LMFDB - Embedded newform 2420.3.f.a.241.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2420,3,Mod(241,2420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2420, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2420.241");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$2420 = 2^{2} \cdot 5 \cdot 11^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	2420.f (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:	$65.9402239752$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 3 x^{15} + 33 x^{14} - 111 x^{13} + 735 x^{12} - 1436 x^{11} + 10633 x^{10} - 25103 x^{9} + \cdots + 75625$ x^16 - 3x^15 + 33x^14 - 111x^13 + 735x^12 - 1436x^11 + 10633x^10 - 25103x^9 + 232616x^8 - 600030x^7 + 1738315x^6 - 2296850x^5 + 2290475x^4 - 1918750x^3 + 1679000x^2 - 543125*x + 75625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{8}\cdot 5$
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.14
Root			$3.61325 - 2.62518i$ of defining polynomial
Character	$\chi$	$=$	2420.241
Dual form			2420.3.f.a.241.13

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84819 q^{3} +2.23607 q^{5} +3.16455i q^{7} -0.887821 q^{9} +1.15125i q^{13} +6.36874 q^{15} +17.5203i q^{17} +20.0285i q^{19} +9.01324i q^{21} -36.9406 q^{23} +5.00000 q^{25} -28.1624 q^{27} -5.00548i q^{29} -61.1323 q^{31} +7.07615i q^{35} -26.0390 q^{37} +3.27896i q^{39} -60.2635i q^{41} -65.3933i q^{43} -1.98523 q^{45} -26.6798 q^{47} +38.9856 q^{49} +49.9012i q^{51} +3.09968 q^{53} +57.0449i q^{57} -9.31135 q^{59} -33.9251i q^{61} -2.80956i q^{63} +2.57426i q^{65} -17.1176 q^{67} -105.214 q^{69} +50.7345 q^{71} +41.3088i q^{73} +14.2409 q^{75} +42.2263i q^{79} -72.2214 q^{81} +80.7279i q^{83} +39.1767i q^{85} -14.2565i q^{87} -109.359 q^{89} -3.64317 q^{91} -174.116 q^{93} +44.7850i q^{95} -20.0382 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 6 q^{3} + 46 q^{9} - 30 q^{15} - 168 q^{23} + 80 q^{25} + 30 q^{27} - 190 q^{31} + 104 q^{37} - 30 q^{45} - 268 q^{47} - 228 q^{49} - 368 q^{53} + 78 q^{59} - 68 q^{67} - 212 q^{69} + 270 q^{71}+ \cdots - 726 q^{97}+O(q^{100})$ 16 * q - 6 * q^3 + 46 * q^9 - 30 * q^15 - 168 * q^23 + 80 * q^25 + 30 * q^27 - 190 * q^31 + 104 * q^37 - 30 * q^45 - 268 * q^47 - 228 * q^49 - 368 * q^53 + 78 * q^59 - 68 * q^67 - 212 * q^69 + 270 * q^71 - 30 * q^75 + 1052 * q^81 - 160 * q^89 - 844 * q^91 - 78 * q^93 - 726 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times$ .

$n$	$1211$	$1937$	$2301$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.84819	0.949396	0.474698	−	0.880149i	$-0.342557\pi$
$3$	2.84819	0.949396	0.474698	+	0.880149i	$0.342557\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	3.16455i	0.452079i	0.974118	+	0.226039i	$0.0725778\pi$
$7$	3.16455i	0.452079i	−0.974118	+	0.226039i	$0.927422\pi$
$8$	0	0
$9$	−0.887821	−0.0986468
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.15125i	0.0885573i	0.999019	+	0.0442787i	$0.0140989\pi$
$13$	1.15125i	0.0885573i	−0.999019	+	0.0442787i	$0.985901\pi$
$14$	0	0
$15$	6.36874	0.424583
$16$	0	0
$17$	17.5203i	1.03061i	0.857007	+	0.515304i	$0.172321\pi$
$17$	17.5203i	1.03061i	−0.857007	+	0.515304i	$0.827679\pi$
$18$	0	0
$19$	20.0285i	1.05413i	0.849825	+	0.527065i	$0.176708\pi$
$19$	20.0285i	1.05413i	−0.849825	+	0.527065i	$0.823292\pi$
$20$	0	0
$21$	9.01324i	0.429202i
$22$	0	0
$23$	−36.9406	−1.60611	−0.803056	−	0.595904i	$-0.796794\pi$
$23$	−36.9406	−1.60611	−0.803056	+	0.595904i	$0.796794\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	−28.1624	−1.04305
$28$	0	0
$29$	− 5.00548i	− 0.172603i	−0.996269	−	0.0863014i	$-0.972495\pi$
$29$	− 5.00548i	− 0.172603i	0.996269	−	0.0863014i	$-0.0275048\pi$
$30$	0	0
$31$	−61.1323	−1.97201	−0.986005	−	0.166714i	$-0.946684\pi$
$31$	−61.1323	−1.97201	−0.986005	+	0.166714i	$0.946684\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.07615i	0.202176i
$36$	0	0
$37$	−26.0390	−0.703757	−0.351879	−	0.936046i	$-0.614457\pi$
$37$	−26.0390	−0.703757	−0.351879	+	0.936046i	$0.614457\pi$
$38$	0	0
$39$	3.27896i	0.0840760i
$40$	0	0
$41$	− 60.2635i	− 1.46984i	−0.678153	−	0.734921i	$-0.737219\pi$
$41$	− 60.2635i	− 1.46984i	0.678153	−	0.734921i	$-0.262781\pi$
$42$	0	0
$43$	− 65.3933i	− 1.52077i	−0.649470	−	0.760387i	$-0.725009\pi$
$43$	− 65.3933i	− 1.52077i	0.649470	−	0.760387i	$-0.274991\pi$
$44$	0	0
$45$	−1.98523	−0.0441162
$46$	0	0
$47$	−26.6798	−0.567655	−0.283827	−	0.958875i	$-0.591604\pi$
$47$	−26.6798	−0.567655	−0.283827	+	0.958875i	$0.591604\pi$
$48$	0	0
$49$	38.9856	0.795625
$50$	0	0
$51$	49.9012i	0.978455i
$52$	0	0
$53$	3.09968	0.0584845	0.0292423	−	0.999572i	$-0.490691\pi$
$53$	3.09968	0.0584845	0.0292423	+	0.999572i	$0.490691\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	57.0449i	1.00079i
$58$	0	0
$59$	−9.31135	−0.157820	−0.0789098	−	0.996882i	$-0.525144\pi$
$59$	−9.31135	−0.157820	−0.0789098	+	0.996882i	$0.525144\pi$
$60$	0	0
$61$	− 33.9251i	− 0.556149i	−0.960559	−	0.278075i	$-0.910304\pi$
$61$	− 33.9251i	− 0.556149i	0.960559	−	0.278075i	$-0.0896962\pi$
$62$	0	0
$63$	− 2.80956i	− 0.0445961i
$64$	0	0
$65$	2.57426i	0.0396040i
$66$	0	0
$67$	−17.1176	−0.255487	−0.127743	−	0.991807i	$-0.540773\pi$
$67$	−17.1176	−0.255487	−0.127743	+	0.991807i	$0.540773\pi$
$68$	0	0
$69$	−105.214	−1.52484
$70$	0	0
$71$	50.7345	0.714571	0.357285	−	0.933995i	$-0.383702\pi$
$71$	50.7345	0.714571	0.357285	+	0.933995i	$0.383702\pi$
$72$	0	0
$73$	41.3088i	0.565873i	0.959139	+	0.282937i	$0.0913086\pi$
$73$	41.3088i	0.565873i	−0.959139	+	0.282937i	$0.908691\pi$
$74$	0	0
$75$	14.2409	0.189879
$76$	0	0
$77$	0	0
$78$	0	0
$79$	42.2263i	0.534510i	0.963626	+	0.267255i	$0.0861166\pi$
$79$	42.2263i	0.534510i	−0.963626	+	0.267255i	$0.913883\pi$
$80$	0	0
$81$	−72.2214	−0.891622
$82$	0	0
$83$	80.7279i	0.972626i	0.873785	+	0.486313i	$0.161658\pi$
$83$	80.7279i	0.972626i	−0.873785	+	0.486313i	$0.838342\pi$
$84$	0	0
$85$	39.1767i	0.460902i
$86$	0	0
$87$	− 14.2565i	− 0.163868i
$88$	0	0
$89$	−109.359	−1.22875	−0.614377	−	0.789013i	$-0.710592\pi$
$89$	−109.359	−1.22875	−0.614377	+	0.789013i	$0.710592\pi$
$90$	0	0
$91$	−3.64317	−0.0400349
$92$	0	0
$93$	−174.116	−1.87222
$94$	0	0
$95$	44.7850i	0.471421i
$96$	0	0
$97$	−20.0382	−0.206580	−0.103290	−	0.994651i	$-0.532937\pi$
$97$	−20.0382	−0.206580	−0.103290	+	0.994651i	$0.532937\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	88.9047i	0.880245i	0.897938	+	0.440123i	$0.145065\pi$
$101$	88.9047i	0.880245i	−0.897938	+	0.440123i	$0.854935\pi$
$102$	0	0
$103$	12.4284	0.120664	0.0603318	−	0.998178i	$-0.480784\pi$
$103$	12.4284	0.120664	0.0603318	+	0.998178i	$0.480784\pi$
$104$	0	0
$105$	20.1542i	0.191945i
$106$	0	0
$107$	− 206.960i	− 1.93421i	−0.254385	−	0.967103i	$-0.581873\pi$
$107$	− 206.960i	− 1.93421i	0.254385	−	0.967103i	$-0.418127\pi$
$108$	0	0
$109$	36.8795i	0.338344i	0.985587	+	0.169172i	$0.0541093\pi$
$109$	36.8795i	0.338344i	−0.985587	+	0.169172i	$0.945891\pi$
$110$	0	0
$111$	−74.1641	−0.668145
$112$	0	0
$113$	−95.7678	−0.847503	−0.423751	−	0.905779i	$-0.639287\pi$
$113$	−95.7678	−0.847503	−0.423751	+	0.905779i	$0.639287\pi$
$114$	0	0
$115$	−82.6016	−0.718275
$116$	0	0
$117$	− 1.02210i	− 0.00873590i
$118$	0	0
$119$	−55.4440	−0.465916
$120$	0	0
$121$	0	0
$122$	0	0
$123$	− 171.642i	− 1.39546i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	142.853i	1.12483i	0.826857	+	0.562413i	$0.190127\pi$
$127$	142.853i	1.12483i	−0.826857	+	0.562413i	$0.809873\pi$
$128$	0	0
$129$	− 186.252i	− 1.44382i
$130$	0	0
$131$	53.7502i	0.410307i	0.978730	+	0.205153i	$0.0657693\pi$
$131$	53.7502i	0.410307i	−0.978730	+	0.205153i	$0.934231\pi$
$132$	0	0
$133$	−63.3811	−0.476550
$134$	0	0
$135$	−62.9730	−0.466467
$136$	0	0
$137$	212.127	1.54838	0.774188	−	0.632956i	$-0.218159\pi$
$137$	212.127	1.54838	0.774188	+	0.632956i	$0.218159\pi$
$138$	0	0
$139$	144.697i	1.04099i	0.853866	+	0.520493i	$0.174252\pi$
$139$	144.697i	1.04099i	−0.853866	+	0.520493i	$0.825748\pi$
$140$	0	0
$141$	−75.9890	−0.538929
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 11.1926i	− 0.0771903i
$146$	0	0
$147$	111.038	0.755363
$148$	0	0
$149$	205.272i	1.37766i	0.724921	+	0.688832i	$0.241876\pi$
$149$	205.272i	1.37766i	−0.724921	+	0.688832i	$0.758124\pi$
$150$	0	0
$151$	− 113.281i	− 0.750203i	−0.926984	−	0.375101i	$-0.877608\pi$
$151$	− 113.281i	− 0.750203i	0.926984	−	0.375101i	$-0.122392\pi$
$152$	0	0
$153$	− 15.5549i	− 0.101666i
$154$	0	0
$155$	−136.696	−0.881910
$156$	0	0
$157$	−28.2169	−0.179726	−0.0898628	−	0.995954i	$-0.528643\pi$
$157$	−28.2169	−0.179726	−0.0898628	+	0.995954i	$0.528643\pi$
$158$	0	0
$159$	8.82847	0.0555250
$160$	0	0
$161$	− 116.900i	− 0.726089i
$162$	0	0
$163$	−15.4153	−0.0945726	−0.0472863	−	0.998881i	$-0.515057\pi$
$163$	−15.4153	−0.0945726	−0.0472863	+	0.998881i	$0.515057\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 152.413i	− 0.912652i	−0.889813	−	0.456326i	$-0.849165\pi$
$167$	− 152.413i	− 0.912652i	0.889813	−	0.456326i	$-0.150835\pi$
$168$	0	0
$169$	167.675	0.992158
$170$	0	0
$171$	− 17.7817i	− 0.103987i
$172$	0	0
$173$	37.6018i	0.217351i	0.994077	+	0.108676i	$0.0346610\pi$
$173$	37.6018i	0.217351i	−0.994077	+	0.108676i	$0.965339\pi$
$174$	0	0
$175$	15.8228i	0.0904158i
$176$	0	0
$177$	−26.5205	−0.149833
$178$	0	0
$179$	75.4978	0.421775	0.210888	−	0.977510i	$-0.432365\pi$
$179$	75.4978	0.421775	0.210888	+	0.977510i	$0.432365\pi$
$180$	0	0
$181$	−274.131	−1.51453	−0.757267	−	0.653105i	$-0.773466\pi$
$181$	−274.131	−1.51453	−0.757267	+	0.653105i	$0.773466\pi$
$182$	0	0
$183$	− 96.6251i	− 0.528006i
$184$	0	0
$185$	−58.2250	−0.314730
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 89.1213i	− 0.471541i
$190$	0	0
$191$	19.4820	0.102000	0.0509999	−	0.998699i	$-0.483759\pi$
$191$	19.4820	0.102000	0.0509999	+	0.998699i	$0.483759\pi$
$192$	0	0
$193$	120.152i	0.622547i	0.950320	+	0.311274i	$0.100756\pi$
$193$	120.152i	0.622547i	−0.950320	+	0.311274i	$0.899244\pi$
$194$	0	0
$195$	7.33199i	0.0375999i
$196$	0	0
$197$	47.0754i	0.238962i	0.992837	+	0.119481i	$0.0381230\pi$
$197$	47.0754i	0.238962i	−0.992837	+	0.119481i	$0.961877\pi$
$198$	0	0
$199$	−17.6716	−0.0888022	−0.0444011	−	0.999014i	$-0.514138\pi$
$199$	−17.6716	−0.0888022	−0.0444011	+	0.999014i	$0.514138\pi$
$200$	0	0
$201$	−48.7542	−0.242558
$202$	0	0
$203$	15.8401	0.0780300
$204$	0	0
$205$	− 134.753i	− 0.657333i
$206$	0	0
$207$	32.7966	0.158438
$208$	0	0
$209$	0	0
$210$	0	0
$211$	99.9396i	0.473647i	0.971553	+	0.236824i	$0.0761064\pi$
$211$	99.9396i	0.473647i	−0.971553	+	0.236824i	$0.923894\pi$
$212$	0	0
$213$	144.501	0.678411
$214$	0	0
$215$	− 146.224i	− 0.680111i
$216$	0	0
$217$	− 193.456i	− 0.891504i
$218$	0	0
$219$	117.655i	0.537238i
$220$	0	0
$221$	−20.1702	−0.0912679
$222$	0	0
$223$	−238.035	−1.06742	−0.533711	−	0.845667i	$-0.679203\pi$
$223$	−238.035	−1.06742	−0.533711	+	0.845667i	$0.679203\pi$
$224$	0	0
$225$	−4.43911	−0.0197294
$226$	0	0
$227$	366.538i	1.61471i	0.590069	+	0.807353i	$0.299100\pi$
$227$	366.538i	1.61471i	−0.590069	+	0.807353i	$0.700900\pi$
$228$	0	0
$229$	262.545	1.14649	0.573243	−	0.819385i	$-0.305685\pi$
$229$	262.545	1.14649	0.573243	+	0.819385i	$0.305685\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	375.852i	1.61310i	0.591168	+	0.806548i	$0.298667\pi$
$233$	375.852i	1.61310i	−0.591168	+	0.806548i	$0.701333\pi$
$234$	0	0
$235$	−59.6578	−0.253863
$236$	0	0
$237$	120.268i	0.507462i
$238$	0	0
$239$	− 300.967i	− 1.25928i	−0.776888	−	0.629639i	$-0.783203\pi$
$239$	− 300.967i	− 1.25928i	0.776888	−	0.629639i	$-0.216797\pi$
$240$	0	0
$241$	478.069i	1.98369i	0.127452	+	0.991845i	$0.459320\pi$
$241$	478.069i	1.98369i	−0.127452	+	0.991845i	$0.540680\pi$
$242$	0	0
$243$	47.7613	0.196549
$244$	0	0
$245$	87.1745	0.355814
$246$	0	0
$247$	−23.0577	−0.0933510
$248$	0	0
$249$	229.928i	0.923407i
$250$	0	0
$251$	−412.053	−1.64165	−0.820824	−	0.571182i	$-0.806485\pi$
$251$	−412.053	−1.64165	−0.820824	+	0.571182i	$0.806485\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	111.583i	0.437579i
$256$	0	0
$257$	299.519	1.16544	0.582722	−	0.812672i	$-0.301988\pi$
$257$	299.519	1.16544	0.582722	+	0.812672i	$0.301988\pi$
$258$	0	0
$259$	− 82.4018i	− 0.318154i
$260$	0	0
$261$	4.44397i	0.0170267i
$262$	0	0
$263$	283.233i	1.07693i	0.842648	+	0.538465i	$0.180996\pi$
$263$	283.233i	1.07693i	−0.842648	+	0.538465i	$0.819004\pi$
$264$	0	0
$265$	6.93109	0.0261551
$266$	0	0
$267$	−311.475	−1.16657
$268$	0	0
$269$	−393.125	−1.46143	−0.730715	−	0.682682i	$-0.760813\pi$
$269$	−393.125	−1.46143	−0.730715	+	0.682682i	$0.760813\pi$
$270$	0	0
$271$	539.473i	1.99068i	0.0964507	+	0.995338i	$0.469251\pi$
$271$	539.473i	1.99068i	−0.0964507	+	0.995338i	$0.530749\pi$
$272$	0	0
$273$	−10.3764	−0.0380090
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 279.440i	− 1.00881i	−0.863468	−	0.504404i	$-0.831712\pi$
$277$	− 279.440i	− 1.00881i	0.863468	−	0.504404i	$-0.168288\pi$
$278$	0	0
$279$	54.2746	0.194533
$280$	0	0
$281$	− 274.373i	− 0.976417i	−0.872727	−	0.488208i	$-0.837651\pi$
$281$	− 274.373i	− 0.976417i	0.872727	−	0.488208i	$-0.162349\pi$
$282$	0	0
$283$	− 523.135i	− 1.84853i	−0.381749	−	0.924266i	$-0.624678\pi$
$283$	− 523.135i	− 1.84853i	0.381749	−	0.924266i	$-0.375322\pi$
$284$	0	0
$285$	127.556i	0.447566i
$286$	0	0
$287$	190.707	0.664484
$288$	0	0
$289$	−17.9622	−0.0621531
$290$	0	0
$291$	−57.0727	−0.196126
$292$	0	0
$293$	− 403.706i	− 1.37784i	−0.724838	−	0.688919i	$-0.758086\pi$
$293$	− 403.706i	− 1.37784i	0.724838	−	0.688919i	$-0.241914\pi$
$294$	0	0
$295$	−20.8208	−0.0705791
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 42.5276i	− 0.142233i
$300$	0	0
$301$	206.940	0.687510
$302$	0	0
$303$	253.217i	0.835701i
$304$	0	0
$305$	− 75.8588i	− 0.248717i
$306$	0	0
$307$	− 290.845i	− 0.947378i	−0.880692	−	0.473689i	$-0.842922\pi$
$307$	− 290.845i	− 0.947378i	0.880692	−	0.473689i	$-0.157078\pi$
$308$	0	0
$309$	35.3983	0.114558
$310$	0	0
$311$	53.4925	0.172002	0.0860008	−	0.996295i	$-0.472591\pi$
$311$	53.4925	0.172002	0.0860008	+	0.996295i	$0.472591\pi$
$312$	0	0
$313$	−204.622	−0.653744	−0.326872	−	0.945069i	$-0.605995\pi$
$313$	−204.622	−0.653744	−0.326872	+	0.945069i	$0.605995\pi$
$314$	0	0
$315$	− 6.28236i	− 0.0199440i
$316$	0	0
$317$	206.669	0.651952	0.325976	−	0.945378i	$-0.394307\pi$
$317$	206.669	0.651952	0.325976	+	0.945378i	$0.394307\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	− 589.461i	− 1.83633i
$322$	0	0
$323$	−350.906	−1.08640
$324$	0	0
$325$	5.75623i	0.0177115i
$326$	0	0
$327$	105.040i	0.321222i
$328$	0	0
$329$	− 84.4295i	− 0.256625i
$330$	0	0
$331$	−311.605	−0.941405	−0.470702	−	0.882292i	$-0.655999\pi$
$331$	−311.605	−0.941405	−0.470702	+	0.882292i	$0.655999\pi$
$332$	0	0
$333$	23.1180	0.0694234
$334$	0	0
$335$	−38.2761	−0.114257
$336$	0	0
$337$	− 491.215i	− 1.45761i	−0.684721	−	0.728805i	$-0.740076\pi$
$337$	− 491.215i	− 1.45761i	0.684721	−	0.728805i	$-0.259924\pi$
$338$	0	0
$339$	−272.765	−0.804616
$340$	0	0
$341$	0	0
$342$	0	0
$343$	278.435i	0.811764i
$344$	0	0
$345$	−235.265	−0.681927
$346$	0	0
$347$	293.183i	0.844908i	0.906384	+	0.422454i	$0.138831\pi$
$347$	293.183i	0.844908i	−0.906384	+	0.422454i	$0.861169\pi$
$348$	0	0
$349$	352.915i	1.01122i	0.862763	+	0.505609i	$0.168732\pi$
$349$	352.915i	1.01122i	−0.862763	+	0.505609i	$0.831268\pi$
$350$	0	0
$351$	− 32.4218i	− 0.0923698i
$352$	0	0
$353$	−203.115	−0.575397	−0.287698	−	0.957721i	$-0.592890\pi$
$353$	−203.115	−0.575397	−0.287698	+	0.957721i	$0.592890\pi$
$354$	0	0
$355$	113.446	0.319566
$356$	0	0
$357$	−157.915	−0.442339
$358$	0	0
$359$	− 413.072i	− 1.15062i	−0.817936	−	0.575309i	$-0.804882\pi$
$359$	− 413.072i	− 1.15062i	0.817936	−	0.575309i	$-0.195118\pi$
$360$	0	0
$361$	−40.1398	−0.111191
$362$	0	0
$363$	0	0
$364$	0	0
$365$	92.3692i	0.253066i
$366$	0	0
$367$	209.158	0.569913	0.284957	−	0.958540i	$-0.408021\pi$
$367$	209.158	0.569913	0.284957	+	0.958540i	$0.408021\pi$
$368$	0	0
$369$	53.5032i	0.144995i
$370$	0	0
$371$	9.80910i	0.0264396i
$372$	0	0
$373$	475.071i	1.27365i	0.771009	+	0.636825i	$0.219752\pi$
$373$	475.071i	1.27365i	−0.771009	+	0.636825i	$0.780248\pi$
$374$	0	0
$375$	31.8437	0.0849166
$376$	0	0
$377$	5.76253	0.0152852
$378$	0	0
$379$	674.152	1.77876	0.889382	−	0.457164i	$-0.151135\pi$
$379$	674.152	1.77876	0.889382	+	0.457164i	$0.151135\pi$
$380$	0	0
$381$	406.872i	1.06790i
$382$	0	0
$383$	−2.60807	−0.00680958	−0.00340479	−	0.999994i	$-0.501084\pi$
$383$	−2.60807	−0.00680958	−0.00340479	+	0.999994i	$0.501084\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	58.0575i	0.150019i
$388$	0	0
$389$	131.846	0.338936	0.169468	−	0.985536i	$-0.445795\pi$
$389$	131.846	0.338936	0.169468	+	0.985536i	$0.445795\pi$
$390$	0	0
$391$	− 647.211i	− 1.65527i
$392$	0	0
$393$	153.091i	0.389544i
$394$	0	0
$395$	94.4209i	0.239040i
$396$	0	0
$397$	−313.373	−0.789352	−0.394676	−	0.918820i	$-0.629143\pi$
$397$	−313.373	−0.789352	−0.394676	+	0.918820i	$0.629143\pi$
$398$	0	0
$399$	−180.521	−0.452435
$400$	0	0
$401$	−100.781	−0.251323	−0.125662	−	0.992073i	$-0.540105\pi$
$401$	−100.781	−0.251323	−0.125662	+	0.992073i	$0.540105\pi$
$402$	0	0
$403$	− 70.3783i	− 0.174636i
$404$	0	0
$405$	−161.492	−0.398745
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 117.329i	− 0.286867i	−0.989660	−	0.143434i	$-0.954186\pi$
$409$	− 117.329i	− 0.286867i	0.989660	−	0.143434i	$-0.0458144\pi$
$410$	0	0
$411$	604.179	1.47002
$412$	0	0
$413$	− 29.4663i	− 0.0713469i
$414$	0	0
$415$	180.513i	0.434971i
$416$	0	0
$417$	412.124i	0.988307i
$418$	0	0
$419$	−256.275	−0.611636	−0.305818	−	0.952090i	$-0.598930\pi$
$419$	−256.275	−0.611636	−0.305818	+	0.952090i	$0.598930\pi$
$420$	0	0
$421$	538.682	1.27953	0.639765	−	0.768570i	$-0.279032\pi$
$421$	538.682	1.27953	0.639765	+	0.768570i	$0.279032\pi$
$422$	0	0
$423$	23.6869	0.0559973
$424$	0	0
$425$	87.6017i	0.206122i
$426$	0	0
$427$	107.358	0.251423
$428$	0	0
$429$	0	0
$430$	0	0
$431$	139.593i	0.323881i	0.986801	+	0.161941i	$0.0517753\pi$
$431$	139.593i	0.323881i	−0.986801	+	0.161941i	$0.948225\pi$
$432$	0	0
$433$	−622.694	−1.43809	−0.719046	−	0.694962i	$-0.755421\pi$
$433$	−622.694	−1.43809	−0.719046	+	0.694962i	$0.755421\pi$
$434$	0	0
$435$	− 31.8786i	− 0.0732842i
$436$	0	0
$437$	− 739.863i	− 1.69305i
$438$	0	0
$439$	− 314.896i	− 0.717303i	−0.933471	−	0.358652i	$-0.883237\pi$
$439$	− 314.896i	− 0.717303i	0.933471	−	0.358652i	$-0.116763\pi$
$440$	0	0
$441$	−34.6123	−0.0784858
$442$	0	0
$443$	−210.951	−0.476188	−0.238094	−	0.971242i	$-0.576523\pi$
$443$	−210.951	−0.476188	−0.238094	+	0.971242i	$0.576523\pi$
$444$	0	0
$445$	−244.534	−0.549515
$446$	0	0
$447$	584.653i	1.30795i
$448$	0	0
$449$	195.081	0.434478	0.217239	−	0.976118i	$-0.430295\pi$
$449$	195.081	0.434478	0.217239	+	0.976118i	$0.430295\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 322.645i	− 0.712240i
$454$	0	0
$455$	−8.14639	−0.0179041
$456$	0	0
$457$	553.264i	1.21064i	0.795981	+	0.605321i	$0.206955\pi$
$457$	553.264i	1.21064i	−0.795981	+	0.605321i	$0.793045\pi$
$458$	0	0
$459$	− 493.414i	− 1.07498i
$460$	0	0
$461$	− 632.590i	− 1.37221i	−0.727501	−	0.686106i	$-0.759319\pi$
$461$	− 632.590i	− 1.37221i	0.727501	−	0.686106i	$-0.240681\pi$
$462$	0	0
$463$	−322.161	−0.695812	−0.347906	−	0.937529i	$-0.613107\pi$
$463$	−322.161	−0.695812	−0.347906	+	0.937529i	$0.613107\pi$
$464$	0	0
$465$	−389.336	−0.837282
$466$	0	0
$467$	199.185	0.426521	0.213260	−	0.976995i	$-0.431592\pi$
$467$	199.185	0.426521	0.213260	+	0.976995i	$0.431592\pi$
$468$	0	0
$469$	− 54.1696i	− 0.115500i
$470$	0	0
$471$	−80.3671	−0.170631
$472$	0	0
$473$	0	0
$474$	0	0
$475$	100.142i	0.210826i
$476$	0	0
$477$	−2.75196	−0.00576931
$478$	0	0
$479$	925.847i	1.93287i	0.256902	+	0.966437i	$0.417298\pi$
$479$	925.847i	1.93287i	−0.256902	+	0.966437i	$0.582702\pi$
$480$	0	0
$481$	− 29.9773i	− 0.0623229i
$482$	0	0
$483$	− 332.954i	− 0.689346i
$484$	0	0
$485$	−44.8068	−0.0923852
$486$	0	0
$487$	472.329	0.969874	0.484937	−	0.874549i	$-0.338842\pi$
$487$	472.329	0.969874	0.484937	+	0.874549i	$0.338842\pi$
$488$	0	0
$489$	−43.9058	−0.0897869
$490$	0	0
$491$	248.799i	0.506720i	0.967372	+	0.253360i	$0.0815357\pi$
$491$	248.799i	0.506720i	−0.967372	+	0.253360i	$0.918464\pi$
$492$	0	0
$493$	87.6977	0.177886
$494$	0	0
$495$	0	0
$496$	0	0
$497$	160.552i	0.323042i
$498$	0	0
$499$	−172.816	−0.346325	−0.173163	−	0.984893i	$-0.555399\pi$
$499$	−172.816	−0.346325	−0.173163	+	0.984893i	$0.555399\pi$
$500$	0	0
$501$	− 434.101i	− 0.866469i
$502$	0	0
$503$	777.548i	1.54582i	0.634515	+	0.772911i	$0.281200\pi$
$503$	777.548i	1.54582i	−0.634515	+	0.772911i	$0.718800\pi$
$504$	0	0
$505$	198.797i	0.393658i
$506$	0	0
$507$	477.569	0.941951
$508$	0	0
$509$	−509.332	−1.00065	−0.500326	−	0.865837i	$-0.666786\pi$
$509$	−509.332	−1.00065	−0.500326	+	0.865837i	$0.666786\pi$
$510$	0	0
$511$	−130.724	−0.255819
$512$	0	0
$513$	− 564.050i	− 1.09951i
$514$	0	0
$515$	27.7906	0.0539624
$516$	0	0
$517$	0	0
$518$	0	0
$519$	107.097i	0.206353i
$520$	0	0
$521$	−113.956	−0.218726	−0.109363	−	0.994002i	$-0.534881\pi$
$521$	−113.956	−0.218726	−0.109363	+	0.994002i	$0.534881\pi$
$522$	0	0
$523$	− 73.3632i	− 0.140274i	−0.997537	−	0.0701369i	$-0.977656\pi$
$523$	− 73.3632i	− 0.140274i	0.997537	−	0.0701369i	$-0.0223436\pi$
$524$	0	0
$525$	45.0662i	0.0858404i
$526$	0	0
$527$	− 1071.06i	− 2.03237i
$528$	0	0
$529$	835.605	1.57959
$530$	0	0
$531$	8.26682	0.0155684
$532$	0	0
$533$	69.3781	0.130165
$534$	0	0
$535$	− 462.777i	− 0.865003i
$536$	0	0
$537$	215.032	0.400432
$538$	0	0
$539$	0	0
$540$	0	0
$541$	809.615i	1.49652i	0.663408	+	0.748258i	$0.269110\pi$
$541$	809.615i	1.49652i	−0.663408	+	0.748258i	$0.730890\pi$
$542$	0	0
$543$	−780.776	−1.43789
$544$	0	0
$545$	82.4650i	0.151312i
$546$	0	0
$547$	516.237i	0.943761i	0.881663	+	0.471880i	$0.156425\pi$
$547$	516.237i	0.943761i	−0.881663	+	0.471880i	$0.843575\pi$
$548$	0	0
$549$	30.1194i	0.0548623i
$550$	0	0
$551$	100.252	0.181946
$552$	0	0
$553$	−133.627	−0.241641
$554$	0	0
$555$	−165.836	−0.298803
$556$	0	0
$557$	515.448i	0.925400i	0.886515	+	0.462700i	$0.153119\pi$
$557$	515.448i	0.925400i	−0.886515	+	0.462700i	$0.846881\pi$
$558$	0	0
$559$	75.2837	0.134676
$560$	0	0
$561$	0	0
$562$	0	0
$563$	380.772i	0.676326i	0.941088	+	0.338163i	$0.109806\pi$
$563$	380.772i	0.676326i	−0.941088	+	0.338163i	$0.890194\pi$
$564$	0	0
$565$	−214.143	−0.379015
$566$	0	0
$567$	− 228.548i	− 0.403083i
$568$	0	0
$569$	323.742i	0.568967i	0.958681	+	0.284483i	$0.0918221\pi$
$569$	323.742i	0.568967i	−0.958681	+	0.284483i	$0.908178\pi$
$570$	0	0
$571$	99.2705i	0.173854i	0.996215	+	0.0869268i	$0.0277046\pi$
$571$	99.2705i	0.173854i	−0.996215	+	0.0869268i	$0.972295\pi$
$572$	0	0
$573$	55.4883	0.0968382
$574$	0	0
$575$	−184.703	−0.321222
$576$	0	0
$577$	964.957	1.67237	0.836184	−	0.548449i	$-0.184781\pi$
$577$	964.957	1.67237	0.836184	+	0.548449i	$0.184781\pi$
$578$	0	0
$579$	342.214i	0.591044i
$580$	0	0
$581$	−255.468	−0.439703
$582$	0	0
$583$	0	0
$584$	0	0
$585$	− 2.28548i	− 0.00390681i
$586$	0	0
$587$	−235.369	−0.400970	−0.200485	−	0.979697i	$-0.564252\pi$
$587$	−235.369	−0.400970	−0.200485	+	0.979697i	$0.564252\pi$
$588$	0	0
$589$	− 1224.39i	− 2.07876i
$590$	0	0
$591$	134.080i	0.226869i
$592$	0	0
$593$	− 1004.87i	− 1.69456i	−0.531149	−	0.847279i	$-0.678239\pi$
$593$	− 1004.87i	− 1.69456i	0.531149	−	0.847279i	$-0.321761\pi$
$594$	0	0
$595$	−123.977	−0.208364
$596$	0	0
$597$	−50.3322	−0.0843085
$598$	0	0
$599$	605.756	1.01128	0.505640	−	0.862745i	$-0.331257\pi$
$599$	605.756	1.01128	0.505640	+	0.862745i	$0.331257\pi$
$600$	0	0
$601$	298.391i	0.496491i	0.968697	+	0.248245i	$0.0798539\pi$
$601$	298.391i	0.496491i	−0.968697	+	0.248245i	$0.920146\pi$
$602$	0	0
$603$	15.1974	0.0252030
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 359.345i	− 0.592002i	−0.955188	−	0.296001i	$-0.904347\pi$
$607$	− 359.345i	− 0.592002i	0.955188	−	0.296001i	$-0.0956532\pi$
$608$	0	0
$609$	45.1156	0.0740814
$610$	0	0
$611$	− 30.7150i	− 0.0502700i
$612$	0	0
$613$	466.311i	0.760704i	0.924842	+	0.380352i	$0.124197\pi$
$613$	466.311i	0.760704i	−0.924842	+	0.380352i	$0.875803\pi$
$614$	0	0
$615$	− 383.803i	− 0.624070i
$616$	0	0
$617$	1068.85	1.73233	0.866166	−	0.499757i	$-0.166577\pi$
$617$	1068.85	1.73233	0.866166	+	0.499757i	$0.166577\pi$
$618$	0	0
$619$	383.688	0.619851	0.309926	−	0.950761i	$-0.399696\pi$
$619$	383.688	0.619851	0.309926	+	0.950761i	$0.399696\pi$
$620$	0	0
$621$	1040.33	1.67526
$622$	0	0
$623$	− 346.072i	− 0.555493i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 456.213i	− 0.725298i
$630$	0	0
$631$	−985.448	−1.56172	−0.780862	−	0.624704i	$-0.785220\pi$
$631$	−985.448	−1.56172	−0.780862	+	0.624704i	$0.785220\pi$
$632$	0	0
$633$	284.647i	0.449679i
$634$	0	0
$635$	319.429i	0.503037i
$636$	0	0
$637$	44.8820i	0.0704584i
$638$	0	0
$639$	−45.0432	−0.0704901
$640$	0	0
$641$	622.897	0.971758	0.485879	−	0.874026i	$-0.338500\pi$
$641$	622.897	0.971758	0.485879	+	0.874026i	$0.338500\pi$
$642$	0	0
$643$	−800.440	−1.24485	−0.622426	−	0.782679i	$-0.713853\pi$
$643$	−800.440	−1.24485	−0.622426	+	0.782679i	$0.713853\pi$
$644$	0	0
$645$	− 416.473i	− 0.645695i
$646$	0	0
$647$	220.659	0.341050	0.170525	−	0.985353i	$-0.445454\pi$
$647$	220.659	0.341050	0.170525	+	0.985353i	$0.445454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 551.000i	− 0.846391i
$652$	0	0
$653$	737.192	1.12893	0.564466	−	0.825456i	$-0.309082\pi$
$653$	737.192	1.12893	0.564466	+	0.825456i	$0.309082\pi$
$654$	0	0
$655$	120.189i	0.183495i
$656$	0	0
$657$	− 36.6748i	− 0.0558216i
$658$	0	0
$659$	− 613.844i	− 0.931479i	−0.884922	−	0.465739i	$-0.845788\pi$
$659$	− 613.844i	− 0.931479i	0.884922	−	0.465739i	$-0.154212\pi$
$660$	0	0
$661$	410.333	0.620776	0.310388	−	0.950610i	$-0.399541\pi$
$661$	410.333	0.620776	0.310388	+	0.950610i	$0.399541\pi$
$662$	0	0
$663$	−57.4485	−0.0866494
$664$	0	0
$665$	−141.725	−0.213120
$666$	0	0
$667$	184.905i	0.277219i
$668$	0	0
$669$	−677.968	−1.01341
$670$	0	0
$671$	0	0
$672$	0	0
$673$	913.162i	1.35685i	0.734669	+	0.678426i	$0.237338\pi$
$673$	913.162i	1.35685i	−0.734669	+	0.678426i	$0.762662\pi$
$674$	0	0
$675$	−140.812	−0.208610
$676$	0	0
$677$	− 645.161i	− 0.952970i	−0.879183	−	0.476485i	$-0.841911\pi$
$677$	− 645.161i	− 0.952970i	0.879183	−	0.476485i	$-0.158089\pi$
$678$	0	0
$679$	− 63.4120i	− 0.0933903i
$680$	0	0
$681$	1043.97i	1.53300i
$682$	0	0
$683$	−918.844	−1.34531	−0.672653	−	0.739958i	$-0.734845\pi$
$683$	−918.844	−1.34531	−0.672653	+	0.739958i	$0.734845\pi$
$684$	0	0
$685$	474.331	0.692455
$686$	0	0
$687$	747.779	1.08847
$688$	0	0
$689$	3.56849i	0.00517923i
$690$	0	0
$691$	−723.035	−1.04636	−0.523180	−	0.852222i	$-0.675255\pi$
$691$	−723.035	−1.04636	−0.523180	+	0.852222i	$0.675255\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	323.552i	0.465543i
$696$	0	0
$697$	1055.84	1.51483
$698$	0	0
$699$	1070.50i	1.53147i
$700$	0	0
$701$	268.703i	0.383314i	0.981462	+	0.191657i	$0.0613860\pi$
$701$	268.703i	0.383314i	−0.981462	+	0.191657i	$0.938614\pi$
$702$	0	0
$703$	− 521.522i	− 0.741852i
$704$	0	0
$705$	−169.917	−0.241017
$706$	0	0
$707$	−281.344	−0.397940
$708$	0	0
$709$	−652.123	−0.919779	−0.459889	−	0.887976i	$-0.652111\pi$
$709$	−652.123	−0.919779	−0.459889	+	0.887976i	$0.652111\pi$
$710$	0	0
$711$	− 37.4894i	− 0.0527277i
$712$	0	0
$713$	2258.26	3.16727
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 857.212i	− 1.19555i
$718$	0	0
$719$	233.292	0.324467	0.162234	−	0.986752i	$-0.448130\pi$
$719$	233.292	0.324467	0.162234	+	0.986752i	$0.448130\pi$
$720$	0	0
$721$	39.3302i	0.0545495i
$722$	0	0
$723$	1361.63i	1.88331i
$724$	0	0
$725$	− 25.0274i	− 0.0345205i
$726$	0	0
$727$	−1255.99	−1.72764	−0.863818	−	0.503804i	$-0.831933\pi$
$727$	−1255.99	−1.72764	−0.863818	+	0.503804i	$0.831933\pi$
$728$	0	0
$729$	786.026	1.07822
$730$	0	0
$731$	1145.71	1.56732
$732$	0	0
$733$	− 1341.16i	− 1.82969i	−0.403807	−	0.914844i	$-0.632313\pi$
$733$	− 1341.16i	− 1.82969i	0.403807	−	0.914844i	$-0.367687\pi$
$734$	0	0
$735$	248.289	0.337809
$736$	0	0
$737$	0	0
$738$	0	0
$739$	985.254i	1.33323i	0.745404	+	0.666613i	$0.232257\pi$
$739$	985.254i	1.33323i	−0.745404	+	0.666613i	$0.767743\pi$
$740$	0	0
$741$	−65.6726	−0.0886270
$742$	0	0
$743$	849.373i	1.14317i	0.820544	+	0.571584i	$0.193671\pi$
$743$	849.373i	1.14317i	−0.820544	+	0.571584i	$0.806329\pi$
$744$	0	0
$745$	459.002i	0.616110i
$746$	0	0
$747$	− 71.6720i	− 0.0959464i
$748$	0	0
$749$	654.936	0.874414
$750$	0	0
$751$	148.178	0.197308	0.0986540	−	0.995122i	$-0.468546\pi$
$751$	148.178	0.197308	0.0986540	+	0.995122i	$0.468546\pi$
$752$	0	0
$753$	−1173.61	−1.55857
$754$	0	0
$755$	− 253.303i	− 0.335501i
$756$	0	0
$757$	−1155.79	−1.52680	−0.763400	−	0.645926i	$-0.776471\pi$
$757$	−1155.79	−1.52680	−0.763400	+	0.645926i	$0.776471\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 812.807i	− 1.06808i	−0.845460	−	0.534039i	$-0.820674\pi$
$761$	− 812.807i	− 1.06808i	0.845460	−	0.534039i	$-0.179326\pi$
$762$	0	0
$763$	−116.707	−0.152958
$764$	0	0
$765$	− 34.7819i	− 0.0454665i
$766$	0	0
$767$	− 10.7197i	− 0.0139761i
$768$	0	0
$769$	298.253i	0.387845i	0.981017	+	0.193923i	$0.0621211\pi$
$769$	298.253i	0.387845i	−0.981017	+	0.193923i	$0.937879\pi$
$770$	0	0
$771$	853.087	1.10647
$772$	0	0
$773$	459.500	0.594437	0.297219	−	0.954809i	$-0.403941\pi$
$773$	459.500	0.594437	0.297219	+	0.954809i	$0.403941\pi$
$774$	0	0
$775$	−305.662	−0.394402
$776$	0	0
$777$	− 234.696i	− 0.302054i
$778$	0	0
$779$	1206.99	1.54940
$780$	0	0
$781$	0	0
$782$	0	0
$783$	140.966i	0.180033i
$784$	0	0
$785$	−63.0950	−0.0803758
$786$	0	0
$787$	− 1141.52i	− 1.45047i	−0.688502	−	0.725235i	$-0.741731\pi$
$787$	− 1141.52i	− 1.45047i	0.688502	−	0.725235i	$-0.258269\pi$
$788$	0	0
$789$	806.700i	1.02243i
$790$	0	0
$791$	− 303.062i	− 0.383138i
$792$	0	0
$793$	39.0561	0.0492511
$794$	0	0
$795$	19.7411	0.0248315
$796$	0	0
$797$	41.3311	0.0518584	0.0259292	−	0.999664i	$-0.491746\pi$
$797$	41.3311	0.0518584	0.0259292	+	0.999664i	$0.491746\pi$
$798$	0	0
$799$	− 467.439i	− 0.585030i
$800$	0	0
$801$	97.0913	0.121213
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 261.397i	− 0.324717i
$806$	0	0
$807$	−1119.69	−1.38748
$808$	0	0
$809$	− 1123.31i	− 1.38852i	−0.719725	−	0.694259i	$-0.755732\pi$
$809$	− 1123.31i	− 1.38852i	0.719725	−	0.694259i	$-0.244268\pi$
$810$	0	0
$811$	− 864.705i	− 1.06622i	−0.846046	−	0.533111i	$-0.821023\pi$
$811$	− 864.705i	− 1.06622i	0.846046	−	0.533111i	$-0.178977\pi$
$812$	0	0
$813$	1536.52i	1.88994i
$814$	0	0
$815$	−34.4697	−0.0422941
$816$	0	0
$817$	1309.73	1.60309
$818$	0	0
$819$	3.23449	0.00394931
$820$	0	0
$821$	933.419i	1.13693i	0.822708	+	0.568465i	$0.192462\pi$
$821$	933.419i	1.13693i	−0.822708	+	0.568465i	$0.807538\pi$
$822$	0	0
$823$	−314.613	−0.382276	−0.191138	−	0.981563i	$-0.561218\pi$
$823$	−314.613	−0.382276	−0.191138	+	0.981563i	$0.561218\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	528.653i	0.639242i	0.947545	+	0.319621i	$0.103556\pi$
$827$	528.653i	0.639242i	−0.947545	+	0.319621i	$0.896444\pi$
$828$	0	0
$829$	−988.626	−1.19255	−0.596276	−	0.802779i	$-0.703354\pi$
$829$	−988.626	−1.19255	−0.596276	+	0.802779i	$0.703354\pi$
$830$	0	0
$831$	− 795.898i	− 0.957759i
$832$	0	0
$833$	683.041i	0.819977i
$834$	0	0
$835$	− 340.806i	− 0.408150i
$836$	0	0
$837$	1721.63	2.05691
$838$	0	0
$839$	1232.81	1.46938	0.734692	−	0.678401i	$-0.237327\pi$
$839$	1232.81	1.46938	0.734692	+	0.678401i	$0.237327\pi$
$840$	0	0
$841$	815.945	0.970208
$842$	0	0
$843$	− 781.466i	− 0.927006i
$844$	0	0
$845$	374.932	0.443706
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 1489.99i	− 1.75499i
$850$	0	0
$851$	961.896	1.13031
$852$	0	0
$853$	− 607.063i	− 0.711679i	−0.934547	−	0.355840i	$-0.884195\pi$
$853$	− 607.063i	− 0.711679i	0.934547	−	0.355840i	$-0.115805\pi$
$854$	0	0
$855$	− 39.7611i	− 0.0465042i
$856$	0	0
$857$	237.750i	0.277422i	0.990333	+	0.138711i	$0.0442959\pi$
$857$	237.750i	0.277422i	−0.990333	+	0.138711i	$0.955704\pi$
$858$	0	0
$859$	792.390	0.922456	0.461228	−	0.887282i	$-0.347409\pi$
$859$	792.390	0.922456	0.461228	+	0.887282i	$0.347409\pi$
$860$	0	0
$861$	543.169	0.630859
$862$	0	0
$863$	−777.416	−0.900829	−0.450415	−	0.892819i	$-0.648724\pi$
$863$	−777.416	−0.900829	−0.450415	+	0.892819i	$0.648724\pi$
$864$	0	0
$865$	84.0801i	0.0972025i
$866$	0	0
$867$	−51.1599	−0.0590079
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 19.7066i	− 0.0226252i
$872$	0	0
$873$	17.7904	0.0203784
$874$	0	0
$875$	35.3808i	0.0404352i
$876$	0	0
$877$	651.290i	0.742634i	0.928506	+	0.371317i	$0.121094\pi$
$877$	651.290i	0.742634i	−0.928506	+	0.371317i	$0.878906\pi$
$878$	0	0
$879$	− 1149.83i	− 1.30811i
$880$	0	0
$881$	1499.78	1.70236	0.851182	−	0.524870i	$-0.175886\pi$
$881$	1499.78	1.70236	0.851182	+	0.524870i	$0.175886\pi$
$882$	0	0
$883$	−1209.15	−1.36936	−0.684681	−	0.728843i	$-0.740058\pi$
$883$	−1209.15	−1.36936	−0.684681	+	0.728843i	$0.740058\pi$
$884$	0	0
$885$	−59.3016	−0.0670075
$886$	0	0
$887$	− 955.521i	− 1.07725i	−0.842545	−	0.538625i	$-0.818944\pi$
$887$	− 955.521i	− 1.07725i	0.842545	−	0.538625i	$-0.181056\pi$
$888$	0	0
$889$	−452.065	−0.508510
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 534.355i	− 0.598382i
$894$	0	0
$895$	168.818	0.188624
$896$	0	0
$897$	− 121.127i	− 0.135035i
$898$	0	0
$899$	305.997i	0.340374i
$900$	0	0
$901$	54.3074i	0.0602746i
$902$	0	0
$903$	589.405	0.652719
$904$	0	0
$905$	−612.975	−0.677320
$906$	0	0
$907$	188.645	0.207988	0.103994	−	0.994578i	$-0.466838\pi$
$907$	188.645	0.207988	0.103994	+	0.994578i	$0.466838\pi$
$908$	0	0
$909$	− 78.9315i	− 0.0868334i
$910$	0	0
$911$	230.077	0.252554	0.126277	−	0.991995i	$-0.459697\pi$
$911$	230.077	0.252554	0.126277	+	0.991995i	$0.459697\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 216.060i	− 0.236131i
$916$	0	0
$917$	−170.095	−0.185491
$918$	0	0
$919$	− 1645.90i	− 1.79097i	−0.445088	−	0.895487i	$-0.646828\pi$
$919$	− 1645.90i	− 1.79097i	0.445088	−	0.895487i	$-0.353172\pi$
$920$	0	0
$921$	− 828.381i	− 0.899437i
$922$	0	0
$923$	58.4079i	0.0632805i
$924$	0	0
$925$	−130.195	−0.140751
$926$	0	0
$927$	−11.0342	−0.0119031
$928$	0	0
$929$	622.194	0.669746	0.334873	−	0.942263i	$-0.391307\pi$
$929$	622.194	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$930$	0	0
$931$	780.822i	0.838692i
$932$	0	0
$933$	152.357	0.163298
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.0297i	0.0459229i	0.999736	+	0.0229614i	$0.00730950\pi$
$937$	43.0297i	0.0459229i	−0.999736	+	0.0229614i	$0.992691\pi$
$938$	0	0
$939$	−582.801	−0.620662
$940$	0	0
$941$	− 1184.75i	− 1.25903i	−0.776988	−	0.629516i	$-0.783253\pi$
$941$	− 1184.75i	− 1.25903i	0.776988	−	0.629516i	$-0.216747\pi$
$942$	0	0
$943$	2226.17i	2.36073i
$944$	0	0
$945$	− 199.281i	− 0.210880i
$946$	0	0
$947$	−486.719	−0.513959	−0.256979	−	0.966417i	$-0.582727\pi$
$947$	−486.719	−0.513959	−0.256979	+	0.966417i	$0.582727\pi$
$948$	0	0
$949$	−47.5565	−0.0501122
$950$	0	0
$951$	588.632	0.618961
$952$	0	0
$953$	− 765.779i	− 0.803546i	−0.915739	−	0.401773i	$-0.868394\pi$
$953$	− 765.779i	− 0.803546i	0.915739	−	0.401773i	$-0.131606\pi$
$954$	0	0
$955$	43.5630	0.0456157
$956$	0	0
$957$	0	0
$958$	0	0
$959$	671.288i	0.699988i
$960$	0	0
$961$	2776.16	2.88883
$962$	0	0
$963$	183.744i	0.190803i
$964$	0	0
$965$	268.667i	0.278412i
$966$	0	0
$967$	1843.46i	1.90637i	0.302386	+	0.953186i	$0.402217\pi$
$967$	1843.46i	1.90637i	−0.302386	+	0.953186i	$0.597783\pi$
$968$	0	0
$969$	−999.445	−1.03142
$970$	0	0
$971$	518.715	0.534207	0.267104	−	0.963668i	$-0.413933\pi$
$971$	518.715	0.534207	0.267104	+	0.963668i	$0.413933\pi$
$972$	0	0
$973$	−457.901	−0.470607
$974$	0	0
$975$	16.3948i	0.0168152i
$976$	0	0
$977$	1587.15	1.62451	0.812256	−	0.583301i	$-0.198239\pi$
$977$	1587.15	1.62451	0.812256	+	0.583301i	$0.198239\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 32.7424i	− 0.0333765i
$982$	0	0
$983$	−1923.05	−1.95631	−0.978153	−	0.207887i	$-0.933341\pi$
$983$	−1923.05	−1.95631	−0.978153	+	0.207887i	$0.933341\pi$
$984$	0	0
$985$	105.264i	0.106867i
$986$	0	0
$987$	− 240.471i	− 0.243638i
$988$	0	0
$989$	2415.66i	2.44253i
$990$	0	0
$991$	722.553	0.729115	0.364558	−	0.931181i	$-0.381220\pi$
$991$	722.553	0.729115	0.364558	+	0.931181i	$0.381220\pi$
$992$	0	0
$993$	−887.510	−0.893766
$994$	0	0
$995$	−39.5150	−0.0397136
$996$	0	0
$997$	74.3117i	0.0745353i	0.999305	+	0.0372676i	$0.0118654\pi$
$997$	74.3117i	0.0745353i	−0.999305	+	0.0372676i	$0.988135\pi$
$998$	0	0
$999$	733.321	0.734055

Display

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a_n

instead) (See

a_n

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a_n

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n

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.3.f.a.241.14		16
11.3	even	5		220.3.p.b.101.1	yes	16
11.7	odd	10		220.3.p.b.61.1	✓	16
11.10	odd	2	inner	2420.3.f.a.241.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.p.b.61.1	✓	16	11.7	odd	10
220.3.p.b.101.1	yes	16	11.3	even	5
2420.3.f.a.241.13		16	11.10	odd	2	inner
2420.3.f.a.241.14		16	1.1	even	1	trivial

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2420.3.f.a.241.14

Level	$2420$
Weight	$3$
Character	2420.241
Analytic conductor	$65.940$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$