LMFDB - Embedded newform 1764.2.k.m.1549.3

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			1549.3
Root			$1.28897 + 2.23256i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.2.k.m.361.3

$q$-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	$-0.517838\pi$
$5$	1.87083	−	3.24037i	0.836660	−	1.44914i	0.892672	−	0.450708i	$-0.148828\pi$
$6$		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	$-0.700222\pi$
$13$	−4.24264			−1.17670			−0.588348	+	0.808608i	$0.700222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.87083	+	3.24037i	0.453743	+	0.785905i	0.998615	−	0.0526138i	$-0.0167552\pi$
$17$	1.87083	+	3.24037i	0.453743	+	0.785905i	−0.544872	+	0.838519i	$0.683422\pi$
$18$		0			0
$19$	−1.41421	+	2.44949i	−0.324443	+	0.561951i	−0.981399	−	0.191977i	$-0.938510\pi$
$19$	−1.41421	+	2.44949i	−0.324443	+	0.561951i	0.656957	+	0.753928i	$0.271843\pi$
$20$		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	$-0.890884\pi$
$31$	−4.24264	−	7.34847i	−0.762001	−	1.31982i	0.179817	−	0.983700i	$-0.442449\pi$
$32$		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	$-0.594379\pi$
$41$	−3.74166			−0.584349			−0.292174	+	0.956365i	$0.594379\pi$
$42$		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	$0.350432\pi$
$47$	−3.74166	+	6.48074i	−0.545777	+	0.945313i	−0.998558	+	0.0536913i	$0.982901\pi$
$48$		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	0.512768	−	0.858527i	$-0.328620\pi$
$59$	−3.74166	−	6.48074i	−0.487122	−	0.843721i	−0.999890	+	0.0148066i	$0.995287\pi$
$60$		0			0
$61$	−4.94975	+	8.57321i	−0.633750	+	1.09769i	0.353028	+	0.935613i	$0.385152\pi$
$61$	−4.94975	+	8.57321i	−0.633750	+	1.09769i	−0.986778	+	0.162075i	$0.948181\pi$
$62$		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	0.821674	−	0.569958i	$-0.193040\pi$
$73$	−0.707107	−	1.22474i	−0.0827606	−	0.143346i	−0.904435	+	0.426612i	$0.859707\pi$
$74$		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	$-0.193193\pi$
$83$	14.9666			1.64280			0.821401	+	0.570352i	$0.193193\pi$
$84$		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	−0.947980	−	0.318331i	$-0.896878\pi$
$89$	−1.87083	+	3.24037i	−0.198307	+	0.343479i	0.749672	+	0.661809i	$0.230211\pi$
$90$		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	$-0.667612\pi$
$97$	−9.89949			−1.00514			−0.502571	+	0.864536i	$0.667612\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.61249	−	9.72111i	−0.558463	−	0.967287i	−0.997625	−	0.0688787i	$-0.978058\pi$
$101$	−5.61249	−	9.72111i	−0.558463	−	0.967287i	0.439162	−	0.898408i	$-0.355275\pi$
$102$		0			0
$103$	1.41421	−	2.44949i	0.139347	−	0.241355i	−0.787903	−	0.615800i	$-0.788833\pi$
$103$	1.41421	−	2.44949i	0.139347	−	0.241355i	0.927249	+	0.374444i	$0.122166\pi$
$104$		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	$-0.577116\pi$
$139$	−5.65685			−0.479808			−0.239904	+	0.970797i	$0.577116\pi$
$140$		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	0.993105	−	0.117225i	$-0.0373999\pi$
$157$	4.94975	+	8.57321i	0.395033	+	0.684217i	−0.598073	+	0.801442i	$0.704067\pi$
$158$		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	$-0.593502\pi$
$167$	−7.48331			−0.579076			−0.289538	+	0.957166i	$0.593502\pi$
$168$		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	−0.569869	−	0.821736i	$-0.693006\pi$
$173$	5.61249	−	9.72111i	0.426709	−	0.739082i	0.996578	+	0.0826532i	$0.0263394\pi$
$174$		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	$-0.449600\pi$
$181$	4.24264			0.315353			0.157676	+	0.987491i	$0.449600\pi$
$182$		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	$-0.899103\pi$
$227$	−11.2250	−	19.4422i	−0.745028	−	1.29043i	0.205154	−	0.978730i	$-0.434230\pi$
$228$		0			0
$229$	6.36396	−	11.0227i	0.420542	−	0.728401i	−0.575450	−	0.817837i	$-0.695173\pi$
$229$	6.36396	−	11.0227i	0.420542	−	0.728401i	0.995993	+	0.0894361i	$0.0285065\pi$
$230$		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	$0.239128\pi$
$241$	14.8492	+	25.7196i	0.956524	+	1.65675i	0.225682	+	0.974201i	$0.427539\pi$
$242$		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	$-0.575893\pi$
$251$	−7.48331			−0.472343			−0.236171	+	0.971711i	$0.575893\pi$
$252$		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	−0.986266	−	0.165164i	$-0.947185\pi$
$257$	−5.61249	+	9.72111i	−0.350097	+	0.606386i	0.636169	+	0.771550i	$0.280518\pi$
$258$		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	0.803340	−	0.595521i	$-0.203054\pi$
$269$	−1.87083	−	3.24037i	−0.114066	−	0.197569i	−0.917406	+	0.397952i	$0.869721\pi$
$270$		0			0
$271$	−9.89949	+	17.1464i	−0.601351	+	1.04157i	0.391265	+	0.920278i	$0.372038\pi$
$271$	−9.89949	+	17.1464i	−0.601351	+	1.04157i	−0.992617	+	0.121293i	$0.961296\pi$
$272$		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	$-0.790958\pi$
$283$	−15.5563	−	26.9444i	−0.924729	−	1.60168i	−0.132733	−	0.991152i	$-0.542375\pi$
$284$		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	$-0.222707\pi$
$293$	26.1916			1.53013			0.765065	+	0.643953i	$0.222707\pi$
$294$		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	$-0.367770\pi$
$307$	14.1421			0.807134			0.403567	+	0.914950i	$0.367770\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.74166	+	6.48074i	0.212170	+	0.367489i	0.952393	−	0.304872i	$-0.0986136\pi$
$311$	3.74166	+	6.48074i	0.212170	+	0.367489i	−0.740223	+	0.672361i	$0.765280\pi$
$312$		0			0
$313$	6.36396	−	11.0227i	0.359712	−	0.623040i	−0.628200	−	0.778052i	$-0.716208\pi$
$313$	6.36396	−	11.0227i	0.359712	−	0.623040i	0.987913	+	0.155012i	$0.0495415\pi$
$314$		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	$-0.207543\pi$
$349$	29.6985			1.58972			0.794862	+	0.606791i	$0.207543\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.35414	−	16.2019i	−0.497871	−	0.862338i	0.502126	−	0.864794i	$-0.332551\pi$
$353$	−9.35414	−	16.2019i	−0.497871	−	0.862338i	−0.999997	+	0.00245682i	$0.999218\pi$
$354$		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	0.975051	−	0.221980i	$-0.0712519\pi$
$367$	5.65685	+	9.79796i	0.295285	+	0.511449i	−0.679766	+	0.733429i	$0.737919\pi$
$368$		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	−0.991402	−	0.130852i	$-0.958229\pi$
$383$	−7.48331	+	12.9615i	−0.382380	+	0.662301i	0.609022	+	0.793153i	$0.291562\pi$
$384$		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	$0.470612\pi$
$397$	−16.2635	+	28.1691i	−0.816239	+	1.41377i	−0.908434	+	0.418027i	$0.862722\pi$
$398$		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	$0.00906711\pi$
$409$	10.6066	+	18.3712i	0.524463	+	0.908396i	−0.475132	+	0.879915i	$0.657600\pi$
$410$		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	$-0.441487\pi$
$419$	7.48331			0.365584			0.182792	+	0.983152i	$0.441487\pi$
$420$		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	$-0.532506\pi$
$433$	−4.24264			−0.203888			−0.101944	+	0.994790i	$0.532506\pi$
$434$		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	$-0.472229\pi$
$461$	3.74166			0.174266			0.0871332	+	0.996197i	$0.472229\pi$
$462$		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	$-0.500197\pi$
$467$	18.7083	−	32.4037i	0.865716	−	1.49946i	0.866334	−	0.499465i	$-0.166470\pi$
$468$		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	$0.159656\pi$
$479$	18.7083	+	32.4037i	0.854803	+	1.48056i	−0.0220238	+	0.999757i	$0.507011\pi$
$480$		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	$-0.391715\pi$
$503$	14.9666			0.667329			0.333665	+	0.942692i	$0.391715\pi$
$504$		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	$-0.467752\pi$
$509$	20.5791	−	35.6441i	0.912153	−	1.57990i	0.811017	−	0.585023i	$-0.198915\pi$
$510$		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	0.585056	−	0.810993i	$-0.301073\pi$
$521$	−9.35414	−	16.2019i	−0.409812	−	0.709816i	−0.994868	+	0.101177i	$0.967739\pi$
$522$		0			0
$523$	8.48528	−	14.6969i	0.371035	−	0.642652i	−0.618690	−	0.785635i	$-0.712336\pi$
$523$	8.48528	−	14.6969i	0.371035	−	0.642652i	0.989725	+	0.142983i	$0.0456695\pi$
$524$		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	0.776344	−	0.630309i	$-0.217072\pi$
$563$	−3.74166	−	6.48074i	−0.157692	−	0.273131i	−0.934036	+	0.357179i	$0.883739\pi$
$564$		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	0.997805	−	0.0662152i	$-0.0210924\pi$
$577$	10.6066	+	18.3712i	0.441559	+	0.764802i	−0.556247	+	0.831017i	$0.687759\pi$
$578$		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	$-0.346664\pi$
$587$	22.4499			0.926608			0.463304	+	0.886199i	$0.346664\pi$
$588$		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	$0.347404\pi$
$593$	−13.0958	+	22.6826i	−0.537780	+	0.931462i	−0.999023	+	0.0441886i	$0.985930\pi$
$594$		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	$-0.472422\pi$
$601$	4.24264			0.173061			0.0865305	+	0.996249i	$0.472422\pi$
$602$		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	−0.985246	−	0.171145i	$-0.945253\pi$
$607$	−8.48528	+	14.6969i	−0.344407	+	0.596530i	0.640839	+	0.767675i	$0.278587\pi$
$608$		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	0.729659	−	0.683811i	$-0.239679\pi$
$619$	−5.65685	−	9.79796i	−0.227368	−	0.393813i	−0.957027	+	0.289998i	$0.906345\pi$
$620$		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	$-0.589955\pi$
$643$	−14.1421			−0.557711			−0.278856	+	0.960333i	$0.589955\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.2250	−	19.4422i	−0.441299	−	0.764353i	0.556487	−	0.830856i	$-0.312149\pi$
$647$	−11.2250	−	19.4422i	−0.441299	−	0.764353i	−0.997786	+	0.0665037i	$0.978816\pi$
$648$		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	0.995168	−	0.0981898i	$-0.0313052\pi$
$661$	10.6066	+	18.3712i	0.412549	+	0.714556i	−0.582619	+	0.812746i	$0.697972\pi$
$662$		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	$0.390692\pi$
$677$	−16.8375	+	29.1633i	−0.647116	+	1.12084i	−0.983808	+	0.179223i	$0.942642\pi$
$678$		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	−0.658268	−	0.752783i	$-0.728711\pi$
$691$	8.48528	−	14.6969i	0.322795	−	0.559098i	0.981064	+	0.193685i	$0.0620441\pi$
$692$		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	$0.355160\pi$
$719$	−14.9666	+	25.9230i	−0.558161	+	0.966763i	−0.997650	+	0.0685154i	$0.978174\pi$
$720$		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	$-0.584465\pi$
$727$	−14.1421			−0.524503			−0.262251	+	0.965000i	$0.584465\pi$
$728$		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	$-0.502620\pi$
$733$	23.3345	−	40.4166i	0.861880	−	1.49282i	0.870112	−	0.492854i	$-0.164046\pi$
$734$		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	$-0.565309\pi$
$761$	20.5791	−	35.6441i	0.745992	−	1.29210i	0.949730	−	0.313071i	$-0.101358\pi$
$762$		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	$-0.524373\pi$
$769$	−4.24264			−0.152994			−0.0764968	+	0.997070i	$0.524373\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.3208	−	42.1248i	−0.874757	−	1.51512i	−0.857021	−	0.515282i	$-0.827687\pi$
$773$	−24.3208	−	42.1248i	−0.874757	−	1.51512i	−0.0177365	−	0.999843i	$-0.505646\pi$
$774$		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	$-0.917273\pi$
$787$	−19.7990	−	34.2929i	−0.705758	−	1.22241i	0.260660	−	0.965431i	$-0.416060\pi$
$788$		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	$-0.436294\pi$
$797$	11.2250			0.397609			0.198804	+	0.980039i	$0.436294\pi$
$798$		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	0.661070	−	0.750324i	$-0.270103\pi$
$829$	−9.19239	−	15.9217i	−0.319265	−	0.552983i	−0.980335	+	0.197341i	$0.936769\pi$
$830$		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	$-0.458767\pi$
$839$	7.48331			0.258353			0.129176	+	0.991622i	$0.458767\pi$
$840$		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	$-0.893186\pi$
$853$	−55.1543			−1.88845			−0.944224	+	0.329304i	$0.893186\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.5791	+	35.6441i	0.702969	+	1.21758i	0.967419	+	0.253179i	$0.0814762\pi$
$857$	20.5791	+	35.6441i	0.702969	+	1.21758i	−0.264450	+	0.964399i	$0.585190\pi$
$858$		0			0
$859$	9.89949	−	17.1464i	0.337766	−	0.585029i	−0.646246	−	0.763129i	$-0.723662\pi$
$859$	9.89949	−	17.1464i	0.337766	−	0.585029i	0.984012	+	0.178101i	$0.0569953\pi$
$860$		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	$-0.105625\pi$
$881$	56.1249			1.89089			0.945447	+	0.325775i	$0.105625\pi$
$882$		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	$0.382859\pi$
$887$	−18.7083	+	32.4037i	−0.628163	+	1.08801i	−0.987920	+	0.154964i	$0.950474\pi$
$888$		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	−0.977682	−	0.210088i	$-0.932625\pi$
$929$	−9.35414	+	16.2019i	−0.306899	+	0.531566i	0.670783	+	0.741654i	$0.265958\pi$
$930$		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	$-0.695973\pi$
$937$	−35.3553			−1.15501			−0.577504	+	0.816388i	$0.695973\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.0958	+	22.6826i	0.426911	+	0.739431i	0.996597	−	0.0824312i	$-0.0262685\pi$
$941$	13.0958	+	22.6826i	0.426911	+	0.739431i	−0.569686	+	0.821863i	$0.692935\pi$
$942$		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	−0.720606	−	0.693345i	$-0.756136\pi$
$971$	7.48331	−	12.9615i	0.240151	−	0.415954i	0.960757	+	0.277391i	$0.0894697\pi$
$972$		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	0.721655	−	0.692253i	$-0.243382\pi$
$983$	−7.48331	−	12.9615i	−0.238681	−	0.413407i	−0.960336	+	0.278846i	$0.910048\pi$
$984$		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	0.854611	−	0.519269i	$-0.173796\pi$
$997$	−0.707107	−	1.22474i	−0.0223943	−	0.0387881i	−0.877005	+	0.480481i	$0.840462\pi$
$998$		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.1549.3		8
3.2	odd	2	inner	1764.2.k.m.1549.2		8
7.2	even	3		1764.2.a.m.1.2	yes	4
7.3	odd	6	inner	1764.2.k.m.361.1		8
7.4	even	3	inner	1764.2.k.m.361.3		8
7.5	odd	6		1764.2.a.m.1.4	yes	4
7.6	odd	2	inner	1764.2.k.m.1549.1		8
21.2	odd	6		1764.2.a.m.1.3	yes	4
21.5	even	6		1764.2.a.m.1.1	✓	4
21.11	odd	6	inner	1764.2.k.m.361.2		8
21.17	even	6	inner	1764.2.k.m.361.4		8
21.20	even	2	inner	1764.2.k.m.1549.4		8
28.19	even	6		7056.2.a.cy.1.3		4
28.23	odd	6		7056.2.a.cy.1.1		4
84.23	even	6		7056.2.a.cy.1.4		4
84.47	odd	6		7056.2.a.cy.1.2		4

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

Related objects

Downloads

Learn more