LMFDB - Embedded newform 1950.2.f.h.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.h.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +1.00000 q^{8} -1.00000 q^{9} -6.00000i q^{11} +1.00000i q^{12} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{16} -6.00000i q^{17} -1.00000 q^{18} +6.00000i q^{19} -6.00000i q^{22} -6.00000i q^{23} +1.00000i q^{24} +(-3.00000 - 2.00000i) q^{26} -1.00000i q^{27} +6.00000 q^{29} +1.00000 q^{32} +6.00000 q^{33} -6.00000i q^{34} -1.00000 q^{36} -6.00000 q^{37} +6.00000i q^{38} +(2.00000 - 3.00000i) q^{39} -12.0000i q^{41} -8.00000i q^{43} -6.00000i q^{44} -6.00000i q^{46} +1.00000i q^{48} -7.00000 q^{49} +6.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} +12.0000i q^{53} -1.00000i q^{54} -6.00000 q^{57} +6.00000 q^{58} +6.00000i q^{59} +10.0000 q^{61} +1.00000 q^{64} +6.00000 q^{66} -6.00000i q^{68} +6.00000 q^{69} +12.0000i q^{71} -1.00000 q^{72} +6.00000 q^{73} -6.00000 q^{74} +6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{82} -8.00000i q^{86} +6.00000i q^{87} -6.00000i q^{88} -6.00000i q^{92} +1.00000i q^{96} -6.00000 q^{97} -7.00000 q^{98} +6.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{13} + 2 q^{16} - 2 q^{18} - 6 q^{26} + 12 q^{29} + 2 q^{32} + 12 q^{33} - 2 q^{36} - 12 q^{37} + 4 q^{39} - 14 q^{49} + 12 q^{51} - 6 q^{52} - 12 q^{57} + 12 q^{58} + 20 q^{61} + 2 q^{64} + 12 q^{66} + 12 q^{69} - 2 q^{72} + 12 q^{73} - 12 q^{74} + 4 q^{78} + 16 q^{79} + 2 q^{81} - 12 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 6 * q^13 + 2 * q^16 - 2 * q^18 - 6 * q^26 + 12 * q^29 + 2 * q^32 + 12 * q^33 - 2 * q^36 - 12 * q^37 + 4 * q^39 - 14 * q^49 + 12 * q^51 - 6 * q^52 - 12 * q^57 + 12 * q^58 + 20 * q^61 + 2 * q^64 + 12 * q^66 + 12 * q^69 - 2 * q^72 + 12 * q^73 - 12 * q^74 + 4 * q^78 + 16 * q^79 + 2 * q^81 - 12 * 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
$11$		−	6.00000i		−	1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$		−	6.00000i		−	1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$			1.00000i			0.288675i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	−1.00000			−0.235702
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		0			0
$22$		−	6.00000i		−	1.27920i
$23$		−	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
$26$	−3.00000	−	2.00000i	−0.588348	−	0.392232i
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	1.00000			0.176777
$33$	6.00000			1.04447
$34$		−	6.00000i		−	1.02899i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$			6.00000i			0.973329i
$39$	2.00000	−	3.00000i	0.320256	−	0.480384i
$40$		0			0
$41$		−	12.0000i		−	1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$		−	12.0000i		−	1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		−	6.00000i		−	0.904534i
$45$		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
$51$	6.00000			0.840168
$52$	−3.00000	−	2.00000i	−0.416025	−	0.277350i
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$		0			0
$57$	−6.00000			−0.794719
$58$	6.00000			0.787839
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$	6.00000			0.738549
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	6.00000i		−	0.727607i
$69$	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	$0.385800\pi$
$74$	−6.00000			−0.697486
$75$		0			0
$76$			6.00000i			0.688247i
$77$		0			0
$78$	2.00000	−	3.00000i	0.226455	−	0.339683i
$79$	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
$82$		−	12.0000i		−	1.32518i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		−	8.00000i		−	0.862662i
$87$			6.00000i			0.643268i
$88$		−	6.00000i		−	0.639602i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		−	6.00000i		−	0.625543i
$93$		0			0
$94$		0			0
$95$		0			0
$96$			1.00000i			0.102062i
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$	−7.00000			−0.707107
$99$			6.00000i			0.603023i
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$	6.00000			0.594089
$103$		−	14.0000i		−	1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$		−	14.0000i		−	1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	−3.00000	−	2.00000i	−0.294174	−	0.196116i
$105$		0			0
$106$			12.0000i			1.16554i
$107$		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
$111$		−	6.00000i		−	0.569495i
$112$		0			0
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	−6.00000			−0.561951
$115$		0			0
$116$	6.00000			0.557086
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$			6.00000i			0.552345i
$119$		0			0
$120$		0			0
$121$	−25.0000			−2.27273
$122$	10.0000			0.905357
$123$	12.0000			1.08200
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	1.00000			0.0883883
$129$	8.00000			0.704361
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$	6.00000			0.522233
$133$		0			0
$134$		0			0
$135$		0			0
$136$		−	6.00000i		−	0.514496i
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$	6.00000			0.510754
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$			12.0000i			1.00702i
$143$	−12.0000	+	18.0000i	−1.00349	+	1.50524i
$144$	−1.00000			−0.0833333
$145$		0			0
$146$	6.00000			0.496564
$147$		−	7.00000i		−	0.577350i
$148$	−6.00000			−0.493197
$149$		−	18.0000i		−	1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$		−	18.0000i		−	1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$			6.00000i			0.486664i
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$	2.00000	−	3.00000i	0.160128	−	0.240192i
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	8.00000			0.636446
$159$	−12.0000			−0.951662
$160$		0			0
$161$		0			0
$162$	1.00000			0.0785674
$163$	−12.0000			−0.939913			−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000			−0.939913			−0.469956	+	0.882690i	$0.655730\pi$
$164$		−	12.0000i		−	0.937043i
$165$		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
$171$		−	6.00000i		−	0.458831i
$172$		−	8.00000i		−	0.609994i
$173$		−	12.0000i		−	0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$		−	12.0000i		−	0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$			6.00000i			0.454859i
$175$		0			0
$176$		−	6.00000i		−	0.452267i
$177$	−6.00000			−0.450988
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$			10.0000i			0.739221i
$184$		−	6.00000i		−	0.442326i
$185$		0			0
$186$		0			0
$187$	−36.0000			−2.63258
$188$		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
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$	−6.00000			−0.430775
$195$		0			0
$196$	−7.00000			−0.500000
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$			6.00000i			0.426401i
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		0			0
$202$	18.0000			1.26648
$203$		0			0
$204$	6.00000			0.420084
$205$		0			0
$206$		−	14.0000i		−	0.975426i
$207$			6.00000i			0.417029i
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$	36.0000			2.49017
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$			12.0000i			0.824163i
$213$	−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	+	18.0000i	−0.807207	+	1.21081i
$222$		−	6.00000i		−	0.402694i
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$			18.0000i			1.19734i
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$	−6.00000			−0.397360
$229$			18.0000i			1.18947i	0.803921	+	0.594737i	$0.202744\pi$
$229$			18.0000i			1.18947i	−0.803921	+	0.594737i	$0.797256\pi$
$230$		0			0
$231$		0			0
$232$	6.00000			0.393919
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	3.00000	+	2.00000i	0.196116	+	0.130744i
$235$		0			0
$236$			6.00000i			0.390567i
$237$			8.00000i			0.519656i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			24.0000i			1.54598i	0.634421	+	0.772988i	$0.281239\pi$
$241$			24.0000i			1.54598i	−0.634421	+	0.772988i	$0.718761\pi$
$242$	−25.0000			−1.60706
$243$			1.00000i			0.0641500i
$244$	10.0000			0.640184
$245$		0			0
$246$	12.0000			0.765092
$247$	12.0000	−	18.0000i	0.763542	−	1.14531i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−36.0000			−2.26330
$254$		−	2.00000i		−	0.125491i
$255$		0			0
$256$	1.00000			0.0625000
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	8.00000			0.498058
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$	12.0000			0.741362
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	6.00000			0.369274
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−30.0000			−1.82913			−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000			−1.82913			−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$		−	12.0000i		−	0.728948i	−0.931214	−	0.364474i	$-0.881249\pi$
$271$		−	12.0000i		−	0.728948i	0.931214	−	0.364474i	$-0.118751\pi$
$272$		−	6.00000i		−	0.363803i
$273$		0			0
$274$	−18.0000			−1.08742
$275$		0			0
$276$	6.00000			0.361158
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	4.00000			0.239904
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		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	+	18.0000i	−0.709575	+	1.06436i
$287$		0			0
$288$	−1.00000			−0.0589256
$289$	−19.0000			−1.11765
$290$		0			0
$291$		−	6.00000i		−	0.351726i
$292$	6.00000			0.351123
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		−	7.00000i		−	0.408248i
$295$		0			0
$296$	−6.00000			−0.348743
$297$	−6.00000			−0.348155
$298$		−	18.0000i		−	1.04271i
$299$	−12.0000	+	18.0000i	−0.693978	+	1.04097i
$300$		0			0
$301$		0			0
$302$		−	12.0000i		−	0.690522i
$303$			18.0000i			1.03407i
$304$			6.00000i			0.344124i
$305$		0			0
$306$			6.00000i			0.342997i
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$	2.00000	−	3.00000i	0.113228	−	0.169842i
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		−	4.00000i		−	0.225733i
$315$		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	$0.553891\pi$
$318$	−12.0000			−0.672927
$319$		−	36.0000i		−	2.01561i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	36.0000			2.00309
$324$	1.00000			0.0555556
$325$		0			0
$326$	−12.0000			−0.664619
$327$	6.00000			0.331801
$328$		−	12.0000i		−	0.662589i
$329$		0			0
$330$		0			0
$331$			18.0000i			0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$			18.0000i			0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$		0			0
$333$	6.00000			0.328798
$334$		0			0
$335$		0			0
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	5.00000	+	12.0000i	0.271964	+	0.652714i
$339$	−18.0000			−0.977626
$340$		0			0
$341$		0			0
$342$		−	6.00000i		−	0.324443i
$343$		0			0
$344$		−	8.00000i		−	0.431331i
$345$		0			0
$346$		−	12.0000i		−	0.645124i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$			6.00000i			0.321634i
$349$			6.00000i			0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$			6.00000i			0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$		0			0
$351$	−2.00000	+	3.00000i	−0.106752	+	0.160128i
$352$		−	6.00000i		−	0.319801i
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$	−6.00000			−0.318896
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−12.0000			−0.634220
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$	2.00000			0.105118
$363$		−	25.0000i		−	1.31216i
$364$		0			0
$365$		0			0
$366$			10.0000i			0.522708i
$367$			26.0000i			1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$			26.0000i			1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$		−	6.00000i		−	0.312772i
$369$			12.0000i			0.624695i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	−36.0000			−1.86152
$375$		0			0
$376$		0			0
$377$	−18.0000	−	12.0000i	−0.927047	−	0.618031i
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	2.00000			0.102463
$382$	−12.0000			−0.613973
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$	−6.00000			−0.305392
$387$			8.00000i			0.406663i
$388$	−6.00000			−0.304604
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$	−7.00000			−0.353553
$393$			12.0000i			0.605320i
$394$	6.00000			0.302276
$395$		0			0
$396$			6.00000i			0.301511i
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$	20.0000			1.00251
$399$		0			0
$400$		0			0
$401$			24.0000i			1.19850i	0.800561	+	0.599251i	$0.204535\pi$
$401$			24.0000i			1.19850i	−0.800561	+	0.599251i	$0.795465\pi$
$402$		0			0
$403$		0			0
$404$	18.0000			0.895533
$405$		0			0
$406$		0			0
$407$			36.0000i			1.78445i
$408$	6.00000			0.297044
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$		−	18.0000i		−	0.887875i
$412$		−	14.0000i		−	0.689730i
$413$		0			0
$414$			6.00000i			0.294884i
$415$		0			0
$416$	−3.00000	−	2.00000i	−0.147087	−	0.0980581i
$417$			4.00000i			0.195881i
$418$	36.0000			1.76082
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$			6.00000i			0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$			6.00000i			0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	4.00000			0.194717
$423$		0			0
$424$			12.0000i			0.582772i
$425$		0			0
$426$	−12.0000			−0.581402
$427$		0			0
$428$		0			0
$429$	−18.0000	−	12.0000i	−0.869048	−	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
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		−	6.00000i		−	0.287348i
$437$	36.0000			1.72211
$438$			6.00000i			0.286691i
$439$	−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
$441$	7.00000			0.333333
$442$	−12.0000	+	18.0000i	−0.570782	+	0.856173i
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		−	6.00000i		−	0.284747i
$445$		0			0
$446$		0			0
$447$	18.0000			0.851371
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$	−72.0000			−3.39035
$452$			18.0000i			0.846649i
$453$	12.0000			0.563809
$454$	24.0000			1.12638
$455$		0			0
$456$	−6.00000			−0.280976
$457$	−6.00000			−0.280668			−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000			−0.280668			−0.140334	+	0.990104i	$0.544818\pi$
$458$			18.0000i			0.841085i
$459$	−6.00000			−0.280056
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	6.00000			0.278543
$465$		0			0
$466$			6.00000i			0.277945i
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	3.00000	+	2.00000i	0.138675	+	0.0924500i
$469$		0			0
$470$		0			0
$471$	4.00000			0.184310
$472$			6.00000i			0.276172i
$473$	−48.0000			−2.20704
$474$			8.00000i			0.367452i
$475$		0			0
$476$		0			0
$477$		−	12.0000i		−	0.549442i
$478$		0			0
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$		0			0
$481$	18.0000	+	12.0000i	0.820729	+	0.547153i
$482$			24.0000i			1.09317i
$483$		0			0
$484$	−25.0000			−1.13636
$485$		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
$489$		−	12.0000i		−	0.542659i
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$	12.0000			0.541002
$493$		−	36.0000i		−	1.62136i
$494$	12.0000	−	18.0000i	0.539906	−	0.809858i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$		0			0
$502$	12.0000			0.535586
$503$		−	18.0000i		−	0.802580i	−0.915951	−	0.401290i	$-0.868562\pi$
$503$		−	18.0000i		−	0.802580i	0.915951	−	0.401290i	$-0.131438\pi$
$504$		0			0
$505$		0			0
$506$	−36.0000			−1.60040
$507$	−12.0000	+	5.00000i	−0.532939	+	0.222058i
$508$		−	2.00000i		−	0.0887357i
$509$			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
$513$	6.00000			0.264906
$514$			18.0000i			0.793946i
$515$		0			0
$516$	8.00000			0.352180
$517$		0			0
$518$		0			0
$519$	12.0000			0.526742
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$	−6.00000			−0.262613
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$		−	6.00000i		−	0.261612i
$527$		0			0
$528$	6.00000			0.261116
$529$	−13.0000			−0.565217
$530$		0			0
$531$		−	6.00000i		−	0.260378i
$532$		0			0
$533$	−24.0000	+	36.0000i	−1.03956	+	1.55933i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	12.0000i		−	0.517838i
$538$	−30.0000			−1.29339
$539$			42.0000i			1.80907i
$540$		0			0
$541$		−	18.0000i		−	0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$		−	18.0000i		−	0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$		−	12.0000i		−	0.515444i
$543$			2.00000i			0.0858282i
$544$		−	6.00000i		−	0.257248i
$545$		0			0
$546$		0			0
$547$		−	28.0000i		−	1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$		−	28.0000i		−	1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	−18.0000			−0.768922
$549$	−10.0000			−0.426790
$550$		0			0
$551$			36.0000i			1.53365i
$552$	6.00000			0.255377
$553$		0			0
$554$			8.00000i			0.339887i
$555$		0			0
$556$	4.00000			0.169638
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$	−16.0000	+	24.0000i	−0.676728	+	1.01509i
$560$		0			0
$561$		−	36.0000i		−	1.51992i
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		−	4.00000i		−	0.168133i
$567$		0			0
$568$			12.0000i			0.503509i
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$	−12.0000	+	18.0000i	−0.501745	+	0.752618i
$573$		−	12.0000i		−	0.501307i
$574$		0			0
$575$		0			0
$576$	−1.00000			−0.0416667
$577$	30.0000			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$578$	−19.0000			−0.790296
$579$		−	6.00000i		−	0.249351i
$580$		0			0
$581$		0			0
$582$		−	6.00000i		−	0.248708i
$583$	72.0000			2.98194
$584$	6.00000			0.248282
$585$		0			0
$586$	−6.00000			−0.247858
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$		−	7.00000i		−	0.288675i
$589$		0			0
$590$		0			0
$591$			6.00000i			0.246807i
$592$	−6.00000			−0.246598
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$	−6.00000			−0.246183
$595$		0			0
$596$		−	18.0000i		−	0.737309i
$597$			20.0000i			0.818546i
$598$	−12.0000	+	18.0000i	−0.490716	+	0.736075i
$599$	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
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$		−	12.0000i		−	0.488273i
$605$		0			0
$606$			18.0000i			0.731200i
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$			6.00000i			0.243332i
$609$		0			0
$610$		0			0
$611$		0			0
$612$			6.00000i			0.242536i
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$	12.0000			0.484281
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$	14.0000			0.563163
$619$		−	30.0000i		−	1.20580i	−0.797816	−	0.602901i	$-0.794011\pi$
$619$		−	30.0000i		−	1.20580i	0.797816	−	0.602901i	$-0.205989\pi$
$620$		0			0
$621$	−6.00000			−0.240772
$622$	12.0000			0.481156
$623$		0			0
$624$	2.00000	−	3.00000i	0.0800641	−	0.120096i
$625$		0			0
$626$		−	10.0000i		−	0.399680i
$627$			36.0000i			1.43770i
$628$		−	4.00000i		−	0.159617i
$629$			36.0000i			1.43541i
$630$		0			0
$631$			12.0000i			0.477712i	0.971055	+	0.238856i	$0.0767725\pi$
$631$			12.0000i			0.477712i	−0.971055	+	0.238856i	$0.923228\pi$
$632$	8.00000			0.318223
$633$			4.00000i			0.158986i
$634$	−6.00000			−0.238290
$635$		0			0
$636$	−12.0000			−0.475831
$637$	21.0000	+	14.0000i	0.832050	+	0.554700i
$638$		−	36.0000i		−	1.42525i
$639$		−	12.0000i		−	0.474713i
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	36.0000			1.41640
$647$		−	42.0000i		−	1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$		−	42.0000i		−	1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$	1.00000			0.0392837
$649$	36.0000			1.41312
$650$		0			0
$651$		0			0
$652$	−12.0000			−0.469956
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$	6.00000			0.234619
$655$		0			0
$656$		−	12.0000i		−	0.468521i
$657$	−6.00000			−0.234082
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$		−	42.0000i		−	1.63361i	−0.576913	−	0.816805i	$-0.695743\pi$
$661$		−	42.0000i		−	1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$			18.0000i			0.699590i
$663$	−18.0000	−	12.0000i	−0.699062	−	0.466041i
$664$		0			0
$665$		0			0
$666$	6.00000			0.232495
$667$		−	36.0000i		−	1.39393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	60.0000i		−	2.31627i
$672$		0			0
$673$			46.0000i			1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$			46.0000i			1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$			22.0000i			0.847408i
$675$		0			0
$676$	5.00000	+	12.0000i	0.192308	+	0.461538i
$677$			24.0000i			0.922395i	0.887298	+	0.461197i	$0.152580\pi$
$677$			24.0000i			0.922395i	−0.887298	+	0.461197i	$0.847420\pi$
$678$	−18.0000			−0.691286
$679$		0			0
$680$		0			0
$681$			24.0000i			0.919682i
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		−	6.00000i		−	0.229416i
$685$		0			0
$686$		0			0
$687$	−18.0000			−0.686743
$688$		−	8.00000i		−	0.304997i
$689$	24.0000	−	36.0000i	0.914327	−	1.37149i
$690$		0			0
$691$			42.0000i			1.59776i	0.601494	+	0.798878i	$0.294573\pi$
$691$			42.0000i			1.59776i	−0.601494	+	0.798878i	$0.705427\pi$
$692$		−	12.0000i		−	0.456172i
$693$		0			0
$694$		0			0
$695$		0			0
$696$			6.00000i			0.227429i
$697$	−72.0000			−2.72719
$698$			6.00000i			0.227103i
$699$	−6.00000			−0.226941
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$	−2.00000	+	3.00000i	−0.0754851	+	0.113228i
$703$		−	36.0000i		−	1.35777i
$704$		−	6.00000i		−	0.226134i
$705$		0			0
$706$	−6.00000			−0.225813
$707$		0			0
$708$	−6.00000			−0.225494
$709$		−	30.0000i		−	1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$		−	30.0000i		−	1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−12.0000			−0.448461
$717$		0			0
$718$		0			0
$719$	−12.0000			−0.447524			−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000			−0.447524			−0.223762	+	0.974644i	$0.571834\pi$
$720$		0			0
$721$		0			0
$722$	−17.0000			−0.632674
$723$	−24.0000			−0.892570
$724$	2.00000			0.0743294
$725$		0			0
$726$		−	25.0000i		−	0.927837i
$727$		−	26.0000i		−	0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$		−	26.0000i		−	0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−48.0000			−1.77534
$732$			10.0000i			0.369611i
$733$	−30.0000			−1.10808			−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000			−1.10808			−0.554038	+	0.832492i	$0.686914\pi$
$734$			26.0000i			0.959678i
$735$		0			0
$736$		−	6.00000i		−	0.221163i
$737$		0			0
$738$			12.0000i			0.441726i
$739$		−	42.0000i		−	1.54499i	−0.635018	−	0.772497i	$-0.719007\pi$
$739$		−	42.0000i		−	1.54499i	0.635018	−	0.772497i	$-0.280993\pi$
$740$		0			0
$741$	18.0000	+	12.0000i	0.661247	+	0.440831i
$742$		0			0
$743$	48.0000			1.76095			0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000			1.76095			0.880475	+	0.474093i	$0.157224\pi$
$744$		0			0
$745$		0			0
$746$		−	4.00000i		−	0.146450i
$747$		0			0
$748$	−36.0000			−1.31629
$749$		0			0
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$			12.0000i			0.437304i
$754$	−18.0000	−	12.0000i	−0.655521	−	0.437014i
$755$		0			0
$756$		0			0
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$			6.00000i			0.217930i
$759$		−	36.0000i		−	1.30672i
$760$		0			0
$761$			48.0000i			1.74000i	0.493053	+	0.869999i	$0.335881\pi$
$761$			48.0000i			1.74000i	−0.493053	+	0.869999i	$0.664119\pi$
$762$	2.00000			0.0724524
$763$		0			0
$764$	−12.0000			−0.434145
$765$		0			0
$766$	24.0000			0.867155
$767$	12.0000	−	18.0000i	0.433295	−	0.649942i
$768$			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
$771$	−18.0000			−0.648254
$772$	−6.00000			−0.215945
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$			8.00000i			0.287554i
$775$		0			0
$776$	−6.00000			−0.215387
$777$		0			0
$778$	−30.0000			−1.07555
$779$	72.0000			2.57967
$780$		0			0
$781$	72.0000			2.57636
$782$	−36.0000			−1.28736
$783$		−	6.00000i		−	0.214423i
$784$	−7.00000			−0.250000
$785$		0			0
$786$			12.0000i			0.428026i
$787$	−12.0000			−0.427754			−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000			−0.427754			−0.213877	+	0.976861i	$0.568609\pi$
$788$	6.00000			0.213741
$789$	6.00000			0.213606
$790$		0			0
$791$		0			0
$792$			6.00000i			0.213201i
$793$	−30.0000	−	20.0000i	−1.06533	−	0.710221i
$794$	−18.0000			−0.638796
$795$		0			0
$796$	20.0000			0.708881
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$			24.0000i			0.847469i
$803$		−	36.0000i		−	1.27041i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	30.0000i		−	1.05605i
$808$	18.0000			0.633238
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$		−	18.0000i		−	0.632065i	−0.948748	−	0.316033i	$-0.897649\pi$
$811$		−	18.0000i		−	0.632065i	0.948748	−	0.316033i	$-0.102351\pi$
$812$		0			0
$813$	12.0000			0.420858
$814$			36.0000i			1.26180i
$815$		0			0
$816$	6.00000			0.210042
$817$	48.0000			1.67931
$818$		−	12.0000i		−	0.419570i
$819$		0			0
$820$		0			0
$821$			6.00000i			0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$			6.00000i			0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$		−	18.0000i		−	0.627822i
$823$			22.0000i			0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$			22.0000i			0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$		−	14.0000i		−	0.487713i
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$			6.00000i			0.208514i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	−8.00000			−0.277517
$832$	−3.00000	−	2.00000i	−0.104006	−	0.0693375i
$833$			42.0000i			1.45521i
$834$			4.00000i			0.138509i
$835$		0			0
$836$	36.0000			1.24509
$837$		0			0
$838$	36.0000			1.24360
$839$		−	12.0000i		−	0.414286i	−0.978311	−	0.207143i	$-0.933583\pi$
$839$		−	12.0000i		−	0.414286i	0.978311	−	0.207143i	$-0.0664165\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$			6.00000i			0.206774i
$843$		0			0
$844$	4.00000			0.137686
$845$		0			0
$846$		0			0
$847$		0			0
$848$			12.0000i			0.412082i
$849$	4.00000			0.137280
$850$		0			0
$851$			36.0000i			1.23406i
$852$	−12.0000			−0.411113
$853$	54.0000			1.84892			0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000			1.84892			0.924462	+	0.381273i	$0.124514\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	42.0000i		−	1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$		−	42.0000i		−	1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	−18.0000	−	12.0000i	−0.614510	−	0.409673i
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$		0			0
$862$		−	12.0000i		−	0.408722i
$863$	−48.0000			−1.63394			−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000			−1.63394			−0.816970	+	0.576681i	$0.804348\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		0			0
$866$			2.00000i			0.0679628i
$867$		−	19.0000i		−	0.645274i
$868$		0			0
$869$		−	48.0000i		−	1.62829i
$870$		0			0
$871$		0			0
$872$		−	6.00000i		−	0.203186i
$873$	6.00000			0.203069
$874$	36.0000			1.21772
$875$		0			0
$876$			6.00000i			0.202721i
$877$	−18.0000			−0.607817			−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000			−0.607817			−0.303908	+	0.952701i	$0.598292\pi$
$878$	−8.00000			−0.269987
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$	7.00000			0.235702
$883$			56.0000i			1.88455i	0.334840	+	0.942275i	$0.391318\pi$
$883$			56.0000i			1.88455i	−0.334840	+	0.942275i	$0.608682\pi$
$884$	−12.0000	+	18.0000i	−0.403604	+	0.605406i
$885$		0			0
$886$		−	24.0000i		−	0.806296i
$887$		−	30.0000i		−	1.00730i	−0.863907	−	0.503651i	$-0.831990\pi$
$887$		−	30.0000i		−	1.00730i	0.863907	−	0.503651i	$-0.168010\pi$
$888$		−	6.00000i		−	0.201347i
$889$		0			0
$890$		0			0
$891$		−	6.00000i		−	0.201008i
$892$		0			0
$893$		0			0
$894$	18.0000			0.602010
$895$		0			0
$896$		0			0
$897$	−18.0000	−	12.0000i	−0.601003	−	0.400668i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	72.0000			2.39867
$902$	−72.0000			−2.39734
$903$		0			0
$904$			18.0000i			0.598671i
$905$		0			0
$906$	12.0000			0.398673
$907$		−	8.00000i		−	0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$		−	8.00000i		−	0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	24.0000			0.796468
$909$	−18.0000			−0.597022
$910$		0			0
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$	−6.00000			−0.198680
$913$		0			0
$914$	−6.00000			−0.198462
$915$		0			0
$916$			18.0000i			0.594737i
$917$		0			0
$918$	−6.00000			−0.198030
$919$	−52.0000			−1.71532			−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000			−1.71532			−0.857661	+	0.514216i	$0.828083\pi$
$920$		0			0
$921$			12.0000i			0.395413i
$922$			6.00000i			0.197599i
$923$	24.0000	−	36.0000i	0.789970	−	1.18495i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			14.0000i			0.459820i
$928$	6.00000			0.196960
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		−	42.0000i		−	1.37649i
$932$			6.00000i			0.196537i
$933$			12.0000i			0.392862i
$934$		−	24.0000i		−	0.785304i
$935$		0			0
$936$	3.00000	+	2.00000i	0.0980581	+	0.0653720i
$937$			34.0000i			1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$			34.0000i			1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$		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	$0.905216\pi$
$942$	4.00000			0.130327
$943$	−72.0000			−2.34464
$944$			6.00000i			0.195283i
$945$		0			0
$946$	−48.0000			−1.56061
$947$	−12.0000			−0.389948			−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000			−0.389948			−0.194974	+	0.980808i	$0.562462\pi$
$948$			8.00000i			0.259828i
$949$	−18.0000	−	12.0000i	−0.584305	−	0.389536i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$		−	54.0000i		−	1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$		−	54.0000i		−	1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$		−	12.0000i		−	0.388514i
$955$		0			0
$956$		0			0
$957$	36.0000			1.16371
$958$			24.0000i			0.775405i
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	18.0000	+	12.0000i	0.580343	+	0.386896i
$963$		0			0
$964$			24.0000i			0.772988i
$965$		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	$0.780637\pi$
$968$	−25.0000			−0.803530
$969$			36.0000i			1.15649i
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$			1.00000i			0.0320750i
$973$		0			0
$974$	24.0000			0.769010
$975$		0			0
$976$	10.0000			0.320092
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		−	12.0000i		−	0.383718i
$979$		0			0
$980$		0			0
$981$			6.00000i			0.191565i
$982$	12.0000			0.382935
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	12.0000			0.382546
$985$		0			0
$986$		−	36.0000i		−	1.14647i
$987$		0			0
$988$	12.0000	−	18.0000i	0.381771	−	0.572656i
$989$	−48.0000			−1.52631
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$	−18.0000			−0.571213
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	28.0000i		−	0.886769i	−0.896332	−	0.443384i	$-0.853778\pi$
$997$		−	28.0000i		−	0.886769i	0.896332	−	0.443384i	$-0.146222\pi$
$998$		−	6.00000i		−	0.189927i
$999$			6.00000i			0.189832i

Twists

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

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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