LMFDB - Embedded newform 399.3.ch.a.44.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,3,Mod(23,399)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(399, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 6, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("399.23");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	399.ch (of order $18$, degree $6$, 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:	$10.8719625480$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			44.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	399.44
Dual form			399.3.ch.a.263.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.520945 - 2.95442i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(6.99660 - 0.218262i) q^{7} +(-8.45723 + 3.07818i) q^{9} +O(q^{10})$	$q+(-0.520945 - 2.95442i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(6.99660 - 0.218262i) q^{7} +(-8.45723 + 3.07818i) q^{9} +(6.00000 + 10.3923i) q^{12} +(-23.8799 + 8.69156i) q^{13} +(12.2567 - 10.2846i) q^{16} +(5.50000 + 18.1865i) q^{19} +(-4.28968 - 20.5572i) q^{21} +(19.1511 + 16.0697i) q^{25} +(13.5000 + 23.3827i) q^{27} +(-26.0000 + 10.3923i) q^{28} +57.4434 q^{31} +(27.5776 - 23.1404i) q^{36} +(-36.3722 + 62.9984i) q^{37} +(38.1186 + 66.0234i) q^{39} +(13.5541 + 76.8693i) q^{43} +(-36.7701 - 30.8538i) q^{48} +(48.9047 - 3.05418i) q^{49} +(77.8681 - 65.3391i) q^{52} +(50.8655 - 25.7235i) q^{57} +(-63.3299 - 53.1401i) q^{61} +(-58.5000 + 23.3827i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(13.1571 - 74.6176i) q^{67} +(-0.723391 - 4.10255i) q^{73} +(37.5000 - 64.9519i) q^{75} +(-45.5539 - 60.8346i) q^{76} +(-117.444 + 98.5470i) q^{79} +(62.0496 - 52.0658i) q^{81} +(44.2478 + 71.4012i) q^{84} +(-165.181 + 66.0234i) q^{91} +(-29.9248 - 169.712i) q^{93} +(-129.462 + 108.631i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 36 q^{12} - 66 q^{13} + 33 q^{19} + 81 q^{27} - 156 q^{28} - 66 q^{43} + 276 q^{52} - 363 q^{61} - 351 q^{63} - 192 q^{64} + 366 q^{67} + 429 q^{73} + 225 q^{75} - 426 q^{79} + 117 q^{93}+O(q^{100}) $ 6 * q + 36 * q^12 - 66 * q^13 + 33 * q^19 + 81 * q^27 - 156 * q^28 - 66 * q^43 + 276 * q^52 - 363 * q^61 - 351 * q^63 - 192 * q^64 + 366 * q^67 + 429 * q^73 + 225 * q^75 - 426 * q^79 + 117 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{7}{9}\right)$

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		−0.173648	−	0.984808i	$-0.555556\pi$
$2$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$3$	−0.520945	−	2.95442i	−0.173648	−	0.984808i
$3$	−0.520945	−	2.95442i	−0.173648	−	0.984808i
$4$	−3.75877	+	1.36808i	−0.939693	+	0.342020i
$5$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$5$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$6$		0			0
$7$	6.99660	−	0.218262i	0.999514	−	0.0311803i
$7$	6.99660	−	0.218262i	0.999514	−	0.0311803i
$8$		0			0
$9$	−8.45723	+	3.07818i	−0.939693	+	0.342020i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	6.00000	+	10.3923i	0.500000	+	0.866025i
$13$	−23.8799	+	8.69156i	−1.83691	+	0.668581i	−0.846154	+	0.532939i	$0.821088\pi$
$13$	−23.8799	+	8.69156i	−1.83691	+	0.668581i	−0.990758	+	0.135642i	$0.956690\pi$
$14$		0			0
$15$		0			0
$16$	12.2567	−	10.2846i	0.766044	−	0.642788i
$17$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$17$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$18$		0			0
$19$	5.50000	+	18.1865i	0.289474	+	0.957186i
$19$	5.50000	+	18.1865i	0.289474	+	0.957186i
$20$		0			0
$21$	−4.28968	−	20.5572i	−0.204270	−	0.978915i
$22$		0			0
$23$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$23$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$24$		0			0
$25$	19.1511	+	16.0697i	0.766044	+	0.642788i
$26$		0			0
$27$	13.5000	+	23.3827i	0.500000	+	0.866025i
$28$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		0			0
$31$	57.4434			1.85301			0.926506	−	0.376280i	$-0.122797\pi$
$31$	57.4434			1.85301			0.926506	+	0.376280i	$0.122797\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	27.5776	−	23.1404i	0.766044	−	0.642788i
$37$	−36.3722	+	62.9984i	−0.983032	+	1.70266i	−0.332655	+	0.943049i	$0.607944\pi$
$37$	−36.3722	+	62.9984i	−0.983032	+	1.70266i	−0.650376	+	0.759612i	$0.725389\pi$
$38$		0			0
$39$	38.1186	+	66.0234i	0.977400	+	1.69291i
$40$		0			0
$41$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$41$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$42$		0			0
$43$	13.5541	+	76.8693i	0.315212	+	1.78766i	0.571026	+	0.820932i	$0.306545\pi$
$43$	13.5541	+	76.8693i	0.315212	+	1.78766i	−0.255814	+	0.966726i	$0.582343\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$47$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$48$	−36.7701	−	30.8538i	−0.766044	−	0.642788i
$49$	48.9047	−	3.05418i	0.998056	−	0.0623303i
$50$		0			0
$51$		0			0
$52$	77.8681	−	65.3391i	1.49746	−	1.25652i
$53$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$53$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	50.8655	−	25.7235i	0.892378	−	0.451290i
$58$		0			0
$59$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$59$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$60$		0			0
$61$	−63.3299	−	53.1401i	−1.03820	−	0.871149i	−0.0463918	−	0.998923i	$-0.514772\pi$
$61$	−63.3299	−	53.1401i	−1.03820	−	0.871149i	−0.991803	+	0.127774i	$0.959217\pi$
$62$		0			0
$63$	−58.5000	+	23.3827i	−0.928571	+	0.371154i
$64$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	13.1571	−	74.6176i	0.196374	−	1.11370i	−0.714073	−	0.700071i	$-0.753152\pi$
$67$	13.1571	−	74.6176i	0.196374	−	1.11370i	0.910448	−	0.413624i	$-0.135737\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$71$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$72$		0			0
$73$	−0.723391	−	4.10255i	−0.00990946	−	0.0561993i	0.979452	−	0.201677i	$-0.0646392\pi$
$73$	−0.723391	−	4.10255i	−0.00990946	−	0.0561993i	−0.989362	+	0.145478i	$0.953528\pi$
$74$		0			0
$75$	37.5000	−	64.9519i	0.500000	−	0.866025i
$76$	−45.5539	−	60.8346i	−0.599393	−	0.800455i
$77$		0			0
$78$		0			0
$79$	−117.444	+	98.5470i	−1.48663	+	1.24743i	−0.587896	+	0.808937i	$0.700043\pi$
$79$	−117.444	+	98.5470i	−1.48663	+	1.24743i	−0.898734	+	0.438494i	$0.855512\pi$
$80$		0			0
$81$	62.0496	−	52.0658i	0.766044	−	0.642788i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$	44.2478	+	71.4012i	0.526760	+	0.850014i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$89$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$90$		0			0
$91$	−165.181	+	66.0234i	−1.81517	+	0.725532i
$92$		0			0
$93$	−29.9248	−	169.712i	−0.321772	−	1.82486i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−129.462	+	108.631i	−1.33465	+	1.11991i	−0.351690	+	0.936117i	$0.614393\pi$
$97$	−129.462	+	108.631i	−1.33465	+	1.11991i	−0.982965	+	0.183792i	$0.941163\pi$
$98$		0			0
$99$		0			0
$100$	−93.9693	−	34.2020i	−0.939693	−	0.342020i
$101$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$101$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$102$		0			0
$103$	193.146			1.87521			0.937603	−	0.347707i	$-0.113040\pi$
$103$	193.146			1.87521			0.937603	+	0.347707i	$0.113040\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−82.7328	−	69.4211i	−0.766044	−	0.642788i
$109$	−163.934	+	137.557i	−1.50398	+	1.26199i	−0.629419	+	0.777066i	$0.716707\pi$
$109$	−163.934	+	137.557i	−1.50398	+	1.26199i	−0.874558	+	0.484921i	$0.838849\pi$
$110$		0			0
$111$	205.072	+	74.6401i	1.84749	+	0.672433i
$112$	83.5105	−	74.6324i	0.745630	−	0.666361i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	175.203	−	147.013i	1.49746	−	1.25652i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−215.916	+	78.5872i	−1.74126	+	0.633767i
$125$		0			0
$126$		0			0
$127$	−100.547	+	36.5962i	−0.791710	+	0.288159i	−0.706047	−	0.708165i	$-0.749523\pi$
$127$	−100.547	+	36.5962i	−0.791710	+	0.288159i	−0.0856630	+	0.996324i	$0.527301\pi$
$128$		0			0
$129$	220.043	−	80.0893i	1.70576	−	0.620847i
$130$		0			0
$131$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$131$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$132$		0			0
$133$	42.4507	+	126.043i	0.319178	+	0.947695i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$137$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$138$		0			0
$139$	−151.555	+	55.1615i	−1.09032	+	0.396846i	−0.823741	−	0.566966i	$-0.808117\pi$
$139$	−151.555	+	55.1615i	−1.09032	+	0.396846i	−0.266583	+	0.963812i	$0.585895\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−72.0000	+	124.708i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−34.5000	−	142.894i	−0.234694	−	0.972069i
$148$	50.5277	−	286.557i	0.341403	−	1.93619i
$149$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$149$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$150$		0			0
$151$	−113.500	−	196.588i	−0.751656	−	1.30191i	−0.947020	−	0.321175i	$-0.895922\pi$
$151$	−113.500	−	196.588i	−0.751656	−	1.30191i	0.195364	−	0.980731i	$-0.437411\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−233.604	−	196.017i	−1.49746	−	1.25652i
$157$	161.181	−	135.247i	1.02663	−	0.861443i	0.0361820	−	0.999345i	$-0.488480\pi$
$157$	161.181	−	135.247i	1.02663	−	0.861443i	0.990446	+	0.137902i	$0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	145.547			0.892925			0.446462	−	0.894802i	$-0.352684\pi$
$163$	145.547			0.892925			0.446462	+	0.894802i	$0.352684\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$167$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$168$		0			0
$169$	365.243	−	306.475i	2.16120	−	1.81346i
$170$		0			0
$171$	−102.496	−	136.878i	−0.599393	−	0.800455i
$172$	−156.110	−	270.391i	−0.907618	−	1.57204i
$173$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$173$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$174$		0			0
$175$	137.500	+	108.253i	0.785714	+	0.618590i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	240.538	+	201.835i	1.32894	+	1.11511i	0.984324	+	0.176368i	$0.0564349\pi$
$181$	240.538	+	201.835i	1.32894	+	1.11511i	0.344615	+	0.938744i	$0.388010\pi$
$182$		0			0
$183$	−124.007	+	214.786i	−0.677634	+	1.17370i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	99.5576	+	160.653i	0.526760	+	0.850014i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	180.421	+	65.6679i	0.939693	+	0.342020i
$193$	59.0414	+	334.840i	0.305914	+	1.73492i	0.619171	+	0.785256i	$0.287469\pi$
$193$	59.0414	+	334.840i	0.305914	+	1.73492i	−0.313257	+	0.949668i	$0.601420\pi$
$194$		0			0
$195$		0			0
$196$	−179.643	+	78.3856i	−0.916547	+	0.399926i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	40.5201	−	229.801i	0.203619	−	1.15478i	−0.695980	−	0.718061i	$-0.745030\pi$
$199$	40.5201	−	229.801i	0.203619	−	1.15478i	0.899599	−	0.436718i	$-0.143859\pi$
$200$		0			0
$201$	−227.306			−1.13088
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−203.299	+	352.125i	−0.977400	+	1.69291i
$209$		0			0
$210$		0			0
$211$	−288.455	+	242.043i	−1.36709	+	1.14712i	−0.393365	+	0.919382i	$0.628689\pi$
$211$	−288.455	+	242.043i	−1.36709	+	1.14712i	−0.973722	+	0.227740i	$0.926866\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	401.908	−	12.5377i	1.85211	−	0.0577775i
$218$		0			0
$219$	−11.7438	+	4.27440i	−0.0536248	+	0.0195178i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−37.8715	−	214.780i	−0.169828	−	0.963140i	−0.943946	−	0.330099i	$-0.892918\pi$
$223$	−37.8715	−	214.780i	−0.169828	−	0.963140i	0.774119	−	0.633041i	$-0.218193\pi$
$224$		0			0
$225$	−211.431	−	76.9545i	−0.939693	−	0.342020i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−156.000	+	166.277i	−0.684211	+	0.729285i
$229$	−65.9803	+	114.281i	−0.288124	+	0.499045i	−0.973362	−	0.229274i	$-0.926365\pi$
$229$	−65.9803	+	114.281i	−0.288124	+	0.499045i	0.685238	+	0.728319i	$0.259698\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$233$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	352.331	+	295.641i	1.48663	+	1.24743i
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	104.224	+	87.4547i	0.432467	+	0.362882i	0.832881	−	0.553451i	$-0.186690\pi$
$241$	104.224	+	87.4547i	0.432467	+	0.362882i	−0.400415	+	0.916334i	$0.631134\pi$
$242$		0			0
$243$	−186.149	−	156.197i	−0.766044	−	0.642788i
$244$	310.742	+	113.101i	1.27353	+	0.463529i
$245$		0			0
$246$		0			0
$247$	−289.408	−	386.488i	−1.17169	−	1.56473i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$251$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$252$	187.899	−	167.923i	0.745630	−	0.666361i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	44.4539	−	252.111i	0.173648	−	0.984808i
$257$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$257$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$258$		0			0
$259$	−240.731	+	448.713i	−0.929464	+	1.73248i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$263$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	52.6284	+	298.470i	0.196374	+	1.11370i
$269$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$269$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$270$		0			0
$271$	−280.968	−	102.264i	−1.03678	−	0.377358i	−0.233123	−	0.972447i	$-0.574894\pi$
$271$	−280.968	−	102.264i	−1.03678	−	0.377358i	−0.803660	+	0.595089i	$0.797117\pi$
$272$		0			0
$273$	281.111	+	453.619i	1.02971	+	1.66161i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−61.0000	+	105.655i	−0.220217	+	0.381426i	−0.954874	−	0.297012i	$-0.904010\pi$
$277$	−61.0000	+	105.655i	−0.220217	+	0.381426i	0.734657	+	0.678439i	$0.237343\pi$
$278$		0			0
$279$	−485.812	+	176.821i	−1.74126	+	0.633767i
$280$		0			0
$281$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$281$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$282$		0			0
$283$	485.821	+	176.824i	1.71668	+	0.624821i	0.997544	−	0.0700455i	$-0.0223145\pi$
$283$	485.821	+	176.824i	1.71668	+	0.624821i	0.719138	+	0.694867i	$0.244537\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	221.387	+	185.766i	0.766044	+	0.642788i
$290$		0			0
$291$	388.385	+	325.893i	1.33465	+	1.11991i
$292$	8.33168	+	14.4309i	0.0285332	+	0.0494209i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−52.0945	+	295.442i	−0.173648	+	0.984808i
$301$	111.610	+	534.865i	0.370799	+	1.77696i
$302$		0			0
$303$		0			0
$304$	254.453	+	166.342i	0.837017	+	0.547177i
$305$		0			0
$306$		0			0
$307$	336.410	+	122.443i	1.09580	+	0.398838i	0.825766	−	0.564013i	$-0.190743\pi$
$307$	336.410	+	122.443i	1.09580	+	0.398838i	0.270032	+	0.962851i	$0.412966\pi$
$308$		0			0
$309$	−100.618	−	570.636i	−0.325626	−	1.84672i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	133.436	−	48.5669i	0.426314	−	0.155166i	−0.119947	−	0.992780i	$-0.538273\pi$
$313$	133.436	−	48.5669i	0.426314	−	0.155166i	0.546262	+	0.837615i	$0.316050\pi$
$314$		0			0
$315$		0			0
$316$	306.624	−	531.088i	0.970329	−	1.68066i
$317$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$317$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−162.000	+	280.592i	−0.500000	+	0.866025i
$325$	−596.996	−	217.289i	−1.83691	−	0.668581i
$326$		0			0
$327$	491.801	+	412.670i	1.50398	+	1.26199i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−150.602			−0.454991			−0.227495	−	0.973779i	$-0.573054\pi$
$331$	−150.602			−0.454991			−0.227495	+	0.973779i	$0.573054\pi$
$332$		0			0
$333$	113.687	−	644.753i	0.341403	−	1.93619i
$334$		0			0
$335$		0			0
$336$	−264.000	−	207.846i	−0.785714	−	0.618590i
$337$	−106.631	−	604.735i	−0.316413	−	1.79446i	−0.564187	−	0.825647i	$-0.690810\pi$
$337$	−106.631	−	604.735i	−0.316413	−	1.79446i	0.247774	−	0.968818i	$-0.420301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	341.500	−	32.0429i	0.995627	−	0.0934196i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$347$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$348$		0			0
$349$	348.145	−	603.005i	0.997550	−	1.72781i	0.438187	−	0.898884i	$-0.355621\pi$
$349$	348.145	−	603.005i	0.997550	−	1.72781i	0.559362	−	0.828923i	$-0.311046\pi$
$350$		0			0
$351$	−525.610	−	441.039i	−1.49746	−	1.25652i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$359$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$360$		0			0
$361$	−300.500	+	200.052i	−0.832410	+	0.554160i
$362$		0			0
$363$	−278.074	+	233.332i	−0.766044	+	0.642788i
$364$	530.551	−	474.147i	1.45756	−	1.30260i
$365$		0			0
$366$		0			0
$367$	476.915	+	400.179i	1.29950	+	1.09041i	0.990232	+	0.139431i	$0.0445272\pi$
$367$	476.915	+	400.179i	1.29950	+	1.09041i	0.309264	+	0.950976i	$0.399917\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	344.660	+	596.969i	0.926506	+	1.60476i
$373$	60.5000	+	104.789i	0.162198	+	0.280936i	0.935657	−	0.352911i	$-0.114808\pi$
$373$	60.5000	+	104.789i	0.162198	+	0.280936i	−0.773458	+	0.633847i	$0.781475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−727.583			−1.91974			−0.959872	−	0.280439i	$-0.909520\pi$
$379$	−727.583			−1.91974			−0.959872	+	0.280439i	$0.909520\pi$
$380$		0			0
$381$	160.500	+	277.994i	0.421260	+	0.729643i
$382$		0			0
$383$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$383$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−351.248	−	608.379i	−0.907618	−	1.57204i
$388$	338.000	−	585.433i	0.871134	−	1.50885i
$389$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$389$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−30.7244	−	174.247i	−0.0773915	−	0.438909i	−0.998741	−	0.0501728i	$-0.984023\pi$
$397$	−30.7244	−	174.247i	−0.0773915	−	0.438909i	0.921349	−	0.388736i	$-0.127088\pi$
$398$		0			0
$399$	350.271	−	191.079i	0.877872	−	0.478895i
$400$	400.000			1.00000
$401$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$401$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$402$		0			0
$403$	−1371.74	+	499.272i	−3.40382	+	1.23889i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	24.8317	−	140.828i	0.0607132	−	0.344322i	−0.939286	−	0.343135i	$-0.888511\pi$
$409$	24.8317	−	140.828i	0.0607132	−	0.344322i	0.999999	−	0.00118660i	$-0.000377706\pi$
$410$		0			0
$411$		0			0
$412$	−725.992	+	264.240i	−1.76212	+	0.641358i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	241.922	+	419.022i	0.580149	+	1.00485i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	451.992	+	164.512i	1.07362	+	0.390764i	0.817528	−	0.575889i	$-0.195344\pi$
$421$	451.992	+	164.512i	1.07362	+	0.390764i	0.256088	+	0.966653i	$0.417566\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−454.692	−	357.977i	−1.06485	−	0.838354i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$431$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$432$	405.947	+	147.753i	0.939693	+	0.342020i
$433$	−123.945	+	702.927i	−0.286247	+	1.62339i	0.414550	+	0.910027i	$0.363939\pi$
$433$	−123.945	+	702.927i	−0.286247	+	1.62339i	−0.700797	+	0.713361i	$0.747172\pi$
$434$		0			0
$435$		0			0
$436$	428.000	−	741.318i	0.981651	−	1.70027i
$437$		0			0
$438$		0			0
$439$	−152.118	−	862.706i	−0.346511	−	1.96516i	−0.239449	−	0.970909i	$-0.576967\pi$
$439$	−152.118	−	862.706i	−0.346511	−	1.96516i	−0.107062	−	0.994252i	$-0.534144\pi$
$440$		0			0
$441$	−404.197	+	176.368i	−0.916547	+	0.399926i
$442$		0			0
$443$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$443$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$444$	−872.932			−1.96606
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−211.794	+	394.775i	−0.472754	+	0.881194i
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−521.676	+	437.738i	−1.15160	+	0.966310i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−134.014	+	232.118i	−0.293246	+	0.507918i	−0.974575	−	0.224060i	$-0.928069\pi$
$457$	−134.014	+	232.118i	−0.293246	+	0.507918i	0.681329	+	0.731977i	$0.261402\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$461$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$462$		0			0
$463$	−43.4653	+	75.2841i	−0.0938775	+	0.162601i	−0.909140	−	0.416492i	$-0.863260\pi$
$463$	−43.4653	+	75.2841i	−0.0938775	+	0.162601i	0.815262	+	0.579092i	$0.196593\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−457.423	+	792.280i	−0.977400	+	1.69291i
$469$	75.7687	−	524.941i	0.161554	−	1.11928i
$470$		0			0
$471$	−483.542	−	405.740i	−1.02663	−	0.861443i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−186.921	+	436.676i	−0.393518	+	0.919317i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$479$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$480$		0			0
$481$	321.008	−	1820.52i	0.667375	−	3.78487i
$482$		0			0
$483$		0			0
$484$	370.766	+	311.109i	0.766044	+	0.642788i
$485$		0			0
$486$		0			0
$487$	962.000			1.97536			0.987680	−	0.156489i	$-0.0500176\pi$
$487$	962.000			1.97536			0.987680	+	0.156489i	$0.0500176\pi$
$488$		0			0
$489$	−75.8218	−	430.007i	−0.155055	−	0.879359i
$490$		0			0
$491$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$491$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	704.067	−	590.782i	1.41949	−	1.19109i
$497$		0			0
$498$		0			0
$499$	583.894	+	212.520i	1.17013	+	0.425892i	0.852705	−	0.522392i	$-0.174960\pi$
$499$	583.894	+	212.520i	1.17013	+	0.425892i	0.317423	+	0.948284i	$0.397182\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$503$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1095.73	−	919.425i	−2.16120	−	1.81346i
$508$	327.867	−	275.113i	0.645408	−	0.541561i
$509$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$509$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$510$		0			0
$511$	−5.95670	−	28.5460i	−0.0116570	−	0.0558630i
$512$		0			0
$513$	−351.000	+	374.123i	−0.684211	+	0.729285i
$514$		0			0
$515$		0			0
$516$	−717.524	+	602.074i	−1.39055	+	1.16681i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−173.158	+	63.0243i	−0.331086	+	0.120505i	−0.502214	−	0.864743i	$-0.667481\pi$
$523$	−173.158	+	63.0243i	−0.331086	+	0.120505i	0.171128	+	0.985249i	$0.445259\pi$
$524$		0			0
$525$	248.196	−	462.627i	0.472754	−	0.881194i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	91.8599	+	520.963i	0.173648	+	0.984808i
$530$		0			0
$531$		0			0
$532$	−332.000	−	415.692i	−0.624060	−	0.781376i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−463.652	+	168.755i	−0.857027	+	0.311932i	−0.732902	−	0.680334i	$-0.761835\pi$
$541$	−463.652	+	168.755i	−0.857027	+	0.311932i	−0.124125	+	0.992267i	$0.539612\pi$
$542$		0			0
$543$	471.000	−	815.796i	0.867403	−	1.50239i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−24.9606	+	141.559i	−0.0456319	+	0.258791i	−0.999086	−	0.0427471i	$-0.986389\pi$
$547$	−24.9606	+	141.559i	−0.0456319	+	0.258791i	0.953454	+	0.301538i	$0.0975001\pi$
$548$		0			0
$549$	699.171	+	254.477i	1.27353	+	0.463529i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−800.198	+	715.127i	−1.44701	+	1.29318i
$554$		0			0
$555$		0			0
$556$	494.195	−	414.679i	0.888841	−	0.745826i
$557$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$557$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$558$		0			0
$559$	−991.784	−	1717.82i	−1.77421	−	3.07302i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	422.772	−	377.826i	0.745630	−	0.666361i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−539.502	−	934.445i	−0.944838	−	1.63651i	−0.756076	−	0.654484i	$-0.772886\pi$
$571$	−539.502	−	934.445i	−0.944838	−	1.63651i	−0.188761	−	0.982023i	$-0.560447\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	100.021	−	567.249i	0.173648	−	0.984808i
$577$	71.0000			0.123050			0.0615251	−	0.998106i	$-0.480404\pi$
$577$	71.0000			0.123050			0.0615251	+	0.998106i	$0.480404\pi$
$578$		0			0
$579$	958.503	−	348.867i	1.65545	−	0.602533i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$587$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$588$	325.168	+	489.908i	0.553007	+	0.833176i
$589$	315.939	+	1044.70i	0.536398	+	1.77368i
$590$		0			0
$591$		0			0
$592$	202.111	+	1146.23i	0.341403	+	1.93619i
$593$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$593$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−700.038			−1.17259
$598$		0			0
$599$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$599$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$600$		0			0
$601$	548.832	−	950.605i	0.913198	−	1.58171i	0.103680	−	0.994611i	$-0.466938\pi$
$601$	548.832	−	950.605i	0.913198	−	1.58171i	0.809518	−	0.587095i	$-0.199728\pi$
$602$		0			0
$603$	118.414	+	671.558i	0.196374	+	1.11370i
$604$	695.568	+	583.651i	1.15160	+	0.966310i
$605$		0			0
$606$		0			0
$607$	22.3115	−	38.6446i	0.0367570	−	0.0636649i	−0.847062	−	0.531495i	$-0.821631\pi$
$607$	22.3115	−	38.6446i	0.0367570	−	0.0636649i	0.883819	+	0.467830i	$0.154964\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	753.022	−	631.860i	1.22842	−	1.03077i	0.230080	−	0.973172i	$-0.426101\pi$
$613$	753.022	−	631.860i	1.22842	−	1.03077i	0.998340	−	0.0575954i	$-0.0183434\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$617$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$618$		0			0
$619$	−309.929	+	536.813i	−0.500693	+	0.867226i	0.499306	+	0.866426i	$0.333588\pi$
$619$	−309.929	+	536.813i	−0.500693	+	0.867226i	−1.00000		0.000800838i	$0.999745\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1146.23	+	417.195i	1.83691	+	0.668581i
$625$	108.530	+	615.505i	0.173648	+	0.984808i
$626$		0			0
$627$		0			0
$628$	−420.813	+	728.869i	−0.670084	+	1.16062i
$629$		0			0
$630$		0			0
$631$	−29.4341	+	166.929i	−0.0466467	+	0.264546i	−0.999208	−	0.0398015i	$-0.987327\pi$
$631$	−29.4341	+	166.929i	−0.0466467	+	0.264546i	0.952561	+	0.304348i	$0.0984386\pi$
$632$		0			0
$633$	865.366	+	726.128i	1.36709	+	1.14712i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1141.29	+	497.992i	−1.79167	+	0.781776i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$641$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$642$		0			0
$643$	1102.83	+	401.397i	1.71513	+	0.624256i	0.997400	−	0.0720661i	$-0.0229593\pi$
$643$	1102.83	+	401.397i	1.71513	+	0.624256i	0.717729	+	0.696322i	$0.245181\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−246.414	−	1180.88i	−0.378515	−	1.81394i
$652$	−547.077	+	199.120i	−0.839075	+	0.305398i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	18.7463	+	32.4695i	0.0285332	+	0.0494209i
$658$		0			0
$659$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$659$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$660$		0			0
$661$	1128.57	−	410.766i	1.70737	−	0.621431i	0.710739	−	0.703456i	$-0.248361\pi$
$661$	1128.57	−	410.766i	1.70737	−	0.621431i	0.996630	+	0.0820248i	$0.0261387\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−614.823	+	223.777i	−0.919017	+	0.334495i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−664.676	−	1151.25i	−0.987631	−	1.71063i	−0.629604	−	0.776916i	$-0.716783\pi$
$673$	−664.676	−	1151.25i	−0.987631	−	1.71063i	−0.358027	−	0.933711i	$-0.616551\pi$
$674$		0			0
$675$	−117.213	+	664.745i	−0.173648	+	0.984808i
$676$	−953.581	+	1651.65i	−1.41062	+	2.44327i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−882.080	+	788.305i	−1.29909	+	1.16098i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	572.520	+	374.269i	0.837017	+	0.547177i
$685$		0			0
$686$		0			0
$687$	372.007	+	135.400i	0.541495	+	0.197088i
$688$	956.699	+	802.766i	1.39055	+	1.16681i
$689$		0			0
$690$		0			0
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.737337	+	0.675525i	$0.236083\pi$
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.216353	+	0.976315i	$0.430584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−664.930	−	218.788i	−0.949900	−	0.312554i
$701$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$701$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$702$		0			0
$703$	−1345.77	−	314.992i	−1.91432	−	0.448069i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−210.622	+	1194.49i	−0.297069	+	1.68476i	0.361606	+	0.932331i	$0.382229\pi$
$709$	−210.622	+	1194.49i	−0.297069	+	1.68476i	−0.658674	+	0.752428i	$0.728882\pi$
$710$		0			0
$711$	689.904	−	1194.95i	0.970329	−	1.68066i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$719$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$720$		0			0
$721$	1351.37	−	42.1565i	1.87429	−	0.0584695i
$722$		0			0
$723$	204.083	−	353.482i	0.282272	−	0.488910i
$724$	−1180.25	−	429.577i	−1.63018	−	0.593339i
$725$		0			0
$726$		0			0
$727$	2.82109	+	15.9992i	0.00388045	+	0.0220071i	0.986687	−	0.162634i	$-0.0519989\pi$
$727$	2.82109	+	15.9992i	0.00388045	+	0.0220071i	−0.982806	+	0.184641i	$0.940888\pi$
$728$		0			0
$729$	−364.500	+	631.333i	−0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$	172.269	−	976.984i	0.235340	−	1.33468i
$733$	1034.00			1.41064			0.705321	−	0.708888i	$-0.250803\pi$
$733$	1034.00			1.41064			0.705321	+	0.708888i	$0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	377.477	+	137.390i	0.510794	+	0.185914i	0.584543	−	0.811363i	$-0.301274\pi$
$739$	377.477	+	137.390i	0.510794	+	0.185914i	−0.0737483	+	0.997277i	$0.523496\pi$
$740$		0			0
$741$	−991.084	+	1056.37i	−1.33750	+	1.42561i
$742$		0			0
$743$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$743$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−321.232	+	116.919i	−0.427739	+	0.155684i	−0.546914	−	0.837189i	$-0.684197\pi$
$751$	−321.232	+	116.919i	−0.427739	+	0.155684i	0.119174	+	0.992873i	$0.461975\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−594.000	−	467.654i	−0.785714	−	0.618590i
$757$	−158.401	−	57.6533i	−0.209248	−	0.0761602i	0.235269	−	0.971930i	$-0.424403\pi$
$757$	−158.401	−	57.6533i	−0.209248	−	0.0761602i	−0.444518	+	0.895770i	$0.646625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−1116.95	+	998.208i	−1.46390	+	1.30827i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−768.000			−1.00000
$769$	−954.772	−	801.149i	−1.24158	−	1.04181i	−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−954.772	−	801.149i	−1.24158	−	1.04181i	−0.244177	−	0.969731i	$-0.578518\pi$
$770$		0			0
$771$		0			0
$772$	−680.012	−	1177.81i	−0.880844	−	1.52567i
$773$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$773$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$774$		0			0
$775$	1100.10	+	923.097i	1.41949	+	1.19109i
$776$		0			0
$777$	1451.10	+	477.467i	1.86756	+	0.614501i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	568.000	−	540.400i	0.724490	−	0.689286i
$785$		0			0
$786$		0			0
$787$	−231.141	+	400.347i	−0.293698	+	0.508700i	−0.974681	−	0.223599i	$-0.928219\pi$
$787$	−231.141	+	400.347i	−0.293698	+	0.508700i	0.680983	+	0.732299i	$0.261553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1974.18	+	718.542i	2.48951	+	0.906106i
$794$		0			0
$795$		0			0
$796$	162.080	+	919.204i	0.203619	+	1.15478i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	854.391	−	310.973i	1.06268	−	0.386782i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	237.742	−	86.5311i	0.293147	−	0.106697i	−0.191260	−	0.981539i	$-0.561257\pi$
$811$	237.742	−	86.5311i	0.293147	−	0.106697i	0.484407	+	0.874843i	$0.339035\pi$
$812$		0			0
$813$	−155.762	+	883.373i	−0.191590	+	1.08656i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1323.44	+	669.284i	−1.61988	+	0.819197i
$818$		0			0
$819$	1193.74	−	1066.83i	1.45756	−	1.30260i
$820$		0			0
$821$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$821$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$822$		0			0
$823$	−1241.76	−	1041.96i	−1.50882	−	1.26605i	−0.866014	−	0.500020i	$-0.833326\pi$
$823$	−1241.76	−	1041.96i	−1.50882	−	1.26605i	−0.642805	−	0.766030i	$-0.722229\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$827$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$828$		0			0
$829$	−673.425			−0.812334			−0.406167	−	0.913799i	$-0.633135\pi$
$829$	−673.425			−0.812334			−0.406167	+	0.913799i	$0.633135\pi$
$830$		0			0
$831$	343.927	+	125.179i	0.413872	+	0.150637i
$832$	282.420	−	1601.69i	0.339448	−	1.92510i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	775.486	+	1343.18i	0.926506	+	1.60476i
$838$		0			0
$839$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$839$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$840$		0			0
$841$	146.038	+	828.223i	0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$	753.103	−	1304.41i	0.892303	−	1.54551i
$845$		0			0
$846$		0			0
$847$	−446.166	−	719.962i	−0.526760	−	0.850014i
$848$		0			0
$849$	269.328	−	1527.44i	0.317230	−	1.79910i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−82.5967	+	69.3069i	−0.0968308	+	0.0812507i	−0.689918	−	0.723888i	$-0.742353\pi$
$853$	−82.5967	+	69.3069i	−0.0968308	+	0.0812507i	0.593087	+	0.805138i	$0.297909\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$857$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$858$		0			0
$859$	−766.613	−	643.265i	−0.892448	−	0.748853i	0.0762515	−	0.997089i	$-0.475705\pi$
$859$	−766.613	−	643.265i	−0.892448	−	0.748853i	−0.968700	+	0.248236i	$0.920149\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	433.500	−	750.844i	0.500000	−	0.866025i
$868$	−1493.53	+	596.969i	−1.72065	+	0.687752i
$869$		0			0
$870$		0			0
$871$	334.353	+	1896.21i	0.383873	+	2.17705i
$872$		0			0
$873$	760.500	−	1317.22i	0.871134	−	1.50885i
$874$		0			0
$875$		0			0
$876$	38.2946	−	32.1330i	0.0437153	−	0.0366815i
$877$	−1.55128	+	1.30168i	−0.00176885	+	0.00148424i	−0.643672	−	0.765302i	$-0.722590\pi$
$877$	−1.55128	+	1.30168i	−0.00176885	+	0.00148424i	0.641903	+	0.766786i	$0.278145\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−581.818	−	488.203i	−0.658911	−	0.552892i	0.250849	−	0.968026i	$-0.419290\pi$
$883$	−581.818	−	488.203i	−0.658911	−	0.552892i	−0.909760	+	0.415134i	$0.863735\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$887$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$888$		0			0
$889$	−695.500	+	277.994i	−0.782340	+	0.312704i
$890$		0			0
$891$		0			0
$892$	436.187	+	755.498i	0.488999	+	0.846971i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	900.000			1.00000
$901$		0			0
$902$		0			0
$903$	1522.07	−	608.379i	1.68558	−	0.673731i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−163.934	+	137.557i	−0.180743	+	0.151661i	−0.728670	−	0.684865i	$-0.759861\pi$
$907$	−163.934	+	137.557i	−0.180743	+	0.151661i	0.547928	+	0.836526i	$0.315417\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	358.888	−	838.417i	0.393518	−	0.919317i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	91.6589	−	519.823i	0.100064	−	0.567493i
$917$		0			0
$918$		0			0
$919$	129.645	+	224.553i	0.141072	+	0.244344i	0.927901	−	0.372827i	$-0.121612\pi$
$919$	129.645	+	224.553i	0.141072	+	0.244344i	−0.786828	+	0.617172i	$0.788278\pi$
$920$		0			0
$921$	186.498	−	1057.68i	0.202495	−	1.14841i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1708.93	+	622.001i	−1.84749	+	0.672433i
$926$		0			0
$927$	−1633.48	+	594.539i	−1.76212	+	0.641358i
$928$		0			0
$929$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$929$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$930$		0			0
$931$	324.521	+	872.609i	0.348572	+	0.937282i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1732.81	−	630.692i	1.84932	−	0.673097i	0.863728	−	0.503959i	$-0.168124\pi$
$937$	1732.81	−	630.692i	1.84932	−	0.673097i	0.985592	−	0.169138i	$-0.0540985\pi$
$938$		0			0
$939$	−213.000	−	368.927i	−0.226837	−	0.392893i
$940$		0			0
$941$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$941$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$947$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$948$	−1728.79	−	629.229i	−1.82362	−	0.663744i
$949$	52.9320	+	91.6809i	0.0557766	+	0.0966080i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$953$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2338.74			2.43365
$962$		0			0
$963$		0			0
$964$	−511.401	−	186.135i	−0.530499	−	0.193086i
$965$		0			0
$966$		0			0
$967$	795.659	−	667.637i	0.822812	−	0.690421i	−0.130817	−	0.991407i	$-0.541760\pi$
$967$	795.659	−	667.637i	0.822812	−	0.690421i	0.953629	+	0.300985i	$0.0973156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$971$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$972$	913.381	+	332.444i	0.939693	+	0.342020i
$973$	−1048.33	+	419.022i	−1.07742	+	0.430649i
$974$		0			0
$975$	−330.961	+	1876.98i	−0.339448	+	1.92510i
$976$	−1322.74			−1.35527
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	963.000	−	1667.96i	0.981651	−	1.70027i
$982$		0			0
$983$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$983$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1616.57	+	1056.79i	1.63620	+	1.06962i
$989$		0			0
$990$		0			0
$991$	−110.176	−	624.836i	−0.111176	−	0.630511i	−0.988573	−	0.150745i	$-0.951833\pi$
$991$	−110.176	−	624.836i	−0.111176	−	0.630511i	0.877397	−	0.479766i	$-0.159278\pi$
$992$		0			0
$993$	78.4552	+	444.942i	0.0790083	+	0.448078i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	49.5202	−	280.843i	0.0496692	−	0.281688i	−0.949850	−	0.312707i	$-0.898764\pi$
$997$	49.5202	−	280.843i	0.0496692	−	0.281688i	0.999519	+	0.0310190i	$0.00987523\pi$
$998$		0			0
$999$	−1964.10			−1.96606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.3.ch.a.44.1	yes	6
3.2	odd	2	CM	399.3.ch.a.44.1	yes	6
7.4	even	3		399.3.ca.a.158.1	yes	6
19.16	even	9		399.3.ca.a.149.1	✓	6
21.11	odd	6		399.3.ca.a.158.1	yes	6
57.35	odd	18		399.3.ca.a.149.1	✓	6
133.130	even	9	inner	399.3.ch.a.263.1	yes	6
399.263	odd	18	inner	399.3.ch.a.263.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.3.ca.a.149.1	✓	6	19.16	even	9
399.3.ca.a.149.1	✓	6	57.35	odd	18
399.3.ca.a.158.1	yes	6	7.4	even	3
399.3.ca.a.158.1	yes	6	21.11	odd	6
399.3.ch.a.44.1	yes	6	1.1	even	1	trivial
399.3.ch.a.44.1	yes	6	3.2	odd	2	CM
399.3.ch.a.263.1	yes	6	133.130	even	9	inner
399.3.ch.a.263.1	yes	6	399.263	odd	18	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:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	399.ch (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.520945 - 2.95442i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(6.99660 - 0.218262i) q^{7} +(-8.45723 + 3.07818i) q^{9} +O(q^{10})\)	\(q+(-0.520945 - 2.95442i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(6.99660 - 0.218262i) q^{7} +(-8.45723 + 3.07818i) q^{9} +(6.00000 + 10.3923i) q^{12} +(-23.8799 + 8.69156i) q^{13} +(12.2567 - 10.2846i) q^{16} +(5.50000 + 18.1865i) q^{19} +(-4.28968 - 20.5572i) q^{21} +(19.1511 + 16.0697i) q^{25} +(13.5000 + 23.3827i) q^{27} +(-26.0000 + 10.3923i) q^{28} +57.4434 q^{31} +(27.5776 - 23.1404i) q^{36} +(-36.3722 + 62.9984i) q^{37} +(38.1186 + 66.0234i) q^{39} +(13.5541 + 76.8693i) q^{43} +(-36.7701 - 30.8538i) q^{48} +(48.9047 - 3.05418i) q^{49} +(77.8681 - 65.3391i) q^{52} +(50.8655 - 25.7235i) q^{57} +(-63.3299 - 53.1401i) q^{61} +(-58.5000 + 23.3827i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(13.1571 - 74.6176i) q^{67} +(-0.723391 - 4.10255i) q^{73} +(37.5000 - 64.9519i) q^{75} +(-45.5539 - 60.8346i) q^{76} +(-117.444 + 98.5470i) q^{79} +(62.0496 - 52.0658i) q^{81} +(44.2478 + 71.4012i) q^{84} +(-165.181 + 66.0234i) q^{91} +(-29.9248 - 169.712i) q^{93} +(-129.462 + 108.631i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 36 q^{12} - 66 q^{13} + 33 q^{19} + 81 q^{27} - 156 q^{28} - 66 q^{43} + 276 q^{52} - 363 q^{61} - 351 q^{63} - 192 q^{64} + 366 q^{67} + 429 q^{73} + 225 q^{75} - 426 q^{79} + 117 q^{93}+O(q^{100}) \) 6 * q + 36 * q^12 - 66 * q^13 + 33 * q^19 + 81 * q^27 - 156 * q^28 - 66 * q^43 + 276 * q^52 - 363 * q^61 - 351 * q^63 - 192 * q^64 + 366 * q^67 + 429 * q^73 + 225 * q^75 - 426 * q^79 + 117 * q^93

Introduction

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