	10.2111i	0.346787i
$868$	− 15.6333i	− 0.530629i
$869$	0	0
$870$	0	0
$871$	− 40.4222i	− 1.36965i
$872$	8.60555i	0.291421i
$873$	8.60555	0.291254
$874$	−22.4222	−0.758442
$875$	0	0
$876$	− 8.60555i	− 0.290755i
$877$	−32.8444	−1.10908	−0.554538	−	0.832158i	$-0.687105\pi$
$877$	−32.8444	−1.10908	−0.554538	+	0.832158i	$0.687105\pi$
$878$	8.00000	0.269987
$879$	18.0000i	0.607125i
$880$	0	0
$881$	24.7889	0.835159	0.417580	−	0.908640i	$-0.362878\pi$
$881$	24.7889	0.835159	0.417580	+	0.908640i	$0.362878\pi$
$882$	−0.211103	−0.00710819
$883$	− 38.4222i	− 1.29301i	−0.762910	−	0.646505i	$-0.776230\pi$
$883$	− 38.4222i	− 1.29301i	0.762910	−	0.646505i	$-0.223770\pi$
$884$	9.39445	0.315970
$885$	0	0
$886$	− 27.6333i	− 0.928359i
$887$	43.0278i	1.44473i	0.691512	+	0.722365i	$0.256945\pi$
$887$	43.0278i	1.44473i	−0.691512	+	0.722365i	$0.743055\pi$
$888$	5.21110i	0.174873i
$889$	34.4222i	1.15448i
$890$	0	0
$891$	0	0
$892$	−19.8167	−0.663511
$893$	13.5778i	0.454364i
$894$	−0.788897	−0.0263847
$895$	0	0
$896$	2.60555	0.0870454
$897$	− 31.0278i	− 1.03599i
$898$	− 9.63331i	− 0.321468i
$899$	15.6333i	0.521400i
$900$	0	0
$901$	−15.6333	−0.520821
$902$	0	0
$903$	−20.8444	−0.693659
$904$	7.81665i	0.259978i
$905$	0	0
$906$	6.00000	0.199337
$907$	50.4222i	1.67424i	0.547018	+	0.837121i	$0.315763\pi$
$907$	50.4222i	1.67424i	−0.547018	+	0.837121i	$0.684237\pi$
$908$	−24.0000	−0.796468
$909$	14.6056	0.484436
$910$	0	0
$911$	15.6333	0.517955	0.258977	−	0.965883i	$-0.416615\pi$
$911$	15.6333	0.517955	0.258977	+	0.965883i	$0.416615\pi$
$912$	2.60555	0.0862784
$913$	0	0
$914$	−12.2389	−0.404825
$915$	0	0
$916$	− 1.81665i	− 0.0600239i
$917$	−40.1110	−1.32458
$918$	2.60555	0.0859960
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	23.2111i	0.764832i
$922$	− 9.63331i	− 0.317256i
$923$	18.7889	0.618444
$924$	0	0
$925$	0	0
$926$	38.6056	1.26866
$927$	− 4.00000i	− 0.131377i
$928$	−2.60555	−0.0855314
$929$	− 24.7889i	− 0.813297i	−0.913585	−	0.406649i	$-0.866697\pi$
$929$	− 24.7889i	− 0.813297i	0.913585	−	0.406649i	$-0.133303\pi$
$930$	0	0
$931$	0.550039i	0.0180268i
$932$	− 19.8167i	− 0.649116i
$933$	12.0000i	0.392862i
$934$	1.57779i	0.0516270i
$935$	0	0
$936$	3.60555i	0.117851i
$937$	− 53.6333i	− 1.75212i	−0.482199	−	0.876062i	$-0.660162\pi$
$937$	− 53.6333i	− 1.75212i	0.482199	−	0.876062i	$-0.339838\pi$
$938$	−29.2111	−0.953776
$939$	32.4222	1.05806
$940$	0	0
$941$	− 54.0000i	− 1.76035i	−0.474650	−	0.880175i	$-0.657425\pi$
$941$	− 54.0000i	− 1.76035i	0.474650	−	0.880175i	$-0.342575\pi$
$942$	−8.42221	−0.274410
$943$	96.4777	3.14175
$944$	− 5.21110i	− 0.169607i
$945$	0	0
$946$	0	0
$947$	−27.6333	−0.897962	−0.448981	−	0.893541i	$-0.648213\pi$
$947$	−27.6333	−0.897962	−0.448981	+	0.893541i	$0.648213\pi$
$948$	− 14.4222i	− 0.468411i
$949$	− 31.0278i	− 1.00720i
$950$	0	0
$951$	18.0000i	0.583690i
$952$	− 6.78890i	− 0.220029i
$953$	− 30.2389i	− 0.979533i	−0.871854	−	0.489766i	$-0.837082\pi$
$953$	− 30.2389i	− 0.979533i	0.871854	−	0.489766i	$-0.162918\pi$
$954$	− 6.00000i	− 0.194257i
$955$	0	0
$956$	0	0
$957$	0	0
$958$	34.4222i	1.11213i
$959$	−29.2111	−0.943276
$960$	0	0
$961$	−5.00000	−0.161290
$962$	18.7889i	0.605778i
$963$	0	0
$964$	22.4222i	0.722171i
$965$	0	0
$966$	−22.4222	−0.721423
$967$	−42.2389	−1.35831	−0.679155	−	0.733995i	$-0.737654\pi$
$967$	−42.2389	−1.35831	−0.679155	+	0.733995i	$0.737654\pi$
$968$	−11.0000	−0.353553
$969$	− 6.78890i	− 0.218091i
$970$	0	0
$971$	−16.9722	−0.544665	−0.272333	−	0.962203i	$-0.587795\pi$
$971$	−16.9722	−0.544665	−0.272333	+	0.962203i	$0.587795\pi$
$972$	1.00000i	0.0320750i
$973$	7.26662	0.232957
$974$	−37.0278	−1.18645
$975$	0	0
$976$	3.21110	0.102785
$977$	38.8444	1.24274	0.621371	−	0.783516i	$-0.286576\pi$
$977$	38.8444	1.24274	0.621371	+	0.783516i	$0.286576\pi$
$978$	4.42221i	0.141407i
$979$	0	0
$980$	0	0
$981$	8.60555i	0.274754i
$982$	13.8167	0.440907
$983$	−13.5778	−0.433064	−0.216532	−	0.976275i	$-0.569475\pi$
$983$	−13.5778	−0.433064	−0.216532	+	0.976275i	$0.569475\pi$
$984$	−11.2111	−0.357397
$985$	0	0
$986$	6.78890i	0.216203i
$987$	13.5778i	0.432186i
$988$	9.39445	0.298877
$989$	68.8444	2.18912
$990$	0	0
$991$	6.42221	0.204008	0.102004	−	0.994784i	$-0.467475\pi$
$991$	6.42221	0.204008	0.102004	+	0.994784i	$0.467475\pi$
$992$	− 6.00000i	− 0.190500i
$993$	9.39445	0.298124
$994$	− 13.5778i	− 0.430662i
$995$	0	0
$996$	17.2111i	0.545355i
$997$	12.4222i	0.393415i	0.980462	+	0.196708i	$0.0630250\pi$
$997$	12.4222i	0.393415i	−0.980462	+	0.196708i	$0.936975\pi$
$998$	− 13.0278i	− 0.412386i
$999$	5.21110i	0.164872i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.f.m.649.3		4
5.2	odd	4		1950.2.b.k.1351.1		4
5.3	odd	4		390.2.b.c.181.4	yes	4
5.4	even	2		1950.2.f.n.649.2		4
13.12	even	2		1950.2.f.n.649.4		4
15.8	even	4		1170.2.b.d.181.2		4
20.3	even	4		3120.2.g.q.961.3		4
65.8	even	4		5070.2.a.z.1.2		2
65.12	odd	4		1950.2.b.k.1351.4		4
65.18	even	4		5070.2.a.bf.1.1		2
65.38	odd	4		390.2.b.c.181.1	✓	4
65.64	even	2	inner	1950.2.f.m.649.1		4
195.38	even	4		1170.2.b.d.181.3		4
260.103	even	4		3120.2.g.q.961.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.c.181.1	✓	4	65.38	odd	4
390.2.b.c.181.4	yes	4	5.3	odd	4
1170.2.b.d.181.2		4	15.8	even	4
1170.2.b.d.181.3		4	195.38	even	4
1950.2.b.k.1351.1		4	5.2	odd	4
1950.2.b.k.1351.4		4	65.12	odd	4
1950.2.f.m.649.1		4	65.64	even	2	inner
1950.2.f.m.649.3		4	1.1	even	1	trivial
1950.2.f.n.649.2		4	5.4	even	2
1950.2.f.n.649.4		4	13.12	even	2
3120.2.g.q.961.2		4	260.103	even	4
3120.2.g.q.961.3		4	20.3	even	4
5070.2.a.z.1.2		2	65.8	even	4
5070.2.a.bf.1.1		2	65.18	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +1.00000i q^{12} +3.60555i q^{13} +2.60555 q^{14} +1.00000 q^{16} -2.60555i q^{17} +1.00000 q^{18} -2.60555i q^{19} -2.60555i q^{21} +8.60555i q^{23} -1.00000i q^{24} -3.60555i q^{26} -1.00000i q^{27} -2.60555 q^{28} +2.60555 q^{29} +6.00000i q^{31} -1.00000 q^{32} +2.60555i q^{34} -1.00000 q^{36} -5.21110 q^{37} +2.60555i q^{38} -3.60555 q^{39} -11.2111i q^{41} +2.60555i q^{42} -8.00000i q^{43} -8.60555i q^{46} -5.21110 q^{47} +1.00000i q^{48} -0.211103 q^{49} +2.60555 q^{51} +3.60555i q^{52} -6.00000i q^{53} +1.00000i q^{54} +2.60555 q^{56} +2.60555 q^{57} -2.60555 q^{58} -5.21110i q^{59} +3.21110 q^{61} -6.00000i q^{62} +2.60555 q^{63} +1.00000 q^{64} -11.2111 q^{67} -2.60555i q^{68} -8.60555 q^{69} -5.21110i q^{71} +1.00000 q^{72} -8.60555 q^{73} +5.21110 q^{74} -2.60555i q^{76} +3.60555 q^{78} -14.4222 q^{79} +1.00000 q^{81} +11.2111i q^{82} +17.2111 q^{83} -2.60555i q^{84} +8.00000i q^{86} +2.60555i q^{87} -0.788897i q^{89} -9.39445i q^{91} +8.60555i q^{92} -6.00000 q^{93} +5.21110 q^{94} -1.00000i q^{96} -8.60555 q^{97} +0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9} - 4 q^{14} + 4 q^{16} + 4 q^{18} + 4 q^{28} - 4 q^{29} - 4 q^{32} - 4 q^{36} + 8 q^{37} + 8 q^{47} + 28 q^{49} - 4 q^{51} - 4 q^{56} - 4 q^{57} + 4 q^{58} - 16 q^{61} - 4 q^{63} + 4 q^{64} - 16 q^{67} - 20 q^{69} + 4 q^{72} - 20 q^{73} - 8 q^{74} + 4 q^{81} + 40 q^{83} - 24 q^{93} - 8 q^{94} - 20 q^{97} - 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 4 * q^9 - 4 * q^14 + 4 * q^16 + 4 * q^18 + 4 * q^28 - 4 * q^29 - 4 * q^32 - 4 * q^36 + 8 * q^37 + 8 * q^47 + 28 * q^49 - 4 * q^51 - 4 * q^56 - 4 * q^57 + 4 * q^58 - 16 * q^61 - 4 * q^63 + 4 * q^64 - 16 * q^67 - 20 * q^69 + 4 * q^72 - 20 * q^73 - 8 * q^74 + 4 * q^81 + 40 * q^83 - 24 * q^93 - 8 * q^94 - 20 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

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Database

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