LMFDB - Embedded newform 1764.2.k.m.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, 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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.12745506816.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 55x^{4} + 72x^{2} + 81 $ x^8 + 8x^6 + 55x^4 + 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$-1.28897 - 2.23256i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.2.k.m.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.87083 + 3.24037i) q^{5} +O(q^{10})$	$q+(-1.87083 + 3.24037i) q^{5} +(-2.64575 - 4.58258i) q^{11} +4.24264 q^{13} +(-1.87083 - 3.24037i) q^{17} +(1.41421 - 2.44949i) q^{19} +(2.64575 - 4.58258i) q^{23} +(-4.50000 - 7.79423i) q^{25} -5.29150 q^{29} +(4.24264 + 7.34847i) q^{31} +(-2.00000 + 3.46410i) q^{37} +3.74166 q^{41} +8.00000 q^{43} +(3.74166 - 6.48074i) q^{47} +(5.29150 + 9.16515i) q^{53} +19.7990 q^{55} +(3.74166 + 6.48074i) q^{59} +(4.94975 - 8.57321i) q^{61} +(-7.93725 + 13.7477i) q^{65} +(-6.00000 - 10.3923i) q^{67} +15.8745 q^{71} +(0.707107 + 1.22474i) q^{73} +(2.00000 - 3.46410i) q^{79} -14.9666 q^{83} +14.0000 q^{85} +(1.87083 - 3.24037i) q^{89} +(5.29150 + 9.16515i) q^{95} +9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 36 q^{25} - 16 q^{37} + 64 q^{43} - 48 q^{67} + 16 q^{79} + 112 q^{85}+O(q^{100}) $ 8 * q - 36 * q^25 - 16 * q^37 + 64 * q^43 - 48 * q^67 + 16 * q^79 + 112 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	−1.87083	+	3.24037i	−0.836660	+	1.44914i	0.0560116	+	0.998430i	$0.482162\pi$
$5$	−1.87083	+	3.24037i	−0.836660	+	1.44914i	−0.892672	+	0.450708i	$0.851172\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.64575	−	4.58258i	−0.797724	−	1.38170i	−0.921095	−	0.389338i	$-0.872704\pi$
$11$	−2.64575	−	4.58258i	−0.797724	−	1.38170i	0.123371	−	0.992361i	$-0.460630\pi$
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.87083	−	3.24037i	−0.453743	−	0.785905i	0.544872	−	0.838519i	$-0.316578\pi$
$17$	−1.87083	−	3.24037i	−0.453743	−	0.785905i	−0.998615	+	0.0526138i	$0.983245\pi$
$18$		0			0
$19$	1.41421	−	2.44949i	0.324443	−	0.561951i	−0.656957	−	0.753928i	$-0.728157\pi$
$19$	1.41421	−	2.44949i	0.324443	−	0.561951i	0.981399	+	0.191977i	$0.0614899\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.64575	−	4.58258i	0.551677	−	0.955533i	−0.446476	−	0.894795i	$-0.647321\pi$
$23$	2.64575	−	4.58258i	0.551677	−	0.955533i	0.998154	−	0.0607377i	$-0.0193453\pi$
$24$		0			0
$25$	−4.50000	−	7.79423i	−0.900000	−	1.55885i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.29150			−0.982607			−0.491304	−	0.870988i	$-0.663479\pi$
$29$	−5.29150			−0.982607			−0.491304	+	0.870988i	$0.663479\pi$
$30$		0			0
$31$	4.24264	+	7.34847i	0.762001	+	1.31982i	0.941818	+	0.336124i	$0.109116\pi$
$31$	4.24264	+	7.34847i	0.762001	+	1.31982i	−0.179817	+	0.983700i	$0.557551\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.00000	+	3.46410i	−0.328798	+	0.569495i	−0.982274	−	0.187453i	$-0.939977\pi$
$37$	−2.00000	+	3.46410i	−0.328798	+	0.569495i	0.653476	+	0.756948i	$0.273310\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.74166			0.584349			0.292174	−	0.956365i	$-0.405621\pi$
$41$	3.74166			0.584349			0.292174	+	0.956365i	$0.405621\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.74166	−	6.48074i	0.545777	−	0.945313i	−0.452781	−	0.891622i	$-0.649568\pi$
$47$	3.74166	−	6.48074i	0.545777	−	0.945313i	0.998558	−	0.0536913i	$-0.0170987\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.29150	+	9.16515i	0.726844	+	1.25893i	0.958211	+	0.286064i	$0.0923469\pi$
$53$	5.29150	+	9.16515i	0.726844	+	1.25893i	−0.231367	+	0.972867i	$0.574320\pi$
$54$		0			0
$55$	19.7990			2.66970
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.74166	+	6.48074i	0.487122	+	0.843721i	0.999890	−	0.0148066i	$-0.00471326\pi$
$59$	3.74166	+	6.48074i	0.487122	+	0.843721i	−0.512768	+	0.858527i	$0.671380\pi$
$60$		0			0
$61$	4.94975	−	8.57321i	0.633750	−	1.09769i	−0.353028	−	0.935613i	$-0.614848\pi$
$61$	4.94975	−	8.57321i	0.633750	−	1.09769i	0.986778	−	0.162075i	$-0.0518186\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.93725	+	13.7477i	−0.984495	+	1.70520i
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.8745			1.88396			0.941979	−	0.335673i	$-0.108964\pi$
$71$	15.8745			1.88396			0.941979	+	0.335673i	$0.108964\pi$
$72$		0			0
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	0.904435	−	0.426612i	$-0.140293\pi$
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	−0.821674	+	0.569958i	$0.806960\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.9666			−1.64280			−0.821401	−	0.570352i	$-0.806807\pi$
$83$	−14.9666			−1.64280			−0.821401	+	0.570352i	$0.806807\pi$
$84$		0			0
$85$	14.0000			1.51851
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.87083	−	3.24037i	0.198307	−	0.343479i	−0.749672	−	0.661809i	$-0.769789\pi$
$89$	1.87083	−	3.24037i	0.198307	−	0.343479i	0.947980	+	0.318331i	$0.103122\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.29150	+	9.16515i	0.542897	+	0.940325i
$96$		0			0
$97$	9.89949			1.00514			0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949			1.00514			0.502571	+	0.864536i	$0.332388\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.61249	+	9.72111i	0.558463	+	0.967287i	0.997625	+	0.0688787i	$0.0219421\pi$
$101$	5.61249	+	9.72111i	0.558463	+	0.967287i	−0.439162	+	0.898408i	$0.644725\pi$
$102$		0			0
$103$	−1.41421	+	2.44949i	−0.139347	+	0.241355i	−0.927249	−	0.374444i	$-0.877834\pi$
$103$	−1.41421	+	2.44949i	−0.139347	+	0.241355i	0.787903	+	0.615800i	$0.211167\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.64575	−	4.58258i	0.255774	−	0.443014i	−0.709331	−	0.704875i	$-0.751003\pi$
$107$	2.64575	−	4.58258i	0.255774	−	0.443014i	0.965106	+	0.261861i	$0.0843362\pi$
$108$		0			0
$109$	−8.00000	−	13.8564i	−0.766261	−	1.32720i	−0.939577	−	0.342337i	$-0.888782\pi$
$109$	−8.00000	−	13.8564i	−0.766261	−	1.32720i	0.173316	−	0.984866i	$-0.444552\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	9.89949	+	17.1464i	0.923133	+	1.59891i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.50000	+	14.7224i	−0.772727	+	1.33840i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	14.9666			1.33866
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.93725	−	13.7477i	−0.678125	−	1.17455i	−0.975545	−	0.219801i	$-0.929459\pi$
$137$	−7.93725	−	13.7477i	−0.678125	−	1.17455i	0.297419	−	0.954747i	$-0.403874\pi$
$138$		0			0
$139$	5.65685			0.479808			0.239904	−	0.970797i	$-0.422884\pi$
$139$	5.65685			0.479808			0.239904	+	0.970797i	$0.422884\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.2250	−	19.4422i	−0.938679	−	1.62584i
$144$		0			0
$145$	9.89949	−	17.1464i	0.822108	−	1.42393i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.29150	−	9.16515i	0.433497	−	0.750838i	−0.563675	−	0.825997i	$-0.690613\pi$
$149$	5.29150	−	9.16515i	0.433497	−	0.750838i	0.997172	+	0.0751583i	$0.0239462\pi$
$150$		0			0
$151$	2.00000	+	3.46410i	0.162758	+	0.281905i	0.935857	−	0.352381i	$-0.114628\pi$
$151$	2.00000	+	3.46410i	0.162758	+	0.281905i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−31.7490			−2.55014
$156$		0			0
$157$	−4.94975	−	8.57321i	−0.395033	−	0.684217i	0.598073	−	0.801442i	$-0.295933\pi$
$157$	−4.94975	−	8.57321i	−0.395033	−	0.684217i	−0.993105	+	0.117225i	$0.962600\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	−0.146772	−	0.989170i	$-0.546888\pi$
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	0.930033	−	0.367477i	$-0.119778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.48331			0.579076			0.289538	−	0.957166i	$-0.406498\pi$
$167$	7.48331			0.579076			0.289538	+	0.957166i	$0.406498\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.61249	+	9.72111i	−0.426709	+	0.739082i	−0.996578	−	0.0826532i	$-0.973661\pi$
$173$	−5.61249	+	9.72111i	−0.426709	+	0.739082i	0.569869	+	0.821736i	$0.306994\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.93725	−	13.7477i	−0.593258	−	1.02755i	−0.993790	−	0.111271i	$-0.964508\pi$
$179$	−7.93725	−	13.7477i	−0.593258	−	1.02755i	0.400532	−	0.916283i	$-0.368825\pi$
$180$		0			0
$181$	−4.24264			−0.315353			−0.157676	−	0.987491i	$-0.550400\pi$
$181$	−4.24264			−0.315353			−0.157676	+	0.987491i	$0.550400\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.48331	−	12.9615i	−0.550184	−	0.952947i
$186$		0			0
$187$	−9.89949	+	17.1464i	−0.723923	+	1.25387i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.64575	−	4.58258i	0.191440	−	0.331584i	−0.754288	−	0.656544i	$-0.772018\pi$
$191$	2.64575	−	4.58258i	0.191440	−	0.331584i	0.945728	+	0.324960i	$0.105351\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.5830			0.754008			0.377004	−	0.926212i	$-0.376954\pi$
$197$	10.5830			0.754008			0.377004	+	0.926212i	$0.376954\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−7.00000	+	12.1244i	−0.488901	+	0.846802i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.9666			−1.03526
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.9666	+	25.9230i	−1.02072	+	1.76793i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.93725	−	13.7477i	−0.533917	−	0.924772i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.2250	+	19.4422i	0.745028	+	1.29043i	0.950182	+	0.311696i	$0.100897\pi$
$227$	11.2250	+	19.4422i	0.745028	+	1.29043i	−0.205154	+	0.978730i	$0.565770\pi$
$228$		0			0
$229$	−6.36396	+	11.0227i	−0.420542	+	0.728401i	−0.995993	−	0.0894361i	$-0.971494\pi$
$229$	−6.36396	+	11.0227i	−0.420542	+	0.728401i	0.575450	+	0.817837i	$0.304827\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.93725	−	13.7477i	0.519987	−	0.900644i	−0.479743	−	0.877409i	$-0.659270\pi$
$233$	7.93725	−	13.7477i	0.519987	−	0.900644i	0.999730	−	0.0232346i	$-0.00739648\pi$
$234$		0			0
$235$	14.0000	+	24.2487i	0.913259	+	1.58181i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.8745			1.02684			0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745			1.02684			0.513418	+	0.858138i	$0.328379\pi$
$240$		0			0
$241$	−14.8492	−	25.7196i	−0.956524	−	1.65675i	−0.730842	−	0.682547i	$-0.760872\pi$
$241$	−14.8492	−	25.7196i	−0.956524	−	1.65675i	−0.225682	−	0.974201i	$-0.572461\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.00000	−	10.3923i	0.381771	−	0.661247i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.48331			0.472343			0.236171	−	0.971711i	$-0.424107\pi$
$251$	7.48331			0.472343			0.236171	+	0.971711i	$0.424107\pi$
$252$		0			0
$253$	−28.0000			−1.76034
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.61249	−	9.72111i	0.350097	−	0.606386i	−0.636169	−	0.771550i	$-0.719482\pi$
$257$	5.61249	−	9.72111i	0.350097	−	0.606386i	0.986266	+	0.165164i	$0.0528152\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.93725	+	13.7477i	0.489432	+	0.847721i	0.999926	−	0.0121601i	$-0.00387079\pi$
$263$	7.93725	+	13.7477i	0.489432	+	0.847721i	−0.510494	+	0.859881i	$0.670537\pi$
$264$		0			0
$265$	−39.5980			−2.43248
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.87083	+	3.24037i	0.114066	+	0.197569i	0.917406	−	0.397952i	$-0.130279\pi$
$269$	1.87083	+	3.24037i	0.114066	+	0.197569i	−0.803340	+	0.595521i	$0.796946\pi$
$270$		0			0
$271$	9.89949	−	17.1464i	0.601351	−	1.04157i	−0.391265	−	0.920278i	$-0.627962\pi$
$271$	9.89949	−	17.1464i	0.601351	−	1.04157i	0.992617	−	0.121293i	$-0.0387042\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.8118	+	41.2432i	−1.43590	+	2.48706i
$276$		0			0
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	0.894503	−	0.447062i	$-0.147530\pi$
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	−0.834419	+	0.551131i	$0.814196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.29150			0.315665			0.157832	−	0.987466i	$-0.449549\pi$
$281$	5.29150			0.315665			0.157832	+	0.987466i	$0.449549\pi$
$282$		0			0
$283$	15.5563	+	26.9444i	0.924729	+	1.60168i	0.791996	+	0.610526i	$0.209042\pi$
$283$	15.5563	+	26.9444i	0.924729	+	1.60168i	0.132733	+	0.991152i	$0.457625\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.50000	−	2.59808i	0.0882353	−	0.152828i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−26.1916			−1.53013			−0.765065	−	0.643953i	$-0.777293\pi$
$293$	−26.1916			−1.53013			−0.765065	+	0.643953i	$0.777293\pi$
$294$		0			0
$295$	−28.0000			−1.63022
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.2250	−	19.4422i	0.649157	−	1.12437i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.5203	+	32.0780i	1.06047	+	1.83678i
$306$		0			0
$307$	−14.1421			−0.807134			−0.403567	−	0.914950i	$-0.632230\pi$
$307$	−14.1421			−0.807134			−0.403567	+	0.914950i	$0.632230\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.74166	−	6.48074i	−0.212170	−	0.367489i	0.740223	−	0.672361i	$-0.234720\pi$
$311$	−3.74166	−	6.48074i	−0.212170	−	0.367489i	−0.952393	+	0.304872i	$0.901386\pi$
$312$		0			0
$313$	−6.36396	+	11.0227i	−0.359712	+	0.623040i	−0.987913	−	0.155012i	$-0.950459\pi$
$313$	−6.36396	+	11.0227i	−0.359712	+	0.623040i	0.628200	+	0.778052i	$0.283792\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.29150	−	9.16515i	0.297200	−	0.514766i	−0.678294	−	0.734791i	$-0.737280\pi$
$317$	5.29150	−	9.16515i	0.297200	−	0.514766i	0.975494	+	0.220024i	$0.0706137\pi$
$318$		0			0
$319$	14.0000	+	24.2487i	0.783850	+	1.35767i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.5830			−0.588854
$324$		0			0
$325$	−19.0919	−	33.0681i	−1.05903	−	1.83429i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	44.8999			2.45314
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	22.4499	−	38.8844i	1.21573	−	2.10571i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.93725	+	13.7477i	0.426094	+	0.738017i	0.996522	−	0.0833311i	$-0.0265559\pi$
$347$	7.93725	+	13.7477i	0.426094	+	0.738017i	−0.570428	+	0.821348i	$0.693223\pi$
$348$		0			0
$349$	−29.6985			−1.58972			−0.794862	−	0.606791i	$-0.792457\pi$
$349$	−29.6985			−1.58972			−0.794862	+	0.606791i	$0.792457\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.35414	+	16.2019i	0.497871	+	0.862338i	0.999997	−	0.00245682i	$-0.000782032\pi$
$353$	9.35414	+	16.2019i	0.497871	+	0.862338i	−0.502126	+	0.864794i	$0.667449\pi$
$354$		0			0
$355$	−29.6985	+	51.4393i	−1.57623	+	2.73011i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.64575	+	4.58258i	−0.139637	+	0.241859i	−0.927359	−	0.374172i	$-0.877927\pi$
$359$	−2.64575	+	4.58258i	−0.139637	+	0.241859i	0.787722	+	0.616031i	$0.211260\pi$
$360$		0			0
$361$	5.50000	+	9.52628i	0.289474	+	0.501383i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.29150			−0.276970
$366$		0			0
$367$	−5.65685	−	9.79796i	−0.295285	−	0.511449i	0.679766	−	0.733429i	$-0.262081\pi$
$367$	−5.65685	−	9.79796i	−0.295285	−	0.511449i	−0.975051	+	0.221980i	$0.928748\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	−0.965945	−	0.258748i	$-0.916690\pi$
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	0.707055	+	0.707159i	$0.250023\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.4499			−1.15623
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.48331	−	12.9615i	0.382380	−	0.662301i	−0.609022	−	0.793153i	$-0.708438\pi$
$383$	7.48331	−	12.9615i	0.382380	−	0.662301i	0.991402	+	0.130852i	$0.0417714\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.64575	+	4.58258i	0.134145	+	0.232346i	0.925270	−	0.379308i	$-0.123838\pi$
$389$	2.64575	+	4.58258i	0.134145	+	0.232346i	−0.791126	+	0.611654i	$0.790505\pi$
$390$		0			0
$391$	−19.7990			−1.00128
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.48331	+	12.9615i	0.376526	+	0.652163i
$396$		0			0
$397$	16.2635	−	28.1691i	0.816239	−	1.41377i	−0.0921950	−	0.995741i	$-0.529388\pi$
$397$	16.2635	−	28.1691i	0.816239	−	1.41377i	0.908434	−	0.418027i	$-0.137278\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.93725	−	13.7477i	0.396368	−	0.686529i	−0.596907	−	0.802310i	$-0.703604\pi$
$401$	7.93725	−	13.7477i	0.396368	−	0.686529i	0.993275	+	0.115782i	$0.0369373\pi$
$402$		0			0
$403$	18.0000	+	31.1769i	0.896644	+	1.55303i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	21.1660			1.04916
$408$		0			0
$409$	−10.6066	−	18.3712i	−0.524463	−	0.908396i	−0.999594	−	0.0284813i	$-0.990933\pi$
$409$	−10.6066	−	18.3712i	−0.524463	−	0.908396i	0.475132	−	0.879915i	$-0.342400\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	28.0000	−	48.4974i	1.37447	−	2.38064i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.48331			−0.365584			−0.182792	−	0.983152i	$-0.558513\pi$
$419$	−7.48331			−0.365584			−0.182792	+	0.983152i	$0.558513\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.8375	+	29.1633i	−0.816737	+	1.41463i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.93725	−	13.7477i	−0.382324	−	0.662205i	0.609070	−	0.793117i	$-0.291543\pi$
$431$	−7.93725	−	13.7477i	−0.382324	−	0.662205i	−0.991394	+	0.130912i	$0.958210\pi$
$432$		0			0
$433$	4.24264			0.203888			0.101944	−	0.994790i	$-0.467494\pi$
$433$	4.24264			0.203888			0.101944	+	0.994790i	$0.467494\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.48331	−	12.9615i	−0.357975	−	0.620032i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.64575	−	4.58258i	0.125703	−	0.217725i	−0.796304	−	0.604896i	$-0.793215\pi$
$443$	2.64575	−	4.58258i	0.125703	−	0.217725i	0.922008	+	0.387172i	$0.126548\pi$
$444$		0			0
$445$	7.00000	+	12.1244i	0.331832	+	0.574750i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	−9.89949	−	17.1464i	−0.466149	−	0.807394i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.00000	+	15.5885i	−0.421002	+	0.729197i	−0.996038	−	0.0889312i	$-0.971655\pi$
$457$	−9.00000	+	15.5885i	−0.421002	+	0.729197i	0.575036	+	0.818128i	$0.304988\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.74166			−0.174266			−0.0871332	−	0.996197i	$-0.527771\pi$
$461$	−3.74166			−0.174266			−0.0871332	+	0.996197i	$0.527771\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.7083	+	32.4037i	−0.865716	+	1.49946i	0.000617928		1.00000i	$0.499803\pi$
$467$	−18.7083	+	32.4037i	−0.865716	+	1.49946i	−0.866334	+	0.499465i	$0.833530\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21.1660	−	36.6606i	−0.973214	−	1.68566i
$474$		0			0
$475$	−25.4558			−1.16799
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.7083	−	32.4037i	−0.854803	−	1.48056i	−0.876827	−	0.480806i	$-0.840344\pi$
$479$	−18.7083	−	32.4037i	−0.854803	−	1.48056i	0.0220238	−	0.999757i	$-0.492989\pi$
$480$		0			0
$481$	−8.48528	+	14.6969i	−0.386896	+	0.670123i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.5203	+	32.0780i	−0.840962	+	1.45659i
$486$		0			0
$487$	6.00000	+	10.3923i	0.271886	+	0.470920i	0.969345	−	0.245705i	$-0.0790193\pi$
$487$	6.00000	+	10.3923i	0.271886	+	0.470920i	−0.697459	+	0.716625i	$0.745686\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.8745			−0.716407			−0.358203	−	0.933644i	$-0.616611\pi$
$491$	−15.8745			−0.716407			−0.358203	+	0.933644i	$0.616611\pi$
$492$		0			0
$493$	9.89949	+	17.1464i	0.445851	+	0.772236i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	0.246214	+	0.969216i	$0.420813\pi$
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.9666			−0.667329			−0.333665	−	0.942692i	$-0.608285\pi$
$503$	−14.9666			−0.667329			−0.333665	+	0.942692i	$0.608285\pi$
$504$		0			0
$505$	−42.0000			−1.86898
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−20.5791	+	35.6441i	−0.912153	+	1.57990i	−0.101137	+	0.994873i	$0.532248\pi$
$509$	−20.5791	+	35.6441i	−0.912153	+	1.57990i	−0.811017	+	0.585023i	$0.801085\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.29150	−	9.16515i	−0.233171	−	0.403865i
$516$		0			0
$517$	−39.5980			−1.74152
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.35414	+	16.2019i	0.409812	+	0.709816i	0.994868	−	0.101177i	$-0.0322608\pi$
$521$	9.35414	+	16.2019i	0.409812	+	0.709816i	−0.585056	+	0.810993i	$0.698927\pi$
$522$		0			0
$523$	−8.48528	+	14.6969i	−0.371035	+	0.642652i	−0.989725	−	0.142983i	$-0.954331\pi$
$523$	−8.48528	+	14.6969i	−0.371035	+	0.642652i	0.618690	+	0.785635i	$0.287664\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.8745	−	27.4955i	0.691504	−	1.19772i
$528$		0			0
$529$	−2.50000	−	4.33013i	−0.108696	−	0.188266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	15.8745			0.687601
$534$		0			0
$535$	9.89949	+	17.1464i	0.427992	+	0.741305i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.00000	+	8.66025i	−0.214967	+	0.372333i	−0.953262	−	0.302144i	$-0.902298\pi$
$541$	−5.00000	+	8.66025i	−0.214967	+	0.372333i	0.738296	+	0.674477i	$0.235631\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	59.8665			2.56440
$546$		0			0
$547$	16.0000			0.684111			0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000			0.684111			0.342055	+	0.939680i	$0.388877\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.48331	+	12.9615i	−0.318800	+	0.552178i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.8745	−	27.4955i	−0.672624	−	1.16502i	−0.977157	−	0.212518i	$-0.931834\pi$
$557$	−15.8745	−	27.4955i	−0.672624	−	1.16502i	0.304533	−	0.952502i	$-0.401500\pi$
$558$		0			0
$559$	33.9411			1.43556
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.74166	+	6.48074i	0.157692	+	0.273131i	0.934036	−	0.357179i	$-0.116261\pi$
$563$	3.74166	+	6.48074i	0.157692	+	0.273131i	−0.776344	+	0.630309i	$0.782928\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.2288	−	22.9129i	0.554578	−	0.960558i	−0.443358	−	0.896345i	$-0.646213\pi$
$569$	13.2288	−	22.9129i	0.554578	−	0.960558i	0.997936	−	0.0642132i	$-0.0204538\pi$
$570$		0			0
$571$	−20.0000	−	34.6410i	−0.836974	−	1.44968i	−0.892413	−	0.451219i	$-0.850989\pi$
$571$	−20.0000	−	34.6410i	−0.836974	−	1.44968i	0.0554391	−	0.998462i	$-0.482344\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−47.6235			−1.98604
$576$		0			0
$577$	−10.6066	−	18.3712i	−0.441559	−	0.764802i	0.556247	−	0.831017i	$-0.312241\pi$
$577$	−10.6066	−	18.3712i	−0.441559	−	0.764802i	−0.997805	+	0.0662152i	$0.978908\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	28.0000	−	48.4974i	1.15964	−	2.00856i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.4499			−0.926608			−0.463304	−	0.886199i	$-0.653336\pi$
$587$	−22.4499			−0.926608			−0.463304	+	0.886199i	$0.653336\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.0958	−	22.6826i	0.537780	−	0.931462i	−0.461243	−	0.887274i	$-0.652596\pi$
$593$	13.0958	−	22.6826i	0.537780	−	0.931462i	0.999023	−	0.0441886i	$-0.0140703\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.64575	+	4.58258i	0.108102	+	0.187239i	0.915002	−	0.403450i	$-0.132189\pi$
$599$	2.64575	+	4.58258i	0.108102	+	0.187239i	−0.806899	+	0.590689i	$0.798856\pi$
$600$		0			0
$601$	−4.24264			−0.173061			−0.0865305	−	0.996249i	$-0.527578\pi$
$601$	−4.24264			−0.173061			−0.0865305	+	0.996249i	$0.527578\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31.8041	−	55.0863i	−1.29302	−	2.23958i
$606$		0			0
$607$	8.48528	−	14.6969i	0.344407	−	0.596530i	−0.640839	−	0.767675i	$-0.721413\pi$
$607$	8.48528	−	14.6969i	0.344407	−	0.596530i	0.985246	+	0.171145i	$0.0547467\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.8745	−	27.4955i	0.642214	−	1.11235i
$612$		0			0
$613$	12.0000	+	20.7846i	0.484675	+	0.839482i	0.999845	−	0.0176058i	$-0.00560439\pi$
$613$	12.0000	+	20.7846i	0.484675	+	0.839482i	−0.515170	+	0.857088i	$0.672271\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.4575			−1.06514			−0.532570	−	0.846386i	$-0.678774\pi$
$617$	−26.4575			−1.06514			−0.532570	+	0.846386i	$0.678774\pi$
$618$		0			0
$619$	5.65685	+	9.79796i	0.227368	+	0.393813i	0.957027	−	0.289998i	$-0.0936546\pi$
$619$	5.65685	+	9.79796i	0.227368	+	0.393813i	−0.729659	+	0.683811i	$0.760321\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	14.9666			0.596759
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.9666	−	25.9230i	0.593933	−	1.02872i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.5203	+	32.0780i	0.731506	+	1.26701i	0.956239	+	0.292586i	$0.0945157\pi$
$641$	18.5203	+	32.0780i	0.731506	+	1.26701i	−0.224733	+	0.974420i	$0.572151\pi$
$642$		0			0
$643$	14.1421			0.557711			0.278856	−	0.960333i	$-0.410045\pi$
$643$	14.1421			0.557711			0.278856	+	0.960333i	$0.410045\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.2250	+	19.4422i	0.441299	+	0.764353i	0.997786	−	0.0665037i	$-0.0211844\pi$
$647$	11.2250	+	19.4422i	0.441299	+	0.764353i	−0.556487	+	0.830856i	$0.687851\pi$
$648$		0			0
$649$	19.7990	−	34.2929i	0.777178	−	1.34611i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.93725	+	13.7477i	−0.310609	+	0.537990i	−0.978494	−	0.206274i	$-0.933866\pi$
$653$	−7.93725	+	13.7477i	−0.310609	+	0.537990i	0.667886	+	0.744264i	$0.267200\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	47.6235			1.85515			0.927575	−	0.373638i	$-0.121890\pi$
$659$	47.6235			1.85515			0.927575	+	0.373638i	$0.121890\pi$
$660$		0			0
$661$	−10.6066	−	18.3712i	−0.412549	−	0.714556i	0.582619	−	0.812746i	$-0.302028\pi$
$661$	−10.6066	−	18.3712i	−0.412549	−	0.714556i	−0.995168	+	0.0981898i	$0.968695\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−14.0000	+	24.2487i	−0.542082	+	0.938914i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−52.3832			−2.02223
$672$		0			0
$673$	36.0000			1.38770			0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000			1.38770			0.693849	+	0.720121i	$0.255914\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.8375	−	29.1633i	0.647116	−	1.12084i	−0.336692	−	0.941615i	$-0.609308\pi$
$677$	16.8375	−	29.1633i	0.647116	−	1.12084i	0.983808	−	0.179223i	$-0.0573584\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.93725	−	13.7477i	−0.303711	−	0.526042i	0.673263	−	0.739403i	$-0.264892\pi$
$683$	−7.93725	−	13.7477i	−0.303711	−	0.526042i	−0.976973	+	0.213361i	$0.931559\pi$
$684$		0			0
$685$	59.3970			2.26944
$686$		0			0
$687$		0			0
$688$		0			0
$689$	22.4499	+	38.8844i	0.855275	+	1.48138i
$690$		0			0
$691$	−8.48528	+	14.6969i	−0.322795	+	0.559098i	−0.981064	−	0.193685i	$-0.937956\pi$
$691$	−8.48528	+	14.6969i	−0.322795	+	0.559098i	0.658268	+	0.752783i	$0.271289\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.5830	+	18.3303i	−0.401436	+	0.695308i
$696$		0			0
$697$	−7.00000	−	12.1244i	−0.265144	−	0.459243i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.8745			−0.599572			−0.299786	−	0.954006i	$-0.596915\pi$
$701$	−15.8745			−0.599572			−0.299786	+	0.954006i	$0.596915\pi$
$702$		0			0
$703$	5.65685	+	9.79796i	0.213352	+	0.369537i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.00000	−	6.92820i	0.150223	−	0.260194i	−0.781086	−	0.624423i	$-0.785334\pi$
$709$	4.00000	−	6.92820i	0.150223	−	0.260194i	0.931309	+	0.364229i	$0.118667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	44.8999			1.68151
$714$		0			0
$715$	84.0000			3.14142
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.9666	−	25.9230i	0.558161	−	0.966763i	−0.439489	−	0.898248i	$-0.644840\pi$
$719$	14.9666	−	25.9230i	0.558161	−	0.966763i	0.997650	−	0.0685154i	$-0.0218262\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	23.8118	+	41.2432i	0.884347	+	1.53173i
$726$		0			0
$727$	14.1421			0.524503			0.262251	−	0.965000i	$-0.415535\pi$
$727$	14.1421			0.524503			0.262251	+	0.965000i	$0.415535\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.9666	−	25.9230i	−0.553561	−	0.958795i
$732$		0			0
$733$	−23.3345	+	40.4166i	−0.861880	+	1.49282i	0.00823152	+	0.999966i	$0.497380\pi$
$733$	−23.3345	+	40.4166i	−0.861880	+	1.49282i	−0.870112	+	0.492854i	$0.835954\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31.7490	+	54.9909i	−1.16949	+	2.02562i
$738$		0			0
$739$	−6.00000	−	10.3923i	−0.220714	−	0.382287i	0.734311	−	0.678813i	$-0.237505\pi$
$739$	−6.00000	−	10.3923i	−0.220714	−	0.382287i	−0.955025	+	0.296526i	$0.904172\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	37.0405			1.35888			0.679442	−	0.733729i	$-0.262222\pi$
$743$	37.0405			1.35888			0.679442	+	0.733729i	$0.262222\pi$
$744$		0			0
$745$	19.7990	+	34.2929i	0.725379	+	1.25639i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	0.411170	+	0.911559i	$0.365120\pi$
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	−0.995018	+	0.0996961i	$0.968213\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−14.9666			−0.544691
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20.5791	+	35.6441i	−0.745992	+	1.29210i	0.203737	+	0.979026i	$0.434691\pi$
$761$	−20.5791	+	35.6441i	−0.745992	+	1.29210i	−0.949730	+	0.313071i	$0.898642\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	15.8745	+	27.4955i	0.573195	+	0.992803i
$768$		0			0
$769$	4.24264			0.152994			0.0764968	−	0.997070i	$-0.475627\pi$
$769$	4.24264			0.152994			0.0764968	+	0.997070i	$0.475627\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.3208	+	42.1248i	0.874757	+	1.51512i	0.857021	+	0.515282i	$0.172313\pi$
$773$	24.3208	+	42.1248i	0.874757	+	1.51512i	0.0177365	+	0.999843i	$0.494354\pi$
$774$		0			0
$775$	38.1838	−	66.1362i	1.37160	−	2.37568i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.29150	−	9.16515i	0.189588	−	0.328376i
$780$		0			0
$781$	−42.0000	−	72.7461i	−1.50288	−	2.60306i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	37.0405			1.32203
$786$		0			0
$787$	19.7990	+	34.2929i	0.705758	+	1.22241i	0.966417	+	0.256978i	$0.0827266\pi$
$787$	19.7990	+	34.2929i	0.705758	+	1.22241i	−0.260660	+	0.965431i	$0.583940\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	21.0000	−	36.3731i	0.745732	−	1.29165i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.2250			−0.397609			−0.198804	−	0.980039i	$-0.563706\pi$
$797$	−11.2250			−0.397609			−0.198804	+	0.980039i	$0.563706\pi$
$798$		0			0
$799$	−28.0000			−0.990569
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.74166	−	6.48074i	0.132040	−	0.228700i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10.5830	+	18.3303i	0.372079	+	0.644459i	0.989885	−	0.141871i	$-0.0453118\pi$
$809$	10.5830	+	18.3303i	0.372079	+	0.644459i	−0.617806	+	0.786330i	$0.711978\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	37.4166	+	64.8074i	1.31065	+	2.27010i
$816$		0			0
$817$	11.3137	−	19.5959i	0.395817	−	0.685574i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.8745	+	27.4955i	−0.554024	+	0.959598i	0.443955	+	0.896049i	$0.353575\pi$
$821$	−15.8745	+	27.4955i	−0.554024	+	0.959598i	−0.997979	+	0.0635487i	$0.979758\pi$
$822$		0			0
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	0.829060	−	0.559159i	$-0.188876\pi$
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	−0.898776	+	0.438408i	$0.855543\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.8745			0.552011			0.276005	−	0.961156i	$-0.410989\pi$
$827$	15.8745			0.552011			0.276005	+	0.961156i	$0.410989\pi$
$828$		0			0
$829$	9.19239	+	15.9217i	0.319265	+	0.552983i	0.980335	−	0.197341i	$-0.0632307\pi$
$829$	9.19239	+	15.9217i	0.319265	+	0.552983i	−0.661070	+	0.750324i	$0.729897\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−14.0000	+	24.2487i	−0.484490	+	0.839161i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.48331			−0.258353			−0.129176	−	0.991622i	$-0.541233\pi$
$839$	−7.48331			−0.258353			−0.129176	+	0.991622i	$0.541233\pi$
$840$		0			0
$841$	−1.00000			−0.0344828
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.35414	+	16.2019i	−0.321792	+	0.557361i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.5830	+	18.3303i	0.362781	+	0.628355i
$852$		0			0
$853$	55.1543			1.88845			0.944224	−	0.329304i	$-0.106814\pi$
$853$	55.1543			1.88845			0.944224	+	0.329304i	$0.106814\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.5791	−	35.6441i	−0.702969	−	1.21758i	−0.967419	−	0.253179i	$-0.918524\pi$
$857$	−20.5791	−	35.6441i	−0.702969	−	1.21758i	0.264450	−	0.964399i	$-0.414810\pi$
$858$		0			0
$859$	−9.89949	+	17.1464i	−0.337766	+	0.585029i	−0.984012	−	0.178101i	$-0.943005\pi$
$859$	−9.89949	+	17.1464i	−0.337766	+	0.585029i	0.646246	+	0.763129i	$0.276338\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.64575	−	4.58258i	0.0900624	−	0.155993i	−0.817475	−	0.575964i	$-0.804627\pi$
$863$	2.64575	−	4.58258i	0.0900624	−	0.155993i	0.907537	+	0.419972i	$0.137960\pi$
$864$		0			0
$865$	−21.0000	−	36.3731i	−0.714021	−	1.23672i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.1660			−0.718008
$870$		0			0
$871$	−25.4558	−	44.0908i	−0.862538	−	1.49396i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.00000	+	6.92820i	−0.135070	+	0.233949i	−0.925624	−	0.378444i	$-0.876459\pi$
$877$	−4.00000	+	6.92820i	−0.135070	+	0.233949i	0.790554	+	0.612392i	$0.209793\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−56.1249			−1.89089			−0.945447	−	0.325775i	$-0.894375\pi$
$881$	−56.1249			−1.89089			−0.945447	+	0.325775i	$0.894375\pi$
$882$		0			0
$883$	8.00000			0.269221			0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.7083	−	32.4037i	0.628163	−	1.08801i	−0.359757	−	0.933046i	$-0.617141\pi$
$887$	18.7083	−	32.4037i	0.628163	−	1.08801i	0.987920	−	0.154964i	$-0.0495261\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.5830	−	18.3303i	−0.354147	−	0.613400i
$894$		0			0
$895$	59.3970			1.98542
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.4499	−	38.8844i	−0.748748	−	1.29687i
$900$		0			0
$901$	19.7990	−	34.2929i	0.659600	−	1.14246i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.93725	−	13.7477i	0.263843	−	0.456990i
$906$		0			0
$907$	−8.00000	−	13.8564i	−0.265636	−	0.460094i	0.702094	−	0.712084i	$-0.252248\pi$
$907$	−8.00000	−	13.8564i	−0.265636	−	0.460094i	−0.967730	+	0.251990i	$0.918915\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.29150			−0.175315			−0.0876577	−	0.996151i	$-0.527938\pi$
$911$	−5.29150			−0.175315			−0.0876577	+	0.996151i	$0.527938\pi$
$912$		0			0
$913$	39.5980	+	68.5857i	1.31050	+	2.26986i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.00000	+	3.46410i	−0.0659739	+	0.114270i	−0.897126	−	0.441776i	$-0.854349\pi$
$919$	−2.00000	+	3.46410i	−0.0659739	+	0.114270i	0.831152	+	0.556046i	$0.187682\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	67.3498			2.21685
$924$		0			0
$925$	36.0000			1.18367
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.35414	−	16.2019i	0.306899	−	0.531566i	−0.670783	−	0.741654i	$-0.734042\pi$
$929$	9.35414	−	16.2019i	0.306899	−	0.531566i	0.977682	+	0.210088i	$0.0673751\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−37.0405	−	64.1561i	−1.21135	−	2.09813i
$936$		0			0
$937$	35.3553			1.15501			0.577504	−	0.816388i	$-0.304027\pi$
$937$	35.3553			1.15501			0.577504	+	0.816388i	$0.304027\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.0958	−	22.6826i	−0.426911	−	0.739431i	0.569686	−	0.821863i	$-0.307065\pi$
$941$	−13.0958	−	22.6826i	−0.426911	−	0.739431i	−0.996597	+	0.0824312i	$0.973732\pi$
$942$		0			0
$943$	9.89949	−	17.1464i	0.322372	−	0.558365i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.8118	−	41.2432i	0.773778	−	1.34022i	−0.161700	−	0.986840i	$-0.551698\pi$
$947$	23.8118	−	41.2432i	0.773778	−	1.34022i	0.935479	−	0.353383i	$-0.114969\pi$
$948$		0			0
$949$	3.00000	+	5.19615i	0.0973841	+	0.168674i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−42.3320			−1.37127			−0.685634	−	0.727946i	$-0.740475\pi$
$953$	−42.3320			−1.37127			−0.685634	+	0.727946i	$0.740475\pi$
$954$		0			0
$955$	9.89949	+	17.1464i	0.320340	+	0.554845i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.5000	+	35.5070i	−0.661290	+	1.14539i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.48331			0.240896
$966$		0			0
$967$	48.0000			1.54358			0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000			1.54358			0.771788	+	0.635880i	$0.219363\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.48331	+	12.9615i	−0.240151	+	0.415954i	−0.960757	−	0.277391i	$-0.910530\pi$
$971$	−7.48331	+	12.9615i	−0.240151	+	0.415954i	0.720606	+	0.693345i	$0.243864\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.93725	−	13.7477i	−0.253935	−	0.439829i	0.710671	−	0.703525i	$-0.248392\pi$
$977$	−7.93725	−	13.7477i	−0.253935	−	0.439829i	−0.964606	+	0.263696i	$0.915058\pi$
$978$		0			0
$979$	−19.7990			−0.632778
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.48331	+	12.9615i	0.238681	+	0.413407i	0.960336	−	0.278846i	$-0.0899517\pi$
$983$	7.48331	+	12.9615i	0.238681	+	0.413407i	−0.721655	+	0.692253i	$0.756618\pi$
$984$		0			0
$985$	−19.7990	+	34.2929i	−0.630848	+	1.09266i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.1660	−	36.6606i	0.673040	−	1.16574i
$990$		0			0
$991$	−22.0000	−	38.1051i	−0.698853	−	1.21045i	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−22.0000	−	38.1051i	−0.698853	−	1.21045i	0.270011	−	0.962857i	$-0.412973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.707107	+	1.22474i	0.0223943	+	0.0387881i	0.877005	−	0.480481i	$-0.159538\pi$
$997$	0.707107	+	1.22474i	0.0223943	+	0.0387881i	−0.854611	+	0.519269i	$0.826204\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.k.m.1549.1		8
3.2	odd	2	inner	1764.2.k.m.1549.4		8
7.2	even	3		1764.2.a.m.1.4	yes	4
7.3	odd	6	inner	1764.2.k.m.361.3		8
7.4	even	3	inner	1764.2.k.m.361.1		8
7.5	odd	6		1764.2.a.m.1.2	yes	4
7.6	odd	2	inner	1764.2.k.m.1549.3		8
21.2	odd	6		1764.2.a.m.1.1	✓	4
21.5	even	6		1764.2.a.m.1.3	yes	4
21.11	odd	6	inner	1764.2.k.m.361.4		8
21.17	even	6	inner	1764.2.k.m.361.2		8
21.20	even	2	inner	1764.2.k.m.1549.2		8
28.19	even	6		7056.2.a.cy.1.1		4
28.23	odd	6		7056.2.a.cy.1.3		4
84.23	even	6		7056.2.a.cy.1.2		4
84.47	odd	6		7056.2.a.cy.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.a.m.1.1	✓	4	21.2	odd	6
1764.2.a.m.1.2	yes	4	7.5	odd	6
1764.2.a.m.1.3	yes	4	21.5	even	6
1764.2.a.m.1.4	yes	4	7.2	even	3
1764.2.k.m.361.1		8	7.4	even	3	inner
1764.2.k.m.361.2		8	21.17	even	6	inner
1764.2.k.m.361.3		8	7.3	odd	6	inner
1764.2.k.m.361.4		8	21.11	odd	6	inner
1764.2.k.m.1549.1		8	1.1	even	1	trivial
1764.2.k.m.1549.2		8	21.20	even	2	inner
1764.2.k.m.1549.3		8	7.6	odd	2	inner
1764.2.k.m.1549.4		8	3.2	odd	2	inner
7056.2.a.cy.1.1		4	28.19	even	6
7056.2.a.cy.1.2		4	84.23	even	6
7056.2.a.cy.1.3		4	28.23	odd	6
7056.2.a.cy.1.4		4	84.47	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.87083 + 3.24037i) q^{5} +O(q^{10})\)	\(q+(-1.87083 + 3.24037i) q^{5} +(-2.64575 - 4.58258i) q^{11} +4.24264 q^{13} +(-1.87083 - 3.24037i) q^{17} +(1.41421 - 2.44949i) q^{19} +(2.64575 - 4.58258i) q^{23} +(-4.50000 - 7.79423i) q^{25} -5.29150 q^{29} +(4.24264 + 7.34847i) q^{31} +(-2.00000 + 3.46410i) q^{37} +3.74166 q^{41} +8.00000 q^{43} +(3.74166 - 6.48074i) q^{47} +(5.29150 + 9.16515i) q^{53} +19.7990 q^{55} +(3.74166 + 6.48074i) q^{59} +(4.94975 - 8.57321i) q^{61} +(-7.93725 + 13.7477i) q^{65} +(-6.00000 - 10.3923i) q^{67} +15.8745 q^{71} +(0.707107 + 1.22474i) q^{73} +(2.00000 - 3.46410i) q^{79} -14.9666 q^{83} +14.0000 q^{85} +(1.87083 - 3.24037i) q^{89} +(5.29150 + 9.16515i) q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 36 q^{25} - 16 q^{37} + 64 q^{43} - 48 q^{67} + 16 q^{79} + 112 q^{85}+O(q^{100}) \) 8 * q - 36 * q^25 - 16 * q^37 + 64 * q^43 - 48 * q^67 + 16 * q^79 + 112 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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