LMFDB - Embedded newform 49.10.c.a.30.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [49,10,Mod(18,49)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49.18"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$N$	$=$	$49 = 7^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	49.c (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,5,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$25.2367559720$
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			30.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	49.30
Dual form			49.10.c.a.18.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{2} +(243.500 + 421.754i) q^{4} +4995.00 q^{8} +(9841.50 - 17046.0i) q^{9} +(21454.0 + 37159.4i) q^{11} +(-112184. + 194309. i) q^{16} +(-49207.5 - 85229.9i) q^{18} +214540. q^{22} +(-698020. + 1.20901e6i) q^{23} +(976562. + 1.69146e6i) q^{25} +7.57143e6 q^{29} +(1.83964e6 + 3.18635e6i) q^{32} +9.58562e6 q^{36} +(-1.13717e7 + 1.96964e7i) q^{37} +3.71018e7 q^{43} +(-1.04481e7 + 1.80966e7i) q^{44} +(3.49010e6 + 6.04503e6i) q^{46} +9.76562e6 q^{50} +(2.90666e7 + 5.03449e7i) q^{53} +(1.89286e7 - 3.27852e7i) q^{58} -9.64805e7 q^{64} +(-1.31235e8 - 2.27306e8i) q^{67} -4.13068e8 q^{71} +(4.91583e7 - 8.51447e7i) q^{72} +(5.68586e7 + 9.84820e7i) q^{74} +(3.01947e8 - 5.22987e8i) q^{79} +(-1.93710e8 - 3.35516e8i) q^{81} +(9.27544e7 - 1.60655e8i) q^{86} +(1.07163e8 + 1.85611e8i) q^{88} -6.79871e8 q^{92} +8.44558e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 5 q^{2} + 487 q^{4} + 9990 q^{8} + 19683 q^{9} + 42908 q^{11} - 224369 q^{16} - 98415 q^{18} + 429080 q^{22} - 1396040 q^{23} + 1953125 q^{25} + 15142852 q^{29} + 3679285 q^{32} + 19171242 q^{36}+ \cdots + 1689116328 q^{99}+O(q^{100})$ 2 * q + 5 * q^2 + 487 * q^4 + 9990 * q^8 + 19683 * q^9 + 42908 * q^11 - 224369 * q^16 - 98415 * q^18 + 429080 * q^22 - 1396040 * q^23 + 1953125 * q^25 + 15142852 * q^29 + 3679285 * q^32 + 19171242 * q^36 - 22743450 * q^37 + 74203560 * q^43 - 20896196 * q^44 + 6980200 * q^46 + 19531250 * q^50 + 58133290 * q^53 + 37857130 * q^58 - 192961006 * q^64 - 262469860 * q^67 - 826135264 * q^71 + 98316585 * q^72 + 113717250 * q^74 + 603893176 * q^79 - 387420489 * q^81 + 185508900 * q^86 + 214325460 * q^88 - 1359742960 * q^92 + 1689116328 * q^99

Character values

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

$n$	$3$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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	0.110485	−	0.191366i	−0.805481	−	0.592622i	$-0.798093\pi$
$2$	2.50000	−	4.33013i	0.110485	−	0.191366i	0.915966	+	0.401256i	$0.131426\pi$
$3$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$	243.500	+	421.754i	0.475586	+	0.823739i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	4995.00			0.431152
$9$	9841.50	−	17046.0i	0.500000	−	0.866025i
$10$		0			0
$11$	21454.0	+	37159.4i	0.441816	+	0.765247i	0.997824	−	0.0659292i	$-0.0210012\pi$
$11$	21454.0	+	37159.4i	0.441816	+	0.765247i	−0.556009	+	0.831177i	$0.687668\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−112184.	+	194309.i	−0.427950	+	0.741231i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	−49207.5	−	85229.9i	−0.110485	−	0.191366i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		0			0
$22$	214540.			0.195257
$23$	−698020.	+	1.20901e6i	−0.520107	+	0.900852i	0.479620	+	0.877476i	$0.340775\pi$
$23$	−698020.	+	1.20901e6i	−0.520107	+	0.900852i	−0.999727	+	0.0233752i	$0.992559\pi$
$24$		0			0
$25$	976562.	+	1.69146e6i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.57143e6			1.98786			0.993932	−	0.109996i	$-0.0350837\pi$
$29$	7.57143e6			1.98786			0.993932	+	0.109996i	$0.0350837\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	1.83964e6	+	3.18635e6i	0.310141	+	0.537179i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	9.58562e6			0.951172
$37$	−1.13717e7	+	1.96964e7i	−0.997513	+	1.72774i	−0.437717	+	0.899113i	$0.644213\pi$
$37$	−1.13717e7	+	1.96964e7i	−0.997513	+	1.72774i	−0.559796	+	0.828631i	$0.689120\pi$
$38$		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
$43$	3.71018e7			1.65496			0.827478	−	0.561497i	$-0.189775\pi$
$43$	3.71018e7			1.65496			0.827478	+	0.561497i	$0.189775\pi$
$44$	−1.04481e7	+	1.80966e7i	−0.420243	+	0.727882i
$45$		0			0
$46$	3.49010e6	+	6.04503e6i	0.114928	+	0.199062i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$	9.76562e6			0.220971
$51$		0			0
$52$		0			0
$53$	2.90666e7	+	5.03449e7i	0.506003	+	0.876424i	0.999976	+	0.00694614i	$0.00221104\pi$
$53$	2.90666e7	+	5.03449e7i	0.506003	+	0.876424i	−0.493972	+	0.869478i	$0.664456\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.89286e7	−	3.27852e7i	0.219630	−	0.380410i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	−9.64805e7			−0.718836
$65$		0			0
$66$		0			0
$67$	−1.31235e8	−	2.27306e8i	−0.795633	−	1.37808i	−0.922436	−	0.386149i	$-0.873805\pi$
$67$	−1.31235e8	−	2.27306e8i	−0.795633	−	1.37808i	0.126803	−	0.991928i	$-0.459528\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.13068e8			−1.92912			−0.964559	−	0.263869i	$-0.915001\pi$
$71$	−4.13068e8			−1.92912			−0.964559	+	0.263869i	$0.915001\pi$
$72$	4.91583e7	−	8.51447e7i	0.215576	−	0.373389i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$	5.68586e7	+	9.84820e7i	0.220421	+	0.381781i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.01947e8	−	5.22987e8i	0.872184	−	1.51067i	0.0124519	−	0.999922i	$-0.496036\pi$
$79$	3.01947e8	−	5.22987e8i	0.872184	−	1.51067i	0.859732	−	0.510745i	$-0.170630\pi$
$80$		0			0
$81$	−1.93710e8	−	3.35516e8i	−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
$86$	9.27544e7	−	1.60655e8i	0.182849	−	0.316703i
$87$		0			0
$88$	1.07163e8	+	1.85611e8i	0.190490	+	0.329938i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$	−6.79871e8			−0.989422
$93$		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
$99$	8.44558e8			0.883631
$100$	−4.75586e8	+	8.23739e8i	−0.475586	+	0.823739i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$	2.90666e8			0.223624
$107$	−9.31196e8	+	1.61288e9i	−0.686774	+	1.18953i	0.286101	+	0.958199i	$0.407641\pi$
$107$	−9.31196e8	+	1.61288e9i	−0.686774	+	1.18953i	−0.972876	+	0.231329i	$0.925693\pi$
$108$		0			0
$109$	−1.04819e8	−	1.81551e8i	−0.0711245	−	0.123191i	0.828270	−	0.560329i	$-0.189325\pi$
$109$	−1.04819e8	−	1.81551e8i	−0.0711245	−	0.123191i	−0.899394	+	0.437138i	$0.855992\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.59946e9			1.49979			0.749894	−	0.661557i	$-0.230104\pi$
$113$	2.59946e9			1.49979			0.749894	+	0.661557i	$0.230104\pi$
$114$		0			0
$115$		0			0
$116$	1.84364e9	+	3.19328e9i	0.945400	+	1.63748i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.58426e8	−	4.47606e8i	0.109598	−	0.189829i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.29030e9			1.46343			0.731713	−	0.681613i	$-0.238721\pi$
$127$	4.29030e9			1.46343			0.731713	+	0.681613i	$0.238721\pi$
$128$	−1.18310e9	+	2.04919e9i	−0.389561	+	0.674740i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$	−1.31235e9			−0.351623
$135$		0			0
$136$		0			0
$137$	−3.04475e9	−	5.27367e9i	−0.738430	−	1.27900i	−0.953202	−	0.302335i	$-0.902234\pi$
$137$	−3.04475e9	−	5.27367e9i	−0.738430	−	1.27900i	0.214771	−	0.976664i	$-0.431099\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−1.03267e9	+	1.78864e9i	−0.213139	+	0.369168i
$143$		0			0
$144$	2.20813e9	+	3.82459e9i	0.427950	+	0.741231i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.10761e10			−1.89761
$149$	3.76933e9	−	6.52867e9i	0.626507	−	1.08514i	−0.361741	−	0.932279i	$-0.617817\pi$
$149$	3.76933e9	−	6.52867e9i	0.626507	−	1.08514i	0.988247	−	0.152863i	$-0.0488493\pi$
$150$		0			0
$151$	−2.38307e9	−	4.12761e9i	−0.373028	−	0.646103i	0.617002	−	0.786962i	$-0.288347\pi$
$151$	−2.38307e9	−	4.12761e9i	−0.373028	−	0.646103i	−0.990030	+	0.140858i	$0.955014\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	−1.50973e9	−	2.61493e9i	−0.192727	−	0.333813i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1.93710e9			−0.220971
$163$	8.74322e9	−	1.51437e10i	0.970124	−	1.68030i	0.274955	−	0.961457i	$-0.411337\pi$
$163$	8.74322e9	−	1.51437e10i	0.970124	−	1.68030i	0.695169	−	0.718846i	$-0.255330\pi$
$164$		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
$169$	−1.06045e10			−1.00000
$170$		0			0
$171$		0			0
$172$	9.03428e9	+	1.56478e10i	0.787074	+	1.36325i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	−9.62723e9			−0.756300
$177$		0			0
$178$		0			0
$179$	−1.34032e10	−	2.32150e10i	−0.975821	−	1.69017i	−0.677200	−	0.735799i	$-0.736807\pi$
$179$	−1.34032e10	−	2.32150e10i	−0.975821	−	1.69017i	−0.298621	−	0.954372i	$-0.596527\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−3.48661e9	+	6.03899e9i	−0.224245	+	0.388404i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.80047e9	+	1.52429e10i	−0.478471	+	0.828736i	−0.999695	−	0.0246838i	$-0.992142\pi$
$191$	−8.80047e9	+	1.52429e10i	−0.478471	+	0.828736i	0.521224	+	0.853420i	$0.325475\pi$
$192$		0			0
$193$	−1.14537e9	−	1.98384e9i	−0.0594206	−	0.102920i	0.834785	−	0.550576i	$-0.185592\pi$
$193$	−1.14537e9	−	1.98384e9i	−0.0594206	−	0.102920i	−0.894205	+	0.447657i	$0.852259\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.98950e10			1.88721			0.943605	−	0.331075i	$-0.107411\pi$
$197$	3.98950e10			1.88721			0.943605	+	0.331075i	$0.107411\pi$
$198$	2.11140e9	−	3.65704e9i	0.0976284	−	0.169097i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	4.87793e9	+	8.44882e9i	0.215576	+	0.373389i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.37391e10	+	2.37969e10i	0.520107	+	0.900852i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	3.66846e10			1.27413			0.637063	−	0.770812i	$-0.280149\pi$
$211$	3.66846e10			1.27413			0.637063	+	0.770812i	$0.280149\pi$
$212$	−1.41555e10	+	2.45180e10i	−0.481296	+	0.833630i
$213$		0			0
$214$	4.65598e9	+	8.06439e9i	0.151757	+	0.262851i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.04819e9			−0.0314329
$219$		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
$225$	3.84434e10			1.00000
$226$	6.49865e9	−	1.12560e10i	0.165705	−	0.287009i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	3.78193e10			0.857072
$233$	−3.68351e10	+	6.38003e10i	−0.818768	+	1.41815i	0.0878231	+	0.996136i	$0.472009\pi$
$233$	−3.68351e10	+	6.38003e10i	−0.818768	+	1.41815i	−0.906591	+	0.422011i	$0.861324\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.92044e10			−1.96671			−0.983355	−	0.181695i	$-0.941842\pi$
$239$	−9.92044e10			−1.96671			−0.983355	+	0.181695i	$0.941842\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	−1.29213e9	−	2.23803e9i	−0.0242179	−	0.0419466i
$243$		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
$253$	−5.99013e10			−0.919166
$254$	1.07257e10	−	1.85775e10i	0.161687	−	0.280051i
$255$		0			0
$256$	−1.87835e10	−	3.25340e10i	−0.273336	−	0.473432i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.45142e10	−	1.29062e11i	0.993932	−	1.72154i
$262$		0			0
$263$	−7.42998e9	−	1.28691e10i	−0.0957606	−	0.165862i	0.814165	−	0.580633i	$-0.197195\pi$
$263$	−7.42998e9	−	1.28691e10i	−0.0957606	−	0.165862i	−0.909926	+	0.414771i	$0.863862\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	6.39114e10	−	1.10698e11i	0.756784	−	1.31079i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$	−3.04475e10			−0.326343
$275$	−4.19023e10	+	7.25770e10i	−0.441816	+	0.765247i
$276$		0			0
$277$	3.77404e10	+	6.53683e10i	0.385166	+	0.667127i	0.991792	−	0.127860i	$-0.0408109\pi$
$277$	3.77404e10	+	6.53683e10i	0.385166	+	0.667127i	−0.606626	+	0.794987i	$0.707478\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.07162e11			−1.98213			−0.991063	−	0.133393i	$-0.957413\pi$
$281$	−2.07162e11			−1.98213			−0.991063	+	0.133393i	$0.957413\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$	−1.00582e11	−	1.74213e11i	−0.917461	−	1.58909i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	7.24194e10			0.620281
$289$	5.92939e10	−	1.02700e11i	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
$296$	−5.68018e10	+	9.83835e10i	−0.430080	+	0.744920i
$297$		0			0
$298$	−1.88466e10	−	3.26433e10i	−0.138440	−	0.239785i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−2.38307e10			−0.164857
$303$		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	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$		0			0
$315$		0			0
$316$	2.94096e11			1.65919
$317$	−1.63740e11	+	2.83606e11i	−0.910725	+	1.57742i	−0.0976831	+	0.995218i	$0.531143\pi$
$317$	−1.63740e11	+	2.83606e11i	−0.910725	+	1.57742i	−0.813042	+	0.582205i	$0.802190\pi$
$318$		0			0
$319$	1.62437e11	+	2.81350e11i	0.878270	+	1.52121i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.43369e10	−	1.63396e11i	0.475586	−	0.823739i
$325$		0			0
$326$	−4.37161e10	−	7.57185e10i	−0.214369	−	0.371298i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.55784e10	+	9.62646e10i	−0.254495	+	0.440799i	−0.964758	−	0.263137i	$-0.915243\pi$
$331$	−5.55784e10	+	9.62646e10i	−0.254495	+	0.440799i	0.710263	+	0.703937i	$0.248576\pi$
$332$		0			0
$333$	2.23830e11	+	3.87684e11i	0.997513	+	1.72774i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.45419e11			1.45885			0.729426	−	0.684059i	$-0.239787\pi$
$337$	3.45419e11			1.45885			0.729426	+	0.684059i	$0.239787\pi$
$338$	−2.65112e10	+	4.59188e10i	−0.110485	+	0.191366i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.85323e11			0.713538
$345$		0			0
$346$		0			0
$347$	−2.22498e11	−	3.85378e11i	−0.823842	−	1.42694i	−0.902801	−	0.430059i	$-0.858493\pi$
$347$	−2.22498e11	−	3.85378e11i	−0.823842	−	1.42694i	0.0789585	−	0.996878i	$-0.474841\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−7.89354e10	+	1.36720e11i	−0.274050	+	0.474668i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.34032e11			−0.431256
$359$	−2.96323e11	+	5.13246e11i	−0.941543	+	1.63080i	−0.179013	+	0.983847i	$0.557291\pi$
$359$	−2.96323e11	+	5.13246e11i	−0.941543	+	1.63080i	−0.762529	+	0.646954i	$0.776043\pi$
$360$		0			0
$361$	1.61344e11	+	2.79456e11i	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	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−1.56614e11	−	2.71263e11i	−0.445159	−	0.771039i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2.75737e11	−	4.77590e11i	0.737573	−	1.27751i	−0.216013	−	0.976391i	$-0.569305\pi$
$373$	2.75737e11	−	4.77590e11i	0.737573	−	1.27751i	0.953585	−	0.301123i	$-0.0973614\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.54154e11			−0.632732			−0.316366	−	0.948637i	$-0.602463\pi$
$379$	−2.54154e11			−0.632732			−0.316366	+	0.948637i	$0.602463\pi$
$380$		0			0
$381$		0			0
$382$	4.40023e10	+	7.62143e10i	0.105728	+	0.183126i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$	−1.14537e10			−0.0262604
$387$	3.65137e11	−	6.32436e11i	0.827478	−	1.43323i
$388$		0			0
$389$	−3.46279e11	−	5.99772e11i	−0.766748	−	1.32805i	−0.939318	−	0.343049i	$-0.888540\pi$
$389$	−3.46279e11	−	5.99772e11i	−0.766748	−	1.32805i	0.172570	−	0.984997i	$-0.444793\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	9.97374e10	−	1.72750e11i	0.208509	−	0.361148i
$395$		0			0
$396$	2.05650e11	+	3.56196e11i	0.420243	+	0.727882i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−4.38221e11			−0.855900
$401$	1.52521e11	−	2.64174e11i	0.294564	−	0.510201i	−0.680319	−	0.732916i	$-0.738159\pi$
$401$	1.52521e11	−	2.64174e11i	0.294564	−	0.510201i	0.974883	+	0.222716i	$0.0714921\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.75876e11			−1.76287
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.37391e11			0.229857
$415$		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
$421$	1.38476e11			0.214835			0.107418	−	0.994214i	$-0.465742\pi$
$421$	1.38476e11			0.214835			0.107418	+	0.994214i	$0.465742\pi$
$422$	9.17114e10	−	1.58849e11i	0.140772	−	0.243825i
$423$		0			0
$424$	1.45188e11	+	2.51473e11i	0.218164	+	0.377872i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−9.06985e11			−1.30648
$429$		0			0
$430$		0			0
$431$	−7.14857e11	−	1.23817e12i	−0.997864	−	1.72835i	−0.555503	−	0.831515i	$-0.687474\pi$
$431$	−7.14857e11	−	1.23817e12i	−0.997864	−	1.72835i	−0.442362	−	0.896837i	$-0.645859\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	5.10467e10	−	8.84154e10i	0.0676516	−	0.117176i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.84090e11	+	1.35808e12i	−0.967273	+	1.67537i	−0.263892	+	0.964552i	$0.585006\pi$
$443$	−7.84090e11	+	1.35808e12i	−0.967273	+	1.67537i	−0.703380	+	0.710814i	$0.748327\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.10022e11			0.824449			0.412224	−	0.911082i	$-0.364752\pi$
$449$	7.10022e11			0.824449			0.412224	+	0.911082i	$0.364752\pi$
$450$	9.61084e10	−	1.66465e11i	0.110485	−	0.191366i
$451$		0			0
$452$	6.32969e11	+	1.09633e12i	0.713279	+	1.23543i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8.89798e11	−	1.54118e12i	0.954264	−	1.65283i	0.218222	−	0.975899i	$-0.429974\pi$
$457$	8.89798e11	−	1.54118e12i	0.954264	−	1.65283i	0.736043	−	0.676935i	$-0.236692\pi$
$458$		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
$463$	1.91187e12			1.93350			0.966750	−	0.255725i	$-0.0823141\pi$
$463$	1.91187e12			1.93350			0.966750	+	0.255725i	$0.0823141\pi$
$464$	−8.49397e11	+	1.47120e12i	−0.850706	+	1.47347i
$465$		0			0
$466$	1.84176e11	+	3.19002e11i	0.180924	+	0.313369i
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.95982e11	+	1.37868e12i	0.731186	+	1.26645i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.14424e12			1.01201
$478$	−2.48011e11	+	4.29568e11i	−0.217293	+	0.376362i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.51707e11			0.208492
$485$		0			0
$486$		0			0
$487$	−1.11121e12	−	1.92467e12i	−0.895191	−	1.55052i	−0.833568	−	0.552417i	$-0.813706\pi$
$487$	−1.11121e12	−	1.92467e12i	−0.895191	−	1.55052i	−0.0616231	−	0.998099i	$-0.519628\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.22041e12			0.947628			0.473814	−	0.880625i	$-0.342877\pi$
$491$	1.22041e12			0.947628			0.473814	+	0.880625i	$0.342877\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.15919e12	+	2.00777e12i	−0.836954	+	1.44965i	0.0554756	+	0.998460i	$0.482332\pi$
$499$	−1.15919e12	+	2.00777e12i	−0.836954	+	1.44965i	−0.892430	+	0.451187i	$0.851001\pi$
$500$		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
$506$	−1.49753e11	+	2.59380e11i	−0.101554	+	0.175897i
$507$		0			0
$508$	1.04469e12	+	1.80945e12i	0.695985	+	1.20548i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$	−1.39933e12			−0.899921
$513$		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	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$	−3.72571e11	−	6.45312e11i	−0.219630	−	0.380410i
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$	−7.42998e10			−0.0423206
$527$		0			0
$528$		0			0
$529$	−7.38875e10	−	1.27977e11i	−0.0410223	−	0.0710528i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−6.55518e11	−	1.13539e12i	−0.343039	−	0.594161i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.76969e12	−	3.06519e12i	0.888195	−	1.53840i	0.0461880	−	0.998933i	$-0.485293\pi$
$541$	1.76969e12	−	3.06519e12i	0.888195	−	1.53840i	0.842007	−	0.539466i	$-0.181374\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.18633e12			−1.99936			−0.999679	−	0.0253427i	$-0.991932\pi$
$547$	−4.18633e12			−1.99936			−0.999679	+	0.0253427i	$0.991932\pi$
$548$	1.48280e12	−	2.56828e12i	0.702374	−	1.21655i
$549$		0			0
$550$	2.09512e11	+	3.62885e11i	0.0976284	+	0.169097i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	3.77404e11			0.170221
$555$		0			0
$556$		0			0
$557$	1.01741e12	+	1.76221e12i	0.447866	+	0.775727i	0.998247	−	0.0591866i	$-0.0188507\pi$
$557$	1.01741e12	+	1.76221e12i	0.447866	+	0.775727i	−0.550381	+	0.834914i	$0.685517\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−5.17905e11	+	8.97037e11i	−0.218996	+	0.379312i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−2.06327e12			−0.831743
$569$	2.30413e12	−	3.99088e12i	0.921515	−	1.59611i	0.124443	−	0.992227i	$-0.460286\pi$
$569$	2.30413e12	−	3.99088e12i	0.921515	−	1.59611i	0.797072	−	0.603884i	$-0.206381\pi$
$570$		0			0
$571$	−1.73919e12	−	3.01237e12i	−0.684676	−	1.18589i	−0.973539	−	0.228523i	$-0.926610\pi$
$571$	−1.73919e12	−	3.01237e12i	−0.684676	−	1.18589i	0.288863	−	0.957371i	$-0.406723\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.72664e12			−1.04021
$576$	−9.49513e11	+	1.64460e12i	−0.359418	+	0.622530i
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	−2.96470e11	−	5.13501e11i	−0.110485	−	0.191366i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.24719e12	+	2.16020e12i	−0.447121	+	0.774436i
$584$		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
$592$	−2.55146e12	−	4.41926e12i	−0.853771	−	1.47878i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$	3.67133e12			1.19183
$597$		0			0
$598$		0			0
$599$	−4.06854e11	−	7.04692e11i	−0.129127	−	0.223655i	0.794211	−	0.607641i	$-0.207884\pi$
$599$	−4.06854e11	−	7.04692e11i	−0.129127	−	0.223655i	−0.923339	+	0.383987i	$0.874551\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	−5.16619e12			−1.59127
$604$	1.16056e12	−	2.01014e12i	0.354814	−	0.614555i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	3.49537e12	+	6.05416e12i	0.999819	+	1.73174i	0.516400	+	0.856348i	$0.327272\pi$
$613$	3.49537e12	+	6.05416e12i	0.999819	+	1.73174i	0.483419	+	0.875389i	$0.339395\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.98517e12			1.94041			0.970206	−	0.242283i	$-0.0778962\pi$
$617$	6.98517e12			1.94041			0.970206	+	0.242283i	$0.0778962\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.90735e12	+	3.30362e12i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.77314e12			0.445258			0.222629	−	0.974903i	$-0.428536\pi$
$631$	1.77314e12			0.445258			0.222629	+	0.974903i	$0.428536\pi$
$632$	1.50822e12	−	2.61232e12i	0.376044	−	0.651327i
$633$		0			0
$634$	8.18699e11	+	1.41803e12i	0.201244	+	0.348564i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.62437e12			0.388144
$639$	−4.06521e12	+	7.04114e12i	−0.964559	+	1.67066i
$640$		0			0
$641$	3.15748e12	+	5.46892e12i	0.738720	+	1.27950i	0.953072	+	0.302745i	$0.0979029\pi$
$641$	3.15748e12	+	5.46892e12i	0.738720	+	1.27950i	−0.214351	+	0.976757i	$0.568764\pi$
$642$		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	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	−9.67583e11	−	1.67590e12i	−0.215576	−	0.373389i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	8.51589e12			1.84551
$653$	1.40163e12	−	2.42770e12i	0.301665	−	0.522499i	−0.674848	−	0.737957i	$-0.735791\pi$
$653$	1.40163e12	−	2.42770e12i	0.301665	−	0.522499i	0.976513	+	0.215457i	$0.0691243\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.54300e12			0.318699			0.159349	−	0.987222i	$-0.449060\pi$
$659$	1.54300e12			0.318699			0.159349	+	0.987222i	$0.449060\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	2.77892e11	+	4.81323e11i	0.0562361	+	0.0974037i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	2.23830e12			0.440843
$667$	−5.28501e12	+	9.15390e12i	−1.03390	+	1.79077i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−7.13300e12			−1.34031			−0.670153	−	0.742223i	$-0.733772\pi$
$673$	−7.13300e12			−1.34031			−0.670153	+	0.742223i	$0.733772\pi$
$674$	8.63547e11	−	1.49571e12i	0.161182	−	0.279175i
$675$		0			0
$676$	−2.58220e12	−	4.47249e12i	−0.475586	−	0.823739i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3.47834e12	+	6.02466e12i	0.611616	+	1.05935i	0.990968	+	0.134097i	$0.0428134\pi$
$683$	3.47834e12	+	6.02466e12i	0.611616	+	1.05935i	−0.379353	+	0.925252i	$0.623853\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−4.16224e12	+	7.20922e12i	−0.708239	+	1.22671i
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$	−2.22498e12			−0.364090
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.26430e12			0.666986			0.333493	−	0.942753i	$-0.391773\pi$
$701$	4.26430e12			0.666986			0.333493	+	0.942753i	$0.391773\pi$
$702$		0			0
$703$		0			0
$704$	−2.06989e12	−	3.58516e12i	−0.317593	−	0.550087i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.71972e12	−	9.90685e12i	0.850094	−	1.47241i	−0.0310286	−	0.999518i	$-0.509878\pi$
$709$	5.71972e12	−	9.90685e12i	0.850094	−	1.47241i	0.881123	−	0.472888i	$-0.156788\pi$
$710$		0			0
$711$	−5.94321e12	−	1.02939e13i	−0.872184	−	1.51067i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	6.52736e12	−	1.13057e13i	0.928173	−	1.60764i
$717$		0			0
$718$	1.48161e12	+	2.56623e12i	0.208054	+	0.360359i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$	1.61344e12			0.220971
$723$		0			0
$724$		0			0
$725$	7.39397e12	+	1.28067e13i	0.993932	+	1.72154i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−7.62560e12			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$	−5.13643e12			−0.645225
$737$	5.63103e12	−	9.75323e12i	0.703046	−	1.21771i
$738$		0			0
$739$	−7.21032e12	−	1.24886e13i	−0.889312	−	1.54033i	−0.840690	−	0.541517i	$-0.817850\pi$
$739$	−7.21032e12	−	1.24886e13i	−0.889312	−	1.54033i	−0.0486226	−	0.998817i	$-0.515483\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.50329e13			−1.80964			−0.904821	−	0.425792i	$-0.859996\pi$
$743$	−1.50329e13			−1.80964			−0.904821	+	0.425792i	$0.859996\pi$
$744$		0			0
$745$		0			0
$746$	−1.37868e12	−	2.38795e12i	−0.162982	−	0.282293i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.54799e12	+	2.68121e12i	−0.177578	+	0.307575i	−0.941051	−	0.338266i	$-0.890160\pi$
$751$	−1.54799e12	+	2.68121e12i	−0.177578	+	0.307575i	0.763472	+	0.645841i	$0.223493\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	3.07428e12			0.340261			0.170131	−	0.985422i	$-0.445581\pi$
$757$	3.07428e12			0.340261			0.170131	+	0.985422i	$0.445581\pi$
$758$	−6.35384e11	+	1.10052e12i	−0.0699077	+	0.121084i
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$	−8.57165e12			−0.910216
$765$		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
$772$	5.57794e11	−	9.66128e11i	0.0565192	−	0.0978942i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	−1.82569e12	−	3.16218e12i	−0.182849	−	0.316703i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−3.46279e12			−0.338858
$779$		0			0
$780$		0			0
$781$	−8.86195e12	−	1.53494e13i	−0.852314	−	1.47625i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	9.71442e12	+	1.68259e13i	0.897530	+	1.55457i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	4.21857e12			0.380980
$793$		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
$800$	−3.59305e12	+	6.22335e12i	−0.310141	+	0.537179i
$801$		0			0
$802$	−7.62606e11	−	1.32087e12i	−0.0650902	−	0.112739i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.60660e12	+	6.24682e12i	0.296026	+	0.512732i	0.975223	−	0.221224i	$-0.0710051\pi$
$809$	3.60660e12	+	6.24682e12i	0.296026	+	0.512732i	−0.679197	+	0.733956i	$0.737672\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−2.43969e12	+	4.22567e12i	−0.194771	+	0.337354i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.12430e12	−	8.87555e12i	0.393632	−	0.681791i	−0.599293	−	0.800529i	$-0.704552\pi$
$821$	5.12430e12	−	8.87555e12i	0.393632	−	0.681791i	0.992926	+	0.118739i	$0.0378851\pi$
$822$		0			0
$823$	1.07696e13	+	1.86535e13i	0.818275	+	1.41729i	0.906952	+	0.421234i	$0.138403\pi$
$823$	1.07696e13	+	1.86535e13i	0.818275	+	1.41729i	−0.0886765	+	0.996060i	$0.528264\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.69005e13			1.99979			0.999896	−	0.0144001i	$-0.00458387\pi$
$827$	2.69005e13			1.99979			0.999896	+	0.0144001i	$0.00458387\pi$
$828$	−6.69096e12	+	1.15891e13i	−0.494711	+	0.856865i
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		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
$841$	4.28193e13			2.95160
$842$	3.46190e11	−	5.99619e11i	0.0237362	−	0.0411122i
$843$		0			0
$844$	8.93269e12	+	1.54719e13i	0.605956	+	1.04955i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.30433e13			−0.866177
$849$		0			0
$850$		0			0
$851$	−1.58754e13	−	2.74970e13i	−1.03763	−	1.79722i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−4.65132e12	+	8.05633e12i	−0.296104	+	0.512868i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$	−7.14857e12			−0.440998
$863$	−1.53537e13	+	2.65934e13i	−0.942246	+	1.63202i	−0.181071	+	0.983470i	$0.557956\pi$
$863$	−1.53537e13	+	2.65934e13i	−0.942246	+	1.63202i	−0.761175	+	0.648547i	$0.775377\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.59118e13			1.54138
$870$		0			0
$871$		0			0
$872$	−5.23569e11	−	9.06848e11i	−0.0306655	−	0.0531142i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.48671e12	+	1.12353e13i	−0.370277	+	0.641338i	−0.989608	−	0.143792i	$-0.954071\pi$
$877$	−6.48671e12	+	1.12353e13i	−0.370277	+	0.641338i	0.619331	+	0.785130i	$0.287404\pi$
$878$		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
$883$	−3.49235e13			−1.93328			−0.966639	−	0.256142i	$-0.917549\pi$
$883$	−3.49235e13			−1.93328			−0.966639	+	0.256142i	$0.917549\pi$
$884$		0			0
$885$		0			0
$886$	3.92045e12	+	6.79042e12i	0.213739	+	0.370207i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	8.31172e12	−	1.43963e13i	0.441816	−	0.765247i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.77506e12	−	3.07449e12i	0.0910896	−	0.157772i
$899$		0			0
$900$	9.36096e12	+	1.62137e13i	0.475586	+	0.823739i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.29843e13			0.646637
$905$		0			0
$906$		0			0
$907$	−1.43210e13	−	2.48047e13i	−0.702653	−	1.21703i	−0.967532	−	0.252749i	$-0.918665\pi$
$907$	−1.43210e13	−	2.48047e13i	−0.702653	−	1.21703i	0.264879	−	0.964282i	$-0.414668\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.76425e13			1.32967			0.664836	−	0.746990i	$-0.268502\pi$
$911$	2.76425e13			1.32967			0.664836	+	0.746990i	$0.268502\pi$
$912$		0			0
$913$		0			0
$914$	−4.44899e12	−	7.70588e12i	−0.210865	−	0.365228i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	2.01747e13	−	3.49435e13i	0.933010	−	1.61602i	0.154866	−	0.987935i	$-0.450505\pi$
$919$	2.01747e13	−	3.49435e13i	0.933010	−	1.61602i	0.778144	−	0.628086i	$-0.216161\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.44208e13			−1.99503
$926$	4.77968e12	−	8.27864e12i	0.213624	−	0.370007i
$927$		0			0
$928$	1.39287e13	+	2.41252e13i	0.616517	+	1.06784i
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$		0			0
$932$	−3.58774e13			−1.55758
$933$		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	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	7.95982e12			0.323142
$947$	4.01403e12	−	6.95250e12i	0.162183	−	0.280909i	−0.773468	−	0.633835i	$-0.781480\pi$
$947$	4.01403e12	−	6.95250e12i	0.162183	−	0.280909i	0.935651	+	0.352926i	$0.114813\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.09255e13			−1.99994			−0.999971	−	0.00763467i	$-0.997570\pi$
$953$	−5.09255e13			−1.99994			−0.999971	+	0.00763467i	$0.997570\pi$
$954$	2.86059e12	−	4.95469e12i	0.111812	−	0.193664i
$955$		0			0
$956$	−2.41563e13	−	4.18399e13i	−0.935340	−	1.62006i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.32198e13	−	2.28974e13i	0.500000	−	0.866025i
$962$		0			0
$963$	1.83287e13	+	3.17463e13i	0.686774	+	1.18953i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.35824e11			−0.00867299			−0.00433649	−	0.999991i	$-0.501380\pi$
$967$	−2.35824e11			−0.00867299			−0.00433649	+	0.999991i	$0.501380\pi$
$968$	1.29084e12	−	2.23579e12i	0.0472533	−	0.0818451i
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$		0			0
$974$	−1.11121e13			−0.395622
$975$		0			0
$976$		0			0
$977$	−2.60972e13	−	4.52018e13i	−0.916366	−	1.58719i	−0.804888	−	0.593426i	$-0.797775\pi$
$977$	−2.60972e13	−	4.52018e13i	−0.916366	−	1.58719i	−0.111478	−	0.993767i	$-0.535558\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−4.12629e12			−0.142249
$982$	3.05102e12	−	5.28452e12i	0.104699	−	0.181344i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.58978e13	+	4.48563e13i	−0.860754	+	1.49087i
$990$		0			0
$991$	−1.12106e13	−	1.94173e13i	−0.369230	−	0.639526i	0.620215	−	0.784432i	$-0.287045\pi$
$991$	−1.12106e13	−	1.94173e13i	−0.369230	−	0.639526i	−0.989445	+	0.144906i	$0.953712\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$	5.79594e12	+	1.00389e13i	0.184942	+	0.320330i
$999$		0			0

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.10.c.a.30.1		2
7.2	even	3		49.10.a.a.1.1	✓	1
7.3	odd	6	inner	49.10.c.a.18.1		2
7.4	even	3	inner	49.10.c.a.18.1		2
7.5	odd	6		49.10.a.a.1.1	✓	1
7.6	odd	2	CM	49.10.c.a.30.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.10.a.a.1.1	✓	1	7.2	even	3
49.10.a.a.1.1	✓	1	7.5	odd	6
49.10.c.a.18.1		2	7.3	odd	6	inner
49.10.c.a.18.1		2	7.4	even	3	inner
49.10.c.a.30.1		2	1.1	even	1	trivial
49.10.c.a.30.1		2	7.6	odd	2	CM

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	49.10.c.a.30.1

Level	$49$
Weight	$10$
Character	49.30
Analytic conductor	$25.237$
Analytic rank	$0$
Dimension	$2$
CM discriminant	-7
Inner twists	$4$