LMFDB - Embedded newform 441.6.a.r.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.44622$ 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)

$f(q)$	$=$	$q-5.44622 q^{2} -2.33867 q^{4} +36.0000 q^{5} +187.016 q^{8} +O(q^{10})$	$q-5.44622 q^{2} -2.33867 q^{4} +36.0000 q^{5} +187.016 q^{8} -196.064 q^{10} -184.430 q^{11} -147.872 q^{13} -943.693 q^{16} +1968.38 q^{17} -1892.51 q^{19} -84.1920 q^{20} +1004.45 q^{22} -136.988 q^{23} -1829.00 q^{25} +805.344 q^{26} +1259.58 q^{29} -8969.02 q^{31} -844.949 q^{32} -10720.3 q^{34} +12897.2 q^{37} +10307.0 q^{38} +6732.58 q^{40} +8975.62 q^{41} +13538.9 q^{43} +431.321 q^{44} +746.069 q^{46} -20046.1 q^{47} +9961.14 q^{50} +345.823 q^{52} -9334.33 q^{53} -6639.49 q^{55} -6859.96 q^{58} +8866.46 q^{59} +41148.3 q^{61} +48847.3 q^{62} +34800.0 q^{64} -5323.39 q^{65} -55351.5 q^{67} -4603.39 q^{68} +63866.8 q^{71} -41299.3 q^{73} -70241.1 q^{74} +4425.95 q^{76} +16963.5 q^{79} -33973.0 q^{80} -48883.2 q^{82} -101693. q^{83} +70861.8 q^{85} -73736.0 q^{86} -34491.4 q^{88} +87102.5 q^{89} +320.370 q^{92} +109176. q^{94} -68130.4 q^{95} +118107. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8	$ 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 + 108 * q^10 - 480 * q^11 - 1296 * q^13 - 1679 * q^16 + 936 * q^17 + 216 * q^19 + 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 - 8892 * q^26 - 6372 * q^29 - 9936 * q^31 - 9039 * q^32 - 19440 * q^34 + 11124 * q^37 + 28116 * q^38 + 8964 * q^40 + 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 + 7920 * q^47 - 5487 * q^50 - 44820 * q^52 - 2220 * q^53 - 17280 * q^55 - 71318 * q^58 + 29736 * q^59 + 17280 * q^61 + 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 - 45216 * q^68 + 92280 * q^71 - 56592 * q^73 - 85218 * q^74 + 87372 * q^76 - 56096 * q^79 - 60444 * q^80 + 52272 * q^82 - 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 + 123192 * q^89 + 25536 * q^92 + 345384 * q^94 + 7776 * q^95 - 35856 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.44622	−0.962765	−0.481383	−	0.876511i	$-0.659865\pi$
$2$	−5.44622	−0.962765	−0.481383	+	0.876511i	$0.659865\pi$
$3$	0	0
$3$	0	0
$4$	−2.33867	−0.0730833
$5$	36.0000	0.643988	0.321994	−	0.946742i	$-0.395647\pi$
$5$	36.0000	0.643988	0.321994	+	0.946742i	$0.395647\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	187.016	1.03313
$9$	0	0
$10$	−196.064	−0.620009
$11$	−184.430	−0.459569	−0.229784	−	0.973242i	$-0.573802\pi$
$11$	−184.430	−0.459569	−0.229784	+	0.973242i	$0.573802\pi$
$12$	0	0
$13$	−147.872	−0.242676	−0.121338	−	0.992611i	$-0.538719\pi$
$13$	−147.872	−0.242676	−0.121338	+	0.992611i	$0.538719\pi$
$14$	0	0
$15$	0	0
$16$	−943.693	−0.921576
$17$	1968.38	1.65191	0.825957	−	0.563733i	$-0.190635\pi$
$17$	1968.38	1.65191	0.825957	+	0.563733i	$0.190635\pi$
$18$	0	0
$19$	−1892.51	−1.20269	−0.601346	−	0.798989i	$-0.705369\pi$
$19$	−1892.51	−1.20269	−0.601346	+	0.798989i	$0.705369\pi$
$20$	−84.1920	−0.0470647
$21$	0	0
$22$	1004.45	0.442457
$23$	−136.988	−0.0539963	−0.0269982	−	0.999635i	$-0.508595\pi$
$23$	−136.988	−0.0539963	−0.0269982	+	0.999635i	$0.508595\pi$
$24$	0	0
$25$	−1829.00	−0.585280
$26$	805.344	0.233640
$27$	0	0
$28$	0	0
$29$	1259.58	0.278120	0.139060	−	0.990284i	$-0.455592\pi$
$29$	1259.58	0.278120	0.139060	+	0.990284i	$0.455592\pi$
$30$	0	0
$31$	−8969.02	−1.67626	−0.838129	−	0.545472i	$-0.816350\pi$
$31$	−8969.02	−1.67626	−0.838129	+	0.545472i	$0.816350\pi$
$32$	−844.949	−0.145866
$33$	0	0
$34$	−10720.3	−1.59041
$35$	0	0
$36$	0	0
$37$	12897.2	1.54879	0.774393	−	0.632705i	$-0.218055\pi$
$37$	12897.2	1.54879	0.774393	+	0.632705i	$0.218055\pi$
$38$	10307.0	1.15791
$39$	0	0
$40$	6732.58	0.665321
$41$	8975.62	0.833882	0.416941	−	0.908934i	$-0.363102\pi$
$41$	8975.62	0.833882	0.416941	+	0.908934i	$0.363102\pi$
$42$	0	0
$43$	13538.9	1.11664	0.558320	−	0.829626i	$-0.311446\pi$
$43$	13538.9	1.11664	0.558320	+	0.829626i	$0.311446\pi$
$44$	431.321	0.0335868
$45$	0	0
$46$	746.069	0.0519858
$47$	−20046.1	−1.32369	−0.661845	−	0.749641i	$-0.730226\pi$
$47$	−20046.1	−1.32369	−0.661845	+	0.749641i	$0.730226\pi$
$48$	0	0
$49$	0	0
$50$	9961.14	0.563487
$51$	0	0
$52$	345.823	0.0177356
$53$	−9334.33	−0.456450	−0.228225	−	0.973608i	$-0.573292\pi$
$53$	−9334.33	−0.456450	−0.228225	+	0.973608i	$0.573292\pi$
$54$	0	0
$55$	−6639.49	−0.295956
$56$	0	0
$57$	0	0
$58$	−6859.96	−0.267764
$59$	8866.46	0.331605	0.165802	−	0.986159i	$-0.446979\pi$
$59$	8866.46	0.331605	0.165802	+	0.986159i	$0.446979\pi$
$60$	0	0
$61$	41148.3	1.41588	0.707942	−	0.706271i	$-0.249624\pi$
$61$	41148.3	1.41588	0.707942	+	0.706271i	$0.249624\pi$
$62$	48847.3	1.61384
$63$	0	0
$64$	34800.0	1.06201
$65$	−5323.39	−0.156281
$66$	0	0
$67$	−55351.5	−1.50641	−0.753204	−	0.657787i	$-0.771493\pi$
$67$	−55351.5	−1.50641	−0.753204	+	0.657787i	$0.771493\pi$
$68$	−4603.39	−0.120727
$69$	0	0
$70$	0	0
$71$	63866.8	1.50359	0.751794	−	0.659398i	$-0.229189\pi$
$71$	63866.8	1.50359	0.751794	+	0.659398i	$0.229189\pi$
$72$	0	0
$73$	−41299.3	−0.907060	−0.453530	−	0.891241i	$-0.649835\pi$
$73$	−41299.3	−0.907060	−0.453530	+	0.891241i	$0.649835\pi$
$74$	−70241.1	−1.49112
$75$	0	0
$76$	4425.95	0.0878968
$77$	0	0
$78$	0	0
$79$	16963.5	0.305808	0.152904	−	0.988241i	$-0.451138\pi$
$79$	16963.5	0.305808	0.152904	+	0.988241i	$0.451138\pi$
$80$	−33973.0	−0.593483
$81$	0	0
$82$	−48883.2	−0.802833
$83$	−101693.	−1.62030	−0.810150	−	0.586223i	$-0.800614\pi$
$83$	−101693.	−1.62030	−0.810150	+	0.586223i	$0.800614\pi$
$84$	0	0
$85$	70861.8	1.06381
$86$	−73736.0	−1.07506
$87$	0	0
$88$	−34491.4	−0.474793
$89$	87102.5	1.16562	0.582808	−	0.812610i	$-0.301954\pi$
$89$	87102.5	1.16562	0.582808	+	0.812610i	$0.301954\pi$
$90$	0	0
$91$	0	0
$92$	320.370	0.00394623
$93$	0	0
$94$	109176.	1.27440
$95$	−68130.4	−0.774519
$96$	0	0
$97$	118107.	1.27452	0.637258	−	0.770650i	$-0.280068\pi$
$97$	118107.	1.27452	0.637258	+	0.770650i	$0.280068\pi$
$98$	0	0
$99$	0	0
$100$	4277.42	0.0427742
$101$	22232.7	0.216865	0.108432	−	0.994104i	$-0.465417\pi$
$101$	22232.7	0.216865	0.108432	+	0.994104i	$0.465417\pi$
$102$	0	0
$103$	−135931.	−1.26248	−0.631239	−	0.775588i	$-0.717453\pi$
$103$	−135931.	−1.26248	−0.631239	+	0.775588i	$0.717453\pi$
$104$	−27654.4	−0.250716
$105$	0	0
$106$	50836.8	0.439454
$107$	−117626.	−0.993218	−0.496609	−	0.867974i	$-0.665422\pi$
$107$	−117626.	−0.993218	−0.496609	+	0.867974i	$0.665422\pi$
$108$	0	0
$109$	−34664.2	−0.279457	−0.139729	−	0.990190i	$-0.544623\pi$
$109$	−34664.2	−0.279457	−0.139729	+	0.990190i	$0.544623\pi$
$110$	36160.1	0.284937
$111$	0	0
$112$	0	0
$113$	26584.5	0.195854	0.0979269	−	0.995194i	$-0.468779\pi$
$113$	26584.5	0.195854	0.0979269	+	0.995194i	$0.468779\pi$
$114$	0	0
$115$	−4931.58	−0.0347730
$116$	−2945.74	−0.0203259
$117$	0	0
$118$	−48288.7	−0.319257
$119$	0	0
$120$	0	0
$121$	−127036.	−0.788797
$122$	−224103.	−1.36316
$123$	0	0
$124$	20975.6	0.122507
$125$	−178344.	−1.02090
$126$	0	0
$127$	−137111.	−0.754334	−0.377167	−	0.926145i	$-0.623102\pi$
$127$	−137111.	−0.754334	−0.377167	+	0.926145i	$0.623102\pi$
$128$	−162490.	−0.876600
$129$	0	0
$130$	28992.4	0.150462
$131$	−54089.2	−0.275380	−0.137690	−	0.990475i	$-0.543968\pi$
$131$	−54089.2	−0.275380	−0.137690	+	0.990475i	$0.543968\pi$
$132$	0	0
$133$	0	0
$134$	301457.	1.45032
$135$	0	0
$136$	368119.	1.70664
$137$	−422849.	−1.92479	−0.962395	−	0.271653i	$-0.912430\pi$
$137$	−422849.	−1.92479	−0.962395	+	0.271653i	$0.912430\pi$
$138$	0	0
$139$	9913.38	0.0435196	0.0217598	−	0.999763i	$-0.493073\pi$
$139$	9913.38	0.0435196	0.0217598	+	0.999763i	$0.493073\pi$
$140$	0	0
$141$	0	0
$142$	−347833.	−1.44760
$143$	27272.1	0.111526
$144$	0	0
$145$	45345.0	0.179106
$146$	224925.	0.873285
$147$	0	0
$148$	−30162.3	−0.113190
$149$	−505462.	−1.86519	−0.932595	−	0.360925i	$-0.882461\pi$
$149$	−505462.	−1.86519	−0.932595	+	0.360925i	$0.882461\pi$
$150$	0	0
$151$	−193103.	−0.689201	−0.344601	−	0.938749i	$-0.611986\pi$
$151$	−193103.	−0.689201	−0.344601	+	0.938749i	$0.611986\pi$
$152$	−353930.	−1.24253
$153$	0	0
$154$	0	0
$155$	−322885.	−1.07949
$156$	0	0
$157$	−264923.	−0.857770	−0.428885	−	0.903359i	$-0.641093\pi$
$157$	−264923.	−0.857770	−0.428885	+	0.903359i	$0.641093\pi$
$158$	−92387.1	−0.294421
$159$	0	0
$160$	−30418.1	−0.0939362
$161$	0	0
$162$	0	0
$163$	−539093.	−1.58926	−0.794629	−	0.607095i	$-0.792335\pi$
$163$	−539093.	−1.58926	−0.794629	+	0.607095i	$0.792335\pi$
$164$	−20991.0	−0.0609429
$165$	0	0
$166$	553842.	1.55997
$167$	218748.	0.606949	0.303475	−	0.952840i	$-0.401853\pi$
$167$	218748.	0.606949	0.303475	+	0.952840i	$0.401853\pi$
$168$	0	0
$169$	−349427.	−0.941108
$170$	−385929.	−1.02420
$171$	0	0
$172$	−31663.0	−0.0816078
$173$	590201.	1.49929	0.749644	−	0.661842i	$-0.230225\pi$
$173$	590201.	1.49929	0.749644	+	0.661842i	$0.230225\pi$
$174$	0	0
$175$	0	0
$176$	174046.	0.423527
$177$	0	0
$178$	−474380.	−1.12222
$179$	217352.	0.507026	0.253513	−	0.967332i	$-0.418414\pi$
$179$	217352.	0.507026	0.253513	+	0.967332i	$0.418414\pi$
$180$	0	0
$181$	−188109.	−0.426790	−0.213395	−	0.976966i	$-0.568452\pi$
$181$	−188109.	−0.426790	−0.213395	+	0.976966i	$0.568452\pi$
$182$	0	0
$183$	0	0
$184$	−25619.0	−0.0557851
$185$	464300.	0.997399
$186$	0	0
$187$	−363029.	−0.759168
$188$	46881.2	0.0967396
$189$	0	0
$190$	371053.	0.745680
$191$	−173523.	−0.344170	−0.172085	−	0.985082i	$-0.555050\pi$
$191$	−173523.	−0.344170	−0.172085	+	0.985082i	$0.555050\pi$
$192$	0	0
$193$	117781.	0.227606	0.113803	−	0.993503i	$-0.463697\pi$
$193$	117781.	0.227606	0.113803	+	0.993503i	$0.463697\pi$
$194$	−643236.	−1.22706
$195$	0	0
$196$	0	0
$197$	−224734.	−0.412575	−0.206288	−	0.978491i	$-0.566138\pi$
$197$	−224734.	−0.412575	−0.206288	+	0.978491i	$0.566138\pi$
$198$	0	0
$199$	−740273.	−1.32513	−0.662567	−	0.749003i	$-0.730533\pi$
$199$	−740273.	−1.32513	−0.662567	+	0.749003i	$0.730533\pi$
$200$	−342052.	−0.604669
$201$	0	0
$202$	−121084.	−0.208790
$203$	0	0
$204$	0	0
$205$	323122.	0.537010
$206$	740308.	1.21547
$207$	0	0
$208$	139546.	0.223645
$209$	349036.	0.552720
$210$	0	0
$211$	−705896.	−1.09153	−0.545763	−	0.837939i	$-0.683760\pi$
$211$	−705896.	−1.09153	−0.545763	+	0.837939i	$0.683760\pi$
$212$	21829.9	0.0333589
$213$	0	0
$214$	640618.	0.956236
$215$	487402.	0.719102
$216$	0	0
$217$	0	0
$218$	188789.	0.269052
$219$	0	0
$220$	15527.5	0.0216295
$221$	−291069.	−0.400881
$222$	0	0
$223$	−42214.3	−0.0568456	−0.0284228	−	0.999596i	$-0.509048\pi$
$223$	−42214.3	−0.0568456	−0.0284228	+	0.999596i	$0.509048\pi$
$224$	0	0
$225$	0	0
$226$	−144785.	−0.188561
$227$	742403.	0.956258	0.478129	−	0.878290i	$-0.341315\pi$
$227$	742403.	0.956258	0.478129	+	0.878290i	$0.341315\pi$
$228$	0	0
$229$	−941236.	−1.18607	−0.593034	−	0.805177i	$-0.702070\pi$
$229$	−941236.	−1.18607	−0.593034	+	0.805177i	$0.702070\pi$
$230$	26858.5	0.0334782
$231$	0	0
$232$	235562.	0.287333
$233$	512331.	0.618246	0.309123	−	0.951022i	$-0.399965\pi$
$233$	512331.	0.618246	0.309123	+	0.951022i	$0.399965\pi$
$234$	0	0
$235$	−721661.	−0.852440
$236$	−20735.7	−0.0242348
$237$	0	0
$238$	0	0
$239$	115323.	0.130593	0.0652966	−	0.997866i	$-0.479201\pi$
$239$	115323.	0.130593	0.0652966	+	0.997866i	$0.479201\pi$
$240$	0	0
$241$	909864.	1.00910	0.504549	−	0.863383i	$-0.331659\pi$
$241$	909864.	1.00910	0.504549	+	0.863383i	$0.331659\pi$
$242$	691869.	0.759426
$243$	0	0
$244$	−96232.2	−0.103477
$245$	0	0
$246$	0	0
$247$	279850.	0.291865
$248$	−1.67735e6	−1.73179
$249$	0	0
$250$	971301.	0.982888
$251$	−321264.	−0.321868	−0.160934	−	0.986965i	$-0.551451\pi$
$251$	−321264.	−0.321868	−0.160934	+	0.986965i	$0.551451\pi$
$252$	0	0
$253$	25264.8	0.0248150
$254$	746738.	0.726246
$255$	0	0
$256$	−228642.	−0.218050
$257$	−556492.	−0.525565	−0.262782	−	0.964855i	$-0.584640\pi$
$257$	−556492.	−0.525565	−0.262782	+	0.964855i	$0.584640\pi$
$258$	0	0
$259$	0	0
$260$	12449.6	0.0114215
$261$	0	0
$262$	294582.	0.265126
$263$	233379.	0.208052	0.104026	−	0.994575i	$-0.466827\pi$
$263$	233379.	0.208052	0.104026	+	0.994575i	$0.466827\pi$
$264$	0	0
$265$	−336036.	−0.293948
$266$	0	0
$267$	0	0
$268$	129449.	0.110093
$269$	−1.47857e6	−1.24584	−0.622919	−	0.782286i	$-0.714053\pi$
$269$	−1.47857e6	−1.24584	−0.622919	+	0.782286i	$0.714053\pi$
$270$	0	0
$271$	−177762.	−0.147033	−0.0735165	−	0.997294i	$-0.523422\pi$
$271$	−177762.	−0.147033	−0.0735165	+	0.997294i	$0.523422\pi$
$272$	−1.85755e6	−1.52236
$273$	0	0
$274$	2.30293e6	1.85312
$275$	337323.	0.268976
$276$	0	0
$277$	−1.22252e6	−0.957318	−0.478659	−	0.878001i	$-0.658877\pi$
$277$	−1.22252e6	−0.957318	−0.478659	+	0.878001i	$0.658877\pi$
$278$	−53990.5	−0.0418991
$279$	0	0
$280$	0	0
$281$	−177799.	−0.134327	−0.0671636	−	0.997742i	$-0.521395\pi$
$281$	−177799.	−0.134327	−0.0671636	+	0.997742i	$0.521395\pi$
$282$	0	0
$283$	1.09052e6	0.809406	0.404703	−	0.914448i	$-0.367375\pi$
$283$	1.09052e6	0.809406	0.404703	+	0.914448i	$0.367375\pi$
$284$	−149363.	−0.109887
$285$	0	0
$286$	−148530.	−0.107374
$287$	0	0
$288$	0	0
$289$	2.45468e6	1.72882
$290$	−246959.	−0.172437
$291$	0	0
$292$	96585.3	0.0662909
$293$	545742.	0.371380	0.185690	−	0.982608i	$-0.440548\pi$
$293$	545742.	0.371380	0.185690	+	0.982608i	$0.440548\pi$
$294$	0	0
$295$	319193.	0.213549
$296$	2.41198e6	1.60009
$297$	0	0
$298$	2.75286e6	1.79574
$299$	20256.8	0.0131036
$300$	0	0
$301$	0	0
$302$	1.05168e6	0.663539
$303$	0	0
$304$	1.78595e6	1.10837
$305$	1.48134e6	0.911811
$306$	0	0
$307$	3.29000e6	1.99228	0.996138	−	0.0878052i	$-0.0279853\pi$
$307$	3.29000e6	1.99228	0.996138	+	0.0878052i	$0.0279853\pi$
$308$	0	0
$309$	0	0
$310$	1.75850e6	1.03929
$311$	−699712.	−0.410221	−0.205111	−	0.978739i	$-0.565755\pi$
$311$	−699712.	−0.410221	−0.205111	+	0.978739i	$0.565755\pi$
$312$	0	0
$313$	−623219.	−0.359567	−0.179783	−	0.983706i	$-0.557540\pi$
$313$	−623219.	−0.359567	−0.179783	+	0.983706i	$0.557540\pi$
$314$	1.44283e6	0.825831
$315$	0	0
$316$	−39672.0	−0.0223494
$317$	639244.	0.357288	0.178644	−	0.983914i	$-0.442829\pi$
$317$	639244.	0.357288	0.178644	+	0.983914i	$0.442829\pi$
$318$	0	0
$319$	−232305.	−0.127815
$320$	1.25280e6	0.683922
$321$	0	0
$322$	0	0
$323$	−3.72519e6	−1.98675
$324$	0	0
$325$	270458.	0.142034
$326$	2.93602e6	1.53008
$327$	0	0
$328$	1.67858e6	0.861506
$329$	0	0
$330$	0	0
$331$	1.40963e6	0.707190	0.353595	−	0.935399i	$-0.384959\pi$
$331$	1.40963e6	0.707190	0.353595	+	0.935399i	$0.384959\pi$
$332$	237826.	0.118417
$333$	0	0
$334$	−1.19135e6	−0.584350
$335$	−1.99265e6	−0.970108
$336$	0	0
$337$	−1.55677e6	−0.746704	−0.373352	−	0.927690i	$-0.621792\pi$
$337$	−1.55677e6	−0.746704	−0.373352	+	0.927690i	$0.621792\pi$
$338$	1.90306e6	0.906066
$339$	0	0
$340$	−165722.	−0.0777469
$341$	1.65416e6	0.770356
$342$	0	0
$343$	0	0
$344$	2.53200e6	1.15363
$345$	0	0
$346$	−3.21437e6	−1.44346
$347$	−248297.	−0.110700	−0.0553500	−	0.998467i	$-0.517627\pi$
$347$	−248297.	−0.110700	−0.0553500	+	0.998467i	$0.517627\pi$
$348$	0	0
$349$	−1.86169e6	−0.818171	−0.409086	−	0.912496i	$-0.634152\pi$
$349$	−1.86169e6	−0.818171	−0.409086	+	0.912496i	$0.634152\pi$
$350$	0	0
$351$	0	0
$352$	155834.	0.0670356
$353$	1.77128e6	0.756574	0.378287	−	0.925688i	$-0.376513\pi$
$353$	1.77128e6	0.756574	0.378287	+	0.925688i	$0.376513\pi$
$354$	0	0
$355$	2.29920e6	0.968292
$356$	−203704.	−0.0851871
$357$	0	0
$358$	−1.18375e6	−0.488147
$359$	4.47560e6	1.83280	0.916401	−	0.400262i	$-0.131081\pi$
$359$	4.47560e6	1.83280	0.916401	+	0.400262i	$0.131081\pi$
$360$	0	0
$361$	1.10550e6	0.446469
$362$	1.02449e6	0.410898
$363$	0	0
$364$	0	0
$365$	−1.48678e6	−0.584135
$366$	0	0
$367$	−3.17097e6	−1.22893	−0.614465	−	0.788944i	$-0.710628\pi$
$367$	−3.17097e6	−1.22893	−0.614465	+	0.788944i	$0.710628\pi$
$368$	129275.	0.0497617
$369$	0	0
$370$	−2.52868e6	−0.960261
$371$	0	0
$372$	0	0
$373$	−2.31179e6	−0.860353	−0.430177	−	0.902745i	$-0.641549\pi$
$373$	−2.31179e6	−0.860353	−0.430177	+	0.902745i	$0.641549\pi$
$374$	1.97714e6	0.730900
$375$	0	0
$376$	−3.74895e6	−1.36754
$377$	−186257.	−0.0674931
$378$	0	0
$379$	−591840.	−0.211644	−0.105822	−	0.994385i	$-0.533747\pi$
$379$	−591840.	−0.211644	−0.105822	+	0.994385i	$0.533747\pi$
$380$	159334.	0.0566044
$381$	0	0
$382$	945042.	0.331354
$383$	−3.59766e6	−1.25321	−0.626605	−	0.779337i	$-0.715556\pi$
$383$	−3.59766e6	−1.25321	−0.626605	+	0.779337i	$0.715556\pi$
$384$	0	0
$385$	0	0
$386$	−641464.	−0.219131
$387$	0	0
$388$	−276212.	−0.0931459
$389$	2.24746e6	0.753040	0.376520	−	0.926409i	$-0.377121\pi$
$389$	2.24746e6	0.753040	0.376520	+	0.926409i	$0.377121\pi$
$390$	0	0
$391$	−269646.	−0.0891973
$392$	0	0
$393$	0	0
$394$	1.22395e6	0.397213
$395$	610687.	0.196936
$396$	0	0
$397$	4.58670e6	1.46058	0.730288	−	0.683139i	$-0.239386\pi$
$397$	4.58670e6	1.46058	0.730288	+	0.683139i	$0.239386\pi$
$398$	4.03169e6	1.27579
$399$	0	0
$400$	1.72602e6	0.539380
$401$	−6.00432e6	−1.86468	−0.932338	−	0.361589i	$-0.882234\pi$
$401$	−6.00432e6	−1.86468	−0.932338	+	0.361589i	$0.882234\pi$
$402$	0	0
$403$	1.32627e6	0.406788
$404$	−51994.9	−0.0158492
$405$	0	0
$406$	0	0
$407$	−2.37864e6	−0.711774
$408$	0	0
$409$	3.55340e6	1.05035	0.525177	−	0.850993i	$-0.323999\pi$
$409$	3.55340e6	1.05035	0.525177	+	0.850993i	$0.323999\pi$
$410$	−1.75980e6	−0.517014
$411$	0	0
$412$	317896.	0.0922661
$413$	0	0
$414$	0	0
$415$	−3.66094e6	−1.04345
$416$	124944.	0.0353983
$417$	0	0
$418$	−1.90093e6	−0.532139
$419$	2.01375e6	0.560365	0.280182	−	0.959947i	$-0.409605\pi$
$419$	2.01375e6	0.560365	0.280182	+	0.959947i	$0.409605\pi$
$420$	0	0
$421$	−5.89987e6	−1.62232	−0.811161	−	0.584823i	$-0.801164\pi$
$421$	−5.89987e6	−1.62232	−0.811161	+	0.584823i	$0.801164\pi$
$422$	3.84446e6	1.05088
$423$	0	0
$424$	−1.74567e6	−0.471571
$425$	−3.60017e6	−0.966832
$426$	0	0
$427$	0	0
$428$	275088.	0.0725877
$429$	0	0
$430$	−2.65450e6	−0.692327
$431$	3.81048e6	0.988066	0.494033	−	0.869443i	$-0.335522\pi$
$431$	3.81048e6	0.988066	0.494033	+	0.869443i	$0.335522\pi$
$432$	0	0
$433$	−6.59449e6	−1.69029	−0.845146	−	0.534536i	$-0.820487\pi$
$433$	−6.59449e6	−1.69029	−0.845146	+	0.534536i	$0.820487\pi$
$434$	0	0
$435$	0	0
$436$	81068.1	0.0204237
$437$	259252.	0.0649410
$438$	0	0
$439$	4.55028e6	1.12688	0.563438	−	0.826158i	$-0.309478\pi$
$439$	4.55028e6	1.12688	0.563438	+	0.826158i	$0.309478\pi$
$440$	−1.24169e6	−0.305761
$441$	0	0
$442$	1.58523e6	0.385954
$443$	−5.11537e6	−1.23842	−0.619210	−	0.785225i	$-0.712547\pi$
$443$	−5.11537e6	−1.23842	−0.619210	+	0.785225i	$0.712547\pi$
$444$	0	0
$445$	3.13569e6	0.750643
$446$	229908.	0.0547290
$447$	0	0
$448$	0	0
$449$	−5.95600e6	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$449$	−5.95600e6	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$450$	0	0
$451$	−1.65537e6	−0.383226
$452$	−62172.2	−0.0143136
$453$	0	0
$454$	−4.04329e6	−0.920652
$455$	0	0
$456$	0	0
$457$	−2.59834e6	−0.581976	−0.290988	−	0.956727i	$-0.593984\pi$
$457$	−2.59834e6	−0.581976	−0.290988	+	0.956727i	$0.593984\pi$
$458$	5.12618e6	1.14191
$459$	0	0
$460$	11533.3	0.00254132
$461$	4.51513e6	0.989505	0.494752	−	0.869034i	$-0.335259\pi$
$461$	4.51513e6	0.989505	0.494752	+	0.869034i	$0.335259\pi$
$462$	0	0
$463$	−5.55129e6	−1.20349	−0.601744	−	0.798689i	$-0.705527\pi$
$463$	−5.55129e6	−1.20349	−0.601744	+	0.798689i	$0.705527\pi$
$464$	−1.18866e6	−0.256308
$465$	0	0
$466$	−2.79027e6	−0.595225
$467$	−7.95350e6	−1.68759	−0.843794	−	0.536668i	$-0.819683\pi$
$467$	−7.95350e6	−1.68759	−0.843794	+	0.536668i	$0.819683\pi$
$468$	0	0
$469$	0	0
$470$	3.93033e6	0.820699
$471$	0	0
$472$	1.65817e6	0.342590
$473$	−2.49699e6	−0.513173
$474$	0	0
$475$	3.46140e6	0.703912
$476$	0	0
$477$	0	0
$478$	−628074.	−0.125731
$479$	−6.56370e6	−1.30710	−0.653552	−	0.756882i	$-0.726722\pi$
$479$	−6.56370e6	−1.30710	−0.653552	+	0.756882i	$0.726722\pi$
$480$	0	0
$481$	−1.90714e6	−0.375854
$482$	−4.95532e6	−0.971525
$483$	0	0
$484$	297096.	0.0576479
$485$	4.25185e6	0.820773
$486$	0	0
$487$	4.66370e6	0.891062	0.445531	−	0.895266i	$-0.353015\pi$
$487$	4.66370e6	0.891062	0.445531	+	0.895266i	$0.353015\pi$
$488$	7.69539e6	1.46279
$489$	0	0
$490$	0	0
$491$	917227.	0.171701	0.0858506	−	0.996308i	$-0.472639\pi$
$491$	917227.	0.171701	0.0858506	+	0.996308i	$0.472639\pi$
$492$	0	0
$493$	2.47934e6	0.459430
$494$	−1.52412e6	−0.280998
$495$	0	0
$496$	8.46401e6	1.54480
$497$	0	0
$498$	0	0
$499$	313436.	0.0563504	0.0281752	−	0.999603i	$-0.491030\pi$
$499$	313436.	0.0563504	0.0281752	+	0.999603i	$0.491030\pi$
$500$	417087.	0.0746108
$501$	0	0
$502$	1.74967e6	0.309883
$503$	−3.74110e6	−0.659294	−0.329647	−	0.944104i	$-0.606930\pi$
$503$	−3.74110e6	−0.659294	−0.329647	+	0.944104i	$0.606930\pi$
$504$	0	0
$505$	800378.	0.139658
$506$	−137598.	−0.0238910
$507$	0	0
$508$	320657.	0.0551292
$509$	1.03017e7	1.76244	0.881220	−	0.472707i	$-0.156723\pi$
$509$	1.03017e7	1.76244	0.881220	+	0.472707i	$0.156723\pi$
$510$	0	0
$511$	0	0
$512$	6.44492e6	1.08653
$513$	0	0
$514$	3.03078e6	0.505995
$515$	−4.89350e6	−0.813021
$516$	0	0
$517$	3.69711e6	0.608326
$518$	0	0
$519$	0	0
$520$	−995560.	−0.161458
$521$	−6.57325e6	−1.06093	−0.530464	−	0.847707i	$-0.677982\pi$
$521$	−6.57325e6	−1.06093	−0.530464	+	0.847707i	$0.677982\pi$
$522$	0	0
$523$	−8.36532e6	−1.33730	−0.668650	−	0.743578i	$-0.733127\pi$
$523$	−8.36532e6	−1.33730	−0.668650	+	0.743578i	$0.733127\pi$
$524$	126497.	0.0201257
$525$	0	0
$526$	−1.27103e6	−0.200306
$527$	−1.76545e7	−2.76904
$528$	0	0
$529$	−6.41758e6	−0.997084
$530$	1.83013e6	0.283003
$531$	0	0
$532$	0	0
$533$	−1.32724e6	−0.202364
$534$	0	0
$535$	−4.23454e6	−0.639620
$536$	−1.03516e7	−1.55631
$537$	0	0
$538$	8.05263e6	1.19945
$539$	0	0
$540$	0	0
$541$	−8.06623e6	−1.18489	−0.592444	−	0.805612i	$-0.701837\pi$
$541$	−8.06623e6	−1.18489	−0.592444	+	0.805612i	$0.701837\pi$
$542$	968129.	0.141558
$543$	0	0
$544$	−1.66318e6	−0.240959
$545$	−1.24791e6	−0.179967
$546$	0	0
$547$	3.90775e6	0.558416	0.279208	−	0.960231i	$-0.409928\pi$
$547$	3.90775e6	0.558416	0.279208	+	0.960231i	$0.409928\pi$
$548$	988902.	0.140670
$549$	0	0
$550$	−1.83714e6	−0.258961
$551$	−2.38377e6	−0.334492
$552$	0	0
$553$	0	0
$554$	6.65811e6	0.921672
$555$	0	0
$556$	−23184.1	−0.00318056
$557$	1.15862e7	1.58235	0.791177	−	0.611588i	$-0.209469\pi$
$557$	1.15862e7	1.58235	0.791177	+	0.611588i	$0.209469\pi$
$558$	0	0
$559$	−2.00203e6	−0.270982
$560$	0	0
$561$	0	0
$562$	968334.	0.129326
$563$	1.91176e6	0.254192	0.127096	−	0.991890i	$-0.459434\pi$
$563$	1.91176e6	0.254192	0.127096	+	0.991890i	$0.459434\pi$
$564$	0	0
$565$	957041.	0.126127
$566$	−5.93920e6	−0.779268
$567$	0	0
$568$	1.19441e7	1.55340
$569$	−8.16817e6	−1.05766	−0.528828	−	0.848729i	$-0.677368\pi$
$569$	−8.16817e6	−1.05766	−0.528828	+	0.848729i	$0.677368\pi$
$570$	0	0
$571$	−7.46593e6	−0.958283	−0.479141	−	0.877738i	$-0.659052\pi$
$571$	−7.46593e6	−0.958283	−0.479141	+	0.877738i	$0.659052\pi$
$572$	−63780.3	−0.00815072
$573$	0	0
$574$	0	0
$575$	250552.	0.0316030
$576$	0	0
$577$	6.88438e6	0.860845	0.430423	−	0.902627i	$-0.358365\pi$
$577$	6.88438e6	0.860845	0.430423	+	0.902627i	$0.358365\pi$
$578$	−1.33687e7	−1.66445
$579$	0	0
$580$	−106047.	−0.0130896
$581$	0	0
$582$	0	0
$583$	1.72153e6	0.209770
$584$	−7.72363e6	−0.937108
$585$	0	0
$586$	−2.97223e6	−0.357551
$587$	−8.91086e6	−1.06739	−0.533696	−	0.845676i	$-0.679197\pi$
$587$	−8.91086e6	−1.06739	−0.533696	+	0.845676i	$0.679197\pi$
$588$	0	0
$589$	1.69740e7	2.01602
$590$	−1.73839e6	−0.205598
$591$	0	0
$592$	−1.21710e7	−1.42732
$593$	1.23430e7	1.44140	0.720700	−	0.693247i	$-0.243820\pi$
$593$	1.23430e7	1.44140	0.720700	+	0.693247i	$0.243820\pi$
$594$	0	0
$595$	0	0
$596$	1.18211e6	0.136314
$597$	0	0
$598$	−110323.	−0.0126157
$599$	−9.05732e6	−1.03141	−0.515707	−	0.856765i	$-0.672471\pi$
$599$	−9.05732e6	−1.03141	−0.515707	+	0.856765i	$0.672471\pi$
$600$	0	0
$601$	−7.41700e6	−0.837611	−0.418805	−	0.908076i	$-0.637551\pi$
$601$	−7.41700e6	−0.837611	−0.418805	+	0.908076i	$0.637551\pi$
$602$	0	0
$603$	0	0
$604$	451603.	0.0503691
$605$	−4.57331e6	−0.507975
$606$	0	0
$607$	5.77813e6	0.636525	0.318263	−	0.948003i	$-0.396901\pi$
$607$	5.77813e6	0.636525	0.318263	+	0.948003i	$0.396901\pi$
$608$	1.59908e6	0.175432
$609$	0	0
$610$	−8.06770e6	−0.877860
$611$	2.96426e6	0.321228
$612$	0	0
$613$	6.29264e6	0.676366	0.338183	−	0.941080i	$-0.390188\pi$
$613$	6.29264e6	0.676366	0.338183	+	0.941080i	$0.390188\pi$
$614$	−1.79180e7	−1.91809
$615$	0	0
$616$	0	0
$617$	9.79133e6	1.03545	0.517725	−	0.855547i	$-0.326779\pi$
$617$	9.79133e6	1.03545	0.517725	+	0.855547i	$0.326779\pi$
$618$	0	0
$619$	1.32677e7	1.39178	0.695889	−	0.718150i	$-0.255011\pi$
$619$	1.32677e7	1.39178	0.695889	+	0.718150i	$0.255011\pi$
$620$	755120.	0.0788927
$621$	0	0
$622$	3.81079e6	0.394947
$623$	0	0
$624$	0	0
$625$	−704759.	−0.0721673
$626$	3.39419e6	0.346178
$627$	0	0
$628$	619567.	0.0626886
$629$	2.53867e7	2.55846
$630$	0	0
$631$	4.41233e6	0.441158	0.220579	−	0.975369i	$-0.429205\pi$
$631$	4.41233e6	0.441158	0.220579	+	0.975369i	$0.429205\pi$
$632$	3.17245e6	0.315938
$633$	0	0
$634$	−3.48146e6	−0.343984
$635$	−4.93600e6	−0.485782
$636$	0	0
$637$	0	0
$638$	1.26518e6	0.123056
$639$	0	0
$640$	−5.84964e6	−0.564520
$641$	−8.13035e6	−0.781564	−0.390782	−	0.920483i	$-0.627795\pi$
$641$	−8.13035e6	−0.781564	−0.390782	+	0.920483i	$0.627795\pi$
$642$	0	0
$643$	3.12961e6	0.298513	0.149256	−	0.988799i	$-0.452312\pi$
$643$	3.12961e6	0.298513	0.149256	+	0.988799i	$0.452312\pi$
$644$	0	0
$645$	0	0
$646$	2.02882e7	1.91277
$647$	1.34660e7	1.26467	0.632336	−	0.774695i	$-0.282096\pi$
$647$	1.34660e7	1.26467	0.632336	+	0.774695i	$0.282096\pi$
$648$	0	0
$649$	−1.63524e6	−0.152395
$650$	−1.47297e6	−0.136745
$651$	0	0
$652$	1.26076e6	0.116148
$653$	−1.44772e7	−1.32862	−0.664312	−	0.747455i	$-0.731275\pi$
$653$	−1.44772e7	−1.32862	−0.664312	+	0.747455i	$0.731275\pi$
$654$	0	0
$655$	−1.94721e6	−0.177341
$656$	−8.47023e6	−0.768485
$657$	0	0
$658$	0	0
$659$	−432708.	−0.0388134	−0.0194067	−	0.999812i	$-0.506178\pi$
$659$	−432708.	−0.0388134	−0.0194067	+	0.999812i	$0.506178\pi$
$660$	0	0
$661$	29965.2	0.00266756	0.00133378	−	0.999999i	$-0.499575\pi$
$661$	29965.2	0.00266756	0.00133378	+	0.999999i	$0.499575\pi$
$662$	−7.67718e6	−0.680858
$663$	0	0
$664$	−1.90182e7	−1.67398
$665$	0	0
$666$	0	0
$667$	−172548.	−0.0150174
$668$	−511578.	−0.0443579
$669$	0	0
$670$	1.08524e7	0.933986
$671$	−7.58899e6	−0.650696
$672$	0	0
$673$	−6.71329e6	−0.571344	−0.285672	−	0.958327i	$-0.592217\pi$
$673$	−6.71329e6	−0.571344	−0.285672	+	0.958327i	$0.592217\pi$
$674$	8.47849e6	0.718901
$675$	0	0
$676$	817193.	0.0687793
$677$	−4.45929e6	−0.373933	−0.186967	−	0.982366i	$-0.559866\pi$
$677$	−4.45929e6	−0.373933	−0.186967	+	0.982366i	$0.559866\pi$
$678$	0	0
$679$	0	0
$680$	1.32523e7	1.09905
$681$	0	0
$682$	−9.00892e6	−0.741672
$683$	1.79839e7	1.47514	0.737569	−	0.675272i	$-0.235974\pi$
$683$	1.79839e7	1.47514	0.737569	+	0.675272i	$0.235974\pi$
$684$	0	0
$685$	−1.52225e7	−1.23954
$686$	0	0
$687$	0	0
$688$	−1.27766e7	−1.02907
$689$	1.38029e6	0.110770
$690$	0	0
$691$	−2.50935e6	−0.199925	−0.0999624	−	0.994991i	$-0.531872\pi$
$691$	−2.50935e6	−0.199925	−0.0999624	+	0.994991i	$0.531872\pi$
$692$	−1.38028e6	−0.109573
$693$	0	0
$694$	1.35228e6	0.106578
$695$	356882.	0.0280261
$696$	0	0
$697$	1.76675e7	1.37750
$698$	1.01392e7	0.787707
$699$	0	0
$700$	0	0
$701$	−9.68649e6	−0.744512	−0.372256	−	0.928130i	$-0.621416\pi$
$701$	−9.68649e6	−0.744512	−0.372256	+	0.928130i	$0.621416\pi$
$702$	0	0
$703$	−2.44081e7	−1.86271
$704$	−6.41817e6	−0.488067
$705$	0	0
$706$	−9.64681e6	−0.728403
$707$	0	0
$708$	0	0
$709$	−9.52720e6	−0.711786	−0.355893	−	0.934527i	$-0.615823\pi$
$709$	−9.52720e6	−0.711786	−0.355893	+	0.934527i	$0.615823\pi$
$710$	−1.25220e7	−0.932238
$711$	0	0
$712$	1.62896e7	1.20423
$713$	1.22865e6	0.0905118
$714$	0	0
$715$	981794.	0.0718217
$716$	−508313.	−0.0370552
$717$	0	0
$718$	−2.43751e7	−1.76456
$719$	757176.	0.0546229	0.0273114	−	0.999627i	$-0.491305\pi$
$719$	757176.	0.0546229	0.0273114	+	0.999627i	$0.491305\pi$
$720$	0	0
$721$	0	0
$722$	−6.02081e6	−0.429845
$723$	0	0
$724$	439925.	0.0311912
$725$	−2.30378e6	−0.162778
$726$	0	0
$727$	2.66570e7	1.87058	0.935288	−	0.353888i	$-0.115141\pi$
$727$	2.66570e7	1.87058	0.935288	+	0.353888i	$0.115141\pi$
$728$	0	0
$729$	0	0
$730$	8.09731e6	0.562385
$731$	2.66498e7	1.84459
$732$	0	0
$733$	−2.70857e7	−1.86200	−0.931001	−	0.365017i	$-0.881063\pi$
$733$	−2.70857e7	−1.86200	−0.931001	+	0.365017i	$0.881063\pi$
$734$	1.72698e7	1.18317
$735$	0	0
$736$	115748.	0.00787625
$737$	1.02085e7	0.692298
$738$	0	0
$739$	2.19507e7	1.47855	0.739276	−	0.673402i	$-0.235168\pi$
$739$	2.19507e7	1.47855	0.739276	+	0.673402i	$0.235168\pi$
$740$	−1.08584e6	−0.0728932
$741$	0	0
$742$	0	0
$743$	−5.77370e6	−0.383691	−0.191846	−	0.981425i	$-0.561447\pi$
$743$	−5.77370e6	−0.383691	−0.191846	+	0.981425i	$0.561447\pi$
$744$	0	0
$745$	−1.81966e7	−1.20116
$746$	1.25905e7	0.828318
$747$	0	0
$748$	849005.	0.0554825
$749$	0	0
$750$	0	0
$751$	−8.37749e6	−0.542018	−0.271009	−	0.962577i	$-0.587357\pi$
$751$	−8.37749e6	−0.542018	−0.271009	+	0.962577i	$0.587357\pi$
$752$	1.89174e7	1.21988
$753$	0	0
$754$	1.01440e6	0.0649800
$755$	−6.95170e6	−0.443837
$756$	0	0
$757$	1.18828e7	0.753665	0.376833	−	0.926281i	$-0.377013\pi$
$757$	1.18828e7	0.753665	0.376833	+	0.926281i	$0.377013\pi$
$758$	3.22329e6	0.203764
$759$	0	0
$760$	−1.27415e7	−0.800177
$761$	1.76940e7	1.10755	0.553777	−	0.832665i	$-0.313186\pi$
$761$	1.76940e7	1.10755	0.553777	+	0.832665i	$0.313186\pi$
$762$	0	0
$763$	0	0
$764$	405811.	0.0251531
$765$	0	0
$766$	1.95937e7	1.20655
$767$	−1.31110e6	−0.0804726
$768$	0	0
$769$	4.12006e6	0.251239	0.125620	−	0.992078i	$-0.459908\pi$
$769$	4.12006e6	0.251239	0.125620	+	0.992078i	$0.459908\pi$
$770$	0	0
$771$	0	0
$772$	−275452.	−0.0166342
$773$	1.26864e7	0.763642	0.381821	−	0.924236i	$-0.375297\pi$
$773$	1.26864e7	0.763642	0.381821	+	0.924236i	$0.375297\pi$
$774$	0	0
$775$	1.64043e7	0.981080
$776$	2.20879e7	1.31674
$777$	0	0
$778$	−1.22402e7	−0.725001
$779$	−1.69865e7	−1.00290
$780$	0	0
$781$	−1.17790e7	−0.691002
$782$	1.46855e6	0.0858761
$783$	0	0
$784$	0	0
$785$	−9.53723e6	−0.552393
$786$	0	0
$787$	−9.14809e6	−0.526494	−0.263247	−	0.964728i	$-0.584793\pi$
$787$	−9.14809e6	−0.526494	−0.263247	+	0.964728i	$0.584793\pi$
$788$	525578.	0.0301524
$789$	0	0
$790$	−3.32594e6	−0.189603
$791$	0	0
$792$	0	0
$793$	−6.08468e6	−0.343602
$794$	−2.49802e7	−1.40619
$795$	0	0
$796$	1.73125e6	0.0968451
$797$	−1.10180e7	−0.614408	−0.307204	−	0.951644i	$-0.599393\pi$
$797$	−1.10180e7	−0.614408	−0.307204	+	0.951644i	$0.599393\pi$
$798$	0	0
$799$	−3.94585e7	−2.18662
$800$	1.54541e6	0.0853727
$801$	0	0
$802$	3.27009e7	1.79524
$803$	7.61684e6	0.416856
$804$	0	0
$805$	0	0
$806$	−7.22315e6	−0.391642
$807$	0	0
$808$	4.15788e6	0.224049
$809$	3.10273e7	1.66676	0.833378	−	0.552703i	$-0.186404\pi$
$809$	3.10273e7	1.66676	0.833378	+	0.552703i	$0.186404\pi$
$810$	0	0
$811$	2.94456e7	1.57206	0.786028	−	0.618191i	$-0.212134\pi$
$811$	2.94456e7	1.57206	0.786028	+	0.618191i	$0.212134\pi$
$812$	0	0
$813$	0	0
$814$	1.29546e7	0.685271
$815$	−1.94073e7	−1.02346
$816$	0	0
$817$	−2.56226e7	−1.34297
$818$	−1.93526e7	−1.01124
$819$	0	0
$820$	−755675.	−0.0392464
$821$	5.23311e6	0.270958	0.135479	−	0.990780i	$-0.456743\pi$
$821$	5.23311e6	0.270958	0.135479	+	0.990780i	$0.456743\pi$
$822$	0	0
$823$	1.41631e7	0.728884	0.364442	−	0.931226i	$-0.381260\pi$
$823$	1.41631e7	0.728884	0.364442	+	0.931226i	$0.381260\pi$
$824$	−2.54212e7	−1.30430
$825$	0	0
$826$	0	0
$827$	−2.98006e7	−1.51517	−0.757585	−	0.652737i	$-0.773621\pi$
$827$	−2.98006e7	−1.51517	−0.757585	+	0.652737i	$0.773621\pi$
$828$	0	0
$829$	1.99304e7	1.00723	0.503617	−	0.863927i	$-0.332002\pi$
$829$	1.99304e7	1.00723	0.503617	+	0.863927i	$0.332002\pi$
$830$	1.99383e7	1.00460
$831$	0	0
$832$	−5.14594e6	−0.257725
$833$	0	0
$834$	0	0
$835$	7.87492e6	0.390868
$836$	−816280.	−0.0403946
$837$	0	0
$838$	−1.09673e7	−0.539500
$839$	−1.88103e7	−0.922552	−0.461276	−	0.887257i	$-0.652608\pi$
$839$	−1.88103e7	−0.922552	−0.461276	+	0.887257i	$0.652608\pi$
$840$	0	0
$841$	−1.89246e7	−0.922650
$842$	3.21320e7	1.56191
$843$	0	0
$844$	1.65085e6	0.0797724
$845$	−1.25794e7	−0.606062
$846$	0	0
$847$	0	0
$848$	8.80874e6	0.420653
$849$	0	0
$850$	1.96073e7	0.930833
$851$	−1.76677e6	−0.0836288
$852$	0	0
$853$	−2.05980e7	−0.969285	−0.484643	−	0.874712i	$-0.661050\pi$
$853$	−2.05980e7	−0.969285	−0.484643	+	0.874712i	$0.661050\pi$
$854$	0	0
$855$	0	0
$856$	−2.19980e7	−1.02612
$857$	−2.47572e7	−1.15146	−0.575731	−	0.817639i	$-0.695282\pi$
$857$	−2.47572e7	−1.15146	−0.575731	+	0.817639i	$0.695282\pi$
$858$	0	0
$859$	3.91065e7	1.80828	0.904141	−	0.427234i	$-0.140512\pi$
$859$	3.91065e7	1.80828	0.904141	+	0.427234i	$0.140512\pi$
$860$	−1.13987e6	−0.0525544
$861$	0	0
$862$	−2.07527e7	−0.951275
$863$	3.93363e7	1.79790	0.898952	−	0.438048i	$-0.144330\pi$
$863$	3.93363e7	1.79790	0.898952	+	0.438048i	$0.144330\pi$
$864$	0	0
$865$	2.12472e7	0.965522
$866$	3.59151e7	1.62735
$867$	0	0
$868$	0	0
$869$	−3.12859e6	−0.140540
$870$	0	0
$871$	8.18494e6	0.365570
$872$	−6.48277e6	−0.288715
$873$	0	0
$874$	−1.41195e6	−0.0625229
$875$	0	0
$876$	0	0
$877$	8.09253e6	0.355292	0.177646	−	0.984094i	$-0.443152\pi$
$877$	8.09253e6	0.355292	0.177646	+	0.984094i	$0.443152\pi$
$878$	−2.47818e7	−1.08492
$879$	0	0
$880$	6.26564e6	0.272746
$881$	−4.05755e6	−0.176126	−0.0880631	−	0.996115i	$-0.528068\pi$
$881$	−4.05755e6	−0.176126	−0.0880631	+	0.996115i	$0.528068\pi$
$882$	0	0
$883$	1.79813e7	0.776102	0.388051	−	0.921638i	$-0.373148\pi$
$883$	1.79813e7	0.776102	0.388051	+	0.921638i	$0.373148\pi$
$884$	680713.	0.0292977
$885$	0	0
$886$	2.78595e7	1.19231
$887$	−1.53139e7	−0.653547	−0.326773	−	0.945103i	$-0.605961\pi$
$887$	−1.53139e7	−0.653547	−0.326773	+	0.945103i	$0.605961\pi$
$888$	0	0
$889$	0	0
$890$	−1.70777e7	−0.722693
$891$	0	0
$892$	98725.0	0.00415447
$893$	3.79376e7	1.59199
$894$	0	0
$895$	7.82466e6	0.326519
$896$	0	0
$897$	0	0
$898$	3.24377e7	1.34233
$899$	−1.12972e7	−0.466200
$900$	0	0
$901$	−1.83735e7	−0.754016
$902$	9.01554e6	0.368957
$903$	0	0
$904$	4.97172e6	0.202342
$905$	−6.77194e6	−0.274847
$906$	0	0
$907$	−2.17757e7	−0.878930	−0.439465	−	0.898260i	$-0.644832\pi$
$907$	−2.17757e7	−0.878930	−0.439465	+	0.898260i	$0.644832\pi$
$908$	−1.73623e6	−0.0698865
$909$	0	0
$910$	0	0
$911$	5.35112e6	0.213623	0.106812	−	0.994279i	$-0.465936\pi$
$911$	5.35112e6	0.213623	0.106812	+	0.994279i	$0.465936\pi$
$912$	0	0
$913$	1.87552e7	0.744639
$914$	1.41511e7	0.560306
$915$	0	0
$916$	2.20124e6	0.0866818
$917$	0	0
$918$	0	0
$919$	−2.67858e7	−1.04620	−0.523102	−	0.852270i	$-0.675225\pi$
$919$	−2.67858e7	−1.04620	−0.523102	+	0.852270i	$0.675225\pi$
$920$	−922285.	−0.0359249
$921$	0	0
$922$	−2.45904e7	−0.952661
$923$	−9.44411e6	−0.364886
$924$	0	0
$925$	−2.35890e7	−0.906474
$926$	3.02336e7	1.15868
$927$	0	0
$928$	−1.06428e6	−0.0405683
$929$	739974.	0.0281305	0.0140652	−	0.999901i	$-0.495523\pi$
$929$	739974.	0.0281305	0.0140652	+	0.999901i	$0.495523\pi$
$930$	0	0
$931$	0	0
$932$	−1.19817e6	−0.0451834
$933$	0	0
$934$	4.33166e7	1.62475
$935$	−1.30691e7	−0.488895
$936$	0	0
$937$	122654.	0.00456385	0.00228193	−	0.999997i	$-0.499274\pi$
$937$	122654.	0.00456385	0.00228193	+	0.999997i	$0.499274\pi$
$938$	0	0
$939$	0	0
$940$	1.68772e6	0.0622991
$941$	1.22466e7	0.450861	0.225431	−	0.974259i	$-0.427621\pi$
$941$	1.22466e7	0.450861	0.225431	+	0.974259i	$0.427621\pi$
$942$	0	0
$943$	−1.22956e6	−0.0450266
$944$	−8.36722e6	−0.305599
$945$	0	0
$946$	1.35992e7	0.494065
$947$	3.71174e6	0.134494	0.0672470	−	0.997736i	$-0.478578\pi$
$947$	3.71174e6	0.134494	0.0672470	+	0.997736i	$0.478578\pi$
$948$	0	0
$949$	6.10701e6	0.220122
$950$	−1.88516e7	−0.677702
$951$	0	0
$952$	0	0
$953$	2.32857e7	0.830533	0.415266	−	0.909700i	$-0.363688\pi$
$953$	2.32857e7	0.830533	0.415266	+	0.909700i	$0.363688\pi$
$954$	0	0
$955$	−6.24681e6	−0.221641
$956$	−269702.	−0.00954418
$957$	0	0
$958$	3.57474e7	1.25843
$959$	0	0
$960$	0	0
$961$	5.18142e7	1.80984
$962$	1.03867e7	0.361859
$963$	0	0
$964$	−2.12787e6	−0.0737483
$965$	4.24013e6	0.146575
$966$	0	0
$967$	2.25257e7	0.774661	0.387330	−	0.921941i	$-0.373397\pi$
$967$	2.25257e7	0.774661	0.387330	+	0.921941i	$0.373397\pi$
$968$	−2.37579e7	−0.814927
$969$	0	0
$970$	−2.31565e7	−0.790212
$971$	1.99456e7	0.678890	0.339445	−	0.940626i	$-0.389761\pi$
$971$	1.99456e7	0.678890	0.339445	+	0.940626i	$0.389761\pi$
$972$	0	0
$973$	0	0
$974$	−2.53995e7	−0.857884
$975$	0	0
$976$	−3.88314e7	−1.30484
$977$	959342.	0.0321542	0.0160771	−	0.999871i	$-0.494882\pi$
$977$	959342.	0.0321542	0.0160771	+	0.999871i	$0.494882\pi$
$978$	0	0
$979$	−1.60643e7	−0.535681
$980$	0	0
$981$	0	0
$982$	−4.99542e6	−0.165308
$983$	5.22097e7	1.72333	0.861663	−	0.507481i	$-0.169423\pi$
$983$	5.22097e7	1.72333	0.861663	+	0.507481i	$0.169423\pi$
$984$	0	0
$985$	−8.09042e6	−0.265693
$986$	−1.35030e7	−0.442323
$987$	0	0
$988$	−654475.	−0.0213305
$989$	−1.85468e6	−0.0602945
$990$	0	0
$991$	−1.76305e7	−0.570269	−0.285134	−	0.958488i	$-0.592038\pi$
$991$	−1.76305e7	−0.570269	−0.285134	+	0.958488i	$0.592038\pi$
$992$	7.57836e6	0.244510
$993$	0	0
$994$	0	0
$995$	−2.66498e7	−0.853369
$996$	0	0
$997$	−4.04875e7	−1.28998	−0.644990	−	0.764191i	$-0.723138\pi$
$997$	−4.04875e7	−1.28998	−0.644990	+	0.764191i	$0.723138\pi$
$998$	−1.70704e6	−0.0542522
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.r.1.1		2
3.2	odd	2		147.6.a.j.1.2	yes	2
7.6	odd	2		441.6.a.q.1.1		2
21.2	odd	6		147.6.e.m.67.1		4
21.5	even	6		147.6.e.n.67.1		4
21.11	odd	6		147.6.e.m.79.1		4
21.17	even	6		147.6.e.n.79.1		4
21.20	even	2		147.6.a.h.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.h.1.2	✓	2	21.20	even	2
147.6.a.j.1.2	yes	2	3.2	odd	2
147.6.e.m.67.1		4	21.2	odd	6
147.6.e.m.79.1		4	21.11	odd	6
147.6.e.n.67.1		4	21.5	even	6
147.6.e.n.79.1		4	21.17	even	6
441.6.a.q.1.1		2	7.6	odd	2
441.6.a.r.1.1		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-5.44622 q^{2} -2.33867 q^{4} +36.0000 q^{5} +187.016 q^{8} +O(q^{10})\)	\(q-5.44622 q^{2} -2.33867 q^{4} +36.0000 q^{5} +187.016 q^{8} -196.064 q^{10} -184.430 q^{11} -147.872 q^{13} -943.693 q^{16} +1968.38 q^{17} -1892.51 q^{19} -84.1920 q^{20} +1004.45 q^{22} -136.988 q^{23} -1829.00 q^{25} +805.344 q^{26} +1259.58 q^{29} -8969.02 q^{31} -844.949 q^{32} -10720.3 q^{34} +12897.2 q^{37} +10307.0 q^{38} +6732.58 q^{40} +8975.62 q^{41} +13538.9 q^{43} +431.321 q^{44} +746.069 q^{46} -20046.1 q^{47} +9961.14 q^{50} +345.823 q^{52} -9334.33 q^{53} -6639.49 q^{55} -6859.96 q^{58} +8866.46 q^{59} +41148.3 q^{61} +48847.3 q^{62} +34800.0 q^{64} -5323.39 q^{65} -55351.5 q^{67} -4603.39 q^{68} +63866.8 q^{71} -41299.3 q^{73} -70241.1 q^{74} +4425.95 q^{76} +16963.5 q^{79} -33973.0 q^{80} -48883.2 q^{82} -101693. q^{83} +70861.8 q^{85} -73736.0 q^{86} -34491.4 q^{88} +87102.5 q^{89} +320.370 q^{92} +109176. q^{94} -68130.4 q^{95} +118107. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8	\( 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 + 108 * q^10 - 480 * q^11 - 1296 * q^13 - 1679 * q^16 + 936 * q^17 + 216 * q^19 + 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 - 8892 * q^26 - 6372 * q^29 - 9936 * q^31 - 9039 * q^32 - 19440 * q^34 + 11124 * q^37 + 28116 * q^38 + 8964 * q^40 + 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 + 7920 * q^47 - 5487 * q^50 - 44820 * q^52 - 2220 * q^53 - 17280 * q^55 - 71318 * q^58 + 29736 * q^59 + 17280 * q^61 + 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 - 45216 * q^68 + 92280 * q^71 - 56592 * q^73 - 85218 * q^74 + 87372 * q^76 - 56096 * q^79 - 60444 * q^80 + 52272 * q^82 - 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 + 123192 * q^89 + 25536 * q^92 + 345384 * q^94 + 7776 * q^95 - 35856 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more