LMFDB - Embedded newform 58.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [58,4,Mod(1,58)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, 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("58.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 58 = 2 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	58.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:	$3.42211078033$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	58.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.44949 q^{3} +4.00000 q^{4} +9.69694 q^{5} +6.89898 q^{6} -27.5959 q^{7} -8.00000 q^{8} -15.1010 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.44949 q^{3} +4.00000 q^{4} +9.69694 q^{5} +6.89898 q^{6} -27.5959 q^{7} -8.00000 q^{8} -15.1010 q^{9} -19.3939 q^{10} -52.3485 q^{11} -13.7980 q^{12} -5.40408 q^{13} +55.1918 q^{14} -33.4495 q^{15} +16.0000 q^{16} +17.1918 q^{17} +30.2020 q^{18} +44.2020 q^{19} +38.7878 q^{20} +95.1918 q^{21} +104.697 q^{22} -205.060 q^{23} +27.5959 q^{24} -30.9694 q^{25} +10.8082 q^{26} +145.227 q^{27} -110.384 q^{28} +29.0000 q^{29} +66.8990 q^{30} +299.994 q^{31} -32.0000 q^{32} +180.576 q^{33} -34.3837 q^{34} -267.596 q^{35} -60.4041 q^{36} +29.7980 q^{37} -88.4041 q^{38} +18.6413 q^{39} -77.5755 q^{40} -43.9592 q^{41} -190.384 q^{42} +64.8230 q^{43} -209.394 q^{44} -146.434 q^{45} +410.120 q^{46} -499.499 q^{47} -55.1918 q^{48} +418.535 q^{49} +61.9388 q^{50} -59.3031 q^{51} -21.6163 q^{52} -351.627 q^{53} -290.454 q^{54} -507.620 q^{55} +220.767 q^{56} -152.474 q^{57} -58.0000 q^{58} +522.372 q^{59} -133.798 q^{60} +484.606 q^{61} -599.989 q^{62} +416.727 q^{63} +64.0000 q^{64} -52.4031 q^{65} -361.151 q^{66} -504.990 q^{67} +68.7673 q^{68} +707.353 q^{69} +535.192 q^{70} +481.283 q^{71} +120.808 q^{72} +3.11019 q^{73} -59.5959 q^{74} +106.829 q^{75} +176.808 q^{76} +1444.60 q^{77} -37.2827 q^{78} -1043.27 q^{79} +155.151 q^{80} -93.2316 q^{81} +87.9184 q^{82} -1007.08 q^{83} +380.767 q^{84} +166.708 q^{85} -129.646 q^{86} -100.035 q^{87} +418.788 q^{88} -295.637 q^{89} +292.867 q^{90} +149.131 q^{91} -820.241 q^{92} -1034.83 q^{93} +998.999 q^{94} +428.624 q^{95} +110.384 q^{96} +428.949 q^{97} -837.069 q^{98} +790.515 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9} + 20 q^{10} - 90 q^{11} - 8 q^{12} - 50 q^{13} + 32 q^{14} - 62 q^{15} + 32 q^{16} - 44 q^{17} + 80 q^{18} + 108 q^{19} - 40 q^{20} + 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} + 100 q^{26} + 70 q^{27} - 64 q^{28} + 58 q^{29} + 124 q^{30} + 66 q^{31} - 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} + 40 q^{37} - 216 q^{38} - 46 q^{39} + 80 q^{40} + 304 q^{41} - 224 q^{42} - 130 q^{43} - 360 q^{44} + 344 q^{45} + 56 q^{46} - 514 q^{47} - 32 q^{48} + 210 q^{49} - 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} + 234 q^{55} + 128 q^{56} - 60 q^{57} - 116 q^{58} - 180 q^{59} - 248 q^{60} + 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} - 252 q^{66} - 912 q^{67} - 176 q^{68} + 964 q^{69} + 992 q^{70} + 796 q^{71} + 320 q^{72} - 856 q^{73} - 80 q^{74} + 488 q^{75} + 432 q^{76} + 1008 q^{77} + 92 q^{78} - 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} + 448 q^{84} + 1372 q^{85} + 260 q^{86} - 58 q^{87} + 720 q^{88} - 944 q^{89} - 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} - 828 q^{95} + 64 q^{96} + 368 q^{97} - 420 q^{98} + 1728 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9 + 20 * q^10 - 90 * q^11 - 8 * q^12 - 50 * q^13 + 32 * q^14 - 62 * q^15 + 32 * q^16 - 44 * q^17 + 80 * q^18 + 108 * q^19 - 40 * q^20 + 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 + 100 * q^26 + 70 * q^27 - 64 * q^28 + 58 * q^29 + 124 * q^30 + 66 * q^31 - 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 + 40 * q^37 - 216 * q^38 - 46 * q^39 + 80 * q^40 + 304 * q^41 - 224 * q^42 - 130 * q^43 - 360 * q^44 + 344 * q^45 + 56 * q^46 - 514 * q^47 - 32 * q^48 + 210 * q^49 - 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 + 234 * q^55 + 128 * q^56 - 60 * q^57 - 116 * q^58 - 180 * q^59 - 248 * q^60 + 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 - 252 * q^66 - 912 * q^67 - 176 * q^68 + 964 * q^69 + 992 * q^70 + 796 * q^71 + 320 * q^72 - 856 * q^73 - 80 * q^74 + 488 * q^75 + 432 * q^76 + 1008 * q^77 + 92 * q^78 - 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 + 448 * q^84 + 1372 * q^85 + 260 * q^86 - 58 * q^87 + 720 * q^88 - 944 * q^89 - 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 - 828 * q^95 + 64 * q^96 + 368 * q^97 - 420 * q^98 + 1728 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.44949	−0.663855	−0.331927	−	0.943305i	$-0.607699\pi$
$3$	−3.44949	−0.663855	−0.331927	+	0.943305i	$0.607699\pi$
$4$	4.00000	0.500000
$5$	9.69694	0.867321	0.433660	−	0.901076i	$-0.357222\pi$
$5$	9.69694	0.867321	0.433660	+	0.901076i	$0.357222\pi$
$6$	6.89898	0.469416
$7$	−27.5959	−1.49004	−0.745020	−	0.667042i	$-0.767560\pi$
$7$	−27.5959	−1.49004	−0.745020	+	0.667042i	$0.767560\pi$
$8$	−8.00000	−0.353553
$9$	−15.1010	−0.559297
$10$	−19.3939	−0.613288
$11$	−52.3485	−1.43488	−0.717439	−	0.696621i	$-0.754686\pi$
$11$	−52.3485	−1.43488	−0.717439	+	0.696621i	$0.754686\pi$
$12$	−13.7980	−0.331927
$13$	−5.40408	−0.115294	−0.0576470	−	0.998337i	$-0.518360\pi$
$13$	−5.40408	−0.115294	−0.0576470	+	0.998337i	$0.518360\pi$
$14$	55.1918	1.05362
$15$	−33.4495	−0.575775
$16$	16.0000	0.250000
$17$	17.1918	0.245273	0.122636	−	0.992452i	$-0.460865\pi$
$17$	17.1918	0.245273	0.122636	+	0.992452i	$0.460865\pi$
$18$	30.2020	0.395483
$19$	44.2020	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$19$	44.2020	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$20$	38.7878	0.433660
$21$	95.1918	0.989170
$22$	104.697	1.01461
$23$	−205.060	−1.85904	−0.929522	−	0.368767i	$-0.879780\pi$
$23$	−205.060	−1.85904	−0.929522	+	0.368767i	$0.879780\pi$
$24$	27.5959	0.234708
$25$	−30.9694	−0.247755
$26$	10.8082	0.0815252
$27$	145.227	1.03515
$28$	−110.384	−0.745020
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	66.8990	0.407134
$31$	299.994	1.73808	0.869042	−	0.494739i	$-0.164736\pi$
$31$	299.994	1.73808	0.869042	+	0.494739i	$0.164736\pi$
$32$	−32.0000	−0.176777
$33$	180.576	0.952550
$34$	−34.3837	−0.173434
$35$	−267.596	−1.29234
$36$	−60.4041	−0.279649
$37$	29.7980	0.132399	0.0661994	−	0.997806i	$-0.478913\pi$
$37$	29.7980	0.132399	0.0661994	+	0.997806i	$0.478913\pi$
$38$	−88.4041	−0.377396
$39$	18.6413	0.0765385
$40$	−77.5755	−0.306644
$41$	−43.9592	−0.167446	−0.0837228	−	0.996489i	$-0.526681\pi$
$41$	−43.9592	−0.167446	−0.0837228	+	0.996489i	$0.526681\pi$
$42$	−190.384	−0.699449
$43$	64.8230	0.229893	0.114947	−	0.993372i	$-0.463330\pi$
$43$	64.8230	0.229893	0.114947	+	0.993372i	$0.463330\pi$
$44$	−209.394	−0.717439
$45$	−146.434	−0.485090
$46$	410.120	1.31454
$47$	−499.499	−1.55020	−0.775101	−	0.631837i	$-0.782301\pi$
$47$	−499.499	−1.55020	−0.775101	+	0.631837i	$0.782301\pi$
$48$	−55.1918	−0.165964
$49$	418.535	1.22022
$50$	61.9388	0.175189
$51$	−59.3031	−0.162825
$52$	−21.6163	−0.0576470
$53$	−351.627	−0.911314	−0.455657	−	0.890156i	$-0.650596\pi$
$53$	−351.627	−0.911314	−0.455657	+	0.890156i	$0.650596\pi$
$54$	−290.454	−0.731959
$55$	−507.620	−1.24450
$56$	220.767	0.526809
$57$	−152.474	−0.354311
$58$	−58.0000	−0.131306
$59$	522.372	1.15266	0.576331	−	0.817216i	$-0.304484\pi$
$59$	522.372	1.15266	0.576331	+	0.817216i	$0.304484\pi$
$60$	−133.798	−0.287887
$61$	484.606	1.01717	0.508586	−	0.861011i	$-0.330168\pi$
$61$	484.606	1.01717	0.508586	+	0.861011i	$0.330168\pi$
$62$	−599.989	−1.22901
$63$	416.727	0.833375
$64$	64.0000	0.125000
$65$	−52.4031	−0.0999969
$66$	−361.151	−0.673555
$67$	−504.990	−0.920811	−0.460405	−	0.887709i	$-0.652296\pi$
$67$	−504.990	−0.920811	−0.460405	+	0.887709i	$0.652296\pi$
$68$	68.7673	0.122636
$69$	707.353	1.23413
$70$	535.192	0.913824
$71$	481.283	0.804475	0.402238	−	0.915535i	$-0.368233\pi$
$71$	481.283	0.804475	0.402238	+	0.915535i	$0.368233\pi$
$72$	120.808	0.197741
$73$	3.11019	0.00498659	0.00249329	−	0.999997i	$-0.499206\pi$
$73$	3.11019	0.00498659	0.00249329	+	0.999997i	$0.499206\pi$
$74$	−59.5959	−0.0936201
$75$	106.829	0.164473
$76$	176.808	0.266859
$77$	1444.60	2.13802
$78$	−37.2827	−0.0541209
$79$	−1043.27	−1.48578	−0.742890	−	0.669414i	$-0.766545\pi$
$79$	−1043.27	−1.48578	−0.742890	+	0.669414i	$0.766545\pi$
$80$	155.151	0.216830
$81$	−93.2316	−0.127890
$82$	87.9184	0.118402
$83$	−1007.08	−1.33182	−0.665912	−	0.746030i	$-0.731958\pi$
$83$	−1007.08	−1.33182	−0.665912	+	0.746030i	$0.731958\pi$
$84$	380.767	0.494585
$85$	166.708	0.212730
$86$	−129.646	−0.162559
$87$	−100.035	−0.123275
$88$	418.788	0.507306
$89$	−295.637	−0.352106	−0.176053	−	0.984381i	$-0.556333\pi$
$89$	−295.637	−0.352106	−0.176053	+	0.984381i	$0.556333\pi$
$90$	292.867	0.343010
$91$	149.131	0.171793
$92$	−820.241	−0.929522
$93$	−1034.83	−1.15383
$94$	998.999	1.09616
$95$	428.624	0.462905
$96$	110.384	0.117354
$97$	428.949	0.449002	0.224501	−	0.974474i	$-0.427925\pi$
$97$	428.949	0.449002	0.224501	+	0.974474i	$0.427925\pi$
$98$	−837.069	−0.862824
$99$	790.515	0.802523
$100$	−123.878	−0.123878
$101$	−1212.16	−1.19420	−0.597101	−	0.802166i	$-0.703681\pi$
$101$	−1212.16	−1.19420	−0.597101	+	0.802166i	$0.703681\pi$
$102$	118.606	0.115135
$103$	1032.07	0.987310	0.493655	−	0.869658i	$-0.335661\pi$
$103$	1032.07	0.987310	0.493655	+	0.869658i	$0.335661\pi$
$104$	43.2327	0.0407626
$105$	923.069	0.857927
$106$	703.253	0.644396
$107$	−1176.06	−1.06256	−0.531280	−	0.847196i	$-0.678289\pi$
$107$	−1176.06	−1.06256	−0.531280	+	0.847196i	$0.678289\pi$
$108$	580.908	0.517573
$109$	−2167.86	−1.90498	−0.952491	−	0.304568i	$-0.901488\pi$
$109$	−2167.86	−1.90498	−0.952491	+	0.304568i	$0.901488\pi$
$110$	1015.24	0.879994
$111$	−102.788	−0.0878935
$112$	−441.535	−0.372510
$113$	1623.65	1.35169	0.675843	−	0.737046i	$-0.263780\pi$
$113$	1623.65	1.35169	0.675843	+	0.737046i	$0.263780\pi$
$114$	304.949	0.250536
$115$	−1988.46	−1.61239
$116$	116.000	0.0928477
$117$	81.6072	0.0644836
$118$	−1044.74	−0.815056
$119$	−474.424	−0.365466
$120$	267.596	0.203567
$121$	1409.36	1.05887
$122$	−969.212	−0.719249
$123$	151.637	0.111160
$124$	1199.98	0.869042
$125$	−1512.43	−1.08220
$126$	−833.453	−0.589285
$127$	333.473	0.233000	0.116500	−	0.993191i	$-0.462833\pi$
$127$	333.473	0.233000	0.116500	+	0.993191i	$0.462833\pi$
$128$	−128.000	−0.0883883
$129$	−223.606	−0.152616
$130$	104.806	0.0707085
$131$	−933.492	−0.622592	−0.311296	−	0.950313i	$-0.600763\pi$
$131$	−933.492	−0.622592	−0.311296	+	0.950313i	$0.600763\pi$
$132$	722.302	0.476275
$133$	−1219.80	−0.795261
$134$	1009.98	0.651112
$135$	1408.26	0.897804
$136$	−137.535	−0.0867169
$137$	−1090.32	−0.679944	−0.339972	−	0.940436i	$-0.610418\pi$
$137$	−1090.32	−0.679944	−0.339972	+	0.940436i	$0.610418\pi$
$138$	−1414.71	−0.872665
$139$	−1924.60	−1.17440	−0.587202	−	0.809440i	$-0.699771\pi$
$139$	−1924.60	−1.17440	−0.587202	+	0.809440i	$0.699771\pi$
$140$	−1070.38	−0.646171
$141$	1723.02	1.02911
$142$	−962.565	−0.568850
$143$	282.895	0.165433
$144$	−241.616	−0.139824
$145$	281.211	0.161057
$146$	−6.22039	−0.00352605
$147$	−1443.73	−0.810047
$148$	119.192	0.0661994
$149$	−1703.62	−0.936687	−0.468343	−	0.883546i	$-0.655149\pi$
$149$	−1703.62	−0.936687	−0.468343	+	0.883546i	$0.655149\pi$
$150$	−213.657	−0.116300
$151$	1082.81	0.583560	0.291780	−	0.956486i	$-0.405753\pi$
$151$	1082.81	0.583560	0.291780	+	0.956486i	$0.405753\pi$
$152$	−353.616	−0.188698
$153$	−259.614	−0.137180
$154$	−2889.21	−1.51181
$155$	2909.03	1.50748
$156$	74.5653	0.0382692
$157$	2407.90	1.22402	0.612010	−	0.790850i	$-0.290361\pi$
$157$	2407.90	1.22402	0.612010	+	0.790850i	$0.290361\pi$
$158$	2086.53	1.05060
$159$	1212.93	0.604980
$160$	−310.302	−0.153322
$161$	5658.82	2.77005
$162$	186.463	0.0904317
$163$	1061.59	0.510123	0.255061	−	0.966925i	$-0.417904\pi$
$163$	1061.59	0.510123	0.255061	+	0.966925i	$0.417904\pi$
$164$	−175.837	−0.0837228
$165$	1751.03	0.826166
$166$	2014.16	0.941742
$167$	1315.63	0.609618	0.304809	−	0.952414i	$-0.401407\pi$
$167$	1315.63	0.609618	0.304809	+	0.952414i	$0.401407\pi$
$168$	−761.535	−0.349724
$169$	−2167.80	−0.986707
$170$	−333.416	−0.150423
$171$	−667.496	−0.298507
$172$	259.292	0.114947
$173$	−653.800	−0.287327	−0.143663	−	0.989627i	$-0.545888\pi$
$173$	−653.800	−0.287327	−0.143663	+	0.989627i	$0.545888\pi$
$174$	200.070	0.0871684
$175$	854.629	0.369165
$176$	−837.576	−0.358719
$177$	−1801.92	−0.765200
$178$	591.273	0.248977
$179$	2055.28	0.858208	0.429104	−	0.903255i	$-0.358829\pi$
$179$	2055.28	0.858208	0.429104	+	0.903255i	$0.358829\pi$
$180$	−585.735	−0.242545
$181$	3398.36	1.39557	0.697785	−	0.716307i	$-0.254169\pi$
$181$	3398.36	1.39557	0.697785	+	0.716307i	$0.254169\pi$
$182$	−298.261	−0.121476
$183$	−1671.64	−0.675254
$184$	1640.48	0.657271
$185$	288.949	0.114832
$186$	2069.66	0.815884
$187$	−899.966	−0.351936
$188$	−1998.00	−0.775101
$189$	−4007.67	−1.54241
$190$	−857.249	−0.327323
$191$	4853.92	1.83883	0.919417	−	0.393285i	$-0.128661\pi$
$191$	4853.92	1.83883	0.919417	+	0.393285i	$0.128661\pi$
$192$	−220.767	−0.0829818
$193$	−4877.24	−1.81902	−0.909512	−	0.415677i	$-0.863545\pi$
$193$	−4877.24	−1.81902	−0.909512	+	0.415677i	$0.863545\pi$
$194$	−857.898	−0.317492
$195$	180.764	0.0663834
$196$	1674.14	0.610109
$197$	−1031.42	−0.373022	−0.186511	−	0.982453i	$-0.559718\pi$
$197$	−1031.42	−0.373022	−0.186511	+	0.982453i	$0.559718\pi$
$198$	−1581.03	−0.567469
$199$	−1167.70	−0.415960	−0.207980	−	0.978133i	$-0.566689\pi$
$199$	−1167.70	−0.415960	−0.207980	+	0.978133i	$0.566689\pi$
$200$	247.755	0.0875946
$201$	1741.96	0.611284
$202$	2424.32	0.844428
$203$	−800.282	−0.276693
$204$	−237.212	−0.0814126
$205$	−426.269	−0.145229
$206$	−2064.14	−0.698134
$207$	3096.62	1.03976
$208$	−86.4653	−0.0288235
$209$	−2313.91	−0.765820
$210$	−1846.14	−0.606646
$211$	794.341	0.259169	0.129585	−	0.991568i	$-0.458636\pi$
$211$	794.341	0.259169	0.129585	+	0.991568i	$0.458636\pi$
$212$	−1406.51	−0.455657
$213$	−1660.18	−0.534055
$214$	2352.12	0.751343
$215$	628.584	0.199391
$216$	−1161.82	−0.365980
$217$	−8278.62	−2.58981
$218$	4335.71	1.34703
$219$	−10.7286	−0.00331037
$220$	−2030.48	−0.622250
$221$	−92.9061	−0.0282785
$222$	205.576	0.0621501
$223$	5136.08	1.54232	0.771161	−	0.636641i	$-0.219676\pi$
$223$	5136.08	1.54232	0.771161	+	0.636641i	$0.219676\pi$
$224$	883.069	0.263404
$225$	467.669	0.138569
$226$	−3247.31	−0.955786
$227$	−5032.84	−1.47155	−0.735774	−	0.677228i	$-0.763181\pi$
$227$	−5032.84	−1.47155	−0.735774	+	0.677228i	$0.763181\pi$
$228$	−609.898	−0.177156
$229$	−6213.12	−1.79290	−0.896451	−	0.443142i	$-0.853864\pi$
$229$	−6213.12	−1.79290	−0.896451	+	0.443142i	$0.853864\pi$
$230$	3976.91	1.14013
$231$	−4983.15	−1.41934
$232$	−232.000	−0.0656532
$233$	−3117.59	−0.876566	−0.438283	−	0.898837i	$-0.644413\pi$
$233$	−3117.59	−0.876566	−0.438283	+	0.898837i	$0.644413\pi$
$234$	−163.214	−0.0455968
$235$	−4843.62	−1.34452
$236$	2089.49	0.576331
$237$	3598.73	0.986342
$238$	948.849	0.258423
$239$	7193.09	1.94679	0.973394	−	0.229136i	$-0.0735902\pi$
$239$	7193.09	1.94679	0.973394	+	0.229136i	$0.0735902\pi$
$240$	−535.192	−0.143944
$241$	−837.347	−0.223810	−0.111905	−	0.993719i	$-0.535695\pi$
$241$	−837.347	−0.223810	−0.111905	+	0.993719i	$0.535695\pi$
$242$	−2818.72	−0.748737
$243$	−3599.53	−0.950246
$244$	1938.42	0.508586
$245$	4058.51	1.05832
$246$	−303.273	−0.0786017
$247$	−238.871	−0.0615345
$248$	−2399.96	−0.614505
$249$	3473.91	0.884138
$250$	3024.85	0.765234
$251$	4306.56	1.08298	0.541489	−	0.840708i	$-0.317861\pi$
$251$	4306.56	1.08298	0.541489	+	0.840708i	$0.317861\pi$
$252$	1666.91	0.416687
$253$	10734.6	2.66750
$254$	−666.947	−0.164756
$255$	−575.058	−0.141222
$256$	256.000	0.0625000
$257$	401.238	0.0973873	0.0486936	−	0.998814i	$-0.484494\pi$
$257$	401.238	0.0973873	0.0486936	+	0.998814i	$0.484494\pi$
$258$	447.212	0.107916
$259$	−822.302	−0.197279
$260$	−209.612	−0.0499985
$261$	−437.930	−0.103859
$262$	1866.98	0.440239
$263$	−2682.72	−0.628988	−0.314494	−	0.949260i	$-0.601835\pi$
$263$	−2682.72	−0.628988	−0.314494	+	0.949260i	$0.601835\pi$
$264$	−1444.60	−0.336777
$265$	−3409.70	−0.790401
$266$	2439.59	0.562334
$267$	1019.80	0.233747
$268$	−2019.96	−0.460405
$269$	−4732.38	−1.07263	−0.536316	−	0.844017i	$-0.680184\pi$
$269$	−4732.38	−1.07263	−0.536316	+	0.844017i	$0.680184\pi$
$270$	−2816.52	−0.634843
$271$	4529.66	1.01534	0.507670	−	0.861552i	$-0.330507\pi$
$271$	4529.66	1.01534	0.507670	+	0.861552i	$0.330507\pi$
$272$	275.069	0.0613181
$273$	−514.424	−0.114045
$274$	2180.64	0.480793
$275$	1621.20	0.355498
$276$	2829.41	0.617067
$277$	7158.69	1.55279	0.776397	−	0.630245i	$-0.217045\pi$
$277$	7158.69	1.55279	0.776397	+	0.630245i	$0.217045\pi$
$278$	3849.19	0.830429
$279$	−4530.22	−0.972105
$280$	2140.77	0.456912
$281$	−5769.75	−1.22489	−0.612446	−	0.790512i	$-0.709814\pi$
$281$	−5769.75	−1.22489	−0.612446	+	0.790512i	$0.709814\pi$
$282$	−3446.04	−0.727690
$283$	−3815.25	−0.801389	−0.400694	−	0.916212i	$-0.631231\pi$
$283$	−3815.25	−0.801389	−0.400694	+	0.916212i	$0.631231\pi$
$284$	1925.13	0.402238
$285$	−1478.54	−0.307301
$286$	−565.791	−0.116979
$287$	1213.09	0.249501
$288$	483.233	0.0988707
$289$	−4617.44	−0.939841
$290$	−562.422	−0.113885
$291$	−1479.66	−0.298072
$292$	12.4408	0.00249329
$293$	5416.66	1.08002	0.540008	−	0.841660i	$-0.318421\pi$
$293$	5416.66	1.08002	0.540008	+	0.841660i	$0.318421\pi$
$294$	2887.46	0.572790
$295$	5065.41	0.999728
$296$	−238.384	−0.0468100
$297$	−7602.41	−1.48531
$298$	3407.25	0.662338
$299$	1108.16	0.214337
$300$	427.314	0.0822367
$301$	−1788.85	−0.342550
$302$	−2165.61	−0.412639
$303$	4181.33	0.792776
$304$	707.233	0.133430
$305$	4699.20	0.882214
$306$	519.229	0.0970010
$307$	4929.06	0.916339	0.458169	−	0.888865i	$-0.348505\pi$
$307$	4929.06	0.916339	0.458169	+	0.888865i	$0.348505\pi$
$308$	5778.42	1.06901
$309$	−3560.12	−0.655430
$310$	−5818.05	−1.06595
$311$	1117.93	0.203832	0.101916	−	0.994793i	$-0.467503\pi$
$311$	1117.93	0.203832	0.101916	+	0.994793i	$0.467503\pi$
$312$	−149.131	−0.0270604
$313$	2032.21	0.366988	0.183494	−	0.983021i	$-0.441259\pi$
$313$	2032.21	0.366988	0.183494	+	0.983021i	$0.441259\pi$
$314$	−4815.79	−0.865512
$315$	4040.97	0.722803
$316$	−4173.06	−0.742890
$317$	2051.75	0.363526	0.181763	−	0.983342i	$-0.441820\pi$
$317$	2051.75	0.363526	0.181763	+	0.983342i	$0.441820\pi$
$318$	−2425.86	−0.427785
$319$	−1518.11	−0.266450
$320$	620.604	0.108415
$321$	4056.80	0.705385
$322$	−11317.6	−1.95872
$323$	759.914	0.130906
$324$	−372.927	−0.0639449
$325$	167.361	0.0285647
$326$	−2123.18	−0.360711
$327$	7478.00	1.26463
$328$	351.673	0.0592010
$329$	13784.1	2.30986
$330$	−3502.06	−0.584188
$331$	−9652.92	−1.60294	−0.801469	−	0.598036i	$-0.795948\pi$
$331$	−9652.92	−1.60294	−0.801469	+	0.598036i	$0.795948\pi$
$332$	−4028.32	−0.665912
$333$	−449.980	−0.0740502
$334$	−2631.25	−0.431065
$335$	−4896.85	−0.798638
$336$	1523.07	0.247292
$337$	4277.57	0.691437	0.345719	−	0.938338i	$-0.387635\pi$
$337$	4277.57	0.691437	0.345719	+	0.938338i	$0.387635\pi$
$338$	4335.59	0.697707
$339$	−5600.77	−0.897322
$340$	666.833	0.106365
$341$	−15704.2	−2.49394
$342$	1334.99	0.211076
$343$	−2084.45	−0.328133
$344$	−518.584	−0.0812795
$345$	6859.16	1.07039
$346$	1307.60	0.203171
$347$	−9475.83	−1.46596	−0.732982	−	0.680248i	$-0.761872\pi$
$347$	−9475.83	−1.46596	−0.732982	+	0.680248i	$0.761872\pi$
$348$	−400.141	−0.0616374
$349$	2968.57	0.455313	0.227656	−	0.973742i	$-0.426894\pi$
$349$	2968.57	0.455313	0.227656	+	0.973742i	$0.426894\pi$
$350$	−1709.26	−0.261039
$351$	−784.819	−0.119346
$352$	1675.15	0.253653
$353$	3969.16	0.598461	0.299231	−	0.954181i	$-0.403270\pi$
$353$	3969.16	0.598461	0.299231	+	0.954181i	$0.403270\pi$
$354$	3603.84	0.541078
$355$	4666.97	0.697738
$356$	−1182.55	−0.176053
$357$	1636.52	0.242616
$358$	−4110.57	−0.606845
$359$	−3503.03	−0.514994	−0.257497	−	0.966279i	$-0.582898\pi$
$359$	−3503.03	−0.514994	−0.257497	+	0.966279i	$0.582898\pi$
$360$	1171.47	0.171505
$361$	−4905.18	−0.715145
$362$	−6796.72	−0.986817
$363$	−4861.58	−0.702939
$364$	596.522	0.0858963
$365$	30.1594	0.00432497
$366$	3343.29	0.477477
$367$	−8253.53	−1.17393	−0.586963	−	0.809614i	$-0.699677\pi$
$367$	−8253.53	−1.17393	−0.586963	+	0.809614i	$0.699677\pi$
$368$	−3280.96	−0.464761
$369$	663.828	0.0936518
$370$	−577.898	−0.0811986
$371$	9703.46	1.35789
$372$	−4139.31	−0.576917
$373$	1253.95	0.174067	0.0870335	−	0.996205i	$-0.472261\pi$
$373$	1253.95	0.174067	0.0870335	+	0.996205i	$0.472261\pi$
$374$	1799.93	0.248856
$375$	5217.10	0.718426
$376$	3996.00	0.548079
$377$	−156.718	−0.0214096
$378$	8015.35	1.09065
$379$	−6966.52	−0.944185	−0.472092	−	0.881549i	$-0.656501\pi$
$379$	−6966.52	−0.944185	−0.472092	+	0.881549i	$0.656501\pi$
$380$	1714.50	0.231452
$381$	−1150.31	−0.154678
$382$	−9707.84	−1.30025
$383$	−9821.84	−1.31037	−0.655186	−	0.755467i	$-0.727410\pi$
$383$	−9821.84	−1.31037	−0.655186	+	0.755467i	$0.727410\pi$
$384$	441.535	0.0586770
$385$	14008.2	1.85435
$386$	9754.49	1.28624
$387$	−978.893	−0.128579
$388$	1715.80	0.224501
$389$	3942.96	0.513923	0.256962	−	0.966422i	$-0.417279\pi$
$389$	3942.96	0.513923	0.256962	+	0.966422i	$0.417279\pi$
$390$	−361.528	−0.0469402
$391$	−3525.36	−0.455972
$392$	−3348.28	−0.431412
$393$	3220.07	0.413311
$394$	2062.83	0.263767
$395$	−10116.5	−1.28865
$396$	3162.06	0.401262
$397$	8725.50	1.10307	0.551537	−	0.834150i	$-0.314041\pi$
$397$	8725.50	1.10307	0.551537	+	0.834150i	$0.314041\pi$
$398$	2335.40	0.294128
$399$	4207.67	0.527938
$400$	−495.510	−0.0619388
$401$	3986.86	0.496494	0.248247	−	0.968697i	$-0.420146\pi$
$401$	3986.86	0.496494	0.248247	+	0.968697i	$0.420146\pi$
$402$	−3483.91	−0.432243
$403$	−1621.19	−0.200391
$404$	−4848.64	−0.597101
$405$	−904.061	−0.110921
$406$	1600.56	0.195652
$407$	−1559.88	−0.189976
$408$	474.424	0.0575674
$409$	−704.847	−0.0852138	−0.0426069	−	0.999092i	$-0.513566\pi$
$409$	−704.847	−0.0852138	−0.0426069	+	0.999092i	$0.513566\pi$
$410$	852.539	0.102692
$411$	3761.05	0.451384
$412$	4128.28	0.493655
$413$	−14415.3	−1.71751
$414$	−6193.24	−0.735220
$415$	−9765.60	−1.15512
$416$	172.931	0.0203813
$417$	6638.88	0.779634
$418$	4627.82	0.541517
$419$	6705.33	0.781807	0.390903	−	0.920432i	$-0.372163\pi$
$419$	6705.33	0.781807	0.390903	+	0.920432i	$0.372163\pi$
$420$	3692.28	0.428964
$421$	8996.55	1.04148	0.520742	−	0.853714i	$-0.325655\pi$
$421$	8996.55	1.04148	0.520742	+	0.853714i	$0.325655\pi$
$422$	−1588.68	−0.183260
$423$	7542.95	0.867023
$424$	2813.01	0.322198
$425$	−532.421	−0.0607675
$426$	3320.36	0.377634
$427$	−13373.2	−1.51563
$428$	−4704.24	−0.531280
$429$	−975.845	−0.109823
$430$	−1257.17	−0.140991
$431$	5383.16	0.601619	0.300810	−	0.953684i	$-0.402743\pi$
$431$	5383.16	0.601619	0.300810	+	0.953684i	$0.402743\pi$
$432$	2323.63	0.258787
$433$	−6901.64	−0.765985	−0.382993	−	0.923751i	$-0.625107\pi$
$433$	−6901.64	−0.765985	−0.382993	+	0.923751i	$0.625107\pi$
$434$	16557.2	1.83127
$435$	−970.035	−0.106919
$436$	−8671.42	−0.952491
$437$	−9064.08	−0.992205
$438$	21.4572	0.00234078
$439$	12119.4	1.31760	0.658802	−	0.752316i	$-0.271063\pi$
$439$	12119.4	1.31760	0.658802	+	0.752316i	$0.271063\pi$
$440$	4060.96	0.439997
$441$	−6320.30	−0.682464
$442$	185.812	0.0199959
$443$	5229.40	0.560849	0.280425	−	0.959876i	$-0.409525\pi$
$443$	5229.40	0.560849	0.280425	+	0.959876i	$0.409525\pi$
$444$	−411.151	−0.0439468
$445$	−2866.77	−0.305389
$446$	−10272.2	−1.09059
$447$	5876.64	0.621824
$448$	−1766.14	−0.186255
$449$	−10220.3	−1.07422	−0.537112	−	0.843511i	$-0.680485\pi$
$449$	−10220.3	−1.07422	−0.537112	+	0.843511i	$0.680485\pi$
$450$	−935.339	−0.0979829
$451$	2301.20	0.240264
$452$	6494.61	0.675843
$453$	−3735.13	−0.387399
$454$	10065.7	1.04054
$455$	1446.11	0.148999
$456$	1219.80	0.125268
$457$	12030.6	1.23143	0.615717	−	0.787967i	$-0.288866\pi$
$457$	12030.6	1.23143	0.615717	+	0.787967i	$0.288866\pi$
$458$	12426.2	1.26777
$459$	2496.72	0.253893
$460$	−7953.82	−0.806193
$461$	1063.24	0.107418	0.0537092	−	0.998557i	$-0.482896\pi$
$461$	1063.24	0.107418	0.0537092	+	0.998557i	$0.482896\pi$
$462$	9966.29	1.00362
$463$	15498.8	1.55570	0.777851	−	0.628449i	$-0.216310\pi$
$463$	15498.8	1.55570	0.777851	+	0.628449i	$0.216310\pi$
$464$	464.000	0.0464238
$465$	−10034.7	−1.00074
$466$	6235.17	0.619826
$467$	13215.6	1.30952	0.654759	−	0.755838i	$-0.272770\pi$
$467$	13215.6	1.30952	0.654759	+	0.755838i	$0.272770\pi$
$468$	326.429	0.0322418
$469$	13935.7	1.37204
$470$	9687.23	0.950721
$471$	−8306.01	−0.812571
$472$	−4178.98	−0.407528
$473$	−3393.38	−0.329869
$474$	−7197.47	−0.697449
$475$	−1368.91	−0.132231
$476$	−1897.70	−0.182733
$477$	5309.92	0.509695
$478$	−14386.2	−1.37659
$479$	−19058.5	−1.81796	−0.908981	−	0.416837i	$-0.863139\pi$
$479$	−19058.5	−1.81796	−0.908981	+	0.416837i	$0.863139\pi$
$480$	1070.38	0.101784
$481$	−161.031	−0.0152648
$482$	1674.69	0.158258
$483$	−19520.1	−1.83891
$484$	5637.45	0.529437
$485$	4159.49	0.389428
$486$	7199.06	0.671926
$487$	−487.214	−0.0453343	−0.0226671	−	0.999743i	$-0.507216\pi$
$487$	−487.214	−0.0453343	−0.0226671	+	0.999743i	$0.507216\pi$
$488$	−3876.85	−0.359624
$489$	−3661.94	−0.338647
$490$	−8117.01	−0.748345
$491$	−6462.89	−0.594025	−0.297013	−	0.954874i	$-0.595990\pi$
$491$	−6462.89	−0.594025	−0.297013	+	0.954874i	$0.595990\pi$
$492$	606.547	0.0555798
$493$	498.563	0.0455460
$494$	477.743	0.0435115
$495$	7665.58	0.696045
$496$	4799.91	0.434521
$497$	−13281.4	−1.19870
$498$	−6947.83	−0.625180
$499$	−7947.79	−0.713010	−0.356505	−	0.934293i	$-0.616032\pi$
$499$	−7947.79	−0.713010	−0.356505	+	0.934293i	$0.616032\pi$
$500$	−6049.70	−0.541102
$501$	−4538.24	−0.404698
$502$	−8613.12	−0.765781
$503$	−2307.81	−0.204573	−0.102287	−	0.994755i	$-0.532616\pi$
$503$	−2307.81	−0.204573	−0.102287	+	0.994755i	$0.532616\pi$
$504$	−3333.81	−0.294642
$505$	−11754.2	−1.03576
$506$	−21469.2	−1.88621
$507$	7477.79	0.655030
$508$	1333.89	0.116500
$509$	−16496.2	−1.43651	−0.718253	−	0.695782i	$-0.755058\pi$
$509$	−16496.2	−1.43651	−0.718253	+	0.695782i	$0.755058\pi$
$510$	1150.12	0.0998588
$511$	−85.8287	−0.00743021
$512$	−512.000	−0.0441942
$513$	6419.33	0.552476
$514$	−802.476	−0.0688632
$515$	10007.9	0.856314
$516$	−894.424	−0.0763078
$517$	26148.0	2.22435
$518$	1644.60	0.139498
$519$	2255.28	0.190743
$520$	419.224	0.0353542
$521$	−10386.5	−0.873398	−0.436699	−	0.899608i	$-0.643853\pi$
$521$	−10386.5	−0.873398	−0.436699	+	0.899608i	$0.643853\pi$
$522$	875.859	0.0734393
$523$	−918.174	−0.0767666	−0.0383833	−	0.999263i	$-0.512221\pi$
$523$	−918.174	−0.0767666	−0.0383833	+	0.999263i	$0.512221\pi$
$524$	−3733.97	−0.311296
$525$	−2948.03	−0.245072
$526$	5365.44	0.444761
$527$	5157.45	0.426304
$528$	2889.21	0.238138
$529$	29882.7	2.45604
$530$	6819.40	0.558898
$531$	−7888.36	−0.644681
$532$	−4879.18	−0.397631
$533$	237.559	0.0193055
$534$	−2039.59	−0.165284
$535$	−11404.2	−0.921580
$536$	4039.92	0.325556
$537$	−7089.68	−0.569725
$538$	9464.75	0.758465
$539$	−21909.7	−1.75086
$540$	5633.03	0.448902
$541$	−12860.2	−1.02200	−0.511001	−	0.859580i	$-0.670725\pi$
$541$	−12860.2	−1.02200	−0.511001	+	0.859580i	$0.670725\pi$
$542$	−9059.32	−0.717954
$543$	−11722.6	−0.926456
$544$	−550.139	−0.0433585
$545$	−21021.6	−1.65223
$546$	1028.85	0.0806423
$547$	4471.84	0.349547	0.174774	−	0.984609i	$-0.444081\pi$
$547$	4471.84	0.349547	0.174774	+	0.984609i	$0.444081\pi$
$548$	−4361.28	−0.339972
$549$	−7318.05	−0.568901
$550$	−3242.40	−0.251375
$551$	1281.86	0.0991090
$552$	−5658.82	−0.436333
$553$	28789.9	2.21387
$554$	−14317.4	−1.09799
$555$	−996.727	−0.0762319
$556$	−7698.39	−0.587202
$557$	2649.40	0.201542	0.100771	−	0.994910i	$-0.467869\pi$
$557$	2649.40	0.201542	0.100771	+	0.994910i	$0.467869\pi$
$558$	9060.44	0.687382
$559$	−350.309	−0.0265053
$560$	−4281.53	−0.323085
$561$	3104.42	0.233634
$562$	11539.5	0.866130
$563$	12615.8	0.944395	0.472198	−	0.881493i	$-0.343461\pi$
$563$	12615.8	0.944395	0.472198	+	0.881493i	$0.343461\pi$
$564$	6892.07	0.514554
$565$	15744.5	1.17234
$566$	7630.50	0.566667
$567$	2572.81	0.190561
$568$	−3850.26	−0.284425
$569$	617.443	0.0454913	0.0227456	−	0.999741i	$-0.492759\pi$
$569$	617.443	0.0454913	0.0227456	+	0.999741i	$0.492759\pi$
$570$	2957.07	0.217295
$571$	8005.50	0.586725	0.293362	−	0.956001i	$-0.405226\pi$
$571$	8005.50	0.586725	0.293362	+	0.956001i	$0.405226\pi$
$572$	1131.58	0.0827164
$573$	−16743.5	−1.22072
$574$	−2426.19	−0.176424
$575$	6350.59	0.460588
$576$	−966.465	−0.0699121
$577$	11761.1	0.848565	0.424282	−	0.905530i	$-0.360526\pi$
$577$	11761.1	0.848565	0.424282	+	0.905530i	$0.360526\pi$
$578$	9234.88	0.664568
$579$	16824.0	1.20757
$580$	1124.84	0.0805287
$581$	27791.3	1.98447
$582$	2959.31	0.210769
$583$	18407.1	1.30762
$584$	−24.8816	−0.00176302
$585$	791.340	0.0559280
$586$	−10833.3	−0.763686
$587$	−1211.03	−0.0851528	−0.0425764	−	0.999093i	$-0.513557\pi$
$587$	−1211.03	−0.0851528	−0.0425764	+	0.999093i	$0.513557\pi$
$588$	−5774.92	−0.405024
$589$	13260.4	0.927646
$590$	−10130.8	−0.706914
$591$	3557.86	0.247633
$592$	476.767	0.0330997
$593$	459.376	0.0318117	0.0159058	−	0.999873i	$-0.494937\pi$
$593$	459.376	0.0318117	0.0159058	+	0.999873i	$0.494937\pi$
$594$	15204.8	1.05027
$595$	−4600.47	−0.316976
$596$	−6814.50	−0.468343
$597$	4027.97	0.276137
$598$	−2216.32	−0.151559
$599$	16686.2	1.13820	0.569099	−	0.822269i	$-0.307292\pi$
$599$	16686.2	1.13820	0.569099	+	0.822269i	$0.307292\pi$
$600$	−854.629	−0.0581501
$601$	11625.9	0.789072	0.394536	−	0.918881i	$-0.370905\pi$
$601$	11625.9	0.789072	0.394536	+	0.918881i	$0.370905\pi$
$602$	3577.70	0.242219
$603$	7625.86	0.515007
$604$	4331.22	0.291780
$605$	13666.5	0.918384
$606$	−8362.66	−0.560577
$607$	−21556.9	−1.44146	−0.720731	−	0.693215i	$-0.756194\pi$
$607$	−21556.9	−1.44146	−0.720731	+	0.693215i	$0.756194\pi$
$608$	−1414.47	−0.0943489
$609$	2760.56	0.183684
$610$	−9398.39	−0.623819
$611$	2699.34	0.178729
$612$	−1038.46	−0.0685901
$613$	−19273.2	−1.26988	−0.634940	−	0.772561i	$-0.718975\pi$
$613$	−19273.2	−1.26988	−0.634940	+	0.772561i	$0.718975\pi$
$614$	−9858.11	−0.647949
$615$	1470.41	0.0964110
$616$	−11556.8	−0.755906
$617$	−3270.70	−0.213409	−0.106704	−	0.994291i	$-0.534030\pi$
$617$	−3270.70	−0.213409	−0.106704	+	0.994291i	$0.534030\pi$
$618$	7120.23	0.463459
$619$	75.8068	0.00492234	0.00246117	−	0.999997i	$-0.499217\pi$
$619$	75.8068	0.00492234	0.00246117	+	0.999997i	$0.499217\pi$
$620$	11636.1	0.753738
$621$	−29780.3	−1.92438
$622$	−2235.85	−0.144131
$623$	8158.37	0.524652
$624$	298.261	0.0191346
$625$	−10794.7	−0.690862
$626$	−4064.41	−0.259499
$627$	7981.81	0.508393
$628$	9631.58	0.612010
$629$	512.282	0.0324738
$630$	−8081.94	−0.511099
$631$	−4915.71	−0.310129	−0.155064	−	0.987904i	$-0.549559\pi$
$631$	−4915.71	−0.310129	−0.155064	+	0.987904i	$0.549559\pi$
$632$	8346.13	0.525302
$633$	−2740.07	−0.172051
$634$	−4103.50	−0.257052
$635$	3233.67	0.202086
$636$	4851.73	0.302490
$637$	−2261.80	−0.140684
$638$	3036.21	0.188409
$639$	−7267.86	−0.449941
$640$	−1241.21	−0.0766610
$641$	3317.63	0.204428	0.102214	−	0.994762i	$-0.467407\pi$
$641$	3317.63	0.204428	0.102214	+	0.994762i	$0.467407\pi$
$642$	−8113.61	−0.498783
$643$	−27222.7	−1.66961	−0.834805	−	0.550546i	$-0.814420\pi$
$643$	−27222.7	−1.66961	−0.834805	+	0.550546i	$0.814420\pi$
$644$	22635.3	1.38502
$645$	−2168.29	−0.132367
$646$	−1519.83	−0.0925648
$647$	24433.6	1.48468	0.742338	−	0.670025i	$-0.233717\pi$
$647$	24433.6	1.48468	0.742338	+	0.670025i	$0.233717\pi$
$648$	745.853	0.0452159
$649$	−27345.4	−1.65393
$650$	−334.722	−0.0201983
$651$	28557.0	1.71926
$652$	4246.35	0.255061
$653$	5755.36	0.344907	0.172454	−	0.985018i	$-0.444830\pi$
$653$	5755.36	0.344907	0.172454	+	0.985018i	$0.444830\pi$
$654$	−14956.0	−0.894229
$655$	−9052.01	−0.539987
$656$	−703.347	−0.0418614
$657$	−46.9671	−0.00278898
$658$	−27568.3	−1.63332
$659$	965.300	0.0570603	0.0285301	−	0.999593i	$-0.490917\pi$
$659$	965.300	0.0570603	0.0285301	+	0.999593i	$0.490917\pi$
$660$	7004.12	0.413083
$661$	−22791.0	−1.34110	−0.670549	−	0.741866i	$-0.733941\pi$
$661$	−22791.0	−1.34110	−0.670549	+	0.741866i	$0.733941\pi$
$662$	19305.8	1.13345
$663$	320.479	0.0187728
$664$	8056.64	0.470871
$665$	−11828.3	−0.689746
$666$	899.959	0.0523614
$667$	−5946.75	−0.345216
$668$	5262.50	0.304809
$669$	−17716.9	−1.02388
$670$	9793.71	0.564722
$671$	−25368.4	−1.45952
$672$	−3046.14	−0.174862
$673$	12412.1	0.710925	0.355463	−	0.934690i	$-0.384323\pi$
$673$	12412.1	0.710925	0.355463	+	0.934690i	$0.384323\pi$
$674$	−8555.15	−0.488920
$675$	−4497.59	−0.256463
$676$	−8671.18	−0.493354
$677$	8998.03	0.510816	0.255408	−	0.966833i	$-0.417790\pi$
$677$	8998.03	0.510816	0.255408	+	0.966833i	$0.417790\pi$
$678$	11201.5	0.634503
$679$	−11837.2	−0.669030
$680$	−1333.67	−0.0752114
$681$	17360.7	0.976893
$682$	31408.5	1.76348
$683$	13439.1	0.752904	0.376452	−	0.926436i	$-0.377144\pi$
$683$	13439.1	0.752904	0.376452	+	0.926436i	$0.377144\pi$
$684$	−2669.98	−0.149253
$685$	−10572.8	−0.589730
$686$	4168.90	0.232025
$687$	21432.1	1.19023
$688$	1037.17	0.0574733
$689$	1900.22	0.105069
$690$	−13718.3	−0.756880
$691$	−16220.4	−0.892983	−0.446492	−	0.894788i	$-0.647327\pi$
$691$	−16220.4	−0.892983	−0.446492	+	0.894788i	$0.647327\pi$
$692$	−2615.20	−0.143663
$693$	−21815.0	−1.19579
$694$	18951.7	1.03659
$695$	−18662.7	−1.01858
$696$	800.282	0.0435842
$697$	−755.739	−0.0410698
$698$	−5937.15	−0.321955
$699$	10754.1	0.581912
$700$	3418.51	0.184582
$701$	−18919.6	−1.01938	−0.509689	−	0.860359i	$-0.670239\pi$
$701$	−18919.6	−1.01938	−0.509689	+	0.860359i	$0.670239\pi$
$702$	1569.64	0.0843905
$703$	1317.13	0.0706636
$704$	−3350.30	−0.179360
$705$	16708.0	0.892567
$706$	−7938.31	−0.423176
$707$	33450.6	1.77941
$708$	−7207.67	−0.382600
$709$	−4781.99	−0.253302	−0.126651	−	0.991947i	$-0.540423\pi$
$709$	−4781.99	−0.253302	−0.126651	+	0.991947i	$0.540423\pi$
$710$	−9333.94	−0.493375
$711$	15754.4	0.830992
$712$	2365.09	0.124488
$713$	−61516.9	−3.23117
$714$	−3273.04	−0.171555
$715$	2743.22	0.143483
$716$	8221.14	0.429104
$717$	−24812.5	−1.29238
$718$	7006.06	0.364156
$719$	13020.1	0.675336	0.337668	−	0.941265i	$-0.390362\pi$
$719$	13020.1	0.675336	0.337668	+	0.941265i	$0.390362\pi$
$720$	−2342.94	−0.121272
$721$	−28480.9	−1.47113
$722$	9810.36	0.505684
$723$	2888.42	0.148577
$724$	13593.4	0.697785
$725$	−898.112	−0.0460070
$726$	9723.16	0.497053
$727$	−8738.68	−0.445804	−0.222902	−	0.974841i	$-0.571553\pi$
$727$	−8738.68	−0.445804	−0.222902	+	0.974841i	$0.571553\pi$
$728$	−1193.04	−0.0607379
$729$	14933.8	0.758715
$730$	−60.3187	−0.00305821
$731$	1114.43	0.0563865
$732$	−6686.58	−0.337627
$733$	−12775.2	−0.643744	−0.321872	−	0.946783i	$-0.604312\pi$
$733$	−12775.2	−0.643744	−0.321872	+	0.946783i	$0.604312\pi$
$734$	16507.1	0.830091
$735$	−13999.8	−0.702571
$736$	6561.93	0.328636
$737$	26435.4	1.32125
$738$	−1327.66	−0.0662219
$739$	5397.68	0.268683	0.134342	−	0.990935i	$-0.457108\pi$
$739$	5397.68	0.268683	0.134342	+	0.990935i	$0.457108\pi$
$740$	1155.80	0.0574161
$741$	823.985	0.0408500
$742$	−19406.9	−0.960176
$743$	−37984.2	−1.87551	−0.937757	−	0.347293i	$-0.887101\pi$
$743$	−37984.2	−1.87551	−0.937757	+	0.347293i	$0.887101\pi$
$744$	8278.62	0.407942
$745$	−16519.9	−0.812408
$746$	−2507.90	−0.123084
$747$	15207.9	0.744886
$748$	−3599.87	−0.175968
$749$	32454.4	1.58326
$750$	−10434.2	−0.508004
$751$	26384.8	1.28202	0.641009	−	0.767534i	$-0.278516\pi$
$751$	26384.8	1.28202	0.641009	+	0.767534i	$0.278516\pi$
$752$	−7991.99	−0.387550
$753$	−14855.4	−0.718940
$754$	313.437	0.0151389
$755$	10499.9	0.506133
$756$	−16030.7	−0.771205
$757$	6889.95	0.330805	0.165403	−	0.986226i	$-0.447108\pi$
$757$	6889.95	0.330805	0.165403	+	0.986226i	$0.447108\pi$
$758$	13933.0	0.667639
$759$	−37028.8	−1.77083
$760$	−3429.00	−0.163662
$761$	31579.3	1.50427	0.752134	−	0.659010i	$-0.229025\pi$
$761$	31579.3	1.50427	0.752134	+	0.659010i	$0.229025\pi$
$762$	2300.63	0.109374
$763$	59824.0	2.83850
$764$	19415.7	0.919417
$765$	−2517.46	−0.118979
$766$	19643.7	0.926574
$767$	−2822.94	−0.132895
$768$	−883.069	−0.0414909
$769$	−12232.7	−0.573632	−0.286816	−	0.957986i	$-0.592597\pi$
$769$	−12232.7	−0.573632	−0.286816	+	0.957986i	$0.592597\pi$
$770$	−28016.5	−1.31123
$771$	−1384.07	−0.0646510
$772$	−19509.0	−0.909512
$773$	9958.46	0.463365	0.231682	−	0.972792i	$-0.425577\pi$
$773$	9958.46	0.463365	0.231682	+	0.972792i	$0.425577\pi$
$774$	1957.79	0.0909188
$775$	−9290.64	−0.430619
$776$	−3431.59	−0.158746
$777$	2836.52	0.130965
$778$	−7885.92	−0.363398
$779$	−1943.09	−0.0893688
$780$	723.055	0.0331917
$781$	−25194.4	−1.15432
$782$	7050.72	0.322421
$783$	4211.58	0.192222
$784$	6696.55	0.305054
$785$	23349.2	1.06162
$786$	−6440.14	−0.292255
$787$	103.911	0.00470653	0.00235326	−	0.999997i	$-0.499251\pi$
$787$	103.911	0.00470653	0.00235326	+	0.999997i	$0.499251\pi$
$788$	−4125.67	−0.186511
$789$	9254.02	0.417556
$790$	20233.0	0.911211
$791$	−44806.2	−2.01406
$792$	−6324.12	−0.283735
$793$	−2618.85	−0.117274
$794$	−17451.0	−0.779991
$795$	11761.7	0.524711
$796$	−4670.80	−0.207980
$797$	30624.4	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$797$	30624.4	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$798$	−8415.35	−0.373308
$799$	−8587.31	−0.380222
$800$	991.020	0.0437973
$801$	4464.42	0.196932
$802$	−7973.71	−0.351074
$803$	−162.814	−0.00715514
$804$	6967.83	0.305642
$805$	54873.3	2.40252
$806$	3242.39	0.141698
$807$	16324.3	0.712072
$808$	9697.27	0.422214
$809$	−7737.50	−0.336262	−0.168131	−	0.985765i	$-0.553773\pi$
$809$	−7737.50	−0.336262	−0.168131	+	0.985765i	$0.553773\pi$
$810$	1808.12	0.0784333
$811$	−12974.1	−0.561756	−0.280878	−	0.959744i	$-0.590626\pi$
$811$	−12974.1	−0.561756	−0.280878	+	0.959744i	$0.590626\pi$
$812$	−3201.13	−0.138347
$813$	−15625.0	−0.674038
$814$	3119.76	0.134333
$815$	10294.2	0.442440
$816$	−948.849	−0.0407063
$817$	2865.31	0.122698
$818$	1409.69	0.0602552
$819$	−2252.02	−0.0960831
$820$	−1705.08	−0.0726145
$821$	3264.26	0.138762	0.0693809	−	0.997590i	$-0.477898\pi$
$821$	3264.26	0.138762	0.0693809	+	0.997590i	$0.477898\pi$
$822$	−7522.10	−0.319177
$823$	23509.7	0.995745	0.497872	−	0.867250i	$-0.334115\pi$
$823$	23509.7	0.995745	0.497872	+	0.867250i	$0.334115\pi$
$824$	−8256.56	−0.349067
$825$	−5592.31	−0.235999
$826$	28830.7	1.21447
$827$	27116.5	1.14019	0.570093	−	0.821580i	$-0.306907\pi$
$827$	27116.5	1.14019	0.570093	+	0.821580i	$0.306907\pi$
$828$	12386.5	0.519879
$829$	15393.6	0.644923	0.322462	−	0.946583i	$-0.395490\pi$
$829$	15393.6	0.644923	0.322462	+	0.946583i	$0.395490\pi$
$830$	19531.2	0.816792
$831$	−24693.8	−1.03083
$832$	−345.861	−0.0144118
$833$	7195.38	0.299286
$834$	−13277.8	−0.551284
$835$	12757.5	0.528734
$836$	−9255.64	−0.382910
$837$	43567.3	1.79917
$838$	−13410.7	−0.552821
$839$	−7906.51	−0.325343	−0.162672	−	0.986680i	$-0.552011\pi$
$839$	−7906.51	−0.325343	−0.162672	+	0.986680i	$0.552011\pi$
$840$	−7384.55	−0.303323
$841$	841.000	0.0344828
$842$	−17993.1	−0.736441
$843$	19902.7	0.813150
$844$	3177.37	0.129585
$845$	−21021.0	−0.855791
$846$	−15085.9	−0.613078
$847$	−38892.6	−1.57777
$848$	−5626.02	−0.227828
$849$	13160.7	0.532005
$850$	1064.84	0.0429691
$851$	−6110.38	−0.246135
$852$	−6640.72	−0.267027
$853$	−11887.1	−0.477146	−0.238573	−	0.971125i	$-0.576680\pi$
$853$	−11887.1	−0.477146	−0.238573	+	0.971125i	$0.576680\pi$
$854$	26746.3	1.07171
$855$	−6472.67	−0.258901
$856$	9408.47	0.375672
$857$	−655.328	−0.0261208	−0.0130604	−	0.999915i	$-0.504157\pi$
$857$	−655.328	−0.0261208	−0.0130604	+	0.999915i	$0.504157\pi$
$858$	1951.69	0.0776569
$859$	−48758.4	−1.93669	−0.968345	−	0.249617i	$-0.919695\pi$
$859$	−48758.4	−1.93669	−0.968345	+	0.249617i	$0.919695\pi$
$860$	2514.34	0.0996956
$861$	−4184.55	−0.165632
$862$	−10766.3	−0.425409
$863$	−10398.2	−0.410150	−0.205075	−	0.978746i	$-0.565744\pi$
$863$	−10398.2	−0.410150	−0.205075	+	0.978746i	$0.565744\pi$
$864$	−4647.27	−0.182990
$865$	−6339.86	−0.249204
$866$	13803.3	0.541633
$867$	15927.8	0.623918
$868$	−33114.5	−1.29491
$869$	54613.4	2.13191
$870$	1940.07	0.0756029
$871$	2729.01	0.106164
$872$	17342.8	0.673513
$873$	−6477.57	−0.251125
$874$	18128.2	0.701595
$875$	41736.8	1.61253
$876$	−42.9143	−0.00165518
$877$	−31024.2	−1.19454	−0.597272	−	0.802039i	$-0.703749\pi$
$877$	−31024.2	−1.19454	−0.597272	+	0.802039i	$0.703749\pi$
$878$	−24238.8	−0.931687
$879$	−18684.7	−0.716973
$880$	−8121.92	−0.311125
$881$	−1510.50	−0.0577641	−0.0288821	−	0.999583i	$-0.509195\pi$
$881$	−1510.50	−0.0577641	−0.0288821	+	0.999583i	$0.509195\pi$
$882$	12640.6	0.482575
$883$	−32376.4	−1.23392	−0.616961	−	0.786993i	$-0.711637\pi$
$883$	−32376.4	−1.23392	−0.616961	+	0.786993i	$0.711637\pi$
$884$	−371.624	−0.0141392
$885$	−17473.1	−0.663674
$886$	−10458.8	−0.396580
$887$	10474.6	0.396510	0.198255	−	0.980150i	$-0.436473\pi$
$887$	10474.6	0.396510	0.198255	+	0.980150i	$0.436473\pi$
$888$	822.302	0.0310751
$889$	−9202.51	−0.347179
$890$	5733.54	0.215942
$891$	4880.53	0.183506
$892$	20544.3	0.771161
$893$	−22078.9	−0.827371
$894$	−11753.3	−0.439696
$895$	19930.0	0.744341
$896$	3532.28	0.131702
$897$	−3822.59	−0.142288
$898$	20440.6	0.759591
$899$	8699.84	0.322754
$900$	1870.68	0.0692843
$901$	−6045.11	−0.223520
$902$	−4602.39	−0.169892
$903$	6170.62	0.227403
$904$	−12989.2	−0.477893
$905$	32953.7	1.21041
$906$	7470.26	0.273932
$907$	−34634.3	−1.26793	−0.633965	−	0.773362i	$-0.718574\pi$
$907$	−34634.3	−1.26793	−0.633965	+	0.773362i	$0.718574\pi$
$908$	−20131.4	−0.735774
$909$	18304.8	0.667913
$910$	−2892.22	−0.105358
$911$	1186.14	0.0431377	0.0215688	−	0.999767i	$-0.493134\pi$
$911$	1186.14	0.0431377	0.0215688	+	0.999767i	$0.493134\pi$
$912$	−2439.59	−0.0885778
$913$	52719.1	1.91101
$914$	−24061.1	−0.870755
$915$	−16209.8	−0.585662
$916$	−24852.5	−0.896451
$917$	25760.6	0.927687
$918$	−4993.44	−0.179529
$919$	−5685.58	−0.204080	−0.102040	−	0.994780i	$-0.532537\pi$
$919$	−5685.58	−0.204080	−0.102040	+	0.994780i	$0.532537\pi$
$920$	15907.6	0.570065
$921$	−17002.7	−0.608316
$922$	−2126.47	−0.0759563
$923$	−2600.89	−0.0927512
$924$	−19932.6	−0.709669
$925$	−922.824	−0.0328025
$926$	−30997.6	−1.10005
$927$	−15585.3	−0.552200
$928$	−928.000	−0.0328266
$929$	−19143.7	−0.676087	−0.338044	−	0.941130i	$-0.609765\pi$
$929$	−19143.7	−0.676087	−0.338044	+	0.941130i	$0.609765\pi$
$930$	20069.3	0.707633
$931$	18500.1	0.651252
$932$	−12470.3	−0.438283
$933$	−3856.28	−0.135315
$934$	−26431.2	−0.925969
$935$	−8726.92	−0.305241
$936$	−652.857	−0.0227984
$937$	−53623.5	−1.86959	−0.934794	−	0.355191i	$-0.884416\pi$
$937$	−53623.5	−1.86959	−0.934794	+	0.355191i	$0.884416\pi$
$938$	−27871.3	−0.970182
$939$	−7010.08	−0.243626
$940$	−19374.5	−0.672261
$941$	−7947.57	−0.275328	−0.137664	−	0.990479i	$-0.543959\pi$
$941$	−7947.57	−0.275328	−0.137664	+	0.990479i	$0.543959\pi$
$942$	16612.0	0.574574
$943$	9014.28	0.311289
$944$	8357.96	0.288166
$945$	−38862.2	−1.33776
$946$	6786.77	0.233252
$947$	−41047.8	−1.40853	−0.704263	−	0.709939i	$-0.748723\pi$
$947$	−41047.8	−1.40853	−0.704263	+	0.709939i	$0.748723\pi$
$948$	14394.9	0.493171
$949$	−16.8077	−0.000574924 0
$950$	2737.82	0.0935017
$951$	−7077.49	−0.241328
$952$	3795.40	0.129212
$953$	−12000.8	−0.407915	−0.203957	−	0.978980i	$-0.565380\pi$
$953$	−12000.8	−0.407915	−0.203957	+	0.978980i	$0.565380\pi$
$954$	−10619.8	−0.360409
$955$	47068.1	1.59486
$956$	28772.4	0.973394
$957$	5236.69	0.176884
$958$	38117.0	1.28549
$959$	30088.4	1.01314
$960$	−2140.77	−0.0719718
$961$	60205.6	2.02093
$962$	322.061	0.0107938
$963$	17759.7	0.594287
$964$	−3349.39	−0.111905
$965$	−47294.3	−1.57768
$966$	39040.1	1.30031
$967$	−55391.0	−1.84204	−0.921021	−	0.389514i	$-0.872643\pi$
$967$	−55391.0	−1.84204	−0.921021	+	0.389514i	$0.872643\pi$
$968$	−11274.9	−0.374369
$969$	−2621.32	−0.0869028
$970$	−8318.98	−0.275367
$971$	−23937.0	−0.791117	−0.395558	−	0.918441i	$-0.629449\pi$
$971$	−23937.0	−0.791117	−0.395558	+	0.918441i	$0.629449\pi$
$972$	−14398.1	−0.475123
$973$	53111.0	1.74991
$974$	974.429	0.0320562
$975$	−577.310	−0.0189628
$976$	7753.70	0.254293
$977$	−43982.3	−1.44024	−0.720122	−	0.693847i	$-0.755914\pi$
$977$	−43982.3	−1.44024	−0.720122	+	0.693847i	$0.755914\pi$
$978$	7323.88	0.239460
$979$	15476.1	0.505229
$980$	16234.0	0.529160
$981$	32736.8	1.06545
$982$	12925.8	0.420039
$983$	2724.78	0.0884100	0.0442050	−	0.999022i	$-0.485925\pi$
$983$	2724.78	0.0884100	0.0442050	+	0.999022i	$0.485925\pi$
$984$	−1213.09	−0.0393008
$985$	−10001.6	−0.323530
$986$	−997.126	−0.0322059
$987$	−47548.3	−1.53341
$988$	−955.486	−0.0307673
$989$	−13292.6	−0.427382
$990$	−15331.2	−0.492178
$991$	−23748.6	−0.761250	−0.380625	−	0.924729i	$-0.624291\pi$
$991$	−23748.6	−0.761250	−0.380625	+	0.924729i	$0.624291\pi$
$992$	−9599.82	−0.307253
$993$	33297.7	1.06412
$994$	26562.9	0.847609
$995$	−11323.1	−0.360771
$996$	13895.7	0.442069
$997$	24718.0	0.785182	0.392591	−	0.919713i	$-0.371579\pi$
$997$	24718.0	0.785182	0.392591	+	0.919713i	$0.371579\pi$
$998$	15895.6	0.504174
$999$	4327.47	0.137052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.4.a.c.1.1	✓	2
3.2	odd	2		522.4.a.j.1.1		2
4.3	odd	2		464.4.a.e.1.2		2
5.4	even	2		1450.4.a.g.1.2		2
8.3	odd	2		1856.4.a.i.1.1		2
8.5	even	2		1856.4.a.l.1.2		2
29.28	even	2		1682.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.c.1.1	✓	2	1.1	even	1	trivial
464.4.a.e.1.2		2	4.3	odd	2
522.4.a.j.1.1		2	3.2	odd	2
1450.4.a.g.1.2		2	5.4	even	2
1682.4.a.c.1.2		2	29.28	even	2
1856.4.a.i.1.1		2	8.3	odd	2
1856.4.a.l.1.2		2	8.5	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:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	58.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.44949 q^{3} +4.00000 q^{4} +9.69694 q^{5} +6.89898 q^{6} -27.5959 q^{7} -8.00000 q^{8} -15.1010 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.44949 q^{3} +4.00000 q^{4} +9.69694 q^{5} +6.89898 q^{6} -27.5959 q^{7} -8.00000 q^{8} -15.1010 q^{9} -19.3939 q^{10} -52.3485 q^{11} -13.7980 q^{12} -5.40408 q^{13} +55.1918 q^{14} -33.4495 q^{15} +16.0000 q^{16} +17.1918 q^{17} +30.2020 q^{18} +44.2020 q^{19} +38.7878 q^{20} +95.1918 q^{21} +104.697 q^{22} -205.060 q^{23} +27.5959 q^{24} -30.9694 q^{25} +10.8082 q^{26} +145.227 q^{27} -110.384 q^{28} +29.0000 q^{29} +66.8990 q^{30} +299.994 q^{31} -32.0000 q^{32} +180.576 q^{33} -34.3837 q^{34} -267.596 q^{35} -60.4041 q^{36} +29.7980 q^{37} -88.4041 q^{38} +18.6413 q^{39} -77.5755 q^{40} -43.9592 q^{41} -190.384 q^{42} +64.8230 q^{43} -209.394 q^{44} -146.434 q^{45} +410.120 q^{46} -499.499 q^{47} -55.1918 q^{48} +418.535 q^{49} +61.9388 q^{50} -59.3031 q^{51} -21.6163 q^{52} -351.627 q^{53} -290.454 q^{54} -507.620 q^{55} +220.767 q^{56} -152.474 q^{57} -58.0000 q^{58} +522.372 q^{59} -133.798 q^{60} +484.606 q^{61} -599.989 q^{62} +416.727 q^{63} +64.0000 q^{64} -52.4031 q^{65} -361.151 q^{66} -504.990 q^{67} +68.7673 q^{68} +707.353 q^{69} +535.192 q^{70} +481.283 q^{71} +120.808 q^{72} +3.11019 q^{73} -59.5959 q^{74} +106.829 q^{75} +176.808 q^{76} +1444.60 q^{77} -37.2827 q^{78} -1043.27 q^{79} +155.151 q^{80} -93.2316 q^{81} +87.9184 q^{82} -1007.08 q^{83} +380.767 q^{84} +166.708 q^{85} -129.646 q^{86} -100.035 q^{87} +418.788 q^{88} -295.637 q^{89} +292.867 q^{90} +149.131 q^{91} -820.241 q^{92} -1034.83 q^{93} +998.999 q^{94} +428.624 q^{95} +110.384 q^{96} +428.949 q^{97} -837.069 q^{98} +790.515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9} + 20 q^{10} - 90 q^{11} - 8 q^{12} - 50 q^{13} + 32 q^{14} - 62 q^{15} + 32 q^{16} - 44 q^{17} + 80 q^{18} + 108 q^{19} - 40 q^{20} + 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} + 100 q^{26} + 70 q^{27} - 64 q^{28} + 58 q^{29} + 124 q^{30} + 66 q^{31} - 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} + 40 q^{37} - 216 q^{38} - 46 q^{39} + 80 q^{40} + 304 q^{41} - 224 q^{42} - 130 q^{43} - 360 q^{44} + 344 q^{45} + 56 q^{46} - 514 q^{47} - 32 q^{48} + 210 q^{49} - 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} + 234 q^{55} + 128 q^{56} - 60 q^{57} - 116 q^{58} - 180 q^{59} - 248 q^{60} + 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} - 252 q^{66} - 912 q^{67} - 176 q^{68} + 964 q^{69} + 992 q^{70} + 796 q^{71} + 320 q^{72} - 856 q^{73} - 80 q^{74} + 488 q^{75} + 432 q^{76} + 1008 q^{77} + 92 q^{78} - 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} + 448 q^{84} + 1372 q^{85} + 260 q^{86} - 58 q^{87} + 720 q^{88} - 944 q^{89} - 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} - 828 q^{95} + 64 q^{96} + 368 q^{97} - 420 q^{98} + 1728 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9 + 20 * q^10 - 90 * q^11 - 8 * q^12 - 50 * q^13 + 32 * q^14 - 62 * q^15 + 32 * q^16 - 44 * q^17 + 80 * q^18 + 108 * q^19 - 40 * q^20 + 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 + 100 * q^26 + 70 * q^27 - 64 * q^28 + 58 * q^29 + 124 * q^30 + 66 * q^31 - 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 + 40 * q^37 - 216 * q^38 - 46 * q^39 + 80 * q^40 + 304 * q^41 - 224 * q^42 - 130 * q^43 - 360 * q^44 + 344 * q^45 + 56 * q^46 - 514 * q^47 - 32 * q^48 + 210 * q^49 - 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 + 234 * q^55 + 128 * q^56 - 60 * q^57 - 116 * q^58 - 180 * q^59 - 248 * q^60 + 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 - 252 * q^66 - 912 * q^67 - 176 * q^68 + 964 * q^69 + 992 * q^70 + 796 * q^71 + 320 * q^72 - 856 * q^73 - 80 * q^74 + 488 * q^75 + 432 * q^76 + 1008 * q^77 + 92 * q^78 - 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 + 448 * q^84 + 1372 * q^85 + 260 * q^86 - 58 * q^87 + 720 * q^88 - 944 * q^89 - 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 - 828 * q^95 + 64 * q^96 + 368 * q^97 - 420 * q^98 + 1728 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more