LMFDB - Embedded newform 168.2.i.b.125.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(125,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("168.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	168.i (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:	$1.34148675396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			125.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	168.125
Dual form			168.2.i.b.125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -3.46410i q^{5} -2.44949i q^{6} +(1.00000 - 2.44949i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -3.46410i q^{5} -2.44949i q^{6} +(1.00000 - 2.44949i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +4.89898i q^{10} +5.65685 q^{11} +3.46410i q^{12} +(-1.41421 + 3.46410i) q^{14} +6.00000 q^{15} +4.00000 q^{16} +4.24264 q^{18} -6.92820i q^{20} +(4.24264 + 1.73205i) q^{21} -8.00000 q^{22} -4.89898i q^{24} -7.00000 q^{25} -5.19615i q^{27} +(2.00000 - 4.89898i) q^{28} +2.82843 q^{29} -8.48528 q^{30} +4.89898i q^{31} -5.65685 q^{32} +9.79796i q^{33} +(-8.48528 - 3.46410i) q^{35} -6.00000 q^{36} +9.79796i q^{40} +(-6.00000 - 2.44949i) q^{42} +11.3137 q^{44} +10.3923i q^{45} +6.92820i q^{48} +(-5.00000 - 4.89898i) q^{49} +9.89949 q^{50} -14.1421 q^{53} +7.34847i q^{54} -19.5959i q^{55} +(-2.82843 + 6.92820i) q^{56} -4.00000 q^{58} +10.3923i q^{59} +12.0000 q^{60} -6.92820i q^{62} +(-3.00000 + 7.34847i) q^{63} +8.00000 q^{64} -13.8564i q^{66} +(12.0000 + 4.89898i) q^{70} +8.48528 q^{72} -9.79796i q^{73} -12.1244i q^{75} +(5.65685 - 13.8564i) q^{77} +10.0000 q^{79} -13.8564i q^{80} +9.00000 q^{81} +17.3205i q^{83} +(8.48528 + 3.46410i) q^{84} +4.89898i q^{87} -16.0000 q^{88} -14.6969i q^{90} -8.48528 q^{93} -9.79796i q^{96} +19.5959i q^{97} +(7.07107 + 6.92820i) q^{98} -16.9706 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 4 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 + 4 * q^7 - 12 * q^9	$ 4 q + 8 q^{4} + 4 q^{7} - 12 q^{9} + 24 q^{15} + 16 q^{16} - 32 q^{22} - 28 q^{25} + 8 q^{28} - 24 q^{36} - 24 q^{42} - 20 q^{49} - 16 q^{58} + 48 q^{60} - 12 q^{63} + 32 q^{64} + 48 q^{70} + 40 q^{79} + 36 q^{81} - 64 q^{88}+O(q^{100}) $ 4 * q + 8 * q^4 + 4 * q^7 - 12 * q^9 + 24 * q^15 + 16 * q^16 - 32 * q^22 - 28 * q^25 + 8 * q^28 - 24 * q^36 - 24 * q^42 - 20 * q^49 - 16 * q^58 + 48 * q^60 - 12 * q^63 + 32 * q^64 + 48 * q^70 + 40 * q^79 + 36 * q^81 - 64 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	−1.41421			−1.00000
$2$	−1.41421			−1.00000
$3$			1.73205i			1.00000i
$3$			1.73205i			1.00000i
$4$	2.00000			1.00000
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		−	2.44949i		−	1.00000i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$8$	−2.82843			−1.00000
$9$	−3.00000			−1.00000
$10$			4.89898i			1.54919i
$11$	5.65685			1.70561			0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685			1.70561			0.852803	+	0.522233i	$0.174901\pi$
$12$			3.46410i			1.00000i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−1.41421	+	3.46410i	−0.377964	+	0.925820i
$15$	6.00000			1.54919
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	4.24264			1.00000
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		−	6.92820i		−	1.54919i
$21$	4.24264	+	1.73205i	0.925820	+	0.377964i
$22$	−8.00000			−1.70561
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		−	4.89898i		−	1.00000i
$25$	−7.00000			−1.40000
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$	2.00000	−	4.89898i	0.377964	−	0.925820i
$29$	2.82843			0.525226			0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843			0.525226			0.262613	+	0.964901i	$0.415416\pi$
$30$	−8.48528			−1.54919
$31$			4.89898i			0.879883i	0.898027	+	0.439941i	$0.145001\pi$
$31$			4.89898i			0.879883i	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−5.65685			−1.00000
$33$			9.79796i			1.70561i
$34$		0			0
$35$	−8.48528	−	3.46410i	−1.43427	−	0.585540i
$36$	−6.00000			−1.00000
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$			9.79796i			1.54919i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−6.00000	−	2.44949i	−0.925820	−	0.377964i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	11.3137			1.70561
$45$			10.3923i			1.54919i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$			6.92820i			1.00000i
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$	9.89949			1.40000
$51$		0			0
$52$		0			0
$53$	−14.1421			−1.94257			−0.971286	−	0.237915i	$-0.923536\pi$
$53$	−14.1421			−1.94257			−0.971286	+	0.237915i	$0.923536\pi$
$54$			7.34847i			1.00000i
$55$		−	19.5959i		−	2.64231i
$56$	−2.82843	+	6.92820i	−0.377964	+	0.925820i
$57$		0			0
$58$	−4.00000			−0.525226
$59$			10.3923i			1.35296i	0.736460	+	0.676481i	$0.236496\pi$
$59$			10.3923i			1.35296i	−0.736460	+	0.676481i	$0.763504\pi$
$60$	12.0000			1.54919
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		−	6.92820i		−	0.879883i
$63$	−3.00000	+	7.34847i	−0.377964	+	0.925820i
$64$	8.00000			1.00000
$65$		0			0
$66$		−	13.8564i		−	1.70561i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$	12.0000	+	4.89898i	1.43427	+	0.585540i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	8.48528			1.00000
$73$		−	9.79796i		−	1.14676i	−0.819288	−	0.573382i	$-0.805631\pi$
$73$		−	9.79796i		−	1.14676i	0.819288	−	0.573382i	$-0.194369\pi$
$74$		0			0
$75$		−	12.1244i		−	1.40000i
$76$		0			0
$77$	5.65685	−	13.8564i	0.644658	−	1.57908i
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		−	13.8564i		−	1.54919i
$81$	9.00000			1.00000
$82$		0			0
$83$			17.3205i			1.90117i	0.310460	+	0.950586i	$0.399517\pi$
$83$			17.3205i			1.90117i	−0.310460	+	0.950586i	$0.600483\pi$
$84$	8.48528	+	3.46410i	0.925820	+	0.377964i
$85$		0			0
$86$		0			0
$87$			4.89898i			0.525226i
$88$	−16.0000			−1.70561
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		−	14.6969i		−	1.54919i
$91$		0			0
$92$		0			0
$93$	−8.48528			−0.879883
$94$		0			0
$95$		0			0
$96$		−	9.79796i		−	1.00000i
$97$			19.5959i			1.98966i	0.101535	+	0.994832i	$0.467625\pi$
$97$			19.5959i			1.98966i	−0.101535	+	0.994832i	$0.532375\pi$
$98$	7.07107	+	6.92820i	0.714286	+	0.699854i
$99$	−16.9706			−1.70561
$100$	−14.0000			−1.40000
$101$			3.46410i			0.344691i	0.985037	+	0.172345i	$0.0551346\pi$
$101$			3.46410i			0.344691i	−0.985037	+	0.172345i	$0.944865\pi$
$102$		0			0
$103$			14.6969i			1.44813i	0.689730	+	0.724066i	$0.257729\pi$
$103$			14.6969i			1.44813i	−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$	6.00000	−	14.6969i	0.585540	−	1.43427i
$106$	20.0000			1.94257
$107$	11.3137			1.09374			0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137			1.09374			0.546869	+	0.837218i	$0.315820\pi$
$108$		−	10.3923i		−	1.00000i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$			27.7128i			2.64231i
$111$		0			0
$112$	4.00000	−	9.79796i	0.377964	−	0.925820i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	5.65685			0.525226
$117$		0			0
$118$		−	14.6969i		−	1.35296i
$119$		0			0
$120$	−16.9706			−1.54919
$121$	21.0000			1.90909
$122$		0			0
$123$		0			0
$124$			9.79796i			0.879883i
$125$			6.92820i			0.619677i
$126$	4.24264	−	10.3923i	0.377964	−	0.925820i
$127$	−22.0000			−1.95218			−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000			−1.95218			−0.976092	+	0.217357i	$0.930256\pi$
$128$	−11.3137			−1.00000
$129$		0			0
$130$		0			0
$131$		−	3.46410i		−	0.302660i	−0.988483	−	0.151330i	$-0.951644\pi$
$131$		−	3.46410i		−	0.302660i	0.988483	−	0.151330i	$-0.0483556\pi$
$132$			19.5959i			1.70561i
$133$		0			0
$134$		0			0
$135$	−18.0000			−1.54919
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−16.9706	−	6.92820i	−1.43427	−	0.585540i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		−	9.79796i		−	0.813676i
$146$			13.8564i			1.14676i
$147$	8.48528	−	8.66025i	0.699854	−	0.714286i
$148$		0			0
$149$	−2.82843			−0.231714			−0.115857	−	0.993266i	$-0.536961\pi$
$149$	−2.82843			−0.231714			−0.115857	+	0.993266i	$0.536961\pi$
$150$			17.1464i			1.40000i
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$	−8.00000	+	19.5959i	−0.644658	+	1.57908i
$155$	16.9706			1.36311
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−14.1421			−1.12509
$159$		−	24.4949i		−	1.94257i
$160$			19.5959i			1.54919i
$161$		0			0
$162$	−12.7279			−1.00000
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	33.9411			2.64231
$166$		−	24.4949i		−	1.90117i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−12.0000	−	4.89898i	−0.925820	−	0.377964i
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	17.3205i		−	1.31685i	−0.752645	−	0.658427i	$-0.771222\pi$
$173$		−	17.3205i		−	1.31685i	0.752645	−	0.658427i	$-0.228778\pi$
$174$		−	6.92820i		−	0.525226i
$175$	−7.00000	+	17.1464i	−0.529150	+	1.29615i
$176$	22.6274			1.70561
$177$	−18.0000			−1.35296
$178$		0			0
$179$	−11.3137			−0.845626			−0.422813	−	0.906217i	$-0.638957\pi$
$179$	−11.3137			−0.845626			−0.422813	+	0.906217i	$0.638957\pi$
$180$			20.7846i			1.54919i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$	12.0000			0.879883
$187$		0			0
$188$		0			0
$189$	−12.7279	−	5.19615i	−0.925820	−	0.377964i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			13.8564i			1.00000i
$193$	26.0000			1.87152			0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000			1.87152			0.935760	+	0.352636i	$0.114715\pi$
$194$		−	27.7128i		−	1.98966i
$195$		0			0
$196$	−10.0000	−	9.79796i	−0.714286	−	0.699854i
$197$	14.1421			1.00759			0.503793	−	0.863825i	$-0.331938\pi$
$197$	14.1421			1.00759			0.503793	+	0.863825i	$0.331938\pi$
$198$	24.0000			1.70561
$199$			24.4949i			1.73640i	0.496217	+	0.868199i	$0.334722\pi$
$199$			24.4949i			1.73640i	−0.496217	+	0.868199i	$0.665278\pi$
$200$	19.7990			1.40000
$201$		0			0
$202$		−	4.89898i		−	0.344691i
$203$	2.82843	−	6.92820i	0.198517	−	0.486265i
$204$		0			0
$205$		0			0
$206$		−	20.7846i		−	1.44813i
$207$		0			0
$208$		0			0
$209$		0			0
$210$	−8.48528	+	20.7846i	−0.585540	+	1.43427i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−28.2843			−1.94257
$213$		0			0
$214$	−16.0000			−1.09374
$215$		0			0
$216$			14.6969i			1.00000i
$217$	12.0000	+	4.89898i	0.814613	+	0.332564i
$218$		0			0
$219$	16.9706			1.14676
$220$		−	39.1918i		−	2.64231i
$221$		0			0
$222$		0			0
$223$		−	14.6969i		−	0.984180i	−0.870544	−	0.492090i	$-0.836233\pi$
$223$		−	14.6969i		−	0.984180i	0.870544	−	0.492090i	$-0.163767\pi$
$224$	−5.65685	+	13.8564i	−0.377964	+	0.925820i
$225$	21.0000			1.40000
$226$		0			0
$227$			10.3923i			0.689761i	0.938647	+	0.344881i	$0.112081\pi$
$227$			10.3923i			0.689761i	−0.938647	+	0.344881i	$0.887919\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$	24.0000	+	9.79796i	1.57908	+	0.644658i
$232$	−8.00000			−0.525226
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$			20.7846i			1.35296i
$237$			17.3205i			1.12509i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	24.0000			1.54919
$241$		−	29.3939i		−	1.89343i	−0.322078	−	0.946713i	$-0.604381\pi$
$241$		−	29.3939i		−	1.89343i	0.322078	−	0.946713i	$-0.395619\pi$
$242$	−29.6985			−1.90909
$243$			15.5885i			1.00000i
$244$		0			0
$245$	−16.9706	+	17.3205i	−1.08421	+	1.10657i
$246$		0			0
$247$		0			0
$248$		−	13.8564i		−	0.879883i
$249$	−30.0000			−1.90117
$250$		−	9.79796i		−	0.619677i
$251$		−	31.1769i		−	1.96787i	−0.178529	−	0.983935i	$-0.557134\pi$
$251$		−	31.1769i		−	1.96787i	0.178529	−	0.983935i	$-0.442866\pi$
$252$	−6.00000	+	14.6969i	−0.377964	+	0.925820i
$253$		0			0
$254$	31.1127			1.95218
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−8.48528			−0.525226
$262$			4.89898i			0.302660i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		−	27.7128i		−	1.70561i
$265$			48.9898i			3.00942i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.3923i			0.633630i	0.948487	+	0.316815i	$0.102613\pi$
$269$			10.3923i			0.633630i	−0.948487	+	0.316815i	$0.897387\pi$
$270$	25.4558			1.54919
$271$		−	24.4949i		−	1.48796i	−0.668202	−	0.743980i	$-0.732936\pi$
$271$		−	24.4949i		−	1.48796i	0.668202	−	0.743980i	$-0.267064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−39.5980			−2.38785
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		−	14.6969i		−	0.879883i
$280$	24.0000	+	9.79796i	1.43427	+	0.585540i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.9706			1.00000
$289$	−17.0000			−1.00000
$290$			13.8564i			0.813676i
$291$	−33.9411			−1.98966
$292$		−	19.5959i		−	1.14676i
$293$		−	31.1769i		−	1.82137i	−0.413096	−	0.910687i	$-0.635553\pi$
$293$		−	31.1769i		−	1.82137i	0.413096	−	0.910687i	$-0.364447\pi$
$294$	−12.0000	+	12.2474i	−0.699854	+	0.714286i
$295$	36.0000			2.09600
$296$		0			0
$297$		−	29.3939i		−	1.70561i
$298$	4.00000			0.231714
$299$		0			0
$300$		−	24.2487i		−	1.40000i
$301$		0			0
$302$	2.82843			0.162758
$303$	−6.00000			−0.344691
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	11.3137	−	27.7128i	0.644658	−	1.57908i
$309$	−25.4558			−1.44813
$310$	−24.0000			−1.36311
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			9.79796i			0.553813i	0.960897	+	0.276907i	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$	25.4558	+	10.3923i	1.43427	+	0.585540i
$316$	20.0000			1.12509
$317$	31.1127			1.74746			0.873732	−	0.486408i	$-0.161693\pi$
$317$	31.1127			1.74746			0.873732	+	0.486408i	$0.161693\pi$
$318$			34.6410i			1.94257i
$319$	16.0000			0.895828
$320$		−	27.7128i		−	1.54919i
$321$			19.5959i			1.09374i
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−48.0000			−2.64231
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			34.6410i			1.90117i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	16.9706	+	6.92820i	0.925820	+	0.377964i
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$	18.3848			1.00000
$339$		0			0
$340$		0			0
$341$			27.7128i			1.50073i
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$		0			0
$345$		0			0
$346$			24.4949i			1.31685i
$347$	28.2843			1.51838			0.759190	−	0.650870i	$-0.225596\pi$
$347$	28.2843			1.51838			0.759190	+	0.650870i	$0.225596\pi$
$348$			9.79796i			0.525226i
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	9.89949	−	24.2487i	0.529150	−	1.29615i
$351$		0			0
$352$	−32.0000			−1.70561
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	25.4558			1.35296
$355$		0			0
$356$		0			0
$357$		0			0
$358$	16.0000			0.845626
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		−	29.3939i		−	1.54919i
$361$	−19.0000			−1.00000
$362$		0			0
$363$			36.3731i			1.90909i
$364$		0			0
$365$	−33.9411			−1.77656
$366$		0			0
$367$			4.89898i			0.255725i	0.991792	+	0.127862i	$0.0408116\pi$
$367$			4.89898i			0.255725i	−0.991792	+	0.127862i	$0.959188\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.1421	+	34.6410i	−0.734223	+	1.79847i
$372$	−16.9706			−0.879883
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−12.0000			−0.619677
$376$		0			0
$377$		0			0
$378$	18.0000	+	7.34847i	0.925820	+	0.377964i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		−	38.1051i		−	1.95218i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		−	19.5959i		−	1.00000i
$385$	−48.0000	−	19.5959i	−2.44631	−	0.998700i
$386$	−36.7696			−1.87152
$387$		0			0
$388$			39.1918i			1.98966i
$389$	−31.1127			−1.57748			−0.788738	−	0.614729i	$-0.789265\pi$
$389$	−31.1127			−1.57748			−0.788738	+	0.614729i	$0.789265\pi$
$390$		0			0
$391$		0			0
$392$	14.1421	+	13.8564i	0.714286	+	0.699854i
$393$	6.00000			0.302660
$394$	−20.0000			−1.00759
$395$		−	34.6410i		−	1.74298i
$396$	−33.9411			−1.70561
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		−	34.6410i		−	1.73640i
$399$		0			0
$400$	−28.0000			−1.40000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$			6.92820i			0.344691i
$405$		−	31.1769i		−	1.54919i
$406$	−4.00000	+	9.79796i	−0.198517	+	0.486265i
$407$		0			0
$408$		0			0
$409$			39.1918i			1.93791i	0.247234	+	0.968956i	$0.420478\pi$
$409$			39.1918i			1.93791i	−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$			29.3939i			1.44813i
$413$	25.4558	+	10.3923i	1.25260	+	0.511372i
$414$		0			0
$415$	60.0000			2.94528
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	10.3923i		−	0.507697i	−0.967244	−	0.253849i	$-0.918303\pi$
$419$		−	10.3923i		−	0.507697i	0.967244	−	0.253849i	$-0.0816965\pi$
$420$	12.0000	−	29.3939i	0.585540	−	1.43427i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	40.0000			1.94257
$425$		0			0
$426$		0			0
$427$		0			0
$428$	22.6274			1.09374
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	20.7846i		−	1.00000i
$433$		−	39.1918i		−	1.88344i	−0.336399	−	0.941720i	$-0.609209\pi$
$433$		−	39.1918i		−	1.88344i	0.336399	−	0.941720i	$-0.390791\pi$
$434$	−16.9706	−	6.92820i	−0.814613	−	0.332564i
$435$	16.9706			0.813676
$436$		0			0
$437$		0			0
$438$	−24.0000			−1.14676
$439$		−	24.4949i		−	1.16908i	−0.811366	−	0.584539i	$-0.801275\pi$
$439$		−	24.4949i		−	1.16908i	0.811366	−	0.584539i	$-0.198725\pi$
$440$			55.4256i			2.64231i
$441$	15.0000	+	14.6969i	0.714286	+	0.699854i
$442$		0			0
$443$	−28.2843			−1.34383			−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−28.2843			−1.34383			−0.671913	+	0.740630i	$0.734527\pi$
$444$		0			0
$445$		0			0
$446$			20.7846i			0.984180i
$447$		−	4.89898i		−	0.231714i
$448$	8.00000	−	19.5959i	0.377964	−	0.925820i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−29.6985			−1.40000
$451$		0			0
$452$		0			0
$453$		−	3.46410i		−	0.162758i
$454$		−	14.6969i		−	0.689761i
$455$		0			0
$456$		0			0
$457$	38.0000			1.77757			0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000			1.77757			0.888783	+	0.458329i	$0.151552\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			38.1051i			1.77473i	0.461065	+	0.887366i	$0.347467\pi$
$461$			38.1051i			1.77473i	−0.461065	+	0.887366i	$0.652533\pi$
$462$	−33.9411	−	13.8564i	−1.57908	−	0.644658i
$463$	−26.0000			−1.20832			−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000			−1.20832			−0.604161	+	0.796862i	$0.706492\pi$
$464$	11.3137			0.525226
$465$			29.3939i			1.36311i
$466$		0			0
$467$		−	17.3205i		−	0.801498i	−0.916188	−	0.400749i	$-0.868750\pi$
$467$		−	17.3205i		−	0.801498i	0.916188	−	0.400749i	$-0.131250\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	29.3939i		−	1.35296i
$473$		0			0
$474$		−	24.4949i		−	1.12509i
$475$		0			0
$476$		0			0
$477$	42.4264			1.94257
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−33.9411			−1.54919
$481$		0			0
$482$			41.5692i			1.89343i
$483$		0			0
$484$	42.0000			1.90909
$485$	67.8823			3.08237
$486$		−	22.0454i		−	1.00000i
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$		0			0
$490$	24.0000	−	24.4949i	1.08421	−	1.10657i
$491$	22.6274			1.02116			0.510581	−	0.859830i	$-0.329431\pi$
$491$	22.6274			1.02116			0.510581	+	0.859830i	$0.329431\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$			58.7878i			2.64231i
$496$			19.5959i			0.879883i
$497$		0			0
$498$	42.4264			1.90117
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$			13.8564i			0.619677i
$501$		0			0
$502$			44.0908i			1.96787i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	8.48528	−	20.7846i	0.377964	−	0.925820i
$505$	12.0000			0.533993
$506$		0			0
$507$		−	22.5167i		−	1.00000i
$508$	−44.0000			−1.95218
$509$			45.0333i			1.99607i	0.0626839	+	0.998033i	$0.480034\pi$
$509$			45.0333i			1.99607i	−0.0626839	+	0.998033i	$0.519966\pi$
$510$		0			0
$511$	−24.0000	−	9.79796i	−1.06170	−	0.433436i
$512$	−22.6274			−1.00000
$513$		0			0
$514$		0			0
$515$	50.9117			2.24344
$516$		0			0
$517$		0			0
$518$		0			0
$519$	30.0000			1.31685
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	12.0000			0.525226
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		−	6.92820i		−	0.302660i
$525$	−29.6985	−	12.1244i	−1.29615	−	0.529150i
$526$		0			0
$527$		0			0
$528$			39.1918i			1.70561i
$529$	23.0000			1.00000
$530$		−	69.2820i		−	3.00942i
$531$		−	31.1769i		−	1.35296i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	39.1918i		−	1.69441i
$536$		0			0
$537$		−	19.5959i		−	0.845626i
$538$		−	14.6969i		−	0.633630i
$539$	−28.2843	−	27.7128i	−1.21829	−	1.19368i
$540$	−36.0000			−1.54919
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$			34.6410i			1.48796i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	56.0000			2.38785
$551$		0			0
$552$		0			0
$553$	10.0000	−	24.4949i	0.425243	−	1.04163i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.1421			−0.599222			−0.299611	−	0.954062i	$-0.596857\pi$
$557$	−14.1421			−0.599222			−0.299611	+	0.954062i	$0.596857\pi$
$558$			20.7846i			0.879883i
$559$		0			0
$560$	−33.9411	−	13.8564i	−1.43427	−	0.585540i
$561$		0			0
$562$		0			0
$563$		−	38.1051i		−	1.60594i	−0.596020	−	0.802970i	$-0.703252\pi$
$563$		−	38.1051i		−	1.60594i	0.596020	−	0.802970i	$-0.296748\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9.00000	−	22.0454i	0.377964	−	0.925820i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$	24.0416			1.00000
$579$			45.0333i			1.87152i
$580$		−	19.5959i		−	0.813676i
$581$	42.4264	+	17.3205i	1.76014	+	0.718576i
$582$	48.0000			1.98966
$583$	−80.0000			−3.31326
$584$			27.7128i			1.14676i
$585$		0			0
$586$			44.0908i			1.82137i
$587$			17.3205i			0.714894i	0.933933	+	0.357447i	$0.116353\pi$
$587$			17.3205i			0.714894i	−0.933933	+	0.357447i	$0.883647\pi$
$588$	16.9706	−	17.3205i	0.699854	−	0.714286i
$589$		0			0
$590$	−50.9117			−2.09600
$591$			24.4949i			1.00759i
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$			41.5692i			1.70561i
$595$		0			0
$596$	−5.65685			−0.231714
$597$	−42.4264			−1.73640
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$			34.2929i			1.40000i
$601$		−	48.9898i		−	1.99834i	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$		−	48.9898i		−	1.99834i	0.0407909	−	0.999168i	$-0.487012\pi$
$602$		0			0
$603$		0			0
$604$	−4.00000			−0.162758
$605$		−	72.7461i		−	2.95755i
$606$	8.48528			0.344691
$607$			44.0908i			1.78959i	0.446476	+	0.894795i	$0.352679\pi$
$607$			44.0908i			1.78959i	−0.446476	+	0.894795i	$0.647321\pi$
$608$		0			0
$609$	12.0000	+	4.89898i	0.486265	+	0.198517i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$	−16.0000	+	39.1918i	−0.644658	+	1.57908i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	36.0000			1.44813
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	33.9411			1.36311
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−11.0000			−0.440000
$626$		−	13.8564i		−	0.553813i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−36.0000	−	14.6969i	−1.43427	−	0.585540i
$631$	−50.0000			−1.99047			−0.995234	−	0.0975126i	$-0.968911\pi$
$631$	−50.0000			−1.99047			−0.995234	+	0.0975126i	$0.968911\pi$
$632$	−28.2843			−1.12509
$633$		0			0
$634$	−44.0000			−1.74746
$635$			76.2102i			3.02431i
$636$		−	48.9898i		−	1.94257i
$637$		0			0
$638$	−22.6274			−0.895828
$639$		0			0
$640$			39.1918i			1.54919i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		−	27.7128i		−	1.09374i
$643$		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			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−25.4558			−1.00000
$649$			58.7878i			2.30762i
$650$		0			0
$651$	−8.48528	+	20.7846i	−0.332564	+	0.814613i
$652$		0			0
$653$	−48.0833			−1.88164			−0.940822	−	0.338902i	$-0.889945\pi$
$653$	−48.0833			−1.88164			−0.940822	+	0.338902i	$0.889945\pi$
$654$		0			0
$655$	−12.0000			−0.468879
$656$		0			0
$657$			29.3939i			1.14676i
$658$		0			0
$659$	−45.2548			−1.76288			−0.881439	−	0.472298i	$-0.843425\pi$
$659$	−45.2548			−1.76288			−0.881439	+	0.472298i	$0.843425\pi$
$660$	67.8823			2.64231
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		−	48.9898i		−	1.90117i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	25.4558			0.984180
$670$		0			0
$671$		0			0
$672$	−24.0000	−	9.79796i	−0.925820	−	0.377964i
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$	31.1127			1.19842
$675$			36.3731i			1.40000i
$676$	−26.0000			−1.00000
$677$		−	51.9615i		−	1.99704i	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$		−	51.9615i		−	1.99704i	0.0543526	−	0.998522i	$-0.482690\pi$
$678$		0			0
$679$	48.0000	+	19.5959i	1.84207	+	0.752022i
$680$		0			0
$681$	−18.0000			−0.689761
$682$		−	39.1918i		−	1.50073i
$683$	5.65685			0.216454			0.108227	−	0.994126i	$-0.465483\pi$
$683$	5.65685			0.216454			0.108227	+	0.994126i	$0.465483\pi$
$684$		0			0
$685$		0			0
$686$	24.0416	−	10.3923i	0.917914	−	0.396780i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		−	34.6410i		−	1.31685i
$693$	−16.9706	+	41.5692i	−0.644658	+	1.57908i
$694$	−40.0000			−1.51838
$695$		0			0
$696$		−	13.8564i		−	0.525226i
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−14.0000	+	34.2929i	−0.529150	+	1.29615i
$701$	−36.7696			−1.38877			−0.694383	−	0.719605i	$-0.744323\pi$
$701$	−36.7696			−1.38877			−0.694383	+	0.719605i	$0.744323\pi$
$702$		0			0
$703$		0			0
$704$	45.2548			1.70561
$705$		0			0
$706$		0			0
$707$	8.48528	+	3.46410i	0.319122	+	0.130281i
$708$	−36.0000			−1.35296
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$	−30.0000			−1.12509
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−22.6274			−0.845626
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$			41.5692i			1.54919i
$721$	36.0000	+	14.6969i	1.34071	+	0.547343i
$722$	26.8701			1.00000
$723$	50.9117			1.89343
$724$		0			0
$725$	−19.7990			−0.735316
$726$		−	51.4393i		−	1.90909i
$727$			53.8888i			1.99862i	0.0370879	+	0.999312i	$0.488192\pi$
$727$			53.8888i			1.99862i	−0.0370879	+	0.999312i	$0.511808\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$	48.0000			1.77656
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		−	6.92820i		−	0.255725i
$735$	−30.0000	−	29.3939i	−1.10657	−	1.08421i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$	20.0000	−	48.9898i	0.734223	−	1.79847i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$	24.0000			0.879883
$745$			9.79796i			0.358969i
$746$		0			0
$747$		−	51.9615i		−	1.90117i
$748$		0			0
$749$	11.3137	−	27.7128i	0.413394	−	1.01260i
$750$	16.9706			0.619677
$751$	10.0000			0.364905			0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000			0.364905			0.182453	+	0.983215i	$0.441596\pi$
$752$		0			0
$753$	54.0000			1.96787
$754$		0			0
$755$			6.92820i			0.252143i
$756$	−25.4558	−	10.3923i	−0.925820	−	0.377964i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$			53.8888i			1.95218i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$			27.7128i			1.00000i
$769$		−	48.9898i		−	1.76662i	−0.468792	−	0.883309i	$-0.655311\pi$
$769$		−	48.9898i		−	1.76662i	0.468792	−	0.883309i	$-0.344689\pi$
$770$	67.8823	+	27.7128i	2.44631	+	0.998700i
$771$		0			0
$772$	52.0000			1.87152
$773$			51.9615i			1.86893i	0.356060	+	0.934463i	$0.384120\pi$
$773$			51.9615i			1.86893i	−0.356060	+	0.934463i	$0.615880\pi$
$774$		0			0
$775$		−	34.2929i		−	1.23184i
$776$		−	55.4256i		−	1.98966i
$777$		0			0
$778$	44.0000			1.57748
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	14.6969i		−	0.525226i
$784$	−20.0000	−	19.5959i	−0.714286	−	0.699854i
$785$		0			0
$786$	−8.48528			−0.302660
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	28.2843			1.00759
$789$		0			0
$790$			48.9898i			1.74298i
$791$		0			0
$792$	48.0000			1.70561
$793$		0			0
$794$		0			0
$795$	−84.8528			−3.00942
$796$			48.9898i			1.73640i
$797$			17.3205i			0.613524i	0.951786	+	0.306762i	$0.0992455\pi$
$797$			17.3205i			0.613524i	−0.951786	+	0.306762i	$0.900754\pi$
$798$		0			0
$799$		0			0
$800$	39.5980			1.40000
$801$		0			0
$802$		0			0
$803$		−	55.4256i		−	1.95593i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−18.0000			−0.633630
$808$		−	9.79796i		−	0.344691i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$			44.0908i			1.54919i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	5.65685	−	13.8564i	0.198517	−	0.486265i
$813$	42.4264			1.48796
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	55.4256i		−	1.93791i
$819$		0			0
$820$		0			0
$821$	−48.0833			−1.67812			−0.839059	−	0.544041i	$-0.816894\pi$
$821$	−48.0833			−1.67812			−0.839059	+	0.544041i	$0.816894\pi$
$822$		0			0
$823$	46.0000			1.60346			0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000			1.60346			0.801730	+	0.597687i	$0.203913\pi$
$824$		−	41.5692i		−	1.44813i
$825$		−	68.5857i		−	2.38785i
$826$	−36.0000	−	14.6969i	−1.25260	−	0.511372i
$827$	−56.5685			−1.96708			−0.983540	−	0.180688i	$-0.942168\pi$
$827$	−56.5685			−1.96708			−0.983540	+	0.180688i	$0.942168\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$	−84.8528			−2.94528
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	25.4558			0.879883
$838$			14.6969i			0.507697i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−16.9706	+	41.5692i	−0.585540	+	1.43427i
$841$	−21.0000			−0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$			45.0333i			1.54919i
$846$		0			0
$847$	21.0000	−	51.4393i	0.721569	−	1.76747i
$848$	−56.5685			−1.94257
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−32.0000			−1.09374
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$			29.3939i			1.00000i
$865$	−60.0000			−2.04006
$866$			55.4256i			1.88344i
$867$		−	29.4449i		−	1.00000i
$868$	24.0000	+	9.79796i	0.814613	+	0.332564i
$869$	56.5685			1.91896
$870$	−24.0000			−0.813676
$871$		0			0
$872$		0			0
$873$		−	58.7878i		−	1.98966i
$874$		0			0
$875$	16.9706	+	6.92820i	0.573710	+	0.234216i
$876$	33.9411			1.14676
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$			34.6410i			1.16908i
$879$	54.0000			1.82137
$880$		−	78.3837i		−	2.64231i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−21.2132	−	20.7846i	−0.714286	−	0.699854i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$			62.3538i			2.09600i
$886$	40.0000			1.34383
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−22.0000	+	53.8888i	−0.737856	+	1.80737i
$890$		0			0
$891$	50.9117			1.70561
$892$		−	29.3939i		−	0.984180i
$893$		0			0
$894$			6.92820i			0.231714i
$895$			39.1918i			1.31004i
$896$	−11.3137	+	27.7128i	−0.377964	+	0.925820i
$897$		0			0
$898$		0			0
$899$			13.8564i			0.462137i
$900$	42.0000			1.40000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$			4.89898i			0.162758i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$			20.7846i			0.689761i
$909$		−	10.3923i		−	0.344691i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$			97.9796i			3.24265i
$914$	−53.7401			−1.77757
$915$		0			0
$916$		0			0
$917$	−8.48528	−	3.46410i	−0.280209	−	0.114395i
$918$		0			0
$919$	50.0000			1.64935			0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000			1.64935			0.824674	+	0.565608i	$0.191359\pi$
$920$		0			0
$921$		0			0
$922$		−	53.8888i		−	1.77473i
$923$		0			0
$924$	48.0000	+	19.5959i	1.57908	+	0.644658i
$925$		0			0
$926$	36.7696			1.20832
$927$		−	44.0908i		−	1.44813i
$928$	−16.0000			−0.525226
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		−	41.5692i		−	1.36311i
$931$		0			0
$932$		0			0
$933$		0			0
$934$			24.4949i			0.801498i
$935$		0			0
$936$		0			0
$937$			19.5959i			0.640171i	0.947389	+	0.320085i	$0.103712\pi$
$937$			19.5959i			0.640171i	−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$	−16.9706			−0.553813
$940$		0			0
$941$		−	38.1051i		−	1.24219i	−0.783735	−	0.621096i	$-0.786688\pi$
$941$		−	38.1051i		−	1.24219i	0.783735	−	0.621096i	$-0.213312\pi$
$942$		0			0
$943$		0			0
$944$			41.5692i			1.35296i
$945$	−18.0000	+	44.0908i	−0.585540	+	1.43427i
$946$		0			0
$947$	56.5685			1.83823			0.919115	−	0.393989i	$-0.128905\pi$
$947$	56.5685			1.83823			0.919115	+	0.393989i	$0.128905\pi$
$948$			34.6410i			1.12509i
$949$		0			0
$950$		0			0
$951$			53.8888i			1.74746i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−60.0000			−1.94257
$955$		0			0
$956$		0			0
$957$			27.7128i			0.895828i
$958$		0			0
$959$		0			0
$960$	48.0000			1.54919
$961$	7.00000			0.225806
$962$		0			0
$963$	−33.9411			−1.09374
$964$		−	58.7878i		−	1.89343i
$965$		−	90.0666i		−	2.89935i
$966$		0			0
$967$	62.0000			1.99379			0.996893	−	0.0787703i	$-0.0250994\pi$
$967$	62.0000			1.99379			0.996893	+	0.0787703i	$0.0250994\pi$
$968$	−59.3970			−1.90909
$969$		0			0
$970$	−96.0000			−3.08237
$971$		−	3.46410i		−	0.111168i	−0.998454	−	0.0555842i	$-0.982298\pi$
$971$		−	3.46410i		−	0.111168i	0.998454	−	0.0555842i	$-0.0177021\pi$
$972$			31.1769i			1.00000i
$973$		0			0
$974$	2.82843			0.0906287
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−33.9411	+	34.6410i	−1.08421	+	1.10657i
$981$		0			0
$982$	−32.0000			−1.02116
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		−	48.9898i		−	1.56094i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		−	83.1384i		−	2.64231i
$991$	−58.0000			−1.84243			−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000			−1.84243			−0.921215	+	0.389053i	$0.872802\pi$
$992$		−	27.7128i		−	0.879883i
$993$		0			0
$994$		0			0
$995$	84.8528			2.69002
$996$	−60.0000			−1.90117
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.i.b.125.2	yes	4
3.2	odd	2	inner	168.2.i.b.125.3	yes	4
4.3	odd	2		672.2.i.b.209.2		4
7.6	odd	2	inner	168.2.i.b.125.1	✓	4
8.3	odd	2		672.2.i.b.209.4		4
8.5	even	2	inner	168.2.i.b.125.3	yes	4
12.11	even	2		672.2.i.b.209.4		4
21.20	even	2	inner	168.2.i.b.125.4	yes	4
24.5	odd	2	CM	168.2.i.b.125.2	yes	4
24.11	even	2		672.2.i.b.209.2		4
28.27	even	2		672.2.i.b.209.3		4
56.13	odd	2	inner	168.2.i.b.125.4	yes	4
56.27	even	2		672.2.i.b.209.1		4
84.83	odd	2		672.2.i.b.209.1		4
168.83	odd	2		672.2.i.b.209.3		4
168.125	even	2	inner	168.2.i.b.125.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.b.125.1	✓	4	7.6	odd	2	inner
168.2.i.b.125.1	✓	4	168.125	even	2	inner
168.2.i.b.125.2	yes	4	1.1	even	1	trivial
168.2.i.b.125.2	yes	4	24.5	odd	2	CM
168.2.i.b.125.3	yes	4	3.2	odd	2	inner
168.2.i.b.125.3	yes	4	8.5	even	2	inner
168.2.i.b.125.4	yes	4	21.20	even	2	inner
168.2.i.b.125.4	yes	4	56.13	odd	2	inner
672.2.i.b.209.1		4	56.27	even	2
672.2.i.b.209.1		4	84.83	odd	2
672.2.i.b.209.2		4	4.3	odd	2
672.2.i.b.209.2		4	24.11	even	2
672.2.i.b.209.3		4	28.27	even	2
672.2.i.b.209.3		4	168.83	odd	2
672.2.i.b.209.4		4	8.3	odd	2
672.2.i.b.209.4		4	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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -3.46410i q^{5} -2.44949i q^{6} +(1.00000 - 2.44949i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -3.46410i q^{5} -2.44949i q^{6} +(1.00000 - 2.44949i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +4.89898i q^{10} +5.65685 q^{11} +3.46410i q^{12} +(-1.41421 + 3.46410i) q^{14} +6.00000 q^{15} +4.00000 q^{16} +4.24264 q^{18} -6.92820i q^{20} +(4.24264 + 1.73205i) q^{21} -8.00000 q^{22} -4.89898i q^{24} -7.00000 q^{25} -5.19615i q^{27} +(2.00000 - 4.89898i) q^{28} +2.82843 q^{29} -8.48528 q^{30} +4.89898i q^{31} -5.65685 q^{32} +9.79796i q^{33} +(-8.48528 - 3.46410i) q^{35} -6.00000 q^{36} +9.79796i q^{40} +(-6.00000 - 2.44949i) q^{42} +11.3137 q^{44} +10.3923i q^{45} +6.92820i q^{48} +(-5.00000 - 4.89898i) q^{49} +9.89949 q^{50} -14.1421 q^{53} +7.34847i q^{54} -19.5959i q^{55} +(-2.82843 + 6.92820i) q^{56} -4.00000 q^{58} +10.3923i q^{59} +12.0000 q^{60} -6.92820i q^{62} +(-3.00000 + 7.34847i) q^{63} +8.00000 q^{64} -13.8564i q^{66} +(12.0000 + 4.89898i) q^{70} +8.48528 q^{72} -9.79796i q^{73} -12.1244i q^{75} +(5.65685 - 13.8564i) q^{77} +10.0000 q^{79} -13.8564i q^{80} +9.00000 q^{81} +17.3205i q^{83} +(8.48528 + 3.46410i) q^{84} +4.89898i q^{87} -16.0000 q^{88} -14.6969i q^{90} -8.48528 q^{93} -9.79796i q^{96} +19.5959i q^{97} +(7.07107 + 6.92820i) q^{98} -16.9706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 4 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 + 4 * q^7 - 12 * q^9	\( 4 q + 8 q^{4} + 4 q^{7} - 12 q^{9} + 24 q^{15} + 16 q^{16} - 32 q^{22} - 28 q^{25} + 8 q^{28} - 24 q^{36} - 24 q^{42} - 20 q^{49} - 16 q^{58} + 48 q^{60} - 12 q^{63} + 32 q^{64} + 48 q^{70} + 40 q^{79} + 36 q^{81} - 64 q^{88}+O(q^{100}) \) 4 * q + 8 * q^4 + 4 * q^7 - 12 * q^9 + 24 * q^15 + 16 * q^16 - 32 * q^22 - 28 * q^25 + 8 * q^28 - 24 * q^36 - 24 * q^42 - 20 * q^49 - 16 * q^58 + 48 * q^60 - 12 * q^63 + 32 * q^64 + 48 * q^70 + 40 * q^79 + 36 * q^81 - 64 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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