LMFDB - Embedded newform 670.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [670,2,Mod(1,670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(670, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 670 = 2 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	670.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:	$5.34997693543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.20025$ of defining polynomial
Character	$\chi$	$=$	670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.37505 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.37505 q^{6} +3.86657 q^{7} -1.00000 q^{8} -1.10924 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.37505 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.37505 q^{6} +3.86657 q^{7} -1.00000 q^{8} -1.10924 q^{9} +1.00000 q^{10} +6.49152 q^{11} -1.37505 q^{12} -5.66632 q^{13} -3.86657 q^{14} +1.37505 q^{15} +1.00000 q^{16} +0.442924 q^{17} +1.10924 q^{18} -6.08273 q^{19} -1.00000 q^{20} -5.31672 q^{21} -6.49152 q^{22} +0.182027 q^{23} +1.37505 q^{24} +1.00000 q^{25} +5.66632 q^{26} +5.65041 q^{27} +3.86657 q^{28} +6.73419 q^{29} -1.37505 q^{30} +8.06682 q^{31} -1.00000 q^{32} -8.92616 q^{33} -0.442924 q^{34} -3.86657 q^{35} -1.10924 q^{36} -0.643176 q^{37} +6.08273 q^{38} +7.79146 q^{39} +1.00000 q^{40} +10.5257 q^{41} +5.31672 q^{42} +1.77555 q^{43} +6.49152 q^{44} +1.10924 q^{45} -0.182027 q^{46} -8.64935 q^{47} -1.37505 q^{48} +7.95035 q^{49} -1.00000 q^{50} -0.609042 q^{51} -5.66632 q^{52} +11.7490 q^{53} -5.65041 q^{54} -6.49152 q^{55} -3.86657 q^{56} +8.36405 q^{57} -6.73419 q^{58} +8.09333 q^{59} +1.37505 q^{60} -8.33986 q^{61} -8.06682 q^{62} -4.28895 q^{63} +1.00000 q^{64} +5.66632 q^{65} +8.92616 q^{66} +1.00000 q^{67} +0.442924 q^{68} -0.250296 q^{69} +3.86657 q^{70} +15.8401 q^{71} +1.10924 q^{72} +3.19302 q^{73} +0.643176 q^{74} -1.37505 q^{75} -6.08273 q^{76} +25.0999 q^{77} -7.79146 q^{78} +2.08273 q^{79} -1.00000 q^{80} -4.44187 q^{81} -10.5257 q^{82} +1.63218 q^{83} -5.31672 q^{84} -0.442924 q^{85} -1.77555 q^{86} -9.25984 q^{87} -6.49152 q^{88} +8.49152 q^{89} -1.10924 q^{90} -21.9092 q^{91} +0.182027 q^{92} -11.0923 q^{93} +8.64935 q^{94} +6.08273 q^{95} +1.37505 q^{96} +1.59121 q^{97} -7.95035 q^{98} -7.20065 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9} + 4 q^{10} + 9 q^{11} - 2 q^{12} - 12 q^{13} + 5 q^{14} + 2 q^{15} + 4 q^{16} - 8 q^{18} + 4 q^{19} - 4 q^{20} + 2 q^{21} - 9 q^{22} + 6 q^{23} + 2 q^{24} + 4 q^{25} + 12 q^{26} + 10 q^{27} - 5 q^{28} + 18 q^{29} - 2 q^{30} + 2 q^{31} - 4 q^{32} + 14 q^{33} + 5 q^{35} + 8 q^{36} + 9 q^{37} - 4 q^{38} + 10 q^{39} + 4 q^{40} + 12 q^{41} - 2 q^{42} - 16 q^{43} + 9 q^{44} - 8 q^{45} - 6 q^{46} + 10 q^{47} - 2 q^{48} + 15 q^{49} - 4 q^{50} - 4 q^{51} - 12 q^{52} + 8 q^{53} - 10 q^{54} - 9 q^{55} + 5 q^{56} + 44 q^{57} - 18 q^{58} + 18 q^{59} + 2 q^{60} - 11 q^{61} - 2 q^{62} - 27 q^{63} + 4 q^{64} + 12 q^{65} - 14 q^{66} + 4 q^{67} + 20 q^{69} - 5 q^{70} + 27 q^{71} - 8 q^{72} + 4 q^{73} - 9 q^{74} - 2 q^{75} + 4 q^{76} + 25 q^{77} - 10 q^{78} - 20 q^{79} - 4 q^{80} + 16 q^{81} - 12 q^{82} + 9 q^{83} + 2 q^{84} + 16 q^{86} + 2 q^{87} - 9 q^{88} + 17 q^{89} + 8 q^{90} - 4 q^{91} + 6 q^{92} + 2 q^{93} - 10 q^{94} - 4 q^{95} + 2 q^{96} - 5 q^{97} - 15 q^{98} + 9 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9 + 4 * q^10 + 9 * q^11 - 2 * q^12 - 12 * q^13 + 5 * q^14 + 2 * q^15 + 4 * q^16 - 8 * q^18 + 4 * q^19 - 4 * q^20 + 2 * q^21 - 9 * q^22 + 6 * q^23 + 2 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 5 * q^28 + 18 * q^29 - 2 * q^30 + 2 * q^31 - 4 * q^32 + 14 * q^33 + 5 * q^35 + 8 * q^36 + 9 * q^37 - 4 * q^38 + 10 * q^39 + 4 * q^40 + 12 * q^41 - 2 * q^42 - 16 * q^43 + 9 * q^44 - 8 * q^45 - 6 * q^46 + 10 * q^47 - 2 * q^48 + 15 * q^49 - 4 * q^50 - 4 * q^51 - 12 * q^52 + 8 * q^53 - 10 * q^54 - 9 * q^55 + 5 * q^56 + 44 * q^57 - 18 * q^58 + 18 * q^59 + 2 * q^60 - 11 * q^61 - 2 * q^62 - 27 * q^63 + 4 * q^64 + 12 * q^65 - 14 * q^66 + 4 * q^67 + 20 * q^69 - 5 * q^70 + 27 * q^71 - 8 * q^72 + 4 * q^73 - 9 * q^74 - 2 * q^75 + 4 * q^76 + 25 * q^77 - 10 * q^78 - 20 * q^79 - 4 * q^80 + 16 * q^81 - 12 * q^82 + 9 * q^83 + 2 * q^84 + 16 * q^86 + 2 * q^87 - 9 * q^88 + 17 * q^89 + 8 * q^90 - 4 * q^91 + 6 * q^92 + 2 * q^93 - 10 * q^94 - 4 * q^95 + 2 * q^96 - 5 * q^97 - 15 * q^98 + 9 * 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
$3$	−1.37505	−0.793885	−0.396943	−	0.917843i	$-0.629929\pi$
$3$	−1.37505	−0.793885	−0.396943	+	0.917843i	$0.629929\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.37505	0.561362
$7$	3.86657	1.46143	0.730713	−	0.682685i	$-0.239188\pi$
$7$	3.86657	1.46143	0.730713	+	0.682685i	$0.239188\pi$
$8$	−1.00000	−0.353553
$9$	−1.10924	−0.369746
$10$	1.00000	0.316228
$11$	6.49152	1.95727	0.978633	−	0.205613i	$-0.0659189\pi$
$11$	6.49152	1.95727	0.978633	+	0.205613i	$0.0659189\pi$
$12$	−1.37505	−0.396943
$13$	−5.66632	−1.57155	−0.785777	−	0.618510i	$-0.787736\pi$
$13$	−5.66632	−1.57155	−0.785777	+	0.618510i	$0.787736\pi$
$14$	−3.86657	−1.03338
$15$	1.37505	0.355036
$16$	1.00000	0.250000
$17$	0.442924	0.107425	0.0537124	−	0.998556i	$-0.482895\pi$
$17$	0.442924	0.107425	0.0537124	+	0.998556i	$0.482895\pi$
$18$	1.10924	0.261450
$19$	−6.08273	−1.39547	−0.697737	−	0.716354i	$-0.745810\pi$
$19$	−6.08273	−1.39547	−0.697737	+	0.716354i	$0.745810\pi$
$20$	−1.00000	−0.223607
$21$	−5.31672	−1.16020
$22$	−6.49152	−1.38400
$23$	0.182027	0.0379553	0.0189776	−	0.999820i	$-0.493959\pi$
$23$	0.182027	0.0379553	0.0189776	+	0.999820i	$0.493959\pi$
$24$	1.37505	0.280681
$25$	1.00000	0.200000
$26$	5.66632	1.11126
$27$	5.65041	1.08742
$28$	3.86657	0.730713
$29$	6.73419	1.25051	0.625254	−	0.780421i	$-0.284995\pi$
$29$	6.73419	1.25051	0.625254	+	0.780421i	$0.284995\pi$
$30$	−1.37505	−0.251049
$31$	8.06682	1.44884	0.724422	−	0.689357i	$-0.242107\pi$
$31$	8.06682	1.44884	0.724422	+	0.689357i	$0.242107\pi$
$32$	−1.00000	−0.176777
$33$	−8.92616	−1.55384
$34$	−0.442924	−0.0759608
$35$	−3.86657	−0.653569
$36$	−1.10924	−0.184873
$37$	−0.643176	−0.105738	−0.0528688	−	0.998601i	$-0.516837\pi$
$37$	−0.643176	−0.105738	−0.0528688	+	0.998601i	$0.516837\pi$
$38$	6.08273	0.986749
$39$	7.79146	1.24763
$40$	1.00000	0.158114
$41$	10.5257	1.64383	0.821915	−	0.569609i	$-0.192906\pi$
$41$	10.5257	1.64383	0.821915	+	0.569609i	$0.192906\pi$
$42$	5.31672	0.820388
$43$	1.77555	0.270770	0.135385	−	0.990793i	$-0.456773\pi$
$43$	1.77555	0.270770	0.135385	+	0.990793i	$0.456773\pi$
$44$	6.49152	0.978633
$45$	1.10924	0.165356
$46$	−0.182027	−0.0268384
$47$	−8.64935	−1.26164	−0.630819	−	0.775930i	$-0.717281\pi$
$47$	−8.64935	−1.26164	−0.630819	+	0.775930i	$0.717281\pi$
$48$	−1.37505	−0.198471
$49$	7.95035	1.13576
$50$	−1.00000	−0.141421
$51$	−0.609042	−0.0852829
$52$	−5.66632	−0.785777
$53$	11.7490	1.61386	0.806928	−	0.590650i	$-0.201129\pi$
$53$	11.7490	1.61386	0.806928	+	0.590650i	$0.201129\pi$
$54$	−5.65041	−0.768923
$55$	−6.49152	−0.875316
$56$	−3.86657	−0.516692
$57$	8.36405	1.10785
$58$	−6.73419	−0.884242
$59$	8.09333	1.05366	0.526831	−	0.849970i	$-0.323380\pi$
$59$	8.09333	1.05366	0.526831	+	0.849970i	$0.323380\pi$
$60$	1.37505	0.177518
$61$	−8.33986	−1.06781	−0.533905	−	0.845545i	$-0.679276\pi$
$61$	−8.33986	−1.06781	−0.533905	+	0.845545i	$0.679276\pi$
$62$	−8.06682	−1.02449
$63$	−4.28895	−0.540357
$64$	1.00000	0.125000
$65$	5.66632	0.702820
$66$	8.92616	1.09873
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0.442924	0.0537124
$69$	−0.250296	−0.0301321
$70$	3.86657	0.462143
$71$	15.8401	1.87987	0.939935	−	0.341355i	$-0.110886\pi$
$71$	15.8401	1.87987	0.939935	+	0.341355i	$0.110886\pi$
$72$	1.10924	0.130725
$73$	3.19302	0.373715	0.186857	−	0.982387i	$-0.440170\pi$
$73$	3.19302	0.373715	0.186857	+	0.982387i	$0.440170\pi$
$74$	0.643176	0.0747677
$75$	−1.37505	−0.158777
$76$	−6.08273	−0.697737
$77$	25.0999	2.86040
$78$	−7.79146	−0.882210
$79$	2.08273	0.234325	0.117163	−	0.993113i	$-0.462620\pi$
$79$	2.08273	0.234325	0.117163	+	0.993113i	$0.462620\pi$
$80$	−1.00000	−0.111803
$81$	−4.44187	−0.493541
$82$	−10.5257	−1.16236
$83$	1.63218	0.179155	0.0895776	−	0.995980i	$-0.471448\pi$
$83$	1.63218	0.179155	0.0895776	+	0.995980i	$0.471448\pi$
$84$	−5.31672	−0.580102
$85$	−0.442924	−0.0480418
$86$	−1.77555	−0.191463
$87$	−9.25984	−0.992759
$88$	−6.49152	−0.691998
$89$	8.49152	0.900099	0.450050	−	0.893004i	$-0.351406\pi$
$89$	8.49152	0.900099	0.450050	+	0.893004i	$0.351406\pi$
$90$	−1.10924	−0.116924
$91$	−21.9092	−2.29671
$92$	0.182027	0.0189776
$93$	−11.0923	−1.15022
$94$	8.64935	0.892113
$95$	6.08273	0.624075
$96$	1.37505	0.140340
$97$	1.59121	0.161563	0.0807815	−	0.996732i	$-0.474258\pi$
$97$	1.59121	0.161563	0.0807815	+	0.996732i	$0.474258\pi$
$98$	−7.95035	−0.803107
$99$	−7.20065	−0.723692
$100$	1.00000	0.100000
$101$	−14.8496	−1.47759	−0.738796	−	0.673930i	$-0.764605\pi$
$101$	−14.8496	−1.47759	−0.738796	+	0.673930i	$0.764605\pi$
$102$	0.609042	0.0603041
$103$	−3.93318	−0.387548	−0.193774	−	0.981046i	$-0.562073\pi$
$103$	−3.93318	−0.387548	−0.193774	+	0.981046i	$0.562073\pi$
$104$	5.66632	0.555628
$105$	5.31672	0.518859
$106$	−11.7490	−1.14117
$107$	−16.1071	−1.55714	−0.778568	−	0.627561i	$-0.784053\pi$
$107$	−16.1071	−1.55714	−0.778568	+	0.627561i	$0.784053\pi$
$108$	5.65041	0.543711
$109$	−0.398189	−0.0381396	−0.0190698	−	0.999818i	$-0.506070\pi$
$109$	−0.398189	−0.0381396	−0.0190698	+	0.999818i	$0.506070\pi$
$110$	6.49152	0.618942
$111$	0.884399	0.0839435
$112$	3.86657	0.365356
$113$	4.33600	0.407897	0.203948	−	0.978982i	$-0.434623\pi$
$113$	4.33600	0.407897	0.203948	+	0.978982i	$0.434623\pi$
$114$	−8.36405	−0.783365
$115$	−0.182027	−0.0169741
$116$	6.73419	0.625254
$117$	6.28530	0.581076
$118$	−8.09333	−0.745051
$119$	1.71259	0.156993
$120$	−1.37505	−0.125524
$121$	31.1398	2.83089
$122$	8.33986	0.755056
$123$	−14.4733	−1.30501
$124$	8.06682	0.724422
$125$	−1.00000	−0.0894427
$126$	4.28895	0.382090
$127$	16.6858	1.48063	0.740313	−	0.672262i	$-0.234677\pi$
$127$	16.6858	1.48063	0.740313	+	0.672262i	$0.234677\pi$
$128$	−1.00000	−0.0883883
$129$	−2.44148	−0.214960
$130$	−5.66632	−0.496969
$131$	7.91516	0.691551	0.345776	−	0.938317i	$-0.387616\pi$
$131$	7.91516	0.691551	0.345776	+	0.938317i	$0.387616\pi$
$132$	−8.92616	−0.776922
$133$	−23.5193	−2.03938
$134$	−1.00000	−0.0863868
$135$	−5.65041	−0.486310
$136$	−0.442924	−0.0379804
$137$	−3.18329	−0.271967	−0.135983	−	0.990711i	$-0.543419\pi$
$137$	−3.18329	−0.271967	−0.135983	+	0.990711i	$0.543419\pi$
$138$	0.250296	0.0213066
$139$	−10.3277	−0.875986	−0.437993	−	0.898978i	$-0.644310\pi$
$139$	−10.3277	−0.875986	−0.437993	+	0.898978i	$0.644310\pi$
$140$	−3.86657	−0.326785
$141$	11.8933	1.00160
$142$	−15.8401	−1.32927
$143$	−36.7830	−3.07595
$144$	−1.10924	−0.0924366
$145$	−6.73419	−0.559244
$146$	−3.19302	−0.264256
$147$	−10.9321	−0.901667
$148$	−0.643176	−0.0528688
$149$	−5.11415	−0.418968	−0.209484	−	0.977812i	$-0.567178\pi$
$149$	−5.11415	−0.418968	−0.209484	+	0.977812i	$0.567178\pi$
$150$	1.37505	0.112272
$151$	−15.4760	−1.25942	−0.629710	−	0.776831i	$-0.716826\pi$
$151$	−15.4760	−1.25942	−0.629710	+	0.776831i	$0.716826\pi$
$152$	6.08273	0.493375
$153$	−0.491308	−0.0397199
$154$	−25.0999	−2.02261
$155$	−8.06682	−0.647943
$156$	7.79146	0.623816
$157$	15.1750	1.21110	0.605549	−	0.795808i	$-0.292954\pi$
$157$	15.1750	1.21110	0.605549	+	0.795808i	$0.292954\pi$
$158$	−2.08273	−0.165693
$159$	−16.1555	−1.28122
$160$	1.00000	0.0790569
$161$	0.703820	0.0554688
$162$	4.44187	0.348986
$163$	−6.30949	−0.494198	−0.247099	−	0.968990i	$-0.579477\pi$
$163$	−6.30949	−0.494198	−0.247099	+	0.968990i	$0.579477\pi$
$164$	10.5257	0.821915
$165$	8.92616	0.694901
$166$	−1.63218	−0.126682
$167$	−12.4977	−0.967101	−0.483550	−	0.875317i	$-0.660653\pi$
$167$	−12.4977	−0.967101	−0.483550	+	0.875317i	$0.660653\pi$
$168$	5.31672	0.410194
$169$	19.1071	1.46978
$170$	0.442924	0.0339707
$171$	6.74720	0.515972
$172$	1.77555	0.135385
$173$	7.24017	0.550460	0.275230	−	0.961378i	$-0.411246\pi$
$173$	7.24017	0.550460	0.275230	+	0.961378i	$0.411246\pi$
$174$	9.25984	0.701987
$175$	3.86657	0.292285
$176$	6.49152	0.489317
$177$	−11.1287	−0.836486
$178$	−8.49152	−0.636466
$179$	−4.82752	−0.360826	−0.180413	−	0.983591i	$-0.557743\pi$
$179$	−4.82752	−0.360826	−0.180413	+	0.983591i	$0.557743\pi$
$180$	1.10924	0.0826778
$181$	6.77064	0.503258	0.251629	−	0.967824i	$-0.419034\pi$
$181$	6.77064	0.503258	0.251629	+	0.967824i	$0.419034\pi$
$182$	21.9092	1.62402
$183$	11.4677	0.847718
$184$	−0.182027	−0.0134192
$185$	0.643176	0.0472873
$186$	11.0923	0.813325
$187$	2.87525	0.210259
$188$	−8.64935	−0.630819
$189$	21.8477	1.58919
$190$	−6.08273	−0.441288
$191$	−16.8678	−1.22051	−0.610257	−	0.792204i	$-0.708934\pi$
$191$	−16.8678	−1.22051	−0.610257	+	0.792204i	$0.708934\pi$
$192$	−1.37505	−0.0992356
$193$	−8.60838	−0.619645	−0.309823	−	0.950794i	$-0.600270\pi$
$193$	−8.60838	−0.619645	−0.309823	+	0.950794i	$0.600270\pi$
$194$	−1.59121	−0.114242
$195$	−7.79146	−0.557958
$196$	7.95035	0.567882
$197$	−23.1485	−1.64926	−0.824631	−	0.565671i	$-0.808617\pi$
$197$	−23.1485	−1.64926	−0.824631	+	0.565671i	$0.808617\pi$
$198$	7.20065	0.511728
$199$	2.30100	0.163113	0.0815567	−	0.996669i	$-0.474011\pi$
$199$	2.30100	0.163113	0.0815567	+	0.996669i	$0.474011\pi$
$200$	−1.00000	−0.0707107
$201$	−1.37505	−0.0969885
$202$	14.8496	1.04481
$203$	26.0382	1.82752
$204$	−0.609042	−0.0426415
$205$	−10.5257	−0.735144
$206$	3.93318	0.274038
$207$	−0.201912	−0.0140338
$208$	−5.66632	−0.392888
$209$	−39.4862	−2.73131
$210$	−5.31672	−0.366889
$211$	−9.25840	−0.637374	−0.318687	−	0.947860i	$-0.603242\pi$
$211$	−9.25840	−0.637374	−0.318687	+	0.947860i	$0.603242\pi$
$212$	11.7490	0.806928
$213$	−21.7809	−1.49240
$214$	16.1071	1.10106
$215$	−1.77555	−0.121092
$216$	−5.65041	−0.384461
$217$	31.1909	2.11738
$218$	0.398189	0.0269687
$219$	−4.39056	−0.296687
$220$	−6.49152	−0.437658
$221$	−2.50974	−0.168824
$222$	−0.884399	−0.0593570
$223$	−1.51676	−0.101570	−0.0507850	−	0.998710i	$-0.516172\pi$
$223$	−1.51676	−0.101570	−0.0507850	+	0.998710i	$0.516172\pi$
$224$	−3.86657	−0.258346
$225$	−1.10924	−0.0739493
$226$	−4.33600	−0.288427
$227$	16.4688	1.09307	0.546535	−	0.837436i	$-0.315946\pi$
$227$	16.4688	1.09307	0.546535	+	0.837436i	$0.315946\pi$
$228$	8.36405	0.553923
$229$	−14.9444	−0.987553	−0.493776	−	0.869589i	$-0.664384\pi$
$229$	−14.9444	−0.987553	−0.493776	+	0.869589i	$0.664384\pi$
$230$	0.182027	0.0120025
$231$	−34.5136	−2.27083
$232$	−6.73419	−0.442121
$233$	13.8097	0.904703	0.452351	−	0.891840i	$-0.350585\pi$
$233$	13.8097	0.904703	0.452351	+	0.891840i	$0.350585\pi$
$234$	−6.28530	−0.410883
$235$	8.64935	0.564222
$236$	8.09333	0.526831
$237$	−2.86386	−0.186027
$238$	−1.71259	−0.111011
$239$	29.8205	1.92893	0.964464	−	0.264215i	$-0.0851127\pi$
$239$	29.8205	1.92893	0.964464	+	0.264215i	$0.0851127\pi$
$240$	1.37505	0.0887591
$241$	−12.1663	−0.783702	−0.391851	−	0.920029i	$-0.628165\pi$
$241$	−12.1663	−0.783702	−0.391851	+	0.920029i	$0.628165\pi$
$242$	−31.1398	−2.00174
$243$	−10.8434	−0.695606
$244$	−8.33986	−0.533905
$245$	−7.95035	−0.507929
$246$	14.4733	0.922783
$247$	34.4667	2.19306
$248$	−8.06682	−0.512244
$249$	−2.24433	−0.142229
$250$	1.00000	0.0632456
$251$	−6.54985	−0.413423	−0.206711	−	0.978402i	$-0.566276\pi$
$251$	−6.54985	−0.413423	−0.206711	+	0.978402i	$0.566276\pi$
$252$	−4.28895	−0.270178
$253$	1.18163	0.0742886
$254$	−16.6858	−1.04696
$255$	0.609042	0.0381397
$256$	1.00000	0.0625000
$257$	−4.99364	−0.311495	−0.155747	−	0.987797i	$-0.549779\pi$
$257$	−4.99364	−0.311495	−0.155747	+	0.987797i	$0.549779\pi$
$258$	2.44148	0.152000
$259$	−2.48689	−0.154528
$260$	5.66632	0.351410
$261$	−7.46983	−0.462371
$262$	−7.91516	−0.489000
$263$	22.5136	1.38825	0.694124	−	0.719855i	$-0.255792\pi$
$263$	22.5136	1.38825	0.694124	+	0.719855i	$0.255792\pi$
$264$	8.92616	0.549367
$265$	−11.7490	−0.721738
$266$	23.5193	1.44206
$267$	−11.6763	−0.714575
$268$	1.00000	0.0610847
$269$	−5.41747	−0.330309	−0.165154	−	0.986268i	$-0.552812\pi$
$269$	−5.41747	−0.330309	−0.165154	+	0.986268i	$0.552812\pi$
$270$	5.65041	0.343873
$271$	−12.7162	−0.772453	−0.386226	−	0.922404i	$-0.626222\pi$
$271$	−12.7162	−0.772453	−0.386226	+	0.922404i	$0.626222\pi$
$272$	0.442924	0.0268562
$273$	30.1262	1.82332
$274$	3.18329	0.192310
$275$	6.49152	0.391453
$276$	−0.250296	−0.0150661
$277$	20.3180	1.22079	0.610395	−	0.792097i	$-0.291011\pi$
$277$	20.3180	1.22079	0.610395	+	0.792097i	$0.291011\pi$
$278$	10.3277	0.619415
$279$	−8.94803	−0.535705
$280$	3.86657	0.231072
$281$	−3.70343	−0.220928	−0.110464	−	0.993880i	$-0.535234\pi$
$281$	−3.70343	−0.220928	−0.110464	+	0.993880i	$0.535234\pi$
$282$	−11.8933	−0.708235
$283$	21.8341	1.29790	0.648951	−	0.760830i	$-0.275208\pi$
$283$	21.8341	1.29790	0.648951	+	0.760830i	$0.275208\pi$
$284$	15.8401	0.939935
$285$	−8.36405	−0.495444
$286$	36.7830	2.17502
$287$	40.6982	2.40234
$288$	1.10924	0.0653625
$289$	−16.8038	−0.988460
$290$	6.73419	0.395445
$291$	−2.18799	−0.128262
$292$	3.19302	0.186857
$293$	−16.8351	−0.983520	−0.491760	−	0.870731i	$-0.663646\pi$
$293$	−16.8351	−0.983520	−0.491760	+	0.870731i	$0.663646\pi$
$294$	10.9321	0.637575
$295$	−8.09333	−0.471212
$296$	0.643176	0.0373839
$297$	36.6797	2.12837
$298$	5.11415	0.296255
$299$	−1.03142	−0.0596487
$300$	−1.37505	−0.0793885
$301$	6.86530	0.395710
$302$	15.4760	0.890544
$303$	20.4189	1.17304
$304$	−6.08273	−0.348868
$305$	8.33986	0.477539
$306$	0.491308	0.0280862
$307$	−16.4505	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$307$	−16.4505	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$308$	25.0999	1.43020
$309$	5.40831	0.307668
$310$	8.06682	0.458165
$311$	−5.13006	−0.290899	−0.145450	−	0.989366i	$-0.546463\pi$
$311$	−5.13006	−0.290899	−0.145450	+	0.989366i	$0.546463\pi$
$312$	−7.79146	−0.441105
$313$	10.5036	0.593697	0.296848	−	0.954925i	$-0.404064\pi$
$313$	10.5036	0.593697	0.296848	+	0.954925i	$0.404064\pi$
$314$	−15.1750	−0.856375
$315$	4.28895	0.241655
$316$	2.08273	0.117163
$317$	18.9177	1.06252	0.531261	−	0.847208i	$-0.321718\pi$
$317$	18.9177	1.06252	0.531261	+	0.847208i	$0.321718\pi$
$318$	16.1555	0.905956
$319$	43.7151	2.44758
$320$	−1.00000	−0.0559017
$321$	22.1481	1.23619
$322$	−0.703820	−0.0392224
$323$	−2.69418	−0.149908
$324$	−4.44187	−0.246771
$325$	−5.66632	−0.314311
$326$	6.30949	0.349450
$327$	0.547529	0.0302784
$328$	−10.5257	−0.581182
$329$	−33.4433	−1.84379
$330$	−8.92616	−0.491369
$331$	−27.8196	−1.52911	−0.764553	−	0.644561i	$-0.777040\pi$
$331$	−27.8196	−1.52911	−0.764553	+	0.644561i	$0.777040\pi$
$332$	1.63218	0.0895776
$333$	0.713436	0.0390961
$334$	12.4977	0.683844
$335$	−1.00000	−0.0546358
$336$	−5.31672	−0.290051
$337$	3.49489	0.190379	0.0951894	−	0.995459i	$-0.469654\pi$
$337$	3.49489	0.190379	0.0951894	+	0.995459i	$0.469654\pi$
$338$	−19.1071	−1.03929
$339$	−5.96222	−0.323823
$340$	−0.442924	−0.0240209
$341$	52.3659	2.83577
$342$	−6.74720	−0.364847
$343$	3.67460	0.198410
$344$	−1.77555	−0.0957315
$345$	0.250296	0.0134755
$346$	−7.24017	−0.389234
$347$	−21.7187	−1.16592	−0.582960	−	0.812501i	$-0.698106\pi$
$347$	−21.7187	−1.16592	−0.582960	+	0.812501i	$0.698106\pi$
$348$	−9.25984	−0.496380
$349$	−2.21384	−0.118504	−0.0592522	−	0.998243i	$-0.518872\pi$
$349$	−2.21384	−0.118504	−0.0592522	+	0.998243i	$0.518872\pi$
$350$	−3.86657	−0.206677
$351$	−32.0170	−1.70894
$352$	−6.49152	−0.345999
$353$	−6.70160	−0.356690	−0.178345	−	0.983968i	$-0.557074\pi$
$353$	−6.70160	−0.356690	−0.178345	+	0.983968i	$0.557074\pi$
$354$	11.1287	0.591485
$355$	−15.8401	−0.840703
$356$	8.49152	0.450050
$357$	−2.35490	−0.124635
$358$	4.82752	0.255142
$359$	12.3254	0.650510	0.325255	−	0.945626i	$-0.394550\pi$
$359$	12.3254	0.650510	0.325255	+	0.945626i	$0.394550\pi$
$360$	−1.10924	−0.0584620
$361$	17.9996	0.947348
$362$	−6.77064	−0.355857
$363$	−42.8188	−2.24740
$364$	−21.9092	−1.14835
$365$	−3.19302	−0.167130
$366$	−11.4677	−0.599427
$367$	−1.31286	−0.0685308	−0.0342654	−	0.999413i	$-0.510909\pi$
$367$	−1.31286	−0.0685308	−0.0342654	+	0.999413i	$0.510909\pi$
$368$	0.182027	0.00948882
$369$	−11.6755	−0.607801
$370$	−0.643176	−0.0334371
$371$	45.4285	2.35853
$372$	−11.0923	−0.575108
$373$	35.3959	1.83273	0.916365	−	0.400344i	$-0.131109\pi$
$373$	35.3959	1.83273	0.916365	+	0.400344i	$0.131109\pi$
$374$	−2.87525	−0.148675
$375$	1.37505	0.0710072
$376$	8.64935	0.446056
$377$	−38.1580	−1.96524
$378$	−21.8477	−1.12372
$379$	13.6723	0.702298	0.351149	−	0.936320i	$-0.385791\pi$
$379$	13.6723	0.702298	0.351149	+	0.936320i	$0.385791\pi$
$380$	6.08273	0.312037
$381$	−22.9438	−1.17545
$382$	16.8678	0.863033
$383$	−4.57280	−0.233659	−0.116830	−	0.993152i	$-0.537273\pi$
$383$	−4.57280	−0.233659	−0.116830	+	0.993152i	$0.537273\pi$
$384$	1.37505	0.0701702
$385$	−25.0999	−1.27921
$386$	8.60838	0.438155
$387$	−1.96952	−0.100116
$388$	1.59121	0.0807815
$389$	−26.2945	−1.33318	−0.666591	−	0.745423i	$-0.732247\pi$
$389$	−26.2945	−1.33318	−0.666591	+	0.745423i	$0.732247\pi$
$390$	7.79146	0.394536
$391$	0.0806241	0.00407733
$392$	−7.95035	−0.401553
$393$	−10.8837	−0.549012
$394$	23.1485	1.16620
$395$	−2.08273	−0.104794
$396$	−7.20065	−0.361846
$397$	−16.3014	−0.818145	−0.409072	−	0.912502i	$-0.634148\pi$
$397$	−16.3014	−0.818145	−0.409072	+	0.912502i	$0.634148\pi$
$398$	−2.30100	−0.115339
$399$	32.3402	1.61903
$400$	1.00000	0.0500000
$401$	2.54222	0.126952	0.0634762	−	0.997983i	$-0.479781\pi$
$401$	2.54222	0.126952	0.0634762	+	0.997983i	$0.479781\pi$
$402$	1.37505	0.0685812
$403$	−45.7092	−2.27694
$404$	−14.8496	−0.738796
$405$	4.44187	0.220718
$406$	−26.0382	−1.29225
$407$	−4.17519	−0.206957
$408$	0.609042	0.0301521
$409$	−6.42557	−0.317724	−0.158862	−	0.987301i	$-0.550782\pi$
$409$	−6.42557	−0.317724	−0.158862	+	0.987301i	$0.550782\pi$
$410$	10.5257	0.519825
$411$	4.37718	0.215910
$412$	−3.93318	−0.193774
$413$	31.2934	1.53985
$414$	0.201912	0.00992341
$415$	−1.63218	−0.0801206
$416$	5.66632	0.277814
$417$	14.2011	0.695432
$418$	39.4862	1.93133
$419$	21.1485	1.03317	0.516586	−	0.856235i	$-0.327203\pi$
$419$	21.1485	1.03317	0.516586	+	0.856235i	$0.327203\pi$
$420$	5.31672	0.259429
$421$	−25.8000	−1.25741	−0.628707	−	0.777643i	$-0.716415\pi$
$421$	−25.8000	−1.25741	−0.628707	+	0.777643i	$0.716415\pi$
$422$	9.25840	0.450692
$423$	9.59420	0.466486
$424$	−11.7490	−0.570584
$425$	0.442924	0.0214849
$426$	21.7809	1.05529
$427$	−32.2466	−1.56052
$428$	−16.1071	−0.778568
$429$	50.5784	2.44195
$430$	1.77555	0.0856249
$431$	−21.8380	−1.05190	−0.525949	−	0.850516i	$-0.676290\pi$
$431$	−21.8380	−1.05190	−0.525949	+	0.850516i	$0.676290\pi$
$432$	5.65041	0.271855
$433$	−16.2429	−0.780583	−0.390292	−	0.920691i	$-0.627626\pi$
$433$	−16.2429	−0.780583	−0.390292	+	0.920691i	$0.627626\pi$
$434$	−31.1909	−1.49721
$435$	9.25984	0.443976
$436$	−0.398189	−0.0190698
$437$	−1.10722	−0.0529656
$438$	4.39056	0.209789
$439$	4.23062	0.201917	0.100958	−	0.994891i	$-0.467809\pi$
$439$	4.23062	0.201917	0.100958	+	0.994891i	$0.467809\pi$
$440$	6.49152	0.309471
$441$	−8.81884	−0.419945
$442$	2.50974	0.119376
$443$	1.55322	0.0737955	0.0368978	−	0.999319i	$-0.488252\pi$
$443$	1.55322	0.0737955	0.0368978	+	0.999319i	$0.488252\pi$
$444$	0.884399	0.0419717
$445$	−8.49152	−0.402537
$446$	1.51676	0.0718209
$447$	7.03221	0.332612
$448$	3.86657	0.182678
$449$	29.5771	1.39583	0.697914	−	0.716182i	$-0.254112\pi$
$449$	29.5771	1.39583	0.697914	+	0.716182i	$0.254112\pi$
$450$	1.10924	0.0522900
$451$	68.3275	3.21742
$452$	4.33600	0.203948
$453$	21.2803	0.999834
$454$	−16.4688	−0.772918
$455$	21.9092	1.02712
$456$	−8.36405	−0.391683
$457$	−2.94659	−0.137836	−0.0689178	−	0.997622i	$-0.521955\pi$
$457$	−2.94659	−0.137836	−0.0689178	+	0.997622i	$0.521955\pi$
$458$	14.9444	0.698305
$459$	2.50270	0.116816
$460$	−0.182027	−0.00848706
$461$	−20.7866	−0.968126	−0.484063	−	0.875033i	$-0.660839\pi$
$461$	−20.7866	−0.968126	−0.484063	+	0.875033i	$0.660839\pi$
$462$	34.5136	1.60572
$463$	22.8519	1.06202	0.531009	−	0.847366i	$-0.321813\pi$
$463$	22.8519	1.06202	0.531009	+	0.847366i	$0.321813\pi$
$464$	6.73419	0.312627
$465$	11.0923	0.514392
$466$	−13.8097	−0.639722
$467$	2.14934	0.0994596	0.0497298	−	0.998763i	$-0.484164\pi$
$467$	2.14934	0.0994596	0.0497298	+	0.998763i	$0.484164\pi$
$468$	6.28530	0.290538
$469$	3.86657	0.178542
$470$	−8.64935	−0.398965
$471$	−20.8664	−0.961472
$472$	−8.09333	−0.372526
$473$	11.5260	0.529968
$474$	2.86386	0.131541
$475$	−6.08273	−0.279095
$476$	1.71259	0.0784966
$477$	−13.0325	−0.596717
$478$	−29.8205	−1.36396
$479$	−9.51698	−0.434842	−0.217421	−	0.976078i	$-0.569764\pi$
$479$	−9.51698	−0.434842	−0.217421	+	0.976078i	$0.569764\pi$
$480$	−1.37505	−0.0627621
$481$	3.64444	0.166172
$482$	12.1663	0.554161
$483$	−0.967787	−0.0440358
$484$	31.1398	1.41545
$485$	−1.59121	−0.0722532
$486$	10.8434	0.491868
$487$	−20.6676	−0.936537	−0.468269	−	0.883586i	$-0.655122\pi$
$487$	−20.6676	−0.936537	−0.468269	+	0.883586i	$0.655122\pi$
$488$	8.33986	0.377528
$489$	8.67586	0.392336
$490$	7.95035	0.359160
$491$	−20.4620	−0.923438	−0.461719	−	0.887026i	$-0.652767\pi$
$491$	−20.4620	−0.923438	−0.461719	+	0.887026i	$0.652767\pi$
$492$	−14.4733	−0.652506
$493$	2.98273	0.134335
$494$	−34.4667	−1.55073
$495$	7.20065	0.323645
$496$	8.06682	0.362211
$497$	61.2467	2.74729
$498$	2.24433	0.100571
$499$	−26.9238	−1.20528	−0.602638	−	0.798014i	$-0.705884\pi$
$499$	−26.9238	−1.20528	−0.602638	+	0.798014i	$0.705884\pi$
$500$	−1.00000	−0.0447214
$501$	17.1849	0.767767
$502$	6.54985	0.292334
$503$	−24.3407	−1.08530	−0.542648	−	0.839960i	$-0.682578\pi$
$503$	−24.3407	−1.08530	−0.542648	+	0.839960i	$0.682578\pi$
$504$	4.28895	0.191045
$505$	14.8496	0.660799
$506$	−1.18163	−0.0525300
$507$	−26.2733	−1.16684
$508$	16.6858	0.740313
$509$	20.2149	0.896011	0.448005	−	0.894031i	$-0.352135\pi$
$509$	20.2149	0.896011	0.448005	+	0.894031i	$0.352135\pi$
$510$	−0.609042	−0.0269688
$511$	12.3460	0.546157
$512$	−1.00000	−0.0441942
$513$	−34.3699	−1.51747
$514$	4.99364	0.220260
$515$	3.93318	0.173317
$516$	−2.44148	−0.107480
$517$	−56.1474	−2.46936
$518$	2.48689	0.109267
$519$	−9.95559	−0.437002
$520$	−5.66632	−0.248484
$521$	31.6059	1.38468	0.692339	−	0.721572i	$-0.256580\pi$
$521$	31.6059	1.38468	0.692339	+	0.721572i	$0.256580\pi$
$522$	7.46983	0.326945
$523$	23.4775	1.02660	0.513299	−	0.858210i	$-0.328423\pi$
$523$	23.4775	1.02660	0.513299	+	0.858210i	$0.328423\pi$
$524$	7.91516	0.345776
$525$	−5.31672	−0.232041
$526$	−22.5136	−0.981640
$527$	3.57299	0.155642
$528$	−8.92616	−0.388461
$529$	−22.9669	−0.998559
$530$	11.7490	0.510346
$531$	−8.97744	−0.389588
$532$	−23.5193	−1.01969
$533$	−59.6417	−2.58337
$534$	11.6763	0.505281
$535$	16.1071	0.696372
$536$	−1.00000	−0.0431934
$537$	6.63808	0.286454
$538$	5.41747	0.233564
$539$	51.6099	2.22299
$540$	−5.65041	−0.243155
$541$	−25.3397	−1.08944	−0.544719	−	0.838619i	$-0.683364\pi$
$541$	−25.3397	−1.08944	−0.544719	+	0.838619i	$0.683364\pi$
$542$	12.7162	0.546206
$543$	−9.30997	−0.399529
$544$	−0.442924	−0.0189902
$545$	0.398189	0.0170565
$546$	−30.1262	−1.28928
$547$	−24.8936	−1.06437	−0.532186	−	0.846627i	$-0.678629\pi$
$547$	−24.8936	−1.06437	−0.532186	+	0.846627i	$0.678629\pi$
$548$	−3.18329	−0.135983
$549$	9.25090	0.394819
$550$	−6.49152	−0.276799
$551$	−40.9623	−1.74505
$552$	0.250296	0.0106533
$553$	8.05302	0.342449
$554$	−20.3180	−0.863229
$555$	−0.884399	−0.0375407
$556$	−10.3277	−0.437993
$557$	−28.3012	−1.19916	−0.599580	−	0.800315i	$-0.704666\pi$
$557$	−28.3012	−1.19916	−0.599580	+	0.800315i	$0.704666\pi$
$558$	8.94803	0.378801
$559$	−10.0609	−0.425529
$560$	−3.86657	−0.163392
$561$	−3.95361	−0.166921
$562$	3.70343	0.156220
$563$	−1.14464	−0.0482407	−0.0241204	−	0.999709i	$-0.507678\pi$
$563$	−1.14464	−0.0482407	−0.0241204	+	0.999709i	$0.507678\pi$
$564$	11.8933	0.500798
$565$	−4.33600	−0.182417
$566$	−21.8341	−0.917755
$567$	−17.1748	−0.721274
$568$	−15.8401	−0.664634
$569$	−14.7463	−0.618198	−0.309099	−	0.951030i	$-0.600028\pi$
$569$	−14.7463	−0.618198	−0.309099	+	0.951030i	$0.600028\pi$
$570$	8.36405	0.350332
$571$	−14.5613	−0.609372	−0.304686	−	0.952453i	$-0.598552\pi$
$571$	−14.5613	−0.609372	−0.304686	+	0.952453i	$0.598552\pi$
$572$	−36.7830	−1.53797
$573$	23.1941	0.968948
$574$	−40.6982	−1.69871
$575$	0.182027	0.00759105
$576$	−1.10924	−0.0462183
$577$	16.5804	0.690252	0.345126	−	0.938556i	$-0.387836\pi$
$577$	16.5804	0.690252	0.345126	+	0.938556i	$0.387836\pi$
$578$	16.8038	0.698947
$579$	11.8370	0.491927
$580$	−6.73419	−0.279622
$581$	6.31094	0.261822
$582$	2.18799	0.0906953
$583$	76.2692	3.15875
$584$	−3.19302	−0.132128
$585$	−6.28530	−0.259865
$586$	16.8351	0.695453
$587$	30.0467	1.24016	0.620080	−	0.784539i	$-0.287100\pi$
$587$	30.0467	1.24016	0.620080	+	0.784539i	$0.287100\pi$
$588$	−10.9321	−0.450833
$589$	−49.0683	−2.02182
$590$	8.09333	0.333197
$591$	31.8303	1.30932
$592$	−0.643176	−0.0264344
$593$	30.1907	1.23978	0.619892	−	0.784687i	$-0.287176\pi$
$593$	30.1907	1.23978	0.619892	+	0.784687i	$0.287176\pi$
$594$	−36.6797	−1.50499
$595$	−1.71259	−0.0702095
$596$	−5.11415	−0.209484
$597$	−3.16399	−0.129493
$598$	1.03142	0.0421780
$599$	−18.3557	−0.749993	−0.374996	−	0.927026i	$-0.622356\pi$
$599$	−18.3557	−0.749993	−0.374996	+	0.927026i	$0.622356\pi$
$600$	1.37505	0.0561362
$601$	34.8800	1.42278	0.711392	−	0.702795i	$-0.248065\pi$
$601$	34.8800	1.42278	0.711392	+	0.702795i	$0.248065\pi$
$602$	−6.86530	−0.279809
$603$	−1.10924	−0.0451717
$604$	−15.4760	−0.629710
$605$	−31.1398	−1.26601
$606$	−20.4189	−0.829463
$607$	−21.9296	−0.890096	−0.445048	−	0.895507i	$-0.646813\pi$
$607$	−21.9296	−0.890096	−0.445048	+	0.895507i	$0.646813\pi$
$608$	6.08273	0.246687
$609$	−35.8038	−1.45084
$610$	−8.33986	−0.337671
$611$	49.0100	1.98273
$612$	−0.491308	−0.0198600
$613$	29.5730	1.19444	0.597221	−	0.802077i	$-0.296272\pi$
$613$	29.5730	1.19444	0.597221	+	0.802077i	$0.296272\pi$
$614$	16.4505	0.663890
$615$	14.4733	0.583620
$616$	−25.0999	−1.01130
$617$	16.8269	0.677424	0.338712	−	0.940890i	$-0.390009\pi$
$617$	16.8269	0.677424	0.338712	+	0.940890i	$0.390009\pi$
$618$	−5.40831	−0.217554
$619$	−31.3649	−1.26066	−0.630330	−	0.776327i	$-0.717081\pi$
$619$	−31.3649	−1.26066	−0.630330	+	0.776327i	$0.717081\pi$
$620$	−8.06682	−0.323971
$621$	1.02853	0.0412734
$622$	5.13006	0.205697
$623$	32.8330	1.31543
$624$	7.79146	0.311908
$625$	1.00000	0.0400000
$626$	−10.5036	−0.419807
$627$	54.2954	2.16835
$628$	15.1750	0.605549
$629$	−0.284878	−0.0113588
$630$	−4.28895	−0.170876
$631$	−16.8314	−0.670047	−0.335023	−	0.942210i	$-0.608744\pi$
$631$	−16.8314	−0.670047	−0.335023	+	0.942210i	$0.608744\pi$
$632$	−2.08273	−0.0828466
$633$	12.7308	0.506002
$634$	−18.9177	−0.751316
$635$	−16.6858	−0.662156
$636$	−16.1555	−0.640608
$637$	−45.0492	−1.78491
$638$	−43.7151	−1.73070
$639$	−17.5704	−0.695075
$640$	1.00000	0.0395285
$641$	−2.40437	−0.0949668	−0.0474834	−	0.998872i	$-0.515120\pi$
$641$	−2.40437	−0.0949668	−0.0474834	+	0.998872i	$0.515120\pi$
$642$	−22.1481	−0.874116
$643$	−37.4907	−1.47849	−0.739244	−	0.673438i	$-0.764817\pi$
$643$	−37.4907	−1.47849	−0.739244	+	0.673438i	$0.764817\pi$
$644$	0.703820	0.0277344
$645$	2.44148	0.0961330
$646$	2.69418	0.106001
$647$	−23.5202	−0.924674	−0.462337	−	0.886704i	$-0.652989\pi$
$647$	−23.5202	−0.924674	−0.462337	+	0.886704i	$0.652989\pi$
$648$	4.44187	0.174493
$649$	52.5380	2.06230
$650$	5.66632	0.222251
$651$	−42.8890	−1.68095
$652$	−6.30949	−0.247099
$653$	−10.4020	−0.407060	−0.203530	−	0.979069i	$-0.565241\pi$
$653$	−10.4020	−0.407060	−0.203530	+	0.979069i	$0.565241\pi$
$654$	−0.547529	−0.0214101
$655$	−7.91516	−0.309271
$656$	10.5257	0.410958
$657$	−3.54183	−0.138180
$658$	33.4433	1.30376
$659$	22.7056	0.884484	0.442242	−	0.896896i	$-0.354183\pi$
$659$	22.7056	0.884484	0.442242	+	0.896896i	$0.354183\pi$
$660$	8.92616	0.347450
$661$	−17.5511	−0.682659	−0.341330	−	0.939944i	$-0.610877\pi$
$661$	−17.5511	−0.682659	−0.341330	+	0.939944i	$0.610877\pi$
$662$	27.8196	1.08124
$663$	3.45102	0.134027
$664$	−1.63218	−0.0633409
$665$	23.5193	0.912039
$666$	−0.713436	−0.0276451
$667$	1.22580	0.0474634
$668$	−12.4977	−0.483550
$669$	2.08563	0.0806350
$670$	1.00000	0.0386334
$671$	−54.1384	−2.08999
$672$	5.31672	0.205097
$673$	−1.07992	−0.0416280	−0.0208140	−	0.999783i	$-0.506626\pi$
$673$	−1.07992	−0.0416280	−0.0208140	+	0.999783i	$0.506626\pi$
$674$	−3.49489	−0.134618
$675$	5.65041	0.217484
$676$	19.1071	0.734890
$677$	24.7625	0.951698	0.475849	−	0.879527i	$-0.342141\pi$
$677$	24.7625	0.951698	0.475849	+	0.879527i	$0.342141\pi$
$678$	5.96222	0.228978
$679$	6.15253	0.236112
$680$	0.442924	0.0169853
$681$	−22.6454	−0.867773
$682$	−52.3659	−2.00519
$683$	26.9686	1.03192	0.515962	−	0.856611i	$-0.327435\pi$
$683$	26.9686	1.03192	0.515962	+	0.856611i	$0.327435\pi$
$684$	6.74720	0.257986
$685$	3.18329	0.121627
$686$	−3.67460	−0.140297
$687$	20.5493	0.784003
$688$	1.77555	0.0676924
$689$	−66.5738	−2.53626
$690$	−0.250296	−0.00952861
$691$	36.2422	1.37872	0.689360	−	0.724419i	$-0.257892\pi$
$691$	36.2422	1.37872	0.689360	+	0.724419i	$0.257892\pi$
$692$	7.24017	0.275230
$693$	−27.8418	−1.05762
$694$	21.7187	0.824430
$695$	10.3277	0.391753
$696$	9.25984	0.350993
$697$	4.66206	0.176588
$698$	2.21384	0.0837953
$699$	−18.9890	−0.718230
$700$	3.86657	0.146143
$701$	−31.1894	−1.17801	−0.589003	−	0.808131i	$-0.700479\pi$
$701$	−31.1894	−1.17801	−0.589003	+	0.808131i	$0.700479\pi$
$702$	32.0170	1.20840
$703$	3.91227	0.147554
$704$	6.49152	0.244658
$705$	−11.8933	−0.447927
$706$	6.70160	0.252218
$707$	−57.4170	−2.15939
$708$	−11.1287	−0.418243
$709$	0.530172	0.0199110	0.00995552	−	0.999950i	$-0.496831\pi$
$709$	0.530172	0.0199110	0.00995552	+	0.999950i	$0.496831\pi$
$710$	15.8401	0.594467
$711$	−2.31025	−0.0866410
$712$	−8.49152	−0.318233
$713$	1.46838	0.0549913
$714$	2.35490	0.0881300
$715$	36.7830	1.37561
$716$	−4.82752	−0.180413
$717$	−41.0047	−1.53135
$718$	−12.3254	−0.459980
$719$	−33.0153	−1.23126	−0.615631	−	0.788035i	$-0.711099\pi$
$719$	−33.0153	−1.23126	−0.615631	+	0.788035i	$0.711099\pi$
$720$	1.10924	0.0413389
$721$	−15.2079	−0.566372
$722$	−17.9996	−0.669876
$723$	16.7293	0.622169
$724$	6.77064	0.251629
$725$	6.73419	0.250102
$726$	42.8188	1.58915
$727$	16.0311	0.594562	0.297281	−	0.954790i	$-0.403920\pi$
$727$	16.0311	0.594562	0.297281	+	0.954790i	$0.403920\pi$
$728$	21.9092	0.812009
$729$	28.2359	1.04577
$730$	3.19302	0.118179
$731$	0.786435	0.0290874
$732$	11.4677	0.423859
$733$	7.53057	0.278148	0.139074	−	0.990282i	$-0.455587\pi$
$733$	7.53057	0.278148	0.139074	+	0.990282i	$0.455587\pi$
$734$	1.31286	0.0484586
$735$	10.9321	0.403238
$736$	−0.182027	−0.00670961
$737$	6.49152	0.239118
$738$	11.6755	0.429780
$739$	41.5267	1.52758	0.763792	−	0.645462i	$-0.223335\pi$
$739$	41.5267	1.52758	0.763792	+	0.645462i	$0.223335\pi$
$740$	0.643176	0.0236436
$741$	−47.3934	−1.74104
$742$	−45.4285	−1.66773
$743$	−47.6184	−1.74695	−0.873474	−	0.486871i	$-0.838138\pi$
$743$	−47.6184	−1.74695	−0.873474	+	0.486871i	$0.838138\pi$
$744$	11.0923	0.406663
$745$	5.11415	0.187368
$746$	−35.3959	−1.29594
$747$	−1.81048	−0.0662420
$748$	2.87525	0.105129
$749$	−62.2793	−2.27564
$750$	−1.37505	−0.0502097
$751$	4.83283	0.176352	0.0881762	−	0.996105i	$-0.471896\pi$
$751$	4.83283	0.176352	0.0881762	+	0.996105i	$0.471896\pi$
$752$	−8.64935	−0.315409
$753$	9.00636	0.328210
$754$	38.1580	1.38963
$755$	15.4760	0.563229
$756$	21.8477	0.794593
$757$	0.197936	0.00719410	0.00359705	−	0.999994i	$-0.498855\pi$
$757$	0.197936	0.00719410	0.00359705	+	0.999994i	$0.498855\pi$
$758$	−13.6723	−0.496600
$759$	−1.62480	−0.0589766
$760$	−6.08273	−0.220644
$761$	4.69301	0.170121	0.0850607	−	0.996376i	$-0.472892\pi$
$761$	4.69301	0.170121	0.0850607	+	0.996376i	$0.472892\pi$
$762$	22.9438	0.831166
$763$	−1.53962	−0.0557381
$764$	−16.8678	−0.610257
$765$	0.491308	0.0177633
$766$	4.57280	0.165222
$767$	−45.8594	−1.65589
$768$	−1.37505	−0.0496178
$769$	53.2397	1.91987	0.959937	−	0.280217i	$-0.0904063\pi$
$769$	53.2397	1.91987	0.959937	+	0.280217i	$0.0904063\pi$
$770$	25.0999	0.904538
$771$	6.86650	0.247291
$772$	−8.60838	−0.309823
$773$	0.121383	0.00436585	0.00218293	−	0.999998i	$-0.499305\pi$
$773$	0.121383	0.00436585	0.00218293	+	0.999998i	$0.499305\pi$
$774$	1.96952	0.0707928
$775$	8.06682	0.289769
$776$	−1.59121	−0.0571211
$777$	3.41959	0.122677
$778$	26.2945	0.942702
$779$	−64.0247	−2.29392
$780$	−7.79146	−0.278979
$781$	102.826	3.67941
$782$	−0.0806241	−0.00288311
$783$	38.0509	1.35983
$784$	7.95035	0.283941
$785$	−15.1750	−0.541619
$786$	10.8837	0.388210
$787$	−34.3462	−1.22431	−0.612154	−	0.790738i	$-0.709697\pi$
$787$	−34.3462	−1.22431	−0.612154	+	0.790738i	$0.709697\pi$
$788$	−23.1485	−0.824631
$789$	−30.9573	−1.10211
$790$	2.08273	0.0741002
$791$	16.7654	0.596111
$792$	7.20065	0.255864
$793$	47.2563	1.67812
$794$	16.3014	0.578516
$795$	16.1555	0.572977
$796$	2.30100	0.0815567
$797$	15.7209	0.556862	0.278431	−	0.960456i	$-0.410186\pi$
$797$	15.7209	0.556862	0.278431	+	0.960456i	$0.410186\pi$
$798$	−32.3402	−1.14483
$799$	−3.83100	−0.135531
$800$	−1.00000	−0.0353553
$801$	−9.41913	−0.332808
$802$	−2.54222	−0.0897689
$803$	20.7276	0.731460
$804$	−1.37505	−0.0484943
$805$	−0.703820	−0.0248064
$806$	45.7092	1.61004
$807$	7.44929	0.262227
$808$	14.8496	0.522407
$809$	7.06401	0.248357	0.124179	−	0.992260i	$-0.460370\pi$
$809$	7.06401	0.248357	0.124179	+	0.992260i	$0.460370\pi$
$810$	−4.44187	−0.156071
$811$	−16.9359	−0.594700	−0.297350	−	0.954769i	$-0.596103\pi$
$811$	−16.9359	−0.594700	−0.297350	+	0.954769i	$0.596103\pi$
$812$	26.0382	0.913762
$813$	17.4854	0.613239
$814$	4.17519	0.146340
$815$	6.30949	0.221012
$816$	−0.609042	−0.0213207
$817$	−10.8002	−0.377852
$818$	6.42557	0.224665
$819$	24.3025	0.849200
$820$	−10.5257	−0.367572
$821$	16.3104	0.569236	0.284618	−	0.958641i	$-0.408133\pi$
$821$	16.3104	0.569236	0.284618	+	0.958641i	$0.408133\pi$
$822$	−4.37718	−0.152672
$823$	−18.6515	−0.650149	−0.325075	−	0.945688i	$-0.605389\pi$
$823$	−18.6515	−0.650149	−0.325075	+	0.945688i	$0.605389\pi$
$824$	3.93318	0.137019
$825$	−8.92616	−0.310769
$826$	−31.2934	−1.08884
$827$	8.80929	0.306329	0.153165	−	0.988201i	$-0.451054\pi$
$827$	8.80929	0.306329	0.153165	+	0.988201i	$0.451054\pi$
$828$	−0.201912	−0.00701691
$829$	33.5074	1.16376	0.581879	−	0.813275i	$-0.302318\pi$
$829$	33.5074	1.16376	0.581879	+	0.813275i	$0.302318\pi$
$830$	1.63218	0.0566538
$831$	−27.9382	−0.969167
$832$	−5.66632	−0.196444
$833$	3.52140	0.122009
$834$	−14.2011	−0.491745
$835$	12.4977	0.432501
$836$	−39.4862	−1.36566
$837$	45.5808	1.57550
$838$	−21.1485	−0.730563
$839$	−27.4952	−0.949240	−0.474620	−	0.880191i	$-0.657414\pi$
$839$	−27.4952	−0.949240	−0.474620	+	0.880191i	$0.657414\pi$
$840$	−5.31672	−0.183444
$841$	16.3493	0.563769
$842$	25.8000	0.889125
$843$	5.09239	0.175391
$844$	−9.25840	−0.318687
$845$	−19.1071	−0.657305
$846$	−9.59420	−0.329855
$847$	120.404	4.13714
$848$	11.7490	0.403464
$849$	−30.0230	−1.03039
$850$	−0.442924	−0.0151922
$851$	−0.117075	−0.00401330
$852$	−21.7809	−0.746200
$853$	28.5197	0.976495	0.488248	−	0.872705i	$-0.337636\pi$
$853$	28.5197	0.976495	0.488248	+	0.872705i	$0.337636\pi$
$854$	32.2466	1.10346
$855$	−6.74720	−0.230749
$856$	16.1071	0.550530
$857$	−23.4042	−0.799472	−0.399736	−	0.916630i	$-0.630898\pi$
$857$	−23.4042	−0.799472	−0.399736	+	0.916630i	$0.630898\pi$
$858$	−50.5784	−1.72672
$859$	52.1450	1.77917	0.889583	−	0.456774i	$-0.150995\pi$
$859$	52.1450	1.77917	0.889583	+	0.456774i	$0.150995\pi$
$860$	−1.77555	−0.0605459
$861$	−55.9620	−1.90718
$862$	21.8380	0.743804
$863$	−51.5303	−1.75411	−0.877056	−	0.480387i	$-0.840496\pi$
$863$	−51.5303	−1.75411	−0.877056	+	0.480387i	$0.840496\pi$
$864$	−5.65041	−0.192231
$865$	−7.24017	−0.246173
$866$	16.2429	0.551956
$867$	23.1061	0.784724
$868$	31.1909	1.05869
$869$	13.5201	0.458637
$870$	−9.25984	−0.313938
$871$	−5.66632	−0.191996
$872$	0.398189	0.0134844
$873$	−1.76503	−0.0597373
$874$	1.10722	0.0374523
$875$	−3.86657	−0.130714
$876$	−4.39056	−0.148343
$877$	1.52991	0.0516614	0.0258307	−	0.999666i	$-0.491777\pi$
$877$	1.52991	0.0516614	0.0258307	+	0.999666i	$0.491777\pi$
$878$	−4.23062	−0.142777
$879$	23.1492	0.780802
$880$	−6.49152	−0.218829
$881$	17.2978	0.582779	0.291389	−	0.956604i	$-0.405882\pi$
$881$	17.2978	0.582779	0.291389	+	0.956604i	$0.405882\pi$
$882$	8.81884	0.296946
$883$	−50.6823	−1.70559	−0.852797	−	0.522243i	$-0.825095\pi$
$883$	−50.6823	−1.70559	−0.852797	+	0.522243i	$0.825095\pi$
$884$	−2.50974	−0.0844119
$885$	11.1287	0.374088
$886$	−1.55322	−0.0521813
$887$	43.6696	1.46628	0.733141	−	0.680076i	$-0.238053\pi$
$887$	43.6696	1.46628	0.733141	+	0.680076i	$0.238053\pi$
$888$	−0.884399	−0.0296785
$889$	64.5168	2.16382
$890$	8.49152	0.284636
$891$	−28.8345	−0.965992
$892$	−1.51676	−0.0507850
$893$	52.6117	1.76058
$894$	−7.03221	−0.235192
$895$	4.82752	0.161366
$896$	−3.86657	−0.129173
$897$	1.41826	0.0473542
$898$	−29.5771	−0.986999
$899$	54.3235	1.81179
$900$	−1.10924	−0.0369746
$901$	5.20393	0.173368
$902$	−68.3275	−2.27506
$903$	−9.44013	−0.314148
$904$	−4.33600	−0.144213
$905$	−6.77064	−0.225064
$906$	−21.2803	−0.706990
$907$	−1.99913	−0.0663801	−0.0331900	−	0.999449i	$-0.510567\pi$
$907$	−1.99913	−0.0663801	−0.0331900	+	0.999449i	$0.510567\pi$
$908$	16.4688	0.546535
$909$	16.4718	0.546334
$910$	−21.9092	−0.726283
$911$	59.2215	1.96210	0.981048	−	0.193766i	$-0.0620701\pi$
$911$	59.2215	1.96210	0.981048	+	0.193766i	$0.0620701\pi$
$912$	8.36405	0.276962
$913$	10.5953	0.350654
$914$	2.94659	0.0974644
$915$	−11.4677	−0.379111
$916$	−14.9444	−0.493776
$917$	30.6045	1.01065
$918$	−2.50270	−0.0826014
$919$	15.2432	0.502826	0.251413	−	0.967880i	$-0.419105\pi$
$919$	15.2432	0.502826	0.251413	+	0.967880i	$0.419105\pi$
$920$	0.182027	0.00600125
$921$	22.6203	0.745365
$922$	20.7866	0.684569
$923$	−89.7548	−2.95431
$924$	−34.5136	−1.13541
$925$	−0.643176	−0.0211475
$926$	−22.8519	−0.750961
$927$	4.36284	0.143294
$928$	−6.73419	−0.221061
$929$	−18.4688	−0.605941	−0.302970	−	0.953000i	$-0.597978\pi$
$929$	−18.4688	−0.605941	−0.302970	+	0.953000i	$0.597978\pi$
$930$	−11.0923	−0.363730
$931$	−48.3598	−1.58493
$932$	13.8097	0.452351
$933$	7.05409	0.230940
$934$	−2.14934	−0.0703286
$935$	−2.87525	−0.0940306
$936$	−6.28530	−0.205441
$937$	−32.5556	−1.06354	−0.531772	−	0.846887i	$-0.678474\pi$
$937$	−32.5556	−1.06354	−0.531772	+	0.846887i	$0.678474\pi$
$938$	−3.86657	−0.126248
$939$	−14.4429	−0.471327
$940$	8.64935	0.282111
$941$	−13.0293	−0.424744	−0.212372	−	0.977189i	$-0.568119\pi$
$941$	−13.0293	−0.424744	−0.212372	+	0.977189i	$0.568119\pi$
$942$	20.8664	0.679864
$943$	1.91595	0.0623920
$944$	8.09333	0.263415
$945$	−21.8477	−0.710705
$946$	−11.5260	−0.374744
$947$	36.4658	1.18498	0.592490	−	0.805578i	$-0.298145\pi$
$947$	36.4658	1.18498	0.592490	+	0.805578i	$0.298145\pi$
$948$	−2.86386	−0.0930137
$949$	−18.0927	−0.587313
$950$	6.08273	0.197350
$951$	−26.0127	−0.843520
$952$	−1.71259	−0.0555055
$953$	45.9063	1.48705	0.743526	−	0.668707i	$-0.233152\pi$
$953$	45.9063	1.48705	0.743526	+	0.668707i	$0.233152\pi$
$954$	13.0325	0.421943
$955$	16.8678	0.545830
$956$	29.8205	0.964464
$957$	−60.1105	−1.94309
$958$	9.51698	0.307480
$959$	−12.3084	−0.397459
$960$	1.37505	0.0443795
$961$	34.0736	1.09915
$962$	−3.64444	−0.117501
$963$	17.8667	0.575745
$964$	−12.1663	−0.391851
$965$	8.60838	0.277114
$966$	0.967787	0.0311380
$967$	−9.83388	−0.316236	−0.158118	−	0.987420i	$-0.550543\pi$
$967$	−9.83388	−0.316236	−0.158118	+	0.987420i	$0.550543\pi$
$968$	−31.1398	−1.00087
$969$	3.70464	0.119010
$970$	1.59121	0.0510907
$971$	38.9809	1.25096	0.625479	−	0.780241i	$-0.284904\pi$
$971$	38.9809	1.25096	0.625479	+	0.780241i	$0.284904\pi$
$972$	−10.8434	−0.347803
$973$	−39.9328	−1.28019
$974$	20.6676	0.662232
$975$	7.79146	0.249527
$976$	−8.33986	−0.266952
$977$	47.4362	1.51762	0.758809	−	0.651313i	$-0.225781\pi$
$977$	47.4362	1.51762	0.758809	+	0.651313i	$0.225781\pi$
$978$	−8.67586	−0.277424
$979$	55.1229	1.76173
$980$	−7.95035	−0.253965
$981$	0.441687	0.0141020
$982$	20.4620	0.652969
$983$	40.9453	1.30595	0.652976	−	0.757379i	$-0.273520\pi$
$983$	40.9453	1.30595	0.652976	+	0.757379i	$0.273520\pi$
$984$	14.4733	0.461392
$985$	23.1485	0.737572
$986$	−2.98273	−0.0949895
$987$	45.9862	1.46376
$988$	34.4667	1.09653
$989$	0.323199	0.0102771
$990$	−7.20065	−0.228852
$991$	−46.3818	−1.47337	−0.736684	−	0.676237i	$-0.763609\pi$
$991$	−46.3818	−1.47337	−0.736684	+	0.676237i	$0.763609\pi$
$992$	−8.06682	−0.256122
$993$	38.2534	1.21393
$994$	−61.2467	−1.94263
$995$	−2.30100	−0.0729465
$996$	−2.24433	−0.0711143
$997$	13.0324	0.412741	0.206370	−	0.978474i	$-0.433835\pi$
$997$	13.0324	0.412741	0.206370	+	0.978474i	$0.433835\pi$
$998$	26.9238	0.852259
$999$	−3.63421	−0.114981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	670.2.a.j.1.2	✓	4
3.2	odd	2		6030.2.a.bt.1.4		4
4.3	odd	2		5360.2.a.be.1.3		4
5.2	odd	4		3350.2.c.m.2949.3		8
5.3	odd	4		3350.2.c.m.2949.6		8
5.4	even	2		3350.2.a.n.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
670.2.a.j.1.2	✓	4	1.1	even	1	trivial
3350.2.a.n.1.3		4	5.4	even	2
3350.2.c.m.2949.3		8	5.2	odd	4
3350.2.c.m.2949.6		8	5.3	odd	4
5360.2.a.be.1.3		4	4.3	odd	2
6030.2.a.bt.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 670 = 2 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.37505 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.37505 q^{6} +3.86657 q^{7} -1.00000 q^{8} -1.10924 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.37505 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.37505 q^{6} +3.86657 q^{7} -1.00000 q^{8} -1.10924 q^{9} +1.00000 q^{10} +6.49152 q^{11} -1.37505 q^{12} -5.66632 q^{13} -3.86657 q^{14} +1.37505 q^{15} +1.00000 q^{16} +0.442924 q^{17} +1.10924 q^{18} -6.08273 q^{19} -1.00000 q^{20} -5.31672 q^{21} -6.49152 q^{22} +0.182027 q^{23} +1.37505 q^{24} +1.00000 q^{25} +5.66632 q^{26} +5.65041 q^{27} +3.86657 q^{28} +6.73419 q^{29} -1.37505 q^{30} +8.06682 q^{31} -1.00000 q^{32} -8.92616 q^{33} -0.442924 q^{34} -3.86657 q^{35} -1.10924 q^{36} -0.643176 q^{37} +6.08273 q^{38} +7.79146 q^{39} +1.00000 q^{40} +10.5257 q^{41} +5.31672 q^{42} +1.77555 q^{43} +6.49152 q^{44} +1.10924 q^{45} -0.182027 q^{46} -8.64935 q^{47} -1.37505 q^{48} +7.95035 q^{49} -1.00000 q^{50} -0.609042 q^{51} -5.66632 q^{52} +11.7490 q^{53} -5.65041 q^{54} -6.49152 q^{55} -3.86657 q^{56} +8.36405 q^{57} -6.73419 q^{58} +8.09333 q^{59} +1.37505 q^{60} -8.33986 q^{61} -8.06682 q^{62} -4.28895 q^{63} +1.00000 q^{64} +5.66632 q^{65} +8.92616 q^{66} +1.00000 q^{67} +0.442924 q^{68} -0.250296 q^{69} +3.86657 q^{70} +15.8401 q^{71} +1.10924 q^{72} +3.19302 q^{73} +0.643176 q^{74} -1.37505 q^{75} -6.08273 q^{76} +25.0999 q^{77} -7.79146 q^{78} +2.08273 q^{79} -1.00000 q^{80} -4.44187 q^{81} -10.5257 q^{82} +1.63218 q^{83} -5.31672 q^{84} -0.442924 q^{85} -1.77555 q^{86} -9.25984 q^{87} -6.49152 q^{88} +8.49152 q^{89} -1.10924 q^{90} -21.9092 q^{91} +0.182027 q^{92} -11.0923 q^{93} +8.64935 q^{94} +6.08273 q^{95} +1.37505 q^{96} +1.59121 q^{97} -7.95035 q^{98} -7.20065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} - 5 q^{7} - 4 q^{8} + 8 q^{9} + 4 q^{10} + 9 q^{11} - 2 q^{12} - 12 q^{13} + 5 q^{14} + 2 q^{15} + 4 q^{16} - 8 q^{18} + 4 q^{19} - 4 q^{20} + 2 q^{21} - 9 q^{22} + 6 q^{23} + 2 q^{24} + 4 q^{25} + 12 q^{26} + 10 q^{27} - 5 q^{28} + 18 q^{29} - 2 q^{30} + 2 q^{31} - 4 q^{32} + 14 q^{33} + 5 q^{35} + 8 q^{36} + 9 q^{37} - 4 q^{38} + 10 q^{39} + 4 q^{40} + 12 q^{41} - 2 q^{42} - 16 q^{43} + 9 q^{44} - 8 q^{45} - 6 q^{46} + 10 q^{47} - 2 q^{48} + 15 q^{49} - 4 q^{50} - 4 q^{51} - 12 q^{52} + 8 q^{53} - 10 q^{54} - 9 q^{55} + 5 q^{56} + 44 q^{57} - 18 q^{58} + 18 q^{59} + 2 q^{60} - 11 q^{61} - 2 q^{62} - 27 q^{63} + 4 q^{64} + 12 q^{65} - 14 q^{66} + 4 q^{67} + 20 q^{69} - 5 q^{70} + 27 q^{71} - 8 q^{72} + 4 q^{73} - 9 q^{74} - 2 q^{75} + 4 q^{76} + 25 q^{77} - 10 q^{78} - 20 q^{79} - 4 q^{80} + 16 q^{81} - 12 q^{82} + 9 q^{83} + 2 q^{84} + 16 q^{86} + 2 q^{87} - 9 q^{88} + 17 q^{89} + 8 q^{90} - 4 q^{91} + 6 q^{92} + 2 q^{93} - 10 q^{94} - 4 q^{95} + 2 q^{96} - 5 q^{97} - 15 q^{98} + 9 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 - 5 * q^7 - 4 * q^8 + 8 * q^9 + 4 * q^10 + 9 * q^11 - 2 * q^12 - 12 * q^13 + 5 * q^14 + 2 * q^15 + 4 * q^16 - 8 * q^18 + 4 * q^19 - 4 * q^20 + 2 * q^21 - 9 * q^22 + 6 * q^23 + 2 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 5 * q^28 + 18 * q^29 - 2 * q^30 + 2 * q^31 - 4 * q^32 + 14 * q^33 + 5 * q^35 + 8 * q^36 + 9 * q^37 - 4 * q^38 + 10 * q^39 + 4 * q^40 + 12 * q^41 - 2 * q^42 - 16 * q^43 + 9 * q^44 - 8 * q^45 - 6 * q^46 + 10 * q^47 - 2 * q^48 + 15 * q^49 - 4 * q^50 - 4 * q^51 - 12 * q^52 + 8 * q^53 - 10 * q^54 - 9 * q^55 + 5 * q^56 + 44 * q^57 - 18 * q^58 + 18 * q^59 + 2 * q^60 - 11 * q^61 - 2 * q^62 - 27 * q^63 + 4 * q^64 + 12 * q^65 - 14 * q^66 + 4 * q^67 + 20 * q^69 - 5 * q^70 + 27 * q^71 - 8 * q^72 + 4 * q^73 - 9 * q^74 - 2 * q^75 + 4 * q^76 + 25 * q^77 - 10 * q^78 - 20 * q^79 - 4 * q^80 + 16 * q^81 - 12 * q^82 + 9 * q^83 + 2 * q^84 + 16 * q^86 + 2 * q^87 - 9 * q^88 + 17 * q^89 + 8 * q^90 - 4 * q^91 + 6 * q^92 + 2 * q^93 - 10 * q^94 - 4 * q^95 + 2 * q^96 - 5 * q^97 - 15 * q^98 + 9 * q^99

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