LMFDB - Embedded newform 115.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,4,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	115.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:	$6.78521965066$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 $ x^5 - x^4 - 27x^3 + 7x^2 + 168*x + 92
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			$-3.93900$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93900 q^{2} -3.85751 q^{3} +0.637693 q^{4} -5.00000 q^{5} +11.3372 q^{6} -23.5932 q^{7} +21.6378 q^{8} -12.1196 q^{9} +O(q^{10})$	\(q-2.93900 q^{2} -3.85751 q^{3} +0.637693 q^{4} -5.00000 q^{5} +11.3372 q^{6} -23.5932 q^{7} +21.6378 q^{8} -12.1196 q^{9} +14.6950 q^{10} -58.2387 q^{11} -2.45991 q^{12} +68.5380 q^{13} +69.3404 q^{14} +19.2876 q^{15} -68.6949 q^{16} +101.409 q^{17} +35.6194 q^{18} -7.02478 q^{19} -3.18846 q^{20} +91.0112 q^{21} +171.163 q^{22} -23.0000 q^{23} -83.4681 q^{24} +25.0000 q^{25} -201.433 q^{26} +150.904 q^{27} -15.0452 q^{28} +206.062 q^{29} -56.6861 q^{30} +54.8201 q^{31} +28.7917 q^{32} +224.657 q^{33} -298.040 q^{34} +117.966 q^{35} -7.72857 q^{36} -241.694 q^{37} +20.6458 q^{38} -264.386 q^{39} -108.189 q^{40} -122.510 q^{41} -267.482 q^{42} -320.358 q^{43} -37.1384 q^{44} +60.5979 q^{45} +67.5969 q^{46} +107.032 q^{47} +264.991 q^{48} +213.641 q^{49} -73.4749 q^{50} -391.186 q^{51} +43.7062 q^{52} +127.122 q^{53} -443.507 q^{54} +291.194 q^{55} -510.506 q^{56} +27.0982 q^{57} -605.615 q^{58} +693.766 q^{59} +12.2995 q^{60} -899.524 q^{61} -161.116 q^{62} +285.940 q^{63} +464.941 q^{64} -342.690 q^{65} -660.265 q^{66} +110.435 q^{67} +64.6677 q^{68} +88.7228 q^{69} -346.702 q^{70} +225.992 q^{71} -262.241 q^{72} +746.207 q^{73} +710.336 q^{74} -96.4378 q^{75} -4.47965 q^{76} +1374.04 q^{77} +777.030 q^{78} -1094.02 q^{79} +343.474 q^{80} -254.887 q^{81} +360.055 q^{82} +1289.81 q^{83} +58.0372 q^{84} -507.045 q^{85} +941.531 q^{86} -794.887 q^{87} -1260.16 q^{88} +1208.07 q^{89} -178.097 q^{90} -1617.03 q^{91} -14.6669 q^{92} -211.469 q^{93} -314.567 q^{94} +35.1239 q^{95} -111.064 q^{96} +903.864 q^{97} -627.890 q^{98} +705.829 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) $ 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9 - 30 * q^10 + 23 * q^11 + 47 * q^12 + 132 * q^13 + 93 * q^14 - 20 * q^15 + 282 * q^16 + 23 * q^17 - 15 * q^18 - 161 * q^19 - 110 * q^20 - 60 * q^21 + 193 * q^22 - 115 * q^23 + 105 * q^24 + 125 * q^25 - 257 * q^26 + 577 * q^27 + 17 * q^28 + 401 * q^29 - 95 * q^30 + 32 * q^31 + 670 * q^32 + 189 * q^33 - 663 * q^34 + 15 * q^35 - 659 * q^36 - 38 * q^37 - 875 * q^38 + 335 * q^39 - 690 * q^40 - 12 * q^41 - 798 * q^42 - 566 * q^43 + 47 * q^44 - 385 * q^45 - 138 * q^46 + 919 * q^47 - 773 * q^48 - 738 * q^49 + 150 * q^50 - 993 * q^51 - 305 * q^52 + 1156 * q^53 - 8 * q^54 - 115 * q^55 + 343 * q^56 + 114 * q^57 - 1042 * q^58 + 1324 * q^59 - 235 * q^60 - 1673 * q^61 + 565 * q^62 + 270 * q^63 + 2466 * q^64 - 660 * q^65 - 2781 * q^66 + 558 * q^67 - 2267 * q^68 - 92 * q^69 - 465 * q^70 - 108 * q^71 - 789 * q^72 + 1173 * q^73 + 1458 * q^74 + 100 * q^75 - 3477 * q^76 + 2608 * q^77 + 704 * q^78 + 656 * q^79 - 1410 * q^80 - 319 * q^81 + 3505 * q^82 - 82 * q^83 - 718 * q^84 - 115 * q^85 + 112 * q^86 + 2389 * q^87 + 2397 * q^88 + 570 * q^89 + 75 * q^90 - 1589 * q^91 - 506 * q^92 + 911 * q^93 - 948 * q^94 + 805 * q^95 - 5991 * q^96 + 633 * q^97 - 2555 * q^98 + 2021 * 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 $

$2$	−2.93900	−1.03909	−0.519546	−	0.854443i	$-0.673899\pi$
$2$	−2.93900	−1.03909	−0.519546	+	0.854443i	$0.673899\pi$
$3$	−3.85751	−0.742379	−0.371189	−	0.928557i	$-0.621050\pi$
$3$	−3.85751	−0.742379	−0.371189	+	0.928557i	$0.621050\pi$
$4$	0.637693	0.0797116
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	11.3372	0.771400
$7$	−23.5932	−1.27392	−0.636958	−	0.770899i	$-0.719807\pi$
$7$	−23.5932	−1.27392	−0.636958	+	0.770899i	$0.719807\pi$
$8$	21.6378	0.956264
$9$	−12.1196	−0.448874
$10$	14.6950	0.464696
$11$	−58.2387	−1.59633	−0.798165	−	0.602439i	$-0.794196\pi$
$11$	−58.2387	−1.59633	−0.798165	+	0.602439i	$0.794196\pi$
$12$	−2.45991	−0.0591762
$13$	68.5380	1.46223	0.731116	−	0.682253i	$-0.239000\pi$
$13$	68.5380	1.46223	0.731116	+	0.682253i	$0.239000\pi$
$14$	69.3404	1.32371
$15$	19.2876	0.332002
$16$	−68.6949	−1.07336
$17$	101.409	1.44678	0.723391	−	0.690439i	$-0.242583\pi$
$17$	101.409	1.44678	0.723391	+	0.690439i	$0.242583\pi$
$18$	35.6194	0.466421
$19$	−7.02478	−0.0848208	−0.0424104	−	0.999100i	$-0.513504\pi$
$19$	−7.02478	−0.0848208	−0.0424104	+	0.999100i	$0.513504\pi$
$20$	−3.18846	−0.0356481
$21$	91.0112	0.945728
$22$	171.163	1.65873
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−83.4681	−0.709910
$25$	25.0000	0.200000
$26$	−201.433	−1.51939
$27$	150.904	1.07561
$28$	−15.0452	−0.101546
$29$	206.062	1.31947	0.659737	−	0.751497i	$-0.270668\pi$
$29$	206.062	1.31947	0.659737	+	0.751497i	$0.270668\pi$
$30$	−56.6861	−0.344980
$31$	54.8201	0.317612	0.158806	−	0.987310i	$-0.449236\pi$
$31$	54.8201	0.317612	0.158806	+	0.987310i	$0.449236\pi$
$32$	28.7917	0.159053
$33$	224.657	1.18508
$34$	−298.040	−1.50334
$35$	117.966	0.569712
$36$	−7.72857	−0.0357804
$37$	−241.694	−1.07390	−0.536948	−	0.843615i	$-0.680423\pi$
$37$	−241.694	−1.07390	−0.536948	+	0.843615i	$0.680423\pi$
$38$	20.6458	0.0881366
$39$	−264.386	−1.08553
$40$	−108.189	−0.427654
$41$	−122.510	−0.466653	−0.233327	−	0.972398i	$-0.574961\pi$
$41$	−122.510	−0.466653	−0.233327	+	0.972398i	$0.574961\pi$
$42$	−267.482	−0.982698
$43$	−320.358	−1.13614	−0.568071	−	0.822979i	$-0.692310\pi$
$43$	−320.358	−1.13614	−0.568071	+	0.822979i	$0.692310\pi$
$44$	−37.1384	−0.127246
$45$	60.5979	0.200742
$46$	67.5969	0.216666
$47$	107.032	0.332175	0.166088	−	0.986111i	$-0.446887\pi$
$47$	107.032	0.332175	0.166088	+	0.986111i	$0.446887\pi$
$48$	264.991	0.796838
$49$	213.641	0.622860
$50$	−73.4749	−0.207818
$51$	−391.186	−1.07406
$52$	43.7062	0.116557
$53$	127.122	0.329463	0.164732	−	0.986338i	$-0.447324\pi$
$53$	127.122	0.329463	0.164732	+	0.986338i	$0.447324\pi$
$54$	−443.507	−1.11766
$55$	291.194	0.713901
$56$	−510.506	−1.21820
$57$	27.0982	0.0629692
$58$	−605.615	−1.37105
$59$	693.766	1.53086	0.765429	−	0.643521i	$-0.222527\pi$
$59$	693.766	1.53086	0.765429	+	0.643521i	$0.222527\pi$
$60$	12.2995	0.0264644
$61$	−899.524	−1.88807	−0.944034	−	0.329847i	$-0.893003\pi$
$61$	−899.524	−1.88807	−0.944034	+	0.329847i	$0.893003\pi$
$62$	−161.116	−0.330028
$63$	285.940	0.571827
$64$	464.941	0.908087
$65$	−342.690	−0.653930
$66$	−660.265	−1.23141
$67$	110.435	0.201369	0.100685	−	0.994918i	$-0.467897\pi$
$67$	110.435	0.201369	0.100685	+	0.994918i	$0.467897\pi$
$68$	64.6677	0.115325
$69$	88.7228	0.154797
$70$	−346.702	−0.591983
$71$	225.992	0.377752	0.188876	−	0.982001i	$-0.439516\pi$
$71$	225.992	0.377752	0.188876	+	0.982001i	$0.439516\pi$
$72$	−262.241	−0.429242
$73$	746.207	1.19640	0.598198	−	0.801348i	$-0.295884\pi$
$73$	746.207	1.19640	0.598198	+	0.801348i	$0.295884\pi$
$74$	710.336	1.11588
$75$	−96.4378	−0.148476
$76$	−4.47965	−0.00676120
$77$	1374.04	2.03359
$78$	777.030	1.12797
$79$	−1094.02	−1.55807	−0.779033	−	0.626983i	$-0.784290\pi$
$79$	−1094.02	−1.55807	−0.779033	+	0.626983i	$0.784290\pi$
$80$	343.474	0.480020
$81$	−254.887	−0.349639
$82$	360.055	0.484896
$83$	1289.81	1.70572	0.852862	−	0.522137i	$-0.174865\pi$
$83$	1289.81	1.70572	0.852862	+	0.522137i	$0.174865\pi$
$84$	58.0372	0.0753854
$85$	−507.045	−0.647020
$86$	941.531	1.18056
$87$	−794.887	−0.979550
$88$	−1260.16	−1.52651
$89$	1208.07	1.43883	0.719413	−	0.694582i	$-0.244411\pi$
$89$	1208.07	1.43883	0.719413	+	0.694582i	$0.244411\pi$
$90$	−178.097	−0.208590
$91$	−1617.03	−1.86276
$92$	−14.6669	−0.0166210
$93$	−211.469	−0.235788
$94$	−314.567	−0.345161
$95$	35.1239	0.0379330
$96$	−111.064	−0.118078
$97$	903.864	0.946119	0.473059	−	0.881031i	$-0.343150\pi$
$97$	903.864	0.946119	0.473059	+	0.881031i	$0.343150\pi$
$98$	−627.890	−0.647209
$99$	705.829	0.716551
$100$	15.9423	0.0159423
$101$	630.076	0.620742	0.310371	−	0.950616i	$-0.399547\pi$
$101$	630.076	0.620742	0.310371	+	0.950616i	$0.399547\pi$
$102$	1149.69	1.11605
$103$	1573.71	1.50546	0.752729	−	0.658331i	$-0.228737\pi$
$103$	1573.71	1.50546	0.752729	+	0.658331i	$0.228737\pi$
$104$	1483.01	1.39828
$105$	−455.056	−0.422942
$106$	−373.611	−0.342343
$107$	−406.908	−0.367639	−0.183819	−	0.982960i	$-0.558846\pi$
$107$	−406.908	−0.367639	−0.183819	+	0.982960i	$0.558846\pi$
$108$	96.2306	0.0857388
$109$	439.432	0.386147	0.193073	−	0.981184i	$-0.438154\pi$
$109$	439.432	0.386147	0.193073	+	0.981184i	$0.438154\pi$
$110$	−855.817	−0.741808
$111$	932.336	0.797238
$112$	1620.74	1.36737
$113$	−1670.86	−1.39099	−0.695494	−	0.718532i	$-0.744814\pi$
$113$	−1670.86	−1.39099	−0.695494	+	0.718532i	$0.744814\pi$
$114$	−79.6415	−0.0654308
$115$	115.000	0.0932505
$116$	131.404	0.105177
$117$	−830.652	−0.656358
$118$	−2038.97	−1.59070
$119$	−2392.57	−1.84308
$120$	417.340	0.317481
$121$	2060.75	1.54827
$122$	2643.70	1.96188
$123$	472.582	0.346434
$124$	34.9583	0.0253174
$125$	−125.000	−0.0894427
$126$	−840.377	−0.594181
$127$	−1141.67	−0.797689	−0.398845	−	0.917018i	$-0.630589\pi$
$127$	−1141.67	−0.797689	−0.398845	+	0.917018i	$0.630589\pi$
$128$	−1596.79	−1.10264
$129$	1235.79	0.843449
$130$	1007.16	0.679493
$131$	493.101	0.328874	0.164437	−	0.986388i	$-0.447419\pi$
$131$	493.101	0.328874	0.164437	+	0.986388i	$0.447419\pi$
$132$	143.262	0.0944647
$133$	165.737	0.108055
$134$	−324.567	−0.209241
$135$	−754.522	−0.481029
$136$	2194.26	1.38350
$137$	−1760.32	−1.09777	−0.548884	−	0.835899i	$-0.684947\pi$
$137$	−1760.32	−1.09777	−0.548884	+	0.835899i	$0.684947\pi$
$138$	−260.756	−0.160848
$139$	−1453.48	−0.886928	−0.443464	−	0.896292i	$-0.646251\pi$
$139$	−1453.48	−0.886928	−0.443464	+	0.896292i	$0.646251\pi$
$140$	75.2262	0.0454127
$141$	−412.878	−0.246600
$142$	−664.191	−0.392519
$143$	−3991.57	−2.33421
$144$	832.554	0.481802
$145$	−1030.31	−0.590087
$146$	−2193.10	−1.24316
$147$	−824.123	−0.462398
$148$	−154.126	−0.0856020
$149$	1969.16	1.08269	0.541343	−	0.840802i	$-0.317916\pi$
$149$	1969.16	1.08269	0.541343	+	0.840802i	$0.317916\pi$
$150$	283.430	0.154280
$151$	−1211.72	−0.653037	−0.326518	−	0.945191i	$-0.605875\pi$
$151$	−1211.72	−0.653037	−0.326518	+	0.945191i	$0.605875\pi$
$152$	−152.001	−0.0811111
$153$	−1229.03	−0.649422
$154$	−4038.30	−2.11309
$155$	−274.100	−0.142040
$156$	−168.597	−0.0865293
$157$	2488.41	1.26495	0.632475	−	0.774581i	$-0.282039\pi$
$157$	2488.41	1.26495	0.632475	+	0.774581i	$0.282039\pi$
$158$	3215.33	1.61897
$159$	−490.375	−0.244587
$160$	−143.958	−0.0711306
$161$	542.645	0.265630
$162$	749.111	0.363307
$163$	2599.31	1.24904	0.624519	−	0.781009i	$-0.285295\pi$
$163$	2599.31	1.24904	0.624519	+	0.781009i	$0.285295\pi$
$164$	−78.1235	−0.0371977
$165$	−1123.28	−0.529985
$166$	−3790.74	−1.77240
$167$	1594.18	0.738690	0.369345	−	0.929292i	$-0.379582\pi$
$167$	1594.18	0.738690	0.369345	+	0.929292i	$0.379582\pi$
$168$	1969.28	0.904365
$169$	2500.46	1.13812
$170$	1490.20	0.672313
$171$	85.1375	0.0380738
$172$	−204.290	−0.0905637
$173$	1616.96	0.710608	0.355304	−	0.934751i	$-0.384377\pi$
$173$	1616.96	0.710608	0.355304	+	0.934751i	$0.384377\pi$
$174$	2336.17	1.01784
$175$	−589.831	−0.254783
$176$	4000.70	1.71343
$177$	−2676.21	−1.13648
$178$	−3550.52	−1.49507
$179$	−1710.33	−0.714167	−0.357084	−	0.934072i	$-0.616229\pi$
$179$	−1710.33	−0.714167	−0.357084	+	0.934072i	$0.616229\pi$
$180$	38.6429	0.0160015
$181$	−1594.53	−0.654811	−0.327405	−	0.944884i	$-0.606174\pi$
$181$	−1594.53	−0.654811	−0.327405	+	0.944884i	$0.606174\pi$
$182$	4752.45	1.93558
$183$	3469.92	1.40166
$184$	−497.669	−0.199395
$185$	1208.47	0.480261
$186$	621.507	0.245006
$187$	−5905.93	−2.30954
$188$	68.2536	0.0264782
$189$	−3560.32	−1.37024
$190$	−103.229	−0.0394159
$191$	2432.84	0.921644	0.460822	−	0.887493i	$-0.347555\pi$
$191$	2432.84	0.921644	0.460822	+	0.887493i	$0.347555\pi$
$192$	−1793.51	−0.674145
$193$	1730.75	0.645504	0.322752	−	0.946484i	$-0.395392\pi$
$193$	1730.75	0.645504	0.322752	+	0.946484i	$0.395392\pi$
$194$	−2656.45	−0.983104
$195$	1321.93	0.485464
$196$	136.237	0.0496492
$197$	1652.36	0.597592	0.298796	−	0.954317i	$-0.403415\pi$
$197$	1652.36	0.597592	0.298796	+	0.954317i	$0.403415\pi$
$198$	−2074.43	−0.744562
$199$	1379.31	0.491341	0.245670	−	0.969353i	$-0.420992\pi$
$199$	1379.31	0.491341	0.245670	+	0.969353i	$0.420992\pi$
$200$	540.945	0.191253
$201$	−426.003	−0.149492
$202$	−1851.79	−0.645008
$203$	−4861.67	−1.68090
$204$	−249.457	−0.0856150
$205$	612.548	0.208694
$206$	−4625.12	−1.56431
$207$	278.751	0.0935966
$208$	−4708.21	−1.56950
$209$	409.114	0.135402
$210$	1337.41	0.439476
$211$	−843.617	−0.275246	−0.137623	−	0.990485i	$-0.543946\pi$
$211$	−843.617	−0.275246	−0.137623	+	0.990485i	$0.543946\pi$
$212$	81.0648	0.0262620
$213$	−871.769	−0.280435
$214$	1195.90	0.382010
$215$	1601.79	0.508099
$216$	3265.24	1.02857
$217$	−1293.38	−0.404611
$218$	−1291.49	−0.401242
$219$	−2878.50	−0.888179
$220$	185.692	0.0569061
$221$	6950.36	2.11553
$222$	−2740.13	−0.828404
$223$	3016.30	0.905767	0.452884	−	0.891570i	$-0.350395\pi$
$223$	3016.30	0.905767	0.452884	+	0.891570i	$0.350395\pi$
$224$	−679.288	−0.202620
$225$	−302.990	−0.0897747
$226$	4910.66	1.44536
$227$	3394.79	0.992601	0.496301	−	0.868151i	$-0.334691\pi$
$227$	3394.79	0.992601	0.496301	+	0.868151i	$0.334691\pi$
$228$	17.2803	0.00501937
$229$	1319.77	0.380841	0.190420	−	0.981703i	$-0.439015\pi$
$229$	1319.77	0.380841	0.190420	+	0.981703i	$0.439015\pi$
$230$	−337.984	−0.0968958
$231$	−5300.38	−1.50969
$232$	4458.72	1.26177
$233$	5928.22	1.66683	0.833413	−	0.552651i	$-0.186384\pi$
$233$	5928.22	1.66683	0.833413	+	0.552651i	$0.186384\pi$
$234$	2441.28	0.682016
$235$	−535.161	−0.148553
$236$	442.409	0.122027
$237$	4220.21	1.15668
$238$	7031.74	1.91513
$239$	−3732.22	−1.01011	−0.505057	−	0.863086i	$-0.668528\pi$
$239$	−3732.22	−1.01011	−0.505057	+	0.863086i	$0.668528\pi$
$240$	−1324.96	−0.356357
$241$	−1286.66	−0.343906	−0.171953	−	0.985105i	$-0.555008\pi$
$241$	−1286.66	−0.343906	−0.171953	+	0.985105i	$0.555008\pi$
$242$	−6056.53	−1.60880
$243$	−3091.19	−0.816049
$244$	−573.619	−0.150501
$245$	−1068.21	−0.278552
$246$	−1388.92	−0.359976
$247$	−481.465	−0.124028
$248$	1186.18	0.303721
$249$	−4975.46	−1.26629
$250$	367.374	0.0929392
$251$	2450.05	0.616117	0.308059	−	0.951367i	$-0.400321\pi$
$251$	2450.05	0.616117	0.308059	+	0.951367i	$0.400321\pi$
$252$	182.342	0.0455812
$253$	1339.49	0.332858
$254$	3355.35	0.828872
$255$	1955.93	0.480334
$256$	973.437	0.237656
$257$	−7516.44	−1.82437	−0.912184	−	0.409780i	$-0.865605\pi$
$257$	−7516.44	−1.82437	−0.912184	+	0.409780i	$0.865605\pi$
$258$	−3631.97	−0.876420
$259$	5702.34	1.36805
$260$	−218.531	−0.0521258
$261$	−2497.39	−0.592277
$262$	−1449.22	−0.341730
$263$	932.166	0.218554	0.109277	−	0.994011i	$-0.465146\pi$
$263$	932.166	0.218554	0.109277	+	0.994011i	$0.465146\pi$
$264$	4861.07	1.13325
$265$	−635.610	−0.147340
$266$	−487.101	−0.112279
$267$	−4660.16	−1.06815
$268$	70.4233	0.0160514
$269$	−1918.86	−0.434924	−0.217462	−	0.976069i	$-0.569778\pi$
$269$	−1918.86	−0.434924	−0.217462	+	0.976069i	$0.569778\pi$
$270$	2217.54	0.499833
$271$	3253.61	0.729309	0.364655	−	0.931143i	$-0.381187\pi$
$271$	3253.61	0.729309	0.364655	+	0.931143i	$0.381187\pi$
$272$	−6966.28	−1.55291
$273$	6237.73	1.38287
$274$	5173.57	1.14068
$275$	−1455.97	−0.319266
$276$	56.5779	0.0123391
$277$	−3073.34	−0.666639	−0.333319	−	0.942814i	$-0.608169\pi$
$277$	−3073.34	−0.666639	−0.333319	+	0.942814i	$0.608169\pi$
$278$	4271.79	0.921599
$279$	−664.397	−0.142568
$280$	2552.53	0.544795
$281$	3551.92	0.754056	0.377028	−	0.926202i	$-0.376946\pi$
$281$	3551.92	0.754056	0.377028	+	0.926202i	$0.376946\pi$
$282$	1213.45	0.256240
$283$	7229.12	1.51847	0.759234	−	0.650818i	$-0.225574\pi$
$283$	7229.12	1.51847	0.759234	+	0.650818i	$0.225574\pi$
$284$	144.114	0.0301112
$285$	−135.491	−0.0281607
$286$	11731.2	2.42545
$287$	2890.40	0.594477
$288$	−348.943	−0.0713947
$289$	5370.77	1.09318
$290$	3028.08	0.613154
$291$	−3486.67	−0.702378
$292$	475.850	0.0953666
$293$	3354.09	0.668764	0.334382	−	0.942438i	$-0.391472\pi$
$293$	3354.09	0.668764	0.334382	+	0.942438i	$0.391472\pi$
$294$	2422.09	0.480474
$295$	−3468.83	−0.684620
$296$	−5229.71	−1.02693
$297$	−8788.48	−1.71703
$298$	−5787.36	−1.12501
$299$	−1576.37	−0.304897
$300$	−61.4977	−0.0118352
$301$	7558.29	1.44735
$302$	3561.24	0.678565
$303$	−2430.53	−0.460826
$304$	482.567	0.0910431
$305$	4497.62	0.844370
$306$	3612.13	0.674809
$307$	−3983.31	−0.740519	−0.370259	−	0.928928i	$-0.620731\pi$
$307$	−3983.31	−0.740519	−0.370259	+	0.928928i	$0.620731\pi$
$308$	876.215	0.162101
$309$	−6070.60	−1.11762
$310$	805.579	0.147593
$311$	−5150.10	−0.939021	−0.469510	−	0.882927i	$-0.655570\pi$
$311$	−5150.10	−0.939021	−0.469510	+	0.882927i	$0.655570\pi$
$312$	−5720.73	−1.03805
$313$	−8536.61	−1.54159	−0.770795	−	0.637083i	$-0.780141\pi$
$313$	−8536.61	−1.54159	−0.770795	+	0.637083i	$0.780141\pi$
$314$	−7313.44	−1.31440
$315$	−1429.70	−0.255729
$316$	−697.650	−0.124196
$317$	8416.99	1.49131	0.745655	−	0.666332i	$-0.232137\pi$
$317$	8416.99	1.49131	0.745655	+	0.666332i	$0.232137\pi$
$318$	1441.21	0.254148
$319$	−12000.8	−2.10632
$320$	−2324.70	−0.406109
$321$	1569.65	0.272927
$322$	−1594.83	−0.276014
$323$	−712.376	−0.122717
$324$	−162.539	−0.0278703
$325$	1713.45	0.292446
$326$	−7639.35	−1.29787
$327$	−1695.12	−0.286667
$328$	−2650.84	−0.446244
$329$	−2525.24	−0.423163
$330$	3301.32	0.550703
$331$	5921.39	0.983290	0.491645	−	0.870796i	$-0.336396\pi$
$331$	5921.39	0.983290	0.491645	+	0.870796i	$0.336396\pi$
$332$	822.502	0.135966
$333$	2929.23	0.482044
$334$	−4685.28	−0.767566
$335$	−552.173	−0.0900550
$336$	−6252.01	−1.01510
$337$	−3472.56	−0.561313	−0.280656	−	0.959808i	$-0.590552\pi$
$337$	−3472.56	−0.561313	−0.280656	+	0.959808i	$0.590552\pi$
$338$	−7348.83	−1.18261
$339$	6445.38	1.03264
$340$	−323.339	−0.0515750
$341$	−3192.65	−0.507014
$342$	−250.219	−0.0395622
$343$	3052.00	0.480444
$344$	−6931.84	−1.08645
$345$	−443.614	−0.0692272
$346$	−4752.24	−0.738387
$347$	−7262.19	−1.12350	−0.561750	−	0.827307i	$-0.689872\pi$
$347$	−7262.19	−1.12350	−0.561750	+	0.827307i	$0.689872\pi$
$348$	−506.893	−0.0780814
$349$	−5933.27	−0.910031	−0.455015	−	0.890484i	$-0.650366\pi$
$349$	−5933.27	−0.910031	−0.455015	+	0.890484i	$0.650366\pi$
$350$	1733.51	0.264743
$351$	10342.7	1.57280
$352$	−1676.79	−0.253901
$353$	−6743.43	−1.01676	−0.508380	−	0.861133i	$-0.669756\pi$
$353$	−6743.43	−1.01676	−0.508380	+	0.861133i	$0.669756\pi$
$354$	7865.37	1.18090
$355$	−1129.96	−0.168936
$356$	770.380	0.114691
$357$	9229.35	1.36826
$358$	5026.65	0.742085
$359$	−9796.77	−1.44026	−0.720131	−	0.693838i	$-0.755918\pi$
$359$	−9796.77	−1.44026	−0.720131	+	0.693838i	$0.755918\pi$
$360$	1311.21	0.191963
$361$	−6809.65	−0.992805
$362$	4686.33	0.680408
$363$	−7949.37	−1.14940
$364$	−1031.17	−0.148484
$365$	−3731.03	−0.535044
$366$	−10198.1	−1.45646
$367$	4388.07	0.624130	0.312065	−	0.950061i	$-0.398979\pi$
$367$	4388.07	0.624130	0.312065	+	0.950061i	$0.398979\pi$
$368$	1579.98	0.223811
$369$	1484.77	0.209468
$370$	−3551.68	−0.499035
$371$	−2999.22	−0.419708
$372$	−134.852	−0.0187951
$373$	−902.507	−0.125282	−0.0626408	−	0.998036i	$-0.519952\pi$
$373$	−902.507	−0.125282	−0.0626408	+	0.998036i	$0.519952\pi$
$374$	17357.5	2.39982
$375$	482.189	0.0664004
$376$	2315.94	0.317647
$377$	14123.1	1.92938
$378$	10463.8	1.42381
$379$	−2833.86	−0.384079	−0.192039	−	0.981387i	$-0.561510\pi$
$379$	−2833.86	−0.384079	−0.192039	+	0.981387i	$0.561510\pi$
$380$	22.3983	0.00302370
$381$	4404.00	0.592188
$382$	−7150.10	−0.957672
$383$	−10428.5	−1.39130	−0.695652	−	0.718379i	$-0.744884\pi$
$383$	−10428.5	−1.39130	−0.695652	+	0.718379i	$0.744884\pi$
$384$	6159.64	0.818576
$385$	−6870.20	−0.909449
$386$	−5086.67	−0.670738
$387$	3882.61	0.509985
$388$	576.387	0.0754166
$389$	5082.96	0.662510	0.331255	−	0.943541i	$-0.392528\pi$
$389$	5082.96	0.662510	0.331255	+	0.943541i	$0.392528\pi$
$390$	−3885.15	−0.504442
$391$	−2332.41	−0.301675
$392$	4622.72	0.595619
$393$	−1902.14	−0.244149
$394$	−4856.27	−0.620953
$395$	5470.12	0.696788
$396$	450.102	0.0571174
$397$	3935.35	0.497506	0.248753	−	0.968567i	$-0.419979\pi$
$397$	3935.35	0.497506	0.248753	+	0.968567i	$0.419979\pi$
$398$	−4053.79	−0.510548
$399$	−639.334	−0.0802174
$400$	−1717.37	−0.214672
$401$	1317.42	0.164062	0.0820309	−	0.996630i	$-0.473859\pi$
$401$	1317.42	0.164062	0.0820309	+	0.996630i	$0.473859\pi$
$402$	1252.02	0.155336
$403$	3757.26	0.464423
$404$	401.795	0.0494803
$405$	1274.43	0.156363
$406$	14288.4	1.74661
$407$	14075.9	1.71429
$408$	−8464.41	−1.02708
$409$	872.011	0.105423	0.0527117	−	0.998610i	$-0.483214\pi$
$409$	872.011	0.105423	0.0527117	+	0.998610i	$0.483214\pi$
$410$	−1800.28	−0.216852
$411$	6790.45	0.814959
$412$	1003.54	0.120002
$413$	−16368.2	−1.95018
$414$	−819.247	−0.0972555
$415$	−6449.05	−0.762823
$416$	1973.32	0.232572
$417$	5606.84	0.658437
$418$	−1202.39	−0.140695
$419$	10151.9	1.18366	0.591832	−	0.806062i	$-0.298405\pi$
$419$	10151.9	1.18366	0.591832	+	0.806062i	$0.298405\pi$
$420$	−290.186	−0.0337134
$421$	16152.5	1.86989	0.934944	−	0.354795i	$-0.115449\pi$
$421$	16152.5	1.86989	0.934944	+	0.354795i	$0.115449\pi$
$422$	2479.39	0.286006
$423$	−1297.19	−0.149105
$424$	2750.64	0.315054
$425$	2535.22	0.289356
$426$	2562.12	0.291397
$427$	21222.7	2.40524
$428$	−259.482	−0.0293050
$429$	15397.5	1.73287
$430$	−4707.65	−0.527961
$431$	772.507	0.0863349	0.0431675	−	0.999068i	$-0.486255\pi$
$431$	772.507	0.0863349	0.0431675	+	0.999068i	$0.486255\pi$
$432$	−10366.4	−1.15452
$433$	3170.57	0.351889	0.175945	−	0.984400i	$-0.443702\pi$
$433$	3170.57	0.351889	0.175945	+	0.984400i	$0.443702\pi$
$434$	3801.25	0.420428
$435$	3974.43	0.438068
$436$	280.223	0.0307803
$437$	161.570	0.0176864
$438$	8459.91	0.922899
$439$	3565.50	0.387636	0.193818	−	0.981037i	$-0.437913\pi$
$439$	3565.50	0.387636	0.193818	+	0.981037i	$0.437913\pi$
$440$	6300.79	0.682678
$441$	−2589.24	−0.279586
$442$	−20427.1	−2.19823
$443$	−3798.32	−0.407368	−0.203684	−	0.979037i	$-0.565291\pi$
$443$	−3798.32	−0.407368	−0.203684	+	0.979037i	$0.565291\pi$
$444$	594.544	0.0635491
$445$	−6040.37	−0.643463
$446$	−8864.88	−0.941175
$447$	−7596.08	−0.803763
$448$	−10969.5	−1.15683
$449$	−8912.91	−0.936807	−0.468403	−	0.883515i	$-0.655171\pi$
$449$	−8912.91	−0.936807	−0.468403	+	0.883515i	$0.655171\pi$
$450$	890.485	0.0932842
$451$	7134.80	0.744933
$452$	−1065.50	−0.110878
$453$	4674.23	0.484801
$454$	−9977.29	−1.03140
$455$	8085.17	0.833052
$456$	586.345	0.0602152
$457$	−554.301	−0.0567377	−0.0283688	−	0.999598i	$-0.509031\pi$
$457$	−554.301	−0.0567377	−0.0283688	+	0.999598i	$0.509031\pi$
$458$	−3878.79	−0.395729
$459$	15303.0	1.55618
$460$	73.3346	0.00743314
$461$	9432.67	0.952978	0.476489	−	0.879180i	$-0.341909\pi$
$461$	9432.67	0.952978	0.476489	+	0.879180i	$0.341909\pi$
$462$	15577.8	1.56871
$463$	17280.6	1.73455	0.867276	−	0.497827i	$-0.165868\pi$
$463$	17280.6	1.73455	0.867276	+	0.497827i	$0.165868\pi$
$464$	−14155.4	−1.41627
$465$	1057.35	0.105448
$466$	−17423.0	−1.73198
$467$	−169.114	−0.0167573	−0.00837867	−	0.999965i	$-0.502667\pi$
$467$	−169.114	−0.0167573	−0.00837867	+	0.999965i	$0.502667\pi$
$468$	−529.701	−0.0523193
$469$	−2605.51	−0.256527
$470$	1572.84	0.154361
$471$	−9599.09	−0.939072
$472$	15011.6	1.46390
$473$	18657.2	1.81366
$474$	−12403.2	−1.20189
$475$	−175.620	−0.0169642
$476$	−1525.72	−0.146915
$477$	−1540.67	−0.147887
$478$	10969.0	1.04960
$479$	−11201.3	−1.06848	−0.534240	−	0.845333i	$-0.679402\pi$
$479$	−11201.3	−1.06848	−0.534240	+	0.845333i	$0.679402\pi$
$480$	555.321	0.0528059
$481$	−16565.2	−1.57029
$482$	3781.50	0.357350
$483$	−2093.26	−0.197198
$484$	1314.12	0.123415
$485$	−4519.32	−0.423117
$486$	9084.99	0.847949
$487$	−6350.96	−0.590944	−0.295472	−	0.955351i	$-0.595477\pi$
$487$	−6350.96	−0.590944	−0.295472	+	0.955351i	$0.595477\pi$
$488$	−19463.7	−1.80549
$489$	−10026.9	−0.927260
$490$	3139.45	0.289441
$491$	2142.21	0.196897	0.0984487	−	0.995142i	$-0.468612\pi$
$491$	2142.21	0.196897	0.0984487	+	0.995142i	$0.468612\pi$
$492$	301.362	0.0276148
$493$	20896.5	1.90899
$494$	1415.02	0.128876
$495$	−3529.15	−0.320451
$496$	−3765.86	−0.340911
$497$	−5331.89	−0.481224
$498$	14622.8	1.31579
$499$	−3588.65	−0.321944	−0.160972	−	0.986959i	$-0.551463\pi$
$499$	−3588.65	−0.321944	−0.160972	+	0.986959i	$0.551463\pi$
$500$	−79.7116	−0.00712962
$501$	−6149.56	−0.548388
$502$	−7200.67	−0.640203
$503$	−8129.30	−0.720611	−0.360306	−	0.932834i	$-0.617328\pi$
$503$	−8129.30	−0.720611	−0.360306	+	0.932834i	$0.617328\pi$
$504$	6187.12	0.546818
$505$	−3150.38	−0.277604
$506$	−3936.76	−0.345870
$507$	−9645.55	−0.844919
$508$	−728.032	−0.0635851
$509$	18273.6	1.59128	0.795640	−	0.605770i	$-0.207135\pi$
$509$	18273.6	1.59128	0.795640	+	0.605770i	$0.207135\pi$
$510$	−5748.47	−0.499111
$511$	−17605.4	−1.52411
$512$	9913.40	0.855693
$513$	−1060.07	−0.0912344
$514$	22090.8	1.89569
$515$	−7868.54	−0.673261
$516$	788.051	0.0672326
$517$	−6233.42	−0.530262
$518$	−16759.1	−1.42153
$519$	−6237.45	−0.527540
$520$	−7415.05	−0.625330
$521$	311.180	0.0261671	0.0130835	−	0.999914i	$-0.495835\pi$
$521$	311.180	0.0261671	0.0130835	+	0.999914i	$0.495835\pi$
$522$	7339.81	0.615430
$523$	−2817.90	−0.235599	−0.117799	−	0.993037i	$-0.537584\pi$
$523$	−2817.90	−0.235599	−0.117799	+	0.993037i	$0.537584\pi$
$524$	314.447	0.0262150
$525$	2275.28	0.189146
$526$	−2739.63	−0.227098
$527$	5559.24	0.459515
$528$	−15432.8	−1.27202
$529$	529.000	0.0434783
$530$	1868.06	0.153100
$531$	−8408.15	−0.687162
$532$	105.690	0.00861320
$533$	−8396.56	−0.682356
$534$	13696.2	1.10991
$535$	2034.54	0.164413
$536$	2389.56	0.192562
$537$	6597.61	0.530183
$538$	5639.51	0.451926
$539$	−12442.2	−0.994291
$540$	−481.153	−0.0383436
$541$	−4143.98	−0.329323	−0.164661	−	0.986350i	$-0.552653\pi$
$541$	−4143.98	−0.329323	−0.164661	+	0.986350i	$0.552653\pi$
$542$	−9562.35	−0.757819
$543$	6150.94	0.486118
$544$	2919.73	0.230115
$545$	−2197.16	−0.172690
$546$	−18332.7	−1.43693
$547$	19749.7	1.54376	0.771881	−	0.635767i	$-0.219316\pi$
$547$	19749.7	1.54376	0.771881	+	0.635767i	$0.219316\pi$
$548$	−1122.54	−0.0875047
$549$	10901.9	0.847504
$550$	4279.08	0.331747
$551$	−1447.54	−0.111919
$552$	1919.77	0.148027
$553$	25811.6	1.98484
$554$	9032.52	0.692699
$555$	−4661.68	−0.356536
$556$	−926.876	−0.0706984
$557$	−4587.27	−0.348957	−0.174478	−	0.984661i	$-0.555824\pi$
$557$	−4587.27	−0.348957	−0.174478	+	0.984661i	$0.555824\pi$
$558$	1952.66	0.148141
$559$	−21956.7	−1.66130
$560$	−8103.68	−0.611505
$561$	22782.2	1.71455
$562$	−10439.1	−0.783533
$563$	15344.5	1.14866	0.574328	−	0.818625i	$-0.305263\pi$
$563$	15344.5	1.14866	0.574328	+	0.818625i	$0.305263\pi$
$564$	−263.289	−0.0196569
$565$	8354.31	0.622068
$566$	−21246.3	−1.57783
$567$	6013.60	0.445410
$568$	4889.98	0.361230
$569$	2862.67	0.210913	0.105456	−	0.994424i	$-0.466370\pi$
$569$	2862.67	0.210913	0.105456	+	0.994424i	$0.466370\pi$
$570$	398.207	0.0292615
$571$	−7228.79	−0.529800	−0.264900	−	0.964276i	$-0.585339\pi$
$571$	−7228.79	−0.529800	−0.264900	+	0.964276i	$0.585339\pi$
$572$	−2545.39	−0.186063
$573$	−9384.71	−0.684209
$574$	−8494.87	−0.617716
$575$	−575.000	−0.0417029
$576$	−5634.89	−0.407616
$577$	−18524.6	−1.33655	−0.668277	−	0.743913i	$-0.732968\pi$
$577$	−18524.6	−1.33655	−0.668277	+	0.743913i	$0.732968\pi$
$578$	−15784.7	−1.13591
$579$	−6676.40	−0.479209
$580$	−657.021	−0.0470367
$581$	−30430.8	−2.17295
$582$	10247.3	0.729836
$583$	−7403.43	−0.525932
$584$	16146.3	1.14407
$585$	4153.26	0.293532
$586$	−9857.64	−0.694907
$587$	−6691.07	−0.470477	−0.235238	−	0.971938i	$-0.575587\pi$
$587$	−6691.07	−0.470477	−0.235238	+	0.971938i	$0.575587\pi$
$588$	−525.537	−0.0368585
$589$	−385.099	−0.0269401
$590$	10194.9	0.711383
$591$	−6373.99	−0.443640
$592$	16603.1	1.15268
$593$	26194.2	1.81394	0.906970	−	0.421195i	$-0.138389\pi$
$593$	26194.2	1.81394	0.906970	+	0.421195i	$0.138389\pi$
$594$	25829.3	1.78416
$595$	11962.8	0.824249
$596$	1255.72	0.0863026
$597$	−5320.71	−0.364761
$598$	4632.96	0.316815
$599$	−26899.9	−1.83489	−0.917444	−	0.397864i	$-0.869752\pi$
$599$	−26899.9	−1.83489	−0.917444	+	0.397864i	$0.869752\pi$
$600$	−2086.70	−0.141982
$601$	−11361.8	−0.771143	−0.385572	−	0.922678i	$-0.625996\pi$
$601$	−11361.8	−0.771143	−0.385572	+	0.922678i	$0.625996\pi$
$602$	−22213.8	−1.50393
$603$	−1338.42	−0.0903893
$604$	−772.706	−0.0520546
$605$	−10303.7	−0.692408
$606$	7143.31	0.478840
$607$	−6946.27	−0.464482	−0.232241	−	0.972658i	$-0.574606\pi$
$607$	−6946.27	−0.464482	−0.232241	+	0.972658i	$0.574606\pi$
$608$	−202.255	−0.0134910
$609$	18754.0	1.24786
$610$	−13218.5	−0.877378
$611$	7335.77	0.485718
$612$	−783.746	−0.0517664
$613$	−15845.6	−1.04404	−0.522022	−	0.852932i	$-0.674822\pi$
$613$	−15845.6	−1.04404	−0.522022	+	0.852932i	$0.674822\pi$
$614$	11706.9	0.769467
$615$	−2362.91	−0.154930
$616$	29731.2	1.94465
$617$	17851.3	1.16478	0.582388	−	0.812911i	$-0.302119\pi$
$617$	17851.3	1.16478	0.582388	+	0.812911i	$0.302119\pi$
$618$	17841.5	1.16131
$619$	2567.73	0.166730	0.0833650	−	0.996519i	$-0.473433\pi$
$619$	2567.73	0.166730	0.0833650	+	0.996519i	$0.473433\pi$
$620$	−174.792	−0.0113223
$621$	−3470.80	−0.224281
$622$	15136.1	0.975729
$623$	−28502.4	−1.83294
$624$	18162.0	1.16516
$625$	625.000	0.0400000
$626$	25089.1	1.60185
$627$	−1578.16	−0.100520
$628$	1586.84	0.100831
$629$	−24509.9	−1.55369
$630$	4201.89	0.265726
$631$	10264.7	0.647590	0.323795	−	0.946127i	$-0.395041\pi$
$631$	10264.7	0.647590	0.323795	+	0.946127i	$0.395041\pi$
$632$	−23672.2	−1.48992
$633$	3254.26	0.204337
$634$	−24737.5	−1.54961
$635$	5708.33	0.356738
$636$	−312.708	−0.0194964
$637$	14642.5	0.910766
$638$	35270.2	2.18866
$639$	−2738.94	−0.169563
$640$	7983.96	0.493115
$641$	−26763.6	−1.64914	−0.824570	−	0.565760i	$-0.808583\pi$
$641$	−26763.6	−1.64914	−0.824570	+	0.565760i	$0.808583\pi$
$642$	−4613.21	−0.283596
$643$	−167.791	−0.0102909	−0.00514544	−	0.999987i	$-0.501638\pi$
$643$	−167.791	−0.0102909	−0.00514544	+	0.999987i	$0.501638\pi$
$644$	346.040	0.0211738
$645$	−6178.93	−0.377202
$646$	2093.67	0.127514
$647$	8505.30	0.516813	0.258406	−	0.966036i	$-0.416803\pi$
$647$	8505.30	0.516813	0.258406	+	0.966036i	$0.416803\pi$
$648$	−5515.18	−0.334347
$649$	−40404.0	−2.44375
$650$	−5035.82	−0.303879
$651$	4989.24	0.300375
$652$	1657.56	0.0995628
$653$	21348.4	1.27937	0.639685	−	0.768637i	$-0.279065\pi$
$653$	21348.4	1.27937	0.639685	+	0.768637i	$0.279065\pi$
$654$	4981.94	0.297873
$655$	−2465.51	−0.147077
$656$	8415.78	0.500886
$657$	−9043.72	−0.537031
$658$	7421.66	0.439706
$659$	19464.9	1.15060	0.575299	−	0.817943i	$-0.304886\pi$
$659$	19464.9	1.15060	0.575299	+	0.817943i	$0.304886\pi$
$660$	−716.309	−0.0422459
$661$	1036.19	0.0609729	0.0304865	−	0.999535i	$-0.490294\pi$
$661$	1036.19	0.0609729	0.0304865	+	0.999535i	$0.490294\pi$
$662$	−17402.9	−1.02173
$663$	−26811.1	−1.57052
$664$	27908.6	1.63112
$665$	−828.687	−0.0483235
$666$	−8608.98	−0.500888
$667$	−4739.42	−0.275129
$668$	1016.60	0.0588821
$669$	−11635.4	−0.672423
$670$	1622.83	0.0935754
$671$	52387.1	3.01398
$672$	2620.36	0.150421
$673$	9758.92	0.558958	0.279479	−	0.960152i	$-0.409838\pi$
$673$	9758.92	0.558958	0.279479	+	0.960152i	$0.409838\pi$
$674$	10205.8	0.583255
$675$	3772.61	0.215123
$676$	1594.52	0.0907216
$677$	10610.4	0.602350	0.301175	−	0.953569i	$-0.402621\pi$
$677$	10610.4	0.602350	0.301175	+	0.953569i	$0.402621\pi$
$678$	−18942.9	−1.07301
$679$	−21325.1	−1.20528
$680$	−10971.3	−0.618722
$681$	−13095.5	−0.736886
$682$	9383.18	0.526834
$683$	9119.45	0.510902	0.255451	−	0.966822i	$-0.417776\pi$
$683$	9119.45	0.510902	0.255451	+	0.966822i	$0.417776\pi$
$684$	54.2915	0.00303493
$685$	8801.59	0.490936
$686$	−8969.81	−0.499226
$687$	−5091.01	−0.282728
$688$	22007.0	1.21949
$689$	8712.69	0.481752
$690$	1303.78	0.0719334
$691$	−11834.9	−0.651547	−0.325774	−	0.945448i	$-0.605625\pi$
$691$	−11834.9	−0.651547	−0.325774	+	0.945448i	$0.605625\pi$
$692$	1031.12	0.0566437
$693$	−16652.8	−0.912825
$694$	21343.5	1.16742
$695$	7267.42	0.396646
$696$	−17199.6	−0.936708
$697$	−12423.6	−0.675145
$698$	17437.9	0.945606
$699$	−22868.2	−1.23742
$700$	−376.131	−0.0203092
$701$	11488.9	0.619014	0.309507	−	0.950897i	$-0.399836\pi$
$701$	11488.9	0.619014	0.309507	+	0.950897i	$0.399836\pi$
$702$	−30397.1	−1.63428
$703$	1697.85	0.0910888
$704$	−27077.5	−1.44961
$705$	2064.39	0.110283
$706$	19818.9	1.05651
$707$	−14865.5	−0.790773
$708$	−1706.60	−0.0905903
$709$	18427.4	0.976103	0.488051	−	0.872815i	$-0.337708\pi$
$709$	18427.4	0.976103	0.488051	+	0.872815i	$0.337708\pi$
$710$	3320.95	0.175540
$711$	13259.1	0.699375
$712$	26140.0	1.37590
$713$	−1260.86	−0.0662267
$714$	−27125.0	−1.42175
$715$	19957.8	1.04389
$716$	−1090.66	−0.0569274
$717$	14397.1	0.749887
$718$	28792.7	1.49656
$719$	33712.4	1.74862	0.874312	−	0.485364i	$-0.161313\pi$
$719$	33712.4	1.74862	0.874312	+	0.485364i	$0.161313\pi$
$720$	−4162.77	−0.215468
$721$	−37128.9	−1.91782
$722$	20013.5	1.03162
$723$	4963.33	0.255308
$724$	−1016.82	−0.0521960
$725$	5151.55	0.263895
$726$	23363.1	1.19434
$727$	10026.4	0.511496	0.255748	−	0.966743i	$-0.417678\pi$
$727$	10026.4	0.511496	0.255748	+	0.966743i	$0.417678\pi$
$728$	−34989.0	−1.78129
$729$	18806.2	0.955456
$730$	10965.5	0.555960
$731$	−32487.2	−1.64375
$732$	2212.74	0.111729
$733$	−4337.96	−0.218589	−0.109295	−	0.994009i	$-0.534859\pi$
$733$	−4337.96	−0.218589	−0.109295	+	0.994009i	$0.534859\pi$
$734$	−12896.5	−0.648528
$735$	4120.62	0.206791
$736$	−662.208	−0.0331648
$737$	−6431.57	−0.321452
$738$	−4363.72	−0.217657
$739$	22794.1	1.13463	0.567316	−	0.823500i	$-0.307982\pi$
$739$	22794.1	1.13463	0.567316	+	0.823500i	$0.307982\pi$
$740$	770.631	0.0382824
$741$	1857.26	0.0920756
$742$	8814.70	0.436115
$743$	−8437.24	−0.416598	−0.208299	−	0.978065i	$-0.566793\pi$
$743$	−8437.24	−0.416598	−0.208299	+	0.978065i	$0.566793\pi$
$744$	−4575.72	−0.225476
$745$	−9845.82	−0.484192
$746$	2652.46	0.130179
$747$	−15632.0	−0.765654
$748$	−3766.17	−0.184097
$749$	9600.29	0.468340
$750$	−1417.15	−0.0689961
$751$	29990.4	1.45721	0.728604	−	0.684935i	$-0.240169\pi$
$751$	29990.4	1.45721	0.728604	+	0.684935i	$0.240169\pi$
$752$	−7352.56	−0.356543
$753$	−9451.08	−0.457393
$754$	−41507.6	−2.00480
$755$	6058.61	0.292047
$756$	−2270.39	−0.109224
$757$	−30698.4	−1.47391	−0.736956	−	0.675941i	$-0.763737\pi$
$757$	−30698.4	−1.47391	−0.736956	+	0.675941i	$0.763737\pi$
$758$	8328.71	0.399093
$759$	−5167.10	−0.247107
$760$	760.004	0.0362740
$761$	−27818.7	−1.32513	−0.662567	−	0.749003i	$-0.730533\pi$
$761$	−27818.7	−1.32513	−0.662567	+	0.749003i	$0.730533\pi$
$762$	−12943.3	−0.615337
$763$	−10367.6	−0.491918
$764$	1551.40	0.0734657
$765$	6145.17	0.290430
$766$	30649.2	1.44569
$767$	47549.3	2.23847
$768$	−3755.05	−0.176430
$769$	−29897.9	−1.40201	−0.701006	−	0.713156i	$-0.747265\pi$
$769$	−29897.9	−1.40201	−0.701006	+	0.713156i	$0.747265\pi$
$770$	20191.5	0.945001
$771$	28994.8	1.35437
$772$	1103.69	0.0514541
$773$	5866.92	0.272986	0.136493	−	0.990641i	$-0.456417\pi$
$773$	5866.92	0.272986	0.136493	+	0.990641i	$0.456417\pi$
$774$	−11411.0	−0.529921
$775$	1370.50	0.0635224
$776$	19557.6	0.904739
$777$	−21996.8	−1.01561
$778$	−14938.8	−0.688409
$779$	860.603	0.0395819
$780$	842.986	0.0386971
$781$	−13161.5	−0.603016
$782$	6854.93	0.313468
$783$	31095.6	1.41924
$784$	−14676.0	−0.668552
$785$	−12442.1	−0.565703
$786$	5590.39	0.253693
$787$	26221.2	1.18766	0.593828	−	0.804592i	$-0.297616\pi$
$787$	26221.2	1.18766	0.593828	+	0.804592i	$0.297616\pi$
$788$	1053.70	0.0476350
$789$	−3595.84	−0.162250
$790$	−16076.6	−0.724027
$791$	39421.1	1.77200
$792$	15272.6	0.685212
$793$	−61651.5	−2.76080
$794$	−11566.0	−0.516954
$795$	2451.88	0.109382
$796$	879.576	0.0391655
$797$	6554.56	0.291310	0.145655	−	0.989335i	$-0.453471\pi$
$797$	6554.56	0.291310	0.145655	+	0.989335i	$0.453471\pi$
$798$	1879.00	0.0833533
$799$	10854.0	0.480585
$800$	719.791	0.0318106
$801$	−14641.4	−0.645851
$802$	−3871.89	−0.170475
$803$	−43458.1	−1.90984
$804$	−271.659	−0.0119163
$805$	−2713.22	−0.118793
$806$	−11042.6	−0.482578
$807$	7402.01	0.322879
$808$	13633.5	0.593593
$809$	34120.3	1.48282	0.741412	−	0.671050i	$-0.234156\pi$
$809$	34120.3	1.48282	0.741412	+	0.671050i	$0.234156\pi$
$810$	−3745.55	−0.162476
$811$	39998.5	1.73186	0.865930	−	0.500165i	$-0.166727\pi$
$811$	39998.5	1.73186	0.865930	+	0.500165i	$0.166727\pi$
$812$	−3100.25	−0.133987
$813$	−12550.8	−0.541424
$814$	−41369.1	−1.78131
$815$	−12996.5	−0.558587
$816$	26872.5	1.15285
$817$	2250.45	0.0963686
$818$	−2562.83	−0.109545
$819$	19597.8	0.836144
$820$	390.617	0.0166353
$821$	−33991.8	−1.44497	−0.722487	−	0.691385i	$-0.757001\pi$
$821$	−33991.8	−1.44497	−0.722487	+	0.691385i	$0.757001\pi$
$822$	−19957.1	−0.846817
$823$	12213.1	0.517281	0.258641	−	0.965974i	$-0.416725\pi$
$823$	12213.1	0.517281	0.258641	+	0.965974i	$0.416725\pi$
$824$	34051.6	1.43961
$825$	5616.42	0.237016
$826$	48106.0	2.02642
$827$	−19385.2	−0.815102	−0.407551	−	0.913182i	$-0.633617\pi$
$827$	−19385.2	−0.815102	−0.407551	+	0.913182i	$0.633617\pi$
$828$	177.757	0.00746073
$829$	−7575.54	−0.317382	−0.158691	−	0.987328i	$-0.550727\pi$
$829$	−7575.54	−0.317382	−0.158691	+	0.987328i	$0.550727\pi$
$830$	18953.7	0.792643
$831$	11855.4	0.494898
$832$	31866.1	1.32783
$833$	21665.1	0.901142
$834$	−16478.5	−0.684176
$835$	−7970.89	−0.330352
$836$	260.889	0.0107931
$837$	8272.59	0.341628
$838$	−29836.5	−1.22993
$839$	17543.7	0.721904	0.360952	−	0.932584i	$-0.382452\pi$
$839$	17543.7	0.721904	0.360952	+	0.932584i	$0.382452\pi$
$840$	−9846.41	−0.404445
$841$	18072.5	0.741011
$842$	−47472.0	−1.94299
$843$	−13701.6	−0.559795
$844$	−537.968	−0.0219403
$845$	−12502.3	−0.508984
$846$	3812.42	0.154934
$847$	−48619.7	−1.97237
$848$	−8732.63	−0.353632
$849$	−27886.4	−1.12728
$850$	−7451.01	−0.300668
$851$	5558.95	0.223923
$852$	−555.920	−0.0223539
$853$	23708.5	0.951658	0.475829	−	0.879538i	$-0.342148\pi$
$853$	23708.5	0.951658	0.475829	+	0.879538i	$0.342148\pi$
$854$	−62373.3	−2.49926
$855$	−425.687	−0.0170271
$856$	−8804.60	−0.351560
$857$	−14313.6	−0.570528	−0.285264	−	0.958449i	$-0.592081\pi$
$857$	−14313.6	−0.570528	−0.285264	+	0.958449i	$0.592081\pi$
$858$	−45253.2	−1.80061
$859$	7863.30	0.312331	0.156166	−	0.987731i	$-0.450087\pi$
$859$	7863.30	0.312331	0.156166	+	0.987731i	$0.450087\pi$
$860$	1021.45	0.0405013
$861$	−11149.8	−0.441327
$862$	−2270.39	−0.0897099
$863$	11316.2	0.446360	0.223180	−	0.974777i	$-0.428356\pi$
$863$	11316.2	0.446360	0.223180	+	0.974777i	$0.428356\pi$
$864$	4344.79	0.171079
$865$	−8084.80	−0.317794
$866$	−9318.30	−0.365645
$867$	−20717.8	−0.811550
$868$	−824.781	−0.0322522
$869$	63714.5	2.48719
$870$	−11680.8	−0.455193
$871$	7568.96	0.294448
$872$	9508.34	0.369258
$873$	−10954.5	−0.424688
$874$	−474.854	−0.0183778
$875$	2949.16	0.113942
$876$	−1835.60	−0.0707981
$877$	−4318.10	−0.166262	−0.0831311	−	0.996539i	$-0.526492\pi$
$877$	−4318.10	−0.166262	−0.0831311	+	0.996539i	$0.526492\pi$
$878$	−10479.0	−0.402789
$879$	−12938.4	−0.496476
$880$	−20003.5	−0.766271
$881$	−24898.0	−0.952141	−0.476071	−	0.879407i	$-0.657939\pi$
$881$	−24898.0	−0.952141	−0.476071	+	0.879407i	$0.657939\pi$
$882$	7609.77	0.290515
$883$	30026.7	1.14437	0.572185	−	0.820125i	$-0.306096\pi$
$883$	30026.7	1.14437	0.572185	+	0.820125i	$0.306096\pi$
$884$	4432.20	0.168632
$885$	13381.0	0.508248
$886$	11163.3	0.423292
$887$	12725.0	0.481697	0.240848	−	0.970563i	$-0.422574\pi$
$887$	12725.0	0.481697	0.240848	+	0.970563i	$0.422574\pi$
$888$	20173.7	0.762370
$889$	26935.6	1.01619
$890$	17752.6	0.668617
$891$	14844.3	0.558139
$892$	1923.47	0.0722001
$893$	−751.878	−0.0281754
$894$	22324.8	0.835184
$895$	8551.64	0.319385
$896$	37673.5	1.40467
$897$	6080.88	0.226349
$898$	26195.0	0.973428
$899$	11296.3	0.419081
$900$	−193.214	−0.00715608
$901$	12891.3	0.476661
$902$	−20969.2	−0.774054
$903$	−29156.2	−1.07448
$904$	−36153.8	−1.33015
$905$	7972.67	0.292840
$906$	−13737.6	−0.503752
$907$	−38323.1	−1.40298	−0.701488	−	0.712681i	$-0.747481\pi$
$907$	−38323.1	−1.40298	−0.701488	+	0.712681i	$0.747481\pi$
$908$	2164.84	0.0791218
$909$	−7636.27	−0.278635
$910$	−23762.3	−0.865617
$911$	47436.8	1.72520	0.862598	−	0.505891i	$-0.168836\pi$
$911$	47436.8	1.72520	0.862598	+	0.505891i	$0.168836\pi$
$912$	−1861.51	−0.0675885
$913$	−75116.9	−2.72290
$914$	1629.09	0.0589556
$915$	−17349.6	−0.626842
$916$	841.605	0.0303574
$917$	−11633.9	−0.418957
$918$	−44975.6	−1.61701
$919$	5246.48	0.188319	0.0941596	−	0.995557i	$-0.469984\pi$
$919$	5246.48	0.188319	0.0941596	+	0.995557i	$0.469984\pi$
$920$	2488.35	0.0891721
$921$	15365.7	0.549746
$922$	−27722.6	−0.990232
$923$	15489.1	0.552361
$924$	−3380.01	−0.120340
$925$	−6042.34	−0.214779
$926$	−50787.6	−1.80236
$927$	−19072.7	−0.675760
$928$	5932.86	0.209866
$929$	−40545.9	−1.43194	−0.715968	−	0.698133i	$-0.754015\pi$
$929$	−40545.9	−1.43194	−0.715968	+	0.698133i	$0.754015\pi$
$930$	−3107.53	−0.109570
$931$	−1500.78	−0.0528315
$932$	3780.38	0.132865
$933$	19866.6	0.697109
$934$	497.026	0.0174124
$935$	29529.6	1.03286
$936$	−17973.5	−0.627651
$937$	15437.5	0.538228	0.269114	−	0.963108i	$-0.413269\pi$
$937$	15437.5	0.538228	0.269114	+	0.963108i	$0.413269\pi$
$938$	7657.58	0.266555
$939$	32930.1	1.14444
$940$	−341.268	−0.0118414
$941$	26090.2	0.903843	0.451922	−	0.892058i	$-0.350739\pi$
$941$	26090.2	0.903843	0.451922	+	0.892058i	$0.350739\pi$
$942$	28211.7	0.975782
$943$	2817.72	0.0973039
$944$	−47658.1	−1.64316
$945$	17801.6	0.612790
$946$	−54833.6	−1.88456
$947$	−46278.9	−1.58803	−0.794015	−	0.607899i	$-0.792013\pi$
$947$	−46278.9	−1.58803	−0.794015	+	0.607899i	$0.792013\pi$
$948$	2691.20	0.0922004
$949$	51143.5	1.74941
$950$	516.145	0.0176273
$951$	−32468.7	−1.10712
$952$	−51769.8	−1.76247
$953$	−35978.1	−1.22292	−0.611461	−	0.791275i	$-0.709418\pi$
$953$	−35978.1	−1.22292	−0.611461	+	0.791275i	$0.709418\pi$
$954$	4528.01	0.153669
$955$	−12164.2	−0.412172
$956$	−2380.01	−0.0805177
$957$	46293.2	1.56368
$958$	32920.7	1.11025
$959$	41531.6	1.39846
$960$	8967.57	0.301487
$961$	−26785.8	−0.899123
$962$	48685.0	1.63167
$963$	4931.56	0.165023
$964$	−820.496	−0.0274133
$965$	−8653.76	−0.288678
$966$	6152.08	0.204907
$967$	9175.70	0.305140	0.152570	−	0.988293i	$-0.451245\pi$
$967$	9175.70	0.305140	0.152570	+	0.988293i	$0.451245\pi$
$968$	44590.0	1.48056
$969$	2748.00	0.0911026
$970$	13282.3	0.439657
$971$	−11637.4	−0.384616	−0.192308	−	0.981335i	$-0.561597\pi$
$971$	−11637.4	−0.384616	−0.192308	+	0.981335i	$0.561597\pi$
$972$	−1971.23	−0.0650485
$973$	34292.4	1.12987
$974$	18665.4	0.614045
$975$	−6609.66	−0.217106
$976$	61792.7	2.02657
$977$	22627.5	0.740961	0.370481	−	0.928840i	$-0.379193\pi$
$977$	22627.5	0.740961	0.370481	+	0.928840i	$0.379193\pi$
$978$	29468.9	0.963508
$979$	−70356.7	−2.29684
$980$	−681.186	−0.0222038
$981$	−5325.74	−0.173331
$982$	−6295.95	−0.204595
$983$	24515.4	0.795444	0.397722	−	0.917506i	$-0.369801\pi$
$983$	24515.4	0.795444	0.397722	+	0.917506i	$0.369801\pi$
$984$	10225.6	0.331282
$985$	−8261.79	−0.267251
$986$	−61414.8	−1.98362
$987$	9741.13	0.314148
$988$	−307.026	−0.00988645
$989$	7368.24	0.236902
$990$	10372.1	0.332978
$991$	−17259.7	−0.553250	−0.276625	−	0.960978i	$-0.589216\pi$
$991$	−17259.7	−0.553250	−0.276625	+	0.960978i	$0.589216\pi$
$992$	1578.36	0.0505171
$993$	−22841.8	−0.729973
$994$	15670.4	0.500035
$995$	−6896.56	−0.219734
$996$	−3172.81	−0.100938
$997$	38645.5	1.22760	0.613799	−	0.789462i	$-0.289640\pi$
$997$	38645.5	1.22760	0.613799	+	0.789462i	$0.289640\pi$
$998$	10547.0	0.334529
$999$	−36472.6	−1.15510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.a.e.1.1	✓	5
3.2	odd	2		1035.4.a.k.1.5		5
4.3	odd	2		1840.4.a.n.1.4		5
5.2	odd	4		575.4.b.i.24.3		10
5.3	odd	4		575.4.b.i.24.8		10
5.4	even	2		575.4.a.j.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.e.1.1	✓	5	1.1	even	1	trivial
575.4.a.j.1.5		5	5.4	even	2
575.4.b.i.24.3		10	5.2	odd	4
575.4.b.i.24.8		10	5.3	odd	4
1035.4.a.k.1.5		5	3.2	odd	2
1840.4.a.n.1.4		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93900 q^{2} -3.85751 q^{3} +0.637693 q^{4} -5.00000 q^{5} +11.3372 q^{6} -23.5932 q^{7} +21.6378 q^{8} -12.1196 q^{9} +O(q^{10})\)	\(q-2.93900 q^{2} -3.85751 q^{3} +0.637693 q^{4} -5.00000 q^{5} +11.3372 q^{6} -23.5932 q^{7} +21.6378 q^{8} -12.1196 q^{9} +14.6950 q^{10} -58.2387 q^{11} -2.45991 q^{12} +68.5380 q^{13} +69.3404 q^{14} +19.2876 q^{15} -68.6949 q^{16} +101.409 q^{17} +35.6194 q^{18} -7.02478 q^{19} -3.18846 q^{20} +91.0112 q^{21} +171.163 q^{22} -23.0000 q^{23} -83.4681 q^{24} +25.0000 q^{25} -201.433 q^{26} +150.904 q^{27} -15.0452 q^{28} +206.062 q^{29} -56.6861 q^{30} +54.8201 q^{31} +28.7917 q^{32} +224.657 q^{33} -298.040 q^{34} +117.966 q^{35} -7.72857 q^{36} -241.694 q^{37} +20.6458 q^{38} -264.386 q^{39} -108.189 q^{40} -122.510 q^{41} -267.482 q^{42} -320.358 q^{43} -37.1384 q^{44} +60.5979 q^{45} +67.5969 q^{46} +107.032 q^{47} +264.991 q^{48} +213.641 q^{49} -73.4749 q^{50} -391.186 q^{51} +43.7062 q^{52} +127.122 q^{53} -443.507 q^{54} +291.194 q^{55} -510.506 q^{56} +27.0982 q^{57} -605.615 q^{58} +693.766 q^{59} +12.2995 q^{60} -899.524 q^{61} -161.116 q^{62} +285.940 q^{63} +464.941 q^{64} -342.690 q^{65} -660.265 q^{66} +110.435 q^{67} +64.6677 q^{68} +88.7228 q^{69} -346.702 q^{70} +225.992 q^{71} -262.241 q^{72} +746.207 q^{73} +710.336 q^{74} -96.4378 q^{75} -4.47965 q^{76} +1374.04 q^{77} +777.030 q^{78} -1094.02 q^{79} +343.474 q^{80} -254.887 q^{81} +360.055 q^{82} +1289.81 q^{83} +58.0372 q^{84} -507.045 q^{85} +941.531 q^{86} -794.887 q^{87} -1260.16 q^{88} +1208.07 q^{89} -178.097 q^{90} -1617.03 q^{91} -14.6669 q^{92} -211.469 q^{93} -314.567 q^{94} +35.1239 q^{95} -111.064 q^{96} +903.864 q^{97} -627.890 q^{98} +705.829 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9	\( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 5 * q + 6 * q^2 + 4 * q^3 + 22 * q^4 - 25 * q^5 + 19 * q^6 - 3 * q^7 + 138 * q^8 + 77 * q^9 - 30 * q^10 + 23 * q^11 + 47 * q^12 + 132 * q^13 + 93 * q^14 - 20 * q^15 + 282 * q^16 + 23 * q^17 - 15 * q^18 - 161 * q^19 - 110 * q^20 - 60 * q^21 + 193 * q^22 - 115 * q^23 + 105 * q^24 + 125 * q^25 - 257 * q^26 + 577 * q^27 + 17 * q^28 + 401 * q^29 - 95 * q^30 + 32 * q^31 + 670 * q^32 + 189 * q^33 - 663 * q^34 + 15 * q^35 - 659 * q^36 - 38 * q^37 - 875 * q^38 + 335 * q^39 - 690 * q^40 - 12 * q^41 - 798 * q^42 - 566 * q^43 + 47 * q^44 - 385 * q^45 - 138 * q^46 + 919 * q^47 - 773 * q^48 - 738 * q^49 + 150 * q^50 - 993 * q^51 - 305 * q^52 + 1156 * q^53 - 8 * q^54 - 115 * q^55 + 343 * q^56 + 114 * q^57 - 1042 * q^58 + 1324 * q^59 - 235 * q^60 - 1673 * q^61 + 565 * q^62 + 270 * q^63 + 2466 * q^64 - 660 * q^65 - 2781 * q^66 + 558 * q^67 - 2267 * q^68 - 92 * q^69 - 465 * q^70 - 108 * q^71 - 789 * q^72 + 1173 * q^73 + 1458 * q^74 + 100 * q^75 - 3477 * q^76 + 2608 * q^77 + 704 * q^78 + 656 * q^79 - 1410 * q^80 - 319 * q^81 + 3505 * q^82 - 82 * q^83 - 718 * q^84 - 115 * q^85 + 112 * q^86 + 2389 * q^87 + 2397 * q^88 + 570 * q^89 + 75 * q^90 - 1589 * q^91 - 506 * q^92 + 911 * q^93 - 948 * q^94 + 805 * q^95 - 5991 * q^96 + 633 * q^97 - 2555 * q^98 + 2021 * q^99

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