LMFDB - Embedded newform 585.6.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,6,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("585.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	585.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:	$93.8245345906$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-5.93318$ of defining polynomial
Character	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.93318 q^{2} +3.20258 q^{4} +25.0000 q^{5} -185.746 q^{7} -170.860 q^{8} +148.329 q^{10} +353.912 q^{11} +169.000 q^{13} -1102.06 q^{14} -1116.23 q^{16} -634.652 q^{17} +1118.29 q^{19} +80.0646 q^{20} +2099.83 q^{22} -3509.85 q^{23} +625.000 q^{25} +1002.71 q^{26} -594.867 q^{28} +3765.67 q^{29} +2906.63 q^{31} -1155.24 q^{32} -3765.50 q^{34} -4643.65 q^{35} +283.305 q^{37} +6635.04 q^{38} -4271.50 q^{40} +13563.6 q^{41} -5184.47 q^{43} +1133.43 q^{44} -20824.6 q^{46} +6781.50 q^{47} +17694.6 q^{49} +3708.24 q^{50} +541.237 q^{52} -7664.43 q^{53} +8847.81 q^{55} +31736.6 q^{56} +22342.4 q^{58} -2806.29 q^{59} -13764.7 q^{61} +17245.5 q^{62} +28865.0 q^{64} +4225.00 q^{65} +67744.1 q^{67} -2032.53 q^{68} -27551.6 q^{70} +66519.0 q^{71} +75902.7 q^{73} +1680.90 q^{74} +3581.43 q^{76} -65737.9 q^{77} +101641. q^{79} -27905.7 q^{80} +80475.1 q^{82} +50882.7 q^{83} -15866.3 q^{85} -30760.4 q^{86} -60469.5 q^{88} +52439.2 q^{89} -31391.1 q^{91} -11240.6 q^{92} +40235.8 q^{94} +27957.4 q^{95} -142557. q^{97} +104985. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 134 q^{4} + 150 q^{5} + 220 q^{7} - 24 q^{8} + 170 q^{11} + 1014 q^{13} + 1440 q^{14} + 3506 q^{16} - 728 q^{17} + 1218 q^{19} + 3350 q^{20} + 5154 q^{22} - 8954 q^{23} + 3750 q^{25} + 13212 q^{28}+ \cdots + 4736 q^{98}+O(q^{100}) $ 6 * q + 134 * q^4 + 150 * q^5 + 220 * q^7 - 24 * q^8 + 170 * q^11 + 1014 * q^13 + 1440 * q^14 + 3506 * q^16 - 728 * q^17 + 1218 * q^19 + 3350 * q^20 + 5154 * q^22 - 8954 * q^23 + 3750 * q^25 + 13212 * q^28 - 8364 * q^29 + 2862 * q^31 + 29672 * q^32 - 21484 * q^34 + 5500 * q^35 + 13840 * q^37 + 27706 * q^38 - 600 * q^40 - 2248 * q^41 + 3822 * q^43 + 73482 * q^44 - 39886 * q^46 + 10860 * q^47 + 85902 * q^49 + 22646 * q^52 - 6584 * q^53 + 4250 * q^55 + 188800 * q^56 - 88500 * q^58 + 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 14022 * q^64 + 25350 * q^65 + 60040 * q^67 - 3504 * q^68 + 36000 * q^70 + 74174 * q^71 + 48960 * q^73 - 226112 * q^74 - 93422 * q^76 - 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 58732 * q^82 - 123260 * q^83 - 18200 * q^85 + 304666 * q^86 + 127622 * q^88 - 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 125920 * q^94 + 30450 * q^95 + 246284 * q^97 + 4736 * q^98

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 $

$2$	5.93318	1.04885	0.524424	−	0.851457i	$-0.324281\pi$
$2$	5.93318	1.04885	0.524424	+	0.851457i	$0.324281\pi$
$3$	0	0
$3$	0	0
$4$	3.20258	0.100081
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−185.746	−1.43276	−0.716382	−	0.697708i	$-0.754203\pi$
$7$	−185.746	−1.43276	−0.716382	+	0.697708i	$0.754203\pi$
$8$	−170.860	−0.943878
$9$	0	0
$10$	148.329	0.469059
$11$	353.912	0.881889	0.440945	−	0.897534i	$-0.354643\pi$
$11$	353.912	0.881889	0.440945	+	0.897534i	$0.354643\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−1102.06	−1.50275
$15$	0	0
$16$	−1116.23	−1.09006
$17$	−634.652	−0.532615	−0.266308	−	0.963888i	$-0.585804\pi$
$17$	−634.652	−0.532615	−0.266308	+	0.963888i	$0.585804\pi$
$18$	0	0
$19$	1118.29	0.710677	0.355338	−	0.934738i	$-0.384366\pi$
$19$	1118.29	0.710677	0.355338	+	0.934738i	$0.384366\pi$
$20$	80.0646	0.0447575
$21$	0	0
$22$	2099.83	0.924967
$23$	−3509.85	−1.38347	−0.691734	−	0.722153i	$-0.743153\pi$
$23$	−3509.85	−1.38347	−0.691734	+	0.722153i	$0.743153\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	1002.71	0.290898
$27$	0	0
$28$	−594.867	−0.143392
$29$	3765.67	0.831471	0.415735	−	0.909486i	$-0.363524\pi$
$29$	3765.67	0.831471	0.415735	+	0.909486i	$0.363524\pi$
$30$	0	0
$31$	2906.63	0.543232	0.271616	−	0.962406i	$-0.412442\pi$
$31$	2906.63	0.543232	0.271616	+	0.962406i	$0.412442\pi$
$32$	−1155.24	−0.199433
$33$	0	0
$34$	−3765.50	−0.558632
$35$	−4643.65	−0.640751
$36$	0	0
$37$	283.305	0.0340213	0.0170106	−	0.999855i	$-0.494585\pi$
$37$	283.305	0.0340213	0.0170106	+	0.999855i	$0.494585\pi$
$38$	6635.04	0.745391
$39$	0	0
$40$	−4271.50	−0.422115
$41$	13563.6	1.26013	0.630064	−	0.776544i	$-0.283029\pi$
$41$	13563.6	1.26013	0.630064	+	0.776544i	$0.283029\pi$
$42$	0	0
$43$	−5184.47	−0.427596	−0.213798	−	0.976878i	$-0.568583\pi$
$43$	−5184.47	−0.427596	−0.213798	+	0.976878i	$0.568583\pi$
$44$	1133.43	0.0882601
$45$	0	0
$46$	−20824.6	−1.45105
$47$	6781.50	0.447797	0.223898	−	0.974612i	$-0.428122\pi$
$47$	6781.50	0.447797	0.223898	+	0.974612i	$0.428122\pi$
$48$	0	0
$49$	17694.6	1.05281
$50$	3708.24	0.209769
$51$	0	0
$52$	541.237	0.0277574
$53$	−7664.43	−0.374792	−0.187396	−	0.982284i	$-0.560005\pi$
$53$	−7664.43	−0.374792	−0.187396	+	0.982284i	$0.560005\pi$
$54$	0	0
$55$	8847.81	0.394393
$56$	31736.6	1.35235
$57$	0	0
$58$	22342.4	0.872086
$59$	−2806.29	−0.104955	−0.0524773	−	0.998622i	$-0.516712\pi$
$59$	−2806.29	−0.104955	−0.0524773	+	0.998622i	$0.516712\pi$
$60$	0	0
$61$	−13764.7	−0.473634	−0.236817	−	0.971554i	$-0.576104\pi$
$61$	−13764.7	−0.473634	−0.236817	+	0.971554i	$0.576104\pi$
$62$	17245.5	0.569768
$63$	0	0
$64$	28865.0	0.880889
$65$	4225.00	0.124035
$66$	0	0
$67$	67744.1	1.84368	0.921838	−	0.387576i	$-0.126687\pi$
$67$	67744.1	1.84368	0.921838	+	0.387576i	$0.126687\pi$
$68$	−2032.53	−0.0533045
$69$	0	0
$70$	−27551.6	−0.672050
$71$	66519.0	1.56603	0.783014	−	0.622004i	$-0.213681\pi$
$71$	66519.0	1.56603	0.783014	+	0.622004i	$0.213681\pi$
$72$	0	0
$73$	75902.7	1.66706	0.833528	−	0.552478i	$-0.186318\pi$
$73$	75902.7	1.66706	0.833528	+	0.552478i	$0.186318\pi$
$74$	1680.90	0.0356831
$75$	0	0
$76$	3581.43	0.0711250
$77$	−65737.9	−1.26354
$78$	0	0
$79$	101641.	1.83233	0.916163	−	0.400806i	$-0.131270\pi$
$79$	101641.	1.83233	0.916163	+	0.400806i	$0.131270\pi$
$80$	−27905.7	−0.487492
$81$	0	0
$82$	80475.1	1.32168
$83$	50882.7	0.810727	0.405363	−	0.914156i	$-0.367145\pi$
$83$	50882.7	0.810727	0.405363	+	0.914156i	$0.367145\pi$
$84$	0	0
$85$	−15866.3	−0.238193
$86$	−30760.4	−0.448483
$87$	0	0
$88$	−60469.5	−0.832396
$89$	52439.2	0.701748	0.350874	−	0.936423i	$-0.385885\pi$
$89$	52439.2	0.701748	0.350874	+	0.936423i	$0.385885\pi$
$90$	0	0
$91$	−31391.1	−0.397377
$92$	−11240.6	−0.138458
$93$	0	0
$94$	40235.8	0.469671
$95$	27957.4	0.317824
$96$	0	0
$97$	−142557.	−1.53837	−0.769183	−	0.639028i	$-0.779337\pi$
$97$	−142557.	−1.53837	−0.769183	+	0.639028i	$0.779337\pi$
$98$	104985.	1.10424
$99$	0	0
$100$	2001.61	0.0200161
$101$	−4751.74	−0.0463499	−0.0231750	−	0.999731i	$-0.507377\pi$
$101$	−4751.74	−0.0463499	−0.0231750	+	0.999731i	$0.507377\pi$
$102$	0	0
$103$	−59290.6	−0.550672	−0.275336	−	0.961348i	$-0.588789\pi$
$103$	−59290.6	−0.550672	−0.275336	+	0.961348i	$0.588789\pi$
$104$	−28875.4	−0.261785
$105$	0	0
$106$	−45474.4	−0.393100
$107$	−157927.	−1.33351	−0.666756	−	0.745276i	$-0.732318\pi$
$107$	−157927.	−1.33351	−0.666756	+	0.745276i	$0.732318\pi$
$108$	0	0
$109$	58878.4	0.474668	0.237334	−	0.971428i	$-0.423726\pi$
$109$	58878.4	0.474668	0.237334	+	0.971428i	$0.423726\pi$
$110$	52495.6	0.413658
$111$	0	0
$112$	207335.	1.56180
$113$	−179734.	−1.32414	−0.662069	−	0.749443i	$-0.730322\pi$
$113$	−179734.	−1.32414	−0.662069	+	0.749443i	$0.730322\pi$
$114$	0	0
$115$	−87746.2	−0.618705
$116$	12059.9	0.0832142
$117$	0	0
$118$	−16650.2	−0.110081
$119$	117884.	0.763112
$120$	0	0
$121$	−35797.0	−0.222271
$122$	−81668.5	−0.496770
$123$	0	0
$124$	9308.72	0.0543671
$125$	15625.0	0.0894427
$126$	0	0
$127$	−123741.	−0.680774	−0.340387	−	0.940286i	$-0.610558\pi$
$127$	−123741.	−0.680774	−0.340387	+	0.940286i	$0.610558\pi$
$128$	208229.	1.12335
$129$	0	0
$130$	25067.7	0.130094
$131$	43205.0	0.219966	0.109983	−	0.993933i	$-0.464920\pi$
$131$	43205.0	0.219966	0.109983	+	0.993933i	$0.464920\pi$
$132$	0	0
$133$	−207719.	−1.01823
$134$	401938.	1.93373
$135$	0	0
$136$	108437.	0.502724
$137$	188517.	0.858120	0.429060	−	0.903276i	$-0.358845\pi$
$137$	188517.	0.858120	0.429060	+	0.903276i	$0.358845\pi$
$138$	0	0
$139$	344148.	1.51081	0.755403	−	0.655260i	$-0.227441\pi$
$139$	344148.	1.51081	0.755403	+	0.655260i	$0.227441\pi$
$140$	−14871.7	−0.0641269
$141$	0	0
$142$	394669.	1.64253
$143$	59811.2	0.244592
$144$	0	0
$145$	94141.7	0.371845
$146$	450344.	1.74849
$147$	0	0
$148$	907.309	0.00340487
$149$	177809.	0.656126	0.328063	−	0.944656i	$-0.393604\pi$
$149$	177809.	0.656126	0.328063	+	0.944656i	$0.393604\pi$
$150$	0	0
$151$	554784.	1.98008	0.990038	−	0.140803i	$-0.0449685\pi$
$151$	554784.	1.98008	0.990038	+	0.140803i	$0.0449685\pi$
$152$	−191072.	−0.670792
$153$	0	0
$154$	−390034.	−1.32526
$155$	72665.7	0.242941
$156$	0	0
$157$	255896.	0.828542	0.414271	−	0.910154i	$-0.364037\pi$
$157$	255896.	0.828542	0.414271	+	0.910154i	$0.364037\pi$
$158$	603056.	1.92183
$159$	0	0
$160$	−28881.0	−0.0891893
$161$	651941.	1.98218
$162$	0	0
$163$	−262686.	−0.774404	−0.387202	−	0.921995i	$-0.626558\pi$
$163$	−262686.	−0.774404	−0.387202	+	0.921995i	$0.626558\pi$
$164$	43438.5	0.126114
$165$	0	0
$166$	301896.	0.850329
$167$	−287069.	−0.796517	−0.398259	−	0.917273i	$-0.630385\pi$
$167$	−287069.	−0.796517	−0.398259	+	0.917273i	$0.630385\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	−94137.6	−0.249828
$171$	0	0
$172$	−16603.7	−0.0427941
$173$	−719663.	−1.82816	−0.914080	−	0.405534i	$-0.867085\pi$
$173$	−719663.	−1.82816	−0.914080	+	0.405534i	$0.867085\pi$
$174$	0	0
$175$	−116091.	−0.286553
$176$	−395046.	−0.961316
$177$	0	0
$178$	311131.	0.736026
$179$	−779772.	−1.81901	−0.909505	−	0.415693i	$-0.863539\pi$
$179$	−779772.	−1.81901	−0.909505	+	0.415693i	$0.863539\pi$
$180$	0	0
$181$	336065.	0.762478	0.381239	−	0.924477i	$-0.375498\pi$
$181$	336065.	0.762478	0.381239	+	0.924477i	$0.375498\pi$
$182$	−186249.	−0.416788
$183$	0	0
$184$	599693.	1.30582
$185$	7082.63	0.0152148
$186$	0	0
$187$	−224611.	−0.469708
$188$	21718.3	0.0448158
$189$	0	0
$190$	165876.	0.333349
$191$	−409479.	−0.812173	−0.406086	−	0.913835i	$-0.633107\pi$
$191$	−409479.	−0.812173	−0.406086	+	0.913835i	$0.633107\pi$
$192$	0	0
$193$	339565.	0.656189	0.328095	−	0.944645i	$-0.393593\pi$
$193$	339565.	0.656189	0.328095	+	0.944645i	$0.393593\pi$
$194$	−845817.	−1.61351
$195$	0	0
$196$	56668.4	0.105366
$197$	871469.	1.59988	0.799938	−	0.600082i	$-0.204865\pi$
$197$	871469.	1.59988	0.799938	+	0.600082i	$0.204865\pi$
$198$	0	0
$199$	270952.	0.485019	0.242510	−	0.970149i	$-0.422029\pi$
$199$	270952.	0.485019	0.242510	+	0.970149i	$0.422029\pi$
$200$	−106788.	−0.188776
$201$	0	0
$202$	−28192.9	−0.0486140
$203$	−699458.	−1.19130
$204$	0	0
$205$	339089.	0.563546
$206$	−351781.	−0.577570
$207$	0	0
$208$	−188642.	−0.302330
$209$	395778.	0.626738
$210$	0	0
$211$	181455.	0.280583	0.140292	−	0.990110i	$-0.455196\pi$
$211$	181455.	0.280583	0.140292	+	0.990110i	$0.455196\pi$
$212$	−24546.0	−0.0375095
$213$	0	0
$214$	−937009.	−1.39865
$215$	−129612.	−0.191227
$216$	0	0
$217$	−539895.	−0.778323
$218$	349336.	0.497854
$219$	0	0
$220$	28335.9	0.0394711
$221$	−107256.	−0.147721
$222$	0	0
$223$	1.38761e6	1.86855	0.934276	−	0.356552i	$-0.116048\pi$
$223$	1.38761e6	1.86855	0.934276	+	0.356552i	$0.116048\pi$
$224$	214582.	0.285741
$225$	0	0
$226$	−1.06639e6	−1.38882
$227$	690397.	0.889271	0.444636	−	0.895711i	$-0.353333\pi$
$227$	690397.	0.889271	0.444636	+	0.895711i	$0.353333\pi$
$228$	0	0
$229$	−1.38257e6	−1.74221	−0.871104	−	0.491099i	$-0.836595\pi$
$229$	−1.38257e6	−1.74221	−0.871104	+	0.491099i	$0.836595\pi$
$230$	−520614.	−0.648927
$231$	0	0
$232$	−643403.	−0.784807
$233$	71911.6	0.0867780	0.0433890	−	0.999058i	$-0.486185\pi$
$233$	71911.6	0.0867780	0.0433890	+	0.999058i	$0.486185\pi$
$234$	0	0
$235$	169537.	0.200261
$236$	−8987.36	−0.0105039
$237$	0	0
$238$	699427.	0.800387
$239$	825442.	0.934743	0.467371	−	0.884061i	$-0.345201\pi$
$239$	825442.	0.934743	0.467371	+	0.884061i	$0.345201\pi$
$240$	0	0
$241$	−615086.	−0.682171	−0.341086	−	0.940032i	$-0.610795\pi$
$241$	−615086.	−0.682171	−0.341086	+	0.940032i	$0.610795\pi$
$242$	−212390.	−0.233128
$243$	0	0
$244$	−44082.7	−0.0474016
$245$	442365.	0.470832
$246$	0	0
$247$	188992.	0.197106
$248$	−496627.	−0.512745
$249$	0	0
$250$	92705.9	0.0938118
$251$	−622589.	−0.623759	−0.311879	−	0.950122i	$-0.600959\pi$
$251$	−622589.	−0.623759	−0.311879	+	0.950122i	$0.600959\pi$
$252$	0	0
$253$	−1.24218e6	−1.22007
$254$	−734174.	−0.714028
$255$	0	0
$256$	311779.	0.297335
$257$	1.04334e6	0.985359	0.492680	−	0.870211i	$-0.336017\pi$
$257$	1.04334e6	0.985359	0.492680	+	0.870211i	$0.336017\pi$
$258$	0	0
$259$	−52622.9	−0.0487444
$260$	13530.9	0.0124135
$261$	0	0
$262$	256343.	0.230711
$263$	1.31940e6	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$263$	1.31940e6	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$264$	0	0
$265$	−191611.	−0.167612
$266$	−1.23243e6	−1.06797
$267$	0	0
$268$	216956.	0.184516
$269$	−369633.	−0.311452	−0.155726	−	0.987800i	$-0.549772\pi$
$269$	−369633.	−0.311452	−0.155726	+	0.987800i	$0.549772\pi$
$270$	0	0
$271$	749291.	0.619765	0.309883	−	0.950775i	$-0.399710\pi$
$271$	749291.	0.619765	0.309883	+	0.950775i	$0.399710\pi$
$272$	708415.	0.580585
$273$	0	0
$274$	1.11850e6	0.900037
$275$	221195.	0.176378
$276$	0	0
$277$	1.64757e6	1.29016	0.645081	−	0.764114i	$-0.276824\pi$
$277$	1.64757e6	1.29016	0.645081	+	0.764114i	$0.276824\pi$
$278$	2.04189e6	1.58461
$279$	0	0
$280$	793415.	0.604791
$281$	−1.21917e6	−0.921080	−0.460540	−	0.887639i	$-0.652344\pi$
$281$	−1.21917e6	−0.921080	−0.460540	+	0.887639i	$0.652344\pi$
$282$	0	0
$283$	−997517.	−0.740379	−0.370190	−	0.928956i	$-0.620707\pi$
$283$	−997517.	−0.740379	−0.370190	+	0.928956i	$0.620707\pi$
$284$	213033.	0.156729
$285$	0	0
$286$	354870.	0.256540
$287$	−2.51938e6	−1.80546
$288$	0	0
$289$	−1.01707e6	−0.716321
$290$	558559.	0.390009
$291$	0	0
$292$	243085.	0.166840
$293$	−1.80793e6	−1.23031	−0.615153	−	0.788408i	$-0.710906\pi$
$293$	−1.80793e6	−1.23031	−0.615153	+	0.788408i	$0.710906\pi$
$294$	0	0
$295$	−70157.1	−0.0469372
$296$	−48405.6	−0.0321119
$297$	0	0
$298$	1.05497e6	0.688176
$299$	−593165.	−0.383705
$300$	0	0
$301$	962995.	0.612644
$302$	3.29163e6	2.07680
$303$	0	0
$304$	−1.24827e6	−0.774683
$305$	−344118.	−0.211816
$306$	0	0
$307$	−24494.5	−0.0148328	−0.00741638	−	0.999972i	$-0.502361\pi$
$307$	−24494.5	−0.0148328	−0.00741638	+	0.999972i	$0.502361\pi$
$308$	−210531.	−0.126456
$309$	0	0
$310$	431139.	0.254808
$311$	−1.48212e6	−0.868924	−0.434462	−	0.900690i	$-0.643061\pi$
$311$	−1.48212e6	−0.868924	−0.434462	+	0.900690i	$0.643061\pi$
$312$	0	0
$313$	−348766.	−0.201221	−0.100611	−	0.994926i	$-0.532080\pi$
$313$	−348766.	−0.201221	−0.100611	+	0.994926i	$0.532080\pi$
$314$	1.51828e6	0.869014
$315$	0	0
$316$	325515.	0.183380
$317$	406486.	0.227194	0.113597	−	0.993527i	$-0.463763\pi$
$317$	406486.	0.227194	0.113597	+	0.993527i	$0.463763\pi$
$318$	0	0
$319$	1.33272e6	0.733265
$320$	721625.	0.393946
$321$	0	0
$322$	3.86808e6	2.07901
$323$	−709728.	−0.378517
$324$	0	0
$325$	105625.	0.0554700
$326$	−1.55856e6	−0.812231
$327$	0	0
$328$	−2.31747e6	−1.18941
$329$	−1.25964e6	−0.641587
$330$	0	0
$331$	1.58614e6	0.795743	0.397871	−	0.917441i	$-0.369749\pi$
$331$	1.58614e6	0.795743	0.397871	+	0.917441i	$0.369749\pi$
$332$	162956.	0.0811381
$333$	0	0
$334$	−1.70323e6	−0.835425
$335$	1.69360e6	0.824517
$336$	0	0
$337$	2.43587e6	1.16837	0.584185	−	0.811621i	$-0.301414\pi$
$337$	2.43587e6	1.16837	0.584185	+	0.811621i	$0.301414\pi$
$338$	169457.	0.0806806
$339$	0	0
$340$	−50813.2	−0.0238385
$341$	1.02869e6	0.479071
$342$	0	0
$343$	−164869.	−0.0756664
$344$	885820.	0.403598
$345$	0	0
$346$	−4.26989e6	−1.91746
$347$	1.17786e6	0.525132	0.262566	−	0.964914i	$-0.415431\pi$
$347$	1.17786e6	0.525132	0.262566	+	0.964914i	$0.415431\pi$
$348$	0	0
$349$	−338854.	−0.148919	−0.0744594	−	0.997224i	$-0.523723\pi$
$349$	−338854.	−0.148919	−0.0744594	+	0.997224i	$0.523723\pi$
$350$	−688790.	−0.300550
$351$	0	0
$352$	−408854.	−0.175878
$353$	−3.25607e6	−1.39077	−0.695387	−	0.718635i	$-0.744767\pi$
$353$	−3.25607e6	−1.39077	−0.695387	+	0.718635i	$0.744767\pi$
$354$	0	0
$355$	1.66297e6	0.700349
$356$	167941.	0.0702314
$357$	0	0
$358$	−4.62652e6	−1.90786
$359$	−2.81818e6	−1.15407	−0.577036	−	0.816719i	$-0.695791\pi$
$359$	−2.81818e6	−1.15407	−0.577036	+	0.816719i	$0.695791\pi$
$360$	0	0
$361$	−1.22552e6	−0.494939
$362$	1.99393e6	0.799723
$363$	0	0
$364$	−100533.	−0.0397698
$365$	1.89757e6	0.745530
$366$	0	0
$367$	3.09661e6	1.20011	0.600056	−	0.799958i	$-0.295145\pi$
$367$	3.09661e6	1.20011	0.600056	+	0.799958i	$0.295145\pi$
$368$	3.91779e6	1.50807
$369$	0	0
$370$	42022.5	0.0159580
$371$	1.42364e6	0.536988
$372$	0	0
$373$	−4.21455e6	−1.56848	−0.784240	−	0.620457i	$-0.786947\pi$
$373$	−4.21455e6	−1.56848	−0.784240	+	0.620457i	$0.786947\pi$
$374$	−1.33266e6	−0.492652
$375$	0	0
$376$	−1.15869e6	−0.422666
$377$	636398.	0.230609
$378$	0	0
$379$	−1.26649e6	−0.452903	−0.226452	−	0.974022i	$-0.572712\pi$
$379$	−1.26649e6	−0.452903	−0.226452	+	0.974022i	$0.572712\pi$
$380$	89535.8	0.0318081
$381$	0	0
$382$	−2.42951e6	−0.851845
$383$	−5.66939e6	−1.97487	−0.987436	−	0.158017i	$-0.949490\pi$
$383$	−5.66939e6	−1.97487	−0.987436	+	0.158017i	$0.949490\pi$
$384$	0	0
$385$	−1.64345e6	−0.565072
$386$	2.01470e6	0.688242
$387$	0	0
$388$	−456551.	−0.153961
$389$	5.49114e6	1.83988	0.919938	−	0.392063i	$-0.128238\pi$
$389$	5.49114e6	1.83988	0.919938	+	0.392063i	$0.128238\pi$
$390$	0	0
$391$	2.22753e6	0.736856
$392$	−3.02330e6	−0.993726
$393$	0	0
$394$	5.17058e6	1.67803
$395$	2.54103e6	0.819441
$396$	0	0
$397$	−1.53688e6	−0.489398	−0.244699	−	0.969599i	$-0.578689\pi$
$397$	−1.53688e6	−0.489398	−0.244699	+	0.969599i	$0.578689\pi$
$398$	1.60760e6	0.508711
$399$	0	0
$400$	−697641.	−0.218013
$401$	−30028.4	−0.00932547	−0.00466273	−	0.999989i	$-0.501484\pi$
$401$	−30028.4	−0.00932547	−0.00466273	+	0.999989i	$0.501484\pi$
$402$	0	0
$403$	491220.	0.150666
$404$	−15217.8	−0.00463873
$405$	0	0
$406$	−4.15001e6	−1.24949
$407$	100265.	0.0300030
$408$	0	0
$409$	−5.54413e6	−1.63880	−0.819398	−	0.573225i	$-0.805692\pi$
$409$	−5.54413e6	−1.63880	−0.819398	+	0.573225i	$0.805692\pi$
$410$	2.01188e6	0.591074
$411$	0	0
$412$	−189883.	−0.0551116
$413$	521257.	0.150375
$414$	0	0
$415$	1.27207e6	0.362568
$416$	−195236.	−0.0553129
$417$	0	0
$418$	2.34822e6	0.657353
$419$	1.29360e6	0.359968	0.179984	−	0.983670i	$-0.442395\pi$
$419$	1.29360e6	0.359968	0.179984	+	0.983670i	$0.442395\pi$
$420$	0	0
$421$	−3.68620e6	−1.01362	−0.506809	−	0.862058i	$-0.669175\pi$
$421$	−3.68620e6	−1.01362	−0.506809	+	0.862058i	$0.669175\pi$
$422$	1.07660e6	0.294289
$423$	0	0
$424$	1.30955e6	0.353758
$425$	−396658.	−0.106523
$426$	0	0
$427$	2.55674e6	0.678606
$428$	−505774.	−0.133459
$429$	0	0
$430$	−769010.	−0.200568
$431$	−1.22645e6	−0.318021	−0.159010	−	0.987277i	$-0.550830\pi$
$431$	−1.22645e6	−0.318021	−0.159010	+	0.987277i	$0.550830\pi$
$432$	0	0
$433$	1.02459e6	0.262621	0.131311	−	0.991341i	$-0.458081\pi$
$433$	1.02459e6	0.262621	0.131311	+	0.991341i	$0.458081\pi$
$434$	−3.20329e6	−0.816342
$435$	0	0
$436$	188563.	0.0475051
$437$	−3.92504e6	−0.983198
$438$	0	0
$439$	−5.04951e6	−1.25051	−0.625256	−	0.780420i	$-0.715005\pi$
$439$	−5.04951e6	−1.25051	−0.625256	+	0.780420i	$0.715005\pi$
$440$	−1.51174e6	−0.372259
$441$	0	0
$442$	−636370.	−0.154937
$443$	−6.30848e6	−1.52727	−0.763634	−	0.645649i	$-0.776587\pi$
$443$	−6.30848e6	−1.52727	−0.763634	+	0.645649i	$0.776587\pi$
$444$	0	0
$445$	1.31098e6	0.313831
$446$	8.23293e6	1.95983
$447$	0	0
$448$	−5.36156e6	−1.26211
$449$	−1.16391e6	−0.272461	−0.136231	−	0.990677i	$-0.543499\pi$
$449$	−1.16391e6	−0.272461	−0.136231	+	0.990677i	$0.543499\pi$
$450$	0	0
$451$	4.80032e6	1.11129
$452$	−575612.	−0.132521
$453$	0	0
$454$	4.09625e6	0.932710
$455$	−784777.	−0.177712
$456$	0	0
$457$	1.48156e6	0.331840	0.165920	−	0.986139i	$-0.446941\pi$
$457$	1.48156e6	0.331840	0.165920	+	0.986139i	$0.446941\pi$
$458$	−8.20306e6	−1.82731
$459$	0	0
$460$	−281015.	−0.0619205
$461$	5.65392e6	1.23908	0.619538	−	0.784967i	$-0.287320\pi$
$461$	5.65392e6	1.23908	0.619538	+	0.784967i	$0.287320\pi$
$462$	0	0
$463$	2.09215e6	0.453566	0.226783	−	0.973945i	$-0.427179\pi$
$463$	2.09215e6	0.453566	0.226783	+	0.973945i	$0.427179\pi$
$464$	−4.20334e6	−0.906357
$465$	0	0
$466$	426664.	0.0910168
$467$	7.48481e6	1.58814	0.794069	−	0.607827i	$-0.207959\pi$
$467$	7.48481e6	1.58814	0.794069	+	0.607827i	$0.207959\pi$
$468$	0	0
$469$	−1.25832e7	−2.64155
$470$	1.00590e6	0.210043
$471$	0	0
$472$	479482.	0.0990644
$473$	−1.83485e6	−0.377092
$474$	0	0
$475$	698934.	0.142135
$476$	377534.	0.0763728
$477$	0	0
$478$	4.89750e6	0.980402
$479$	−2.54779e6	−0.507371	−0.253685	−	0.967287i	$-0.581643\pi$
$479$	−2.54779e6	−0.507371	−0.253685	+	0.967287i	$0.581643\pi$
$480$	0	0
$481$	47878.6	0.00943580
$482$	−3.64941e6	−0.715493
$483$	0	0
$484$	−114643.	−0.0222450
$485$	−3.56393e6	−0.687978
$486$	0	0
$487$	3.67779e6	0.702692	0.351346	−	0.936246i	$-0.385724\pi$
$487$	3.67779e6	0.702692	0.351346	+	0.936246i	$0.385724\pi$
$488$	2.35184e6	0.447053
$489$	0	0
$490$	2.62463e6	0.493830
$491$	7.23294e6	1.35398	0.676988	−	0.735994i	$-0.263285\pi$
$491$	7.23294e6	1.35398	0.676988	+	0.735994i	$0.263285\pi$
$492$	0	0
$493$	−2.38989e6	−0.442854
$494$	1.12132e6	0.206734
$495$	0	0
$496$	−3.24446e6	−0.592158
$497$	−1.23556e7	−2.24375
$498$	0	0
$499$	875124.	0.157332	0.0786662	−	0.996901i	$-0.474934\pi$
$499$	875124.	0.157332	0.0786662	+	0.996901i	$0.474934\pi$
$500$	50040.4	0.00895149
$501$	0	0
$502$	−3.69393e6	−0.654228
$503$	2.95982e6	0.521609	0.260805	−	0.965392i	$-0.416012\pi$
$503$	2.95982e6	0.521609	0.260805	+	0.965392i	$0.416012\pi$
$504$	0	0
$505$	−118793.	−0.0207283
$506$	−7.37007e6	−1.27966
$507$	0	0
$508$	−396289.	−0.0681323
$509$	1.12208e7	1.91968	0.959842	−	0.280540i	$-0.0905134\pi$
$509$	1.12208e7	1.91968	0.959842	+	0.280540i	$0.0905134\pi$
$510$	0	0
$511$	−1.40986e7	−2.38850
$512$	−4.81348e6	−0.811493
$513$	0	0
$514$	6.19034e6	1.03349
$515$	−1.48226e6	−0.246268
$516$	0	0
$517$	2.40006e6	0.394907
$518$	−312221.	−0.0511255
$519$	0	0
$520$	−721884.	−0.117074
$521$	−8.92586e6	−1.44064	−0.720321	−	0.693641i	$-0.756005\pi$
$521$	−8.92586e6	−1.44064	−0.720321	+	0.693641i	$0.756005\pi$
$522$	0	0
$523$	6.44897e6	1.03095	0.515473	−	0.856906i	$-0.327616\pi$
$523$	6.44897e6	1.03095	0.515473	+	0.856906i	$0.327616\pi$
$524$	138368.	0.0220144
$525$	0	0
$526$	7.82821e6	1.23367
$527$	−1.84470e6	−0.289334
$528$	0	0
$529$	5.88270e6	0.913982
$530$	−1.13686e6	−0.175800
$531$	0	0
$532$	−665237.	−0.101905
$533$	2.29224e6	0.349496
$534$	0	0
$535$	−3.94818e6	−0.596365
$536$	−1.15748e7	−1.74020
$537$	0	0
$538$	−2.19310e6	−0.326665
$539$	6.26234e6	0.928463
$540$	0	0
$541$	6.01652e6	0.883796	0.441898	−	0.897065i	$-0.354305\pi$
$541$	6.01652e6	0.883796	0.441898	+	0.897065i	$0.354305\pi$
$542$	4.44567e6	0.650039
$543$	0	0
$544$	733177.	0.106221
$545$	1.47196e6	0.212278
$546$	0	0
$547$	928354.	0.132662	0.0663308	−	0.997798i	$-0.478871\pi$
$547$	928354.	0.132662	0.0663308	+	0.997798i	$0.478871\pi$
$548$	603740.	0.0858813
$549$	0	0
$550$	1.31239e6	0.184993
$551$	4.21113e6	0.590907
$552$	0	0
$553$	−1.88795e7	−2.62529
$554$	9.77532e6	1.35318
$555$	0	0
$556$	1.10216e6	0.151203
$557$	−652347.	−0.0890924	−0.0445462	−	0.999007i	$-0.514184\pi$
$557$	−652347.	−0.0890924	−0.0445462	+	0.999007i	$0.514184\pi$
$558$	0	0
$559$	−876176.	−0.118594
$560$	5.18337e6	0.698460
$561$	0	0
$562$	−7.23353e6	−0.966072
$563$	992674.	0.131988	0.0659942	−	0.997820i	$-0.478978\pi$
$563$	992674.	0.131988	0.0659942	+	0.997820i	$0.478978\pi$
$564$	0	0
$565$	−4.49334e6	−0.592173
$566$	−5.91844e6	−0.776545
$567$	0	0
$568$	−1.13654e7	−1.47814
$569$	−8.79703e6	−1.13908	−0.569541	−	0.821963i	$-0.692879\pi$
$569$	−8.79703e6	−1.13908	−0.569541	+	0.821963i	$0.692879\pi$
$570$	0	0
$571$	−6.18261e6	−0.793563	−0.396782	−	0.917913i	$-0.629873\pi$
$571$	−6.18261e6	−0.793563	−0.396782	+	0.917913i	$0.629873\pi$
$572$	191550.	0.0244790
$573$	0	0
$574$	−1.49479e7	−1.89366
$575$	−2.19366e6	−0.276693
$576$	0	0
$577$	1.42671e7	1.78401	0.892003	−	0.452030i	$-0.149300\pi$
$577$	1.42671e7	1.78401	0.892003	+	0.452030i	$0.149300\pi$
$578$	−6.03448e6	−0.751312
$579$	0	0
$580$	301497.	0.0372145
$581$	−9.45125e6	−1.16158
$582$	0	0
$583$	−2.71254e6	−0.330525
$584$	−1.29687e7	−1.57350
$585$	0	0
$586$	−1.07268e7	−1.29040
$587$	7.84422e6	0.939625	0.469812	−	0.882766i	$-0.344322\pi$
$587$	7.84422e6	0.939625	0.469812	+	0.882766i	$0.344322\pi$
$588$	0	0
$589$	3.25047e6	0.386062
$590$	−416255.	−0.0492299
$591$	0	0
$592$	−316233.	−0.0370854
$593$	1.85555e6	0.216689	0.108344	−	0.994113i	$-0.465445\pi$
$593$	1.85555e6	0.216689	0.108344	+	0.994113i	$0.465445\pi$
$594$	0	0
$595$	2.94710e6	0.341274
$596$	569447.	0.0656656
$597$	0	0
$598$	−3.51935e6	−0.402448
$599$	1.54479e7	1.75915	0.879573	−	0.475764i	$-0.157828\pi$
$599$	1.54479e7	1.75915	0.879573	+	0.475764i	$0.157828\pi$
$600$	0	0
$601$	431785.	0.0487619	0.0243810	−	0.999703i	$-0.492239\pi$
$601$	431785.	0.0487619	0.0243810	+	0.999703i	$0.492239\pi$
$602$	5.71362e6	0.642570
$603$	0	0
$604$	1.77674e6	0.198167
$605$	−894924.	−0.0994026
$606$	0	0
$607$	1.21071e7	1.33373	0.666863	−	0.745180i	$-0.267637\pi$
$607$	1.21071e7	1.33373	0.666863	+	0.745180i	$0.267637\pi$
$608$	−1.29190e6	−0.141733
$609$	0	0
$610$	−2.04171e6	−0.222162
$611$	1.14607e6	0.124197
$612$	0	0
$613$	9.44151e6	1.01482	0.507411	−	0.861704i	$-0.330602\pi$
$613$	9.44151e6	1.01482	0.507411	+	0.861704i	$0.330602\pi$
$614$	−145330.	−0.0155573
$615$	0	0
$616$	1.12320e7	1.19263
$617$	9.98389e6	1.05581	0.527906	−	0.849303i	$-0.322977\pi$
$617$	9.98389e6	1.05581	0.527906	+	0.849303i	$0.322977\pi$
$618$	0	0
$619$	−6.44689e6	−0.676275	−0.338138	−	0.941097i	$-0.609797\pi$
$619$	−6.44689e6	−0.676275	−0.338138	+	0.941097i	$0.609797\pi$
$620$	232718.	0.0243137
$621$	0	0
$622$	−8.79367e6	−0.911369
$623$	−9.74037e6	−1.00544
$624$	0	0
$625$	390625.	0.0400000
$626$	−2.06929e6	−0.211050
$627$	0	0
$628$	819528.	0.0829211
$629$	−179800.	−0.0181202
$630$	0	0
$631$	4.35897e6	0.435823	0.217912	−	0.975969i	$-0.430076\pi$
$631$	4.35897e6	0.435823	0.217912	+	0.975969i	$0.430076\pi$
$632$	−1.73665e7	−1.72949
$633$	0	0
$634$	2.41176e6	0.238292
$635$	−3.09351e6	−0.304451
$636$	0	0
$637$	2.99039e6	0.291997
$638$	7.90725e6	0.769084
$639$	0	0
$640$	5.20572e6	0.502378
$641$	−1.63272e7	−1.56952	−0.784758	−	0.619803i	$-0.787213\pi$
$641$	−1.63272e7	−1.56952	−0.784758	+	0.619803i	$0.787213\pi$
$642$	0	0
$643$	1.31929e7	1.25838	0.629192	−	0.777250i	$-0.283386\pi$
$643$	1.31929e7	1.25838	0.629192	+	0.777250i	$0.283386\pi$
$644$	2.08789e6	0.198378
$645$	0	0
$646$	−4.21094e6	−0.397007
$647$	9.42830e6	0.885468	0.442734	−	0.896653i	$-0.354009\pi$
$647$	9.42830e6	0.885468	0.442734	+	0.896653i	$0.354009\pi$
$648$	0	0
$649$	−993180.	−0.0925584
$650$	626692.	0.0581796
$651$	0	0
$652$	−841273.	−0.0775029
$653$	−1.60701e7	−1.47481	−0.737406	−	0.675450i	$-0.763950\pi$
$653$	−1.60701e7	−1.47481	−0.737406	+	0.675450i	$0.763950\pi$
$654$	0	0
$655$	1.08012e6	0.0983718
$656$	−1.51400e7	−1.37362
$657$	0	0
$658$	−7.47365e6	−0.672927
$659$	−6.63639e6	−0.595276	−0.297638	−	0.954679i	$-0.596199\pi$
$659$	−6.63639e6	−0.595276	−0.297638	+	0.954679i	$0.596199\pi$
$660$	0	0
$661$	2.01198e7	1.79110	0.895552	−	0.444956i	$-0.146781\pi$
$661$	2.01198e7	1.79110	0.895552	+	0.444956i	$0.146781\pi$
$662$	9.41087e6	0.834613
$663$	0	0
$664$	−8.69382e6	−0.765227
$665$	−5.19297e6	−0.455367
$666$	0	0
$667$	−1.32169e7	−1.15031
$668$	−919363.	−0.0797160
$669$	0	0
$670$	1.00484e7	0.864792
$671$	−4.87151e6	−0.417693
$672$	0	0
$673$	−9.52533e6	−0.810667	−0.405334	−	0.914169i	$-0.632845\pi$
$673$	−9.52533e6	−0.810667	−0.405334	+	0.914169i	$0.632845\pi$
$674$	1.44525e7	1.22544
$675$	0	0
$676$	91469.0	0.00769852
$677$	3.37825e6	0.283283	0.141641	−	0.989918i	$-0.454762\pi$
$677$	3.37825e6	0.283283	0.141641	+	0.989918i	$0.454762\pi$
$678$	0	0
$679$	2.64794e7	2.20412
$680$	2.71092e6	0.224825
$681$	0	0
$682$	6.10341e6	0.502472
$683$	8.86253e6	0.726953	0.363476	−	0.931603i	$-0.381590\pi$
$683$	8.86253e6	0.726953	0.363476	+	0.931603i	$0.381590\pi$
$684$	0	0
$685$	4.71291e6	0.383763
$686$	−978195.	−0.0793625
$687$	0	0
$688$	5.78704e6	0.466107
$689$	−1.29529e6	−0.103949
$690$	0	0
$691$	25025.8	0.00199385	0.000996925	−	1.00000i	$-0.499683\pi$
$691$	25025.8	0.00199385	0.000996925		1.00000i	$0.499683\pi$
$692$	−2.30478e6	−0.182964
$693$	0	0
$694$	6.98842e6	0.550783
$695$	8.60371e6	0.675653
$696$	0	0
$697$	−8.60815e6	−0.671163
$698$	−2.01048e6	−0.156193
$699$	0	0
$700$	−371792.	−0.0286784
$701$	2.15506e7	1.65640	0.828198	−	0.560436i	$-0.189366\pi$
$701$	2.15506e7	1.65640	0.828198	+	0.560436i	$0.189366\pi$
$702$	0	0
$703$	316819.	0.0241781
$704$	1.02157e7	0.776847
$705$	0	0
$706$	−1.93188e7	−1.45871
$707$	882617.	0.0664085
$708$	0	0
$709$	2.07938e7	1.55352	0.776761	−	0.629796i	$-0.216861\pi$
$709$	2.07938e7	1.55352	0.776761	+	0.629796i	$0.216861\pi$
$710$	9.86672e6	0.734560
$711$	0	0
$712$	−8.95977e6	−0.662364
$713$	−1.02018e7	−0.751544
$714$	0	0
$715$	1.49528e6	0.109385
$716$	−2.49728e6	−0.182048
$717$	0	0
$718$	−1.67208e7	−1.21045
$719$	3.65717e6	0.263829	0.131915	−	0.991261i	$-0.457887\pi$
$719$	3.65717e6	0.263829	0.131915	+	0.991261i	$0.457887\pi$
$720$	0	0
$721$	1.10130e7	0.788982
$722$	−7.27121e6	−0.519115
$723$	0	0
$724$	1.07628e6	0.0763093
$725$	2.35354e6	0.166294
$726$	0	0
$727$	−8.36880e6	−0.587256	−0.293628	−	0.955920i	$-0.594863\pi$
$727$	−8.36880e6	−0.587256	−0.293628	+	0.955920i	$0.594863\pi$
$728$	5.36349e6	0.375075
$729$	0	0
$730$	1.12586e7	0.781947
$731$	3.29034e6	0.227744
$732$	0	0
$733$	1.81111e7	1.24504	0.622522	−	0.782602i	$-0.286108\pi$
$733$	1.81111e7	1.24504	0.622522	+	0.782602i	$0.286108\pi$
$734$	1.83727e7	1.25873
$735$	0	0
$736$	4.05472e6	0.275910
$737$	2.39755e7	1.62592
$738$	0	0
$739$	7.31705e6	0.492861	0.246431	−	0.969160i	$-0.420742\pi$
$739$	7.31705e6	0.492861	0.246431	+	0.969160i	$0.420742\pi$
$740$	22682.7	0.00152271
$741$	0	0
$742$	8.44670e6	0.563219
$743$	−1.04179e7	−0.692322	−0.346161	−	0.938175i	$-0.612515\pi$
$743$	−1.04179e7	−0.692322	−0.346161	+	0.938175i	$0.612515\pi$
$744$	0	0
$745$	4.44522e6	0.293428
$746$	−2.50057e7	−1.64510
$747$	0	0
$748$	−719336.	−0.0470087
$749$	2.93343e7	1.91061
$750$	0	0
$751$	−1.16729e6	−0.0755229	−0.0377615	−	0.999287i	$-0.512023\pi$
$751$	−1.16729e6	−0.0755229	−0.0377615	+	0.999287i	$0.512023\pi$
$752$	−7.56969e6	−0.488128
$753$	0	0
$754$	3.77586e6	0.241873
$755$	1.38696e7	0.885517
$756$	0	0
$757$	4.75104e6	0.301334	0.150667	−	0.988585i	$-0.451858\pi$
$757$	4.75104e6	0.301334	0.150667	+	0.988585i	$0.451858\pi$
$758$	−7.51433e6	−0.475026
$759$	0	0
$760$	−4.77680e6	−0.299987
$761$	5.92209e6	0.370692	0.185346	−	0.982673i	$-0.440659\pi$
$761$	5.92209e6	0.370692	0.185346	+	0.982673i	$0.440659\pi$
$762$	0	0
$763$	−1.09364e7	−0.680087
$764$	−1.31139e6	−0.0812828
$765$	0	0
$766$	−3.36375e7	−2.07134
$767$	−474262.	−0.0291092
$768$	0	0
$769$	−5.07027e6	−0.309183	−0.154591	−	0.987979i	$-0.549406\pi$
$769$	−5.07027e6	−0.309183	−0.154591	+	0.987979i	$0.549406\pi$
$770$	−9.75086e6	−0.592674
$771$	0	0
$772$	1.08748e6	0.0656719
$773$	−2.31839e7	−1.39552	−0.697761	−	0.716330i	$-0.745820\pi$
$773$	−2.31839e7	−1.39552	−0.697761	+	0.716330i	$0.745820\pi$
$774$	0	0
$775$	1.81664e6	0.108646
$776$	2.43573e7	1.45203
$777$	0	0
$778$	3.25799e7	1.92975
$779$	1.51681e7	0.895543
$780$	0	0
$781$	2.35419e7	1.38106
$782$	1.32163e7	0.772849
$783$	0	0
$784$	−1.97512e7	−1.14763
$785$	6.39740e6	0.370535
$786$	0	0
$787$	−6.07650e6	−0.349717	−0.174858	−	0.984594i	$-0.555947\pi$
$787$	−6.07650e6	−0.349717	−0.174858	+	0.984594i	$0.555947\pi$
$788$	2.79095e6	0.160117
$789$	0	0
$790$	1.50764e7	0.859469
$791$	3.33848e7	1.89718
$792$	0	0
$793$	−2.32624e6	−0.131362
$794$	−9.11855e6	−0.513304
$795$	0	0
$796$	867745.	0.0485411
$797$	−2.78805e7	−1.55473	−0.777363	−	0.629052i	$-0.783443\pi$
$797$	−2.78805e7	−1.55473	−0.777363	+	0.629052i	$0.783443\pi$
$798$	0	0
$799$	−4.30389e6	−0.238503
$800$	−722026.	−0.0398867
$801$	0	0
$802$	−178164.	−0.00978099
$803$	2.68629e7	1.47016
$804$	0	0
$805$	1.62985e7	0.886459
$806$	2.91450e6	0.158025
$807$	0	0
$808$	811883.	0.0437487
$809$	−6.10438e6	−0.327922	−0.163961	−	0.986467i	$-0.552427\pi$
$809$	−6.10438e6	−0.327922	−0.163961	+	0.986467i	$0.552427\pi$
$810$	0	0
$811$	−2.23956e7	−1.19567	−0.597835	−	0.801619i	$-0.703972\pi$
$811$	−2.23956e7	−1.19567	−0.597835	+	0.801619i	$0.703972\pi$
$812$	−2.24007e6	−0.119226
$813$	0	0
$814$	594892.	0.0314686
$815$	−6.56714e6	−0.346324
$816$	0	0
$817$	−5.79777e6	−0.303882
$818$	−3.28943e7	−1.71885
$819$	0	0
$820$	1.08596e6	0.0564001
$821$	2.42967e7	1.25803	0.629014	−	0.777394i	$-0.283459\pi$
$821$	2.42967e7	1.25803	0.629014	+	0.777394i	$0.283459\pi$
$822$	0	0
$823$	3.64578e7	1.87625	0.938127	−	0.346293i	$-0.112560\pi$
$823$	3.64578e7	1.87625	0.938127	+	0.346293i	$0.112560\pi$
$824$	1.01304e7	0.519767
$825$	0	0
$826$	3.09271e6	0.157721
$827$	2.81247e7	1.42996	0.714981	−	0.699143i	$-0.246435\pi$
$827$	2.81247e7	1.42996	0.714981	+	0.699143i	$0.246435\pi$
$828$	0	0
$829$	2.68734e7	1.35812	0.679058	−	0.734085i	$-0.262389\pi$
$829$	2.68734e7	1.35812	0.679058	+	0.734085i	$0.262389\pi$
$830$	7.54739e6	0.380279
$831$	0	0
$832$	4.87818e6	0.244315
$833$	−1.12299e7	−0.560743
$834$	0	0
$835$	−7.17673e6	−0.356213
$836$	1.26751e6	0.0627244
$837$	0	0
$838$	7.67514e6	0.377552
$839$	−3.46774e7	−1.70076	−0.850378	−	0.526172i	$-0.823627\pi$
$839$	−3.46774e7	−1.70076	−0.850378	+	0.526172i	$0.823627\pi$
$840$	0	0
$841$	−6.33089e6	−0.308656
$842$	−2.18709e7	−1.06313
$843$	0	0
$844$	581123.	0.0280810
$845$	714025.	0.0344010
$846$	0	0
$847$	6.64914e6	0.318462
$848$	8.55524e6	0.408548
$849$	0	0
$850$	−2.35344e6	−0.111726
$851$	−994359.	−0.0470673
$852$	0	0
$853$	1.54571e6	0.0727368	0.0363684	−	0.999338i	$-0.488421\pi$
$853$	1.54571e6	0.0727368	0.0363684	+	0.999338i	$0.488421\pi$
$854$	1.51696e7	0.711754
$855$	0	0
$856$	2.69834e7	1.25867
$857$	1.27926e7	0.594987	0.297493	−	0.954724i	$-0.403849\pi$
$857$	1.27926e7	0.594987	0.297493	+	0.954724i	$0.403849\pi$
$858$	0	0
$859$	2.66940e6	0.123433	0.0617165	−	0.998094i	$-0.480343\pi$
$859$	2.66940e6	0.123433	0.0617165	+	0.998094i	$0.480343\pi$
$860$	−415093.	−0.0191381
$861$	0	0
$862$	−7.27673e6	−0.333555
$863$	3.37798e7	1.54394	0.771970	−	0.635659i	$-0.219272\pi$
$863$	3.37798e7	1.54394	0.771970	+	0.635659i	$0.219272\pi$
$864$	0	0
$865$	−1.79916e7	−0.817578
$866$	6.07906e6	0.275449
$867$	0	0
$868$	−1.72906e6	−0.0778952
$869$	3.59721e7	1.61591
$870$	0	0
$871$	1.14488e7	0.511344
$872$	−1.00600e7	−0.448029
$873$	0	0
$874$	−2.32880e7	−1.03122
$875$	−2.90228e6	−0.128150
$876$	0	0
$877$	3.97308e7	1.74433	0.872164	−	0.489214i	$-0.162716\pi$
$877$	3.97308e7	1.74433	0.872164	+	0.489214i	$0.162716\pi$
$878$	−2.99596e7	−1.31160
$879$	0	0
$880$	−9.87616e6	−0.429914
$881$	1.67058e7	0.725149	0.362574	−	0.931955i	$-0.381898\pi$
$881$	1.67058e7	0.725149	0.362574	+	0.931955i	$0.381898\pi$
$882$	0	0
$883$	−1.67930e7	−0.724816	−0.362408	−	0.932020i	$-0.618045\pi$
$883$	−1.67930e7	−0.724816	−0.362408	+	0.932020i	$0.618045\pi$
$884$	−343497.	−0.0147840
$885$	0	0
$886$	−3.74293e7	−1.60187
$887$	−8.61531e6	−0.367673	−0.183837	−	0.982957i	$-0.558852\pi$
$887$	−8.61531e6	−0.367673	−0.183837	+	0.982957i	$0.558852\pi$
$888$	0	0
$889$	2.29843e7	0.975388
$890$	7.77828e6	0.329161
$891$	0	0
$892$	4.44393e6	0.187006
$893$	7.58371e6	0.318239
$894$	0	0
$895$	−1.94943e7	−0.813486
$896$	−3.86777e7	−1.60950
$897$	0	0
$898$	−6.90571e6	−0.285770
$899$	1.09454e7	0.451682
$900$	0	0
$901$	4.86425e6	0.199620
$902$	2.84811e7	1.16558
$903$	0	0
$904$	3.07093e7	1.24983
$905$	8.40163e6	0.340990
$906$	0	0
$907$	7.34436e6	0.296439	0.148220	−	0.988954i	$-0.452646\pi$
$907$	7.34436e6	0.296439	0.148220	+	0.988954i	$0.452646\pi$
$908$	2.21105e6	0.0889989
$909$	0	0
$910$	−4.65622e6	−0.186393
$911$	−3.63225e7	−1.45004	−0.725019	−	0.688729i	$-0.758169\pi$
$911$	−3.63225e7	−1.45004	−0.725019	+	0.688729i	$0.758169\pi$
$912$	0	0
$913$	1.80080e7	0.714971
$914$	8.79036e6	0.348049
$915$	0	0
$916$	−4.42781e6	−0.174361
$917$	−8.02515e6	−0.315159
$918$	0	0
$919$	2.25278e7	0.879892	0.439946	−	0.898024i	$-0.354998\pi$
$919$	2.25278e7	0.879892	0.439946	+	0.898024i	$0.354998\pi$
$920$	1.49923e7	0.583982
$921$	0	0
$922$	3.35457e7	1.29960
$923$	1.12417e7	0.434338
$924$	0	0
$925$	177066.	0.00680425
$926$	1.24131e7	0.475721
$927$	0	0
$928$	−4.35026e6	−0.165823
$929$	−3.93312e7	−1.49520	−0.747598	−	0.664152i	$-0.768793\pi$
$929$	−3.93312e7	−1.49520	−0.747598	+	0.664152i	$0.768793\pi$
$930$	0	0
$931$	1.97878e7	0.748209
$932$	230303.	0.00868480
$933$	0	0
$934$	4.44087e7	1.66571
$935$	−5.61528e6	−0.210060
$936$	0	0
$937$	−1.36354e7	−0.507362	−0.253681	−	0.967288i	$-0.581641\pi$
$937$	−1.36354e7	−0.507362	−0.253681	+	0.967288i	$0.581641\pi$
$938$	−7.46584e7	−2.77058
$939$	0	0
$940$	542958.	0.0200423
$941$	−2.28438e7	−0.840995	−0.420498	−	0.907294i	$-0.638144\pi$
$941$	−2.28438e7	−0.840995	−0.420498	+	0.907294i	$0.638144\pi$
$942$	0	0
$943$	−4.76061e7	−1.74334
$944$	3.13245e6	0.114407
$945$	0	0
$946$	−1.08865e7	−0.395512
$947$	6.46944e6	0.234418	0.117209	−	0.993107i	$-0.462605\pi$
$947$	6.46944e6	0.234418	0.117209	+	0.993107i	$0.462605\pi$
$948$	0	0
$949$	1.28276e7	0.462358
$950$	4.14690e6	0.149078
$951$	0	0
$952$	−2.01417e7	−0.720284
$953$	−2.58068e6	−0.0920452	−0.0460226	−	0.998940i	$-0.514655\pi$
$953$	−2.58068e6	−0.0920452	−0.0460226	+	0.998940i	$0.514655\pi$
$954$	0	0
$955$	−1.02370e7	−0.363215
$956$	2.64355e6	0.0935497
$957$	0	0
$958$	−1.51165e7	−0.532154
$959$	−3.50162e7	−1.22948
$960$	0	0
$961$	−2.01807e7	−0.704899
$962$	284072.	0.00989672
$963$	0	0
$964$	−1.96986e6	−0.0682722
$965$	8.48912e6	0.293457
$966$	0	0
$967$	−1.88591e6	−0.0648568	−0.0324284	−	0.999474i	$-0.510324\pi$
$967$	−1.88591e6	−0.0648568	−0.0324284	+	0.999474i	$0.510324\pi$
$968$	6.11627e6	0.209797
$969$	0	0
$970$	−2.11454e7	−0.721584
$971$	2.49003e7	0.847534	0.423767	−	0.905771i	$-0.360708\pi$
$971$	2.49003e7	0.847534	0.423767	+	0.905771i	$0.360708\pi$
$972$	0	0
$973$	−6.39242e7	−2.16463
$974$	2.18210e7	0.737016
$975$	0	0
$976$	1.53645e7	0.516292
$977$	−1.01689e7	−0.340829	−0.170415	−	0.985372i	$-0.554511\pi$
$977$	−1.01689e7	−0.340829	−0.170415	+	0.985372i	$0.554511\pi$
$978$	0	0
$979$	1.85589e7	0.618864
$980$	1.41671e6	0.0471212
$981$	0	0
$982$	4.29143e7	1.42011
$983$	−4.29289e6	−0.141699	−0.0708494	−	0.997487i	$-0.522571\pi$
$983$	−4.29289e6	−0.141699	−0.0708494	+	0.997487i	$0.522571\pi$
$984$	0	0
$985$	2.17867e7	0.715487
$986$	−1.41796e7	−0.464486
$987$	0	0
$988$	605262.	0.0197265
$989$	1.81967e7	0.591565
$990$	0	0
$991$	−2.64765e7	−0.856398	−0.428199	−	0.903684i	$-0.640852\pi$
$991$	−2.64765e7	−0.856398	−0.428199	+	0.903684i	$0.640852\pi$
$992$	−3.35786e6	−0.108339
$993$	0	0
$994$	−7.33082e7	−2.35335
$995$	6.77379e6	0.216907
$996$	0	0
$997$	4.21089e6	0.134164	0.0670820	−	0.997747i	$-0.478631\pi$
$997$	4.21089e6	0.134164	0.0670820	+	0.997747i	$0.478631\pi$
$998$	5.19227e6	0.165018
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.6.a.m.1.5		6
3.2	odd	2		65.6.a.d.1.2	✓	6
12.11	even	2		1040.6.a.q.1.5		6
15.2	even	4		325.6.b.g.274.4		12
15.8	even	4		325.6.b.g.274.9		12
15.14	odd	2		325.6.a.g.1.5		6
39.38	odd	2		845.6.a.h.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.2	✓	6	3.2	odd	2
325.6.a.g.1.5		6	15.14	odd	2
325.6.b.g.274.4		12	15.2	even	4
325.6.b.g.274.9		12	15.8	even	4
585.6.a.m.1.5		6	1.1	even	1	trivial
845.6.a.h.1.5		6	39.38	odd	2
1040.6.a.q.1.5		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

\(f(q)\)	\(=\)	\(q+5.93318 q^{2} +3.20258 q^{4} +25.0000 q^{5} -185.746 q^{7} -170.860 q^{8} +148.329 q^{10} +353.912 q^{11} +169.000 q^{13} -1102.06 q^{14} -1116.23 q^{16} -634.652 q^{17} +1118.29 q^{19} +80.0646 q^{20} +2099.83 q^{22} -3509.85 q^{23} +625.000 q^{25} +1002.71 q^{26} -594.867 q^{28} +3765.67 q^{29} +2906.63 q^{31} -1155.24 q^{32} -3765.50 q^{34} -4643.65 q^{35} +283.305 q^{37} +6635.04 q^{38} -4271.50 q^{40} +13563.6 q^{41} -5184.47 q^{43} +1133.43 q^{44} -20824.6 q^{46} +6781.50 q^{47} +17694.6 q^{49} +3708.24 q^{50} +541.237 q^{52} -7664.43 q^{53} +8847.81 q^{55} +31736.6 q^{56} +22342.4 q^{58} -2806.29 q^{59} -13764.7 q^{61} +17245.5 q^{62} +28865.0 q^{64} +4225.00 q^{65} +67744.1 q^{67} -2032.53 q^{68} -27551.6 q^{70} +66519.0 q^{71} +75902.7 q^{73} +1680.90 q^{74} +3581.43 q^{76} -65737.9 q^{77} +101641. q^{79} -27905.7 q^{80} +80475.1 q^{82} +50882.7 q^{83} -15866.3 q^{85} -30760.4 q^{86} -60469.5 q^{88} +52439.2 q^{89} -31391.1 q^{91} -11240.6 q^{92} +40235.8 q^{94} +27957.4 q^{95} -142557. q^{97} +104985. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 134 q^{4} + 150 q^{5} + 220 q^{7} - 24 q^{8} + 170 q^{11} + 1014 q^{13} + 1440 q^{14} + 3506 q^{16} - 728 q^{17} + 1218 q^{19} + 3350 q^{20} + 5154 q^{22} - 8954 q^{23} + 3750 q^{25} + 13212 q^{28}+ \cdots + 4736 q^{98}+O(q^{100}) \) 6 * q + 134 * q^4 + 150 * q^5 + 220 * q^7 - 24 * q^8 + 170 * q^11 + 1014 * q^13 + 1440 * q^14 + 3506 * q^16 - 728 * q^17 + 1218 * q^19 + 3350 * q^20 + 5154 * q^22 - 8954 * q^23 + 3750 * q^25 + 13212 * q^28 - 8364 * q^29 + 2862 * q^31 + 29672 * q^32 - 21484 * q^34 + 5500 * q^35 + 13840 * q^37 + 27706 * q^38 - 600 * q^40 - 2248 * q^41 + 3822 * q^43 + 73482 * q^44 - 39886 * q^46 + 10860 * q^47 + 85902 * q^49 + 22646 * q^52 - 6584 * q^53 + 4250 * q^55 + 188800 * q^56 - 88500 * q^58 + 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 14022 * q^64 + 25350 * q^65 + 60040 * q^67 - 3504 * q^68 + 36000 * q^70 + 74174 * q^71 + 48960 * q^73 - 226112 * q^74 - 93422 * q^76 - 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 58732 * q^82 - 123260 * q^83 - 18200 * q^85 + 304666 * q^86 + 127622 * q^88 - 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 125920 * q^94 + 30450 * q^95 + 246284 * q^97 + 4736 * q^98

Introduction

L-functions

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