LMFDB - Embedded newform 1045.4.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-3.55600$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.55600 q^{2} +6.62353 q^{3} +4.64511 q^{4} -5.00000 q^{5} +23.5532 q^{6} -2.03165 q^{7} -11.9300 q^{8} +16.8711 q^{9} +O(q^{10})$	\(q+3.55600 q^{2} +6.62353 q^{3} +4.64511 q^{4} -5.00000 q^{5} +23.5532 q^{6} -2.03165 q^{7} -11.9300 q^{8} +16.8711 q^{9} -17.7800 q^{10} -11.0000 q^{11} +30.7670 q^{12} -18.7788 q^{13} -7.22452 q^{14} -33.1176 q^{15} -79.5838 q^{16} -128.340 q^{17} +59.9936 q^{18} +19.0000 q^{19} -23.2255 q^{20} -13.4567 q^{21} -39.1160 q^{22} +68.3956 q^{23} -79.0185 q^{24} +25.0000 q^{25} -66.7772 q^{26} -67.0891 q^{27} -9.43722 q^{28} -89.8328 q^{29} -117.766 q^{30} -43.4357 q^{31} -187.560 q^{32} -72.8588 q^{33} -456.376 q^{34} +10.1582 q^{35} +78.3681 q^{36} -142.330 q^{37} +67.5639 q^{38} -124.382 q^{39} +59.6499 q^{40} +367.030 q^{41} -47.8518 q^{42} -298.680 q^{43} -51.0962 q^{44} -84.3555 q^{45} +243.215 q^{46} -51.7860 q^{47} -527.126 q^{48} -338.872 q^{49} +88.8999 q^{50} -850.062 q^{51} -87.2294 q^{52} +35.2083 q^{53} -238.568 q^{54} +55.0000 q^{55} +24.2375 q^{56} +125.847 q^{57} -319.445 q^{58} +374.758 q^{59} -153.835 q^{60} +470.097 q^{61} -154.457 q^{62} -34.2761 q^{63} -30.2919 q^{64} +93.8938 q^{65} -259.086 q^{66} +682.651 q^{67} -596.152 q^{68} +453.020 q^{69} +36.1226 q^{70} -636.206 q^{71} -201.272 q^{72} -400.595 q^{73} -506.126 q^{74} +165.588 q^{75} +88.2571 q^{76} +22.3481 q^{77} -442.300 q^{78} -830.373 q^{79} +397.919 q^{80} -899.886 q^{81} +1305.16 q^{82} +1257.69 q^{83} -62.5076 q^{84} +641.699 q^{85} -1062.11 q^{86} -595.010 q^{87} +131.230 q^{88} -512.429 q^{89} -299.968 q^{90} +38.1518 q^{91} +317.705 q^{92} -287.697 q^{93} -184.151 q^{94} -95.0000 q^{95} -1242.31 q^{96} -575.971 q^{97} -1205.03 q^{98} -185.582 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * 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 $

$2$	3.55600	1.25723	0.628617	−	0.777715i	$-0.283621\pi$
$2$	3.55600	1.25723	0.628617	+	0.777715i	$0.283621\pi$
$3$	6.62353	1.27470	0.637349	−	0.770575i	$-0.280031\pi$
$3$	6.62353	1.27470	0.637349	+	0.770575i	$0.280031\pi$
$4$	4.64511	0.580639
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	23.5532	1.60259
$7$	−2.03165	−0.109699	−0.0548493	−	0.998495i	$-0.517468\pi$
$7$	−2.03165	−0.109699	−0.0548493	+	0.998495i	$0.517468\pi$
$8$	−11.9300	−0.527236
$9$	16.8711	0.624855
$10$	−17.7800	−0.562252
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	30.7670	0.740139
$13$	−18.7788	−0.400638	−0.200319	−	0.979731i	$-0.564198\pi$
$13$	−18.7788	−0.400638	−0.200319	+	0.979731i	$0.564198\pi$
$14$	−7.22452	−0.137917
$15$	−33.1176	−0.570062
$16$	−79.5838	−1.24350
$17$	−128.340	−1.83100	−0.915499	−	0.402320i	$-0.868204\pi$
$17$	−128.340	−1.83100	−0.915499	+	0.402320i	$0.868204\pi$
$18$	59.9936	0.785590
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−23.2255	−0.259670
$21$	−13.4567	−0.139833
$22$	−39.1160	−0.379070
$23$	68.3956	0.620064	0.310032	−	0.950726i	$-0.399660\pi$
$23$	68.3956	0.620064	0.310032	+	0.950726i	$0.399660\pi$
$24$	−79.0185	−0.672066
$25$	25.0000	0.200000
$26$	−66.7772	−0.503696
$27$	−67.0891	−0.478196
$28$	−9.43722	−0.0636952
$29$	−89.8328	−0.575225	−0.287613	−	0.957747i	$-0.592862\pi$
$29$	−89.8328	−0.575225	−0.287613	+	0.957747i	$0.592862\pi$
$30$	−117.766	−0.716702
$31$	−43.4357	−0.251654	−0.125827	−	0.992052i	$-0.540158\pi$
$31$	−43.4357	−0.251654	−0.125827	+	0.992052i	$0.540158\pi$
$32$	−187.560	−1.03613
$33$	−72.8588	−0.384336
$34$	−456.376	−2.30199
$35$	10.1582	0.0490587
$36$	78.3681	0.362815
$37$	−142.330	−0.632405	−0.316202	−	0.948692i	$-0.602408\pi$
$37$	−142.330	−0.632405	−0.316202	+	0.948692i	$0.602408\pi$
$38$	67.5639	0.288429
$39$	−124.382	−0.510692
$40$	59.6499	0.235787
$41$	367.030	1.39806	0.699029	−	0.715093i	$-0.253616\pi$
$41$	367.030	1.39806	0.699029	+	0.715093i	$0.253616\pi$
$42$	−47.8518	−0.175802
$43$	−298.680	−1.05926	−0.529631	−	0.848228i	$-0.677670\pi$
$43$	−298.680	−1.05926	−0.529631	+	0.848228i	$0.677670\pi$
$44$	−51.0962	−0.175069
$45$	−84.3555	−0.279444
$46$	243.215	0.779566
$47$	−51.7860	−0.160718	−0.0803592	−	0.996766i	$-0.525607\pi$
$47$	−51.7860	−0.160718	−0.0803592	+	0.996766i	$0.525607\pi$
$48$	−527.126	−1.58508
$49$	−338.872	−0.987966
$50$	88.8999	0.251447
$51$	−850.062	−2.33397
$52$	−87.2294	−0.232626
$53$	35.2083	0.0912497	0.0456249	−	0.998959i	$-0.485472\pi$
$53$	35.2083	0.0912497	0.0456249	+	0.998959i	$0.485472\pi$
$54$	−238.568	−0.601205
$55$	55.0000	0.134840
$56$	24.2375	0.0578370
$57$	125.847	0.292436
$58$	−319.445	−0.723193
$59$	374.758	0.826938	0.413469	−	0.910518i	$-0.364317\pi$
$59$	374.758	0.826938	0.413469	+	0.910518i	$0.364317\pi$
$60$	−153.835	−0.331000
$61$	470.097	0.986717	0.493358	−	0.869826i	$-0.335769\pi$
$61$	470.097	0.986717	0.493358	+	0.869826i	$0.335769\pi$
$62$	−154.457	−0.316388
$63$	−34.2761	−0.0685457
$64$	−30.2919	−0.0591639
$65$	93.8938	0.179171
$66$	−259.086	−0.483200
$67$	682.651	1.24476	0.622381	−	0.782714i	$-0.286165\pi$
$67$	682.651	1.24476	0.622381	+	0.782714i	$0.286165\pi$
$68$	−596.152	−1.06315
$69$	453.020	0.790394
$70$	36.1226	0.0616783
$71$	−636.206	−1.06343	−0.531717	−	0.846922i	$-0.678453\pi$
$71$	−636.206	−1.06343	−0.531717	+	0.846922i	$0.678453\pi$
$72$	−201.272	−0.329446
$73$	−400.595	−0.642275	−0.321138	−	0.947033i	$-0.604065\pi$
$73$	−400.595	−0.642275	−0.321138	+	0.947033i	$0.604065\pi$
$74$	−506.126	−0.795081
$75$	165.588	0.254940
$76$	88.2571	0.133208
$77$	22.3481	0.0330754
$78$	−442.300	−0.642060
$79$	−830.373	−1.18259	−0.591293	−	0.806457i	$-0.701382\pi$
$79$	−830.373	−1.18259	−0.591293	+	0.806457i	$0.701382\pi$
$80$	397.919	0.556109
$81$	−899.886	−1.23441
$82$	1305.16	1.75769
$83$	1257.69	1.66324	0.831622	−	0.555342i	$-0.187413\pi$
$83$	1257.69	1.66324	0.831622	+	0.555342i	$0.187413\pi$
$84$	−62.5076	−0.0811922
$85$	641.699	0.818847
$86$	−1062.11	−1.33174
$87$	−595.010	−0.733239
$88$	131.230	0.158968
$89$	−512.429	−0.610307	−0.305154	−	0.952303i	$-0.598708\pi$
$89$	−512.429	−0.610307	−0.305154	+	0.952303i	$0.598708\pi$
$90$	−299.968	−0.351326
$91$	38.1518	0.0439494
$92$	317.705	0.360033
$93$	−287.697	−0.320783
$94$	−184.151	−0.202061
$95$	−95.0000	−0.102598
$96$	−1242.31	−1.32076
$97$	−575.971	−0.602897	−0.301448	−	0.953483i	$-0.597470\pi$
$97$	−575.971	−0.602897	−0.301448	+	0.953483i	$0.597470\pi$
$98$	−1205.03	−1.24211
$99$	−185.582	−0.188401
$100$	116.128	0.116128
$101$	−972.694	−0.958284	−0.479142	−	0.877737i	$-0.659052\pi$
$101$	−972.694	−0.958284	−0.479142	+	0.877737i	$0.659052\pi$
$102$	−3022.82	−2.93435
$103$	113.627	0.108699	0.0543493	−	0.998522i	$-0.482692\pi$
$103$	113.627	0.108699	0.0543493	+	0.998522i	$0.482692\pi$
$104$	224.030	0.211230
$105$	67.2833	0.0625350
$106$	125.201	0.114722
$107$	388.179	0.350717	0.175359	−	0.984505i	$-0.443892\pi$
$107$	388.179	0.350717	0.175359	+	0.984505i	$0.443892\pi$
$108$	−311.636	−0.277659
$109$	683.095	0.600263	0.300132	−	0.953898i	$-0.402969\pi$
$109$	683.095	0.600263	0.300132	+	0.953898i	$0.402969\pi$
$110$	195.580	0.169525
$111$	−942.729	−0.806125
$112$	161.686	0.136410
$113$	−413.578	−0.344302	−0.172151	−	0.985071i	$-0.555072\pi$
$113$	−413.578	−0.344302	−0.172151	+	0.985071i	$0.555072\pi$
$114$	447.511	0.367660
$115$	−341.978	−0.277301
$116$	−417.283	−0.333998
$117$	−316.818	−0.250341
$118$	1332.64	1.03966
$119$	260.741	0.200858
$120$	395.093	0.300557
$121$	121.000	0.0909091
$122$	1671.66	1.24053
$123$	2431.03	1.78210
$124$	−201.763	−0.146120
$125$	−125.000	−0.0894427
$126$	−121.886	−0.0861781
$127$	1361.99	0.951629	0.475814	−	0.879546i	$-0.342153\pi$
$127$	1361.99	0.951629	0.475814	+	0.879546i	$0.342153\pi$
$128$	1392.76	0.961749
$129$	−1978.32	−1.35024
$130$	333.886	0.225260
$131$	−796.587	−0.531283	−0.265642	−	0.964072i	$-0.585584\pi$
$131$	−796.587	−0.531283	−0.265642	+	0.964072i	$0.585584\pi$
$132$	−338.437	−0.223160
$133$	−38.6013	−0.0251666
$134$	2427.50	1.56496
$135$	335.445	0.213856
$136$	1531.09	0.965368
$137$	−1610.58	−1.00439	−0.502193	−	0.864755i	$-0.667473\pi$
$137$	−1610.58	−1.00439	−0.502193	+	0.864755i	$0.667473\pi$
$138$	1610.94	0.993711
$139$	1431.53	0.873528	0.436764	−	0.899576i	$-0.356124\pi$
$139$	1431.53	0.873528	0.436764	+	0.899576i	$0.356124\pi$
$140$	47.1861	0.0284854
$141$	−343.006	−0.204867
$142$	−2262.35	−1.33699
$143$	206.566	0.120797
$144$	−1342.67	−0.777006
$145$	449.164	0.257249
$146$	−1424.51	−0.807490
$147$	−2244.53	−1.25936
$148$	−661.140	−0.367199
$149$	180.642	0.0993205	0.0496602	−	0.998766i	$-0.484186\pi$
$149$	180.642	0.0993205	0.0496602	+	0.998766i	$0.484186\pi$
$150$	588.831	0.320519
$151$	−1550.01	−0.835349	−0.417675	−	0.908597i	$-0.637155\pi$
$151$	−1550.01	−0.835349	−0.417675	+	0.908597i	$0.637155\pi$
$152$	−226.670	−0.120956
$153$	−2165.23	−1.14411
$154$	79.4698	0.0415835
$155$	217.178	0.112543
$156$	−577.766	−0.296528
$157$	454.288	0.230931	0.115465	−	0.993312i	$-0.463164\pi$
$157$	454.288	0.230931	0.115465	+	0.993312i	$0.463164\pi$
$158$	−2952.80	−1.48679
$159$	233.203	0.116316
$160$	937.800	0.463372
$161$	−138.956	−0.0680201
$162$	−3199.99	−1.55194
$163$	611.534	0.293859	0.146930	−	0.989147i	$-0.453061\pi$
$163$	611.534	0.293859	0.146930	+	0.989147i	$0.453061\pi$
$164$	1704.89	0.811767
$165$	364.294	0.171880
$166$	4472.34	2.09109
$167$	−376.046	−0.174248	−0.0871238	−	0.996197i	$-0.527768\pi$
$167$	−376.046	−0.174248	−0.0871238	+	0.996197i	$0.527768\pi$
$168$	160.538	0.0737247
$169$	−1844.36	−0.839489
$170$	2281.88	1.02948
$171$	320.551	0.143352
$172$	−1387.40	−0.615049
$173$	1418.39	0.623342	0.311671	−	0.950190i	$-0.399111\pi$
$173$	1418.39	0.623342	0.311671	+	0.950190i	$0.399111\pi$
$174$	−2115.85	−0.921853
$175$	−50.7911	−0.0219397
$176$	875.422	0.374929
$177$	2482.22	1.05410
$178$	−1822.19	−0.767299
$179$	1469.38	0.613555	0.306777	−	0.951781i	$-0.400749\pi$
$179$	1469.38	0.613555	0.306777	+	0.951781i	$0.400749\pi$
$180$	−391.840	−0.162256
$181$	−3885.53	−1.59563	−0.797815	−	0.602902i	$-0.794011\pi$
$181$	−3885.53	−1.59563	−0.797815	+	0.602902i	$0.794011\pi$
$182$	135.668	0.0552547
$183$	3113.70	1.25777
$184$	−815.958	−0.326920
$185$	711.652	0.282820
$186$	−1023.05	−0.403300
$187$	1411.74	0.552067
$188$	−240.552	−0.0933193
$189$	136.301	0.0524574
$190$	−337.820	−0.128990
$191$	−1138.46	−0.431290	−0.215645	−	0.976472i	$-0.569185\pi$
$191$	−1138.46	−0.431290	−0.215645	+	0.976472i	$0.569185\pi$
$192$	−200.639	−0.0754161
$193$	−313.065	−0.116761	−0.0583806	−	0.998294i	$-0.518594\pi$
$193$	−313.065	−0.116761	−0.0583806	+	0.998294i	$0.518594\pi$
$194$	−2048.15	−0.757982
$195$	621.908	0.228388
$196$	−1574.10	−0.573651
$197$	4160.28	1.50461	0.752303	−	0.658817i	$-0.228943\pi$
$197$	4160.28	1.50461	0.752303	+	0.658817i	$0.228943\pi$
$198$	−659.929	−0.236864
$199$	−449.741	−0.160207	−0.0801037	−	0.996787i	$-0.525525\pi$
$199$	−449.741	−0.160207	−0.0801037	+	0.996787i	$0.525525\pi$
$200$	−298.249	−0.105447
$201$	4521.56	1.58670
$202$	−3458.90	−1.20479
$203$	182.508	0.0631014
$204$	−3948.63	−1.35519
$205$	−1835.15	−0.625231
$206$	404.056	0.136660
$207$	1153.91	0.387450
$208$	1494.49	0.498192
$209$	−209.000	−0.0691714
$210$	239.259	0.0786212
$211$	2981.68	0.972831	0.486415	−	0.873728i	$-0.338304\pi$
$211$	2981.68	0.972831	0.486415	+	0.873728i	$0.338304\pi$
$212$	163.547	0.0529831
$213$	−4213.93	−1.35556
$214$	1380.36	0.440934
$215$	1493.40	0.473717
$216$	800.371	0.252122
$217$	88.2459	0.0276061
$218$	2429.08	0.754671
$219$	−2653.35	−0.818707
$220$	255.481	0.0782933
$221$	2410.06	0.733567
$222$	−3352.34	−1.01349
$223$	2932.71	0.880667	0.440333	−	0.897834i	$-0.354860\pi$
$223$	2932.71	0.880667	0.440333	+	0.897834i	$0.354860\pi$
$224$	381.055	0.113662
$225$	421.777	0.124971
$226$	−1470.68	−0.432868
$227$	−1015.09	−0.296800	−0.148400	−	0.988927i	$-0.547412\pi$
$227$	−1015.09	−0.296800	−0.148400	+	0.988927i	$0.547412\pi$
$228$	584.573	0.169800
$229$	3284.66	0.947845	0.473922	−	0.880567i	$-0.342838\pi$
$229$	3284.66	0.947845	0.473922	+	0.880567i	$0.342838\pi$
$230$	−1216.07	−0.348632
$231$	148.023	0.0421611
$232$	1071.70	0.303279
$233$	−2449.22	−0.688644	−0.344322	−	0.938852i	$-0.611891\pi$
$233$	−2449.22	−0.688644	−0.344322	+	0.938852i	$0.611891\pi$
$234$	−1126.60	−0.314737
$235$	258.930	0.0718754
$236$	1740.79	0.480152
$237$	−5500.00	−1.50744
$238$	927.194	0.252525
$239$	−1926.68	−0.521450	−0.260725	−	0.965413i	$-0.583962\pi$
$239$	−1926.68	−0.521450	−0.260725	+	0.965413i	$0.583962\pi$
$240$	2635.63	0.708871
$241$	2237.67	0.598095	0.299048	−	0.954238i	$-0.403331\pi$
$241$	2237.67	0.598095	0.299048	+	0.954238i	$0.403331\pi$
$242$	430.276	0.114294
$243$	−4149.01	−1.09531
$244$	2183.65	0.572926
$245$	1694.36	0.441832
$246$	8644.74	2.24052
$247$	−356.796	−0.0919126
$248$	518.187	0.132681
$249$	8330.33	2.12013
$250$	−444.500	−0.112450
$251$	−6247.82	−1.57115	−0.785575	−	0.618766i	$-0.787633\pi$
$251$	−6247.82	−1.57115	−0.785575	+	0.618766i	$0.787633\pi$
$252$	−159.216	−0.0398003
$253$	−752.352	−0.186956
$254$	4843.22	1.19642
$255$	4250.31	1.04378
$256$	5194.99	1.26831
$257$	−2215.15	−0.537655	−0.268827	−	0.963188i	$-0.586636\pi$
$257$	−2215.15	−0.537655	−0.268827	+	0.963188i	$0.586636\pi$
$258$	−7034.89	−1.69757
$259$	289.165	0.0693739
$260$	436.147	0.104033
$261$	−1515.58	−0.359433
$262$	−2832.66	−0.667948
$263$	3788.00	0.888130	0.444065	−	0.895995i	$-0.353536\pi$
$263$	3788.00	0.888130	0.444065	+	0.895995i	$0.353536\pi$
$264$	869.204	0.202636
$265$	−176.042	−0.0408081
$266$	−137.266	−0.0316403
$267$	−3394.08	−0.777957
$268$	3170.99	0.722757
$269$	−3992.71	−0.904981	−0.452490	−	0.891769i	$-0.649464\pi$
$269$	−3992.71	−0.904981	−0.452490	+	0.891769i	$0.649464\pi$
$270$	1192.84	0.268867
$271$	1481.05	0.331983	0.165992	−	0.986127i	$-0.446918\pi$
$271$	1481.05	0.331983	0.165992	+	0.986127i	$0.446918\pi$
$272$	10213.8	2.27684
$273$	252.699	0.0560222
$274$	−5727.21	−1.26275
$275$	−275.000	−0.0603023
$276$	2104.33	0.458934
$277$	64.3176	0.0139512	0.00697558	−	0.999976i	$-0.497780\pi$
$277$	64.3176	0.0139512	0.00697558	+	0.999976i	$0.497780\pi$
$278$	5090.50	1.09823
$279$	−732.807	−0.157247
$280$	−121.187	−0.0258655
$281$	−4851.92	−1.03004	−0.515020	−	0.857178i	$-0.672216\pi$
$281$	−4851.92	−1.03004	−0.515020	+	0.857178i	$0.672216\pi$
$282$	−1219.73	−0.257566
$283$	3689.34	0.774943	0.387471	−	0.921882i	$-0.373349\pi$
$283$	3689.34	0.774943	0.387471	+	0.921882i	$0.373349\pi$
$284$	−2955.25	−0.617471
$285$	−629.235	−0.130781
$286$	734.549	0.151870
$287$	−745.674	−0.153365
$288$	−3164.34	−0.647433
$289$	11558.1	2.35256
$290$	1597.23	0.323422
$291$	−3814.96	−0.768511
$292$	−1860.81	−0.372930
$293$	622.035	0.124026	0.0620131	−	0.998075i	$-0.480248\pi$
$293$	622.035	0.124026	0.0620131	+	0.998075i	$0.480248\pi$
$294$	−7981.54	−1.58331
$295$	−1873.79	−0.369818
$296$	1698.00	0.333426
$297$	737.980	0.144182
$298$	642.362	0.124869
$299$	−1284.38	−0.248421
$300$	769.175	0.148028
$301$	606.812	0.116200
$302$	−5511.82	−1.05023
$303$	−6442.66	−1.22152
$304$	−1512.09	−0.285278
$305$	−2350.48	−0.441273
$306$	−7699.56	−1.43841
$307$	−1729.87	−0.321593	−0.160796	−	0.986988i	$-0.551406\pi$
$307$	−1729.87	−0.321593	−0.160796	+	0.986988i	$0.551406\pi$
$308$	103.809	0.0192048
$309$	752.609	0.138558
$310$	772.286	0.141493
$311$	1861.95	0.339490	0.169745	−	0.985488i	$-0.445706\pi$
$311$	1861.95	0.339490	0.169745	+	0.985488i	$0.445706\pi$
$312$	1483.87	0.269255
$313$	−1044.72	−0.188662	−0.0943309	−	0.995541i	$-0.530071\pi$
$313$	−1044.72	−0.188662	−0.0943309	+	0.995541i	$0.530071\pi$
$314$	1615.45	0.290334
$315$	171.380	0.0306546
$316$	−3857.17	−0.686655
$317$	−6228.94	−1.10363	−0.551817	−	0.833965i	$-0.686065\pi$
$317$	−6228.94	−1.10363	−0.551817	+	0.833965i	$0.686065\pi$
$318$	829.270	0.146236
$319$	988.161	0.173437
$320$	151.460	0.0264589
$321$	2571.12	0.447058
$322$	−494.126	−0.0855173
$323$	−2438.46	−0.420060
$324$	−4180.07	−0.716747
$325$	−469.469	−0.0801275
$326$	2174.61	0.369450
$327$	4524.50	0.765154
$328$	−4378.66	−0.737106
$329$	105.211	0.0176306
$330$	1295.43	0.216094
$331$	−755.428	−0.125444	−0.0627221	−	0.998031i	$-0.519978\pi$
$331$	−755.428	−0.125444	−0.0627221	+	0.998031i	$0.519978\pi$
$332$	5842.10	0.965744
$333$	−2401.27	−0.395161
$334$	−1337.22	−0.219070
$335$	−3413.25	−0.556675
$336$	1070.93	0.173881
$337$	721.457	0.116618	0.0583090	−	0.998299i	$-0.481429\pi$
$337$	721.457	0.116618	0.0583090	+	0.998299i	$0.481429\pi$
$338$	−6558.53	−1.05544
$339$	−2739.34	−0.438881
$340$	2980.76	0.475454
$341$	477.792	0.0758766
$342$	1139.88	0.180227
$343$	1385.32	0.218077
$344$	3563.25	0.558481
$345$	−2265.10	−0.353475
$346$	5043.79	0.783687
$347$	−10349.7	−1.60115	−0.800575	−	0.599233i	$-0.795472\pi$
$347$	−10349.7	−1.60115	−0.800575	+	0.599233i	$0.795472\pi$
$348$	−2763.89	−0.425747
$349$	−6974.56	−1.06974	−0.534870	−	0.844934i	$-0.679640\pi$
$349$	−6974.56	−1.06974	−0.534870	+	0.844934i	$0.679640\pi$
$350$	−180.613	−0.0275834
$351$	1259.85	0.191583
$352$	2063.16	0.312406
$353$	−4958.12	−0.747575	−0.373788	−	0.927514i	$-0.621941\pi$
$353$	−4958.12	−0.747575	−0.373788	+	0.927514i	$0.621941\pi$
$354$	8826.76	1.32525
$355$	3181.03	0.475582
$356$	−2380.29	−0.354368
$357$	1727.02	0.256033
$358$	5225.10	0.771382
$359$	2029.28	0.298332	0.149166	−	0.988812i	$-0.452341\pi$
$359$	2029.28	0.298332	0.149166	+	0.988812i	$0.452341\pi$
$360$	1006.36	0.147333
$361$	361.000	0.0526316
$362$	−13816.9	−2.00608
$363$	801.447	0.115882
$364$	177.219	0.0255187
$365$	2002.97	0.287234
$366$	11072.3	1.58131
$367$	−11859.7	−1.68684	−0.843422	−	0.537251i	$-0.819463\pi$
$367$	−11859.7	−1.68684	−0.843422	+	0.537251i	$0.819463\pi$
$368$	−5443.19	−0.771048
$369$	6192.19	0.873585
$370$	2530.63	0.355571
$371$	−71.5309	−0.0100100
$372$	−1336.39	−0.186259
$373$	−9368.52	−1.30049	−0.650246	−	0.759724i	$-0.725334\pi$
$373$	−9368.52	−1.30049	−0.650246	+	0.759724i	$0.725334\pi$
$374$	5020.13	0.694077
$375$	−827.941	−0.114012
$376$	617.806	0.0847364
$377$	1686.95	0.230457
$378$	484.687	0.0659513
$379$	13120.7	1.77828	0.889139	−	0.457637i	$-0.151304\pi$
$379$	13120.7	1.77828	0.889139	+	0.457637i	$0.151304\pi$
$380$	−441.285	−0.0595723
$381$	9021.16	1.21304
$382$	−4048.38	−0.542233
$383$	−10579.2	−1.41142	−0.705709	−	0.708502i	$-0.749371\pi$
$383$	−10579.2	−1.41142	−0.705709	+	0.708502i	$0.749371\pi$
$384$	9224.99	1.22594
$385$	−111.741	−0.0147918
$386$	−1113.26	−0.146796
$387$	−5039.06	−0.661886
$388$	−2675.45	−0.350065
$389$	4479.17	0.583812	0.291906	−	0.956447i	$-0.405711\pi$
$389$	4479.17	0.583812	0.291906	+	0.956447i	$0.405711\pi$
$390$	2211.50	0.287138
$391$	−8777.88	−1.13534
$392$	4042.74	0.520891
$393$	−5276.21	−0.677226
$394$	14793.9	1.89164
$395$	4151.86	0.528868
$396$	−862.049	−0.109393
$397$	−1849.10	−0.233763	−0.116881	−	0.993146i	$-0.537290\pi$
$397$	−1849.10	−0.233763	−0.116881	+	0.993146i	$0.537290\pi$
$398$	−1599.28	−0.201418
$399$	−255.677	−0.0320798
$400$	−1989.60	−0.248699
$401$	1261.89	0.157147	0.0785733	−	0.996908i	$-0.474964\pi$
$401$	1261.89	0.157147	0.0785733	+	0.996908i	$0.474964\pi$
$402$	16078.6	1.99485
$403$	815.668	0.100822
$404$	−4518.27	−0.556417
$405$	4499.43	0.552045
$406$	648.999	0.0793332
$407$	1565.63	0.190677
$408$	10141.2	1.23055
$409$	−3119.09	−0.377089	−0.188544	−	0.982065i	$-0.560377\pi$
$409$	−3119.09	−0.377089	−0.188544	+	0.982065i	$0.560377\pi$
$410$	−6525.78	−0.786062
$411$	−10667.7	−1.28029
$412$	527.808	0.0631146
$413$	−761.376	−0.0907139
$414$	4103.30	0.487116
$415$	−6288.44	−0.743826
$416$	3522.14	0.415114
$417$	9481.75	1.11348
$418$	−743.203	−0.0869647
$419$	1487.06	0.173383	0.0866916	−	0.996235i	$-0.472371\pi$
$419$	1487.06	0.173383	0.0866916	+	0.996235i	$0.472371\pi$
$420$	312.538	0.0363102
$421$	−4548.51	−0.526557	−0.263279	−	0.964720i	$-0.584804\pi$
$421$	−4548.51	−0.526557	−0.263279	+	0.964720i	$0.584804\pi$
$422$	10602.8	1.22308
$423$	−873.686	−0.100426
$424$	−420.035	−0.0481101
$425$	−3208.50	−0.366200
$426$	−14984.7	−1.70425
$427$	−955.070	−0.108241
$428$	1803.14	0.203640
$429$	1368.20	0.153979
$430$	5310.53	0.595573
$431$	11111.9	1.24186	0.620930	−	0.783866i	$-0.286755\pi$
$431$	11111.9	1.24186	0.620930	+	0.783866i	$0.286755\pi$
$432$	5339.20	0.594636
$433$	−4335.51	−0.481181	−0.240591	−	0.970627i	$-0.577341\pi$
$433$	−4335.51	−0.481181	−0.240591	+	0.970627i	$0.577341\pi$
$434$	313.802	0.0347073
$435$	2975.05	0.327914
$436$	3173.05	0.348536
$437$	1299.52	0.142252
$438$	−9435.30	−1.02931
$439$	9210.53	1.00135	0.500677	−	0.865634i	$-0.333084\pi$
$439$	9210.53	1.00135	0.500677	+	0.865634i	$0.333084\pi$
$440$	−656.149	−0.0710924
$441$	−5717.15	−0.617336
$442$	8570.17	0.922266
$443$	−86.1483	−0.00923935	−0.00461967	−	0.999989i	$-0.501470\pi$
$443$	−86.1483	−0.00923935	−0.00461967	+	0.999989i	$0.501470\pi$
$444$	−4379.08	−0.468067
$445$	2562.14	0.272938
$446$	10428.7	1.10720
$447$	1196.49	0.126604
$448$	61.5425	0.00649020
$449$	3806.18	0.400055	0.200027	−	0.979790i	$-0.435897\pi$
$449$	3806.18	0.400055	0.200027	+	0.979790i	$0.435897\pi$
$450$	1499.84	0.157118
$451$	−4037.33	−0.421531
$452$	−1921.11	−0.199915
$453$	−10266.5	−1.06482
$454$	−3609.64	−0.373147
$455$	−190.759	−0.0196548
$456$	−1501.35	−0.154183
$457$	−14610.5	−1.49551	−0.747756	−	0.663974i	$-0.768869\pi$
$457$	−14610.5	−1.49551	−0.747756	+	0.663974i	$0.768869\pi$
$458$	11680.2	1.19166
$459$	8610.20	0.875576
$460$	−1588.53	−0.161012
$461$	−10506.5	−1.06147	−0.530735	−	0.847538i	$-0.678084\pi$
$461$	−10506.5	−1.06147	−0.530735	+	0.847538i	$0.678084\pi$
$462$	526.370	0.0530064
$463$	12658.3	1.27058	0.635292	−	0.772272i	$-0.280880\pi$
$463$	12658.3	1.27058	0.635292	+	0.772272i	$0.280880\pi$
$464$	7149.24	0.715291
$465$	1438.49	0.143459
$466$	−8709.43	−0.865787
$467$	−9572.46	−0.948523	−0.474262	−	0.880384i	$-0.657285\pi$
$467$	−9572.46	−0.948523	−0.474262	+	0.880384i	$0.657285\pi$
$468$	−1471.66	−0.145357
$469$	−1386.90	−0.136549
$470$	920.754	0.0903643
$471$	3008.99	0.294367
$472$	−4470.86	−0.435991
$473$	3285.48	0.319380
$474$	−19558.0	−1.89520
$475$	475.000	0.0458831
$476$	1211.17	0.116626
$477$	594.003	0.0570179
$478$	−6851.26	−0.655585
$479$	9171.36	0.874843	0.437422	−	0.899256i	$-0.355892\pi$
$479$	9171.36	0.874843	0.437422	+	0.899256i	$0.355892\pi$
$480$	6211.54	0.590660
$481$	2672.79	0.253365
$482$	7957.15	0.751946
$483$	−920.377	−0.0867051
$484$	562.058	0.0527853
$485$	2879.85	0.269624
$486$	−14753.9	−1.37706
$487$	14170.5	1.31854	0.659269	−	0.751907i	$-0.270866\pi$
$487$	14170.5	1.31854	0.659269	+	0.751907i	$0.270866\pi$
$488$	−5608.24	−0.520232
$489$	4050.51	0.374582
$490$	6025.15	0.555486
$491$	−12909.8	−1.18658	−0.593292	−	0.804987i	$-0.702172\pi$
$491$	−12909.8	−1.18658	−0.593292	+	0.804987i	$0.702172\pi$
$492$	11292.4	1.03476
$493$	11529.1	1.05324
$494$	−1268.77	−0.115556
$495$	927.910	0.0842555
$496$	3456.78	0.312931
$497$	1292.55	0.116657
$498$	29622.6	2.66551
$499$	6754.72	0.605977	0.302989	−	0.952994i	$-0.402016\pi$
$499$	6754.72	0.605977	0.302989	+	0.952994i	$0.402016\pi$
$500$	−580.639	−0.0519339
$501$	−2490.75	−0.222113
$502$	−22217.2	−1.97530
$503$	−4202.12	−0.372492	−0.186246	−	0.982503i	$-0.559632\pi$
$503$	−4202.12	−0.372492	−0.186246	+	0.982503i	$0.559632\pi$
$504$	408.913	0.0361398
$505$	4863.47	0.428558
$506$	−2675.36	−0.235048
$507$	−12216.2	−1.07010
$508$	6326.58	0.552552
$509$	15491.8	1.34905	0.674523	−	0.738254i	$-0.264350\pi$
$509$	15491.8	1.34905	0.674523	+	0.738254i	$0.264350\pi$
$510$	15114.1	1.31228
$511$	813.867	0.0704566
$512$	7331.28	0.632812
$513$	−1274.69	−0.109706
$514$	−7877.07	−0.675958
$515$	−568.133	−0.0486115
$516$	−9189.50	−0.784002
$517$	569.646	0.0484584
$518$	1028.27	0.0872192
$519$	9394.74	0.794573
$520$	−1120.15	−0.0944651
$521$	5852.76	0.492157	0.246079	−	0.969250i	$-0.420858\pi$
$521$	5852.76	0.492157	0.246079	+	0.969250i	$0.420858\pi$
$522$	−5389.39	−0.451891
$523$	22907.7	1.91527	0.957633	−	0.287990i	$-0.0929870\pi$
$523$	22907.7	1.91527	0.957633	+	0.287990i	$0.0929870\pi$
$524$	−3700.23	−0.308484
$525$	−336.416	−0.0279665
$526$	13470.1	1.11659
$527$	5574.53	0.460778
$528$	5798.38	0.477921
$529$	−7489.04	−0.615521
$530$	−626.004	−0.0513054
$531$	6322.58	0.516717
$532$	−179.307	−0.0146127
$533$	−6892.36	−0.560115
$534$	−12069.4	−0.978075
$535$	−1940.90	−0.156845
$536$	−8144.01	−0.656283
$537$	9732.45	0.782097
$538$	−14198.1	−1.13777
$539$	3727.60	0.297883
$540$	1558.18	0.124173
$541$	−18829.9	−1.49642	−0.748208	−	0.663464i	$-0.769086\pi$
$541$	−18829.9	−1.49642	−0.748208	+	0.663464i	$0.769086\pi$
$542$	5266.61	0.417381
$543$	−25735.9	−2.03395
$544$	24071.4	1.89716
$545$	−3415.48	−0.268446
$546$	898.598	0.0704330
$547$	−7921.96	−0.619230	−0.309615	−	0.950862i	$-0.600200\pi$
$547$	−7921.96	−0.619230	−0.309615	+	0.950862i	$0.600200\pi$
$548$	−7481.31	−0.583186
$549$	7931.05	0.616555
$550$	−977.899	−0.0758141
$551$	−1706.82	−0.131966
$552$	−5404.52	−0.416724
$553$	1687.02	0.129728
$554$	228.713	0.0175399
$555$	4713.64	0.360510
$556$	6649.59	0.507204
$557$	−17173.0	−1.30636	−0.653181	−	0.757201i	$-0.726566\pi$
$557$	−17173.0	−1.30636	−0.653181	+	0.757201i	$0.726566\pi$
$558$	−2605.86	−0.197697
$559$	5608.84	0.424381
$560$	−808.431	−0.0610044
$561$	9350.68	0.703719
$562$	−17253.4	−1.29500
$563$	25463.5	1.90614	0.953072	−	0.302743i	$-0.0979025\pi$
$563$	25463.5	1.90614	0.953072	+	0.302743i	$0.0979025\pi$
$564$	−1593.30	−0.118954
$565$	2067.89	0.153976
$566$	13119.3	0.974285
$567$	1828.25	0.135413
$568$	7589.93	0.560680
$569$	−12332.0	−0.908585	−0.454293	−	0.890853i	$-0.650108\pi$
$569$	−12332.0	−0.908585	−0.454293	+	0.890853i	$0.650108\pi$
$570$	−2237.56	−0.164423
$571$	5342.66	0.391565	0.195782	−	0.980647i	$-0.437275\pi$
$571$	5342.66	0.391565	0.195782	+	0.980647i	$0.437275\pi$
$572$	959.523	0.0701393
$573$	−7540.65	−0.549765
$574$	−2651.62	−0.192816
$575$	1709.89	0.124013
$576$	−511.058	−0.0369689
$577$	18443.8	1.33072	0.665358	−	0.746524i	$-0.268279\pi$
$577$	18443.8	1.33072	0.665358	+	0.746524i	$0.268279\pi$
$578$	41100.6	2.95771
$579$	−2073.59	−0.148835
$580$	2086.42	0.149368
$581$	−2555.18	−0.182456
$582$	−13566.0	−0.966199
$583$	−387.292	−0.0275128
$584$	4779.09	0.338630
$585$	1584.09	0.111956
$586$	2211.95	0.155930
$587$	27202.4	1.91271	0.956356	−	0.292203i	$-0.0943883\pi$
$587$	27202.4	1.91271	0.956356	+	0.292203i	$0.0943883\pi$
$588$	−10426.1	−0.731232
$589$	−825.278	−0.0577334
$590$	−6663.19	−0.464948
$591$	27555.7	1.91792
$592$	11327.2	0.786393
$593$	12203.9	0.845115	0.422557	−	0.906336i	$-0.361133\pi$
$593$	12203.9	0.845115	0.422557	+	0.906336i	$0.361133\pi$
$594$	2624.25	0.181270
$595$	−1303.71	−0.0898264
$596$	839.101	0.0576693
$597$	−2978.87	−0.204216
$598$	−4567.27	−0.312324
$599$	−9661.42	−0.659023	−0.329512	−	0.944152i	$-0.606884\pi$
$599$	−9661.42	−0.659023	−0.329512	+	0.944152i	$0.606884\pi$
$600$	−1975.46	−0.134413
$601$	−12317.1	−0.835981	−0.417990	−	0.908451i	$-0.637265\pi$
$601$	−12317.1	−0.835981	−0.417990	+	0.908451i	$0.637265\pi$
$602$	2157.82	0.146090
$603$	11517.1	0.777797
$604$	−7199.95	−0.485036
$605$	−605.000	−0.0406558
$606$	−22910.1	−1.53574
$607$	−1513.16	−0.101181	−0.0505907	−	0.998719i	$-0.516110\pi$
$607$	−1513.16	−0.101181	−0.0505907	+	0.998719i	$0.516110\pi$
$608$	−3563.64	−0.237705
$609$	1208.85	0.0804352
$610$	−8358.31	−0.554784
$611$	972.476	0.0643898
$612$	−10057.7	−0.664314
$613$	18887.0	1.24444	0.622218	−	0.782844i	$-0.286232\pi$
$613$	18887.0	1.24444	0.622218	+	0.782844i	$0.286232\pi$
$614$	−6151.42	−0.404318
$615$	−12155.2	−0.796981
$616$	−266.612	−0.0174385
$617$	−27905.7	−1.82081	−0.910405	−	0.413718i	$-0.864230\pi$
$617$	−27905.7	−1.82081	−0.910405	+	0.413718i	$0.864230\pi$
$618$	2676.27	0.174200
$619$	−9303.29	−0.604089	−0.302044	−	0.953294i	$-0.597669\pi$
$619$	−9303.29	−0.604089	−0.302044	+	0.953294i	$0.597669\pi$
$620$	1008.82	0.0653469
$621$	−4588.60	−0.296512
$622$	6621.08	0.426819
$623$	1041.07	0.0669498
$624$	9898.76	0.635044
$625$	625.000	0.0400000
$626$	−3715.03	−0.237192
$627$	−1384.32	−0.0881727
$628$	2110.22	0.134087
$629$	18266.7	1.15793
$630$	609.428	0.0385400
$631$	−15151.5	−0.955895	−0.477948	−	0.878388i	$-0.658619\pi$
$631$	−15151.5	−0.955895	−0.477948	+	0.878388i	$0.658619\pi$
$632$	9906.33	0.623501
$633$	19749.2	1.24007
$634$	−22150.1	−1.38753
$635$	−6809.94	−0.425581
$636$	1083.25	0.0675375
$637$	6363.60	0.395817
$638$	3513.90	0.218051
$639$	−10733.5	−0.664492
$640$	−6963.81	−0.430107
$641$	−19587.3	−1.20694	−0.603471	−	0.797385i	$-0.706216\pi$
$641$	−19587.3	−1.20694	−0.603471	+	0.797385i	$0.706216\pi$
$642$	9142.88	0.562057
$643$	−3234.59	−0.198382	−0.0991912	−	0.995068i	$-0.531626\pi$
$643$	−3234.59	−0.198382	−0.0991912	+	0.995068i	$0.531626\pi$
$644$	−645.464	−0.0394951
$645$	9891.58	0.603846
$646$	−8671.14	−0.528114
$647$	−192.466	−0.0116949	−0.00584746	−	0.999983i	$-0.501861\pi$
$647$	−192.466	−0.0116949	−0.00584746	+	0.999983i	$0.501861\pi$
$648$	10735.6	0.650825
$649$	−4122.34	−0.249331
$650$	−1669.43	−0.100739
$651$	584.499	0.0351894
$652$	2840.64	0.170626
$653$	−3595.41	−0.215466	−0.107733	−	0.994180i	$-0.534359\pi$
$653$	−3595.41	−0.215466	−0.107733	+	0.994180i	$0.534359\pi$
$654$	16089.1	0.961978
$655$	3982.93	0.237597
$656$	−29209.6	−1.73848
$657$	−6758.47	−0.401329
$658$	374.129	0.0221658
$659$	10575.7	0.625146	0.312573	−	0.949894i	$-0.398809\pi$
$659$	10575.7	0.625146	0.312573	+	0.949894i	$0.398809\pi$
$660$	1692.19	0.0998003
$661$	15026.1	0.884189	0.442095	−	0.896968i	$-0.354236\pi$
$661$	15026.1	0.884189	0.442095	+	0.896968i	$0.354236\pi$
$662$	−2686.30	−0.157713
$663$	15963.1	0.935077
$664$	−15004.2	−0.876922
$665$	193.006	0.0112548
$666$	−8538.90	−0.496811
$667$	−6144.17	−0.356677
$668$	−1746.78	−0.101175
$669$	19424.9	1.12258
$670$	−12137.5	−0.699871
$671$	−5171.06	−0.297506
$672$	2523.93	0.144885
$673$	−32280.0	−1.84889	−0.924444	−	0.381318i	$-0.875470\pi$
$673$	−32280.0	−1.84889	−0.924444	+	0.381318i	$0.875470\pi$
$674$	2565.50	0.146616
$675$	−1677.23	−0.0956392
$676$	−8567.25	−0.487440
$677$	−7504.25	−0.426015	−0.213007	−	0.977051i	$-0.568326\pi$
$677$	−7504.25	−0.426015	−0.213007	+	0.977051i	$0.568326\pi$
$678$	−9741.09	−0.551776
$679$	1170.17	0.0661369
$680$	−7655.46	−0.431725
$681$	−6723.45	−0.378330
$682$	1699.03	0.0953947
$683$	−16560.3	−0.927761	−0.463880	−	0.885898i	$-0.653543\pi$
$683$	−16560.3	−0.927761	−0.463880	+	0.885898i	$0.653543\pi$
$684$	1488.99	0.0832355
$685$	8052.89	0.449175
$686$	4926.20	0.274174
$687$	21756.0	1.20822
$688$	23770.1	1.31719
$689$	−661.169	−0.0365581
$690$	−8054.69	−0.444401
$691$	−30146.7	−1.65968	−0.829838	−	0.558004i	$-0.811567\pi$
$691$	−30146.7	−1.65968	−0.829838	+	0.558004i	$0.811567\pi$
$692$	6588.58	0.361937
$693$	377.037	0.0206673
$694$	−36803.3	−2.01302
$695$	−7157.63	−0.390654
$696$	7098.46	0.386589
$697$	−47104.5	−2.55984
$698$	−24801.5	−1.34491
$699$	−16222.5	−0.877813
$700$	−235.930	−0.0127390
$701$	−16632.4	−0.896143	−0.448071	−	0.893998i	$-0.647889\pi$
$701$	−16632.4	−0.896143	−0.448071	+	0.893998i	$0.647889\pi$
$702$	4480.02	0.240865
$703$	−2704.28	−0.145084
$704$	333.211	0.0178386
$705$	1715.03	0.0916195
$706$	−17631.1	−0.939878
$707$	1976.17	0.105122
$708$	11530.2	0.612049
$709$	23519.2	1.24582	0.622908	−	0.782295i	$-0.285951\pi$
$709$	23519.2	1.24582	0.622908	+	0.782295i	$0.285951\pi$
$710$	11311.7	0.597918
$711$	−14009.3	−0.738945
$712$	6113.26	0.321776
$713$	−2970.81	−0.156042
$714$	6141.29	0.321894
$715$	−1032.83	−0.0540220
$716$	6825.41	0.356254
$717$	−12761.4	−0.664691
$718$	7216.11	0.375074
$719$	17047.8	0.884249	0.442124	−	0.896954i	$-0.354225\pi$
$719$	17047.8	0.884249	0.442124	+	0.896954i	$0.354225\pi$
$720$	6713.33	0.347488
$721$	−230.849	−0.0119241
$722$	1283.71	0.0661702
$723$	14821.3	0.762391
$724$	−18048.7	−0.926484
$725$	−2245.82	−0.115045
$726$	2849.94	0.145690
$727$	9340.76	0.476520	0.238260	−	0.971201i	$-0.423423\pi$
$727$	9340.76	0.476520	0.238260	+	0.971201i	$0.423423\pi$
$728$	−455.150	−0.0231717
$729$	−3184.17	−0.161773
$730$	7122.57	0.361121
$731$	38332.6	1.93951
$732$	14463.5	0.730308
$733$	31801.8	1.60249	0.801246	−	0.598335i	$-0.204171\pi$
$733$	31801.8	1.60249	0.801246	+	0.598335i	$0.204171\pi$
$734$	−42173.1	−2.12076
$735$	11222.7	0.563202
$736$	−12828.3	−0.642468
$737$	−7509.16	−0.375310
$738$	22019.4	1.09830
$739$	14361.5	0.714880	0.357440	−	0.933936i	$-0.383650\pi$
$739$	14361.5	0.714880	0.357440	+	0.933936i	$0.383650\pi$
$740$	3305.70	0.164216
$741$	−2363.25	−0.117161
$742$	−254.363	−0.0125849
$743$	22926.6	1.13203	0.566014	−	0.824396i	$-0.308485\pi$
$743$	22926.6	1.13203	0.566014	+	0.824396i	$0.308485\pi$
$744$	3432.22	0.169128
$745$	−903.209	−0.0444175
$746$	−33314.4	−1.63502
$747$	21218.6	1.03929
$748$	6557.68	0.320551
$749$	−788.643	−0.0384732
$750$	−2944.15	−0.143340
$751$	−16355.6	−0.794704	−0.397352	−	0.917666i	$-0.630071\pi$
$751$	−16355.6	−0.794704	−0.397352	+	0.917666i	$0.630071\pi$
$752$	4121.33	0.199853
$753$	−41382.6	−2.00274
$754$	5998.78	0.289738
$755$	7750.03	0.373580
$756$	633.134	0.0304588
$757$	−14584.0	−0.700215	−0.350108	−	0.936709i	$-0.613855\pi$
$757$	−14584.0	−0.700215	−0.350108	+	0.936709i	$0.613855\pi$
$758$	46657.3	2.23571
$759$	−4983.22	−0.238313
$760$	1133.35	0.0540932
$761$	−26442.2	−1.25956	−0.629782	−	0.776772i	$-0.716856\pi$
$761$	−26442.2	−1.25956	−0.629782	+	0.776772i	$0.716856\pi$
$762$	32079.2	1.52507
$763$	−1387.81	−0.0658480
$764$	−5288.29	−0.250424
$765$	10826.2	0.511661
$766$	−37619.7	−1.77448
$767$	−7037.49	−0.331303
$768$	34409.2	1.61671
$769$	2313.64	0.108494	0.0542470	−	0.998528i	$-0.482724\pi$
$769$	2313.64	0.108494	0.0542470	+	0.998528i	$0.482724\pi$
$770$	−397.349	−0.0185967
$771$	−14672.1	−0.685348
$772$	−1454.22	−0.0677960
$773$	19113.5	0.889346	0.444673	−	0.895693i	$-0.353320\pi$
$773$	19113.5	0.889346	0.444673	+	0.895693i	$0.353320\pi$
$774$	−17918.9	−0.832146
$775$	−1085.89	−0.0503308
$776$	6871.32	0.317868
$777$	1915.29	0.0884307
$778$	15927.9	0.733988
$779$	6973.56	0.320737
$780$	2888.83	0.132611
$781$	6998.27	0.320637
$782$	−31214.1	−1.42738
$783$	6026.80	0.275071
$784$	26968.8	1.22853
$785$	−2271.44	−0.103275
$786$	−18762.2	−0.851432
$787$	19512.0	0.883771	0.441886	−	0.897071i	$-0.354310\pi$
$787$	19512.0	0.883771	0.441886	+	0.897071i	$0.354310\pi$
$788$	19324.9	0.873633
$789$	25089.9	1.13210
$790$	14764.0	0.664911
$791$	840.243	0.0377694
$792$	2213.99	0.0993317
$793$	−8827.83	−0.395316
$794$	−6575.40	−0.293895
$795$	−1166.02	−0.0520180
$796$	−2089.09	−0.0930226
$797$	−34672.5	−1.54098	−0.770492	−	0.637450i	$-0.779989\pi$
$797$	−34672.5	−1.54098	−0.770492	+	0.637450i	$0.779989\pi$
$798$	−909.185	−0.0403318
$799$	6646.20	0.294275
$800$	−4689.00	−0.207226
$801$	−8645.23	−0.381354
$802$	4487.28	0.197570
$803$	4406.54	0.193653
$804$	21003.1	0.921297
$805$	694.778	0.0304195
$806$	2900.51	0.126757
$807$	−26445.8	−1.15358
$808$	11604.2	0.505241
$809$	−13939.7	−0.605802	−0.302901	−	0.953022i	$-0.597955\pi$
$809$	−13939.7	−0.605802	−0.302901	+	0.953022i	$0.597955\pi$
$810$	16000.0	0.694051
$811$	−18909.7	−0.818755	−0.409377	−	0.912365i	$-0.634254\pi$
$811$	−18909.7	−0.818755	−0.409377	+	0.912365i	$0.634254\pi$
$812$	847.772	0.0366391
$813$	9809.78	0.423178
$814$	5567.39	0.239726
$815$	−3057.67	−0.131418
$816$	67651.2	2.90229
$817$	−5674.92	−0.243012
$818$	−11091.5	−0.474089
$819$	643.662	0.0274620
$820$	−8524.47	−0.363033
$821$	3660.58	0.155609	0.0778045	−	0.996969i	$-0.475209\pi$
$821$	3660.58	0.155609	0.0778045	+	0.996969i	$0.475209\pi$
$822$	−37934.3	−1.60962
$823$	190.131	0.00805292	0.00402646	−	0.999992i	$-0.498718\pi$
$823$	190.131	0.00805292	0.00402646	+	0.999992i	$0.498718\pi$
$824$	−1355.56	−0.0573098
$825$	−1821.47	−0.0768672
$826$	−2707.45	−0.114049
$827$	42596.9	1.79110	0.895550	−	0.444960i	$-0.146782\pi$
$827$	42596.9	1.79110	0.895550	+	0.444960i	$0.146782\pi$
$828$	5360.03	0.224969
$829$	−31088.8	−1.30248	−0.651242	−	0.758870i	$-0.725752\pi$
$829$	−31088.8	−1.30248	−0.651242	+	0.758870i	$0.725752\pi$
$830$	−22361.7	−0.935163
$831$	426.009	0.0177835
$832$	568.845	0.0237033
$833$	43490.8	1.80896
$834$	33717.0	1.39991
$835$	1880.23	0.0779259
$836$	−970.828	−0.0401636
$837$	2914.06	0.120340
$838$	5287.98	0.217983
$839$	−12089.2	−0.497456	−0.248728	−	0.968573i	$-0.580013\pi$
$839$	−12089.2	−0.497456	−0.248728	+	0.968573i	$0.580013\pi$
$840$	−802.688	−0.0329707
$841$	−16319.1	−0.669116
$842$	−16174.5	−0.662006
$843$	−32136.8	−1.31299
$844$	13850.2	0.564863
$845$	9221.79	0.375431
$846$	−3106.82	−0.126259
$847$	−245.829	−0.00997260
$848$	−2802.01	−0.113469
$849$	24436.5	0.987818
$850$	−11409.4	−0.460399
$851$	−9734.77	−0.392131
$852$	−19574.2	−0.787089
$853$	−7367.29	−0.295722	−0.147861	−	0.989008i	$-0.547239\pi$
$853$	−7367.29	−0.295722	−0.147861	+	0.989008i	$0.547239\pi$
$854$	−3396.23	−0.136085
$855$	−1602.75	−0.0641088
$856$	−4630.97	−0.184911
$857$	35901.9	1.43102	0.715510	−	0.698602i	$-0.246194\pi$
$857$	35901.9	1.43102	0.715510	+	0.698602i	$0.246194\pi$
$858$	4865.31	0.193588
$859$	11900.5	0.472691	0.236345	−	0.971669i	$-0.424050\pi$
$859$	11900.5	0.472691	0.236345	+	0.971669i	$0.424050\pi$
$860$	6937.01	0.275058
$861$	−4938.99	−0.195494
$862$	39513.9	1.56131
$863$	−19339.6	−0.762836	−0.381418	−	0.924403i	$-0.624564\pi$
$863$	−19339.6	−0.762836	−0.381418	+	0.924403i	$0.624564\pi$
$864$	12583.2	0.495474
$865$	−7091.95	−0.278767
$866$	−15417.1	−0.604958
$867$	76555.4	2.99880
$868$	409.912	0.0160292
$869$	9134.10	0.356563
$870$	10579.3	0.412265
$871$	−12819.3	−0.498699
$872$	−8149.31	−0.316480
$873$	−9717.26	−0.376723
$874$	4621.08	0.178845
$875$	253.956	0.00981174
$876$	−12325.1	−0.475373
$877$	−50436.3	−1.94198	−0.970988	−	0.239130i	$-0.923138\pi$
$877$	−50436.3	−1.94198	−0.970988	+	0.239130i	$0.923138\pi$
$878$	32752.6	1.25894
$879$	4120.06	0.158096
$880$	−4377.11	−0.167673
$881$	−13741.6	−0.525500	−0.262750	−	0.964864i	$-0.584629\pi$
$881$	−13741.6	−0.525500	−0.262750	+	0.964864i	$0.584629\pi$
$882$	−20330.2	−0.776136
$883$	−27411.6	−1.04470	−0.522352	−	0.852730i	$-0.674945\pi$
$883$	−27411.6	−1.04470	−0.522352	+	0.852730i	$0.674945\pi$
$884$	11195.0	0.425937
$885$	−12411.1	−0.471406
$886$	−306.343	−0.0116160
$887$	−31762.6	−1.20235	−0.601174	−	0.799118i	$-0.705300\pi$
$887$	−31762.6	−1.20235	−0.601174	+	0.799118i	$0.705300\pi$
$888$	11246.7	0.425018
$889$	−2767.08	−0.104392
$890$	9110.97	0.343147
$891$	9898.74	0.372189
$892$	13622.7	0.511349
$893$	−983.933	−0.0368713
$894$	4254.70	0.159170
$895$	−7346.88	−0.274390
$896$	−2829.60	−0.105503
$897$	−8507.16	−0.316662
$898$	13534.8	0.502963
$899$	3901.95	0.144758
$900$	1959.20	0.0725630
$901$	−4518.63	−0.167078
$902$	−14356.7	−0.529963
$903$	4019.24	0.148119
$904$	4933.97	0.181528
$905$	19427.6	0.713587
$906$	−36507.7	−1.33873
$907$	−31664.4	−1.15920	−0.579602	−	0.814900i	$-0.696792\pi$
$907$	−31664.4	−1.15920	−0.579602	+	0.814900i	$0.696792\pi$
$908$	−4715.18	−0.172334
$909$	−16410.4	−0.598789
$910$	−678.338	−0.0247106
$911$	10601.3	0.385551	0.192775	−	0.981243i	$-0.438251\pi$
$911$	10601.3	0.385551	0.192775	+	0.981243i	$0.438251\pi$
$912$	−10015.4	−0.363643
$913$	−13834.6	−0.501487
$914$	−51954.8	−1.88021
$915$	−15568.5	−0.562490
$916$	15257.6	0.550355
$917$	1618.38	0.0582810
$918$	30617.8	1.10080
$919$	30143.7	1.08199	0.540995	−	0.841026i	$-0.318048\pi$
$919$	30143.7	1.08199	0.540995	+	0.841026i	$0.318048\pi$
$920$	4079.79	0.146203
$921$	−11457.8	−0.409934
$922$	−37361.1	−1.33452
$923$	11947.2	0.426052
$924$	687.584	0.0244804
$925$	−3558.26	−0.126481
$926$	45012.8	1.59742
$927$	1917.01	0.0679209
$928$	16849.0	0.596009
$929$	11299.3	0.399051	0.199525	−	0.979893i	$-0.436060\pi$
$929$	11299.3	0.399051	0.199525	+	0.979893i	$0.436060\pi$
$930$	5115.25	0.180361
$931$	−6438.58	−0.226655
$932$	−11376.9	−0.399853
$933$	12332.7	0.432748
$934$	−34039.6	−1.19252
$935$	−7058.69	−0.246892
$936$	3779.63	0.131988
$937$	17889.6	0.623722	0.311861	−	0.950128i	$-0.399048\pi$
$937$	17889.6	0.623722	0.311861	+	0.950128i	$0.399048\pi$
$938$	−4931.83	−0.171674
$939$	−6919.74	−0.240487
$940$	1202.76	0.0417336
$941$	−8311.62	−0.287940	−0.143970	−	0.989582i	$-0.545987\pi$
$941$	−8311.62	−0.287940	−0.143970	+	0.989582i	$0.545987\pi$
$942$	10700.0	0.370088
$943$	25103.2	0.866886
$944$	−29824.7	−1.02830
$945$	−681.506	−0.0234597
$946$	11683.2	0.401535
$947$	6398.75	0.219569	0.109784	−	0.993955i	$-0.464984\pi$
$947$	6398.75	0.219569	0.109784	+	0.993955i	$0.464984\pi$
$948$	−25548.1	−0.875278
$949$	7522.67	0.257320
$950$	1689.10	0.0576859
$951$	−41257.6	−1.40680
$952$	−3110.63	−0.105899
$953$	−36676.8	−1.24667	−0.623335	−	0.781955i	$-0.714223\pi$
$953$	−36676.8	−1.24667	−0.623335	+	0.781955i	$0.714223\pi$
$954$	2112.27	0.0716849
$955$	5692.32	0.192879
$956$	−8949.63	−0.302774
$957$	6545.11	0.221080
$958$	32613.3	1.09988
$959$	3272.12	0.110180
$960$	1003.20	0.0337271
$961$	−27904.3	−0.936670
$962$	9504.42	0.318539
$963$	6549.01	0.219147
$964$	10394.2	0.347277
$965$	1565.32	0.0522172
$966$	−3272.86	−0.109009
$967$	54104.1	1.79925	0.899623	−	0.436668i	$-0.143842\pi$
$967$	54104.1	1.79925	0.899623	+	0.436668i	$0.143842\pi$
$968$	−1443.53	−0.0479305
$969$	−16151.2	−0.535450
$970$	10240.7	0.338980
$971$	−37068.9	−1.22513	−0.612564	−	0.790421i	$-0.709862\pi$
$971$	−37068.9	−1.22513	−0.612564	+	0.790421i	$0.709862\pi$
$972$	−19272.6	−0.635977
$973$	−2908.35	−0.0958248
$974$	50390.4	1.65771
$975$	−3109.54	−0.102138
$976$	−37412.1	−1.22698
$977$	18502.7	0.605890	0.302945	−	0.953008i	$-0.402030\pi$
$977$	18502.7	0.605890	0.302945	+	0.953008i	$0.402030\pi$
$978$	14403.6	0.470937
$979$	5636.72	0.184015
$980$	7870.50	0.256545
$981$	11524.6	0.375078
$982$	−45907.3	−1.49181
$983$	−58984.3	−1.91384	−0.956922	−	0.290347i	$-0.906229\pi$
$983$	−58984.3	−1.91384	−0.956922	+	0.290347i	$0.906229\pi$
$984$	−29002.1	−0.939588
$985$	−20801.4	−0.672880
$986$	40997.5	1.32417
$987$	696.866	0.0224737
$988$	−1657.36	−0.0533680
$989$	−20428.4	−0.656811
$990$	3299.65	0.105929
$991$	−33096.5	−1.06089	−0.530447	−	0.847718i	$-0.677976\pi$
$991$	−33096.5	−1.06089	−0.530447	+	0.847718i	$0.677976\pi$
$992$	8146.80	0.260747
$993$	−5003.59	−0.159904
$994$	4596.29	0.146665
$995$	2248.70	0.0716469
$996$	38695.3	1.23103
$997$	6392.68	0.203067	0.101534	−	0.994832i	$-0.467625\pi$
$997$	6392.68	0.203067	0.101534	+	0.994832i	$0.467625\pi$
$998$	24019.8	0.761856
$999$	9548.81	0.302413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.16	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.16	✓	20	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+3.55600 q^{2} +6.62353 q^{3} +4.64511 q^{4} -5.00000 q^{5} +23.5532 q^{6} -2.03165 q^{7} -11.9300 q^{8} +16.8711 q^{9} +O(q^{10})\)	\(q+3.55600 q^{2} +6.62353 q^{3} +4.64511 q^{4} -5.00000 q^{5} +23.5532 q^{6} -2.03165 q^{7} -11.9300 q^{8} +16.8711 q^{9} -17.7800 q^{10} -11.0000 q^{11} +30.7670 q^{12} -18.7788 q^{13} -7.22452 q^{14} -33.1176 q^{15} -79.5838 q^{16} -128.340 q^{17} +59.9936 q^{18} +19.0000 q^{19} -23.2255 q^{20} -13.4567 q^{21} -39.1160 q^{22} +68.3956 q^{23} -79.0185 q^{24} +25.0000 q^{25} -66.7772 q^{26} -67.0891 q^{27} -9.43722 q^{28} -89.8328 q^{29} -117.766 q^{30} -43.4357 q^{31} -187.560 q^{32} -72.8588 q^{33} -456.376 q^{34} +10.1582 q^{35} +78.3681 q^{36} -142.330 q^{37} +67.5639 q^{38} -124.382 q^{39} +59.6499 q^{40} +367.030 q^{41} -47.8518 q^{42} -298.680 q^{43} -51.0962 q^{44} -84.3555 q^{45} +243.215 q^{46} -51.7860 q^{47} -527.126 q^{48} -338.872 q^{49} +88.8999 q^{50} -850.062 q^{51} -87.2294 q^{52} +35.2083 q^{53} -238.568 q^{54} +55.0000 q^{55} +24.2375 q^{56} +125.847 q^{57} -319.445 q^{58} +374.758 q^{59} -153.835 q^{60} +470.097 q^{61} -154.457 q^{62} -34.2761 q^{63} -30.2919 q^{64} +93.8938 q^{65} -259.086 q^{66} +682.651 q^{67} -596.152 q^{68} +453.020 q^{69} +36.1226 q^{70} -636.206 q^{71} -201.272 q^{72} -400.595 q^{73} -506.126 q^{74} +165.588 q^{75} +88.2571 q^{76} +22.3481 q^{77} -442.300 q^{78} -830.373 q^{79} +397.919 q^{80} -899.886 q^{81} +1305.16 q^{82} +1257.69 q^{83} -62.5076 q^{84} +641.699 q^{85} -1062.11 q^{86} -595.010 q^{87} +131.230 q^{88} -512.429 q^{89} -299.968 q^{90} +38.1518 q^{91} +317.705 q^{92} -287.697 q^{93} -184.151 q^{94} -95.0000 q^{95} -1242.31 q^{96} -575.971 q^{97} -1205.03 q^{98} -185.582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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