LMFDB - Embedded newform 1045.4.a.i.1.13

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			1.13
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)

$f(q)$	$=$	$q+0.109799 q^{2} +6.64848 q^{3} -7.98794 q^{4} -5.00000 q^{5} +0.729999 q^{6} +28.5341 q^{7} -1.75547 q^{8} +17.2022 q^{9} +O(q^{10})$	\(q+0.109799 q^{2} +6.64848 q^{3} -7.98794 q^{4} -5.00000 q^{5} +0.729999 q^{6} +28.5341 q^{7} -1.75547 q^{8} +17.2022 q^{9} -0.548997 q^{10} +11.0000 q^{11} -53.1077 q^{12} +18.1430 q^{13} +3.13303 q^{14} -33.2424 q^{15} +63.7108 q^{16} +0.876794 q^{17} +1.88880 q^{18} +19.0000 q^{19} +39.9397 q^{20} +189.709 q^{21} +1.20779 q^{22} -88.8409 q^{23} -11.6712 q^{24} +25.0000 q^{25} +1.99209 q^{26} -65.1402 q^{27} -227.929 q^{28} +43.2463 q^{29} -3.64999 q^{30} +75.3898 q^{31} +21.0391 q^{32} +73.1332 q^{33} +0.0962715 q^{34} -142.671 q^{35} -137.411 q^{36} +178.888 q^{37} +2.08619 q^{38} +120.623 q^{39} +8.77734 q^{40} +367.116 q^{41} +20.8299 q^{42} -223.021 q^{43} -87.8674 q^{44} -86.0112 q^{45} -9.75468 q^{46} +366.924 q^{47} +423.580 q^{48} +471.197 q^{49} +2.74499 q^{50} +5.82935 q^{51} -144.925 q^{52} -543.307 q^{53} -7.15235 q^{54} -55.0000 q^{55} -50.0907 q^{56} +126.321 q^{57} +4.74842 q^{58} +331.736 q^{59} +265.538 q^{60} -202.336 q^{61} +8.27776 q^{62} +490.851 q^{63} -507.376 q^{64} -90.7151 q^{65} +8.02999 q^{66} +31.2297 q^{67} -7.00379 q^{68} -590.657 q^{69} -15.6652 q^{70} +566.971 q^{71} -30.1980 q^{72} -1219.75 q^{73} +19.6418 q^{74} +166.212 q^{75} -151.771 q^{76} +313.876 q^{77} +13.2444 q^{78} +1198.51 q^{79} -318.554 q^{80} -897.543 q^{81} +40.3092 q^{82} +1124.31 q^{83} -1515.38 q^{84} -4.38397 q^{85} -24.4876 q^{86} +287.522 q^{87} -19.3101 q^{88} +1324.18 q^{89} -9.44398 q^{90} +517.695 q^{91} +709.656 q^{92} +501.228 q^{93} +40.2880 q^{94} -95.0000 q^{95} +139.878 q^{96} +216.768 q^{97} +51.7372 q^{98} +189.225 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.109799	0.0388200	0.0194100	−	0.999812i	$-0.493821\pi$
$2$	0.109799	0.0388200	0.0194100	+	0.999812i	$0.493821\pi$
$3$	6.64848	1.27950	0.639750	−	0.768583i	$-0.279038\pi$
$3$	6.64848	1.27950	0.639750	+	0.768583i	$0.279038\pi$
$4$	−7.98794	−0.998493
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0.729999	0.0496701
$7$	28.5341	1.54070	0.770349	−	0.637622i	$-0.220082\pi$
$7$	28.5341	1.54070	0.770349	+	0.637622i	$0.220082\pi$
$8$	−1.75547	−0.0775814
$9$	17.2022	0.637120
$10$	−0.548997	−0.0173608
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−53.1077	−1.27757
$13$	18.1430	0.387074	0.193537	−	0.981093i	$-0.438004\pi$
$13$	18.1430	0.387074	0.193537	+	0.981093i	$0.438004\pi$
$14$	3.13303	0.0598099
$15$	−33.2424	−0.572210
$16$	63.7108	0.995481
$17$	0.876794	0.0125091	0.00625453	−	0.999980i	$-0.498009\pi$
$17$	0.876794	0.0125091	0.00625453	+	0.999980i	$0.498009\pi$
$18$	1.88880	0.0247330
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	39.9397	0.446540
$21$	189.709	1.97132
$22$	1.20779	0.0117047
$23$	−88.8409	−0.805418	−0.402709	−	0.915328i	$-0.631931\pi$
$23$	−88.8409	−0.805418	−0.402709	+	0.915328i	$0.631931\pi$
$24$	−11.6712	−0.0992654
$25$	25.0000	0.200000
$26$	1.99209	0.0150262
$27$	−65.1402	−0.464305
$28$	−227.929	−1.53838
$29$	43.2463	0.276918	0.138459	−	0.990368i	$-0.455785\pi$
$29$	43.2463	0.276918	0.138459	+	0.990368i	$0.455785\pi$
$30$	−3.64999	−0.0222132
$31$	75.3898	0.436788	0.218394	−	0.975861i	$-0.429918\pi$
$31$	75.3898	0.436788	0.218394	+	0.975861i	$0.429918\pi$
$32$	21.0391	0.116226
$33$	73.1332	0.385784
$34$	0.0962715	0.000485601 0
$35$	−142.671	−0.689021
$36$	−137.411	−0.636160
$37$	178.888	0.794839	0.397419	−	0.917637i	$-0.369906\pi$
$37$	178.888	0.794839	0.397419	+	0.917637i	$0.369906\pi$
$38$	2.08619	0.00890591
$39$	120.623	0.495262
$40$	8.77734	0.0346955
$41$	367.116	1.39839	0.699194	−	0.714932i	$-0.253542\pi$
$41$	367.116	1.39839	0.699194	+	0.714932i	$0.253542\pi$
$42$	20.8299	0.0765267
$43$	−223.021	−0.790940	−0.395470	−	0.918479i	$-0.629418\pi$
$43$	−223.021	−0.790940	−0.395470	+	0.918479i	$0.629418\pi$
$44$	−87.8674	−0.301057
$45$	−86.0112	−0.284929
$46$	−9.75468	−0.0312663
$47$	366.924	1.13875	0.569376	−	0.822077i	$-0.307185\pi$
$47$	366.924	1.13875	0.569376	+	0.822077i	$0.307185\pi$
$48$	423.580	1.27372
$49$	471.197	1.37375
$50$	2.74499	0.00776399
$51$	5.82935	0.0160053
$52$	−144.925	−0.386491
$53$	−543.307	−1.40809	−0.704047	−	0.710154i	$-0.748625\pi$
$53$	−543.307	−1.40809	−0.704047	+	0.710154i	$0.748625\pi$
$54$	−7.15235	−0.0180243
$55$	−55.0000	−0.134840
$56$	−50.0907	−0.119530
$57$	126.321	0.293537
$58$	4.74842	0.0107500
$59$	331.736	0.732007	0.366003	−	0.930614i	$-0.380726\pi$
$59$	331.736	0.732007	0.366003	+	0.930614i	$0.380726\pi$
$60$	265.538	0.571347
$61$	−202.336	−0.424697	−0.212348	−	0.977194i	$-0.568111\pi$
$61$	−202.336	−0.424697	−0.212348	+	0.977194i	$0.568111\pi$
$62$	8.27776	0.0169561
$63$	490.851	0.981610
$64$	−507.376	−0.990969
$65$	−90.7151	−0.173105
$66$	8.02999	0.0149761
$67$	31.2297	0.0569450	0.0284725	−	0.999595i	$-0.490936\pi$
$67$	31.2297	0.0569450	0.0284725	+	0.999595i	$0.490936\pi$
$68$	−7.00379	−0.0124902
$69$	−590.657	−1.03053
$70$	−15.6652	−0.0267478
$71$	566.971	0.947705	0.473852	−	0.880604i	$-0.342863\pi$
$71$	566.971	0.947705	0.473852	+	0.880604i	$0.342863\pi$
$72$	−30.1980	−0.0494287
$73$	−1219.75	−1.95564	−0.977819	−	0.209454i	$-0.932831\pi$
$73$	−1219.75	−1.95564	−0.977819	+	0.209454i	$0.932831\pi$
$74$	19.6418	0.0308556
$75$	166.212	0.255900
$76$	−151.771	−0.229070
$77$	313.876	0.464538
$78$	13.2444	0.0192260
$79$	1198.51	1.70687	0.853433	−	0.521202i	$-0.174516\pi$
$79$	1198.51	1.70687	0.853433	+	0.521202i	$0.174516\pi$
$80$	−318.554	−0.445193
$81$	−897.543	−1.23120
$82$	40.3092	0.0542854
$83$	1124.31	1.48686	0.743430	−	0.668814i	$-0.233198\pi$
$83$	1124.31	1.48686	0.743430	+	0.668814i	$0.233198\pi$
$84$	−1515.38	−1.96835
$85$	−4.38397	−0.00559422
$86$	−24.4876	−0.0307043
$87$	287.522	0.354317
$88$	−19.3101	−0.0233917
$89$	1324.18	1.57711	0.788554	−	0.614966i	$-0.210830\pi$
$89$	1324.18	1.57711	0.788554	+	0.614966i	$0.210830\pi$
$90$	−9.44398	−0.0110609
$91$	517.695	0.596365
$92$	709.656	0.804204
$93$	501.228	0.558870
$94$	40.2880	0.0442063
$95$	−95.0000	−0.102598
$96$	139.878	0.148711
$97$	216.768	0.226902	0.113451	−	0.993544i	$-0.463809\pi$
$97$	216.768	0.226902	0.113451	+	0.993544i	$0.463809\pi$
$98$	51.7372	0.0533290
$99$	189.225	0.192099
$100$	−199.699	−0.199699
$101$	−1068.40	−1.05258	−0.526288	−	0.850306i	$-0.676417\pi$
$101$	−1068.40	−1.05258	−0.526288	+	0.850306i	$0.676417\pi$
$102$	0.640059	0.000621326 0
$103$	1844.81	1.76481	0.882403	−	0.470494i	$-0.155924\pi$
$103$	1844.81	1.76481	0.882403	+	0.470494i	$0.155924\pi$
$104$	−31.8495	−0.0300298
$105$	−948.543	−0.881603
$106$	−59.6548	−0.0546621
$107$	1618.55	1.46235	0.731174	−	0.682191i	$-0.238973\pi$
$107$	1618.55	1.46235	0.731174	+	0.682191i	$0.238973\pi$
$108$	520.336	0.463605
$109$	907.329	0.797306	0.398653	−	0.917102i	$-0.369478\pi$
$109$	907.329	0.797306	0.398653	+	0.917102i	$0.369478\pi$
$110$	−6.03897	−0.00523448
$111$	1189.33	1.01700
$112$	1817.93	1.53374
$113$	1618.27	1.34721	0.673603	−	0.739094i	$-0.264746\pi$
$113$	1618.27	1.34721	0.673603	+	0.739094i	$0.264746\pi$
$114$	13.8700	0.0113951
$115$	444.205	0.360194
$116$	−345.449	−0.276501
$117$	312.100	0.246613
$118$	36.4245	0.0284165
$119$	25.0186	0.0192727
$120$	58.3559	0.0443928
$121$	121.000	0.0909091
$122$	−22.2164	−0.0164867
$123$	2440.76	1.78924
$124$	−602.210	−0.436129
$125$	−125.000	−0.0894427
$126$	53.8952	0.0381061
$127$	−1411.17	−0.985996	−0.492998	−	0.870030i	$-0.664099\pi$
$127$	−1411.17	−0.985996	−0.492998	+	0.870030i	$0.664099\pi$
$128$	−224.023	−0.154695
$129$	−1482.75	−1.01201
$130$	−9.96046	−0.00671993
$131$	468.367	0.312377	0.156189	−	0.987727i	$-0.450079\pi$
$131$	468.367	0.312377	0.156189	+	0.987727i	$0.450079\pi$
$132$	−584.184	−0.385202
$133$	542.149	0.353460
$134$	3.42900	0.00221060
$135$	325.701	0.207643
$136$	−1.53918	−0.000970470 0
$137$	339.370	0.211637	0.105819	−	0.994385i	$-0.466254\pi$
$137$	339.370	0.211637	0.105819	+	0.994385i	$0.466254\pi$
$138$	−64.8538	−0.0400052
$139$	−1181.05	−0.720685	−0.360342	−	0.932820i	$-0.617340\pi$
$139$	−1181.05	−0.720685	−0.360342	+	0.932820i	$0.617340\pi$
$140$	1139.65	0.687983
$141$	2439.48	1.45703
$142$	62.2531	0.0367899
$143$	199.573	0.116707
$144$	1095.97	0.634241
$145$	−216.231	−0.123842
$146$	−133.928	−0.0759178
$147$	3132.74	1.75772
$148$	−1428.95	−0.793641
$149$	1234.72	0.678873	0.339437	−	0.940629i	$-0.389764\pi$
$149$	1234.72	0.678873	0.339437	+	0.940629i	$0.389764\pi$
$150$	18.2500	0.00993403
$151$	−1865.23	−1.00523	−0.502616	−	0.864510i	$-0.667629\pi$
$151$	−1865.23	−1.00523	−0.502616	+	0.864510i	$0.667629\pi$
$152$	−33.3539	−0.0177984
$153$	15.0828	0.00796977
$154$	34.4634	0.0180334
$155$	−376.949	−0.195337
$156$	−963.533	−0.494515
$157$	1433.51	0.728702	0.364351	−	0.931262i	$-0.381291\pi$
$157$	1433.51	0.728702	0.364351	+	0.931262i	$0.381291\pi$
$158$	131.595	0.0662605
$159$	−3612.16	−1.80165
$160$	−105.196	−0.0519778
$161$	−2535.00	−1.24091
$162$	−98.5498	−0.0477951
$163$	−2621.84	−1.25987	−0.629933	−	0.776650i	$-0.716918\pi$
$163$	−2621.84	−1.25987	−0.629933	+	0.776650i	$0.716918\pi$
$164$	−2932.50	−1.39628
$165$	−365.666	−0.172528
$166$	123.449	0.0577198
$167$	−2427.30	−1.12473	−0.562366	−	0.826888i	$-0.690109\pi$
$167$	−2427.30	−1.12473	−0.562366	+	0.826888i	$0.690109\pi$
$168$	−333.027	−0.152938
$169$	−1867.83	−0.850173
$170$	−0.481358	−0.000217167 0
$171$	326.843	0.146165
$172$	1781.48	0.789748
$173$	2290.92	1.00679	0.503397	−	0.864055i	$-0.332083\pi$
$173$	2290.92	1.00679	0.503397	+	0.864055i	$0.332083\pi$
$174$	31.5697	0.0137546
$175$	713.353	0.308140
$176$	700.819	0.300149
$177$	2205.54	0.936603
$178$	145.394	0.0612232
$179$	3232.92	1.34994	0.674972	−	0.737843i	$-0.264156\pi$
$179$	3232.92	1.34994	0.674972	+	0.737843i	$0.264156\pi$
$180$	687.053	0.284499
$181$	1151.86	0.473022	0.236511	−	0.971629i	$-0.423996\pi$
$181$	1151.86	0.473022	0.236511	+	0.971629i	$0.423996\pi$
$182$	56.8426	0.0231509
$183$	−1345.23	−0.543399
$184$	155.957	0.0624855
$185$	−894.441	−0.355463
$186$	55.0345	0.0216953
$187$	9.64474	0.00377162
$188$	−2930.97	−1.13704
$189$	−1858.72	−0.715354
$190$	−10.4309	−0.00398284
$191$	2392.98	0.906546	0.453273	−	0.891372i	$-0.350256\pi$
$191$	2392.98	0.906546	0.453273	+	0.891372i	$0.350256\pi$
$192$	−3373.28	−1.26795
$193$	683.958	0.255090	0.127545	−	0.991833i	$-0.459290\pi$
$193$	683.958	0.255090	0.127545	+	0.991833i	$0.459290\pi$
$194$	23.8011	0.00880833
$195$	−603.117	−0.221488
$196$	−3763.90	−1.37168
$197$	2734.38	0.988916	0.494458	−	0.869202i	$-0.335367\pi$
$197$	2734.38	0.988916	0.494458	+	0.869202i	$0.335367\pi$
$198$	20.7768	0.00745727
$199$	−169.261	−0.0602945	−0.0301472	−	0.999545i	$-0.509598\pi$
$199$	−169.261	−0.0602945	−0.0301472	+	0.999545i	$0.509598\pi$
$200$	−43.8867	−0.0155163
$201$	207.630	0.0728611
$202$	−117.310	−0.0408609
$203$	1233.99	0.426648
$204$	−46.5645	−0.0159812
$205$	−1835.58	−0.625378
$206$	202.560	0.0685097
$207$	−1528.26	−0.513148
$208$	1155.91	0.385325
$209$	209.000	0.0691714
$210$	−104.149	−0.0342238
$211$	−1151.33	−0.375644	−0.187822	−	0.982203i	$-0.560143\pi$
$211$	−1151.33	−0.375644	−0.187822	+	0.982203i	$0.560143\pi$
$212$	4339.90	1.40597
$213$	3769.49	1.21259
$214$	177.716	0.0567683
$215$	1115.11	0.353719
$216$	114.351	0.0360214
$217$	2151.18	0.672958
$218$	99.6242	0.0309514
$219$	−8109.51	−2.50224
$220$	439.337	0.134637
$221$	15.9077	0.00484193
$222$	130.588	0.0394797
$223$	−5272.55	−1.58330	−0.791651	−	0.610974i	$-0.790778\pi$
$223$	−5272.55	−1.58330	−0.791651	+	0.610974i	$0.790778\pi$
$224$	600.334	0.179069
$225$	430.056	0.127424
$226$	177.685	0.0522985
$227$	−2057.32	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$227$	−2057.32	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$228$	−1009.05	−0.293095
$229$	1176.61	0.339531	0.169766	−	0.985484i	$-0.445699\pi$
$229$	1176.61	0.339531	0.169766	+	0.985484i	$0.445699\pi$
$230$	48.7734	0.0139827
$231$	2086.79	0.594376
$232$	−75.9174	−0.0214837
$233$	−3524.60	−0.991007	−0.495503	−	0.868606i	$-0.665016\pi$
$233$	−3524.60	−0.991007	−0.495503	+	0.868606i	$0.665016\pi$
$234$	34.2685	0.00957350
$235$	−1834.62	−0.509265
$236$	−2649.89	−0.730904
$237$	7968.24	2.18394
$238$	2.74703	0.000748165 0
$239$	1384.74	0.374775	0.187388	−	0.982286i	$-0.439998\pi$
$239$	1384.74	0.374775	0.187388	+	0.982286i	$0.439998\pi$
$240$	−2117.90	−0.569624
$241$	−1871.49	−0.500220	−0.250110	−	0.968217i	$-0.580467\pi$
$241$	−1871.49	−0.500220	−0.250110	+	0.968217i	$0.580467\pi$
$242$	13.2857	0.00352909
$243$	−4208.51	−1.11101
$244$	1616.25	0.424057
$245$	−2355.98	−0.614361
$246$	267.994	0.0694581
$247$	344.717	0.0888010
$248$	−132.344	−0.0338866
$249$	7474.97	1.90244
$250$	−13.7249	−0.00347216
$251$	241.506	0.0607320	0.0303660	−	0.999539i	$-0.490333\pi$
$251$	241.506	0.0607320	0.0303660	+	0.999539i	$0.490333\pi$
$252$	−3920.89	−0.980131
$253$	−977.250	−0.242843
$254$	−154.946	−0.0382763
$255$	−29.1467	−0.00715780
$256$	4034.41	0.984964
$257$	−1677.73	−0.407213	−0.203606	−	0.979053i	$-0.565266\pi$
$257$	−1677.73	−0.407213	−0.203606	+	0.979053i	$0.565266\pi$
$258$	−162.805	−0.0392861
$259$	5104.42	1.22461
$260$	724.627	0.172844
$261$	743.933	0.176430
$262$	51.4264	0.0121265
$263$	−6256.31	−1.46685	−0.733423	−	0.679772i	$-0.762079\pi$
$263$	−6256.31	−1.46685	−0.733423	+	0.679772i	$0.762079\pi$
$264$	−128.383	−0.0299297
$265$	2716.53	0.629718
$266$	59.5276	0.0137213
$267$	8803.76	2.01791
$268$	−249.461	−0.0568592
$269$	−4514.66	−1.02329	−0.511643	−	0.859198i	$-0.670963\pi$
$269$	−4514.66	−1.02329	−0.511643	+	0.859198i	$0.670963\pi$
$270$	35.7618	0.00806071
$271$	−3461.12	−0.775822	−0.387911	−	0.921697i	$-0.626803\pi$
$271$	−3461.12	−0.775822	−0.387911	+	0.921697i	$0.626803\pi$
$272$	55.8613	0.0124525
$273$	3441.88	0.763049
$274$	37.2626	0.00821576
$275$	275.000	0.0603023
$276$	4718.13	1.02898
$277$	7878.30	1.70889	0.854443	−	0.519545i	$-0.173899\pi$
$277$	7878.30	1.70889	0.854443	+	0.519545i	$0.173899\pi$
$278$	−129.678	−0.0279770
$279$	1296.87	0.278286
$280$	250.454	0.0534553
$281$	−2863.74	−0.607960	−0.303980	−	0.952678i	$-0.598316\pi$
$281$	−2863.74	−0.607960	−0.303980	+	0.952678i	$0.598316\pi$
$282$	267.854	0.0565619
$283$	8110.23	1.70355	0.851773	−	0.523912i	$-0.175528\pi$
$283$	8110.23	1.70355	0.851773	+	0.523912i	$0.175528\pi$
$284$	−4528.93	−0.946277
$285$	−631.605	−0.131274
$286$	21.9130	0.00453057
$287$	10475.3	2.15450
$288$	361.920	0.0740499
$289$	−4912.23	−0.999844
$290$	−23.7421	−0.00480753
$291$	1441.18	0.290321
$292$	9743.33	1.95269
$293$	3510.33	0.699916	0.349958	−	0.936765i	$-0.386196\pi$
$293$	3510.33	0.699916	0.349958	+	0.936765i	$0.386196\pi$
$294$	343.973	0.0682345
$295$	−1658.68	−0.327363
$296$	−314.032	−0.0616647
$297$	−716.542	−0.139993
$298$	135.571	0.0263538
$299$	−1611.84	−0.311757
$300$	−1327.69	−0.255514
$301$	−6363.72	−1.21860
$302$	−204.801	−0.0390231
$303$	−7103.26	−1.34677
$304$	1210.51	0.228379
$305$	1011.68	0.189930
$306$	1.65609	0.000309386 0
$307$	−4922.68	−0.915154	−0.457577	−	0.889170i	$-0.651283\pi$
$307$	−4922.68	−0.915154	−0.457577	+	0.889170i	$0.651283\pi$
$308$	−2507.22	−0.463838
$309$	12265.2	2.25807
$310$	−41.3888	−0.00758299
$311$	−2287.64	−0.417106	−0.208553	−	0.978011i	$-0.566875\pi$
$311$	−2287.64	−0.417106	−0.208553	+	0.978011i	$0.566875\pi$
$312$	−211.750	−0.0384231
$313$	−4500.28	−0.812687	−0.406343	−	0.913720i	$-0.633196\pi$
$313$	−4500.28	−0.812687	−0.406343	+	0.913720i	$0.633196\pi$
$314$	157.398	0.0282882
$315$	−2454.26	−0.438989
$316$	−9573.60	−1.70429
$317$	8185.84	1.45035	0.725177	−	0.688562i	$-0.241758\pi$
$317$	8185.84	1.45035	0.725177	+	0.688562i	$0.241758\pi$
$318$	−396.613	−0.0699402
$319$	475.709	0.0834940
$320$	2536.88	0.443175
$321$	10760.9	1.87107
$322$	−278.341	−0.0481719
$323$	16.6591	0.00286977
$324$	7169.53	1.22934
$325$	453.575	0.0774149
$326$	−287.876	−0.0489079
$327$	6032.36	1.02015
$328$	−644.461	−0.108489
$329$	10469.8	1.75447
$330$	−40.1499	−0.00669752
$331$	−9354.54	−1.55339	−0.776695	−	0.629877i	$-0.783105\pi$
$331$	−9354.54	−1.55339	−0.776695	+	0.629877i	$0.783105\pi$
$332$	−8980.95	−1.48462
$333$	3077.28	0.506408
$334$	−266.516	−0.0436621
$335$	−156.149	−0.0254666
$336$	12086.5	1.96242
$337$	−8179.43	−1.32214	−0.661071	−	0.750324i	$-0.729898\pi$
$337$	−8179.43	−1.32214	−0.661071	+	0.750324i	$0.729898\pi$
$338$	−205.087	−0.0330037
$339$	10759.0	1.72375
$340$	35.0189	0.00558579
$341$	829.288	0.131696
$342$	35.8871	0.00567413
$343$	3657.99	0.575839
$344$	391.506	0.0613622
$345$	2953.28	0.460868
$346$	251.542	0.0390837
$347$	4786.71	0.740530	0.370265	−	0.928926i	$-0.379267\pi$
$347$	4786.71	0.740530	0.370265	+	0.928926i	$0.379267\pi$
$348$	−2296.71	−0.353783
$349$	−1156.63	−0.177401	−0.0887007	−	0.996058i	$-0.528271\pi$
$349$	−1156.63	−0.177401	−0.0887007	+	0.996058i	$0.528271\pi$
$350$	78.3258	0.0119620
$351$	−1181.84	−0.179721
$352$	231.431	0.0350434
$353$	−4207.63	−0.634418	−0.317209	−	0.948356i	$-0.602746\pi$
$353$	−4207.63	−0.634418	−0.317209	+	0.948356i	$0.602746\pi$
$354$	242.167	0.0363589
$355$	−2834.85	−0.423826
$356$	−10577.5	−1.57473
$357$	166.335	0.0246594
$358$	354.973	0.0524048
$359$	−10823.7	−1.59123	−0.795617	−	0.605800i	$-0.792853\pi$
$359$	−10823.7	−1.59123	−0.795617	+	0.605800i	$0.792853\pi$
$360$	150.990	0.0221052
$361$	361.000	0.0526316
$362$	126.473	0.0183627
$363$	804.466	0.116318
$364$	−4135.32	−0.595466
$365$	6098.77	0.874588
$366$	−147.705	−0.0210947
$367$	12202.3	1.73557	0.867783	−	0.496944i	$-0.165544\pi$
$367$	12202.3	1.73557	0.867783	+	0.496944i	$0.165544\pi$
$368$	−5660.13	−0.801779
$369$	6315.22	0.890941
$370$	−98.2091	−0.0137990
$371$	−15502.8	−2.16945
$372$	−4003.78	−0.558028
$373$	−10886.3	−1.51119	−0.755593	−	0.655042i	$-0.772651\pi$
$373$	−10886.3	−1.51119	−0.755593	+	0.655042i	$0.772651\pi$
$374$	1.05899	0.000146414 0
$375$	−831.060	−0.114442
$376$	−644.122	−0.0883459
$377$	784.618	0.107188
$378$	−204.086	−0.0277700
$379$	6485.65	0.879012	0.439506	−	0.898240i	$-0.355154\pi$
$379$	6485.65	0.879012	0.439506	+	0.898240i	$0.355154\pi$
$380$	758.855	0.102443
$381$	−9382.16	−1.26158
$382$	262.748	0.0351921
$383$	−4967.62	−0.662751	−0.331376	−	0.943499i	$-0.607513\pi$
$383$	−4967.62	−0.662751	−0.331376	+	0.943499i	$0.607513\pi$
$384$	−1489.41	−0.197933
$385$	−1569.38	−0.207748
$386$	75.0982	0.00990258
$387$	−3836.46	−0.503924
$388$	−1731.53	−0.226560
$389$	4194.91	0.546762	0.273381	−	0.961906i	$-0.411858\pi$
$389$	4194.91	0.546762	0.273381	+	0.961906i	$0.411858\pi$
$390$	−66.2219	−0.00859815
$391$	−77.8952	−0.0100750
$392$	−827.171	−0.106578
$393$	3113.93	0.399687
$394$	300.233	0.0383897
$395$	−5992.53	−0.763334
$396$	−1511.52	−0.191809
$397$	−8474.21	−1.07131	−0.535653	−	0.844438i	$-0.679934\pi$
$397$	−8474.21	−1.07131	−0.535653	+	0.844438i	$0.679934\pi$
$398$	−18.5848	−0.00234063
$399$	3604.46	0.452253
$400$	1592.77	0.199096
$401$	11106.5	1.38312	0.691559	−	0.722320i	$-0.256924\pi$
$401$	11106.5	1.38312	0.691559	+	0.722320i	$0.256924\pi$
$402$	22.7977	0.00282847
$403$	1367.80	0.169069
$404$	8534.35	1.05099
$405$	4487.72	0.550609
$406$	135.492	0.0165624
$407$	1967.77	0.239653
$408$	−10.2332	−0.00124172
$409$	2760.77	0.333769	0.166884	−	0.985976i	$-0.446629\pi$
$409$	2760.77	0.333769	0.166884	+	0.985976i	$0.446629\pi$
$410$	−201.546	−0.0242772
$411$	2256.29	0.270790
$412$	−14736.3	−1.76215
$413$	9465.81	1.12780
$414$	−167.802	−0.0199204
$415$	−5621.56	−0.664944
$416$	381.714	0.0449881
$417$	−7852.17	−0.922116
$418$	22.9481	0.00268523
$419$	−8422.61	−0.982032	−0.491016	−	0.871151i	$-0.663374\pi$
$419$	−8422.61	−0.982032	−0.491016	+	0.871151i	$0.663374\pi$
$420$	7576.91	0.880274
$421$	6222.15	0.720307	0.360154	−	0.932893i	$-0.382724\pi$
$421$	6222.15	0.720307	0.360154	+	0.932893i	$0.382724\pi$
$422$	−126.416	−0.0145825
$423$	6311.91	0.725521
$424$	953.757	0.109242
$425$	21.9199	0.00250181
$426$	413.888	0.0470726
$427$	−5773.49	−0.654330
$428$	−12928.9	−1.46014
$429$	1326.86	0.149327
$430$	122.438	0.0137314
$431$	12909.7	1.44278	0.721392	−	0.692527i	$-0.243503\pi$
$431$	12909.7	1.44278	0.721392	+	0.692527i	$0.243503\pi$
$432$	−4150.13	−0.462207
$433$	13771.9	1.52849	0.764244	−	0.644927i	$-0.223112\pi$
$433$	13771.9	1.52849	0.764244	+	0.644927i	$0.223112\pi$
$434$	236.199	0.0261242
$435$	−1437.61	−0.158455
$436$	−7247.70	−0.796105
$437$	−1687.98	−0.184776
$438$	−890.420	−0.0971368
$439$	12088.4	1.31423	0.657114	−	0.753792i	$-0.271777\pi$
$439$	12088.4	1.31423	0.657114	+	0.753792i	$0.271777\pi$
$440$	96.5507	0.0104611
$441$	8105.64	0.875245
$442$	1.74666	0.000187964 0
$443$	−14749.8	−1.58190	−0.790950	−	0.611880i	$-0.790413\pi$
$443$	−14749.8	−1.58190	−0.790950	+	0.611880i	$0.790413\pi$
$444$	−9500.33	−1.01546
$445$	−6620.89	−0.705304
$446$	−578.923	−0.0614637
$447$	8209.00	0.868618
$448$	−14477.5	−1.52679
$449$	−14447.3	−1.51851	−0.759256	−	0.650792i	$-0.774437\pi$
$449$	−14447.3	−1.51851	−0.759256	+	0.650792i	$0.774437\pi$
$450$	47.2199	0.00494659
$451$	4038.28	0.421630
$452$	−12926.7	−1.34518
$453$	−12400.9	−1.28619
$454$	−225.893	−0.0233517
$455$	−2588.48	−0.266703
$456$	−221.752	−0.0227730
$457$	14276.8	1.46136	0.730681	−	0.682719i	$-0.239203\pi$
$457$	14276.8	1.46136	0.730681	+	0.682719i	$0.239203\pi$
$458$	129.191	0.0131806
$459$	−57.1145	−0.00580801
$460$	−3548.28	−0.359651
$461$	−16285.1	−1.64528	−0.822640	−	0.568562i	$-0.807500\pi$
$461$	−16285.1	−1.64528	−0.822640	+	0.568562i	$0.807500\pi$
$462$	229.129	0.0230737
$463$	−18778.6	−1.88491	−0.942455	−	0.334332i	$-0.891489\pi$
$463$	−18778.6	−1.88491	−0.942455	+	0.334332i	$0.891489\pi$
$464$	2755.25	0.275667
$465$	−2506.14	−0.249934
$466$	−387.000	−0.0384708
$467$	2789.43	0.276401	0.138200	−	0.990404i	$-0.455868\pi$
$467$	2789.43	0.276401	0.138200	+	0.990404i	$0.455868\pi$
$468$	−2493.04	−0.246241
$469$	891.113	0.0877351
$470$	−201.440	−0.0197696
$471$	9530.64	0.932375
$472$	−582.352	−0.0567901
$473$	−2453.23	−0.238477
$474$	874.908	0.0847803
$475$	475.000	0.0458831
$476$	−199.847	−0.0192436
$477$	−9346.09	−0.897124
$478$	152.043	0.0145488
$479$	−2584.39	−0.246521	−0.123261	−	0.992374i	$-0.539335\pi$
$479$	−2584.39	−0.246521	−0.123261	+	0.992374i	$0.539335\pi$
$480$	−699.391	−0.0665056
$481$	3245.57	0.307662
$482$	−205.488	−0.0194185
$483$	−16853.9	−1.58774
$484$	−966.541	−0.0907721
$485$	−1083.84	−0.101474
$486$	−462.092	−0.0431295
$487$	−4731.20	−0.440228	−0.220114	−	0.975474i	$-0.570643\pi$
$487$	−4731.20	−0.440228	−0.220114	+	0.975474i	$0.570643\pi$
$488$	355.195	0.0329486
$489$	−17431.2	−1.61200
$490$	−258.686	−0.0238495
$491$	4587.53	0.421654	0.210827	−	0.977523i	$-0.432384\pi$
$491$	4587.53	0.421654	0.210827	+	0.977523i	$0.432384\pi$
$492$	−19496.7	−1.78654
$493$	37.9181	0.00346399
$494$	37.8498	0.00344725
$495$	−946.123	−0.0859092
$496$	4803.15	0.434814
$497$	16178.0	1.46013
$498$	820.747	0.0738525
$499$	−13544.7	−1.21512	−0.607561	−	0.794273i	$-0.707852\pi$
$499$	−13544.7	−1.21512	−0.607561	+	0.794273i	$0.707852\pi$
$500$	998.493	0.0893079
$501$	−16137.9	−1.43909
$502$	26.5172	0.00235761
$503$	−16170.8	−1.43344	−0.716718	−	0.697363i	$-0.754357\pi$
$503$	−16170.8	−1.43344	−0.716718	+	0.697363i	$0.754357\pi$
$504$	−861.673	−0.0761547
$505$	5342.02	0.470726
$506$	−107.302	−0.00942714
$507$	−12418.2	−1.08780
$508$	11272.4	0.984510
$509$	4421.46	0.385025	0.192513	−	0.981295i	$-0.438336\pi$
$509$	4421.46	0.385025	0.192513	+	0.981295i	$0.438336\pi$
$510$	−3.20030	−0.000277866 0
$511$	−34804.7	−3.01305
$512$	2235.16	0.192932
$513$	−1237.66	−0.106519
$514$	−184.213	−0.0158080
$515$	−9224.07	−0.789245
$516$	11844.1	1.01048
$517$	4036.16	0.343346
$518$	560.462	0.0475392
$519$	15231.1	1.28819
$520$	159.247	0.0134297
$521$	10660.3	0.896424	0.448212	−	0.893927i	$-0.352061\pi$
$521$	10660.3	0.896424	0.448212	+	0.893927i	$0.352061\pi$
$522$	81.6834	0.00684901
$523$	−16175.5	−1.35240	−0.676201	−	0.736717i	$-0.736375\pi$
$523$	−16175.5	−1.35240	−0.676201	+	0.736717i	$0.736375\pi$
$524$	−3741.29	−0.311907
$525$	4742.71	0.394265
$526$	−686.939	−0.0569429
$527$	66.1014	0.00546380
$528$	4659.38	0.384040
$529$	−4274.29	−0.351302
$530$	298.274	0.0244456
$531$	5706.61	0.466376
$532$	−4330.65	−0.352928
$533$	6660.60	0.541280
$534$	966.648	0.0783351
$535$	−8092.75	−0.653982
$536$	−54.8227	−0.00441787
$537$	21494.0	1.72725
$538$	−495.707	−0.0397239
$539$	5183.17	0.414202
$540$	−2601.68	−0.207331
$541$	14787.5	1.17516	0.587581	−	0.809165i	$-0.300080\pi$
$541$	14787.5	1.17516	0.587581	+	0.809165i	$0.300080\pi$
$542$	−380.029	−0.0301174
$543$	7658.11	0.605232
$544$	18.4470	0.00145388
$545$	−4536.65	−0.356566
$546$	377.917	0.0296215
$547$	−3330.42	−0.260326	−0.130163	−	0.991493i	$-0.541550\pi$
$547$	−3330.42	−0.260326	−0.130163	+	0.991493i	$0.541550\pi$
$548$	−2710.87	−0.211319
$549$	−3480.64	−0.270583
$550$	30.1948	0.00234093
$551$	821.679	0.0635294
$552$	1036.88	0.0799502
$553$	34198.3	2.62977
$554$	865.033	0.0663389
$555$	−5946.67	−0.454814
$556$	9434.15	0.719599
$557$	−24727.1	−1.88101	−0.940504	−	0.339784i	$-0.889646\pi$
$557$	−24727.1	−1.88101	−0.940504	+	0.339784i	$0.889646\pi$
$558$	142.396	0.0108031
$559$	−4046.28	−0.306153
$560$	−9089.66	−0.685908
$561$	64.1228	0.00482579
$562$	−314.437	−0.0236010
$563$	−6276.46	−0.469842	−0.234921	−	0.972014i	$-0.575483\pi$
$563$	−6276.46	−0.469842	−0.234921	+	0.972014i	$0.575483\pi$
$564$	−19486.5	−1.45484
$565$	−8091.36	−0.602489
$566$	890.499	0.0661316
$567$	−25610.6	−1.89691
$568$	−995.298	−0.0735243
$569$	−15552.6	−1.14587	−0.572935	−	0.819601i	$-0.694195\pi$
$569$	−15552.6	−1.14587	−0.572935	+	0.819601i	$0.694195\pi$
$570$	−69.3499	−0.00509605
$571$	−1842.11	−0.135008	−0.0675042	−	0.997719i	$-0.521504\pi$
$571$	−1842.11	−0.135008	−0.0675042	+	0.997719i	$0.521504\pi$
$572$	−1594.18	−0.116531
$573$	15909.7	1.15993
$574$	1150.19	0.0836374
$575$	−2221.02	−0.161084
$576$	−8728.01	−0.631366
$577$	−21360.4	−1.54116	−0.770578	−	0.637346i	$-0.780032\pi$
$577$	−21360.4	−1.54116	−0.770578	+	0.637346i	$0.780032\pi$
$578$	−539.360	−0.0388139
$579$	4547.28	0.326387
$580$	1727.24	0.123655
$581$	32081.3	2.29080
$582$	158.241	0.0112703
$583$	−5976.38	−0.424556
$584$	2141.24	0.151721
$585$	−1560.50	−0.110289
$586$	385.432	0.0271707
$587$	−3086.91	−0.217054	−0.108527	−	0.994094i	$-0.534613\pi$
$587$	−3086.91	−0.217054	−0.108527	+	0.994094i	$0.534613\pi$
$588$	−25024.2	−1.75507
$589$	1432.41	0.100206
$590$	−182.122	−0.0127082
$591$	18179.5	1.26532
$592$	11397.1	0.791247
$593$	−8167.94	−0.565627	−0.282814	−	0.959175i	$-0.591268\pi$
$593$	−8167.94	−0.565627	−0.282814	+	0.959175i	$0.591268\pi$
$594$	−78.6759	−0.00543453
$595$	−125.093	−0.00861900
$596$	−9862.87	−0.677850
$597$	−1125.33	−0.0771467
$598$	−176.979	−0.0121024
$599$	−18318.4	−1.24953	−0.624765	−	0.780813i	$-0.714805\pi$
$599$	−18318.4	−1.24953	−0.624765	+	0.780813i	$0.714805\pi$
$600$	−291.780	−0.0198531
$601$	−7252.40	−0.492232	−0.246116	−	0.969240i	$-0.579154\pi$
$601$	−7252.40	−0.492232	−0.246116	+	0.969240i	$0.579154\pi$
$602$	−698.733	−0.0473060
$603$	537.221	0.0362808
$604$	14899.3	1.00372
$605$	−605.000	−0.0406558
$606$	−779.934	−0.0522816
$607$	10844.0	0.725111	0.362556	−	0.931962i	$-0.381904\pi$
$607$	10844.0	0.725111	0.362556	+	0.931962i	$0.381904\pi$
$608$	399.744	0.0266641
$609$	8204.19	0.545896
$610$	111.082	0.00737308
$611$	6657.10	0.440781
$612$	−120.481	−0.00795776
$613$	8182.85	0.539155	0.269578	−	0.962979i	$-0.413116\pi$
$613$	8182.85	0.539155	0.269578	+	0.962979i	$0.413116\pi$
$614$	−540.508	−0.0355262
$615$	−12203.8	−0.800172
$616$	−550.998	−0.0360395
$617$	−14598.8	−0.952554	−0.476277	−	0.879295i	$-0.658014\pi$
$617$	−14598.8	−0.952554	−0.476277	+	0.879295i	$0.658014\pi$
$618$	1346.71	0.0876582
$619$	−14471.2	−0.939656	−0.469828	−	0.882758i	$-0.655684\pi$
$619$	−14471.2	−0.939656	−0.469828	+	0.882758i	$0.655684\pi$
$620$	3011.05	0.195043
$621$	5787.11	0.373960
$622$	−251.181	−0.0161920
$623$	37784.3	2.42985
$624$	7685.01	0.493024
$625$	625.000	0.0400000
$626$	−494.128	−0.0315485
$627$	1389.53	0.0885049
$628$	−11450.8	−0.727604
$629$	156.848	0.00994268
$630$	−269.476	−0.0170415
$631$	1037.29	0.0654421	0.0327211	−	0.999465i	$-0.489583\pi$
$631$	1037.29	0.0654421	0.0327211	+	0.999465i	$0.489583\pi$
$632$	−2103.94	−0.132421
$633$	−7654.60	−0.480637
$634$	898.800	0.0563027
$635$	7055.87	0.440951
$636$	28853.8	1.79894
$637$	8548.93	0.531744
$638$	52.2326	0.00324123
$639$	9753.17	0.603802
$640$	1120.11	0.0691819
$641$	18847.1	1.16134	0.580668	−	0.814140i	$-0.302791\pi$
$641$	18847.1	1.16134	0.580668	+	0.814140i	$0.302791\pi$
$642$	1181.54	0.0726350
$643$	20785.6	1.27481	0.637405	−	0.770529i	$-0.280008\pi$
$643$	20785.6	1.27481	0.637405	+	0.770529i	$0.280008\pi$
$644$	20249.4	1.23904
$645$	7413.76	0.452583
$646$	1.82916	0.000111404 0
$647$	25518.6	1.55061	0.775303	−	0.631590i	$-0.217597\pi$
$647$	25518.6	1.55061	0.775303	+	0.631590i	$0.217597\pi$
$648$	1575.61	0.0955181
$649$	3649.10	0.220708
$650$	49.8023	0.00300524
$651$	14302.1	0.861050
$652$	20943.1	1.25797
$653$	−21140.3	−1.26690	−0.633450	−	0.773784i	$-0.718362\pi$
$653$	−21140.3	−1.26690	−0.633450	+	0.773784i	$0.718362\pi$
$654$	662.349	0.0396023
$655$	−2341.84	−0.139699
$656$	23389.3	1.39207
$657$	−20982.5	−1.24598
$658$	1149.58	0.0681086
$659$	8646.93	0.511133	0.255567	−	0.966791i	$-0.417738\pi$
$659$	8646.93	0.511133	0.255567	+	0.966791i	$0.417738\pi$
$660$	2920.92	0.172268
$661$	−4784.42	−0.281532	−0.140766	−	0.990043i	$-0.544956\pi$
$661$	−4784.42	−0.281532	−0.140766	+	0.990043i	$0.544956\pi$
$662$	−1027.12	−0.0603025
$663$	105.762	0.00619525
$664$	−1973.69	−0.115353
$665$	−2710.74	−0.158072
$666$	337.883	0.0196587
$667$	−3842.04	−0.223035
$668$	19389.2	1.12304
$669$	−35054.4	−2.02583
$670$	−17.1450	−0.000988612 0
$671$	−2225.70	−0.128051
$672$	3991.31	0.229119
$673$	4539.23	0.259992	0.129996	−	0.991515i	$-0.458504\pi$
$673$	4539.23	0.259992	0.129996	+	0.991515i	$0.458504\pi$
$674$	−898.096	−0.0513255
$675$	−1628.50	−0.0928610
$676$	14920.1	0.848892
$677$	5897.06	0.334775	0.167387	−	0.985891i	$-0.446467\pi$
$677$	5897.06	0.334775	0.167387	+	0.985891i	$0.446467\pi$
$678$	1181.34	0.0669159
$679$	6185.30	0.349588
$680$	7.69592	0.000434007 0
$681$	−13678.0	−0.769668
$682$	91.0554	0.00511245
$683$	−7085.57	−0.396957	−0.198479	−	0.980105i	$-0.563600\pi$
$683$	−7085.57	−0.396957	−0.198479	+	0.980105i	$0.563600\pi$
$684$	−2610.80	−0.145945
$685$	−1696.85	−0.0946472
$686$	401.645	0.0223541
$687$	7822.67	0.434430
$688$	−14208.9	−0.787366
$689$	−9857.22	−0.545037
$690$	324.269	0.0178909
$691$	2459.11	0.135382	0.0676910	−	0.997706i	$-0.478437\pi$
$691$	2459.11	0.135382	0.0676910	+	0.997706i	$0.478437\pi$
$692$	−18299.8	−1.00528
$693$	5399.36	0.295967
$694$	525.578	0.0287473
$695$	5905.24	0.322300
$696$	−504.735	−0.0274884
$697$	321.886	0.0174925
$698$	−126.998	−0.00688671
$699$	−23433.3	−1.26799
$700$	−5698.23	−0.307675
$701$	20306.5	1.09410	0.547052	−	0.837098i	$-0.315750\pi$
$701$	20306.5	1.09410	0.547052	+	0.837098i	$0.315750\pi$
$702$	−129.765	−0.00697674
$703$	3398.88	0.182349
$704$	−5581.14	−0.298789
$705$	−12197.4	−0.651605
$706$	−461.996	−0.0246281
$707$	−30486.0	−1.62170
$708$	−17617.7	−0.935191
$709$	−19144.6	−1.01409	−0.507045	−	0.861919i	$-0.669262\pi$
$709$	−19144.6	−1.01409	−0.507045	+	0.861919i	$0.669262\pi$
$710$	−311.265	−0.0164529
$711$	20617.0	1.08748
$712$	−2324.55	−0.122354
$713$	−6697.70	−0.351797
$714$	18.2635	0.000957277 0
$715$	−997.866	−0.0521931
$716$	−25824.4	−1.34791
$717$	9206.40	0.479525
$718$	−1188.44	−0.0617716
$719$	11228.0	0.582383	0.291192	−	0.956665i	$-0.405948\pi$
$719$	11228.0	0.582383	0.291192	+	0.956665i	$0.405948\pi$
$720$	−5479.84	−0.283641
$721$	52640.2	2.71903
$722$	39.6376	0.00204316
$723$	−12442.5	−0.640032
$724$	−9200.99	−0.472309
$725$	1081.16	0.0553837
$726$	88.3299	0.00451547
$727$	−15363.2	−0.783754	−0.391877	−	0.920018i	$-0.628174\pi$
$727$	−15363.2	−0.783754	−0.391877	+	0.920018i	$0.628174\pi$
$728$	−908.797	−0.0462668
$729$	−3746.52	−0.190343
$730$	669.642	0.0339515
$731$	−195.544	−0.00989391
$732$	10745.6	0.542580
$733$	2676.90	0.134889	0.0674443	−	0.997723i	$-0.478515\pi$
$733$	2676.90	0.134889	0.0674443	+	0.997723i	$0.478515\pi$
$734$	1339.80	0.0673746
$735$	−15663.7	−0.786074
$736$	−1869.14	−0.0936105
$737$	343.527	0.0171696
$738$	693.408	0.0345863
$739$	6667.78	0.331906	0.165953	−	0.986134i	$-0.446930\pi$
$739$	6667.78	0.331906	0.165953	+	0.986134i	$0.446930\pi$
$740$	7144.74	0.354927
$741$	2291.84	0.113621
$742$	−1702.20	−0.0842179
$743$	−5348.07	−0.264067	−0.132034	−	0.991245i	$-0.542151\pi$
$743$	−5348.07	−0.264067	−0.132034	+	0.991245i	$0.542151\pi$
$744$	−879.889	−0.0433579
$745$	−6173.59	−0.303601
$746$	−1195.31	−0.0586642
$747$	19340.7	0.947308
$748$	−77.0416	−0.00376594
$749$	46183.9	2.25304
$750$	−91.2499	−0.00444263
$751$	−21741.0	−1.05638	−0.528190	−	0.849126i	$-0.677129\pi$
$751$	−21741.0	−1.05638	−0.528190	+	0.849126i	$0.677129\pi$
$752$	23377.0	1.13361
$753$	1605.65	0.0777065
$754$	86.1506	0.00416103
$755$	9326.13	0.449553
$756$	14847.3	0.714276
$757$	13493.2	0.647846	0.323923	−	0.946083i	$-0.394998\pi$
$757$	13493.2	0.647846	0.323923	+	0.946083i	$0.394998\pi$
$758$	712.121	0.0341232
$759$	−6497.22	−0.310717
$760$	166.769	0.00795969
$761$	6567.00	0.312817	0.156408	−	0.987692i	$-0.450008\pi$
$761$	6567.00	0.312817	0.156408	+	0.987692i	$0.450008\pi$
$762$	−1030.16	−0.0489746
$763$	25889.9	1.22841
$764$	−19115.0	−0.905180
$765$	−75.4141	−0.00356419
$766$	−545.442	−0.0257280
$767$	6018.70	0.283341
$768$	26822.7	1.26026
$769$	11823.8	0.554457	0.277228	−	0.960804i	$-0.410584\pi$
$769$	11823.8	0.554457	0.277228	+	0.960804i	$0.410584\pi$
$770$	−172.317	−0.00806476
$771$	−11154.3	−0.521029
$772$	−5463.41	−0.254705
$773$	6449.98	0.300116	0.150058	−	0.988677i	$-0.452054\pi$
$773$	6449.98	0.300116	0.150058	+	0.988677i	$0.452054\pi$
$774$	−421.242	−0.0195623
$775$	1884.75	0.0873575
$776$	−380.530	−0.0176034
$777$	33936.6	1.56688
$778$	460.599	0.0212253
$779$	6975.21	0.320812
$780$	4817.67	0.221154
$781$	6236.68	0.285744
$782$	−8.55285	−0.000391112 0
$783$	−2817.07	−0.128575
$784$	30020.3	1.36754
$785$	−7167.53	−0.325886
$786$	341.908	0.0155158
$787$	1831.37	0.0829495	0.0414748	−	0.999140i	$-0.486794\pi$
$787$	1831.37	0.0829495	0.0414748	+	0.999140i	$0.486794\pi$
$788$	−21842.1	−0.987425
$789$	−41594.9	−1.87683
$790$	−657.976	−0.0296326
$791$	46176.0	2.07564
$792$	−332.178	−0.0149033
$793$	−3670.99	−0.164389
$794$	−930.463	−0.0415880
$795$	18060.8	0.805725
$796$	1352.05	0.0602036
$797$	−24772.8	−1.10100	−0.550500	−	0.834835i	$-0.685563\pi$
$797$	−24772.8	−1.10100	−0.550500	+	0.834835i	$0.685563\pi$
$798$	395.768	0.0175564
$799$	321.717	0.0142447
$800$	525.979	0.0232452
$801$	22778.8	1.00481
$802$	1219.48	0.0536926
$803$	−13417.3	−0.589647
$804$	−1658.54	−0.0727513
$805$	12675.0	0.554950
$806$	150.184	0.00656326
$807$	−30015.6	−1.30929
$808$	1875.55	0.0816603
$809$	17543.7	0.762429	0.381215	−	0.924487i	$-0.375506\pi$
$809$	17543.7	0.762429	0.381215	+	0.924487i	$0.375506\pi$
$810$	492.749	0.0213746
$811$	6181.81	0.267661	0.133830	−	0.991004i	$-0.457272\pi$
$811$	6181.81	0.267661	0.133830	+	0.991004i	$0.457272\pi$
$812$	−9857.08	−0.426005
$813$	−23011.1	−0.992664
$814$	216.060	0.00930332
$815$	13109.2	0.563429
$816$	371.392	0.0159330
$817$	−4237.40	−0.181454
$818$	303.131	0.0129569
$819$	8905.52	0.379956
$820$	14662.5	0.624436
$821$	−11995.4	−0.509916	−0.254958	−	0.966952i	$-0.582062\pi$
$821$	−11995.4	−0.509916	−0.254958	+	0.966952i	$0.582062\pi$
$822$	247.740	0.0105121
$823$	36578.4	1.54926	0.774632	−	0.632413i	$-0.217935\pi$
$823$	36578.4	1.54926	0.774632	+	0.632413i	$0.217935\pi$
$824$	−3238.51	−0.136916
$825$	1828.33	0.0771567
$826$	1039.34	0.0437812
$827$	28177.9	1.18482	0.592408	−	0.805638i	$-0.298177\pi$
$827$	28177.9	1.18482	0.592408	+	0.805638i	$0.298177\pi$
$828$	12207.7	0.512375
$829$	−22794.3	−0.954982	−0.477491	−	0.878637i	$-0.658454\pi$
$829$	−22794.3	−0.954982	−0.477491	+	0.878637i	$0.658454\pi$
$830$	−617.244	−0.0258131
$831$	52378.7	2.18652
$832$	−9205.34	−0.383579
$833$	413.143	0.0171843
$834$	−862.164	−0.0357965
$835$	12136.5	0.502995
$836$	−1669.48	−0.0690672
$837$	−4910.91	−0.202803
$838$	−924.797	−0.0381224
$839$	5737.18	0.236078	0.118039	−	0.993009i	$-0.462339\pi$
$839$	5737.18	0.236078	0.118039	+	0.993009i	$0.462339\pi$
$840$	1665.14	0.0683960
$841$	−22518.8	−0.923316
$842$	683.189	0.0279623
$843$	−19039.5	−0.777884
$844$	9196.77	0.375078
$845$	9339.16	0.380209
$846$	693.044	0.0281647
$847$	3452.63	0.140064
$848$	−34614.5	−1.40173
$849$	53920.7	2.17969
$850$	2.40679	9.71202e−5 0
$851$	−15892.6	−0.640177
$852$	−30110.5	−1.21076
$853$	−26740.2	−1.07335	−0.536674	−	0.843789i	$-0.680320\pi$
$853$	−26740.2	−1.07335	−0.536674	+	0.843789i	$0.680320\pi$
$854$	−633.926	−0.0254010
$855$	−1634.21	−0.0653671
$856$	−2841.31	−0.113451
$857$	28971.4	1.15478	0.577388	−	0.816470i	$-0.304072\pi$
$857$	28971.4	1.15478	0.577388	+	0.816470i	$0.304072\pi$
$858$	145.688	0.00579687
$859$	−6979.52	−0.277227	−0.138614	−	0.990347i	$-0.544265\pi$
$859$	−6979.52	−0.277227	−0.138614	+	0.990347i	$0.544265\pi$
$860$	−8907.40	−0.353186
$861$	69645.1	2.75668
$862$	1417.48	0.0560088
$863$	−13241.4	−0.522299	−0.261149	−	0.965298i	$-0.584101\pi$
$863$	−13241.4	−0.522299	−0.261149	+	0.965298i	$0.584101\pi$
$864$	−1370.49	−0.0539643
$865$	−11454.6	−0.450252
$866$	1512.15	0.0593359
$867$	−32658.9	−1.27930
$868$	−17183.5	−0.671944
$869$	13183.6	0.514640
$870$	−157.849	−0.00615123
$871$	566.601	0.0220420
$872$	−1592.79	−0.0618562
$873$	3728.90	0.144564
$874$	−185.339	−0.00717298
$875$	−3566.77	−0.137804
$876$	64778.3	2.49847
$877$	5558.26	0.214013	0.107006	−	0.994258i	$-0.465873\pi$
$877$	5558.26	0.214013	0.107006	+	0.994258i	$0.465873\pi$
$878$	1327.29	0.0510182
$879$	23338.3	0.895543
$880$	−3504.09	−0.134231
$881$	−20189.3	−0.772073	−0.386036	−	0.922484i	$-0.626156\pi$
$881$	−20189.3	−0.772073	−0.386036	+	0.922484i	$0.626156\pi$
$882$	889.995	0.0339770
$883$	5218.87	0.198900	0.0994501	−	0.995043i	$-0.468292\pi$
$883$	5218.87	0.198900	0.0994501	+	0.995043i	$0.468292\pi$
$884$	−127.070	−0.00483464
$885$	−11027.7	−0.418861
$886$	−1619.51	−0.0614093
$887$	−35234.3	−1.33377	−0.666883	−	0.745162i	$-0.732372\pi$
$887$	−35234.3	−1.33377	−0.666883	+	0.745162i	$0.732372\pi$
$888$	−2087.84	−0.0789000
$889$	−40266.7	−1.51912
$890$	−726.970	−0.0273799
$891$	−9872.98	−0.371220
$892$	42116.9	1.58092
$893$	6971.55	0.261247
$894$	901.344	0.0337197
$895$	−16164.6	−0.603713
$896$	−6392.30	−0.238339
$897$	−10716.3	−0.398893
$898$	−1586.31	−0.0589486
$899$	3260.33	0.120954
$900$	−3435.26	−0.127232
$901$	−476.368	−0.0176139
$902$	443.401	0.0163677
$903$	−42309.0	−1.55920
$904$	−2840.82	−0.104518
$905$	−5759.30	−0.211542
$906$	−1361.61	−0.0499300
$907$	11907.9	0.435937	0.217969	−	0.975956i	$-0.430057\pi$
$907$	11907.9	0.435937	0.217969	+	0.975956i	$0.430057\pi$
$908$	16433.8	0.600632
$909$	−18378.9	−0.670617
$910$	−284.213	−0.0103534
$911$	11298.4	0.410905	0.205452	−	0.978667i	$-0.434133\pi$
$911$	11298.4	0.410905	0.205452	+	0.978667i	$0.434133\pi$
$912$	8048.02	0.292211
$913$	12367.4	0.448305
$914$	1567.59	0.0567300
$915$	6726.14	0.243016
$916$	−9398.70	−0.339019
$917$	13364.5	0.481279
$918$	−6.27114	−0.000225467 0
$919$	5254.02	0.188590	0.0942950	−	0.995544i	$-0.469940\pi$
$919$	5254.02	0.188590	0.0942950	+	0.995544i	$0.469940\pi$
$920$	−779.787	−0.0279444
$921$	−32728.3	−1.17094
$922$	−1788.10	−0.0638697
$923$	10286.6	0.366832
$924$	−16669.2	−0.593481
$925$	4472.20	0.158968
$926$	−2061.88	−0.0731722
$927$	31734.9	1.12439
$928$	909.865	0.0321851
$929$	48410.5	1.70969	0.854843	−	0.518887i	$-0.173653\pi$
$929$	48410.5	1.70969	0.854843	+	0.518887i	$0.173653\pi$
$930$	−275.173	−0.00970243
$931$	8952.74	0.315160
$932$	28154.3	0.989513
$933$	−15209.3	−0.533687
$934$	306.277	0.0107299
$935$	−48.2237	−0.00168672
$936$	−547.882	−0.0191326
$937$	40457.7	1.41056	0.705281	−	0.708928i	$-0.250821\pi$
$937$	40457.7	1.41056	0.705281	+	0.708928i	$0.250821\pi$
$938$	97.8437	0.00340587
$939$	−29920.0	−1.03983
$940$	14654.8	0.508498
$941$	20435.6	0.707950	0.353975	−	0.935255i	$-0.384830\pi$
$941$	20435.6	0.707950	0.353975	+	0.935255i	$0.384830\pi$
$942$	1046.46	0.0361947
$943$	−32614.9	−1.12629
$944$	21135.2	0.728699
$945$	9293.59	0.319916
$946$	−269.364	−0.00925768
$947$	−10108.9	−0.346881	−0.173441	−	0.984844i	$-0.555489\pi$
$947$	−10108.9	−0.346881	−0.173441	+	0.984844i	$0.555489\pi$
$948$	−63649.9	−2.18064
$949$	−22130.0	−0.756977
$950$	52.1547	0.00178118
$951$	54423.4	1.85573
$952$	−43.9193	−0.00149520
$953$	24002.2	0.815853	0.407927	−	0.913015i	$-0.366252\pi$
$953$	24002.2	0.815853	0.407927	+	0.913015i	$0.366252\pi$
$954$	−1026.20	−0.0348263
$955$	−11964.9	−0.405420
$956$	−11061.2	−0.374210
$957$	3162.74	0.106831
$958$	−283.764	−0.00956995
$959$	9683.63	0.326070
$960$	16866.4	0.567042
$961$	−24107.4	−0.809217
$962$	356.362	0.0119434
$963$	27842.7	0.931691
$964$	14949.3	0.499467
$965$	−3419.79	−0.114080
$966$	−1850.55	−0.0616360
$967$	−6237.92	−0.207444	−0.103722	−	0.994606i	$-0.533075\pi$
$967$	−6237.92	−0.207444	−0.103722	+	0.994606i	$0.533075\pi$
$968$	−212.412	−0.00705286
$969$	110.758	0.00367187
$970$	−119.005	−0.00393920
$971$	−13230.5	−0.437268	−0.218634	−	0.975807i	$-0.570160\pi$
$971$	−13230.5	−0.437268	−0.218634	+	0.975807i	$0.570160\pi$
$972$	33617.4	1.10934
$973$	−33700.2	−1.11036
$974$	−519.483	−0.0170896
$975$	3015.59	0.0990523
$976$	−12891.0	−0.422778
$977$	11842.5	0.387794	0.193897	−	0.981022i	$-0.437887\pi$
$977$	11842.5	0.387794	0.193897	+	0.981022i	$0.437887\pi$
$978$	−1913.94	−0.0625777
$979$	14566.0	0.475516
$980$	18819.5	0.613435
$981$	15608.1	0.507980
$982$	503.708	0.0163686
$983$	−55716.8	−1.80782	−0.903911	−	0.427720i	$-0.859317\pi$
$983$	−55716.8	−1.80782	−0.903911	+	0.427720i	$0.859317\pi$
$984$	−4284.68	−0.138812
$985$	−13671.9	−0.442257
$986$	4.16338	0.000134472 0
$987$	69608.5	2.24485
$988$	−2753.58	−0.0886671
$989$	19813.4	0.637037
$990$	−103.884	−0.00333499
$991$	−3142.33	−0.100726	−0.0503630	−	0.998731i	$-0.516038\pi$
$991$	−3142.33	−0.100726	−0.0503630	+	0.998731i	$0.516038\pi$
$992$	1586.14	0.0507661
$993$	−62193.4	−1.98756
$994$	1776.34	0.0566821
$995$	846.305	0.0269645
$996$	−59709.6	−1.89957
$997$	−48810.1	−1.55048	−0.775242	−	0.631665i	$-0.782372\pi$
$997$	−48810.1	−1.55048	−0.775242	+	0.631665i	$0.782372\pi$
$998$	−1487.20	−0.0471710
$999$	−11652.8	−0.369048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.i.1.13	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.i.1.13	✓	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)

\(f(q)\)	\(=\)	\(q+0.109799 q^{2} +6.64848 q^{3} -7.98794 q^{4} -5.00000 q^{5} +0.729999 q^{6} +28.5341 q^{7} -1.75547 q^{8} +17.2022 q^{9} +O(q^{10})\)	\(q+0.109799 q^{2} +6.64848 q^{3} -7.98794 q^{4} -5.00000 q^{5} +0.729999 q^{6} +28.5341 q^{7} -1.75547 q^{8} +17.2022 q^{9} -0.548997 q^{10} +11.0000 q^{11} -53.1077 q^{12} +18.1430 q^{13} +3.13303 q^{14} -33.2424 q^{15} +63.7108 q^{16} +0.876794 q^{17} +1.88880 q^{18} +19.0000 q^{19} +39.9397 q^{20} +189.709 q^{21} +1.20779 q^{22} -88.8409 q^{23} -11.6712 q^{24} +25.0000 q^{25} +1.99209 q^{26} -65.1402 q^{27} -227.929 q^{28} +43.2463 q^{29} -3.64999 q^{30} +75.3898 q^{31} +21.0391 q^{32} +73.1332 q^{33} +0.0962715 q^{34} -142.671 q^{35} -137.411 q^{36} +178.888 q^{37} +2.08619 q^{38} +120.623 q^{39} +8.77734 q^{40} +367.116 q^{41} +20.8299 q^{42} -223.021 q^{43} -87.8674 q^{44} -86.0112 q^{45} -9.75468 q^{46} +366.924 q^{47} +423.580 q^{48} +471.197 q^{49} +2.74499 q^{50} +5.82935 q^{51} -144.925 q^{52} -543.307 q^{53} -7.15235 q^{54} -55.0000 q^{55} -50.0907 q^{56} +126.321 q^{57} +4.74842 q^{58} +331.736 q^{59} +265.538 q^{60} -202.336 q^{61} +8.27776 q^{62} +490.851 q^{63} -507.376 q^{64} -90.7151 q^{65} +8.02999 q^{66} +31.2297 q^{67} -7.00379 q^{68} -590.657 q^{69} -15.6652 q^{70} +566.971 q^{71} -30.1980 q^{72} -1219.75 q^{73} +19.6418 q^{74} +166.212 q^{75} -151.771 q^{76} +313.876 q^{77} +13.2444 q^{78} +1198.51 q^{79} -318.554 q^{80} -897.543 q^{81} +40.3092 q^{82} +1124.31 q^{83} -1515.38 q^{84} -4.38397 q^{85} -24.4876 q^{86} +287.522 q^{87} -19.3101 q^{88} +1324.18 q^{89} -9.44398 q^{90} +517.695 q^{91} +709.656 q^{92} +501.228 q^{93} +40.2880 q^{94} -95.0000 q^{95} +139.878 q^{96} +216.768 q^{97} +51.7372 q^{98} +189.225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) 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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