LMFDB - Embedded newform 441.6.a.be.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$70.7292645375$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 146x^{6} + 5453x^{4} - 40868x^{2} + 3844 $ x^8 - 146x^6 + 5453x^4 - 40868*x^2 + 3844
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-6.71178$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.72287 q^{2} -29.0317 q^{4} -73.1337 q^{5} -105.150 q^{8} +O(q^{10})$	$q+1.72287 q^{2} -29.0317 q^{4} -73.1337 q^{5} -105.150 q^{8} -126.000 q^{10} +193.070 q^{11} +151.500 q^{13} +747.857 q^{16} +1760.12 q^{17} +1126.79 q^{19} +2123.20 q^{20} +332.635 q^{22} +2413.33 q^{23} +2223.54 q^{25} +261.015 q^{26} -7729.11 q^{29} -8176.75 q^{31} +4653.24 q^{32} +3032.45 q^{34} +5932.38 q^{37} +1941.31 q^{38} +7689.98 q^{40} +4975.84 q^{41} -5652.51 q^{43} -5605.17 q^{44} +4157.84 q^{46} +4720.93 q^{47} +3830.86 q^{50} -4398.32 q^{52} -32072.1 q^{53} -14120.0 q^{55} -13316.2 q^{58} -28566.9 q^{59} +29003.7 q^{61} -14087.5 q^{62} -15914.5 q^{64} -11079.8 q^{65} -30517.4 q^{67} -51099.3 q^{68} +56944.5 q^{71} +9726.79 q^{73} +10220.7 q^{74} -32712.7 q^{76} -98483.5 q^{79} -54693.5 q^{80} +8572.71 q^{82} +50637.0 q^{83} -128724. q^{85} -9738.52 q^{86} -20301.3 q^{88} +59655.8 q^{89} -70063.0 q^{92} +8133.54 q^{94} -82406.4 q^{95} -134206. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{4}+O(q^{10}) $ 8 * q + 20 * q^4	$ 8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100}) $ 8 * q + 20 * q^4 - 828 * q^16 - 2384 * q^22 - 2392 * q^25 - 19136 * q^37 - 41184 * q^43 - 13152 * q^46 - 88872 * q^58 - 210812 * q^64 - 42336 * q^67 - 251072 * q^79 - 567664 * q^85 - 88752 * q^88

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.72287	0.304563	0.152281	−	0.988337i	$-0.451338\pi$
$2$	1.72287	0.304563	0.152281	+	0.988337i	$0.451338\pi$
$3$	0	0
$3$	0	0
$4$	−29.0317	−0.907242
$5$	−73.1337	−1.30826	−0.654128	−	0.756384i	$-0.726964\pi$
$5$	−73.1337	−1.30826	−0.654128	+	0.756384i	$0.726964\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−105.150	−0.580875
$9$	0	0
$10$	−126.000	−0.398446
$11$	193.070	0.481099	0.240549	−	0.970637i	$-0.422672\pi$
$11$	193.070	0.481099	0.240549	+	0.970637i	$0.422672\pi$
$12$	0	0
$13$	151.500	0.248631	0.124315	−	0.992243i	$-0.460327\pi$
$13$	151.500	0.248631	0.124315	+	0.992243i	$0.460327\pi$
$14$	0	0
$15$	0	0
$16$	747.857	0.730329
$17$	1760.12	1.47713	0.738567	−	0.674180i	$-0.235503\pi$
$17$	1760.12	1.47713	0.738567	+	0.674180i	$0.235503\pi$
$18$	0	0
$19$	1126.79	0.716076	0.358038	−	0.933707i	$-0.383446\pi$
$19$	1126.79	0.716076	0.358038	+	0.933707i	$0.383446\pi$
$20$	2123.20	1.18690
$21$	0	0
$22$	332.635	0.146525
$23$	2413.33	0.951254	0.475627	−	0.879647i	$-0.342221\pi$
$23$	2413.33	0.951254	0.475627	+	0.879647i	$0.342221\pi$
$24$	0	0
$25$	2223.54	0.711532
$26$	261.015	0.0757237
$27$	0	0
$28$	0	0
$29$	−7729.11	−1.70661	−0.853305	−	0.521412i	$-0.825405\pi$
$29$	−7729.11	−1.70661	−0.853305	+	0.521412i	$0.825405\pi$
$30$	0	0
$31$	−8176.75	−1.52819	−0.764094	−	0.645105i	$-0.776814\pi$
$31$	−8176.75	−1.52819	−0.764094	+	0.645105i	$0.776814\pi$
$32$	4653.24	0.803306
$33$	0	0
$34$	3032.45	0.449880
$35$	0	0
$36$	0	0
$37$	5932.38	0.712401	0.356200	−	0.934410i	$-0.384072\pi$
$37$	5932.38	0.712401	0.356200	+	0.934410i	$0.384072\pi$
$38$	1941.31	0.218090
$39$	0	0
$40$	7689.98	0.759932
$41$	4975.84	0.462282	0.231141	−	0.972920i	$-0.425754\pi$
$41$	4975.84	0.462282	0.231141	+	0.972920i	$0.425754\pi$
$42$	0	0
$43$	−5652.51	−0.466198	−0.233099	−	0.972453i	$-0.574887\pi$
$43$	−5652.51	−0.466198	−0.233099	+	0.972453i	$0.574887\pi$
$44$	−5605.17	−0.436473
$45$	0	0
$46$	4157.84	0.289716
$47$	4720.93	0.311733	0.155867	−	0.987778i	$-0.450183\pi$
$47$	4720.93	0.311733	0.155867	+	0.987778i	$0.450183\pi$
$48$	0	0
$49$	0	0
$50$	3830.86	0.216706
$51$	0	0
$52$	−4398.32	−0.225568
$53$	−32072.1	−1.56833	−0.784166	−	0.620551i	$-0.786909\pi$
$53$	−32072.1	−1.56833	−0.784166	+	0.620551i	$0.786909\pi$
$54$	0	0
$55$	−14120.0	−0.629400
$56$	0	0
$57$	0	0
$58$	−13316.2	−0.519770
$59$	−28566.9	−1.06840	−0.534200	−	0.845358i	$-0.679387\pi$
$59$	−28566.9	−1.06840	−0.534200	+	0.845358i	$0.679387\pi$
$60$	0	0
$61$	29003.7	0.997994	0.498997	−	0.866604i	$-0.333702\pi$
$61$	29003.7	0.997994	0.498997	+	0.866604i	$0.333702\pi$
$62$	−14087.5	−0.465429
$63$	0	0
$64$	−15914.5	−0.485672
$65$	−11079.8	−0.325273
$66$	0	0
$67$	−30517.4	−0.830540	−0.415270	−	0.909698i	$-0.636313\pi$
$67$	−30517.4	−0.830540	−0.415270	+	0.909698i	$0.636313\pi$
$68$	−51099.3	−1.34012
$69$	0	0
$70$	0	0
$71$	56944.5	1.34062	0.670310	−	0.742081i	$-0.266161\pi$
$71$	56944.5	1.34062	0.670310	+	0.742081i	$0.266161\pi$
$72$	0	0
$73$	9726.79	0.213630	0.106815	−	0.994279i	$-0.465935\pi$
$73$	9726.79	0.213630	0.106815	+	0.994279i	$0.465935\pi$
$74$	10220.7	0.216971
$75$	0	0
$76$	−32712.7	−0.649654
$77$	0	0
$78$	0	0
$79$	−98483.5	−1.77540	−0.887699	−	0.460424i	$-0.847697\pi$
$79$	−98483.5	−1.77540	−0.887699	+	0.460424i	$0.847697\pi$
$80$	−54693.5	−0.955457
$81$	0	0
$82$	8572.71	0.140794
$83$	50637.0	0.806813	0.403406	−	0.915021i	$-0.367826\pi$
$83$	50637.0	0.806813	0.403406	+	0.915021i	$0.367826\pi$
$84$	0	0
$85$	−128724.	−1.93247
$86$	−9738.52	−0.141986
$87$	0	0
$88$	−20301.3	−0.279458
$89$	59655.8	0.798321	0.399161	−	0.916881i	$-0.369302\pi$
$89$	59655.8	0.798321	0.399161	+	0.916881i	$0.369302\pi$
$90$	0	0
$91$	0	0
$92$	−70063.0	−0.863017
$93$	0	0
$94$	8133.54	0.0949423
$95$	−82406.4	−0.936811
$96$	0	0
$97$	−134206.	−1.44825	−0.724124	−	0.689670i	$-0.757756\pi$
$97$	−134206.	−1.44825	−0.724124	+	0.689670i	$0.757756\pi$
$98$	0	0
$99$	0	0
$100$	−64553.2	−0.645532
$101$	135281.	1.31957	0.659786	−	0.751454i	$-0.270647\pi$
$101$	135281.	1.31957	0.659786	+	0.751454i	$0.270647\pi$
$102$	0	0
$103$	199410.	1.85205	0.926025	−	0.377461i	$-0.123203\pi$
$103$	199410.	1.85205	0.926025	+	0.377461i	$0.123203\pi$
$104$	−15930.2	−0.144423
$105$	0	0
$106$	−55256.0	−0.477655
$107$	93298.1	0.787795	0.393898	−	0.919154i	$-0.371126\pi$
$107$	93298.1	0.787795	0.393898	+	0.919154i	$0.371126\pi$
$108$	0	0
$109$	−41818.7	−0.337136	−0.168568	−	0.985690i	$-0.553914\pi$
$109$	−41818.7	−0.337136	−0.168568	+	0.985690i	$0.553914\pi$
$110$	−24326.8	−0.191692
$111$	0	0
$112$	0	0
$113$	−170560.	−1.25655	−0.628277	−	0.777990i	$-0.716240\pi$
$113$	−170560.	−1.25655	−0.628277	+	0.777990i	$0.716240\pi$
$114$	0	0
$115$	−176495.	−1.24448
$116$	224389.	1.54831
$117$	0	0
$118$	−49217.0	−0.325395
$119$	0	0
$120$	0	0
$121$	−123775.	−0.768544
$122$	49969.4	0.303952
$123$	0	0
$124$	237385.	1.38644
$125$	65927.2	0.377389
$126$	0	0
$127$	−314872.	−1.73231	−0.866154	−	0.499777i	$-0.833415\pi$
$127$	−314872.	−1.73231	−0.866154	+	0.499777i	$0.833415\pi$
$128$	−176322.	−0.951223
$129$	0	0
$130$	−19089.0	−0.0990660
$131$	91423.9	0.465459	0.232729	−	0.972542i	$-0.425234\pi$
$131$	91423.9	0.465459	0.232729	+	0.972542i	$0.425234\pi$
$132$	0	0
$133$	0	0
$134$	−52577.4	−0.252951
$135$	0	0
$136$	−185076.	−0.858030
$137$	−5959.51	−0.0271275	−0.0135637	−	0.999908i	$-0.504318\pi$
$137$	−5959.51	−0.0271275	−0.0135637	+	0.999908i	$0.504318\pi$
$138$	0	0
$139$	−13792.3	−0.0605479	−0.0302740	−	0.999542i	$-0.509638\pi$
$139$	−13792.3	−0.0605479	−0.0302740	+	0.999542i	$0.509638\pi$
$140$	0	0
$141$	0	0
$142$	98107.8	0.408303
$143$	29250.2	0.119616
$144$	0	0
$145$	565258.	2.23268
$146$	16758.0	0.0650637
$147$	0	0
$148$	−172227.	−0.646320
$149$	400637.	1.47838	0.739189	−	0.673498i	$-0.235209\pi$
$149$	400637.	1.47838	0.739189	+	0.673498i	$0.235209\pi$
$150$	0	0
$151$	91744.7	0.327445	0.163723	−	0.986506i	$-0.447650\pi$
$151$	91744.7	0.327445	0.163723	+	0.986506i	$0.447650\pi$
$152$	−118482.	−0.415951
$153$	0	0
$154$	0	0
$155$	597996.	1.99926
$156$	0	0
$157$	−498424.	−1.61380	−0.806900	−	0.590688i	$-0.798857\pi$
$157$	−498424.	−1.61380	−0.806900	+	0.590688i	$0.798857\pi$
$158$	−169674.	−0.540720
$159$	0	0
$160$	−340309.	−1.05093
$161$	0	0
$162$	0	0
$163$	−242531.	−0.714988	−0.357494	−	0.933915i	$-0.616369\pi$
$163$	−242531.	−0.714988	−0.357494	+	0.933915i	$0.616369\pi$
$164$	−144457.	−0.419401
$165$	0	0
$166$	87240.8	0.245725
$167$	−525601.	−1.45836	−0.729181	−	0.684321i	$-0.760099\pi$
$167$	−525601.	−1.45836	−0.729181	+	0.684321i	$0.760099\pi$
$168$	0	0
$169$	−348341.	−0.938183
$170$	−221775.	−0.588558
$171$	0	0
$172$	164102.	0.422954
$173$	96423.4	0.244944	0.122472	−	0.992472i	$-0.460918\pi$
$173$	96423.4	0.244944	0.122472	+	0.992472i	$0.460918\pi$
$174$	0	0
$175$	0	0
$176$	144389.	0.351360
$177$	0	0
$178$	102779.	0.243139
$179$	178701.	0.416865	0.208432	−	0.978037i	$-0.433164\pi$
$179$	178701.	0.416865	0.208432	+	0.978037i	$0.433164\pi$
$180$	0	0
$181$	−224533.	−0.509428	−0.254714	−	0.967016i	$-0.581981\pi$
$181$	−224533.	−0.509428	−0.254714	+	0.967016i	$0.581981\pi$
$182$	0	0
$183$	0	0
$184$	−253760.	−0.552559
$185$	−433857.	−0.932002
$186$	0	0
$187$	339827.	0.710647
$188$	−137057.	−0.282817
$189$	0	0
$190$	−141975.	−0.285318
$191$	−905785.	−1.79656	−0.898280	−	0.439423i	$-0.855183\pi$
$191$	−905785.	−1.79656	−0.898280	+	0.439423i	$0.855183\pi$
$192$	0	0
$193$	−717637.	−1.38679	−0.693396	−	0.720556i	$-0.743886\pi$
$193$	−717637.	−1.38679	−0.693396	+	0.720556i	$0.743886\pi$
$194$	−231219.	−0.441082
$195$	0	0
$196$	0	0
$197$	−57173.0	−0.104960	−0.0524802	−	0.998622i	$-0.516713\pi$
$197$	−57173.0	−0.104960	−0.0524802	+	0.998622i	$0.516713\pi$
$198$	0	0
$199$	263037.	0.470853	0.235426	−	0.971892i	$-0.424351\pi$
$199$	263037.	0.470853	0.235426	+	0.971892i	$0.424351\pi$
$200$	−233804.	−0.413311
$201$	0	0
$202$	233071.	0.401892
$203$	0	0
$204$	0	0
$205$	−363902.	−0.604782
$206$	343556.	0.564066
$207$	0	0
$208$	113301.	0.181582
$209$	217550.	0.344503
$210$	0	0
$211$	−495237.	−0.765786	−0.382893	−	0.923793i	$-0.625072\pi$
$211$	−495237.	−0.765786	−0.382893	+	0.923793i	$0.625072\pi$
$212$	931109.	1.42286
$213$	0	0
$214$	160740.	0.239933
$215$	413389.	0.609906
$216$	0	0
$217$	0	0
$218$	−72048.1	−0.102679
$219$	0	0
$220$	409927.	0.571018
$221$	266659.	0.367261
$222$	0	0
$223$	−373295.	−0.502679	−0.251339	−	0.967899i	$-0.580871\pi$
$223$	−373295.	−0.502679	−0.251339	+	0.967899i	$0.580871\pi$
$224$	0	0
$225$	0	0
$226$	−293852.	−0.382700
$227$	284573.	0.366547	0.183273	−	0.983062i	$-0.441331\pi$
$227$	284573.	0.366547	0.183273	+	0.983062i	$0.441331\pi$
$228$	0	0
$229$	−989364.	−1.24672	−0.623358	−	0.781937i	$-0.714232\pi$
$229$	−989364.	−1.24672	−0.623358	+	0.781937i	$0.714232\pi$
$230$	−304078.	−0.379023
$231$	0	0
$232$	812712.	0.991327
$233$	602273.	0.726781	0.363391	−	0.931637i	$-0.381619\pi$
$233$	602273.	0.726781	0.363391	+	0.931637i	$0.381619\pi$
$234$	0	0
$235$	−345259.	−0.407827
$236$	829347.	0.969296
$237$	0	0
$238$	0	0
$239$	−1.44372e6	−1.63489	−0.817447	−	0.576004i	$-0.804611\pi$
$239$	−1.44372e6	−1.63489	−0.817447	+	0.576004i	$0.804611\pi$
$240$	0	0
$241$	538796.	0.597560	0.298780	−	0.954322i	$-0.403420\pi$
$241$	538796.	0.597560	0.298780	+	0.954322i	$0.403420\pi$
$242$	−213248.	−0.234070
$243$	0	0
$244$	−842026.	−0.905422
$245$	0	0
$246$	0	0
$247$	170709.	0.178039
$248$	859782.	0.887685
$249$	0	0
$250$	113584.	0.114939
$251$	1.63982e6	1.64290	0.821449	−	0.570282i	$-0.193166\pi$
$251$	1.63982e6	1.64290	0.821449	+	0.570282i	$0.193166\pi$
$252$	0	0
$253$	465942.	0.457647
$254$	−542483.	−0.527596
$255$	0	0
$256$	205484.	0.195965
$257$	−1.82127e6	−1.72005	−0.860026	−	0.510251i	$-0.829553\pi$
$257$	−1.82127e6	−1.72005	−0.860026	+	0.510251i	$0.829553\pi$
$258$	0	0
$259$	0	0
$260$	321665.	0.295101
$261$	0	0
$262$	157511.	0.141761
$263$	1.77477e6	1.58217	0.791085	−	0.611706i	$-0.209517\pi$
$263$	1.77477e6	1.58217	0.791085	+	0.611706i	$0.209517\pi$
$264$	0	0
$265$	2.34555e6	2.05178
$266$	0	0
$267$	0	0
$268$	885972.	0.753500
$269$	−769025.	−0.647977	−0.323989	−	0.946061i	$-0.605024\pi$
$269$	−769025.	−0.647977	−0.323989	+	0.946061i	$0.605024\pi$
$270$	0	0
$271$	−1.81374e6	−1.50021	−0.750106	−	0.661317i	$-0.769998\pi$
$271$	−1.81374e6	−1.50021	−0.750106	+	0.661317i	$0.769998\pi$
$272$	1.31632e6	1.07879
$273$	0	0
$274$	−10267.4	−0.00826202
$275$	429299.	0.342317
$276$	0	0
$277$	1.12526e6	0.881157	0.440578	−	0.897714i	$-0.354773\pi$
$277$	1.12526e6	0.881157	0.440578	+	0.897714i	$0.354773\pi$
$278$	−23762.3	−0.0184406
$279$	0	0
$280$	0	0
$281$	−243202.	−0.183739	−0.0918696	−	0.995771i	$-0.529284\pi$
$281$	−243202.	−0.183739	−0.0918696	+	0.995771i	$0.529284\pi$
$282$	0	0
$283$	−1.06625e6	−0.791396	−0.395698	−	0.918381i	$-0.629497\pi$
$283$	−1.06625e6	−0.791396	−0.395698	+	0.918381i	$0.629497\pi$
$284$	−1.65320e6	−1.21627
$285$	0	0
$286$	50394.3	0.0364306
$287$	0	0
$288$	0	0
$289$	1.67817e6	1.18193
$290$	973865.	0.679992
$291$	0	0
$292$	−282385.	−0.193814
$293$	−2.32478e6	−1.58203	−0.791013	−	0.611799i	$-0.790446\pi$
$293$	−2.32478e6	−1.58203	−0.791013	+	0.611799i	$0.790446\pi$
$294$	0	0
$295$	2.08921e6	1.39774
$296$	−623787.	−0.413816
$297$	0	0
$298$	690244.	0.450259
$299$	365620.	0.236511
$300$	0	0
$301$	0	0
$302$	158064.	0.0997275
$303$	0	0
$304$	842678.	0.522971
$305$	−2.12114e6	−1.30563
$306$	0	0
$307$	2.37882e6	1.44051	0.720255	−	0.693710i	$-0.244025\pi$
$307$	2.37882e6	1.44051	0.720255	+	0.693710i	$0.244025\pi$
$308$	0	0
$309$	0	0
$310$	1.03027e6	0.608900
$311$	−2.22414e6	−1.30395	−0.651976	−	0.758240i	$-0.726060\pi$
$311$	−2.22414e6	−1.30395	−0.651976	+	0.758240i	$0.726060\pi$
$312$	0	0
$313$	−1.03945e6	−0.599713	−0.299857	−	0.953984i	$-0.596939\pi$
$313$	−1.03945e6	−0.599713	−0.299857	+	0.953984i	$0.596939\pi$
$314$	−858718.	−0.491503
$315$	0	0
$316$	2.85915e6	1.61072
$317$	−428476.	−0.239485	−0.119743	−	0.992805i	$-0.538207\pi$
$317$	−428476.	−0.239485	−0.119743	+	0.992805i	$0.538207\pi$
$318$	0	0
$319$	−1.49226e6	−0.821048
$320$	1.16389e6	0.635383
$321$	0	0
$322$	0	0
$323$	1.98329e6	1.05774
$324$	0	0
$325$	336867.	0.176909
$326$	−417849.	−0.217759
$327$	0	0
$328$	−523207.	−0.268528
$329$	0	0
$330$	0	0
$331$	2.79057e6	1.39998	0.699991	−	0.714152i	$-0.253187\pi$
$331$	2.79057e6	1.39998	0.699991	+	0.714152i	$0.253187\pi$
$332$	−1.47008e6	−0.731974
$333$	0	0
$334$	−905541.	−0.444163
$335$	2.23185e6	1.08656
$336$	0	0
$337$	1.30809e6	0.627427	0.313714	−	0.949518i	$-0.398427\pi$
$337$	1.30809e6	0.627427	0.313714	+	0.949518i	$0.398427\pi$
$338$	−600145.	−0.285735
$339$	0	0
$340$	3.73708e6	1.75322
$341$	−1.57869e6	−0.735209
$342$	0	0
$343$	0	0
$344$	594359.	0.270802
$345$	0	0
$346$	166125.	0.0746009
$347$	664031.	0.296050	0.148025	−	0.988984i	$-0.452708\pi$
$347$	664031.	0.296050	0.148025	+	0.988984i	$0.452708\pi$
$348$	0	0
$349$	2.27119e6	0.998136	0.499068	−	0.866563i	$-0.333676\pi$
$349$	2.27119e6	0.998136	0.499068	+	0.866563i	$0.333676\pi$
$350$	0	0
$351$	0	0
$352$	898404.	0.386469
$353$	−1.96361e6	−0.838723	−0.419361	−	0.907819i	$-0.637746\pi$
$353$	−1.96361e6	−0.838723	−0.419361	+	0.907819i	$0.637746\pi$
$354$	0	0
$355$	−4.16456e6	−1.75387
$356$	−1.73191e6	−0.724270
$357$	0	0
$358$	307879.	0.126961
$359$	−2.09222e6	−0.856783	−0.428391	−	0.903593i	$-0.640920\pi$
$359$	−2.09222e6	−0.856783	−0.428391	+	0.903593i	$0.640920\pi$
$360$	0	0
$361$	−1.20644e6	−0.487235
$362$	−386840.	−0.155153
$363$	0	0
$364$	0	0
$365$	−711356.	−0.279483
$366$	0	0
$367$	1.33844e6	0.518719	0.259360	−	0.965781i	$-0.416488\pi$
$367$	1.33844e6	0.518719	0.259360	+	0.965781i	$0.416488\pi$
$368$	1.80482e6	0.694728
$369$	0	0
$370$	−747477.	−0.283853
$371$	0	0
$372$	0	0
$373$	−1.32657e6	−0.493694	−0.246847	−	0.969054i	$-0.579394\pi$
$373$	−1.32657e6	−0.493694	−0.246847	+	0.969054i	$0.579394\pi$
$374$	585477.	0.216437
$375$	0	0
$376$	−496404.	−0.181078
$377$	−1.17096e6	−0.424316
$378$	0	0
$379$	863514.	0.308796	0.154398	−	0.988009i	$-0.450656\pi$
$379$	863514.	0.308796	0.154398	+	0.988009i	$0.450656\pi$
$380$	2.39240e6	0.849914
$381$	0	0
$382$	−1.56055e6	−0.547165
$383$	2.53709e6	0.883769	0.441884	−	0.897072i	$-0.354310\pi$
$383$	2.53709e6	0.883769	0.441884	+	0.897072i	$0.354310\pi$
$384$	0	0
$385$	0	0
$386$	−1.23639e6	−0.422365
$387$	0	0
$388$	3.89624e6	1.31391
$389$	−253814.	−0.0850434	−0.0425217	−	0.999096i	$-0.513539\pi$
$389$	−253814.	−0.0850434	−0.0425217	+	0.999096i	$0.513539\pi$
$390$	0	0
$391$	4.24774e6	1.40513
$392$	0	0
$393$	0	0
$394$	−98501.4	−0.0319670
$395$	7.20246e6	2.32267
$396$	0	0
$397$	−4.57767e6	−1.45770	−0.728850	−	0.684673i	$-0.759945\pi$
$397$	−4.57767e6	−1.45770	−0.728850	+	0.684673i	$0.759945\pi$
$398$	453179.	0.143404
$399$	0	0
$400$	1.66289e6	0.519653
$401$	4.38670e6	1.36231	0.681156	−	0.732138i	$-0.261477\pi$
$401$	4.38670e6	1.36231	0.681156	+	0.732138i	$0.261477\pi$
$402$	0	0
$403$	−1.23878e6	−0.379955
$404$	−3.92744e6	−1.19717
$405$	0	0
$406$	0	0
$407$	1.14537e6	0.342735
$408$	0	0
$409$	222245.	0.0656936	0.0328468	−	0.999460i	$-0.489543\pi$
$409$	222245.	0.0656936	0.0328468	+	0.999460i	$0.489543\pi$
$410$	−626954.	−0.184194
$411$	0	0
$412$	−5.78920e6	−1.68026
$413$	0	0
$414$	0	0
$415$	−3.70327e6	−1.05552
$416$	704968.	0.199727
$417$	0	0
$418$	374810.	0.104923
$419$	4.33564e6	1.20647	0.603237	−	0.797562i	$-0.293877\pi$
$419$	4.33564e6	1.20647	0.603237	+	0.797562i	$0.293877\pi$
$420$	0	0
$421$	572536.	0.157434	0.0787168	−	0.996897i	$-0.474918\pi$
$421$	572536.	0.157434	0.0787168	+	0.996897i	$0.474918\pi$
$422$	−853228.	−0.233230
$423$	0	0
$424$	3.37237e6	0.911004
$425$	3.91370e6	1.05103
$426$	0	0
$427$	0	0
$428$	−2.70860e6	−0.714720
$429$	0	0
$430$	712214.	0.185754
$431$	3.30224e6	0.856279	0.428139	−	0.903713i	$-0.359169\pi$
$431$	3.30224e6	0.856279	0.428139	+	0.903713i	$0.359169\pi$
$432$	0	0
$433$	−19285.5	−0.00494322	−0.00247161	−	0.999997i	$-0.500787\pi$
$433$	−19285.5	−0.00494322	−0.00247161	+	0.999997i	$0.500787\pi$
$434$	0	0
$435$	0	0
$436$	1.21407e6	0.305863
$437$	2.71931e6	0.681170
$438$	0	0
$439$	376815.	0.0933182	0.0466591	−	0.998911i	$-0.485143\pi$
$439$	376815.	0.0933182	0.0466591	+	0.998911i	$0.485143\pi$
$440$	1.48471e6	0.365602
$441$	0	0
$442$	459418.	0.111854
$443$	−1.24690e6	−0.301871	−0.150936	−	0.988544i	$-0.548229\pi$
$443$	−1.24690e6	−0.301871	−0.150936	+	0.988544i	$0.548229\pi$
$444$	0	0
$445$	−4.36285e6	−1.04441
$446$	−643138.	−0.153097
$447$	0	0
$448$	0	0
$449$	−2.85298e6	−0.667857	−0.333928	−	0.942598i	$-0.608374\pi$
$449$	−2.85298e6	−0.667857	−0.333928	+	0.942598i	$0.608374\pi$
$450$	0	0
$451$	960687.	0.222403
$452$	4.95165e6	1.14000
$453$	0	0
$454$	490282.	0.111636
$455$	0	0
$456$	0	0
$457$	2.10513e6	0.471506	0.235753	−	0.971813i	$-0.424244\pi$
$457$	2.10513e6	0.471506	0.235753	+	0.971813i	$0.424244\pi$
$458$	−1.70454e6	−0.379703
$459$	0	0
$460$	5.12397e6	1.12905
$461$	−8.96535e6	−1.96478	−0.982392	−	0.186832i	$-0.940178\pi$
$461$	−8.96535e6	−1.96478	−0.982392	+	0.186832i	$0.940178\pi$
$462$	0	0
$463$	−7.01066e6	−1.51987	−0.759935	−	0.649999i	$-0.774769\pi$
$463$	−7.01066e6	−1.51987	−0.759935	+	0.649999i	$0.774769\pi$
$464$	−5.78026e6	−1.24639
$465$	0	0
$466$	1.03764e6	0.221350
$467$	3.26840e6	0.693495	0.346747	−	0.937959i	$-0.387286\pi$
$467$	3.26840e6	0.693495	0.346747	+	0.937959i	$0.387286\pi$
$468$	0	0
$469$	0	0
$470$	−594836.	−0.124209
$471$	0	0
$472$	3.00380e6	0.620606
$473$	−1.09133e6	−0.224287
$474$	0	0
$475$	2.50546e6	0.509511
$476$	0	0
$477$	0	0
$478$	−2.48734e6	−0.497928
$479$	−4.89425e6	−0.974646	−0.487323	−	0.873222i	$-0.662027\pi$
$479$	−4.89425e6	−0.974646	−0.487323	+	0.873222i	$0.662027\pi$
$480$	0	0
$481$	898757.	0.177125
$482$	928274.	0.181995
$483$	0	0
$484$	3.59340e6	0.697255
$485$	9.81499e6	1.89468
$486$	0	0
$487$	−4.60536e6	−0.879916	−0.439958	−	0.898018i	$-0.645007\pi$
$487$	−4.60536e6	−0.879916	−0.439958	+	0.898018i	$0.645007\pi$
$488$	−3.04972e6	−0.579710
$489$	0	0
$490$	0	0
$491$	5.58108e6	1.04475	0.522377	−	0.852714i	$-0.325045\pi$
$491$	5.58108e6	1.04475	0.522377	+	0.852714i	$0.325045\pi$
$492$	0	0
$493$	−1.36042e7	−2.52089
$494$	294109.	0.0542240
$495$	0	0
$496$	−6.11504e6	−1.11608
$497$	0	0
$498$	0	0
$499$	−1.49822e6	−0.269354	−0.134677	−	0.990890i	$-0.543000\pi$
$499$	−1.49822e6	−0.269354	−0.134677	+	0.990890i	$0.543000\pi$
$500$	−1.91398e6	−0.342383
$501$	0	0
$502$	2.82518e6	0.500366
$503$	−479890.	−0.0845710	−0.0422855	−	0.999106i	$-0.513464\pi$
$503$	−479890.	−0.0845710	−0.0422855	+	0.999106i	$0.513464\pi$
$504$	0	0
$505$	−9.89359e6	−1.72634
$506$	802756.	0.139382
$507$	0	0
$508$	9.14129e6	1.57162
$509$	1.21062e6	0.207117	0.103558	−	0.994623i	$-0.466977\pi$
$509$	1.21062e6	0.207117	0.103558	+	0.994623i	$0.466977\pi$
$510$	0	0
$511$	0	0
$512$	5.99634e6	1.01091
$513$	0	0
$514$	−3.13780e6	−0.523863
$515$	−1.45836e7	−2.42296
$516$	0	0
$517$	911472.	0.149974
$518$	0	0
$519$	0	0
$520$	1.16503e6	0.188943
$521$	69333.1	0.0111904	0.00559521	−	0.999984i	$-0.498219\pi$
$521$	69333.1	0.0111904	0.00559521	+	0.999984i	$0.498219\pi$
$522$	0	0
$523$	8.95346e6	1.43132	0.715660	−	0.698449i	$-0.246126\pi$
$523$	8.95346e6	1.43132	0.715660	+	0.698449i	$0.246126\pi$
$524$	−2.65419e6	−0.422284
$525$	0	0
$526$	3.05770e6	0.481870
$527$	−1.43921e7	−2.25734
$528$	0	0
$529$	−612203.	−0.0951166
$530$	4.04107e6	0.624895
$531$	0	0
$532$	0	0
$533$	753841.	0.114938
$534$	0	0
$535$	−6.82323e6	−1.03064
$536$	3.20889e6	0.482439
$537$	0	0
$538$	−1.32493e6	−0.197350
$539$	0	0
$540$	0	0
$541$	−2.38913e6	−0.350951	−0.175476	−	0.984484i	$-0.556146\pi$
$541$	−2.38913e6	−0.350951	−0.175476	+	0.984484i	$0.556146\pi$
$542$	−3.12484e6	−0.456909
$543$	0	0
$544$	8.19027e6	1.18659
$545$	3.05836e6	0.441059
$546$	0	0
$547$	−7.19431e6	−1.02807	−0.514033	−	0.857770i	$-0.671849\pi$
$547$	−7.19431e6	−1.02807	−0.514033	+	0.857770i	$0.671849\pi$
$548$	173015.	0.0246112
$549$	0	0
$550$	739626.	0.104257
$551$	−8.70909e6	−1.22206
$552$	0	0
$553$	0	0
$554$	1.93867e6	0.268367
$555$	0	0
$556$	400414.	0.0549316
$557$	753706.	0.102935	0.0514676	−	0.998675i	$-0.483610\pi$
$557$	753706.	0.102935	0.0514676	+	0.998675i	$0.483610\pi$
$558$	0	0
$559$	−856357.	−0.115911
$560$	0	0
$561$	0	0
$562$	−419005.	−0.0559601
$563$	−1.22890e6	−0.163397	−0.0816985	−	0.996657i	$-0.526034\pi$
$563$	−1.22890e6	−0.163397	−0.0816985	+	0.996657i	$0.526034\pi$
$564$	0	0
$565$	1.24737e7	1.64389
$566$	−1.83701e6	−0.241030
$567$	0	0
$568$	−5.98769e6	−0.778733
$569$	4.30545e6	0.557491	0.278745	−	0.960365i	$-0.410081\pi$
$569$	4.30545e6	0.557491	0.278745	+	0.960365i	$0.410081\pi$
$570$	0	0
$571$	−3.96799e6	−0.509307	−0.254654	−	0.967032i	$-0.581961\pi$
$571$	−3.96799e6	−0.509307	−0.254654	+	0.967032i	$0.581961\pi$
$572$	−849185.	−0.108521
$573$	0	0
$574$	0	0
$575$	5.36612e6	0.676848
$576$	0	0
$577$	−596027.	−0.0745292	−0.0372646	−	0.999305i	$-0.511864\pi$
$577$	−596027.	−0.0745292	−0.0372646	+	0.999305i	$0.511864\pi$
$578$	2.89126e6	0.359971
$579$	0	0
$580$	−1.64104e7	−2.02558
$581$	0	0
$582$	0	0
$583$	−6.19218e6	−0.754522
$584$	−1.02277e6	−0.124092
$585$	0	0
$586$	−4.00529e6	−0.481826
$587$	2.47516e6	0.296489	0.148244	−	0.988951i	$-0.452638\pi$
$587$	2.47516e6	0.296489	0.148244	+	0.988951i	$0.452638\pi$
$588$	0	0
$589$	−9.21349e6	−1.09430
$590$	3.59942e6	0.425699
$591$	0	0
$592$	4.43657e6	0.520287
$593$	−1.14108e7	−1.33253	−0.666267	−	0.745713i	$-0.732109\pi$
$593$	−1.14108e7	−1.33253	−0.666267	+	0.745713i	$0.732109\pi$
$594$	0	0
$595$	0	0
$596$	−1.16312e7	−1.34125
$597$	0	0
$598$	629914.	0.0720325
$599$	8.60394e6	0.979784	0.489892	−	0.871783i	$-0.337036\pi$
$599$	8.60394e6	0.979784	0.489892	+	0.871783i	$0.337036\pi$
$600$	0	0
$601$	1.00463e7	1.13454	0.567270	−	0.823532i	$-0.307999\pi$
$601$	1.00463e7	1.13454	0.567270	+	0.823532i	$0.307999\pi$
$602$	0	0
$603$	0	0
$604$	−2.66351e6	−0.297072
$605$	9.05211e6	1.00545
$606$	0	0
$607$	2.22979e6	0.245636	0.122818	−	0.992429i	$-0.460807\pi$
$607$	2.22979e6	0.245636	0.122818	+	0.992429i	$0.460807\pi$
$608$	5.24323e6	0.575228
$609$	0	0
$610$	−3.65445e6	−0.397647
$611$	715223.	0.0775065
$612$	0	0
$613$	−1.63670e7	−1.75921	−0.879607	−	0.475701i	$-0.842194\pi$
$613$	−1.63670e7	−1.75921	−0.879607	+	0.475701i	$0.842194\pi$
$614$	4.09840e6	0.438726
$615$	0	0
$616$	0	0
$617$	−1.21424e7	−1.28407	−0.642037	−	0.766674i	$-0.721910\pi$
$617$	−1.21424e7	−1.28407	−0.642037	+	0.766674i	$0.721910\pi$
$618$	0	0
$619$	−1.12279e7	−1.17780	−0.588898	−	0.808207i	$-0.700438\pi$
$619$	−1.12279e7	−1.17780	−0.588898	+	0.808207i	$0.700438\pi$
$620$	−1.73609e7	−1.81381
$621$	0	0
$622$	−3.83190e6	−0.397135
$623$	0	0
$624$	0	0
$625$	−1.17701e7	−1.20525
$626$	−1.79084e6	−0.182650
$627$	0	0
$628$	1.44701e7	1.46411
$629$	1.04417e7	1.05231
$630$	0	0
$631$	−5.69644e6	−0.569548	−0.284774	−	0.958595i	$-0.591918\pi$
$631$	−5.69644e6	−0.569548	−0.284774	+	0.958595i	$0.591918\pi$
$632$	1.03555e7	1.03128
$633$	0	0
$634$	−738208.	−0.0729383
$635$	2.30278e7	2.26630
$636$	0	0
$637$	0	0
$638$	−2.57097e6	−0.250060
$639$	0	0
$640$	1.28951e7	1.24444
$641$	2.96921e6	0.285427	0.142714	−	0.989764i	$-0.454417\pi$
$641$	2.96921e6	0.285427	0.142714	+	0.989764i	$0.454417\pi$
$642$	0	0
$643$	−1.22682e7	−1.17018	−0.585091	−	0.810968i	$-0.698941\pi$
$643$	−1.22682e7	−1.17018	−0.585091	+	0.810968i	$0.698941\pi$
$644$	0	0
$645$	0	0
$646$	3.41694e6	0.322149
$647$	−1.21684e6	−0.114281	−0.0571404	−	0.998366i	$-0.518198\pi$
$647$	−1.21684e6	−0.114281	−0.0571404	+	0.998366i	$0.518198\pi$
$648$	0	0
$649$	−5.51543e6	−0.514005
$650$	580377.	0.0538799
$651$	0	0
$652$	7.04110e6	0.648667
$653$	−1.10265e7	−1.01194	−0.505969	−	0.862552i	$-0.668865\pi$
$653$	−1.10265e7	−1.01194	−0.505969	+	0.862552i	$0.668865\pi$
$654$	0	0
$655$	−6.68617e6	−0.608939
$656$	3.72121e6	0.337618
$657$	0	0
$658$	0	0
$659$	−1.32297e7	−1.18668	−0.593342	−	0.804951i	$-0.702192\pi$
$659$	−1.32297e7	−1.18668	−0.593342	+	0.804951i	$0.702192\pi$
$660$	0	0
$661$	6.60257e6	0.587773	0.293886	−	0.955840i	$-0.405051\pi$
$661$	6.60257e6	0.587773	0.293886	+	0.955840i	$0.405051\pi$
$662$	4.80777e6	0.426382
$663$	0	0
$664$	−5.32446e6	−0.468657
$665$	0	0
$666$	0	0
$667$	−1.86529e7	−1.62342
$668$	1.52591e7	1.32309
$669$	0	0
$670$	3.84518e6	0.330925
$671$	5.59975e6	0.480134
$672$	0	0
$673$	−1.42013e7	−1.20862	−0.604310	−	0.796749i	$-0.706551\pi$
$673$	−1.42013e7	−1.20862	−0.604310	+	0.796749i	$0.706551\pi$
$674$	2.25367e6	0.191091
$675$	0	0
$676$	1.01129e7	0.851158
$677$	−1.55878e7	−1.30712	−0.653559	−	0.756876i	$-0.726725\pi$
$677$	−1.55878e7	−1.30712	−0.653559	+	0.756876i	$0.726725\pi$
$678$	0	0
$679$	0	0
$680$	1.35353e7	1.12252
$681$	0	0
$682$	−2.71987e6	−0.223917
$683$	1.37838e7	1.13063	0.565313	−	0.824877i	$-0.308756\pi$
$683$	1.37838e7	1.13063	0.565313	+	0.824877i	$0.308756\pi$
$684$	0	0
$685$	435841.	0.0354897
$686$	0	0
$687$	0	0
$688$	−4.22727e6	−0.340478
$689$	−4.85894e6	−0.389936
$690$	0	0
$691$	−1.13345e7	−0.903043	−0.451522	−	0.892260i	$-0.649119\pi$
$691$	−1.13345e7	−0.903043	−0.451522	+	0.892260i	$0.649119\pi$
$692$	−2.79934e6	−0.222224
$693$	0	0
$694$	1.14404e6	0.0901657
$695$	1.00868e6	0.0792121
$696$	0	0
$697$	8.75808e6	0.682852
$698$	3.91296e6	0.303995
$699$	0	0
$700$	0	0
$701$	−4.21970e6	−0.324330	−0.162165	−	0.986764i	$-0.551848\pi$
$701$	−4.21970e6	−0.324330	−0.162165	+	0.986764i	$0.551848\pi$
$702$	0	0
$703$	6.68455e6	0.510133
$704$	−3.07262e6	−0.233656
$705$	0	0
$706$	−3.38304e6	−0.255444
$707$	0	0
$708$	0	0
$709$	1.25786e7	0.939757	0.469878	−	0.882731i	$-0.344298\pi$
$709$	1.25786e7	0.939757	0.469878	+	0.882731i	$0.344298\pi$
$710$	−7.17499e6	−0.534165
$711$	0	0
$712$	−6.27278e6	−0.463724
$713$	−1.97332e7	−1.45369
$714$	0	0
$715$	−2.13918e6	−0.156488
$716$	−5.18801e6	−0.378197
$717$	0	0
$718$	−3.60461e6	−0.260944
$719$	−2.55041e7	−1.83988	−0.919938	−	0.392065i	$-0.871761\pi$
$719$	−2.55041e7	−1.83988	−0.919938	+	0.392065i	$0.871761\pi$
$720$	0	0
$721$	0	0
$722$	−2.07854e6	−0.148393
$723$	0	0
$724$	6.51857e6	0.462175
$725$	−1.71860e7	−1.21431
$726$	0	0
$727$	2.25405e7	1.58171	0.790857	−	0.612001i	$-0.209635\pi$
$727$	2.25405e7	1.58171	0.790857	+	0.612001i	$0.209635\pi$
$728$	0	0
$729$	0	0
$730$	−1.22557e6	−0.0851200
$731$	−9.94909e6	−0.688637
$732$	0	0
$733$	1.07168e7	0.736724	0.368362	−	0.929682i	$-0.379919\pi$
$733$	1.07168e7	0.736724	0.368362	+	0.929682i	$0.379919\pi$
$734$	2.30595e6	0.157983
$735$	0	0
$736$	1.12298e7	0.764147
$737$	−5.89200e6	−0.399571
$738$	0	0
$739$	−1.67756e7	−1.12997	−0.564987	−	0.825100i	$-0.691119\pi$
$739$	−1.67756e7	−1.12997	−0.564987	+	0.825100i	$0.691119\pi$
$740$	1.25956e7	0.845551
$741$	0	0
$742$	0	0
$743$	8.93625e6	0.593859	0.296929	−	0.954899i	$-0.404037\pi$
$743$	8.93625e6	0.593859	0.296929	+	0.954899i	$0.404037\pi$
$744$	0	0
$745$	−2.93001e7	−1.93409
$746$	−2.28550e6	−0.150361
$747$	0	0
$748$	−9.86577e6	−0.644729
$749$	0	0
$750$	0	0
$751$	1.36875e7	0.885571	0.442785	−	0.896628i	$-0.353990\pi$
$751$	1.36875e7	0.885571	0.442785	+	0.896628i	$0.353990\pi$
$752$	3.53058e6	0.227668
$753$	0	0
$754$	−2.01741e6	−0.129231
$755$	−6.70963e6	−0.428382
$756$	0	0
$757$	1.97877e7	1.25504	0.627518	−	0.778602i	$-0.284071\pi$
$757$	1.97877e7	1.25504	0.627518	+	0.778602i	$0.284071\pi$
$758$	1.48772e6	0.0940477
$759$	0	0
$760$	8.66499e6	0.544170
$761$	5.23103e6	0.327435	0.163718	−	0.986507i	$-0.447651\pi$
$761$	5.23103e6	0.327435	0.163718	+	0.986507i	$0.447651\pi$
$762$	0	0
$763$	0	0
$764$	2.62965e7	1.62991
$765$	0	0
$766$	4.37107e6	0.269163
$767$	−4.32790e6	−0.265637
$768$	0	0
$769$	1.04244e7	0.635673	0.317837	−	0.948145i	$-0.397044\pi$
$769$	1.04244e7	0.635673	0.317837	+	0.948145i	$0.397044\pi$
$770$	0	0
$771$	0	0
$772$	2.08342e7	1.25816
$773$	−8.04863e6	−0.484477	−0.242238	−	0.970217i	$-0.577882\pi$
$773$	−8.04863e6	−0.484477	−0.242238	+	0.970217i	$0.577882\pi$
$774$	0	0
$775$	−1.81813e7	−1.08735
$776$	1.41117e7	0.841251
$777$	0	0
$778$	−437287.	−0.0259011
$779$	5.60673e6	0.331029
$780$	0	0
$781$	1.09943e7	0.644971
$782$	7.31830e6	0.427950
$783$	0	0
$784$	0	0
$785$	3.64516e7	2.11126
$786$	0	0
$787$	1.51619e7	0.872602	0.436301	−	0.899801i	$-0.356288\pi$
$787$	1.51619e7	0.872602	0.436301	+	0.899801i	$0.356288\pi$
$788$	1.65983e6	0.0952244
$789$	0	0
$790$	1.24089e7	0.707400
$791$	0	0
$792$	0	0
$793$	4.39406e6	0.248132
$794$	−7.88672e6	−0.443961
$795$	0	0
$796$	−7.63643e6	−0.427177
$797$	−1.70525e7	−0.950915	−0.475458	−	0.879739i	$-0.657717\pi$
$797$	−1.70525e7	−0.950915	−0.475458	+	0.879739i	$0.657717\pi$
$798$	0	0
$799$	8.30941e6	0.460472
$800$	1.03467e7	0.571578
$801$	0	0
$802$	7.55770e6	0.414910
$803$	1.87795e6	0.102777
$804$	0	0
$805$	0	0
$806$	−2.13425e6	−0.115720
$807$	0	0
$808$	−1.42247e7	−0.766506
$809$	1.81012e7	0.972381	0.486190	−	0.873853i	$-0.338386\pi$
$809$	1.81012e7	0.972381	0.486190	+	0.873853i	$0.338386\pi$
$810$	0	0
$811$	1.37346e7	0.733271	0.366636	−	0.930365i	$-0.380510\pi$
$811$	1.37346e7	0.733271	0.366636	+	0.930365i	$0.380510\pi$
$812$	0	0
$813$	0	0
$814$	1.97331e6	0.104384
$815$	1.77372e7	0.935387
$816$	0	0
$817$	−6.36919e6	−0.333833
$818$	382898.	0.0200078
$819$	0	0
$820$	1.05647e7	0.548684
$821$	−2.27546e7	−1.17818	−0.589089	−	0.808068i	$-0.700513\pi$
$821$	−2.27546e7	−1.17818	−0.589089	+	0.808068i	$0.700513\pi$
$822$	0	0
$823$	8.91287e6	0.458689	0.229344	−	0.973345i	$-0.426342\pi$
$823$	8.91287e6	0.458689	0.229344	+	0.973345i	$0.426342\pi$
$824$	−2.09678e7	−1.07581
$825$	0	0
$826$	0	0
$827$	−2.85147e7	−1.44979	−0.724896	−	0.688859i	$-0.758112\pi$
$827$	−2.85147e7	−1.44979	−0.724896	+	0.688859i	$0.758112\pi$
$828$	0	0
$829$	1.53299e7	0.774733	0.387366	−	0.921926i	$-0.373385\pi$
$829$	1.53299e7	0.774733	0.387366	+	0.921926i	$0.373385\pi$
$830$	−6.38024e6	−0.321471
$831$	0	0
$832$	−2.41105e6	−0.120753
$833$	0	0
$834$	0	0
$835$	3.84392e7	1.90791
$836$	−6.31585e6	−0.312548
$837$	0	0
$838$	7.46973e6	0.367447
$839$	−2.17291e7	−1.06571	−0.532853	−	0.846208i	$-0.678880\pi$
$839$	−2.17291e7	−1.06571	−0.532853	+	0.846208i	$0.678880\pi$
$840$	0	0
$841$	3.92279e7	1.91252
$842$	986403.	0.0479484
$843$	0	0
$844$	1.43776e7	0.694753
$845$	2.54754e7	1.22738
$846$	0	0
$847$	0	0
$848$	−2.39853e7	−1.14540
$849$	0	0
$850$	6.74278e6	0.320104
$851$	1.43168e7	0.677674
$852$	0	0
$853$	2.60934e7	1.22789	0.613943	−	0.789350i	$-0.289582\pi$
$853$	2.60934e7	1.22789	0.613943	+	0.789350i	$0.289582\pi$
$854$	0	0
$855$	0	0
$856$	−9.81025e6	−0.457610
$857$	−1.07218e7	−0.498675	−0.249337	−	0.968417i	$-0.580213\pi$
$857$	−1.07218e7	−0.498675	−0.249337	+	0.968417i	$0.580213\pi$
$858$	0	0
$859$	−3.54531e7	−1.63935	−0.819675	−	0.572829i	$-0.805846\pi$
$859$	−3.54531e7	−1.63935	−0.819675	+	0.572829i	$0.805846\pi$
$860$	−1.20014e7	−0.553332
$861$	0	0
$862$	5.68932e6	0.260791
$863$	1.63278e7	0.746278	0.373139	−	0.927775i	$-0.378281\pi$
$863$	1.63278e7	0.746278	0.373139	+	0.927775i	$0.378281\pi$
$864$	0	0
$865$	−7.05180e6	−0.320450
$866$	−33226.3	−0.00150552
$867$	0	0
$868$	0	0
$869$	−1.90143e7	−0.854142
$870$	0	0
$871$	−4.62339e6	−0.206498
$872$	4.39722e6	0.195834
$873$	0	0
$874$	4.68501e6	0.207459
$875$	0	0
$876$	0	0
$877$	1.94566e7	0.854216	0.427108	−	0.904201i	$-0.359532\pi$
$877$	1.94566e7	0.854216	0.427108	+	0.904201i	$0.359532\pi$
$878$	649201.	0.0284213
$879$	0	0
$880$	−1.05597e7	−0.459669
$881$	4.31688e7	1.87383	0.936916	−	0.349555i	$-0.113667\pi$
$881$	4.31688e7	1.87383	0.936916	+	0.349555i	$0.113667\pi$
$882$	0	0
$883$	−7.63478e6	−0.329530	−0.164765	−	0.986333i	$-0.552687\pi$
$883$	−7.63478e6	−0.329530	−0.164765	+	0.986333i	$0.552687\pi$
$884$	−7.74157e6	−0.333195
$885$	0	0
$886$	−2.14824e6	−0.0919387
$887$	−4.08512e6	−0.174340	−0.0871698	−	0.996193i	$-0.527782\pi$
$887$	−4.08512e6	−0.174340	−0.0871698	+	0.996193i	$0.527782\pi$
$888$	0	0
$889$	0	0
$890$	−7.51661e6	−0.318088
$891$	0	0
$892$	1.08374e7	0.456051
$893$	5.31950e6	0.223225
$894$	0	0
$895$	−1.30691e7	−0.545366
$896$	0	0
$897$	0	0
$898$	−4.91531e6	−0.203404
$899$	6.31990e7	2.60802
$900$	0	0
$901$	−5.64508e7	−2.31664
$902$	1.65514e6	0.0677357
$903$	0	0
$904$	1.79343e7	0.729900
$905$	1.64209e7	0.666462
$906$	0	0
$907$	9.00583e6	0.363501	0.181751	−	0.983345i	$-0.441824\pi$
$907$	9.00583e6	0.363501	0.181751	+	0.983345i	$0.441824\pi$
$908$	−8.26165e6	−0.332546
$909$	0	0
$910$	0	0
$911$	2.17438e7	0.868041	0.434020	−	0.900903i	$-0.357095\pi$
$911$	2.17438e7	0.868041	0.434020	+	0.900903i	$0.357095\pi$
$912$	0	0
$913$	9.77650e6	0.388156
$914$	3.62685e6	0.143603
$915$	0	0
$916$	2.87229e7	1.13107
$917$	0	0
$918$	0	0
$919$	−3.84235e7	−1.50075	−0.750374	−	0.661013i	$-0.770127\pi$
$919$	−3.84235e7	−1.50075	−0.750374	+	0.661013i	$0.770127\pi$
$920$	1.85584e7	0.722888
$921$	0	0
$922$	−1.54461e7	−0.598400
$923$	8.62711e6	0.333320
$924$	0	0
$925$	1.31909e7	0.506896
$926$	−1.20784e7	−0.462896
$927$	0	0
$928$	−3.59654e7	−1.37093
$929$	3.67005e7	1.39519	0.697594	−	0.716493i	$-0.254254\pi$
$929$	3.67005e7	1.39519	0.697594	+	0.716493i	$0.254254\pi$
$930$	0	0
$931$	0	0
$932$	−1.74850e7	−0.659366
$933$	0	0
$934$	5.63102e6	0.211213
$935$	−2.48528e7	−0.929708
$936$	0	0
$937$	−1.37847e7	−0.512919	−0.256459	−	0.966555i	$-0.582556\pi$
$937$	−1.37847e7	−0.512919	−0.256459	+	0.966555i	$0.582556\pi$
$938$	0	0
$939$	0	0
$940$	1.00235e7	0.369997
$941$	−6.91322e6	−0.254511	−0.127255	−	0.991870i	$-0.540617\pi$
$941$	−6.91322e6	−0.254511	−0.127255	+	0.991870i	$0.540617\pi$
$942$	0	0
$943$	1.20083e7	0.439747
$944$	−2.13640e7	−0.780283
$945$	0	0
$946$	−1.88022e6	−0.0683094
$947$	1.67481e7	0.606863	0.303431	−	0.952853i	$-0.401868\pi$
$947$	1.67481e7	0.606863	0.303431	+	0.952853i	$0.401868\pi$
$948$	0	0
$949$	1.47361e6	0.0531150
$950$	4.31658e6	0.155178
$951$	0	0
$952$	0	0
$953$	1.09625e7	0.390999	0.195500	−	0.980704i	$-0.437367\pi$
$953$	1.09625e7	0.390999	0.195500	+	0.980704i	$0.437367\pi$
$954$	0	0
$955$	6.62434e7	2.35036
$956$	4.19138e7	1.48324
$957$	0	0
$958$	−8.43213e6	−0.296841
$959$	0	0
$960$	0	0
$961$	3.82302e7	1.33536
$962$	1.54844e6	0.0539456
$963$	0	0
$964$	−1.56422e7	−0.542132
$965$	5.24835e7	1.81428
$966$	0	0
$967$	2.98311e7	1.02589	0.512947	−	0.858420i	$-0.328554\pi$
$967$	2.98311e7	1.02589	0.512947	+	0.858420i	$0.328554\pi$
$968$	1.30149e7	0.446428
$969$	0	0
$970$	1.69099e7	0.577049
$971$	−1.74542e7	−0.594090	−0.297045	−	0.954864i	$-0.596001\pi$
$971$	−1.74542e7	−0.594090	−0.297045	+	0.954864i	$0.596001\pi$
$972$	0	0
$973$	0	0
$974$	−7.93442e6	−0.267989
$975$	0	0
$976$	2.16906e7	0.728864
$977$	−2.21922e7	−0.743812	−0.371906	−	0.928270i	$-0.621296\pi$
$977$	−2.21922e7	−0.743812	−0.371906	+	0.928270i	$0.621296\pi$
$978$	0	0
$979$	1.15178e7	0.384071
$980$	0	0
$981$	0	0
$982$	9.61545e6	0.318193
$983$	−5.05328e7	−1.66798	−0.833988	−	0.551783i	$-0.813948\pi$
$983$	−5.05328e7	−1.66798	−0.833988	+	0.551783i	$0.813948\pi$
$984$	0	0
$985$	4.18127e6	0.137315
$986$	−2.34382e7	−0.767770
$987$	0	0
$988$	−4.95598e6	−0.161524
$989$	−1.36413e7	−0.443472
$990$	0	0
$991$	−7.91699e6	−0.256080	−0.128040	−	0.991769i	$-0.540869\pi$
$991$	−7.91699e6	−0.256080	−0.128040	+	0.991769i	$0.540869\pi$
$992$	−3.80484e7	−1.22760
$993$	0	0
$994$	0	0
$995$	−1.92369e7	−0.615995
$996$	0	0
$997$	−3.28160e7	−1.04556	−0.522778	−	0.852469i	$-0.675104\pi$
$997$	−3.28160e7	−1.04556	−0.522778	+	0.852469i	$0.675104\pi$
$998$	−2.58123e6	−0.0820352
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.be.1.5	yes	8
3.2	odd	2	inner	441.6.a.be.1.4	yes	8
7.6	odd	2	inner	441.6.a.be.1.6	yes	8
21.20	even	2	inner	441.6.a.be.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.6.a.be.1.3	✓	8	21.20	even	2	inner
441.6.a.be.1.4	yes	8	3.2	odd	2	inner
441.6.a.be.1.5	yes	8	1.1	even	1	trivial
441.6.a.be.1.6	yes	8	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72287 q^{2} -29.0317 q^{4} -73.1337 q^{5} -105.150 q^{8} +O(q^{10})\)	\(q+1.72287 q^{2} -29.0317 q^{4} -73.1337 q^{5} -105.150 q^{8} -126.000 q^{10} +193.070 q^{11} +151.500 q^{13} +747.857 q^{16} +1760.12 q^{17} +1126.79 q^{19} +2123.20 q^{20} +332.635 q^{22} +2413.33 q^{23} +2223.54 q^{25} +261.015 q^{26} -7729.11 q^{29} -8176.75 q^{31} +4653.24 q^{32} +3032.45 q^{34} +5932.38 q^{37} +1941.31 q^{38} +7689.98 q^{40} +4975.84 q^{41} -5652.51 q^{43} -5605.17 q^{44} +4157.84 q^{46} +4720.93 q^{47} +3830.86 q^{50} -4398.32 q^{52} -32072.1 q^{53} -14120.0 q^{55} -13316.2 q^{58} -28566.9 q^{59} +29003.7 q^{61} -14087.5 q^{62} -15914.5 q^{64} -11079.8 q^{65} -30517.4 q^{67} -51099.3 q^{68} +56944.5 q^{71} +9726.79 q^{73} +10220.7 q^{74} -32712.7 q^{76} -98483.5 q^{79} -54693.5 q^{80} +8572.71 q^{82} +50637.0 q^{83} -128724. q^{85} -9738.52 q^{86} -20301.3 q^{88} +59655.8 q^{89} -70063.0 q^{92} +8133.54 q^{94} -82406.4 q^{95} -134206. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{4}+O(q^{10}) \) 8 * q + 20 * q^4	\( 8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100}) \) 8 * q + 20 * q^4 - 828 * q^16 - 2384 * q^22 - 2392 * q^25 - 19136 * q^37 - 41184 * q^43 - 13152 * q^46 - 88872 * q^58 - 210812 * q^64 - 42336 * q^67 - 251072 * q^79 - 567664 * q^85 - 88752 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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