LMFDB - Embedded newform 572.2.f.c.441.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (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:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 $ x^8 + 21x^6 + 136x^4 + 309*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.5
Root			$1.25126i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.c.441.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25126 q^{3} -2.18309i q^{5} +0.251260i q^{7} -1.43435 q^{9} +O(q^{10})$	$q+1.25126 q^{3} -2.18309i q^{5} +0.251260i q^{7} -1.43435 q^{9} -1.00000i q^{11} +(3.07148 - 1.88839i) q^{13} -2.73161i q^{15} +5.77678 q^{17} -7.21113i q^{19} +0.314392i q^{21} +1.93183 q^{23} +0.234131 q^{25} -5.54852 q^{27} -5.64044 q^{29} +3.81691i q^{31} -1.25126i q^{33} +0.548523 q^{35} -6.68561i q^{37} +(3.84322 - 2.36287i) q^{39} +10.9427i q^{41} -7.64044 q^{43} +3.13131i q^{45} -6.50252i q^{47} +6.93687 q^{49} +7.22826 q^{51} +9.98287 q^{53} -2.18309 q^{55} -9.02300i q^{57} +13.6003i q^{59} -9.64044 q^{61} -0.360395i q^{63} +(-4.12252 - 6.70530i) q^{65} +11.7367i q^{67} +2.41722 q^{69} +8.73161i q^{71} -11.9369i q^{73} +0.292959 q^{75} +0.251260 q^{77} +14.6915 q^{79} -2.63961 q^{81} +6.39422i q^{83} -12.6112i q^{85} -7.05765 q^{87} +6.82195i q^{89} +(0.474478 + 0.771741i) q^{91} +4.77595i q^{93} -15.7425 q^{95} +10.1831i q^{97} +1.43435i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 + 18 * q^9	$ 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) $ 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.25126			0.722416			0.361208	−	0.932485i	$-0.382365\pi$
$3$	1.25126			0.722416			0.361208	+	0.932485i	$0.382365\pi$
$4$		0			0
$5$		−	2.18309i		−	0.976306i	−0.872758	−	0.488153i	$-0.837671\pi$
$5$		−	2.18309i		−	0.976306i	0.872758	−	0.488153i	$-0.162329\pi$
$6$		0			0
$7$			0.251260i			0.0949675i	0.998872	+	0.0474838i	$0.0151202\pi$
$7$			0.251260i			0.0949675i	−0.998872	+	0.0474838i	$0.984880\pi$
$8$		0			0
$9$	−1.43435			−0.478116
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	3.07148	−	1.88839i	0.851875	−	0.523745i
$13$	3.07148	−	1.88839i	0.851875	−	0.523745i
$14$		0			0
$15$		−	2.73161i		−	0.705299i
$16$		0			0
$17$	5.77678			1.40108			0.700538	−	0.713615i	$-0.252944\pi$
$17$	5.77678			1.40108			0.700538	+	0.713615i	$0.252944\pi$
$18$		0			0
$19$		−	7.21113i		−	1.65435i	−0.561947	−	0.827173i	$-0.689947\pi$
$19$		−	7.21113i		−	1.65435i	0.561947	−	0.827173i	$-0.310053\pi$
$20$		0			0
$21$			0.314392i			0.0686060i
$22$		0			0
$23$	1.93183			0.402814			0.201407	−	0.979508i	$-0.435449\pi$
$23$	1.93183			0.402814			0.201407	+	0.979508i	$0.435449\pi$
$24$		0			0
$25$	0.234131			0.0468262
$26$		0			0
$27$	−5.54852			−1.06781
$28$		0			0
$29$	−5.64044			−1.04740			−0.523701	−	0.851902i	$-0.675449\pi$
$29$	−5.64044			−1.04740			−0.523701	+	0.851902i	$0.675449\pi$
$30$		0			0
$31$			3.81691i			0.685538i	0.939420	+	0.342769i	$0.111365\pi$
$31$			3.81691i			0.685538i	−0.939420	+	0.342769i	$0.888635\pi$
$32$		0			0
$33$		−	1.25126i		−	0.217816i
$34$		0			0
$35$	0.548523			0.0927174
$36$		0			0
$37$		−	6.68561i		−	1.09911i	−0.835458	−	0.549554i	$-0.814798\pi$
$37$		−	6.68561i		−	1.09911i	0.835458	−	0.549554i	$-0.185202\pi$
$38$		0			0
$39$	3.84322	−	2.36287i	0.615408	−	0.378362i
$40$		0			0
$41$			10.9427i			1.70897i	0.519477	+	0.854484i	$0.326127\pi$
$41$			10.9427i			1.70897i	−0.519477	+	0.854484i	$0.673873\pi$
$42$		0			0
$43$	−7.64044			−1.16516			−0.582578	−	0.812775i	$-0.697956\pi$
$43$	−7.64044			−1.16516			−0.582578	+	0.812775i	$0.697956\pi$
$44$		0			0
$45$			3.13131i			0.466787i
$46$		0			0
$47$		−	6.50252i		−	0.948490i	−0.880393	−	0.474245i	$-0.842721\pi$
$47$		−	6.50252i		−	0.948490i	0.880393	−	0.474245i	$-0.157279\pi$
$48$		0			0
$49$	6.93687			0.990981
$50$		0			0
$51$	7.22826			1.01216
$52$		0			0
$53$	9.98287			1.37125			0.685626	−	0.727954i	$-0.259528\pi$
$53$	9.98287			1.37125			0.685626	+	0.727954i	$0.259528\pi$
$54$		0			0
$55$	−2.18309			−0.294367
$56$		0			0
$57$		−	9.02300i		−	1.19513i
$58$		0			0
$59$			13.6003i			1.77061i	0.465012	+	0.885304i	$0.346050\pi$
$59$			13.6003i			1.77061i	−0.465012	+	0.885304i	$0.653950\pi$
$60$		0			0
$61$	−9.64044			−1.23433			−0.617166	−	0.786833i	$-0.711719\pi$
$61$	−9.64044			−1.23433			−0.617166	+	0.786833i	$0.711719\pi$
$62$		0			0
$63$		−	0.360395i		−	0.0454055i
$64$		0			0
$65$	−4.12252	−	6.70530i	−0.511336	−	0.831691i
$66$		0			0
$67$			11.7367i			1.43386i	0.697145	+	0.716930i	$0.254453\pi$
$67$			11.7367i			1.43386i	−0.697145	+	0.716930i	$0.745547\pi$
$68$		0			0
$69$	2.41722			0.290999
$70$		0			0
$71$			8.73161i			1.03625i	0.855304	+	0.518126i	$0.173370\pi$
$71$			8.73161i			1.03625i	−0.855304	+	0.518126i	$0.826630\pi$
$72$		0			0
$73$		−	11.9369i		−	1.39710i	−0.715559	−	0.698552i	$-0.753828\pi$
$73$		−	11.9369i		−	1.39710i	0.715559	−	0.698552i	$-0.246172\pi$
$74$		0			0
$75$	0.292959			0.0338280
$76$		0			0
$77$	0.251260			0.0286338
$78$		0			0
$79$	14.6915			1.65292			0.826460	−	0.562995i	$-0.190351\pi$
$79$	14.6915			1.65292			0.826460	+	0.562995i	$0.190351\pi$
$80$		0			0
$81$	−2.63961			−0.293289
$82$		0			0
$83$			6.39422i			0.701856i	0.936402	+	0.350928i	$0.114134\pi$
$83$			6.39422i			0.701856i	−0.936402	+	0.350928i	$0.885866\pi$
$84$		0			0
$85$		−	12.6112i		−	1.36788i
$86$		0			0
$87$	−7.05765			−0.756660
$88$		0			0
$89$			6.82195i			0.723126i	0.932348	+	0.361563i	$0.117757\pi$
$89$			6.82195i			0.723126i	−0.932348	+	0.361563i	$0.882243\pi$
$90$		0			0
$91$	0.474478	+	0.771741i	0.0497388	+	0.0809004i
$92$		0			0
$93$			4.77595i			0.495243i
$94$		0			0
$95$	−15.7425			−1.61515
$96$		0			0
$97$			10.1831i			1.03394i	0.856005	+	0.516968i	$0.172939\pi$
$97$			10.1831i			1.03394i	−0.856005	+	0.516968i	$0.827061\pi$
$98$		0			0
$99$			1.43435i			0.144157i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	−12.7195			−1.25329			−0.626646	−	0.779304i	$-0.715573\pi$
$103$	−12.7195			−1.25329			−0.626646	+	0.779304i	$0.715573\pi$
$104$		0			0
$105$	0.686346			0.0669805
$106$		0			0
$107$	−2.72574			−0.263507			−0.131754	−	0.991283i	$-0.542061\pi$
$107$	−2.72574			−0.263507			−0.131754	+	0.991283i	$0.542061\pi$
$108$		0			0
$109$		−	4.70861i		−	0.451003i	−0.974243	−	0.225501i	$-0.927598\pi$
$109$		−	4.70861i		−	0.451003i	0.974243	−	0.225501i	$-0.0724020\pi$
$110$		0			0
$111$		−	8.36544i		−	0.794012i
$112$		0			0
$113$	16.1200			1.51644			0.758219	−	0.652000i	$-0.226070\pi$
$113$	16.1200			1.51644			0.758219	+	0.652000i	$0.226070\pi$
$114$		0			0
$115$		−	4.21735i		−	0.393270i
$116$		0			0
$117$	−4.40557	+	2.70861i	−0.407295	+	0.250411i
$118$		0			0
$119$			1.45148i			0.133057i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			13.6922i			1.23459i
$124$		0			0
$125$		−	11.4266i		−	1.02202i
$126$		0			0
$127$	12.2859			1.09020			0.545099	−	0.838372i	$-0.316492\pi$
$127$	12.2859			1.09020			0.545099	+	0.838372i	$0.316492\pi$
$128$		0			0
$129$	−9.56017			−0.841726
$130$		0			0
$131$	−17.3712			−1.51773			−0.758865	−	0.651248i	$-0.774246\pi$
$131$	−17.3712			−1.51773			−0.758865	+	0.651248i	$0.774246\pi$
$132$		0			0
$133$	1.81187			0.157109
$134$		0			0
$135$			12.1129i			1.04251i
$136$		0			0
$137$			15.1999i			1.29861i	0.760527	+	0.649306i	$0.224941\pi$
$137$			15.1999i			1.29861i	−0.760527	+	0.649306i	$0.775059\pi$
$138$		0			0
$139$	−7.45148			−0.632026			−0.316013	−	0.948755i	$-0.602344\pi$
$139$	−7.45148			−0.632026			−0.316013	+	0.948755i	$0.602344\pi$
$140$		0			0
$141$		−	8.13635i		−	0.685204i
$142$		0			0
$143$	−1.88839	−	3.07148i	−0.157915	−	0.256850i
$144$		0			0
$145$			12.3136i			1.02259i
$146$		0			0
$147$	8.67983			0.715900
$148$		0			0
$149$		−	5.61239i		−	0.459785i	−0.973216	−	0.229893i	$-0.926163\pi$
$149$		−	5.61239i		−	0.459785i	0.973216	−	0.229893i	$-0.0738375\pi$
$150$		0			0
$151$			15.6062i			1.27001i	0.772507	+	0.635006i	$0.219003\pi$
$151$			15.6062i			1.27001i	−0.772507	+	0.635006i	$0.780997\pi$
$152$		0			0
$153$	−8.28591			−0.669876
$154$		0			0
$155$	8.33265			0.669295
$156$		0			0
$157$	2.48035			0.197953			0.0989767	−	0.995090i	$-0.468443\pi$
$157$	2.48035			0.197953			0.0989767	+	0.995090i	$0.468443\pi$
$158$		0			0
$159$	12.4912			0.990614
$160$		0			0
$161$			0.485392i			0.0382542i
$162$		0			0
$163$			9.55356i			0.748293i	0.927370	+	0.374146i	$0.122064\pi$
$163$			9.55356i			0.748293i	−0.927370	+	0.374146i	$0.877936\pi$
$164$		0			0
$165$	−2.73161			−0.212656
$166$		0			0
$167$			6.89757i			0.533750i	0.963731	+	0.266875i	$0.0859911\pi$
$167$			6.89757i			0.533750i	−0.963731	+	0.266875i	$0.914009\pi$
$168$		0			0
$169$	5.86796	−	11.6003i	0.451381	−	0.892331i
$170$		0			0
$171$			10.3433i			0.790969i
$172$		0			0
$173$	−2.78182			−0.211498			−0.105749	−	0.994393i	$-0.533724\pi$
$173$	−2.78182			−0.211498			−0.105749	+	0.994393i	$0.533724\pi$
$174$		0			0
$175$			0.0588279i			0.00444697i
$176$		0			0
$177$			17.0175i			1.27912i
$178$		0			0
$179$	−11.1539			−0.833679			−0.416840	−	0.908980i	$-0.636862\pi$
$179$	−11.1539			−0.833679			−0.416840	+	0.908980i	$0.636862\pi$
$180$		0			0
$181$	15.3533			1.14121			0.570603	−	0.821226i	$-0.306710\pi$
$181$	15.3533			1.14121			0.570603	+	0.821226i	$0.306710\pi$
$182$		0			0
$183$	−12.0627			−0.891700
$184$		0			0
$185$	−14.5953			−1.07307
$186$		0			0
$187$		−	5.77678i		−	0.422440i
$188$		0			0
$189$		−	1.39412i		−	0.101408i
$190$		0			0
$191$	−14.3023			−1.03488			−0.517439	−	0.855720i	$-0.673115\pi$
$191$	−14.3023			−1.03488			−0.517439	+	0.855720i	$0.673115\pi$
$192$		0			0
$193$		−	13.0296i		−	0.937892i	−0.883227	−	0.468946i	$-0.844634\pi$
$193$		−	13.0296i		−	0.937892i	0.883227	−	0.468946i	$-0.155366\pi$
$194$		0			0
$195$	−5.15835	−	8.39008i	−0.369397	−	0.600826i
$196$		0			0
$197$			2.88582i			0.205606i	0.994702	+	0.102803i	$0.0327812\pi$
$197$			2.88582i			0.205606i	−0.994702	+	0.102803i	$0.967219\pi$
$198$		0			0
$199$	16.0390			1.13697			0.568486	−	0.822693i	$-0.307529\pi$
$199$	16.0390			1.13697			0.568486	+	0.822693i	$0.307529\pi$
$200$		0			0
$201$			14.6856i			1.03584i
$202$		0			0
$203$		−	1.41722i		−	0.0994692i
$204$		0			0
$205$	23.8890			1.66848
$206$		0			0
$207$	−2.77091			−0.192592
$208$		0			0
$209$	−7.21113			−0.498804
$210$		0			0
$211$	−5.18739			−0.357115			−0.178557	−	0.983930i	$-0.557143\pi$
$211$	−5.18739			−0.357115			−0.178557	+	0.983930i	$0.557143\pi$
$212$		0			0
$213$			10.9255i			0.748604i
$214$		0			0
$215$			16.6797i			1.13755i
$216$		0			0
$217$	−0.959039			−0.0651038
$218$		0			0
$219$		−	14.9361i		−	1.00929i
$220$		0			0
$221$	17.7433	−	10.9088i	1.19354	−	0.733807i
$222$		0			0
$223$			11.9190i			0.798155i	0.916917	+	0.399078i	$0.130670\pi$
$223$			11.9190i			0.798155i	−0.916917	+	0.399078i	$0.869330\pi$
$224$		0			0
$225$	−0.335825			−0.0223884
$226$		0			0
$227$			13.1550i			0.873131i	0.899672	+	0.436565i	$0.143805\pi$
$227$			13.1550i			0.873131i	−0.899672	+	0.436565i	$0.856195\pi$
$228$		0			0
$229$		−	27.3837i		−	1.80957i	−0.425873	−	0.904783i	$-0.640033\pi$
$229$		−	27.3837i		−	1.80957i	0.425873	−	0.904783i	$-0.359967\pi$
$230$		0			0
$231$	0.314392			0.0206855
$232$		0			0
$233$	−4.05609			−0.265723			−0.132862	−	0.991135i	$-0.542417\pi$
$233$	−4.05609			−0.265723			−0.132862	+	0.991135i	$0.542417\pi$
$234$		0			0
$235$	−14.1956			−0.926017
$236$		0			0
$237$	18.3829			1.19410
$238$		0			0
$239$		−	26.4051i		−	1.70801i	−0.520268	−	0.854003i	$-0.674168\pi$
$239$		−	26.4051i		−	1.70801i	0.520268	−	0.854003i	$-0.325832\pi$
$240$		0			0
$241$		−	19.4453i		−	1.25258i	−0.779590	−	0.626290i	$-0.784573\pi$
$241$		−	19.4453i		−	1.25258i	0.779590	−	0.626290i	$-0.215427\pi$
$242$		0			0
$243$	13.3427			0.855937
$244$		0			0
$245$		−	15.1438i		−	0.967501i
$246$		0			0
$247$	−13.6174	−	22.1488i	−0.866457	−	1.40930i
$248$		0			0
$249$			8.00083i			0.507032i
$250$		0			0
$251$	1.85157			0.116870			0.0584349	−	0.998291i	$-0.481389\pi$
$251$	1.85157			0.116870			0.0584349	+	0.998291i	$0.481389\pi$
$252$		0			0
$253$		−	1.93183i		−	0.121453i
$254$		0			0
$255$		−	15.7799i		−	0.988177i
$256$		0			0
$257$	−24.4008			−1.52208			−0.761041	−	0.648704i	$-0.775311\pi$
$257$	−24.4008			−1.52208			−0.761041	+	0.648704i	$0.775311\pi$
$258$		0			0
$259$	1.67983			0.104379
$260$		0			0
$261$	8.09034			0.500780
$262$		0			0
$263$	8.45652			0.521451			0.260726	−	0.965413i	$-0.416038\pi$
$263$	8.45652			0.521451			0.260726	+	0.965413i	$0.416038\pi$
$264$		0			0
$265$		−	21.7935i		−	1.33876i
$266$		0			0
$267$			8.53604i			0.522397i
$268$		0			0
$269$	18.9493			1.15536			0.577681	−	0.816263i	$-0.303958\pi$
$269$	18.9493			1.15536			0.577681	+	0.816263i	$0.303958\pi$
$270$		0			0
$271$		−	24.5875i		−	1.49358i	−0.665058	−	0.746792i	$-0.731593\pi$
$271$		−	24.5875i		−	1.49358i	0.665058	−	0.746792i	$-0.268407\pi$
$272$		0			0
$273$	0.593695	+	0.965649i	0.0359321	+	0.0584437i
$274$		0			0
$275$		−	0.234131i		−	0.0141186i
$276$		0			0
$277$	2.53678			0.152420			0.0762101	−	0.997092i	$-0.475718\pi$
$277$	2.53678			0.152420			0.0762101	+	0.997092i	$0.475718\pi$
$278$		0			0
$279$		−	5.47478i		−	0.327766i
$280$		0			0
$281$			32.5547i			1.94205i	0.238977	+	0.971025i	$0.423188\pi$
$281$			32.5547i			1.94205i	−0.238977	+	0.971025i	$0.576812\pi$
$282$		0			0
$283$	−6.26922			−0.372666			−0.186333	−	0.982487i	$-0.559660\pi$
$283$	−6.26922			−0.372666			−0.186333	+	0.982487i	$0.559660\pi$
$284$		0			0
$285$	−19.6980			−1.16681
$286$		0			0
$287$	−2.74948			−0.162297
$288$		0			0
$289$	16.3712			0.963013
$290$		0			0
$291$			12.7417i			0.746931i
$292$		0			0
$293$		−	6.05261i		−	0.353597i	−0.984247	−	0.176799i	$-0.943426\pi$
$293$		−	6.05261i		−	0.353597i	0.984247	−	0.176799i	$-0.0565742\pi$
$294$		0			0
$295$	29.6906			1.72866
$296$		0			0
$297$			5.54852i			0.321958i
$298$		0			0
$299$	5.93356	−	3.64804i	0.343147	−	0.210972i
$300$		0			0
$301$		−	1.91974i		−	0.110652i
$302$		0			0
$303$	−7.50756			−0.431298
$304$		0			0
$305$			21.0459i			1.20509i
$306$		0			0
$307$		−	30.8344i		−	1.75981i	−0.475146	−	0.879907i	$-0.657605\pi$
$307$		−	30.8344i		−	1.75981i	0.475146	−	0.879907i	$-0.342395\pi$
$308$		0			0
$309$	−15.9154			−0.905397
$310$		0			0
$311$	10.6692			0.604996			0.302498	−	0.953150i	$-0.402179\pi$
$311$	10.6692			0.604996			0.302498	+	0.953150i	$0.402179\pi$
$312$		0			0
$313$	28.0011			1.58272			0.791359	−	0.611352i	$-0.209374\pi$
$313$	28.0011			1.58272			0.791359	+	0.611352i	$0.209374\pi$
$314$		0			0
$315$	−0.786773			−0.0443296
$316$		0			0
$317$		−	14.2734i		−	0.801676i	−0.916149	−	0.400838i	$-0.868719\pi$
$317$		−	14.2734i		−	0.801676i	0.916149	−	0.400838i	$-0.131281\pi$
$318$		0			0
$319$			5.64044i			0.315804i
$320$		0			0
$321$	−3.41061			−0.190362
$322$		0			0
$323$		−	41.6571i		−	2.31786i
$324$		0			0
$325$	0.719129	−	0.442131i	0.0398901	−	0.0245250i
$326$		0			0
$327$		−	5.89170i		−	0.325812i
$328$		0			0
$329$	1.63383			0.0900757
$330$		0			0
$331$			2.45578i			0.134982i	0.997720	+	0.0674910i	$0.0214994\pi$
$331$			2.45578i			0.134982i	−0.997720	+	0.0674910i	$0.978501\pi$
$332$		0			0
$333$			9.58948i			0.525500i
$334$		0			0
$335$	25.6221			1.39989
$336$		0			0
$337$	10.0627			0.548150			0.274075	−	0.961708i	$-0.411628\pi$
$337$	10.0627			0.548150			0.274075	+	0.961708i	$0.411628\pi$
$338$		0			0
$339$	20.1703			1.09550
$340$		0			0
$341$	3.81691			0.206697
$342$		0			0
$343$			3.50178i			0.189079i
$344$		0			0
$345$		−	5.27700i		−	0.284104i
$346$		0			0
$347$	17.6404			0.946988			0.473494	−	0.880797i	$-0.342993\pi$
$347$	17.6404			0.946988			0.473494	+	0.880797i	$0.342993\pi$
$348$		0			0
$349$			2.79317i			0.149515i	0.997202	+	0.0747576i	$0.0238183\pi$
$349$			2.79317i			0.149515i	−0.997202	+	0.0747576i	$0.976182\pi$
$350$		0			0
$351$	−17.0422	+	10.4778i	−0.909644	+	0.559263i
$352$		0			0
$353$			0.639605i			0.0340428i	0.999855	+	0.0170214i	$0.00541833\pi$
$353$			0.639605i			0.0340428i	−0.999855	+	0.0170214i	$0.994582\pi$
$354$		0			0
$355$	19.0619			1.01170
$356$		0			0
$357$			1.81618i			0.0961222i
$358$		0			0
$359$		−	2.20178i		−	0.116206i	−0.998311	−	0.0581029i	$-0.981495\pi$
$359$		−	2.20178i		−	0.116206i	0.998311	−	0.0581029i	$-0.0185051\pi$
$360$		0			0
$361$	−33.0004			−1.73686
$362$		0			0
$363$	−1.25126			−0.0656741
$364$		0			0
$365$	−26.0592			−1.36400
$366$		0			0
$367$	−14.2680			−0.744786			−0.372393	−	0.928075i	$-0.621463\pi$
$367$	−14.2680			−0.744786			−0.372393	+	0.928075i	$0.621463\pi$
$368$		0			0
$369$		−	15.6957i		−	0.817085i
$370$		0			0
$371$			2.50830i			0.130224i
$372$		0			0
$373$	−4.31009			−0.223168			−0.111584	−	0.993755i	$-0.535592\pi$
$373$	−4.31009			−0.223168			−0.111584	+	0.993755i	$0.535592\pi$
$374$		0			0
$375$		−	14.2976i		−	0.738325i
$376$		0			0
$377$	−17.3245	+	10.6513i	−0.892256	+	0.548572i
$378$		0			0
$379$			5.22239i			0.268256i	0.990964	+	0.134128i	$0.0428233\pi$
$379$			5.22239i			0.268256i	−0.990964	+	0.134128i	$0.957177\pi$
$380$		0			0
$381$	15.3729			0.787576
$382$		0			0
$383$			11.9665i			0.611459i	0.952118	+	0.305729i	$0.0989002\pi$
$383$			11.9665i			0.611459i	−0.952118	+	0.305729i	$0.901100\pi$
$384$		0			0
$385$		−	0.548523i		−	0.0279553i
$386$		0			0
$387$	10.9590			0.557079
$388$		0			0
$389$	3.75378			0.190324			0.0951621	−	0.995462i	$-0.469663\pi$
$389$	3.75378			0.190324			0.0951621	+	0.995462i	$0.469663\pi$
$390$		0			0
$391$	11.1597			0.564372
$392$		0			0
$393$	−21.7359			−1.09643
$394$		0			0
$395$		−	32.0728i		−	1.61376i
$396$		0			0
$397$			30.8344i			1.54754i	0.633469	+	0.773768i	$0.281630\pi$
$397$			30.8344i			1.54754i	−0.633469	+	0.773768i	$0.718370\pi$
$398$		0			0
$399$	2.26712			0.113498
$400$		0			0
$401$			26.3880i			1.31775i	0.752251	+	0.658877i	$0.228968\pi$
$401$			26.3880i			1.31775i	−0.752251	+	0.658877i	$0.771032\pi$
$402$		0			0
$403$	7.20783	+	11.7236i	0.359047	+	0.583992i
$404$		0			0
$405$			5.76249i			0.286340i
$406$		0			0
$407$	−6.68561			−0.331393
$408$		0			0
$409$			22.2859i			1.10197i	0.834516	+	0.550984i	$0.185747\pi$
$409$			22.2859i			1.10197i	−0.834516	+	0.550984i	$0.814253\pi$
$410$		0			0
$411$			19.0190i			0.938138i
$412$		0			0
$413$	−3.41722			−0.168150
$414$		0			0
$415$	13.9591			0.685227
$416$		0			0
$417$	−9.32374			−0.456585
$418$		0			0
$419$	7.48465			0.365649			0.182825	−	0.983146i	$-0.441476\pi$
$419$	7.48465			0.365649			0.182825	+	0.983146i	$0.441476\pi$
$420$		0			0
$421$		−	23.7374i		−	1.15689i	−0.815722	−	0.578445i	$-0.803660\pi$
$421$		−	23.7374i		−	1.15689i	0.815722	−	0.578445i	$-0.196340\pi$
$422$		0			0
$423$			9.32687i			0.453488i
$424$		0			0
$425$	1.35252			0.0656071
$426$		0			0
$427$		−	2.42226i		−	0.117221i
$428$		0			0
$429$	−2.36287	−	3.84322i	−0.114080	−	0.185552i
$430$		0			0
$431$		−	10.6669i		−	0.513807i	−0.966437	−	0.256903i	$-0.917298\pi$
$431$		−	10.6669i		−	0.513807i	0.966437	−	0.256903i	$-0.0827023\pi$
$432$		0			0
$433$	−35.6853			−1.71492			−0.857462	−	0.514547i	$-0.827960\pi$
$433$	−35.6853			−1.71492			−0.857462	+	0.514547i	$0.827960\pi$
$434$		0			0
$435$			15.4075i			0.738732i
$436$		0			0
$437$		−	13.9307i		−	0.666393i
$438$		0			0
$439$	−17.1020			−0.816234			−0.408117	−	0.912930i	$-0.633814\pi$
$439$	−17.1020			−0.816234			−0.408117	+	0.912930i	$0.633814\pi$
$440$		0			0
$441$	−9.94988			−0.473804
$442$		0			0
$443$	−26.7763			−1.27218			−0.636091	−	0.771614i	$-0.719450\pi$
$443$	−26.7763			−1.27218			−0.636091	+	0.771614i	$0.719450\pi$
$444$		0			0
$445$	14.8929			0.705992
$446$		0			0
$447$		−	7.02256i		−	0.332156i
$448$		0			0
$449$		−	24.5953i		−	1.16072i	−0.814359	−	0.580361i	$-0.802911\pi$
$449$		−	24.5953i		−	1.16072i	0.814359	−	0.580361i	$-0.197089\pi$
$450$		0			0
$451$	10.9427			0.515274
$452$		0			0
$453$			19.5274i			0.917477i
$454$		0			0
$455$	1.68478	−	1.03583i	0.0789836	−	0.0485603i
$456$		0			0
$457$			9.92678i			0.464355i	0.972673	+	0.232178i	$0.0745851\pi$
$457$			9.92678i			0.464355i	−0.972673	+	0.232178i	$0.925415\pi$
$458$		0			0
$459$	−32.0526			−1.49609
$460$		0			0
$461$			33.0748i			1.54045i	0.637775	+	0.770223i	$0.279855\pi$
$461$			33.0748i			1.54045i	−0.637775	+	0.770223i	$0.720145\pi$
$462$		0			0
$463$			3.22239i			0.149757i	0.997193	+	0.0748785i	$0.0238569\pi$
$463$			3.22239i			0.149757i	−0.997193	+	0.0748785i	$0.976143\pi$
$464$		0			0
$465$	10.4263			0.483509
$466$		0			0
$467$	2.21735			0.102607			0.0513033	−	0.998683i	$-0.483662\pi$
$467$	2.21735			0.102607			0.0513033	+	0.998683i	$0.483662\pi$
$468$		0			0
$469$	−2.94896			−0.136170
$470$		0			0
$471$	3.10356			0.143005
$472$		0			0
$473$			7.64044i			0.351308i
$474$		0			0
$475$		−	1.68835i		−	0.0774668i
$476$		0			0
$477$	−14.3189			−0.655618
$478$		0			0
$479$		−	30.7424i		−	1.40466i	−0.711853	−	0.702329i	$-0.752144\pi$
$479$		−	30.7424i		−	1.40466i	0.711853	−	0.702329i	$-0.247856\pi$
$480$		0			0
$481$	−12.6250	−	20.5347i	−0.575652	−	0.936302i
$482$		0			0
$483$			0.607351i			0.0276354i
$484$		0			0
$485$	22.2306			1.00944
$486$		0			0
$487$		−	1.32448i		−	0.0600177i	−0.999550	−	0.0300089i	$-0.990446\pi$
$487$		−	1.32448i		−	0.0600177i	0.999550	−	0.0300089i	$-0.00955355\pi$
$488$		0			0
$489$			11.9540i			0.540578i
$490$		0			0
$491$	−1.70313			−0.0768612			−0.0384306	−	0.999261i	$-0.512236\pi$
$491$	−1.70313			−0.0768612			−0.0384306	+	0.999261i	$0.512236\pi$
$492$		0			0
$493$	−32.5836			−1.46749
$494$		0			0
$495$	3.13131			0.140742
$496$		0			0
$497$	−2.19391			−0.0984102
$498$		0			0
$499$		−	10.5586i		−	0.472668i	−0.971672	−	0.236334i	$-0.924054\pi$
$499$		−	10.5586i		−	0.472668i	0.971672	−	0.236334i	$-0.0759460\pi$
$500$		0			0
$501$			8.63065i			0.385589i
$502$		0			0
$503$	−15.7592			−0.702669			−0.351334	−	0.936250i	$-0.614272\pi$
$503$	−15.7592			−0.702669			−0.351334	+	0.936250i	$0.614272\pi$
$504$		0			0
$505$			13.0985i			0.582877i
$506$		0			0
$507$	7.34234	−	14.5150i	0.326085	−	0.644634i
$508$		0			0
$509$			25.4740i			1.12912i	0.825393	+	0.564559i	$0.190954\pi$
$509$			25.4740i			1.12912i	−0.825393	+	0.564559i	$0.809046\pi$
$510$		0			0
$511$	2.99926			0.132680
$512$		0			0
$513$			40.0111i			1.76653i
$514$		0			0
$515$			27.7678i			1.22360i
$516$		0			0
$517$	−6.50252			−0.285981
$518$		0			0
$519$	−3.48079			−0.152790
$520$		0			0
$521$	23.0390			1.00936			0.504678	−	0.863308i	$-0.331611\pi$
$521$	23.0390			1.00936			0.504678	+	0.863308i	$0.331611\pi$
$522$		0			0
$523$	19.6077			0.857383			0.428691	−	0.903451i	$-0.358975\pi$
$523$	19.6077			0.857383			0.428691	+	0.903451i	$0.358975\pi$
$524$		0			0
$525$			0.0736090i			0.00321256i
$526$		0			0
$527$			22.0495i			0.960490i
$528$		0			0
$529$	−19.2680			−0.837741
$530$		0			0
$531$		−	19.5076i		−	0.846556i
$532$		0			0
$533$	20.6642	+	33.6104i	0.895065	+	1.45583i
$534$		0			0
$535$			5.95052i			0.257264i
$536$		0			0
$537$	−13.9564			−0.602263
$538$		0			0
$539$		−	6.93687i		−	0.298792i
$540$		0			0
$541$		−	18.5780i		−	0.798732i	−0.916791	−	0.399366i	$-0.869230\pi$
$541$		−	18.5780i		−	0.798732i	0.916791	−	0.399366i	$-0.130770\pi$
$542$		0			0
$543$	19.2110			0.824425
$544$		0			0
$545$	−10.2793			−0.440317
$546$		0			0
$547$	26.6521			1.13956			0.569780	−	0.821797i	$-0.307028\pi$
$547$	26.6521			1.13956			0.569780	+	0.821797i	$0.307028\pi$
$548$		0			0
$549$	13.8277			0.590153
$550$		0			0
$551$			40.6739i			1.73277i
$552$		0			0
$553$			3.69139i			0.156974i
$554$		0			0
$555$	−18.2625			−0.775199
$556$		0			0
$557$			8.76039i			0.371190i	0.982626	+	0.185595i	$0.0594212\pi$
$557$			8.76039i			0.371190i	−0.982626	+	0.185595i	$0.940579\pi$
$558$		0			0
$559$	−23.4674	+	14.4281i	−0.992567	+	0.610245i
$560$		0			0
$561$		−	7.22826i		−	0.305177i
$562$		0			0
$563$	24.1204			1.01656			0.508278	−	0.861193i	$-0.330282\pi$
$563$	24.1204			1.01656			0.508278	+	0.861193i	$0.330282\pi$
$564$		0			0
$565$		−	35.1913i		−	1.48051i
$566$		0			0
$567$		−	0.663228i		−	0.0278530i
$568$		0			0
$569$	18.1363			0.760315			0.380158	−	0.924922i	$-0.375870\pi$
$569$	18.1363			0.760315			0.380158	+	0.924922i	$0.375870\pi$
$570$		0			0
$571$	−10.9248			−0.457188			−0.228594	−	0.973522i	$-0.573413\pi$
$571$	−10.9248			−0.457188			−0.228594	+	0.973522i	$0.573413\pi$
$572$		0			0
$573$	−17.8959			−0.747612
$574$		0			0
$575$	0.452301			0.0188622
$576$		0			0
$577$		−	14.5018i		−	0.603717i	−0.953353	−	0.301859i	$-0.902393\pi$
$577$		−	14.5018i		−	0.603717i	0.953353	−	0.301859i	$-0.0976070\pi$
$578$		0			0
$579$		−	16.3034i		−	0.677548i
$580$		0			0
$581$	−1.60661			−0.0666536
$582$		0			0
$583$		−	9.98287i		−	0.413448i
$584$		0			0
$585$	5.91313	+	9.61774i	0.244478	+	0.397644i
$586$		0			0
$587$			1.42730i			0.0589110i	0.999566	+	0.0294555i	$0.00937734\pi$
$587$			1.42730i			0.0589110i	−0.999566	+	0.0294555i	$0.990623\pi$
$588$		0			0
$589$	27.5243			1.13412
$590$		0			0
$591$			3.61092i			0.148533i
$592$		0			0
$593$			8.35335i			0.343031i	0.985181	+	0.171515i	$0.0548663\pi$
$593$			8.35335i			0.343031i	−0.985181	+	0.171515i	$0.945134\pi$
$594$		0			0
$595$	3.16870			0.129904
$596$		0			0
$597$	20.0689			0.821366
$598$		0			0
$599$	3.48035			0.142203			0.0711016	−	0.997469i	$-0.477349\pi$
$599$	3.48035			0.142203			0.0711016	+	0.997469i	$0.477349\pi$
$600$		0			0
$601$	−15.2910			−0.623731			−0.311866	−	0.950126i	$-0.600954\pi$
$601$	−15.2910			−0.623731			−0.311866	+	0.950126i	$0.600954\pi$
$602$		0			0
$603$		−	16.8344i		−	0.685551i
$604$		0			0
$605$			2.18309i			0.0887551i
$606$		0			0
$607$	−32.4783			−1.31826			−0.659128	−	0.752031i	$-0.729074\pi$
$607$	−32.4783			−1.31826			−0.659128	+	0.752031i	$0.729074\pi$
$608$		0			0
$609$		−	1.77331i		−	0.0718581i
$610$		0			0
$611$	−12.2793	−	19.9724i	−0.496767	−	0.807995i
$612$		0			0
$613$			20.3557i			0.822157i	0.911600	+	0.411079i	$0.134848\pi$
$613$			20.3557i			0.822157i	−0.911600	+	0.411079i	$0.865152\pi$
$614$		0			0
$615$	29.8913			1.20533
$616$		0			0
$617$		−	2.40043i		−	0.0966378i	−0.998832	−	0.0483189i	$-0.984614\pi$
$617$		−	2.40043i		−	0.0966378i	0.998832	−	0.0483189i	$-0.0153864\pi$
$618$		0			0
$619$			2.87804i			0.115678i	0.998326	+	0.0578391i	$0.0184210\pi$
$619$			2.87804i			0.115678i	−0.998326	+	0.0578391i	$0.981579\pi$
$620$		0			0
$621$	−10.7188			−0.430130
$622$		0			0
$623$	−1.71409			−0.0686734
$624$		0			0
$625$	−23.7745			−0.950981
$626$		0			0
$627$	−9.02300			−0.360344
$628$		0			0
$629$		−	38.6213i		−	1.53993i
$630$		0			0
$631$			33.4180i			1.33035i	0.746688	+	0.665174i	$0.231643\pi$
$631$			33.4180i			1.33035i	−0.746688	+	0.665174i	$0.768357\pi$
$632$		0			0
$633$	−6.49078			−0.257985
$634$		0			0
$635$		−	26.8212i		−	1.06437i
$636$		0			0
$637$	21.3064	−	13.0995i	0.844192	−	0.519022i
$638$		0			0
$639$		−	12.5242i		−	0.495448i
$640$		0			0
$641$	−23.0522			−0.910506			−0.455253	−	0.890362i	$-0.650451\pi$
$641$	−23.0522			−0.910506			−0.455253	+	0.890362i	$0.650451\pi$
$642$		0			0
$643$			18.0467i			0.711694i	0.934544	+	0.355847i	$0.115808\pi$
$643$			18.0467i			0.711694i	−0.934544	+	0.355847i	$0.884192\pi$
$644$		0			0
$645$			20.8707i			0.821783i
$646$		0			0
$647$	3.35335			0.131834			0.0659169	−	0.997825i	$-0.479003\pi$
$647$	3.35335			0.131834			0.0659169	+	0.997825i	$0.479003\pi$
$648$		0			0
$649$	13.6003			0.533859
$650$		0			0
$651$	−1.20001			−0.0470320
$652$		0			0
$653$	−36.3545			−1.42266			−0.711330	−	0.702858i	$-0.751907\pi$
$653$	−36.3545			−1.42266			−0.711330	+	0.702858i	$0.751907\pi$
$654$		0			0
$655$			37.9229i			1.48177i
$656$		0			0
$657$			17.1216i			0.667978i
$658$		0			0
$659$	−4.53331			−0.176593			−0.0882963	−	0.996094i	$-0.528142\pi$
$659$	−4.53331			−0.176593			−0.0882963	+	0.996094i	$0.528142\pi$
$660$		0			0
$661$		−	13.4398i		−	0.522747i	−0.965238	−	0.261373i	$-0.915825\pi$
$661$		−	13.4398i		−	0.522747i	0.965238	−	0.261373i	$-0.0841754\pi$
$662$		0			0
$663$	22.2014	−	13.6498i	0.862233	−	0.530114i
$664$		0			0
$665$		−	3.95547i		−	0.153387i
$666$		0			0
$667$	−10.8963			−0.421908
$668$		0			0
$669$			14.9138i			0.576600i
$670$		0			0
$671$			9.64044i			0.372165i
$672$		0			0
$673$	28.0167			1.07996			0.539982	−	0.841677i	$-0.318431\pi$
$673$	28.0167			1.07996			0.539982	+	0.841677i	$0.318431\pi$
$674$		0			0
$675$	−1.29908			−0.0500017
$676$		0			0
$677$	−27.8294			−1.06957			−0.534785	−	0.844988i	$-0.679607\pi$
$677$	−27.8294			−1.06957			−0.534785	+	0.844988i	$0.679607\pi$
$678$		0			0
$679$	−2.55861			−0.0981903
$680$		0			0
$681$			16.4604i			0.630763i
$682$		0			0
$683$			35.5996i			1.36218i	0.732200	+	0.681090i	$0.238494\pi$
$683$			35.5996i			1.36218i	−0.732200	+	0.681090i	$0.761506\pi$
$684$		0			0
$685$	33.1826			1.26784
$686$		0			0
$687$		−	34.2641i		−	1.30726i
$688$		0			0
$689$	30.6622	−	18.8516i	1.16814	−	0.718187i
$690$		0			0
$691$			1.22239i			0.0465018i	0.999730	+	0.0232509i	$0.00740166\pi$
$691$			1.22239i			0.0465018i	−0.999730	+	0.0232509i	$0.992598\pi$
$692$		0			0
$693$	−0.360395			−0.0136903
$694$		0			0
$695$			16.2672i			0.617051i
$696$		0			0
$697$			63.2138i			2.39439i
$698$		0			0
$699$	−5.07522			−0.191962
$700$		0			0
$701$	−5.17565			−0.195481			−0.0977407	−	0.995212i	$-0.531162\pi$
$701$	−5.17565			−0.195481			−0.0977407	+	0.995212i	$0.531162\pi$
$702$		0			0
$703$	−48.2108			−1.81830
$704$		0			0
$705$	−17.7624			−0.668969
$706$		0			0
$707$		−	1.50756i		−	0.0566977i
$708$		0			0
$709$		−	5.55596i		−	0.208659i	−0.994543	−	0.104329i	$-0.966730\pi$
$709$		−	5.55596i		−	0.208659i	0.994543	−	0.104329i	$-0.0332696\pi$
$710$		0			0
$711$	−21.0727			−0.790287
$712$		0			0
$713$			7.37361i			0.276144i
$714$		0			0
$715$	−6.70530	+	4.12252i	−0.250764	+	0.154174i
$716$		0			0
$717$		−	33.0397i		−	1.23389i
$718$		0			0
$719$	−0.846130			−0.0315553			−0.0157777	−	0.999876i	$-0.505022\pi$
$719$	−0.846130			−0.0315553			−0.0157777	+	0.999876i	$0.505022\pi$
$720$		0			0
$721$		−	3.19591i		−	0.119022i
$722$		0			0
$723$		−	24.3311i		−	0.904883i
$724$		0			0
$725$	−1.32060			−0.0490459
$726$		0			0
$727$	7.61166			0.282301			0.141150	−	0.989988i	$-0.454920\pi$
$727$	7.61166			0.282301			0.141150	+	0.989988i	$0.454920\pi$
$728$		0			0
$729$	24.6141			0.911632
$730$		0			0
$731$	−44.1371			−1.63247
$732$		0			0
$733$		−	23.7997i		−	0.879062i	−0.898228	−	0.439531i	$-0.855145\pi$
$733$		−	23.7997i		−	0.879062i	0.898228	−	0.439531i	$-0.144855\pi$
$734$		0			0
$735$		−	18.9488i		−	0.698938i
$736$		0			0
$737$	11.7367			0.432325
$738$		0			0
$739$		−	24.0907i		−	0.886192i	−0.896474	−	0.443096i	$-0.853880\pi$
$739$		−	24.0907i		−	0.886192i	0.896474	−	0.443096i	$-0.146120\pi$
$740$		0			0
$741$	−17.0390	−	27.7140i	−0.625942	−	1.01810i
$742$		0			0
$743$		−	29.1581i		−	1.06971i	−0.844945	−	0.534853i	$-0.820367\pi$
$743$		−	29.1581i		−	1.06971i	0.844945	−	0.534853i	$-0.179633\pi$
$744$		0			0
$745$	−12.2523			−0.448891
$746$		0			0
$747$		−	9.17153i		−	0.335569i
$748$		0			0
$749$		−	0.684870i		−	0.0250246i
$750$		0			0
$751$	−18.8952			−0.689495			−0.344747	−	0.938696i	$-0.612035\pi$
$751$	−18.8952			−0.689495			−0.344747	+	0.938696i	$0.612035\pi$
$752$		0			0
$753$	2.31679			0.0844285
$754$		0			0
$755$	34.0696			1.23992
$756$		0			0
$757$	24.7655			0.900118			0.450059	−	0.892999i	$-0.351403\pi$
$757$	24.7655			0.900118			0.450059	+	0.892999i	$0.351403\pi$
$758$		0			0
$759$		−	2.41722i		−	0.0877395i
$760$		0			0
$761$		−	36.2696i		−	1.31477i	−0.753554	−	0.657386i	$-0.771662\pi$
$761$		−	36.2696i		−	1.31477i	0.753554	−	0.657386i	$-0.228338\pi$
$762$		0			0
$763$	1.18309			0.0428306
$764$		0			0
$765$			18.0889i			0.654004i
$766$		0			0
$767$	25.6827	+	41.7730i	0.927348	+	1.50834i
$768$		0			0
$769$		−	5.81804i		−	0.209804i	−0.994483	−	0.104902i	$-0.966547\pi$
$769$		−	5.81804i		−	0.209804i	0.994483	−	0.104902i	$-0.0334529\pi$
$770$		0			0
$771$	−30.5318			−1.09958
$772$		0			0
$773$			7.81765i			0.281181i	0.990068	+	0.140591i	$0.0449002\pi$
$773$			7.81765i			0.281181i	−0.990068	+	0.140591i	$0.955100\pi$
$774$		0			0
$775$			0.893658i			0.0321011i
$776$		0			0
$777$	2.10190			0.0754053
$778$		0			0
$779$	78.9095			2.82723
$780$		0			0
$781$	8.73161			0.312442
$782$		0			0
$783$	31.2961			1.11843
$784$		0			0
$785$		−	5.41482i		−	0.193263i
$786$		0			0
$787$		−	2.14770i		−	0.0765571i	−0.999267	−	0.0382786i	$-0.987813\pi$
$787$		−	2.14770i		−	0.0765571i	0.999267	−	0.0382786i	$-0.0121874\pi$
$788$		0			0
$789$	10.5813			0.376704
$790$		0			0
$791$			4.05031i			0.144012i
$792$		0			0
$793$	−29.6104	+	18.2049i	−1.05150	+	0.646475i
$794$		0			0
$795$		−	27.2693i		−	0.967143i
$796$		0			0
$797$	−16.7558			−0.593520			−0.296760	−	0.954952i	$-0.595906\pi$
$797$	−16.7558			−0.593520			−0.296760	+	0.954952i	$0.595906\pi$
$798$		0			0
$799$		−	37.5636i		−	1.32891i
$800$		0			0
$801$		−	9.78505i		−	0.345738i
$802$		0			0
$803$	−11.9369			−0.421243
$804$		0			0
$805$	1.05965			0.0373478
$806$		0			0
$807$	23.7106			0.834652
$808$		0			0
$809$	−6.12774			−0.215440			−0.107720	−	0.994181i	$-0.534355\pi$
$809$	−6.12774			−0.215440			−0.107720	+	0.994181i	$0.534355\pi$
$810$		0			0
$811$			46.3022i			1.62589i	0.582340	+	0.812945i	$0.302137\pi$
$811$			46.3022i			1.62589i	−0.582340	+	0.812945i	$0.697863\pi$
$812$		0			0
$813$		−	30.7653i		−	1.07899i
$814$		0			0
$815$	20.8563			0.730563
$816$		0			0
$817$			55.0962i			1.92757i
$818$		0			0
$819$	−0.680566	−	1.10694i	−0.0237809	−	0.0386798i
$820$		0			0
$821$			21.8671i			0.763168i	0.924334	+	0.381584i	$0.124621\pi$
$821$			21.8671i			0.763168i	−0.924334	+	0.381584i	$0.875379\pi$
$822$		0			0
$823$	−39.3591			−1.37197			−0.685986	−	0.727614i	$-0.740629\pi$
$823$	−39.3591			−1.37197			−0.685986	+	0.727614i	$0.740629\pi$
$824$		0			0
$825$		−	0.292959i		−	0.0101995i
$826$		0			0
$827$		−	17.1223i		−	0.595399i	−0.954660	−	0.297700i	$-0.903781\pi$
$827$		−	17.1223i		−	0.595399i	0.954660	−	0.297700i	$-0.0962194\pi$
$828$		0			0
$829$	54.3642			1.88815			0.944073	−	0.329737i	$-0.106960\pi$
$829$	54.3642			1.88815			0.944073	+	0.329737i	$0.106960\pi$
$830$		0			0
$831$	3.17417			0.110111
$832$		0			0
$833$	40.0728			1.38844
$834$		0			0
$835$	15.0580			0.521103
$836$		0			0
$837$		−	21.1782i		−	0.732027i
$838$		0			0
$839$		−	44.5133i		−	1.53677i	−0.639987	−	0.768386i	$-0.721060\pi$
$839$		−	44.5133i		−	1.53677i	0.639987	−	0.768386i	$-0.278940\pi$
$840$		0			0
$841$	2.81451			0.0970522
$842$		0			0
$843$			40.7344i			1.40297i
$844$		0			0
$845$	−25.3245	−	12.8103i	−0.871188	−	0.440686i
$846$		0			0
$847$		−	0.251260i		−	0.00863341i
$848$		0			0
$849$	−7.84443			−0.269220
$850$		0			0
$851$		−	12.9154i		−	0.442735i
$852$		0			0
$853$		−	13.1343i		−	0.449711i	−0.974392	−	0.224856i	$-0.927809\pi$
$853$		−	13.1343i		−	0.449711i	0.974392	−	0.224856i	$-0.0721911\pi$
$854$		0			0
$855$	22.5802			0.772228
$856$		0			0
$857$	−8.14809			−0.278333			−0.139167	−	0.990269i	$-0.544442\pi$
$857$	−8.14809			−0.278333			−0.139167	+	0.990269i	$0.544442\pi$
$858$		0			0
$859$	−5.58318			−0.190496			−0.0952478	−	0.995454i	$-0.530364\pi$
$859$	−5.58318			−0.190496			−0.0952478	+	0.995454i	$0.530364\pi$
$860$		0			0
$861$	−3.44031			−0.117246
$862$		0			0
$863$		−	7.77331i		−	0.264607i	−0.991209	−	0.132303i	$-0.957763\pi$
$863$		−	7.77331i		−	0.264607i	0.991209	−	0.132303i	$-0.0422373\pi$
$864$		0			0
$865$			6.07296i			0.206487i
$866$		0			0
$867$	20.4847			0.695695
$868$		0			0
$869$		−	14.6915i		−	0.498374i
$870$		0			0
$871$	22.1634	+	36.0489i	0.750978	+	1.22147i
$872$		0			0
$873$		−	14.6061i		−	0.494341i
$874$		0			0
$875$	2.87104			0.0970590
$876$		0			0
$877$		−	17.3475i		−	0.585783i	−0.956146	−	0.292891i	$-0.905383\pi$
$877$		−	17.3475i		−	0.585783i	0.956146	−	0.292891i	$-0.0946174\pi$
$878$		0			0
$879$		−	7.57339i		−	0.255444i
$880$		0			0
$881$	32.2781			1.08748			0.543739	−	0.839254i	$-0.317008\pi$
$881$	32.2781			1.08748			0.543739	+	0.839254i	$0.317008\pi$
$882$		0			0
$883$	−57.1694			−1.92390			−0.961952	−	0.273220i	$-0.911911\pi$
$883$	−57.1694			−1.92390			−0.961952	+	0.273220i	$0.911911\pi$
$884$		0			0
$885$	37.1507			1.24881
$886$		0			0
$887$	−49.0778			−1.64787			−0.823936	−	0.566683i	$-0.808226\pi$
$887$	−49.0778			−1.64787			−0.823936	+	0.566683i	$0.808226\pi$
$888$		0			0
$889$			3.08696i			0.103533i
$890$		0			0
$891$			2.63961i			0.0884301i
$892$		0			0
$893$	−46.8905			−1.56913
$894$		0			0
$895$			24.3499i			0.813926i
$896$		0			0
$897$	7.42443	−	4.56465i	0.247895	−	0.152409i
$898$		0			0
$899$		−	21.5291i		−	0.718034i
$900$		0			0
$901$	57.6689			1.92123
$902$		0			0
$903$		−	2.40209i		−	0.0799367i
$904$		0			0
$905$		−	33.5177i		−	1.11417i
$906$		0			0
$907$	−24.1367			−0.801447			−0.400724	−	0.916199i	$-0.631241\pi$
$907$	−24.1367			−0.801447			−0.400724	+	0.916199i	$0.631241\pi$
$908$		0			0
$909$	8.60608			0.285446
$910$		0			0
$911$	33.3958			1.10645			0.553226	−	0.833031i	$-0.313397\pi$
$911$	33.3958			1.10645			0.553226	+	0.833031i	$0.313397\pi$
$912$		0			0
$913$	6.39422			0.211618
$914$		0			0
$915$			26.3339i			0.870572i
$916$		0			0
$917$		−	4.36470i		−	0.144135i
$918$		0			0
$919$	−19.4581			−0.641863			−0.320932	−	0.947102i	$-0.603996\pi$
$919$	−19.4581			−0.641863			−0.320932	+	0.947102i	$0.603996\pi$
$920$		0			0
$921$		−	38.5819i		−	1.27132i
$922$		0			0
$923$	16.4887	+	26.8190i	0.542732	+	0.882757i
$924$		0			0
$925$		−	1.56531i		−	0.0514670i
$926$		0			0
$927$	18.2442			0.599219
$928$		0			0
$929$		−	20.0592i		−	0.658122i	−0.944309	−	0.329061i	$-0.893268\pi$
$929$		−	20.0592i		−	0.658122i	0.944309	−	0.329061i	$-0.106732\pi$
$930$		0			0
$931$		−	50.0227i		−	1.63943i
$932$		0			0
$933$	13.3500			0.437058
$934$		0			0
$935$	−12.6112			−0.412431
$936$		0			0
$937$	−15.0494			−0.491642			−0.245821	−	0.969315i	$-0.579058\pi$
$937$	−15.0494			−0.491642			−0.245821	+	0.969315i	$0.579058\pi$
$938$		0			0
$939$	35.0367			1.14338
$940$		0			0
$941$		−	51.0624i		−	1.66459i	−0.554337	−	0.832293i	$-0.687028\pi$
$941$		−	51.0624i		−	1.66459i	0.554337	−	0.832293i	$-0.312972\pi$
$942$		0			0
$943$			21.1395i			0.688396i
$944$		0			0
$945$	−3.04349			−0.0990049
$946$		0			0
$947$		−	26.3545i		−	0.856406i	−0.903683	−	0.428203i	$-0.859147\pi$
$947$		−	26.3545i		−	0.856406i	0.903683	−	0.428203i	$-0.140853\pi$
$948$		0			0
$949$	−22.5415	−	36.6638i	−0.731727	−	1.19016i
$950$		0			0
$951$		−	17.8598i		−	0.579143i
$952$		0			0
$953$	5.54035			0.179469			0.0897347	−	0.995966i	$-0.471398\pi$
$953$	5.54035			0.179469			0.0897347	+	0.995966i	$0.471398\pi$
$954$		0			0
$955$			31.2232i			1.01036i
$956$		0			0
$957$			7.05765i			0.228142i
$958$		0			0
$959$	−3.81913			−0.123326
$960$		0			0
$961$	16.4312			0.530038
$962$		0			0
$963$	3.90966			0.125987
$964$		0			0
$965$	−28.4448			−0.915670
$966$		0			0
$967$			25.2278i			0.811272i	0.914035	+	0.405636i	$0.132950\pi$
$967$			25.2278i			0.811272i	−0.914035	+	0.405636i	$0.867050\pi$
$968$		0			0
$969$		−	52.1239i		−	1.67446i
$970$		0			0
$971$	9.47795			0.304162			0.152081	−	0.988368i	$-0.451403\pi$
$971$	9.47795			0.304162			0.152081	+	0.988368i	$0.451403\pi$
$972$		0			0
$973$		−	1.87226i		−	0.0600219i
$974$		0			0
$975$	0.899817	−	0.553221i	0.0288172	−	0.0177173i
$976$		0			0
$977$			10.6974i			0.342239i	0.985250	+	0.171119i	$0.0547384\pi$
$977$			10.6974i			0.342239i	−0.985250	+	0.171119i	$0.945262\pi$
$978$		0			0
$979$	6.82195			0.218031
$980$		0			0
$981$			6.75378i			0.215632i
$982$		0			0
$983$			3.04170i			0.0970151i	0.998823	+	0.0485076i	$0.0154465\pi$
$983$			3.04170i			0.0970151i	−0.998823	+	0.0485076i	$0.984554\pi$
$984$		0			0
$985$	6.30001			0.200735
$986$		0			0
$987$	2.04434			0.0650721
$988$		0			0
$989$	−14.7600			−0.469341
$990$		0			0
$991$	6.12235			0.194483			0.0972415	−	0.995261i	$-0.468998\pi$
$991$	6.12235			0.194483			0.0972415	+	0.995261i	$0.468998\pi$
$992$		0			0
$993$			3.07282i			0.0975130i
$994$		0			0
$995$		−	35.0144i		−	1.11003i
$996$		0			0
$997$	−11.1531			−0.353223			−0.176612	−	0.984281i	$-0.556514\pi$
$997$	−11.1531			−0.353223			−0.176612	+	0.984281i	$0.556514\pi$
$998$		0			0
$999$			37.0953i			1.17364i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.c.441.5	✓	8
3.2	odd	2		5148.2.e.c.1585.5		8
4.3	odd	2		2288.2.j.i.1585.3		8
13.5	odd	4		7436.2.a.p.1.3		4
13.8	odd	4		7436.2.a.o.1.3		4
13.12	even	2	inner	572.2.f.c.441.6	yes	8
39.38	odd	2		5148.2.e.c.1585.4		8
52.51	odd	2		2288.2.j.i.1585.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.c.441.5	✓	8	1.1	even	1	trivial
572.2.f.c.441.6	yes	8	13.12	even	2	inner
2288.2.j.i.1585.3		8	4.3	odd	2
2288.2.j.i.1585.4		8	52.51	odd	2
5148.2.e.c.1585.4		8	39.38	odd	2
5148.2.e.c.1585.5		8	3.2	odd	2
7436.2.a.o.1.3		4	13.8	odd	4
7436.2.a.p.1.3		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.25126 q^{3} -2.18309i q^{5} +0.251260i q^{7} -1.43435 q^{9} +O(q^{10})\)	\(q+1.25126 q^{3} -2.18309i q^{5} +0.251260i q^{7} -1.43435 q^{9} -1.00000i q^{11} +(3.07148 - 1.88839i) q^{13} -2.73161i q^{15} +5.77678 q^{17} -7.21113i q^{19} +0.314392i q^{21} +1.93183 q^{23} +0.234131 q^{25} -5.54852 q^{27} -5.64044 q^{29} +3.81691i q^{31} -1.25126i q^{33} +0.548523 q^{35} -6.68561i q^{37} +(3.84322 - 2.36287i) q^{39} +10.9427i q^{41} -7.64044 q^{43} +3.13131i q^{45} -6.50252i q^{47} +6.93687 q^{49} +7.22826 q^{51} +9.98287 q^{53} -2.18309 q^{55} -9.02300i q^{57} +13.6003i q^{59} -9.64044 q^{61} -0.360395i q^{63} +(-4.12252 - 6.70530i) q^{65} +11.7367i q^{67} +2.41722 q^{69} +8.73161i q^{71} -11.9369i q^{73} +0.292959 q^{75} +0.251260 q^{77} +14.6915 q^{79} -2.63961 q^{81} +6.39422i q^{83} -12.6112i q^{85} -7.05765 q^{87} +6.82195i q^{89} +(0.474478 + 0.771741i) q^{91} +4.77595i q^{93} -15.7425 q^{95} +10.1831i q^{97} +1.43435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 + 18 * q^9	\( 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) \) 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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