LMFDB - Embedded newform 1360.2.e.b.1089.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1360,2,Mod(1089,1360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1360, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1360.1089");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1360 = 2^{4} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1360.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.8596546749$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1089.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1360.1089
Dual form			1360.2.e.b.1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{9} +6.00000 q^{11} -3.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +1.00000i q^{17} -7.00000 q^{19} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +5.00000i q^{27} +5.00000 q^{29} -5.00000 q^{31} +6.00000i q^{33} +8.00000i q^{37} +3.00000 q^{39} -4.00000i q^{43} +(2.00000 - 4.00000i) q^{45} -3.00000i q^{47} +7.00000 q^{49} -1.00000 q^{51} -9.00000i q^{53} +(6.00000 - 12.0000i) q^{55} -7.00000i q^{57} +5.00000 q^{59} -3.00000 q^{61} +(-6.00000 - 3.00000i) q^{65} +2.00000i q^{67} +8.00000 q^{69} +15.0000 q^{71} +11.0000i q^{73} +(4.00000 - 3.00000i) q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +(2.00000 + 1.00000i) q^{85} +5.00000i q^{87} +1.00000 q^{89} -5.00000i q^{93} +(-7.00000 + 14.0000i) q^{95} -9.00000i q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 + 4 * q^9	$ 2 q + 2 q^{5} + 4 q^{9} + 12 q^{11} + 4 q^{15} - 14 q^{19} - 6 q^{25} + 10 q^{29} - 10 q^{31} + 6 q^{39} + 4 q^{45} + 14 q^{49} - 2 q^{51} + 12 q^{55} + 10 q^{59} - 6 q^{61} - 12 q^{65} + 16 q^{69} + 30 q^{71} + 8 q^{75} + 16 q^{79} + 2 q^{81} + 4 q^{85} + 2 q^{89} - 14 q^{95} + 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 + 4 * q^9 + 12 * q^11 + 4 * q^15 - 14 * q^19 - 6 * q^25 + 10 * q^29 - 10 * q^31 + 6 * q^39 + 4 * q^45 + 14 * q^49 - 2 * q^51 + 12 * q^55 + 10 * q^59 - 6 * q^61 - 12 * q^65 + 16 * q^69 + 30 * q^71 + 8 * q^75 + 16 * q^79 + 2 * q^81 + 4 * q^85 + 2 * q^89 - 14 * q^95 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$		−	3.00000i		−	0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$		−	3.00000i		−	0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$		0			0
$15$	2.00000	+	1.00000i	0.516398	+	0.258199i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	−7.00000			−1.60591			−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000			−1.60591			−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$			6.00000i			1.04447i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$	3.00000			0.480384
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	2.00000	−	4.00000i	0.298142	−	0.596285i
$46$		0			0
$47$		−	3.00000i		−	0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$		−	3.00000i		−	0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$	6.00000	−	12.0000i	0.809040	−	1.61808i
$56$		0			0
$57$		−	7.00000i		−	0.927173i
$58$		0			0
$59$	5.00000			0.650945			0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000			0.650945			0.325472	+	0.945552i	$0.394477\pi$
$60$		0			0
$61$	−3.00000			−0.384111			−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000			−0.384111			−0.192055	+	0.981384i	$0.561515\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	−	3.00000i	−0.744208	−	0.372104i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	15.0000			1.78017			0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000			1.78017			0.890086	+	0.455792i	$0.150644\pi$
$72$		0			0
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$		0			0
$75$	4.00000	−	3.00000i	0.461880	−	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		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
$85$	2.00000	+	1.00000i	0.216930	+	0.108465i
$86$		0			0
$87$			5.00000i			0.536056i
$88$		0			0
$89$	1.00000			0.106000			0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000			0.106000			0.0529999	+	0.998595i	$0.483122\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	5.00000i		−	0.518476i
$94$		0			0
$95$	−7.00000	+	14.0000i	−0.718185	+	1.43637i
$96$		0			0
$97$		−	9.00000i		−	0.913812i	−0.889515	−	0.456906i	$-0.848958\pi$
$97$		−	9.00000i		−	0.913812i	0.889515	−	0.456906i	$-0.151042\pi$
$98$		0			0
$99$	12.0000			1.20605
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	−9.00000			−0.862044			−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000			−0.862044			−0.431022	+	0.902342i	$0.641847\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$			13.0000i			1.22294i	0.791269	+	0.611469i	$0.209421\pi$
$113$			13.0000i			1.22294i	−0.791269	+	0.611469i	$0.790579\pi$
$114$		0			0
$115$	−16.0000	−	8.00000i	−1.49201	−	0.746004i
$116$		0			0
$117$		−	6.00000i		−	0.554700i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	1.00000i		−	0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$		−	1.00000i		−	0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	−6.00000			−0.524222			−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000			−0.524222			−0.262111	+	0.965038i	$0.584419\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.0000	+	5.00000i	0.860663	+	0.430331i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$		0			0
$143$		−	18.0000i		−	1.50524i
$144$		0			0
$145$	5.00000	−	10.0000i	0.415227	−	0.830455i
$146$		0			0
$147$			7.00000i			0.577350i
$148$		0			0
$149$	2.00000			0.163846			0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000			0.163846			0.0819232	+	0.996639i	$0.473894\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$	−5.00000	+	10.0000i	−0.401610	+	0.803219i
$156$		0			0
$157$			14.0000i			1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$			14.0000i			1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$		0			0
$159$	9.00000			0.713746
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	12.0000	+	6.00000i	0.934199	+	0.467099i
$166$		0			0
$167$			6.00000i			0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$			6.00000i			0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$		0			0
$171$	−14.0000			−1.07061
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			5.00000i			0.375823i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		−	3.00000i		−	0.221766i
$184$		0			0
$185$	16.0000	+	8.00000i	1.17634	+	0.588172i
$186$		0			0
$187$			6.00000i			0.438763i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	26.0000i		−	1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$		−	26.0000i		−	1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$		0			0
$195$	3.00000	−	6.00000i	0.214834	−	0.429669i
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	7.00000			0.496217			0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000			0.496217			0.248108	+	0.968732i	$0.420191\pi$
$200$		0			0
$201$	−2.00000			−0.141069
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	16.0000i		−	1.11208i
$208$		0			0
$209$	−42.0000			−2.90520
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$			15.0000i			1.02778i
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−11.0000			−0.743311
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$		−	15.0000i		−	1.00447i	−0.864730	−	0.502237i	$-0.832510\pi$
$223$		−	15.0000i		−	1.00447i	0.864730	−	0.502237i	$-0.167490\pi$
$224$		0			0
$225$	−6.00000	−	8.00000i	−0.400000	−	0.533333i
$226$		0			0
$227$			9.00000i			0.597351i	0.954355	+	0.298675i	$0.0965448\pi$
$227$			9.00000i			0.597351i	−0.954355	+	0.298675i	$0.903455\pi$
$228$		0			0
$229$	−18.0000			−1.18947			−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000			−1.18947			−0.594737	+	0.803921i	$0.702744\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	27.0000i		−	1.76883i	−0.466702	−	0.884414i	$-0.654558\pi$
$233$		−	27.0000i		−	1.76883i	0.466702	−	0.884414i	$-0.345442\pi$
$234$		0			0
$235$	−6.00000	−	3.00000i	−0.391397	−	0.195698i
$236$		0			0
$237$			8.00000i			0.519656i
$238$		0			0
$239$	10.0000			0.646846			0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000			0.646846			0.323423	+	0.946254i	$0.395166\pi$
$240$		0			0
$241$	−20.0000			−1.28831			−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000			−1.28831			−0.644157	+	0.764894i	$0.722792\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	7.00000	−	14.0000i	0.447214	−	0.894427i
$246$		0			0
$247$			21.0000i			1.33620i
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		−	48.0000i		−	3.01773i
$254$		0			0
$255$	−1.00000	+	2.00000i	−0.0626224	+	0.125245i
$256$		0			0
$257$			28.0000i			1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$			28.0000i			1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	10.0000			0.618984
$262$		0			0
$263$			27.0000i			1.66489i	0.554107	+	0.832446i	$0.313060\pi$
$263$			27.0000i			1.66489i	−0.554107	+	0.832446i	$0.686940\pi$
$264$		0			0
$265$	−18.0000	−	9.00000i	−1.10573	−	0.552866i
$266$		0			0
$267$			1.00000i			0.0611990i
$268$		0			0
$269$	19.0000			1.15845			0.579225	−	0.815168i	$-0.303355\pi$
$269$	19.0000			1.15845			0.579225	+	0.815168i	$0.303355\pi$
$270$		0			0
$271$	6.00000			0.364474			0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000			0.364474			0.182237	+	0.983255i	$0.441666\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−18.0000	−	24.0000i	−1.08544	−	1.44725i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		0			0
$279$	−10.0000			−0.598684
$280$		0			0
$281$	−33.0000			−1.96861			−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000			−1.96861			−0.984307	+	0.176462i	$0.943535\pi$
$282$		0			0
$283$			11.0000i			0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$			11.0000i			0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$		0			0
$285$	−14.0000	−	7.00000i	−0.829288	−	0.414644i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	9.00000			0.527589
$292$		0			0
$293$			19.0000i			1.10999i	0.831853	+	0.554996i	$0.187280\pi$
$293$			19.0000i			1.10999i	−0.831853	+	0.554996i	$0.812720\pi$
$294$		0			0
$295$	5.00000	−	10.0000i	0.291111	−	0.582223i
$296$		0			0
$297$			30.0000i			1.74078i
$298$		0			0
$299$	−24.0000			−1.38796
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	12.0000i		−	0.689382i
$304$		0			0
$305$	−3.00000	+	6.00000i	−0.171780	+	0.343559i
$306$		0			0
$307$			24.0000i			1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$			24.0000i			1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$		0			0
$313$			22.0000i			1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$			22.0000i			1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	30.0000			1.67968
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		−	7.00000i		−	0.389490i
$324$		0			0
$325$	−12.0000	+	9.00000i	−0.665640	+	0.499230i
$326$		0			0
$327$		−	9.00000i		−	0.497701i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.00000			−0.384755			−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000			−0.384755			−0.192377	+	0.981321i	$0.561620\pi$
$332$		0			0
$333$			16.0000i			0.876795i
$334$		0			0
$335$	4.00000	+	2.00000i	0.218543	+	0.109272i
$336$		0			0
$337$			13.0000i			0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$			13.0000i			0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$		0			0
$339$	−13.0000			−0.706063
$340$		0			0
$341$	−30.0000			−1.62459
$342$		0			0
$343$		0			0
$344$		0			0
$345$	8.00000	−	16.0000i	0.430706	−	0.861411i
$346$		0			0
$347$		−	5.00000i		−	0.268414i	−0.990953	−	0.134207i	$-0.957151\pi$
$347$		−	5.00000i		−	0.268414i	0.990953	−	0.134207i	$-0.0428487\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	15.0000			0.800641
$352$		0			0
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	15.0000	−	30.0000i	0.796117	−	1.59223i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−28.0000			−1.47778			−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000			−1.47778			−0.738892	+	0.673824i	$0.764651\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$			25.0000i			1.31216i
$364$		0			0
$365$	22.0000	+	11.0000i	1.15153	+	0.575766i
$366$		0			0
$367$			10.0000i			0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$			10.0000i			0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$	−2.00000	−	11.0000i	−0.103280	−	0.568038i
$376$		0			0
$377$		−	15.0000i		−	0.772539i
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	1.00000			0.0512316
$382$		0			0
$383$		−	27.0000i		−	1.37964i	−0.723983	−	0.689818i	$-0.757691\pi$
$383$		−	27.0000i		−	1.37964i	0.723983	−	0.689818i	$-0.242309\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		−	6.00000i		−	0.302660i
$394$		0			0
$395$	8.00000	−	16.0000i	0.402524	−	0.805047i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$			15.0000i			0.747203i
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$			48.0000i			2.37927i
$408$		0			0
$409$	31.0000			1.53285			0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000			1.53285			0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.00000	+	4.00000i	0.392705	+	0.196352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$		0			0
$423$		−	6.00000i		−	0.291730i
$424$		0			0
$425$	4.00000	−	3.00000i	0.194029	−	0.145521i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	18.0000			0.869048
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$		−	20.0000i		−	0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$		−	20.0000i		−	0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$		0			0
$435$	10.0000	+	5.00000i	0.479463	+	0.239732i
$436$		0			0
$437$			56.0000i			2.67884i
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	14.0000			0.666667
$442$		0			0
$443$			10.0000i			0.475114i	0.971374	+	0.237557i	$0.0763467\pi$
$443$			10.0000i			0.475114i	−0.971374	+	0.237557i	$0.923653\pi$
$444$		0			0
$445$	1.00000	−	2.00000i	0.0474045	−	0.0948091i
$446$		0			0
$447$			2.00000i			0.0945968i
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$	−5.00000			−0.233380
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$			31.0000i			1.44069i	0.693615	+	0.720346i	$0.256017\pi$
$463$			31.0000i			1.44069i	−0.693615	+	0.720346i	$0.743983\pi$
$464$		0			0
$465$	−10.0000	−	5.00000i	−0.463739	−	0.231869i
$466$		0			0
$467$			14.0000i			0.647843i	0.946084	+	0.323921i	$0.105001\pi$
$467$			14.0000i			0.647843i	−0.946084	+	0.323921i	$0.894999\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−14.0000			−0.645086
$472$		0			0
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$	21.0000	+	28.0000i	0.963546	+	1.28473i
$476$		0			0
$477$		−	18.0000i		−	0.824163i
$478$		0			0
$479$	−37.0000			−1.69057			−0.845287	−	0.534313i	$-0.820570\pi$
$479$	−37.0000			−1.69057			−0.845287	+	0.534313i	$0.820570\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.0000	−	9.00000i	−0.817338	−	0.408669i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	13.0000			0.586682			0.293341	−	0.956008i	$-0.405233\pi$
$491$	13.0000			0.586682			0.293341	+	0.956008i	$0.405233\pi$
$492$		0			0
$493$			5.00000i			0.225189i
$494$		0			0
$495$	12.0000	−	24.0000i	0.539360	−	1.07872i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−28.0000			−1.25345			−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000			−1.25345			−0.626726	+	0.779240i	$0.715605\pi$
$500$		0			0
$501$	−6.00000			−0.268060
$502$		0			0
$503$		−	42.0000i		−	1.87269i	−0.351085	−	0.936344i	$-0.614187\pi$
$503$		−	42.0000i		−	1.87269i	0.351085	−	0.936344i	$-0.385813\pi$
$504$		0			0
$505$	−12.0000	+	24.0000i	−0.533993	+	1.06799i
$506$		0			0
$507$			4.00000i			0.177646i
$508$		0			0
$509$	12.0000			0.531891			0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000			0.531891			0.265945	+	0.963988i	$0.414316\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	35.0000i		−	1.54529i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	18.0000i		−	0.791639i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.0000			1.22670			0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000			1.22670			0.613351	+	0.789810i	$0.289821\pi$
$522$		0			0
$523$			14.0000i			0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$			14.0000i			0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	5.00000i		−	0.217803i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	10.0000			0.433963
$532$		0			0
$533$		0			0
$534$		0			0
$535$	8.00000	+	4.00000i	0.345870	+	0.172935i
$536$		0			0
$537$		−	12.0000i		−	0.517838i
$538$		0			0
$539$	42.0000			1.80907
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$			2.00000i			0.0858282i
$544$		0			0
$545$	−9.00000	+	18.0000i	−0.385518	+	0.771035i
$546$		0			0
$547$		−	13.0000i		−	0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$		−	13.0000i		−	0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$		0			0
$549$	−6.00000			−0.256074
$550$		0			0
$551$	−35.0000			−1.49105
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−8.00000	+	16.0000i	−0.339581	+	0.679162i
$556$		0			0
$557$			11.0000i			0.466085i	0.972467	+	0.233042i	$0.0748681\pi$
$557$			11.0000i			0.466085i	−0.972467	+	0.233042i	$0.925132\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$	−6.00000			−0.253320
$562$		0			0
$563$			32.0000i			1.34864i	0.738440	+	0.674320i	$0.235563\pi$
$563$			32.0000i			1.34864i	−0.738440	+	0.674320i	$0.764437\pi$
$564$		0			0
$565$	26.0000	+	13.0000i	1.09383	+	0.546914i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.0000			0.880366			0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000			0.880366			0.440183	+	0.897908i	$0.354914\pi$
$570$		0			0
$571$	10.0000			0.418487			0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000			0.418487			0.209243	+	0.977864i	$0.432900\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−32.0000	+	24.0000i	−1.33449	+	1.00087i
$576$		0			0
$577$			16.0000i			0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$			16.0000i			0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$		0			0
$579$	26.0000			1.08052
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	54.0000i		−	2.23645i
$584$		0			0
$585$	−12.0000	−	6.00000i	−0.496139	−	0.248069i
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$		0			0
$591$	−2.00000			−0.0822690
$592$		0			0
$593$		−	22.0000i		−	0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$		−	22.0000i		−	0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			7.00000i			0.286491i
$598$		0			0
$599$	−32.0000			−1.30748			−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000			−1.30748			−0.653742	+	0.756717i	$0.726802\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$	25.0000	−	50.0000i	1.01639	−	2.03279i
$606$		0			0
$607$			12.0000i			0.487065i	0.969893	+	0.243532i	$0.0783062\pi$
$607$			12.0000i			0.487065i	−0.969893	+	0.243532i	$0.921694\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.00000			−0.364101
$612$		0			0
$613$			17.0000i			0.686624i	0.939222	+	0.343312i	$0.111549\pi$
$613$			17.0000i			0.686624i	−0.939222	+	0.343312i	$0.888451\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	29.0000i		−	1.16750i	−0.811935	−	0.583748i	$-0.801586\pi$
$617$		−	29.0000i		−	1.16750i	0.811935	−	0.583748i	$-0.198414\pi$
$618$		0			0
$619$	14.0000			0.562708			0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000			0.562708			0.281354	+	0.959604i	$0.409217\pi$
$620$		0			0
$621$	40.0000			1.60514
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		−	42.0000i		−	1.67732i
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$	42.0000			1.67199			0.835997	−	0.548734i	$-0.184890\pi$
$631$	42.0000			1.67199			0.835997	+	0.548734i	$0.184890\pi$
$632$		0			0
$633$		−	16.0000i		−	0.635943i
$634$		0			0
$635$	−2.00000	−	1.00000i	−0.0793676	−	0.0396838i
$636$		0			0
$637$		−	21.0000i		−	0.832050i
$638$		0			0
$639$	30.0000			1.18678
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$		0			0
$645$	4.00000	−	8.00000i	0.157500	−	0.315000i
$646$		0			0
$647$		−	17.0000i		−	0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$		−	17.0000i		−	0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	30.0000i		−	1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$		−	30.0000i		−	1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$		0			0
$655$	−6.00000	+	12.0000i	−0.234439	+	0.468879i
$656$		0			0
$657$			22.0000i			0.858302i
$658$		0			0
$659$	19.0000			0.740135			0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000			0.740135			0.370067	+	0.929005i	$0.379335\pi$
$660$		0			0
$661$	−16.0000			−0.622328			−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000			−0.622328			−0.311164	+	0.950356i	$0.600719\pi$
$662$		0			0
$663$			3.00000i			0.116510i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	40.0000i		−	1.54881i
$668$		0			0
$669$	15.0000			0.579934
$670$		0			0
$671$	−18.0000			−0.694882
$672$		0			0
$673$			35.0000i			1.34915i	0.738206	+	0.674575i	$0.235673\pi$
$673$			35.0000i			1.34915i	−0.738206	+	0.674575i	$0.764327\pi$
$674$		0			0
$675$	20.0000	−	15.0000i	0.769800	−	0.577350i
$676$		0			0
$677$		−	24.0000i		−	0.922395i	−0.887298	−	0.461197i	$-0.847420\pi$
$677$		−	24.0000i		−	0.922395i	0.887298	−	0.461197i	$-0.152580\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−9.00000			−0.344881
$682$		0			0
$683$			45.0000i			1.72188i	0.508709	+	0.860939i	$0.330123\pi$
$683$			45.0000i			1.72188i	−0.508709	+	0.860939i	$0.669877\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$		0			0
$687$		−	18.0000i		−	0.686743i
$688$		0			0
$689$	−27.0000			−1.02862
$690$		0			0
$691$	16.0000			0.608669			0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000			0.608669			0.304334	+	0.952565i	$0.401566\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	27.0000			1.02123
$700$		0			0
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$		−	56.0000i		−	2.11208i
$704$		0			0
$705$	3.00000	−	6.00000i	0.112987	−	0.225973i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.0000			−0.863783			−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000			−0.863783			−0.431892	+	0.901926i	$0.642154\pi$
$710$		0			0
$711$	16.0000			0.600047
$712$		0			0
$713$			40.0000i			1.49801i
$714$		0			0
$715$	−36.0000	−	18.0000i	−1.34632	−	0.673162i
$716$		0			0
$717$			10.0000i			0.373457i
$718$		0			0
$719$	−27.0000			−1.00693			−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000			−1.00693			−0.503465	+	0.864016i	$0.667942\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	20.0000i		−	0.743808i
$724$		0			0
$725$	−15.0000	−	20.0000i	−0.557086	−	0.742781i
$726$		0			0
$727$			29.0000i			1.07555i	0.843088	+	0.537775i	$0.180735\pi$
$727$			29.0000i			1.07555i	−0.843088	+	0.537775i	$0.819265\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	4.00000			0.147945
$732$		0			0
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$		0			0
$735$	14.0000	+	7.00000i	0.516398	+	0.258199i
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$	−1.00000			−0.0367856			−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000			−0.0367856			−0.0183928	+	0.999831i	$0.505855\pi$
$740$		0			0
$741$	−21.0000			−0.771454
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	2.00000	−	4.00000i	0.0732743	−	0.146549i
$746$		0			0
$747$			8.00000i			0.292705i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−11.0000			−0.401396			−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−11.0000			−0.401396			−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	45.0000i		−	1.63555i	−0.575536	−	0.817776i	$-0.695207\pi$
$757$		−	45.0000i		−	1.63555i	0.575536	−	0.817776i	$-0.304793\pi$
$758$		0			0
$759$	48.0000			1.74229
$760$		0			0
$761$	14.0000			0.507500			0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000			0.507500			0.253750	+	0.967270i	$0.418336\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	4.00000	+	2.00000i	0.144620	+	0.0723102i
$766$		0			0
$767$		−	15.0000i		−	0.541619i
$768$		0			0
$769$	−37.0000			−1.33425			−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000			−1.33425			−0.667127	+	0.744944i	$0.732476\pi$
$770$		0			0
$771$	−28.0000			−1.00840
$772$		0			0
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$	15.0000	+	20.0000i	0.538816	+	0.718421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	90.0000			3.22045
$782$		0			0
$783$			25.0000i			0.893427i
$784$		0			0
$785$	28.0000	+	14.0000i	0.999363	+	0.499681i
$786$		0			0
$787$			47.0000i			1.67537i	0.546154	+	0.837685i	$0.316091\pi$
$787$			47.0000i			1.67537i	−0.546154	+	0.837685i	$0.683909\pi$
$788$		0			0
$789$	−27.0000			−0.961225
$790$		0			0
$791$		0			0
$792$		0			0
$793$			9.00000i			0.319599i
$794$		0			0
$795$	9.00000	−	18.0000i	0.319197	−	0.638394i
$796$		0			0
$797$			54.0000i			1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$			54.0000i			1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$		0			0
$799$	3.00000			0.106132
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$			66.0000i			2.32909i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			19.0000i			0.668832i
$808$		0			0
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$		0			0
$813$			6.00000i			0.210429i
$814$		0			0
$815$	−8.00000	−	4.00000i	−0.280228	−	0.140114i
$816$		0			0
$817$			28.0000i			0.979596i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−27.0000			−0.942306			−0.471153	−	0.882051i	$-0.656162\pi$
$821$	−27.0000			−0.942306			−0.471153	+	0.882051i	$0.656162\pi$
$822$		0			0
$823$		−	54.0000i		−	1.88232i	−0.337959	−	0.941161i	$-0.609737\pi$
$823$		−	54.0000i		−	1.88232i	0.337959	−	0.941161i	$-0.390263\pi$
$824$		0			0
$825$	24.0000	−	18.0000i	0.835573	−	0.626680i
$826$		0			0
$827$			32.0000i			1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$			32.0000i			1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$		0			0
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$		0			0
$831$	2.00000			0.0693792
$832$		0			0
$833$			7.00000i			0.242536i
$834$		0			0
$835$	12.0000	+	6.00000i	0.415277	+	0.207639i
$836$		0			0
$837$		−	25.0000i		−	0.864126i
$838$		0			0
$839$	31.0000			1.07024			0.535119	−	0.844776i	$-0.320267\pi$
$839$	31.0000			1.07024			0.535119	+	0.844776i	$0.320267\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		−	33.0000i		−	1.13658i
$844$		0			0
$845$	4.00000	−	8.00000i	0.137604	−	0.275208i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−11.0000			−0.377519
$850$		0			0
$851$	64.0000			2.19389
$852$		0			0
$853$		−	30.0000i		−	1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$		−	30.0000i		−	1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$		0			0
$855$	−14.0000	+	28.0000i	−0.478790	+	0.957580i
$856$		0			0
$857$		−	57.0000i		−	1.94708i	−0.228510	−	0.973541i	$-0.573386\pi$
$857$		−	57.0000i		−	1.94708i	0.228510	−	0.973541i	$-0.426614\pi$
$858$		0			0
$859$	33.0000			1.12595			0.562973	−	0.826475i	$-0.309658\pi$
$859$	33.0000			1.12595			0.562973	+	0.826475i	$0.309658\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	48.0000			1.62829
$870$		0			0
$871$	6.00000			0.203302
$872$		0			0
$873$		−	18.0000i		−	0.609208i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	10.0000i		−	0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$		−	10.0000i		−	0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$		0			0
$879$	−19.0000			−0.640854
$880$		0			0
$881$	38.0000			1.28025			0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000			1.28025			0.640126	+	0.768270i	$0.278882\pi$
$882$		0			0
$883$		−	26.0000i		−	0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$		−	26.0000i		−	0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$		0			0
$885$	10.0000	+	5.00000i	0.336146	+	0.168073i
$886$		0			0
$887$			14.0000i			0.470074i	0.971986	+	0.235037i	$0.0755211\pi$
$887$			14.0000i			0.470074i	−0.971986	+	0.235037i	$0.924479\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	6.00000			0.201008
$892$		0			0
$893$			21.0000i			0.702738i
$894$		0			0
$895$	−12.0000	+	24.0000i	−0.401116	+	0.802232i
$896$		0			0
$897$		−	24.0000i		−	0.801337i
$898$		0			0
$899$	−25.0000			−0.833797
$900$		0			0
$901$	9.00000			0.299833
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.00000	−	4.00000i	0.0664822	−	0.132964i
$906$		0			0
$907$		−	27.0000i		−	0.896520i	−0.893903	−	0.448260i	$-0.852044\pi$
$907$		−	27.0000i		−	0.896520i	0.893903	−	0.448260i	$-0.147956\pi$
$908$		0			0
$909$	−24.0000			−0.796030
$910$		0			0
$911$	−8.00000			−0.265052			−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000			−0.265052			−0.132526	+	0.991180i	$0.542309\pi$
$912$		0			0
$913$			24.0000i			0.794284i
$914$		0			0
$915$	−6.00000	−	3.00000i	−0.198354	−	0.0991769i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−6.00000			−0.197922			−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000			−0.197922			−0.0989609	+	0.995091i	$0.531552\pi$
$920$		0			0
$921$	−24.0000			−0.790827
$922$		0			0
$923$		−	45.0000i		−	1.48119i
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	34.0000			1.11550			0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000			1.11550			0.557752	+	0.830008i	$0.311664\pi$
$930$		0			0
$931$	−49.0000			−1.60591
$932$		0			0
$933$			4.00000i			0.130954i
$934$		0			0
$935$	12.0000	+	6.00000i	0.392442	+	0.196221i
$936$		0			0
$937$		−	44.0000i		−	1.43742i	−0.695311	−	0.718709i	$-0.744734\pi$
$937$		−	44.0000i		−	1.43742i	0.695311	−	0.718709i	$-0.255266\pi$
$938$		0			0
$939$	−22.0000			−0.717943
$940$		0			0
$941$	−5.00000			−0.162995			−0.0814977	−	0.996674i	$-0.525970\pi$
$941$	−5.00000			−0.162995			−0.0814977	+	0.996674i	$0.525970\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			9.00000i			0.292461i	0.989251	+	0.146230i	$0.0467141\pi$
$947$			9.00000i			0.292461i	−0.989251	+	0.146230i	$0.953286\pi$
$948$		0			0
$949$	33.0000			1.07123
$950$		0			0
$951$		0			0
$952$		0			0
$953$			44.0000i			1.42530i	0.701520	+	0.712650i	$0.252505\pi$
$953$			44.0000i			1.42530i	−0.701520	+	0.712650i	$0.747495\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			30.0000i			0.969762i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$			8.00000i			0.257796i
$964$		0			0
$965$	−52.0000	−	26.0000i	−1.67394	−	0.836970i
$966$		0			0
$967$			16.0000i			0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$			16.0000i			0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$		0			0
$969$	7.00000			0.224872
$970$		0			0
$971$	−37.0000			−1.18739			−0.593693	−	0.804691i	$-0.702331\pi$
$971$	−37.0000			−1.18739			−0.593693	+	0.804691i	$0.702331\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−9.00000	−	12.0000i	−0.288231	−	0.384308i
$976$		0			0
$977$			38.0000i			1.21573i	0.794041	+	0.607864i	$0.207973\pi$
$977$			38.0000i			1.21573i	−0.794041	+	0.607864i	$0.792027\pi$
$978$		0			0
$979$	6.00000			0.191761
$980$		0			0
$981$	−18.0000			−0.574696
$982$		0			0
$983$			30.0000i			0.956851i	0.878128	+	0.478426i	$0.158792\pi$
$983$			30.0000i			0.956851i	−0.878128	+	0.478426i	$0.841208\pi$
$984$		0			0
$985$	4.00000	+	2.00000i	0.127451	+	0.0637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	19.0000			0.603555			0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000			0.603555			0.301777	+	0.953378i	$0.402420\pi$
$992$		0			0
$993$		−	7.00000i		−	0.222138i
$994$		0			0
$995$	7.00000	−	14.0000i	0.221915	−	0.443830i
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$		0			0
$999$	−40.0000			−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1360.2.e.b.1089.2		2
4.3	odd	2		170.2.c.a.69.1	✓	2
5.2	odd	4		6800.2.a.r.1.1		1
5.3	odd	4		6800.2.a.g.1.1		1
5.4	even	2	inner	1360.2.e.b.1089.1		2
12.11	even	2		1530.2.d.b.919.2		2
20.3	even	4		850.2.a.d.1.1		1
20.7	even	4		850.2.a.h.1.1		1
20.19	odd	2		170.2.c.a.69.2	yes	2
60.23	odd	4		7650.2.a.cb.1.1		1
60.47	odd	4		7650.2.a.s.1.1		1
60.59	even	2		1530.2.d.b.919.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.c.a.69.1	✓	2	4.3	odd	2
170.2.c.a.69.2	yes	2	20.19	odd	2
850.2.a.d.1.1		1	20.3	even	4
850.2.a.h.1.1		1	20.7	even	4
1360.2.e.b.1089.1		2	5.4	even	2	inner
1360.2.e.b.1089.2		2	1.1	even	1	trivial
1530.2.d.b.919.1		2	60.59	even	2
1530.2.d.b.919.2		2	12.11	even	2
6800.2.a.g.1.1		1	5.3	odd	4
6800.2.a.r.1.1		1	5.2	odd	4
7650.2.a.s.1.1		1	60.47	odd	4
7650.2.a.cb.1.1		1	60.23	odd	4

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:	\( N \)	\(=\)	\( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1360.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{9} +6.00000 q^{11} -3.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +1.00000i q^{17} -7.00000 q^{19} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +5.00000i q^{27} +5.00000 q^{29} -5.00000 q^{31} +6.00000i q^{33} +8.00000i q^{37} +3.00000 q^{39} -4.00000i q^{43} +(2.00000 - 4.00000i) q^{45} -3.00000i q^{47} +7.00000 q^{49} -1.00000 q^{51} -9.00000i q^{53} +(6.00000 - 12.0000i) q^{55} -7.00000i q^{57} +5.00000 q^{59} -3.00000 q^{61} +(-6.00000 - 3.00000i) q^{65} +2.00000i q^{67} +8.00000 q^{69} +15.0000 q^{71} +11.0000i q^{73} +(4.00000 - 3.00000i) q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +(2.00000 + 1.00000i) q^{85} +5.00000i q^{87} +1.00000 q^{89} -5.00000i q^{93} +(-7.00000 + 14.0000i) q^{95} -9.00000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 + 4 * q^9	\( 2 q + 2 q^{5} + 4 q^{9} + 12 q^{11} + 4 q^{15} - 14 q^{19} - 6 q^{25} + 10 q^{29} - 10 q^{31} + 6 q^{39} + 4 q^{45} + 14 q^{49} - 2 q^{51} + 12 q^{55} + 10 q^{59} - 6 q^{61} - 12 q^{65} + 16 q^{69} + 30 q^{71} + 8 q^{75} + 16 q^{79} + 2 q^{81} + 4 q^{85} + 2 q^{89} - 14 q^{95} + 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 + 4 * q^9 + 12 * q^11 + 4 * q^15 - 14 * q^19 - 6 * q^25 + 10 * q^29 - 10 * q^31 + 6 * q^39 + 4 * q^45 + 14 * q^49 - 2 * q^51 + 12 * q^55 + 10 * q^59 - 6 * q^61 - 12 * q^65 + 16 * q^69 + 30 * q^71 + 8 * q^75 + 16 * q^79 + 2 * q^81 + 4 * q^85 + 2 * q^89 - 14 * q^95 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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