LMFDB - Embedded newform 225.3.d.b.224.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(224,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("225.224");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.2
Root			$-0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.3.d.b.224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65028 q^{2} +9.32456 q^{4} +7.16228i q^{7} -19.4361 q^{8} +5.42736i q^{11} -9.81139i q^{13} -26.1443i q^{14} +33.6491 q^{16} +12.2317 q^{17} -6.32456 q^{19} -19.8114i q^{22} -12.0394 q^{23} +35.8143i q^{26} +66.7851i q^{28} +44.9881i q^{29} -58.2719 q^{31} -45.0842 q^{32} -44.6491 q^{34} +66.4605i q^{37} +23.0864 q^{38} +16.4743i q^{41} +43.6228i q^{43} +50.6077i q^{44} +43.9473 q^{46} -40.0570 q^{47} -2.29822 q^{49} -91.4868i q^{52} +13.2242 q^{53} -139.207i q^{56} -164.219i q^{58} -25.1519i q^{59} -35.6754 q^{61} +212.709 q^{62} +29.9737 q^{64} +26.7018i q^{67} +114.055 q^{68} +92.7301i q^{71} -60.3246i q^{73} -242.600i q^{74} -58.9737 q^{76} -38.8723 q^{77} +96.2192 q^{79} -60.1359i q^{82} -79.1215 q^{83} -159.235i q^{86} -105.487i q^{88} +107.443i q^{89} +70.2719 q^{91} -112.262 q^{92} +146.219 q^{94} -1.07900i q^{97} +8.38915 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4} + 168 q^{16} - 112 q^{31} - 256 q^{34} + 48 q^{46} + 184 q^{49} - 336 q^{61} + 88 q^{64} - 320 q^{76} + 112 q^{79} + 208 q^{91} + 512 q^{94}+O(q^{100}) $ 8 * q + 24 * q^4 + 168 * q^16 - 112 * q^31 - 256 * q^34 + 48 * q^46 + 184 * q^49 - 336 * q^61 + 88 * q^64 - 320 * q^76 + 112 * q^79 + 208 * q^91 + 512 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−3.65028	−1.82514	−0.912570	−	0.408920i	$-0.865906\pi$
$2$	−3.65028	−1.82514	−0.912570	+	0.408920i	$0.865906\pi$
$3$	0	0
$3$	0	0
$4$	9.32456	2.33114
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.16228i	1.02318i	0.859229	+	0.511591i	$0.170944\pi$
$7$	7.16228i	1.02318i	−0.859229	+	0.511591i	$0.829056\pi$
$8$	−19.4361	−2.42952
$9$	0	0
$10$	0	0
$11$	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
$13$	− 9.81139i	− 0.754722i	−0.926066	−	0.377361i	$-0.876832\pi$
$13$	− 9.81139i	− 0.754722i	0.926066	−	0.377361i	$-0.123168\pi$
$14$	− 26.1443i	− 1.86745i
$15$	0	0
$16$	33.6491	2.10307
$17$	12.2317	0.719511	0.359756	−	0.933047i	$-0.382860\pi$
$17$	12.2317	0.719511	0.359756	+	0.933047i	$0.382860\pi$
$18$	0	0
$19$	−6.32456	−0.332871	−0.166436	−	0.986052i	$-0.553226\pi$
$19$	−6.32456	−0.332871	−0.166436	+	0.986052i	$0.553226\pi$
$20$	0	0
$21$	0	0
$22$	− 19.8114i	− 0.900518i
$23$	−12.0394	−0.523454	−0.261727	−	0.965142i	$-0.584292\pi$
$23$	−12.0394	−0.523454	−0.261727	+	0.965142i	$0.584292\pi$
$24$	0	0
$25$	0	0
$26$	35.8143i	1.37747i
$27$	0	0
$28$	66.7851i	2.38518i
$29$	44.9881i	1.55131i	0.631155	+	0.775657i	$0.282581\pi$
$29$	44.9881i	1.55131i	−0.631155	+	0.775657i	$0.717419\pi$
$30$	0	0
$31$	−58.2719	−1.87974	−0.939869	−	0.341535i	$-0.889053\pi$
$31$	−58.2719	−1.87974	−0.939869	+	0.341535i	$0.889053\pi$
$32$	−45.0842	−1.40888
$33$	0	0
$34$	−44.6491	−1.31321
$35$	0	0
$36$	0	0
$37$	66.4605i	1.79623i	0.439761	+	0.898115i	$0.355063\pi$
$37$	66.4605i	1.79623i	−0.439761	+	0.898115i	$0.644937\pi$
$38$	23.0864	0.607537
$39$	0	0
$40$	0	0
$41$	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
$43$	43.6228i	1.01448i	0.861804	+	0.507242i	$0.169335\pi$
$43$	43.6228i	1.01448i	−0.861804	+	0.507242i	$0.830665\pi$
$44$	50.6077i	1.15018i
$45$	0	0
$46$	43.9473	0.955377
$47$	−40.0570	−0.852276	−0.426138	−	0.904658i	$-0.640126\pi$
$47$	−40.0570	−0.852276	−0.426138	+	0.904658i	$0.640126\pi$
$48$	0	0
$49$	−2.29822	−0.0469025
$50$	0	0
$51$	0	0
$52$	− 91.4868i	− 1.75936i
$53$	13.2242	0.249512	0.124756	−	0.992187i	$-0.460185\pi$
$53$	13.2242	0.249512	0.124756	+	0.992187i	$0.460185\pi$
$54$	0	0
$55$	0	0
$56$	− 139.207i	− 2.48584i
$57$	0	0
$58$	− 164.219i	− 2.83137i
$59$	− 25.1519i	− 0.426303i	−0.977019	−	0.213151i	$-0.931627\pi$
$59$	− 25.1519i	− 0.426303i	0.977019	−	0.213151i	$-0.0683728\pi$
$60$	0	0
$61$	−35.6754	−0.584843	−0.292422	−	0.956289i	$-0.594461\pi$
$61$	−35.6754	−0.584843	−0.292422	+	0.956289i	$0.594461\pi$
$62$	212.709	3.43079
$63$	0	0
$64$	29.9737	0.468339
$65$	0	0
$66$	0	0
$67$	26.7018i	0.398534i	0.979945	+	0.199267i	$0.0638561\pi$
$67$	26.7018i	0.398534i	−0.979945	+	0.199267i	$0.936144\pi$
$68$	114.055	1.67728
$69$	0	0
$70$	0	0
$71$	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
$73$	− 60.3246i	− 0.826364i	−0.910649	−	0.413182i	$-0.864417\pi$
$73$	− 60.3246i	− 0.826364i	0.910649	−	0.413182i	$-0.135583\pi$
$74$	− 242.600i	− 3.27837i
$75$	0	0
$76$	−58.9737	−0.775969
$77$	−38.8723	−0.504834
$78$	0	0
$79$	96.2192	1.21796	0.608982	−	0.793184i	$-0.291578\pi$
$79$	96.2192	1.21796	0.608982	+	0.793184i	$0.291578\pi$
$80$	0	0
$81$	0	0
$82$	− 60.1359i	− 0.733365i
$83$	−79.1215	−0.953271	−0.476635	−	0.879101i	$-0.658144\pi$
$83$	−79.1215	−0.953271	−0.476635	+	0.879101i	$0.658144\pi$
$84$	0	0
$85$	0	0
$86$	− 159.235i	− 1.85157i
$87$	0	0
$88$	− 105.487i	− 1.19871i
$89$	107.443i	1.20722i	0.797278	+	0.603612i	$0.206272\pi$
$89$	107.443i	1.20722i	−0.797278	+	0.603612i	$0.793728\pi$
$90$	0	0
$91$	70.2719	0.772219
$92$	−112.262	−1.22024
$93$	0	0
$94$	146.219	1.55552
$95$	0	0
$96$	0	0
$97$	− 1.07900i	− 0.0111237i	−0.999985	−	0.00556187i	$-0.998230\pi$
$97$	− 1.07900i	− 0.0111237i	0.999985	−	0.00556187i	$-0.00177041\pi$
$98$	8.38915	0.0856036
$99$	0	0
$100$	0	0
$101$	− 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
$103$	128.460i	1.24719i	0.781748	+	0.623595i	$0.214328\pi$
$103$	128.460i	1.24719i	−0.781748	+	0.623595i	$0.785672\pi$
$104$	190.695i	1.83361i
$105$	0	0
$106$	−48.2719	−0.455395
$107$	−76.3675	−0.713715	−0.356858	−	0.934159i	$-0.616152\pi$
$107$	−76.3675	−0.713715	−0.356858	+	0.934159i	$0.616152\pi$
$108$	0	0
$109$	13.0790	0.119991	0.0599954	−	0.998199i	$-0.480891\pi$
$109$	13.0790	0.119991	0.0599954	+	0.998199i	$0.480891\pi$
$110$	0	0
$111$	0	0
$112$	241.004i	2.15182i
$113$	−20.7170	−0.183336	−0.0916680	−	0.995790i	$-0.529220\pi$
$113$	−20.7170	−0.183336	−0.0916680	+	0.995790i	$0.529220\pi$
$114$	0	0
$115$	0	0
$116$	419.494i	3.61633i
$117$	0	0
$118$	91.8114i	0.778063i
$119$	87.6068i	0.736191i
$120$	0	0
$121$	91.5438	0.756560
$122$	130.225	1.06742
$123$	0	0
$124$	−543.359	−4.38193
$125$	0	0
$126$	0	0
$127$	− 38.0306i	− 0.299454i	−0.988727	−	0.149727i	$-0.952161\pi$
$127$	− 38.0306i	− 0.299454i	0.988727	−	0.149727i	$-0.0478394\pi$
$128$	70.9246	0.554098
$129$	0	0
$130$	0	0
$131$	− 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
$133$	− 45.2982i	− 0.340588i
$134$	− 97.4690i	− 0.727381i
$135$	0	0
$136$	−237.737	−1.74806
$137$	−15.5936	−0.113822	−0.0569109	−	0.998379i	$-0.518125\pi$
$137$	−15.5936	−0.113822	−0.0569109	+	0.998379i	$0.518125\pi$
$138$	0	0
$139$	−67.8420	−0.488072	−0.244036	−	0.969766i	$-0.578471\pi$
$139$	−67.8420	−0.488072	−0.244036	+	0.969766i	$0.578471\pi$
$140$	0	0
$141$	0	0
$142$	− 338.491i	− 2.38374i
$143$	53.2499	0.372377
$144$	0	0
$145$	0	0
$146$	220.202i	1.50823i
$147$	0	0
$148$	619.715i	4.18726i
$149$	− 233.426i	− 1.56662i	−0.621634	−	0.783308i	$-0.713531\pi$
$149$	− 233.426i	− 1.56662i	0.621634	−	0.783308i	$-0.286469\pi$
$150$	0	0
$151$	185.351	1.22749	0.613745	−	0.789505i	$-0.289662\pi$
$151$	185.351	1.22749	0.613745	+	0.789505i	$0.289662\pi$
$152$	122.925	0.808716
$153$	0	0
$154$	141.895	0.921394
$155$	0	0
$156$	0	0
$157$	− 111.276i	− 0.708765i	−0.935100	−	0.354383i	$-0.884691\pi$
$157$	− 111.276i	− 0.708765i	0.935100	−	0.354383i	$-0.115309\pi$
$158$	−351.227	−2.22296
$159$	0	0
$160$	0	0
$161$	− 86.2298i	− 0.535589i
$162$	0	0
$163$	− 118.763i	− 0.728607i	−0.931280	−	0.364304i	$-0.881307\pi$
$163$	− 118.763i	− 0.728607i	0.931280	−	0.364304i	$-0.118693\pi$
$164$	153.616i	0.936682i
$165$	0	0
$166$	288.816	1.73985
$167$	−221.194	−1.32452	−0.662258	−	0.749276i	$-0.730402\pi$
$167$	−221.194	−1.32452	−0.662258	+	0.749276i	$0.730402\pi$
$168$	0	0
$169$	72.7367	0.430394
$170$	0	0
$171$	0	0
$172$	406.763i	2.36490i
$173$	190.807	1.10293	0.551466	−	0.834198i	$-0.314069\pi$
$173$	190.807	1.10293	0.551466	+	0.834198i	$0.314069\pi$
$174$	0	0
$175$	0	0
$176$	182.626i	1.03765i
$177$	0	0
$178$	− 392.197i	− 2.20335i
$179$	58.1005i	0.324584i	0.986743	+	0.162292i	$0.0518886\pi$
$179$	58.1005i	0.324584i	−0.986743	+	0.162292i	$0.948111\pi$
$180$	0	0
$181$	−162.921	−0.900116	−0.450058	−	0.892999i	$-0.648597\pi$
$181$	−162.921	−0.900116	−0.450058	+	0.892999i	$0.648597\pi$
$182$	−256.512	−1.40941
$183$	0	0
$184$	234.000	1.27174
$185$	0	0
$186$	0	0
$187$	66.3858i	0.355004i
$188$	−373.513	−1.98677
$189$	0	0
$190$	0	0
$191$	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
$193$	61.8947i	0.320698i	0.987060	+	0.160349i	$0.0512619\pi$
$193$	61.8947i	0.320698i	−0.987060	+	0.160349i	$0.948738\pi$
$194$	3.93866i	0.0203024i
$195$	0	0
$196$	−21.4299	−0.109336
$197$	24.8791	0.126290	0.0631449	−	0.998004i	$-0.479887\pi$
$197$	24.8791	0.126290	0.0631449	+	0.998004i	$0.479887\pi$
$198$	0	0
$199$	156.491	0.786387	0.393194	−	0.919456i	$-0.371370\pi$
$199$	156.491	0.786387	0.393194	+	0.919456i	$0.371370\pi$
$200$	0	0
$201$	0	0
$202$	621.579i	3.07712i
$203$	−322.217	−1.58728
$204$	0	0
$205$	0	0
$206$	− 468.917i	− 2.27630i
$207$	0	0
$208$	− 330.144i	− 1.58723i
$209$	− 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	$0.309461\pi$
$212$	123.309	0.581648
$213$	0	0
$214$	278.763	1.30263
$215$	0	0
$216$	0	0
$217$	− 417.359i	− 1.92332i
$218$	−47.7420	−0.219000
$219$	0	0
$220$	0	0
$221$	− 120.010i	− 0.543031i
$222$	0	0
$223$	182.302i	0.817500i	0.912646	+	0.408750i	$0.134035\pi$
$223$	182.302i	0.817500i	−0.912646	+	0.408750i	$0.865965\pi$
$224$	− 322.906i	− 1.44154i
$225$	0	0
$226$	75.6228	0.334614
$227$	406.078	1.78889	0.894444	−	0.447180i	$-0.147572\pi$
$227$	406.078	1.78889	0.894444	+	0.447180i	$0.147572\pi$
$228$	0	0
$229$	27.2982	0.119206	0.0596031	−	0.998222i	$-0.481016\pi$
$229$	27.2982	0.119206	0.0596031	+	0.998222i	$0.481016\pi$
$230$	0	0
$231$	0	0
$232$	− 874.394i	− 3.76894i
$233$	356.382	1.52954	0.764768	−	0.644306i	$-0.222854\pi$
$233$	356.382	1.52954	0.764768	+	0.644306i	$0.222854\pi$
$234$	0	0
$235$	0	0
$236$	− 234.530i	− 0.993771i
$237$	0	0
$238$	− 319.789i	− 1.34365i
$239$	− 271.690i	− 1.13678i	−0.822760	−	0.568389i	$-0.807567\pi$
$239$	− 271.690i	− 1.13678i	0.822760	−	0.568389i	$-0.192433\pi$
$240$	0	0
$241$	−224.438	−0.931280	−0.465640	−	0.884974i	$-0.654176\pi$
$241$	−224.438	−0.931280	−0.465640	+	0.884974i	$0.654176\pi$
$242$	−334.161	−1.38083
$243$	0	0
$244$	−332.658	−1.36335
$245$	0	0
$246$	0	0
$247$	62.0527i	0.251225i
$248$	1132.58	4.56685
$249$	0	0
$250$	0	0
$251$	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
$253$	− 65.3423i	− 0.258270i
$254$	138.822i	0.546545i
$255$	0	0
$256$	−378.789	−1.47965
$257$	371.975	1.44738	0.723688	−	0.690128i	$-0.242446\pi$
$257$	371.975	1.44738	0.723688	+	0.690128i	$0.242446\pi$
$258$	0	0
$259$	−476.009	−1.83787
$260$	0	0
$261$	0	0
$262$	306.408i	1.16950i
$263$	−238.549	−0.907031	−0.453515	−	0.891248i	$-0.649830\pi$
$263$	−238.549	−0.907031	−0.453515	+	0.891248i	$0.649830\pi$
$264$	0	0
$265$	0	0
$266$	165.351i	0.621621i
$267$	0	0
$268$	248.982i	0.929038i
$269$	125.871i	0.467922i	0.972246	+	0.233961i	$0.0751688\pi$
$269$	125.871i	0.467922i	−0.972246	+	0.233961i	$0.924831\pi$
$270$	0	0
$271$	−258.649	−0.954425	−0.477212	−	0.878788i	$-0.658353\pi$
$271$	−258.649	−0.954425	−0.477212	+	0.878788i	$0.658353\pi$
$272$	411.585	1.51318
$273$	0	0
$274$	56.9210	0.207741
$275$	0	0
$276$	0	0
$277$	227.715i	0.822074i	0.911619	+	0.411037i	$0.134833\pi$
$277$	227.715i	0.822074i	−0.911619	+	0.411037i	$0.865167\pi$
$278$	247.642	0.890800
$279$	0	0
$280$	0	0
$281$	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
$283$	− 208.333i	− 0.736159i	−0.929794	−	0.368080i	$-0.880015\pi$
$283$	− 208.333i	− 0.736159i	0.929794	−	0.368080i	$-0.119985\pi$
$284$	864.667i	3.04460i
$285$	0	0
$286$	−194.377	−0.679641
$287$	−117.994	−0.411128
$288$	0	0
$289$	−139.386	−0.482304
$290$	0	0
$291$	0	0
$292$	− 562.500i	− 1.92637i
$293$	201.693	0.688372	0.344186	−	0.938901i	$-0.388155\pi$
$293$	201.693	0.688372	0.344186	+	0.938901i	$0.388155\pi$
$294$	0	0
$295$	0	0
$296$	− 1291.73i	− 4.36397i
$297$	0	0
$298$	852.070i	2.85929i
$299$	118.124i	0.395062i
$300$	0	0
$301$	−312.438	−1.03800
$302$	−676.583	−2.24034
$303$	0	0
$304$	−212.816	−0.700052
$305$	0	0
$306$	0	0
$307$	342.824i	1.11669i	0.829608	+	0.558346i	$0.188564\pi$
$307$	342.824i	1.11669i	−0.829608	+	0.558346i	$0.811436\pi$
$308$	−362.466	−1.17684
$309$	0	0
$310$	0	0
$311$	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
$313$	− 281.895i	− 0.900622i	−0.892872	−	0.450311i	$-0.851313\pi$
$313$	− 281.895i	− 0.900622i	0.892872	−	0.450311i	$-0.148687\pi$
$314$	406.189i	1.29360i
$315$	0	0
$316$	897.201	2.83925
$317$	15.4013	0.0485847	0.0242923	−	0.999705i	$-0.492267\pi$
$317$	15.4013	0.0485847	0.0242923	+	0.999705i	$0.492267\pi$
$318$	0	0
$319$	−244.167	−0.765412
$320$	0	0
$321$	0	0
$322$	314.763i	0.977525i
$323$	−77.3600	−0.239505
$324$	0	0
$325$	0	0
$326$	433.518i	1.32981i
$327$	0	0
$328$	− 320.197i	− 0.976211i
$329$	− 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	$0.308009\pi$
$332$	−737.773	−2.22221
$333$	0	0
$334$	807.421	2.41743
$335$	0	0
$336$	0	0
$337$	188.114i	0.558201i	0.960262	+	0.279101i	$0.0900363\pi$
$337$	188.114i	0.558201i	−0.960262	+	0.279101i	$0.909964\pi$
$338$	−265.509	−0.785530
$339$	0	0
$340$	0	0
$341$	− 316.262i	− 0.927456i
$342$	0	0
$343$	334.491i	0.975193i
$344$	− 847.858i	− 2.46470i
$345$	0	0
$346$	−696.500	−2.01300
$347$	513.793	1.48067	0.740336	−	0.672237i	$-0.234666\pi$
$347$	513.793	1.48067	0.740336	+	0.672237i	$0.234666\pi$
$348$	0	0
$349$	−112.535	−0.322451	−0.161225	−	0.986918i	$-0.551545\pi$
$349$	−112.535	−0.322451	−0.161225	+	0.986918i	$0.551545\pi$
$350$	0	0
$351$	0	0
$352$	− 244.688i	− 0.695137i
$353$	428.172	1.21295	0.606475	−	0.795102i	$-0.292583\pi$
$353$	428.172	1.21295	0.606475	+	0.795102i	$0.292583\pi$
$354$	0	0
$355$	0	0
$356$	1001.86i	2.81421i
$357$	0	0
$358$	− 212.083i	− 0.592411i
$359$	− 56.1961i	− 0.156535i	−0.996932	−	0.0782676i	$-0.975061\pi$
$359$	− 56.1961i	− 0.156535i	0.996932	−	0.0782676i	$-0.0249389\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	594.708	1.64284
$363$	0	0
$364$	655.254	1.80015
$365$	0	0
$366$	0	0
$367$	− 154.364i	− 0.420610i	−0.977636	−	0.210305i	$-0.932554\pi$
$367$	− 154.364i	− 0.420610i	0.977636	−	0.210305i	$-0.0674456\pi$
$368$	−405.116	−1.10086
$369$	0	0
$370$	0	0
$371$	94.7151i	0.255297i
$372$	0	0
$373$	− 557.285i	− 1.49406i	−0.664790	−	0.747030i	$-0.731479\pi$
$373$	− 557.285i	− 1.49406i	0.664790	−	0.747030i	$-0.268521\pi$
$374$	− 242.327i	− 0.647932i
$375$	0	0
$376$	778.552	2.07062
$377$	441.396	1.17081
$378$	0	0
$379$	−147.404	−0.388928	−0.194464	−	0.980910i	$-0.562297\pi$
$379$	−147.404	−0.388928	−0.194464	+	0.980910i	$0.562297\pi$
$380$	0	0
$381$	0	0
$382$	− 365.254i	− 0.956163i
$383$	−736.619	−1.92329	−0.961644	−	0.274302i	$-0.911553\pi$
$383$	−736.619	−1.92329	−0.961644	+	0.274302i	$0.911553\pi$
$384$	0	0
$385$	0	0
$386$	− 225.933i	− 0.585319i
$387$	0	0
$388$	− 10.0612i	− 0.0259310i
$389$	296.408i	0.761975i	0.924580	+	0.380987i	$0.124416\pi$
$389$	296.408i	0.761975i	−0.924580	+	0.380987i	$0.875584\pi$
$390$	0	0
$391$	−147.263	−0.376631
$392$	44.6685	0.113950
$393$	0	0
$394$	−90.8157	−0.230497
$395$	0	0
$396$	0	0
$397$	− 457.057i	− 1.15128i	−0.817704	−	0.575638i	$-0.804754\pi$
$397$	− 457.057i	− 1.15128i	0.817704	−	0.575638i	$-0.195246\pi$
$398$	−571.237	−1.43527
$399$	0	0
$400$	0	0
$401$	− 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
$403$	571.728i	1.41868i
$404$	− 1587.81i	− 3.93022i
$405$	0	0
$406$	1176.18	2.89700
$407$	−360.705	−0.886253
$408$	0	0
$409$	411.842	1.00695	0.503474	−	0.864010i	$-0.332055\pi$
$409$	411.842	1.00695	0.503474	+	0.864010i	$0.332055\pi$
$410$	0	0
$411$	0	0
$412$	1197.84i	2.90737i
$413$	180.145	0.436186
$414$	0	0
$415$	0	0
$416$	442.339i	1.06331i
$417$	0	0
$418$	125.298i	0.299757i
$419$	− 653.447i	− 1.55954i	−0.626066	−	0.779770i	$-0.715336\pi$
$419$	− 653.447i	− 1.55954i	0.626066	−	0.779770i	$-0.284664\pi$
$420$	0	0
$421$	125.035	0.296995	0.148497	−	0.988913i	$-0.452556\pi$
$421$	125.035	0.296995	0.148497	+	0.988913i	$0.452556\pi$
$422$	−867.998	−2.05687
$423$	0	0
$424$	−257.026	−0.606194
$425$	0	0
$426$	0	0
$427$	− 255.517i	− 0.598401i
$428$	−712.093	−1.66377
$429$	0	0
$430$	0	0
$431$	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
$433$	− 560.114i	− 1.29357i	−0.762674	−	0.646783i	$-0.776114\pi$
$433$	− 560.114i	− 1.29357i	0.762674	−	0.646783i	$-0.223886\pi$
$434$	1523.48i	3.51032i
$435$	0	0
$436$	121.956	0.279715
$437$	76.1441	0.174243
$438$	0	0
$439$	664.386	1.51341	0.756704	−	0.653758i	$-0.226809\pi$
$439$	664.386	1.51341	0.756704	+	0.653758i	$0.226809\pi$
$440$	0	0
$441$	0	0
$442$	438.070i	0.991108i
$443$	371.305	0.838160	0.419080	−	0.907949i	$-0.362353\pi$
$443$	371.305	0.838160	0.419080	+	0.907949i	$0.362353\pi$
$444$	0	0
$445$	0	0
$446$	− 665.455i	− 1.49205i
$447$	0	0
$448$	214.680i	0.479196i
$449$	585.471i	1.30395i	0.758243	+	0.651973i	$0.226058\pi$
$449$	585.471i	1.30395i	−0.758243	+	0.651973i	$0.773942\pi$
$450$	0	0
$451$	−89.4121	−0.198253
$452$	−193.177	−0.427382
$453$	0	0
$454$	−1482.30	−3.26497
$455$	0	0
$456$	0	0
$457$	168.641i	0.369017i	0.982831	+	0.184508i	$0.0590693\pi$
$457$	168.641i	0.369017i	−0.982831	+	0.184508i	$0.940931\pi$
$458$	−99.6462	−0.217568
$459$	0	0
$460$	0	0
$461$	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
$463$	595.285i	1.28571i	0.765987	+	0.642856i	$0.222251\pi$
$463$	595.285i	1.28571i	−0.765987	+	0.642856i	$0.777749\pi$
$464$	1513.81i	3.26252i
$465$	0	0
$466$	−1300.89	−2.79162
$467$	−623.655	−1.33545	−0.667725	−	0.744408i	$-0.732732\pi$
$467$	−623.655	−1.33545	−0.667725	+	0.744408i	$0.732732\pi$
$468$	0	0
$469$	−191.246	−0.407773
$470$	0	0
$471$	0	0
$472$	488.855i	1.03571i
$473$	−236.756	−0.500542
$474$	0	0
$475$	0	0
$476$	816.894i	1.71616i
$477$	0	0
$478$	991.745i	2.07478i
$479$	131.857i	0.275276i	0.990483	+	0.137638i	$0.0439510\pi$
$479$	131.857i	0.275276i	−0.990483	+	0.137638i	$0.956049\pi$
$480$	0	0
$481$	652.070	1.35565
$482$	819.263	1.69972
$483$	0	0
$484$	853.605	1.76365
$485$	0	0
$486$	0	0
$487$	41.8028i	0.0858375i	0.999079	+	0.0429187i	$0.0136657\pi$
$487$	41.8028i	0.0858375i	−0.999079	+	0.0429187i	$0.986334\pi$
$488$	693.392	1.42089
$489$	0	0
$490$	0	0
$491$	− 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
$493$	550.280i	1.11619i
$494$	− 226.510i	− 0.458522i
$495$	0	0
$496$	−1960.80	−3.95322
$497$	−664.159	−1.33634
$498$	0	0
$499$	39.0961	0.0783489	0.0391744	−	0.999232i	$-0.487527\pi$
$499$	39.0961	0.0783489	0.0391744	+	0.999232i	$0.487527\pi$
$500$	0	0
$501$	0	0
$502$	− 1163.62i	− 2.31796i
$503$	578.698	1.15049	0.575247	−	0.817980i	$-0.304906\pi$
$503$	578.698	1.15049	0.575247	+	0.817980i	$0.304906\pi$
$504$	0	0
$505$	0	0
$506$	238.518i	0.471379i
$507$	0	0
$508$	− 354.619i	− 0.698068i
$509$	− 355.743i	− 0.698905i	−0.936954	−	0.349453i	$-0.886368\pi$
$509$	− 355.743i	− 0.698905i	0.936954	−	0.349453i	$-0.113632\pi$
$510$	0	0
$511$	432.061	0.845521
$512$	1098.99	2.14646
$513$	0	0
$514$	−1357.81	−2.64166
$515$	0	0
$516$	0	0
$517$	− 217.404i	− 0.420510i
$518$	1737.57	3.35437
$519$	0	0
$520$	0	0
$521$	− 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
$523$	− 720.483i	− 1.37760i	−0.724953	−	0.688798i	$-0.758139\pi$
$523$	− 720.483i	− 1.37760i	0.724953	−	0.688798i	$-0.241861\pi$
$524$	− 782.711i	− 1.49372i
$525$	0	0
$526$	870.772	1.65546
$527$	−712.764	−1.35249
$528$	0	0
$529$	−384.052	−0.725996
$530$	0	0
$531$	0	0
$532$	− 422.386i	− 0.793958i
$533$	161.636	0.303257
$534$	0	0
$535$	0	0
$536$	− 518.979i	− 0.968245i
$537$	0	0
$538$	− 459.465i	− 0.854024i
$539$	− 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	$0.396035\pi$
$542$	944.142	1.74196
$543$	0	0
$544$	−551.456	−1.01371
$545$	0	0
$546$	0	0
$547$	− 720.833i	− 1.31779i	−0.752234	−	0.658896i	$-0.771024\pi$
$547$	− 720.833i	− 1.31779i	0.752234	−	0.658896i	$-0.228976\pi$
$548$	−145.403	−0.265334
$549$	0	0
$550$	0	0
$551$	− 284.530i	− 0.516388i
$552$	0	0
$553$	689.149i	1.24620i
$554$	− 831.222i	− 1.50040i
$555$	0	0
$556$	−632.596	−1.13776
$557$	429.102	0.770380	0.385190	−	0.922837i	$-0.374136\pi$
$557$	429.102	0.770380	0.385190	+	0.922837i	$0.374136\pi$
$558$	0	0
$559$	428.000	0.765653
$560$	0	0
$561$	0	0
$562$	− 881.118i	− 1.56783i
$563$	−670.820	−1.19151	−0.595755	−	0.803166i	$-0.703147\pi$
$563$	−670.820	−1.19151	−0.595755	+	0.803166i	$0.703147\pi$
$564$	0	0
$565$	0	0
$566$	760.474i	1.34359i
$567$	0	0
$568$	− 1802.31i	− 3.17309i
$569$	− 368.663i	− 0.647914i	−0.946072	−	0.323957i	$-0.894987\pi$
$569$	− 368.663i	− 0.647914i	0.946072	−	0.323957i	$-0.105013\pi$
$570$	0	0
$571$	124.289	0.217669	0.108834	−	0.994060i	$-0.465288\pi$
$571$	124.289	0.217669	0.108834	+	0.994060i	$0.465288\pi$
$572$	496.532	0.868063
$573$	0	0
$574$	430.710	0.750366
$575$	0	0
$576$	0	0
$577$	504.236i	0.873893i	0.899487	+	0.436947i	$0.143940\pi$
$577$	504.236i	0.873893i	−0.899487	+	0.436947i	$0.856060\pi$
$578$	508.797	0.880272
$579$	0	0
$580$	0	0
$581$	− 566.690i	− 0.975370i
$582$	0	0
$583$	71.7722i	0.123108i
$584$	1172.48i	2.00766i
$585$	0	0
$586$	−736.236	−1.25638
$587$	−39.2256	−0.0668238	−0.0334119	−	0.999442i	$-0.510637\pi$
$587$	−39.2256	−0.0668238	−0.0334119	+	0.999442i	$0.510637\pi$
$588$	0	0
$589$	368.544	0.625711
$590$	0	0
$591$	0	0
$592$	2236.34i	3.77760i
$593$	−621.670	−1.04835	−0.524174	−	0.851611i	$-0.675626\pi$
$593$	−621.670	−1.04835	−0.524174	+	0.851611i	$0.675626\pi$
$594$	0	0
$595$	0	0
$596$	− 2176.59i	− 3.65200i
$597$	0	0
$598$	− 431.184i	− 0.721044i
$599$	1119.77i	1.86940i	0.355436	+	0.934701i	$0.384332\pi$
$599$	1119.77i	1.86940i	−0.355436	+	0.934701i	$0.615668\pi$
$600$	0	0
$601$	−323.789	−0.538751	−0.269375	−	0.963035i	$-0.586817\pi$
$601$	−323.789	−0.538751	−0.269375	+	0.963035i	$0.586817\pi$
$602$	1140.49	1.89450
$603$	0	0
$604$	1728.31	2.86145
$605$	0	0
$606$	0	0
$607$	1025.63i	1.68966i	0.535031	+	0.844832i	$0.320300\pi$
$607$	1025.63i	1.68966i	−0.535031	+	0.844832i	$0.679700\pi$
$608$	285.138	0.468976
$609$	0	0
$610$	0	0
$611$	393.014i	0.643232i
$612$	0	0
$613$	− 904.153i	− 1.47496i	−0.675367	−	0.737482i	$-0.736015\pi$
$613$	− 904.153i	− 1.47496i	0.675367	−	0.737482i	$-0.263985\pi$
$614$	− 1251.40i	− 2.03812i
$615$	0	0
$616$	755.526	1.22650
$617$	−710.716	−1.15189	−0.575945	−	0.817488i	$-0.695366\pi$
$617$	−710.716	−1.15189	−0.575945	+	0.817488i	$0.695366\pi$
$618$	0	0
$619$	583.737	0.943032	0.471516	−	0.881858i	$-0.343707\pi$
$619$	583.737	0.943032	0.471516	+	0.881858i	$0.343707\pi$
$620$	0	0
$621$	0	0
$622$	− 794.447i	− 1.27725i
$623$	−769.537	−1.23521
$624$	0	0
$625$	0	0
$626$	1028.99i	1.64376i
$627$	0	0
$628$	− 1037.60i	− 1.65223i
$629$	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	$0.505069\pi$
$632$	−1870.13	−2.95906
$633$	0	0
$634$	−56.2192	−0.0886738
$635$	0	0
$636$	0	0
$637$	22.5487i	0.0353983i
$638$	891.277	1.39699
$639$	0	0
$640$	0	0
$641$	− 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
$643$	469.693i	0.730471i	0.930915	+	0.365235i	$0.119011\pi$
$643$	469.693i	0.730471i	−0.930915	+	0.365235i	$0.880989\pi$
$644$	− 804.054i	− 1.24853i
$645$	0	0
$646$	282.386	0.437130
$647$	−572.099	−0.884234	−0.442117	−	0.896957i	$-0.645772\pi$
$647$	−572.099	−0.884234	−0.442117	+	0.896957i	$0.645772\pi$
$648$	0	0
$649$	136.508	0.210336
$650$	0	0
$651$	0	0
$652$	− 1107.41i	− 1.69848i
$653$	−213.540	−0.327014	−0.163507	−	0.986542i	$-0.552281\pi$
$653$	−213.540	−0.327014	−0.163507	+	0.986542i	$0.552281\pi$
$654$	0	0
$655$	0	0
$656$	554.347i	0.845040i
$657$	0	0
$658$	1047.26i	1.59158i
$659$	− 420.983i	− 0.638820i	−0.947617	−	0.319410i	$-0.896515\pi$
$659$	− 420.983i	− 0.638820i	0.947617	−	0.319410i	$-0.103485\pi$
$660$	0	0
$661$	434.272	0.656992	0.328496	−	0.944505i	$-0.393458\pi$
$661$	434.272	0.656992	0.328496	+	0.944505i	$0.393458\pi$
$662$	−1370.74	−2.07061
$663$	0	0
$664$	1537.81	2.31599
$665$	0	0
$666$	0	0
$667$	− 541.631i	− 0.812041i
$668$	−2062.54	−3.08763
$669$	0	0
$670$	0	0
$671$	− 193.623i	− 0.288560i
$672$	0	0
$673$	− 72.7801i	− 0.108143i	−0.998537	−	0.0540714i	$-0.982780\pi$
$673$	− 72.7801i	− 0.108143i	0.998537	−	0.0540714i	$-0.0172199\pi$
$674$	− 686.669i	− 1.01880i
$675$	0	0
$676$	678.237	1.00331
$677$	172.106	0.254219	0.127109	−	0.991889i	$-0.459430\pi$
$677$	172.106	0.254219	0.127109	+	0.991889i	$0.459430\pi$
$678$	0	0
$679$	7.72811	0.0113816
$680$	0	0
$681$	0	0
$682$	1154.45i	1.69274i
$683$	792.592	1.16046	0.580228	−	0.814454i	$-0.302963\pi$
$683$	792.592	1.16046	0.580228	+	0.814454i	$0.302963\pi$
$684$	0	0
$685$	0	0
$686$	− 1220.99i	− 1.77986i
$687$	0	0
$688$	1467.87i	2.13353i
$689$	− 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	$0.464259\pi$
$692$	1779.19	2.57109
$693$	0	0
$694$	−1875.49	−2.70244
$695$	0	0
$696$	0	0
$697$	201.509i	0.289109i
$698$	410.785	0.588518
$699$	0	0
$700$	0	0
$701$	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
$703$	− 420.333i	− 0.597913i
$704$	162.678i	0.231076i
$705$	0	0
$706$	−1562.95	−2.21381
$707$	1219.61	1.72505
$708$	0	0
$709$	−390.350	−0.550564	−0.275282	−	0.961363i	$-0.588771\pi$
$709$	−390.350	−0.550564	−0.275282	+	0.961363i	$0.588771\pi$
$710$	0	0
$711$	0	0
$712$	− 2088.28i	− 2.93297i
$713$	701.561	0.983956
$714$	0	0
$715$	0	0
$716$	541.762i	0.756650i
$717$	0	0
$718$	205.132i	0.285699i
$719$	655.227i	0.911303i	0.890158	+	0.455651i	$0.150594\pi$
$719$	655.227i	0.911303i	−0.890158	+	0.455651i	$0.849406\pi$
$720$	0	0
$721$	−920.070	−1.27610
$722$	1171.74	1.62291
$723$	0	0
$724$	−1519.17	−2.09830
$725$	0	0
$726$	0	0
$727$	1424.25i	1.95908i	0.201254	+	0.979539i	$0.435498\pi$
$727$	1424.25i	1.95908i	−0.201254	+	0.979539i	$0.564502\pi$
$728$	−1365.81	−1.87612
$729$	0	0
$730$	0	0
$731$	533.580i	0.729932i
$732$	0	0
$733$	946.749i	1.29161i	0.763503	+	0.645805i	$0.223478\pi$
$733$	946.749i	1.29161i	−0.763503	+	0.645805i	$0.776522\pi$
$734$	563.471i	0.767672i
$735$	0	0
$736$	542.789	0.737485
$737$	−144.920	−0.196635
$738$	0	0
$739$	−591.429	−0.800310	−0.400155	−	0.916447i	$-0.631044\pi$
$739$	−591.429	−0.800310	−0.400155	+	0.916447i	$0.631044\pi$
$740$	0	0
$741$	0	0
$742$	− 345.737i	− 0.465952i
$743$	732.202	0.985467	0.492734	−	0.870180i	$-0.335998\pi$
$743$	732.202	0.985467	0.492734	+	0.870180i	$0.335998\pi$
$744$	0	0
$745$	0	0
$746$	2034.25i	2.72687i
$747$	0	0
$748$	619.018i	0.827564i
$749$	− 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	$0.545798\pi$
$752$	−1347.88	−1.79240
$753$	0	0
$754$	−1611.22	−2.13689
$755$	0	0
$756$	0	0
$757$	276.258i	0.364938i	0.983212	+	0.182469i	$0.0584090\pi$
$757$	276.258i	0.364938i	−0.983212	+	0.182469i	$0.941591\pi$
$758$	538.064	0.709848
$759$	0	0
$760$	0	0
$761$	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
$763$	93.6754i	0.122773i
$764$	933.033i	1.22125i
$765$	0	0
$766$	2688.87	3.51027
$767$	−246.775	−0.321740
$768$	0	0
$769$	−284.316	−0.369722	−0.184861	−	0.982765i	$-0.559183\pi$
$769$	−284.316	−0.369722	−0.184861	+	0.982765i	$0.559183\pi$
$770$	0	0
$771$	0	0
$772$	577.140i	0.747591i
$773$	−1059.64	−1.37081	−0.685405	−	0.728162i	$-0.740375\pi$
$773$	−1059.64	−1.37081	−0.685405	+	0.728162i	$0.740375\pi$
$774$	0	0
$775$	0	0
$776$	20.9716i	0.0270253i
$777$	0	0
$778$	− 1081.97i	− 1.39071i
$779$	− 104.193i	− 0.133752i
$780$	0	0
$781$	−503.280	−0.644404
$782$	537.550	0.687404
$783$	0	0
$784$	−77.3331	−0.0986392
$785$	0	0
$786$	0	0
$787$	− 875.517i	− 1.11247i	−0.831024	−	0.556237i	$-0.812245\pi$
$787$	− 875.517i	− 1.11247i	0.831024	−	0.556237i	$-0.187755\pi$
$788$	231.986	0.294399
$789$	0	0
$790$	0	0
$791$	− 148.381i	− 0.187586i
$792$	0	0
$793$	350.026i	0.441394i
$794$	1668.39i	2.10124i
$795$	0	0
$796$	1459.21	1.83318
$797$	−742.449	−0.931555	−0.465777	−	0.884902i	$-0.654225\pi$
$797$	−742.449	−0.931555	−0.465777	+	0.884902i	$0.654225\pi$
$798$	0	0
$799$	−489.964	−0.613222
$800$	0	0
$801$	0	0
$802$	1427.78i	1.78027i
$803$	327.403	0.407725
$804$	0	0
$805$	0	0
$806$	− 2086.97i	− 2.58929i
$807$	0	0
$808$	3309.63i	4.09608i
$809$	− 113.720i	− 0.140568i	−0.997527	−	0.0702842i	$-0.977609\pi$
$809$	− 113.720i	− 0.140568i	0.997527	−	0.0702842i	$-0.0223906\pi$
$810$	0	0
$811$	−1466.03	−1.80769	−0.903844	−	0.427863i	$-0.859267\pi$
$811$	−1466.03	−1.80769	−0.903844	+	0.427863i	$0.859267\pi$
$812$	−3004.53	−3.70016
$813$	0	0
$814$	1316.67	1.61754
$815$	0	0
$816$	0	0
$817$	− 275.895i	− 0.337692i
$818$	−1503.34	−1.83782
$819$	0	0
$820$	0	0
$821$	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
$823$	− 1392.51i	− 1.69199i	−0.533187	−	0.845997i	$-0.679006\pi$
$823$	− 1392.51i	− 1.69199i	0.533187	−	0.845997i	$-0.320994\pi$
$824$	− 2496.77i	− 3.03007i
$825$	0	0
$826$	−657.579	−0.796100
$827$	955.922	1.15589	0.577945	−	0.816075i	$-0.303855\pi$
$827$	955.922	1.15589	0.577945	+	0.816075i	$0.303855\pi$
$828$	0	0
$829$	1652.69	1.99360	0.996798	−	0.0799552i	$-0.0254777\pi$
$829$	1652.69	1.99360	0.996798	+	0.0799552i	$0.0254777\pi$
$830$	0	0
$831$	0	0
$832$	− 294.083i	− 0.353465i
$833$	−28.1111	−0.0337469
$834$	0	0
$835$	0	0
$836$	− 320.071i	− 0.382860i
$837$	0	0
$838$	2385.27i	2.84638i
$839$	1568.11i	1.86903i	0.355927	+	0.934514i	$0.384165\pi$
$839$	1568.11i	1.86903i	−0.355927	+	0.934514i	$0.615835\pi$
$840$	0	0
$841$	−1182.93	−1.40657
$842$	−456.413	−0.542058
$843$	0	0
$844$	2217.28	2.62711
$845$	0	0
$846$	0	0
$847$	655.662i	0.774099i
$848$	444.981	0.524742
$849$	0	0
$850$	0	0
$851$	− 800.147i	− 0.940243i
$852$	0	0
$853$	651.232i	0.763461i	0.924274	+	0.381730i	$0.124672\pi$
$853$	651.232i	0.763461i	−0.924274	+	0.381730i	$0.875328\pi$
$854$	932.711i	1.09217i
$855$	0	0
$856$	1484.29	1.73398
$857$	299.131	0.349044	0.174522	−	0.984653i	$-0.444162\pi$
$857$	299.131	0.349044	0.174522	+	0.984653i	$0.444162\pi$
$858$	0	0
$859$	−1095.72	−1.27558	−0.637788	−	0.770212i	$-0.720150\pi$
$859$	−1095.72	−1.27558	−0.637788	+	0.770212i	$0.720150\pi$
$860$	0	0
$861$	0	0
$862$	− 1449.92i	− 1.68204i
$863$	−1221.95	−1.41593	−0.707965	−	0.706247i	$-0.750387\pi$
$863$	−1221.95	−1.41593	−0.707965	+	0.706247i	$0.750387\pi$
$864$	0	0
$865$	0	0
$866$	2044.57i	2.36094i
$867$	0	0
$868$	− 3891.69i	− 4.48352i
$869$	522.216i	0.600939i
$870$	0	0
$871$	261.982	0.300782
$872$	−254.205	−0.291520
$873$	0	0
$874$	−277.947	−0.318018
$875$	0	0
$876$	0	0
$877$	766.399i	0.873887i	0.899489	+	0.436944i	$0.143939\pi$
$877$	766.399i	0.873887i	−0.899489	+	0.436944i	$0.856061\pi$
$878$	−2425.20	−2.76218
$879$	0	0
$880$	0	0
$881$	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
$883$	122.236i	0.138433i	0.997602	+	0.0692165i	$0.0220499\pi$
$883$	122.236i	0.138433i	−0.997602	+	0.0692165i	$0.977950\pi$
$884$	− 1119.04i	− 1.26588i
$885$	0	0
$886$	−1355.37	−1.52976
$887$	265.444	0.299261	0.149630	−	0.988742i	$-0.452192\pi$
$887$	265.444	0.299261	0.149630	+	0.988742i	$0.452192\pi$
$888$	0	0
$889$	272.386	0.306396
$890$	0	0
$891$	0	0
$892$	1699.89i	1.90571i
$893$	253.343	0.283698
$894$	0	0
$895$	0	0
$896$	507.981i	0.566944i
$897$	0	0
$898$	− 2137.14i	− 2.37988i
$899$	− 2621.54i	− 2.91606i
$900$	0	0
$901$	161.754	0.179527
$902$	326.379	0.361840
$903$	0	0
$904$	402.658	0.445418
$905$	0	0
$906$	0	0
$907$	672.622i	0.741590i	0.928715	+	0.370795i	$0.120915\pi$
$907$	672.622i	0.741590i	−0.928715	+	0.370795i	$0.879085\pi$
$908$	3786.49	4.17015
$909$	0	0
$910$	0	0
$911$	− 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
$913$	− 429.421i	− 0.470340i
$914$	− 615.586i	− 0.673507i
$915$	0	0
$916$	254.544	0.277886
$917$	601.208	0.655625
$918$	0	0
$919$	338.255	0.368068	0.184034	−	0.982920i	$-0.441084\pi$
$919$	338.255	0.368068	0.184034	+	0.982920i	$0.441084\pi$
$920$	0	0
$921$	0	0
$922$	− 1089.58i	− 1.18176i
$923$	909.811	0.985711
$924$	0	0
$925$	0	0
$926$	− 2172.96i	− 2.34661i
$927$	0	0
$928$	− 2028.25i	− 2.18562i
$929$	148.207i	0.159533i	0.996814	+	0.0797667i	$0.0254175\pi$
$929$	148.207i	0.159533i	−0.996814	+	0.0797667i	$0.974582\pi$
$930$	0	0
$931$	14.5352	0.0156125
$932$	3323.10	3.56556
$933$	0	0
$934$	2276.52	2.43738
$935$	0	0
$936$	0	0
$937$	− 1416.72i	− 1.51197i	−0.654587	−	0.755987i	$-0.727157\pi$
$937$	− 1416.72i	− 1.51197i	0.654587	−	0.755987i	$-0.272843\pi$
$938$	698.100	0.744243
$939$	0	0
$940$	0	0
$941$	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
$943$	− 198.342i	− 0.210330i
$944$	− 846.338i	− 0.896544i
$945$	0	0
$946$	864.228	0.913560
$947$	1050.57	1.10937	0.554686	−	0.832060i	$-0.312839\pi$
$947$	1050.57	1.10937	0.554686	+	0.832060i	$0.312839\pi$
$948$	0	0
$949$	−591.868	−0.623675
$950$	0	0
$951$	0	0
$952$	− 1702.74i	− 1.78859i
$953$	551.928	0.579148	0.289574	−	0.957156i	$-0.406486\pi$
$953$	551.928	0.579148	0.289574	+	0.957156i	$0.406486\pi$
$954$	0	0
$955$	0	0
$956$	− 2533.39i	− 2.64999i
$957$	0	0
$958$	− 481.315i	− 0.502417i
$959$	− 111.686i	− 0.116461i
$960$	0	0
$961$	2434.61	2.53342
$962$	−2380.24	−2.47426
$963$	0	0
$964$	−2092.79	−2.17094
$965$	0	0
$966$	0	0
$967$	− 357.093i	− 0.369279i	−0.982806	−	0.184639i	$-0.940888\pi$
$967$	− 357.093i	− 0.369279i	0.982806	−	0.184639i	$-0.0591117\pi$
$968$	−1779.26	−1.83807
$969$	0	0
$970$	0	0
$971$	− 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
$973$	− 485.903i	− 0.499387i
$974$	− 152.592i	− 0.156665i
$975$	0	0
$976$	−1200.45	−1.22997
$977$	253.280	0.259243	0.129621	−	0.991564i	$-0.458624\pi$
$977$	253.280	0.259243	0.129621	+	0.991564i	$0.458624\pi$
$978$	0	0
$979$	−583.132	−0.595640
$980$	0	0
$981$	0	0
$982$	652.732i	0.664697i
$983$	−1068.73	−1.08722	−0.543609	−	0.839339i	$-0.682942\pi$
$983$	−1068.73	−1.08722	−0.543609	+	0.839339i	$0.682942\pi$
$984$	0	0
$985$	0	0
$986$	− 2008.68i	− 2.03720i
$987$	0	0
$988$	578.614i	0.585641i
$989$	− 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	$0.454779\pi$
$992$	2627.14	2.64833
$993$	0	0
$994$	2424.37	2.43900
$995$	0	0
$996$	0	0
$997$	− 356.574i	− 0.357647i	−0.983881	−	0.178824i	$-0.942771\pi$
$997$	− 356.574i	− 0.357647i	0.983881	−	0.178824i	$-0.0572292\pi$
$998$	−142.712	−0.142998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.d.b.224.2		8
3.2	odd	2	inner	225.3.d.b.224.8		8
4.3	odd	2		3600.3.c.i.449.1		8
5.2	odd	4		225.3.c.c.26.1		4
5.3	odd	4		45.3.c.a.26.4	yes	4
5.4	even	2	inner	225.3.d.b.224.7		8
12.11	even	2		3600.3.c.i.449.2		8
15.2	even	4		225.3.c.c.26.4		4
15.8	even	4		45.3.c.a.26.1	✓	4
15.14	odd	2	inner	225.3.d.b.224.1		8
20.3	even	4		720.3.l.a.161.3		4
20.7	even	4		3600.3.l.v.1601.3		4
20.19	odd	2		3600.3.c.i.449.7		8
40.3	even	4		2880.3.l.c.1601.1		4
40.13	odd	4		2880.3.l.g.1601.2		4
45.13	odd	12		405.3.i.d.26.1		8
45.23	even	12		405.3.i.d.26.4		8
45.38	even	12		405.3.i.d.296.1		8
45.43	odd	12		405.3.i.d.296.4		8
60.23	odd	4		720.3.l.a.161.1		4
60.47	odd	4		3600.3.l.v.1601.4		4
60.59	even	2		3600.3.c.i.449.8		8
120.53	even	4		2880.3.l.g.1601.4		4
120.83	odd	4		2880.3.l.c.1601.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.1	✓	4	15.8	even	4
45.3.c.a.26.4	yes	4	5.3	odd	4
225.3.c.c.26.1		4	5.2	odd	4
225.3.c.c.26.4		4	15.2	even	4
225.3.d.b.224.1		8	15.14	odd	2	inner
225.3.d.b.224.2		8	1.1	even	1	trivial
225.3.d.b.224.7		8	5.4	even	2	inner
225.3.d.b.224.8		8	3.2	odd	2	inner
405.3.i.d.26.1		8	45.13	odd	12
405.3.i.d.26.4		8	45.23	even	12
405.3.i.d.296.1		8	45.38	even	12
405.3.i.d.296.4		8	45.43	odd	12
720.3.l.a.161.1		4	60.23	odd	4
720.3.l.a.161.3		4	20.3	even	4
2880.3.l.c.1601.1		4	40.3	even	4
2880.3.l.c.1601.3		4	120.83	odd	4
2880.3.l.g.1601.2		4	40.13	odd	4
2880.3.l.g.1601.4		4	120.53	even	4
3600.3.c.i.449.1		8	4.3	odd	2
3600.3.c.i.449.2		8	12.11	even	2
3600.3.c.i.449.7		8	20.19	odd	2
3600.3.c.i.449.8		8	60.59	even	2
3600.3.l.v.1601.3		4	20.7	even	4
3600.3.l.v.1601.4		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.65028 q^{2} +9.32456 q^{4} +7.16228i q^{7} -19.4361 q^{8} +5.42736i q^{11} -9.81139i q^{13} -26.1443i q^{14} +33.6491 q^{16} +12.2317 q^{17} -6.32456 q^{19} -19.8114i q^{22} -12.0394 q^{23} +35.8143i q^{26} +66.7851i q^{28} +44.9881i q^{29} -58.2719 q^{31} -45.0842 q^{32} -44.6491 q^{34} +66.4605i q^{37} +23.0864 q^{38} +16.4743i q^{41} +43.6228i q^{43} +50.6077i q^{44} +43.9473 q^{46} -40.0570 q^{47} -2.29822 q^{49} -91.4868i q^{52} +13.2242 q^{53} -139.207i q^{56} -164.219i q^{58} -25.1519i q^{59} -35.6754 q^{61} +212.709 q^{62} +29.9737 q^{64} +26.7018i q^{67} +114.055 q^{68} +92.7301i q^{71} -60.3246i q^{73} -242.600i q^{74} -58.9737 q^{76} -38.8723 q^{77} +96.2192 q^{79} -60.1359i q^{82} -79.1215 q^{83} -159.235i q^{86} -105.487i q^{88} +107.443i q^{89} +70.2719 q^{91} -112.262 q^{92} +146.219 q^{94} -1.07900i q^{97} +8.38915 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4} + 168 q^{16} - 112 q^{31} - 256 q^{34} + 48 q^{46} + 184 q^{49} - 336 q^{61} + 88 q^{64} - 320 q^{76} + 112 q^{79} + 208 q^{91} + 512 q^{94}+O(q^{100}) \) 8 * q + 24 * q^4 + 168 * q^16 - 112 * q^31 - 256 * q^34 + 48 * q^46 + 184 * q^49 - 336 * q^61 + 88 * q^64 - 320 * q^76 + 112 * q^79 + 208 * q^91 + 512 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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