LMFDB - Embedded newform 1088.4.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.550476.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 15x^{2} + 19x + 20 $ x^4 - x^3 - 15x^2 + 19x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.698199$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.12911 q^{3} -20.1066 q^{5} +21.0725 q^{7} -0.692234 q^{9} +O(q^{10})$	$q-5.12911 q^{3} -20.1066 q^{5} +21.0725 q^{7} -0.692234 q^{9} +17.0861 q^{11} -78.4238 q^{13} +103.129 q^{15} -17.0000 q^{17} +139.741 q^{19} -108.083 q^{21} -138.464 q^{23} +279.277 q^{25} +142.037 q^{27} +203.057 q^{29} -42.6601 q^{31} -87.6363 q^{33} -423.698 q^{35} +39.9480 q^{37} +402.244 q^{39} +313.527 q^{41} -229.478 q^{43} +13.9185 q^{45} +322.172 q^{47} +101.052 q^{49} +87.1949 q^{51} +213.671 q^{53} -343.543 q^{55} -716.747 q^{57} +488.327 q^{59} -382.584 q^{61} -14.5871 q^{63} +1576.84 q^{65} -426.513 q^{67} +710.196 q^{69} +483.945 q^{71} -637.051 q^{73} -1432.44 q^{75} +360.047 q^{77} +120.822 q^{79} -709.830 q^{81} +465.209 q^{83} +341.813 q^{85} -1041.50 q^{87} +1346.34 q^{89} -1652.59 q^{91} +218.808 q^{93} -2809.72 q^{95} -1117.14 q^{97} -11.8276 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 8 q^{5} + 22 q^{7} + 100 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 8 * q^5 + 22 * q^7 + 100 * q^9	$ 4 q - 2 q^{3} - 8 q^{5} + 22 q^{7} + 100 q^{9} + 70 q^{11} - 120 q^{13} - 140 q^{15} - 68 q^{17} - 44 q^{19} - 488 q^{21} - 158 q^{23} + 548 q^{25} - 392 q^{27} - 264 q^{29} - 122 q^{31} + 136 q^{33} - 44 q^{35} - 256 q^{37} + 528 q^{39} + 240 q^{41} - 1100 q^{43} + 880 q^{45} + 800 q^{47} + 12 q^{49} + 34 q^{51} - 432 q^{53} + 532 q^{55} + 472 q^{57} - 148 q^{59} + 728 q^{61} + 1450 q^{63} - 72 q^{65} - 1032 q^{67} + 1024 q^{69} - 798 q^{71} - 1544 q^{73} - 2974 q^{75} - 656 q^{77} - 758 q^{79} - 20 q^{81} + 244 q^{83} + 136 q^{85} - 1524 q^{87} + 1440 q^{89} - 1104 q^{91} + 2464 q^{93} - 7016 q^{95} - 1344 q^{97} - 86 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 8 * q^5 + 22 * q^7 + 100 * q^9 + 70 * q^11 - 120 * q^13 - 140 * q^15 - 68 * q^17 - 44 * q^19 - 488 * q^21 - 158 * q^23 + 548 * q^25 - 392 * q^27 - 264 * q^29 - 122 * q^31 + 136 * q^33 - 44 * q^35 - 256 * q^37 + 528 * q^39 + 240 * q^41 - 1100 * q^43 + 880 * q^45 + 800 * q^47 + 12 * q^49 + 34 * q^51 - 432 * q^53 + 532 * q^55 + 472 * q^57 - 148 * q^59 + 728 * q^61 + 1450 * q^63 - 72 * q^65 - 1032 * q^67 + 1024 * q^69 - 798 * q^71 - 1544 * q^73 - 2974 * q^75 - 656 * q^77 - 758 * q^79 - 20 * q^81 + 244 * q^83 + 136 * q^85 - 1524 * q^87 + 1440 * q^89 - 1104 * q^91 + 2464 * q^93 - 7016 * q^95 - 1344 * q^97 - 86 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−5.12911	−0.987098	−0.493549	−	0.869718i	$-0.664301\pi$
$3$	−5.12911	−0.987098	−0.493549	+	0.869718i	$0.664301\pi$
$4$	0	0
$5$	−20.1066	−1.79839	−0.899196	−	0.437545i	$-0.855848\pi$
$5$	−20.1066	−1.79839	−0.899196	+	0.437545i	$0.855848\pi$
$6$	0	0
$7$	21.0725	1.13781	0.568905	−	0.822403i	$-0.307367\pi$
$7$	21.0725	1.13781	0.568905	+	0.822403i	$0.307367\pi$
$8$	0	0
$9$	−0.692234	−0.0256383
$10$	0	0
$11$	17.0861	0.468331	0.234165	−	0.972197i	$-0.424764\pi$
$11$	17.0861	0.468331	0.234165	+	0.972197i	$0.424764\pi$
$12$	0	0
$13$	−78.4238	−1.67314	−0.836572	−	0.547858i	$-0.815444\pi$
$13$	−78.4238	−1.67314	−0.836572	+	0.547858i	$0.815444\pi$
$14$	0	0
$15$	103.129	1.77519
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	139.741	1.68730	0.843652	−	0.536890i	$-0.180401\pi$
$19$	139.741	1.68730	0.843652	+	0.536890i	$0.180401\pi$
$20$	0	0
$21$	−108.083	−1.12313
$22$	0	0
$23$	−138.464	−1.25529	−0.627645	−	0.778500i	$-0.715981\pi$
$23$	−138.464	−1.25529	−0.627645	+	0.778500i	$0.715981\pi$
$24$	0	0
$25$	279.277	2.23422
$26$	0	0
$27$	142.037	1.01241
$28$	0	0
$29$	203.057	1.30024	0.650118	−	0.759834i	$-0.274720\pi$
$29$	203.057	1.30024	0.650118	+	0.759834i	$0.274720\pi$
$30$	0	0
$31$	−42.6601	−0.247161	−0.123580	−	0.992335i	$-0.539438\pi$
$31$	−42.6601	−0.247161	−0.123580	+	0.992335i	$0.539438\pi$
$32$	0	0
$33$	−87.6363	−0.462288
$34$	0	0
$35$	−423.698	−2.04623
$36$	0	0
$37$	39.9480	0.177498	0.0887488	−	0.996054i	$-0.471713\pi$
$37$	39.9480	0.177498	0.0887488	+	0.996054i	$0.471713\pi$
$38$	0	0
$39$	402.244	1.65156
$40$	0	0
$41$	313.527	1.19426	0.597130	−	0.802145i	$-0.296308\pi$
$41$	313.527	1.19426	0.597130	+	0.802145i	$0.296308\pi$
$42$	0	0
$43$	−229.478	−0.813839	−0.406920	−	0.913464i	$-0.633397\pi$
$43$	−229.478	−0.813839	−0.406920	+	0.913464i	$0.633397\pi$
$44$	0	0
$45$	13.9185	0.0461078
$46$	0	0
$47$	322.172	0.999864	0.499932	−	0.866065i	$-0.333358\pi$
$47$	322.172	0.999864	0.499932	+	0.866065i	$0.333358\pi$
$48$	0	0
$49$	101.052	0.294612
$50$	0	0
$51$	87.1949	0.239406
$52$	0	0
$53$	213.671	0.553774	0.276887	−	0.960903i	$-0.410697\pi$
$53$	213.671	0.553774	0.276887	+	0.960903i	$0.410697\pi$
$54$	0	0
$55$	−343.543	−0.842243
$56$	0	0
$57$	−716.747	−1.66553
$58$	0	0
$59$	488.327	1.07754	0.538769	−	0.842454i	$-0.318890\pi$
$59$	488.327	1.07754	0.538769	+	0.842454i	$0.318890\pi$
$60$	0	0
$61$	−382.584	−0.803031	−0.401516	−	0.915852i	$-0.631516\pi$
$61$	−382.584	−0.803031	−0.401516	+	0.915852i	$0.631516\pi$
$62$	0	0
$63$	−14.5871	−0.0291715
$64$	0	0
$65$	1576.84	3.00897
$66$	0	0
$67$	−426.513	−0.777714	−0.388857	−	0.921298i	$-0.627130\pi$
$67$	−426.513	−0.777714	−0.388857	+	0.921298i	$0.627130\pi$
$68$	0	0
$69$	710.196	1.23909
$70$	0	0
$71$	483.945	0.808925	0.404462	−	0.914555i	$-0.367459\pi$
$71$	483.945	0.808925	0.404462	+	0.914555i	$0.367459\pi$
$72$	0	0
$73$	−637.051	−1.02139	−0.510693	−	0.859763i	$-0.670611\pi$
$73$	−637.051	−1.02139	−0.510693	+	0.859763i	$0.670611\pi$
$74$	0	0
$75$	−1432.44	−2.20539
$76$	0	0
$77$	360.047	0.532872
$78$	0	0
$79$	120.822	0.172071	0.0860354	−	0.996292i	$-0.472580\pi$
$79$	120.822	0.172071	0.0860354	+	0.996292i	$0.472580\pi$
$80$	0	0
$81$	−709.830	−0.973704
$82$	0	0
$83$	465.209	0.615221	0.307611	−	0.951512i	$-0.400471\pi$
$83$	465.209	0.615221	0.307611	+	0.951512i	$0.400471\pi$
$84$	0	0
$85$	341.813	0.436174
$86$	0	0
$87$	−1041.50	−1.28346
$88$	0	0
$89$	1346.34	1.60350	0.801751	−	0.597658i	$-0.203902\pi$
$89$	1346.34	1.60350	0.801751	+	0.597658i	$0.203902\pi$
$90$	0	0
$91$	−1652.59	−1.90372
$92$	0	0
$93$	218.808	0.243972
$94$	0	0
$95$	−2809.72	−3.03444
$96$	0	0
$97$	−1117.14	−1.16936	−0.584680	−	0.811264i	$-0.698780\pi$
$97$	−1117.14	−1.16936	−0.584680	+	0.811264i	$0.698780\pi$
$98$	0	0
$99$	−11.8276	−0.0120072
$100$	0	0
$101$	−24.1892	−0.0238308	−0.0119154	−	0.999929i	$-0.503793\pi$
$101$	−24.1892	−0.0238308	−0.0119154	+	0.999929i	$0.503793\pi$
$102$	0	0
$103$	−797.676	−0.763081	−0.381541	−	0.924352i	$-0.624606\pi$
$103$	−797.676	−0.763081	−0.381541	+	0.924352i	$0.624606\pi$
$104$	0	0
$105$	2173.19	2.01983
$106$	0	0
$107$	−1202.56	−1.08651	−0.543254	−	0.839568i	$-0.682808\pi$
$107$	−1202.56	−1.08651	−0.543254	+	0.839568i	$0.682808\pi$
$108$	0	0
$109$	−214.814	−0.188766	−0.0943829	−	0.995536i	$-0.530088\pi$
$109$	−214.814	−0.188766	−0.0943829	+	0.995536i	$0.530088\pi$
$110$	0	0
$111$	−204.898	−0.175207
$112$	0	0
$113$	−1535.38	−1.27820	−0.639100	−	0.769124i	$-0.720693\pi$
$113$	−1535.38	−1.27820	−0.639100	+	0.769124i	$0.720693\pi$
$114$	0	0
$115$	2784.04	2.25751
$116$	0	0
$117$	54.2877	0.0428966
$118$	0	0
$119$	−358.233	−0.275959
$120$	0	0
$121$	−1039.07	−0.780666
$122$	0	0
$123$	−1608.11	−1.17885
$124$	0	0
$125$	−3102.00	−2.21961
$126$	0	0
$127$	−659.153	−0.460554	−0.230277	−	0.973125i	$-0.573963\pi$
$127$	−659.153	−0.460554	−0.230277	+	0.973125i	$0.573963\pi$
$128$	0	0
$129$	1177.02	0.803339
$130$	0	0
$131$	1087.80	0.725509	0.362754	−	0.931885i	$-0.381836\pi$
$131$	1087.80	0.725509	0.362754	+	0.931885i	$0.381836\pi$
$132$	0	0
$133$	2944.70	1.91983
$134$	0	0
$135$	−2855.88	−1.82070
$136$	0	0
$137$	−1762.20	−1.09894	−0.549472	−	0.835512i	$-0.685171\pi$
$137$	−1762.20	−1.09894	−0.549472	+	0.835512i	$0.685171\pi$
$138$	0	0
$139$	−992.512	−0.605639	−0.302819	−	0.953048i	$-0.597928\pi$
$139$	−992.512	−0.605639	−0.302819	+	0.953048i	$0.597928\pi$
$140$	0	0
$141$	−1652.45	−0.986963
$142$	0	0
$143$	−1339.95	−0.783585
$144$	0	0
$145$	−4082.80	−2.33833
$146$	0	0
$147$	−518.306	−0.290811
$148$	0	0
$149$	−1305.14	−0.717590	−0.358795	−	0.933416i	$-0.616812\pi$
$149$	−1305.14	−0.717590	−0.358795	+	0.933416i	$0.616812\pi$
$150$	0	0
$151$	−1566.45	−0.844210	−0.422105	−	0.906547i	$-0.638709\pi$
$151$	−1566.45	−0.844210	−0.422105	+	0.906547i	$0.638709\pi$
$152$	0	0
$153$	11.7680	0.00621820
$154$	0	0
$155$	857.751	0.444492
$156$	0	0
$157$	−3409.68	−1.73326	−0.866630	−	0.498951i	$-0.833719\pi$
$157$	−3409.68	−1.73326	−0.866630	+	0.498951i	$0.833719\pi$
$158$	0	0
$159$	−1095.94	−0.546629
$160$	0	0
$161$	−2917.78	−1.42828
$162$	0	0
$163$	−1775.65	−0.853251	−0.426625	−	0.904428i	$-0.640298\pi$
$163$	−1775.65	−0.853251	−0.426625	+	0.904428i	$0.640298\pi$
$164$	0	0
$165$	1762.07	0.831376
$166$	0	0
$167$	651.188	0.301739	0.150870	−	0.988554i	$-0.451793\pi$
$167$	651.188	0.301739	0.150870	+	0.988554i	$0.451793\pi$
$168$	0	0
$169$	3953.30	1.79941
$170$	0	0
$171$	−96.7335	−0.0432596
$172$	0	0
$173$	1737.90	0.763758	0.381879	−	0.924212i	$-0.375277\pi$
$173$	1737.90	0.763758	0.381879	+	0.924212i	$0.375277\pi$
$174$	0	0
$175$	5885.08	2.54212
$176$	0	0
$177$	−2504.68	−1.06363
$178$	0	0
$179$	2890.53	1.20698	0.603488	−	0.797372i	$-0.293777\pi$
$179$	2890.53	1.20698	0.603488	+	0.797372i	$0.293777\pi$
$180$	0	0
$181$	1617.76	0.664349	0.332175	−	0.943218i	$-0.392218\pi$
$181$	1617.76	0.664349	0.332175	+	0.943218i	$0.392218\pi$
$182$	0	0
$183$	1962.32	0.792670
$184$	0	0
$185$	−803.220	−0.319210
$186$	0	0
$187$	−290.463	−0.113587
$188$	0	0
$189$	2993.07	1.15192
$190$	0	0
$191$	209.705	0.0794436	0.0397218	−	0.999211i	$-0.487353\pi$
$191$	209.705	0.0794436	0.0397218	+	0.999211i	$0.487353\pi$
$192$	0	0
$193$	2714.19	1.01229	0.506144	−	0.862449i	$-0.331070\pi$
$193$	2714.19	1.01229	0.506144	+	0.862449i	$0.331070\pi$
$194$	0	0
$195$	−8087.79	−2.97015
$196$	0	0
$197$	1932.09	0.698760	0.349380	−	0.936981i	$-0.386392\pi$
$197$	1932.09	0.698760	0.349380	+	0.936981i	$0.386392\pi$
$198$	0	0
$199$	2114.22	0.753132	0.376566	−	0.926390i	$-0.377105\pi$
$199$	2114.22	0.753132	0.376566	+	0.926390i	$0.377105\pi$
$200$	0	0
$201$	2187.63	0.767680
$202$	0	0
$203$	4278.94	1.47942
$204$	0	0
$205$	−6303.97	−2.14775
$206$	0	0
$207$	95.8493	0.0321835
$208$	0	0
$209$	2387.62	0.790217
$210$	0	0
$211$	−3828.43	−1.24910	−0.624550	−	0.780985i	$-0.714718\pi$
$211$	−3828.43	−1.24910	−0.624550	+	0.780985i	$0.714718\pi$
$212$	0	0
$213$	−2482.20	−0.798488
$214$	0	0
$215$	4614.04	1.46360
$216$	0	0
$217$	−898.956	−0.281222
$218$	0	0
$219$	3267.51	1.00821
$220$	0	0
$221$	1333.21	0.405797
$222$	0	0
$223$	3318.07	0.996387	0.498194	−	0.867066i	$-0.333997\pi$
$223$	3318.07	0.996387	0.498194	+	0.867066i	$0.333997\pi$
$224$	0	0
$225$	−193.325	−0.0572816
$226$	0	0
$227$	1265.77	0.370097	0.185048	−	0.982729i	$-0.440756\pi$
$227$	1265.77	0.370097	0.185048	+	0.982729i	$0.440756\pi$
$228$	0	0
$229$	−470.289	−0.135710	−0.0678550	−	0.997695i	$-0.521616\pi$
$229$	−470.289	−0.135710	−0.0678550	+	0.997695i	$0.521616\pi$
$230$	0	0
$231$	−1846.72	−0.525996
$232$	0	0
$233$	−598.736	−0.168345	−0.0841727	−	0.996451i	$-0.526825\pi$
$233$	−598.736	−0.168345	−0.0841727	+	0.996451i	$0.526825\pi$
$234$	0	0
$235$	−6477.80	−1.79815
$236$	0	0
$237$	−619.712	−0.169851
$238$	0	0
$239$	−5431.34	−1.46997	−0.734987	−	0.678081i	$-0.762812\pi$
$239$	−5431.34	−1.46997	−0.734987	+	0.678081i	$0.762812\pi$
$240$	0	0
$241$	3929.62	1.05033	0.525164	−	0.851001i	$-0.324004\pi$
$241$	3929.62	1.05033	0.525164	+	0.851001i	$0.324004\pi$
$242$	0	0
$243$	−194.187	−0.0512639
$244$	0	0
$245$	−2031.81	−0.529828
$246$	0	0
$247$	−10959.0	−2.82310
$248$	0	0
$249$	−2386.11	−0.607283
$250$	0	0
$251$	−5275.18	−1.32656	−0.663280	−	0.748371i	$-0.730836\pi$
$251$	−5275.18	−1.32656	−0.663280	+	0.748371i	$0.730836\pi$
$252$	0	0
$253$	−2365.80	−0.587891
$254$	0	0
$255$	−1753.20	−0.430547
$256$	0	0
$257$	3811.20	0.925043	0.462522	−	0.886608i	$-0.346945\pi$
$257$	3811.20	0.925043	0.462522	+	0.886608i	$0.346945\pi$
$258$	0	0
$259$	841.806	0.201959
$260$	0	0
$261$	−140.563	−0.0333358
$262$	0	0
$263$	−5112.71	−1.19872	−0.599360	−	0.800479i	$-0.704578\pi$
$263$	−5112.71	−1.19872	−0.599360	+	0.800479i	$0.704578\pi$
$264$	0	0
$265$	−4296.21	−0.995903
$266$	0	0
$267$	−6905.52	−1.58281
$268$	0	0
$269$	−975.150	−0.221026	−0.110513	−	0.993875i	$-0.535249\pi$
$269$	−975.150	−0.221026	−0.110513	+	0.993875i	$0.535249\pi$
$270$	0	0
$271$	4216.45	0.945133	0.472567	−	0.881295i	$-0.343328\pi$
$271$	4216.45	0.945133	0.472567	+	0.881295i	$0.343328\pi$
$272$	0	0
$273$	8476.31	1.87916
$274$	0	0
$275$	4771.75	1.04635
$276$	0	0
$277$	8128.34	1.76312	0.881560	−	0.472072i	$-0.156494\pi$
$277$	8128.34	1.76312	0.881560	+	0.472072i	$0.156494\pi$
$278$	0	0
$279$	29.5308	0.00633678
$280$	0	0
$281$	−7222.06	−1.53321	−0.766605	−	0.642119i	$-0.778056\pi$
$281$	−7222.06	−1.53321	−0.766605	+	0.642119i	$0.778056\pi$
$282$	0	0
$283$	393.419	0.0826373	0.0413186	−	0.999146i	$-0.486844\pi$
$283$	393.419	0.0826373	0.0413186	+	0.999146i	$0.486844\pi$
$284$	0	0
$285$	14411.4	2.99528
$286$	0	0
$287$	6606.80	1.35884
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	5729.91	1.15427
$292$	0	0
$293$	−2276.65	−0.453937	−0.226968	−	0.973902i	$-0.572881\pi$
$293$	−2276.65	−0.453937	−0.226968	+	0.973902i	$0.572881\pi$
$294$	0	0
$295$	−9818.61	−1.93784
$296$	0	0
$297$	2426.84	0.474141
$298$	0	0
$299$	10858.9	2.10028
$300$	0	0
$301$	−4835.69	−0.925995
$302$	0	0
$303$	124.069	0.0235233
$304$	0	0
$305$	7692.49	1.44417
$306$	0	0
$307$	−8281.24	−1.53953	−0.769764	−	0.638328i	$-0.779626\pi$
$307$	−8281.24	−1.53953	−0.769764	+	0.638328i	$0.779626\pi$
$308$	0	0
$309$	4091.37	0.753235
$310$	0	0
$311$	−846.944	−0.154424	−0.0772119	−	0.997015i	$-0.524602\pi$
$311$	−846.944	−0.154424	−0.0772119	+	0.997015i	$0.524602\pi$
$312$	0	0
$313$	−1769.07	−0.319469	−0.159734	−	0.987160i	$-0.551064\pi$
$313$	−1769.07	−0.319469	−0.159734	+	0.987160i	$0.551064\pi$
$314$	0	0
$315$	293.298	0.0524619
$316$	0	0
$317$	−4119.16	−0.729826	−0.364913	−	0.931042i	$-0.618901\pi$
$317$	−4119.16	−0.729826	−0.364913	+	0.931042i	$0.618901\pi$
$318$	0	0
$319$	3469.45	0.608940
$320$	0	0
$321$	6168.09	1.07249
$322$	0	0
$323$	−2375.60	−0.409231
$324$	0	0
$325$	−21902.0	−3.73817
$326$	0	0
$327$	1101.81	0.186330
$328$	0	0
$329$	6788.98	1.13766
$330$	0	0
$331$	−10126.2	−1.68152	−0.840761	−	0.541407i	$-0.817892\pi$
$331$	−10126.2	−1.68152	−0.840761	+	0.541407i	$0.817892\pi$
$332$	0	0
$333$	−27.6534	−0.00455074
$334$	0	0
$335$	8575.75	1.39864
$336$	0	0
$337$	−2869.47	−0.463828	−0.231914	−	0.972736i	$-0.574499\pi$
$337$	−2869.47	−0.463828	−0.231914	+	0.972736i	$0.574499\pi$
$338$	0	0
$339$	7875.14	1.26171
$340$	0	0
$341$	−728.893	−0.115753
$342$	0	0
$343$	−5098.46	−0.802598
$344$	0	0
$345$	−14279.6	−2.22838
$346$	0	0
$347$	6322.40	0.978110	0.489055	−	0.872253i	$-0.337342\pi$
$347$	6322.40	0.978110	0.489055	+	0.872253i	$0.337342\pi$
$348$	0	0
$349$	−4459.97	−0.684060	−0.342030	−	0.939689i	$-0.611114\pi$
$349$	−4459.97	−0.684060	−0.342030	+	0.939689i	$0.611114\pi$
$350$	0	0
$351$	−11139.0	−1.69390
$352$	0	0
$353$	−2585.23	−0.389795	−0.194898	−	0.980824i	$-0.562437\pi$
$353$	−2585.23	−0.389795	−0.194898	+	0.980824i	$0.562437\pi$
$354$	0	0
$355$	−9730.50	−1.45476
$356$	0	0
$357$	1837.42	0.272399
$358$	0	0
$359$	−3840.82	−0.564654	−0.282327	−	0.959318i	$-0.591106\pi$
$359$	−3840.82	−0.564654	−0.282327	+	0.959318i	$0.591106\pi$
$360$	0	0
$361$	12668.5	1.84699
$362$	0	0
$363$	5329.49	0.770594
$364$	0	0
$365$	12809.0	1.83685
$366$	0	0
$367$	9773.07	1.39006	0.695028	−	0.718983i	$-0.255392\pi$
$367$	9773.07	1.39006	0.695028	+	0.718983i	$0.255392\pi$
$368$	0	0
$369$	−217.034	−0.0306188
$370$	0	0
$371$	4502.60	0.630089
$372$	0	0
$373$	11983.1	1.66344	0.831719	−	0.555196i	$-0.187357\pi$
$373$	11983.1	1.66344	0.831719	+	0.555196i	$0.187357\pi$
$374$	0	0
$375$	15910.5	2.19097
$376$	0	0
$377$	−15924.5	−2.17548
$378$	0	0
$379$	−7730.36	−1.04771	−0.523855	−	0.851808i	$-0.675507\pi$
$379$	−7730.36	−1.04771	−0.523855	+	0.851808i	$0.675507\pi$
$380$	0	0
$381$	3380.87	0.454612
$382$	0	0
$383$	2206.48	0.294375	0.147188	−	0.989109i	$-0.452978\pi$
$383$	2206.48	0.294375	0.147188	+	0.989109i	$0.452978\pi$
$384$	0	0
$385$	−7239.33	−0.958313
$386$	0	0
$387$	158.853	0.0208655
$388$	0	0
$389$	2565.02	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$389$	2565.02	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$390$	0	0
$391$	2353.88	0.304453
$392$	0	0
$393$	−5579.45	−0.716148
$394$	0	0
$395$	−2429.33	−0.309451
$396$	0	0
$397$	7974.54	1.00814	0.504069	−	0.863664i	$-0.331836\pi$
$397$	7974.54	1.00814	0.504069	+	0.863664i	$0.331836\pi$
$398$	0	0
$399$	−15103.7	−1.89506
$400$	0	0
$401$	9035.21	1.12518	0.562590	−	0.826736i	$-0.309805\pi$
$401$	9035.21	1.12518	0.562590	+	0.826736i	$0.309805\pi$
$402$	0	0
$403$	3345.57	0.413535
$404$	0	0
$405$	14272.3	1.75110
$406$	0	0
$407$	682.554	0.0831276
$408$	0	0
$409$	−10355.1	−1.25190	−0.625950	−	0.779863i	$-0.715289\pi$
$409$	−10355.1	−1.25190	−0.625950	+	0.779863i	$0.715289\pi$
$410$	0	0
$411$	9038.54	1.08476
$412$	0	0
$413$	10290.3	1.22603
$414$	0	0
$415$	−9353.80	−1.10641
$416$	0	0
$417$	5090.70	0.597824
$418$	0	0
$419$	11297.2	1.31720	0.658598	−	0.752495i	$-0.271150\pi$
$419$	11297.2	1.31720	0.658598	+	0.752495i	$0.271150\pi$
$420$	0	0
$421$	6109.03	0.707211	0.353605	−	0.935395i	$-0.384956\pi$
$421$	6109.03	0.707211	0.353605	+	0.935395i	$0.384956\pi$
$422$	0	0
$423$	−223.018	−0.0256348
$424$	0	0
$425$	−4747.71	−0.541877
$426$	0	0
$427$	−8062.02	−0.913697
$428$	0	0
$429$	6872.77	0.773475
$430$	0	0
$431$	12011.6	1.34240	0.671202	−	0.741274i	$-0.265778\pi$
$431$	12011.6	1.34240	0.671202	+	0.741274i	$0.265778\pi$
$432$	0	0
$433$	10677.2	1.18501	0.592507	−	0.805565i	$-0.298138\pi$
$433$	10677.2	1.18501	0.592507	+	0.805565i	$0.298138\pi$
$434$	0	0
$435$	20941.1	2.30816
$436$	0	0
$437$	−19349.1	−2.11806
$438$	0	0
$439$	−6934.54	−0.753912	−0.376956	−	0.926231i	$-0.623029\pi$
$439$	−6934.54	−0.753912	−0.376956	+	0.926231i	$0.623029\pi$
$440$	0	0
$441$	−69.9516	−0.00755335
$442$	0	0
$443$	9431.64	1.01154	0.505768	−	0.862669i	$-0.331209\pi$
$443$	9431.64	1.01154	0.505768	+	0.862669i	$0.331209\pi$
$444$	0	0
$445$	−27070.4	−2.88373
$446$	0	0
$447$	6694.18	0.708331
$448$	0	0
$449$	−7280.77	−0.765258	−0.382629	−	0.923902i	$-0.624981\pi$
$449$	−7280.77	−0.765258	−0.382629	+	0.923902i	$0.624981\pi$
$450$	0	0
$451$	5356.94	0.559309
$452$	0	0
$453$	8034.48	0.833318
$454$	0	0
$455$	33228.0	3.42364
$456$	0	0
$457$	−391.607	−0.0400845	−0.0200422	−	0.999799i	$-0.506380\pi$
$457$	−391.607	−0.0400845	−0.0200422	+	0.999799i	$0.506380\pi$
$458$	0	0
$459$	−2414.62	−0.245544
$460$	0	0
$461$	−16052.7	−1.62179	−0.810897	−	0.585190i	$-0.801020\pi$
$461$	−16052.7	−1.62179	−0.810897	+	0.585190i	$0.801020\pi$
$462$	0	0
$463$	−9109.91	−0.914413	−0.457207	−	0.889360i	$-0.651150\pi$
$463$	−9109.91	−0.914413	−0.457207	+	0.889360i	$0.651150\pi$
$464$	0	0
$465$	−4399.50	−0.438757
$466$	0	0
$467$	−15478.7	−1.53376	−0.766882	−	0.641788i	$-0.778193\pi$
$467$	−15478.7	−1.53376	−0.766882	+	0.641788i	$0.778193\pi$
$468$	0	0
$469$	−8987.71	−0.884891
$470$	0	0
$471$	17488.6	1.71090
$472$	0	0
$473$	−3920.88	−0.381146
$474$	0	0
$475$	39026.5	3.76980
$476$	0	0
$477$	−147.911	−0.0141978
$478$	0	0
$479$	16337.4	1.55840	0.779201	−	0.626774i	$-0.215625\pi$
$479$	16337.4	1.55840	0.779201	+	0.626774i	$0.215625\pi$
$480$	0	0
$481$	−3132.88	−0.296979
$482$	0	0
$483$	14965.6	1.40985
$484$	0	0
$485$	22461.8	2.10297
$486$	0	0
$487$	5914.12	0.550296	0.275148	−	0.961402i	$-0.411273\pi$
$487$	5914.12	0.550296	0.275148	+	0.961402i	$0.411273\pi$
$488$	0	0
$489$	9107.52	0.842242
$490$	0	0
$491$	2100.85	0.193096	0.0965478	−	0.995328i	$-0.469220\pi$
$491$	2100.85	0.193096	0.0965478	+	0.995328i	$0.469220\pi$
$492$	0	0
$493$	−3451.98	−0.315353
$494$	0	0
$495$	237.813	0.0215937
$496$	0	0
$497$	10197.9	0.920403
$498$	0	0
$499$	−17306.7	−1.55261	−0.776305	−	0.630357i	$-0.782909\pi$
$499$	−17306.7	−1.55261	−0.776305	+	0.630357i	$0.782909\pi$
$500$	0	0
$501$	−3340.01	−0.297846
$502$	0	0
$503$	−7147.42	−0.633574	−0.316787	−	0.948497i	$-0.602604\pi$
$503$	−7147.42	−0.633574	−0.316787	+	0.948497i	$0.602604\pi$
$504$	0	0
$505$	486.363	0.0428572
$506$	0	0
$507$	−20276.9	−1.77619
$508$	0	0
$509$	7892.80	0.687313	0.343656	−	0.939095i	$-0.388334\pi$
$509$	7892.80	0.687313	0.343656	+	0.939095i	$0.388334\pi$
$510$	0	0
$511$	−13424.3	−1.16214
$512$	0	0
$513$	19848.3	1.70824
$514$	0	0
$515$	16038.6	1.37232
$516$	0	0
$517$	5504.65	0.468267
$518$	0	0
$519$	−8913.88	−0.753904
$520$	0	0
$521$	−4246.71	−0.357105	−0.178552	−	0.983930i	$-0.557141\pi$
$521$	−4246.71	−0.357105	−0.178552	+	0.983930i	$0.557141\pi$
$522$	0	0
$523$	9208.75	0.769925	0.384962	−	0.922932i	$-0.374214\pi$
$523$	9208.75	0.769925	0.384962	+	0.922932i	$0.374214\pi$
$524$	0	0
$525$	−30185.2	−2.50932
$526$	0	0
$527$	725.222	0.0599452
$528$	0	0
$529$	7005.20	0.575754
$530$	0	0
$531$	−338.037	−0.0276262
$532$	0	0
$533$	−24588.0	−1.99817
$534$	0	0
$535$	24179.5	1.95397
$536$	0	0
$537$	−14825.9	−1.19140
$538$	0	0
$539$	1726.58	0.137976
$540$	0	0
$541$	−19505.1	−1.55007	−0.775036	−	0.631917i	$-0.782269\pi$
$541$	−19505.1	−1.55007	−0.775036	+	0.631917i	$0.782269\pi$
$542$	0	0
$543$	−8297.67	−0.655778
$544$	0	0
$545$	4319.20	0.339475
$546$	0	0
$547$	10605.7	0.829005	0.414503	−	0.910048i	$-0.363956\pi$
$547$	10605.7	0.829005	0.414503	+	0.910048i	$0.363956\pi$
$548$	0	0
$549$	264.838	0.0205884
$550$	0	0
$551$	28375.4	2.19389
$552$	0	0
$553$	2546.04	0.195784
$554$	0	0
$555$	4119.80	0.315092
$556$	0	0
$557$	−5623.77	−0.427804	−0.213902	−	0.976855i	$-0.568617\pi$
$557$	−5623.77	−0.427804	−0.213902	+	0.976855i	$0.568617\pi$
$558$	0	0
$559$	17996.6	1.36167
$560$	0	0
$561$	1489.82	0.112121
$562$	0	0
$563$	−16606.1	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$563$	−16606.1	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$564$	0	0
$565$	30871.4	2.29871
$566$	0	0
$567$	−14957.9	−1.10789
$568$	0	0
$569$	−12583.8	−0.927135	−0.463568	−	0.886062i	$-0.653431\pi$
$569$	−12583.8	−0.927135	−0.463568	+	0.886062i	$0.653431\pi$
$570$	0	0
$571$	13239.1	0.970299	0.485149	−	0.874431i	$-0.338765\pi$
$571$	13239.1	0.970299	0.485149	+	0.874431i	$0.338765\pi$
$572$	0	0
$573$	−1075.60	−0.0784186
$574$	0	0
$575$	−38669.8	−2.80459
$576$	0	0
$577$	−13563.1	−0.978580	−0.489290	−	0.872121i	$-0.662744\pi$
$577$	−13563.1	−0.978580	−0.489290	+	0.872121i	$0.662744\pi$
$578$	0	0
$579$	−13921.4	−0.999227
$580$	0	0
$581$	9803.14	0.700005
$582$	0	0
$583$	3650.80	0.259349
$584$	0	0
$585$	−1091.54	−0.0771449
$586$	0	0
$587$	−22754.1	−1.59993	−0.799967	−	0.600044i	$-0.795150\pi$
$587$	−22754.1	−1.59993	−0.799967	+	0.600044i	$0.795150\pi$
$588$	0	0
$589$	−5961.36	−0.417035
$590$	0	0
$591$	−9909.90	−0.689744
$592$	0	0
$593$	14368.2	0.994991	0.497496	−	0.867466i	$-0.334253\pi$
$593$	14368.2	0.994991	0.497496	+	0.867466i	$0.334253\pi$
$594$	0	0
$595$	7202.87	0.496284
$596$	0	0
$597$	−10844.1	−0.743414
$598$	0	0
$599$	−26564.5	−1.81201	−0.906007	−	0.423262i	$-0.860885\pi$
$599$	−26564.5	−1.81201	−0.906007	+	0.423262i	$0.860885\pi$
$600$	0	0
$601$	10021.8	0.680194	0.340097	−	0.940390i	$-0.389540\pi$
$601$	10021.8	0.680194	0.340097	+	0.940390i	$0.389540\pi$
$602$	0	0
$603$	295.247	0.0199393
$604$	0	0
$605$	20892.1	1.40394
$606$	0	0
$607$	25176.7	1.68351	0.841756	−	0.539857i	$-0.181522\pi$
$607$	25176.7	1.68351	0.841756	+	0.539857i	$0.181522\pi$
$608$	0	0
$609$	−21947.1	−1.46033
$610$	0	0
$611$	−25266.0	−1.67292
$612$	0	0
$613$	−8018.82	−0.528347	−0.264174	−	0.964475i	$-0.585099\pi$
$613$	−8018.82	−0.528347	−0.264174	+	0.964475i	$0.585099\pi$
$614$	0	0
$615$	32333.8	2.12004
$616$	0	0
$617$	−3141.96	−0.205009	−0.102504	−	0.994733i	$-0.532686\pi$
$617$	−3141.96	−0.205009	−0.102504	+	0.994733i	$0.532686\pi$
$618$	0	0
$619$	−103.440	−0.00671663	−0.00335832	−	0.999994i	$-0.501069\pi$
$619$	−103.440	−0.00671663	−0.00335832	+	0.999994i	$0.501069\pi$
$620$	0	0
$621$	−19666.9	−1.27086
$622$	0	0
$623$	28370.8	1.82448
$624$	0	0
$625$	27461.1	1.75751
$626$	0	0
$627$	−12246.4	−0.780021
$628$	0	0
$629$	−679.116	−0.0430495
$630$	0	0
$631$	−1125.52	−0.0710085	−0.0355043	−	0.999370i	$-0.511304\pi$
$631$	−1125.52	−0.0710085	−0.0355043	+	0.999370i	$0.511304\pi$
$632$	0	0
$633$	19636.4	1.23298
$634$	0	0
$635$	13253.4	0.828257
$636$	0	0
$637$	−7924.88	−0.492928
$638$	0	0
$639$	−335.003	−0.0207395
$640$	0	0
$641$	−27474.5	−1.69295	−0.846473	−	0.532432i	$-0.821278\pi$
$641$	−27474.5	−1.69295	−0.846473	+	0.532432i	$0.821278\pi$
$642$	0	0
$643$	−27502.5	−1.68677	−0.843385	−	0.537310i	$-0.819441\pi$
$643$	−27502.5	−1.68677	−0.843385	+	0.537310i	$0.819441\pi$
$644$	0	0
$645$	−23665.9	−1.44472
$646$	0	0
$647$	−19119.2	−1.16175	−0.580876	−	0.813992i	$-0.697290\pi$
$647$	−19119.2	−1.16175	−0.580876	+	0.813992i	$0.697290\pi$
$648$	0	0
$649$	8343.58	0.504644
$650$	0	0
$651$	4610.85	0.277593
$652$	0	0
$653$	323.964	0.0194145	0.00970726	−	0.999953i	$-0.496910\pi$
$653$	323.964	0.0194145	0.00970726	+	0.999953i	$0.496910\pi$
$654$	0	0
$655$	−21872.0	−1.30475
$656$	0	0
$657$	440.989	0.0261866
$658$	0	0
$659$	9944.76	0.587850	0.293925	−	0.955829i	$-0.405038\pi$
$659$	9944.76	0.587850	0.293925	+	0.955829i	$0.405038\pi$
$660$	0	0
$661$	−7481.89	−0.440260	−0.220130	−	0.975471i	$-0.570648\pi$
$661$	−7481.89	−0.440260	−0.220130	+	0.975471i	$0.570648\pi$
$662$	0	0
$663$	−6838.16	−0.400561
$664$	0	0
$665$	−59208.0	−3.45261
$666$	0	0
$667$	−28116.1	−1.63217
$668$	0	0
$669$	−17018.7	−0.983532
$670$	0	0
$671$	−6536.86	−0.376084
$672$	0	0
$673$	196.143	0.0112344	0.00561719	−	0.999984i	$-0.498212\pi$
$673$	196.143	0.0112344	0.00561719	+	0.999984i	$0.498212\pi$
$674$	0	0
$675$	39667.6	2.26193
$676$	0	0
$677$	19213.7	1.09076	0.545379	−	0.838190i	$-0.316386\pi$
$677$	19213.7	1.09076	0.545379	+	0.838190i	$0.316386\pi$
$678$	0	0
$679$	−23540.9	−1.33051
$680$	0	0
$681$	−6492.26	−0.365322
$682$	0	0
$683$	−26093.2	−1.46183	−0.730913	−	0.682471i	$-0.760905\pi$
$683$	−26093.2	−1.46183	−0.730913	+	0.682471i	$0.760905\pi$
$684$	0	0
$685$	35432.0	1.97633
$686$	0	0
$687$	2412.16	0.133959
$688$	0	0
$689$	−16756.9	−0.926543
$690$	0	0
$691$	−15862.5	−0.873283	−0.436642	−	0.899636i	$-0.643832\pi$
$691$	−15862.5	−0.873283	−0.436642	+	0.899636i	$0.643832\pi$
$692$	0	0
$693$	−249.237	−0.0136619
$694$	0	0
$695$	19956.1	1.08918
$696$	0	0
$697$	−5329.95	−0.289651
$698$	0	0
$699$	3070.98	0.166173
$700$	0	0
$701$	−15929.2	−0.858256	−0.429128	−	0.903244i	$-0.641179\pi$
$701$	−15929.2	−0.858256	−0.429128	+	0.903244i	$0.641179\pi$
$702$	0	0
$703$	5582.37	0.299492
$704$	0	0
$705$	33225.3	1.77495
$706$	0	0
$707$	−509.727	−0.0271150
$708$	0	0
$709$	21143.4	1.11997	0.559984	−	0.828504i	$-0.310807\pi$
$709$	21143.4	1.11997	0.559984	+	0.828504i	$0.310807\pi$
$710$	0	0
$711$	−83.6375	−0.00441160
$712$	0	0
$713$	5906.87	0.310258
$714$	0	0
$715$	26942.0	1.40919
$716$	0	0
$717$	27857.9	1.45101
$718$	0	0
$719$	−33525.4	−1.73892	−0.869462	−	0.493999i	$-0.835535\pi$
$719$	−33525.4	−1.73892	−0.869462	+	0.493999i	$0.835535\pi$
$720$	0	0
$721$	−16809.1	−0.868241
$722$	0	0
$723$	−20155.5	−1.03678
$724$	0	0
$725$	56709.3	2.90501
$726$	0	0
$727$	2772.68	0.141448	0.0707242	−	0.997496i	$-0.477469\pi$
$727$	2772.68	0.141448	0.0707242	+	0.997496i	$0.477469\pi$
$728$	0	0
$729$	20161.4	1.02431
$730$	0	0
$731$	3901.13	0.197385
$732$	0	0
$733$	31779.0	1.60134	0.800671	−	0.599105i	$-0.204477\pi$
$733$	31779.0	1.60134	0.800671	+	0.599105i	$0.204477\pi$
$734$	0	0
$735$	10421.4	0.522992
$736$	0	0
$737$	−7287.43	−0.364228
$738$	0	0
$739$	779.798	0.0388164	0.0194082	−	0.999812i	$-0.493822\pi$
$739$	779.798	0.0388164	0.0194082	+	0.999812i	$0.493822\pi$
$740$	0	0
$741$	56210.0	2.78668
$742$	0	0
$743$	9130.73	0.450840	0.225420	−	0.974262i	$-0.427625\pi$
$743$	9130.73	0.450840	0.225420	+	0.974262i	$0.427625\pi$
$744$	0	0
$745$	26241.9	1.29051
$746$	0	0
$747$	−322.034	−0.0157732
$748$	0	0
$749$	−25341.1	−1.23624
$750$	0	0
$751$	30497.8	1.48186	0.740932	−	0.671580i	$-0.234384\pi$
$751$	30497.8	1.48186	0.740932	+	0.671580i	$0.234384\pi$
$752$	0	0
$753$	27057.0	1.30944
$754$	0	0
$755$	31496.0	1.51822
$756$	0	0
$757$	−22888.3	−1.09893	−0.549466	−	0.835516i	$-0.685169\pi$
$757$	−22888.3	−1.09893	−0.549466	+	0.835516i	$0.685169\pi$
$758$	0	0
$759$	12134.4	0.580306
$760$	0	0
$761$	1765.87	0.0841164	0.0420582	−	0.999115i	$-0.486609\pi$
$761$	1765.87	0.0841164	0.0420582	+	0.999115i	$0.486609\pi$
$762$	0	0
$763$	−4526.68	−0.214780
$764$	0	0
$765$	−236.615	−0.0111828
$766$	0	0
$767$	−38296.5	−1.80287
$768$	0	0
$769$	19453.3	0.912228	0.456114	−	0.889921i	$-0.349241\pi$
$769$	19453.3	0.912228	0.456114	+	0.889921i	$0.349241\pi$
$770$	0	0
$771$	−19548.1	−0.913108
$772$	0	0
$773$	−24322.6	−1.13173	−0.565863	−	0.824499i	$-0.691457\pi$
$773$	−24322.6	−1.13173	−0.565863	+	0.824499i	$0.691457\pi$
$774$	0	0
$775$	−11914.0	−0.552210
$776$	0	0
$777$	−4317.71	−0.199353
$778$	0	0
$779$	43812.5	2.01508
$780$	0	0
$781$	8268.71	0.378844
$782$	0	0
$783$	28841.6	1.31636
$784$	0	0
$785$	68557.2	3.11708
$786$	0	0
$787$	38992.7	1.76612	0.883062	−	0.469256i	$-0.155478\pi$
$787$	38992.7	1.76612	0.883062	+	0.469256i	$0.155478\pi$
$788$	0	0
$789$	26223.7	1.18325
$790$	0	0
$791$	−32354.4	−1.45435
$792$	0	0
$793$	30003.7	1.34359
$794$	0	0
$795$	22035.7	0.983053
$796$	0	0
$797$	−1275.33	−0.0566806	−0.0283403	−	0.999598i	$-0.509022\pi$
$797$	−1275.33	−0.0566806	−0.0283403	+	0.999598i	$0.509022\pi$
$798$	0	0
$799$	−5476.92	−0.242503
$800$	0	0
$801$	−931.982	−0.0411111
$802$	0	0
$803$	−10884.7	−0.478347
$804$	0	0
$805$	58666.8	2.56861
$806$	0	0
$807$	5001.65	0.218174
$808$	0	0
$809$	−21147.8	−0.919056	−0.459528	−	0.888163i	$-0.651981\pi$
$809$	−21147.8	−0.919056	−0.459528	+	0.888163i	$0.651981\pi$
$810$	0	0
$811$	−14387.6	−0.622955	−0.311477	−	0.950254i	$-0.600824\pi$
$811$	−14387.6	−0.622955	−0.311477	+	0.950254i	$0.600824\pi$
$812$	0	0
$813$	−21626.6	−0.932939
$814$	0	0
$815$	35702.4	1.53448
$816$	0	0
$817$	−32067.5	−1.37319
$818$	0	0
$819$	1143.98	0.0488081
$820$	0	0
$821$	−18679.9	−0.794070	−0.397035	−	0.917803i	$-0.629961\pi$
$821$	−18679.9	−0.794070	−0.397035	+	0.917803i	$0.629961\pi$
$822$	0	0
$823$	−19564.0	−0.828623	−0.414311	−	0.910135i	$-0.635978\pi$
$823$	−19564.0	−0.828623	−0.414311	+	0.910135i	$0.635978\pi$
$824$	0	0
$825$	−24474.8	−1.03285
$826$	0	0
$827$	23623.7	0.993323	0.496661	−	0.867944i	$-0.334559\pi$
$827$	23623.7	0.993323	0.496661	+	0.867944i	$0.334559\pi$
$828$	0	0
$829$	−34397.8	−1.44111	−0.720557	−	0.693396i	$-0.756114\pi$
$829$	−34397.8	−1.44111	−0.720557	+	0.693396i	$0.756114\pi$
$830$	0	0
$831$	−41691.1	−1.74037
$832$	0	0
$833$	−1717.88	−0.0714539
$834$	0	0
$835$	−13093.2	−0.542645
$836$	0	0
$837$	−6059.29	−0.250227
$838$	0	0
$839$	−11412.1	−0.469593	−0.234797	−	0.972045i	$-0.575442\pi$
$839$	−11412.1	−0.469593	−0.234797	+	0.972045i	$0.575442\pi$
$840$	0	0
$841$	16843.3	0.690611
$842$	0	0
$843$	37042.7	1.51343
$844$	0	0
$845$	−79487.6	−3.23604
$846$	0	0
$847$	−21895.8	−0.888250
$848$	0	0
$849$	−2017.89	−0.0815711
$850$	0	0
$851$	−5531.35	−0.222811
$852$	0	0
$853$	−15920.1	−0.639030	−0.319515	−	0.947581i	$-0.603520\pi$
$853$	−15920.1	−0.639030	−0.319515	+	0.947581i	$0.603520\pi$
$854$	0	0
$855$	1944.99	0.0777978
$856$	0	0
$857$	173.324	0.00690854	0.00345427	−	0.999994i	$-0.498900\pi$
$857$	173.324	0.00690854	0.00345427	+	0.999994i	$0.498900\pi$
$858$	0	0
$859$	−9092.59	−0.361159	−0.180579	−	0.983560i	$-0.557797\pi$
$859$	−9092.59	−0.361159	−0.180579	+	0.983560i	$0.557797\pi$
$860$	0	0
$861$	−33887.0	−1.34131
$862$	0	0
$863$	−1032.20	−0.0407145	−0.0203572	−	0.999793i	$-0.506480\pi$
$863$	−1032.20	−0.0407145	−0.0203572	+	0.999793i	$0.506480\pi$
$864$	0	0
$865$	−34943.4	−1.37354
$866$	0	0
$867$	−1482.31	−0.0580646
$868$	0	0
$869$	2064.38	0.0805861
$870$	0	0
$871$	33448.8	1.30123
$872$	0	0
$873$	773.319	0.0299804
$874$	0	0
$875$	−65366.9	−2.52549
$876$	0	0
$877$	−26483.1	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$877$	−26483.1	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$878$	0	0
$879$	11677.2	0.448080
$880$	0	0
$881$	11437.8	0.437402	0.218701	−	0.975792i	$-0.429818\pi$
$881$	11437.8	0.437402	0.218701	+	0.975792i	$0.429818\pi$
$882$	0	0
$883$	−31024.5	−1.18240	−0.591200	−	0.806525i	$-0.701345\pi$
$883$	−31024.5	−1.18240	−0.591200	+	0.806525i	$0.701345\pi$
$884$	0	0
$885$	50360.7	1.91283
$886$	0	0
$887$	8871.64	0.335829	0.167915	−	0.985802i	$-0.446297\pi$
$887$	8871.64	0.335829	0.167915	+	0.985802i	$0.446297\pi$
$888$	0	0
$889$	−13890.0	−0.524023
$890$	0	0
$891$	−12128.2	−0.456016
$892$	0	0
$893$	45020.6	1.68707
$894$	0	0
$895$	−58118.9	−2.17062
$896$	0	0
$897$	−55696.3	−2.07318
$898$	0	0
$899$	−8662.45	−0.321367
$900$	0	0
$901$	−3632.41	−0.134310
$902$	0	0
$903$	24802.8	0.914047
$904$	0	0
$905$	−32527.7	−1.19476
$906$	0	0
$907$	19665.1	0.719921	0.359961	−	0.932967i	$-0.382790\pi$
$907$	19665.1	0.719921	0.359961	+	0.932967i	$0.382790\pi$
$908$	0	0
$909$	16.7446	0.000610982 0
$910$	0	0
$911$	−33849.7	−1.23105	−0.615527	−	0.788116i	$-0.711057\pi$
$911$	−33849.7	−1.23105	−0.615527	+	0.788116i	$0.711057\pi$
$912$	0	0
$913$	7948.60	0.288127
$914$	0	0
$915$	−39455.6	−1.42553
$916$	0	0
$917$	22922.7	0.825491
$918$	0	0
$919$	−39175.1	−1.40617	−0.703083	−	0.711108i	$-0.748194\pi$
$919$	−39175.1	−1.40617	−0.703083	+	0.711108i	$0.748194\pi$
$920$	0	0
$921$	42475.4	1.51967
$922$	0	0
$923$	−37952.8	−1.35345
$924$	0	0
$925$	11156.6	0.396568
$926$	0	0
$927$	552.179	0.0195641
$928$	0	0
$929$	25575.4	0.903230	0.451615	−	0.892213i	$-0.350848\pi$
$929$	25575.4	0.903230	0.451615	+	0.892213i	$0.350848\pi$
$930$	0	0
$931$	14121.1	0.497100
$932$	0	0
$933$	4344.07	0.152431
$934$	0	0
$935$	5840.24	0.204274
$936$	0	0
$937$	−2451.58	−0.0854745	−0.0427373	−	0.999086i	$-0.513608\pi$
$937$	−2451.58	−0.0854745	−0.0427373	+	0.999086i	$0.513608\pi$
$938$	0	0
$939$	9073.76	0.315347
$940$	0	0
$941$	−3901.72	−0.135167	−0.0675837	−	0.997714i	$-0.521529\pi$
$941$	−3901.72	−0.135167	−0.0675837	+	0.997714i	$0.521529\pi$
$942$	0	0
$943$	−43412.1	−1.49914
$944$	0	0
$945$	−60180.6	−2.07161
$946$	0	0
$947$	14785.0	0.507335	0.253668	−	0.967291i	$-0.418363\pi$
$947$	14785.0	0.507335	0.253668	+	0.967291i	$0.418363\pi$
$948$	0	0
$949$	49960.0	1.70893
$950$	0	0
$951$	21127.6	0.720410
$952$	0	0
$953$	7120.38	0.242027	0.121014	−	0.992651i	$-0.461386\pi$
$953$	7120.38	0.242027	0.121014	+	0.992651i	$0.461386\pi$
$954$	0	0
$955$	−4216.46	−0.142871
$956$	0	0
$957$	−17795.2	−0.601084
$958$	0	0
$959$	−37134.1	−1.25039
$960$	0	0
$961$	−27971.1	−0.938912
$962$	0	0
$963$	832.457	0.0278562
$964$	0	0
$965$	−54573.2	−1.82049
$966$	0	0
$967$	18022.9	0.599355	0.299677	−	0.954041i	$-0.403121\pi$
$967$	18022.9	0.599355	0.299677	+	0.954041i	$0.403121\pi$
$968$	0	0
$969$	12184.7	0.403951
$970$	0	0
$971$	30934.5	1.02238	0.511192	−	0.859467i	$-0.329204\pi$
$971$	30934.5	1.02238	0.511192	+	0.859467i	$0.329204\pi$
$972$	0	0
$973$	−20914.7	−0.689102
$974$	0	0
$975$	112338.	3.68993
$976$	0	0
$977$	9224.43	0.302063	0.151032	−	0.988529i	$-0.451740\pi$
$977$	9224.43	0.302063	0.151032	+	0.988529i	$0.451740\pi$
$978$	0	0
$979$	23003.6	0.750970
$980$	0	0
$981$	148.702	0.00483964
$982$	0	0
$983$	−13784.1	−0.447248	−0.223624	−	0.974675i	$-0.571789\pi$
$983$	−13784.1	−0.447248	−0.223624	+	0.974675i	$0.571789\pi$
$984$	0	0
$985$	−38847.9	−1.25665
$986$	0	0
$987$	−34821.4	−1.12298
$988$	0	0
$989$	31774.4	1.02160
$990$	0	0
$991$	−22391.2	−0.717740	−0.358870	−	0.933388i	$-0.616838\pi$
$991$	−22391.2	−0.717740	−0.358870	+	0.933388i	$0.616838\pi$
$992$	0	0
$993$	51938.1	1.65983
$994$	0	0
$995$	−42509.9	−1.35443
$996$	0	0
$997$	−36575.1	−1.16183	−0.580916	−	0.813963i	$-0.697306\pi$
$997$	−36575.1	−1.16183	−0.580916	+	0.813963i	$0.697306\pi$
$998$	0	0
$999$	5674.07	0.179699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bb.1.2		4
4.3	odd	2		1088.4.a.be.1.3		4
8.3	odd	2		136.4.a.d.1.2	✓	4
8.5	even	2		272.4.a.k.1.3		4
24.5	odd	2		2448.4.a.bq.1.1		4
24.11	even	2		1224.4.a.l.1.1		4
136.67	odd	2		2312.4.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.d.1.2	✓	4	8.3	odd	2
272.4.a.k.1.3		4	8.5	even	2
1088.4.a.bb.1.2		4	1.1	even	1	trivial
1088.4.a.be.1.3		4	4.3	odd	2
1224.4.a.l.1.1		4	24.11	even	2
2312.4.a.e.1.3		4	136.67	odd	2
2448.4.a.bq.1.1		4	24.5	odd	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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-5.12911 q^{3} -20.1066 q^{5} +21.0725 q^{7} -0.692234 q^{9} +O(q^{10})\)	\(q-5.12911 q^{3} -20.1066 q^{5} +21.0725 q^{7} -0.692234 q^{9} +17.0861 q^{11} -78.4238 q^{13} +103.129 q^{15} -17.0000 q^{17} +139.741 q^{19} -108.083 q^{21} -138.464 q^{23} +279.277 q^{25} +142.037 q^{27} +203.057 q^{29} -42.6601 q^{31} -87.6363 q^{33} -423.698 q^{35} +39.9480 q^{37} +402.244 q^{39} +313.527 q^{41} -229.478 q^{43} +13.9185 q^{45} +322.172 q^{47} +101.052 q^{49} +87.1949 q^{51} +213.671 q^{53} -343.543 q^{55} -716.747 q^{57} +488.327 q^{59} -382.584 q^{61} -14.5871 q^{63} +1576.84 q^{65} -426.513 q^{67} +710.196 q^{69} +483.945 q^{71} -637.051 q^{73} -1432.44 q^{75} +360.047 q^{77} +120.822 q^{79} -709.830 q^{81} +465.209 q^{83} +341.813 q^{85} -1041.50 q^{87} +1346.34 q^{89} -1652.59 q^{91} +218.808 q^{93} -2809.72 q^{95} -1117.14 q^{97} -11.8276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 8 q^{5} + 22 q^{7} + 100 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 8 * q^5 + 22 * q^7 + 100 * q^9	\( 4 q - 2 q^{3} - 8 q^{5} + 22 q^{7} + 100 q^{9} + 70 q^{11} - 120 q^{13} - 140 q^{15} - 68 q^{17} - 44 q^{19} - 488 q^{21} - 158 q^{23} + 548 q^{25} - 392 q^{27} - 264 q^{29} - 122 q^{31} + 136 q^{33} - 44 q^{35} - 256 q^{37} + 528 q^{39} + 240 q^{41} - 1100 q^{43} + 880 q^{45} + 800 q^{47} + 12 q^{49} + 34 q^{51} - 432 q^{53} + 532 q^{55} + 472 q^{57} - 148 q^{59} + 728 q^{61} + 1450 q^{63} - 72 q^{65} - 1032 q^{67} + 1024 q^{69} - 798 q^{71} - 1544 q^{73} - 2974 q^{75} - 656 q^{77} - 758 q^{79} - 20 q^{81} + 244 q^{83} + 136 q^{85} - 1524 q^{87} + 1440 q^{89} - 1104 q^{91} + 2464 q^{93} - 7016 q^{95} - 1344 q^{97} - 86 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 8 * q^5 + 22 * q^7 + 100 * q^9 + 70 * q^11 - 120 * q^13 - 140 * q^15 - 68 * q^17 - 44 * q^19 - 488 * q^21 - 158 * q^23 + 548 * q^25 - 392 * q^27 - 264 * q^29 - 122 * q^31 + 136 * q^33 - 44 * q^35 - 256 * q^37 + 528 * q^39 + 240 * q^41 - 1100 * q^43 + 880 * q^45 + 800 * q^47 + 12 * q^49 + 34 * q^51 - 432 * q^53 + 532 * q^55 + 472 * q^57 - 148 * q^59 + 728 * q^61 + 1450 * q^63 - 72 * q^65 - 1032 * q^67 + 1024 * q^69 - 798 * q^71 - 1544 * q^73 - 2974 * q^75 - 656 * q^77 - 758 * q^79 - 20 * q^81 + 244 * q^83 + 136 * q^85 - 1524 * q^87 + 1440 * q^89 - 1104 * q^91 + 2464 * q^93 - 7016 * q^95 - 1344 * q^97 - 86 * q^99

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