LMFDB - Embedded newform 1045.4.a.i.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.51141 q^{2} +8.82864 q^{3} -1.69283 q^{4} -5.00000 q^{5} -22.1723 q^{6} +6.85719 q^{7} +24.3426 q^{8} +50.9449 q^{9} +O(q^{10})$	\(q-2.51141 q^{2} +8.82864 q^{3} -1.69283 q^{4} -5.00000 q^{5} -22.1723 q^{6} +6.85719 q^{7} +24.3426 q^{8} +50.9449 q^{9} +12.5570 q^{10} +11.0000 q^{11} -14.9453 q^{12} +0.255142 q^{13} -17.2212 q^{14} -44.1432 q^{15} -47.5917 q^{16} -78.4943 q^{17} -127.943 q^{18} +19.0000 q^{19} +8.46413 q^{20} +60.5397 q^{21} -27.6255 q^{22} +52.4469 q^{23} +214.912 q^{24} +25.0000 q^{25} -0.640765 q^{26} +211.401 q^{27} -11.6080 q^{28} +164.477 q^{29} +110.862 q^{30} +304.136 q^{31} -75.2189 q^{32} +97.1150 q^{33} +197.131 q^{34} -34.2860 q^{35} -86.2408 q^{36} -292.276 q^{37} -47.7168 q^{38} +2.25255 q^{39} -121.713 q^{40} +189.236 q^{41} -152.040 q^{42} +386.238 q^{43} -18.6211 q^{44} -254.724 q^{45} -131.716 q^{46} -319.521 q^{47} -420.170 q^{48} -295.979 q^{49} -62.7852 q^{50} -692.998 q^{51} -0.431910 q^{52} +517.414 q^{53} -530.914 q^{54} -55.0000 q^{55} +166.922 q^{56} +167.744 q^{57} -413.069 q^{58} +57.5472 q^{59} +74.7267 q^{60} -466.915 q^{61} -763.811 q^{62} +349.339 q^{63} +569.639 q^{64} -1.27571 q^{65} -243.896 q^{66} -231.952 q^{67} +132.877 q^{68} +463.035 q^{69} +86.1060 q^{70} +281.634 q^{71} +1240.13 q^{72} +289.367 q^{73} +734.026 q^{74} +220.716 q^{75} -32.1637 q^{76} +75.4291 q^{77} -5.65708 q^{78} -313.046 q^{79} +237.959 q^{80} +490.870 q^{81} -475.249 q^{82} -85.8906 q^{83} -102.483 q^{84} +392.471 q^{85} -970.002 q^{86} +1452.11 q^{87} +267.769 q^{88} +934.911 q^{89} +639.717 q^{90} +1.74955 q^{91} -88.7835 q^{92} +2685.11 q^{93} +802.448 q^{94} -95.0000 q^{95} -664.080 q^{96} +1589.56 q^{97} +743.324 q^{98} +560.394 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.51141	−0.887917	−0.443959	−	0.896047i	$-0.646426\pi$
$2$	−2.51141	−0.887917	−0.443959	+	0.896047i	$0.646426\pi$
$3$	8.82864	1.69907	0.849536	−	0.527530i	$-0.176882\pi$
$3$	8.82864	1.69907	0.849536	+	0.527530i	$0.176882\pi$
$4$	−1.69283	−0.211603
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−22.1723	−1.50864
$7$	6.85719	0.370253	0.185127	−	0.982715i	$-0.440730\pi$
$7$	6.85719	0.370253	0.185127	+	0.982715i	$0.440730\pi$
$8$	24.3426	1.07580
$9$	50.9449	1.88685
$10$	12.5570	0.397089
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−14.9453	−0.359529
$13$	0.255142	0.00544335	0.00272167	−	0.999996i	$-0.499134\pi$
$13$	0.255142	0.00544335	0.00272167	+	0.999996i	$0.499134\pi$
$14$	−17.2212	−0.328754
$15$	−44.1432	−0.759848
$16$	−47.5917	−0.743621
$17$	−78.4943	−1.11986	−0.559931	−	0.828539i	$-0.689172\pi$
$17$	−78.4943	−1.11986	−0.559931	+	0.828539i	$0.689172\pi$
$18$	−127.943	−1.67536
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	8.46413	0.0946318
$21$	60.5397	0.629088
$22$	−27.6255	−0.267717
$23$	52.4469	0.475476	0.237738	−	0.971329i	$-0.423594\pi$
$23$	52.4469	0.475476	0.237738	+	0.971329i	$0.423594\pi$
$24$	214.912	1.82787
$25$	25.0000	0.200000
$26$	−0.640765	−0.00483324
$27$	211.401	1.50682
$28$	−11.6080	−0.0783468
$29$	164.477	1.05319	0.526597	−	0.850115i	$-0.323468\pi$
$29$	164.477	1.05319	0.526597	+	0.850115i	$0.323468\pi$
$30$	110.862	0.674682
$31$	304.136	1.76208	0.881040	−	0.473041i	$-0.156844\pi$
$31$	304.136	1.76208	0.881040	+	0.473041i	$0.156844\pi$
$32$	−75.2189	−0.415529
$33$	97.1150	0.512290
$34$	197.131	0.994345
$35$	−34.2860	−0.165582
$36$	−86.2408	−0.399263
$37$	−292.276	−1.29865	−0.649324	−	0.760512i	$-0.724948\pi$
$37$	−292.276	−1.29865	−0.649324	+	0.760512i	$0.724948\pi$
$38$	−47.7168	−0.203702
$39$	2.25255	0.00924865
$40$	−121.713	−0.481114
$41$	189.236	0.720822	0.360411	−	0.932794i	$-0.382637\pi$
$41$	189.236	0.720822	0.360411	+	0.932794i	$0.382637\pi$
$42$	−152.040	−0.558578
$43$	386.238	1.36978	0.684892	−	0.728644i	$-0.259849\pi$
$43$	386.238	1.36978	0.684892	+	0.728644i	$0.259849\pi$
$44$	−18.6211	−0.0638008
$45$	−254.724	−0.843824
$46$	−131.716	−0.422183
$47$	−319.521	−0.991637	−0.495819	−	0.868426i	$-0.665132\pi$
$47$	−319.521	−0.991637	−0.495819	+	0.868426i	$0.665132\pi$
$48$	−420.170	−1.26347
$49$	−295.979	−0.862912
$50$	−62.7852	−0.177583
$51$	−692.998	−1.90273
$52$	−0.431910	−0.00115183
$53$	517.414	1.34099	0.670493	−	0.741915i	$-0.266082\pi$
$53$	517.414	1.34099	0.670493	+	0.741915i	$0.266082\pi$
$54$	−530.914	−1.33793
$55$	−55.0000	−0.134840
$56$	166.922	0.398320
$57$	167.744	0.389794
$58$	−413.069	−0.935148
$59$	57.5472	0.126983	0.0634916	−	0.997982i	$-0.479776\pi$
$59$	57.5472	0.126983	0.0634916	+	0.997982i	$0.479776\pi$
$60$	74.7267	0.160786
$61$	−466.915	−0.980039	−0.490019	−	0.871712i	$-0.663010\pi$
$61$	−466.915	−0.980039	−0.490019	+	0.871712i	$0.663010\pi$
$62$	−763.811	−1.56458
$63$	349.339	0.698612
$64$	569.639	1.11258
$65$	−1.27571	−0.00243434
$66$	−243.896	−0.454871
$67$	−231.952	−0.422946	−0.211473	−	0.977384i	$-0.567826\pi$
$67$	−231.952	−0.422946	−0.211473	+	0.977384i	$0.567826\pi$
$68$	132.877	0.236966
$69$	463.035	0.807868
$70$	86.1060	0.147023
$71$	281.634	0.470758	0.235379	−	0.971904i	$-0.424367\pi$
$71$	281.634	0.470758	0.235379	+	0.971904i	$0.424367\pi$
$72$	1240.13	2.02988
$73$	289.367	0.463943	0.231971	−	0.972723i	$-0.425482\pi$
$73$	289.367	0.463943	0.231971	+	0.972723i	$0.425482\pi$
$74$	734.026	1.15309
$75$	220.716	0.339815
$76$	−32.1637	−0.0485451
$77$	75.4291	0.111636
$78$	−5.65708	−0.00821203
$79$	−313.046	−0.445828	−0.222914	−	0.974838i	$-0.571557\pi$
$79$	−313.046	−0.445828	−0.222914	+	0.974838i	$0.571557\pi$
$80$	237.959	0.332557
$81$	490.870	0.673348
$82$	−475.249	−0.640030
$83$	−85.8906	−0.113587	−0.0567935	−	0.998386i	$-0.518088\pi$
$83$	−85.8906	−0.113587	−0.0567935	+	0.998386i	$0.518088\pi$
$84$	−102.483	−0.133117
$85$	392.471	0.500818
$86$	−970.002	−1.21626
$87$	1452.11	1.78945
$88$	267.769	0.324367
$89$	934.911	1.11349	0.556744	−	0.830684i	$-0.312051\pi$
$89$	934.911	1.11349	0.556744	+	0.830684i	$0.312051\pi$
$90$	639.717	0.749246
$91$	1.74955	0.00201542
$92$	−88.7835	−0.100612
$93$	2685.11	2.99390
$94$	802.448	0.880492
$95$	−95.0000	−0.102598
$96$	−664.080	−0.706015
$97$	1589.56	1.66387	0.831936	−	0.554871i	$-0.187232\pi$
$97$	1589.56	1.66387	0.831936	+	0.554871i	$0.187232\pi$
$98$	743.324	0.766195
$99$	560.394	0.568906
$100$	−42.3206	−0.0423206
$101$	1167.13	1.14984	0.574921	−	0.818209i	$-0.305033\pi$
$101$	1167.13	1.14984	0.574921	+	0.818209i	$0.305033\pi$
$102$	1740.40	1.68946
$103$	301.230	0.288166	0.144083	−	0.989566i	$-0.453977\pi$
$103$	301.230	0.288166	0.144083	+	0.989566i	$0.453977\pi$
$104$	6.21082	0.00585597
$105$	−302.698	−0.281337
$106$	−1299.44	−1.19069
$107$	−1012.52	−0.914802	−0.457401	−	0.889260i	$-0.651220\pi$
$107$	−1012.52	−0.914802	−0.457401	+	0.889260i	$0.651220\pi$
$108$	−357.865	−0.318848
$109$	−3.42523	−0.00300989	−0.00150494	−	0.999999i	$-0.500479\pi$
$109$	−3.42523	−0.00300989	−0.00150494	+	0.999999i	$0.500479\pi$
$110$	138.127	0.119727
$111$	−2580.40	−2.20650
$112$	−326.346	−0.275328
$113$	−1131.31	−0.941812	−0.470906	−	0.882183i	$-0.656073\pi$
$113$	−1131.31	−0.941812	−0.470906	+	0.882183i	$0.656073\pi$
$114$	−421.274	−0.346105
$115$	−262.235	−0.212639
$116$	−278.431	−0.222859
$117$	12.9982	0.0102708
$118$	−144.525	−0.112751
$119$	−538.250	−0.414633
$120$	−1074.56	−0.817447
$121$	121.000	0.0909091
$122$	1172.61	0.870193
$123$	1670.70	1.22473
$124$	−514.850	−0.372862
$125$	−125.000	−0.0894427
$126$	−877.333	−0.620310
$127$	1877.15	1.31158	0.655790	−	0.754944i	$-0.272336\pi$
$127$	1877.15	1.31158	0.655790	+	0.754944i	$0.272336\pi$
$128$	−828.846	−0.572346
$129$	3409.96	2.32736
$130$	3.20382	0.00216149
$131$	1510.02	1.00711	0.503555	−	0.863963i	$-0.332025\pi$
$131$	1510.02	1.00711	0.503555	+	0.863963i	$0.332025\pi$
$132$	−164.399	−0.108402
$133$	130.287	0.0849420
$134$	582.526	0.375541
$135$	−1057.00	−0.673870
$136$	−1910.76	−1.20475
$137$	−824.680	−0.514286	−0.257143	−	0.966373i	$-0.582781\pi$
$137$	−824.680	−0.514286	−0.257143	+	0.966373i	$0.582781\pi$
$138$	−1162.87	−0.717320
$139$	1537.45	0.938161	0.469081	−	0.883155i	$-0.344585\pi$
$139$	1537.45	0.938161	0.469081	+	0.883155i	$0.344585\pi$
$140$	58.0401	0.0350378
$141$	−2820.94	−1.68486
$142$	−707.298	−0.417994
$143$	2.80656	0.00164123
$144$	−2424.56	−1.40310
$145$	−822.385	−0.471002
$146$	−726.719	−0.411943
$147$	−2613.09	−1.46615
$148$	494.773	0.274798
$149$	−2047.62	−1.12582	−0.562910	−	0.826518i	$-0.690318\pi$
$149$	−2047.62	−1.12582	−0.562910	+	0.826518i	$0.690318\pi$
$150$	−554.308	−0.301727
$151$	1918.99	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$151$	1918.99	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$152$	462.510	0.246806
$153$	−3998.88	−2.11301
$154$	−189.433	−0.0991232
$155$	−1520.68	−0.788026
$156$	−3.81318	−0.00195704
$157$	−398.426	−0.202534	−0.101267	−	0.994859i	$-0.532290\pi$
$157$	−398.426	−0.202534	−0.101267	+	0.994859i	$0.532290\pi$
$158$	786.185	0.395858
$159$	4568.06	2.27843
$160$	376.094	0.185830
$161$	359.639	0.176047
$162$	−1232.78	−0.597877
$163$	3678.16	1.76746	0.883730	−	0.467997i	$-0.155024\pi$
$163$	3678.16	1.76746	0.883730	+	0.467997i	$0.155024\pi$
$164$	−320.343	−0.152528
$165$	−485.575	−0.229103
$166$	215.706	0.100856
$167$	−103.673	−0.0480385	−0.0240192	−	0.999711i	$-0.507646\pi$
$167$	−103.673	−0.0480385	−0.0240192	+	0.999711i	$0.507646\pi$
$168$	1473.70	0.676774
$169$	−2196.93	−0.999970
$170$	−985.656	−0.444685
$171$	967.953	0.432873
$172$	−653.834	−0.289851
$173$	4354.70	1.91377	0.956885	−	0.290468i	$-0.0938110\pi$
$173$	4354.70	1.91377	0.956885	+	0.290468i	$0.0938110\pi$
$174$	−3646.84	−1.58888
$175$	171.430	0.0740507
$176$	−523.509	−0.224210
$177$	508.064	0.215754
$178$	−2347.94	−0.988685
$179$	1598.28	0.667379	0.333689	−	0.942683i	$-0.391706\pi$
$179$	1598.28	0.667379	0.333689	+	0.942683i	$0.391706\pi$
$180$	431.204	0.178556
$181$	−1797.13	−0.738009	−0.369004	−	0.929428i	$-0.620301\pi$
$181$	−1797.13	−0.738009	−0.369004	+	0.929428i	$0.620301\pi$
$182$	−4.39385	−0.00178953
$183$	−4122.23	−1.66516
$184$	1276.70	0.511518
$185$	1461.38	0.580773
$186$	−6743.41	−2.65834
$187$	−863.437	−0.337651
$188$	540.894	0.209834
$189$	1449.62	0.557905
$190$	238.584	0.0910984
$191$	−763.945	−0.289409	−0.144705	−	0.989475i	$-0.546223\pi$
$191$	−763.945	−0.289409	−0.144705	+	0.989475i	$0.546223\pi$
$192$	5029.14	1.89035
$193$	1503.37	0.560697	0.280349	−	0.959898i	$-0.409550\pi$
$193$	1503.37	0.560697	0.280349	+	0.959898i	$0.409550\pi$
$194$	−3992.04	−1.47738
$195$	−11.2628	−0.00413612
$196$	501.041	0.182595
$197$	4289.27	1.55126	0.775630	−	0.631188i	$-0.217432\pi$
$197$	4289.27	1.55126	0.775630	+	0.631188i	$0.217432\pi$
$198$	−1407.38	−0.505141
$199$	1458.09	0.519402	0.259701	−	0.965689i	$-0.416376\pi$
$199$	1458.09	0.519402	0.259701	+	0.965689i	$0.416376\pi$
$200$	608.566	0.215161
$201$	−2047.82	−0.718617
$202$	−2931.15	−1.02096
$203$	1127.85	0.389948
$204$	1173.12	0.402623
$205$	−946.180	−0.322361
$206$	−756.513	−0.255868
$207$	2671.90	0.897150
$208$	−12.1426	−0.00404779
$209$	209.000	0.0691714
$210$	760.199	0.249804
$211$	1637.03	0.534112	0.267056	−	0.963681i	$-0.413949\pi$
$211$	1637.03	0.534112	0.267056	+	0.963681i	$0.413949\pi$
$212$	−875.892	−0.283757
$213$	2486.45	0.799852
$214$	2542.85	0.812269
$215$	−1931.19	−0.612586
$216$	5146.06	1.62104
$217$	2085.52	0.652416
$218$	8.60216	0.00267253
$219$	2554.72	0.788273
$220$	93.1054	0.0285326
$221$	−20.0272	−0.00609580
$222$	6480.45	1.95919
$223$	−3383.86	−1.01614	−0.508072	−	0.861315i	$-0.669642\pi$
$223$	−3383.86	−1.01614	−0.508072	+	0.861315i	$0.669642\pi$
$224$	−515.790	−0.153851
$225$	1273.62	0.377370
$226$	2841.18	0.836251
$227$	−306.927	−0.0897422	−0.0448711	−	0.998993i	$-0.514288\pi$
$227$	−306.927	−0.0897422	−0.0448711	+	0.998993i	$0.514288\pi$
$228$	−283.962	−0.0824817
$229$	−2665.32	−0.769123	−0.384562	−	0.923099i	$-0.625647\pi$
$229$	−2665.32	−0.769123	−0.384562	+	0.923099i	$0.625647\pi$
$230$	658.578	0.188806
$231$	665.936	0.189677
$232$	4003.80	1.13303
$233$	2923.41	0.821970	0.410985	−	0.911642i	$-0.365185\pi$
$233$	2923.41	0.821970	0.410985	+	0.911642i	$0.365185\pi$
$234$	−32.6437	−0.00911960
$235$	1597.61	0.443474
$236$	−97.4174	−0.0268701
$237$	−2763.77	−0.757493
$238$	1351.77	0.368160
$239$	4145.26	1.12190	0.560951	−	0.827849i	$-0.310436\pi$
$239$	4145.26	1.12190	0.560951	+	0.827849i	$0.310436\pi$
$240$	2100.85	0.565039
$241$	−2628.98	−0.702686	−0.351343	−	0.936247i	$-0.614275\pi$
$241$	−2628.98	−0.702686	−0.351343	+	0.936247i	$0.614275\pi$
$242$	−303.880	−0.0807197
$243$	−1374.11	−0.362753
$244$	790.406	0.207379
$245$	1479.89	0.385906
$246$	−4195.80	−1.08746
$247$	4.84769	0.00124879
$248$	7403.48	1.89565
$249$	−758.297	−0.192993
$250$	313.926	0.0794177
$251$	−1475.95	−0.371160	−0.185580	−	0.982629i	$-0.559416\pi$
$251$	−1475.95	−0.371160	−0.185580	+	0.982629i	$0.559416\pi$
$252$	−591.370	−0.147829
$253$	576.916	0.143361
$254$	−4714.30	−1.16457
$255$	3464.99	0.850926
$256$	−2475.54	−0.604380
$257$	7105.77	1.72469	0.862346	−	0.506319i	$-0.168994\pi$
$257$	7105.77	1.72469	0.862346	+	0.506319i	$0.168994\pi$
$258$	−8563.80	−2.06651
$259$	−2004.20	−0.480829
$260$	2.15955	0.000515114 0
$261$	8379.26	1.98722
$262$	−3792.28	−0.894229
$263$	−4790.45	−1.12316	−0.561581	−	0.827421i	$-0.689807\pi$
$263$	−4790.45	−1.12316	−0.561581	+	0.827421i	$0.689807\pi$
$264$	2364.04	0.551123
$265$	−2587.07	−0.599708
$266$	−327.203	−0.0754214
$267$	8254.00	1.89190
$268$	392.654	0.0894968
$269$	5800.83	1.31481	0.657403	−	0.753539i	$-0.271655\pi$
$269$	5800.83	1.31481	0.657403	+	0.753539i	$0.271655\pi$
$270$	2654.57	0.598341
$271$	−3148.78	−0.705811	−0.352905	−	0.935659i	$-0.614806\pi$
$271$	−3148.78	−0.705811	−0.352905	+	0.935659i	$0.614806\pi$
$272$	3735.68	0.832753
$273$	15.4462	0.00342434
$274$	2071.11	0.456643
$275$	275.000	0.0603023
$276$	−783.838	−0.170947
$277$	3708.62	0.804439	0.402220	−	0.915543i	$-0.368239\pi$
$277$	3708.62	0.804439	0.402220	+	0.915543i	$0.368239\pi$
$278$	−3861.15	−0.833009
$279$	15494.2	3.32478
$280$	−834.611	−0.178134
$281$	8257.17	1.75296	0.876480	−	0.481439i	$-0.159886\pi$
$281$	8257.17	1.75296	0.876480	+	0.481439i	$0.159886\pi$
$282$	7084.53	1.49602
$283$	−7213.13	−1.51511	−0.757555	−	0.652772i	$-0.773606\pi$
$283$	−7213.13	−1.51511	−0.757555	+	0.652772i	$0.773606\pi$
$284$	−476.757	−0.0996139
$285$	−838.721	−0.174321
$286$	−7.04841	−0.00145728
$287$	1297.63	0.266887
$288$	−3832.02	−0.784041
$289$	1248.35	0.254092
$290$	2065.34	0.418211
$291$	14033.7	2.82704
$292$	−489.848	−0.0981718
$293$	−4024.21	−0.802378	−0.401189	−	0.915995i	$-0.631403\pi$
$293$	−4024.21	−0.802378	−0.401189	+	0.915995i	$0.631403\pi$
$294$	6562.54	1.30182
$295$	−287.736	−0.0567886
$296$	−7114.78	−1.39709
$297$	2325.41	0.454323
$298$	5142.40	0.999635
$299$	13.3814	0.00258818
$300$	−373.634	−0.0719058
$301$	2648.51	0.507168
$302$	−4819.36	−0.918287
$303$	10304.2	1.95366
$304$	−904.243	−0.170598
$305$	2334.58	0.438287
$306$	10042.8	1.87618
$307$	5250.27	0.976055	0.488028	−	0.872828i	$-0.337716\pi$
$307$	5250.27	0.976055	0.488028	+	0.872828i	$0.337716\pi$
$308$	−127.688	−0.0236225
$309$	2659.46	0.489615
$310$	3819.05	0.699702
$311$	−2049.11	−0.373616	−0.186808	−	0.982396i	$-0.559814\pi$
$311$	−2049.11	−0.373616	−0.186808	+	0.982396i	$0.559814\pi$
$312$	54.8331	0.00994972
$313$	−3165.46	−0.571637	−0.285818	−	0.958284i	$-0.592265\pi$
$313$	−3165.46	−0.571637	−0.285818	+	0.958284i	$0.592265\pi$
$314$	1000.61	0.179834
$315$	−1746.69	−0.312429
$316$	529.931	0.0943385
$317$	−6289.36	−1.11434	−0.557170	−	0.830398i	$-0.688113\pi$
$317$	−6289.36	−1.11434	−0.557170	+	0.830398i	$0.688113\pi$
$318$	−11472.3	−2.02306
$319$	1809.25	0.317550
$320$	−2848.20	−0.497559
$321$	−8939.16	−1.55432
$322$	−903.200	−0.156315
$323$	−1491.39	−0.256914
$324$	−830.958	−0.142483
$325$	6.37854	0.00108867
$326$	−9237.37	−1.56936
$327$	−30.2402	−0.00511402
$328$	4606.50	0.775462
$329$	−2191.02	−0.367157
$330$	1219.48	0.203424
$331$	−5904.39	−0.980468	−0.490234	−	0.871591i	$-0.663089\pi$
$331$	−5904.39	−0.980468	−0.490234	+	0.871591i	$0.663089\pi$
$332$	145.398	0.0240354
$333$	−14890.0	−2.45035
$334$	260.364	0.0426542
$335$	1159.76	0.189147
$336$	−2881.19	−0.467803
$337$	7894.32	1.27606	0.638028	−	0.770013i	$-0.279750\pi$
$337$	7894.32	1.27606	0.638028	+	0.770013i	$0.279750\pi$
$338$	5517.40	0.887891
$339$	−9987.93	−1.60021
$340$	−664.386	−0.105975
$341$	3345.50	0.531287
$342$	−2430.93	−0.384355
$343$	−4381.60	−0.689750
$344$	9402.06	1.47362
$345$	−2315.18	−0.361289
$346$	−10936.4	−1.69927
$347$	4618.16	0.714455	0.357228	−	0.934017i	$-0.383722\pi$
$347$	4618.16	0.714455	0.357228	+	0.934017i	$0.383722\pi$
$348$	−2458.17	−0.378654
$349$	−5950.40	−0.912658	−0.456329	−	0.889811i	$-0.650836\pi$
$349$	−5950.40	−0.912658	−0.456329	+	0.889811i	$0.650836\pi$
$350$	−430.530	−0.0657509
$351$	53.9372	0.00820214
$352$	−827.408	−0.125287
$353$	−7459.80	−1.12477	−0.562387	−	0.826874i	$-0.690117\pi$
$353$	−7459.80	−1.12477	−0.562387	+	0.826874i	$0.690117\pi$
$354$	−1275.96	−0.191571
$355$	−1408.17	−0.210529
$356$	−1582.64	−0.235618
$357$	−4752.02	−0.704492
$358$	−4013.93	−0.592577
$359$	−682.864	−0.100391	−0.0501953	−	0.998739i	$-0.515984\pi$
$359$	−682.864	−0.100391	−0.0501953	+	0.998739i	$0.515984\pi$
$360$	−6200.67	−0.907789
$361$	361.000	0.0526316
$362$	4513.33	0.655291
$363$	1068.27	0.154461
$364$	−2.96169	−0.000426469 0
$365$	−1446.83	−0.207482
$366$	10352.6	1.47852
$367$	−5226.16	−0.743333	−0.371667	−	0.928366i	$-0.621214\pi$
$367$	−5226.16	−0.743333	−0.371667	+	0.928366i	$0.621214\pi$
$368$	−2496.04	−0.353574
$369$	9640.60	1.36008
$370$	−3670.13	−0.515678
$371$	3548.01	0.496505
$372$	−4545.42	−0.633519
$373$	−9247.95	−1.28376	−0.641878	−	0.766807i	$-0.721844\pi$
$373$	−9247.95	−1.28376	−0.641878	+	0.766807i	$0.721844\pi$
$374$	2168.44	0.299806
$375$	−1103.58	−0.151970
$376$	−7777.99	−1.06681
$377$	41.9649	0.00573290
$378$	−3640.58	−0.495374
$379$	−1915.59	−0.259624	−0.129812	−	0.991539i	$-0.541437\pi$
$379$	−1915.59	−0.259624	−0.129812	+	0.991539i	$0.541437\pi$
$380$	160.818	0.0217100
$381$	16572.7	2.22847
$382$	1918.58	0.256971
$383$	−3640.37	−0.485677	−0.242838	−	0.970067i	$-0.578078\pi$
$383$	−3640.37	−0.485677	−0.242838	+	0.970067i	$0.578078\pi$
$384$	−7317.58	−0.972458
$385$	−377.145	−0.0499250
$386$	−3775.56	−0.497853
$387$	19676.9	2.58458
$388$	−2690.85	−0.352081
$389$	−8694.50	−1.13324	−0.566618	−	0.823981i	$-0.691748\pi$
$389$	−8694.50	−1.13324	−0.566618	+	0.823981i	$0.691748\pi$
$390$	28.2854	0.00367253
$391$	−4116.78	−0.532467
$392$	−7204.91	−0.928324
$393$	13331.4	1.71115
$394$	−10772.1	−1.37739
$395$	1565.23	0.199380
$396$	−948.649	−0.120382
$397$	−5869.10	−0.741969	−0.370984	−	0.928639i	$-0.620980\pi$
$397$	−5869.10	−0.741969	−0.370984	+	0.928639i	$0.620980\pi$
$398$	−3661.85	−0.461186
$399$	1150.25	0.144323
$400$	−1189.79	−0.148724
$401$	1835.29	0.228554	0.114277	−	0.993449i	$-0.463545\pi$
$401$	1835.29	0.228554	0.114277	+	0.993449i	$0.463545\pi$
$402$	5142.91	0.638072
$403$	77.5978	0.00959162
$404$	−1975.75	−0.243310
$405$	−2454.35	−0.301130
$406$	−2832.49	−0.346242
$407$	−3215.04	−0.391557
$408$	−16869.4	−2.04696
$409$	5106.08	0.617309	0.308654	−	0.951174i	$-0.400121\pi$
$409$	5106.08	0.617309	0.308654	+	0.951174i	$0.400121\pi$
$410$	2376.24	0.286230
$411$	−7280.80	−0.873809
$412$	−509.931	−0.0609769
$413$	394.612	0.0470160
$414$	−6710.24	−0.796595
$415$	429.453	0.0507976
$416$	−19.1915	−0.00226187
$417$	13573.6	1.59400
$418$	−524.884	−0.0614185
$419$	1689.61	0.197000	0.0985001	−	0.995137i	$-0.468596\pi$
$419$	1689.61	0.197000	0.0985001	+	0.995137i	$0.468596\pi$
$420$	512.415	0.0595317
$421$	11302.8	1.30846	0.654231	−	0.756295i	$-0.272992\pi$
$421$	11302.8	1.30846	0.654231	+	0.756295i	$0.272992\pi$
$422$	−4111.25	−0.474247
$423$	−16278.0	−1.87107
$424$	12595.2	1.44264
$425$	−1962.36	−0.223972
$426$	−6244.48	−0.710202
$427$	−3201.73	−0.362863
$428$	1714.02	0.193575
$429$	24.7781	0.00278857
$430$	4850.01	0.543926
$431$	−789.519	−0.0882361	−0.0441181	−	0.999026i	$-0.514048\pi$
$431$	−789.519	−0.0882361	−0.0441181	+	0.999026i	$0.514048\pi$
$432$	−10060.9	−1.12050
$433$	−5468.67	−0.606945	−0.303473	−	0.952840i	$-0.598146\pi$
$433$	−5468.67	−0.606945	−0.303473	+	0.952840i	$0.598146\pi$
$434$	−5237.59	−0.579292
$435$	−7260.54	−0.800267
$436$	5.79832	0.000636902 0
$437$	996.492	0.109082
$438$	−6415.94	−0.699921
$439$	9607.91	1.04456	0.522278	−	0.852775i	$-0.325082\pi$
$439$	9607.91	1.04456	0.522278	+	0.852775i	$0.325082\pi$
$440$	−1338.85	−0.145061
$441$	−15078.6	−1.62818
$442$	50.2964	0.00541257
$443$	−723.039	−0.0775455	−0.0387727	−	0.999248i	$-0.512345\pi$
$443$	−723.039	−0.0775455	−0.0387727	+	0.999248i	$0.512345\pi$
$444$	4368.17	0.466902
$445$	−4674.56	−0.497967
$446$	8498.26	0.902252
$447$	−18077.7	−1.91285
$448$	3906.12	0.411935
$449$	6602.21	0.693936	0.346968	−	0.937877i	$-0.387211\pi$
$449$	6602.21	0.693936	0.346968	+	0.937877i	$0.387211\pi$
$450$	−3198.59	−0.335073
$451$	2081.60	0.217336
$452$	1915.11	0.199290
$453$	16942.0	1.75719
$454$	770.820	0.0796836
$455$	−8.74777	−0.000901323 0
$456$	4083.34	0.419342
$457$	−9773.47	−1.00040	−0.500201	−	0.865909i	$-0.666741\pi$
$457$	−9773.47	−1.00040	−0.500201	+	0.865909i	$0.666741\pi$
$458$	6693.71	0.682918
$459$	−16593.8	−1.68743
$460$	443.917	0.0449951
$461$	8864.38	0.895565	0.447782	−	0.894143i	$-0.352214\pi$
$461$	8864.38	0.895565	0.447782	+	0.894143i	$0.352214\pi$
$462$	−1672.44	−0.168418
$463$	−13925.1	−1.39774	−0.698869	−	0.715250i	$-0.746313\pi$
$463$	−13925.1	−1.39774	−0.698869	+	0.715250i	$0.746313\pi$
$464$	−7827.74	−0.783176
$465$	−13425.6	−1.33891
$466$	−7341.88	−0.729842
$467$	−17262.5	−1.71052	−0.855261	−	0.518198i	$-0.826603\pi$
$467$	−17262.5	−1.71052	−0.855261	+	0.518198i	$0.826603\pi$
$468$	−22.0036	−0.00217333
$469$	−1590.54	−0.156597
$470$	−4012.24	−0.393768
$471$	−3517.56	−0.344121
$472$	1400.85	0.136609
$473$	4248.62	0.413006
$474$	6940.95	0.672591
$475$	475.000	0.0458831
$476$	911.164	0.0877376
$477$	26359.6	2.53024
$478$	−10410.4	−0.996155
$479$	13352.8	1.27370	0.636851	−	0.770987i	$-0.280237\pi$
$479$	13352.8	1.27370	0.636851	+	0.770987i	$0.280237\pi$
$480$	3320.40	0.315739
$481$	−74.5719	−0.00706899
$482$	6602.44	0.623927
$483$	3175.12	0.299116
$484$	−204.832	−0.0192367
$485$	−7947.81	−0.744106
$486$	3450.94	0.322094
$487$	−1845.65	−0.171734	−0.0858671	−	0.996307i	$-0.527366\pi$
$487$	−1845.65	−0.171734	−0.0858671	+	0.996307i	$0.527366\pi$
$488$	−11366.0	−1.05433
$489$	32473.2	3.00304
$490$	−3716.62	−0.342653
$491$	3389.00	0.311493	0.155747	−	0.987797i	$-0.450222\pi$
$491$	3389.00	0.311493	0.155747	+	0.987797i	$0.450222\pi$
$492$	−2828.20	−0.259156
$493$	−12910.5	−1.17943
$494$	−12.1745	−0.00110882
$495$	−2801.97	−0.254423
$496$	−14474.4	−1.31032
$497$	1931.22	0.174300
$498$	1904.39	0.171361
$499$	13110.5	1.17616	0.588081	−	0.808802i	$-0.299884\pi$
$499$	13110.5	1.17616	0.588081	+	0.808802i	$0.299884\pi$
$500$	211.603	0.0189264
$501$	−915.288	−0.0816209
$502$	3706.71	0.329559
$503$	−18456.4	−1.63605	−0.818023	−	0.575185i	$-0.804930\pi$
$503$	−18456.4	−1.63605	−0.818023	+	0.575185i	$0.804930\pi$
$504$	8503.83	0.751569
$505$	−5835.66	−0.514225
$506$	−1448.87	−0.127293
$507$	−19395.9	−1.69902
$508$	−3177.70	−0.277534
$509$	3442.99	0.299819	0.149910	−	0.988700i	$-0.452102\pi$
$509$	3442.99	0.299819	0.149910	+	0.988700i	$0.452102\pi$
$510$	−8702.00	−0.755551
$511$	1984.24	0.171776
$512$	12847.9	1.10899
$513$	4016.62	0.345688
$514$	−17845.5	−1.53138
$515$	−1506.15	−0.128872
$516$	−5772.46	−0.492478
$517$	−3514.73	−0.298990
$518$	5033.35	0.426936
$519$	38446.1	3.25163
$520$	−31.0541	−0.00261887
$521$	2094.37	0.176115	0.0880574	−	0.996115i	$-0.471934\pi$
$521$	2094.37	0.176115	0.0880574	+	0.996115i	$0.471934\pi$
$522$	−21043.8	−1.76448
$523$	−1534.96	−0.128335	−0.0641676	−	0.997939i	$-0.520439\pi$
$523$	−1534.96	−0.128335	−0.0641676	+	0.997939i	$0.520439\pi$
$524$	−2556.20	−0.213107
$525$	1513.49	0.125818
$526$	12030.8	0.997276
$527$	−23873.0	−1.97329
$528$	−4621.87	−0.380949
$529$	−9416.32	−0.773923
$530$	6497.19	0.532491
$531$	2931.74	0.239598
$532$	−220.553	−0.0179740
$533$	48.2820	0.00392368
$534$	−20729.2	−1.67985
$535$	5062.59	0.409112
$536$	−5646.32	−0.455007
$537$	14110.6	1.13393
$538$	−14568.3	−1.16744
$539$	−3255.77	−0.260178
$540$	1789.32	0.142593
$541$	−21329.1	−1.69503	−0.847514	−	0.530773i	$-0.821902\pi$
$541$	−21329.1	−1.69503	−0.847514	+	0.530773i	$0.821902\pi$
$542$	7907.87	0.626702
$543$	−15866.2	−1.25393
$544$	5904.25	0.465336
$545$	17.1262	0.00134606
$546$	−38.7917	−0.00304053
$547$	−16896.3	−1.32072	−0.660358	−	0.750951i	$-0.729596\pi$
$547$	−16896.3	−1.32072	−0.660358	+	0.750951i	$0.729596\pi$
$548$	1396.04	0.108825
$549$	−23786.9	−1.84918
$550$	−690.637	−0.0535434
$551$	3125.06	0.241619
$552$	11271.5	0.869107
$553$	−2146.61	−0.165069
$554$	−9313.87	−0.714275
$555$	12902.0	0.986775
$556$	−2602.63	−0.198518
$557$	7562.01	0.575247	0.287624	−	0.957743i	$-0.407135\pi$
$557$	7562.01	0.575247	0.287624	+	0.957743i	$0.407135\pi$
$558$	−38912.3	−2.95213
$559$	98.5454	0.00745622
$560$	1631.73	0.123131
$561$	−7622.98	−0.573694
$562$	−20737.1	−1.55648
$563$	11902.3	0.890984	0.445492	−	0.895286i	$-0.353029\pi$
$563$	11902.3	0.890984	0.445492	+	0.895286i	$0.353029\pi$
$564$	4775.35	0.356523
$565$	5656.55	0.421191
$566$	18115.1	1.34529
$567$	3365.99	0.249309
$568$	6855.72	0.506443
$569$	4798.49	0.353538	0.176769	−	0.984252i	$-0.443435\pi$
$569$	4798.49	0.353538	0.176769	+	0.984252i	$0.443435\pi$
$570$	2106.37	0.154783
$571$	−21631.9	−1.58541	−0.792703	−	0.609608i	$-0.791327\pi$
$571$	−21631.9	−1.58541	−0.792703	+	0.609608i	$0.791327\pi$
$572$	−4.75101	−0.000347290 0
$573$	−6744.60	−0.491727
$574$	−3258.87	−0.236973
$575$	1311.17	0.0950951
$576$	29020.2	2.09926
$577$	−20953.5	−1.51179	−0.755896	−	0.654692i	$-0.772798\pi$
$577$	−20953.5	−1.51179	−0.755896	+	0.654692i	$0.772798\pi$
$578$	−3135.12	−0.225612
$579$	13272.7	0.952666
$580$	1392.15	0.0996656
$581$	−588.968	−0.0420560
$582$	−35244.3	−2.51018
$583$	5691.56	0.404323
$584$	7043.96	0.499111
$585$	−64.9908	−0.00459323
$586$	10106.4	0.712445
$587$	−13607.6	−0.956806	−0.478403	−	0.878140i	$-0.658784\pi$
$587$	−13607.6	−0.956806	−0.478403	+	0.878140i	$0.658784\pi$
$588$	4423.51	0.310242
$589$	5778.59	0.404249
$590$	722.623	0.0504236
$591$	37868.5	2.63570
$592$	13909.9	0.965701
$593$	9539.02	0.660574	0.330287	−	0.943881i	$-0.392854\pi$
$593$	9539.02	0.660574	0.330287	+	0.943881i	$0.392854\pi$
$594$	−5840.06	−0.403401
$595$	2691.25	0.185429
$596$	3466.26	0.238227
$597$	12872.9	0.882501
$598$	−33.6061	−0.00229809
$599$	−22574.3	−1.53983	−0.769917	−	0.638144i	$-0.779702\pi$
$599$	−22574.3	−1.53983	−0.769917	+	0.638144i	$0.779702\pi$
$600$	5372.81	0.365574
$601$	3863.81	0.262243	0.131121	−	0.991366i	$-0.458142\pi$
$601$	3863.81	0.262243	0.131121	+	0.991366i	$0.458142\pi$
$602$	−6651.49	−0.450323
$603$	−11816.8	−0.798036
$604$	−3248.51	−0.218841
$605$	−605.000	−0.0406558
$606$	−25878.0	−1.73469
$607$	13757.8	0.919953	0.459976	−	0.887931i	$-0.347858\pi$
$607$	13757.8	0.919953	0.459976	+	0.887931i	$0.347858\pi$
$608$	−1429.16	−0.0953290
$609$	9957.38	0.662551
$610$	−5863.07	−0.389162
$611$	−81.5231	−0.00539783
$612$	6769.41	0.447120
$613$	−27669.1	−1.82307	−0.911537	−	0.411218i	$-0.865104\pi$
$613$	−27669.1	−1.82307	−0.911537	+	0.411218i	$0.865104\pi$
$614$	−13185.6	−0.866656
$615$	−8353.48	−0.547715
$616$	1836.14	0.120098
$617$	14693.5	0.958735	0.479368	−	0.877614i	$-0.340866\pi$
$617$	14693.5	0.958735	0.479368	+	0.877614i	$0.340866\pi$
$618$	−6678.98	−0.434738
$619$	−26485.6	−1.71978	−0.859891	−	0.510477i	$-0.829469\pi$
$619$	−26485.6	−1.71978	−0.859891	+	0.510477i	$0.829469\pi$
$620$	2574.25	0.166749
$621$	11087.3	0.716456
$622$	5146.16	0.331740
$623$	6410.87	0.412273
$624$	−107.203	−0.00687749
$625$	625.000	0.0400000
$626$	7949.76	0.507566
$627$	1845.19	0.117527
$628$	674.466	0.0428569
$629$	22942.0	1.45431
$630$	4386.66	0.277411
$631$	−22444.2	−1.41599	−0.707996	−	0.706217i	$-0.750400\pi$
$631$	−22444.2	−1.41599	−0.707996	+	0.706217i	$0.750400\pi$
$632$	−7620.36	−0.479623
$633$	14452.7	0.907495
$634$	15795.2	0.989442
$635$	−9385.77	−0.586556
$636$	−7732.94	−0.482124
$637$	−75.5165	−0.00469713
$638$	−4543.76	−0.281958
$639$	14347.8	0.888248
$640$	4144.23	0.255961
$641$	12625.0	0.777939	0.388969	−	0.921251i	$-0.372831\pi$
$641$	12625.0	0.777939	0.388969	+	0.921251i	$0.372831\pi$
$642$	22449.9	1.38010
$643$	−1206.65	−0.0740055	−0.0370028	−	0.999315i	$-0.511781\pi$
$643$	−1206.65	−0.0740055	−0.0370028	+	0.999315i	$0.511781\pi$
$644$	−608.805	−0.0372520
$645$	−17049.8	−1.04083
$646$	3745.49	0.228118
$647$	−26185.1	−1.59110	−0.795551	−	0.605886i	$-0.792819\pi$
$647$	−26185.1	−1.59110	−0.795551	+	0.605886i	$0.792819\pi$
$648$	11949.1	0.724390
$649$	633.019	0.0382869
$650$	−16.0191	−0.000966649 0
$651$	18412.3	1.10850
$652$	−6226.49	−0.374000
$653$	23849.4	1.42925	0.714623	−	0.699510i	$-0.246598\pi$
$653$	23849.4	1.42925	0.714623	+	0.699510i	$0.246598\pi$
$654$	75.9454	0.00454083
$655$	−7550.11	−0.450393
$656$	−9006.07	−0.536018
$657$	14741.8	0.875390
$658$	5502.54	0.326005
$659$	7404.36	0.437683	0.218841	−	0.975760i	$-0.429772\pi$
$659$	7404.36	0.437683	0.218841	+	0.975760i	$0.429772\pi$
$660$	821.994	0.0484789
$661$	−10396.6	−0.611773	−0.305887	−	0.952068i	$-0.598953\pi$
$661$	−10396.6	−0.611773	−0.305887	+	0.952068i	$0.598953\pi$
$662$	14828.3	0.870574
$663$	−176.813	−0.0103572
$664$	−2090.81	−0.122197
$665$	−651.433	−0.0379872
$666$	37394.9	2.17571
$667$	8626.31	0.500768
$668$	175.500	0.0101651
$669$	−29874.9	−1.72650
$670$	−2912.63	−0.167947
$671$	−5136.07	−0.295493
$672$	−4553.73	−0.261404
$673$	11538.6	0.660892	0.330446	−	0.943825i	$-0.392801\pi$
$673$	11538.6	0.660892	0.330446	+	0.943825i	$0.392801\pi$
$674$	−19825.9	−1.13303
$675$	5285.02	0.301364
$676$	3719.03	0.211597
$677$	−30709.2	−1.74336	−0.871678	−	0.490080i	$-0.836968\pi$
$677$	−30709.2	−1.74336	−0.871678	+	0.490080i	$0.836968\pi$
$678$	25083.8	1.42085
$679$	10899.9	0.616055
$680$	9553.79	0.538781
$681$	−2709.75	−0.152478
$682$	−8401.92	−0.471739
$683$	−669.906	−0.0375304	−0.0187652	−	0.999824i	$-0.505973\pi$
$683$	−669.906	−0.0375304	−0.0187652	+	0.999824i	$0.505973\pi$
$684$	−1638.58	−0.0915972
$685$	4123.40	0.229996
$686$	11004.0	0.612441
$687$	−23531.1	−1.30680
$688$	−18381.7	−1.01860
$689$	132.014	0.00729946
$690$	5814.35	0.320795
$691$	3293.80	0.181334	0.0906671	−	0.995881i	$-0.471100\pi$
$691$	3293.80	0.181334	0.0906671	+	0.995881i	$0.471100\pi$
$692$	−7371.76	−0.404960
$693$	3842.73	0.210639
$694$	−11598.1	−0.634377
$695$	−7687.23	−0.419558
$696$	35348.1	1.92510
$697$	−14853.9	−0.807221
$698$	14943.9	0.810365
$699$	25809.8	1.39659
$700$	−290.201	−0.0156694
$701$	21296.5	1.14744	0.573722	−	0.819050i	$-0.305499\pi$
$701$	21296.5	1.14744	0.573722	+	0.819050i	$0.305499\pi$
$702$	−135.458	−0.00728282
$703$	−5553.25	−0.297930
$704$	6266.03	0.335454
$705$	14104.7	0.753494
$706$	18734.6	0.998706
$707$	8003.25	0.425733
$708$	−860.063	−0.0456542
$709$	−28168.8	−1.49210	−0.746051	−	0.665888i	$-0.768053\pi$
$709$	−28168.8	−1.49210	−0.746051	+	0.665888i	$0.768053\pi$
$710$	3536.49	0.186933
$711$	−15948.1	−0.841209
$712$	22758.2	1.19789
$713$	15951.0	0.837826
$714$	11934.3	0.625530
$715$	−14.0328	−0.000733981 0
$716$	−2705.60	−0.141219
$717$	36597.0	1.90619
$718$	1714.95	0.0891385
$719$	1160.89	0.0602139	0.0301070	−	0.999547i	$-0.490415\pi$
$719$	1160.89	0.0602139	0.0301070	+	0.999547i	$0.490415\pi$
$720$	12122.8	0.627485
$721$	2065.59	0.106695
$722$	−906.619	−0.0467325
$723$	−23210.3	−1.19391
$724$	3042.23	0.156165
$725$	4111.92	0.210639
$726$	−2682.85	−0.137149
$727$	30699.0	1.56611	0.783056	−	0.621951i	$-0.213660\pi$
$727$	30699.0	1.56611	0.783056	+	0.621951i	$0.213660\pi$
$728$	42.5888	0.00216819
$729$	−25385.0	−1.28969
$730$	3633.59	0.184226
$731$	−30317.5	−1.53397
$732$	6978.21	0.352352
$733$	14213.1	0.716198	0.358099	−	0.933684i	$-0.383425\pi$
$733$	14213.1	0.716198	0.358099	+	0.933684i	$0.383425\pi$
$734$	13125.0	0.660018
$735$	13065.5	0.655683
$736$	−3945.00	−0.197574
$737$	−2551.47	−0.127523
$738$	−24211.5	−1.20764
$739$	−11786.8	−0.586718	−0.293359	−	0.956002i	$-0.594773\pi$
$739$	−11786.8	−0.586718	−0.293359	+	0.956002i	$0.594773\pi$
$740$	−2473.86	−0.122893
$741$	42.7985	0.00212178
$742$	−8910.50	−0.440855
$743$	12102.1	0.597553	0.298777	−	0.954323i	$-0.403421\pi$
$743$	12102.1	0.597553	0.298777	+	0.954323i	$0.403421\pi$
$744$	65362.7	3.22085
$745$	10238.1	0.503482
$746$	23225.4	1.13987
$747$	−4375.69	−0.214321
$748$	1461.65	0.0714481
$749$	−6943.03	−0.338709
$750$	2771.54	0.134936
$751$	−27673.7	−1.34465	−0.672323	−	0.740258i	$-0.734703\pi$
$751$	−27673.7	−1.34465	−0.672323	+	0.740258i	$0.734703\pi$
$752$	15206.6	0.737402
$753$	−13030.6	−0.630627
$754$	−105.391	−0.00509034
$755$	−9594.93	−0.462510
$756$	−2453.95	−0.118055
$757$	12443.1	0.597427	0.298713	−	0.954343i	$-0.403443\pi$
$757$	12443.1	0.597427	0.298713	+	0.954343i	$0.403443\pi$
$758$	4810.84	0.230525
$759$	5093.39	0.243581
$760$	−2312.55	−0.110375
$761$	−7049.72	−0.335811	−0.167906	−	0.985803i	$-0.553700\pi$
$761$	−7049.72	−0.335811	−0.167906	+	0.985803i	$0.553700\pi$
$762$	−41620.9	−1.97870
$763$	−23.4875	−0.00111442
$764$	1293.23	0.0612399
$765$	19994.4	0.944967
$766$	9142.46	0.431241
$767$	14.6827	0.000691214 0
$768$	−21855.7	−1.02689
$769$	−7023.94	−0.329375	−0.164688	−	0.986346i	$-0.552662\pi$
$769$	−7023.94	−0.329375	−0.164688	+	0.986346i	$0.552662\pi$
$770$	947.167	0.0443292
$771$	62734.3	2.93038
$772$	−2544.93	−0.118645
$773$	42112.2	1.95947	0.979735	−	0.200300i	$-0.0641918\pi$
$773$	42112.2	1.95947	0.979735	+	0.200300i	$0.0641918\pi$
$774$	−49416.6	−2.29489
$775$	7603.41	0.352416
$776$	38694.2	1.79000
$777$	−17694.3	−0.816963
$778$	21835.5	1.00622
$779$	3595.48	0.165368
$780$	19.0659	0.000875216 0
$781$	3097.97	0.141939
$782$	10338.9	0.472787
$783$	34770.6	1.58697
$784$	14086.2	0.641680
$785$	1992.13	0.0905761
$786$	−33480.7	−1.51936
$787$	8623.17	0.390575	0.195288	−	0.980746i	$-0.437436\pi$
$787$	8623.17	0.390575	0.195288	+	0.980746i	$0.437436\pi$
$788$	−7260.99	−0.328251
$789$	−42293.2	−1.90834
$790$	−3930.93	−0.177033
$791$	−7757.61	−0.348709
$792$	13641.5	0.612031
$793$	−119.129	−0.00533469
$794$	14739.7	0.658807
$795$	−22840.3	−1.01895
$796$	−2468.28	−0.109907
$797$	−7845.94	−0.348704	−0.174352	−	0.984683i	$-0.555783\pi$
$797$	−7845.94	−0.348704	−0.174352	+	0.984683i	$0.555783\pi$
$798$	−2888.76	−0.128147
$799$	25080.6	1.11050
$800$	−1880.47	−0.0831059
$801$	47629.0	2.10098
$802$	−4609.17	−0.202937
$803$	3183.04	0.139884
$804$	3466.60	0.152062
$805$	−1798.19	−0.0787304
$806$	−194.880	−0.00851656
$807$	51213.5	2.23395
$808$	28411.1	1.23700
$809$	38.9755	0.00169383	0.000846914	−	1.00000i	$-0.499730\pi$
$809$	38.9755	0.00169383	0.000846914		1.00000i	$0.499730\pi$
$810$	6163.88	0.267379
$811$	−8405.09	−0.363924	−0.181962	−	0.983306i	$-0.558245\pi$
$811$	−8405.09	−0.363924	−0.181962	+	0.983306i	$0.558245\pi$
$812$	−1909.25	−0.0825143
$813$	−27799.4	−1.19922
$814$	8074.28	0.347670
$815$	−18390.8	−0.790432
$816$	32981.0	1.41491
$817$	7338.52	0.314250
$818$	−12823.4	−0.548119
$819$	89.1309	0.00380279
$820$	1601.72	0.0682127
$821$	21310.1	0.905882	0.452941	−	0.891540i	$-0.350375\pi$
$821$	21310.1	0.905882	0.452941	+	0.891540i	$0.350375\pi$
$822$	18285.1	0.775870
$823$	−34193.5	−1.44825	−0.724125	−	0.689669i	$-0.757756\pi$
$823$	−34193.5	−1.44825	−0.724125	+	0.689669i	$0.757756\pi$
$824$	7332.75	0.310010
$825$	2427.88	0.102458
$826$	−991.033	−0.0417463
$827$	−33266.1	−1.39876	−0.699380	−	0.714750i	$-0.746540\pi$
$827$	−33266.1	−1.39876	−0.699380	+	0.714750i	$0.746540\pi$
$828$	−4523.07	−0.189840
$829$	31146.6	1.30491	0.652453	−	0.757829i	$-0.273740\pi$
$829$	31146.6	1.30491	0.652453	+	0.757829i	$0.273740\pi$
$830$	−1078.53	−0.0451041
$831$	32742.1	1.36680
$832$	145.339	0.00605614
$833$	23232.7	0.966343
$834$	−34088.7	−1.41534
$835$	518.363	0.0214835
$836$	−353.801	−0.0146369
$837$	64294.7	2.65514
$838$	−4243.31	−0.174920
$839$	−27722.3	−1.14074	−0.570370	−	0.821388i	$-0.693200\pi$
$839$	−27722.3	−1.14074	−0.570370	+	0.821388i	$0.693200\pi$
$840$	−7368.48	−0.302663
$841$	2663.66	0.109216
$842$	−28385.8	−1.16181
$843$	72899.6	2.97841
$844$	−2771.20	−0.113020
$845$	10984.7	0.447200
$846$	40880.6	1.66135
$847$	829.720	0.0336594
$848$	−24624.6	−0.997186
$849$	−63682.1	−2.57428
$850$	4928.28	0.198869
$851$	−15329.0	−0.617475
$852$	−4209.12	−0.169251
$853$	6677.22	0.268023	0.134011	−	0.990980i	$-0.457214\pi$
$853$	6677.22	0.268023	0.134011	+	0.990980i	$0.457214\pi$
$854$	8040.84	0.322192
$855$	−4839.77	−0.193587
$856$	−24647.4	−0.984147
$857$	21113.3	0.841560	0.420780	−	0.907163i	$-0.361756\pi$
$857$	21113.3	0.841560	0.420780	+	0.907163i	$0.361756\pi$
$858$	−62.2279	−0.00247602
$859$	−23769.5	−0.944127	−0.472063	−	0.881565i	$-0.656491\pi$
$859$	−23769.5	−0.944127	−0.472063	+	0.881565i	$0.656491\pi$
$860$	3269.17	0.129625
$861$	11456.3	0.453460
$862$	1982.80	0.0783464
$863$	10576.0	0.417164	0.208582	−	0.978005i	$-0.433115\pi$
$863$	10576.0	0.417164	0.208582	+	0.978005i	$0.433115\pi$
$864$	−15901.3	−0.626128
$865$	−21773.5	−0.855864
$866$	13734.1	0.538917
$867$	11021.3	0.431720
$868$	−3530.42	−0.138053
$869$	−3443.50	−0.134422
$870$	18234.2	0.710571
$871$	−59.1805	−0.00230225
$872$	−83.3793	−0.00323805
$873$	80980.1	3.13947
$874$	−2502.60	−0.0968554
$875$	−857.149	−0.0331165
$876$	−4324.69	−0.166801
$877$	43447.8	1.67290	0.836448	−	0.548047i	$-0.184628\pi$
$877$	43447.8	1.67290	0.836448	+	0.548047i	$0.184628\pi$
$878$	−24129.4	−0.927480
$879$	−35528.3	−1.36330
$880$	2617.55	0.100270
$881$	30306.8	1.15898	0.579491	−	0.814978i	$-0.303251\pi$
$881$	30306.8	1.15898	0.579491	+	0.814978i	$0.303251\pi$
$882$	37868.6	1.44569
$883$	−48086.9	−1.83268	−0.916338	−	0.400405i	$-0.868870\pi$
$883$	−48086.9	−1.83268	−0.916338	+	0.400405i	$0.868870\pi$
$884$	33.9025	0.00128989
$885$	−2540.32	−0.0964880
$886$	1815.85	0.0688539
$887$	−38942.2	−1.47413	−0.737065	−	0.675822i	$-0.763789\pi$
$887$	−38942.2	−1.47413	−0.737065	+	0.675822i	$0.763789\pi$
$888$	−62813.9	−2.37376
$889$	12872.0	0.485617
$890$	11739.7	0.442153
$891$	5399.57	0.203022
$892$	5728.29	0.215019
$893$	−6070.90	−0.227497
$894$	45400.4	1.69845
$895$	−7991.38	−0.298461
$896$	−5683.56	−0.211913
$897$	118.140	0.00439751
$898$	−16580.8	−0.616158
$899$	50023.4	1.85581
$900$	−2156.02	−0.0798526
$901$	−40614.1	−1.50172
$902$	−5227.74	−0.192976
$903$	23382.7	0.861715
$904$	−27539.1	−1.01320
$905$	8985.65	0.330048
$906$	−42548.4	−1.56024
$907$	−41102.3	−1.50472	−0.752360	−	0.658753i	$-0.771084\pi$
$907$	−41102.3	−1.50472	−0.752360	+	0.658753i	$0.771084\pi$
$908$	519.574	0.0189897
$909$	59459.5	2.16958
$910$	21.9692	0.000800300 0
$911$	−11576.2	−0.421007	−0.210504	−	0.977593i	$-0.567510\pi$
$911$	−11576.2	−0.421007	−0.210504	+	0.977593i	$0.567510\pi$
$912$	−7983.24	−0.289859
$913$	−944.797	−0.0342478
$914$	24545.2	0.888274
$915$	20611.1	0.744681
$916$	4511.92	0.162749
$917$	10354.5	0.372886
$918$	41673.7	1.49830
$919$	−14363.5	−0.515569	−0.257785	−	0.966202i	$-0.582992\pi$
$919$	−14363.5	−0.515569	−0.257785	+	0.966202i	$0.582992\pi$
$920$	−6383.49	−0.228758
$921$	46352.8	1.65839
$922$	−22262.1	−0.795187
$923$	71.8565	0.00256250
$924$	−1127.31	−0.0401363
$925$	−7306.91	−0.259729
$926$	34971.5	1.24107
$927$	15346.2	0.543726
$928$	−12371.8	−0.437633
$929$	−53124.3	−1.87616	−0.938080	−	0.346419i	$-0.887398\pi$
$929$	−53124.3	−1.87616	−0.938080	+	0.346419i	$0.887398\pi$
$930$	33717.0	1.18884
$931$	−5623.60	−0.197966
$932$	−4948.83	−0.173932
$933$	−18090.9	−0.634801
$934$	43353.2	1.51880
$935$	4317.19	0.151002
$936$	316.410	0.0110493
$937$	−5274.07	−0.183881	−0.0919404	−	0.995765i	$-0.529307\pi$
$937$	−5274.07	−0.183881	−0.0919404	+	0.995765i	$0.529307\pi$
$938$	3994.49	0.139046
$939$	−27946.7	−0.971253
$940$	−2704.47	−0.0938404
$941$	−38045.0	−1.31799	−0.658997	−	0.752146i	$-0.729019\pi$
$941$	−38045.0	−1.31799	−0.658997	+	0.752146i	$0.729019\pi$
$942$	8834.04	0.305551
$943$	9924.84	0.342733
$944$	−2738.77	−0.0944274
$945$	−7248.08	−0.249503
$946$	−10670.0	−0.366715
$947$	6252.32	0.214544	0.107272	−	0.994230i	$-0.465788\pi$
$947$	6252.32	0.214544	0.107272	+	0.994230i	$0.465788\pi$
$948$	4678.57	0.160288
$949$	73.8295	0.00252540
$950$	−1192.92	−0.0407404
$951$	−55526.5	−1.89334
$952$	−13102.4	−0.446063
$953$	5129.59	0.174359	0.0871794	−	0.996193i	$-0.472215\pi$
$953$	5129.59	0.174359	0.0871794	+	0.996193i	$0.472215\pi$
$954$	−66199.8	−2.24664
$955$	3819.73	0.129428
$956$	−7017.20	−0.237398
$957$	15973.2	0.539540
$958$	−33534.3	−1.13094
$959$	−5654.99	−0.190416
$960$	−25145.7	−0.845390
$961$	62707.9	2.10493
$962$	187.280	0.00627668
$963$	−51582.7	−1.72609
$964$	4450.40	0.148691
$965$	−7516.83	−0.250752
$966$	−7974.02	−0.265590
$967$	−44170.7	−1.46891	−0.734454	−	0.678659i	$-0.762561\pi$
$967$	−44170.7	−1.46891	−0.734454	+	0.678659i	$0.762561\pi$
$968$	2945.46	0.0978003
$969$	−13167.0	−0.436516
$970$	19960.2	0.660705
$971$	26452.9	0.874268	0.437134	−	0.899396i	$-0.355994\pi$
$971$	26452.9	0.874268	0.437134	+	0.899396i	$0.355994\pi$
$972$	2326.12	0.0767596
$973$	10542.6	0.347357
$974$	4635.19	0.152486
$975$	56.3138	0.00184973
$976$	22221.3	0.728777
$977$	−33826.0	−1.10767	−0.553833	−	0.832628i	$-0.686835\pi$
$977$	−33826.0	−1.10767	−0.553833	+	0.832628i	$0.686835\pi$
$978$	−81553.4	−2.66645
$979$	10284.0	0.335729
$980$	−2505.20	−0.0816590
$981$	−174.498	−0.00567920
$982$	−8511.15	−0.276580
$983$	40057.7	1.29974	0.649869	−	0.760046i	$-0.274824\pi$
$983$	40057.7	1.29974	0.649869	+	0.760046i	$0.274824\pi$
$984$	40669.2	1.31757
$985$	−21446.4	−0.693744
$986$	32423.5	1.04724
$987$	−19343.7	−0.623827
$988$	−8.20629	−0.000264248 0
$989$	20257.0	0.651299
$990$	7036.89	0.225906
$991$	15648.7	0.501613	0.250806	−	0.968037i	$-0.419304\pi$
$991$	15648.7	0.501613	0.250806	+	0.968037i	$0.419304\pi$
$992$	−22876.8	−0.732196
$993$	−52127.8	−1.66589
$994$	−4850.08	−0.154764
$995$	−7290.43	−0.232283
$996$	1283.67	0.0408378
$997$	−39922.7	−1.26817	−0.634084	−	0.773264i	$-0.718623\pi$
$997$	−39922.7	−1.26817	−0.634084	+	0.773264i	$0.718623\pi$
$998$	−32925.7	−1.04433
$999$	−61787.5	−1.95683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.i.1.9	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.i.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-2.51141 q^{2} +8.82864 q^{3} -1.69283 q^{4} -5.00000 q^{5} -22.1723 q^{6} +6.85719 q^{7} +24.3426 q^{8} +50.9449 q^{9} +O(q^{10})\)	\(q-2.51141 q^{2} +8.82864 q^{3} -1.69283 q^{4} -5.00000 q^{5} -22.1723 q^{6} +6.85719 q^{7} +24.3426 q^{8} +50.9449 q^{9} +12.5570 q^{10} +11.0000 q^{11} -14.9453 q^{12} +0.255142 q^{13} -17.2212 q^{14} -44.1432 q^{15} -47.5917 q^{16} -78.4943 q^{17} -127.943 q^{18} +19.0000 q^{19} +8.46413 q^{20} +60.5397 q^{21} -27.6255 q^{22} +52.4469 q^{23} +214.912 q^{24} +25.0000 q^{25} -0.640765 q^{26} +211.401 q^{27} -11.6080 q^{28} +164.477 q^{29} +110.862 q^{30} +304.136 q^{31} -75.2189 q^{32} +97.1150 q^{33} +197.131 q^{34} -34.2860 q^{35} -86.2408 q^{36} -292.276 q^{37} -47.7168 q^{38} +2.25255 q^{39} -121.713 q^{40} +189.236 q^{41} -152.040 q^{42} +386.238 q^{43} -18.6211 q^{44} -254.724 q^{45} -131.716 q^{46} -319.521 q^{47} -420.170 q^{48} -295.979 q^{49} -62.7852 q^{50} -692.998 q^{51} -0.431910 q^{52} +517.414 q^{53} -530.914 q^{54} -55.0000 q^{55} +166.922 q^{56} +167.744 q^{57} -413.069 q^{58} +57.5472 q^{59} +74.7267 q^{60} -466.915 q^{61} -763.811 q^{62} +349.339 q^{63} +569.639 q^{64} -1.27571 q^{65} -243.896 q^{66} -231.952 q^{67} +132.877 q^{68} +463.035 q^{69} +86.1060 q^{70} +281.634 q^{71} +1240.13 q^{72} +289.367 q^{73} +734.026 q^{74} +220.716 q^{75} -32.1637 q^{76} +75.4291 q^{77} -5.65708 q^{78} -313.046 q^{79} +237.959 q^{80} +490.870 q^{81} -475.249 q^{82} -85.8906 q^{83} -102.483 q^{84} +392.471 q^{85} -970.002 q^{86} +1452.11 q^{87} +267.769 q^{88} +934.911 q^{89} +639.717 q^{90} +1.74955 q^{91} -88.7835 q^{92} +2685.11 q^{93} +802.448 q^{94} -95.0000 q^{95} -664.080 q^{96} +1589.56 q^{97} +743.324 q^{98} +560.394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) 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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