LMFDB - Embedded newform 2394.2.b.a.1709.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1709,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.1709");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1709.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2394.1709
Dual form			2394.2.b.a.1709.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -4.24264i q^{13} +1.00000 q^{14} +1.00000 q^{16} +(-1.00000 + 4.24264i) q^{19} +8.48528i q^{23} +5.00000 q^{25} +4.24264i q^{26} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{32} -8.48528i q^{37} +(1.00000 - 4.24264i) q^{38} +8.00000 q^{43} -8.48528i q^{46} +4.24264i q^{47} +1.00000 q^{49} -5.00000 q^{50} -4.24264i q^{52} +6.00000 q^{53} +1.00000 q^{56} +6.00000 q^{58} +10.0000 q^{61} +1.00000 q^{64} -4.24264i q^{67} +6.00000 q^{71} +2.00000 q^{73} +8.48528i q^{74} +(-1.00000 + 4.24264i) q^{76} +4.24264i q^{79} +4.24264i q^{83} -8.00000 q^{86} -6.00000 q^{89} +4.24264i q^{91} +8.48528i q^{92} -4.24264i q^{94} +4.24264i q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 2 q^{14} + 2 q^{16} - 2 q^{19} + 10 q^{25} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 2 q^{38} + 16 q^{43} + 2 q^{49} - 10 q^{50} + 12 q^{53} + 2 q^{56} + 12 q^{58} + 20 q^{61} + 2 q^{64} + 12 q^{71} + 4 q^{73} - 2 q^{76} - 16 q^{86} - 12 q^{89} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 2 * q^14 + 2 * q^16 - 2 * q^19 + 10 * q^25 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 2 * q^38 + 16 * q^43 + 2 * q^49 - 10 * q^50 + 12 * q^53 + 2 * q^56 + 12 * q^58 + 20 * q^61 + 2 * q^64 + 12 * q^71 + 4 * q^73 - 2 * q^76 - 16 * q^86 - 12 * q^89 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	4.24264i		−	1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$		−	4.24264i		−	1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.48528i			1.76930i	0.466252	+	0.884652i	$0.345604\pi$
$23$			8.48528i			1.76930i	−0.466252	+	0.884652i	$0.654396\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$			4.24264i			0.832050i
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$	1.00000	−	4.24264i	0.162221	−	0.688247i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		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
$46$		−	8.48528i		−	1.25109i
$47$			4.24264i			0.618853i	0.950923	+	0.309426i	$0.100137\pi$
$47$			4.24264i			0.618853i	−0.950923	+	0.309426i	$0.899863\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$	−5.00000			−0.707107
$51$		0			0
$52$		−	4.24264i		−	0.588348i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$	1.00000			0.133631
$57$		0			0
$58$	6.00000			0.787839
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$		−	4.24264i		−	0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$		−	4.24264i		−	0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$			8.48528i			0.986394i
$75$		0			0
$76$	−1.00000	+	4.24264i	−0.114708	+	0.486664i
$77$		0			0
$78$		0			0
$79$			4.24264i			0.477334i	0.971101	+	0.238667i	$0.0767105\pi$
$79$			4.24264i			0.477334i	−0.971101	+	0.238667i	$0.923290\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.24264i			0.465690i	0.972514	+	0.232845i	$0.0748035\pi$
$83$			4.24264i			0.465690i	−0.972514	+	0.232845i	$0.925196\pi$
$84$		0			0
$85$		0			0
$86$	−8.00000			−0.862662
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$			4.24264i			0.444750i
$92$			8.48528i			0.884652i
$93$		0			0
$94$		−	4.24264i		−	0.437595i
$95$		0			0
$96$		0			0
$97$			4.24264i			0.430775i	0.976529	+	0.215387i	$0.0691014\pi$
$97$			4.24264i			0.430775i	−0.976529	+	0.215387i	$0.930899\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	5.00000			0.500000
$101$			8.48528i			0.844317i	0.906522	+	0.422159i	$0.138727\pi$
$101$			8.48528i			0.844317i	−0.906522	+	0.422159i	$0.861273\pi$
$102$		0			0
$103$			8.48528i			0.836080i	0.908429	+	0.418040i	$0.137283\pi$
$103$			8.48528i			0.836080i	−0.908429	+	0.418040i	$0.862717\pi$
$104$			4.24264i			0.416025i
$105$		0			0
$106$	−6.00000			−0.582772
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000			−0.0944911
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$	−6.00000			−0.557086
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−10.0000			−0.905357
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	12.7279i		−	1.12942i	−0.825289	−	0.564710i	$-0.808988\pi$
$127$		−	12.7279i		−	1.12942i	0.825289	−	0.564710i	$-0.191012\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		0			0
$131$		−	12.7279i		−	1.11204i	−0.831168	−	0.556022i	$-0.812327\pi$
$131$		−	12.7279i		−	1.11204i	0.831168	−	0.556022i	$-0.187673\pi$
$132$		0			0
$133$	1.00000	−	4.24264i	0.0867110	−	0.367884i
$134$			4.24264i			0.366508i
$135$		0			0
$136$		0			0
$137$			4.24264i			0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$			4.24264i			0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$		0			0
$139$	22.0000			1.86602			0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000			1.86602			0.933008	+	0.359856i	$0.117174\pi$
$140$		0			0
$141$		0			0
$142$	−6.00000			−0.503509
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−2.00000			−0.165521
$147$		0			0
$148$		−	8.48528i		−	0.697486i
$149$		−	4.24264i		−	0.347571i	−0.984784	−	0.173785i	$-0.944400\pi$
$149$		−	4.24264i		−	0.347571i	0.984784	−	0.173785i	$-0.0555999\pi$
$150$		0			0
$151$			4.24264i			0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$			4.24264i			0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$	1.00000	−	4.24264i	0.0811107	−	0.344124i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		−	4.24264i		−	0.337526i
$159$		0			0
$160$		0			0
$161$		−	8.48528i		−	0.668734i
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$		0			0
$166$		−	4.24264i		−	0.329293i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$	−5.00000			−0.377964
$176$		0			0
$177$		0			0
$178$	6.00000			0.449719
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$			4.24264i			0.315353i	0.987491	+	0.157676i	$0.0504003\pi$
$181$			4.24264i			0.315353i	−0.987491	+	0.157676i	$0.949600\pi$
$182$		−	4.24264i		−	0.314485i
$183$		0			0
$184$		−	8.48528i		−	0.625543i
$185$		0			0
$186$		0			0
$187$		0			0
$188$			4.24264i			0.309426i
$189$		0			0
$190$		0			0
$191$			16.9706i			1.22795i	0.789327	+	0.613973i	$0.210430\pi$
$191$			16.9706i			1.22795i	−0.789327	+	0.613973i	$0.789570\pi$
$192$		0			0
$193$			16.9706i			1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$			16.9706i			1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$		−	4.24264i		−	0.304604i
$195$		0			0
$196$	1.00000			0.0714286
$197$			21.2132i			1.51138i	0.654931	+	0.755689i	$0.272698\pi$
$197$			21.2132i			1.51138i	−0.654931	+	0.755689i	$0.727302\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	−5.00000			−0.353553
$201$		0			0
$202$		−	8.48528i		−	0.597022i
$203$	6.00000			0.421117
$204$		0			0
$205$		0			0
$206$		−	8.48528i		−	0.591198i
$207$		0			0
$208$		−	4.24264i		−	0.294174i
$209$		0			0
$210$		0			0
$211$			4.24264i			0.292075i	0.989279	+	0.146038i	$0.0466521\pi$
$211$			4.24264i			0.292075i	−0.989279	+	0.146038i	$0.953348\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$	−18.0000			−1.23045
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	6.00000			0.399114
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$	6.00000			0.393919
$233$			4.24264i			0.277945i	0.990296	+	0.138972i	$0.0443799\pi$
$233$			4.24264i			0.277945i	−0.990296	+	0.138972i	$0.955620\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	16.9706i		−	1.09773i	−0.835910	−	0.548867i	$-0.815059\pi$
$239$		−	16.9706i		−	1.09773i	0.835910	−	0.548867i	$-0.184941\pi$
$240$		0			0
$241$			21.2132i			1.36646i	0.730202	+	0.683231i	$0.239426\pi$
$241$			21.2132i			1.36646i	−0.730202	+	0.683231i	$0.760574\pi$
$242$	−11.0000			−0.707107
$243$		0			0
$244$	10.0000			0.640184
$245$		0			0
$246$		0			0
$247$	18.0000	+	4.24264i	1.14531	+	0.269953i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	12.7279i		−	0.803379i	−0.915776	−	0.401690i	$-0.868423\pi$
$251$		−	12.7279i		−	0.803379i	0.915776	−	0.401690i	$-0.131577\pi$
$252$		0			0
$253$		0			0
$254$			12.7279i			0.798621i
$255$		0			0
$256$	1.00000			0.0625000
$257$	30.0000			1.87135			0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000			1.87135			0.935674	+	0.352865i	$0.114792\pi$
$258$		0			0
$259$			8.48528i			0.527250i
$260$		0			0
$261$		0			0
$262$			12.7279i			0.786334i
$263$			8.48528i			0.523225i	0.965173	+	0.261612i	$0.0842542\pi$
$263$			8.48528i			0.523225i	−0.965173	+	0.261612i	$0.915746\pi$
$264$		0			0
$265$		0			0
$266$	−1.00000	+	4.24264i	−0.0613139	+	0.260133i
$267$		0			0
$268$		−	4.24264i		−	0.259161i
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$		0			0
$273$		0			0
$274$		−	4.24264i		−	0.256307i
$275$		0			0
$276$		0			0
$277$	28.0000			1.68236			0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000			1.68236			0.841178	+	0.540758i	$0.181862\pi$
$278$	−22.0000			−1.31947
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	2.00000			0.117041
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$		0			0
$296$			8.48528i			0.493197i
$297$		0			0
$298$			4.24264i			0.245770i
$299$	36.0000			2.08193
$300$		0			0
$301$	−8.00000			−0.461112
$302$		−	4.24264i		−	0.244137i
$303$		0			0
$304$	−1.00000	+	4.24264i	−0.0573539	+	0.243332i
$305$		0			0
$306$		0			0
$307$			25.4558i			1.45284i	0.687250	+	0.726421i	$0.258818\pi$
$307$			25.4558i			1.45284i	−0.687250	+	0.726421i	$0.741182\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	21.2132i		−	1.20289i	−0.798914	−	0.601445i	$-0.794592\pi$
$311$		−	21.2132i		−	1.20289i	0.798914	−	0.601445i	$-0.205408\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$			4.24264i			0.238667i
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$			8.48528i			0.472866i
$323$		0			0
$324$		0			0
$325$		−	21.2132i		−	1.17670i
$326$	−16.0000			−0.886158
$327$		0			0
$328$		0			0
$329$		−	4.24264i		−	0.233904i
$330$		0			0
$331$			21.2132i			1.16598i	0.812478	+	0.582992i	$0.198118\pi$
$331$			21.2132i			1.16598i	−0.812478	+	0.582992i	$0.801882\pi$
$332$			4.24264i			0.232845i
$333$		0			0
$334$	−12.0000			−0.656611
$335$		0			0
$336$		0			0
$337$		−	16.9706i		−	0.924445i	−0.886764	−	0.462223i	$-0.847052\pi$
$337$		−	16.9706i		−	0.924445i	0.886764	−	0.462223i	$-0.152948\pi$
$338$	5.00000			0.271964
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	−8.00000			−0.431331
$345$		0			0
$346$	−18.0000			−0.967686
$347$		−	25.4558i		−	1.36654i	−0.730165	−	0.683271i	$-0.760557\pi$
$347$		−	25.4558i		−	1.36654i	0.730165	−	0.683271i	$-0.239443\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$	5.00000			0.267261
$351$		0			0
$352$		0			0
$353$		−	16.9706i		−	0.903252i	−0.892207	−	0.451626i	$-0.850844\pi$
$353$		−	16.9706i		−	0.903252i	0.892207	−	0.451626i	$-0.149156\pi$
$354$		0			0
$355$		0			0
$356$	−6.00000			−0.317999
$357$		0			0
$358$	−6.00000			−0.317110
$359$			8.48528i			0.447836i	0.974608	+	0.223918i	$0.0718848\pi$
$359$			8.48528i			0.447836i	−0.974608	+	0.223918i	$0.928115\pi$
$360$		0			0
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$		−	4.24264i		−	0.222988i
$363$		0			0
$364$			4.24264i			0.222375i
$365$		0			0
$366$		0			0
$367$	−10.0000			−0.521996			−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000			−0.521996			−0.260998	+	0.965339i	$0.584052\pi$
$368$			8.48528i			0.442326i
$369$		0			0
$370$		0			0
$371$	−6.00000			−0.311504
$372$		0			0
$373$			16.9706i			0.878702i	0.898315	+	0.439351i	$0.144792\pi$
$373$			16.9706i			0.878702i	−0.898315	+	0.439351i	$0.855208\pi$
$374$		0			0
$375$		0			0
$376$		−	4.24264i		−	0.218797i
$377$			25.4558i			1.31104i
$378$		0			0
$379$		−	21.2132i		−	1.08965i	−0.838550	−	0.544825i	$-0.816596\pi$
$379$		−	21.2132i		−	1.08965i	0.838550	−	0.544825i	$-0.183404\pi$
$380$		0			0
$381$		0			0
$382$		−	16.9706i		−	0.868290i
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$		0			0
$386$		−	16.9706i		−	0.863779i
$387$		0			0
$388$			4.24264i			0.215387i
$389$		−	4.24264i		−	0.215110i	−0.994199	−	0.107555i	$-0.965698\pi$
$389$		−	4.24264i		−	0.215110i	0.994199	−	0.107555i	$-0.0343022\pi$
$390$		0			0
$391$		0			0
$392$	−1.00000			−0.0505076
$393$		0			0
$394$		−	21.2132i		−	1.06871i
$395$		0			0
$396$		0			0
$397$	−34.0000			−1.70641			−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000			−1.70641			−0.853206	+	0.521575i	$0.825345\pi$
$398$	16.0000			0.802008
$399$		0			0
$400$	5.00000			0.250000
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$		0			0
$404$			8.48528i			0.422159i
$405$		0			0
$406$	−6.00000			−0.297775
$407$		0			0
$408$		0			0
$409$		−	21.2132i		−	1.04893i	−0.851433	−	0.524463i	$-0.824266\pi$
$409$		−	21.2132i		−	1.04893i	0.851433	−	0.524463i	$-0.175734\pi$
$410$		0			0
$411$		0			0
$412$			8.48528i			0.418040i
$413$		0			0
$414$		0			0
$415$		0			0
$416$			4.24264i			0.208013i
$417$		0			0
$418$		0			0
$419$		−	12.7279i		−	0.621800i	−0.950443	−	0.310900i	$-0.899370\pi$
$419$		−	12.7279i		−	0.621800i	0.950443	−	0.310900i	$-0.100630\pi$
$420$		0			0
$421$			8.48528i			0.413547i	0.978389	+	0.206774i	$0.0662964\pi$
$421$			8.48528i			0.413547i	−0.978389	+	0.206774i	$0.933704\pi$
$422$		−	4.24264i		−	0.206529i
$423$		0			0
$424$	−6.00000			−0.291386
$425$		0			0
$426$		0			0
$427$	−10.0000			−0.483934
$428$	18.0000			0.870063
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$		−	38.1838i		−	1.83499i	−0.397742	−	0.917497i	$-0.630206\pi$
$433$		−	38.1838i		−	1.83499i	0.397742	−	0.917497i	$-0.369794\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−36.0000	−	8.48528i	−1.72211	−	0.405906i
$438$		0			0
$439$			25.4558i			1.21494i	0.794342	+	0.607471i	$0.207816\pi$
$439$			25.4558i			1.21494i	−0.794342	+	0.607471i	$0.792184\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			16.9706i			0.806296i	0.915135	+	0.403148i	$0.132084\pi$
$443$			16.9706i			0.806296i	−0.915135	+	0.403148i	$0.867916\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.00000			−0.0472456
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$	−6.00000			−0.282216
$453$		0			0
$454$	−12.0000			−0.563188
$455$		0			0
$456$		0			0
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$	14.0000			0.654177
$459$		0			0
$460$		0			0
$461$			8.48528i			0.395199i	0.980283	+	0.197599i	$0.0633145\pi$
$461$			8.48528i			0.395199i	−0.980283	+	0.197599i	$0.936685\pi$
$462$		0			0
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−6.00000			−0.278543
$465$		0			0
$466$		−	4.24264i		−	0.196537i
$467$		−	29.6985i		−	1.37428i	−0.726524	−	0.687141i	$-0.758865\pi$
$467$		−	29.6985i		−	1.37428i	0.726524	−	0.687141i	$-0.241135\pi$
$468$		0			0
$469$			4.24264i			0.195907i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−5.00000	+	21.2132i	−0.229416	+	0.973329i
$476$		0			0
$477$		0			0
$478$			16.9706i			0.776215i
$479$		−	38.1838i		−	1.74466i	−0.488917	−	0.872330i	$-0.662608\pi$
$479$		−	38.1838i		−	1.74466i	0.488917	−	0.872330i	$-0.337392\pi$
$480$		0			0
$481$	−36.0000			−1.64146
$482$		−	21.2132i		−	0.966235i
$483$		0			0
$484$	11.0000			0.500000
$485$		0			0
$486$		0			0
$487$			29.6985i			1.34577i	0.739749	+	0.672883i	$0.234944\pi$
$487$			29.6985i			1.34577i	−0.739749	+	0.672883i	$0.765056\pi$
$488$	−10.0000			−0.452679
$489$		0			0
$490$		0			0
$491$			8.48528i			0.382935i	0.981499	+	0.191468i	$0.0613247\pi$
$491$			8.48528i			0.382935i	−0.981499	+	0.191468i	$0.938675\pi$
$492$		0			0
$493$		0			0
$494$	−18.0000	−	4.24264i	−0.809858	−	0.190885i
$495$		0			0
$496$		0			0
$497$	−6.00000			−0.269137
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$		0			0
$502$			12.7279i			0.568075i
$503$			4.24264i			0.189170i	0.995517	+	0.0945850i	$0.0301524\pi$
$503$			4.24264i			0.189170i	−0.995517	+	0.0945850i	$0.969848\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		−	12.7279i		−	0.564710i
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$	−2.00000			−0.0884748
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−30.0000			−1.32324
$515$		0			0
$516$		0			0
$517$		0			0
$518$		−	8.48528i		−	0.372822i
$519$		0			0
$520$		0			0
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		0			0
$523$			33.9411i			1.48414i	0.670321	+	0.742071i	$0.266156\pi$
$523$			33.9411i			1.48414i	−0.670321	+	0.742071i	$0.733844\pi$
$524$		−	12.7279i		−	0.556022i
$525$		0			0
$526$		−	8.48528i		−	0.369976i
$527$		0			0
$528$		0			0
$529$	−49.0000			−2.13043
$530$		0			0
$531$		0			0
$532$	1.00000	−	4.24264i	0.0433555	−	0.183942i
$533$		0			0
$534$		0			0
$535$		0			0
$536$			4.24264i			0.183254i
$537$		0			0
$538$	24.0000			1.03471
$539$		0			0
$540$		0			0
$541$	20.0000			0.859867			0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000			0.859867			0.429934	+	0.902861i	$0.358537\pi$
$542$	−2.00000			−0.0859074
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	4.24264i		−	0.181402i	−0.995878	−	0.0907011i	$-0.971089\pi$
$547$		−	4.24264i		−	0.181402i	0.995878	−	0.0907011i	$-0.0289108\pi$
$548$			4.24264i			0.181237i
$549$		0			0
$550$		0			0
$551$	6.00000	−	25.4558i	0.255609	−	1.08446i
$552$		0			0
$553$		−	4.24264i		−	0.180415i
$554$	−28.0000			−1.18961
$555$		0			0
$556$	22.0000			0.933008
$557$		−	46.6690i		−	1.97743i	−0.149805	−	0.988716i	$-0.547865\pi$
$557$		−	46.6690i		−	1.97743i	0.149805	−	0.988716i	$-0.452135\pi$
$558$		0			0
$559$		−	33.9411i		−	1.43556i
$560$		0			0
$561$		0			0
$562$	18.0000			0.759284
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$		0			0
$566$	4.00000			0.168133
$567$		0			0
$568$	−6.00000			−0.251754
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			42.4264i			1.76930i
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$	−17.0000			−0.707107
$579$		0			0
$580$		0			0
$581$		−	4.24264i		−	0.176014i
$582$		0			0
$583$		0			0
$584$	−2.00000			−0.0827606
$585$		0			0
$586$	−12.0000			−0.495715
$587$			21.2132i			0.875563i	0.899081	+	0.437781i	$0.144236\pi$
$587$			21.2132i			0.875563i	−0.899081	+	0.437781i	$0.855764\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	8.48528i		−	0.348743i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		−	4.24264i		−	0.173785i
$597$		0			0
$598$	−36.0000			−1.47215
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$			12.7279i			0.519183i	0.965719	+	0.259591i	$0.0835879\pi$
$601$			12.7279i			0.519183i	−0.965719	+	0.259591i	$0.916412\pi$
$602$	8.00000			0.326056
$603$		0			0
$604$			4.24264i			0.172631i
$605$		0			0
$606$		0			0
$607$		−	8.48528i		−	0.344407i	−0.985061	−	0.172203i	$-0.944911\pi$
$607$		−	8.48528i		−	0.344407i	0.985061	−	0.172203i	$-0.0550887\pi$
$608$	1.00000	−	4.24264i	0.0405554	−	0.172062i
$609$		0			0
$610$		0			0
$611$	18.0000			0.728202
$612$		0			0
$613$	−20.0000			−0.807792			−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000			−0.807792			−0.403896	+	0.914805i	$0.632344\pi$
$614$		−	25.4558i		−	1.02731i
$615$		0			0
$616$		0			0
$617$		−	21.2132i		−	0.854011i	−0.904249	−	0.427006i	$-0.859568\pi$
$617$		−	21.2132i		−	0.854011i	0.904249	−	0.427006i	$-0.140432\pi$
$618$		0			0
$619$	−44.0000			−1.76851			−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000			−1.76851			−0.884255	+	0.467005i	$0.845333\pi$
$620$		0			0
$621$		0			0
$622$			21.2132i			0.850572i
$623$	6.00000			0.240385
$624$		0			0
$625$	25.0000			1.00000
$626$	−10.0000			−0.399680
$627$		0			0
$628$	−14.0000			−0.558661
$629$		0			0
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		−	4.24264i		−	0.168763i
$633$		0			0
$634$	−18.0000			−0.714871
$635$		0			0
$636$		0			0
$637$		−	4.24264i		−	0.168100i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.00000			0.236986			0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000			0.236986			0.118493	+	0.992955i	$0.462194\pi$
$642$		0			0
$643$	14.0000			0.552106			0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000			0.552106			0.276053	+	0.961142i	$0.410973\pi$
$644$		−	8.48528i		−	0.334367i
$645$		0			0
$646$		0			0
$647$			46.6690i			1.83475i	0.398024	+	0.917375i	$0.369696\pi$
$647$			46.6690i			1.83475i	−0.398024	+	0.917375i	$0.630304\pi$
$648$		0			0
$649$		0			0
$650$			21.2132i			0.832050i
$651$		0			0
$652$	16.0000			0.626608
$653$		−	4.24264i		−	0.166027i	−0.996548	−	0.0830137i	$-0.973545\pi$
$653$		−	4.24264i		−	0.166027i	0.996548	−	0.0830137i	$-0.0264545\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$			4.24264i			0.165395i
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$		−	29.6985i		−	1.15514i	−0.816342	−	0.577569i	$-0.804002\pi$
$661$		−	29.6985i		−	1.15514i	0.816342	−	0.577569i	$-0.195998\pi$
$662$		−	21.2132i		−	0.824475i
$663$		0			0
$664$		−	4.24264i		−	0.164646i
$665$		0			0
$666$		0			0
$667$		−	50.9117i		−	1.97131i
$668$	12.0000			0.464294
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			8.48528i			0.327084i	0.986536	+	0.163542i	$0.0522919\pi$
$673$			8.48528i			0.327084i	−0.986536	+	0.163542i	$0.947708\pi$
$674$			16.9706i			0.653682i
$675$		0			0
$676$	−5.00000			−0.192308
$677$	−42.0000			−1.61419			−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000			−1.61419			−0.807096	+	0.590421i	$0.798962\pi$
$678$		0			0
$679$		−	4.24264i		−	0.162818i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		0			0
$686$	1.00000			0.0381802
$687$		0			0
$688$	8.00000			0.304997
$689$		−	25.4558i		−	0.969790i
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000			0.684257
$693$		0			0
$694$			25.4558i			0.966291i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	10.0000			0.378506
$699$		0			0
$700$	−5.00000			−0.188982
$701$			29.6985i			1.12170i	0.827919	+	0.560848i	$0.189525\pi$
$701$			29.6985i			1.12170i	−0.827919	+	0.560848i	$0.810475\pi$
$702$		0			0
$703$	36.0000	+	8.48528i	1.35777	+	0.320028i
$704$		0			0
$705$		0			0
$706$			16.9706i			0.638696i
$707$		−	8.48528i		−	0.319122i
$708$		0			0
$709$	8.00000			0.300446			0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000			0.300446			0.150223	+	0.988652i	$0.452001\pi$
$710$		0			0
$711$		0			0
$712$	6.00000			0.224860
$713$		0			0
$714$		0			0
$715$		0			0
$716$	6.00000			0.224231
$717$		0			0
$718$		−	8.48528i		−	0.316668i
$719$		−	29.6985i		−	1.10757i	−0.832661	−	0.553783i	$-0.813184\pi$
$719$		−	29.6985i		−	1.10757i	0.832661	−	0.553783i	$-0.186816\pi$
$720$		0			0
$721$		−	8.48528i		−	0.316008i
$722$	17.0000	+	8.48528i	0.632674	+	0.315789i
$723$		0			0
$724$			4.24264i			0.157676i
$725$	−30.0000			−1.11417
$726$		0			0
$727$	26.0000			0.964287			0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000			0.964287			0.482143	+	0.876092i	$0.339858\pi$
$728$		−	4.24264i		−	0.157243i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$	10.0000			0.369107
$735$		0			0
$736$		−	8.48528i		−	0.312772i
$737$		0			0
$738$		0			0
$739$	16.0000			0.588570			0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000			0.588570			0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$		0			0
$742$	6.00000			0.220267
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$		0			0
$746$		−	16.9706i		−	0.621336i
$747$		0			0
$748$		0			0
$749$	−18.0000			−0.657706
$750$		0			0
$751$		−	21.2132i		−	0.774081i	−0.922063	−	0.387040i	$-0.873497\pi$
$751$		−	21.2132i		−	0.774081i	0.922063	−	0.387040i	$-0.126503\pi$
$752$			4.24264i			0.154713i
$753$		0			0
$754$		−	25.4558i		−	0.927047i
$755$		0			0
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$			21.2132i			0.770498i
$759$		0			0
$760$		0			0
$761$		−	25.4558i		−	0.922774i	−0.887199	−	0.461387i	$-0.847352\pi$
$761$		−	25.4558i		−	0.922774i	0.887199	−	0.461387i	$-0.152648\pi$
$762$		0			0
$763$		0			0
$764$			16.9706i			0.613973i
$765$		0			0
$766$	12.0000			0.433578
$767$		0			0
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$		0			0
$772$			16.9706i			0.610784i
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$		0			0
$776$		−	4.24264i		−	0.152302i
$777$		0			0
$778$			4.24264i			0.152106i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			0.0357143
$785$		0			0
$786$		0			0
$787$			33.9411i			1.20987i	0.796275	+	0.604935i	$0.206801\pi$
$787$			33.9411i			1.20987i	−0.796275	+	0.604935i	$0.793199\pi$
$788$			21.2132i			0.755689i
$789$		0			0
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$		−	42.4264i		−	1.50661i
$794$	34.0000			1.20661
$795$		0			0
$796$	−16.0000			−0.567105
$797$	−36.0000			−1.27519			−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000			−1.27519			−0.637593	+	0.770374i	$0.720070\pi$
$798$		0			0
$799$		0			0
$800$	−5.00000			−0.176777
$801$		0			0
$802$	6.00000			0.211867
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		−	8.48528i		−	0.298511i
$809$			12.7279i			0.447490i	0.974648	+	0.223745i	$0.0718283\pi$
$809$			12.7279i			0.447490i	−0.974648	+	0.223745i	$0.928172\pi$
$810$		0			0
$811$		−	25.4558i		−	0.893876i	−0.894565	−	0.446938i	$-0.852515\pi$
$811$		−	25.4558i		−	0.893876i	0.894565	−	0.446938i	$-0.147485\pi$
$812$	6.00000			0.210559
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.00000	+	33.9411i	−0.279885	+	1.18745i
$818$			21.2132i			0.741702i
$819$		0			0
$820$		0			0
$821$		−	12.7279i		−	0.444208i	−0.975023	−	0.222104i	$-0.928708\pi$
$821$		−	12.7279i		−	0.444208i	0.975023	−	0.222104i	$-0.0712924\pi$
$822$		0			0
$823$	−32.0000			−1.11545			−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000			−1.11545			−0.557725	+	0.830026i	$0.688326\pi$
$824$		−	8.48528i		−	0.295599i
$825$		0			0
$826$		0			0
$827$	−18.0000			−0.625921			−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000			−0.625921			−0.312961	+	0.949766i	$0.601321\pi$
$828$		0			0
$829$			4.24264i			0.147353i	0.997282	+	0.0736765i	$0.0234732\pi$
$829$			4.24264i			0.147353i	−0.997282	+	0.0736765i	$0.976527\pi$
$830$		0			0
$831$		0			0
$832$		−	4.24264i		−	0.147087i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$			12.7279i			0.439679i
$839$	−36.0000			−1.24286			−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000			−1.24286			−0.621429	+	0.783470i	$0.713448\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	8.48528i		−	0.292422i
$843$		0			0
$844$			4.24264i			0.146038i
$845$		0			0
$846$		0			0
$847$	−11.0000			−0.377964
$848$	6.00000			0.206041
$849$		0			0
$850$		0			0
$851$	72.0000			2.46813
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$	10.0000			0.342193
$855$		0			0
$856$	−18.0000			−0.615227
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$		0			0
$862$	−30.0000			−1.02180
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$		0			0
$866$			38.1838i			1.29754i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−18.0000			−0.609907
$872$		0			0
$873$		0			0
$874$	36.0000	+	8.48528i	1.21772	+	0.287019i
$875$		0			0
$876$		0			0
$877$			8.48528i			0.286528i	0.989685	+	0.143264i	$0.0457597\pi$
$877$			8.48528i			0.286528i	−0.989685	+	0.143264i	$0.954240\pi$
$878$		−	25.4558i		−	0.859093i
$879$		0			0
$880$		0			0
$881$			50.9117i			1.71526i	0.514269	+	0.857629i	$0.328063\pi$
$881$			50.9117i			1.71526i	−0.514269	+	0.857629i	$0.671937\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$		0			0
$886$		−	16.9706i		−	0.570137i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$			12.7279i			0.426881i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.0000	−	4.24264i	−0.602347	−	0.141975i
$894$		0			0
$895$		0			0
$896$	1.00000			0.0334077
$897$		0			0
$898$	30.0000			1.00111
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	6.00000			0.199557
$905$		0			0
$906$		0			0
$907$			21.2132i			0.704373i	0.935930	+	0.352186i	$0.114562\pi$
$907$			21.2132i			0.704373i	−0.935930	+	0.352186i	$0.885438\pi$
$908$	12.0000			0.398234
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$	−8.00000			−0.264616
$915$		0			0
$916$	−14.0000			−0.462573
$917$			12.7279i			0.420313i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		−	8.48528i		−	0.279448i
$923$		−	25.4558i		−	0.837889i
$924$		0			0
$925$		−	42.4264i		−	1.39497i
$926$	4.00000			0.131448
$927$		0			0
$928$	6.00000			0.196960
$929$			42.4264i			1.39197i	0.718059	+	0.695983i	$0.245031\pi$
$929$			42.4264i			1.39197i	−0.718059	+	0.695983i	$0.754969\pi$
$930$		0			0
$931$	−1.00000	+	4.24264i	−0.0327737	+	0.139047i
$932$			4.24264i			0.138972i
$933$		0			0
$934$			29.6985i			0.971764i
$935$		0			0
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		−	4.24264i		−	0.138527i
$939$		0			0
$940$		0			0
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			42.4264i			1.37867i	0.724441	+	0.689336i	$0.242098\pi$
$947$			42.4264i			1.37867i	−0.724441	+	0.689336i	$0.757902\pi$
$948$		0			0
$949$		−	8.48528i		−	0.275444i
$950$	5.00000	−	21.2132i	0.162221	−	0.688247i
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		0			0
$956$		−	16.9706i		−	0.548867i
$957$		0			0
$958$			38.1838i			1.23366i
$959$		−	4.24264i		−	0.137002i
$960$		0			0
$961$	31.0000			1.00000
$962$	36.0000			1.16069
$963$		0			0
$964$			21.2132i			0.683231i
$965$		0			0
$966$		0			0
$967$	4.00000			0.128631			0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000			0.128631			0.0643157	+	0.997930i	$0.479514\pi$
$968$	−11.0000			−0.353553
$969$		0			0
$970$		0			0
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$	−22.0000			−0.705288
$974$		−	29.6985i		−	0.951601i
$975$		0			0
$976$	10.0000			0.320092
$977$	−6.00000			−0.191957			−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000			−0.191957			−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		−	8.48528i		−	0.270776i
$983$	12.0000			0.382741			0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000			0.382741			0.191370	+	0.981518i	$0.438707\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	18.0000	+	4.24264i	0.572656	+	0.134976i
$989$			67.8823i			2.15853i
$990$		0			0
$991$			55.1543i			1.75203i	0.482279	+	0.876017i	$0.339809\pi$
$991$			55.1543i			1.75203i	−0.482279	+	0.876017i	$0.660191\pi$
$992$		0			0
$993$		0			0
$994$	6.00000			0.190308
$995$		0			0
$996$		0			0
$997$	10.0000			0.316703			0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000			0.316703			0.158352	+	0.987383i	$0.449382\pi$
$998$	−4.00000			−0.126618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.b.a.1709.1	✓	2
3.2	odd	2		2394.2.b.d.1709.1	yes	2
19.18	odd	2		2394.2.b.d.1709.2	yes	2
57.56	even	2	inner	2394.2.b.a.1709.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.b.a.1709.1	✓	2	1.1	even	1	trivial
2394.2.b.a.1709.2	yes	2	57.56	even	2	inner
2394.2.b.d.1709.1	yes	2	3.2	odd	2
2394.2.b.d.1709.2	yes	2	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -4.24264i q^{13} +1.00000 q^{14} +1.00000 q^{16} +(-1.00000 + 4.24264i) q^{19} +8.48528i q^{23} +5.00000 q^{25} +4.24264i q^{26} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{32} -8.48528i q^{37} +(1.00000 - 4.24264i) q^{38} +8.00000 q^{43} -8.48528i q^{46} +4.24264i q^{47} +1.00000 q^{49} -5.00000 q^{50} -4.24264i q^{52} +6.00000 q^{53} +1.00000 q^{56} +6.00000 q^{58} +10.0000 q^{61} +1.00000 q^{64} -4.24264i q^{67} +6.00000 q^{71} +2.00000 q^{73} +8.48528i q^{74} +(-1.00000 + 4.24264i) q^{76} +4.24264i q^{79} +4.24264i q^{83} -8.00000 q^{86} -6.00000 q^{89} +4.24264i q^{91} +8.48528i q^{92} -4.24264i q^{94} +4.24264i q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 2 q^{14} + 2 q^{16} - 2 q^{19} + 10 q^{25} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 2 q^{38} + 16 q^{43} + 2 q^{49} - 10 q^{50} + 12 q^{53} + 2 q^{56} + 12 q^{58} + 20 q^{61} + 2 q^{64} + 12 q^{71} + 4 q^{73} - 2 q^{76} - 16 q^{86} - 12 q^{89} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 2 * q^14 + 2 * q^16 - 2 * q^19 + 10 * q^25 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 2 * q^38 + 16 * q^43 + 2 * q^49 - 10 * q^50 + 12 * q^53 + 2 * q^56 + 12 * q^58 + 20 * q^61 + 2 * q^64 + 12 * q^71 + 4 * q^73 - 2 * q^76 - 16 * q^86 - 12 * q^89 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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