LMFDB - Embedded newform 2442.4.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2442,4,Mod(1,2442)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2442.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2442, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2442 = 2 \cdot 3 \cdot 11 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2442.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,24,-36,48,15] 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:	$144.082664234$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 987 x^{10} + 1990 x^{9} + 356253 x^{8} - 389456 x^{7} - 58119076 x^{6} + \cdots - 100856276127 $ x^12 - 3x^11 - 987x^10 + 1990x^9 + 356253x^8 - 389456x^7 - 58119076x^6 + 33226526x^5 + 4183756273x^4 - 3584251341x^3 - 106356359697x^2 + 218229482620*x - 100856276127
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-11.9335$ of defining polynomial
Character	$\chi$	$=$	2442.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -10.9335 q^{5} -6.00000 q^{6} -16.0653 q^{7} +8.00000 q^{8} +9.00000 q^{9} -21.8671 q^{10} +11.0000 q^{11} -12.0000 q^{12} -13.5335 q^{13} -32.1306 q^{14} +32.8006 q^{15} +16.0000 q^{16} -9.81869 q^{17} +18.0000 q^{18} -11.9414 q^{19} -43.7342 q^{20} +48.1959 q^{21} +22.0000 q^{22} +3.58079 q^{23} -24.0000 q^{24} -5.45769 q^{25} -27.0670 q^{26} -27.0000 q^{27} -64.2613 q^{28} +17.9232 q^{29} +65.6012 q^{30} -158.084 q^{31} +32.0000 q^{32} -33.0000 q^{33} -19.6374 q^{34} +175.651 q^{35} +36.0000 q^{36} -37.0000 q^{37} -23.8828 q^{38} +40.6005 q^{39} -87.4683 q^{40} -317.405 q^{41} +96.3919 q^{42} +77.0272 q^{43} +44.0000 q^{44} -98.4019 q^{45} +7.16157 q^{46} -300.514 q^{47} -48.0000 q^{48} -84.9056 q^{49} -10.9154 q^{50} +29.4561 q^{51} -54.1340 q^{52} -393.890 q^{53} -54.0000 q^{54} -120.269 q^{55} -128.523 q^{56} +35.8242 q^{57} +35.8464 q^{58} -217.312 q^{59} +131.202 q^{60} +786.549 q^{61} -316.168 q^{62} -144.588 q^{63} +64.0000 q^{64} +147.969 q^{65} -66.0000 q^{66} +341.912 q^{67} -39.2748 q^{68} -10.7424 q^{69} +351.302 q^{70} +590.068 q^{71} +72.0000 q^{72} -1132.43 q^{73} -74.0000 q^{74} +16.3731 q^{75} -47.7655 q^{76} -176.718 q^{77} +81.2009 q^{78} +62.5044 q^{79} -174.937 q^{80} +81.0000 q^{81} -634.811 q^{82} +668.540 q^{83} +192.784 q^{84} +107.353 q^{85} +154.054 q^{86} -53.7697 q^{87} +88.0000 q^{88} -490.110 q^{89} -196.804 q^{90} +217.420 q^{91} +14.3231 q^{92} +474.252 q^{93} -601.029 q^{94} +130.562 q^{95} -96.0000 q^{96} +1237.68 q^{97} -169.811 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{2} - 36 q^{3} + 48 q^{4} + 15 q^{5} - 72 q^{6} + 31 q^{7} + 96 q^{8} + 108 q^{9} + 30 q^{10} + 132 q^{11} - 144 q^{12} + 58 q^{13} + 62 q^{14} - 45 q^{15} + 192 q^{16} + 193 q^{17} + 216 q^{18}+ \cdots + 1188 q^{99}+O(q^{100}) $ 12 * q + 24 * q^2 - 36 * q^3 + 48 * q^4 + 15 * q^5 - 72 * q^6 + 31 * q^7 + 96 * q^8 + 108 * q^9 + 30 * q^10 + 132 * q^11 - 144 * q^12 + 58 * q^13 + 62 * q^14 - 45 * q^15 + 192 * q^16 + 193 * q^17 + 216 * q^18 + 49 * q^19 + 60 * q^20 - 93 * q^21 + 264 * q^22 + 341 * q^23 - 288 * q^24 + 501 * q^25 + 116 * q^26 - 324 * q^27 + 124 * q^28 - 104 * q^29 - 90 * q^30 + 274 * q^31 + 384 * q^32 - 396 * q^33 + 386 * q^34 + 573 * q^35 + 432 * q^36 - 444 * q^37 + 98 * q^38 - 174 * q^39 + 120 * q^40 + 259 * q^41 - 186 * q^42 + 680 * q^43 + 528 * q^44 + 135 * q^45 + 682 * q^46 - 127 * q^47 - 576 * q^48 + 1111 * q^49 + 1002 * q^50 - 579 * q^51 + 232 * q^52 + 322 * q^53 - 648 * q^54 + 165 * q^55 + 248 * q^56 - 147 * q^57 - 208 * q^58 - 206 * q^59 - 180 * q^60 + 59 * q^61 + 548 * q^62 + 279 * q^63 + 768 * q^64 + 741 * q^65 - 792 * q^66 + 272 * q^67 + 772 * q^68 - 1023 * q^69 + 1146 * q^70 + 236 * q^71 + 864 * q^72 + 773 * q^73 - 888 * q^74 - 1503 * q^75 + 196 * q^76 + 341 * q^77 - 348 * q^78 - 809 * q^79 + 240 * q^80 + 972 * q^81 + 518 * q^82 - 395 * q^83 - 372 * q^84 + 1972 * q^85 + 1360 * q^86 + 312 * q^87 + 1056 * q^88 + 1515 * q^89 + 270 * q^90 - 124 * q^91 + 1364 * q^92 - 822 * q^93 - 254 * q^94 + 253 * q^95 - 1152 * q^96 + 43 * q^97 + 2222 * q^98 + 1188 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−10.9335	−0.977926	−0.488963	−	0.872305i	$-0.662625\pi$
$5$	−10.9335	−0.977926	−0.488963	+	0.872305i	$0.662625\pi$
$6$	−6.00000	−0.408248
$7$	−16.0653	−0.867446	−0.433723	−	0.901046i	$-0.642800\pi$
$7$	−16.0653	−0.867446	−0.433723	+	0.901046i	$0.642800\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−21.8671	−0.691498
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−12.0000	−0.288675
$13$	−13.5335	−0.288732	−0.144366	−	0.989524i	$-0.546114\pi$
$13$	−13.5335	−0.288732	−0.144366	+	0.989524i	$0.546114\pi$
$14$	−32.1306	−0.613377
$15$	32.8006	0.564606
$16$	16.0000	0.250000
$17$	−9.81869	−0.140081	−0.0700407	−	0.997544i	$-0.522313\pi$
$17$	−9.81869	−0.140081	−0.0700407	+	0.997544i	$0.522313\pi$
$18$	18.0000	0.235702
$19$	−11.9414	−0.144186	−0.0720932	−	0.997398i	$-0.522968\pi$
$19$	−11.9414	−0.144186	−0.0720932	+	0.997398i	$0.522968\pi$
$20$	−43.7342	−0.488963
$21$	48.1959	0.500820
$22$	22.0000	0.213201
$23$	3.58079	0.0324629	0.0162314	−	0.999868i	$-0.494833\pi$
$23$	3.58079	0.0324629	0.0162314	+	0.999868i	$0.494833\pi$
$24$	−24.0000	−0.204124
$25$	−5.45769	−0.0436615
$26$	−27.0670	−0.204164
$27$	−27.0000	−0.192450
$28$	−64.2613	−0.433723
$29$	17.9232	0.114768	0.0573838	−	0.998352i	$-0.481724\pi$
$29$	17.9232	0.114768	0.0573838	+	0.998352i	$0.481724\pi$
$30$	65.6012	0.399236
$31$	−158.084	−0.915895	−0.457947	−	0.888979i	$-0.651415\pi$
$31$	−158.084	−0.915895	−0.457947	+	0.888979i	$0.651415\pi$
$32$	32.0000	0.176777
$33$	−33.0000	−0.174078
$34$	−19.6374	−0.0990524
$35$	175.651	0.848297
$36$	36.0000	0.166667
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	−23.8828	−0.101955
$39$	40.6005	0.166699
$40$	−87.4683	−0.345749
$41$	−317.405	−1.20903	−0.604517	−	0.796592i	$-0.706634\pi$
$41$	−317.405	−1.20903	−0.604517	+	0.796592i	$0.706634\pi$
$42$	96.3919	0.354133
$43$	77.0272	0.273175	0.136588	−	0.990628i	$-0.456386\pi$
$43$	77.0272	0.273175	0.136588	+	0.990628i	$0.456386\pi$
$44$	44.0000	0.150756
$45$	−98.4019	−0.325975
$46$	7.16157	0.0229547
$47$	−300.514	−0.932650	−0.466325	−	0.884614i	$-0.654422\pi$
$47$	−300.514	−0.932650	−0.466325	+	0.884614i	$0.654422\pi$
$48$	−48.0000	−0.144338
$49$	−84.9056	−0.247538
$50$	−10.9154	−0.0308733
$51$	29.4561	0.0808760
$52$	−54.1340	−0.144366
$53$	−393.890	−1.02085	−0.510424	−	0.859923i	$-0.670512\pi$
$53$	−393.890	−1.02085	−0.510424	+	0.859923i	$0.670512\pi$
$54$	−54.0000	−0.136083
$55$	−120.269	−0.294856
$56$	−128.523	−0.306688
$57$	35.8242	0.0832461
$58$	35.8464	0.0811529
$59$	−217.312	−0.479519	−0.239759	−	0.970832i	$-0.577069\pi$
$59$	−217.312	−0.479519	−0.239759	+	0.970832i	$0.577069\pi$
$60$	131.202	0.282303
$61$	786.549	1.65094	0.825469	−	0.564447i	$-0.190911\pi$
$61$	786.549	1.65094	0.825469	+	0.564447i	$0.190911\pi$
$62$	−316.168	−0.647635
$63$	−144.588	−0.289149
$64$	64.0000	0.125000
$65$	147.969	0.282358
$66$	−66.0000	−0.123091
$67$	341.912	0.623451	0.311725	−	0.950172i	$-0.399093\pi$
$67$	341.912	0.623451	0.311725	+	0.950172i	$0.399093\pi$
$68$	−39.2748	−0.0700407
$69$	−10.7424	−0.0187424
$70$	351.302	0.599837
$71$	590.068	0.986312	0.493156	−	0.869941i	$-0.335843\pi$
$71$	590.068	0.986312	0.493156	+	0.869941i	$0.335843\pi$
$72$	72.0000	0.117851
$73$	−1132.43	−1.81562	−0.907811	−	0.419379i	$-0.862248\pi$
$73$	−1132.43	−1.81562	−0.907811	+	0.419379i	$0.862248\pi$
$74$	−74.0000	−0.116248
$75$	16.3731	0.0252080
$76$	−47.7655	−0.0720932
$77$	−176.718	−0.261545
$78$	81.2009	0.117874
$79$	62.5044	0.0890163	0.0445082	−	0.999009i	$-0.485828\pi$
$79$	62.5044	0.0890163	0.0445082	+	0.999009i	$0.485828\pi$
$80$	−174.937	−0.244481
$81$	81.0000	0.111111
$82$	−634.811	−0.854916
$83$	668.540	0.884118	0.442059	−	0.896986i	$-0.354248\pi$
$83$	668.540	0.884118	0.442059	+	0.896986i	$0.354248\pi$
$84$	192.784	0.250410
$85$	107.353	0.136989
$86$	154.054	0.193164
$87$	−53.7697	−0.0662611
$88$	88.0000	0.106600
$89$	−490.110	−0.583726	−0.291863	−	0.956460i	$-0.594275\pi$
$89$	−490.110	−0.583726	−0.291863	+	0.956460i	$0.594275\pi$
$90$	−196.804	−0.230499
$91$	217.420	0.250459
$92$	14.3231	0.0162314
$93$	474.252	0.528792
$94$	−601.029	−0.659483
$95$	130.562	0.141004
$96$	−96.0000	−0.102062
$97$	1237.68	1.29554	0.647769	−	0.761837i	$-0.275702\pi$
$97$	1237.68	1.29554	0.647769	+	0.761837i	$0.275702\pi$
$98$	−169.811	−0.175036
$99$	99.0000	0.100504
$100$	−21.8307	−0.0218307
$101$	265.995	0.262055	0.131027	−	0.991379i	$-0.458172\pi$
$101$	265.995	0.262055	0.131027	+	0.991379i	$0.458172\pi$
$102$	58.9121	0.0571880
$103$	301.977	0.288881	0.144440	−	0.989514i	$-0.453862\pi$
$103$	301.977	0.288881	0.144440	+	0.989514i	$0.453862\pi$
$104$	−108.268	−0.102082
$105$	−526.952	−0.489765
$106$	−787.780	−0.721849
$107$	−1143.84	−1.03345	−0.516725	−	0.856151i	$-0.672849\pi$
$107$	−1143.84	−1.03345	−0.516725	+	0.856151i	$0.672849\pi$
$108$	−108.000	−0.0962250
$109$	−284.939	−0.250387	−0.125193	−	0.992132i	$-0.539955\pi$
$109$	−284.939	−0.250387	−0.125193	+	0.992132i	$0.539955\pi$
$110$	−240.538	−0.208494
$111$	111.000	0.0949158
$112$	−257.045	−0.216861
$113$	2265.41	1.88594	0.942971	−	0.332876i	$-0.108019\pi$
$113$	2265.41	1.88594	0.942971	+	0.332876i	$0.108019\pi$
$114$	71.6483	0.0588639
$115$	−39.1507	−0.0317463
$116$	71.6929	0.0573838
$117$	−121.801	−0.0962440
$118$	−434.624	−0.339071
$119$	157.740	0.121513
$120$	262.405	0.199618
$121$	121.000	0.0909091
$122$	1573.10	1.16739
$123$	952.216	0.698036
$124$	−632.336	−0.457947
$125$	1426.36	1.02062
$126$	−289.176	−0.204459
$127$	−261.354	−0.182610	−0.0913048	−	0.995823i	$-0.529104\pi$
$127$	−261.354	−0.182610	−0.0913048	+	0.995823i	$0.529104\pi$
$128$	128.000	0.0883883
$129$	−231.082	−0.157718
$130$	295.938	0.199657
$131$	−1406.58	−0.938119	−0.469060	−	0.883167i	$-0.655407\pi$
$131$	−1406.58	−0.938119	−0.469060	+	0.883167i	$0.655407\pi$
$132$	−132.000	−0.0870388
$133$	191.842	0.125074
$134$	683.824	0.440846
$135$	295.206	0.188202
$136$	−78.5495	−0.0495262
$137$	391.946	0.244425	0.122212	−	0.992504i	$-0.461001\pi$
$137$	391.946	0.244425	0.122212	+	0.992504i	$0.461001\pi$
$138$	−21.4847	−0.0132529
$139$	1608.61	0.981587	0.490793	−	0.871276i	$-0.336707\pi$
$139$	1608.61	0.981587	0.490793	+	0.871276i	$0.336707\pi$
$140$	702.603	0.424149
$141$	901.543	0.538466
$142$	1180.14	0.697428
$143$	−148.868	−0.0870559
$144$	144.000	0.0833333
$145$	−195.964	−0.112234
$146$	−2264.85	−1.28384
$147$	254.717	0.142916
$148$	−148.000	−0.0821995
$149$	−346.803	−0.190679	−0.0953397	−	0.995445i	$-0.530394\pi$
$149$	−346.803	−0.190679	−0.0953397	+	0.995445i	$0.530394\pi$
$150$	32.7461	0.0178247
$151$	3283.85	1.76977	0.884887	−	0.465805i	$-0.154235\pi$
$151$	3283.85	1.76977	0.884887	+	0.465805i	$0.154235\pi$
$152$	−95.5311	−0.0509776
$153$	−88.3682	−0.0466938
$154$	−353.437	−0.184940
$155$	1728.42	0.895677
$156$	162.402	0.0833497
$157$	2682.69	1.36371	0.681853	−	0.731490i	$-0.261175\pi$
$157$	2682.69	1.36371	0.681853	+	0.731490i	$0.261175\pi$
$158$	125.009	0.0629441
$159$	1181.67	0.589387
$160$	−349.873	−0.172874
$161$	−57.5265	−0.0281598
$162$	162.000	0.0785674
$163$	640.480	0.307768	0.153884	−	0.988089i	$-0.450822\pi$
$163$	640.480	0.307768	0.153884	+	0.988089i	$0.450822\pi$
$164$	−1269.62	−0.604517
$165$	360.807	0.170235
$166$	1337.08	0.625166
$167$	−3290.20	−1.52457	−0.762286	−	0.647240i	$-0.775923\pi$
$167$	−3290.20	−1.52457	−0.762286	+	0.647240i	$0.775923\pi$
$168$	385.568	0.177067
$169$	−2013.84	−0.916634
$170$	214.706	0.0968659
$171$	−107.472	−0.0480621
$172$	308.109	0.136588
$173$	2519.04	1.10705	0.553524	−	0.832833i	$-0.313283\pi$
$173$	2519.04	1.10705	0.553524	+	0.832833i	$0.313283\pi$
$174$	−107.539	−0.0468536
$175$	87.6795	0.0378740
$176$	176.000	0.0753778
$177$	651.936	0.276850
$178$	−980.221	−0.412756
$179$	3340.79	1.39499	0.697493	−	0.716592i	$-0.254299\pi$
$179$	3340.79	1.39499	0.697493	+	0.716592i	$0.254299\pi$
$180$	−393.607	−0.162988
$181$	−1785.35	−0.733173	−0.366586	−	0.930384i	$-0.619474\pi$
$181$	−1785.35	−0.733173	−0.366586	+	0.930384i	$0.619474\pi$
$182$	434.840	0.177101
$183$	−2359.65	−0.953170
$184$	28.6463	0.0114774
$185$	404.541	0.160770
$186$	948.504	0.373912
$187$	−108.006	−0.0422361
$188$	−1202.06	−0.466325
$189$	433.764	0.166940
$190$	261.123	0.0997046
$191$	−777.765	−0.294644	−0.147322	−	0.989089i	$-0.547065\pi$
$191$	−777.765	−0.294644	−0.147322	+	0.989089i	$0.547065\pi$
$192$	−192.000	−0.0721688
$193$	4833.70	1.80278	0.901391	−	0.433005i	$-0.142547\pi$
$193$	4833.70	1.80278	0.901391	+	0.433005i	$0.142547\pi$
$194$	2475.36	0.916083
$195$	−443.907	−0.163020
$196$	−339.622	−0.123769
$197$	−1041.42	−0.376640	−0.188320	−	0.982108i	$-0.560304\pi$
$197$	−1041.42	−0.376640	−0.188320	+	0.982108i	$0.560304\pi$
$198$	198.000	0.0710669
$199$	−3545.04	−1.26282	−0.631410	−	0.775449i	$-0.717524\pi$
$199$	−3545.04	−1.26282	−0.631410	+	0.775449i	$0.717524\pi$
$200$	−43.6615	−0.0154367
$201$	−1025.74	−0.359949
$202$	531.990	0.185301
$203$	−287.942	−0.0995546
$204$	117.824	0.0404380
$205$	3470.36	1.18235
$206$	603.955	0.204270
$207$	32.2271	0.0108210
$208$	−216.536	−0.0721830
$209$	−131.355	−0.0434738
$210$	−1053.90	−0.346316
$211$	3613.88	1.17910	0.589548	−	0.807733i	$-0.299306\pi$
$211$	3613.88	1.17910	0.589548	+	0.807733i	$0.299306\pi$
$212$	−1575.56	−0.510424
$213$	−1770.20	−0.569447
$214$	−2287.68	−0.730759
$215$	−842.180	−0.267145
$216$	−216.000	−0.0680414
$217$	2539.67	0.794489
$218$	−569.877	−0.177050
$219$	3397.28	1.04825
$220$	−481.076	−0.147428
$221$	132.881	0.0404459
$222$	222.000	0.0671156
$223$	5934.28	1.78201	0.891007	−	0.453990i	$-0.150000\pi$
$223$	5934.28	1.78201	0.891007	+	0.453990i	$0.150000\pi$
$224$	−514.090	−0.153344
$225$	−49.1192	−0.0145538
$226$	4530.81	1.33356
$227$	2981.33	0.871707	0.435854	−	0.900018i	$-0.356447\pi$
$227$	2981.33	0.871707	0.435854	+	0.900018i	$0.356447\pi$
$228$	143.297	0.0416230
$229$	−2396.83	−0.691647	−0.345824	−	0.938300i	$-0.612400\pi$
$229$	−2396.83	−0.691647	−0.345824	+	0.938300i	$0.612400\pi$
$230$	−78.3014	−0.0224480
$231$	530.155	0.151003
$232$	143.386	0.0405765
$233$	4593.00	1.29141	0.645703	−	0.763589i	$-0.276564\pi$
$233$	4593.00	1.29141	0.645703	+	0.763589i	$0.276564\pi$
$234$	−243.603	−0.0680548
$235$	3285.69	0.912062
$236$	−869.248	−0.239759
$237$	−187.513	−0.0513936
$238$	315.481	0.0859226
$239$	2641.69	0.714965	0.357482	−	0.933920i	$-0.383635\pi$
$239$	2641.69	0.714965	0.357482	+	0.933920i	$0.383635\pi$
$240$	524.810	0.141151
$241$	−3549.78	−0.948803	−0.474402	−	0.880309i	$-0.657336\pi$
$241$	−3549.78	−0.948803	−0.474402	+	0.880309i	$0.657336\pi$
$242$	242.000	0.0642824
$243$	−243.000	−0.0641500
$244$	3146.19	0.825469
$245$	928.319	0.242074
$246$	1904.43	0.493586
$247$	161.609	0.0416312
$248$	−1264.67	−0.323818
$249$	−2005.62	−0.510446
$250$	2852.73	0.721690
$251$	−1648.83	−0.414635	−0.207318	−	0.978274i	$-0.566473\pi$
$251$	−1648.83	−0.414635	−0.207318	+	0.978274i	$0.566473\pi$
$252$	−578.351	−0.144574
$253$	39.3887	0.00978792
$254$	−522.708	−0.129125
$255$	−322.059	−0.0790907
$256$	256.000	0.0625000
$257$	6174.78	1.49872	0.749362	−	0.662161i	$-0.230360\pi$
$257$	6174.78	1.49872	0.749362	+	0.662161i	$0.230360\pi$
$258$	−462.163	−0.111523
$259$	594.417	0.142607
$260$	591.876	0.141179
$261$	161.309	0.0382558
$262$	−2813.16	−0.663350
$263$	922.879	0.216377	0.108189	−	0.994130i	$-0.465495\pi$
$263$	922.879	0.216377	0.108189	+	0.994130i	$0.465495\pi$
$264$	−264.000	−0.0615457
$265$	4306.61	0.998314
$266$	383.684	0.0884406
$267$	1470.33	0.337014
$268$	1367.65	0.311725
$269$	7299.94	1.65459	0.827296	−	0.561766i	$-0.189878\pi$
$269$	7299.94	1.65459	0.827296	+	0.561766i	$0.189878\pi$
$270$	590.411	0.133079
$271$	−2751.56	−0.616772	−0.308386	−	0.951261i	$-0.599789\pi$
$271$	−2751.56	−0.616772	−0.308386	+	0.951261i	$0.599789\pi$
$272$	−157.099	−0.0350203
$273$	−652.259	−0.144603
$274$	783.892	0.172834
$275$	−60.0345	−0.0131644
$276$	−42.9694	−0.00937122
$277$	−7450.39	−1.61607	−0.808033	−	0.589137i	$-0.799468\pi$
$277$	−7450.39	−1.61607	−0.808033	+	0.589137i	$0.799468\pi$
$278$	3217.22	0.694087
$279$	−1422.76	−0.305298
$280$	1405.21	0.299918
$281$	−3284.70	−0.697326	−0.348663	−	0.937248i	$-0.613364\pi$
$281$	−3284.70	−0.697326	−0.348663	+	0.937248i	$0.613364\pi$
$282$	1803.09	0.380753
$283$	−8610.72	−1.80867	−0.904336	−	0.426821i	$-0.859634\pi$
$283$	−8610.72	−1.80867	−0.904336	+	0.426821i	$0.859634\pi$
$284$	2360.27	0.493156
$285$	−391.685	−0.0814085
$286$	−297.737	−0.0615578
$287$	5099.22	1.04877
$288$	288.000	0.0589256
$289$	−4816.59	−0.980377
$290$	−391.929	−0.0793615
$291$	−3713.03	−0.747979
$292$	−4529.70	−0.907811
$293$	9597.27	1.91358	0.956789	−	0.290784i	$-0.0939160\pi$
$293$	9597.27	1.91358	0.956789	+	0.290784i	$0.0939160\pi$
$294$	509.434	0.101057
$295$	2375.99	0.468934
$296$	−296.000	−0.0581238
$297$	−297.000	−0.0580259
$298$	−693.606	−0.134831
$299$	−48.4605	−0.00937306
$300$	65.4922	0.0126040
$301$	−1237.47	−0.236965
$302$	6567.70	1.25142
$303$	−797.986	−0.151297
$304$	−191.062	−0.0360466
$305$	−8599.76	−1.61449
$306$	−176.736	−0.0330175
$307$	1029.92	0.191469	0.0957344	−	0.995407i	$-0.469480\pi$
$307$	1029.92	0.191469	0.0957344	+	0.995407i	$0.469480\pi$
$308$	−706.874	−0.130772
$309$	−905.932	−0.166785
$310$	3456.84	0.633339
$311$	7018.15	1.27962	0.639812	−	0.768532i	$-0.279012\pi$
$311$	7018.15	1.27962	0.639812	+	0.768532i	$0.279012\pi$
$312$	324.804	0.0589371
$313$	−4474.84	−0.808092	−0.404046	−	0.914739i	$-0.632397\pi$
$313$	−4474.84	−0.808092	−0.404046	+	0.914739i	$0.632397\pi$
$314$	5365.37	0.964285
$315$	1580.86	0.282766
$316$	250.017	0.0445082
$317$	5850.46	1.03658	0.518288	−	0.855206i	$-0.326570\pi$
$317$	5850.46	1.03658	0.518288	+	0.855206i	$0.326570\pi$
$318$	2363.34	0.416760
$319$	197.155	0.0346037
$320$	−699.747	−0.122241
$321$	3431.52	0.596663
$322$	−115.053	−0.0199120
$323$	117.249	0.0201978
$324$	324.000	0.0555556
$325$	73.8615	0.0126065
$326$	1280.96	0.217625
$327$	854.816	0.144561
$328$	−2539.24	−0.427458
$329$	4827.86	0.809023
$330$	721.614	0.120374
$331$	3949.28	0.655806	0.327903	−	0.944711i	$-0.393658\pi$
$331$	3949.28	0.655806	0.327903	+	0.944711i	$0.393658\pi$
$332$	2674.16	0.442059
$333$	−333.000	−0.0547997
$334$	−6580.40	−1.07804
$335$	−3738.31	−0.609688
$336$	771.135	0.125205
$337$	1671.16	0.270131	0.135065	−	0.990837i	$-0.456876\pi$
$337$	1671.16	0.270131	0.135065	+	0.990837i	$0.456876\pi$
$338$	−4027.69	−0.648158
$339$	−6796.22	−1.08885
$340$	429.412	0.0684946
$341$	−1738.92	−0.276153
$342$	−214.945	−0.0339851
$343$	6874.44	1.08217
$344$	616.218	0.0965821
$345$	117.452	0.0183287
$346$	5038.09	0.782801
$347$	−5153.29	−0.797242	−0.398621	−	0.917116i	$-0.630511\pi$
$347$	−5153.29	−0.797242	−0.398621	+	0.917116i	$0.630511\pi$
$348$	−215.079	−0.0331305
$349$	4917.61	0.754252	0.377126	−	0.926162i	$-0.376912\pi$
$349$	4917.61	0.754252	0.377126	+	0.926162i	$0.376912\pi$
$350$	175.359	0.0267809
$351$	365.404	0.0555665
$352$	352.000	0.0533002
$353$	−9736.55	−1.46806	−0.734029	−	0.679119i	$-0.762362\pi$
$353$	−9736.55	−1.46806	−0.734029	+	0.679119i	$0.762362\pi$
$354$	1303.87	0.195763
$355$	−6451.53	−0.964540
$356$	−1960.44	−0.291863
$357$	−473.221	−0.0701555
$358$	6681.58	0.986404
$359$	272.185	0.0400150	0.0200075	−	0.999800i	$-0.493631\pi$
$359$	272.185	0.0400150	0.0200075	+	0.999800i	$0.493631\pi$
$360$	−787.215	−0.115250
$361$	−6716.40	−0.979210
$362$	−3570.71	−0.518432
$363$	−363.000	−0.0524864
$364$	869.679	0.125230
$365$	12381.4	1.77554
$366$	−4719.29	−0.673993
$367$	−11252.1	−1.60042	−0.800210	−	0.599720i	$-0.795279\pi$
$367$	−11252.1	−1.60042	−0.800210	+	0.599720i	$0.795279\pi$
$368$	57.2926	0.00811571
$369$	−2856.65	−0.403011
$370$	809.082	0.113682
$371$	6327.97	0.885531
$372$	1897.01	0.264396
$373$	−4371.85	−0.606878	−0.303439	−	0.952851i	$-0.598135\pi$
$373$	−4371.85	−0.606878	−0.303439	+	0.952851i	$0.598135\pi$
$374$	−216.011	−0.0298654
$375$	−4279.09	−0.589257
$376$	−2404.12	−0.329742
$377$	−242.564	−0.0331370
$378$	867.527	0.118044
$379$	7062.94	0.957253	0.478626	−	0.878019i	$-0.341135\pi$
$379$	7062.94	0.957253	0.478626	+	0.878019i	$0.341135\pi$
$380$	522.247	0.0705018
$381$	784.062	0.105430
$382$	−1555.53	−0.208345
$383$	−6166.74	−0.822730	−0.411365	−	0.911471i	$-0.634948\pi$
$383$	−6166.74	−0.822730	−0.411365	+	0.911471i	$0.634948\pi$
$384$	−384.000	−0.0510310
$385$	1932.16	0.255771
$386$	9667.39	1.27476
$387$	693.245	0.0910584
$388$	4950.71	0.647769
$389$	2084.49	0.271691	0.135845	−	0.990730i	$-0.456625\pi$
$389$	2084.49	0.271691	0.135845	+	0.990730i	$0.456625\pi$
$390$	−887.814	−0.115272
$391$	−35.1586	−0.00454744
$392$	−679.245	−0.0875180
$393$	4219.74	0.541623
$394$	−2082.84	−0.266324
$395$	−683.394	−0.0870514
$396$	396.000	0.0502519
$397$	13060.2	1.65107	0.825534	−	0.564352i	$-0.190874\pi$
$397$	13060.2	1.65107	0.825534	+	0.564352i	$0.190874\pi$
$398$	−7090.08	−0.892949
$399$	−575.526	−0.0722114
$400$	−87.3230	−0.0109154
$401$	−12346.9	−1.53759	−0.768795	−	0.639495i	$-0.779144\pi$
$401$	−12346.9	−1.53759	−0.768795	+	0.639495i	$0.779144\pi$
$402$	−2051.47	−0.254523
$403$	2139.43	0.264448
$404$	1063.98	0.131027
$405$	−885.617	−0.108658
$406$	−575.884	−0.0703957
$407$	−407.000	−0.0495682
$408$	235.649	0.0285940
$409$	6959.79	0.841417	0.420708	−	0.907196i	$-0.361782\pi$
$409$	6959.79	0.841417	0.420708	+	0.907196i	$0.361782\pi$
$410$	6940.73	0.836044
$411$	−1175.84	−0.141119
$412$	1207.91	0.144440
$413$	3491.19	0.415957
$414$	64.4542	0.00765157
$415$	−7309.51	−0.864602
$416$	−433.072	−0.0510411
$417$	−4825.83	−0.566719
$418$	−262.710	−0.0307406
$419$	4604.08	0.536811	0.268406	−	0.963306i	$-0.413503\pi$
$419$	4604.08	0.536811	0.268406	+	0.963306i	$0.413503\pi$
$420$	−2107.81	−0.244882
$421$	−8610.76	−0.996824	−0.498412	−	0.866940i	$-0.666083\pi$
$421$	−8610.76	−0.996824	−0.498412	+	0.866940i	$0.666083\pi$
$422$	7227.75	0.833747
$423$	−2704.63	−0.310883
$424$	−3151.12	−0.360924
$425$	53.5873	0.00611616
$426$	−3540.41	−0.402660
$427$	−12636.2	−1.43210
$428$	−4575.36	−0.516725
$429$	446.605	0.0502618
$430$	−1684.36	−0.188900
$431$	13409.1	1.49860	0.749299	−	0.662232i	$-0.230390\pi$
$431$	13409.1	1.49860	0.749299	+	0.662232i	$0.230390\pi$
$432$	−432.000	−0.0481125
$433$	2097.08	0.232746	0.116373	−	0.993206i	$-0.462873\pi$
$433$	2097.08	0.232746	0.116373	+	0.993206i	$0.462873\pi$
$434$	5079.34	0.561788
$435$	587.893	0.0647984
$436$	−1139.75	−0.125193
$437$	−42.7596	−0.00468070
$438$	6794.56	0.741225
$439$	−3377.68	−0.367217	−0.183608	−	0.982999i	$-0.558778\pi$
$439$	−3377.68	−0.367217	−0.183608	+	0.982999i	$0.558778\pi$
$440$	−962.152	−0.104247
$441$	−764.150	−0.0825127
$442$	265.762	0.0285996
$443$	8334.25	0.893842	0.446921	−	0.894573i	$-0.352520\pi$
$443$	8334.25	0.893842	0.446921	+	0.894573i	$0.352520\pi$
$444$	444.000	0.0474579
$445$	5358.64	0.570840
$446$	11868.6	1.26007
$447$	1040.41	0.110089
$448$	−1028.18	−0.108431
$449$	2914.02	0.306283	0.153141	−	0.988204i	$-0.451061\pi$
$449$	2914.02	0.306283	0.153141	+	0.988204i	$0.451061\pi$
$450$	−98.2383	−0.0102911
$451$	−3491.46	−0.364537
$452$	9061.62	0.942971
$453$	−9851.55	−1.02178
$454$	5962.65	0.616390
$455$	−2377.17	−0.244930
$456$	286.593	0.0294319
$457$	2488.19	0.254689	0.127344	−	0.991859i	$-0.459355\pi$
$457$	2488.19	0.254689	0.127344	+	0.991859i	$0.459355\pi$
$458$	−4793.67	−0.489068
$459$	265.105	0.0269587
$460$	−156.603	−0.0158731
$461$	−1457.32	−0.147232	−0.0736160	−	0.997287i	$-0.523454\pi$
$461$	−1457.32	−0.147232	−0.0736160	+	0.997287i	$0.523454\pi$
$462$	1060.31	0.106775
$463$	−11232.9	−1.12751	−0.563754	−	0.825943i	$-0.690643\pi$
$463$	−11232.9	−1.12751	−0.563754	+	0.825943i	$0.690643\pi$
$464$	286.772	0.0286919
$465$	−5185.25	−0.517119
$466$	9186.00	0.913162
$467$	527.735	0.0522926	0.0261463	−	0.999658i	$-0.491676\pi$
$467$	527.735	0.0522926	0.0261463	+	0.999658i	$0.491676\pi$
$468$	−487.206	−0.0481220
$469$	−5492.92	−0.540810
$470$	6571.37	0.644925
$471$	−8048.06	−0.787336
$472$	−1738.50	−0.169536
$473$	847.299	0.0823655
$474$	−375.026	−0.0363408
$475$	65.1723	0.00629539
$476$	630.961	0.0607565
$477$	−3545.01	−0.340283
$478$	5283.37	0.505556
$479$	1647.77	0.157178	0.0785892	−	0.996907i	$-0.474958\pi$
$479$	1647.77	0.157178	0.0785892	+	0.996907i	$0.474958\pi$
$480$	1049.62	0.0998091
$481$	500.739	0.0474672
$482$	−7099.56	−0.670905
$483$	172.579	0.0162580
$484$	484.000	0.0454545
$485$	−13532.2	−1.26694
$486$	−486.000	−0.0453609
$487$	20901.7	1.94486	0.972429	−	0.233197i	$-0.0749188\pi$
$487$	20901.7	1.94486	0.972429	+	0.233197i	$0.0749188\pi$
$488$	6292.39	0.583695
$489$	−1921.44	−0.177690
$490$	1856.64	0.171172
$491$	661.192	0.0607722	0.0303861	−	0.999538i	$-0.490326\pi$
$491$	661.192	0.0607722	0.0303861	+	0.999538i	$0.490326\pi$
$492$	3808.86	0.349018
$493$	−175.983	−0.0160768
$494$	323.217	0.0294377
$495$	−1082.42	−0.0982852
$496$	−2529.34	−0.228974
$497$	−9479.62	−0.855572
$498$	−4011.24	−0.360940
$499$	19224.5	1.72466	0.862330	−	0.506347i	$-0.169004\pi$
$499$	19224.5	1.72466	0.862330	+	0.506347i	$0.169004\pi$
$500$	5705.46	0.510312
$501$	9870.61	0.880212
$502$	−3297.67	−0.293191
$503$	−15957.6	−1.41454	−0.707269	−	0.706945i	$-0.750073\pi$
$503$	−15957.6	−1.41454	−0.707269	+	0.706945i	$0.750073\pi$
$504$	−1156.70	−0.102229
$505$	−2908.27	−0.256270
$506$	78.7773	0.00692110
$507$	6041.53	0.529219
$508$	−1045.42	−0.0913048
$509$	−11729.1	−1.02138	−0.510692	−	0.859763i	$-0.670611\pi$
$509$	−11729.1	−1.02138	−0.510692	+	0.859763i	$0.670611\pi$
$510$	−644.118	−0.0559256
$511$	18192.8	1.57495
$512$	512.000	0.0441942
$513$	322.417	0.0277487
$514$	12349.6	1.05976
$515$	−3301.68	−0.282504
$516$	−924.327	−0.0788589
$517$	−3305.66	−0.281204
$518$	1188.83	0.100838
$519$	−7557.13	−0.639154
$520$	1183.75	0.0998287
$521$	12893.1	1.08418	0.542091	−	0.840320i	$-0.317633\pi$
$521$	12893.1	1.08418	0.542091	+	0.840320i	$0.317633\pi$
$522$	322.618	0.0270510
$523$	−19397.1	−1.62175	−0.810874	−	0.585221i	$-0.801008\pi$
$523$	−19397.1	−1.62175	−0.810874	+	0.585221i	$0.801008\pi$
$524$	−5626.33	−0.469060
$525$	−263.038	−0.0218665
$526$	1845.76	0.153002
$527$	1552.18	0.128300
$528$	−528.000	−0.0435194
$529$	−12154.2	−0.998946
$530$	8613.23	0.705915
$531$	−1955.81	−0.159840
$532$	767.369	0.0625369
$533$	4295.60	0.349087
$534$	2940.66	0.238305
$535$	12506.2	1.01064
$536$	2735.30	0.220423
$537$	−10022.4	−0.805395
$538$	14599.9	1.16997
$539$	−933.962	−0.0746356
$540$	1180.82	0.0941009
$541$	−17804.2	−1.41490	−0.707449	−	0.706764i	$-0.750154\pi$
$541$	−17804.2	−1.41490	−0.707449	+	0.706764i	$0.750154\pi$
$542$	−5503.11	−0.436123
$543$	5356.06	0.423298
$544$	−314.198	−0.0247631
$545$	3115.39	0.244860
$546$	−1304.52	−0.102250
$547$	3625.95	0.283427	0.141713	−	0.989908i	$-0.454739\pi$
$547$	3625.95	0.283427	0.141713	+	0.989908i	$0.454739\pi$
$548$	1567.78	0.122212
$549$	7078.94	0.550313
$550$	−120.069	−0.00930866
$551$	−214.028	−0.0165479
$552$	−85.9389	−0.00662645
$553$	−1004.15	−0.0772168
$554$	−14900.8	−1.14273
$555$	−1213.62	−0.0928206
$556$	6434.44	0.490793
$557$	8041.23	0.611702	0.305851	−	0.952079i	$-0.401059\pi$
$557$	8041.23	0.611702	0.305851	+	0.952079i	$0.401059\pi$
$558$	−2845.51	−0.215878
$559$	−1042.45	−0.0788744
$560$	2810.41	0.212074
$561$	324.017	0.0243850
$562$	−6569.40	−0.493084
$563$	10.2659	0.000768482 0	0.000384241	−	1.00000i	$-0.499878\pi$
$563$	10.2659	0.000768482 0	0.000384241		1.00000i	$0.499878\pi$
$564$	3606.17	0.269233
$565$	−24768.9	−1.84431
$566$	−17221.4	−1.27892
$567$	−1301.29	−0.0963828
$568$	4720.54	0.348714
$569$	19679.7	1.44994	0.724971	−	0.688779i	$-0.241853\pi$
$569$	19679.7	1.44994	0.724971	+	0.688779i	$0.241853\pi$
$570$	−783.370	−0.0575645
$571$	18775.0	1.37602	0.688011	−	0.725700i	$-0.258484\pi$
$571$	18775.0	1.37602	0.688011	+	0.725700i	$0.258484\pi$
$572$	−595.473	−0.0435280
$573$	2333.29	0.170113
$574$	10198.4	0.741593
$575$	−19.5428	−0.00141738
$576$	576.000	0.0416667
$577$	22172.3	1.59973	0.799867	−	0.600177i	$-0.204903\pi$
$577$	22172.3	1.59973	0.799867	+	0.600177i	$0.204903\pi$
$578$	−9633.19	−0.693231
$579$	−14501.1	−1.04084
$580$	−783.857	−0.0561171
$581$	−10740.3	−0.766924
$582$	−7426.07	−0.528901
$583$	−4332.79	−0.307797
$584$	−9059.41	−0.641919
$585$	1331.72	0.0941194
$586$	19194.5	1.35310
$587$	2017.04	0.141826	0.0709131	−	0.997482i	$-0.477409\pi$
$587$	2017.04	0.141826	0.0709131	+	0.997482i	$0.477409\pi$
$588$	1018.87	0.0714581
$589$	1887.74	0.132060
$590$	4751.98	0.331586
$591$	3124.26	0.217453
$592$	−592.000	−0.0410997
$593$	5067.42	0.350918	0.175459	−	0.984487i	$-0.443859\pi$
$593$	5067.42	0.350918	0.175459	+	0.984487i	$0.443859\pi$
$594$	−594.000	−0.0410305
$595$	−1724.66	−0.118831
$596$	−1387.21	−0.0953397
$597$	10635.1	0.729090
$598$	−96.9211	−0.00662775
$599$	2237.92	0.152653	0.0763264	−	0.997083i	$-0.475681\pi$
$599$	2237.92	0.152653	0.0763264	+	0.997083i	$0.475681\pi$
$600$	130.984	0.00891236
$601$	21595.8	1.46574	0.732871	−	0.680368i	$-0.238180\pi$
$601$	21595.8	1.46574	0.732871	+	0.680368i	$0.238180\pi$
$602$	−2474.93	−0.167559
$603$	3077.21	0.207817
$604$	13135.4	0.884887
$605$	−1322.96	−0.0889023
$606$	−1595.97	−0.106983
$607$	8000.48	0.534974	0.267487	−	0.963561i	$-0.413807\pi$
$607$	8000.48	0.534974	0.267487	+	0.963561i	$0.413807\pi$
$608$	−382.124	−0.0254888
$609$	863.827	0.0574779
$610$	−17199.5	−1.14162
$611$	4067.01	0.269286
$612$	−353.473	−0.0233469
$613$	15369.7	1.01268	0.506342	−	0.862333i	$-0.330997\pi$
$613$	15369.7	1.01268	0.506342	+	0.862333i	$0.330997\pi$
$614$	2059.85	0.135389
$615$	−10411.1	−0.682627
$616$	−1413.75	−0.0924700
$617$	24121.7	1.57391	0.786954	−	0.617011i	$-0.211657\pi$
$617$	24121.7	1.57391	0.786954	+	0.617011i	$0.211657\pi$
$618$	−1811.86	−0.117935
$619$	−27287.9	−1.77188	−0.885941	−	0.463798i	$-0.846486\pi$
$619$	−27287.9	−1.77188	−0.885941	+	0.463798i	$0.846486\pi$
$620$	6913.67	0.447838
$621$	−96.6812	−0.00624748
$622$	14036.3	0.904830
$623$	7873.78	0.506350
$624$	649.607	0.0416749
$625$	−14913.0	−0.954432
$626$	−8949.68	−0.571408
$627$	394.066	0.0250996
$628$	10730.7	0.681853
$629$	363.292	0.0230292
$630$	3161.71	0.199946
$631$	−19680.5	−1.24163	−0.620816	−	0.783956i	$-0.713199\pi$
$631$	−19680.5	−1.24163	−0.620816	+	0.783956i	$0.713199\pi$
$632$	500.035	0.0314720
$633$	−10841.6	−0.680752
$634$	11700.9	0.732970
$635$	2857.53	0.178579
$636$	4726.68	0.294694
$637$	1149.07	0.0714722
$638$	394.311	0.0244685
$639$	5310.61	0.328771
$640$	−1399.49	−0.0864372
$641$	28859.6	1.77829	0.889147	−	0.457622i	$-0.151299\pi$
$641$	28859.6	1.77829	0.889147	+	0.457622i	$0.151299\pi$
$642$	6863.04	0.421904
$643$	−21497.3	−1.31846	−0.659232	−	0.751939i	$-0.729119\pi$
$643$	−21497.3	−1.31846	−0.659232	+	0.751939i	$0.729119\pi$
$644$	−230.106	−0.0140799
$645$	2526.54	0.154236
$646$	234.498	0.0142820
$647$	−30378.2	−1.84589	−0.922945	−	0.384932i	$-0.874225\pi$
$647$	−30378.2	−1.84589	−0.922945	+	0.384932i	$0.874225\pi$
$648$	648.000	0.0392837
$649$	−2390.43	−0.144580
$650$	147.723	0.00891412
$651$	−7619.01	−0.458698
$652$	2561.92	0.153884
$653$	17274.5	1.03522	0.517612	−	0.855615i	$-0.326821\pi$
$653$	17274.5	1.03522	0.517612	+	0.855615i	$0.326821\pi$
$654$	1709.63	0.102220
$655$	15378.9	0.917411
$656$	−5078.49	−0.302259
$657$	−10191.8	−0.605207
$658$	9655.72	0.572066
$659$	−29879.4	−1.76621	−0.883107	−	0.469172i	$-0.844552\pi$
$659$	−29879.4	−1.76621	−0.883107	+	0.469172i	$0.844552\pi$
$660$	1443.23	0.0851175
$661$	19911.1	1.17164	0.585818	−	0.810443i	$-0.300773\pi$
$661$	19911.1	1.17164	0.585818	+	0.810443i	$0.300773\pi$
$662$	7898.55	0.463725
$663$	−398.643	−0.0233515
$664$	5348.32	0.312583
$665$	−2097.51	−0.122313
$666$	−666.000	−0.0387492
$667$	64.1792	0.00372568
$668$	−13160.8	−0.762286
$669$	−17802.8	−1.02885
$670$	−7476.62	−0.431115
$671$	8652.04	0.497777
$672$	1542.27	0.0885333
$673$	28656.4	1.64134	0.820670	−	0.571403i	$-0.193601\pi$
$673$	28656.4	1.64134	0.820670	+	0.571403i	$0.193601\pi$
$674$	3342.33	0.191011
$675$	147.358	0.00840266
$676$	−8055.38	−0.458317
$677$	−14001.2	−0.794842	−0.397421	−	0.917636i	$-0.630095\pi$
$677$	−14001.2	−0.794842	−0.397421	+	0.917636i	$0.630095\pi$
$678$	−13592.4	−0.769932
$679$	−19883.7	−1.12381
$680$	858.824	0.0484330
$681$	−8943.98	−0.503280
$682$	−3477.85	−0.195269
$683$	2831.74	0.158643	0.0793217	−	0.996849i	$-0.474725\pi$
$683$	2831.74	0.158643	0.0793217	+	0.996849i	$0.474725\pi$
$684$	−429.890	−0.0240311
$685$	−4285.36	−0.239029
$686$	13748.9	0.765211
$687$	7190.50	0.399323
$688$	1232.44	0.0682938
$689$	5330.71	0.294752
$690$	234.904	0.0129604
$691$	−18470.6	−1.01686	−0.508432	−	0.861102i	$-0.669775\pi$
$691$	−18470.6	−1.01686	−0.508432	+	0.861102i	$0.669775\pi$
$692$	10076.2	0.553524
$693$	−1590.47	−0.0871816
$694$	−10306.6	−0.563735
$695$	−17587.8	−0.959919
$696$	−430.157	−0.0234268
$697$	3116.51	0.169363
$698$	9835.23	0.533336
$699$	−13779.0	−0.745593
$700$	350.718	0.0189370
$701$	16189.8	0.872295	0.436147	−	0.899875i	$-0.356343\pi$
$701$	16189.8	0.872295	0.436147	+	0.899875i	$0.356343\pi$
$702$	730.808	0.0392914
$703$	441.831	0.0237041
$704$	704.000	0.0376889
$705$	−9857.06	−0.526579
$706$	−19473.1	−1.03807
$707$	−4273.30	−0.227318
$708$	2607.74	0.138425
$709$	−7521.26	−0.398402	−0.199201	−	0.979959i	$-0.563835\pi$
$709$	−7521.26	−0.398402	−0.199201	+	0.979959i	$0.563835\pi$
$710$	−12903.1	−0.682033
$711$	562.539	0.0296721
$712$	−3920.88	−0.206378
$713$	−566.065	−0.0297326
$714$	−946.442	−0.0496074
$715$	1627.66	0.0851342
$716$	13363.2	0.697493
$717$	−7925.06	−0.412785
$718$	544.370	0.0282949
$719$	−11287.2	−0.585455	−0.292727	−	0.956196i	$-0.594563\pi$
$719$	−11287.2	−0.585455	−0.292727	+	0.956196i	$0.594563\pi$
$720$	−1574.43	−0.0814938
$721$	−4851.36	−0.250588
$722$	−13432.8	−0.692406
$723$	10649.3	0.547792
$724$	−7141.42	−0.366586
$725$	−97.8193	−0.00501092
$726$	−726.000	−0.0371135
$727$	23970.1	1.22284	0.611418	−	0.791308i	$-0.290599\pi$
$727$	23970.1	1.22284	0.611418	+	0.791308i	$0.290599\pi$
$728$	1739.36	0.0885507
$729$	729.000	0.0370370
$730$	24762.9	1.25550
$731$	−756.306	−0.0382668
$732$	−9438.58	−0.476585
$733$	1783.83	0.0898872	0.0449436	−	0.998990i	$-0.485689\pi$
$733$	1783.83	0.0898872	0.0449436	+	0.998990i	$0.485689\pi$
$734$	−22504.2	−1.13167
$735$	−2784.96	−0.139761
$736$	114.585	0.00573868
$737$	3761.03	0.187977
$738$	−5713.30	−0.284972
$739$	−35016.9	−1.74305	−0.871526	−	0.490349i	$-0.836869\pi$
$739$	−35016.9	−1.74305	−0.871526	+	0.490349i	$0.836869\pi$
$740$	1618.16	0.0803850
$741$	−484.826	−0.0240358
$742$	12655.9	0.626165
$743$	27471.1	1.35642	0.678208	−	0.734870i	$-0.262757\pi$
$743$	27471.1	1.35642	0.678208	+	0.734870i	$0.262757\pi$
$744$	3794.02	0.186956
$745$	3791.79	0.186470
$746$	−8743.69	−0.429128
$747$	6016.86	0.294706
$748$	−432.022	−0.0211181
$749$	18376.1	0.896462
$750$	−8558.19	−0.416668
$751$	27502.9	1.33634	0.668172	−	0.744007i	$-0.267077\pi$
$751$	27502.9	1.33634	0.668172	+	0.744007i	$0.267077\pi$
$752$	−4808.23	−0.233162
$753$	4946.50	0.239390
$754$	−485.127	−0.0234314
$755$	−35904.1	−1.73071
$756$	1735.05	0.0834700
$757$	−20274.0	−0.973408	−0.486704	−	0.873567i	$-0.661801\pi$
$757$	−20274.0	−0.973408	−0.486704	+	0.873567i	$0.661801\pi$
$758$	14125.9	0.676880
$759$	−118.166	−0.00565106
$760$	1044.49	0.0498523
$761$	−18725.1	−0.891965	−0.445982	−	0.895042i	$-0.647146\pi$
$761$	−18725.1	−0.891965	−0.445982	+	0.895042i	$0.647146\pi$
$762$	1568.12	0.0745501
$763$	4577.63	0.217197
$764$	−3111.06	−0.147322
$765$	966.177	0.0456630
$766$	−12333.5	−0.581758
$767$	2940.99	0.138452
$768$	−768.000	−0.0360844
$769$	−18962.7	−0.889224	−0.444612	−	0.895723i	$-0.646658\pi$
$769$	−18962.7	−0.889224	−0.444612	+	0.895723i	$0.646658\pi$
$770$	3864.32	0.180858
$771$	−18524.3	−0.865288
$772$	19334.8	0.901391
$773$	5471.66	0.254595	0.127298	−	0.991865i	$-0.459370\pi$
$773$	5471.66	0.254595	0.127298	+	0.991865i	$0.459370\pi$
$774$	1386.49	0.0643880
$775$	862.773	0.0399893
$776$	9901.42	0.458042
$777$	−1783.25	−0.0823343
$778$	4168.97	0.192114
$779$	3790.26	0.174326
$780$	−1775.63	−0.0815098
$781$	6490.74	0.297384
$782$	−70.3173	−0.00321553
$783$	−483.927	−0.0220870
$784$	−1358.49	−0.0618846
$785$	−29331.3	−1.33360
$786$	8439.49	0.382986
$787$	−8383.96	−0.379741	−0.189870	−	0.981809i	$-0.560807\pi$
$787$	−8383.96	−0.379741	−0.189870	+	0.981809i	$0.560807\pi$
$788$	−4165.67	−0.188320
$789$	−2768.64	−0.124925
$790$	−1366.79	−0.0615546
$791$	−36394.5	−1.63595
$792$	792.000	0.0355335
$793$	−10644.7	−0.476679
$794$	26120.5	1.16748
$795$	−12919.8	−0.576377
$796$	−14180.2	−0.631410
$797$	716.910	0.0318623	0.0159311	−	0.999873i	$-0.494929\pi$
$797$	716.910	0.0318623	0.0159311	+	0.999873i	$0.494929\pi$
$798$	−1151.05	−0.0510612
$799$	2950.66	0.130647
$800$	−174.646	−0.00771833
$801$	−4410.99	−0.194575
$802$	−24693.8	−1.08724
$803$	−12456.7	−0.547431
$804$	−4102.94	−0.179975
$805$	628.968	0.0275382
$806$	4278.86	0.186993
$807$	−21899.8	−0.955279
$808$	2127.96	0.0926503
$809$	−35809.3	−1.55623	−0.778115	−	0.628122i	$-0.783824\pi$
$809$	−35809.3	−1.55623	−0.778115	+	0.628122i	$0.783824\pi$
$810$	−1771.23	−0.0768331
$811$	20095.8	0.870110	0.435055	−	0.900404i	$-0.356729\pi$
$811$	20095.8	0.870110	0.435055	+	0.900404i	$0.356729\pi$
$812$	−1151.77	−0.0497773
$813$	8254.67	0.356093
$814$	−814.000	−0.0350500
$815$	−7002.71	−0.300975
$816$	471.297	0.0202190
$817$	−919.812	−0.0393882
$818$	13919.6	0.594972
$819$	1956.78	0.0834864
$820$	13881.5	0.591173
$821$	−39766.0	−1.69043	−0.845216	−	0.534425i	$-0.820528\pi$
$821$	−39766.0	−1.69043	−0.845216	+	0.534425i	$0.820528\pi$
$822$	−2351.67	−0.0997860
$823$	−43505.2	−1.84264	−0.921322	−	0.388800i	$-0.872890\pi$
$823$	−43505.2	−1.84264	−0.921322	+	0.388800i	$0.872890\pi$
$824$	2415.82	0.102135
$825$	180.104	0.00760049
$826$	6982.37	0.294126
$827$	−10669.9	−0.448642	−0.224321	−	0.974515i	$-0.572016\pi$
$827$	−10669.9	−0.448642	−0.224321	+	0.974515i	$0.572016\pi$
$828$	128.908	0.00541048
$829$	16961.4	0.710609	0.355304	−	0.934751i	$-0.384377\pi$
$829$	16961.4	0.710609	0.355304	+	0.934751i	$0.384377\pi$
$830$	−14619.0	−0.611366
$831$	22351.2	0.933036
$832$	−866.143	−0.0360915
$833$	833.662	0.0346755
$834$	−9651.66	−0.400731
$835$	35973.6	1.49092
$836$	−525.421	−0.0217369
$837$	4268.27	0.176264
$838$	9208.15	0.379583
$839$	−12511.8	−0.514845	−0.257422	−	0.966299i	$-0.582873\pi$
$839$	−12511.8	−0.514845	−0.257422	+	0.966299i	$0.582873\pi$
$840$	−4215.62	−0.173158
$841$	−24067.8	−0.986828
$842$	−17221.5	−0.704861
$843$	9854.09	0.402601
$844$	14455.5	0.589548
$845$	22018.5	0.896400
$846$	−5409.26	−0.219828
$847$	−1943.90	−0.0788587
$848$	−6302.24	−0.255212
$849$	25832.2	1.04424
$850$	107.175	0.00432478
$851$	−132.489	−0.00533686
$852$	−7080.81	−0.284724
$853$	11339.9	0.455183	0.227591	−	0.973757i	$-0.426915\pi$
$853$	11339.9	0.455183	0.227591	+	0.973757i	$0.426915\pi$
$854$	−25272.3	−1.01265
$855$	1175.05	0.0470012
$856$	−9150.72	−0.365380
$857$	36159.9	1.44131	0.720653	−	0.693296i	$-0.243842\pi$
$857$	36159.9	1.44131	0.720653	+	0.693296i	$0.243842\pi$
$858$	893.210	0.0355404
$859$	8136.73	0.323192	0.161596	−	0.986857i	$-0.448336\pi$
$859$	8136.73	0.323192	0.161596	+	0.986857i	$0.448336\pi$
$860$	−3368.72	−0.133573
$861$	−15297.7	−0.605508
$862$	26818.3	1.05967
$863$	24275.8	0.957543	0.478771	−	0.877940i	$-0.341082\pi$
$863$	24275.8	0.957543	0.478771	+	0.877940i	$0.341082\pi$
$864$	−864.000	−0.0340207
$865$	−27542.1	−1.08261
$866$	4194.15	0.164576
$867$	14449.8	0.566021
$868$	10158.7	0.397244
$869$	687.548	0.0268394
$870$	1175.79	0.0458194
$871$	−4627.26	−0.180010
$872$	−2279.51	−0.0885251
$873$	11139.1	0.431846
$874$	−85.5191	−0.00330976
$875$	−22915.0	−0.885335
$876$	13589.1	0.524125
$877$	−27741.5	−1.06815	−0.534073	−	0.845438i	$-0.679339\pi$
$877$	−27741.5	−1.06815	−0.534073	+	0.845438i	$0.679339\pi$
$878$	−6755.37	−0.259661
$879$	−28791.8	−1.10480
$880$	−1924.30	−0.0737139
$881$	10558.1	0.403758	0.201879	−	0.979410i	$-0.435295\pi$
$881$	10558.1	0.403758	0.201879	+	0.979410i	$0.435295\pi$
$882$	−1528.30	−0.0583453
$883$	−24625.3	−0.938512	−0.469256	−	0.883062i	$-0.655478\pi$
$883$	−24625.3	−0.938512	−0.469256	+	0.883062i	$0.655478\pi$
$884$	531.525	0.0202230
$885$	−7127.97	−0.270739
$886$	16668.5	0.632042
$887$	32180.1	1.21815	0.609077	−	0.793111i	$-0.291540\pi$
$887$	32180.1	1.21815	0.609077	+	0.793111i	$0.291540\pi$
$888$	888.000	0.0335578
$889$	4198.74	0.158404
$890$	10717.3	0.403645
$891$	891.000	0.0335013
$892$	23737.1	0.891007
$893$	3588.56	0.134475
$894$	2080.82	0.0778445
$895$	−36526.7	−1.36419
$896$	−2056.36	−0.0766721
$897$	145.382	0.00541154
$898$	5828.03	0.216575
$899$	−2833.37	−0.105115
$900$	−196.477	−0.00727691
$901$	3867.49	0.143002
$902$	−6982.92	−0.257767
$903$	3712.40	0.136812
$904$	18123.2	0.666781
$905$	19520.2	0.716989
$906$	−19703.1	−0.722508
$907$	−35394.5	−1.29576	−0.647880	−	0.761743i	$-0.724344\pi$
$907$	−35394.5	−1.29576	−0.647880	+	0.761743i	$0.724344\pi$
$908$	11925.3	0.435854
$909$	2393.96	0.0873515
$910$	−4754.34	−0.173192
$911$	−11137.1	−0.405039	−0.202519	−	0.979278i	$-0.564913\pi$
$911$	−11137.1	−0.405039	−0.202519	+	0.979278i	$0.564913\pi$
$912$	573.187	0.0208115
$913$	7353.94	0.266572
$914$	4976.39	0.180092
$915$	25799.3	0.932129
$916$	−9587.33	−0.345824
$917$	22597.2	0.813767
$918$	530.209	0.0190627
$919$	−25258.6	−0.906642	−0.453321	−	0.891347i	$-0.649761\pi$
$919$	−25258.6	−0.906642	−0.453321	+	0.891347i	$0.649761\pi$
$920$	−313.205	−0.0112240
$921$	−3089.77	−0.110545
$922$	−2914.63	−0.104109
$923$	−7985.67	−0.284780
$924$	2120.62	0.0755014
$925$	201.934	0.00717790
$926$	−22465.8	−0.797268
$927$	2717.80	0.0962936
$928$	573.543	0.0202882
$929$	−39068.1	−1.37975	−0.689873	−	0.723931i	$-0.742333\pi$
$929$	−39068.1	−1.37975	−0.689873	+	0.723931i	$0.742333\pi$
$930$	−10370.5	−0.365658
$931$	1013.89	0.0356916
$932$	18372.0	0.645703
$933$	−21054.5	−0.738791
$934$	1055.47	0.0369765
$935$	1180.88	0.0413038
$936$	−974.411	−0.0340274
$937$	7059.61	0.246134	0.123067	−	0.992398i	$-0.460727\pi$
$937$	7059.61	0.246134	0.123067	+	0.992398i	$0.460727\pi$
$938$	−10985.8	−0.382410
$939$	13424.5	0.466552
$940$	13142.7	0.456031
$941$	−30205.1	−1.04640	−0.523198	−	0.852211i	$-0.675261\pi$
$941$	−30205.1	−1.04640	−0.523198	+	0.852211i	$0.675261\pi$
$942$	−16096.1	−0.556730
$943$	−1136.56	−0.0392487
$944$	−3476.99	−0.119880
$945$	−4742.57	−0.163255
$946$	1694.60	0.0582412
$947$	−14988.2	−0.514308	−0.257154	−	0.966370i	$-0.582785\pi$
$947$	−14988.2	−0.514308	−0.257154	+	0.966370i	$0.582785\pi$
$948$	−750.052	−0.0256968
$949$	15325.7	0.524228
$950$	130.345	0.00445152
$951$	−17551.4	−0.598468
$952$	1261.92	0.0429613
$953$	−49715.9	−1.68988	−0.844940	−	0.534861i	$-0.820364\pi$
$953$	−49715.9	−1.68988	−0.844940	+	0.534861i	$0.820364\pi$
$954$	−7090.02	−0.240616
$955$	8503.72	0.288140
$956$	10566.7	0.357482
$957$	−591.466	−0.0199785
$958$	3295.54	0.111142
$959$	−6296.73	−0.212025
$960$	2099.24	0.0705757
$961$	−4800.44	−0.161137
$962$	1001.48	0.0335644
$963$	−10294.6	−0.344483
$964$	−14199.1	−0.474402
$965$	−52849.4	−1.76299
$966$	345.159	0.0114962
$967$	28000.7	0.931170	0.465585	−	0.885003i	$-0.345844\pi$
$967$	28000.7	0.931170	0.465585	+	0.885003i	$0.345844\pi$
$968$	968.000	0.0321412
$969$	−351.746	−0.0116612
$970$	−27064.4	−0.895861
$971$	27496.8	0.908770	0.454385	−	0.890805i	$-0.349859\pi$
$971$	27496.8	0.908770	0.454385	+	0.890805i	$0.349859\pi$
$972$	−972.000	−0.0320750
$973$	−25842.8	−0.851473
$974$	41803.4	1.37522
$975$	−221.585	−0.00727834
$976$	12584.8	0.412735
$977$	2327.76	0.0762248	0.0381124	−	0.999273i	$-0.487866\pi$
$977$	2327.76	0.0762248	0.0381124	+	0.999273i	$0.487866\pi$
$978$	−3842.88	−0.125646
$979$	−5391.21	−0.176000
$980$	3713.28	0.121037
$981$	−2564.45	−0.0834623
$982$	1322.38	0.0429724
$983$	38486.6	1.24876	0.624380	−	0.781120i	$-0.285352\pi$
$983$	38486.6	1.24876	0.624380	+	0.781120i	$0.285352\pi$
$984$	7617.73	0.246793
$985$	11386.4	0.368325
$986$	−351.965	−0.0113680
$987$	−14483.6	−0.467090
$988$	646.434	0.0208156
$989$	275.818	0.00886805
$990$	−2164.84	−0.0694981
$991$	24411.0	0.782482	0.391241	−	0.920288i	$-0.372046\pi$
$991$	24411.0	0.782482	0.391241	+	0.920288i	$0.372046\pi$
$992$	−5058.69	−0.161909
$993$	−11847.8	−0.378630
$994$	−18959.2	−0.604981
$995$	38759.9	1.23494
$996$	−8022.48	−0.255223
$997$	−27824.6	−0.883866	−0.441933	−	0.897048i	$-0.645707\pi$
$997$	−27824.6	−0.883866	−0.441933	+	0.897048i	$0.645707\pi$
$998$	38448.9	1.21952
$999$	999.000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2442.4.a.o.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2442.4.a.o.1.3	✓	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2442 = 2 \cdot 3 \cdot 11 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2442.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -10.9335 q^{5} -6.00000 q^{6} -16.0653 q^{7} +8.00000 q^{8} +9.00000 q^{9} -21.8671 q^{10} +11.0000 q^{11} -12.0000 q^{12} -13.5335 q^{13} -32.1306 q^{14} +32.8006 q^{15} +16.0000 q^{16} -9.81869 q^{17} +18.0000 q^{18} -11.9414 q^{19} -43.7342 q^{20} +48.1959 q^{21} +22.0000 q^{22} +3.58079 q^{23} -24.0000 q^{24} -5.45769 q^{25} -27.0670 q^{26} -27.0000 q^{27} -64.2613 q^{28} +17.9232 q^{29} +65.6012 q^{30} -158.084 q^{31} +32.0000 q^{32} -33.0000 q^{33} -19.6374 q^{34} +175.651 q^{35} +36.0000 q^{36} -37.0000 q^{37} -23.8828 q^{38} +40.6005 q^{39} -87.4683 q^{40} -317.405 q^{41} +96.3919 q^{42} +77.0272 q^{43} +44.0000 q^{44} -98.4019 q^{45} +7.16157 q^{46} -300.514 q^{47} -48.0000 q^{48} -84.9056 q^{49} -10.9154 q^{50} +29.4561 q^{51} -54.1340 q^{52} -393.890 q^{53} -54.0000 q^{54} -120.269 q^{55} -128.523 q^{56} +35.8242 q^{57} +35.8464 q^{58} -217.312 q^{59} +131.202 q^{60} +786.549 q^{61} -316.168 q^{62} -144.588 q^{63} +64.0000 q^{64} +147.969 q^{65} -66.0000 q^{66} +341.912 q^{67} -39.2748 q^{68} -10.7424 q^{69} +351.302 q^{70} +590.068 q^{71} +72.0000 q^{72} -1132.43 q^{73} -74.0000 q^{74} +16.3731 q^{75} -47.7655 q^{76} -176.718 q^{77} +81.2009 q^{78} +62.5044 q^{79} -174.937 q^{80} +81.0000 q^{81} -634.811 q^{82} +668.540 q^{83} +192.784 q^{84} +107.353 q^{85} +154.054 q^{86} -53.7697 q^{87} +88.0000 q^{88} -490.110 q^{89} -196.804 q^{90} +217.420 q^{91} +14.3231 q^{92} +474.252 q^{93} -601.029 q^{94} +130.562 q^{95} -96.0000 q^{96} +1237.68 q^{97} -169.811 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{2} - 36 q^{3} + 48 q^{4} + 15 q^{5} - 72 q^{6} + 31 q^{7} + 96 q^{8} + 108 q^{9} + 30 q^{10} + 132 q^{11} - 144 q^{12} + 58 q^{13} + 62 q^{14} - 45 q^{15} + 192 q^{16} + 193 q^{17} + 216 q^{18}+ \cdots + 1188 q^{99}+O(q^{100}) \) 12 * q + 24 * q^2 - 36 * q^3 + 48 * q^4 + 15 * q^5 - 72 * q^6 + 31 * q^7 + 96 * q^8 + 108 * q^9 + 30 * q^10 + 132 * q^11 - 144 * q^12 + 58 * q^13 + 62 * q^14 - 45 * q^15 + 192 * q^16 + 193 * q^17 + 216 * q^18 + 49 * q^19 + 60 * q^20 - 93 * q^21 + 264 * q^22 + 341 * q^23 - 288 * q^24 + 501 * q^25 + 116 * q^26 - 324 * q^27 + 124 * q^28 - 104 * q^29 - 90 * q^30 + 274 * q^31 + 384 * q^32 - 396 * q^33 + 386 * q^34 + 573 * q^35 + 432 * q^36 - 444 * q^37 + 98 * q^38 - 174 * q^39 + 120 * q^40 + 259 * q^41 - 186 * q^42 + 680 * q^43 + 528 * q^44 + 135 * q^45 + 682 * q^46 - 127 * q^47 - 576 * q^48 + 1111 * q^49 + 1002 * q^50 - 579 * q^51 + 232 * q^52 + 322 * q^53 - 648 * q^54 + 165 * q^55 + 248 * q^56 - 147 * q^57 - 208 * q^58 - 206 * q^59 - 180 * q^60 + 59 * q^61 + 548 * q^62 + 279 * q^63 + 768 * q^64 + 741 * q^65 - 792 * q^66 + 272 * q^67 + 772 * q^68 - 1023 * q^69 + 1146 * q^70 + 236 * q^71 + 864 * q^72 + 773 * q^73 - 888 * q^74 - 1503 * q^75 + 196 * q^76 + 341 * q^77 - 348 * q^78 - 809 * q^79 + 240 * q^80 + 972 * q^81 + 518 * q^82 - 395 * q^83 - 372 * q^84 + 1972 * q^85 + 1360 * q^86 + 312 * q^87 + 1056 * q^88 + 1515 * q^89 + 270 * q^90 - 124 * q^91 + 1364 * q^92 - 822 * q^93 - 254 * q^94 + 253 * q^95 - 1152 * q^96 + 43 * q^97 + 2222 * q^98 + 1188 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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