LMFDB - Embedded newform 49.6.c.a.18.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,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, 6, names="a")

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$7.85880717084$
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			49.6.c.a.30.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +O(q^{10})$	$q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +(38.0000 - 65.8179i) q^{11} +(-2024.50 - 3506.54i) q^{16} +(1336.50 - 2314.89i) q^{18} -836.000 q^{22} +(2476.00 + 4288.56i) q^{23} +(1562.50 - 2706.33i) q^{25} +7282.00 q^{29} +(-12237.5 + 21196.0i) q^{32} -21627.0 q^{36} +(4443.00 + 7695.50i) q^{37} +11748.0 q^{43} +(3382.00 + 5857.80i) q^{44} +(27236.0 - 47174.1i) q^{46} -34375.0 q^{50} +(-12275.0 + 21260.9i) q^{53} +(-40051.0 - 69370.4i) q^{58} +139657. q^{64} +(-34682.0 + 60071.0i) q^{67} -2224.00 q^{71} +(76180.5 + 131948. i) q^{72} +(48873.0 - 84650.5i) q^{74} +(-40084.0 - 69427.5i) q^{79} +(-29524.5 + 51137.9i) q^{81} +(-64614.0 - 111915. i) q^{86} +(23826.0 - 41267.8i) q^{88} -440728. q^{92} +18468.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9}+O(q^{10}) $ 2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9	$ 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9} + 76 q^{11} - 4049 q^{16} + 2673 q^{18} - 1672 q^{22} + 4952 q^{23} + 3125 q^{25} + 14564 q^{29} - 24475 q^{32} - 43254 q^{36} + 8886 q^{37} + 23496 q^{43} + 6764 q^{44} + 54472 q^{46} - 68750 q^{50} - 24550 q^{53} - 80102 q^{58} + 279314 q^{64} - 69364 q^{67} - 4448 q^{71} + 152361 q^{72} + 97746 q^{74} - 80168 q^{79} - 59049 q^{81} - 129228 q^{86} + 47652 q^{88} - 881456 q^{92} + 36936 q^{99}+O(q^{100}) $ 2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9 + 76 * q^11 - 4049 * q^16 + 2673 * q^18 - 1672 * q^22 + 4952 * q^23 + 3125 * q^25 + 14564 * q^29 - 24475 * q^32 - 43254 * q^36 + 8886 * q^37 + 23496 * q^43 + 6764 * q^44 + 54472 * q^46 - 68750 * q^50 - 24550 * q^53 - 80102 * q^58 + 279314 * q^64 - 69364 * q^67 - 4448 * q^71 + 152361 * q^72 + 97746 * q^74 - 80168 * q^79 - 59049 * q^81 - 129228 * q^86 + 47652 * q^88 - 881456 * q^92 + 36936 * 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 $

$2$	−5.50000	−	9.52628i	−0.972272	−	1.68402i	−0.688659	−	0.725085i	$-0.741800\pi$
$2$	−5.50000	−	9.52628i	−0.972272	−	1.68402i	−0.283613	−	0.958939i	$-0.591533\pi$
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$	−44.5000	+	77.0763i	−1.39062	+	2.40863i
$5$		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
$8$	627.000			3.46372
$9$	121.500	+	210.444i	0.500000	+	0.866025i
$10$		0			0
$11$	38.0000	−	65.8179i	0.0946895	−	0.164007i	−0.814789	−	0.579757i	$-0.803148\pi$
$11$	38.0000	−	65.8179i	0.0946895	−	0.164007i	0.909479	+	0.415750i	$0.136481\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−2024.50	−	3506.54i	−1.97705	−	3.42435i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	1336.50	−	2314.89i	0.972272	−	1.68402i
$19$		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
$22$	−836.000			−0.368256
$23$	2476.00	+	4288.56i	0.975958	+	1.69041i	0.676737	+	0.736225i	$0.263393\pi$
$23$	2476.00	+	4288.56i	0.975958	+	1.69041i	0.299220	+	0.954184i	$0.403273\pi$
$24$		0			0
$25$	1562.50	−	2706.33i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7282.00			1.60789			0.803944	−	0.594705i	$-0.202731\pi$
$29$	7282.00			1.60789			0.803944	+	0.594705i	$0.202731\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	−12237.5	+	21196.0i	−2.11260	+	3.65913i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−21627.0			−2.78125
$37$	4443.00	+	7695.50i	0.533546	+	0.924129i	0.999232	+	0.0391791i	$0.0124743\pi$
$37$	4443.00	+	7695.50i	0.533546	+	0.924129i	−0.465686	+	0.884950i	$0.654192\pi$
$38$		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
$43$	11748.0			0.968931			0.484465	−	0.874810i	$-0.339014\pi$
$43$	11748.0			0.968931			0.484465	+	0.874810i	$0.339014\pi$
$44$	3382.00	+	5857.80i	0.263355	+	0.456145i
$45$		0			0
$46$	27236.0	−	47174.1i	1.89779	−	3.28707i
$47$		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
$50$	−34375.0			−1.94454
$51$		0			0
$52$		0			0
$53$	−12275.0	+	21260.9i	−0.600250	+	1.03966i	0.392533	+	0.919738i	$0.371599\pi$
$53$	−12275.0	+	21260.9i	−0.600250	+	1.03966i	−0.992783	+	0.119925i	$0.961735\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−40051.0	−	69370.4i	−1.56330	−	2.70772i
$59$		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
$64$	139657.			4.26199
$65$		0			0
$66$		0			0
$67$	−34682.0	+	60071.0i	−0.943881	+	1.63485i	−0.185904	+	0.982568i	$0.559521\pi$
$67$	−34682.0	+	60071.0i	−0.943881	+	1.63485i	−0.757977	+	0.652281i	$0.773812\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2224.00			−0.0523587			−0.0261794	−	0.999657i	$-0.508334\pi$
$71$	−2224.00			−0.0523587			−0.0261794	+	0.999657i	$0.508334\pi$
$72$	76180.5	+	131948.i	1.73186	+	2.99967i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$	48873.0	−	84650.5i	1.03750	−	1.79701i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−40084.0	−	69427.5i	−0.722609	−	1.25160i	−0.959951	−	0.280169i	$-0.909609\pi$
$79$	−40084.0	−	69427.5i	−0.722609	−	1.25160i	0.237342	−	0.971426i	$-0.423724\pi$
$80$		0			0
$81$	−29524.5	+	51137.9i	−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
$86$	−64614.0	−	111915.i	−0.942064	−	1.63170i
$87$		0			0
$88$	23826.0	−	41267.8i	0.327978	−	0.568074i
$89$		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
$92$	−440728.			−5.42877
$93$		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
$99$	18468.0			0.189379
$100$	139062.	+	240863.i	1.39062	+	2.40863i
$101$		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
$106$	270050.			2.33442
$107$	32450.0	+	56205.0i	0.274003	+	0.474587i	0.969883	−	0.243571i	$-0.0783189\pi$
$107$	32450.0	+	56205.0i	0.274003	+	0.474587i	−0.695880	+	0.718158i	$0.744986\pi$
$108$		0			0
$109$	109791.	−	190164.i	0.885117	−	1.53307i	0.0395367	−	0.999218i	$-0.487412\pi$
$109$	109791.	−	190164.i	0.885117	−	1.53307i	0.845580	−	0.533849i	$-0.179255\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	123202.			0.907657			0.453828	−	0.891089i	$-0.350058\pi$
$113$	123202.			0.907657			0.453828	+	0.891089i	$0.350058\pi$
$114$		0			0
$115$		0			0
$116$	−324049.	+	561269.i	−2.23597	+	3.87281i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	77637.5	+	134472.i	0.482068	+	0.834966i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−262064.			−1.44178			−0.720888	−	0.693051i	$-0.756266\pi$
$127$	−262064.			−1.44178			−0.720888	+	0.693051i	$0.756266\pi$
$128$	−376514.	−	652141.i	−2.03121	−	3.51817i
$129$		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
$134$	763004.			3.67083
$135$		0			0
$136$		0			0
$137$	176725.	−	306097.i	0.804445	−	1.39334i	−0.112219	−	0.993683i	$-0.535796\pi$
$137$	176725.	−	306097.i	0.804445	−	1.39334i	0.916665	−	0.399657i	$-0.130871\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	12232.0	+	21186.4i	0.0509069	+	0.0881733i
$143$		0			0
$144$	491954.	−	852088.i	1.97705	−	3.42435i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−790854.			−2.96785
$149$	168509.	+	291866.i	0.621810	+	1.07701i	0.989149	+	0.146919i	$0.0469356\pi$
$149$	168509.	+	291866.i	0.621810	+	1.07701i	−0.367339	+	0.930087i	$0.619731\pi$
$150$		0			0
$151$	130812.	−	226573.i	0.466880	−	0.808660i	−0.532404	−	0.846490i	$-0.678711\pi$
$151$	130812.	−	226573.i	0.466880	−	0.808660i	0.999284	+	0.0378305i	$0.0120447\pi$
$152$		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	$0.166667\pi$
$158$	−440924.	+	763703.i	−1.40514	+	2.43378i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	649539.			1.94454
$163$	−331550.	−	574261.i	−0.977417	−	1.69294i	−0.671716	−	0.740809i	$-0.734442\pi$
$163$	−331550.	−	574261.i	−0.977417	−	1.69294i	−0.305702	−	0.952127i	$-0.598891\pi$
$164$		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
$169$	−371293.			−1.00000
$170$		0			0
$171$		0			0
$172$	−522786.	+	905492.i	−1.34742	+	2.33380i
$173$		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
$176$	−307724.			−0.748824
$177$		0			0
$178$		0			0
$179$	−292282.	+	506247.i	−0.681820	+	1.18095i	0.292605	+	0.956233i	$0.405478\pi$
$179$	−292282.	+	506247.i	−0.681820	+	1.18095i	−0.974425	+	0.224713i	$0.927856\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	1.55245e6	+	2.68893e6i	3.38044	+	5.85510i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−501524.	−	868665.i	−0.994737	−	1.72294i	−0.586102	−	0.810237i	$-0.699338\pi$
$191$	−501524.	−	868665.i	−0.994737	−	1.72294i	−0.408635	−	0.912698i	$-0.633995\pi$
$192$		0			0
$193$	192951.	−	334201.i	0.372867	−	0.645824i	−0.617138	−	0.786855i	$-0.711708\pi$
$193$	192951.	−	334201.i	0.372867	−	0.645824i	0.990005	+	0.141030i	$0.0450415\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	385814.			0.708292			0.354146	−	0.935190i	$-0.384772\pi$
$197$	385814.			0.708292			0.354146	+	0.935190i	$0.384772\pi$
$198$	−101574.	−	175931.i	−0.184128	−	0.318919i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	979688.	−	1.69687e6i	1.73186	−	2.99967i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−601668.	+	1.04212e6i	−0.975958	+	1.69041i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.09705e6			−1.69637			−0.848186	−	0.529699i	$-0.822305\pi$
$211$	−1.09705e6			−1.69637			−0.848186	+	0.529699i	$0.822305\pi$
$212$	−1.09248e6	−	1.89222e6i	−1.66944	−	2.89156i
$213$		0			0
$214$	356950.	−	618256.i	0.532811	−	0.922855i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.41540e6			−3.44230
$219$		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
$225$	759375.			1.00000
$226$	−677611.	−	1.17366e6i	−0.882489	−	1.52852i
$227$		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
$232$	4.56581e6			5.56927
$233$	639749.	+	1.10808e6i	0.772004	+	1.33715i	0.936463	+	0.350767i	$0.114079\pi$
$233$	639749.	+	1.10808e6i	0.772004	+	1.33715i	−0.164458	+	0.986384i	$0.552588\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	723536.			0.819342			0.409671	−	0.912233i	$-0.365643\pi$
$239$	723536.			0.819342			0.409671	+	0.912233i	$0.365643\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	854012.	−	1.47919e6i	0.937402	−	1.62363i
$243$		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
$253$	376352.			0.369652
$254$	1.44135e6	+	2.49649e6i	1.40180	+	2.42799i
$255$		0			0
$256$	−1.90714e6	+	3.30326e6i	−1.81879	+	3.15023i
$257$		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
$261$	884763.	+	1.53245e6i	0.803944	+	1.39247i
$262$		0			0
$263$	−765776.	+	1.32636e6i	−0.682672	+	1.18242i	0.291490	+	0.956574i	$0.405849\pi$
$263$	−765776.	+	1.32636e6i	−0.682672	+	1.18242i	−0.974162	+	0.225849i	$0.927484\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3.08670e6	−	5.34632e6i	−2.62517	−	4.54692i
$269$		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
$274$	−3.88795e6			−3.12856
$275$	−118750.	−	205681.i	−0.0946895	−	0.164007i
$276$		0			0
$277$	1.27572e6	−	2.20962e6i	0.998982	−	1.73029i	0.460423	−	0.887700i	$-0.347698\pi$
$277$	1.27572e6	−	2.20962e6i	0.998982	−	1.73029i	0.538559	−	0.842588i	$-0.318969\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.54797e6			1.92499			0.962497	−	0.271294i	$-0.0874515\pi$
$281$	2.54797e6			1.92499			0.962497	+	0.271294i	$0.0874515\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	98968.0	−	171418.i	0.0728113	−	0.126113i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−5.94742e6			−4.22520
$289$	709928.	+	1.22963e6i	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
$296$	2.78576e6	+	4.82508e6i	1.84805	+	3.20092i
$297$		0			0
$298$	1.85360e6	−	3.21053e6i	1.20914	−	2.09428i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−2.87786e6			−1.81574
$303$		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
$316$	7.13495e6			4.01951
$317$	110857.	+	192010.i	0.0619605	+	0.107319i	0.895342	−	0.445380i	$-0.146931\pi$
$317$	110857.	+	192010.i	0.0619605	+	0.107319i	−0.833381	+	0.552699i	$0.813598\pi$
$318$		0			0
$319$	276716.	−	479286.i	0.152250	−	0.263705i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.62768e6	−	4.55128e6i	−1.39062	−	2.40863i
$325$		0			0
$326$	−3.64705e6	+	6.31688e6i	−1.90063	+	3.29199i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.48574e6	−	2.57337e6i	−0.745371	−	1.29102i	−0.950021	−	0.312185i	$-0.898939\pi$
$331$	−1.48574e6	−	2.57337e6i	−0.745371	−	1.29102i	0.204651	−	0.978835i	$-0.434394\pi$
$332$		0			0
$333$	−1.07965e6	+	1.87001e6i	−0.533546	+	0.924129i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.15965e6			−1.99518			−0.997590	−	0.0693859i	$-0.977896\pi$
$337$	−4.15965e6			−1.99518			−0.997590	+	0.0693859i	$0.977896\pi$
$338$	2.04211e6	+	3.53704e6i	0.972272	+	1.68402i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	7.36600e6			3.35610
$345$		0			0
$346$		0			0
$347$	−1.14908e6	+	1.99027e6i	−0.512304	+	0.887336i	0.487595	+	0.873070i	$0.337874\pi$
$347$	−1.14908e6	+	1.99027e6i	−0.512304	+	0.887336i	−0.999898	+	0.0142656i	$0.995459\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	930050.	+	1.61089e6i	0.400083	+	0.692963i
$353$		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
$358$	6.43020e6			2.65166
$359$	−2.13448e6	−	3.69703e6i	−0.874091	−	1.51397i	−0.857728	−	0.514104i	$-0.828124\pi$
$359$	−2.13448e6	−	3.69703e6i	−0.874091	−	1.51397i	−0.0163636	−	0.999866i	$-0.505209\pi$
$360$		0			0
$361$	1.23805e6	−	2.14436e6i	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	$0.166667\pi$
$368$	1.00253e7	−	1.73644e7i	3.85904	−	6.68405i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−299651.	−	519011.i	−0.111518	−	0.193154i	0.804865	−	0.593458i	$-0.202238\pi$
$373$	−299651.	−	519011.i	−0.111518	−	0.193154i	−0.916382	+	0.400304i	$0.868904\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−5.59273e6			−1.99998			−0.999991	−	0.00429827i	$-0.998632\pi$
$379$	−5.59273e6			−1.99998			−0.999991	+	0.00429827i	$0.998632\pi$
$380$		0			0
$381$		0			0
$382$	−5.51676e6	+	9.55532e6i	−1.93431	+	3.35032i
$383$		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
$386$	−4.24492e6			−1.45011
$387$	1.42738e6	+	2.47230e6i	0.484465	+	0.839119i
$388$		0			0
$389$	630059.	−	1.09129e6i	0.211109	−	0.365652i	−0.740953	−	0.671557i	$-0.765626\pi$
$389$	630059.	−	1.09129e6i	0.211109	−	0.365652i	0.952062	+	0.305905i	$0.0989591\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−2.12198e6	−	3.67537e6i	−0.688652	−	1.19278i
$395$		0			0
$396$	−821826.	+	1.42344e6i	−0.263355	+	0.456145i
$397$		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
$400$	−1.26531e7			−3.95410
$401$	−3.01646e6	−	5.22467e6i	−0.936779	−	1.62255i	−0.771431	−	0.636313i	$-0.780459\pi$
$401$	−3.01646e6	−	5.22467e6i	−0.936779	−	1.62255i	−0.165348	−	0.986235i	$-0.552875\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	675336.			0.202085
$408$		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
$414$	1.32367e7			3.79559
$415$		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
$421$	2.07477e6			0.570513			0.285257	−	0.958451i	$-0.407921\pi$
$421$	2.07477e6			0.570513			0.285257	+	0.958451i	$0.407921\pi$
$422$	6.03379e6	+	1.04508e7i	1.64933	+	2.85673i
$423$		0			0
$424$	−7.69643e6	+	1.33306e7i	−2.07910	+	3.60110i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−5.77610e6			−1.52414
$429$		0			0
$430$		0			0
$431$	3.66942e6	−	6.35563e6i	0.951491	−	1.64803i	0.209290	−	0.977854i	$-0.432885\pi$
$431$	3.66942e6	−	6.35563e6i	0.951491	−	1.64803i	0.742201	−	0.670177i	$-0.233782\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	9.77140e6	+	1.69246e7i	2.46173	+	4.26384i
$437$		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
$443$	2.55245e6	+	4.42097e6i	0.617942	+	1.07031i	0.989861	+	0.142042i	$0.0453668\pi$
$443$	2.55245e6	+	4.42097e6i	0.617942	+	1.07031i	−0.371918	+	0.928265i	$0.621300\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.99808e6			0.467732			0.233866	−	0.972269i	$-0.424862\pi$
$449$	1.99808e6			0.467732			0.233866	+	0.972269i	$0.424862\pi$
$450$	−4.17656e6	−	7.23402e6i	−0.972272	−	1.68402i
$451$		0			0
$452$	−5.48249e6	+	9.49595e6i	−1.26221	+	2.18621i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.88984e6	+	5.00535e6i	0.647267	+	1.12110i	0.983773	+	0.179419i	$0.0574216\pi$
$457$	2.88984e6	+	5.00535e6i	0.647267	+	1.12110i	−0.336505	+	0.941682i	$0.609245\pi$
$458$		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
$463$	2.88620e6			0.625711			0.312856	−	0.949801i	$-0.398714\pi$
$463$	2.88620e6			0.625711			0.312856	+	0.949801i	$0.398714\pi$
$464$	−1.47424e7	−	2.55346e7i	−3.17888	−	5.50597i
$465$		0			0
$466$	7.03724e6	−	1.21889e7i	1.50120	−	2.60015i
$467$		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
$473$	446424.	−	773229.i	0.0917476	−	0.158911i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.96565e6			−1.20050
$478$	−3.97945e6	−	6.89261e6i	−0.796623	−	1.37979i
$479$		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
$484$	−1.38195e7			−2.68150
$485$		0			0
$486$		0			0
$487$	1.38073e6	−	2.39150e6i	0.263807	−	0.456928i	−0.703443	−	0.710752i	$-0.748355\pi$
$487$	1.38073e6	−	2.39150e6i	0.263807	−	0.456928i	0.967250	+	0.253824i	$0.0816883\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.82732e6			1.65244			0.826219	−	0.563349i	$-0.190487\pi$
$491$	8.82732e6			1.65244			0.826219	+	0.563349i	$0.190487\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	5.56018e6	+	9.63052e6i	0.999626	+	1.73140i	0.523489	+	0.852032i	$0.324630\pi$
$499$	5.56018e6	+	9.63052e6i	0.999626	+	1.73140i	0.476137	+	0.879371i	$0.342037\pi$
$500$		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
$506$	−2.06994e6	−	3.58523e6i	−0.359402	−	0.622503i
$507$		0			0
$508$	1.16618e7	−	2.01989e7i	2.00497	−	3.47271i
$509$		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
$512$	1.78601e7			3.01099
$513$		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	$0.166667\pi$
$522$	9.73239e6	−	1.68570e7i	1.56330	−	2.70772i
$523$		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
$526$	1.68471e7			2.65497
$527$		0			0
$528$		0			0
$529$	−9.04298e6	+	1.56629e7i	−1.40499	+	2.43351i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−2.17456e7	+	3.76645e7i	−3.26934	+	5.66266i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.56306e6	−	9.63551e6i	−0.817186	−	1.41541i	−0.907748	−	0.419517i	$-0.862200\pi$
$541$	−5.56306e6	−	9.63551e6i	−0.817186	−	1.41541i	0.0905619	−	0.995891i	$-0.471134\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.23604e6			−0.319529			−0.159765	−	0.987155i	$-0.551074\pi$
$547$	−2.23604e6			−0.319529			−0.159765	+	0.987155i	$0.551074\pi$
$548$	1.57285e7	+	2.72426e7i	2.23736	+	3.87523i
$549$		0			0
$550$	−1.30625e6	+	2.26249e6i	−0.184128	+	0.318919i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−2.80660e7			−3.88513
$555$		0			0
$556$		0			0
$557$	−3.11974e6	+	5.40355e6i	−0.426070	+	0.737975i	−0.996520	−	0.0833578i	$-0.973436\pi$
$557$	−3.11974e6	+	5.40355e6i	−0.426070	+	0.737975i	0.570450	+	0.821332i	$0.306769\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.40139e7	−	2.42727e7i	−1.87162	−	3.24174i
$563$		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
$568$	−1.39445e6			−0.181356
$569$	−5.23689e6	−	9.07056e6i	−0.678099	−	1.17450i	−0.975553	−	0.219765i	$-0.929471\pi$
$569$	−5.23689e6	−	9.07056e6i	−0.678099	−	1.17450i	0.297454	−	0.954736i	$-0.403862\pi$
$570$		0			0
$571$	−3.16956e6	+	5.48984e6i	−0.406826	+	0.704644i	−0.994532	−	0.104431i	$-0.966698\pi$
$571$	−3.16956e6	+	5.48984e6i	−0.406826	+	0.704644i	0.587706	+	0.809075i	$0.300031\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.54750e7			1.95192
$576$	1.69683e7	+	2.93900e7i	2.13100	+	3.69099i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	7.80921e6	−	1.35260e7i	0.972272	−	1.68402i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	932900.	+	1.61583e6i	0.113675	+	0.196890i
$584$		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
$592$	1.79897e7	−	3.11591e7i	2.10970	−	3.65410i
$593$		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
$596$	−2.99946e7			−3.45882
$597$		0			0
$598$		0			0
$599$	3.58878e6	−	6.21596e6i	0.408677	−	0.707849i	−0.586065	−	0.810264i	$-0.699324\pi$
$599$	3.58878e6	−	6.21596e6i	0.408677	−	0.707849i	0.994742	+	0.102415i	$0.0326569\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	−1.68555e7			−1.88776
$604$	1.16423e7	+	2.01650e7i	1.29851	+	2.24909i
$605$		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
$613$	−8.77570e6	+	1.52000e7i	−0.943258	+	1.63377i	−0.184057	+	0.982916i	$0.558923\pi$
$613$	−8.77570e6	+	1.52000e7i	−0.943258	+	1.63377i	−0.759201	+	0.650856i	$0.774410\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.14899e6			−0.756017			−0.378008	−	0.925802i	$-0.623391\pi$
$617$	−7.14899e6			−0.756017			−0.378008	+	0.925802i	$0.623391\pi$
$618$		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
$625$	−4.88281e6	−	8.45728e6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.99786e7			1.99752			0.998760	−	0.0497844i	$-0.0158534\pi$
$631$	1.99786e7			1.99752			0.998760	+	0.0497844i	$0.0158534\pi$
$632$	−2.51327e7	−	4.35311e7i	−2.50291	−	4.33517i
$633$		0			0
$634$	1.21943e6	−	2.11211e6i	0.120485	−	0.208686i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−6.08775e6			−0.592114
$639$	−270216.	−	468028.i	−0.0261794	−	0.0453440i
$640$		0			0
$641$	5.74814e6	−	9.95607e6i	0.552563	−	0.957068i	−0.445525	−	0.895269i	$-0.646983\pi$
$641$	5.74814e6	−	9.95607e6i	0.552563	−	0.957068i	0.998089	−	0.0617984i	$-0.0196836\pi$
$642$		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	$0.166667\pi$
$648$	−1.85119e7	+	3.20635e7i	−1.73186	+	2.99967i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	5.90159e7			5.43688
$653$	−5.48862e6	−	9.50658e6i	−0.503710	−	0.872451i	−0.999991	−	0.00428929i	$-0.998635\pi$
$653$	−5.48862e6	−	9.50658e6i	−0.503710	−	0.872451i	0.496281	−	0.868162i	$-0.334699\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.02232e7			−1.81400			−0.907000	−	0.421131i	$-0.861633\pi$
$659$	−2.02232e7			−1.81400			−0.907000	+	0.421131i	$0.861633\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	−1.63431e7	+	2.83071e7i	−1.44941	+	2.51044i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	2.37523e7			2.07501
$667$	1.80302e7	+	3.12293e7i	1.56923	+	2.71799i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.38435e6			0.117817			0.0589085	−	0.998263i	$-0.481238\pi$
$673$	1.38435e6			0.117817			0.0589085	+	0.998263i	$0.481238\pi$
$674$	2.28781e7	+	3.96260e7i	1.93986	+	3.35993i
$675$		0			0
$676$	1.65225e7	−	2.86179e7i	1.39062	−	2.40863i
$677$		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
$683$	1.06574e7	−	1.84592e7i	0.874179	−	1.51412i	0.0165451	−	0.999863i	$-0.494733\pi$
$683$	1.06574e7	−	1.84592e7i	0.874179	−	1.51412i	0.857634	−	0.514260i	$-0.171933\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−2.37838e7	−	4.11948e7i	−1.91563	−	3.31796i
$689$		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
$694$	2.52798e7			1.99239
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.88600e6			0.375542			0.187771	−	0.982213i	$-0.439874\pi$
$701$	4.88600e6			0.375542			0.187771	+	0.982213i	$0.439874\pi$
$702$		0			0
$703$		0			0
$704$	5.30697e6	−	9.19193e6i	0.403566	−	0.698997i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.16124e6	+	1.24036e7i	0.535023	+	0.926688i	0.999162	+	0.0409253i	$0.0130306\pi$
$709$	7.16124e6	+	1.24036e7i	0.535023	+	0.926688i	−0.464139	+	0.885763i	$0.653636\pi$
$710$		0			0
$711$	9.74041e6	−	1.68709e7i	0.722609	−	1.25160i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−2.60131e7	−	4.50560e7i	−1.89631	−	3.28451i
$717$		0			0
$718$	−2.34793e7	+	4.06674e7i	−1.69971	+	2.94398i
$719$		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
$722$	−2.72371e7			−1.94454
$723$		0			0
$724$		0			0
$725$	1.13781e7	−	1.97075e7i	0.803944	−	1.39247i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.43489e7			−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
$736$	−1.21200e8			−8.24724
$737$	2.63583e6	+	4.56539e6i	0.178751	+	0.309606i
$738$		0			0
$739$	1.32447e6	−	2.29404e6i	0.0892133	−	0.154522i	−0.817966	−	0.575267i	$-0.804898\pi$
$739$	1.32447e6	−	2.29404e6i	0.0892133	−	0.154522i	0.907179	+	0.420745i	$0.138231\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.49502e7			1.65807			0.829033	−	0.559199i	$-0.188891\pi$
$743$	2.49502e7			1.65807			0.829033	+	0.559199i	$0.188891\pi$
$744$		0			0
$745$		0			0
$746$	−3.29616e6	+	5.70912e6i	−0.216851	+	0.375597i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.25544e7	−	2.17448e7i	−0.812260	−	1.40688i	−0.911278	−	0.411791i	$-0.864903\pi$
$751$	−1.25544e7	−	2.17448e7i	−0.812260	−	1.40688i	0.0990181	−	0.995086i	$-0.468430\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.78870e7			−1.13448			−0.567242	−	0.823551i	$-0.691989\pi$
$757$	−1.78870e7			−1.13448			−0.567242	+	0.823551i	$0.691989\pi$
$758$	3.07600e7	+	5.32779e7i	1.94453	+	3.36802i
$759$		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
$764$	8.92713e7			5.53322
$765$		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
$772$	1.71726e7	+	2.97439e7i	1.03704	+	1.79620i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	1.57012e7	−	2.71953e7i	0.942064	−	1.63170i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.38613e7			−0.821022
$779$		0			0
$780$		0			0
$781$	−84512.0	+	146379.i	−0.00495782	+	0.00858720i
$782$		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	$0.166667\pi$
$788$	−1.71687e7	+	2.97371e7i	−0.984969	+	1.70602i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.15794e7			0.655956
$793$		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
$800$	3.82422e7	+	6.62374e7i	2.11260	+	3.65913i
$801$		0			0
$802$	−3.31811e7	+	5.74713e7i	−1.82161	+	3.15512i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.18534e6	+	1.59095e7i	−0.493428	+	0.854643i	−0.999971	−	0.00757189i	$-0.997590\pi$
$809$	−9.18534e6	+	1.59095e7i	−0.493428	+	0.854643i	0.506543	+	0.862215i	$0.330923\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−3.71435e6	−	6.43344e6i	−0.196481	−	0.340316i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.77941e7	−	3.08202e7i	−0.921334	−	1.59580i	−0.797354	−	0.603512i	$-0.793768\pi$
$821$	−1.77941e7	−	3.08202e7i	−0.921334	−	1.59580i	−0.123980	−	0.992285i	$-0.539566\pi$
$822$		0			0
$823$	−3.54338e6	+	6.13731e6i	−0.182355	+	0.315848i	−0.942682	−	0.333692i	$-0.891705\pi$
$823$	−3.54338e6	+	6.13731e6i	−0.182355	+	0.315848i	0.760327	+	0.649540i	$0.225039\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.68552e7			−1.87385			−0.936926	−	0.349527i	$-0.886342\pi$
$827$	−3.68552e7			−1.87385			−0.936926	+	0.349527i	$0.886342\pi$
$828$	−5.35485e7	−	9.27486e7i	−2.71438	−	4.70145i
$829$		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
$841$	3.25164e7			1.58530
$842$	−1.14113e7	−	1.97649e7i	−0.554694	−	0.960758i
$843$		0			0
$844$	4.88188e7	−	8.45567e7i	2.35902	−	4.08594i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	9.94030e7			4.74690
$849$		0			0
$850$		0			0
$851$	−2.20017e7	+	3.81081e7i	−1.04144	+	1.80382i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	2.03461e7	+	3.52406e7i	0.949069	+	1.64384i
$857$		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
$862$	−8.07273e7			−3.70043
$863$	−1.38071e7	−	2.39146e7i	−0.631068	−	1.09304i	−0.987334	−	0.158657i	$-0.949284\pi$
$863$	−1.38071e7	−	2.39146e7i	−0.631068	−	1.09304i	0.356266	−	0.934385i	$-0.384050\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.09277e6			−0.273694
$870$		0			0
$871$		0			0
$872$	6.88390e7	−	1.19233e8i	3.06579	−	5.31011i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.16714e7	+	3.75359e7i	0.951453	+	1.64797i	0.742283	+	0.670086i	$0.233743\pi$
$877$	2.16714e7	+	3.75359e7i	0.951453	+	1.64797i	0.209170	+	0.977879i	$0.432924\pi$
$878$		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
$883$	−3.94011e7			−1.70062			−0.850308	−	0.526286i	$-0.823584\pi$
$883$	−3.94011e7			−1.70062			−0.850308	+	0.526286i	$0.823584\pi$
$884$		0			0
$885$		0			0
$886$	2.80770e7	−	4.86307e7i	1.20162	−	2.08126i
$887$		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
$891$	2.24386e6	+	3.88648e6i	0.0946895	+	0.164007i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.09895e7	−	1.90343e7i	−0.454763	−	0.787673i
$899$		0			0
$900$	−3.37922e7	+	5.85298e7i	−1.39062	+	2.40863i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	7.72477e7			3.14387
$905$		0			0
$906$		0			0
$907$	−2.24173e7	+	3.88280e7i	−0.904828	+	1.56721i	−0.0836802	+	0.996493i	$0.526667\pi$
$907$	−2.24173e7	+	3.88280e7i	−0.904828	+	1.56721i	−0.821148	+	0.570715i	$0.806666\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.87850e7			1.94756			0.973778	−	0.227498i	$-0.0730547\pi$
$911$	4.87850e7			1.94756			0.973778	+	0.227498i	$0.0730547\pi$
$912$		0			0
$913$		0			0
$914$	3.17883e7	−	5.50589e7i	1.25864	−	2.18003i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.53358e7	−	4.38829e7i	−0.989569	−	1.71398i	−0.619543	−	0.784962i	$-0.712682\pi$
$919$	−2.53358e7	−	4.38829e7i	−0.989569	−	1.71398i	−0.370026	−	0.929022i	$-0.620651\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.77688e7			1.06709
$926$	−1.58741e7	−	2.74947e7i	−0.608361	−	1.05371i
$927$		0			0
$928$	−8.91135e7	+	1.54349e8i	−3.39683	+	5.88348i
$929$		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
$932$	−1.13875e8			−4.29427
$933$		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
$946$	−9.82133e6			−0.356814
$947$	2.75004e7	+	4.76322e7i	0.996471	+	1.72594i	0.570923	+	0.821003i	$0.306585\pi$
$947$	2.75004e7	+	4.76322e7i	0.996471	+	1.72594i	0.425548	+	0.904936i	$0.360081\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.97130e6			−0.355647			−0.177824	−	0.984062i	$-0.556906\pi$
$953$	−9.97130e6			−0.355647			−0.177824	+	0.984062i	$0.556906\pi$
$954$	3.28111e7	+	5.68304e7i	1.16721	+	2.02167i
$955$		0			0
$956$	−3.21974e7	+	5.57674e7i	−1.13940	+	1.97350i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.43146e7	+	2.47936e7i	0.500000	+	0.866025i
$962$		0			0
$963$	−7.88535e6	+	1.36578e7i	−0.274003	+	0.474587i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.00222e7			0.688566			0.344283	−	0.938866i	$-0.388122\pi$
$967$	2.00222e7			0.688566			0.344283	+	0.938866i	$0.388122\pi$
$968$	4.86787e7	+	8.43140e7i	1.66975	+	2.89209i
$969$		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
$974$	−3.03761e7			−1.02597
$975$		0			0
$976$		0			0
$977$	2.49909e7	−	4.32854e7i	0.837616	−	1.45079i	−0.0542674	−	0.998526i	$-0.517282\pi$
$977$	2.49909e7	−	4.32854e7i	0.837616	−	1.45079i	0.891883	−	0.452266i	$-0.149384\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	5.33584e7			1.77023
$982$	−4.85503e7	−	8.40916e7i	−1.60662	−	2.78275i
$983$		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
$989$	2.90880e7	+	5.03820e7i	0.945636	+	1.63789i
$990$		0			0
$991$	2.86572e7	−	4.96357e7i	0.926936	−	1.60550i	0.138519	−	0.990360i	$-0.455766\pi$
$991$	2.86572e7	−	4.96357e7i	0.926936	−	1.60550i	0.788417	−	0.615141i	$-0.210901\pi$
$992$		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	$0.166667\pi$
$998$	6.11620e7	−	1.05936e8i	1.94382	−	3.36679i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.c.a.18.1		2
7.2	even	3	inner	49.6.c.a.30.1		2
7.3	odd	6		49.6.a.b.1.1	✓	1
7.4	even	3		49.6.a.b.1.1	✓	1
7.5	odd	6	inner	49.6.c.a.30.1		2
7.6	odd	2	CM	49.6.c.a.18.1		2
21.11	odd	6		441.6.a.a.1.1		1
21.17	even	6		441.6.a.a.1.1		1
28.3	even	6		784.6.a.g.1.1		1
28.11	odd	6		784.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.b.1.1	✓	1	7.3	odd	6
49.6.a.b.1.1	✓	1	7.4	even	3
49.6.c.a.18.1		2	1.1	even	1	trivial
49.6.c.a.18.1		2	7.6	odd	2	CM
49.6.c.a.30.1		2	7.2	even	3	inner
49.6.c.a.30.1		2	7.5	odd	6	inner
441.6.a.a.1.1		1	21.11	odd	6
441.6.a.a.1.1		1	21.17	even	6
784.6.a.g.1.1		1	28.3	even	6
784.6.a.g.1.1		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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +O(q^{10})\)	\(q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +(38.0000 - 65.8179i) q^{11} +(-2024.50 - 3506.54i) q^{16} +(1336.50 - 2314.89i) q^{18} -836.000 q^{22} +(2476.00 + 4288.56i) q^{23} +(1562.50 - 2706.33i) q^{25} +7282.00 q^{29} +(-12237.5 + 21196.0i) q^{32} -21627.0 q^{36} +(4443.00 + 7695.50i) q^{37} +11748.0 q^{43} +(3382.00 + 5857.80i) q^{44} +(27236.0 - 47174.1i) q^{46} -34375.0 q^{50} +(-12275.0 + 21260.9i) q^{53} +(-40051.0 - 69370.4i) q^{58} +139657. q^{64} +(-34682.0 + 60071.0i) q^{67} -2224.00 q^{71} +(76180.5 + 131948. i) q^{72} +(48873.0 - 84650.5i) q^{74} +(-40084.0 - 69427.5i) q^{79} +(-29524.5 + 51137.9i) q^{81} +(-64614.0 - 111915. i) q^{86} +(23826.0 - 41267.8i) q^{88} -440728. q^{92} +18468.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9}+O(q^{10}) \) 2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9	\( 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9} + 76 q^{11} - 4049 q^{16} + 2673 q^{18} - 1672 q^{22} + 4952 q^{23} + 3125 q^{25} + 14564 q^{29} - 24475 q^{32} - 43254 q^{36} + 8886 q^{37} + 23496 q^{43} + 6764 q^{44} + 54472 q^{46} - 68750 q^{50} - 24550 q^{53} - 80102 q^{58} + 279314 q^{64} - 69364 q^{67} - 4448 q^{71} + 152361 q^{72} + 97746 q^{74} - 80168 q^{79} - 59049 q^{81} - 129228 q^{86} + 47652 q^{88} - 881456 q^{92} + 36936 q^{99}+O(q^{100}) \) 2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9 + 76 * q^11 - 4049 * q^16 + 2673 * q^18 - 1672 * q^22 + 4952 * q^23 + 3125 * q^25 + 14564 * q^29 - 24475 * q^32 - 43254 * q^36 + 8886 * q^37 + 23496 * q^43 + 6764 * q^44 + 54472 * q^46 - 68750 * q^50 - 24550 * q^53 - 80102 * q^58 + 279314 * q^64 - 69364 * q^67 - 4448 * q^71 + 152361 * q^72 + 97746 * q^74 - 80168 * q^79 - 59049 * q^81 - 129228 * q^86 + 47652 * q^88 - 881456 * q^92 + 36936 * q^99

Introduction

L-functions

Modular forms

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Database

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