LMFDB - Embedded newform 242.4.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,4,Mod(1,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("242.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	242.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:	$14.2784622214$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.19378$ of defining polynomial
Character	$\chi$	$=$	242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +7.57575 q^{3} +4.00000 q^{4} +5.40810 q^{5} +15.1515 q^{6} +22.1498 q^{7} +8.00000 q^{8} +30.3919 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +7.57575 q^{3} +4.00000 q^{4} +5.40810 q^{5} +15.1515 q^{6} +22.1498 q^{7} +8.00000 q^{8} +30.3919 q^{9} +10.8162 q^{10} +30.3030 q^{12} -76.8269 q^{13} +44.2996 q^{14} +40.9704 q^{15} +16.0000 q^{16} -59.2883 q^{17} +60.7839 q^{18} -95.2626 q^{19} +21.6324 q^{20} +167.801 q^{21} +142.484 q^{23} +60.6060 q^{24} -95.7524 q^{25} -153.654 q^{26} +25.6965 q^{27} +88.5993 q^{28} +20.4183 q^{29} +81.9408 q^{30} +213.304 q^{31} +32.0000 q^{32} -118.577 q^{34} +119.788 q^{35} +121.568 q^{36} -145.578 q^{37} -190.525 q^{38} -582.021 q^{39} +43.2648 q^{40} -82.8326 q^{41} +335.603 q^{42} +151.373 q^{43} +164.363 q^{45} +284.968 q^{46} +90.3643 q^{47} +121.212 q^{48} +147.615 q^{49} -191.505 q^{50} -449.153 q^{51} -307.308 q^{52} -234.849 q^{53} +51.3931 q^{54} +177.199 q^{56} -721.685 q^{57} +40.8366 q^{58} +302.497 q^{59} +163.882 q^{60} +149.279 q^{61} +426.608 q^{62} +673.176 q^{63} +64.0000 q^{64} -415.488 q^{65} +826.236 q^{67} -237.153 q^{68} +1079.42 q^{69} +239.577 q^{70} -898.965 q^{71} +243.136 q^{72} +137.993 q^{73} -291.155 q^{74} -725.396 q^{75} -381.050 q^{76} -1164.04 q^{78} +304.459 q^{79} +86.5296 q^{80} -625.912 q^{81} -165.665 q^{82} +764.294 q^{83} +671.206 q^{84} -320.637 q^{85} +302.746 q^{86} +154.684 q^{87} -313.100 q^{89} +328.725 q^{90} -1701.70 q^{91} +569.936 q^{92} +1615.94 q^{93} +180.729 q^{94} -515.189 q^{95} +242.424 q^{96} -582.505 q^{97} +295.229 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9} + 50 q^{10} + 16 q^{12} - 41 q^{13} + 6 q^{14} + 68 q^{15} + 64 q^{16} + 52 q^{17} + 204 q^{18} + 16 q^{19} + 100 q^{20} + 25 q^{21} + 314 q^{23} + 32 q^{24} - 21 q^{25} - 82 q^{26} + 286 q^{27} + 12 q^{28} + 561 q^{29} + 136 q^{30} + 199 q^{31} + 128 q^{32} + 104 q^{34} - 714 q^{35} + 408 q^{36} + 357 q^{37} + 32 q^{38} - 1038 q^{39} + 200 q^{40} + 32 q^{41} + 50 q^{42} - 721 q^{43} + 1326 q^{45} + 628 q^{46} + 403 q^{47} + 64 q^{48} + 823 q^{49} - 42 q^{50} - 174 q^{51} - 164 q^{52} - 133 q^{53} + 572 q^{54} + 24 q^{56} - 1031 q^{57} + 1122 q^{58} + 1016 q^{59} + 272 q^{60} - 919 q^{61} + 398 q^{62} - 1367 q^{63} + 256 q^{64} + 69 q^{65} + 289 q^{67} + 208 q^{68} - 1620 q^{69} - 1428 q^{70} - 1205 q^{71} + 816 q^{72} - 1234 q^{73} + 714 q^{74} - 911 q^{75} + 64 q^{76} - 2076 q^{78} - 603 q^{79} + 400 q^{80} - 1400 q^{81} + 64 q^{82} + 1514 q^{83} + 100 q^{84} + 717 q^{85} - 1442 q^{86} - 1061 q^{87} - 1101 q^{89} + 2652 q^{90} - 2306 q^{91} + 1256 q^{92} - 2298 q^{93} + 806 q^{94} - 1766 q^{95} + 128 q^{96} + 2116 q^{97} + 1646 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9 + 50 * q^10 + 16 * q^12 - 41 * q^13 + 6 * q^14 + 68 * q^15 + 64 * q^16 + 52 * q^17 + 204 * q^18 + 16 * q^19 + 100 * q^20 + 25 * q^21 + 314 * q^23 + 32 * q^24 - 21 * q^25 - 82 * q^26 + 286 * q^27 + 12 * q^28 + 561 * q^29 + 136 * q^30 + 199 * q^31 + 128 * q^32 + 104 * q^34 - 714 * q^35 + 408 * q^36 + 357 * q^37 + 32 * q^38 - 1038 * q^39 + 200 * q^40 + 32 * q^41 + 50 * q^42 - 721 * q^43 + 1326 * q^45 + 628 * q^46 + 403 * q^47 + 64 * q^48 + 823 * q^49 - 42 * q^50 - 174 * q^51 - 164 * q^52 - 133 * q^53 + 572 * q^54 + 24 * q^56 - 1031 * q^57 + 1122 * q^58 + 1016 * q^59 + 272 * q^60 - 919 * q^61 + 398 * q^62 - 1367 * q^63 + 256 * q^64 + 69 * q^65 + 289 * q^67 + 208 * q^68 - 1620 * q^69 - 1428 * q^70 - 1205 * q^71 + 816 * q^72 - 1234 * q^73 + 714 * q^74 - 911 * q^75 + 64 * q^76 - 2076 * q^78 - 603 * q^79 + 400 * q^80 - 1400 * q^81 + 64 * q^82 + 1514 * q^83 + 100 * q^84 + 717 * q^85 - 1442 * q^86 - 1061 * q^87 - 1101 * q^89 + 2652 * q^90 - 2306 * q^91 + 1256 * q^92 - 2298 * q^93 + 806 * q^94 - 1766 * q^95 + 128 * q^96 + 2116 * q^97 + 1646 * q^98

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	7.57575	1.45795	0.728977	−	0.684539i	$-0.239996\pi$
$3$	7.57575	1.45795	0.728977	+	0.684539i	$0.239996\pi$
$4$	4.00000	0.500000
$5$	5.40810	0.483715	0.241858	−	0.970312i	$-0.422243\pi$
$5$	5.40810	0.483715	0.241858	+	0.970312i	$0.422243\pi$
$6$	15.1515	1.03093
$7$	22.1498	1.19598	0.597989	−	0.801504i	$-0.295967\pi$
$7$	22.1498	1.19598	0.597989	+	0.801504i	$0.295967\pi$
$8$	8.00000	0.353553
$9$	30.3919	1.12563
$10$	10.8162	0.342038
$11$	0	0
$11$	0	0
$12$	30.3030	0.728977
$13$	−76.8269	−1.63907	−0.819537	−	0.573027i	$-0.805769\pi$
$13$	−76.8269	−1.63907	−0.819537	+	0.573027i	$0.805769\pi$
$14$	44.2996	0.845684
$15$	40.9704	0.705234
$16$	16.0000	0.250000
$17$	−59.2883	−0.845855	−0.422927	−	0.906164i	$-0.638997\pi$
$17$	−59.2883	−0.845855	−0.422927	+	0.906164i	$0.638997\pi$
$18$	60.7839	0.795939
$19$	−95.2626	−1.15025	−0.575124	−	0.818066i	$-0.695046\pi$
$19$	−95.2626	−1.15025	−0.575124	+	0.818066i	$0.695046\pi$
$20$	21.6324	0.241858
$21$	167.801	1.74368
$22$	0	0
$23$	142.484	1.29174	0.645868	−	0.763449i	$-0.276495\pi$
$23$	142.484	1.29174	0.645868	+	0.763449i	$0.276495\pi$
$24$	60.6060	0.515464
$25$	−95.7524	−0.766020
$26$	−153.654	−1.15900
$27$	25.6965	0.183159
$28$	88.5993	0.597989
$29$	20.4183	0.130744	0.0653721	−	0.997861i	$-0.479177\pi$
$29$	20.4183	0.130744	0.0653721	+	0.997861i	$0.479177\pi$
$30$	81.9408	0.498676
$31$	213.304	1.23582	0.617912	−	0.786247i	$-0.287979\pi$
$31$	213.304	1.23582	0.617912	+	0.786247i	$0.287979\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−118.577	−0.598110
$35$	119.788	0.578513
$36$	121.568	0.562814
$37$	−145.578	−0.646832	−0.323416	−	0.946257i	$-0.604831\pi$
$37$	−145.578	−0.646832	−0.323416	+	0.946257i	$0.604831\pi$
$38$	−190.525	−0.813349
$39$	−582.021	−2.38969
$40$	43.2648	0.171019
$41$	−82.8326	−0.315519	−0.157759	−	0.987478i	$-0.550427\pi$
$41$	−82.8326	−0.315519	−0.157759	+	0.987478i	$0.550427\pi$
$42$	335.603	1.23297
$43$	151.373	0.536841	0.268420	−	0.963302i	$-0.413498\pi$
$43$	151.373	0.536841	0.268420	+	0.963302i	$0.413498\pi$
$44$	0	0
$45$	164.363	0.544483
$46$	284.968	0.913396
$47$	90.3643	0.280447	0.140223	−	0.990120i	$-0.455218\pi$
$47$	90.3643	0.280447	0.140223	+	0.990120i	$0.455218\pi$
$48$	121.212	0.364488
$49$	147.615	0.430363
$50$	−191.505	−0.541658
$51$	−449.153	−1.23322
$52$	−307.308	−0.819537
$53$	−234.849	−0.608660	−0.304330	−	0.952567i	$-0.598433\pi$
$53$	−234.849	−0.608660	−0.304330	+	0.952567i	$0.598433\pi$
$54$	51.3931	0.129513
$55$	0	0
$56$	177.199	0.422842
$57$	−721.685	−1.67701
$58$	40.8366	0.0924502
$59$	302.497	0.667488	0.333744	−	0.942664i	$-0.391688\pi$
$59$	302.497	0.667488	0.333744	+	0.942664i	$0.391688\pi$
$60$	163.882	0.352617
$61$	149.279	0.313332	0.156666	−	0.987652i	$-0.449925\pi$
$61$	149.279	0.313332	0.156666	+	0.987652i	$0.449925\pi$
$62$	426.608	0.873860
$63$	673.176	1.34623
$64$	64.0000	0.125000
$65$	−415.488	−0.792845
$66$	0	0
$67$	826.236	1.50658	0.753290	−	0.657689i	$-0.228466\pi$
$67$	826.236	1.50658	0.753290	+	0.657689i	$0.228466\pi$
$68$	−237.153	−0.422927
$69$	1079.42	1.88329
$70$	239.577	0.409070
$71$	−898.965	−1.50264	−0.751321	−	0.659937i	$-0.770583\pi$
$71$	−898.965	−1.50264	−0.751321	+	0.659937i	$0.770583\pi$
$72$	243.136	0.397969
$73$	137.993	0.221245	0.110623	−	0.993862i	$-0.464716\pi$
$73$	137.993	0.221245	0.110623	+	0.993862i	$0.464716\pi$
$74$	−291.155	−0.457380
$75$	−725.396	−1.11682
$76$	−381.050	−0.575124
$77$	0	0
$78$	−1164.04	−1.68977
$79$	304.459	0.433598	0.216799	−	0.976216i	$-0.430438\pi$
$79$	304.459	0.433598	0.216799	+	0.976216i	$0.430438\pi$
$80$	86.5296	0.120929
$81$	−625.912	−0.858590
$82$	−165.665	−0.223106
$83$	764.294	1.01075	0.505374	−	0.862900i	$-0.331354\pi$
$83$	764.294	1.01075	0.505374	+	0.862900i	$0.331354\pi$
$84$	671.206	0.871840
$85$	−320.637	−0.409153
$86$	302.746	0.379604
$87$	154.684	0.190619
$88$	0	0
$89$	−313.100	−0.372905	−0.186452	−	0.982464i	$-0.559699\pi$
$89$	−313.100	−0.372905	−0.186452	+	0.982464i	$0.559699\pi$
$90$	328.725	0.385008
$91$	−1701.70	−1.96030
$92$	569.936	0.645868
$93$	1615.94	1.80177
$94$	180.729	0.198306
$95$	−515.189	−0.556393
$96$	242.424	0.257732
$97$	−582.505	−0.609737	−0.304868	−	0.952394i	$-0.598612\pi$
$97$	−582.505	−0.609737	−0.304868	+	0.952394i	$0.598612\pi$
$98$	295.229	0.304313
$99$	0	0
$100$	−383.010	−0.383010
$101$	141.554	0.139457	0.0697286	−	0.997566i	$-0.477787\pi$
$101$	141.554	0.139457	0.0697286	+	0.997566i	$0.477787\pi$
$102$	−898.307	−0.872016
$103$	841.380	0.804890	0.402445	−	0.915444i	$-0.368161\pi$
$103$	841.380	0.804890	0.402445	+	0.915444i	$0.368161\pi$
$104$	−614.615	−0.579500
$105$	907.487	0.843444
$106$	−469.698	−0.430387
$107$	64.4319	0.0582137	0.0291069	−	0.999576i	$-0.490734\pi$
$107$	64.4319	0.0582137	0.0291069	+	0.999576i	$0.490734\pi$
$108$	102.786	0.0915797
$109$	−1559.04	−1.36999	−0.684995	−	0.728547i	$-0.740196\pi$
$109$	−1559.04	−1.36999	−0.684995	+	0.728547i	$0.740196\pi$
$110$	0	0
$111$	−1102.86	−0.943052
$112$	354.397	0.298994
$113$	−2239.14	−1.86408	−0.932039	−	0.362359i	$-0.881971\pi$
$113$	−2239.14	−1.86408	−0.932039	+	0.362359i	$0.881971\pi$
$114$	−1443.37	−1.18582
$115$	770.567	0.624833
$116$	81.6732	0.0653721
$117$	−2334.92	−1.84499
$118$	604.995	0.471986
$119$	−1313.23	−1.01162
$120$	327.763	0.249338
$121$	0	0
$122$	298.559	0.221559
$123$	−627.519	−0.460012
$124$	853.217	0.617912
$125$	−1193.85	−0.854251
$126$	1346.35	0.951925
$127$	−1093.46	−0.764004	−0.382002	−	0.924162i	$-0.624765\pi$
$127$	−1093.46	−0.764004	−0.382002	+	0.924162i	$0.624765\pi$
$128$	128.000	0.0883883
$129$	1146.76	0.782688
$130$	−830.976	−0.560626
$131$	2466.16	1.64481	0.822403	−	0.568906i	$-0.192633\pi$
$131$	2466.16	1.64481	0.822403	+	0.568906i	$0.192633\pi$
$132$	0	0
$133$	−2110.05	−1.37567
$134$	1652.47	1.06531
$135$	138.970	0.0885970
$136$	−474.307	−0.299055
$137$	−973.286	−0.606959	−0.303480	−	0.952838i	$-0.598148\pi$
$137$	−973.286	−0.606959	−0.303480	+	0.952838i	$0.598148\pi$
$138$	2158.84	1.33169
$139$	1263.04	0.770717	0.385358	−	0.922767i	$-0.374078\pi$
$139$	1263.04	0.770717	0.385358	+	0.922767i	$0.374078\pi$
$140$	479.154	0.289256
$141$	684.577	0.408878
$142$	−1797.93	−1.06253
$143$	0	0
$144$	486.271	0.281407
$145$	110.424	0.0632430
$146$	275.987	0.156444
$147$	1118.29	0.627449
$148$	−582.310	−0.323416
$149$	489.340	0.269049	0.134525	−	0.990910i	$-0.457049\pi$
$149$	489.340	0.269049	0.134525	+	0.990910i	$0.457049\pi$
$150$	−1450.79	−0.789712
$151$	−1707.77	−0.920376	−0.460188	−	0.887822i	$-0.652218\pi$
$151$	−1707.77	−0.920376	−0.460188	+	0.887822i	$0.652218\pi$
$152$	−762.100	−0.406674
$153$	−1801.89	−0.952118
$154$	0	0
$155$	1153.57	0.597787
$156$	−2328.09	−1.19485
$157$	2561.73	1.30222	0.651108	−	0.758985i	$-0.274304\pi$
$157$	2561.73	1.30222	0.651108	+	0.758985i	$0.274304\pi$
$158$	608.917	0.306600
$159$	−1779.16	−0.887397
$160$	173.059	0.0855096
$161$	3155.99	1.54489
$162$	−1251.82	−0.607115
$163$	1816.09	0.872681	0.436340	−	0.899782i	$-0.356274\pi$
$163$	1816.09	0.872681	0.436340	+	0.899782i	$0.356274\pi$
$164$	−331.330	−0.157759
$165$	0	0
$166$	1528.59	0.714707
$167$	−378.925	−0.175581	−0.0877907	−	0.996139i	$-0.527981\pi$
$167$	−378.925	−0.175581	−0.0877907	+	0.996139i	$0.527981\pi$
$168$	1342.41	0.616484
$169$	3705.38	1.68656
$170$	−641.275	−0.289315
$171$	−2895.21	−1.29475
$172$	605.491	0.268420
$173$	−50.7220	−0.0222909	−0.0111454	−	0.999938i	$-0.503548\pi$
$173$	−50.7220	−0.0222909	−0.0111454	+	0.999938i	$0.503548\pi$
$174$	309.368	0.134788
$175$	−2120.90	−0.916142
$176$	0	0
$177$	2291.64	0.973167
$178$	−626.199	−0.263683
$179$	20.8678	0.00871359	0.00435679	−	0.999991i	$-0.498613\pi$
$179$	20.8678	0.00871359	0.00435679	+	0.999991i	$0.498613\pi$
$180$	657.451	0.272242
$181$	−176.153	−0.0723391	−0.0361695	−	0.999346i	$-0.511516\pi$
$181$	−176.153	−0.0723391	−0.0361695	+	0.999346i	$0.511516\pi$
$182$	−3403.41	−1.38614
$183$	1130.90	0.456824
$184$	1139.87	0.456698
$185$	−787.298	−0.312883
$186$	3231.88	1.27405
$187$	0	0
$188$	361.457	0.140223
$189$	569.174	0.219055
$190$	−1030.38	−0.393429
$191$	−1159.22	−0.439153	−0.219576	−	0.975595i	$-0.570468\pi$
$191$	−1159.22	−0.439153	−0.219576	+	0.975595i	$0.570468\pi$
$192$	484.848	0.182244
$193$	1284.94	0.479232	0.239616	−	0.970868i	$-0.422978\pi$
$193$	1284.94	0.479232	0.239616	+	0.970868i	$0.422978\pi$
$194$	−1165.01	−0.431149
$195$	−3147.63	−1.15593
$196$	590.458	0.215181
$197$	2685.06	0.971078	0.485539	−	0.874215i	$-0.338623\pi$
$197$	2685.06	0.971078	0.485539	+	0.874215i	$0.338623\pi$
$198$	0	0
$199$	−1333.54	−0.475036	−0.237518	−	0.971383i	$-0.576334\pi$
$199$	−1333.54	−0.475036	−0.237518	+	0.971383i	$0.576334\pi$
$200$	−766.020	−0.270829
$201$	6259.36	2.19652
$202$	283.109	0.0986112
$203$	452.262	0.156367
$204$	−1796.61	−0.616608
$205$	−447.967	−0.152621
$206$	1682.76	0.569143
$207$	4330.36	1.45401
$208$	−1229.23	−0.409768
$209$	0	0
$210$	1814.97	0.596405
$211$	1628.17	0.531221	0.265611	−	0.964080i	$-0.414426\pi$
$211$	1628.17	0.531221	0.265611	+	0.964080i	$0.414426\pi$
$212$	−939.395	−0.304330
$213$	−6810.33	−2.19078
$214$	128.864	0.0411633
$215$	818.640	0.259678
$216$	205.572	0.0647566
$217$	4724.65	1.47802
$218$	−3118.08	−0.968730
$219$	1045.40	0.322565
$220$	0	0
$221$	4554.94	1.38642
$222$	−2205.72	−0.666838
$223$	−2628.19	−0.789224	−0.394612	−	0.918848i	$-0.629121\pi$
$223$	−2628.19	−0.789224	−0.394612	+	0.918848i	$0.629121\pi$
$224$	708.794	0.211421
$225$	−2910.10	−0.862253
$226$	−4478.28	−1.31810
$227$	−1702.56	−0.497809	−0.248904	−	0.968528i	$-0.580070\pi$
$227$	−1702.56	−0.497809	−0.248904	+	0.968528i	$0.580070\pi$
$228$	−2886.74	−0.838505
$229$	5395.95	1.55709	0.778546	−	0.627587i	$-0.215957\pi$
$229$	5395.95	1.55709	0.778546	+	0.627587i	$0.215957\pi$
$230$	1541.13	0.441823
$231$	0	0
$232$	163.346	0.0462251
$233$	6054.18	1.70224	0.851122	−	0.524969i	$-0.175923\pi$
$233$	6054.18	1.70224	0.851122	+	0.524969i	$0.175923\pi$
$234$	−4669.84	−1.30460
$235$	488.699	0.135656
$236$	1209.99	0.333744
$237$	2306.50	0.632166
$238$	−2626.45	−0.715326
$239$	−3136.60	−0.848911	−0.424456	−	0.905449i	$-0.639534\pi$
$239$	−3136.60	−0.848911	−0.424456	+	0.905449i	$0.639534\pi$
$240$	655.526	0.176309
$241$	6499.26	1.73715	0.868577	−	0.495555i	$-0.165035\pi$
$241$	6499.26	1.73715	0.868577	+	0.495555i	$0.165035\pi$
$242$	0	0
$243$	−5435.56	−1.43494
$244$	597.118	0.156666
$245$	798.314	0.208173
$246$	−1255.04	−0.325277
$247$	7318.73	1.88534
$248$	1706.43	0.436930
$249$	5790.10	1.47362
$250$	−2387.70	−0.604046
$251$	5565.79	1.39964	0.699820	−	0.714320i	$-0.253264\pi$
$251$	5565.79	1.39964	0.699820	+	0.714320i	$0.253264\pi$
$252$	2692.70	0.673113
$253$	0	0
$254$	−2186.91	−0.540232
$255$	−2429.07	−0.596526
$256$	256.000	0.0625000
$257$	−6036.31	−1.46512	−0.732558	−	0.680705i	$-0.761673\pi$
$257$	−6036.31	−1.46512	−0.732558	+	0.680705i	$0.761673\pi$
$258$	2293.53	0.553444
$259$	−3224.52	−0.773597
$260$	−1661.95	−0.396422
$261$	620.552	0.147169
$262$	4932.32	1.16305
$263$	708.442	0.166100	0.0830502	−	0.996545i	$-0.473534\pi$
$263$	708.442	0.166100	0.0830502	+	0.996545i	$0.473534\pi$
$264$	0	0
$265$	−1270.09	−0.294418
$266$	−4220.10	−0.972747
$267$	−2371.96	−0.543677
$268$	3304.95	0.753290
$269$	−4809.71	−1.09016	−0.545080	−	0.838384i	$-0.683501\pi$
$269$	−4809.71	−1.09016	−0.545080	+	0.838384i	$0.683501\pi$
$270$	277.939	0.0626475
$271$	5240.88	1.17476	0.587382	−	0.809310i	$-0.300159\pi$
$271$	5240.88	1.17476	0.587382	+	0.809310i	$0.300159\pi$
$272$	−948.613	−0.211464
$273$	−12891.7	−2.85802
$274$	−1946.57	−0.429185
$275$	0	0
$276$	4317.69	0.941646
$277$	5339.75	1.15825	0.579123	−	0.815240i	$-0.303395\pi$
$277$	5339.75	1.15825	0.579123	+	0.815240i	$0.303395\pi$
$278$	2526.08	0.544979
$279$	6482.73	1.39108
$280$	958.308	0.204535
$281$	3279.97	0.696322	0.348161	−	0.937435i	$-0.386806\pi$
$281$	3279.97	0.696322	0.348161	+	0.937435i	$0.386806\pi$
$282$	1369.15	0.289120
$283$	4895.49	1.02829	0.514146	−	0.857703i	$-0.328109\pi$
$283$	4895.49	1.02829	0.514146	+	0.857703i	$0.328109\pi$
$284$	−3595.86	−0.751321
$285$	−3902.95	−0.811195
$286$	0	0
$287$	−1834.73	−0.377354
$288$	972.542	0.198985
$289$	−1397.89	−0.284529
$290$	220.848	0.0447195
$291$	−4412.91	−0.888968
$292$	551.974	0.110623
$293$	−6705.80	−1.33705	−0.668527	−	0.743688i	$-0.733075\pi$
$293$	−6705.80	−1.33705	−0.668527	+	0.743688i	$0.733075\pi$
$294$	2236.58	0.443674
$295$	1635.94	0.322874
$296$	−1164.62	−0.228690
$297$	0	0
$298$	978.680	0.190246
$299$	−10946.6	−2.11725
$300$	−2901.59	−0.558410
$301$	3352.88	0.642049
$302$	−3415.55	−0.650804
$303$	1072.38	0.203322
$304$	−1524.20	−0.287562
$305$	807.318	0.151564
$306$	−3603.78	−0.673249
$307$	−9507.29	−1.76746	−0.883730	−	0.467998i	$-0.844975\pi$
$307$	−9507.29	−1.76746	−0.883730	+	0.467998i	$0.844975\pi$
$308$	0	0
$309$	6374.08	1.17349
$310$	2307.14	0.422699
$311$	5768.07	1.05170	0.525848	−	0.850579i	$-0.323748\pi$
$311$	5768.07	1.05170	0.525848	+	0.850579i	$0.323748\pi$
$312$	−4656.17	−0.844884
$313$	5977.75	1.07950	0.539748	−	0.841826i	$-0.318519\pi$
$313$	5977.75	1.07950	0.539748	+	0.841826i	$0.318519\pi$
$314$	5123.45	0.920806
$315$	3640.60	0.651190
$316$	1217.83	0.216799
$317$	5231.93	0.926987	0.463493	−	0.886100i	$-0.346596\pi$
$317$	5231.93	0.926987	0.463493	+	0.886100i	$0.346596\pi$
$318$	−3558.31	−0.627485
$319$	0	0
$320$	346.118	0.0604644
$321$	488.120	0.0848729
$322$	6311.99	1.09240
$323$	5647.96	0.972944
$324$	−2503.65	−0.429295
$325$	7356.37	1.25556
$326$	3632.17	0.617078
$327$	−11810.9	−1.99738
$328$	−662.661	−0.111553
$329$	2001.55	0.335408
$330$	0	0
$331$	−3963.72	−0.658205	−0.329102	−	0.944294i	$-0.606746\pi$
$331$	−3963.72	−0.658205	−0.329102	+	0.944294i	$0.606746\pi$
$332$	3057.17	0.505374
$333$	−4424.39	−0.728093
$334$	−757.850	−0.124155
$335$	4468.37	0.728756
$336$	2684.82	0.435920
$337$	−5106.60	−0.825443	−0.412721	−	0.910857i	$-0.635422\pi$
$337$	−5106.60	−0.825443	−0.412721	+	0.910857i	$0.635422\pi$
$338$	7410.75	1.19258
$339$	−16963.2	−2.71774
$340$	−1282.55	−0.204576
$341$	0	0
$342$	−5790.43	−0.915528
$343$	−4327.75	−0.681273
$344$	1210.98	0.189802
$345$	5837.62	0.910977
$346$	−101.444	−0.0157620
$347$	−7545.75	−1.16737	−0.583685	−	0.811980i	$-0.698390\pi$
$347$	−7545.75	−1.16737	−0.583685	+	0.811980i	$0.698390\pi$
$348$	618.736	0.0953095
$349$	3256.71	0.499507	0.249753	−	0.968309i	$-0.419650\pi$
$349$	3256.71	0.499507	0.249753	+	0.968309i	$0.419650\pi$
$350$	−4241.80	−0.647811
$351$	−1974.19	−0.300212
$352$	0	0
$353$	−6506.83	−0.981087	−0.490543	−	0.871417i	$-0.663202\pi$
$353$	−6506.83	−0.981087	−0.490543	+	0.871417i	$0.663202\pi$
$354$	4583.29	0.688133
$355$	−4861.69	−0.726851
$356$	−1252.40	−0.186452
$357$	−9948.67	−1.47490
$358$	41.7356	0.00616144
$359$	−4066.21	−0.597789	−0.298895	−	0.954286i	$-0.596618\pi$
$359$	−4066.21	−0.597789	−0.298895	+	0.954286i	$0.596618\pi$
$360$	1314.90	0.192504
$361$	2215.95	0.323073
$362$	−352.307	−0.0511514
$363$	0	0
$364$	−6806.81	−0.980148
$365$	746.282	0.107020
$366$	2261.81	0.323023
$367$	−1392.27	−0.198028	−0.0990138	−	0.995086i	$-0.531569\pi$
$367$	−1392.27	−0.198028	−0.0990138	+	0.995086i	$0.531569\pi$
$368$	2279.74	0.322934
$369$	−2517.44	−0.355157
$370$	−1574.60	−0.221241
$371$	−5201.86	−0.727944
$372$	6463.75	0.900887
$373$	−9440.30	−1.31046	−0.655228	−	0.755431i	$-0.727428\pi$
$373$	−9440.30	−1.31046	−0.655228	+	0.755431i	$0.727428\pi$
$374$	0	0
$375$	−9044.32	−1.24546
$376$	722.915	0.0991529
$377$	−1568.68	−0.214299
$378$	1138.35	0.154895
$379$	1290.35	0.174884	0.0874418	−	0.996170i	$-0.472131\pi$
$379$	1290.35	0.174884	0.0874418	+	0.996170i	$0.472131\pi$
$380$	−2060.76	−0.278196
$381$	−8283.74	−1.11388
$382$	−2318.44	−0.310528
$383$	−5652.89	−0.754176	−0.377088	−	0.926177i	$-0.623075\pi$
$383$	−5652.89	−0.754176	−0.377088	+	0.926177i	$0.623075\pi$
$384$	969.696	0.128866
$385$	0	0
$386$	2569.88	0.338868
$387$	4600.52	0.604283
$388$	−2330.02	−0.304868
$389$	−12838.0	−1.67330	−0.836648	−	0.547741i	$-0.815488\pi$
$389$	−12838.0	−1.67330	−0.836648	+	0.547741i	$0.815488\pi$
$390$	−6295.26	−0.817366
$391$	−8447.63	−1.09262
$392$	1180.92	0.152156
$393$	18683.0	2.39805
$394$	5370.12	0.686656
$395$	1646.54	0.209738
$396$	0	0
$397$	7691.26	0.972326	0.486163	−	0.873868i	$-0.338396\pi$
$397$	7691.26	0.972326	0.486163	+	0.873868i	$0.338396\pi$
$398$	−2667.08	−0.335901
$399$	−15985.2	−2.00567
$400$	−1532.04	−0.191505
$401$	7786.17	0.969633	0.484817	−	0.874616i	$-0.338886\pi$
$401$	7786.17	0.969633	0.484817	+	0.874616i	$0.338886\pi$
$402$	12518.7	1.55318
$403$	−16387.5	−2.02561
$404$	566.217	0.0697286
$405$	−3385.00	−0.415313
$406$	904.523	0.110568
$407$	0	0
$408$	−3593.23	−0.436008
$409$	8859.88	1.07113	0.535566	−	0.844493i	$-0.320098\pi$
$409$	8859.88	1.07113	0.535566	+	0.844493i	$0.320098\pi$
$410$	−895.934	−0.107920
$411$	−7373.37	−0.884918
$412$	3365.52	0.402445
$413$	6700.26	0.798301
$414$	8660.73	1.02814
$415$	4133.38	0.488914
$416$	−2458.46	−0.289750
$417$	9568.47	1.12367
$418$	0	0
$419$	9469.08	1.10405	0.552023	−	0.833829i	$-0.313856\pi$
$419$	9469.08	1.10405	0.552023	+	0.833829i	$0.313856\pi$
$420$	3629.95	0.421722
$421$	5177.94	0.599424	0.299712	−	0.954030i	$-0.403109\pi$
$421$	5177.94	0.599424	0.299712	+	0.954030i	$0.403109\pi$
$422$	3256.34	0.375630
$423$	2746.35	0.315678
$424$	−1878.79	−0.215194
$425$	5677.00	0.647941
$426$	−13620.7	−1.54912
$427$	3306.51	0.374739
$428$	257.728	0.0291069
$429$	0	0
$430$	1637.28	0.183620
$431$	−10463.6	−1.16940	−0.584700	−	0.811249i	$-0.698788\pi$
$431$	−10463.6	−1.16940	−0.584700	+	0.811249i	$0.698788\pi$
$432$	411.145	0.0457898
$433$	15838.2	1.75782	0.878911	−	0.476986i	$-0.158271\pi$
$433$	15838.2	1.75782	0.878911	+	0.476986i	$0.158271\pi$
$434$	9449.30	1.04512
$435$	836.546	0.0922053
$436$	−6236.16	−0.684995
$437$	−13573.4	−1.48582
$438$	2090.81	0.228088
$439$	4824.70	0.524534	0.262267	−	0.964995i	$-0.415530\pi$
$439$	4824.70	0.524534	0.262267	+	0.964995i	$0.415530\pi$
$440$	0	0
$441$	4486.29	0.484429
$442$	9109.88	0.980346
$443$	3583.38	0.384315	0.192158	−	0.981364i	$-0.438451\pi$
$443$	3583.38	0.384315	0.192158	+	0.981364i	$0.438451\pi$
$444$	−4411.44	−0.471526
$445$	−1693.27	−0.180380
$446$	−5256.39	−0.558066
$447$	3707.12	0.392261
$448$	1417.59	0.149497
$449$	−7795.98	−0.819410	−0.409705	−	0.912218i	$-0.634368\pi$
$449$	−7795.98	−0.819410	−0.409705	+	0.912218i	$0.634368\pi$
$450$	−5820.21	−0.609705
$451$	0	0
$452$	−8956.57	−0.932039
$453$	−12937.7	−1.34186
$454$	−3405.11	−0.352004
$455$	−9202.98	−0.948225
$456$	−5773.48	−0.592912
$457$	−3348.02	−0.342700	−0.171350	−	0.985210i	$-0.554813\pi$
$457$	−3348.02	−0.342700	−0.171350	+	0.985210i	$0.554813\pi$
$458$	10791.9	1.10103
$459$	−1523.51	−0.154926
$460$	3082.27	0.312416
$461$	−7171.96	−0.724580	−0.362290	−	0.932065i	$-0.618005\pi$
$461$	−7171.96	−0.724580	−0.362290	+	0.932065i	$0.618005\pi$
$462$	0	0
$463$	4034.74	0.404990	0.202495	−	0.979283i	$-0.435095\pi$
$463$	4034.74	0.404990	0.202495	+	0.979283i	$0.435095\pi$
$464$	326.693	0.0326861
$465$	8739.16	0.871546
$466$	12108.4	1.20367
$467$	−11821.8	−1.17141	−0.585706	−	0.810523i	$-0.699183\pi$
$467$	−11821.8	−1.17141	−0.585706	+	0.810523i	$0.699183\pi$
$468$	−9339.68	−0.922493
$469$	18301.0	1.80184
$470$	977.399	0.0959235
$471$	19407.0	1.89857
$472$	2419.98	0.235993
$473$	0	0
$474$	4613.00	0.447009
$475$	9121.62	0.881113
$476$	−5252.90	−0.505812
$477$	−7137.51	−0.685124
$478$	−6273.20	−0.600271
$479$	8659.94	0.826060	0.413030	−	0.910717i	$-0.364470\pi$
$479$	8659.94	0.826060	0.413030	+	0.910717i	$0.364470\pi$
$480$	1311.05	0.124669
$481$	11184.3	1.06021
$482$	12998.5	1.22835
$483$	23909.0	2.25238
$484$	0	0
$485$	−3150.25	−0.294939
$486$	−10871.1	−1.01466
$487$	7427.66	0.691128	0.345564	−	0.938395i	$-0.387688\pi$
$487$	7427.66	0.691128	0.345564	+	0.938395i	$0.387688\pi$
$488$	1194.24	0.110780
$489$	13758.2	1.27233
$490$	1596.63	0.147201
$491$	5543.89	0.509556	0.254778	−	0.967000i	$-0.417998\pi$
$491$	5543.89	0.509556	0.254778	+	0.967000i	$0.417998\pi$
$492$	−2510.07	−0.230006
$493$	−1210.57	−0.110591
$494$	14637.5	1.33314
$495$	0	0
$496$	3412.87	0.308956
$497$	−19911.9	−1.79713
$498$	11580.2	1.04201
$499$	20412.0	1.83120	0.915599	−	0.402093i	$-0.131717\pi$
$499$	20412.0	1.83120	0.915599	+	0.402093i	$0.131717\pi$
$500$	−4775.41	−0.427125
$501$	−2870.64	−0.255990
$502$	11131.6	0.989694
$503$	182.742	0.0161989	0.00809946	−	0.999967i	$-0.497422\pi$
$503$	182.742	0.0161989	0.00809946	+	0.999967i	$0.497422\pi$
$504$	5385.41	0.475963
$505$	765.540	0.0674576
$506$	0	0
$507$	28071.0	2.45893
$508$	−4373.82	−0.382002
$509$	4755.29	0.414095	0.207048	−	0.978331i	$-0.433614\pi$
$509$	4755.29	0.414095	0.207048	+	0.978331i	$0.433614\pi$
$510$	−4858.13	−0.421807
$511$	3056.53	0.264604
$512$	512.000	0.0441942
$513$	−2447.92	−0.210679
$514$	−12072.6	−1.03599
$515$	4550.27	0.389337
$516$	4587.05	0.391344
$517$	0	0
$518$	−6449.03	−0.547016
$519$	−384.257	−0.0324990
$520$	−3323.90	−0.280313
$521$	−11731.1	−0.986470	−0.493235	−	0.869896i	$-0.664186\pi$
$521$	−11731.1	−0.986470	−0.493235	+	0.869896i	$0.664186\pi$
$522$	1241.10	0.104064
$523$	343.839	0.0287477	0.0143738	−	0.999897i	$-0.495425\pi$
$523$	343.839	0.0287477	0.0143738	+	0.999897i	$0.495425\pi$
$524$	9864.65	0.822403
$525$	−16067.4	−1.33569
$526$	1416.88	0.117451
$527$	−12646.4	−1.04533
$528$	0	0
$529$	8134.66	0.668584
$530$	−2540.17	−0.208185
$531$	9193.49	0.751343
$532$	−8440.19	−0.687836
$533$	6363.77	0.517159
$534$	−4743.93	−0.384438
$535$	348.454	0.0281589
$536$	6609.89	0.532656
$537$	158.089	0.0127040
$538$	−9619.41	−0.770859
$539$	0	0
$540$	555.878	0.0442985
$541$	17243.8	1.37037	0.685183	−	0.728371i	$-0.259722\pi$
$541$	17243.8	1.37037	0.685183	+	0.728371i	$0.259722\pi$
$542$	10481.8	0.830683
$543$	−1334.49	−0.105467
$544$	−1897.23	−0.149527
$545$	−8431.45	−0.662685
$546$	−25783.3	−2.02092
$547$	7643.21	0.597441	0.298720	−	0.954341i	$-0.403440\pi$
$547$	7643.21	0.597441	0.298720	+	0.954341i	$0.403440\pi$
$548$	−3893.14	−0.303480
$549$	4536.89	0.352696
$550$	0	0
$551$	−1945.10	−0.150388
$552$	8635.38	0.665844
$553$	6743.70	0.518574
$554$	10679.5	0.819004
$555$	−5964.37	−0.456168
$556$	5052.16	0.385358
$557$	−26106.2	−1.98591	−0.992957	−	0.118471i	$-0.962201\pi$
$557$	−26106.2	−1.98591	−0.992957	+	0.118471i	$0.962201\pi$
$558$	12965.5	0.983641
$559$	−11629.5	−0.879921
$560$	1916.62	0.144628
$561$	0	0
$562$	6559.94	0.492374
$563$	18073.9	1.35297	0.676487	−	0.736455i	$-0.263502\pi$
$563$	18073.9	1.35297	0.676487	+	0.736455i	$0.263502\pi$
$564$	2738.31	0.204439
$565$	−12109.5	−0.901682
$566$	9790.98	0.727112
$567$	−13863.8	−1.02685
$568$	−7191.72	−0.531264
$569$	−7805.16	−0.575060	−0.287530	−	0.957772i	$-0.592834\pi$
$569$	−7805.16	−0.575060	−0.287530	+	0.957772i	$0.592834\pi$
$570$	−7805.89	−0.573601
$571$	13039.6	0.955672	0.477836	−	0.878449i	$-0.341421\pi$
$571$	13039.6	0.955672	0.477836	+	0.878449i	$0.341421\pi$
$572$	0	0
$573$	−8781.96	−0.640264
$574$	−3669.45	−0.266829
$575$	−13643.2	−0.989496
$576$	1945.08	0.140703
$577$	−9569.43	−0.690434	−0.345217	−	0.938523i	$-0.612195\pi$
$577$	−9569.43	−0.690434	−0.345217	+	0.938523i	$0.612195\pi$
$578$	−2795.79	−0.201193
$579$	9734.36	0.698698
$580$	441.697	0.0316215
$581$	16929.0	1.20883
$582$	−8825.83	−0.628595
$583$	0	0
$584$	1103.95	0.0782220
$585$	−12627.5	−0.892448
$586$	−13411.6	−0.945440
$587$	9727.00	0.683946	0.341973	−	0.939710i	$-0.388905\pi$
$587$	9727.00	0.683946	0.341973	+	0.939710i	$0.388905\pi$
$588$	4473.16	0.313725
$589$	−20319.9	−1.42151
$590$	3271.87	0.228307
$591$	20341.3	1.41579
$592$	−2329.24	−0.161708
$593$	−6926.77	−0.479677	−0.239838	−	0.970813i	$-0.577094\pi$
$593$	−6926.77	−0.479677	−0.239838	+	0.970813i	$0.577094\pi$
$594$	0	0
$595$	−7102.06	−0.489338
$596$	1957.36	0.134525
$597$	−10102.6	−0.692580
$598$	−21893.2	−1.49712
$599$	−21984.7	−1.49961	−0.749807	−	0.661656i	$-0.769854\pi$
$599$	−21984.7	−1.49961	−0.749807	+	0.661656i	$0.769854\pi$
$600$	−5803.17	−0.394856
$601$	7363.27	0.499757	0.249878	−	0.968277i	$-0.419609\pi$
$601$	7363.27	0.499757	0.249878	+	0.968277i	$0.419609\pi$
$602$	6705.76	0.453998
$603$	25110.9	1.69585
$604$	−6831.10	−0.460188
$605$	0	0
$606$	2144.76	0.143771
$607$	−10168.4	−0.679938	−0.339969	−	0.940437i	$-0.610417\pi$
$607$	−10168.4	−0.679938	−0.339969	+	0.940437i	$0.610417\pi$
$608$	−3048.40	−0.203337
$609$	3426.22	0.227976
$610$	1614.64	0.107172
$611$	−6942.41	−0.459673
$612$	−7207.55	−0.476059
$613$	−25084.5	−1.65278	−0.826389	−	0.563099i	$-0.809609\pi$
$613$	−25084.5	−1.65278	−0.826389	+	0.563099i	$0.809609\pi$
$614$	−19014.6	−1.24978
$615$	−3393.68	−0.222515
$616$	0	0
$617$	−26335.5	−1.71836	−0.859178	−	0.511677i	$-0.829024\pi$
$617$	−26335.5	−1.71836	−0.859178	+	0.511677i	$0.829024\pi$
$618$	12748.2	0.829784
$619$	−24643.2	−1.60015	−0.800075	−	0.599901i	$-0.795207\pi$
$619$	−24643.2	−1.60015	−0.800075	+	0.599901i	$0.795207\pi$
$620$	4614.28	0.298894
$621$	3661.34	0.236594
$622$	11536.1	0.743661
$623$	−6935.10	−0.445986
$624$	−9312.34	−0.597423
$625$	5512.59	0.352806
$626$	11955.5	0.763319
$627$	0	0
$628$	10246.9	0.651108
$629$	8631.05	0.547126
$630$	7281.21	0.460461
$631$	7455.32	0.470351	0.235176	−	0.971953i	$-0.424433\pi$
$631$	7455.32	0.470351	0.235176	+	0.971953i	$0.424433\pi$
$632$	2435.67	0.153300
$633$	12334.6	0.774496
$634$	10463.9	0.655478
$635$	−5913.52	−0.369560
$636$	−7116.62	−0.443699
$637$	−11340.8	−0.705397
$638$	0	0
$639$	−27321.3	−1.69141
$640$	692.237	0.0427548
$641$	−2778.67	−0.171218	−0.0856090	−	0.996329i	$-0.527284\pi$
$641$	−2778.67	−0.171218	−0.0856090	+	0.996329i	$0.527284\pi$
$642$	976.240	0.0600142
$643$	7557.71	0.463526	0.231763	−	0.972772i	$-0.425551\pi$
$643$	7557.71	0.463526	0.231763	+	0.972772i	$0.425551\pi$
$644$	12624.0	0.772444
$645$	6201.81	0.378598
$646$	11295.9	0.687975
$647$	−27430.7	−1.66679	−0.833394	−	0.552679i	$-0.813605\pi$
$647$	−27430.7	−1.66679	−0.833394	+	0.552679i	$0.813605\pi$
$648$	−5007.30	−0.303557
$649$	0	0
$650$	14712.7	0.887817
$651$	35792.7	2.15488
$652$	7264.35	0.436340
$653$	284.728	0.0170632	0.00853160	−	0.999964i	$-0.497284\pi$
$653$	284.728	0.0170632	0.00853160	+	0.999964i	$0.497284\pi$
$654$	−23621.8	−1.41236
$655$	13337.2	0.795617
$656$	−1325.32	−0.0788797
$657$	4193.89	0.249040
$658$	4003.11	0.237169
$659$	−20747.0	−1.22639	−0.613193	−	0.789933i	$-0.710115\pi$
$659$	−20747.0	−1.22639	−0.613193	+	0.789933i	$0.710115\pi$
$660$	0	0
$661$	−18908.8	−1.11266	−0.556328	−	0.830963i	$-0.687790\pi$
$661$	−18908.8	−1.11266	−0.556328	+	0.830963i	$0.687790\pi$
$662$	−7927.44	−0.465421
$663$	34507.1	2.02133
$664$	6114.35	0.357354
$665$	−11411.4	−0.665434
$666$	−8848.77	−0.514839
$667$	2909.28	0.168887
$668$	−1515.70	−0.0877907
$669$	−19910.5	−1.15065
$670$	8936.74	0.515308
$671$	0	0
$672$	5369.65	0.308242
$673$	−29865.9	−1.71062	−0.855309	−	0.518118i	$-0.826633\pi$
$673$	−29865.9	−1.71062	−0.855309	+	0.518118i	$0.826633\pi$
$674$	−10213.2	−0.583676
$675$	−2460.51	−0.140304
$676$	14821.5	0.843281
$677$	17280.7	0.981024	0.490512	−	0.871435i	$-0.336810\pi$
$677$	17280.7	0.981024	0.490512	+	0.871435i	$0.336810\pi$
$678$	−33926.3	−1.92173
$679$	−12902.4	−0.729232
$680$	−2565.10	−0.144657
$681$	−12898.1	−0.725782
$682$	0	0
$683$	25104.0	1.40641	0.703204	−	0.710988i	$-0.251752\pi$
$683$	25104.0	1.40641	0.703204	+	0.710988i	$0.251752\pi$
$684$	−11580.9	−0.647376
$685$	−5263.63	−0.293595
$686$	−8655.51	−0.481733
$687$	40878.3	2.27017
$688$	2421.97	0.134210
$689$	18042.7	0.997638
$690$	11675.2	0.644158
$691$	−26163.8	−1.44040	−0.720201	−	0.693765i	$-0.755950\pi$
$691$	−26163.8	−1.44040	−0.720201	+	0.693765i	$0.755950\pi$
$692$	−202.888	−0.0111454
$693$	0	0
$694$	−15091.5	−0.825455
$695$	6830.64	0.372807
$696$	1237.47	0.0673940
$697$	4911.00	0.266883
$698$	6513.42	0.353205
$699$	45865.0	2.48179
$700$	−8483.60	−0.458071
$701$	1519.45	0.0818669	0.0409334	−	0.999162i	$-0.486967\pi$
$701$	1519.45	0.0818669	0.0409334	+	0.999162i	$0.486967\pi$
$702$	−3948.37	−0.212282
$703$	13868.1	0.744018
$704$	0	0
$705$	3702.26	0.197781
$706$	−13013.7	−0.693733
$707$	3135.40	0.166788
$708$	9166.58	0.486583
$709$	−21733.9	−1.15125	−0.575623	−	0.817715i	$-0.695240\pi$
$709$	−21733.9	−1.15125	−0.575623	+	0.817715i	$0.695240\pi$
$710$	−9723.39	−0.513961
$711$	9253.09	0.488070
$712$	−2504.80	−0.131842
$713$	30392.4	1.59636
$714$	−19897.3	−1.04291
$715$	0	0
$716$	83.4712	0.00435679
$717$	−23762.1	−1.23767
$718$	−8132.42	−0.422701
$719$	−14071.8	−0.729890	−0.364945	−	0.931029i	$-0.618912\pi$
$719$	−14071.8	−0.729890	−0.364945	+	0.931029i	$0.618912\pi$
$720$	2629.80	0.136121
$721$	18636.4	0.962630
$722$	4431.91	0.228447
$723$	49236.7	2.53269
$724$	−704.613	−0.0361695
$725$	−1955.10	−0.100153
$726$	0	0
$727$	10409.2	0.531027	0.265513	−	0.964107i	$-0.414459\pi$
$727$	10409.2	0.531027	0.265513	+	0.964107i	$0.414459\pi$
$728$	−13613.6	−0.693069
$729$	−24278.8	−1.23349
$730$	1492.56	0.0756744
$731$	−8974.65	−0.454089
$732$	4523.61	0.228412
$733$	19394.9	0.977307	0.488653	−	0.872478i	$-0.337488\pi$
$733$	19394.9	0.977307	0.488653	+	0.872478i	$0.337488\pi$
$734$	−2784.55	−0.140027
$735$	6047.83	0.303507
$736$	4559.48	0.228349
$737$	0	0
$738$	−5034.89	−0.251134
$739$	6672.24	0.332128	0.166064	−	0.986115i	$-0.446894\pi$
$739$	6672.24	0.332128	0.166064	+	0.986115i	$0.446894\pi$
$740$	−3149.19	−0.156441
$741$	55444.8	2.74874
$742$	−10403.7	−0.514734
$743$	−6565.18	−0.324163	−0.162082	−	0.986777i	$-0.551821\pi$
$743$	−6565.18	−0.324163	−0.162082	+	0.986777i	$0.551821\pi$
$744$	12927.5	0.637023
$745$	2646.40	0.130143
$746$	−18880.6	−0.926633
$747$	23228.4	1.13773
$748$	0	0
$749$	1427.16	0.0696223
$750$	−18088.6	−0.880671
$751$	18938.7	0.920216	0.460108	−	0.887863i	$-0.347811\pi$
$751$	18938.7	0.920216	0.460108	+	0.887863i	$0.347811\pi$
$752$	1445.83	0.0701117
$753$	42165.0	2.04061
$754$	−3137.35	−0.151533
$755$	−9235.82	−0.445200
$756$	2276.70	0.109527
$757$	−10958.4	−0.526141	−0.263070	−	0.964777i	$-0.584735\pi$
$757$	−10958.4	−0.526141	−0.263070	+	0.964777i	$0.584735\pi$
$758$	2580.70	0.123661
$759$	0	0
$760$	−4121.52	−0.196715
$761$	35442.2	1.68828	0.844138	−	0.536126i	$-0.180113\pi$
$761$	35442.2	1.68828	0.844138	+	0.536126i	$0.180113\pi$
$762$	−16567.5	−0.787633
$763$	−34532.5	−1.63848
$764$	−4636.88	−0.219576
$765$	−9744.79	−0.460554
$766$	−11305.8	−0.533283
$767$	−23240.0	−1.09406
$768$	1939.39	0.0911221
$769$	−4444.21	−0.208404	−0.104202	−	0.994556i	$-0.533229\pi$
$769$	−4444.21	−0.208404	−0.104202	+	0.994556i	$0.533229\pi$
$770$	0	0
$771$	−45729.5	−2.13607
$772$	5139.75	0.239616
$773$	10160.2	0.472750	0.236375	−	0.971662i	$-0.424041\pi$
$773$	10160.2	0.472750	0.236375	+	0.971662i	$0.424041\pi$
$774$	9201.03	0.427292
$775$	−20424.4	−0.946666
$776$	−4660.04	−0.215575
$777$	−24428.1	−1.12787
$778$	−25676.0	−1.18320
$779$	7890.84	0.362925
$780$	−12590.5	−0.577965
$781$	0	0
$782$	−16895.3	−0.772600
$783$	524.680	0.0239470
$784$	2361.83	0.107591
$785$	13854.1	0.629902
$786$	37366.0	1.69568
$787$	18270.1	0.827520	0.413760	−	0.910386i	$-0.364215\pi$
$787$	18270.1	0.827520	0.413760	+	0.910386i	$0.364215\pi$
$788$	10740.2	0.485539
$789$	5366.98	0.242167
$790$	3293.09	0.148307
$791$	−49596.6	−2.22939
$792$	0	0
$793$	−11468.7	−0.513575
$794$	15382.5	0.687538
$795$	−9621.85	−0.429248
$796$	−5334.16	−0.237518
$797$	17984.0	0.799279	0.399639	−	0.916672i	$-0.369135\pi$
$797$	17984.0	0.799279	0.399639	+	0.916672i	$0.369135\pi$
$798$	−31970.4	−1.41822
$799$	−5357.55	−0.237217
$800$	−3064.08	−0.135414
$801$	−9515.71	−0.419752
$802$	15572.3	0.685634
$803$	0	0
$804$	25037.4	1.09826
$805$	17067.9	0.747286
$806$	−32775.0	−1.43232
$807$	−36437.1	−1.58940
$808$	1132.43	0.0493056
$809$	−5144.24	−0.223562	−0.111781	−	0.993733i	$-0.535656\pi$
$809$	−5144.24	−0.223562	−0.111781	+	0.993733i	$0.535656\pi$
$810$	−6769.99	−0.293671
$811$	8255.99	0.357469	0.178734	−	0.983897i	$-0.442800\pi$
$811$	8255.99	0.357469	0.178734	+	0.983897i	$0.442800\pi$
$812$	1809.05	0.0781836
$813$	39703.6	1.71275
$814$	0	0
$815$	9821.58	0.422129
$816$	−7186.46	−0.308304
$817$	−14420.2	−0.617500
$818$	17719.8	0.757405
$819$	−51718.1	−2.20656
$820$	−1791.87	−0.0763106
$821$	−247.250	−0.0105105	−0.00525523	−	0.999986i	$-0.501673\pi$
$821$	−247.250	−0.0105105	−0.00525523	+	0.999986i	$0.501673\pi$
$822$	−14746.7	−0.625732
$823$	11794.8	0.499564	0.249782	−	0.968302i	$-0.419641\pi$
$823$	11794.8	0.499564	0.249782	+	0.968302i	$0.419641\pi$
$824$	6731.04	0.284571
$825$	0	0
$826$	13400.5	0.564484
$827$	2771.66	0.116542	0.0582709	−	0.998301i	$-0.481441\pi$
$827$	2771.66	0.116542	0.0582709	+	0.998301i	$0.481441\pi$
$828$	17321.5	0.727007
$829$	−27298.0	−1.14367	−0.571834	−	0.820370i	$-0.693768\pi$
$829$	−27298.0	−1.14367	−0.571834	+	0.820370i	$0.693768\pi$
$830$	8266.75	0.345715
$831$	40452.6	1.68867
$832$	−4916.92	−0.204884
$833$	−8751.82	−0.364025
$834$	19136.9	0.794554
$835$	−2049.26	−0.0849314
$836$	0	0
$837$	5481.18	0.226353
$838$	18938.2	0.780678
$839$	8754.74	0.360247	0.180123	−	0.983644i	$-0.442350\pi$
$839$	8754.74	0.360247	0.180123	+	0.983644i	$0.442350\pi$
$840$	7259.90	0.298203
$841$	−23972.1	−0.982906
$842$	10355.9	0.423857
$843$	24848.2	1.01521
$844$	6512.67	0.265611
$845$	20039.1	0.815816
$846$	5492.70	0.223218
$847$	0	0
$848$	−3757.58	−0.152165
$849$	37087.0	1.49920
$850$	11354.0	0.458164
$851$	−20742.5	−0.835537
$852$	−27241.3	−1.09539
$853$	−41184.3	−1.65313	−0.826566	−	0.562839i	$-0.809709\pi$
$853$	−41184.3	−1.65313	−0.826566	+	0.562839i	$0.809709\pi$
$854$	6613.03	0.264980
$855$	−15657.6	−0.626291
$856$	515.455	0.0205817
$857$	18020.0	0.718265	0.359132	−	0.933287i	$-0.383073\pi$
$857$	18020.0	0.718265	0.359132	+	0.933287i	$0.383073\pi$
$858$	0	0
$859$	6159.11	0.244640	0.122320	−	0.992491i	$-0.460967\pi$
$859$	6159.11	0.244640	0.122320	+	0.992491i	$0.460967\pi$
$860$	3274.56	0.129839
$861$	−13899.4	−0.550164
$862$	−20927.1	−0.826891
$863$	28932.6	1.14122	0.570612	−	0.821220i	$-0.306706\pi$
$863$	28932.6	1.14122	0.570612	+	0.821220i	$0.306706\pi$
$864$	822.289	0.0323783
$865$	−274.309	−0.0107824
$866$	31676.5	1.24297
$867$	−10590.1	−0.414831
$868$	18898.6	0.739009
$869$	0	0
$870$	1673.09	0.0651990
$871$	−63477.2	−2.46940
$872$	−12472.3	−0.484365
$873$	−17703.5	−0.686337
$874$	−27146.8	−1.05063
$875$	−26443.6	−1.02166
$876$	4181.61	0.161283
$877$	45152.6	1.73854	0.869268	−	0.494341i	$-0.164591\pi$
$877$	45152.6	1.73854	0.869268	+	0.494341i	$0.164591\pi$
$878$	9649.40	0.370902
$879$	−50801.4	−1.94936
$880$	0	0
$881$	22263.5	0.851391	0.425696	−	0.904866i	$-0.360029\pi$
$881$	22263.5	0.851391	0.425696	+	0.904866i	$0.360029\pi$
$882$	8972.58	0.342543
$883$	38215.5	1.45646	0.728230	−	0.685332i	$-0.240343\pi$
$883$	38215.5	1.45646	0.728230	+	0.685332i	$0.240343\pi$
$884$	18219.8	0.693209
$885$	12393.4	0.470736
$886$	7166.77	0.271752
$887$	−48160.6	−1.82308	−0.911541	−	0.411208i	$-0.865107\pi$
$887$	−48160.6	−1.82308	−0.911541	+	0.411208i	$0.865107\pi$
$888$	−8822.87	−0.333419
$889$	−24219.8	−0.913731
$890$	−3386.55	−0.127548
$891$	0	0
$892$	−10512.8	−0.394612
$893$	−8608.34	−0.322583
$894$	7414.23	0.277370
$895$	112.855	0.00421490
$896$	2835.18	0.105711
$897$	−82928.7	−3.08685
$898$	−15592.0	−0.579411
$899$	4355.31	0.161577
$900$	−11640.4	−0.431126
$901$	13923.8	0.514838
$902$	0	0
$903$	25400.6	0.936078
$904$	−17913.1	−0.659051
$905$	−952.655	−0.0349915
$906$	−25875.3	−0.948842
$907$	8509.88	0.311539	0.155770	−	0.987793i	$-0.450214\pi$
$907$	8509.88	0.311539	0.155770	+	0.987793i	$0.450214\pi$
$908$	−6810.22	−0.248904
$909$	4302.11	0.156977
$910$	−18406.0	−0.670496
$911$	11283.5	0.410360	0.205180	−	0.978724i	$-0.434222\pi$
$911$	11283.5	0.410360	0.205180	+	0.978724i	$0.434222\pi$
$912$	−11547.0	−0.419252
$913$	0	0
$914$	−6696.04	−0.242325
$915$	6116.04	0.220973
$916$	21583.8	0.778546
$917$	54625.0	1.96715
$918$	−3047.01	−0.109549
$919$	−26556.9	−0.953245	−0.476622	−	0.879108i	$-0.658139\pi$
$919$	−26556.9	−0.953245	−0.476622	+	0.879108i	$0.658139\pi$
$920$	6164.54	0.220912
$921$	−72024.9	−2.57687
$922$	−14343.9	−0.512355
$923$	69064.7	2.46294
$924$	0	0
$925$	13939.4	0.495486
$926$	8069.49	0.286371
$927$	25571.2	0.906006
$928$	653.386	0.0231125
$929$	−47913.1	−1.69212	−0.846059	−	0.533089i	$-0.821031\pi$
$929$	−47913.1	−1.69212	−0.846059	+	0.533089i	$0.821031\pi$
$930$	17478.3	0.616276
$931$	−14062.1	−0.495025
$932$	24216.7	0.851122
$933$	43697.5	1.53332
$934$	−23643.7	−0.828314
$935$	0	0
$936$	−18679.4	−0.652301
$937$	−5718.95	−0.199391	−0.0996957	−	0.995018i	$-0.531787\pi$
$937$	−5718.95	−0.199391	−0.0996957	+	0.995018i	$0.531787\pi$
$938$	36602.0	1.27409
$939$	45285.9	1.57386
$940$	1954.80	0.0678281
$941$	11105.6	0.384732	0.192366	−	0.981323i	$-0.438384\pi$
$941$	11105.6	0.384732	0.192366	+	0.981323i	$0.438384\pi$
$942$	38814.0	1.34249
$943$	−11802.3	−0.407567
$944$	4839.96	0.166872
$945$	3078.15	0.105960
$946$	0	0
$947$	39759.1	1.36430	0.682152	−	0.731210i	$-0.261044\pi$
$947$	39759.1	1.36430	0.682152	+	0.731210i	$0.261044\pi$
$948$	9226.01	0.316083
$949$	−10601.6	−0.362637
$950$	18243.2	0.623041
$951$	39635.8	1.35150
$952$	−10505.8	−0.357663
$953$	15932.0	0.541539	0.270770	−	0.962644i	$-0.412722\pi$
$953$	15932.0	0.541539	0.270770	+	0.962644i	$0.412722\pi$
$954$	−14275.0	−0.484456
$955$	−6269.18	−0.212425
$956$	−12546.4	−0.424456
$957$	0	0
$958$	17319.9	0.584113
$959$	−21558.1	−0.725910
$960$	2622.11	0.0881543
$961$	15707.7	0.527262
$962$	22368.6	0.749679
$963$	1958.21	0.0655270
$964$	25997.0	0.868577
$965$	6949.07	0.231812
$966$	47818.0	1.59267
$967$	−14952.9	−0.497264	−0.248632	−	0.968598i	$-0.579981\pi$
$967$	−14952.9	−0.497264	−0.248632	+	0.968598i	$0.579981\pi$
$968$	0	0
$969$	42787.5	1.41851
$970$	−6300.50	−0.208553
$971$	27937.3	0.923327	0.461663	−	0.887055i	$-0.347253\pi$
$971$	27937.3	0.923327	0.461663	+	0.887055i	$0.347253\pi$
$972$	−21742.2	−0.717472
$973$	27976.1	0.921760
$974$	14855.3	0.488701
$975$	55730.0	1.83055
$976$	2388.47	0.0783331
$977$	33359.2	1.09238	0.546191	−	0.837661i	$-0.316077\pi$
$977$	33359.2	1.09238	0.546191	+	0.837661i	$0.316077\pi$
$978$	27516.4	0.899671
$979$	0	0
$980$	3193.26	0.104087
$981$	−47382.3	−1.54210
$982$	11087.8	0.360311
$983$	12350.1	0.400718	0.200359	−	0.979723i	$-0.435789\pi$
$983$	12350.1	0.400718	0.200359	+	0.979723i	$0.435789\pi$
$984$	−5020.15	−0.162639
$985$	14521.1	0.469725
$986$	−2421.13	−0.0781994
$987$	15163.3	0.489009
$988$	29274.9	0.942671
$989$	21568.2	0.693457
$990$	0	0
$991$	−30154.0	−0.966571	−0.483286	−	0.875463i	$-0.660557\pi$
$991$	−30154.0	−0.966571	−0.483286	+	0.875463i	$0.660557\pi$
$992$	6825.73	0.218465
$993$	−30028.1	−0.959632
$994$	−39823.8	−1.27076
$995$	−7211.91	−0.229782
$996$	23160.4	0.736812
$997$	−2093.09	−0.0664884	−0.0332442	−	0.999447i	$-0.510584\pi$
$997$	−2093.09	−0.0664884	−0.0332442	+	0.999447i	$0.510584\pi$
$998$	40824.0	1.29485
$999$	−3740.84	−0.118473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.4.a.o.1.3		4
3.2	odd	2		2178.4.a.bt.1.3		4
4.3	odd	2		1936.4.a.bm.1.2		4
11.2	odd	10		242.4.c.r.81.2		8
11.3	even	5		242.4.c.q.9.1		8
11.4	even	5		242.4.c.q.27.1		8
11.5	even	5		242.4.c.n.3.2		8
11.6	odd	10		242.4.c.r.3.2		8
11.7	odd	10		22.4.c.b.5.1	✓	8
11.8	odd	10		22.4.c.b.9.1	yes	8
11.9	even	5		242.4.c.n.81.2		8
11.10	odd	2		242.4.a.n.1.3		4
33.8	even	10		198.4.f.d.163.2		8
33.29	even	10		198.4.f.d.181.2		8
33.32	even	2		2178.4.a.by.1.3		4
44.7	even	10		176.4.m.b.49.2		8
44.19	even	10		176.4.m.b.97.2		8
44.43	even	2		1936.4.a.bn.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.1	✓	8	11.7	odd	10
22.4.c.b.9.1	yes	8	11.8	odd	10
176.4.m.b.49.2		8	44.7	even	10
176.4.m.b.97.2		8	44.19	even	10
198.4.f.d.163.2		8	33.8	even	10
198.4.f.d.181.2		8	33.29	even	10
242.4.a.n.1.3		4	11.10	odd	2
242.4.a.o.1.3		4	1.1	even	1	trivial
242.4.c.n.3.2		8	11.5	even	5
242.4.c.n.81.2		8	11.9	even	5
242.4.c.q.9.1		8	11.3	even	5
242.4.c.q.27.1		8	11.4	even	5
242.4.c.r.3.2		8	11.6	odd	10
242.4.c.r.81.2		8	11.2	odd	10
1936.4.a.bm.1.2		4	4.3	odd	2
1936.4.a.bn.1.2		4	44.43	even	2
2178.4.a.bt.1.3		4	3.2	odd	2
2178.4.a.by.1.3		4	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +7.57575 q^{3} +4.00000 q^{4} +5.40810 q^{5} +15.1515 q^{6} +22.1498 q^{7} +8.00000 q^{8} +30.3919 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +7.57575 q^{3} +4.00000 q^{4} +5.40810 q^{5} +15.1515 q^{6} +22.1498 q^{7} +8.00000 q^{8} +30.3919 q^{9} +10.8162 q^{10} +30.3030 q^{12} -76.8269 q^{13} +44.2996 q^{14} +40.9704 q^{15} +16.0000 q^{16} -59.2883 q^{17} +60.7839 q^{18} -95.2626 q^{19} +21.6324 q^{20} +167.801 q^{21} +142.484 q^{23} +60.6060 q^{24} -95.7524 q^{25} -153.654 q^{26} +25.6965 q^{27} +88.5993 q^{28} +20.4183 q^{29} +81.9408 q^{30} +213.304 q^{31} +32.0000 q^{32} -118.577 q^{34} +119.788 q^{35} +121.568 q^{36} -145.578 q^{37} -190.525 q^{38} -582.021 q^{39} +43.2648 q^{40} -82.8326 q^{41} +335.603 q^{42} +151.373 q^{43} +164.363 q^{45} +284.968 q^{46} +90.3643 q^{47} +121.212 q^{48} +147.615 q^{49} -191.505 q^{50} -449.153 q^{51} -307.308 q^{52} -234.849 q^{53} +51.3931 q^{54} +177.199 q^{56} -721.685 q^{57} +40.8366 q^{58} +302.497 q^{59} +163.882 q^{60} +149.279 q^{61} +426.608 q^{62} +673.176 q^{63} +64.0000 q^{64} -415.488 q^{65} +826.236 q^{67} -237.153 q^{68} +1079.42 q^{69} +239.577 q^{70} -898.965 q^{71} +243.136 q^{72} +137.993 q^{73} -291.155 q^{74} -725.396 q^{75} -381.050 q^{76} -1164.04 q^{78} +304.459 q^{79} +86.5296 q^{80} -625.912 q^{81} -165.665 q^{82} +764.294 q^{83} +671.206 q^{84} -320.637 q^{85} +302.746 q^{86} +154.684 q^{87} -313.100 q^{89} +328.725 q^{90} -1701.70 q^{91} +569.936 q^{92} +1615.94 q^{93} +180.729 q^{94} -515.189 q^{95} +242.424 q^{96} -582.505 q^{97} +295.229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9	\( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} + 8 q^{6} + 3 q^{7} + 32 q^{8} + 102 q^{9} + 50 q^{10} + 16 q^{12} - 41 q^{13} + 6 q^{14} + 68 q^{15} + 64 q^{16} + 52 q^{17} + 204 q^{18} + 16 q^{19} + 100 q^{20} + 25 q^{21} + 314 q^{23} + 32 q^{24} - 21 q^{25} - 82 q^{26} + 286 q^{27} + 12 q^{28} + 561 q^{29} + 136 q^{30} + 199 q^{31} + 128 q^{32} + 104 q^{34} - 714 q^{35} + 408 q^{36} + 357 q^{37} + 32 q^{38} - 1038 q^{39} + 200 q^{40} + 32 q^{41} + 50 q^{42} - 721 q^{43} + 1326 q^{45} + 628 q^{46} + 403 q^{47} + 64 q^{48} + 823 q^{49} - 42 q^{50} - 174 q^{51} - 164 q^{52} - 133 q^{53} + 572 q^{54} + 24 q^{56} - 1031 q^{57} + 1122 q^{58} + 1016 q^{59} + 272 q^{60} - 919 q^{61} + 398 q^{62} - 1367 q^{63} + 256 q^{64} + 69 q^{65} + 289 q^{67} + 208 q^{68} - 1620 q^{69} - 1428 q^{70} - 1205 q^{71} + 816 q^{72} - 1234 q^{73} + 714 q^{74} - 911 q^{75} + 64 q^{76} - 2076 q^{78} - 603 q^{79} + 400 q^{80} - 1400 q^{81} + 64 q^{82} + 1514 q^{83} + 100 q^{84} + 717 q^{85} - 1442 q^{86} - 1061 q^{87} - 1101 q^{89} + 2652 q^{90} - 2306 q^{91} + 1256 q^{92} - 2298 q^{93} + 806 q^{94} - 1766 q^{95} + 128 q^{96} + 2116 q^{97} + 1646 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 + 8 * q^6 + 3 * q^7 + 32 * q^8 + 102 * q^9 + 50 * q^10 + 16 * q^12 - 41 * q^13 + 6 * q^14 + 68 * q^15 + 64 * q^16 + 52 * q^17 + 204 * q^18 + 16 * q^19 + 100 * q^20 + 25 * q^21 + 314 * q^23 + 32 * q^24 - 21 * q^25 - 82 * q^26 + 286 * q^27 + 12 * q^28 + 561 * q^29 + 136 * q^30 + 199 * q^31 + 128 * q^32 + 104 * q^34 - 714 * q^35 + 408 * q^36 + 357 * q^37 + 32 * q^38 - 1038 * q^39 + 200 * q^40 + 32 * q^41 + 50 * q^42 - 721 * q^43 + 1326 * q^45 + 628 * q^46 + 403 * q^47 + 64 * q^48 + 823 * q^49 - 42 * q^50 - 174 * q^51 - 164 * q^52 - 133 * q^53 + 572 * q^54 + 24 * q^56 - 1031 * q^57 + 1122 * q^58 + 1016 * q^59 + 272 * q^60 - 919 * q^61 + 398 * q^62 - 1367 * q^63 + 256 * q^64 + 69 * q^65 + 289 * q^67 + 208 * q^68 - 1620 * q^69 - 1428 * q^70 - 1205 * q^71 + 816 * q^72 - 1234 * q^73 + 714 * q^74 - 911 * q^75 + 64 * q^76 - 2076 * q^78 - 603 * q^79 + 400 * q^80 - 1400 * q^81 + 64 * q^82 + 1514 * q^83 + 100 * q^84 + 717 * q^85 - 1442 * q^86 - 1061 * q^87 - 1101 * q^89 + 2652 * q^90 - 2306 * q^91 + 1256 * q^92 - 2298 * q^93 + 806 * q^94 - 1766 * q^95 + 128 * q^96 + 2116 * q^97 + 1646 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more