LMFDB - Embedded newform 21.10.c.a.20.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,10,Mod(20,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("21.20");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	21.c (of order $2$, degree $1$, 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:	$10.8157525594$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			20.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.20
Dual form			21.10.c.a.20.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+140.296i q^{3} +512.000 q^{4} +(6290.00 + 888.542i) q^{7} -19683.0 q^{9} +O(q^{10})$	$q+140.296i q^{3} +512.000 q^{4} +(6290.00 + 888.542i) q^{7} -19683.0 q^{9} +71831.6i q^{12} +168542. i q^{13} +262144. q^{16} +580358. i q^{19} +(-124659. + 882463. i) q^{21} -1.95312e6 q^{25} -2.76145e6i q^{27} +(3.22048e6 + 454934. i) q^{28} -1.01439e7i q^{31} -1.00777e7 q^{36} +1.53845e7 q^{37} -2.36458e7 q^{39} +1.65771e7 q^{43} +3.67778e7i q^{48} +(3.87746e7 + 1.11779e7i) q^{49} +8.62937e7i q^{52} -8.14220e7 q^{57} -1.81316e8i q^{61} +(-1.23806e8 - 1.74892e7i) q^{63} +1.34218e8 q^{64} +1.12542e8 q^{67} -3.84255e8i q^{73} -2.74016e8i q^{75} +2.97143e8i q^{76} -6.16732e8 q^{79} +3.87420e8 q^{81} +(-6.38254e7 + 4.51821e8i) q^{84} +(-1.49757e8 + 1.06013e9i) q^{91} +1.42315e9 q^{93} +1.17456e9i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1024 q^{4} + 12580 q^{7} - 39366 q^{9}+O(q^{10}) $ 2 * q + 1024 * q^4 + 12580 * q^7 - 39366 * q^9	$ 2 q + 1024 q^{4} + 12580 q^{7} - 39366 q^{9} + 524288 q^{16} - 249318 q^{21} - 3906250 q^{25} + 6440960 q^{28} - 20155392 q^{36} + 30768980 q^{37} - 47291688 q^{39} + 33154160 q^{43} + 77549186 q^{49} - 162844020 q^{57} - 247612140 q^{63} + 268435456 q^{64} + 225084640 q^{67} - 1233464648 q^{79} + 774840978 q^{81} - 127650816 q^{84} - 299514024 q^{91} + 2846293020 q^{93}+O(q^{100}) $ 2 * q + 1024 * q^4 + 12580 * q^7 - 39366 * q^9 + 524288 * q^16 - 249318 * q^21 - 3906250 * q^25 + 6440960 * q^28 - 20155392 * q^36 + 30768980 * q^37 - 47291688 * q^39 + 33154160 * q^43 + 77549186 * q^49 - 162844020 * q^57 - 247612140 * q^63 + 268435456 * q^64 + 225084640 * q^67 - 1233464648 * q^79 + 774840978 * q^81 - 127650816 * q^84 - 299514024 * q^91 + 2846293020 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$-1$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$			140.296i			1.00000i
$3$			140.296i			1.00000i
$4$	512.000			1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	6290.00	+	888.542i	0.990169	+	0.139874i
$7$	6290.00	+	888.542i	0.990169	+	0.139874i
$8$		0			0
$9$	−19683.0			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			71831.6i			1.00000i
$13$			168542.i			1.63668i	0.574734	+	0.818341i	$0.305106\pi$
$13$			168542.i			1.63668i	−0.574734	+	0.818341i	$0.694894\pi$
$14$		0			0
$15$		0			0
$16$	262144.			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			580358.i			1.02166i	0.859683	+	0.510828i	$0.170661\pi$
$19$			580358.i			1.02166i	−0.859683	+	0.510828i	$0.829339\pi$
$20$		0			0
$21$	−124659.	+	882463.i	−0.139874	+	0.990169i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−1.95312e6			−1.00000
$26$		0			0
$27$		−	2.76145e6i		−	1.00000i
$28$	3.22048e6	+	454934.i	0.990169	+	0.139874i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	1.01439e7i		−	1.97277i	−0.164454	−	0.986385i	$-0.552586\pi$
$31$		−	1.01439e7i		−	1.97277i	0.164454	−	0.986385i	$-0.447414\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.00777e7			−1.00000
$37$	1.53845e7			1.34951			0.674754	−	0.738043i	$-0.264250\pi$
$37$	1.53845e7			1.34951			0.674754	+	0.738043i	$0.264250\pi$
$38$		0			0
$39$	−2.36458e7			−1.63668
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	1.65771e7			0.739435			0.369717	−	0.929144i	$-0.379454\pi$
$43$	1.65771e7			0.739435			0.369717	+	0.929144i	$0.379454\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$			3.67778e7i			1.00000i
$49$	3.87746e7	+	1.11779e7i	0.960871	+	0.276998i
$50$		0			0
$51$		0			0
$52$			8.62937e7i			1.63668i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.14220e7			−1.02166
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	1.81316e8i		−	1.67669i	−0.545143	−	0.838343i	$-0.683525\pi$
$61$		−	1.81316e8i		−	1.67669i	0.545143	−	0.838343i	$-0.316475\pi$
$62$		0			0
$63$	−1.23806e8	−	1.74892e7i	−0.990169	−	0.139874i
$64$	1.34218e8			1.00000
$65$		0			0
$66$		0			0
$67$	1.12542e8			0.682306			0.341153	−	0.940008i	$-0.389183\pi$
$67$	1.12542e8			0.682306			0.341153	+	0.940008i	$0.389183\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	3.84255e8i		−	1.58368i	−0.610729	−	0.791839i	$-0.709124\pi$
$73$		−	3.84255e8i		−	1.58368i	0.610729	−	0.791839i	$-0.290876\pi$
$74$		0			0
$75$		−	2.74016e8i		−	1.00000i
$76$			2.97143e8i			1.02166i
$77$		0			0
$78$		0			0
$79$	−6.16732e8			−1.78145			−0.890727	−	0.454538i	$-0.849804\pi$
$79$	−6.16732e8			−1.78145			−0.890727	+	0.454538i	$0.849804\pi$
$80$		0			0
$81$	3.87420e8			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−6.38254e7	+	4.51821e8i	−0.139874	+	0.990169i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−1.49757e8	+	1.06013e9i	−0.228929	+	1.62059i
$92$		0			0
$93$	1.42315e9			1.97277
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.17456e9i			1.34711i	0.739139	+	0.673553i	$0.235233\pi$
$97$			1.17456e9i			1.34711i	−0.739139	+	0.673553i	$0.764767\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000e9			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	2.28368e9i		−	1.99926i	−0.0272610	−	0.999628i	$-0.508679\pi$
$103$		−	2.28368e9i		−	1.99926i	0.0272610	−	0.999628i	$-0.491321\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		−	1.41386e9i		−	1.00000i
$109$	−2.24018e9			−1.52007			−0.760035	−	0.649882i	$-0.774818\pi$
$109$	−2.24018e9			−1.52007			−0.760035	+	0.649882i	$0.774818\pi$
$110$		0			0
$111$			2.15838e9i			1.34951i
$112$	1.64889e9	+	2.32926e8i	0.990169	+	0.139874i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	3.31742e9i		−	1.63668i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.35795e9			1.00000
$122$		0			0
$123$		0			0
$124$		−	5.19366e9i		−	1.97277i
$125$		0			0
$126$		0			0
$127$	−2.28028e9			−0.777808			−0.388904	−	0.921278i	$-0.627146\pi$
$127$	−2.28028e9			−0.777808			−0.388904	+	0.921278i	$0.627146\pi$
$128$		0			0
$129$			2.32570e9i			0.739435i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−5.15673e8	+	3.65045e9i	−0.142903	+	1.01161i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$			8.00499e9i			1.81884i	0.415880	+	0.909419i	$0.363474\pi$
$139$			8.00499e9i			1.81884i	−0.415880	+	0.909419i	$0.636526\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−5.15978e9			−1.00000
$145$		0			0
$146$		0			0
$147$	−1.56821e9	+	5.43992e9i	−0.276998	+	0.960871i
$148$	7.87686e9			1.34951
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	1.27138e10			1.99012			0.995061	−	0.0992678i	$-0.0316501\pi$
$151$	1.27138e10			1.99012			0.995061	+	0.0992678i	$0.0316501\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.21067e10			−1.63668
$157$		−	8.87818e9i		−	1.16621i	−0.812398	−	0.583103i	$-0.801838\pi$
$157$		−	8.87818e9i		−	1.16621i	0.812398	−	0.583103i	$-0.198162\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.86962e9			0.540320			0.270160	−	0.962815i	$-0.412923\pi$
$163$	4.86962e9			0.540320			0.270160	+	0.962815i	$0.412923\pi$
$164$		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
$169$	−1.78020e10			−1.67873
$170$		0			0
$171$		−	1.14232e10i		−	1.02166i
$172$	8.48746e9			0.739435
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−1.22852e10	−	1.73543e9i	−0.990169	−	0.139874i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	2.06729e10i		−	1.43168i	−0.698262	−	0.715842i	$-0.746043\pi$
$181$		−	2.06729e10i		−	1.43168i	0.698262	−	0.715842i	$-0.253957\pi$
$182$		0			0
$183$	2.54379e10			1.67669
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.45366e9	−	1.73695e10i	0.139874	−	0.990169i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			1.88302e10i			1.00000i
$193$	−2.32814e10			−1.20782			−0.603908	−	0.797054i	$-0.706391\pi$
$193$	−2.32814e10			−1.20782			−0.603908	+	0.797054i	$0.706391\pi$
$194$		0			0
$195$		0			0
$196$	1.98526e10	+	5.72306e9i	0.960871	+	0.276998i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			3.95879e10i			1.78947i	0.446602	+	0.894733i	$0.352634\pi$
$199$			3.95879e10i			1.78947i	−0.446602	+	0.894733i	$0.647366\pi$
$200$		0			0
$201$			1.57893e10i			0.682306i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			4.41824e10i			1.63668i
$209$		0			0
$210$		0			0
$211$	−4.94819e10			−1.71860			−0.859301	−	0.511471i	$-0.829101\pi$
$211$	−4.94819e10			−1.71860			−0.859301	+	0.511471i	$0.829101\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.01326e9	−	6.38050e10i	0.275939	−	1.95338i
$218$		0			0
$219$	5.39095e10			1.58368
$220$		0			0
$221$		0			0
$222$		0			0
$223$			4.20446e9i			0.113851i	0.998378	+	0.0569257i	$0.0181298\pi$
$223$			4.20446e9i			0.113851i	−0.998378	+	0.0569257i	$0.981870\pi$
$224$		0			0
$225$	3.84434e10			1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−4.16881e10			−1.02166
$229$			4.20623e10i			1.01073i	0.862907	+	0.505363i	$0.168641\pi$
$229$			4.20623e10i			1.01073i	−0.862907	+	0.505363i	$0.831359\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	8.65251e10i		−	1.78145i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	5.04285e10i		−	0.962940i	−0.876463	−	0.481470i	$-0.840103\pi$
$241$		−	5.04285e10i		−	0.962940i	0.876463	−	0.481470i	$-0.159897\pi$
$242$		0			0
$243$			5.43536e10i			1.00000i
$244$		−	9.28337e10i		−	1.67669i
$245$		0			0
$246$		0			0
$247$	−9.78150e10			−1.67213
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−6.33887e10	−	8.95446e9i	−0.990169	−	0.139874i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	6.87195e10			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	9.67684e10	+	1.36698e10i	1.33624	+	0.188761i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	5.76217e10			0.682306
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			1.70984e11i			1.92572i	0.270004	+	0.962859i	$0.412975\pi$
$271$			1.70984e11i			1.92572i	−0.270004	+	0.962859i	$0.587025\pi$
$272$		0			0
$273$	−1.48732e11	−	2.10103e10i	−1.62059	−	0.228929i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.91509e11			1.95448			0.977239	−	0.212142i	$-0.0680439\pi$
$277$	1.91509e11			1.95448			0.977239	+	0.212142i	$0.0680439\pi$
$278$		0			0
$279$			1.99662e11i			1.97277i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			6.69561e10i			0.620514i	0.950653	+	0.310257i	$0.100415\pi$
$283$			6.69561e10i			0.620514i	−0.950653	+	0.310257i	$0.899585\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.18588e11			−1.00000
$290$		0			0
$291$	−1.64786e11			−1.34711
$292$		−	1.96739e11i		−	1.58368i
$293$		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
$300$		−	1.40296e11i		−	1.00000i
$301$	1.04270e11	+	1.47294e10i	0.732166	+	0.103428i
$302$		0			0
$303$		0			0
$304$			1.52137e11i			1.02166i
$305$		0			0
$306$		0			0
$307$			1.33999e11i			0.860954i	0.902601	+	0.430477i	$0.141655\pi$
$307$			1.33999e11i			0.860954i	−0.902601	+	0.430477i	$0.858345\pi$
$308$		0			0
$309$	3.20392e11			1.99926
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	3.29523e11i		−	1.94060i	−0.241908	−	0.970299i	$-0.577773\pi$
$313$		−	3.29523e11i		−	1.94060i	0.241908	−	0.970299i	$-0.422227\pi$
$314$		0			0
$315$		0			0
$316$	−3.15767e11			−1.78145
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.98359e11			1.00000
$325$		−	3.29184e11i		−	1.63668i
$326$		0			0
$327$		−	3.14289e11i		−	1.52007i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.06948e11			0.489717			0.244858	−	0.969559i	$-0.421259\pi$
$331$	1.06948e11			0.489717			0.244858	+	0.969559i	$0.421259\pi$
$332$		0			0
$333$	−3.02813e11			−1.34951
$334$		0			0
$335$		0			0
$336$	−3.26786e10	+	2.31332e11i	−0.139874	+	0.990169i
$337$	−4.46231e11			−1.88463			−0.942313	−	0.334734i	$-0.891354\pi$
$337$	−4.46231e11			−1.88463			−0.942313	+	0.334734i	$0.891354\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2.33960e11	+	1.04762e11i	0.912680	+	0.408675i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$			5.27012e11i			1.90154i	0.309892	+	0.950772i	$0.399707\pi$
$349$			5.27012e11i			1.90154i	−0.309892	+	0.950772i	$0.600293\pi$
$350$		0			0
$351$	4.65421e11			1.63668
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−1.41280e10			−0.0437823
$362$		0			0
$363$			3.30811e11i			1.00000i
$364$	−7.66756e10	+	5.42787e11i	−0.228929	+	1.62059i
$365$		0			0
$366$		0			0
$367$		−	4.09208e11i		−	1.17746i	−0.808329	−	0.588732i	$-0.799628\pi$
$367$		−	4.09208e11i		−	1.17746i	0.808329	−	0.588732i	$-0.200372\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	7.28651e11			1.97277
$373$	−3.90612e10			−0.104485			−0.0522427	−	0.998634i	$-0.516637\pi$
$373$	−3.90612e10			−0.104485			−0.0522427	+	0.998634i	$0.516637\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−7.69425e11			−1.91553			−0.957767	−	0.287546i	$-0.907160\pi$
$379$	−7.69425e11			−1.91553			−0.957767	+	0.287546i	$0.907160\pi$
$380$		0			0
$381$		−	3.19915e11i		−	0.777808i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.26287e11			−0.739435
$388$			6.01374e11i			1.34711i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	9.64747e11i		−	1.94920i	−0.223957	−	0.974599i	$-0.571897\pi$
$397$		−	9.64747e11i		−	1.94920i	0.223957	−	0.974599i	$-0.428103\pi$
$398$		0			0
$399$	−5.12144e11	−	7.23469e10i	−1.01161	−	0.142903i
$400$	−5.12000e11			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	1.70967e12			3.22879
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			6.04364e9i			0.0106793i	0.999986	+	0.00533967i	$0.00169968\pi$
$409$			6.04364e9i			0.0106793i	−0.999986	+	0.00533967i	$0.998300\pi$
$410$		0			0
$411$		0			0
$412$		−	1.16925e12i		−	1.99926i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.12307e12			−1.81884
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	1.19719e12			1.85734			0.928672	−	0.370902i	$-0.120951\pi$
$421$	1.19719e12			1.85734			0.928672	+	0.370902i	$0.120951\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.61107e11	−	1.14048e12i	0.234525	−	1.66020i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	7.23897e11i		−	1.00000i
$433$			1.32812e12i			1.81569i	0.419304	+	0.907846i	$0.362274\pi$
$433$			1.32812e12i			1.81569i	−0.419304	+	0.907846i	$0.637726\pi$
$434$		0			0
$435$		0			0
$436$	−1.14697e12			−1.52007
$437$		0			0
$438$		0			0
$439$			1.48564e12i			1.90907i	0.298095	+	0.954536i	$0.403649\pi$
$439$			1.48564e12i			1.90907i	−0.298095	+	0.954536i	$0.596351\pi$
$440$		0			0
$441$	−7.63200e11	−	2.20014e11i	−0.960871	−	0.276998i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$			1.10509e12i			1.34951i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	8.44230e11	+	1.19258e11i	0.990169	+	0.139874i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			1.78370e12i			1.99012i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.58605e12			1.70096			0.850482	−	0.526004i	$-0.176310\pi$
$457$	1.58605e12			1.70096			0.850482	+	0.526004i	$0.176310\pi$
$458$		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
$463$	−1.84923e12			−1.87015			−0.935075	−	0.354451i	$-0.884668\pi$
$463$	−1.84923e12			−1.87015			−0.935075	+	0.354451i	$0.884668\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		−	1.69852e12i		−	1.63668i
$469$	7.07891e11	+	9.99986e10i	0.675599	+	0.0954368i
$470$		0			0
$471$	1.24557e12			1.16621
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	1.13351e12i		−	1.02166i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			2.59294e12i			2.20871i
$482$		0			0
$483$		0			0
$484$	1.20727e12			1.00000
$485$		0			0
$486$		0			0
$487$	−1.88740e12			−1.52049			−0.760244	−	0.649638i	$-0.774921\pi$
$487$	−1.88740e12			−1.52049			−0.760244	+	0.649638i	$0.774921\pi$
$488$		0			0
$489$			6.83189e11i			0.540320i
$490$		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
$496$		−	2.65916e12i		−	1.97277i
$497$		0			0
$498$		0			0
$499$	−2.17438e12			−1.56994			−0.784969	−	0.619535i	$-0.787321\pi$
$499$	−2.17438e12			−1.56994			−0.784969	+	0.619535i	$0.787321\pi$
$500$		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
$507$		−	2.49756e12i		−	1.67873i
$508$	−1.16751e12			−0.777808
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	3.41427e11	−	2.41697e12i	0.221515	−	1.56811i
$512$		0			0
$513$	1.60263e12			1.02166
$514$		0			0
$515$		0			0
$516$			1.19076e12i			0.739435i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		−	3.93962e10i		−	0.0230248i	−0.999934	−	0.0115124i	$-0.996335\pi$
$523$		−	3.93962e10i		−	0.0230248i	0.999934	−	0.0115124i	$-0.00366460\pi$
$524$		0			0
$525$	2.43475e11	−	1.72356e12i	0.139874	−	0.990169i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.80115e12			1.00000
$530$		0			0
$531$		0			0
$532$	−2.64024e11	+	1.86903e12i	−0.142903	+	1.01161i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.90575e11			−0.446974			−0.223487	−	0.974707i	$-0.571744\pi$
$541$	−8.90575e11			−0.446974			−0.223487	+	0.974707i	$0.571744\pi$
$542$		0			0
$543$	2.90032e12			1.43168
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.00832e11			−0.382471			−0.191236	−	0.981544i	$-0.561249\pi$
$547$	−8.00832e11			−0.382471			−0.191236	+	0.981544i	$0.561249\pi$
$548$		0			0
$549$			3.56884e12i			1.67669i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.87925e12	−	5.47993e11i	−1.76394	−	0.249179i
$554$		0			0
$555$		0			0
$556$			4.09856e12i			1.81884i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$			2.79394e12i			1.21022i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.43687e12	+	3.44239e11i	0.990169	+	0.139874i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−3.51053e11			−0.138201			−0.0691003	−	0.997610i	$-0.522013\pi$
$571$	−3.51053e11			−0.138201			−0.0691003	+	0.997610i	$0.522013\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2.64181e12			−1.00000
$577$		−	2.59193e12i		−	0.973491i	−0.873544	−	0.486745i	$-0.838184\pi$
$577$		−	2.59193e12i		−	0.973491i	0.873544	−	0.486745i	$-0.161816\pi$
$578$		0			0
$579$		−	3.26629e12i		−	1.20782i
$580$		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	$0.5\pi$
$588$	−8.02924e11	+	2.78524e12i	−0.276998	+	0.960871i
$589$	5.88708e12			2.01549
$590$		0			0
$591$		0			0
$592$	4.03295e12			1.34951
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.55402e12			−1.78947
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			2.00008e12i			0.625334i	0.949863	+	0.312667i	$0.101222\pi$
$601$			2.00008e12i			0.625334i	−0.949863	+	0.312667i	$0.898778\pi$
$602$		0			0
$603$	−2.21517e12			−0.682306
$604$	6.50947e12			1.99012
$605$		0			0
$606$		0			0
$607$		−	5.44443e12i		−	1.62781i	−0.580998	−	0.813905i	$-0.697338\pi$
$607$		−	5.44443e12i		−	1.62781i	0.580998	−	0.813905i	$-0.302662\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.75823e12			−1.93313			−0.966564	−	0.256426i	$-0.917455\pi$
$613$	−6.75823e12			−1.93313			−0.966564	+	0.256426i	$0.917455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		−	7.30525e12i		−	1.99999i	−0.00360376	−	0.999994i	$-0.501147\pi$
$619$		−	7.30525e12i		−	1.99999i	0.00360376	−	0.999994i	$-0.498853\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−6.19862e12			−1.63668
$625$	3.81470e12			1.00000
$626$		0			0
$627$		0			0
$628$		−	4.54563e12i		−	1.16621i
$629$		0			0
$630$		0			0
$631$	−1.41926e12			−0.356394			−0.178197	−	0.983995i	$-0.557026\pi$
$631$	−1.41926e12			−0.356394			−0.178197	+	0.983995i	$0.557026\pi$
$632$		0			0
$633$		−	6.94212e12i		−	1.71860i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.88394e12	+	6.53516e12i	−0.453357	+	1.57264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			2.76470e12i			0.637820i	0.947785	+	0.318910i	$0.103317\pi$
$643$			2.76470e12i			0.637820i	−0.947785	+	0.318910i	$0.896683\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.95159e12	+	1.26453e12i	1.95338	+	0.275939i
$652$	2.49325e12			0.540320
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			7.56330e12i			1.58368i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			3.54530e12i			0.722348i	0.932498	+	0.361174i	$0.117624\pi$
$661$			3.54530e12i			0.722348i	−0.932498	+	0.361174i	$0.882376\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−5.89870e11			−0.113851
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.08227e12			1.51868			0.759339	−	0.650695i	$-0.225523\pi$
$673$	8.08227e12			1.51868			0.759339	+	0.650695i	$0.225523\pi$
$674$		0			0
$675$			5.39345e12i			1.00000i
$676$	−9.11465e12			−1.67873
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−1.04364e12	+	7.38797e12i	−0.188425	+	1.33386i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		−	5.84867e12i		−	1.02166i
$685$		0			0
$686$		0			0
$687$	−5.90118e12			−1.01073
$688$	4.34558e12			0.739435
$689$		0			0
$690$		0			0
$691$		−	2.33370e12i		−	0.389399i	−0.980863	−	0.194699i	$-0.937627\pi$
$691$		−	2.33370e12i		−	0.389399i	0.980863	−	0.194699i	$-0.0623731\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−6.29000e12	−	8.88542e11i	−0.990169	−	0.139874i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$			8.92852e12i			1.37873i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8.57856e12			−1.27499			−0.637494	−	0.770455i	$-0.720029\pi$
$709$	−8.57856e12			−1.27499			−0.637494	+	0.770455i	$0.720029\pi$
$710$		0			0
$711$	1.21391e13			1.78145
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	2.02915e12	−	1.43644e13i	0.279644	−	1.97960i
$722$		0			0
$723$	7.07492e12			0.962940
$724$		−	1.05845e13i		−	1.43168i
$725$		0			0
$726$		0			0
$727$			1.01029e13i			1.34135i	0.741752	+	0.670674i	$0.233995\pi$
$727$			1.01029e13i			1.34135i	−0.741752	+	0.670674i	$0.766005\pi$
$728$		0			0
$729$	−7.62560e12			−1.00000
$730$		0			0
$731$		0			0
$732$	1.30242e13			1.67669
$733$		−	6.14544e12i		−	0.786294i	−0.919476	−	0.393147i	$-0.871386\pi$
$733$		−	6.14544e12i		−	0.786294i	0.919476	−	0.393147i	$-0.128614\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−7.10029e12			−0.875742			−0.437871	−	0.899038i	$-0.644267\pi$
$739$	−7.10029e12			−0.875742			−0.437871	+	0.899038i	$0.644267\pi$
$740$		0			0
$741$		−	1.37231e13i		−	1.67213i
$742$		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
$751$	−1.69018e13			−1.93889			−0.969446	−	0.245306i	$-0.921111\pi$
$751$	−1.69018e13			−1.93889			−0.969446	+	0.245306i	$0.921111\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.25628e12	−	8.89319e12i	0.139874	−	0.990169i
$757$	1.73497e13			1.92026			0.960129	−	0.279557i	$-0.0901876\pi$
$757$	1.73497e13			1.92026			0.960129	+	0.279557i	$0.0901876\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−1.40907e13	−	1.99049e12i	−1.50513	−	0.212618i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$			9.64108e12i			1.00000i
$769$		−	1.83824e13i		−	1.89554i	−0.318947	−	0.947772i	$-0.603329\pi$
$769$		−	1.83824e13i		−	1.89554i	0.318947	−	0.947772i	$-0.396671\pi$
$770$		0			0
$771$		0			0
$772$	−1.19201e13			−1.20782
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$			1.98123e13i			1.97277i
$776$		0			0
$777$	−1.91782e12	+	1.35762e13i	−0.188761	+	1.33624i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.01645e13	+	2.93021e12i	0.960871	+	0.276998i
$785$		0			0
$786$		0			0
$787$			1.13604e13i			1.05562i	0.849361	+	0.527812i	$0.176987\pi$
$787$			1.13604e13i			1.05562i	−0.849361	+	0.527812i	$0.823013\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	3.05594e13			2.74420
$794$		0			0
$795$		0			0
$796$			2.02690e13i			1.78947i
$797$		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
$804$			8.08410e12i			0.682306i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		−	2.45590e13i		−	1.99351i	−0.0805218	−	0.996753i	$-0.525659\pi$
$811$		−	2.45590e13i		−	1.99351i	0.0805218	−	0.996753i	$-0.474341\pi$
$812$		0			0
$813$	−2.39883e13			−1.92572
$814$		0			0
$815$		0			0
$816$		0			0
$817$			9.62065e12i			0.755449i
$818$		0			0
$819$	2.94767e12	−	2.08666e13i	0.228929	−	1.62059i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	1.86110e13			1.41407			0.707035	−	0.707178i	$-0.250032\pi$
$823$	1.86110e13			1.41407			0.707035	+	0.707178i	$0.250032\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	1.24152e13i		−	0.912975i	−0.889730	−	0.456488i	$-0.849107\pi$
$829$		−	1.24152e13i		−	0.912975i	0.889730	−	0.456488i	$-0.150893\pi$
$830$		0			0
$831$			2.68680e13i			1.95448i
$832$			2.26214e13i			1.63668i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.80118e13			−1.97277
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.45071e13			1.00000
$842$		0			0
$843$		0			0
$844$	−2.53347e13			−1.71860
$845$		0			0
$846$		0			0
$847$	1.48315e13	+	2.09514e12i	0.990169	+	0.139874i
$848$		0			0
$849$	−9.39368e12			−0.620514
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	2.71268e13i		−	1.75440i	−0.480128	−	0.877199i	$-0.659410\pi$
$853$		−	2.71268e13i		−	1.75440i	0.480128	−	0.877199i	$-0.340590\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		−	1.36498e13i		−	0.855376i	−0.903927	−	0.427688i	$-0.859328\pi$
$859$		−	1.36498e13i		−	0.855376i	0.903927	−	0.427688i	$-0.140672\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.66374e13i		−	1.00000i
$868$	4.61479e12	−	3.26682e13i	0.275939	−	1.95338i
$869$		0			0
$870$		0			0
$871$			1.89682e13i			1.11672i
$872$		0			0
$873$		−	2.31188e13i		−	1.34711i
$874$		0			0
$875$		0			0
$876$	2.76017e13			1.58368
$877$	2.46459e13			1.40685			0.703424	−	0.710771i	$-0.251654\pi$
$877$	2.46459e13			1.40685			0.703424	+	0.710771i	$0.251654\pi$
$878$		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
$883$	−3.38683e13			−1.87487			−0.937434	−	0.348163i	$-0.886806\pi$
$883$	−3.38683e13			−1.87487			−0.937434	+	0.348163i	$0.886806\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−1.43430e13	−	2.02613e12i	−0.770161	−	0.108795i
$890$		0			0
$891$		0			0
$892$			2.15269e12i			0.113851i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.96830e13			1.00000
$901$		0			0
$902$		0			0
$903$	−2.06648e12	+	1.46287e13i	−0.103428	+	0.732166i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.02147e13			0.501177			0.250588	−	0.968094i	$-0.419376\pi$
$907$	1.02147e13			0.501177			0.250588	+	0.968094i	$0.419376\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−2.13443e13			−1.02166
$913$		0			0
$914$		0			0
$915$		0			0
$916$			2.15359e13i			1.01073i
$917$		0			0
$918$		0			0
$919$	6.39629e12			0.295807			0.147903	−	0.989002i	$-0.452748\pi$
$919$	6.39629e12			0.295807			0.147903	+	0.989002i	$0.452748\pi$
$920$		0			0
$921$	−1.87996e13			−0.860954
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−3.00478e13			−1.34951
$926$		0			0
$927$			4.49498e13i			1.99926i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−6.48716e12	+	2.25032e13i	−0.282997	+	0.981680i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			3.26746e13i			1.38478i	0.721521	+	0.692392i	$0.243443\pi$
$937$			3.26746e13i			1.38478i	−0.721521	+	0.692392i	$0.756557\pi$
$938$		0			0
$939$	4.62307e13			1.94060
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		−	4.43009e13i		−	1.78145i
$949$	6.47633e13			2.59198
$950$		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
$961$	−7.64586e13			−2.89182
$962$		0			0
$963$		0			0
$964$		−	2.58194e13i		−	0.962940i
$965$		0			0
$966$		0			0
$967$	1.05389e13			0.387591			0.193796	−	0.981042i	$-0.437920\pi$
$967$	1.05389e13			0.387591			0.193796	+	0.981042i	$0.437920\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$			2.78290e13i			1.00000i
$973$	−7.11277e12	+	5.03514e13i	−0.254408	+	1.80096i
$974$		0			0
$975$	4.61833e13			1.63668
$976$		−	4.75309e13i		−	1.67669i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	4.40935e13			1.52007
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−5.00813e13			−1.67213
$989$		0			0
$990$		0			0
$991$	−3.82277e13			−1.25906			−0.629531	−	0.776976i	$-0.716753\pi$
$991$	−3.82277e13			−1.25906			−0.629531	+	0.776976i	$0.716753\pi$
$992$		0			0
$993$			1.50043e13i			0.489717i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	8.68273e12i		−	0.278309i	−0.990271	−	0.139155i	$-0.955561\pi$
$997$		−	8.68273e12i		−	0.278309i	0.990271	−	0.139155i	$-0.0444385\pi$
$998$		0			0
$999$		−	4.24835e13i		−	1.34951i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.10.c.a.20.2	yes	2
3.2	odd	2	CM	21.10.c.a.20.2	yes	2
7.6	odd	2	inner	21.10.c.a.20.1	✓	2
21.20	even	2	inner	21.10.c.a.20.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.10.c.a.20.1	✓	2	7.6	odd	2	inner
21.10.c.a.20.1	✓	2	21.20	even	2	inner
21.10.c.a.20.2	yes	2	1.1	even	1	trivial
21.10.c.a.20.2	yes	2	3.2	odd	2	CM

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	21.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+140.296i q^{3} +512.000 q^{4} +(6290.00 + 888.542i) q^{7} -19683.0 q^{9} +O(q^{10})\)	\(q+140.296i q^{3} +512.000 q^{4} +(6290.00 + 888.542i) q^{7} -19683.0 q^{9} +71831.6i q^{12} +168542. i q^{13} +262144. q^{16} +580358. i q^{19} +(-124659. + 882463. i) q^{21} -1.95312e6 q^{25} -2.76145e6i q^{27} +(3.22048e6 + 454934. i) q^{28} -1.01439e7i q^{31} -1.00777e7 q^{36} +1.53845e7 q^{37} -2.36458e7 q^{39} +1.65771e7 q^{43} +3.67778e7i q^{48} +(3.87746e7 + 1.11779e7i) q^{49} +8.62937e7i q^{52} -8.14220e7 q^{57} -1.81316e8i q^{61} +(-1.23806e8 - 1.74892e7i) q^{63} +1.34218e8 q^{64} +1.12542e8 q^{67} -3.84255e8i q^{73} -2.74016e8i q^{75} +2.97143e8i q^{76} -6.16732e8 q^{79} +3.87420e8 q^{81} +(-6.38254e7 + 4.51821e8i) q^{84} +(-1.49757e8 + 1.06013e9i) q^{91} +1.42315e9 q^{93} +1.17456e9i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1024 q^{4} + 12580 q^{7} - 39366 q^{9}+O(q^{10}) \) 2 * q + 1024 * q^4 + 12580 * q^7 - 39366 * q^9	\( 2 q + 1024 q^{4} + 12580 q^{7} - 39366 q^{9} + 524288 q^{16} - 249318 q^{21} - 3906250 q^{25} + 6440960 q^{28} - 20155392 q^{36} + 30768980 q^{37} - 47291688 q^{39} + 33154160 q^{43} + 77549186 q^{49} - 162844020 q^{57} - 247612140 q^{63} + 268435456 q^{64} + 225084640 q^{67} - 1233464648 q^{79} + 774840978 q^{81} - 127650816 q^{84} - 299514024 q^{91} + 2846293020 q^{93}+O(q^{100}) \) 2 * q + 1024 * q^4 + 12580 * q^7 - 39366 * q^9 + 524288 * q^16 - 249318 * q^21 - 3906250 * q^25 + 6440960 * q^28 - 20155392 * q^36 + 30768980 * q^37 - 47291688 * q^39 + 33154160 * q^43 + 77549186 * q^49 - 162844020 * q^57 - 247612140 * q^63 + 268435456 * q^64 + 225084640 * q^67 - 1233464648 * q^79 + 774840978 * q^81 - 127650816 * q^84 - 299514024 * q^91 + 2846293020 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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