LMFDB - Embedded newform 1045.4.a.i.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+1.75191 q^{2} +3.91466 q^{3} -4.93083 q^{4} -5.00000 q^{5} +6.85812 q^{6} -21.2936 q^{7} -22.6536 q^{8} -11.6754 q^{9} +O(q^{10})$	\(q+1.75191 q^{2} +3.91466 q^{3} -4.93083 q^{4} -5.00000 q^{5} +6.85812 q^{6} -21.2936 q^{7} -22.6536 q^{8} -11.6754 q^{9} -8.75953 q^{10} +11.0000 q^{11} -19.3025 q^{12} -4.48744 q^{13} -37.3044 q^{14} -19.5733 q^{15} -0.240318 q^{16} -10.0838 q^{17} -20.4542 q^{18} +19.0000 q^{19} +24.6541 q^{20} -83.3573 q^{21} +19.2710 q^{22} -38.4102 q^{23} -88.6812 q^{24} +25.0000 q^{25} -7.86156 q^{26} -151.401 q^{27} +104.995 q^{28} +293.991 q^{29} -34.2906 q^{30} +25.6408 q^{31} +180.808 q^{32} +43.0613 q^{33} -17.6659 q^{34} +106.468 q^{35} +57.5695 q^{36} +159.605 q^{37} +33.2862 q^{38} -17.5668 q^{39} +113.268 q^{40} -148.857 q^{41} -146.034 q^{42} -161.369 q^{43} -54.2391 q^{44} +58.3771 q^{45} -67.2910 q^{46} +186.576 q^{47} -0.940763 q^{48} +110.418 q^{49} +43.7976 q^{50} -39.4747 q^{51} +22.1268 q^{52} -509.472 q^{53} -265.241 q^{54} -55.0000 q^{55} +482.377 q^{56} +74.3786 q^{57} +515.044 q^{58} +616.923 q^{59} +96.5126 q^{60} +778.108 q^{61} +44.9203 q^{62} +248.612 q^{63} +318.680 q^{64} +22.4372 q^{65} +75.4393 q^{66} +872.934 q^{67} +49.7215 q^{68} -150.363 q^{69} +186.522 q^{70} +846.715 q^{71} +264.490 q^{72} -332.719 q^{73} +279.613 q^{74} +97.8666 q^{75} -93.6857 q^{76} -234.230 q^{77} -30.7754 q^{78} -23.4401 q^{79} +1.20159 q^{80} -277.448 q^{81} -260.783 q^{82} -1264.68 q^{83} +411.020 q^{84} +50.4190 q^{85} -282.704 q^{86} +1150.87 q^{87} -249.189 q^{88} -46.4275 q^{89} +102.271 q^{90} +95.5537 q^{91} +189.394 q^{92} +100.375 q^{93} +326.863 q^{94} -95.0000 q^{95} +707.801 q^{96} +625.523 q^{97} +193.441 q^{98} -128.430 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) $ 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9} - 10 q^{10} + 275 q^{11} + 44 q^{12} + 53 q^{13} - 51 q^{14} - 45 q^{15} + 438 q^{16} + 153 q^{17} + 9 q^{18} + 475 q^{19} - 610 q^{20} + 259 q^{21} + 22 q^{22} - 7 q^{23} + 186 q^{24} + 625 q^{25} + 543 q^{26} + 495 q^{27} - 525 q^{28} + 169 q^{29} - 55 q^{30} + 102 q^{31} + 327 q^{32} + 99 q^{33} - 879 q^{34} + 75 q^{35} + 2293 q^{36} - 46 q^{37} + 38 q^{38} + 233 q^{39} - 300 q^{40} + 1190 q^{41} - 684 q^{42} - 408 q^{43} + 1342 q^{44} - 1500 q^{45} + 757 q^{46} + 1068 q^{47} + 715 q^{48} + 1930 q^{49} + 50 q^{50} + 1655 q^{51} - 94 q^{52} + 143 q^{53} + 1970 q^{54} - 1375 q^{55} - 1397 q^{56} + 171 q^{57} + 1366 q^{58} + 2945 q^{59} - 220 q^{60} + 1160 q^{61} + 194 q^{62} + 1804 q^{63} + 3000 q^{64} - 265 q^{65} + 121 q^{66} - 353 q^{67} + 5452 q^{68} + 3289 q^{69} + 255 q^{70} + 230 q^{71} + 196 q^{72} + 1357 q^{73} + 4379 q^{74} + 225 q^{75} + 2318 q^{76} - 165 q^{77} + 2008 q^{78} + 1266 q^{79} - 2190 q^{80} + 1709 q^{81} + 1010 q^{82} + 3856 q^{83} + 9354 q^{84} - 765 q^{85} + 6746 q^{86} + 3113 q^{87} + 660 q^{88} + 3562 q^{89} - 45 q^{90} - 833 q^{91} + 4276 q^{92} + 1312 q^{93} + 5124 q^{94} - 2375 q^{95} + 3828 q^{96} - 914 q^{97} + 2478 q^{98} + 3300 q^{99}+O(q^{100}) \) 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9 - 10 * q^10 + 275 * q^11 + 44 * q^12 + 53 * q^13 - 51 * q^14 - 45 * q^15 + 438 * q^16 + 153 * q^17 + 9 * q^18 + 475 * q^19 - 610 * q^20 + 259 * q^21 + 22 * q^22 - 7 * q^23 + 186 * q^24 + 625 * q^25 + 543 * q^26 + 495 * q^27 - 525 * q^28 + 169 * q^29 - 55 * q^30 + 102 * q^31 + 327 * q^32 + 99 * q^33 - 879 * q^34 + 75 * q^35 + 2293 * q^36 - 46 * q^37 + 38 * q^38 + 233 * q^39 - 300 * q^40 + 1190 * q^41 - 684 * q^42 - 408 * q^43 + 1342 * q^44 - 1500 * q^45 + 757 * q^46 + 1068 * q^47 + 715 * q^48 + 1930 * q^49 + 50 * q^50 + 1655 * q^51 - 94 * q^52 + 143 * q^53 + 1970 * q^54 - 1375 * q^55 - 1397 * q^56 + 171 * q^57 + 1366 * q^58 + 2945 * q^59 - 220 * q^60 + 1160 * q^61 + 194 * q^62 + 1804 * q^63 + 3000 * q^64 - 265 * q^65 + 121 * q^66 - 353 * q^67 + 5452 * q^68 + 3289 * q^69 + 255 * q^70 + 230 * q^71 + 196 * q^72 + 1357 * q^73 + 4379 * q^74 + 225 * q^75 + 2318 * q^76 - 165 * q^77 + 2008 * q^78 + 1266 * q^79 - 2190 * q^80 + 1709 * q^81 + 1010 * q^82 + 3856 * q^83 + 9354 * q^84 - 765 * q^85 + 6746 * q^86 + 3113 * q^87 + 660 * q^88 + 3562 * q^89 - 45 * q^90 - 833 * q^91 + 4276 * q^92 + 1312 * q^93 + 5124 * q^94 - 2375 * q^95 + 3828 * q^96 - 914 * q^97 + 2478 * q^98 + 3300 * 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 $

$2$	1.75191	0.619392	0.309696	−	0.950836i	$-0.399773\pi$
$2$	1.75191	0.619392	0.309696	+	0.950836i	$0.399773\pi$
$3$	3.91466	0.753377	0.376689	−	0.926340i	$-0.377063\pi$
$3$	3.91466	0.753377	0.376689	+	0.926340i	$0.377063\pi$
$4$	−4.93083	−0.616353
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.85812	0.466636
$7$	−21.2936	−1.14975	−0.574873	−	0.818242i	$-0.694949\pi$
$7$	−21.2936	−1.14975	−0.574873	+	0.818242i	$0.694949\pi$
$8$	−22.6536	−1.00116
$9$	−11.6754	−0.432423
$10$	−8.75953	−0.277001
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−19.3025	−0.464347
$13$	−4.48744	−0.0957378	−0.0478689	−	0.998854i	$-0.515243\pi$
$13$	−4.48744	−0.0957378	−0.0478689	+	0.998854i	$0.515243\pi$
$14$	−37.3044	−0.712144
$15$	−19.5733	−0.336921
$16$	−0.240318	−0.00375497
$17$	−10.0838	−0.143864	−0.0719318	−	0.997410i	$-0.522916\pi$
$17$	−10.0838	−0.143864	−0.0719318	+	0.997410i	$0.522916\pi$
$18$	−20.4542	−0.267839
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	24.6541	0.275642
$21$	−83.3573	−0.866193
$22$	19.2710	0.186754
$23$	−38.4102	−0.348221	−0.174110	−	0.984726i	$-0.555705\pi$
$23$	−38.4102	−0.348221	−0.174110	+	0.984726i	$0.555705\pi$
$24$	−88.6812	−0.754249
$25$	25.0000	0.200000
$26$	−7.86156	−0.0592992
$27$	−151.401	−1.07915
$28$	104.995	0.708650
$29$	293.991	1.88251	0.941254	−	0.337700i	$-0.109649\pi$
$29$	293.991	1.88251	0.941254	+	0.337700i	$0.109649\pi$
$30$	−34.2906	−0.208686
$31$	25.6408	0.148556	0.0742779	−	0.997238i	$-0.476335\pi$
$31$	25.6408	0.148556	0.0742779	+	0.997238i	$0.476335\pi$
$32$	180.808	0.998831
$33$	43.0613	0.227152
$34$	−17.6659	−0.0891079
$35$	106.468	0.514182
$36$	57.5695	0.266525
$37$	159.605	0.709159	0.354580	−	0.935026i	$-0.384624\pi$
$37$	159.605	0.709159	0.354580	+	0.935026i	$0.384624\pi$
$38$	33.2862	0.142098
$39$	−17.5668	−0.0721267
$40$	113.268	0.447731
$41$	−148.857	−0.567012	−0.283506	−	0.958970i	$-0.591498\pi$
$41$	−148.857	−0.567012	−0.283506	+	0.958970i	$0.591498\pi$
$42$	−146.034	−0.536513
$43$	−161.369	−0.572293	−0.286147	−	0.958186i	$-0.592374\pi$
$43$	−161.369	−0.572293	−0.286147	+	0.958186i	$0.592374\pi$
$44$	−54.2391	−0.185838
$45$	58.3771	0.193385
$46$	−67.2910	−0.215685
$47$	186.576	0.579040	0.289520	−	0.957172i	$-0.406504\pi$
$47$	186.576	0.579040	0.289520	+	0.957172i	$0.406504\pi$
$48$	−0.940763	−0.00282891
$49$	110.418	0.321918
$50$	43.7976	0.123878
$51$	−39.4747	−0.108383
$52$	22.1268	0.0590083
$53$	−509.472	−1.32040	−0.660201	−	0.751089i	$-0.729529\pi$
$53$	−509.472	−1.32040	−0.660201	+	0.751089i	$0.729529\pi$
$54$	−265.241	−0.668420
$55$	−55.0000	−0.134840
$56$	482.377	1.15108
$57$	74.3786	0.172837
$58$	515.044	1.16601
$59$	616.923	1.36130	0.680648	−	0.732610i	$-0.261698\pi$
$59$	616.923	1.36130	0.680648	+	0.732610i	$0.261698\pi$
$60$	96.5126	0.207662
$61$	778.108	1.63322	0.816611	−	0.577189i	$-0.195850\pi$
$61$	778.108	1.63322	0.816611	+	0.577189i	$0.195850\pi$
$62$	44.9203	0.0920143
$63$	248.612	0.497177
$64$	318.680	0.622423
$65$	22.4372	0.0428152
$66$	75.4393	0.140696
$67$	872.934	1.59173	0.795865	−	0.605474i	$-0.207016\pi$
$67$	872.934	1.59173	0.795865	+	0.605474i	$0.207016\pi$
$68$	49.7215	0.0886708
$69$	−150.363	−0.262341
$70$	186.522	0.318480
$71$	846.715	1.41530	0.707652	−	0.706561i	$-0.249754\pi$
$71$	846.715	1.41530	0.707652	+	0.706561i	$0.249754\pi$
$72$	264.490	0.432923
$73$	−332.719	−0.533450	−0.266725	−	0.963773i	$-0.585942\pi$
$73$	−332.719	−0.533450	−0.266725	+	0.963773i	$0.585942\pi$
$74$	279.613	0.439248
$75$	97.8666	0.150675
$76$	−93.6857	−0.141401
$77$	−234.230	−0.346662
$78$	−30.7754	−0.0446747
$79$	−23.4401	−0.0333825	−0.0166912	−	0.999861i	$-0.505313\pi$
$79$	−23.4401	−0.0333825	−0.0166912	+	0.999861i	$0.505313\pi$
$80$	1.20159	0.00167927
$81$	−277.448	−0.380588
$82$	−260.783	−0.351203
$83$	−1264.68	−1.67249	−0.836243	−	0.548358i	$-0.815253\pi$
$83$	−1264.68	−1.67249	−0.836243	+	0.548358i	$0.815253\pi$
$84$	411.020	0.533881
$85$	50.4190	0.0643377
$86$	−282.704	−0.354474
$87$	1150.87	1.41824
$88$	−249.189	−0.301860
$89$	−46.4275	−0.0552956	−0.0276478	−	0.999618i	$-0.508802\pi$
$89$	−46.4275	−0.0552956	−0.0276478	+	0.999618i	$0.508802\pi$
$90$	102.271	0.119781
$91$	95.5537	0.110074
$92$	189.394	0.214627
$93$	100.375	0.111919
$94$	326.863	0.358653
$95$	−95.0000	−0.102598
$96$	707.801	0.752496
$97$	625.523	0.654765	0.327383	−	0.944892i	$-0.393833\pi$
$97$	625.523	0.654765	0.327383	+	0.944892i	$0.393833\pi$
$98$	193.441	0.199393
$99$	−128.430	−0.130380
$100$	−123.271	−0.123271
$101$	1485.18	1.46317	0.731587	−	0.681748i	$-0.238780\pi$
$101$	1485.18	1.46317	0.731587	+	0.681748i	$0.238780\pi$
$102$	−69.1559	−0.0671319
$103$	−975.253	−0.932956	−0.466478	−	0.884533i	$-0.654477\pi$
$103$	−975.253	−0.932956	−0.466478	+	0.884533i	$0.654477\pi$
$104$	101.657	0.0958485
$105$	416.786	0.387373
$106$	−892.546	−0.817846
$107$	−55.1067	−0.0497884	−0.0248942	−	0.999690i	$-0.507925\pi$
$107$	−55.1067	−0.0497884	−0.0248942	+	0.999690i	$0.507925\pi$
$108$	746.533	0.665141
$109$	−1305.69	−1.14736	−0.573679	−	0.819080i	$-0.694484\pi$
$109$	−1305.69	−1.14736	−0.573679	+	0.819080i	$0.694484\pi$
$110$	−96.3548	−0.0835188
$111$	624.799	0.534264
$112$	5.11723	0.00431726
$113$	918.100	0.764315	0.382158	−	0.924097i	$-0.375181\pi$
$113$	918.100	0.764315	0.382158	+	0.924097i	$0.375181\pi$
$114$	130.304	0.107054
$115$	192.051	0.155729
$116$	−1449.62	−1.16029
$117$	52.3927	0.0413992
$118$	1080.79	0.843176
$119$	214.720	0.165407
$120$	443.406	0.337310
$121$	121.000	0.0909091
$122$	1363.17	1.01160
$123$	−582.723	−0.427174
$124$	−126.431	−0.0915629
$125$	−125.000	−0.0894427
$126$	435.544	0.307947
$127$	74.4766	0.0520372	0.0260186	−	0.999661i	$-0.491717\pi$
$127$	74.4766	0.0520372	0.0260186	+	0.999661i	$0.491717\pi$
$128$	−888.163	−0.613307
$129$	−631.707	−0.431153
$130$	39.3078	0.0265194
$131$	687.299	0.458394	0.229197	−	0.973380i	$-0.426390\pi$
$131$	687.299	0.458394	0.229197	+	0.973380i	$0.426390\pi$
$132$	−212.328	−0.140006
$133$	−404.579	−0.263770
$134$	1529.30	0.985905
$135$	757.006	0.482613
$136$	228.434	0.144030
$137$	−1163.23	−0.725409	−0.362705	−	0.931904i	$-0.618147\pi$
$137$	−1163.23	−0.725409	−0.362705	+	0.931904i	$0.618147\pi$
$138$	−263.421	−0.162492
$139$	2770.25	1.69043	0.845215	−	0.534426i	$-0.179472\pi$
$139$	2770.25	1.69043	0.845215	+	0.534426i	$0.179472\pi$
$140$	−524.976	−0.316918
$141$	730.382	0.436236
$142$	1483.36	0.876628
$143$	−49.3618	−0.0288660
$144$	2.80581	0.00162373
$145$	−1469.95	−0.841883
$146$	−582.893	−0.330415
$147$	432.248	0.242525
$148$	−786.984	−0.437093
$149$	219.428	0.120646	0.0603231	−	0.998179i	$-0.480787\pi$
$149$	219.428	0.120646	0.0603231	+	0.998179i	$0.480787\pi$
$150$	171.453	0.0933272
$151$	533.382	0.287457	0.143729	−	0.989617i	$-0.454091\pi$
$151$	533.382	0.287457	0.143729	+	0.989617i	$0.454091\pi$
$152$	−430.418	−0.229681
$153$	117.733	0.0622099
$154$	−410.348	−0.214720
$155$	−128.204	−0.0664362
$156$	86.6189	0.0444555
$157$	−2147.61	−1.09170	−0.545852	−	0.837882i	$-0.683794\pi$
$157$	−2147.61	−1.09170	−0.545852	+	0.837882i	$0.683794\pi$
$158$	−41.0648	−0.0206768
$159$	−1994.41	−0.994761
$160$	−904.038	−0.446691
$161$	817.891	0.400365
$162$	−486.063	−0.235733
$163$	1072.27	0.515256	0.257628	−	0.966244i	$-0.417059\pi$
$163$	1072.27	0.515256	0.257628	+	0.966244i	$0.417059\pi$
$164$	733.986	0.349480
$165$	−215.306	−0.101585
$166$	−2215.60	−1.03593
$167$	1984.53	0.919566	0.459783	−	0.888031i	$-0.347927\pi$
$167$	1984.53	0.919566	0.459783	+	0.888031i	$0.347927\pi$
$168$	1888.34	0.867195
$169$	−2176.86	−0.990834
$170$	88.3293	0.0398503
$171$	−221.833	−0.0992046
$172$	795.685	0.352735
$173$	−2561.66	−1.12578	−0.562889	−	0.826533i	$-0.690310\pi$
$173$	−2561.66	−1.12578	−0.562889	+	0.826533i	$0.690310\pi$
$174$	2016.22	0.878446
$175$	−532.340	−0.229949
$176$	−2.64350	−0.00113216
$177$	2415.04	1.02557
$178$	−81.3366	−0.0342496
$179$	−440.974	−0.184134	−0.0920668	−	0.995753i	$-0.529347\pi$
$179$	−440.974	−0.184134	−0.0920668	+	0.995753i	$0.529347\pi$
$180$	−287.847	−0.119194
$181$	139.603	0.0573293	0.0286646	−	0.999589i	$-0.490875\pi$
$181$	139.603	0.0573293	0.0286646	+	0.999589i	$0.490875\pi$
$182$	167.401	0.0681791
$183$	3046.03	1.23043
$184$	870.128	0.348623
$185$	−798.025	−0.317146
$186$	175.848	0.0693215
$187$	−110.922	−0.0433765
$188$	−919.974	−0.356893
$189$	3223.88	1.24075
$190$	−166.431	−0.0635483
$191$	−4181.82	−1.58422	−0.792109	−	0.610379i	$-0.791017\pi$
$191$	−4181.82	−1.58422	−0.792109	+	0.610379i	$0.791017\pi$
$192$	1247.53	0.468919
$193$	205.592	0.0766779	0.0383390	−	0.999265i	$-0.487793\pi$
$193$	205.592	0.0766779	0.0383390	+	0.999265i	$0.487793\pi$
$194$	1095.86	0.405557
$195$	87.8340	0.0322560
$196$	−544.451	−0.198415
$197$	2886.19	1.04382	0.521910	−	0.853001i	$-0.325220\pi$
$197$	2886.19	1.04382	0.521910	+	0.853001i	$0.325220\pi$
$198$	−224.996	−0.0807566
$199$	−1075.67	−0.383176	−0.191588	−	0.981475i	$-0.561364\pi$
$199$	−1075.67	−0.383176	−0.191588	+	0.981475i	$0.561364\pi$
$200$	−566.340	−0.200231
$201$	3417.24	1.19917
$202$	2601.89	0.906278
$203$	−6260.12	−2.16441
$204$	194.643	0.0668025
$205$	744.283	0.253576
$206$	−1708.55	−0.577866
$207$	448.455	0.150578
$208$	1.07841	0.000359492 0
$209$	209.000	0.0691714
$210$	730.170	0.239936
$211$	1399.41	0.456586	0.228293	−	0.973592i	$-0.426686\pi$
$211$	1399.41	0.456586	0.228293	+	0.973592i	$0.426686\pi$
$212$	2512.12	0.813834
$213$	3314.60	1.06626
$214$	−96.5416	−0.0308386
$215$	806.847	0.255937
$216$	3429.78	1.08040
$217$	−545.986	−0.170802
$218$	−2287.44	−0.710665
$219$	−1302.48	−0.401889
$220$	271.196	0.0831091
$221$	45.2504	0.0137732
$222$	1094.59	0.330919
$223$	4847.50	1.45566	0.727831	−	0.685757i	$-0.240529\pi$
$223$	4847.50	1.45566	0.727831	+	0.685757i	$0.240529\pi$
$224$	−3850.05	−1.14840
$225$	−291.885	−0.0864846
$226$	1608.42	0.473411
$227$	−5912.61	−1.72878	−0.864392	−	0.502819i	$-0.832296\pi$
$227$	−5912.61	−1.72878	−0.864392	+	0.502819i	$0.832296\pi$
$228$	−366.748	−0.106528
$229$	4038.09	1.16526	0.582629	−	0.812738i	$-0.302024\pi$
$229$	4038.09	1.16526	0.582629	+	0.812738i	$0.302024\pi$
$230$	336.455	0.0964573
$231$	−916.930	−0.261167
$232$	−6659.95	−1.88468
$233$	3852.08	1.08308	0.541541	−	0.840674i	$-0.317841\pi$
$233$	3852.08	1.08308	0.541541	+	0.840674i	$0.317841\pi$
$234$	91.7870	0.0256423
$235$	−932.879	−0.258955
$236$	−3041.94	−0.839040
$237$	−91.7600	−0.0251496
$238$	376.170	0.102452
$239$	−3690.81	−0.998906	−0.499453	−	0.866341i	$-0.666466\pi$
$239$	−3690.81	−0.998906	−0.499453	+	0.866341i	$0.666466\pi$
$240$	4.70382	0.00126513
$241$	3858.00	1.03119	0.515593	−	0.856834i	$-0.327572\pi$
$241$	3858.00	1.03119	0.515593	+	0.856834i	$0.327572\pi$
$242$	211.981	0.0563084
$243$	3001.72	0.792429
$244$	−3836.72	−1.00664
$245$	−552.089	−0.143966
$246$	−1020.88	−0.264588
$247$	−85.2613	−0.0219638
$248$	−580.857	−0.148728
$249$	−4950.79	−1.26001
$250$	−218.988	−0.0554001
$251$	−1120.16	−0.281689	−0.140844	−	0.990032i	$-0.544982\pi$
$251$	−1120.16	−0.281689	−0.140844	+	0.990032i	$0.544982\pi$
$252$	−1225.86	−0.306437
$253$	−422.512	−0.104992
$254$	130.476	0.0322314
$255$	197.373	0.0484706
$256$	−4105.42	−1.00230
$257$	6266.52	1.52099	0.760495	−	0.649344i	$-0.224956\pi$
$257$	6266.52	1.52099	0.760495	+	0.649344i	$0.224956\pi$
$258$	−1106.69	−0.267053
$259$	−3398.56	−0.815353
$260$	−110.634	−0.0263893
$261$	−3432.46	−0.814039
$262$	1204.08	0.283926
$263$	2997.84	0.702870	0.351435	−	0.936212i	$-0.385694\pi$
$263$	2997.84	0.702870	0.351435	+	0.936212i	$0.385694\pi$
$264$	−975.493	−0.227414
$265$	2547.36	0.590502
$266$	−708.783	−0.163377
$267$	−181.748	−0.0416584
$268$	−4304.29	−0.981068
$269$	6760.35	1.53229	0.766145	−	0.642668i	$-0.222172\pi$
$269$	6760.35	1.53229	0.766145	+	0.642668i	$0.222172\pi$
$270$	1326.20	0.298926
$271$	−3177.08	−0.712154	−0.356077	−	0.934457i	$-0.615886\pi$
$271$	−3177.08	−0.712154	−0.356077	+	0.934457i	$0.615886\pi$
$272$	2.42332	0.000540203 0
$273$	374.061	0.0829274
$274$	−2037.86	−0.449313
$275$	275.000	0.0603023
$276$	741.413	0.161695
$277$	8739.71	1.89573	0.947867	−	0.318665i	$-0.103235\pi$
$277$	8739.71	1.89573	0.947867	+	0.318665i	$0.103235\pi$
$278$	4853.22	1.04704
$279$	−299.367	−0.0642389
$280$	−2411.88	−0.514777
$281$	−3894.23	−0.826726	−0.413363	−	0.910566i	$-0.635646\pi$
$281$	−3894.23	−0.826726	−0.413363	+	0.910566i	$0.635646\pi$
$282$	1279.56	0.270201
$283$	−2978.12	−0.625550	−0.312775	−	0.949827i	$-0.601259\pi$
$283$	−2978.12	−0.625550	−0.312775	+	0.949827i	$0.601259\pi$
$284$	−4175.01	−0.872328
$285$	−371.893	−0.0772949
$286$	−86.4772	−0.0178794
$287$	3169.69	0.651920
$288$	−2111.00	−0.431917
$289$	−4811.32	−0.979303
$290$	−2575.22	−0.521456
$291$	2448.71	0.493285
$292$	1640.58	0.328794
$293$	−7863.26	−1.56784	−0.783919	−	0.620864i	$-0.786782\pi$
$293$	−7863.26	−1.56784	−0.783919	+	0.620864i	$0.786782\pi$
$294$	757.258	0.150218
$295$	−3084.61	−0.608790
$296$	−3615.62	−0.709979
$297$	−1665.41	−0.325377
$298$	384.418	0.0747273
$299$	172.363	0.0333379
$300$	−482.563	−0.0928693
$301$	3436.14	0.657992
$302$	934.436	0.178049
$303$	5813.96	1.10232
$304$	−4.56604	−0.000861448 0
$305$	−3890.54	−0.730399
$306$	206.256	0.0385323
$307$	3338.33	0.620615	0.310307	−	0.950636i	$-0.399568\pi$
$307$	3338.33	0.620615	0.310307	+	0.950636i	$0.399568\pi$
$308$	1154.95	0.213666
$309$	−3817.79	−0.702868
$310$	−224.602	−0.0411501
$311$	5778.76	1.05365	0.526823	−	0.849975i	$-0.323383\pi$
$311$	5778.76	1.05365	0.526823	+	0.849975i	$0.323383\pi$
$312$	397.951	0.0722101
$313$	7039.78	1.27128	0.635642	−	0.771984i	$-0.280735\pi$
$313$	7039.78	1.27128	0.635642	+	0.771984i	$0.280735\pi$
$314$	−3762.40	−0.676193
$315$	−1243.06	−0.222344
$316$	115.579	0.0205754
$317$	−6380.05	−1.13041	−0.565204	−	0.824951i	$-0.691203\pi$
$317$	−6380.05	−1.13041	−0.565204	+	0.824951i	$0.691203\pi$
$318$	−3494.02	−0.616147
$319$	3233.90	0.567597
$320$	−1593.40	−0.278356
$321$	−215.724	−0.0375095
$322$	1432.87	0.247983
$323$	−191.592	−0.0330046
$324$	1368.05	0.234577
$325$	−112.186	−0.0191476
$326$	1878.52	0.319146
$327$	−5111.32	−0.864394
$328$	3372.14	0.567668
$329$	−3972.87	−0.665750
$330$	−377.197	−0.0629212
$331$	5305.77	0.881061	0.440531	−	0.897738i	$-0.354790\pi$
$331$	5305.77	0.881061	0.440531	+	0.897738i	$0.354790\pi$
$332$	6235.91	1.03084
$333$	−1863.45	−0.306657
$334$	3476.71	0.569572
$335$	−4364.67	−0.711843
$336$	20.0322	0.00325253
$337$	10112.2	1.63457	0.817283	−	0.576237i	$-0.195479\pi$
$337$	10112.2	1.63457	0.817283	+	0.576237i	$0.195479\pi$
$338$	−3813.66	−0.613715
$339$	3594.05	0.575818
$340$	−248.607	−0.0396548
$341$	282.049	0.0447913
$342$	−388.630	−0.0614465
$343$	4952.52	0.779623
$344$	3655.60	0.572955
$345$	751.814	0.117323
$346$	−4487.79	−0.697298
$347$	9411.88	1.45607	0.728035	−	0.685540i	$-0.240434\pi$
$347$	9411.88	1.45607	0.728035	+	0.685540i	$0.240434\pi$
$348$	−5674.77	−0.874136
$349$	3357.19	0.514917	0.257458	−	0.966289i	$-0.417115\pi$
$349$	3357.19	0.514917	0.257458	+	0.966289i	$0.417115\pi$
$350$	−932.610	−0.142429
$351$	679.403	0.103316
$352$	1988.88	0.301159
$353$	4601.65	0.693828	0.346914	−	0.937897i	$-0.387230\pi$
$353$	4601.65	0.693828	0.346914	+	0.937897i	$0.387230\pi$
$354$	4230.93	0.635230
$355$	−4233.58	−0.632943
$356$	228.926	0.0340816
$357$	840.558	0.124614
$358$	−772.544	−0.114051
$359$	5375.17	0.790225	0.395112	−	0.918633i	$-0.370706\pi$
$359$	5375.17	0.790225	0.395112	+	0.918633i	$0.370706\pi$
$360$	−1322.45	−0.193609
$361$	361.000	0.0526316
$362$	244.571	0.0355093
$363$	473.674	0.0684888
$364$	−471.159	−0.0678446
$365$	1663.60	0.238566
$366$	5336.36	0.762120
$367$	−8161.70	−1.16086	−0.580432	−	0.814309i	$-0.697116\pi$
$367$	−8161.70	−1.16086	−0.580432	+	0.814309i	$0.697116\pi$
$368$	9.23065	0.00130756
$369$	1737.96	0.245189
$370$	−1398.06	−0.196437
$371$	10848.5	1.51813
$372$	−494.933	−0.0689814
$373$	−5877.03	−0.815821	−0.407910	−	0.913022i	$-0.633742\pi$
$373$	−5877.03	−0.815821	−0.407910	+	0.913022i	$0.633742\pi$
$374$	−194.324	−0.0268671
$375$	−489.333	−0.0673841
$376$	−4226.61	−0.579710
$377$	−1319.27	−0.180227
$378$	5647.93	0.768514
$379$	−9421.25	−1.27688	−0.638440	−	0.769672i	$-0.720420\pi$
$379$	−9421.25	−1.27688	−0.638440	+	0.769672i	$0.720420\pi$
$380$	468.429	0.0632365
$381$	291.551	0.0392037
$382$	−7326.15	−0.981253
$383$	8988.94	1.19925	0.599626	−	0.800280i	$-0.295316\pi$
$383$	8988.94	1.19925	0.599626	+	0.800280i	$0.295316\pi$
$384$	−3476.86	−0.462051
$385$	1171.15	0.155032
$386$	360.178	0.0474937
$387$	1884.06	0.247473
$388$	−3084.35	−0.403567
$389$	−13057.5	−1.70191	−0.850953	−	0.525241i	$-0.823975\pi$
$389$	−13057.5	−1.70191	−0.850953	+	0.525241i	$0.823975\pi$
$390$	153.877	0.0199791
$391$	387.320	0.0500962
$392$	−2501.36	−0.322290
$393$	2690.54	0.345344
$394$	5056.33	0.646534
$395$	117.200	0.0149291
$396$	633.264	0.0803604
$397$	2149.58	0.271749	0.135874	−	0.990726i	$-0.456616\pi$
$397$	2149.58	0.271749	0.135874	+	0.990726i	$0.456616\pi$
$398$	−1884.47	−0.237336
$399$	−1583.79	−0.198718
$400$	−6.00795	−0.000750993 0
$401$	−15187.0	−1.89127	−0.945636	−	0.325226i	$-0.894560\pi$
$401$	−15187.0	−1.89127	−0.945636	+	0.325226i	$0.894560\pi$
$402$	5986.69	0.742758
$403$	−115.062	−0.0142224
$404$	−7323.15	−0.901832
$405$	1387.24	0.170204
$406$	−10967.1	−1.34062
$407$	1755.65	0.213820
$408$	894.243	0.108509
$409$	5010.43	0.605746	0.302873	−	0.953031i	$-0.402054\pi$
$409$	5010.43	0.605746	0.302873	+	0.953031i	$0.402054\pi$
$410$	1303.91	0.157063
$411$	−4553.64	−0.546507
$412$	4808.80	0.575031
$413$	−13136.5	−1.56515
$414$	785.650	0.0932671
$415$	6323.39	0.747959
$416$	−811.363	−0.0956258
$417$	10844.6	1.27353
$418$	366.148	0.0428442
$419$	7612.89	0.887623	0.443811	−	0.896120i	$-0.353626\pi$
$419$	7612.89	0.887623	0.443811	+	0.896120i	$0.353626\pi$
$420$	−2055.10	−0.238759
$421$	13906.5	1.60988	0.804940	−	0.593357i	$-0.202198\pi$
$421$	13906.5	1.60988	0.804940	+	0.593357i	$0.202198\pi$
$422$	2451.64	0.282806
$423$	−2178.35	−0.250390
$424$	11541.4	1.32193
$425$	−252.095	−0.0287727
$426$	5806.87	0.660432
$427$	−16568.7	−1.87779
$428$	271.721	0.0306873
$429$	−193.235	−0.0217470
$430$	1413.52	0.158526
$431$	7975.92	0.891384	0.445692	−	0.895186i	$-0.352958\pi$
$431$	7975.92	0.891384	0.445692	+	0.895186i	$0.352958\pi$
$432$	36.3844	0.00405219
$433$	−6356.61	−0.705495	−0.352747	−	0.935719i	$-0.614753\pi$
$433$	−6356.61	−0.705495	−0.352747	+	0.935719i	$0.614753\pi$
$434$	−956.516	−0.105793
$435$	−5754.37	−0.634255
$436$	6438.11	0.707178
$437$	−729.793	−0.0798873
$438$	−2281.83	−0.248927
$439$	−9109.59	−0.990380	−0.495190	−	0.868785i	$-0.664902\pi$
$439$	−9109.59	−0.990380	−0.495190	+	0.868785i	$0.664902\pi$
$440$	1245.95	0.134996
$441$	−1289.17	−0.139204
$442$	79.2744	0.00853099
$443$	14848.4	1.59248	0.796241	−	0.604979i	$-0.206819\pi$
$443$	14848.4	1.59248	0.796241	+	0.604979i	$0.206819\pi$
$444$	−3080.78	−0.329296
$445$	232.138	0.0247289
$446$	8492.36	0.901625
$447$	858.988	0.0908921
$448$	−6785.86	−0.715629
$449$	18071.6	1.89945	0.949725	−	0.313087i	$-0.101363\pi$
$449$	18071.6	1.89945	0.949725	+	0.313087i	$0.101363\pi$
$450$	−511.356	−0.0535678
$451$	−1637.42	−0.170961
$452$	−4526.99	−0.471088
$453$	2088.01	0.216564
$454$	−10358.3	−1.07079
$455$	−477.769	−0.0492267
$456$	−1684.94	−0.173036
$457$	−2507.85	−0.256701	−0.128351	−	0.991729i	$-0.540968\pi$
$457$	−2507.85	−0.256701	−0.128351	+	0.991729i	$0.540968\pi$
$458$	7074.35	0.721752
$459$	1526.70	0.155251
$460$	−946.969	−0.0959841
$461$	−16886.6	−1.70604	−0.853022	−	0.521875i	$-0.825233\pi$
$461$	−16886.6	−1.70604	−0.853022	+	0.521875i	$0.825233\pi$
$462$	−1606.37	−0.161765
$463$	−10623.1	−1.06630	−0.533149	−	0.846021i	$-0.678991\pi$
$463$	−10623.1	−1.06630	−0.533149	+	0.846021i	$0.678991\pi$
$464$	−70.6512	−0.00706875
$465$	−501.876	−0.0500515
$466$	6748.48	0.670852
$467$	−8704.29	−0.862497	−0.431249	−	0.902233i	$-0.641927\pi$
$467$	−8704.29	−0.862497	−0.431249	+	0.902233i	$0.641927\pi$
$468$	−258.339	−0.0255165
$469$	−18587.9	−1.83009
$470$	−1634.32	−0.160394
$471$	−8407.15	−0.822465
$472$	−13975.5	−1.36287
$473$	−1775.06	−0.172553
$474$	−160.755	−0.0155775
$475$	475.000	0.0458831
$476$	−1058.75	−0.101949
$477$	5948.29	0.570972
$478$	−6465.95	−0.618714
$479$	−13510.4	−1.28874	−0.644368	−	0.764715i	$-0.722880\pi$
$479$	−13510.4	−1.28874	−0.644368	+	0.764715i	$0.722880\pi$
$480$	−3539.01	−0.336527
$481$	−716.217	−0.0678933
$482$	6758.86	0.638709
$483$	3201.77	0.301626
$484$	−596.630	−0.0560321
$485$	−3127.62	−0.292820
$486$	5258.72	0.490824
$487$	15476.5	1.44006	0.720030	−	0.693943i	$-0.244128\pi$
$487$	15476.5	1.44006	0.720030	+	0.693943i	$0.244128\pi$
$488$	−17626.9	−1.63511
$489$	4197.58	0.388183
$490$	−967.207	−0.0891713
$491$	−7283.44	−0.669444	−0.334722	−	0.942317i	$-0.608642\pi$
$491$	−7283.44	−0.669444	−0.334722	+	0.942317i	$0.608642\pi$
$492$	2873.31	0.263290
$493$	−2964.54	−0.270824
$494$	−149.370	−0.0136042
$495$	642.148	0.0583079
$496$	−6.16195	−0.000557822 0
$497$	−18029.6	−1.62724
$498$	−8673.31	−0.780442
$499$	−15045.0	−1.34971	−0.674856	−	0.737949i	$-0.735794\pi$
$499$	−15045.0	−1.34971	−0.674856	+	0.737949i	$0.735794\pi$
$500$	616.353	0.0551283
$501$	7768.76	0.692780
$502$	−1962.41	−0.174476
$503$	−12653.2	−1.12162	−0.560812	−	0.827943i	$-0.689511\pi$
$503$	−12653.2	−1.12162	−0.560812	+	0.827943i	$0.689511\pi$
$504$	−5631.95	−0.497752
$505$	−7425.88	−0.654351
$506$	−740.201	−0.0650315
$507$	−8521.68	−0.746472
$508$	−367.231	−0.0320733
$509$	−1838.83	−0.160127	−0.0800634	−	0.996790i	$-0.525512\pi$
$509$	−1838.83	−0.160127	−0.0800634	+	0.996790i	$0.525512\pi$
$510$	345.779	0.0300223
$511$	7084.80	0.613333
$512$	−87.0038	−0.00750988
$513$	−2876.62	−0.247575
$514$	10978.3	0.942089
$515$	4876.26	0.417231
$516$	3114.84	0.265743
$517$	2052.33	0.174587
$518$	−5953.96	−0.505023
$519$	−10028.0	−0.848135
$520$	−508.283	−0.0428647
$521$	14256.2	1.19880	0.599402	−	0.800448i	$-0.295405\pi$
$521$	14256.2	1.19880	0.599402	+	0.800448i	$0.295405\pi$
$522$	−6013.35	−0.504209
$523$	21102.7	1.76435	0.882175	−	0.470921i	$-0.156078\pi$
$523$	21102.7	1.76435	0.882175	+	0.470921i	$0.156078\pi$
$524$	−3388.95	−0.282533
$525$	−2083.93	−0.173239
$526$	5251.94	0.435352
$527$	−258.557	−0.0213718
$528$	−10.3484	−0.000852947 0
$529$	−10691.7	−0.878742
$530$	4462.73	0.365752
$531$	−7202.83	−0.588656
$532$	1994.91	0.162576
$533$	667.985	0.0542845
$534$	−318.405	−0.0258029
$535$	275.533	0.0222661
$536$	−19775.1	−1.59357
$537$	−1726.26	−0.138722
$538$	11843.5	0.949088
$539$	1214.59	0.0970618
$540$	−3732.67	−0.297460
$541$	6279.61	0.499042	0.249521	−	0.968369i	$-0.419727\pi$
$541$	6279.61	0.499042	0.249521	+	0.968369i	$0.419727\pi$
$542$	−5565.94	−0.441103
$543$	546.498	0.0431906
$544$	−1823.23	−0.143695
$545$	6528.43	0.513114
$546$	655.319	0.0513646
$547$	8446.36	0.660220	0.330110	−	0.943943i	$-0.392914\pi$
$547$	8446.36	0.660220	0.330110	+	0.943943i	$0.392914\pi$
$548$	5735.67	0.447109
$549$	−9084.73	−0.706242
$550$	481.774	0.0373507
$551$	5585.82	0.431877
$552$	3406.26	0.262645
$553$	499.124	0.0383814
$554$	15311.2	1.17420
$555$	−3124.00	−0.238930
$556$	−13659.6	−1.04190
$557$	−11288.1	−0.858692	−0.429346	−	0.903140i	$-0.641256\pi$
$557$	−11288.1	−0.858692	−0.429346	+	0.903140i	$0.641256\pi$
$558$	−524.463	−0.0397891
$559$	724.135	0.0547901
$560$	−25.5862	−0.00193074
$561$	−434.221	−0.0326789
$562$	−6822.32	−0.512068
$563$	3274.86	0.245149	0.122575	−	0.992459i	$-0.460885\pi$
$563$	3274.86	0.245149	0.122575	+	0.992459i	$0.460885\pi$
$564$	−3601.39	−0.268875
$565$	−4590.50	−0.341812
$566$	−5217.38	−0.387461
$567$	5907.88	0.437580
$568$	−19181.1	−1.41694
$569$	15974.5	1.17695	0.588476	−	0.808515i	$-0.299728\pi$
$569$	15974.5	1.17695	0.588476	+	0.808515i	$0.299728\pi$
$570$	−651.521	−0.0478758
$571$	17077.2	1.25159	0.625796	−	0.779987i	$-0.284774\pi$
$571$	17077.2	1.25159	0.625796	+	0.779987i	$0.284774\pi$
$572$	243.395	0.0177917
$573$	−16370.4	−1.19351
$574$	5553.00	0.403794
$575$	−960.254	−0.0696441
$576$	−3720.73	−0.269150
$577$	−17287.6	−1.24730	−0.623651	−	0.781703i	$-0.714351\pi$
$577$	−17287.6	−1.24730	−0.623651	+	0.781703i	$0.714351\pi$
$578$	−8428.97	−0.606573
$579$	804.823	0.0577674
$580$	7248.09	0.518897
$581$	26929.6	1.92294
$582$	4289.91	0.305537
$583$	−5604.19	−0.398116
$584$	7537.29	0.534067
$585$	−261.963	−0.0185143
$586$	−13775.7	−0.971106
$587$	−15844.1	−1.11407	−0.557033	−	0.830491i	$-0.688060\pi$
$587$	−15844.1	−1.11407	−0.557033	+	0.830491i	$0.688060\pi$
$588$	−2131.34	−0.149481
$589$	487.176	0.0340810
$590$	−5403.95	−0.377080
$591$	11298.5	0.786390
$592$	−38.3559	−0.00266287
$593$	−4108.11	−0.284485	−0.142243	−	0.989832i	$-0.545431\pi$
$593$	−4108.11	−0.284485	−0.142243	+	0.989832i	$0.545431\pi$
$594$	−2917.65	−0.201536
$595$	−1073.60	−0.0739721
$596$	−1081.96	−0.0743607
$597$	−4210.88	−0.288676
$598$	301.964	0.0206492
$599$	−3104.88	−0.211790	−0.105895	−	0.994377i	$-0.533771\pi$
$599$	−3104.88	−0.211790	−0.105895	+	0.994377i	$0.533771\pi$
$600$	−2217.03	−0.150850
$601$	10484.1	0.711570	0.355785	−	0.934568i	$-0.384214\pi$
$601$	10484.1	0.711570	0.355785	+	0.934568i	$0.384214\pi$
$602$	6019.79	0.407555
$603$	−10191.9	−0.688300
$604$	−2630.02	−0.177175
$605$	−605.000	−0.0406558
$606$	10185.5	0.682769
$607$	−7086.01	−0.473826	−0.236913	−	0.971531i	$-0.576136\pi$
$607$	−7086.01	−0.473826	−0.236913	+	0.971531i	$0.576136\pi$
$608$	3435.35	0.229147
$609$	−24506.3	−1.63061
$610$	−6815.86	−0.452403
$611$	−837.247	−0.0554360
$612$	−580.519	−0.0383433
$613$	3871.26	0.255072	0.127536	−	0.991834i	$-0.459293\pi$
$613$	3871.26	0.255072	0.127536	+	0.991834i	$0.459293\pi$
$614$	5848.44	0.384404
$615$	2913.62	0.191038
$616$	5306.14	0.347063
$617$	−16357.6	−1.06732	−0.533658	−	0.845700i	$-0.679183\pi$
$617$	−16357.6	−1.06732	−0.533658	+	0.845700i	$0.679183\pi$
$618$	−6688.40	−0.435351
$619$	11432.1	0.742316	0.371158	−	0.928570i	$-0.378961\pi$
$619$	11432.1	0.742316	0.371158	+	0.928570i	$0.378961\pi$
$620$	632.153	0.0409482
$621$	5815.34	0.375784
$622$	10123.8	0.652620
$623$	988.609	0.0635759
$624$	4.22162	0.000270833 0
$625$	625.000	0.0400000
$626$	12333.0	0.787424
$627$	818.165	0.0521122
$628$	10589.5	0.672876
$629$	−1609.42	−0.102022
$630$	−2177.72	−0.137718
$631$	3860.28	0.243542	0.121771	−	0.992558i	$-0.461143\pi$
$631$	3860.28	0.243542	0.121771	+	0.992558i	$0.461143\pi$
$632$	531.002	0.0334211
$633$	5478.24	0.343982
$634$	−11177.2	−0.700166
$635$	−372.383	−0.0232718
$636$	9834.09	0.613124
$637$	−495.493	−0.0308197
$638$	5665.48	0.351565
$639$	−9885.75	−0.612010
$640$	4440.82	0.274279
$641$	17212.5	1.06061	0.530307	−	0.847806i	$-0.322077\pi$
$641$	17212.5	1.06061	0.530307	+	0.847806i	$0.322077\pi$
$642$	−377.928	−0.0232331
$643$	−11398.6	−0.699092	−0.349546	−	0.936919i	$-0.613664\pi$
$643$	−11398.6	−0.699092	−0.349546	+	0.936919i	$0.613664\pi$
$644$	−4032.88	−0.246767
$645$	3158.54	0.192817
$646$	−335.651	−0.0204428
$647$	5713.41	0.347167	0.173584	−	0.984819i	$-0.444465\pi$
$647$	5713.41	0.347167	0.173584	+	0.984819i	$0.444465\pi$
$648$	6285.20	0.381028
$649$	6786.15	0.410446
$650$	−196.539	−0.0118598
$651$	−2137.35	−0.128678
$652$	−5287.19	−0.317580
$653$	−21572.2	−1.29278	−0.646389	−	0.763008i	$-0.723722\pi$
$653$	−21572.2	−1.29278	−0.646389	+	0.763008i	$0.723722\pi$
$654$	−8954.55	−0.535399
$655$	−3436.50	−0.205000
$656$	35.7729	0.00212911
$657$	3884.64	0.230676
$658$	−6960.10	−0.412360
$659$	6603.36	0.390334	0.195167	−	0.980770i	$-0.437475\pi$
$659$	6603.36	0.390334	0.195167	+	0.980770i	$0.437475\pi$
$660$	1061.64	0.0626125
$661$	26250.6	1.54468	0.772338	−	0.635211i	$-0.219087\pi$
$661$	26250.6	1.54468	0.772338	+	0.635211i	$0.219087\pi$
$662$	9295.20	0.545723
$663$	177.140	0.0103764
$664$	28649.5	1.67442
$665$	2022.89	0.117962
$666$	−3264.59	−0.189941
$667$	−11292.2	−0.655528
$668$	−9785.37	−0.566778
$669$	18976.3	1.09666
$670$	−7646.49	−0.440910
$671$	8559.19	0.492435
$672$	−15071.6	−0.865180
$673$	−16800.1	−0.962256	−0.481128	−	0.876650i	$-0.659773\pi$
$673$	−16800.1	−0.962256	−0.481128	+	0.876650i	$0.659773\pi$
$674$	17715.7	1.01244
$675$	−3785.03	−0.215831
$676$	10733.7	0.610704
$677$	13912.7	0.789821	0.394910	−	0.918720i	$-0.370776\pi$
$677$	13912.7	0.789821	0.394910	+	0.918720i	$0.370776\pi$
$678$	6296.44	0.356657
$679$	−13319.6	−0.752814
$680$	−1142.17	−0.0644121
$681$	−23145.9	−1.30243
$682$	494.124	0.0277434
$683$	14558.6	0.815619	0.407809	−	0.913067i	$-0.366293\pi$
$683$	14558.6	0.815619	0.407809	+	0.913067i	$0.366293\pi$
$684$	1093.82	0.0611451
$685$	5816.13	0.324413
$686$	8676.34	0.482892
$687$	15807.7	0.877880
$688$	38.7800	0.00214894
$689$	2286.22	0.126412
$690$	1317.11	0.0726687
$691$	6482.93	0.356906	0.178453	−	0.983948i	$-0.442891\pi$
$691$	6482.93	0.356906	0.178453	+	0.983948i	$0.442891\pi$
$692$	12631.1	0.693877
$693$	2734.73	0.149904
$694$	16488.7	0.901878
$695$	−13851.3	−0.755984
$696$	−26071.4	−1.41988
$697$	1501.04	0.0815724
$698$	5881.47	0.318936
$699$	15079.6	0.815969
$700$	2624.88	0.141730
$701$	−9552.82	−0.514700	−0.257350	−	0.966318i	$-0.582849\pi$
$701$	−9552.82	−0.514700	−0.257350	+	0.966318i	$0.582849\pi$
$702$	1190.25	0.0639930
$703$	3032.49	0.162692
$704$	3505.49	0.187668
$705$	−3651.91	−0.195091
$706$	8061.66	0.429751
$707$	−31624.8	−1.68228
$708$	−11908.2	−0.632114
$709$	4549.00	0.240961	0.120480	−	0.992716i	$-0.461557\pi$
$709$	4549.00	0.240961	0.120480	+	0.992716i	$0.461557\pi$
$710$	−7416.82	−0.392040
$711$	273.673	0.0144353
$712$	1051.75	0.0553595
$713$	−984.869	−0.0517302
$714$	1472.58	0.0771847
$715$	246.809	0.0129093
$716$	2174.36	0.113491
$717$	−14448.3	−0.752553
$718$	9416.79	0.489459
$719$	2251.68	0.116792	0.0583960	−	0.998293i	$-0.481401\pi$
$719$	2251.68	0.116792	0.0583960	+	0.998293i	$0.481401\pi$
$720$	−14.0291	−0.000726155 0
$721$	20766.6	1.07266
$722$	632.438	0.0325996
$723$	15102.8	0.776872
$724$	−688.357	−0.0353351
$725$	7349.77	0.376501
$726$	829.832	0.0424214
$727$	−5928.78	−0.302457	−0.151228	−	0.988499i	$-0.548323\pi$
$727$	−5928.78	−0.302457	−0.151228	+	0.988499i	$0.548323\pi$
$728$	−2164.63	−0.110201
$729$	19241.8	0.977585
$730$	2914.46	0.147766
$731$	1627.22	0.0823321
$732$	−15019.4	−0.758381
$733$	−31414.3	−1.58297	−0.791483	−	0.611191i	$-0.790691\pi$
$733$	−31414.3	−1.58297	−0.791483	+	0.611191i	$0.790691\pi$
$734$	−14298.5	−0.719030
$735$	−2161.24	−0.108461
$736$	−6944.85	−0.347813
$737$	9602.28	0.479925
$738$	3044.75	0.151868
$739$	7772.23	0.386883	0.193441	−	0.981112i	$-0.438035\pi$
$739$	7772.23	0.386883	0.193441	+	0.981112i	$0.438035\pi$
$740$	3934.92	0.195474
$741$	−333.769	−0.0165470
$742$	19005.5	0.940316
$743$	11555.7	0.570573	0.285287	−	0.958442i	$-0.407911\pi$
$743$	11555.7	0.570573	0.285287	+	0.958442i	$0.407911\pi$
$744$	−2273.86	−0.112048
$745$	−1097.14	−0.0539546
$746$	−10296.0	−0.505313
$747$	14765.6	0.723221
$748$	546.936	0.0267352
$749$	1173.42	0.0572441
$750$	−857.265	−0.0417372
$751$	17510.7	0.850834	0.425417	−	0.904998i	$-0.360127\pi$
$751$	17510.7	0.850834	0.425417	+	0.904998i	$0.360127\pi$
$752$	−44.8375	−0.00217428
$753$	−4385.05	−0.212218
$754$	−2311.23	−0.111631
$755$	−2666.91	−0.128555
$756$	−15896.4	−0.764743
$757$	29129.6	1.39859	0.699297	−	0.714832i	$-0.253497\pi$
$757$	29129.6	1.39859	0.699297	+	0.714832i	$0.253497\pi$
$758$	−16505.1	−0.790889
$759$	−1653.99	−0.0790989
$760$	2152.09	0.102716
$761$	35912.1	1.71066	0.855330	−	0.518083i	$-0.173354\pi$
$761$	35912.1	1.71066	0.855330	+	0.518083i	$0.173354\pi$
$762$	510.769	0.0242824
$763$	27802.8	1.31917
$764$	20619.8	0.976439
$765$	−588.663	−0.0278211
$766$	15747.8	0.742807
$767$	−2768.40	−0.130328
$768$	−16071.3	−0.755110
$769$	16571.2	0.777078	0.388539	−	0.921432i	$-0.372980\pi$
$769$	16571.2	0.777078	0.388539	+	0.921432i	$0.372980\pi$
$770$	2051.74	0.0960255
$771$	24531.3	1.14588
$772$	−1013.74	−0.0472607
$773$	−11595.3	−0.539529	−0.269764	−	0.962926i	$-0.586946\pi$
$773$	−11595.3	−0.539529	−0.269764	+	0.962926i	$0.586946\pi$
$774$	3300.69	0.153283
$775$	641.021	0.0297112
$776$	−14170.3	−0.655523
$777$	−13304.2	−0.614269
$778$	−22875.5	−1.05415
$779$	−2828.28	−0.130082
$780$	−433.094	−0.0198811
$781$	9313.87	0.426730
$782$	678.548	0.0310292
$783$	−44510.6	−2.03152
$784$	−26.5353	−0.00120879
$785$	10738.0	0.488225
$786$	4713.58	0.213903
$787$	−627.435	−0.0284188	−0.0142094	−	0.999899i	$-0.504523\pi$
$787$	−627.435	−0.0284188	−0.0142094	+	0.999899i	$0.504523\pi$
$788$	−14231.3	−0.643362
$789$	11735.5	0.529527
$790$	205.324	0.00924696
$791$	−19549.7	−0.878769
$792$	2909.39	0.130531
$793$	−3491.71	−0.156361
$794$	3765.86	0.168319
$795$	9972.05	0.444870
$796$	5303.93	0.236172
$797$	−17834.0	−0.792612	−0.396306	−	0.918119i	$-0.629708\pi$
$797$	−17834.0	−0.792612	−0.396306	+	0.918119i	$0.629708\pi$
$798$	−2774.65	−0.123085
$799$	−1881.39	−0.0833028
$800$	4520.19	0.199766
$801$	542.060	0.0239111
$802$	−26606.1	−1.17144
$803$	−3659.91	−0.160841
$804$	−16849.8	−0.739115
$805$	−4089.45	−0.179049
$806$	−201.577	−0.00880925
$807$	26464.5	1.15439
$808$	−33644.6	−1.46487
$809$	45144.7	1.96193	0.980967	−	0.194174i	$-0.0622025\pi$
$809$	45144.7	1.96193	0.980967	+	0.194174i	$0.0622025\pi$
$810$	2430.32	0.105423
$811$	−20516.0	−0.888304	−0.444152	−	0.895952i	$-0.646495\pi$
$811$	−20516.0	−0.888304	−0.444152	+	0.895952i	$0.646495\pi$
$812$	30867.6	1.33404
$813$	−12437.2	−0.536521
$814$	3075.74	0.132438
$815$	−5361.36	−0.230430
$816$	9.48647	0.000406976 0
$817$	−3066.02	−0.131293
$818$	8777.81	0.375194
$819$	−1115.63	−0.0475986
$820$	−3669.93	−0.156292
$821$	−16230.5	−0.689948	−0.344974	−	0.938612i	$-0.612112\pi$
$821$	−16230.5	−0.689948	−0.344974	+	0.938612i	$0.612112\pi$
$822$	−7977.54	−0.338502
$823$	−23658.0	−1.00202	−0.501011	−	0.865441i	$-0.667039\pi$
$823$	−23658.0	−1.00202	−0.501011	+	0.865441i	$0.667039\pi$
$824$	22093.0	0.934035
$825$	1076.53	0.0454304
$826$	−23013.9	−0.969439
$827$	15296.3	0.643174	0.321587	−	0.946880i	$-0.395784\pi$
$827$	15296.3	0.643174	0.321587	+	0.946880i	$0.395784\pi$
$828$	−2211.25	−0.0928096
$829$	−20328.0	−0.851653	−0.425827	−	0.904805i	$-0.640017\pi$
$829$	−20328.0	−0.851653	−0.425827	+	0.904805i	$0.640017\pi$
$830$	11078.0	0.463280
$831$	34213.0	1.42820
$832$	−1430.06	−0.0595894
$833$	−1113.43	−0.0463122
$834$	18998.7	0.788816
$835$	−9922.65	−0.411242
$836$	−1030.54	−0.0426341
$837$	−3882.05	−0.160315
$838$	13337.1	0.549786
$839$	−1290.94	−0.0531208	−0.0265604	−	0.999647i	$-0.508455\pi$
$839$	−1290.94	−0.0531208	−0.0265604	+	0.999647i	$0.508455\pi$
$840$	−9441.71	−0.387821
$841$	62041.6	2.54383
$842$	24362.8	0.997146
$843$	−15244.6	−0.622837
$844$	−6900.27	−0.281419
$845$	10884.3	0.443115
$846$	−3816.26	−0.155090
$847$	−2576.53	−0.104522
$848$	122.435	0.00495806
$849$	−11658.3	−0.471275
$850$	−441.646	−0.0178216
$851$	−6130.45	−0.246944
$852$	−16343.7	−0.657192
$853$	36598.9	1.46908	0.734539	−	0.678567i	$-0.237399\pi$
$853$	36598.9	1.46908	0.734539	+	0.678567i	$0.237399\pi$
$854$	−29026.8	−1.16309
$855$	1109.16	0.0443656
$856$	1248.36	0.0498460
$857$	33703.1	1.34338	0.671690	−	0.740832i	$-0.265569\pi$
$857$	33703.1	1.34338	0.671690	+	0.740832i	$0.265569\pi$
$858$	−338.529	−0.0134699
$859$	−11060.0	−0.439306	−0.219653	−	0.975578i	$-0.570492\pi$
$859$	−11060.0	−0.439306	−0.219653	+	0.975578i	$0.570492\pi$
$860$	−3978.43	−0.157748
$861$	12408.3	0.491142
$862$	13973.1	0.552116
$863$	−8886.40	−0.350518	−0.175259	−	0.984522i	$-0.556076\pi$
$863$	−8886.40	−0.350518	−0.175259	+	0.984522i	$0.556076\pi$
$864$	−27374.5	−1.07789
$865$	12808.3	0.503463
$866$	−11136.2	−0.436978
$867$	−18834.7	−0.737785
$868$	2692.16	0.105274
$869$	−257.841	−0.0100652
$870$	−10081.1	−0.392853
$871$	−3917.24	−0.152389
$872$	29578.5	1.14869
$873$	−7303.24	−0.283135
$874$	−1278.53	−0.0494815
$875$	2661.70	0.102836
$876$	6422.33	0.247706
$877$	45688.4	1.75917	0.879583	−	0.475746i	$-0.157822\pi$
$877$	45688.4	1.75917	0.879583	+	0.475746i	$0.157822\pi$
$878$	−15959.1	−0.613433
$879$	−30782.0	−1.18117
$880$	13.2175	0.000506320 0
$881$	27861.1	1.06545	0.532727	−	0.846287i	$-0.321167\pi$
$881$	27861.1	1.06545	0.532727	+	0.846287i	$0.321167\pi$
$882$	−2258.51	−0.0862222
$883$	−44435.9	−1.69353	−0.846766	−	0.531966i	$-0.821453\pi$
$883$	−44435.9	−1.69353	−0.846766	+	0.531966i	$0.821453\pi$
$884$	−223.122	−0.00848914
$885$	−12075.2	−0.458649
$886$	26013.0	0.986371
$887$	29514.6	1.11725	0.558627	−	0.829419i	$-0.311329\pi$
$887$	29514.6	1.11725	0.558627	+	0.829419i	$0.311329\pi$
$888$	−14153.9	−0.534882
$889$	−1585.88	−0.0598296
$890$	406.683	0.0153169
$891$	−3051.93	−0.114752
$892$	−23902.2	−0.897202
$893$	3544.94	0.132841
$894$	1504.87	0.0562978
$895$	2204.87	0.0823470
$896$	18912.2	0.705148
$897$	674.744	0.0251160
$898$	31659.8	1.17650
$899$	7538.17	0.279657
$900$	1439.24	0.0533051
$901$	5137.41	0.189958
$902$	−2868.61	−0.105892
$903$	13451.3	0.495717
$904$	−20798.3	−0.765199
$905$	−698.014	−0.0256384
$906$	3658.00	0.134138
$907$	13969.3	0.511403	0.255702	−	0.966756i	$-0.417694\pi$
$907$	13969.3	0.511403	0.255702	+	0.966756i	$0.417694\pi$
$908$	29154.1	1.06554
$909$	−17340.0	−0.632710
$910$	−837.005	−0.0304906
$911$	8915.00	0.324223	0.162112	−	0.986772i	$-0.448170\pi$
$911$	8915.00	0.324223	0.162112	+	0.986772i	$0.448170\pi$
$912$	−17.8745	−0.000648996 0
$913$	−13911.5	−0.504274
$914$	−4393.52	−0.158999
$915$	−15230.1	−0.550266
$916$	−19911.1	−0.718211
$917$	−14635.1	−0.527037
$918$	2674.63	0.0961612
$919$	−12664.0	−0.454566	−0.227283	−	0.973829i	$-0.572984\pi$
$919$	−12664.0	−0.454566	−0.227283	+	0.973829i	$0.572984\pi$
$920$	−4350.64	−0.155909
$921$	13068.4	0.467557
$922$	−29583.7	−1.05671
$923$	−3799.58	−0.135498
$924$	4521.22	0.160971
$925$	3990.12	0.141832
$926$	−18610.6	−0.660456
$927$	11386.5	0.403432
$928$	53155.8	1.88031
$929$	−29490.7	−1.04150	−0.520752	−	0.853708i	$-0.674349\pi$
$929$	−29490.7	−1.04150	−0.520752	+	0.853708i	$0.674349\pi$
$930$	−879.240	−0.0310015
$931$	2097.94	0.0738530
$932$	−18993.9	−0.667561
$933$	22621.9	0.793792
$934$	−15249.1	−0.534224
$935$	554.609	0.0193986
$936$	−1186.88	−0.0414471
$937$	296.579	0.0103402	0.00517012	−	0.999987i	$-0.498354\pi$
$937$	296.579	0.0103402	0.00517012	+	0.999987i	$0.498354\pi$
$938$	−32564.3	−1.13354
$939$	27558.4	0.957757
$940$	4599.87	0.159608
$941$	−26831.5	−0.929524	−0.464762	−	0.885436i	$-0.653860\pi$
$941$	−26831.5	−0.929524	−0.464762	+	0.885436i	$0.653860\pi$
$942$	−14728.5	−0.509428
$943$	5717.61	0.197445
$944$	−148.258	−0.00511162
$945$	−16119.4	−0.554882
$946$	−3109.74	−0.106878
$947$	28371.0	0.973531	0.486766	−	0.873533i	$-0.338177\pi$
$947$	28371.0	0.973531	0.486766	+	0.873533i	$0.338177\pi$
$948$	452.453	0.0155010
$949$	1493.06	0.0510713
$950$	832.155	0.0284197
$951$	−24975.8	−0.851624
$952$	−4864.19	−0.165598
$953$	50869.5	1.72909	0.864545	−	0.502555i	$-0.167607\pi$
$953$	50869.5	1.72909	0.864545	+	0.502555i	$0.167607\pi$
$954$	10420.8	0.353655
$955$	20909.1	0.708484
$956$	18198.7	0.615679
$957$	12659.6	0.427615
$958$	−23668.9	−0.798233
$959$	24769.3	0.834037
$960$	−6237.63	−0.209707
$961$	−29133.5	−0.977931
$962$	−1254.74	−0.0420526
$963$	643.393	0.0215296
$964$	−19023.2	−0.635575
$965$	−1027.96	−0.0342914
$966$	5609.19	0.186825
$967$	−4395.06	−0.146159	−0.0730795	−	0.997326i	$-0.523283\pi$
$967$	−4395.06	−0.146159	−0.0730795	+	0.997326i	$0.523283\pi$
$968$	−2741.08	−0.0910142
$969$	−750.019	−0.0248649
$970$	−5479.29	−0.181370
$971$	39921.6	1.31941	0.659704	−	0.751526i	$-0.270682\pi$
$971$	39921.6	1.31941	0.659704	+	0.751526i	$0.270682\pi$
$972$	−14800.9	−0.488416
$973$	−58988.7	−1.94357
$974$	27113.4	0.891961
$975$	−439.170	−0.0144253
$976$	−186.993	−0.00613269
$977$	56637.5	1.85465	0.927325	−	0.374258i	$-0.122102\pi$
$977$	56637.5	1.85465	0.927325	+	0.374258i	$0.122102\pi$
$978$	7353.77	0.240437
$979$	−510.703	−0.0166722
$980$	2722.25	0.0887339
$981$	15244.4	0.496144
$982$	−12759.9	−0.414648
$983$	21544.8	0.699057	0.349528	−	0.936926i	$-0.386342\pi$
$983$	21544.8	0.699057	0.349528	+	0.936926i	$0.386342\pi$
$984$	13200.8	0.427668
$985$	−14431.0	−0.466811
$986$	−5193.60	−0.167746
$987$	−15552.5	−0.501561
$988$	420.409	0.0135374
$989$	6198.23	0.199284
$990$	1124.98	0.0361154
$991$	−2446.66	−0.0784266	−0.0392133	−	0.999231i	$-0.512485\pi$
$991$	−2446.66	−0.0784266	−0.0392133	+	0.999231i	$0.512485\pi$
$992$	4636.06	0.148382
$993$	20770.3	0.663772
$994$	−31586.2	−1.00790
$995$	5378.34	0.171362
$996$	24411.5	0.776614
$997$	−23153.7	−0.735491	−0.367746	−	0.929926i	$-0.619870\pi$
$997$	−23153.7	−0.735491	−0.367746	+	0.929926i	$0.619870\pi$
$998$	−26357.4	−0.836001
$999$	−24164.4	−0.765292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.i.1.16	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.i.1.16	✓	25	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)

\(f(q)\)	\(=\)	\(q+1.75191 q^{2} +3.91466 q^{3} -4.93083 q^{4} -5.00000 q^{5} +6.85812 q^{6} -21.2936 q^{7} -22.6536 q^{8} -11.6754 q^{9} +O(q^{10})\)	\(q+1.75191 q^{2} +3.91466 q^{3} -4.93083 q^{4} -5.00000 q^{5} +6.85812 q^{6} -21.2936 q^{7} -22.6536 q^{8} -11.6754 q^{9} -8.75953 q^{10} +11.0000 q^{11} -19.3025 q^{12} -4.48744 q^{13} -37.3044 q^{14} -19.5733 q^{15} -0.240318 q^{16} -10.0838 q^{17} -20.4542 q^{18} +19.0000 q^{19} +24.6541 q^{20} -83.3573 q^{21} +19.2710 q^{22} -38.4102 q^{23} -88.6812 q^{24} +25.0000 q^{25} -7.86156 q^{26} -151.401 q^{27} +104.995 q^{28} +293.991 q^{29} -34.2906 q^{30} +25.6408 q^{31} +180.808 q^{32} +43.0613 q^{33} -17.6659 q^{34} +106.468 q^{35} +57.5695 q^{36} +159.605 q^{37} +33.2862 q^{38} -17.5668 q^{39} +113.268 q^{40} -148.857 q^{41} -146.034 q^{42} -161.369 q^{43} -54.2391 q^{44} +58.3771 q^{45} -67.2910 q^{46} +186.576 q^{47} -0.940763 q^{48} +110.418 q^{49} +43.7976 q^{50} -39.4747 q^{51} +22.1268 q^{52} -509.472 q^{53} -265.241 q^{54} -55.0000 q^{55} +482.377 q^{56} +74.3786 q^{57} +515.044 q^{58} +616.923 q^{59} +96.5126 q^{60} +778.108 q^{61} +44.9203 q^{62} +248.612 q^{63} +318.680 q^{64} +22.4372 q^{65} +75.4393 q^{66} +872.934 q^{67} +49.7215 q^{68} -150.363 q^{69} +186.522 q^{70} +846.715 q^{71} +264.490 q^{72} -332.719 q^{73} +279.613 q^{74} +97.8666 q^{75} -93.6857 q^{76} -234.230 q^{77} -30.7754 q^{78} -23.4401 q^{79} +1.20159 q^{80} -277.448 q^{81} -260.783 q^{82} -1264.68 q^{83} +411.020 q^{84} +50.4190 q^{85} -282.704 q^{86} +1150.87 q^{87} -249.189 q^{88} -46.4275 q^{89} +102.271 q^{90} +95.5537 q^{91} +189.394 q^{92} +100.375 q^{93} +326.863 q^{94} -95.0000 q^{95} +707.801 q^{96} +625.523 q^{97} +193.441 q^{98} -128.430 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9} - 10 q^{10} + 275 q^{11} + 44 q^{12} + 53 q^{13} - 51 q^{14} - 45 q^{15} + 438 q^{16} + 153 q^{17} + 9 q^{18} + 475 q^{19} - 610 q^{20} + 259 q^{21} + 22 q^{22} - 7 q^{23} + 186 q^{24} + 625 q^{25} + 543 q^{26} + 495 q^{27} - 525 q^{28} + 169 q^{29} - 55 q^{30} + 102 q^{31} + 327 q^{32} + 99 q^{33} - 879 q^{34} + 75 q^{35} + 2293 q^{36} - 46 q^{37} + 38 q^{38} + 233 q^{39} - 300 q^{40} + 1190 q^{41} - 684 q^{42} - 408 q^{43} + 1342 q^{44} - 1500 q^{45} + 757 q^{46} + 1068 q^{47} + 715 q^{48} + 1930 q^{49} + 50 q^{50} + 1655 q^{51} - 94 q^{52} + 143 q^{53} + 1970 q^{54} - 1375 q^{55} - 1397 q^{56} + 171 q^{57} + 1366 q^{58} + 2945 q^{59} - 220 q^{60} + 1160 q^{61} + 194 q^{62} + 1804 q^{63} + 3000 q^{64} - 265 q^{65} + 121 q^{66} - 353 q^{67} + 5452 q^{68} + 3289 q^{69} + 255 q^{70} + 230 q^{71} + 196 q^{72} + 1357 q^{73} + 4379 q^{74} + 225 q^{75} + 2318 q^{76} - 165 q^{77} + 2008 q^{78} + 1266 q^{79} - 2190 q^{80} + 1709 q^{81} + 1010 q^{82} + 3856 q^{83} + 9354 q^{84} - 765 q^{85} + 6746 q^{86} + 3113 q^{87} + 660 q^{88} + 3562 q^{89} - 45 q^{90} - 833 q^{91} + 4276 q^{92} + 1312 q^{93} + 5124 q^{94} - 2375 q^{95} + 3828 q^{96} - 914 q^{97} + 2478 q^{98} + 3300 q^{99}+O(q^{100}) \) 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9 - 10 * q^10 + 275 * q^11 + 44 * q^12 + 53 * q^13 - 51 * q^14 - 45 * q^15 + 438 * q^16 + 153 * q^17 + 9 * q^18 + 475 * q^19 - 610 * q^20 + 259 * q^21 + 22 * q^22 - 7 * q^23 + 186 * q^24 + 625 * q^25 + 543 * q^26 + 495 * q^27 - 525 * q^28 + 169 * q^29 - 55 * q^30 + 102 * q^31 + 327 * q^32 + 99 * q^33 - 879 * q^34 + 75 * q^35 + 2293 * q^36 - 46 * q^37 + 38 * q^38 + 233 * q^39 - 300 * q^40 + 1190 * q^41 - 684 * q^42 - 408 * q^43 + 1342 * q^44 - 1500 * q^45 + 757 * q^46 + 1068 * q^47 + 715 * q^48 + 1930 * q^49 + 50 * q^50 + 1655 * q^51 - 94 * q^52 + 143 * q^53 + 1970 * q^54 - 1375 * q^55 - 1397 * q^56 + 171 * q^57 + 1366 * q^58 + 2945 * q^59 - 220 * q^60 + 1160 * q^61 + 194 * q^62 + 1804 * q^63 + 3000 * q^64 - 265 * q^65 + 121 * q^66 - 353 * q^67 + 5452 * q^68 + 3289 * q^69 + 255 * q^70 + 230 * q^71 + 196 * q^72 + 1357 * q^73 + 4379 * q^74 + 225 * q^75 + 2318 * q^76 - 165 * q^77 + 2008 * q^78 + 1266 * q^79 - 2190 * q^80 + 1709 * q^81 + 1010 * q^82 + 3856 * q^83 + 9354 * q^84 - 765 * q^85 + 6746 * q^86 + 3113 * q^87 + 660 * q^88 + 3562 * q^89 - 45 * q^90 - 833 * q^91 + 4276 * q^92 + 1312 * q^93 + 5124 * q^94 - 2375 * q^95 + 3828 * q^96 - 914 * q^97 + 2478 * q^98 + 3300 * q^99

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