LMFDB - Embedded newform 3789.1.du.a.1972.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3789,1,Mod(10,3789)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3789, base_ring=CyclotomicField(140))

chi = DirichletCharacter(H, H._module([0, 93]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3789.10");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3789 = 3^{2} \cdot 421 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3789.du (of order $140$, degree $48$, 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:	$1.89095733287$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{140})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{38} - x^{36} - x^{34} - x^{32} + x^{28} + x^{26} + x^{24} + x^{22} + x^{20} + \cdots + 1 $ x^48 + x^46 - x^38 - x^36 - x^34 - x^32 + x^28 + x^26 + x^24 + x^22 + x^20 - x^16 - x^14 - x^12 - x^10 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{140}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{140} - \cdots)$

Embedding invariants

Embedding label			1972.1
Root			$-0.834573 + 0.550897i$ of defining polynomial
Character	$\chi$	$=$	3789.1972
Dual form			3789.1.du.a.3349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.919528 + 0.393025i) q^{4} +(-0.00804330 - 0.0893684i) q^{7} +O(q^{10})$	$q+(-0.919528 + 0.393025i) q^{4} +(-0.00804330 - 0.0893684i) q^{7} +(-1.26597 + 0.200510i) q^{13} +(0.691063 - 0.722795i) q^{16} +(-0.487571 + 0.445659i) q^{19} +(-0.691063 - 0.722795i) q^{25} +(0.0425201 + 0.0790155i) q^{28} +(0.202174 - 1.49251i) q^{31} +(-0.130460 - 0.0328731i) q^{37} +(-1.69302 + 0.0379977i) q^{43} +(0.976008 - 0.177119i) q^{49} +(1.08529 - 0.681933i) q^{52} +(-0.0435159 - 1.93889i) q^{61} +(-0.351375 + 0.936235i) q^{64} +(-0.312745 - 0.430457i) q^{67} +(-0.809017 + 1.58779i) q^{73} +(0.273179 - 0.601423i) q^{76} +(-1.15348 - 1.20645i) q^{79} +(0.0281019 + 0.111525i) q^{91} +(-0.0790155 - 1.75942i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{25} + 6 q^{28} + 4 q^{31} + 12 q^{37} - 2 q^{43} - 6 q^{49} + 2 q^{52} + 8 q^{61} - 10 q^{67} - 12 q^{73} - 2 q^{76} + 4 q^{91}+O(q^{100}) $ 48 * q - 2 * q^13 - 2 * q^16 - 2 * q^19 + 2 * q^25 + 6 * q^28 + 4 * q^31 + 12 * q^37 - 2 * q^43 - 6 * q^49 + 2 * q^52 + 8 * q^61 - 10 * q^67 - 12 * q^73 - 2 * q^76 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3789\mathbb{Z}\right)^\times$.

$n$	$1685$	$2107$
$\chi(n)$	$1$	$e\left(\frac{131}{140}\right)$

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		−0.200589	−	0.979675i	$-0.564286\pi$
$2$		0			0		0.200589	+	0.979675i	$0.435714\pi$
$3$		0			0
$3$		0			0
$4$	−0.919528	+	0.393025i	−0.919528	+	0.393025i
$5$		0			0		0.393025	−	0.919528i	$-0.371429\pi$
$5$		0			0		−0.393025	+	0.919528i	$0.628571\pi$
$6$		0			0
$7$	−0.00804330	−	0.0893684i	−0.00804330	−	0.0893684i	0.990950	−	0.134233i	$-0.0428571\pi$
$7$	−0.00804330	−	0.0893684i	−0.00804330	−	0.0893684i	−0.998993	+	0.0448648i	$0.985714\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.512899	−	0.858449i	$-0.671429\pi$
$11$		0			0		0.512899	+	0.858449i	$0.328571\pi$
$12$		0			0
$13$	−1.26597	+	0.200510i	−1.26597	+	0.200510i	−0.753071	−	0.657939i	$-0.771429\pi$
$13$	−1.26597	+	0.200510i	−1.26597	+	0.200510i	−0.512899	+	0.858449i	$0.671429\pi$
$14$		0			0
$15$		0			0
$16$	0.691063	−	0.722795i	0.691063	−	0.722795i
$17$		0			0		−0.936235	−	0.351375i	$-0.885714\pi$
$17$		0			0		0.936235	+	0.351375i	$0.114286\pi$
$18$		0			0
$19$	−0.487571	+	0.445659i	−0.487571	+	0.445659i	−0.880596	−	0.473869i	$-0.842857\pi$
$19$	−0.487571	+	0.445659i	−0.487571	+	0.445659i	0.393025	+	0.919528i	$0.371429\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.0672690	−	0.997735i	$-0.521429\pi$
$23$		0			0		0.0672690	+	0.997735i	$0.478571\pi$
$24$		0			0
$25$	−0.691063	−	0.722795i	−0.691063	−	0.722795i
$26$		0			0
$27$		0			0
$28$	0.0425201	+	0.0790155i	0.0425201	+	0.0790155i
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	0.202174	−	1.49251i	0.202174	−	1.49251i	−0.550897	−	0.834573i	$-0.685714\pi$
$31$	0.202174	−	1.49251i	0.202174	−	1.49251i	0.753071	−	0.657939i	$-0.228571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.130460	−	0.0328731i	−0.130460	−	0.0328731i	0.178557	−	0.983930i	$-0.442857\pi$
$37$	−0.130460	−	0.0328731i	−0.130460	−	0.0328731i	−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.738119	−	0.674671i	$-0.235714\pi$
$41$		0			0		−0.738119	+	0.674671i	$0.764286\pi$
$42$		0			0
$43$	−1.69302	+	0.0379977i	−1.69302	+	0.0379977i	−0.858449	−	0.512899i	$-0.828571\pi$
$43$	−1.69302	+	0.0379977i	−1.69302	+	0.0379977i	−0.834573	+	0.550897i	$0.814286\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.372294	−	0.928115i	$-0.621429\pi$
$47$		0			0		0.372294	+	0.928115i	$0.378571\pi$
$48$		0			0
$49$	0.976008	−	0.177119i	0.976008	−	0.177119i
$50$		0			0
$51$		0			0
$52$	1.08529	−	0.681933i	1.08529	−	0.681933i
$53$		0			0		0.640876	−	0.767645i	$-0.278571\pi$
$53$		0			0		−0.640876	+	0.767645i	$0.721429\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.493508	−	0.869741i	$-0.664286\pi$
$59$		0			0		0.493508	+	0.869741i	$0.335714\pi$
$60$		0			0
$61$	−0.0435159	−	1.93889i	−0.0435159	−	1.93889i	−0.266037	−	0.963963i	$-0.585714\pi$
$61$	−0.0435159	−	1.93889i	−0.0435159	−	1.93889i	0.222521	−	0.974928i	$-0.428571\pi$
$62$		0			0
$63$		0			0
$64$	−0.351375	+	0.936235i	−0.351375	+	0.936235i
$65$		0			0
$66$		0			0
$67$	−0.312745	−	0.430457i	−0.312745	−	0.430457i	0.623490	−	0.781831i	$-0.285714\pi$
$67$	−0.312745	−	0.430457i	−0.312745	−	0.430457i	−0.936235	+	0.351375i	$0.885714\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.493508	−	0.869741i	$-0.335714\pi$
$71$		0			0		−0.493508	+	0.869741i	$0.664286\pi$
$72$		0			0
$73$	−0.809017	+	1.58779i	−0.809017	+	1.58779i			1.00000i	$0.5\pi$
$73$	−0.809017	+	1.58779i	−0.809017	+	1.58779i	−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$		0			0
$76$	0.273179	−	0.601423i	0.273179	−	0.601423i
$77$		0			0
$78$		0			0
$79$	−1.15348	−	1.20645i	−1.15348	−	1.20645i	−0.974928	−	0.222521i	$-0.928571\pi$
$79$	−1.15348	−	1.20645i	−1.15348	−	1.20645i	−0.178557	−	0.983930i	$-0.557143\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.869741	−	0.493508i	$-0.164286\pi$
$83$		0			0		−0.869741	+	0.493508i	$0.835714\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.795625	−	0.605790i	$-0.792857\pi$
$89$		0			0		0.795625	+	0.605790i	$0.207143\pi$
$90$		0			0
$91$	0.0281019	+	0.111525i	0.0281019	+	0.111525i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.0790155	−	1.75942i	−0.0790155	−	1.75942i	−0.512899	−	0.858449i	$-0.671429\pi$
$97$	−0.0790155	−	1.75942i	−0.0790155	−	1.75942i	0.433884	−	0.900969i	$-0.357143\pi$
$98$		0			0
$99$		0			0
$100$	0.919528	+	0.393025i	0.919528	+	0.393025i
$101$		0			0		0.691063	−	0.722795i	$-0.257143\pi$
$101$		0			0		−0.691063	+	0.722795i	$0.742857\pi$
$102$		0			0
$103$	0.340473	−	0.515795i	0.340473	−	0.515795i	−0.623490	−	0.781831i	$-0.714286\pi$
$103$	0.340473	−	0.515795i	0.340473	−	0.515795i	0.963963	+	0.266037i	$0.0857143\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$		0			0
$109$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$109$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$110$		0			0
$111$		0			0
$112$	−0.0701535	−	0.0559455i	−0.0701535	−	0.0559455i
$113$		0			0		−0.858449	−	0.512899i	$-0.828571\pi$
$113$		0			0		0.858449	+	0.512899i	$0.171429\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.473869	+	0.880596i	−0.473869	+	0.880596i
$122$		0			0
$123$		0			0
$124$	0.400690	+	1.45187i	0.400690	+	1.45187i
$125$		0			0
$126$		0			0
$127$	0.563508	+	1.87657i	0.563508	+	1.87657i	0.473869	+	0.880596i	$0.342857\pi$
$127$	0.563508	+	1.87657i	0.563508	+	1.87657i	0.0896393	+	0.995974i	$0.471429\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.550897	−	0.834573i	$-0.685714\pi$
$131$		0			0		0.550897	+	0.834573i	$0.314286\pi$
$132$		0			0
$133$	0.0437495	+	0.0399888i	0.0437495	+	0.0399888i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$137$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$138$		0			0
$139$	−0.939160	−	1.74525i	−0.939160	−	1.74525i	−0.587785	−	0.809017i	$-0.700000\pi$
$139$	−0.939160	−	1.74525i	−0.939160	−	1.74525i	−0.351375	−	0.936235i	$-0.614286\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.132882	−	0.0210464i	0.132882	−	0.0210464i
$149$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$149$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$150$		0			0
$151$	0.746362	−	1.46482i	0.746362	−	1.46482i	−0.134233	−	0.990950i	$-0.542857\pi$
$151$	0.746362	−	1.46482i	0.746362	−	1.46482i	0.880596	−	0.473869i	$-0.157143\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.272115	−	0.154403i	−0.272115	−	0.154403i	0.351375	−	0.936235i	$-0.385714\pi$
$157$	−0.272115	−	0.154403i	−0.272115	−	0.154403i	−0.623490	+	0.781831i	$0.714286\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.0597394	−	0.441014i	−0.0597394	−	0.441014i	−0.995974	−	0.0896393i	$-0.971429\pi$
$163$	−0.0597394	−	0.441014i	−0.0597394	−	0.441014i	0.936235	−	0.351375i	$-0.114286\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.473869	−	0.880596i	$-0.657143\pi$
$167$		0			0		0.473869	+	0.880596i	$0.342857\pi$
$168$		0			0
$169$	0.611421	−	0.198663i	0.611421	−	0.198663i
$170$		0			0
$171$		0			0
$172$	1.54185	−	0.700340i	1.54185	−	0.700340i
$173$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$173$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$174$		0			0
$175$	−0.0590366	+	0.0675728i	−0.0590366	+	0.0675728i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.287599	−	0.957751i	$-0.407143\pi$
$179$		0			0		−0.287599	+	0.957751i	$0.592857\pi$
$180$		0			0
$181$	−0.217162	−	0.668355i	−0.217162	−	0.668355i	−0.998993	−	0.0448648i	$-0.985714\pi$
$181$	−0.217162	−	0.668355i	−0.217162	−	0.668355i	0.781831	−	0.623490i	$-0.214286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.990950	−	0.134233i	$-0.957143\pi$
$191$		0			0		0.990950	+	0.134233i	$0.0428571\pi$
$192$		0			0
$193$	−0.194760	−	0.110511i	−0.194760	−	0.110511i	0.393025	−	0.919528i	$-0.371429\pi$
$193$	−0.194760	−	0.110511i	−0.194760	−	0.110511i	−0.587785	+	0.809017i	$0.700000\pi$
$194$		0			0
$195$		0			0
$196$	−0.827854	+	0.546461i	−0.827854	+	0.546461i
$197$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$197$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$198$		0			0
$199$	−0.105514	+	0.778936i	−0.105514	+	0.778936i	0.858449	+	0.512899i	$0.171429\pi$
$199$	−0.105514	+	0.778936i	−0.105514	+	0.778936i	−0.963963	+	0.266037i	$0.914286\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.729937	+	1.05360i	−0.729937	+	1.05360i
$209$		0			0
$210$		0			0
$211$	−1.78394	−	0.365264i	−1.78394	−	0.365264i	−0.809017	−	0.587785i	$-0.800000\pi$
$211$	−1.78394	−	0.365264i	−1.78394	−	0.365264i	−0.974928	+	0.222521i	$0.928571\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.135010	−	0.00606329i	−0.135010	−	0.00606329i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.752407	+	1.19745i	0.752407	+	1.19745i	0.974928	+	0.222521i	$0.0714286\pi$
$223$	0.752407	+	1.19745i	0.752407	+	1.19745i	−0.222521	+	0.974928i	$0.571429\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.244340	−	0.969690i	$-0.578571\pi$
$227$		0			0		0.244340	+	0.969690i	$0.421429\pi$
$228$		0			0
$229$	1.65693	+	0.797936i	1.65693	+	0.797936i	0.998993	+	0.0448648i	$0.0142857\pi$
$229$	1.65693	+	0.797936i	1.65693	+	0.797936i	0.657939	+	0.753071i	$0.271429\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$239$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$240$		0			0
$241$	−0.209906	+	1.15668i	−0.209906	+	1.15668i	0.691063	+	0.722795i	$0.257143\pi$
$241$	−0.209906	+	1.15668i	−0.209906	+	1.15668i	−0.900969	+	0.433884i	$0.857143\pi$
$242$		0			0
$243$		0			0
$244$	0.802047	+	1.76576i	0.802047	+	1.76576i
$245$		0			0
$246$		0			0
$247$	0.527891	−	0.661954i	0.527891	−	0.661954i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$251$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.0448648	−	0.998993i	−0.0448648	−	0.998993i
$257$		0			0		−0.910478	−	0.413559i	$-0.864286\pi$
$257$		0			0		0.910478	+	0.413559i	$0.135714\pi$
$258$		0			0
$259$	−0.00188848	+	0.0119234i	−0.00188848	+	0.0119234i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.822002	−	0.569484i	$-0.807143\pi$
$263$		0			0		0.822002	+	0.569484i	$0.192857\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.456758	+	0.272900i	0.456758	+	0.272900i
$269$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$269$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$270$		0			0
$271$	−0.530079	+	0.634932i	−0.530079	+	0.634932i	−0.963963	−	0.266037i	$-0.914286\pi$
$271$	−0.530079	+	0.634932i	−0.530079	+	0.634932i	0.433884	+	0.900969i	$0.357143\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.951057	−	1.30902i	−0.951057	−	1.30902i	−0.951057	−	0.309017i	$-0.900000\pi$
$277$	−0.951057	−	1.30902i	−0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.0896393	−	0.995974i	$-0.528571\pi$
$281$		0			0		0.0896393	+	0.995974i	$0.471429\pi$
$282$		0			0
$283$	0.258120	+	1.42236i	0.258120	+	1.42236i	0.809017	+	0.587785i	$0.200000\pi$
$283$	0.258120	+	1.42236i	0.258120	+	1.42236i	−0.550897	+	0.834573i	$0.685714\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.753071	+	0.657939i	0.753071	+	0.657939i
$290$		0			0
$291$		0			0
$292$	0.119874	−	1.77798i	0.119874	−	1.77798i
$293$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$293$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.0170133	+	0.150997i	0.0170133	+	0.150997i
$302$		0			0
$303$		0			0
$304$	−0.0148216	+	0.660392i	−0.0148216	+	0.660392i
$305$		0			0
$306$		0			0
$307$	−1.29295	−	1.12961i	−1.29295	−	1.12961i	−0.983930	−	0.178557i	$-0.942857\pi$
$307$	−1.29295	−	1.12961i	−1.29295	−	1.12961i	−0.309017	−	0.951057i	$-0.600000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$311$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$312$		0			0
$313$	0.264152	+	1.95005i	0.264152	+	1.95005i	0.309017	+	0.951057i	$0.400000\pi$
$313$	0.264152	+	1.95005i	0.264152	+	1.95005i	−0.0448648	+	0.998993i	$0.514286\pi$
$314$		0			0
$315$		0			0
$316$	1.53483	+	0.656016i	1.53483	+	0.656016i
$317$		0			0		0.722795	−	0.691063i	$-0.242857\pi$
$317$		0			0		−0.722795	+	0.691063i	$0.757143\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.01979	+	0.776472i	1.01979	+	0.776472i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0167071	−	0.0416502i	0.0167071	−	0.0416502i	−0.919528	−	0.393025i	$-0.871429\pi$
$331$	0.0167071	−	0.0416502i	0.0167071	−	0.0416502i	0.936235	+	0.351375i	$0.114286\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.79295	−	0.766342i	1.79295	−	0.766342i	0.809017	−	0.587785i	$-0.200000\pi$
$337$	1.79295	−	0.766342i	1.79295	−	0.766342i	0.983930	−	0.178557i	$-0.0571429\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.0475506	−	0.172296i	−0.0475506	−	0.172296i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.998993	−	0.0448648i	$-0.985714\pi$
$347$		0			0		0.998993	+	0.0448648i	$0.0142857\pi$
$348$		0			0
$349$	0.307930	+	0.677930i	0.307930	+	0.677930i	0.998993	−	0.0448648i	$-0.0142857\pi$
$349$	0.307930	+	0.677930i	0.307930	+	0.677930i	−0.691063	+	0.722795i	$0.742857\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.0448648	−	0.998993i	$-0.514286\pi$
$353$		0			0		0.0448648	+	0.998993i	$0.485714\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.0672690	−	0.997735i	$-0.521429\pi$
$359$		0			0		0.0672690	+	0.997735i	$0.478571\pi$
$360$		0			0
$361$	−0.0505264	+	0.561394i	−0.0505264	+	0.561394i
$362$		0			0
$363$		0			0
$364$	−0.0696726	−	0.0915057i	−0.0696726	−	0.0915057i
$365$		0			0
$366$		0			0
$367$	−0.117013	+	1.73554i	−0.117013	+	1.73554i	0.433884	+	0.900969i	$0.357143\pi$
$367$	−0.117013	+	1.73554i	−0.117013	+	1.73554i	−0.550897	+	0.834573i	$0.685714\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.02879	+	1.17755i	1.02879	+	1.17755i	0.983930	+	0.178557i	$0.0571429\pi$
$373$	1.02879	+	1.17755i	1.02879	+	1.17755i	0.0448648	+	0.998993i	$0.485714\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.70413	+	0.348922i	−1.70413	+	0.348922i	−0.951057	−	0.309017i	$-0.900000\pi$
$379$	−1.70413	+	0.348922i	−1.70413	+	0.348922i	−0.753071	+	0.657939i	$0.771429\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$383$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.764152	+	1.58678i	0.764152	+	1.58678i
$389$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$389$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.974865	+	1.71807i	0.974865	+	1.71807i	0.623490	+	0.781831i	$0.285714\pi$
$397$	0.974865	+	1.71807i	0.974865	+	1.71807i	0.351375	+	0.936235i	$0.385714\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	0.0433167	+	1.93001i	0.0433167	+	1.93001i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.268155	−	0.825296i	−0.268155	−	0.825296i	−0.990950	−	0.134233i	$-0.957143\pi$
$409$	−0.268155	−	0.825296i	−0.268155	−	0.825296i	0.722795	−	0.691063i	$-0.242857\pi$
$410$		0			0
$411$		0			0
$412$	−0.110354	+	0.608102i	−0.110354	+	0.608102i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.979675	−	0.200589i	$-0.0642857\pi$
$419$		0			0		−0.979675	+	0.200589i	$0.935714\pi$
$420$		0			0
$421$	−0.0896393	+	0.995974i	−0.0896393	+	0.995974i
$421$	−0.0896393	+	0.995974i	−0.0896393	+	0.995974i
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.172926	+	0.0194840i	−0.172926	+	0.0194840i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.200589	−	0.979675i	$-0.435714\pi$
$431$		0			0		−0.200589	+	0.979675i	$0.564286\pi$
$432$		0			0
$433$	1.63287	−	0.530551i	1.63287	−	0.530551i	0.657939	−	0.753071i	$-0.271429\pi$
$433$	1.63287	−	0.530551i	1.63287	−	0.530551i	0.974928	+	0.222521i	$0.0714286\pi$
$434$		0			0
$435$		0			0
$436$	1.58678	+	0.853882i	1.58678	+	0.853882i
$437$		0			0
$438$		0			0
$439$	−0.575054	+	0.0129063i	−0.575054	+	0.0129063i	−0.309017	−	0.951057i	$-0.600000\pi$
$439$	−0.575054	+	0.0129063i	−0.575054	+	0.0129063i	−0.266037	+	0.963963i	$0.585714\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.674671	−	0.738119i	$-0.735714\pi$
$443$		0			0		0.674671	+	0.738119i	$0.264286\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0864961	+	0.0238714i	0.0864961	+	0.0238714i
$449$		0			0		0.880596	−	0.473869i	$-0.157143\pi$
$449$		0			0		−0.880596	+	0.473869i	$0.842857\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.923976	−	0.210891i	0.923976	−	0.210891i	0.266037	−	0.963963i	$-0.414286\pi$
$457$	0.923976	−	0.210891i	0.923976	−	0.210891i	0.657939	+	0.753071i	$0.271429\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.200589	−	0.979675i	$-0.564286\pi$
$461$		0			0		0.200589	+	0.979675i	$0.435714\pi$
$462$		0			0
$463$	−0.0804722	−	0.393025i	−0.0804722	−	0.393025i	0.919528	−	0.393025i	$-0.128571\pi$
$463$	−0.0804722	−	0.393025i	−0.0804722	−	0.393025i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.266037	−	0.963963i	$-0.585714\pi$
$467$		0			0		0.266037	+	0.963963i	$0.414286\pi$
$468$		0			0
$469$	−0.0359537	+	0.0314118i	−0.0359537	+	0.0314118i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.659062	+	0.0444351i	0.659062	+	0.0444351i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.550897	−	0.834573i	$-0.314286\pi$
$479$		0			0		−0.550897	+	0.834573i	$0.685714\pi$
$480$		0			0
$481$	0.171750	+	0.0154578i	0.171750	+	0.0154578i
$482$		0			0
$483$		0			0
$484$	0.0896393	−	0.995974i	0.0896393	−	0.995974i
$485$		0			0
$486$		0			0
$487$	−1.07357	+	0.817417i	−1.07357	+	0.817417i	−0.983930	−	0.178557i	$-0.942857\pi$
$487$	−1.07357	+	0.817417i	−1.07357	+	0.817417i	−0.0896393	+	0.995974i	$0.528571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$491$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.939065	−	1.17755i	−0.939065	−	1.17755i
$497$		0			0
$498$		0			0
$499$	−1.89694	+	0.523523i	−1.89694	+	0.523523i	−0.900969	+	0.433884i	$0.857143\pi$
$499$	−1.89694	+	0.523523i	−1.89694	+	0.523523i	−0.995974	+	0.0896393i	$0.971429\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$503$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.25570	−	1.50409i	−1.25570	−	1.50409i
$509$		0			0		−0.910478	−	0.413559i	$-0.864286\pi$
$509$		0			0		0.910478	+	0.413559i	$0.135714\pi$
$510$		0			0
$511$	0.148405	+	0.0595295i	0.148405	+	0.0595295i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.393025	−	0.919528i	$-0.371429\pi$
$521$		0			0		−0.393025	+	0.919528i	$0.628571\pi$
$522$		0			0
$523$	−0.241168	+	0.957099i	−0.241168	+	0.957099i	0.722795	+	0.691063i	$0.242857\pi$
$523$	−0.241168	+	0.957099i	−0.241168	+	0.957099i	−0.963963	+	0.266037i	$0.914286\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.990950	+	0.134233i	−0.990950	+	0.134233i
$530$		0			0
$531$		0			0
$532$	−0.0559455	−	0.0195762i	−0.0559455	−	0.0195762i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.20396	−	0.135654i	1.20396	−	0.135654i	0.512899	−	0.858449i	$-0.328571\pi$
$541$	1.20396	−	0.135654i	1.20396	−	0.135654i	0.691063	+	0.722795i	$0.257143\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.223873	−	1.98692i	−0.223873	−	1.98692i	−0.134233	−	0.990950i	$-0.542857\pi$
$547$	−0.223873	−	1.98692i	−0.223873	−	1.98692i	−0.0896393	−	0.995974i	$-0.528571\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.0985408	+	0.112789i	−0.0985408	+	0.112789i
$554$		0			0
$555$		0			0
$556$	1.54951	+	1.23569i	1.54951	+	1.23569i
$557$		0			0		0.969690	−	0.244340i	$-0.0785714\pi$
$557$		0			0		−0.969690	+	0.244340i	$0.921429\pi$
$558$		0			0
$559$	2.13570	−	0.387572i	2.13570	−	0.387572i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$569$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$570$		0			0
$571$	−0.930063	−	0.776472i	−0.930063	−	0.776472i	0.0448648	−	0.998993i	$-0.485714\pi$
$571$	−0.930063	−	0.776472i	−0.930063	−	0.776472i	−0.974928	+	0.222521i	$0.928571\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.77081	−	0.902272i	1.77081	−	0.902272i	0.834573	−	0.550897i	$-0.185714\pi$
$577$	1.77081	−	0.902272i	1.77081	−	0.902272i	0.936235	−	0.351375i	$-0.114286\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.753071	−	0.657939i	$-0.228571\pi$
$587$		0			0		−0.753071	+	0.657939i	$0.771429\pi$
$588$		0			0
$589$	0.566577	+	0.817806i	0.566577	+	0.817806i
$590$		0			0
$591$		0			0
$592$	−0.113917	+	0.0715785i	−0.113917	+	0.0715785i
$593$		0			0		−0.413559	−	0.910478i	$-0.635714\pi$
$593$		0			0		0.413559	+	0.910478i	$0.364286\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$599$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$600$		0			0
$601$	−1.08409	−	0.196733i	−1.08409	−	0.196733i	−0.393025	−	0.919528i	$-0.628571\pi$
$601$	−1.08409	−	0.196733i	−1.08409	−	0.196733i	−0.691063	+	0.722795i	$0.742857\pi$
$602$		0			0
$603$		0			0
$604$	−0.110591	+	1.64028i	−0.110591	+	1.64028i
$605$		0			0
$606$		0			0
$607$	−0.0203733	+	0.907752i	−0.0203733	+	0.907752i	0.880596	+	0.473869i	$0.157143\pi$
$607$	−0.0203733	+	0.907752i	−0.0203733	+	0.907752i	−0.900969	+	0.433884i	$0.857143\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.599682	−	1.24525i	0.599682	−	1.24525i	−0.351375	−	0.936235i	$-0.614286\pi$
$613$	0.599682	−	1.24525i	0.599682	−	1.24525i	0.951057	−	0.309017i	$-0.100000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.550897	−	0.834573i	$-0.685714\pi$
$617$		0			0		0.550897	+	0.834573i	$0.314286\pi$
$618$		0			0
$619$	1.69302	−	1.06380i	1.69302	−	1.06380i	0.834573	−	0.550897i	$-0.185714\pi$
$619$	1.69302	−	1.06380i	1.69302	−	1.06380i	0.858449	−	0.512899i	$-0.171429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0448648	+	0.998993i	−0.0448648	+	0.998993i
$626$		0			0
$627$		0			0
$628$	0.310902	+	0.0350302i	0.310902	+	0.0350302i
$629$		0			0
$630$		0			0
$631$	0.365264	−	1.78394i	0.365264	−	1.78394i	−0.222521	−	0.974928i	$-0.571429\pi$
$631$	0.365264	−	1.78394i	0.365264	−	1.78394i	0.587785	−	0.809017i	$-0.300000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.20008	+	0.419927i	−1.20008	+	0.419927i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.512899	−	0.858449i	$-0.671429\pi$
$641$		0			0		0.512899	+	0.858449i	$0.328571\pi$
$642$		0			0
$643$	−1.09182	−	0.147897i	−1.09182	−	0.147897i	−0.433884	−	0.900969i	$-0.642857\pi$
$643$	−1.09182	−	0.147897i	−1.09182	−	0.147897i	−0.657939	+	0.753071i	$0.728571\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$647$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.228262	+	0.382046i	0.228262	+	0.382046i
$653$		0			0		0.178557	−	0.983930i	$-0.442857\pi$
$653$		0			0		−0.178557	+	0.983930i	$0.557143\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.0672690	−	0.997735i	$-0.478571\pi$
$659$		0			0		−0.0672690	+	0.997735i	$0.521429\pi$
$660$		0			0
$661$	1.18595	−	0.385338i	1.18595	−	0.385338i	0.351375	−	0.936235i	$-0.385714\pi$
$661$	1.18595	−	0.385338i	1.18595	−	0.385338i	0.834573	+	0.550897i	$0.185714\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.385338	−	1.18595i	−0.385338	−	1.18595i	−0.936235	−	0.351375i	$-0.885714\pi$
$673$	−0.385338	−	1.18595i	−0.385338	−	1.18595i	0.550897	−	0.834573i	$-0.314286\pi$
$674$		0			0
$675$		0			0
$676$	−0.484139	+	0.422980i	−0.484139	+	0.422980i
$677$		0			0		0.998993	−	0.0448648i	$-0.0142857\pi$
$677$		0			0		−0.998993	+	0.0448648i	$0.985714\pi$
$678$		0			0
$679$	−0.156601	+	0.0212130i	−0.156601	+	0.0212130i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.14252	+	1.24997i	−1.14252	+	1.24997i
$689$		0			0
$690$		0			0
$691$	−0.0399849	−	0.0203733i	−0.0399849	−	0.0203733i	0.433884	−	0.900969i	$-0.357143\pi$
$691$	−0.0399849	−	0.0203733i	−0.0399849	−	0.0203733i	−0.473869	+	0.880596i	$0.657143\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.0277280	−	0.0853380i	0.0277280	−	0.0853380i
$701$		0			0		0.957751	−	0.287599i	$-0.0928571\pi$
$701$		0			0		−0.957751	+	0.287599i	$0.907143\pi$
$702$		0			0
$703$	0.0782587	−	0.0421128i	0.0782587	−	0.0421128i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.151078	−	0.165286i	0.151078	−	0.165286i	−0.657939	−	0.753071i	$-0.728571\pi$
$709$	0.151078	−	0.165286i	0.151078	−	0.165286i	0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.605790	−	0.795625i	$-0.707143\pi$
$719$		0			0		0.605790	+	0.795625i	$0.292857\pi$
$720$		0			0
$721$	−0.0488343	−	0.0262789i	−0.0488343	−	0.0262789i
$722$		0			0
$723$		0			0
$724$	0.462366	+	0.529221i	0.462366	+	0.529221i
$725$		0			0
$726$		0			0
$727$	0.161401	−	0.461258i	0.161401	−	0.461258i	−0.834573	−	0.550897i	$-0.814286\pi$
$727$	0.161401	−	0.461258i	0.161401	−	0.461258i	0.995974	+	0.0896393i	$0.0285714\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.740971	−	0.590905i	0.740971	−	0.590905i	−0.178557	−	0.983930i	$-0.557143\pi$
$733$	0.740971	−	0.590905i	0.740971	−	0.590905i	0.919528	+	0.393025i	$0.128571\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.43288	−	0.945834i	−1.43288	−	0.945834i	−0.998993	−	0.0448648i	$-0.985714\pi$
$739$	−1.43288	−	0.945834i	−1.43288	−	0.945834i	−0.433884	−	0.900969i	$-0.642857\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.936235	−	0.351375i	$-0.114286\pi$
$743$		0			0		−0.936235	+	0.351375i	$0.885714\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.30844	−	0.329699i	1.30844	−	0.329699i	0.473869	−	0.880596i	$-0.342857\pi$
$751$	1.30844	−	0.329699i	1.30844	−	0.329699i	0.834573	+	0.550897i	$0.185714\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.849696	+	0.812393i	0.849696	+	0.812393i	0.983930	−	0.178557i	$-0.0571429\pi$
$757$	0.849696	+	0.812393i	0.849696	+	0.812393i	−0.134233	+	0.990950i	$0.542857\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.936235	−	0.351375i	$-0.885714\pi$
$761$		0			0		0.936235	+	0.351375i	$0.114286\pi$
$762$		0			0
$763$	−0.116867	+	0.111736i	−0.116867	+	0.111736i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.20227	−	0.612588i	−1.20227	−	0.612588i	−0.266037	−	0.963963i	$-0.585714\pi$
$769$	−1.20227	−	0.612588i	−1.20227	−	0.612588i	−0.936235	+	0.351375i	$0.885714\pi$
$770$		0			0
$771$		0			0
$772$	0.222521	+	0.0250721i	0.222521	+	0.0250721i
$773$		0			0		−0.999748	−	0.0224381i	$-0.992857\pi$
$773$		0			0		0.999748	+	0.0224381i	$0.00714286\pi$
$774$		0			0
$775$	−1.21850	+	0.885289i	−1.21850	+	0.885289i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.546461	−	0.827854i	0.546461	−	0.827854i
$785$		0			0
$786$		0			0
$787$	0.0479366	+	0.353882i	0.0479366	+	0.353882i	0.998993	+	0.0448648i	$0.0142857\pi$
$787$	0.0479366	+	0.353882i	0.0479366	+	0.353882i	−0.951057	+	0.309017i	$0.900000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.443857	+	2.44585i	0.443857	+	2.44585i
$794$		0			0
$795$		0			0
$796$	−0.209118	−	0.757723i	−0.209118	−	0.757723i
$797$		0			0		−0.834573	−	0.550897i	$-0.814286\pi$
$797$		0			0		0.834573	+	0.550897i	$0.185714\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$809$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$810$		0			0
$811$	−1.36962	−	0.185527i	−1.36962	−	0.185527i	−0.587785	−	0.809017i	$-0.700000\pi$
$811$	−1.36962	−	0.185527i	−1.36962	−	0.185527i	−0.781831	+	0.623490i	$0.785714\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.808534	−	0.773037i	0.808534	−	0.773037i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	0.612588	+	0.670198i	0.612588	+	0.670198i	0.963963	−	0.266037i	$-0.0857143\pi$
$823$	0.612588	+	0.670198i	0.612588	+	0.670198i	−0.351375	+	0.936235i	$0.614286\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.880596	−	0.473869i	$-0.842857\pi$
$827$		0			0		0.880596	+	0.473869i	$0.157143\pi$
$828$		0			0
$829$	0.142040	+	0.896802i	0.142040	+	0.896802i	0.951057	+	0.309017i	$0.100000\pi$
$829$	0.142040	+	0.896802i	0.142040	+	0.896802i	−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$		0			0
$832$	0.257106	−	1.25570i	0.257106	−	1.25570i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.995974	−	0.0896393i	$-0.971429\pi$
$839$		0			0		0.995974	+	0.0896393i	$0.0285714\pi$
$840$		0			0
$841$		−	1.00000i		−	1.00000i
$842$		0			0
$843$		0			0
$844$	1.78394	−	0.365264i	1.78394	−	0.365264i
$845$		0			0
$846$		0			0
$847$	0.0825089	+	0.0352660i	0.0825089	+	0.0352660i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.603275	+	0.360440i	−0.603275	+	0.360440i	−0.781831	−	0.623490i	$-0.785714\pi$
$853$	−0.603275	+	0.360440i	−0.603275	+	0.360440i	0.178557	+	0.983930i	$0.442857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.473869	−	0.880596i	$-0.342857\pi$
$857$		0			0		−0.473869	+	0.880596i	$0.657143\pi$
$858$		0			0
$859$	−0.657939	+	1.75307i	−0.657939	+	1.75307i			1.00000i	$0.5\pi$
$859$	−0.657939	+	1.75307i	−0.657939	+	1.75307i	−0.657939	+	0.753071i	$0.728571\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.126528	−	0.0474868i	0.126528	−	0.0474868i
$869$		0			0
$870$		0			0
$871$	0.482237	+	0.482237i	0.482237	+	0.482237i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.446712	+	1.37484i	−0.446712	+	1.37484i	0.433884	+	0.900969i	$0.357143\pi$
$877$	−0.446712	+	1.37484i	−0.446712	+	1.37484i	−0.880596	+	0.473869i	$0.842857\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.0672690	−	0.997735i	$-0.521429\pi$
$881$		0			0		0.0672690	+	0.997735i	$0.478571\pi$
$882$		0			0
$883$	−0.211083	−	0.230935i	−0.211083	−	0.230935i	0.623490	−	0.781831i	$-0.285714\pi$
$883$	−0.211083	−	0.230935i	−0.211083	−	0.230935i	−0.834573	+	0.550897i	$0.814286\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.550897	−	0.834573i	$-0.314286\pi$
$887$		0			0		−0.550897	+	0.834573i	$0.685714\pi$
$888$		0			0
$889$	0.163174	−	0.0654536i	0.163174	−	0.0654536i
$890$		0			0
$891$		0			0
$892$	−1.16249	−	0.805373i	−1.16249	−	0.805373i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.722019	+	0.181933i	0.722019	+	0.181933i	0.587785	−	0.809017i	$-0.300000\pi$
$907$	0.722019	+	0.181933i	0.722019	+	0.181933i	0.134233	+	0.990950i	$0.457143\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.0224381	−	0.999748i	$-0.492857\pi$
$911$		0			0		−0.0224381	+	0.999748i	$0.507143\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.83720	−	0.0825089i	−1.83720	−	0.0825089i
$917$		0			0
$918$		0			0
$919$	−0.105377	−	0.145039i	−0.105377	−	0.145039i	0.753071	−	0.657939i	$-0.228571\pi$
$919$	−0.105377	−	0.145039i	−0.105377	−	0.145039i	−0.858449	+	0.512899i	$0.828571\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.0663956	+	0.117013i	0.0663956	+	0.117013i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.767645	−	0.640876i	$-0.221429\pi$
$929$		0			0		−0.767645	+	0.640876i	$0.778571\pi$
$930$		0			0
$931$	−0.396938	+	0.521325i	−0.396938	+	0.521325i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.55175	+	0.542980i	−1.55175	+	0.542980i	−0.963963	−	0.266037i	$-0.914286\pi$
$937$	−1.55175	+	0.542980i	−1.55175	+	0.542980i	−0.587785	+	0.809017i	$0.700000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.351375	−	0.936235i	$-0.614286\pi$
$941$		0			0		0.351375	+	0.936235i	$0.385714\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.919528	−	0.393025i	$-0.871429\pi$
$947$		0			0		0.919528	+	0.393025i	$0.128571\pi$
$948$		0			0
$949$	0.705825	−	2.17231i	0.705825	−	2.17231i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$953$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.22275	−	0.337459i	−1.22275	−	0.337459i
$962$		0			0
$963$		0			0
$964$	−0.261589	−	1.14610i	−0.261589	−	1.14610i
$965$		0			0
$966$		0			0
$967$	−0.696390	−	1.85552i	−0.696390	−	1.85552i	−0.473869	−	0.880596i	$-0.657143\pi$
$967$	−0.696390	−	1.85552i	−0.696390	−	1.85552i	−0.222521	−	0.974928i	$-0.571429\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.0672690	−	0.997735i	$-0.478571\pi$
$971$		0			0		−0.0672690	+	0.997735i	$0.521429\pi$
$972$		0			0
$973$	−0.148416	+	0.0979689i	−0.148416	+	0.0979689i
$974$		0			0
$975$		0			0
$976$	−1.43149	−	1.30844i	−1.43149	−	1.30844i
$977$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$977$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.287599	−	0.957751i	$-0.592857\pi$
$983$		0			0		0.287599	+	0.957751i	$0.407143\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.225246	+	0.816160i	−0.225246	+	0.816160i
$989$		0			0
$990$		0			0
$991$	−0.633156	+	0.304912i	−0.633156	+	0.304912i	−0.722795	−	0.691063i	$-0.757143\pi$
$991$	−0.633156	+	0.304912i	−0.633156	+	0.304912i	0.0896393	+	0.995974i	$0.471429\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.975592	−	1.63287i	0.975592	−	1.63287i	0.222521	−	0.974928i	$-0.428571\pi$
$997$	0.975592	−	1.63287i	0.975592	−	1.63287i	0.753071	−	0.657939i	$-0.228571\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3789.1.du.a.1972.1	✓	48
3.2	odd	2	CM	3789.1.du.a.1972.1	✓	48
421.402	odd	140	inner	3789.1.du.a.3349.1	yes	48
1263.1244	even	140	inner	3789.1.du.a.3349.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3789.1.du.a.1972.1	✓	48	1.1	even	1	trivial
3789.1.du.a.1972.1	✓	48	3.2	odd	2	CM
3789.1.du.a.3349.1	yes	48	421.402	odd	140	inner
3789.1.du.a.3349.1	yes	48	1263.1244	even	140	inner

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:	\( N \)	\(=\)	\( 3789 = 3^{2} \cdot 421 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3789.du (of order \(140\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.919528 + 0.393025i) q^{4} +(-0.00804330 - 0.0893684i) q^{7} +O(q^{10})\)	\(q+(-0.919528 + 0.393025i) q^{4} +(-0.00804330 - 0.0893684i) q^{7} +(-1.26597 + 0.200510i) q^{13} +(0.691063 - 0.722795i) q^{16} +(-0.487571 + 0.445659i) q^{19} +(-0.691063 - 0.722795i) q^{25} +(0.0425201 + 0.0790155i) q^{28} +(0.202174 - 1.49251i) q^{31} +(-0.130460 - 0.0328731i) q^{37} +(-1.69302 + 0.0379977i) q^{43} +(0.976008 - 0.177119i) q^{49} +(1.08529 - 0.681933i) q^{52} +(-0.0435159 - 1.93889i) q^{61} +(-0.351375 + 0.936235i) q^{64} +(-0.312745 - 0.430457i) q^{67} +(-0.809017 + 1.58779i) q^{73} +(0.273179 - 0.601423i) q^{76} +(-1.15348 - 1.20645i) q^{79} +(0.0281019 + 0.111525i) q^{91} +(-0.0790155 - 1.75942i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{25} + 6 q^{28} + 4 q^{31} + 12 q^{37} - 2 q^{43} - 6 q^{49} + 2 q^{52} + 8 q^{61} - 10 q^{67} - 12 q^{73} - 2 q^{76} + 4 q^{91}+O(q^{100}) \) 48 * q - 2 * q^13 - 2 * q^16 - 2 * q^19 + 2 * q^25 + 6 * q^28 + 4 * q^31 + 12 * q^37 - 2 * q^43 - 6 * q^49 + 2 * q^52 + 8 * q^61 - 10 * q^67 - 12 * q^73 - 2 * q^76 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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