LMFDB - Embedded newform 1458.4.a.i.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1458,4,Mod(1,1458)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1458, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1458.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1458.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-30,0,60,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.0247847884$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 339 x^{13} - 1151 x^{12} + 35865 x^{11} + 180141 x^{10} - 1644266 x^{9} - 10786662 x^{8} + \cdots + 33185995624 $ x^15 - 339x^13 - 1151x^12 + 35865x^11 + 180141x^10 - 1644266x^9 - 10786662x^8 + 31430970x^7 + 304598356x^6 - 45265230x^5 - 3933959154x^4 - 5634080801x^3 + 16258950054x^2 + 47732969400*x + 33185995624
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{24} $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-4.70587$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -3.62116 q^{5} -20.2477 q^{7} -8.00000 q^{8} +7.24232 q^{10} -52.1506 q^{11} +26.4418 q^{13} +40.4955 q^{14} +16.0000 q^{16} +47.5095 q^{17} +122.735 q^{19} -14.4846 q^{20} +104.301 q^{22} -100.464 q^{23} -111.887 q^{25} -52.8835 q^{26} -80.9910 q^{28} -87.0110 q^{29} -227.936 q^{31} -32.0000 q^{32} -95.0189 q^{34} +73.3204 q^{35} -120.659 q^{37} -245.469 q^{38} +28.9693 q^{40} -32.0233 q^{41} -284.285 q^{43} -208.602 q^{44} +200.928 q^{46} +251.077 q^{47} +66.9712 q^{49} +223.774 q^{50} +105.767 q^{52} -747.772 q^{53} +188.846 q^{55} +161.982 q^{56} +174.022 q^{58} -610.068 q^{59} +147.508 q^{61} +455.872 q^{62} +64.0000 q^{64} -95.7499 q^{65} +888.870 q^{67} +190.038 q^{68} -146.641 q^{70} +373.659 q^{71} +631.992 q^{73} +241.317 q^{74} +490.938 q^{76} +1055.93 q^{77} +1061.08 q^{79} -57.9386 q^{80} +64.0467 q^{82} -1216.10 q^{83} -172.039 q^{85} +568.570 q^{86} +417.205 q^{88} +301.617 q^{89} -535.386 q^{91} -401.855 q^{92} -502.154 q^{94} -444.442 q^{95} -451.471 q^{97} -133.942 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 30 q^{2} + 60 q^{4} - 15 q^{5} + 42 q^{7} - 120 q^{8} + 30 q^{10} - 33 q^{11} + 117 q^{13} - 84 q^{14} + 240 q^{16} - 102 q^{17} + 171 q^{19} - 60 q^{20} + 66 q^{22} - 174 q^{23} + 600 q^{25} - 234 q^{26}+ \cdots - 4002 q^{98}+O(q^{100}) $ 15 * q - 30 * q^2 + 60 * q^4 - 15 * q^5 + 42 * q^7 - 120 * q^8 + 30 * q^10 - 33 * q^11 + 117 * q^13 - 84 * q^14 + 240 * q^16 - 102 * q^17 + 171 * q^19 - 60 * q^20 + 66 * q^22 - 174 * q^23 + 600 * q^25 - 234 * q^26 + 168 * q^28 - 573 * q^29 + 372 * q^31 - 480 * q^32 + 204 * q^34 - 624 * q^35 + 555 * q^37 - 342 * q^38 + 120 * q^40 - 852 * q^41 + 1002 * q^43 - 132 * q^44 + 348 * q^46 - 306 * q^47 + 2001 * q^49 - 1200 * q^50 + 468 * q^52 - 897 * q^53 + 1953 * q^55 - 336 * q^56 + 1146 * q^58 - 795 * q^59 + 1587 * q^61 - 744 * q^62 + 960 * q^64 - 1020 * q^65 + 1884 * q^67 - 408 * q^68 + 1248 * q^70 - 120 * q^71 + 2604 * q^73 - 1110 * q^74 + 684 * q^76 - 984 * q^77 + 1944 * q^79 - 240 * q^80 + 1704 * q^82 - 2079 * q^83 + 2916 * q^85 - 2004 * q^86 + 264 * q^88 - 2604 * q^89 + 3399 * q^91 - 696 * q^92 + 612 * q^94 - 1038 * q^95 + 3690 * q^97 - 4002 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−3.62116	−0.323887	−0.161943	−	0.986800i	$-0.551776\pi$
$5$	−3.62116	−0.323887	−0.161943	+	0.986800i	$0.551776\pi$
$6$	0	0
$7$	−20.2477	−1.09328	−0.546638	−	0.837369i	$-0.684093\pi$
$7$	−20.2477	−1.09328	−0.546638	+	0.837369i	$0.684093\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	7.24232	0.229022
$11$	−52.1506	−1.42945	−0.714727	−	0.699403i	$-0.753449\pi$
$11$	−52.1506	−1.42945	−0.714727	+	0.699403i	$0.753449\pi$
$12$	0	0
$13$	26.4418	0.564125	0.282062	−	0.959396i	$-0.408981\pi$
$13$	26.4418	0.564125	0.282062	+	0.959396i	$0.408981\pi$
$14$	40.4955	0.773063
$15$	0	0
$16$	16.0000	0.250000
$17$	47.5095	0.677808	0.338904	−	0.940821i	$-0.389944\pi$
$17$	47.5095	0.677808	0.338904	+	0.940821i	$0.389944\pi$
$18$	0	0
$19$	122.735	1.48196	0.740980	−	0.671527i	$-0.234361\pi$
$19$	122.735	1.48196	0.740980	+	0.671527i	$0.234361\pi$
$20$	−14.4846	−0.161943
$21$	0	0
$22$	104.301	1.01078
$23$	−100.464	−0.910789	−0.455395	−	0.890290i	$-0.650502\pi$
$23$	−100.464	−0.910789	−0.455395	+	0.890290i	$0.650502\pi$
$24$	0	0
$25$	−111.887	−0.895097
$26$	−52.8835	−0.398897
$27$	0	0
$28$	−80.9910	−0.546638
$29$	−87.0110	−0.557156	−0.278578	−	0.960414i	$-0.589863\pi$
$29$	−87.0110	−0.557156	−0.278578	+	0.960414i	$0.589863\pi$
$30$	0	0
$31$	−227.936	−1.32060	−0.660299	−	0.751003i	$-0.729570\pi$
$31$	−227.936	−1.32060	−0.660299	+	0.751003i	$0.729570\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−95.0189	−0.479283
$35$	73.3204	0.354097
$36$	0	0
$37$	−120.659	−0.536113	−0.268056	−	0.963403i	$-0.586381\pi$
$37$	−120.659	−0.536113	−0.268056	+	0.963403i	$0.586381\pi$
$38$	−245.469	−1.04790
$39$	0	0
$40$	28.9693	0.114511
$41$	−32.0233	−0.121981	−0.0609903	−	0.998138i	$-0.519426\pi$
$41$	−32.0233	−0.121981	−0.0609903	+	0.998138i	$0.519426\pi$
$42$	0	0
$43$	−284.285	−1.00821	−0.504105	−	0.863642i	$-0.668178\pi$
$43$	−284.285	−1.00821	−0.504105	+	0.863642i	$0.668178\pi$
$44$	−208.602	−0.714727
$45$	0	0
$46$	200.928	0.644025
$47$	251.077	0.779221	0.389610	−	0.920980i	$-0.372610\pi$
$47$	251.077	0.779221	0.389610	+	0.920980i	$0.372610\pi$
$48$	0	0
$49$	66.9712	0.195251
$50$	223.774	0.632930
$51$	0	0
$52$	105.767	0.282062
$53$	−747.772	−1.93801	−0.969004	−	0.247046i	$-0.920540\pi$
$53$	−747.772	−1.93801	−0.969004	+	0.247046i	$0.920540\pi$
$54$	0	0
$55$	188.846	0.462981
$56$	161.982	0.386531
$57$	0	0
$58$	174.022	0.393969
$59$	−610.068	−1.34617	−0.673086	−	0.739564i	$-0.735032\pi$
$59$	−610.068	−1.34617	−0.673086	+	0.739564i	$0.735032\pi$
$60$	0	0
$61$	147.508	0.309614	0.154807	−	0.987945i	$-0.450524\pi$
$61$	147.508	0.309614	0.154807	+	0.987945i	$0.450524\pi$
$62$	455.872	0.933803
$63$	0	0
$64$	64.0000	0.125000
$65$	−95.7499	−0.182712
$66$	0	0
$67$	888.870	1.62079	0.810394	−	0.585886i	$-0.199253\pi$
$67$	888.870	1.62079	0.810394	+	0.585886i	$0.199253\pi$
$68$	190.038	0.338904
$69$	0	0
$70$	−146.641	−0.250385
$71$	373.659	0.624580	0.312290	−	0.949987i	$-0.398904\pi$
$71$	373.659	0.624580	0.312290	+	0.949987i	$0.398904\pi$
$72$	0	0
$73$	631.992	1.01327	0.506637	−	0.862159i	$-0.330888\pi$
$73$	631.992	1.01327	0.506637	+	0.862159i	$0.330888\pi$
$74$	241.317	0.379089
$75$	0	0
$76$	490.938	0.740980
$77$	1055.93	1.56279
$78$	0	0
$79$	1061.08	1.51115	0.755577	−	0.655060i	$-0.227357\pi$
$79$	1061.08	1.51115	0.755577	+	0.655060i	$0.227357\pi$
$80$	−57.9386	−0.0809716
$81$	0	0
$82$	64.0467	0.0862533
$83$	−1216.10	−1.60825	−0.804125	−	0.594460i	$-0.797366\pi$
$83$	−1216.10	−1.60825	−0.804125	+	0.594460i	$0.797366\pi$
$84$	0	0
$85$	−172.039	−0.219533
$86$	568.570	0.712912
$87$	0	0
$88$	417.205	0.505388
$89$	301.617	0.359229	0.179614	−	0.983737i	$-0.442515\pi$
$89$	301.617	0.359229	0.179614	+	0.983737i	$0.442515\pi$
$90$	0	0
$91$	−535.386	−0.616744
$92$	−401.855	−0.455395
$93$	0	0
$94$	−502.154	−0.550992
$95$	−444.442	−0.479987
$96$	0	0
$97$	−451.471	−0.472576	−0.236288	−	0.971683i	$-0.575931\pi$
$97$	−451.471	−0.472576	−0.236288	+	0.971683i	$0.575931\pi$
$98$	−133.942	−0.138064
$99$	0	0
$100$	−447.549	−0.447549
$101$	−1207.88	−1.18999	−0.594993	−	0.803731i	$-0.702845\pi$
$101$	−1207.88	−1.18999	−0.594993	+	0.803731i	$0.702845\pi$
$102$	0	0
$103$	1241.27	1.18743	0.593717	−	0.804674i	$-0.297660\pi$
$103$	1241.27	1.18743	0.593717	+	0.804674i	$0.297660\pi$
$104$	−211.534	−0.199448
$105$	0	0
$106$	1495.54	1.37038
$107$	1309.46	1.18309	0.591543	−	0.806273i	$-0.298519\pi$
$107$	1309.46	1.18309	0.591543	+	0.806273i	$0.298519\pi$
$108$	0	0
$109$	−1059.01	−0.930590	−0.465295	−	0.885156i	$-0.654052\pi$
$109$	−1059.01	−0.930590	−0.465295	+	0.885156i	$0.654052\pi$
$110$	−377.692	−0.327377
$111$	0	0
$112$	−323.964	−0.273319
$113$	−1385.26	−1.15322	−0.576612	−	0.817018i	$-0.695625\pi$
$113$	−1385.26	−1.15322	−0.576612	+	0.817018i	$0.695625\pi$
$114$	0	0
$115$	363.796	0.294992
$116$	−348.044	−0.278578
$117$	0	0
$118$	1220.14	0.951887
$119$	−961.959	−0.741031
$120$	0	0
$121$	1388.69	1.04334
$122$	−295.016	−0.218930
$123$	0	0
$124$	−911.744	−0.660299
$125$	857.807	0.613797
$126$	0	0
$127$	1190.27	0.831648	0.415824	−	0.909445i	$-0.363493\pi$
$127$	1190.27	0.831648	0.415824	+	0.909445i	$0.363493\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	191.500	0.129197
$131$	343.987	0.229422	0.114711	−	0.993399i	$-0.463406\pi$
$131$	343.987	0.229422	0.114711	+	0.993399i	$0.463406\pi$
$132$	0	0
$133$	−2485.10	−1.62019
$134$	−1777.74	−1.14607
$135$	0	0
$136$	−380.076	−0.239641
$137$	−1760.66	−1.09798	−0.548991	−	0.835829i	$-0.684988\pi$
$137$	−1760.66	−1.09798	−0.548991	+	0.835829i	$0.684988\pi$
$138$	0	0
$139$	3179.31	1.94004	0.970019	−	0.243029i	$-0.0781409\pi$
$139$	3179.31	1.94004	0.970019	+	0.243029i	$0.0781409\pi$
$140$	293.281	0.177049
$141$	0	0
$142$	−747.318	−0.441644
$143$	−1378.95	−0.806391
$144$	0	0
$145$	315.081	0.180455
$146$	−1263.98	−0.716493
$147$	0	0
$148$	−482.635	−0.268056
$149$	1150.64	0.632643	0.316322	−	0.948652i	$-0.397552\pi$
$149$	1150.64	0.632643	0.316322	+	0.948652i	$0.397552\pi$
$150$	0	0
$151$	2862.97	1.54295	0.771474	−	0.636261i	$-0.219520\pi$
$151$	2862.97	1.54295	0.771474	+	0.636261i	$0.219520\pi$
$152$	−981.876	−0.523952
$153$	0	0
$154$	−2111.86	−1.10506
$155$	825.393	0.427724
$156$	0	0
$157$	2973.35	1.51146	0.755731	−	0.654882i	$-0.227282\pi$
$157$	2973.35	1.51146	0.755731	+	0.654882i	$0.227282\pi$
$158$	−2122.17	−1.06855
$159$	0	0
$160$	115.877	0.0572556
$161$	2034.17	0.995743
$162$	0	0
$163$	−644.242	−0.309576	−0.154788	−	0.987948i	$-0.549469\pi$
$163$	−644.242	−0.309576	−0.154788	+	0.987948i	$0.549469\pi$
$164$	−128.093	−0.0609903
$165$	0	0
$166$	2432.21	1.13720
$167$	1266.91	0.587043	0.293522	−	0.955952i	$-0.405173\pi$
$167$	1266.91	0.587043	0.293522	+	0.955952i	$0.405173\pi$
$168$	0	0
$169$	−1497.83	−0.681763
$170$	344.079	0.155233
$171$	0	0
$172$	−1137.14	−0.504105
$173$	1374.97	0.604262	0.302131	−	0.953266i	$-0.402302\pi$
$173$	1374.97	0.604262	0.302131	+	0.953266i	$0.402302\pi$
$174$	0	0
$175$	2265.46	0.978588
$176$	−834.410	−0.357364
$177$	0	0
$178$	−603.235	−0.254013
$179$	1495.97	0.624660	0.312330	−	0.949974i	$-0.398891\pi$
$179$	1495.97	0.624660	0.312330	+	0.949974i	$0.398891\pi$
$180$	0	0
$181$	−2246.24	−0.922439	−0.461219	−	0.887286i	$-0.652588\pi$
$181$	−2246.24	−0.922439	−0.461219	+	0.887286i	$0.652588\pi$
$182$	1070.77	0.436104
$183$	0	0
$184$	803.710	0.322013
$185$	436.925	0.173640
$186$	0	0
$187$	−2477.65	−0.968896
$188$	1004.31	0.389610
$189$	0	0
$190$	888.883	0.339402
$191$	4579.17	1.73475	0.867375	−	0.497655i	$-0.165805\pi$
$191$	4579.17	1.73475	0.867375	+	0.497655i	$0.165805\pi$
$192$	0	0
$193$	1394.34	0.520034	0.260017	−	0.965604i	$-0.416272\pi$
$193$	1394.34	0.520034	0.260017	+	0.965604i	$0.416272\pi$
$194$	902.942	0.334162
$195$	0	0
$196$	267.885	0.0976257
$197$	87.4216	0.0316169	0.0158085	−	0.999875i	$-0.494968\pi$
$197$	87.4216	0.0316169	0.0158085	+	0.999875i	$0.494968\pi$
$198$	0	0
$199$	−4136.77	−1.47361	−0.736803	−	0.676107i	$-0.763666\pi$
$199$	−4136.77	−1.47361	−0.736803	+	0.676107i	$0.763666\pi$
$200$	895.097	0.316465
$201$	0	0
$202$	2415.76	0.841447
$203$	1761.78	0.609125
$204$	0	0
$205$	115.962	0.0395079
$206$	−2482.53	−0.839642
$207$	0	0
$208$	423.068	0.141031
$209$	−6400.68	−2.11839
$210$	0	0
$211$	−4378.88	−1.42869	−0.714347	−	0.699791i	$-0.753276\pi$
$211$	−4378.88	−1.42869	−0.714347	+	0.699791i	$0.753276\pi$
$212$	−2991.09	−0.969004
$213$	0	0
$214$	−2618.92	−0.836569
$215$	1029.44	0.326546
$216$	0	0
$217$	4615.19	1.44378
$218$	2118.01	0.658027
$219$	0	0
$220$	755.383	0.231491
$221$	1256.23	0.382368
$222$	0	0
$223$	1142.73	0.343153	0.171576	−	0.985171i	$-0.445114\pi$
$223$	1142.73	0.343153	0.171576	+	0.985171i	$0.445114\pi$
$224$	647.928	0.193266
$225$	0	0
$226$	2770.52	0.815452
$227$	5245.67	1.53378	0.766889	−	0.641780i	$-0.221804\pi$
$227$	5245.67	1.53378	0.766889	+	0.641780i	$0.221804\pi$
$228$	0	0
$229$	2794.80	0.806486	0.403243	−	0.915093i	$-0.367883\pi$
$229$	2794.80	0.806486	0.403243	+	0.915093i	$0.367883\pi$
$230$	−727.591	−0.208591
$231$	0	0
$232$	696.088	0.196985
$233$	−920.848	−0.258913	−0.129457	−	0.991585i	$-0.541323\pi$
$233$	−920.848	−0.258913	−0.129457	+	0.991585i	$0.541323\pi$
$234$	0	0
$235$	−909.191	−0.252379
$236$	−2440.27	−0.673086
$237$	0	0
$238$	1923.92	0.523988
$239$	−2271.82	−0.614861	−0.307431	−	0.951571i	$-0.599469\pi$
$239$	−2271.82	−0.614861	−0.307431	+	0.951571i	$0.599469\pi$
$240$	0	0
$241$	−508.396	−0.135887	−0.0679433	−	0.997689i	$-0.521644\pi$
$241$	−508.396	−0.135887	−0.0679433	+	0.997689i	$0.521644\pi$
$242$	−2777.37	−0.737753
$243$	0	0
$244$	590.032	0.154807
$245$	−242.514	−0.0632393
$246$	0	0
$247$	3245.32	0.836010
$248$	1823.49	0.466902
$249$	0	0
$250$	−1715.61	−0.434020
$251$	2021.42	0.508331	0.254165	−	0.967161i	$-0.418199\pi$
$251$	2021.42	0.508331	0.254165	+	0.967161i	$0.418199\pi$
$252$	0	0
$253$	5239.25	1.30193
$254$	−2380.54	−0.588064
$255$	0	0
$256$	256.000	0.0625000
$257$	−4920.22	−1.19422	−0.597111	−	0.802159i	$-0.703685\pi$
$257$	−4920.22	−1.19422	−0.597111	+	0.802159i	$0.703685\pi$
$258$	0	0
$259$	2443.07	0.586119
$260$	−383.000	−0.0913562
$261$	0	0
$262$	−687.974	−0.162226
$263$	2950.26	0.691715	0.345858	−	0.938287i	$-0.387588\pi$
$263$	2950.26	0.691715	0.345858	+	0.938287i	$0.387588\pi$
$264$	0	0
$265$	2707.80	0.627695
$266$	4970.20	1.14565
$267$	0	0
$268$	3555.48	0.810394
$269$	1216.81	0.275800	0.137900	−	0.990446i	$-0.455965\pi$
$269$	1216.81	0.275800	0.137900	+	0.990446i	$0.455965\pi$
$270$	0	0
$271$	746.032	0.167226	0.0836129	−	0.996498i	$-0.473354\pi$
$271$	746.032	0.167226	0.0836129	+	0.996498i	$0.473354\pi$
$272$	760.151	0.169452
$273$	0	0
$274$	3521.32	0.776390
$275$	5834.98	1.27950
$276$	0	0
$277$	1445.92	0.313634	0.156817	−	0.987628i	$-0.449877\pi$
$277$	1445.92	0.313634	0.156817	+	0.987628i	$0.449877\pi$
$278$	−6358.61	−1.37181
$279$	0	0
$280$	−586.563	−0.125192
$281$	−1678.49	−0.356336	−0.178168	−	0.984000i	$-0.557017\pi$
$281$	−1678.49	−0.356336	−0.178168	+	0.984000i	$0.557017\pi$
$282$	0	0
$283$	6931.71	1.45600	0.727999	−	0.685578i	$-0.240450\pi$
$283$	6931.71	1.45600	0.727999	+	0.685578i	$0.240450\pi$
$284$	1494.64	0.312290
$285$	0	0
$286$	2757.91	0.570204
$287$	648.400	0.133358
$288$	0	0
$289$	−2655.85	−0.540576
$290$	−630.162	−0.127601
$291$	0	0
$292$	2527.97	0.506637
$293$	−2544.58	−0.507359	−0.253679	−	0.967288i	$-0.581641\pi$
$293$	−2544.58	−0.507359	−0.253679	+	0.967288i	$0.581641\pi$
$294$	0	0
$295$	2209.16	0.436007
$296$	965.270	0.189544
$297$	0	0
$298$	−2301.27	−0.447346
$299$	−2656.44	−0.513799
$300$	0	0
$301$	5756.13	1.10225
$302$	−5725.94	−1.09103
$303$	0	0
$304$	1963.75	0.370490
$305$	−534.150	−0.100280
$306$	0	0
$307$	−1787.40	−0.332288	−0.166144	−	0.986101i	$-0.553132\pi$
$307$	−1787.40	−0.332288	−0.166144	+	0.986101i	$0.553132\pi$
$308$	4223.73	0.781394
$309$	0	0
$310$	−1650.79	−0.302446
$311$	4551.43	0.829864	0.414932	−	0.909852i	$-0.363805\pi$
$311$	4551.43	0.829864	0.414932	+	0.909852i	$0.363805\pi$
$312$	0	0
$313$	306.354	0.0553232	0.0276616	−	0.999617i	$-0.491194\pi$
$313$	306.354	0.0553232	0.0276616	+	0.999617i	$0.491194\pi$
$314$	−5946.71	−1.06876
$315$	0	0
$316$	4244.33	0.755577
$317$	4423.31	0.783715	0.391857	−	0.920026i	$-0.371833\pi$
$317$	4423.31	0.783715	0.391857	+	0.920026i	$0.371833\pi$
$318$	0	0
$319$	4537.68	0.796430
$320$	−231.754	−0.0404858
$321$	0	0
$322$	−4068.33	−0.704097
$323$	5831.05	1.00448
$324$	0	0
$325$	−2958.49	−0.504947
$326$	1288.48	0.218903
$327$	0	0
$328$	256.187	0.0431267
$329$	−5083.75	−0.851903
$330$	0	0
$331$	3467.73	0.575842	0.287921	−	0.957654i	$-0.407036\pi$
$331$	3467.73	0.575842	0.287921	+	0.957654i	$0.407036\pi$
$332$	−4864.42	−0.804125
$333$	0	0
$334$	−2533.81	−0.415102
$335$	−3218.74	−0.524951
$336$	0	0
$337$	−6166.97	−0.996844	−0.498422	−	0.866935i	$-0.666087\pi$
$337$	−6166.97	−0.996844	−0.498422	+	0.866935i	$0.666087\pi$
$338$	2995.67	0.482079
$339$	0	0
$340$	−688.158	−0.109766
$341$	11887.0	1.88773
$342$	0	0
$343$	5588.96	0.879812
$344$	2274.28	0.356456
$345$	0	0
$346$	−2749.95	−0.427278
$347$	867.635	0.134228	0.0671139	−	0.997745i	$-0.478621\pi$
$347$	867.635	0.134228	0.0671139	+	0.997745i	$0.478621\pi$
$348$	0	0
$349$	10191.6	1.56316	0.781582	−	0.623803i	$-0.214413\pi$
$349$	10191.6	1.56316	0.781582	+	0.623803i	$0.214413\pi$
$350$	−4530.93	−0.691966
$351$	0	0
$352$	1668.82	0.252694
$353$	1965.19	0.296308	0.148154	−	0.988964i	$-0.452667\pi$
$353$	1965.19	0.296308	0.148154	+	0.988964i	$0.452667\pi$
$354$	0	0
$355$	−1353.08	−0.202293
$356$	1206.47	0.179614
$357$	0	0
$358$	−2991.94	−0.441701
$359$	6668.35	0.980339	0.490170	−	0.871627i	$-0.336935\pi$
$359$	6668.35	0.980339	0.490170	+	0.871627i	$0.336935\pi$
$360$	0	0
$361$	8204.77	1.19620
$362$	4492.47	0.652263
$363$	0	0
$364$	−2141.54	−0.308372
$365$	−2288.54	−0.328186
$366$	0	0
$367$	−8035.15	−1.14286	−0.571432	−	0.820649i	$-0.693612\pi$
$367$	−8035.15	−1.14286	−0.571432	+	0.820649i	$0.693612\pi$
$368$	−1607.42	−0.227697
$369$	0	0
$370$	−873.850	−0.122782
$371$	15140.7	2.11878
$372$	0	0
$373$	12857.8	1.78486	0.892431	−	0.451183i	$-0.148998\pi$
$373$	12857.8	1.78486	0.892431	+	0.451183i	$0.148998\pi$
$374$	4955.29	0.685113
$375$	0	0
$376$	−2008.62	−0.275496
$377$	−2300.72	−0.314306
$378$	0	0
$379$	−2336.31	−0.316644	−0.158322	−	0.987388i	$-0.550608\pi$
$379$	−2336.31	−0.316644	−0.158322	+	0.987388i	$0.550608\pi$
$380$	−1777.77	−0.239993
$381$	0	0
$382$	−9158.35	−1.22665
$383$	−4666.17	−0.622533	−0.311267	−	0.950323i	$-0.600753\pi$
$383$	−4666.17	−0.622533	−0.311267	+	0.950323i	$0.600753\pi$
$384$	0	0
$385$	−3823.70	−0.506166
$386$	−2788.67	−0.367720
$387$	0	0
$388$	−1805.88	−0.236288
$389$	9639.91	1.25646	0.628230	−	0.778028i	$-0.283780\pi$
$389$	9639.91	1.25646	0.628230	+	0.778028i	$0.283780\pi$
$390$	0	0
$391$	−4772.98	−0.617340
$392$	−535.770	−0.0690318
$393$	0	0
$394$	−174.843	−0.0223565
$395$	−3842.35	−0.489443
$396$	0	0
$397$	−885.563	−0.111952	−0.0559762	−	0.998432i	$-0.517827\pi$
$397$	−885.563	−0.111952	−0.0559762	+	0.998432i	$0.517827\pi$
$398$	8273.54	1.04200
$399$	0	0
$400$	−1790.19	−0.223774
$401$	1629.75	0.202957	0.101479	−	0.994838i	$-0.467643\pi$
$401$	1629.75	0.202957	0.101479	+	0.994838i	$0.467643\pi$
$402$	0	0
$403$	−6027.03	−0.744982
$404$	−4831.52	−0.594993
$405$	0	0
$406$	−3523.55	−0.430717
$407$	6292.42	0.766349
$408$	0	0
$409$	−2781.76	−0.336305	−0.168153	−	0.985761i	$-0.553780\pi$
$409$	−2781.76	−0.336305	−0.168153	+	0.985761i	$0.553780\pi$
$410$	−231.923	−0.0279363
$411$	0	0
$412$	4965.07	0.593717
$413$	12352.5	1.47174
$414$	0	0
$415$	4403.71	0.520891
$416$	−846.136	−0.0997241
$417$	0	0
$418$	12801.4	1.49793
$419$	−10878.0	−1.26832	−0.634161	−	0.773201i	$-0.718654\pi$
$419$	−10878.0	−1.26832	−0.634161	+	0.773201i	$0.718654\pi$
$420$	0	0
$421$	−5262.79	−0.609246	−0.304623	−	0.952473i	$-0.598530\pi$
$421$	−5262.79	−0.609246	−0.304623	+	0.952473i	$0.598530\pi$
$422$	8757.76	1.01024
$423$	0	0
$424$	5982.18	0.685189
$425$	−5315.70	−0.606704
$426$	0	0
$427$	−2986.70	−0.338493
$428$	5237.84	0.591543
$429$	0	0
$430$	−2058.88	−0.230903
$431$	−507.234	−0.0566881	−0.0283441	−	0.999598i	$-0.509023\pi$
$431$	−507.234	−0.0566881	−0.0283441	+	0.999598i	$0.509023\pi$
$432$	0	0
$433$	1544.74	0.171444	0.0857222	−	0.996319i	$-0.472680\pi$
$433$	1544.74	0.171444	0.0857222	+	0.996319i	$0.472680\pi$
$434$	−9230.38	−1.02090
$435$	0	0
$436$	−4236.02	−0.465295
$437$	−12330.4	−1.34975
$438$	0	0
$439$	−8267.85	−0.898868	−0.449434	−	0.893314i	$-0.648374\pi$
$439$	−8267.85	−0.898868	−0.449434	+	0.893314i	$0.648374\pi$
$440$	−1510.77	−0.163689
$441$	0	0
$442$	−2512.47	−0.270375
$443$	−15491.0	−1.66140	−0.830698	−	0.556724i	$-0.812058\pi$
$443$	−15491.0	−1.66140	−0.830698	+	0.556724i	$0.812058\pi$
$444$	0	0
$445$	−1092.21	−0.116349
$446$	−2285.47	−0.242646
$447$	0	0
$448$	−1295.86	−0.136659
$449$	3727.10	0.391743	0.195871	−	0.980630i	$-0.437246\pi$
$449$	3727.10	0.391743	0.195871	+	0.980630i	$0.437246\pi$
$450$	0	0
$451$	1670.04	0.174366
$452$	−5541.04	−0.576612
$453$	0	0
$454$	−10491.3	−1.08454
$455$	1938.72	0.199755
$456$	0	0
$457$	15863.9	1.62381	0.811906	−	0.583789i	$-0.198430\pi$
$457$	15863.9	1.62381	0.811906	+	0.583789i	$0.198430\pi$
$458$	−5589.59	−0.570272
$459$	0	0
$460$	1455.18	0.147496
$461$	7045.89	0.711843	0.355922	−	0.934516i	$-0.384167\pi$
$461$	7045.89	0.711843	0.355922	+	0.934516i	$0.384167\pi$
$462$	0	0
$463$	−4614.93	−0.463227	−0.231614	−	0.972808i	$-0.574400\pi$
$463$	−4614.93	−0.463227	−0.231614	+	0.972808i	$0.574400\pi$
$464$	−1392.18	−0.139289
$465$	0	0
$466$	1841.70	0.183079
$467$	16591.2	1.64401	0.822003	−	0.569483i	$-0.192857\pi$
$467$	16591.2	1.64401	0.822003	+	0.569483i	$0.192857\pi$
$468$	0	0
$469$	−17997.6	−1.77197
$470$	1818.38	0.178459
$471$	0	0
$472$	4880.55	0.475944
$473$	14825.6	1.44119
$474$	0	0
$475$	−13732.4	−1.32650
$476$	−3847.84	−0.370515
$477$	0	0
$478$	4543.64	0.434772
$479$	782.869	0.0746768	0.0373384	−	0.999303i	$-0.488112\pi$
$479$	782.869	0.0746768	0.0373384	+	0.999303i	$0.488112\pi$
$480$	0	0
$481$	−3190.43	−0.302435
$482$	1016.79	0.0960864
$483$	0	0
$484$	5554.74	0.521670
$485$	1634.85	0.153061
$486$	0	0
$487$	−6863.12	−0.638599	−0.319300	−	0.947654i	$-0.603448\pi$
$487$	−6863.12	−0.638599	−0.319300	+	0.947654i	$0.603448\pi$
$488$	−1180.06	−0.109465
$489$	0	0
$490$	485.027	0.0447169
$491$	18986.5	1.74511	0.872557	−	0.488513i	$-0.162461\pi$
$491$	18986.5	1.74511	0.872557	+	0.488513i	$0.162461\pi$
$492$	0	0
$493$	−4133.85	−0.377645
$494$	−6490.63	−0.591149
$495$	0	0
$496$	−3646.98	−0.330149
$497$	−7565.75	−0.682838
$498$	0	0
$499$	1889.80	0.169537	0.0847686	−	0.996401i	$-0.472985\pi$
$499$	1889.80	0.169537	0.0847686	+	0.996401i	$0.472985\pi$
$500$	3431.23	0.306898
$501$	0	0
$502$	−4042.85	−0.359444
$503$	−14200.1	−1.25875	−0.629374	−	0.777102i	$-0.716689\pi$
$503$	−14200.1	−1.25875	−0.629374	+	0.777102i	$0.716689\pi$
$504$	0	0
$505$	4373.93	0.385421
$506$	−10478.5	−0.920605
$507$	0	0
$508$	4761.07	0.415824
$509$	−3810.78	−0.331847	−0.165923	−	0.986139i	$-0.553060\pi$
$509$	−3810.78	−0.331847	−0.165923	+	0.986139i	$0.553060\pi$
$510$	0	0
$511$	−12796.4	−1.10779
$512$	−512.000	−0.0441942
$513$	0	0
$514$	9840.44	0.844442
$515$	−4494.83	−0.384594
$516$	0	0
$517$	−13093.8	−1.11386
$518$	−4886.13	−0.414449
$519$	0	0
$520$	765.999	0.0645986
$521$	−2856.90	−0.240236	−0.120118	−	0.992760i	$-0.538327\pi$
$521$	−2856.90	−0.240236	−0.120118	+	0.992760i	$0.538327\pi$
$522$	0	0
$523$	−4295.63	−0.359148	−0.179574	−	0.983744i	$-0.557472\pi$
$523$	−4295.63	−0.359148	−0.179574	+	0.983744i	$0.557472\pi$
$524$	1375.95	0.114711
$525$	0	0
$526$	−5900.53	−0.489116
$527$	−10829.1	−0.895111
$528$	0	0
$529$	−2074.03	−0.170463
$530$	−5415.61	−0.443847
$531$	0	0
$532$	−9940.39	−0.810095
$533$	−846.753	−0.0688123
$534$	0	0
$535$	−4741.77	−0.383186
$536$	−7110.96	−0.573035
$537$	0	0
$538$	−2433.62	−0.195020
$539$	−3492.59	−0.279103
$540$	0	0
$541$	−10906.1	−0.866713	−0.433357	−	0.901223i	$-0.642671\pi$
$541$	−10906.1	−0.866713	−0.433357	+	0.901223i	$0.642671\pi$
$542$	−1492.06	−0.118247
$543$	0	0
$544$	−1520.30	−0.119821
$545$	3834.83	0.301406
$546$	0	0
$547$	8766.70	0.685260	0.342630	−	0.939471i	$-0.388682\pi$
$547$	8766.70	0.685260	0.342630	+	0.939471i	$0.388682\pi$
$548$	−7042.64	−0.548991
$549$	0	0
$550$	−11670.0	−0.904744
$551$	−10679.3	−0.825683
$552$	0	0
$553$	−21484.5	−1.65211
$554$	−2891.83	−0.221773
$555$	0	0
$556$	12717.2	0.970019
$557$	17472.2	1.32912	0.664562	−	0.747233i	$-0.268618\pi$
$557$	17472.2	1.32912	0.664562	+	0.747233i	$0.268618\pi$
$558$	0	0
$559$	−7516.99	−0.568756
$560$	1173.13	0.0885243
$561$	0	0
$562$	3356.98	0.251967
$563$	10249.2	0.767236	0.383618	−	0.923492i	$-0.374678\pi$
$563$	10249.2	0.767236	0.383618	+	0.923492i	$0.374678\pi$
$564$	0	0
$565$	5016.25	0.373514
$566$	−13863.4	−1.02955
$567$	0	0
$568$	−2989.27	−0.220822
$569$	−9368.76	−0.690262	−0.345131	−	0.938555i	$-0.612165\pi$
$569$	−9368.76	−0.690262	−0.345131	+	0.938555i	$0.612165\pi$
$570$	0	0
$571$	−20528.3	−1.50452	−0.752262	−	0.658864i	$-0.771037\pi$
$571$	−20528.3	−1.50452	−0.752262	+	0.658864i	$0.771037\pi$
$572$	−5515.81	−0.403195
$573$	0	0
$574$	−1296.80	−0.0942986
$575$	11240.6	0.815245
$576$	0	0
$577$	18795.5	1.35609	0.678047	−	0.735018i	$-0.262826\pi$
$577$	18795.5	1.35609	0.678047	+	0.735018i	$0.262826\pi$
$578$	5311.70	0.382245
$579$	0	0
$580$	1260.32	0.0902277
$581$	24623.4	1.75826
$582$	0	0
$583$	38996.8	2.77029
$584$	−5055.93	−0.358247
$585$	0	0
$586$	5089.16	0.358757
$587$	−12756.4	−0.896955	−0.448478	−	0.893794i	$-0.648034\pi$
$587$	−12756.4	−0.896955	−0.448478	+	0.893794i	$0.648034\pi$
$588$	0	0
$589$	−27975.6	−1.95707
$590$	−4418.31	−0.308304
$591$	0	0
$592$	−1930.54	−0.134028
$593$	4347.81	0.301085	0.150542	−	0.988604i	$-0.451898\pi$
$593$	4347.81	0.301085	0.150542	+	0.988604i	$0.451898\pi$
$594$	0	0
$595$	3483.41	0.240010
$596$	4602.55	0.316322
$597$	0	0
$598$	5312.88	0.363311
$599$	7730.40	0.527305	0.263652	−	0.964618i	$-0.415073\pi$
$599$	7730.40	0.527305	0.263652	+	0.964618i	$0.415073\pi$
$600$	0	0
$601$	5990.34	0.406574	0.203287	−	0.979119i	$-0.434838\pi$
$601$	5990.34	0.406574	0.203287	+	0.979119i	$0.434838\pi$
$602$	−11512.3	−0.779409
$603$	0	0
$604$	11451.9	0.771474
$605$	−5028.66	−0.337924
$606$	0	0
$607$	−8643.22	−0.577953	−0.288977	−	0.957336i	$-0.593315\pi$
$607$	−8643.22	−0.577953	−0.288977	+	0.957336i	$0.593315\pi$
$608$	−3927.51	−0.261976
$609$	0	0
$610$	1068.30	0.0709086
$611$	6638.92	0.439578
$612$	0	0
$613$	−15902.2	−1.04777	−0.523887	−	0.851788i	$-0.675518\pi$
$613$	−15902.2	−1.04777	−0.523887	+	0.851788i	$0.675518\pi$
$614$	3574.81	0.234963
$615$	0	0
$616$	−8447.46	−0.552529
$617$	−16869.3	−1.10070	−0.550350	−	0.834934i	$-0.685506\pi$
$617$	−16869.3	−1.10070	−0.550350	+	0.834934i	$0.685506\pi$
$618$	0	0
$619$	−11336.2	−0.736090	−0.368045	−	0.929808i	$-0.619973\pi$
$619$	−11336.2	−0.736090	−0.368045	+	0.929808i	$0.619973\pi$
$620$	3301.57	0.213862
$621$	0	0
$622$	−9102.85	−0.586802
$623$	−6107.07	−0.392736
$624$	0	0
$625$	10879.6	0.696297
$626$	−612.708	−0.0391194
$627$	0	0
$628$	11893.4	0.755731
$629$	−5732.43	−0.363382
$630$	0	0
$631$	−4497.55	−0.283748	−0.141874	−	0.989885i	$-0.545313\pi$
$631$	−4497.55	−0.283748	−0.141874	+	0.989885i	$0.545313\pi$
$632$	−8488.66	−0.534274
$633$	0	0
$634$	−8846.61	−0.554170
$635$	−4310.16	−0.269360
$636$	0	0
$637$	1770.84	0.110146
$638$	−9075.35	−0.563161
$639$	0	0
$640$	463.509	0.0286278
$641$	1828.88	0.112693	0.0563467	−	0.998411i	$-0.482055\pi$
$641$	1828.88	0.112693	0.0563467	+	0.998411i	$0.482055\pi$
$642$	0	0
$643$	−7174.69	−0.440034	−0.220017	−	0.975496i	$-0.570611\pi$
$643$	−7174.69	−0.440034	−0.220017	+	0.975496i	$0.570611\pi$
$644$	8136.66	0.497872
$645$	0	0
$646$	−11662.1	−0.710278
$647$	25129.0	1.52693	0.763465	−	0.645849i	$-0.223496\pi$
$647$	25129.0	1.52693	0.763465	+	0.645849i	$0.223496\pi$
$648$	0	0
$649$	31815.4	1.92429
$650$	5916.99	0.357051
$651$	0	0
$652$	−2576.97	−0.154788
$653$	−11805.4	−0.707475	−0.353737	−	0.935345i	$-0.615089\pi$
$653$	−11805.4	−0.707475	−0.353737	+	0.935345i	$0.615089\pi$
$654$	0	0
$655$	−1245.63	−0.0743067
$656$	−512.373	−0.0304951
$657$	0	0
$658$	10167.5	0.602386
$659$	22560.4	1.33358	0.666791	−	0.745245i	$-0.267668\pi$
$659$	22560.4	1.33358	0.666791	+	0.745245i	$0.267668\pi$
$660$	0	0
$661$	−24301.4	−1.42997	−0.714987	−	0.699137i	$-0.753568\pi$
$661$	−24301.4	−1.42997	−0.714987	+	0.699137i	$0.753568\pi$
$662$	−6935.46	−0.407182
$663$	0	0
$664$	9728.84	0.568602
$665$	8998.94	0.524758
$666$	0	0
$667$	8741.45	0.507452
$668$	5067.63	0.293522
$669$	0	0
$670$	6437.49	0.371197
$671$	−7692.63	−0.442579
$672$	0	0
$673$	13809.4	0.790956	0.395478	−	0.918475i	$-0.370579\pi$
$673$	13809.4	0.790956	0.395478	+	0.918475i	$0.370579\pi$
$674$	12333.9	0.704875
$675$	0	0
$676$	−5991.33	−0.340882
$677$	1886.30	0.107085	0.0535423	−	0.998566i	$-0.482949\pi$
$677$	1886.30	0.107085	0.0535423	+	0.998566i	$0.482949\pi$
$678$	0	0
$679$	9141.27	0.516656
$680$	1376.32	0.0776166
$681$	0	0
$682$	−23774.0	−1.33483
$683$	−19956.8	−1.11804	−0.559022	−	0.829153i	$-0.688823\pi$
$683$	−19956.8	−1.11804	−0.559022	+	0.829153i	$0.688823\pi$
$684$	0	0
$685$	6375.64	0.355621
$686$	−11177.9	−0.622121
$687$	0	0
$688$	−4548.56	−0.252053
$689$	−19772.4	−1.09328
$690$	0	0
$691$	−19652.6	−1.08194	−0.540970	−	0.841042i	$-0.681943\pi$
$691$	−19652.6	−1.08194	−0.540970	+	0.841042i	$0.681943\pi$
$692$	5499.90	0.302131
$693$	0	0
$694$	−1735.27	−0.0949134
$695$	−11512.8	−0.628352
$696$	0	0
$697$	−1521.41	−0.0826794
$698$	−20383.2	−1.10532
$699$	0	0
$700$	9061.85	0.489294
$701$	18729.0	1.00911	0.504554	−	0.863380i	$-0.331657\pi$
$701$	18729.0	1.00911	0.504554	+	0.863380i	$0.331657\pi$
$702$	0	0
$703$	−14809.0	−0.794497
$704$	−3337.64	−0.178682
$705$	0	0
$706$	−3930.39	−0.209521
$707$	24456.9	1.30098
$708$	0	0
$709$	−7655.20	−0.405497	−0.202748	−	0.979231i	$-0.564987\pi$
$709$	−7655.20	−0.405497	−0.202748	+	0.979231i	$0.564987\pi$
$710$	2706.16	0.143043
$711$	0	0
$712$	−2412.94	−0.127007
$713$	22899.3	1.20279
$714$	0	0
$715$	4993.41	0.261179
$716$	5983.88	0.312330
$717$	0	0
$718$	−13336.7	−0.693205
$719$	−18128.9	−0.940326	−0.470163	−	0.882580i	$-0.655805\pi$
$719$	−18128.9	−0.940326	−0.470163	+	0.882580i	$0.655805\pi$
$720$	0	0
$721$	−25132.9	−1.29819
$722$	−16409.5	−0.845844
$723$	0	0
$724$	−8984.95	−0.461219
$725$	9735.41	0.498709
$726$	0	0
$727$	−12803.2	−0.653158	−0.326579	−	0.945170i	$-0.605896\pi$
$727$	−12803.2	−0.653158	−0.326579	+	0.945170i	$0.605896\pi$
$728$	4283.09	0.218052
$729$	0	0
$730$	4577.09	0.232063
$731$	−13506.2	−0.683373
$732$	0	0
$733$	−339.698	−0.0171174	−0.00855868	−	0.999963i	$-0.502724\pi$
$733$	−339.698	−0.0171174	−0.00855868	+	0.999963i	$0.502724\pi$
$734$	16070.3	0.808127
$735$	0	0
$736$	3214.84	0.161006
$737$	−46355.1	−2.31684
$738$	0	0
$739$	12426.9	0.618582	0.309291	−	0.950967i	$-0.399908\pi$
$739$	12426.9	0.618582	0.309291	+	0.950967i	$0.399908\pi$
$740$	1747.70	0.0868199
$741$	0	0
$742$	−30281.4	−1.49820
$743$	15019.3	0.741592	0.370796	−	0.928714i	$-0.379085\pi$
$743$	15019.3	0.741592	0.370796	+	0.928714i	$0.379085\pi$
$744$	0	0
$745$	−4166.64	−0.204905
$746$	−25715.7	−1.26209
$747$	0	0
$748$	−9910.59	−0.484448
$749$	−26513.6	−1.29344
$750$	0	0
$751$	−30388.6	−1.47656	−0.738280	−	0.674494i	$-0.764362\pi$
$751$	−30388.6	−1.47656	−0.738280	+	0.674494i	$0.764362\pi$
$752$	4017.24	0.194805
$753$	0	0
$754$	4601.45	0.222248
$755$	−10367.3	−0.499740
$756$	0	0
$757$	19500.3	0.936262	0.468131	−	0.883659i	$-0.344928\pi$
$757$	19500.3	0.936262	0.468131	+	0.883659i	$0.344928\pi$
$758$	4672.61	0.223901
$759$	0	0
$760$	3555.53	0.169701
$761$	−3455.08	−0.164581	−0.0822907	−	0.996608i	$-0.526224\pi$
$761$	−3455.08	−0.164581	−0.0822907	+	0.996608i	$0.526224\pi$
$762$	0	0
$763$	21442.5	1.01739
$764$	18316.7	0.867375
$765$	0	0
$766$	9332.34	0.440197
$767$	−16131.3	−0.759409
$768$	0	0
$769$	4888.93	0.229258	0.114629	−	0.993408i	$-0.463432\pi$
$769$	4888.93	0.229258	0.114629	+	0.993408i	$0.463432\pi$
$770$	7647.40	0.357913
$771$	0	0
$772$	5577.35	0.260017
$773$	8683.60	0.404046	0.202023	−	0.979381i	$-0.435248\pi$
$773$	8683.60	0.404046	0.202023	+	0.979381i	$0.435248\pi$
$774$	0	0
$775$	25503.1	1.18206
$776$	3611.77	0.167081
$777$	0	0
$778$	−19279.8	−0.888451
$779$	−3930.37	−0.180770
$780$	0	0
$781$	−19486.5	−0.892808
$782$	9545.96	0.436525
$783$	0	0
$784$	1071.54	0.0488128
$785$	−10767.0	−0.489542
$786$	0	0
$787$	−14830.2	−0.671716	−0.335858	−	0.941913i	$-0.609026\pi$
$787$	−14830.2	−0.671716	−0.335858	+	0.941913i	$0.609026\pi$
$788$	349.686	0.0158085
$789$	0	0
$790$	7684.71	0.346088
$791$	28048.4	1.26079
$792$	0	0
$793$	3900.37	0.174661
$794$	1771.13	0.0791623
$795$	0	0
$796$	−16547.1	−0.736803
$797$	−38300.7	−1.70223	−0.851116	−	0.524977i	$-0.824074\pi$
$797$	−38300.7	−1.70223	−0.851116	+	0.524977i	$0.824074\pi$
$798$	0	0
$799$	11928.5	0.528162
$800$	3580.39	0.158232
$801$	0	0
$802$	−3259.50	−0.143512
$803$	−32958.8	−1.44843
$804$	0	0
$805$	−7366.04	−0.322508
$806$	12054.1	0.526782
$807$	0	0
$808$	9663.04	0.420724
$809$	−28776.2	−1.25058	−0.625290	−	0.780393i	$-0.715019\pi$
$809$	−28776.2	−1.25058	−0.625290	+	0.780393i	$0.715019\pi$
$810$	0	0
$811$	4006.85	0.173489	0.0867444	−	0.996231i	$-0.472354\pi$
$811$	4006.85	0.173489	0.0867444	+	0.996231i	$0.472354\pi$
$812$	7047.11	0.304563
$813$	0	0
$814$	−12584.8	−0.541890
$815$	2332.90	0.100268
$816$	0	0
$817$	−34891.6	−1.49413
$818$	5563.51	0.237804
$819$	0	0
$820$	463.847	0.0197539
$821$	12196.4	0.518464	0.259232	−	0.965815i	$-0.416531\pi$
$821$	12196.4	0.518464	0.259232	+	0.965815i	$0.416531\pi$
$822$	0	0
$823$	25125.7	1.06419	0.532093	−	0.846686i	$-0.321406\pi$
$823$	25125.7	1.06419	0.532093	+	0.846686i	$0.321406\pi$
$824$	−9930.14	−0.419821
$825$	0	0
$826$	−24705.0	−1.04068
$827$	21592.3	0.907904	0.453952	−	0.891026i	$-0.350014\pi$
$827$	21592.3	0.907904	0.453952	+	0.891026i	$0.350014\pi$
$828$	0	0
$829$	−4498.87	−0.188483	−0.0942414	−	0.995549i	$-0.530043\pi$
$829$	−4498.87	−0.188483	−0.0942414	+	0.995549i	$0.530043\pi$
$830$	−8807.42	−0.368325
$831$	0	0
$832$	1692.27	0.0705156
$833$	3181.77	0.132343
$834$	0	0
$835$	−4587.67	−0.190135
$836$	−25602.7	−1.05920
$837$	0	0
$838$	21756.1	0.896839
$839$	−4651.63	−0.191409	−0.0957044	−	0.995410i	$-0.530510\pi$
$839$	−4651.63	−0.191409	−0.0957044	+	0.995410i	$0.530510\pi$
$840$	0	0
$841$	−16818.1	−0.689577
$842$	10525.6	0.430802
$843$	0	0
$844$	−17515.5	−0.714347
$845$	5423.90	0.220814
$846$	0	0
$847$	−28117.8	−1.14066
$848$	−11964.4	−0.484502
$849$	0	0
$850$	10631.4	0.429005
$851$	12121.8	0.488286
$852$	0	0
$853$	−526.648	−0.0211396	−0.0105698	−	0.999944i	$-0.503365\pi$
$853$	−526.648	−0.0211396	−0.0105698	+	0.999944i	$0.503365\pi$
$854$	5973.41	0.239351
$855$	0	0
$856$	−10475.7	−0.418284
$857$	33910.0	1.35162	0.675812	−	0.737074i	$-0.263793\pi$
$857$	33910.0	1.35162	0.675812	+	0.737074i	$0.263793\pi$
$858$	0	0
$859$	−35076.1	−1.39323	−0.696614	−	0.717447i	$-0.745311\pi$
$859$	−35076.1	−1.39323	−0.696614	+	0.717447i	$0.745311\pi$
$860$	4117.77	0.163273
$861$	0	0
$862$	1014.47	0.0400846
$863$	41934.5	1.65408	0.827039	−	0.562145i	$-0.190024\pi$
$863$	41934.5	1.65408	0.827039	+	0.562145i	$0.190024\pi$
$864$	0	0
$865$	−4979.00	−0.195712
$866$	−3089.48	−0.121230
$867$	0	0
$868$	18460.8	0.721888
$869$	−55336.1	−2.16013
$870$	0	0
$871$	23503.3	0.914327
$872$	8472.05	0.329013
$873$	0	0
$874$	24660.8	0.954419
$875$	−17368.7	−0.671049
$876$	0	0
$877$	−5432.70	−0.209178	−0.104589	−	0.994516i	$-0.533353\pi$
$877$	−5432.70	−0.209178	−0.104589	+	0.994516i	$0.533353\pi$
$878$	16535.7	0.635596
$879$	0	0
$880$	3021.53	0.115745
$881$	−30295.4	−1.15854	−0.579272	−	0.815134i	$-0.696663\pi$
$881$	−30295.4	−1.15854	−0.579272	+	0.815134i	$0.696663\pi$
$882$	0	0
$883$	−982.972	−0.0374628	−0.0187314	−	0.999825i	$-0.505963\pi$
$883$	−982.972	−0.0374628	−0.0187314	+	0.999825i	$0.505963\pi$
$884$	5024.93	0.191184
$885$	0	0
$886$	30981.9	1.17478
$887$	43878.6	1.66099	0.830496	−	0.557024i	$-0.188057\pi$
$887$	43878.6	1.66099	0.830496	+	0.557024i	$0.188057\pi$
$888$	0	0
$889$	−24100.3	−0.909220
$890$	2184.41	0.0822715
$891$	0	0
$892$	4570.93	0.171576
$893$	30815.8	1.15477
$894$	0	0
$895$	−5417.15	−0.202319
$896$	2591.71	0.0966328
$897$	0	0
$898$	−7454.19	−0.277004
$899$	19832.9	0.735779
$900$	0	0
$901$	−35526.2	−1.31360
$902$	−3340.07	−0.123295
$903$	0	0
$904$	11082.1	0.407726
$905$	8133.99	0.298766
$906$	0	0
$907$	26008.0	0.952128	0.476064	−	0.879411i	$-0.342063\pi$
$907$	26008.0	0.952128	0.476064	+	0.879411i	$0.342063\pi$
$908$	20982.7	0.766889
$909$	0	0
$910$	−3877.44	−0.141248
$911$	39452.4	1.43481	0.717407	−	0.696654i	$-0.245329\pi$
$911$	39452.4	1.43481	0.717407	+	0.696654i	$0.245329\pi$
$912$	0	0
$913$	63420.6	2.29892
$914$	−31727.8	−1.14821
$915$	0	0
$916$	11179.2	0.403243
$917$	−6964.96	−0.250821
$918$	0	0
$919$	1286.40	0.0461745	0.0230872	−	0.999733i	$-0.492650\pi$
$919$	1286.40	0.0461745	0.0230872	+	0.999733i	$0.492650\pi$
$920$	−2910.37	−0.104296
$921$	0	0
$922$	−14091.8	−0.503349
$923$	9880.20	0.352341
$924$	0	0
$925$	13500.2	0.479873
$926$	9229.87	0.327551
$927$	0	0
$928$	2784.35	0.0984923
$929$	35390.7	1.24987	0.624937	−	0.780675i	$-0.285125\pi$
$929$	35390.7	1.24987	0.624937	+	0.780675i	$0.285125\pi$
$930$	0	0
$931$	8219.68	0.289355
$932$	−3683.39	−0.129457
$933$	0	0
$934$	−33182.5	−1.16249
$935$	8971.96	0.313812
$936$	0	0
$937$	43283.5	1.50908	0.754541	−	0.656253i	$-0.227860\pi$
$937$	43283.5	1.50908	0.754541	+	0.656253i	$0.227860\pi$
$938$	35995.2	1.25297
$939$	0	0
$940$	−3636.76	−0.126190
$941$	−49221.1	−1.70517	−0.852583	−	0.522592i	$-0.824965\pi$
$941$	−49221.1	−1.70517	−0.852583	+	0.522592i	$0.824965\pi$
$942$	0	0
$943$	3217.19	0.111099
$944$	−9761.10	−0.336543
$945$	0	0
$946$	−29651.3	−1.01908
$947$	5252.21	0.180226	0.0901129	−	0.995932i	$-0.471277\pi$
$947$	5252.21	0.180226	0.0901129	+	0.995932i	$0.471277\pi$
$948$	0	0
$949$	16711.0	0.571613
$950$	27464.8	0.937976
$951$	0	0
$952$	7695.68	0.261994
$953$	24759.7	0.841599	0.420800	−	0.907154i	$-0.361750\pi$
$953$	24759.7	0.841599	0.420800	+	0.907154i	$0.361750\pi$
$954$	0	0
$955$	−16581.9	−0.561862
$956$	−9087.28	−0.307431
$957$	0	0
$958$	−1565.74	−0.0528045
$959$	35649.4	1.20040
$960$	0	0
$961$	22163.8	0.743977
$962$	6380.86	0.213854
$963$	0	0
$964$	−2033.58	−0.0679433
$965$	−5049.12	−0.168432
$966$	0	0
$967$	24572.7	0.817171	0.408586	−	0.912720i	$-0.366022\pi$
$967$	24572.7	0.817171	0.408586	+	0.912720i	$0.366022\pi$
$968$	−11109.5	−0.368876
$969$	0	0
$970$	−3269.70	−0.108231
$971$	−48144.1	−1.59116	−0.795580	−	0.605849i	$-0.792834\pi$
$971$	−48144.1	−1.59116	−0.795580	+	0.605849i	$0.792834\pi$
$972$	0	0
$973$	−64373.8	−2.12100
$974$	13726.2	0.451558
$975$	0	0
$976$	2360.13	0.0774035
$977$	−28410.3	−0.930323	−0.465161	−	0.885226i	$-0.654004\pi$
$977$	−28410.3	−0.930323	−0.465161	+	0.885226i	$0.654004\pi$
$978$	0	0
$979$	−15729.5	−0.513501
$980$	−970.055	−0.0316197
$981$	0	0
$982$	−37973.1	−1.23398
$983$	−8667.36	−0.281227	−0.140613	−	0.990065i	$-0.544907\pi$
$983$	−8667.36	−0.281227	−0.140613	+	0.990065i	$0.544907\pi$
$984$	0	0
$985$	−316.568	−0.0102403
$986$	8267.69	0.267035
$987$	0	0
$988$	12981.3	0.418005
$989$	28560.3	0.918267
$990$	0	0
$991$	50964.8	1.63365	0.816826	−	0.576884i	$-0.195732\pi$
$991$	50964.8	1.63365	0.816826	+	0.576884i	$0.195732\pi$
$992$	7293.95	0.233451
$993$	0	0
$994$	15131.5	0.482839
$995$	14979.9	0.477281
$996$	0	0
$997$	36469.5	1.15848	0.579238	−	0.815159i	$-0.303350\pi$
$997$	36469.5	1.15848	0.579238	+	0.815159i	$0.303350\pi$
$998$	−3779.60	−0.119881
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.4.a.i.1.7		15
3.2	odd	2		1458.4.a.j.1.9		15
27.4	even	9		54.4.e.b.43.5	✓	30
27.7	even	9		54.4.e.b.49.5	yes	30
27.20	odd	18		162.4.e.b.37.3		30
27.23	odd	18		162.4.e.b.127.3		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.e.b.43.5	✓	30	27.4	even	9
54.4.e.b.49.5	yes	30	27.7	even	9
162.4.e.b.37.3		30	27.20	odd	18
162.4.e.b.127.3		30	27.23	odd	18
1458.4.a.i.1.7		15	1.1	even	1	trivial
1458.4.a.j.1.9		15	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -3.62116 q^{5} -20.2477 q^{7} -8.00000 q^{8} +7.24232 q^{10} -52.1506 q^{11} +26.4418 q^{13} +40.4955 q^{14} +16.0000 q^{16} +47.5095 q^{17} +122.735 q^{19} -14.4846 q^{20} +104.301 q^{22} -100.464 q^{23} -111.887 q^{25} -52.8835 q^{26} -80.9910 q^{28} -87.0110 q^{29} -227.936 q^{31} -32.0000 q^{32} -95.0189 q^{34} +73.3204 q^{35} -120.659 q^{37} -245.469 q^{38} +28.9693 q^{40} -32.0233 q^{41} -284.285 q^{43} -208.602 q^{44} +200.928 q^{46} +251.077 q^{47} +66.9712 q^{49} +223.774 q^{50} +105.767 q^{52} -747.772 q^{53} +188.846 q^{55} +161.982 q^{56} +174.022 q^{58} -610.068 q^{59} +147.508 q^{61} +455.872 q^{62} +64.0000 q^{64} -95.7499 q^{65} +888.870 q^{67} +190.038 q^{68} -146.641 q^{70} +373.659 q^{71} +631.992 q^{73} +241.317 q^{74} +490.938 q^{76} +1055.93 q^{77} +1061.08 q^{79} -57.9386 q^{80} +64.0467 q^{82} -1216.10 q^{83} -172.039 q^{85} +568.570 q^{86} +417.205 q^{88} +301.617 q^{89} -535.386 q^{91} -401.855 q^{92} -502.154 q^{94} -444.442 q^{95} -451.471 q^{97} -133.942 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 30 q^{2} + 60 q^{4} - 15 q^{5} + 42 q^{7} - 120 q^{8} + 30 q^{10} - 33 q^{11} + 117 q^{13} - 84 q^{14} + 240 q^{16} - 102 q^{17} + 171 q^{19} - 60 q^{20} + 66 q^{22} - 174 q^{23} + 600 q^{25} - 234 q^{26}+ \cdots - 4002 q^{98}+O(q^{100}) \) 15 * q - 30 * q^2 + 60 * q^4 - 15 * q^5 + 42 * q^7 - 120 * q^8 + 30 * q^10 - 33 * q^11 + 117 * q^13 - 84 * q^14 + 240 * q^16 - 102 * q^17 + 171 * q^19 - 60 * q^20 + 66 * q^22 - 174 * q^23 + 600 * q^25 - 234 * q^26 + 168 * q^28 - 573 * q^29 + 372 * q^31 - 480 * q^32 + 204 * q^34 - 624 * q^35 + 555 * q^37 - 342 * q^38 + 120 * q^40 - 852 * q^41 + 1002 * q^43 - 132 * q^44 + 348 * q^46 - 306 * q^47 + 2001 * q^49 - 1200 * q^50 + 468 * q^52 - 897 * q^53 + 1953 * q^55 - 336 * q^56 + 1146 * q^58 - 795 * q^59 + 1587 * q^61 - 744 * q^62 + 960 * q^64 - 1020 * q^65 + 1884 * q^67 - 408 * q^68 + 1248 * q^70 - 120 * q^71 + 2604 * q^73 - 1110 * q^74 + 684 * q^76 - 984 * q^77 + 1944 * q^79 - 240 * q^80 + 1704 * q^82 - 2079 * q^83 + 2916 * q^85 - 2004 * q^86 + 264 * q^88 - 2604 * q^89 + 3399 * q^91 - 696 * q^92 + 612 * q^94 - 1038 * q^95 + 3690 * q^97 - 4002 * q^98

Introduction

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