LMFDB - Embedded newform 8027.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8027,2,Mod(1,8027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8027, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8027 = 23 \cdot 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8027.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:	$64.0959177025$
Analytic rank:	$0$
Dimension:	$176$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.81854 q^{2} -0.267983 q^{3} +5.94415 q^{4} -3.79404 q^{5} +0.755319 q^{6} -0.324133 q^{7} -11.1167 q^{8} -2.92819 q^{9} +O(q^{10})$	\(q-2.81854 q^{2} -0.267983 q^{3} +5.94415 q^{4} -3.79404 q^{5} +0.755319 q^{6} -0.324133 q^{7} -11.1167 q^{8} -2.92819 q^{9} +10.6936 q^{10} +1.85803 q^{11} -1.59293 q^{12} +5.15183 q^{13} +0.913581 q^{14} +1.01674 q^{15} +19.4446 q^{16} +0.693605 q^{17} +8.25320 q^{18} +1.82537 q^{19} -22.5523 q^{20} +0.0868621 q^{21} -5.23694 q^{22} +1.00000 q^{23} +2.97909 q^{24} +9.39471 q^{25} -14.5206 q^{26} +1.58865 q^{27} -1.92669 q^{28} -6.16549 q^{29} -2.86571 q^{30} -2.72225 q^{31} -32.5718 q^{32} -0.497921 q^{33} -1.95495 q^{34} +1.22977 q^{35} -17.4056 q^{36} +2.45632 q^{37} -5.14487 q^{38} -1.38060 q^{39} +42.1772 q^{40} -0.589626 q^{41} -0.244824 q^{42} -0.677822 q^{43} +11.0444 q^{44} +11.1096 q^{45} -2.81854 q^{46} +11.0837 q^{47} -5.21081 q^{48} -6.89494 q^{49} -26.4793 q^{50} -0.185874 q^{51} +30.6232 q^{52} -1.42941 q^{53} -4.47767 q^{54} -7.04945 q^{55} +3.60329 q^{56} -0.489168 q^{57} +17.3777 q^{58} +5.83870 q^{59} +6.04363 q^{60} +13.0099 q^{61} +7.67276 q^{62} +0.949122 q^{63} +52.9156 q^{64} -19.5462 q^{65} +1.40341 q^{66} -4.11044 q^{67} +4.12289 q^{68} -0.267983 q^{69} -3.46616 q^{70} -0.762409 q^{71} +32.5518 q^{72} -3.96717 q^{73} -6.92321 q^{74} -2.51762 q^{75} +10.8503 q^{76} -0.602250 q^{77} +3.89128 q^{78} +14.5022 q^{79} -73.7734 q^{80} +8.35882 q^{81} +1.66188 q^{82} +13.7861 q^{83} +0.516321 q^{84} -2.63156 q^{85} +1.91047 q^{86} +1.65224 q^{87} -20.6552 q^{88} +7.51916 q^{89} -31.3129 q^{90} -1.66988 q^{91} +5.94415 q^{92} +0.729516 q^{93} -31.2399 q^{94} -6.92552 q^{95} +8.72868 q^{96} +0.659690 q^{97} +19.4336 q^{98} -5.44067 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) $ 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * 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 $

$2$	−2.81854	−1.99301	−0.996503	−	0.0835574i	$-0.973372\pi$
$2$	−2.81854	−1.99301	−0.996503	+	0.0835574i	$0.973372\pi$
$3$	−0.267983	−0.154720	−0.0773600	−	0.997003i	$-0.524649\pi$
$3$	−0.267983	−0.154720	−0.0773600	+	0.997003i	$0.524649\pi$
$4$	5.94415	2.97207
$5$	−3.79404	−1.69674	−0.848372	−	0.529400i	$-0.822417\pi$
$5$	−3.79404	−1.69674	−0.848372	+	0.529400i	$0.822417\pi$
$6$	0.755319	0.308358
$7$	−0.324133	−0.122511	−0.0612554	−	0.998122i	$-0.519510\pi$
$7$	−0.324133	−0.122511	−0.0612554	+	0.998122i	$0.519510\pi$
$8$	−11.1167	−3.93035
$9$	−2.92819	−0.976062
$10$	10.6936	3.38162
$11$	1.85803	0.560218	0.280109	−	0.959968i	$-0.409629\pi$
$11$	1.85803	0.560218	0.280109	+	0.959968i	$0.409629\pi$
$12$	−1.59293	−0.459839
$13$	5.15183	1.42886	0.714431	−	0.699706i	$-0.246686\pi$
$13$	5.15183	1.42886	0.714431	+	0.699706i	$0.246686\pi$
$14$	0.913581	0.244165
$15$	1.01674	0.262520
$16$	19.4446	4.86114
$17$	0.693605	0.168224	0.0841120	−	0.996456i	$-0.473195\pi$
$17$	0.693605	0.168224	0.0841120	+	0.996456i	$0.473195\pi$
$18$	8.25320	1.94530
$19$	1.82537	0.418769	0.209384	−	0.977833i	$-0.432854\pi$
$19$	1.82537	0.418769	0.209384	+	0.977833i	$0.432854\pi$
$20$	−22.5523	−5.04285
$21$	0.0868621	0.0189549
$22$	−5.23694	−1.11652
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	2.97909	0.608104
$25$	9.39471	1.87894
$26$	−14.5206	−2.84773
$27$	1.58865	0.305736
$28$	−1.92669	−0.364111
$29$	−6.16549	−1.14490	−0.572451	−	0.819939i	$-0.694007\pi$
$29$	−6.16549	−1.14490	−0.572451	+	0.819939i	$0.694007\pi$
$30$	−2.86571	−0.523204
$31$	−2.72225	−0.488930	−0.244465	−	0.969658i	$-0.578612\pi$
$31$	−2.72225	−0.488930	−0.244465	+	0.969658i	$0.578612\pi$
$32$	−32.5718	−5.75793
$33$	−0.497921	−0.0866769
$34$	−1.95495	−0.335271
$35$	1.22977	0.207869
$36$	−17.4056	−2.90093
$37$	2.45632	0.403816	0.201908	−	0.979405i	$-0.435286\pi$
$37$	2.45632	0.403816	0.201908	+	0.979405i	$0.435286\pi$
$38$	−5.14487	−0.834608
$39$	−1.38060	−0.221073
$40$	42.1772	6.66880
$41$	−0.589626	−0.0920842	−0.0460421	−	0.998940i	$-0.514661\pi$
$41$	−0.589626	−0.0920842	−0.0460421	+	0.998940i	$0.514661\pi$
$42$	−0.244824	−0.0377771
$43$	−0.677822	−0.103367	−0.0516835	−	0.998664i	$-0.516459\pi$
$43$	−0.677822	−0.103367	−0.0516835	+	0.998664i	$0.516459\pi$
$44$	11.0444	1.66501
$45$	11.1096	1.65613
$46$	−2.81854	−0.415570
$47$	11.0837	1.61673	0.808364	−	0.588683i	$-0.200353\pi$
$47$	11.0837	1.61673	0.808364	+	0.588683i	$0.200353\pi$
$48$	−5.21081	−0.752116
$49$	−6.89494	−0.984991
$50$	−26.4793	−3.74474
$51$	−0.185874	−0.0260276
$52$	30.6232	4.24668
$53$	−1.42941	−0.196345	−0.0981726	−	0.995169i	$-0.531300\pi$
$53$	−1.42941	−0.196345	−0.0981726	+	0.995169i	$0.531300\pi$
$54$	−4.47767	−0.609334
$55$	−7.04945	−0.950547
$56$	3.60329	0.481510
$57$	−0.489168	−0.0647919
$58$	17.3777	2.28180
$59$	5.83870	0.760134	0.380067	−	0.924959i	$-0.375901\pi$
$59$	5.83870	0.760134	0.380067	+	0.924959i	$0.375901\pi$
$60$	6.04363	0.780229
$61$	13.0099	1.66574	0.832872	−	0.553466i	$-0.186695\pi$
$61$	13.0099	1.66574	0.832872	+	0.553466i	$0.186695\pi$
$62$	7.67276	0.974441
$63$	0.949122	0.119578
$64$	52.9156	6.61445
$65$	−19.5462	−2.42441
$66$	1.40341	0.172748
$67$	−4.11044	−0.502171	−0.251085	−	0.967965i	$-0.580787\pi$
$67$	−4.11044	−0.502171	−0.251085	+	0.967965i	$0.580787\pi$
$68$	4.12289	0.499974
$69$	−0.267983	−0.0322613
$70$	−3.46616	−0.414285
$71$	−0.762409	−0.0904814	−0.0452407	−	0.998976i	$-0.514405\pi$
$71$	−0.762409	−0.0904814	−0.0452407	+	0.998976i	$0.514405\pi$
$72$	32.5518	3.83627
$73$	−3.96717	−0.464322	−0.232161	−	0.972677i	$-0.574580\pi$
$73$	−3.96717	−0.464322	−0.232161	+	0.972677i	$0.574580\pi$
$74$	−6.92321	−0.804807
$75$	−2.51762	−0.290710
$76$	10.8503	1.24461
$77$	−0.602250	−0.0686328
$78$	3.89128	0.440600
$79$	14.5022	1.63162	0.815812	−	0.578317i	$-0.196290\pi$
$79$	14.5022	1.63162	0.815812	+	0.578317i	$0.196290\pi$
$80$	−73.7734	−8.24812
$81$	8.35882	0.928758
$82$	1.66188	0.183524
$83$	13.7861	1.51322	0.756608	−	0.653868i	$-0.226855\pi$
$83$	13.7861	1.51322	0.756608	+	0.653868i	$0.226855\pi$
$84$	0.516321	0.0563352
$85$	−2.63156	−0.285433
$86$	1.91047	0.206011
$87$	1.65224	0.177139
$88$	−20.6552	−2.20186
$89$	7.51916	0.797030	0.398515	−	0.917162i	$-0.369526\pi$
$89$	7.51916	0.797030	0.398515	+	0.917162i	$0.369526\pi$
$90$	−31.3129	−3.30067
$91$	−1.66988	−0.175051
$92$	5.94415	0.619720
$93$	0.729516	0.0756473
$94$	−31.2399	−3.22215
$95$	−6.92552	−0.710543
$96$	8.72868	0.890867
$97$	0.659690	0.0669814	0.0334907	−	0.999439i	$-0.489338\pi$
$97$	0.659690	0.0669814	0.0334907	+	0.999439i	$0.489338\pi$
$98$	19.4336	1.96309
$99$	−5.44067	−0.546808
$100$	55.8435	5.58435
$101$	−5.69095	−0.566270	−0.283135	−	0.959080i	$-0.591375\pi$
$101$	−5.69095	−0.566270	−0.283135	+	0.959080i	$0.591375\pi$
$102$	0.523893	0.0518732
$103$	−1.20101	−0.118339	−0.0591695	−	0.998248i	$-0.518845\pi$
$103$	−1.20101	−0.118339	−0.0591695	+	0.998248i	$0.518845\pi$
$104$	−57.2715	−5.61593
$105$	−0.329558	−0.0321615
$106$	4.02886	0.391317
$107$	−5.55369	−0.536896	−0.268448	−	0.963294i	$-0.586511\pi$
$107$	−5.55369	−0.536896	−0.268448	+	0.963294i	$0.586511\pi$
$108$	9.44317	0.908670
$109$	−5.86430	−0.561698	−0.280849	−	0.959752i	$-0.590616\pi$
$109$	−5.86430	−0.561698	−0.280849	+	0.959752i	$0.590616\pi$
$110$	19.8691	1.89445
$111$	−0.658250	−0.0624783
$112$	−6.30263	−0.595542
$113$	−0.943382	−0.0887459	−0.0443729	−	0.999015i	$-0.514129\pi$
$113$	−0.943382	−0.0887459	−0.0443729	+	0.999015i	$0.514129\pi$
$114$	1.37874	0.129131
$115$	−3.79404	−0.353796
$116$	−36.6486	−3.40273
$117$	−15.0855	−1.39466
$118$	−16.4566	−1.51495
$119$	−0.224820	−0.0206092
$120$	−11.3028	−1.03180
$121$	−7.54771	−0.686155
$122$	−36.6688	−3.31984
$123$	0.158010	0.0142473
$124$	−16.1814	−1.45314
$125$	−16.6737	−1.49134
$126$	−2.67513	−0.238320
$127$	−6.29139	−0.558270	−0.279135	−	0.960252i	$-0.590048\pi$
$127$	−6.29139	−0.558270	−0.279135	+	0.960252i	$0.590048\pi$
$128$	−84.0010	−7.42471
$129$	0.181645	0.0159929
$130$	55.0918	4.83187
$131$	6.29335	0.549852	0.274926	−	0.961465i	$-0.411347\pi$
$131$	6.29335	0.549852	0.274926	+	0.961465i	$0.411347\pi$
$132$	−2.95972	−0.257610
$133$	−0.591663	−0.0513037
$134$	11.5854	1.00083
$135$	−6.02740	−0.518756
$136$	−7.71061	−0.661180
$137$	6.84600	0.584893	0.292446	−	0.956282i	$-0.405531\pi$
$137$	6.84600	0.584893	0.292446	+	0.956282i	$0.405531\pi$
$138$	0.755319	0.0642970
$139$	−0.469084	−0.0397871	−0.0198936	−	0.999802i	$-0.506333\pi$
$139$	−0.469084	−0.0397871	−0.0198936	+	0.999802i	$0.506333\pi$
$140$	7.30994	0.617803
$141$	−2.97025	−0.250140
$142$	2.14888	0.180330
$143$	9.57228	0.800474
$144$	−56.9373	−4.74478
$145$	23.3921	1.94261
$146$	11.1816	0.925397
$147$	1.84772	0.152398
$148$	14.6007	1.20017
$149$	−18.0840	−1.48150	−0.740748	−	0.671783i	$-0.765529\pi$
$149$	−18.0840	−1.48150	−0.740748	+	0.671783i	$0.765529\pi$
$150$	7.09600	0.579386
$151$	−4.47189	−0.363918	−0.181959	−	0.983306i	$-0.558244\pi$
$151$	−4.47189	−0.363918	−0.181959	+	0.983306i	$0.558244\pi$
$152$	−20.2921	−1.64591
$153$	−2.03100	−0.164197
$154$	1.69746	0.136786
$155$	10.3283	0.829590
$156$	−8.20650	−0.657046
$157$	−21.0337	−1.67867	−0.839336	−	0.543614i	$-0.817056\pi$
$157$	−21.0337	−1.67867	−0.839336	+	0.543614i	$0.817056\pi$
$158$	−40.8750	−3.25184
$159$	0.383058	0.0303785
$160$	123.579	9.76974
$161$	−0.324133	−0.0255453
$162$	−23.5596	−1.85102
$163$	−6.52007	−0.510692	−0.255346	−	0.966850i	$-0.582189\pi$
$163$	−6.52007	−0.510692	−0.255346	+	0.966850i	$0.582189\pi$
$164$	−3.50483	−0.273681
$165$	1.88913	0.147069
$166$	−38.8565	−3.01585
$167$	10.5852	0.819108	0.409554	−	0.912286i	$-0.365684\pi$
$167$	10.5852	0.819108	0.409554	+	0.912286i	$0.365684\pi$
$168$	−0.965621	−0.0744993
$169$	13.5414	1.04165
$170$	7.41715	0.568870
$171$	−5.34502	−0.408744
$172$	−4.02907	−0.307214
$173$	−23.9742	−1.82272	−0.911361	−	0.411608i	$-0.864967\pi$
$173$	−23.9742	−1.82272	−0.911361	+	0.411608i	$0.864967\pi$
$174$	−4.65691	−0.353040
$175$	−3.04513	−0.230191
$176$	36.1287	2.72330
$177$	−1.56467	−0.117608
$178$	−21.1930	−1.58848
$179$	−7.46864	−0.558232	−0.279116	−	0.960257i	$-0.590041\pi$
$179$	−7.46864	−0.558232	−0.279116	+	0.960257i	$0.590041\pi$
$180$	66.0373	4.92213
$181$	−5.56778	−0.413850	−0.206925	−	0.978357i	$-0.566346\pi$
$181$	−5.56778	−0.413850	−0.206925	+	0.978357i	$0.566346\pi$
$182$	4.70661	0.348878
$183$	−3.48642	−0.257724
$184$	−11.1167	−0.819535
$185$	−9.31935	−0.685172
$186$	−2.05617	−0.150765
$187$	1.28874	0.0942422
$188$	65.8833	4.80503
$189$	−0.514934	−0.0374560
$190$	19.5198	1.41612
$191$	−18.1693	−1.31469	−0.657343	−	0.753592i	$-0.728320\pi$
$191$	−18.1693	−1.31469	−0.657343	+	0.753592i	$0.728320\pi$
$192$	−14.1805	−1.02339
$193$	−0.703618	−0.0506475	−0.0253238	−	0.999679i	$-0.508062\pi$
$193$	−0.703618	−0.0506475	−0.0253238	+	0.999679i	$0.508062\pi$
$194$	−1.85936	−0.133494
$195$	5.23805	0.375105
$196$	−40.9845	−2.92747
$197$	18.5015	1.31818	0.659090	−	0.752064i	$-0.270942\pi$
$197$	18.5015	1.31818	0.659090	+	0.752064i	$0.270942\pi$
$198$	15.3347	1.08979
$199$	2.09032	0.148179	0.0740893	−	0.997252i	$-0.476395\pi$
$199$	2.09032	0.148179	0.0740893	+	0.997252i	$0.476395\pi$
$200$	−104.438	−7.38490
$201$	1.10153	0.0776958
$202$	16.0401	1.12858
$203$	1.99844	0.140263
$204$	−1.10486	−0.0773559
$205$	2.23706	0.156243
$206$	3.38509	0.235850
$207$	−2.92819	−0.203523
$208$	100.175	6.94590
$209$	3.39160	0.234602
$210$	0.928870	0.0640981
$211$	−20.4856	−1.41028	−0.705142	−	0.709066i	$-0.749117\pi$
$211$	−20.4856	−1.41028	−0.705142	+	0.709066i	$0.749117\pi$
$212$	−8.49665	−0.583552
$213$	0.204313	0.0139993
$214$	15.6533	1.07004
$215$	2.57168	0.175387
$216$	−17.6606	−1.20165
$217$	0.882371	0.0598992
$218$	16.5287	1.11947
$219$	1.06313	0.0718399
$220$	−41.9029	−2.82510
$221$	3.57334	0.240369
$222$	1.85530	0.124520
$223$	18.6326	1.24773	0.623864	−	0.781533i	$-0.285562\pi$
$223$	18.6326	1.24773	0.623864	+	0.781533i	$0.285562\pi$
$224$	10.5576	0.705409
$225$	−27.5094	−1.83396
$226$	2.65896	0.176871
$227$	11.6564	0.773665	0.386832	−	0.922150i	$-0.373569\pi$
$227$	11.6564	0.773665	0.386832	+	0.922150i	$0.373569\pi$
$228$	−2.90768	−0.192566
$229$	16.1122	1.06472	0.532362	−	0.846517i	$-0.321305\pi$
$229$	16.1122	1.06472	0.532362	+	0.846517i	$0.321305\pi$
$230$	10.6936	0.705117
$231$	0.161393	0.0106189
$232$	68.5400	4.49987
$233$	18.0067	1.17966	0.589830	−	0.807528i	$-0.299195\pi$
$233$	18.0067	1.17966	0.589830	+	0.807528i	$0.299195\pi$
$234$	42.5191	2.77956
$235$	−42.0521	−2.74317
$236$	34.7061	2.25917
$237$	−3.88634	−0.252445
$238$	0.633664	0.0410744
$239$	−21.5384	−1.39320	−0.696602	−	0.717458i	$-0.745306\pi$
$239$	−21.5384	−1.39320	−0.696602	+	0.717458i	$0.745306\pi$
$240$	19.7700	1.27615
$241$	−11.5565	−0.744422	−0.372211	−	0.928148i	$-0.621400\pi$
$241$	−11.5565	−0.744422	−0.372211	+	0.928148i	$0.621400\pi$
$242$	21.2735	1.36751
$243$	−7.00597	−0.449433
$244$	77.3326	4.95071
$245$	26.1596	1.67128
$246$	−0.445356	−0.0283949
$247$	9.40400	0.598362
$248$	30.2625	1.92167
$249$	−3.69443	−0.234125
$250$	46.9954	2.97225
$251$	−13.1944	−0.832825	−0.416412	−	0.909176i	$-0.636713\pi$
$251$	−13.1944	−0.832825	−0.416412	+	0.909176i	$0.636713\pi$
$252$	5.64172	0.355395
$253$	1.85803	0.116814
$254$	17.7325	1.11264
$255$	0.705213	0.0441622
$256$	130.929	8.18304
$257$	25.0317	1.56143	0.780717	−	0.624885i	$-0.214854\pi$
$257$	25.0317	1.56143	0.780717	+	0.624885i	$0.214854\pi$
$258$	−0.511972	−0.0318740
$259$	−0.796173	−0.0494718
$260$	−116.186	−7.20553
$261$	18.0537	1.11750
$262$	−17.7380	−1.09586
$263$	−8.74632	−0.539321	−0.269660	−	0.962955i	$-0.586911\pi$
$263$	−8.74632	−0.539321	−0.269660	+	0.962955i	$0.586911\pi$
$264$	5.53525	0.340671
$265$	5.42325	0.333148
$266$	1.66762	0.102249
$267$	−2.01501	−0.123316
$268$	−24.4331	−1.49249
$269$	14.4746	0.882534	0.441267	−	0.897376i	$-0.354529\pi$
$269$	14.4746	0.882534	0.441267	+	0.897376i	$0.354529\pi$
$270$	16.9884	1.03388
$271$	−27.4726	−1.66884	−0.834421	−	0.551128i	$-0.814198\pi$
$271$	−27.4726	−1.66884	−0.834421	+	0.551128i	$0.814198\pi$
$272$	13.4869	0.817761
$273$	0.447499	0.0270839
$274$	−19.2957	−1.16570
$275$	17.4557	1.05262
$276$	−1.59293	−0.0958830
$277$	6.00630	0.360884	0.180442	−	0.983586i	$-0.442247\pi$
$277$	6.00630	0.360884	0.180442	+	0.983586i	$0.442247\pi$
$278$	1.32213	0.0792960
$279$	7.97125	0.477226
$280$	−13.6710	−0.817000
$281$	−19.8236	−1.18257	−0.591287	−	0.806461i	$-0.701380\pi$
$281$	−19.8236	−1.18257	−0.591287	+	0.806461i	$0.701380\pi$
$282$	8.37175	0.498531
$283$	−25.6160	−1.52271	−0.761357	−	0.648333i	$-0.775466\pi$
$283$	−25.6160	−1.52271	−0.761357	+	0.648333i	$0.775466\pi$
$284$	−4.53187	−0.268917
$285$	1.85592	0.109935
$286$	−26.9798	−1.59535
$287$	0.191117	0.0112813
$288$	95.3763	5.62010
$289$	−16.5189	−0.971701
$290$	−65.9314	−3.87163
$291$	−0.176786	−0.0103634
$292$	−23.5814	−1.38000
$293$	11.0535	0.645753	0.322876	−	0.946441i	$-0.395350\pi$
$293$	11.0535	0.645753	0.322876	+	0.946441i	$0.395350\pi$
$294$	−5.20788	−0.303730
$295$	−22.1522	−1.28975
$296$	−27.3062	−1.58714
$297$	2.95177	0.171279
$298$	50.9703	2.95263
$299$	5.15183	0.297938
$300$	−14.9651	−0.864010
$301$	0.219705	0.0126636
$302$	12.6042	0.725290
$303$	1.52508	0.0876133
$304$	35.4935	2.03569
$305$	−49.3599	−2.82634
$306$	5.72446	0.327246
$307$	32.5870	1.85984	0.929920	−	0.367763i	$-0.119876\pi$
$307$	32.5870	1.85984	0.929920	+	0.367763i	$0.119876\pi$
$308$	−3.57986	−0.203982
$309$	0.321850	0.0183094
$310$	−29.1107	−1.65338
$311$	−9.57836	−0.543139	−0.271570	−	0.962419i	$-0.587543\pi$
$311$	−9.57836	−0.543139	−0.271570	+	0.962419i	$0.587543\pi$
$312$	15.3478	0.868896
$313$	−33.2840	−1.88132	−0.940662	−	0.339346i	$-0.889794\pi$
$313$	−33.2840	−1.88132	−0.940662	+	0.339346i	$0.889794\pi$
$314$	59.2842	3.34560
$315$	−3.60100	−0.202893
$316$	86.2032	4.84931
$317$	0.529211	0.0297235	0.0148617	−	0.999890i	$-0.495269\pi$
$317$	0.529211	0.0297235	0.0148617	+	0.999890i	$0.495269\pi$
$318$	−1.07966	−0.0605446
$319$	−11.4557	−0.641396
$320$	−200.764	−11.2230
$321$	1.48829	0.0830684
$322$	0.913581	0.0509119
$323$	1.26609	0.0704469
$324$	49.6861	2.76034
$325$	48.4000	2.68475
$326$	18.3771	1.01781
$327$	1.57153	0.0869059
$328$	6.55471	0.361923
$329$	−3.59260	−0.198067
$330$	−5.32458	−0.293109
$331$	29.9481	1.64609	0.823047	−	0.567973i	$-0.192272\pi$
$331$	29.9481	1.64609	0.823047	+	0.567973i	$0.192272\pi$
$332$	81.9463	4.49739
$333$	−7.19255	−0.394149
$334$	−29.8348	−1.63249
$335$	15.5952	0.852055
$336$	1.68900	0.0921423
$337$	2.52615	0.137608	0.0688041	−	0.997630i	$-0.478082\pi$
$337$	2.52615	0.137608	0.0688041	+	0.997630i	$0.478082\pi$
$338$	−38.1669	−2.07600
$339$	0.252810	0.0137308
$340$	−15.6424	−0.848328
$341$	−5.05803	−0.273908
$342$	15.0651	0.814629
$343$	4.50381	0.243183
$344$	7.53516	0.406268
$345$	1.01674	0.0547392
$346$	67.5720	3.63270
$347$	−9.84212	−0.528353	−0.264176	−	0.964474i	$-0.585100\pi$
$347$	−9.84212	−0.528353	−0.264176	+	0.964474i	$0.585100\pi$
$348$	9.82118	0.526471
$349$	−1.00000	−0.0535288
$349$	−1.00000	−0.0535288
$350$	8.58282	0.458771
$351$	8.18447	0.436855
$352$	−60.5195	−3.22570
$353$	6.68229	0.355662	0.177831	−	0.984061i	$-0.443092\pi$
$353$	6.68229	0.355662	0.177831	+	0.984061i	$0.443092\pi$
$354$	4.41008	0.234393
$355$	2.89261	0.153524
$356$	44.6950	2.36883
$357$	0.0602480	0.00318866
$358$	21.0506	1.11256
$359$	3.13552	0.165486	0.0827431	−	0.996571i	$-0.473632\pi$
$359$	3.13552	0.165486	0.0827431	+	0.996571i	$0.473632\pi$
$360$	−123.503	−6.50916
$361$	−15.6680	−0.824633
$362$	15.6930	0.824806
$363$	2.02266	0.106162
$364$	−9.92601	−0.520264
$365$	15.0516	0.787836
$366$	9.82660	0.513645
$367$	23.2393	1.21308	0.606541	−	0.795052i	$-0.292556\pi$
$367$	23.2393	1.21308	0.606541	+	0.795052i	$0.292556\pi$
$368$	19.4446	1.01362
$369$	1.72654	0.0898798
$370$	26.2669	1.36555
$371$	0.463321	0.0240544
$372$	4.33635	0.224829
$373$	33.0523	1.71138	0.855691	−	0.517488i	$-0.173133\pi$
$373$	33.0523	1.71138	0.855691	+	0.517488i	$0.173133\pi$
$374$	−3.63237	−0.187825
$375$	4.46826	0.230740
$376$	−123.215	−6.35431
$377$	−31.7636	−1.63591
$378$	1.45136	0.0746500
$379$	21.0593	1.08175	0.540873	−	0.841105i	$-0.318094\pi$
$379$	21.0593	1.08175	0.540873	+	0.841105i	$0.318094\pi$
$380$	−41.1663	−2.11179
$381$	1.68598	0.0863756
$382$	51.2109	2.62018
$383$	−19.3439	−0.988429	−0.494214	−	0.869340i	$-0.664544\pi$
$383$	−19.3439	−0.988429	−0.494214	+	0.869340i	$0.664544\pi$
$384$	22.5108	1.14875
$385$	2.28496	0.116452
$386$	1.98317	0.100941
$387$	1.98479	0.100892
$388$	3.92130	0.199074
$389$	−21.0726	−1.06842	−0.534211	−	0.845351i	$-0.679391\pi$
$389$	−21.0726	−1.06842	−0.534211	+	0.845351i	$0.679391\pi$
$390$	−14.7636	−0.747586
$391$	0.693605	0.0350771
$392$	76.6491	3.87136
$393$	−1.68651	−0.0850731
$394$	−52.1472	−2.62714
$395$	−55.0218	−2.76845
$396$	−32.3401	−1.62515
$397$	4.08912	0.205227	0.102614	−	0.994721i	$-0.467279\pi$
$397$	4.08912	0.205227	0.102614	+	0.994721i	$0.467279\pi$
$398$	−5.89163	−0.295321
$399$	0.158555	0.00793770
$400$	182.676	9.13380
$401$	24.8241	1.23966	0.619828	−	0.784738i	$-0.287202\pi$
$401$	24.8241	1.23966	0.619828	+	0.784738i	$0.287202\pi$
$402$	−3.10470	−0.154848
$403$	−14.0246	−0.698614
$404$	−33.8278	−1.68300
$405$	−31.7137	−1.57587
$406$	−5.63267	−0.279545
$407$	4.56392	0.226225
$408$	2.06631	0.102298
$409$	−32.6092	−1.61242	−0.806210	−	0.591630i	$-0.798485\pi$
$409$	−32.6092	−1.61242	−0.806210	+	0.591630i	$0.798485\pi$
$410$	−6.30524	−0.311394
$411$	−1.83461	−0.0904946
$412$	−7.13897	−0.351712
$413$	−1.89252	−0.0931247
$414$	8.25320	0.405622
$415$	−52.3048	−2.56754
$416$	−167.804	−8.22729
$417$	0.125706	0.00615586
$418$	−9.55935	−0.467563
$419$	16.7037	0.816027	0.408014	−	0.912976i	$-0.366222\pi$
$419$	16.7037	0.816027	0.408014	+	0.912976i	$0.366222\pi$
$420$	−1.95894	−0.0955864
$421$	5.53715	0.269864	0.134932	−	0.990855i	$-0.456918\pi$
$421$	5.53715	0.269864	0.134932	+	0.990855i	$0.456918\pi$
$422$	57.7393	2.81071
$423$	−32.4552	−1.57803
$424$	15.8904	0.771706
$425$	6.51622	0.316083
$426$	−0.575862	−0.0279006
$427$	−4.21693	−0.204072
$428$	−33.0119	−1.59569
$429$	−2.56521	−0.123849
$430$	−7.24838	−0.349548
$431$	25.0882	1.20845	0.604227	−	0.796812i	$-0.293482\pi$
$431$	25.0882	1.20845	0.604227	+	0.796812i	$0.293482\pi$
$432$	30.8906	1.48623
$433$	−12.3063	−0.591401	−0.295700	−	0.955281i	$-0.595553\pi$
$433$	−12.3063	−0.591401	−0.295700	+	0.955281i	$0.595553\pi$
$434$	−2.48699	−0.119380
$435$	−6.26868	−0.300560
$436$	−34.8582	−1.66941
$437$	1.82537	0.0873193
$438$	−2.99648	−0.143177
$439$	15.5771	0.743456	0.371728	−	0.928342i	$-0.378765\pi$
$439$	15.5771	0.743456	0.371728	+	0.928342i	$0.378765\pi$
$440$	78.3667	3.73599
$441$	20.1897	0.961412
$442$	−10.0716	−0.479056
$443$	−6.67638	−0.317204	−0.158602	−	0.987343i	$-0.550699\pi$
$443$	−6.67638	−0.317204	−0.158602	+	0.987343i	$0.550699\pi$
$444$	−3.91274	−0.185690
$445$	−28.5280	−1.35236
$446$	−52.5166	−2.48673
$447$	4.84619	0.229217
$448$	−17.1517	−0.810342
$449$	−34.4039	−1.62362	−0.811811	−	0.583920i	$-0.801518\pi$
$449$	−34.4039	−1.62362	−0.811811	+	0.583920i	$0.801518\pi$
$450$	77.5364	3.65510
$451$	−1.09555	−0.0515872
$452$	−5.60760	−0.263759
$453$	1.19839	0.0563053
$454$	−32.8541	−1.54192
$455$	6.33558	0.297017
$456$	5.43794	0.254655
$457$	6.97757	0.326397	0.163199	−	0.986593i	$-0.447819\pi$
$457$	6.97757	0.326397	0.163199	+	0.986593i	$0.447819\pi$
$458$	−45.4128	−2.12200
$459$	1.10190	0.0514321
$460$	−22.5523	−1.05151
$461$	8.59245	0.400190	0.200095	−	0.979776i	$-0.435875\pi$
$461$	8.59245	0.400190	0.200095	+	0.979776i	$0.435875\pi$
$462$	−0.454891	−0.0211634
$463$	17.5167	0.814068	0.407034	−	0.913413i	$-0.366563\pi$
$463$	17.5167	0.814068	0.407034	+	0.913413i	$0.366563\pi$
$464$	−119.885	−5.56554
$465$	−2.76781	−0.128354
$466$	−50.7526	−2.35107
$467$	7.44382	0.344459	0.172229	−	0.985057i	$-0.444903\pi$
$467$	7.44382	0.344459	0.172229	+	0.985057i	$0.444903\pi$
$468$	−89.6705	−4.14502
$469$	1.33233	0.0615213
$470$	118.525	5.46716
$471$	5.63667	0.259724
$472$	−64.9072	−2.98760
$473$	−1.25942	−0.0579080
$474$	10.9538	0.503124
$475$	17.1488	0.786842
$476$	−1.33636	−0.0612522
$477$	4.18559	0.191645
$478$	60.7068	2.77666
$479$	35.7964	1.63558	0.817790	−	0.575516i	$-0.195199\pi$
$479$	35.7964	1.63558	0.817790	+	0.575516i	$0.195199\pi$
$480$	−33.1169	−1.51157
$481$	12.6545	0.576997
$482$	32.5725	1.48364
$483$	0.0868621	0.00395236
$484$	−44.8647	−2.03930
$485$	−2.50289	−0.113650
$486$	19.7466	0.895724
$487$	33.0024	1.49548	0.747741	−	0.663990i	$-0.231138\pi$
$487$	33.0024	1.49548	0.747741	+	0.663990i	$0.231138\pi$
$488$	−144.627	−6.54696
$489$	1.74727	0.0790142
$490$	−73.7319	−3.33087
$491$	27.4658	1.23951	0.619756	−	0.784794i	$-0.287231\pi$
$491$	27.4658	1.23951	0.619756	+	0.784794i	$0.287231\pi$
$492$	0.939233	0.0423439
$493$	−4.27642	−0.192600
$494$	−26.5055	−1.19254
$495$	20.6421	0.927793
$496$	−52.9330	−2.37676
$497$	0.247122	0.0110849
$498$	10.4129	0.466612
$499$	−37.8760	−1.69556	−0.847782	−	0.530345i	$-0.822063\pi$
$499$	−37.8760	−1.69556	−0.847782	+	0.530345i	$0.822063\pi$
$500$	−99.1107	−4.43237
$501$	−2.83665	−0.126732
$502$	37.1890	1.65982
$503$	42.2944	1.88582	0.942908	−	0.333054i	$-0.108079\pi$
$503$	42.2944	1.88582	0.942908	+	0.333054i	$0.108079\pi$
$504$	−10.5511	−0.469984
$505$	21.5917	0.960816
$506$	−5.23694	−0.232810
$507$	−3.62886	−0.161163
$508$	−37.3969	−1.65922
$509$	−17.8524	−0.791292	−0.395646	−	0.918403i	$-0.629479\pi$
$509$	−17.8524	−0.791292	−0.395646	+	0.918403i	$0.629479\pi$
$510$	−1.98767	−0.0880155
$511$	1.28589	0.0568844
$512$	−201.025	−8.88414
$513$	2.89988	0.128033
$514$	−70.5527	−3.11195
$515$	4.55667	0.200791
$516$	1.07972	0.0475321
$517$	20.5940	0.905721
$518$	2.24404	0.0985976
$519$	6.42466	0.282011
$520$	217.290	9.52880
$521$	−23.7181	−1.03911	−0.519554	−	0.854438i	$-0.673902\pi$
$521$	−23.7181	−1.03911	−0.519554	+	0.854438i	$0.673902\pi$
$522$	−50.8850	−2.22718
$523$	−1.14530	−0.0500803	−0.0250402	−	0.999686i	$-0.507971\pi$
$523$	−1.14530	−0.0500803	−0.0250402	+	0.999686i	$0.507971\pi$
$524$	37.4086	1.63420
$525$	0.816044	0.0356151
$526$	24.6518	1.07487
$527$	−1.88817	−0.0822498
$528$	−9.68186	−0.421349
$529$	1.00000	0.0434783
$530$	−15.2856	−0.663965
$531$	−17.0968	−0.741938
$532$	−3.51693	−0.152478
$533$	−3.03766	−0.131576
$534$	5.67937	0.245770
$535$	21.0709	0.910975
$536$	45.6946	1.97371
$537$	2.00147	0.0863696
$538$	−40.7973	−1.75890
$539$	−12.8110	−0.551810
$540$	−35.8277	−1.54178
$541$	−3.93514	−0.169185	−0.0845924	−	0.996416i	$-0.526959\pi$
$541$	−3.93514	−0.169185	−0.0845924	+	0.996416i	$0.526959\pi$
$542$	77.4325	3.32601
$543$	1.49207	0.0640308
$544$	−22.5920	−0.968623
$545$	22.2494	0.953058
$546$	−1.26129	−0.0539783
$547$	−9.28526	−0.397009	−0.198504	−	0.980100i	$-0.563608\pi$
$547$	−9.28526	−0.397009	−0.198504	+	0.980100i	$0.563608\pi$
$548$	40.6936	1.73834
$549$	−38.0953	−1.62587
$550$	−49.1995	−2.09787
$551$	−11.2543	−0.479449
$552$	2.97909	0.126798
$553$	−4.70064	−0.199892
$554$	−16.9290	−0.719243
$555$	2.49742	0.106010
$556$	−2.78830	−0.118250
$557$	25.1594	1.06604	0.533020	−	0.846103i	$-0.321057\pi$
$557$	25.1594	1.06604	0.533020	+	0.846103i	$0.321057\pi$
$558$	−22.4673	−0.951115
$559$	−3.49203	−0.147697
$560$	23.9124	1.01048
$561$	−0.345361	−0.0145811
$562$	55.8734	2.35688
$563$	38.3463	1.61610	0.808052	−	0.589111i	$-0.200522\pi$
$563$	38.3463	1.61610	0.808052	+	0.589111i	$0.200522\pi$
$564$	−17.6556	−0.743434
$565$	3.57922	0.150579
$566$	72.1996	3.03478
$567$	−2.70937	−0.113783
$568$	8.47549	0.355624
$569$	40.5380	1.69944	0.849721	−	0.527232i	$-0.176770\pi$
$569$	40.5380	1.69944	0.849721	+	0.527232i	$0.176770\pi$
$570$	−5.23098	−0.219102
$571$	21.8284	0.913491	0.456746	−	0.889597i	$-0.349015\pi$
$571$	21.8284	0.913491	0.456746	+	0.889597i	$0.349015\pi$
$572$	56.8990	2.37907
$573$	4.86906	0.203408
$574$	−0.538671	−0.0224837
$575$	9.39471	0.391786
$576$	−154.947	−6.45612
$577$	3.31806	0.138133	0.0690663	−	0.997612i	$-0.477998\pi$
$577$	3.31806	0.138133	0.0690663	+	0.997612i	$0.477998\pi$
$578$	46.5591	1.93661
$579$	0.188557	0.00783618
$580$	139.046	5.77357
$581$	−4.46852	−0.185385
$582$	0.498277	0.0206542
$583$	−2.65590	−0.109996
$584$	44.1019	1.82495
$585$	57.2350	2.36638
$586$	−31.1547	−1.28699
$587$	37.1140	1.53186	0.765929	−	0.642926i	$-0.222280\pi$
$587$	37.1140	1.53186	0.765929	+	0.642926i	$0.222280\pi$
$588$	10.9831	0.452937
$589$	−4.96911	−0.204749
$590$	62.4369	2.57049
$591$	−4.95809	−0.203949
$592$	47.7620	1.96301
$593$	−2.21354	−0.0908992	−0.0454496	−	0.998967i	$-0.514472\pi$
$593$	−2.21354	−0.0908992	−0.0454496	+	0.998967i	$0.514472\pi$
$594$	−8.31967	−0.341360
$595$	0.852976	0.0349686
$596$	−107.494	−4.40311
$597$	−0.560169	−0.0229262
$598$	−14.5206	−0.593793
$599$	33.6880	1.37645	0.688227	−	0.725495i	$-0.258389\pi$
$599$	33.6880	1.37645	0.688227	+	0.725495i	$0.258389\pi$
$600$	27.9877	1.14259
$601$	26.8841	1.09663	0.548313	−	0.836273i	$-0.315270\pi$
$601$	26.8841	1.09663	0.548313	+	0.836273i	$0.315270\pi$
$602$	−0.619245	−0.0252386
$603$	12.0361	0.490150
$604$	−26.5816	−1.08159
$605$	28.6363	1.16423
$606$	−4.29848	−0.174614
$607$	−18.9227	−0.768047	−0.384023	−	0.923323i	$-0.625462\pi$
$607$	−18.9227	−0.768047	−0.384023	+	0.923323i	$0.625462\pi$
$608$	−59.4556	−2.41124
$609$	−0.535547	−0.0217015
$610$	139.123	5.63291
$611$	57.1015	2.31008
$612$	−12.0726	−0.488005
$613$	−16.1660	−0.652938	−0.326469	−	0.945208i	$-0.605859\pi$
$613$	−16.1660	−0.652938	−0.326469	+	0.945208i	$0.605859\pi$
$614$	−91.8477	−3.70667
$615$	−0.599494	−0.0241740
$616$	6.69504	0.269751
$617$	14.9047	0.600042	0.300021	−	0.953933i	$-0.403006\pi$
$617$	14.9047	0.600042	0.300021	+	0.953933i	$0.403006\pi$
$618$	−0.907145	−0.0364907
$619$	7.58313	0.304792	0.152396	−	0.988320i	$-0.451301\pi$
$619$	7.58313	0.304792	0.152396	+	0.988320i	$0.451301\pi$
$620$	61.3930	2.46560
$621$	1.58865	0.0637504
$622$	26.9970	1.08248
$623$	−2.43721	−0.0976447
$624$	−26.8452	−1.07467
$625$	16.2870	0.651479
$626$	93.8122	3.74949
$627$	−0.908890	−0.0362976
$628$	−125.027	−4.98913
$629$	1.70371	0.0679315
$630$	10.1496	0.404368
$631$	−26.7709	−1.06573	−0.532865	−	0.846200i	$-0.678885\pi$
$631$	−26.7709	−1.06573	−0.532865	+	0.846200i	$0.678885\pi$
$632$	−161.217	−6.41286
$633$	5.48978	0.218199
$634$	−1.49160	−0.0592390
$635$	23.8698	0.947242
$636$	2.27696	0.0902871
$637$	−35.5216	−1.40742
$638$	32.2883	1.27831
$639$	2.23248	0.0883154
$640$	318.703	12.5978
$641$	−29.3579	−1.15957	−0.579784	−	0.814770i	$-0.696863\pi$
$641$	−29.3579	−1.15957	−0.579784	+	0.814770i	$0.696863\pi$
$642$	−4.19481	−0.165556
$643$	36.6279	1.44446	0.722232	−	0.691651i	$-0.243116\pi$
$643$	36.6279	1.44446	0.722232	+	0.691651i	$0.243116\pi$
$644$	−1.92669	−0.0759224
$645$	−0.689166	−0.0271359
$646$	−3.56851	−0.140401
$647$	−1.55802	−0.0612522	−0.0306261	−	0.999531i	$-0.509750\pi$
$647$	−1.55802	−0.0612522	−0.0306261	+	0.999531i	$0.509750\pi$
$648$	−92.9227	−3.65035
$649$	10.8485	0.425841
$650$	−136.417	−5.35072
$651$	−0.236460	−0.00926760
$652$	−38.7563	−1.51781
$653$	10.9090	0.426902	0.213451	−	0.976954i	$-0.431530\pi$
$653$	10.9090	0.426902	0.213451	+	0.976954i	$0.431530\pi$
$654$	−4.42942	−0.173204
$655$	−23.8772	−0.932959
$656$	−11.4650	−0.447634
$657$	11.6166	0.453207
$658$	10.1259	0.394748
$659$	−13.7070	−0.533949	−0.266974	−	0.963704i	$-0.586024\pi$
$659$	−13.7070	−0.533949	−0.266974	+	0.963704i	$0.586024\pi$
$660$	11.2293	0.437099
$661$	−8.52238	−0.331482	−0.165741	−	0.986169i	$-0.553002\pi$
$661$	−8.52238	−0.331482	−0.165741	+	0.986169i	$0.553002\pi$
$662$	−84.4097	−3.28068
$663$	−0.957593	−0.0371898
$664$	−153.256	−5.94747
$665$	2.24479	0.0870492
$666$	20.2725	0.785542
$667$	−6.16549	−0.238729
$668$	62.9200	2.43445
$669$	−4.99321	−0.193048
$670$	−43.9556	−1.69815
$671$	24.1728	0.933180
$672$	−2.82925	−0.109141
$673$	50.4719	1.94555	0.972774	−	0.231754i	$-0.0744464\pi$
$673$	50.4719	1.94555	0.972774	+	0.231754i	$0.0744464\pi$
$674$	−7.12004	−0.274254
$675$	14.9249	0.574460
$676$	80.4920	3.09584
$677$	5.86499	0.225410	0.112705	−	0.993628i	$-0.464049\pi$
$677$	5.86499	0.225410	0.112705	+	0.993628i	$0.464049\pi$
$678$	−0.712554	−0.0273655
$679$	−0.213827	−0.00820594
$680$	29.2543	1.12185
$681$	−3.12372	−0.119701
$682$	14.2562	0.545900
$683$	31.9577	1.22283	0.611414	−	0.791311i	$-0.290601\pi$
$683$	31.9577	1.22283	0.611414	+	0.791311i	$0.290601\pi$
$684$	−31.7716	−1.21482
$685$	−25.9740	−0.992414
$686$	−12.6941	−0.484665
$687$	−4.31779	−0.164734
$688$	−13.1800	−0.502481
$689$	−7.36411	−0.280550
$690$	−2.86571	−0.109096
$691$	0.550965	0.0209597	0.0104799	−	0.999945i	$-0.496664\pi$
$691$	0.550965	0.0209597	0.0104799	+	0.999945i	$0.496664\pi$
$692$	−142.506	−5.41726
$693$	1.76350	0.0669898
$694$	27.7404	1.05301
$695$	1.77972	0.0675086
$696$	−18.3675	−0.696220
$697$	−0.408968	−0.0154908
$698$	2.81854	0.106683
$699$	−4.82549	−0.182517
$700$	−18.1007	−0.684143
$701$	−34.8659	−1.31687	−0.658434	−	0.752639i	$-0.728781\pi$
$701$	−34.8659	−1.31687	−0.658434	+	0.752639i	$0.728781\pi$
$702$	−23.0682	−0.870654
$703$	4.48369	0.169105
$704$	98.3191	3.70554
$705$	11.2692	0.424424
$706$	−18.8343	−0.708837
$707$	1.84462	0.0693742
$708$	−9.30064	−0.349539
$709$	−22.6223	−0.849597	−0.424798	−	0.905288i	$-0.639655\pi$
$709$	−22.6223	−0.849597	−0.424798	+	0.905288i	$0.639655\pi$
$710$	−8.15292	−0.305974
$711$	−42.4651	−1.59257
$712$	−83.5884	−3.13261
$713$	−2.72225	−0.101949
$714$	−0.169811	−0.00635502
$715$	−36.3176	−1.35820
$716$	−44.3947	−1.65911
$717$	5.77192	0.215556
$718$	−8.83756	−0.329815
$719$	32.7072	1.21977	0.609887	−	0.792489i	$-0.291215\pi$
$719$	32.7072	1.21977	0.609887	+	0.792489i	$0.291215\pi$
$720$	216.022	8.05067
$721$	0.389287	0.0144978
$722$	44.1609	1.64350
$723$	3.09695	0.115177
$724$	−33.0957	−1.22999
$725$	−57.9230	−2.15121
$726$	−5.70093	−0.211581
$727$	21.6151	0.801659	0.400830	−	0.916153i	$-0.368722\pi$
$727$	21.6151	0.801659	0.400830	+	0.916153i	$0.368722\pi$
$728$	18.5636	0.688012
$729$	−23.1990	−0.859222
$730$	−42.4234	−1.57016
$731$	−0.470141	−0.0173888
$732$	−20.7238	−0.765974
$733$	7.34628	0.271341	0.135670	−	0.990754i	$-0.456681\pi$
$733$	7.34628	0.271341	0.135670	+	0.990754i	$0.456681\pi$
$734$	−65.5009	−2.41768
$735$	−7.01033	−0.258580
$736$	−32.5718	−1.20061
$737$	−7.63735	−0.281325
$738$	−4.86630	−0.179131
$739$	6.53886	0.240536	0.120268	−	0.992741i	$-0.461625\pi$
$739$	6.53886	0.240536	0.120268	+	0.992741i	$0.461625\pi$
$740$	−55.3956	−2.03638
$741$	−2.52011	−0.0925786
$742$	−1.30589	−0.0479406
$743$	−13.3405	−0.489416	−0.244708	−	0.969597i	$-0.578692\pi$
$743$	−13.3405	−0.489416	−0.244708	+	0.969597i	$0.578692\pi$
$744$	−8.10982	−0.297320
$745$	68.6112	2.51372
$746$	−93.1590	−3.41079
$747$	−40.3681	−1.47699
$748$	7.66047	0.280095
$749$	1.80013	0.0657755
$750$	−12.5939	−0.459866
$751$	−12.4598	−0.454664	−0.227332	−	0.973817i	$-0.573000\pi$
$751$	−12.4598	−0.454664	−0.227332	+	0.973817i	$0.573000\pi$
$752$	215.518	7.85915
$753$	3.53588	0.128855
$754$	89.5268	3.26037
$755$	16.9665	0.617475
$756$	−3.06084	−0.111322
$757$	14.5537	0.528965	0.264482	−	0.964391i	$-0.414799\pi$
$757$	14.5537	0.528965	0.264482	+	0.964391i	$0.414799\pi$
$758$	−59.3565	−2.15592
$759$	−0.497921	−0.0180734
$760$	76.9890	2.79269
$761$	21.6147	0.783532	0.391766	−	0.920065i	$-0.371864\pi$
$761$	21.6147	0.783532	0.391766	+	0.920065i	$0.371864\pi$
$762$	−4.75201	−0.172147
$763$	1.90081	0.0688141
$764$	−108.001	−3.90734
$765$	7.70570	0.278600
$766$	54.5216	1.96994
$767$	30.0800	1.08613
$768$	−35.0866	−1.26608
$769$	17.7367	0.639603	0.319801	−	0.947485i	$-0.396384\pi$
$769$	17.7367	0.639603	0.319801	+	0.947485i	$0.396384\pi$
$770$	−6.44024	−0.232090
$771$	−6.70806	−0.241585
$772$	−4.18241	−0.150528
$773$	−48.9806	−1.76171	−0.880856	−	0.473385i	$-0.843032\pi$
$773$	−48.9806	−1.76171	−0.880856	+	0.473385i	$0.843032\pi$
$774$	−5.59420	−0.201079
$775$	−25.5747	−0.918671
$776$	−7.33359	−0.263261
$777$	0.213361	0.00765427
$778$	59.3938	2.12937
$779$	−1.07629	−0.0385620
$780$	31.1358	1.11484
$781$	−1.41658	−0.0506893
$782$	−1.95495	−0.0699089
$783$	−9.79481	−0.350038
$784$	−134.069	−4.78818
$785$	79.8026	2.84828
$786$	4.75348	0.169551
$787$	−22.6953	−0.809001	−0.404501	−	0.914538i	$-0.632555\pi$
$787$	−22.6953	−0.809001	−0.404501	+	0.914538i	$0.632555\pi$
$788$	109.976	3.91773
$789$	2.34386	0.0834437
$790$	155.081	5.51754
$791$	0.305781	0.0108723
$792$	60.4824	2.14915
$793$	67.0247	2.38012
$794$	−11.5253	−0.409019
$795$	−1.45334	−0.0515446
$796$	12.4251	0.440397
$797$	−20.2681	−0.717935	−0.358967	−	0.933350i	$-0.616871\pi$
$797$	−20.2681	−0.717935	−0.358967	+	0.933350i	$0.616871\pi$
$798$	−0.446894	−0.0158199
$799$	7.68773	0.271972
$800$	−306.002	−10.8188
$801$	−22.0175	−0.777950
$802$	−69.9676	−2.47064
$803$	−7.37114	−0.260122
$804$	6.54764	0.230918
$805$	1.22977	0.0433438
$806$	39.5288	1.39234
$807$	−3.87895	−0.136546
$808$	63.2646	2.22564
$809$	−25.0165	−0.879534	−0.439767	−	0.898112i	$-0.644939\pi$
$809$	−25.0165	−0.879534	−0.439767	+	0.898112i	$0.644939\pi$
$810$	89.3861	3.14071
$811$	30.3600	1.06608	0.533041	−	0.846089i	$-0.321049\pi$
$811$	30.3600	1.06608	0.533041	+	0.846089i	$0.321049\pi$
$812$	11.8790	0.416872
$813$	7.36218	0.258203
$814$	−12.8636	−0.450868
$815$	24.7374	0.866513
$816$	−3.61424	−0.126524
$817$	−1.23728	−0.0432868
$818$	91.9101	3.21356
$819$	4.88972	0.170860
$820$	13.2974	0.464366
$821$	−29.1515	−1.01739	−0.508697	−	0.860945i	$-0.669873\pi$
$821$	−29.1515	−1.01739	−0.508697	+	0.860945i	$0.669873\pi$
$822$	5.17091	0.180356
$823$	−6.30843	−0.219898	−0.109949	−	0.993937i	$-0.535069\pi$
$823$	−6.30843	−0.219898	−0.109949	+	0.993937i	$0.535069\pi$
$824$	13.3513	0.465114
$825$	−4.67782	−0.162861
$826$	5.33413	0.185598
$827$	−40.3336	−1.40254	−0.701269	−	0.712897i	$-0.747383\pi$
$827$	−40.3336	−1.40254	−0.701269	+	0.712897i	$0.747383\pi$
$828$	−17.4056	−0.604885
$829$	2.83129	0.0983347	0.0491673	−	0.998791i	$-0.484343\pi$
$829$	2.83129	0.0983347	0.0491673	+	0.998791i	$0.484343\pi$
$830$	147.423	5.11713
$831$	−1.60958	−0.0558359
$832$	272.613	9.45114
$833$	−4.78236	−0.165699
$834$	−0.354308	−0.0122687
$835$	−40.1607	−1.38982
$836$	20.1602	0.697254
$837$	−4.32470	−0.149484
$838$	−47.0799	−1.62635
$839$	42.8674	1.47995	0.739973	−	0.672637i	$-0.234838\pi$
$839$	42.8674	1.47995	0.739973	+	0.672637i	$0.234838\pi$
$840$	3.66360	0.126406
$841$	9.01327	0.310802
$842$	−15.6067	−0.537841
$843$	5.31237	0.182968
$844$	−121.769	−4.19147
$845$	−51.3765	−1.76741
$846$	91.4762	3.14502
$847$	2.44646	0.0840614
$848$	−27.7944	−0.954462
$849$	6.86465	0.235594
$850$	−18.3662	−0.629955
$851$	2.45632	0.0842014
$852$	1.21446	0.0416068
$853$	35.7999	1.22577	0.612883	−	0.790174i	$-0.290010\pi$
$853$	35.7999	1.22577	0.612883	+	0.790174i	$0.290010\pi$
$854$	11.8856	0.406716
$855$	20.2792	0.693534
$856$	61.7388	2.11019
$857$	−13.7457	−0.469543	−0.234772	−	0.972051i	$-0.575434\pi$
$857$	−13.7457	−0.469543	−0.234772	+	0.972051i	$0.575434\pi$
$858$	7.23013	0.246832
$859$	−28.2863	−0.965118	−0.482559	−	0.875864i	$-0.660293\pi$
$859$	−28.2863	−0.965118	−0.482559	+	0.875864i	$0.660293\pi$
$860$	15.2865	0.521264
$861$	−0.0512162	−0.00174544
$862$	−70.7119	−2.40846
$863$	37.7777	1.28597	0.642983	−	0.765880i	$-0.277696\pi$
$863$	37.7777	1.28597	0.642983	+	0.765880i	$0.277696\pi$
$864$	−51.7452	−1.76041
$865$	90.9588	3.09269
$866$	34.6856	1.17867
$867$	4.42678	0.150341
$868$	5.24494	0.178025
$869$	26.9456	0.914066
$870$	17.6685	0.599018
$871$	−21.1763	−0.717532
$872$	65.1918	2.20767
$873$	−1.93170	−0.0653780
$874$	−5.14487	−0.174028
$875$	5.40449	0.182705
$876$	6.31942	0.213513
$877$	27.0297	0.912729	0.456365	−	0.889793i	$-0.349151\pi$
$877$	27.0297	0.912729	0.456365	+	0.889793i	$0.349151\pi$
$878$	−43.9047	−1.48171
$879$	−2.96215	−0.0999108
$880$	−137.073	−4.62075
$881$	−5.56956	−0.187643	−0.0938215	−	0.995589i	$-0.529908\pi$
$881$	−5.56956	−0.187643	−0.0938215	+	0.995589i	$0.529908\pi$
$882$	−56.9053	−1.91610
$883$	−20.4513	−0.688241	−0.344121	−	0.938925i	$-0.611823\pi$
$883$	−20.4513	−0.688241	−0.344121	+	0.938925i	$0.611823\pi$
$884$	21.2404	0.714393
$885$	5.93642	0.199551
$886$	18.8176	0.632190
$887$	49.0006	1.64528	0.822639	−	0.568564i	$-0.192501\pi$
$887$	49.0006	1.64528	0.822639	+	0.568564i	$0.192501\pi$
$888$	7.31758	0.245562
$889$	2.03925	0.0683941
$890$	80.4071	2.69525
$891$	15.5310	0.520307
$892$	110.755	3.70834
$893$	20.2319	0.677035
$894$	−13.6592	−0.456831
$895$	28.3363	0.947177
$896$	27.2275	0.909607
$897$	−1.38060	−0.0460970
$898$	96.9687	3.23589
$899$	16.7840	0.559778
$900$	−163.520	−5.45067
$901$	−0.991449	−0.0330300
$902$	3.08784	0.102814
$903$	−0.0588770	−0.00195930
$904$	10.4873	0.348803
$905$	21.1244	0.702198
$906$	−3.37771	−0.112217
$907$	32.0713	1.06491	0.532454	−	0.846459i	$-0.321270\pi$
$907$	32.0713	1.06491	0.532454	+	0.846459i	$0.321270\pi$
$908$	69.2875	2.29939
$909$	16.6641	0.552715
$910$	−17.8571	−0.591956
$911$	8.18820	0.271287	0.135644	−	0.990758i	$-0.456690\pi$
$911$	8.18820	0.271287	0.135644	+	0.990758i	$0.456690\pi$
$912$	−9.51166	−0.314962
$913$	25.6150	0.847732
$914$	−19.6665	−0.650511
$915$	13.2276	0.437291
$916$	95.7732	3.16443
$917$	−2.03988	−0.0673628
$918$	−3.10574	−0.102505
$919$	18.5535	0.612025	0.306012	−	0.952028i	$-0.401005\pi$
$919$	18.5535	0.612025	0.306012	+	0.952028i	$0.401005\pi$
$920$	42.1772	1.39054
$921$	−8.73276	−0.287754
$922$	−24.2181	−0.797582
$923$	−3.92781	−0.129285
$924$	0.959341	0.0315600
$925$	23.0764	0.758746
$926$	−49.3713	−1.62244
$927$	3.51678	0.115506
$928$	200.821	6.59228
$929$	17.3367	0.568797	0.284399	−	0.958706i	$-0.408206\pi$
$929$	17.3367	0.568797	0.284399	+	0.958706i	$0.408206\pi$
$930$	7.80117	0.255810
$931$	−12.5858	−0.412483
$932$	107.035	3.50603
$933$	2.56684	0.0840345
$934$	−20.9807	−0.686508
$935$	−4.88953	−0.159905
$936$	167.701	5.48149
$937$	50.4371	1.64771	0.823855	−	0.566801i	$-0.191819\pi$
$937$	50.4371	1.64771	0.823855	+	0.566801i	$0.191819\pi$
$938$	−3.75522	−0.122612
$939$	8.91954	0.291078
$940$	−249.964	−8.15292
$941$	14.9229	0.486471	0.243236	−	0.969967i	$-0.421791\pi$
$941$	14.9229	0.486471	0.243236	+	0.969967i	$0.421791\pi$
$942$	−15.8871	−0.517631
$943$	−0.589626	−0.0192009
$944$	113.531	3.69512
$945$	1.95368	0.0635532
$946$	3.54971	0.115411
$947$	7.22419	0.234755	0.117377	−	0.993087i	$-0.462551\pi$
$947$	7.22419	0.234755	0.117377	+	0.993087i	$0.462551\pi$
$948$	−23.1010	−0.750284
$949$	−20.4382	−0.663452
$950$	−48.3346	−1.56818
$951$	−0.141819	−0.00459881
$952$	2.49926	0.0810016
$953$	−49.1250	−1.59132	−0.795658	−	0.605747i	$-0.792875\pi$
$953$	−49.1250	−1.59132	−0.795658	+	0.605747i	$0.792875\pi$
$954$	−11.7972	−0.381950
$955$	68.9350	2.23069
$956$	−128.027	−4.14071
$957$	3.06993	0.0992367
$958$	−100.894	−3.25972
$959$	−2.21901	−0.0716557
$960$	53.8012	1.73643
$961$	−23.5894	−0.760947
$962$	−35.6672	−1.14996
$963$	16.2622	0.524043
$964$	−68.6937	−2.21247
$965$	2.66955	0.0859359
$966$	−0.244824	−0.00787708
$967$	34.3913	1.10595	0.552975	−	0.833198i	$-0.313492\pi$
$967$	34.3913	1.10595	0.552975	+	0.833198i	$0.313492\pi$
$968$	83.9057	2.69683
$969$	−0.339289	−0.0108995
$970$	7.05448	0.226506
$971$	23.1930	0.744298	0.372149	−	0.928173i	$-0.378621\pi$
$971$	23.1930	0.744298	0.372149	+	0.928173i	$0.378621\pi$
$972$	−41.6445	−1.33575
$973$	0.152045	0.00487435
$974$	−93.0185	−2.98051
$975$	−12.9704	−0.415384
$976$	252.971	8.09742
$977$	20.7243	0.663028	0.331514	−	0.943450i	$-0.392441\pi$
$977$	20.7243	0.663028	0.331514	+	0.943450i	$0.392441\pi$
$978$	−4.92474	−0.157476
$979$	13.9709	0.446511
$980$	155.497	4.96716
$981$	17.1718	0.548252
$982$	−77.4132	−2.47036
$983$	15.1307	0.482593	0.241297	−	0.970451i	$-0.422427\pi$
$983$	15.1307	0.482593	0.241297	+	0.970451i	$0.422427\pi$
$984$	−1.75655	−0.0559967
$985$	−70.1955	−2.23661
$986$	12.0532	0.383853
$987$	0.962756	0.0306449
$988$	55.8988	1.77838
$989$	−0.677822	−0.0215535
$990$	−58.1805	−1.84910
$991$	−40.9248	−1.30002	−0.650010	−	0.759926i	$-0.725235\pi$
$991$	−40.9248	−1.30002	−0.650010	+	0.759926i	$0.725235\pi$
$992$	88.6685	2.81523
$993$	−8.02556	−0.254684
$994$	−0.696523	−0.0220924
$995$	−7.93073	−0.251421
$996$	−21.9602	−0.695836
$997$	15.9243	0.504329	0.252165	−	0.967684i	$-0.418858\pi$
$997$	15.9243	0.504329	0.252165	+	0.967684i	$0.418858\pi$
$998$	106.755	3.37927
$999$	3.90223	0.123461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8027.2.a.f.1.1	✓	176

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8027.2.a.f.1.1	✓	176	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8027 = 23 \cdot 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.81854 q^{2} -0.267983 q^{3} +5.94415 q^{4} -3.79404 q^{5} +0.755319 q^{6} -0.324133 q^{7} -11.1167 q^{8} -2.92819 q^{9} +O(q^{10})\)	\(q-2.81854 q^{2} -0.267983 q^{3} +5.94415 q^{4} -3.79404 q^{5} +0.755319 q^{6} -0.324133 q^{7} -11.1167 q^{8} -2.92819 q^{9} +10.6936 q^{10} +1.85803 q^{11} -1.59293 q^{12} +5.15183 q^{13} +0.913581 q^{14} +1.01674 q^{15} +19.4446 q^{16} +0.693605 q^{17} +8.25320 q^{18} +1.82537 q^{19} -22.5523 q^{20} +0.0868621 q^{21} -5.23694 q^{22} +1.00000 q^{23} +2.97909 q^{24} +9.39471 q^{25} -14.5206 q^{26} +1.58865 q^{27} -1.92669 q^{28} -6.16549 q^{29} -2.86571 q^{30} -2.72225 q^{31} -32.5718 q^{32} -0.497921 q^{33} -1.95495 q^{34} +1.22977 q^{35} -17.4056 q^{36} +2.45632 q^{37} -5.14487 q^{38} -1.38060 q^{39} +42.1772 q^{40} -0.589626 q^{41} -0.244824 q^{42} -0.677822 q^{43} +11.0444 q^{44} +11.1096 q^{45} -2.81854 q^{46} +11.0837 q^{47} -5.21081 q^{48} -6.89494 q^{49} -26.4793 q^{50} -0.185874 q^{51} +30.6232 q^{52} -1.42941 q^{53} -4.47767 q^{54} -7.04945 q^{55} +3.60329 q^{56} -0.489168 q^{57} +17.3777 q^{58} +5.83870 q^{59} +6.04363 q^{60} +13.0099 q^{61} +7.67276 q^{62} +0.949122 q^{63} +52.9156 q^{64} -19.5462 q^{65} +1.40341 q^{66} -4.11044 q^{67} +4.12289 q^{68} -0.267983 q^{69} -3.46616 q^{70} -0.762409 q^{71} +32.5518 q^{72} -3.96717 q^{73} -6.92321 q^{74} -2.51762 q^{75} +10.8503 q^{76} -0.602250 q^{77} +3.89128 q^{78} +14.5022 q^{79} -73.7734 q^{80} +8.35882 q^{81} +1.66188 q^{82} +13.7861 q^{83} +0.516321 q^{84} -2.63156 q^{85} +1.91047 q^{86} +1.65224 q^{87} -20.6552 q^{88} +7.51916 q^{89} -31.3129 q^{90} -1.66988 q^{91} +5.94415 q^{92} +0.729516 q^{93} -31.2399 q^{94} -6.92552 q^{95} +8.72868 q^{96} +0.659690 q^{97} +19.4336 q^{98} -5.44067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

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