LMFDB - Embedded newform 3360.2.z.b.1231.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1231,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1231"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.z (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.4
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3360.1231
Dual form			3360.2.z.b.1231.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000 q^{5} +(1.00000 + 2.44949i) q^{7} -1.00000 q^{9} -2.00000 q^{11} +6.89898 q^{13} +1.00000i q^{15} -4.89898i q^{17} -6.89898i q^{19} +(-2.44949 + 1.00000i) q^{21} -6.00000i q^{23} +1.00000 q^{25} -1.00000i q^{27} -8.89898i q^{29} -2.00000 q^{31} -2.00000i q^{33} +(1.00000 + 2.44949i) q^{35} -7.79796i q^{37} +6.89898i q^{39} -2.89898i q^{41} +8.89898 q^{43} -1.00000 q^{45} -10.8990 q^{47} +(-5.00000 + 4.89898i) q^{49} +4.89898 q^{51} -9.79796i q^{53} -2.00000 q^{55} +6.89898 q^{57} +3.10102i q^{59} -0.898979 q^{61} +(-1.00000 - 2.44949i) q^{63} +6.89898 q^{65} +4.89898 q^{67} +6.00000 q^{69} +12.8990i q^{71} +8.89898i q^{73} +1.00000i q^{75} +(-2.00000 - 4.89898i) q^{77} -4.00000i q^{79} +1.00000 q^{81} +4.00000i q^{83} -4.89898i q^{85} +8.89898 q^{87} +1.10102i q^{89} +(6.89898 + 16.8990i) q^{91} -2.00000i q^{93} -6.89898i q^{95} +0.898979i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7} - 4 q^{9} - 8 q^{11} + 8 q^{13} + 4 q^{25} - 8 q^{31} + 4 q^{35} + 16 q^{43} - 4 q^{45} - 24 q^{47} - 20 q^{49} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 4 q^{63} + 8 q^{65} + 24 q^{69}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 - 4 * q^9 - 8 * q^11 + 8 * q^13 + 4 * q^25 - 8 * q^31 + 4 * q^35 + 16 * q^43 - 4 * q^45 - 24 * q^47 - 20 * q^49 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 4 * q^63 + 8 * q^65 + 24 * q^69 - 8 * q^77 + 4 * q^81 + 16 * q^87 + 8 * q^91 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	1.00000	+	2.44949i	0.377964	+	0.925820i
$7$	1.00000	+	2.44949i	0.377964	+	0.925820i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	6.89898			1.91343			0.956716	−	0.291022i	$-0.0939953\pi$
$13$	6.89898			1.91343			0.956716	+	0.291022i	$0.0939953\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$		−	6.89898i		−	1.58273i	−0.611341	−	0.791367i	$-0.709370\pi$
$19$		−	6.89898i		−	1.58273i	0.611341	−	0.791367i	$-0.290630\pi$
$20$		0			0
$21$	−2.44949	+	1.00000i	−0.534522	+	0.218218i
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	8.89898i		−	1.65250i	−0.563304	−	0.826250i	$-0.690470\pi$
$29$		−	8.89898i		−	1.65250i	0.563304	−	0.826250i	$-0.309530\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	1.00000	+	2.44949i	0.169031	+	0.414039i
$36$		0			0
$37$		−	7.79796i		−	1.28198i	−0.767551	−	0.640988i	$-0.778525\pi$
$37$		−	7.79796i		−	1.28198i	0.767551	−	0.640988i	$-0.221475\pi$
$38$		0			0
$39$			6.89898i			1.10472i
$40$		0			0
$41$		−	2.89898i		−	0.452745i	−0.974041	−	0.226372i	$-0.927313\pi$
$41$		−	2.89898i		−	0.452745i	0.974041	−	0.226372i	$-0.0726866\pi$
$42$		0			0
$43$	8.89898			1.35708			0.678541	−	0.734563i	$-0.262613\pi$
$43$	8.89898			1.35708			0.678541	+	0.734563i	$0.262613\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$	−10.8990			−1.58978			−0.794890	−	0.606754i	$-0.792471\pi$
$47$	−10.8990			−1.58978			−0.794890	+	0.606754i	$0.792471\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$	4.89898			0.685994
$52$		0			0
$53$		−	9.79796i		−	1.34585i	−0.739709	−	0.672927i	$-0.765037\pi$
$53$		−	9.79796i		−	1.34585i	0.739709	−	0.672927i	$-0.234963\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$	6.89898			0.913792
$58$		0			0
$59$			3.10102i			0.403718i	0.979415	+	0.201859i	$0.0646984\pi$
$59$			3.10102i			0.403718i	−0.979415	+	0.201859i	$0.935302\pi$
$60$		0			0
$61$	−0.898979			−0.115103			−0.0575513	−	0.998343i	$-0.518329\pi$
$61$	−0.898979			−0.115103			−0.0575513	+	0.998343i	$0.518329\pi$
$62$		0			0
$63$	−1.00000	−	2.44949i	−0.125988	−	0.308607i
$64$		0			0
$65$	6.89898			0.855713
$66$		0			0
$67$	4.89898			0.598506			0.299253	−	0.954174i	$-0.403263\pi$
$67$	4.89898			0.598506			0.299253	+	0.954174i	$0.403263\pi$
$68$		0			0
$69$	6.00000			0.722315
$70$		0			0
$71$			12.8990i			1.53083i	0.643539	+	0.765414i	$0.277466\pi$
$71$			12.8990i			1.53083i	−0.643539	+	0.765414i	$0.722534\pi$
$72$		0			0
$73$			8.89898i			1.04155i	0.853695	+	0.520773i	$0.174356\pi$
$73$			8.89898i			1.04155i	−0.853695	+	0.520773i	$0.825644\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	−2.00000	−	4.89898i	−0.227921	−	0.558291i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$		−	4.89898i		−	0.531369i
$86$		0			0
$87$	8.89898			0.954071
$88$		0			0
$89$			1.10102i			0.116708i	0.998296	+	0.0583540i	$0.0185852\pi$
$89$			1.10102i			0.116708i	−0.998296	+	0.0583540i	$0.981415\pi$
$90$		0			0
$91$	6.89898	+	16.8990i	0.723210	+	1.77149i
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$		−	6.89898i		−	0.707820i
$96$		0			0
$97$			0.898979i			0.0912775i	0.998958	+	0.0456388i	$0.0145323\pi$
$97$			0.898979i			0.0912775i	−0.998958	+	0.0456388i	$0.985468\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−3.79796			−0.377911			−0.188956	−	0.981986i	$-0.560510\pi$
$101$	−3.79796			−0.377911			−0.188956	+	0.981986i	$0.560510\pi$
$102$		0			0
$103$	0.202041			0.0199077			0.00995385	−	0.999950i	$-0.496832\pi$
$103$	0.202041			0.0199077			0.00995385	+	0.999950i	$0.496832\pi$
$104$		0			0
$105$	−2.44949	+	1.00000i	−0.239046	+	0.0975900i
$106$		0			0
$107$	7.79796			0.753857			0.376929	−	0.926242i	$-0.376980\pi$
$107$	7.79796			0.753857			0.376929	+	0.926242i	$0.376980\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$	7.79796			0.740150
$112$		0			0
$113$	−1.79796			−0.169138			−0.0845689	−	0.996418i	$-0.526951\pi$
$113$	−1.79796			−0.169138			−0.0845689	+	0.996418i	$0.526951\pi$
$114$		0			0
$115$		−	6.00000i		−	0.559503i
$116$		0			0
$117$	−6.89898			−0.637811
$118$		0			0
$119$	12.0000	−	4.89898i	1.10004	−	0.449089i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	2.89898			0.261392
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	0.898979i		−	0.0797715i	−0.999204	−	0.0398858i	$-0.987301\pi$
$127$		−	0.898979i		−	0.0797715i	0.999204	−	0.0398858i	$-0.0126994\pi$
$128$		0			0
$129$			8.89898i			0.783511i
$130$		0			0
$131$		−	7.10102i		−	0.620419i	−0.950668	−	0.310210i	$-0.899601\pi$
$131$		−	7.10102i		−	0.620419i	0.950668	−	0.310210i	$-0.100399\pi$
$132$		0			0
$133$	16.8990	−	6.89898i	1.46533	−	0.598217i
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	13.7980			1.17884			0.589420	−	0.807827i	$-0.299356\pi$
$137$	13.7980			1.17884			0.589420	+	0.807827i	$0.299356\pi$
$138$		0			0
$139$		−	6.89898i		−	0.585164i	−0.956240	−	0.292582i	$-0.905486\pi$
$139$		−	6.89898i		−	0.585164i	0.956240	−	0.292582i	$-0.0945144\pi$
$140$		0			0
$141$		−	10.8990i		−	0.917860i
$142$		0			0
$143$	−13.7980			−1.15384
$144$		0			0
$145$		−	8.89898i		−	0.739020i
$146$		0			0
$147$	−4.89898	−	5.00000i	−0.404061	−	0.412393i
$148$		0			0
$149$			16.8990i			1.38442i	0.721697	+	0.692209i	$0.243362\pi$
$149$			16.8990i			1.38442i	−0.721697	+	0.692209i	$0.756638\pi$
$150$		0			0
$151$		−	5.79796i		−	0.471831i	−0.971774	−	0.235916i	$-0.924191\pi$
$151$		−	5.79796i		−	0.471831i	0.971774	−	0.235916i	$-0.0758089\pi$
$152$		0			0
$153$			4.89898i			0.396059i
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	16.6969			1.33256			0.666280	−	0.745701i	$-0.267885\pi$
$157$	16.6969			1.33256			0.666280	+	0.745701i	$0.267885\pi$
$158$		0			0
$159$	9.79796			0.777029
$160$		0			0
$161$	14.6969	−	6.00000i	1.15828	−	0.472866i
$162$		0			0
$163$	−22.6969			−1.77776			−0.888881	−	0.458139i	$-0.848516\pi$
$163$	−22.6969			−1.77776			−0.888881	+	0.458139i	$0.848516\pi$
$164$		0			0
$165$		−	2.00000i		−	0.155700i
$166$		0			0
$167$	−10.8990			−0.843388			−0.421694	−	0.906738i	$-0.638564\pi$
$167$	−10.8990			−0.843388			−0.421694	+	0.906738i	$0.638564\pi$
$168$		0			0
$169$	34.5959			2.66122
$170$		0			0
$171$			6.89898i			0.527578i
$172$		0			0
$173$	−11.7980			−0.896982			−0.448491	−	0.893787i	$-0.648038\pi$
$173$	−11.7980			−0.896982			−0.448491	+	0.893787i	$0.648038\pi$
$174$		0			0
$175$	1.00000	+	2.44949i	0.0755929	+	0.185164i
$176$		0			0
$177$	−3.10102			−0.233087
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	−20.8990			−1.55341			−0.776704	−	0.629865i	$-0.783110\pi$
$181$	−20.8990			−1.55341			−0.776704	+	0.629865i	$0.783110\pi$
$182$		0			0
$183$		−	0.898979i		−	0.0664545i
$184$		0			0
$185$		−	7.79796i		−	0.573317i
$186$		0			0
$187$			9.79796i			0.716498i
$188$		0			0
$189$	2.44949	−	1.00000i	0.178174	−	0.0727393i
$190$		0			0
$191$		−	7.10102i		−	0.513812i	−0.966436	−	0.256906i	$-0.917297\pi$
$191$		−	7.10102i		−	0.513812i	0.966436	−	0.256906i	$-0.0827030\pi$
$192$		0			0
$193$	−0.202041			−0.0145432			−0.00727162	−	0.999974i	$-0.502315\pi$
$193$	−0.202041			−0.0145432			−0.00727162	+	0.999974i	$0.502315\pi$
$194$		0			0
$195$			6.89898i			0.494046i
$196$		0			0
$197$			13.7980i			0.983064i	0.870860	+	0.491532i	$0.163563\pi$
$197$			13.7980i			0.983064i	−0.870860	+	0.491532i	$0.836437\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$		0			0
$201$			4.89898i			0.345547i
$202$		0			0
$203$	21.7980	−	8.89898i	1.52992	−	0.624586i
$204$		0			0
$205$		−	2.89898i		−	0.202474i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$			13.7980i			0.954425i
$210$		0			0
$211$	−22.0000			−1.51454			−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000			−1.51454			−0.757271	+	0.653101i	$0.773468\pi$
$212$		0			0
$213$	−12.8990			−0.883824
$214$		0			0
$215$	8.89898			0.606905
$216$		0			0
$217$	−2.00000	−	4.89898i	−0.135769	−	0.332564i
$218$		0			0
$219$	−8.89898			−0.601337
$220$		0			0
$221$		−	33.7980i		−	2.27350i
$222$		0			0
$223$	6.00000			0.401790			0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000			0.401790			0.200895	+	0.979613i	$0.435615\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	24.4949			1.61867			0.809334	−	0.587348i	$-0.199828\pi$
$229$	24.4949			1.61867			0.809334	+	0.587348i	$0.199828\pi$
$230$		0			0
$231$	4.89898	−	2.00000i	0.322329	−	0.131590i
$232$		0			0
$233$	−16.0000			−1.04819			−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000			−1.04819			−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$	−10.8990			−0.710971
$236$		0			0
$237$	4.00000			0.259828
$238$		0			0
$239$		−	2.69694i		−	0.174450i	−0.996189	−	0.0872252i	$-0.972200\pi$
$239$		−	2.69694i		−	0.174450i	0.996189	−	0.0872252i	$-0.0278000\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−5.00000	+	4.89898i	−0.319438	+	0.312984i
$246$		0			0
$247$		−	47.5959i		−	3.02846i
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$			12.8990i			0.814176i	0.913389	+	0.407088i	$0.133456\pi$
$251$			12.8990i			0.814176i	−0.913389	+	0.407088i	$0.866544\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$	4.89898			0.306786
$256$		0			0
$257$			15.1010i			0.941976i	0.882140	+	0.470988i	$0.156102\pi$
$257$			15.1010i			0.941976i	−0.882140	+	0.470988i	$0.843898\pi$
$258$		0			0
$259$	19.1010	−	7.79796i	1.18688	−	0.484542i
$260$		0			0
$261$			8.89898i			0.550833i
$262$		0			0
$263$		−	11.7980i		−	0.727493i	−0.931498	−	0.363747i	$-0.881497\pi$
$263$		−	11.7980i		−	0.727493i	0.931498	−	0.363747i	$-0.118503\pi$
$264$		0			0
$265$		−	9.79796i		−	0.601884i
$266$		0			0
$267$	−1.10102			−0.0673814
$268$		0			0
$269$	15.7980			0.963219			0.481609	−	0.876386i	$-0.340052\pi$
$269$	15.7980			0.963219			0.481609	+	0.876386i	$0.340052\pi$
$270$		0			0
$271$	23.7980			1.44562			0.722812	−	0.691045i	$-0.242849\pi$
$271$	23.7980			1.44562			0.722812	+	0.691045i	$0.242849\pi$
$272$		0			0
$273$	−16.8990	+	6.89898i	−1.02277	+	0.417545i
$274$		0			0
$275$	−2.00000			−0.120605
$276$		0			0
$277$			3.79796i			0.228197i	0.993469	+	0.114099i	$0.0363980\pi$
$277$			3.79796i			0.228197i	−0.993469	+	0.114099i	$0.963602\pi$
$278$		0			0
$279$	2.00000			0.119737
$280$		0			0
$281$	27.7980			1.65829			0.829144	−	0.559036i	$-0.188829\pi$
$281$	27.7980			1.65829			0.829144	+	0.559036i	$0.188829\pi$
$282$		0			0
$283$			9.79796i			0.582428i	0.956658	+	0.291214i	$0.0940592\pi$
$283$			9.79796i			0.582428i	−0.956658	+	0.291214i	$0.905941\pi$
$284$		0			0
$285$	6.89898			0.408660
$286$		0			0
$287$	7.10102	−	2.89898i	0.419160	−	0.171121i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−0.898979			−0.0526991
$292$		0			0
$293$	−17.5959			−1.02796			−0.513982	−	0.857801i	$-0.671830\pi$
$293$	−17.5959			−1.02796			−0.513982	+	0.857801i	$0.671830\pi$
$294$		0			0
$295$			3.10102i			0.180548i
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$		−	41.3939i		−	2.39387i
$300$		0			0
$301$	8.89898	+	21.7980i	0.512929	+	1.25641i
$302$		0			0
$303$		−	3.79796i		−	0.218187i
$304$		0			0
$305$	−0.898979			−0.0514754
$306$		0			0
$307$			23.5959i			1.34669i	0.739328	+	0.673345i	$0.235143\pi$
$307$			23.5959i			1.34669i	−0.739328	+	0.673345i	$0.764857\pi$
$308$		0			0
$309$			0.202041i			0.0114937i
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$			20.4949i			1.15844i	0.815171	+	0.579220i	$0.196643\pi$
$313$			20.4949i			1.15844i	−0.815171	+	0.579220i	$0.803357\pi$
$314$		0			0
$315$	−1.00000	−	2.44949i	−0.0563436	−	0.138013i
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$			17.7980i			0.996494i
$320$		0			0
$321$			7.79796i			0.435240i
$322$		0			0
$323$	−33.7980			−1.88057
$324$		0			0
$325$	6.89898			0.382687
$326$		0			0
$327$	−4.00000			−0.221201
$328$		0			0
$329$	−10.8990	−	26.6969i	−0.600880	−	1.47185i
$330$		0			0
$331$	35.3939			1.94542			0.972712	−	0.232017i	$-0.0745325\pi$
$331$	35.3939			1.94542			0.972712	+	0.232017i	$0.0745325\pi$
$332$		0			0
$333$			7.79796i			0.427326i
$334$		0			0
$335$	4.89898			0.267660
$336$		0			0
$337$	12.2020			0.664688			0.332344	−	0.943158i	$-0.392161\pi$
$337$	12.2020			0.664688			0.332344	+	0.943158i	$0.392161\pi$
$338$		0			0
$339$		−	1.79796i		−	0.0976517i
$340$		0			0
$341$	4.00000			0.216612
$342$		0			0
$343$	−17.0000	−	7.34847i	−0.917914	−	0.396780i
$344$		0			0
$345$	6.00000			0.323029
$346$		0			0
$347$	13.5959			0.729867			0.364934	−	0.931034i	$-0.381092\pi$
$347$	13.5959			0.729867			0.364934	+	0.931034i	$0.381092\pi$
$348$		0			0
$349$	−1.30306			−0.0697513			−0.0348756	−	0.999392i	$-0.511104\pi$
$349$	−1.30306			−0.0697513			−0.0348756	+	0.999392i	$0.511104\pi$
$350$		0			0
$351$		−	6.89898i		−	0.368240i
$352$		0			0
$353$		−	5.30306i		−	0.282253i	−0.989992	−	0.141127i	$-0.954927\pi$
$353$		−	5.30306i		−	0.282253i	0.989992	−	0.141127i	$-0.0450725\pi$
$354$		0			0
$355$			12.8990i			0.684607i
$356$		0			0
$357$	4.89898	+	12.0000i	0.259281	+	0.635107i
$358$		0			0
$359$		−	16.8990i		−	0.891894i	−0.895059	−	0.445947i	$-0.852867\pi$
$359$		−	16.8990i		−	0.891894i	0.895059	−	0.445947i	$-0.147133\pi$
$360$		0			0
$361$	−28.5959			−1.50505
$362$		0			0
$363$		−	7.00000i		−	0.367405i
$364$		0			0
$365$			8.89898i			0.465794i
$366$		0			0
$367$	−3.79796			−0.198252			−0.0991259	−	0.995075i	$-0.531605\pi$
$367$	−3.79796			−0.198252			−0.0991259	+	0.995075i	$0.531605\pi$
$368$		0			0
$369$			2.89898i			0.150915i
$370$		0			0
$371$	24.0000	−	9.79796i	1.24602	−	0.508685i
$372$		0			0
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$		−	61.3939i		−	3.16195i
$378$		0			0
$379$	27.3939			1.40713			0.703564	−	0.710631i	$-0.251591\pi$
$379$	27.3939			1.40713			0.703564	+	0.710631i	$0.251591\pi$
$380$		0			0
$381$	0.898979			0.0460561
$382$		0			0
$383$	18.8990			0.965693			0.482846	−	0.875705i	$-0.339603\pi$
$383$	18.8990			0.965693			0.482846	+	0.875705i	$0.339603\pi$
$384$		0			0
$385$	−2.00000	−	4.89898i	−0.101929	−	0.249675i
$386$		0			0
$387$	−8.89898			−0.452361
$388$		0			0
$389$			2.69694i			0.136740i	0.997660	+	0.0683701i	$0.0217799\pi$
$389$			2.69694i			0.136740i	−0.997660	+	0.0683701i	$0.978220\pi$
$390$		0			0
$391$	−29.3939			−1.48651
$392$		0			0
$393$	7.10102			0.358199
$394$		0			0
$395$		−	4.00000i		−	0.201262i
$396$		0			0
$397$	16.6969			0.837995			0.418998	−	0.907987i	$-0.362382\pi$
$397$	16.6969			0.837995			0.418998	+	0.907987i	$0.362382\pi$
$398$		0			0
$399$	6.89898	+	16.8990i	0.345381	+	0.846007i
$400$		0			0
$401$	7.79796			0.389411			0.194706	−	0.980862i	$-0.437625\pi$
$401$	7.79796			0.389411			0.194706	+	0.980862i	$0.437625\pi$
$402$		0			0
$403$	−13.7980			−0.687325
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$			15.5959i			0.773061i
$408$		0			0
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		0			0
$411$			13.7980i			0.680603i
$412$		0			0
$413$	−7.59592	+	3.10102i	−0.373771	+	0.152591i
$414$		0			0
$415$			4.00000i			0.196352i
$416$		0			0
$417$	6.89898			0.337844
$418$		0			0
$419$		−	34.2929i		−	1.67532i	−0.546195	−	0.837658i	$-0.683924\pi$
$419$		−	34.2929i		−	1.67532i	0.546195	−	0.837658i	$-0.316076\pi$
$420$		0			0
$421$			25.7980i			1.25732i	0.777682	+	0.628658i	$0.216395\pi$
$421$			25.7980i			1.25732i	−0.777682	+	0.628658i	$0.783605\pi$
$422$		0			0
$423$	10.8990			0.529927
$424$		0			0
$425$		−	4.89898i		−	0.237635i
$426$		0			0
$427$	−0.898979	−	2.20204i	−0.0435047	−	0.106564i
$428$		0			0
$429$		−	13.7980i		−	0.666172i
$430$		0			0
$431$		−	12.8990i		−	0.621322i	−0.950521	−	0.310661i	$-0.899450\pi$
$431$		−	12.8990i		−	0.621322i	0.950521	−	0.310661i	$-0.100550\pi$
$432$		0			0
$433$			34.6969i			1.66743i	0.552196	+	0.833714i	$0.313790\pi$
$433$			34.6969i			1.66743i	−0.552196	+	0.833714i	$0.686210\pi$
$434$		0			0
$435$	8.89898			0.426673
$436$		0			0
$437$	−41.3939			−1.98014
$438$		0			0
$439$	−4.20204			−0.200552			−0.100276	−	0.994960i	$-0.531973\pi$
$439$	−4.20204			−0.200552			−0.100276	+	0.994960i	$0.531973\pi$
$440$		0			0
$441$	5.00000	−	4.89898i	0.238095	−	0.233285i
$442$		0			0
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$			1.10102i			0.0521934i
$446$		0			0
$447$	−16.8990			−0.799294
$448$		0			0
$449$	4.20204			0.198307			0.0991533	−	0.995072i	$-0.468387\pi$
$449$	4.20204			0.198307			0.0991533	+	0.995072i	$0.468387\pi$
$450$		0			0
$451$			5.79796i			0.273015i
$452$		0			0
$453$	5.79796			0.272412
$454$		0			0
$455$	6.89898	+	16.8990i	0.323429	+	0.792236i
$456$		0			0
$457$	−7.79796			−0.364773			−0.182387	−	0.983227i	$-0.558382\pi$
$457$	−7.79796			−0.364773			−0.182387	+	0.983227i	$0.558382\pi$
$458$		0			0
$459$	−4.89898			−0.228665
$460$		0			0
$461$	−23.7980			−1.10838			−0.554191	−	0.832390i	$-0.686972\pi$
$461$	−23.7980			−1.10838			−0.554191	+	0.832390i	$0.686972\pi$
$462$		0			0
$463$			2.69694i			0.125337i	0.998034	+	0.0626687i	$0.0199611\pi$
$463$			2.69694i			0.125337i	−0.998034	+	0.0626687i	$0.980039\pi$
$464$		0			0
$465$		−	2.00000i		−	0.0927478i
$466$		0			0
$467$		−	2.20204i		−	0.101898i	−0.998701	−	0.0509492i	$-0.983775\pi$
$467$		−	2.20204i		−	0.101898i	0.998701	−	0.0509492i	$-0.0162246\pi$
$468$		0			0
$469$	4.89898	+	12.0000i	0.226214	+	0.554109i
$470$		0			0
$471$			16.6969i			0.769354i
$472$		0			0
$473$	−17.7980			−0.818351
$474$		0			0
$475$		−	6.89898i		−	0.316547i
$476$		0			0
$477$			9.79796i			0.448618i
$478$		0			0
$479$	11.5959			0.529831			0.264916	−	0.964272i	$-0.414656\pi$
$479$	11.5959			0.529831			0.264916	+	0.964272i	$0.414656\pi$
$480$		0			0
$481$		−	53.7980i		−	2.45298i
$482$		0			0
$483$	6.00000	+	14.6969i	0.273009	+	0.668734i
$484$		0			0
$485$			0.898979i			0.0408206i
$486$		0			0
$487$			24.8990i			1.12828i	0.825679	+	0.564140i	$0.190792\pi$
$487$			24.8990i			1.12828i	−0.825679	+	0.564140i	$0.809208\pi$
$488$		0			0
$489$		−	22.6969i		−	1.02639i
$490$		0			0
$491$	−13.5959			−0.613575			−0.306788	−	0.951778i	$-0.599254\pi$
$491$	−13.5959			−0.613575			−0.306788	+	0.951778i	$0.599254\pi$
$492$		0			0
$493$	−43.5959			−1.96346
$494$		0			0
$495$	2.00000			0.0898933
$496$		0			0
$497$	−31.5959	+	12.8990i	−1.41727	+	0.578598i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		0			0
$501$		−	10.8990i		−	0.486930i
$502$		0			0
$503$	−38.4949			−1.71640			−0.858201	−	0.513313i	$-0.828418\pi$
$503$	−38.4949			−1.71640			−0.858201	+	0.513313i	$0.828418\pi$
$504$		0			0
$505$	−3.79796			−0.169007
$506$		0			0
$507$			34.5959i			1.53646i
$508$		0			0
$509$	−1.59592			−0.0707378			−0.0353689	−	0.999374i	$-0.511261\pi$
$509$	−1.59592			−0.0707378			−0.0353689	+	0.999374i	$0.511261\pi$
$510$		0			0
$511$	−21.7980	+	8.89898i	−0.964285	+	0.393668i
$512$		0			0
$513$	−6.89898			−0.304597
$514$		0			0
$515$	0.202041			0.00890299
$516$		0			0
$517$	21.7980			0.958673
$518$		0			0
$519$		−	11.7980i		−	0.517873i
$520$		0			0
$521$			42.8990i			1.87944i	0.341947	+	0.939719i	$0.388914\pi$
$521$			42.8990i			1.87944i	−0.341947	+	0.939719i	$0.611086\pi$
$522$		0			0
$523$			35.5959i			1.55650i	0.627954	+	0.778250i	$0.283893\pi$
$523$			35.5959i			1.55650i	−0.627954	+	0.778250i	$0.716107\pi$
$524$		0			0
$525$	−2.44949	+	1.00000i	−0.106904	+	0.0436436i
$526$		0			0
$527$			9.79796i			0.426806i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		−	3.10102i		−	0.134573i
$532$		0			0
$533$		−	20.0000i		−	0.866296i
$534$		0			0
$535$	7.79796			0.337135
$536$		0			0
$537$			10.0000i			0.431532i
$538$		0			0
$539$	10.0000	−	9.79796i	0.430730	−	0.422028i
$540$		0			0
$541$			17.3939i			0.747821i	0.927465	+	0.373911i	$0.121983\pi$
$541$			17.3939i			0.747821i	−0.927465	+	0.373911i	$0.878017\pi$
$542$		0			0
$543$		−	20.8990i		−	0.896861i
$544$		0			0
$545$			4.00000i			0.171341i
$546$		0			0
$547$	−38.2929			−1.63728			−0.818642	−	0.574304i	$-0.805273\pi$
$547$	−38.2929			−1.63728			−0.818642	+	0.574304i	$0.805273\pi$
$548$		0			0
$549$	0.898979			0.0383675
$550$		0			0
$551$	−61.3939			−2.61547
$552$		0			0
$553$	9.79796	−	4.00000i	0.416652	−	0.170097i
$554$		0			0
$555$	7.79796			0.331005
$556$		0			0
$557$			13.7980i			0.584638i	0.956321	+	0.292319i	$0.0944269\pi$
$557$			13.7980i			0.584638i	−0.956321	+	0.292319i	$0.905573\pi$
$558$		0			0
$559$	61.3939			2.59668
$560$		0			0
$561$	−9.79796			−0.413670
$562$		0			0
$563$		−	21.7980i		−	0.918674i	−0.888262	−	0.459337i	$-0.848087\pi$
$563$		−	21.7980i		−	0.918674i	0.888262	−	0.459337i	$-0.151913\pi$
$564$		0			0
$565$	−1.79796			−0.0756407
$566$		0			0
$567$	1.00000	+	2.44949i	0.0419961	+	0.102869i
$568$		0			0
$569$	−21.5959			−0.905348			−0.452674	−	0.891676i	$-0.649530\pi$
$569$	−21.5959			−0.905348			−0.452674	+	0.891676i	$0.649530\pi$
$570$		0			0
$571$	3.79796			0.158940			0.0794698	−	0.996837i	$-0.474677\pi$
$571$	3.79796			0.158940			0.0794698	+	0.996837i	$0.474677\pi$
$572$		0			0
$573$	7.10102			0.296649
$574$		0			0
$575$		−	6.00000i		−	0.250217i
$576$		0			0
$577$			15.1010i			0.628664i	0.949313	+	0.314332i	$0.101780\pi$
$577$			15.1010i			0.628664i	−0.949313	+	0.314332i	$0.898220\pi$
$578$		0			0
$579$		−	0.202041i		−	0.00839654i
$580$		0			0
$581$	−9.79796	+	4.00000i	−0.406488	+	0.165948i
$582$		0			0
$583$			19.5959i			0.811580i
$584$		0			0
$585$	−6.89898			−0.285238
$586$		0			0
$587$			26.2020i			1.08147i	0.841192	+	0.540737i	$0.181855\pi$
$587$			26.2020i			1.08147i	−0.841192	+	0.540737i	$0.818145\pi$
$588$		0			0
$589$			13.7980i			0.568535i
$590$		0			0
$591$	−13.7980			−0.567572
$592$		0			0
$593$			3.10102i			0.127344i	0.997971	+	0.0636718i	$0.0202811\pi$
$593$			3.10102i			0.127344i	−0.997971	+	0.0636718i	$0.979719\pi$
$594$		0			0
$595$	12.0000	−	4.89898i	0.491952	−	0.200839i
$596$		0			0
$597$		−	10.0000i		−	0.409273i
$598$		0			0
$599$			8.89898i			0.363602i	0.983335	+	0.181801i	$0.0581927\pi$
$599$			8.89898i			0.363602i	−0.983335	+	0.181801i	$0.941807\pi$
$600$		0			0
$601$			5.79796i			0.236504i	0.992984	+	0.118252i	$0.0377290\pi$
$601$			5.79796i			0.236504i	−0.992984	+	0.118252i	$0.962271\pi$
$602$		0			0
$603$	−4.89898			−0.199502
$604$		0			0
$605$	−7.00000			−0.284590
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$	8.89898	+	21.7980i	0.360605	+	0.883298i
$610$		0			0
$611$	−75.1918			−3.04194
$612$		0			0
$613$		−	19.7980i		−	0.799632i	−0.916595	−	0.399816i	$-0.869074\pi$
$613$		−	19.7980i		−	0.799632i	0.916595	−	0.399816i	$-0.130926\pi$
$614$		0			0
$615$	2.89898			0.116898
$616$		0			0
$617$	22.2020			0.893821			0.446910	−	0.894579i	$-0.352524\pi$
$617$	22.2020			0.893821			0.446910	+	0.894579i	$0.352524\pi$
$618$		0			0
$619$		−	24.2929i		−	0.976412i	−0.872728	−	0.488206i	$-0.837651\pi$
$619$		−	24.2929i		−	0.976412i	0.872728	−	0.488206i	$-0.162349\pi$
$620$		0			0
$621$	−6.00000			−0.240772
$622$		0			0
$623$	−2.69694	+	1.10102i	−0.108051	+	0.0441115i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−13.7980			−0.551037
$628$		0			0
$629$	−38.2020			−1.52322
$630$		0			0
$631$		−	5.79796i		−	0.230813i	−0.993318	−	0.115407i	$-0.963183\pi$
$631$		−	5.79796i		−	0.230813i	0.993318	−	0.115407i	$-0.0368171\pi$
$632$		0			0
$633$		−	22.0000i		−	0.874421i
$634$		0			0
$635$		−	0.898979i		−	0.0356749i
$636$		0			0
$637$	−34.4949	+	33.7980i	−1.36674	+	1.33912i
$638$		0			0
$639$		−	12.8990i		−	0.510276i
$640$		0			0
$641$	−6.40408			−0.252946			−0.126473	−	0.991970i	$-0.540366\pi$
$641$	−6.40408			−0.252946			−0.126473	+	0.991970i	$0.540366\pi$
$642$		0			0
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$		0			0
$645$			8.89898i			0.350397i
$646$		0			0
$647$	3.30306			0.129857			0.0649284	−	0.997890i	$-0.479318\pi$
$647$	3.30306			0.129857			0.0649284	+	0.997890i	$0.479318\pi$
$648$		0			0
$649$		−	6.20204i		−	0.243451i
$650$		0			0
$651$	4.89898	−	2.00000i	0.192006	−	0.0783862i
$652$		0			0
$653$		−	44.0000i		−	1.72185i	−0.508729	−	0.860927i	$-0.669885\pi$
$653$		−	44.0000i		−	1.72185i	0.508729	−	0.860927i	$-0.330115\pi$
$654$		0			0
$655$		−	7.10102i		−	0.277460i
$656$		0			0
$657$		−	8.89898i		−	0.347182i
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$	30.6969			1.19397			0.596986	−	0.802251i	$-0.296365\pi$
$661$	30.6969			1.19397			0.596986	+	0.802251i	$0.296365\pi$
$662$		0			0
$663$	33.7980			1.31260
$664$		0			0
$665$	16.8990	−	6.89898i	0.655314	−	0.267531i
$666$		0			0
$667$	−53.3939			−2.06742
$668$		0			0
$669$			6.00000i			0.231973i
$670$		0			0
$671$	1.79796			0.0694094
$672$		0			0
$673$	25.5959			0.986650			0.493325	−	0.869845i	$-0.335781\pi$
$673$	25.5959			0.986650			0.493325	+	0.869845i	$0.335781\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$	12.2020			0.468963			0.234481	−	0.972121i	$-0.424661\pi$
$677$	12.2020			0.468963			0.234481	+	0.972121i	$0.424661\pi$
$678$		0			0
$679$	−2.20204	+	0.898979i	−0.0845066	+	0.0344997i
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$	31.7980			1.21672			0.608358	−	0.793663i	$-0.291829\pi$
$683$	31.7980			1.21672			0.608358	+	0.793663i	$0.291829\pi$
$684$		0			0
$685$	13.7980			0.527193
$686$		0			0
$687$			24.4949i			0.934539i
$688$		0			0
$689$		−	67.5959i		−	2.57520i
$690$		0			0
$691$		−	14.4949i		−	0.551412i	−0.961242	−	0.275706i	$-0.911088\pi$
$691$		−	14.4949i		−	0.551412i	0.961242	−	0.275706i	$-0.0889116\pi$
$692$		0			0
$693$	2.00000	+	4.89898i	0.0759737	+	0.186097i
$694$		0			0
$695$		−	6.89898i		−	0.261693i
$696$		0			0
$697$	−14.2020			−0.537941
$698$		0			0
$699$		−	16.0000i		−	0.605176i
$700$		0			0
$701$			32.8990i			1.24258i	0.783582	+	0.621289i	$0.213391\pi$
$701$			32.8990i			1.24258i	−0.783582	+	0.621289i	$0.786609\pi$
$702$		0			0
$703$	−53.7980			−2.02903
$704$		0			0
$705$		−	10.8990i		−	0.410479i
$706$		0			0
$707$	−3.79796	−	9.30306i	−0.142837	−	0.349878i
$708$		0			0
$709$		−	10.2020i		−	0.383146i	−0.981478	−	0.191573i	$-0.938641\pi$
$709$		−	10.2020i		−	0.383146i	0.981478	−	0.191573i	$-0.0613588\pi$
$710$		0			0
$711$			4.00000i			0.150012i
$712$		0			0
$713$			12.0000i			0.449404i
$714$		0			0
$715$	−13.7980			−0.516014
$716$		0			0
$717$	2.69694			0.100719
$718$		0			0
$719$	−45.7980			−1.70798			−0.853988	−	0.520293i	$-0.825823\pi$
$719$	−45.7980			−1.70798			−0.853988	+	0.520293i	$0.825823\pi$
$720$		0			0
$721$	0.202041	+	0.494897i	0.00752440	+	0.0184309i
$722$		0			0
$723$	−20.0000			−0.743808
$724$		0			0
$725$		−	8.89898i		−	0.330500i
$726$		0			0
$727$	19.3939			0.719279			0.359640	−	0.933091i	$-0.382900\pi$
$727$	19.3939			0.719279			0.359640	+	0.933091i	$0.382900\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	43.5959i		−	1.61245i
$732$		0			0
$733$	−50.4949			−1.86507			−0.932536	−	0.361078i	$-0.882409\pi$
$733$	−50.4949			−1.86507			−0.932536	+	0.361078i	$0.882409\pi$
$734$		0			0
$735$	−4.89898	−	5.00000i	−0.180702	−	0.184428i
$736$		0			0
$737$	−9.79796			−0.360912
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$	47.5959			1.74848
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$			16.8990i			0.619131i
$746$		0			0
$747$		−	4.00000i		−	0.146352i
$748$		0			0
$749$	7.79796	+	19.1010i	0.284931	+	0.697936i
$750$		0			0
$751$			5.79796i			0.211571i	0.994389	+	0.105785i	$0.0337356\pi$
$751$			5.79796i			0.211571i	−0.994389	+	0.105785i	$0.966264\pi$
$752$		0			0
$753$	−12.8990			−0.470065
$754$		0			0
$755$		−	5.79796i		−	0.211009i
$756$		0			0
$757$		−	33.5959i		−	1.22106i	−0.791991	−	0.610532i	$-0.790956\pi$
$757$		−	33.5959i		−	1.22106i	0.791991	−	0.610532i	$-0.209044\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$			28.6969i			1.04026i	0.854086	+	0.520132i	$0.174117\pi$
$761$			28.6969i			1.04026i	−0.854086	+	0.520132i	$0.825883\pi$
$762$		0			0
$763$	−9.79796	+	4.00000i	−0.354710	+	0.144810i
$764$		0			0
$765$			4.89898i			0.177123i
$766$		0			0
$767$			21.3939i			0.772488i
$768$		0			0
$769$			12.4041i			0.447303i	0.974669	+	0.223651i	$0.0717977\pi$
$769$			12.4041i			0.447303i	−0.974669	+	0.223651i	$0.928202\pi$
$770$		0			0
$771$	−15.1010			−0.543850
$772$		0			0
$773$	−0.202041			−0.00726691			−0.00363346	−	0.999993i	$-0.501157\pi$
$773$	−0.202041			−0.00726691			−0.00363346	+	0.999993i	$0.501157\pi$
$774$		0			0
$775$	−2.00000			−0.0718421
$776$		0			0
$777$	7.79796	+	19.1010i	0.279750	+	0.685245i
$778$		0			0
$779$	−20.0000			−0.716574
$780$		0			0
$781$		−	25.7980i		−	0.923124i
$782$		0			0
$783$	−8.89898			−0.318024
$784$		0			0
$785$	16.6969			0.595939
$786$		0			0
$787$			3.59592i			0.128181i	0.997944	+	0.0640903i	$0.0204146\pi$
$787$			3.59592i			0.128181i	−0.997944	+	0.0640903i	$0.979585\pi$
$788$		0			0
$789$	11.7980			0.420018
$790$		0			0
$791$	−1.79796	−	4.40408i	−0.0639281	−	0.156591i
$792$		0			0
$793$	−6.20204			−0.220241
$794$		0			0
$795$	9.79796			0.347498
$796$		0			0
$797$	−13.5959			−0.481592			−0.240796	−	0.970576i	$-0.577408\pi$
$797$	−13.5959			−0.481592			−0.240796	+	0.970576i	$0.577408\pi$
$798$		0			0
$799$			53.3939i			1.88894i
$800$		0			0
$801$		−	1.10102i		−	0.0389026i
$802$		0			0
$803$		−	17.7980i		−	0.628076i
$804$		0			0
$805$	14.6969	−	6.00000i	0.517999	−	0.211472i
$806$		0			0
$807$			15.7980i			0.556114i
$808$		0			0
$809$	4.20204			0.147736			0.0738679	−	0.997268i	$-0.476466\pi$
$809$	4.20204			0.147736			0.0738679	+	0.997268i	$0.476466\pi$
$810$		0			0
$811$		−	22.8990i		−	0.804092i	−0.915619	−	0.402046i	$-0.868299\pi$
$811$		−	22.8990i		−	0.804092i	0.915619	−	0.402046i	$-0.131701\pi$
$812$		0			0
$813$			23.7980i			0.834631i
$814$		0			0
$815$	−22.6969			−0.795039
$816$		0			0
$817$		−	61.3939i		−	2.14790i
$818$		0			0
$819$	−6.89898	−	16.8990i	−0.241070	−	0.590498i
$820$		0			0
$821$		−	18.6969i		−	0.652528i	−0.945279	−	0.326264i	$-0.894210\pi$
$821$		−	18.6969i		−	0.652528i	0.945279	−	0.326264i	$-0.105790\pi$
$822$		0			0
$823$			8.49490i			0.296114i	0.988979	+	0.148057i	$0.0473018\pi$
$823$			8.49490i			0.296114i	−0.988979	+	0.148057i	$0.952698\pi$
$824$		0			0
$825$		−	2.00000i		−	0.0696311i
$826$		0			0
$827$	45.1918			1.57147			0.785737	−	0.618561i	$-0.212284\pi$
$827$	45.1918			1.57147			0.785737	+	0.618561i	$0.212284\pi$
$828$		0			0
$829$	38.6969			1.34400			0.672000	−	0.740551i	$-0.265435\pi$
$829$	38.6969			1.34400			0.672000	+	0.740551i	$0.265435\pi$
$830$		0			0
$831$	−3.79796			−0.131750
$832$		0			0
$833$	24.0000	+	24.4949i	0.831551	+	0.848698i
$834$		0			0
$835$	−10.8990			−0.377175
$836$		0			0
$837$			2.00000i			0.0691301i
$838$		0			0
$839$	45.7980			1.58112			0.790561	−	0.612384i	$-0.209789\pi$
$839$	45.7980			1.58112			0.790561	+	0.612384i	$0.209789\pi$
$840$		0			0
$841$	−50.1918			−1.73075
$842$		0			0
$843$			27.7980i			0.957413i
$844$		0			0
$845$	34.5959			1.19014
$846$		0			0
$847$	−7.00000	−	17.1464i	−0.240523	−	0.589158i
$848$		0			0
$849$	−9.79796			−0.336265
$850$		0			0
$851$	−46.7878			−1.60386
$852$		0			0
$853$	15.3031			0.523967			0.261983	−	0.965072i	$-0.415623\pi$
$853$	15.3031			0.523967			0.261983	+	0.965072i	$0.415623\pi$
$854$		0			0
$855$			6.89898i			0.235940i
$856$		0			0
$857$		−	36.4949i		−	1.24664i	−0.781966	−	0.623321i	$-0.785783\pi$
$857$		−	36.4949i		−	1.24664i	0.781966	−	0.623321i	$-0.214217\pi$
$858$		0			0
$859$			18.8990i			0.644825i	0.946599	+	0.322412i	$0.104494\pi$
$859$			18.8990i			0.644825i	−0.946599	+	0.322412i	$0.895506\pi$
$860$		0			0
$861$	2.89898	+	7.10102i	0.0987970	+	0.242002i
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	−11.7980			−0.401143
$866$		0			0
$867$		−	7.00000i		−	0.237732i
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$	33.7980			1.14520
$872$		0			0
$873$		−	0.898979i		−	0.0304258i
$874$		0			0
$875$	1.00000	+	2.44949i	0.0338062	+	0.0828079i
$876$		0			0
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$		0			0
$879$		−	17.5959i		−	0.593496i
$880$		0			0
$881$			28.6969i			0.966824i	0.875393	+	0.483412i	$0.160603\pi$
$881$			28.6969i			0.966824i	−0.875393	+	0.483412i	$0.839397\pi$
$882$		0			0
$883$	−8.49490			−0.285876			−0.142938	−	0.989732i	$-0.545655\pi$
$883$	−8.49490			−0.285876			−0.142938	+	0.989732i	$0.545655\pi$
$884$		0			0
$885$	−3.10102			−0.104240
$886$		0			0
$887$	6.49490			0.218077			0.109039	−	0.994038i	$-0.465223\pi$
$887$	6.49490			0.218077			0.109039	+	0.994038i	$0.465223\pi$
$888$		0			0
$889$	2.20204	−	0.898979i	0.0738541	−	0.0301508i
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$			75.1918i			2.51620i
$894$		0			0
$895$	10.0000			0.334263
$896$		0			0
$897$	41.3939			1.38210
$898$		0			0
$899$			17.7980i			0.593595i
$900$		0			0
$901$	−48.0000			−1.59911
$902$		0			0
$903$	−21.7980	+	8.89898i	−0.725391	+	0.296139i
$904$		0			0
$905$	−20.8990			−0.694706
$906$		0			0
$907$	−12.4949			−0.414886			−0.207443	−	0.978247i	$-0.566514\pi$
$907$	−12.4949			−0.414886			−0.207443	+	0.978247i	$0.566514\pi$
$908$		0			0
$909$	3.79796			0.125970
$910$		0			0
$911$			32.8990i			1.08999i	0.838439	+	0.544996i	$0.183469\pi$
$911$			32.8990i			1.08999i	−0.838439	+	0.544996i	$0.816531\pi$
$912$		0			0
$913$		−	8.00000i		−	0.264761i
$914$		0			0
$915$		−	0.898979i		−	0.0297193i
$916$		0			0
$917$	17.3939	−	7.10102i	0.574396	−	0.234496i
$918$		0			0
$919$			33.3939i			1.10156i	0.834650	+	0.550781i	$0.185670\pi$
$919$			33.3939i			1.10156i	−0.834650	+	0.550781i	$0.814330\pi$
$920$		0			0
$921$	−23.5959			−0.777512
$922$		0			0
$923$			88.9898i			2.92913i
$924$		0			0
$925$		−	7.79796i		−	0.256395i
$926$		0			0
$927$	−0.202041			−0.00663590
$928$		0			0
$929$		−	10.4949i		−	0.344326i	−0.985068	−	0.172163i	$-0.944924\pi$
$929$		−	10.4949i		−	0.344326i	0.985068	−	0.172163i	$-0.0550756\pi$
$930$		0			0
$931$	33.7980	+	34.4949i	1.10768	+	1.13052i
$932$		0			0
$933$			8.00000i			0.261908i
$934$		0			0
$935$			9.79796i			0.320428i
$936$		0			0
$937$			12.4949i			0.408191i	0.978951	+	0.204095i	$0.0654252\pi$
$937$			12.4949i			0.408191i	−0.978951	+	0.204095i	$0.934575\pi$
$938$		0			0
$939$	−20.4949			−0.668826
$940$		0			0
$941$	−3.79796			−0.123810			−0.0619050	−	0.998082i	$-0.519718\pi$
$941$	−3.79796			−0.123810			−0.0619050	+	0.998082i	$0.519718\pi$
$942$		0			0
$943$	−17.3939			−0.566423
$944$		0			0
$945$	2.44949	−	1.00000i	0.0796819	−	0.0325300i
$946$		0			0
$947$	−12.2020			−0.396513			−0.198257	−	0.980150i	$-0.563528\pi$
$947$	−12.2020			−0.396513			−0.198257	+	0.980150i	$0.563528\pi$
$948$		0			0
$949$			61.3939i			1.99293i
$950$		0			0
$951$	12.0000			0.389127
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$		−	7.10102i		−	0.229784i
$956$		0			0
$957$	−17.7980			−0.575326
$958$		0			0
$959$	13.7980	+	33.7980i	0.445559	+	1.09139i
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	−7.79796			−0.251286
$964$		0			0
$965$	−0.202041			−0.00650393
$966$		0			0
$967$			7.50510i			0.241348i	0.992692	+	0.120674i	$0.0385055\pi$
$967$			7.50510i			0.241348i	−0.992692	+	0.120674i	$0.961494\pi$
$968$		0			0
$969$		−	33.7980i		−	1.08575i
$970$		0			0
$971$			12.8990i			0.413948i	0.978346	+	0.206974i	$0.0663615\pi$
$971$			12.8990i			0.413948i	−0.978346	+	0.206974i	$0.933638\pi$
$972$		0			0
$973$	16.8990	−	6.89898i	0.541756	−	0.221171i
$974$		0			0
$975$			6.89898i			0.220944i
$976$		0			0
$977$	−55.1918			−1.76574			−0.882872	−	0.469614i	$-0.844393\pi$
$977$	−55.1918			−1.76574			−0.882872	+	0.469614i	$0.844393\pi$
$978$		0			0
$979$		−	2.20204i		−	0.0703775i
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		0			0
$983$	−6.89898			−0.220043			−0.110022	−	0.993929i	$-0.535092\pi$
$983$	−6.89898			−0.220043			−0.110022	+	0.993929i	$0.535092\pi$
$984$		0			0
$985$			13.7980i			0.439640i
$986$		0			0
$987$	26.6969	−	10.8990i	0.849773	−	0.346918i
$988$		0			0
$989$		−	53.3939i		−	1.69783i
$990$		0			0
$991$		−	20.0000i		−	0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$		−	20.0000i		−	0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$		0			0
$993$			35.3939i			1.12319i
$994$		0			0
$995$	−10.0000			−0.317021
$996$		0			0
$997$	−52.2929			−1.65613			−0.828066	−	0.560631i	$-0.810559\pi$
$997$	−52.2929			−1.65613			−0.828066	+	0.560631i	$0.810559\pi$
$998$		0			0
$999$	−7.79796			−0.246717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.z.b.1231.4		4
4.3	odd	2		840.2.z.b.811.1	yes	4
7.6	odd	2		3360.2.z.a.1231.2		4
8.3	odd	2		3360.2.z.a.1231.3		4
8.5	even	2		840.2.z.a.811.4	yes	4
28.27	even	2		840.2.z.a.811.1	✓	4
56.13	odd	2		840.2.z.b.811.4	yes	4
56.27	even	2	inner	3360.2.z.b.1231.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.z.a.811.1	✓	4	28.27	even	2
840.2.z.a.811.4	yes	4	8.5	even	2
840.2.z.b.811.1	yes	4	4.3	odd	2
840.2.z.b.811.4	yes	4	56.13	odd	2
3360.2.z.a.1231.2		4	7.6	odd	2
3360.2.z.a.1231.3		4	8.3	odd	2
3360.2.z.b.1231.1		4	56.27	even	2	inner
3360.2.z.b.1231.4		4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.z (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000 q^{5} +(1.00000 + 2.44949i) q^{7} -1.00000 q^{9} -2.00000 q^{11} +6.89898 q^{13} +1.00000i q^{15} -4.89898i q^{17} -6.89898i q^{19} +(-2.44949 + 1.00000i) q^{21} -6.00000i q^{23} +1.00000 q^{25} -1.00000i q^{27} -8.89898i q^{29} -2.00000 q^{31} -2.00000i q^{33} +(1.00000 + 2.44949i) q^{35} -7.79796i q^{37} +6.89898i q^{39} -2.89898i q^{41} +8.89898 q^{43} -1.00000 q^{45} -10.8990 q^{47} +(-5.00000 + 4.89898i) q^{49} +4.89898 q^{51} -9.79796i q^{53} -2.00000 q^{55} +6.89898 q^{57} +3.10102i q^{59} -0.898979 q^{61} +(-1.00000 - 2.44949i) q^{63} +6.89898 q^{65} +4.89898 q^{67} +6.00000 q^{69} +12.8990i q^{71} +8.89898i q^{73} +1.00000i q^{75} +(-2.00000 - 4.89898i) q^{77} -4.00000i q^{79} +1.00000 q^{81} +4.00000i q^{83} -4.89898i q^{85} +8.89898 q^{87} +1.10102i q^{89} +(6.89898 + 16.8990i) q^{91} -2.00000i q^{93} -6.89898i q^{95} +0.898979i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7} - 4 q^{9} - 8 q^{11} + 8 q^{13} + 4 q^{25} - 8 q^{31} + 4 q^{35} + 16 q^{43} - 4 q^{45} - 24 q^{47} - 20 q^{49} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 4 q^{63} + 8 q^{65} + 24 q^{69}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 - 4 * q^9 - 8 * q^11 + 8 * q^13 + 4 * q^25 - 8 * q^31 + 4 * q^35 + 16 * q^43 - 4 * q^45 - 24 * q^47 - 20 * q^49 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 4 * q^63 + 8 * q^65 + 24 * q^69 - 8 * q^77 + 4 * q^81 + 16 * q^87 + 8 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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