LMFDB - Embedded newform 45.3.c.a.26.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,3,Mod(26,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	45.c (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:	$1.22616118962$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.4
Root			$1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	45.26
Dual form			45.3.c.a.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65028i q^{2} -9.32456 q^{4} +2.23607i q^{5} +7.16228 q^{7} -19.4361i q^{8} +O(q^{10})$	\(q+3.65028i q^{2} -9.32456 q^{4} +2.23607i q^{5} +7.16228 q^{7} -19.4361i q^{8} -8.16228 q^{10} +5.42736i q^{11} +9.81139 q^{13} +26.1443i q^{14} +33.6491 q^{16} -12.2317i q^{17} +6.32456 q^{19} -20.8503i q^{20} -19.8114 q^{22} -12.0394i q^{23} -5.00000 q^{25} +35.8143i q^{26} -66.7851 q^{28} -44.9881i q^{29} -58.2719 q^{31} +45.0842i q^{32} +44.6491 q^{34} +16.0153i q^{35} +66.4605 q^{37} +23.0864i q^{38} +43.4605 q^{40} +16.4743i q^{41} -43.6228 q^{43} -50.6077i q^{44} +43.9473 q^{46} +40.0570i q^{47} +2.29822 q^{49} -18.2514i q^{50} -91.4868 q^{52} +13.2242i q^{53} -12.1359 q^{55} -139.207i q^{56} +164.219 q^{58} +25.1519i q^{59} -35.6754 q^{61} -212.709i q^{62} -29.9737 q^{64} +21.9389i q^{65} +26.7018 q^{67} +114.055i q^{68} -58.4605 q^{70} +92.7301i q^{71} +60.3246 q^{73} +242.600i q^{74} -58.9737 q^{76} +38.8723i q^{77} -96.2192 q^{79} +75.2417i q^{80} -60.1359 q^{82} -79.1215i q^{83} +27.3509 q^{85} -159.235i q^{86} +105.487 q^{88} -107.443i q^{89} +70.2719 q^{91} +112.262i q^{92} -146.219 q^{94} +14.1421i q^{95} -1.07900 q^{97} +8.38915i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4} + 16 q^{7}+O(q^{10}) $ 4 * q - 12 * q^4 + 16 * q^7	$ 4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55} + 328 q^{58} - 168 q^{61} - 44 q^{64} + 208 q^{67} - 120 q^{70} + 216 q^{73} - 160 q^{76} - 56 q^{79} - 152 q^{82} + 160 q^{85} + 384 q^{88} + 104 q^{91} - 256 q^{94} - 232 q^{97}+O(q^{100}) $ 4 * q - 12 * q^4 + 16 * q^7 - 20 * q^10 - 24 * q^13 + 84 * q^16 - 16 * q^22 - 20 * q^25 - 128 * q^28 - 56 * q^31 + 128 * q^34 + 152 * q^37 + 60 * q^40 - 48 * q^43 + 24 * q^46 - 92 * q^49 - 328 * q^52 + 40 * q^55 + 328 * q^58 - 168 * q^61 - 44 * q^64 + 208 * q^67 - 120 * q^70 + 216 * q^73 - 160 * q^76 - 56 * q^79 - 152 * q^82 + 160 * q^85 + 384 * q^88 + 104 * q^91 - 256 * q^94 - 232 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.65028i	1.82514i	0.408920	+	0.912570i	$0.365906\pi$
$2$	3.65028i	1.82514i	−0.408920	+	0.912570i	$0.634094\pi$
$3$	0	0
$3$	0	0
$4$	−9.32456	−2.33114
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	7.16228	1.02318	0.511591	−	0.859229i	$-0.329056\pi$
$7$	7.16228	1.02318	0.511591	+	0.859229i	$0.329056\pi$
$8$	− 19.4361i	− 2.42952i
$9$	0	0
$10$	−8.16228	−0.816228
$11$	5.42736i	0.493396i	0.969092	+	0.246698i	$0.0793456\pi$
$11$	5.42736i	0.493396i	−0.969092	+	0.246698i	$0.920654\pi$
$12$	0	0
$13$	9.81139	0.754722	0.377361	−	0.926066i	$-0.376832\pi$
$13$	9.81139	0.754722	0.377361	+	0.926066i	$0.376832\pi$
$14$	26.1443i	1.86745i
$15$	0	0
$16$	33.6491	2.10307
$17$	− 12.2317i	− 0.719511i	−0.933047	−	0.359756i	$-0.882860\pi$
$17$	− 12.2317i	− 0.719511i	0.933047	−	0.359756i	$-0.117140\pi$
$18$	0	0
$19$	6.32456	0.332871	0.166436	−	0.986052i	$-0.446774\pi$
$19$	6.32456	0.332871	0.166436	+	0.986052i	$0.446774\pi$
$20$	− 20.8503i	− 1.04252i
$21$	0	0
$22$	−19.8114	−0.900518
$23$	− 12.0394i	− 0.523454i	−0.965142	−	0.261727i	$-0.915708\pi$
$23$	− 12.0394i	− 0.523454i	0.965142	−	0.261727i	$-0.0842920\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	35.8143i	1.37747i
$27$	0	0
$28$	−66.7851	−2.38518
$29$	− 44.9881i	− 1.55131i	−0.631155	−	0.775657i	$-0.717419\pi$
$29$	− 44.9881i	− 1.55131i	0.631155	−	0.775657i	$-0.282581\pi$
$30$	0	0
$31$	−58.2719	−1.87974	−0.939869	−	0.341535i	$-0.889053\pi$
$31$	−58.2719	−1.87974	−0.939869	+	0.341535i	$0.889053\pi$
$32$	45.0842i	1.40888i
$33$	0	0
$34$	44.6491	1.31321
$35$	16.0153i	0.457581i
$36$	0	0
$37$	66.4605	1.79623	0.898115	−	0.439761i	$-0.144937\pi$
$37$	66.4605	1.79623	0.898115	+	0.439761i	$0.144937\pi$
$38$	23.0864i	0.607537i
$39$	0	0
$40$	43.4605	1.08651
$41$	16.4743i	0.401813i	0.979610	+	0.200906i	$0.0643887\pi$
$41$	16.4743i	0.401813i	−0.979610	+	0.200906i	$0.935611\pi$
$42$	0	0
$43$	−43.6228	−1.01448	−0.507242	−	0.861804i	$-0.669335\pi$
$43$	−43.6228	−1.01448	−0.507242	+	0.861804i	$0.669335\pi$
$44$	− 50.6077i	− 1.15018i
$45$	0	0
$46$	43.9473	0.955377
$47$	40.0570i	0.852276i	0.904658	+	0.426138i	$0.140126\pi$
$47$	40.0570i	0.852276i	−0.904658	+	0.426138i	$0.859874\pi$
$48$	0	0
$49$	2.29822	0.0469025
$50$	− 18.2514i	− 0.365028i
$51$	0	0
$52$	−91.4868	−1.75936
$53$	13.2242i	0.249512i	0.992187	+	0.124756i	$0.0398149\pi$
$53$	13.2242i	0.249512i	−0.992187	+	0.124756i	$0.960185\pi$
$54$	0	0
$55$	−12.1359	−0.220654
$56$	− 139.207i	− 2.48584i
$57$	0	0
$58$	164.219	2.83137
$59$	25.1519i	0.426303i	0.977019	+	0.213151i	$0.0683728\pi$
$59$	25.1519i	0.426303i	−0.977019	+	0.213151i	$0.931627\pi$
$60$	0	0
$61$	−35.6754	−0.584843	−0.292422	−	0.956289i	$-0.594461\pi$
$61$	−35.6754	−0.584843	−0.292422	+	0.956289i	$0.594461\pi$
$62$	− 212.709i	− 3.43079i
$63$	0	0
$64$	−29.9737	−0.468339
$65$	21.9389i	0.337522i
$66$	0	0
$67$	26.7018	0.398534	0.199267	−	0.979945i	$-0.436144\pi$
$67$	26.7018	0.398534	0.199267	+	0.979945i	$0.436144\pi$
$68$	114.055i	1.67728i
$69$	0	0
$70$	−58.4605	−0.835150
$71$	92.7301i	1.30606i	0.757333	+	0.653029i	$0.226502\pi$
$71$	92.7301i	1.30606i	−0.757333	+	0.653029i	$0.773498\pi$
$72$	0	0
$73$	60.3246	0.826364	0.413182	−	0.910649i	$-0.364417\pi$
$73$	60.3246	0.826364	0.413182	+	0.910649i	$0.364417\pi$
$74$	242.600i	3.27837i
$75$	0	0
$76$	−58.9737	−0.775969
$77$	38.8723i	0.504834i
$78$	0	0
$79$	−96.2192	−1.21796	−0.608982	−	0.793184i	$-0.708422\pi$
$79$	−96.2192	−1.21796	−0.608982	+	0.793184i	$0.708422\pi$
$80$	75.2417i	0.940521i
$81$	0	0
$82$	−60.1359	−0.733365
$83$	− 79.1215i	− 0.953271i	−0.879101	−	0.476635i	$-0.841856\pi$
$83$	− 79.1215i	− 0.953271i	0.879101	−	0.476635i	$-0.158144\pi$
$84$	0	0
$85$	27.3509	0.321775
$86$	− 159.235i	− 1.85157i
$87$	0	0
$88$	105.487	1.19871
$89$	− 107.443i	− 1.20722i	−0.797278	−	0.603612i	$-0.793728\pi$
$89$	− 107.443i	− 1.20722i	0.797278	−	0.603612i	$-0.206272\pi$
$90$	0	0
$91$	70.2719	0.772219
$92$	112.262i	1.22024i
$93$	0	0
$94$	−146.219	−1.55552
$95$	14.1421i	0.148865i
$96$	0	0
$97$	−1.07900	−0.0111237	−0.00556187	−	0.999985i	$-0.501770\pi$
$97$	−1.07900	−0.0111237	−0.00556187	+	0.999985i	$0.501770\pi$
$98$	8.38915i	0.0856036i
$99$	0	0
$100$	46.6228	0.466228
$101$	− 170.282i	− 1.68596i	−0.537942	−	0.842982i	$-0.680798\pi$
$101$	− 170.282i	− 1.68596i	0.537942	−	0.842982i	$-0.319202\pi$
$102$	0	0
$103$	−128.460	−1.24719	−0.623595	−	0.781748i	$-0.714328\pi$
$103$	−128.460	−1.24719	−0.623595	+	0.781748i	$0.714328\pi$
$104$	− 190.695i	− 1.83361i
$105$	0	0
$106$	−48.2719	−0.455395
$107$	76.3675i	0.713715i	0.934159	+	0.356858i	$0.116152\pi$
$107$	76.3675i	0.713715i	−0.934159	+	0.356858i	$0.883848\pi$
$108$	0	0
$109$	−13.0790	−0.119991	−0.0599954	−	0.998199i	$-0.519109\pi$
$109$	−13.0790	−0.119991	−0.0599954	+	0.998199i	$0.519109\pi$
$110$	− 44.2996i	− 0.402724i
$111$	0	0
$112$	241.004	2.15182
$113$	− 20.7170i	− 0.183336i	−0.995790	−	0.0916680i	$-0.970780\pi$
$113$	− 20.7170i	− 0.183336i	0.995790	−	0.0916680i	$-0.0292199\pi$
$114$	0	0
$115$	26.9210	0.234096
$116$	419.494i	3.61633i
$117$	0	0
$118$	−91.8114	−0.778063
$119$	− 87.6068i	− 0.736191i
$120$	0	0
$121$	91.5438	0.756560
$122$	− 130.225i	− 1.06742i
$123$	0	0
$124$	543.359	4.38193
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−38.0306	−0.299454	−0.149727	−	0.988727i	$-0.547839\pi$
$127$	−38.0306	−0.299454	−0.149727	+	0.988727i	$0.547839\pi$
$128$	70.9246i	0.554098i
$129$	0	0
$130$	−80.0833	−0.616025
$131$	− 83.9409i	− 0.640770i	−0.947287	−	0.320385i	$-0.896188\pi$
$131$	− 83.9409i	− 0.640770i	0.947287	−	0.320385i	$-0.103812\pi$
$132$	0	0
$133$	45.2982	0.340588
$134$	97.4690i	0.727381i
$135$	0	0
$136$	−237.737	−1.74806
$137$	15.5936i	0.113822i	0.998379	+	0.0569109i	$0.0181251\pi$
$137$	15.5936i	0.113822i	−0.998379	+	0.0569109i	$0.981875\pi$
$138$	0	0
$139$	67.8420	0.488072	0.244036	−	0.969766i	$-0.421529\pi$
$139$	67.8420	0.488072	0.244036	+	0.969766i	$0.421529\pi$
$140$	− 149.336i	− 1.06669i
$141$	0	0
$142$	−338.491	−2.38374
$143$	53.2499i	0.372377i
$144$	0	0
$145$	100.596	0.693769
$146$	220.202i	1.50823i
$147$	0	0
$148$	−619.715	−4.18726
$149$	233.426i	1.56662i	0.621634	+	0.783308i	$0.286469\pi$
$149$	233.426i	1.56662i	−0.621634	+	0.783308i	$0.713531\pi$
$150$	0	0
$151$	185.351	1.22749	0.613745	−	0.789505i	$-0.289662\pi$
$151$	185.351	1.22749	0.613745	+	0.789505i	$0.289662\pi$
$152$	− 122.925i	− 0.808716i
$153$	0	0
$154$	−141.895	−0.921394
$155$	− 130.300i	− 0.840645i
$156$	0	0
$157$	−111.276	−0.708765	−0.354383	−	0.935100i	$-0.615309\pi$
$157$	−111.276	−0.708765	−0.354383	+	0.935100i	$0.615309\pi$
$158$	− 351.227i	− 2.22296i
$159$	0	0
$160$	−100.811	−0.630071
$161$	− 86.2298i	− 0.535589i
$162$	0	0
$163$	118.763	0.728607	0.364304	−	0.931280i	$-0.381307\pi$
$163$	118.763	0.728607	0.364304	+	0.931280i	$0.381307\pi$
$164$	− 153.616i	− 0.936682i
$165$	0	0
$166$	288.816	1.73985
$167$	221.194i	1.32452i	0.749276	+	0.662258i	$0.230402\pi$
$167$	221.194i	1.32452i	−0.749276	+	0.662258i	$0.769598\pi$
$168$	0	0
$169$	−72.7367	−0.430394
$170$	99.8384i	0.587285i
$171$	0	0
$172$	406.763	2.36490
$173$	190.807i	1.10293i	0.834198	+	0.551466i	$0.185931\pi$
$173$	190.807i	1.10293i	−0.834198	+	0.551466i	$0.814069\pi$
$174$	0	0
$175$	−35.8114	−0.204637
$176$	182.626i	1.03765i
$177$	0	0
$178$	392.197	2.20335
$179$	− 58.1005i	− 0.324584i	−0.986743	−	0.162292i	$-0.948111\pi$
$179$	− 58.1005i	− 0.324584i	0.986743	−	0.162292i	$-0.0518886\pi$
$180$	0	0
$181$	−162.921	−0.900116	−0.450058	−	0.892999i	$-0.648597\pi$
$181$	−162.921	−0.900116	−0.450058	+	0.892999i	$0.648597\pi$
$182$	256.512i	1.40941i
$183$	0	0
$184$	−234.000	−1.27174
$185$	148.610i	0.803298i
$186$	0	0
$187$	66.3858	0.355004
$188$	− 373.513i	− 1.98677i
$189$	0	0
$190$	−51.6228	−0.271699
$191$	100.062i	0.523884i	0.965084	+	0.261942i	$0.0843630\pi$
$191$	100.062i	0.523884i	−0.965084	+	0.261942i	$0.915637\pi$
$192$	0	0
$193$	−61.8947	−0.320698	−0.160349	−	0.987060i	$-0.551262\pi$
$193$	−61.8947	−0.320698	−0.160349	+	0.987060i	$0.551262\pi$
$194$	− 3.93866i	− 0.0203024i
$195$	0	0
$196$	−21.4299	−0.109336
$197$	− 24.8791i	− 0.126290i	−0.998004	−	0.0631449i	$-0.979887\pi$
$197$	− 24.8791i	− 0.126290i	0.998004	−	0.0631449i	$-0.0201130\pi$
$198$	0	0
$199$	−156.491	−0.786387	−0.393194	−	0.919456i	$-0.628630\pi$
$199$	−156.491	−0.786387	−0.393194	+	0.919456i	$0.628630\pi$
$200$	97.1806i	0.485903i
$201$	0	0
$202$	621.579	3.07712
$203$	− 322.217i	− 1.58728i
$204$	0	0
$205$	−36.8377	−0.179696
$206$	− 468.917i	− 2.27630i
$207$	0	0
$208$	330.144	1.58723
$209$	34.3256i	0.164237i
$210$	0	0
$211$	237.789	1.12696	0.563482	−	0.826128i	$-0.309461\pi$
$211$	237.789	1.12696	0.563482	+	0.826128i	$0.309461\pi$
$212$	− 123.309i	− 0.581648i
$213$	0	0
$214$	−278.763	−1.30263
$215$	− 97.5435i	− 0.453691i
$216$	0	0
$217$	−417.359	−1.92332
$218$	− 47.7420i	− 0.219000i
$219$	0	0
$220$	113.162	0.514374
$221$	− 120.010i	− 0.543031i
$222$	0	0
$223$	−182.302	−0.817500	−0.408750	−	0.912646i	$-0.634035\pi$
$223$	−182.302	−0.817500	−0.408750	+	0.912646i	$0.634035\pi$
$224$	322.906i	1.44154i
$225$	0	0
$226$	75.6228	0.334614
$227$	− 406.078i	− 1.78889i	−0.447180	−	0.894444i	$-0.647572\pi$
$227$	− 406.078i	− 1.78889i	0.447180	−	0.894444i	$-0.352428\pi$
$228$	0	0
$229$	−27.2982	−0.119206	−0.0596031	−	0.998222i	$-0.518984\pi$
$229$	−27.2982	−0.119206	−0.0596031	+	0.998222i	$0.518984\pi$
$230$	98.2692i	0.427257i
$231$	0	0
$232$	−874.394	−3.76894
$233$	356.382i	1.52954i	0.644306	+	0.764768i	$0.277146\pi$
$233$	356.382i	1.52954i	−0.644306	+	0.764768i	$0.722854\pi$
$234$	0	0
$235$	−89.5701	−0.381149
$236$	− 234.530i	− 0.993771i
$237$	0	0
$238$	319.789	1.34365
$239$	271.690i	1.13678i	0.822760	+	0.568389i	$0.192433\pi$
$239$	271.690i	1.13678i	−0.822760	+	0.568389i	$0.807567\pi$
$240$	0	0
$241$	−224.438	−0.931280	−0.465640	−	0.884974i	$-0.654176\pi$
$241$	−224.438	−0.931280	−0.465640	+	0.884974i	$0.654176\pi$
$242$	334.161i	1.38083i
$243$	0	0
$244$	332.658	1.36335
$245$	5.13898i	0.0209754i
$246$	0	0
$247$	62.0527	0.251225
$248$	1132.58i	4.56685i
$249$	0	0
$250$	40.8114	0.163246
$251$	318.775i	1.27002i	0.772504	+	0.635010i	$0.219004\pi$
$251$	318.775i	1.27002i	−0.772504	+	0.635010i	$0.780996\pi$
$252$	0	0
$253$	65.3423	0.258270
$254$	− 138.822i	− 0.546545i
$255$	0	0
$256$	−378.789	−1.47965
$257$	− 371.975i	− 1.44738i	−0.690128	−	0.723688i	$-0.742446\pi$
$257$	− 371.975i	− 1.44738i	0.690128	−	0.723688i	$-0.257554\pi$
$258$	0	0
$259$	476.009	1.83787
$260$	− 204.571i	− 0.786811i
$261$	0	0
$262$	306.408	1.16950
$263$	− 238.549i	− 0.907031i	−0.891248	−	0.453515i	$-0.850170\pi$
$263$	− 238.549i	− 0.907031i	0.891248	−	0.453515i	$-0.149830\pi$
$264$	0	0
$265$	−29.5701	−0.111585
$266$	165.351i	0.621621i
$267$	0	0
$268$	−248.982	−0.929038
$269$	− 125.871i	− 0.467922i	−0.972246	−	0.233961i	$-0.924831\pi$
$269$	− 125.871i	− 0.467922i	0.972246	−	0.233961i	$-0.0751688\pi$
$270$	0	0
$271$	−258.649	−0.954425	−0.477212	−	0.878788i	$-0.658353\pi$
$271$	−258.649	−0.954425	−0.477212	+	0.878788i	$0.658353\pi$
$272$	− 411.585i	− 1.51318i
$273$	0	0
$274$	−56.9210	−0.207741
$275$	− 27.1368i	− 0.0986793i
$276$	0	0
$277$	227.715	0.822074	0.411037	−	0.911619i	$-0.365167\pi$
$277$	227.715	0.822074	0.411037	+	0.911619i	$0.365167\pi$
$278$	247.642i	0.890800i
$279$	0	0
$280$	311.276	1.11170
$281$	241.384i	0.859016i	0.903063	+	0.429508i	$0.141313\pi$
$281$	241.384i	0.859016i	−0.903063	+	0.429508i	$0.858687\pi$
$282$	0	0
$283$	208.333	0.736159	0.368080	−	0.929794i	$-0.380015\pi$
$283$	208.333	0.736159	0.368080	+	0.929794i	$0.380015\pi$
$284$	− 864.667i	− 3.04460i
$285$	0	0
$286$	−194.377	−0.679641
$287$	117.994i	0.411128i
$288$	0	0
$289$	139.386	0.482304
$290$	367.205i	1.26623i
$291$	0	0
$292$	−562.500	−1.92637
$293$	201.693i	0.688372i	0.938901	+	0.344186i	$0.111845\pi$
$293$	201.693i	0.688372i	−0.938901	+	0.344186i	$0.888155\pi$
$294$	0	0
$295$	−56.2413	−0.190648
$296$	− 1291.73i	− 4.36397i
$297$	0	0
$298$	−852.070	−2.85929
$299$	− 118.124i	− 0.395062i
$300$	0	0
$301$	−312.438	−1.03800
$302$	676.583i	2.24034i
$303$	0	0
$304$	212.816	0.700052
$305$	− 79.7727i	− 0.261550i
$306$	0	0
$307$	342.824	1.11669	0.558346	−	0.829608i	$-0.311436\pi$
$307$	342.824	1.11669	0.558346	+	0.829608i	$0.311436\pi$
$308$	− 362.466i	− 1.17684i
$309$	0	0
$310$	475.631	1.53429
$311$	217.640i	0.699807i	0.936786	+	0.349903i	$0.113786\pi$
$311$	217.640i	0.699807i	−0.936786	+	0.349903i	$0.886214\pi$
$312$	0	0
$313$	281.895	0.900622	0.450311	−	0.892872i	$-0.351313\pi$
$313$	281.895	0.900622	0.450311	+	0.892872i	$0.351313\pi$
$314$	− 406.189i	− 1.29360i
$315$	0	0
$316$	897.201	2.83925
$317$	− 15.4013i	− 0.0485847i	−0.999705	−	0.0242923i	$-0.992267\pi$
$317$	− 15.4013i	− 0.0485847i	0.999705	−	0.0242923i	$-0.00773325\pi$
$318$	0	0
$319$	244.167	0.765412
$320$	− 67.0232i	− 0.209447i
$321$	0	0
$322$	314.763	0.977525
$323$	− 77.3600i	− 0.239505i
$324$	0	0
$325$	−49.0569	−0.150944
$326$	433.518i	1.32981i
$327$	0	0
$328$	320.197	0.976211
$329$	286.899i	0.872034i
$330$	0	0
$331$	375.517	1.13449	0.567247	−	0.823548i	$-0.308009\pi$
$331$	375.517	1.13449	0.567247	+	0.823548i	$0.308009\pi$
$332$	737.773i	2.22221i
$333$	0	0
$334$	−807.421	−2.41743
$335$	59.7070i	0.178230i
$336$	0	0
$337$	188.114	0.558201	0.279101	−	0.960262i	$-0.409964\pi$
$337$	188.114	0.558201	0.279101	+	0.960262i	$0.409964\pi$
$338$	− 265.509i	− 0.785530i
$339$	0	0
$340$	−255.035	−0.750103
$341$	− 316.262i	− 0.927456i
$342$	0	0
$343$	−334.491	−0.975193
$344$	847.858i	2.46470i
$345$	0	0
$346$	−696.500	−2.01300
$347$	− 513.793i	− 1.48067i	−0.672237	−	0.740336i	$-0.734666\pi$
$347$	− 513.793i	− 1.48067i	0.672237	−	0.740336i	$-0.265334\pi$
$348$	0	0
$349$	112.535	0.322451	0.161225	−	0.986918i	$-0.448455\pi$
$349$	112.535	0.322451	0.161225	+	0.986918i	$0.448455\pi$
$350$	− 130.722i	− 0.373490i
$351$	0	0
$352$	−244.688	−0.695137
$353$	428.172i	1.21295i	0.795102	+	0.606475i	$0.207417\pi$
$353$	428.172i	1.21295i	−0.795102	+	0.606475i	$0.792583\pi$
$354$	0	0
$355$	−207.351	−0.584087
$356$	1001.86i	2.81421i
$357$	0	0
$358$	212.083	0.592411
$359$	56.1961i	0.156535i	0.996932	+	0.0782676i	$0.0249389\pi$
$359$	56.1961i	0.156535i	−0.996932	+	0.0782676i	$0.975061\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	− 594.708i	− 1.64284i
$363$	0	0
$364$	−655.254	−1.80015
$365$	134.890i	0.369561i
$366$	0	0
$367$	−154.364	−0.420610	−0.210305	−	0.977636i	$-0.567446\pi$
$367$	−154.364	−0.420610	−0.210305	+	0.977636i	$0.567446\pi$
$368$	− 405.116i	− 1.10086i
$369$	0	0
$370$	−542.469	−1.46613
$371$	94.7151i	0.255297i
$372$	0	0
$373$	557.285	1.49406	0.747030	−	0.664790i	$-0.231479\pi$
$373$	557.285	1.49406	0.747030	+	0.664790i	$0.231479\pi$
$374$	242.327i	0.647932i
$375$	0	0
$376$	778.552	2.07062
$377$	− 441.396i	− 1.17081i
$378$	0	0
$379$	147.404	0.388928	0.194464	−	0.980910i	$-0.437703\pi$
$379$	147.404	0.388928	0.194464	+	0.980910i	$0.437703\pi$
$380$	− 131.869i	− 0.347024i
$381$	0	0
$382$	−365.254	−0.956163
$383$	− 736.619i	− 1.92329i	−0.274302	−	0.961644i	$-0.588447\pi$
$383$	− 736.619i	− 1.92329i	0.274302	−	0.961644i	$-0.411553\pi$
$384$	0	0
$385$	−86.9210	−0.225769
$386$	− 225.933i	− 0.585319i
$387$	0	0
$388$	10.0612	0.0259310
$389$	− 296.408i	− 0.761975i	−0.924580	−	0.380987i	$-0.875584\pi$
$389$	− 296.408i	− 0.761975i	0.924580	−	0.380987i	$-0.124416\pi$
$390$	0	0
$391$	−147.263	−0.376631
$392$	− 44.6685i	− 0.113950i
$393$	0	0
$394$	90.8157	0.230497
$395$	− 215.153i	− 0.544690i
$396$	0	0
$397$	−457.057	−1.15128	−0.575638	−	0.817704i	$-0.695246\pi$
$397$	−457.057	−1.15128	−0.575638	+	0.817704i	$0.695246\pi$
$398$	− 571.237i	− 1.43527i
$399$	0	0
$400$	−168.246	−0.420614
$401$	− 391.141i	− 0.975415i	−0.873007	−	0.487707i	$-0.837833\pi$
$401$	− 391.141i	− 0.975415i	0.873007	−	0.487707i	$-0.162167\pi$
$402$	0	0
$403$	−571.728	−1.41868
$404$	1587.81i	3.93022i
$405$	0	0
$406$	1176.18	2.89700
$407$	360.705i	0.886253i
$408$	0	0
$409$	−411.842	−1.00695	−0.503474	−	0.864010i	$-0.667945\pi$
$409$	−411.842	−1.00695	−0.503474	+	0.864010i	$0.667945\pi$
$410$	− 134.468i	− 0.327971i
$411$	0	0
$412$	1197.84	2.90737
$413$	180.145i	0.436186i
$414$	0	0
$415$	176.921	0.426316
$416$	442.339i	1.06331i
$417$	0	0
$418$	−125.298	−0.299757
$419$	653.447i	1.55954i	0.626066	+	0.779770i	$0.284664\pi$
$419$	653.447i	1.55954i	−0.626066	+	0.779770i	$0.715336\pi$
$420$	0	0
$421$	125.035	0.296995	0.148497	−	0.988913i	$-0.452556\pi$
$421$	125.035	0.296995	0.148497	+	0.988913i	$0.452556\pi$
$422$	867.998i	2.05687i
$423$	0	0
$424$	257.026	0.606194
$425$	61.1584i	0.143902i
$426$	0	0
$427$	−255.517	−0.598401
$428$	− 712.093i	− 1.66377i
$429$	0	0
$430$	356.061	0.828049
$431$	397.208i	0.921596i	0.887505	+	0.460798i	$0.152437\pi$
$431$	397.208i	0.921596i	−0.887505	+	0.460798i	$0.847563\pi$
$432$	0	0
$433$	560.114	1.29357	0.646783	−	0.762674i	$-0.276114\pi$
$433$	560.114	1.29357	0.646783	+	0.762674i	$0.276114\pi$
$434$	− 1523.48i	− 3.51032i
$435$	0	0
$436$	121.956	0.279715
$437$	− 76.1441i	− 0.174243i
$438$	0	0
$439$	−664.386	−1.51341	−0.756704	−	0.653758i	$-0.773191\pi$
$439$	−664.386	−1.51341	−0.756704	+	0.653758i	$0.773191\pi$
$440$	235.876i	0.536081i
$441$	0	0
$442$	438.070	0.991108
$443$	371.305i	0.838160i	0.907949	+	0.419080i	$0.137647\pi$
$443$	371.305i	0.838160i	−0.907949	+	0.419080i	$0.862353\pi$
$444$	0	0
$445$	240.250	0.539887
$446$	− 665.455i	− 1.49205i
$447$	0	0
$448$	−214.680	−0.479196
$449$	− 585.471i	− 1.30395i	−0.758243	−	0.651973i	$-0.773942\pi$
$449$	− 585.471i	− 1.30395i	0.758243	−	0.651973i	$-0.226058\pi$
$450$	0	0
$451$	−89.4121	−0.198253
$452$	193.177i	0.427382i
$453$	0	0
$454$	1482.30	3.26497
$455$	157.133i	0.345347i
$456$	0	0
$457$	168.641	0.369017	0.184508	−	0.982831i	$-0.440931\pi$
$457$	168.641	0.369017	0.184508	+	0.982831i	$0.440931\pi$
$458$	− 99.6462i	− 0.217568i
$459$	0	0
$460$	−251.026	−0.545709
$461$	298.492i	0.647487i	0.946145	+	0.323744i	$0.104942\pi$
$461$	298.492i	0.647487i	−0.946145	+	0.323744i	$0.895058\pi$
$462$	0	0
$463$	−595.285	−1.28571	−0.642856	−	0.765987i	$-0.722251\pi$
$463$	−595.285	−1.28571	−0.642856	+	0.765987i	$0.722251\pi$
$464$	− 1513.81i	− 3.26252i
$465$	0	0
$466$	−1300.89	−2.79162
$467$	623.655i	1.33545i	0.744408	+	0.667725i	$0.232732\pi$
$467$	623.655i	1.33545i	−0.744408	+	0.667725i	$0.767268\pi$
$468$	0	0
$469$	191.246	0.407773
$470$	− 326.956i	− 0.695651i
$471$	0	0
$472$	488.855	1.03571
$473$	− 236.756i	− 0.500542i
$474$	0	0
$475$	−31.6228	−0.0665743
$476$	816.894i	1.71616i
$477$	0	0
$478$	−991.745	−2.07478
$479$	− 131.857i	− 0.275276i	−0.990483	−	0.137638i	$-0.956049\pi$
$479$	− 131.857i	− 0.275276i	0.990483	−	0.137638i	$-0.0439510\pi$
$480$	0	0
$481$	652.070	1.35565
$482$	− 819.263i	− 1.69972i
$483$	0	0
$484$	−853.605	−1.76365
$485$	− 2.41272i	− 0.00497468i
$486$	0	0
$487$	41.8028	0.0858375	0.0429187	−	0.999079i	$-0.486334\pi$
$487$	41.8028	0.0858375	0.0429187	+	0.999079i	$0.486334\pi$
$488$	693.392i	1.42089i
$489$	0	0
$490$	−18.7587	−0.0382831
$491$	− 178.817i	− 0.364189i	−0.983281	−	0.182095i	$-0.941712\pi$
$491$	− 178.817i	− 0.364189i	0.983281	−	0.182095i	$-0.0582877\pi$
$492$	0	0
$493$	−550.280	−1.11619
$494$	226.510i	0.458522i
$495$	0	0
$496$	−1960.80	−3.95322
$497$	664.159i	1.33634i
$498$	0	0
$499$	−39.0961	−0.0783489	−0.0391744	−	0.999232i	$-0.512473\pi$
$499$	−39.0961	−0.0783489	−0.0391744	+	0.999232i	$0.512473\pi$
$500$	104.252i	0.208503i
$501$	0	0
$502$	−1163.62	−2.31796
$503$	578.698i	1.15049i	0.817980	+	0.575247i	$0.195094\pi$
$503$	578.698i	1.15049i	−0.817980	+	0.575247i	$0.804906\pi$
$504$	0	0
$505$	380.763	0.753986
$506$	238.518i	0.471379i
$507$	0	0
$508$	354.619	0.698068
$509$	355.743i	0.698905i	0.936954	+	0.349453i	$0.113632\pi$
$509$	355.743i	0.698905i	−0.936954	+	0.349453i	$0.886368\pi$
$510$	0	0
$511$	432.061	0.845521
$512$	− 1098.99i	− 2.14646i
$513$	0	0
$514$	1357.81	2.64166
$515$	− 287.246i	− 0.557760i
$516$	0	0
$517$	−217.404	−0.420510
$518$	1737.57i	3.35437i
$519$	0	0
$520$	426.408	0.820015
$521$	− 810.952i	− 1.55653i	−0.627936	−	0.778265i	$-0.716100\pi$
$521$	− 810.952i	− 1.55653i	0.627936	−	0.778265i	$-0.283900\pi$
$522$	0	0
$523$	720.483	1.37760	0.688798	−	0.724953i	$-0.258139\pi$
$523$	720.483	1.37760	0.688798	+	0.724953i	$0.258139\pi$
$524$	782.711i	1.49372i
$525$	0	0
$526$	870.772	1.65546
$527$	712.764i	1.35249i
$528$	0	0
$529$	384.052	0.725996
$530$	− 107.939i	− 0.203659i
$531$	0	0
$532$	−422.386	−0.793958
$533$	161.636i	0.303257i
$534$	0	0
$535$	−170.763	−0.319183
$536$	− 518.979i	− 0.968245i
$537$	0	0
$538$	459.465	0.854024
$539$	12.4733i	0.0231415i
$540$	0	0
$541$	347.149	0.641680	0.320840	−	0.947133i	$-0.396035\pi$
$541$	347.149	0.641680	0.320840	+	0.947133i	$0.396035\pi$
$542$	− 944.142i	− 1.74196i
$543$	0	0
$544$	551.456	1.01371
$545$	− 29.2455i	− 0.0536615i
$546$	0	0
$547$	−720.833	−1.31779	−0.658896	−	0.752234i	$-0.728976\pi$
$547$	−720.833	−1.31779	−0.658896	+	0.752234i	$0.728976\pi$
$548$	− 145.403i	− 0.265334i
$549$	0	0
$550$	99.0569	0.180104
$551$	− 284.530i	− 0.516388i
$552$	0	0
$553$	−689.149	−1.24620
$554$	831.222i	1.50040i
$555$	0	0
$556$	−632.596	−1.13776
$557$	− 429.102i	− 0.770380i	−0.922837	−	0.385190i	$-0.874136\pi$
$557$	− 429.102i	− 0.770380i	0.922837	−	0.385190i	$-0.125864\pi$
$558$	0	0
$559$	−428.000	−0.765653
$560$	538.902i	0.962325i
$561$	0	0
$562$	−881.118	−1.56783
$563$	− 670.820i	− 1.19151i	−0.803166	−	0.595755i	$-0.796853\pi$
$563$	− 670.820i	− 1.19151i	0.803166	−	0.595755i	$-0.203147\pi$
$564$	0	0
$565$	46.3246	0.0819904
$566$	760.474i	1.34359i
$567$	0	0
$568$	1802.31	3.17309
$569$	368.663i	0.647914i	0.946072	+	0.323957i	$0.105013\pi$
$569$	368.663i	0.647914i	−0.946072	+	0.323957i	$0.894987\pi$
$570$	0	0
$571$	124.289	0.217669	0.108834	−	0.994060i	$-0.465288\pi$
$571$	124.289	0.217669	0.108834	+	0.994060i	$0.465288\pi$
$572$	− 496.532i	− 0.868063i
$573$	0	0
$574$	−430.710	−0.750366
$575$	60.1972i	0.104691i
$576$	0	0
$577$	504.236	0.873893	0.436947	−	0.899487i	$-0.356060\pi$
$577$	504.236	0.873893	0.436947	+	0.899487i	$0.356060\pi$
$578$	508.797i	0.880272i
$579$	0	0
$580$	−938.017	−1.61727
$581$	− 566.690i	− 0.975370i
$582$	0	0
$583$	−71.7722	−0.123108
$584$	− 1172.48i	− 2.00766i
$585$	0	0
$586$	−736.236	−1.25638
$587$	39.2256i	0.0668238i	0.999442	+	0.0334119i	$0.0106373\pi$
$587$	39.2256i	0.0668238i	−0.999442	+	0.0334119i	$0.989363\pi$
$588$	0	0
$589$	−368.544	−0.625711
$590$	− 205.297i	− 0.347960i
$591$	0	0
$592$	2236.34	3.77760
$593$	− 621.670i	− 1.04835i	−0.851611	−	0.524174i	$-0.824374\pi$
$593$	− 621.670i	− 1.04835i	0.851611	−	0.524174i	$-0.175626\pi$
$594$	0	0
$595$	195.895	0.329235
$596$	− 2176.59i	− 3.65200i
$597$	0	0
$598$	431.184	0.721044
$599$	− 1119.77i	− 1.86940i	−0.355436	−	0.934701i	$-0.615668\pi$
$599$	− 1119.77i	− 1.86940i	0.355436	−	0.934701i	$-0.384332\pi$
$600$	0	0
$601$	−323.789	−0.538751	−0.269375	−	0.963035i	$-0.586817\pi$
$601$	−323.789	−0.538751	−0.269375	+	0.963035i	$0.586817\pi$
$602$	− 1140.49i	− 1.89450i
$603$	0	0
$604$	−1728.31	−2.86145
$605$	204.698i	0.338344i
$606$	0	0
$607$	1025.63	1.68966	0.844832	−	0.535031i	$-0.179700\pi$
$607$	1025.63	1.68966	0.844832	+	0.535031i	$0.179700\pi$
$608$	285.138i	0.468976i
$609$	0	0
$610$	291.193	0.477365
$611$	393.014i	0.643232i
$612$	0	0
$613$	904.153	1.47496	0.737482	−	0.675367i	$-0.236015\pi$
$613$	904.153	1.47496	0.737482	+	0.675367i	$0.236015\pi$
$614$	1251.40i	2.03812i
$615$	0	0
$616$	755.526	1.22650
$617$	710.716i	1.15189i	0.817488	+	0.575945i	$0.195366\pi$
$617$	710.716i	1.15189i	−0.817488	+	0.575945i	$0.804634\pi$
$618$	0	0
$619$	−583.737	−0.943032	−0.471516	−	0.881858i	$-0.656293\pi$
$619$	−583.737	−0.943032	−0.471516	+	0.881858i	$0.656293\pi$
$620$	1214.99i	1.95966i
$621$	0	0
$622$	−794.447	−1.27725
$623$	− 769.537i	− 1.23521i
$624$	0	0
$625$	25.0000	0.0400000
$626$	1028.99i	1.64376i
$627$	0	0
$628$	1037.60	1.65223
$629$	− 812.924i	− 1.29241i
$630$	0	0
$631$	−20.0968	−0.0318491	−0.0159246	−	0.999873i	$-0.505069\pi$
$631$	−20.0968	−0.0318491	−0.0159246	+	0.999873i	$0.505069\pi$
$632$	1870.13i	2.95906i
$633$	0	0
$634$	56.2192	0.0886738
$635$	− 85.0390i	− 0.133920i
$636$	0	0
$637$	22.5487	0.0353983
$638$	891.277i	1.39699i
$639$	0	0
$640$	−158.592	−0.247800
$641$	− 341.607i	− 0.532928i	−0.963845	−	0.266464i	$-0.914145\pi$
$641$	− 341.607i	− 0.532928i	0.963845	−	0.266464i	$-0.0858553\pi$
$642$	0	0
$643$	−469.693	−0.730471	−0.365235	−	0.930915i	$-0.619011\pi$
$643$	−469.693	−0.730471	−0.365235	+	0.930915i	$0.619011\pi$
$644$	804.054i	1.24853i
$645$	0	0
$646$	282.386	0.437130
$647$	572.099i	0.884234i	0.896957	+	0.442117i	$0.145772\pi$
$647$	572.099i	0.884234i	−0.896957	+	0.442117i	$0.854228\pi$
$648$	0	0
$649$	−136.508	−0.210336
$650$	− 179.072i	− 0.275495i
$651$	0	0
$652$	−1107.41	−1.69848
$653$	− 213.540i	− 0.327014i	−0.986542	−	0.163507i	$-0.947719\pi$
$653$	− 213.540i	− 0.327014i	0.986542	−	0.163507i	$-0.0522806\pi$
$654$	0	0
$655$	187.698	0.286561
$656$	554.347i	0.845040i
$657$	0	0
$658$	−1047.26	−1.59158
$659$	420.983i	0.638820i	0.947617	+	0.319410i	$0.103485\pi$
$659$	420.983i	0.638820i	−0.947617	+	0.319410i	$0.896515\pi$
$660$	0	0
$661$	434.272	0.656992	0.328496	−	0.944505i	$-0.393458\pi$
$661$	434.272	0.656992	0.328496	+	0.944505i	$0.393458\pi$
$662$	1370.74i	2.07061i
$663$	0	0
$664$	−1537.81	−2.31599
$665$	101.290i	0.152316i
$666$	0	0
$667$	−541.631	−0.812041
$668$	− 2062.54i	− 3.08763i
$669$	0	0
$670$	−217.947	−0.325295
$671$	− 193.623i	− 0.288560i
$672$	0	0
$673$	72.7801	0.108143	0.0540714	−	0.998537i	$-0.482780\pi$
$673$	72.7801	0.108143	0.0540714	+	0.998537i	$0.482780\pi$
$674$	686.669i	1.01880i
$675$	0	0
$676$	678.237	1.00331
$677$	− 172.106i	− 0.254219i	−0.991889	−	0.127109i	$-0.959430\pi$
$677$	− 172.106i	− 0.254219i	0.991889	−	0.127109i	$-0.0405700\pi$
$678$	0	0
$679$	−7.72811	−0.0113816
$680$	− 531.595i	− 0.781758i
$681$	0	0
$682$	1154.45	1.69274
$683$	792.592i	1.16046i	0.814454	+	0.580228i	$0.197037\pi$
$683$	792.592i	1.16046i	−0.814454	+	0.580228i	$0.802963\pi$
$684$	0	0
$685$	−34.8683	−0.0509027
$686$	− 1220.99i	− 1.77986i
$687$	0	0
$688$	−1467.87	−2.13353
$689$	129.747i	0.188313i
$690$	0	0
$691$	154.851	0.224097	0.112049	−	0.993703i	$-0.464259\pi$
$691$	154.851	0.224097	0.112049	+	0.993703i	$0.464259\pi$
$692$	− 1779.19i	− 2.57109i
$693$	0	0
$694$	1875.49	2.70244
$695$	151.699i	0.218272i
$696$	0	0
$697$	201.509	0.289109
$698$	410.785i	0.588518i
$699$	0	0
$700$	333.925	0.477036
$701$	950.544i	1.35598i	0.735070	+	0.677991i	$0.237149\pi$
$701$	950.544i	1.35598i	−0.735070	+	0.677991i	$0.762851\pi$
$702$	0	0
$703$	420.333	0.597913
$704$	− 162.678i	− 0.231076i
$705$	0	0
$706$	−1562.95	−2.21381
$707$	− 1219.61i	− 1.72505i
$708$	0	0
$709$	390.350	0.550564	0.275282	−	0.961363i	$-0.411229\pi$
$709$	390.350	0.550564	0.275282	+	0.961363i	$0.411229\pi$
$710$	− 756.889i	− 1.06604i
$711$	0	0
$712$	−2088.28	−2.93297
$713$	701.561i	0.983956i
$714$	0	0
$715$	−119.070	−0.166532
$716$	541.762i	0.756650i
$717$	0	0
$718$	−205.132	−0.285699
$719$	− 655.227i	− 0.911303i	−0.890158	−	0.455651i	$-0.849406\pi$
$719$	− 655.227i	− 0.911303i	0.890158	−	0.455651i	$-0.150594\pi$
$720$	0	0
$721$	−920.070	−1.27610
$722$	− 1171.74i	− 1.62291i
$723$	0	0
$724$	1519.17	2.09830
$725$	224.940i	0.310263i
$726$	0	0
$727$	1424.25	1.95908	0.979539	−	0.201254i	$-0.0645017\pi$
$727$	1424.25	1.95908	0.979539	+	0.201254i	$0.0645017\pi$
$728$	− 1365.81i	− 1.87612i
$729$	0	0
$730$	−492.386	−0.674501
$731$	533.580i	0.729932i
$732$	0	0
$733$	−946.749	−1.29161	−0.645805	−	0.763503i	$-0.723478\pi$
$733$	−946.749	−1.29161	−0.645805	+	0.763503i	$0.723478\pi$
$734$	− 563.471i	− 0.767672i
$735$	0	0
$736$	542.789	0.737485
$737$	144.920i	0.196635i
$738$	0	0
$739$	591.429	0.800310	0.400155	−	0.916447i	$-0.368956\pi$
$739$	591.429	0.800310	0.400155	+	0.916447i	$0.368956\pi$
$740$	− 1385.72i	− 1.87260i
$741$	0	0
$742$	−345.737	−0.465952
$743$	732.202i	0.985467i	0.870180	+	0.492734i	$0.164002\pi$
$743$	732.202i	0.985467i	−0.870180	+	0.492734i	$0.835998\pi$
$744$	0	0
$745$	−521.956	−0.700612
$746$	2034.25i	2.72687i
$747$	0	0
$748$	−619.018	−0.827564
$749$	546.965i	0.730261i
$750$	0	0
$751$	−215.359	−0.286764	−0.143382	−	0.989667i	$-0.545798\pi$
$751$	−215.359	−0.286764	−0.143382	+	0.989667i	$0.545798\pi$
$752$	1347.88i	1.79240i
$753$	0	0
$754$	1611.22	2.13689
$755$	414.457i	0.548950i
$756$	0	0
$757$	276.258	0.364938	0.182469	−	0.983212i	$-0.441591\pi$
$757$	276.258	0.364938	0.182469	+	0.983212i	$0.441591\pi$
$758$	538.064i	0.709848i
$759$	0	0
$760$	274.868	0.361669
$761$	893.373i	1.17395i	0.809606	+	0.586973i	$0.199681\pi$
$761$	893.373i	1.17395i	−0.809606	+	0.586973i	$0.800319\pi$
$762$	0	0
$763$	−93.6754	−0.122773
$764$	− 933.033i	− 1.22125i
$765$	0	0
$766$	2688.87	3.51027
$767$	246.775i	0.321740i
$768$	0	0
$769$	284.316	0.369722	0.184861	−	0.982765i	$-0.440817\pi$
$769$	284.316	0.369722	0.184861	+	0.982765i	$0.440817\pi$
$770$	− 317.286i	− 0.412060i
$771$	0	0
$772$	577.140	0.747591
$773$	− 1059.64i	− 1.37081i	−0.728162	−	0.685405i	$-0.759625\pi$
$773$	− 1059.64i	− 1.37081i	0.728162	−	0.685405i	$-0.240375\pi$
$774$	0	0
$775$	291.359	0.375948
$776$	20.9716i	0.0270253i
$777$	0	0
$778$	1081.97	1.39071
$779$	104.193i	0.133752i
$780$	0	0
$781$	−503.280	−0.644404
$782$	− 537.550i	− 0.687404i
$783$	0	0
$784$	77.3331	0.0986392
$785$	− 248.821i	− 0.316970i
$786$	0	0
$787$	−875.517	−1.11247	−0.556237	−	0.831024i	$-0.687755\pi$
$787$	−875.517	−1.11247	−0.556237	+	0.831024i	$0.687755\pi$
$788$	231.986i	0.294399i
$789$	0	0
$790$	785.368	0.994137
$791$	− 148.381i	− 0.187586i
$792$	0	0
$793$	−350.026	−0.441394
$794$	− 1668.39i	− 2.10124i
$795$	0	0
$796$	1459.21	1.83318
$797$	742.449i	0.931555i	0.884902	+	0.465777i	$0.154225\pi$
$797$	742.449i	0.931555i	−0.884902	+	0.465777i	$0.845775\pi$
$798$	0	0
$799$	489.964	0.613222
$800$	− 225.421i	− 0.281776i
$801$	0	0
$802$	1427.78	1.78027
$803$	327.403i	0.407725i
$804$	0	0
$805$	192.816	0.239523
$806$	− 2086.97i	− 2.58929i
$807$	0	0
$808$	−3309.63	−4.09608
$809$	113.720i	0.140568i	0.997527	+	0.0702842i	$0.0223906\pi$
$809$	113.720i	0.140568i	−0.997527	+	0.0702842i	$0.977609\pi$
$810$	0	0
$811$	−1466.03	−1.80769	−0.903844	−	0.427863i	$-0.859267\pi$
$811$	−1466.03	−1.80769	−0.903844	+	0.427863i	$0.859267\pi$
$812$	3004.53i	3.70016i
$813$	0	0
$814$	−1316.67	−1.61754
$815$	265.562i	0.325843i
$816$	0	0
$817$	−275.895	−0.337692
$818$	− 1503.34i	− 1.83782i
$819$	0	0
$820$	343.495	0.418897
$821$	550.073i	0.670003i	0.942218	+	0.335002i	$0.108737\pi$
$821$	550.073i	0.670003i	−0.942218	+	0.335002i	$0.891263\pi$
$822$	0	0
$823$	1392.51	1.69199	0.845997	−	0.533187i	$-0.179006\pi$
$823$	1392.51	1.69199	0.845997	+	0.533187i	$0.179006\pi$
$824$	2496.77i	3.03007i
$825$	0	0
$826$	−657.579	−0.796100
$827$	− 955.922i	− 1.15589i	−0.816075	−	0.577945i	$-0.803855\pi$
$827$	− 955.922i	− 1.15589i	0.816075	−	0.577945i	$-0.196145\pi$
$828$	0	0
$829$	−1652.69	−1.99360	−0.996798	−	0.0799552i	$-0.974522\pi$
$829$	−1652.69	−1.99360	−0.996798	+	0.0799552i	$0.974522\pi$
$830$	645.811i	0.778086i
$831$	0	0
$832$	−294.083	−0.353465
$833$	− 28.1111i	− 0.0337469i
$834$	0	0
$835$	−494.605	−0.592341
$836$	− 320.071i	− 0.382860i
$837$	0	0
$838$	−2385.27	−2.84638
$839$	− 1568.11i	− 1.86903i	−0.355927	−	0.934514i	$-0.615835\pi$
$839$	− 1568.11i	− 1.86903i	0.355927	−	0.934514i	$-0.384165\pi$
$840$	0	0
$841$	−1182.93	−1.40657
$842$	456.413i	0.542058i
$843$	0	0
$844$	−2217.28	−2.62711
$845$	− 162.644i	− 0.192478i
$846$	0	0
$847$	655.662	0.774099
$848$	444.981i	0.524742i
$849$	0	0
$850$	−223.246	−0.262642
$851$	− 800.147i	− 0.940243i
$852$	0	0
$853$	−651.232	−0.763461	−0.381730	−	0.924274i	$-0.624672\pi$
$853$	−651.232	−0.763461	−0.381730	+	0.924274i	$0.624672\pi$
$854$	− 932.711i	− 1.09217i
$855$	0	0
$856$	1484.29	1.73398
$857$	− 299.131i	− 0.349044i	−0.984653	−	0.174522i	$-0.944162\pi$
$857$	− 299.131i	− 0.349044i	0.984653	−	0.174522i	$-0.0558380\pi$
$858$	0	0
$859$	1095.72	1.27558	0.637788	−	0.770212i	$-0.279850\pi$
$859$	1095.72	1.27558	0.637788	+	0.770212i	$0.279850\pi$
$860$	909.550i	1.05762i
$861$	0	0
$862$	−1449.92	−1.68204
$863$	− 1221.95i	− 1.41593i	−0.706247	−	0.707965i	$-0.749613\pi$
$863$	− 1221.95i	− 1.41593i	0.706247	−	0.707965i	$-0.250387\pi$
$864$	0	0
$865$	−426.658	−0.493246
$866$	2044.57i	2.36094i
$867$	0	0
$868$	3891.69	4.48352
$869$	− 522.216i	− 0.600939i
$870$	0	0
$871$	261.982	0.300782
$872$	254.205i	0.291520i
$873$	0	0
$874$	277.947	0.318018
$875$	− 80.0767i	− 0.0915162i
$876$	0	0
$877$	766.399	0.873887	0.436944	−	0.899489i	$-0.356061\pi$
$877$	766.399	0.873887	0.436944	+	0.899489i	$0.356061\pi$
$878$	− 2425.20i	− 2.76218i
$879$	0	0
$880$	−408.364	−0.464050
$881$	310.097i	0.351983i	0.984392	+	0.175992i	$0.0563132\pi$
$881$	310.097i	0.351983i	−0.984392	+	0.175992i	$0.943687\pi$
$882$	0	0
$883$	−122.236	−0.138433	−0.0692165	−	0.997602i	$-0.522050\pi$
$883$	−122.236	−0.138433	−0.0692165	+	0.997602i	$0.522050\pi$
$884$	1119.04i	1.26588i
$885$	0	0
$886$	−1355.37	−1.52976
$887$	− 265.444i	− 0.299261i	−0.988742	−	0.149630i	$-0.952192\pi$
$887$	− 265.444i	− 0.299261i	0.988742	−	0.149630i	$-0.0478084\pi$
$888$	0	0
$889$	−272.386	−0.306396
$890$	876.980i	0.985370i
$891$	0	0
$892$	1699.89	1.90571
$893$	253.343i	0.283698i
$894$	0	0
$895$	129.917	0.145158
$896$	507.981i	0.566944i
$897$	0	0
$898$	2137.14	2.37988
$899$	2621.54i	2.91606i
$900$	0	0
$901$	161.754	0.179527
$902$	− 326.379i	− 0.361840i
$903$	0	0
$904$	−402.658	−0.445418
$905$	− 364.302i	− 0.402544i
$906$	0	0
$907$	672.622	0.741590	0.370795	−	0.928715i	$-0.379085\pi$
$907$	672.622	0.741590	0.370795	+	0.928715i	$0.379085\pi$
$908$	3786.49i	4.17015i
$909$	0	0
$910$	−573.579	−0.630306
$911$	− 1402.48i	− 1.53949i	−0.638350	−	0.769746i	$-0.720383\pi$
$911$	− 1402.48i	− 1.53949i	0.638350	−	0.769746i	$-0.279617\pi$
$912$	0	0
$913$	429.421	0.470340
$914$	615.586i	0.673507i
$915$	0	0
$916$	254.544	0.277886
$917$	− 601.208i	− 0.655625i
$918$	0	0
$919$	−338.255	−0.368068	−0.184034	−	0.982920i	$-0.558916\pi$
$919$	−338.255	−0.368068	−0.184034	+	0.982920i	$0.558916\pi$
$920$	− 523.240i	− 0.568739i
$921$	0	0
$922$	−1089.58	−1.18176
$923$	909.811i	0.985711i
$924$	0	0
$925$	−332.302	−0.359246
$926$	− 2172.96i	− 2.34661i
$927$	0	0
$928$	2028.25	2.18562
$929$	− 148.207i	− 0.159533i	−0.996814	−	0.0797667i	$-0.974582\pi$
$929$	− 148.207i	− 0.159533i	0.996814	−	0.0797667i	$-0.0254175\pi$
$930$	0	0
$931$	14.5352	0.0156125
$932$	− 3323.10i	− 3.56556i
$933$	0	0
$934$	−2276.52	−2.43738
$935$	148.443i	0.158763i
$936$	0	0
$937$	−1416.72	−1.51197	−0.755987	−	0.654587i	$-0.772843\pi$
$937$	−1416.72	−1.51197	−0.755987	+	0.654587i	$0.772843\pi$
$938$	698.100i	0.744243i
$939$	0	0
$940$	835.201	0.888512
$941$	1398.92i	1.48663i	0.668939	+	0.743317i	$0.266749\pi$
$941$	1398.92i	1.48663i	−0.668939	+	0.743317i	$0.733251\pi$
$942$	0	0
$943$	198.342	0.210330
$944$	846.338i	0.896544i
$945$	0	0
$946$	864.228	0.913560
$947$	− 1050.57i	− 1.10937i	−0.832060	−	0.554686i	$-0.812839\pi$
$947$	− 1050.57i	− 1.10937i	0.832060	−	0.554686i	$-0.187161\pi$
$948$	0	0
$949$	591.868	0.623675
$950$	− 115.432i	− 0.121507i
$951$	0	0
$952$	−1702.74	−1.78859
$953$	551.928i	0.579148i	0.957156	+	0.289574i	$0.0935136\pi$
$953$	551.928i	0.579148i	−0.957156	+	0.289574i	$0.906486\pi$
$954$	0	0
$955$	−223.745	−0.234288
$956$	− 2533.39i	− 2.64999i
$957$	0	0
$958$	481.315	0.502417
$959$	111.686i	0.116461i
$960$	0	0
$961$	2434.61	2.53342
$962$	2380.24i	2.47426i
$963$	0	0
$964$	2092.79	2.17094
$965$	− 138.401i	− 0.143420i
$966$	0	0
$967$	−357.093	−0.369279	−0.184639	−	0.982806i	$-0.559112\pi$
$967$	−357.093	−0.369279	−0.184639	+	0.982806i	$0.559112\pi$
$968$	− 1779.26i	− 1.83807i
$969$	0	0
$970$	8.80711	0.00907950
$971$	− 308.206i	− 0.317411i	−0.987326	−	0.158705i	$-0.949268\pi$
$971$	− 308.206i	− 0.317411i	0.987326	−	0.158705i	$-0.0507320\pi$
$972$	0	0
$973$	485.903	0.499387
$974$	152.592i	0.156665i
$975$	0	0
$976$	−1200.45	−1.22997
$977$	− 253.280i	− 0.259243i	−0.991564	−	0.129621i	$-0.958624\pi$
$977$	− 253.280i	− 0.259243i	0.991564	−	0.129621i	$-0.0413762\pi$
$978$	0	0
$979$	583.132	0.595640
$980$	− 47.9187i	− 0.0488966i
$981$	0	0
$982$	652.732	0.664697
$983$	− 1068.73i	− 1.08722i	−0.839339	−	0.543609i	$-0.817058\pi$
$983$	− 1068.73i	− 1.08722i	0.839339	−	0.543609i	$-0.182942\pi$
$984$	0	0
$985$	55.6313	0.0564785
$986$	− 2008.68i	− 2.03720i
$987$	0	0
$988$	−578.614	−0.585641
$989$	525.194i	0.531035i
$990$	0	0
$991$	280.631	0.283179	0.141590	−	0.989925i	$-0.454779\pi$
$991$	280.631	0.283179	0.141590	+	0.989925i	$0.454779\pi$
$992$	− 2627.14i	− 2.64833i
$993$	0	0
$994$	−2424.37	−2.43900
$995$	− 349.925i	− 0.351683i
$996$	0	0
$997$	−356.574	−0.357647	−0.178824	−	0.983881i	$-0.557229\pi$
$997$	−356.574	−0.357647	−0.178824	+	0.983881i	$0.557229\pi$
$998$	− 142.712i	− 0.142998i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.3.c.a.26.4	yes	4
3.2	odd	2	inner	45.3.c.a.26.1	✓	4
4.3	odd	2		720.3.l.a.161.3		4
5.2	odd	4		225.3.d.b.224.2		8
5.3	odd	4		225.3.d.b.224.7		8
5.4	even	2		225.3.c.c.26.1		4
8.3	odd	2		2880.3.l.c.1601.1		4
8.5	even	2		2880.3.l.g.1601.2		4
9.2	odd	6		405.3.i.d.296.1		8
9.4	even	3		405.3.i.d.26.1		8
9.5	odd	6		405.3.i.d.26.4		8
9.7	even	3		405.3.i.d.296.4		8
12.11	even	2		720.3.l.a.161.1		4
15.2	even	4		225.3.d.b.224.8		8
15.8	even	4		225.3.d.b.224.1		8
15.14	odd	2		225.3.c.c.26.4		4
20.3	even	4		3600.3.c.i.449.7		8
20.7	even	4		3600.3.c.i.449.1		8
20.19	odd	2		3600.3.l.v.1601.3		4
24.5	odd	2		2880.3.l.g.1601.4		4
24.11	even	2		2880.3.l.c.1601.3		4
60.23	odd	4		3600.3.c.i.449.8		8
60.47	odd	4		3600.3.c.i.449.2		8
60.59	even	2		3600.3.l.v.1601.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.1	✓	4	3.2	odd	2	inner
45.3.c.a.26.4	yes	4	1.1	even	1	trivial
225.3.c.c.26.1		4	5.4	even	2
225.3.c.c.26.4		4	15.14	odd	2
225.3.d.b.224.1		8	15.8	even	4
225.3.d.b.224.2		8	5.2	odd	4
225.3.d.b.224.7		8	5.3	odd	4
225.3.d.b.224.8		8	15.2	even	4
405.3.i.d.26.1		8	9.4	even	3
405.3.i.d.26.4		8	9.5	odd	6
405.3.i.d.296.1		8	9.2	odd	6
405.3.i.d.296.4		8	9.7	even	3
720.3.l.a.161.1		4	12.11	even	2
720.3.l.a.161.3		4	4.3	odd	2
2880.3.l.c.1601.1		4	8.3	odd	2
2880.3.l.c.1601.3		4	24.11	even	2
2880.3.l.g.1601.2		4	8.5	even	2
2880.3.l.g.1601.4		4	24.5	odd	2
3600.3.c.i.449.1		8	20.7	even	4
3600.3.c.i.449.2		8	60.47	odd	4
3600.3.c.i.449.7		8	20.3	even	4
3600.3.c.i.449.8		8	60.23	odd	4
3600.3.l.v.1601.3		4	20.19	odd	2
3600.3.l.v.1601.4		4	60.59	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	45.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.65028i q^{2} -9.32456 q^{4} +2.23607i q^{5} +7.16228 q^{7} -19.4361i q^{8} +O(q^{10})\)	\(q+3.65028i q^{2} -9.32456 q^{4} +2.23607i q^{5} +7.16228 q^{7} -19.4361i q^{8} -8.16228 q^{10} +5.42736i q^{11} +9.81139 q^{13} +26.1443i q^{14} +33.6491 q^{16} -12.2317i q^{17} +6.32456 q^{19} -20.8503i q^{20} -19.8114 q^{22} -12.0394i q^{23} -5.00000 q^{25} +35.8143i q^{26} -66.7851 q^{28} -44.9881i q^{29} -58.2719 q^{31} +45.0842i q^{32} +44.6491 q^{34} +16.0153i q^{35} +66.4605 q^{37} +23.0864i q^{38} +43.4605 q^{40} +16.4743i q^{41} -43.6228 q^{43} -50.6077i q^{44} +43.9473 q^{46} +40.0570i q^{47} +2.29822 q^{49} -18.2514i q^{50} -91.4868 q^{52} +13.2242i q^{53} -12.1359 q^{55} -139.207i q^{56} +164.219 q^{58} +25.1519i q^{59} -35.6754 q^{61} -212.709i q^{62} -29.9737 q^{64} +21.9389i q^{65} +26.7018 q^{67} +114.055i q^{68} -58.4605 q^{70} +92.7301i q^{71} +60.3246 q^{73} +242.600i q^{74} -58.9737 q^{76} +38.8723i q^{77} -96.2192 q^{79} +75.2417i q^{80} -60.1359 q^{82} -79.1215i q^{83} +27.3509 q^{85} -159.235i q^{86} +105.487 q^{88} -107.443i q^{89} +70.2719 q^{91} +112.262i q^{92} -146.219 q^{94} +14.1421i q^{95} -1.07900 q^{97} +8.38915i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} + 16 q^{7}+O(q^{10}) \) 4 * q - 12 * q^4 + 16 * q^7	\( 4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55} + 328 q^{58} - 168 q^{61} - 44 q^{64} + 208 q^{67} - 120 q^{70} + 216 q^{73} - 160 q^{76} - 56 q^{79} - 152 q^{82} + 160 q^{85} + 384 q^{88} + 104 q^{91} - 256 q^{94} - 232 q^{97}+O(q^{100}) \) 4 * q - 12 * q^4 + 16 * q^7 - 20 * q^10 - 24 * q^13 + 84 * q^16 - 16 * q^22 - 20 * q^25 - 128 * q^28 - 56 * q^31 + 128 * q^34 + 152 * q^37 + 60 * q^40 - 48 * q^43 + 24 * q^46 - 92 * q^49 - 328 * q^52 + 40 * q^55 + 328 * q^58 - 168 * q^61 - 44 * q^64 + 208 * q^67 - 120 * q^70 + 216 * q^73 - 160 * q^76 - 56 * q^79 - 152 * q^82 + 160 * q^85 + 384 * q^88 + 104 * q^91 - 256 * q^94 - 232 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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