LMFDB - Embedded newform 1950.2.b.e.1351.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1950,2,Mod(1351,1950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1950.1351"); 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.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,-2,0,0,0,0,2,0,0,-2,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1351
Dual form			1950.2.b.e.1351.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		−	1.00000i		−	0.408248i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			1.00000i			0.353553i
$9$	1.00000			0.333333
$10$		0			0
$11$			6.00000i			1.80907i	0.426401	+	0.904534i	$0.359781\pi$
$11$			6.00000i			1.80907i	−0.426401	+	0.904534i	$0.640219\pi$
$12$	−1.00000			−0.288675
$13$	−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
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		−	1.00000i		−	0.235702i
$19$			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
$22$	6.00000			1.27920
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$			1.00000i			0.204124i
$25$		0			0
$26$	−3.00000	+	2.00000i	−0.588348	+	0.392232i
$27$	1.00000			0.192450
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	1.00000i		−	0.176777i
$33$			6.00000i			1.04447i
$34$		−	6.00000i		−	1.02899i
$35$		0			0
$36$	−1.00000			−0.166667
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	6.00000			0.973329
$39$	−2.00000	−	3.00000i	−0.320256	−	0.480384i
$40$		0			0
$41$			12.0000i			1.87409i	0.349215	+	0.937043i	$0.386448\pi$
$41$			12.0000i			1.87409i	−0.349215	+	0.937043i	$0.613552\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		−	6.00000i		−	0.904534i
$45$		0			0
$46$			6.00000i			0.884652i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	1.00000			0.144338
$49$	7.00000			1.00000
$50$		0			0
$51$	6.00000			0.840168
$52$	2.00000	+	3.00000i	0.277350	+	0.416025i
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$		0			0
$57$			6.00000i			0.794719i
$58$			6.00000i			0.787839i
$59$			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		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−6.00000			−0.727607
$69$	−6.00000			−0.722315
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$			1.00000i			0.117851i
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	6.00000			0.697486
$75$		0			0
$76$		−	6.00000i		−	0.688247i
$77$		0			0
$78$	−3.00000	+	2.00000i	−0.339683	+	0.226455i
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	12.0000			1.32518
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$			8.00000i			0.862662i
$87$	−6.00000			−0.643268
$88$	−6.00000			−0.639602
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	6.00000			0.625543
$93$		0			0
$94$		0			0
$95$		0			0
$96$		−	1.00000i		−	0.102062i
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		−	7.00000i		−	0.707107i
$99$			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	$0.146794\pi$
$102$		−	6.00000i		−	0.594089i
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$	3.00000	−	2.00000i	0.294174	−	0.196116i
$105$		0			0
$106$		−	12.0000i		−	1.16554i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−1.00000			−0.0962250
$109$		−	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
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$	6.00000			0.561951
$115$		0			0
$116$	6.00000			0.557086
$117$	−2.00000	−	3.00000i	−0.184900	−	0.277350i
$118$	6.00000			0.552345
$119$		0			0
$120$		0			0
$121$	−25.0000			−2.27273
$122$		−	10.0000i		−	0.905357i
$123$			12.0000i			1.08200i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$			1.00000i			0.0883883i
$129$	−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	$0.324356\pi$
$132$		−	6.00000i		−	0.522233i
$133$		0			0
$134$		0			0
$135$		0			0
$136$			6.00000i			0.514496i
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$			6.00000i			0.510754i
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$	−12.0000			−1.00702
$143$	18.0000	−	12.0000i	1.50524	−	1.00349i
$144$	1.00000			0.0833333
$145$		0			0
$146$	6.00000			0.496564
$147$	7.00000			0.577350
$148$		−	6.00000i		−	0.493197i
$149$		−	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	$0.162373\pi$
$151$			12.0000i			0.976546i	−0.872691	+	0.488273i	$0.837627\pi$
$152$	−6.00000			−0.486664
$153$	6.00000			0.485071
$154$		0			0
$155$		0			0
$156$	2.00000	+	3.00000i	0.160128	+	0.240192i
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$			8.00000i			0.636446i
$159$	12.0000			0.951662
$160$		0			0
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$		−	12.0000i		−	0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$		−	12.0000i		−	0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$		−	12.0000i		−	0.937043i
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$			6.00000i			0.458831i
$172$	8.00000			0.609994
$173$	−12.0000			−0.912343			−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000			−0.912343			−0.456172	+	0.889892i	$0.650780\pi$
$174$			6.00000i			0.454859i
$175$		0			0
$176$			6.00000i			0.452267i
$177$			6.00000i			0.450988i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		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
$183$	10.0000			0.739221
$184$		−	6.00000i		−	0.442326i
$185$		0			0
$186$		0			0
$187$			36.0000i			2.63258i
$188$		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	$0.642949\pi$
$192$	−1.00000			−0.0721688
$193$		−	6.00000i		−	0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$		−	6.00000i		−	0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	6.00000			0.430775
$195$		0			0
$196$	−7.00000			−0.500000
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	6.00000			0.426401
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		0			0
$202$		−	18.0000i		−	1.26648i
$203$		0			0
$204$	−6.00000			−0.420084
$205$		0			0
$206$			14.0000i			0.975426i
$207$	−6.00000			−0.417029
$208$	−2.00000	−	3.00000i	−0.138675	−	0.208013i
$209$	−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	$0.456034\pi$
$212$	−12.0000			−0.824163
$213$		−	12.0000i		−	0.822226i
$214$		0			0
$215$		0			0
$216$			1.00000i			0.0680414i
$217$		0			0
$218$	−6.00000			−0.406371
$219$			6.00000i			0.405442i
$220$		0			0
$221$	−12.0000	−	18.0000i	−0.807207	−	1.21081i
$222$	6.00000			0.402694
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		−	18.0000i		−	1.19734i
$227$		−	24.0000i		−	1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$		−	24.0000i		−	1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$		−	6.00000i		−	0.397360i
$229$			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
$232$		−	6.00000i		−	0.393919i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$	−3.00000	+	2.00000i	−0.196116	+	0.130744i
$235$		0			0
$236$		−	6.00000i		−	0.390567i
$237$	−8.00000			−0.519656
$238$		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	$-0.718761\pi$
$241$		−	24.0000i		−	1.54598i	0.634421	−	0.772988i	$-0.281239\pi$
$242$			25.0000i			1.60706i
$243$	1.00000			0.0641500
$244$	−10.0000			−0.640184
$245$		0			0
$246$	12.0000			0.765092
$247$	18.0000	−	12.0000i	1.14531	−	0.763542i
$248$		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
$253$		−	36.0000i		−	2.26330i
$254$		−	2.00000i		−	0.125491i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$			8.00000i			0.498058i
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		−	12.0000i		−	0.741362i
$263$	−6.00000			−0.369976			−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000			−0.369976			−0.184988	+	0.982741i	$0.559225\pi$
$264$	−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	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$			12.0000i			0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$			12.0000i			0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$	18.0000			1.08742
$275$		0			0
$276$	6.00000			0.361158
$277$	−8.00000			−0.480673			−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000			−0.480673			−0.240337	+	0.970690i	$0.577258\pi$
$278$			4.00000i			0.239904i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$			12.0000i			0.712069i
$285$		0			0
$286$	−12.0000	−	18.0000i	−0.709575	−	1.06436i
$287$		0			0
$288$		−	1.00000i		−	0.0589256i
$289$	19.0000			1.11765
$290$		0			0
$291$			6.00000i			0.351726i
$292$		−	6.00000i		−	0.351123i
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		−	7.00000i		−	0.408248i
$295$		0			0
$296$	−6.00000			−0.348743
$297$			6.00000i			0.348155i
$298$	−18.0000			−1.04271
$299$	12.0000	+	18.0000i	0.693978	+	1.04097i
$300$		0			0
$301$		0			0
$302$	12.0000			0.690522
$303$	18.0000			1.03407
$304$			6.00000i			0.344124i
$305$		0			0
$306$		−	6.00000i		−	0.342997i
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		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	$0.389495\pi$
$312$	3.00000	−	2.00000i	0.169842	−	0.113228i
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		−	4.00000i		−	0.225733i
$315$		0			0
$316$	8.00000			0.450035
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		−	12.0000i		−	0.672927i
$319$		−	36.0000i		−	2.01561i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			36.0000i			2.00309i
$324$	−1.00000			−0.0555556
$325$		0			0
$326$	−12.0000			−0.664619
$327$		−	6.00000i		−	0.331801i
$328$	−12.0000			−0.662589
$329$		0			0
$330$		0			0
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$	12.0000	+	5.00000i	0.652714	+	0.271964i
$339$	18.0000			0.977626
$340$		0			0
$341$		0			0
$342$	6.00000			0.324443
$343$		0			0
$344$		−	8.00000i		−	0.431331i
$345$		0			0
$346$			12.0000i			0.645124i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	6.00000			0.321634
$349$			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
$352$	6.00000			0.319801
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	6.00000			0.318896
$355$		0			0
$356$		0			0
$357$		0			0
$358$		−	12.0000i		−	0.634220i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		−	2.00000i		−	0.105118i
$363$	−25.0000			−1.31216
$364$		0			0
$365$		0			0
$366$		−	10.0000i		−	0.522708i
$367$	−26.0000			−1.35719			−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000			−1.35719			−0.678594	+	0.734513i	$0.737411\pi$
$368$	−6.00000			−0.312772
$369$			12.0000i			0.624695i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$	36.0000			1.86152
$375$		0			0
$376$		0			0
$377$	12.0000	+	18.0000i	0.618031	+	0.927047i
$378$		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
$382$			12.0000i			0.613973i
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$	−6.00000			−0.305392
$387$	−8.00000			−0.406663
$388$		−	6.00000i		−	0.304604i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$			7.00000i			0.353553i
$393$	12.0000			0.605320
$394$	−6.00000			−0.302276
$395$		0			0
$396$		−	6.00000i		−	0.301511i
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$			20.0000i			1.00251i
$399$		0			0
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$		0			0
$403$		0			0
$404$	−18.0000			−0.895533
$405$		0			0
$406$		0			0
$407$	−36.0000			−1.78445
$408$			6.00000i			0.297044i
$409$		−	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
$412$	14.0000			0.689730
$413$		0			0
$414$			6.00000i			0.294884i
$415$		0			0
$416$	−3.00000	+	2.00000i	−0.147087	+	0.0980581i
$417$	−4.00000			−0.195881
$418$			36.0000i			1.76082i
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$		−	6.00000i		−	0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$		−	6.00000i		−	0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$		−	4.00000i		−	0.194717i
$423$		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	$0.0933260\pi$
$431$			12.0000i			0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	1.00000			0.0481125
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$			6.00000i			0.287348i
$437$		−	36.0000i		−	1.72211i
$438$	6.00000			0.286691
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$	−18.0000	+	12.0000i	−0.856173	+	0.570782i
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		−	6.00000i		−	0.284747i
$445$		0			0
$446$		0			0
$447$		−	18.0000i		−	0.851371i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$	−72.0000			−3.39035
$452$	−18.0000			−0.846649
$453$			12.0000i			0.563809i
$454$	−24.0000			−1.12638
$455$		0			0
$456$	−6.00000			−0.280976
$457$			6.00000i			0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$			6.00000i			0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	18.0000			0.841085
$459$	6.00000			0.280056
$460$		0			0
$461$		−	6.00000i		−	0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$		−	6.00000i		−	0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−6.00000			−0.278543
$465$		0			0
$466$		−	6.00000i		−	0.277945i
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$	2.00000	+	3.00000i	0.0924500	+	0.138675i
$469$		0			0
$470$		0			0
$471$	4.00000			0.184310
$472$	−6.00000			−0.276172
$473$		−	48.0000i		−	2.20704i
$474$			8.00000i			0.367452i
$475$		0			0
$476$		0			0
$477$	12.0000			0.549442
$478$		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
$482$	−24.0000			−1.09317
$483$		0			0
$484$	25.0000			1.13636
$485$		0			0
$486$		−	1.00000i		−	0.0453609i
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$			10.0000i			0.452679i
$489$		−	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	$0.412720\pi$
$492$		−	12.0000i		−	0.541002i
$493$	−36.0000			−1.62136
$494$	−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
$502$		−	12.0000i		−	0.535586i
$503$	−18.0000			−0.802580			−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000			−0.802580			−0.401290	+	0.915951i	$0.631438\pi$
$504$		0			0
$505$		0			0
$506$	−36.0000			−1.60040
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$	−2.00000			−0.0887357
$509$			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
$512$		−	1.00000i		−	0.0441942i
$513$			6.00000i			0.264906i
$514$			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	$0.541958\pi$
$522$			6.00000i			0.262613i
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$			6.00000i			0.261612i
$527$		0			0
$528$			6.00000i			0.261116i
$529$	13.0000			0.565217
$530$		0			0
$531$			6.00000i			0.260378i
$532$		0			0
$533$	36.0000	−	24.0000i	1.55933	−	1.03956i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.0000			0.517838
$538$		−	30.0000i		−	1.29339i
$539$			42.0000i			1.80907i
$540$		0			0
$541$			18.0000i			0.773880i	0.922105	+	0.386940i	$0.126468\pi$
$541$			18.0000i			0.773880i	−0.922105	+	0.386940i	$0.873532\pi$
$542$	12.0000			0.515444
$543$	2.00000			0.0858282
$544$		−	6.00000i		−	0.257248i
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		−	18.0000i		−	0.768922i
$549$	10.0000			0.426790
$550$		0			0
$551$		−	36.0000i		−	1.53365i
$552$		−	6.00000i		−	0.255377i
$553$		0			0
$554$			8.00000i			0.339887i
$555$		0			0
$556$	4.00000			0.169638
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		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			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$			4.00000i			0.168133i
$567$		0			0
$568$	12.0000			0.503509
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$	−18.0000	+	12.0000i	−0.752618	+	0.501745i
$573$	−12.0000			−0.501307
$574$		0			0
$575$		0			0
$576$	−1.00000			−0.0416667
$577$		−	30.0000i		−	1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$		−	30.0000i		−	1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$		−	19.0000i		−	0.790296i
$579$		−	6.00000i		−	0.249351i
$580$		0			0
$581$		0			0
$582$	6.00000			0.248708
$583$			72.0000i			2.98194i
$584$	−6.00000			−0.248282
$585$		0			0
$586$	−6.00000			−0.247858
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	−7.00000			−0.288675
$589$		0			0
$590$		0			0
$591$		−	6.00000i		−	0.246807i
$592$			6.00000i			0.246598i
$593$			30.0000i			1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$			30.0000i			1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	6.00000			0.246183
$595$		0			0
$596$			18.0000i			0.737309i
$597$	−20.0000			−0.818546
$598$	18.0000	−	12.0000i	0.736075	−	0.490716i
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		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
$607$	−14.0000			−0.568242			−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000			−0.568242			−0.284121	+	0.958788i	$0.591702\pi$
$608$	6.00000			0.243332
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−6.00000			−0.242536
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$			14.0000i			0.563163i
$619$		−	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
$622$		−	12.0000i		−	0.481156i
$623$		0			0
$624$	−2.00000	−	3.00000i	−0.0800641	−	0.120096i
$625$		0			0
$626$			10.0000i			0.399680i
$627$	−36.0000			−1.43770
$628$	−4.00000			−0.159617
$629$			36.0000i			1.43541i
$630$		0			0
$631$		−	12.0000i		−	0.477712i	−0.971055	−	0.238856i	$-0.923228\pi$
$631$		−	12.0000i		−	0.477712i	0.971055	−	0.238856i	$-0.0767725\pi$
$632$		−	8.00000i		−	0.318223i
$633$	4.00000			0.158986
$634$	6.00000			0.238290
$635$		0			0
$636$	−12.0000			−0.475831
$637$	−14.0000	−	21.0000i	−0.554700	−	0.832050i
$638$	−36.0000			−1.42525
$639$		−	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		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	36.0000			1.41640
$647$	42.0000			1.65119			0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000			1.65119			0.825595	+	0.564263i	$0.190840\pi$
$648$			1.00000i			0.0392837i
$649$	−36.0000			−1.41312
$650$		0			0
$651$		0			0
$652$			12.0000i			0.469956i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$	−6.00000			−0.234619
$655$		0			0
$656$			12.0000i			0.468521i
$657$			6.00000i			0.234082i
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$			42.0000i			1.63361i	0.576913	+	0.816805i	$0.304257\pi$
$661$			42.0000i			1.63361i	−0.576913	+	0.816805i	$0.695743\pi$
$662$	−18.0000			−0.699590
$663$	−12.0000	−	18.0000i	−0.466041	−	0.699062i
$664$		0			0
$665$		0			0
$666$	6.00000			0.232495
$667$	36.0000			1.39393
$668$		0			0
$669$		0			0
$670$		0			0
$671$			60.0000i			2.31627i
$672$		0			0
$673$	46.0000			1.77317			0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000			1.77317			0.886585	+	0.462566i	$0.153071\pi$
$674$			22.0000i			0.847408i
$675$		0			0
$676$	5.00000	−	12.0000i	0.192308	−	0.461538i
$677$	−24.0000			−0.922395			−0.461197	−	0.887298i	$-0.652580\pi$
$677$	−24.0000			−0.922395			−0.461197	+	0.887298i	$0.652580\pi$
$678$		−	18.0000i		−	0.691286i
$679$		0			0
$680$		0			0
$681$		−	24.0000i		−	0.919682i
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		−	6.00000i		−	0.229416i
$685$		0			0
$686$		0			0
$687$			18.0000i			0.686743i
$688$	−8.00000			−0.304997
$689$	−24.0000	−	36.0000i	−0.914327	−	1.37149i
$690$		0			0
$691$		−	42.0000i		−	1.59776i	−0.601494	−	0.798878i	$-0.705427\pi$
$691$		−	42.0000i		−	1.59776i	0.601494	−	0.798878i	$-0.294573\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$		0			0
$695$		0			0
$696$		−	6.00000i		−	0.227429i
$697$			72.0000i			2.72719i
$698$	6.00000			0.227103
$699$	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	$0.536145\pi$
$702$	−3.00000	+	2.00000i	−0.113228	+	0.0754851i
$703$	−36.0000			−1.35777
$704$		−	6.00000i		−	0.226134i
$705$		0			0
$706$	−6.00000			−0.225813
$707$		0			0
$708$		−	6.00000i		−	0.225494i
$709$		−	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	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$		0			0
$722$			17.0000i			0.632674i
$723$		−	24.0000i		−	0.892570i
$724$	−2.00000			−0.0743294
$725$		0			0
$726$			25.0000i			0.927837i
$727$	26.0000			0.964287			0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000			0.964287			0.482143	+	0.876092i	$0.339858\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−48.0000			−1.77534
$732$	−10.0000			−0.369611
$733$		−	30.0000i		−	1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$		−	30.0000i		−	1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$			26.0000i			0.959678i
$735$		0			0
$736$			6.00000i			0.221163i
$737$		0			0
$738$	12.0000			0.441726
$739$		−	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
$743$			48.0000i			1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$			48.0000i			1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$		0			0
$745$		0			0
$746$			4.00000i			0.146450i
$747$		0			0
$748$		−	36.0000i		−	1.31629i
$749$		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
$753$	12.0000			0.437304
$754$	18.0000	−	12.0000i	0.655521	−	0.437014i
$755$		0			0
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$	6.00000			0.217930
$759$		−	36.0000i		−	1.30672i
$760$		0			0
$761$		−	48.0000i		−	1.74000i	−0.493053	−	0.869999i	$-0.664119\pi$
$761$		−	48.0000i		−	1.74000i	0.493053	−	0.869999i	$-0.335881\pi$
$762$		−	2.00000i		−	0.0724524i
$763$		0			0
$764$	12.0000			0.434145
$765$		0			0
$766$	24.0000			0.867155
$767$	18.0000	−	12.0000i	0.649942	−	0.433295i
$768$	1.00000			0.0360844
$769$			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
$772$			6.00000i			0.215945i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$			8.00000i			0.287554i
$775$		0			0
$776$	−6.00000			−0.215387
$777$		0			0
$778$		−	30.0000i		−	1.07555i
$779$	−72.0000			−2.57967
$780$		0			0
$781$	72.0000			2.57636
$782$			36.0000i			1.28736i
$783$	−6.00000			−0.214423
$784$	7.00000			0.250000
$785$		0			0
$786$		−	12.0000i		−	0.428026i
$787$			12.0000i			0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$			12.0000i			0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$			6.00000i			0.213741i
$789$	−6.00000			−0.213606
$790$		0			0
$791$		0			0
$792$	−6.00000			−0.213201
$793$	−20.0000	−	30.0000i	−0.710221	−	1.06533i
$794$	18.0000			0.638796
$795$		0			0
$796$	20.0000			0.708881
$797$	−12.0000			−0.425062			−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000			−0.425062			−0.212531	+	0.977154i	$0.568171\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−24.0000			−0.847469
$803$	−36.0000			−1.27041
$804$		0			0
$805$		0			0
$806$		0			0
$807$	30.0000			1.05605
$808$			18.0000i			0.633238i
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$			18.0000i			0.632065i	0.948748	+	0.316033i	$0.102351\pi$
$811$			18.0000i			0.632065i	−0.948748	+	0.316033i	$0.897649\pi$
$812$		0			0
$813$			12.0000i			0.420858i
$814$			36.0000i			1.26180i
$815$		0			0
$816$	6.00000			0.210042
$817$		−	48.0000i		−	1.67931i
$818$	−12.0000			−0.419570
$819$		0			0
$820$		0			0
$821$		−	6.00000i		−	0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$		−	6.00000i		−	0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	18.0000			0.627822
$823$	22.0000			0.766872			0.383436	−	0.923567i	$-0.374741\pi$
$823$	22.0000			0.766872			0.383436	+	0.923567i	$0.374741\pi$
$824$		−	14.0000i		−	0.487713i
$825$		0			0
$826$		0			0
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	6.00000			0.208514
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	−8.00000			−0.277517
$832$	2.00000	+	3.00000i	0.0693375	+	0.104006i
$833$	42.0000			1.45521
$834$			4.00000i			0.138509i
$835$		0			0
$836$	36.0000			1.24509
$837$		0			0
$838$			36.0000i			1.24360i
$839$		−	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
$842$	−6.00000			−0.206774
$843$		0			0
$844$	−4.00000			−0.137686
$845$		0			0
$846$		0			0
$847$		0			0
$848$	12.0000			0.412082
$849$	−4.00000			−0.137280
$850$		0			0
$851$		−	36.0000i		−	1.23406i
$852$			12.0000i			0.411113i
$853$			54.0000i			1.84892i	0.381273	+	0.924462i	$0.375486\pi$
$853$			54.0000i			1.84892i	−0.381273	+	0.924462i	$0.624514\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.0000			1.43469			0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000			1.43469			0.717346	+	0.696717i	$0.245357\pi$
$858$	−12.0000	−	18.0000i	−0.409673	−	0.614510i
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$		0			0
$862$	12.0000			0.408722
$863$		−	48.0000i		−	1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$		−	48.0000i		−	1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		0			0
$866$		−	2.00000i		−	0.0679628i
$867$	19.0000			0.645274
$868$		0			0
$869$		−	48.0000i		−	1.62829i
$870$		0			0
$871$		0			0
$872$	6.00000			0.203186
$873$			6.00000i			0.203069i
$874$	−36.0000			−1.21772
$875$		0			0
$876$		−	6.00000i		−	0.202721i
$877$			18.0000i			0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$			18.0000i			0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$		−	8.00000i		−	0.269987i
$879$		−	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	$0.136347\pi$
$882$		−	7.00000i		−	0.235702i
$883$	56.0000			1.88455			0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000			1.88455			0.942275	+	0.334840i	$0.108682\pi$
$884$	12.0000	+	18.0000i	0.403604	+	0.605406i
$885$		0			0
$886$			24.0000i			0.806296i
$887$	30.0000			1.00730			0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000			1.00730			0.503651	+	0.863907i	$0.331990\pi$
$888$	−6.00000			−0.201347
$889$		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
$897$	12.0000	+	18.0000i	0.400668	+	0.601003i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	72.0000			2.39867
$902$			72.0000i			2.39734i
$903$		0			0
$904$			18.0000i			0.598671i
$905$		0			0
$906$	12.0000			0.398673
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			0.132818	+	0.991140i	$0.457597\pi$
$908$			24.0000i			0.796468i
$909$	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	$0.296610\pi$
$912$			6.00000i			0.198680i
$913$		0			0
$914$	6.00000			0.198462
$915$		0			0
$916$		−	18.0000i		−	0.594737i
$917$		0			0
$918$		−	6.00000i		−	0.198030i
$919$	52.0000			1.71532			0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000			1.71532			0.857661	+	0.514216i	$0.171917\pi$
$920$		0			0
$921$		−	12.0000i		−	0.395413i
$922$	−6.00000			−0.197599
$923$	−36.0000	+	24.0000i	−1.18495	+	0.789970i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−14.0000			−0.459820
$928$			6.00000i			0.196960i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$			42.0000i			1.37649i
$932$	−6.00000			−0.196537
$933$	12.0000			0.392862
$934$		−	24.0000i		−	0.785304i
$935$		0			0
$936$	3.00000	−	2.00000i	0.0980581	−	0.0653720i
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		0			0
$941$		−	18.0000i		−	0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$		−	18.0000i		−	0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$		−	4.00000i		−	0.130327i
$943$		−	72.0000i		−	2.34464i
$944$			6.00000i			0.195283i
$945$		0			0
$946$	−48.0000			−1.56061
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	8.00000			0.259828
$949$	18.0000	−	12.0000i	0.584305	−	0.389536i
$950$		0			0
$951$			6.00000i			0.194563i
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		−	12.0000i		−	0.388514i
$955$		0			0
$956$		0			0
$957$		−	36.0000i		−	1.16371i
$958$	24.0000			0.775405
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	−12.0000	−	18.0000i	−0.386896	−	0.580343i
$963$		0			0
$964$			24.0000i			0.772988i
$965$		0			0
$966$		0			0
$967$			48.0000i			1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$			48.0000i			1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$		−	25.0000i		−	0.803530i
$969$			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	$0.303971\pi$
$972$	−1.00000			−0.0320750
$973$		0			0
$974$	−24.0000			−0.769010
$975$		0			0
$976$	10.0000			0.320092
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	−12.0000			−0.383718
$979$		0			0
$980$		0			0
$981$		−	6.00000i		−	0.191565i
$982$		−	12.0000i		−	0.382935i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−12.0000			−0.382546
$985$		0			0
$986$			36.0000i			1.14647i
$987$		0			0
$988$	−18.0000	+	12.0000i	−0.572656	+	0.381771i
$989$	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
$993$		−	18.0000i		−	0.571213i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$	−6.00000			−0.189927
$999$			6.00000i			0.189832i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.b.181.1	✓	2	65.64	even	2
390.2.b.b.181.2	yes	2	5.4	even	2
1170.2.b.c.181.1		2	15.14	odd	2
1170.2.b.c.181.2		2	195.194	odd	2
1950.2.b.e.1351.1		2	1.1	even	1	trivial
1950.2.b.e.1351.2		2	13.12	even	2	inner
1950.2.f.c.649.1		2	65.12	odd	4
1950.2.f.c.649.2		2	5.3	odd	4
1950.2.f.h.649.1		2	5.2	odd	4
1950.2.f.h.649.2		2	65.38	odd	4
3120.2.g.i.961.1		2	20.19	odd	2
3120.2.g.i.961.2		2	260.259	odd	2
5070.2.a.h.1.1		1	65.34	odd	4
5070.2.a.l.1.1		1	65.44	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.b (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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