LMFDB - Embedded newform 5070.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.131472.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 19x^{2} + 20x + 52 $ x^4 - 2x^3 - 19x^2 + 20*x + 52
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.32258$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.32258 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.32258 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.34541 q^{11} +1.00000 q^{12} +2.32258 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.02283 q^{19} +1.00000 q^{20} -2.32258 q^{21} +5.34541 q^{22} -4.93593 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -2.32258 q^{28} +4.29078 q^{29} -1.00000 q^{30} -3.47183 q^{31} -1.00000 q^{32} -5.34541 q^{33} -4.00000 q^{34} -2.32258 q^{35} +1.00000 q^{36} +3.14925 q^{37} -4.02283 q^{38} -1.00000 q^{40} +2.64516 q^{41} +2.32258 q^{42} +12.2508 q^{43} -5.34541 q^{44} +1.00000 q^{45} +4.93593 q^{46} -1.81894 q^{47} +1.00000 q^{48} -1.60562 q^{49} -1.00000 q^{50} +4.00000 q^{51} -5.48693 q^{53} -1.00000 q^{54} -5.34541 q^{55} +2.32258 q^{56} +4.02283 q^{57} -4.29078 q^{58} +6.78668 q^{59} +1.00000 q^{60} +0.535898 q^{61} +3.47183 q^{62} -2.32258 q^{63} +1.00000 q^{64} +5.34541 q^{66} +4.10926 q^{67} +4.00000 q^{68} -4.93593 q^{69} +2.32258 q^{70} -15.8719 q^{71} -1.00000 q^{72} -13.5734 q^{73} -3.14925 q^{74} +1.00000 q^{75} +4.02283 q^{76} +12.4151 q^{77} -7.96774 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.64516 q^{82} +11.3360 q^{83} -2.32258 q^{84} +4.00000 q^{85} -12.2508 q^{86} +4.29078 q^{87} +5.34541 q^{88} +1.73978 q^{89} -1.00000 q^{90} -4.93593 q^{92} -3.47183 q^{93} +1.81894 q^{94} +4.02283 q^{95} -1.00000 q^{96} +16.1093 q^{97} +1.60562 q^{98} -5.34541 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 16 q^{17} - 4 q^{18} + 4 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27} - 2 q^{28} + 8 q^{29} - 4 q^{30} - 4 q^{31} - 4 q^{32} + 2 q^{33} - 16 q^{34} - 2 q^{35} + 4 q^{36} + 10 q^{37} - 4 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 2 q^{44} + 4 q^{45} - 4 q^{46} - 8 q^{47} + 4 q^{48} + 14 q^{49} - 4 q^{50} + 16 q^{51} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{56} - 8 q^{58} + 6 q^{59} + 4 q^{60} + 16 q^{61} + 4 q^{62} - 2 q^{63} + 4 q^{64} - 2 q^{66} - 12 q^{67} + 16 q^{68} + 4 q^{69} + 2 q^{70} - 16 q^{71} - 4 q^{72} - 12 q^{73} - 10 q^{74} + 4 q^{75} - 8 q^{77} - 10 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{82} - 16 q^{83} - 2 q^{84} + 16 q^{85} - 14 q^{86} + 8 q^{87} - 2 q^{88} + 4 q^{89} - 4 q^{90} + 4 q^{92} - 4 q^{93} + 8 q^{94} - 4 q^{96} + 36 q^{97} - 14 q^{98} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 2 * q^14 + 4 * q^15 + 4 * q^16 + 16 * q^17 - 4 * q^18 + 4 * q^20 - 2 * q^21 - 2 * q^22 + 4 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 - 2 * q^28 + 8 * q^29 - 4 * q^30 - 4 * q^31 - 4 * q^32 + 2 * q^33 - 16 * q^34 - 2 * q^35 + 4 * q^36 + 10 * q^37 - 4 * q^40 - 4 * q^41 + 2 * q^42 + 14 * q^43 + 2 * q^44 + 4 * q^45 - 4 * q^46 - 8 * q^47 + 4 * q^48 + 14 * q^49 - 4 * q^50 + 16 * q^51 + 8 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^56 - 8 * q^58 + 6 * q^59 + 4 * q^60 + 16 * q^61 + 4 * q^62 - 2 * q^63 + 4 * q^64 - 2 * q^66 - 12 * q^67 + 16 * q^68 + 4 * q^69 + 2 * q^70 - 16 * q^71 - 4 * q^72 - 12 * q^73 - 10 * q^74 + 4 * q^75 - 8 * q^77 - 10 * q^79 + 4 * q^80 + 4 * q^81 + 4 * q^82 - 16 * q^83 - 2 * q^84 + 16 * q^85 - 14 * q^86 + 8 * q^87 - 2 * q^88 + 4 * q^89 - 4 * q^90 + 4 * q^92 - 4 * q^93 + 8 * q^94 - 4 * q^96 + 36 * q^97 - 14 * q^98 + 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−2.32258	−0.877853	−0.438926	−	0.898523i	$-0.644641\pi$
$7$	−2.32258	−0.877853	−0.438926	+	0.898523i	$0.644641\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−5.34541	−1.61170	−0.805850	−	0.592119i	$-0.798291\pi$
$11$	−5.34541	−1.61170	−0.805850	+	0.592119i	$0.798291\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	2.32258	0.620736
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	4.02283	0.922900	0.461450	−	0.887166i	$-0.347329\pi$
$19$	4.02283	0.922900	0.461450	+	0.887166i	$0.347329\pi$
$20$	1.00000	0.223607
$21$	−2.32258	−0.506828
$22$	5.34541	1.13964
$23$	−4.93593	−1.02921	−0.514607	−	0.857426i	$-0.672062\pi$
$23$	−4.93593	−1.02921	−0.514607	+	0.857426i	$0.672062\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−2.32258	−0.438926
$29$	4.29078	0.796777	0.398388	−	0.917217i	$-0.369570\pi$
$29$	4.29078	0.796777	0.398388	+	0.917217i	$0.369570\pi$
$30$	−1.00000	−0.182574
$31$	−3.47183	−0.623560	−0.311780	−	0.950154i	$-0.600925\pi$
$31$	−3.47183	−0.623560	−0.311780	+	0.950154i	$0.600925\pi$
$32$	−1.00000	−0.176777
$33$	−5.34541	−0.930516
$34$	−4.00000	−0.685994
$35$	−2.32258	−0.392588
$36$	1.00000	0.166667
$37$	3.14925	0.517734	0.258867	−	0.965913i	$-0.416651\pi$
$37$	3.14925	0.517734	0.258867	+	0.965913i	$0.416651\pi$
$38$	−4.02283	−0.652589
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.64516	0.413104	0.206552	−	0.978436i	$-0.433776\pi$
$41$	2.64516	0.413104	0.206552	+	0.978436i	$0.433776\pi$
$42$	2.32258	0.358382
$43$	12.2508	1.86823	0.934113	−	0.356976i	$-0.116192\pi$
$43$	12.2508	1.86823	0.934113	+	0.356976i	$0.116192\pi$
$44$	−5.34541	−0.805850
$45$	1.00000	0.149071
$46$	4.93593	0.727764
$47$	−1.81894	−0.265320	−0.132660	−	0.991162i	$-0.542352\pi$
$47$	−1.81894	−0.265320	−0.132660	+	0.991162i	$0.542352\pi$
$48$	1.00000	0.144338
$49$	−1.60562	−0.229375
$50$	−1.00000	−0.141421
$51$	4.00000	0.560112
$52$	0	0
$53$	−5.48693	−0.753687	−0.376844	−	0.926277i	$-0.622991\pi$
$53$	−5.48693	−0.753687	−0.376844	+	0.926277i	$0.622991\pi$
$54$	−1.00000	−0.136083
$55$	−5.34541	−0.720774
$56$	2.32258	0.310368
$57$	4.02283	0.532836
$58$	−4.29078	−0.563406
$59$	6.78668	0.883551	0.441775	−	0.897126i	$-0.354349\pi$
$59$	6.78668	0.883551	0.441775	+	0.897126i	$0.354349\pi$
$60$	1.00000	0.129099
$61$	0.535898	0.0686148	0.0343074	−	0.999411i	$-0.489077\pi$
$61$	0.535898	0.0686148	0.0343074	+	0.999411i	$0.489077\pi$
$62$	3.47183	0.440923
$63$	−2.32258	−0.292618
$64$	1.00000	0.125000
$65$	0	0
$66$	5.34541	0.657974
$67$	4.10926	0.502026	0.251013	−	0.967984i	$-0.419236\pi$
$67$	4.10926	0.502026	0.251013	+	0.967984i	$0.419236\pi$
$68$	4.00000	0.485071
$69$	−4.93593	−0.594217
$70$	2.32258	0.277601
$71$	−15.8719	−1.88364	−0.941822	−	0.336112i	$-0.890888\pi$
$71$	−15.8719	−1.88364	−0.941822	+	0.336112i	$0.890888\pi$
$72$	−1.00000	−0.117851
$73$	−13.5734	−1.58864	−0.794321	−	0.607498i	$-0.792173\pi$
$73$	−13.5734	−1.58864	−0.794321	+	0.607498i	$0.792173\pi$
$74$	−3.14925	−0.366093
$75$	1.00000	0.115470
$76$	4.02283	0.461450
$77$	12.4151	1.41484
$78$	0	0
$79$	−7.96774	−0.896441	−0.448220	−	0.893923i	$-0.647942\pi$
$79$	−7.96774	−0.896441	−0.448220	+	0.893923i	$0.647942\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.64516	−0.292109
$83$	11.3360	1.24428	0.622142	−	0.782904i	$-0.286263\pi$
$83$	11.3360	1.24428	0.622142	+	0.782904i	$0.286263\pi$
$84$	−2.32258	−0.253414
$85$	4.00000	0.433861
$86$	−12.2508	−1.32104
$87$	4.29078	0.460019
$88$	5.34541	0.569822
$89$	1.73978	0.184417	0.0922083	−	0.995740i	$-0.470607\pi$
$89$	1.73978	0.184417	0.0922083	+	0.995740i	$0.470607\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.93593	−0.514607
$93$	−3.47183	−0.360012
$94$	1.81894	0.187610
$95$	4.02283	0.412733
$96$	−1.00000	−0.102062
$97$	16.1093	1.63565	0.817824	−	0.575469i	$-0.195180\pi$
$97$	16.1093	1.63565	0.817824	+	0.575469i	$0.195180\pi$
$98$	1.60562	0.162192
$99$	−5.34541	−0.537233
$100$	1.00000	0.100000
$101$	12.6908	1.26278	0.631391	−	0.775464i	$-0.282484\pi$
$101$	12.6908	1.26278	0.631391	+	0.775464i	$0.282484\pi$
$102$	−4.00000	−0.396059
$103$	4.79612	0.472575	0.236288	−	0.971683i	$-0.424069\pi$
$103$	4.79612	0.472575	0.236288	+	0.971683i	$0.424069\pi$
$104$	0	0
$105$	−2.32258	−0.226661
$106$	5.48693	0.532938
$107$	3.07180	0.296962	0.148481	−	0.988915i	$-0.452562\pi$
$107$	3.07180	0.296962	0.148481	+	0.988915i	$0.452562\pi$
$108$	1.00000	0.0962250
$109$	6.69081	0.640864	0.320432	−	0.947272i	$-0.396172\pi$
$109$	6.69081	0.640864	0.320432	+	0.947272i	$0.396172\pi$
$110$	5.34541	0.509664
$111$	3.14925	0.298914
$112$	−2.32258	−0.219463
$113$	−7.10972	−0.668826	−0.334413	−	0.942427i	$-0.608538\pi$
$113$	−7.10972	−0.668826	−0.334413	+	0.942427i	$0.608538\pi$
$114$	−4.02283	−0.376772
$115$	−4.93593	−0.460278
$116$	4.29078	0.398388
$117$	0	0
$118$	−6.78668	−0.624765
$119$	−9.29032	−0.851642
$120$	−1.00000	−0.0912871
$121$	17.5734	1.59758
$122$	−0.535898	−0.0485180
$123$	2.64516	0.238506
$124$	−3.47183	−0.311780
$125$	1.00000	0.0894427
$126$	2.32258	0.206912
$127$	−2.13209	−0.189192	−0.0945961	−	0.995516i	$-0.530156\pi$
$127$	−2.13209	−0.189192	−0.0945961	+	0.995516i	$0.530156\pi$
$128$	−1.00000	−0.0883883
$129$	12.2508	1.07862
$130$	0	0
$131$	1.25851	0.109957	0.0549785	−	0.998488i	$-0.482491\pi$
$131$	1.25851	0.109957	0.0549785	+	0.998488i	$0.482491\pi$
$132$	−5.34541	−0.465258
$133$	−9.34333	−0.810170
$134$	−4.10926	−0.354986
$135$	1.00000	0.0860663
$136$	−4.00000	−0.342997
$137$	19.7149	1.68436	0.842178	−	0.539199i	$-0.181273\pi$
$137$	19.7149	1.68436	0.842178	+	0.539199i	$0.181273\pi$
$138$	4.93593	0.420175
$139$	5.67742	0.481553	0.240776	−	0.970581i	$-0.422598\pi$
$139$	5.67742	0.481553	0.240776	+	0.970581i	$0.422598\pi$
$140$	−2.32258	−0.196294
$141$	−1.81894	−0.153183
$142$	15.8719	1.33194
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.29078	0.356329
$146$	13.5734	1.12334
$147$	−1.60562	−0.132430
$148$	3.14925	0.258867
$149$	19.4775	1.59566	0.797829	−	0.602884i	$-0.205982\pi$
$149$	19.4775	1.59566	0.797829	+	0.602884i	$0.205982\pi$
$150$	−1.00000	−0.0816497
$151$	14.5170	1.18138	0.590690	−	0.806899i	$-0.298856\pi$
$151$	14.5170	1.18138	0.590690	+	0.806899i	$0.298856\pi$
$152$	−4.02283	−0.326294
$153$	4.00000	0.323381
$154$	−12.4151	−1.00044
$155$	−3.47183	−0.278864
$156$	0	0
$157$	24.3829	1.94596	0.972982	−	0.230879i	$-0.0741602\pi$
$157$	24.3829	1.94596	0.972982	+	0.230879i	$0.0741602\pi$
$158$	7.96774	0.633879
$159$	−5.48693	−0.435142
$160$	−1.00000	−0.0790569
$161$	11.4641	0.903498
$162$	−1.00000	−0.0785674
$163$	23.6267	1.85059	0.925295	−	0.379249i	$-0.123818\pi$
$163$	23.6267	1.85059	0.925295	+	0.379249i	$0.123818\pi$
$164$	2.64516	0.206552
$165$	−5.34541	−0.416139
$166$	−11.3360	−0.879842
$167$	−16.7471	−1.29593	−0.647967	−	0.761669i	$-0.724380\pi$
$167$	−16.7471	−1.29593	−0.647967	+	0.761669i	$0.724380\pi$
$168$	2.32258	0.179191
$169$	0	0
$170$	−4.00000	−0.306786
$171$	4.02283	0.307633
$172$	12.2508	0.934113
$173$	16.1321	1.22650	0.613250	−	0.789889i	$-0.289862\pi$
$173$	16.1321	1.22650	0.613250	+	0.789889i	$0.289862\pi$
$174$	−4.29078	−0.325283
$175$	−2.32258	−0.175571
$176$	−5.34541	−0.402925
$177$	6.78668	0.510118
$178$	−1.73978	−0.130402
$179$	7.32824	0.547738	0.273869	−	0.961767i	$-0.411696\pi$
$179$	7.32824	0.547738	0.273869	+	0.961767i	$0.411696\pi$
$180$	1.00000	0.0745356
$181$	19.5734	1.45488	0.727438	−	0.686173i	$-0.240711\pi$
$181$	19.5734	1.45488	0.727438	+	0.686173i	$0.240711\pi$
$182$	0	0
$183$	0.535898	0.0396147
$184$	4.93593	0.363882
$185$	3.14925	0.231538
$186$	3.47183	0.254567
$187$	−21.3816	−1.56358
$188$	−1.81894	−0.132660
$189$	−2.32258	−0.168943
$190$	−4.02283	−0.291846
$191$	−17.0375	−1.23279	−0.616394	−	0.787438i	$-0.711407\pi$
$191$	−17.0375	−1.23279	−0.616394	+	0.787438i	$0.711407\pi$
$192$	1.00000	0.0721688
$193$	−6.15491	−0.443040	−0.221520	−	0.975156i	$-0.571102\pi$
$193$	−6.15491	−0.443040	−0.221520	+	0.975156i	$0.571102\pi$
$194$	−16.1093	−1.15658
$195$	0	0
$196$	−1.60562	−0.114687
$197$	−13.7867	−0.982260	−0.491130	−	0.871086i	$-0.663416\pi$
$197$	−13.7867	−0.982260	−0.491130	+	0.871086i	$0.663416\pi$
$198$	5.34541	0.379881
$199$	2.28304	0.161841	0.0809203	−	0.996721i	$-0.474214\pi$
$199$	2.28304	0.161841	0.0809203	+	0.996721i	$0.474214\pi$
$200$	−1.00000	−0.0707107
$201$	4.10926	0.289845
$202$	−12.6908	−0.892922
$203$	−9.96567	−0.699453
$204$	4.00000	0.280056
$205$	2.64516	0.184746
$206$	−4.79612	−0.334161
$207$	−4.93593	−0.343071
$208$	0	0
$209$	−21.5036	−1.48744
$210$	2.32258	0.160273
$211$	22.4775	1.54741	0.773707	−	0.633543i	$-0.218400\pi$
$211$	22.4775	1.54741	0.773707	+	0.633543i	$0.218400\pi$
$212$	−5.48693	−0.376844
$213$	−15.8719	−1.08752
$214$	−3.07180	−0.209984
$215$	12.2508	0.835496
$216$	−1.00000	−0.0680414
$217$	8.06361	0.547393
$218$	−6.69081	−0.453159
$219$	−13.5734	−0.917203
$220$	−5.34541	−0.360387
$221$	0	0
$222$	−3.14925	−0.211364
$223$	1.37891	0.0923389	0.0461694	−	0.998934i	$-0.485299\pi$
$223$	1.37891	0.0923389	0.0461694	+	0.998934i	$0.485299\pi$
$224$	2.32258	0.155184
$225$	1.00000	0.0666667
$226$	7.10972	0.472931
$227$	19.3205	1.28235	0.641174	−	0.767396i	$-0.278448\pi$
$227$	19.3205	1.28235	0.641174	+	0.767396i	$0.278448\pi$
$228$	4.02283	0.266418
$229$	−15.7626	−1.04162	−0.520811	−	0.853672i	$-0.674370\pi$
$229$	−15.7626	−1.04162	−0.520811	+	0.853672i	$0.674370\pi$
$230$	4.93593	0.325466
$231$	12.4151	0.816856
$232$	−4.29078	−0.281703
$233$	16.5549	1.08455	0.542275	−	0.840201i	$-0.317563\pi$
$233$	16.5549	1.08455	0.542275	+	0.840201i	$0.317563\pi$
$234$	0	0
$235$	−1.81894	−0.118655
$236$	6.78668	0.441775
$237$	−7.96774	−0.517560
$238$	9.29032	0.602202
$239$	−26.2006	−1.69477	−0.847387	−	0.530976i	$-0.821825\pi$
$239$	−26.2006	−1.69477	−0.847387	+	0.530976i	$0.821825\pi$
$240$	1.00000	0.0645497
$241$	−15.6496	−1.00808	−0.504039	−	0.863681i	$-0.668153\pi$
$241$	−15.6496	−1.00808	−0.504039	+	0.863681i	$0.668153\pi$
$242$	−17.5734	−1.12966
$243$	1.00000	0.0641500
$244$	0.535898	0.0343074
$245$	−1.60562	−0.102580
$246$	−2.64516	−0.168649
$247$	0	0
$248$	3.47183	0.220462
$249$	11.3360	0.718388
$250$	−1.00000	−0.0632456
$251$	−28.3416	−1.78891	−0.894454	−	0.447160i	$-0.852435\pi$
$251$	−28.3416	−1.78891	−0.894454	+	0.447160i	$0.852435\pi$
$252$	−2.32258	−0.146309
$253$	26.3846	1.65878
$254$	2.13209	0.133779
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−12.5093	−0.780309	−0.390154	−	0.920750i	$-0.627578\pi$
$257$	−12.5093	−0.780309	−0.390154	+	0.920750i	$0.627578\pi$
$258$	−12.2508	−0.762700
$259$	−7.31439	−0.454494
$260$	0	0
$261$	4.29078	0.265592
$262$	−1.25851	−0.0777513
$263$	−10.7059	−0.660154	−0.330077	−	0.943954i	$-0.607075\pi$
$263$	−10.7059	−0.660154	−0.330077	+	0.943954i	$0.607075\pi$
$264$	5.34541	0.328987
$265$	−5.48693	−0.337059
$266$	9.34333	0.572877
$267$	1.73978	0.106473
$268$	4.10926	0.251013
$269$	7.52522	0.458821	0.229410	−	0.973330i	$-0.426320\pi$
$269$	7.52522	0.458821	0.229410	+	0.973330i	$0.426320\pi$
$270$	−1.00000	−0.0608581
$271$	−9.84868	−0.598264	−0.299132	−	0.954212i	$-0.596697\pi$
$271$	−9.84868	−0.598264	−0.299132	+	0.954212i	$0.596697\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−19.7149	−1.19102
$275$	−5.34541	−0.322340
$276$	−4.93593	−0.297108
$277$	0.953101	0.0572663	0.0286331	−	0.999590i	$-0.490885\pi$
$277$	0.953101	0.0572663	0.0286331	+	0.999590i	$0.490885\pi$
$278$	−5.67742	−0.340509
$279$	−3.47183	−0.207853
$280$	2.32258	0.138801
$281$	−5.57336	−0.332479	−0.166239	−	0.986085i	$-0.553162\pi$
$281$	−5.57336	−0.332479	−0.166239	+	0.986085i	$0.553162\pi$
$282$	1.81894	0.108316
$283$	22.0134	1.30856	0.654280	−	0.756252i	$-0.272972\pi$
$283$	22.0134	1.30856	0.654280	+	0.756252i	$0.272972\pi$
$284$	−15.8719	−0.941822
$285$	4.02283	0.238292
$286$	0	0
$287$	−6.14359	−0.362645
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−4.29078	−0.251963
$291$	16.1093	0.944342
$292$	−13.5734	−0.794321
$293$	6.84302	0.399773	0.199887	−	0.979819i	$-0.435943\pi$
$293$	6.84302	0.399773	0.199887	+	0.979819i	$0.435943\pi$
$294$	1.60562	0.0936419
$295$	6.78668	0.395136
$296$	−3.14925	−0.183047
$297$	−5.34541	−0.310172
$298$	−19.4775	−1.12830
$299$	0	0
$300$	1.00000	0.0577350
$301$	−28.4534	−1.64003
$302$	−14.5170	−0.835361
$303$	12.6908	0.729068
$304$	4.02283	0.230725
$305$	0.535898	0.0306855
$306$	−4.00000	−0.228665
$307$	−4.75442	−0.271349	−0.135675	−	0.990753i	$-0.543320\pi$
$307$	−4.75442	−0.271349	−0.135675	+	0.990753i	$0.543320\pi$
$308$	12.4151	0.707418
$309$	4.79612	0.272842
$310$	3.47183	0.197187
$311$	−1.93639	−0.109803	−0.0549013	−	0.998492i	$-0.517484\pi$
$311$	−1.93639	−0.109803	−0.0549013	+	0.998492i	$0.517484\pi$
$312$	0	0
$313$	25.5545	1.44443	0.722213	−	0.691671i	$-0.243125\pi$
$313$	25.5545	1.44443	0.722213	+	0.691671i	$0.243125\pi$
$314$	−24.3829	−1.37600
$315$	−2.32258	−0.130863
$316$	−7.96774	−0.448220
$317$	12.6667	0.711435	0.355717	−	0.934594i	$-0.384237\pi$
$317$	12.6667	0.711435	0.355717	+	0.934594i	$0.384237\pi$
$318$	5.48693	0.307692
$319$	−22.9359	−1.28417
$320$	1.00000	0.0559017
$321$	3.07180	0.171451
$322$	−11.4641	−0.638869
$323$	16.0913	0.895344
$324$	1.00000	0.0555556
$325$	0	0
$326$	−23.6267	−1.30856
$327$	6.69081	0.370003
$328$	−2.64516	−0.146054
$329$	4.22464	0.232912
$330$	5.34541	0.294255
$331$	23.6981	1.30256	0.651282	−	0.758836i	$-0.274231\pi$
$331$	23.6981	1.30256	0.651282	+	0.758836i	$0.274231\pi$
$332$	11.3360	0.622142
$333$	3.14925	0.172578
$334$	16.7471	0.916363
$335$	4.10926	0.224513
$336$	−2.32258	−0.126707
$337$	−19.5554	−1.06525	−0.532625	−	0.846351i	$-0.678795\pi$
$337$	−19.5554	−1.06525	−0.532625	+	0.846351i	$0.678795\pi$
$338$	0	0
$339$	−7.10972	−0.386147
$340$	4.00000	0.216930
$341$	18.5584	1.00499
$342$	−4.02283	−0.217530
$343$	19.9872	1.07921
$344$	−12.2508	−0.660518
$345$	−4.93593	−0.265742
$346$	−16.1321	−0.867266
$347$	−23.0375	−1.23672	−0.618358	−	0.785897i	$-0.712202\pi$
$347$	−23.0375	−1.23672	−0.618358	+	0.785897i	$0.712202\pi$
$348$	4.29078	0.230010
$349$	−15.3205	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$349$	−15.3205	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$350$	2.32258	0.124147
$351$	0	0
$352$	5.34541	0.284911
$353$	28.4078	1.51199	0.755996	−	0.654576i	$-0.227153\pi$
$353$	28.4078	1.51199	0.755996	+	0.654576i	$0.227153\pi$
$354$	−6.78668	−0.360708
$355$	−15.8719	−0.842391
$356$	1.73978	0.0922083
$357$	−9.29032	−0.491696
$358$	−7.32824	−0.387309
$359$	23.5734	1.24415	0.622077	−	0.782956i	$-0.286289\pi$
$359$	23.5734	1.24415	0.622077	+	0.782956i	$0.286289\pi$
$360$	−1.00000	−0.0527046
$361$	−2.81687	−0.148256
$362$	−19.5734	−1.02875
$363$	17.5734	0.922362
$364$	0	0
$365$	−13.5734	−0.710462
$366$	−0.535898	−0.0280119
$367$	27.4909	1.43501	0.717506	−	0.696552i	$-0.245283\pi$
$367$	27.4909	1.43501	0.717506	+	0.696552i	$0.245283\pi$
$368$	−4.93593	−0.257303
$369$	2.64516	0.137701
$370$	−3.14925	−0.163722
$371$	12.7438	0.661627
$372$	−3.47183	−0.180006
$373$	26.3421	1.36394	0.681971	−	0.731379i	$-0.261123\pi$
$373$	26.3421	1.36394	0.681971	+	0.731379i	$0.261123\pi$
$374$	21.3816	1.10562
$375$	1.00000	0.0516398
$376$	1.81894	0.0938048
$377$	0	0
$378$	2.32258	0.119461
$379$	0.448551	0.0230405	0.0115202	−	0.999934i	$-0.496333\pi$
$379$	0.448551	0.0230405	0.0115202	+	0.999934i	$0.496333\pi$
$380$	4.02283	0.206367
$381$	−2.13209	−0.109230
$382$	17.0375	0.871713
$383$	−12.1227	−0.619439	−0.309719	−	0.950828i	$-0.600235\pi$
$383$	−12.1227	−0.619439	−0.309719	+	0.950828i	$0.600235\pi$
$384$	−1.00000	−0.0510310
$385$	12.4151	0.632734
$386$	6.15491	0.313277
$387$	12.2508	0.622742
$388$	16.1093	0.817824
$389$	−18.6195	−0.944045	−0.472022	−	0.881587i	$-0.656476\pi$
$389$	−18.6195	−0.944045	−0.472022	+	0.881587i	$0.656476\pi$
$390$	0	0
$391$	−19.7437	−0.998484
$392$	1.60562	0.0810962
$393$	1.25851	0.0634837
$394$	13.7867	0.694563
$395$	−7.96774	−0.400900
$396$	−5.34541	−0.268617
$397$	11.3042	0.567340	0.283670	−	0.958922i	$-0.408448\pi$
$397$	11.3042	0.567340	0.283670	+	0.958922i	$0.408448\pi$
$398$	−2.28304	−0.114439
$399$	−9.34333	−0.467752
$400$	1.00000	0.0500000
$401$	1.50580	0.0751959	0.0375980	−	0.999293i	$-0.488029\pi$
$401$	1.50580	0.0751959	0.0375980	+	0.999293i	$0.488029\pi$
$402$	−4.10926	−0.204951
$403$	0	0
$404$	12.6908	0.631391
$405$	1.00000	0.0496904
$406$	9.96567	0.494588
$407$	−16.8340	−0.834432
$408$	−4.00000	−0.198030
$409$	−11.1394	−0.550806	−0.275403	−	0.961329i	$-0.588811\pi$
$409$	−11.1394	−0.550806	−0.275403	+	0.961329i	$0.588811\pi$
$410$	−2.64516	−0.130635
$411$	19.7149	0.972464
$412$	4.79612	0.236288
$413$	−15.7626	−0.775627
$414$	4.93593	0.242588
$415$	11.3360	0.556461
$416$	0	0
$417$	5.67742	0.278024
$418$	21.5036	1.05178
$419$	−15.5098	−0.757701	−0.378851	−	0.925458i	$-0.623681\pi$
$419$	−15.5098	−0.757701	−0.378851	+	0.925458i	$0.623681\pi$
$420$	−2.32258	−0.113330
$421$	39.4452	1.92244	0.961221	−	0.275778i	$-0.0889353\pi$
$421$	39.4452	1.92244	0.961221	+	0.275778i	$0.0889353\pi$
$422$	−22.4775	−1.09419
$423$	−1.81894	−0.0884400
$424$	5.48693	0.266469
$425$	4.00000	0.194029
$426$	15.8719	0.768995
$427$	−1.24467	−0.0602336
$428$	3.07180	0.148481
$429$	0	0
$430$	−12.2508	−0.590785
$431$	2.15491	0.103799	0.0518993	−	0.998652i	$-0.483473\pi$
$431$	2.15491	0.103799	0.0518993	+	0.998652i	$0.483473\pi$
$432$	1.00000	0.0481125
$433$	1.33938	0.0643664	0.0321832	−	0.999482i	$-0.489754\pi$
$433$	1.33938	0.0643664	0.0321832	+	0.999482i	$0.489754\pi$
$434$	−8.06361	−0.387066
$435$	4.29078	0.205727
$436$	6.69081	0.320432
$437$	−19.8564	−0.949861
$438$	13.5734	0.648560
$439$	−31.6981	−1.51287	−0.756434	−	0.654071i	$-0.773060\pi$
$439$	−31.6981	−1.51287	−0.756434	+	0.654071i	$0.773060\pi$
$440$	5.34541	0.254832
$441$	−1.60562	−0.0764583
$442$	0	0
$443$	−28.0904	−1.33461	−0.667307	−	0.744782i	$-0.732553\pi$
$443$	−28.0904	−1.33461	−0.667307	+	0.744782i	$0.732553\pi$
$444$	3.14925	0.149457
$445$	1.73978	0.0824736
$446$	−1.37891	−0.0652935
$447$	19.4775	0.921254
$448$	−2.32258	−0.109732
$449$	28.9167	1.36466	0.682332	−	0.731043i	$-0.260966\pi$
$449$	28.9167	1.36466	0.682332	+	0.731043i	$0.260966\pi$
$450$	−1.00000	−0.0471405
$451$	−14.1395	−0.665801
$452$	−7.10972	−0.334413
$453$	14.5170	0.682070
$454$	−19.3205	−0.906756
$455$	0	0
$456$	−4.02283	−0.188386
$457$	9.69900	0.453700	0.226850	−	0.973930i	$-0.427157\pi$
$457$	9.69900	0.453700	0.226850	+	0.973930i	$0.427157\pi$
$458$	15.7626	0.736538
$459$	4.00000	0.186704
$460$	−4.93593	−0.230139
$461$	−11.9979	−0.558799	−0.279400	−	0.960175i	$-0.590135\pi$
$461$	−11.9979	−0.558799	−0.279400	+	0.960175i	$0.590135\pi$
$462$	−12.4151	−0.577604
$463$	6.42664	0.298671	0.149336	−	0.988787i	$-0.452287\pi$
$463$	6.42664	0.298671	0.149336	+	0.988787i	$0.452287\pi$
$464$	4.29078	0.199194
$465$	−3.47183	−0.161002
$466$	−16.5549	−0.766893
$467$	−26.2642	−1.21536	−0.607681	−	0.794182i	$-0.707900\pi$
$467$	−26.2642	−1.21536	−0.607681	+	0.794182i	$0.707900\pi$
$468$	0	0
$469$	−9.54409	−0.440705
$470$	1.81894	0.0839015
$471$	24.3829	1.12350
$472$	−6.78668	−0.312382
$473$	−65.4854	−3.01102
$474$	7.96774	0.365970
$475$	4.02283	0.184580
$476$	−9.29032	−0.425821
$477$	−5.48693	−0.251229
$478$	26.2006	1.19839
$479$	25.6826	1.17347	0.586735	−	0.809779i	$-0.300413\pi$
$479$	25.6826	1.17347	0.586735	+	0.809779i	$0.300413\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	15.6496	0.712818
$483$	11.4641	0.521635
$484$	17.5734	0.798789
$485$	16.1093	0.731484
$486$	−1.00000	−0.0453609
$487$	9.37891	0.424999	0.212500	−	0.977161i	$-0.431840\pi$
$487$	9.37891	0.424999	0.212500	+	0.977161i	$0.431840\pi$
$488$	−0.535898	−0.0242590
$489$	23.6267	1.06844
$490$	1.60562	0.0725347
$491$	−7.73854	−0.349235	−0.174618	−	0.984636i	$-0.555869\pi$
$491$	−7.73854	−0.349235	−0.174618	+	0.984636i	$0.555869\pi$
$492$	2.64516	0.119253
$493$	17.1631	0.772987
$494$	0	0
$495$	−5.34541	−0.240258
$496$	−3.47183	−0.155890
$497$	36.8637	1.65356
$498$	−11.3360	−0.507977
$499$	−3.18106	−0.142404	−0.0712019	−	0.997462i	$-0.522683\pi$
$499$	−3.18106	−0.142404	−0.0712019	+	0.997462i	$0.522683\pi$
$500$	1.00000	0.0447214
$501$	−16.7471	−0.748207
$502$	28.3416	1.26495
$503$	−35.0758	−1.56395	−0.781975	−	0.623309i	$-0.785788\pi$
$503$	−35.0758	−1.56395	−0.781975	+	0.623309i	$0.785788\pi$
$504$	2.32258	0.103456
$505$	12.6908	0.564734
$506$	−26.3846	−1.17294
$507$	0	0
$508$	−2.13209	−0.0945961
$509$	−18.7867	−0.832705	−0.416353	−	0.909203i	$-0.636692\pi$
$509$	−18.7867	−0.832705	−0.416353	+	0.909203i	$0.636692\pi$
$510$	−4.00000	−0.177123
$511$	31.5252	1.39459
$512$	−1.00000	−0.0441942
$513$	4.02283	0.177612
$514$	12.5093	0.551761
$515$	4.79612	0.211342
$516$	12.2508	0.539311
$517$	9.72298	0.427616
$518$	7.31439	0.321376
$519$	16.1321	0.708120
$520$	0	0
$521$	5.28512	0.231545	0.115773	−	0.993276i	$-0.463066\pi$
$521$	5.28512	0.231545	0.115773	+	0.993276i	$0.463066\pi$
$522$	−4.29078	−0.187802
$523$	−20.7523	−0.907437	−0.453718	−	0.891145i	$-0.649903\pi$
$523$	−20.7523	−0.907437	−0.453718	+	0.891145i	$0.649903\pi$
$524$	1.25851	0.0549785
$525$	−2.32258	−0.101366
$526$	10.7059	0.466800
$527$	−13.8873	−0.604942
$528$	−5.34541	−0.232629
$529$	1.36345	0.0592805
$530$	5.48693	0.238337
$531$	6.78668	0.294517
$532$	−9.34333	−0.405085
$533$	0	0
$534$	−1.73978	−0.0752877
$535$	3.07180	0.132805
$536$	−4.10926	−0.177493
$537$	7.32824	0.316237
$538$	−7.52522	−0.324435
$539$	8.58271	0.369683
$540$	1.00000	0.0430331
$541$	−28.7365	−1.23548	−0.617739	−	0.786384i	$-0.711951\pi$
$541$	−28.7365	−1.23548	−0.617739	+	0.786384i	$0.711951\pi$
$542$	9.84868	0.423037
$543$	19.5734	0.839973
$544$	−4.00000	−0.171499
$545$	6.69081	0.286603
$546$	0	0
$547$	18.3768	0.785737	0.392869	−	0.919595i	$-0.371483\pi$
$547$	18.3768	0.785737	0.392869	+	0.919595i	$0.371483\pi$
$548$	19.7149	0.842178
$549$	0.535898	0.0228716
$550$	5.34541	0.227929
$551$	17.2610	0.735345
$552$	4.93593	0.210087
$553$	18.5057	0.786943
$554$	−0.953101	−0.0404934
$555$	3.14925	0.133678
$556$	5.67742	0.240776
$557$	−26.5386	−1.12448	−0.562238	−	0.826975i	$-0.690060\pi$
$557$	−26.5386	−1.12448	−0.562238	+	0.826975i	$0.690060\pi$
$558$	3.47183	0.146974
$559$	0	0
$560$	−2.32258	−0.0981469
$561$	−21.3816	−0.902733
$562$	5.57336	0.235098
$563$	−6.06111	−0.255446	−0.127723	−	0.991810i	$-0.540767\pi$
$563$	−6.06111	−0.255446	−0.127723	+	0.991810i	$0.540767\pi$
$564$	−1.81894	−0.0765913
$565$	−7.10972	−0.299108
$566$	−22.0134	−0.925292
$567$	−2.32258	−0.0975392
$568$	15.8719	0.665969
$569$	−14.4964	−0.607719	−0.303860	−	0.952717i	$-0.598275\pi$
$569$	−14.4964	−0.607719	−0.303860	+	0.952717i	$0.598275\pi$
$570$	−4.02283	−0.168498
$571$	2.90413	0.121534	0.0607670	−	0.998152i	$-0.480645\pi$
$571$	2.90413	0.121534	0.0607670	+	0.998152i	$0.480645\pi$
$572$	0	0
$573$	−17.0375	−0.711750
$574$	6.14359	0.256429
$575$	−4.93593	−0.205843
$576$	1.00000	0.0416667
$577$	18.8180	0.783405	0.391702	−	0.920092i	$-0.371886\pi$
$577$	18.8180	0.783405	0.391702	+	0.920092i	$0.371886\pi$
$578$	1.00000	0.0415945
$579$	−6.15491	−0.255789
$580$	4.29078	0.178165
$581$	−26.3287	−1.09230
$582$	−16.1093	−0.667750
$583$	29.3299	1.21472
$584$	13.5734	0.561670
$585$	0	0
$586$	−6.84302	−0.282682
$587$	−3.93639	−0.162472	−0.0812361	−	0.996695i	$-0.525887\pi$
$587$	−3.93639	−0.162472	−0.0812361	+	0.996695i	$0.525887\pi$
$588$	−1.60562	−0.0662148
$589$	−13.9666	−0.575483
$590$	−6.78668	−0.279403
$591$	−13.7867	−0.567108
$592$	3.14925	0.129434
$593$	20.1227	0.826338	0.413169	−	0.910654i	$-0.364422\pi$
$593$	20.1227	0.826338	0.413169	+	0.910654i	$0.364422\pi$
$594$	5.34541	0.219325
$595$	−9.29032	−0.380866
$596$	19.4775	0.797829
$597$	2.28304	0.0934388
$598$	0	0
$599$	48.1172	1.96601	0.983007	−	0.183567i	$-0.0587644\pi$
$599$	48.1172	1.96601	0.983007	+	0.183567i	$0.0587644\pi$
$600$	−1.00000	−0.0408248
$601$	−4.24217	−0.173042	−0.0865209	−	0.996250i	$-0.527575\pi$
$601$	−4.24217	−0.173042	−0.0865209	+	0.996250i	$0.527575\pi$
$602$	28.4534	1.15967
$603$	4.10926	0.167342
$604$	14.5170	0.590690
$605$	17.5734	0.714459
$606$	−12.6908	−0.515529
$607$	12.0685	0.489844	0.244922	−	0.969543i	$-0.421238\pi$
$607$	12.0685	0.489844	0.244922	+	0.969543i	$0.421238\pi$
$608$	−4.02283	−0.163147
$609$	−9.96567	−0.403829
$610$	−0.535898	−0.0216979
$611$	0	0
$612$	4.00000	0.161690
$613$	41.7421	1.68595	0.842974	−	0.537954i	$-0.180803\pi$
$613$	41.7421	1.68595	0.842974	+	0.537954i	$0.180803\pi$
$614$	4.75442	0.191873
$615$	2.64516	0.106663
$616$	−12.4151	−0.500220
$617$	33.4586	1.34699	0.673497	−	0.739190i	$-0.264792\pi$
$617$	33.4586	1.34699	0.673497	+	0.739190i	$0.264792\pi$
$618$	−4.79612	−0.192928
$619$	38.0978	1.53128	0.765639	−	0.643270i	$-0.222423\pi$
$619$	38.0978	1.53128	0.765639	+	0.643270i	$0.222423\pi$
$620$	−3.47183	−0.139432
$621$	−4.93593	−0.198072
$622$	1.93639	0.0776422
$623$	−4.04078	−0.161891
$624$	0	0
$625$	1.00000	0.0400000
$626$	−25.5545	−1.02136
$627$	−21.5036	−0.858773
$628$	24.3829	0.972982
$629$	12.5970	0.502276
$630$	2.32258	0.0925338
$631$	−24.8001	−0.987275	−0.493638	−	0.869668i	$-0.664333\pi$
$631$	−24.8001	−0.987275	−0.493638	+	0.869668i	$0.664333\pi$
$632$	7.96774	0.316940
$633$	22.4775	0.893400
$634$	−12.6667	−0.503060
$635$	−2.13209	−0.0846093
$636$	−5.48693	−0.217571
$637$	0	0
$638$	22.9359	0.908042
$639$	−15.8719	−0.627881
$640$	−1.00000	−0.0395285
$641$	4.08519	0.161355	0.0806776	−	0.996740i	$-0.474292\pi$
$641$	4.08519	0.161355	0.0806776	+	0.996740i	$0.474292\pi$
$642$	−3.07180	−0.121234
$643$	36.5816	1.44264	0.721318	−	0.692604i	$-0.243537\pi$
$643$	36.5816	1.44264	0.721318	+	0.692604i	$0.243537\pi$
$644$	11.4641	0.451749
$645$	12.2508	0.482374
$646$	−16.0913	−0.633104
$647$	17.2341	0.677541	0.338771	−	0.940869i	$-0.389989\pi$
$647$	17.2341	0.677541	0.338771	+	0.940869i	$0.389989\pi$
$648$	−1.00000	−0.0392837
$649$	−36.2776	−1.42402
$650$	0	0
$651$	8.06361	0.316038
$652$	23.6267	0.925295
$653$	−21.9274	−0.858085	−0.429042	−	0.903284i	$-0.641149\pi$
$653$	−21.9274	−0.858085	−0.429042	+	0.903284i	$0.641149\pi$
$654$	−6.69081	−0.261631
$655$	1.25851	0.0491742
$656$	2.64516	0.103276
$657$	−13.5734	−0.529547
$658$	−4.22464	−0.164694
$659$	−48.4759	−1.88835	−0.944176	−	0.329441i	$-0.893140\pi$
$659$	−48.4759	−1.88835	−0.944176	+	0.329441i	$0.893140\pi$
$660$	−5.34541	−0.208070
$661$	29.8221	1.15994	0.579972	−	0.814636i	$-0.303063\pi$
$661$	29.8221	1.15994	0.579972	+	0.814636i	$0.303063\pi$
$662$	−23.6981	−0.921052
$663$	0	0
$664$	−11.3360	−0.439921
$665$	−9.34333	−0.362319
$666$	−3.14925	−0.122031
$667$	−21.1790	−0.820054
$668$	−16.7471	−0.647967
$669$	1.37891	0.0533119
$670$	−4.10926	−0.158755
$671$	−2.86459	−0.110586
$672$	2.32258	0.0895955
$673$	−30.6719	−1.18232	−0.591158	−	0.806556i	$-0.701329\pi$
$673$	−30.6719	−1.18232	−0.591158	+	0.806556i	$0.701329\pi$
$674$	19.5554	0.753246
$675$	1.00000	0.0384900
$676$	0	0
$677$	21.0831	0.810290	0.405145	−	0.914252i	$-0.367221\pi$
$677$	21.0831	0.810290	0.405145	+	0.914252i	$0.367221\pi$
$678$	7.10972	0.273047
$679$	−37.4150	−1.43586
$680$	−4.00000	−0.153393
$681$	19.3205	0.740363
$682$	−18.5584	−0.710636
$683$	−13.3360	−0.510287	−0.255143	−	0.966903i	$-0.582123\pi$
$683$	−13.3360	−0.510287	−0.255143	+	0.966903i	$0.582123\pi$
$684$	4.02283	0.153817
$685$	19.7149	0.753267
$686$	−19.9872	−0.763117
$687$	−15.7626	−0.601381
$688$	12.2508	0.467057
$689$	0	0
$690$	4.93593	0.187908
$691$	−34.9054	−1.32786	−0.663932	−	0.747793i	$-0.731113\pi$
$691$	−34.9054	−1.32786	−0.663932	+	0.747793i	$0.731113\pi$
$692$	16.1321	0.613250
$693$	12.4151	0.471612
$694$	23.0375	0.874490
$695$	5.67742	0.215357
$696$	−4.29078	−0.162641
$697$	10.5806	0.400770
$698$	15.3205	0.579890
$699$	16.5549	0.626166
$700$	−2.32258	−0.0877853
$701$	−39.6715	−1.49837	−0.749186	−	0.662360i	$-0.769555\pi$
$701$	−39.6715	−1.49837	−0.749186	+	0.662360i	$0.769555\pi$
$702$	0	0
$703$	12.6689	0.477817
$704$	−5.34541	−0.201463
$705$	−1.81894	−0.0685053
$706$	−28.4078	−1.06914
$707$	−29.4754	−1.10854
$708$	6.78668	0.255059
$709$	2.83441	0.106448	0.0532242	−	0.998583i	$-0.483050\pi$
$709$	2.83441	0.106448	0.0532242	+	0.998583i	$0.483050\pi$
$710$	15.8719	0.595661
$711$	−7.96774	−0.298814
$712$	−1.73978	−0.0652011
$713$	17.1367	0.641776
$714$	9.29032	0.347681
$715$	0	0
$716$	7.32824	0.273869
$717$	−26.2006	−0.978478
$718$	−23.5734	−0.879750
$719$	−43.7128	−1.63021	−0.815106	−	0.579311i	$-0.803322\pi$
$719$	−43.7128	−1.63021	−0.815106	+	0.579311i	$0.803322\pi$
$720$	1.00000	0.0372678
$721$	−11.1394	−0.414852
$722$	2.81687	0.104833
$723$	−15.6496	−0.582014
$724$	19.5734	0.727438
$725$	4.29078	0.159355
$726$	−17.5734	−0.652209
$727$	16.2568	0.602932	0.301466	−	0.953477i	$-0.402524\pi$
$727$	16.2568	0.602932	0.301466	+	0.953477i	$0.402524\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	13.5734	0.502373
$731$	49.0031	1.81245
$732$	0.535898	0.0198074
$733$	−4.96774	−0.183488	−0.0917438	−	0.995783i	$-0.529244\pi$
$733$	−4.96774	−0.183488	−0.0917438	+	0.995783i	$0.529244\pi$
$734$	−27.4909	−1.01471
$735$	−1.60562	−0.0592243
$736$	4.93593	0.181941
$737$	−21.9657	−0.809116
$738$	−2.64516	−0.0973697
$739$	−49.6875	−1.82778	−0.913892	−	0.405957i	$-0.866938\pi$
$739$	−49.6875	−1.82778	−0.913892	+	0.405957i	$0.866938\pi$
$740$	3.14925	0.115769
$741$	0	0
$742$	−12.7438	−0.467841
$743$	21.5420	0.790300	0.395150	−	0.918617i	$-0.370693\pi$
$743$	21.5420	0.790300	0.395150	+	0.918617i	$0.370693\pi$
$744$	3.47183	0.127284
$745$	19.4775	0.713600
$746$	−26.3421	−0.964452
$747$	11.3360	0.414761
$748$	−21.3816	−0.781790
$749$	−7.13449	−0.260689
$750$	−1.00000	−0.0365148
$751$	−6.34872	−0.231668	−0.115834	−	0.993269i	$-0.536954\pi$
$751$	−6.34872	−0.231668	−0.115834	+	0.993269i	$0.536954\pi$
$752$	−1.81894	−0.0663300
$753$	−28.3416	−1.03283
$754$	0	0
$755$	14.5170	0.528329
$756$	−2.32258	−0.0844714
$757$	−48.1107	−1.74861	−0.874307	−	0.485373i	$-0.838684\pi$
$757$	−48.1107	−1.74861	−0.874307	+	0.485373i	$0.838684\pi$
$758$	−0.448551	−0.0162921
$759$	26.3846	0.957699
$760$	−4.02283	−0.145923
$761$	33.6866	1.22114	0.610569	−	0.791963i	$-0.290941\pi$
$761$	33.6866	1.22114	0.610569	+	0.791963i	$0.290941\pi$
$762$	2.13209	0.0772374
$763$	−15.5399	−0.562584
$764$	−17.0375	−0.616394
$765$	4.00000	0.144620
$766$	12.1227	0.438009
$767$	0	0
$768$	1.00000	0.0360844
$769$	−5.98886	−0.215964	−0.107982	−	0.994153i	$-0.534439\pi$
$769$	−5.98886	−0.215964	−0.107982	+	0.994153i	$0.534439\pi$
$770$	−12.4151	−0.447410
$771$	−12.5093	−0.450511
$772$	−6.15491	−0.221520
$773$	−13.4882	−0.485136	−0.242568	−	0.970134i	$-0.577990\pi$
$773$	−13.4882	−0.485136	−0.242568	+	0.970134i	$0.577990\pi$
$774$	−12.2508	−0.440345
$775$	−3.47183	−0.124712
$776$	−16.1093	−0.578289
$777$	−7.31439	−0.262402
$778$	18.6195	0.667540
$779$	10.6410	0.381254
$780$	0	0
$781$	84.8416	3.03587
$782$	19.7437	0.706035
$783$	4.29078	0.153340
$784$	−1.60562	−0.0573437
$785$	24.3829	0.870262
$786$	−1.25851	−0.0448897
$787$	−54.6487	−1.94802	−0.974009	−	0.226510i	$-0.927268\pi$
$787$	−54.6487	−1.94802	−0.974009	+	0.226510i	$0.927268\pi$
$788$	−13.7867	−0.491130
$789$	−10.7059	−0.381140
$790$	7.96774	0.283479
$791$	16.5129	0.587131
$792$	5.34541	0.189941
$793$	0	0
$794$	−11.3042	−0.401170
$795$	−5.48693	−0.194601
$796$	2.28304	0.0809203
$797$	−38.3244	−1.35752	−0.678760	−	0.734361i	$-0.737482\pi$
$797$	−38.3244	−1.35752	−0.678760	+	0.734361i	$0.737482\pi$
$798$	9.34333	0.330750
$799$	−7.27577	−0.257398
$800$	−1.00000	−0.0353553
$801$	1.73978	0.0614722
$802$	−1.50580	−0.0531716
$803$	72.5551	2.56041
$804$	4.10926	0.144922
$805$	11.4641	0.404056
$806$	0	0
$807$	7.52522	0.264900
$808$	−12.6908	−0.446461
$809$	11.3816	0.400157	0.200078	−	0.979780i	$-0.435880\pi$
$809$	11.3816	0.400157	0.200078	+	0.979780i	$0.435880\pi$
$810$	−1.00000	−0.0351364
$811$	10.8899	0.382397	0.191198	−	0.981551i	$-0.438763\pi$
$811$	10.8899	0.382397	0.191198	+	0.981551i	$0.438763\pi$
$812$	−9.96567	−0.349726
$813$	−9.84868	−0.345408
$814$	16.8340	0.590033
$815$	23.6267	0.827609
$816$	4.00000	0.140028
$817$	49.2828	1.72419
$818$	11.1394	0.389479
$819$	0	0
$820$	2.64516	0.0923730
$821$	−1.68810	−0.0589152	−0.0294576	−	0.999566i	$-0.509378\pi$
$821$	−1.68810	−0.0589152	−0.0294576	+	0.999566i	$0.509378\pi$
$822$	−19.7149	−0.687636
$823$	6.77816	0.236272	0.118136	−	0.992997i	$-0.462308\pi$
$823$	6.77816	0.236272	0.118136	+	0.992997i	$0.462308\pi$
$824$	−4.79612	−0.167081
$825$	−5.34541	−0.186103
$826$	15.7626	0.548451
$827$	−30.2298	−1.05119	−0.525597	−	0.850733i	$-0.676158\pi$
$827$	−30.2298	−1.05119	−0.525597	+	0.850733i	$0.676158\pi$
$828$	−4.93593	−0.171536
$829$	−10.4405	−0.362612	−0.181306	−	0.983427i	$-0.558032\pi$
$829$	−10.4405	−0.362612	−0.181306	+	0.983427i	$0.558032\pi$
$830$	−11.3360	−0.393477
$831$	0.953101	0.0330627
$832$	0	0
$833$	−6.42249	−0.222526
$834$	−5.67742	−0.196593
$835$	−16.7471	−0.579559
$836$	−21.5036	−0.743719
$837$	−3.47183	−0.120004
$838$	15.5098	0.535776
$839$	11.1965	0.386547	0.193273	−	0.981145i	$-0.438090\pi$
$839$	11.1965	0.386547	0.193273	+	0.981145i	$0.438090\pi$
$840$	2.32258	0.0801366
$841$	−10.5892	−0.365146
$842$	−39.4452	−1.35937
$843$	−5.57336	−0.191957
$844$	22.4775	0.773707
$845$	0	0
$846$	1.81894	0.0625365
$847$	−40.8155	−1.40244
$848$	−5.48693	−0.188422
$849$	22.0134	0.755498
$850$	−4.00000	−0.137199
$851$	−15.5445	−0.532859
$852$	−15.8719	−0.543761
$853$	21.8185	0.747051	0.373525	−	0.927620i	$-0.378149\pi$
$853$	21.8185	0.747051	0.373525	+	0.927620i	$0.378149\pi$
$854$	1.24467	0.0425916
$855$	4.02283	0.137578
$856$	−3.07180	−0.104992
$857$	9.42369	0.321907	0.160954	−	0.986962i	$-0.448543\pi$
$857$	9.42369	0.321907	0.160954	+	0.986962i	$0.448543\pi$
$858$	0	0
$859$	24.4775	0.835161	0.417581	−	0.908640i	$-0.362878\pi$
$859$	24.4775	0.835161	0.417581	+	0.908640i	$0.362878\pi$
$860$	12.2508	0.417748
$861$	−6.14359	−0.209373
$862$	−2.15491	−0.0733966
$863$	29.7873	1.01397	0.506986	−	0.861954i	$-0.330760\pi$
$863$	29.7873	1.01397	0.506986	+	0.861954i	$0.330760\pi$
$864$	−1.00000	−0.0340207
$865$	16.1321	0.548507
$866$	−1.33938	−0.0455139
$867$	−1.00000	−0.0339618
$868$	8.06361	0.273697
$869$	42.5908	1.44479
$870$	−4.29078	−0.145471
$871$	0	0
$872$	−6.69081	−0.226579
$873$	16.1093	0.545216
$874$	19.8564	0.671653
$875$	−2.32258	−0.0785175
$876$	−13.5734	−0.458601
$877$	−1.90979	−0.0644890	−0.0322445	−	0.999480i	$-0.510266\pi$
$877$	−1.90979	−0.0644890	−0.0322445	+	0.999480i	$0.510266\pi$
$878$	31.6981	1.06976
$879$	6.84302	0.230809
$880$	−5.34541	−0.180194
$881$	6.36823	0.214551	0.107276	−	0.994229i	$-0.465787\pi$
$881$	6.36823	0.214551	0.107276	+	0.994229i	$0.465787\pi$
$882$	1.60562	0.0540642
$883$	34.9897	1.17750	0.588749	−	0.808316i	$-0.299621\pi$
$883$	34.9897	1.17750	0.588749	+	0.808316i	$0.299621\pi$
$884$	0	0
$885$	6.78668	0.228132
$886$	28.0904	0.943715
$887$	21.4001	0.718546	0.359273	−	0.933233i	$-0.383025\pi$
$887$	21.4001	0.718546	0.359273	+	0.933233i	$0.383025\pi$
$888$	−3.14925	−0.105682
$889$	4.95194	0.166083
$890$	−1.73978	−0.0583176
$891$	−5.34541	−0.179078
$892$	1.37891	0.0461694
$893$	−7.31729	−0.244864
$894$	−19.4775	−0.651425
$895$	7.32824	0.244956
$896$	2.32258	0.0775919
$897$	0	0
$898$	−28.9167	−0.964963
$899$	−14.8969	−0.496838
$900$	1.00000	0.0333333
$901$	−21.9477	−0.731184
$902$	14.1395	0.470792
$903$	−28.4534	−0.946871
$904$	7.10972	0.236466
$905$	19.5734	0.650641
$906$	−14.5170	−0.482296
$907$	32.4693	1.07813	0.539063	−	0.842266i	$-0.318779\pi$
$907$	32.4693	1.07813	0.539063	+	0.842266i	$0.318779\pi$
$908$	19.3205	0.641174
$909$	12.6908	0.420928
$910$	0	0
$911$	34.4380	1.14098	0.570490	−	0.821304i	$-0.306753\pi$
$911$	34.4380	1.14098	0.570490	+	0.821304i	$0.306753\pi$
$912$	4.02283	0.133209
$913$	−60.5954	−2.00541
$914$	−9.69900	−0.320814
$915$	0.535898	0.0177163
$916$	−15.7626	−0.520811
$917$	−2.92300	−0.0965260
$918$	−4.00000	−0.132020
$919$	36.4827	1.20345	0.601727	−	0.798702i	$-0.294480\pi$
$919$	36.4827	1.20345	0.601727	+	0.798702i	$0.294480\pi$
$920$	4.93593	0.162733
$921$	−4.75442	−0.156663
$922$	11.9979	0.395131
$923$	0	0
$924$	12.4151	0.408428
$925$	3.14925	0.103547
$926$	−6.42664	−0.211192
$927$	4.79612	0.157525
$928$	−4.29078	−0.140852
$929$	−49.9100	−1.63749	−0.818747	−	0.574155i	$-0.805331\pi$
$929$	−49.9100	−1.63749	−0.818747	+	0.574155i	$0.805331\pi$
$930$	3.47183	0.113846
$931$	−6.45914	−0.211690
$932$	16.5549	0.542275
$933$	−1.93639	−0.0633946
$934$	26.2642	0.859390
$935$	−21.3816	−0.699254
$936$	0	0
$937$	−15.8873	−0.519017	−0.259508	−	0.965741i	$-0.583560\pi$
$937$	−15.8873	−0.519017	−0.259508	+	0.965741i	$0.583560\pi$
$938$	9.54409	0.311625
$939$	25.5545	0.833939
$940$	−1.81894	−0.0593274
$941$	−6.99273	−0.227956	−0.113978	−	0.993483i	$-0.536359\pi$
$941$	−6.99273	−0.227956	−0.113978	+	0.993483i	$0.536359\pi$
$942$	−24.3829	−0.794437
$943$	−13.0563	−0.425173
$944$	6.78668	0.220888
$945$	−2.32258	−0.0755535
$946$	65.4854	2.12911
$947$	52.3064	1.69973	0.849865	−	0.527000i	$-0.176683\pi$
$947$	52.3064	1.69973	0.849865	+	0.527000i	$0.176683\pi$
$948$	−7.96774	−0.258780
$949$	0	0
$950$	−4.02283	−0.130518
$951$	12.6667	0.410747
$952$	9.29032	0.301101
$953$	28.8827	0.935603	0.467802	−	0.883833i	$-0.345046\pi$
$953$	28.8827	0.935603	0.467802	+	0.883833i	$0.345046\pi$
$954$	5.48693	0.177646
$955$	−17.0375	−0.551319
$956$	−26.2006	−0.847387
$957$	−22.9359	−0.741413
$958$	−25.6826	−0.829768
$959$	−45.7894	−1.47862
$960$	1.00000	0.0322749
$961$	−18.9464	−0.611173
$962$	0	0
$963$	3.07180	0.0989873
$964$	−15.6496	−0.504039
$965$	−6.15491	−0.198134
$966$	−11.4641	−0.368851
$967$	−10.8610	−0.349265	−0.174633	−	0.984634i	$-0.555874\pi$
$967$	−10.8610	−0.349265	−0.174633	+	0.984634i	$0.555874\pi$
$968$	−17.5734	−0.564829
$969$	16.0913	0.516927
$970$	−16.1093	−0.517237
$971$	−24.9979	−0.802222	−0.401111	−	0.916030i	$-0.631376\pi$
$971$	−24.9979	−0.802222	−0.401111	+	0.916030i	$0.631376\pi$
$972$	1.00000	0.0320750
$973$	−13.1863	−0.422732
$974$	−9.37891	−0.300520
$975$	0	0
$976$	0.535898	0.0171537
$977$	−42.7653	−1.36818	−0.684092	−	0.729396i	$-0.739801\pi$
$977$	−42.7653	−1.36818	−0.684092	+	0.729396i	$0.739801\pi$
$978$	−23.6267	−0.755500
$979$	−9.29984	−0.297224
$980$	−1.60562	−0.0512898
$981$	6.69081	0.213621
$982$	7.73854	0.246947
$983$	−10.1288	−0.323058	−0.161529	−	0.986868i	$-0.551642\pi$
$983$	−10.1288	−0.323058	−0.161529	+	0.986868i	$0.551642\pi$
$984$	−2.64516	−0.0843246
$985$	−13.7867	−0.439280
$986$	−17.1631	−0.546584
$987$	4.22464	0.134472
$988$	0	0
$989$	−60.4691	−1.92280
$990$	5.34541	0.169888
$991$	20.2483	0.643208	0.321604	−	0.946874i	$-0.395778\pi$
$991$	20.2483	0.643208	0.321604	+	0.946874i	$0.395778\pi$
$992$	3.47183	0.110231
$993$	23.6981	0.752036
$994$	−36.8637	−1.16924
$995$	2.28304	0.0723774
$996$	11.3360	0.359194
$997$	−20.7961	−0.658620	−0.329310	−	0.944222i	$-0.606816\pi$
$997$	−20.7961	−0.658620	−0.329310	+	0.944222i	$0.606816\pi$
$998$	3.18106	0.100695
$999$	3.14925	0.0996380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bz.1.2		4
13.2	odd	12		390.2.bb.c.121.1	✓	8
13.5	odd	4		5070.2.b.ba.1351.6		8
13.7	odd	12		390.2.bb.c.361.1	yes	8
13.8	odd	4		5070.2.b.ba.1351.3		8
13.12	even	2		5070.2.a.ca.1.3		4
39.2	even	12		1170.2.bs.f.901.3		8
39.20	even	12		1170.2.bs.f.361.3		8
65.2	even	12		1950.2.y.k.199.3		8
65.7	even	12		1950.2.y.j.49.2		8
65.28	even	12		1950.2.y.j.199.2		8
65.33	even	12		1950.2.y.k.49.3		8
65.54	odd	12		1950.2.bc.g.901.4		8
65.59	odd	12		1950.2.bc.g.751.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.c.121.1	✓	8	13.2	odd	12
390.2.bb.c.361.1	yes	8	13.7	odd	12
1170.2.bs.f.361.3		8	39.20	even	12
1170.2.bs.f.901.3		8	39.2	even	12
1950.2.y.j.49.2		8	65.7	even	12
1950.2.y.j.199.2		8	65.28	even	12
1950.2.y.k.49.3		8	65.33	even	12
1950.2.y.k.199.3		8	65.2	even	12
1950.2.bc.g.751.4		8	65.59	odd	12
1950.2.bc.g.901.4		8	65.54	odd	12
5070.2.a.bz.1.2		4	1.1	even	1	trivial
5070.2.a.ca.1.3		4	13.12	even	2
5070.2.b.ba.1351.3		8	13.8	odd	4
5070.2.b.ba.1351.6		8	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.32258 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.32258 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.34541 q^{11} +1.00000 q^{12} +2.32258 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.02283 q^{19} +1.00000 q^{20} -2.32258 q^{21} +5.34541 q^{22} -4.93593 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -2.32258 q^{28} +4.29078 q^{29} -1.00000 q^{30} -3.47183 q^{31} -1.00000 q^{32} -5.34541 q^{33} -4.00000 q^{34} -2.32258 q^{35} +1.00000 q^{36} +3.14925 q^{37} -4.02283 q^{38} -1.00000 q^{40} +2.64516 q^{41} +2.32258 q^{42} +12.2508 q^{43} -5.34541 q^{44} +1.00000 q^{45} +4.93593 q^{46} -1.81894 q^{47} +1.00000 q^{48} -1.60562 q^{49} -1.00000 q^{50} +4.00000 q^{51} -5.48693 q^{53} -1.00000 q^{54} -5.34541 q^{55} +2.32258 q^{56} +4.02283 q^{57} -4.29078 q^{58} +6.78668 q^{59} +1.00000 q^{60} +0.535898 q^{61} +3.47183 q^{62} -2.32258 q^{63} +1.00000 q^{64} +5.34541 q^{66} +4.10926 q^{67} +4.00000 q^{68} -4.93593 q^{69} +2.32258 q^{70} -15.8719 q^{71} -1.00000 q^{72} -13.5734 q^{73} -3.14925 q^{74} +1.00000 q^{75} +4.02283 q^{76} +12.4151 q^{77} -7.96774 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.64516 q^{82} +11.3360 q^{83} -2.32258 q^{84} +4.00000 q^{85} -12.2508 q^{86} +4.29078 q^{87} +5.34541 q^{88} +1.73978 q^{89} -1.00000 q^{90} -4.93593 q^{92} -3.47183 q^{93} +1.81894 q^{94} +4.02283 q^{95} -1.00000 q^{96} +16.1093 q^{97} +1.60562 q^{98} -5.34541 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 16 q^{17} - 4 q^{18} + 4 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27} - 2 q^{28} + 8 q^{29} - 4 q^{30} - 4 q^{31} - 4 q^{32} + 2 q^{33} - 16 q^{34} - 2 q^{35} + 4 q^{36} + 10 q^{37} - 4 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 2 q^{44} + 4 q^{45} - 4 q^{46} - 8 q^{47} + 4 q^{48} + 14 q^{49} - 4 q^{50} + 16 q^{51} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{56} - 8 q^{58} + 6 q^{59} + 4 q^{60} + 16 q^{61} + 4 q^{62} - 2 q^{63} + 4 q^{64} - 2 q^{66} - 12 q^{67} + 16 q^{68} + 4 q^{69} + 2 q^{70} - 16 q^{71} - 4 q^{72} - 12 q^{73} - 10 q^{74} + 4 q^{75} - 8 q^{77} - 10 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{82} - 16 q^{83} - 2 q^{84} + 16 q^{85} - 14 q^{86} + 8 q^{87} - 2 q^{88} + 4 q^{89} - 4 q^{90} + 4 q^{92} - 4 q^{93} + 8 q^{94} - 4 q^{96} + 36 q^{97} - 14 q^{98} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 2 * q^14 + 4 * q^15 + 4 * q^16 + 16 * q^17 - 4 * q^18 + 4 * q^20 - 2 * q^21 - 2 * q^22 + 4 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 - 2 * q^28 + 8 * q^29 - 4 * q^30 - 4 * q^31 - 4 * q^32 + 2 * q^33 - 16 * q^34 - 2 * q^35 + 4 * q^36 + 10 * q^37 - 4 * q^40 - 4 * q^41 + 2 * q^42 + 14 * q^43 + 2 * q^44 + 4 * q^45 - 4 * q^46 - 8 * q^47 + 4 * q^48 + 14 * q^49 - 4 * q^50 + 16 * q^51 + 8 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^56 - 8 * q^58 + 6 * q^59 + 4 * q^60 + 16 * q^61 + 4 * q^62 - 2 * q^63 + 4 * q^64 - 2 * q^66 - 12 * q^67 + 16 * q^68 + 4 * q^69 + 2 * q^70 - 16 * q^71 - 4 * q^72 - 12 * q^73 - 10 * q^74 + 4 * q^75 - 8 * q^77 - 10 * q^79 + 4 * q^80 + 4 * q^81 + 4 * q^82 - 16 * q^83 - 2 * q^84 + 16 * q^85 - 14 * q^86 + 8 * q^87 - 2 * q^88 + 4 * q^89 - 4 * q^90 + 4 * q^92 - 4 * q^93 + 8 * q^94 - 4 * q^96 + 36 * q^97 - 14 * q^98 + 2 * q^99

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