LMFDB - Embedded newform 1521.4.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 82x^{10} + 2137x^{8} - 22488x^{6} + 89784x^{4} - 67392x^{2} + 11664 $ x^12 - 82x^10 + 2137x^8 - 22488x^6 + 89784x^4 - 67392*x^2 + 11664
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.63568$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.90363 q^{2} +16.0456 q^{4} +5.61325 q^{5} -3.21345 q^{7} -39.4526 q^{8} +O(q^{10})$	$q-4.90363 q^{2} +16.0456 q^{4} +5.61325 q^{5} -3.21345 q^{7} -39.4526 q^{8} -27.5253 q^{10} +43.6466 q^{11} +15.7576 q^{14} +65.0962 q^{16} -65.8757 q^{17} -3.06427 q^{19} +90.0680 q^{20} -214.027 q^{22} +163.764 q^{23} -93.4914 q^{25} -51.5617 q^{28} -155.449 q^{29} +240.463 q^{31} -3.58708 q^{32} +323.030 q^{34} -18.0379 q^{35} -317.662 q^{37} +15.0261 q^{38} -221.457 q^{40} +127.910 q^{41} -379.001 q^{43} +700.336 q^{44} -803.039 q^{46} +79.4626 q^{47} -332.674 q^{49} +458.447 q^{50} +571.746 q^{53} +244.999 q^{55} +126.779 q^{56} +762.263 q^{58} -105.538 q^{59} -724.638 q^{61} -1179.14 q^{62} -503.180 q^{64} +302.222 q^{67} -1057.01 q^{68} +88.4512 q^{70} -1003.39 q^{71} +277.988 q^{73} +1557.70 q^{74} -49.1681 q^{76} -140.256 q^{77} -959.137 q^{79} +365.402 q^{80} -627.224 q^{82} +410.957 q^{83} -369.777 q^{85} +1858.48 q^{86} -1721.97 q^{88} -226.199 q^{89} +2627.69 q^{92} -389.655 q^{94} -17.2005 q^{95} -888.458 q^{97} +1631.31 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 32 q^{4}+O(q^{10}) $ 12 * q + 32 * q^4	$ 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) $ 12 * q + 32 * q^4 - 112 * q^10 + 184 * q^16 - 584 * q^22 + 92 * q^25 - 448 * q^40 - 1620 * q^43 - 2136 * q^49 - 920 * q^55 - 2588 * q^61 + 184 * q^64 - 6380 * q^79 - 2536 * q^82 - 10280 * q^88 - 3320 * q^94

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 $

$2$	−4.90363	−1.73370	−0.866848	−	0.498573i	$-0.833857\pi$
$2$	−4.90363	−1.73370	−0.866848	+	0.498573i	$0.833857\pi$
$3$	0	0
$3$	0	0
$4$	16.0456	2.00570
$5$	5.61325	0.502065	0.251032	−	0.967979i	$-0.419230\pi$
$5$	5.61325	0.502065	0.251032	+	0.967979i	$0.419230\pi$
$6$	0	0
$7$	−3.21345	−0.173510	−0.0867550	−	0.996230i	$-0.527650\pi$
$7$	−3.21345	−0.173510	−0.0867550	+	0.996230i	$0.527650\pi$
$8$	−39.4526	−1.74357
$9$	0	0
$10$	−27.5253	−0.870427
$11$	43.6466	1.19636	0.598179	−	0.801362i	$-0.295891\pi$
$11$	43.6466	1.19636	0.598179	+	0.801362i	$0.295891\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	15.7576	0.300813
$15$	0	0
$16$	65.0962	1.01713
$17$	−65.8757	−0.939835	−0.469918	−	0.882710i	$-0.655716\pi$
$17$	−65.8757	−0.939835	−0.469918	+	0.882710i	$0.655716\pi$
$18$	0	0
$19$	−3.06427	−0.0369996	−0.0184998	−	0.999829i	$-0.505889\pi$
$19$	−3.06427	−0.0369996	−0.0184998	+	0.999829i	$0.505889\pi$
$20$	90.0680	1.00699
$21$	0	0
$22$	−214.027	−2.07412
$23$	163.764	1.48466	0.742330	−	0.670034i	$-0.233721\pi$
$23$	163.764	1.48466	0.742330	+	0.670034i	$0.233721\pi$
$24$	0	0
$25$	−93.4914	−0.747931
$26$	0	0
$27$	0	0
$28$	−51.5617	−0.348009
$29$	−155.449	−0.995382	−0.497691	−	0.867354i	$-0.665819\pi$
$29$	−155.449	−0.995382	−0.497691	+	0.867354i	$0.665819\pi$
$30$	0	0
$31$	240.463	1.39318	0.696588	−	0.717471i	$-0.254701\pi$
$31$	240.463	1.39318	0.696588	+	0.717471i	$0.254701\pi$
$32$	−3.58708	−0.0198160
$33$	0	0
$34$	323.030	1.62939
$35$	−18.0379	−0.0871132
$36$	0	0
$37$	−317.662	−1.41144	−0.705721	−	0.708490i	$-0.749377\pi$
$37$	−317.662	−1.41144	−0.705721	+	0.708490i	$0.749377\pi$
$38$	15.0261	0.0641460
$39$	0	0
$40$	−221.457	−0.875387
$41$	127.910	0.487224	0.243612	−	0.969873i	$-0.421668\pi$
$41$	127.910	0.487224	0.243612	+	0.969873i	$0.421668\pi$
$42$	0	0
$43$	−379.001	−1.34412	−0.672060	−	0.740497i	$-0.734590\pi$
$43$	−379.001	−1.34412	−0.672060	+	0.740497i	$0.734590\pi$
$44$	700.336	2.39954
$45$	0	0
$46$	−803.039	−2.57395
$47$	79.4626	0.246613	0.123306	−	0.992369i	$-0.460650\pi$
$47$	79.4626	0.246613	0.123306	+	0.992369i	$0.460650\pi$
$48$	0	0
$49$	−332.674	−0.969894
$50$	458.447	1.29668
$51$	0	0
$52$	0	0
$53$	571.746	1.48180	0.740899	−	0.671616i	$-0.234400\pi$
$53$	571.746	1.48180	0.740899	+	0.671616i	$0.234400\pi$
$54$	0	0
$55$	244.999	0.600649
$56$	126.779	0.302528
$57$	0	0
$58$	762.263	1.72569
$59$	−105.538	−0.232879	−0.116440	−	0.993198i	$-0.537148\pi$
$59$	−105.538	−0.232879	−0.116440	+	0.993198i	$0.537148\pi$
$60$	0	0
$61$	−724.638	−1.52099	−0.760495	−	0.649344i	$-0.775043\pi$
$61$	−724.638	−1.52099	−0.760495	+	0.649344i	$0.775043\pi$
$62$	−1179.14	−2.41534
$63$	0	0
$64$	−503.180	−0.982774
$65$	0	0
$66$	0	0
$67$	302.222	0.551078	0.275539	−	0.961290i	$-0.411144\pi$
$67$	302.222	0.551078	0.275539	+	0.961290i	$0.411144\pi$
$68$	−1057.01	−1.88503
$69$	0	0
$70$	88.4512	0.151028
$71$	−1003.39	−1.67719	−0.838594	−	0.544757i	$-0.816622\pi$
$71$	−1003.39	−1.67719	−0.838594	+	0.544757i	$0.816622\pi$
$72$	0	0
$73$	277.988	0.445699	0.222850	−	0.974853i	$-0.428464\pi$
$73$	277.988	0.445699	0.222850	+	0.974853i	$0.428464\pi$
$74$	1557.70	2.44701
$75$	0	0
$76$	−49.1681	−0.0742101
$77$	−140.256	−0.207580
$78$	0	0
$79$	−959.137	−1.36597	−0.682983	−	0.730434i	$-0.739318\pi$
$79$	−959.137	−1.36597	−0.682983	+	0.730434i	$0.739318\pi$
$80$	365.402	0.510664
$81$	0	0
$82$	−627.224	−0.844699
$83$	410.957	0.543475	0.271737	−	0.962371i	$-0.412402\pi$
$83$	410.957	0.543475	0.271737	+	0.962371i	$0.412402\pi$
$84$	0	0
$85$	−369.777	−0.471858
$86$	1858.48	2.33029
$87$	0	0
$88$	−1721.97	−2.08594
$89$	−226.199	−0.269405	−0.134703	−	0.990886i	$-0.543008\pi$
$89$	−226.199	−0.269405	−0.134703	+	0.990886i	$0.543008\pi$
$90$	0	0
$91$	0	0
$92$	2627.69	2.97778
$93$	0	0
$94$	−389.655	−0.427552
$95$	−17.2005	−0.0185762
$96$	0	0
$97$	−888.458	−0.929992	−0.464996	−	0.885313i	$-0.653944\pi$
$97$	−888.458	−0.929992	−0.464996	+	0.885313i	$0.653944\pi$
$98$	1631.31	1.68150
$99$	0	0
$100$	−1500.12	−1.50012
$101$	−1860.36	−1.83280	−0.916400	−	0.400264i	$-0.868918\pi$
$101$	−1860.36	−1.83280	−0.916400	+	0.400264i	$0.868918\pi$
$102$	0	0
$103$	328.512	0.314265	0.157132	−	0.987578i	$-0.449775\pi$
$103$	328.512	0.314265	0.157132	+	0.987578i	$0.449775\pi$
$104$	0	0
$105$	0	0
$106$	−2803.63	−2.56899
$107$	328.774	0.297045	0.148522	−	0.988909i	$-0.452548\pi$
$107$	328.774	0.297045	0.148522	+	0.988909i	$0.452548\pi$
$108$	0	0
$109$	201.673	0.177218	0.0886092	−	0.996066i	$-0.471758\pi$
$109$	201.673	0.177218	0.0886092	+	0.996066i	$0.471758\pi$
$110$	−1201.39	−1.04134
$111$	0	0
$112$	−209.183	−0.176482
$113$	289.984	0.241411	0.120705	−	0.992688i	$-0.461484\pi$
$113$	289.984	0.241411	0.120705	+	0.992688i	$0.461484\pi$
$114$	0	0
$115$	919.250	0.745395
$116$	−2494.27	−1.99644
$117$	0	0
$118$	517.520	0.403742
$119$	211.688	0.163071
$120$	0	0
$121$	574.026	0.431274
$122$	3553.36	2.63693
$123$	0	0
$124$	3858.37	2.79429
$125$	−1226.45	−0.877574
$126$	0	0
$127$	752.567	0.525823	0.262911	−	0.964820i	$-0.415317\pi$
$127$	752.567	0.525823	0.262911	+	0.964820i	$0.415317\pi$
$128$	2496.11	1.72365
$129$	0	0
$130$	0	0
$131$	2200.41	1.46756	0.733782	−	0.679385i	$-0.237753\pi$
$131$	2200.41	1.46756	0.733782	+	0.679385i	$0.237753\pi$
$132$	0	0
$133$	9.84689	0.00641980
$134$	−1481.98	−0.955402
$135$	0	0
$136$	2598.97	1.63867
$137$	364.076	0.227044	0.113522	−	0.993535i	$-0.463787\pi$
$137$	364.076	0.227044	0.113522	+	0.993535i	$0.463787\pi$
$138$	0	0
$139$	2453.59	1.49720	0.748600	−	0.663022i	$-0.230727\pi$
$139$	2453.59	1.49720	0.748600	+	0.663022i	$0.230727\pi$
$140$	−289.429	−0.174723
$141$	0	0
$142$	4920.25	2.90773
$143$	0	0
$144$	0	0
$145$	−872.573	−0.499746
$146$	−1363.15	−0.772706
$147$	0	0
$148$	−5097.08	−2.83093
$149$	−65.1448	−0.0358179	−0.0179090	−	0.999840i	$-0.505701\pi$
$149$	−65.1448	−0.0358179	−0.0179090	+	0.999840i	$0.505701\pi$
$150$	0	0
$151$	−250.204	−0.134843	−0.0674216	−	0.997725i	$-0.521477\pi$
$151$	−250.204	−0.134843	−0.0674216	+	0.997725i	$0.521477\pi$
$152$	120.894	0.0645116
$153$	0	0
$154$	687.764	0.359881
$155$	1349.78	0.699464
$156$	0	0
$157$	1768.99	0.899242	0.449621	−	0.893219i	$-0.351559\pi$
$157$	1768.99	0.899242	0.449621	+	0.893219i	$0.351559\pi$
$158$	4703.25	2.36817
$159$	0	0
$160$	−20.1352	−0.00994890
$161$	−526.248	−0.257603
$162$	0	0
$163$	2707.32	1.30094	0.650470	−	0.759532i	$-0.274572\pi$
$163$	2707.32	1.30094	0.650470	+	0.759532i	$0.274572\pi$
$164$	2052.39	0.977226
$165$	0	0
$166$	−2015.18	−0.942220
$167$	3492.29	1.61821	0.809107	−	0.587661i	$-0.199951\pi$
$167$	3492.29	1.61821	0.809107	+	0.587661i	$0.199951\pi$
$168$	0	0
$169$	0	0
$170$	1813.25	0.818058
$171$	0	0
$172$	−6081.30	−2.69590
$173$	−3981.96	−1.74996	−0.874979	−	0.484160i	$-0.839125\pi$
$173$	−3981.96	−1.74996	−0.874979	+	0.484160i	$0.839125\pi$
$174$	0	0
$175$	300.430	0.129774
$176$	2841.23	1.21685
$177$	0	0
$178$	1109.20	0.467066
$179$	2255.05	0.941624	0.470812	−	0.882234i	$-0.343961\pi$
$179$	2255.05	0.941624	0.470812	+	0.882234i	$0.343961\pi$
$180$	0	0
$181$	−890.108	−0.365532	−0.182766	−	0.983156i	$-0.558505\pi$
$181$	−890.108	−0.365532	−0.182766	+	0.983156i	$0.558505\pi$
$182$	0	0
$183$	0	0
$184$	−6460.92	−2.58862
$185$	−1783.12	−0.708635
$186$	0	0
$187$	−2875.25	−1.12438
$188$	1275.02	0.494631
$189$	0	0
$190$	84.3451	0.0322055
$191$	799.774	0.302982	0.151491	−	0.988459i	$-0.451593\pi$
$191$	799.774	0.302982	0.151491	+	0.988459i	$0.451593\pi$
$192$	0	0
$193$	2693.40	1.00454	0.502268	−	0.864712i	$-0.332499\pi$
$193$	2693.40	1.00454	0.502268	+	0.864712i	$0.332499\pi$
$194$	4356.67	1.61232
$195$	0	0
$196$	−5337.95	−1.94532
$197$	−2735.70	−0.989395	−0.494698	−	0.869065i	$-0.664721\pi$
$197$	−2735.70	−0.989395	−0.494698	+	0.869065i	$0.664721\pi$
$198$	0	0
$199$	−1176.44	−0.419072	−0.209536	−	0.977801i	$-0.567195\pi$
$199$	−1176.44	−0.419072	−0.209536	+	0.977801i	$0.567195\pi$
$200$	3688.48	1.30407
$201$	0	0
$202$	9122.52	3.17752
$203$	499.526	0.172709
$204$	0	0
$205$	717.992	0.244618
$206$	−1610.90	−0.544839
$207$	0	0
$208$	0	0
$209$	−133.745	−0.0442648
$210$	0	0
$211$	−2914.65	−0.950961	−0.475481	−	0.879726i	$-0.657726\pi$
$211$	−2914.65	−0.950961	−0.475481	+	0.879726i	$0.657726\pi$
$212$	9174.00	2.97204
$213$	0	0
$214$	−1612.19	−0.514985
$215$	−2127.43	−0.674835
$216$	0	0
$217$	−772.716	−0.241730
$218$	−988.932	−0.307243
$219$	0	0
$220$	3931.16	1.20472
$221$	0	0
$222$	0	0
$223$	1481.39	0.444848	0.222424	−	0.974950i	$-0.428603\pi$
$223$	1481.39	0.444848	0.222424	+	0.974950i	$0.428603\pi$
$224$	11.5269	0.00343827
$225$	0	0
$226$	−1421.98	−0.418533
$227$	−4.38339	−0.00128166	−0.000640828	−	1.00000i	$-0.500204\pi$
$227$	−4.38339	−0.00128166	−0.000640828		1.00000i	$0.500204\pi$
$228$	0	0
$229$	−4689.58	−1.35326	−0.676629	−	0.736324i	$-0.736560\pi$
$229$	−4689.58	−1.35326	−0.676629	+	0.736324i	$0.736560\pi$
$230$	−4507.66	−1.29229
$231$	0	0
$232$	6132.85	1.73552
$233$	−4124.52	−1.15968	−0.579842	−	0.814729i	$-0.696886\pi$
$233$	−4124.52	−1.15968	−0.579842	+	0.814729i	$0.696886\pi$
$234$	0	0
$235$	446.044	0.123816
$236$	−1693.42	−0.467086
$237$	0	0
$238$	−1038.04	−0.282715
$239$	−3455.41	−0.935196	−0.467598	−	0.883941i	$-0.654880\pi$
$239$	−3455.41	−0.935196	−0.467598	+	0.883941i	$0.654880\pi$
$240$	0	0
$241$	−4109.67	−1.09845	−0.549226	−	0.835674i	$-0.685077\pi$
$241$	−4109.67	−1.09845	−0.549226	+	0.835674i	$0.685077\pi$
$242$	−2814.81	−0.747698
$243$	0	0
$244$	−11627.2	−3.05065
$245$	−1867.38	−0.486950
$246$	0	0
$247$	0	0
$248$	−9486.89	−2.42911
$249$	0	0
$250$	6014.05	1.52145
$251$	−5032.21	−1.26546	−0.632730	−	0.774372i	$-0.718066\pi$
$251$	−5032.21	−1.26546	−0.632730	+	0.774372i	$0.718066\pi$
$252$	0	0
$253$	7147.75	1.77619
$254$	−3690.31	−0.911617
$255$	0	0
$256$	−8214.54	−2.00550
$257$	−5633.97	−1.36746	−0.683731	−	0.729735i	$-0.739644\pi$
$257$	−5633.97	−1.36746	−0.683731	+	0.729735i	$0.739644\pi$
$258$	0	0
$259$	1020.79	0.244899
$260$	0	0
$261$	0	0
$262$	−10790.0	−2.54431
$263$	5112.62	1.19870	0.599350	−	0.800487i	$-0.295426\pi$
$263$	5112.62	1.19870	0.599350	+	0.800487i	$0.295426\pi$
$264$	0	0
$265$	3209.35	0.743959
$266$	−48.2855	−0.0111300
$267$	0	0
$268$	4849.32	1.10530
$269$	6562.49	1.48744	0.743721	−	0.668490i	$-0.233059\pi$
$269$	6562.49	1.48744	0.743721	+	0.668490i	$0.233059\pi$
$270$	0	0
$271$	−8161.31	−1.82939	−0.914694	−	0.404146i	$-0.867569\pi$
$271$	−8161.31	−1.82939	−0.914694	+	0.404146i	$0.867569\pi$
$272$	−4288.26	−0.955934
$273$	0	0
$274$	−1785.29	−0.393626
$275$	−4080.58	−0.894794
$276$	0	0
$277$	2726.57	0.591421	0.295710	−	0.955278i	$-0.404444\pi$
$277$	2726.57	0.591421	0.295710	+	0.955278i	$0.404444\pi$
$278$	−12031.5	−2.59569
$279$	0	0
$280$	711.642	0.151888
$281$	−1216.67	−0.258294	−0.129147	−	0.991625i	$-0.541224\pi$
$281$	−1216.67	−0.258294	−0.129147	+	0.991625i	$0.541224\pi$
$282$	0	0
$283$	−3248.32	−0.682306	−0.341153	−	0.940008i	$-0.610817\pi$
$283$	−3248.32	−0.682306	−0.341153	+	0.940008i	$0.610817\pi$
$284$	−16100.0	−3.36393
$285$	0	0
$286$	0	0
$287$	−411.033	−0.0845383
$288$	0	0
$289$	−573.393	−0.116709
$290$	4278.77	0.866408
$291$	0	0
$292$	4460.48	0.893938
$293$	−9222.59	−1.83887	−0.919435	−	0.393241i	$-0.871354\pi$
$293$	−9222.59	−1.83887	−0.919435	+	0.393241i	$0.871354\pi$
$294$	0	0
$295$	−592.412	−0.116920
$296$	12532.6	2.46095
$297$	0	0
$298$	319.446	0.0620974
$299$	0	0
$300$	0	0
$301$	1217.90	0.233218
$302$	1226.91	0.233777
$303$	0	0
$304$	−199.473	−0.0376334
$305$	−4067.58	−0.763635
$306$	0	0
$307$	2993.33	0.556477	0.278238	−	0.960512i	$-0.410249\pi$
$307$	2993.33	0.556477	0.278238	+	0.960512i	$0.410249\pi$
$308$	−2250.49	−0.416343
$309$	0	0
$310$	−6618.82	−1.21266
$311$	7261.04	1.32391	0.661955	−	0.749544i	$-0.269727\pi$
$311$	7261.04	1.32391	0.661955	+	0.749544i	$0.269727\pi$
$312$	0	0
$313$	−3903.02	−0.704829	−0.352415	−	0.935844i	$-0.614639\pi$
$313$	−3903.02	−0.704829	−0.352415	+	0.935844i	$0.614639\pi$
$314$	−8674.49	−1.55901
$315$	0	0
$316$	−15389.9	−2.73972
$317$	−4592.24	−0.813646	−0.406823	−	0.913507i	$-0.633363\pi$
$317$	−4592.24	−0.813646	−0.406823	+	0.913507i	$0.633363\pi$
$318$	0	0
$319$	−6784.81	−1.19083
$320$	−2824.48	−0.493416
$321$	0	0
$322$	2580.52	0.446606
$323$	201.861	0.0347735
$324$	0	0
$325$	0	0
$326$	−13275.7	−2.25543
$327$	0	0
$328$	−5046.39	−0.849512
$329$	−255.349	−0.0427898
$330$	0	0
$331$	−5115.67	−0.849495	−0.424747	−	0.905312i	$-0.639637\pi$
$331$	−5115.67	−0.849495	−0.424747	+	0.905312i	$0.639637\pi$
$332$	6594.05	1.09005
$333$	0	0
$334$	−17124.9	−2.80549
$335$	1696.45	0.276677
$336$	0	0
$337$	−6808.37	−1.10052	−0.550260	−	0.834993i	$-0.685471\pi$
$337$	−6808.37	−1.10052	−0.550260	+	0.834993i	$0.685471\pi$
$338$	0	0
$339$	0	0
$340$	−5933.29	−0.946405
$341$	10495.4	1.66674
$342$	0	0
$343$	2171.24	0.341796
$344$	14952.6	2.34357
$345$	0	0
$346$	19526.1	3.03389
$347$	−10918.2	−1.68910	−0.844551	−	0.535475i	$-0.820133\pi$
$347$	−10918.2	−1.68910	−0.844551	+	0.535475i	$0.820133\pi$
$348$	0	0
$349$	10239.2	1.57047	0.785234	−	0.619199i	$-0.212543\pi$
$349$	10239.2	1.57047	0.785234	+	0.619199i	$0.212543\pi$
$350$	−1473.20	−0.224988
$351$	0	0
$352$	−156.564	−0.0237070
$353$	−864.031	−0.130277	−0.0651384	−	0.997876i	$-0.520749\pi$
$353$	−864.031	−0.130277	−0.0651384	+	0.997876i	$0.520749\pi$
$354$	0	0
$355$	−5632.27	−0.842057
$356$	−3629.50	−0.540346
$357$	0	0
$358$	−11058.0	−1.63249
$359$	−11430.0	−1.68036	−0.840182	−	0.542305i	$-0.817552\pi$
$359$	−11430.0	−1.68036	−0.840182	+	0.542305i	$0.817552\pi$
$360$	0	0
$361$	−6849.61	−0.998631
$362$	4364.76	0.633720
$363$	0	0
$364$	0	0
$365$	1560.42	0.223770
$366$	0	0
$367$	9212.03	1.31026	0.655128	−	0.755518i	$-0.272615\pi$
$367$	9212.03	1.31026	0.655128	+	0.755518i	$0.272615\pi$
$368$	10660.4	1.51009
$369$	0	0
$370$	8743.75	1.22856
$371$	−1837.28	−0.257107
$372$	0	0
$373$	−1118.13	−0.155213	−0.0776065	−	0.996984i	$-0.524728\pi$
$373$	−1118.13	−0.155213	−0.0776065	+	0.996984i	$0.524728\pi$
$374$	14099.2	1.94933
$375$	0	0
$376$	−3135.00	−0.429988
$377$	0	0
$378$	0	0
$379$	8067.99	1.09347	0.546735	−	0.837306i	$-0.315871\pi$
$379$	8067.99	1.09347	0.546735	+	0.837306i	$0.315871\pi$
$380$	−275.993	−0.0372583
$381$	0	0
$382$	−3921.79	−0.525279
$383$	2023.73	0.269995	0.134997	−	0.990846i	$-0.456897\pi$
$383$	2023.73	0.269995	0.134997	+	0.990846i	$0.456897\pi$
$384$	0	0
$385$	−787.293	−0.104219
$386$	−13207.5	−1.74156
$387$	0	0
$388$	−14255.8	−1.86528
$389$	9697.21	1.26393	0.631964	−	0.774998i	$-0.282249\pi$
$389$	9697.21	1.26393	0.631964	+	0.774998i	$0.282249\pi$
$390$	0	0
$391$	−10788.1	−1.39534
$392$	13124.8	1.69108
$393$	0	0
$394$	13414.9	1.71531
$395$	−5383.88	−0.685803
$396$	0	0
$397$	3396.25	0.429353	0.214676	−	0.976685i	$-0.431130\pi$
$397$	3396.25	0.429353	0.214676	+	0.976685i	$0.431130\pi$
$398$	5768.80	0.726543
$399$	0	0
$400$	−6085.94	−0.760742
$401$	6649.91	0.828131	0.414066	−	0.910247i	$-0.364108\pi$
$401$	6649.91	0.828131	0.414066	+	0.910247i	$0.364108\pi$
$402$	0	0
$403$	0	0
$404$	−29850.6	−3.67604
$405$	0	0
$406$	−2449.49	−0.299424
$407$	−13864.9	−1.68859
$408$	0	0
$409$	−2515.44	−0.304109	−0.152055	−	0.988372i	$-0.548589\pi$
$409$	−2515.44	−0.304109	−0.152055	+	0.988372i	$0.548589\pi$
$410$	−3520.77	−0.424093
$411$	0	0
$412$	5271.17	0.630320
$413$	339.141	0.0404069
$414$	0	0
$415$	2306.81	0.272860
$416$	0	0
$417$	0	0
$418$	655.837	0.0767417
$419$	−11907.4	−1.38835	−0.694173	−	0.719808i	$-0.744230\pi$
$419$	−11907.4	−1.38835	−0.694173	+	0.719808i	$0.744230\pi$
$420$	0	0
$421$	2382.88	0.275854	0.137927	−	0.990442i	$-0.455956\pi$
$421$	2382.88	0.275854	0.137927	+	0.990442i	$0.455956\pi$
$422$	14292.4	1.64868
$423$	0	0
$424$	−22556.9	−2.58363
$425$	6158.81	0.702932
$426$	0	0
$427$	2328.59	0.263907
$428$	5275.37	0.595782
$429$	0	0
$430$	10432.1	1.16996
$431$	286.545	0.0320241	0.0160121	−	0.999872i	$-0.494903\pi$
$431$	286.545	0.0320241	0.0160121	+	0.999872i	$0.494903\pi$
$432$	0	0
$433$	−848.354	−0.0941555	−0.0470777	−	0.998891i	$-0.514991\pi$
$433$	−848.354	−0.0941555	−0.0470777	+	0.998891i	$0.514991\pi$
$434$	3789.11	0.419086
$435$	0	0
$436$	3235.97	0.355447
$437$	−501.818	−0.0549318
$438$	0	0
$439$	69.6932	0.00757694	0.00378847	−	0.999993i	$-0.498794\pi$
$439$	69.6932	0.00757694	0.00378847	+	0.999993i	$0.498794\pi$
$440$	−9665.87	−1.04728
$441$	0	0
$442$	0	0
$443$	7331.02	0.786247	0.393123	−	0.919486i	$-0.371395\pi$
$443$	7331.02	0.786247	0.393123	+	0.919486i	$0.371395\pi$
$444$	0	0
$445$	−1269.71	−0.135259
$446$	−7264.18	−0.771231
$447$	0	0
$448$	1616.94	0.170521
$449$	−14213.2	−1.49390	−0.746950	−	0.664880i	$-0.768483\pi$
$449$	−14213.2	−1.49390	−0.746950	+	0.664880i	$0.768483\pi$
$450$	0	0
$451$	5582.84	0.582895
$452$	4652.97	0.484197
$453$	0	0
$454$	21.4945	0.00222200
$455$	0	0
$456$	0	0
$457$	−5903.20	−0.604246	−0.302123	−	0.953269i	$-0.597695\pi$
$457$	−5903.20	−0.604246	−0.302123	+	0.953269i	$0.597695\pi$
$458$	22996.0	2.34614
$459$	0	0
$460$	14749.9	1.49504
$461$	563.806	0.0569610	0.0284805	−	0.999594i	$-0.490933\pi$
$461$	563.806	0.0569610	0.0284805	+	0.999594i	$0.490933\pi$
$462$	0	0
$463$	−17543.8	−1.76097	−0.880487	−	0.474070i	$-0.842784\pi$
$463$	−17543.8	−1.76097	−0.880487	+	0.474070i	$0.842784\pi$
$464$	−10119.1	−1.01243
$465$	0	0
$466$	20225.1	2.01054
$467$	9125.81	0.904266	0.452133	−	0.891951i	$-0.350663\pi$
$467$	9125.81	0.904266	0.452133	+	0.891951i	$0.350663\pi$
$468$	0	0
$469$	−971.174	−0.0956176
$470$	−2187.23	−0.214659
$471$	0	0
$472$	4163.75	0.406043
$473$	−16542.1	−1.60805
$474$	0	0
$475$	286.483	0.0276732
$476$	3396.66	0.327071
$477$	0	0
$478$	16944.0	1.62134
$479$	4970.15	0.474096	0.237048	−	0.971498i	$-0.423820\pi$
$479$	4970.15	0.474096	0.237048	+	0.971498i	$0.423820\pi$
$480$	0	0
$481$	0	0
$482$	20152.3	1.90438
$483$	0	0
$484$	9210.59	0.865006
$485$	−4987.14	−0.466916
$486$	0	0
$487$	−5716.80	−0.531936	−0.265968	−	0.963982i	$-0.585692\pi$
$487$	−5716.80	−0.531936	−0.265968	+	0.963982i	$0.585692\pi$
$488$	28588.8	2.65196
$489$	0	0
$490$	9156.95	0.844222
$491$	−6044.21	−0.555543	−0.277771	−	0.960647i	$-0.589596\pi$
$491$	−6044.21	−0.555543	−0.277771	+	0.960647i	$0.589596\pi$
$492$	0	0
$493$	10240.3	0.935496
$494$	0	0
$495$	0	0
$496$	15653.2	1.41704
$497$	3224.34	0.291009
$498$	0	0
$499$	1101.16	0.0987870	0.0493935	−	0.998779i	$-0.484271\pi$
$499$	1101.16	0.0987870	0.0493935	+	0.998779i	$0.484271\pi$
$500$	−19679.1	−1.76015
$501$	0	0
$502$	24676.1	2.19392
$503$	−1341.97	−0.118957	−0.0594787	−	0.998230i	$-0.518944\pi$
$503$	−1341.97	−0.118957	−0.0594787	+	0.998230i	$0.518944\pi$
$504$	0	0
$505$	−10442.7	−0.920184
$506$	−35049.9	−3.07937
$507$	0	0
$508$	12075.4	1.05464
$509$	−13473.6	−1.17329	−0.586647	−	0.809843i	$-0.699553\pi$
$509$	−13473.6	−1.17329	−0.586647	+	0.809843i	$0.699553\pi$
$510$	0	0
$511$	−893.300	−0.0773332
$512$	20312.2	1.75328
$513$	0	0
$514$	27626.9	2.37076
$515$	1844.02	0.157781
$516$	0	0
$517$	3468.27	0.295038
$518$	−5005.58	−0.424581
$519$	0	0
$520$	0	0
$521$	10374.9	0.872423	0.436211	−	0.899844i	$-0.356320\pi$
$521$	10374.9	0.872423	0.436211	+	0.899844i	$0.356320\pi$
$522$	0	0
$523$	−22295.4	−1.86407	−0.932037	−	0.362364i	$-0.881970\pi$
$523$	−22295.4	−1.86407	−0.932037	+	0.362364i	$0.881970\pi$
$524$	35306.9	2.94349
$525$	0	0
$526$	−25070.4	−2.07818
$527$	−15840.7	−1.30936
$528$	0	0
$529$	14651.7	1.20422
$530$	−15737.5	−1.28980
$531$	0	0
$532$	157.999	0.0128762
$533$	0	0
$534$	0	0
$535$	1845.49	0.149136
$536$	−11923.4	−0.960846
$537$	0	0
$538$	−32180.0	−2.57877
$539$	−14520.1	−1.16034
$540$	0	0
$541$	−10201.9	−0.810747	−0.405373	−	0.914151i	$-0.632859\pi$
$541$	−10201.9	−0.810747	−0.405373	+	0.914151i	$0.632859\pi$
$542$	40020.1	3.17160
$543$	0	0
$544$	236.301	0.0186238
$545$	1132.04	0.0889751
$546$	0	0
$547$	−9312.12	−0.727893	−0.363947	−	0.931420i	$-0.618571\pi$
$547$	−9312.12	−0.727893	−0.363947	+	0.931420i	$0.618571\pi$
$548$	5841.81	0.455383
$549$	0	0
$550$	20009.7	1.55130
$551$	476.337	0.0368288
$552$	0	0
$553$	3082.14	0.237009
$554$	−13370.1	−1.02534
$555$	0	0
$556$	39369.3	3.00293
$557$	−144.324	−0.0109788	−0.00548940	−	0.999985i	$-0.501747\pi$
$557$	−144.324	−0.0109788	−0.00548940	+	0.999985i	$0.501747\pi$
$558$	0	0
$559$	0	0
$560$	−1174.20	−0.0886054
$561$	0	0
$562$	5966.11	0.447802
$563$	−14667.0	−1.09794	−0.548968	−	0.835843i	$-0.684979\pi$
$563$	−14667.0	−1.09794	−0.548968	+	0.835843i	$0.684979\pi$
$564$	0	0
$565$	1627.75	0.121204
$566$	15928.6	1.18291
$567$	0	0
$568$	39586.3	2.92430
$569$	−18754.1	−1.38174	−0.690871	−	0.722978i	$-0.742773\pi$
$569$	−18754.1	−1.38174	−0.690871	+	0.722978i	$0.742773\pi$
$570$	0	0
$571$	−13886.4	−1.01773	−0.508867	−	0.860845i	$-0.669935\pi$
$571$	−13886.4	−1.01773	−0.508867	+	0.860845i	$0.669935\pi$
$572$	0	0
$573$	0	0
$574$	2015.55	0.146564
$575$	−15310.5	−1.11042
$576$	0	0
$577$	−9338.81	−0.673795	−0.336898	−	0.941541i	$-0.609378\pi$
$577$	−9338.81	−0.673795	−0.336898	+	0.941541i	$0.609378\pi$
$578$	2811.71	0.202338
$579$	0	0
$580$	−14000.9	−1.00234
$581$	−1320.59	−0.0942983
$582$	0	0
$583$	24954.8	1.77276
$584$	−10967.3	−0.777110
$585$	0	0
$586$	45224.1	3.18804
$587$	3724.13	0.261859	0.130930	−	0.991392i	$-0.458204\pi$
$587$	3724.13	0.261859	0.130930	+	0.991392i	$0.458204\pi$
$588$	0	0
$589$	−736.845	−0.0515470
$590$	2904.97	0.202704
$591$	0	0
$592$	−20678.6	−1.43562
$593$	19910.2	1.37877	0.689387	−	0.724393i	$-0.257880\pi$
$593$	19910.2	1.37877	0.689387	+	0.724393i	$0.257880\pi$
$594$	0	0
$595$	1188.26	0.0818721
$596$	−1045.29	−0.0718400
$597$	0	0
$598$	0	0
$599$	14756.0	1.00653	0.503265	−	0.864132i	$-0.332132\pi$
$599$	14756.0	1.00653	0.503265	+	0.864132i	$0.332132\pi$
$600$	0	0
$601$	−7939.05	−0.538837	−0.269418	−	0.963023i	$-0.586831\pi$
$601$	−7939.05	−0.538837	−0.269418	+	0.963023i	$0.586831\pi$
$602$	−5972.14	−0.404329
$603$	0	0
$604$	−4014.67	−0.270455
$605$	3222.15	0.216528
$606$	0	0
$607$	−2563.88	−0.171441	−0.0857205	−	0.996319i	$-0.527319\pi$
$607$	−2563.88	−0.171441	−0.0857205	+	0.996319i	$0.527319\pi$
$608$	10.9918	0.000733184 0
$609$	0	0
$610$	19945.9	1.32391
$611$	0	0
$612$	0	0
$613$	17593.4	1.15920	0.579602	−	0.814900i	$-0.303208\pi$
$613$	17593.4	1.15920	0.579602	+	0.814900i	$0.303208\pi$
$614$	−14678.2	−0.964761
$615$	0	0
$616$	5533.47	0.361932
$617$	−24479.7	−1.59727	−0.798635	−	0.601816i	$-0.794444\pi$
$617$	−24479.7	−1.59727	−0.798635	+	0.601816i	$0.794444\pi$
$618$	0	0
$619$	2927.43	0.190086	0.0950431	−	0.995473i	$-0.469701\pi$
$619$	2927.43	0.190086	0.0950431	+	0.995473i	$0.469701\pi$
$620$	21658.0	1.40291
$621$	0	0
$622$	−35605.5	−2.29526
$623$	726.879	0.0467445
$624$	0	0
$625$	4802.06	0.307332
$626$	19138.9	1.22196
$627$	0	0
$628$	28384.5	1.80361
$629$	20926.2	1.32652
$630$	0	0
$631$	10183.0	0.642439	0.321220	−	0.947005i	$-0.395907\pi$
$631$	10183.0	0.642439	0.321220	+	0.947005i	$0.395907\pi$
$632$	37840.4	2.38166
$633$	0	0
$634$	22518.6	1.41061
$635$	4224.35	0.263997
$636$	0	0
$637$	0	0
$638$	33270.2	2.06454
$639$	0	0
$640$	14011.3	0.865382
$641$	−17103.4	−1.05389	−0.526946	−	0.849899i	$-0.676663\pi$
$641$	−17103.4	−1.05389	−0.526946	+	0.849899i	$0.676663\pi$
$642$	0	0
$643$	21733.9	1.33297	0.666485	−	0.745518i	$-0.267798\pi$
$643$	21733.9	1.33297	0.666485	+	0.745518i	$0.267798\pi$
$644$	−8443.96	−0.516675
$645$	0	0
$646$	−989.853	−0.0602867
$647$	−22302.3	−1.35517	−0.677584	−	0.735446i	$-0.736973\pi$
$647$	−22302.3	−1.35517	−0.677584	+	0.735446i	$0.736973\pi$
$648$	0	0
$649$	−4606.38	−0.278607
$650$	0	0
$651$	0	0
$652$	43440.5	2.60930
$653$	6977.15	0.418127	0.209063	−	0.977902i	$-0.432958\pi$
$653$	6977.15	0.418127	0.209063	+	0.977902i	$0.432958\pi$
$654$	0	0
$655$	12351.5	0.736812
$656$	8326.47	0.495570
$657$	0	0
$658$	1252.14	0.0741845
$659$	10026.2	0.592662	0.296331	−	0.955085i	$-0.404237\pi$
$659$	10026.2	0.592662	0.296331	+	0.955085i	$0.404237\pi$
$660$	0	0
$661$	4710.64	0.277190	0.138595	−	0.990349i	$-0.455741\pi$
$661$	4710.64	0.277190	0.138595	+	0.990349i	$0.455741\pi$
$662$	25085.4	1.47277
$663$	0	0
$664$	−16213.3	−0.947589
$665$	55.2731	0.00322316
$666$	0	0
$667$	−25456.9	−1.47780
$668$	56035.9	3.24565
$669$	0	0
$670$	−8318.75	−0.479674
$671$	−31628.0	−1.81965
$672$	0	0
$673$	−14727.2	−0.843523	−0.421762	−	0.906707i	$-0.638588\pi$
$673$	−14727.2	−0.843523	−0.421762	+	0.906707i	$0.638588\pi$
$674$	33385.7	1.90797
$675$	0	0
$676$	0	0
$677$	2118.31	0.120256	0.0601279	−	0.998191i	$-0.480849\pi$
$677$	2118.31	0.120256	0.0601279	+	0.998191i	$0.480849\pi$
$678$	0	0
$679$	2855.02	0.161363
$680$	14588.7	0.822720
$681$	0	0
$682$	−51465.6	−2.88962
$683$	−7070.72	−0.396125	−0.198063	−	0.980189i	$-0.563465\pi$
$683$	−7070.72	−0.396125	−0.198063	+	0.980189i	$0.563465\pi$
$684$	0	0
$685$	2043.65	0.113991
$686$	−10647.0	−0.592571
$687$	0	0
$688$	−24671.6	−1.36714
$689$	0	0
$690$	0	0
$691$	−15584.0	−0.857947	−0.428974	−	0.903317i	$-0.641125\pi$
$691$	−15584.0	−0.857947	−0.428974	+	0.903317i	$0.641125\pi$
$692$	−63892.9	−3.50989
$693$	0	0
$694$	53538.7	2.92839
$695$	13772.6	0.751691
$696$	0	0
$697$	−8426.17	−0.457911
$698$	−50209.4	−2.72271
$699$	0	0
$700$	4820.57	0.260287
$701$	13190.6	0.710703	0.355352	−	0.934733i	$-0.384361\pi$
$701$	13190.6	0.710703	0.355352	+	0.934733i	$0.384361\pi$
$702$	0	0
$703$	973.404	0.0522228
$704$	−21962.1	−1.17575
$705$	0	0
$706$	4236.89	0.225860
$707$	5978.17	0.318009
$708$	0	0
$709$	−31565.9	−1.67205	−0.836025	−	0.548692i	$-0.815126\pi$
$709$	−31565.9	−1.67205	−0.836025	+	0.548692i	$0.815126\pi$
$710$	27618.6	1.45987
$711$	0	0
$712$	8924.14	0.469728
$713$	39379.2	2.06839
$714$	0	0
$715$	0	0
$716$	36183.7	1.88861
$717$	0	0
$718$	56048.3	2.91324
$719$	15535.3	0.805800	0.402900	−	0.915244i	$-0.368002\pi$
$719$	15535.3	0.805800	0.402900	+	0.915244i	$0.368002\pi$
$720$	0	0
$721$	−1055.66	−0.0545281
$722$	33588.0	1.73132
$723$	0	0
$724$	−14282.3	−0.733146
$725$	14533.1	0.744477
$726$	0	0
$727$	−27639.6	−1.41004	−0.705018	−	0.709189i	$-0.749061\pi$
$727$	−27639.6	−1.41004	−0.705018	+	0.709189i	$0.749061\pi$
$728$	0	0
$729$	0	0
$730$	−7651.71	−0.387949
$731$	24967.0	1.26325
$732$	0	0
$733$	1318.97	0.0664629	0.0332314	−	0.999448i	$-0.489420\pi$
$733$	1318.97	0.0664629	0.0332314	+	0.999448i	$0.489420\pi$
$734$	−45172.4	−2.27158
$735$	0	0
$736$	−587.434	−0.0294200
$737$	13190.9	0.659287
$738$	0	0
$739$	2970.75	0.147876	0.0739382	−	0.997263i	$-0.476443\pi$
$739$	2970.75	0.147876	0.0739382	+	0.997263i	$0.476443\pi$
$740$	−28611.2	−1.42131
$741$	0	0
$742$	9009.32	0.445745
$743$	28792.6	1.42167	0.710833	−	0.703361i	$-0.248318\pi$
$743$	28792.6	1.42167	0.710833	+	0.703361i	$0.248318\pi$
$744$	0	0
$745$	−365.675	−0.0179829
$746$	5482.88	0.269092
$747$	0	0
$748$	−46135.1	−2.25517
$749$	−1056.50	−0.0515402
$750$	0	0
$751$	−7839.45	−0.380913	−0.190456	−	0.981696i	$-0.560997\pi$
$751$	−7839.45	−0.380913	−0.190456	+	0.981696i	$0.560997\pi$
$752$	5172.71	0.250837
$753$	0	0
$754$	0	0
$755$	−1404.46	−0.0677000
$756$	0	0
$757$	−3192.42	−0.153277	−0.0766383	−	0.997059i	$-0.524419\pi$
$757$	−3192.42	−0.153277	−0.0766383	+	0.997059i	$0.524419\pi$
$758$	−39562.5	−1.89574
$759$	0	0
$760$	678.606	0.0323890
$761$	25421.9	1.21096	0.605481	−	0.795860i	$-0.292981\pi$
$761$	25421.9	1.21096	0.605481	+	0.795860i	$0.292981\pi$
$762$	0	0
$763$	−648.067	−0.0307492
$764$	12832.8	0.607691
$765$	0	0
$766$	−9923.64	−0.468089
$767$	0	0
$768$	0	0
$769$	33930.6	1.59112	0.795559	−	0.605876i	$-0.207177\pi$
$769$	33930.6	1.59112	0.795559	+	0.605876i	$0.207177\pi$
$770$	3860.60	0.180683
$771$	0	0
$772$	43217.2	2.01480
$773$	40596.9	1.88897	0.944483	−	0.328560i	$-0.106563\pi$
$773$	40596.9	1.88897	0.944483	+	0.328560i	$0.106563\pi$
$774$	0	0
$775$	−22481.2	−1.04200
$776$	35052.0	1.62151
$777$	0	0
$778$	−47551.5	−2.19127
$779$	−391.952	−0.0180271
$780$	0	0
$781$	−43794.5	−2.00652
$782$	52900.7	2.41909
$783$	0	0
$784$	−21655.8	−0.986507
$785$	9929.81	0.451478
$786$	0	0
$787$	−39046.2	−1.76855	−0.884274	−	0.466969i	$-0.845346\pi$
$787$	−39046.2	−1.76855	−0.884274	+	0.466969i	$0.845346\pi$
$788$	−43896.0	−1.98443
$789$	0	0
$790$	26400.5	1.18897
$791$	−931.850	−0.0418872
$792$	0	0
$793$	0	0
$794$	−16654.0	−0.744367
$795$	0	0
$796$	−18876.6	−0.840532
$797$	−8074.79	−0.358876	−0.179438	−	0.983769i	$-0.557428\pi$
$797$	−8074.79	−0.358876	−0.179438	+	0.983769i	$0.557428\pi$
$798$	0	0
$799$	−5234.65	−0.231776
$800$	335.361	0.0148210
$801$	0	0
$802$	−32608.7	−1.43573
$803$	12133.2	0.533216
$804$	0	0
$805$	−2953.96	−0.129334
$806$	0	0
$807$	0	0
$808$	73396.1	3.19562
$809$	−30128.0	−1.30933	−0.654663	−	0.755921i	$-0.727189\pi$
$809$	−30128.0	−1.30933	−0.654663	+	0.755921i	$0.727189\pi$
$810$	0	0
$811$	707.812	0.0306469	0.0153235	−	0.999883i	$-0.495122\pi$
$811$	707.812	0.0306469	0.0153235	+	0.999883i	$0.495122\pi$
$812$	8015.20	0.346402
$813$	0	0
$814$	67988.2	2.92750
$815$	15196.9	0.653156
$816$	0	0
$817$	1161.36	0.0497319
$818$	12334.8	0.527233
$819$	0	0
$820$	11520.6	0.490630
$821$	−15620.3	−0.664009	−0.332004	−	0.943278i	$-0.607725\pi$
$821$	−15620.3	−0.664009	−0.332004	+	0.943278i	$0.607725\pi$
$822$	0	0
$823$	32710.1	1.38542	0.692710	−	0.721216i	$-0.256417\pi$
$823$	32710.1	1.38542	0.692710	+	0.721216i	$0.256417\pi$
$824$	−12960.7	−0.547944
$825$	0	0
$826$	−1663.02	−0.0700532
$827$	28246.1	1.18768	0.593842	−	0.804582i	$-0.297611\pi$
$827$	28246.1	1.18768	0.593842	+	0.804582i	$0.297611\pi$
$828$	0	0
$829$	−14099.2	−0.590694	−0.295347	−	0.955390i	$-0.595435\pi$
$829$	−14099.2	−0.590694	−0.295347	+	0.955390i	$0.595435\pi$
$830$	−11311.7	−0.473055
$831$	0	0
$832$	0	0
$833$	21915.1	0.911541
$834$	0	0
$835$	19603.1	0.812449
$836$	−2146.02	−0.0887819
$837$	0	0
$838$	58389.7	2.40697
$839$	−11485.3	−0.472605	−0.236302	−	0.971680i	$-0.575936\pi$
$839$	−11485.3	−0.472605	−0.236302	+	0.971680i	$0.575936\pi$
$840$	0	0
$841$	−224.714	−0.00921373
$842$	−11684.8	−0.478247
$843$	0	0
$844$	−46767.3	−1.90734
$845$	0	0
$846$	0	0
$847$	−1844.60	−0.0748304
$848$	37218.5	1.50718
$849$	0	0
$850$	−30200.5	−1.21867
$851$	−52021.7	−2.09551
$852$	0	0
$853$	−11836.8	−0.475128	−0.237564	−	0.971372i	$-0.576349\pi$
$853$	−11836.8	−0.475128	−0.237564	+	0.971372i	$0.576349\pi$
$854$	−11418.5	−0.457534
$855$	0	0
$856$	−12971.0	−0.517920
$857$	−31156.2	−1.24186	−0.620931	−	0.783865i	$-0.713245\pi$
$857$	−31156.2	−1.24186	−0.620931	+	0.783865i	$0.713245\pi$
$858$	0	0
$859$	−1584.88	−0.0629515	−0.0314758	−	0.999505i	$-0.510021\pi$
$859$	−1584.88	−0.0629515	−0.0314758	+	0.999505i	$0.510021\pi$
$860$	−34135.9	−1.35352
$861$	0	0
$862$	−1405.11	−0.0555200
$863$	−19884.8	−0.784340	−0.392170	−	0.919893i	$-0.628276\pi$
$863$	−19884.8	−0.784340	−0.392170	+	0.919893i	$0.628276\pi$
$864$	0	0
$865$	−22351.8	−0.878592
$866$	4160.02	0.163237
$867$	0	0
$868$	−12398.7	−0.484837
$869$	−41863.1	−1.63419
$870$	0	0
$871$	0	0
$872$	−7956.54	−0.308994
$873$	0	0
$874$	2460.73	0.0952351
$875$	3941.13	0.152268
$876$	0	0
$877$	5483.43	0.211132	0.105566	−	0.994412i	$-0.466335\pi$
$877$	5483.43	0.211132	0.105566	+	0.994412i	$0.466335\pi$
$878$	−341.750	−0.0131361
$879$	0	0
$880$	15948.5	0.610938
$881$	−6581.12	−0.251672	−0.125836	−	0.992051i	$-0.540161\pi$
$881$	−6581.12	−0.251672	−0.125836	+	0.992051i	$0.540161\pi$
$882$	0	0
$883$	6942.60	0.264595	0.132297	−	0.991210i	$-0.457765\pi$
$883$	6942.60	0.264595	0.132297	+	0.991210i	$0.457765\pi$
$884$	0	0
$885$	0	0
$886$	−35948.6	−1.36311
$887$	6562.68	0.248425	0.124213	−	0.992256i	$-0.460360\pi$
$887$	6562.68	0.248425	0.124213	+	0.992256i	$0.460360\pi$
$888$	0	0
$889$	−2418.34	−0.0912355
$890$	6226.20	0.234498
$891$	0	0
$892$	23769.7	0.892231
$893$	−243.495	−0.00912458
$894$	0	0
$895$	12658.2	0.472756
$896$	−8021.11	−0.299070
$897$	0	0
$898$	69696.2	2.58997
$899$	−37379.7	−1.38674
$900$	0	0
$901$	−37664.1	−1.39265
$902$	−27376.2	−1.01056
$903$	0	0
$904$	−11440.6	−0.420918
$905$	−4996.40	−0.183521
$906$	0	0
$907$	−1652.33	−0.0604902	−0.0302451	−	0.999543i	$-0.509629\pi$
$907$	−1652.33	−0.0604902	−0.0302451	+	0.999543i	$0.509629\pi$
$908$	−70.3341	−0.00257061
$909$	0	0
$910$	0	0
$911$	−506.240	−0.0184110	−0.00920552	−	0.999958i	$-0.502930\pi$
$911$	−506.240	−0.0184110	−0.00920552	+	0.999958i	$0.502930\pi$
$912$	0	0
$913$	17936.9	0.650191
$914$	28947.1	1.04758
$915$	0	0
$916$	−75247.1	−2.71423
$917$	−7070.91	−0.254637
$918$	0	0
$919$	−2469.29	−0.0886336	−0.0443168	−	0.999018i	$-0.514111\pi$
$919$	−2469.29	−0.0886336	−0.0443168	+	0.999018i	$0.514111\pi$
$920$	−36266.8	−1.29965
$921$	0	0
$922$	−2764.69	−0.0987531
$923$	0	0
$924$	0	0
$925$	29698.7	1.05566
$926$	86028.5	3.05299
$927$	0	0
$928$	557.606	0.0197245
$929$	41645.7	1.47078	0.735388	−	0.677646i	$-0.237000\pi$
$929$	41645.7	1.47078	0.735388	+	0.677646i	$0.237000\pi$
$930$	0	0
$931$	1019.40	0.0358857
$932$	−66180.3	−2.32597
$933$	0	0
$934$	−44749.6	−1.56772
$935$	−16139.5	−0.564512
$936$	0	0
$937$	−33891.2	−1.18162	−0.590810	−	0.806811i	$-0.701192\pi$
$937$	−33891.2	−1.18162	−0.590810	+	0.806811i	$0.701192\pi$
$938$	4762.28	0.165772
$939$	0	0
$940$	7157.03	0.248337
$941$	−28351.3	−0.982173	−0.491087	−	0.871111i	$-0.663400\pi$
$941$	−28351.3	−0.982173	−0.491087	+	0.871111i	$0.663400\pi$
$942$	0	0
$943$	20947.1	0.723363
$944$	−6870.13	−0.236868
$945$	0	0
$946$	81116.5	2.78787
$947$	−3732.71	−0.128085	−0.0640427	−	0.997947i	$-0.520399\pi$
$947$	−3732.71	−0.128085	−0.0640427	+	0.997947i	$0.520399\pi$
$948$	0	0
$949$	0	0
$950$	−1404.81	−0.0479768
$951$	0	0
$952$	−8351.65	−0.284326
$953$	9652.09	0.328082	0.164041	−	0.986454i	$-0.447547\pi$
$953$	9652.09	0.328082	0.164041	+	0.986454i	$0.447547\pi$
$954$	0	0
$955$	4489.33	0.152117
$956$	−55444.1	−1.87572
$957$	0	0
$958$	−24371.8	−0.821938
$959$	−1169.94	−0.0393945
$960$	0	0
$961$	28031.5	0.940939
$962$	0	0
$963$	0	0
$964$	−65942.1	−2.20316
$965$	15118.8	0.504342
$966$	0	0
$967$	36196.6	1.20373	0.601864	−	0.798598i	$-0.294425\pi$
$967$	36196.6	1.20373	0.601864	+	0.798598i	$0.294425\pi$
$968$	−22646.8	−0.751959
$969$	0	0
$970$	24455.1	0.809490
$971$	−5250.98	−0.173545	−0.0867723	−	0.996228i	$-0.527655\pi$
$971$	−5250.98	−0.173545	−0.0867723	+	0.996228i	$0.527655\pi$
$972$	0	0
$973$	−7884.48	−0.259779
$974$	28033.1	0.922215
$975$	0	0
$976$	−47171.2	−1.54704
$977$	23174.8	0.758881	0.379440	−	0.925216i	$-0.376117\pi$
$977$	23174.8	0.758881	0.379440	+	0.925216i	$0.376117\pi$
$978$	0	0
$979$	−9872.82	−0.322305
$980$	−29963.2	−0.976674
$981$	0	0
$982$	29638.6	0.963142
$983$	33285.5	1.08000	0.540001	−	0.841664i	$-0.318424\pi$
$983$	33285.5	1.08000	0.540001	+	0.841664i	$0.318424\pi$
$984$	0	0
$985$	−15356.2	−0.496740
$986$	−50214.6	−1.62186
$987$	0	0
$988$	0	0
$989$	−62066.8	−1.99556
$990$	0	0
$991$	−15535.4	−0.497979	−0.248990	−	0.968506i	$-0.580098\pi$
$991$	−15535.4	−0.497979	−0.248990	+	0.968506i	$0.580098\pi$
$992$	−862.559	−0.0276071
$993$	0	0
$994$	−15811.0	−0.504520
$995$	−6603.63	−0.210401
$996$	0	0
$997$	26302.4	0.835511	0.417756	−	0.908559i	$-0.362817\pi$
$997$	26302.4	0.835511	0.417756	+	0.908559i	$0.362817\pi$
$998$	−5399.69	−0.171267
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bl.1.1		12
3.2	odd	2	inner	1521.4.a.bl.1.11		12
13.6	odd	12		117.4.q.f.10.6	yes	12
13.11	odd	12		117.4.q.f.82.6	yes	12
13.12	even	2	inner	1521.4.a.bl.1.12		12
39.11	even	12		117.4.q.f.82.1	yes	12
39.32	even	12		117.4.q.f.10.1	✓	12
39.38	odd	2	inner	1521.4.a.bl.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.q.f.10.1	✓	12	39.32	even	12
117.4.q.f.10.6	yes	12	13.6	odd	12
117.4.q.f.82.1	yes	12	39.11	even	12
117.4.q.f.82.6	yes	12	13.11	odd	12
1521.4.a.bl.1.1		12	1.1	even	1	trivial
1521.4.a.bl.1.2		12	39.38	odd	2	inner
1521.4.a.bl.1.11		12	3.2	odd	2	inner
1521.4.a.bl.1.12		12	13.12	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-4.90363 q^{2} +16.0456 q^{4} +5.61325 q^{5} -3.21345 q^{7} -39.4526 q^{8} +O(q^{10})\)	\(q-4.90363 q^{2} +16.0456 q^{4} +5.61325 q^{5} -3.21345 q^{7} -39.4526 q^{8} -27.5253 q^{10} +43.6466 q^{11} +15.7576 q^{14} +65.0962 q^{16} -65.8757 q^{17} -3.06427 q^{19} +90.0680 q^{20} -214.027 q^{22} +163.764 q^{23} -93.4914 q^{25} -51.5617 q^{28} -155.449 q^{29} +240.463 q^{31} -3.58708 q^{32} +323.030 q^{34} -18.0379 q^{35} -317.662 q^{37} +15.0261 q^{38} -221.457 q^{40} +127.910 q^{41} -379.001 q^{43} +700.336 q^{44} -803.039 q^{46} +79.4626 q^{47} -332.674 q^{49} +458.447 q^{50} +571.746 q^{53} +244.999 q^{55} +126.779 q^{56} +762.263 q^{58} -105.538 q^{59} -724.638 q^{61} -1179.14 q^{62} -503.180 q^{64} +302.222 q^{67} -1057.01 q^{68} +88.4512 q^{70} -1003.39 q^{71} +277.988 q^{73} +1557.70 q^{74} -49.1681 q^{76} -140.256 q^{77} -959.137 q^{79} +365.402 q^{80} -627.224 q^{82} +410.957 q^{83} -369.777 q^{85} +1858.48 q^{86} -1721.97 q^{88} -226.199 q^{89} +2627.69 q^{92} -389.655 q^{94} -17.2005 q^{95} -888.458 q^{97} +1631.31 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 32 q^{4}+O(q^{10}) \) 12 * q + 32 * q^4	\( 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) \) 12 * q + 32 * q^4 - 112 * q^10 + 184 * q^16 - 584 * q^22 + 92 * q^25 - 448 * q^40 - 1620 * q^43 - 2136 * q^49 - 920 * q^55 - 2588 * q^61 + 184 * q^64 - 6380 * q^79 - 2536 * q^82 - 10280 * q^88 - 3320 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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