LMFDB - Embedded newform 2004.1.g.e.2003.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2004,1,Mod(2003,2004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2004, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2004.2003");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2004 = 2^{2} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2004.g (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:	$1.00012628532$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} + \cdots)$

Embedding invariants

Embedding label			2003.7
Root			$0.654861 + 0.755750i$ of defining polynomial
Character	$\chi$	$=$	2004.2003
Dual form			2004.1.g.e.2003.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.415415 - 0.909632i) q^{2} +(-0.142315 + 0.989821i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(0.841254 + 0.540641i) q^{6} +1.08128i q^{7} +(-0.959493 + 0.281733i) q^{8} +(-0.959493 - 0.281733i) q^{9} +O(q^{10})$	\(q+(0.415415 - 0.909632i) q^{2} +(-0.142315 + 0.989821i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(0.841254 + 0.540641i) q^{6} +1.08128i q^{7} +(-0.959493 + 0.281733i) q^{8} +(-0.959493 - 0.281733i) q^{9} +1.30972 q^{11} +(0.841254 - 0.540641i) q^{12} +(0.983568 + 0.449181i) q^{14} +(-0.142315 + 0.989821i) q^{16} +(-0.654861 + 0.755750i) q^{18} +0.563465i q^{19} +(-1.07028 - 0.153882i) q^{21} +(0.544078 - 1.19136i) q^{22} +(-0.142315 - 0.989821i) q^{24} -1.00000 q^{25} +(0.415415 - 0.909632i) q^{27} +(0.817178 - 0.708089i) q^{28} +1.08128i q^{29} +1.81926i q^{31} +(0.841254 + 0.540641i) q^{32} +(-0.186393 + 1.29639i) q^{33} +(0.415415 + 0.909632i) q^{36} +(0.512546 + 0.234072i) q^{38} +(-0.584585 + 0.909632i) q^{42} +(-0.857685 - 0.989821i) q^{44} +1.91899 q^{47} +(-0.959493 - 0.281733i) q^{48} -0.169170 q^{49} +(-0.415415 + 0.909632i) q^{50} +(-0.654861 - 0.755750i) q^{54} +(-0.304632 - 1.03748i) q^{56} +(-0.557730 - 0.0801894i) q^{57} +(0.983568 + 0.449181i) q^{58} +0.284630 q^{61} +(1.65486 + 0.755750i) q^{62} +(0.304632 - 1.03748i) q^{63} +(0.841254 - 0.540641i) q^{64} +(1.10181 + 0.708089i) q^{66} +1.00000 q^{72} +(0.142315 - 0.989821i) q^{75} +(0.425839 - 0.368991i) q^{76} +1.41618i q^{77} +(0.841254 + 0.540641i) q^{81} +(0.584585 + 0.909632i) q^{84} +(-1.07028 - 0.153882i) q^{87} +(-1.25667 + 0.368991i) q^{88} -1.97964i q^{89} +(-1.80075 - 0.258908i) q^{93} +(0.797176 - 1.74557i) q^{94} +(-0.654861 + 0.755750i) q^{96} -1.30972 q^{97} +(-0.0702757 + 0.153882i) q^{98} +(-1.25667 - 0.368991i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} - q^{3} - q^{4} - q^{6} - q^{8} - q^{9}+O(q^{10}) $ 10 * q - q^2 - q^3 - q^4 - q^6 - q^8 - q^9	$ 10 q - q^{2} - q^{3} - q^{4} - q^{6} - q^{8} - q^{9} + 2 q^{11} - q^{12} - q^{16} - q^{18} + 2 q^{22} - q^{24} - 10 q^{25} - q^{27} - q^{32} + 2 q^{33} - q^{36} - 11 q^{42} - 9 q^{44} + 2 q^{47} - q^{48} - 12 q^{49} + q^{50} - q^{54} + 2 q^{61} + 11 q^{62} - q^{64} + 2 q^{66} + 10 q^{72} + q^{75} - q^{81} + 11 q^{84} + 2 q^{88} + 2 q^{94} - q^{96} - 2 q^{97} + 10 q^{98} + 2 q^{99}+O(q^{100}) $ 10 * q - q^2 - q^3 - q^4 - q^6 - q^8 - q^9 + 2 * q^11 - q^12 - q^16 - q^18 + 2 * q^22 - q^24 - 10 * q^25 - q^27 - q^32 + 2 * q^33 - q^36 - 11 * q^42 - 9 * q^44 + 2 * q^47 - q^48 - 12 * q^49 + q^50 - q^54 + 2 * q^61 + 11 * q^62 - q^64 + 2 * q^66 + 10 * q^72 + q^75 - q^81 + 11 * q^84 + 2 * q^88 + 2 * q^94 - q^96 - 2 * q^97 + 10 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2004\mathbb{Z}\right)^\times$.

$n$	$673$	$1003$	$1337$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0.415415	−	0.909632i	0.415415	−	0.909632i
$2$	0.415415	−	0.909632i	0.415415	−	0.909632i
$3$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$3$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$4$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	0.841254	+	0.540641i	0.841254	+	0.540641i
$7$			1.08128i			1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$7$			1.08128i			1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$8$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$9$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$10$		0			0
$11$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$11$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$12$	0.841254	−	0.540641i	0.841254	−	0.540641i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	0.983568	+	0.449181i	0.983568	+	0.449181i
$15$		0			0
$16$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$19$			0.563465i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$19$			0.563465i			0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$20$		0			0
$21$	−1.07028	−	0.153882i	−1.07028	−	0.153882i
$22$	0.544078	−	1.19136i	0.544078	−	1.19136i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$25$	−1.00000			−1.00000
$26$		0			0
$27$	0.415415	−	0.909632i	0.415415	−	0.909632i
$28$	0.817178	−	0.708089i	0.817178	−	0.708089i
$29$			1.08128i			1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$29$			1.08128i			1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$30$		0			0
$31$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$31$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$32$	0.841254	+	0.540641i	0.841254	+	0.540641i
$33$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$34$		0			0
$35$		0			0
$36$	0.415415	+	0.909632i	0.415415	+	0.909632i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	0.512546	+	0.234072i	0.512546	+	0.234072i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−0.584585	+	0.909632i	−0.584585	+	0.909632i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$45$		0			0
$46$		0			0
$47$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$47$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$48$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$49$	−0.169170			−0.169170
$50$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$55$		0			0
$56$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$57$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$58$	0.983568	+	0.449181i	0.983568	+	0.449181i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$61$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$62$	1.65486	+	0.755750i	1.65486	+	0.755750i
$63$	0.304632	−	1.03748i	0.304632	−	1.03748i
$64$	0.841254	−	0.540641i	0.841254	−	0.540641i
$65$		0			0
$66$	1.10181	+	0.708089i	1.10181	+	0.708089i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1.00000			1.00000
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.142315	−	0.989821i	0.142315	−	0.989821i
$76$	0.425839	−	0.368991i	0.425839	−	0.368991i
$77$			1.41618i			1.41618i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	0.841254	+	0.540641i	0.841254	+	0.540641i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	0.584585	+	0.909632i	0.584585	+	0.909632i
$85$		0			0
$86$		0			0
$87$	−1.07028	−	0.153882i	−1.07028	−	0.153882i
$88$	−1.25667	+	0.368991i	−1.25667	+	0.368991i
$89$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$89$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.80075	−	0.258908i	−1.80075	−	0.258908i
$94$	0.797176	−	1.74557i	0.797176	−	1.74557i
$95$		0			0
$96$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$97$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$97$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$98$	−0.0702757	+	0.153882i	−0.0702757	+	0.153882i
$99$	−1.25667	−	0.368991i	−1.25667	−	0.368991i
$100$	0.654861	+	0.755750i	0.654861	+	0.755750i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$107$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$108$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−1.07028	−	0.153882i	−1.07028	−	0.153882i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$	−0.304632	+	0.474017i	−0.304632	+	0.474017i
$115$		0			0
$116$	0.817178	−	0.708089i	0.817178	−	0.708089i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.715370			0.715370
$122$	0.118239	−	0.258908i	0.118239	−	0.258908i
$123$		0			0
$124$	1.37491	−	1.19136i	1.37491	−	1.19136i
$125$		0			0
$126$	−0.817178	−	0.708089i	−0.817178	−	0.708089i
$127$		−	1.51150i		−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$127$		−	1.51150i		−	1.51150i	0.654861	−	0.755750i	$-0.272727\pi$
$128$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	1.10181	−	0.708089i	1.10181	−	0.708089i
$133$	−0.609264			−0.609264
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.51150i		−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$137$		−	1.51150i		−	1.51150i	0.654861	−	0.755750i	$-0.272727\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−0.273100	+	1.89945i	−0.273100	+	1.89945i
$142$		0			0
$143$		0			0
$144$	0.415415	−	0.909632i	0.415415	−	0.909632i
$145$		0			0
$146$		0			0
$147$	0.0240754	−	0.167448i	0.0240754	−	0.167448i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−0.158746	−	0.540641i	−0.158746	−	0.540641i
$153$		0			0
$154$	1.28820	+	0.588302i	1.28820	+	0.588302i
$155$		0			0
$156$		0			0
$157$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$157$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.841254	−	0.540641i	0.841254	−	0.540641i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000			1.00000
$167$	1.00000			1.00000
$168$	1.07028	−	0.153882i	1.07028	−	0.153882i
$169$	1.00000			1.00000
$170$		0			0
$171$	0.158746	−	0.540641i	0.158746	−	0.540641i
$172$		0			0
$173$		−	1.51150i		−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$173$		−	1.51150i		−	1.51150i	0.654861	−	0.755750i	$-0.272727\pi$
$174$	−0.584585	+	0.909632i	−0.584585	+	0.909632i
$175$		−	1.08128i		−	1.08128i
$176$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$177$		0			0
$178$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$179$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$179$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$180$		0			0
$181$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$181$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$182$		0			0
$183$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$184$		0			0
$185$		0			0
$186$	−0.983568	+	1.53046i	−0.983568	+	1.53046i
$187$		0			0
$188$	−1.25667	−	1.45027i	−1.25667	−	1.45027i
$189$	0.983568	+	0.449181i	0.983568	+	0.449181i
$190$		0			0
$191$	−0.830830			−0.830830			−0.415415	−	0.909632i	$-0.636364\pi$
$191$	−0.830830			−0.830830			−0.415415	+	0.909632i	$0.636364\pi$
$192$	0.415415	+	0.909632i	0.415415	+	0.909632i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$195$		0			0
$196$	0.110783	+	0.127850i	0.110783	+	0.127850i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−0.857685	+	0.989821i	−0.857685	+	0.989821i
$199$			1.51150i			1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$199$			1.51150i			1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$200$	0.959493	−	0.281733i	0.959493	−	0.281733i
$201$		0			0
$202$		0			0
$203$	−1.16917			−1.16917
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.737982i			0.737982i
$210$		0			0
$211$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$211$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$212$		0			0
$213$		0			0
$214$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$215$		0			0
$216$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$217$	−1.96714			−1.96714
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$223$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$224$	−0.584585	+	0.909632i	−0.584585	+	0.909632i
$225$	0.959493	+	0.281733i	0.959493	+	0.281733i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$	0.304632	+	0.474017i	0.304632	+	0.474017i
$229$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$229$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$230$		0			0
$231$	−1.40176	−	0.201543i	−1.40176	−	0.201543i
$232$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$233$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$233$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$239$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.297176	−	0.650724i	0.297176	−	0.650724i
$243$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$244$	−0.186393	−	0.215109i	−0.186393	−	0.215109i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−0.512546	−	1.74557i	−0.512546	−	1.74557i
$249$		0			0
$250$		0			0
$251$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$251$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$252$	−0.983568	+	0.449181i	−0.983568	+	0.449181i
$253$		0			0
$254$	−1.37491	−	0.627899i	−1.37491	−	0.627899i
$255$		0			0
$256$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.304632	−	1.03748i	0.304632	−	1.03748i
$262$		0			0
$263$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$263$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$264$	−0.186393	−	1.29639i	−0.186393	−	1.29639i
$265$		0			0
$266$	−0.253098	+	0.554206i	−0.253098	+	0.554206i
$267$	1.95949	+	0.281733i	1.95949	+	0.281733i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	−1.37491	−	0.627899i	−1.37491	−	0.627899i
$275$	−1.30972			−1.30972
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	0.512546	−	1.74557i	0.512546	−	1.74557i
$280$		0			0
$281$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$281$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$282$	1.61435	+	1.03748i	1.61435	+	1.03748i
$283$			1.51150i			1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$283$			1.51150i			1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$289$	−1.00000			−1.00000
$290$		0			0
$291$	0.186393	−	1.29639i	0.186393	−	1.29639i
$292$		0			0
$293$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$293$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$294$	−0.142315	−	0.0914602i	−0.142315	−	0.0914602i
$295$		0			0
$296$		0			0
$297$	0.544078	−	1.19136i	0.544078	−	1.19136i
$298$		0			0
$299$		0			0
$300$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	1.07028	−	0.927399i	1.07028	−	0.927399i
$309$		0			0
$310$		0			0
$311$	2.00000			2.00000			1.00000			$0$
$311$	2.00000			2.00000			1.00000			$0$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	0.345139	−	0.755750i	0.345139	−	0.755750i
$315$		0			0
$316$		0			0
$317$			1.97964i			1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$317$			1.97964i			1.97964i	−0.142315	+	0.989821i	$0.545455\pi$
$318$		0			0
$319$			1.41618i			1.41618i
$320$		0			0
$321$	0.239446	−	1.66538i	0.239446	−	1.66538i
$322$		0			0
$323$		0			0
$324$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2.07496i			2.07496i
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	0.415415	−	0.909632i	0.415415	−	0.909632i
$335$		0			0
$336$	0.304632	−	1.03748i	0.304632	−	1.03748i
$337$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$337$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$338$	0.415415	−	0.909632i	0.415415	−	0.909632i
$339$		0			0
$340$		0			0
$341$			2.38273i			2.38273i
$342$	−0.425839	−	0.368991i	−0.425839	−	0.368991i
$343$			0.898361i			0.898361i
$344$		0			0
$345$		0			0
$346$	−1.37491	−	0.627899i	−1.37491	−	0.627899i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$	0.584585	+	0.909632i	0.584585	+	0.909632i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−0.983568	−	0.449181i	−0.983568	−	0.449181i
$351$		0			0
$352$	1.10181	+	0.708089i	1.10181	+	0.708089i
$353$		−	1.08128i		−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$353$		−	1.08128i		−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$354$		0			0
$355$		0			0
$356$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$357$		0			0
$358$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$359$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$359$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$360$		0			0
$361$	0.682507			0.682507
$362$	0.797176	−	1.74557i	0.797176	−	1.74557i
$363$	−0.101808	+	0.708089i	−0.101808	+	0.708089i
$364$		0			0
$365$		0			0
$366$	0.239446	+	0.153882i	0.239446	+	0.153882i
$367$		−	0.563465i		−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$367$		−	0.563465i		−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.983568	+	1.53046i	0.983568	+	1.53046i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−1.84125	+	0.540641i	−1.84125	+	0.540641i
$377$		0			0
$378$	0.817178	−	0.708089i	0.817178	−	0.708089i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	1.49611	+	0.215109i	1.49611	+	0.215109i
$382$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$383$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$383$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$384$	1.00000			1.00000
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.857685	+	0.989821i	0.857685	+	0.989821i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	0.162317	−	0.0476607i	0.162317	−	0.0476607i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.544078	+	1.19136i	0.544078	+	1.19136i
$397$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$397$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$398$	1.37491	+	0.627899i	1.37491	+	0.627899i
$399$	0.0867074	−	0.603063i	0.0867074	−	0.603063i
$400$	0.142315	−	0.989821i	0.142315	−	0.989821i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.485691	+	1.06351i	−0.485691	+	1.06351i
$407$		0			0
$408$		0			0
$409$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$409$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$410$		0			0
$411$	1.49611	+	0.215109i	1.49611	+	0.215109i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.671292	+	0.306569i	0.671292	+	0.306569i
$419$	−0.830830			−0.830830			−0.415415	−	0.909632i	$-0.636364\pi$
$419$	−0.830830			−0.830830			−0.415415	+	0.909632i	$0.636364\pi$
$420$		0			0
$421$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$421$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$422$	1.65486	+	0.755750i	1.65486	+	0.755750i
$423$	−1.84125	−	0.540641i	−1.84125	−	0.540641i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			0.307765i			0.307765i
$428$	1.10181	+	1.27155i	1.10181	+	1.27155i
$429$		0			0
$430$		0			0
$431$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$431$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$432$	0.841254	+	0.540641i	0.841254	+	0.540641i
$433$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$433$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$434$	−0.817178	+	1.78937i	−0.817178	+	1.78937i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	0.162317	+	0.0476607i	0.162317	+	0.0476607i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$447$		0			0
$448$	0.584585	+	0.909632i	0.584585	+	0.909632i
$449$		−	0.563465i		−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$449$		−	0.563465i		−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$450$	0.654861	−	0.755750i	0.654861	−	0.755750i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	0.557730	−	0.0801894i	0.557730	−	0.0801894i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$459$		0			0
$460$		0			0
$461$			0.563465i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$461$			0.563465i			0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$462$	−0.765644	+	1.19136i	−0.765644	+	1.19136i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	−1.07028	−	0.153882i	−1.07028	−	0.153882i
$465$		0			0
$466$	−1.65486	−	0.755750i	−1.65486	−	0.755750i
$467$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$467$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	0.563465i		−	0.563465i
$476$		0			0
$477$		0			0
$478$	0.118239	−	0.258908i	0.118239	−	0.258908i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.468468	−	0.540641i	−0.468468	−	0.540641i
$485$		0			0
$486$	0.415415	+	0.909632i	0.415415	+	0.909632i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i
$489$		0			0
$490$		0			0
$491$	−2.00000			−2.00000			−1.00000			$\pi$
$491$	−2.00000			−2.00000			−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.80075	−	0.258908i	−1.80075	−	0.258908i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$502$	0.698939	−	1.53046i	0.698939	−	1.53046i
$503$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$503$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$504$			1.08128i			1.08128i
$505$		0			0
$506$		0			0
$507$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$508$	−1.14231	+	0.989821i	−1.14231	+	0.989821i
$509$			1.51150i			1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$509$			1.51150i			1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$510$		0			0
$511$		0			0
$512$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$513$	0.512546	+	0.234072i	0.512546	+	0.234072i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.51334			2.51334
$518$		0			0
$519$	1.49611	+	0.215109i	1.49611	+	0.215109i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−0.817178	−	0.708089i	−0.817178	−	0.708089i
$523$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$523$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$524$		0			0
$525$	1.07028	+	0.153882i	1.07028	+	0.153882i
$526$	−0.118239	+	0.258908i	−0.118239	+	0.258908i
$527$		0			0
$528$	−1.25667	−	0.368991i	−1.25667	−	0.368991i
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$	0.398983	+	0.460451i	0.398983	+	0.460451i
$533$		0			0
$534$	1.07028	−	1.66538i	1.07028	−	1.66538i
$535$		0			0
$536$		0			0
$537$	0.273100	−	1.89945i	0.273100	−	1.89945i
$538$		0			0
$539$	−0.221566			−0.221566
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−0.273100	+	1.89945i	−0.273100	+	1.89945i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−1.14231	+	0.989821i	−1.14231	+	0.989821i
$549$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$550$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$551$	−0.609264			−0.609264
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			1.51150i			1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$557$			1.51150i			1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$558$	−1.37491	−	1.19136i	−1.37491	−	1.19136i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$563$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$563$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$564$	1.61435	−	1.03748i	1.61435	−	1.03748i
$565$		0			0
$566$	1.37491	+	0.627899i	1.37491	+	0.627899i
$567$	−0.584585	+	0.909632i	−0.584585	+	0.909632i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	0.118239	−	0.822373i	0.118239	−	0.822373i
$574$		0			0
$575$		0			0
$576$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$577$	−0.830830			−0.830830			−0.415415	−	0.909632i	$-0.636364\pi$
$577$	−0.830830			−0.830830			−0.415415	+	0.909632i	$0.636364\pi$
$578$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−1.65486	−	0.755750i	−1.65486	−	0.755750i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−0.142315	+	0.0914602i	−0.142315	+	0.0914602i
$589$	−1.02509			−1.02509
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$595$		0			0
$596$		0			0
$597$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$598$		0			0
$599$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$599$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$600$	0.142315	+	0.989821i	0.142315	+	0.989821i
$601$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$601$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−0.304632	+	0.474017i	−0.304632	+	0.474017i
$609$	0.166390	−	1.15727i	0.166390	−	1.15727i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.830830			−0.830830			−0.415415	−	0.909632i	$-0.636364\pi$
$613$	−0.830830			−0.830830			−0.415415	+	0.909632i	$0.636364\pi$
$614$		0			0
$615$		0			0
$616$	−0.398983	−	1.35881i	−0.398983	−	1.35881i
$617$			1.08128i			1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$617$			1.08128i			1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	0.830830	−	1.81926i	0.830830	−	1.81926i
$623$	2.14055			2.14055
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$	−0.730471	−	0.105026i	−0.730471	−	0.105026i
$628$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$629$		0			0
$630$		0			0
$631$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$631$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$632$		0			0
$633$	−1.80075	−	0.258908i	−1.80075	−	0.258908i
$634$	1.80075	+	0.822373i	1.80075	+	0.822373i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.28820	+	0.588302i	1.28820	+	0.588302i
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$	−1.41542	−	0.909632i	−1.41542	−	0.909632i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$649$		0			0
$650$		0			0
$651$	0.279953	−	1.94711i	0.279953	−	1.94711i
$652$		0			0
$653$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$653$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	1.88745	+	0.861971i	1.88745	+	0.861971i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$669$	1.95949	+	0.281733i	1.95949	+	0.281733i
$670$		0			0
$671$	0.372786			0.372786
$672$	−0.817178	−	0.708089i	−0.817178	−	0.708089i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$675$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$676$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		−	1.41618i		−	1.41618i
$680$		0			0
$681$		0			0
$682$	2.16741	+	0.989821i	2.16741	+	0.989821i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−0.512546	+	0.234072i	−0.512546	+	0.234072i
$685$		0			0
$686$	0.817178	+	0.373193i	0.817178	+	0.373193i
$687$	0.239446	−	1.66538i	0.239446	−	1.66538i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−1.14231	+	0.989821i	−1.14231	+	0.989821i
$693$	0.398983	−	1.35881i	0.398983	−	1.35881i
$694$		0			0
$695$		0			0
$696$	1.07028	−	0.153882i	1.07028	−	0.153882i
$697$		0			0
$698$		0			0
$699$	1.80075	+	0.258908i	1.80075	+	0.258908i
$700$	−0.817178	+	0.708089i	−0.817178	+	0.708089i
$701$			1.97964i			1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$701$			1.97964i			1.97964i	−0.142315	+	0.989821i	$0.545455\pi$
$702$		0			0
$703$		0			0
$704$	1.10181	−	0.708089i	1.10181	−	0.708089i
$705$		0			0
$706$	−0.983568	−	0.449181i	−0.983568	−	0.449181i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	0.557730	+	1.89945i	0.557730	+	1.89945i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.25667	+	1.45027i	1.25667	+	1.45027i
$717$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$718$	0.345139	−	0.755750i	0.345139	−	0.755750i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	0.283524	−	0.620830i	0.283524	−	0.620830i
$723$		0			0
$724$	−1.25667	−	1.45027i	−1.25667	−	1.45027i
$725$		−	1.08128i		−	1.08128i
$726$	0.601808	+	0.386758i	0.601808	+	0.386758i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$730$		0			0
$731$		0			0
$732$	0.239446	−	0.153882i	0.239446	−	0.153882i
$733$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$733$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$734$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.00000			2.00000			1.00000			$0$
$743$	2.00000			2.00000			1.00000			$0$
$744$	1.80075	−	0.258908i	1.80075	−	0.258908i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1.81926i		−	1.81926i
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	−0.273100	+	1.89945i	−0.273100	+	1.89945i
$753$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$754$		0			0
$755$		0			0
$756$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$757$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$757$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$761$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$762$	0.817178	−	1.27155i	0.817178	−	1.27155i
$763$		0			0
$764$	0.544078	+	0.627899i	0.544078	+	0.627899i
$765$		0			0
$766$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$767$		0			0
$768$	0.415415	−	0.909632i	0.415415	−	0.909632i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		−	1.81926i		−	1.81926i
$776$	1.25667	−	0.368991i	1.25667	−	0.368991i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.983568	+	0.449181i	0.983568	+	0.449181i
$784$	0.0240754	−	0.167448i	0.0240754	−	0.167448i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	0.0405070	−	0.281733i	0.0405070	−	0.281733i
$790$		0			0
$791$		0			0
$792$	1.30972			1.30972
$793$		0			0
$794$	0.118239	−	0.258908i	0.118239	−	0.258908i
$795$		0			0
$796$	1.14231	−	0.989821i	1.14231	−	0.989821i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$	−0.512546	−	0.329393i	−0.512546	−	0.329393i
$799$		0			0
$800$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$801$	−0.557730	+	1.89945i	−0.557730	+	1.89945i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.51150i			1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$809$			1.51150i			1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	0.765644	+	0.883600i	0.765644	+	0.883600i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.544078	−	1.19136i	0.544078	−	1.19136i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	0.817178	−	1.27155i	0.817178	−	1.27155i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	0.186393	−	1.29639i	0.186393	−	1.29639i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.557730	−	0.483276i	0.557730	−	0.483276i
$837$	1.65486	+	0.755750i	1.65486	+	0.755750i
$838$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$839$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$839$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$840$		0			0
$841$	−0.169170			−0.169170
$842$	0.698939	−	1.53046i	0.698939	−	1.53046i
$843$	1.95949	+	0.281733i	1.95949	+	0.281733i
$844$	1.37491	−	1.19136i	1.37491	−	1.19136i
$845$		0			0
$846$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$847$			0.773517i			0.773517i
$848$		0			0
$849$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$853$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$854$	0.279953	+	0.127850i	0.279953	+	0.127850i
$855$		0			0
$856$	1.61435	−	0.474017i	1.61435	−	0.474017i
$857$		−	0.563465i		−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$857$		−	0.563465i		−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$858$		0			0
$859$			1.08128i			1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$859$			1.08128i			1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$860$		0			0
$861$		0			0
$862$	−0.118239	+	0.258908i	−0.118239	+	0.258908i
$863$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$863$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$864$	0.841254	−	0.540641i	0.841254	−	0.540641i
$865$		0			0
$866$	0.797176	−	1.74557i	0.797176	−	1.74557i
$867$	0.142315	−	0.989821i	0.142315	−	0.989821i
$868$	1.28820	+	1.48666i	1.28820	+	1.48666i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.25667	+	0.368991i	1.25667	+	0.368991i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$877$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$878$		0			0
$879$	1.80075	+	0.258908i	1.80075	+	0.258908i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	0.110783	−	0.127850i	0.110783	−	0.127850i
$883$			1.08128i			1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$883$			1.08128i			1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	1.63436			1.63436
$890$		0			0
$891$	1.10181	+	0.708089i	1.10181	+	0.708089i
$892$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$893$			1.08128i			1.08128i
$894$		0			0
$895$		0			0
$896$	1.07028	−	0.153882i	1.07028	−	0.153882i
$897$		0			0
$898$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$899$	−1.96714			−1.96714
$900$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$907$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$911$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$912$	0.158746	−	0.540641i	0.158746	−	0.540641i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.10181	+	1.27155i	1.10181	+	1.27155i
$917$		0			0
$918$		0			0
$919$			0.563465i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$919$			0.563465i			0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$920$		0			0
$921$		0			0
$922$	0.512546	+	0.234072i	0.512546	+	0.234072i
$923$		0			0
$924$	0.765644	+	1.19136i	0.765644	+	1.19136i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.584585	+	0.909632i	−0.584585	+	0.909632i
$929$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$929$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$930$		0			0
$931$		−	0.0953214i		−	0.0953214i
$932$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$933$	−0.284630	+	1.97964i	−0.284630	+	1.97964i
$934$	0.698939	−	1.53046i	0.698939	−	1.53046i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$	0.698939	+	0.449181i	0.698939	+	0.449181i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$947$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$948$		0			0
$949$		0			0
$950$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$951$	−1.95949	−	0.281733i	−1.95949	−	0.281733i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	−0.186393	−	0.215109i	−0.186393	−	0.215109i
$957$	−1.40176	−	0.201543i	−1.40176	−	0.201543i
$958$		0			0
$959$	1.63436			1.63436
$960$		0			0
$961$	−2.30972			−2.30972
$962$		0			0
$963$	1.61435	+	0.474017i	1.61435	+	0.474017i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$967$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$968$	−0.686393	+	0.201543i	−0.686393	+	0.201543i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	1.00000			1.00000
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		−	2.59278i		−	2.59278i
$980$		0			0
$981$		0			0
$982$	−0.830830	+	1.81926i	−0.830830	+	1.81926i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.05384	−	0.295298i	−2.05384	−	0.295298i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−0.983568	+	1.53046i	−0.983568	+	1.53046i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$997$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.1.g.e.2003.7	✓	10
3.2	odd	2		2004.1.g.f.2003.4	yes	10
4.3	odd	2		2004.1.g.f.2003.3	yes	10
12.11	even	2	inner	2004.1.g.e.2003.8	yes	10
167.166	odd	2	CM	2004.1.g.e.2003.7	✓	10
501.500	even	2		2004.1.g.f.2003.4	yes	10
668.667	even	2		2004.1.g.f.2003.3	yes	10
2004.2003	odd	2	inner	2004.1.g.e.2003.8	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.1.g.e.2003.7	✓	10	1.1	even	1	trivial
2004.1.g.e.2003.7	✓	10	167.166	odd	2	CM
2004.1.g.e.2003.8	yes	10	12.11	even	2	inner
2004.1.g.e.2003.8	yes	10	2004.2003	odd	2	inner
2004.1.g.f.2003.3	yes	10	4.3	odd	2
2004.1.g.f.2003.3	yes	10	668.667	even	2
2004.1.g.f.2003.4	yes	10	3.2	odd	2
2004.1.g.f.2003.4	yes	10	501.500	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:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2004.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.415415 - 0.909632i) q^{2} +(-0.142315 + 0.989821i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(0.841254 + 0.540641i) q^{6} +1.08128i q^{7} +(-0.959493 + 0.281733i) q^{8} +(-0.959493 - 0.281733i) q^{9} +O(q^{10})\)	\(q+(0.415415 - 0.909632i) q^{2} +(-0.142315 + 0.989821i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(0.841254 + 0.540641i) q^{6} +1.08128i q^{7} +(-0.959493 + 0.281733i) q^{8} +(-0.959493 - 0.281733i) q^{9} +1.30972 q^{11} +(0.841254 - 0.540641i) q^{12} +(0.983568 + 0.449181i) q^{14} +(-0.142315 + 0.989821i) q^{16} +(-0.654861 + 0.755750i) q^{18} +0.563465i q^{19} +(-1.07028 - 0.153882i) q^{21} +(0.544078 - 1.19136i) q^{22} +(-0.142315 - 0.989821i) q^{24} -1.00000 q^{25} +(0.415415 - 0.909632i) q^{27} +(0.817178 - 0.708089i) q^{28} +1.08128i q^{29} +1.81926i q^{31} +(0.841254 + 0.540641i) q^{32} +(-0.186393 + 1.29639i) q^{33} +(0.415415 + 0.909632i) q^{36} +(0.512546 + 0.234072i) q^{38} +(-0.584585 + 0.909632i) q^{42} +(-0.857685 - 0.989821i) q^{44} +1.91899 q^{47} +(-0.959493 - 0.281733i) q^{48} -0.169170 q^{49} +(-0.415415 + 0.909632i) q^{50} +(-0.654861 - 0.755750i) q^{54} +(-0.304632 - 1.03748i) q^{56} +(-0.557730 - 0.0801894i) q^{57} +(0.983568 + 0.449181i) q^{58} +0.284630 q^{61} +(1.65486 + 0.755750i) q^{62} +(0.304632 - 1.03748i) q^{63} +(0.841254 - 0.540641i) q^{64} +(1.10181 + 0.708089i) q^{66} +1.00000 q^{72} +(0.142315 - 0.989821i) q^{75} +(0.425839 - 0.368991i) q^{76} +1.41618i q^{77} +(0.841254 + 0.540641i) q^{81} +(0.584585 + 0.909632i) q^{84} +(-1.07028 - 0.153882i) q^{87} +(-1.25667 + 0.368991i) q^{88} -1.97964i q^{89} +(-1.80075 - 0.258908i) q^{93} +(0.797176 - 1.74557i) q^{94} +(-0.654861 + 0.755750i) q^{96} -1.30972 q^{97} +(-0.0702757 + 0.153882i) q^{98} +(-1.25667 - 0.368991i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} - q^{3} - q^{4} - q^{6} - q^{8} - q^{9}+O(q^{10}) \) 10 * q - q^2 - q^3 - q^4 - q^6 - q^8 - q^9	\( 10 q - q^{2} - q^{3} - q^{4} - q^{6} - q^{8} - q^{9} + 2 q^{11} - q^{12} - q^{16} - q^{18} + 2 q^{22} - q^{24} - 10 q^{25} - q^{27} - q^{32} + 2 q^{33} - q^{36} - 11 q^{42} - 9 q^{44} + 2 q^{47} - q^{48} - 12 q^{49} + q^{50} - q^{54} + 2 q^{61} + 11 q^{62} - q^{64} + 2 q^{66} + 10 q^{72} + q^{75} - q^{81} + 11 q^{84} + 2 q^{88} + 2 q^{94} - q^{96} - 2 q^{97} + 10 q^{98} + 2 q^{99}+O(q^{100}) \) 10 * q - q^2 - q^3 - q^4 - q^6 - q^8 - q^9 + 2 * q^11 - q^12 - q^16 - q^18 + 2 * q^22 - q^24 - 10 * q^25 - q^27 - q^32 + 2 * q^33 - q^36 - 11 * q^42 - 9 * q^44 + 2 * q^47 - q^48 - 12 * q^49 + q^50 - q^54 + 2 * q^61 + 11 * q^62 - q^64 + 2 * q^66 + 10 * q^72 + q^75 - q^81 + 11 * q^84 + 2 * q^88 + 2 * q^94 - q^96 - 2 * q^97 + 10 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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