LMFDB - Embedded newform 1045.4.a.g.1.1

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:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-5.41975 q^{2} +4.17361 q^{3} +21.3737 q^{4} -5.00000 q^{5} -22.6199 q^{6} -25.0607 q^{7} -72.4822 q^{8} -9.58100 q^{9} +O(q^{10})$	\(q-5.41975 q^{2} +4.17361 q^{3} +21.3737 q^{4} -5.00000 q^{5} -22.6199 q^{6} -25.0607 q^{7} -72.4822 q^{8} -9.58100 q^{9} +27.0988 q^{10} -11.0000 q^{11} +89.2055 q^{12} -82.0427 q^{13} +135.823 q^{14} -20.8680 q^{15} +221.846 q^{16} -5.77877 q^{17} +51.9267 q^{18} -19.0000 q^{19} -106.869 q^{20} -104.594 q^{21} +59.6173 q^{22} -202.864 q^{23} -302.512 q^{24} +25.0000 q^{25} +444.651 q^{26} -152.675 q^{27} -535.641 q^{28} -147.937 q^{29} +113.100 q^{30} +9.64993 q^{31} -622.492 q^{32} -45.9097 q^{33} +31.3195 q^{34} +125.304 q^{35} -204.782 q^{36} -294.770 q^{37} +102.975 q^{38} -342.414 q^{39} +362.411 q^{40} +106.299 q^{41} +566.872 q^{42} -268.084 q^{43} -235.111 q^{44} +47.9050 q^{45} +1099.47 q^{46} +379.568 q^{47} +925.898 q^{48} +285.040 q^{49} -135.494 q^{50} -24.1183 q^{51} -1753.56 q^{52} +566.101 q^{53} +827.459 q^{54} +55.0000 q^{55} +1816.46 q^{56} -79.2985 q^{57} +801.781 q^{58} -202.796 q^{59} -446.027 q^{60} -360.835 q^{61} -52.3002 q^{62} +240.107 q^{63} +1598.99 q^{64} +410.214 q^{65} +248.819 q^{66} +523.425 q^{67} -123.514 q^{68} -846.674 q^{69} -679.115 q^{70} -454.152 q^{71} +694.452 q^{72} -744.750 q^{73} +1597.58 q^{74} +104.340 q^{75} -406.101 q^{76} +275.668 q^{77} +1855.80 q^{78} +210.822 q^{79} -1109.23 q^{80} -378.517 q^{81} -576.116 q^{82} +908.973 q^{83} -2235.55 q^{84} +28.8939 q^{85} +1452.95 q^{86} -617.431 q^{87} +797.304 q^{88} +1202.84 q^{89} -259.633 q^{90} +2056.05 q^{91} -4335.95 q^{92} +40.2750 q^{93} -2057.16 q^{94} +95.0000 q^{95} -2598.04 q^{96} +80.6757 q^{97} -1544.85 q^{98} +105.391 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * 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 $

$2$	−5.41975	−1.91617	−0.958086	−	0.286481i	$-0.907514\pi$
$2$	−5.41975	−1.91617	−0.958086	+	0.286481i	$0.907514\pi$
$3$	4.17361	0.803211	0.401606	−	0.915813i	$-0.368452\pi$
$3$	4.17361	0.803211	0.401606	+	0.915813i	$0.368452\pi$
$4$	21.3737	2.67171
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−22.6199	−1.53909
$7$	−25.0607	−1.35315	−0.676576	−	0.736373i	$-0.736537\pi$
$7$	−25.0607	−1.35315	−0.676576	+	0.736373i	$0.736537\pi$
$8$	−72.4822	−3.20329
$9$	−9.58100	−0.354852
$10$	27.0988	0.856938
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	89.2055	2.14595
$13$	−82.0427	−1.75035	−0.875175	−	0.483806i	$-0.839254\pi$
$13$	−82.0427	−1.75035	−0.875175	+	0.483806i	$0.839254\pi$
$14$	135.823	2.59287
$15$	−20.8680	−0.359207
$16$	221.846	3.46634
$17$	−5.77877	−0.0824446	−0.0412223	−	0.999150i	$-0.513125\pi$
$17$	−5.77877	−0.0824446	−0.0412223	+	0.999150i	$0.513125\pi$
$18$	51.9267	0.679957
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−106.869	−1.19483
$21$	−104.594	−1.08687
$22$	59.6173	0.577748
$23$	−202.864	−1.83913	−0.919566	−	0.392937i	$-0.871459\pi$
$23$	−202.864	−1.83913	−0.919566	+	0.392937i	$0.871459\pi$
$24$	−302.512	−2.57292
$25$	25.0000	0.200000
$26$	444.651	3.35397
$27$	−152.675	−1.08823
$28$	−535.641	−3.61524
$29$	−147.937	−0.947283	−0.473641	−	0.880718i	$-0.657061\pi$
$29$	−147.937	−0.947283	−0.473641	+	0.880718i	$0.657061\pi$
$30$	113.100	0.688302
$31$	9.64993	0.0559090	0.0279545	−	0.999609i	$-0.491101\pi$
$31$	9.64993	0.0559090	0.0279545	+	0.999609i	$0.491101\pi$
$32$	−622.492	−3.43882
$33$	−45.9097	−0.242177
$34$	31.3195	0.157978
$35$	125.304	0.605148
$36$	−204.782	−0.948063
$37$	−294.770	−1.30973	−0.654864	−	0.755747i	$-0.727274\pi$
$37$	−294.770	−1.30973	−0.654864	+	0.755747i	$0.727274\pi$
$38$	102.975	0.439600
$39$	−342.414	−1.40590
$40$	362.411	1.43256
$41$	106.299	0.404907	0.202453	−	0.979292i	$-0.435109\pi$
$41$	106.299	0.404907	0.202453	+	0.979292i	$0.435109\pi$
$42$	566.872	2.08262
$43$	−268.084	−0.950755	−0.475378	−	0.879782i	$-0.657689\pi$
$43$	−268.084	−0.950755	−0.475378	+	0.879782i	$0.657689\pi$
$44$	−235.111	−0.805552
$45$	47.9050	0.158695
$46$	1099.47	3.52409
$47$	379.568	1.17799	0.588997	−	0.808136i	$-0.299523\pi$
$47$	379.568	1.17799	0.588997	+	0.808136i	$0.299523\pi$
$48$	925.898	2.78420
$49$	285.040	0.831021
$50$	−135.494	−0.383234
$51$	−24.1183	−0.0662204
$52$	−1753.56	−4.67644
$53$	566.101	1.46717	0.733585	−	0.679598i	$-0.237846\pi$
$53$	566.101	1.46717	0.733585	+	0.679598i	$0.237846\pi$
$54$	827.459	2.08524
$55$	55.0000	0.134840
$56$	1816.46	4.33454
$57$	−79.2985	−0.184269
$58$	801.781	1.81516
$59$	−202.796	−0.447487	−0.223744	−	0.974648i	$-0.571828\pi$
$59$	−202.796	−0.447487	−0.223744	+	0.974648i	$0.571828\pi$
$60$	−446.027	−0.959698
$61$	−360.835	−0.757380	−0.378690	−	0.925524i	$-0.623625\pi$
$61$	−360.835	−0.757380	−0.378690	+	0.925524i	$0.623625\pi$
$62$	−52.3002	−0.107131
$63$	240.107	0.480169
$64$	1598.99	3.12302
$65$	410.214	0.782781
$66$	248.819	0.464053
$67$	523.425	0.954426	0.477213	−	0.878788i	$-0.341647\pi$
$67$	523.425	0.954426	0.477213	+	0.878788i	$0.341647\pi$
$68$	−123.514	−0.220268
$69$	−846.674	−1.47721
$70$	−679.115	−1.15957
$71$	−454.152	−0.759126	−0.379563	−	0.925166i	$-0.623926\pi$
$71$	−454.152	−0.759126	−0.379563	+	0.925166i	$0.623926\pi$
$72$	694.452	1.13669
$73$	−744.750	−1.19406	−0.597030	−	0.802219i	$-0.703653\pi$
$73$	−744.750	−1.19406	−0.597030	+	0.802219i	$0.703653\pi$
$74$	1597.58	2.50966
$75$	104.340	0.160642
$76$	−406.101	−0.612933
$77$	275.668	0.407991
$78$	1855.80	2.69395
$79$	210.822	0.300244	0.150122	−	0.988667i	$-0.452033\pi$
$79$	210.822	0.300244	0.150122	+	0.988667i	$0.452033\pi$
$80$	−1109.23	−1.55020
$81$	−378.517	−0.519228
$82$	−576.116	−0.775871
$83$	908.973	1.20208	0.601040	−	0.799219i	$-0.294753\pi$
$83$	908.973	1.20208	0.601040	+	0.799219i	$0.294753\pi$
$84$	−2235.55	−2.90380
$85$	28.8939	0.0368703
$86$	1452.95	1.82181
$87$	−617.431	−0.760868
$88$	797.304	0.965829
$89$	1202.84	1.43259	0.716295	−	0.697797i	$-0.245836\pi$
$89$	1202.84	1.43259	0.716295	+	0.697797i	$0.245836\pi$
$90$	−259.633	−0.304086
$91$	2056.05	2.36849
$92$	−4335.95	−4.91363
$93$	40.2750	0.0449067
$94$	−2057.16	−2.25724
$95$	95.0000	0.102598
$96$	−2598.04	−2.76209
$97$	80.6757	0.0844472	0.0422236	−	0.999108i	$-0.486556\pi$
$97$	80.6757	0.0844472	0.0422236	+	0.999108i	$0.486556\pi$
$98$	−1544.85	−1.59238
$99$	105.391	0.106992
$100$	534.343	0.534343
$101$	−232.107	−0.228668	−0.114334	−	0.993442i	$-0.536473\pi$
$101$	−232.107	−0.228668	−0.114334	+	0.993442i	$0.536473\pi$
$102$	130.715	0.126890
$103$	−640.944	−0.613146	−0.306573	−	0.951847i	$-0.599182\pi$
$103$	−640.944	−0.613146	−0.306573	+	0.951847i	$0.599182\pi$
$104$	5946.64	5.60688
$105$	522.968	0.486062
$106$	−3068.13	−2.81135
$107$	−1467.07	−1.32549	−0.662743	−	0.748847i	$-0.730608\pi$
$107$	−1467.07	−1.32549	−0.662743	+	0.748847i	$0.730608\pi$
$108$	−3263.23	−2.90744
$109$	−1584.85	−1.39267	−0.696335	−	0.717717i	$-0.745187\pi$
$109$	−1584.85	−1.39267	−0.696335	+	0.717717i	$0.745187\pi$
$110$	−298.086	−0.258377
$111$	−1230.25	−1.05199
$112$	−5559.62	−4.69049
$113$	278.263	0.231653	0.115826	−	0.993269i	$-0.463048\pi$
$113$	278.263	0.231653	0.115826	+	0.993269i	$0.463048\pi$
$114$	429.778	0.353092
$115$	1014.32	0.822484
$116$	−3161.96	−2.53087
$117$	786.052	0.621116
$118$	1099.10	0.857462
$119$	144.820	0.111560
$120$	1512.56	1.15064
$121$	121.000	0.0909091
$122$	1955.64	1.45127
$123$	443.652	0.325225
$124$	206.255	0.149373
$125$	−125.000	−0.0894427
$126$	−1301.32	−0.920086
$127$	−2238.12	−1.56379	−0.781894	−	0.623411i	$-0.785746\pi$
$127$	−2238.12	−1.56379	−0.781894	+	0.623411i	$0.785746\pi$
$128$	−3686.17	−2.54542
$129$	−1118.88	−0.763657
$130$	−2223.26	−1.49994
$131$	−1063.48	−0.709286	−0.354643	−	0.935002i	$-0.615398\pi$
$131$	−1063.48	−0.709286	−0.354643	+	0.935002i	$0.615398\pi$
$132$	−981.260	−0.647028
$133$	476.154	0.310434
$134$	−2836.84	−1.82884
$135$	763.374	0.486672
$136$	418.858	0.264094
$137$	−1006.07	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$137$	−1006.07	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$138$	4588.76	2.83059
$139$	1050.99	0.641324	0.320662	−	0.947194i	$-0.396095\pi$
$139$	1050.99	0.641324	0.320662	+	0.947194i	$0.396095\pi$
$140$	2678.20	1.61678
$141$	1584.17	0.946177
$142$	2461.39	1.45462
$143$	902.470	0.527751
$144$	−2125.51	−1.23004
$145$	739.685	0.423638
$146$	4036.36	2.28802
$147$	1189.65	0.667485
$148$	−6300.33	−3.49922
$149$	713.659	0.392384	0.196192	−	0.980565i	$-0.437142\pi$
$149$	713.659	0.392384	0.196192	+	0.980565i	$0.437142\pi$
$150$	−565.498	−0.307818
$151$	3069.42	1.65421	0.827107	−	0.562045i	$-0.189985\pi$
$151$	3069.42	1.65421	0.827107	+	0.562045i	$0.189985\pi$
$152$	1377.16	0.734885
$153$	55.3664	0.0292556
$154$	−1494.05	−0.781780
$155$	−48.2497	−0.0250033
$156$	−7318.66	−3.75617
$157$	−2561.60	−1.30215	−0.651077	−	0.759011i	$-0.725683\pi$
$157$	−2561.60	−1.30215	−0.651077	+	0.759011i	$0.725683\pi$
$158$	−1142.60	−0.575320
$159$	2362.68	1.17845
$160$	3112.46	1.53788
$161$	5083.91	2.48862
$162$	2051.47	0.994930
$163$	2082.25	1.00058	0.500289	−	0.865859i	$-0.333227\pi$
$163$	2082.25	1.00058	0.500289	+	0.865859i	$0.333227\pi$
$164$	2272.01	1.08179
$165$	229.548	0.108305
$166$	−4926.41	−2.30339
$167$	−1298.09	−0.601493	−0.300746	−	0.953704i	$-0.597236\pi$
$167$	−1298.09	−0.601493	−0.300746	+	0.953704i	$0.597236\pi$
$168$	7581.18	3.48155
$169$	4534.01	2.06373
$170$	−156.598	−0.0706499
$171$	182.039	0.0814086
$172$	−5729.96	−2.54015
$173$	1663.03	0.730854	0.365427	−	0.930840i	$-0.380923\pi$
$173$	1663.03	0.730854	0.365427	+	0.930840i	$0.380923\pi$
$174$	3346.32	1.45795
$175$	−626.518	−0.270630
$176$	−2440.30	−1.04514
$177$	−846.389	−0.359427
$178$	−6519.08	−2.74509
$179$	−2741.86	−1.14489	−0.572447	−	0.819942i	$-0.694006\pi$
$179$	−2741.86	−1.14489	−0.572447	+	0.819942i	$0.694006\pi$
$180$	1023.91	0.423987
$181$	−2709.69	−1.11276	−0.556381	−	0.830927i	$-0.687811\pi$
$181$	−2709.69	−1.11276	−0.556381	+	0.830927i	$0.687811\pi$
$182$	−11143.3	−4.53844
$183$	−1505.98	−0.608336
$184$	14704.0	5.89127
$185$	1473.85	0.585728
$186$	−218.281	−0.0860490
$187$	63.5665	0.0248580
$188$	8112.78	3.14726
$189$	3826.14	1.47254
$190$	−514.876	−0.196595
$191$	−3677.72	−1.39325	−0.696624	−	0.717437i	$-0.745315\pi$
$191$	−3677.72	−1.39325	−0.696624	+	0.717437i	$0.745315\pi$
$192$	6673.54	2.50844
$193$	−3183.27	−1.18724	−0.593619	−	0.804747i	$-0.702301\pi$
$193$	−3183.27	−1.18724	−0.593619	+	0.804747i	$0.702301\pi$
$194$	−437.242	−0.161815
$195$	1712.07	0.628738
$196$	6092.36	2.22025
$197$	3608.65	1.30510	0.652552	−	0.757744i	$-0.273699\pi$
$197$	3608.65	1.30510	0.652552	+	0.757744i	$0.273699\pi$
$198$	−571.193	−0.205015
$199$	2698.25	0.961177	0.480588	−	0.876946i	$-0.340423\pi$
$199$	2698.25	0.961177	0.480588	+	0.876946i	$0.340423\pi$
$200$	−1812.06	−0.640658
$201$	2184.57	0.766606
$202$	1257.96	0.438168
$203$	3707.41	1.28182
$204$	−515.498	−0.176922
$205$	−531.497	−0.181080
$206$	3473.76	1.17489
$207$	1943.64	0.652619
$208$	−18200.8	−6.06731
$209$	209.000	0.0691714
$210$	−2834.36	−0.931377
$211$	−2591.90	−0.845656	−0.422828	−	0.906210i	$-0.638963\pi$
$211$	−2591.90	−0.845656	−0.422828	+	0.906210i	$0.638963\pi$
$212$	12099.7	3.91986
$213$	−1895.45	−0.609738
$214$	7951.15	2.53986
$215$	1340.42	0.425191
$216$	11066.2	3.48592
$217$	−241.834	−0.0756534
$218$	8589.49	2.66859
$219$	−3108.29	−0.959082
$220$	1175.55	0.360254
$221$	474.106	0.144307
$222$	6667.67	2.01579
$223$	−2100.75	−0.630836	−0.315418	−	0.948953i	$-0.602145\pi$
$223$	−2100.75	−0.630836	−0.315418	+	0.948953i	$0.602145\pi$
$224$	15600.1	4.65324
$225$	−239.525	−0.0709704
$226$	−1508.12	−0.443887
$227$	2155.42	0.630221	0.315111	−	0.949055i	$-0.397958\pi$
$227$	2155.42	0.630221	0.315111	+	0.949055i	$0.397958\pi$
$228$	−1694.90	−0.492315
$229$	2165.43	0.624873	0.312436	−	0.949939i	$-0.398855\pi$
$229$	2165.43	0.624873	0.312436	+	0.949939i	$0.398855\pi$
$230$	−5497.36	−1.57602
$231$	1150.53	0.327703
$232$	10722.8	3.03442
$233$	4674.76	1.31439	0.657196	−	0.753719i	$-0.271742\pi$
$233$	4674.76	1.31439	0.657196	+	0.753719i	$0.271742\pi$
$234$	−4260.21	−1.19016
$235$	−1897.84	−0.526815
$236$	−4334.50	−1.19556
$237$	879.888	0.241160
$238$	−784.890	−0.213768
$239$	1183.25	0.320242	0.160121	−	0.987097i	$-0.448812\pi$
$239$	1183.25	0.320242	0.160121	+	0.987097i	$0.448812\pi$
$240$	−4629.49	−1.24513
$241$	−3838.30	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$241$	−3838.30	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$242$	−655.790	−0.174197
$243$	2542.44	0.671182
$244$	−7712.38	−2.02350
$245$	−1425.20	−0.371644
$246$	−2404.48	−0.623188
$247$	1558.81	0.401558
$248$	−699.448	−0.179093
$249$	3793.69	0.965525
$250$	677.469	0.171388
$251$	−1165.86	−0.293180	−0.146590	−	0.989197i	$-0.546830\pi$
$251$	−1165.86	−0.293180	−0.146590	+	0.989197i	$0.546830\pi$
$252$	5131.98	1.28287
$253$	2231.50	0.554519
$254$	12130.1	2.99649
$255$	120.592	0.0296147
$256$	7186.23	1.75445
$257$	2414.50	0.586040	0.293020	−	0.956106i	$-0.405340\pi$
$257$	2414.50	0.586040	0.293020	+	0.956106i	$0.405340\pi$
$258$	6064.04	1.46330
$259$	7387.15	1.77226
$260$	8767.79	2.09137
$261$	1417.38	0.336145
$262$	5763.78	1.35911
$263$	−1353.52	−0.317346	−0.158673	−	0.987331i	$-0.550722\pi$
$263$	−1353.52	−0.317346	−0.158673	+	0.987331i	$0.550722\pi$
$264$	3327.63	0.775764
$265$	−2830.51	−0.656138
$266$	−2580.64	−0.594846
$267$	5020.17	1.15067
$268$	11187.5	2.54995
$269$	1473.66	0.334017	0.167008	−	0.985955i	$-0.446589\pi$
$269$	1473.66	0.334017	0.167008	+	0.985955i	$0.446589\pi$
$270$	−4137.30	−0.932547
$271$	−8027.92	−1.79949	−0.899745	−	0.436417i	$-0.856247\pi$
$271$	−8027.92	−1.79949	−0.899745	+	0.436417i	$0.856247\pi$
$272$	−1282.00	−0.285781
$273$	8581.15	1.90240
$274$	5452.62	1.20221
$275$	−275.000	−0.0603023
$276$	−18096.6	−3.94668
$277$	−5703.14	−1.23707	−0.618535	−	0.785757i	$-0.712273\pi$
$277$	−5703.14	−1.23707	−0.618535	+	0.785757i	$0.712273\pi$
$278$	−5696.12	−1.22889
$279$	−92.4560	−0.0198394
$280$	−9082.28	−1.93847
$281$	1355.85	0.287840	0.143920	−	0.989589i	$-0.454029\pi$
$281$	1355.85	0.287840	0.143920	+	0.989589i	$0.454029\pi$
$282$	−8585.80	−1.81304
$283$	2717.39	0.570785	0.285392	−	0.958411i	$-0.407876\pi$
$283$	2717.39	0.570785	0.285392	+	0.958411i	$0.407876\pi$
$284$	−9706.92	−2.02817
$285$	396.493	0.0824077
$286$	−4891.16	−1.01126
$287$	−2663.94	−0.547900
$288$	5964.10	1.22027
$289$	−4879.61	−0.993203
$290$	−4008.91	−0.811763
$291$	336.709	0.0678289
$292$	−15918.1	−3.19019
$293$	2400.50	0.478630	0.239315	−	0.970942i	$-0.423077\pi$
$293$	2400.50	0.478630	0.239315	+	0.970942i	$0.423077\pi$
$294$	−6447.58	−1.27902
$295$	1013.98	0.200122
$296$	21365.6	4.19544
$297$	1679.42	0.328114
$298$	−3867.85	−0.751875
$299$	16643.5	3.21912
$300$	2230.14	0.429190
$301$	6718.39	1.28652
$302$	−16635.5	−3.16976
$303$	−968.723	−0.183669
$304$	−4215.07	−0.795233
$305$	1804.17	0.338711
$306$	−300.072	−0.0560588
$307$	−5425.60	−1.00865	−0.504325	−	0.863514i	$-0.668258\pi$
$307$	−5425.60	−1.00865	−0.504325	+	0.863514i	$0.668258\pi$
$308$	5892.05	1.09003
$309$	−2675.05	−0.492486
$310$	261.501	0.0479105
$311$	−7323.81	−1.33535	−0.667677	−	0.744451i	$-0.732711\pi$
$311$	−7323.81	−1.33535	−0.667677	+	0.744451i	$0.732711\pi$
$312$	24818.9	4.50351
$313$	−8744.54	−1.57914	−0.789570	−	0.613661i	$-0.789696\pi$
$313$	−8744.54	−1.57914	−0.789570	+	0.613661i	$0.789696\pi$
$314$	13883.3	2.49515
$315$	−1200.53	−0.214738
$316$	4506.05	0.802167
$317$	−2117.78	−0.375226	−0.187613	−	0.982243i	$-0.560075\pi$
$317$	−2117.78	−0.375226	−0.187613	+	0.982243i	$0.560075\pi$
$318$	−12805.2	−2.25811
$319$	1627.31	0.285616
$320$	−7994.93	−1.39666
$321$	−6122.97	−1.06464
$322$	−27553.5	−4.76863
$323$	109.797	0.0189141
$324$	−8090.32	−1.38723
$325$	−2051.07	−0.350070
$326$	−11285.3	−1.91728
$327$	−6614.54	−1.11861
$328$	−7704.81	−1.29703
$329$	−9512.25	−1.59400
$330$	−1244.10	−0.207531
$331$	8936.92	1.48404	0.742021	−	0.670377i	$-0.233868\pi$
$331$	8936.92	1.48404	0.742021	+	0.670377i	$0.233868\pi$
$332$	19428.1	3.21162
$333$	2824.19	0.464759
$334$	7035.33	1.15256
$335$	−2617.13	−0.426832
$336$	−23203.7	−3.76745
$337$	−3405.61	−0.550491	−0.275246	−	0.961374i	$-0.588759\pi$
$337$	−3405.61	−0.550491	−0.275246	+	0.961374i	$0.588759\pi$
$338$	−24573.2	−3.95446
$339$	1161.36	0.186066
$340$	617.569	0.0985070
$341$	−106.149	−0.0168572
$342$	−986.607	−0.155993
$343$	1452.52	0.228655
$344$	19431.3	3.04555
$345$	4233.37	0.660629
$346$	−9013.20	−1.40044
$347$	−211.625	−0.0327395	−0.0163698	−	0.999866i	$-0.505211\pi$
$347$	−211.625	−0.0327395	−0.0163698	+	0.999866i	$0.505211\pi$
$348$	−13196.8	−2.03282
$349$	6062.57	0.929863	0.464931	−	0.885347i	$-0.346079\pi$
$349$	6062.57	0.929863	0.464931	+	0.885347i	$0.346079\pi$
$350$	3395.57	0.518574
$351$	12525.9	1.90479
$352$	6847.41	1.03684
$353$	−9739.87	−1.46856	−0.734279	−	0.678847i	$-0.762480\pi$
$353$	−9739.87	−1.46856	−0.734279	+	0.678847i	$0.762480\pi$
$354$	4587.22	0.688723
$355$	2270.76	0.339491
$356$	25709.1	3.82747
$357$	604.423	0.0896063
$358$	14860.2	2.19381
$359$	11248.7	1.65372	0.826858	−	0.562411i	$-0.190126\pi$
$359$	11248.7	1.65372	0.826858	+	0.562411i	$0.190126\pi$
$360$	−3472.26	−0.508345
$361$	361.000	0.0526316
$362$	14685.9	2.13224
$363$	505.006	0.0730192
$364$	43945.4	6.32793
$365$	3723.75	0.534000
$366$	8162.06	1.16568
$367$	−12617.0	−1.79456	−0.897278	−	0.441466i	$-0.854459\pi$
$367$	−12617.0	−1.79456	−0.897278	+	0.441466i	$0.854459\pi$
$368$	−45004.5	−6.37506
$369$	−1018.45	−0.143682
$370$	−7987.90	−1.12236
$371$	−14186.9	−1.98530
$372$	860.827	0.119978
$373$	11800.8	1.63813	0.819063	−	0.573703i	$-0.194494\pi$
$373$	11800.8	1.63813	0.819063	+	0.573703i	$0.194494\pi$
$374$	−344.515	−0.0476322
$375$	−521.701	−0.0718414
$376$	−27511.9	−3.77346
$377$	12137.2	1.65808
$378$	−20736.7	−2.82165
$379$	−4533.42	−0.614422	−0.307211	−	0.951641i	$-0.599396\pi$
$379$	−4533.42	−0.614422	−0.307211	+	0.951641i	$0.599396\pi$
$380$	2030.50	0.274112
$381$	−9341.04	−1.25605
$382$	19932.3	2.66970
$383$	9538.41	1.27256	0.636279	−	0.771459i	$-0.280473\pi$
$383$	9538.41	1.27256	0.636279	+	0.771459i	$0.280473\pi$
$384$	−15384.6	−2.04451
$385$	−1378.34	−0.182459
$386$	17252.5	2.27495
$387$	2568.52	0.337377
$388$	1724.34	0.225619
$389$	−6670.41	−0.869417	−0.434709	−	0.900571i	$-0.643149\pi$
$389$	−6670.41	−0.869417	−0.434709	+	0.900571i	$0.643149\pi$
$390$	−9279.00	−1.20477
$391$	1172.30	0.151626
$392$	−20660.3	−2.66200
$393$	−4438.53	−0.569706
$394$	−19558.0	−2.50080
$395$	−1054.11	−0.134273
$396$	2252.60	0.285852
$397$	−5265.41	−0.665651	−0.332825	−	0.942988i	$-0.608002\pi$
$397$	−5265.41	−0.665651	−0.332825	+	0.942988i	$0.608002\pi$
$398$	−14623.9	−1.84178
$399$	1987.28	0.249344
$400$	5546.15	0.693268
$401$	7172.70	0.893236	0.446618	−	0.894725i	$-0.352628\pi$
$401$	7172.70	0.893236	0.446618	+	0.894725i	$0.352628\pi$
$402$	−11839.8	−1.46895
$403$	−791.707	−0.0978604
$404$	−4960.99	−0.610936
$405$	1892.59	0.232206
$406$	−20093.2	−2.45618
$407$	3242.47	0.394898
$408$	1748.15	0.212123
$409$	−2181.75	−0.263767	−0.131883	−	0.991265i	$-0.542102\pi$
$409$	−2181.75	−0.263767	−0.131883	+	0.991265i	$0.542102\pi$
$410$	2880.58	0.346980
$411$	−4198.92	−0.503936
$412$	−13699.3	−1.63815
$413$	5082.21	0.605518
$414$	−10534.0	−1.25053
$415$	−4544.86	−0.537587
$416$	51071.0	6.01913
$417$	4386.43	0.515118
$418$	−1132.73	−0.132544
$419$	10253.3	1.19548	0.597740	−	0.801690i	$-0.296066\pi$
$419$	10253.3	1.19548	0.597740	+	0.801690i	$0.296066\pi$
$420$	11177.8	1.29862
$421$	11282.9	1.30617	0.653083	−	0.757286i	$-0.273475\pi$
$421$	11282.9	1.30617	0.653083	+	0.757286i	$0.273475\pi$
$422$	14047.4	1.62042
$423$	−3636.64	−0.418013
$424$	−41032.3	−4.69977
$425$	−144.469	−0.0164889
$426$	10272.9	1.16836
$427$	9042.79	1.02485
$428$	−31356.7	−3.54132
$429$	3766.56	0.423895
$430$	−7264.75	−0.814738
$431$	13060.7	1.45965	0.729826	−	0.683633i	$-0.239601\pi$
$431$	13060.7	1.45965	0.729826	+	0.683633i	$0.239601\pi$
$432$	−33870.3	−3.77218
$433$	15225.2	1.68978	0.844890	−	0.534940i	$-0.179666\pi$
$433$	15225.2	1.68978	0.844890	+	0.534940i	$0.179666\pi$
$434$	1310.68	0.144965
$435$	3087.15	0.340270
$436$	−33874.1	−3.72082
$437$	3854.41	0.421926
$438$	16846.2	1.83777
$439$	−3785.28	−0.411529	−0.205765	−	0.978602i	$-0.565968\pi$
$439$	−3785.28	−0.411529	−0.205765	+	0.978602i	$0.565968\pi$
$440$	−3986.52	−0.431932
$441$	−2730.97	−0.294889
$442$	−2569.54	−0.276517
$443$	1806.49	0.193745	0.0968723	−	0.995297i	$-0.469116\pi$
$443$	1806.49	0.193745	0.0968723	+	0.995297i	$0.469116\pi$
$444$	−26295.1	−2.81061
$445$	−6014.19	−0.640674
$446$	11385.5	1.20879
$447$	2978.53	0.315167
$448$	−40071.7	−4.22592
$449$	9696.52	1.01917	0.509585	−	0.860421i	$-0.329799\pi$
$449$	9696.52	1.01917	0.509585	+	0.860421i	$0.329799\pi$
$450$	1298.17	0.135991
$451$	−1169.29	−0.122084
$452$	5947.51	0.618910
$453$	12810.6	1.32868
$454$	−11681.8	−1.20761
$455$	−10280.3	−1.05922
$456$	5747.73	0.590268
$457$	−9250.03	−0.946823	−0.473412	−	0.880841i	$-0.656978\pi$
$457$	−9250.03	−0.946823	−0.473412	+	0.880841i	$0.656978\pi$
$458$	−11736.1	−1.19736
$459$	882.273	0.0897189
$460$	21679.8	2.19744
$461$	3414.12	0.344928	0.172464	−	0.985016i	$-0.444827\pi$
$461$	3414.12	0.344928	0.172464	+	0.985016i	$0.444827\pi$
$462$	−6235.59	−0.627935
$463$	−9312.51	−0.934750	−0.467375	−	0.884059i	$-0.654800\pi$
$463$	−9312.51	−0.934750	−0.467375	+	0.884059i	$0.654800\pi$
$464$	−32819.2	−3.28361
$465$	−201.375	−0.0200829
$466$	−25336.0	−2.51860
$467$	−14125.0	−1.39963	−0.699814	−	0.714325i	$-0.746734\pi$
$467$	−14125.0	−1.39963	−0.699814	+	0.714325i	$0.746734\pi$
$468$	16800.8	1.65944
$469$	−13117.4	−1.29148
$470$	10285.8	1.00947
$471$	−10691.1	−1.04591
$472$	14699.1	1.43343
$473$	2948.93	0.286663
$474$	−4768.77	−0.462103
$475$	−475.000	−0.0458831
$476$	3095.35	0.298057
$477$	−5423.82	−0.520628
$478$	−6412.90	−0.613638
$479$	−16576.8	−1.58124	−0.790620	−	0.612307i	$-0.790242\pi$
$479$	−16576.8	−1.58124	−0.790620	+	0.612307i	$0.790242\pi$
$480$	12990.2	1.23525
$481$	24183.7	2.29248
$482$	20802.6	1.96584
$483$	21218.3	1.99889
$484$	2586.22	0.242883
$485$	−403.379	−0.0377659
$486$	−13779.4	−1.28610
$487$	−4579.41	−0.426104	−0.213052	−	0.977041i	$-0.568340\pi$
$487$	−4579.41	−0.426104	−0.213052	+	0.977041i	$0.568340\pi$
$488$	26154.1	2.42611
$489$	8690.48	0.803675
$490$	7724.23	0.712133
$491$	15360.1	1.41179	0.705896	−	0.708315i	$-0.250545\pi$
$491$	15360.1	1.41179	0.705896	+	0.708315i	$0.250545\pi$
$492$	9482.49	0.868909
$493$	854.894	0.0780983
$494$	−8448.37	−0.769454
$495$	−526.955	−0.0478482
$496$	2140.80	0.193800
$497$	11381.4	1.02721
$498$	−20560.9	−1.85011
$499$	−18617.5	−1.67021	−0.835103	−	0.550093i	$-0.814592\pi$
$499$	−18617.5	−1.67021	−0.835103	+	0.550093i	$0.814592\pi$
$500$	−2671.71	−0.238965
$501$	−5417.72	−0.483126
$502$	6318.66	0.561784
$503$	−4537.85	−0.402252	−0.201126	−	0.979565i	$-0.564460\pi$
$503$	−4537.85	−0.402252	−0.201126	+	0.979565i	$0.564460\pi$
$504$	−17403.5	−1.53812
$505$	1160.53	0.102264
$506$	−12094.2	−1.06255
$507$	18923.2	1.65761
$508$	−47836.9	−4.17799
$509$	−5543.45	−0.482729	−0.241365	−	0.970435i	$-0.577595\pi$
$509$	−5543.45	−0.482729	−0.241365	+	0.970435i	$0.577595\pi$
$510$	−653.577	−0.0567468
$511$	18664.0	1.61574
$512$	−9458.26	−0.816406
$513$	2900.82	0.249658
$514$	−13086.0	−1.12295
$515$	3204.72	0.274207
$516$	−23914.6	−2.04027
$517$	−4175.25	−0.355178
$518$	−40036.5	−3.39595
$519$	6940.83	0.587030
$520$	−29733.2	−2.50747
$521$	12872.6	1.08245	0.541225	−	0.840878i	$-0.317961\pi$
$521$	12872.6	1.08245	0.541225	+	0.840878i	$0.317961\pi$
$522$	−7681.87	−0.644112
$523$	20304.1	1.69759	0.848793	−	0.528726i	$-0.177330\pi$
$523$	20304.1	1.69759	0.848793	+	0.528726i	$0.177330\pi$
$524$	−22730.4	−1.89501
$525$	−2614.84	−0.217373
$526$	7335.77	0.608089
$527$	−55.7648	−0.00460939
$528$	−10184.9	−0.839469
$529$	28986.7	2.38240
$530$	15340.6	1.25727
$531$	1942.99	0.158792
$532$	10177.2	0.829392
$533$	−8721.09	−0.708729
$534$	−27208.1	−2.20489
$535$	7335.35	0.592775
$536$	−37939.0	−3.05731
$537$	−11443.4	−0.919592
$538$	−7986.86	−0.640033
$539$	−3135.44	−0.250562
$540$	16316.1	1.30025
$541$	−5429.60	−0.431491	−0.215746	−	0.976450i	$-0.569218\pi$
$541$	−5429.60	−0.431491	−0.215746	+	0.976450i	$0.569218\pi$
$542$	43509.4	3.44813
$543$	−11309.2	−0.893783
$544$	3597.24	0.283512
$545$	7924.24	0.622821
$546$	−46507.7	−3.64532
$547$	−2500.25	−0.195435	−0.0977176	−	0.995214i	$-0.531154\pi$
$547$	−2500.25	−0.195435	−0.0977176	+	0.995214i	$0.531154\pi$
$548$	−21503.4	−1.67624
$549$	3457.16	0.268758
$550$	1490.43	0.115550
$551$	2810.80	0.217322
$552$	61368.8	4.73194
$553$	−5283.35	−0.406276
$554$	30909.6	2.37044
$555$	6151.27	0.470463
$556$	22463.6	1.71343
$557$	−10360.2	−0.788108	−0.394054	−	0.919087i	$-0.628928\pi$
$557$	−10360.2	−0.788108	−0.394054	+	0.919087i	$0.628928\pi$
$558$	501.089	0.0380157
$559$	21994.4	1.66416
$560$	27798.1	2.09765
$561$	265.302	0.0199662
$562$	−7348.36	−0.551551
$563$	19791.2	1.48152	0.740762	−	0.671768i	$-0.234465\pi$
$563$	19791.2	1.48152	0.740762	+	0.671768i	$0.234465\pi$
$564$	33859.5	2.52791
$565$	−1391.31	−0.103598
$566$	−14727.6	−1.09372
$567$	9485.92	0.702595
$568$	32917.9	2.43170
$569$	3962.11	0.291916	0.145958	−	0.989291i	$-0.453374\pi$
$569$	3962.11	0.291916	0.145958	+	0.989291i	$0.453374\pi$
$570$	−2148.89	−0.157907
$571$	15509.1	1.13667	0.568334	−	0.822798i	$-0.307588\pi$
$571$	15509.1	1.13667	0.568334	+	0.822798i	$0.307588\pi$
$572$	19289.1	1.41000
$573$	−15349.3	−1.11907
$574$	14437.9	1.04987
$575$	−5071.59	−0.367826
$576$	−15319.9	−1.10821
$577$	−12131.3	−0.875275	−0.437637	−	0.899152i	$-0.644185\pi$
$577$	−12131.3	−0.875275	−0.437637	+	0.899152i	$0.644185\pi$
$578$	26446.3	1.90315
$579$	−13285.7	−0.953602
$580$	15809.8	1.13184
$581$	−22779.5	−1.62660
$582$	−1824.88	−0.129972
$583$	−6227.11	−0.442368
$584$	53981.1	3.82492
$585$	−3930.26	−0.277771
$586$	−13010.1	−0.917137
$587$	3648.79	0.256562	0.128281	−	0.991738i	$-0.459054\pi$
$587$	3648.79	0.256562	0.128281	+	0.991738i	$0.459054\pi$
$588$	25427.1	1.78333
$589$	−183.349	−0.0128264
$590$	−5495.51	−0.383469
$591$	15061.1	1.04827
$592$	−65393.5	−4.53996
$593$	−6325.40	−0.438032	−0.219016	−	0.975721i	$-0.570285\pi$
$593$	−6325.40	−0.438032	−0.219016	+	0.975721i	$0.570285\pi$
$594$	−9102.05	−0.628723
$595$	−724.101	−0.0498912
$596$	15253.5	1.04834
$597$	11261.5	0.772028
$598$	−90203.6	−6.16840
$599$	−4687.27	−0.319727	−0.159864	−	0.987139i	$-0.551105\pi$
$599$	−4687.27	−0.319727	−0.159864	+	0.987139i	$0.551105\pi$
$600$	−7562.81	−0.514584
$601$	−16420.7	−1.11450	−0.557250	−	0.830345i	$-0.688143\pi$
$601$	−16420.7	−1.11450	−0.557250	+	0.830345i	$0.688143\pi$
$602$	−36412.0	−2.46519
$603$	−5014.94	−0.338680
$604$	65605.0	4.41959
$605$	−605.000	−0.0406558
$606$	5250.24	0.351941
$607$	1105.17	0.0739004	0.0369502	−	0.999317i	$-0.488236\pi$
$607$	1105.17	0.0739004	0.0369502	+	0.999317i	$0.488236\pi$
$608$	11827.3	0.788918
$609$	15473.3	1.02957
$610$	−9778.18	−0.649028
$611$	−31140.8	−2.06190
$612$	1183.39	0.0781627
$613$	−307.862	−0.0202845	−0.0101423	−	0.999949i	$-0.503228\pi$
$613$	−307.862	−0.0202845	−0.0101423	+	0.999949i	$0.503228\pi$
$614$	29405.4	1.93274
$615$	−2218.26	−0.145445
$616$	−19981.0	−1.30691
$617$	10862.9	0.708793	0.354396	−	0.935095i	$-0.384686\pi$
$617$	10862.9	0.708793	0.354396	+	0.935095i	$0.384686\pi$
$618$	14498.1	0.943687
$619$	−15516.9	−1.00755	−0.503776	−	0.863834i	$-0.668056\pi$
$619$	−15516.9	−1.00755	−0.503776	+	0.863834i	$0.668056\pi$
$620$	−1031.27	−0.0668016
$621$	30972.2	2.00140
$622$	39693.2	2.55877
$623$	−30144.0	−1.93851
$624$	−75963.2	−4.87333
$625$	625.000	0.0400000
$626$	47393.3	3.02590
$627$	872.284	0.0555593
$628$	−54751.0	−3.47899
$629$	1703.41	0.107980
$630$	6506.60	0.411475
$631$	−11712.3	−0.738923	−0.369461	−	0.929246i	$-0.620458\pi$
$631$	−11712.3	−0.738923	−0.369461	+	0.929246i	$0.620458\pi$
$632$	−15280.8	−0.961771
$633$	−10817.6	−0.679240
$634$	11477.9	0.718997
$635$	11190.6	0.699347
$636$	50499.3	3.14847
$637$	−23385.5	−1.45458
$638$	−8819.60	−0.547290
$639$	4351.23	0.269377
$640$	18430.8	1.13835
$641$	−20010.3	−1.23301	−0.616504	−	0.787352i	$-0.711451\pi$
$641$	−20010.3	−1.23301	−0.616504	+	0.787352i	$0.711451\pi$
$642$	33185.0	2.04004
$643$	25584.2	1.56912	0.784559	−	0.620055i	$-0.212890\pi$
$643$	25584.2	1.56912	0.784559	+	0.620055i	$0.212890\pi$
$644$	108662.	6.64889
$645$	5594.39	0.341518
$646$	−595.071	−0.0362426
$647$	−9891.54	−0.601046	−0.300523	−	0.953775i	$-0.597161\pi$
$647$	−9891.54	−0.601046	−0.300523	+	0.953775i	$0.597161\pi$
$648$	27435.8	1.66324
$649$	2230.75	0.134922
$650$	11116.3	0.670795
$651$	−1009.32	−0.0607656
$652$	44505.3	2.67326
$653$	−20123.5	−1.20596	−0.602982	−	0.797755i	$-0.706021\pi$
$653$	−20123.5	−1.20596	−0.602982	+	0.797755i	$0.706021\pi$
$654$	35849.1	2.14344
$655$	5317.38	0.317202
$656$	23582.1	1.40354
$657$	7135.45	0.423715
$658$	51554.0	3.05439
$659$	−27421.6	−1.62093	−0.810464	−	0.585788i	$-0.800785\pi$
$659$	−27421.6	−1.62093	−0.810464	+	0.585788i	$0.800785\pi$
$660$	4906.30	0.289360
$661$	−11446.5	−0.673553	−0.336776	−	0.941585i	$-0.609337\pi$
$661$	−11446.5	−0.673553	−0.336776	+	0.941585i	$0.609337\pi$
$662$	−48435.9	−2.84368
$663$	1978.73	0.115909
$664$	−65884.3	−3.85061
$665$	−2380.77	−0.138830
$666$	−15306.4	−0.890559
$667$	30011.0	1.74218
$668$	−27745.0	−1.60702
$669$	−8767.69	−0.506695
$670$	14184.2	0.817884
$671$	3969.18	0.228359
$672$	65108.7	3.73753
$673$	22210.5	1.27214	0.636070	−	0.771631i	$-0.280559\pi$
$673$	22210.5	1.27214	0.636070	+	0.771631i	$0.280559\pi$
$674$	18457.6	1.05484
$675$	−3816.87	−0.217646
$676$	96908.6	5.51369
$677$	−34262.3	−1.94506	−0.972530	−	0.232777i	$-0.925219\pi$
$677$	−34262.3	−1.94506	−0.972530	+	0.232777i	$0.925219\pi$
$678$	−6294.29	−0.356535
$679$	−2021.79	−0.114270
$680$	−2094.29	−0.118106
$681$	8995.88	0.506201
$682$	575.303	0.0323013
$683$	22189.4	1.24312	0.621562	−	0.783365i	$-0.286498\pi$
$683$	22189.4	1.24312	0.621562	+	0.783365i	$0.286498\pi$
$684$	3890.85	0.217501
$685$	5030.33	0.280582
$686$	−7872.29	−0.438142
$687$	9037.67	0.501905
$688$	−59473.4	−3.29564
$689$	−46444.5	−2.56806
$690$	−22943.8	−1.26588
$691$	−13233.6	−0.728552	−0.364276	−	0.931291i	$-0.618684\pi$
$691$	−13233.6	−0.728552	−0.364276	+	0.931291i	$0.618684\pi$
$692$	35545.1	1.95263
$693$	−2641.18	−0.144776
$694$	1146.95	0.0627346
$695$	−5254.96	−0.286809
$696$	44752.7	2.43728
$697$	−614.280	−0.0333824
$698$	−32857.6	−1.78178
$699$	19510.6	1.05573
$700$	−13391.0	−0.723047
$701$	−22430.8	−1.20856	−0.604279	−	0.796773i	$-0.706539\pi$
$701$	−22430.8	−1.20856	−0.604279	+	0.796773i	$0.706539\pi$
$702$	−67887.0	−3.64990
$703$	5600.63	0.300472
$704$	−17588.8	−0.941625
$705$	−7920.84	−0.423143
$706$	52787.7	2.81401
$707$	5816.77	0.309423
$708$	−18090.5	−0.960285
$709$	26194.6	1.38753	0.693764	−	0.720202i	$-0.255951\pi$
$709$	26194.6	1.38753	0.693764	+	0.720202i	$0.255951\pi$
$710$	−12307.0	−0.650524
$711$	−2019.89	−0.106542
$712$	−87184.3	−4.58900
$713$	−1957.62	−0.102824
$714$	−3275.82	−0.171701
$715$	−4512.35	−0.236017
$716$	−58603.7	−3.05883
$717$	4938.40	0.257222
$718$	−60965.2	−3.16880
$719$	−31525.6	−1.63520	−0.817598	−	0.575789i	$-0.804695\pi$
$719$	−31525.6	−1.63520	−0.817598	+	0.575789i	$0.804695\pi$
$720$	10627.5	0.550090
$721$	16062.5	0.829680
$722$	−1956.53	−0.100851
$723$	−16019.5	−0.824029
$724$	−57916.2	−2.97298
$725$	−3698.42	−0.189457
$726$	−2737.01	−0.139917
$727$	−4509.63	−0.230059	−0.115030	−	0.993362i	$-0.536696\pi$
$727$	−4509.63	−0.230059	−0.115030	+	0.993362i	$0.536696\pi$
$728$	−149027.	−7.58697
$729$	20831.1	1.05833
$730$	−20181.8	−1.02324
$731$	1549.20	0.0783846
$732$	−32188.5	−1.62530
$733$	21605.2	1.08868	0.544341	−	0.838864i	$-0.316780\pi$
$733$	21605.2	1.08868	0.544341	+	0.838864i	$0.316780\pi$
$734$	68381.0	3.43868
$735$	−5948.23	−0.298508
$736$	126281.	6.32443
$737$	−5757.68	−0.287770
$738$	5519.77	0.275319
$739$	−16247.4	−0.808756	−0.404378	−	0.914592i	$-0.632512\pi$
$739$	−16247.4	−0.808756	−0.404378	+	0.914592i	$0.632512\pi$
$740$	31501.7	1.56490
$741$	6505.87	0.322536
$742$	76889.5	3.80418
$743$	−10800.5	−0.533286	−0.266643	−	0.963795i	$-0.585915\pi$
$743$	−10800.5	−0.533286	−0.266643	+	0.963795i	$0.585915\pi$
$744$	−2919.22	−0.143849
$745$	−3568.29	−0.175479
$746$	−63957.3	−3.13893
$747$	−8708.87	−0.426561
$748$	1358.65	0.0664134
$749$	36765.8	1.79358
$750$	2827.49	0.137660
$751$	36543.5	1.77562	0.887810	−	0.460209i	$-0.152226\pi$
$751$	36543.5	1.77562	0.887810	+	0.460209i	$0.152226\pi$
$752$	84205.6	4.08333
$753$	−4865.83	−0.235486
$754$	−65780.3	−3.17716
$755$	−15347.1	−0.739787
$756$	81778.8	3.93422
$757$	19631.4	0.942556	0.471278	−	0.881985i	$-0.343793\pi$
$757$	19631.4	0.942556	0.471278	+	0.881985i	$0.343793\pi$
$758$	24570.0	1.17734
$759$	9313.41	0.445396
$760$	−6885.81	−0.328651
$761$	−9207.23	−0.438583	−0.219291	−	0.975659i	$-0.570375\pi$
$761$	−9207.23	−0.438583	−0.219291	+	0.975659i	$0.570375\pi$
$762$	50626.1	2.40681
$763$	39717.5	1.88449
$764$	−78606.4	−3.72236
$765$	−276.832	−0.0130835
$766$	−51695.8	−2.43844
$767$	16637.9	0.783260
$768$	29992.5	1.40919
$769$	−17546.7	−0.822823	−0.411411	−	0.911450i	$-0.634964\pi$
$769$	−17546.7	−0.822823	−0.411411	+	0.911450i	$0.634964\pi$
$770$	7470.26	0.349623
$771$	10077.2	0.470714
$772$	−68038.3	−3.17196
$773$	−23903.1	−1.11220	−0.556102	−	0.831114i	$-0.687704\pi$
$773$	−23903.1	−1.11220	−0.556102	+	0.831114i	$0.687704\pi$
$774$	−13920.7	−0.646473
$775$	241.248	0.0111818
$776$	−5847.56	−0.270509
$777$	30831.1	1.42350
$778$	36152.0	1.66595
$779$	−2019.69	−0.0928920
$780$	36593.3	1.67981
$781$	4995.67	0.228885
$782$	−6353.59	−0.290542
$783$	22586.2	1.03086
$784$	63235.0	2.88060
$785$	12808.0	0.582341
$786$	24055.8	1.09165
$787$	−7756.81	−0.351335	−0.175667	−	0.984450i	$-0.556208\pi$
$787$	−7756.81	−0.351335	−0.175667	+	0.984450i	$0.556208\pi$
$788$	77130.2	3.48686
$789$	−5649.08	−0.254896
$790$	5713.01	0.257291
$791$	−6973.47	−0.313462
$792$	−7638.98	−0.342726
$793$	29603.9	1.32568
$794$	28537.2	1.27550
$795$	−11813.4	−0.527018
$796$	57671.7	2.56799
$797$	−13080.9	−0.581366	−0.290683	−	0.956819i	$-0.593882\pi$
$797$	−13080.9	−0.581366	−0.290683	+	0.956819i	$0.593882\pi$
$798$	−10770.6	−0.477787
$799$	−2193.44	−0.0971192
$800$	−15562.3	−0.687763
$801$	−11524.4	−0.508358
$802$	−38874.3	−1.71159
$803$	8192.25	0.360023
$804$	46692.4	2.04815
$805$	−25419.6	−1.11295
$806$	4290.85	0.187517
$807$	6150.47	0.268286
$808$	16823.6	0.732491
$809$	−19766.4	−0.859023	−0.429511	−	0.903061i	$-0.641314\pi$
$809$	−19766.4	−0.859023	−0.429511	+	0.903061i	$0.641314\pi$
$810$	−10257.3	−0.444946
$811$	−659.366	−0.0285493	−0.0142746	−	0.999898i	$-0.504544\pi$
$811$	−659.366	−0.0285493	−0.0142746	+	0.999898i	$0.504544\pi$
$812$	79241.1	3.42465
$813$	−33505.4	−1.44537
$814$	−17573.4	−0.756692
$815$	−10411.2	−0.447472
$816$	−5350.55	−0.229543
$817$	5093.60	0.218118
$818$	11824.6	0.505423
$819$	−19699.0	−0.840464
$820$	−11360.1	−0.483793
$821$	7497.14	0.318699	0.159350	−	0.987222i	$-0.449060\pi$
$821$	7497.14	0.318699	0.159350	+	0.987222i	$0.449060\pi$
$822$	22757.1	0.965627
$823$	28078.0	1.18923	0.594616	−	0.804010i	$-0.297304\pi$
$823$	28078.0	1.18923	0.594616	+	0.804010i	$0.297304\pi$
$824$	46457.0	1.96409
$825$	−1147.74	−0.0484354
$826$	−27544.3	−1.16028
$827$	−19705.6	−0.828575	−0.414288	−	0.910146i	$-0.635969\pi$
$827$	−19705.6	−0.828575	−0.414288	+	0.910146i	$0.635969\pi$
$828$	41542.8	1.74361
$829$	−4954.58	−0.207575	−0.103787	−	0.994599i	$-0.533096\pi$
$829$	−4954.58	−0.207575	−0.103787	+	0.994599i	$0.533096\pi$
$830$	24632.0	1.03011
$831$	−23802.7	−0.993628
$832$	−131185.	−5.46638
$833$	−1647.18	−0.0685132
$834$	−23773.4	−0.987055
$835$	6490.46	0.268996
$836$	4467.11	0.184806
$837$	−1473.30	−0.0608420
$838$	−55570.3	−2.29074
$839$	−41933.5	−1.72551	−0.862756	−	0.505620i	$-0.831264\pi$
$839$	−41933.5	−1.72551	−0.862756	+	0.505620i	$0.831264\pi$
$840$	−37905.9	−1.55700
$841$	−2503.67	−0.102656
$842$	−61150.6	−2.50284
$843$	5658.78	0.231196
$844$	−55398.4	−2.25935
$845$	−22670.1	−0.922927
$846$	19709.7	0.800985
$847$	−3032.35	−0.123014
$848$	125587.	5.08571
$849$	11341.3	0.458461
$850$	782.988	0.0315956
$851$	59798.2	2.40876
$852$	−40512.8	−1.62905
$853$	−36593.1	−1.46885	−0.734423	−	0.678692i	$-0.762547\pi$
$853$	−36593.1	−1.46885	−0.734423	+	0.678692i	$0.762547\pi$
$854$	−49009.7	−1.96379
$855$	−910.195	−0.0364070
$856$	106336.	4.24591
$857$	39541.0	1.57607	0.788036	−	0.615629i	$-0.211098\pi$
$857$	39541.0	1.57607	0.788036	+	0.615629i	$0.211098\pi$
$858$	−20413.8	−0.812256
$859$	−12280.5	−0.487784	−0.243892	−	0.969802i	$-0.578424\pi$
$859$	−12280.5	−0.487784	−0.243892	+	0.969802i	$0.578424\pi$
$860$	28649.8	1.13599
$861$	−11118.2	−0.440080
$862$	−70785.6	−2.79694
$863$	−27126.3	−1.06998	−0.534988	−	0.844860i	$-0.679684\pi$
$863$	−27126.3	−1.06998	−0.534988	+	0.844860i	$0.679684\pi$
$864$	95038.8	3.74223
$865$	−8315.14	−0.326848
$866$	−82516.7	−3.23791
$867$	−20365.6	−0.797752
$868$	−5168.90	−0.202124
$869$	−2319.04	−0.0905271
$870$	−16731.6	−0.652017
$871$	−42943.2	−1.67058
$872$	114873.	4.46113
$873$	−772.955	−0.0299663
$874$	−20890.0	−0.808482
$875$	3132.59	0.121030
$876$	−66435.8	−2.56239
$877$	15583.8	0.600029	0.300015	−	0.953935i	$-0.403008\pi$
$877$	15583.8	0.600029	0.300015	+	0.953935i	$0.403008\pi$
$878$	20515.3	0.788561
$879$	10018.7	0.384441
$880$	12201.5	0.467401
$881$	−44504.6	−1.70193	−0.850963	−	0.525226i	$-0.823981\pi$
$881$	−44504.6	−1.70193	−0.850963	+	0.525226i	$0.823981\pi$
$882$	14801.2	0.565059
$883$	−36634.1	−1.39619	−0.698095	−	0.716005i	$-0.745969\pi$
$883$	−36634.1	−1.39619	−0.698095	+	0.716005i	$0.745969\pi$
$884$	10133.4	0.385547
$885$	4231.95	0.160740
$886$	−9790.72	−0.371248
$887$	−1863.14	−0.0705278	−0.0352639	−	0.999378i	$-0.511227\pi$
$887$	−1863.14	−0.0705278	−0.0352639	+	0.999378i	$0.511227\pi$
$888$	89171.6	3.36982
$889$	56088.9	2.11604
$890$	32595.4	1.22764
$891$	4163.69	0.156553
$892$	−44900.8	−1.68541
$893$	−7211.79	−0.270250
$894$	−16142.9	−0.603914
$895$	13709.3	0.512012
$896$	92378.1	3.44435
$897$	69463.4	2.58564
$898$	−52552.7	−1.95290
$899$	−1427.58	−0.0529616
$900$	−5119.54	−0.189613
$901$	−3271.37	−0.120960
$902$	6337.28	0.233934
$903$	28039.9	1.03334
$904$	−20169.1	−0.742052
$905$	13548.5	0.497642
$906$	−69430.1	−2.54598
$907$	8050.51	0.294722	0.147361	−	0.989083i	$-0.452922\pi$
$907$	8050.51	0.294722	0.147361	+	0.989083i	$0.452922\pi$
$908$	46069.3	1.68377
$909$	2223.82	0.0811434
$910$	55716.4	2.02965
$911$	−6835.04	−0.248579	−0.124289	−	0.992246i	$-0.539665\pi$
$911$	−6835.04	−0.248579	−0.124289	+	0.992246i	$0.539665\pi$
$912$	−17592.1	−0.638740
$913$	−9998.70	−0.362441
$914$	50132.9	1.81428
$915$	7529.92	0.272056
$916$	46283.3	1.66948
$917$	26651.5	0.959771
$918$	−4781.70	−0.171917
$919$	−49646.7	−1.78204	−0.891020	−	0.453964i	$-0.850009\pi$
$919$	−49646.7	−1.78204	−0.891020	+	0.453964i	$0.850009\pi$
$920$	−73520.1	−2.63466
$921$	−22644.3	−0.810158
$922$	−18503.7	−0.660941
$923$	37259.9	1.32874
$924$	24591.1	0.875528
$925$	−7369.25	−0.261945
$926$	50471.5	1.79114
$927$	6140.88	0.217576
$928$	92089.6	3.25753
$929$	12513.4	0.441929	0.220965	−	0.975282i	$-0.429079\pi$
$929$	12513.4	0.441929	0.220965	+	0.975282i	$0.429079\pi$
$930$	1091.40	0.0384823
$931$	−5415.76	−0.190649
$932$	99916.9	3.51168
$933$	−30566.7	−1.07257
$934$	76553.9	2.68193
$935$	−317.832	−0.0111168
$936$	−56974.8	−1.98961
$937$	−4556.01	−0.158846	−0.0794229	−	0.996841i	$-0.525308\pi$
$937$	−4556.01	−0.158846	−0.0794229	+	0.996841i	$0.525308\pi$
$938$	71093.1	2.47471
$939$	−36496.3	−1.26838
$940$	−40563.9	−1.40750
$941$	−43231.4	−1.49767	−0.748833	−	0.662758i	$-0.769386\pi$
$941$	−43231.4	−1.49767	−0.748833	+	0.662758i	$0.769386\pi$
$942$	57943.3	2.00413
$943$	−21564.3	−0.744676
$944$	−44989.4	−1.55114
$945$	−19130.7	−0.658541
$946$	−15982.5	−0.549296
$947$	−23295.7	−0.799376	−0.399688	−	0.916651i	$-0.630882\pi$
$947$	−23295.7	−0.799376	−0.399688	+	0.916651i	$0.630882\pi$
$948$	18806.5	0.644310
$949$	61101.3	2.09002
$950$	2574.38	0.0879200
$951$	−8838.80	−0.301386
$952$	−10496.9	−0.357359
$953$	28259.9	0.960577	0.480288	−	0.877111i	$-0.340532\pi$
$953$	28259.9	0.960577	0.480288	+	0.877111i	$0.340532\pi$
$954$	29395.8	0.997613
$955$	18388.6	0.623079
$956$	25290.3	0.855594
$957$	6791.74	0.229410
$958$	89842.3	3.02993
$959$	25212.7	0.848969
$960$	−33367.7	−1.12181
$961$	−29697.9	−0.996874
$962$	−131070.	−4.39279
$963$	14056.0	0.470351
$964$	−82038.6	−2.74096
$965$	15916.3	0.530949
$966$	−114998.	−3.83022
$967$	6294.97	0.209341	0.104670	−	0.994507i	$-0.466621\pi$
$967$	6294.97	0.209341	0.104670	+	0.994507i	$0.466621\pi$
$968$	−8770.35	−0.291208
$969$	458.248	0.0151920
$970$	2186.21	0.0723660
$971$	25713.3	0.849824	0.424912	−	0.905235i	$-0.360305\pi$
$971$	25713.3	0.849824	0.424912	+	0.905235i	$0.360305\pi$
$972$	54341.3	1.79321
$973$	−26338.6	−0.867808
$974$	24819.2	0.816489
$975$	−8560.35	−0.281180
$976$	−80049.7	−2.62534
$977$	39488.7	1.29310	0.646548	−	0.762873i	$-0.276212\pi$
$977$	39488.7	1.29310	0.646548	+	0.762873i	$0.276212\pi$
$978$	−47100.2	−1.53998
$979$	−13231.2	−0.431942
$980$	−30461.8	−0.992926
$981$	15184.4	0.494192
$982$	−83247.7	−2.70524
$983$	−28569.0	−0.926970	−0.463485	−	0.886105i	$-0.653401\pi$
$983$	−28569.0	−0.926970	−0.463485	+	0.886105i	$0.653401\pi$
$984$	−32156.9	−1.04179
$985$	−18043.2	−0.583660
$986$	−4633.31	−0.149650
$987$	−39700.4	−1.28032
$988$	33317.6	1.07285
$989$	54384.6	1.74856
$990$	2855.97	0.0916854
$991$	−32929.0	−1.05552	−0.527762	−	0.849392i	$-0.676969\pi$
$991$	−32929.0	−1.05552	−0.527762	+	0.849392i	$0.676969\pi$
$992$	−6007.01	−0.192261
$993$	37299.2	1.19200
$994$	−61684.3	−1.96832
$995$	−13491.3	−0.429851
$996$	81085.3	2.57961
$997$	59599.7	1.89322	0.946610	−	0.322382i	$-0.104483\pi$
$997$	59599.7	1.89322	0.946610	+	0.322382i	$0.104483\pi$
$998$	100902.	3.20040
$999$	45003.9	1.42529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.1	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.1	✓	23	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)

\(f(q)\)	\(=\)	\(q-5.41975 q^{2} +4.17361 q^{3} +21.3737 q^{4} -5.00000 q^{5} -22.6199 q^{6} -25.0607 q^{7} -72.4822 q^{8} -9.58100 q^{9} +O(q^{10})\)	\(q-5.41975 q^{2} +4.17361 q^{3} +21.3737 q^{4} -5.00000 q^{5} -22.6199 q^{6} -25.0607 q^{7} -72.4822 q^{8} -9.58100 q^{9} +27.0988 q^{10} -11.0000 q^{11} +89.2055 q^{12} -82.0427 q^{13} +135.823 q^{14} -20.8680 q^{15} +221.846 q^{16} -5.77877 q^{17} +51.9267 q^{18} -19.0000 q^{19} -106.869 q^{20} -104.594 q^{21} +59.6173 q^{22} -202.864 q^{23} -302.512 q^{24} +25.0000 q^{25} +444.651 q^{26} -152.675 q^{27} -535.641 q^{28} -147.937 q^{29} +113.100 q^{30} +9.64993 q^{31} -622.492 q^{32} -45.9097 q^{33} +31.3195 q^{34} +125.304 q^{35} -204.782 q^{36} -294.770 q^{37} +102.975 q^{38} -342.414 q^{39} +362.411 q^{40} +106.299 q^{41} +566.872 q^{42} -268.084 q^{43} -235.111 q^{44} +47.9050 q^{45} +1099.47 q^{46} +379.568 q^{47} +925.898 q^{48} +285.040 q^{49} -135.494 q^{50} -24.1183 q^{51} -1753.56 q^{52} +566.101 q^{53} +827.459 q^{54} +55.0000 q^{55} +1816.46 q^{56} -79.2985 q^{57} +801.781 q^{58} -202.796 q^{59} -446.027 q^{60} -360.835 q^{61} -52.3002 q^{62} +240.107 q^{63} +1598.99 q^{64} +410.214 q^{65} +248.819 q^{66} +523.425 q^{67} -123.514 q^{68} -846.674 q^{69} -679.115 q^{70} -454.152 q^{71} +694.452 q^{72} -744.750 q^{73} +1597.58 q^{74} +104.340 q^{75} -406.101 q^{76} +275.668 q^{77} +1855.80 q^{78} +210.822 q^{79} -1109.23 q^{80} -378.517 q^{81} -576.116 q^{82} +908.973 q^{83} -2235.55 q^{84} +28.8939 q^{85} +1452.95 q^{86} -617.431 q^{87} +797.304 q^{88} +1202.84 q^{89} -259.633 q^{90} +2056.05 q^{91} -4335.95 q^{92} +40.2750 q^{93} -2057.16 q^{94} +95.0000 q^{95} -2598.04 q^{96} +80.6757 q^{97} -1544.85 q^{98} +105.391 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

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