LMFDB - Embedded newform 1312.2.l.e.993.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1312,2,Mod(993,1312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1312, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

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

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

chi := DirichletCharacter("1312.993");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1312 = 2^{5} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1312.l (of order $4$, degree $2$, 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:	$10.4763727452$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			993.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1312.993
Dual form			1312.2.l.e.1057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} +(3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+2.00000i q^{5} +(3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +(4.00000 + 4.00000i) q^{13} +(-3.00000 + 3.00000i) q^{17} +(6.00000 - 6.00000i) q^{19} -6.00000 q^{23} +1.00000 q^{25} +(2.00000 + 2.00000i) q^{29} -6.00000 q^{31} +(-6.00000 + 6.00000i) q^{35} +6.00000 q^{37} +(-4.00000 + 5.00000i) q^{41} +6.00000 q^{45} +(3.00000 - 3.00000i) q^{47} +11.0000i q^{49} -12.0000 q^{59} +6.00000i q^{61} +(9.00000 - 9.00000i) q^{63} +(-8.00000 + 8.00000i) q^{65} +(6.00000 - 6.00000i) q^{67} +(9.00000 + 9.00000i) q^{71} +12.0000i q^{73} +(-9.00000 - 9.00000i) q^{79} -9.00000 q^{81} +(-6.00000 - 6.00000i) q^{85} +(5.00000 + 5.00000i) q^{89} +24.0000i q^{91} +(12.0000 + 12.0000i) q^{95} +(5.00000 - 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7}+O(q^{10}) $ 2 * q + 6 * q^7	$ 2 q + 6 q^{7} + 8 q^{13} - 6 q^{17} + 12 q^{19} - 12 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 12 q^{35} + 12 q^{37} - 8 q^{41} + 12 q^{45} + 6 q^{47} - 24 q^{59} + 18 q^{63} - 16 q^{65} + 12 q^{67} + 18 q^{71} - 18 q^{79} - 18 q^{81} - 12 q^{85} + 10 q^{89} + 24 q^{95} + 10 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 + 8 * q^13 - 6 * q^17 + 12 * q^19 - 12 * q^23 + 2 * q^25 + 4 * q^29 - 12 * q^31 - 12 * q^35 + 12 * q^37 - 8 * q^41 + 12 * q^45 + 6 * q^47 - 24 * q^59 + 18 * q^63 - 16 * q^65 + 12 * q^67 + 18 * q^71 - 18 * q^79 - 18 * q^81 - 12 * q^85 + 10 * q^89 + 24 * q^95 + 10 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1312\mathbb{Z}\right)^\times$.

$n$	$129$	$165$	$575$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.989524	+	0.144370i	$0.0461154\pi$
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.144370	+	0.989524i	$0.453885\pi$
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$	4.00000	+	4.00000i	1.10940	+	1.10940i	0.993229	+	0.116171i	$0.0370621\pi$
$13$	4.00000	+	4.00000i	1.10940	+	1.10940i	0.116171	+	0.993229i	$0.462938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	+	3.00000i	−0.727607	+	0.727607i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−3.00000	+	3.00000i	−0.727607	+	0.727607i	0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	6.00000	−	6.00000i	1.37649	−	1.37649i	0.526026	−	0.850469i	$-0.323682\pi$
$19$	6.00000	−	6.00000i	1.37649	−	1.37649i	0.850469	−	0.526026i	$-0.176318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.00000	+	2.00000i	0.371391	+	0.371391i	0.867984	−	0.496593i	$-0.165416\pi$
$29$	2.00000	+	2.00000i	0.371391	+	0.371391i	−0.496593	+	0.867984i	$0.665416\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000	+	6.00000i	−1.01419	+	1.01419i
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.00000	+	5.00000i	−0.624695	+	0.780869i
$41$	−4.00000	+	5.00000i	−0.624695	+	0.780869i
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	6.00000			0.894427
$46$		0			0
$47$	3.00000	−	3.00000i	0.437595	−	0.437595i	−0.453607	−	0.891202i	$-0.649863\pi$
$47$	3.00000	−	3.00000i	0.437595	−	0.437595i	0.891202	+	0.453607i	$0.149863\pi$
$48$		0			0
$49$			11.0000i			1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$		0			0
$63$	9.00000	−	9.00000i	1.13389	−	1.13389i
$64$		0			0
$65$	−8.00000	+	8.00000i	−0.992278	+	0.992278i
$66$		0			0
$67$	6.00000	−	6.00000i	0.733017	−	0.733017i	−0.238200	−	0.971216i	$-0.576557\pi$
$67$	6.00000	−	6.00000i	0.733017	−	0.733017i	0.971216	+	0.238200i	$0.0765572\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.00000	+	9.00000i	1.06810	+	1.06810i	0.997505	+	0.0705987i	$0.0224910\pi$
$71$	9.00000	+	9.00000i	1.06810	+	1.06810i	0.0705987	+	0.997505i	$0.477509\pi$
$72$		0			0
$73$			12.0000i			1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$			12.0000i			1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−9.00000	−	9.00000i	−1.01258	−	1.01258i	−0.999920	−	0.0126592i	$-0.995970\pi$
$79$	−9.00000	−	9.00000i	−1.01258	−	1.01258i	−0.0126592	−	0.999920i	$-0.504030\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−6.00000	−	6.00000i	−0.650791	−	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.00000	+	5.00000i	0.529999	+	0.529999i	0.920572	−	0.390573i	$-0.127723\pi$
$89$	5.00000	+	5.00000i	0.529999	+	0.529999i	−0.390573	+	0.920572i	$0.627723\pi$
$90$		0			0
$91$			24.0000i			2.51588i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.0000	+	12.0000i	1.23117	+	1.23117i
$96$		0			0
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−4.00000	+	4.00000i	−0.383131	+	0.383131i	−0.872229	−	0.489098i	$-0.837326\pi$
$109$	−4.00000	+	4.00000i	−0.383131	+	0.383131i	0.489098	+	0.872229i	$0.337326\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		−	12.0000i		−	1.11901i
$116$		0			0
$117$	12.0000	−	12.0000i	1.10940	−	1.10940i
$118$		0			0
$119$	−18.0000			−1.65006
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			12.0000i			1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$			12.0000i			1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$		0			0
$133$	36.0000			3.12160
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.00000	−	3.00000i	0.256307	−	0.256307i	−0.567243	−	0.823550i	$-0.691990\pi$
$137$	3.00000	−	3.00000i	0.256307	−	0.256307i	0.823550	+	0.567243i	$0.191990\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−4.00000	+	4.00000i	−0.332182	+	0.332182i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.00000	−	8.00000i	0.655386	−	0.655386i	−0.298899	−	0.954285i	$-0.596619\pi$
$149$	8.00000	−	8.00000i	0.655386	−	0.655386i	0.954285	+	0.298899i	$0.0966194\pi$
$150$		0			0
$151$	−9.00000	−	9.00000i	−0.732410	−	0.732410i	0.238687	−	0.971097i	$-0.423283\pi$
$151$	−9.00000	−	9.00000i	−0.732410	−	0.732410i	−0.971097	+	0.238687i	$0.923283\pi$
$152$		0			0
$153$	9.00000	+	9.00000i	0.727607	+	0.727607i
$154$		0			0
$155$		−	12.0000i		−	0.963863i
$156$		0			0
$157$	4.00000	+	4.00000i	0.319235	+	0.319235i	0.848473	−	0.529238i	$-0.177522\pi$
$157$	4.00000	+	4.00000i	0.319235	+	0.319235i	−0.529238	+	0.848473i	$0.677522\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.0000	−	18.0000i	−1.41860	−	1.41860i
$162$		0			0
$163$	−24.0000			−1.87983			−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000			−1.87983			−0.939913	+	0.341415i	$0.889094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.00000	+	3.00000i	0.232147	+	0.232147i	0.813588	−	0.581441i	$-0.197511\pi$
$167$	3.00000	+	3.00000i	0.232147	+	0.232147i	−0.581441	+	0.813588i	$0.697511\pi$
$168$		0			0
$169$			19.0000i			1.46154i
$170$		0			0
$171$	−18.0000	−	18.0000i	−1.37649	−	1.37649i
$172$		0			0
$173$		−	22.0000i		−	1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$		−	22.0000i		−	1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$		0			0
$175$	3.00000	+	3.00000i	0.226779	+	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000	−	12.0000i	0.896922	−	0.896922i	−0.0982406	−	0.995163i	$-0.531321\pi$
$179$	12.0000	−	12.0000i	0.896922	−	0.896922i	0.995163	+	0.0982406i	$0.0313215\pi$
$180$		0			0
$181$	−10.0000	+	10.0000i	−0.743294	+	0.743294i	−0.973210	−	0.229916i	$-0.926155\pi$
$181$	−10.0000	+	10.0000i	−0.743294	+	0.743294i	0.229916	+	0.973210i	$0.426155\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			12.0000i			0.882258i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	−	3.00000i	0.217072	−	0.217072i	−0.590191	−	0.807264i	$-0.700948\pi$
$191$	3.00000	−	3.00000i	0.217072	−	0.217072i	0.807264	+	0.590191i	$0.200948\pi$
$192$		0			0
$193$	−17.0000	−	17.0000i	−1.22369	−	1.22369i	−0.966312	−	0.257375i	$-0.917142\pi$
$193$	−17.0000	−	17.0000i	−1.22369	−	1.22369i	−0.257375	−	0.966312i	$-0.582858\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	−15.0000	+	15.0000i	−1.06332	+	1.06332i	−0.0654671	+	0.997855i	$0.520854\pi$
$199$	−15.0000	+	15.0000i	−1.06332	+	1.06332i	−0.997855	+	0.0654671i	$0.979146\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			12.0000i			0.842235i
$204$		0			0
$205$	−10.0000	−	8.00000i	−0.698430	−	0.558744i
$206$		0			0
$207$			18.0000i			1.25109i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6.00000	−	6.00000i	0.413057	−	0.413057i	−0.469745	−	0.882802i	$-0.655654\pi$
$211$	6.00000	−	6.00000i	0.413057	−	0.413057i	0.882802	+	0.469745i	$0.155654\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−18.0000	−	18.0000i	−1.22192	−	1.22192i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.0000			−1.61441
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$		−	3.00000i		−	0.200000i
$226$		0			0
$227$	−6.00000	+	6.00000i	−0.398234	+	0.398234i	−0.877610	−	0.479376i	$-0.840863\pi$
$227$	−6.00000	+	6.00000i	−0.398234	+	0.398234i	0.479376	+	0.877610i	$0.340863\pi$
$228$		0			0
$229$	14.0000	−	14.0000i	0.925146	−	0.925146i	−0.0722412	−	0.997387i	$-0.523015\pi$
$229$	14.0000	−	14.0000i	0.925146	−	0.925146i	0.997387	+	0.0722412i	$0.0230151\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.0000	−	13.0000i	−0.851658	−	0.851658i	0.138679	−	0.990337i	$-0.455714\pi$
$233$	−13.0000	−	13.0000i	−0.851658	−	0.851658i	−0.990337	+	0.138679i	$0.955714\pi$
$234$		0			0
$235$	6.00000	+	6.00000i	0.391397	+	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.00000	−	3.00000i	−0.194054	−	0.194054i	0.603391	−	0.797445i	$-0.293816\pi$
$239$	−3.00000	−	3.00000i	−0.194054	−	0.194054i	−0.797445	+	0.603391i	$0.793816\pi$
$240$		0			0
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−22.0000			−1.40553
$246$		0			0
$247$	48.0000			3.05417
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	24.0000i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$		−	24.0000i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000	−	3.00000i	−0.187135	−	0.187135i	0.607321	−	0.794456i	$-0.292244\pi$
$257$	−3.00000	−	3.00000i	−0.187135	−	0.187135i	−0.794456	+	0.607321i	$0.792244\pi$
$258$		0			0
$259$	18.0000	+	18.0000i	1.11847	+	1.11847i
$260$		0			0
$261$	6.00000	−	6.00000i	0.371391	−	0.371391i
$262$		0			0
$263$	15.0000	−	15.0000i	0.924940	−	0.924940i	−0.0724336	−	0.997373i	$-0.523077\pi$
$263$	15.0000	−	15.0000i	0.924940	−	0.924940i	0.997373	+	0.0724336i	$0.0230765\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$	−6.00000			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$			18.0000i			1.07763i
$280$		0			0
$281$	21.0000	−	21.0000i	1.25275	−	1.25275i	0.298275	−	0.954480i	$-0.403589\pi$
$281$	21.0000	−	21.0000i	1.25275	−	1.25275i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$		0			0
$283$	24.0000			1.42665			0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000			1.42665			0.713326	+	0.700832i	$0.247188\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−27.0000	+	3.00000i	−1.59376	+	0.177084i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.0000	+	10.0000i	−0.584206	+	0.584206i	−0.936056	−	0.351850i	$-0.885553\pi$
$293$	−10.0000	+	10.0000i	−0.584206	+	0.584206i	0.351850	+	0.936056i	$0.385553\pi$
$294$		0			0
$295$		−	24.0000i		−	1.39733i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.0000	−	24.0000i	−1.38796	−	1.38796i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	21.0000	−	21.0000i	1.19080	−	1.19080i	0.213958	−	0.976843i	$-0.431364\pi$
$311$	21.0000	−	21.0000i	1.19080	−	1.19080i	0.976843	−	0.213958i	$-0.0686355\pi$
$312$		0			0
$313$	17.0000	−	17.0000i	0.960897	−	0.960897i	−0.0383669	−	0.999264i	$-0.512216\pi$
$313$	17.0000	−	17.0000i	0.960897	−	0.960897i	0.999264	+	0.0383669i	$0.0122156\pi$
$314$		0			0
$315$	18.0000	+	18.0000i	1.01419	+	1.01419i
$316$		0			0
$317$	12.0000	+	12.0000i	0.673987	+	0.673987i	0.958633	−	0.284646i	$-0.0918759\pi$
$317$	12.0000	+	12.0000i	0.673987	+	0.673987i	−0.284646	+	0.958633i	$0.591876\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			36.0000i			2.00309i
$324$		0			0
$325$	4.00000	+	4.00000i	0.221880	+	0.221880i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.0000			0.992372
$330$		0			0
$331$	−6.00000	−	6.00000i	−0.329790	−	0.329790i	0.522717	−	0.852506i	$-0.324919\pi$
$331$	−6.00000	−	6.00000i	−0.329790	−	0.329790i	−0.852506	+	0.522717i	$0.824919\pi$
$332$		0			0
$333$		−	18.0000i		−	0.986394i
$334$		0			0
$335$	12.0000	+	12.0000i	0.655630	+	0.655630i
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	6.00000i	−0.322097	+	0.322097i	−0.849571	−	0.527474i	$-0.823139\pi$
$347$	−6.00000	+	6.00000i	−0.322097	+	0.322097i	0.527474	+	0.849571i	$0.323139\pi$
$348$		0			0
$349$		−	10.0000i		−	0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$		−	10.0000i		−	0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.0000			−0.851594			−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000			−0.851594			−0.425797	+	0.904819i	$0.640006\pi$
$354$		0			0
$355$	−18.0000	+	18.0000i	−0.955341	+	0.955341i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$		−	53.0000i		−	2.78947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−24.0000			−1.25622
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	15.0000	+	12.0000i	0.780869	+	0.624695i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$	−24.0000			−1.23280			−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000			−1.23280			−0.616399	+	0.787434i	$0.711409\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.0000	−	15.0000i	0.766464	−	0.766464i	−0.211018	−	0.977482i	$-0.567678\pi$
$383$	15.0000	−	15.0000i	0.766464	−	0.766464i	0.977482	+	0.211018i	$0.0676779\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2.00000i		−	0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$		−	2.00000i		−	0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$		0			0
$391$	18.0000	−	18.0000i	0.910299	−	0.910299i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	18.0000	−	18.0000i	0.905678	−	0.905678i
$396$		0			0
$397$	2.00000	+	2.00000i	0.100377	+	0.100377i	0.755512	−	0.655135i	$-0.227388\pi$
$397$	2.00000	+	2.00000i	0.100377	+	0.100377i	−0.655135	+	0.755512i	$0.727388\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.0000i			0.799002i	0.916733	+	0.399501i	$0.130817\pi$
$401$			16.0000i			0.799002i	−0.916733	+	0.399501i	$0.869183\pi$
$402$		0			0
$403$	−24.0000	−	24.0000i	−1.19553	−	1.19553i
$404$		0			0
$405$		−	18.0000i		−	0.894427i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4.00000			−0.197787			−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000			−0.197787			−0.0988936	+	0.995098i	$0.531530\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−36.0000	−	36.0000i	−1.77144	−	1.77144i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	36.0000i		−	1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$		−	36.0000i		−	1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$		0			0
$421$	2.00000	+	2.00000i	0.0974740	+	0.0974740i	0.754162	−	0.656688i	$-0.228043\pi$
$421$	2.00000	+	2.00000i	0.0974740	+	0.0974740i	−0.656688	+	0.754162i	$0.728043\pi$
$422$		0			0
$423$	−9.00000	−	9.00000i	−0.437595	−	0.437595i
$424$		0			0
$425$	−3.00000	+	3.00000i	−0.145521	+	0.145521i
$426$		0			0
$427$	−18.0000	+	18.0000i	−0.871081	+	0.871081i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	24.0000i		−	1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$		−	24.0000i		−	1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−36.0000	+	36.0000i	−1.72211	+	1.72211i
$438$		0			0
$439$	3.00000	+	3.00000i	0.143182	+	0.143182i	0.775064	−	0.631882i	$-0.217717\pi$
$439$	3.00000	+	3.00000i	0.143182	+	0.143182i	−0.631882	+	0.775064i	$0.717717\pi$
$440$		0			0
$441$	33.0000			1.57143
$442$		0			0
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		0			0
$445$	−10.0000	+	10.0000i	−0.474045	+	0.474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.00000i		−	0.188772i	−0.995536	−	0.0943858i	$-0.969911\pi$
$449$		−	4.00000i		−	0.188772i	0.995536	−	0.0943858i	$-0.0300887\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−48.0000			−2.25027
$456$		0			0
$457$	17.0000	−	17.0000i	0.795226	−	0.795226i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	17.0000	−	17.0000i	0.795226	−	0.795226i	0.982339	+	0.187112i	$0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$	15.0000	+	15.0000i	0.697109	+	0.697109i	0.963786	−	0.266677i	$-0.0859256\pi$
$463$	15.0000	+	15.0000i	0.697109	+	0.697109i	−0.266677	+	0.963786i	$0.585926\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.00000	−	6.00000i	0.275299	−	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	27.0000	+	27.0000i	1.23366	+	1.23366i	0.962547	+	0.271114i	$0.0873922\pi$
$479$	27.0000	+	27.0000i	1.23366	+	1.23366i	0.271114	+	0.962547i	$0.412608\pi$
$480$		0			0
$481$	24.0000	+	24.0000i	1.09431	+	1.09431i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.0000	+	10.0000i	0.454077	+	0.454077i
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−12.0000			−0.540453
$494$		0			0
$495$		0			0
$496$		0			0
$497$			54.0000i			2.42223i
$498$		0			0
$499$	−6.00000	−	6.00000i	−0.268597	−	0.268597i	0.559938	−	0.828535i	$-0.310825\pi$
$499$	−6.00000	−	6.00000i	−0.268597	−	0.268597i	−0.828535	+	0.559938i	$0.810825\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.00000	−	3.00000i	−0.133763	−	0.133763i	0.637055	−	0.770818i	$-0.280152\pi$
$503$	−3.00000	−	3.00000i	−0.133763	−	0.133763i	−0.770818	+	0.637055i	$0.780152\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−26.0000	+	26.0000i	−1.15243	+	1.15243i	−0.166366	+	0.986064i	$0.553203\pi$
$509$	−26.0000	+	26.0000i	−1.15243	+	1.15243i	−0.986064	+	0.166366i	$0.946797\pi$
$510$		0			0
$511$	−36.0000	+	36.0000i	−1.59255	+	1.59255i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.0000			0.528783
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.0000	+	27.0000i	1.18289	+	1.18289i	0.978992	+	0.203900i	$0.0653616\pi$
$521$	27.0000	+	27.0000i	1.18289	+	1.18289i	0.203900	+	0.978992i	$0.434638\pi$
$522$		0			0
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.0000	−	18.0000i	0.784092	−	0.784092i
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$			36.0000i			1.56227i
$532$		0			0
$533$	−36.0000	+	4.00000i	−1.55933	+	0.173259i
$534$		0			0
$535$			24.0000i			1.03761i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	6.00000i		−	0.257960i	−0.991647	−	0.128980i	$-0.958830\pi$
$541$		−	6.00000i		−	0.257960i	0.991647	−	0.128980i	$-0.0411703\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.00000	−	8.00000i	−0.342682	−	0.342682i
$546$		0			0
$547$	12.0000	−	12.0000i	0.513083	−	0.513083i	−0.402387	−	0.915470i	$-0.631819\pi$
$547$	12.0000	−	12.0000i	0.513083	−	0.513083i	0.915470	+	0.402387i	$0.131819\pi$
$548$		0			0
$549$	18.0000			0.768221
$550$		0			0
$551$	24.0000			1.02243
$552$		0			0
$553$		−	54.0000i		−	2.29631i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.00000	−	6.00000i	0.254228	−	0.254228i	−0.568473	−	0.822702i	$-0.692466\pi$
$557$	6.00000	−	6.00000i	0.254228	−	0.254228i	0.822702	+	0.568473i	$0.192466\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.00000	+	6.00000i	0.252870	+	0.252870i	0.822146	−	0.569276i	$-0.192777\pi$
$563$	6.00000	+	6.00000i	0.252870	+	0.252870i	−0.569276	+	0.822146i	$0.692777\pi$
$564$		0			0
$565$		−	4.00000i		−	0.168281i
$566$		0			0
$567$	−27.0000	−	27.0000i	−1.13389	−	1.13389i
$568$		0			0
$569$			8.00000i			0.335377i	0.985840	+	0.167689i	$0.0536304\pi$
$569$			8.00000i			0.335377i	−0.985840	+	0.167689i	$0.946370\pi$
$570$		0			0
$571$	6.00000	+	6.00000i	0.251092	+	0.251092i	0.821418	−	0.570326i	$-0.193183\pi$
$571$	6.00000	+	6.00000i	0.251092	+	0.251092i	−0.570326	+	0.821418i	$0.693183\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	13.0000	+	13.0000i	0.541197	+	0.541197i	0.923880	−	0.382683i	$-0.125000\pi$
$577$	13.0000	+	13.0000i	0.541197	+	0.541197i	−0.382683	+	0.923880i	$0.625000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	24.0000	+	24.0000i	0.992278	+	0.992278i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$	−36.0000	+	36.0000i	−1.48335	+	1.48335i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.00000	+	5.00000i	−0.205325	+	0.205325i	−0.802277	−	0.596952i	$-0.796378\pi$
$593$	−5.00000	+	5.00000i	−0.205325	+	0.205325i	0.596952	+	0.802277i	$0.296378\pi$
$594$		0			0
$595$		−	36.0000i		−	1.47586i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.0000			0.735460			0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000			0.735460			0.367730	+	0.929933i	$0.380135\pi$
$600$		0			0
$601$	17.0000	−	17.0000i	0.693444	−	0.693444i	−0.269544	−	0.962988i	$-0.586873\pi$
$601$	17.0000	−	17.0000i	0.693444	−	0.693444i	0.962988	+	0.269544i	$0.0868729\pi$
$602$		0			0
$603$	−18.0000	−	18.0000i	−0.733017	−	0.733017i
$604$		0			0
$605$	22.0000			0.894427
$606$		0			0
$607$		−	24.0000i		−	0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$		−	24.0000i		−	0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000			0.970936
$612$		0			0
$613$		−	2.00000i		−	0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$		−	2.00000i		−	0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.0000i		−	1.28827i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$		−	32.0000i		−	1.28827i	0.764911	−	0.644136i	$-0.222783\pi$
$618$		0			0
$619$	12.0000			0.482321			0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000			0.482321			0.241160	+	0.970485i	$0.422472\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			30.0000i			1.20192i
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.0000	+	18.0000i	−0.717707	+	0.717707i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			12.0000i			0.476205i
$636$		0			0
$637$	−44.0000	+	44.0000i	−1.74334	+	1.74334i
$638$		0			0
$639$	27.0000	−	27.0000i	1.06810	−	1.06810i
$640$		0			0
$641$	−21.0000	+	21.0000i	−0.829450	+	0.829450i	−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−21.0000	+	21.0000i	−0.829450	+	0.829450i	0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$	−18.0000	−	18.0000i	−0.709851	−	0.709851i	0.256653	−	0.966504i	$-0.417380\pi$
$643$	−18.0000	−	18.0000i	−0.709851	−	0.709851i	−0.966504	+	0.256653i	$0.917380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			42.0000i			1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$			42.0000i			1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000	+	18.0000i	0.704394	+	0.704394i	0.965351	−	0.260956i	$-0.0840378\pi$
$653$	18.0000	+	18.0000i	0.704394	+	0.704394i	−0.260956	+	0.965351i	$0.584038\pi$
$654$		0			0
$655$	−24.0000			−0.937758
$656$		0			0
$657$	36.0000			1.40449
$658$		0			0
$659$	−18.0000	−	18.0000i	−0.701180	−	0.701180i	0.263483	−	0.964664i	$-0.415129\pi$
$659$	−18.0000	−	18.0000i	−0.701180	−	0.701180i	−0.964664	+	0.263483i	$0.915129\pi$
$660$		0			0
$661$			22.0000i			0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$			22.0000i			0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			72.0000i			2.79204i
$666$		0			0
$667$	−12.0000	−	12.0000i	−0.464642	−	0.464642i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−7.00000	+	7.00000i	−0.269830	+	0.269830i	−0.829032	−	0.559202i	$-0.811108\pi$
$673$	−7.00000	+	7.00000i	−0.269830	+	0.269830i	0.559202	+	0.829032i	$0.311108\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$		0			0
$679$	30.0000			1.15129
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.0000	+	24.0000i	−0.918334	+	0.918334i	−0.996908	−	0.0785738i	$-0.974963\pi$
$683$	−24.0000	+	24.0000i	−0.918334	+	0.918334i	0.0785738	+	0.996908i	$0.474963\pi$
$684$		0			0
$685$	6.00000	+	6.00000i	0.229248	+	0.229248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−24.0000	+	24.0000i	−0.913003	+	0.913003i	−0.996507	−	0.0835044i	$-0.973389\pi$
$691$	−24.0000	+	24.0000i	−0.913003	+	0.913003i	0.0835044	+	0.996507i	$0.473389\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			24.0000i			0.910372i
$696$		0			0
$697$	−3.00000	−	27.0000i	−0.113633	−	1.02270i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.0000			−1.43524			−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000			−1.43524			−0.717620	+	0.696435i	$0.754769\pi$
$702$		0			0
$703$	36.0000	−	36.0000i	1.35777	−	1.35777i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.0000	+	26.0000i	0.976450	+	0.976450i	0.999729	−	0.0232785i	$-0.00741045\pi$
$709$	26.0000	+	26.0000i	0.976450	+	0.976450i	−0.0232785	+	0.999729i	$0.507410\pi$
$710$		0			0
$711$	−27.0000	+	27.0000i	−1.01258	+	1.01258i
$712$		0			0
$713$	36.0000			1.34821
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33.0000	+	33.0000i	−1.23069	+	1.23069i	−0.266994	+	0.963698i	$0.586030\pi$
$719$	−33.0000	+	33.0000i	−1.23069	+	1.23069i	−0.963698	+	0.266994i	$0.913970\pi$
$720$		0			0
$721$	18.0000	−	18.0000i	0.670355	−	0.670355i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.00000	+	2.00000i	0.0742781	+	0.0742781i
$726$		0			0
$727$	−9.00000	−	9.00000i	−0.333792	−	0.333792i	0.520233	−	0.854024i	$-0.325845\pi$
$727$	−9.00000	−	9.00000i	−0.333792	−	0.333792i	−0.854024	+	0.520233i	$0.825845\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−12.0000			−0.441427			−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	42.0000i		−	1.54083i	−0.637542	−	0.770415i	$-0.720049\pi$
$743$		−	42.0000i		−	1.54083i	0.637542	−	0.770415i	$-0.279951\pi$
$744$		0			0
$745$	16.0000	+	16.0000i	0.586195	+	0.586195i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.0000	+	36.0000i	1.31541	+	1.31541i
$750$		0			0
$751$	−15.0000	−	15.0000i	−0.547358	−	0.547358i	0.378318	−	0.925676i	$-0.376503\pi$
$751$	−15.0000	−	15.0000i	−0.547358	−	0.547358i	−0.925676	+	0.378318i	$0.876503\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.0000	−	18.0000i	0.655087	−	0.655087i
$756$		0			0
$757$	−10.0000	+	10.0000i	−0.363456	+	0.363456i	−0.865084	−	0.501628i	$-0.832735\pi$
$757$	−10.0000	+	10.0000i	−0.363456	+	0.363456i	0.501628	+	0.865084i	$0.332735\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.0000			0.507500			0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000			0.507500			0.253750	+	0.967270i	$0.418336\pi$
$762$		0			0
$763$	−24.0000			−0.868858
$764$		0			0
$765$	−18.0000	+	18.0000i	−0.650791	+	0.650791i
$766$		0			0
$767$	−48.0000	−	48.0000i	−1.73318	−	1.73318i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.0000	−	14.0000i	0.503545	−	0.503545i	−0.408993	−	0.912538i	$-0.634120\pi$
$773$	14.0000	−	14.0000i	0.503545	−	0.503545i	0.912538	+	0.408993i	$0.134120\pi$
$774$		0			0
$775$	−6.00000			−0.215526
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.00000	+	54.0000i	0.214972	+	1.93475i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.00000	+	8.00000i	−0.285532	+	0.285532i
$786$		0			0
$787$			24.0000i			0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$			24.0000i			0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000	−	6.00000i	−0.213335	−	0.213335i
$792$		0			0
$793$	−24.0000	+	24.0000i	−0.852265	+	0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.0000			0.779280			0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000			0.779280			0.389640	+	0.920967i	$0.372599\pi$
$798$		0			0
$799$			18.0000i			0.636794i
$800$		0			0
$801$	15.0000	−	15.0000i	0.529999	−	0.529999i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	36.0000	−	36.0000i	1.26883	−	1.26883i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13.0000	+	13.0000i	0.457056	+	0.457056i	0.897688	−	0.440632i	$-0.145246\pi$
$809$	13.0000	+	13.0000i	0.457056	+	0.457056i	−0.440632	+	0.897688i	$0.645246\pi$
$810$		0			0
$811$		−	48.0000i		−	1.68551i	−0.538299	−	0.842754i	$-0.680933\pi$
$811$		−	48.0000i		−	1.68551i	0.538299	−	0.842754i	$-0.319067\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	48.0000i		−	1.68137i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	72.0000			2.51588
$820$		0			0
$821$	46.0000			1.60541			0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000			1.60541			0.802706	+	0.596376i	$0.203393\pi$
$822$		0			0
$823$	21.0000	+	21.0000i	0.732014	+	0.732014i	0.971018	−	0.239004i	$-0.0768211\pi$
$823$	21.0000	+	21.0000i	0.732014	+	0.732014i	−0.239004	+	0.971018i	$0.576821\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.00000	+	6.00000i	0.208640	+	0.208640i	0.803689	−	0.595049i	$-0.202867\pi$
$827$	6.00000	+	6.00000i	0.208640	+	0.208640i	−0.595049	+	0.803689i	$0.702867\pi$
$828$		0			0
$829$			54.0000i			1.87550i	0.347314	+	0.937749i	$0.387094\pi$
$829$			54.0000i			1.87550i	−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−33.0000	−	33.0000i	−1.14338	−	1.14338i
$834$		0			0
$835$	−6.00000	+	6.00000i	−0.207639	+	0.207639i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.00000	−	9.00000i	0.310715	−	0.310715i	−0.534472	−	0.845186i	$-0.679489\pi$
$839$	9.00000	−	9.00000i	0.310715	−	0.310715i	0.845186	+	0.534472i	$0.179489\pi$
$840$		0			0
$841$		−	21.0000i		−	0.724138i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−38.0000			−1.30724
$846$		0			0
$847$	33.0000	−	33.0000i	1.13389	−	1.13389i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−36.0000			−1.23406
$852$		0			0
$853$		−	18.0000i		−	0.616308i	−0.951336	−	0.308154i	$-0.900289\pi$
$853$		−	18.0000i		−	0.616308i	0.951336	−	0.308154i	$-0.0997113\pi$
$854$		0			0
$855$	36.0000	−	36.0000i	1.23117	−	1.23117i
$856$		0			0
$857$	−28.0000			−0.956462			−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000			−0.956462			−0.478231	+	0.878234i	$0.658722\pi$
$858$		0			0
$859$		−	12.0000i		−	0.409435i	−0.978821	−	0.204717i	$-0.934372\pi$
$859$		−	12.0000i		−	0.409435i	0.978821	−	0.204717i	$-0.0656275\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$	44.0000			1.49604
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$	−15.0000	−	15.0000i	−0.507673	−	0.507673i
$874$		0			0
$875$	−36.0000	+	36.0000i	−1.21702	+	1.21702i
$876$		0			0
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	50.0000i		−	1.68454i	−0.539054	−	0.842271i	$-0.681218\pi$
$881$		−	50.0000i		−	1.68454i	0.539054	−	0.842271i	$-0.318782\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.0000	+	21.0000i	−0.705111	+	0.705111i	−0.965503	−	0.260392i	$-0.916148\pi$
$887$	−21.0000	+	21.0000i	−0.705111	+	0.705111i	0.260392	+	0.965503i	$0.416148\pi$
$888$		0			0
$889$	18.0000	+	18.0000i	0.603701	+	0.603701i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	36.0000i		−	1.20469i
$894$		0			0
$895$	24.0000	+	24.0000i	0.802232	+	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000	−	12.0000i	−0.400222	−	0.400222i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0000	−	20.0000i	−0.664822	−	0.664822i
$906$		0			0
$907$		−	12.0000i		−	0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$		−	12.0000i		−	0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			24.0000i			0.795155i	0.917568	+	0.397578i	$0.130149\pi$
$911$			24.0000i			0.795155i	−0.917568	+	0.397578i	$0.869851\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−36.0000	+	36.0000i	−1.18882	+	1.18882i
$918$		0			0
$919$	−21.0000	+	21.0000i	−0.692726	+	0.692726i	−0.962831	−	0.270105i	$-0.912942\pi$
$919$	−21.0000	+	21.0000i	−0.692726	+	0.692726i	0.270105	+	0.962831i	$0.412942\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			72.0000i			2.36991i
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	−18.0000			−0.591198
$928$		0			0
$929$	−33.0000	+	33.0000i	−1.08269	+	1.08269i	−0.0864376	+	0.996257i	$0.527548\pi$
$929$	−33.0000	+	33.0000i	−1.08269	+	1.08269i	−0.996257	+	0.0864376i	$0.972452\pi$
$930$		0			0
$931$	66.0000	+	66.0000i	2.16306	+	2.16306i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	7.00000	−	7.00000i	0.228680	−	0.228680i	−0.583461	−	0.812141i	$-0.698302\pi$
$937$	7.00000	−	7.00000i	0.228680	−	0.228680i	0.812141	+	0.583461i	$0.198302\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	2.00000i		−	0.0651981i	−0.999469	−	0.0325991i	$-0.989622\pi$
$941$		−	2.00000i		−	0.0651981i	0.999469	−	0.0325991i	$-0.0103784\pi$
$942$		0			0
$943$	24.0000	−	30.0000i	0.781548	−	0.976934i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$		0			0
$949$	−48.0000	+	48.0000i	−1.55815	+	1.55815i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−52.0000			−1.68445			−0.842223	−	0.539130i	$-0.818753\pi$
$953$	−52.0000			−1.68445			−0.842223	+	0.539130i	$0.818753\pi$
$954$		0			0
$955$	6.00000	+	6.00000i	0.194155	+	0.194155i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000			0.581250
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		−	36.0000i		−	1.16008i
$964$		0			0
$965$	34.0000	−	34.0000i	1.09450	−	1.09450i
$966$		0			0
$967$	−33.0000	+	33.0000i	−1.06121	+	1.06121i	−0.0632081	+	0.998000i	$0.520133\pi$
$967$	−33.0000	+	33.0000i	−1.06121	+	1.06121i	−0.998000	+	0.0632081i	$0.979867\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000	−	12.0000i	−0.385098	−	0.385098i	0.487837	−	0.872935i	$-0.337786\pi$
$971$	−12.0000	−	12.0000i	−0.385098	−	0.385098i	−0.872935	+	0.487837i	$0.837786\pi$
$972$		0			0
$973$	36.0000	+	36.0000i	1.15411	+	1.15411i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.00000	−	9.00000i	−0.287936	−	0.287936i	0.548328	−	0.836263i	$-0.315265\pi$
$977$	−9.00000	−	9.00000i	−0.287936	−	0.287936i	−0.836263	+	0.548328i	$0.815265\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	12.0000	+	12.0000i	0.383131	+	0.383131i
$982$		0			0
$983$	−18.0000			−0.574111			−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000			−0.574111			−0.287055	+	0.957914i	$0.592676\pi$
$984$		0			0
$985$	20.0000			0.637253
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	15.0000	+	15.0000i	0.476491	+	0.476491i	0.904007	−	0.427517i	$-0.140611\pi$
$991$	15.0000	+	15.0000i	0.476491	+	0.476491i	−0.427517	+	0.904007i	$0.640611\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−30.0000	−	30.0000i	−0.951064	−	0.951064i
$996$		0			0
$997$	−20.0000	−	20.0000i	−0.633406	−	0.633406i	0.315514	−	0.948921i	$-0.397823\pi$
$997$	−20.0000	−	20.0000i	−0.633406	−	0.633406i	−0.948921	+	0.315514i	$0.897823\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1312.2.l.e.993.1	yes	2
4.3	odd	2		1312.2.l.b.993.1	✓	2
41.32	even	4	inner	1312.2.l.e.1057.1	yes	2
164.155	odd	4		1312.2.l.b.1057.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1312.2.l.b.993.1	✓	2	4.3	odd	2
1312.2.l.b.1057.1	yes	2	164.155	odd	4
1312.2.l.e.993.1	yes	2	1.1	even	1	trivial
1312.2.l.e.1057.1	yes	2	41.32	even	4	inner

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:	\( N \)	\(=\)	\( 1312 = 2^{5} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1312.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} +(3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+2.00000i q^{5} +(3.00000 + 3.00000i) q^{7} -3.00000i q^{9} +(4.00000 + 4.00000i) q^{13} +(-3.00000 + 3.00000i) q^{17} +(6.00000 - 6.00000i) q^{19} -6.00000 q^{23} +1.00000 q^{25} +(2.00000 + 2.00000i) q^{29} -6.00000 q^{31} +(-6.00000 + 6.00000i) q^{35} +6.00000 q^{37} +(-4.00000 + 5.00000i) q^{41} +6.00000 q^{45} +(3.00000 - 3.00000i) q^{47} +11.0000i q^{49} -12.0000 q^{59} +6.00000i q^{61} +(9.00000 - 9.00000i) q^{63} +(-8.00000 + 8.00000i) q^{65} +(6.00000 - 6.00000i) q^{67} +(9.00000 + 9.00000i) q^{71} +12.0000i q^{73} +(-9.00000 - 9.00000i) q^{79} -9.00000 q^{81} +(-6.00000 - 6.00000i) q^{85} +(5.00000 + 5.00000i) q^{89} +24.0000i q^{91} +(12.0000 + 12.0000i) q^{95} +(5.00000 - 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7}+O(q^{10}) \) 2 * q + 6 * q^7	\( 2 q + 6 q^{7} + 8 q^{13} - 6 q^{17} + 12 q^{19} - 12 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 12 q^{35} + 12 q^{37} - 8 q^{41} + 12 q^{45} + 6 q^{47} - 24 q^{59} + 18 q^{63} - 16 q^{65} + 12 q^{67} + 18 q^{71} - 18 q^{79} - 18 q^{81} - 12 q^{85} + 10 q^{89} + 24 q^{95} + 10 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 + 8 * q^13 - 6 * q^17 + 12 * q^19 - 12 * q^23 + 2 * q^25 + 4 * q^29 - 12 * q^31 - 12 * q^35 + 12 * q^37 - 8 * q^41 + 12 * q^45 + 6 * q^47 - 24 * q^59 + 18 * q^63 - 16 * q^65 + 12 * q^67 + 18 * q^71 - 18 * q^79 - 18 * q^81 - 12 * q^85 + 10 * q^89 + 24 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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