LMFDB - Embedded newform 21.5.h.a.2.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,5,Mod(2,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	21.h (of order $6$, degree $2$, 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:	$2.17076922476$
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:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			2.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.2
Dual form			21.5.h.a.11.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(35.5000 - 33.7750i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	$q+(-4.50000 - 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(35.5000 - 33.7750i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-72.0000 + 124.708i) q^{12} +191.000 q^{13} +(-128.000 + 221.703i) q^{16} +(300.500 - 520.481i) q^{19} +(-423.000 - 124.708i) q^{21} +(-312.500 - 541.266i) q^{25} +729.000 q^{27} +(-752.000 - 221.703i) q^{28} +(876.500 + 1518.14i) q^{31} +1296.00 q^{36} +(-1295.50 + 2243.87i) q^{37} +(-859.500 - 1488.70i) q^{39} +23.0000 q^{43} +2304.00 q^{48} +(119.500 - 2398.02i) q^{49} +(-1528.00 - 2646.57i) q^{52} -5409.00 q^{57} +(983.000 - 1702.61i) q^{61} +(931.500 + 3858.14i) q^{63} +4096.00 q^{64} +(4404.50 + 7628.82i) q^{67} +(624.500 + 1081.67i) q^{73} +(-2812.50 + 4871.39i) q^{75} -9616.00 q^{76} +(6180.50 - 10704.9i) q^{79} +(-3280.50 - 5681.99i) q^{81} +(1656.00 + 6858.92i) q^{84} +(6780.50 - 6451.02i) q^{91} +(7888.50 - 13663.3i) q^{93} -18814.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{3} - 16 q^{4} + 71 q^{7} - 81 q^{9}+O(q^{10}) $ 2 * q - 9 * q^3 - 16 * q^4 + 71 * q^7 - 81 * q^9	$ 2 q - 9 q^{3} - 16 q^{4} + 71 q^{7} - 81 q^{9} - 144 q^{12} + 382 q^{13} - 256 q^{16} + 601 q^{19} - 846 q^{21} - 625 q^{25} + 1458 q^{27} - 1504 q^{28} + 1753 q^{31} + 2592 q^{36} - 2591 q^{37} - 1719 q^{39} + 46 q^{43} + 4608 q^{48} + 239 q^{49} - 3056 q^{52} - 10818 q^{57} + 1966 q^{61} + 1863 q^{63} + 8192 q^{64} + 8809 q^{67} + 1249 q^{73} - 5625 q^{75} - 19232 q^{76} + 12361 q^{79} - 6561 q^{81} + 3312 q^{84} + 13561 q^{91} + 15777 q^{93} - 37628 q^{97}+O(q^{100}) $ 2 * q - 9 * q^3 - 16 * q^4 + 71 * q^7 - 81 * q^9 - 144 * q^12 + 382 * q^13 - 256 * q^16 + 601 * q^19 - 846 * q^21 - 625 * q^25 + 1458 * q^27 - 1504 * q^28 + 1753 * q^31 + 2592 * q^36 - 2591 * q^37 - 1719 * q^39 + 46 * q^43 + 4608 * q^48 + 239 * q^49 - 3056 * q^52 - 10818 * q^57 + 1966 * q^61 + 1863 * q^63 + 8192 * q^64 + 8809 * q^67 + 1249 * q^73 - 5625 * q^75 - 19232 * q^76 + 12361 * q^79 - 6561 * q^81 + 3312 * q^84 + 13561 * q^91 + 15777 * q^93 - 37628 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$3$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$4$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	35.5000	−	33.7750i	0.724490	−	0.689286i
$7$	35.5000	−	33.7750i	0.724490	−	0.689286i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−72.0000	+	124.708i	−0.500000	+	0.866025i
$13$	191.000			1.13018			0.565089	−	0.825030i	$-0.308842\pi$
$13$	191.000			1.13018			0.565089	+	0.825030i	$0.308842\pi$
$14$		0			0
$15$		0			0
$16$	−128.000	+	221.703i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	300.500	−	520.481i	0.832410	−	1.44178i	−0.0637119	−	0.997968i	$-0.520294\pi$
$19$	300.500	−	520.481i	0.832410	−	1.44178i	0.896122	−	0.443808i	$-0.146373\pi$
$20$		0			0
$21$	−423.000	−	124.708i	−0.959184	−	0.282784i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−312.500	−	541.266i	−0.500000	−	0.866025i
$26$		0			0
$27$	729.000			1.00000
$28$	−752.000	−	221.703i	−0.959184	−	0.282784i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	876.500	+	1518.14i	0.912071	+	1.57975i	0.811134	+	0.584860i	$0.198851\pi$
$31$	876.500	+	1518.14i	0.912071	+	1.57975i	0.100937	+	0.994893i	$0.467816\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1296.00			1.00000
$37$	−1295.50	+	2243.87i	−0.946311	+	1.63906i	−0.193207	+	0.981158i	$0.561889\pi$
$37$	−1295.50	+	2243.87i	−0.946311	+	1.63906i	−0.753104	+	0.657901i	$0.771445\pi$
$38$		0			0
$39$	−859.500	−	1488.70i	−0.565089	−	0.978762i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	23.0000			0.0124392			0.00621958	−	0.999981i	$-0.498020\pi$
$43$	23.0000			0.0124392			0.00621958	+	0.999981i	$0.498020\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	2304.00			1.00000
$49$	119.500	−	2398.02i	0.0497709	−	0.998761i
$50$		0			0
$51$		0			0
$52$	−1528.00	−	2646.57i	−0.565089	−	0.978762i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5409.00			−1.66482
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	983.000	−	1702.61i	0.264176	−	0.457567i	−0.703171	−	0.711021i	$-0.748233\pi$
$61$	983.000	−	1702.61i	0.264176	−	0.457567i	0.967347	+	0.253454i	$0.0815666\pi$
$62$		0			0
$63$	931.500	+	3858.14i	0.234694	+	0.972069i
$64$	4096.00			1.00000
$65$		0			0
$66$		0			0
$67$	4404.50	+	7628.82i	0.981176	+	1.69945i	0.657830	+	0.753166i	$0.271474\pi$
$67$	4404.50	+	7628.82i	0.981176	+	1.69945i	0.323346	+	0.946281i	$0.395192\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	624.500	+	1081.67i	0.117189	+	0.202977i	0.918653	−	0.395066i	$-0.129278\pi$
$73$	624.500	+	1081.67i	0.117189	+	0.202977i	−0.801464	+	0.598043i	$0.795945\pi$
$74$		0			0
$75$	−2812.50	+	4871.39i	−0.500000	+	0.866025i
$76$	−9616.00			−1.66482
$77$		0			0
$78$		0			0
$79$	6180.50	−	10704.9i	0.990306	−	1.71526i	0.374860	−	0.927082i	$-0.377691\pi$
$79$	6180.50	−	10704.9i	0.990306	−	1.71526i	0.615446	−	0.788179i	$-0.288976\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	1656.00	+	6858.92i	0.234694	+	0.972069i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	6780.50	−	6451.02i	0.818802	−	0.779015i
$92$		0			0
$93$	7888.50	−	13663.3i	0.912071	−	1.57975i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18814.0			−1.99957			−0.999787	−	0.0206175i	$-0.993437\pi$
$97$	−18814.0			−1.99957			−0.999787	+	0.0206175i	$0.993437\pi$
$98$		0			0
$99$		0			0
$100$	−5000.00	+	8660.25i	−0.500000	+	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−1715.50	+	2971.33i	−0.161702	+	0.280077i	−0.935479	−	0.353381i	$-0.885032\pi$
$103$	−1715.50	+	2971.33i	−0.161702	+	0.280077i	0.773777	+	0.633458i	$0.218365\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−5832.00	−	10101.3i	−0.500000	−	0.866025i
$109$	9360.50	+	16212.9i	0.787855	+	1.36460i	0.927279	+	0.374371i	$0.122141\pi$
$109$	9360.50	+	16212.9i	0.787855	+	1.36460i	−0.139424	+	0.990233i	$0.544525\pi$
$110$		0			0
$111$	23319.0			1.89262
$112$	2944.00	+	12193.6i	0.234694	+	0.972069i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7735.50	+	13398.3i	−0.565089	+	0.978762i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7320.50	+	12679.5i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	14024.0	−	24290.3i	0.912071	−	1.57975i
$125$		0			0
$126$		0			0
$127$	−20809.0			−1.29016			−0.645080	−	0.764115i	$-0.723176\pi$
$127$	−20809.0			−1.29016			−0.645080	+	0.764115i	$0.723176\pi$
$128$		0			0
$129$	−103.500	−	179.267i	−0.00621958	−	0.0107726i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−6911.50	−	28626.5i	−0.390723	−	1.61832i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	13799.0			0.714197			0.357098	−	0.934067i	$-0.383766\pi$
$139$	13799.0			0.714197			0.357098	+	0.934067i	$0.383766\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−10368.0	−	17957.9i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−19228.5	+	9859.70i	−0.889838	+	0.456277i
$148$	41456.0			1.89262
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−18097.0	−	31344.9i	−0.793693	−	1.37472i	−0.923666	−	0.383199i	$-0.874822\pi$
$151$	−18097.0	−	31344.9i	−0.793693	−	1.37472i	0.129972	−	0.991518i	$-0.458511\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−13752.0	+	23819.2i	−0.565089	+	0.978762i
$157$	17687.0	+	30634.8i	0.717554	+	1.24284i	0.961966	+	0.273169i	$0.0880719\pi$
$157$	17687.0	+	30634.8i	0.717554	+	1.24284i	−0.244412	+	0.969672i	$0.578595\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−7753.00	+	13428.6i	−0.291806	+	0.505423i	−0.974237	−	0.225527i	$-0.927590\pi$
$163$	−7753.00	+	13428.6i	−0.291806	+	0.505423i	0.682431	+	0.730950i	$0.260923\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	7920.00			0.277301
$170$		0			0
$171$	24340.5	+	42159.0i	0.832410	+	1.44178i
$172$	−184.000	−	318.697i	−0.00621958	−	0.0107726i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	−29375.0	−	8660.25i	−0.959184	−	0.282784i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	32447.0			0.990415			0.495208	−	0.868775i	$-0.335092\pi$
$181$	32447.0			0.990415			0.495208	+	0.868775i	$0.335092\pi$
$182$		0			0
$183$	−17694.0			−0.528353
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	25879.5	−	24622.0i	0.724490	−	0.689286i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−18432.0	−	31925.2i	−0.500000	−	0.866025i
$193$	8688.50	+	15048.9i	0.233255	+	0.404009i	0.958764	−	0.284203i	$-0.0917291\pi$
$193$	8688.50	+	15048.9i	0.233255	+	0.404009i	−0.725509	+	0.688212i	$0.758396\pi$
$194$		0			0
$195$		0			0
$196$	−34184.0	+	17528.4i	−0.889838	+	0.456277i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−34897.0	−	60443.4i	−0.881215	−	1.52631i	−0.849991	−	0.526797i	$-0.823393\pi$
$199$	−34897.0	−	60443.4i	−0.881215	−	1.52631i	−0.0312240	−	0.999512i	$-0.509941\pi$
$200$		0			0
$201$	39640.5	−	68659.4i	0.981176	−	1.69945i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−24448.0	+	42345.2i	−0.565089	+	0.978762i
$209$		0			0
$210$		0			0
$211$	−61486.0			−1.38106			−0.690528	−	0.723306i	$-0.742622\pi$
$211$	−61486.0			−1.38106			−0.690528	+	0.723306i	$0.742622\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	82391.0	+	24290.3i	1.74969	+	0.515838i
$218$		0			0
$219$	5620.50	−	9734.99i	0.117189	−	0.202977i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	14786.0			0.297332			0.148666	−	0.988888i	$-0.452502\pi$
$223$	14786.0			0.297332			0.148666	+	0.988888i	$0.452502\pi$
$224$		0			0
$225$	50625.0			1.00000
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	43272.0	+	74949.3i	0.832410	+	1.44178i
$229$	−31199.5	+	54039.1i	−0.594945	+	1.03047i	0.398610	+	0.917121i	$0.369493\pi$
$229$	−31199.5	+	54039.1i	−0.594945	+	1.03047i	−0.993555	+	0.113354i	$0.963841\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−111249.			−1.98061
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	17183.0	+	29761.8i	0.295845	+	0.512419i	0.975181	−	0.221408i	$-0.0710653\pi$
$241$	17183.0	+	29761.8i	0.295845	+	0.512419i	−0.679336	+	0.733828i	$0.737732\pi$
$242$		0			0
$243$	−29524.5	+	51137.9i	−0.500000	+	0.866025i
$244$	−31456.0			−0.528353
$245$		0			0
$246$		0			0
$247$	57395.5	−	99411.9i	0.940771	−	1.62946i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	46008.0	−	43772.4i	0.724490	−	0.689286i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−32768.0	−	56755.8i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	29796.5	+	123413.i	0.444187	+	1.83976i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	70472.0	−	122061.i	0.981176	−	1.69945i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	44159.0	−	76485.6i	0.601285	−	1.04146i	−0.391341	−	0.920246i	$-0.627989\pi$
$271$	44159.0	−	76485.6i	0.601285	−	1.04146i	0.992627	−	0.121211i	$-0.0386778\pi$
$272$		0			0
$273$	−80793.0	−	23819.2i	−1.08405	−	0.319596i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6095.50	−	10557.7i	−0.0794419	−	0.137597i	0.823567	−	0.567218i	$-0.191980\pi$
$277$	−6095.50	−	10557.7i	−0.0794419	−	0.137597i	−0.903009	+	0.429621i	$0.858647\pi$
$278$		0			0
$279$	−141993.			−1.82414
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−53555.5	−	92760.8i	−0.668700	−	1.15822i	−0.978268	−	0.207345i	$-0.933518\pi$
$283$	−53555.5	−	92760.8i	−0.668700	−	1.15822i	0.309568	−	0.950877i	$-0.399816\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−41760.5	+	72331.3i	−0.500000	+	0.866025i
$290$		0			0
$291$	84663.0	+	146641.i	0.999787	+	1.73168i
$292$	9992.00	−	17306.7i	0.117189	−	0.202977i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	90000.0			1.00000
$301$	816.500	−	776.825i	0.00901204	−	0.00857413i
$302$		0			0
$303$		0			0
$304$	76928.0	+	133243.i	0.832410	+	1.44178i
$305$		0			0
$306$		0			0
$307$	184823.			1.96101			0.980504	−	0.196500i	$-0.0629577\pi$
$307$	184823.			1.96101			0.980504	+	0.196500i	$0.0629577\pi$
$308$		0			0
$309$	30879.0			0.323405
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−81431.5	+	141043.i	−0.831197	+	1.43967i	0.0658933	+	0.997827i	$0.479010\pi$
$313$	−81431.5	+	141043.i	−0.831197	+	1.43967i	−0.897090	+	0.441848i	$0.854323\pi$
$314$		0			0
$315$		0			0
$316$	−197776.			−1.98061
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−52488.0	+	90911.9i	−0.500000	+	0.866025i
$325$	−59687.5	−	103382.i	−0.565089	−	0.978762i
$326$		0			0
$327$	84244.5	−	145916.i	0.787855	−	1.36460i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−108900.	+	188619.i	−0.993962	+	1.72159i	−0.401959	+	0.915658i	$0.631670\pi$
$331$	−108900.	+	188619.i	−0.993962	+	1.72159i	−0.592004	+	0.805935i	$0.701663\pi$
$332$		0			0
$333$	−104936.	−	181754.i	−0.946311	−	1.63906i
$334$		0			0
$335$		0			0
$336$	81792.0	−	77817.6i	0.724490	−	0.689286i
$337$	194063.			1.70877			0.854384	−	0.519643i	$-0.173935\pi$
$337$	194063.			1.70877			0.854384	+	0.519643i	$0.173935\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−76751.0	−	89166.0i	−0.652373	−	0.757898i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	8402.00			0.0689814			0.0344907	−	0.999405i	$-0.489019\pi$
$349$	8402.00			0.0689814			0.0344907	+	0.999405i	$0.489019\pi$
$350$		0			0
$351$	139239.			1.13018
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−115440.	−	199948.i	−0.885813	−	1.53427i
$362$		0			0
$363$	131769.			1.00000
$364$	−143632.	−	42345.2i	−1.08405	−	0.319596i
$365$		0			0
$366$		0			0
$367$	14148.5	+	24505.9i	0.105046	+	0.181944i	0.913757	−	0.406261i	$-0.133168\pi$
$367$	14148.5	+	24505.9i	0.105046	+	0.181944i	−0.808711	+	0.588206i	$0.799834\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−252432.			−1.82414
$373$	−27335.5	+	47346.5i	−0.196476	+	0.340306i	−0.947383	−	0.320101i	$-0.896283\pi$
$373$	−27335.5	+	47346.5i	−0.196476	+	0.340306i	0.750907	+	0.660407i	$0.229616\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	86039.0			0.598986			0.299493	−	0.954098i	$-0.403182\pi$
$379$	86039.0			0.598986			0.299493	+	0.954098i	$0.403182\pi$
$380$		0			0
$381$	93640.5	+	162190.i	0.645080	+	1.11731i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−931.500	+	1613.41i	−0.00621958	+	0.0107726i
$388$	150512.	+	260694.i	0.999787	+	1.73168i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	64728.5	−	112113.i	0.410690	−	0.711337i	−0.584275	−	0.811556i	$-0.698621\pi$
$397$	64728.5	−	112113.i	0.410690	−	0.711337i	0.994965	+	0.100219i	$0.0319544\pi$
$398$		0			0
$399$	−192020.	+	182689.i	−1.20615	+	1.14754i
$400$	160000.			1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	167412.	+	289965.i	1.03080	+	1.78540i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−128400.	−	222394.i	−0.767568	−	1.32947i	−0.938878	−	0.344249i	$-0.888133\pi$
$409$	−128400.	−	222394.i	−0.767568	−	1.32947i	0.171311	−	0.985217i	$-0.445200\pi$
$410$		0			0
$411$		0			0
$412$	54896.0			0.323405
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−62095.5	−	107553.i	−0.357098	−	0.618513i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−123121.			−0.694653			−0.347327	−	0.937744i	$-0.612910\pi$
$421$	−123121.			−0.694653			−0.347327	+	0.937744i	$0.612910\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−22609.0	−	93643.3i	−0.124001	−	0.513595i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−93312.0	+	161621.i	−0.500000	+	0.866025i
$433$	−246097.			−1.31259			−0.656297	−	0.754503i	$-0.727878\pi$
$433$	−246097.			−1.31259			−0.656297	+	0.754503i	$0.727878\pi$
$434$		0			0
$435$		0			0
$436$	149768.	−	259406.i	0.787855	−	1.36460i
$437$		0			0
$438$		0			0
$439$	188303.	−	326150.i	0.977076	−	1.69234i	0.304168	−	0.952619i	$-0.401622\pi$
$439$	188303.	−	326150.i	0.977076	−	1.69234i	0.672908	−	0.739726i	$-0.265045\pi$
$440$		0			0
$441$	163377.	+	105503.i	0.840067	+	0.542483i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−186552.	−	323118.i	−0.946311	−	1.63906i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	145408.	−	138342.i	0.724490	−	0.689286i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−162873.	+	282104.i	−0.793693	+	1.37472i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−85295.5	+	147736.i	−0.408408	+	0.707383i	−0.994711	−	0.102709i	$-0.967249\pi$
$457$	−85295.5	+	147736.i	−0.408408	+	0.707383i	0.586304	+	0.810091i	$0.300582\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−271129.			−1.26478			−0.632389	−	0.774651i	$-0.717925\pi$
$463$	−271129.			−1.26478			−0.632389	+	0.774651i	$0.717925\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	247536.			1.13018
$469$	414023.	+	122061.i	1.88226	+	0.554921i
$470$		0			0
$471$	159183.	−	275713.i	0.717554	−	1.24284i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−375625.			−1.66482
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−247440.	+	428580.i	−1.06950	+	1.85243i
$482$		0			0
$483$		0			0
$484$	234256.			1.00000
$485$		0			0
$486$		0			0
$487$	176268.	+	305306.i	0.743219	+	1.28729i	0.951022	+	0.309122i	$0.100035\pi$
$487$	176268.	+	305306.i	0.743219	+	1.28729i	−0.207803	+	0.978171i	$0.566631\pi$
$488$		0			0
$489$	139554.			0.583612
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−448768.			−1.82414
$497$		0			0
$498$		0			0
$499$	−113100.	+	195894.i	−0.454213	+	0.786720i	−0.998643	−	0.0520865i	$-0.983413\pi$
$499$	−113100.	+	195894.i	−0.454213	+	0.786720i	0.544430	+	0.838807i	$0.316746\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−35640.0	−	61730.3i	−0.138651	−	0.240150i
$508$	166472.	+	288338.i	0.645080	+	1.11731i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	58703.0	+	17306.7i	0.224811	+	0.0662783i
$512$		0			0
$513$	219064.	−	379431.i	0.832410	−	1.44178i
$514$		0			0
$515$		0			0
$516$	−1656.00	+	2868.28i	−0.00621958	+	0.0107726i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	257508.	−	446018.i	0.941430	−	1.63061i	0.178685	−	0.983906i	$-0.442816\pi$
$523$	257508.	−	446018.i	0.941430	−	1.63061i	0.762745	−	0.646699i	$-0.223851\pi$
$524$		0			0
$525$	64687.5	+	267927.i	0.234694	+	0.972069i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−139920.	−	242349.i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−341368.	+	324780.i	−1.20615	+	1.14754i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	263640.	−	456639.i	0.900778	−	1.56019i	0.0742908	−	0.997237i	$-0.476331\pi$
$541$	263640.	−	456639.i	0.900778	−	1.56019i	0.826487	−	0.562956i	$-0.190336\pi$
$542$		0			0
$543$	−146012.	−	252899.i	−0.495208	−	0.857725i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−342382.			−1.14429			−0.572145	−	0.820152i	$-0.693889\pi$
$547$	−342382.			−1.14429			−0.572145	+	0.820152i	$0.693889\pi$
$548$		0			0
$549$	79623.0	+	137911.i	0.264176	+	0.457567i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−142152.	−	588772.i	−0.464838	−	1.92529i
$554$		0			0
$555$		0			0
$556$	−110392.	−	191205.i	−0.357098	−	0.618513i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	4393.00			0.0140585
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−308367.	−	90911.9i	−0.959184	−	0.282784i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	309660.	+	536348.i	0.949759	+	1.64503i	0.745929	+	0.666025i	$0.232006\pi$
$571$	309660.	+	536348.i	0.949759	+	1.64503i	0.203830	+	0.979006i	$0.434661\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−165888.	+	287326.i	−0.500000	+	0.866025i
$577$	330408.	+	572284.i	0.992429	+	1.71894i	0.602577	+	0.798060i	$0.294140\pi$
$577$	330408.	+	572284.i	0.992429	+	1.71894i	0.389852	+	0.920878i	$0.372526\pi$
$578$		0			0
$579$	78196.5	−	135440.i	0.233255	−	0.404009i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	290448.	+	187560.i	0.840067	+	0.542483i
$589$	1.05355e6			3.03687
$590$		0			0
$591$		0			0
$592$	−331648.	−	574431.i	−0.946311	−	1.63906i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−314073.	+	543990.i	−0.881215	+	1.52631i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	715199.			1.98006			0.990029	−	0.140863i	$-0.0449877\pi$
$601$	715199.			1.98006			0.990029	+	0.140863i	$0.0449877\pi$
$602$		0			0
$603$	−713529.			−1.96235
$604$	−289552.	+	501519.i	−0.793693	+	1.37472i
$605$		0			0
$606$		0			0
$607$	−336036.	+	582031.i	−0.912027	+	1.57968i	−0.100831	+	0.994904i	$0.532150\pi$
$607$	−336036.	+	582031.i	−0.912027	+	1.57968i	−0.811196	+	0.584774i	$0.801183\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−258169.	−	447162.i	−0.687042	−	1.18999i	−0.972790	−	0.231687i	$-0.925576\pi$
$613$	−258169.	−	447162.i	−0.687042	−	1.18999i	0.285749	−	0.958305i	$-0.407758\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−293124.	−	507705.i	−0.765014	−	1.32504i	−0.940239	−	0.340515i	$-0.889399\pi$
$619$	−293124.	−	507705.i	−0.765014	−	1.32504i	0.175225	−	0.984528i	$-0.443935\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	440064.			1.13018
$625$	−195312.	+	338291.i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	282992.	−	490157.i	0.717554	−	1.24284i
$629$		0			0
$630$		0			0
$631$	−342046.			−0.859065			−0.429532	−	0.903052i	$-0.641322\pi$
$631$	−342046.			−0.859065			−0.429532	+	0.903052i	$0.641322\pi$
$632$		0			0
$633$	276687.	+	479236.i	0.690528	+	1.19603i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22824.5	−	458023.i	0.0562500	−	1.12878i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	703271.			1.70099			0.850493	−	0.525986i	$-0.176304\pi$
$643$	703271.			1.70099			0.850493	+	0.525986i	$0.176304\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−181436.	−	751481.i	−0.428115	−	1.77319i
$652$	248096.			0.583612
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−101169.			−0.234378
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−145200.	−	251493.i	−0.332324	−	0.575603i	0.650643	−	0.759384i	$-0.274500\pi$
$661$	−145200.	−	251493.i	−0.332324	−	0.575603i	−0.982967	+	0.183781i	$0.941166\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−66537.0	−	115245.i	−0.148666	−	0.257497i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−905329.			−1.99883			−0.999416	−	0.0341703i	$-0.989121\pi$
$673$	−905329.			−1.99883			−0.999416	+	0.0341703i	$0.989121\pi$
$674$		0			0
$675$	−227812.	−	394583.i	−0.500000	−	0.866025i
$676$	−63360.0	−	109743.i	−0.138651	−	0.240150i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−667897.	+	635443.i	−1.44867	+	1.37828i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	389448.	−	674544.i	0.832410	−	1.44178i
$685$		0			0
$686$		0			0
$687$	561591.			1.18989
$688$	−2944.00	+	5099.16i	−0.00621958	+	0.0107726i
$689$		0			0
$690$		0			0
$691$	−41699.5	+	72225.7i	−0.0873323	+	0.151264i	−0.906383	−	0.422458i	$-0.861168\pi$
$691$	−41699.5	+	72225.7i	−0.0873323	+	0.151264i	0.819050	+	0.573722i	$0.194501\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	115000.	+	476314.i	0.234694	+	0.972069i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	778596.	+	1.34857e6i	1.57544	+	2.72874i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	66503.0	−	115187.i	0.132297	−	0.229144i	−0.792265	−	0.610177i	$-0.791098\pi$
$709$	66503.0	−	115187.i	0.132297	−	0.229144i	0.924562	+	0.381033i	$0.124432\pi$
$710$		0			0
$711$	500620.	+	867100.i	0.990306	+	1.71526i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	39456.5	+	163423.i	0.0759011	+	0.314372i
$722$		0			0
$723$	154647.	−	267856.i	0.295845	−	0.512419i
$724$	−259576.	−	449599.i	−0.495208	−	0.857725i
$725$		0			0
$726$		0			0
$727$	−160249.			−0.303198			−0.151599	−	0.988442i	$-0.548442\pi$
$727$	−160249.			−0.303198			−0.151599	+	0.988442i	$0.548442\pi$
$728$		0			0
$729$	531441.			1.00000
$730$		0			0
$731$		0			0
$732$	141552.	+	245175.i	0.264176	+	0.457567i
$733$	−466656.	+	808271.i	−0.868537	+	1.50435i	−0.00504570	+	0.999987i	$0.501606\pi$
$733$	−466656.	+	808271.i	−0.868537	+	1.50435i	−0.863492	+	0.504363i	$0.831727\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−339660.	−	588308.i	−0.621949	−	1.07725i	−0.989122	−	0.147095i	$-0.953008\pi$
$739$	−339660.	−	588308.i	−0.621949	−	1.07725i	0.367173	−	0.930153i	$-0.380326\pi$
$740$		0			0
$741$	−1.03312e6			−1.88154
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	547980.	−	949130.i	0.971595	−	1.68285i	0.280852	−	0.959751i	$-0.409383\pi$
$751$	547980.	−	949130.i	0.971595	−	1.68285i	0.690743	−	0.723101i	$-0.257284\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−548208.	−	161621.i	−0.959184	−	0.282784i
$757$	−443854.			−0.774548			−0.387274	−	0.921965i	$-0.626583\pi$
$757$	−443854.			−0.774548			−0.387274	+	0.921965i	$0.626583\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	879887.	+	259406.i	1.51139	+	0.445585i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−294912.	+	510803.i	−0.500000	+	0.866025i
$769$	−437953.			−0.740585			−0.370292	−	0.928915i	$-0.620743\pi$
$769$	−437953.			−0.740585			−0.370292	+	0.928915i	$0.620743\pi$
$770$		0			0
$771$		0			0
$772$	139016.	−	240783.i	0.233255	−	0.404009i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	547812.	−	948839.i	0.912071	−	1.57975i
$776$		0			0
$777$	827824.	−	787599.i	1.37119	−	1.30456i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	516352.	+	333441.i	0.840067	+	0.542483i
$785$		0			0
$786$		0			0
$787$	−600553.	−	1.04019e6i	−0.969621	−	1.67943i	−0.696652	−	0.717409i	$-0.745328\pi$
$787$	−600553.	−	1.04019e6i	−0.969621	−	1.67943i	−0.272969	−	0.962023i	$-0.588006\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	187753.	−	325198.i	0.298566	−	0.517132i
$794$		0			0
$795$		0			0
$796$	−558352.	+	967094.i	−0.881215	+	1.52631i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.26850e6			−1.96235
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	976754.			1.48506			0.742529	−	0.669814i	$-0.233626\pi$
$811$	976754.			1.48506			0.742529	+	0.669814i	$0.233626\pi$
$812$		0			0
$813$	−794862.			−1.20257
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6911.50	−	11971.1i	0.0103545	−	0.0179345i
$818$		0			0
$819$	177916.	+	736905.i	0.265246	+	1.09861i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	117647.	+	203771.i	0.173693	+	0.300844i	0.939708	−	0.341978i	$-0.111097\pi$
$823$	117647.	+	203771.i	0.173693	+	0.300844i	−0.766015	+	0.642822i	$0.777763\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−607200.	−	1.05170e6i	−0.883532	−	1.53032i	−0.847387	−	0.530976i	$-0.821825\pi$
$829$	−607200.	−	1.05170e6i	−0.883532	−	1.53032i	−0.0361453	−	0.999347i	$-0.511508\pi$
$830$		0			0
$831$	−54859.5	+	95019.4i	−0.0794419	+	0.137597i
$832$	782336.			1.13018
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	638968.	+	1.10673e6i	0.912071	+	1.57975i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	707281.			1.00000
$842$		0			0
$843$		0			0
$844$	491888.	+	851975.i	0.690528	+	1.19603i
$845$		0			0
$846$		0			0
$847$	168372.	+	697371.i	0.234694	+	0.972069i
$848$		0			0
$849$	−482000.	+	834848.i	−0.668700	+	1.15822i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.22386e6			−1.68203			−0.841013	−	0.541015i	$-0.818040\pi$
$853$	−1.22386e6			−1.68203			−0.841013	+	0.541015i	$0.818040\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−267481.	+	463291.i	−0.362499	+	0.627866i	−0.988371	−	0.152059i	$-0.951410\pi$
$859$	−267481.	+	463291.i	−0.362499	+	0.627866i	0.625873	+	0.779925i	$0.284743\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	751689.			1.00000
$868$	−322552.	−	1.33597e6i	−0.428115	−	1.77319i
$869$		0			0
$870$		0			0
$871$	841260.	+	1.45710e6i	1.10890	+	1.92068i
$872$		0			0
$873$	761967.	−	1.31977e6i	0.999787	−	1.73168i
$874$		0			0
$875$		0			0
$876$	−179856.			−0.234378
$877$	590327.	−	1.02248e6i	0.767527	−	1.32940i	−0.171374	−	0.985206i	$-0.554821\pi$
$877$	590327.	−	1.02248e6i	0.767527	−	1.32940i	0.938900	−	0.344189i	$-0.111846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1.36313e6			−1.74830			−0.874149	−	0.485658i	$-0.838580\pi$
$883$	−1.36313e6			−1.74830			−0.874149	+	0.485658i	$0.838580\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−738720.	+	702824.i	−0.934708	+	0.889289i
$890$		0			0
$891$		0			0
$892$	−118288.	−	204881.i	−0.148666	−	0.257497i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−405000.	−	701481.i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−9729.00	−	2868.28i	−0.0119314	−	0.00351759i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−566796.	−	981719.i	−0.688988	−	1.19336i	−0.972166	−	0.234295i	$-0.924722\pi$
$907$	−566796.	−	981719.i	−0.688988	−	1.19336i	0.283177	−	0.959068i	$-0.408612\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	692352.	−	1.19919e6i	0.832410	−	1.44178i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	998384.			1.18989
$917$		0			0
$918$		0			0
$919$	−842724.	+	1.45964e6i	−0.997824	+	1.72828i	−0.441816	+	0.897106i	$0.645666\pi$
$919$	−842724.	+	1.45964e6i	−0.997824	+	1.72828i	−0.556008	+	0.831177i	$0.687668\pi$
$920$		0			0
$921$	−831704.	−	1.44055e6i	−0.980504	−	1.69828i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.61938e6			1.89262
$926$		0			0
$927$	−138956.	−	240678.i	−0.161702	−	0.280077i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	−1.21222e6	−	782804.i	−1.39856	−	0.903137i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.33474e6			−1.52026			−0.760128	−	0.649774i	$-0.774864\pi$
$937$	−1.33474e6			−1.52026			−0.760128	+	0.649774i	$0.774864\pi$
$938$		0			0
$939$	1.46577e6			1.66239
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	889992.	+	1.54151e6i	0.990306	+	1.71526i
$949$	119280.	+	206598.i	0.132444	+	0.229400i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.07474e6	+	1.86151e6i	−1.16375	+	2.01567i
$962$		0			0
$963$		0			0
$964$	274928.	−	476189.i	0.295845	−	0.512419i
$965$		0			0
$966$		0			0
$967$	1.32319e6			1.41504			0.707521	−	0.706692i	$-0.249813\pi$
$967$	1.32319e6			1.41504			0.707521	+	0.706692i	$0.249813\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	944784.			1.00000
$973$	489864.	−	466061.i	0.517428	−	0.492286i
$974$		0			0
$975$	−537188.	+	930436.i	−0.565089	+	0.978762i
$976$	251648.	+	435867.i	0.264176	+	0.457567i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.51640e6			−1.57571
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.83666e6			−1.88154
$989$		0			0
$990$		0			0
$991$	−451044.	−	781230.i	−0.459273	−	0.795485i	0.539649	−	0.841890i	$-0.318557\pi$
$991$	−451044.	−	781230.i	−0.459273	−	0.795485i	−0.998923	+	0.0464053i	$0.985223\pi$
$992$		0			0
$993$	1.96019e6			1.98792
$994$		0			0
$995$		0			0
$996$		0			0
$997$	911184.	+	1.57822e6i	0.916676	+	1.58773i	0.804428	+	0.594050i	$0.202472\pi$
$997$	911184.	+	1.57822e6i	0.916676	+	1.58773i	0.112248	+	0.993680i	$0.464195\pi$
$998$		0			0
$999$	−944420.	+	1.63578e6i	−0.946311	+	1.63906i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.h.a.2.1	✓	2
3.2	odd	2	CM	21.5.h.a.2.1	✓	2
7.2	even	3		147.5.b.b.50.1		1
7.3	odd	6		147.5.h.a.116.1		2
7.4	even	3	inner	21.5.h.a.11.1	yes	2
7.5	odd	6		147.5.b.a.50.1		1
7.6	odd	2		147.5.h.a.128.1		2
21.2	odd	6		147.5.b.b.50.1		1
21.5	even	6		147.5.b.a.50.1		1
21.11	odd	6	inner	21.5.h.a.11.1	yes	2
21.17	even	6		147.5.h.a.116.1		2
21.20	even	2		147.5.h.a.128.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.h.a.2.1	✓	2	1.1	even	1	trivial
21.5.h.a.2.1	✓	2	3.2	odd	2	CM
21.5.h.a.11.1	yes	2	7.4	even	3	inner
21.5.h.a.11.1	yes	2	21.11	odd	6	inner
147.5.b.a.50.1		1	7.5	odd	6
147.5.b.a.50.1		1	21.5	even	6
147.5.b.b.50.1		1	7.2	even	3
147.5.b.b.50.1		1	21.2	odd	6
147.5.h.a.116.1		2	7.3	odd	6
147.5.h.a.116.1		2	21.17	even	6
147.5.h.a.128.1		2	7.6	odd	2
147.5.h.a.128.1		2	21.20	even	2

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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(35.5000 - 33.7750i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 - 7.79423i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(35.5000 - 33.7750i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-72.0000 + 124.708i) q^{12} +191.000 q^{13} +(-128.000 + 221.703i) q^{16} +(300.500 - 520.481i) q^{19} +(-423.000 - 124.708i) q^{21} +(-312.500 - 541.266i) q^{25} +729.000 q^{27} +(-752.000 - 221.703i) q^{28} +(876.500 + 1518.14i) q^{31} +1296.00 q^{36} +(-1295.50 + 2243.87i) q^{37} +(-859.500 - 1488.70i) q^{39} +23.0000 q^{43} +2304.00 q^{48} +(119.500 - 2398.02i) q^{49} +(-1528.00 - 2646.57i) q^{52} -5409.00 q^{57} +(983.000 - 1702.61i) q^{61} +(931.500 + 3858.14i) q^{63} +4096.00 q^{64} +(4404.50 + 7628.82i) q^{67} +(624.500 + 1081.67i) q^{73} +(-2812.50 + 4871.39i) q^{75} -9616.00 q^{76} +(6180.50 - 10704.9i) q^{79} +(-3280.50 - 5681.99i) q^{81} +(1656.00 + 6858.92i) q^{84} +(6780.50 - 6451.02i) q^{91} +(7888.50 - 13663.3i) q^{93} -18814.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{3} - 16 q^{4} + 71 q^{7} - 81 q^{9}+O(q^{10}) \) 2 * q - 9 * q^3 - 16 * q^4 + 71 * q^7 - 81 * q^9	\( 2 q - 9 q^{3} - 16 q^{4} + 71 q^{7} - 81 q^{9} - 144 q^{12} + 382 q^{13} - 256 q^{16} + 601 q^{19} - 846 q^{21} - 625 q^{25} + 1458 q^{27} - 1504 q^{28} + 1753 q^{31} + 2592 q^{36} - 2591 q^{37} - 1719 q^{39} + 46 q^{43} + 4608 q^{48} + 239 q^{49} - 3056 q^{52} - 10818 q^{57} + 1966 q^{61} + 1863 q^{63} + 8192 q^{64} + 8809 q^{67} + 1249 q^{73} - 5625 q^{75} - 19232 q^{76} + 12361 q^{79} - 6561 q^{81} + 3312 q^{84} + 13561 q^{91} + 15777 q^{93} - 37628 q^{97}+O(q^{100}) \) 2 * q - 9 * q^3 - 16 * q^4 + 71 * q^7 - 81 * q^9 - 144 * q^12 + 382 * q^13 - 256 * q^16 + 601 * q^19 - 846 * q^21 - 625 * q^25 + 1458 * q^27 - 1504 * q^28 + 1753 * q^31 + 2592 * q^36 - 2591 * q^37 - 1719 * q^39 + 46 * q^43 + 4608 * q^48 + 239 * q^49 - 3056 * q^52 - 10818 * q^57 + 1966 * q^61 + 1863 * q^63 + 8192 * q^64 + 8809 * q^67 + 1249 * q^73 - 5625 * q^75 - 19232 * q^76 + 12361 * q^79 - 6561 * q^81 + 3312 * q^84 + 13561 * q^91 + 15777 * q^93 - 37628 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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