LMFDB - Embedded newform 4030.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.6550837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 17x^{2} - 9x - 5 $ x^6 - x^5 - 8x^4 + 6x^3 + 17x^2 - 9x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.920227$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.92023 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.92023 q^{6} -3.60391 q^{7} +1.00000 q^{8} +0.687271 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.92023 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.92023 q^{6} -3.60391 q^{7} +1.00000 q^{8} +0.687271 q^{9} +1.00000 q^{10} +0.0845362 q^{11} -1.92023 q^{12} +1.00000 q^{13} -3.60391 q^{14} -1.92023 q^{15} +1.00000 q^{16} +4.00227 q^{17} +0.687271 q^{18} +0.208127 q^{19} +1.00000 q^{20} +6.92032 q^{21} +0.0845362 q^{22} -4.53069 q^{23} -1.92023 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.44096 q^{27} -3.60391 q^{28} +5.22918 q^{29} -1.92023 q^{30} +1.00000 q^{31} +1.00000 q^{32} -0.162329 q^{33} +4.00227 q^{34} -3.60391 q^{35} +0.687271 q^{36} -10.3530 q^{37} +0.208127 q^{38} -1.92023 q^{39} +1.00000 q^{40} -0.177400 q^{41} +6.92032 q^{42} +2.02931 q^{43} +0.0845362 q^{44} +0.687271 q^{45} -4.53069 q^{46} -8.10231 q^{47} -1.92023 q^{48} +5.98816 q^{49} +1.00000 q^{50} -7.68526 q^{51} +1.00000 q^{52} +8.39317 q^{53} +4.44096 q^{54} +0.0845362 q^{55} -3.60391 q^{56} -0.399650 q^{57} +5.22918 q^{58} -1.10053 q^{59} -1.92023 q^{60} -9.68873 q^{61} +1.00000 q^{62} -2.47686 q^{63} +1.00000 q^{64} +1.00000 q^{65} -0.162329 q^{66} -5.41187 q^{67} +4.00227 q^{68} +8.69995 q^{69} -3.60391 q^{70} +15.0695 q^{71} +0.687271 q^{72} -4.10062 q^{73} -10.3530 q^{74} -1.92023 q^{75} +0.208127 q^{76} -0.304661 q^{77} -1.92023 q^{78} -12.5377 q^{79} +1.00000 q^{80} -10.5895 q^{81} -0.177400 q^{82} -15.7793 q^{83} +6.92032 q^{84} +4.00227 q^{85} +2.02931 q^{86} -10.0412 q^{87} +0.0845362 q^{88} +3.83140 q^{89} +0.687271 q^{90} -3.60391 q^{91} -4.53069 q^{92} -1.92023 q^{93} -8.10231 q^{94} +0.208127 q^{95} -1.92023 q^{96} -0.487698 q^{97} +5.98816 q^{98} +0.0580993 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 7 q^{3} + 6 q^{4} + 6 q^{5} - 7 q^{6} - 8 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 7 * q^3 + 6 * q^4 + 6 * q^5 - 7 * q^6 - 8 * q^7 + 6 * q^8 + 7 * q^9	\( 6 q + 6 q^{2} - 7 q^{3} + 6 q^{4} + 6 q^{5} - 7 q^{6} - 8 q^{7} + 6 q^{8} + 7 q^{9} + 6 q^{10} - 10 q^{11} - 7 q^{12} + 6 q^{13} - 8 q^{14} - 7 q^{15} + 6 q^{16} - 14 q^{17} + 7 q^{18} - 3 q^{19} + 6 q^{20} + 3 q^{21} - 10 q^{22} - 13 q^{23} - 7 q^{24} + 6 q^{25} + 6 q^{26} - 25 q^{27} - 8 q^{28} - 8 q^{29} - 7 q^{30} + 6 q^{31} + 6 q^{32} + 18 q^{33} - 14 q^{34} - 8 q^{35} + 7 q^{36} - 16 q^{37} - 3 q^{38} - 7 q^{39} + 6 q^{40} - 10 q^{41} + 3 q^{42} - 17 q^{43} - 10 q^{44} + 7 q^{45} - 13 q^{46} - 14 q^{47} - 7 q^{48} + 18 q^{49} + 6 q^{50} + q^{51} + 6 q^{52} - 22 q^{53} - 25 q^{54} - 10 q^{55} - 8 q^{56} + 9 q^{57} - 8 q^{58} - 23 q^{59} - 7 q^{60} + 17 q^{61} + 6 q^{62} - 5 q^{63} + 6 q^{64} + 6 q^{65} + 18 q^{66} - 16 q^{67} - 14 q^{68} - 8 q^{70} + 2 q^{71} + 7 q^{72} - 7 q^{73} - 16 q^{74} - 7 q^{75} - 3 q^{76} - 13 q^{77} - 7 q^{78} - 8 q^{79} + 6 q^{80} + 38 q^{81} - 10 q^{82} - 16 q^{83} + 3 q^{84} - 14 q^{85} - 17 q^{86} - 3 q^{87} - 10 q^{88} - 32 q^{89} + 7 q^{90} - 8 q^{91} - 13 q^{92} - 7 q^{93} - 14 q^{94} - 3 q^{95} - 7 q^{96} + 31 q^{97} + 18 q^{98} - 17 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 7 * q^3 + 6 * q^4 + 6 * q^5 - 7 * q^6 - 8 * q^7 + 6 * q^8 + 7 * q^9 + 6 * q^10 - 10 * q^11 - 7 * q^12 + 6 * q^13 - 8 * q^14 - 7 * q^15 + 6 * q^16 - 14 * q^17 + 7 * q^18 - 3 * q^19 + 6 * q^20 + 3 * q^21 - 10 * q^22 - 13 * q^23 - 7 * q^24 + 6 * q^25 + 6 * q^26 - 25 * q^27 - 8 * q^28 - 8 * q^29 - 7 * q^30 + 6 * q^31 + 6 * q^32 + 18 * q^33 - 14 * q^34 - 8 * q^35 + 7 * q^36 - 16 * q^37 - 3 * q^38 - 7 * q^39 + 6 * q^40 - 10 * q^41 + 3 * q^42 - 17 * q^43 - 10 * q^44 + 7 * q^45 - 13 * q^46 - 14 * q^47 - 7 * q^48 + 18 * q^49 + 6 * q^50 + q^51 + 6 * q^52 - 22 * q^53 - 25 * q^54 - 10 * q^55 - 8 * q^56 + 9 * q^57 - 8 * q^58 - 23 * q^59 - 7 * q^60 + 17 * q^61 + 6 * q^62 - 5 * q^63 + 6 * q^64 + 6 * q^65 + 18 * q^66 - 16 * q^67 - 14 * q^68 - 8 * q^70 + 2 * q^71 + 7 * q^72 - 7 * q^73 - 16 * q^74 - 7 * q^75 - 3 * q^76 - 13 * q^77 - 7 * q^78 - 8 * q^79 + 6 * q^80 + 38 * q^81 - 10 * q^82 - 16 * q^83 + 3 * q^84 - 14 * q^85 - 17 * q^86 - 3 * q^87 - 10 * q^88 - 32 * q^89 + 7 * q^90 - 8 * q^91 - 13 * q^92 - 7 * q^93 - 14 * q^94 - 3 * q^95 - 7 * q^96 + 31 * q^97 + 18 * q^98 - 17 * 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
$3$	−1.92023	−1.10864	−0.554322	−	0.832302i	$-0.687022\pi$
$3$	−1.92023	−1.10864	−0.554322	+	0.832302i	$0.687022\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.92023	−0.783929
$7$	−3.60391	−1.36215	−0.681075	−	0.732214i	$-0.738487\pi$
$7$	−3.60391	−1.36215	−0.681075	+	0.732214i	$0.738487\pi$
$8$	1.00000	0.353553
$9$	0.687271	0.229090
$10$	1.00000	0.316228
$11$	0.0845362	0.0254886	0.0127443	−	0.999919i	$-0.495943\pi$
$11$	0.0845362	0.0254886	0.0127443	+	0.999919i	$0.495943\pi$
$12$	−1.92023	−0.554322
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.60391	−0.963185
$15$	−1.92023	−0.495800
$16$	1.00000	0.250000
$17$	4.00227	0.970692	0.485346	−	0.874322i	$-0.338694\pi$
$17$	4.00227	0.970692	0.485346	+	0.874322i	$0.338694\pi$
$18$	0.687271	0.161991
$19$	0.208127	0.0477475	0.0238738	−	0.999715i	$-0.492400\pi$
$19$	0.208127	0.0477475	0.0238738	+	0.999715i	$0.492400\pi$
$20$	1.00000	0.223607
$21$	6.92032	1.51014
$22$	0.0845362	0.0180232
$23$	−4.53069	−0.944714	−0.472357	−	0.881407i	$-0.656597\pi$
$23$	−4.53069	−0.944714	−0.472357	+	0.881407i	$0.656597\pi$
$24$	−1.92023	−0.391965
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	4.44096	0.854664
$28$	−3.60391	−0.681075
$29$	5.22918	0.971035	0.485518	−	0.874227i	$-0.338631\pi$
$29$	5.22918	0.971035	0.485518	+	0.874227i	$0.338631\pi$
$30$	−1.92023	−0.350584
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−0.162329	−0.0282578
$34$	4.00227	0.686383
$35$	−3.60391	−0.609172
$36$	0.687271	0.114545
$37$	−10.3530	−1.70202	−0.851010	−	0.525150i	$-0.824009\pi$
$37$	−10.3530	−1.70202	−0.851010	+	0.525150i	$0.824009\pi$
$38$	0.208127	0.0337626
$39$	−1.92023	−0.307482
$40$	1.00000	0.158114
$41$	−0.177400	−0.0277053	−0.0138526	−	0.999904i	$-0.504410\pi$
$41$	−0.177400	−0.0277053	−0.0138526	+	0.999904i	$0.504410\pi$
$42$	6.92032	1.06783
$43$	2.02931	0.309467	0.154733	−	0.987956i	$-0.450548\pi$
$43$	2.02931	0.309467	0.154733	+	0.987956i	$0.450548\pi$
$44$	0.0845362	0.0127443
$45$	0.687271	0.102452
$46$	−4.53069	−0.668014
$47$	−8.10231	−1.18184	−0.590922	−	0.806729i	$-0.701236\pi$
$47$	−8.10231	−1.18184	−0.590922	+	0.806729i	$0.701236\pi$
$48$	−1.92023	−0.277161
$49$	5.98816	0.855451
$50$	1.00000	0.141421
$51$	−7.68526	−1.07615
$52$	1.00000	0.138675
$53$	8.39317	1.15289	0.576445	−	0.817136i	$-0.304439\pi$
$53$	8.39317	1.15289	0.576445	+	0.817136i	$0.304439\pi$
$54$	4.44096	0.604339
$55$	0.0845362	0.0113989
$56$	−3.60391	−0.481593
$57$	−0.399650	−0.0529350
$58$	5.22918	0.686625
$59$	−1.10053	−0.143277	−0.0716383	−	0.997431i	$-0.522823\pi$
$59$	−1.10053	−0.143277	−0.0716383	+	0.997431i	$0.522823\pi$
$60$	−1.92023	−0.247900
$61$	−9.68873	−1.24051	−0.620257	−	0.784398i	$-0.712972\pi$
$61$	−9.68873	−1.24051	−0.620257	+	0.784398i	$0.712972\pi$
$62$	1.00000	0.127000
$63$	−2.47686	−0.312055
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−0.162329	−0.0199813
$67$	−5.41187	−0.661165	−0.330583	−	0.943777i	$-0.607245\pi$
$67$	−5.41187	−0.661165	−0.330583	+	0.943777i	$0.607245\pi$
$68$	4.00227	0.485346
$69$	8.69995	1.04735
$70$	−3.60391	−0.430750
$71$	15.0695	1.78842	0.894211	−	0.447645i	$-0.147737\pi$
$71$	15.0695	1.78842	0.894211	+	0.447645i	$0.147737\pi$
$72$	0.687271	0.0809957
$73$	−4.10062	−0.479942	−0.239971	−	0.970780i	$-0.577138\pi$
$73$	−4.10062	−0.479942	−0.239971	+	0.970780i	$0.577138\pi$
$74$	−10.3530	−1.20351
$75$	−1.92023	−0.221729
$76$	0.208127	0.0238738
$77$	−0.304661	−0.0347193
$78$	−1.92023	−0.217423
$79$	−12.5377	−1.41060	−0.705302	−	0.708907i	$-0.749189\pi$
$79$	−12.5377	−1.41060	−0.705302	+	0.708907i	$0.749189\pi$
$80$	1.00000	0.111803
$81$	−10.5895	−1.17661
$82$	−0.177400	−0.0195906
$83$	−15.7793	−1.73200	−0.865999	−	0.500045i	$-0.833317\pi$
$83$	−15.7793	−1.73200	−0.865999	+	0.500045i	$0.833317\pi$
$84$	6.92032	0.755069
$85$	4.00227	0.434107
$86$	2.02931	0.218826
$87$	−10.0412	−1.07653
$88$	0.0845362	0.00901158
$89$	3.83140	0.406127	0.203064	−	0.979166i	$-0.434910\pi$
$89$	3.83140	0.406127	0.203064	+	0.979166i	$0.434910\pi$
$90$	0.687271	0.0724447
$91$	−3.60391	−0.377792
$92$	−4.53069	−0.472357
$93$	−1.92023	−0.199118
$94$	−8.10231	−0.835690
$95$	0.208127	0.0213533
$96$	−1.92023	−0.195982
$97$	−0.487698	−0.0495183	−0.0247591	−	0.999693i	$-0.507882\pi$
$97$	−0.487698	−0.0495183	−0.0247591	+	0.999693i	$0.507882\pi$
$98$	5.98816	0.604896
$99$	0.0580993	0.00583920
$100$	1.00000	0.100000
$101$	−10.6491	−1.05962	−0.529811	−	0.848116i	$-0.677737\pi$
$101$	−10.6491	−1.05962	−0.529811	+	0.848116i	$0.677737\pi$
$102$	−7.68526	−0.760954
$103$	−11.2053	−1.10409	−0.552047	−	0.833813i	$-0.686153\pi$
$103$	−11.2053	−1.10409	−0.552047	+	0.833813i	$0.686153\pi$
$104$	1.00000	0.0980581
$105$	6.92032	0.675354
$106$	8.39317	0.815217
$107$	−7.88504	−0.762276	−0.381138	−	0.924518i	$-0.624468\pi$
$107$	−7.88504	−0.762276	−0.381138	+	0.924518i	$0.624468\pi$
$108$	4.44096	0.427332
$109$	5.96414	0.571261	0.285631	−	0.958340i	$-0.407797\pi$
$109$	5.96414	0.571261	0.285631	+	0.958340i	$0.407797\pi$
$110$	0.0845362	0.00806021
$111$	19.8801	1.88693
$112$	−3.60391	−0.340537
$113$	−15.8751	−1.49340	−0.746702	−	0.665159i	$-0.768364\pi$
$113$	−15.8751	−1.49340	−0.746702	+	0.665159i	$0.768364\pi$
$114$	−0.399650	−0.0374307
$115$	−4.53069	−0.422489
$116$	5.22918	0.485518
$117$	0.687271	0.0635382
$118$	−1.10053	−0.101312
$119$	−14.4238	−1.32223
$120$	−1.92023	−0.175292
$121$	−10.9929	−0.999350
$122$	−9.68873	−0.877177
$123$	0.340649	0.0307153
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	−2.47686	−0.220656
$127$	−15.3821	−1.36494	−0.682472	−	0.730912i	$-0.739095\pi$
$127$	−15.3821	−1.36494	−0.682472	+	0.730912i	$0.739095\pi$
$128$	1.00000	0.0883883
$129$	−3.89673	−0.343088
$130$	1.00000	0.0877058
$131$	−22.4474	−1.96123	−0.980617	−	0.195933i	$-0.937226\pi$
$131$	−22.4474	−1.96123	−0.980617	+	0.195933i	$0.937226\pi$
$132$	−0.162329	−0.0141289
$133$	−0.750069	−0.0650392
$134$	−5.41187	−0.467514
$135$	4.44096	0.382217
$136$	4.00227	0.343192
$137$	17.8271	1.52307	0.761537	−	0.648121i	$-0.224445\pi$
$137$	17.8271	1.52307	0.761537	+	0.648121i	$0.224445\pi$
$138$	8.69995	0.740589
$139$	20.9373	1.77588	0.887939	−	0.459961i	$-0.152136\pi$
$139$	20.9373	1.77588	0.887939	+	0.459961i	$0.152136\pi$
$140$	−3.60391	−0.304586
$141$	15.5583	1.31024
$142$	15.0695	1.26461
$143$	0.0845362	0.00706927
$144$	0.687271	0.0572726
$145$	5.22918	0.434260
$146$	−4.10062	−0.339370
$147$	−11.4986	−0.948391
$148$	−10.3530	−0.851010
$149$	2.23186	0.182841	0.0914207	−	0.995812i	$-0.470859\pi$
$149$	2.23186	0.182841	0.0914207	+	0.995812i	$0.470859\pi$
$150$	−1.92023	−0.156786
$151$	−20.5846	−1.67515	−0.837573	−	0.546325i	$-0.816026\pi$
$151$	−20.5846	−1.67515	−0.837573	+	0.546325i	$0.816026\pi$
$152$	0.208127	0.0168813
$153$	2.75064	0.222376
$154$	−0.304661	−0.0245503
$155$	1.00000	0.0803219
$156$	−1.92023	−0.153741
$157$	15.7190	1.25452	0.627258	−	0.778812i	$-0.284177\pi$
$157$	15.7190	1.25452	0.627258	+	0.778812i	$0.284177\pi$
$158$	−12.5377	−0.997447
$159$	−16.1168	−1.27814
$160$	1.00000	0.0790569
$161$	16.3282	1.28684
$162$	−10.5895	−0.831987
$163$	10.1189	0.792573	0.396286	−	0.918127i	$-0.370299\pi$
$163$	10.1189	0.792573	0.396286	+	0.918127i	$0.370299\pi$
$164$	−0.177400	−0.0138526
$165$	−0.162329	−0.0126373
$166$	−15.7793	−1.22471
$167$	4.66955	0.361341	0.180670	−	0.983544i	$-0.442173\pi$
$167$	4.66955	0.361341	0.180670	+	0.983544i	$0.442173\pi$
$168$	6.92032	0.533915
$169$	1.00000	0.0769231
$170$	4.00227	0.306960
$171$	0.143039	0.0109385
$172$	2.02931	0.154733
$173$	−1.18414	−0.0900288	−0.0450144	−	0.998986i	$-0.514333\pi$
$173$	−1.18414	−0.0900288	−0.0450144	+	0.998986i	$0.514333\pi$
$174$	−10.0412	−0.761223
$175$	−3.60391	−0.272430
$176$	0.0845362	0.00637215
$177$	2.11326	0.158843
$178$	3.83140	0.287175
$179$	−4.36624	−0.326348	−0.163174	−	0.986597i	$-0.552173\pi$
$179$	−4.36624	−0.326348	−0.163174	+	0.986597i	$0.552173\pi$
$180$	0.687271	0.0512262
$181$	7.60738	0.565452	0.282726	−	0.959201i	$-0.408761\pi$
$181$	7.60738	0.565452	0.282726	+	0.959201i	$0.408761\pi$
$182$	−3.60391	−0.267140
$183$	18.6046	1.37529
$184$	−4.53069	−0.334007
$185$	−10.3530	−0.761166
$186$	−1.92023	−0.140798
$187$	0.338336	0.0247416
$188$	−8.10231	−0.590922
$189$	−16.0048	−1.16418
$190$	0.208127	0.0150991
$191$	15.9018	1.15062	0.575308	−	0.817937i	$-0.304882\pi$
$191$	15.9018	1.15062	0.575308	+	0.817937i	$0.304882\pi$
$192$	−1.92023	−0.138580
$193$	−16.5213	−1.18923	−0.594615	−	0.804010i	$-0.702696\pi$
$193$	−16.5213	−1.18923	−0.594615	+	0.804010i	$0.702696\pi$
$194$	−0.487698	−0.0350147
$195$	−1.92023	−0.137510
$196$	5.98816	0.427726
$197$	10.0618	0.716877	0.358438	−	0.933553i	$-0.383309\pi$
$197$	10.0618	0.716877	0.358438	+	0.933553i	$0.383309\pi$
$198$	0.0580993	0.00412893
$199$	−8.63101	−0.611836	−0.305918	−	0.952058i	$-0.598963\pi$
$199$	−8.63101	−0.611836	−0.305918	+	0.952058i	$0.598963\pi$
$200$	1.00000	0.0707107
$201$	10.3920	0.732996
$202$	−10.6491	−0.749266
$203$	−18.8455	−1.32270
$204$	−7.68526	−0.538076
$205$	−0.177400	−0.0123902
$206$	−11.2053	−0.780712
$207$	−3.11381	−0.216425
$208$	1.00000	0.0693375
$209$	0.0175942	0.00121702
$210$	6.92032	0.477548
$211$	15.8817	1.09334	0.546669	−	0.837349i	$-0.315895\pi$
$211$	15.8817	1.09334	0.546669	+	0.837349i	$0.315895\pi$
$212$	8.39317	0.576445
$213$	−28.9369	−1.98272
$214$	−7.88504	−0.539010
$215$	2.02931	0.138398
$216$	4.44096	0.302169
$217$	−3.60391	−0.244649
$218$	5.96414	0.403943
$219$	7.87413	0.532084
$220$	0.0845362	0.00569943
$221$	4.00227	0.269222
$222$	19.8801	1.33426
$223$	−24.8754	−1.66578	−0.832890	−	0.553439i	$-0.813315\pi$
$223$	−24.8754	−1.66578	−0.832890	+	0.553439i	$0.813315\pi$
$224$	−3.60391	−0.240796
$225$	0.687271	0.0458181
$226$	−15.8751	−1.05600
$227$	18.9628	1.25861	0.629303	−	0.777160i	$-0.283340\pi$
$227$	18.9628	1.25861	0.629303	+	0.777160i	$0.283340\pi$
$228$	−0.399650	−0.0264675
$229$	−13.5063	−0.892519	−0.446259	−	0.894904i	$-0.647244\pi$
$229$	−13.5063	−0.892519	−0.446259	+	0.894904i	$0.647244\pi$
$230$	−4.53069	−0.298745
$231$	0.585017	0.0384913
$232$	5.22918	0.343313
$233$	11.9733	0.784398	0.392199	−	0.919880i	$-0.371715\pi$
$233$	11.9733	0.784398	0.392199	+	0.919880i	$0.371715\pi$
$234$	0.687271	0.0449283
$235$	−8.10231	−0.528537
$236$	−1.10053	−0.0716383
$237$	24.0753	1.56386
$238$	−14.4238	−0.934956
$239$	−7.07060	−0.457359	−0.228679	−	0.973502i	$-0.573441\pi$
$239$	−7.07060	−0.457359	−0.228679	+	0.973502i	$0.573441\pi$
$240$	−1.92023	−0.123950
$241$	19.7777	1.27399	0.636996	−	0.770867i	$-0.280177\pi$
$241$	19.7777	1.27399	0.636996	+	0.770867i	$0.280177\pi$
$242$	−10.9929	−0.706647
$243$	7.01130	0.449775
$244$	−9.68873	−0.620257
$245$	5.98816	0.382570
$246$	0.340649	0.0217190
$247$	0.208127	0.0132428
$248$	1.00000	0.0635001
$249$	30.2998	1.92017
$250$	1.00000	0.0632456
$251$	−2.49802	−0.157674	−0.0788369	−	0.996888i	$-0.525121\pi$
$251$	−2.49802	−0.157674	−0.0788369	+	0.996888i	$0.525121\pi$
$252$	−2.47686	−0.156028
$253$	−0.383007	−0.0240794
$254$	−15.3821	−0.965161
$255$	−7.68526	−0.481270
$256$	1.00000	0.0625000
$257$	−14.0483	−0.876310	−0.438155	−	0.898899i	$-0.644368\pi$
$257$	−14.0483	−0.876310	−0.438155	+	0.898899i	$0.644368\pi$
$258$	−3.89673	−0.242600
$259$	37.3112	2.31841
$260$	1.00000	0.0620174
$261$	3.59387	0.222455
$262$	−22.4474	−1.38680
$263$	−11.6971	−0.721275	−0.360638	−	0.932706i	$-0.617441\pi$
$263$	−11.6971	−0.721275	−0.360638	+	0.932706i	$0.617441\pi$
$264$	−0.162329	−0.00999063
$265$	8.39317	0.515588
$266$	−0.750069	−0.0459897
$267$	−7.35715	−0.450250
$268$	−5.41187	−0.330583
$269$	1.00288	0.0611469	0.0305734	−	0.999533i	$-0.490267\pi$
$269$	1.00288	0.0611469	0.0305734	+	0.999533i	$0.490267\pi$
$270$	4.44096	0.270268
$271$	−26.1579	−1.58898	−0.794491	−	0.607276i	$-0.792262\pi$
$271$	−26.1579	−1.58898	−0.794491	+	0.607276i	$0.792262\pi$
$272$	4.00227	0.242673
$273$	6.92032	0.418837
$274$	17.8271	1.07698
$275$	0.0845362	0.00509772
$276$	8.69995	0.523676
$277$	−30.9679	−1.86068	−0.930340	−	0.366698i	$-0.880488\pi$
$277$	−30.9679	−1.86068	−0.930340	+	0.366698i	$0.880488\pi$
$278$	20.9373	1.25574
$279$	0.687271	0.0411459
$280$	−3.60391	−0.215375
$281$	−3.80555	−0.227020	−0.113510	−	0.993537i	$-0.536209\pi$
$281$	−3.80555	−0.227020	−0.113510	+	0.993537i	$0.536209\pi$
$282$	15.5583	0.926482
$283$	−31.2796	−1.85938	−0.929690	−	0.368342i	$-0.879926\pi$
$283$	−31.2796	−1.85938	−0.929690	+	0.368342i	$0.879926\pi$
$284$	15.0695	0.894211
$285$	−0.399650	−0.0236732
$286$	0.0845362	0.00499873
$287$	0.639334	0.0377387
$288$	0.687271	0.0404978
$289$	−0.981859	−0.0577564
$290$	5.22918	0.307068
$291$	0.936491	0.0548981
$292$	−4.10062	−0.239971
$293$	−24.3949	−1.42517	−0.712583	−	0.701587i	$-0.752475\pi$
$293$	−24.3949	−1.42517	−0.712583	+	0.701587i	$0.752475\pi$
$294$	−11.4986	−0.670613
$295$	−1.10053	−0.0640752
$296$	−10.3530	−0.601755
$297$	0.375422	0.0217842
$298$	2.23186	0.129288
$299$	−4.53069	−0.262017
$300$	−1.92023	−0.110864
$301$	−7.31344	−0.421540
$302$	−20.5846	−1.18451
$303$	20.4486	1.17474
$304$	0.208127	0.0119369
$305$	−9.68873	−0.554775
$306$	2.75064	0.157244
$307$	−8.03795	−0.458750	−0.229375	−	0.973338i	$-0.573668\pi$
$307$	−8.03795	−0.458750	−0.229375	+	0.973338i	$0.573668\pi$
$308$	−0.304661	−0.0173596
$309$	21.5168	1.22405
$310$	1.00000	0.0567962
$311$	−20.5480	−1.16517	−0.582585	−	0.812769i	$-0.697959\pi$
$311$	−20.5480	−1.16517	−0.582585	+	0.812769i	$0.697959\pi$
$312$	−1.92023	−0.108711
$313$	−10.1538	−0.573925	−0.286962	−	0.957942i	$-0.592646\pi$
$313$	−10.1538	−0.573925	−0.286962	+	0.957942i	$0.592646\pi$
$314$	15.7190	0.887077
$315$	−2.47686	−0.139555
$316$	−12.5377	−0.705302
$317$	17.5130	0.983626	0.491813	−	0.870701i	$-0.336334\pi$
$317$	17.5130	0.983626	0.491813	+	0.870701i	$0.336334\pi$
$318$	−16.1168	−0.903785
$319$	0.442055	0.0247503
$320$	1.00000	0.0559017
$321$	15.1411	0.845092
$322$	16.3282	0.909935
$323$	0.832978	0.0463481
$324$	−10.5895	−0.588304
$325$	1.00000	0.0554700
$326$	10.1189	0.560434
$327$	−11.4525	−0.633325
$328$	−0.177400	−0.00979529
$329$	29.2000	1.60985
$330$	−0.162329	−0.00893589
$331$	10.8893	0.598528	0.299264	−	0.954170i	$-0.403259\pi$
$331$	10.8893	0.598528	0.299264	+	0.954170i	$0.403259\pi$
$332$	−15.7793	−0.865999
$333$	−7.11531	−0.389916
$334$	4.66955	0.255507
$335$	−5.41187	−0.295682
$336$	6.92032	0.377535
$337$	−17.0399	−0.928221	−0.464111	−	0.885777i	$-0.653626\pi$
$337$	−17.0399	−0.928221	−0.464111	+	0.885777i	$0.653626\pi$
$338$	1.00000	0.0543928
$339$	30.4838	1.65565
$340$	4.00227	0.217053
$341$	0.0845362	0.00457789
$342$	0.143039	0.00773468
$343$	3.64658	0.196897
$344$	2.02931	0.109413
$345$	8.69995	0.468390
$346$	−1.18414	−0.0636600
$347$	−24.3037	−1.30469	−0.652346	−	0.757921i	$-0.726215\pi$
$347$	−24.3037	−1.30469	−0.652346	+	0.757921i	$0.726215\pi$
$348$	−10.0412	−0.538266
$349$	25.5954	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$349$	25.5954	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$350$	−3.60391	−0.192637
$351$	4.44096	0.237041
$352$	0.0845362	0.00450579
$353$	28.9862	1.54278	0.771390	−	0.636362i	$-0.219562\pi$
$353$	28.9862	1.54278	0.771390	+	0.636362i	$0.219562\pi$
$354$	2.11326	0.112319
$355$	15.0695	0.799807
$356$	3.83140	0.203064
$357$	27.6970	1.46588
$358$	−4.36624	−0.230763
$359$	−0.998497	−0.0526987	−0.0263493	−	0.999653i	$-0.508388\pi$
$359$	−0.998497	−0.0526987	−0.0263493	+	0.999653i	$0.508388\pi$
$360$	0.687271	0.0362224
$361$	−18.9567	−0.997720
$362$	7.60738	0.399835
$363$	21.1088	1.10792
$364$	−3.60391	−0.188896
$365$	−4.10062	−0.214636
$366$	18.6046	0.972476
$367$	−1.11288	−0.0580920	−0.0290460	−	0.999578i	$-0.509247\pi$
$367$	−1.11288	−0.0580920	−0.0290460	+	0.999578i	$0.509247\pi$
$368$	−4.53069	−0.236179
$369$	−0.121922	−0.00634701
$370$	−10.3530	−0.538226
$371$	−30.2482	−1.57041
$372$	−1.92023	−0.0995591
$373$	−34.9834	−1.81137	−0.905685	−	0.423951i	$-0.860643\pi$
$373$	−34.9834	−1.81137	−0.905685	+	0.423951i	$0.860643\pi$
$374$	0.338336	0.0174950
$375$	−1.92023	−0.0991601
$376$	−8.10231	−0.417845
$377$	5.22918	0.269317
$378$	−16.0048	−0.823200
$379$	19.6093	1.00726	0.503630	−	0.863920i	$-0.331998\pi$
$379$	19.6093	1.00726	0.503630	+	0.863920i	$0.331998\pi$
$380$	0.208127	0.0106767
$381$	29.5372	1.51324
$382$	15.9018	0.813609
$383$	−7.68692	−0.392783	−0.196392	−	0.980526i	$-0.562922\pi$
$383$	−7.68692	−0.392783	−0.196392	+	0.980526i	$0.562922\pi$
$384$	−1.92023	−0.0979912
$385$	−0.304661	−0.0155269
$386$	−16.5213	−0.840913
$387$	1.39469	0.0708958
$388$	−0.487698	−0.0247591
$389$	−14.0947	−0.714628	−0.357314	−	0.933984i	$-0.616307\pi$
$389$	−14.0947	−0.714628	−0.357314	+	0.933984i	$0.616307\pi$
$390$	−1.92023	−0.0972345
$391$	−18.1330	−0.917027
$392$	5.98816	0.302448
$393$	43.1040	2.17431
$394$	10.0618	0.506909
$395$	−12.5377	−0.630841
$396$	0.0580993	0.00291960
$397$	2.07834	0.104309	0.0521545	−	0.998639i	$-0.483391\pi$
$397$	2.07834	0.104309	0.0521545	+	0.998639i	$0.483391\pi$
$398$	−8.63101	−0.432634
$399$	1.44030	0.0721053
$400$	1.00000	0.0500000
$401$	−19.4715	−0.972360	−0.486180	−	0.873859i	$-0.661610\pi$
$401$	−19.4715	−0.972360	−0.486180	+	0.873859i	$0.661610\pi$
$402$	10.3920	0.518307
$403$	1.00000	0.0498135
$404$	−10.6491	−0.529811
$405$	−10.5895	−0.526195
$406$	−18.8455	−0.935287
$407$	−0.875201	−0.0433821
$408$	−7.68526	−0.380477
$409$	26.8376	1.32704	0.663518	−	0.748160i	$-0.269063\pi$
$409$	26.8376	1.32704	0.663518	+	0.748160i	$0.269063\pi$
$410$	−0.177400	−0.00876117
$411$	−34.2321	−1.68855
$412$	−11.2053	−0.552047
$413$	3.96620	0.195164
$414$	−3.11381	−0.153036
$415$	−15.7793	−0.774573
$416$	1.00000	0.0490290
$417$	−40.2044	−1.96882
$418$	0.0175942	0.000860561 0
$419$	35.6221	1.74025	0.870126	−	0.492830i	$-0.164038\pi$
$419$	35.6221	1.74025	0.870126	+	0.492830i	$0.164038\pi$
$420$	6.92032	0.337677
$421$	1.73974	0.0847898	0.0423949	−	0.999101i	$-0.486501\pi$
$421$	1.73974	0.0847898	0.0423949	+	0.999101i	$0.486501\pi$
$422$	15.8817	0.773107
$423$	−5.56849	−0.270749
$424$	8.39317	0.407608
$425$	4.00227	0.194138
$426$	−28.9369	−1.40200
$427$	34.9173	1.68977
$428$	−7.88504	−0.381138
$429$	−0.162329	−0.00783730
$430$	2.02931	0.0978619
$431$	26.9292	1.29713	0.648566	−	0.761158i	$-0.275369\pi$
$431$	26.9292	1.29713	0.648566	+	0.761158i	$0.275369\pi$
$432$	4.44096	0.213666
$433$	0.0763143	0.00366743	0.00183372	−	0.999998i	$-0.499416\pi$
$433$	0.0763143	0.00366743	0.00183372	+	0.999998i	$0.499416\pi$
$434$	−3.60391	−0.172993
$435$	−10.0412	−0.481440
$436$	5.96414	0.285631
$437$	−0.942957	−0.0451077
$438$	7.87413	0.376240
$439$	37.1534	1.77323	0.886617	−	0.462504i	$-0.153049\pi$
$439$	37.1534	1.77323	0.886617	+	0.462504i	$0.153049\pi$
$440$	0.0845362	0.00403010
$441$	4.11549	0.195976
$442$	4.00227	0.190368
$443$	14.5089	0.689339	0.344670	−	0.938724i	$-0.387991\pi$
$443$	14.5089	0.689339	0.344670	+	0.938724i	$0.387991\pi$
$444$	19.8801	0.943466
$445$	3.83140	0.181626
$446$	−24.8754	−1.17788
$447$	−4.28568	−0.202706
$448$	−3.60391	−0.170269
$449$	−2.27975	−0.107588	−0.0537939	−	0.998552i	$-0.517131\pi$
$449$	−2.27975	−0.107588	−0.0537939	+	0.998552i	$0.517131\pi$
$450$	0.687271	0.0323983
$451$	−0.0149967	−0.000706169 0
$452$	−15.8751	−0.746702
$453$	39.5270	1.85714
$454$	18.9628	0.889969
$455$	−3.60391	−0.168954
$456$	−0.399650	−0.0187153
$457$	−33.3130	−1.55832	−0.779158	−	0.626827i	$-0.784353\pi$
$457$	−33.3130	−1.55832	−0.779158	+	0.626827i	$0.784353\pi$
$458$	−13.5063	−0.631106
$459$	17.7739	0.829616
$460$	−4.53069	−0.211245
$461$	37.9379	1.76695	0.883473	−	0.468482i	$-0.155199\pi$
$461$	37.9379	1.76695	0.883473	+	0.468482i	$0.155199\pi$
$462$	0.585017	0.0272175
$463$	−15.7948	−0.734046	−0.367023	−	0.930212i	$-0.619623\pi$
$463$	−15.7948	−0.734046	−0.367023	+	0.930212i	$0.619623\pi$
$464$	5.22918	0.242759
$465$	−1.92023	−0.0890484
$466$	11.9733	0.554653
$467$	34.2977	1.58711	0.793554	−	0.608500i	$-0.208228\pi$
$467$	34.2977	1.58711	0.793554	+	0.608500i	$0.208228\pi$
$468$	0.687271	0.0317691
$469$	19.5039	0.900606
$470$	−8.10231	−0.373732
$471$	−30.1841	−1.39081
$472$	−1.10053	−0.0506559
$473$	0.171550	0.00788787
$474$	24.0753	1.10581
$475$	0.208127	0.00954950
$476$	−14.4238	−0.661114
$477$	5.76838	0.264116
$478$	−7.07060	−0.323402
$479$	25.4663	1.16359	0.581793	−	0.813337i	$-0.302351\pi$
$479$	25.4663	1.16359	0.581793	+	0.813337i	$0.302351\pi$
$480$	−1.92023	−0.0876460
$481$	−10.3530	−0.472055
$482$	19.7777	0.900848
$483$	−31.3538	−1.42665
$484$	−10.9929	−0.499675
$485$	−0.487698	−0.0221452
$486$	7.01130	0.318039
$487$	25.2048	1.14214	0.571070	−	0.820902i	$-0.306529\pi$
$487$	25.2048	1.14214	0.571070	+	0.820902i	$0.306529\pi$
$488$	−9.68873	−0.438588
$489$	−19.4306	−0.878681
$490$	5.98816	0.270517
$491$	−28.8389	−1.30148	−0.650741	−	0.759299i	$-0.725542\pi$
$491$	−28.8389	−1.30148	−0.650741	+	0.759299i	$0.725542\pi$
$492$	0.340649	0.0153576
$493$	20.9286	0.942576
$494$	0.208127	0.00936406
$495$	0.0580993	0.00261137
$496$	1.00000	0.0449013
$497$	−54.3092	−2.43610
$498$	30.2998	1.35776
$499$	35.7033	1.59830	0.799149	−	0.601133i	$-0.205284\pi$
$499$	35.7033	1.59830	0.799149	+	0.601133i	$0.205284\pi$
$500$	1.00000	0.0447214
$501$	−8.96660	−0.400598
$502$	−2.49802	−0.111492
$503$	23.2277	1.03567	0.517836	−	0.855480i	$-0.326738\pi$
$503$	23.2277	1.03567	0.517836	+	0.855480i	$0.326738\pi$
$504$	−2.47686	−0.110328
$505$	−10.6491	−0.473878
$506$	−0.383007	−0.0170267
$507$	−1.92023	−0.0852803
$508$	−15.3821	−0.682472
$509$	8.99893	0.398871	0.199435	−	0.979911i	$-0.436089\pi$
$509$	8.99893	0.398871	0.199435	+	0.979911i	$0.436089\pi$
$510$	−7.68526	−0.340309
$511$	14.7783	0.653752
$512$	1.00000	0.0441942
$513$	0.924283	0.0408081
$514$	−14.0483	−0.619645
$515$	−11.2053	−0.493765
$516$	−3.89673	−0.171544
$517$	−0.684938	−0.0301236
$518$	37.3112	1.63936
$519$	2.27383	0.0998098
$520$	1.00000	0.0438529
$521$	15.7072	0.688144	0.344072	−	0.938943i	$-0.388194\pi$
$521$	15.7072	0.688144	0.344072	+	0.938943i	$0.388194\pi$
$522$	3.59387	0.157299
$523$	−1.69271	−0.0740170	−0.0370085	−	0.999315i	$-0.511783\pi$
$523$	−1.69271	−0.0740170	−0.0370085	+	0.999315i	$0.511783\pi$
$524$	−22.4474	−0.980617
$525$	6.92032	0.302028
$526$	−11.6971	−0.510019
$527$	4.00227	0.174341
$528$	−0.162329	−0.00706445
$529$	−2.47285	−0.107515
$530$	8.39317	0.364576
$531$	−0.756361	−0.0328233
$532$	−0.750069	−0.0325196
$533$	−0.177400	−0.00768406
$534$	−7.35715	−0.318375
$535$	−7.88504	−0.340900
$536$	−5.41187	−0.233757
$537$	8.38417	0.361804
$538$	1.00288	0.0432374
$539$	0.506216	0.0218043
$540$	4.44096	0.191109
$541$	−32.2669	−1.38726	−0.693631	−	0.720331i	$-0.743990\pi$
$541$	−32.2669	−1.38726	−0.693631	+	0.720331i	$0.743990\pi$
$542$	−26.1579	−1.12358
$543$	−14.6079	−0.626885
$544$	4.00227	0.171596
$545$	5.96414	0.255476
$546$	6.92032	0.296162
$547$	−43.5958	−1.86402	−0.932010	−	0.362432i	$-0.881947\pi$
$547$	−43.5958	−1.86402	−0.932010	+	0.362432i	$0.881947\pi$
$548$	17.8271	0.761537
$549$	−6.65879	−0.284190
$550$	0.0845362	0.00360463
$551$	1.08833	0.0463645
$552$	8.69995	0.370295
$553$	45.1848	1.92145
$554$	−30.9679	−1.31570
$555$	19.8801	0.843862
$556$	20.9373	0.887939
$557$	4.20596	0.178212	0.0891062	−	0.996022i	$-0.471599\pi$
$557$	4.20596	0.178212	0.0891062	+	0.996022i	$0.471599\pi$
$558$	0.687271	0.0290945
$559$	2.02931	0.0858306
$560$	−3.60391	−0.152293
$561$	−0.649682	−0.0274296
$562$	−3.80555	−0.160527
$563$	23.8263	1.00416	0.502080	−	0.864821i	$-0.332568\pi$
$563$	23.8263	1.00416	0.502080	+	0.864821i	$0.332568\pi$
$564$	15.5583	0.655122
$565$	−15.8751	−0.667870
$566$	−31.2796	−1.31478
$567$	38.1635	1.60272
$568$	15.0695	0.632303
$569$	25.8219	1.08251	0.541255	−	0.840858i	$-0.317949\pi$
$569$	25.8219	1.08251	0.541255	+	0.840858i	$0.317949\pi$
$570$	−0.399650	−0.0167395
$571$	17.8577	0.747321	0.373660	−	0.927566i	$-0.378103\pi$
$571$	17.8577	0.747321	0.373660	+	0.927566i	$0.378103\pi$
$572$	0.0845362	0.00353463
$573$	−30.5351	−1.27562
$574$	0.639334	0.0266853
$575$	−4.53069	−0.188943
$576$	0.687271	0.0286363
$577$	3.49487	0.145493	0.0727467	−	0.997350i	$-0.476824\pi$
$577$	3.49487	0.145493	0.0727467	+	0.997350i	$0.476824\pi$
$578$	−0.981859	−0.0408399
$579$	31.7247	1.31843
$580$	5.22918	0.217130
$581$	56.8670	2.35924
$582$	0.936491	0.0388188
$583$	0.709526	0.0293856
$584$	−4.10062	−0.169685
$585$	0.687271	0.0284152
$586$	−24.3949	−1.00775
$587$	−21.6727	−0.894527	−0.447264	−	0.894402i	$-0.647601\pi$
$587$	−21.6727	−0.894527	−0.447264	+	0.894402i	$0.647601\pi$
$588$	−11.4986	−0.474195
$589$	0.208127	0.00857571
$590$	−1.10053	−0.0453080
$591$	−19.3210	−0.794761
$592$	−10.3530	−0.425505
$593$	8.92510	0.366510	0.183255	−	0.983065i	$-0.441337\pi$
$593$	8.92510	0.366510	0.183255	+	0.983065i	$0.441337\pi$
$594$	0.375422	0.0154038
$595$	−14.4238	−0.591318
$596$	2.23186	0.0914207
$597$	16.5735	0.678308
$598$	−4.53069	−0.185274
$599$	−33.6969	−1.37682	−0.688410	−	0.725322i	$-0.741691\pi$
$599$	−33.6969	−1.37682	−0.688410	+	0.725322i	$0.741691\pi$
$600$	−1.92023	−0.0783929
$601$	11.1419	0.454489	0.227244	−	0.973838i	$-0.427028\pi$
$601$	11.1419	0.454489	0.227244	+	0.973838i	$0.427028\pi$
$602$	−7.31344	−0.298074
$603$	−3.71942	−0.151467
$604$	−20.5846	−0.837573
$605$	−10.9929	−0.446923
$606$	20.4486	0.830669
$607$	−15.1198	−0.613695	−0.306848	−	0.951759i	$-0.599274\pi$
$607$	−15.1198	−0.613695	−0.306848	+	0.951759i	$0.599274\pi$
$608$	0.208127	0.00844065
$609$	36.1876	1.46640
$610$	−9.68873	−0.392285
$611$	−8.10231	−0.327784
$612$	2.75064	0.111188
$613$	−15.1847	−0.613302	−0.306651	−	0.951822i	$-0.599209\pi$
$613$	−15.1847	−0.613302	−0.306651	+	0.951822i	$0.599209\pi$
$614$	−8.03795	−0.324385
$615$	0.340649	0.0137363
$616$	−0.304661	−0.0122751
$617$	43.1575	1.73746	0.868728	−	0.495290i	$-0.164938\pi$
$617$	43.1575	1.73746	0.868728	+	0.495290i	$0.164938\pi$
$618$	21.5168	0.865531
$619$	18.9295	0.760841	0.380421	−	0.924814i	$-0.375779\pi$
$619$	18.9295	0.760841	0.380421	+	0.924814i	$0.375779\pi$
$620$	1.00000	0.0401610
$621$	−20.1206	−0.807413
$622$	−20.5480	−0.823900
$623$	−13.8080	−0.553206
$624$	−1.92023	−0.0768706
$625$	1.00000	0.0400000
$626$	−10.1538	−0.405826
$627$	−0.0337849	−0.00134924
$628$	15.7190	0.627258
$629$	−41.4354	−1.65214
$630$	−2.47686	−0.0986806
$631$	−26.0088	−1.03539	−0.517696	−	0.855565i	$-0.673210\pi$
$631$	−26.0088	−1.03539	−0.517696	+	0.855565i	$0.673210\pi$
$632$	−12.5377	−0.498724
$633$	−30.4964	−1.21212
$634$	17.5130	0.695528
$635$	−15.3821	−0.610421
$636$	−16.1168	−0.639072
$637$	5.98816	0.237260
$638$	0.442055	0.0175011
$639$	10.3568	0.409711
$640$	1.00000	0.0395285
$641$	−0.651791	−0.0257442	−0.0128721	−	0.999917i	$-0.504097\pi$
$641$	−0.651791	−0.0257442	−0.0128721	+	0.999917i	$0.504097\pi$
$642$	15.1411	0.597570
$643$	−44.5930	−1.75858	−0.879288	−	0.476291i	$-0.841981\pi$
$643$	−44.5930	−1.75858	−0.879288	+	0.476291i	$0.841981\pi$
$644$	16.3282	0.643421
$645$	−3.89673	−0.153434
$646$	0.832978	0.0327731
$647$	−29.5519	−1.16180	−0.580902	−	0.813974i	$-0.697300\pi$
$647$	−29.5519	−1.16180	−0.580902	+	0.813974i	$0.697300\pi$
$648$	−10.5895	−0.415994
$649$	−0.0930344	−0.00365192
$650$	1.00000	0.0392232
$651$	6.92032	0.271229
$652$	10.1189	0.396286
$653$	36.1911	1.41627	0.708134	−	0.706078i	$-0.249537\pi$
$653$	36.1911	1.41627	0.708134	+	0.706078i	$0.249537\pi$
$654$	−11.4525	−0.447828
$655$	−22.4474	−0.877091
$656$	−0.177400	−0.00692632
$657$	−2.81824	−0.109950
$658$	29.2000	1.13833
$659$	−20.8487	−0.812148	−0.406074	−	0.913840i	$-0.633103\pi$
$659$	−20.8487	−0.812148	−0.406074	+	0.913840i	$0.633103\pi$
$660$	−0.162329	−0.00631863
$661$	−42.7596	−1.66316	−0.831578	−	0.555407i	$-0.812562\pi$
$661$	−42.7596	−1.66316	−0.831578	+	0.555407i	$0.812562\pi$
$662$	10.8893	0.423223
$663$	−7.68526	−0.298471
$664$	−15.7793	−0.612354
$665$	−0.750069	−0.0290864
$666$	−7.11531	−0.275713
$667$	−23.6918	−0.917351
$668$	4.66955	0.180670
$669$	47.7664	1.84676
$670$	−5.41187	−0.209079
$671$	−0.819048	−0.0316190
$672$	6.92032	0.266957
$673$	35.1084	1.35333	0.676665	−	0.736291i	$-0.263424\pi$
$673$	35.1084	1.35333	0.676665	+	0.736291i	$0.263424\pi$
$674$	−17.0399	−0.656352
$675$	4.44096	0.170933
$676$	1.00000	0.0384615
$677$	−17.8384	−0.685585	−0.342792	−	0.939411i	$-0.611373\pi$
$677$	−17.8384	−0.685585	−0.342792	+	0.939411i	$0.611373\pi$
$678$	30.4838	1.17072
$679$	1.75762	0.0674513
$680$	4.00227	0.153480
$681$	−36.4129	−1.39535
$682$	0.0845362	0.00323706
$683$	4.17013	0.159565	0.0797827	−	0.996812i	$-0.474577\pi$
$683$	4.17013	0.159565	0.0797827	+	0.996812i	$0.474577\pi$
$684$	0.143039	0.00546925
$685$	17.8271	0.681139
$686$	3.64658	0.139227
$687$	25.9351	0.989485
$688$	2.02931	0.0773666
$689$	8.39317	0.319754
$690$	8.69995	0.331202
$691$	6.16884	0.234674	0.117337	−	0.993092i	$-0.462564\pi$
$691$	6.16884	0.234674	0.117337	+	0.993092i	$0.462564\pi$
$692$	−1.18414	−0.0450144
$693$	−0.209384	−0.00795386
$694$	−24.3037	−0.922557
$695$	20.9373	0.794197
$696$	−10.0412	−0.380611
$697$	−0.710003	−0.0268933
$698$	25.5954	0.968798
$699$	−22.9915	−0.869617
$700$	−3.60391	−0.136215
$701$	−11.7232	−0.442779	−0.221390	−	0.975185i	$-0.571059\pi$
$701$	−11.7232	−0.442779	−0.221390	+	0.975185i	$0.571059\pi$
$702$	4.44096	0.167613
$703$	−2.15473	−0.0812672
$704$	0.0845362	0.00318608
$705$	15.5583	0.585959
$706$	28.9862	1.09091
$707$	38.3783	1.44336
$708$	2.11326	0.0794213
$709$	−6.78640	−0.254868	−0.127434	−	0.991847i	$-0.540674\pi$
$709$	−6.78640	−0.254868	−0.127434	+	0.991847i	$0.540674\pi$
$710$	15.0695	0.565549
$711$	−8.61681	−0.323156
$712$	3.83140	0.143588
$713$	−4.53069	−0.169676
$714$	27.6970	1.03653
$715$	0.0845362	0.00316147
$716$	−4.36624	−0.163174
$717$	13.5772	0.507048
$718$	−0.998497	−0.0372636
$719$	23.4644	0.875073	0.437536	−	0.899201i	$-0.355851\pi$
$719$	23.4644	0.875073	0.437536	+	0.899201i	$0.355851\pi$
$720$	0.687271	0.0256131
$721$	40.3830	1.50394
$722$	−18.9567	−0.705495
$723$	−37.9776	−1.41240
$724$	7.60738	0.282726
$725$	5.22918	0.194207
$726$	21.1088	0.783420
$727$	37.7114	1.39864	0.699318	−	0.714810i	$-0.253487\pi$
$727$	37.7114	1.39864	0.699318	+	0.714810i	$0.253487\pi$
$728$	−3.60391	−0.133570
$729$	18.3051	0.677968
$730$	−4.10062	−0.151771
$731$	8.12183	0.300397
$732$	18.6046	0.687644
$733$	45.7349	1.68926	0.844629	−	0.535352i	$-0.179821\pi$
$733$	45.7349	1.68926	0.844629	+	0.535352i	$0.179821\pi$
$734$	−1.11288	−0.0410773
$735$	−11.4986	−0.424133
$736$	−4.53069	−0.167003
$737$	−0.457499	−0.0168522
$738$	−0.121922	−0.00448801
$739$	52.4457	1.92925	0.964624	−	0.263630i	$-0.0849198\pi$
$739$	52.4457	1.92925	0.964624	+	0.263630i	$0.0849198\pi$
$740$	−10.3530	−0.380583
$741$	−0.399650	−0.0146815
$742$	−30.2482	−1.11045
$743$	12.0755	0.443006	0.221503	−	0.975160i	$-0.428904\pi$
$743$	12.0755	0.443006	0.221503	+	0.975160i	$0.428904\pi$
$744$	−1.92023	−0.0703989
$745$	2.23186	0.0817691
$746$	−34.9834	−1.28083
$747$	−10.8446	−0.396784
$748$	0.338336	0.0123708
$749$	28.4170	1.03833
$750$	−1.92023	−0.0701168
$751$	−32.0042	−1.16785	−0.583925	−	0.811808i	$-0.698484\pi$
$751$	−32.0042	−1.16785	−0.583925	+	0.811808i	$0.698484\pi$
$752$	−8.10231	−0.295461
$753$	4.79677	0.174804
$754$	5.22918	0.190436
$755$	−20.5846	−0.749148
$756$	−16.0048	−0.582090
$757$	6.43349	0.233829	0.116915	−	0.993142i	$-0.462700\pi$
$757$	6.43349	0.233829	0.116915	+	0.993142i	$0.462700\pi$
$758$	19.6093	0.712240
$759$	0.735460	0.0266955
$760$	0.208127	0.00754954
$761$	−19.9346	−0.722630	−0.361315	−	0.932444i	$-0.617672\pi$
$761$	−19.9346	−0.722630	−0.361315	+	0.932444i	$0.617672\pi$
$762$	29.5372	1.07002
$763$	−21.4942	−0.778143
$764$	15.9018	0.575308
$765$	2.75064	0.0994497
$766$	−7.68692	−0.277740
$767$	−1.10053	−0.0397378
$768$	−1.92023	−0.0692902
$769$	1.65958	0.0598460	0.0299230	−	0.999552i	$-0.490474\pi$
$769$	1.65958	0.0598460	0.0299230	+	0.999552i	$0.490474\pi$
$770$	−0.304661	−0.0109792
$771$	26.9760	0.971516
$772$	−16.5213	−0.594615
$773$	17.6234	0.633870	0.316935	−	0.948447i	$-0.397346\pi$
$773$	17.6234	0.633870	0.316935	+	0.948447i	$0.397346\pi$
$774$	1.39469	0.0501309
$775$	1.00000	0.0359211
$776$	−0.487698	−0.0175073
$777$	−71.6460	−2.57028
$778$	−14.0947	−0.505319
$779$	−0.0369217	−0.00132286
$780$	−1.92023	−0.0687552
$781$	1.27392	0.0455844
$782$	−18.1330	−0.648436
$783$	23.2226	0.829909
$784$	5.98816	0.213863
$785$	15.7190	0.561036
$786$	43.1040	1.53747
$787$	2.31175	0.0824050	0.0412025	−	0.999151i	$-0.486881\pi$
$787$	2.31175	0.0824050	0.0412025	+	0.999151i	$0.486881\pi$
$788$	10.0618	0.358438
$789$	22.4611	0.799637
$790$	−12.5377	−0.446072
$791$	57.2124	2.03424
$792$	0.0580993	0.00206447
$793$	−9.68873	−0.344057
$794$	2.07834	0.0737576
$795$	−16.1168	−0.571604
$796$	−8.63101	−0.305918
$797$	−15.5519	−0.550876	−0.275438	−	0.961319i	$-0.588823\pi$
$797$	−15.5519	−0.550876	−0.275438	+	0.961319i	$0.588823\pi$
$798$	1.44030	0.0509862
$799$	−32.4276	−1.14721
$800$	1.00000	0.0353553
$801$	2.63321	0.0930398
$802$	−19.4715	−0.687562
$803$	−0.346651	−0.0122330
$804$	10.3920	0.366498
$805$	16.3282	0.575493
$806$	1.00000	0.0352235
$807$	−1.92576	−0.0677901
$808$	−10.6491	−0.374633
$809$	−12.7365	−0.447792	−0.223896	−	0.974613i	$-0.571878\pi$
$809$	−12.7365	−0.447792	−0.223896	+	0.974613i	$0.571878\pi$
$810$	−10.5895	−0.372076
$811$	49.9067	1.75246	0.876231	−	0.481891i	$-0.160050\pi$
$811$	49.9067	1.75246	0.876231	+	0.481891i	$0.160050\pi$
$812$	−18.8455	−0.661348
$813$	50.2292	1.76161
$814$	−0.875201	−0.0306758
$815$	10.1189	0.354449
$816$	−7.68526	−0.269038
$817$	0.422353	0.0147763
$818$	26.8376	0.938356
$819$	−2.47686	−0.0865486
$820$	−0.177400	−0.00619509
$821$	−14.9906	−0.523176	−0.261588	−	0.965180i	$-0.584246\pi$
$821$	−14.9906	−0.523176	−0.261588	+	0.965180i	$0.584246\pi$
$822$	−34.2321	−1.19398
$823$	51.1094	1.78156	0.890781	−	0.454433i	$-0.150158\pi$
$823$	51.1094	1.78156	0.890781	+	0.454433i	$0.150158\pi$
$824$	−11.2053	−0.390356
$825$	−0.162329	−0.00565156
$826$	3.96620	0.138002
$827$	−24.9075	−0.866119	−0.433059	−	0.901365i	$-0.642566\pi$
$827$	−24.9075	−0.866119	−0.433059	+	0.901365i	$0.642566\pi$
$828$	−3.11381	−0.108212
$829$	−37.0558	−1.28700	−0.643501	−	0.765446i	$-0.722519\pi$
$829$	−37.0558	−1.28700	−0.643501	+	0.765446i	$0.722519\pi$
$830$	−15.7793	−0.547706
$831$	59.4654	2.06283
$832$	1.00000	0.0346688
$833$	23.9662	0.830380
$834$	−40.2044	−1.39216
$835$	4.66955	0.161597
$836$	0.0175942	0.000608509 0
$837$	4.44096	0.153502
$838$	35.6221	1.23054
$839$	−16.8920	−0.583177	−0.291589	−	0.956544i	$-0.594184\pi$
$839$	−16.8920	−0.583177	−0.291589	+	0.956544i	$0.594184\pi$
$840$	6.92032	0.238774
$841$	−1.65564	−0.0570909
$842$	1.73974	0.0599554
$843$	7.30752	0.251684
$844$	15.8817	0.546669
$845$	1.00000	0.0344010
$846$	−5.56849	−0.191449
$847$	39.6172	1.36126
$848$	8.39317	0.288223
$849$	60.0640	2.06139
$850$	4.00227	0.137277
$851$	46.9061	1.60792
$852$	−28.9369	−0.991362
$853$	41.4044	1.41766	0.708830	−	0.705380i	$-0.249223\pi$
$853$	41.4044	1.41766	0.708830	+	0.705380i	$0.249223\pi$
$854$	34.9173	1.19485
$855$	0.143039	0.00489184
$856$	−7.88504	−0.269505
$857$	32.4017	1.10682	0.553410	−	0.832909i	$-0.313326\pi$
$857$	32.4017	1.10682	0.553410	+	0.832909i	$0.313326\pi$
$858$	−0.162329	−0.00554181
$859$	35.3551	1.20630	0.603151	−	0.797627i	$-0.293912\pi$
$859$	35.3551	1.20630	0.603151	+	0.797627i	$0.293912\pi$
$860$	2.02931	0.0691988
$861$	−1.22767	−0.0418388
$862$	26.9292	0.917211
$863$	−20.4499	−0.696124	−0.348062	−	0.937472i	$-0.613160\pi$
$863$	−20.4499	−0.696124	−0.348062	+	0.937472i	$0.613160\pi$
$864$	4.44096	0.151085
$865$	−1.18414	−0.0402621
$866$	0.0763143	0.00259327
$867$	1.88539	0.0640313
$868$	−3.60391	−0.122325
$869$	−1.05989	−0.0359543
$870$	−10.0412	−0.340429
$871$	−5.41187	−0.183374
$872$	5.96414	0.201971
$873$	−0.335181	−0.0113442
$874$	−0.942957	−0.0318960
$875$	−3.60391	−0.121834
$876$	7.87413	0.266042
$877$	8.02665	0.271041	0.135520	−	0.990775i	$-0.456729\pi$
$877$	8.02665	0.271041	0.135520	+	0.990775i	$0.456729\pi$
$878$	37.1534	1.25387
$879$	46.8438	1.58000
$880$	0.0845362	0.00284971
$881$	−4.99206	−0.168187	−0.0840934	−	0.996458i	$-0.526799\pi$
$881$	−4.99206	−0.168187	−0.0840934	+	0.996458i	$0.526799\pi$
$882$	4.11549	0.138576
$883$	−30.1459	−1.01449	−0.507245	−	0.861802i	$-0.669336\pi$
$883$	−30.1459	−1.01449	−0.507245	+	0.861802i	$0.669336\pi$
$884$	4.00227	0.134611
$885$	2.11326	0.0710366
$886$	14.5089	0.487436
$887$	18.0827	0.607159	0.303579	−	0.952806i	$-0.401818\pi$
$887$	18.0827	0.607159	0.303579	+	0.952806i	$0.401818\pi$
$888$	19.8801	0.667132
$889$	55.4358	1.85926
$890$	3.83140	0.128429
$891$	−0.895193	−0.0299901
$892$	−24.8754	−0.832890
$893$	−1.68631	−0.0564301
$894$	−4.28568	−0.143335
$895$	−4.36624	−0.145947
$896$	−3.60391	−0.120398
$897$	8.69995	0.290483
$898$	−2.27975	−0.0760761
$899$	5.22918	0.174403
$900$	0.687271	0.0229090
$901$	33.5917	1.11910
$902$	−0.0149967	−0.000499337 0
$903$	14.0435	0.467337
$904$	−15.8751	−0.527998
$905$	7.60738	0.252878
$906$	39.5270	1.31320
$907$	−37.1638	−1.23400	−0.617002	−	0.786962i	$-0.711653\pi$
$907$	−37.1638	−1.23400	−0.617002	+	0.786962i	$0.711653\pi$
$908$	18.9628	0.629303
$909$	−7.31880	−0.242749
$910$	−3.60391	−0.119468
$911$	−1.22589	−0.0406157	−0.0203078	−	0.999794i	$-0.506465\pi$
$911$	−1.22589	−0.0406157	−0.0203078	+	0.999794i	$0.506465\pi$
$912$	−0.399650	−0.0132337
$913$	−1.33392	−0.0441462
$914$	−33.3130	−1.10190
$915$	18.6046	0.615048
$916$	−13.5063	−0.446259
$917$	80.8982	2.67149
$918$	17.7739	0.586627
$919$	45.7413	1.50886	0.754432	−	0.656378i	$-0.227912\pi$
$919$	45.7413	1.50886	0.754432	+	0.656378i	$0.227912\pi$
$920$	−4.53069	−0.149372
$921$	15.4347	0.508590
$922$	37.9379	1.24942
$923$	15.0695	0.496019
$924$	0.585017	0.0192457
$925$	−10.3530	−0.340404
$926$	−15.7948	−0.519049
$927$	−7.70110	−0.252937
$928$	5.22918	0.171656
$929$	−25.0752	−0.822691	−0.411345	−	0.911480i	$-0.634941\pi$
$929$	−25.0752	−0.822691	−0.411345	+	0.911480i	$0.634941\pi$
$930$	−1.92023	−0.0629667
$931$	1.24630	0.0408457
$932$	11.9733	0.392199
$933$	39.4568	1.29176
$934$	34.2977	1.12225
$935$	0.338336	0.0110648
$936$	0.687271	0.0224642
$937$	46.9820	1.53484	0.767418	−	0.641147i	$-0.221541\pi$
$937$	46.9820	1.53484	0.767418	+	0.641147i	$0.221541\pi$
$938$	19.5039	0.636824
$939$	19.4975	0.636278
$940$	−8.10231	−0.264268
$941$	−27.9342	−0.910628	−0.455314	−	0.890331i	$-0.650473\pi$
$941$	−27.9342	−0.910628	−0.455314	+	0.890331i	$0.650473\pi$
$942$	−30.1841	−0.983452
$943$	0.803746	0.0261736
$944$	−1.10053	−0.0358191
$945$	−16.0048	−0.520637
$946$	0.171550	0.00557757
$947$	−43.2381	−1.40505	−0.702525	−	0.711659i	$-0.747944\pi$
$947$	−43.2381	−1.40505	−0.702525	+	0.711659i	$0.747944\pi$
$948$	24.0753	0.781928
$949$	−4.10062	−0.133112
$950$	0.208127	0.00675252
$951$	−33.6288	−1.09049
$952$	−14.4238	−0.467478
$953$	−35.7849	−1.15919	−0.579593	−	0.814906i	$-0.696788\pi$
$953$	−35.7849	−1.15919	−0.579593	+	0.814906i	$0.696788\pi$
$954$	5.76838	0.186758
$955$	15.9018	0.514571
$956$	−7.07060	−0.228679
$957$	−0.848846	−0.0274393
$958$	25.4663	0.822779
$959$	−64.2473	−2.07465
$960$	−1.92023	−0.0619751
$961$	1.00000	0.0322581
$962$	−10.3530	−0.333794
$963$	−5.41916	−0.174630
$964$	19.7777	0.636996
$965$	−16.5213	−0.531840
$966$	−31.3538	−1.00879
$967$	−48.0445	−1.54501	−0.772503	−	0.635011i	$-0.780996\pi$
$967$	−48.0445	−1.54501	−0.772503	+	0.635011i	$0.780996\pi$
$968$	−10.9929	−0.353324
$969$	−1.59951	−0.0513836
$970$	−0.487698	−0.0156590
$971$	−44.1215	−1.41593	−0.707964	−	0.706249i	$-0.750386\pi$
$971$	−44.1215	−1.41593	−0.707964	+	0.706249i	$0.750386\pi$
$972$	7.01130	0.224887
$973$	−75.4561	−2.41901
$974$	25.2048	0.807614
$975$	−1.92023	−0.0614965
$976$	−9.68873	−0.310129
$977$	33.4121	1.06895	0.534473	−	0.845185i	$-0.320510\pi$
$977$	33.4121	1.06895	0.534473	+	0.845185i	$0.320510\pi$
$978$	−19.4306	−0.621321
$979$	0.323891	0.0103516
$980$	5.98816	0.191285
$981$	4.09898	0.130870
$982$	−28.8389	−0.920287
$983$	−42.6002	−1.35874	−0.679368	−	0.733798i	$-0.737746\pi$
$983$	−42.6002	−1.35874	−0.679368	+	0.733798i	$0.737746\pi$
$984$	0.340649	0.0108595
$985$	10.0618	0.320597
$986$	20.9286	0.666502
$987$	−56.0706	−1.78475
$988$	0.208127	0.00662139
$989$	−9.19417	−0.292357
$990$	0.0580993	0.00184652
$991$	18.7792	0.596541	0.298271	−	0.954481i	$-0.403590\pi$
$991$	18.7792	0.596541	0.298271	+	0.954481i	$0.403590\pi$
$992$	1.00000	0.0317500
$993$	−20.9098	−0.663554
$994$	−54.3092	−1.72258
$995$	−8.63101	−0.273621
$996$	30.2998	0.960085
$997$	61.0136	1.93232	0.966160	−	0.257943i	$-0.0830446\pi$
$997$	61.0136	1.93232	0.966160	+	0.257943i	$0.0830446\pi$
$998$	35.7033	1.13017
$999$	−45.9772	−1.45465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.e.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.e.1.3	✓	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.92023 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.92023 q^{6} -3.60391 q^{7} +1.00000 q^{8} +0.687271 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.92023 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.92023 q^{6} -3.60391 q^{7} +1.00000 q^{8} +0.687271 q^{9} +1.00000 q^{10} +0.0845362 q^{11} -1.92023 q^{12} +1.00000 q^{13} -3.60391 q^{14} -1.92023 q^{15} +1.00000 q^{16} +4.00227 q^{17} +0.687271 q^{18} +0.208127 q^{19} +1.00000 q^{20} +6.92032 q^{21} +0.0845362 q^{22} -4.53069 q^{23} -1.92023 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.44096 q^{27} -3.60391 q^{28} +5.22918 q^{29} -1.92023 q^{30} +1.00000 q^{31} +1.00000 q^{32} -0.162329 q^{33} +4.00227 q^{34} -3.60391 q^{35} +0.687271 q^{36} -10.3530 q^{37} +0.208127 q^{38} -1.92023 q^{39} +1.00000 q^{40} -0.177400 q^{41} +6.92032 q^{42} +2.02931 q^{43} +0.0845362 q^{44} +0.687271 q^{45} -4.53069 q^{46} -8.10231 q^{47} -1.92023 q^{48} +5.98816 q^{49} +1.00000 q^{50} -7.68526 q^{51} +1.00000 q^{52} +8.39317 q^{53} +4.44096 q^{54} +0.0845362 q^{55} -3.60391 q^{56} -0.399650 q^{57} +5.22918 q^{58} -1.10053 q^{59} -1.92023 q^{60} -9.68873 q^{61} +1.00000 q^{62} -2.47686 q^{63} +1.00000 q^{64} +1.00000 q^{65} -0.162329 q^{66} -5.41187 q^{67} +4.00227 q^{68} +8.69995 q^{69} -3.60391 q^{70} +15.0695 q^{71} +0.687271 q^{72} -4.10062 q^{73} -10.3530 q^{74} -1.92023 q^{75} +0.208127 q^{76} -0.304661 q^{77} -1.92023 q^{78} -12.5377 q^{79} +1.00000 q^{80} -10.5895 q^{81} -0.177400 q^{82} -15.7793 q^{83} +6.92032 q^{84} +4.00227 q^{85} +2.02931 q^{86} -10.0412 q^{87} +0.0845362 q^{88} +3.83140 q^{89} +0.687271 q^{90} -3.60391 q^{91} -4.53069 q^{92} -1.92023 q^{93} -8.10231 q^{94} +0.208127 q^{95} -1.92023 q^{96} -0.487698 q^{97} +5.98816 q^{98} +0.0580993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 7 q^{3} + 6 q^{4} + 6 q^{5} - 7 q^{6} - 8 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 7 * q^3 + 6 * q^4 + 6 * q^5 - 7 * q^6 - 8 * q^7 + 6 * q^8 + 7 * q^9	\( 6 q + 6 q^{2} - 7 q^{3} + 6 q^{4} + 6 q^{5} - 7 q^{6} - 8 q^{7} + 6 q^{8} + 7 q^{9} + 6 q^{10} - 10 q^{11} - 7 q^{12} + 6 q^{13} - 8 q^{14} - 7 q^{15} + 6 q^{16} - 14 q^{17} + 7 q^{18} - 3 q^{19} + 6 q^{20} + 3 q^{21} - 10 q^{22} - 13 q^{23} - 7 q^{24} + 6 q^{25} + 6 q^{26} - 25 q^{27} - 8 q^{28} - 8 q^{29} - 7 q^{30} + 6 q^{31} + 6 q^{32} + 18 q^{33} - 14 q^{34} - 8 q^{35} + 7 q^{36} - 16 q^{37} - 3 q^{38} - 7 q^{39} + 6 q^{40} - 10 q^{41} + 3 q^{42} - 17 q^{43} - 10 q^{44} + 7 q^{45} - 13 q^{46} - 14 q^{47} - 7 q^{48} + 18 q^{49} + 6 q^{50} + q^{51} + 6 q^{52} - 22 q^{53} - 25 q^{54} - 10 q^{55} - 8 q^{56} + 9 q^{57} - 8 q^{58} - 23 q^{59} - 7 q^{60} + 17 q^{61} + 6 q^{62} - 5 q^{63} + 6 q^{64} + 6 q^{65} + 18 q^{66} - 16 q^{67} - 14 q^{68} - 8 q^{70} + 2 q^{71} + 7 q^{72} - 7 q^{73} - 16 q^{74} - 7 q^{75} - 3 q^{76} - 13 q^{77} - 7 q^{78} - 8 q^{79} + 6 q^{80} + 38 q^{81} - 10 q^{82} - 16 q^{83} + 3 q^{84} - 14 q^{85} - 17 q^{86} - 3 q^{87} - 10 q^{88} - 32 q^{89} + 7 q^{90} - 8 q^{91} - 13 q^{92} - 7 q^{93} - 14 q^{94} - 3 q^{95} - 7 q^{96} + 31 q^{97} + 18 q^{98} - 17 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 7 * q^3 + 6 * q^4 + 6 * q^5 - 7 * q^6 - 8 * q^7 + 6 * q^8 + 7 * q^9 + 6 * q^10 - 10 * q^11 - 7 * q^12 + 6 * q^13 - 8 * q^14 - 7 * q^15 + 6 * q^16 - 14 * q^17 + 7 * q^18 - 3 * q^19 + 6 * q^20 + 3 * q^21 - 10 * q^22 - 13 * q^23 - 7 * q^24 + 6 * q^25 + 6 * q^26 - 25 * q^27 - 8 * q^28 - 8 * q^29 - 7 * q^30 + 6 * q^31 + 6 * q^32 + 18 * q^33 - 14 * q^34 - 8 * q^35 + 7 * q^36 - 16 * q^37 - 3 * q^38 - 7 * q^39 + 6 * q^40 - 10 * q^41 + 3 * q^42 - 17 * q^43 - 10 * q^44 + 7 * q^45 - 13 * q^46 - 14 * q^47 - 7 * q^48 + 18 * q^49 + 6 * q^50 + q^51 + 6 * q^52 - 22 * q^53 - 25 * q^54 - 10 * q^55 - 8 * q^56 + 9 * q^57 - 8 * q^58 - 23 * q^59 - 7 * q^60 + 17 * q^61 + 6 * q^62 - 5 * q^63 + 6 * q^64 + 6 * q^65 + 18 * q^66 - 16 * q^67 - 14 * q^68 - 8 * q^70 + 2 * q^71 + 7 * q^72 - 7 * q^73 - 16 * q^74 - 7 * q^75 - 3 * q^76 - 13 * q^77 - 7 * q^78 - 8 * q^79 + 6 * q^80 + 38 * q^81 - 10 * q^82 - 16 * q^83 + 3 * q^84 - 14 * q^85 - 17 * q^86 - 3 * q^87 - 10 * q^88 - 32 * q^89 + 7 * q^90 - 8 * q^91 - 13 * q^92 - 7 * q^93 - 14 * q^94 - 3 * q^95 - 7 * q^96 + 31 * q^97 + 18 * q^98 - 17 * q^99

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