LMFDB - Embedded newform 446.2.a.f.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [446,2,Mod(1,446)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(446, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("446.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 446 = 2 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	446.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:	$3.56132793015$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 12x^{6} + 46x^{5} + 54x^{4} - 148x^{3} - 98x^{2} + 126x + 69 $ x^8 - 4x^7 - 12x^6 + 46x^5 + 54x^4 - 148x^3 - 98x^2 + 126*x + 69
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.36314$ of defining polynomial
Character	$\chi$	$=$	446.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.36314 q^{3} +1.00000 q^{4} -2.80312 q^{5} -3.36314 q^{6} +1.62712 q^{7} -1.00000 q^{8} +8.31074 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.36314 q^{3} +1.00000 q^{4} -2.80312 q^{5} -3.36314 q^{6} +1.62712 q^{7} -1.00000 q^{8} +8.31074 q^{9} +2.80312 q^{10} -4.88403 q^{11} +3.36314 q^{12} +3.62240 q^{13} -1.62712 q^{14} -9.42730 q^{15} +1.00000 q^{16} +7.41973 q^{17} -8.31074 q^{18} +5.70702 q^{19} -2.80312 q^{20} +5.47224 q^{21} +4.88403 q^{22} +1.28060 q^{23} -3.36314 q^{24} +2.85748 q^{25} -3.62240 q^{26} +17.8608 q^{27} +1.62712 q^{28} -4.72629 q^{29} +9.42730 q^{30} -9.52740 q^{31} -1.00000 q^{32} -16.4257 q^{33} -7.41973 q^{34} -4.56101 q^{35} +8.31074 q^{36} +2.72697 q^{37} -5.70702 q^{38} +12.1827 q^{39} +2.80312 q^{40} -7.35248 q^{41} -5.47224 q^{42} -5.14186 q^{43} -4.88403 q^{44} -23.2960 q^{45} -1.28060 q^{46} -3.87590 q^{47} +3.36314 q^{48} -4.35248 q^{49} -2.85748 q^{50} +24.9536 q^{51} +3.62240 q^{52} +1.89922 q^{53} -17.8608 q^{54} +13.6905 q^{55} -1.62712 q^{56} +19.1935 q^{57} +4.72629 q^{58} +2.18450 q^{59} -9.42730 q^{60} -8.49027 q^{61} +9.52740 q^{62} +13.5226 q^{63} +1.00000 q^{64} -10.1540 q^{65} +16.4257 q^{66} -7.77136 q^{67} +7.41973 q^{68} +4.30685 q^{69} +4.56101 q^{70} +1.80780 q^{71} -8.31074 q^{72} +1.22647 q^{73} -2.72697 q^{74} +9.61013 q^{75} +5.70702 q^{76} -7.94690 q^{77} -12.1827 q^{78} -6.91055 q^{79} -2.80312 q^{80} +35.1362 q^{81} +7.35248 q^{82} +14.8463 q^{83} +5.47224 q^{84} -20.7984 q^{85} +5.14186 q^{86} -15.8952 q^{87} +4.88403 q^{88} +7.10986 q^{89} +23.2960 q^{90} +5.89409 q^{91} +1.28060 q^{92} -32.0420 q^{93} +3.87590 q^{94} -15.9975 q^{95} -3.36314 q^{96} +2.46564 q^{97} +4.35248 q^{98} -40.5899 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 8 q^{8} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 - 8 * q^8 + 16 * q^9	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 8 q^{8} + 16 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 10 q^{13} - 4 q^{14} - 4 q^{15} + 8 q^{16} + 8 q^{17} - 16 q^{18} + 16 q^{19} + 4 q^{20} - 2 q^{22} - 4 q^{23} - 4 q^{24} + 24 q^{25} - 10 q^{26} + 22 q^{27} + 4 q^{28} + 8 q^{29} + 4 q^{30} + 12 q^{31} - 8 q^{32} - 8 q^{34} - 24 q^{35} + 16 q^{36} + 16 q^{37} - 16 q^{38} - 4 q^{40} + 16 q^{41} + 12 q^{43} + 2 q^{44} + 34 q^{45} + 4 q^{46} - 4 q^{47} + 4 q^{48} + 40 q^{49} - 24 q^{50} + 26 q^{51} + 10 q^{52} - 8 q^{53} - 22 q^{54} + 8 q^{55} - 4 q^{56} + 16 q^{57} - 8 q^{58} - 4 q^{60} + 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} - 24 q^{65} - 10 q^{67} + 8 q^{68} - 28 q^{69} + 24 q^{70} + 8 q^{71} - 16 q^{72} + 8 q^{73} - 16 q^{74} + 6 q^{75} + 16 q^{76} - 56 q^{77} + 16 q^{79} + 4 q^{80} + 24 q^{81} - 16 q^{82} - 44 q^{83} - 18 q^{85} - 12 q^{86} - 72 q^{87} - 2 q^{88} - 34 q^{90} + 4 q^{91} - 4 q^{92} - 62 q^{93} + 4 q^{94} - 48 q^{95} - 4 q^{96} + 12 q^{97} - 40 q^{98} - 40 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 - 8 * q^8 + 16 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 10 * q^13 - 4 * q^14 - 4 * q^15 + 8 * q^16 + 8 * q^17 - 16 * q^18 + 16 * q^19 + 4 * q^20 - 2 * q^22 - 4 * q^23 - 4 * q^24 + 24 * q^25 - 10 * q^26 + 22 * q^27 + 4 * q^28 + 8 * q^29 + 4 * q^30 + 12 * q^31 - 8 * q^32 - 8 * q^34 - 24 * q^35 + 16 * q^36 + 16 * q^37 - 16 * q^38 - 4 * q^40 + 16 * q^41 + 12 * q^43 + 2 * q^44 + 34 * q^45 + 4 * q^46 - 4 * q^47 + 4 * q^48 + 40 * q^49 - 24 * q^50 + 26 * q^51 + 10 * q^52 - 8 * q^53 - 22 * q^54 + 8 * q^55 - 4 * q^56 + 16 * q^57 - 8 * q^58 - 4 * q^60 + 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 - 24 * q^65 - 10 * q^67 + 8 * q^68 - 28 * q^69 + 24 * q^70 + 8 * q^71 - 16 * q^72 + 8 * q^73 - 16 * q^74 + 6 * q^75 + 16 * q^76 - 56 * q^77 + 16 * q^79 + 4 * q^80 + 24 * q^81 - 16 * q^82 - 44 * q^83 - 18 * q^85 - 12 * q^86 - 72 * q^87 - 2 * q^88 - 34 * q^90 + 4 * q^91 - 4 * q^92 - 62 * q^93 + 4 * q^94 - 48 * q^95 - 4 * q^96 + 12 * q^97 - 40 * q^98 - 40 * 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
$3$	3.36314	1.94171	0.970856	−	0.239663i	$-0.0770368\pi$
$3$	3.36314	1.94171	0.970856	+	0.239663i	$0.0770368\pi$
$4$	1.00000	0.500000
$5$	−2.80312	−1.25359	−0.626797	−	0.779183i	$-0.715634\pi$
$5$	−2.80312	−1.25359	−0.626797	+	0.779183i	$0.715634\pi$
$6$	−3.36314	−1.37300
$7$	1.62712	0.614994	0.307497	−	0.951549i	$-0.400509\pi$
$7$	1.62712	0.614994	0.307497	+	0.951549i	$0.400509\pi$
$8$	−1.00000	−0.353553
$9$	8.31074	2.77025
$10$	2.80312	0.886424
$11$	−4.88403	−1.47259	−0.736295	−	0.676661i	$-0.763426\pi$
$11$	−4.88403	−1.47259	−0.736295	+	0.676661i	$0.763426\pi$
$12$	3.36314	0.970856
$13$	3.62240	1.00467	0.502337	−	0.864672i	$-0.332474\pi$
$13$	3.62240	1.00467	0.502337	+	0.864672i	$0.332474\pi$
$14$	−1.62712	−0.434866
$15$	−9.42730	−2.43412
$16$	1.00000	0.250000
$17$	7.41973	1.79955	0.899774	−	0.436356i	$-0.143731\pi$
$17$	7.41973	1.79955	0.899774	+	0.436356i	$0.143731\pi$
$18$	−8.31074	−1.95886
$19$	5.70702	1.30928	0.654640	−	0.755941i	$-0.272820\pi$
$19$	5.70702	1.30928	0.654640	+	0.755941i	$0.272820\pi$
$20$	−2.80312	−0.626797
$21$	5.47224	1.19414
$22$	4.88403	1.04128
$23$	1.28060	0.267024	0.133512	−	0.991047i	$-0.457375\pi$
$23$	1.28060	0.267024	0.133512	+	0.991047i	$0.457375\pi$
$24$	−3.36314	−0.686499
$25$	2.85748	0.571496
$26$	−3.62240	−0.710412
$27$	17.8608	3.43731
$28$	1.62712	0.307497
$29$	−4.72629	−0.877650	−0.438825	−	0.898573i	$-0.644605\pi$
$29$	−4.72629	−0.877650	−0.438825	+	0.898573i	$0.644605\pi$
$30$	9.42730	1.72118
$31$	−9.52740	−1.71117	−0.855585	−	0.517662i	$-0.826803\pi$
$31$	−9.52740	−1.71117	−0.855585	+	0.517662i	$0.826803\pi$
$32$	−1.00000	−0.176777
$33$	−16.4257	−2.85935
$34$	−7.41973	−1.27247
$35$	−4.56101	−0.770952
$36$	8.31074	1.38512
$37$	2.72697	0.448311	0.224156	−	0.974553i	$-0.428038\pi$
$37$	2.72697	0.448311	0.224156	+	0.974553i	$0.428038\pi$
$38$	−5.70702	−0.925801
$39$	12.1827	1.95079
$40$	2.80312	0.443212
$41$	−7.35248	−1.14826	−0.574132	−	0.818763i	$-0.694660\pi$
$41$	−7.35248	−1.14826	−0.574132	+	0.818763i	$0.694660\pi$
$42$	−5.47224	−0.844385
$43$	−5.14186	−0.784127	−0.392063	−	0.919938i	$-0.628239\pi$
$43$	−5.14186	−0.784127	−0.392063	+	0.919938i	$0.628239\pi$
$44$	−4.88403	−0.736295
$45$	−23.2960	−3.47276
$46$	−1.28060	−0.188814
$47$	−3.87590	−0.565358	−0.282679	−	0.959215i	$-0.591223\pi$
$47$	−3.87590	−0.565358	−0.282679	+	0.959215i	$0.591223\pi$
$48$	3.36314	0.485428
$49$	−4.35248	−0.621783
$50$	−2.85748	−0.404109
$51$	24.9536	3.49420
$52$	3.62240	0.502337
$53$	1.89922	0.260878	0.130439	−	0.991456i	$-0.458361\pi$
$53$	1.89922	0.260878	0.130439	+	0.991456i	$0.458361\pi$
$54$	−17.8608	−2.43055
$55$	13.6905	1.84603
$56$	−1.62712	−0.217433
$57$	19.1935	2.54225
$58$	4.72629	0.620592
$59$	2.18450	0.284398	0.142199	−	0.989838i	$-0.454583\pi$
$59$	2.18450	0.284398	0.142199	+	0.989838i	$0.454583\pi$
$60$	−9.42730	−1.21706
$61$	−8.49027	−1.08707	−0.543534	−	0.839387i	$-0.682914\pi$
$61$	−8.49027	−1.08707	−0.543534	+	0.839387i	$0.682914\pi$
$62$	9.52740	1.20998
$63$	13.5226	1.70368
$64$	1.00000	0.125000
$65$	−10.1540	−1.25945
$66$	16.4257	2.02186
$67$	−7.77136	−0.949423	−0.474711	−	0.880142i	$-0.657448\pi$
$67$	−7.77136	−0.949423	−0.474711	+	0.880142i	$0.657448\pi$
$68$	7.41973	0.899774
$69$	4.30685	0.518484
$70$	4.56101	0.545145
$71$	1.80780	0.214546	0.107273	−	0.994230i	$-0.465788\pi$
$71$	1.80780	0.214546	0.107273	+	0.994230i	$0.465788\pi$
$72$	−8.31074	−0.979430
$73$	1.22647	0.143547	0.0717737	−	0.997421i	$-0.477134\pi$
$73$	1.22647	0.143547	0.0717737	+	0.997421i	$0.477134\pi$
$74$	−2.72697	−0.317004
$75$	9.61013	1.10968
$76$	5.70702	0.654640
$77$	−7.94690	−0.905633
$78$	−12.1827	−1.37942
$79$	−6.91055	−0.777497	−0.388749	−	0.921344i	$-0.627093\pi$
$79$	−6.91055	−0.777497	−0.388749	+	0.921344i	$0.627093\pi$
$80$	−2.80312	−0.313398
$81$	35.1362	3.90402
$82$	7.35248	0.811946
$83$	14.8463	1.62960	0.814799	−	0.579744i	$-0.196847\pi$
$83$	14.8463	1.62960	0.814799	+	0.579744i	$0.196847\pi$
$84$	5.47224	0.597070
$85$	−20.7984	−2.25590
$86$	5.14186	0.554461
$87$	−15.8952	−1.70414
$88$	4.88403	0.520639
$89$	7.10986	0.753644	0.376822	−	0.926286i	$-0.377017\pi$
$89$	7.10986	0.753644	0.376822	+	0.926286i	$0.377017\pi$
$90$	23.2960	2.45561
$91$	5.89409	0.617868
$92$	1.28060	0.133512
$93$	−32.0420	−3.32260
$94$	3.87590	0.399768
$95$	−15.9975	−1.64130
$96$	−3.36314	−0.343250
$97$	2.46564	0.250348	0.125174	−	0.992135i	$-0.460051\pi$
$97$	2.46564	0.250348	0.125174	+	0.992135i	$0.460051\pi$
$98$	4.35248	0.439667
$99$	−40.5899	−4.07944
$100$	2.85748	0.285748
$101$	−2.76208	−0.274837	−0.137419	−	0.990513i	$-0.543881\pi$
$101$	−2.76208	−0.274837	−0.137419	+	0.990513i	$0.543881\pi$
$102$	−24.9536	−2.47078
$103$	1.58040	0.155722	0.0778608	−	0.996964i	$-0.475191\pi$
$103$	1.58040	0.155722	0.0778608	+	0.996964i	$0.475191\pi$
$104$	−3.62240	−0.355206
$105$	−15.3393	−1.49697
$106$	−1.89922	−0.184469
$107$	−17.4350	−1.68551	−0.842754	−	0.538299i	$-0.819067\pi$
$107$	−17.4350	−1.68551	−0.842754	+	0.538299i	$0.819067\pi$
$108$	17.8608	1.71866
$109$	−12.4801	−1.19538	−0.597688	−	0.801728i	$-0.703914\pi$
$109$	−12.4801	−1.19538	−0.597688	+	0.801728i	$0.703914\pi$
$110$	−13.6905	−1.30534
$111$	9.17120	0.870491
$112$	1.62712	0.153748
$113$	5.00641	0.470963	0.235482	−	0.971879i	$-0.424333\pi$
$113$	5.00641	0.470963	0.235482	+	0.971879i	$0.424333\pi$
$114$	−19.1935	−1.79764
$115$	−3.58968	−0.334739
$116$	−4.72629	−0.438825
$117$	30.1049	2.78320
$118$	−2.18450	−0.201100
$119$	12.0728	1.10671
$120$	9.42730	0.860591
$121$	12.8537	1.16852
$122$	8.49027	0.768673
$123$	−24.7275	−2.22960
$124$	−9.52740	−0.855585
$125$	6.00574	0.537169
$126$	−13.5226	−1.20469
$127$	0.506922	0.0449821	0.0224910	−	0.999747i	$-0.492840\pi$
$127$	0.506922	0.0449821	0.0224910	+	0.999747i	$0.492840\pi$
$128$	−1.00000	−0.0883883
$129$	−17.2928	−1.52255
$130$	10.1540	0.890568
$131$	−10.7751	−0.941429	−0.470714	−	0.882286i	$-0.656004\pi$
$131$	−10.7751	−0.941429	−0.470714	+	0.882286i	$0.656004\pi$
$132$	−16.4257	−1.42967
$133$	9.28601	0.805199
$134$	7.77136	0.671343
$135$	−50.0659	−4.30899
$136$	−7.41973	−0.636236
$137$	8.36832	0.714954	0.357477	−	0.933922i	$-0.383637\pi$
$137$	8.36832	0.714954	0.357477	+	0.933922i	$0.383637\pi$
$138$	−4.30685	−0.366623
$139$	−0.475069	−0.0402948	−0.0201474	−	0.999797i	$-0.506414\pi$
$139$	−0.475069	−0.0402948	−0.0201474	+	0.999797i	$0.506414\pi$
$140$	−4.56101	−0.385476
$141$	−13.0352	−1.09776
$142$	−1.80780	−0.151707
$143$	−17.6919	−1.47947
$144$	8.31074	0.692562
$145$	13.2484	1.10022
$146$	−1.22647	−0.101503
$147$	−14.6380	−1.20732
$148$	2.72697	0.224156
$149$	−16.2885	−1.33440	−0.667201	−	0.744878i	$-0.732508\pi$
$149$	−16.2885	−1.33440	−0.667201	+	0.744878i	$0.732508\pi$
$150$	−9.61013	−0.784663
$151$	0.513029	0.0417497	0.0208749	−	0.999782i	$-0.493355\pi$
$151$	0.513029	0.0417497	0.0208749	+	0.999782i	$0.493355\pi$
$152$	−5.70702	−0.462900
$153$	61.6634	4.98519
$154$	7.94690	0.640379
$155$	26.7064	2.14511
$156$	12.1827	0.975394
$157$	6.63432	0.529476	0.264738	−	0.964320i	$-0.414715\pi$
$157$	6.63432	0.529476	0.264738	+	0.964320i	$0.414715\pi$
$158$	6.91055	0.549774
$159$	6.38735	0.506550
$160$	2.80312	0.221606
$161$	2.08369	0.164218
$162$	−35.1362	−2.76056
$163$	−10.1213	−0.792759	−0.396379	−	0.918087i	$-0.629733\pi$
$163$	−10.1213	−0.792759	−0.396379	+	0.918087i	$0.629733\pi$
$164$	−7.35248	−0.574132
$165$	46.0432	3.58446
$166$	−14.8463	−1.15230
$167$	18.5167	1.43286	0.716432	−	0.697657i	$-0.245774\pi$
$167$	18.5167	1.43286	0.716432	+	0.697657i	$0.245774\pi$
$168$	−5.47224	−0.422192
$169$	0.121818	0.00937058
$170$	20.7984	1.59516
$171$	47.4296	3.62703
$172$	−5.14186	−0.392063
$173$	−7.89063	−0.599913	−0.299957	−	0.953953i	$-0.596972\pi$
$173$	−7.89063	−0.599913	−0.299957	+	0.953953i	$0.596972\pi$
$174$	15.8952	1.20501
$175$	4.64947	0.351467
$176$	−4.88403	−0.368147
$177$	7.34679	0.552219
$178$	−7.10986	−0.532907
$179$	−11.3018	−0.844737	−0.422369	−	0.906424i	$-0.638801\pi$
$179$	−11.3018	−0.844737	−0.422369	+	0.906424i	$0.638801\pi$
$180$	−23.2960	−1.73638
$181$	6.66659	0.495524	0.247762	−	0.968821i	$-0.420305\pi$
$181$	6.66659	0.495524	0.247762	+	0.968821i	$0.420305\pi$
$182$	−5.89409	−0.436899
$183$	−28.5540	−2.11077
$184$	−1.28060	−0.0944072
$185$	−7.64403	−0.562000
$186$	32.0420	2.34943
$187$	−36.2381	−2.65000
$188$	−3.87590	−0.282679
$189$	29.0617	2.11392
$190$	15.9975	1.16058
$191$	9.93188	0.718645	0.359323	−	0.933213i	$-0.383008\pi$
$191$	9.93188	0.718645	0.359323	+	0.933213i	$0.383008\pi$
$192$	3.36314	0.242714
$193$	12.3928	0.892052	0.446026	−	0.895020i	$-0.352839\pi$
$193$	12.3928	0.892052	0.446026	+	0.895020i	$0.352839\pi$
$194$	−2.46564	−0.177023
$195$	−34.1495	−2.44550
$196$	−4.35248	−0.310891
$197$	−5.56956	−0.396814	−0.198407	−	0.980120i	$-0.563577\pi$
$197$	−5.56956	−0.396814	−0.198407	+	0.980120i	$0.563577\pi$
$198$	40.5899	2.88460
$199$	21.3068	1.51040	0.755199	−	0.655496i	$-0.227540\pi$
$199$	21.3068	1.51040	0.755199	+	0.655496i	$0.227540\pi$
$200$	−2.85748	−0.202054
$201$	−26.1362	−1.84351
$202$	2.76208	0.194339
$203$	−7.69024	−0.539749
$204$	24.9536	1.74710
$205$	20.6099	1.43946
$206$	−1.58040	−0.110112
$207$	10.6428	0.739722
$208$	3.62240	0.251169
$209$	−27.8732	−1.92803
$210$	15.3393	1.05852
$211$	4.56855	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$211$	4.56855	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$212$	1.89922	0.130439
$213$	6.07989	0.416587
$214$	17.4350	1.19183
$215$	14.4133	0.982976
$216$	−17.8608	−1.21527
$217$	−15.5022	−1.05236
$218$	12.4801	0.845259
$219$	4.12479	0.278728
$220$	13.6905	0.923014
$221$	26.8773	1.80796
$222$	−9.17120	−0.615530
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−1.62712	−0.108717
$225$	23.7478	1.58319
$226$	−5.00641	−0.333021
$227$	−1.37598	−0.0913268	−0.0456634	−	0.998957i	$-0.514540\pi$
$227$	−1.37598	−0.0913268	−0.0456634	+	0.998957i	$0.514540\pi$
$228$	19.1935	1.27112
$229$	−10.4332	−0.689442	−0.344721	−	0.938705i	$-0.612026\pi$
$229$	−10.4332	−0.689442	−0.344721	+	0.938705i	$0.612026\pi$
$230$	3.58968	0.236697
$231$	−26.7266	−1.75848
$232$	4.72629	0.310296
$233$	−3.28891	−0.215464	−0.107732	−	0.994180i	$-0.534359\pi$
$233$	−3.28891	−0.215464	−0.107732	+	0.994180i	$0.534359\pi$
$234$	−30.1049	−1.96802
$235$	10.8646	0.708729
$236$	2.18450	0.142199
$237$	−23.2412	−1.50968
$238$	−12.0728	−0.782562
$239$	−17.5274	−1.13375	−0.566877	−	0.823803i	$-0.691848\pi$
$239$	−17.5274	−1.13375	−0.566877	+	0.823803i	$0.691848\pi$
$240$	−9.42730	−0.608529
$241$	−0.255243	−0.0164417	−0.00822083	−	0.999966i	$-0.502617\pi$
$241$	−0.255243	−0.0164417	−0.00822083	+	0.999966i	$0.502617\pi$
$242$	−12.8537	−0.826268
$243$	64.5858	4.14318
$244$	−8.49027	−0.543534
$245$	12.2005	0.779463
$246$	24.7275	1.57656
$247$	20.6731	1.31540
$248$	9.52740	0.604990
$249$	49.9304	3.16421
$250$	−6.00574	−0.379836
$251$	5.33942	0.337021	0.168510	−	0.985700i	$-0.446104\pi$
$251$	5.33942	0.337021	0.168510	+	0.985700i	$0.446104\pi$
$252$	13.5226	0.851842
$253$	−6.25449	−0.393217
$254$	−0.506922	−0.0318071
$255$	−69.9480	−4.38031
$256$	1.00000	0.0625000
$257$	15.3493	0.957462	0.478731	−	0.877962i	$-0.341097\pi$
$257$	15.3493	0.957462	0.478731	+	0.877962i	$0.341097\pi$
$258$	17.2928	1.07660
$259$	4.43711	0.275709
$260$	−10.1540	−0.629727
$261$	−39.2790	−2.43131
$262$	10.7751	0.665691
$263$	−26.6345	−1.64235	−0.821177	−	0.570673i	$-0.806682\pi$
$263$	−26.6345	−1.64235	−0.821177	+	0.570673i	$0.806682\pi$
$264$	16.4257	1.01093
$265$	−5.32374	−0.327035
$266$	−9.28601	−0.569362
$267$	23.9115	1.46336
$268$	−7.77136	−0.474711
$269$	8.87441	0.541083	0.270541	−	0.962708i	$-0.412797\pi$
$269$	8.87441	0.541083	0.270541	+	0.962708i	$0.412797\pi$
$270$	50.0659	3.04692
$271$	14.9484	0.908051	0.454025	−	0.890989i	$-0.349988\pi$
$271$	14.9484	0.908051	0.454025	+	0.890989i	$0.349988\pi$
$272$	7.41973	0.449887
$273$	19.8227	1.19972
$274$	−8.36832	−0.505549
$275$	−13.9560	−0.841580
$276$	4.30685	0.259242
$277$	−10.4679	−0.628957	−0.314479	−	0.949265i	$-0.601830\pi$
$277$	−10.4679	−0.628957	−0.314479	+	0.949265i	$0.601830\pi$
$278$	0.475069	0.0284927
$279$	−79.1797	−4.74037
$280$	4.56101	0.272573
$281$	−3.55084	−0.211826	−0.105913	−	0.994375i	$-0.533776\pi$
$281$	−3.55084	−0.211826	−0.105913	+	0.994375i	$0.533776\pi$
$282$	13.0352	0.776235
$283$	32.0270	1.90381	0.951903	−	0.306400i	$-0.0991245\pi$
$283$	32.0270	1.90381	0.951903	+	0.306400i	$0.0991245\pi$
$284$	1.80780	0.107273
$285$	−53.8018	−3.18694
$286$	17.6919	1.04615
$287$	−11.9634	−0.706175
$288$	−8.31074	−0.489715
$289$	38.0523	2.23837
$290$	−13.2484	−0.777970
$291$	8.29230	0.486103
$292$	1.22647	0.0717737
$293$	33.2339	1.94154	0.970772	−	0.240003i	$-0.0771484\pi$
$293$	33.2339	1.94154	0.970772	+	0.240003i	$0.0771484\pi$
$294$	14.6380	0.853707
$295$	−6.12342	−0.356519
$296$	−2.72697	−0.158502
$297$	−87.2326	−5.06175
$298$	16.2885	0.943565
$299$	4.63886	0.268272
$300$	9.61013	0.554841
$301$	−8.36643	−0.482233
$302$	−0.513029	−0.0295215
$303$	−9.28928	−0.533655
$304$	5.70702	0.327320
$305$	23.7992	1.36274
$306$	−61.6634	−3.52506
$307$	−20.4035	−1.16449	−0.582244	−	0.813014i	$-0.697825\pi$
$307$	−20.4035	−1.16449	−0.582244	+	0.813014i	$0.697825\pi$
$308$	−7.94690	−0.452817
$309$	5.31512	0.302366
$310$	−26.7064	−1.51682
$311$	−8.99243	−0.509914	−0.254957	−	0.966952i	$-0.582061\pi$
$311$	−8.99243	−0.509914	−0.254957	+	0.966952i	$0.582061\pi$
$312$	−12.1827	−0.689708
$313$	−4.30737	−0.243467	−0.121734	−	0.992563i	$-0.538845\pi$
$313$	−4.30737	−0.243467	−0.121734	+	0.992563i	$0.538845\pi$
$314$	−6.63432	−0.374396
$315$	−37.9054	−2.13573
$316$	−6.91055	−0.388749
$317$	−3.84895	−0.216179	−0.108089	−	0.994141i	$-0.534473\pi$
$317$	−3.84895	−0.216179	−0.108089	+	0.994141i	$0.534473\pi$
$318$	−6.38735	−0.358185
$319$	23.0833	1.29242
$320$	−2.80312	−0.156699
$321$	−58.6365	−3.27277
$322$	−2.08369	−0.116120
$323$	42.3445	2.35611
$324$	35.1362	1.95201
$325$	10.3510	0.574168
$326$	10.1213	0.560565
$327$	−41.9724	−2.32108
$328$	7.35248	0.405973
$329$	−6.30655	−0.347692
$330$	−46.0432	−2.53459
$331$	−8.07055	−0.443598	−0.221799	−	0.975092i	$-0.571193\pi$
$331$	−8.07055	−0.443598	−0.221799	+	0.975092i	$0.571193\pi$
$332$	14.8463	0.814799
$333$	22.6631	1.24193
$334$	−18.5167	−1.01319
$335$	21.7841	1.19019
$336$	5.47224	0.298535
$337$	10.2899	0.560525	0.280263	−	0.959923i	$-0.409578\pi$
$337$	10.2899	0.560525	0.280263	+	0.959923i	$0.409578\pi$
$338$	−0.121818	−0.00662600
$339$	16.8373	0.914476
$340$	−20.7984	−1.12795
$341$	46.5321	2.51985
$342$	−47.4296	−2.56470
$343$	−18.4718	−0.997386
$344$	5.14186	0.277231
$345$	−12.0726	−0.649968
$346$	7.89063	0.424203
$347$	−13.7951	−0.740560	−0.370280	−	0.928920i	$-0.620738\pi$
$347$	−13.7951	−0.740560	−0.370280	+	0.928920i	$0.620738\pi$
$348$	−15.8952	−0.852072
$349$	−11.7721	−0.630145	−0.315072	−	0.949068i	$-0.602029\pi$
$349$	−11.7721	−0.630145	−0.315072	+	0.949068i	$0.602029\pi$
$350$	−4.64947	−0.248524
$351$	64.6990	3.45338
$352$	4.88403	0.260319
$353$	19.8217	1.05501	0.527503	−	0.849553i	$-0.323129\pi$
$353$	19.8217	1.05501	0.527503	+	0.849553i	$0.323129\pi$
$354$	−7.34679	−0.390478
$355$	−5.06748	−0.268954
$356$	7.10986	0.376822
$357$	40.6025	2.14891
$358$	11.3018	0.597320
$359$	−6.89554	−0.363933	−0.181966	−	0.983305i	$-0.558246\pi$
$359$	−6.89554	−0.363933	−0.181966	+	0.983305i	$0.558246\pi$
$360$	23.2960	1.22781
$361$	13.5701	0.714214
$362$	−6.66659	−0.350388
$363$	43.2289	2.26893
$364$	5.89409	0.308934
$365$	−3.43794	−0.179950
$366$	28.5540	1.49254
$367$	11.9728	0.624973	0.312486	−	0.949922i	$-0.398838\pi$
$367$	11.9728	0.624973	0.312486	+	0.949922i	$0.398838\pi$
$368$	1.28060	0.0667560
$369$	−61.1046	−3.18098
$370$	7.64403	0.397394
$371$	3.09026	0.160438
$372$	−32.0420	−1.66130
$373$	−17.7096	−0.916970	−0.458485	−	0.888702i	$-0.651608\pi$
$373$	−17.7096	−0.916970	−0.458485	+	0.888702i	$0.651608\pi$
$374$	36.2381	1.87383
$375$	20.1982	1.04303
$376$	3.87590	0.199884
$377$	−17.1205	−0.881752
$378$	−29.0617	−1.49477
$379$	6.18691	0.317800	0.158900	−	0.987295i	$-0.449205\pi$
$379$	6.18691	0.317800	0.158900	+	0.987295i	$0.449205\pi$
$380$	−15.9975	−0.820652
$381$	1.70485	0.0873422
$382$	−9.93188	−0.508159
$383$	12.2112	0.623963	0.311982	−	0.950088i	$-0.399007\pi$
$383$	12.2112	0.623963	0.311982	+	0.950088i	$0.399007\pi$
$384$	−3.36314	−0.171625
$385$	22.2761	1.13530
$386$	−12.3928	−0.630776
$387$	−42.7327	−2.17222
$388$	2.46564	0.125174
$389$	20.3603	1.03231	0.516154	−	0.856496i	$-0.327363\pi$
$389$	20.3603	1.03231	0.516154	+	0.856496i	$0.327363\pi$
$390$	34.1495	1.72923
$391$	9.50171	0.480522
$392$	4.35248	0.219833
$393$	−36.2384	−1.82798
$394$	5.56956	0.280590
$395$	19.3711	0.974665
$396$	−40.5899	−2.03972
$397$	−26.7165	−1.34087	−0.670433	−	0.741970i	$-0.733891\pi$
$397$	−26.7165	−1.34087	−0.670433	+	0.741970i	$0.733891\pi$
$398$	−21.3068	−1.06801
$399$	31.2302	1.56346
$400$	2.85748	0.142874
$401$	20.2413	1.01080	0.505400	−	0.862885i	$-0.331345\pi$
$401$	20.2413	1.01080	0.505400	+	0.862885i	$0.331345\pi$
$402$	26.1362	1.30356
$403$	−34.5121	−1.71917
$404$	−2.76208	−0.137419
$405$	−98.4910	−4.89406
$406$	7.69024	0.381660
$407$	−13.3186	−0.660178
$408$	−24.9536	−1.23539
$409$	29.0470	1.43628	0.718140	−	0.695899i	$-0.244994\pi$
$409$	29.0470	1.43628	0.718140	+	0.695899i	$0.244994\pi$
$410$	−20.6099	−1.01785
$411$	28.1439	1.38823
$412$	1.58040	0.0778608
$413$	3.55445	0.174903
$414$	−10.6428	−0.523063
$415$	−41.6161	−2.04285
$416$	−3.62240	−0.177603
$417$	−1.59772	−0.0782409
$418$	27.8732	1.36332
$419$	−9.10873	−0.444990	−0.222495	−	0.974934i	$-0.571420\pi$
$419$	−9.10873	−0.444990	−0.222495	+	0.974934i	$0.571420\pi$
$420$	−15.3393	−0.748483
$421$	29.6256	1.44386	0.721931	−	0.691966i	$-0.243255\pi$
$421$	29.6256	1.44386	0.721931	+	0.691966i	$0.243255\pi$
$422$	−4.56855	−0.222394
$423$	−32.2116	−1.56618
$424$	−1.89922	−0.0922343
$425$	21.2017	1.02844
$426$	−6.07989	−0.294572
$427$	−13.8147	−0.668539
$428$	−17.4350	−0.842754
$429$	−59.5005	−2.87271
$430$	−14.4133	−0.695069
$431$	−9.05749	−0.436284	−0.218142	−	0.975917i	$-0.570000\pi$
$431$	−9.05749	−0.436284	−0.218142	+	0.975917i	$0.570000\pi$
$432$	17.8608	0.859328
$433$	−15.1574	−0.728418	−0.364209	−	0.931317i	$-0.618661\pi$
$433$	−15.1574	−0.728418	−0.364209	+	0.931317i	$0.618661\pi$
$434$	15.5022	0.744130
$435$	44.5561	2.13630
$436$	−12.4801	−0.597688
$437$	7.30842	0.349609
$438$	−4.12479	−0.197090
$439$	22.8847	1.09223	0.546114	−	0.837711i	$-0.316106\pi$
$439$	22.8847	1.09223	0.546114	+	0.837711i	$0.316106\pi$
$440$	−13.6905	−0.652670
$441$	−36.1723	−1.72249
$442$	−26.8773	−1.27842
$443$	−14.8447	−0.705295	−0.352647	−	0.935756i	$-0.614718\pi$
$443$	−14.8447	−0.705295	−0.352647	+	0.935756i	$0.614718\pi$
$444$	9.17120	0.435246
$445$	−19.9298	−0.944763
$446$	−1.00000	−0.0473514
$447$	−54.7804	−2.59103
$448$	1.62712	0.0768742
$449$	13.8348	0.652905	0.326453	−	0.945214i	$-0.394147\pi$
$449$	13.8348	0.652905	0.326453	+	0.945214i	$0.394147\pi$
$450$	−23.7478	−1.11948
$451$	35.9097	1.69092
$452$	5.00641	0.235482
$453$	1.72539	0.0810659
$454$	1.37598	0.0645778
$455$	−16.5218	−0.774556
$456$	−19.1935	−0.898819
$457$	35.0332	1.63879	0.819393	−	0.573232i	$-0.194311\pi$
$457$	35.0332	1.63879	0.819393	+	0.573232i	$0.194311\pi$
$458$	10.4332	0.487509
$459$	132.522	6.18561
$460$	−3.58968	−0.167370
$461$	34.9345	1.62706	0.813532	−	0.581520i	$-0.197542\pi$
$461$	34.9345	1.62706	0.813532	+	0.581520i	$0.197542\pi$
$462$	26.7266	1.24343
$463$	−9.21484	−0.428250	−0.214125	−	0.976806i	$-0.568690\pi$
$463$	−9.21484	−0.428250	−0.214125	+	0.976806i	$0.568690\pi$
$464$	−4.72629	−0.219412
$465$	89.8176	4.16519
$466$	3.28891	0.152356
$467$	−7.43435	−0.344021	−0.172010	−	0.985095i	$-0.555026\pi$
$467$	−7.43435	−0.344021	−0.172010	+	0.985095i	$0.555026\pi$
$468$	30.1049	1.39160
$469$	−12.6449	−0.583889
$470$	−10.8646	−0.501147
$471$	22.3122	1.02809
$472$	−2.18450	−0.100550
$473$	25.1130	1.15470
$474$	23.2412	1.06750
$475$	16.3077	0.748249
$476$	12.0728	0.553355
$477$	15.7839	0.722697
$478$	17.5274	0.801684
$479$	−0.658732	−0.0300982	−0.0150491	−	0.999887i	$-0.504790\pi$
$479$	−0.658732	−0.0300982	−0.0150491	+	0.999887i	$0.504790\pi$
$480$	9.42730	0.430295
$481$	9.87819	0.450407
$482$	0.255243	0.0116260
$483$	7.00776	0.318864
$484$	12.8537	0.584260
$485$	−6.91148	−0.313834
$486$	−64.5858	−2.92967
$487$	41.8859	1.89803	0.949015	−	0.315231i	$-0.102082\pi$
$487$	41.8859	1.89803	0.949015	+	0.315231i	$0.102082\pi$
$488$	8.49027	0.384336
$489$	−34.0393	−1.53931
$490$	−12.2005	−0.551164
$491$	−27.2206	−1.22845	−0.614225	−	0.789131i	$-0.710531\pi$
$491$	−27.2206	−1.22845	−0.614225	+	0.789131i	$0.710531\pi$
$492$	−24.7275	−1.11480
$493$	−35.0678	−1.57937
$494$	−20.6731	−0.930128
$495$	113.778	5.11396
$496$	−9.52740	−0.427793
$497$	2.94151	0.131945
$498$	−49.9304	−2.23743
$499$	−38.6603	−1.73067	−0.865336	−	0.501192i	$-0.832895\pi$
$499$	−38.6603	−1.73067	−0.865336	+	0.501192i	$0.832895\pi$
$500$	6.00574	0.268585
$501$	62.2743	2.78221
$502$	−5.33942	−0.238310
$503$	−24.2740	−1.08233	−0.541163	−	0.840918i	$-0.682016\pi$
$503$	−24.2740	−1.08233	−0.541163	+	0.840918i	$0.682016\pi$
$504$	−13.5226	−0.602343
$505$	7.74245	0.344534
$506$	6.25449	0.278046
$507$	0.409690	0.0181950
$508$	0.506922	0.0224910
$509$	−4.37168	−0.193771	−0.0968856	−	0.995296i	$-0.530888\pi$
$509$	−4.37168	−0.193771	−0.0968856	+	0.995296i	$0.530888\pi$
$510$	69.9480	3.09735
$511$	1.99561	0.0882807
$512$	−1.00000	−0.0441942
$513$	101.932	4.50040
$514$	−15.3493	−0.677028
$515$	−4.43005	−0.195212
$516$	−17.2928	−0.761274
$517$	18.9300	0.832540
$518$	−4.43711	−0.194955
$519$	−26.5373	−1.16486
$520$	10.1540	0.445284
$521$	−39.8560	−1.74612	−0.873061	−	0.487611i	$-0.837869\pi$
$521$	−39.8560	−1.74612	−0.873061	+	0.487611i	$0.837869\pi$
$522$	39.2790	1.71919
$523$	−33.6725	−1.47240	−0.736198	−	0.676766i	$-0.763381\pi$
$523$	−33.6725	−1.47240	−0.736198	+	0.676766i	$0.763381\pi$
$524$	−10.7751	−0.470714
$525$	15.6368	0.682447
$526$	26.6345	1.16132
$527$	−70.6907	−3.07933
$528$	−16.4257	−0.714836
$529$	−21.3601	−0.928698
$530$	5.32374	0.231249
$531$	18.1548	0.787852
$532$	9.28601	0.402599
$533$	−26.6337	−1.15363
$534$	−23.9115	−1.03475
$535$	48.8725	2.11294
$536$	7.77136	0.335672
$537$	−38.0096	−1.64024
$538$	−8.87441	−0.382603
$539$	21.2576	0.915631
$540$	−50.0659	−2.15450
$541$	−10.9916	−0.472566	−0.236283	−	0.971684i	$-0.575929\pi$
$541$	−10.9916	−0.472566	−0.236283	+	0.971684i	$0.575929\pi$
$542$	−14.9484	−0.642089
$543$	22.4207	0.962164
$544$	−7.41973	−0.318118
$545$	34.9832	1.49852
$546$	−19.8227	−0.848332
$547$	−2.76693	−0.118305	−0.0591527	−	0.998249i	$-0.518840\pi$
$547$	−2.76693	−0.118305	−0.0591527	+	0.998249i	$0.518840\pi$
$548$	8.36832	0.357477
$549$	−70.5604	−3.01145
$550$	13.9560	0.595087
$551$	−26.9730	−1.14909
$552$	−4.30685	−0.183312
$553$	−11.2443	−0.478156
$554$	10.4679	0.444740
$555$	−25.7080	−1.09124
$556$	−0.475069	−0.0201474
$557$	34.5169	1.46253	0.731265	−	0.682094i	$-0.238930\pi$
$557$	34.5169	1.46253	0.731265	+	0.682094i	$0.238930\pi$
$558$	79.1797	3.35195
$559$	−18.6259	−0.787792
$560$	−4.56101	−0.192738
$561$	−121.874	−5.14553
$562$	3.55084	0.149783
$563$	−21.6499	−0.912435	−0.456217	−	0.889868i	$-0.650796\pi$
$563$	−21.6499	−0.912435	−0.456217	+	0.889868i	$0.650796\pi$
$564$	−13.0352	−0.548881
$565$	−14.0336	−0.590397
$566$	−32.0270	−1.34619
$567$	57.1708	2.40095
$568$	−1.80780	−0.0758536
$569$	−24.5140	−1.02768	−0.513840	−	0.857886i	$-0.671777\pi$
$569$	−24.5140	−1.02768	−0.513840	+	0.857886i	$0.671777\pi$
$570$	53.8018	2.25351
$571$	34.8592	1.45881	0.729406	−	0.684081i	$-0.239796\pi$
$571$	34.8592	1.45881	0.729406	+	0.684081i	$0.239796\pi$
$572$	−17.6919	−0.739736
$573$	33.4023	1.39540
$574$	11.9634	0.499341
$575$	3.65930	0.152603
$576$	8.31074	0.346281
$577$	−2.20534	−0.0918096	−0.0459048	−	0.998946i	$-0.514617\pi$
$577$	−2.20534	−0.0918096	−0.0459048	+	0.998946i	$0.514617\pi$
$578$	−38.0523	−1.58277
$579$	41.6787	1.73211
$580$	13.2484	0.550108
$581$	24.1568	1.00219
$582$	−8.29230	−0.343727
$583$	−9.27584	−0.384166
$584$	−1.22647	−0.0507517
$585$	−84.3876	−3.48900
$586$	−33.2339	−1.37288
$587$	−24.4096	−1.00749	−0.503746	−	0.863852i	$-0.668045\pi$
$587$	−24.4096	−1.00749	−0.503746	+	0.863852i	$0.668045\pi$
$588$	−14.6380	−0.603662
$589$	−54.3730	−2.24040
$590$	6.12342	0.252097
$591$	−18.7312	−0.770500
$592$	2.72697	0.112078
$593$	20.2832	0.832930	0.416465	−	0.909152i	$-0.363269\pi$
$593$	20.2832	0.832930	0.416465	+	0.909152i	$0.363269\pi$
$594$	87.2326	3.57920
$595$	−33.8415	−1.38736
$596$	−16.2885	−0.667201
$597$	71.6578	2.93276
$598$	−4.63886	−0.189697
$599$	21.2505	0.868274	0.434137	−	0.900847i	$-0.357053\pi$
$599$	21.2505	0.868274	0.434137	+	0.900847i	$0.357053\pi$
$600$	−9.61013	−0.392332
$601$	1.51881	0.0619535	0.0309768	−	0.999520i	$-0.490138\pi$
$601$	1.51881	0.0619535	0.0309768	+	0.999520i	$0.490138\pi$
$602$	8.36643	0.340990
$603$	−64.5858	−2.63014
$604$	0.513029	0.0208749
$605$	−36.0305	−1.46485
$606$	9.28928	0.377351
$607$	−46.9670	−1.90633	−0.953165	−	0.302450i	$-0.902195\pi$
$607$	−46.9670	−1.90633	−0.953165	+	0.302450i	$0.902195\pi$
$608$	−5.70702	−0.231450
$609$	−25.8634	−1.04804
$610$	−23.7992	−0.963603
$611$	−14.0401	−0.568001
$612$	61.6634	2.49260
$613$	−31.0349	−1.25349	−0.626744	−	0.779225i	$-0.715613\pi$
$613$	−31.0349	−1.25349	−0.626744	+	0.779225i	$0.715613\pi$
$614$	20.4035	0.823417
$615$	69.3140	2.79501
$616$	7.94690	0.320190
$617$	32.8342	1.32185	0.660927	−	0.750450i	$-0.270163\pi$
$617$	32.8342	1.32185	0.660927	+	0.750450i	$0.270163\pi$
$618$	−5.31512	−0.213805
$619$	24.7996	0.996780	0.498390	−	0.866953i	$-0.333925\pi$
$619$	24.7996	0.996780	0.498390	+	0.866953i	$0.333925\pi$
$620$	26.7064	1.07256
$621$	22.8726	0.917844
$622$	8.99243	0.360564
$623$	11.5686	0.463486
$624$	12.1827	0.487697
$625$	−31.1222	−1.24489
$626$	4.30737	0.172157
$627$	−93.7417	−3.74368
$628$	6.63432	0.264738
$629$	20.2334	0.806757
$630$	37.9054	1.51019
$631$	5.87503	0.233881	0.116941	−	0.993139i	$-0.462691\pi$
$631$	5.87503	0.233881	0.116941	+	0.993139i	$0.462691\pi$
$632$	6.91055	0.274887
$633$	15.3647	0.610692
$634$	3.84895	0.152861
$635$	−1.42096	−0.0563892
$636$	6.38735	0.253275
$637$	−15.7664	−0.624689
$638$	−23.0833	−0.913877
$639$	15.0242	0.594346
$640$	2.80312	0.110803
$641$	12.9994	0.513446	0.256723	−	0.966485i	$-0.417357\pi$
$641$	12.9994	0.513446	0.256723	+	0.966485i	$0.417357\pi$
$642$	58.6365	2.31420
$643$	−30.1448	−1.18879	−0.594397	−	0.804172i	$-0.702609\pi$
$643$	−30.1448	−1.18879	−0.594397	+	0.804172i	$0.702609\pi$
$644$	2.08369	0.0821090
$645$	48.4739	1.90866
$646$	−42.3445	−1.66602
$647$	5.45361	0.214404	0.107202	−	0.994237i	$-0.465811\pi$
$647$	5.45361	0.214404	0.107202	+	0.994237i	$0.465811\pi$
$648$	−35.1362	−1.38028
$649$	−10.6692	−0.418801
$650$	−10.3510	−0.405998
$651$	−52.1362	−2.04338
$652$	−10.1213	−0.396379
$653$	−19.4342	−0.760520	−0.380260	−	0.924880i	$-0.624165\pi$
$653$	−19.4342	−0.760520	−0.380260	+	0.924880i	$0.624165\pi$
$654$	41.9724	1.64125
$655$	30.2040	1.18017
$656$	−7.35248	−0.287066
$657$	10.1929	0.397662
$658$	6.30655	0.245855
$659$	17.1998	0.670008	0.335004	−	0.942217i	$-0.391262\pi$
$659$	17.1998	0.670008	0.335004	+	0.942217i	$0.391262\pi$
$660$	46.0432	1.79223
$661$	24.5129	0.953442	0.476721	−	0.879055i	$-0.341825\pi$
$661$	24.5129	0.953442	0.476721	+	0.879055i	$0.341825\pi$
$662$	8.07055	0.313671
$663$	90.3921	3.51054
$664$	−14.8463	−0.576150
$665$	−26.0298	−1.00939
$666$	−22.6631	−0.878179
$667$	−6.05249	−0.234354
$668$	18.5167	0.716432
$669$	3.36314	0.130027
$670$	−21.7841	−0.841591
$671$	41.4667	1.60080
$672$	−5.47224	−0.211096
$673$	−15.1588	−0.584329	−0.292164	−	0.956368i	$-0.594375\pi$
$673$	−15.1588	−0.584329	−0.292164	+	0.956368i	$0.594375\pi$
$674$	−10.2899	−0.396351
$675$	51.0369	1.96441
$676$	0.121818	0.00468529
$677$	29.3114	1.12653	0.563264	−	0.826277i	$-0.309545\pi$
$677$	29.3114	1.12653	0.563264	+	0.826277i	$0.309545\pi$
$678$	−16.8373	−0.646632
$679$	4.01189	0.153962
$680$	20.7984	0.797582
$681$	−4.62761	−0.177330
$682$	−46.5321	−1.78180
$683$	28.4148	1.08726	0.543630	−	0.839325i	$-0.317049\pi$
$683$	28.4148	1.08726	0.543630	+	0.839325i	$0.317049\pi$
$684$	47.4296	1.81351
$685$	−23.4574	−0.896262
$686$	18.4718	0.705258
$687$	−35.0882	−1.33870
$688$	−5.14186	−0.196032
$689$	6.87975	0.262097
$690$	12.0726	0.459597
$691$	−16.2985	−0.620026	−0.310013	−	0.950732i	$-0.600333\pi$
$691$	−16.2985	−0.620026	−0.310013	+	0.950732i	$0.600333\pi$
$692$	−7.89063	−0.299957
$693$	−66.0446	−2.50883
$694$	13.7951	0.523655
$695$	1.33167	0.0505133
$696$	15.8952	0.602506
$697$	−54.5534	−2.06636
$698$	11.7721	0.445580
$699$	−11.0611	−0.418369
$700$	4.64947	0.175733
$701$	46.1845	1.74437	0.872183	−	0.489180i	$-0.162704\pi$
$701$	46.1845	1.74437	0.872183	+	0.489180i	$0.162704\pi$
$702$	−64.6990	−2.44191
$703$	15.5629	0.586965
$704$	−4.88403	−0.184074
$705$	36.5393	1.37615
$706$	−19.8217	−0.746001
$707$	−4.49424	−0.169023
$708$	7.34679	0.276109
$709$	27.1290	1.01885	0.509426	−	0.860515i	$-0.329858\pi$
$709$	27.1290	1.01885	0.509426	+	0.860515i	$0.329858\pi$
$710$	5.06748	0.190179
$711$	−57.4318	−2.15386
$712$	−7.10986	−0.266453
$713$	−12.2008	−0.456924
$714$	−40.6025	−1.51951
$715$	49.5926	1.85466
$716$	−11.3018	−0.422369
$717$	−58.9472	−2.20142
$718$	6.89554	0.257339
$719$	6.14447	0.229150	0.114575	−	0.993415i	$-0.463449\pi$
$719$	6.14447	0.229150	0.114575	+	0.993415i	$0.463449\pi$
$720$	−23.2960	−0.868191
$721$	2.57150	0.0957677
$722$	−13.5701	−0.505026
$723$	−0.858420	−0.0319250
$724$	6.66659	0.247762
$725$	−13.5053	−0.501574
$726$	−43.2289	−1.60438
$727$	35.1476	1.30355	0.651776	−	0.758411i	$-0.274024\pi$
$727$	35.1476	1.30355	0.651776	+	0.758411i	$0.274024\pi$
$728$	−5.89409	−0.218449
$729$	111.803	4.14084
$730$	3.43794	0.127244
$731$	−38.1512	−1.41107
$732$	−28.5540	−1.05539
$733$	12.3156	0.454887	0.227443	−	0.973791i	$-0.426963\pi$
$733$	12.3156	0.454887	0.227443	+	0.973791i	$0.426963\pi$
$734$	−11.9728	−0.441923
$735$	41.0321	1.51349
$736$	−1.28060	−0.0472036
$737$	37.9555	1.39811
$738$	61.1046	2.24929
$739$	−1.26153	−0.0464063	−0.0232031	−	0.999731i	$-0.507386\pi$
$739$	−1.26153	−0.0464063	−0.0232031	+	0.999731i	$0.507386\pi$
$740$	−7.64403	−0.281000
$741$	69.5267	2.55413
$742$	−3.09026	−0.113447
$743$	43.6588	1.60169	0.800843	−	0.598874i	$-0.204385\pi$
$743$	43.6588	1.60169	0.800843	+	0.598874i	$0.204385\pi$
$744$	32.0420	1.17472
$745$	45.6585	1.67280
$746$	17.7096	0.648396
$747$	123.384	4.51439
$748$	−36.2381	−1.32500
$749$	−28.3689	−1.03658
$750$	−20.1982	−0.737532
$751$	44.3812	1.61949	0.809746	−	0.586780i	$-0.199605\pi$
$751$	44.3812	1.61949	0.809746	+	0.586780i	$0.199605\pi$
$752$	−3.87590	−0.141340
$753$	17.9572	0.654398
$754$	17.1205	0.623493
$755$	−1.43808	−0.0523372
$756$	29.0617	1.05696
$757$	24.7579	0.899840	0.449920	−	0.893069i	$-0.351452\pi$
$757$	24.7579	0.899840	0.449920	+	0.893069i	$0.351452\pi$
$758$	−6.18691	−0.224719
$759$	−21.0348	−0.763514
$760$	15.9975	0.580289
$761$	17.9077	0.649154	0.324577	−	0.945859i	$-0.394778\pi$
$761$	17.9077	0.649154	0.324577	+	0.945859i	$0.394778\pi$
$762$	−1.70485	−0.0617603
$763$	−20.3066	−0.735149
$764$	9.93188	0.359323
$765$	−172.850	−6.24940
$766$	−12.2112	−0.441209
$767$	7.91315	0.285727
$768$	3.36314	0.121357
$769$	−16.2476	−0.585903	−0.292952	−	0.956127i	$-0.594638\pi$
$769$	−16.2476	−0.585903	−0.292952	+	0.956127i	$0.594638\pi$
$770$	−22.2761	−0.802775
$771$	51.6219	1.85912
$772$	12.3928	0.446026
$773$	−45.7237	−1.64457	−0.822284	−	0.569077i	$-0.807300\pi$
$773$	−45.7237	−1.64457	−0.822284	+	0.569077i	$0.807300\pi$
$774$	42.7327	1.53599
$775$	−27.2244	−0.977928
$776$	−2.46564	−0.0885113
$777$	14.9226	0.535347
$778$	−20.3603	−0.729952
$779$	−41.9607	−1.50340
$780$	−34.1495	−1.22275
$781$	−8.82934	−0.315939
$782$	−9.50171	−0.339781
$783$	−84.4153	−3.01676
$784$	−4.35248	−0.155446
$785$	−18.5968	−0.663748
$786$	36.2384	1.29258
$787$	−27.8093	−0.991293	−0.495647	−	0.868524i	$-0.665069\pi$
$787$	−27.8093	−0.991293	−0.495647	+	0.868524i	$0.665069\pi$
$788$	−5.56956	−0.198407
$789$	−89.5757	−3.18898
$790$	−19.3711	−0.689192
$791$	8.14603	0.289640
$792$	40.5899	1.44230
$793$	−30.7552	−1.09215
$794$	26.7165	0.948135
$795$	−17.9045	−0.635008
$796$	21.3068	0.755199
$797$	54.7954	1.94095	0.970476	−	0.241197i	$-0.0775398\pi$
$797$	54.7954	1.94095	0.970476	+	0.241197i	$0.0775398\pi$
$798$	−31.2302	−1.10554
$799$	−28.7581	−1.01739
$800$	−2.85748	−0.101027
$801$	59.0882	2.08778
$802$	−20.2413	−0.714744
$803$	−5.99011	−0.211386
$804$	−26.1362	−0.921753
$805$	−5.84084	−0.205863
$806$	34.5121	1.21564
$807$	29.8459	1.05063
$808$	2.76208	0.0971697
$809$	10.9965	0.386617	0.193309	−	0.981138i	$-0.438078\pi$
$809$	10.9965	0.386617	0.193309	+	0.981138i	$0.438078\pi$
$810$	98.4910	3.46062
$811$	21.3521	0.749773	0.374887	−	0.927071i	$-0.377682\pi$
$811$	21.3521	0.749773	0.374887	+	0.927071i	$0.377682\pi$
$812$	−7.69024	−0.269875
$813$	50.2736	1.76317
$814$	13.3186	0.466817
$815$	28.3711	0.993797
$816$	24.9536	0.873551
$817$	−29.3447	−1.02664
$818$	−29.0470	−1.01560
$819$	48.9842	1.71165
$820$	20.6099	0.719728
$821$	34.6645	1.20980	0.604900	−	0.796301i	$-0.293213\pi$
$821$	34.6645	1.20980	0.604900	+	0.796301i	$0.293213\pi$
$822$	−28.1439	−0.981630
$823$	−43.5483	−1.51800	−0.758999	−	0.651092i	$-0.774311\pi$
$823$	−43.5483	−1.51800	−0.758999	+	0.651092i	$0.774311\pi$
$824$	−1.58040	−0.0550559
$825$	−46.9361	−1.63411
$826$	−3.55445	−0.123675
$827$	18.3929	0.639585	0.319792	−	0.947488i	$-0.396387\pi$
$827$	18.3929	0.639585	0.319792	+	0.947488i	$0.396387\pi$
$828$	10.6428	0.369861
$829$	24.2452	0.842071	0.421036	−	0.907044i	$-0.361667\pi$
$829$	24.2452	0.842071	0.421036	+	0.907044i	$0.361667\pi$
$830$	41.6161	1.44452
$831$	−35.2052	−1.22125
$832$	3.62240	0.125584
$833$	−32.2942	−1.11893
$834$	1.59772	0.0553247
$835$	−51.9045	−1.79623
$836$	−27.8732	−0.964016
$837$	−170.167	−5.88183
$838$	9.10873	0.314656
$839$	29.6480	1.02356	0.511781	−	0.859116i	$-0.328986\pi$
$839$	29.6480	1.02356	0.511781	+	0.859116i	$0.328986\pi$
$840$	15.3393	0.529258
$841$	−6.66219	−0.229731
$842$	−29.6256	−1.02096
$843$	−11.9420	−0.411304
$844$	4.56855	0.157256
$845$	−0.341469	−0.0117469
$846$	32.2116	1.10746
$847$	20.9145	0.718632
$848$	1.89922	0.0652195
$849$	107.711	3.69664
$850$	−21.2017	−0.727213
$851$	3.49216	0.119710
$852$	6.07989	0.208294
$853$	−7.06609	−0.241938	−0.120969	−	0.992656i	$-0.538600\pi$
$853$	−7.06609	−0.241938	−0.120969	+	0.992656i	$0.538600\pi$
$854$	13.8147	0.472729
$855$	−132.951	−4.54682
$856$	17.4350	0.595917
$857$	−3.29313	−0.112491	−0.0562455	−	0.998417i	$-0.517913\pi$
$857$	−3.29313	−0.112491	−0.0562455	+	0.998417i	$0.517913\pi$
$858$	59.5005	2.03131
$859$	−14.7315	−0.502633	−0.251317	−	0.967905i	$-0.580864\pi$
$859$	−14.7315	−0.502633	−0.251317	+	0.967905i	$0.580864\pi$
$860$	14.4133	0.491488
$861$	−40.2345	−1.37119
$862$	9.05749	0.308499
$863$	−6.02541	−0.205107	−0.102554	−	0.994727i	$-0.532701\pi$
$863$	−6.02541	−0.205107	−0.102554	+	0.994727i	$0.532701\pi$
$864$	−17.8608	−0.607637
$865$	22.1184	0.752047
$866$	15.1574	0.515069
$867$	127.975	4.34628
$868$	−15.5022	−0.526180
$869$	33.7513	1.14493
$870$	−44.5561	−1.51059
$871$	−28.1510	−0.953861
$872$	12.4801	0.422630
$873$	20.4913	0.693525
$874$	−7.30842	−0.247211
$875$	9.77205	0.330356
$876$	4.12479	0.139364
$877$	−22.0686	−0.745203	−0.372601	−	0.927991i	$-0.621534\pi$
$877$	−22.0686	−0.745203	−0.372601	+	0.927991i	$0.621534\pi$
$878$	−22.8847	−0.772322
$879$	111.770	3.76992
$880$	13.6905	0.461507
$881$	1.19776	0.0403534	0.0201767	−	0.999796i	$-0.493577\pi$
$881$	1.19776	0.0403534	0.0201767	+	0.999796i	$0.493577\pi$
$882$	36.1723	1.21799
$883$	50.4064	1.69631	0.848155	−	0.529748i	$-0.177713\pi$
$883$	50.4064	1.69631	0.848155	+	0.529748i	$0.177713\pi$
$884$	26.8773	0.903980
$885$	−20.5939	−0.692258
$886$	14.8447	0.498719
$887$	−8.22912	−0.276307	−0.138153	−	0.990411i	$-0.544117\pi$
$887$	−8.22912	−0.276307	−0.138153	+	0.990411i	$0.544117\pi$
$888$	−9.17120	−0.307765
$889$	0.824823	0.0276637
$890$	19.9298	0.668049
$891$	−171.606	−5.74902
$892$	1.00000	0.0334825
$893$	−22.1198	−0.740212
$894$	54.7804	1.83213
$895$	31.6804	1.05896
$896$	−1.62712	−0.0543583
$897$	15.6012	0.520907
$898$	−13.8348	−0.461674
$899$	45.0292	1.50181
$900$	23.7478	0.791593
$901$	14.0917	0.469462
$902$	−35.9097	−1.19566
$903$	−28.1375	−0.936357
$904$	−5.00641	−0.166511
$905$	−18.6872	−0.621185
$906$	−1.72539	−0.0573223
$907$	−11.0264	−0.366124	−0.183062	−	0.983101i	$-0.558601\pi$
$907$	−11.0264	−0.366124	−0.183062	+	0.983101i	$0.558601\pi$
$908$	−1.37598	−0.0456634
$909$	−22.9549	−0.761367
$910$	16.5218	0.547694
$911$	41.2046	1.36517	0.682585	−	0.730806i	$-0.260856\pi$
$911$	41.2046	1.36517	0.682585	+	0.730806i	$0.260856\pi$
$912$	19.1935	0.635561
$913$	−72.5099	−2.39973
$914$	−35.0332	−1.15880
$915$	80.0403	2.64605
$916$	−10.4332	−0.344721
$917$	−17.5325	−0.578973
$918$	−132.522	−4.37388
$919$	35.3908	1.16743	0.583717	−	0.811957i	$-0.301598\pi$
$919$	35.3908	1.16743	0.583717	+	0.811957i	$0.301598\pi$
$920$	3.58968	0.118348
$921$	−68.6198	−2.26110
$922$	−34.9345	−1.15051
$923$	6.54858	0.215549
$924$	−26.7266	−0.879239
$925$	7.79227	0.256208
$926$	9.21484	0.302819
$927$	13.1343	0.431387
$928$	4.72629	0.155148
$929$	−35.5904	−1.16768	−0.583842	−	0.811867i	$-0.698451\pi$
$929$	−35.5904	−1.16768	−0.583842	+	0.811867i	$0.698451\pi$
$930$	−89.8176	−2.94524
$931$	−24.8397	−0.814088
$932$	−3.28891	−0.107732
$933$	−30.2428	−0.990106
$934$	7.43435	0.243259
$935$	101.580	3.32202
$936$	−30.1049	−0.984009
$937$	35.0053	1.14357	0.571786	−	0.820403i	$-0.306251\pi$
$937$	35.0053	1.14357	0.571786	+	0.820403i	$0.306251\pi$
$938$	12.6449	0.412872
$939$	−14.4863	−0.472743
$940$	10.8646	0.354365
$941$	−48.1829	−1.57072	−0.785359	−	0.619041i	$-0.787522\pi$
$941$	−48.1829	−1.57072	−0.785359	+	0.619041i	$0.787522\pi$
$942$	−22.3122	−0.726970
$943$	−9.41560	−0.306614
$944$	2.18450	0.0710995
$945$	−81.4633	−2.65000
$946$	−25.1130	−0.816494
$947$	−3.93084	−0.127735	−0.0638675	−	0.997958i	$-0.520344\pi$
$947$	−3.93084	−0.127735	−0.0638675	+	0.997958i	$0.520344\pi$
$948$	−23.2412	−0.754838
$949$	4.44277	0.144218
$950$	−16.3077	−0.529092
$951$	−12.9446	−0.419757
$952$	−12.0728	−0.391281
$953$	16.8623	0.546223	0.273112	−	0.961982i	$-0.411947\pi$
$953$	16.8623	0.546223	0.273112	+	0.961982i	$0.411947\pi$
$954$	−15.7839	−0.511024
$955$	−27.8402	−0.900889
$956$	−17.5274	−0.566877
$957$	77.6326	2.50950
$958$	0.658732	0.0212827
$959$	13.6163	0.439692
$960$	−9.42730	−0.304265
$961$	59.7713	1.92811
$962$	−9.87819	−0.318486
$963$	−144.898	−4.66927
$964$	−0.255243	−0.00822083
$965$	−34.7384	−1.11827
$966$	−7.00776	−0.225471
$967$	−18.3545	−0.590242	−0.295121	−	0.955460i	$-0.595360\pi$
$967$	−18.3545	−0.590242	−0.295121	+	0.955460i	$0.595360\pi$
$968$	−12.8537	−0.413134
$969$	142.411	4.57489
$970$	6.91148	0.221914
$971$	−57.6781	−1.85098	−0.925489	−	0.378774i	$-0.876346\pi$
$971$	−57.6781	−1.85098	−0.925489	+	0.378774i	$0.876346\pi$
$972$	64.5858	2.07159
$973$	−0.772994	−0.0247810
$974$	−41.8859	−1.34211
$975$	34.8118	1.11487
$976$	−8.49027	−0.271767
$977$	−34.9645	−1.11861	−0.559306	−	0.828961i	$-0.688932\pi$
$977$	−34.9645	−1.11861	−0.559306	+	0.828961i	$0.688932\pi$
$978$	34.0393	1.08846
$979$	−34.7248	−1.10981
$980$	12.2005	0.389731
$981$	−103.719	−3.31149
$982$	27.2206	0.868645
$983$	10.6590	0.339968	0.169984	−	0.985447i	$-0.445628\pi$
$983$	10.6590	0.339968	0.169984	+	0.985447i	$0.445628\pi$
$984$	24.7275	0.788282
$985$	15.6121	0.497444
$986$	35.0678	1.11679
$987$	−21.2099	−0.675117
$988$	20.6731	0.657700
$989$	−6.58468	−0.209381
$990$	−113.778	−3.61611
$991$	−30.3268	−0.963363	−0.481682	−	0.876346i	$-0.659974\pi$
$991$	−30.3268	−0.963363	−0.481682	+	0.876346i	$0.659974\pi$
$992$	9.52740	0.302495
$993$	−27.1424	−0.861339
$994$	−2.94151	−0.0932989
$995$	−59.7255	−1.89343
$996$	49.9304	1.58211
$997$	−30.9818	−0.981204	−0.490602	−	0.871384i	$-0.663223\pi$
$997$	−30.9818	−0.981204	−0.490602	+	0.871384i	$0.663223\pi$
$998$	38.6603	1.22377
$999$	48.7059	1.54099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	446.2.a.f.1.8	✓	8
3.2	odd	2		4014.2.a.z.1.7		8
4.3	odd	2		3568.2.a.n.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.f.1.8	✓	8	1.1	even	1	trivial
3568.2.a.n.1.1		8	4.3	odd	2
4014.2.a.z.1.7		8	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:	\( N \)	\(=\)	\( 446 = 2 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	446.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.36314 q^{3} +1.00000 q^{4} -2.80312 q^{5} -3.36314 q^{6} +1.62712 q^{7} -1.00000 q^{8} +8.31074 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.36314 q^{3} +1.00000 q^{4} -2.80312 q^{5} -3.36314 q^{6} +1.62712 q^{7} -1.00000 q^{8} +8.31074 q^{9} +2.80312 q^{10} -4.88403 q^{11} +3.36314 q^{12} +3.62240 q^{13} -1.62712 q^{14} -9.42730 q^{15} +1.00000 q^{16} +7.41973 q^{17} -8.31074 q^{18} +5.70702 q^{19} -2.80312 q^{20} +5.47224 q^{21} +4.88403 q^{22} +1.28060 q^{23} -3.36314 q^{24} +2.85748 q^{25} -3.62240 q^{26} +17.8608 q^{27} +1.62712 q^{28} -4.72629 q^{29} +9.42730 q^{30} -9.52740 q^{31} -1.00000 q^{32} -16.4257 q^{33} -7.41973 q^{34} -4.56101 q^{35} +8.31074 q^{36} +2.72697 q^{37} -5.70702 q^{38} +12.1827 q^{39} +2.80312 q^{40} -7.35248 q^{41} -5.47224 q^{42} -5.14186 q^{43} -4.88403 q^{44} -23.2960 q^{45} -1.28060 q^{46} -3.87590 q^{47} +3.36314 q^{48} -4.35248 q^{49} -2.85748 q^{50} +24.9536 q^{51} +3.62240 q^{52} +1.89922 q^{53} -17.8608 q^{54} +13.6905 q^{55} -1.62712 q^{56} +19.1935 q^{57} +4.72629 q^{58} +2.18450 q^{59} -9.42730 q^{60} -8.49027 q^{61} +9.52740 q^{62} +13.5226 q^{63} +1.00000 q^{64} -10.1540 q^{65} +16.4257 q^{66} -7.77136 q^{67} +7.41973 q^{68} +4.30685 q^{69} +4.56101 q^{70} +1.80780 q^{71} -8.31074 q^{72} +1.22647 q^{73} -2.72697 q^{74} +9.61013 q^{75} +5.70702 q^{76} -7.94690 q^{77} -12.1827 q^{78} -6.91055 q^{79} -2.80312 q^{80} +35.1362 q^{81} +7.35248 q^{82} +14.8463 q^{83} +5.47224 q^{84} -20.7984 q^{85} +5.14186 q^{86} -15.8952 q^{87} +4.88403 q^{88} +7.10986 q^{89} +23.2960 q^{90} +5.89409 q^{91} +1.28060 q^{92} -32.0420 q^{93} +3.87590 q^{94} -15.9975 q^{95} -3.36314 q^{96} +2.46564 q^{97} +4.35248 q^{98} -40.5899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 8 q^{8} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 - 8 * q^8 + 16 * q^9	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 8 q^{8} + 16 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 10 q^{13} - 4 q^{14} - 4 q^{15} + 8 q^{16} + 8 q^{17} - 16 q^{18} + 16 q^{19} + 4 q^{20} - 2 q^{22} - 4 q^{23} - 4 q^{24} + 24 q^{25} - 10 q^{26} + 22 q^{27} + 4 q^{28} + 8 q^{29} + 4 q^{30} + 12 q^{31} - 8 q^{32} - 8 q^{34} - 24 q^{35} + 16 q^{36} + 16 q^{37} - 16 q^{38} - 4 q^{40} + 16 q^{41} + 12 q^{43} + 2 q^{44} + 34 q^{45} + 4 q^{46} - 4 q^{47} + 4 q^{48} + 40 q^{49} - 24 q^{50} + 26 q^{51} + 10 q^{52} - 8 q^{53} - 22 q^{54} + 8 q^{55} - 4 q^{56} + 16 q^{57} - 8 q^{58} - 4 q^{60} + 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} - 24 q^{65} - 10 q^{67} + 8 q^{68} - 28 q^{69} + 24 q^{70} + 8 q^{71} - 16 q^{72} + 8 q^{73} - 16 q^{74} + 6 q^{75} + 16 q^{76} - 56 q^{77} + 16 q^{79} + 4 q^{80} + 24 q^{81} - 16 q^{82} - 44 q^{83} - 18 q^{85} - 12 q^{86} - 72 q^{87} - 2 q^{88} - 34 q^{90} + 4 q^{91} - 4 q^{92} - 62 q^{93} + 4 q^{94} - 48 q^{95} - 4 q^{96} + 12 q^{97} - 40 q^{98} - 40 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 - 8 * q^8 + 16 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 10 * q^13 - 4 * q^14 - 4 * q^15 + 8 * q^16 + 8 * q^17 - 16 * q^18 + 16 * q^19 + 4 * q^20 - 2 * q^22 - 4 * q^23 - 4 * q^24 + 24 * q^25 - 10 * q^26 + 22 * q^27 + 4 * q^28 + 8 * q^29 + 4 * q^30 + 12 * q^31 - 8 * q^32 - 8 * q^34 - 24 * q^35 + 16 * q^36 + 16 * q^37 - 16 * q^38 - 4 * q^40 + 16 * q^41 + 12 * q^43 + 2 * q^44 + 34 * q^45 + 4 * q^46 - 4 * q^47 + 4 * q^48 + 40 * q^49 - 24 * q^50 + 26 * q^51 + 10 * q^52 - 8 * q^53 - 22 * q^54 + 8 * q^55 - 4 * q^56 + 16 * q^57 - 8 * q^58 - 4 * q^60 + 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 - 24 * q^65 - 10 * q^67 + 8 * q^68 - 28 * q^69 + 24 * q^70 + 8 * q^71 - 16 * q^72 + 8 * q^73 - 16 * q^74 + 6 * q^75 + 16 * q^76 - 56 * q^77 + 16 * q^79 + 4 * q^80 + 24 * q^81 - 16 * q^82 - 44 * q^83 - 18 * q^85 - 12 * q^86 - 72 * q^87 - 2 * q^88 - 34 * q^90 + 4 * q^91 - 4 * q^92 - 62 * q^93 + 4 * q^94 - 48 * q^95 - 4 * q^96 + 12 * q^97 - 40 * q^98 - 40 * q^99

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