LMFDB - Embedded newform 672.2.j.d.239.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(239,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.j (of order $2$, degree $1$, not 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.2593100598870016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.8
Root			$1.37027 - 0.349801i$ of defining polynomial
Character	$\chi$	$=$	672.239
Dual form			672.2.j.d.239.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.203364 + 1.72007i) q^{3} +3.44014 q^{5} +1.00000i q^{7} +(-2.91729 + 0.699602i) q^{9} +O(q^{10})$	$q+(0.203364 + 1.72007i) q^{3} +3.44014 q^{5} +1.00000i q^{7} +(-2.91729 + 0.699602i) q^{9} +5.42784i q^{13} +(0.699602 + 5.91729i) q^{15} -3.15559i q^{17} +4.40673 q^{19} +(-1.72007 + 0.203364i) q^{21} -4.55480 q^{23} +6.83457 q^{25} +(-1.79664 - 4.87566i) q^{27} +4.08188 q^{29} +3.44014i q^{35} +3.18654i q^{37} +(-9.33627 + 1.10383i) q^{39} -5.95400i q^{41} -9.83457 q^{43} +(-10.0359 + 2.40673i) q^{45} +10.9622 q^{47} -1.00000 q^{49} +(5.42784 - 0.641735i) q^{51} -6.31119 q^{53} +(0.896171 + 7.57988i) q^{57} +0.641735i q^{59} +10.2413i q^{61} +(-0.699602 - 2.91729i) q^{63} +18.6725i q^{65} +9.02112 q^{67} +(-0.926283 - 7.83457i) q^{69} +3.15559 q^{71} -6.00000 q^{73} +(1.38991 + 11.7559i) q^{75} +1.62691i q^{79} +(8.02112 - 4.08188i) q^{81} -12.5497i q^{83} -10.8557i q^{85} +(0.830108 + 7.02112i) q^{87} -7.80657i q^{89} -5.42784 q^{91} +15.1598 q^{95} -16.0422 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{9}+O(q^{10}) $ 12 * q + 4 * q^9	$ 12 q + 4 q^{9} + 48 q^{19} + 4 q^{25} - 24 q^{27} - 40 q^{43} - 12 q^{49} - 8 q^{51} + 40 q^{57} + 40 q^{67} - 72 q^{73} + 24 q^{75} + 28 q^{81} + 8 q^{91} - 56 q^{97}+O(q^{100}) $ 12 * q + 4 * q^9 + 48 * q^19 + 4 * q^25 - 24 * q^27 - 40 * q^43 - 12 * q^49 - 8 * q^51 + 40 * q^57 + 40 * q^67 - 72 * q^73 + 24 * q^75 + 28 * q^81 + 8 * q^91 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.203364	+	1.72007i	0.117412	+	0.993083i
$3$	0.203364	+	1.72007i	0.117412	+	0.993083i
$4$		0			0
$5$	3.44014			1.53848			0.769239	−	0.638961i	$-0.220636\pi$
$5$	3.44014			1.53848			0.769239	+	0.638961i	$0.220636\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.91729	+	0.699602i	−0.972429	+	0.233201i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$			5.42784i			1.50541i	0.658356	+	0.752706i	$0.271252\pi$
$13$			5.42784i			1.50541i	−0.658356	+	0.752706i	$0.728748\pi$
$14$		0			0
$15$	0.699602	+	5.91729i	0.180636	+	1.52784i
$16$		0			0
$17$		−	3.15559i		−	0.765344i	−0.923884	−	0.382672i	$-0.875004\pi$
$17$		−	3.15559i		−	0.765344i	0.923884	−	0.382672i	$-0.124996\pi$
$18$		0			0
$19$	4.40673			1.01097			0.505486	−	0.862835i	$-0.331313\pi$
$19$	4.40673			1.01097			0.505486	+	0.862835i	$0.331313\pi$
$20$		0			0
$21$	−1.72007	+	0.203364i	−0.375350	+	0.0443777i
$22$		0			0
$23$	−4.55480			−0.949741			−0.474870	−	0.880056i	$-0.657505\pi$
$23$	−4.55480			−0.949741			−0.474870	+	0.880056i	$0.657505\pi$
$24$		0			0
$25$	6.83457			1.36691
$26$		0			0
$27$	−1.79664	−	4.87566i	−0.345763	−	0.938322i
$28$		0			0
$29$	4.08188			0.757985			0.378993	−	0.925400i	$-0.376271\pi$
$29$	4.08188			0.757985			0.378993	+	0.925400i	$0.376271\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.44014i			0.581490i
$36$		0			0
$37$			3.18654i			0.523864i	0.965086	+	0.261932i	$0.0843597\pi$
$37$			3.18654i			0.523864i	−0.965086	+	0.261932i	$0.915640\pi$
$38$		0			0
$39$	−9.33627	+	1.10383i	−1.49500	+	0.176754i
$40$		0			0
$41$		−	5.95400i		−	0.929859i	−0.885348	−	0.464929i	$-0.846080\pi$
$41$		−	5.95400i		−	0.929859i	0.885348	−	0.464929i	$-0.153920\pi$
$42$		0			0
$43$	−9.83457			−1.49976			−0.749879	−	0.661575i	$-0.769888\pi$
$43$	−9.83457			−1.49976			−0.749879	+	0.661575i	$0.769888\pi$
$44$		0			0
$45$	−10.0359	+	2.40673i	−1.49606	+	0.358774i
$46$		0			0
$47$	10.9622			1.59900			0.799498	−	0.600669i	$-0.205099\pi$
$47$	10.9622			1.59900			0.799498	+	0.600669i	$0.205099\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	5.42784	−	0.641735i	0.760050	−	0.0898608i
$52$		0			0
$53$	−6.31119			−0.866908			−0.433454	−	0.901176i	$-0.642705\pi$
$53$	−6.31119			−0.866908			−0.433454	+	0.901176i	$0.642705\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.896171	+	7.57988i	0.118701	+	1.00398i
$58$		0			0
$59$			0.641735i			0.0835468i	0.999127	+	0.0417734i	$0.0133007\pi$
$59$			0.641735i			0.0835468i	−0.999127	+	0.0417734i	$0.986699\pi$
$60$		0			0
$61$			10.2413i			1.31126i	0.755081	+	0.655632i	$0.227598\pi$
$61$			10.2413i			1.31126i	−0.755081	+	0.655632i	$0.772402\pi$
$62$		0			0
$63$	−0.699602	−	2.91729i	−0.0881415	−	0.367543i
$64$		0			0
$65$			18.6725i			2.31604i
$66$		0			0
$67$	9.02112			1.10210			0.551052	−	0.834471i	$-0.314226\pi$
$67$	9.02112			1.10210			0.551052	+	0.834471i	$0.314226\pi$
$68$		0			0
$69$	−0.926283	−	7.83457i	−0.111511	−	0.943172i
$70$		0			0
$71$	3.15559			0.374500			0.187250	−	0.982312i	$-0.440043\pi$
$71$	3.15559			0.374500			0.187250	+	0.982312i	$0.440043\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	1.38991	+	11.7559i	0.160493	+	1.35746i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			1.62691i			0.183042i	0.995803	+	0.0915210i	$0.0291729\pi$
$79$			1.62691i			0.183042i	−0.995803	+	0.0915210i	$0.970827\pi$
$80$		0			0
$81$	8.02112	−	4.08188i	0.891235	−	0.453542i
$82$		0			0
$83$		−	12.5497i		−	1.37751i	−0.724993	−	0.688756i	$-0.758157\pi$
$83$		−	12.5497i		−	1.37751i	0.724993	−	0.688756i	$-0.241843\pi$
$84$		0			0
$85$		−	10.8557i		−	1.17746i
$86$		0			0
$87$	0.830108	+	7.02112i	0.0889969	+	0.752743i
$88$		0			0
$89$		−	7.80657i		−	0.827494i	−0.910392	−	0.413747i	$-0.864220\pi$
$89$		−	7.80657i		−	0.827494i	0.910392	−	0.413747i	$-0.135780\pi$
$90$		0			0
$91$	−5.42784			−0.568993
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.1598			1.55536
$96$		0			0
$97$	−16.0422			−1.62884			−0.814421	−	0.580275i	$-0.802945\pi$
$97$	−16.0422			−1.62884			−0.814421	+	0.580275i	$0.802945\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.75133			0.970293			0.485147	−	0.874433i	$-0.338766\pi$
$101$	9.75133			0.970293			0.485147	+	0.874433i	$0.338766\pi$
$102$		0			0
$103$		−	1.62691i		−	0.160305i	−0.996783	−	0.0801523i	$-0.974459\pi$
$103$		−	1.62691i		−	0.160305i	0.996783	−	0.0801523i	$-0.0255407\pi$
$104$		0			0
$105$	−5.91729	+	0.699602i	−0.577468	+	0.0682741i
$106$		0			0
$107$		−	6.88028i		−	0.665142i	−0.943078	−	0.332571i	$-0.892084\pi$
$107$		−	6.88028i		−	0.665142i	0.943078	−	0.332571i	$-0.107916\pi$
$108$		0			0
$109$		−	12.8135i		−	1.22731i	−0.789576	−	0.613653i	$-0.789699\pi$
$109$		−	12.8135i		−	1.22731i	0.789576	−	0.613653i	$-0.210301\pi$
$110$		0			0
$111$	−5.48108	+	0.648029i	−0.520241	+	0.0615082i
$112$		0			0
$113$		−	1.39920i		−	0.131626i	−0.997832	−	0.0658130i	$-0.979036\pi$
$113$		−	1.39920i		−	0.131626i	0.997832	−	0.0658130i	$-0.0209641\pi$
$114$		0			0
$115$	−15.6691			−1.46116
$116$		0			0
$117$	−3.79733	−	15.8346i	−0.351063	−	1.46391i
$118$		0			0
$119$	3.15559			0.289273
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	10.2413	−	1.21083i	0.923427	−	0.109177i
$124$		0			0
$125$	6.31119			0.564490
$126$		0			0
$127$		−	10.2077i		−	0.905783i	−0.891566	−	0.452892i	$-0.850392\pi$
$127$		−	10.2077i		−	0.905783i	0.891566	−	0.452892i	$-0.149608\pi$
$128$		0			0
$129$	−2.00000	−	16.9162i	−0.176090	−	1.48938i
$130$		0			0
$131$		−	1.21083i		−	0.105791i	−0.998600	−	0.0528954i	$-0.983155\pi$
$131$		−	1.21083i		−	0.105791i	0.998600	−	0.0528954i	$-0.0168450\pi$
$132$		0			0
$133$			4.40673i			0.382112i
$134$		0			0
$135$	−6.18068	−	16.7730i	−0.531948	−	1.44359i
$136$		0			0
$137$		−	7.44938i		−	0.636443i	−0.948016	−	0.318222i	$-0.896914\pi$
$137$		−	7.44938i		−	0.636443i	0.948016	−	0.318222i	$-0.103086\pi$
$138$		0			0
$139$	7.26242			0.615990			0.307995	−	0.951388i	$-0.400342\pi$
$139$	7.26242			0.615990			0.307995	+	0.951388i	$0.400342\pi$
$140$		0			0
$141$	2.22931	+	18.8557i	0.187742	+	1.58794i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	14.0422			1.16614
$146$		0			0
$147$	−0.203364	−	1.72007i	−0.0167732	−	0.141869i
$148$		0			0
$149$	20.0718			1.64434			0.822171	−	0.569241i	$-0.192763\pi$
$149$	20.0718			1.64434			0.822171	+	0.569241i	$0.192763\pi$
$150$		0			0
$151$		−	13.4615i		−	1.09548i	−0.836649	−	0.547740i	$-0.815488\pi$
$151$		−	13.4615i		−	1.09548i	0.836649	−	0.547740i	$-0.184512\pi$
$152$		0			0
$153$	2.20766	+	9.20577i	0.178479	+	0.744242i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	11.8009i		−	0.941817i	−0.882182	−	0.470908i	$-0.843926\pi$
$157$		−	11.8009i		−	0.941817i	0.882182	−	0.470908i	$-0.156074\pi$
$158$		0			0
$159$	−1.28347	−	10.8557i	−0.101786	−	0.860912i
$160$		0			0
$161$		−	4.55480i		−	0.358968i
$162$		0			0
$163$	1.83457			0.143695			0.0718474	−	0.997416i	$-0.477111\pi$
$163$	1.83457			0.143695			0.0718474	+	0.997416i	$0.477111\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.44938			−0.576450			−0.288225	−	0.957563i	$-0.593065\pi$
$167$	−7.44938			−0.576450			−0.288225	+	0.957563i	$0.593065\pi$
$168$		0			0
$169$	−16.4615			−1.26627
$170$		0			0
$171$	−12.8557	+	3.08295i	−0.983099	+	0.235759i
$172$		0			0
$173$	−15.3481			−1.16690			−0.583449	−	0.812150i	$-0.698297\pi$
$173$	−15.3481			−1.16690			−0.583449	+	0.812150i	$0.698297\pi$
$174$		0			0
$175$			6.83457i			0.516645i
$176$		0			0
$177$	−1.10383	+	0.130506i	−0.0829689	+	0.00980942i
$178$		0			0
$179$			19.5027i			1.45770i	0.684675	+	0.728848i	$0.259944\pi$
$179$			19.5027i			1.45770i	−0.684675	+	0.728848i	$0.740056\pi$
$180$		0			0
$181$			10.2413i			0.761230i	0.924734	+	0.380615i	$0.124288\pi$
$181$			10.2413i			0.761230i	−0.924734	+	0.380615i	$0.875712\pi$
$182$		0			0
$183$	−17.6158	+	2.08271i	−1.30219	+	0.153959i
$184$		0			0
$185$			10.9622i			0.805954i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.87566	−	1.79664i	0.354652	−	0.130686i
$190$		0			0
$191$	13.4034			0.969834			0.484917	−	0.874560i	$-0.338850\pi$
$191$	13.4034			0.969834			0.484917	+	0.874560i	$0.338850\pi$
$192$		0			0
$193$	12.2077			0.878726			0.439363	−	0.898309i	$-0.355204\pi$
$193$	12.2077			0.878726			0.439363	+	0.898309i	$0.355204\pi$
$194$		0			0
$195$	−32.1181	+	3.79733i	−2.30003	+	0.271932i
$196$		0			0
$197$	−10.9622			−0.781021			−0.390511	−	0.920598i	$-0.627702\pi$
$197$	−10.9622			−0.781021			−0.390511	+	0.920598i	$0.627702\pi$
$198$		0			0
$199$		−	7.66914i		−	0.543651i	−0.962347	−	0.271826i	$-0.912373\pi$
$199$		−	7.66914i		−	0.543651i	0.962347	−	0.271826i	$-0.0876274\pi$
$200$		0			0
$201$	1.83457	+	15.5170i	0.129401	+	1.09448i
$202$		0			0
$203$			4.08188i			0.286492i
$204$		0			0
$205$		−	20.4826i		−	1.43057i
$206$		0			0
$207$	13.2876	−	3.18654i	0.923555	−	0.221480i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.64803			0.182298			0.0911489	−	0.995837i	$-0.470946\pi$
$211$	2.64803			0.182298			0.0911489	+	0.995837i	$0.470946\pi$
$212$		0			0
$213$	0.641735	+	5.42784i	0.0439709	+	0.371910i
$214$		0			0
$215$	−33.8323			−2.30734
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−1.22019	−	10.3204i	−0.0824525	−	0.697390i
$220$		0			0
$221$	17.1281			1.15216
$222$		0			0
$223$			1.95777i			0.131102i	0.997849	+	0.0655510i	$0.0208805\pi$
$223$			1.95777i			0.131102i	−0.997849	+	0.0655510i	$0.979120\pi$
$224$		0			0
$225$	−19.9384	+	4.78148i	−1.32923	+	0.318765i
$226$		0			0
$227$		−	1.58758i		−	0.105371i	−0.998611	−	0.0526856i	$-0.983222\pi$
$227$		−	1.58758i		−	0.105371i	0.998611	−	0.0526856i	$-0.0167781\pi$
$228$		0			0
$229$		−	25.9104i		−	1.71221i	−0.516802	−	0.856105i	$-0.672878\pi$
$229$		−	25.9104i		−	1.71221i	0.516802	−	0.856105i	$-0.327122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.31119i		−	0.413460i	−0.978398	−	0.206730i	$-0.933718\pi$
$233$		−	6.31119i		−	0.413460i	0.978398	−	0.206730i	$-0.0662821\pi$
$234$		0			0
$235$	37.7114			2.46002
$236$		0			0
$237$	−2.79841	+	0.330856i	−0.181776	+	0.0214914i
$238$		0			0
$239$	−1.04202			−0.0674024			−0.0337012	−	0.999432i	$-0.510729\pi$
$239$	−1.04202			−0.0674024			−0.0337012	+	0.999432i	$0.510729\pi$
$240$		0			0
$241$	−12.0422			−0.775708			−0.387854	−	0.921721i	$-0.626784\pi$
$241$	−12.0422			−0.775708			−0.387854	+	0.921721i	$0.626784\pi$
$242$		0			0
$243$	8.65232	+	12.9668i	0.555047	+	0.831819i
$244$		0			0
$245$	−3.44014			−0.219783
$246$		0			0
$247$			23.9190i			1.52193i
$248$		0			0
$249$	21.5864	−	2.55217i	1.36798	−	0.161737i
$250$		0			0
$251$			9.03695i			0.570407i	0.958467	+	0.285204i	$0.0920613\pi$
$251$			9.03695i			0.570407i	−0.958467	+	0.285204i	$0.907939\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	18.6725	−	2.20766i	1.16932	−	0.138249i
$256$		0			0
$257$		−	11.3193i		−	0.706081i	−0.935608	−	0.353041i	$-0.885148\pi$
$257$		−	11.3193i		−	0.706081i	0.935608	−	0.353041i	$-0.114852\pi$
$258$		0			0
$259$	−3.18654			−0.198002
$260$		0			0
$261$	−11.9080	+	2.85569i	−0.737087	+	0.176763i
$262$		0			0
$263$	−9.46678			−0.583747			−0.291873	−	0.956457i	$-0.594279\pi$
$263$	−9.46678			−0.583747			−0.291873	+	0.956457i	$0.594279\pi$
$264$		0			0
$265$	−21.7114			−1.33372
$266$		0			0
$267$	13.4278	−	1.58758i	0.821771	−	0.0971581i
$268$		0			0
$269$	−5.29271			−0.322702			−0.161351	−	0.986897i	$-0.551585\pi$
$269$	−5.29271			−0.322702			−0.161351	+	0.986897i	$0.551585\pi$
$270$		0			0
$271$			23.3383i			1.41770i	0.705359	+	0.708850i	$0.250786\pi$
$271$			23.3383i			1.41770i	−0.705359	+	0.708850i	$0.749214\pi$
$272$		0			0
$273$	−1.10383	−	9.33627i	−0.0668068	−	0.565057i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			23.6691i			1.42214i	0.703121	+	0.711071i	$0.251789\pi$
$277$			23.6691i			1.42214i	−0.703121	+	0.711071i	$0.748211\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.0718i		−	1.19738i	−0.800981	−	0.598690i	$-0.795688\pi$
$281$		−	20.0718i		−	1.19738i	0.800981	−	0.598690i	$-0.204312\pi$
$282$		0			0
$283$	12.0759			0.717836			0.358918	−	0.933369i	$-0.383146\pi$
$283$	12.0759			0.717836			0.358918	+	0.933369i	$0.383146\pi$
$284$		0			0
$285$	3.08295	+	26.0759i	0.182618	+	1.54460i
$286$		0			0
$287$	5.95400			0.351454
$288$		0			0
$289$	7.04223			0.414249
$290$		0			0
$291$	−3.26242	−	27.5938i	−0.191246	−	1.61758i
$292$		0			0
$293$	−15.3481			−0.896648			−0.448324	−	0.893871i	$-0.647979\pi$
$293$	−15.3481			−0.896648			−0.448324	+	0.893871i	$0.647979\pi$
$294$		0			0
$295$			2.20766i			0.128535i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	24.7227i		−	1.42975i
$300$		0			0
$301$		−	9.83457i		−	0.566855i
$302$		0			0
$303$	1.98307	+	16.7730i	0.113924	+	0.963582i
$304$		0			0
$305$			35.2315i			2.01735i
$306$		0			0
$307$	−24.5585			−1.40163			−0.700813	−	0.713345i	$-0.747179\pi$
$307$	−24.5585			−1.40163			−0.700813	+	0.713345i	$0.747179\pi$
$308$		0			0
$309$	2.79841	−	0.330856i	0.159196	−	0.0188217i
$310$		0			0
$311$	−2.79841			−0.158683			−0.0793415	−	0.996847i	$-0.525282\pi$
$311$	−2.79841			−0.158683			−0.0793415	+	0.996847i	$0.525282\pi$
$312$		0			0
$313$	23.2961			1.31677			0.658386	−	0.752681i	$-0.271240\pi$
$313$	23.2961			1.31677			0.658386	+	0.752681i	$0.271240\pi$
$314$		0			0
$315$	−2.40673	−	10.0359i	−0.135604	−	0.565458i
$316$		0			0
$317$	−0.714374			−0.0401232			−0.0200616	−	0.999799i	$-0.506386\pi$
$317$	−0.714374			−0.0401232			−0.0200616	+	0.999799i	$0.506386\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	11.8346	−	1.39920i	0.660541	−	0.0780959i
$322$		0			0
$323$		−	13.9058i		−	0.773742i
$324$		0			0
$325$			37.0970i			2.05777i
$326$		0			0
$327$	22.0401	−	2.60580i	1.21882	−	0.144101i
$328$		0			0
$329$			10.9622i			0.604363i
$330$		0			0
$331$	−11.8768			−0.652808			−0.326404	−	0.945230i	$-0.605837\pi$
$331$	−11.8768			−0.652808			−0.326404	+	0.945230i	$0.605837\pi$
$332$		0			0
$333$	−2.22931	−	9.29606i	−0.122165	−	0.509421i
$334$		0			0
$335$	31.0339			1.69556
$336$		0			0
$337$	11.4615			0.624347			0.312173	−	0.950025i	$-0.398943\pi$
$337$	11.4615			0.624347			0.312173	+	0.950025i	$0.398943\pi$
$338$		0			0
$339$	2.40673	−	0.284548i	0.130716	−	0.0154545i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−3.18654	−	26.9520i	−0.171558	−	1.45105i
$346$		0			0
$347$			19.3574i			1.03916i	0.854422	+	0.519579i	$0.173911\pi$
$347$			19.3574i			1.03916i	−0.854422	+	0.519579i	$0.826089\pi$
$348$		0			0
$349$			3.86821i			0.207061i	0.994626	+	0.103530i	$0.0330139\pi$
$349$			3.86821i			0.207061i	−0.994626	+	0.103530i	$0.966986\pi$
$350$		0			0
$351$	26.4643	−	9.75186i	1.41256	−	0.520516i
$352$		0			0
$353$			7.80657i			0.415502i	0.978182	+	0.207751i	$0.0666143\pi$
$353$			7.80657i			0.415502i	−0.978182	+	0.207751i	$0.933386\pi$
$354$		0			0
$355$	10.8557			0.576160
$356$		0			0
$357$	0.641735	+	5.42784i	0.0339642	+	0.287272i
$358$		0			0
$359$	−17.3695			−0.916728			−0.458364	−	0.888765i	$-0.651564\pi$
$359$	−17.3695			−0.916728			−0.458364	+	0.888765i	$0.651564\pi$
$360$		0			0
$361$	0.419255			0.0220661
$362$		0			0
$363$	2.23701	+	18.9208i	0.117412	+	0.993083i
$364$		0			0
$365$	−20.6408			−1.08039
$366$		0			0
$367$			17.2961i			0.902847i	0.892310	+	0.451423i	$0.149084\pi$
$367$			17.2961i			0.902847i	−0.892310	+	0.451423i	$0.850916\pi$
$368$		0			0
$369$	4.16543	+	17.3695i	0.216844	+	0.904221i
$370$		0			0
$371$		−	6.31119i		−	0.327660i
$372$		0			0
$373$			33.2961i			1.72400i	0.506904	+	0.862002i	$0.330790\pi$
$373$			33.2961i			1.72400i	−0.506904	+	0.862002i	$0.669210\pi$
$374$		0			0
$375$	1.28347	+	10.8557i	0.0662781	+	0.560585i
$376$		0			0
$377$			22.1558i			1.14108i
$378$		0			0
$379$	−31.5459			−1.62041			−0.810203	−	0.586149i	$-0.800643\pi$
$379$	−31.5459			−1.62041			−0.810203	+	0.586149i	$0.800643\pi$
$380$		0			0
$381$	17.5579	−	2.07587i	0.899518	−	0.106350i
$382$		0			0
$383$	4.65097			0.237654			0.118827	−	0.992915i	$-0.462087\pi$
$383$	4.65097			0.237654			0.118827	+	0.992915i	$0.462087\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	28.6903	−	6.88028i	1.45841	−	0.349744i
$388$		0			0
$389$	4.08188			0.206959			0.103480	−	0.994632i	$-0.467002\pi$
$389$	4.08188			0.206959			0.103480	+	0.994632i	$0.467002\pi$
$390$		0			0
$391$			14.3731i			0.726878i
$392$		0			0
$393$	2.08271	−	0.246240i	0.105059	−	0.0124211i
$394$		0			0
$395$			5.59681i			0.281606i
$396$		0			0
$397$			6.98747i			0.350691i	0.984507	+	0.175346i	$0.0561043\pi$
$397$			6.98747i			0.350691i	−0.984507	+	0.175346i	$0.943896\pi$
$398$		0			0
$399$	−7.57988	+	0.896171i	−0.379469	+	0.0448647i
$400$		0			0
$401$		−	4.91198i		−	0.245293i	−0.992450	−	0.122646i	$-0.960862\pi$
$401$		−	4.91198i		−	0.245293i	0.992450	−	0.122646i	$-0.0391381\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	27.5938	−	14.0422i	1.37115	−	0.697764i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−30.9652			−1.53113			−0.765565	−	0.643358i	$-0.777541\pi$
$409$	−30.9652			−1.53113			−0.765565	+	0.643358i	$0.777541\pi$
$410$		0			0
$411$	12.8135	−	1.51494i	0.632041	−	0.0747263i
$412$		0			0
$413$	−0.641735			−0.0315777
$414$		0			0
$415$		−	43.1729i		−	2.11927i
$416$		0			0
$417$	1.47692	+	12.4919i	0.0723249	+	0.611729i
$418$		0			0
$419$			20.7135i			1.01192i	0.862557	+	0.505960i	$0.168862\pi$
$419$			20.7135i			1.01192i	−0.862557	+	0.505960i	$0.831138\pi$
$420$		0			0
$421$			14.0422i			0.684376i	0.939631	+	0.342188i	$0.111168\pi$
$421$			14.0422i			0.684376i	−0.939631	+	0.342188i	$0.888832\pi$
$422$		0			0
$423$	−31.9798	+	7.66914i	−1.55491	+	0.372887i
$424$		0			0
$425$		−	21.5671i		−	1.04616i
$426$		0			0
$427$	−10.2413			−0.495611
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.9919			−1.44466			−0.722329	−	0.691549i	$-0.756929\pi$
$431$	−29.9919			−1.44466			−0.722329	+	0.691549i	$0.756929\pi$
$432$		0			0
$433$	−8.37309			−0.402385			−0.201192	−	0.979552i	$-0.564482\pi$
$433$	−8.37309			−0.402385			−0.201192	+	0.979552i	$0.564482\pi$
$434$		0			0
$435$	2.85569	+	24.1536i	0.136920	+	1.15808i
$436$		0			0
$437$	−20.0718			−0.960162
$438$		0			0
$439$		−	12.4153i		−	0.592551i	−0.955103	−	0.296275i	$-0.904255\pi$
$439$		−	12.4153i		−	0.592551i	0.955103	−	0.296275i	$-0.0957446\pi$
$440$		0			0
$441$	2.91729	−	0.699602i	0.138918	−	0.0333144i
$442$		0			0
$443$			36.9684i			1.75642i	0.478276	+	0.878210i	$0.341262\pi$
$443$			36.9684i			1.75642i	−0.478276	+	0.878210i	$0.658738\pi$
$444$		0			0
$445$		−	26.8557i		−	1.27308i
$446$		0			0
$447$	4.08188	+	34.5248i	0.193066	+	1.63297i
$448$		0			0
$449$		−	21.9243i		−	1.03467i	−0.855782	−	0.517336i	$-0.826924\pi$
$449$		−	21.9243i		−	1.03467i	0.855782	−	0.517336i	$-0.173076\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	23.1547	−	2.73758i	1.08790	−	0.128623i
$454$		0			0
$455$	−18.6725			−0.875383
$456$		0			0
$457$	19.5459			0.914321			0.457160	−	0.889384i	$-0.348867\pi$
$457$	19.5459			0.914321			0.457160	+	0.889384i	$0.348867\pi$
$458$		0			0
$459$	−15.3856	+	5.66945i	−0.718139	+	0.264627i
$460$		0			0
$461$	−10.3204			−0.480670			−0.240335	−	0.970690i	$-0.577257\pi$
$461$	−10.3204			−0.480670			−0.240335	+	0.970690i	$0.577257\pi$
$462$		0			0
$463$			39.3383i			1.82821i	0.405483	+	0.914103i	$0.367103\pi$
$463$			39.3383i			1.82821i	−0.405483	+	0.914103i	$0.632897\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	26.0788i		−	1.20678i	−0.797445	−	0.603392i	$-0.793815\pi$
$467$		−	26.0788i		−	1.20678i	0.797445	−	0.603392i	$-0.206185\pi$
$468$		0			0
$469$			9.02112i			0.416556i
$470$		0			0
$471$	20.2984	−	2.39989i	0.935302	−	0.110581i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	30.1181			1.38191
$476$		0			0
$477$	18.4115	−	4.41532i	0.843006	−	0.202163i
$478$		0			0
$479$	−8.16375			−0.373011			−0.186506	−	0.982454i	$-0.559716\pi$
$479$	−8.16375			−0.373011			−0.186506	+	0.982454i	$0.559716\pi$
$480$		0			0
$481$	−17.2961			−0.788632
$482$		0			0
$483$	7.83457	−	0.926283i	0.356485	−	0.0421473i
$484$		0			0
$485$	−55.1875			−2.50594
$486$		0			0
$487$		−	1.87680i		−	0.0850460i	−0.999095	−	0.0425230i	$-0.986460\pi$
$487$		−	1.87680i		−	0.0850460i	0.999095	−	0.0425230i	$-0.0135396\pi$
$488$		0			0
$489$	0.373086	+	3.15559i	0.0168715	+	0.142701i
$490$		0			0
$491$		−	26.2377i		−	1.18409i	−0.805905	−	0.592045i	$-0.798321\pi$
$491$		−	26.2377i		−	1.18409i	0.805905	−	0.592045i	$-0.201679\pi$
$492$		0			0
$493$		−	12.8807i		−	0.580119i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.15559i			0.141548i
$498$		0			0
$499$	12.2749			0.549502			0.274751	−	0.961515i	$-0.411405\pi$
$499$	12.2749			0.549502			0.274751	+	0.961515i	$0.411405\pi$
$500$		0			0
$501$	−1.51494	−	12.8135i	−0.0676824	−	0.572463i
$502$		0			0
$503$	−33.8323			−1.50851			−0.754254	−	0.656582i	$-0.772001\pi$
$503$	−33.8323			−1.50851			−0.754254	+	0.656582i	$0.772001\pi$
$504$		0			0
$505$	33.5459			1.49278
$506$		0			0
$507$	−3.34768	−	28.3149i	−0.148676	−	1.25751i
$508$		0			0
$509$	31.1065			1.37877			0.689387	−	0.724393i	$-0.257880\pi$
$509$	31.1065			1.37877			0.689387	+	0.724393i	$0.257880\pi$
$510$		0			0
$511$		−	6.00000i		−	0.265424i
$512$		0			0
$513$	−7.91729	−	21.4857i	−0.349557	−	0.948618i
$514$		0			0
$515$		−	5.59681i		−	0.246625i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−3.12126	−	26.3999i	−0.137008	−	1.15883i
$520$		0			0
$521$			36.2735i			1.58917i	0.607151	+	0.794586i	$0.292312\pi$
$521$			36.2735i			1.58917i	−0.607151	+	0.794586i	$0.707688\pi$
$522$		0			0
$523$	−34.1181			−1.49188			−0.745940	−	0.666013i	$-0.768000\pi$
$523$	−34.1181			−1.49188			−0.745940	+	0.666013i	$0.768000\pi$
$524$		0			0
$525$	−11.7559	+	1.38991i	−0.513072	+	0.0606605i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−2.25383			−0.0979925
$530$		0			0
$531$	−0.448959	−	1.87212i	−0.0194831	−	0.0812433i
$532$		0			0
$533$	32.3174			1.39982
$534$		0			0
$535$		−	23.6691i		−	1.02331i
$536$		0			0
$537$	−33.5459	+	3.96614i	−1.44761	+	0.171152i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	36.4153i		−	1.56562i	−0.622263	−	0.782808i	$-0.713786\pi$
$541$		−	36.4153i		−	1.56562i	0.622263	−	0.782808i	$-0.286214\pi$
$542$		0			0
$543$	−17.6158	+	2.08271i	−0.755965	+	0.0893778i
$544$		0			0
$545$		−	44.0801i		−	1.88818i
$546$		0			0
$547$	17.8346			0.762551			0.381276	−	0.924461i	$-0.375485\pi$
$547$	17.8346			0.762551			0.381276	+	0.924461i	$0.375485\pi$
$548$		0			0
$549$	−7.16483	−	29.8768i	−0.305788	−	1.27511i
$550$		0			0
$551$	17.9877			0.766303
$552$		0			0
$553$	−1.62691			−0.0691834
$554$		0			0
$555$	−18.8557	+	2.22931i	−0.800379	+	0.0946290i
$556$		0			0
$557$	−37.5374			−1.59051			−0.795256	−	0.606273i	$-0.792664\pi$
$557$	−37.5374			−1.59051			−0.795256	+	0.606273i	$0.792664\pi$
$558$		0			0
$559$		−	53.3805i		−	2.25776i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	0.641735i		−	0.0270459i	−0.999909	−	0.0135229i	$-0.995695\pi$
$563$		−	0.641735i		−	0.0270459i	0.999909	−	0.0135229i	$-0.00430462\pi$
$564$		0			0
$565$		−	4.81346i		−	0.202504i
$566$		0			0
$567$	4.08188	+	8.02112i	0.171423	+	0.336855i
$568$		0			0
$569$		−	0.684829i		−	0.0287095i	−0.999897	−	0.0143548i	$-0.995431\pi$
$569$		−	0.684829i		−	0.0287095i	0.999897	−	0.0143548i	$-0.00456942\pi$
$570$		0			0
$571$	−6.16543			−0.258015			−0.129008	−	0.991644i	$-0.541179\pi$
$571$	−6.16543			−0.258015			−0.129008	+	0.991644i	$0.541179\pi$
$572$		0			0
$573$	2.72577	+	23.0548i	0.113871	+	0.963126i
$574$		0			0
$575$	−31.1301			−1.29821
$576$		0			0
$577$	24.9230			1.03756			0.518778	−	0.854909i	$-0.326387\pi$
$577$	24.9230			1.03756			0.518778	+	0.854909i	$0.326387\pi$
$578$		0			0
$579$	2.48260	+	20.9980i	0.103173	+	0.872649i
$580$		0			0
$581$	12.5497			0.520651
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−13.0633	−	54.4732i	−0.540103	−	2.25219i
$586$		0			0
$587$		−	12.1730i		−	0.502433i	−0.967931	−	0.251217i	$-0.919169\pi$
$587$		−	12.1730i		−	0.502433i	0.967931	−	0.251217i	$-0.0808306\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−2.22931	−	18.8557i	−0.0917016	−	0.775619i
$592$		0			0
$593$			1.30303i			0.0535090i	0.999642	+	0.0267545i	$0.00851723\pi$
$593$			1.30303i			0.0535090i	−0.999642	+	0.0267545i	$0.991483\pi$
$594$		0			0
$595$	10.8557			0.445040
$596$		0			0
$597$	13.1915	−	1.55963i	0.539891	−	0.0638314i
$598$		0			0
$599$	37.9338			1.54993			0.774966	−	0.632003i	$-0.217767\pi$
$599$	37.9338			1.54993			0.774966	+	0.632003i	$0.217767\pi$
$600$		0			0
$601$	−1.58468			−0.0646406			−0.0323203	−	0.999478i	$-0.510290\pi$
$601$	−1.58468			−0.0646406			−0.0323203	+	0.999478i	$0.510290\pi$
$602$		0			0
$603$	−26.3172	+	6.31119i	−1.07172	+	0.257011i
$604$		0			0
$605$	37.8416			1.53848
$606$		0			0
$607$		−	37.7114i		−	1.53066i	−0.643640	−	0.765329i	$-0.722576\pi$
$607$		−	37.7114i		−	1.53066i	0.643640	−	0.765329i	$-0.277424\pi$
$608$		0			0
$609$	−7.02112	+	0.830108i	−0.284510	+	0.0336377i
$610$		0			0
$611$			59.5009i			2.40715i
$612$		0			0
$613$			40.8979i			1.65185i	0.563779	+	0.825926i	$0.309347\pi$
$613$			40.8979i			1.65185i	−0.563779	+	0.825926i	$0.690653\pi$
$614$		0			0
$615$	35.2315	−	4.16543i	1.42067	−	0.167966i
$616$		0			0
$617$			38.0299i			1.53103i	0.643420	+	0.765514i	$0.277515\pi$
$617$			38.0299i			1.53103i	−0.643420	+	0.765514i	$0.722485\pi$
$618$		0			0
$619$	−2.03364			−0.0817390			−0.0408695	−	0.999164i	$-0.513013\pi$
$619$	−2.03364			−0.0817390			−0.0408695	+	0.999164i	$0.513013\pi$
$620$		0			0
$621$	8.18331	+	22.2077i	0.328385	+	0.891163i
$622$		0			0
$623$	7.80657			0.312763
$624$		0			0
$625$	−12.4615			−0.498459
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.0554			0.400936
$630$		0			0
$631$			23.3383i			0.929083i	0.885551	+	0.464541i	$0.153781\pi$
$631$			23.3383i			0.929083i	−0.885551	+	0.464541i	$0.846219\pi$
$632$		0			0
$633$	0.538514	+	4.55480i	0.0214040	+	0.181037i
$634$		0			0
$635$		−	35.1158i		−	1.39353i
$636$		0			0
$637$		−	5.42784i		−	0.215059i
$638$		0			0
$639$	−9.20577	+	2.20766i	−0.364175	+	0.0873336i
$640$		0			0
$641$			41.7351i			1.64844i	0.566273	+	0.824218i	$0.308385\pi$
$641$			41.7351i			1.64844i	−0.566273	+	0.824218i	$0.691615\pi$
$642$		0			0
$643$	−0.0758724			−0.00299212			−0.00149606	−	0.999999i	$-0.500476\pi$
$643$	−0.0758724			−0.00299212			−0.00149606	+	0.999999i	$0.500476\pi$
$644$		0			0
$645$	−6.88028	−	58.1940i	−0.270911	−	2.29139i
$646$		0			0
$647$	6.54265			0.257218			0.128609	−	0.991695i	$-0.458949\pi$
$647$	6.54265			0.257218			0.128609	+	0.991695i	$0.458949\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.569096			−0.0222704			−0.0111352	−	0.999938i	$-0.503545\pi$
$653$	−0.569096			−0.0222704			−0.0111352	+	0.999938i	$0.503545\pi$
$654$		0			0
$655$		−	4.16543i		−	0.162757i
$656$		0			0
$657$	17.5037	−	4.19761i	0.682885	−	0.163764i
$658$		0			0
$659$			17.0810i			0.665381i	0.943036	+	0.332691i	$0.107956\pi$
$659$			17.0810i			0.665381i	−0.943036	+	0.332691i	$0.892044\pi$
$660$		0			0
$661$		−	6.65662i		−	0.258912i	−0.991585	−	0.129456i	$-0.958677\pi$
$661$		−	6.65662i		−	0.258912i	0.991585	−	0.129456i	$-0.0413232\pi$
$662$		0			0
$663$	3.48324	+	29.4615i	0.135278	+	1.14419i
$664$		0			0
$665$			15.1598i			0.587871i
$666$		0			0
$667$	−18.5921			−0.719890
$668$		0			0
$669$	−3.36750	+	0.398140i	−0.130195	+	0.0153930i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0422303			−0.00162786			−0.000813928	−	1.00000i	$-0.500259\pi$
$673$	−0.0422303			−0.00162786			−0.000813928		1.00000i	$0.500259\pi$
$674$		0			0
$675$	−12.2792	−	33.3231i	−0.472628	−	1.28261i
$676$		0			0
$677$	−24.0810			−0.925508			−0.462754	−	0.886487i	$-0.653139\pi$
$677$	−24.0810			−0.925508			−0.462754	+	0.886487i	$0.653139\pi$
$678$		0			0
$679$		−	16.0422i		−	0.615644i
$680$		0			0
$681$	2.73074	−	0.322856i	0.104642	−	0.0123719i
$682$		0			0
$683$		−	11.3389i		−	0.433871i	−0.976186	−	0.216936i	$-0.930394\pi$
$683$		−	11.3389i		−	0.433871i	0.976186	−	0.216936i	$-0.0696061\pi$
$684$		0			0
$685$		−	25.6269i		−	0.979154i
$686$		0			0
$687$	44.5678	−	5.26926i	1.70037	−	0.201035i
$688$		0			0
$689$		−	34.2561i		−	1.30505i
$690$		0			0
$691$	26.4490			1.00617			0.503083	−	0.864238i	$-0.332199\pi$
$691$	26.4490			1.00617			0.503083	+	0.864238i	$0.332199\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	24.9837			0.947687
$696$		0			0
$697$	−18.7884			−0.711662
$698$		0			0
$699$	10.8557	−	1.28347i	0.410600	−	0.0485453i
$700$		0			0
$701$	−13.1915			−0.498235			−0.249117	−	0.968473i	$-0.580141\pi$
$701$	−13.1915			−0.498235			−0.249117	+	0.968473i	$0.580141\pi$
$702$		0			0
$703$			14.0422i			0.529613i
$704$		0			0
$705$	7.66914	+	64.8662i	0.288837	+	2.44300i
$706$		0			0
$707$			9.75133i			0.366736i
$708$		0			0
$709$		−	18.5248i		−	0.695715i	−0.937547	−	0.347857i	$-0.886909\pi$
$709$		−	18.5248i		−	0.695715i	0.937547	−	0.347857i	$-0.113091\pi$
$710$		0			0
$711$	−1.13819	−	4.74617i	−0.0426855	−	0.177995i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.211909	−	1.79234i	−0.00791388	−	0.0669362i
$718$		0			0
$719$	−10.2478			−0.382178			−0.191089	−	0.981573i	$-0.561202\pi$
$719$	−10.2478			−0.382178			−0.191089	+	0.981573i	$0.561202\pi$
$720$		0			0
$721$	1.62691			0.0605894
$722$		0			0
$723$	−2.44896	−	20.7135i	−0.0910778	−	0.770343i
$724$		0			0
$725$	27.8979			1.03610
$726$		0			0
$727$		−	41.2961i		−	1.53159i	−0.643087	−	0.765793i	$-0.722347\pi$
$727$		−	41.2961i		−	1.53159i	0.643087	−	0.765793i	$-0.277653\pi$
$728$		0			0
$729$	−20.5442	+	17.5196i	−0.760896	+	0.648874i
$730$		0			0
$731$			31.0339i			1.14783i
$732$		0			0
$733$		−	13.3606i		−	0.493484i	−0.969081	−	0.246742i	$-0.920640\pi$
$733$		−	13.3606i		−	0.493484i	0.969081	−	0.246742i	$-0.0793599\pi$
$734$		0			0
$735$	−0.699602	−	5.91729i	−0.0258052	−	0.218262i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−49.1729			−1.80885			−0.904426	−	0.426630i	$-0.859701\pi$
$739$	−49.1729			−1.80885			−0.904426	+	0.426630i	$0.859701\pi$
$740$		0			0
$741$	−41.1424	+	4.86428i	−1.51140	+	0.178694i
$742$		0			0
$743$	32.2683			1.18381			0.591904	−	0.806008i	$-0.298376\pi$
$743$	32.2683			1.18381			0.591904	+	0.806008i	$0.298376\pi$
$744$		0			0
$745$	69.0497			2.52978
$746$		0			0
$747$	8.77981	+	36.6112i	0.321237	+	1.33953i
$748$		0			0
$749$	6.88028			0.251400
$750$		0			0
$751$			49.8768i			1.82003i	0.414575	+	0.910015i	$0.363930\pi$
$751$			49.8768i			1.82003i	−0.414575	+	0.910015i	$0.636070\pi$
$752$		0			0
$753$	−15.5442	+	1.83779i	−0.566462	+	0.0669729i
$754$		0			0
$755$		−	46.3094i		−	1.68537i
$756$		0			0
$757$		−	2.52483i		−	0.0917665i	−0.998947	−	0.0458833i	$-0.985390\pi$
$757$		−	2.52483i		−	0.0917665i	0.998947	−	0.0458833i	$-0.0146102\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	34.1895i		−	1.23937i	−0.784851	−	0.619684i	$-0.787261\pi$
$761$		−	34.1895i		−	1.23937i	0.784851	−	0.619684i	$-0.212739\pi$
$762$		0			0
$763$	12.8135			0.463878
$764$		0			0
$765$	7.59466	+	31.6691i	0.274585	+	1.14500i
$766$		0			0
$767$	−3.48324			−0.125772
$768$		0			0
$769$	19.2961			0.695834			0.347917	−	0.937525i	$-0.386889\pi$
$769$	19.2961			0.695834			0.347917	+	0.937525i	$0.386889\pi$
$770$		0			0
$771$	19.4701	−	2.30195i	0.701198	−	0.0829027i
$772$		0			0
$773$	−9.18223			−0.330262			−0.165131	−	0.986272i	$-0.552805\pi$
$773$	−9.18223			−0.330262			−0.165131	+	0.986272i	$0.552805\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.648029	−	5.48108i	−0.0232479	−	0.196633i
$778$		0			0
$779$		−	26.2377i		−	0.940062i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−7.33364	−	19.9019i	−0.262083	−	0.711234i
$784$		0			0
$785$		−	40.5969i		−	1.44896i
$786$		0			0
$787$	−0.737584			−0.0262920			−0.0131460	−	0.999914i	$-0.504185\pi$
$787$	−0.737584			−0.0262920			−0.0131460	+	0.999914i	$0.504185\pi$
$788$		0			0
$789$	−1.92520	−	16.2835i	−0.0685391	−	0.579709i
$790$		0			0
$791$	1.39920			0.0497499
$792$		0			0
$793$	−55.5882			−1.97399
$794$		0			0
$795$	−4.41532	−	37.3451i	−0.156595	−	1.32449i
$796$		0			0
$797$	−7.32967			−0.259630			−0.129815	−	0.991538i	$-0.541438\pi$
$797$	−7.32967			−0.259630			−0.129815	+	0.991538i	$0.541438\pi$
$798$		0			0
$799$		−	34.5921i		−	1.22378i
$800$		0			0
$801$	5.46149	+	22.7740i	0.192972	+	0.804679i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	15.6691i		−	0.552265i
$806$		0			0
$807$	−1.07635	−	9.10383i	−0.0378892	−	0.320470i
$808$		0			0
$809$		−	22.4168i		−	0.788133i	−0.919082	−	0.394066i	$-0.871068\pi$
$809$		−	22.4168i		−	0.788133i	0.919082	−	0.394066i	$-0.128932\pi$
$810$		0			0
$811$	39.0833			1.37240			0.686200	−	0.727413i	$-0.259277\pi$
$811$	39.0833			1.37240			0.686200	+	0.727413i	$0.259277\pi$
$812$		0			0
$813$	−40.1435	+	4.74617i	−1.40789	+	0.166456i
$814$		0			0
$815$	6.31119			0.221071
$816$		0			0
$817$	−43.3383			−1.51621
$818$		0			0
$819$	15.8346	−	3.79733i	0.553305	−	0.132689i
$820$		0			0
$821$	29.3737			1.02515			0.512575	−	0.858643i	$-0.328692\pi$
$821$	29.3737			1.02515			0.512575	+	0.858643i	$0.328692\pi$
$822$		0			0
$823$			16.9652i			0.591370i	0.955286	+	0.295685i	$0.0955478\pi$
$823$			16.9652i			0.591370i	−0.955286	+	0.295685i	$0.904452\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			25.0995i			0.872794i	0.899754	+	0.436397i	$0.143746\pi$
$827$			25.0995i			0.872794i	−0.899754	+	0.436397i	$0.856254\pi$
$828$		0			0
$829$			9.57959i			0.332713i	0.986066	+	0.166356i	$0.0532002\pi$
$829$			9.57959i			0.332713i	−0.986066	+	0.166356i	$0.946800\pi$
$830$		0			0
$831$	−40.7126	+	4.81346i	−1.41230	+	0.166977i
$832$		0			0
$833$			3.15559i			0.109335i
$834$		0			0
$835$	−25.6269			−0.886856
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.5682			−0.468427			−0.234213	−	0.972185i	$-0.575251\pi$
$839$	−13.5682			−0.468427			−0.234213	+	0.972185i	$0.575251\pi$
$840$		0			0
$841$	−12.3383			−0.425458
$842$		0			0
$843$	34.5248	−	4.08188i	1.18910	−	0.140587i
$844$		0			0
$845$	−56.6298			−1.94813
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$	2.45580	+	20.7714i	0.0842828	+	0.712871i
$850$		0			0
$851$		−	14.5141i		−	0.497535i
$852$		0			0
$853$			5.42784i			0.185846i	0.995673	+	0.0929229i	$0.0296210\pi$
$853$			5.42784i			0.185846i	−0.995673	+	0.0929229i	$0.970379\pi$
$854$		0			0
$855$	−44.2254	+	10.6058i	−1.51248	+	0.362711i
$856$		0			0
$857$			1.26391i			0.0431744i	0.999767	+	0.0215872i	$0.00687195\pi$
$857$			1.26391i			0.0431744i	−0.999767	+	0.0215872i	$0.993128\pi$
$858$		0			0
$859$	−33.2202			−1.13346			−0.566729	−	0.823904i	$-0.691791\pi$
$859$	−33.2202			−1.13346			−0.566729	+	0.823904i	$0.691791\pi$
$860$		0			0
$861$	1.21083	+	10.2413i	0.0412650	+	0.349023i
$862$		0			0
$863$	32.3369			1.10076			0.550381	−	0.834914i	$-0.314482\pi$
$863$	32.3369			1.10076			0.550381	+	0.834914i	$0.314482\pi$
$864$		0			0
$865$	−52.7998			−1.79525
$866$		0			0
$867$	1.43214	+	12.1131i	0.0486379	+	0.411384i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			48.9652i			1.65912i
$872$		0			0
$873$	46.7998	−	11.2232i	1.58393	−	0.379847i
$874$		0			0
$875$			6.31119i			0.213357i
$876$		0			0
$877$			31.2710i			1.05595i	0.849261	+	0.527973i	$0.177048\pi$
$877$			31.2710i			1.05595i	−0.849261	+	0.527973i	$0.822952\pi$
$878$		0			0
$879$	−3.12126	−	26.3999i	−0.105278	−	0.890446i
$880$		0			0
$881$		−	42.3533i		−	1.42692i	−0.700697	−	0.713459i	$-0.747128\pi$
$881$		−	42.3533i		−	1.42692i	0.700697	−	0.713459i	$-0.252872\pi$
$882$		0			0
$883$	22.3172			0.751033			0.375516	−	0.926816i	$-0.377465\pi$
$883$	22.3172			0.751033			0.375516	+	0.926816i	$0.377465\pi$
$884$		0			0
$885$	−3.79733	+	0.448959i	−0.127646	+	0.0150916i
$886$		0			0
$887$	−31.2654			−1.04979			−0.524894	−	0.851167i	$-0.675895\pi$
$887$	−31.2654			−1.04979			−0.524894	+	0.851167i	$0.675895\pi$
$888$		0			0
$889$	10.2077			0.342354
$890$		0			0
$891$		0			0
$892$		0			0
$893$	48.3073			1.61654
$894$		0			0
$895$			67.0919i			2.24263i
$896$		0			0
$897$	42.5248	−	5.02772i	1.41986	−	0.167871i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			19.9155i			0.663483i
$902$		0			0
$903$	16.9162	−	2.00000i	0.562935	−	0.0665558i
$904$		0			0
$905$			35.2315i			1.17114i
$906$		0			0
$907$	−1.43643			−0.0476959			−0.0238480	−	0.999716i	$-0.507592\pi$
$907$	−1.43643			−0.0476959			−0.0238480	+	0.999716i	$0.507592\pi$
$908$		0			0
$909$	−28.4474	+	6.82204i	−0.943541	+	0.226273i
$910$		0			0
$911$	−55.8919			−1.85178			−0.925891	−	0.377791i	$-0.876684\pi$
$911$	−55.8919			−1.85178			−0.925891	+	0.377791i	$0.876684\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−60.6007	+	7.16483i	−2.00340	+	0.236862i
$916$		0			0
$917$	1.21083			0.0399851
$918$		0			0
$919$		−	26.4575i		−	0.872754i	−0.899764	−	0.436377i	$-0.856261\pi$
$919$		−	26.4575i		−	0.872754i	0.899764	−	0.436377i	$-0.143739\pi$
$920$		0			0
$921$	−4.99431	−	42.2423i	−0.164568	−	1.39193i
$922$		0			0
$923$			17.1281i			0.563777i
$924$		0			0
$925$			21.7787i			0.716078i
$926$		0			0
$927$	1.13819	+	4.74617i	0.0373831	+	0.155885i
$928$		0			0
$929$			18.7687i			0.615782i	0.951422	+	0.307891i	$0.0996232\pi$
$929$			18.7687i			0.615782i	−0.951422	+	0.307891i	$0.900377\pi$
$930$		0			0
$931$	−4.40673			−0.144425
$932$		0			0
$933$	−0.569096	−	4.81346i	−0.0186314	−	0.157586i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	30.8306			1.00719			0.503596	−	0.863939i	$-0.332010\pi$
$937$	30.8306			1.00719			0.503596	+	0.863939i	$0.332010\pi$
$938$		0			0
$939$	4.73758	+	40.0709i	0.154605	+	1.30766i
$940$		0			0
$941$	−19.8068			−0.645682			−0.322841	−	0.946453i	$-0.604638\pi$
$941$	−19.8068			−0.645682			−0.322841	+	0.946453i	$0.604638\pi$
$942$		0			0
$943$			27.1193i			0.883125i
$944$		0			0
$945$	16.7730	−	6.18068i	0.545625	−	0.201058i
$946$		0			0
$947$		−	12.6224i		−	0.410172i	−0.978744	−	0.205086i	$-0.934253\pi$
$947$		−	12.6224i		−	0.410172i	0.978744	−	0.205086i	$-0.0657474\pi$
$948$		0			0
$949$		−	32.5671i		−	1.05717i
$950$		0			0
$951$	−0.145278	−	1.22877i	−0.00471096	−	0.0398457i
$952$		0			0
$953$			45.7403i			1.48167i	0.671685	+	0.740837i	$0.265571\pi$
$953$			45.7403i			1.48167i	−0.671685	+	0.740837i	$0.734429\pi$
$954$		0			0
$955$	46.1095			1.49207
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.44938			0.240553
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	4.81346	+	20.0718i	0.155111	+	0.646803i
$964$		0			0
$965$	41.9961			1.35190
$966$		0			0
$967$		−	1.87680i		−	0.0603539i	−0.999545	−	0.0301769i	$-0.990393\pi$
$967$		−	1.87680i		−	0.0603539i	0.999545	−	0.0301769i	$-0.00960708\pi$
$968$		0			0
$969$	23.9190	−	2.82795i	0.768390	−	0.0908469i
$970$		0			0
$971$			37.2725i			1.19613i	0.801448	+	0.598065i	$0.204064\pi$
$971$			37.2725i			1.19613i	−0.801448	+	0.598065i	$0.795936\pi$
$972$		0			0
$973$			7.26242i			0.232822i
$974$		0			0
$975$	−63.8094	+	7.54420i	−2.04354	+	0.241608i
$976$		0			0
$977$			37.5374i			1.20093i	0.799651	+	0.600465i	$0.205018\pi$
$977$			37.5374i			1.20093i	−0.799651	+	0.600465i	$0.794982\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	8.96432	+	37.3805i	0.286209	+	1.19347i
$982$		0			0
$983$	53.3820			1.70262			0.851311	−	0.524661i	$-0.175808\pi$
$983$	53.3820			1.70262			0.851311	+	0.524661i	$0.175808\pi$
$984$		0			0
$985$	−37.7114			−1.20158
$986$		0			0
$987$	−18.8557	+	2.22931i	−0.600183	+	0.0709597i
$988$		0			0
$989$	44.7945			1.42438
$990$		0			0
$991$			45.7114i			1.45207i	0.687658	+	0.726035i	$0.258639\pi$
$991$			45.7114i			1.45207i	−0.687658	+	0.726035i	$0.741361\pi$
$992$		0			0
$993$	−2.41532	−	20.4289i	−0.0766478	−	0.648293i
$994$		0			0
$995$		−	26.3829i		−	0.836395i
$996$		0			0
$997$		−	13.4951i		−	0.427395i	−0.976900	−	0.213697i	$-0.931449\pi$
$997$		−	13.4951i		−	0.427395i	0.976900	−	0.213697i	$-0.0685507\pi$
$998$		0			0
$999$	15.5365	−	5.72506i	0.491554	−	0.181133i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.j.d.239.8		12
3.2	odd	2	inner	672.2.j.d.239.5		12
4.3	odd	2		168.2.j.d.155.7	yes	12
8.3	odd	2	inner	672.2.j.d.239.7		12
8.5	even	2		168.2.j.d.155.5	✓	12
12.11	even	2		168.2.j.d.155.6	yes	12
24.5	odd	2		168.2.j.d.155.8	yes	12
24.11	even	2	inner	672.2.j.d.239.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.j.d.155.5	✓	12	8.5	even	2
168.2.j.d.155.6	yes	12	12.11	even	2
168.2.j.d.155.7	yes	12	4.3	odd	2
168.2.j.d.155.8	yes	12	24.5	odd	2
672.2.j.d.239.5		12	3.2	odd	2	inner
672.2.j.d.239.6		12	24.11	even	2	inner
672.2.j.d.239.7		12	8.3	odd	2	inner
672.2.j.d.239.8		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.203364 + 1.72007i) q^{3} +3.44014 q^{5} +1.00000i q^{7} +(-2.91729 + 0.699602i) q^{9} +O(q^{10})\)	\(q+(0.203364 + 1.72007i) q^{3} +3.44014 q^{5} +1.00000i q^{7} +(-2.91729 + 0.699602i) q^{9} +5.42784i q^{13} +(0.699602 + 5.91729i) q^{15} -3.15559i q^{17} +4.40673 q^{19} +(-1.72007 + 0.203364i) q^{21} -4.55480 q^{23} +6.83457 q^{25} +(-1.79664 - 4.87566i) q^{27} +4.08188 q^{29} +3.44014i q^{35} +3.18654i q^{37} +(-9.33627 + 1.10383i) q^{39} -5.95400i q^{41} -9.83457 q^{43} +(-10.0359 + 2.40673i) q^{45} +10.9622 q^{47} -1.00000 q^{49} +(5.42784 - 0.641735i) q^{51} -6.31119 q^{53} +(0.896171 + 7.57988i) q^{57} +0.641735i q^{59} +10.2413i q^{61} +(-0.699602 - 2.91729i) q^{63} +18.6725i q^{65} +9.02112 q^{67} +(-0.926283 - 7.83457i) q^{69} +3.15559 q^{71} -6.00000 q^{73} +(1.38991 + 11.7559i) q^{75} +1.62691i q^{79} +(8.02112 - 4.08188i) q^{81} -12.5497i q^{83} -10.8557i q^{85} +(0.830108 + 7.02112i) q^{87} -7.80657i q^{89} -5.42784 q^{91} +15.1598 q^{95} -16.0422 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{9}+O(q^{10}) \) 12 * q + 4 * q^9	\( 12 q + 4 q^{9} + 48 q^{19} + 4 q^{25} - 24 q^{27} - 40 q^{43} - 12 q^{49} - 8 q^{51} + 40 q^{57} + 40 q^{67} - 72 q^{73} + 24 q^{75} + 28 q^{81} + 8 q^{91} - 56 q^{97}+O(q^{100}) \) 12 * q + 4 * q^9 + 48 * q^19 + 4 * q^25 - 24 * q^27 - 40 * q^43 - 12 * q^49 - 8 * q^51 + 40 * q^57 + 40 * q^67 - 72 * q^73 + 24 * q^75 + 28 * q^81 + 8 * q^91 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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