LMFDB - Embedded newform 21.6.g.a.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,6,Mod(5,21)]

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, 5]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	21.g (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:	$3.36806021607$
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			5.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.5
Dual form			21.6.g.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.5000 - 7.79423i) q^{3} +(-16.0000 - 27.7128i) q^{4} +(105.500 - 75.3442i) q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})$	$q+(13.5000 - 7.79423i) q^{3} +(-16.0000 - 27.7128i) q^{4} +(105.500 - 75.3442i) q^{7} +(121.500 - 210.444i) q^{9} +(-432.000 - 249.415i) q^{12} +1141.42i q^{13} +(-512.000 + 886.810i) q^{16} +(-139.500 - 80.5404i) q^{19} +(837.000 - 1839.44i) q^{21} +(1562.50 + 2706.33i) q^{25} -3788.00i q^{27} +(-3776.00 - 1718.19i) q^{28} +(8962.50 - 5174.50i) q^{31} -7776.00 q^{36} +(-3330.50 + 5768.60i) q^{37} +(8896.50 + 15409.2i) q^{39} -22475.0 q^{43} +15962.6i q^{48} +(5453.50 - 15897.6i) q^{49} +(31632.0 - 18262.7i) q^{52} -2511.00 q^{57} +(-37614.0 - 21716.5i) q^{61} +(-3037.50 - 31356.2i) q^{63} +32768.0 q^{64} +(18969.5 + 32856.1i) q^{67} +(-40513.5 + 23390.5i) q^{73} +(42187.5 + 24357.0i) q^{75} +5154.58i q^{76} +(-45428.5 + 78684.5i) q^{79} +(-29524.5 - 51137.9i) q^{81} +(-64368.0 + 6235.38i) q^{84} +(85999.5 + 120420. i) q^{91} +(80662.5 - 139712. i) q^{93} -127631. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9}+O(q^{10}) $ 2 * q + 27 * q^3 - 32 * q^4 + 211 * q^7 + 243 * q^9	$ 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9} - 864 q^{12} - 1024 q^{16} - 279 q^{19} + 1674 q^{21} + 3125 q^{25} - 7552 q^{28} + 17925 q^{31} - 15552 q^{36} - 6661 q^{37} + 17793 q^{39} - 44950 q^{43} + 10907 q^{49} + 63264 q^{52} - 5022 q^{57} - 75228 q^{61} - 6075 q^{63} + 65536 q^{64} + 37939 q^{67} - 81027 q^{73} + 84375 q^{75} - 90857 q^{79} - 59049 q^{81} - 128736 q^{84} + 171999 q^{91} + 161325 q^{93}+O(q^{100}) $ 2 * q + 27 * q^3 - 32 * q^4 + 211 * q^7 + 243 * q^9 - 864 * q^12 - 1024 * q^16 - 279 * q^19 + 1674 * q^21 + 3125 * q^25 - 7552 * q^28 + 17925 * q^31 - 15552 * q^36 - 6661 * q^37 + 17793 * q^39 - 44950 * q^43 + 10907 * q^49 + 63264 * q^52 - 5022 * q^57 - 75228 * q^61 - 6075 * q^63 + 65536 * q^64 + 37939 * q^67 - 81027 * q^73 + 84375 * q^75 - 90857 * q^79 - 59049 * q^81 - 128736 * q^84 + 171999 * q^91 + 161325 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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	$0.666667\pi$
$3$	13.5000	−	7.79423i	0.866025	−	0.500000i
$3$	13.5000	−	7.79423i	0.866025	−	0.500000i
$4$	−16.0000	−	27.7128i	−0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	105.500	−	75.3442i	0.813781	−	0.581172i
$7$	105.500	−	75.3442i	0.813781	−	0.581172i
$8$		0			0
$9$	121.500	−	210.444i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−432.000	−	249.415i	−0.866025	−	0.500000i
$13$			1141.42i			1.87322i	0.350380	+	0.936608i	$0.386052\pi$
$13$			1141.42i			1.87322i	−0.350380	+	0.936608i	$0.613948\pi$
$14$		0			0
$15$		0			0
$16$	−512.000	+	886.810i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−139.500	−	80.5404i	−0.0886523	−	0.0511835i	0.455018	−	0.890482i	$-0.349633\pi$
$19$	−139.500	−	80.5404i	−0.0886523	−	0.0511835i	−0.543671	+	0.839299i	$0.682966\pi$
$20$		0			0
$21$	837.000	−	1839.44i	0.414169	−	0.910200i
$22$		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
$25$	1562.50	+	2706.33i	0.500000	+	0.866025i
$26$		0			0
$27$		−	3788.00i		−	1.00000i
$28$	−3776.00	−	1718.19i	−0.910200	−	0.414169i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	8962.50	−	5174.50i	1.67504	−	0.967084i	0.710291	−	0.703908i	$-0.248563\pi$
$31$	8962.50	−	5174.50i	1.67504	−	0.967084i	0.964748	−	0.263176i	$-0.0847701\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−7776.00			−1.00000
$37$	−3330.50	+	5768.60i	−0.399949	+	0.692733i	−0.993719	−	0.111902i	$-0.964306\pi$
$37$	−3330.50	+	5768.60i	−0.399949	+	0.692733i	0.593770	+	0.804635i	$0.297639\pi$
$38$		0			0
$39$	8896.50	+	15409.2i	0.936608	+	1.62225i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−22475.0			−1.85365			−0.926827	−	0.375489i	$-0.877475\pi$
$43$	−22475.0			−1.85365			−0.926827	+	0.375489i	$0.877475\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$			15962.6i			1.00000i
$49$	5453.50	−	15897.6i	0.324478	−	0.945893i
$50$		0			0
$51$		0			0
$52$	31632.0	−	18262.7i	1.62225	−	0.936608i
$53$		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
$57$	−2511.00			−0.102367
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−37614.0	−	21716.5i	−1.29427	−	0.747247i	−0.314862	−	0.949137i	$-0.601958\pi$
$61$	−37614.0	−	21716.5i	−1.29427	−	0.747247i	−0.979408	+	0.201890i	$0.935292\pi$
$62$		0			0
$63$	−3037.50	−	31356.2i	−0.0964195	−	0.995341i
$64$	32768.0			1.00000
$65$		0			0
$66$		0			0
$67$	18969.5	+	32856.1i	0.516260	+	0.894189i	0.999822	+	0.0188789i	$0.00600969\pi$
$67$	18969.5	+	32856.1i	0.516260	+	0.894189i	−0.483561	+	0.875310i	$0.660657\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−40513.5	+	23390.5i	−0.889800	+	0.513727i	−0.873877	−	0.486147i	$-0.838402\pi$
$73$	−40513.5	+	23390.5i	−0.889800	+	0.513727i	−0.0159232	+	0.999873i	$0.505069\pi$
$74$		0			0
$75$	42187.5	+	24357.0i	0.866025	+	0.500000i
$76$			5154.58i			0.102367i
$77$		0			0
$78$		0			0
$79$	−45428.5	+	78684.5i	−0.818956	+	1.41847i	0.0874958	+	0.996165i	$0.472114\pi$
$79$	−45428.5	+	78684.5i	−0.818956	+	1.41847i	−0.906452	+	0.422309i	$0.861220\pi$
$80$		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	$0.5\pi$
$84$	−64368.0	+	6235.38i	−0.995341	+	0.0964195i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	85999.5	+	120420.i	1.08866	+	1.52439i
$92$		0			0
$93$	80662.5	−	139712.i	0.967084	−	1.67504i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	127631.i		−	1.37730i	−0.725095	−	0.688649i	$-0.758204\pi$
$97$		−	127631.i		−	1.37730i	0.725095	−	0.688649i	$-0.241796\pi$
$98$		0			0
$99$		0			0
$100$	50000.0	−	86602.5i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	35161.5	+	20300.5i	0.326569	+	0.188544i	0.654317	−	0.756221i	$-0.272956\pi$
$103$	35161.5	+	20300.5i	0.326569	+	0.188544i	−0.327748	+	0.944765i	$0.606290\pi$
$104$		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	$0.666667\pi$
$108$	−104976.	+	60607.9i	−0.866025	+	0.500000i
$109$	−123922.	−	214638.i	−0.999034	−	1.73038i	−0.537567	−	0.843221i	$-0.680656\pi$
$109$	−123922.	−	214638.i	−0.999034	−	1.73038i	−0.461467	−	0.887157i	$-0.652677\pi$
$110$		0			0
$111$			103835.i			0.799899i
$112$	12800.0	+	132135.i	0.0964195	+	0.995341i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	240206.	+	138683.i	1.62225	+	0.936608i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−80525.5	+	139474.i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−286800.	−	165584.i	−1.67504	−	0.967084i
$125$		0			0
$126$		0			0
$127$	347111.			1.90967			0.954837	−	0.297131i	$-0.0960299\pi$
$127$	347111.			1.90967			0.954837	+	0.297131i	$0.0960299\pi$
$128$		0			0
$129$	−303412.	+	175175.i	−1.60531	+	0.926827i
$130$		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
$133$	−20785.5	+	2013.51i	−0.101890	+	0.00987017i
$134$		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
$139$		−	29024.0i		−	0.127415i	−0.997969	−	0.0637074i	$-0.979708\pi$
$139$		−	29024.0i		−	0.127415i	0.997969	−	0.0637074i	$-0.0202924\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	124416.	+	215495.i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−50287.5	−	257124.i	−0.191941	−	0.981407i
$148$	213152.			0.799899
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−204362.	−	353965.i	−0.729387	−	1.26333i	−0.957143	−	0.289616i	$-0.906472\pi$
$151$	−204362.	−	353965.i	−0.729387	−	1.26333i	0.227756	−	0.973718i	$-0.426861\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	284688.	−	493094.i	0.936608	−	1.62225i
$157$	526518.	−	303985.i	1.70476	−	0.984246i	0.763978	−	0.645242i	$-0.223243\pi$
$157$	526518.	−	303985.i	1.70476	−	0.984246i	0.940785	−	0.339004i	$-0.110090\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−339124.	+	587380.i	−0.999746	+	1.73161i	−0.480341	+	0.877082i	$0.659487\pi$
$163$	−339124.	+	587380.i	−0.999746	+	1.73161i	−0.519405	+	0.854528i	$0.673846\pi$
$164$		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
$169$	−931550.			−2.50893
$170$		0			0
$171$	−33898.5	+	19571.3i	−0.0886523	+	0.0511835i
$172$	359600.	+	622845.i	0.926827	+	1.60531i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	368750.	+	167792.i	0.910200	+	0.414169i
$176$		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
$181$			222265.i			0.504284i	0.967690	+	0.252142i	$0.0811351\pi$
$181$			222265.i			0.504284i	−0.967690	+	0.252142i	$0.918865\pi$
$182$		0			0
$183$	−677052.			−1.49449
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−285404.	−	399633.i	−0.581172	−	0.813781i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	442368.	−	255401.i	0.866025	−	0.500000i
$193$	328188.	+	568437.i	0.634204	+	1.09847i	0.986683	+	0.162653i	$0.0520053\pi$
$193$	328188.	+	568437.i	0.634204	+	1.09847i	−0.352480	+	0.935820i	$0.614661\pi$
$194$		0			0
$195$		0			0
$196$	−527824.	+	103230.i	−0.981407	+	0.191941i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	558627.	−	322523.i	0.999975	−	0.577336i	0.0917343	−	0.995784i	$-0.470759\pi$
$199$	558627.	−	322523.i	0.999975	−	0.577336i	0.908241	+	0.418448i	$0.137426\pi$
$200$		0			0
$201$	512176.	+	295705.i	0.894189	+	0.516260i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.01222e6	−	584408.i	−1.62225	−	0.936608i
$209$		0			0
$210$		0			0
$211$	288976.			0.446844			0.223422	−	0.974722i	$-0.428277\pi$
$211$	288976.			0.446844			0.223422	+	0.974722i	$0.428277\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	555675.	−	1.22118e6i	0.801072	−	1.76048i
$218$		0			0
$219$	−364622.	+	631543.i	−0.513727	+	0.889800i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	1.45377e6i		−	1.95764i	−0.204718	−	0.978821i	$-0.565628\pi$
$223$		−	1.45377e6i		−	1.95764i	0.204718	−	0.978821i	$-0.434372\pi$
$224$		0			0
$225$	759375.			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	40176.0	+	69586.9i	0.0511835	+	0.0886523i
$229$	540064.	+	311806.i	0.680545	+	0.392913i	0.800060	−	0.599919i	$-0.204801\pi$
$229$	540064.	+	311806.i	0.680545	+	0.392913i	−0.119515	+	0.992832i	$0.538134\pi$
$230$		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
$237$			1.41632e6i			1.63791i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1.08506e6	+	626462.i	−1.20341	+	0.694788i	−0.961311	−	0.275465i	$-0.911168\pi$
$241$	−1.08506e6	+	626462.i	−1.20341	+	0.694788i	−0.242096	+	0.970252i	$0.577835\pi$
$242$		0			0
$243$	−797161.	−	460241.i	−0.866025	−	0.500000i
$244$			1.38985e6i			1.49449i
$245$		0			0
$246$		0			0
$247$	91930.5	−	159228.i	0.0958776	−	0.166065i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−820368.	+	585877.i	−0.813781	+	0.581172i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−524288.	−	908093.i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	83262.5	+	859521.i	0.0771259	+	0.796172i
$260$		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
$268$	607024.	−	1.05140e6i	0.516260	−	0.894189i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	760575.	+	439118.i	0.629099	+	0.363210i	0.780403	−	0.625277i	$-0.215014\pi$
$271$	760575.	+	439118.i	0.629099	+	0.363210i	−0.151304	+	0.988487i	$0.548347\pi$
$272$		0			0
$273$	2.09957e6	+	955370.i	1.70500	+	0.775827i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−369194.	−	639464.i	−0.289105	−	0.500745i	0.684491	−	0.729021i	$-0.260024\pi$
$277$	−369194.	−	639464.i	−0.289105	−	0.500745i	−0.973596	+	0.228276i	$0.926691\pi$
$278$		0			0
$279$		−	2.51481e6i		−	1.93417i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−1.16534e6	+	672807.i	−0.864939	+	0.499373i	−0.865663	−	0.500627i	$-0.833103\pi$
$283$	−1.16534e6	+	672807.i	−0.864939	+	0.499373i	0.000724409		1.00000i	$0.499769\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	709928.	−	1.22963e6i	0.500000	−	0.866025i
$290$		0			0
$291$	−994788.	−	1.72302e6i	−0.688649	−	1.19278i
$292$	1.29643e6	+	748495.i	0.889800	+	0.513727i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	1.55885e6i		−	1.00000i
$301$	−2.37111e6	+	1.69336e6i	−1.50847	+	1.07729i
$302$		0			0
$303$		0			0
$304$	142848.	−	82473.3i	0.0886523	−	0.0511835i
$305$		0			0
$306$		0			0
$307$			808260.i			0.489446i	0.969593	+	0.244723i	$0.0786971\pi$
$307$			808260.i			0.489446i	−0.969593	+	0.244723i	$0.921303\pi$
$308$		0			0
$309$	632907.			0.377089
$310$		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
$313$	2.01744e6	+	1.16477e6i	1.16396	+	0.672014i	0.952250	−	0.305318i	$-0.0987629\pi$
$313$	2.01744e6	+	1.16477e6i	1.16396	+	0.672014i	0.211712	+	0.977332i	$0.432096\pi$
$314$		0			0
$315$		0			0
$316$	2.90742e6			1.63791
$317$		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
$324$	−944784.	+	1.63641e6i	−0.500000	+	0.866025i
$325$	−3.08906e6	+	1.78347e6i	−1.62225	+	0.936608i
$326$		0			0
$327$	−3.34588e6	−	1.93175e6i	−1.73038	−	0.999034i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−273076.	+	472981.i	−0.136998	+	0.237287i	−0.926359	−	0.376642i	$-0.877079\pi$
$331$	−273076.	+	472981.i	−0.136998	+	0.237287i	0.789361	+	0.613929i	$0.210412\pi$
$332$		0			0
$333$	809312.	+	1.40177e6i	0.399949	+	0.692733i
$334$		0			0
$335$		0			0
$336$	1.20269e6	+	1.68405e6i	0.581172	+	0.813781i
$337$	−2.63172e6			−1.26231			−0.631155	−	0.775657i	$-0.717419\pi$
$337$	−2.63172e6			−1.26231			−0.631155	+	0.775657i	$0.717419\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−622450.	−	2.08809e6i	−0.285673	−	0.958327i
$344$		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
$349$			1.55867e6i			0.685001i	0.939518	+	0.342501i	$0.111274\pi$
$349$			1.55867e6i			0.685001i	−0.939518	+	0.342501i	$0.888726\pi$
$350$		0			0
$351$	4.32370e6			1.87322
$352$		0			0
$353$		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
$358$		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
$361$	−1.22508e6	−	2.12189e6i	−0.494761	−	0.856950i
$362$		0			0
$363$			2.51054e6i			1.00000i
$364$	1.96118e6	−	4.31001e6i	0.775827	−	1.70500i
$365$		0			0
$366$		0			0
$367$	3.86895e6	−	2.23374e6i	1.49944	−	0.865700i	0.499437	−	0.866350i	$-0.333540\pi$
$367$	3.86895e6	−	2.23374e6i	1.49944	−	0.865700i	1.00000		0.000650122i	$0.000206940\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−5.16240e6			−1.93417
$373$	2.65864e6	−	4.60489e6i	0.989434	−	1.71375i	0.369158	−	0.929367i	$-0.379646\pi$
$373$	2.65864e6	−	4.60489e6i	0.989434	−	1.71375i	0.620276	−	0.784384i	$-0.287021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−69893.0			−0.0249940			−0.0124970	−	0.999922i	$-0.503978\pi$
$379$	−69893.0			−0.0249940			−0.0124970	+	0.999922i	$0.503978\pi$
$380$		0			0
$381$	4.68600e6	−	2.70546e6i	1.65383	−	0.954837i
$382$		0			0
$383$		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
$386$		0			0
$387$	−2.73071e6	+	4.72973e6i	−0.926827	+	1.60531i
$388$	−3.53702e6	+	2.04210e6i	−1.19278	+	0.688649i
$389$		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
$397$	−3.28777e6	−	1.89820e6i	−1.04695	−	0.604456i	−0.125156	−	0.992137i	$-0.539943\pi$
$397$	−3.28777e6	−	1.89820e6i	−1.04695	−	0.604456i	−0.921794	+	0.387681i	$0.873276\pi$
$398$		0			0
$399$	−264910.	+	189189.i	−0.0833042	+	0.0594928i
$400$	−3.20000e6			−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
$403$	5.90629e6	+	1.02300e7i	1.81156	+	3.13771i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	3.77979e6	−	2.18226e6i	1.11727	−	0.645057i	0.176569	−	0.984288i	$-0.443500\pi$
$409$	3.77979e6	−	2.18226e6i	1.11727	−	0.645057i	0.940703	+	0.339231i	$0.110167\pi$
$410$		0			0
$411$		0			0
$412$		−	1.29923e6i		−	0.377089i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−226220.	−	391824.i	−0.0637074	−	0.110344i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−4.87580e6			−1.34073			−0.670364	−	0.742033i	$-0.733862\pi$
$421$	−4.87580e6			−1.34073			−0.670364	+	0.742033i	$0.733862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5.60449e6	+	542911.i	−1.48753	+	0.144098i
$428$		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	$0.333333\pi$
$432$	3.35923e6	+	1.93945e6i	0.866025	+	0.500000i
$433$		−	2.18049e6i		−	0.558901i	−0.960160	−	0.279450i	$-0.909848\pi$
$433$		−	2.18049e6i		−	0.558901i	0.960160	−	0.279450i	$-0.0901522\pi$
$434$		0			0
$435$		0			0
$436$	−3.96549e6	+	6.86843e6i	−0.999034	+	1.73038i
$437$		0			0
$438$		0			0
$439$	6.07908e6	+	3.50976e6i	1.50548	+	0.869192i	0.999980	+	0.00636830i	$0.00202711\pi$
$439$	6.07908e6	+	3.50976e6i	1.50548	+	0.869192i	0.505505	+	0.862824i	$0.331306\pi$
$440$		0			0
$441$	−2.68296e6	−	3.07922e6i	−0.656929	−	0.753953i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	2.87755e6	−	1.66136e6i	0.692733	−	0.399949i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3.45702e6	−	2.46888e6i	0.813781	−	0.581172i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−5.51777e6	−	3.18569e6i	−1.26333	−	0.729387i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	592880.	−	1.02690e6i	0.132793	−	0.230005i	−0.791959	−	0.610574i	$-0.790939\pi$
$457$	592880.	−	1.02690e6i	0.132793	−	0.230005i	0.924752	+	0.380569i	$0.124272\pi$
$458$		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
$463$	2.92217e6			0.633510			0.316755	−	0.948507i	$-0.397407\pi$
$463$	2.92217e6			0.633510			0.316755	+	0.948507i	$0.397407\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		−	8.87569e6i		−	1.87322i
$469$	4.47680e6	+	2.03708e6i	0.939801	+	0.427638i
$470$		0			0
$471$	4.73866e6	−	8.20760e6i	0.984246	−	1.70476i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	503377.i		−	0.102367i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−6.58440e6	−	3.80150e6i	−1.29764	−	0.749191i
$482$		0			0
$483$		0			0
$484$	5.15363e6			1.00000
$485$		0			0
$486$		0			0
$487$	−682437.	−	1.18202e6i	−0.130389	−	0.225840i	0.793438	−	0.608652i	$-0.208289\pi$
$487$	−682437.	−	1.18202e6i	−0.130389	−	0.225840i	−0.923827	+	0.382811i	$0.874956\pi$
$488$		0			0
$489$			1.05728e7i			1.99949i
$490$		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
$496$			1.05974e7i			1.93417i
$497$		0			0
$498$		0			0
$499$	−1.08685e6	+	1.88247e6i	−0.195397	+	0.338437i	−0.947030	−	0.321144i	$-0.895933\pi$
$499$	−1.08685e6	+	1.88247e6i	−0.195397	+	0.338437i	0.751634	+	0.659581i	$0.229266\pi$
$500$		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
$506$		0			0
$507$	−1.25759e7	+	7.26071e6i	−2.17280	+	1.25447i
$508$	−5.55378e6	−	9.61942e6i	−0.954837	−	1.65383i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−2.51184e6	+	5.52015e6i	−0.425539	+	0.935188i
$512$		0			0
$513$	−305086.	+	528425.i	−0.0511835	+	0.0886523i
$514$		0			0
$515$		0			0
$516$	9.70920e6	+	5.60561e6i	1.60531	+	0.926827i
$517$		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	$0.833333\pi$
$522$		0			0
$523$	−3.77071e6	−	2.17702e6i	−0.602794	−	0.348023i	0.167346	−	0.985898i	$-0.446480\pi$
$523$	−3.77071e6	−	2.17702e6i	−0.602794	−	0.348023i	−0.770140	+	0.637875i	$0.779814\pi$
$524$		0			0
$525$	6.28594e6	−	608924.i	0.995341	−	0.0964195i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.21817e6	−	5.57404e6i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	388368.	+	543809.i	0.0594928	+	0.0833042i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.51118e6	−	2.61743e6i	0.221984	−	0.384488i	−0.733426	−	0.679769i	$-0.762080\pi$
$541$	1.51118e6	−	2.61743e6i	0.221984	−	0.384488i	0.955410	+	0.295281i	$0.0954134\pi$
$542$		0			0
$543$	1.73239e6	+	3.00058e6i	0.252142	+	0.436723i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.27982e7			1.82886			0.914430	−	0.404744i	$-0.132639\pi$
$547$	1.27982e7			1.82886			0.914430	+	0.404744i	$0.132639\pi$
$548$		0			0
$549$	−9.14020e6	+	5.27710e6i	−1.29427	+	0.747247i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.13571e6	+	1.17240e7i	0.157927	+	1.63028i
$554$		0			0
$555$		0			0
$556$	−804336.	+	464384.i	−0.110344	+	0.0637074i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		−	2.56534e7i		−	3.47229i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		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
$567$	−6.96778e6	−	3.17055e6i	−0.910200	−	0.414169i
$568$		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
$571$	−2.94390e6	−	5.09898e6i	−0.377862	−	0.654475i	0.612889	−	0.790169i	$-0.290007\pi$
$571$	−2.94390e6	−	5.09898e6i	−0.377862	−	0.654475i	−0.990751	+	0.135693i	$0.956674\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	3.98131e6	−	6.89583e6i	0.500000	−	0.866025i
$577$	8.17725e6	−	4.72114e6i	1.02251	−	0.590347i	0.107680	−	0.994186i	$-0.465658\pi$
$577$	8.17725e6	−	4.72114e6i	1.02251	−	0.590347i	0.914830	+	0.403839i	$0.132324\pi$
$578$		0			0
$579$	8.86106e6	+	5.11594e6i	1.09847	+	0.634204i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−6.32102e6	+	5.50759e6i	−0.753953	+	0.656929i
$589$	−1.66702e6			−0.197995
$590$		0			0
$591$		0			0
$592$	−3.41043e6	−	5.90704e6i	−0.399949	−	0.692733i
$593$		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
$596$		0			0
$597$	5.02764e6	−	8.70813e6i	0.577336	−	0.999975i
$598$		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
$601$			3.11249e6i			0.351498i	0.984435	+	0.175749i	$0.0562346\pi$
$601$			3.11249e6i			0.351498i	−0.984435	+	0.175749i	$0.943765\pi$
$602$		0			0
$603$	9.21918e6			1.03252
$604$	−6.53958e6	+	1.13269e7i	−0.729387	+	1.26333i
$605$		0			0
$606$		0			0
$607$	7.92463e6	+	4.57529e6i	0.872986	+	0.504019i	0.868339	−	0.495970i	$-0.165188\pi$
$607$	7.92463e6	+	4.57529e6i	0.872986	+	0.504019i	0.00464665	+	0.999989i	$0.498521\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7.91172e6	+	1.37035e7i	0.850394	+	1.47292i	0.880853	+	0.473389i	$0.156969\pi$
$613$	7.91172e6	+	1.37035e7i	0.850394	+	1.47292i	−0.0304599	+	0.999536i	$0.509697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−1.26698e7	+	7.31491e6i	−1.32905	+	0.767330i	−0.985153	−	0.171676i	$-0.945082\pi$
$619$	−1.26698e7	+	7.31491e6i	−1.32905	+	0.767330i	−0.343901	+	0.939006i	$0.611748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.82200e7			−1.87322
$625$	−4.88281e6	+	8.45728e6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−1.68486e7	−	9.72753e6i	−1.70476	−	0.984246i
$629$		0			0
$630$		0			0
$631$	−1.62439e7			−1.62411			−0.812057	−	0.583579i	$-0.801652\pi$
$631$	−1.62439e7			−1.62411			−0.812057	+	0.583579i	$0.801652\pi$
$632$		0			0
$633$	3.90118e6	−	2.25235e6i	0.386978	−	0.223422i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.81459e7	+	6.22474e6i	1.77186	+	0.607817i
$638$		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
$643$			1.61402e7i			1.53951i	0.638342	+	0.769753i	$0.279620\pi$
$643$			1.61402e7i			1.53951i	−0.638342	+	0.769753i	$0.720380\pi$
$644$		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	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.01656e6	−	2.08170e7i	−0.186492	−	1.92516i
$652$	2.17039e7			1.99949
$653$		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
$657$			1.13678e7i			1.02745i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	1.94473e7	−	1.12279e7i	1.73123	−	0.999527i	0.850238	−	0.526398i	$-0.176458\pi$
$661$	1.94473e7	−	1.12279e7i	1.73123	−	0.999527i	0.880993	−	0.473129i	$-0.156876\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.13310e7	−	1.96259e7i	−0.978821	−	1.69537i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.73115e7			−1.47332			−0.736661	−	0.676262i	$-0.763599\pi$
$673$	−1.73115e7			−1.47332			−0.736661	+	0.676262i	$0.763599\pi$
$674$		0			0
$675$	1.02516e7	−	5.91874e6i	0.866025	−	0.500000i
$676$	1.49048e7	+	2.58159e7i	1.25447	+	2.17280i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−9.61628e6	−	1.34651e7i	−0.800447	−	1.12082i
$680$		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	$0.333333\pi$
$684$	1.08475e6	+	626282.i	0.0886523	+	0.0511835i
$685$		0			0
$686$		0			0
$687$	9.72116e6			0.785826
$688$	1.15072e7	−	1.99311e7i	0.926827	−	1.60531i
$689$		0			0
$690$		0			0
$691$	−2.08726e7	−	1.20508e7i	−1.66296	−	0.960112i	−0.971289	−	0.237901i	$-0.923541\pi$
$691$	−2.08726e7	−	1.20508e7i	−1.66296	−	0.960112i	−0.691673	−	0.722211i	$-0.743126\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.25000e6	−	1.29038e7i	−0.0964195	−	0.995341i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	929210.	−	536479.i	0.0709129	−	0.0409416i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.13507e7	+	1.96600e7i	−0.848021	+	1.46882i	0.0349502	+	0.999389i	$0.488873\pi$
$709$	−1.13507e7	+	1.96600e7i	−0.848021	+	1.46882i	−0.882971	+	0.469427i	$0.844461\pi$
$710$		0			0
$711$	1.10391e7	+	1.91203e7i	0.818956	+	1.41847i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	5.23906e6	−	507513.i	0.375332	−	0.0363587i
$722$		0			0
$723$	−9.76558e6	+	1.69145e7i	−0.694788	+	1.20341i
$724$	6.15960e6	−	3.55625e6i	0.436723	−	0.252142i
$725$		0			0
$726$		0			0
$727$		−	2.37945e7i		−	1.66971i	−0.550474	−	0.834853i	$-0.685553\pi$
$727$		−	2.37945e7i		−	1.66971i	0.550474	−	0.834853i	$-0.314447\pi$
$728$		0			0
$729$	−1.43489e7			−1.00000
$730$		0			0
$731$		0			0
$732$	1.08328e7	+	1.87630e7i	0.747247	+	1.29427i
$733$	2.00852e7	+	1.15962e7i	1.38075	+	0.797177i	0.992248	−	0.124270i	$-0.0396590\pi$
$733$	2.00852e7	+	1.15962e7i	1.38075	+	0.797177i	0.388503	+	0.921448i	$0.372992\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.40852e6	−	1.10999e7i	−0.431665	−	0.747666i	0.565352	−	0.824850i	$-0.308740\pi$
$739$	−6.40852e6	−	1.10999e7i	−0.431665	−	0.747666i	−0.997017	+	0.0771842i	$0.975407\pi$
$740$		0			0
$741$		−	2.86611e6i		−	0.191755i
$742$		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
$751$	−1.36610e7	+	2.36615e7i	−0.883857	+	1.53088i	−0.0368381	+	0.999321i	$0.511729\pi$
$751$	−1.36610e7	+	2.36615e7i	−0.883857	+	1.53088i	−0.847019	+	0.531563i	$0.821605\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−6.50851e6	+	1.43035e7i	−0.414169	+	0.910200i
$757$	−1.98526e7			−1.25915			−0.629575	−	0.776940i	$-0.716771\pi$
$757$	−1.98526e7			−1.25915			−0.629575	+	0.776940i	$0.716771\pi$
$758$		0			0
$759$		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
$763$	−2.92455e7	−	1.33076e7i	−1.81864	−	0.827537i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.41558e7	−	8.17284e6i	−0.866025	−	0.500000i
$769$			2.12277e7i			1.29446i	0.762296	+	0.647228i	$0.224072\pi$
$769$			2.12277e7i			1.29446i	−0.762296	+	0.647228i	$0.775928\pi$
$770$		0			0
$771$		0			0
$772$	1.05020e7	−	1.81900e7i	0.634204	−	1.09847i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	2.80078e7	+	1.61703e7i	1.67504	+	0.967084i
$776$		0			0
$777$	7.82334e6	+	1.09546e7i	0.464879	+	0.650942i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.13060e7	+	1.29758e7i	0.656929	+	0.753953i
$785$		0			0
$786$		0			0
$787$	9.14931e6	−	5.28235e6i	0.526564	−	0.304012i	−0.213052	−	0.977041i	$-0.568340\pi$
$787$	9.14931e6	−	5.28235e6i	0.526564	−	0.304012i	0.739616	+	0.673029i	$0.235007\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.47876e7	−	4.29334e7i	1.39975	−	2.42445i
$794$		0			0
$795$		0			0
$796$	−1.78761e7	−	1.03208e7i	−0.999975	−	0.577336i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	1.89251e7i		−	1.03252i
$805$		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
$811$		−	2.97462e7i		−	1.58811i	−0.607849	−	0.794053i	$-0.707967\pi$
$811$		−	2.97462e7i		−	1.58811i	0.607849	−	0.794053i	$-0.292033\pi$
$812$		0			0
$813$	1.36904e7			0.726421
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.13526e6	+	1.81014e6i	0.164331	+	0.0948764i
$818$		0			0
$819$	3.57906e7	−	3.46707e6i	1.86449	−	0.180614i
$820$		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
$823$	−1.11462e7	−	1.93057e7i	−0.573623	−	0.993543i	−0.996190	−	0.0872118i	$-0.972204\pi$
$823$	−1.11462e7	−	1.93057e7i	−0.573623	−	0.993543i	0.422567	−	0.906332i	$-0.361129\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−2.81051e7	+	1.62265e7i	−1.42036	+	0.820045i	−0.996329	−	0.0856034i	$-0.972718\pi$
$829$	−2.81051e7	+	1.62265e7i	−1.42036	+	0.820045i	−0.424030	+	0.905648i	$0.639385\pi$
$830$		0			0
$831$	−9.96825e6	−	5.75517e6i	−0.500745	−	0.289105i
$832$			3.74021e7i			1.87322i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.96010e7	−	3.39499e7i	−0.967084	−	1.67504i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	2.05111e7			1.00000
$842$		0			0
$843$		0			0
$844$	−4.62362e6	−	8.00834e6i	−0.223422	−	0.386978i
$845$		0			0
$846$		0			0
$847$	2.01314e6	+	2.07817e7i	0.0964195	+	0.995341i
$848$		0			0
$849$	−1.04880e7	+	1.81658e7i	−0.499373	+	0.864939i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			4.23947e7i			1.99498i	0.0707842	+	0.997492i	$0.477450\pi$
$853$			4.23947e7i			1.99498i	−0.0707842	+	0.997492i	$0.522550\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	3.73636e7	+	2.15719e7i	1.72769	+	0.997482i	0.899307	+	0.437317i	$0.144071\pi$
$859$	3.73636e7	+	2.15719e7i	1.72769	+	0.997482i	0.828381	+	0.560164i	$0.189262\pi$
$860$		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
$867$		−	2.21334e7i		−	1.00000i
$868$	−4.27332e7	+	4.13960e6i	−1.92516	+	0.186492i
$869$		0			0
$870$		0			0
$871$	−3.75027e7	+	2.16522e7i	−1.67501	+	0.967067i
$872$		0			0
$873$	−2.68593e7	−	1.55072e7i	−1.19278	−	0.688649i
$874$		0			0
$875$		0			0
$876$	2.33358e7			1.02745
$877$	1.91966e6	−	3.32494e6i	0.0842800	−	0.145977i	−0.820804	−	0.571210i	$-0.806474\pi$
$877$	1.91966e6	−	3.32494e6i	0.0842800	−	0.145977i	0.905084	+	0.425232i	$0.139808\pi$
$878$		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
$883$	4.58066e7			1.97709			0.988545	−	0.150925i	$-0.0482253\pi$
$883$	4.58066e7			1.97709			0.988545	+	0.150925i	$0.0482253\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	3.66202e7	−	2.61528e7i	1.55406	−	1.10985i
$890$		0			0
$891$		0			0
$892$	−4.02880e7	+	2.32603e7i	−1.69537	+	0.978821i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.21500e7	−	2.10444e7i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−1.88116e7	+	4.13414e7i	−0.767725	+	1.68720i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.31060e7	+	2.27002e7i	0.528995	+	0.916247i	0.999428	+	0.0338109i	$0.0107644\pi$
$907$	1.31060e7	+	2.27002e7i	0.528995	+	0.916247i	−0.470433	+	0.882436i	$0.655902\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	1.28563e6	−	2.22678e6i	0.0511835	−	0.0886523i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	1.99556e7i		−	0.785826i
$917$		0			0
$918$		0			0
$919$	−2.10911e6	+	3.65308e6i	−0.0823777	+	0.142682i	−0.904271	−	0.426959i	$-0.859585\pi$
$919$	−2.10911e6	+	3.65308e6i	−0.0823777	+	0.142682i	0.821893	+	0.569642i	$0.192918\pi$
$920$		0			0
$921$	6.29976e6	+	1.09115e7i	0.244723	+	0.423873i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.08156e7			−0.799899
$926$		0			0
$927$	8.54424e6	−	4.93302e6i	0.326569	−	0.188544i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	−2.04116e6	+	1.77849e6i	−0.0771798	+	0.0672478i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	5.34884e7i		−	1.99026i	−0.0985501	−	0.995132i	$-0.531420\pi$
$937$		−	5.34884e7i		−	1.99026i	0.0985501	−	0.995132i	$-0.468580\pi$
$938$		0			0
$939$	3.63138e7			1.34403
$940$		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
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	3.92502e7	−	2.26611e7i	1.41847	−	0.818956i
$949$	−2.66984e7	−	4.62430e7i	−0.962320	−	1.66679i
$950$		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
$961$	3.92364e7	−	6.79594e7i	1.37050	−	2.37378i
$962$		0			0
$963$		0			0
$964$	3.47220e7	+	2.00468e7i	1.20341	+	0.694788i
$965$		0			0
$966$		0			0
$967$	−3.23453e7			−1.11236			−0.556180	−	0.831062i	$-0.687733\pi$
$967$	−3.23453e7			−1.11236			−0.556180	+	0.831062i	$0.687733\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$			2.94555e7i			1.00000i
$973$	−2.18679e6	−	3.06203e6i	−0.0740499	−	0.103688i
$974$		0			0
$975$	−2.78016e7	+	4.81537e7i	−0.936608	+	1.62225i
$976$	3.85167e7	−	2.22376e7i	1.29427	−	0.747247i
$977$		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
$981$	−6.02258e7			−1.99807
$982$		0			0
$983$		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
$988$	−5.88355e6			−0.191755
$989$		0			0
$990$		0			0
$991$	3.02764e7	+	5.24403e7i	0.979310	+	1.69622i	0.664908	+	0.746925i	$0.268471\pi$
$991$	3.02764e7	+	5.24403e7i	0.979310	+	1.69622i	0.314402	+	0.949290i	$0.398196\pi$
$992$		0			0
$993$			8.51365e6i			0.273995i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.45790e7	+	8.41722e6i	−0.464506	+	0.268183i	−0.713937	−	0.700210i	$-0.753090\pi$
$997$	−1.45790e7	+	8.41722e6i	−0.464506	+	0.268183i	0.249431	+	0.968393i	$0.419756\pi$
$998$		0			0
$999$	2.18514e7	+	1.26159e7i	0.692733	+	0.399949i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.6.g.a.5.1	✓	2
3.2	odd	2	CM	21.6.g.a.5.1	✓	2
7.2	even	3		147.6.c.a.146.2		2
7.3	odd	6	inner	21.6.g.a.17.1	yes	2
7.5	odd	6		147.6.c.a.146.1		2
21.2	odd	6		147.6.c.a.146.2		2
21.5	even	6		147.6.c.a.146.1		2
21.17	even	6	inner	21.6.g.a.17.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.g.a.5.1	✓	2	1.1	even	1	trivial
21.6.g.a.5.1	✓	2	3.2	odd	2	CM
21.6.g.a.17.1	yes	2	7.3	odd	6	inner
21.6.g.a.17.1	yes	2	21.17	even	6	inner
147.6.c.a.146.1		2	7.5	odd	6
147.6.c.a.146.1		2	21.5	even	6
147.6.c.a.146.2		2	7.2	even	3
147.6.c.a.146.2		2	21.2	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(13.5000 - 7.79423i) q^{3} +(-16.0000 - 27.7128i) q^{4} +(105.500 - 75.3442i) q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})\)	\(q+(13.5000 - 7.79423i) q^{3} +(-16.0000 - 27.7128i) q^{4} +(105.500 - 75.3442i) q^{7} +(121.500 - 210.444i) q^{9} +(-432.000 - 249.415i) q^{12} +1141.42i q^{13} +(-512.000 + 886.810i) q^{16} +(-139.500 - 80.5404i) q^{19} +(837.000 - 1839.44i) q^{21} +(1562.50 + 2706.33i) q^{25} -3788.00i q^{27} +(-3776.00 - 1718.19i) q^{28} +(8962.50 - 5174.50i) q^{31} -7776.00 q^{36} +(-3330.50 + 5768.60i) q^{37} +(8896.50 + 15409.2i) q^{39} -22475.0 q^{43} +15962.6i q^{48} +(5453.50 - 15897.6i) q^{49} +(31632.0 - 18262.7i) q^{52} -2511.00 q^{57} +(-37614.0 - 21716.5i) q^{61} +(-3037.50 - 31356.2i) q^{63} +32768.0 q^{64} +(18969.5 + 32856.1i) q^{67} +(-40513.5 + 23390.5i) q^{73} +(42187.5 + 24357.0i) q^{75} +5154.58i q^{76} +(-45428.5 + 78684.5i) q^{79} +(-29524.5 - 51137.9i) q^{81} +(-64368.0 + 6235.38i) q^{84} +(85999.5 + 120420. i) q^{91} +(80662.5 - 139712. i) q^{93} -127631. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9}+O(q^{10}) \) 2 * q + 27 * q^3 - 32 * q^4 + 211 * q^7 + 243 * q^9	\( 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9} - 864 q^{12} - 1024 q^{16} - 279 q^{19} + 1674 q^{21} + 3125 q^{25} - 7552 q^{28} + 17925 q^{31} - 15552 q^{36} - 6661 q^{37} + 17793 q^{39} - 44950 q^{43} + 10907 q^{49} + 63264 q^{52} - 5022 q^{57} - 75228 q^{61} - 6075 q^{63} + 65536 q^{64} + 37939 q^{67} - 81027 q^{73} + 84375 q^{75} - 90857 q^{79} - 59049 q^{81} - 128736 q^{84} + 171999 q^{91} + 161325 q^{93}+O(q^{100}) \) 2 * q + 27 * q^3 - 32 * q^4 + 211 * q^7 + 243 * q^9 - 864 * q^12 - 1024 * q^16 - 279 * q^19 + 1674 * q^21 + 3125 * q^25 - 7552 * q^28 + 17925 * q^31 - 15552 * q^36 - 6661 * q^37 + 17793 * q^39 - 44950 * q^43 + 10907 * q^49 + 63264 * q^52 - 5022 * q^57 - 75228 * q^61 - 6075 * q^63 + 65536 * q^64 + 37939 * q^67 - 81027 * q^73 + 84375 * q^75 - 90857 * q^79 - 59049 * q^81 - 128736 * q^84 + 171999 * q^91 + 161325 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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