LMFDB - Embedded newform 1075.6.a.k.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$1$
Dimension:	$52$
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.21304 q^{2} -4.58150 q^{3} +52.8802 q^{4} +42.2096 q^{6} -256.470 q^{7} -192.370 q^{8} -222.010 q^{9} +O(q^{10})$	\(q-9.21304 q^{2} -4.58150 q^{3} +52.8802 q^{4} +42.2096 q^{6} -256.470 q^{7} -192.370 q^{8} -222.010 q^{9} +465.471 q^{11} -242.271 q^{12} +971.715 q^{13} +2362.87 q^{14} +80.1485 q^{16} -1619.21 q^{17} +2045.39 q^{18} -665.148 q^{19} +1175.02 q^{21} -4288.41 q^{22} -3625.67 q^{23} +881.345 q^{24} -8952.45 q^{26} +2130.44 q^{27} -13562.2 q^{28} +4201.01 q^{29} +3410.38 q^{31} +5417.43 q^{32} -2132.56 q^{33} +14917.9 q^{34} -11739.9 q^{36} -11742.7 q^{37} +6128.04 q^{38} -4451.92 q^{39} -13665.4 q^{41} -10825.5 q^{42} +1849.00 q^{43} +24614.2 q^{44} +33403.5 q^{46} +12849.5 q^{47} -367.201 q^{48} +48970.0 q^{49} +7418.43 q^{51} +51384.5 q^{52} -13941.6 q^{53} -19627.9 q^{54} +49337.2 q^{56} +3047.38 q^{57} -38704.1 q^{58} +16559.1 q^{59} +16949.3 q^{61} -31420.0 q^{62} +56938.9 q^{63} -52475.8 q^{64} +19647.4 q^{66} -18209.5 q^{67} -85624.3 q^{68} +16611.0 q^{69} +18749.6 q^{71} +42708.1 q^{72} +7603.69 q^{73} +108186. q^{74} -35173.2 q^{76} -119380. q^{77} +41015.7 q^{78} -41654.9 q^{79} +44187.7 q^{81} +125900. q^{82} +24823.0 q^{83} +62135.3 q^{84} -17034.9 q^{86} -19246.9 q^{87} -89542.8 q^{88} +62891.0 q^{89} -249216. q^{91} -191726. q^{92} -15624.7 q^{93} -118383. q^{94} -24820.0 q^{96} -72473.7 q^{97} -451163. q^{98} -103339. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9}+O(q^{10}) $ 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9	\( 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9} - 664 q^{11} + 523 q^{12} - 2704 q^{13} + 150 q^{14} + 13474 q^{16} - 7266 q^{17} - 4860 q^{18} - 1970 q^{19} + 800 q^{21} - 14477 q^{22} - 9522 q^{23} + 314 q^{24} + 5514 q^{26} - 22926 q^{27} - 9408 q^{28} - 7188 q^{29} - 11556 q^{31} - 48390 q^{32} - 26136 q^{33} + 16774 q^{34} + 51872 q^{36} - 42558 q^{37} - 46208 q^{38} + 4682 q^{39} - 7746 q^{41} - 174265 q^{42} + 96148 q^{43} - 48600 q^{44} + 16182 q^{46} - 87136 q^{47} + 2912 q^{48} + 142286 q^{49} - 3710 q^{51} - 146868 q^{52} - 127034 q^{53} - 49563 q^{54} - 2849 q^{56} - 101594 q^{57} - 9480 q^{58} - 55924 q^{59} + 73702 q^{61} - 186016 q^{62} - 50120 q^{63} + 157750 q^{64} + 58211 q^{66} - 131996 q^{67} - 298560 q^{68} + 128436 q^{69} - 56284 q^{71} - 343775 q^{72} - 128620 q^{73} - 17721 q^{74} - 170410 q^{76} - 448438 q^{77} - 237616 q^{78} + 106204 q^{79} + 478568 q^{81} - 249596 q^{82} - 348616 q^{83} - 131855 q^{84} - 36980 q^{86} - 267478 q^{87} - 525216 q^{88} + 80410 q^{89} + 226376 q^{91} - 581456 q^{92} - 902902 q^{93} + 180980 q^{94} + 38543 q^{96} - 316148 q^{97} - 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9 - 664 * q^11 + 523 * q^12 - 2704 * q^13 + 150 * q^14 + 13474 * q^16 - 7266 * q^17 - 4860 * q^18 - 1970 * q^19 + 800 * q^21 - 14477 * q^22 - 9522 * q^23 + 314 * q^24 + 5514 * q^26 - 22926 * q^27 - 9408 * q^28 - 7188 * q^29 - 11556 * q^31 - 48390 * q^32 - 26136 * q^33 + 16774 * q^34 + 51872 * q^36 - 42558 * q^37 - 46208 * q^38 + 4682 * q^39 - 7746 * q^41 - 174265 * q^42 + 96148 * q^43 - 48600 * q^44 + 16182 * q^46 - 87136 * q^47 + 2912 * q^48 + 142286 * q^49 - 3710 * q^51 - 146868 * q^52 - 127034 * q^53 - 49563 * q^54 - 2849 * q^56 - 101594 * q^57 - 9480 * q^58 - 55924 * q^59 + 73702 * q^61 - 186016 * q^62 - 50120 * q^63 + 157750 * q^64 + 58211 * q^66 - 131996 * q^67 - 298560 * q^68 + 128436 * q^69 - 56284 * q^71 - 343775 * q^72 - 128620 * q^73 - 17721 * q^74 - 170410 * q^76 - 448438 * q^77 - 237616 * q^78 + 106204 * q^79 + 478568 * q^81 - 249596 * q^82 - 348616 * q^83 - 131855 * q^84 - 36980 * q^86 - 267478 * q^87 - 525216 * q^88 + 80410 * q^89 + 226376 * q^91 - 581456 * q^92 - 902902 * q^93 + 180980 * q^94 + 38543 * q^96 - 316148 * q^97 - 295095 * q^98 + 68428 * 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 $

$2$	−9.21304	−1.62865	−0.814326	−	0.580408i	$-0.802893\pi$
$2$	−9.21304	−1.62865	−0.814326	+	0.580408i	$0.802893\pi$
$3$	−4.58150	−0.293904	−0.146952	−	0.989144i	$-0.546946\pi$
$3$	−4.58150	−0.293904	−0.146952	+	0.989144i	$0.546946\pi$
$4$	52.8802	1.65251
$5$	0	0
$5$	0	0
$6$	42.2096	0.478667
$7$	−256.470	−1.97830	−0.989150	−	0.146912i	$-0.953066\pi$
$7$	−256.470	−1.97830	−0.989150	+	0.146912i	$0.953066\pi$
$8$	−192.370	−1.06270
$9$	−222.010	−0.913621
$10$	0	0
$11$	465.471	1.15987	0.579937	−	0.814661i	$-0.303077\pi$
$11$	465.471	1.15987	0.579937	+	0.814661i	$0.303077\pi$
$12$	−242.271	−0.485678
$13$	971.715	1.59471	0.797353	−	0.603514i	$-0.206233\pi$
$13$	971.715	1.59471	0.797353	+	0.603514i	$0.206233\pi$
$14$	2362.87	3.22196
$15$	0	0
$16$	80.1485	0.0782700
$17$	−1619.21	−1.35888	−0.679441	−	0.733730i	$-0.737778\pi$
$17$	−1619.21	−1.35888	−0.679441	+	0.733730i	$0.737778\pi$
$18$	2045.39	1.48797
$19$	−665.148	−0.422702	−0.211351	−	0.977410i	$-0.567786\pi$
$19$	−665.148	−0.422702	−0.211351	+	0.977410i	$0.567786\pi$
$20$	0	0
$21$	1175.02	0.581429
$22$	−4288.41	−1.88903
$23$	−3625.67	−1.42912	−0.714560	−	0.699574i	$-0.753373\pi$
$23$	−3625.67	−1.42912	−0.714560	+	0.699574i	$0.753373\pi$
$24$	881.345	0.312333
$25$	0	0
$26$	−8952.45	−2.59722
$27$	2130.44	0.562420
$28$	−13562.2	−3.26915
$29$	4201.01	0.927595	0.463797	−	0.885941i	$-0.346487\pi$
$29$	4201.01	0.927595	0.463797	+	0.885941i	$0.346487\pi$
$30$	0	0
$31$	3410.38	0.637380	0.318690	−	0.947859i	$-0.396757\pi$
$31$	3410.38	0.637380	0.318690	+	0.947859i	$0.396757\pi$
$32$	5417.43	0.935230
$33$	−2132.56	−0.340891
$34$	14917.9	2.21315
$35$	0	0
$36$	−11739.9	−1.50976
$37$	−11742.7	−1.41014	−0.705070	−	0.709137i	$-0.749085\pi$
$37$	−11742.7	−1.41014	−0.705070	+	0.709137i	$0.749085\pi$
$38$	6128.04	0.688435
$39$	−4451.92	−0.468690
$40$	0	0
$41$	−13665.4	−1.26959	−0.634796	−	0.772680i	$-0.718916\pi$
$41$	−13665.4	−1.26959	−0.634796	+	0.772680i	$0.718916\pi$
$42$	−10825.5	−0.946946
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	24614.2	1.91670
$45$	0	0
$46$	33403.5	2.32754
$47$	12849.5	0.848481	0.424240	−	0.905550i	$-0.360541\pi$
$47$	12849.5	0.848481	0.424240	+	0.905550i	$0.360541\pi$
$48$	−367.201	−0.0230038
$49$	48970.0	2.91367
$50$	0	0
$51$	7418.43	0.399380
$52$	51384.5	2.63526
$53$	−13941.6	−0.681748	−0.340874	−	0.940109i	$-0.610723\pi$
$53$	−13941.6	−0.681748	−0.340874	+	0.940109i	$0.610723\pi$
$54$	−19627.9	−0.915986
$55$	0	0
$56$	49337.2	2.10235
$57$	3047.38	0.124234
$58$	−38704.1	−1.51073
$59$	16559.1	0.619306	0.309653	−	0.950850i	$-0.399787\pi$
$59$	16559.1	0.619306	0.309653	+	0.950850i	$0.399787\pi$
$60$	0	0
$61$	16949.3	0.583211	0.291606	−	0.956539i	$-0.405810\pi$
$61$	16949.3	0.583211	0.291606	+	0.956539i	$0.405810\pi$
$62$	−31420.0	−1.03807
$63$	56938.9	1.80741
$64$	−52475.8	−1.60143
$65$	0	0
$66$	19647.4	0.555193
$67$	−18209.5	−0.495576	−0.247788	−	0.968814i	$-0.579704\pi$
$67$	−18209.5	−0.495576	−0.247788	+	0.968814i	$0.579704\pi$
$68$	−85624.3	−2.24556
$69$	16611.0	0.420024
$70$	0	0
$71$	18749.6	0.441415	0.220707	−	0.975340i	$-0.429163\pi$
$71$	18749.6	0.441415	0.220707	+	0.975340i	$0.429163\pi$
$72$	42708.1	0.970909
$73$	7603.69	0.167000	0.0835001	−	0.996508i	$-0.473390\pi$
$73$	7603.69	0.167000	0.0835001	+	0.996508i	$0.473390\pi$
$74$	108186.	2.29663
$75$	0	0
$76$	−35173.2	−0.698518
$77$	−119380.	−2.29458
$78$	41015.7	0.763332
$79$	−41654.9	−0.750928	−0.375464	−	0.926837i	$-0.622517\pi$
$79$	−41654.9	−0.750928	−0.375464	+	0.926837i	$0.622517\pi$
$80$	0	0
$81$	44187.7	0.748323
$82$	125900.	2.06772
$83$	24823.0	0.395511	0.197755	−	0.980251i	$-0.436635\pi$
$83$	24823.0	0.395511	0.197755	+	0.980251i	$0.436635\pi$
$84$	62135.3	0.960815
$85$	0	0
$86$	−17034.9	−0.248367
$87$	−19246.9	−0.272623
$88$	−89542.8	−1.23260
$89$	62891.0	0.841616	0.420808	−	0.907150i	$-0.361747\pi$
$89$	62891.0	0.841616	0.420808	+	0.907150i	$0.361747\pi$
$90$	0	0
$91$	−249216.	−3.15480
$92$	−191726.	−2.36163
$93$	−15624.7	−0.187328
$94$	−118383.	−1.38188
$95$	0	0
$96$	−24820.0	−0.274868
$97$	−72473.7	−0.782080	−0.391040	−	0.920374i	$-0.627885\pi$
$97$	−72473.7	−0.782080	−0.391040	+	0.920374i	$0.627885\pi$
$98$	−451163.	−4.74535
$99$	−103339.	−1.05969
$100$	0	0
$101$	63919.0	0.623486	0.311743	−	0.950167i	$-0.399087\pi$
$101$	63919.0	0.623486	0.311743	+	0.950167i	$0.399087\pi$
$102$	−68346.4	−0.650452
$103$	−126426.	−1.17420	−0.587102	−	0.809513i	$-0.699731\pi$
$103$	−126426.	−1.17420	−0.587102	+	0.809513i	$0.699731\pi$
$104$	−186929.	−1.69470
$105$	0	0
$106$	128445.	1.11033
$107$	−1136.02	−0.00959236	−0.00479618	−	0.999988i	$-0.501527\pi$
$107$	−1136.02	−0.00959236	−0.00479618	+	0.999988i	$0.501527\pi$
$108$	112658.	0.929403
$109$	−28323.4	−0.228339	−0.114169	−	0.993461i	$-0.536421\pi$
$109$	−28323.4	−0.228339	−0.114169	+	0.993461i	$0.536421\pi$
$110$	0	0
$111$	53799.1	0.414446
$112$	−20555.7	−0.154841
$113$	252933.	1.86341	0.931706	−	0.363214i	$-0.118321\pi$
$113$	252933.	1.86341	0.931706	+	0.363214i	$0.118321\pi$
$114$	−28075.7	−0.202333
$115$	0	0
$116$	222150.	1.53286
$117$	−215730.	−1.45696
$118$	−152559.	−1.00863
$119$	415280.	2.68828
$120$	0	0
$121$	55612.4	0.345309
$122$	−156154.	−0.949848
$123$	62608.3	0.373138
$124$	180341.	1.05327
$125$	0	0
$126$	−524581.	−2.94365
$127$	318701.	1.75337	0.876686	−	0.481064i	$-0.159749\pi$
$127$	318701.	1.75337	0.876686	+	0.481064i	$0.159749\pi$
$128$	310104.	1.67295
$129$	−8471.20	−0.0448199
$130$	0	0
$131$	−1513.18	−0.00770392	−0.00385196	−	0.999993i	$-0.501226\pi$
$131$	−1513.18	−0.00770392	−0.00385196	+	0.999993i	$0.501226\pi$
$132$	−112770.	−0.563325
$133$	170591.	0.836231
$134$	167765.	0.807121
$135$	0	0
$136$	311488.	1.44409
$137$	400540.	1.82324	0.911620	−	0.411033i	$-0.134832\pi$
$137$	400540.	1.82324	0.911620	+	0.411033i	$0.134832\pi$
$138$	−153038.	−0.684073
$139$	86118.0	0.378057	0.189028	−	0.981972i	$-0.439466\pi$
$139$	86118.0	0.378057	0.189028	+	0.981972i	$0.439466\pi$
$140$	0	0
$141$	−58870.1	−0.249372
$142$	−172741.	−0.718911
$143$	452305.	1.84966
$144$	−17793.7	−0.0715091
$145$	0	0
$146$	−70053.1	−0.271985
$147$	−224356.	−0.856337
$148$	−620955.	−2.33027
$149$	−112164.	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$149$	−112164.	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$150$	0	0
$151$	35447.4	0.126515	0.0632575	−	0.997997i	$-0.479851\pi$
$151$	35447.4	0.126515	0.0632575	+	0.997997i	$0.479851\pi$
$152$	127955.	0.449208
$153$	359481.	1.24150
$154$	1.09985e6	3.73707
$155$	0	0
$156$	−235418.	−0.774513
$157$	34838.3	0.112800	0.0563998	−	0.998408i	$-0.482038\pi$
$157$	34838.3	0.112800	0.0563998	+	0.998408i	$0.482038\pi$
$158$	383768.	1.22300
$159$	63873.7	0.200368
$160$	0	0
$161$	929877.	2.82723
$162$	−407104.	−1.21876
$163$	256299.	0.755574	0.377787	−	0.925892i	$-0.376685\pi$
$163$	256299.	0.755574	0.377787	+	0.925892i	$0.376685\pi$
$164$	−722631.	−2.09801
$165$	0	0
$166$	−228695.	−0.644149
$167$	−147475.	−0.409193	−0.204597	−	0.978846i	$-0.565588\pi$
$167$	−147475.	−0.409193	−0.204597	+	0.978846i	$0.565588\pi$
$168$	−226039.	−0.617888
$169$	572937.	1.54309
$170$	0	0
$171$	147669.	0.386189
$172$	97775.5	0.252005
$173$	−342843.	−0.870925	−0.435462	−	0.900207i	$-0.643415\pi$
$173$	−342843.	−0.870925	−0.435462	+	0.900207i	$0.643415\pi$
$174$	177323.	0.444009
$175$	0	0
$176$	37306.8	0.0907834
$177$	−75865.4	−0.182016
$178$	−579418.	−1.37070
$179$	228584.	0.533229	0.266615	−	0.963803i	$-0.414095\pi$
$179$	228584.	0.533229	0.266615	+	0.963803i	$0.414095\pi$
$180$	0	0
$181$	−155330.	−0.352418	−0.176209	−	0.984353i	$-0.556383\pi$
$181$	−155330.	−0.352418	−0.176209	+	0.984353i	$0.556383\pi$
$182$	2.29604e6	5.13808
$183$	−77653.1	−0.171408
$184$	697471.	1.51873
$185$	0	0
$186$	143951.	0.305093
$187$	−753697.	−1.57613
$188$	679485.	1.40212
$189$	−546396.	−1.11264
$190$	0	0
$191$	184274.	0.365494	0.182747	−	0.983160i	$-0.441501\pi$
$191$	184274.	0.365494	0.182747	+	0.983160i	$0.441501\pi$
$192$	240418.	0.470667
$193$	267816.	0.517539	0.258769	−	0.965939i	$-0.416683\pi$
$193$	267816.	0.517539	0.258769	+	0.965939i	$0.416683\pi$
$194$	667704.	1.27374
$195$	0	0
$196$	2.58954e6	4.81485
$197$	589398.	1.08204	0.541020	−	0.841010i	$-0.318038\pi$
$197$	589398.	1.08204	0.541020	+	0.841010i	$0.318038\pi$
$198$	952068.	1.72586
$199$	184875.	0.330938	0.165469	−	0.986215i	$-0.447086\pi$
$199$	184875.	0.330938	0.165469	+	0.986215i	$0.447086\pi$
$200$	0	0
$201$	83426.8	0.145652
$202$	−588888.	−1.01544
$203$	−1.07743e6	−1.83506
$204$	392288.	0.659979
$205$	0	0
$206$	1.16477e6	1.91237
$207$	804935.	1.30567
$208$	77881.5	0.124818
$209$	−309607.	−0.490282
$210$	0	0
$211$	133546.	0.206502	0.103251	−	0.994655i	$-0.467075\pi$
$211$	133546.	0.206502	0.103251	+	0.994655i	$0.467075\pi$
$212$	−737237.	−1.12659
$213$	−85901.5	−0.129733
$214$	10466.2	0.0156226
$215$	0	0
$216$	−409834.	−0.597687
$217$	−874661.	−1.26093
$218$	260945.	0.371884
$219$	−34836.3	−0.0490820
$220$	0	0
$221$	−1.57341e6	−2.16702
$222$	−495653.	−0.674987
$223$	−304916.	−0.410600	−0.205300	−	0.978699i	$-0.565817\pi$
$223$	−304916.	−0.410600	−0.205300	+	0.978699i	$0.565817\pi$
$224$	−1.38941e6	−1.85017
$225$	0	0
$226$	−2.33028e6	−3.03485
$227$	−245927.	−0.316769	−0.158384	−	0.987378i	$-0.550629\pi$
$227$	−245927.	−0.316769	−0.158384	+	0.987378i	$0.550629\pi$
$228$	161146.	0.205297
$229$	1.27564e6	1.60745	0.803727	−	0.594998i	$-0.202847\pi$
$229$	1.27564e6	1.60745	0.803727	+	0.594998i	$0.202847\pi$
$230$	0	0
$231$	546938.	0.674385
$232$	−808148.	−0.985759
$233$	−930644.	−1.12304	−0.561518	−	0.827465i	$-0.689782\pi$
$233$	−930644.	−1.12304	−0.561518	+	0.827465i	$0.689782\pi$
$234$	1.98753e6	2.37287
$235$	0	0
$236$	875646.	1.02341
$237$	190842.	0.220700
$238$	−3.82599e6	−4.37826
$239$	−469691.	−0.531885	−0.265943	−	0.963989i	$-0.585683\pi$
$239$	−469691.	−0.531885	−0.265943	+	0.963989i	$0.585683\pi$
$240$	0	0
$241$	590289.	0.654669	0.327335	−	0.944908i	$-0.393850\pi$
$241$	590289.	0.654669	0.327335	+	0.944908i	$0.393850\pi$
$242$	−512360.	−0.562389
$243$	−720144.	−0.782355
$244$	896280.	0.963760
$245$	0	0
$246$	−576813.	−0.607711
$247$	−646335.	−0.674086
$248$	−656055.	−0.677347
$249$	−113727.	−0.116242
$250$	0	0
$251$	705112.	0.706437	0.353219	−	0.935541i	$-0.385087\pi$
$251$	705112.	0.706437	0.353219	+	0.935541i	$0.385087\pi$
$252$	3.01094e6	2.98676
$253$	−1.68765e6	−1.65760
$254$	−2.93620e6	−2.85563
$255$	0	0
$256$	−1.17778e6	−1.12322
$257$	−1.72516e6	−1.62928	−0.814642	−	0.579965i	$-0.803066\pi$
$257$	−1.72516e6	−1.62928	−0.814642	+	0.579965i	$0.803066\pi$
$258$	78045.6	0.0729960
$259$	3.01165e6	2.78968
$260$	0	0
$261$	−932664.	−0.847470
$262$	13941.0	0.0125470
$263$	1.64106e6	1.46297	0.731484	−	0.681858i	$-0.238828\pi$
$263$	1.64106e6	1.46297	0.731484	+	0.681858i	$0.238828\pi$
$264$	410241.	0.362267
$265$	0	0
$266$	−1.57166e6	−1.36193
$267$	−288136.	−0.247354
$268$	−962921.	−0.818943
$269$	−1.25631e6	−1.05856	−0.529279	−	0.848448i	$-0.677538\pi$
$269$	−1.25631e6	−1.05856	−0.529279	+	0.848448i	$0.677538\pi$
$270$	0	0
$271$	983561.	0.813538	0.406769	−	0.913531i	$-0.366655\pi$
$271$	983561.	0.813538	0.406769	+	0.913531i	$0.366655\pi$
$272$	−129777.	−0.106360
$273$	1.14178e6	0.927209
$274$	−3.69019e6	−2.96942
$275$	0	0
$276$	878395.	0.694092
$277$	1.26732e6	0.992398	0.496199	−	0.868209i	$-0.334729\pi$
$277$	1.26732e6	0.992398	0.496199	+	0.868209i	$0.334729\pi$
$278$	−793409.	−0.615723
$279$	−757137.	−0.582323
$280$	0	0
$281$	−107091.	−0.0809072	−0.0404536	−	0.999181i	$-0.512880\pi$
$281$	−107091.	−0.0809072	−0.0404536	+	0.999181i	$0.512880\pi$
$282$	542373.	0.406139
$283$	2.03193e6	1.50814	0.754071	−	0.656793i	$-0.228088\pi$
$283$	2.03193e6	1.50814	0.754071	+	0.656793i	$0.228088\pi$
$284$	991484.	0.729441
$285$	0	0
$286$	−4.16711e6	−3.01245
$287$	3.50478e6	2.51163
$288$	−1.20272e6	−0.854446
$289$	1.20200e6	0.846561
$290$	0	0
$291$	332039.	0.229856
$292$	402084.	0.275969
$293$	−134124.	−0.0912720	−0.0456360	−	0.998958i	$-0.514531\pi$
$293$	−134124.	−0.0912720	−0.0456360	+	0.998958i	$0.514531\pi$
$294$	2.06700e6	1.39468
$295$	0	0
$296$	2.25894e6	1.49856
$297$	991661.	0.652337
$298$	1.03337e6	0.674086
$299$	−3.52312e6	−2.27903
$300$	0	0
$301$	−474214.	−0.301688
$302$	−326579.	−0.206049
$303$	−292845.	−0.183245
$304$	−53310.6	−0.0330849
$305$	0	0
$306$	−3.31192e6	−2.02198
$307$	−1.11378e6	−0.674456	−0.337228	−	0.941423i	$-0.609489\pi$
$307$	−1.11378e6	−0.674456	−0.337228	+	0.941423i	$0.609489\pi$
$308$	−6.31281e6	−3.79181
$309$	579221.	0.345103
$310$	0	0
$311$	−1.54255e6	−0.904352	−0.452176	−	0.891929i	$-0.649352\pi$
$311$	−1.54255e6	−0.904352	−0.452176	+	0.891929i	$0.649352\pi$
$312$	856416.	0.498079
$313$	2.68483e6	1.54902	0.774509	−	0.632563i	$-0.217997\pi$
$313$	2.68483e6	1.54902	0.774509	+	0.632563i	$0.217997\pi$
$314$	−320967.	−0.183711
$315$	0	0
$316$	−2.20272e6	−1.24091
$317$	−519936.	−0.290604	−0.145302	−	0.989387i	$-0.546415\pi$
$317$	−519936.	−0.290604	−0.145302	+	0.989387i	$0.546415\pi$
$318$	−588471.	−0.326330
$319$	1.95545e6	1.07589
$320$	0	0
$321$	5204.67	0.00281923
$322$	−8.56700e6	−4.60457
$323$	1.07702e6	0.574403
$324$	2.33666e6	1.23661
$325$	0	0
$326$	−2.36129e6	−1.23057
$327$	129764.	0.0671096
$328$	2.62882e6	1.34920
$329$	−3.29552e6	−1.67855
$330$	0	0
$331$	1.30767e6	0.656038	0.328019	−	0.944671i	$-0.393619\pi$
$331$	1.30767e6	0.656038	0.328019	+	0.944671i	$0.393619\pi$
$332$	1.31264e6	0.653584
$333$	2.60699e6	1.28833
$334$	1.35870e6	0.666433
$335$	0	0
$336$	94176.0	0.0455085
$337$	−2.37449e6	−1.13893	−0.569464	−	0.822017i	$-0.692849\pi$
$337$	−2.37449e6	−1.13893	−0.569464	+	0.822017i	$0.692849\pi$
$338$	−5.27849e6	−2.51315
$339$	−1.15881e6	−0.547663
$340$	0	0
$341$	1.58743e6	0.739281
$342$	−1.36049e6	−0.628968
$343$	−8.24885e6	−3.78581
$344$	−355692.	−0.162061
$345$	0	0
$346$	3.15863e6	1.41843
$347$	−1.26079e6	−0.562105	−0.281053	−	0.959692i	$-0.590684\pi$
$347$	−1.26079e6	−0.562105	−0.281053	+	0.959692i	$0.590684\pi$
$348$	−1.01778e6	−0.450512
$349$	−3.06942e6	−1.34894	−0.674471	−	0.738301i	$-0.735628\pi$
$349$	−3.06942e6	−1.34894	−0.674471	+	0.738301i	$0.735628\pi$
$350$	0	0
$351$	2.07018e6	0.896894
$352$	2.52166e6	1.08475
$353$	−2.88788e6	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$353$	−2.88788e6	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$354$	698951.	0.296441
$355$	0	0
$356$	3.32569e6	1.39077
$357$	−1.90261e6	−0.790094
$358$	−2.10596e6	−0.868445
$359$	−1.97243e6	−0.807730	−0.403865	−	0.914819i	$-0.632334\pi$
$359$	−1.97243e6	−0.807730	−0.403865	+	0.914819i	$0.632334\pi$
$360$	0	0
$361$	−2.03368e6	−0.821323
$362$	1.43106e6	0.573966
$363$	−254789.	−0.101488
$364$	−1.31786e7	−5.21333
$365$	0	0
$366$	715421.	0.279164
$367$	−2.04851e6	−0.793914	−0.396957	−	0.917837i	$-0.629934\pi$
$367$	−2.04851e6	−0.793914	−0.396957	+	0.917837i	$0.629934\pi$
$368$	−290592.	−0.111857
$369$	3.03386e6	1.15993
$370$	0	0
$371$	3.57562e6	1.34870
$372$	−826235.	−0.309561
$373$	4.66217e6	1.73507	0.867533	−	0.497380i	$-0.165705\pi$
$373$	4.66217e6	1.73507	0.867533	+	0.497380i	$0.165705\pi$
$374$	6.94385e6	2.56697
$375$	0	0
$376$	−2.47186e6	−0.901685
$377$	4.08218e6	1.47924
$378$	5.03397e6	1.81210
$379$	−1.65003e6	−0.590057	−0.295028	−	0.955488i	$-0.595329\pi$
$379$	−1.65003e6	−0.590057	−0.295028	+	0.955488i	$0.595329\pi$
$380$	0	0
$381$	−1.46013e6	−0.515322
$382$	−1.69772e6	−0.595262
$383$	543685.	0.189387	0.0946935	−	0.995506i	$-0.469813\pi$
$383$	543685.	0.189387	0.0946935	+	0.995506i	$0.469813\pi$
$384$	−1.42074e6	−0.491686
$385$	0	0
$386$	−2.46740e6	−0.842890
$387$	−410496.	−0.139326
$388$	−3.83242e6	−1.29239
$389$	−315960.	−0.105866	−0.0529332	−	0.998598i	$-0.516857\pi$
$389$	−315960.	−0.105866	−0.0529332	+	0.998598i	$0.516857\pi$
$390$	0	0
$391$	5.87074e6	1.94201
$392$	−9.42037e6	−3.09637
$393$	6932.64	0.00226421
$394$	−5.43015e6	−1.76227
$395$	0	0
$396$	−5.46460e6	−1.75114
$397$	2.58774e6	0.824031	0.412016	−	0.911177i	$-0.364825\pi$
$397$	2.58774e6	0.824031	0.412016	+	0.911177i	$0.364825\pi$
$398$	−1.70327e6	−0.538982
$399$	−781562.	−0.245771
$400$	0	0
$401$	−616297.	−0.191394	−0.0956972	−	0.995410i	$-0.530508\pi$
$401$	−616297.	−0.191394	−0.0956972	+	0.995410i	$0.530508\pi$
$402$	−768615.	−0.237216
$403$	3.31392e6	1.01643
$404$	3.38005e6	1.03031
$405$	0	0
$406$	9.92644e6	2.98867
$407$	−5.46587e6	−1.63559
$408$	−1.42709e6	−0.424424
$409$	−2.16406e6	−0.639678	−0.319839	−	0.947472i	$-0.603629\pi$
$409$	−2.16406e6	−0.639678	−0.319839	+	0.947472i	$0.603629\pi$
$410$	0	0
$411$	−1.83507e6	−0.535857
$412$	−6.68543e6	−1.94038
$413$	−4.24691e6	−1.22517
$414$	−7.41590e6	−2.12649
$415$	0	0
$416$	5.26420e6	1.49142
$417$	−394550.	−0.111112
$418$	2.85243e6	0.798498
$419$	363871.	0.101254	0.0506271	−	0.998718i	$-0.483878\pi$
$419$	363871.	0.101254	0.0506271	+	0.998718i	$0.483878\pi$
$420$	0	0
$421$	−4.93609e6	−1.35731	−0.678653	−	0.734459i	$-0.737436\pi$
$421$	−4.93609e6	−1.35731	−0.678653	+	0.734459i	$0.737436\pi$
$422$	−1.23037e6	−0.336320
$423$	−2.85272e6	−0.775190
$424$	2.68195e6	0.724497
$425$	0	0
$426$	791415.	0.211291
$427$	−4.34698e6	−1.15377
$428$	−60072.8	−0.0158514
$429$	−2.07224e6	−0.543621
$430$	0	0
$431$	−179319.	−0.0464979	−0.0232489	−	0.999730i	$-0.507401\pi$
$431$	−179319.	−0.0464979	−0.0232489	+	0.999730i	$0.507401\pi$
$432$	170752.	0.0440206
$433$	5.77117e6	1.47926	0.739630	−	0.673014i	$-0.235001\pi$
$433$	5.77117e6	1.47926	0.739630	+	0.673014i	$0.235001\pi$
$434$	8.05829e6	2.05361
$435$	0	0
$436$	−1.49775e6	−0.377331
$437$	2.41161e6	0.604093
$438$	320949.	0.0799374
$439$	−4.17056e6	−1.03284	−0.516421	−	0.856335i	$-0.672736\pi$
$439$	−4.17056e6	−1.03284	−0.516421	+	0.856335i	$0.672736\pi$
$440$	0	0
$441$	−1.08718e7	−2.66199
$442$	1.44959e7	3.52932
$443$	2.99752e6	0.725692	0.362846	−	0.931849i	$-0.381805\pi$
$443$	2.99752e6	0.725692	0.362846	+	0.931849i	$0.381805\pi$
$444$	2.84491e6	0.684874
$445$	0	0
$446$	2.80921e6	0.668724
$447$	513879.	0.121644
$448$	1.34585e7	3.16812
$449$	−5.72247e6	−1.33958	−0.669789	−	0.742552i	$-0.733615\pi$
$449$	−5.72247e6	−1.33958	−0.669789	+	0.742552i	$0.733615\pi$
$450$	0	0
$451$	−6.36087e6	−1.47257
$452$	1.33751e7	3.07930
$453$	−162403.	−0.0371832
$454$	2.26574e6	0.515906
$455$	0	0
$456$	−586225.	−0.132024
$457$	−4.33119e6	−0.970100	−0.485050	−	0.874486i	$-0.661199\pi$
$457$	−4.33119e6	−0.970100	−0.485050	+	0.874486i	$0.661199\pi$
$458$	−1.17525e7	−2.61798
$459$	−3.44964e6	−0.764263
$460$	0	0
$461$	−5.96681e6	−1.30765	−0.653823	−	0.756648i	$-0.726836\pi$
$461$	−5.96681e6	−1.30765	−0.653823	+	0.756648i	$0.726836\pi$
$462$	−5.03896e6	−1.09834
$463$	−5.98996e6	−1.29859	−0.649294	−	0.760538i	$-0.724935\pi$
$463$	−5.98996e6	−1.29859	−0.649294	+	0.760538i	$0.724935\pi$
$464$	336704.	0.0726028
$465$	0	0
$466$	8.57406e6	1.82903
$467$	1.62449e6	0.344688	0.172344	−	0.985037i	$-0.444866\pi$
$467$	1.62449e6	0.344688	0.172344	+	0.985037i	$0.444866\pi$
$468$	−1.14079e7	−2.40763
$469$	4.67019e6	0.980398
$470$	0	0
$471$	−159612.	−0.0331522
$472$	−3.18547e6	−0.658140
$473$	860656.	0.176879
$474$	−1.75824e6	−0.359444
$475$	0	0
$476$	2.19601e7	4.44239
$477$	3.09518e6	0.622859
$478$	4.32729e6	0.866255
$479$	−2.76889e6	−0.551400	−0.275700	−	0.961244i	$-0.588910\pi$
$479$	−2.76889e6	−0.551400	−0.275700	+	0.961244i	$0.588910\pi$
$480$	0	0
$481$	−1.14105e7	−2.24876
$482$	−5.43836e6	−1.06623
$483$	−4.26024e6	−0.830933
$484$	2.94080e6	0.570626
$485$	0	0
$486$	6.63472e6	1.27418
$487$	6.58262e6	1.25770	0.628849	−	0.777527i	$-0.283526\pi$
$487$	6.58262e6	1.25770	0.628849	+	0.777527i	$0.283526\pi$
$488$	−3.26053e6	−0.619782
$489$	−1.17423e6	−0.222066
$490$	0	0
$491$	−9.73855e6	−1.82302	−0.911508	−	0.411282i	$-0.865081\pi$
$491$	−9.73855e6	−1.82302	−0.911508	+	0.411282i	$0.865081\pi$
$492$	3.31074e6	0.616612
$493$	−6.80233e6	−1.26049
$494$	5.95471e6	1.09785
$495$	0	0
$496$	273337.	0.0498877
$497$	−4.80872e6	−0.873251
$498$	1.04777e6	0.189318
$499$	3.51711e6	0.632316	0.316158	−	0.948707i	$-0.397607\pi$
$499$	3.51711e6	0.632316	0.316158	+	0.948707i	$0.397607\pi$
$500$	0	0
$501$	675660.	0.120263
$502$	−6.49622e6	−1.15054
$503$	4.37082e6	0.770269	0.385135	−	0.922860i	$-0.374155\pi$
$503$	4.37082e6	0.770269	0.385135	+	0.922860i	$0.374155\pi$
$504$	−1.09533e7	−1.92075
$505$	0	0
$506$	1.55484e7	2.69965
$507$	−2.62491e6	−0.453519
$508$	1.68530e7	2.89746
$509$	−6.34696e6	−1.08585	−0.542927	−	0.839780i	$-0.682684\pi$
$509$	−6.34696e6	−1.08585	−0.542927	+	0.839780i	$0.682684\pi$
$510$	0	0
$511$	−1.95012e6	−0.330376
$512$	927580.	0.156378
$513$	−1.41706e6	−0.237736
$514$	1.58940e7	2.65353
$515$	0	0
$516$	−447959.	−0.0740651
$517$	5.98108e6	0.984131
$518$	−2.77464e7	−4.54342
$519$	1.57074e6	0.255968
$520$	0	0
$521$	−1.76861e6	−0.285455	−0.142728	−	0.989762i	$-0.545587\pi$
$521$	−1.76861e6	−0.285455	−0.142728	+	0.989762i	$0.545587\pi$
$522$	8.59268e6	1.38023
$523$	5.91067e6	0.944893	0.472446	−	0.881359i	$-0.343371\pi$
$523$	5.91067e6	0.944893	0.472446	+	0.881359i	$0.343371\pi$
$524$	−80017.2	−0.0127308
$525$	0	0
$526$	−1.51192e7	−2.38267
$527$	−5.52213e6	−0.866124
$528$	−170921.	−0.0266816
$529$	6.70916e6	1.04239
$530$	0	0
$531$	−3.67627e6	−0.565811
$532$	9.02087e6	1.38188
$533$	−1.32789e7	−2.02462
$534$	2.65461e6	0.402853
$535$	0	0
$536$	3.50296e6	0.526651
$537$	−1.04726e6	−0.156718
$538$	1.15744e7	1.72402
$539$	2.27941e7	3.37949
$540$	0	0
$541$	−1.81452e6	−0.266543	−0.133272	−	0.991080i	$-0.542548\pi$
$541$	−1.81452e6	−0.266543	−0.133272	+	0.991080i	$0.542548\pi$
$542$	−9.06159e6	−1.32497
$543$	711644.	0.103577
$544$	−8.77198e6	−1.27087
$545$	0	0
$546$	−1.05193e7	−1.51010
$547$	−419554.	−0.0599542	−0.0299771	−	0.999551i	$-0.509543\pi$
$547$	−419554.	−0.0599542	−0.0299771	+	0.999551i	$0.509543\pi$
$548$	2.11806e7	3.01292
$549$	−3.76290e6	−0.532834
$550$	0	0
$551$	−2.79429e6	−0.392096
$552$	−3.19547e6	−0.446361
$553$	1.06832e7	1.48556
$554$	−1.16758e7	−1.61627
$555$	0	0
$556$	4.55394e6	0.624741
$557$	2.71282e6	0.370496	0.185248	−	0.982692i	$-0.440691\pi$
$557$	2.71282e6	0.370496	0.185248	+	0.982692i	$0.440691\pi$
$558$	6.97554e6	0.948402
$559$	1.79670e6	0.243190
$560$	0	0
$561$	3.45307e6	0.463231
$562$	986634.	0.131770
$563$	1.33486e7	1.77486	0.887429	−	0.460944i	$-0.152489\pi$
$563$	1.33486e7	1.77486	0.887429	+	0.460944i	$0.152489\pi$
$564$	−3.11306e6	−0.412088
$565$	0	0
$566$	−1.87202e7	−2.45624
$567$	−1.13328e7	−1.48041
$568$	−3.60687e6	−0.469094
$569$	−1.52081e7	−1.96922	−0.984608	−	0.174778i	$-0.944079\pi$
$569$	−1.52081e7	−1.96922	−0.984608	+	0.174778i	$0.944079\pi$
$570$	0	0
$571$	−1.00970e7	−1.29599	−0.647997	−	0.761643i	$-0.724393\pi$
$571$	−1.00970e7	−1.29599	−0.647997	+	0.761643i	$0.724393\pi$
$572$	2.39180e7	3.05657
$573$	−844251.	−0.107420
$574$	−3.22897e7	−4.09057
$575$	0	0
$576$	1.16501e7	1.46310
$577$	1.58468e6	0.198153	0.0990766	−	0.995080i	$-0.468411\pi$
$577$	1.58468e6	0.198153	0.0990766	+	0.995080i	$0.468411\pi$
$578$	−1.10740e7	−1.37875
$579$	−1.22700e6	−0.152107
$580$	0	0
$581$	−6.36635e6	−0.782439
$582$	−3.05909e6	−0.374356
$583$	−6.48943e6	−0.790743
$584$	−1.46272e6	−0.177472
$585$	0	0
$586$	1.23569e6	0.148650
$587$	−1.66744e6	−0.199736	−0.0998680	−	0.995001i	$-0.531842\pi$
$587$	−1.66744e6	−0.199736	−0.0998680	+	0.995001i	$0.531842\pi$
$588$	−1.18640e7	−1.41510
$589$	−2.26841e6	−0.269422
$590$	0	0
$591$	−2.70033e6	−0.318016
$592$	−941157.	−0.110372
$593$	9.06333e6	1.05840	0.529202	−	0.848496i	$-0.322492\pi$
$593$	9.06333e6	1.05840	0.529202	+	0.848496i	$0.322492\pi$
$594$	−9.13621e6	−1.06243
$595$	0	0
$596$	−5.93124e6	−0.683959
$597$	−847008.	−0.0972638
$598$	3.24587e7	3.71174
$599$	−1.67612e7	−1.90870	−0.954350	−	0.298690i	$-0.903450\pi$
$599$	−1.67612e7	−1.90870	−0.954350	+	0.298690i	$0.903450\pi$
$600$	0	0
$601$	1.21393e7	1.37091	0.685455	−	0.728115i	$-0.259603\pi$
$601$	1.21393e7	1.37091	0.685455	+	0.728115i	$0.259603\pi$
$602$	4.36895e6	0.491344
$603$	4.04268e6	0.452769
$604$	1.87447e6	0.209067
$605$	0	0
$606$	2.69800e6	0.298442
$607$	−624059.	−0.0687471	−0.0343735	−	0.999409i	$-0.510944\pi$
$607$	−624059.	−0.0687471	−0.0343735	+	0.999409i	$0.510944\pi$
$608$	−3.60340e6	−0.395324
$609$	4.93626e6	0.539331
$610$	0	0
$611$	1.24861e7	1.35308
$612$	1.90094e7	2.05159
$613$	−1.77380e7	−1.90658	−0.953288	−	0.302064i	$-0.902324\pi$
$613$	−1.77380e7	−1.90658	−0.953288	+	0.302064i	$0.902324\pi$
$614$	1.02613e7	1.09845
$615$	0	0
$616$	2.29651e7	2.43846
$617$	−1.09395e6	−0.115687	−0.0578436	−	0.998326i	$-0.518422\pi$
$617$	−1.09395e6	−0.115687	−0.0578436	+	0.998326i	$0.518422\pi$
$618$	−5.33639e6	−0.562052
$619$	1.56291e7	1.63948	0.819741	−	0.572734i	$-0.194117\pi$
$619$	1.56291e7	1.63948	0.819741	+	0.572734i	$0.194117\pi$
$620$	0	0
$621$	−7.72429e6	−0.803766
$622$	1.42116e7	1.47287
$623$	−1.61297e7	−1.66497
$624$	−356814.	−0.0366843
$625$	0	0
$626$	−2.47355e7	−2.52281
$627$	1.41847e6	0.144096
$628$	1.84225e6	0.186402
$629$	1.90139e7	1.91622
$630$	0	0
$631$	1.50965e7	1.50940	0.754699	−	0.656071i	$-0.227783\pi$
$631$	1.50965e7	1.50940	0.754699	+	0.656071i	$0.227783\pi$
$632$	8.01315e6	0.798014
$633$	−611842.	−0.0606918
$634$	4.79019e6	0.473293
$635$	0	0
$636$	3.37765e6	0.331110
$637$	4.75849e7	4.64644
$638$	−1.80156e7	−1.75226
$639$	−4.16260e6	−0.403286
$640$	0	0
$641$	−4.37920e6	−0.420969	−0.210484	−	0.977597i	$-0.567504\pi$
$641$	−4.37920e6	−0.420969	−0.210484	+	0.977597i	$0.567504\pi$
$642$	−47950.8	−0.00459154
$643$	−1.70309e7	−1.62446	−0.812232	−	0.583334i	$-0.801748\pi$
$643$	−1.70309e7	−1.62446	−0.812232	+	0.583334i	$0.801748\pi$
$644$	4.91721e7	4.67201
$645$	0	0
$646$	−9.92261e6	−0.935502
$647$	−1.27556e7	−1.19795	−0.598977	−	0.800766i	$-0.704426\pi$
$647$	−1.27556e7	−1.19795	−0.598977	+	0.800766i	$0.704426\pi$
$648$	−8.50040e6	−0.795247
$649$	7.70776e6	0.718318
$650$	0	0
$651$	4.00726e6	0.370591
$652$	1.35531e7	1.24859
$653$	7.96481e6	0.730958	0.365479	−	0.930820i	$-0.380905\pi$
$653$	7.96481e6	0.730958	0.365479	+	0.930820i	$0.380905\pi$
$654$	−1.19552e6	−0.109298
$655$	0	0
$656$	−1.09526e6	−0.0993709
$657$	−1.68809e6	−0.152575
$658$	3.03618e7	2.73377
$659$	5.67581e6	0.509113	0.254557	−	0.967058i	$-0.418070\pi$
$659$	5.67581e6	0.509113	0.254557	+	0.967058i	$0.418070\pi$
$660$	0	0
$661$	2.75203e6	0.244990	0.122495	−	0.992469i	$-0.460910\pi$
$661$	2.75203e6	0.244990	0.122495	+	0.992469i	$0.460910\pi$
$662$	−1.20476e7	−1.06846
$663$	7.20860e6	0.636894
$664$	−4.77520e6	−0.420311
$665$	0	0
$666$	−2.40183e7	−2.09825
$667$	−1.52315e7	−1.32565
$668$	−7.79853e6	−0.676194
$669$	1.39698e6	0.120677
$670$	0	0
$671$	7.88939e6	0.676452
$672$	6.36559e6	0.543770
$673$	−1.10238e7	−0.938192	−0.469096	−	0.883147i	$-0.655420\pi$
$673$	−1.10238e7	−0.938192	−0.469096	+	0.883147i	$0.655420\pi$
$674$	2.18763e7	1.85492
$675$	0	0
$676$	3.02970e7	2.54996
$677$	3.11405e6	0.261129	0.130564	−	0.991440i	$-0.458321\pi$
$677$	3.11405e6	0.261129	0.130564	+	0.991440i	$0.458321\pi$
$678$	1.06762e7	0.891953
$679$	1.85874e7	1.54719
$680$	0	0
$681$	1.12672e6	0.0930995
$682$	−1.46251e7	−1.20403
$683$	−2.33180e7	−1.91267	−0.956335	−	0.292274i	$-0.905588\pi$
$683$	−2.33180e7	−1.91267	−0.956335	+	0.292274i	$0.905588\pi$
$684$	7.80879e6	0.638180
$685$	0	0
$686$	7.59971e7	6.16576
$687$	−5.84434e6	−0.472437
$688$	148195.	0.0119361
$689$	−1.35473e7	−1.08719
$690$	0	0
$691$	−2.20335e7	−1.75545	−0.877725	−	0.479165i	$-0.840939\pi$
$691$	−2.20335e7	−1.75545	−0.877725	+	0.479165i	$0.840939\pi$
$692$	−1.81296e7	−1.43921
$693$	2.65034e7	2.09637
$694$	1.16157e7	0.915474
$695$	0	0
$696$	3.70253e6	0.289718
$697$	2.21273e7	1.72523
$698$	2.82787e7	2.19696
$699$	4.26375e6	0.330064
$700$	0	0
$701$	−2.13110e7	−1.63798	−0.818989	−	0.573809i	$-0.805465\pi$
$701$	−2.13110e7	−1.63798	−0.818989	+	0.573809i	$0.805465\pi$
$702$	−1.90727e7	−1.46073
$703$	7.81062e6	0.596070
$704$	−2.44260e7	−1.85746
$705$	0	0
$706$	2.66062e7	2.00896
$707$	−1.63933e7	−1.23344
$708$	−4.01178e6	−0.300783
$709$	8.64200e6	0.645653	0.322826	−	0.946458i	$-0.395367\pi$
$709$	8.64200e6	0.645653	0.322826	+	0.946458i	$0.395367\pi$
$710$	0	0
$711$	9.24779e6	0.686063
$712$	−1.20984e7	−0.894389
$713$	−1.23649e7	−0.910893
$714$	1.75288e7	1.28679
$715$	0	0
$716$	1.20876e7	0.881165
$717$	2.15189e6	0.156323
$718$	1.81721e7	1.31551
$719$	9.89921e6	0.714131	0.357066	−	0.934079i	$-0.383777\pi$
$719$	9.89921e6	0.714131	0.357066	+	0.934079i	$0.383777\pi$
$720$	0	0
$721$	3.24245e7	2.32292
$722$	1.87364e7	1.33765
$723$	−2.70441e6	−0.192410
$724$	−8.21386e6	−0.582373
$725$	0	0
$726$	2.34738e6	0.165288
$727$	−1.23688e7	−0.867940	−0.433970	−	0.900927i	$-0.642888\pi$
$727$	−1.23688e7	−0.867940	−0.433970	+	0.900927i	$0.642888\pi$
$728$	4.79417e7	3.35263
$729$	−7.43828e6	−0.518386
$730$	0	0
$731$	−2.99393e6	−0.207228
$732$	−4.10631e6	−0.283253
$733$	4.38004e6	0.301105	0.150552	−	0.988602i	$-0.451895\pi$
$733$	4.38004e6	0.301105	0.150552	+	0.988602i	$0.451895\pi$
$734$	1.88730e7	1.29301
$735$	0	0
$736$	−1.96418e7	−1.33656
$737$	−8.47599e6	−0.574806
$738$	−2.79511e7	−1.88911
$739$	8.85977e6	0.596776	0.298388	−	0.954445i	$-0.403551\pi$
$739$	8.85977e6	0.596776	0.298388	+	0.954445i	$0.403551\pi$
$740$	0	0
$741$	2.96118e6	0.198116
$742$	−3.29423e7	−2.19657
$743$	2.83714e6	0.188542	0.0942710	−	0.995547i	$-0.469948\pi$
$743$	2.83714e6	0.188542	0.0942710	+	0.995547i	$0.469948\pi$
$744$	3.00572e6	0.199075
$745$	0	0
$746$	−4.29528e7	−2.82582
$747$	−5.51094e6	−0.361347
$748$	−3.98557e7	−2.60457
$749$	291355.	0.0189765
$750$	0	0
$751$	8.81181e6	0.570119	0.285059	−	0.958510i	$-0.407987\pi$
$751$	8.81181e6	0.570119	0.285059	+	0.958510i	$0.407987\pi$
$752$	1.02987e6	0.0664106
$753$	−3.23047e6	−0.207624
$754$	−3.76093e7	−2.40917
$755$	0	0
$756$	−2.88935e7	−1.83864
$757$	−2.42976e7	−1.54107	−0.770536	−	0.637396i	$-0.780012\pi$
$757$	−2.42976e7	−1.54107	−0.770536	+	0.637396i	$0.780012\pi$
$758$	1.52018e7	0.960997
$759$	7.73196e6	0.487175
$760$	0	0
$761$	−3.03773e7	−1.90146	−0.950730	−	0.310021i	$-0.899664\pi$
$761$	−3.03773e7	−1.90146	−0.950730	+	0.310021i	$0.899664\pi$
$762$	1.34522e7	0.839280
$763$	7.26412e6	0.451723
$764$	9.74443e6	0.603981
$765$	0	0
$766$	−5.00899e6	−0.308446
$767$	1.60907e7	0.987611
$768$	5.39599e6	0.330117
$769$	−2.85943e7	−1.74367	−0.871833	−	0.489804i	$-0.837068\pi$
$769$	−2.85943e7	−1.74367	−0.871833	+	0.489804i	$0.837068\pi$
$770$	0	0
$771$	7.90383e6	0.478852
$772$	1.41621e7	0.855236
$773$	−2.46040e6	−0.148101	−0.0740503	−	0.997255i	$-0.523593\pi$
$773$	−2.46040e6	−0.148101	−0.0740503	+	0.997255i	$0.523593\pi$
$774$	3.78192e6	0.226913
$775$	0	0
$776$	1.39418e7	0.831120
$777$	−1.37979e7	−0.819897
$778$	2.91095e6	0.172420
$779$	9.08955e6	0.536659
$780$	0	0
$781$	8.72742e6	0.511986
$782$	−5.40874e7	−3.16285
$783$	8.95001e6	0.521698
$784$	3.92487e6	0.228053
$785$	0	0
$786$	−63870.7	−0.00368761
$787$	−1.22092e7	−0.702667	−0.351334	−	0.936250i	$-0.614272\pi$
$787$	−1.22092e7	−0.702667	−0.351334	+	0.936250i	$0.614272\pi$
$788$	3.11675e7	1.78808
$789$	−7.51852e6	−0.429972
$790$	0	0
$791$	−6.48697e7	−3.68639
$792$	1.98794e7	1.12613
$793$	1.64698e7	0.930050
$794$	−2.38409e7	−1.34206
$795$	0	0
$796$	9.77625e6	0.546877
$797$	−1.12683e7	−0.628368	−0.314184	−	0.949362i	$-0.601731\pi$
$797$	−1.12683e7	−0.628368	−0.314184	+	0.949362i	$0.601731\pi$
$798$	7.20057e6	0.400276
$799$	−2.08061e7	−1.15299
$800$	0	0
$801$	−1.39624e7	−0.768917
$802$	5.67797e6	0.311715
$803$	3.53930e6	0.193699
$804$	4.41163e6	0.240690
$805$	0	0
$806$	−3.05313e7	−1.65542
$807$	5.75578e6	0.311114
$808$	−1.22961e7	−0.662581
$809$	−1.59272e7	−0.855594	−0.427797	−	0.903875i	$-0.640710\pi$
$809$	−1.59272e7	−0.855594	−0.427797	+	0.903875i	$0.640710\pi$
$810$	0	0
$811$	3.11381e7	1.66242	0.831208	−	0.555962i	$-0.187650\pi$
$811$	3.11381e7	1.66242	0.831208	+	0.555962i	$0.187650\pi$
$812$	−5.69749e7	−3.03245
$813$	−4.50619e6	−0.239102
$814$	5.03573e7	2.66380
$815$	0	0
$816$	594576.	0.0312595
$817$	−1.22986e6	−0.0644615
$818$	1.99376e7	1.04181
$819$	5.53284e7	2.88229
$820$	0	0
$821$	788370.	0.0408199	0.0204100	−	0.999792i	$-0.493503\pi$
$821$	788370.	0.0408199	0.0204100	+	0.999792i	$0.493503\pi$
$822$	1.69066e7	0.872725
$823$	1.25307e7	0.644877	0.322438	−	0.946590i	$-0.395498\pi$
$823$	1.25307e7	0.644877	0.322438	+	0.946590i	$0.395498\pi$
$824$	2.43206e7	1.24783
$825$	0	0
$826$	3.91269e7	1.99538
$827$	2.50390e7	1.27307	0.636537	−	0.771246i	$-0.280366\pi$
$827$	2.50390e7	1.27307	0.636537	+	0.771246i	$0.280366\pi$
$828$	4.25651e7	2.15763
$829$	2.29892e7	1.16182	0.580909	−	0.813969i	$-0.302697\pi$
$829$	2.29892e7	1.16182	0.580909	+	0.813969i	$0.302697\pi$
$830$	0	0
$831$	−5.80622e6	−0.291669
$832$	−5.09915e7	−2.55382
$833$	−7.92929e7	−3.95933
$834$	3.63501e6	0.180963
$835$	0	0
$836$	−1.63721e7	−0.810193
$837$	7.26562e6	0.358475
$838$	−3.35236e6	−0.164908
$839$	1.19050e7	0.583882	0.291941	−	0.956436i	$-0.405699\pi$
$839$	1.19050e7	0.583882	0.291941	+	0.956436i	$0.405699\pi$
$840$	0	0
$841$	−2.86270e6	−0.139568
$842$	4.54764e7	2.21058
$843$	490638.	0.0237789
$844$	7.06194e6	0.341246
$845$	0	0
$846$	2.62822e7	1.26251
$847$	−1.42629e7	−0.683125
$848$	−1.11740e6	−0.0533604
$849$	−9.30928e6	−0.443248
$850$	0	0
$851$	4.25751e7	2.01526
$852$	−4.54249e6	−0.214385
$853$	3.89723e7	1.83393	0.916966	−	0.398966i	$-0.130631\pi$
$853$	3.89723e7	1.83393	0.916966	+	0.398966i	$0.130631\pi$
$854$	4.00489e7	1.87908
$855$	0	0
$856$	218536.	0.0101938
$857$	−1.65243e7	−0.768547	−0.384274	−	0.923219i	$-0.625548\pi$
$857$	−1.65243e7	−0.768547	−0.384274	+	0.923219i	$0.625548\pi$
$858$	1.90916e7	0.885370
$859$	−9.30251e6	−0.430147	−0.215074	−	0.976598i	$-0.568999\pi$
$859$	−9.30251e6	−0.430147	−0.215074	+	0.976598i	$0.568999\pi$
$860$	0	0
$861$	−1.60572e7	−0.738178
$862$	1.65207e6	0.0757288
$863$	−147398.	−0.00673699	−0.00336849	−	0.999994i	$-0.501072\pi$
$863$	−147398.	−0.00673699	−0.00336849	+	0.999994i	$0.501072\pi$
$864$	1.15415e7	0.525992
$865$	0	0
$866$	−5.31701e7	−2.40920
$867$	−5.50695e6	−0.248807
$868$	−4.62522e7	−2.08369
$869$	−1.93891e7	−0.870982
$870$	0	0
$871$	−1.76944e7	−0.790298
$872$	5.44859e6	0.242657
$873$	1.60899e7	0.714524
$874$	−2.22183e7	−0.983856
$875$	0	0
$876$	−1.84215e6	−0.0811083
$877$	3.42750e7	1.50480	0.752399	−	0.658707i	$-0.228896\pi$
$877$	3.42750e7	1.50480	0.752399	+	0.658707i	$0.228896\pi$
$878$	3.84236e7	1.68214
$879$	614490.	0.0268252
$880$	0	0
$881$	3.59944e7	1.56241	0.781205	−	0.624274i	$-0.214605\pi$
$881$	3.59944e7	1.56241	0.781205	+	0.624274i	$0.214605\pi$
$882$	1.00163e8	4.33545
$883$	−2.08344e7	−0.899247	−0.449624	−	0.893218i	$-0.648442\pi$
$883$	−2.08344e7	−0.899247	−0.449624	+	0.893218i	$0.648442\pi$
$884$	−8.32024e7	−3.58101
$885$	0	0
$886$	−2.76163e7	−1.18190
$887$	674157.	0.0287708	0.0143854	−	0.999897i	$-0.495421\pi$
$887$	674157.	0.0287708	0.0143854	+	0.999897i	$0.495421\pi$
$888$	−1.03493e7	−0.440433
$889$	−8.17373e7	−3.46869
$890$	0	0
$891$	2.05681e7	0.867961
$892$	−1.61240e7	−0.678518
$893$	−8.54683e6	−0.358655
$894$	−4.73439e6	−0.198116
$895$	0	0
$896$	−7.95325e7	−3.30959
$897$	1.61412e7	0.669814
$898$	5.27214e7	2.18170
$899$	1.43270e7	0.591230
$900$	0	0
$901$	2.25745e7	0.926416
$902$	5.86030e7	2.39830
$903$	2.17261e6	0.0886671
$904$	−4.86567e7	−1.98026
$905$	0	0
$906$	1.49622e6	0.0605586
$907$	3.98516e7	1.60852	0.804262	−	0.594276i	$-0.202561\pi$
$907$	3.98516e7	1.60852	0.804262	+	0.594276i	$0.202561\pi$
$908$	−1.30047e7	−0.523462
$909$	−1.41906e7	−0.569629
$910$	0	0
$911$	1.85267e7	0.739610	0.369805	−	0.929109i	$-0.379425\pi$
$911$	1.85267e7	0.739610	0.369805	+	0.929109i	$0.379425\pi$
$912$	244243.	0.00972377
$913$	1.15544e7	0.458743
$914$	3.99034e7	1.57996
$915$	0	0
$916$	6.74560e7	2.65633
$917$	388085.	0.0152407
$918$	3.17817e7	1.24472
$919$	−8.90949e6	−0.347988	−0.173994	−	0.984747i	$-0.555667\pi$
$919$	−8.90949e6	−0.347988	−0.173994	+	0.984747i	$0.555667\pi$
$920$	0	0
$921$	5.10279e6	0.198225
$922$	5.49725e7	2.12970
$923$	1.82193e7	0.703927
$924$	2.89222e7	1.11443
$925$	0	0
$926$	5.51857e7	2.11495
$927$	2.80678e7	1.07278
$928$	2.27587e7	0.867515
$929$	−3.39571e7	−1.29089	−0.645447	−	0.763805i	$-0.723329\pi$
$929$	−3.39571e7	−1.29089	−0.645447	+	0.763805i	$0.723329\pi$
$930$	0	0
$931$	−3.25723e7	−1.23161
$932$	−4.92126e7	−1.85582
$933$	7.06719e6	0.265792
$934$	−1.49665e7	−0.561376
$935$	0	0
$936$	4.15001e7	1.54831
$937$	5.59394e6	0.208146	0.104073	−	0.994570i	$-0.466812\pi$
$937$	5.59394e6	0.208146	0.104073	+	0.994570i	$0.466812\pi$
$938$	−4.30267e7	−1.59673
$939$	−1.23006e7	−0.455262
$940$	0	0
$941$	−5.85644e6	−0.215606	−0.107803	−	0.994172i	$-0.534381\pi$
$941$	−5.85644e6	−0.215606	−0.107803	+	0.994172i	$0.534381\pi$
$942$	1.47051e6	0.0539934
$943$	4.95464e7	1.81440
$944$	1.32718e6	0.0484731
$945$	0	0
$946$	−7.92926e6	−0.288075
$947$	−3.96169e7	−1.43551	−0.717755	−	0.696296i	$-0.754830\pi$
$947$	−3.96169e7	−1.43551	−0.717755	+	0.696296i	$0.754830\pi$
$948$	1.00918e7	0.364709
$949$	7.38861e6	0.266316
$950$	0	0
$951$	2.38209e6	0.0854096
$952$	−7.98875e7	−2.85684
$953$	−4.43057e7	−1.58026	−0.790128	−	0.612943i	$-0.789986\pi$
$953$	−4.43057e7	−1.58026	−0.790128	+	0.612943i	$0.789986\pi$
$954$	−2.85160e7	−1.01442
$955$	0	0
$956$	−2.48374e7	−0.878943
$957$	−8.95889e6	−0.316209
$958$	2.55099e7	0.898038
$959$	−1.02726e8	−3.60692
$960$	0	0
$961$	−1.69985e7	−0.593747
$962$	1.05126e8	3.66245
$963$	252207.	0.00876377
$964$	3.12146e7	1.08184
$965$	0	0
$966$	3.92498e7	1.35330
$967$	3.97613e7	1.36740	0.683698	−	0.729765i	$-0.260370\pi$
$967$	3.97613e7	1.36740	0.683698	+	0.729765i	$0.260370\pi$
$968$	−1.06982e7	−0.366962
$969$	−4.93436e6	−0.168819
$970$	0	0
$971$	−2.63035e6	−0.0895292	−0.0447646	−	0.998998i	$-0.514254\pi$
$971$	−2.63035e6	−0.0895292	−0.0447646	+	0.998998i	$0.514254\pi$
$972$	−3.80814e7	−1.29285
$973$	−2.20867e7	−0.747909
$974$	−6.06460e7	−2.04835
$975$	0	0
$976$	1.35846e6	0.0456480
$977$	−1.09614e7	−0.367393	−0.183697	−	0.982983i	$-0.558806\pi$
$977$	−1.09614e7	−0.367393	−0.183697	+	0.982983i	$0.558806\pi$
$978$	1.08183e7	0.361668
$979$	2.92740e7	0.976169
$980$	0	0
$981$	6.28808e6	0.208615
$982$	8.97217e7	2.96906
$983$	2.67152e7	0.881809	0.440905	−	0.897554i	$-0.354658\pi$
$983$	2.67152e7	0.881809	0.440905	+	0.897554i	$0.354658\pi$
$984$	−1.20440e7	−0.396535
$985$	0	0
$986$	6.26701e7	2.05290
$987$	1.50984e7	0.493332
$988$	−3.41783e7	−1.11393
$989$	−6.70387e6	−0.217939
$990$	0	0
$991$	3.20214e7	1.03575	0.517877	−	0.855455i	$-0.326722\pi$
$991$	3.20214e7	1.03575	0.517877	+	0.855455i	$0.326722\pi$
$992$	1.84755e7	0.596097
$993$	−5.99110e6	−0.192812
$994$	4.43030e7	1.42222
$995$	0	0
$996$	−6.01388e6	−0.192091
$997$	−6.48848e6	−0.206731	−0.103365	−	0.994643i	$-0.532961\pi$
$997$	−6.48848e6	−0.206731	−0.103365	+	0.994643i	$0.532961\pi$
$998$	−3.24033e7	−1.02982
$999$	−2.50171e7	−0.793092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.k.1.8		52
5.2	odd	4		215.6.b.a.44.14	✓	104
5.3	odd	4		215.6.b.a.44.91	yes	104
5.4	even	2		1075.6.a.l.1.45		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.b.a.44.14	✓	104	5.2	odd	4
215.6.b.a.44.91	yes	104	5.3	odd	4
1075.6.a.k.1.8		52	1.1	even	1	trivial
1075.6.a.l.1.45		52	5.4	even	2

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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-9.21304 q^{2} -4.58150 q^{3} +52.8802 q^{4} +42.2096 q^{6} -256.470 q^{7} -192.370 q^{8} -222.010 q^{9} +O(q^{10})\)	\(q-9.21304 q^{2} -4.58150 q^{3} +52.8802 q^{4} +42.2096 q^{6} -256.470 q^{7} -192.370 q^{8} -222.010 q^{9} +465.471 q^{11} -242.271 q^{12} +971.715 q^{13} +2362.87 q^{14} +80.1485 q^{16} -1619.21 q^{17} +2045.39 q^{18} -665.148 q^{19} +1175.02 q^{21} -4288.41 q^{22} -3625.67 q^{23} +881.345 q^{24} -8952.45 q^{26} +2130.44 q^{27} -13562.2 q^{28} +4201.01 q^{29} +3410.38 q^{31} +5417.43 q^{32} -2132.56 q^{33} +14917.9 q^{34} -11739.9 q^{36} -11742.7 q^{37} +6128.04 q^{38} -4451.92 q^{39} -13665.4 q^{41} -10825.5 q^{42} +1849.00 q^{43} +24614.2 q^{44} +33403.5 q^{46} +12849.5 q^{47} -367.201 q^{48} +48970.0 q^{49} +7418.43 q^{51} +51384.5 q^{52} -13941.6 q^{53} -19627.9 q^{54} +49337.2 q^{56} +3047.38 q^{57} -38704.1 q^{58} +16559.1 q^{59} +16949.3 q^{61} -31420.0 q^{62} +56938.9 q^{63} -52475.8 q^{64} +19647.4 q^{66} -18209.5 q^{67} -85624.3 q^{68} +16611.0 q^{69} +18749.6 q^{71} +42708.1 q^{72} +7603.69 q^{73} +108186. q^{74} -35173.2 q^{76} -119380. q^{77} +41015.7 q^{78} -41654.9 q^{79} +44187.7 q^{81} +125900. q^{82} +24823.0 q^{83} +62135.3 q^{84} -17034.9 q^{86} -19246.9 q^{87} -89542.8 q^{88} +62891.0 q^{89} -249216. q^{91} -191726. q^{92} -15624.7 q^{93} -118383. q^{94} -24820.0 q^{96} -72473.7 q^{97} -451163. q^{98} -103339. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9}+O(q^{10}) \) 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9	\( 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9} - 664 q^{11} + 523 q^{12} - 2704 q^{13} + 150 q^{14} + 13474 q^{16} - 7266 q^{17} - 4860 q^{18} - 1970 q^{19} + 800 q^{21} - 14477 q^{22} - 9522 q^{23} + 314 q^{24} + 5514 q^{26} - 22926 q^{27} - 9408 q^{28} - 7188 q^{29} - 11556 q^{31} - 48390 q^{32} - 26136 q^{33} + 16774 q^{34} + 51872 q^{36} - 42558 q^{37} - 46208 q^{38} + 4682 q^{39} - 7746 q^{41} - 174265 q^{42} + 96148 q^{43} - 48600 q^{44} + 16182 q^{46} - 87136 q^{47} + 2912 q^{48} + 142286 q^{49} - 3710 q^{51} - 146868 q^{52} - 127034 q^{53} - 49563 q^{54} - 2849 q^{56} - 101594 q^{57} - 9480 q^{58} - 55924 q^{59} + 73702 q^{61} - 186016 q^{62} - 50120 q^{63} + 157750 q^{64} + 58211 q^{66} - 131996 q^{67} - 298560 q^{68} + 128436 q^{69} - 56284 q^{71} - 343775 q^{72} - 128620 q^{73} - 17721 q^{74} - 170410 q^{76} - 448438 q^{77} - 237616 q^{78} + 106204 q^{79} + 478568 q^{81} - 249596 q^{82} - 348616 q^{83} - 131855 q^{84} - 36980 q^{86} - 267478 q^{87} - 525216 q^{88} + 80410 q^{89} + 226376 q^{91} - 581456 q^{92} - 902902 q^{93} + 180980 q^{94} + 38543 q^{96} - 316148 q^{97} - 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9 - 664 * q^11 + 523 * q^12 - 2704 * q^13 + 150 * q^14 + 13474 * q^16 - 7266 * q^17 - 4860 * q^18 - 1970 * q^19 + 800 * q^21 - 14477 * q^22 - 9522 * q^23 + 314 * q^24 + 5514 * q^26 - 22926 * q^27 - 9408 * q^28 - 7188 * q^29 - 11556 * q^31 - 48390 * q^32 - 26136 * q^33 + 16774 * q^34 + 51872 * q^36 - 42558 * q^37 - 46208 * q^38 + 4682 * q^39 - 7746 * q^41 - 174265 * q^42 + 96148 * q^43 - 48600 * q^44 + 16182 * q^46 - 87136 * q^47 + 2912 * q^48 + 142286 * q^49 - 3710 * q^51 - 146868 * q^52 - 127034 * q^53 - 49563 * q^54 - 2849 * q^56 - 101594 * q^57 - 9480 * q^58 - 55924 * q^59 + 73702 * q^61 - 186016 * q^62 - 50120 * q^63 + 157750 * q^64 + 58211 * q^66 - 131996 * q^67 - 298560 * q^68 + 128436 * q^69 - 56284 * q^71 - 343775 * q^72 - 128620 * q^73 - 17721 * q^74 - 170410 * q^76 - 448438 * q^77 - 237616 * q^78 + 106204 * q^79 + 478568 * q^81 - 249596 * q^82 - 348616 * q^83 - 131855 * q^84 - 36980 * q^86 - 267478 * q^87 - 525216 * q^88 + 80410 * q^89 + 226376 * q^91 - 581456 * q^92 - 902902 * q^93 + 180980 * q^94 + 38543 * q^96 - 316148 * q^97 - 295095 * q^98 + 68428 * q^99

Introduction

L-functions

Modular forms

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