LMFDB - Embedded newform 21.4.c.a.20.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,4,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, 4, names="a")

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

chi := DirichletCharacter("21.20");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
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:	$1.23904011012$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			20.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.20
Dual form			21.4.c.a.20.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} +8.00000 q^{4} +(-10.0000 + 15.5885i) q^{7} -27.0000 q^{9} -41.5692i q^{12} +62.3538i q^{13} +64.0000 q^{16} -155.885i q^{19} +(81.0000 + 51.9615i) q^{21} -125.000 q^{25} +140.296i q^{27} +(-80.0000 + 124.708i) q^{28} -155.885i q^{31} -216.000 q^{36} -110.000 q^{37} +324.000 q^{39} +520.000 q^{43} -332.554i q^{48} +(-143.000 - 311.769i) q^{49} +498.831i q^{52} -810.000 q^{57} +935.307i q^{61} +(270.000 - 420.888i) q^{63} +512.000 q^{64} -880.000 q^{67} -374.123i q^{73} +649.519i q^{75} -1247.08i q^{76} +884.000 q^{79} +729.000 q^{81} +(648.000 + 415.692i) q^{84} +(-972.000 - 623.538i) q^{91} -810.000 q^{93} +1371.78i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{4} - 20 q^{7} - 54 q^{9} + 128 q^{16} + 162 q^{21} - 250 q^{25} - 160 q^{28} - 432 q^{36} - 220 q^{37} + 648 q^{39} + 1040 q^{43} - 286 q^{49} - 1620 q^{57} + 540 q^{63} + 1024 q^{64} - 1760 q^{67}+ \cdots - 1620 q^{93}+O(q^{100}) $ 2 * q + 16 * q^4 - 20 * q^7 - 54 * q^9 + 128 * q^16 + 162 * q^21 - 250 * q^25 - 160 * q^28 - 432 * q^36 - 220 * q^37 + 648 * q^39 + 1040 * q^43 - 286 * q^49 - 1620 * q^57 + 540 * q^63 + 1024 * q^64 - 1760 * q^67 + 1768 * q^79 + 1458 * q^81 + 1296 * q^84 - 1944 * q^91 - 1620 * q^93

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			$\pi$
$3$		−	5.19615i		−	1.00000i
$3$		−	5.19615i		−	1.00000i
$4$	8.00000			1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$7$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$8$		0			0
$9$	−27.0000			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	41.5692i		−	1.00000i
$13$			62.3538i			1.33030i	0.746712	+	0.665148i	$0.231631\pi$
$13$			62.3538i			1.33030i	−0.746712	+	0.665148i	$0.768369\pi$
$14$		0			0
$15$		0			0
$16$	64.0000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	155.885i		−	1.88223i	−0.338086	−	0.941115i	$-0.609780\pi$
$19$		−	155.885i		−	1.88223i	0.338086	−	0.941115i	$-0.390220\pi$
$20$		0			0
$21$	81.0000	+	51.9615i	0.841698	+	0.539949i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−125.000			−1.00000
$26$		0			0
$27$			140.296i			1.00000i
$28$	−80.0000	+	124.708i	−0.539949	+	0.841698i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	155.885i		−	0.903151i	−0.892233	−	0.451576i	$-0.850862\pi$
$31$		−	155.885i		−	0.903151i	0.892233	−	0.451576i	$-0.149138\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−216.000			−1.00000
$37$	−110.000			−0.488754			−0.244377	−	0.969680i	$-0.578583\pi$
$37$	−110.000			−0.488754			−0.244377	+	0.969680i	$0.578583\pi$
$38$		0			0
$39$	324.000			1.33030
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	520.000			1.84417			0.922084	−	0.386989i	$-0.126485\pi$
$43$	520.000			1.84417			0.922084	+	0.386989i	$0.126485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		−	332.554i		−	1.00000i
$49$	−143.000	−	311.769i	−0.416910	−	0.908948i
$50$		0			0
$51$		0			0
$52$			498.831i			1.33030i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−810.000			−1.88223
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			935.307i			1.96318i	0.191006	+	0.981589i	$0.438825\pi$
$61$			935.307i			1.96318i	−0.191006	+	0.981589i	$0.561175\pi$
$62$		0			0
$63$	270.000	−	420.888i	0.539949	−	0.841698i
$64$	512.000			1.00000
$65$		0			0
$66$		0			0
$67$	−880.000			−1.60461			−0.802307	−	0.596912i	$-0.796394\pi$
$67$	−880.000			−1.60461			−0.802307	+	0.596912i	$0.796394\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	374.123i		−	0.599833i	−0.953966	−	0.299916i	$-0.903041\pi$
$73$		−	374.123i		−	0.599833i	0.953966	−	0.299916i	$-0.0969588\pi$
$74$		0			0
$75$			649.519i			1.00000i
$76$		−	1247.08i		−	1.88223i
$77$		0			0
$78$		0			0
$79$	884.000			1.25896			0.629480	−	0.777017i	$-0.283268\pi$
$79$	884.000			1.25896			0.629480	+	0.777017i	$0.283268\pi$
$80$		0			0
$81$	729.000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	648.000	+	415.692i	0.841698	+	0.539949i
$85$		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
$91$	−972.000	−	623.538i	−1.11971	−	0.718292i
$92$		0			0
$93$	−810.000			−0.903151
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1371.78i			1.43591i	0.696088	+	0.717957i	$0.254922\pi$
$97$			1371.78i			1.43591i	−0.696088	+	0.717957i	$0.745078\pi$
$98$		0			0
$99$		0			0
$100$	−1000.00			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	1028.84i		−	0.984218i	−0.870534	−	0.492109i	$-0.836226\pi$
$103$		−	1028.84i		−	0.984218i	0.870534	−	0.492109i	$-0.163774\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			1122.37i			1.00000i
$109$	646.000			0.567666			0.283833	−	0.958874i	$-0.408394\pi$
$109$	646.000			0.567666			0.283833	+	0.958874i	$0.408394\pi$
$110$		0			0
$111$			571.577i			0.488754i
$112$	−640.000	+	997.661i	−0.539949	+	0.841698i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	1683.55i		−	1.33030i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$		0			0
$123$		0			0
$124$		−	1247.08i		−	0.903151i
$125$		0			0
$126$		0			0
$127$	380.000			0.265508			0.132754	−	0.991149i	$-0.457618\pi$
$127$	380.000			0.265508			0.132754	+	0.991149i	$0.457618\pi$
$128$		0			0
$129$		−	2702.00i		−	1.84417i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	2430.00	+	1558.85i	1.58427	+	1.01631i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$			2026.50i			1.23659i	0.785948	+	0.618293i	$0.212175\pi$
$139$			2026.50i			1.23659i	−0.785948	+	0.618293i	$0.787825\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1728.00			−1.00000
$145$		0			0
$146$		0			0
$147$	−1620.00	+	743.050i	−0.908948	+	0.416910i
$148$	−880.000			−0.488754
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−1748.00			−0.942054			−0.471027	−	0.882119i	$-0.656117\pi$
$151$	−1748.00			−0.942054			−0.471027	+	0.882119i	$0.656117\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	2592.00			1.33030
$157$		−	810.600i		−	0.412057i	−0.978546	−	0.206028i	$-0.933946\pi$
$157$		−	810.600i		−	0.412057i	0.978546	−	0.206028i	$-0.0660539\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3400.00			−1.63379			−0.816897	−	0.576783i	$-0.804308\pi$
$163$	−3400.00			−1.63379			−0.816897	+	0.576783i	$0.804308\pi$
$164$		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
$169$	−1691.00			−0.769686
$170$		0			0
$171$			4208.88i			1.88223i
$172$	4160.00			1.84417
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	1250.00	−	1948.56i	0.539949	−	0.841698i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	3429.46i		−	1.40834i	−0.710031	−	0.704171i	$-0.751319\pi$
$181$		−	3429.46i		−	1.40834i	0.710031	−	0.704171i	$-0.248681\pi$
$182$		0			0
$183$	4860.00			1.96318
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2187.00	−	1402.96i	−0.841698	−	0.539949i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	2660.43i		−	1.00000i
$193$	1150.00			0.428906			0.214453	−	0.976734i	$-0.431203\pi$
$193$	1150.00			0.428906			0.214453	+	0.976734i	$0.431203\pi$
$194$		0			0
$195$		0			0
$196$	−1144.00	−	2494.15i	−0.416910	−	0.908948i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			2026.50i			0.721883i	0.932588	+	0.360942i	$0.117545\pi$
$199$			2026.50i			0.721883i	−0.932588	+	0.360942i	$0.882455\pi$
$200$		0			0
$201$			4572.61i			1.60461i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			3990.65i			1.33030i
$209$		0			0
$210$		0			0
$211$	−6032.00			−1.96806			−0.984028	−	0.178011i	$-0.943034\pi$
$211$	−6032.00			−1.96806			−0.984028	+	0.178011i	$0.943034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2430.00	+	1558.85i	0.760180	+	0.487656i
$218$		0			0
$219$	−1944.00			−0.599833
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	5830.08i		−	1.75072i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$		−	5830.08i		−	1.75072i	0.483469	−	0.875362i	$-0.339377\pi$
$224$		0			0
$225$	3375.00			1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−6480.00			−1.88223
$229$			5300.08i			1.52943i	0.644370	+	0.764714i	$0.277120\pi$
$229$			5300.08i			1.52943i	−0.644370	+	0.764714i	$0.722880\pi$
$230$		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
$237$		−	4593.40i		−	1.25896i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	1247.08i		−	0.333325i	−0.986014	−	0.166662i	$-0.946701\pi$
$241$		−	1247.08i		−	0.333325i	0.986014	−	0.166662i	$-0.0532990\pi$
$242$		0			0
$243$		−	3788.00i		−	1.00000i
$244$			7482.46i			1.96318i
$245$		0			0
$246$		0			0
$247$	9720.00			2.50392
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	2160.00	−	3367.11i	0.539949	−	0.841698i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4096.00			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	1100.00	−	1714.73i	0.263902	−	0.411383i
$260$		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
$268$	−7040.00			−1.60461
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		−	8885.42i		−	1.99170i	−0.0910064	−	0.995850i	$-0.529008\pi$
$271$		−	8885.42i		−	1.99170i	0.0910064	−	0.995850i	$-0.470992\pi$
$272$		0			0
$273$	−3240.00	+	5050.66i	−0.718292	+	1.11971i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4030.00			−0.874149			−0.437074	−	0.899425i	$-0.643985\pi$
$277$	−4030.00			−0.874149			−0.437074	+	0.899425i	$0.643985\pi$
$278$		0			0
$279$			4208.88i			0.903151i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			7700.70i			1.61752i	0.588137	+	0.808761i	$0.299862\pi$
$283$			7700.70i			1.61752i	−0.588137	+	0.808761i	$0.700138\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4913.00			−1.00000
$290$		0			0
$291$	7128.00			1.43591
$292$		−	2992.98i		−	0.599833i
$293$		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
$300$			5196.15i			1.00000i
$301$	−5200.00	+	8106.00i	−0.995758	+	1.55223i
$302$		0			0
$303$		0			0
$304$		−	9976.61i		−	1.88223i
$305$		0			0
$306$		0			0
$307$			1590.02i			0.295594i	0.989018	+	0.147797i	$0.0472182\pi$
$307$			1590.02i			0.295594i	−0.989018	+	0.147797i	$0.952782\pi$
$308$		0			0
$309$	−5346.00			−0.984218
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	4738.89i		−	0.855776i	−0.903832	−	0.427888i	$-0.859258\pi$
$313$		−	4738.89i		−	0.855776i	0.903832	−	0.427888i	$-0.140742\pi$
$314$		0			0
$315$		0			0
$316$	7072.00			1.25896
$317$		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
$324$	5832.00			1.00000
$325$		−	7794.23i		−	1.33030i
$326$		0			0
$327$		−	3356.71i		−	0.567666i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−992.000			−0.164729			−0.0823644	−	0.996602i	$-0.526247\pi$
$331$	−992.000			−0.164729			−0.0823644	+	0.996602i	$0.526247\pi$
$332$		0			0
$333$	2970.00			0.488754
$334$		0			0
$335$		0			0
$336$	5184.00	+	3325.54i	0.841698	+	0.539949i
$337$	4930.00			0.796897			0.398448	−	0.917191i	$-0.369549\pi$
$337$	4930.00			0.796897			0.398448	+	0.917191i	$0.369549\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	6290.00	+	888.542i	0.990169	+	0.139874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$			5300.08i			0.812913i	0.913670	+	0.406456i	$0.133236\pi$
$349$			5300.08i			0.812913i	−0.913670	+	0.406456i	$0.866764\pi$
$350$		0			0
$351$	−8748.00			−1.33030
$352$		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
$361$	−17441.0			−2.54279
$362$		0			0
$363$		−	6916.08i		−	1.00000i
$364$	−7776.00	−	4988.31i	−1.11971	−	0.718292i
$365$		0			0
$366$		0			0
$367$			13374.9i			1.90235i	0.308646	+	0.951177i	$0.400124\pi$
$367$			13374.9i			1.90235i	−0.308646	+	0.951177i	$0.599876\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−6480.00			−0.903151
$373$	12350.0			1.71437			0.857183	−	0.515011i	$-0.172212\pi$
$373$	12350.0			1.71437			0.857183	+	0.515011i	$0.172212\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8584.00			1.16340			0.581702	−	0.813402i	$-0.302387\pi$
$379$	8584.00			1.16340			0.581702	+	0.813402i	$0.302387\pi$
$380$		0			0
$381$		−	1974.54i		−	0.265508i
$382$		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
$387$	−14040.0			−1.84417
$388$			10974.3i			1.43591i
$389$		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
$397$			15775.5i			1.99433i	0.0752196	+	0.997167i	$0.476034\pi$
$397$			15775.5i			1.99433i	−0.0752196	+	0.997167i	$0.523966\pi$
$398$		0			0
$399$	8100.00	−	12626.7i	1.01631	−	1.58427i
$400$	−8000.00			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	9720.00			1.20146
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	14341.4i		−	1.73383i	−0.498458	−	0.866914i	$-0.666100\pi$
$409$		−	14341.4i		−	1.73383i	0.498458	−	0.866914i	$-0.333900\pi$
$410$		0			0
$411$		0			0
$412$		−	8230.71i		−	0.984218i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	10530.0			1.23659
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	17138.0			1.98398			0.991989	−	0.126322i	$-0.0403172\pi$
$421$	17138.0			1.98398			0.991989	+	0.126322i	$0.0403172\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−14580.0	−	9353.07i	−1.65240	−	1.06002i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			8978.95i			1.00000i
$433$		−	17833.2i		−	1.97923i	−0.143727	−	0.989617i	$-0.545909\pi$
$433$		−	17833.2i		−	1.97923i	0.143727	−	0.989617i	$-0.454091\pi$
$434$		0			0
$435$		0			0
$436$	5168.00			0.567666
$437$		0			0
$438$		0			0
$439$			10756.0i			1.16938i	0.811257	+	0.584690i	$0.198784\pi$
$439$			10756.0i			1.16938i	−0.811257	+	0.584690i	$0.801216\pi$
$440$		0			0
$441$	3861.00	+	8417.77i	0.416910	+	0.908948i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$			4572.61i			0.488754i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−5120.00	+	7981.29i	−0.539949	+	0.841698i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			9082.87i			0.942054i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−12710.0			−1.30098			−0.650491	−	0.759514i	$-0.725437\pi$
$457$	−12710.0			−1.30098			−0.650491	+	0.759514i	$0.725437\pi$
$458$		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
$463$	−19780.0			−1.98543			−0.992716	−	0.120482i	$-0.961556\pi$
$463$	−19780.0			−1.98543			−0.992716	+	0.120482i	$0.961556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		−	13468.4i		−	1.33030i
$469$	8800.00	−	13717.8i	0.866410	−	1.35060i
$470$		0			0
$471$	−4212.00			−0.412057
$472$		0			0
$473$		0			0
$474$		0			0
$475$			19485.6i			1.88223i
$476$		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
$481$		−	6858.92i		−	0.650187i
$482$		0			0
$483$		0			0
$484$	10648.0			1.00000
$485$		0			0
$486$		0			0
$487$	−20900.0			−1.94470			−0.972351	−	0.233526i	$-0.924974\pi$
$487$	−20900.0			−1.94470			−0.972351	+	0.233526i	$0.924974\pi$
$488$		0			0
$489$			17666.9i			1.63379i
$490$		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
$496$		−	9976.61i		−	0.903151i
$497$		0			0
$498$		0			0
$499$	15136.0			1.35788			0.678938	−	0.734195i	$-0.262440\pi$
$499$	15136.0			1.35788			0.678938	+	0.734195i	$0.262440\pi$
$500$		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
$507$			8786.69i			0.769686i
$508$	3040.00			0.265508
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	5832.00	+	3741.23i	0.504878	+	0.323879i
$512$		0			0
$513$	21870.0			1.88223
$514$		0			0
$515$		0			0
$516$		−	21616.0i		−	1.84417i
$517$		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
$523$		−	20670.3i		−	1.72820i	−0.503320	−	0.864100i	$-0.667888\pi$
$523$		−	20670.3i		−	1.72820i	0.503320	−	0.864100i	$-0.332112\pi$
$524$		0			0
$525$	−10125.0	−	6495.19i	−0.841698	−	0.539949i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	12167.0			1.00000
$530$		0			0
$531$		0			0
$532$	19440.0	+	12470.8i	1.58427	+	1.01631i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−22678.0			−1.80222			−0.901112	−	0.433586i	$-0.857248\pi$
$541$	−22678.0			−1.80222			−0.901112	+	0.433586i	$0.857248\pi$
$542$		0			0
$543$	−17820.0			−1.40834
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1640.00			0.128193			0.0640963	−	0.997944i	$-0.479584\pi$
$547$	1640.00			0.128193			0.0640963	+	0.997944i	$0.479584\pi$
$548$		0			0
$549$		−	25253.3i		−	1.96318i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8840.00	+	13780.2i	−0.679774	+	1.05966i
$554$		0			0
$555$		0			0
$556$			16212.0i			1.23659i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$			32424.0i			2.45329i
$560$		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
$567$	−7290.00	+	11364.0i	−0.539949	+	0.841698i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	23312.0			1.70854			0.854270	−	0.519829i	$-0.174004\pi$
$571$	23312.0			1.70854			0.854270	+	0.519829i	$0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−13824.0			−1.00000
$577$		−	21325.0i		−	1.53860i	−0.638888	−	0.769300i	$-0.720605\pi$
$577$		−	21325.0i		−	1.53860i	0.638888	−	0.769300i	$-0.279395\pi$
$578$		0			0
$579$		−	5975.58i		−	0.428906i
$580$		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	$0.5\pi$
$588$	−12960.0	+	5944.40i	−0.908948	+	0.416910i
$589$	−24300.0			−1.69994
$590$		0			0
$591$		0			0
$592$	−7040.00			−0.488754
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	10530.0			0.721883
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			3117.69i			0.211603i	0.994387	+	0.105801i	$0.0337408\pi$
$601$			3117.69i			0.211603i	−0.994387	+	0.105801i	$0.966259\pi$
$602$		0			0
$603$	23760.0			1.60461
$604$	−13984.0			−0.942054
$605$		0			0
$606$		0			0
$607$		−	9321.90i		−	0.623335i	−0.950191	−	0.311667i	$-0.899113\pi$
$607$		−	9321.90i		−	0.623335i	0.950191	−	0.311667i	$-0.100887\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	17390.0			1.14580			0.572900	−	0.819625i	$-0.305818\pi$
$613$	17390.0			1.14580			0.572900	+	0.819625i	$0.305818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		−	15432.6i		−	1.00208i	−0.865424	−	0.501040i	$-0.832951\pi$
$619$		−	15432.6i		−	1.00208i	0.865424	−	0.501040i	$-0.167049\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	20736.0			1.33030
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		−	6484.80i		−	0.412057i
$629$		0			0
$630$		0			0
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		0			0
$633$			31343.2i			1.96806i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	19440.0	−	8916.60i	1.20917	−	0.554613i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	29836.3i		−	1.82991i	−0.403561	−	0.914953i	$-0.632228\pi$
$643$		−	29836.3i		−	1.82991i	0.403561	−	0.914953i	$-0.367772\pi$
$644$		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
$651$	8100.00	−	12626.7i	0.487656	−	0.760180i
$652$	−27200.0			−1.63379
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			10101.3i			0.599833i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			27123.9i			1.59606i	0.602615	+	0.798032i	$0.294125\pi$
$661$			27123.9i			1.59606i	−0.602615	+	0.798032i	$0.705875\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−30294.0			−1.75072
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24050.0			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$673$	−24050.0			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$674$		0			0
$675$		−	17537.0i		−	1.00000i
$676$	−13528.0			−0.769686
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−21384.0	−	13717.8i	−1.20860	−	0.775320i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$			33671.1i			1.88223i
$685$		0			0
$686$		0			0
$687$	27540.0			1.52943
$688$	33280.0			1.84417
$689$		0			0
$690$		0			0
$691$			32579.9i			1.79363i	0.442408	+	0.896814i	$0.354124\pi$
$691$			32579.9i			1.79363i	−0.442408	+	0.896814i	$0.645876\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	10000.0	−	15588.5i	0.539949	−	0.841698i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$			17147.3i			0.919947i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−36146.0			−1.91466			−0.957328	−	0.289003i	$-0.906676\pi$
$709$	−36146.0			−1.91466			−0.957328	+	0.289003i	$0.906676\pi$
$710$		0			0
$711$	−23868.0			−1.25896
$712$		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
$721$	16038.0	+	10288.4i	0.828414	+	0.531428i
$722$		0			0
$723$	−6480.00			−0.333325
$724$		−	27435.7i		−	1.40834i
$725$		0			0
$726$		0			0
$727$		−	37692.9i		−	1.92290i	−0.274971	−	0.961452i	$-0.588668\pi$
$727$		−	37692.9i		−	1.92290i	0.274971	−	0.961452i	$-0.411332\pi$
$728$		0			0
$729$	−19683.0			−1.00000
$730$		0			0
$731$		0			0
$732$	38880.0			1.96318
$733$			36726.4i			1.85064i	0.379184	+	0.925321i	$0.376205\pi$
$733$			36726.4i			1.85064i	−0.379184	+	0.925321i	$0.623795\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	31376.0			1.56182			0.780910	−	0.624644i	$-0.214756\pi$
$739$	31376.0			1.56182			0.780910	+	0.624644i	$0.214756\pi$
$740$		0			0
$741$		−	50506.6i		−	2.50392i
$742$		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
$751$	23452.0			1.13951			0.569757	−	0.821813i	$-0.307037\pi$
$751$	23452.0			1.13951			0.569757	+	0.821813i	$0.307037\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−17496.0	−	11223.7i	−0.841698	−	0.539949i
$757$	41470.0			1.99109			0.995543	−	0.0943039i	$-0.0300625\pi$
$757$	41470.0			1.99109			0.995543	+	0.0943039i	$0.0300625\pi$
$758$		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
$763$	−6460.00	+	10070.1i	−0.306511	+	0.477803i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	21283.4i		−	1.00000i
$769$			42400.6i			1.98830i	0.107995	+	0.994151i	$0.465557\pi$
$769$			42400.6i			1.98830i	−0.107995	+	0.994151i	$0.534443\pi$
$770$		0			0
$771$		0			0
$772$	9200.00			0.428906
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$			19485.6i			0.903151i
$776$		0			0
$777$	−8910.00	−	5715.77i	−0.411383	−	0.263902i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−9152.00	−	19953.2i	−0.416910	−	0.908948i
$785$		0			0
$786$		0			0
$787$			8137.17i			0.368563i	0.982874	+	0.184281i	$0.0589958\pi$
$787$			8137.17i			0.368563i	−0.982874	+	0.184281i	$0.941004\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−58320.0			−2.61161
$794$		0			0
$795$		0			0
$796$			16212.0i			0.721883i
$797$		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
$804$			36580.9i			1.60461i
$805$		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
$811$		−	24162.1i		−	1.04617i	−0.852280	−	0.523087i	$-0.824780\pi$
$811$		−	24162.1i		−	1.04617i	0.852280	−	0.523087i	$-0.175220\pi$
$812$		0			0
$813$	−46170.0			−1.99170
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	81060.0i		−	3.47115i
$818$		0			0
$819$	26244.0	+	16835.5i	1.11971	+	0.718292i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−12220.0			−0.517573			−0.258786	−	0.965935i	$-0.583323\pi$
$823$	−12220.0			−0.517573			−0.258786	+	0.965935i	$0.583323\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$			44583.0i			1.86783i	0.357495	+	0.933915i	$0.383631\pi$
$829$			44583.0i			1.86783i	−0.357495	+	0.933915i	$0.616369\pi$
$830$		0			0
$831$			20940.5i			0.874149i
$832$			31925.2i			1.33030i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	21870.0			0.903151
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	−48256.0			−1.96806
$845$		0			0
$846$		0			0
$847$	−13310.0	+	20748.2i	−0.539949	+	0.841698i
$848$		0			0
$849$	40014.0			1.61752
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	17396.7i		−	0.698303i	−0.937066	−	0.349151i	$-0.886470\pi$
$853$		−	17396.7i		−	0.698303i	0.937066	−	0.349151i	$-0.113530\pi$
$854$		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
$859$		−	39438.8i		−	1.56651i	−0.621699	−	0.783256i	$-0.713557\pi$
$859$		−	39438.8i		−	1.56651i	0.621699	−	0.783256i	$-0.286443\pi$
$860$		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
$867$			25528.7i			1.00000i
$868$	19440.0	+	12470.8i	0.760180	+	0.487656i
$869$		0			0
$870$		0			0
$871$		−	54871.4i		−	2.13461i
$872$		0			0
$873$		−	37038.2i		−	1.43591i
$874$		0			0
$875$		0			0
$876$	−15552.0			−0.599833
$877$	50150.0			1.93095			0.965476	−	0.260491i	$-0.0838846\pi$
$877$	50150.0			1.93095			0.965476	+	0.260491i	$0.0838846\pi$
$878$		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
$883$	20680.0			0.788151			0.394076	−	0.919078i	$-0.371065\pi$
$883$	20680.0			0.788151			0.394076	+	0.919078i	$0.371065\pi$
$884$		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
$889$	−3800.00	+	5923.61i	−0.143361	+	0.223478i
$890$		0			0
$891$		0			0
$892$		−	46640.7i		−	1.75072i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	27000.0			1.00000
$901$		0			0
$902$		0			0
$903$	42120.0	+	27020.0i	1.55223	+	0.995758i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−44840.0			−1.64155			−0.820776	−	0.571250i	$-0.806459\pi$
$907$	−44840.0			−1.64155			−0.820776	+	0.571250i	$0.806459\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−51840.0			−1.88223
$913$		0			0
$914$		0			0
$915$		0			0
$916$			42400.6i			1.52943i
$917$		0			0
$918$		0			0
$919$	−2756.00			−0.0989250			−0.0494625	−	0.998776i	$-0.515751\pi$
$919$	−2756.00			−0.0989250			−0.0494625	+	0.998776i	$0.515751\pi$
$920$		0			0
$921$	8262.00			0.295594
$922$		0			0
$923$		0			0
$924$		0			0
$925$	13750.0			0.488754
$926$		0			0
$927$			27778.6i			0.984218i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−48600.0	+	22291.5i	−1.71085	+	0.784720i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			14466.1i			0.504361i	0.967680	+	0.252181i	$0.0811477\pi$
$937$			14466.1i			0.504361i	−0.967680	+	0.252181i	$0.918852\pi$
$938$		0			0
$939$	−24624.0			−0.855776
$940$		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			$\pi$
$948$		−	36747.2i		−	1.25896i
$949$	23328.0			0.797955
$950$		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
$961$	5491.00			0.184317
$962$		0			0
$963$		0			0
$964$		−	9976.61i		−	0.333325i
$965$		0			0
$966$		0			0
$967$	−50020.0			−1.66343			−0.831714	−	0.555204i	$-0.812640\pi$
$967$	−50020.0			−1.66343			−0.831714	+	0.555204i	$0.812640\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		−	30304.0i		−	1.00000i
$973$	−31590.0	−	20265.0i	−1.04083	−	0.667694i
$974$		0			0
$975$	−40500.0			−1.33030
$976$			59859.7i			1.96318i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−17442.0			−0.567666
$982$		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
$988$	77760.0			2.50392
$989$		0			0
$990$		0			0
$991$	45628.0			1.46258			0.731292	−	0.682064i	$-0.238918\pi$
$991$	45628.0			1.46258			0.731292	+	0.682064i	$0.238918\pi$
$992$		0			0
$993$			5154.58i			0.164729i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			55931.4i			1.77669i	0.459172	+	0.888347i	$0.348146\pi$
$997$			55931.4i			1.77669i	−0.459172	+	0.888347i	$0.651854\pi$
$998$		0			0
$999$		−	15432.6i		−	0.488754i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.4.c.a.20.1	✓	2
3.2	odd	2	CM	21.4.c.a.20.1	✓	2
4.3	odd	2		336.4.k.a.209.2		2
7.2	even	3		147.4.g.b.80.1		2
7.3	odd	6		147.4.g.b.68.1		2
7.4	even	3		147.4.g.a.68.1		2
7.5	odd	6		147.4.g.a.80.1		2
7.6	odd	2	inner	21.4.c.a.20.2	yes	2
12.11	even	2		336.4.k.a.209.2		2
21.2	odd	6		147.4.g.b.80.1		2
21.5	even	6		147.4.g.a.80.1		2
21.11	odd	6		147.4.g.a.68.1		2
21.17	even	6		147.4.g.b.68.1		2
21.20	even	2	inner	21.4.c.a.20.2	yes	2
28.27	even	2		336.4.k.a.209.1		2
84.83	odd	2		336.4.k.a.209.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.c.a.20.1	✓	2	1.1	even	1	trivial
21.4.c.a.20.1	✓	2	3.2	odd	2	CM
21.4.c.a.20.2	yes	2	7.6	odd	2	inner
21.4.c.a.20.2	yes	2	21.20	even	2	inner
147.4.g.a.68.1		2	7.4	even	3
147.4.g.a.68.1		2	21.11	odd	6
147.4.g.a.80.1		2	7.5	odd	6
147.4.g.a.80.1		2	21.5	even	6
147.4.g.b.68.1		2	7.3	odd	6
147.4.g.b.68.1		2	21.17	even	6
147.4.g.b.80.1		2	7.2	even	3
147.4.g.b.80.1		2	21.2	odd	6
336.4.k.a.209.1		2	28.27	even	2
336.4.k.a.209.1		2	84.83	odd	2
336.4.k.a.209.2		2	4.3	odd	2
336.4.k.a.209.2		2	12.11	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	21.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} +8.00000 q^{4} +(-10.0000 + 15.5885i) q^{7} -27.0000 q^{9} -41.5692i q^{12} +62.3538i q^{13} +64.0000 q^{16} -155.885i q^{19} +(81.0000 + 51.9615i) q^{21} -125.000 q^{25} +140.296i q^{27} +(-80.0000 + 124.708i) q^{28} -155.885i q^{31} -216.000 q^{36} -110.000 q^{37} +324.000 q^{39} +520.000 q^{43} -332.554i q^{48} +(-143.000 - 311.769i) q^{49} +498.831i q^{52} -810.000 q^{57} +935.307i q^{61} +(270.000 - 420.888i) q^{63} +512.000 q^{64} -880.000 q^{67} -374.123i q^{73} +649.519i q^{75} -1247.08i q^{76} +884.000 q^{79} +729.000 q^{81} +(648.000 + 415.692i) q^{84} +(-972.000 - 623.538i) q^{91} -810.000 q^{93} +1371.78i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{4} - 20 q^{7} - 54 q^{9} + 128 q^{16} + 162 q^{21} - 250 q^{25} - 160 q^{28} - 432 q^{36} - 220 q^{37} + 648 q^{39} + 1040 q^{43} - 286 q^{49} - 1620 q^{57} + 540 q^{63} + 1024 q^{64} - 1760 q^{67}+ \cdots - 1620 q^{93}+O(q^{100}) \) 2 * q + 16 * q^4 - 20 * q^7 - 54 * q^9 + 128 * q^16 + 162 * q^21 - 250 * q^25 - 160 * q^28 - 432 * q^36 - 220 * q^37 + 648 * q^39 + 1040 * q^43 - 286 * q^49 - 1620 * q^57 + 540 * q^63 + 1024 * q^64 - 1760 * q^67 + 1768 * q^79 + 1458 * q^81 + 1296 * q^84 - 1944 * q^91 - 1620 * q^93

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