LMFDB - Embedded newform 1620.2.d.d.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(649,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1620.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.d (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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.301925376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 14x^{4} + 43x^{2} + 36 $ x^6 + 14x^4 + 43x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$3.17695i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.2.d.d.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.123563 + 2.23265i) q^{5} -1.28835i q^{7} -5.09303 q^{11} -3.56951i q^{13} -0.895796i q^{17} +5.34015 q^{19} -5.06555i q^{23} +(-4.96946 + 0.551747i) q^{25} -3.00000 q^{29} -6.59877 q^{31} +(2.87644 - 0.159193i) q^{35} -7.24970i q^{37} +7.84590 q^{41} -10.9299i q^{43} -2.96925i q^{47} +5.34015 q^{49} +4.78369i q^{53} +(-0.629311 - 11.3710i) q^{55} -5.75287 q^{59} +4.34015 q^{61} +(7.96946 - 0.441060i) q^{65} -8.53805i q^{67} +5.34015 q^{71} +9.34600i q^{73} +6.56160i q^{77} +0.741379 q^{79} +9.21247i q^{83} +(2.00000 - 0.110687i) q^{85} -9.24713 q^{89} -4.59877 q^{91} +(0.659846 + 11.9227i) q^{95} -3.45882i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) $ 6 * q + q^5 + 2 * q^11 + 3 * q^25 - 18 * q^29 - 6 * q^31 + 17 * q^35 + 14 * q^41 - 3 * q^55 - 34 * q^59 - 6 * q^61 + 15 * q^65 + 6 * q^79 + 12 * q^85 - 56 * q^89 + 6 * q^91 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.123563	+	2.23265i	0.0552591	+	0.998472i
$5$	0.123563	+	2.23265i	0.0552591	+	0.998472i
$6$		0			0
$7$		−	1.28835i		−	0.486951i	−0.969907	−	0.243475i	$-0.921713\pi$
$7$		−	1.28835i		−	0.486951i	0.969907	−	0.243475i	$-0.0782875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.09303			−1.53561			−0.767803	−	0.640686i	$-0.778650\pi$
$11$	−5.09303			−1.53561			−0.767803	+	0.640686i	$0.778650\pi$
$12$		0			0
$13$		−	3.56951i		−	0.990003i	−0.868892	−	0.495002i	$-0.835167\pi$
$13$		−	3.56951i		−	0.990003i	0.868892	−	0.495002i	$-0.164833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.895796i		−	0.217262i	−0.994082	−	0.108631i	$-0.965353\pi$
$17$		−	0.895796i		−	0.217262i	0.994082	−	0.108631i	$-0.0346468\pi$
$18$		0			0
$19$	5.34015			1.22512			0.612558	−	0.790426i	$-0.290141\pi$
$19$	5.34015			1.22512			0.612558	+	0.790426i	$0.290141\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.06555i		−	1.05624i	−0.849169	−	0.528121i	$-0.822897\pi$
$23$		−	5.06555i		−	1.05624i	0.849169	−	0.528121i	$-0.177103\pi$
$24$		0			0
$25$	−4.96946	+	0.551747i	−0.993893	+	0.110349i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	−6.59877			−1.18517			−0.592587	−	0.805506i	$-0.701894\pi$
$31$	−6.59877			−1.18517			−0.592587	+	0.805506i	$0.701894\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.87644	−	0.159193i	0.486207	−	0.0269085i
$36$		0			0
$37$		−	7.24970i		−	1.19184i	−0.803043	−	0.595922i	$-0.796787\pi$
$37$		−	7.24970i		−	1.19184i	0.803043	−	0.595922i	$-0.203213\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.84590			1.22532			0.612662	−	0.790345i	$-0.290099\pi$
$41$	7.84590			1.22532			0.612662	+	0.790345i	$0.290099\pi$
$42$		0			0
$43$		−	10.9299i		−	1.66679i	−0.552675	−	0.833397i	$-0.686393\pi$
$43$		−	10.9299i		−	1.66679i	0.552675	−	0.833397i	$-0.313607\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.96925i		−	0.433110i	−0.976270	−	0.216555i	$-0.930518\pi$
$47$		−	2.96925i		−	0.433110i	0.976270	−	0.216555i	$-0.0694821\pi$
$48$		0			0
$49$	5.34015			0.762879
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.78369i			0.657090i	0.944488	+	0.328545i	$0.106558\pi$
$53$			4.78369i			0.657090i	−0.944488	+	0.328545i	$0.893442\pi$
$54$		0			0
$55$	−0.629311	−	11.3710i	−0.0848562	−	1.53326i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.75287			−0.748960			−0.374480	−	0.927235i	$-0.622179\pi$
$59$	−5.75287			−0.748960			−0.374480	+	0.927235i	$0.622179\pi$
$60$		0			0
$61$	4.34015			0.555700			0.277850	−	0.960625i	$-0.410378\pi$
$61$	4.34015			0.555700			0.277850	+	0.960625i	$0.410378\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.96946	−	0.441060i	0.988490	−	0.0547067i
$66$		0			0
$67$		−	8.53805i		−	1.04309i	−0.853224	−	0.521544i	$-0.825356\pi$
$67$		−	8.53805i		−	1.04309i	0.853224	−	0.521544i	$-0.174644\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.34015			0.633760			0.316880	−	0.948466i	$-0.397365\pi$
$71$	5.34015			0.633760			0.316880	+	0.948466i	$0.397365\pi$
$72$		0			0
$73$			9.34600i			1.09387i	0.837176	+	0.546933i	$0.184205\pi$
$73$			9.34600i			1.09387i	−0.837176	+	0.546933i	$0.815795\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.56160i			0.747764i
$78$		0			0
$79$	0.741379			0.0834116			0.0417058	−	0.999130i	$-0.486721\pi$
$79$	0.741379			0.0834116			0.0417058	+	0.999130i	$0.486721\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			9.21247i			1.01120i	0.862768	+	0.505600i	$0.168729\pi$
$83$			9.21247i			1.01120i	−0.862768	+	0.505600i	$0.831271\pi$
$84$		0			0
$85$	2.00000	−	0.110687i	0.216930	−	0.0120057i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.24713			−0.980193			−0.490097	−	0.871668i	$-0.663039\pi$
$89$	−9.24713			−0.980193			−0.490097	+	0.871668i	$0.663039\pi$
$90$		0			0
$91$	−4.59877			−0.482083
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.659846	+	11.9227i	0.0676988	+	1.22324i
$96$		0			0
$97$		−	3.45882i		−	0.351190i	−0.984462	−	0.175595i	$-0.943815\pi$
$97$		−	3.45882i		−	0.351190i	0.984462	−	0.175595i	$-0.0561849\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	16.4332			1.63516			0.817581	−	0.575813i	$-0.195314\pi$
$101$	16.4332			1.63516			0.817581	+	0.575813i	$0.195314\pi$
$102$		0			0
$103$		−	15.8619i		−	1.56292i	−0.623954	−	0.781461i	$-0.714475\pi$
$103$		−	15.8619i		−	1.56292i	0.623954	−	0.781461i	$-0.285525\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.53086i		−	0.921383i	−0.887560	−	0.460691i	$-0.847601\pi$
$107$		−	9.53086i		−	0.921383i	0.887560	−	0.460691i	$-0.152399\pi$
$108$		0			0
$109$	−16.2791			−1.55925			−0.779627	−	0.626245i	$-0.784591\pi$
$109$	−16.2791			−1.55925			−0.779627	+	0.626245i	$0.784591\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	18.1796i		−	1.71019i	−0.518469	−	0.855096i	$-0.673498\pi$
$113$		−	18.1796i		−	1.71019i	0.518469	−	0.855096i	$-0.326502\pi$
$114$		0			0
$115$	11.3096	−	0.625916i	1.05463	−	0.0583670i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.15410			−0.105796
$120$		0			0
$121$	14.9389			1.35808
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.84590	−	11.0269i	−0.165102	−	0.986276i
$126$		0			0
$127$		−	11.9969i		−	1.06455i	−0.846571	−	0.532275i	$-0.821337\pi$
$127$		−	11.9969i		−	1.06455i	0.846571	−	0.532275i	$-0.178663\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.93893			−0.868368			−0.434184	−	0.900824i	$-0.642963\pi$
$131$	−9.93893			−0.868368			−0.434184	+	0.900824i	$0.642963\pi$
$132$		0			0
$133$		−	6.87999i		−	0.596571i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.66581i		−	0.484063i	−0.970268	−	0.242031i	$-0.922186\pi$
$137$		−	5.66581i		−	0.484063i	0.970268	−	0.242031i	$-0.0778137\pi$
$138$		0			0
$139$	−7.93893			−0.673371			−0.336686	−	0.941617i	$-0.609306\pi$
$139$	−7.93893			−0.673371			−0.336686	+	0.941617i	$0.609306\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			18.1796i			1.52025i
$144$		0			0
$145$	−0.370689	−	6.69795i	−0.0307841	−	0.556235i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14.8344			−1.21528			−0.607641	−	0.794212i	$-0.707884\pi$
$149$	−14.8344			−1.21528			−0.607641	+	0.794212i	$0.707884\pi$
$150$		0			0
$151$	−0.598775			−0.0487276			−0.0243638	−	0.999703i	$-0.507756\pi$
$151$	−0.598775			−0.0487276			−0.0243638	+	0.999703i	$0.507756\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.815365	−	14.7328i	−0.0654917	−	1.18336i
$156$		0			0
$157$			17.9582i			1.43322i	0.697473	+	0.716611i	$0.254308\pi$
$157$			17.9582i			1.43322i	−0.697473	+	0.716611i	$0.745692\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.52621			−0.514337
$162$		0			0
$163$			4.67300i			0.366018i	0.983111	+	0.183009i	$0.0585837\pi$
$163$			4.67300i			0.366018i	−0.983111	+	0.183009i	$0.941416\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.96135i		−	0.461303i	−0.973036	−	0.230652i	$-0.925914\pi$
$167$		−	5.96135i		−	0.461303i	0.973036	−	0.230652i	$-0.0740858\pi$
$168$		0			0
$169$	0.258621			0.0198939
$170$		0			0
$171$		0			0
$172$		0			0
$173$			9.81272i			0.746048i	0.927822	+	0.373024i	$0.121679\pi$
$173$			9.81272i			0.746048i	−0.927822	+	0.373024i	$0.878321\pi$
$174$		0			0
$175$	0.710843	+	6.40241i	0.0537347	+	0.483977i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.3516			−1.22218			−0.611090	−	0.791561i	$-0.709269\pi$
$179$	−16.3516			−1.22218			−0.611090	+	0.791561i	$0.709269\pi$
$180$		0			0
$181$	2.93893			0.218449			0.109224	−	0.994017i	$-0.465163\pi$
$181$	2.93893			0.218449			0.109224	+	0.994017i	$0.465163\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.1861	−	0.895796i	1.19002	−	0.0658602i
$186$		0			0
$187$			4.56231i			0.333629i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.59877			0.332756			0.166378	−	0.986062i	$-0.446793\pi$
$191$	4.59877			0.332756			0.166378	+	0.986062i	$0.446793\pi$
$192$		0			0
$193$			1.58389i			0.114011i	0.998374	+	0.0570056i	$0.0181553\pi$
$193$			1.58389i			0.114011i	−0.998374	+	0.0570056i	$0.981845\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.0775i			0.931735i	0.884855	+	0.465867i	$0.154258\pi$
$197$			13.0775i			0.931735i	−0.884855	+	0.465867i	$0.845742\pi$
$198$		0			0
$199$	5.34015			0.378553			0.189277	−	0.981924i	$-0.439386\pi$
$199$	5.34015			0.378553			0.189277	+	0.981924i	$0.439386\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			3.86505i			0.271273i
$204$		0			0
$205$	0.969464	+	17.5172i	0.0677103	+	1.22345i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−27.1975			−1.88129
$210$		0			0
$211$	4.59877			0.316593			0.158296	−	0.987392i	$-0.449400\pi$
$211$	4.59877			0.316593			0.158296	+	0.987392i	$0.449400\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.4026	−	1.35053i	1.66425	−	0.0921055i
$216$		0			0
$217$			8.50153i			0.577122i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.19755			−0.215090
$222$		0			0
$223$		−	8.53805i		−	0.571750i	−0.958267	−	0.285875i	$-0.907716\pi$
$223$		−	8.53805i		−	0.571750i	0.958267	−	0.285875i	$-0.0922842\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			17.2838i			1.14717i	0.819147	+	0.573583i	$0.194447\pi$
$227$			17.2838i			1.14717i	−0.819147	+	0.573583i	$0.805553\pi$
$228$		0			0
$229$	−15.6803			−1.03618			−0.518092	−	0.855325i	$-0.673357\pi$
$229$	−15.6803			−1.03618			−0.518092	+	0.855325i	$0.673357\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.7649i		−	1.03279i	−0.856349	−	0.516397i	$-0.827273\pi$
$233$		−	15.7649i		−	1.03279i	0.856349	−	0.516397i	$-0.172727\pi$
$234$		0			0
$235$	6.62931	−	0.366890i	0.432449	−	0.0239333i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.56682			−0.101349			−0.0506745	−	0.998715i	$-0.516137\pi$
$239$	−1.56682			−0.101349			−0.0506745	+	0.998715i	$0.516137\pi$
$240$		0			0
$241$	−21.0205			−1.35405			−0.677023	−	0.735961i	$-0.736730\pi$
$241$	−21.0205			−1.35405			−0.677023	+	0.735961i	$0.736730\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.659846	+	11.9227i	0.0421560	+	0.761713i
$246$		0			0
$247$		−	19.0617i		−	1.21287i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.85740			0.622193			0.311097	−	0.950378i	$-0.399304\pi$
$251$	9.85740			0.622193			0.311097	+	0.950378i	$0.399304\pi$
$252$		0			0
$253$			25.7990i			1.62197i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			25.9097i			1.61620i	0.589044	+	0.808101i	$0.299505\pi$
$257$			25.9097i			1.61620i	−0.589044	+	0.808101i	$0.700495\pi$
$258$		0			0
$259$	−9.34015			−0.580369
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	19.6391i		−	1.21100i	−0.795845	−	0.605500i	$-0.792973\pi$
$263$		−	19.6391i		−	1.21100i	0.795845	−	0.605500i	$-0.207027\pi$
$264$		0			0
$265$	−10.6803	+	0.591088i	−0.656086	+	0.0363102i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.279082			0.0170160			0.00850798	−	0.999964i	$-0.497292\pi$
$269$	0.279082			0.0170160			0.00850798	+	0.999964i	$0.497292\pi$
$270$		0			0
$271$	9.34015			0.567374			0.283687	−	0.958917i	$-0.408442\pi$
$271$	9.34015			0.567374			0.283687	+	0.958917i	$0.408442\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	25.3096	−	2.81006i	1.52623	−	0.169453i
$276$		0			0
$277$			18.0689i			1.08566i	0.839844	+	0.542828i	$0.182646\pi$
$277$			18.0689i			1.08566i	−0.839844	+	0.542828i	$0.817354\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.02046			−0.418806			−0.209403	−	0.977829i	$-0.567152\pi$
$281$	−7.02046			−0.418806			−0.209403	+	0.977829i	$0.567152\pi$
$282$		0			0
$283$			18.3645i			1.09165i	0.837898	+	0.545827i	$0.183784\pi$
$283$			18.3645i			1.09165i	−0.837898	+	0.545827i	$0.816216\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	10.1083i		−	0.596672i
$288$		0			0
$289$	16.1975			0.952797
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	10.7085i		−	0.625598i	−0.949819	−	0.312799i	$-0.898733\pi$
$293$		−	10.7085i		−	0.625598i	0.949819	−	0.312799i	$-0.101267\pi$
$294$		0			0
$295$	−0.710843	−	12.8442i	−0.0413869	−	0.747816i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−18.0815			−1.04568
$300$		0			0
$301$	−14.0815			−0.811646
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.536283	+	9.69005i	0.0307075	+	0.554851i
$306$		0			0
$307$			14.3145i			0.816974i	0.912764	+	0.408487i	$0.133944\pi$
$307$			14.3145i			0.816974i	−0.912764	+	0.408487i	$0.866056\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.4766			1.38794			0.693971	−	0.720003i	$-0.255859\pi$
$311$	24.4766			1.38794			0.693971	+	0.720003i	$0.255859\pi$
$312$		0			0
$313$			31.9772i			1.80746i	0.428103	+	0.903730i	$0.359182\pi$
$313$			31.9772i			1.80746i	−0.428103	+	0.903730i	$0.640818\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	26.2144i		−	1.47235i	−0.676792	−	0.736174i	$-0.736631\pi$
$317$		−	26.2144i		−	1.47235i	0.676792	−	0.736174i	$-0.263369\pi$
$318$		0			0
$319$	15.2791			0.855464
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.78369i		−	0.266172i
$324$		0			0
$325$	1.96946	+	17.7385i	0.109246	+	0.983957i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.82544			−0.210903
$330$		0			0
$331$	−2.74138			−0.150680			−0.0753399	−	0.997158i	$-0.524004\pi$
$331$	−2.74138			−0.150680			−0.0753399	+	0.997158i	$0.524004\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	19.0625	−	1.05499i	1.04150	−	0.0576402i
$336$		0			0
$337$			32.6790i			1.78014i	0.455825	+	0.890069i	$0.349344\pi$
$337$			32.6790i			1.78014i	−0.455825	+	0.890069i	$0.650656\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	33.6077			1.81996
$342$		0			0
$343$		−	15.8984i		−	0.858435i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	13.1369i		−	0.705225i	−0.935769	−	0.352612i	$-0.885293\pi$
$347$		−	13.1369i		−	0.705225i	0.935769	−	0.352612i	$-0.114707\pi$
$348$		0			0
$349$	14.1975			0.759977			0.379989	−	0.924991i	$-0.375928\pi$
$349$	14.1975			0.759977			0.379989	+	0.924991i	$0.375928\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			5.10207i			0.271556i	0.990739	+	0.135778i	$0.0433534\pi$
$353$			5.10207i			0.271556i	−0.990739	+	0.135778i	$0.956647\pi$
$354$		0			0
$355$	0.659846	+	11.9227i	0.0350210	+	0.632791i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.8229			−0.676767			−0.338384	−	0.941008i	$-0.609880\pi$
$359$	−12.8229			−0.676767			−0.338384	+	0.941008i	$0.609880\pi$
$360$		0			0
$361$	9.51724			0.500907
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.8664	+	1.15482i	−1.09220	+	0.0604461i
$366$		0			0
$367$		−	5.77649i		−	0.301531i	−0.988570	−	0.150765i	$-0.951826\pi$
$367$		−	5.77649i		−	0.301531i	0.988570	−	0.150765i	$-0.0481738\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.16307			0.319970
$372$		0			0
$373$			12.8048i			0.663008i	0.943454	+	0.331504i	$0.107556\pi$
$373$			12.8048i			0.663008i	−0.943454	+	0.331504i	$0.892444\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			10.7085i			0.551517i
$378$		0			0
$379$	7.19755			0.369713			0.184857	−	0.982765i	$-0.440818\pi$
$379$	7.19755			0.369713			0.184857	+	0.982765i	$0.440818\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			2.41469i			0.123385i	0.998095	+	0.0616923i	$0.0196498\pi$
$383$			2.41469i			0.123385i	−0.998095	+	0.0616923i	$0.980350\pi$
$384$		0			0
$385$	−14.6498	+	0.810772i	−0.746622	+	0.0413208i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.6484			−0.539893			−0.269946	−	0.962875i	$-0.587006\pi$
$389$	−10.6484			−0.539893			−0.269946	+	0.962875i	$0.587006\pi$
$390$		0			0
$391$	−4.53770			−0.229482
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.0916071	+	1.65524i	0.00460925	+	0.0832842i
$396$		0			0
$397$			10.3068i			0.517284i	0.965973	+	0.258642i	$0.0832749\pi$
$397$			10.3068i			0.517284i	−0.965973	+	0.258642i	$0.916725\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.7529			0.886536			0.443268	−	0.896389i	$-0.353819\pi$
$401$	17.7529			0.886536			0.443268	+	0.896389i	$0.353819\pi$
$402$		0			0
$403$			23.5544i			1.17333i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			36.9229i			1.83020i
$408$		0			0
$409$	23.1365			1.14403			0.572013	−	0.820245i	$-0.306163\pi$
$409$	23.1365			1.14403			0.572013	+	0.820245i	$0.306163\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7.41172i			0.364707i
$414$		0			0
$415$	−20.5682	+	1.13832i	−1.00966	+	0.0558780i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.2676			−1.38096			−0.690481	−	0.723351i	$-0.742601\pi$
$419$	−28.2676			−1.38096			−0.690481	+	0.723351i	$0.742601\pi$
$420$		0			0
$421$	35.1365			1.71245			0.856224	−	0.516605i	$-0.172805\pi$
$421$	35.1365			1.71245			0.856224	+	0.516605i	$0.172805\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.494253	+	4.45163i	0.0239748	+	0.215936i
$426$		0			0
$427$		−	5.59164i		−	0.270598i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−36.8894			−1.77690			−0.888449	−	0.458976i	$-0.848216\pi$
$431$	−36.8894			−1.77690			−0.888449	+	0.458976i	$0.848216\pi$
$432$		0			0
$433$		−	11.8120i		−	0.567649i	−0.958876	−	0.283825i	$-0.908397\pi$
$433$		−	11.8120i		−	0.567649i	0.958876	−	0.283825i	$-0.0916033\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	27.0508i		−	1.29402i
$438$		0			0
$439$	0.741379			0.0353841			0.0176920	−	0.999843i	$-0.494368\pi$
$439$	0.741379			0.0353841			0.0176920	+	0.999843i	$0.494368\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.37026i		−	0.255149i	−0.991829	−	0.127574i	$-0.959281\pi$
$443$		−	5.37026i		−	0.255149i	0.991829	−	0.127574i	$-0.0407192\pi$
$444$		0			0
$445$	−1.14260	−	20.6456i	−0.0541646	−	0.978696i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.1975			−1.00037			−0.500187	−	0.865917i	$-0.666735\pi$
$449$	−21.1975			−1.00037			−0.500187	+	0.865917i	$0.666735\pi$
$450$		0			0
$451$	−39.9594			−1.88161
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.568239	−	10.2675i	−0.0266395	−	0.481346i
$456$		0			0
$457$		−	0.992806i		−	0.0464415i	−0.999730	−	0.0232208i	$-0.992608\pi$
$457$		−	0.992806i		−	0.0464415i	0.999730	−	0.0232208i	$-0.00739206\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.3197			0.480636			0.240318	−	0.970694i	$-0.422748\pi$
$461$	10.3197			0.480636			0.240318	+	0.970694i	$0.422748\pi$
$462$		0			0
$463$			13.0638i			0.607128i	0.952811	+	0.303564i	$0.0981767\pi$
$463$			13.0638i			0.607128i	−0.952811	+	0.303564i	$0.901823\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			19.3938i			0.897437i	0.893673	+	0.448718i	$0.148119\pi$
$467$			19.3938i			0.897437i	−0.893673	+	0.448718i	$0.851881\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$			55.6663i			2.55954i
$474$		0			0
$475$	−26.5377	+	2.94641i	−1.21763	+	0.135191i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.9159			0.544453			0.272227	−	0.962233i	$-0.412240\pi$
$479$	11.9159			0.544453			0.272227	+	0.962233i	$0.412240\pi$
$480$		0			0
$481$	−25.8779			−1.17993
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.72234	−	0.427383i	0.350653	−	0.0194064i
$486$		0			0
$487$			18.8027i			0.852031i	0.904716	+	0.426016i	$0.140083\pi$
$487$			18.8027i			0.852031i	−0.904716	+	0.426016i	$0.859917\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.93893			−0.448538			−0.224269	−	0.974527i	$-0.571999\pi$
$491$	−9.93893			−0.448538			−0.224269	+	0.974527i	$0.571999\pi$
$492$		0			0
$493$			2.68739i			0.121034i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.87999i		−	0.308610i
$498$		0			0
$499$	9.93893			0.444928			0.222464	−	0.974941i	$-0.428590\pi$
$499$	9.93893			0.444928			0.222464	+	0.974941i	$0.428590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	36.0636i		−	1.60800i	−0.594630	−	0.803999i	$-0.702702\pi$
$503$		−	36.0636i		−	1.60800i	0.594630	−	0.803999i	$-0.297298\pi$
$504$		0			0
$505$	2.03054	+	36.6896i	0.0903577	+	1.63266i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−40.8779			−1.81188			−0.905940	−	0.423407i	$-0.860834\pi$
$509$	−40.8779			−1.81188			−0.905940	+	0.423407i	$0.860834\pi$
$510$		0			0
$511$	12.0409			0.532659
$512$		0			0
$513$		0			0
$514$		0			0
$515$	35.4141	−	1.95995i	1.56053	−	0.0863657i
$516$		0			0
$517$			15.1225i			0.665087i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	17.9504			0.786422			0.393211	−	0.919448i	$-0.371364\pi$
$521$	17.9504			0.786422			0.393211	+	0.919448i	$0.371364\pi$
$522$		0			0
$523$		−	20.9732i		−	0.917092i	−0.888671	−	0.458546i	$-0.848370\pi$
$523$		−	20.9732i		−	0.917092i	0.888671	−	0.458546i	$-0.151630\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.91116i			0.257494i
$528$		0			0
$529$	−2.65985			−0.115645
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	28.0060i		−	1.21307i
$534$		0			0
$535$	21.2791	−	1.17766i	0.919975	−	0.0509148i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−27.1975			−1.17148
$540$		0			0
$541$	−13.6192			−0.585537			−0.292768	−	0.956183i	$-0.594576\pi$
$541$	−13.6192			−0.585537			−0.292768	+	0.956183i	$0.594576\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.01149	−	36.3455i	−0.0861630	−	1.55687i
$546$		0			0
$547$			8.79708i			0.376136i	0.982156	+	0.188068i	$0.0602225\pi$
$547$			8.79708i			0.376136i	−0.982156	+	0.188068i	$0.939777\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.0205			−0.682495
$552$		0			0
$553$		−	0.955156i		−	0.0406174i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	33.9308i		−	1.43770i	−0.695168	−	0.718848i	$-0.744670\pi$
$557$		−	33.9308i		−	1.43770i	0.695168	−	0.718848i	$-0.255330\pi$
$558$		0			0
$559$	−39.0143			−1.65013
$560$		0			0
$561$		0			0
$562$		0			0
$563$			33.3626i			1.40607i	0.711158	+	0.703033i	$0.248171\pi$
$563$			33.3626i			1.40607i	−0.711158	+	0.703033i	$0.751829\pi$
$564$		0			0
$565$	40.5887	−	2.24633i	1.70758	−	0.0945037i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.63073			0.319897			0.159948	−	0.987125i	$-0.448867\pi$
$569$	7.63073			0.319897			0.159948	+	0.987125i	$0.448867\pi$
$570$		0			0
$571$	26.4971			1.10887			0.554434	−	0.832227i	$-0.312935\pi$
$571$	26.4971			1.10887			0.554434	+	0.832227i	$0.312935\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.79490	+	25.1731i	0.116556	+	1.04979i
$576$		0			0
$577$		−	9.93709i		−	0.413686i	−0.978374	−	0.206843i	$-0.933681\pi$
$577$		−	9.93709i		−	0.413686i	0.978374	−	0.206843i	$-0.0663190\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.8689			0.492405
$582$		0			0
$583$		−	24.3635i		−	1.00903i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	23.5179i		−	0.970686i	−0.874324	−	0.485343i	$-0.838695\pi$
$587$		−	23.5179i		−	0.970686i	0.874324	−	0.485343i	$-0.161305\pi$
$588$		0			0
$589$	−35.2385			−1.45198
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	11.6637i		−	0.478970i	−0.970900	−	0.239485i	$-0.923021\pi$
$593$		−	11.6637i		−	0.478970i	0.970900	−	0.239485i	$-0.0769786\pi$
$594$		0			0
$595$	−0.142604	−	2.57670i	−0.00584620	−	0.105634i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.06107			0.0842131			0.0421065	−	0.999113i	$-0.486593\pi$
$599$	2.06107			0.0842131			0.0421065	+	0.999113i	$0.486593\pi$
$600$		0			0
$601$	−13.9389			−0.568581			−0.284290	−	0.958738i	$-0.591758\pi$
$601$	−13.9389			−0.568581			−0.284290	+	0.958738i	$0.591758\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.84590	+	33.3534i	0.0750466	+	1.35601i
$606$		0			0
$607$		−	47.2526i		−	1.91792i	−0.283537	−	0.958961i	$-0.591508\pi$
$607$		−	47.2526i		−	1.91792i	0.283537	−	0.958961i	$-0.408492\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.5988			−0.428781
$612$		0			0
$613$		−	31.4648i		−	1.27085i	−0.772162	−	0.635426i	$-0.780825\pi$
$613$		−	31.4648i		−	1.27085i	0.772162	−	0.635426i	$-0.219175\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	21.0987i		−	0.849400i	−0.905334	−	0.424700i	$-0.860380\pi$
$617$		−	21.0987i		−	0.849400i	0.905334	−	0.424700i	$-0.139620\pi$
$618$		0			0
$619$	3.91847			0.157496			0.0787482	−	0.996895i	$-0.474908\pi$
$619$	3.91847			0.157496			0.0787482	+	0.996895i	$0.474908\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			11.9135i			0.477306i
$624$		0			0
$625$	24.3912	−	5.48377i	0.975646	−	0.219351i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.49425			−0.258943
$630$		0			0
$631$	44.5582			1.77383			0.886916	−	0.461930i	$-0.152843\pi$
$631$	44.5582			1.77383			0.886916	+	0.461930i	$0.152843\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	26.7848	−	1.48237i	1.06292	−	0.0588261i
$636$		0			0
$637$		−	19.0617i		−	0.755253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.3287			0.605446			0.302723	−	0.953079i	$-0.402104\pi$
$641$	15.3287			0.605446			0.302723	+	0.953079i	$0.402104\pi$
$642$		0			0
$643$			5.37026i			0.211783i	0.994378	+	0.105891i	$0.0337695\pi$
$643$			5.37026i			0.211783i	−0.994378	+	0.105891i	$0.966230\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.65087i		−	0.104216i	−0.998641	−	0.0521082i	$-0.983406\pi$
$647$		−	2.65087i		−	0.104216i	0.998641	−	0.0521082i	$-0.0165941\pi$
$648$		0			0
$649$	29.2995			1.15011
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	7.71642i		−	0.301967i	−0.988536	−	0.150984i	$-0.951756\pi$
$653$		−	7.71642i		−	0.301967i	0.988536	−	0.150984i	$-0.0482440\pi$
$654$		0			0
$655$	−1.22809	−	22.1902i	−0.0479853	−	0.867041i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.7529			−0.457827			−0.228913	−	0.973447i	$-0.573517\pi$
$659$	−11.7529			−0.457827			−0.228913	+	0.973447i	$0.573517\pi$
$660$		0			0
$661$	10.0611			0.391330			0.195665	−	0.980671i	$-0.437313\pi$
$661$	10.0611			0.391330			0.195665	+	0.980671i	$0.437313\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.3606	−	0.850113i	0.595659	−	0.0329660i
$666$		0			0
$667$			15.1967i			0.588417i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.1045			−0.853336
$672$		0			0
$673$			32.1986i			1.24116i	0.784141	+	0.620582i	$0.213104\pi$
$673$			32.1986i			1.24116i	−0.784141	+	0.620582i	$0.786896\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1.82360i			0.0700866i	0.999386	+	0.0350433i	$0.0111569\pi$
$677$			1.82360i			0.0700866i	−0.999386	+	0.0350433i	$0.988843\pi$
$678$		0			0
$679$	−4.45617			−0.171012
$680$		0			0
$681$		0			0
$682$		0			0
$683$			36.5909i			1.40011i	0.714089	+	0.700055i	$0.246841\pi$
$683$			36.5909i			1.40011i	−0.714089	+	0.700055i	$0.753159\pi$
$684$		0			0
$685$	12.6498	−	0.700085i	0.483323	−	0.0267489i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17.0754			0.650521
$690$		0			0
$691$	7.38076			0.280777			0.140389	−	0.990096i	$-0.455165\pi$
$691$	7.38076			0.280777			0.140389	+	0.990096i	$0.455165\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.980959	−	17.7249i	−0.0372099	−	0.672342i
$696$		0			0
$697$		−	7.02833i		−	0.266217i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.1045			1.17480			0.587401	−	0.809296i	$-0.300151\pi$
$701$	31.1045			1.17480			0.587401	+	0.809296i	$0.300151\pi$
$702$		0			0
$703$		−	38.7145i		−	1.46015i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	21.1717i		−	0.796243i
$708$		0			0
$709$	−26.0754			−0.979282			−0.489641	−	0.871924i	$-0.662872\pi$
$709$	−26.0754			−0.979282			−0.489641	+	0.871924i	$0.662872\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			33.4265i			1.25183i
$714$		0			0
$715$	−40.5887	+	2.24633i	−1.51793	+	0.0840079i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	43.0524			1.60558			0.802792	−	0.596259i	$-0.203347\pi$
$719$	43.0524			1.60558			0.802792	+	0.596259i	$0.203347\pi$
$720$		0			0
$721$	−20.4357			−0.761066
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.9084	−	1.65524i	0.553684	−	0.0614741i
$726$		0			0
$727$		−	14.2825i		−	0.529710i	−0.964288	−	0.264855i	$-0.914676\pi$
$727$		−	14.2825i		−	0.529710i	0.964288	−	0.264855i	$-0.0853241\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.79096			−0.362132
$732$		0			0
$733$			3.53186i			0.130452i	0.997871	+	0.0652260i	$0.0207768\pi$
$733$			3.53186i			0.130452i	−0.997871	+	0.0652260i	$0.979223\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			43.4845i			1.60177i
$738$		0			0
$739$	−13.3606			−0.491478			−0.245739	−	0.969336i	$-0.579031\pi$
$739$	−13.3606			−0.491478			−0.245739	+	0.969336i	$0.579031\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			21.1625i			0.776378i	0.921580	+	0.388189i	$0.126899\pi$
$743$			21.1625i			0.776378i	−0.921580	+	0.388189i	$0.873101\pi$
$744$		0			0
$745$	−1.83299	−	33.1201i	−0.0671554	−	1.21343i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.2791			−0.448668
$750$		0			0
$751$	8.08153			0.294899			0.147450	−	0.989070i	$-0.452894\pi$
$751$	8.08153			0.294899			0.147450	+	0.989070i	$0.452894\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.0739865	−	1.33686i	−0.00269264	−	0.0486532i
$756$		0			0
$757$		−	26.3114i		−	0.956305i	−0.878277	−	0.478152i	$-0.841307\pi$
$757$		−	26.3114i		−	0.956305i	0.878277	−	0.478152i	$-0.158693\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.1631			0.984660			0.492330	−	0.870409i	$-0.336145\pi$
$761$	27.1631			0.984660			0.492330	+	0.870409i	$0.336145\pi$
$762$		0			0
$763$			20.9732i			0.759279i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			20.5349i			0.741473i
$768$		0			0
$769$	−2.83693			−0.102302			−0.0511512	−	0.998691i	$-0.516289\pi$
$769$	−2.83693			−0.102302			−0.0511512	+	0.998691i	$0.516289\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	32.7924	−	3.64085i	1.17794	−	0.130783i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	41.8983			1.50116
$780$		0			0
$781$	−27.1975			−0.973205
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.0944	+	2.21897i	−1.43103	+	0.0791986i
$786$		0			0
$787$			3.79088i			0.135130i	0.997715	+	0.0675652i	$0.0215231\pi$
$787$			3.79088i			0.135130i	−0.997715	+	0.0675652i	$0.978477\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23.4217			−0.832779
$792$		0			0
$793$		−	15.4922i		−	0.550144i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			44.0756i			1.56124i	0.625007	+	0.780619i	$0.285096\pi$
$797$			44.0756i			1.56124i	−0.625007	+	0.780619i	$0.714904\pi$
$798$		0			0
$799$	−2.65985			−0.0940986
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	47.5994i		−	1.67975i
$804$		0			0
$805$	−0.806399	−	14.5707i	−0.0284218	−	0.513551i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.48276			−0.0521310			−0.0260655	−	0.999660i	$-0.508298\pi$
$809$	−1.48276			−0.0521310			−0.0260655	+	0.999660i	$0.508298\pi$
$810$		0			0
$811$	51.2385			1.79923			0.899613	−	0.436688i	$-0.143849\pi$
$811$	51.2385			1.79923			0.899613	+	0.436688i	$0.143849\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.4332	+	0.577411i	−0.365458	+	0.0202258i
$816$		0			0
$817$		−	58.3673i		−	2.04201i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.7123			0.792663			0.396332	−	0.918107i	$-0.370283\pi$
$821$	22.7123			0.792663			0.396332	+	0.918107i	$0.370283\pi$
$822$		0			0
$823$		−	13.1004i		−	0.456650i	−0.973585	−	0.228325i	$-0.926675\pi$
$823$		−	13.1004i		−	0.456650i	0.973585	−	0.228325i	$-0.0733249\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	34.8631i		−	1.21231i	−0.795346	−	0.606155i	$-0.792711\pi$
$827$		−	34.8631i		−	1.21231i	0.795346	−	0.606155i	$-0.207289\pi$
$828$		0			0
$829$	25.5988			0.889082			0.444541	−	0.895758i	$-0.353367\pi$
$829$	25.5988			0.889082			0.444541	+	0.895758i	$0.353367\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.78369i		−	0.165745i
$834$		0			0
$835$	13.3096	−	0.736603i	0.460598	−	0.0254912i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−40.3315			−1.39240			−0.696199	−	0.717849i	$-0.745127\pi$
$839$	−40.3315			−1.39240			−0.696199	+	0.717849i	$0.745127\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.0319560	+	0.577411i	0.00109932	+	0.0198635i
$846$		0			0
$847$		−	19.2466i		−	0.661320i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−36.7238			−1.25887
$852$		0			0
$853$		−	40.2984i		−	1.37979i	−0.723909	−	0.689896i	$-0.757656\pi$
$853$		−	40.2984i		−	1.37979i	0.723909	−	0.689896i	$-0.242344\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22.0401i		−	0.752877i	−0.926442	−	0.376438i	$-0.877149\pi$
$857$		−	22.0401i		−	0.752877i	0.926442	−	0.376438i	$-0.122851\pi$
$858$		0			0
$859$	−29.2791			−0.998989			−0.499495	−	0.866317i	$-0.666481\pi$
$859$	−29.2791			−0.998989			−0.499495	+	0.866317i	$0.666481\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	46.1491i		−	1.57093i	−0.618904	−	0.785466i	$-0.712423\pi$
$863$		−	46.1491i		−	1.57093i	0.618904	−	0.785466i	$-0.287577\pi$
$864$		0			0
$865$	−21.9084	+	1.21249i	−0.744908	+	0.0412259i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.77586			−0.128087
$870$		0			0
$871$	−30.4766			−1.03266
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.2065	+	2.37817i	−0.480268	+	0.0803967i
$876$		0			0
$877$		−	27.6363i		−	0.933211i	−0.884466	−	0.466606i	$-0.845477\pi$
$877$		−	27.6363i		−	0.933211i	0.884466	−	0.466606i	$-0.154523\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.79632			−0.161592			−0.0807961	−	0.996731i	$-0.525746\pi$
$881$	−4.79632			−0.161592			−0.0807961	+	0.996731i	$0.525746\pi$
$882$		0			0
$883$			21.7126i			0.730687i	0.930873	+	0.365343i	$0.119048\pi$
$883$			21.7126i			0.730687i	−0.930873	+	0.365343i	$0.880952\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	22.0401i		−	0.740035i	−0.929025	−	0.370018i	$-0.879352\pi$
$887$		−	22.0401i		−	0.740035i	0.929025	−	0.370018i	$-0.120648\pi$
$888$		0			0
$889$	−15.4562			−0.518383
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	15.8563i		−	0.530610i
$894$		0			0
$895$	−2.02046	−	36.5075i	−0.0675366	−	1.22031i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19.7963			0.660244
$900$		0			0
$901$	4.28521			0.142761
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.363143	+	6.56160i	0.0120713	+	0.218115i
$906$		0			0
$907$			39.3011i			1.30497i	0.757801	+	0.652486i	$0.226274\pi$
$907$			39.3011i			1.30497i	−0.757801	+	0.652486i	$0.773726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.9389			−0.528080			−0.264040	−	0.964512i	$-0.585055\pi$
$911$	−15.9389			−0.528080			−0.264040	+	0.964512i	$0.585055\pi$
$912$		0			0
$913$		−	46.9194i		−	1.55280i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			12.8048i			0.422852i
$918$		0			0
$919$	6.16307			0.203301			0.101650	−	0.994820i	$-0.467588\pi$
$919$	6.16307			0.203301			0.101650	+	0.994820i	$0.467588\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	19.0617i		−	0.627424i
$924$		0			0
$925$	4.00000	+	36.0271i	0.131519	+	1.18456i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.1020			−0.331436			−0.165718	−	0.986173i	$-0.552994\pi$
$929$	−10.1020			−0.331436			−0.165718	+	0.986173i	$0.552994\pi$
$930$		0			0
$931$	28.5172			0.934615
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.1861	+	0.563734i	−0.333120	+	0.0184361i
$936$		0			0
$937$		−	5.00506i		−	0.163508i	−0.996653	−	0.0817541i	$-0.973948\pi$
$937$		−	5.00506i		−	0.163508i	0.996653	−	0.0817541i	$-0.0260522\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.98851			0.325616			0.162808	−	0.986658i	$-0.447945\pi$
$941$	9.98851			0.325616			0.162808	+	0.986658i	$0.447945\pi$
$942$		0			0
$943$		−	39.7438i		−	1.29424i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			47.6223i			1.54752i	0.633481	+	0.773758i	$0.281625\pi$
$947$			47.6223i			1.54752i	−0.633481	+	0.773758i	$0.718375\pi$
$948$		0			0
$949$	33.3606			1.08293
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	34.2355i		−	1.10900i	−0.832184	−	0.554499i	$-0.812910\pi$
$953$		−	34.2355i		−	1.10900i	0.832184	−	0.554499i	$-0.187090\pi$
$954$		0			0
$955$	0.568239	+	10.2675i	0.0183878	+	0.332247i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.29954			−0.235715
$960$		0			0
$961$	12.5438			0.404640
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.53628	+	0.195711i	−0.113837	+	0.00630016i
$966$		0			0
$967$			3.71671i			0.119521i	0.998213	+	0.0597607i	$0.0190338\pi$
$967$			3.71671i			0.119521i	−0.998213	+	0.0597607i	$0.980966\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	45.0959			1.44720			0.723598	−	0.690222i	$-0.242487\pi$
$971$	45.0959			1.44720			0.723598	+	0.690222i	$0.242487\pi$
$972$		0			0
$973$			10.2281i			0.327898i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	6.58896i		−	0.210799i	−0.994430	−	0.105400i	$-0.966388\pi$
$977$		−	6.58896i		−	0.210799i	0.994430	−	0.105400i	$-0.0336122\pi$
$978$		0			0
$979$	47.0959			1.50519
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	12.2319i		−	0.390138i	−0.980790	−	0.195069i	$-0.937507\pi$
$983$		−	12.2319i		−	0.390138i	0.980790	−	0.195069i	$-0.0624930\pi$
$984$		0			0
$985$	−29.1975	+	1.61590i	−0.930311	+	0.0514869i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−55.3660			−1.76054
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.659846	+	11.9227i	0.0209185	+	0.377975i
$996$		0			0
$997$		−	3.23744i		−	0.102531i	−0.998685	−	0.0512655i	$-0.983675\pi$
$997$		−	3.23744i		−	0.102531i	0.998685	−	0.0512655i	$-0.0163255\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.d.d.649.4		6
3.2	odd	2		1620.2.d.c.649.3		6
5.2	odd	4		8100.2.a.bd.1.4		6
5.3	odd	4		8100.2.a.bd.1.3		6
5.4	even	2	inner	1620.2.d.d.649.3		6
9.2	odd	6		180.2.r.a.49.4	yes	12
9.4	even	3		540.2.r.a.289.5		12
9.5	odd	6		180.2.r.a.169.3	yes	12
9.7	even	3		540.2.r.a.469.1		12
15.2	even	4		8100.2.a.bc.1.4		6
15.8	even	4		8100.2.a.bc.1.3		6
15.14	odd	2		1620.2.d.c.649.4		6
36.7	odd	6		2160.2.by.e.1009.1		12
36.11	even	6		720.2.by.e.49.3		12
36.23	even	6		720.2.by.e.529.4		12
36.31	odd	6		2160.2.by.e.289.5		12
45.2	even	12		900.2.i.f.301.6		12
45.4	even	6		540.2.r.a.289.1		12
45.7	odd	12		2700.2.i.f.901.3		12
45.13	odd	12		2700.2.i.f.1801.4		12
45.14	odd	6		180.2.r.a.169.4	yes	12
45.22	odd	12		2700.2.i.f.1801.3		12
45.23	even	12		900.2.i.f.601.1		12
45.29	odd	6		180.2.r.a.49.3	✓	12
45.32	even	12		900.2.i.f.601.6		12
45.34	even	6		540.2.r.a.469.5		12
45.38	even	12		900.2.i.f.301.1		12
45.43	odd	12		2700.2.i.f.901.4		12
180.59	even	6		720.2.by.e.529.3		12
180.79	odd	6		2160.2.by.e.1009.5		12
180.119	even	6		720.2.by.e.49.4		12
180.139	odd	6		2160.2.by.e.289.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.3	✓	12	45.29	odd	6
180.2.r.a.49.4	yes	12	9.2	odd	6
180.2.r.a.169.3	yes	12	9.5	odd	6
180.2.r.a.169.4	yes	12	45.14	odd	6
540.2.r.a.289.1		12	45.4	even	6
540.2.r.a.289.5		12	9.4	even	3
540.2.r.a.469.1		12	9.7	even	3
540.2.r.a.469.5		12	45.34	even	6
720.2.by.e.49.3		12	36.11	even	6
720.2.by.e.49.4		12	180.119	even	6
720.2.by.e.529.3		12	180.59	even	6
720.2.by.e.529.4		12	36.23	even	6
900.2.i.f.301.1		12	45.38	even	12
900.2.i.f.301.6		12	45.2	even	12
900.2.i.f.601.1		12	45.23	even	12
900.2.i.f.601.6		12	45.32	even	12
1620.2.d.c.649.3		6	3.2	odd	2
1620.2.d.c.649.4		6	15.14	odd	2
1620.2.d.d.649.3		6	5.4	even	2	inner
1620.2.d.d.649.4		6	1.1	even	1	trivial
2160.2.by.e.289.1		12	180.139	odd	6
2160.2.by.e.289.5		12	36.31	odd	6
2160.2.by.e.1009.1		12	36.7	odd	6
2160.2.by.e.1009.5		12	180.79	odd	6
2700.2.i.f.901.3		12	45.7	odd	12
2700.2.i.f.901.4		12	45.43	odd	12
2700.2.i.f.1801.3		12	45.22	odd	12
2700.2.i.f.1801.4		12	45.13	odd	12
8100.2.a.bc.1.3		6	15.8	even	4
8100.2.a.bc.1.4		6	15.2	even	4
8100.2.a.bd.1.3		6	5.3	odd	4
8100.2.a.bd.1.4		6	5.2	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.123563 + 2.23265i) q^{5} -1.28835i q^{7} -5.09303 q^{11} -3.56951i q^{13} -0.895796i q^{17} +5.34015 q^{19} -5.06555i q^{23} +(-4.96946 + 0.551747i) q^{25} -3.00000 q^{29} -6.59877 q^{31} +(2.87644 - 0.159193i) q^{35} -7.24970i q^{37} +7.84590 q^{41} -10.9299i q^{43} -2.96925i q^{47} +5.34015 q^{49} +4.78369i q^{53} +(-0.629311 - 11.3710i) q^{55} -5.75287 q^{59} +4.34015 q^{61} +(7.96946 - 0.441060i) q^{65} -8.53805i q^{67} +5.34015 q^{71} +9.34600i q^{73} +6.56160i q^{77} +0.741379 q^{79} +9.21247i q^{83} +(2.00000 - 0.110687i) q^{85} -9.24713 q^{89} -4.59877 q^{91} +(0.659846 + 11.9227i) q^{95} -3.45882i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) \) 6 * q + q^5 + 2 * q^11 + 3 * q^25 - 18 * q^29 - 6 * q^31 + 17 * q^35 + 14 * q^41 - 3 * q^55 - 34 * q^59 - 6 * q^61 + 15 * q^65 + 6 * q^79 + 12 * q^85 - 56 * q^89 + 6 * q^91 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more