LMFDB - Embedded newform 5220.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.a (trivial)

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:	yes
Analytic conductor:	$41.6819098551$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.32088 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.32088 q^{7} -5.32088 q^{11} -5.02827 q^{13} +6.34916 q^{17} -4.34916 q^{19} +1.70739 q^{23} +1.00000 q^{25} +1.00000 q^{29} +8.34916 q^{31} +1.32088 q^{35} -6.93438 q^{37} +1.02827 q^{41} +10.7357 q^{43} +0.679116 q^{47} -5.25526 q^{49} -2.38650 q^{53} -5.32088 q^{55} -10.4431 q^{59} -6.38650 q^{61} -5.02827 q^{65} -5.70739 q^{67} +3.61350 q^{71} -6.73566 q^{73} -7.02827 q^{77} -11.3774 q^{79} +3.96265 q^{83} +6.34916 q^{85} -2.58522 q^{89} -6.64177 q^{91} -4.34916 q^{95} -15.3209 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 4 * q^7	$ 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 4 * q^7 - 8 * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^19 + 3 * q^25 + 3 * q^29 + 4 * q^31 - 4 * q^35 - 10 * q^37 - 10 * q^41 + 14 * q^43 + 10 * q^47 + 3 * q^49 - 10 * q^53 - 8 * q^55 - 8 * q^59 - 22 * q^61 - 2 * q^65 - 12 * q^67 + 8 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^83 - 2 * q^85 - 18 * q^89 - 4 * q^91 + 8 * q^95 - 38 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.32088	0.499247	0.249624	−	0.968343i	$-0.419693\pi$
$7$	1.32088	0.499247	0.249624	+	0.968343i	$0.419693\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.32088	−1.60431	−0.802154	−	0.597118i	$-0.796312\pi$
$11$	−5.32088	−1.60431	−0.802154	+	0.597118i	$0.796312\pi$
$12$	0	0
$13$	−5.02827	−1.39459	−0.697296	−	0.716783i	$-0.745614\pi$
$13$	−5.02827	−1.39459	−0.697296	+	0.716783i	$0.745614\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.34916	1.53990	0.769949	−	0.638106i	$-0.220282\pi$
$17$	6.34916	1.53990	0.769949	+	0.638106i	$0.220282\pi$
$18$	0	0
$19$	−4.34916	−0.997765	−0.498883	−	0.866670i	$-0.666256\pi$
$19$	−4.34916	−0.997765	−0.498883	+	0.866670i	$0.666256\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.70739	0.356015	0.178008	−	0.984029i	$-0.443035\pi$
$23$	1.70739	0.356015	0.178008	+	0.984029i	$0.443035\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.34916	1.49955	0.749777	−	0.661691i	$-0.230161\pi$
$31$	8.34916	1.49955	0.749777	+	0.661691i	$0.230161\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.32088	0.223270
$36$	0	0
$37$	−6.93438	−1.14000	−0.570002	−	0.821643i	$-0.693058\pi$
$37$	−6.93438	−1.14000	−0.570002	+	0.821643i	$0.693058\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.02827	0.160589	0.0802947	−	0.996771i	$-0.474414\pi$
$41$	1.02827	0.160589	0.0802947	+	0.996771i	$0.474414\pi$
$42$	0	0
$43$	10.7357	1.63717	0.818587	−	0.574383i	$-0.194758\pi$
$43$	10.7357	1.63717	0.818587	+	0.574383i	$0.194758\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.679116	0.0990592	0.0495296	−	0.998773i	$-0.484228\pi$
$47$	0.679116	0.0990592	0.0495296	+	0.998773i	$0.484228\pi$
$48$	0	0
$49$	−5.25526	−0.750752
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.38650	−0.327812	−0.163906	−	0.986476i	$-0.552409\pi$
$53$	−2.38650	−0.327812	−0.163906	+	0.986476i	$0.552409\pi$
$54$	0	0
$55$	−5.32088	−0.717468
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4431	−1.35957	−0.679785	−	0.733412i	$-0.737927\pi$
$59$	−10.4431	−1.35957	−0.679785	+	0.733412i	$0.737927\pi$
$60$	0	0
$61$	−6.38650	−0.817708	−0.408854	−	0.912600i	$-0.634072\pi$
$61$	−6.38650	−0.817708	−0.408854	+	0.912600i	$0.634072\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.02827	−0.623681
$66$	0	0
$67$	−5.70739	−0.697269	−0.348634	−	0.937259i	$-0.613354\pi$
$67$	−5.70739	−0.697269	−0.348634	+	0.937259i	$0.613354\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.61350	0.428843	0.214421	−	0.976741i	$-0.431213\pi$
$71$	3.61350	0.428843	0.214421	+	0.976741i	$0.431213\pi$
$72$	0	0
$73$	−6.73566	−0.788350	−0.394175	−	0.919035i	$-0.628970\pi$
$73$	−6.73566	−0.788350	−0.394175	+	0.919035i	$0.628970\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.02827	−0.800946
$78$	0	0
$79$	−11.3774	−1.28006	−0.640031	−	0.768349i	$-0.721078\pi$
$79$	−11.3774	−1.28006	−0.640031	+	0.768349i	$0.721078\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.96265	0.434958	0.217479	−	0.976065i	$-0.430217\pi$
$83$	3.96265	0.434958	0.217479	+	0.976065i	$0.430217\pi$
$84$	0	0
$85$	6.34916	0.688663
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.58522	−0.274033	−0.137016	−	0.990569i	$-0.543751\pi$
$89$	−2.58522	−0.274033	−0.137016	+	0.990569i	$0.543751\pi$
$90$	0	0
$91$	−6.64177	−0.696247
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.34916	−0.446214
$96$	0	0
$97$	−15.3209	−1.55560	−0.777800	−	0.628512i	$-0.783664\pi$
$97$	−15.3209	−1.55560	−0.777800	+	0.628512i	$0.783664\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.8688	−1.18099	−0.590493	−	0.807043i	$-0.701067\pi$
$101$	−11.8688	−1.18099	−0.590493	+	0.807043i	$0.701067\pi$
$102$	0	0
$103$	−3.65084	−0.359728	−0.179864	−	0.983691i	$-0.557566\pi$
$103$	−3.65084	−0.359728	−0.179864	+	0.983691i	$0.557566\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.32088	0.514389	0.257195	−	0.966360i	$-0.417202\pi$
$107$	5.32088	0.514389	0.257195	+	0.966360i	$0.417202\pi$
$108$	0	0
$109$	−14.3118	−1.37082	−0.685411	−	0.728156i	$-0.740378\pi$
$109$	−14.3118	−1.37082	−0.685411	+	0.728156i	$0.740378\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.03735	−0.191657	−0.0958287	−	0.995398i	$-0.530550\pi$
$113$	−2.03735	−0.191657	−0.0958287	+	0.995398i	$0.530550\pi$
$114$	0	0
$115$	1.70739	0.159215
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.38650	0.768790
$120$	0	0
$121$	17.3118	1.57380
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.26434	−0.112192	−0.0560959	−	0.998425i	$-0.517865\pi$
$127$	−1.26434	−0.112192	−0.0560959	+	0.998425i	$0.517865\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.70739	0.149175	0.0745877	−	0.997214i	$-0.476236\pi$
$131$	1.70739	0.149175	0.0745877	+	0.997214i	$0.476236\pi$
$132$	0	0
$133$	−5.74474	−0.498132
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.7639	−1.85942	−0.929709	−	0.368294i	$-0.879942\pi$
$137$	−21.7639	−1.85942	−0.929709	+	0.368294i	$0.879942\pi$
$138$	0	0
$139$	−16.6983	−1.41633	−0.708166	−	0.706046i	$-0.750477\pi$
$139$	−16.6983	−1.41633	−0.708166	+	0.706046i	$0.750477\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	26.7549	2.23735
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	6.44305	0.524328	0.262164	−	0.965023i	$-0.415564\pi$
$151$	6.44305	0.524328	0.262164	+	0.965023i	$0.415564\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.34916	0.670621
$156$	0	0
$157$	10.7357	0.856799	0.428400	−	0.903589i	$-0.359078\pi$
$157$	10.7357	0.856799	0.428400	+	0.903589i	$0.359078\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.25526	0.177740
$162$	0	0
$163$	3.90611	0.305950	0.152975	−	0.988230i	$-0.451115\pi$
$163$	3.90611	0.305950	0.152975	+	0.988230i	$0.451115\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.51960	0.427120	0.213560	−	0.976930i	$-0.431494\pi$
$167$	5.51960	0.427120	0.213560	+	0.976930i	$0.431494\pi$
$168$	0	0
$169$	12.2835	0.944888
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.80128	−0.441063	−0.220532	−	0.975380i	$-0.570779\pi$
$173$	−5.80128	−0.441063	−0.220532	+	0.975380i	$0.570779\pi$
$174$	0	0
$175$	1.32088	0.0998495
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.0565	−1.05064	−0.525318	−	0.850906i	$-0.676054\pi$
$179$	−14.0565	−1.05064	−0.525318	+	0.850906i	$0.676054\pi$
$180$	0	0
$181$	−7.08482	−0.526611	−0.263305	−	0.964713i	$-0.584813\pi$
$181$	−7.08482	−0.526611	−0.263305	+	0.964713i	$0.584813\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.93438	−0.509826
$186$	0	0
$187$	−33.7831	−2.47047
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.547875	0.0396428	0.0198214	−	0.999804i	$-0.493690\pi$
$191$	0.547875	0.0396428	0.0198214	+	0.999804i	$0.493690\pi$
$192$	0	0
$193$	4.40571	0.317130	0.158565	−	0.987349i	$-0.449313\pi$
$193$	4.40571	0.317130	0.158565	+	0.987349i	$0.449313\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.64177	0.330712	0.165356	−	0.986234i	$-0.447123\pi$
$197$	4.64177	0.330712	0.165356	+	0.986234i	$0.447123\pi$
$198$	0	0
$199$	18.4431	1.30739	0.653697	−	0.756757i	$-0.273217\pi$
$199$	18.4431	1.30739	0.653697	+	0.756757i	$0.273217\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.32088	0.0927079
$204$	0	0
$205$	1.02827	0.0718178
$206$	0	0
$207$	0	0
$208$	0	0
$209$	23.1414	1.60072
$210$	0	0
$211$	−4.54787	−0.313089	−0.156544	−	0.987671i	$-0.550035\pi$
$211$	−4.54787	−0.313089	−0.156544	+	0.987671i	$0.550035\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.7357	0.732166
$216$	0	0
$217$	11.0283	0.748648
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−31.9253	−2.14753
$222$	0	0
$223$	1.12217	0.0751459	0.0375730	−	0.999294i	$-0.488037\pi$
$223$	1.12217	0.0751459	0.0375730	+	0.999294i	$0.488037\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.63270	−0.639345	−0.319672	−	0.947528i	$-0.603573\pi$
$227$	−9.63270	−0.639345	−0.319672	+	0.947528i	$0.603573\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.05655	0.527802	0.263901	−	0.964550i	$-0.414991\pi$
$233$	8.05655	0.527802	0.263901	+	0.964550i	$0.414991\pi$
$234$	0	0
$235$	0.679116	0.0443006
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.6135	−1.00995	−0.504977	−	0.863133i	$-0.668499\pi$
$239$	−15.6135	−1.00995	−0.504977	+	0.863133i	$0.668499\pi$
$240$	0	0
$241$	18.5105	1.19237	0.596184	−	0.802848i	$-0.296683\pi$
$241$	18.5105	1.19237	0.596184	+	0.802848i	$0.296683\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.25526	−0.335747
$246$	0	0
$247$	21.8688	1.39148
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.67912	0.421582	0.210791	−	0.977531i	$-0.432396\pi$
$251$	6.67912	0.421582	0.210791	+	0.977531i	$0.432396\pi$
$252$	0	0
$253$	−9.08482	−0.571158
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.4249	1.64834	0.824170	−	0.566342i	$-0.191642\pi$
$257$	26.4249	1.64834	0.824170	+	0.566342i	$0.191642\pi$
$258$	0	0
$259$	−9.15951	−0.569145
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.4340	−1.69165	−0.845826	−	0.533459i	$-0.820892\pi$
$263$	−27.4340	−1.69165	−0.845826	+	0.533459i	$0.820892\pi$
$264$	0	0
$265$	−2.38650	−0.146602
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.9717	0.668958	0.334479	−	0.942403i	$-0.391440\pi$
$269$	10.9717	0.668958	0.334479	+	0.942403i	$0.391440\pi$
$270$	0	0
$271$	20.1504	1.22405	0.612026	−	0.790838i	$-0.290355\pi$
$271$	20.1504	1.22405	0.612026	+	0.790838i	$0.290355\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.32088	−0.320861
$276$	0	0
$277$	3.15951	0.189837	0.0949184	−	0.995485i	$-0.469741\pi$
$277$	3.15951	0.189837	0.0949184	+	0.995485i	$0.469741\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.3118	0.615151	0.307576	−	0.951524i	$-0.400482\pi$
$281$	10.3118	0.615151	0.307576	+	0.951524i	$0.400482\pi$
$282$	0	0
$283$	0.423851	0.0251953	0.0125977	−	0.999921i	$-0.495990\pi$
$283$	0.423851	0.0251953	0.0125977	+	0.999921i	$0.495990\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.35823	0.0801738
$288$	0	0
$289$	23.3118	1.37128
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.2462	−1.12437	−0.562187	−	0.827010i	$-0.690040\pi$
$293$	−19.2462	−1.12437	−0.562187	+	0.827010i	$0.690040\pi$
$294$	0	0
$295$	−10.4431	−0.608018
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.58522	−0.496496
$300$	0	0
$301$	14.1806	0.817355
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.38650	−0.365690
$306$	0	0
$307$	22.7357	1.29759	0.648796	−	0.760962i	$-0.275273\pi$
$307$	22.7357	1.29759	0.648796	+	0.760962i	$0.275273\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.8031	−0.839409	−0.419704	−	0.907661i	$-0.637866\pi$
$311$	−14.8031	−0.839409	−0.419704	+	0.907661i	$0.637866\pi$
$312$	0	0
$313$	−14.9717	−0.846252	−0.423126	−	0.906071i	$-0.639067\pi$
$313$	−14.9717	−0.846252	−0.423126	+	0.906071i	$0.639067\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.1222	−1.29867	−0.649335	−	0.760502i	$-0.724953\pi$
$317$	−23.1222	−1.29867	−0.649335	+	0.760502i	$0.724953\pi$
$318$	0	0
$319$	−5.32088	−0.297912
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−27.6135	−1.53646
$324$	0	0
$325$	−5.02827	−0.278918
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.897033	0.0494550
$330$	0	0
$331$	−9.82048	−0.539783	−0.269891	−	0.962891i	$-0.586988\pi$
$331$	−9.82048	−0.539783	−0.269891	+	0.962891i	$0.586988\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.70739	−0.311828
$336$	0	0
$337$	−8.09389	−0.440903	−0.220451	−	0.975398i	$-0.570753\pi$
$337$	−8.09389	−0.440903	−0.220451	+	0.975398i	$0.570753\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−44.4249	−2.40574
$342$	0	0
$343$	−16.1878	−0.874058
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.37743	0.396041	0.198021	−	0.980198i	$-0.436549\pi$
$347$	7.37743	0.396041	0.198021	+	0.980198i	$0.436549\pi$
$348$	0	0
$349$	8.82956	0.472635	0.236318	−	0.971676i	$-0.424059\pi$
$349$	8.82956	0.472635	0.236318	+	0.971676i	$0.424059\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.7266	−0.730593	−0.365296	−	0.930891i	$-0.619032\pi$
$353$	−13.7266	−0.730593	−0.365296	+	0.930891i	$0.619032\pi$
$354$	0	0
$355$	3.61350	0.191784
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.70739	0.0901126	0.0450563	−	0.998984i	$-0.485653\pi$
$359$	1.70739	0.0901126	0.0450563	+	0.998984i	$0.485653\pi$
$360$	0	0
$361$	−0.0848216	−0.00446429
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.73566	−0.352561
$366$	0	0
$367$	−25.9627	−1.35524	−0.677620	−	0.735412i	$-0.736988\pi$
$367$	−25.9627	−1.35524	−0.677620	+	0.735412i	$0.736988\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.15230	−0.163659
$372$	0	0
$373$	9.92531	0.513913	0.256956	−	0.966423i	$-0.417280\pi$
$373$	9.92531	0.513913	0.256956	+	0.966423i	$0.417280\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.02827	−0.258969
$378$	0	0
$379$	34.2070	1.75710	0.878548	−	0.477655i	$-0.158513\pi$
$379$	34.2070	1.75710	0.878548	+	0.477655i	$0.158513\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.4340	1.09523	0.547613	−	0.836732i	$-0.315537\pi$
$383$	21.4340	1.09523	0.547613	+	0.836732i	$0.315537\pi$
$384$	0	0
$385$	−7.02827	−0.358194
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.3684	1.84395	0.921975	−	0.387251i	$-0.126575\pi$
$389$	36.3684	1.84395	0.921975	+	0.387251i	$0.126575\pi$
$390$	0	0
$391$	10.8405	0.548227
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.3774	−0.572461
$396$	0	0
$397$	−27.4713	−1.37875	−0.689373	−	0.724406i	$-0.742114\pi$
$397$	−27.4713	−1.37875	−0.689373	+	0.724406i	$0.742114\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.3118	−1.11420	−0.557099	−	0.830446i	$-0.688086\pi$
$401$	−22.3118	−1.11420	−0.557099	+	0.830446i	$0.688086\pi$
$402$	0	0
$403$	−41.9819	−2.09127
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.8970	1.82892
$408$	0	0
$409$	4.95358	0.244939	0.122469	−	0.992472i	$-0.460919\pi$
$409$	4.95358	0.244939	0.122469	+	0.992472i	$0.460919\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.7941	−0.678762
$414$	0	0
$415$	3.96265	0.194519
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.198716	0.00970793	0.00485397	−	0.999988i	$-0.498455\pi$
$419$	0.198716	0.00970793	0.00485397	+	0.999988i	$0.498455\pi$
$420$	0	0
$421$	26.4996	1.29151	0.645756	−	0.763544i	$-0.276542\pi$
$421$	26.4996	1.29151	0.645756	+	0.763544i	$0.276542\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.34916	0.307979
$426$	0	0
$427$	−8.43584	−0.408239
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.2835	1.02519	0.512596	−	0.858630i	$-0.328684\pi$
$431$	21.2835	1.02519	0.512596	+	0.858630i	$0.328684\pi$
$432$	0	0
$433$	−24.8031	−1.19196	−0.595981	−	0.802998i	$-0.703237\pi$
$433$	−24.8031	−1.19196	−0.595981	+	0.802998i	$0.703237\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.42571	−0.355220
$438$	0	0
$439$	−32.3118	−1.54216	−0.771079	−	0.636739i	$-0.780283\pi$
$439$	−32.3118	−1.54216	−0.771079	+	0.636739i	$0.780283\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.3774	−1.20572	−0.602859	−	0.797848i	$-0.705972\pi$
$443$	−25.3774	−1.20572	−0.602859	+	0.797848i	$0.705972\pi$
$444$	0	0
$445$	−2.58522	−0.122551
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.7266	−0.647798	−0.323899	−	0.946092i	$-0.604994\pi$
$449$	−13.7266	−0.647798	−0.323899	+	0.946092i	$0.604994\pi$
$450$	0	0
$451$	−5.47133	−0.257635
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.64177	−0.311371
$456$	0	0
$457$	−15.0101	−0.702144	−0.351072	−	0.936348i	$-0.614183\pi$
$457$	−15.0101	−0.702144	−0.351072	+	0.936348i	$0.614183\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.7549	−1.52555	−0.762773	−	0.646666i	$-0.776163\pi$
$461$	−32.7549	−1.52555	−0.762773	+	0.646666i	$0.776163\pi$
$462$	0	0
$463$	−24.5369	−1.14033	−0.570164	−	0.821531i	$-0.693120\pi$
$463$	−24.5369	−1.14033	−0.570164	+	0.821531i	$0.693120\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.60442	0.398165	0.199083	−	0.979983i	$-0.436204\pi$
$467$	8.60442	0.398165	0.199083	+	0.979983i	$0.436204\pi$
$468$	0	0
$469$	−7.53880	−0.348110
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−57.1232	−2.62653
$474$	0	0
$475$	−4.34916	−0.199553
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.26434	−0.149151	−0.0745757	−	0.997215i	$-0.523760\pi$
$479$	−3.26434	−0.149151	−0.0745757	+	0.997215i	$0.523760\pi$
$480$	0	0
$481$	34.8680	1.58984
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.3209	−0.695686
$486$	0	0
$487$	34.2070	1.55007	0.775033	−	0.631920i	$-0.217733\pi$
$487$	34.2070	1.55007	0.775033	+	0.631920i	$0.217733\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.93438	0.403203	0.201601	−	0.979468i	$-0.435385\pi$
$491$	8.93438	0.403203	0.201601	+	0.979468i	$0.435385\pi$
$492$	0	0
$493$	6.34916	0.285952
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.77301	0.214099
$498$	0	0
$499$	28.6236	1.28137	0.640685	−	0.767804i	$-0.278651\pi$
$499$	28.6236	1.28137	0.640685	+	0.767804i	$0.278651\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.0192012	0.000856140 0	0.000428070	−	1.00000i	$-0.499864\pi$
$503$	0.0192012	0.000856140 0	0.000428070		1.00000i	$0.499864\pi$
$504$	0	0
$505$	−11.8688	−0.528153
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.25526	0.365908	0.182954	−	0.983121i	$-0.441434\pi$
$509$	8.25526	0.365908	0.182954	+	0.983121i	$0.441434\pi$
$510$	0	0
$511$	−8.89703	−0.393582
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.65084	−0.160875
$516$	0	0
$517$	−3.61350	−0.158921
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.2553	0.887399	0.443700	−	0.896176i	$-0.353666\pi$
$521$	20.2553	0.887399	0.443700	+	0.896176i	$0.353666\pi$
$522$	0	0
$523$	1.82048	0.0796042	0.0398021	−	0.999208i	$-0.487327\pi$
$523$	1.82048	0.0796042	0.0398021	+	0.999208i	$0.487327\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	53.0101	2.30916
$528$	0	0
$529$	−20.0848	−0.873253
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.17044	−0.223957
$534$	0	0
$535$	5.32088	0.230042
$536$	0	0
$537$	0	0
$538$	0	0
$539$	27.9627	1.20444
$540$	0	0
$541$	1.03920	0.0446788	0.0223394	−	0.999750i	$-0.492889\pi$
$541$	1.03920	0.0446788	0.0223394	+	0.999750i	$0.492889\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.3118	−0.613051
$546$	0	0
$547$	37.7567	1.61436	0.807180	−	0.590305i	$-0.200992\pi$
$547$	37.7567	1.61436	0.807180	+	0.590305i	$0.200992\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.34916	−0.185280
$552$	0	0
$553$	−15.0283	−0.639067
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−45.8506	−1.94275	−0.971376	−	0.237545i	$-0.923657\pi$
$557$	−45.8506	−1.94275	−0.971376	+	0.237545i	$0.923657\pi$
$558$	0	0
$559$	−53.9819	−2.28319
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.3027	−1.40354	−0.701772	−	0.712402i	$-0.747607\pi$
$563$	−33.3027	−1.40354	−0.701772	+	0.712402i	$0.747607\pi$
$564$	0	0
$565$	−2.03735	−0.0857118
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.6700	0.992300	0.496150	−	0.868237i	$-0.334747\pi$
$569$	23.6700	0.992300	0.496150	+	0.868237i	$0.334747\pi$
$570$	0	0
$571$	14.5671	0.609613	0.304807	−	0.952414i	$-0.401408\pi$
$571$	14.5671	0.609613	0.304807	+	0.952414i	$0.401408\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.70739	0.0712031
$576$	0	0
$577$	−13.7639	−0.573000	−0.286500	−	0.958080i	$-0.592492\pi$
$577$	−13.7639	−0.573000	−0.286500	+	0.958080i	$0.592492\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.23421	0.217152
$582$	0	0
$583$	12.6983	0.525911
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.04748	0.0432339	0.0216170	−	0.999766i	$-0.493119\pi$
$587$	1.04748	0.0432339	0.0216170	+	0.999766i	$0.493119\pi$
$588$	0	0
$589$	−36.3118	−1.49620
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.3865	−0.590783	−0.295391	−	0.955376i	$-0.595450\pi$
$593$	−14.3865	−0.590783	−0.295391	+	0.955376i	$0.595450\pi$
$594$	0	0
$595$	8.38650	0.343813
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.98080	0.244369	0.122184	−	0.992507i	$-0.461010\pi$
$599$	5.98080	0.244369	0.122184	+	0.992507i	$0.461010\pi$
$600$	0	0
$601$	−7.08482	−0.288996	−0.144498	−	0.989505i	$-0.546157\pi$
$601$	−7.08482	−0.288996	−0.144498	+	0.989505i	$0.546157\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17.3118	0.703825
$606$	0	0
$607$	32.7175	1.32796	0.663982	−	0.747749i	$-0.268865\pi$
$607$	32.7175	1.32796	0.663982	+	0.747749i	$0.268865\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.41478	−0.138147
$612$	0	0
$613$	9.22699	0.372675	0.186337	−	0.982486i	$-0.440338\pi$
$613$	9.22699	0.372675	0.186337	+	0.982486i	$0.440338\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.281683	−0.0113401	−0.00567006	−	0.999984i	$-0.501805\pi$
$617$	−0.281683	−0.0113401	−0.00567006	+	0.999984i	$0.501805\pi$
$618$	0	0
$619$	21.2462	0.853957	0.426978	−	0.904262i	$-0.359578\pi$
$619$	21.2462	0.853957	0.426978	+	0.904262i	$0.359578\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.41478	−0.136810
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−44.0275	−1.75549
$630$	0	0
$631$	37.8688	1.50753	0.753766	−	0.657143i	$-0.228235\pi$
$631$	37.8688	1.50753	0.753766	+	0.657143i	$0.228235\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.26434	−0.0501737
$636$	0	0
$637$	26.4249	1.04699
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.7658	1.45216	0.726081	−	0.687609i	$-0.241340\pi$
$641$	36.7658	1.45216	0.726081	+	0.687609i	$0.241340\pi$
$642$	0	0
$643$	−24.9728	−0.984830	−0.492415	−	0.870360i	$-0.663886\pi$
$643$	−24.9728	−0.984830	−0.492415	+	0.870360i	$0.663886\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.56522	0.140163	0.0700816	−	0.997541i	$-0.477674\pi$
$647$	3.56522	0.140163	0.0700816	+	0.997541i	$0.477674\pi$
$648$	0	0
$649$	55.5663	2.18117
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.5652	0.843912	0.421956	−	0.906616i	$-0.361344\pi$
$653$	21.5652	0.843912	0.421956	+	0.906616i	$0.361344\pi$
$654$	0	0
$655$	1.70739	0.0667132
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.4623	1.57619	0.788093	−	0.615556i	$-0.211069\pi$
$659$	40.4623	1.57619	0.788093	+	0.615556i	$0.211069\pi$
$660$	0	0
$661$	26.1987	1.01901	0.509506	−	0.860467i	$-0.329828\pi$
$661$	26.1987	1.01901	0.509506	+	0.860467i	$0.329828\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.74474	−0.222771
$666$	0	0
$667$	1.70739	0.0661104
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33.9819	1.31185
$672$	0	0
$673$	3.35823	0.129450	0.0647251	−	0.997903i	$-0.479383\pi$
$673$	3.35823	0.129450	0.0647251	+	0.997903i	$0.479383\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.6965	0.603264	0.301632	−	0.953424i	$-0.402469\pi$
$677$	15.6965	0.603264	0.301632	+	0.953424i	$0.402469\pi$
$678$	0	0
$679$	−20.2371	−0.776629
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.1998	1.53820	0.769101	−	0.639128i	$-0.220704\pi$
$683$	40.1998	1.53820	0.769101	+	0.639128i	$0.220704\pi$
$684$	0	0
$685$	−21.7639	−0.831557
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	25.7084	0.977995	0.488998	−	0.872285i	$-0.337363\pi$
$691$	25.7084	0.977995	0.488998	+	0.872285i	$0.337363\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.6983	−0.633403
$696$	0	0
$697$	6.52867	0.247291
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.51775	0.321711	0.160855	−	0.986978i	$-0.448575\pi$
$701$	8.51775	0.321711	0.160855	+	0.986978i	$0.448575\pi$
$702$	0	0
$703$	30.1587	1.13746
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.6773	−0.589604
$708$	0	0
$709$	35.0848	1.31764	0.658819	−	0.752301i	$-0.271056\pi$
$709$	35.0848	1.31764	0.658819	+	0.752301i	$0.271056\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.2553	0.533864
$714$	0	0
$715$	26.7549	1.00058
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.3292	−1.16838	−0.584190	−	0.811617i	$-0.698588\pi$
$719$	−31.3292	−1.16838	−0.584190	+	0.811617i	$0.698588\pi$
$720$	0	0
$721$	−4.82234	−0.179593
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−13.9627	−0.517846	−0.258923	−	0.965898i	$-0.583368\pi$
$727$	−13.9627	−0.517846	−0.258923	+	0.965898i	$0.583368\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	68.1624	2.52108
$732$	0	0
$733$	−29.5761	−1.09242	−0.546210	−	0.837648i	$-0.683930\pi$
$733$	−29.5761	−1.09242	−0.546210	+	0.837648i	$0.683930\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.3684	1.11863
$738$	0	0
$739$	−6.66819	−0.245293	−0.122647	−	0.992450i	$-0.539138\pi$
$739$	−6.66819	−0.245293	−0.122647	+	0.992450i	$0.539138\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.7002	1.23634	0.618170	−	0.786045i	$-0.287874\pi$
$743$	33.7002	1.23634	0.618170	+	0.786045i	$0.287874\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.02827	0.256808
$750$	0	0
$751$	−48.3129	−1.76296	−0.881481	−	0.472220i	$-0.843453\pi$
$751$	−48.3129	−1.76296	−0.881481	+	0.472220i	$0.843453\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.44305	0.234487
$756$	0	0
$757$	42.3985	1.54100	0.770500	−	0.637440i	$-0.220007\pi$
$757$	42.3985	1.54100	0.770500	+	0.637440i	$0.220007\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	44.7933	1.62375	0.811877	−	0.583828i	$-0.198446\pi$
$761$	44.7933	1.62375	0.811877	+	0.583828i	$0.198446\pi$
$762$	0	0
$763$	−18.9043	−0.684380
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52.5105	1.89605
$768$	0	0
$769$	−52.2080	−1.88267	−0.941335	−	0.337473i	$-0.890428\pi$
$769$	−52.2080	−1.88267	−0.941335	+	0.337473i	$0.890428\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.2179	0.871058	0.435529	−	0.900175i	$-0.356561\pi$
$773$	24.2179	0.871058	0.435529	+	0.900175i	$0.356561\pi$
$774$	0	0
$775$	8.34916	0.299911
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.47213	−0.160231
$780$	0	0
$781$	−19.2270	−0.687996
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.7357	0.383172
$786$	0	0
$787$	−1.43398	−0.0511159	−0.0255579	−	0.999673i	$-0.508136\pi$
$787$	−1.43398	−0.0511159	−0.0255579	+	0.999673i	$0.508136\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.69110	−0.0956845
$792$	0	0
$793$	32.1131	1.14037
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.4732	0.512666	0.256333	−	0.966588i	$-0.417486\pi$
$797$	14.4732	0.512666	0.256333	+	0.966588i	$0.417486\pi$
$798$	0	0
$799$	4.31181	0.152541
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.8397	1.26476
$804$	0	0
$805$	2.25526	0.0794876
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−49.2654	−1.73208	−0.866039	−	0.499976i	$-0.833342\pi$
$809$	−49.2654	−1.73208	−0.866039	+	0.499976i	$0.833342\pi$
$810$	0	0
$811$	−4.69832	−0.164980	−0.0824901	−	0.996592i	$-0.526287\pi$
$811$	−4.69832	−0.164980	−0.0824901	+	0.996592i	$0.526287\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.90611	0.136825
$816$	0	0
$817$	−46.6911	−1.63351
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.7730	−1.21359	−0.606793	−	0.794860i	$-0.707544\pi$
$821$	−34.7730	−1.21359	−0.606793	+	0.794860i	$0.707544\pi$
$822$	0	0
$823$	−44.1323	−1.53836	−0.769178	−	0.639035i	$-0.779334\pi$
$823$	−44.1323	−1.53836	−0.769178	+	0.639035i	$0.779334\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.6610	−0.787999	−0.394000	−	0.919111i	$-0.628909\pi$
$827$	−22.6610	−0.787999	−0.394000	+	0.919111i	$0.628909\pi$
$828$	0	0
$829$	−37.9144	−1.31682	−0.658410	−	0.752659i	$-0.728771\pi$
$829$	−37.9144	−1.31682	−0.658410	+	0.752659i	$0.728771\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−33.3665	−1.15608
$834$	0	0
$835$	5.51960	0.191014
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.7175	−1.06049	−0.530243	−	0.847846i	$-0.677899\pi$
$839$	−30.7175	−1.06049	−0.530243	+	0.847846i	$0.677899\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.2835	0.422567
$846$	0	0
$847$	22.8669	0.785716
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−11.8397	−0.405859
$852$	0	0
$853$	1.25341	0.0429159	0.0214579	−	0.999770i	$-0.493169\pi$
$853$	1.25341	0.0429159	0.0214579	+	0.999770i	$0.493169\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.85783	0.268418	0.134209	−	0.990953i	$-0.457151\pi$
$857$	7.85783	0.268418	0.134209	+	0.990953i	$0.457151\pi$
$858$	0	0
$859$	−14.4913	−0.494438	−0.247219	−	0.968960i	$-0.579517\pi$
$859$	−14.4913	−0.494438	−0.247219	+	0.968960i	$0.579517\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.69646	0.330071	0.165036	−	0.986288i	$-0.447226\pi$
$863$	9.69646	0.330071	0.165036	+	0.986288i	$0.447226\pi$
$864$	0	0
$865$	−5.80128	−0.197250
$866$	0	0
$867$	0	0
$868$	0	0
$869$	60.5380	2.05361
$870$	0	0
$871$	28.6983	0.972405
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.32088	0.0446540
$876$	0	0
$877$	6.27341	0.211838	0.105919	−	0.994375i	$-0.466222\pi$
$877$	6.27341	0.211838	0.105919	+	0.994375i	$0.466222\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−19.5953	−0.660184	−0.330092	−	0.943949i	$-0.607080\pi$
$881$	−19.5953	−0.660184	−0.330092	+	0.943949i	$0.607080\pi$
$882$	0	0
$883$	−5.24619	−0.176548	−0.0882742	−	0.996096i	$-0.528135\pi$
$883$	−5.24619	−0.176548	−0.0882742	+	0.996096i	$0.528135\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.6044	1.09475	0.547375	−	0.836888i	$-0.315627\pi$
$887$	32.6044	1.09475	0.547375	+	0.836888i	$0.315627\pi$
$888$	0	0
$889$	−1.67004	−0.0560114
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.95358	−0.0988378
$894$	0	0
$895$	−14.0565	−0.469859
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.34916	0.278460
$900$	0	0
$901$	−15.1523	−0.504796
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.08482	−0.235507
$906$	0	0
$907$	−1.96265	−0.0651688	−0.0325844	−	0.999469i	$-0.510374\pi$
$907$	−1.96265	−0.0651688	−0.0325844	+	0.999469i	$0.510374\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50.7066	−1.67998	−0.839992	−	0.542599i	$-0.817440\pi$
$911$	−50.7066	−1.67998	−0.839992	+	0.542599i	$0.817440\pi$
$912$	0	0
$913$	−21.0848	−0.697806
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.25526	0.0744754
$918$	0	0
$919$	56.2371	1.85509	0.927546	−	0.373710i	$-0.121914\pi$
$919$	56.2371	1.85509	0.927546	+	0.373710i	$0.121914\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.1696	−0.598061
$924$	0	0
$925$	−6.93438	−0.228001
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.34009	0.175203	0.0876013	−	0.996156i	$-0.472080\pi$
$929$	5.34009	0.175203	0.0876013	+	0.996156i	$0.472080\pi$
$930$	0	0
$931$	22.8560	0.749074
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−33.7831	−1.10483
$936$	0	0
$937$	−3.01013	−0.0983366	−0.0491683	−	0.998791i	$-0.515657\pi$
$937$	−3.01013	−0.0983366	−0.0491683	+	0.998791i	$0.515657\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.829557	0.0270428	0.0135214	−	0.999909i	$-0.495696\pi$
$941$	0.829557	0.0270428	0.0135214	+	0.999909i	$0.495696\pi$
$942$	0	0
$943$	1.75566	0.0571723
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.3027	−0.692246	−0.346123	−	0.938189i	$-0.612502\pi$
$947$	−21.3027	−0.692246	−0.346123	+	0.938189i	$0.612502\pi$
$948$	0	0
$949$	33.8688	1.09943
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.0203	1.29638	0.648192	−	0.761477i	$-0.275526\pi$
$953$	40.0203	1.29638	0.648192	+	0.761477i	$0.275526\pi$
$954$	0	0
$955$	0.547875	0.0177288
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.7476	−0.928310
$960$	0	0
$961$	38.7084	1.24866
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.40571	0.141825
$966$	0	0
$967$	−31.0365	−0.998068	−0.499034	−	0.866582i	$-0.666312\pi$
$967$	−31.0365	−0.998068	−0.499034	+	0.866582i	$0.666312\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.66819	0.0856262	0.0428131	−	0.999083i	$-0.486368\pi$
$971$	2.66819	0.0856262	0.0428131	+	0.999083i	$0.486368\pi$
$972$	0	0
$973$	−22.0565	−0.707100
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.6519	−1.46053	−0.730267	−	0.683162i	$-0.760604\pi$
$977$	−45.6519	−1.46053	−0.730267	+	0.683162i	$0.760604\pi$
$978$	0	0
$979$	13.7557	0.439633
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.4521	0.556636	0.278318	−	0.960489i	$-0.410223\pi$
$983$	17.4521	0.556636	0.278318	+	0.960489i	$0.410223\pi$
$984$	0	0
$985$	4.64177	0.147899
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.3300	0.582859
$990$	0	0
$991$	−4.49960	−0.142935	−0.0714673	−	0.997443i	$-0.522768\pi$
$991$	−4.49960	−0.142935	−0.0714673	+	0.997443i	$0.522768\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.4431	0.584684
$996$	0	0
$997$	−21.1896	−0.671083	−0.335541	−	0.942025i	$-0.608919\pi$
$997$	−21.1896	−0.671083	−0.335541	+	0.942025i	$0.608919\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.x.1.3		3
3.2	odd	2		580.2.a.c.1.3	✓	3
12.11	even	2		2320.2.a.m.1.1		3
15.2	even	4		2900.2.c.f.349.1		6
15.8	even	4		2900.2.c.f.349.6		6
15.14	odd	2		2900.2.a.g.1.1		3
24.5	odd	2		9280.2.a.bk.1.1		3
24.11	even	2		9280.2.a.bw.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.c.1.3	✓	3	3.2	odd	2
2320.2.a.m.1.1		3	12.11	even	2
2900.2.a.g.1.1		3	15.14	odd	2
2900.2.c.f.349.1		6	15.2	even	4
2900.2.c.f.349.6		6	15.8	even	4
5220.2.a.x.1.3		3	1.1	even	1	trivial
9280.2.a.bk.1.1		3	24.5	odd	2
9280.2.a.bw.1.3		3	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.32088 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.32088 q^{7} -5.32088 q^{11} -5.02827 q^{13} +6.34916 q^{17} -4.34916 q^{19} +1.70739 q^{23} +1.00000 q^{25} +1.00000 q^{29} +8.34916 q^{31} +1.32088 q^{35} -6.93438 q^{37} +1.02827 q^{41} +10.7357 q^{43} +0.679116 q^{47} -5.25526 q^{49} -2.38650 q^{53} -5.32088 q^{55} -10.4431 q^{59} -6.38650 q^{61} -5.02827 q^{65} -5.70739 q^{67} +3.61350 q^{71} -6.73566 q^{73} -7.02827 q^{77} -11.3774 q^{79} +3.96265 q^{83} +6.34916 q^{85} -2.58522 q^{89} -6.64177 q^{91} -4.34916 q^{95} -15.3209 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 4 * q^7	\( 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 4 * q^7 - 8 * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^19 + 3 * q^25 + 3 * q^29 + 4 * q^31 - 4 * q^35 - 10 * q^37 - 10 * q^41 + 14 * q^43 + 10 * q^47 + 3 * q^49 - 10 * q^53 - 8 * q^55 - 8 * q^59 - 22 * q^61 - 2 * q^65 - 12 * q^67 + 8 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^83 - 2 * q^85 - 18 * q^89 - 4 * q^91 + 8 * q^95 - 38 * q^97

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