LMFDB - Embedded newform 49.4.c.d.18.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,Mod(18,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	49.c (of order $3$, 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:	$2.89109359028$
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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			18.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	49.18
Dual form			49.4.c.d.30.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} -45.0000 q^{8} +(13.5000 + 23.3827i) q^{9} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} -45.0000 q^{8} +(13.5000 + 23.3827i) q^{9} +(34.0000 - 58.8897i) q^{11} +(-44.5000 - 77.0763i) q^{16} +(-67.5000 + 116.913i) q^{18} +340.000 q^{22} +(20.0000 + 34.6410i) q^{23} +(62.5000 - 108.253i) q^{25} -166.000 q^{29} +(42.5000 - 73.6122i) q^{32} -459.000 q^{36} +(-225.000 - 389.711i) q^{37} -180.000 q^{43} +(578.000 + 1001.13i) q^{44} +(-100.000 + 173.205i) q^{46} +625.000 q^{50} +(-295.000 + 510.955i) q^{53} +(-415.000 - 718.801i) q^{58} -287.000 q^{64} +(370.000 - 640.859i) q^{67} +688.000 q^{71} +(-607.500 - 1052.22i) q^{72} +(1125.00 - 1948.56i) q^{74} +(692.000 + 1198.58i) q^{79} +(-364.500 + 631.333i) q^{81} +(-450.000 - 779.423i) q^{86} +(-1530.00 + 2650.04i) q^{88} -680.000 q^{92} +1836.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9}+O(q^{10}) $ 2 * q + 5 * q^2 - 17 * q^4 - 90 * q^8 + 27 * q^9	$ 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9} + 68 q^{11} - 89 q^{16} - 135 q^{18} + 680 q^{22} + 40 q^{23} + 125 q^{25} - 332 q^{29} + 85 q^{32} - 918 q^{36} - 450 q^{37} - 360 q^{43} + 1156 q^{44} - 200 q^{46} + 1250 q^{50} - 590 q^{53} - 830 q^{58} - 574 q^{64} + 740 q^{67} + 1376 q^{71} - 1215 q^{72} + 2250 q^{74} + 1384 q^{79} - 729 q^{81} - 900 q^{86} - 3060 q^{88} - 1360 q^{92} + 3672 q^{99}+O(q^{100}) $ 2 * q + 5 * q^2 - 17 * q^4 - 90 * q^8 + 27 * q^9 + 68 * q^11 - 89 * q^16 - 135 * q^18 + 680 * q^22 + 40 * q^23 + 125 * q^25 - 332 * q^29 + 85 * q^32 - 918 * q^36 - 450 * q^37 - 360 * q^43 + 1156 * q^44 - 200 * q^46 + 1250 * q^50 - 590 * q^53 - 830 * q^58 - 574 * q^64 + 740 * q^67 + 1376 * q^71 - 1215 * q^72 + 2250 * q^74 + 1384 * q^79 - 729 * q^81 - 900 * q^86 - 3060 * q^88 - 1360 * q^92 + 3672 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	2.50000	+	4.33013i	0.883883	+	1.53093i	0.846988	+	0.531612i	$0.178414\pi$
$2$	2.50000	+	4.33013i	0.883883	+	1.53093i	0.0368954	+	0.999319i	$0.488253\pi$
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$	−8.50000	+	14.7224i	−1.06250	+	1.84030i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−45.0000			−1.98874
$9$	13.5000	+	23.3827i	0.500000	+	0.866025i
$10$		0			0
$11$	34.0000	−	58.8897i	0.931944	−	1.61417i	0.151950	−	0.988388i	$-0.451445\pi$
$11$	34.0000	−	58.8897i	0.931944	−	1.61417i	0.779994	−	0.625786i	$-0.215222\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−44.5000	−	77.0763i	−0.695312	−	1.20432i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	−67.5000	+	116.913i	−0.883883	+	1.53093i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$	340.000			3.29492
$23$	20.0000	+	34.6410i	0.181317	+	0.314050i	0.942329	−	0.334687i	$-0.108631\pi$
$23$	20.0000	+	34.6410i	0.181317	+	0.314050i	−0.761012	+	0.648737i	$0.775297\pi$
$24$		0			0
$25$	62.5000	−	108.253i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−166.000			−1.06295			−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000			−1.06295			−0.531473	+	0.847075i	$0.678361\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	42.5000	−	73.6122i	0.234782	−	0.406654i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−459.000			−2.12500
$37$	−225.000	−	389.711i	−0.999724	−	1.73157i	−0.520223	−	0.854030i	$-0.674151\pi$
$37$	−225.000	−	389.711i	−0.999724	−	1.73157i	−0.479500	−	0.877542i	$-0.659182\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−180.000			−0.638366			−0.319183	−	0.947693i	$-0.603408\pi$
$43$	−180.000			−0.638366			−0.319183	+	0.947693i	$0.603408\pi$
$44$	578.000	+	1001.13i	1.98038	+	3.43012i
$45$		0			0
$46$	−100.000	+	173.205i	−0.320526	+	0.555167i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$	625.000			1.76777
$51$		0			0
$52$		0			0
$53$	−295.000	+	510.955i	−0.764554	+	1.32425i	0.175928	+	0.984403i	$0.443707\pi$
$53$	−295.000	+	510.955i	−0.764554	+	1.32425i	−0.940482	+	0.339843i	$0.889626\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−415.000	−	718.801i	−0.939520	−	1.62730i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$	−287.000			−0.560547
$65$		0			0
$66$		0			0
$67$	370.000	−	640.859i	0.674667	−	1.16856i	−0.301899	−	0.953340i	$-0.597621\pi$
$67$	370.000	−	640.859i	0.674667	−	1.16856i	0.976566	−	0.215218i	$-0.0690461\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	688.000			1.15001			0.575004	−	0.818151i	$-0.305000\pi$
$71$	688.000			1.15001			0.575004	+	0.818151i	$0.305000\pi$
$72$	−607.500	−	1052.22i	−0.994369	−	1.72230i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$	1125.00	−	1948.56i	1.76728	−	3.06102i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	692.000	+	1198.58i	0.985520	+	1.70697i	0.639602	+	0.768706i	$0.279099\pi$
$79$	692.000	+	1198.58i	0.985520	+	1.70697i	0.345918	+	0.938265i	$0.387568\pi$
$80$		0			0
$81$	−364.500	+	631.333i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−450.000	−	779.423i	−0.564241	−	0.977295i
$87$		0			0
$88$	−1530.00	+	2650.04i	−1.85339	+	3.21017i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$	−680.000			−0.770597
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	1836.00			1.86389
$100$	1062.50	+	1840.30i	1.06250	+	1.84030i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$	−2950.00			−2.70311
$107$	790.000	+	1368.32i	0.713759	+	1.23627i	0.963436	+	0.267937i	$0.0863420\pi$
$107$	790.000	+	1368.32i	0.713759	+	1.23627i	−0.249678	+	0.968329i	$0.580325\pi$
$108$		0			0
$109$	27.0000	−	46.7654i	0.0237260	−	0.0410946i	−0.853919	−	0.520407i	$-0.825780\pi$
$109$	27.0000	−	46.7654i	0.0237260	−	0.0410946i	0.877645	+	0.479312i	$0.159114\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−670.000			−0.557773			−0.278886	−	0.960324i	$-0.589965\pi$
$113$	−670.000			−0.557773			−0.278886	+	0.960324i	$0.589965\pi$
$114$		0			0
$115$		0			0
$116$	1411.00	−	2443.92i	1.12938	−	1.95614i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1646.50	−	2851.82i	−1.23704	−	2.14262i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2000.00			−1.39741			−0.698706	−	0.715409i	$-0.746240\pi$
$127$	−2000.00			−1.39741			−0.698706	+	0.715409i	$0.746240\pi$
$128$	−1057.50	−	1831.64i	−0.730240	−	1.26481i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$	3700.00			2.38531
$135$		0			0
$136$		0			0
$137$	−1555.00	+	2693.34i	−0.969727	+	1.67962i	−0.273388	+	0.961904i	$0.588144\pi$
$137$	−1555.00	+	2693.34i	−0.969727	+	1.67962i	−0.696339	+	0.717713i	$0.745189\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	1720.00	+	2979.13i	1.01647	+	1.76058i
$143$		0			0
$144$	1201.50	−	2081.06i	0.695312	−	1.20432i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	7650.00			4.24883
$149$	−407.000	−	704.945i	−0.223777	−	0.387593i	0.732175	−	0.681117i	$-0.238505\pi$
$149$	−407.000	−	704.945i	−0.223777	−	0.387593i	−0.955952	+	0.293524i	$0.905172\pi$
$150$		0			0
$151$	1476.00	−	2556.51i	0.795465	−	1.37779i	−0.127079	−	0.991893i	$-0.540560\pi$
$151$	1476.00	−	2556.51i	0.795465	−	1.37779i	0.922544	−	0.385893i	$-0.126107\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$	−3460.00	+	5992.90i	−1.74217	+	3.01753i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−3645.00			−1.76777
$163$	−890.000	−	1541.53i	−0.427670	−	0.740746i	0.568996	−	0.822340i	$-0.307332\pi$
$163$	−890.000	−	1541.53i	−0.427670	−	0.740746i	−0.996666	+	0.0815946i	$0.973999\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−2197.00			−1.00000
$170$		0			0
$171$		0			0
$172$	1530.00	−	2650.04i	0.678264	−	1.17479i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	−6052.00			−2.59197
$177$		0			0
$178$		0			0
$179$	1042.00	−	1804.80i	0.435099	−	0.753614i	−0.562205	−	0.826998i	$-0.690047\pi$
$179$	1042.00	−	1804.80i	0.435099	−	0.753614i	0.997304	+	0.0733844i	$0.0233800\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−900.000	−	1558.85i	−0.360592	−	0.624563i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2036.00	+	3526.46i	0.771308	+	1.33594i	0.936846	+	0.349741i	$0.113730\pi$
$191$	2036.00	+	3526.46i	0.771308	+	1.33594i	−0.165539	+	0.986203i	$0.552936\pi$
$192$		0			0
$193$	2295.00	−	3975.06i	0.855947	−	1.48254i	−0.0198172	−	0.999804i	$-0.506308\pi$
$193$	2295.00	−	3975.06i	0.855947	−	1.48254i	0.875764	−	0.482740i	$-0.160358\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2210.00			−0.799269			−0.399634	−	0.916675i	$-0.630863\pi$
$197$	−2210.00			−0.799269			−0.399634	+	0.916675i	$0.630863\pi$
$198$	4590.00	+	7950.11i	1.64746	+	2.85348i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	−2812.50	+	4871.39i	−0.994369	+	1.72230i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−540.000	+	935.307i	−0.181317	+	0.314050i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	5868.00			1.91455			0.957274	−	0.289181i	$-0.0933830\pi$
$211$	5868.00			1.91455			0.957274	+	0.289181i	$0.0933830\pi$
$212$	−5015.00	−	8686.23i	−1.62468	−	2.81402i
$213$		0			0
$214$	−3950.00	+	6841.60i	−1.26176	+	2.18543i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	270.000			0.0838840
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	3375.00			1.00000
$226$	−1675.00	−	2901.19i	−0.493006	−	0.853911i
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$	7470.00			2.11392
$233$	2365.00	+	4096.30i	0.664963	+	1.15175i	0.979295	+	0.202436i	$0.0648859\pi$
$233$	2365.00	+	4096.30i	0.664963	+	1.15175i	−0.314333	+	0.949313i	$0.601781\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7376.00			−1.99629			−0.998146	−	0.0608655i	$-0.980614\pi$
$239$	−7376.00			−1.99629			−0.998146	+	0.0608655i	$0.980614\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	8232.50	−	14259.1i	2.18680	−	3.78765i
$243$		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			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	2720.00			0.675909
$254$	−5000.00	−	8660.25i	−1.23515	−	2.13934i
$255$		0			0
$256$	4139.50	−	7169.82i	1.01062	−	1.75045i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2241.00	−	3881.53i	−0.531473	−	0.920538i
$262$		0			0
$263$	−3760.00	+	6512.51i	−0.881565	+	1.52691i	−0.0319637	+	0.999489i	$0.510176\pi$
$263$	−3760.00	+	6512.51i	−0.881565	+	1.52691i	−0.849601	+	0.527426i	$0.823157\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	6290.00	+	10894.6i	1.43367	+	2.48319i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$	−15550.0			−3.42850
$275$	−4250.00	−	7361.22i	−0.931944	−	1.61417i
$276$		0			0
$277$	−3655.00	+	6330.65i	−0.792807	+	1.37318i	0.131415	+	0.991327i	$0.458048\pi$
$277$	−3655.00	+	6330.65i	−0.792807	+	1.37318i	−0.924222	+	0.381855i	$0.875285\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4342.00			0.921786			0.460893	−	0.887456i	$-0.347529\pi$
$281$	4342.00			0.921786			0.460893	+	0.887456i	$0.347529\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	−5848.00	+	10129.0i	−1.22188	+	2.11636i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2295.00			0.469563
$289$	2456.50	+	4254.78i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	10125.0	+	17537.0i	1.98819	+	3.44364i
$297$		0			0
$298$	2035.00	−	3524.72i	0.395585	−	0.685174i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	14760.0			2.81239
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$		0			0
$316$	−23528.0			−4.18846
$317$	3485.00	+	6036.20i	0.617467	+	1.06948i	0.989946	+	0.141444i	$0.0451745\pi$
$317$	3485.00	+	6036.20i	0.617467	+	1.06948i	−0.372479	+	0.928041i	$0.621492\pi$
$318$		0			0
$319$	−5644.00	+	9775.69i	−0.990606	+	1.71578i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−6196.50	−	10732.7i	−1.06250	−	1.84030i
$325$		0			0
$326$	4450.00	−	7707.63i	0.756021	−	1.30947i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5454.00	−	9446.61i	−0.905677	−	1.56868i	−0.820006	−	0.572354i	$-0.806030\pi$
$331$	−5454.00	−	9446.61i	−0.905677	−	1.56868i	−0.0856702	−	0.996324i	$-0.527303\pi$
$332$		0			0
$333$	6075.00	−	10522.2i	0.999724	−	1.73157i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3330.00			−0.538269			−0.269135	−	0.963103i	$-0.586738\pi$
$337$	−3330.00			−0.538269			−0.269135	+	0.963103i	$0.586738\pi$
$338$	−5492.50	−	9513.29i	−0.883883	−	1.53093i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	8100.00			1.26954
$345$		0			0
$346$		0			0
$347$	2050.00	−	3550.70i	0.317146	−	0.549314i	−0.662745	−	0.748845i	$-0.730609\pi$
$347$	2050.00	−	3550.70i	0.317146	−	0.549314i	0.979891	+	0.199532i	$0.0639420\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−2890.00	−	5005.63i	−0.437607	−	0.757957i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	10420.0			1.53831
$359$	4052.00	+	7018.27i	0.595700	+	1.03178i	0.993448	+	0.114288i	$0.0364587\pi$
$359$	4052.00	+	7018.27i	0.595700	+	1.03178i	−0.397747	+	0.917495i	$0.630208\pi$
$360$		0			0
$361$	3429.50	−	5940.07i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$	1780.00	−	3083.05i	0.252144	−	0.436726i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6985.00	+	12098.4i	0.969624	+	1.67944i	0.696643	+	0.717418i	$0.254676\pi$
$373$	6985.00	+	12098.4i	0.969624	+	1.67944i	0.272980	+	0.962020i	$0.411991\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11916.0			1.61500			0.807498	−	0.589870i	$-0.200821\pi$
$379$	11916.0			1.61500			0.807498	+	0.589870i	$0.200821\pi$
$380$		0			0
$381$		0			0
$382$	−10180.0	+	17632.3i	−1.36349	+	2.36164i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	22950.0			3.02623
$387$	−2430.00	−	4208.88i	−0.319183	−	0.552841i
$388$		0			0
$389$	5263.00	−	9115.78i	0.685976	−	1.18815i	−0.287153	−	0.957885i	$-0.592709\pi$
$389$	5263.00	−	9115.78i	0.685976	−	1.18815i	0.973129	−	0.230261i	$-0.0739579\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−5525.00	−	9569.58i	−0.706461	−	1.22363i
$395$		0			0
$396$	−15606.0	+	27030.4i	−1.98038	+	3.43012i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−11125.0			−1.39062
$401$	−799.000	−	1383.91i	−0.0995016	−	0.172342i	0.811977	−	0.583690i	$-0.198392\pi$
$401$	−799.000	−	1383.91i	−0.0995016	−	0.172342i	−0.911478	+	0.411348i	$0.865058\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−30600.0			−3.72675
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−5400.00			−0.641052
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	15262.0			1.76680			0.883402	−	0.468616i	$-0.155247\pi$
$421$	15262.0			1.76680			0.883402	+	0.468616i	$0.155247\pi$
$422$	14670.0	+	25409.2i	1.69224	+	2.93104i
$423$		0			0
$424$	13275.0	−	22993.0i	1.52050	−	2.63358i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−26860.0			−3.03347
$429$		0			0
$430$		0			0
$431$	4304.00	−	7454.75i	0.481012	−	0.833138i	−0.518750	−	0.854926i	$-0.673602\pi$
$431$	4304.00	−	7454.75i	0.481012	−	0.833138i	0.999763	+	0.0217878i	$0.00693583\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	459.000	+	795.011i	0.0504177	+	0.0873260i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9290.00	−	16090.8i	−0.996346	−	1.72572i	−0.572140	−	0.820156i	$-0.693887\pi$
$443$	−9290.00	−	16090.8i	−0.996346	−	1.72572i	−0.424205	−	0.905566i	$-0.639447\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2686.00			−0.282317			−0.141158	−	0.989987i	$-0.545083\pi$
$449$	−2686.00			−0.282317			−0.141158	+	0.989987i	$0.545083\pi$
$450$	8437.50	+	14614.2i	0.883883	+	1.53093i
$451$		0			0
$452$	5695.00	−	9864.03i	0.592633	−	1.02647i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4005.00	−	6936.86i	−0.409947	−	0.710050i	0.584936	−	0.811079i	$-0.301120\pi$
$457$	−4005.00	−	6936.86i	−0.409947	−	0.710050i	−0.994883	+	0.101030i	$0.967786\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−8440.00			−0.847171			−0.423585	−	0.905856i	$-0.639229\pi$
$463$	−8440.00			−0.847171			−0.423585	+	0.905856i	$0.639229\pi$
$464$	7387.00	+	12794.7i	0.739079	+	1.28012i
$465$		0			0
$466$	−11825.0	+	20481.5i	−1.17550	+	2.03602i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6120.00	+	10600.2i	−0.594922	+	1.03043i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15930.0			−1.52911
$478$	−18440.0	−	31939.0i	−1.76449	−	3.05619i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	55981.0			5.25742
$485$		0			0
$486$		0			0
$487$	−10620.0	+	18394.4i	−0.988169	+	1.71156i	−0.361261	+	0.932465i	$0.617654\pi$
$487$	−10620.0	+	18394.4i	−0.988169	+	1.71156i	−0.626908	+	0.779094i	$0.715680\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20372.0			1.87246			0.936228	−	0.351394i	$-0.114292\pi$
$491$	20372.0			1.87246			0.936228	+	0.351394i	$0.114292\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3618.00	+	6266.56i	0.324577	+	0.562184i	0.981427	−	0.191838i	$-0.0614447\pi$
$499$	3618.00	+	6266.56i	0.324577	+	0.562184i	−0.656850	+	0.754022i	$0.728111\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	6800.00	+	11777.9i	0.597425	+	1.03477i
$507$		0			0
$508$	17000.0	−	29444.9i	1.48475	−	2.57166i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$	24475.0			2.11260
$513$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$	11205.0	−	19407.6i	0.939520	−	1.62730i
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$	−37600.0			−3.11680
$527$		0			0
$528$		0			0
$529$	5283.50	−	9151.29i	0.434248	−	0.752140i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−16650.0	+	28838.6i	−1.34174	+	2.32395i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7939.00	−	13750.8i	−0.630914	−	1.09277i	−0.987365	−	0.158461i	$-0.949347\pi$
$541$	−7939.00	−	13750.8i	−0.630914	−	1.09277i	0.356452	−	0.934314i	$-0.383986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12980.0			1.01460			0.507299	−	0.861770i	$-0.330644\pi$
$547$	12980.0			1.01460			0.507299	+	0.861770i	$0.330644\pi$
$548$	−26435.0	−	45786.8i	−2.06067	−	3.56919i
$549$		0			0
$550$	21250.0	−	36806.1i	1.64746	−	2.85348i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−36550.0			−2.80300
$555$		0			0
$556$		0			0
$557$	−10235.0	+	17727.5i	−0.778583	+	1.34855i	0.154175	+	0.988044i	$0.450728\pi$
$557$	−10235.0	+	17727.5i	−0.778583	+	1.34855i	−0.932758	+	0.360502i	$0.882605\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	10855.0	+	18801.4i	0.814752	+	1.41119i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−30960.0			−2.28706
$569$	13453.0	+	23301.3i	0.991176	+	1.71677i	0.610382	+	0.792107i	$0.291016\pi$
$569$	13453.0	+	23301.3i	0.991176	+	1.71677i	0.380794	+	0.924660i	$0.375651\pi$
$570$		0			0
$571$	3394.00	−	5878.58i	0.248747	−	0.430842i	−0.714431	−	0.699705i	$-0.753315\pi$
$571$	3394.00	−	5878.58i	0.248747	−	0.430842i	0.963178	+	0.268863i	$0.0866479\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5000.00			0.362634
$576$	−3874.50	−	6710.83i	−0.280273	−	0.485448i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	−12282.5	+	21273.9i	−0.883883	+	1.53093i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	20060.0	+	34744.9i	1.42504	+	2.46825i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−20025.0	+	34684.3i	−1.39024	+	2.40797i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$	13838.0			0.951051
$597$		0			0
$598$		0			0
$599$	12368.0	−	21422.0i	0.843644	−	1.46123i	−0.0431495	−	0.999069i	$-0.513739\pi$
$599$	12368.0	−	21422.0i	0.843644	−	1.46123i	0.886794	−	0.462166i	$-0.152927\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	19980.0			1.34933
$604$	25092.0	+	43460.6i	1.69036	+	2.92779i
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7505.00	+	12999.0i	−0.494493	+	0.856487i	−0.999980	−	0.00634752i	$-0.997980\pi$
$613$	−7505.00	+	12999.0i	−0.494493	+	0.856487i	0.505487	+	0.862834i	$0.331313\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30550.0			1.99335			0.996675	−	0.0814823i	$-0.0259654\pi$
$617$	30550.0			1.99335			0.996675	+	0.0814823i	$0.0259654\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−26192.0			−1.65244			−0.826218	−	0.563351i	$-0.809512\pi$
$631$	−26192.0			−1.65244			−0.826218	+	0.563351i	$0.809512\pi$
$632$	−31140.0	−	53936.1i	−1.95994	−	3.39472i
$633$		0			0
$634$	−17425.0	+	30181.0i	−1.09154	+	1.89060i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−56440.0			−3.50232
$639$	9288.00	+	16087.3i	0.575004	+	0.995936i
$640$		0			0
$641$	−4439.00	+	7688.57i	−0.273526	+	0.473760i	−0.969762	−	0.244052i	$-0.921523\pi$
$641$	−4439.00	+	7688.57i	−0.273526	+	0.473760i	0.696236	+	0.717813i	$0.254857\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	16402.5	−	28410.0i	0.994369	−	1.72230i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	30260.0			1.81760
$653$	−13525.0	−	23426.0i	−0.810527	−	1.40387i	−0.912496	−	0.409086i	$-0.865847\pi$
$653$	−13525.0	−	23426.0i	−0.810527	−	1.40387i	0.101969	−	0.994788i	$-0.467486\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1804.00			−0.106637			−0.0533186	−	0.998578i	$-0.516980\pi$
$659$	−1804.00			−0.106637			−0.0533186	+	0.998578i	$0.516980\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	27270.0	−	47233.0i	1.60103	−	2.77306i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	60750.0			3.53456
$667$	−3320.00	−	5750.41i	−0.192730	−	0.333818i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33570.0			−1.92278			−0.961388	−	0.275196i	$-0.911257\pi$
$673$	−33570.0			−1.92278			−0.961388	+	0.275196i	$0.911257\pi$
$674$	−8325.00	−	14419.3i	−0.475767	−	0.824053i
$675$		0			0
$676$	18674.5	−	32345.2i	1.06250	−	1.84030i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17030.0	−	29496.8i	0.954077	−	1.65251i	0.217612	−	0.976035i	$-0.430173\pi$
$683$	17030.0	−	29496.8i	0.954077	−	1.65251i	0.736466	−	0.676475i	$-0.236493\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	8010.00	+	13873.7i	0.443864	+	0.768795i
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$	20500.0			1.12128
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4198.00			−0.226186			−0.113093	−	0.993584i	$-0.536076\pi$
$701$	−4198.00			−0.226186			−0.113093	+	0.993584i	$0.536076\pi$
$702$		0			0
$703$		0			0
$704$	−9758.00	+	16901.4i	−0.522398	+	0.904821i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6273.00	−	10865.2i	−0.332281	−	0.575528i	0.650677	−	0.759354i	$-0.274485\pi$
$709$	−6273.00	−	10865.2i	−0.332281	−	0.575528i	−0.982959	+	0.183826i	$0.941152\pi$
$710$		0			0
$711$	−18684.0	+	32361.6i	−0.985520	+	1.70697i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	17714.0	+	30681.5i	0.924586	+	1.60143i
$717$		0			0
$718$	−20260.0	+	35091.3i	−1.05306	+	1.82395i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$	34295.0			1.76777
$723$		0			0
$724$		0			0
$725$	−10375.0	+	17970.0i	−0.531473	+	0.920538i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−19683.0			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$	3400.00			0.170279
$737$	−25160.0	−	43578.4i	−1.25750	−	2.17806i
$738$		0			0
$739$	12662.0	−	21931.2i	0.630283	−	1.09168i	−0.357211	−	0.934024i	$-0.616272\pi$
$739$	12662.0	−	21931.2i	0.630283	−	1.09168i	0.987494	−	0.157658i	$-0.0503945\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25160.0			1.24230			0.621151	−	0.783691i	$-0.286665\pi$
$743$	25160.0			1.24230			0.621151	+	0.783691i	$0.286665\pi$
$744$		0			0
$745$		0			0
$746$	−34925.0	+	60491.9i	−1.71407	+	2.96885i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1224.00	+	2120.03i	0.0594732	+	0.103011i	0.894229	−	0.447610i	$-0.147725\pi$
$751$	1224.00	+	2120.03i	0.0594732	+	0.103011i	−0.834756	+	0.550620i	$0.814391\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34830.0			−1.67228			−0.836141	−	0.548514i	$-0.815194\pi$
$757$	−34830.0			−1.67228			−0.836141	+	0.548514i	$0.815194\pi$
$758$	29790.0	+	51597.8i	1.42747	+	2.47245i
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$	−69224.0			−3.27806
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	39015.0	+	67576.0i	1.81889	+	3.15040i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	12150.0	−	21044.4i	0.564241	−	0.977295i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	52630.0			2.42529
$779$		0			0
$780$		0			0
$781$	23392.0	−	40516.1i	1.07174	−	1.85631i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	18785.0	−	32536.6i	0.849223	−	1.47090i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−82620.0			−3.70679
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−5312.50	−	9201.52i	−0.234782	−	0.406654i
$801$		0			0
$802$	3995.00	−	6919.54i	0.175896	−	0.304660i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18677.0	+	32349.5i	−0.811679	+	1.40587i	0.100009	+	0.994987i	$0.468113\pi$
$809$	−18677.0	+	32349.5i	−0.811679	+	1.40587i	−0.911688	+	0.410883i	$0.865221\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−76500.0	−	132502.i	−3.29401	−	5.70539i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21769.0	+	37705.0i	0.925388	+	1.60282i	0.790936	+	0.611898i	$0.209594\pi$
$821$	21769.0	+	37705.0i	0.925388	+	1.60282i	0.134451	+	0.990920i	$0.457073\pi$
$822$		0			0
$823$	23120.0	−	40045.0i	0.979238	−	1.69609i	0.314062	−	0.949403i	$-0.398310\pi$
$823$	23120.0	−	40045.0i	0.979238	−	1.69609i	0.665176	−	0.746687i	$-0.268357\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−23980.0			−1.00830			−0.504151	−	0.863615i	$-0.668195\pi$
$827$	−23980.0			−1.00830			−0.504151	+	0.863615i	$0.668195\pi$
$828$	−9180.00	−	15900.2i	−0.385298	−	0.667356i
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		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			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	3167.00			0.129854
$842$	38155.0	+	66086.4i	1.56165	+	2.70485i
$843$		0			0
$844$	−49878.0	+	86391.2i	−2.03421	+	3.52335i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	52510.0			2.12642
$849$		0			0
$850$		0			0
$851$	9000.00	−	15588.5i	0.362534	−	0.627926i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−35550.0	−	61574.4i	−1.41948	−	2.45861i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	43040.0			1.70064
$863$	10100.0	+	17493.7i	0.398387	+	0.690027i	0.993527	−	0.113595i	$-0.0362368\pi$
$863$	10100.0	+	17493.7i	0.398387	+	0.690027i	−0.595140	+	0.803622i	$0.702903\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	94112.0			3.67380
$870$		0			0
$871$		0			0
$872$	−1215.00	+	2104.44i	−0.0471847	+	0.0817264i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3275.00	+	5672.47i	0.126099	+	0.218410i	0.922162	−	0.386804i	$-0.126421\pi$
$877$	3275.00	+	5672.47i	0.126099	+	0.218410i	−0.796063	+	0.605214i	$0.793088\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	30060.0			1.14564			0.572820	−	0.819681i	$-0.305850\pi$
$883$	30060.0			1.14564			0.572820	+	0.819681i	$0.305850\pi$
$884$		0			0
$885$		0			0
$886$	46450.0	−	80453.8i	1.76131	−	3.05067i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	24786.0	+	42930.6i	0.931944	+	1.61417i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−6715.00	−	11630.7i	−0.249535	−	0.432207i
$899$		0			0
$900$	−28687.5	+	49688.2i	−1.06250	+	1.84030i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	30150.0			1.10926
$905$		0			0
$906$		0			0
$907$	−26370.0	+	45674.2i	−0.965382	+	1.67209i	−0.256797	+	0.966465i	$0.582667\pi$
$907$	−26370.0	+	45674.2i	−0.965382	+	1.67209i	−0.708585	+	0.705625i	$0.750666\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−39632.0			−1.44135			−0.720673	−	0.693275i	$-0.756167\pi$
$911$	−39632.0			−1.44135			−0.720673	+	0.693275i	$0.756167\pi$
$912$		0			0
$913$		0			0
$914$	20025.0	−	34684.3i	0.724692	−	1.25520i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−10872.0	−	18830.9i	−0.390244	−	0.675922i	0.602238	−	0.798317i	$-0.294276\pi$
$919$	−10872.0	−	18830.9i	−0.390244	−	0.675922i	−0.992482	+	0.122395i	$0.960943\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−56250.0			−1.99945
$926$	−21100.0	−	36546.3i	−0.748800	−	1.29696i
$927$		0			0
$928$	−7055.00	+	12219.6i	−0.249560	+	0.432251i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$	−80410.0			−2.82609
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−61200.0			−2.10337
$947$	−24410.0	−	42279.4i	−0.837612	−	1.45079i	−0.891886	−	0.452260i	$-0.850618\pi$
$947$	−24410.0	−	42279.4i	−0.837612	−	1.45079i	0.0542742	−	0.998526i	$-0.482715\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29290.0			0.995589			0.497794	−	0.867295i	$-0.334143\pi$
$953$	29290.0			0.995589			0.497794	+	0.867295i	$0.334143\pi$
$954$	−39825.0	−	68978.9i	−1.35155	−	2.34096i
$955$		0			0
$956$	62696.0	−	108593.i	2.12106	−	3.67378i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	14895.5	+	25799.8i	0.500000	+	0.866025i
$962$		0			0
$963$	−21330.0	+	36944.6i	−0.713759	+	1.23627i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	52040.0			1.73060			0.865302	−	0.501251i	$-0.167127\pi$
$967$	52040.0			1.73060			0.865302	+	0.501251i	$0.167127\pi$
$968$	74092.5	+	128332.i	2.46015	+	4.26110i
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$	−106200.			−3.49370
$975$		0			0
$976$		0			0
$977$	18745.0	−	32467.3i	0.613824	−	1.06317i	−0.376766	−	0.926308i	$-0.622964\pi$
$977$	18745.0	−	32467.3i	0.613824	−	1.06317i	0.990590	−	0.136865i	$-0.0437027\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1458.00			0.0474519
$982$	50930.0	+	88213.3i	1.65503	+	2.86660i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3600.00	−	6235.38i	−0.115747	−	0.200479i
$990$		0			0
$991$	−28764.0	+	49820.7i	−0.922017	+	1.59698i	−0.125727	+	0.992065i	$0.540126\pi$
$991$	−28764.0	+	49820.7i	−0.922017	+	1.59698i	−0.796290	+	0.604915i	$0.793207\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$	−18090.0	+	31332.8i	−0.573776	+	0.993810i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.c.d.18.1		2
3.2	odd	2		441.4.e.a.361.1		2
7.2	even	3	inner	49.4.c.d.30.1		2
7.3	odd	6		49.4.a.a.1.1	✓	1
7.4	even	3		49.4.a.a.1.1	✓	1
7.5	odd	6	inner	49.4.c.d.30.1		2
7.6	odd	2	CM	49.4.c.d.18.1		2
21.2	odd	6		441.4.e.a.226.1		2
21.5	even	6		441.4.e.a.226.1		2
21.11	odd	6		441.4.a.m.1.1		1
21.17	even	6		441.4.a.m.1.1		1
21.20	even	2		441.4.e.a.361.1		2
28.3	even	6		784.4.a.k.1.1		1
28.11	odd	6		784.4.a.k.1.1		1
35.4	even	6		1225.4.a.l.1.1		1
35.24	odd	6		1225.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.a.1.1	✓	1	7.3	odd	6
49.4.a.a.1.1	✓	1	7.4	even	3
49.4.c.d.18.1		2	1.1	even	1	trivial
49.4.c.d.18.1		2	7.6	odd	2	CM
49.4.c.d.30.1		2	7.2	even	3	inner
49.4.c.d.30.1		2	7.5	odd	6	inner
441.4.a.m.1.1		1	21.11	odd	6
441.4.a.m.1.1		1	21.17	even	6
441.4.e.a.226.1		2	21.2	odd	6
441.4.e.a.226.1		2	21.5	even	6
441.4.e.a.361.1		2	3.2	odd	2
441.4.e.a.361.1		2	21.20	even	2
784.4.a.k.1.1		1	28.3	even	6
784.4.a.k.1.1		1	28.11	odd	6
1225.4.a.l.1.1		1	35.4	even	6
1225.4.a.l.1.1		1	35.24	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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} -45.0000 q^{8} +(13.5000 + 23.3827i) q^{9} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} -45.0000 q^{8} +(13.5000 + 23.3827i) q^{9} +(34.0000 - 58.8897i) q^{11} +(-44.5000 - 77.0763i) q^{16} +(-67.5000 + 116.913i) q^{18} +340.000 q^{22} +(20.0000 + 34.6410i) q^{23} +(62.5000 - 108.253i) q^{25} -166.000 q^{29} +(42.5000 - 73.6122i) q^{32} -459.000 q^{36} +(-225.000 - 389.711i) q^{37} -180.000 q^{43} +(578.000 + 1001.13i) q^{44} +(-100.000 + 173.205i) q^{46} +625.000 q^{50} +(-295.000 + 510.955i) q^{53} +(-415.000 - 718.801i) q^{58} -287.000 q^{64} +(370.000 - 640.859i) q^{67} +688.000 q^{71} +(-607.500 - 1052.22i) q^{72} +(1125.00 - 1948.56i) q^{74} +(692.000 + 1198.58i) q^{79} +(-364.500 + 631.333i) q^{81} +(-450.000 - 779.423i) q^{86} +(-1530.00 + 2650.04i) q^{88} -680.000 q^{92} +1836.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9}+O(q^{10}) \) 2 * q + 5 * q^2 - 17 * q^4 - 90 * q^8 + 27 * q^9	\( 2 q + 5 q^{2} - 17 q^{4} - 90 q^{8} + 27 q^{9} + 68 q^{11} - 89 q^{16} - 135 q^{18} + 680 q^{22} + 40 q^{23} + 125 q^{25} - 332 q^{29} + 85 q^{32} - 918 q^{36} - 450 q^{37} - 360 q^{43} + 1156 q^{44} - 200 q^{46} + 1250 q^{50} - 590 q^{53} - 830 q^{58} - 574 q^{64} + 740 q^{67} + 1376 q^{71} - 1215 q^{72} + 2250 q^{74} + 1384 q^{79} - 729 q^{81} - 900 q^{86} - 3060 q^{88} - 1360 q^{92} + 3672 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 - 17 * q^4 - 90 * q^8 + 27 * q^9 + 68 * q^11 - 89 * q^16 - 135 * q^18 + 680 * q^22 + 40 * q^23 + 125 * q^25 - 332 * q^29 + 85 * q^32 - 918 * q^36 - 450 * q^37 - 360 * q^43 + 1156 * q^44 - 200 * q^46 + 1250 * q^50 - 590 * q^53 - 830 * q^58 - 574 * q^64 + 740 * q^67 + 1376 * q^71 - 1215 * q^72 + 2250 * q^74 + 1384 * q^79 - 729 * q^81 - 900 * q^86 - 3060 * q^88 - 1360 * q^92 + 3672 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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