LMFDB - Embedded newform 1344.2.p.c.223.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(223,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.p (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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 $ x^12 - 15x^10 + 90x^8 - 247x^6 + 270x^4 + 21*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.3
Root			$2.23871 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1344.223
Dual form			1344.2.p.c.223.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.11575 q^{5} +(-1.37268 - 2.26180i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.11575 q^{5} +(-1.37268 - 2.26180i) q^{7} -1.00000 q^{9} -0.812826 q^{11} -2.00000 q^{13} -1.11575i q^{15} +3.55819i q^{17} -4.52360i q^{19} +(-2.26180 + 1.37268i) q^{21} -3.63935i q^{23} -3.75510 q^{25} +1.00000i q^{27} -8.95482i q^{29} -5.08975 q^{31} +0.812826i q^{33} +(-1.53157 - 2.52360i) q^{35} +0.718741i q^{37} +2.00000i q^{39} +10.4864i q^{41} -1.11971 q^{43} -1.11575 q^{45} -8.55385 q^{47} +(-3.23150 + 6.20946i) q^{49} +3.55819 q^{51} -5.08975i q^{53} -0.906910 q^{55} -4.52360 q^{57} -2.23150i q^{59} -5.27871 q^{61} +(1.37268 + 2.26180i) q^{63} -2.23150 q^{65} +15.1643 q^{67} -3.63935 q^{69} -5.87085i q^{71} -3.86507i q^{73} +3.75510i q^{75} +(1.11575 + 1.83845i) q^{77} -1.70789i q^{79} +1.00000 q^{81} -7.27871i q^{83} +3.97004i q^{85} -8.95482 q^{87} -3.37002i q^{89} +(2.74536 + 4.52360i) q^{91} +5.08975i q^{93} -5.04721i q^{95} +8.55385i q^{97} +0.812826 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} - 24 q^{13} + 36 q^{25} - 12 q^{49} + 72 q^{61} + 24 q^{69} + 12 q^{81}+O(q^{100}) $ 12 * q - 12 * q^9 - 24 * q^13 + 36 * q^25 - 12 * q^49 + 72 * q^61 + 24 * q^69 + 12 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	1.11575			0.498978			0.249489	−	0.968378i	$-0.419737\pi$
$5$	1.11575			0.498978			0.249489	+	0.968378i	$0.419737\pi$
$6$		0			0
$7$	−1.37268	−	2.26180i	−0.518824	−	0.854881i
$7$	−1.37268	−	2.26180i	−0.518824	−	0.854881i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.812826			−0.245076			−0.122538	−	0.992464i	$-0.539103\pi$
$11$	−0.812826			−0.245076			−0.122538	+	0.992464i	$0.539103\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		−	1.11575i		−	0.288085i
$16$		0			0
$17$			3.55819i			0.862987i	0.902116	+	0.431493i	$0.142013\pi$
$17$			3.55819i			0.862987i	−0.902116	+	0.431493i	$0.857987\pi$
$18$		0			0
$19$		−	4.52360i		−	1.03779i	−0.854839	−	0.518893i	$-0.826344\pi$
$19$		−	4.52360i		−	1.03779i	0.854839	−	0.518893i	$-0.173656\pi$
$20$		0			0
$21$	−2.26180	+	1.37268i	−0.493566	+	0.299543i
$22$		0			0
$23$		−	3.63935i		−	0.758858i	−0.925221	−	0.379429i	$-0.876120\pi$
$23$		−	3.63935i		−	0.758858i	0.925221	−	0.379429i	$-0.123880\pi$
$24$		0			0
$25$	−3.75510			−0.751021
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	8.95482i		−	1.66287i	−0.555623	−	0.831434i	$-0.687520\pi$
$29$		−	8.95482i		−	1.66287i	0.555623	−	0.831434i	$-0.312480\pi$
$30$		0			0
$31$	−5.08975			−0.914147			−0.457073	−	0.889429i	$-0.651102\pi$
$31$	−5.08975			−0.914147			−0.457073	+	0.889429i	$0.651102\pi$
$32$		0			0
$33$			0.812826i			0.141495i
$34$		0			0
$35$	−1.53157	−	2.52360i	−0.258882	−	0.426567i
$36$		0			0
$37$			0.718741i			0.118160i	0.998253	+	0.0590802i	$0.0188168\pi$
$37$			0.718741i			0.118160i	−0.998253	+	0.0590802i	$0.981183\pi$
$38$		0			0
$39$			2.00000i			0.320256i
$40$		0			0
$41$			10.4864i			1.63770i	0.574008	+	0.818849i	$0.305388\pi$
$41$			10.4864i			1.63770i	−0.574008	+	0.818849i	$0.694612\pi$
$42$		0			0
$43$	−1.11971			−0.170754			−0.0853770	−	0.996349i	$-0.527209\pi$
$43$	−1.11971			−0.170754			−0.0853770	+	0.996349i	$0.527209\pi$
$44$		0			0
$45$	−1.11575			−0.166326
$46$		0			0
$47$	−8.55385			−1.24771			−0.623854	−	0.781541i	$-0.714434\pi$
$47$	−8.55385			−1.24771			−0.623854	+	0.781541i	$0.714434\pi$
$48$		0			0
$49$	−3.23150	+	6.20946i	−0.461643	+	0.887066i
$50$		0			0
$51$	3.55819			0.498246
$52$		0			0
$53$		−	5.08975i		−	0.699131i	−0.936912	−	0.349566i	$-0.886329\pi$
$53$		−	5.08975i		−	0.699131i	0.936912	−	0.349566i	$-0.113671\pi$
$54$		0			0
$55$	−0.906910			−0.122288
$56$		0			0
$57$	−4.52360			−0.599166
$58$		0			0
$59$		−	2.23150i		−	0.290516i	−0.989394	−	0.145258i	$-0.953599\pi$
$59$		−	2.23150i		−	0.290516i	0.989394	−	0.145258i	$-0.0464013\pi$
$60$		0			0
$61$	−5.27871			−0.675869			−0.337935	−	0.941170i	$-0.609728\pi$
$61$	−5.27871			−0.675869			−0.337935	+	0.941170i	$0.609728\pi$
$62$		0			0
$63$	1.37268	+	2.26180i	0.172941	+	0.284960i
$64$		0			0
$65$	−2.23150			−0.276783
$66$		0			0
$67$	15.1643			1.85261			0.926306	−	0.376772i	$-0.122966\pi$
$67$	15.1643			1.85261			0.926306	+	0.376772i	$0.122966\pi$
$68$		0			0
$69$	−3.63935			−0.438127
$70$		0			0
$71$		−	5.87085i		−	0.696742i	−0.937357	−	0.348371i	$-0.886735\pi$
$71$		−	5.87085i		−	0.696742i	0.937357	−	0.348371i	$-0.113265\pi$
$72$		0			0
$73$		−	3.86507i		−	0.452372i	−0.974084	−	0.226186i	$-0.927374\pi$
$73$		−	3.86507i		−	0.452372i	0.974084	−	0.226186i	$-0.0726257\pi$
$74$		0			0
$75$			3.75510i			0.433602i
$76$		0			0
$77$	1.11575	+	1.83845i	0.127151	+	0.209511i
$78$		0			0
$79$		−	1.70789i		−	0.192153i	−0.995374	−	0.0960766i	$-0.969371\pi$
$79$		−	1.70789i		−	0.192153i	0.995374	−	0.0960766i	$-0.0306294\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.27871i		−	0.798942i	−0.916746	−	0.399471i	$-0.869194\pi$
$83$		−	7.27871i		−	0.798942i	0.916746	−	0.399471i	$-0.130806\pi$
$84$		0			0
$85$			3.97004i			0.430612i
$86$		0			0
$87$	−8.95482			−0.960058
$88$		0			0
$89$		−	3.37002i		−	0.357221i	−0.983920	−	0.178611i	$-0.942840\pi$
$89$		−	3.37002i		−	0.357221i	0.983920	−	0.178611i	$-0.0571602\pi$
$90$		0			0
$91$	2.74536	+	4.52360i	0.287792	+	0.474203i
$92$		0			0
$93$			5.08975i			0.527783i
$94$		0			0
$95$		−	5.04721i		−	0.517833i
$96$		0			0
$97$			8.55385i			0.868512i	0.900789	+	0.434256i	$0.142989\pi$
$97$			8.55385i			0.868512i	−0.900789	+	0.434256i	$0.857011\pi$
$98$		0			0
$99$	0.812826			0.0816920
$100$		0			0
$101$	1.11575			0.111021			0.0555106	−	0.998458i	$-0.482321\pi$
$101$	1.11575			0.111021			0.0555106	+	0.998458i	$0.482321\pi$
$102$		0			0
$103$	−14.2574			−1.40482			−0.702410	−	0.711772i	$-0.747893\pi$
$103$	−14.2574			−1.40482			−0.702410	+	0.711772i	$0.747893\pi$
$104$		0			0
$105$	−2.52360	+	1.53157i	−0.246279	+	0.149466i
$106$		0			0
$107$	−13.2317			−1.27916			−0.639581	−	0.768724i	$-0.720892\pi$
$107$	−13.2317			−1.27916			−0.639581	+	0.768724i	$0.720892\pi$
$108$		0			0
$109$		−	1.43748i		−	0.137686i	−0.997628	−	0.0688429i	$-0.978069\pi$
$109$		−	1.43748i		−	0.137686i	0.997628	−	0.0688429i	$-0.0219307\pi$
$110$		0			0
$111$	0.718741			0.0682199
$112$		0			0
$113$	8.23150			0.774354			0.387177	−	0.922005i	$-0.373450\pi$
$113$	8.23150			0.774354			0.387177	+	0.922005i	$0.373450\pi$
$114$		0			0
$115$		−	4.06061i		−	0.378654i
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$	8.04791	−	4.88425i	0.737751	−	0.447739i
$120$		0			0
$121$	−10.3393			−0.939938
$122$		0			0
$123$	10.4864			0.945526
$124$		0			0
$125$	−9.76850			−0.873721
$126$		0			0
$127$		−	2.75510i		−	0.244476i	−0.992501	−	0.122238i	$-0.960993\pi$
$127$		−	2.75510i		−	0.244476i	0.992501	−	0.122238i	$-0.0390071\pi$
$128$		0			0
$129$			1.11971i			0.0985849i
$130$		0			0
$131$		−	19.2787i		−	1.68439i	−0.539174	−	0.842194i	$-0.681264\pi$
$131$		−	19.2787i		−	1.68439i	0.539174	−	0.842194i	$-0.318736\pi$
$132$		0			0
$133$	−10.2315	+	6.20946i	−0.887183	+	0.538429i
$134$		0			0
$135$			1.11575i			0.0960284i
$136$		0			0
$137$	6.58421			0.562527			0.281264	−	0.959631i	$-0.409246\pi$
$137$	6.58421			0.562527			0.281264	+	0.959631i	$0.409246\pi$
$138$		0			0
$139$			17.5102i			1.48520i	0.669737	+	0.742598i	$0.266407\pi$
$139$			17.5102i			1.48520i	−0.669737	+	0.742598i	$0.733593\pi$
$140$		0			0
$141$			8.55385i			0.720364i
$142$		0			0
$143$	1.62565			0.135944
$144$		0			0
$145$		−	9.99134i		−	0.829735i
$146$		0			0
$147$	6.20946	+	3.23150i	0.512148	+	0.266530i
$148$		0			0
$149$		−	10.5805i		−	0.866786i	−0.901205	−	0.433393i	$-0.857316\pi$
$149$		−	10.5805i		−	0.866786i	0.901205	−	0.433393i	$-0.142684\pi$
$150$		0			0
$151$			12.5236i			1.01916i	0.860424	+	0.509578i	$0.170199\pi$
$151$			12.5236i			1.01916i	−0.860424	+	0.509578i	$0.829801\pi$
$152$		0			0
$153$		−	3.55819i		−	0.287662i
$154$		0			0
$155$	−5.67889			−0.456139
$156$		0			0
$157$	−0.815710			−0.0651008			−0.0325504	−	0.999470i	$-0.510363\pi$
$157$	−0.815710			−0.0651008			−0.0325504	+	0.999470i	$0.510363\pi$
$158$		0			0
$159$	−5.08975			−0.403644
$160$		0			0
$161$	−8.23150	+	4.99567i	−0.648733	+	0.393714i
$162$		0			0
$163$	−4.18284			−0.327626			−0.163813	−	0.986491i	$-0.552379\pi$
$163$	−4.18284			−0.327626			−0.163813	+	0.986491i	$0.552379\pi$
$164$		0			0
$165$			0.906910i			0.0706028i
$166$		0			0
$167$	12.4189			0.961005			0.480503	−	0.876993i	$-0.340454\pi$
$167$	12.4189			0.961005			0.480503	+	0.876993i	$0.340454\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$			4.52360i			0.345929i
$172$		0			0
$173$	25.4417			1.93429			0.967147	−	0.254218i	$-0.0818180\pi$
$173$	25.4417			1.93429			0.967147	+	0.254218i	$0.0818180\pi$
$174$		0			0
$175$	5.15456	+	8.49330i	0.389648	+	0.642033i
$176$		0			0
$177$	−2.23150			−0.167730
$178$		0			0
$179$	−9.36668			−0.700099			−0.350049	−	0.936731i	$-0.613835\pi$
$179$	−9.36668			−0.700099			−0.350049	+	0.936731i	$0.613835\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$			5.27871i			0.390213i
$184$		0			0
$185$			0.801935i			0.0589595i
$186$		0			0
$187$		−	2.89218i		−	0.211497i
$188$		0			0
$189$	2.26180	−	1.37268i	0.164522	−	0.0998478i
$190$		0			0
$191$		−	11.1764i		−	0.808693i	−0.914606	−	0.404346i	$-0.867499\pi$
$191$		−	11.1764i		−	0.808693i	0.914606	−	0.404346i	$-0.132501\pi$
$192$		0			0
$193$	13.2181			0.951460			0.475730	−	0.879591i	$-0.342184\pi$
$193$	13.2181			0.951460			0.475730	+	0.879591i	$0.342184\pi$
$194$		0			0
$195$			2.23150i			0.159801i
$196$		0			0
$197$		−	20.5718i		−	1.46568i	−0.680401	−	0.732840i	$-0.738194\pi$
$197$		−	20.5718i		−	1.46568i	0.680401	−	0.732840i	$-0.261806\pi$
$198$		0			0
$199$	13.7268			0.973067			0.486534	−	0.873662i	$-0.338261\pi$
$199$	13.7268			0.973067			0.486534	+	0.873662i	$0.338261\pi$
$200$		0			0
$201$		−	15.1643i		−	1.06961i
$202$		0			0
$203$	−20.2540	+	12.2921i	−1.42155	+	0.862737i
$204$		0			0
$205$			11.7002i			0.817176i
$206$		0			0
$207$			3.63935i			0.252953i
$208$		0			0
$209$			3.67690i			0.254337i
$210$		0			0
$211$	9.86173			0.678910			0.339455	−	0.940622i	$-0.389757\pi$
$211$	9.86173			0.678910			0.339455	+	0.940622i	$0.389757\pi$
$212$		0			0
$213$	−5.87085			−0.402264
$214$		0			0
$215$	−1.24931			−0.0852026
$216$		0			0
$217$	6.98660	+	11.5120i	0.474281	+	0.781486i
$218$		0			0
$219$	−3.86507			−0.261177
$220$		0			0
$221$		−	7.11637i		−	0.478699i
$222$		0			0
$223$	25.3438			1.69715			0.848573	−	0.529079i	$-0.177462\pi$
$223$	25.3438			1.69715			0.848573	+	0.529079i	$0.177462\pi$
$224$		0			0
$225$	3.75510			0.250340
$226$		0			0
$227$		−	14.5574i		−	0.966210i	−0.875563	−	0.483105i	$-0.839509\pi$
$227$		−	14.5574i		−	0.966210i	0.875563	−	0.483105i	$-0.160491\pi$
$228$		0			0
$229$	14.5842			0.963752			0.481876	−	0.876239i	$-0.339956\pi$
$229$	14.5842			0.963752			0.481876	+	0.876239i	$0.339956\pi$
$230$		0			0
$231$	1.83845	−	1.11575i	0.120961	−	0.0734109i
$232$		0			0
$233$	27.8361			1.82361			0.911803	−	0.410629i	$-0.134691\pi$
$233$	27.8361			1.82361			0.911803	+	0.410629i	$0.134691\pi$
$234$		0			0
$235$	−9.54396			−0.622579
$236$		0			0
$237$	−1.70789			−0.110940
$238$		0			0
$239$			20.1024i			1.30031i	0.759800	+	0.650157i	$0.225297\pi$
$239$			20.1024i			1.30031i	−0.759800	+	0.650157i	$0.774703\pi$
$240$		0			0
$241$		−	16.2840i		−	1.04894i	−0.851428	−	0.524472i	$-0.824263\pi$
$241$		−	16.2840i		−	1.04894i	0.851428	−	0.524472i	$-0.175737\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−3.60554	+	6.92820i	−0.230350	+	0.442627i
$246$		0			0
$247$			9.04721i			0.575660i
$248$		0			0
$249$	−7.27871			−0.461269
$250$		0			0
$251$			12.5842i			0.794308i	0.917752	+	0.397154i	$0.130002\pi$
$251$			12.5842i			0.794308i	−0.917752	+	0.397154i	$0.869998\pi$
$252$		0			0
$253$			2.95816i			0.185978i
$254$		0			0
$255$	3.97004			0.248614
$256$		0			0
$257$		−	28.2079i		−	1.75956i	−0.475383	−	0.879779i	$-0.657690\pi$
$257$		−	28.2079i		−	1.75956i	0.475383	−	0.879779i	$-0.342310\pi$
$258$		0			0
$259$	1.62565	−	0.986602i	0.101013	−	0.0613045i
$260$		0			0
$261$			8.95482i			0.554290i
$262$		0			0
$263$		−	6.12915i		−	0.377939i	−0.981983	−	0.188970i	$-0.939485\pi$
$263$		−	6.12915i		−	0.377939i	0.981983	−	0.188970i	$-0.0605148\pi$
$264$		0			0
$265$		−	5.67889i		−	0.348851i
$266$		0			0
$267$	−3.37002			−0.206242
$268$		0			0
$269$	23.2102			1.41515			0.707574	−	0.706639i	$-0.249789\pi$
$269$	23.2102			1.41515			0.707574	+	0.706639i	$0.249789\pi$
$270$		0			0
$271$	−21.1856			−1.28693			−0.643466	−	0.765475i	$-0.722504\pi$
$271$	−21.1856			−1.28693			−0.643466	+	0.765475i	$0.722504\pi$
$272$		0			0
$273$	4.52360	−	2.74536i	0.273781	−	0.166157i
$274$		0			0
$275$	3.05224			0.184057
$276$		0			0
$277$		−	6.20946i		−	0.373090i	−0.982446	−	0.186545i	$-0.940271\pi$
$277$		−	6.20946i		−	0.373090i	0.982446	−	0.186545i	$-0.0597291\pi$
$278$		0			0
$279$	5.08975			0.304716
$280$		0			0
$281$	25.8629			1.54285			0.771426	−	0.636319i	$-0.219544\pi$
$281$	25.8629			1.54285			0.771426	+	0.636319i	$0.219544\pi$
$282$		0			0
$283$			27.0810i			1.60980i	0.593411	+	0.804900i	$0.297781\pi$
$283$			27.0810i			1.60980i	−0.593411	+	0.804900i	$0.702219\pi$
$284$		0			0
$285$	−5.04721			−0.298971
$286$		0			0
$287$	23.7181	−	14.3945i	1.40004	−	0.849678i
$288$		0			0
$289$	4.33931			0.255254
$290$		0			0
$291$	8.55385			0.501436
$292$		0			0
$293$	25.4417			1.48632			0.743159	−	0.669115i	$-0.233327\pi$
$293$	25.4417			1.48632			0.743159	+	0.669115i	$0.233327\pi$
$294$		0			0
$295$		−	2.48979i		−	0.144961i
$296$		0			0
$297$		−	0.812826i		−	0.0471649i
$298$		0			0
$299$			7.27871i			0.420939i
$300$		0			0
$301$	1.53700	+	2.53256i	0.0885913	+	0.145974i
$302$		0			0
$303$		−	1.11575i		−	0.0640981i
$304$		0			0
$305$	−5.88972			−0.337244
$306$		0			0
$307$		−	10.0338i		−	0.572660i	−0.958131	−	0.286330i	$-0.907565\pi$
$307$		−	10.0338i		−	0.572660i	0.958131	−	0.286330i	$-0.0924353\pi$
$308$		0			0
$309$			14.2574i			0.811074i
$310$		0			0
$311$	−10.7933			−0.612030			−0.306015	−	0.952027i	$-0.598996\pi$
$311$	−10.7933			−0.612030			−0.306015	+	0.952027i	$0.598996\pi$
$312$		0			0
$313$		−	30.9641i		−	1.75020i	−0.483946	−	0.875098i	$-0.660797\pi$
$313$		−	30.9641i		−	1.75020i	0.483946	−	0.875098i	$-0.339203\pi$
$314$		0			0
$315$	1.53157	+	2.52360i	0.0862940	+	0.142189i
$316$		0			0
$317$		−	10.5805i		−	0.594259i	−0.954837	−	0.297129i	$-0.903971\pi$
$317$		−	10.5805i		−	0.594259i	0.954837	−	0.297129i	$-0.0960292\pi$
$318$		0			0
$319$			7.27871i			0.407529i
$320$		0			0
$321$			13.2317i			0.738524i
$322$		0			0
$323$	16.0958			0.895596
$324$		0			0
$325$	7.51021			0.416591
$326$		0			0
$327$	−1.43748			−0.0794930
$328$		0			0
$329$	11.7417	+	19.3471i	0.647341	+	1.06664i
$330$		0			0
$331$	5.80849			0.319264			0.159632	−	0.987177i	$-0.448969\pi$
$331$	5.80849			0.319264			0.159632	+	0.987177i	$0.448969\pi$
$332$		0			0
$333$		−	0.718741i		−	0.0393868i
$334$		0			0
$335$	16.9195			0.924413
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$		−	8.23150i		−	0.447074i
$340$		0			0
$341$	4.13708			0.224035
$342$		0			0
$343$	18.4804	−	1.21459i	0.997847	−	0.0655819i
$344$		0			0
$345$	−4.06061			−0.218616
$346$		0			0
$347$	−24.0250			−1.28973			−0.644865	−	0.764296i	$-0.723087\pi$
$347$	−24.0250			−1.28973			−0.644865	+	0.764296i	$0.723087\pi$
$348$		0			0
$349$	−33.2787			−1.78137			−0.890684	−	0.454623i	$-0.849774\pi$
$349$	−33.2787			−1.78137			−0.890684	+	0.454623i	$0.849774\pi$
$350$		0			0
$351$		−	2.00000i		−	0.106752i
$352$		0			0
$353$		−	24.7191i		−	1.31567i	−0.753164	−	0.657833i	$-0.771473\pi$
$353$		−	24.7191i		−	1.31567i	0.753164	−	0.657833i	$-0.228527\pi$
$354$		0			0
$355$		−	6.55040i		−	0.347659i
$356$		0			0
$357$	−4.88425	−	8.04791i	−0.258502	−	0.425941i
$358$		0			0
$359$			29.8709i			1.57652i	0.615340	+	0.788262i	$0.289019\pi$
$359$			29.8709i			1.57652i	−0.615340	+	0.788262i	$0.710981\pi$
$360$		0			0
$361$	−1.46300			−0.0769999
$362$		0			0
$363$			10.3393i			0.542673i
$364$		0			0
$365$		−	4.31245i		−	0.225724i
$366$		0			0
$367$	−0.505942			−0.0264100			−0.0132050	−	0.999913i	$-0.504203\pi$
$367$	−0.505942			−0.0264100			−0.0132050	+	0.999913i	$0.504203\pi$
$368$		0			0
$369$		−	10.4864i		−	0.545900i
$370$		0			0
$371$	−11.5120	+	6.98660i	−0.597674	+	0.362726i
$372$		0			0
$373$			21.1609i			1.09567i	0.836586	+	0.547836i	$0.184548\pi$
$373$			21.1609i			1.09567i	−0.836586	+	0.547836i	$0.815452\pi$
$374$		0			0
$375$			9.76850i			0.504443i
$376$		0			0
$377$			17.9096i			0.922394i
$378$		0			0
$379$	−21.4787			−1.10329			−0.551644	−	0.834080i	$-0.685999\pi$
$379$	−21.4787			−1.10329			−0.551644	+	0.834080i	$0.685999\pi$
$380$		0			0
$381$	−2.75510			−0.141148
$382$		0			0
$383$	−1.24931			−0.0638370			−0.0319185	−	0.999490i	$-0.510162\pi$
$383$	−1.24931			−0.0638370			−0.0319185	+	0.999490i	$0.510162\pi$
$384$		0			0
$385$	1.24490	+	2.05125i	0.0634458	+	0.104541i
$386$		0			0
$387$	1.11971			0.0569180
$388$		0			0
$389$			20.5718i			1.04303i	0.853241	+	0.521516i	$0.174633\pi$
$389$			20.5718i			1.04303i	−0.853241	+	0.521516i	$0.825367\pi$
$390$		0			0
$391$	12.9495			0.654884
$392$		0			0
$393$	−19.2787			−0.972482
$394$		0			0
$395$		−	1.90558i		−	0.0958802i
$396$		0			0
$397$	22.3259			1.12051			0.560253	−	0.828322i	$-0.310704\pi$
$397$	22.3259			1.12051			0.560253	+	0.828322i	$0.310704\pi$
$398$		0			0
$399$	6.20946	+	10.2315i	0.310862	+	0.512216i
$400$		0			0
$401$	−25.6046			−1.27863			−0.639317	−	0.768943i	$-0.720783\pi$
$401$	−25.6046			−1.27863			−0.639317	+	0.768943i	$0.720783\pi$
$402$		0			0
$403$	10.1795			0.507077
$404$		0			0
$405$	1.11575			0.0554420
$406$		0			0
$407$		−	0.584211i		−	0.0289583i
$408$		0			0
$409$		−	16.2840i		−	0.805192i	−0.915378	−	0.402596i	$-0.868108\pi$
$409$		−	16.2840i		−	0.805192i	0.915378	−	0.402596i	$-0.131892\pi$
$410$		0			0
$411$		−	6.58421i		−	0.324775i
$412$		0			0
$413$	−5.04721	+	3.06313i	−0.248357	+	0.150727i
$414$		0			0
$415$		−	8.12121i		−	0.398655i
$416$		0			0
$417$	17.5102			0.857479
$418$		0			0
$419$			18.6945i			0.913286i	0.889650	+	0.456643i	$0.150948\pi$
$419$			18.6945i			0.913286i	−0.889650	+	0.456643i	$0.849052\pi$
$420$		0			0
$421$			31.0473i			1.51315i	0.653905	+	0.756577i	$0.273130\pi$
$421$			31.0473i			1.51315i	−0.653905	+	0.756577i	$0.726870\pi$
$422$		0			0
$423$	8.55385			0.415903
$424$		0			0
$425$		−	13.3614i		−	0.648121i
$426$		0			0
$427$	7.24598	+	11.9394i	0.350657	+	0.577788i
$428$		0			0
$429$		−	1.62565i		−	0.0784872i
$430$		0			0
$431$			1.99207i			0.0959545i	0.998848	+	0.0479772i	$0.0152775\pi$
$431$			1.99207i			0.0959545i	−0.998848	+	0.0479772i	$0.984723\pi$
$432$		0			0
$433$			10.6051i			0.509649i	0.966987	+	0.254824i	$0.0820177\pi$
$433$			10.6051i			0.509649i	−0.966987	+	0.254824i	$0.917982\pi$
$434$		0			0
$435$	−9.99134			−0.479048
$436$		0			0
$437$	−16.4630			−0.787532
$438$		0			0
$439$	−22.0925			−1.05442			−0.527208	−	0.849736i	$-0.676761\pi$
$439$	−22.0925			−1.05442			−0.527208	+	0.849736i	$0.676761\pi$
$440$		0			0
$441$	3.23150	−	6.20946i	0.153881	−	0.295689i
$442$		0			0
$443$	7.55286			0.358847			0.179424	−	0.983772i	$-0.442577\pi$
$443$	7.55286			0.358847			0.179424	+	0.983772i	$0.442577\pi$
$444$		0			0
$445$		−	3.76010i		−	0.178246i
$446$		0			0
$447$	−10.5805			−0.500439
$448$		0			0
$449$	−18.3259			−0.864853			−0.432427	−	0.901669i	$-0.642343\pi$
$449$	−18.3259			−0.864853			−0.432427	+	0.901669i	$0.642343\pi$
$450$		0			0
$451$		−	8.52360i		−	0.401361i
$452$		0			0
$453$	12.5236			0.588410
$454$		0			0
$455$	3.06313	+	5.04721i	0.143602	+	0.236617i
$456$		0			0
$457$	2.00000			0.0935561			0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000			0.0935561			0.0467780	+	0.998905i	$0.485105\pi$
$458$		0			0
$459$	−3.55819			−0.166082
$460$		0			0
$461$	−29.5787			−1.37762			−0.688810	−	0.724942i	$-0.741866\pi$
$461$	−29.5787			−1.37762			−0.688810	+	0.724942i	$0.741866\pi$
$462$		0			0
$463$			16.2653i			0.755913i	0.925823	+	0.377957i	$0.123373\pi$
$463$			16.2653i			0.755913i	−0.925823	+	0.377957i	$0.876627\pi$
$464$		0			0
$465$			5.67889i			0.263352i
$466$		0			0
$467$		−	21.5102i		−	0.995374i	−0.867357	−	0.497687i	$-0.834183\pi$
$467$		−	21.5102i		−	0.995374i	0.867357	−	0.497687i	$-0.165817\pi$
$468$		0			0
$469$	−20.8157	−	34.2986i	−0.961180	−	1.58376i
$470$		0			0
$471$			0.815710i			0.0375859i
$472$		0			0
$473$	0.910128			0.0418477
$474$		0			0
$475$			16.9866i			0.779399i
$476$		0			0
$477$			5.08975i			0.233044i
$478$		0			0
$479$	−9.99134			−0.456516			−0.228258	−	0.973601i	$-0.573303\pi$
$479$	−9.99134			−0.456516			−0.228258	+	0.973601i	$0.573303\pi$
$480$		0			0
$481$		−	1.43748i		−	0.0655436i
$482$		0			0
$483$	4.99567	+	8.23150i	0.227311	+	0.374546i
$484$		0			0
$485$			9.54396i			0.433369i
$486$		0			0
$487$		−	19.0810i		−	0.864644i	−0.901719	−	0.432322i	$-0.857694\pi$
$487$		−	19.0810i		−	0.864644i	0.901719	−	0.432322i	$-0.142306\pi$
$488$		0			0
$489$			4.18284i			0.189155i
$490$		0			0
$491$	38.0696			1.71806			0.859028	−	0.511928i	$-0.171069\pi$
$491$	38.0696			1.71806			0.859028	+	0.511928i	$0.171069\pi$
$492$		0			0
$493$	31.8629			1.43503
$494$		0			0
$495$	0.906910			0.0407626
$496$		0			0
$497$	−13.2787	+	8.05880i	−0.595631	+	0.361487i
$498$		0			0
$499$	11.2992			0.505822			0.252911	−	0.967490i	$-0.418612\pi$
$499$	11.2992			0.505822			0.252911	+	0.967490i	$0.418612\pi$
$500$		0			0
$501$		−	12.4189i		−	0.554837i
$502$		0			0
$503$	22.4103			0.999224			0.499612	−	0.866249i	$-0.333476\pi$
$503$	22.4103			0.999224			0.499612	+	0.866249i	$0.333476\pi$
$504$		0			0
$505$	1.24490			0.0553972
$506$		0			0
$507$			9.00000i			0.399704i
$508$		0			0
$509$	−11.2102			−0.496882			−0.248441	−	0.968647i	$-0.579918\pi$
$509$	−11.2102			−0.496882			−0.248441	+	0.968647i	$0.579918\pi$
$510$		0			0
$511$	−8.74202	+	5.30550i	−0.386724	+	0.234702i
$512$		0			0
$513$	4.52360			0.199722
$514$		0			0
$515$	−15.9077			−0.700975
$516$		0			0
$517$	6.95279			0.305783
$518$		0			0
$519$		−	25.4417i		−	1.11677i
$520$		0			0
$521$			17.7909i			0.779435i	0.920935	+	0.389717i	$0.127427\pi$
$521$			17.7909i			0.779435i	−0.920935	+	0.389717i	$0.872573\pi$
$522$		0			0
$523$		−	26.4362i		−	1.15597i	−0.816046	−	0.577987i	$-0.803838\pi$
$523$		−	26.4362i		−	1.15597i	0.816046	−	0.577987i	$-0.196162\pi$
$524$		0			0
$525$	8.49330	−	5.15456i	0.370678	−	0.224963i
$526$		0			0
$527$		−	18.1103i		−	0.788896i
$528$		0			0
$529$	9.75510			0.424135
$530$		0			0
$531$			2.23150i			0.0968388i
$532$		0			0
$533$		−	20.9728i		−	0.908432i
$534$		0			0
$535$	−14.7633			−0.638274
$536$		0			0
$537$			9.36668i			0.404202i
$538$		0			0
$539$	2.62664	−	5.04721i	0.113138	−	0.217399i
$540$		0			0
$541$		−	31.4236i		−	1.35101i	−0.737356	−	0.675504i	$-0.763926\pi$
$541$		−	31.4236i		−	1.35101i	0.737356	−	0.675504i	$-0.236074\pi$
$542$		0			0
$543$		−	2.00000i		−	0.0858282i
$544$		0			0
$545$		−	1.60387i		−	0.0687023i
$546$		0			0
$547$	−21.2906			−0.910318			−0.455159	−	0.890410i	$-0.650418\pi$
$547$	−21.2906			−0.910318			−0.455159	+	0.890410i	$0.650418\pi$
$548$		0			0
$549$	5.27871			0.225290
$550$		0			0
$551$	−40.5081			−1.72570
$552$		0			0
$553$	−3.86292	+	2.34439i	−0.164268	+	0.0996937i
$554$		0			0
$555$	0.801935			0.0340403
$556$		0			0
$557$			35.0420i			1.48478i	0.669970	+	0.742388i	$0.266307\pi$
$557$			35.0420i			1.48478i	−0.669970	+	0.742388i	$0.733693\pi$
$558$		0			0
$559$	2.23942			0.0947173
$560$		0			0
$561$	−2.89218			−0.122108
$562$		0			0
$563$			2.81571i			0.118668i	0.998238	+	0.0593340i	$0.0188977\pi$
$563$			2.81571i			0.118668i	−0.998238	+	0.0593340i	$0.981102\pi$
$564$		0			0
$565$	9.18429			0.386386
$566$		0			0
$567$	−1.37268	−	2.26180i	−0.0576471	−	0.0949868i
$568$		0			0
$569$	10.7213			0.449460			0.224730	−	0.974421i	$-0.427850\pi$
$569$	10.7213			0.449460			0.224730	+	0.974421i	$0.427850\pi$
$570$		0			0
$571$	22.4688			0.940291			0.470145	−	0.882589i	$-0.344201\pi$
$571$	22.4688			0.940291			0.470145	+	0.882589i	$0.344201\pi$
$572$		0			0
$573$	−11.1764			−0.466899
$574$		0			0
$575$			13.6661i			0.569918i
$576$		0			0
$577$			25.2142i			1.04968i	0.851201	+	0.524840i	$0.175875\pi$
$577$			25.2142i			1.04968i	−0.851201	+	0.524840i	$0.824125\pi$
$578$		0			0
$579$		−	13.2181i		−	0.549326i
$580$		0			0
$581$	−16.4630	+	9.99134i	−0.683000	+	0.414511i
$582$		0			0
$583$			4.13708i			0.171340i
$584$		0			0
$585$	2.23150			0.0922611
$586$		0			0
$587$		−	9.51021i		−	0.392528i	−0.980551	−	0.196264i	$-0.937119\pi$
$587$		−	9.51021i		−	0.392528i	0.980551	−	0.196264i	$-0.0628810\pi$
$588$		0			0
$589$			23.0240i			0.948689i
$590$		0			0
$591$	−20.5718			−0.846211
$592$		0			0
$593$		−	28.3960i		−	1.16609i	−0.812442	−	0.583043i	$-0.801862\pi$
$593$		−	28.3960i		−	1.16609i	0.812442	−	0.583043i	$-0.198138\pi$
$594$		0			0
$595$	8.97945	−	5.44960i	0.368122	−	0.223412i
$596$		0			0
$597$		−	13.7268i		−	0.561801i
$598$		0			0
$599$		−	39.9653i		−	1.63294i	−0.577390	−	0.816468i	$-0.695929\pi$
$599$		−	39.9653i		−	1.63294i	0.577390	−	0.816468i	$-0.304071\pi$
$600$		0			0
$601$			46.6343i			1.90225i	0.308799	+	0.951127i	$0.400073\pi$
$601$			46.6343i			1.90225i	−0.308799	+	0.951127i	$0.599927\pi$
$602$		0			0
$603$	−15.1643			−0.617537
$604$		0			0
$605$	−11.5361			−0.469009
$606$		0			0
$607$	−30.4582			−1.23626			−0.618130	−	0.786076i	$-0.712109\pi$
$607$	−30.4582			−1.23626			−0.618130	+	0.786076i	$0.712109\pi$
$608$		0			0
$609$	12.2921	+	20.2540i	0.498101	+	0.820735i
$610$		0			0
$611$	17.1077			0.692104
$612$		0			0
$613$		−	14.8683i		−	0.600525i	−0.953857	−	0.300262i	$-0.902926\pi$
$613$		−	14.8683i		−	0.600525i	0.953857	−	0.300262i	$-0.0970742\pi$
$614$		0			0
$615$	11.7002			0.471797
$616$		0			0
$617$	18.5842			0.748172			0.374086	−	0.927394i	$-0.377956\pi$
$617$	18.5842			0.748172			0.374086	+	0.927394i	$0.377956\pi$
$618$		0			0
$619$		−	35.6046i		−	1.43107i	−0.698577	−	0.715535i	$-0.746183\pi$
$619$		−	35.6046i		−	1.43107i	0.698577	−	0.715535i	$-0.253817\pi$
$620$		0			0
$621$	3.63935			0.146042
$622$		0			0
$623$	−7.62231	+	4.62596i	−0.305382	+	0.185335i
$624$		0			0
$625$	7.87632			0.315053
$626$		0			0
$627$	3.67690			0.146841
$628$		0			0
$629$	−2.55742			−0.101971
$630$		0			0
$631$		−	24.2653i		−	0.965987i	−0.875624	−	0.482993i	$-0.839550\pi$
$631$		−	24.2653i		−	0.965987i	0.875624	−	0.482993i	$-0.160450\pi$
$632$		0			0
$633$		−	9.86173i		−	0.391969i
$634$		0			0
$635$		−	3.07400i		−	0.121988i
$636$		0			0
$637$	6.46300	−	12.4189i	0.256073	−	0.492056i
$638$		0			0
$639$			5.87085i			0.232247i
$640$		0			0
$641$	−42.6518			−1.68465			−0.842323	−	0.538973i	$-0.818812\pi$
$641$	−42.6518			−1.68465			−0.842323	+	0.538973i	$0.818812\pi$
$642$		0			0
$643$			8.06061i			0.317879i	0.987288	+	0.158940i	$0.0508075\pi$
$643$			8.06061i			0.317879i	−0.987288	+	0.158940i	$0.949192\pi$
$644$		0			0
$645$			1.24931i			0.0491917i
$646$		0			0
$647$	−21.1609			−0.831923			−0.415961	−	0.909382i	$-0.636555\pi$
$647$	−21.1609			−0.831923			−0.415961	+	0.909382i	$0.636555\pi$
$648$		0			0
$649$			1.81382i			0.0711986i
$650$		0			0
$651$	11.5120	−	6.98660i	0.451191	−	0.273827i
$652$		0			0
$653$		−	4.71342i		−	0.184450i	−0.995738	−	0.0922251i	$-0.970602\pi$
$653$		−	4.71342i		−	0.184450i	0.995738	−	0.0922251i	$-0.0293979\pi$
$654$		0			0
$655$		−	21.5102i		−	0.840473i
$656$		0			0
$657$			3.86507i			0.150791i
$658$		0			0
$659$	44.1959			1.72163			0.860813	−	0.508921i	$-0.169955\pi$
$659$	44.1959			1.72163			0.860813	+	0.508921i	$0.169955\pi$
$660$		0			0
$661$	−43.3731			−1.68702			−0.843510	−	0.537114i	$-0.819514\pi$
$661$	−43.3731			−1.68702			−0.843510	+	0.537114i	$0.819514\pi$
$662$		0			0
$663$	−7.11637			−0.276377
$664$		0			0
$665$	−11.4158	+	6.92820i	−0.442685	+	0.268664i
$666$		0			0
$667$	−32.5898			−1.26188
$668$		0			0
$669$		−	25.3438i		−	0.979847i
$670$		0			0
$671$	4.29067			0.165639
$672$		0			0
$673$	−11.2449			−0.433459			−0.216729	−	0.976232i	$-0.569539\pi$
$673$	−11.2449			−0.433459			−0.216729	+	0.976232i	$0.569539\pi$
$674$		0			0
$675$		−	3.75510i		−	0.144534i
$676$		0			0
$677$	16.5157			0.634749			0.317374	−	0.948300i	$-0.397199\pi$
$677$	16.5157			0.634749			0.317374	+	0.948300i	$0.397199\pi$
$678$		0			0
$679$	19.3471	−	11.7417i	0.742475	−	0.450605i
$680$		0			0
$681$	−14.5574			−0.557841
$682$		0			0
$683$	−23.0349			−0.881407			−0.440703	−	0.897653i	$-0.645271\pi$
$683$	−23.0349			−0.881407			−0.440703	+	0.897653i	$0.645271\pi$
$684$		0			0
$685$	7.34633			0.280689
$686$		0			0
$687$		−	14.5842i		−	0.556422i
$688$		0			0
$689$			10.1795i			0.387808i
$690$		0			0
$691$			11.8788i			0.451890i	0.974140	+	0.225945i	$0.0725470\pi$
$691$			11.8788i			0.451890i	−0.974140	+	0.225945i	$0.927453\pi$
$692$		0			0
$693$	−1.11575	−	1.83845i	−0.0423838	−	0.0698370i
$694$		0			0
$695$			19.5370i			0.741081i
$696$		0			0
$697$	−37.3125			−1.41331
$698$		0			0
$699$		−	27.8361i		−	1.05286i
$700$		0			0
$701$		−	28.9157i		−	1.09213i	−0.837742	−	0.546066i	$-0.816125\pi$
$701$		−	28.9157i		−	1.09213i	0.837742	−	0.546066i	$-0.183875\pi$
$702$		0			0
$703$	3.25130			0.122625
$704$		0			0
$705$			9.54396i			0.359446i
$706$		0			0
$707$	−1.53157	−	2.52360i	−0.0576005	−	0.0949099i
$708$		0			0
$709$		−	38.6943i		−	1.45319i	−0.687064	−	0.726597i	$-0.741101\pi$
$709$		−	38.6943i		−	1.45319i	0.687064	−	0.726597i	$-0.258899\pi$
$710$		0			0
$711$			1.70789i			0.0640510i
$712$		0			0
$713$			18.5234i			0.693707i
$714$		0			0
$715$	1.81382			0.0678330
$716$		0			0
$717$	20.1024			0.750736
$718$		0			0
$719$	−31.9542			−1.19169			−0.595846	−	0.803099i	$-0.703183\pi$
$719$	−31.9542			−1.19169			−0.595846	+	0.803099i	$0.703183\pi$
$720$		0			0
$721$	19.5708	+	32.2474i	0.728855	+	1.20095i
$722$		0			0
$723$	−16.2840			−0.605608
$724$		0			0
$725$			33.6263i			1.24885i
$726$		0			0
$727$	−7.32917			−0.271824			−0.135912	−	0.990721i	$-0.543396\pi$
$727$	−7.32917			−0.271824			−0.135912	+	0.990721i	$0.543396\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	3.98413i		−	0.147358i
$732$		0			0
$733$	−33.6046			−1.24122			−0.620608	−	0.784121i	$-0.713114\pi$
$733$	−33.6046			−1.24122			−0.620608	+	0.784121i	$0.713114\pi$
$734$		0			0
$735$	6.92820	+	3.60554i	0.255551	+	0.132992i
$736$		0			0
$737$	−12.3259			−0.454031
$738$		0			0
$739$	−44.2653			−1.62833			−0.814163	−	0.580636i	$-0.802804\pi$
$739$	−44.2653			−1.62833			−0.814163	+	0.580636i	$0.802804\pi$
$740$		0			0
$741$	9.04721			0.332358
$742$		0			0
$743$			25.7338i			0.944081i	0.881577	+	0.472040i	$0.156482\pi$
$743$			25.7338i			0.944081i	−0.881577	+	0.472040i	$0.843518\pi$
$744$		0			0
$745$		−	11.8052i		−	0.432507i
$746$		0			0
$747$			7.27871i			0.266314i
$748$		0			0
$749$	18.1630	+	29.9276i	0.663660	+	1.09353i
$750$		0			0
$751$		−	18.9598i		−	0.691853i	−0.938262	−	0.345927i	$-0.887565\pi$
$751$		−	18.9598i		−	0.691853i	0.938262	−	0.345927i	$-0.112435\pi$
$752$		0			0
$753$	12.5842			0.458594
$754$		0			0
$755$			13.9732i			0.508537i
$756$		0			0
$757$		−	39.0706i		−	1.42004i	−0.704179	−	0.710022i	$-0.748685\pi$
$757$		−	39.0706i		−	1.42004i	0.704179	−	0.710022i	$-0.251315\pi$
$758$		0			0
$759$	2.95816			0.107374
$760$		0			0
$761$			10.6746i			0.386952i	0.981105	+	0.193476i	$0.0619762\pi$
$761$			10.6746i			0.386952i	−0.981105	+	0.193476i	$0.938024\pi$
$762$		0			0
$763$	−3.25130	+	1.97320i	−0.117705	+	0.0714348i
$764$		0			0
$765$		−	3.97004i		−	0.143537i
$766$		0			0
$767$			4.46300i			0.161150i
$768$		0			0
$769$			12.7953i			0.461409i	0.973024	+	0.230704i	$0.0741030\pi$
$769$			12.7953i			0.461409i	−0.973024	+	0.230704i	$0.925897\pi$
$770$		0			0
$771$	−28.2079			−1.01588
$772$		0			0
$773$	−20.6528			−0.742828			−0.371414	−	0.928467i	$-0.621127\pi$
$773$	−20.6528			−0.742828			−0.371414	+	0.928467i	$0.621127\pi$
$774$		0			0
$775$	19.1125			0.686543
$776$		0			0
$777$	−0.986602	−	1.62565i	−0.0353942	−	0.0583199i
$778$		0			0
$779$	47.4363			1.69958
$780$		0			0
$781$			4.77198i			0.170755i
$782$		0			0
$783$	8.95482			0.320019
$784$		0			0
$785$	−0.910128			−0.0324839
$786$		0			0
$787$		−	4.00000i		−	0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$		−	4.00000i		−	0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$		0			0
$789$	−6.12915			−0.218203
$790$		0			0
$791$	−11.2992	−	18.6180i	−0.401754	−	0.661981i
$792$		0			0
$793$	10.5574			0.374905
$794$		0			0
$795$	−5.67889			−0.201409
$796$		0			0
$797$	−50.6101			−1.79270			−0.896351	−	0.443346i	$-0.853791\pi$
$797$	−50.6101			−1.79270			−0.896351	+	0.443346i	$0.853791\pi$
$798$		0			0
$799$		−	30.4362i		−	1.07676i
$800$		0			0
$801$			3.37002i			0.119074i
$802$		0			0
$803$			3.14163i			0.110866i
$804$		0			0
$805$	−9.18429	+	5.57391i	−0.323704	+	0.196455i
$806$		0			0
$807$		−	23.2102i		−	0.817037i
$808$		0			0
$809$	−27.8361			−0.978666			−0.489333	−	0.872097i	$-0.662760\pi$
$809$	−27.8361			−0.978666			−0.489333	+	0.872097i	$0.662760\pi$
$810$		0			0
$811$		−	23.1416i		−	0.812612i	−0.913737	−	0.406306i	$-0.866817\pi$
$811$		−	23.1416i		−	0.812612i	0.913737	−	0.406306i	$-0.133183\pi$
$812$		0			0
$813$			21.1856i			0.743011i
$814$		0			0
$815$	−4.66700			−0.163478
$816$		0			0
$817$			5.06512i			0.177206i
$818$		0			0
$819$	−2.74536	−	4.52360i	−0.0959306	−	0.158068i
$820$		0			0
$821$		−	9.56859i		−	0.333946i	−0.985961	−	0.166973i	$-0.946601\pi$
$821$		−	9.56859i		−	0.333946i	0.985961	−	0.166973i	$-0.0533993\pi$
$822$		0			0
$823$			50.3597i			1.75543i	0.479183	+	0.877715i	$0.340933\pi$
$823$			50.3597i			1.75543i	−0.479183	+	0.877715i	$0.659067\pi$
$824$		0			0
$825$		−	3.05224i		−	0.106265i
$826$		0			0
$827$	40.3308			1.40244			0.701220	−	0.712945i	$-0.252639\pi$
$827$	40.3308			1.40244			0.701220	+	0.712945i	$0.252639\pi$
$828$		0			0
$829$	17.8788			0.620956			0.310478	−	0.950581i	$-0.399511\pi$
$829$	17.8788			0.620956			0.310478	+	0.950581i	$0.399511\pi$
$830$		0			0
$831$	−6.20946			−0.215404
$832$		0			0
$833$	−22.0944	−	11.4983i	−0.765526	−	0.398392i
$834$		0			0
$835$	13.8564			0.479521
$836$		0			0
$837$		−	5.08975i		−	0.175928i
$838$		0			0
$839$	−38.0587			−1.31393			−0.656966	−	0.753920i	$-0.728161\pi$
$839$	−38.0587			−1.31393			−0.656966	+	0.753920i	$0.728161\pi$
$840$		0			0
$841$	−51.1888			−1.76513
$842$		0			0
$843$		−	25.8629i		−	0.890766i
$844$		0			0
$845$	−10.0417			−0.345447
$846$		0			0
$847$	14.1926	+	23.3855i	0.487663	+	0.803535i
$848$		0			0
$849$	27.0810			0.929418
$850$		0			0
$851$	2.61575			0.0896669
$852$		0			0
$853$	18.7213			0.641005			0.320502	−	0.947248i	$-0.396148\pi$
$853$	18.7213			0.641005			0.320502	+	0.947248i	$0.396148\pi$
$854$		0			0
$855$			5.04721i			0.172611i
$856$		0			0
$857$			5.81938i			0.198786i	0.995048	+	0.0993932i	$0.0316902\pi$
$857$			5.81938i			0.198786i	−0.995048	+	0.0993932i	$0.968310\pi$
$858$		0			0
$859$		−	38.4968i		−	1.31349i	−0.754111	−	0.656747i	$-0.771932\pi$
$859$		−	38.4968i		−	1.31349i	0.754111	−	0.656747i	$-0.228068\pi$
$860$		0			0
$861$	−14.3945	−	23.7181i	−0.490562	−	0.808312i
$862$		0			0
$863$		−	2.25036i		−	0.0766032i	−0.999266	−	0.0383016i	$-0.987805\pi$
$863$		−	2.25036i		−	0.0766032i	0.999266	−	0.0383016i	$-0.0121948\pi$
$864$		0			0
$865$	28.3865			0.965171
$866$		0			0
$867$		−	4.33931i		−	0.147371i
$868$		0			0
$869$			1.38822i			0.0470921i
$870$		0			0
$871$	−30.3286			−1.02764
$872$		0			0
$873$		−	8.55385i		−	0.289504i
$874$		0			0
$875$	13.4090	+	22.0944i	0.453308	+	0.746928i
$876$		0			0
$877$			33.5799i			1.13391i	0.823748	+	0.566956i	$0.191879\pi$
$877$			33.5799i			1.13391i	−0.823748	+	0.566956i	$0.808121\pi$
$878$		0			0
$879$		−	25.4417i		−	0.858126i
$880$		0			0
$881$			6.43315i			0.216738i	0.994111	+	0.108369i	$0.0345629\pi$
$881$			6.43315i			0.216738i	−0.994111	+	0.108369i	$0.965437\pi$
$882$		0			0
$883$	−47.1185			−1.58566			−0.792832	−	0.609440i	$-0.791394\pi$
$883$	−47.1185			−1.58566			−0.792832	+	0.609440i	$0.791394\pi$
$884$		0			0
$885$	−2.48979			−0.0836935
$886$		0			0
$887$	−18.9215			−0.635323			−0.317661	−	0.948204i	$-0.602898\pi$
$887$	−18.9215			−0.635323			−0.317661	+	0.948204i	$0.602898\pi$
$888$		0			0
$889$	−6.23150	+	3.78188i	−0.208998	+	0.126840i
$890$		0			0
$891$	−0.812826			−0.0272307
$892$		0			0
$893$			38.6943i			1.29485i
$894$		0			0
$895$	−10.4509			−0.349334
$896$		0			0
$897$	7.27871			0.243029
$898$		0			0
$899$			45.5778i			1.52011i
$900$		0			0
$901$	18.1103			0.603341
$902$		0			0
$903$	2.53256	−	1.53700i	0.0842783	−	0.0511482i
$904$		0			0
$905$	2.23150			0.0741775
$906$		0			0
$907$	28.1970			0.936265			0.468133	−	0.883658i	$-0.344927\pi$
$907$	28.1970			0.936265			0.468133	+	0.883658i	$0.344927\pi$
$908$		0			0
$909$	−1.11575			−0.0370071
$910$		0			0
$911$			30.1291i			0.998223i	0.866538	+	0.499112i	$0.166340\pi$
$911$			30.1291i			0.998223i	−0.866538	+	0.499112i	$0.833660\pi$
$912$		0			0
$913$			5.91632i			0.195802i
$914$		0			0
$915$			5.88972i			0.194708i
$916$		0			0
$917$	−43.6046	+	26.4635i	−1.43995	+	0.873902i
$918$		0			0
$919$		−	30.4968i		−	1.00600i	−0.864287	−	0.502998i	$-0.832230\pi$
$919$		−	30.4968i		−	1.00600i	0.864287	−	0.502998i	$-0.167770\pi$
$920$		0			0
$921$	−10.0338			−0.330625
$922$		0			0
$923$			11.7417i			0.386483i
$924$		0			0
$925$		−	2.69895i		−	0.0887409i
$926$		0			0
$927$	14.2574			0.468274
$928$		0			0
$929$			6.62132i			0.217238i	0.994083	+	0.108619i	$0.0346429\pi$
$929$			6.62132i			0.217238i	−0.994083	+	0.108619i	$0.965357\pi$
$930$		0			0
$931$	28.0891	+	14.6180i	0.920585	+	0.479086i
$932$		0			0
$933$			10.7933i			0.353356i
$934$		0			0
$935$		−	3.22695i		−	0.105533i
$936$		0			0
$937$		−	18.5452i		−	0.605845i	−0.953015	−	0.302922i	$-0.902038\pi$
$937$		−	18.5452i		−	0.605845i	0.953015	−	0.302922i	$-0.0979623\pi$
$938$		0			0
$939$	−30.9641			−1.01048
$940$		0			0
$941$	32.6528			1.06445			0.532225	−	0.846603i	$-0.321356\pi$
$941$	32.6528			1.06445			0.532225	+	0.846603i	$0.321356\pi$
$942$		0			0
$943$	38.1637			1.24278
$944$		0			0
$945$	2.52360	−	1.53157i	0.0820929	−	0.0498219i
$946$		0			0
$947$	10.9923			0.357203			0.178601	−	0.983922i	$-0.442843\pi$
$947$	10.9923			0.357203			0.178601	+	0.983922i	$0.442843\pi$
$948$		0			0
$949$			7.73014i			0.250931i
$950$		0			0
$951$	−10.5805			−0.343095
$952$		0			0
$953$	43.9732			1.42443			0.712216	−	0.701960i	$-0.247692\pi$
$953$	43.9732			1.42443			0.712216	+	0.701960i	$0.247692\pi$
$954$		0			0
$955$		−	12.4700i		−	0.403520i
$956$		0			0
$957$	7.27871			0.235287
$958$		0			0
$959$	−9.03802	−	14.8922i	−0.291853	−	0.480894i
$960$		0			0
$961$	−5.09442			−0.164336
$962$		0			0
$963$	13.2317			0.426387
$964$		0			0
$965$	14.7481			0.474758
$966$		0			0
$967$		−	27.6811i		−	0.890164i	−0.895490	−	0.445082i	$-0.853175\pi$
$967$		−	27.6811i		−	0.890164i	0.895490	−	0.445082i	$-0.146825\pi$
$968$		0			0
$969$		−	16.0958i		−	0.517072i
$970$		0			0
$971$		−	1.16842i		−	0.0374965i	−0.999824	−	0.0187482i	$-0.994032\pi$
$971$		−	1.16842i		−	0.0374965i	0.999824	−	0.0187482i	$-0.00596810\pi$
$972$		0			0
$973$	39.6046	−	24.0359i	1.26967	−	0.770556i
$974$		0			0
$975$		−	7.51021i		−	0.240519i
$976$		0			0
$977$	−36.6945			−1.17396			−0.586980	−	0.809601i	$-0.699683\pi$
$977$	−36.6945			−1.17396			−0.586980	+	0.809601i	$0.699683\pi$
$978$		0			0
$979$			2.73924i			0.0875464i
$980$		0			0
$981$			1.43748i			0.0458953i
$982$		0			0
$983$	59.2197			1.88881			0.944407	−	0.328779i	$-0.106637\pi$
$983$	59.2197			1.88881			0.944407	+	0.328779i	$0.106637\pi$
$984$		0			0
$985$		−	22.9530i		−	0.731343i
$986$		0			0
$987$	19.3471	−	11.7417i	0.615826	−	0.373743i
$988$		0			0
$989$			4.07502i			0.129578i
$990$		0			0
$991$		−	18.1550i		−	0.576713i	−0.957523	−	0.288357i	$-0.906891\pi$
$991$		−	18.1550i		−	0.576713i	0.957523	−	0.288357i	$-0.0931089\pi$
$992$		0			0
$993$		−	5.80849i		−	0.184327i
$994$		0			0
$995$	15.3157			0.485539
$996$		0			0
$997$	−23.6473			−0.748917			−0.374459	−	0.927244i	$-0.622171\pi$
$997$	−23.6473			−0.748917			−0.374459	+	0.927244i	$0.622171\pi$
$998$		0			0
$999$	−0.718741			−0.0227400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.p.c.223.3	✓	12
3.2	odd	2		4032.2.p.j.1567.5		12
4.3	odd	2	inner	1344.2.p.c.223.10	yes	12
7.6	odd	2		1344.2.p.d.223.10	yes	12
8.3	odd	2		1344.2.p.d.223.4	yes	12
8.5	even	2		1344.2.p.d.223.9	yes	12
12.11	even	2		4032.2.p.j.1567.8		12
21.20	even	2		4032.2.p.k.1567.7		12
24.5	odd	2		4032.2.p.k.1567.5		12
24.11	even	2		4032.2.p.k.1567.8		12
28.27	even	2		1344.2.p.d.223.3	yes	12
56.13	odd	2	inner	1344.2.p.c.223.4	yes	12
56.27	even	2	inner	1344.2.p.c.223.9	yes	12
84.83	odd	2		4032.2.p.k.1567.6		12
168.83	odd	2		4032.2.p.j.1567.6		12
168.125	even	2		4032.2.p.j.1567.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.p.c.223.3	✓	12	1.1	even	1	trivial
1344.2.p.c.223.4	yes	12	56.13	odd	2	inner
1344.2.p.c.223.9	yes	12	56.27	even	2	inner
1344.2.p.c.223.10	yes	12	4.3	odd	2	inner
1344.2.p.d.223.3	yes	12	28.27	even	2
1344.2.p.d.223.4	yes	12	8.3	odd	2
1344.2.p.d.223.9	yes	12	8.5	even	2
1344.2.p.d.223.10	yes	12	7.6	odd	2
4032.2.p.j.1567.5		12	3.2	odd	2
4032.2.p.j.1567.6		12	168.83	odd	2
4032.2.p.j.1567.7		12	168.125	even	2
4032.2.p.j.1567.8		12	12.11	even	2
4032.2.p.k.1567.5		12	24.5	odd	2
4032.2.p.k.1567.6		12	84.83	odd	2
4032.2.p.k.1567.7		12	21.20	even	2
4032.2.p.k.1567.8		12	24.11	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.11575 q^{5} +(-1.37268 - 2.26180i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.11575 q^{5} +(-1.37268 - 2.26180i) q^{7} -1.00000 q^{9} -0.812826 q^{11} -2.00000 q^{13} -1.11575i q^{15} +3.55819i q^{17} -4.52360i q^{19} +(-2.26180 + 1.37268i) q^{21} -3.63935i q^{23} -3.75510 q^{25} +1.00000i q^{27} -8.95482i q^{29} -5.08975 q^{31} +0.812826i q^{33} +(-1.53157 - 2.52360i) q^{35} +0.718741i q^{37} +2.00000i q^{39} +10.4864i q^{41} -1.11971 q^{43} -1.11575 q^{45} -8.55385 q^{47} +(-3.23150 + 6.20946i) q^{49} +3.55819 q^{51} -5.08975i q^{53} -0.906910 q^{55} -4.52360 q^{57} -2.23150i q^{59} -5.27871 q^{61} +(1.37268 + 2.26180i) q^{63} -2.23150 q^{65} +15.1643 q^{67} -3.63935 q^{69} -5.87085i q^{71} -3.86507i q^{73} +3.75510i q^{75} +(1.11575 + 1.83845i) q^{77} -1.70789i q^{79} +1.00000 q^{81} -7.27871i q^{83} +3.97004i q^{85} -8.95482 q^{87} -3.37002i q^{89} +(2.74536 + 4.52360i) q^{91} +5.08975i q^{93} -5.04721i q^{95} +8.55385i q^{97} +0.812826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 24 q^{13} + 36 q^{25} - 12 q^{49} + 72 q^{61} + 24 q^{69} + 12 q^{81}+O(q^{100}) \) 12 * q - 12 * q^9 - 24 * q^13 + 36 * q^25 - 12 * q^49 + 72 * q^61 + 24 * q^69 + 12 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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