LMFDB - Embedded newform 2415.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,2,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2415.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:	$19.2838720881$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2508628.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - 2x^{2} + 23x + 8 $ x^5 - 10x^3 - 2x^2 + 23*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66023$ of defining polynomial
Character	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66023 q^{2} -1.00000 q^{3} +5.07682 q^{4} -1.00000 q^{5} +2.66023 q^{6} -1.00000 q^{7} -8.18506 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.66023 q^{2} -1.00000 q^{3} +5.07682 q^{4} -1.00000 q^{5} +2.66023 q^{6} -1.00000 q^{7} -8.18506 q^{8} +1.00000 q^{9} +2.66023 q^{10} +1.86460 q^{11} -5.07682 q^{12} -0.416595 q^{13} +2.66023 q^{14} +1.00000 q^{15} +11.6205 q^{16} -2.00000 q^{17} -2.66023 q^{18} +0.224791 q^{19} -5.07682 q^{20} +1.00000 q^{21} -4.96027 q^{22} -1.00000 q^{23} +8.18506 q^{24} +1.00000 q^{25} +1.10824 q^{26} -1.00000 q^{27} -5.07682 q^{28} -3.52483 q^{29} -2.66023 q^{30} +2.60225 q^{31} -14.5431 q^{32} -1.86460 q^{33} +5.32046 q^{34} +1.00000 q^{35} +5.07682 q^{36} +9.26189 q^{37} -0.597996 q^{38} +0.416595 q^{39} +8.18506 q^{40} -0.429163 q^{41} -2.66023 q^{42} +1.01884 q^{43} +9.46626 q^{44} -1.00000 q^{45} +2.66023 q^{46} -3.44801 q^{47} -11.6205 q^{48} +1.00000 q^{49} -2.66023 q^{50} +2.00000 q^{51} -2.11498 q^{52} -8.28905 q^{53} +2.66023 q^{54} -1.86460 q^{55} +8.18506 q^{56} -0.224791 q^{57} +9.37687 q^{58} -2.37059 q^{59} +5.07682 q^{60} +2.37844 q^{61} -6.92258 q^{62} -1.00000 q^{63} +15.4470 q^{64} +0.416595 q^{65} +4.96027 q^{66} -2.89602 q^{67} -10.1536 q^{68} +1.00000 q^{69} -2.66023 q^{70} -8.39571 q^{71} -8.18506 q^{72} +0.602250 q^{73} -24.6388 q^{74} -1.00000 q^{75} +1.14123 q^{76} -1.86460 q^{77} -1.10824 q^{78} +11.5327 q^{79} -11.6205 q^{80} +1.00000 q^{81} +1.14167 q^{82} -0.352338 q^{83} +5.07682 q^{84} +2.00000 q^{85} -2.71036 q^{86} +3.52483 q^{87} -15.2619 q^{88} +10.3888 q^{89} +2.66023 q^{90} +0.416595 q^{91} -5.07682 q^{92} -2.60225 q^{93} +9.17249 q^{94} -0.224791 q^{95} +14.5431 q^{96} -14.3957 q^{97} -2.66023 q^{98} +1.86460 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9	$ 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9} + q^{11} - 10 q^{12} + 5 q^{15} + 8 q^{16} - 10 q^{17} + 3 q^{19} - 10 q^{20} + 5 q^{21} + 12 q^{22} - 5 q^{23} + 6 q^{24} + 5 q^{25} - 14 q^{26} - 5 q^{27} - 10 q^{28} + 4 q^{29} + 2 q^{31} - 12 q^{32} - q^{33} + 5 q^{35} + 10 q^{36} - 4 q^{37} - 24 q^{38} + 6 q^{40} - 9 q^{41} - 8 q^{43} + 2 q^{44} - 5 q^{45} - 11 q^{47} - 8 q^{48} + 5 q^{49} + 10 q^{51} - 22 q^{52} - 19 q^{53} - q^{55} + 6 q^{56} - 3 q^{57} + 8 q^{58} + 5 q^{59} + 10 q^{60} - 17 q^{61} - 24 q^{62} - 5 q^{63} - 8 q^{64} - 12 q^{66} - 2 q^{67} - 20 q^{68} + 5 q^{69} + 10 q^{71} - 6 q^{72} - 8 q^{73} - 34 q^{74} - 5 q^{75} - 8 q^{76} - q^{77} + 14 q^{78} + 24 q^{79} - 8 q^{80} + 5 q^{81} - 8 q^{82} - 24 q^{83} + 10 q^{84} + 10 q^{85} - 10 q^{86} - 4 q^{87} - 26 q^{88} - 4 q^{89} - 10 q^{92} - 2 q^{93} + 2 q^{94} - 3 q^{95} + 12 q^{96} - 20 q^{97} + q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9 + q^11 - 10 * q^12 + 5 * q^15 + 8 * q^16 - 10 * q^17 + 3 * q^19 - 10 * q^20 + 5 * q^21 + 12 * q^22 - 5 * q^23 + 6 * q^24 + 5 * q^25 - 14 * q^26 - 5 * q^27 - 10 * q^28 + 4 * q^29 + 2 * q^31 - 12 * q^32 - q^33 + 5 * q^35 + 10 * q^36 - 4 * q^37 - 24 * q^38 + 6 * q^40 - 9 * q^41 - 8 * q^43 + 2 * q^44 - 5 * q^45 - 11 * q^47 - 8 * q^48 + 5 * q^49 + 10 * q^51 - 22 * q^52 - 19 * q^53 - q^55 + 6 * q^56 - 3 * q^57 + 8 * q^58 + 5 * q^59 + 10 * q^60 - 17 * q^61 - 24 * q^62 - 5 * q^63 - 8 * q^64 - 12 * q^66 - 2 * q^67 - 20 * q^68 + 5 * q^69 + 10 * q^71 - 6 * q^72 - 8 * q^73 - 34 * q^74 - 5 * q^75 - 8 * q^76 - q^77 + 14 * q^78 + 24 * q^79 - 8 * q^80 + 5 * q^81 - 8 * q^82 - 24 * q^83 + 10 * q^84 + 10 * q^85 - 10 * q^86 - 4 * q^87 - 26 * q^88 - 4 * q^89 - 10 * q^92 - 2 * q^93 + 2 * q^94 - 3 * q^95 + 12 * q^96 - 20 * q^97 + 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 $

$2$	−2.66023	−1.88107	−0.940533	−	0.339701i	$-0.889674\pi$
$2$	−2.66023	−1.88107	−0.940533	+	0.339701i	$0.889674\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.07682	2.53841
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.66023	1.08603
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−8.18506	−2.89386
$9$	1.00000	0.333333
$10$	2.66023	0.841239
$11$	1.86460	0.562199	0.281099	−	0.959679i	$-0.409301\pi$
$11$	1.86460	0.562199	0.281099	+	0.959679i	$0.409301\pi$
$12$	−5.07682	−1.46555
$13$	−0.416595	−0.115543	−0.0577713	−	0.998330i	$-0.518399\pi$
$13$	−0.416595	−0.115543	−0.0577713	+	0.998330i	$0.518399\pi$
$14$	2.66023	0.710976
$15$	1.00000	0.258199
$16$	11.6205	2.90513
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−2.66023	−0.627022
$19$	0.224791	0.0515706	0.0257853	−	0.999668i	$-0.491791\pi$
$19$	0.224791	0.0515706	0.0257853	+	0.999668i	$0.491791\pi$
$20$	−5.07682	−1.13521
$21$	1.00000	0.218218
$22$	−4.96027	−1.05753
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	8.18506	1.67077
$25$	1.00000	0.200000
$26$	1.10824	0.217343
$27$	−1.00000	−0.192450
$28$	−5.07682	−0.959430
$29$	−3.52483	−0.654545	−0.327272	−	0.944930i	$-0.606130\pi$
$29$	−3.52483	−0.654545	−0.327272	+	0.944930i	$0.606130\pi$
$30$	−2.66023	−0.485689
$31$	2.60225	0.467378	0.233689	−	0.972311i	$-0.424920\pi$
$31$	2.60225	0.467378	0.233689	+	0.972311i	$0.424920\pi$
$32$	−14.5431	−2.57088
$33$	−1.86460	−0.324586
$34$	5.32046	0.912451
$35$	1.00000	0.169031
$36$	5.07682	0.846137
$37$	9.26189	1.52264	0.761322	−	0.648373i	$-0.224550\pi$
$37$	9.26189	1.52264	0.761322	+	0.648373i	$0.224550\pi$
$38$	−0.597996	−0.0970078
$39$	0.416595	0.0667085
$40$	8.18506	1.29417
$41$	−0.429163	−0.0670241	−0.0335120	−	0.999438i	$-0.510669\pi$
$41$	−0.429163	−0.0670241	−0.0335120	+	0.999438i	$0.510669\pi$
$42$	−2.66023	−0.410482
$43$	1.01884	0.155372	0.0776862	−	0.996978i	$-0.475247\pi$
$43$	1.01884	0.155372	0.0776862	+	0.996978i	$0.475247\pi$
$44$	9.46626	1.42709
$45$	−1.00000	−0.149071
$46$	2.66023	0.392230
$47$	−3.44801	−0.502944	−0.251472	−	0.967865i	$-0.580915\pi$
$47$	−3.44801	−0.502944	−0.251472	+	0.967865i	$0.580915\pi$
$48$	−11.6205	−1.67727
$49$	1.00000	0.142857
$50$	−2.66023	−0.376213
$51$	2.00000	0.280056
$52$	−2.11498	−0.293295
$53$	−8.28905	−1.13859	−0.569294	−	0.822134i	$-0.692783\pi$
$53$	−8.28905	−1.13859	−0.569294	+	0.822134i	$0.692783\pi$
$54$	2.66023	0.362011
$55$	−1.86460	−0.251423
$56$	8.18506	1.09378
$57$	−0.224791	−0.0297743
$58$	9.37687	1.23124
$59$	−2.37059	−0.308625	−0.154312	−	0.988022i	$-0.549316\pi$
$59$	−2.37059	−0.308625	−0.154312	+	0.988022i	$0.549316\pi$
$60$	5.07682	0.655415
$61$	2.37844	0.304528	0.152264	−	0.988340i	$-0.451344\pi$
$61$	2.37844	0.304528	0.152264	+	0.988340i	$0.451344\pi$
$62$	−6.92258	−0.879169
$63$	−1.00000	−0.125988
$64$	15.4470	1.93087
$65$	0.416595	0.0516722
$66$	4.96027	0.610567
$67$	−2.89602	−0.353805	−0.176902	−	0.984228i	$-0.556608\pi$
$67$	−2.89602	−0.353805	−0.176902	+	0.984228i	$0.556608\pi$
$68$	−10.1536	−1.23131
$69$	1.00000	0.120386
$70$	−2.66023	−0.317958
$71$	−8.39571	−0.996388	−0.498194	−	0.867066i	$-0.666003\pi$
$71$	−8.39571	−0.996388	−0.498194	+	0.867066i	$0.666003\pi$
$72$	−8.18506	−0.964619
$73$	0.602250	0.0704880	0.0352440	−	0.999379i	$-0.488779\pi$
$73$	0.602250	0.0704880	0.0352440	+	0.999379i	$0.488779\pi$
$74$	−24.6388	−2.86420
$75$	−1.00000	−0.115470
$76$	1.14123	0.130908
$77$	−1.86460	−0.212491
$78$	−1.10824	−0.125483
$79$	11.5327	1.29753	0.648764	−	0.760990i	$-0.275286\pi$
$79$	11.5327	1.29753	0.648764	+	0.760990i	$0.275286\pi$
$80$	−11.6205	−1.29921
$81$	1.00000	0.111111
$82$	1.14167	0.126077
$83$	−0.352338	−0.0386742	−0.0193371	−	0.999813i	$-0.506156\pi$
$83$	−0.352338	−0.0386742	−0.0193371	+	0.999813i	$0.506156\pi$
$84$	5.07682	0.553927
$85$	2.00000	0.216930
$86$	−2.71036	−0.292266
$87$	3.52483	0.377902
$88$	−15.2619	−1.62692
$89$	10.3888	1.10122	0.550608	−	0.834764i	$-0.314396\pi$
$89$	10.3888	1.10122	0.550608	+	0.834764i	$0.314396\pi$
$90$	2.66023	0.280413
$91$	0.416595	0.0436710
$92$	−5.07682	−0.529296
$93$	−2.60225	−0.269841
$94$	9.17249	0.946071
$95$	−0.224791	−0.0230631
$96$	14.5431	1.48430
$97$	−14.3957	−1.46166	−0.730831	−	0.682558i	$-0.760867\pi$
$97$	−14.3957	−1.46166	−0.730831	+	0.682558i	$0.760867\pi$
$98$	−2.66023	−0.268724
$99$	1.86460	0.187400
$100$	5.07682	0.507682
$101$	4.58281	0.456007	0.228003	−	0.973660i	$-0.426780\pi$
$101$	4.58281	0.456007	0.228003	+	0.973660i	$0.426780\pi$
$102$	−5.32046	−0.526804
$103$	1.05798	0.104246	0.0521230	−	0.998641i	$-0.483401\pi$
$103$	1.05798	0.104246	0.0521230	+	0.998641i	$0.483401\pi$
$104$	3.40985	0.334364
$105$	−1.00000	−0.0975900
$106$	22.0508	2.14176
$107$	16.2222	1.56826	0.784128	−	0.620600i	$-0.213111\pi$
$107$	16.2222	1.56826	0.784128	+	0.620600i	$0.213111\pi$
$108$	−5.07682	−0.488518
$109$	0.531573	0.0509155	0.0254577	−	0.999676i	$-0.491896\pi$
$109$	0.531573	0.0509155	0.0254577	+	0.999676i	$0.491896\pi$
$110$	4.96027	0.472943
$111$	−9.26189	−0.879099
$112$	−11.6205	−1.09803
$113$	7.91584	0.744660	0.372330	−	0.928100i	$-0.378559\pi$
$113$	7.91584	0.744660	0.372330	+	0.928100i	$0.378559\pi$
$114$	0.597996	0.0560075
$115$	1.00000	0.0932505
$116$	−17.8950	−1.66151
$117$	−0.416595	−0.0385142
$118$	6.30632	0.580544
$119$	2.00000	0.183340
$120$	−8.18506	−0.747191
$121$	−7.52326	−0.683933
$122$	−6.32720	−0.572838
$123$	0.429163	0.0386964
$124$	13.2112	1.18640
$125$	−1.00000	−0.0894427
$126$	2.66023	0.236992
$127$	17.6073	1.56240	0.781199	−	0.624282i	$-0.214608\pi$
$127$	17.6073	1.56240	0.781199	+	0.624282i	$0.214608\pi$
$128$	−12.0063	−1.06121
$129$	−1.01884	−0.0897042
$130$	−1.10824	−0.0971989
$131$	−6.96499	−0.608534	−0.304267	−	0.952587i	$-0.598412\pi$
$131$	−6.96499	−0.608534	−0.304267	+	0.952587i	$0.598412\pi$
$132$	−9.46626	−0.823932
$133$	−0.224791	−0.0194919
$134$	7.70407	0.665530
$135$	1.00000	0.0860663
$136$	16.3701	1.40373
$137$	7.93311	0.677771	0.338886	−	0.940828i	$-0.389950\pi$
$137$	7.93311	0.677771	0.338886	+	0.940828i	$0.389950\pi$
$138$	−2.66023	−0.226454
$139$	−6.90955	−0.586060	−0.293030	−	0.956103i	$-0.594664\pi$
$139$	−6.90955	−0.586060	−0.293030	+	0.956103i	$0.594664\pi$
$140$	5.07682	0.429070
$141$	3.44801	0.290375
$142$	22.3345	1.87427
$143$	−0.776784	−0.0649579
$144$	11.6205	0.968375
$145$	3.52483	0.292721
$146$	−1.60212	−0.132593
$147$	−1.00000	−0.0824786
$148$	47.0210	3.86510
$149$	16.2869	1.33427	0.667137	−	0.744935i	$-0.267520\pi$
$149$	16.2869	1.33427	0.667137	+	0.744935i	$0.267520\pi$
$150$	2.66023	0.217207
$151$	−5.72449	−0.465852	−0.232926	−	0.972494i	$-0.574830\pi$
$151$	−5.72449	−0.465852	−0.232926	+	0.972494i	$0.574830\pi$
$152$	−1.83993	−0.149238
$153$	−2.00000	−0.161690
$154$	4.96027	0.399710
$155$	−2.60225	−0.209018
$156$	2.11498	0.169334
$157$	−5.12081	−0.408685	−0.204342	−	0.978899i	$-0.565506\pi$
$157$	−5.12081	−0.408685	−0.204342	+	0.978899i	$0.565506\pi$
$158$	−30.6796	−2.44074
$159$	8.28905	0.657364
$160$	14.5431	1.14973
$161$	1.00000	0.0788110
$162$	−2.66023	−0.209007
$163$	−7.67017	−0.600774	−0.300387	−	0.953817i	$-0.597116\pi$
$163$	−7.67017	−0.600774	−0.300387	+	0.953817i	$0.597116\pi$
$164$	−2.17879	−0.170135
$165$	1.86460	0.145159
$166$	0.937301	0.0727487
$167$	3.94096	0.304961	0.152480	−	0.988306i	$-0.451274\pi$
$167$	3.94096	0.304961	0.152480	+	0.988306i	$0.451274\pi$
$168$	−8.18506	−0.631491
$169$	−12.8264	−0.986650
$170$	−5.32046	−0.408061
$171$	0.224791	0.0171902
$172$	5.17249	0.394399
$173$	−4.92258	−0.374257	−0.187129	−	0.982335i	$-0.559918\pi$
$173$	−4.92258	−0.374257	−0.187129	+	0.982335i	$0.559918\pi$
$174$	−9.37687	−0.710858
$175$	−1.00000	−0.0755929
$176$	21.6676	1.63326
$177$	2.37059	0.178185
$178$	−27.6367	−2.07146
$179$	−17.4234	−1.30229	−0.651143	−	0.758955i	$-0.725710\pi$
$179$	−17.4234	−1.30229	−0.651143	+	0.758955i	$0.725710\pi$
$180$	−5.07682	−0.378404
$181$	−21.6529	−1.60945	−0.804724	−	0.593650i	$-0.797687\pi$
$181$	−21.6529	−1.60945	−0.804724	+	0.593650i	$0.797687\pi$
$182$	−1.10824	−0.0821481
$183$	−2.37844	−0.175819
$184$	8.18506	0.603411
$185$	−9.26189	−0.680948
$186$	6.92258	0.507588
$187$	−3.72920	−0.272706
$188$	−17.5049	−1.27668
$189$	1.00000	0.0727393
$190$	0.597996	0.0433832
$191$	−18.2570	−1.32103	−0.660517	−	0.750811i	$-0.729663\pi$
$191$	−18.2570	−1.32103	−0.660517	+	0.750811i	$0.729663\pi$
$192$	−15.4470	−1.11479
$193$	−10.4826	−0.754551	−0.377275	−	0.926101i	$-0.623139\pi$
$193$	−10.4826	−0.754551	−0.377275	+	0.926101i	$0.623139\pi$
$194$	38.2959	2.74949
$195$	−0.416595	−0.0298330
$196$	5.07682	0.362630
$197$	2.42085	0.172478	0.0862391	−	0.996274i	$-0.472515\pi$
$197$	2.42085	0.172478	0.0862391	+	0.996274i	$0.472515\pi$
$198$	−4.96027	−0.352511
$199$	−2.53328	−0.179579	−0.0897896	−	0.995961i	$-0.528619\pi$
$199$	−2.53328	−0.179579	−0.0897896	+	0.995961i	$0.528619\pi$
$200$	−8.18506	−0.578771
$201$	2.89602	0.204269
$202$	−12.1913	−0.857780
$203$	3.52483	0.247395
$204$	10.1536	0.710898
$205$	0.429163	0.0299741
$206$	−2.81447	−0.196094
$207$	−1.00000	−0.0695048
$208$	−4.84104	−0.335666
$209$	0.419146	0.0289929
$210$	2.66023	0.183573
$211$	8.73765	0.601524	0.300762	−	0.953699i	$-0.402759\pi$
$211$	8.73765	0.601524	0.300762	+	0.953699i	$0.402759\pi$
$212$	−42.0820	−2.89021
$213$	8.39571	0.575265
$214$	−43.1547	−2.94999
$215$	−1.01884	−0.0694846
$216$	8.18506	0.556923
$217$	−2.60225	−0.176652
$218$	−1.41411	−0.0957754
$219$	−0.602250	−0.0406963
$220$	−9.46626	−0.638215
$221$	0.833189	0.0560464
$222$	24.6388	1.65364
$223$	−12.7637	−0.854724	−0.427362	−	0.904081i	$-0.640557\pi$
$223$	−12.7637	−0.854724	−0.427362	+	0.904081i	$0.640557\pi$
$224$	14.5431	0.971701
$225$	1.00000	0.0666667
$226$	−21.0580	−1.40075
$227$	−6.33362	−0.420377	−0.210189	−	0.977661i	$-0.567408\pi$
$227$	−6.33362	−0.420377	−0.210189	+	0.977661i	$0.567408\pi$
$228$	−1.14123	−0.0755795
$229$	−27.3738	−1.80891	−0.904455	−	0.426568i	$-0.859722\pi$
$229$	−27.3738	−1.80891	−0.904455	+	0.426568i	$0.859722\pi$
$230$	−2.66023	−0.175410
$231$	1.86460	0.122682
$232$	28.8510	1.89416
$233$	1.19121	0.0780389	0.0390194	−	0.999238i	$-0.487577\pi$
$233$	1.19121	0.0780389	0.0390194	+	0.999238i	$0.487577\pi$
$234$	1.10824	0.0724478
$235$	3.44801	0.224923
$236$	−12.0351	−0.783417
$237$	−11.5327	−0.749128
$238$	−5.32046	−0.344874
$239$	4.29520	0.277833	0.138917	−	0.990304i	$-0.455638\pi$
$239$	4.29520	0.277833	0.138917	+	0.990304i	$0.455638\pi$
$240$	11.6205	0.750100
$241$	−8.85361	−0.570311	−0.285156	−	0.958481i	$-0.592045\pi$
$241$	−8.85361	−0.570311	−0.285156	+	0.958481i	$0.592045\pi$
$242$	20.0136	1.28652
$243$	−1.00000	−0.0641500
$244$	12.0749	0.773018
$245$	−1.00000	−0.0638877
$246$	−1.14167	−0.0727904
$247$	−0.0936468	−0.00595861
$248$	−21.2996	−1.35252
$249$	0.352338	0.0223285
$250$	2.66023	0.168248
$251$	7.04967	0.444971	0.222485	−	0.974936i	$-0.428583\pi$
$251$	7.04967	0.444971	0.222485	+	0.974936i	$0.428583\pi$
$252$	−5.07682	−0.319810
$253$	−1.86460	−0.117227
$254$	−46.8396	−2.93898
$255$	−2.00000	−0.125245
$256$	1.04556	0.0653473
$257$	13.3200	0.830878	0.415439	−	0.909621i	$-0.363628\pi$
$257$	13.3200	0.830878	0.415439	+	0.909621i	$0.363628\pi$
$258$	2.71036	0.168740
$259$	−9.26189	−0.575506
$260$	2.11498	0.131165
$261$	−3.52483	−0.218182
$262$	18.5285	1.14469
$263$	−25.0815	−1.54659	−0.773294	−	0.634047i	$-0.781392\pi$
$263$	−25.0815	−1.54659	−0.773294	+	0.634047i	$0.781392\pi$
$264$	15.2619	0.939304
$265$	8.28905	0.509192
$266$	0.597996	0.0366655
$267$	−10.3888	−0.635787
$268$	−14.7026	−0.898102
$269$	−9.84542	−0.600286	−0.300143	−	0.953894i	$-0.597034\pi$
$269$	−9.84542	−0.600286	−0.300143	+	0.953894i	$0.597034\pi$
$270$	−2.66023	−0.161896
$271$	19.4882	1.18383	0.591913	−	0.806002i	$-0.298373\pi$
$271$	19.4882	1.18383	0.591913	+	0.806002i	$0.298373\pi$
$272$	−23.2410	−1.40919
$273$	−0.416595	−0.0252135
$274$	−21.1039	−1.27493
$275$	1.86460	0.112440
$276$	5.07682	0.305589
$277$	−0.761601	−0.0457601	−0.0228801	−	0.999738i	$-0.507284\pi$
$277$	−0.761601	−0.0457601	−0.0228801	+	0.999738i	$0.507284\pi$
$278$	18.3810	1.10242
$279$	2.60225	0.155793
$280$	−8.18506	−0.489151
$281$	−18.2065	−1.08611	−0.543053	−	0.839698i	$-0.682732\pi$
$281$	−18.2065	−1.08611	−0.543053	+	0.839698i	$0.682732\pi$
$282$	−9.17249	−0.546214
$283$	−30.6279	−1.82064	−0.910319	−	0.413906i	$-0.864164\pi$
$283$	−30.6279	−1.82064	−0.910319	+	0.413906i	$0.864164\pi$
$284$	−42.6236	−2.52924
$285$	0.224791	0.0133155
$286$	2.06642	0.122190
$287$	0.429163	0.0253327
$288$	−14.5431	−0.856960
$289$	−13.0000	−0.764706
$290$	−9.37687	−0.550629
$291$	14.3957	0.843892
$292$	3.05752	0.178928
$293$	0.949151	0.0554500	0.0277250	−	0.999616i	$-0.491174\pi$
$293$	0.949151	0.0554500	0.0277250	+	0.999616i	$0.491174\pi$
$294$	2.66023	0.155148
$295$	2.37059	0.138021
$296$	−75.8091	−4.40632
$297$	−1.86460	−0.108195
$298$	−43.3269	−2.50986
$299$	0.416595	0.0240923
$300$	−5.07682	−0.293111
$301$	−1.01884	−0.0587252
$302$	15.2285	0.876299
$303$	−4.58281	−0.263276
$304$	2.61219	0.149819
$305$	−2.37844	−0.136189
$306$	5.32046	0.304150
$307$	−3.67640	−0.209823	−0.104912	−	0.994482i	$-0.533456\pi$
$307$	−3.67640	−0.209823	−0.104912	+	0.994482i	$0.533456\pi$
$308$	−9.46626	−0.539390
$309$	−1.05798	−0.0601864
$310$	6.92258	0.393176
$311$	−3.57509	−0.202725	−0.101362	−	0.994850i	$-0.532320\pi$
$311$	−3.57509	−0.202725	−0.101362	+	0.994850i	$0.532320\pi$
$312$	−3.40985	−0.193045
$313$	−33.3110	−1.88285	−0.941423	−	0.337227i	$-0.890511\pi$
$313$	−33.3110	−1.88285	−0.941423	+	0.337227i	$0.890511\pi$
$314$	13.6225	0.768764
$315$	1.00000	0.0563436
$316$	58.5494	3.29366
$317$	−28.3822	−1.59410	−0.797052	−	0.603910i	$-0.793609\pi$
$317$	−28.3822	−1.59410	−0.797052	+	0.603910i	$0.793609\pi$
$318$	−22.0508	−1.23655
$319$	−6.57241	−0.367984
$320$	−15.4470	−0.863511
$321$	−16.2222	−0.905433
$322$	−2.66023	−0.148249
$323$	−0.449582	−0.0250154
$324$	5.07682	0.282046
$325$	−0.416595	−0.0231085
$326$	20.4044	1.13010
$327$	−0.531573	−0.0293961
$328$	3.51273	0.193958
$329$	3.44801	0.190095
$330$	−4.96027	−0.273054
$331$	10.8881	0.598467	0.299233	−	0.954180i	$-0.403269\pi$
$331$	10.8881	0.598467	0.299233	+	0.954180i	$0.403269\pi$
$332$	−1.78876	−0.0981710
$333$	9.26189	0.507548
$334$	−10.4839	−0.573652
$335$	2.89602	0.158226
$336$	11.6205	0.633950
$337$	−0.294213	−0.0160268	−0.00801341	−	0.999968i	$-0.502551\pi$
$337$	−0.294213	−0.0160268	−0.00801341	+	0.999968i	$0.502551\pi$
$338$	34.1213	1.85595
$339$	−7.91584	−0.429930
$340$	10.1536	0.550659
$341$	4.85216	0.262759
$342$	−0.597996	−0.0323359
$343$	−1.00000	−0.0539949
$344$	−8.33930	−0.449625
$345$	−1.00000	−0.0538382
$346$	13.0952	0.704003
$347$	0.0408381	0.00219230	0.00109615	−	0.999999i	$-0.499651\pi$
$347$	0.0408381	0.00219230	0.00109615	+	0.999999i	$0.499651\pi$
$348$	17.8950	0.959270
$349$	−3.79236	−0.203000	−0.101500	−	0.994836i	$-0.532364\pi$
$349$	−3.79236	−0.203000	−0.101500	+	0.994836i	$0.532364\pi$
$350$	2.66023	0.142195
$351$	0.416595	0.0222362
$352$	−27.1171	−1.44534
$353$	−11.3413	−0.603639	−0.301819	−	0.953365i	$-0.597594\pi$
$353$	−11.3413	−0.603639	−0.301819	+	0.953365i	$0.597594\pi$
$354$	−6.30632	−0.335177
$355$	8.39571	0.445598
$356$	52.7423	2.79534
$357$	−2.00000	−0.105851
$358$	46.3502	2.44969
$359$	5.16779	0.272746	0.136373	−	0.990658i	$-0.456455\pi$
$359$	5.16779	0.272746	0.136373	+	0.990658i	$0.456455\pi$
$360$	8.18506	0.431391
$361$	−18.9495	−0.997340
$362$	57.6017	3.02748
$363$	7.52326	0.394869
$364$	2.11498	0.110855
$365$	−0.602250	−0.0315232
$366$	6.32720	0.330728
$367$	−11.6085	−0.605960	−0.302980	−	0.952997i	$-0.597982\pi$
$367$	−11.6085	−0.605960	−0.302980	+	0.952997i	$0.597982\pi$
$368$	−11.6205	−0.605761
$369$	−0.429163	−0.0223414
$370$	24.6388	1.28091
$371$	8.28905	0.430346
$372$	−13.2112	−0.684967
$373$	20.6883	1.07120	0.535600	−	0.844472i	$-0.320086\pi$
$373$	20.6883	1.07120	0.535600	+	0.844472i	$0.320086\pi$
$374$	9.92054	0.512979
$375$	1.00000	0.0516398
$376$	28.2222	1.45545
$377$	1.46843	0.0756278
$378$	−2.66023	−0.136827
$379$	−13.9791	−0.718059	−0.359029	−	0.933326i	$-0.616892\pi$
$379$	−13.9791	−0.718059	−0.359029	+	0.933326i	$0.616892\pi$
$380$	−1.14123	−0.0585436
$381$	−17.6073	−0.902051
$382$	48.5679	2.48495
$383$	5.45539	0.278758	0.139379	−	0.990239i	$-0.455489\pi$
$383$	5.45539	0.278758	0.139379	+	0.990239i	$0.455489\pi$
$384$	12.0063	0.612693
$385$	1.86460	0.0950289
$386$	27.8860	1.41936
$387$	1.01884	0.0517908
$388$	−73.0845	−3.71030
$389$	−6.42445	−0.325732	−0.162866	−	0.986648i	$-0.552074\pi$
$389$	−6.42445	−0.325732	−0.162866	+	0.986648i	$0.552074\pi$
$390$	1.10824	0.0561178
$391$	2.00000	0.101144
$392$	−8.18506	−0.413408
$393$	6.96499	0.351337
$394$	−6.44001	−0.324443
$395$	−11.5327	−0.580272
$396$	9.46626	0.475697
$397$	−5.58712	−0.280410	−0.140205	−	0.990123i	$-0.544776\pi$
$397$	−5.58712	−0.280410	−0.140205	+	0.990123i	$0.544776\pi$
$398$	6.73909	0.337800
$399$	0.224791	0.0112536
$400$	11.6205	0.581025
$401$	9.70217	0.484503	0.242252	−	0.970213i	$-0.422114\pi$
$401$	9.70217	0.484503	0.242252	+	0.970213i	$0.422114\pi$
$402$	−7.70407	−0.384244
$403$	−1.08408	−0.0540020
$404$	23.2661	1.15753
$405$	−1.00000	−0.0496904
$406$	−9.37687	−0.465366
$407$	17.2697	0.856029
$408$	−16.3701	−0.810442
$409$	−31.2887	−1.54713	−0.773564	−	0.633719i	$-0.781528\pi$
$409$	−31.2887	−1.54713	−0.773564	+	0.633719i	$0.781528\pi$
$410$	−1.14167	−0.0563832
$411$	−7.93311	−0.391312
$412$	5.37118	0.264619
$413$	2.37059	0.116649
$414$	2.66023	0.130743
$415$	0.352338	0.0172956
$416$	6.05857	0.297046
$417$	6.90955	0.338362
$418$	−1.11503	−0.0545377
$419$	−8.40874	−0.410794	−0.205397	−	0.978679i	$-0.565849\pi$
$419$	−8.40874	−0.410794	−0.205397	+	0.978679i	$0.565849\pi$
$420$	−5.07682	−0.247724
$421$	19.4587	0.948357	0.474179	−	0.880429i	$-0.342745\pi$
$421$	19.4587	0.948357	0.474179	+	0.880429i	$0.342745\pi$
$422$	−23.2442	−1.13151
$423$	−3.44801	−0.167648
$424$	67.8464	3.29491
$425$	−2.00000	−0.0970143
$426$	−22.3345	−1.08211
$427$	−2.37844	−0.115101
$428$	82.3571	3.98088
$429$	0.776784	0.0375035
$430$	2.71036	0.130705
$431$	24.2599	1.16856	0.584278	−	0.811553i	$-0.301378\pi$
$431$	24.2599	1.16856	0.584278	+	0.811553i	$0.301378\pi$
$432$	−11.6205	−0.559092
$433$	−38.4306	−1.84686	−0.923430	−	0.383768i	$-0.874626\pi$
$433$	−38.4306	−1.84686	−0.923430	+	0.383768i	$0.874626\pi$
$434$	6.92258	0.332295
$435$	−3.52483	−0.169003
$436$	2.69870	0.129245
$437$	−0.224791	−0.0107532
$438$	1.60212	0.0765524
$439$	−4.81022	−0.229579	−0.114790	−	0.993390i	$-0.536619\pi$
$439$	−4.81022	−0.229579	−0.114790	+	0.993390i	$0.536619\pi$
$440$	15.2619	0.727582
$441$	1.00000	0.0476190
$442$	−2.21648	−0.105427
$443$	−15.0461	−0.714860	−0.357430	−	0.933940i	$-0.616347\pi$
$443$	−15.0461	−0.714860	−0.357430	+	0.933940i	$0.616347\pi$
$444$	−47.0210	−2.23152
$445$	−10.3888	−0.492478
$446$	33.9545	1.60779
$447$	−16.2869	−0.770343
$448$	−15.4470	−0.729800
$449$	−35.7235	−1.68590	−0.842948	−	0.537995i	$-0.819182\pi$
$449$	−35.7235	−1.68590	−0.842948	+	0.537995i	$0.819182\pi$
$450$	−2.66023	−0.125404
$451$	−0.800219	−0.0376808
$452$	40.1873	1.89025
$453$	5.72449	0.268960
$454$	16.8489	0.790758
$455$	−0.416595	−0.0195303
$456$	1.83993	0.0861626
$457$	16.5449	0.773938	0.386969	−	0.922093i	$-0.373522\pi$
$457$	16.5449	0.773938	0.386969	+	0.922093i	$0.373522\pi$
$458$	72.8206	3.40268
$459$	2.00000	0.0933520
$460$	5.07682	0.236708
$461$	−25.0399	−1.16622	−0.583112	−	0.812392i	$-0.698165\pi$
$461$	−25.0399	−1.16622	−0.583112	+	0.812392i	$0.698165\pi$
$462$	−4.96027	−0.230773
$463$	14.6269	0.679771	0.339885	−	0.940467i	$-0.389612\pi$
$463$	14.6269	0.679771	0.339885	+	0.940467i	$0.389612\pi$
$464$	−40.9603	−1.90154
$465$	2.60225	0.120676
$466$	−3.16890	−0.146796
$467$	10.6774	0.494092	0.247046	−	0.969004i	$-0.420540\pi$
$467$	10.6774	0.494092	0.247046	+	0.969004i	$0.420540\pi$
$468$	−2.11498	−0.0977649
$469$	2.89602	0.133726
$470$	−9.17249	−0.423096
$471$	5.12081	0.235954
$472$	19.4034	0.893115
$473$	1.89974	0.0873501
$474$	30.6796	1.40916
$475$	0.224791	0.0103141
$476$	10.1536	0.465392
$477$	−8.28905	−0.379529
$478$	−11.4262	−0.522623
$479$	−11.8454	−0.541231	−0.270616	−	0.962688i	$-0.587227\pi$
$479$	−11.8454	−0.541231	−0.270616	+	0.962688i	$0.587227\pi$
$480$	−14.5431	−0.663798
$481$	−3.85845	−0.175930
$482$	23.5526	1.07279
$483$	−1.00000	−0.0455016
$484$	−38.1943	−1.73610
$485$	14.3957	0.653676
$486$	2.66023	0.120670
$487$	32.2509	1.46143	0.730714	−	0.682684i	$-0.239187\pi$
$487$	32.2509	1.46143	0.730714	+	0.682684i	$0.239187\pi$
$488$	−19.4677	−0.881261
$489$	7.67017	0.346857
$490$	2.66023	0.120177
$491$	29.0074	1.30909	0.654544	−	0.756024i	$-0.272861\pi$
$491$	29.0074	1.30909	0.654544	+	0.756024i	$0.272861\pi$
$492$	2.17879	0.0982273
$493$	7.04967	0.317501
$494$	0.249122	0.0112085
$495$	−1.86460	−0.0838076
$496$	30.2394	1.35779
$497$	8.39571	0.376599
$498$	−0.937301	−0.0420015
$499$	32.7403	1.46565	0.732827	−	0.680414i	$-0.238200\pi$
$499$	32.7403	1.46565	0.732827	+	0.680414i	$0.238200\pi$
$500$	−5.07682	−0.227043
$501$	−3.94096	−0.176069
$502$	−18.7537	−0.837020
$503$	−35.1554	−1.56750	−0.783751	−	0.621075i	$-0.786696\pi$
$503$	−35.1554	−1.56750	−0.783751	+	0.621075i	$0.786696\pi$
$504$	8.18506	0.364592
$505$	−4.58281	−0.203933
$506$	4.96027	0.220511
$507$	12.8264	0.569643
$508$	89.3894	3.96601
$509$	0.969824	0.0429867	0.0214933	−	0.999769i	$-0.493158\pi$
$509$	0.969824	0.0429867	0.0214933	+	0.999769i	$0.493158\pi$
$510$	5.32046	0.235594
$511$	−0.602250	−0.0266420
$512$	21.2311	0.938292
$513$	−0.224791	−0.00992477
$514$	−35.4343	−1.56294
$515$	−1.05798	−0.0466202
$516$	−5.17249	−0.227706
$517$	−6.42916	−0.282754
$518$	24.6388	1.08256
$519$	4.92258	0.216077
$520$	−3.40985	−0.149532
$521$	31.4303	1.37698	0.688492	−	0.725244i	$-0.258273\pi$
$521$	31.4303	1.37698	0.688492	+	0.725244i	$0.258273\pi$
$522$	9.37687	0.410414
$523$	−20.4659	−0.894912	−0.447456	−	0.894306i	$-0.647670\pi$
$523$	−20.4659	−0.894912	−0.447456	+	0.894306i	$0.647670\pi$
$524$	−35.3600	−1.54471
$525$	1.00000	0.0436436
$526$	66.7224	2.90924
$527$	−5.20450	−0.226712
$528$	−21.6676	−0.942962
$529$	1.00000	0.0434783
$530$	−22.0508	−0.957824
$531$	−2.37059	−0.102875
$532$	−1.14123	−0.0494784
$533$	0.178787	0.00774413
$534$	27.6367	1.19596
$535$	−16.2222	−0.701345
$536$	23.7041	1.02386
$537$	17.4234	0.751875
$538$	26.1911	1.12918
$539$	1.86460	0.0803141
$540$	5.07682	0.218472
$541$	−30.5015	−1.31136	−0.655681	−	0.755038i	$-0.727619\pi$
$541$	−30.5015	−1.31136	−0.655681	+	0.755038i	$0.727619\pi$
$542$	−51.8432	−2.22686
$543$	21.6529	0.929215
$544$	29.0862	1.24706
$545$	−0.531573	−0.0227701
$546$	1.10824	0.0474282
$547$	−11.5922	−0.495645	−0.247822	−	0.968805i	$-0.579715\pi$
$547$	−11.5922	−0.495645	−0.247822	+	0.968805i	$0.579715\pi$
$548$	40.2750	1.72046
$549$	2.37844	0.101509
$550$	−4.96027	−0.211507
$551$	−0.792351	−0.0337553
$552$	−8.18506	−0.348379
$553$	−11.5327	−0.490420
$554$	2.02603	0.0860779
$555$	9.26189	0.393145
$556$	−35.0786	−1.48766
$557$	20.3305	0.861432	0.430716	−	0.902488i	$-0.358261\pi$
$557$	20.3305	0.861432	0.430716	+	0.902488i	$0.358261\pi$
$558$	−6.92258	−0.293056
$559$	−0.424445	−0.0179521
$560$	11.6205	0.491056
$561$	3.72920	0.157447
$562$	48.4334	2.04304
$563$	17.5734	0.740632	0.370316	−	0.928906i	$-0.379249\pi$
$563$	17.5734	0.740632	0.370316	+	0.928906i	$0.379249\pi$
$564$	17.5049	0.737091
$565$	−7.91584	−0.333022
$566$	81.4772	3.42474
$567$	−1.00000	−0.0419961
$568$	68.7194	2.88340
$569$	−0.174999	−0.00733633	−0.00366817	−	0.999993i	$-0.501168\pi$
$569$	−0.174999	−0.00733633	−0.00366817	+	0.999993i	$0.501168\pi$
$570$	−0.597996	−0.0250473
$571$	10.6961	0.447617	0.223809	−	0.974633i	$-0.428151\pi$
$571$	10.6961	0.447617	0.223809	+	0.974633i	$0.428151\pi$
$572$	−3.94359	−0.164890
$573$	18.2570	0.762699
$574$	−1.14167	−0.0476525
$575$	−1.00000	−0.0417029
$576$	15.4470	0.643623
$577$	41.9737	1.74739	0.873695	−	0.486475i	$-0.161717\pi$
$577$	41.9737	1.74739	0.873695	+	0.486475i	$0.161717\pi$
$578$	34.5830	1.43846
$579$	10.4826	0.435640
$580$	17.8950	0.743048
$581$	0.352338	0.0146175
$582$	−38.2959	−1.58742
$583$	−15.4558	−0.640113
$584$	−4.92945	−0.203982
$585$	0.416595	0.0172241
$586$	−2.52496	−0.104305
$587$	23.7641	0.980851	0.490426	−	0.871483i	$-0.336841\pi$
$587$	23.7641	0.980851	0.490426	+	0.871483i	$0.336841\pi$
$588$	−5.07682	−0.209365
$589$	0.584963	0.0241030
$590$	−6.30632	−0.259627
$591$	−2.42085	−0.0995804
$592$	107.628	4.42347
$593$	−23.7240	−0.974229	−0.487114	−	0.873338i	$-0.661951\pi$
$593$	−23.7240	−0.974229	−0.487114	+	0.873338i	$0.661951\pi$
$594$	4.96027	0.203522
$595$	−2.00000	−0.0819920
$596$	82.6856	3.38694
$597$	2.53328	0.103680
$598$	−1.10824	−0.0453192
$599$	−21.7947	−0.890507	−0.445254	−	0.895405i	$-0.646886\pi$
$599$	−21.7947	−0.890507	−0.445254	+	0.895405i	$0.646886\pi$
$600$	8.18506	0.334154
$601$	18.4009	0.750590	0.375295	−	0.926905i	$-0.377541\pi$
$601$	18.4009	0.750590	0.375295	+	0.926905i	$0.377541\pi$
$602$	2.71036	0.110466
$603$	−2.89602	−0.117935
$604$	−29.0622	−1.18252
$605$	7.52326	0.305864
$606$	12.1913	0.495239
$607$	43.6391	1.77126	0.885629	−	0.464394i	$-0.153728\pi$
$607$	43.6391	1.77126	0.885629	+	0.464394i	$0.153728\pi$
$608$	−3.26916	−0.132582
$609$	−3.52483	−0.142833
$610$	6.32720	0.256181
$611$	1.43642	0.0581114
$612$	−10.1536	−0.410437
$613$	−31.6989	−1.28031	−0.640153	−	0.768247i	$-0.721129\pi$
$613$	−31.6989	−1.28031	−0.640153	+	0.768247i	$0.721129\pi$
$614$	9.78006	0.394691
$615$	−0.429163	−0.0173055
$616$	15.2619	0.614919
$617$	−6.98467	−0.281192	−0.140596	−	0.990067i	$-0.544902\pi$
$617$	−6.98467	−0.281192	−0.140596	+	0.990067i	$0.544902\pi$
$618$	2.81447	0.113215
$619$	−4.38198	−0.176126	−0.0880632	−	0.996115i	$-0.528068\pi$
$619$	−4.38198	−0.176126	−0.0880632	+	0.996115i	$0.528068\pi$
$620$	−13.2112	−0.530573
$621$	1.00000	0.0401286
$622$	9.51056	0.381339
$623$	−10.3888	−0.416220
$624$	4.84104	0.193797
$625$	1.00000	0.0400000
$626$	88.6148	3.54176
$627$	−0.419146	−0.0167391
$628$	−25.9974	−1.03741
$629$	−18.5238	−0.738591
$630$	−2.66023	−0.105986
$631$	−32.1227	−1.27879	−0.639393	−	0.768880i	$-0.720814\pi$
$631$	−32.1227	−1.27879	−0.639393	+	0.768880i	$0.720814\pi$
$632$	−94.3957	−3.75486
$633$	−8.73765	−0.347290
$634$	75.5033	2.99862
$635$	−17.6073	−0.698726
$636$	42.0820	1.66866
$637$	−0.416595	−0.0165061
$638$	17.4841	0.692203
$639$	−8.39571	−0.332129
$640$	12.0063	0.474590
$641$	−44.9926	−1.77710	−0.888550	−	0.458779i	$-0.848287\pi$
$641$	−44.9926	−1.77710	−0.888550	+	0.458779i	$0.848287\pi$
$642$	43.1547	1.70318
$643$	11.3099	0.446020	0.223010	−	0.974816i	$-0.428412\pi$
$643$	11.3099	0.446020	0.223010	+	0.974816i	$0.428412\pi$
$644$	5.07682	0.200055
$645$	1.01884	0.0401170
$646$	1.19599	0.0470557
$647$	−46.4905	−1.82773	−0.913865	−	0.406017i	$-0.866917\pi$
$647$	−46.4905	−1.82773	−0.913865	+	0.406017i	$0.866917\pi$
$648$	−8.18506	−0.321540
$649$	−4.42021	−0.173508
$650$	1.10824	0.0434687
$651$	2.60225	0.101990
$652$	−38.9401	−1.52501
$653$	−1.62313	−0.0635179	−0.0317589	−	0.999496i	$-0.510111\pi$
$653$	−1.62313	−0.0635179	−0.0317589	+	0.999496i	$0.510111\pi$
$654$	1.41411	0.0552960
$655$	6.96499	0.272145
$656$	−4.98709	−0.194713
$657$	0.602250	0.0234960
$658$	−9.17249	−0.357581
$659$	−11.8364	−0.461080	−0.230540	−	0.973063i	$-0.574049\pi$
$659$	−11.8364	−0.461080	−0.230540	+	0.973063i	$0.574049\pi$
$660$	9.46626	0.368474
$661$	16.1992	0.630076	0.315038	−	0.949079i	$-0.397983\pi$
$661$	16.1992	0.630076	0.315038	+	0.949079i	$0.397983\pi$
$662$	−28.9650	−1.12576
$663$	−0.833189	−0.0323584
$664$	2.88391	0.111917
$665$	0.224791	0.00871703
$666$	−24.6388	−0.954732
$667$	3.52483	0.136482
$668$	20.0076	0.774116
$669$	12.7637	0.493475
$670$	−7.70407	−0.297634
$671$	4.43485	0.171205
$672$	−14.5431	−0.561012
$673$	7.04495	0.271563	0.135781	−	0.990739i	$-0.456646\pi$
$673$	7.04495	0.271563	0.135781	+	0.990739i	$0.456646\pi$
$674$	0.782676	0.0301475
$675$	−1.00000	−0.0384900
$676$	−65.1176	−2.50452
$677$	−29.4333	−1.13121	−0.565606	−	0.824675i	$-0.691358\pi$
$677$	−29.4333	−1.13121	−0.565606	+	0.824675i	$0.691358\pi$
$678$	21.0580	0.808726
$679$	14.3957	0.552457
$680$	−16.3701	−0.627766
$681$	6.33362	0.242705
$682$	−12.9079	−0.494268
$683$	48.2832	1.84751	0.923753	−	0.382990i	$-0.125106\pi$
$683$	48.2832	1.84751	0.923753	+	0.382990i	$0.125106\pi$
$684$	1.14123	0.0436359
$685$	−7.93311	−0.303109
$686$	2.66023	0.101568
$687$	27.3738	1.04438
$688$	11.8395	0.451376
$689$	3.45317	0.131555
$690$	2.66023	0.101273
$691$	−29.5982	−1.12597	−0.562984	−	0.826468i	$-0.690347\pi$
$691$	−29.5982	−1.12597	−0.562984	+	0.826468i	$0.690347\pi$
$692$	−24.9911	−0.950019
$693$	−1.86460	−0.0708304
$694$	−0.108639	−0.00412387
$695$	6.90955	0.262094
$696$	−28.8510	−1.09359
$697$	0.858327	0.0325114
$698$	10.0885	0.381857
$699$	−1.19121	−0.0450558
$700$	−5.07682	−0.191886
$701$	−7.73678	−0.292214	−0.146107	−	0.989269i	$-0.546674\pi$
$701$	−7.73678	−0.292214	−0.146107	+	0.989269i	$0.546674\pi$
$702$	−1.10824	−0.0418277
$703$	2.08199	0.0785238
$704$	28.8024	1.08553
$705$	−3.44801	−0.129859
$706$	30.1706	1.13549
$707$	−4.58281	−0.172354
$708$	12.0351	0.452306
$709$	10.3305	0.387969	0.193985	−	0.981005i	$-0.437859\pi$
$709$	10.3305	0.387969	0.193985	+	0.981005i	$0.437859\pi$
$710$	−22.3345	−0.838200
$711$	11.5327	0.432509
$712$	−85.0333	−3.18676
$713$	−2.60225	−0.0974550
$714$	5.32046	0.199113
$715$	0.776784	0.0290501
$716$	−88.4555	−3.30574
$717$	−4.29520	−0.160407
$718$	−13.7475	−0.513053
$719$	12.3655	0.461157	0.230579	−	0.973054i	$-0.425938\pi$
$719$	12.3655	0.461157	0.230579	+	0.973054i	$0.425938\pi$
$720$	−11.6205	−0.433071
$721$	−1.05798	−0.0394013
$722$	50.4100	1.87606
$723$	8.85361	0.329269
$724$	−109.928	−4.08544
$725$	−3.52483	−0.130909
$726$	−20.0136	−0.742774
$727$	15.3341	0.568712	0.284356	−	0.958719i	$-0.408220\pi$
$727$	15.3341	0.568712	0.284356	+	0.958719i	$0.408220\pi$
$728$	−3.40985	−0.126378
$729$	1.00000	0.0370370
$730$	1.60212	0.0592972
$731$	−2.03769	−0.0753666
$732$	−12.0749	−0.446302
$733$	11.9578	0.441671	0.220835	−	0.975311i	$-0.429122\pi$
$733$	11.9578	0.441671	0.220835	+	0.975311i	$0.429122\pi$
$734$	30.8814	1.13985
$735$	1.00000	0.0368856
$736$	14.5431	0.536065
$737$	−5.39992	−0.198909
$738$	1.14167	0.0420256
$739$	1.84950	0.0680349	0.0340175	−	0.999421i	$-0.489170\pi$
$739$	1.84950	0.0680349	0.0340175	+	0.999421i	$0.489170\pi$
$740$	−47.0210	−1.72853
$741$	0.0936468	0.00344020
$742$	−22.0508	−0.809509
$743$	33.7375	1.23771	0.618854	−	0.785506i	$-0.287597\pi$
$743$	33.7375	1.23771	0.618854	+	0.785506i	$0.287597\pi$
$744$	21.2996	0.780880
$745$	−16.2869	−0.596705
$746$	−55.0356	−2.01500
$747$	−0.352338	−0.0128914
$748$	−18.9325	−0.692241
$749$	−16.2222	−0.592745
$750$	−2.66023	−0.0971379
$751$	−19.1171	−0.697592	−0.348796	−	0.937199i	$-0.613409\pi$
$751$	−19.1171	−0.697592	−0.348796	+	0.937199i	$0.613409\pi$
$752$	−40.0676	−1.46111
$753$	−7.04967	−0.256904
$754$	−3.90635	−0.142261
$755$	5.72449	0.208335
$756$	5.07682	0.184642
$757$	−6.54269	−0.237798	−0.118899	−	0.992906i	$-0.537937\pi$
$757$	−6.54269	−0.237798	−0.118899	+	0.992906i	$0.537937\pi$
$758$	37.1877	1.35072
$759$	1.86460	0.0676808
$760$	1.83993	0.0667413
$761$	−17.9056	−0.649076	−0.324538	−	0.945873i	$-0.605209\pi$
$761$	−17.9056	−0.649076	−0.324538	+	0.945873i	$0.605209\pi$
$762$	46.8396	1.69682
$763$	−0.531573	−0.0192442
$764$	−92.6878	−3.35333
$765$	2.00000	0.0723102
$766$	−14.5126	−0.524362
$767$	0.987576	0.0356593
$768$	−1.04556	−0.0377283
$769$	−51.0698	−1.84162	−0.920812	−	0.390007i	$-0.872473\pi$
$769$	−51.0698	−1.84162	−0.920812	+	0.390007i	$0.872473\pi$
$770$	−4.96027	−0.178756
$771$	−13.3200	−0.479708
$772$	−53.2181	−1.91536
$773$	−18.2133	−0.655088	−0.327544	−	0.944836i	$-0.606221\pi$
$773$	−18.2133	−0.655088	−0.327544	+	0.944836i	$0.606221\pi$
$774$	−2.71036	−0.0974219
$775$	2.60225	0.0934756
$776$	117.830	4.22984
$777$	9.26189	0.332268
$778$	17.0905	0.612724
$779$	−0.0964721	−0.00345647
$780$	−2.11498	−0.0757284
$781$	−15.6547	−0.560168
$782$	−5.32046	−0.190259
$783$	3.52483	0.125967
$784$	11.6205	0.415018
$785$	5.12081	0.182769
$786$	−18.5285	−0.660889
$787$	47.7103	1.70069	0.850344	−	0.526228i	$-0.176394\pi$
$787$	47.7103	1.70069	0.850344	+	0.526228i	$0.176394\pi$
$788$	12.2902	0.437821
$789$	25.0815	0.892923
$790$	30.6796	1.09153
$791$	−7.91584	−0.281455
$792$	−15.2619	−0.542308
$793$	−0.990846	−0.0351860
$794$	14.8630	0.527469
$795$	−8.28905	−0.293982
$796$	−12.8610	−0.455846
$797$	23.7271	0.840457	0.420228	−	0.907418i	$-0.361950\pi$
$797$	23.7271	0.840457	0.420228	+	0.907418i	$0.361950\pi$
$798$	−0.597996	−0.0211688
$799$	6.89602	0.243963
$800$	−14.5431	−0.514176
$801$	10.3888	0.367072
$802$	−25.8100	−0.911383
$803$	1.12296	0.0396283
$804$	14.7026	0.518519
$805$	−1.00000	−0.0352454
$806$	2.88391	0.101581
$807$	9.84542	0.346575
$808$	−37.5106	−1.31962
$809$	−30.4767	−1.07150	−0.535751	−	0.844376i	$-0.679971\pi$
$809$	−30.4767	−1.07150	−0.535751	+	0.844376i	$0.679971\pi$
$810$	2.66023	0.0934710
$811$	−27.0052	−0.948279	−0.474140	−	0.880450i	$-0.657241\pi$
$811$	−27.0052	−0.948279	−0.474140	+	0.880450i	$0.657241\pi$
$812$	17.8950	0.627990
$813$	−19.4882	−0.683483
$814$	−45.9415	−1.61025
$815$	7.67017	0.268674
$816$	23.2410	0.813598
$817$	0.229027	0.00801265
$818$	83.2352	2.91025
$819$	0.416595	0.0145570
$820$	2.17879	0.0760865
$821$	43.7305	1.52620	0.763102	−	0.646278i	$-0.223675\pi$
$821$	43.7305	1.52620	0.763102	+	0.646278i	$0.223675\pi$
$822$	21.1039	0.736083
$823$	21.2142	0.739479	0.369740	−	0.929135i	$-0.379447\pi$
$823$	21.2142	0.739479	0.369740	+	0.929135i	$0.379447\pi$
$824$	−8.65964	−0.301673
$825$	−1.86460	−0.0649171
$826$	−6.30632	−0.219425
$827$	−16.9124	−0.588103	−0.294051	−	0.955790i	$-0.595004\pi$
$827$	−16.9124	−0.588103	−0.294051	+	0.955790i	$0.595004\pi$
$828$	−5.07682	−0.176432
$829$	7.76003	0.269517	0.134758	−	0.990878i	$-0.456974\pi$
$829$	7.76003	0.269517	0.134758	+	0.990878i	$0.456974\pi$
$830$	−0.937301	−0.0325342
$831$	0.761601	0.0264196
$832$	−6.43512	−0.223098
$833$	−2.00000	−0.0692959
$834$	−18.3810	−0.636482
$835$	−3.94096	−0.136383
$836$	2.12793	0.0735961
$837$	−2.60225	−0.0899469
$838$	22.3692	0.772731
$839$	−3.84975	−0.132908	−0.0664541	−	0.997789i	$-0.521169\pi$
$839$	−3.84975	−0.132908	−0.0664541	+	0.997789i	$0.521169\pi$
$840$	8.18506	0.282412
$841$	−16.5756	−0.571571
$842$	−51.7645	−1.78392
$843$	18.2065	0.627064
$844$	44.3595	1.52692
$845$	12.8264	0.441243
$846$	9.17249	0.315357
$847$	7.52326	0.258502
$848$	−96.3229	−3.30774
$849$	30.6279	1.05115
$850$	5.32046	0.182490
$851$	−9.26189	−0.317493
$852$	42.6236	1.46026
$853$	−28.3376	−0.970261	−0.485130	−	0.874442i	$-0.661228\pi$
$853$	−28.3376	−0.970261	−0.485130	+	0.874442i	$0.661228\pi$
$854$	6.32720	0.216512
$855$	−0.224791	−0.00768770
$856$	−132.779	−4.53831
$857$	13.0084	0.444357	0.222179	−	0.975006i	$-0.428683\pi$
$857$	13.0084	0.444357	0.222179	+	0.975006i	$0.428683\pi$
$858$	−2.06642	−0.0705465
$859$	5.42092	0.184959	0.0924797	−	0.995715i	$-0.470521\pi$
$859$	5.42092	0.184959	0.0924797	+	0.995715i	$0.470521\pi$
$860$	−5.17249	−0.176381
$861$	−0.429163	−0.0146258
$862$	−64.5368	−2.19813
$863$	47.8170	1.62771	0.813855	−	0.581069i	$-0.197365\pi$
$863$	47.8170	1.62771	0.813855	+	0.581069i	$0.197365\pi$
$864$	14.5431	0.494766
$865$	4.92258	0.167373
$866$	102.234	3.47407
$867$	13.0000	0.441503
$868$	−13.2112	−0.448416
$869$	21.5039	0.729469
$870$	9.37687	0.317906
$871$	1.20646	0.0408795
$872$	−4.35096	−0.147342
$873$	−14.3957	−0.487221
$874$	0.597996	0.0202275
$875$	1.00000	0.0338062
$876$	−3.05752	−0.103304
$877$	6.89130	0.232703	0.116351	−	0.993208i	$-0.462880\pi$
$877$	6.89130	0.232703	0.116351	+	0.993208i	$0.462880\pi$
$878$	12.7963	0.431854
$879$	−0.949151	−0.0320141
$880$	−21.6676	−0.730415
$881$	−0.100260	−0.00337786	−0.00168893	−	0.999999i	$-0.500538\pi$
$881$	−0.100260	−0.00337786	−0.00168893	+	0.999999i	$0.500538\pi$
$882$	−2.66023	−0.0895746
$883$	15.8639	0.533862	0.266931	−	0.963716i	$-0.413990\pi$
$883$	15.8639	0.533862	0.266931	+	0.963716i	$0.413990\pi$
$884$	4.22996	0.142269
$885$	−2.37059	−0.0796865
$886$	40.0260	1.34470
$887$	−16.8012	−0.564129	−0.282064	−	0.959396i	$-0.591019\pi$
$887$	−16.8012	−0.564129	−0.282064	+	0.959396i	$0.591019\pi$
$888$	75.8091	2.54399
$889$	−17.6073	−0.590531
$890$	27.6367	0.926385
$891$	1.86460	0.0624665
$892$	−64.7993	−2.16964
$893$	−0.775082	−0.0259371
$894$	43.3269	1.44907
$895$	17.4234	0.582400
$896$	12.0063	0.401102
$897$	−0.416595	−0.0139097
$898$	95.0327	3.17128
$899$	−9.17249	−0.305920
$900$	5.07682	0.169227
$901$	16.5781	0.552296
$902$	2.12877	0.0708802
$903$	1.01884	0.0339050
$904$	−64.7917	−2.15494
$905$	21.6529	0.719767
$906$	−15.2285	−0.505931
$907$	−31.3038	−1.03943	−0.519713	−	0.854341i	$-0.673961\pi$
$907$	−31.3038	−1.03943	−0.519713	+	0.854341i	$0.673961\pi$
$908$	−32.1547	−1.06709
$909$	4.58281	0.152002
$910$	1.10824	0.0367377
$911$	48.5775	1.60945	0.804723	−	0.593651i	$-0.202314\pi$
$911$	48.5775	1.60945	0.804723	+	0.593651i	$0.202314\pi$
$912$	−2.61219	−0.0864981
$913$	−0.656971	−0.0217426
$914$	−44.0133	−1.45583
$915$	2.37844	0.0786288
$916$	−138.972	−4.59176
$917$	6.96499	0.230004
$918$	−5.32046	−0.175601
$919$	20.5644	0.678356	0.339178	−	0.940722i	$-0.389851\pi$
$919$	20.5644	0.678356	0.339178	+	0.940722i	$0.389851\pi$
$920$	−8.18506	−0.269854
$921$	3.67640	0.121141
$922$	66.6119	2.19375
$923$	3.49761	0.115125
$924$	9.46626	0.311417
$925$	9.26189	0.304529
$926$	−38.9110	−1.27869
$927$	1.05798	0.0347486
$928$	51.2619	1.68276
$929$	33.7076	1.10591	0.552956	−	0.833211i	$-0.313500\pi$
$929$	33.7076	1.10591	0.552956	+	0.833211i	$0.313500\pi$
$930$	−6.92258	−0.227000
$931$	0.224791	0.00736723
$932$	6.04757	0.198095
$933$	3.57509	0.117043
$934$	−28.4044	−0.929421
$935$	3.72920	0.121958
$936$	3.40985	0.111455
$937$	−15.1419	−0.494663	−0.247332	−	0.968931i	$-0.579554\pi$
$937$	−15.1419	−0.494663	−0.247332	+	0.968931i	$0.579554\pi$
$938$	−7.70407	−0.251547
$939$	33.3110	1.08706
$940$	17.5049	0.570948
$941$	56.4684	1.84082	0.920409	−	0.390958i	$-0.127856\pi$
$941$	56.4684	1.84082	0.920409	+	0.390958i	$0.127856\pi$
$942$	−13.6225	−0.443846
$943$	0.429163	0.0139755
$944$	−27.5475	−0.896593
$945$	−1.00000	−0.0325300
$946$	−5.05374	−0.164311
$947$	1.30431	0.0423844	0.0211922	−	0.999775i	$-0.493254\pi$
$947$	1.30431	0.0423844	0.0211922	+	0.999775i	$0.493254\pi$
$948$	−58.5494	−1.90160
$949$	−0.250894	−0.00814436
$950$	−0.597996	−0.0194016
$951$	28.3822	0.920357
$952$	−16.3701	−0.530559
$953$	4.64916	0.150601	0.0753005	−	0.997161i	$-0.476008\pi$
$953$	4.64916	0.150601	0.0753005	+	0.997161i	$0.476008\pi$
$954$	22.0508	0.713920
$955$	18.2570	0.590784
$956$	21.8060	0.705255
$957$	6.57241	0.212456
$958$	31.5115	1.01809
$959$	−7.93311	−0.256174
$960$	15.4470	0.498548
$961$	−24.2283	−0.781558
$962$	10.2644	0.330937
$963$	16.2222	0.522752
$964$	−44.9482	−1.44768
$965$	10.4826	0.337445
$966$	2.66023	0.0855915
$967$	−32.6303	−1.04932	−0.524660	−	0.851312i	$-0.675807\pi$
$967$	−32.6303	−1.04932	−0.524660	+	0.851312i	$0.675807\pi$
$968$	61.5783	1.97920
$969$	0.449582	0.0144427
$970$	−38.2959	−1.22961
$971$	17.9290	0.575371	0.287685	−	0.957725i	$-0.407114\pi$
$971$	17.9290	0.575371	0.287685	+	0.957725i	$0.407114\pi$
$972$	−5.07682	−0.162839
$973$	6.90955	0.221510
$974$	−85.7948	−2.74904
$975$	0.416595	0.0133417
$976$	27.6387	0.884693
$977$	−54.8882	−1.75603	−0.878015	−	0.478634i	$-0.841132\pi$
$977$	−54.8882	−1.75603	−0.878015	+	0.478634i	$0.841132\pi$
$978$	−20.4044	−0.652461
$979$	19.3711	0.619102
$980$	−5.07682	−0.162173
$981$	0.531573	0.0169718
$982$	−77.1665	−2.46248
$983$	−14.6386	−0.466898	−0.233449	−	0.972369i	$-0.575001\pi$
$983$	−14.6386	−0.466898	−0.233449	+	0.972369i	$0.575001\pi$
$984$	−3.51273	−0.111982
$985$	−2.42085	−0.0771346
$986$	−18.7537	−0.597241
$987$	−3.44801	−0.109751
$988$	−0.475429	−0.0151254
$989$	−1.01884	−0.0323974
$990$	4.96027	0.157648
$991$	−42.1242	−1.33812	−0.669059	−	0.743209i	$-0.733303\pi$
$991$	−42.1242	−1.33812	−0.669059	+	0.743209i	$0.733303\pi$
$992$	−37.8447	−1.20157
$993$	−10.8881	−0.345525
$994$	−22.3345	−0.708408
$995$	2.53328	0.0803102
$996$	1.78876	0.0566790
$997$	50.7909	1.60856	0.804282	−	0.594248i	$-0.202550\pi$
$997$	50.7909	1.60856	0.804282	+	0.594248i	$0.202550\pi$
$998$	−87.0966	−2.75699
$999$	−9.26189	−0.293033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.2.a.o.1.1	✓	5
3.2	odd	2		7245.2.a.bg.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.o.1.1	✓	5	1.1	even	1	trivial
7245.2.a.bg.1.5		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66023 q^{2} -1.00000 q^{3} +5.07682 q^{4} -1.00000 q^{5} +2.66023 q^{6} -1.00000 q^{7} -8.18506 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.66023 q^{2} -1.00000 q^{3} +5.07682 q^{4} -1.00000 q^{5} +2.66023 q^{6} -1.00000 q^{7} -8.18506 q^{8} +1.00000 q^{9} +2.66023 q^{10} +1.86460 q^{11} -5.07682 q^{12} -0.416595 q^{13} +2.66023 q^{14} +1.00000 q^{15} +11.6205 q^{16} -2.00000 q^{17} -2.66023 q^{18} +0.224791 q^{19} -5.07682 q^{20} +1.00000 q^{21} -4.96027 q^{22} -1.00000 q^{23} +8.18506 q^{24} +1.00000 q^{25} +1.10824 q^{26} -1.00000 q^{27} -5.07682 q^{28} -3.52483 q^{29} -2.66023 q^{30} +2.60225 q^{31} -14.5431 q^{32} -1.86460 q^{33} +5.32046 q^{34} +1.00000 q^{35} +5.07682 q^{36} +9.26189 q^{37} -0.597996 q^{38} +0.416595 q^{39} +8.18506 q^{40} -0.429163 q^{41} -2.66023 q^{42} +1.01884 q^{43} +9.46626 q^{44} -1.00000 q^{45} +2.66023 q^{46} -3.44801 q^{47} -11.6205 q^{48} +1.00000 q^{49} -2.66023 q^{50} +2.00000 q^{51} -2.11498 q^{52} -8.28905 q^{53} +2.66023 q^{54} -1.86460 q^{55} +8.18506 q^{56} -0.224791 q^{57} +9.37687 q^{58} -2.37059 q^{59} +5.07682 q^{60} +2.37844 q^{61} -6.92258 q^{62} -1.00000 q^{63} +15.4470 q^{64} +0.416595 q^{65} +4.96027 q^{66} -2.89602 q^{67} -10.1536 q^{68} +1.00000 q^{69} -2.66023 q^{70} -8.39571 q^{71} -8.18506 q^{72} +0.602250 q^{73} -24.6388 q^{74} -1.00000 q^{75} +1.14123 q^{76} -1.86460 q^{77} -1.10824 q^{78} +11.5327 q^{79} -11.6205 q^{80} +1.00000 q^{81} +1.14167 q^{82} -0.352338 q^{83} +5.07682 q^{84} +2.00000 q^{85} -2.71036 q^{86} +3.52483 q^{87} -15.2619 q^{88} +10.3888 q^{89} +2.66023 q^{90} +0.416595 q^{91} -5.07682 q^{92} -2.60225 q^{93} +9.17249 q^{94} -0.224791 q^{95} +14.5431 q^{96} -14.3957 q^{97} -2.66023 q^{98} +1.86460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9} + q^{11} - 10 q^{12} + 5 q^{15} + 8 q^{16} - 10 q^{17} + 3 q^{19} - 10 q^{20} + 5 q^{21} + 12 q^{22} - 5 q^{23} + 6 q^{24} + 5 q^{25} - 14 q^{26} - 5 q^{27} - 10 q^{28} + 4 q^{29} + 2 q^{31} - 12 q^{32} - q^{33} + 5 q^{35} + 10 q^{36} - 4 q^{37} - 24 q^{38} + 6 q^{40} - 9 q^{41} - 8 q^{43} + 2 q^{44} - 5 q^{45} - 11 q^{47} - 8 q^{48} + 5 q^{49} + 10 q^{51} - 22 q^{52} - 19 q^{53} - q^{55} + 6 q^{56} - 3 q^{57} + 8 q^{58} + 5 q^{59} + 10 q^{60} - 17 q^{61} - 24 q^{62} - 5 q^{63} - 8 q^{64} - 12 q^{66} - 2 q^{67} - 20 q^{68} + 5 q^{69} + 10 q^{71} - 6 q^{72} - 8 q^{73} - 34 q^{74} - 5 q^{75} - 8 q^{76} - q^{77} + 14 q^{78} + 24 q^{79} - 8 q^{80} + 5 q^{81} - 8 q^{82} - 24 q^{83} + 10 q^{84} + 10 q^{85} - 10 q^{86} - 4 q^{87} - 26 q^{88} - 4 q^{89} - 10 q^{92} - 2 q^{93} + 2 q^{94} - 3 q^{95} + 12 q^{96} - 20 q^{97} + q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9 + q^11 - 10 * q^12 + 5 * q^15 + 8 * q^16 - 10 * q^17 + 3 * q^19 - 10 * q^20 + 5 * q^21 + 12 * q^22 - 5 * q^23 + 6 * q^24 + 5 * q^25 - 14 * q^26 - 5 * q^27 - 10 * q^28 + 4 * q^29 + 2 * q^31 - 12 * q^32 - q^33 + 5 * q^35 + 10 * q^36 - 4 * q^37 - 24 * q^38 + 6 * q^40 - 9 * q^41 - 8 * q^43 + 2 * q^44 - 5 * q^45 - 11 * q^47 - 8 * q^48 + 5 * q^49 + 10 * q^51 - 22 * q^52 - 19 * q^53 - q^55 + 6 * q^56 - 3 * q^57 + 8 * q^58 + 5 * q^59 + 10 * q^60 - 17 * q^61 - 24 * q^62 - 5 * q^63 - 8 * q^64 - 12 * q^66 - 2 * q^67 - 20 * q^68 + 5 * q^69 + 10 * q^71 - 6 * q^72 - 8 * q^73 - 34 * q^74 - 5 * q^75 - 8 * q^76 - q^77 + 14 * q^78 + 24 * q^79 - 8 * q^80 + 5 * q^81 - 8 * q^82 - 24 * q^83 + 10 * q^84 + 10 * q^85 - 10 * q^86 - 4 * q^87 - 26 * q^88 - 4 * q^89 - 10 * q^92 - 2 * q^93 + 2 * q^94 - 3 * q^95 + 12 * q^96 - 20 * q^97 + q^99

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