LMFDB - Embedded newform 441.3.m.a.325.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,3,Mod(19,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("441.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	441.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.0163796583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			325.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.325
Dual form			441.3.m.a.19.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{2} +(-2.50000 + 4.33013i) q^{4} +3.00000 q^{8} +O(q^{10})$	$q+(-1.50000 - 2.59808i) q^{2} +(-2.50000 + 4.33013i) q^{4} +3.00000 q^{8} +(-3.00000 + 5.19615i) q^{11} +(5.50000 + 9.52628i) q^{16} +18.0000 q^{22} +(9.00000 + 15.5885i) q^{23} +(-12.5000 + 21.6506i) q^{25} +54.0000 q^{29} +(22.5000 - 38.9711i) q^{32} +(19.0000 + 32.9090i) q^{37} +58.0000 q^{43} +(-15.0000 - 25.9808i) q^{44} +(27.0000 - 46.7654i) q^{46} +75.0000 q^{50} +(-3.00000 + 5.19615i) q^{53} +(-81.0000 - 140.296i) q^{58} -91.0000 q^{64} +(59.0000 - 102.191i) q^{67} -114.000 q^{71} +(57.0000 - 98.7269i) q^{74} +(47.0000 + 81.4064i) q^{79} +(-87.0000 - 150.688i) q^{86} +(-9.00000 + 15.5885i) q^{88} -90.0000 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 5 q^{4} + 6 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 - 5 * q^4 + 6 * q^8	$ 2 q - 3 q^{2} - 5 q^{4} + 6 q^{8} - 6 q^{11} + 11 q^{16} + 36 q^{22} + 18 q^{23} - 25 q^{25} + 108 q^{29} + 45 q^{32} + 38 q^{37} + 116 q^{43} - 30 q^{44} + 54 q^{46} + 150 q^{50} - 6 q^{53} - 162 q^{58} - 182 q^{64} + 118 q^{67} - 228 q^{71} + 114 q^{74} + 94 q^{79} - 174 q^{86} - 18 q^{88} - 180 q^{92}+O(q^{100}) $ 2 * q - 3 * q^2 - 5 * q^4 + 6 * q^8 - 6 * q^11 + 11 * q^16 + 36 * q^22 + 18 * q^23 - 25 * q^25 + 108 * q^29 + 45 * q^32 + 38 * q^37 + 116 * q^43 - 30 * q^44 + 54 * q^46 + 150 * q^50 - 6 * q^53 - 162 * q^58 - 182 * q^64 + 118 * q^67 - 228 * q^71 + 114 * q^74 + 94 * q^79 - 174 * q^86 - 18 * q^88 - 180 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$

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 $

$2$	−1.50000	−	2.59808i	−0.750000	−	1.29904i	−0.947822	−	0.318800i	$-0.896720\pi$
$2$	−1.50000	−	2.59808i	−0.750000	−	1.29904i	0.197822	−	0.980238i	$-0.436613\pi$
$3$		0			0
$3$		0			0
$4$	−2.50000	+	4.33013i	−0.625000	+	1.08253i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	3.00000			0.375000
$9$		0			0
$10$		0			0
$11$	−3.00000	+	5.19615i	−0.272727	+	0.472377i	−0.969559	−	0.244857i	$-0.921259\pi$
$11$	−3.00000	+	5.19615i	−0.272727	+	0.472377i	0.696832	+	0.717234i	$0.254592\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	5.50000	+	9.52628i	0.343750	+	0.595392i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$		0			0
$22$	18.0000			0.818182
$23$	9.00000	+	15.5885i	0.391304	+	0.677759i	0.992622	−	0.121251i	$-0.0386906\pi$
$23$	9.00000	+	15.5885i	0.391304	+	0.677759i	−0.601318	+	0.799010i	$0.705357\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	54.0000			1.86207			0.931034	−	0.364931i	$-0.118907\pi$
$29$	54.0000			1.86207			0.931034	+	0.364931i	$0.118907\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	22.5000	−	38.9711i	0.703125	−	1.21785i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	19.0000	+	32.9090i	0.513514	+	0.889431i	0.999877	+	0.0156750i	$0.00498971\pi$
$37$	19.0000	+	32.9090i	0.513514	+	0.889431i	−0.486364	+	0.873757i	$0.661677\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	58.0000			1.34884			0.674419	−	0.738349i	$-0.264394\pi$
$43$	58.0000			1.34884			0.674419	+	0.738349i	$0.264394\pi$
$44$	−15.0000	−	25.9808i	−0.340909	−	0.590472i
$45$		0			0
$46$	27.0000	−	46.7654i	0.586957	−	1.01664i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$		0			0
$50$	75.0000			1.50000
$51$		0			0
$52$		0			0
$53$	−3.00000	+	5.19615i	−0.0566038	+	0.0980406i	−0.892939	−	0.450178i	$-0.851361\pi$
$53$	−3.00000	+	5.19615i	−0.0566038	+	0.0980406i	0.836335	+	0.548219i	$0.184694\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−81.0000	−	140.296i	−1.39655	−	2.41890i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−91.0000			−1.42188
$65$		0			0
$66$		0			0
$67$	59.0000	−	102.191i	0.880597	−	1.52524i	0.0299186	−	0.999552i	$-0.490475\pi$
$67$	59.0000	−	102.191i	0.880597	−	1.52524i	0.850678	−	0.525686i	$-0.176191\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−114.000			−1.60563			−0.802817	−	0.596226i	$-0.796666\pi$
$71$	−114.000			−1.60563			−0.802817	+	0.596226i	$0.796666\pi$
$72$		0			0
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	57.0000	−	98.7269i	0.770270	−	1.33415i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	47.0000	+	81.4064i	0.594937	+	1.03046i	0.993556	+	0.113344i	$0.0361562\pi$
$79$	47.0000	+	81.4064i	0.594937	+	1.03046i	−0.398619	+	0.917117i	$0.630510\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	−87.0000	−	150.688i	−1.01163	−	1.75219i
$87$		0			0
$88$	−9.00000	+	15.5885i	−0.102273	+	0.177142i
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$	−90.0000			−0.978261
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−62.5000	−	108.253i	−0.625000	−	1.08253i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	18.0000			0.169811
$107$	93.0000	+	161.081i	0.869159	+	1.50543i	0.862858	+	0.505447i	$0.168672\pi$
$107$	93.0000	+	161.081i	0.869159	+	1.50543i	0.00630134	+	0.999980i	$0.497994\pi$
$108$		0			0
$109$	−53.0000	+	91.7987i	−0.486239	+	0.842190i	−0.999875	−	0.0158181i	$-0.994965\pi$
$109$	−53.0000	+	91.7987i	−0.486239	+	0.842190i	0.513636	+	0.858008i	$0.328298\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	222.000			1.96460			0.982301	−	0.187310i	$-0.0599768\pi$
$113$	222.000			1.96460			0.982301	+	0.187310i	$0.0599768\pi$
$114$		0			0
$115$		0			0
$116$	−135.000	+	233.827i	−1.16379	+	2.01575i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	42.5000	+	73.6122i	0.351240	+	0.608365i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.00000			0.0157480			0.00787402	−	0.999969i	$-0.497494\pi$
$127$	2.00000			0.0157480			0.00787402	+	0.999969i	$0.497494\pi$
$128$	46.5000	+	80.5404i	0.363281	+	0.629222i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$	−354.000			−2.64179
$135$		0			0
$136$		0			0
$137$	−87.0000	+	150.688i	−0.635036	+	1.09992i	0.351471	+	0.936199i	$0.385682\pi$
$137$	−87.0000	+	150.688i	−0.635036	+	1.09992i	−0.986507	+	0.163717i	$0.947652\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	171.000	+	296.181i	1.20423	+	2.08578i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−190.000			−1.28378
$149$	93.0000	+	161.081i	0.624161	+	1.08108i	0.988702	+	0.149892i	$0.0478924\pi$
$149$	93.0000	+	161.081i	0.624161	+	1.08108i	−0.364541	+	0.931187i	$0.618774\pi$
$150$		0			0
$151$	−137.000	+	237.291i	−0.907285	+	1.57146i	−0.0894642	+	0.995990i	$0.528515\pi$
$151$	−137.000	+	237.291i	−0.907285	+	1.57146i	−0.817821	+	0.575473i	$0.804818\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	141.000	−	244.219i	0.892405	−	1.54569i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−37.0000	−	64.0859i	−0.226994	−	0.393165i	0.729922	−	0.683531i	$-0.239556\pi$
$163$	−37.0000	−	64.0859i	−0.226994	−	0.393165i	−0.956916	+	0.290366i	$0.906223\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	169.000			1.00000
$170$		0			0
$171$		0			0
$172$	−145.000	+	251.147i	−0.843023	+	1.46016i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$	−66.0000			−0.375000
$177$		0			0
$178$		0			0
$179$	−171.000	+	296.181i	−0.955307	+	1.65464i	−0.221644	+	0.975128i	$0.571142\pi$
$179$	−171.000	+	296.181i	−0.955307	+	1.65464i	−0.733663	+	0.679513i	$0.762191\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	27.0000	+	46.7654i	0.146739	+	0.254160i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−159.000	−	275.396i	−0.832461	−	1.44186i	−0.896081	−	0.443890i	$-0.853598\pi$
$191$	−159.000	−	275.396i	−0.832461	−	1.44186i	0.0636205	−	0.997974i	$-0.479735\pi$
$192$		0			0
$193$	31.0000	−	53.6936i	0.160622	−	0.278205i	−0.774470	−	0.632611i	$-0.781983\pi$
$193$	31.0000	−	53.6936i	0.160622	−	0.278205i	0.935092	+	0.354405i	$0.115317\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−282.000			−1.43147			−0.715736	−	0.698371i	$-0.753909\pi$
$197$	−282.000			−1.43147			−0.715736	+	0.698371i	$0.753909\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−37.5000	+	64.9519i	−0.187500	+	0.324760i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−278.000			−1.31754			−0.658768	−	0.752346i	$-0.728922\pi$
$211$	−278.000			−1.31754			−0.658768	+	0.752346i	$0.728922\pi$
$212$	−15.0000	−	25.9808i	−0.0707547	−	0.122551i
$213$		0			0
$214$	279.000	−	483.242i	1.30374	−	2.25814i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	318.000			1.45872
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−333.000	−	576.773i	−1.47345	−	2.55209i
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	162.000			0.698276
$233$	9.00000	+	15.5885i	0.0386266	+	0.0669033i	0.884692	−	0.466175i	$-0.154368\pi$
$233$	9.00000	+	15.5885i	0.0386266	+	0.0669033i	−0.846066	+	0.533078i	$0.821035\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	222.000			0.928870			0.464435	−	0.885607i	$-0.346257\pi$
$239$	222.000			0.928870			0.464435	+	0.885607i	$0.346257\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	127.500	−	220.836i	0.526860	−	0.912547i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−108.000			−0.426877
$254$	−3.00000	−	5.19615i	−0.0118110	−	0.0204573i
$255$		0			0
$256$	−42.5000	+	73.6122i	−0.166016	+	0.287547i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	249.000	−	431.281i	0.946768	−	1.63985i	0.194596	−	0.980883i	$-0.437660\pi$
$263$	249.000	−	431.281i	0.946768	−	1.63985i	0.752172	−	0.658967i	$-0.229006\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	295.000	+	510.955i	1.10075	+	1.90655i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	522.000			1.90511
$275$	−75.0000	−	129.904i	−0.272727	−	0.472377i
$276$		0			0
$277$	227.000	−	393.176i	0.819495	−	1.41941i	−0.0865605	−	0.996247i	$-0.527588\pi$
$277$	227.000	−	393.176i	0.819495	−	1.41941i	0.906055	−	0.423160i	$-0.139079\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−114.000			−0.405694			−0.202847	−	0.979210i	$-0.565019\pi$
$281$	−114.000			−0.405694			−0.202847	+	0.979210i	$0.565019\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	285.000	−	493.634i	1.00352	−	1.73815i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−144.500	−	250.281i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	57.0000	+	98.7269i	0.192568	+	0.333537i
$297$		0			0
$298$	279.000	−	483.242i	0.936242	−	1.62162i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	822.000			2.72185
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	−470.000			−1.48734
$317$	261.000	+	452.065i	0.823344	+	1.42607i	0.903178	+	0.429265i	$0.141227\pi$
$317$	261.000	+	452.065i	0.823344	+	1.42607i	−0.0798346	+	0.996808i	$0.525439\pi$
$318$		0			0
$319$	−162.000	+	280.592i	−0.507837	+	0.879599i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−111.000	+	192.258i	−0.340491	+	0.589747i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−317.000	−	549.060i	−0.957704	−	1.65879i	−0.728055	−	0.685518i	$-0.759576\pi$
$331$	−317.000	−	549.060i	−0.957704	−	1.65879i	−0.229648	−	0.973274i	$-0.573758\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	226.000			0.670623			0.335312	−	0.942107i	$-0.391158\pi$
$337$	226.000			0.670623			0.335312	+	0.942107i	$0.391158\pi$
$338$	−253.500	−	439.075i	−0.750000	−	1.29904i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	174.000			0.505814
$345$		0			0
$346$		0			0
$347$	−339.000	+	587.165i	−0.976945	+	1.69212i	−0.303585	+	0.952804i	$0.598184\pi$
$347$	−339.000	+	587.165i	−0.976945	+	1.69212i	−0.673360	+	0.739314i	$0.735150\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	135.000	+	233.827i	0.383523	+	0.664281i
$353$		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
$358$	1026.00			2.86592
$359$	−327.000	−	566.381i	−0.910864	−	1.57766i	−0.812847	−	0.582477i	$-0.802084\pi$
$359$	−327.000	−	566.381i	−0.910864	−	1.57766i	−0.0980164	−	0.995185i	$-0.531250\pi$
$360$		0			0
$361$	−180.500	+	312.635i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−99.0000	+	171.473i	−0.269022	+	0.465959i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	131.000	+	226.899i	0.351206	+	0.608307i	0.986461	−	0.163995i	$-0.0524382\pi$
$373$	131.000	+	226.899i	0.351206	+	0.608307i	−0.635255	+	0.772303i	$0.719105\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−614.000			−1.62005			−0.810026	−	0.586393i	$-0.800547\pi$
$379$	−614.000			−1.62005			−0.810026	+	0.586393i	$0.800547\pi$
$380$		0			0
$381$		0			0
$382$	−477.000	+	826.188i	−1.24869	+	2.16280i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$	−186.000			−0.481865
$387$		0			0
$388$		0			0
$389$	333.000	−	576.773i	0.856041	−	1.48271i	−0.0196346	−	0.999807i	$-0.506250\pi$
$389$	333.000	−	576.773i	0.856041	−	1.48271i	0.875676	−	0.482900i	$-0.160416\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	423.000	+	732.657i	1.07360	+	1.85954i
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	−275.000			−0.687500
$401$	177.000	+	306.573i	0.441397	+	0.764521i	0.997793	−	0.0663955i	$-0.0211499\pi$
$401$	177.000	+	306.573i	0.441397	+	0.764521i	−0.556397	+	0.830917i	$0.687817\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−228.000			−0.560197
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−166.000			−0.394299			−0.197150	−	0.980373i	$-0.563168\pi$
$421$	−166.000			−0.394299			−0.197150	+	0.980373i	$0.563168\pi$
$422$	417.000	+	722.265i	0.988152	+	1.71153i
$423$		0			0
$424$	−9.00000	+	15.5885i	−0.0212264	+	0.0367652i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−930.000			−2.17290
$429$		0			0
$430$		0			0
$431$	81.0000	−	140.296i	0.187935	−	0.325513i	−0.756627	−	0.653847i	$-0.773154\pi$
$431$	81.0000	−	140.296i	0.187935	−	0.325513i	0.944562	+	0.328334i	$0.106487\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−265.000	−	458.993i	−0.607798	−	1.05274i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−243.000	−	420.888i	−0.548533	−	0.950087i	−0.998375	−	0.0569787i	$-0.981853\pi$
$443$	−243.000	−	420.888i	−0.548533	−	0.950087i	0.449843	−	0.893108i	$-0.351480\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	894.000			1.99109			0.995546	−	0.0942807i	$-0.0300551\pi$
$449$	894.000			1.99109			0.995546	+	0.0942807i	$0.0300551\pi$
$450$		0			0
$451$		0			0
$452$	−555.000	+	961.288i	−1.22788	+	2.12674i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	439.000	+	760.370i	0.960613	+	1.66383i	0.720967	+	0.692970i	$0.243698\pi$
$457$	439.000	+	760.370i	0.960613	+	1.66383i	0.239646	+	0.970860i	$0.422969\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	674.000			1.45572			0.727862	−	0.685724i	$-0.240514\pi$
$463$	674.000			1.45572			0.727862	+	0.685724i	$0.240514\pi$
$464$	297.000	+	514.419i	0.640086	+	1.10866i
$465$		0			0
$466$	27.0000	−	46.7654i	0.0579399	−	0.100355i
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−174.000	+	301.377i	−0.367865	+	0.637160i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−333.000	−	576.773i	−0.696653	−	1.20664i
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−425.000			−0.878099
$485$		0			0
$486$		0			0
$487$	199.000	−	344.678i	0.408624	−	0.707758i	−0.586112	−	0.810230i	$-0.699342\pi$
$487$	199.000	−	344.678i	0.408624	−	0.707758i	0.994736	+	0.102472i	$0.0326753\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−954.000			−1.94297			−0.971487	−	0.237094i	$-0.923805\pi$
$491$	−954.000			−1.94297			−0.971487	+	0.237094i	$0.923805\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−149.000	−	258.076i	−0.298597	−	0.517186i	0.677218	−	0.735782i	$-0.263185\pi$
$499$	−149.000	−	258.076i	−0.298597	−	0.517186i	−0.975815	+	0.218597i	$0.929852\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$	162.000	+	280.592i	0.320158	+	0.554530i
$507$		0			0
$508$	−5.00000	+	8.66025i	−0.00984252	+	0.0170477i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	627.000			1.22461
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		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
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	−1494.00			−2.84030
$527$		0			0
$528$		0			0
$529$	102.500	−	177.535i	0.193762	−	0.335605i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	177.000	−	306.573i	0.330224	−	0.571965i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−37.0000	−	64.0859i	−0.0683919	−	0.118458i	0.829802	−	0.558058i	$-0.188453\pi$
$541$	−37.0000	−	64.0859i	−0.0683919	−	0.118458i	−0.898194	+	0.439600i	$0.855120\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	842.000			1.53931			0.769653	−	0.638463i	$-0.220429\pi$
$547$	842.000			1.53931			0.769653	+	0.638463i	$0.220429\pi$
$548$	−435.000	−	753.442i	−0.793796	−	1.37489i
$549$		0			0
$550$	−225.000	+	389.711i	−0.409091	+	0.708566i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1362.00			−2.45848
$555$		0			0
$556$		0			0
$557$	501.000	−	867.757i	0.899461	−	1.55791i	0.0712774	−	0.997457i	$-0.477292\pi$
$557$	501.000	−	867.757i	0.899461	−	1.55791i	0.828184	−	0.560456i	$-0.189374\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	171.000	+	296.181i	0.304270	+	0.527012i
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−342.000			−0.602113
$569$	−327.000	−	566.381i	−0.574692	−	0.995397i	−0.996075	−	0.0885135i	$-0.971788\pi$
$569$	−327.000	−	566.381i	−0.574692	−	0.995397i	0.421383	−	0.906883i	$-0.361545\pi$
$570$		0			0
$571$	563.000	−	975.145i	0.985989	−	1.70778i	0.348535	−	0.937296i	$-0.386679\pi$
$571$	563.000	−	975.145i	0.985989	−	1.70778i	0.637454	−	0.770488i	$-0.279988\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−450.000			−0.782609
$576$		0			0
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−433.500	+	750.844i	−0.750000	+	1.29904i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−18.0000	−	31.1769i	−0.0308748	−	0.0534767i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−209.000	+	361.999i	−0.353041	+	0.611484i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	−930.000			−1.56040
$597$		0			0
$598$		0			0
$599$	−87.0000	+	150.688i	−0.145242	+	0.251567i	−0.929463	−	0.368915i	$-0.879729\pi$
$599$	−87.0000	+	150.688i	−0.145242	+	0.251567i	0.784221	+	0.620481i	$0.213063\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−685.000	−	1186.45i	−1.13411	−	1.96433i
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−109.000	+	188.794i	−0.177814	+	0.307983i	−0.941132	−	0.338041i	$-0.890236\pi$
$613$	−109.000	+	188.794i	−0.177814	+	0.307983i	0.763318	+	0.646024i	$0.223569\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	558.000			0.904376			0.452188	−	0.891923i	$-0.350644\pi$
$617$	558.000			0.904376			0.452188	+	0.891923i	$0.350644\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1006.00			−1.59429			−0.797147	−	0.603785i	$-0.793659\pi$
$631$	−1006.00			−1.59429			−0.797147	+	0.603785i	$0.793659\pi$
$632$	141.000	+	244.219i	0.223101	+	0.386423i
$633$		0			0
$634$	783.000	−	1356.20i	1.23502	−	2.13911i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	972.000			1.52351
$639$		0			0
$640$		0			0
$641$	417.000	−	722.265i	0.650546	−	1.12678i	−0.332445	−	0.943123i	$-0.607873\pi$
$641$	417.000	−	722.265i	0.650546	−	1.12678i	0.982991	−	0.183656i	$-0.0587932\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	370.000			0.567485
$653$	597.000	+	1034.03i	0.914242	+	1.58351i	0.808007	+	0.589172i	$0.200546\pi$
$653$	597.000	+	1034.03i	0.914242	+	1.58351i	0.106235	+	0.994341i	$0.466121\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−618.000			−0.937785			−0.468892	−	0.883255i	$-0.655347\pi$
$659$	−618.000			−0.937785			−0.468892	+	0.883255i	$0.655347\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−951.000	+	1647.18i	−1.43656	+	2.48819i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	486.000	+	841.777i	0.728636	+	1.26203i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−446.000			−0.662704			−0.331352	−	0.943507i	$-0.607505\pi$
$673$	−446.000			−0.662704			−0.331352	+	0.943507i	$0.607505\pi$
$674$	−339.000	−	587.165i	−0.502967	−	0.871165i
$675$		0			0
$676$	−422.500	+	731.791i	−0.625000	+	1.08253i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	669.000	−	1158.74i	0.979502	−	1.69655i	0.315305	−	0.948991i	$-0.397893\pi$
$683$	669.000	−	1158.74i	0.979502	−	1.69655i	0.664198	−	0.747557i	$-0.268773\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	319.000	+	552.524i	0.463663	+	0.803088i
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$	2034.00			2.93084
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1398.00			1.99429			0.997147	−	0.0754851i	$-0.0240505\pi$
$701$	1398.00			1.99429			0.997147	+	0.0754851i	$0.0240505\pi$
$702$		0			0
$703$		0			0
$704$	273.000	−	472.850i	0.387784	−	0.671662i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	691.000	+	1196.85i	0.974612	+	1.68808i	0.681209	+	0.732089i	$0.261454\pi$
$709$	691.000	+	1196.85i	0.974612	+	1.68808i	0.293403	+	0.955989i	$0.405212\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−855.000	−	1480.90i	−1.19413	−	2.06830i
$717$		0			0
$718$	−981.000	+	1699.14i	−1.36630	+	2.36649i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	1083.00			1.50000
$723$		0			0
$724$		0			0
$725$	−675.000	+	1169.13i	−0.931034	+	1.61260i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	810.000			1.10054
$737$	354.000	+	613.146i	0.480326	+	0.831948i
$738$		0			0
$739$	−613.000	+	1061.75i	−0.829499	+	1.43673i	0.0689322	+	0.997621i	$0.478041\pi$
$739$	−613.000	+	1061.75i	−0.829499	+	1.43673i	−0.898432	+	0.439114i	$0.855293\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−114.000			−0.153432			−0.0767160	−	0.997053i	$-0.524443\pi$
$743$	−114.000			−0.153432			−0.0767160	+	0.997053i	$0.524443\pi$
$744$		0			0
$745$		0			0
$746$	393.000	−	680.696i	0.526810	−	0.912461i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−401.000	−	694.552i	−0.533955	−	0.924837i	−0.999213	−	0.0396618i	$-0.987372\pi$
$751$	−401.000	−	694.552i	−0.533955	−	0.924837i	0.465258	−	0.885175i	$-0.345961\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1402.00			1.85205			0.926024	−	0.377465i	$-0.123204\pi$
$757$	1402.00			1.85205			0.926024	+	0.377465i	$0.123204\pi$
$758$	921.000	+	1595.22i	1.21504	+	2.10451i
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$	1590.00			2.08115
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	155.000	+	268.468i	0.200777	+	0.347756i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1998.00			−2.56812
$779$		0			0
$780$		0			0
$781$	342.000	−	592.361i	0.437900	−	0.758465i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	705.000	−	1221.10i	0.894670	−	1.54961i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	562.500	+	974.279i	0.703125	+	1.21785i
$801$		0			0
$802$	531.000	−	919.719i	0.662095	−	1.14678i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−87.0000	+	150.688i	−0.107540	+	0.186265i	−0.914773	−	0.403968i	$-0.867631\pi$
$809$	−87.0000	+	150.688i	−0.107540	+	0.186265i	0.807233	+	0.590233i	$0.200964\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	342.000	+	592.361i	0.420147	+	0.727717i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−579.000	−	1002.86i	−0.705238	−	1.22151i	−0.966606	−	0.256268i	$-0.917507\pi$
$821$	−579.000	−	1002.86i	−0.705238	−	1.22151i	0.261368	−	0.965239i	$-0.415826\pi$
$822$		0			0
$823$	311.000	−	538.668i	0.377886	−	0.654517i	−0.612869	−	0.790185i	$-0.709985\pi$
$823$	311.000	−	538.668i	0.377886	−	0.654517i	0.990754	+	0.135667i	$0.0433179\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−282.000			−0.340992			−0.170496	−	0.985358i	$-0.554537\pi$
$827$	−282.000			−0.340992			−0.170496	+	0.985358i	$0.554537\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	2075.00			2.46730
$842$	249.000	+	431.281i	0.295724	+	0.512210i
$843$		0			0
$844$	695.000	−	1203.78i	0.823460	−	1.42627i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−66.0000			−0.0778302
$849$		0			0
$850$		0			0
$851$	−342.000	+	592.361i	−0.401880	+	0.696077i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	279.000	+	483.242i	0.325935	+	0.564535i
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−486.000			−0.563805
$863$	−831.000	−	1439.33i	−0.962920	−	1.66783i	−0.715102	−	0.699020i	$-0.753620\pi$
$863$	−831.000	−	1439.33i	−0.962920	−	1.66783i	−0.247818	−	0.968807i	$-0.579713\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−564.000			−0.649022
$870$		0			0
$871$		0			0
$872$	−159.000	+	275.396i	−0.182339	+	0.315821i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−373.000	−	646.055i	−0.425314	−	0.736665i	0.571136	−	0.820855i	$-0.306503\pi$
$877$	−373.000	−	646.055i	−0.425314	−	0.736665i	−0.996450	+	0.0841907i	$0.973170\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1622.00			−1.83692			−0.918460	−	0.395514i	$-0.870566\pi$
$883$	−1622.00			−1.83692			−0.918460	+	0.395514i	$0.870566\pi$
$884$		0			0
$885$		0			0
$886$	−729.000	+	1262.67i	−0.822799	+	1.42513i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1341.00	−	2322.68i	−1.49332	−	2.58650i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	666.000			0.736726
$905$		0			0
$906$		0			0
$907$	−893.000	+	1546.72i	−0.984564	+	1.70532i	−0.340709	+	0.940169i	$0.610667\pi$
$907$	−893.000	+	1546.72i	−0.984564	+	1.70532i	−0.643856	+	0.765147i	$0.722666\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1566.00			1.71899			0.859495	−	0.511144i	$-0.170778\pi$
$911$	1566.00			1.71899			0.859495	+	0.511144i	$0.170778\pi$
$912$		0			0
$913$		0			0
$914$	1317.00	−	2281.11i	1.44092	−	2.49574i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−233.000	−	403.568i	−0.253536	−	0.439138i	0.710961	−	0.703232i	$-0.248260\pi$
$919$	−233.000	−	403.568i	−0.253536	−	0.439138i	−0.964497	+	0.264094i	$0.914927\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−950.000			−1.02703
$926$	−1011.00	−	1751.10i	−1.09179	−	1.89104i
$927$		0			0
$928$	1215.00	−	2104.44i	1.30927	−	2.26772i
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$	−90.0000			−0.0965665
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	1044.00			1.10359
$947$	−747.000	−	1293.84i	−0.788807	−	1.36625i	−0.926698	−	0.375806i	$-0.877366\pi$
$947$	−747.000	−	1293.84i	−0.788807	−	1.36625i	0.137892	−	0.990447i	$-0.455967\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1458.00			−1.52991			−0.764953	−	0.644086i	$-0.777238\pi$
$953$	−1458.00			−1.52991			−0.764953	+	0.644086i	$0.777238\pi$
$954$		0			0
$955$		0			0
$956$	−555.000	+	961.288i	−0.580544	+	1.00553i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	−	832.250i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−334.000			−0.345398			−0.172699	−	0.984975i	$-0.555249\pi$
$967$	−334.000			−0.345398			−0.172699	+	0.984975i	$0.555249\pi$
$968$	127.500	+	220.836i	0.131715	+	0.228137i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$	−1194.00			−1.22587
$975$		0			0
$976$		0			0
$977$	81.0000	−	140.296i	0.0829069	−	0.143599i	−0.821591	−	0.570078i	$-0.806913\pi$
$977$	81.0000	−	140.296i	0.0829069	−	0.143599i	0.904497	+	0.426479i	$0.140246\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1431.00	+	2478.56i	1.45723	+	2.52400i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	522.000	+	904.131i	0.527806	+	0.914187i
$990$		0			0
$991$	703.000	−	1217.63i	0.709384	−	1.22869i	−0.255701	−	0.966756i	$-0.582306\pi$
$991$	703.000	−	1217.63i	0.709384	−	1.22869i	0.965086	−	0.261934i	$-0.0843603\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−447.000	+	774.227i	−0.447896	+	0.775778i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.m.a.325.1		2
3.2	odd	2		49.3.d.a.31.1		2
7.2	even	3	inner	441.3.m.a.19.1		2
7.3	odd	6		63.3.d.a.55.1		1
7.4	even	3		63.3.d.a.55.1		1
7.5	odd	6	inner	441.3.m.a.19.1		2
7.6	odd	2	CM	441.3.m.a.325.1		2
12.11	even	2		784.3.s.a.129.1		2
21.2	odd	6		49.3.d.a.19.1		2
21.5	even	6		49.3.d.a.19.1		2
21.11	odd	6		7.3.b.a.6.1	✓	1
21.17	even	6		7.3.b.a.6.1	✓	1
21.20	even	2		49.3.d.a.31.1		2
28.3	even	6		1008.3.f.a.433.1		1
28.11	odd	6		1008.3.f.a.433.1		1
84.11	even	6		112.3.c.a.97.1		1
84.23	even	6		784.3.s.a.705.1		2
84.47	odd	6		784.3.s.a.705.1		2
84.59	odd	6		112.3.c.a.97.1		1
84.83	odd	2		784.3.s.a.129.1		2
105.17	odd	12		175.3.c.a.174.1		2
105.32	even	12		175.3.c.a.174.1		2
105.38	odd	12		175.3.c.a.174.2		2
105.53	even	12		175.3.c.a.174.2		2
105.59	even	6		175.3.d.a.76.1		1
105.74	odd	6		175.3.d.a.76.1		1
168.11	even	6		448.3.c.b.321.1		1
168.53	odd	6		448.3.c.a.321.1		1
168.59	odd	6		448.3.c.b.321.1		1
168.101	even	6		448.3.c.a.321.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.3.b.a.6.1	✓	1	21.11	odd	6
7.3.b.a.6.1	✓	1	21.17	even	6
49.3.d.a.19.1		2	21.2	odd	6
49.3.d.a.19.1		2	21.5	even	6
49.3.d.a.31.1		2	3.2	odd	2
49.3.d.a.31.1		2	21.20	even	2
63.3.d.a.55.1		1	7.3	odd	6
63.3.d.a.55.1		1	7.4	even	3
112.3.c.a.97.1		1	84.11	even	6
112.3.c.a.97.1		1	84.59	odd	6
175.3.c.a.174.1		2	105.17	odd	12
175.3.c.a.174.1		2	105.32	even	12
175.3.c.a.174.2		2	105.38	odd	12
175.3.c.a.174.2		2	105.53	even	12
175.3.d.a.76.1		1	105.59	even	6
175.3.d.a.76.1		1	105.74	odd	6
441.3.m.a.19.1		2	7.2	even	3	inner
441.3.m.a.19.1		2	7.5	odd	6	inner
441.3.m.a.325.1		2	1.1	even	1	trivial
441.3.m.a.325.1		2	7.6	odd	2	CM
448.3.c.a.321.1		1	168.53	odd	6
448.3.c.a.321.1		1	168.101	even	6
448.3.c.b.321.1		1	168.11	even	6
448.3.c.b.321.1		1	168.59	odd	6
784.3.s.a.129.1		2	12.11	even	2
784.3.s.a.129.1		2	84.83	odd	2
784.3.s.a.705.1		2	84.23	even	6
784.3.s.a.705.1		2	84.47	odd	6
1008.3.f.a.433.1		1	28.3	even	6
1008.3.f.a.433.1		1	28.11	odd	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{2} +(-2.50000 + 4.33013i) q^{4} +3.00000 q^{8} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{2} +(-2.50000 + 4.33013i) q^{4} +3.00000 q^{8} +(-3.00000 + 5.19615i) q^{11} +(5.50000 + 9.52628i) q^{16} +18.0000 q^{22} +(9.00000 + 15.5885i) q^{23} +(-12.5000 + 21.6506i) q^{25} +54.0000 q^{29} +(22.5000 - 38.9711i) q^{32} +(19.0000 + 32.9090i) q^{37} +58.0000 q^{43} +(-15.0000 - 25.9808i) q^{44} +(27.0000 - 46.7654i) q^{46} +75.0000 q^{50} +(-3.00000 + 5.19615i) q^{53} +(-81.0000 - 140.296i) q^{58} -91.0000 q^{64} +(59.0000 - 102.191i) q^{67} -114.000 q^{71} +(57.0000 - 98.7269i) q^{74} +(47.0000 + 81.4064i) q^{79} +(-87.0000 - 150.688i) q^{86} +(-9.00000 + 15.5885i) q^{88} -90.0000 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 5 q^{4} + 6 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 - 5 * q^4 + 6 * q^8	\( 2 q - 3 q^{2} - 5 q^{4} + 6 q^{8} - 6 q^{11} + 11 q^{16} + 36 q^{22} + 18 q^{23} - 25 q^{25} + 108 q^{29} + 45 q^{32} + 38 q^{37} + 116 q^{43} - 30 q^{44} + 54 q^{46} + 150 q^{50} - 6 q^{53} - 162 q^{58} - 182 q^{64} + 118 q^{67} - 228 q^{71} + 114 q^{74} + 94 q^{79} - 174 q^{86} - 18 q^{88} - 180 q^{92}+O(q^{100}) \) 2 * q - 3 * q^2 - 5 * q^4 + 6 * q^8 - 6 * q^11 + 11 * q^16 + 36 * q^22 + 18 * q^23 - 25 * q^25 + 108 * q^29 + 45 * q^32 + 38 * q^37 + 116 * q^43 - 30 * q^44 + 54 * q^46 + 150 * q^50 - 6 * q^53 - 162 * q^58 - 182 * q^64 + 118 * q^67 - 228 * q^71 + 114 * q^74 + 94 * q^79 - 174 * q^86 - 18 * q^88 - 180 * q^92

Introduction

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