LMFDB - Embedded newform 6845.2.a.k.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6845,2,Mod(1,6845)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6845 = 5 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6845.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,2,8,-7,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.6576001836$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - x^{4} + 35x^{3} + 7x^{2} - 27x - 9 $ x^7 - 11x^5 - x^4 + 35x^3 + 7x^2 - 27x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.97176$ of defining polynomial
Character	$\chi$	$=$	6845.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.97176 q^{2} -2.64931 q^{3} +1.88784 q^{4} -1.00000 q^{5} -5.22380 q^{6} +0.783600 q^{7} -0.221160 q^{8} +4.01883 q^{9} -1.97176 q^{10} -0.568551 q^{11} -5.00146 q^{12} +0.00611121 q^{13} +1.54507 q^{14} +2.64931 q^{15} -4.21175 q^{16} -2.96565 q^{17} +7.92416 q^{18} -1.03553 q^{19} -1.88784 q^{20} -2.07600 q^{21} -1.12105 q^{22} +8.65194 q^{23} +0.585920 q^{24} +1.00000 q^{25} +0.0120498 q^{26} -2.69919 q^{27} +1.47931 q^{28} +4.06655 q^{29} +5.22380 q^{30} +9.38666 q^{31} -7.86223 q^{32} +1.50627 q^{33} -5.84755 q^{34} -0.783600 q^{35} +7.58689 q^{36} -2.04182 q^{38} -0.0161905 q^{39} +0.221160 q^{40} -4.13168 q^{41} -4.09337 q^{42} -7.06722 q^{43} -1.07333 q^{44} -4.01883 q^{45} +17.0596 q^{46} +5.52310 q^{47} +11.1582 q^{48} -6.38597 q^{49} +1.97176 q^{50} +7.85691 q^{51} +0.0115370 q^{52} -9.14251 q^{53} -5.32215 q^{54} +0.568551 q^{55} -0.173301 q^{56} +2.74344 q^{57} +8.01827 q^{58} -9.24422 q^{59} +5.00146 q^{60} +9.98328 q^{61} +18.5082 q^{62} +3.14915 q^{63} -7.07894 q^{64} -0.00611121 q^{65} +2.96999 q^{66} -11.3571 q^{67} -5.59866 q^{68} -22.9217 q^{69} -1.54507 q^{70} +3.92800 q^{71} -0.888803 q^{72} +16.5499 q^{73} -2.64931 q^{75} -1.95491 q^{76} -0.445516 q^{77} -0.0319237 q^{78} -10.9033 q^{79} +4.21175 q^{80} -4.90551 q^{81} -8.14668 q^{82} +0.861082 q^{83} -3.91914 q^{84} +2.96565 q^{85} -13.9349 q^{86} -10.7735 q^{87} +0.125741 q^{88} +0.287249 q^{89} -7.92416 q^{90} +0.00478874 q^{91} +16.3335 q^{92} -24.8681 q^{93} +10.8902 q^{94} +1.03553 q^{95} +20.8295 q^{96} +4.33052 q^{97} -12.5916 q^{98} -2.28491 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 8 q^{4} - 7 q^{5} + 2 q^{6} - 3 q^{8} + 13 q^{9} - q^{11} - 6 q^{12} + 4 q^{13} - 10 q^{14} - 2 q^{15} - 2 q^{16} - 3 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} - q^{21} + 7 q^{22} + 15 q^{24}+ \cdots - 56 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 8 * q^4 - 7 * q^5 + 2 * q^6 - 3 * q^8 + 13 * q^9 - q^11 - 6 * q^12 + 4 * q^13 - 10 * q^14 - 2 * q^15 - 2 * q^16 - 3 * q^17 - 6 * q^18 - 14 * q^19 - 8 * q^20 - q^21 + 7 * q^22 + 15 * q^24 + 7 * q^25 - 11 * q^26 - 7 * q^27 - 9 * q^28 + 6 * q^29 - 2 * q^30 + 4 * q^32 + 11 * q^33 - 33 * q^34 + 20 * q^36 - 2 * q^38 - 25 * q^39 + 3 * q^40 + 7 * q^41 - 29 * q^42 - 19 * q^43 + 19 * q^44 - 13 * q^45 + 14 * q^46 + 4 * q^47 + 18 * q^48 - 5 * q^49 - 29 * q^51 - 8 * q^52 + 6 * q^53 - 9 * q^54 + q^55 - 25 * q^56 - 44 * q^57 + 12 * q^58 + 2 * q^59 + 6 * q^60 - 4 * q^61 + 25 * q^62 + 5 * q^63 - 31 * q^64 - 4 * q^65 + q^66 + 8 * q^67 - 13 * q^68 - 11 * q^69 + 10 * q^70 - 33 * q^71 + 43 * q^72 + 2 * q^75 - 12 * q^76 + 5 * q^77 + 22 * q^78 - 9 * q^79 + 2 * q^80 + 35 * q^81 - 76 * q^82 + 11 * q^83 - 40 * q^84 + 3 * q^85 - 17 * q^86 - 29 * q^87 - 20 * q^88 - 12 * q^89 + 6 * q^90 - 24 * q^91 - 7 * q^92 - 47 * q^93 + 10 * q^94 + 14 * q^95 - 15 * q^96 - 15 * q^97 + 54 * q^98 - 56 * q^99

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 $

$2$	1.97176	1.39424	0.697122	−	0.716952i	$-0.254463\pi$
$2$	1.97176	1.39424	0.697122	+	0.716952i	$0.254463\pi$
$3$	−2.64931	−1.52958	−0.764789	−	0.644281i	$-0.777157\pi$
$3$	−2.64931	−1.52958	−0.764789	+	0.644281i	$0.777157\pi$
$4$	1.88784	0.943918
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−5.22380	−2.13261
$7$	0.783600	0.296173	0.148086	−	0.988974i	$-0.452689\pi$
$7$	0.783600	0.296173	0.148086	+	0.988974i	$0.452689\pi$
$8$	−0.221160	−0.0781918
$9$	4.01883	1.33961
$10$	−1.97176	−0.623525
$11$	−0.568551	−0.171425	−0.0857123	−	0.996320i	$-0.527317\pi$
$11$	−0.568551	−0.171425	−0.0857123	+	0.996320i	$0.527317\pi$
$12$	−5.00146	−1.44380
$13$	0.00611121	0.00169494	0.000847472	−	1.00000i	$-0.499730\pi$
$13$	0.00611121	0.00169494	0.000847472		1.00000i	$0.499730\pi$
$14$	1.54507	0.412937
$15$	2.64931	0.684048
$16$	−4.21175	−1.05294
$17$	−2.96565	−0.719275	−0.359638	−	0.933092i	$-0.617100\pi$
$17$	−2.96565	−0.719275	−0.359638	+	0.933092i	$0.617100\pi$
$18$	7.92416	1.86774
$19$	−1.03553	−0.237567	−0.118783	−	0.992920i	$-0.537899\pi$
$19$	−1.03553	−0.237567	−0.118783	+	0.992920i	$0.537899\pi$
$20$	−1.88784	−0.422133
$21$	−2.07600	−0.453020
$22$	−1.12105	−0.239008
$23$	8.65194	1.80405	0.902027	−	0.431679i	$-0.142079\pi$
$23$	8.65194	1.80405	0.902027	+	0.431679i	$0.142079\pi$
$24$	0.585920	0.119601
$25$	1.00000	0.200000
$26$	0.0120498	0.00236317
$27$	−2.69919	−0.519459
$28$	1.47931	0.279563
$29$	4.06655	0.755140	0.377570	−	0.925981i	$-0.376760\pi$
$29$	4.06655	0.755140	0.377570	+	0.925981i	$0.376760\pi$
$30$	5.22380	0.953730
$31$	9.38666	1.68589	0.842947	−	0.537997i	$-0.180819\pi$
$31$	9.38666	1.68589	0.842947	+	0.537997i	$0.180819\pi$
$32$	−7.86223	−1.38986
$33$	1.50627	0.262207
$34$	−5.84755	−1.00285
$35$	−0.783600	−0.132453
$36$	7.58689	1.26448
$37$	0	0
$37$	0	0
$38$	−2.04182	−0.331227
$39$	−0.0161905	−0.00259255
$40$	0.221160	0.0349685
$41$	−4.13168	−0.645260	−0.322630	−	0.946525i	$-0.604567\pi$
$41$	−4.13168	−0.645260	−0.322630	+	0.946525i	$0.604567\pi$
$42$	−4.09337	−0.631620
$43$	−7.06722	−1.07774	−0.538870	−	0.842389i	$-0.681149\pi$
$43$	−7.06722	−1.07774	−0.538870	+	0.842389i	$0.681149\pi$
$44$	−1.07333	−0.161811
$45$	−4.01883	−0.599091
$46$	17.0596	2.51529
$47$	5.52310	0.805628	0.402814	−	0.915282i	$-0.368032\pi$
$47$	5.52310	0.805628	0.402814	+	0.915282i	$0.368032\pi$
$48$	11.1582	1.61055
$49$	−6.38597	−0.912282
$50$	1.97176	0.278849
$51$	7.85691	1.10019
$52$	0.0115370	0.00159989
$53$	−9.14251	−1.25582	−0.627910	−	0.778286i	$-0.716089\pi$
$53$	−9.14251	−1.25582	−0.627910	+	0.778286i	$0.716089\pi$
$54$	−5.32215	−0.724252
$55$	0.568551	0.0766634
$56$	−0.173301	−0.0231583
$57$	2.74344	0.363377
$58$	8.01827	1.05285
$59$	−9.24422	−1.20349	−0.601747	−	0.798687i	$-0.705529\pi$
$59$	−9.24422	−1.20349	−0.601747	+	0.798687i	$0.705529\pi$
$60$	5.00146	0.645685
$61$	9.98328	1.27823	0.639114	−	0.769112i	$-0.279301\pi$
$61$	9.98328	1.27823	0.639114	+	0.769112i	$0.279301\pi$
$62$	18.5082	2.35055
$63$	3.14915	0.396756
$64$	−7.07894	−0.884867
$65$	−0.00611121	−0.000758002 0
$66$	2.96999	0.365581
$67$	−11.3571	−1.38749	−0.693743	−	0.720222i	$-0.744040\pi$
$67$	−11.3571	−1.38749	−0.693743	+	0.720222i	$0.744040\pi$
$68$	−5.59866	−0.678937
$69$	−22.9217	−2.75944
$70$	−1.54507	−0.184671
$71$	3.92800	0.466167	0.233084	−	0.972457i	$-0.425118\pi$
$71$	3.92800	0.466167	0.233084	+	0.972457i	$0.425118\pi$
$72$	−0.888803	−0.104746
$73$	16.5499	1.93701	0.968507	−	0.248987i	$-0.0800975\pi$
$73$	16.5499	1.93701	0.968507	+	0.248987i	$0.0800975\pi$
$74$	0	0
$75$	−2.64931	−0.305916
$76$	−1.95491	−0.224244
$77$	−0.445516	−0.0507713
$78$	−0.0319237	−0.00361465
$79$	−10.9033	−1.22672	−0.613361	−	0.789803i	$-0.710183\pi$
$79$	−10.9033	−1.22672	−0.613361	+	0.789803i	$0.710183\pi$
$80$	4.21175	0.470888
$81$	−4.90551	−0.545057
$82$	−8.14668	−0.899650
$83$	0.861082	0.0945160	0.0472580	−	0.998883i	$-0.484952\pi$
$83$	0.861082	0.0945160	0.0472580	+	0.998883i	$0.484952\pi$
$84$	−3.91914	−0.427613
$85$	2.96565	0.321670
$86$	−13.9349	−1.50263
$87$	−10.7735	−1.15505
$88$	0.125741	0.0134040
$89$	0.287249	0.0304484	0.0152242	−	0.999884i	$-0.495154\pi$
$89$	0.287249	0.0304484	0.0152242	+	0.999884i	$0.495154\pi$
$90$	−7.92416	−0.835280
$91$	0.00478874	0.000501997 0
$92$	16.3335	1.70288
$93$	−24.8681	−2.57871
$94$	10.8902	1.12324
$95$	1.03553	0.106243
$96$	20.8295	2.12590
$97$	4.33052	0.439697	0.219849	−	0.975534i	$-0.429444\pi$
$97$	4.33052	0.439697	0.219849	+	0.975534i	$0.429444\pi$
$98$	−12.5916	−1.27194
$99$	−2.28491	−0.229642
$100$	1.88784	0.188784
$101$	−5.40523	−0.537840	−0.268920	−	0.963163i	$-0.586667\pi$
$101$	−5.40523	−0.537840	−0.268920	+	0.963163i	$0.586667\pi$
$102$	15.4919	1.53393
$103$	−12.2753	−1.20952	−0.604759	−	0.796409i	$-0.706731\pi$
$103$	−12.2753	−1.20952	−0.604759	+	0.796409i	$0.706731\pi$
$104$	−0.00135155	−0.000132531 0
$105$	2.07600	0.202596
$106$	−18.0268	−1.75092
$107$	13.6927	1.32372	0.661860	−	0.749627i	$-0.269767\pi$
$107$	13.6927	1.32372	0.661860	+	0.749627i	$0.269767\pi$
$108$	−5.09562	−0.490326
$109$	−7.32473	−0.701582	−0.350791	−	0.936454i	$-0.614087\pi$
$109$	−7.32473	−0.701582	−0.350791	+	0.936454i	$0.614087\pi$
$110$	1.12105	0.106888
$111$	0	0
$112$	−3.30032	−0.311851
$113$	4.37525	0.411589	0.205794	−	0.978595i	$-0.434022\pi$
$113$	4.37525	0.411589	0.205794	+	0.978595i	$0.434022\pi$
$114$	5.40940	0.506637
$115$	−8.65194	−0.806798
$116$	7.67699	0.712790
$117$	0.0245599	0.00227056
$118$	−18.2274	−1.67797
$119$	−2.32388	−0.213030
$120$	−0.585920	−0.0534870
$121$	−10.6767	−0.970614
$122$	19.6846	1.78216
$123$	10.9461	0.986975
$124$	17.7205	1.59135
$125$	−1.00000	−0.0894427
$126$	6.20937	0.553175
$127$	19.5518	1.73494	0.867471	−	0.497488i	$-0.165744\pi$
$127$	19.5518	1.73494	0.867471	+	0.497488i	$0.165744\pi$
$128$	1.76650	0.156138
$129$	18.7232	1.64849
$130$	−0.0120498	−0.00105684
$131$	−9.55395	−0.834733	−0.417366	−	0.908738i	$-0.637047\pi$
$131$	−9.55395	−0.834733	−0.417366	+	0.908738i	$0.637047\pi$
$132$	2.84358	0.247502
$133$	−0.811441	−0.0703609
$134$	−22.3934	−1.93450
$135$	2.69919	0.232309
$136$	0.655883	0.0562415
$137$	−5.12953	−0.438245	−0.219123	−	0.975697i	$-0.570319\pi$
$137$	−5.12953	−0.438245	−0.219123	+	0.975697i	$0.570319\pi$
$138$	−45.1960	−3.84734
$139$	−10.3410	−0.877115	−0.438557	−	0.898703i	$-0.644510\pi$
$139$	−10.3410	−0.877115	−0.438557	+	0.898703i	$0.644510\pi$
$140$	−1.47931	−0.125024
$141$	−14.6324	−1.23227
$142$	7.74506	0.649951
$143$	−0.00347454	−0.000290555 0
$144$	−16.9263	−1.41052
$145$	−4.06655	−0.337709
$146$	32.6323	2.70067
$147$	16.9184	1.39541
$148$	0	0
$149$	−10.8779	−0.891149	−0.445575	−	0.895245i	$-0.647001\pi$
$149$	−10.8779	−0.891149	−0.445575	+	0.895245i	$0.647001\pi$
$150$	−5.22380	−0.426521
$151$	0.0154190	0.00125478	0.000627392	−	1.00000i	$-0.499800\pi$
$151$	0.0154190	0.00125478	0.000627392		1.00000i	$0.499800\pi$
$152$	0.229018	0.0185758
$153$	−11.9184	−0.963548
$154$	−0.878451	−0.0707876
$155$	−9.38666	−0.753955
$156$	−0.0305650	−0.00244716
$157$	−14.1321	−1.12786	−0.563930	−	0.825822i	$-0.690711\pi$
$157$	−14.1321	−1.12786	−0.563930	+	0.825822i	$0.690711\pi$
$158$	−21.4988	−1.71035
$159$	24.2213	1.92088
$160$	7.86223	0.621564
$161$	6.77966	0.534312
$162$	−9.67249	−0.759942
$163$	−21.2937	−1.66785	−0.833926	−	0.551876i	$-0.813912\pi$
$163$	−21.2937	−1.66785	−0.833926	+	0.551876i	$0.813912\pi$
$164$	−7.79993	−0.609073
$165$	−1.50627	−0.117263
$166$	1.69785	0.131778
$167$	7.97136	0.616842	0.308421	−	0.951250i	$-0.400199\pi$
$167$	7.97136	0.616842	0.308421	+	0.951250i	$0.400199\pi$
$168$	0.459127	0.0354224
$169$	−13.0000	−0.999997
$170$	5.84755	0.448486
$171$	−4.16162	−0.318247
$172$	−13.3418	−1.01730
$173$	−14.9134	−1.13385	−0.566923	−	0.823771i	$-0.691866\pi$
$173$	−14.9134	−1.13385	−0.566923	+	0.823771i	$0.691866\pi$
$174$	−21.2429	−1.61042
$175$	0.783600	0.0592346
$176$	2.39459	0.180499
$177$	24.4908	1.84084
$178$	0.566387	0.0424525
$179$	7.73854	0.578406	0.289203	−	0.957268i	$-0.406610\pi$
$179$	7.73854	0.578406	0.289203	+	0.957268i	$0.406610\pi$
$180$	−7.58689	−0.565493
$181$	13.1559	0.977869	0.488935	−	0.872320i	$-0.337386\pi$
$181$	13.1559	0.977869	0.488935	+	0.872320i	$0.337386\pi$
$182$	0.00944225	0.000699906 0
$183$	−26.4488	−1.95515
$184$	−1.91346	−0.141062
$185$	0	0
$186$	−49.0340	−3.59535
$187$	1.68612	0.123301
$188$	10.4267	0.760447
$189$	−2.11508	−0.153850
$190$	2.04182	0.148129
$191$	−23.2461	−1.68203	−0.841014	−	0.541013i	$-0.818041\pi$
$191$	−23.2461	−1.68203	−0.841014	+	0.541013i	$0.818041\pi$
$192$	18.7543	1.35347
$193$	−5.38208	−0.387411	−0.193705	−	0.981060i	$-0.562051\pi$
$193$	−5.38208	−0.387411	−0.193705	+	0.981060i	$0.562051\pi$
$194$	8.53874	0.613045
$195$	0.0161905	0.00115942
$196$	−12.0557	−0.861119
$197$	−0.737219	−0.0525246	−0.0262623	−	0.999655i	$-0.508361\pi$
$197$	−0.737219	−0.0525246	−0.0262623	+	0.999655i	$0.508361\pi$
$198$	−4.50529	−0.320177
$199$	−3.52332	−0.249762	−0.124881	−	0.992172i	$-0.539855\pi$
$199$	−3.52332	−0.249762	−0.124881	+	0.992172i	$0.539855\pi$
$200$	−0.221160	−0.0156384
$201$	30.0884	2.12227
$202$	−10.6578	−0.749881
$203$	3.18655	0.223652
$204$	14.8326	1.03849
$205$	4.13168	0.288569
$206$	−24.2039	−1.68636
$207$	34.7707	2.41673
$208$	−0.0257389	−0.00178467
$209$	0.588752	0.0407248
$210$	4.09337	0.282469
$211$	−14.6000	−1.00511	−0.502554	−	0.864546i	$-0.667606\pi$
$211$	−14.6000	−1.00511	−0.502554	+	0.864546i	$0.667606\pi$
$212$	−17.2596	−1.18539
$213$	−10.4065	−0.713039
$214$	26.9987	1.84559
$215$	7.06722	0.481980
$216$	0.596952	0.0406174
$217$	7.35538	0.499316
$218$	−14.4426	−0.978177
$219$	−43.8456	−2.96281
$220$	1.07333	0.0723640
$221$	−0.0181237	−0.00121913
$222$	0	0
$223$	15.9258	1.06647	0.533236	−	0.845967i	$-0.320976\pi$
$223$	15.9258	1.06647	0.533236	+	0.845967i	$0.320976\pi$
$224$	−6.16084	−0.411639
$225$	4.01883	0.267922
$226$	8.62694	0.573855
$227$	−14.9021	−0.989085	−0.494543	−	0.869153i	$-0.664664\pi$
$227$	−14.9021	−0.989085	−0.494543	+	0.869153i	$0.664664\pi$
$228$	5.17916	0.342998
$229$	−6.39246	−0.422426	−0.211213	−	0.977440i	$-0.567741\pi$
$229$	−6.39246	−0.422426	−0.211213	+	0.977440i	$0.567741\pi$
$230$	−17.0596	−1.12487
$231$	1.18031	0.0776587
$232$	−0.899359	−0.0590458
$233$	−11.8008	−0.773094	−0.386547	−	0.922270i	$-0.626332\pi$
$233$	−11.8008	−0.773094	−0.386547	+	0.922270i	$0.626332\pi$
$234$	0.0484262	0.00316572
$235$	−5.52310	−0.360288
$236$	−17.4516	−1.13600
$237$	28.8863	1.87637
$238$	−4.58214	−0.297016
$239$	3.82917	0.247688	0.123844	−	0.992302i	$-0.460478\pi$
$239$	3.82917	0.247688	0.123844	+	0.992302i	$0.460478\pi$
$240$	−11.1582	−0.720259
$241$	19.3203	1.24453	0.622266	−	0.782806i	$-0.286212\pi$
$241$	19.3203	1.24453	0.622266	+	0.782806i	$0.286212\pi$
$242$	−21.0520	−1.35327
$243$	21.0938	1.35317
$244$	18.8468	1.20654
$245$	6.38597	0.407985
$246$	21.5831	1.37609
$247$	−0.00632834	−0.000402663 0
$248$	−2.07595	−0.131823
$249$	−2.28127	−0.144570
$250$	−1.97176	−0.124705
$251$	0.889477	0.0561433	0.0280716	−	0.999606i	$-0.491063\pi$
$251$	0.889477	0.0561433	0.0280716	+	0.999606i	$0.491063\pi$
$252$	5.94508	0.374505
$253$	−4.91907	−0.309259
$254$	38.5515	2.41893
$255$	−7.85691	−0.492019
$256$	17.6410	1.10256
$257$	−14.2509	−0.888947	−0.444474	−	0.895792i	$-0.646609\pi$
$257$	−14.2509	−0.888947	−0.444474	+	0.895792i	$0.646609\pi$
$258$	36.9177	2.29840
$259$	0	0
$260$	−0.0115370	−0.000715492 0
$261$	16.3428	1.01159
$262$	−18.8381	−1.16382
$263$	−24.7839	−1.52824	−0.764120	−	0.645074i	$-0.776826\pi$
$263$	−24.7839	−1.52824	−0.764120	+	0.645074i	$0.776826\pi$
$264$	−0.333126	−0.0205025
$265$	9.14251	0.561620
$266$	−1.59997	−0.0981003
$267$	−0.761012	−0.0465732
$268$	−21.4403	−1.30967
$269$	−9.55497	−0.582577	−0.291289	−	0.956635i	$-0.594084\pi$
$269$	−9.55497	−0.582577	−0.291289	+	0.956635i	$0.594084\pi$
$270$	5.32215	0.323895
$271$	8.47437	0.514781	0.257391	−	0.966307i	$-0.417137\pi$
$271$	8.47437	0.514781	0.257391	+	0.966307i	$0.417137\pi$
$272$	12.4906	0.757351
$273$	−0.0126868	−0.000767843 0
$274$	−10.1142	−0.611021
$275$	−0.568551	−0.0342849
$276$	−43.2723	−2.60469
$277$	−29.2731	−1.75885	−0.879426	−	0.476036i	$-0.842073\pi$
$277$	−29.2731	−1.75885	−0.879426	+	0.476036i	$0.842073\pi$
$278$	−20.3900	−1.22291
$279$	37.7234	2.25844
$280$	0.173301	0.0103567
$281$	14.5887	0.870287	0.435143	−	0.900361i	$-0.356698\pi$
$281$	14.5887	0.870287	0.435143	+	0.900361i	$0.356698\pi$
$282$	−28.8516	−1.71809
$283$	−18.5933	−1.10525	−0.552627	−	0.833429i	$-0.686375\pi$
$283$	−18.5933	−1.10525	−0.552627	+	0.833429i	$0.686375\pi$
$284$	7.41541	0.440024
$285$	−2.74344	−0.162507
$286$	−0.00685095	−0.000405105 0
$287$	−3.23758	−0.191108
$288$	−31.5970	−1.86187
$289$	−8.20493	−0.482643
$290$	−8.01827	−0.470849
$291$	−11.4729	−0.672551
$292$	31.2434	1.82838
$293$	5.80288	0.339008	0.169504	−	0.985530i	$-0.445783\pi$
$293$	5.80288	0.339008	0.169504	+	0.985530i	$0.445783\pi$
$294$	33.3590	1.94554
$295$	9.24422	0.538219
$296$	0	0
$297$	1.53462	0.0890480
$298$	−21.4485	−1.24248
$299$	0.0528738	0.00305777
$300$	−5.00146	−0.288759
$301$	−5.53787	−0.319198
$302$	0.0304027	0.00174948
$303$	14.3201	0.822668
$304$	4.36139	0.250143
$305$	−9.98328	−0.571641
$306$	−23.5003	−1.34342
$307$	−21.6882	−1.23781	−0.618907	−	0.785464i	$-0.712424\pi$
$307$	−21.6882	−1.23781	−0.618907	+	0.785464i	$0.712424\pi$
$308$	−0.841062	−0.0479240
$309$	32.5209	1.85005
$310$	−18.5082	−1.05120
$311$	−7.02589	−0.398402	−0.199201	−	0.979959i	$-0.563835\pi$
$311$	−7.02589	−0.398402	−0.199201	+	0.979959i	$0.563835\pi$
$312$	0.00358068	0.000202716 0
$313$	23.7710	1.34362	0.671808	−	0.740725i	$-0.265518\pi$
$313$	23.7710	1.34362	0.671808	+	0.740725i	$0.265518\pi$
$314$	−27.8650	−1.57251
$315$	−3.14915	−0.177435
$316$	−20.5837	−1.15792
$317$	23.1107	1.29802	0.649012	−	0.760778i	$-0.275182\pi$
$317$	23.1107	1.29802	0.649012	+	0.760778i	$0.275182\pi$
$318$	47.7586	2.67817
$319$	−2.31204	−0.129450
$320$	7.07894	0.395725
$321$	−36.2761	−2.02473
$322$	13.3679	0.744962
$323$	3.07102	0.170876
$324$	−9.26080	−0.514489
$325$	0.00611121	0.000338989 0
$326$	−41.9861	−2.32539
$327$	19.4055	1.07312
$328$	0.913762	0.0504541
$329$	4.32790	0.238605
$330$	−2.96999	−0.163493
$331$	10.4669	0.575314	0.287657	−	0.957734i	$-0.407124\pi$
$331$	10.4669	0.575314	0.287657	+	0.957734i	$0.407124\pi$
$332$	1.62558	0.0892154
$333$	0	0
$334$	15.7176	0.860029
$335$	11.3571	0.620503
$336$	8.74357	0.477001
$337$	−10.5341	−0.573831	−0.286915	−	0.957956i	$-0.592630\pi$
$337$	−10.5341	−0.573831	−0.286915	+	0.957956i	$0.592630\pi$
$338$	−25.6328	−1.39424
$339$	−11.5914	−0.629557
$340$	5.59866	0.303630
$341$	−5.33680	−0.289004
$342$	−8.20571	−0.443714
$343$	−10.4892	−0.566366
$344$	1.56299	0.0842705
$345$	22.9217	1.23406
$346$	−29.4057	−1.58086
$347$	21.5633	1.15758	0.578789	−	0.815478i	$-0.303526\pi$
$347$	21.5633	1.15758	0.578789	+	0.815478i	$0.303526\pi$
$348$	−20.3387	−1.09027
$349$	10.5797	0.566320	0.283160	−	0.959073i	$-0.408617\pi$
$349$	10.5797	0.566320	0.283160	+	0.959073i	$0.408617\pi$
$350$	1.54507	0.0825875
$351$	−0.0164953	−0.000880454 0
$352$	4.47008	0.238256
$353$	−10.9952	−0.585216	−0.292608	−	0.956233i	$-0.594523\pi$
$353$	−10.9952	−0.585216	−0.292608	+	0.956233i	$0.594523\pi$
$354$	48.2899	2.56658
$355$	−3.92800	−0.208476
$356$	0.542280	0.0287408
$357$	6.15668	0.325846
$358$	15.2585	0.806439
$359$	−34.6984	−1.83131	−0.915657	−	0.401961i	$-0.868329\pi$
$359$	−34.6984	−1.83131	−0.915657	+	0.401961i	$0.868329\pi$
$360$	0.888803	0.0468441
$361$	−17.9277	−0.943562
$362$	25.9402	1.36339
$363$	28.2860	1.48463
$364$	0.00904036	0.000473844 0
$365$	−16.5499	−0.866259
$366$	−52.1506	−2.72596
$367$	−11.5572	−0.603279	−0.301639	−	0.953422i	$-0.597534\pi$
$367$	−11.5572	−0.603279	−0.301639	+	0.953422i	$0.597534\pi$
$368$	−36.4398	−1.89956
$369$	−16.6045	−0.864396
$370$	0	0
$371$	−7.16407	−0.371940
$372$	−46.9470	−2.43409
$373$	10.2547	0.530970	0.265485	−	0.964115i	$-0.414468\pi$
$373$	10.2547	0.530970	0.265485	+	0.964115i	$0.414468\pi$
$374$	3.32463	0.171912
$375$	2.64931	0.136810
$376$	−1.22149	−0.0629935
$377$	0.0248516	0.00127992
$378$	−4.17043	−0.214504
$379$	5.35858	0.275252	0.137626	−	0.990484i	$-0.456053\pi$
$379$	5.35858	0.275252	0.137626	+	0.990484i	$0.456053\pi$
$380$	1.95491	0.100285
$381$	−51.7987	−2.65373
$382$	−45.8357	−2.34516
$383$	23.2604	1.18855	0.594276	−	0.804261i	$-0.297439\pi$
$383$	23.2604	1.18855	0.594276	+	0.804261i	$0.297439\pi$
$384$	−4.67999	−0.238825
$385$	0.445516	0.0227056
$386$	−10.6122	−0.540145
$387$	−28.4019	−1.44375
$388$	8.17530	0.415038
$389$	−17.2275	−0.873471	−0.436735	−	0.899590i	$-0.643865\pi$
$389$	−17.2275	−0.873471	−0.436735	+	0.899590i	$0.643865\pi$
$390$	0.0319237	0.00161652
$391$	−25.6586	−1.29761
$392$	1.41232	0.0713330
$393$	25.3114	1.27679
$394$	−1.45362	−0.0732322
$395$	10.9033	0.548606
$396$	−4.31353	−0.216763
$397$	30.2268	1.51704	0.758520	−	0.651650i	$-0.225923\pi$
$397$	30.2268	1.51704	0.758520	+	0.651650i	$0.225923\pi$
$398$	−6.94714	−0.348229
$399$	2.14976	0.107622
$400$	−4.21175	−0.210587
$401$	−12.0733	−0.602909	−0.301455	−	0.953480i	$-0.597472\pi$
$401$	−12.0733	−0.602909	−0.301455	+	0.953480i	$0.597472\pi$
$402$	59.3270	2.95896
$403$	0.0573639	0.00285750
$404$	−10.2042	−0.507677
$405$	4.90551	0.243757
$406$	6.28311	0.311826
$407$	0	0
$408$	−1.73763	−0.0860257
$409$	−0.909485	−0.0449711	−0.0224856	−	0.999747i	$-0.507158\pi$
$409$	−0.909485	−0.0449711	−0.0224856	+	0.999747i	$0.507158\pi$
$410$	8.14668	0.402336
$411$	13.5897	0.670330
$412$	−23.1737	−1.14169
$413$	−7.24377	−0.356442
$414$	68.5594	3.36951
$415$	−0.861082	−0.0422688
$416$	−0.0480478	−0.00235574
$417$	27.3966	1.34162
$418$	1.16088	0.0567804
$419$	5.44038	0.265780	0.132890	−	0.991131i	$-0.457574\pi$
$419$	5.44038	0.265780	0.132890	+	0.991131i	$0.457574\pi$
$420$	3.91914	0.191234
$421$	−32.1650	−1.56762	−0.783812	−	0.620998i	$-0.786727\pi$
$421$	−32.1650	−1.56762	−0.783812	+	0.620998i	$0.786727\pi$
$422$	−28.7878	−1.40137
$423$	22.1964	1.07923
$424$	2.02196	0.0981949
$425$	−2.96565	−0.143855
$426$	−20.5190	−0.994151
$427$	7.82289	0.378576
$428$	25.8495	1.24948
$429$	0.00920511	0.000444427 0
$430$	13.9349	0.671999
$431$	−2.57507	−0.124037	−0.0620184	−	0.998075i	$-0.519754\pi$
$431$	−2.57507	−0.124037	−0.0620184	+	0.998075i	$0.519754\pi$
$432$	11.3683	0.546957
$433$	32.5699	1.56521	0.782605	−	0.622519i	$-0.213891\pi$
$433$	32.5699	1.56521	0.782605	+	0.622519i	$0.213891\pi$
$434$	14.5031	0.696169
$435$	10.7735	0.516552
$436$	−13.8279	−0.662236
$437$	−8.95935	−0.428584
$438$	−86.4531	−4.13089
$439$	−38.0631	−1.81665	−0.908326	−	0.418262i	$-0.862639\pi$
$439$	−38.0631	−1.81665	−0.908326	+	0.418262i	$0.862639\pi$
$440$	−0.125741	−0.00599445
$441$	−25.6641	−1.22210
$442$	−0.0357356	−0.00169977
$443$	−9.79268	−0.465264	−0.232632	−	0.972565i	$-0.574734\pi$
$443$	−9.79268	−0.465264	−0.232632	+	0.972565i	$0.574734\pi$
$444$	0	0
$445$	−0.287249	−0.0136169
$446$	31.4019	1.48692
$447$	28.8188	1.36308
$448$	−5.54706	−0.262074
$449$	−12.2877	−0.579891	−0.289945	−	0.957043i	$-0.593637\pi$
$449$	−12.2877	−0.579891	−0.289945	+	0.957043i	$0.593637\pi$
$450$	7.92416	0.373549
$451$	2.34907	0.110613
$452$	8.25975	0.388506
$453$	−0.0408498	−0.00191929
$454$	−29.3833	−1.37903
$455$	−0.00478874	−0.000224500 0
$456$	−0.606739	−0.0284131
$457$	11.2094	0.524356	0.262178	−	0.965020i	$-0.415559\pi$
$457$	11.2094	0.524356	0.262178	+	0.965020i	$0.415559\pi$
$458$	−12.6044	−0.588965
$459$	8.00484	0.373634
$460$	−16.3335	−0.761551
$461$	−22.0191	−1.02553	−0.512767	−	0.858528i	$-0.671379\pi$
$461$	−22.0191	−1.02553	−0.512767	+	0.858528i	$0.671379\pi$
$462$	2.32729	0.108275
$463$	−20.9473	−0.973504	−0.486752	−	0.873540i	$-0.661818\pi$
$463$	−20.9473	−0.973504	−0.486752	+	0.873540i	$0.661818\pi$
$464$	−17.1273	−0.795115
$465$	24.8681	1.15323
$466$	−23.2683	−1.07788
$467$	1.71804	0.0795016	0.0397508	−	0.999210i	$-0.487344\pi$
$467$	1.71804	0.0795016	0.0397508	+	0.999210i	$0.487344\pi$
$468$	0.0463651	0.00214323
$469$	−8.89940	−0.410936
$470$	−10.8902	−0.502329
$471$	37.4401	1.72515
$472$	2.04445	0.0941034
$473$	4.01808	0.184751
$474$	56.9568	2.61611
$475$	−1.03553	−0.0475134
$476$	−4.38711	−0.201083
$477$	−36.7422	−1.68231
$478$	7.55020	0.345338
$479$	33.0337	1.50935	0.754674	−	0.656100i	$-0.227795\pi$
$479$	33.0337	1.50935	0.754674	+	0.656100i	$0.227795\pi$
$480$	−20.8295	−0.950731
$481$	0	0
$482$	38.0951	1.73518
$483$	−17.9614	−0.817272
$484$	−20.1560	−0.916180
$485$	−4.33052	−0.196639
$486$	41.5918	1.88664
$487$	−13.4557	−0.609734	−0.304867	−	0.952395i	$-0.598612\pi$
$487$	−13.4557	−0.609734	−0.304867	+	0.952395i	$0.598612\pi$
$488$	−2.20790	−0.0999470
$489$	56.4136	2.55111
$490$	12.5916	0.568831
$491$	13.8635	0.625650	0.312825	−	0.949811i	$-0.398725\pi$
$491$	13.8635	0.625650	0.312825	+	0.949811i	$0.398725\pi$
$492$	20.6644	0.931624
$493$	−12.0600	−0.543154
$494$	−0.0124780	−0.000561411 0
$495$	2.28491	0.102699
$496$	−39.5342	−1.77514
$497$	3.07798	0.138066
$498$	−4.49812	−0.201565
$499$	−32.2091	−1.44188	−0.720940	−	0.692998i	$-0.756290\pi$
$499$	−32.2091	−1.44188	−0.720940	+	0.692998i	$0.756290\pi$
$500$	−1.88784	−0.0844266
$501$	−21.1186	−0.943509
$502$	1.75383	0.0782775
$503$	−38.1050	−1.69902	−0.849510	−	0.527573i	$-0.823102\pi$
$503$	−38.1050	−1.69902	−0.849510	+	0.527573i	$0.823102\pi$
$504$	−0.696466	−0.0310231
$505$	5.40523	0.240529
$506$	−9.69923	−0.431183
$507$	34.4409	1.52957
$508$	36.9106	1.63764
$509$	40.4853	1.79448	0.897240	−	0.441544i	$-0.145569\pi$
$509$	40.4853	1.79448	0.897240	+	0.441544i	$0.145569\pi$
$510$	−15.4919	−0.685995
$511$	12.9685	0.573691
$512$	31.2508	1.38110
$513$	2.79509	0.123406
$514$	−28.0994	−1.23941
$515$	12.2753	0.540913
$516$	35.3464	1.55604
$517$	−3.14017	−0.138104
$518$	0	0
$519$	39.5102	1.73431
$520$	0.00135155	5.92696e−5 0
$521$	1.83900	0.0805683	0.0402841	−	0.999188i	$-0.487174\pi$
$521$	1.83900	0.0805683	0.0402841	+	0.999188i	$0.487174\pi$
$522$	32.2240	1.41041
$523$	−12.2306	−0.534807	−0.267403	−	0.963585i	$-0.586166\pi$
$523$	−12.2306	−0.534807	−0.267403	+	0.963585i	$0.586166\pi$
$524$	−18.0363	−0.787919
$525$	−2.07600	−0.0906039
$526$	−48.8679	−2.13074
$527$	−27.8375	−1.21262
$528$	−6.34401	−0.276088
$529$	51.8561	2.25461
$530$	18.0268	0.783036
$531$	−37.1509	−1.61221
$532$	−1.53187	−0.0664149
$533$	−0.0252496	−0.00109368
$534$	−1.50053	−0.0649344
$535$	−13.6927	−0.591986
$536$	2.51173	0.108490
$537$	−20.5018	−0.884717
$538$	−18.8401	−0.812255
$539$	3.63075	0.156388
$540$	5.09562	0.219281
$541$	9.85561	0.423726	0.211863	−	0.977299i	$-0.432047\pi$
$541$	9.85561	0.423726	0.211863	+	0.977299i	$0.432047\pi$
$542$	16.7094	0.717731
$543$	−34.8540	−1.49573
$544$	23.3166	0.999692
$545$	7.32473	0.313757
$546$	−0.0250154	−0.00107056
$547$	22.7670	0.973447	0.486723	−	0.873556i	$-0.338192\pi$
$547$	22.7670	0.973447	0.486723	+	0.873556i	$0.338192\pi$
$548$	−9.68371	−0.413667
$549$	40.1211	1.71233
$550$	−1.12105	−0.0478016
$551$	−4.21104	−0.179396
$552$	5.06935	0.215766
$553$	−8.54385	−0.363321
$554$	−57.7196	−2.45227
$555$	0	0
$556$	−19.5222	−0.827924
$557$	43.6643	1.85012	0.925058	−	0.379825i	$-0.124016\pi$
$557$	43.6643	1.85012	0.925058	+	0.379825i	$0.124016\pi$
$558$	74.3814	3.14882
$559$	−0.0431893	−0.00182671
$560$	3.30032	0.139464
$561$	−4.46706	−0.188599
$562$	28.7653	1.21339
$563$	−24.4057	−1.02858	−0.514288	−	0.857617i	$-0.671944\pi$
$563$	−24.4057	−1.02858	−0.514288	+	0.857617i	$0.671944\pi$
$564$	−27.6236	−1.16316
$565$	−4.37525	−0.184068
$566$	−36.6614	−1.54099
$567$	−3.84396	−0.161431
$568$	−0.868715	−0.0364505
$569$	−41.3282	−1.73257	−0.866285	−	0.499550i	$-0.833499\pi$
$569$	−41.3282	−1.73257	−0.866285	+	0.499550i	$0.833499\pi$
$570$	−5.40940	−0.226575
$571$	18.3668	0.768625	0.384313	−	0.923203i	$-0.374438\pi$
$571$	18.3668	0.768625	0.384313	+	0.923203i	$0.374438\pi$
$572$	−0.00655935	−0.000274260 0
$573$	61.5860	2.57279
$574$	−6.38374	−0.266452
$575$	8.65194	0.360811
$576$	−28.4490	−1.18538
$577$	−19.7486	−0.822146	−0.411073	−	0.911602i	$-0.634846\pi$
$577$	−19.7486	−0.822146	−0.411073	+	0.911602i	$0.634846\pi$
$578$	−16.1781	−0.672922
$579$	14.2588	0.592575
$580$	−7.67699	−0.318770
$581$	0.674744	0.0279931
$582$	−22.6217	−0.937701
$583$	5.19798	0.215278
$584$	−3.66016	−0.151459
$585$	−0.0245599	−0.00101543
$586$	11.4419	0.472660
$587$	32.9271	1.35905	0.679524	−	0.733654i	$-0.262187\pi$
$587$	32.9271	1.35905	0.679524	+	0.733654i	$0.262187\pi$
$588$	31.9392	1.31715
$589$	−9.72017	−0.400513
$590$	18.2274	0.750409
$591$	1.95312	0.0803405
$592$	0	0
$593$	16.7712	0.688712	0.344356	−	0.938839i	$-0.388097\pi$
$593$	16.7712	0.688712	0.344356	+	0.938839i	$0.388097\pi$
$594$	3.02591	0.124155
$595$	2.32388	0.0952698
$596$	−20.5356	−0.841172
$597$	9.33436	0.382030
$598$	0.104255	0.00426328
$599$	11.5132	0.470417	0.235209	−	0.971945i	$-0.424423\pi$
$599$	11.5132	0.470417	0.235209	+	0.971945i	$0.424423\pi$
$600$	0.585920	0.0239201
$601$	9.35707	0.381683	0.190841	−	0.981621i	$-0.438878\pi$
$601$	9.35707	0.381683	0.190841	+	0.981621i	$0.438878\pi$
$602$	−10.9194	−0.445040
$603$	−45.6421	−1.85869
$604$	0.0291086	0.00118441
$605$	10.6767	0.434072
$606$	28.2358	1.14700
$607$	−8.05240	−0.326837	−0.163418	−	0.986557i	$-0.552252\pi$
$607$	−8.05240	−0.326837	−0.163418	+	0.986557i	$0.552252\pi$
$608$	8.14158	0.330185
$609$	−8.44215	−0.342093
$610$	−19.6846	−0.797007
$611$	0.0337529	0.00136549
$612$	−22.5000	−0.909510
$613$	3.42028	0.138144	0.0690720	−	0.997612i	$-0.477996\pi$
$613$	3.42028	0.138144	0.0690720	+	0.997612i	$0.477996\pi$
$614$	−42.7640	−1.72581
$615$	−10.9461	−0.441389
$616$	0.0985304	0.00396990
$617$	−17.8938	−0.720376	−0.360188	−	0.932880i	$-0.617287\pi$
$617$	−17.8938	−0.720376	−0.360188	+	0.932880i	$0.617287\pi$
$618$	64.1235	2.57943
$619$	1.02595	0.0412363	0.0206182	−	0.999787i	$-0.493437\pi$
$619$	1.02595	0.0412363	0.0206182	+	0.999787i	$0.493437\pi$
$620$	−17.7205	−0.711671
$621$	−23.3532	−0.937132
$622$	−13.8534	−0.555469
$623$	0.225089	0.00901798
$624$	0.0681902	0.00272979
$625$	1.00000	0.0400000
$626$	46.8707	1.87333
$627$	−1.55978	−0.0622918
$628$	−26.6790	−1.06461
$629$	0	0
$630$	−6.20937	−0.247387
$631$	−45.6234	−1.81624	−0.908119	−	0.418711i	$-0.862482\pi$
$631$	−45.6234	−1.81624	−0.908119	+	0.418711i	$0.862482\pi$
$632$	2.41138	0.0959196
$633$	38.6800	1.53739
$634$	45.5687	1.80976
$635$	−19.5518	−0.775890
$636$	45.7259	1.81315
$637$	−0.0390260	−0.00154627
$638$	−4.55879	−0.180484
$639$	15.7859	0.624482
$640$	−1.76650	−0.0698269
$641$	7.98108	0.315234	0.157617	−	0.987500i	$-0.449619\pi$
$641$	7.98108	0.315234	0.157617	+	0.987500i	$0.449619\pi$
$642$	−71.5278	−2.82298
$643$	11.7007	0.461432	0.230716	−	0.973021i	$-0.425893\pi$
$643$	11.7007	0.461432	0.230716	+	0.973021i	$0.425893\pi$
$644$	12.7989	0.504347
$645$	−18.7232	−0.737227
$646$	6.05531	0.238243
$647$	−2.01865	−0.0793615	−0.0396807	−	0.999212i	$-0.512634\pi$
$647$	−2.01865	−0.0793615	−0.0396807	+	0.999212i	$0.512634\pi$
$648$	1.08490	0.0426190
$649$	5.25581	0.206309
$650$	0.0120498	0.000472634 0
$651$	−19.4867	−0.763743
$652$	−40.1991	−1.57432
$653$	31.0085	1.21346	0.606728	−	0.794910i	$-0.292482\pi$
$653$	31.0085	1.21346	0.606728	+	0.794910i	$0.292482\pi$
$654$	38.2629	1.49620
$655$	9.55395	0.373304
$656$	17.4016	0.679418
$657$	66.5110	2.59484
$658$	8.53359	0.332674
$659$	−12.9866	−0.505885	−0.252943	−	0.967481i	$-0.581398\pi$
$659$	−12.9866	−0.505885	−0.252943	+	0.967481i	$0.581398\pi$
$660$	−2.84358	−0.110686
$661$	3.27374	0.127334	0.0636668	−	0.997971i	$-0.479720\pi$
$661$	3.27374	0.127334	0.0636668	+	0.997971i	$0.479720\pi$
$662$	20.6383	0.802128
$663$	0.0480152	0.00186476
$664$	−0.190437	−0.00739038
$665$	0.811441	0.0314663
$666$	0	0
$667$	35.1836	1.36231
$668$	15.0486	0.582249
$669$	−42.1924	−1.63125
$670$	22.3934	0.865133
$671$	−5.67600	−0.219120
$672$	16.3220	0.629633
$673$	23.3594	0.900441	0.450220	−	0.892918i	$-0.351345\pi$
$673$	23.3594	0.900441	0.450220	+	0.892918i	$0.351345\pi$
$674$	−20.7708	−0.800060
$675$	−2.69919	−0.103892
$676$	−24.5418	−0.943915
$677$	−15.6182	−0.600255	−0.300127	−	0.953899i	$-0.597029\pi$
$677$	−15.6182	−0.600255	−0.300127	+	0.953899i	$0.597029\pi$
$678$	−22.8554	−0.877756
$679$	3.39339	0.130226
$680$	−0.655883	−0.0251519
$681$	39.4802	1.51288
$682$	−10.5229	−0.402942
$683$	20.2355	0.774290	0.387145	−	0.922019i	$-0.373461\pi$
$683$	20.2355	0.774290	0.387145	+	0.922019i	$0.373461\pi$
$684$	−7.85645	−0.300399
$685$	5.12953	0.195989
$686$	−20.6823	−0.789653
$687$	16.9356	0.646133
$688$	29.7653	1.13479
$689$	−0.0558718	−0.00212855
$690$	45.1960	1.72058
$691$	−6.61621	−0.251693	−0.125846	−	0.992050i	$-0.540165\pi$
$691$	−6.61621	−0.251693	−0.125846	+	0.992050i	$0.540165\pi$
$692$	−28.1541	−1.07026
$693$	−1.79045	−0.0680137
$694$	42.5176	1.61395
$695$	10.3410	0.392258
$696$	2.38268	0.0903151
$697$	12.2531	0.464120
$698$	20.8607	0.789588
$699$	31.2638	1.18251
$700$	1.47931	0.0559126
$701$	36.0667	1.36222	0.681111	−	0.732180i	$-0.261497\pi$
$701$	36.0667	1.36222	0.681111	+	0.732180i	$0.261497\pi$
$702$	−0.0325248	−0.00122757
$703$	0	0
$704$	4.02474	0.151688
$705$	14.6324	0.551088
$706$	−21.6799	−0.815934
$707$	−4.23553	−0.159294
$708$	46.2346	1.73760
$709$	12.5403	0.470960	0.235480	−	0.971879i	$-0.424334\pi$
$709$	12.5403	0.470960	0.235480	+	0.971879i	$0.424334\pi$
$710$	−7.74506	−0.290667
$711$	−43.8186	−1.64333
$712$	−0.0635281	−0.00238081
$713$	81.2129	3.04145
$714$	12.1395	0.454309
$715$	0.00347454	0.000129940 0
$716$	14.6091	0.545968
$717$	−10.1446	−0.378859
$718$	−68.4170	−2.55330
$719$	−16.9908	−0.633651	−0.316825	−	0.948484i	$-0.602617\pi$
$719$	−16.9908	−0.633651	−0.316825	+	0.948484i	$0.602617\pi$
$720$	16.9263	0.630805
$721$	−9.61889	−0.358226
$722$	−35.3491	−1.31556
$723$	−51.1855	−1.90361
$724$	24.8362	0.923028
$725$	4.06655	0.151028
$726$	55.7732	2.06994
$727$	−22.9726	−0.852008	−0.426004	−	0.904721i	$-0.640079\pi$
$727$	−22.9726	−0.852008	−0.426004	+	0.904721i	$0.640079\pi$
$728$	−0.00105908	−3.92520e−5 0
$729$	−41.1673	−1.52472
$730$	−32.6323	−1.20778
$731$	20.9589	0.775193
$732$	−49.9309	−1.84550
$733$	6.20870	0.229323	0.114662	−	0.993405i	$-0.463422\pi$
$733$	6.20870	0.229323	0.114662	+	0.993405i	$0.463422\pi$
$734$	−22.7879	−0.841118
$735$	−16.9184	−0.624045
$736$	−68.0236	−2.50738
$737$	6.45707	0.237849
$738$	−32.7401	−1.20518
$739$	−9.96051	−0.366403	−0.183202	−	0.983075i	$-0.558646\pi$
$739$	−9.96051	−0.366403	−0.183202	+	0.983075i	$0.558646\pi$
$740$	0	0
$741$	0.0167657	0.000615904 0
$742$	−14.1258	−0.518575
$743$	35.1785	1.29057	0.645287	−	0.763941i	$-0.276738\pi$
$743$	35.1785	1.29057	0.645287	+	0.763941i	$0.276738\pi$
$744$	5.49984	0.201634
$745$	10.8779	0.398534
$746$	20.2199	0.740302
$747$	3.46054	0.126615
$748$	3.18312	0.116387
$749$	10.7296	0.392050
$750$	5.22380	0.190746
$751$	−37.7381	−1.37708	−0.688542	−	0.725197i	$-0.741749\pi$
$751$	−37.7381	−1.37708	−0.688542	+	0.725197i	$0.741749\pi$
$752$	−23.2619	−0.848275
$753$	−2.35650	−0.0858755
$754$	0.0490013	0.00178452
$755$	−0.0154190	−0.000561157 0
$756$	−3.99293	−0.145221
$757$	−2.38524	−0.0866930	−0.0433465	−	0.999060i	$-0.513802\pi$
$757$	−2.38524	−0.0866930	−0.0433465	+	0.999060i	$0.513802\pi$
$758$	10.5658	0.383769
$759$	13.0321	0.473036
$760$	−0.229018	−0.00830735
$761$	0.479949	0.0173981	0.00869907	−	0.999962i	$-0.497231\pi$
$761$	0.479949	0.0173981	0.00869907	+	0.999962i	$0.497231\pi$
$762$	−102.135	−3.69995
$763$	−5.73966	−0.207790
$764$	−43.8848	−1.58770
$765$	11.9184	0.430912
$766$	45.8640	1.65713
$767$	−0.0564933	−0.00203986
$768$	−46.7364	−1.68645
$769$	45.6865	1.64750	0.823748	−	0.566956i	$-0.191879\pi$
$769$	45.6865	1.64750	0.823748	+	0.566956i	$0.191879\pi$
$770$	0.878451	0.0316572
$771$	37.7550	1.35971
$772$	−10.1605	−0.365684
$773$	15.0962	0.542974	0.271487	−	0.962442i	$-0.412485\pi$
$773$	15.0962	0.542974	0.271487	+	0.962442i	$0.412485\pi$
$774$	−56.0018	−2.01294
$775$	9.38666	0.337179
$776$	−0.957736	−0.0343807
$777$	0	0
$778$	−33.9686	−1.21783
$779$	4.27848	0.153292
$780$	0.0305650	0.00109440
$781$	−2.23327	−0.0799125
$782$	−50.5926	−1.80919
$783$	−10.9764	−0.392264
$784$	26.8961	0.960575
$785$	14.1321	0.504395
$786$	49.9079	1.78016
$787$	20.9934	0.748334	0.374167	−	0.927361i	$-0.377929\pi$
$787$	20.9934	0.748334	0.374167	+	0.927361i	$0.377929\pi$
$788$	−1.39175	−0.0495790
$789$	65.6601	2.33756
$790$	21.4988	0.764891
$791$	3.42844	0.121901
$792$	0.505330	0.0179561
$793$	0.0610099	0.00216653
$794$	59.6000	2.11512
$795$	−24.2213	−0.859042
$796$	−6.65145	−0.235755
$797$	20.9652	0.742627	0.371314	−	0.928508i	$-0.378907\pi$
$797$	20.9652	0.742627	0.371314	+	0.928508i	$0.378907\pi$
$798$	4.23880	0.150052
$799$	−16.3796	−0.579468
$800$	−7.86223	−0.277972
$801$	1.15441	0.0407889
$802$	−23.8056	−0.840603
$803$	−9.40944	−0.332052
$804$	56.8019	2.00325
$805$	−6.77966	−0.238952
$806$	0.113108	0.00398405
$807$	25.3141	0.891097
$808$	1.19542	0.0420547
$809$	8.19866	0.288249	0.144125	−	0.989560i	$-0.453963\pi$
$809$	8.19866	0.288249	0.144125	+	0.989560i	$0.453963\pi$
$810$	9.67249	0.339857
$811$	27.5998	0.969158	0.484579	−	0.874747i	$-0.338973\pi$
$811$	27.5998	0.969158	0.484579	+	0.874747i	$0.338973\pi$
$812$	6.01569	0.211109
$813$	−22.4512	−0.787398
$814$	0	0
$815$	21.2937	0.745886
$816$	−33.0913	−1.15843
$817$	7.31832	0.256036
$818$	−1.79329	−0.0627008
$819$	0.0192451	0.000672479 0
$820$	7.79993	0.272386
$821$	−10.2184	−0.356626	−0.178313	−	0.983974i	$-0.557064\pi$
$821$	−10.2184	−0.356626	−0.178313	+	0.983974i	$0.557064\pi$
$822$	26.7956	0.934604
$823$	22.3010	0.777363	0.388681	−	0.921372i	$-0.372931\pi$
$823$	22.3010	0.777363	0.388681	+	0.921372i	$0.372931\pi$
$824$	2.71480	0.0945744
$825$	1.50627	0.0524415
$826$	−14.2830	−0.496968
$827$	42.7428	1.48631	0.743157	−	0.669117i	$-0.233328\pi$
$827$	42.7428	1.48631	0.743157	+	0.669117i	$0.233328\pi$
$828$	65.6413	2.28119
$829$	9.55795	0.331961	0.165981	−	0.986129i	$-0.446921\pi$
$829$	9.55795	0.331961	0.165981	+	0.986129i	$0.446921\pi$
$830$	−1.69785	−0.0589331
$831$	77.5535	2.69030
$832$	−0.0432609	−0.00149980
$833$	18.9385	0.656182
$834$	54.0194	1.87054
$835$	−7.97136	−0.275860
$836$	1.11147	0.0384409
$837$	−25.3363	−0.875752
$838$	10.7271	0.370562
$839$	32.9053	1.13602	0.568009	−	0.823022i	$-0.307714\pi$
$839$	32.9053	1.13602	0.568009	+	0.823022i	$0.307714\pi$
$840$	−0.459127	−0.0158414
$841$	−12.4631	−0.429763
$842$	−63.4216	−2.18565
$843$	−38.6498	−1.33117
$844$	−27.5625	−0.948740
$845$	13.0000	0.447212
$846$	43.7660	1.50471
$847$	−8.36630	−0.287469
$848$	38.5059	1.32230
$849$	49.2592	1.69057
$850$	−5.84755	−0.200569
$851$	0	0
$852$	−19.6457	−0.673051
$853$	−24.1424	−0.826621	−0.413310	−	0.910590i	$-0.635628\pi$
$853$	−24.1424	−0.826621	−0.413310	+	0.910590i	$0.635628\pi$
$854$	15.4249	0.527828
$855$	4.16162	0.142324
$856$	−3.02827	−0.103504
$857$	42.5290	1.45276	0.726382	−	0.687292i	$-0.241200\pi$
$857$	42.5290	1.45276	0.726382	+	0.687292i	$0.241200\pi$
$858$	0.0181503	0.000619640 0
$859$	−51.3470	−1.75194	−0.875968	−	0.482370i	$-0.839776\pi$
$859$	−51.3470	−1.75194	−0.875968	+	0.482370i	$0.839776\pi$
$860$	13.3418	0.454950
$861$	8.57735	0.292315
$862$	−5.07742	−0.172938
$863$	−49.5747	−1.68754	−0.843772	−	0.536702i	$-0.819670\pi$
$863$	−49.5747	−1.68754	−0.843772	+	0.536702i	$0.819670\pi$
$864$	21.2216	0.721974
$865$	14.9134	0.507072
$866$	64.2200	2.18228
$867$	21.7374	0.738240
$868$	13.8858	0.471313
$869$	6.19910	0.210290
$870$	21.2429	0.720200
$871$	−0.0694054	−0.00235171
$872$	1.61994	0.0548580
$873$	17.4036	0.589022
$874$	−17.6657	−0.597551
$875$	−0.783600	−0.0264905
$876$	−82.7734	−2.79665
$877$	27.7964	0.938618	0.469309	−	0.883034i	$-0.344503\pi$
$877$	27.7964	0.938618	0.469309	+	0.883034i	$0.344503\pi$
$878$	−75.0513	−2.53286
$879$	−15.3736	−0.518539
$880$	−2.39459	−0.0807217
$881$	49.8350	1.67898	0.839492	−	0.543372i	$-0.182853\pi$
$881$	49.8350	1.67898	0.839492	+	0.543372i	$0.182853\pi$
$882$	−50.6035	−1.70391
$883$	−37.0770	−1.24774	−0.623870	−	0.781528i	$-0.714440\pi$
$883$	−37.0770	−1.24774	−0.623870	+	0.781528i	$0.714440\pi$
$884$	−0.0342146	−0.00115076
$885$	−24.4908	−0.823248
$886$	−19.3088	−0.648692
$887$	37.5487	1.26076	0.630380	−	0.776286i	$-0.282899\pi$
$887$	37.5487	1.26076	0.630380	+	0.776286i	$0.282899\pi$
$888$	0	0
$889$	15.3208	0.513843
$890$	−0.566387	−0.0189853
$891$	2.78903	0.0934361
$892$	30.0653	1.00666
$893$	−5.71934	−0.191391
$894$	56.8237	1.90047
$895$	−7.73854	−0.258671
$896$	1.38423	0.0462438
$897$	−0.140079	−0.00467710
$898$	−24.2283	−0.808510
$899$	38.1714	1.27309
$900$	7.58689	0.252896
$901$	27.1135	0.903281
$902$	4.63180	0.154222
$903$	14.6715	0.488238
$904$	−0.967629	−0.0321829
$905$	−13.1559	−0.437316
$906$	−0.0805460	−0.00267596
$907$	32.0705	1.06488	0.532441	−	0.846467i	$-0.321275\pi$
$907$	32.0705	1.06488	0.532441	+	0.846467i	$0.321275\pi$
$908$	−28.1327	−0.933616
$909$	−21.7227	−0.720495
$910$	−0.00944225	−0.000313008 0
$911$	−26.0546	−0.863229	−0.431614	−	0.902058i	$-0.642056\pi$
$911$	−26.0546	−0.863229	−0.431614	+	0.902058i	$0.642056\pi$
$912$	−11.5547	−0.382613
$913$	−0.489569	−0.0162024
$914$	22.1023	0.731080
$915$	26.4488	0.874369
$916$	−12.0679	−0.398735
$917$	−7.48647	−0.247225
$918$	15.7836	0.520937
$919$	5.23024	0.172530	0.0862648	−	0.996272i	$-0.472507\pi$
$919$	5.23024	0.172530	0.0862648	+	0.996272i	$0.472507\pi$
$920$	1.91346	0.0630850
$921$	57.4588	1.89333
$922$	−43.4165	−1.42985
$923$	0.0240048	0.000790128 0
$924$	2.22823	0.0733034
$925$	0	0
$926$	−41.3031	−1.35730
$927$	−49.3322	−1.62028
$928$	−31.9722	−1.04954
$929$	−3.18876	−0.104620	−0.0523099	−	0.998631i	$-0.516658\pi$
$929$	−3.18876	−0.104620	−0.0523099	+	0.998631i	$0.516658\pi$
$930$	49.0340	1.60789
$931$	6.61287	0.216728
$932$	−22.2779	−0.729737
$933$	18.6137	0.609386
$934$	3.38757	0.110845
$935$	−1.68612	−0.0551421
$936$	−0.00543166	−0.000177540 0
$937$	54.4768	1.77968	0.889840	−	0.456273i	$-0.150816\pi$
$937$	54.4768	1.77968	0.889840	+	0.456273i	$0.150816\pi$
$938$	−17.5475	−0.572945
$939$	−62.9767	−2.05517
$940$	−10.4267	−0.340082
$941$	−44.2180	−1.44147	−0.720733	−	0.693213i	$-0.756194\pi$
$941$	−44.2180	−1.44147	−0.720733	+	0.693213i	$0.756194\pi$
$942$	73.8230	2.40528
$943$	−35.7471	−1.16408
$944$	38.9343	1.26720
$945$	2.11508	0.0688036
$946$	7.92268	0.257589
$947$	−11.0647	−0.359553	−0.179777	−	0.983707i	$-0.557537\pi$
$947$	−11.0647	−0.359553	−0.179777	+	0.983707i	$0.557537\pi$
$948$	54.5326	1.77114
$949$	0.101140	0.00328313
$950$	−2.04182	−0.0662453
$951$	−61.2272	−1.98543
$952$	0.513949	0.0166572
$953$	43.7259	1.41642	0.708210	−	0.706002i	$-0.249503\pi$
$953$	43.7259	1.41642	0.708210	+	0.706002i	$0.249503\pi$
$954$	−72.4467	−2.34555
$955$	23.2461	0.752226
$956$	7.22884	0.233798
$957$	6.12531	0.198003
$958$	65.1345	2.10440
$959$	−4.01950	−0.129796
$960$	−18.7543	−0.605292
$961$	57.1094	1.84224
$962$	0	0
$963$	55.0285	1.77327
$964$	36.4736	1.17474
$965$	5.38208	0.173255
$966$	−35.4156	−1.13948
$967$	29.0347	0.933692	0.466846	−	0.884339i	$-0.345390\pi$
$967$	29.0347	0.933692	0.466846	+	0.884339i	$0.345390\pi$
$968$	2.36127	0.0758941
$969$	−8.13607	−0.261368
$970$	−8.53874	−0.274162
$971$	16.3088	0.523374	0.261687	−	0.965153i	$-0.415721\pi$
$971$	16.3088	0.523374	0.261687	+	0.965153i	$0.415721\pi$
$972$	39.8216	1.27728
$973$	−8.10323	−0.259778
$974$	−26.5313	−0.850118
$975$	−0.0161905	−0.000518510 0
$976$	−42.0470	−1.34589
$977$	19.6114	0.627425	0.313712	−	0.949518i	$-0.398427\pi$
$977$	19.6114	0.627425	0.313712	+	0.949518i	$0.398427\pi$
$978$	111.234	3.55687
$979$	−0.163316	−0.00521960
$980$	12.0557	0.385104
$981$	−29.4368	−0.939846
$982$	27.3354	0.872309
$983$	−53.4039	−1.70332	−0.851660	−	0.524095i	$-0.824404\pi$
$983$	−53.4039	−1.70332	−0.851660	+	0.524095i	$0.824404\pi$
$984$	−2.42084	−0.0771734
$985$	0.737219	0.0234897
$986$	−23.7794	−0.757289
$987$	−11.4659	−0.364965
$988$	−0.0119469	−0.000380081 0
$989$	−61.1452	−1.94430
$990$	4.50529	0.143188
$991$	−47.7930	−1.51820	−0.759098	−	0.650976i	$-0.774360\pi$
$991$	−47.7930	−1.51820	−0.759098	+	0.650976i	$0.774360\pi$
$992$	−73.8001	−2.34316
$993$	−27.7301	−0.879988
$994$	6.06903	0.192498
$995$	3.52332	0.111697
$996$	−4.30666	−0.136462
$997$	−56.6250	−1.79333	−0.896667	−	0.442707i	$-0.854018\pi$
$997$	−56.6250	−1.79333	−0.896667	+	0.442707i	$0.854018\pi$
$998$	−63.5087	−2.01033
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6845.2.a.k.1.6		7
37.11	even	6		185.2.e.a.121.6	yes	14
37.27	even	6		185.2.e.a.26.6	✓	14
37.36	even	2		6845.2.a.l.1.2		7
185.27	odd	12		925.2.o.b.174.4		28
185.48	odd	12		925.2.o.b.824.4		28
185.64	even	6		925.2.e.c.26.2		14
185.122	odd	12		925.2.o.b.824.11		28
185.138	odd	12		925.2.o.b.174.11		28
185.159	even	6		925.2.e.c.676.2		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.e.a.26.6	✓	14	37.27	even	6
185.2.e.a.121.6	yes	14	37.11	even	6
925.2.e.c.26.2		14	185.64	even	6
925.2.e.c.676.2		14	185.159	even	6
925.2.o.b.174.4		28	185.27	odd	12
925.2.o.b.174.11		28	185.138	odd	12
925.2.o.b.824.4		28	185.48	odd	12
925.2.o.b.824.11		28	185.122	odd	12
6845.2.a.k.1.6		7	1.1	even	1	trivial
6845.2.a.l.1.2		7	37.36	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6845 = 5 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6845.a (trivial)

\(f(q)\)	\(=\)	\(q+1.97176 q^{2} -2.64931 q^{3} +1.88784 q^{4} -1.00000 q^{5} -5.22380 q^{6} +0.783600 q^{7} -0.221160 q^{8} +4.01883 q^{9} -1.97176 q^{10} -0.568551 q^{11} -5.00146 q^{12} +0.00611121 q^{13} +1.54507 q^{14} +2.64931 q^{15} -4.21175 q^{16} -2.96565 q^{17} +7.92416 q^{18} -1.03553 q^{19} -1.88784 q^{20} -2.07600 q^{21} -1.12105 q^{22} +8.65194 q^{23} +0.585920 q^{24} +1.00000 q^{25} +0.0120498 q^{26} -2.69919 q^{27} +1.47931 q^{28} +4.06655 q^{29} +5.22380 q^{30} +9.38666 q^{31} -7.86223 q^{32} +1.50627 q^{33} -5.84755 q^{34} -0.783600 q^{35} +7.58689 q^{36} -2.04182 q^{38} -0.0161905 q^{39} +0.221160 q^{40} -4.13168 q^{41} -4.09337 q^{42} -7.06722 q^{43} -1.07333 q^{44} -4.01883 q^{45} +17.0596 q^{46} +5.52310 q^{47} +11.1582 q^{48} -6.38597 q^{49} +1.97176 q^{50} +7.85691 q^{51} +0.0115370 q^{52} -9.14251 q^{53} -5.32215 q^{54} +0.568551 q^{55} -0.173301 q^{56} +2.74344 q^{57} +8.01827 q^{58} -9.24422 q^{59} +5.00146 q^{60} +9.98328 q^{61} +18.5082 q^{62} +3.14915 q^{63} -7.07894 q^{64} -0.00611121 q^{65} +2.96999 q^{66} -11.3571 q^{67} -5.59866 q^{68} -22.9217 q^{69} -1.54507 q^{70} +3.92800 q^{71} -0.888803 q^{72} +16.5499 q^{73} -2.64931 q^{75} -1.95491 q^{76} -0.445516 q^{77} -0.0319237 q^{78} -10.9033 q^{79} +4.21175 q^{80} -4.90551 q^{81} -8.14668 q^{82} +0.861082 q^{83} -3.91914 q^{84} +2.96565 q^{85} -13.9349 q^{86} -10.7735 q^{87} +0.125741 q^{88} +0.287249 q^{89} -7.92416 q^{90} +0.00478874 q^{91} +16.3335 q^{92} -24.8681 q^{93} +10.8902 q^{94} +1.03553 q^{95} +20.8295 q^{96} +4.33052 q^{97} -12.5916 q^{98} -2.28491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 8 q^{4} - 7 q^{5} + 2 q^{6} - 3 q^{8} + 13 q^{9} - q^{11} - 6 q^{12} + 4 q^{13} - 10 q^{14} - 2 q^{15} - 2 q^{16} - 3 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} - q^{21} + 7 q^{22} + 15 q^{24}+ \cdots - 56 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 8 * q^4 - 7 * q^5 + 2 * q^6 - 3 * q^8 + 13 * q^9 - q^11 - 6 * q^12 + 4 * q^13 - 10 * q^14 - 2 * q^15 - 2 * q^16 - 3 * q^17 - 6 * q^18 - 14 * q^19 - 8 * q^20 - q^21 + 7 * q^22 + 15 * q^24 + 7 * q^25 - 11 * q^26 - 7 * q^27 - 9 * q^28 + 6 * q^29 - 2 * q^30 + 4 * q^32 + 11 * q^33 - 33 * q^34 + 20 * q^36 - 2 * q^38 - 25 * q^39 + 3 * q^40 + 7 * q^41 - 29 * q^42 - 19 * q^43 + 19 * q^44 - 13 * q^45 + 14 * q^46 + 4 * q^47 + 18 * q^48 - 5 * q^49 - 29 * q^51 - 8 * q^52 + 6 * q^53 - 9 * q^54 + q^55 - 25 * q^56 - 44 * q^57 + 12 * q^58 + 2 * q^59 + 6 * q^60 - 4 * q^61 + 25 * q^62 + 5 * q^63 - 31 * q^64 - 4 * q^65 + q^66 + 8 * q^67 - 13 * q^68 - 11 * q^69 + 10 * q^70 - 33 * q^71 + 43 * q^72 + 2 * q^75 - 12 * q^76 + 5 * q^77 + 22 * q^78 - 9 * q^79 + 2 * q^80 + 35 * q^81 - 76 * q^82 + 11 * q^83 - 40 * q^84 + 3 * q^85 - 17 * q^86 - 29 * q^87 - 20 * q^88 - 12 * q^89 + 6 * q^90 - 24 * q^91 - 7 * q^92 - 47 * q^93 + 10 * q^94 + 14 * q^95 - 15 * q^96 - 15 * q^97 + 54 * q^98 - 56 * q^99

Introduction

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