LMFDB - Embedded newform 1045.4.a.f.1.14

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

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+1.33604 q^{2} +6.37251 q^{3} -6.21499 q^{4} -5.00000 q^{5} +8.51394 q^{6} +16.1837 q^{7} -18.9918 q^{8} +13.6088 q^{9} +O(q^{10})$	\(q+1.33604 q^{2} +6.37251 q^{3} -6.21499 q^{4} -5.00000 q^{5} +8.51394 q^{6} +16.1837 q^{7} -18.9918 q^{8} +13.6088 q^{9} -6.68022 q^{10} +11.0000 q^{11} -39.6050 q^{12} -55.6263 q^{13} +21.6222 q^{14} -31.8625 q^{15} +24.3460 q^{16} -8.16400 q^{17} +18.1820 q^{18} -19.0000 q^{19} +31.0749 q^{20} +103.131 q^{21} +14.6965 q^{22} +158.522 q^{23} -121.026 q^{24} +25.0000 q^{25} -74.3192 q^{26} -85.3353 q^{27} -100.582 q^{28} +190.636 q^{29} -42.5697 q^{30} -264.051 q^{31} +184.462 q^{32} +70.0976 q^{33} -10.9075 q^{34} -80.9186 q^{35} -84.5787 q^{36} -375.544 q^{37} -25.3848 q^{38} -354.479 q^{39} +94.9592 q^{40} -318.942 q^{41} +137.787 q^{42} -514.113 q^{43} -68.3649 q^{44} -68.0441 q^{45} +211.793 q^{46} -25.3814 q^{47} +155.145 q^{48} -81.0869 q^{49} +33.4011 q^{50} -52.0251 q^{51} +345.717 q^{52} +236.823 q^{53} -114.012 q^{54} -55.0000 q^{55} -307.359 q^{56} -121.078 q^{57} +254.698 q^{58} -183.845 q^{59} +198.025 q^{60} -772.405 q^{61} -352.784 q^{62} +220.242 q^{63} +51.6814 q^{64} +278.132 q^{65} +93.6534 q^{66} -390.379 q^{67} +50.7392 q^{68} +1010.18 q^{69} -108.111 q^{70} +525.311 q^{71} -258.457 q^{72} +693.653 q^{73} -501.743 q^{74} +159.313 q^{75} +118.085 q^{76} +178.021 q^{77} -473.599 q^{78} -1292.39 q^{79} -121.730 q^{80} -911.238 q^{81} -426.120 q^{82} +660.387 q^{83} -640.957 q^{84} +40.8200 q^{85} -686.877 q^{86} +1214.83 q^{87} -208.910 q^{88} +1227.25 q^{89} -90.9099 q^{90} -900.241 q^{91} -985.215 q^{92} -1682.67 q^{93} -33.9106 q^{94} +95.0000 q^{95} +1175.49 q^{96} -35.3173 q^{97} -108.336 q^{98} +149.697 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * 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 $

$2$	1.33604	0.472363	0.236181	−	0.971709i	$-0.424104\pi$
$2$	1.33604	0.472363	0.236181	+	0.971709i	$0.424104\pi$
$3$	6.37251	1.22639	0.613195	−	0.789932i	$-0.289884\pi$
$3$	6.37251	1.22639	0.613195	+	0.789932i	$0.289884\pi$
$4$	−6.21499	−0.776873
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	8.51394	0.579300
$7$	16.1837	0.873839	0.436920	−	0.899501i	$-0.356069\pi$
$7$	16.1837	0.873839	0.436920	+	0.899501i	$0.356069\pi$
$8$	−18.9918	−0.839329
$9$	13.6088	0.504030
$10$	−6.68022	−0.211247
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−39.6050	−0.952749
$13$	−55.6263	−1.18677	−0.593383	−	0.804920i	$-0.702208\pi$
$13$	−55.6263	−1.18677	−0.593383	+	0.804920i	$0.702208\pi$
$14$	21.6222	0.412769
$15$	−31.8625	−0.548458
$16$	24.3460	0.380406
$17$	−8.16400	−0.116474	−0.0582371	−	0.998303i	$-0.518548\pi$
$17$	−8.16400	−0.116474	−0.0582371	+	0.998303i	$0.518548\pi$
$18$	18.1820	0.238085
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	31.0749	0.347428
$21$	103.131	1.07167
$22$	14.6965	0.142423
$23$	158.522	1.43714	0.718570	−	0.695455i	$-0.244797\pi$
$23$	158.522	1.43714	0.718570	+	0.695455i	$0.244797\pi$
$24$	−121.026	−1.02934
$25$	25.0000	0.200000
$26$	−74.3192	−0.560584
$27$	−85.3353	−0.608252
$28$	−100.582	−0.678863
$29$	190.636	1.22070	0.610348	−	0.792133i	$-0.291030\pi$
$29$	190.636	1.22070	0.610348	+	0.792133i	$0.291030\pi$
$30$	−42.5697	−0.259071
$31$	−264.051	−1.52984	−0.764919	−	0.644127i	$-0.777221\pi$
$31$	−264.051	−1.52984	−0.764919	+	0.644127i	$0.777221\pi$
$32$	184.462	1.01902
$33$	70.0976	0.369770
$34$	−10.9075	−0.0550181
$35$	−80.9186	−0.390793
$36$	−84.5787	−0.391568
$37$	−375.544	−1.66862	−0.834311	−	0.551294i	$-0.814134\pi$
$37$	−375.544	−1.66862	−0.834311	+	0.551294i	$0.814134\pi$
$38$	−25.3848	−0.108367
$39$	−354.479	−1.45544
$40$	94.9592	0.375359
$41$	−318.942	−1.21489	−0.607443	−	0.794363i	$-0.707805\pi$
$41$	−318.942	−1.21489	−0.607443	+	0.794363i	$0.707805\pi$
$42$	137.787	0.506216
$43$	−514.113	−1.82329	−0.911645	−	0.410979i	$-0.865187\pi$
$43$	−514.113	−1.82329	−0.911645	+	0.410979i	$0.865187\pi$
$44$	−68.3649	−0.234236
$45$	−68.0441	−0.225409
$46$	211.793	0.678851
$47$	−25.3814	−0.0787714	−0.0393857	−	0.999224i	$-0.512540\pi$
$47$	−25.3814	−0.0787714	−0.0393857	+	0.999224i	$0.512540\pi$
$48$	155.145	0.466526
$49$	−81.0869	−0.236405
$50$	33.4011	0.0944725
$51$	−52.0251	−0.142843
$52$	345.717	0.921968
$53$	236.823	0.613776	0.306888	−	0.951746i	$-0.400712\pi$
$53$	236.823	0.613776	0.306888	+	0.951746i	$0.400712\pi$
$54$	−114.012	−0.287315
$55$	−55.0000	−0.134840
$56$	−307.359	−0.733438
$57$	−121.078	−0.281353
$58$	254.698	0.576611
$59$	−183.845	−0.405670	−0.202835	−	0.979213i	$-0.565016\pi$
$59$	−183.845	−0.405670	−0.202835	+	0.979213i	$0.565016\pi$
$60$	198.025	0.426082
$61$	−772.405	−1.62125	−0.810626	−	0.585564i	$-0.800873\pi$
$61$	−772.405	−1.62125	−0.810626	+	0.585564i	$0.800873\pi$
$62$	−352.784	−0.722638
$63$	220.242	0.440442
$64$	51.6814	0.100940
$65$	278.132	0.530738
$66$	93.6534	0.174666
$67$	−390.379	−0.711827	−0.355913	−	0.934519i	$-0.615830\pi$
$67$	−390.379	−0.711827	−0.355913	+	0.934519i	$0.615830\pi$
$68$	50.7392	0.0904857
$69$	1010.18	1.76249
$70$	−108.111	−0.184596
$71$	525.311	0.878070	0.439035	−	0.898470i	$-0.355320\pi$
$71$	525.311	0.878070	0.439035	+	0.898470i	$0.355320\pi$
$72$	−258.457	−0.423047
$73$	693.653	1.11214	0.556068	−	0.831137i	$-0.312309\pi$
$73$	693.653	1.11214	0.556068	+	0.831137i	$0.312309\pi$
$74$	−501.743	−0.788195
$75$	159.313	0.245278
$76$	118.085	0.178227
$77$	178.021	0.263472
$78$	−473.599	−0.687495
$79$	−1292.39	−1.84057	−0.920284	−	0.391252i	$-0.872042\pi$
$79$	−1292.39	−1.84057	−0.920284	+	0.391252i	$0.872042\pi$
$80$	−121.730	−0.170123
$81$	−911.238	−1.24998
$82$	−426.120	−0.573867
$83$	660.387	0.873337	0.436668	−	0.899623i	$-0.356158\pi$
$83$	660.387	0.873337	0.436668	+	0.899623i	$0.356158\pi$
$84$	−640.957	−0.832550
$85$	40.8200	0.0520888
$86$	−686.877	−0.861254
$87$	1214.83	1.49705
$88$	−208.910	−0.253067
$89$	1227.25	1.46167	0.730833	−	0.682556i	$-0.239132\pi$
$89$	1227.25	1.46167	0.730833	+	0.682556i	$0.239132\pi$
$90$	−90.9099	−0.106475
$91$	−900.241	−1.03704
$92$	−985.215	−1.11648
$93$	−1682.67	−1.87618
$94$	−33.9106	−0.0372087
$95$	95.0000	0.102598
$96$	1175.49	1.24971
$97$	−35.3173	−0.0369683	−0.0184842	−	0.999829i	$-0.505884\pi$
$97$	−35.3173	−0.0369683	−0.0184842	+	0.999829i	$0.505884\pi$
$98$	−108.336	−0.111669
$99$	149.697	0.151971
$100$	−155.375	−0.155375
$101$	−973.146	−0.958729	−0.479365	−	0.877616i	$-0.659133\pi$
$101$	−973.146	−0.958729	−0.479365	+	0.877616i	$0.659133\pi$
$102$	−69.5079	−0.0674736
$103$	−729.674	−0.698028	−0.349014	−	0.937117i	$-0.613483\pi$
$103$	−729.674	−0.698028	−0.349014	+	0.937117i	$0.613483\pi$
$104$	1056.45	0.996087
$105$	−515.655	−0.479264
$106$	316.406	0.289925
$107$	82.6837	0.0747040	0.0373520	−	0.999302i	$-0.488108\pi$
$107$	82.6837	0.0747040	0.0373520	+	0.999302i	$0.488108\pi$
$108$	530.358	0.472535
$109$	211.830	0.186143	0.0930717	−	0.995659i	$-0.470331\pi$
$109$	211.830	0.186143	0.0930717	+	0.995659i	$0.470331\pi$
$110$	−73.4824	−0.0636934
$111$	−2393.15	−2.04638
$112$	394.009	0.332414
$113$	93.8322	0.0781150	0.0390575	−	0.999237i	$-0.487564\pi$
$113$	93.8322	0.0781150	0.0390575	+	0.999237i	$0.487564\pi$
$114$	−161.765	−0.132901
$115$	−792.612	−0.642708
$116$	−1184.80	−0.948326
$117$	−757.009	−0.598167
$118$	−245.625	−0.191624
$119$	−132.124	−0.101780
$120$	605.128	0.460336
$121$	121.000	0.0909091
$122$	−1031.97	−0.765819
$123$	−2032.46	−1.48992
$124$	1641.07	1.18849
$125$	−125.000	−0.0894427
$126$	294.252	0.208048
$127$	389.724	0.272302	0.136151	−	0.990688i	$-0.456527\pi$
$127$	389.724	0.272302	0.136151	+	0.990688i	$0.456527\pi$
$128$	−1406.65	−0.971338
$129$	−3276.19	−2.23606
$130$	371.596	0.250701
$131$	−1255.36	−0.837259	−0.418629	−	0.908157i	$-0.637489\pi$
$131$	−1255.36	−0.837259	−0.418629	+	0.908157i	$0.637489\pi$
$132$	−435.655	−0.287265
$133$	−307.491	−0.200472
$134$	−521.563	−0.336240
$135$	426.677	0.272018
$136$	155.049	0.0977601
$137$	−1296.97	−0.808818	−0.404409	−	0.914578i	$-0.632523\pi$
$137$	−1296.97	−0.808818	−0.404409	+	0.914578i	$0.632523\pi$
$138$	1349.65	0.832536
$139$	−344.563	−0.210255	−0.105127	−	0.994459i	$-0.533525\pi$
$139$	−344.563	−0.210255	−0.105127	+	0.994459i	$0.533525\pi$
$140$	502.908	0.303597
$141$	−161.743	−0.0966044
$142$	701.838	0.414767
$143$	−611.890	−0.357824
$144$	331.320	0.191736
$145$	−953.179	−0.545912
$146$	926.751	0.525332
$147$	−516.727	−0.289924
$148$	2334.00	1.29631
$149$	1881.27	1.03436	0.517179	−	0.855877i	$-0.326982\pi$
$149$	1881.27	1.03436	0.517179	+	0.855877i	$0.326982\pi$
$150$	212.849	0.115860
$151$	2296.86	1.23785	0.618926	−	0.785449i	$-0.287568\pi$
$151$	2296.86	1.23785	0.618926	+	0.785449i	$0.287568\pi$
$152$	360.845	0.192555
$153$	−111.102	−0.0587065
$154$	237.844	0.124455
$155$	1320.26	0.684164
$156$	2203.08	1.13069
$157$	−2479.02	−1.26017	−0.630086	−	0.776525i	$-0.716981\pi$
$157$	−2479.02	−1.26017	−0.630086	+	0.776525i	$0.716981\pi$
$158$	−1726.68	−0.869415
$159$	1509.15	0.752728
$160$	−922.310	−0.455719
$161$	2565.48	1.25583
$162$	−1217.45	−0.590446
$163$	3559.37	1.71038	0.855188	−	0.518317i	$-0.173441\pi$
$163$	3559.37	1.71038	0.855188	+	0.518317i	$0.173441\pi$
$164$	1982.22	0.943813
$165$	−350.488	−0.165366
$166$	882.306	0.412532
$167$	−3197.14	−1.48145	−0.740724	−	0.671810i	$-0.765517\pi$
$167$	−3197.14	−1.48145	−0.740724	+	0.671810i	$0.765517\pi$
$168$	−1958.65	−0.899481
$169$	897.289	0.408415
$170$	54.5373	0.0246048
$171$	−258.568	−0.115633
$172$	3195.21	1.41647
$173$	−3458.07	−1.51972	−0.759862	−	0.650084i	$-0.774734\pi$
$173$	−3458.07	−1.51972	−0.759862	+	0.650084i	$0.774734\pi$
$174$	1623.06	0.707150
$175$	404.593	0.174768
$176$	267.806	0.114697
$177$	−1171.55	−0.497510
$178$	1639.66	0.690436
$179$	−3179.62	−1.32769	−0.663844	−	0.747871i	$-0.731076\pi$
$179$	−3179.62	−1.32769	−0.663844	+	0.747871i	$0.731076\pi$
$180$	422.893	0.175114
$181$	300.453	0.123384	0.0616919	−	0.998095i	$-0.480350\pi$
$181$	300.453	0.123384	0.0616919	+	0.998095i	$0.480350\pi$
$182$	−1202.76	−0.489861
$183$	−4922.16	−1.98829
$184$	−3010.63	−1.20623
$185$	1877.72	0.746230
$186$	−2248.12	−0.886236
$187$	−89.8040	−0.0351183
$188$	157.745	0.0611954
$189$	−1381.04	−0.531514
$190$	126.924	0.0484634
$191$	3613.46	1.36890	0.684451	−	0.729058i	$-0.260042\pi$
$191$	3613.46	1.36890	0.684451	+	0.729058i	$0.260042\pi$
$192$	329.340	0.123792
$193$	2387.10	0.890297	0.445148	−	0.895457i	$-0.353151\pi$
$193$	2387.10	0.890297	0.445148	+	0.895457i	$0.353151\pi$
$194$	−47.1855	−0.0174625
$195$	1772.40	0.650892
$196$	503.954	0.183657
$197$	−3110.45	−1.12493	−0.562463	−	0.826822i	$-0.690146\pi$
$197$	−3110.45	−1.12493	−0.562463	+	0.826822i	$0.690146\pi$
$198$	200.002	0.0717854
$199$	−123.725	−0.0440735	−0.0220368	−	0.999757i	$-0.507015\pi$
$199$	−123.725	−0.0440735	−0.0220368	+	0.999757i	$0.507015\pi$
$200$	−474.796	−0.167866
$201$	−2487.69	−0.872977
$202$	−1300.17	−0.452868
$203$	3085.20	1.06669
$204$	323.336	0.110971
$205$	1594.71	0.543314
$206$	−974.876	−0.329722
$207$	2157.30	0.724362
$208$	−1354.28	−0.451453
$209$	−209.000	−0.0691714
$210$	−688.937	−0.226386
$211$	3809.75	1.24301	0.621503	−	0.783412i	$-0.286522\pi$
$211$	3809.75	1.24301	0.621503	+	0.783412i	$0.286522\pi$
$212$	−1471.85	−0.476826
$213$	3347.55	1.07686
$214$	110.469	0.0352874
$215$	2570.56	0.815400
$216$	1620.68	0.510523
$217$	−4273.33	−1.33683
$218$	283.014	0.0879272
$219$	4420.31	1.36391
$220$	341.824	0.104754
$221$	454.133	0.138228
$222$	−3197.36	−0.966634
$223$	−1854.26	−0.556819	−0.278409	−	0.960463i	$-0.589807\pi$
$223$	−1854.26	−0.556819	−0.278409	+	0.960463i	$0.589807\pi$
$224$	2985.28	0.890458
$225$	340.221	0.100806
$226$	125.364	0.0368986
$227$	−313.399	−0.0916346	−0.0458173	−	0.998950i	$-0.514589\pi$
$227$	−313.399	−0.0916346	−0.0458173	+	0.998950i	$0.514589\pi$
$228$	752.496	0.218576
$229$	3415.74	0.985670	0.492835	−	0.870123i	$-0.335961\pi$
$229$	3415.74	0.985670	0.492835	+	0.870123i	$0.335961\pi$
$230$	−1058.96	−0.303591
$231$	1134.44	0.323120
$232$	−3620.52	−1.02457
$233$	4166.48	1.17148	0.585740	−	0.810499i	$-0.300804\pi$
$233$	4166.48	1.17148	0.585740	+	0.810499i	$0.300804\pi$
$234$	−1011.40	−0.282552
$235$	126.907	0.0352276
$236$	1142.59	0.315155
$237$	−8235.74	−2.25725
$238$	−176.523	−0.0480769
$239$	2655.56	0.718719	0.359359	−	0.933199i	$-0.382995\pi$
$239$	2655.56	0.718719	0.359359	+	0.933199i	$0.382995\pi$
$240$	−775.724	−0.208637
$241$	−210.549	−0.0562767	−0.0281383	−	0.999604i	$-0.508958\pi$
$241$	−210.549	−0.0562767	−0.0281383	+	0.999604i	$0.508958\pi$
$242$	161.661	0.0429421
$243$	−3502.82	−0.924715
$244$	4800.49	1.25951
$245$	405.434	0.105723
$246$	−2715.45	−0.703784
$247$	1056.90	0.272263
$248$	5014.82	1.28404
$249$	4208.32	1.07105
$250$	−167.005	−0.0422494
$251$	−2113.44	−0.531471	−0.265736	−	0.964046i	$-0.585615\pi$
$251$	−2113.44	−0.531471	−0.265736	+	0.964046i	$0.585615\pi$
$252$	−1368.80	−0.342167
$253$	1743.75	0.433314
$254$	520.688	0.128625
$255$	260.126	0.0638812
$256$	−2292.79	−0.559764
$257$	−4717.34	−1.14498	−0.572490	−	0.819912i	$-0.694022\pi$
$257$	−4717.34	−1.14498	−0.572490	+	0.819912i	$0.694022\pi$
$258$	−4377.13	−1.05623
$259$	−6077.70	−1.45811
$260$	−1728.58	−0.412316
$261$	2594.33	0.615268
$262$	−1677.21	−0.395490
$263$	7112.92	1.66769	0.833843	−	0.552002i	$-0.186136\pi$
$263$	7112.92	1.66769	0.833843	+	0.552002i	$0.186136\pi$
$264$	−1331.28	−0.310359
$265$	−1184.11	−0.274489
$266$	−410.821	−0.0946957
$267$	7820.66	1.79257
$268$	2426.20	0.552999
$269$	3644.48	0.826051	0.413025	−	0.910720i	$-0.364472\pi$
$269$	3644.48	0.826051	0.413025	+	0.910720i	$0.364472\pi$
$270$	570.059	0.128491
$271$	7347.81	1.64704	0.823520	−	0.567288i	$-0.192007\pi$
$271$	7347.81	1.64704	0.823520	+	0.567288i	$0.192007\pi$
$272$	−198.761	−0.0443075
$273$	−5736.79	−1.27182
$274$	−1732.81	−0.382055
$275$	275.000	0.0603023
$276$	−6278.29	−1.36923
$277$	−3213.35	−0.697008	−0.348504	−	0.937307i	$-0.613310\pi$
$277$	−3213.35	−0.697008	−0.348504	+	0.937307i	$0.613310\pi$
$278$	−460.351	−0.0993165
$279$	−3593.42	−0.771085
$280$	1536.79	0.328004
$281$	724.016	0.153705	0.0768527	−	0.997042i	$-0.475513\pi$
$281$	724.016	0.153705	0.0768527	+	0.997042i	$0.475513\pi$
$282$	−216.096	−0.0456323
$283$	7087.10	1.48864	0.744319	−	0.667825i	$-0.232774\pi$
$283$	7087.10	1.48864	0.744319	+	0.667825i	$0.232774\pi$
$284$	−3264.80	−0.682149
$285$	605.388	0.125825
$286$	−817.511	−0.169023
$287$	−5161.67	−1.06162
$288$	2510.31	0.513616
$289$	−4846.35	−0.986434
$290$	−1273.49	−0.257868
$291$	−225.060	−0.0453376
$292$	−4311.05	−0.863989
$293$	−2569.95	−0.512416	−0.256208	−	0.966622i	$-0.582473\pi$
$293$	−2569.95	−0.512416	−0.256208	+	0.966622i	$0.582473\pi$
$294$	−690.369	−0.136949
$295$	919.224	0.181421
$296$	7132.27	1.40052
$297$	−938.689	−0.183395
$298$	2513.46	0.488593
$299$	−8818.02	−1.70555
$300$	−990.126	−0.190550
$301$	−8320.26	−1.59326
$302$	3068.70	0.584715
$303$	−6201.38	−1.17578
$304$	−462.574	−0.0872711
$305$	3862.03	0.725046
$306$	−148.438	−0.0277308
$307$	−2238.80	−0.416206	−0.208103	−	0.978107i	$-0.566729\pi$
$307$	−2238.80	−0.416206	−0.208103	+	0.978107i	$0.566729\pi$
$308$	−1106.40	−0.204685
$309$	−4649.85	−0.856054
$310$	1763.92	0.323174
$311$	−6471.38	−1.17993	−0.589965	−	0.807429i	$-0.700858\pi$
$311$	−6471.38	−1.17993	−0.589965	+	0.807429i	$0.700858\pi$
$312$	6732.21	1.22159
$313$	−1910.50	−0.345009	−0.172504	−	0.985009i	$-0.555186\pi$
$313$	−1910.50	−0.345009	−0.172504	+	0.985009i	$0.555186\pi$
$314$	−3312.07	−0.595259
$315$	−1101.21	−0.196971
$316$	8032.16	1.42989
$317$	−993.770	−0.176075	−0.0880374	−	0.996117i	$-0.528059\pi$
$317$	−993.770	−0.176075	−0.0880374	+	0.996117i	$0.528059\pi$
$318$	2016.30	0.355561
$319$	2096.99	0.368054
$320$	−258.407	−0.0451419
$321$	526.902	0.0916162
$322$	3427.60	0.593207
$323$	155.116	0.0267210
$324$	5663.33	0.971079
$325$	−1390.66	−0.237353
$326$	4755.47	0.807918
$327$	1349.89	0.228284
$328$	6057.29	1.01969
$329$	−410.766	−0.0688335
$330$	−468.267	−0.0781129
$331$	8091.85	1.34371	0.671855	−	0.740683i	$-0.265498\pi$
$331$	8091.85	1.34371	0.671855	+	0.740683i	$0.265498\pi$
$332$	−4104.30	−0.678472
$333$	−5110.71	−0.841036
$334$	−4271.51	−0.699781
$335$	1951.90	0.318339
$336$	2510.82	0.407668
$337$	3825.80	0.618412	0.309206	−	0.950995i	$-0.399937\pi$
$337$	3825.80	0.618412	0.309206	+	0.950995i	$0.399937\pi$
$338$	1198.82	0.192920
$339$	597.946	0.0957993
$340$	−253.696	−0.0404664
$341$	−2904.56	−0.461263
$342$	−345.458	−0.0546205
$343$	−6863.31	−1.08042
$344$	9763.95	1.53034
$345$	−5050.92	−0.788211
$346$	−4620.14	−0.717861
$347$	1626.66	0.251653	0.125827	−	0.992052i	$-0.459842\pi$
$347$	1626.66	0.251653	0.125827	+	0.992052i	$0.459842\pi$
$348$	−7550.14	−1.16302
$349$	−4307.34	−0.660649	−0.330325	−	0.943867i	$-0.607158\pi$
$349$	−4307.34	−0.660649	−0.330325	+	0.943867i	$0.607158\pi$
$350$	540.554	0.0825538
$351$	4746.89	0.721853
$352$	2029.08	0.307246
$353$	4790.02	0.722230	0.361115	−	0.932521i	$-0.382396\pi$
$353$	4790.02	0.722230	0.361115	+	0.932521i	$0.382396\pi$
$354$	−1565.24	−0.235005
$355$	−2626.55	−0.392685
$356$	−7627.35	−1.13553
$357$	−841.961	−0.124822
$358$	−4248.12	−0.627151
$359$	−4284.06	−0.629816	−0.314908	−	0.949122i	$-0.601974\pi$
$359$	−4284.06	−0.629816	−0.314908	+	0.949122i	$0.601974\pi$
$360$	1292.28	0.189192
$361$	361.000	0.0526316
$362$	401.418	0.0582819
$363$	771.073	0.111490
$364$	5594.99	0.805652
$365$	−3468.27	−0.497362
$366$	−6576.22	−0.939192
$367$	3382.58	0.481115	0.240558	−	0.970635i	$-0.422670\pi$
$367$	3382.58	0.481115	0.240558	+	0.970635i	$0.422670\pi$
$368$	3859.38	0.546696
$369$	−4340.42	−0.612340
$370$	2508.71	0.352491
$371$	3832.68	0.536341
$372$	10457.8	1.45755
$373$	−274.267	−0.0380725	−0.0190362	−	0.999819i	$-0.506060\pi$
$373$	−274.267	−0.0380725	−0.0190362	+	0.999819i	$0.506060\pi$
$374$	−119.982	−0.0165886
$375$	−796.563	−0.109692
$376$	482.039	0.0661151
$377$	−10604.4	−1.44868
$378$	−1845.13	−0.251067
$379$	−6237.57	−0.845389	−0.422695	−	0.906272i	$-0.638916\pi$
$379$	−6237.57	−0.845389	−0.422695	+	0.906272i	$0.638916\pi$
$380$	−590.424	−0.0797055
$381$	2483.52	0.333948
$382$	4827.73	0.646619
$383$	−7812.09	−1.04224	−0.521121	−	0.853483i	$-0.674486\pi$
$383$	−7812.09	−1.04224	−0.521121	+	0.853483i	$0.674486\pi$
$384$	−8963.87	−1.19124
$385$	−890.105	−0.117828
$386$	3189.27	0.420543
$387$	−6996.47	−0.918994
$388$	219.497	0.0287197
$389$	13766.4	1.79431	0.897153	−	0.441720i	$-0.145632\pi$
$389$	13766.4	1.79431	0.897153	+	0.441720i	$0.145632\pi$
$390$	2368.00	0.307457
$391$	−1294.18	−0.167390
$392$	1539.99	0.198421
$393$	−7999.76	−1.02681
$394$	−4155.70	−0.531373
$395$	6461.93	0.823127
$396$	−930.365	−0.118062
$397$	−1047.29	−0.132398	−0.0661991	−	0.997806i	$-0.521087\pi$
$397$	−1047.29	−0.132398	−0.0661991	+	0.997806i	$0.521087\pi$
$398$	−165.302	−0.0208187
$399$	−1959.49	−0.245857
$400$	608.649	0.0760812
$401$	−1652.58	−0.205800	−0.102900	−	0.994692i	$-0.532812\pi$
$401$	−1652.58	−0.205800	−0.102900	+	0.994692i	$0.532812\pi$
$402$	−3323.67	−0.412362
$403$	14688.2	1.81556
$404$	6048.09	0.744811
$405$	4556.19	0.559010
$406$	4121.96	0.503865
$407$	−4130.98	−0.503109
$408$	988.053	0.119892
$409$	−5896.77	−0.712901	−0.356451	−	0.934314i	$-0.616013\pi$
$409$	−5896.77	−0.712901	−0.356451	+	0.934314i	$0.616013\pi$
$410$	2130.60	0.256641
$411$	−8264.98	−0.991925
$412$	4534.91	0.542280
$413$	−2975.30	−0.354491
$414$	2882.25	0.342162
$415$	−3301.94	−0.390568
$416$	−10260.9	−1.20934
$417$	−2195.73	−0.257854
$418$	−279.233	−0.0326740
$419$	906.271	0.105666	0.0528332	−	0.998603i	$-0.483175\pi$
$419$	906.271	0.105666	0.0528332	+	0.998603i	$0.483175\pi$
$420$	3204.79	0.372328
$421$	8147.97	0.943249	0.471625	−	0.881799i	$-0.343668\pi$
$421$	8147.97	0.943249	0.471625	+	0.881799i	$0.343668\pi$
$422$	5090.00	0.587150
$423$	−345.411	−0.0397032
$424$	−4497.70	−0.515160
$425$	−204.100	−0.0232948
$426$	4472.47	0.508666
$427$	−12500.4	−1.41671
$428$	−513.878	−0.0580356
$429$	−3899.27	−0.438831
$430$	3434.39	0.385165
$431$	1736.38	0.194057	0.0970284	−	0.995282i	$-0.469066\pi$
$431$	1736.38	0.194057	0.0970284	+	0.995282i	$0.469066\pi$
$432$	−2077.57	−0.231383
$433$	−4257.81	−0.472557	−0.236278	−	0.971685i	$-0.575928\pi$
$433$	−4257.81	−0.472557	−0.236278	+	0.971685i	$0.575928\pi$
$434$	−5709.36	−0.631470
$435$	−6074.14	−0.669500
$436$	−1316.52	−0.144610
$437$	−3011.93	−0.329702
$438$	5905.72	0.644261
$439$	16056.0	1.74558	0.872790	−	0.488095i	$-0.162308\pi$
$439$	16056.0	1.74558	0.872790	+	0.488095i	$0.162308\pi$
$440$	1044.55	0.113175
$441$	−1103.50	−0.119155
$442$	606.742	0.0652936
$443$	9468.36	1.01547	0.507737	−	0.861512i	$-0.330482\pi$
$443$	9468.36	1.01547	0.507737	+	0.861512i	$0.330482\pi$
$444$	14873.4	1.58978
$445$	−6136.25	−0.653677
$446$	−2477.37	−0.263020
$447$	11988.4	1.26853
$448$	836.398	0.0882056
$449$	−12227.6	−1.28520	−0.642602	−	0.766200i	$-0.722145\pi$
$449$	−12227.6	−1.28520	−0.642602	+	0.766200i	$0.722145\pi$
$450$	454.549	0.0476170
$451$	−3508.36	−0.366302
$452$	−583.166	−0.0606854
$453$	14636.7	1.51809
$454$	−418.715	−0.0432848
$455$	4501.21	0.463780
$456$	2299.49	0.236148
$457$	14105.9	1.44386	0.721930	−	0.691966i	$-0.243255\pi$
$457$	14105.9	1.44386	0.721930	+	0.691966i	$0.243255\pi$
$458$	4563.58	0.465594
$459$	696.678	0.0708456
$460$	4926.07	0.499303
$461$	−2306.26	−0.233000	−0.116500	−	0.993191i	$-0.537168\pi$
$461$	−2306.26	−0.233000	−0.116500	+	0.993191i	$0.537168\pi$
$462$	1515.66	0.152630
$463$	14582.0	1.46367	0.731837	−	0.681480i	$-0.238663\pi$
$463$	14582.0	1.46367	0.731837	+	0.681480i	$0.238663\pi$
$464$	4641.22	0.464360
$465$	8413.33	0.839052
$466$	5566.59	0.553364
$467$	7890.71	0.781882	0.390941	−	0.920416i	$-0.372150\pi$
$467$	7890.71	0.781882	0.390941	+	0.920416i	$0.372150\pi$
$468$	4704.80	0.464700
$469$	−6317.79	−0.622022
$470$	169.553	0.0166402
$471$	−15797.6	−1.54546
$472$	3491.55	0.340491
$473$	−5655.24	−0.549743
$474$	−11003.3	−1.06624
$475$	−475.000	−0.0458831
$476$	821.149	0.0790700
$477$	3222.88	0.309362
$478$	3547.94	0.339496
$479$	14609.9	1.39362	0.696809	−	0.717257i	$-0.254603\pi$
$479$	14609.9	1.39362	0.696809	+	0.717257i	$0.254603\pi$
$480$	−5877.43	−0.558889
$481$	20890.1	1.98027
$482$	−281.303	−0.0265830
$483$	16348.6	1.54013
$484$	−752.014	−0.0706249
$485$	176.587	0.0165327
$486$	−4679.91	−0.436801
$487$	−10799.7	−1.00489	−0.502445	−	0.864609i	$-0.667566\pi$
$487$	−10799.7	−1.00489	−0.502445	+	0.864609i	$0.667566\pi$
$488$	14669.4	1.36076
$489$	22682.1	2.09759
$490$	541.678	0.0499398
$491$	−14534.9	−1.33594	−0.667972	−	0.744186i	$-0.732838\pi$
$491$	−14534.9	−1.33594	−0.667972	+	0.744186i	$0.732838\pi$
$492$	12631.7	1.15748
$493$	−1556.35	−0.142180
$494$	1412.06	0.128607
$495$	−748.485	−0.0679635
$496$	−6428.58	−0.581959
$497$	8501.49	0.767292
$498$	5622.50	0.505924
$499$	11239.3	1.00829	0.504147	−	0.863618i	$-0.331807\pi$
$499$	11239.3	1.00829	0.504147	+	0.863618i	$0.331807\pi$
$500$	776.873	0.0694857
$501$	−20373.8	−1.81683
$502$	−2823.65	−0.251047
$503$	4334.05	0.384186	0.192093	−	0.981377i	$-0.438472\pi$
$503$	4334.05	0.384186	0.192093	+	0.981377i	$0.438472\pi$
$504$	−4182.79	−0.369675
$505$	4865.73	0.428757
$506$	2329.72	0.204681
$507$	5717.98	0.500876
$508$	−2422.13	−0.211544
$509$	672.374	0.0585510	0.0292755	−	0.999571i	$-0.490680\pi$
$509$	672.374	0.0585510	0.0292755	+	0.999571i	$0.490680\pi$
$510$	347.539	0.0301751
$511$	11225.9	0.971828
$512$	8189.91	0.706926
$513$	1621.37	0.139543
$514$	−6302.58	−0.540846
$515$	3648.37	0.312168
$516$	20361.5	1.73714
$517$	−279.195	−0.0237505
$518$	−8120.07	−0.688756
$519$	−22036.6	−1.86377
$520$	−5282.23	−0.445464
$521$	15470.9	1.30094	0.650471	−	0.759531i	$-0.274571\pi$
$521$	15470.9	1.30094	0.650471	+	0.759531i	$0.274571\pi$
$522$	3466.14	0.290630
$523$	−13609.9	−1.13790	−0.568949	−	0.822373i	$-0.692650\pi$
$523$	−13609.9	−1.13790	−0.568949	+	0.822373i	$0.692650\pi$
$524$	7802.02	0.650444
$525$	2578.27	0.214333
$526$	9503.17	0.787752
$527$	2155.71	0.178187
$528$	1706.59	0.140663
$529$	12962.4	1.06537
$530$	−1582.03	−0.129658
$531$	−2501.91	−0.204470
$532$	1911.05	0.155742
$533$	17741.6	1.44179
$534$	10448.7	0.846744
$535$	−413.418	−0.0334087
$536$	7414.02	0.597457
$537$	−20262.2	−1.62826
$538$	4869.18	0.390195
$539$	−891.956	−0.0712788
$540$	−2651.79	−0.211324
$541$	1509.57	0.119966	0.0599831	−	0.998199i	$-0.480895\pi$
$541$	1509.57	0.119966	0.0599831	+	0.998199i	$0.480895\pi$
$542$	9816.99	0.778000
$543$	1914.64	0.151317
$544$	−1505.95	−0.118689
$545$	−1059.15	−0.0832458
$546$	−7664.61	−0.600760
$547$	−5393.41	−0.421582	−0.210791	−	0.977531i	$-0.567604\pi$
$547$	−5393.41	−0.421582	−0.210791	+	0.977531i	$0.567604\pi$
$548$	8060.68	0.628349
$549$	−10511.5	−0.817161
$550$	367.412	0.0284845
$551$	−3622.08	−0.280047
$552$	−19185.3	−1.47931
$553$	−20915.6	−1.60836
$554$	−4293.17	−0.329241
$555$	11965.8	0.915169
$556$	2141.45	0.163341
$557$	12878.8	0.979695	0.489848	−	0.871808i	$-0.337052\pi$
$557$	12878.8	0.979695	0.489848	+	0.871808i	$0.337052\pi$
$558$	−4800.97	−0.364232
$559$	28598.2	2.16382
$560$	−1970.04	−0.148660
$561$	−572.277	−0.0430687
$562$	967.317	0.0726047
$563$	4508.88	0.337525	0.168762	−	0.985657i	$-0.446023\pi$
$563$	4508.88	0.337525	0.168762	+	0.985657i	$0.446023\pi$
$564$	1005.23	0.0750494
$565$	−469.161	−0.0349341
$566$	9468.67	0.703177
$567$	−14747.2	−1.09228
$568$	−9976.62	−0.736989
$569$	18323.8	1.35004	0.675020	−	0.737799i	$-0.264135\pi$
$569$	18323.8	1.35004	0.675020	+	0.737799i	$0.264135\pi$
$570$	808.825	0.0594350
$571$	−17211.5	−1.26144	−0.630718	−	0.776012i	$-0.717240\pi$
$571$	−17211.5	−1.26144	−0.630718	+	0.776012i	$0.717240\pi$
$572$	3802.89	0.277984
$573$	23026.8	1.67881
$574$	−6896.21	−0.501468
$575$	3963.06	0.287428
$576$	703.323	0.0508770
$577$	−19003.3	−1.37109	−0.685544	−	0.728031i	$-0.740436\pi$
$577$	−19003.3	−1.37109	−0.685544	+	0.728031i	$0.740436\pi$
$578$	−6474.93	−0.465954
$579$	15211.8	1.09185
$580$	5924.00	0.424104
$581$	10687.5	0.763156
$582$	−300.690	−0.0214158
$583$	2605.05	0.185060
$584$	−13173.7	−0.933448
$585$	3785.04	0.267508
$586$	−3433.56	−0.242046
$587$	4312.20	0.303209	0.151604	−	0.988441i	$-0.451556\pi$
$587$	4312.20	0.303209	0.151604	+	0.988441i	$0.451556\pi$
$588$	3211.45	0.225235
$589$	5016.97	0.350969
$590$	1228.12	0.0856967
$591$	−19821.4	−1.37960
$592$	−9142.98	−0.634754
$593$	−25154.7	−1.74195	−0.870977	−	0.491323i	$-0.836513\pi$
$593$	−25154.7	−1.74195	−0.870977	+	0.491323i	$0.836513\pi$
$594$	−1254.13	−0.0866288
$595$	660.620	0.0455173
$596$	−11692.1	−0.803566
$597$	−788.438	−0.0540513
$598$	−11781.3	−0.805638
$599$	−6007.92	−0.409811	−0.204905	−	0.978782i	$-0.565689\pi$
$599$	−6007.92	−0.409811	−0.204905	+	0.978782i	$0.565689\pi$
$600$	−3025.64	−0.205869
$601$	7349.48	0.498821	0.249411	−	0.968398i	$-0.419763\pi$
$601$	7349.48	0.498821	0.249411	+	0.968398i	$0.419763\pi$
$602$	−11116.2	−0.752598
$603$	−5312.60	−0.358782
$604$	−14274.9	−0.961655
$605$	−605.000	−0.0406558
$606$	−8285.31	−0.555392
$607$	4191.05	0.280246	0.140123	−	0.990134i	$-0.455250\pi$
$607$	4191.05	0.280246	0.140123	+	0.990134i	$0.455250\pi$
$608$	−3504.78	−0.233779
$609$	19660.4	1.30818
$610$	5159.84	0.342485
$611$	1411.87	0.0934833
$612$	690.500	0.0456076
$613$	−26945.4	−1.77539	−0.887695	−	0.460433i	$-0.847694\pi$
$613$	−26945.4	−1.77539	−0.887695	+	0.460433i	$0.847694\pi$
$614$	−2991.14	−0.196600
$615$	10162.3	0.666314
$616$	−3380.95	−0.221140
$617$	8286.07	0.540655	0.270328	−	0.962768i	$-0.412868\pi$
$617$	8286.07	0.540655	0.270328	+	0.962768i	$0.412868\pi$
$618$	−6212.40	−0.404368
$619$	16184.0	1.05087	0.525436	−	0.850833i	$-0.323902\pi$
$619$	16184.0	1.05087	0.525436	+	0.850833i	$0.323902\pi$
$620$	−8205.37	−0.531509
$621$	−13527.6	−0.874142
$622$	−8646.04	−0.557355
$623$	19861.5	1.27726
$624$	−8630.14	−0.553657
$625$	625.000	0.0400000
$626$	−2552.51	−0.162969
$627$	−1331.85	−0.0848311
$628$	15407.1	0.978995
$629$	3065.94	0.194351
$630$	−1471.26	−0.0930420
$631$	−27170.9	−1.71420	−0.857098	−	0.515154i	$-0.827735\pi$
$631$	−27170.9	−1.71420	−0.857098	+	0.515154i	$0.827735\pi$
$632$	24544.8	1.54484
$633$	24277.7	1.52441
$634$	−1327.72	−0.0831712
$635$	−1948.62	−0.121777
$636$	−9379.38	−0.584775
$637$	4510.57	0.280558
$638$	2801.67	0.173855
$639$	7148.86	0.442574
$640$	7033.24	0.434396
$641$	−12674.5	−0.780988	−0.390494	−	0.920605i	$-0.627696\pi$
$641$	−12674.5	−0.780988	−0.390494	+	0.920605i	$0.627696\pi$
$642$	703.964	0.0432761
$643$	3268.44	0.200458	0.100229	−	0.994964i	$-0.468042\pi$
$643$	3268.44	0.200458	0.100229	+	0.994964i	$0.468042\pi$
$644$	−15944.5	−0.975620
$645$	16380.9	0.999998
$646$	207.242	0.0126220
$647$	11934.3	0.725173	0.362587	−	0.931950i	$-0.381894\pi$
$647$	11934.3	0.725173	0.362587	+	0.931950i	$0.381894\pi$
$648$	17306.1	1.04915
$649$	−2022.29	−0.122314
$650$	−1857.98	−0.112117
$651$	−27231.8	−1.63948
$652$	−22121.4	−1.32875
$653$	−23562.5	−1.41205	−0.706027	−	0.708185i	$-0.749514\pi$
$653$	−23562.5	−1.41205	−0.706027	+	0.708185i	$0.749514\pi$
$654$	1803.51	0.107833
$655$	6276.78	0.374434
$656$	−7764.95	−0.462150
$657$	9439.80	0.560551
$658$	−548.801	−0.0325144
$659$	−33717.6	−1.99309	−0.996547	−	0.0830268i	$-0.973541\pi$
$659$	−33717.6	−1.99309	−0.996547	+	0.0830268i	$0.973541\pi$
$660$	2178.28	0.128469
$661$	−7939.55	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$661$	−7939.55	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$662$	10811.1	0.634719
$663$	2893.97	0.169521
$664$	−12542.0	−0.733016
$665$	1537.45	0.0896540
$666$	−6828.13	−0.397274
$667$	30220.0	1.75431
$668$	19870.2	1.15090
$669$	−11816.3	−0.682876
$670$	2607.82	0.150371
$671$	−8496.46	−0.488826
$672$	19023.7	1.09205
$673$	−33061.2	−1.89364	−0.946818	−	0.321769i	$-0.895722\pi$
$673$	−33061.2	−1.89364	−0.946818	+	0.321769i	$0.895722\pi$
$674$	5111.44	0.292115
$675$	−2133.38	−0.121650
$676$	−5576.64	−0.317287
$677$	−9384.02	−0.532729	−0.266364	−	0.963872i	$-0.585822\pi$
$677$	−9384.02	−0.532729	−0.266364	+	0.963872i	$0.585822\pi$
$678$	798.882	0.0452520
$679$	−571.566	−0.0323044
$680$	−775.247	−0.0437197
$681$	−1997.14	−0.112380
$682$	−3880.62	−0.217884
$683$	35110.1	1.96699	0.983493	−	0.180946i	$-0.0579159\pi$
$683$	35110.1	1.96699	0.983493	+	0.180946i	$0.0579159\pi$
$684$	1606.99	0.0898318
$685$	6484.87	0.361714
$686$	−9169.68	−0.510350
$687$	21766.8	1.20881
$688$	−12516.6	−0.693590
$689$	−13173.6	−0.728409
$690$	−6748.25	−0.372321
$691$	−29368.8	−1.61685	−0.808423	−	0.588602i	$-0.799678\pi$
$691$	−29368.8	−1.61685	−0.808423	+	0.588602i	$0.799678\pi$
$692$	21491.9	1.18063
$693$	2422.66	0.132798
$694$	2173.29	0.118872
$695$	1722.81	0.0940288
$696$	−23071.8	−1.25652
$697$	2603.84	0.141503
$698$	−5754.79	−0.312066
$699$	26550.9	1.43669
$700$	−2514.54	−0.135773
$701$	−19197.6	−1.03436	−0.517178	−	0.855878i	$-0.673018\pi$
$701$	−19197.6	−1.03436	−0.517178	+	0.855878i	$0.673018\pi$
$702$	6342.05	0.340976
$703$	7135.33	0.382808
$704$	568.496	0.0304346
$705$	808.715	0.0432028
$706$	6399.67	0.341154
$707$	−15749.1	−0.837775
$708$	7281.18	0.386502
$709$	7987.42	0.423094	0.211547	−	0.977368i	$-0.432150\pi$
$709$	7987.42	0.423094	0.211547	+	0.977368i	$0.432150\pi$
$710$	−3509.19	−0.185490
$711$	−17587.9	−0.927702
$712$	−23307.7	−1.22682
$713$	−41858.0	−2.19859
$714$	−1124.90	−0.0589610
$715$	3059.45	0.160024
$716$	19761.3	1.03145
$717$	16922.6	0.881429
$718$	−5723.69	−0.297502
$719$	21182.4	1.09871	0.549354	−	0.835589i	$-0.314874\pi$
$719$	21182.4	1.09871	0.549354	+	0.835589i	$0.314874\pi$
$720$	−1656.60	−0.0857470
$721$	−11808.8	−0.609964
$722$	482.312	0.0248612
$723$	−1341.73	−0.0690171
$724$	−1867.31	−0.0958536
$725$	4765.90	0.244139
$726$	1030.19	0.0526637
$727$	22205.7	1.13283	0.566414	−	0.824121i	$-0.308330\pi$
$727$	22205.7	1.13283	0.566414	+	0.824121i	$0.308330\pi$
$728$	17097.2	0.870420
$729$	2281.72	0.115923
$730$	−4633.75	−0.234935
$731$	4197.22	0.212366
$732$	30591.2	1.54465
$733$	−11700.2	−0.589574	−0.294787	−	0.955563i	$-0.595249\pi$
$733$	−11700.2	−0.589574	−0.294787	+	0.955563i	$0.595249\pi$
$734$	4519.27	0.227261
$735$	2583.63	0.129658
$736$	29241.4	1.46447
$737$	−4294.17	−0.214624
$738$	−5798.99	−0.289246
$739$	−14887.0	−0.741037	−0.370519	−	0.928825i	$-0.620820\pi$
$739$	−14887.0	−0.741037	−0.370519	+	0.928825i	$0.620820\pi$
$740$	−11670.0	−0.579727
$741$	6735.10	0.333900
$742$	5120.62	0.253348
$743$	−4523.46	−0.223351	−0.111675	−	0.993745i	$-0.535622\pi$
$743$	−4523.46	−0.223351	−0.111675	+	0.993745i	$0.535622\pi$
$744$	31956.9	1.57473
$745$	−9406.34	−0.462579
$746$	−366.433	−0.0179840
$747$	8987.10	0.440188
$748$	558.131	0.0272825
$749$	1338.13	0.0652793
$750$	−1064.24	−0.0518142
$751$	1537.89	0.0747247	0.0373623	−	0.999302i	$-0.488104\pi$
$751$	1537.89	0.0747247	0.0373623	+	0.999302i	$0.488104\pi$
$752$	−617.935	−0.0299651
$753$	−13467.9	−0.651791
$754$	−14167.9	−0.684303
$755$	−11484.3	−0.553584
$756$	8583.17	0.412919
$757$	−30832.6	−1.48036	−0.740179	−	0.672410i	$-0.765259\pi$
$757$	−30832.6	−1.48036	−0.740179	+	0.672410i	$0.765259\pi$
$758$	−8333.67	−0.399330
$759$	11112.0	0.531411
$760$	−1804.22	−0.0861133
$761$	−24994.6	−1.19061	−0.595305	−	0.803500i	$-0.702969\pi$
$761$	−24994.6	−1.19061	−0.595305	+	0.803500i	$0.702969\pi$
$762$	3318.08	0.157745
$763$	3428.20	0.162659
$764$	−22457.6	−1.06346
$765$	555.512	0.0262544
$766$	−10437.3	−0.492317
$767$	10226.6	0.481436
$768$	−14610.8	−0.686489
$769$	−41790.5	−1.95969	−0.979847	−	0.199750i	$-0.935987\pi$
$769$	−41790.5	−1.95969	−0.979847	+	0.199750i	$0.935987\pi$
$770$	−1189.22	−0.0556578
$771$	−30061.3	−1.40419
$772$	−14835.8	−0.691648
$773$	2305.03	0.107253	0.0536263	−	0.998561i	$-0.482922\pi$
$773$	2305.03	0.107253	0.0536263	+	0.998561i	$0.482922\pi$
$774$	−9347.59	−0.434098
$775$	−6601.28	−0.305968
$776$	670.741	0.0310286
$777$	−38730.2	−1.78821
$778$	18392.5	0.847563
$779$	6059.89	0.278714
$780$	−11015.4	−0.505660
$781$	5778.42	0.264748
$782$	−1729.08	−0.0790686
$783$	−16268.0	−0.742490
$784$	−1974.14	−0.0899298
$785$	12395.1	0.563566
$786$	−10688.0	−0.485025
$787$	−10552.7	−0.477973	−0.238987	−	0.971023i	$-0.576815\pi$
$787$	−10552.7	−0.477973	−0.238987	+	0.971023i	$0.576815\pi$
$788$	19331.4	0.873925
$789$	45327.1	2.04523
$790$	8633.42	0.388814
$791$	1518.55	0.0682599
$792$	−2843.02	−0.127554
$793$	42966.1	1.92405
$794$	−1399.23	−0.0625400
$795$	−7545.77	−0.336630
$796$	768.949	0.0342396
$797$	−21595.2	−0.959776	−0.479888	−	0.877330i	$-0.659323\pi$
$797$	−21595.2	−0.959776	−0.479888	+	0.877330i	$0.659323\pi$
$798$	−2617.96	−0.116134
$799$	207.214	0.00917484
$800$	4611.55	0.203804
$801$	16701.4	0.736724
$802$	−2207.92	−0.0972124
$803$	7630.18	0.335322
$804$	15461.0	0.678193
$805$	−12827.4	−0.561624
$806$	19624.1	0.857603
$807$	23224.4	1.01306
$808$	18481.8	0.804689
$809$	14589.0	0.634021	0.317010	−	0.948422i	$-0.397321\pi$
$809$	14589.0	0.634021	0.317010	+	0.948422i	$0.397321\pi$
$810$	6087.27	0.264055
$811$	−14548.8	−0.629937	−0.314968	−	0.949102i	$-0.601994\pi$
$811$	−14548.8	−0.629937	−0.314968	+	0.949102i	$0.601994\pi$
$812$	−19174.5	−0.828685
$813$	46823.9	2.01991
$814$	−5519.17	−0.237650
$815$	−17796.9	−0.764904
$816$	−1266.60	−0.0543382
$817$	9768.14	0.418291
$818$	−7878.34	−0.336748
$819$	−12251.2	−0.522701
$820$	−9911.10	−0.422086
$821$	−15330.2	−0.651679	−0.325840	−	0.945425i	$-0.605647\pi$
$821$	−15330.2	−0.651679	−0.325840	+	0.945425i	$0.605647\pi$
$822$	−11042.4	−0.468548
$823$	34569.5	1.46418	0.732089	−	0.681209i	$-0.238545\pi$
$823$	34569.5	1.46418	0.732089	+	0.681209i	$0.238545\pi$
$824$	13857.8	0.585875
$825$	1752.44	0.0739541
$826$	−3975.12	−0.167448
$827$	−28353.0	−1.19218	−0.596088	−	0.802919i	$-0.703279\pi$
$827$	−28353.0	−1.19218	−0.596088	+	0.802919i	$0.703279\pi$
$828$	−13407.6	−0.562738
$829$	−12736.0	−0.533583	−0.266791	−	0.963754i	$-0.585963\pi$
$829$	−12736.0	−0.533583	−0.266791	+	0.963754i	$0.585963\pi$
$830$	−4411.53	−0.184490
$831$	−20477.1	−0.854803
$832$	−2874.85	−0.119793
$833$	661.994	0.0275351
$834$	−2933.59	−0.121801
$835$	15985.7	0.662524
$836$	1298.93	0.0537375
$837$	22532.9	0.930526
$838$	1210.82	0.0499129
$839$	21311.4	0.876937	0.438468	−	0.898747i	$-0.355521\pi$
$839$	21311.4	0.876937	0.438468	+	0.898747i	$0.355521\pi$
$840$	9793.23	0.402260
$841$	11953.0	0.490098
$842$	10886.0	0.445556
$843$	4613.80	0.188503
$844$	−23677.6	−0.965659
$845$	−4486.44	−0.182649
$846$	−461.484	−0.0187543
$847$	1958.23	0.0794399
$848$	5765.68	0.233484
$849$	45162.6	1.82565
$850$	−272.687	−0.0110036
$851$	−59532.1	−2.39804
$852$	−20805.0	−0.836580
$853$	34367.5	1.37951	0.689754	−	0.724044i	$-0.257719\pi$
$853$	34367.5	1.37951	0.689754	+	0.724044i	$0.257719\pi$
$854$	−16701.1	−0.669203
$855$	1292.84	0.0517124
$856$	−1570.31	−0.0627012
$857$	−20520.7	−0.817937	−0.408969	−	0.912548i	$-0.634111\pi$
$857$	−20520.7	−0.817937	−0.408969	+	0.912548i	$0.634111\pi$
$858$	−5209.59	−0.207287
$859$	−29464.4	−1.17033	−0.585165	−	0.810914i	$-0.698970\pi$
$859$	−29464.4	−1.17033	−0.585165	+	0.810914i	$0.698970\pi$
$860$	−15976.0	−0.633463
$861$	−32892.8	−1.30195
$862$	2319.88	0.0916652
$863$	30659.3	1.20933	0.604666	−	0.796479i	$-0.293307\pi$
$863$	30659.3	1.20933	0.604666	+	0.796479i	$0.293307\pi$
$864$	−15741.1	−0.619820
$865$	17290.4	0.679642
$866$	−5688.61	−0.223218
$867$	−30883.4	−1.20975
$868$	26558.7	1.03855
$869$	−14216.2	−0.554952
$870$	−8115.31	−0.316247
$871$	21715.4	0.844772
$872$	−4023.04	−0.156235
$873$	−480.627	−0.0186332
$874$	−4024.06	−0.155739
$875$	−2022.97	−0.0781586
$876$	−27472.2	−1.05959
$877$	24383.9	0.938865	0.469432	−	0.882968i	$-0.344459\pi$
$877$	24383.9	0.938865	0.469432	+	0.882968i	$0.344459\pi$
$878$	21451.5	0.824547
$879$	−16377.0	−0.628421
$880$	−1339.03	−0.0512939
$881$	15756.3	0.602547	0.301274	−	0.953538i	$-0.402588\pi$
$881$	15756.3	0.602547	0.301274	+	0.953538i	$0.402588\pi$
$882$	−1474.32	−0.0562845
$883$	7158.69	0.272830	0.136415	−	0.990652i	$-0.456442\pi$
$883$	7158.69	0.272830	0.136415	+	0.990652i	$0.456442\pi$
$884$	−2822.43	−0.107385
$885$	5857.76	0.222493
$886$	12650.1	0.479672
$887$	−3056.17	−0.115689	−0.0578445	−	0.998326i	$-0.518423\pi$
$887$	−3056.17	−0.115689	−0.0578445	+	0.998326i	$0.518423\pi$
$888$	45450.4	1.71759
$889$	6307.18	0.237948
$890$	−8198.30	−0.308773
$891$	−10023.6	−0.376884
$892$	11524.2	0.432578
$893$	482.246	0.0180714
$894$	16017.0	0.599205
$895$	15898.1	0.593760
$896$	−22764.8	−0.848793
$897$	−56192.9	−2.09167
$898$	−16336.6	−0.607082
$899$	−50337.6	−1.86747
$900$	−2114.47	−0.0783136
$901$	−1933.42	−0.0714891
$902$	−4687.32	−0.173027
$903$	−53020.9	−1.95396
$904$	−1782.05	−0.0655641
$905$	−1502.26	−0.0551789
$906$	19555.3	0.717089
$907$	37209.2	1.36220	0.681098	−	0.732192i	$-0.261503\pi$
$907$	37209.2	1.36220	0.681098	+	0.732192i	$0.261503\pi$
$908$	1947.77	0.0711885
$909$	−13243.4	−0.483229
$910$	6013.81	0.219072
$911$	41987.3	1.52701	0.763503	−	0.645804i	$-0.223478\pi$
$911$	41987.3	1.52701	0.763503	+	0.645804i	$0.223478\pi$
$912$	−2947.75	−0.107028
$913$	7264.26	0.263321
$914$	18846.0	0.682025
$915$	24610.8	0.889189
$916$	−21228.8	−0.765741
$917$	−20316.3	−0.731630
$918$	930.792	0.0334648
$919$	1597.88	0.0573551	0.0286776	−	0.999589i	$-0.490870\pi$
$919$	1597.88	0.0573551	0.0286776	+	0.999589i	$0.490870\pi$
$920$	15053.2	0.539444
$921$	−14266.8	−0.510430
$922$	−3081.26	−0.110061
$923$	−29221.1	−1.04206
$924$	−7050.53	−0.251023
$925$	−9388.59	−0.333724
$926$	19482.1	0.691385
$927$	−9930.00	−0.351827
$928$	35165.1	1.24391
$929$	−19471.3	−0.687655	−0.343828	−	0.939033i	$-0.611724\pi$
$929$	−19471.3	−0.687655	−0.343828	+	0.939033i	$0.611724\pi$
$930$	11240.6	0.396337
$931$	1540.65	0.0542350
$932$	−25894.6	−0.910092
$933$	−41238.9	−1.44705
$934$	10542.3	0.369332
$935$	449.020	0.0157054
$936$	14377.0	0.502058
$937$	−30179.7	−1.05222	−0.526109	−	0.850417i	$-0.676350\pi$
$937$	−30179.7	−1.05222	−0.526109	+	0.850417i	$0.676350\pi$
$938$	−8440.84	−0.293820
$939$	−12174.7	−0.423115
$940$	−788.725	−0.0273674
$941$	−39477.9	−1.36763	−0.683816	−	0.729654i	$-0.739681\pi$
$941$	−39477.9	−1.36763	−0.683816	+	0.729654i	$0.739681\pi$
$942$	−21106.2	−0.730019
$943$	−50559.4	−1.74596
$944$	−4475.88	−0.154319
$945$	6905.22	0.237700
$946$	−7555.65	−0.259678
$947$	51137.3	1.75474	0.877370	−	0.479814i	$-0.159296\pi$
$947$	51137.3	1.75474	0.877370	+	0.479814i	$0.159296\pi$
$948$	51185.0	1.75360
$949$	−38585.4	−1.31985
$950$	−634.621	−0.0216735
$951$	−6332.81	−0.215936
$952$	2509.28	0.0854266
$953$	4988.61	0.169566	0.0847832	−	0.996399i	$-0.472980\pi$
$953$	4988.61	0.169566	0.0847832	+	0.996399i	$0.472980\pi$
$954$	4305.91	0.146131
$955$	−18067.3	−0.612192
$956$	−16504.3	−0.558354
$957$	13363.1	0.451377
$958$	19519.4	0.658293
$959$	−20989.9	−0.706777
$960$	−1646.70	−0.0553615
$961$	39932.0	1.34040
$962$	27910.1	0.935403
$963$	1125.23	0.0376531
$964$	1308.56	0.0437199
$965$	−11935.5	−0.398153
$966$	21842.4	0.727502
$967$	−7236.60	−0.240655	−0.120328	−	0.992734i	$-0.538394\pi$
$967$	−7236.60	−0.240655	−0.120328	+	0.992734i	$0.538394\pi$
$968$	−2298.01	−0.0763026
$969$	988.478	0.0327704
$970$	235.927	0.00780945
$971$	24676.7	0.815564	0.407782	−	0.913079i	$-0.366302\pi$
$971$	24676.7	0.815564	0.407782	+	0.913079i	$0.366302\pi$
$972$	21770.0	0.718387
$973$	−5576.31	−0.183729
$974$	−14428.9	−0.474673
$975$	−8861.98	−0.291088
$976$	−18805.0	−0.616734
$977$	−18854.4	−0.617406	−0.308703	−	0.951159i	$-0.599895\pi$
$977$	−18854.4	−0.617406	−0.308703	+	0.951159i	$0.599895\pi$
$978$	30304.3	0.990822
$979$	13499.8	0.440709
$980$	−2519.77	−0.0821338
$981$	2882.76	0.0938219
$982$	−19419.2	−0.631050
$983$	−4656.66	−0.151093	−0.0755465	−	0.997142i	$-0.524070\pi$
$983$	−4656.66	−0.151093	−0.0755465	+	0.997142i	$0.524070\pi$
$984$	38600.1	1.25054
$985$	15552.3	0.503082
$986$	−2079.35	−0.0671603
$987$	−2617.61	−0.0844167
$988$	−6568.62	−0.211514
$989$	−81498.4	−2.62032
$990$	−1000.01	−0.0321034
$991$	−58115.5	−1.86287	−0.931433	−	0.363913i	$-0.881441\pi$
$991$	−58115.5	−1.86287	−0.931433	+	0.363913i	$0.881441\pi$
$992$	−48707.4	−1.55893
$993$	51565.3	1.64791
$994$	11358.4	0.362440
$995$	618.625	0.0197103
$996$	−26154.7	−0.832071
$997$	−24550.4	−0.779858	−0.389929	−	0.920845i	$-0.627500\pi$
$997$	−24550.4	−0.779858	−0.389929	+	0.920845i	$0.627500\pi$
$998$	15016.1	0.476280
$999$	32047.2	1.01494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.f.1.14	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.1.14	✓	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)

\(f(q)\)	\(=\)	\(q+1.33604 q^{2} +6.37251 q^{3} -6.21499 q^{4} -5.00000 q^{5} +8.51394 q^{6} +16.1837 q^{7} -18.9918 q^{8} +13.6088 q^{9} +O(q^{10})\)	\(q+1.33604 q^{2} +6.37251 q^{3} -6.21499 q^{4} -5.00000 q^{5} +8.51394 q^{6} +16.1837 q^{7} -18.9918 q^{8} +13.6088 q^{9} -6.68022 q^{10} +11.0000 q^{11} -39.6050 q^{12} -55.6263 q^{13} +21.6222 q^{14} -31.8625 q^{15} +24.3460 q^{16} -8.16400 q^{17} +18.1820 q^{18} -19.0000 q^{19} +31.0749 q^{20} +103.131 q^{21} +14.6965 q^{22} +158.522 q^{23} -121.026 q^{24} +25.0000 q^{25} -74.3192 q^{26} -85.3353 q^{27} -100.582 q^{28} +190.636 q^{29} -42.5697 q^{30} -264.051 q^{31} +184.462 q^{32} +70.0976 q^{33} -10.9075 q^{34} -80.9186 q^{35} -84.5787 q^{36} -375.544 q^{37} -25.3848 q^{38} -354.479 q^{39} +94.9592 q^{40} -318.942 q^{41} +137.787 q^{42} -514.113 q^{43} -68.3649 q^{44} -68.0441 q^{45} +211.793 q^{46} -25.3814 q^{47} +155.145 q^{48} -81.0869 q^{49} +33.4011 q^{50} -52.0251 q^{51} +345.717 q^{52} +236.823 q^{53} -114.012 q^{54} -55.0000 q^{55} -307.359 q^{56} -121.078 q^{57} +254.698 q^{58} -183.845 q^{59} +198.025 q^{60} -772.405 q^{61} -352.784 q^{62} +220.242 q^{63} +51.6814 q^{64} +278.132 q^{65} +93.6534 q^{66} -390.379 q^{67} +50.7392 q^{68} +1010.18 q^{69} -108.111 q^{70} +525.311 q^{71} -258.457 q^{72} +693.653 q^{73} -501.743 q^{74} +159.313 q^{75} +118.085 q^{76} +178.021 q^{77} -473.599 q^{78} -1292.39 q^{79} -121.730 q^{80} -911.238 q^{81} -426.120 q^{82} +660.387 q^{83} -640.957 q^{84} +40.8200 q^{85} -686.877 q^{86} +1214.83 q^{87} -208.910 q^{88} +1227.25 q^{89} -90.9099 q^{90} -900.241 q^{91} -985.215 q^{92} -1682.67 q^{93} -33.9106 q^{94} +95.0000 q^{95} +1175.49 q^{96} -35.3173 q^{97} -108.336 q^{98} +149.697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

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