LMFDB - Embedded newform 1449.1.bq.b.1189.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,1,Mod(55,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 10]))

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

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

chi := DirichletCharacter("1449.55");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1449.bq (of order $22$, degree $10$, 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:	$0.723145203305$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
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			1189.2
Root			$-0.989821 + 0.142315i$ of defining polynomial
Character	$\chi$	$=$	1449.1189
Dual form			1449.1.bq.b.496.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.29639 - 1.49611i) q^{2} +(-0.415415 - 2.88927i) q^{4} +(0.959493 - 0.281733i) q^{7} +(-3.19584 - 2.05384i) q^{8} +O(q^{10})$	$q+(1.29639 - 1.49611i) q^{2} +(-0.415415 - 2.88927i) q^{4} +(0.959493 - 0.281733i) q^{7} +(-3.19584 - 2.05384i) q^{8} +(0.708089 + 0.817178i) q^{11} +(0.822373 - 1.80075i) q^{14} +(-4.41510 + 1.29639i) q^{16} +2.14055 q^{22} +(-0.909632 + 0.415415i) q^{23} +(-0.654861 + 0.755750i) q^{25} +(-1.21259 - 2.65520i) q^{28} +(-0.215109 + 1.49611i) q^{29} +(-2.20602 + 4.83052i) q^{32} +(0.797176 - 1.74557i) q^{37} +(-0.698939 + 0.449181i) q^{43} +(2.06690 - 2.38533i) q^{44} +(-0.557730 + 1.89945i) q^{46} +(0.841254 - 0.540641i) q^{49} +(0.281733 + 1.95949i) q^{50} +(-1.45027 + 0.425839i) q^{53} +(-3.64502 - 1.07028i) q^{56} +(1.95949 + 2.26138i) q^{58} +(2.45561 + 5.37703i) q^{64} +(0.186393 - 0.215109i) q^{67} +(0.708089 - 0.817178i) q^{71} +(-1.57812 - 3.45561i) q^{74} +(0.909632 + 0.584585i) q^{77} +(1.25667 + 0.368991i) q^{79} +(-0.234072 + 1.62801i) q^{86} +(-0.584585 - 4.06588i) q^{88} +(1.57812 + 2.45561i) q^{92} +(0.281733 - 1.95949i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{4} + 2 q^{7}+O(q^{10}) $ 20 * q + 2 * q^4 + 2 * q^7	$ 20 q + 2 q^{4} + 2 q^{7} - 24 q^{16} - 2 q^{25} - 2 q^{28} + 4 q^{37} + 4 q^{43} - 2 q^{49} + 22 q^{58} + 2 q^{64} - 4 q^{67} - 4 q^{79} - 22 q^{88}+O(q^{100}) $ 20 * q + 2 * q^4 + 2 * q^7 - 24 * q^16 - 2 * q^25 - 2 * q^28 + 4 * q^37 + 4 * q^43 - 2 * q^49 + 22 * q^58 + 2 * q^64 - 4 * q^67 - 4 * q^79 - 22 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times$.

$n$	$442$	$829$	$1289$
$\chi(n)$	$e\left(\frac{4}{11}\right)$	$-1$	$1$

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 $

$2$	1.29639	−	1.49611i	1.29639	−	1.49611i	0.540641	−	0.841254i	$-0.318182\pi$
$2$	1.29639	−	1.49611i	1.29639	−	1.49611i	0.755750	−	0.654861i	$-0.227273\pi$
$3$		0			0
$3$		0			0
$4$	−0.415415	−	2.88927i	−0.415415	−	2.88927i
$5$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$5$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$6$		0			0
$7$	0.959493	−	0.281733i	0.959493	−	0.281733i
$7$	0.959493	−	0.281733i	0.959493	−	0.281733i
$8$	−3.19584	−	2.05384i	−3.19584	−	2.05384i
$9$		0			0
$10$		0			0
$11$	0.708089	+	0.817178i	0.708089	+	0.817178i	0.989821	−	0.142315i	$-0.0454545\pi$
$11$	0.708089	+	0.817178i	0.708089	+	0.817178i	−0.281733	+	0.959493i	$0.590909\pi$
$12$		0			0
$13$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$13$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$14$	0.822373	−	1.80075i	0.822373	−	1.80075i
$15$		0			0
$16$	−4.41510	+	1.29639i	−4.41510	+	1.29639i
$17$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$17$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$18$		0			0
$19$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$19$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$20$		0			0
$21$		0			0
$22$	2.14055			2.14055
$23$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$23$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$24$		0			0
$25$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$26$		0			0
$27$		0			0
$28$	−1.21259	−	2.65520i	−1.21259	−	2.65520i
$29$	−0.215109	+	1.49611i	−0.215109	+	1.49611i	0.540641	+	0.841254i	$0.318182\pi$
$29$	−0.215109	+	1.49611i	−0.215109	+	1.49611i	−0.755750	+	0.654861i	$0.772727\pi$
$30$		0			0
$31$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$31$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$32$	−2.20602	+	4.83052i	−2.20602	+	4.83052i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$37$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$41$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$42$		0			0
$43$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$43$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$44$	2.06690	−	2.38533i	2.06690	−	2.38533i
$45$		0			0
$46$	−0.557730	+	1.89945i	−0.557730	+	1.89945i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	0.841254	−	0.540641i	0.841254	−	0.540641i
$50$	0.281733	+	1.95949i	0.281733	+	1.95949i
$51$		0			0
$52$		0			0
$53$	−1.45027	+	0.425839i	−1.45027	+	0.425839i	−0.909632	−	0.415415i	$-0.863636\pi$
$53$	−1.45027	+	0.425839i	−1.45027	+	0.425839i	−0.540641	+	0.841254i	$0.681818\pi$
$54$		0			0
$55$		0			0
$56$	−3.64502	−	1.07028i	−3.64502	−	1.07028i
$57$		0			0
$58$	1.95949	+	2.26138i	1.95949	+	2.26138i
$59$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$59$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$60$		0			0
$61$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$61$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$62$		0			0
$63$		0			0
$64$	2.45561	+	5.37703i	2.45561	+	5.37703i
$65$		0			0
$66$		0			0
$67$	0.186393	−	0.215109i	0.186393	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$67$	0.186393	−	0.215109i	0.186393	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.708089	−	0.817178i	0.708089	−	0.817178i	−0.281733	−	0.959493i	$-0.590909\pi$
$71$	0.708089	−	0.817178i	0.708089	−	0.817178i	0.989821	+	0.142315i	$0.0454545\pi$
$72$		0			0
$73$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$73$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$74$	−1.57812	−	3.45561i	−1.57812	−	3.45561i
$75$		0			0
$76$		0			0
$77$	0.909632	+	0.584585i	0.909632	+	0.584585i
$78$		0			0
$79$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$79$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$83$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$84$		0			0
$85$		0			0
$86$	−0.234072	+	1.62801i	−0.234072	+	1.62801i
$87$		0			0
$88$	−0.584585	−	4.06588i	−0.584585	−	4.06588i
$89$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$89$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$90$		0			0
$91$		0			0
$92$	1.57812	+	2.45561i	1.57812	+	2.45561i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$97$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$98$	0.281733	−	1.95949i	0.281733	−	1.95949i
$99$		0			0
$100$	2.45561	+	1.57812i	2.45561	+	1.57812i
$101$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$101$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$102$		0			0
$103$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$103$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$104$		0			0
$105$		0			0
$106$	−1.24302	+	2.72183i	−1.24302	+	2.72183i
$107$	0.474017	+	0.304632i	0.474017	+	0.304632i	0.755750	−	0.654861i	$-0.227273\pi$
$107$	0.474017	+	0.304632i	0.474017	+	0.304632i	−0.281733	+	0.959493i	$0.590909\pi$
$108$		0			0
$109$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$109$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	−1.00000			$1.00000\pi$
$110$		0			0
$111$		0			0
$112$	−3.87102	+	2.48775i	−3.87102	+	2.48775i
$113$	1.19136	−	1.37491i	1.19136	−	1.37491i	0.281733	−	0.959493i	$-0.409091\pi$
$113$	1.19136	−	1.37491i	1.19136	−	1.37491i	0.909632	−	0.415415i	$-0.136364\pi$
$114$		0			0
$115$		0			0
$116$	4.41204			4.41204
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.0240754	+	0.167448i	−0.0240754	+	0.167448i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$127$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$128$	6.13278	+	1.80075i	6.13278	+	1.80075i
$129$		0			0
$130$		0			0
$131$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$131$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$132$		0			0
$133$		0			0
$134$	−0.0801894	−	0.557730i	−0.0801894	−	0.557730i
$135$		0			0
$136$		0			0
$137$	−1.97964			−1.97964			−0.989821	−	0.142315i	$-0.954545\pi$
$137$	−1.97964			−1.97964			−0.989821	+	0.142315i	$0.954545\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−0.304632	−	2.11876i	−0.304632	−	2.11876i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−5.37459	−	1.57812i	−5.37459	−	1.57812i
$149$	−1.19136	−	1.37491i	−1.19136	−	1.37491i	−0.909632	−	0.415415i	$-0.863636\pi$
$149$	−1.19136	−	1.37491i	−1.19136	−	1.37491i	−0.281733	−	0.959493i	$-0.590909\pi$
$150$		0			0
$151$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$151$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$152$		0			0
$153$		0			0
$154$	2.05384	−	0.603063i	2.05384	−	0.603063i
$155$		0			0
$156$		0			0
$157$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$157$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$158$	2.18119	−	1.40176i	2.18119	−	1.40176i
$159$		0			0
$160$		0			0
$161$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$162$		0			0
$163$	0.857685	−	0.989821i	0.857685	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$163$	0.857685	−	0.989821i	0.857685	−	0.989821i	1.00000			$0$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$167$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$168$		0			0
$169$	0.841254	+	0.540641i	0.841254	+	0.540641i
$170$		0			0
$171$		0			0
$172$	1.58816	+	1.83283i	1.58816	+	1.83283i
$173$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$173$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$174$		0			0
$175$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$176$	−4.18567	−	2.68996i	−4.18567	−	2.68996i
$177$		0			0
$178$		0			0
$179$	0.755750	+	1.65486i	0.755750	+	1.65486i	0.755750	+	0.654861i	$0.227273\pi$
$179$	0.755750	+	1.65486i	0.755750	+	1.65486i			1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$181$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$182$		0			0
$183$		0			0
$184$	3.76024	+	0.540641i	3.76024	+	0.540641i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.540641	−	0.158746i	0.540641	−	0.158746i		−	1.00000i	$-0.5\pi$
$191$	0.540641	−	0.158746i	0.540641	−	0.158746i	0.540641	+	0.841254i	$0.318182\pi$
$192$		0			0
$193$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$193$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$194$		0			0
$195$		0			0
$196$	−1.91153	−	2.20602i	−1.91153	−	2.20602i
$197$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.755750	−	0.654861i	$-0.772727\pi$
$197$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.989821	+	0.142315i	$0.954545\pi$
$198$		0			0
$199$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$199$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$200$	3.64502	−	1.07028i	3.64502	−	1.07028i
$201$		0			0
$202$		0			0
$203$	0.215109	+	1.49611i	0.215109	+	1.49611i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.239446	+	1.66538i	0.239446	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$211$	0.239446	+	1.66538i	0.239446	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$212$	1.83283	+	4.01334i	1.83283	+	4.01334i
$213$		0			0
$214$	1.07028	−	0.314261i	1.07028	−	0.314261i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.368991	+	0.425839i	0.368991	+	0.425839i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$223$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$224$	−0.755750	+	5.25635i	−0.755750	+	5.25635i
$225$		0			0
$226$	−0.512546	−	3.56484i	−0.512546	−	3.56484i
$227$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$227$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	3.76024	−	4.33955i	3.76024	−	4.33955i
$233$	−0.474017	+	0.304632i	−0.474017	+	0.304632i	−0.755750	−	0.654861i	$-0.772727\pi$
$233$	−0.474017	+	0.304632i	−0.474017	+	0.304632i	0.281733	+	0.959493i	$0.409091\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.234072	−	0.512546i	0.234072	−	0.512546i	−0.755750	−	0.654861i	$-0.772727\pi$
$239$	0.234072	−	0.512546i	0.234072	−	0.512546i	0.989821	+	0.142315i	$0.0454545\pi$
$240$		0			0
$241$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$241$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$242$	0.219310	+	0.253098i	0.219310	+	0.253098i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$251$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$252$		0			0
$253$	−0.983568	−	0.449181i	−0.983568	−	0.449181i
$254$	−1.64475			−1.64475
$255$		0			0
$256$	5.67177	−	3.64502i	5.67177	−	3.64502i
$257$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$257$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$258$		0			0
$259$	0.273100	−	1.89945i	0.273100	−	1.89945i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.89945	+	0.557730i	1.89945	+	0.557730i	0.989821	+	0.142315i	$0.0454545\pi$
$263$	1.89945	+	0.557730i	1.89945	+	0.557730i	0.909632	+	0.415415i	$0.136364\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.698939	−	0.449181i	−0.698939	−	0.449181i
$269$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$269$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$270$		0			0
$271$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$271$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$272$		0			0
$273$		0			0
$274$	−2.56639	+	2.96177i	−2.56639	+	2.96177i
$275$	−1.08128			−1.08128
$276$		0			0
$277$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$277$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.627899	−	1.37491i	−0.627899	−	1.37491i	−0.909632	−	0.415415i	$-0.863636\pi$
$281$	−0.627899	−	1.37491i	−0.627899	−	1.37491i	0.281733	−	0.959493i	$-0.409091\pi$
$282$		0			0
$283$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$283$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$284$	−2.65520	−	1.70640i	−2.65520	−	1.70640i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$293$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$		0			0
$296$	−6.13278	+	3.94130i	−6.13278	+	3.94130i
$297$		0			0
$298$	−3.60149			−3.60149
$299$		0			0
$300$		0			0
$301$	−0.544078	+	0.627899i	−0.544078	+	0.627899i
$302$	−2.18119	+	1.40176i	−2.18119	+	1.40176i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$307$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$308$	1.31115	−	2.87102i	1.31115	−	2.87102i
$309$		0			0
$310$		0			0
$311$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$311$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$312$		0			0
$313$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$313$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$314$		0			0
$315$		0			0
$316$	0.544078	−	3.78415i	0.544078	−	3.78415i
$317$	0.234072	+	0.512546i	0.234072	+	0.512546i	0.989821	−	0.142315i	$-0.0454545\pi$
$317$	0.234072	+	0.512546i	0.234072	+	0.512546i	−0.755750	+	0.654861i	$0.772727\pi$
$318$		0			0
$319$	−1.37491	+	0.883600i	−1.37491	+	0.883600i
$320$		0			0
$321$		0			0
$322$			1.97964i			1.97964i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−0.368991	−	2.56639i	−0.368991	−	2.56639i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	0.654861	+	0.755750i	$0.272727\pi$
$331$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$337$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−1.00000			$\pi$
$338$	1.89945	−	0.557730i	1.89945	−	0.557730i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.654861	−	0.755750i	0.654861	−	0.755750i
$344$	3.15625			3.15625
$345$		0			0
$346$		0			0
$347$	0.368991	−	0.425839i	0.368991	−	0.425839i	−0.540641	−	0.841254i	$-0.681818\pi$
$347$	0.368991	−	0.425839i	0.368991	−	0.425839i	0.909632	+	0.415415i	$0.136364\pi$
$348$		0			0
$349$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$349$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$350$	0.822373	+	1.80075i	0.822373	+	1.80075i
$351$		0			0
$352$	−5.50945	+	1.61772i	−5.50945	+	1.61772i
$353$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$353$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	3.45561	+	1.01466i	3.45561	+	1.01466i
$359$	0.755750	−	1.65486i	0.755750	−	1.65486i		−	1.00000i	$-0.5\pi$
$359$	0.755750	−	1.65486i	0.755750	−	1.65486i	0.755750	−	0.654861i	$-0.227273\pi$
$360$		0			0
$361$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	3.47758	−	3.01334i	3.47758	−	3.01334i
$369$		0			0
$370$		0			0
$371$	−1.27155	+	0.817178i	−1.27155	+	0.817178i
$372$		0			0
$373$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$373$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.857685	−	0.989821i	−0.857685	−	0.989821i	0.142315	−	0.989821i	$-0.454545\pi$
$379$	−0.857685	−	0.989821i	−0.857685	−	0.989821i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	0.463379	−	1.01466i	0.463379	−	1.01466i
$383$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$383$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$384$		0			0
$385$		0			0
$386$	1.38365	+	3.02977i	1.38365	+	3.02977i
$387$		0			0
$388$		0			0
$389$	−0.708089	+	0.817178i	−0.708089	+	0.817178i	−0.989821	−	0.142315i	$-0.954545\pi$
$389$	−0.708089	+	0.817178i	−0.708089	+	0.817178i	0.281733	+	0.959493i	$0.409091\pi$
$390$		0			0
$391$		0			0
$392$	−3.79891			−3.79891
$393$		0			0
$394$	−3.02977	+	1.94711i	−3.02977	+	1.94711i
$395$		0			0
$396$		0			0
$397$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$397$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$398$		0			0
$399$		0			0
$400$	1.91153	−	4.18567i	1.91153	−	4.18567i
$401$	0.540641	+	0.158746i	0.540641	+	0.158746i	0.540641	−	0.841254i	$-0.318182\pi$
$401$	0.540641	+	0.158746i	0.540641	+	0.158746i			1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	2.51722	+	1.61772i	2.51722	+	1.61772i
$407$	1.99091	−	0.584585i	1.99091	−	0.584585i
$408$		0			0
$409$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$409$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$419$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$420$		0			0
$421$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$421$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$422$	2.80202	+	1.80075i	2.80202	+	1.80075i
$423$		0			0
$424$	5.50945	+	1.61772i	5.50945	+	1.61772i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.683252	−	1.49611i	0.683252	−	1.49611i
$429$		0			0
$430$		0			0
$431$	0.258908	−	1.80075i	0.258908	−	1.80075i	−0.281733	−	0.959493i	$-0.590909\pi$
$431$	0.258908	−	1.80075i	0.258908	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$432$		0			0
$433$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$433$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$434$		0			0
$435$		0			0
$436$	0.830830			0.830830
$437$		0			0
$438$		0			0
$439$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$439$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	0.755750	+	0.654861i	$0.227273\pi$
$443$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	−0.909632	+	0.415415i	$0.863636\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3.87102	+	4.46740i	3.87102	+	4.46740i
$449$	−0.368991	−	0.425839i	−0.368991	−	0.425839i	0.540641	−	0.841254i	$-0.318182\pi$
$449$	−0.368991	−	0.425839i	−0.368991	−	0.425839i	−0.909632	+	0.415415i	$0.863636\pi$
$450$		0			0
$451$		0			0
$452$	−4.46740	−	2.87102i	−4.46740	−	2.87102i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.10181	−	0.708089i	1.10181	−	0.708089i	0.142315	−	0.989821i	$-0.454545\pi$
$457$	1.10181	−	0.708089i	1.10181	−	0.708089i	0.959493	+	0.281733i	$0.0909091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$463$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$464$	−0.989821	−	6.88436i	−0.989821	−	6.88436i
$465$		0			0
$466$	−0.158746	+	1.10411i	−0.158746	+	1.10411i
$467$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$467$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$468$		0			0
$469$	0.118239	−	0.258908i	0.118239	−	0.258908i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.861971	−	0.253098i	−0.861971	−	0.253098i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.463379	−	1.01466i	−0.463379	−	1.01466i
$479$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$479$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.493805			0.493805
$485$		0			0
$486$		0			0
$487$	0.118239	+	0.822373i	0.118239	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$487$	0.118239	+	0.822373i	0.118239	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.66538	+	1.07028i	1.66538	+	1.07028i	0.909632	+	0.415415i	$0.136364\pi$
$491$	1.66538	+	1.07028i	1.66538	+	1.07028i	0.755750	+	0.654861i	$0.227273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.449181	−	0.983568i	0.449181	−	0.983568i
$498$		0			0
$499$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$499$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$503$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$504$		0			0
$505$		0			0
$506$	−1.94711	+	0.889217i	−1.94711	+	0.889217i
$507$		0			0
$508$	−1.58816	+	1.83283i	−1.58816	+	1.83283i
$509$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$509$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$510$		0			0
$511$		0			0
$512$	0.989821	−	6.88436i	0.989821	−	6.88436i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−2.48775	−	2.87102i	−2.48775	−	2.87102i
$519$		0			0
$520$		0			0
$521$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$521$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$522$		0			0
$523$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$523$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$524$		0			0
$525$		0			0
$526$	3.29686	−	2.11876i	3.29686	−	2.11876i
$527$		0			0
$528$		0			0
$529$	0.654861	−	0.755750i	0.654861	−	0.755750i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.03748	+	0.304632i	−1.03748	+	0.304632i
$537$		0			0
$538$		0			0
$539$	1.03748	+	0.304632i	1.03748	+	0.304632i
$540$		0			0
$541$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$541$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$547$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$548$	0.822373	+	5.71973i	0.822373	+	5.71973i
$549$		0			0
$550$	−1.40176	+	1.61772i	−1.40176	+	1.61772i
$551$		0			0
$552$		0			0
$553$	1.30972			1.30972
$554$	−2.48775	+	2.87102i	−2.48775	+	2.87102i
$555$		0			0
$556$		0			0
$557$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$557$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−2.87102	−	0.843008i	−2.87102	−	0.843008i
$563$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$563$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−3.94130	+	1.15727i	−3.94130	+	1.15727i
$569$	−0.0801894	+	0.557730i	−0.0801894	+	0.557730i	0.909632	+	0.415415i	$0.136364\pi$
$569$	−0.0801894	+	0.557730i	−0.0801894	+	0.557730i	−0.989821	+	0.142315i	$0.954545\pi$
$570$		0			0
$571$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$571$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.281733	−	0.959493i	0.281733	−	0.959493i
$576$		0			0
$577$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$577$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$578$	−1.66538	+	1.07028i	−1.66538	+	1.07028i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.37491	−	0.883600i	−1.37491	−	0.883600i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$587$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.25667	+	8.74032i	−1.25667	+	8.74032i
$593$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$593$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$594$		0			0
$595$		0			0
$596$	−3.47758	+	4.01334i	−3.47758	+	4.01334i
$597$		0			0
$598$		0			0
$599$	1.97964			1.97964			0.989821	−	0.142315i	$-0.0454545\pi$
$599$	1.97964			1.97964			0.989821	+	0.142315i	$0.0454545\pi$
$600$		0			0
$601$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$601$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$602$	0.234072	+	1.62801i	0.234072	+	1.62801i
$603$		0			0
$604$	−0.544078	+	3.78415i	−0.544078	+	3.78415i
$605$		0			0
$606$		0			0
$607$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$607$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$613$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$614$		0			0
$615$		0			0
$616$	−1.70640	−	3.73648i	−1.70640	−	3.73648i
$617$	−0.281733	−	1.95949i	−0.281733	−	1.95949i	−0.281733	−	0.959493i	$-0.590909\pi$
$617$	−0.281733	−	1.95949i	−0.281733	−	1.95949i		−	1.00000i	$-0.5\pi$
$618$		0			0
$619$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$619$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$631$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$632$	−3.25827	−	3.76024i	−3.25827	−	3.76024i
$633$		0			0
$634$	1.07028	+	0.314261i	1.07028	+	0.314261i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.460451	+	3.20251i	−0.460451	+	3.20251i
$639$		0			0
$640$		0			0
$641$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$641$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	2.20602	+	1.91153i	2.20602	+	1.91153i
$645$		0			0
$646$		0			0
$647$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$647$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−3.21616	−	2.06690i	−3.21616	−	2.06690i
$653$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.281733	−	0.959493i	$-0.409091\pi$
$653$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.27155	+	0.817178i	1.27155	+	0.817178i	0.989821	−	0.142315i	$-0.0454545\pi$
$659$	1.27155	+	0.817178i	1.27155	+	0.817178i	0.281733	+	0.959493i	$0.409091\pi$
$660$		0			0
$661$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$661$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$662$	0.683252	+	1.49611i	0.683252	+	1.49611i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.425839	−	1.45027i	−0.425839	−	1.45027i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$673$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	−1.00000			$1.00000\pi$
$674$	−3.19584	+	0.938384i	−3.19584	+	0.938384i
$675$		0			0
$676$	1.21259	−	2.65520i	1.21259	−	2.65520i
$677$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$677$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.540641	+	0.158746i	−0.540641	+	0.158746i	−0.540641	−	0.841254i	$-0.681818\pi$
$683$	−0.540641	+	0.158746i	−0.540641	+	0.158746i			1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	−0.281733	−	1.95949i	−0.281733	−	1.95949i
$687$		0			0
$688$	2.50357	−	2.88927i	2.50357	−	2.88927i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−0.158746	−	1.10411i	−0.158746	−	1.10411i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.80075	+	0.822373i	2.80075	+	0.822373i
$701$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$701$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.540641	−	0.841254i	$-0.681818\pi$
$702$		0			0
$703$		0			0
$704$	−2.65520	+	5.81408i	−2.65520	+	5.81408i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.239446	+	1.66538i	0.239446	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$709$	0.239446	+	1.66538i	0.239446	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	4.46740	−	2.87102i	4.46740	−	2.87102i
$717$		0			0
$718$	−1.49611	−	3.27603i	−1.49611	−	3.27603i
$719$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$719$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$720$		0			0
$721$		0			0
$722$	−0.822373	+	1.80075i	−0.822373	+	1.80075i
$723$		0			0
$724$		0			0
$725$	−0.989821	−	1.14231i	−0.989821	−	1.14231i
$726$		0			0
$727$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$727$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$733$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$734$		0			0
$735$		0			0
$736$		−	5.31041i		−	5.31041i
$737$	0.307765			0.307765
$738$		0			0
$739$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$739$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$740$		0			0
$741$		0			0
$742$	−0.425839	+	2.96177i	−0.425839	+	2.96177i
$743$	−1.89945	+	0.557730i	−1.89945	+	0.557730i	−0.909632	+	0.415415i	$0.863636\pi$
$743$	−1.89945	+	0.557730i	−1.89945	+	0.557730i	−0.989821	+	0.142315i	$0.954545\pi$
$744$		0			0
$745$		0			0
$746$	3.19584	+	0.938384i	3.19584	+	0.938384i
$747$		0			0
$748$		0			0
$749$	0.540641	+	0.158746i	0.540641	+	0.158746i
$750$		0			0
$751$	1.10181	+	0.708089i	1.10181	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$751$	1.10181	+	0.708089i	1.10181	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$757$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$758$	−2.59278			−2.59278
$759$		0			0
$760$		0			0
$761$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$761$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$762$		0			0
$763$	0.0405070	+	0.281733i	0.0405070	+	0.281733i
$764$	−0.683252	−	1.49611i	−0.683252	−	1.49611i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$769$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$770$		0			0
$771$		0			0
$772$	4.71228	+	1.38365i	4.71228	+	1.38365i
$773$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$773$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.304632	+	2.11876i	0.304632	+	2.11876i
$779$		0			0
$780$		0			0
$781$	1.16917			1.16917
$782$		0			0
$783$		0			0
$784$	−3.01334	+	3.47758i	−3.01334	+	3.47758i
$785$		0			0
$786$		0			0
$787$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$787$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$788$	−0.755750	+	5.25635i	−0.755750	+	5.25635i
$789$		0			0
$790$		0			0
$791$	0.755750	−	1.65486i	0.755750	−	1.65486i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$797$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$798$		0			0
$799$		0			0
$800$	−2.20602	−	4.83052i	−2.20602	−	4.83052i
$801$		0			0
$802$	0.938384	−	0.603063i	0.938384	−	0.603063i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.215109	+	1.49611i	0.215109	+	1.49611i	0.755750	+	0.654861i	$0.227273\pi$
$809$	0.215109	+	1.49611i	0.215109	+	1.49611i	−0.540641	+	0.841254i	$0.681818\pi$
$810$		0			0
$811$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$811$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$812$	4.23333	−	1.24302i	4.23333	−	1.24302i
$813$		0			0
$814$	1.70640	−	3.73648i	1.70640	−	3.73648i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.03748	−	0.304632i	1.03748	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$821$	1.03748	−	0.304632i	1.03748	−	0.304632i	0.755750	+	0.654861i	$0.227273\pi$
$822$		0			0
$823$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$823$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.563465			−0.563465			−0.281733	−	0.959493i	$-0.590909\pi$
$827$	−0.563465			−0.563465			−0.281733	+	0.959493i	$0.590909\pi$
$828$		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			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$839$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$840$		0			0
$841$	−1.23259	−	0.361922i	−1.23259	−	0.361922i
$842$	−0.683252	+	1.49611i	−0.683252	+	1.49611i
$843$		0			0
$844$	4.71228	−	1.38365i	4.71228	−	1.38365i
$845$		0			0
$846$		0			0
$847$	0.0240754	+	0.167448i	0.0240754	+	0.167448i
$848$	5.85105	−	3.76024i	5.85105	−	3.76024i
$849$		0			0
$850$		0			0
$851$			1.91899i			1.91899i
$852$		0			0
$853$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$853$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$854$		0			0
$855$		0			0
$856$	−0.889217	−	1.94711i	−0.889217	−	1.94711i
$857$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$857$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$858$		0			0
$859$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$859$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$860$		0			0
$861$		0			0
$862$	−2.35848	−	2.72183i	−2.35848	−	2.72183i
$863$	−0.708089	−	0.817178i	−0.708089	−	0.817178i	0.281733	−	0.959493i	$-0.409091\pi$
$863$	−0.708089	−	0.817178i	−0.708089	−	0.817178i	−0.989821	+	0.142315i	$0.954545\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.588302	+	1.28820i	0.588302	+	1.28820i
$870$		0			0
$871$		0			0
$872$	0.708089	−	0.817178i	0.708089	−	0.817178i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$877$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$881$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$882$		0			0
$883$	0.544078	−	1.19136i	0.544078	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$883$	0.544078	−	1.19136i	0.544078	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$884$		0			0
$885$		0			0
$886$	1.40176	+	1.61772i	1.40176	+	1.61772i
$887$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$887$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$888$		0			0
$889$	−0.698939	−	0.449181i	−0.698939	−	0.449181i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	6.39169			6.39169
$897$		0			0
$898$	−1.11546			−1.11546
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−6.63126	+	1.94711i	−6.63126	+	1.94711i
$905$		0			0
$906$		0			0
$907$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	0.142315	−	0.989821i	$-0.454545\pi$
$907$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	−0.415415	+	0.909632i	$0.636364\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.281733	−	0.959493i	$-0.409091\pi$
$911$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$912$		0			0
$913$		0			0
$914$	0.368991	−	2.56639i	0.368991	−	2.56639i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$919$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.797176	+	1.74557i	0.797176	+	1.74557i
$926$	−0.0801894	+	0.557730i	−0.0801894	+	0.557730i
$927$		0			0
$928$	−6.75247	−	4.33955i	−6.75247	−	4.33955i
$929$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$929$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$930$		0			0
$931$		0			0
$932$	1.07708	+	1.24302i	1.07708	+	1.24302i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$937$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$938$	−0.234072	−	0.512546i	−0.234072	−	0.512546i
$939$		0			0
$940$		0			0
$941$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$941$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−1.49611	+	0.961494i	−1.49611	+	0.961494i
$947$	−0.0801894	−	0.557730i	−0.0801894	−	0.557730i	−0.989821	−	0.142315i	$-0.954545\pi$
$947$	−0.0801894	−	0.557730i	−0.0801894	−	0.557730i	0.909632	−	0.415415i	$-0.136364\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.540641	−	0.158746i	−0.540641	−	0.158746i		−	1.00000i	$-0.5\pi$
$953$	−0.540641	−	0.158746i	−0.540641	−	0.158746i	−0.540641	+	0.841254i	$0.681818\pi$
$954$		0			0
$955$		0			0
$956$	−1.57812	−	0.463379i	−1.57812	−	0.463379i
$957$		0			0
$958$		0			0
$959$	−1.89945	+	0.557730i	−1.89945	+	0.557730i
$960$		0			0
$961$	0.415415	+	0.909632i	0.415415	+	0.909632i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$967$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$968$	0.420853	−	0.485691i	0.420853	−	0.485691i
$969$		0			0
$970$		0			0
$971$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$971$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$972$		0			0
$973$		0			0
$974$	1.38365	+	0.889217i	1.38365	+	0.889217i
$975$		0			0
$976$		0			0
$977$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$977$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	3.76024	−	1.10411i	3.76024	−	1.10411i
$983$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$983$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.449181	−	0.698939i	0.449181	−	0.698939i
$990$		0			0
$991$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$991$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$992$		0			0
$993$		0			0
$994$	−0.889217	−	1.94711i	−0.889217	−	1.94711i
$995$		0			0
$996$		0			0
$997$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$997$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$998$	−0.234072	+	0.512546i	−0.234072	+	0.512546i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.1.bq.b.1189.2	yes	20
3.2	odd	2	inner	1449.1.bq.b.1189.1	yes	20
7.6	odd	2	CM	1449.1.bq.b.1189.2	yes	20
21.20	even	2	inner	1449.1.bq.b.1189.1	yes	20
23.13	even	11	inner	1449.1.bq.b.496.2	yes	20
69.59	odd	22	inner	1449.1.bq.b.496.1	✓	20
161.13	odd	22	inner	1449.1.bq.b.496.2	yes	20
483.335	even	22	inner	1449.1.bq.b.496.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1449.1.bq.b.496.1	✓	20	69.59	odd	22	inner
1449.1.bq.b.496.1	✓	20	483.335	even	22	inner
1449.1.bq.b.496.2	yes	20	23.13	even	11	inner
1449.1.bq.b.496.2	yes	20	161.13	odd	22	inner
1449.1.bq.b.1189.1	yes	20	3.2	odd	2	inner
1449.1.bq.b.1189.1	yes	20	21.20	even	2	inner
1449.1.bq.b.1189.2	yes	20	1.1	even	1	trivial
1449.1.bq.b.1189.2	yes	20	7.6	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1449.bq (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(1.29639 - 1.49611i) q^{2} +(-0.415415 - 2.88927i) q^{4} +(0.959493 - 0.281733i) q^{7} +(-3.19584 - 2.05384i) q^{8} +O(q^{10})\)	\(q+(1.29639 - 1.49611i) q^{2} +(-0.415415 - 2.88927i) q^{4} +(0.959493 - 0.281733i) q^{7} +(-3.19584 - 2.05384i) q^{8} +(0.708089 + 0.817178i) q^{11} +(0.822373 - 1.80075i) q^{14} +(-4.41510 + 1.29639i) q^{16} +2.14055 q^{22} +(-0.909632 + 0.415415i) q^{23} +(-0.654861 + 0.755750i) q^{25} +(-1.21259 - 2.65520i) q^{28} +(-0.215109 + 1.49611i) q^{29} +(-2.20602 + 4.83052i) q^{32} +(0.797176 - 1.74557i) q^{37} +(-0.698939 + 0.449181i) q^{43} +(2.06690 - 2.38533i) q^{44} +(-0.557730 + 1.89945i) q^{46} +(0.841254 - 0.540641i) q^{49} +(0.281733 + 1.95949i) q^{50} +(-1.45027 + 0.425839i) q^{53} +(-3.64502 - 1.07028i) q^{56} +(1.95949 + 2.26138i) q^{58} +(2.45561 + 5.37703i) q^{64} +(0.186393 - 0.215109i) q^{67} +(0.708089 - 0.817178i) q^{71} +(-1.57812 - 3.45561i) q^{74} +(0.909632 + 0.584585i) q^{77} +(1.25667 + 0.368991i) q^{79} +(-0.234072 + 1.62801i) q^{86} +(-0.584585 - 4.06588i) q^{88} +(1.57812 + 2.45561i) q^{92} +(0.281733 - 1.95949i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) 20 * q + 2 * q^4 + 2 * q^7	\( 20 q + 2 q^{4} + 2 q^{7} - 24 q^{16} - 2 q^{25} - 2 q^{28} + 4 q^{37} + 4 q^{43} - 2 q^{49} + 22 q^{58} + 2 q^{64} - 4 q^{67} - 4 q^{79} - 22 q^{88}+O(q^{100}) \) 20 * q + 2 * q^4 + 2 * q^7 - 24 * q^16 - 2 * q^25 - 2 * q^28 + 4 * q^37 + 4 * q^43 - 2 * q^49 + 22 * q^58 + 2 * q^64 - 4 * q^67 - 4 * q^79 - 22 * q^88

Introduction

L-functions

Modular forms

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Groups

Database

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