LMFDB - Embedded newform 1075.6.a.b.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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$5.31531$ of defining polynomial
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)

$f(q)$	$=$	$q+4.31531 q^{2} +23.8469 q^{3} -13.3781 q^{4} +102.907 q^{6} +174.859 q^{7} -195.821 q^{8} +325.673 q^{9} +O(q^{10})$	\(q+4.31531 q^{2} +23.8469 q^{3} -13.3781 q^{4} +102.907 q^{6} +174.859 q^{7} -195.821 q^{8} +325.673 q^{9} -447.981 q^{11} -319.027 q^{12} -669.141 q^{13} +754.569 q^{14} -416.925 q^{16} +849.648 q^{17} +1405.38 q^{18} -1288.87 q^{19} +4169.83 q^{21} -1933.18 q^{22} +378.254 q^{23} -4669.71 q^{24} -2887.55 q^{26} +1971.50 q^{27} -2339.28 q^{28} +765.100 q^{29} -7094.59 q^{31} +4467.10 q^{32} -10683.0 q^{33} +3666.49 q^{34} -4356.90 q^{36} +7908.22 q^{37} -5561.87 q^{38} -15956.9 q^{39} +12855.9 q^{41} +17994.1 q^{42} -1849.00 q^{43} +5993.15 q^{44} +1632.28 q^{46} -26785.7 q^{47} -9942.37 q^{48} +13768.5 q^{49} +20261.5 q^{51} +8951.86 q^{52} -30000.7 q^{53} +8507.60 q^{54} -34240.9 q^{56} -30735.5 q^{57} +3301.64 q^{58} +1247.64 q^{59} -48441.4 q^{61} -30615.3 q^{62} +56946.8 q^{63} +32618.5 q^{64} -46100.2 q^{66} -67004.8 q^{67} -11366.7 q^{68} +9020.17 q^{69} -74553.1 q^{71} -63773.5 q^{72} +18066.8 q^{73} +34126.4 q^{74} +17242.7 q^{76} -78333.4 q^{77} -68859.0 q^{78} -63230.5 q^{79} -32124.6 q^{81} +55477.3 q^{82} +88066.7 q^{83} -55784.5 q^{84} -7979.00 q^{86} +18245.2 q^{87} +87724.0 q^{88} +50373.5 q^{89} -117005. q^{91} -5060.33 q^{92} -169184. q^{93} -115588. q^{94} +106526. q^{96} -17502.1 q^{97} +59415.5 q^{98} -145895. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) $ 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * 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 $

$2$	4.31531	0.762846	0.381423	−	0.924401i	$-0.375434\pi$
$2$	4.31531	0.762846	0.381423	+	0.924401i	$0.375434\pi$
$3$	23.8469	1.52978	0.764889	−	0.644163i	$-0.222794\pi$
$3$	23.8469	1.52978	0.764889	+	0.644163i	$0.222794\pi$
$4$	−13.3781	−0.418067
$5$	0	0
$5$	0	0
$6$	102.907	1.16698
$7$	174.859	1.34878	0.674391	−	0.738374i	$-0.264406\pi$
$7$	174.859	1.34878	0.674391	+	0.738374i	$0.264406\pi$
$8$	−195.821	−1.08177
$9$	325.673	1.34022
$10$	0	0
$11$	−447.981	−1.11629	−0.558147	−	0.829742i	$-0.688487\pi$
$11$	−447.981	−1.11629	−0.558147	+	0.829742i	$0.688487\pi$
$12$	−319.027	−0.639549
$13$	−669.141	−1.09814	−0.549072	−	0.835775i	$-0.685019\pi$
$13$	−669.141	−1.09814	−0.549072	+	0.835775i	$0.685019\pi$
$14$	754.569	1.02891
$15$	0	0
$16$	−416.925	−0.407154
$17$	849.648	0.713045	0.356522	−	0.934287i	$-0.383962\pi$
$17$	849.648	0.713045	0.356522	+	0.934287i	$0.383962\pi$
$18$	1405.38	1.02238
$19$	−1288.87	−0.819078	−0.409539	−	0.912293i	$-0.634310\pi$
$19$	−1288.87	−0.819078	−0.409539	+	0.912293i	$0.634310\pi$
$20$	0	0
$21$	4169.83	2.06334
$22$	−1933.18	−0.851559
$23$	378.254	0.149095	0.0745476	−	0.997217i	$-0.476249\pi$
$23$	378.254	0.149095	0.0745476	+	0.997217i	$0.476249\pi$
$24$	−4669.71	−1.65486
$25$	0	0
$26$	−2887.55	−0.837715
$27$	1971.50	0.520459
$28$	−2339.28	−0.563881
$29$	765.100	0.168936	0.0844682	−	0.996426i	$-0.473081\pi$
$29$	765.100	0.168936	0.0844682	+	0.996426i	$0.473081\pi$
$30$	0	0
$31$	−7094.59	−1.32594	−0.662969	−	0.748647i	$-0.730704\pi$
$31$	−7094.59	−1.32594	−0.662969	+	0.748647i	$0.730704\pi$
$32$	4467.10	0.771170
$33$	−10683.0	−1.70768
$34$	3666.49	0.543943
$35$	0	0
$36$	−4356.90	−0.560301
$37$	7908.22	0.949674	0.474837	−	0.880074i	$-0.342507\pi$
$37$	7908.22	0.949674	0.474837	+	0.880074i	$0.342507\pi$
$38$	−5561.87	−0.624830
$39$	−15956.9	−1.67992
$40$	0	0
$41$	12855.9	1.19438	0.597192	−	0.802098i	$-0.296283\pi$
$41$	12855.9	1.19438	0.597192	+	0.802098i	$0.296283\pi$
$42$	17994.1	1.57401
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	5993.15	0.466685
$45$	0	0
$46$	1632.28	0.113737
$47$	−26785.7	−1.76871	−0.884357	−	0.466811i	$-0.845403\pi$
$47$	−26785.7	−1.76871	−0.884357	+	0.466811i	$0.845403\pi$
$48$	−9942.37	−0.622855
$49$	13768.5	0.819215
$50$	0	0
$51$	20261.5	1.09080
$52$	8951.86	0.459098
$53$	−30000.7	−1.46704	−0.733520	−	0.679668i	$-0.762124\pi$
$53$	−30000.7	−1.46704	−0.733520	+	0.679668i	$0.762124\pi$
$54$	8507.60	0.397030
$55$	0	0
$56$	−34240.9	−1.45907
$57$	−30735.5	−1.25301
$58$	3301.64	0.128872
$59$	1247.64	0.0466614	0.0233307	−	0.999728i	$-0.492573\pi$
$59$	1247.64	0.0466614	0.0233307	+	0.999728i	$0.492573\pi$
$60$	0	0
$61$	−48441.4	−1.66683	−0.833417	−	0.552644i	$-0.813619\pi$
$61$	−48441.4	−1.66683	−0.833417	+	0.552644i	$0.813619\pi$
$62$	−30615.3	−1.01149
$63$	56946.8	1.80766
$64$	32618.5	0.995438
$65$	0	0
$66$	−46100.2	−1.30270
$67$	−67004.8	−1.82356	−0.911778	−	0.410683i	$-0.865290\pi$
$67$	−67004.8	−1.82356	−0.911778	+	0.410683i	$0.865290\pi$
$68$	−11366.7	−0.298100
$69$	9020.17	0.228082
$70$	0	0
$71$	−74553.1	−1.75517	−0.877587	−	0.479418i	$-0.840848\pi$
$71$	−74553.1	−1.75517	−0.877587	+	0.479418i	$0.840848\pi$
$72$	−63773.5	−1.44980
$73$	18066.8	0.396802	0.198401	−	0.980121i	$-0.436425\pi$
$73$	18066.8	0.396802	0.198401	+	0.980121i	$0.436425\pi$
$74$	34126.4	0.724454
$75$	0	0
$76$	17242.7	0.342429
$77$	−78333.4	−1.50564
$78$	−68859.0	−1.28152
$79$	−63230.5	−1.13988	−0.569939	−	0.821687i	$-0.693033\pi$
$79$	−63230.5	−1.13988	−0.569939	+	0.821687i	$0.693033\pi$
$80$	0	0
$81$	−32124.6	−0.544033
$82$	55477.3	0.911131
$83$	88066.7	1.40319	0.701595	−	0.712576i	$-0.252472\pi$
$83$	88066.7	1.40319	0.701595	+	0.712576i	$0.252472\pi$
$84$	−55784.5	−0.862612
$85$	0	0
$86$	−7979.00	−0.116333
$87$	18245.2	0.258435
$88$	87724.0	1.20757
$89$	50373.5	0.674104	0.337052	−	0.941486i	$-0.390570\pi$
$89$	50373.5	0.674104	0.337052	+	0.941486i	$0.390570\pi$
$90$	0	0
$91$	−117005.	−1.48116
$92$	−5060.33	−0.0623317
$93$	−169184.	−2.02839
$94$	−115588.	−1.34926
$95$	0	0
$96$	106526.	1.17972
$97$	−17502.1	−0.188869	−0.0944345	−	0.995531i	$-0.530104\pi$
$97$	−17502.1	−0.188869	−0.0944345	+	0.995531i	$0.530104\pi$
$98$	59415.5	0.624934
$99$	−145895.	−1.49608
$100$	0	0
$101$	182669.	1.78181	0.890906	−	0.454188i	$-0.150071\pi$
$101$	182669.	1.78181	0.890906	+	0.454188i	$0.150071\pi$
$102$	87434.4	0.832112
$103$	110578.	1.02702	0.513508	−	0.858085i	$-0.328346\pi$
$103$	110578.	1.02702	0.513508	+	0.858085i	$0.328346\pi$
$104$	131032.	1.18794
$105$	0	0
$106$	−129462.	−1.11913
$107$	43935.4	0.370984	0.185492	−	0.982646i	$-0.440612\pi$
$107$	43935.4	0.370984	0.185492	+	0.982646i	$0.440612\pi$
$108$	−26374.9	−0.217586
$109$	−78170.4	−0.630197	−0.315098	−	0.949059i	$-0.602038\pi$
$109$	−78170.4	−0.630197	−0.315098	+	0.949059i	$0.602038\pi$
$110$	0	0
$111$	188586.	1.45279
$112$	−72903.0	−0.549162
$113$	−23097.5	−0.170165	−0.0850823	−	0.996374i	$-0.527115\pi$
$113$	−23097.5	−0.170165	−0.0850823	+	0.996374i	$0.527115\pi$
$114$	−132633.	−0.955850
$115$	0	0
$116$	−10235.6	−0.0706267
$117$	−217921.	−1.47175
$118$	5383.93	0.0355955
$119$	148568.	0.961743
$120$	0	0
$121$	39636.3	0.246110
$122$	−209040.	−1.27154
$123$	306574.	1.82714
$124$	94912.4	0.554330
$125$	0	0
$126$	245743.	1.37897
$127$	−218127.	−1.20005	−0.600027	−	0.799980i	$-0.704844\pi$
$127$	−218127.	−1.20005	−0.600027	+	0.799980i	$0.704844\pi$
$128$	−2188.25	−0.0118052
$129$	−44092.9	−0.233289
$130$	0	0
$131$	107160.	0.545575	0.272787	−	0.962074i	$-0.412054\pi$
$131$	107160.	0.545575	0.272787	+	0.962074i	$0.412054\pi$
$132$	142918.	0.713924
$133$	−225370.	−1.10476
$134$	−289146.	−1.39109
$135$	0	0
$136$	−166379.	−0.771348
$137$	168770.	0.768234	0.384117	−	0.923284i	$-0.374506\pi$
$137$	168770.	0.768234	0.384117	+	0.923284i	$0.374506\pi$
$138$	38924.8	0.173992
$139$	294636.	1.29345	0.646725	−	0.762723i	$-0.276138\pi$
$139$	294636.	1.29345	0.646725	+	0.762723i	$0.276138\pi$
$140$	0	0
$141$	−638754.	−2.70574
$142$	−321720.	−1.33893
$143$	299763.	1.22585
$144$	−135781.	−0.545675
$145$	0	0
$146$	77963.8	0.302699
$147$	328337.	1.25322
$148$	−105797.	−0.397027
$149$	121995.	0.450169	0.225084	−	0.974339i	$-0.427734\pi$
$149$	121995.	0.450169	0.225084	+	0.974339i	$0.427734\pi$
$150$	0	0
$151$	−515468.	−1.83975	−0.919877	−	0.392208i	$-0.871711\pi$
$151$	−515468.	−1.83975	−0.919877	+	0.392208i	$0.871711\pi$
$152$	252387.	0.886050
$153$	276708.	0.955636
$154$	−338033.	−1.14857
$155$	0	0
$156$	213474.	0.702317
$157$	−178197.	−0.576969	−0.288484	−	0.957485i	$-0.593151\pi$
$157$	−178197.	−0.576969	−0.288484	+	0.957485i	$0.593151\pi$
$158$	−272859.	−0.869552
$159$	−715423.	−2.24424
$160$	0	0
$161$	66140.9	0.201097
$162$	−138627.	−0.415013
$163$	111853.	0.329747	0.164873	−	0.986315i	$-0.447278\pi$
$163$	111853.	0.329747	0.164873	+	0.986315i	$0.447278\pi$
$164$	−171988.	−0.499332
$165$	0	0
$166$	380035.	1.07042
$167$	128666.	0.357003	0.178501	−	0.983940i	$-0.442875\pi$
$167$	128666.	0.357003	0.178501	+	0.983940i	$0.442875\pi$
$168$	−816538.	−2.23205
$169$	76457.2	0.205922
$170$	0	0
$171$	−419750.	−1.09774
$172$	24736.2	0.0637546
$173$	−553314.	−1.40558	−0.702792	−	0.711396i	$-0.748063\pi$
$173$	−553314.	−1.40558	−0.702792	+	0.711396i	$0.748063\pi$
$174$	78733.8	0.197146
$175$	0	0
$176$	186775.	0.454503
$177$	29752.2	0.0713816
$178$	217377.	0.514238
$179$	−302733.	−0.706200	−0.353100	−	0.935586i	$-0.614872\pi$
$179$	−302733.	−0.706200	−0.353100	+	0.935586i	$0.614872\pi$
$180$	0	0
$181$	−702706.	−1.59433	−0.797164	−	0.603763i	$-0.793667\pi$
$181$	−702706.	−1.59433	−0.797164	+	0.603763i	$0.793667\pi$
$182$	−504913.	−1.12990
$183$	−1.15518e6	−2.54989
$184$	−74069.9	−0.161286
$185$	0	0
$186$	−730080.	−1.54735
$187$	−380627.	−0.795967
$188$	358342.	0.739440
$189$	344733.	0.701986
$190$	0	0
$191$	105355.	0.208965	0.104482	−	0.994527i	$-0.466681\pi$
$191$	105355.	0.208965	0.104482	+	0.994527i	$0.466681\pi$
$192$	777849.	1.52280
$193$	−11272.6	−0.0217836	−0.0108918	−	0.999941i	$-0.503467\pi$
$193$	−11272.6	−0.0217836	−0.0108918	+	0.999941i	$0.503467\pi$
$194$	−75526.9	−0.144078
$195$	0	0
$196$	−184197.	−0.342486
$197$	−44453.7	−0.0816098	−0.0408049	−	0.999167i	$-0.512992\pi$
$197$	−44453.7	−0.0816098	−0.0408049	+	0.999167i	$0.512992\pi$
$198$	−629584.	−1.14128
$199$	1.01040e6	1.80867	0.904333	−	0.426827i	$-0.140369\pi$
$199$	1.01040e6	1.80867	0.904333	+	0.426827i	$0.140369\pi$
$200$	0	0
$201$	−1.59786e6	−2.78963
$202$	788273.	1.35925
$203$	133784.	0.227858
$204$	−271060.	−0.456027
$205$	0	0
$206$	477180.	0.783455
$207$	123187.	0.199820
$208$	278982.	0.447114
$209$	577390.	0.914331
$210$	0	0
$211$	758827.	1.17337	0.586687	−	0.809814i	$-0.300432\pi$
$211$	758827.	1.17337	0.586687	+	0.809814i	$0.300432\pi$
$212$	401354.	0.613321
$213$	−1.77786e6	−2.68502
$214$	189595.	0.283003
$215$	0	0
$216$	−386059.	−0.563014
$217$	−1.24055e6	−1.78840
$218$	−337329.	−0.480743
$219$	430837.	0.607019
$220$	0	0
$221$	−568535.	−0.783026
$222$	813807.	1.10825
$223$	517178.	0.696431	0.348215	−	0.937415i	$-0.386788\pi$
$223$	517178.	0.696431	0.348215	+	0.937415i	$0.386788\pi$
$224$	781110.	1.04014
$225$	0	0
$226$	−99672.9	−0.129809
$227$	−627619.	−0.808409	−0.404205	−	0.914669i	$-0.632452\pi$
$227$	−627619.	−0.808409	−0.404205	+	0.914669i	$0.632452\pi$
$228$	411184.	0.523840
$229$	800425.	1.00863	0.504315	−	0.863520i	$-0.331745\pi$
$229$	800425.	1.00863	0.504315	+	0.863520i	$0.331745\pi$
$230$	0	0
$231$	−1.86801e6	−2.30329
$232$	−149822.	−0.182750
$233$	−102393.	−0.123560	−0.0617802	−	0.998090i	$-0.519678\pi$
$233$	−102393.	−0.123560	−0.0617802	+	0.998090i	$0.519678\pi$
$234$	−940397.	−1.12272
$235$	0	0
$236$	−16691.0	−0.0195076
$237$	−1.50785e6	−1.74376
$238$	641118.	0.733661
$239$	−879306.	−0.995739	−0.497869	−	0.867252i	$-0.665884\pi$
$239$	−879306.	−0.995739	−0.497869	+	0.867252i	$0.665884\pi$
$240$	0	0
$241$	196740.	0.218198	0.109099	−	0.994031i	$-0.465203\pi$
$241$	196740.	0.218198	0.109099	+	0.994031i	$0.465203\pi$
$242$	171043.	0.187744
$243$	−1.24514e6	−1.35271
$244$	648056.	0.696848
$245$	0	0
$246$	1.32296e6	1.39383
$247$	862436.	0.899466
$248$	1.38927e6	1.43435
$249$	2.10011e6	2.14657
$250$	0	0
$251$	−798778.	−0.800280	−0.400140	−	0.916454i	$-0.631038\pi$
$251$	−798778.	−0.800280	−0.400140	+	0.916454i	$0.631038\pi$
$252$	−761841.	−0.755724
$253$	−169451.	−0.166434
$254$	−941287.	−0.915456
$255$	0	0
$256$	−1.05324e6	−1.00444
$257$	93933.3	0.0887129	0.0443565	−	0.999016i	$-0.485876\pi$
$257$	93933.3	0.0887129	0.0443565	+	0.999016i	$0.485876\pi$
$258$	−190274.	−0.177963
$259$	1.38282e6	1.28090
$260$	0	0
$261$	249173.	0.226412
$262$	462428.	0.416189
$263$	898502.	0.800995	0.400497	−	0.916298i	$-0.368837\pi$
$263$	898502.	0.800995	0.400497	+	0.916298i	$0.368837\pi$
$264$	2.09194e6	1.84731
$265$	0	0
$266$	−972541.	−0.842760
$267$	1.20125e6	1.03123
$268$	896400.	0.762368
$269$	−921887.	−0.776778	−0.388389	−	0.921495i	$-0.626968\pi$
$269$	−921887.	−0.776778	−0.388389	+	0.921495i	$0.626968\pi$
$270$	0	0
$271$	−227722.	−0.188357	−0.0941784	−	0.995555i	$-0.530022\pi$
$271$	−227722.	−0.188357	−0.0941784	+	0.995555i	$0.530022\pi$
$272$	−354240.	−0.290319
$273$	−2.79021e6	−2.26584
$274$	728294.	0.586044
$275$	0	0
$276$	−120673.	−0.0953537
$277$	−1.76180e6	−1.37961	−0.689806	−	0.723994i	$-0.742304\pi$
$277$	−1.76180e6	−1.37961	−0.689806	+	0.723994i	$0.742304\pi$
$278$	1.27145e6	0.986702
$279$	−2.31052e6	−1.77705
$280$	0	0
$281$	−270410.	−0.204295	−0.102147	−	0.994769i	$-0.532571\pi$
$281$	−270410.	−0.204295	−0.102147	+	0.994769i	$0.532571\pi$
$282$	−2.75642e6	−2.06406
$283$	−2.04589e6	−1.51851	−0.759254	−	0.650795i	$-0.774436\pi$
$283$	−2.04589e6	−1.51851	−0.759254	+	0.650795i	$0.774436\pi$
$284$	997381.	0.733779
$285$	0	0
$286$	1.29357e6	0.935135
$287$	2.24797e6	1.61096
$288$	1.45481e6	1.03354
$289$	−697955.	−0.491567
$290$	0	0
$291$	−417370.	−0.288927
$292$	−241700.	−0.165890
$293$	−1.15172e6	−0.783751	−0.391875	−	0.920018i	$-0.628174\pi$
$293$	−1.15172e6	−0.783751	−0.391875	+	0.920018i	$0.628174\pi$
$294$	1.41687e6	0.956010
$295$	0	0
$296$	−1.54859e6	−1.02732
$297$	−883193.	−0.580984
$298$	526445.	0.343409
$299$	−253105.	−0.163728
$300$	0	0
$301$	−323314.	−0.205687
$302$	−2.22440e6	−1.40345
$303$	4.35609e6	2.72578
$304$	537363.	0.333490
$305$	0	0
$306$	1.19408e6	0.729003
$307$	1.88053e6	1.13877	0.569384	−	0.822072i	$-0.307182\pi$
$307$	1.88053e6	1.13877	0.569384	+	0.822072i	$0.307182\pi$
$308$	1.04795e6	0.629456
$309$	2.63695e6	1.57111
$310$	0	0
$311$	1.94805e6	1.14209	0.571043	−	0.820920i	$-0.306539\pi$
$311$	1.94805e6	1.14209	0.571043	+	0.820920i	$0.306539\pi$
$312$	3.12469e6	1.81728
$313$	944428.	0.544889	0.272445	−	0.962171i	$-0.412168\pi$
$313$	944428.	0.544889	0.272445	+	0.962171i	$0.412168\pi$
$314$	−768977.	−0.440138
$315$	0	0
$316$	845906.	0.476545
$317$	−34149.2	−0.0190868	−0.00954339	−	0.999954i	$-0.503038\pi$
$317$	−34149.2	−0.0190868	−0.00954339	+	0.999954i	$0.503038\pi$
$318$	−3.08727e6	−1.71201
$319$	−342751.	−0.188583
$320$	0	0
$321$	1.04772e6	0.567522
$322$	285418.	0.153406
$323$	−1.09509e6	−0.584039
$324$	429767.	0.227442
$325$	0	0
$326$	482682.	0.251546
$327$	−1.86412e6	−0.964061
$328$	−2.51746e6	−1.29204
$329$	−4.68370e6	−2.38561
$330$	0	0
$331$	1.73996e6	0.872908	0.436454	−	0.899727i	$-0.356234\pi$
$331$	1.73996e6	0.872908	0.436454	+	0.899727i	$0.356234\pi$
$332$	−1.17817e6	−0.586627
$333$	2.57549e6	1.27277
$334$	555232.	0.272338
$335$	0	0
$336$	−1.73851e6	−0.840095
$337$	1.09453e6	0.524990	0.262495	−	0.964933i	$-0.415455\pi$
$337$	1.09453e6	0.524990	0.262495	+	0.964933i	$0.415455\pi$
$338$	329936.	0.157086
$339$	−550804.	−0.260314
$340$	0	0
$341$	3.17825e6	1.48014
$342$	−1.81135e6	−0.837408
$343$	−531301.	−0.243840
$344$	362072.	0.164968
$345$	0	0
$346$	−2.38772e6	−1.07224
$347$	561799.	0.250471	0.125236	−	0.992127i	$-0.460031\pi$
$347$	561799.	0.250471	0.125236	+	0.992127i	$0.460031\pi$
$348$	−244087.	−0.108043
$349$	2.06827e6	0.908957	0.454478	−	0.890758i	$-0.349826\pi$
$349$	2.06827e6	0.908957	0.454478	+	0.890758i	$0.349826\pi$
$350$	0	0
$351$	−1.31921e6	−0.571539
$352$	−2.00118e6	−0.860852
$353$	−568169.	−0.242684	−0.121342	−	0.992611i	$-0.538720\pi$
$353$	−568169.	−0.242684	−0.121342	+	0.992611i	$0.538720\pi$
$354$	128390.	0.0544531
$355$	0	0
$356$	−673903.	−0.281821
$357$	3.54289e6	1.47125
$358$	−1.30639e6	−0.538722
$359$	2.57455e6	1.05430	0.527151	−	0.849771i	$-0.323260\pi$
$359$	2.57455e6	1.05430	0.527151	+	0.849771i	$0.323260\pi$
$360$	0	0
$361$	−814914.	−0.329112
$362$	−3.03239e6	−1.21623
$363$	945202.	0.376494
$364$	1.56531e6	0.619223
$365$	0	0
$366$	−4.98494e6	−1.94517
$367$	−272102.	−0.105455	−0.0527274	−	0.998609i	$-0.516791\pi$
$367$	−272102.	−0.105455	−0.0527274	+	0.998609i	$0.516791\pi$
$368$	−157704.	−0.0607047
$369$	4.18683e6	1.60074
$370$	0	0
$371$	−5.24589e6	−1.97872
$372$	2.26336e6	0.848002
$373$	268967.	0.100098	0.0500491	−	0.998747i	$-0.484062\pi$
$373$	268967.	0.100098	0.0500491	+	0.998747i	$0.484062\pi$
$374$	−1.64252e6	−0.607200
$375$	0	0
$376$	5.24518e6	1.91333
$377$	−511960.	−0.185517
$378$	1.48763e6	0.535507
$379$	−522052.	−0.186688	−0.0933438	−	0.995634i	$-0.529756\pi$
$379$	−522052.	−0.186688	−0.0933438	+	0.995634i	$0.529756\pi$
$380$	0	0
$381$	−5.20166e6	−1.83582
$382$	454640.	0.159408
$383$	−2.00529e6	−0.698523	−0.349262	−	0.937025i	$-0.613568\pi$
$383$	−2.00529e6	−0.698523	−0.349262	+	0.937025i	$0.613568\pi$
$384$	−52183.0	−0.0180593
$385$	0	0
$386$	−48644.5	−0.0166175
$387$	−602170.	−0.204381
$388$	234145.	0.0789598
$389$	−1.10827e6	−0.371340	−0.185670	−	0.982612i	$-0.559446\pi$
$389$	−1.10827e6	−0.371340	−0.185670	+	0.982612i	$0.559446\pi$
$390$	0	0
$391$	321383.	0.106312
$392$	−2.69616e6	−0.886198
$393$	2.55543e6	0.834608
$394$	−191831.	−0.0622557
$395$	0	0
$396$	1.95181e6	0.625460
$397$	3.77816e6	1.20311	0.601554	−	0.798832i	$-0.294548\pi$
$397$	3.77816e6	1.20311	0.601554	+	0.798832i	$0.294548\pi$
$398$	4.36016e6	1.37973
$399$	−5.37437e6	−1.69003
$400$	0	0
$401$	1.64922e6	0.512173	0.256086	−	0.966654i	$-0.417567\pi$
$401$	1.64922e6	0.512173	0.256086	+	0.966654i	$0.417567\pi$
$402$	−6.89524e6	−2.12806
$403$	4.74729e6	1.45607
$404$	−2.44377e6	−0.744916
$405$	0	0
$406$	577320.	0.173821
$407$	−3.54273e6	−1.06011
$408$	−3.96761e6	−1.17999
$409$	−2.14199e6	−0.633153	−0.316577	−	0.948567i	$-0.602533\pi$
$409$	−2.14199e6	−0.633153	−0.316577	+	0.948567i	$0.602533\pi$
$410$	0	0
$411$	4.02463e6	1.17523
$412$	−1.47933e6	−0.429361
$413$	218160.	0.0629361
$414$	531590.	0.152432
$415$	0	0
$416$	−2.98912e6	−0.846857
$417$	7.02616e6	1.97869
$418$	2.49161e6	0.697493
$419$	6.49479e6	1.80730	0.903650	−	0.428272i	$-0.140877\pi$
$419$	6.49479e6	1.80730	0.903650	+	0.428272i	$0.140877\pi$
$420$	0	0
$421$	3.30359e6	0.908408	0.454204	−	0.890898i	$-0.349924\pi$
$421$	3.30359e6	0.908408	0.454204	+	0.890898i	$0.349924\pi$
$422$	3.27457e6	0.895103
$423$	−8.72337e6	−2.37046
$424$	5.87476e6	1.58699
$425$	0	0
$426$	−7.67200e6	−2.04826
$427$	−8.47040e6	−2.24820
$428$	−587773.	−0.155096
$429$	7.14841e6	1.87528
$430$	0	0
$431$	−2.23343e6	−0.579133	−0.289567	−	0.957158i	$-0.593511\pi$
$431$	−2.23343e6	−0.579133	−0.289567	+	0.957158i	$0.593511\pi$
$432$	−821966.	−0.211907
$433$	3.81775e6	0.978561	0.489280	−	0.872127i	$-0.337259\pi$
$433$	3.81775e6	0.978561	0.489280	+	0.872127i	$0.337259\pi$
$434$	−5.35336e6	−1.36427
$435$	0	0
$436$	1.04577e6	0.263464
$437$	−487520.	−0.122121
$438$	1.85919e6	0.463062
$439$	4.94254e6	1.22402	0.612011	−	0.790850i	$-0.290361\pi$
$439$	4.94254e6	1.22402	0.612011	+	0.790850i	$0.290361\pi$
$440$	0	0
$441$	4.48404e6	1.09793
$442$	−2.45340e6	−0.597328
$443$	740033.	0.179160	0.0895802	−	0.995980i	$-0.471447\pi$
$443$	740033.	0.179160	0.0895802	+	0.995980i	$0.471447\pi$
$444$	−2.52293e6	−0.607363
$445$	0	0
$446$	2.23178e6	0.531269
$447$	2.90919e6	0.688658
$448$	5.70363e6	1.34263
$449$	−1.75356e6	−0.410492	−0.205246	−	0.978710i	$-0.565799\pi$
$449$	−1.75356e6	−0.410492	−0.205246	+	0.978710i	$0.565799\pi$
$450$	0	0
$451$	−5.75922e6	−1.33328
$452$	309002.	0.0711402
$453$	−1.22923e7	−2.81441
$454$	−2.70837e6	−0.616692
$455$	0	0
$456$	6.01864e6	1.35546
$457$	−1.43140e6	−0.320605	−0.160302	−	0.987068i	$-0.551247\pi$
$457$	−1.43140e6	−0.320605	−0.160302	+	0.987068i	$0.551247\pi$
$458$	3.45408e6	0.769429
$459$	1.67508e6	0.371110
$460$	0	0
$461$	−8.02421e6	−1.75853	−0.879265	−	0.476332i	$-0.841966\pi$
$461$	−8.02421e6	−1.75853	−0.879265	+	0.476332i	$0.841966\pi$
$462$	−8.06102e6	−1.75705
$463$	5.42639e6	1.17641	0.588205	−	0.808712i	$-0.299835\pi$
$463$	5.42639e6	1.17641	0.588205	+	0.808712i	$0.299835\pi$
$464$	−318990.	−0.0687831
$465$	0	0
$466$	−441856.	−0.0942575
$467$	4.68842e6	0.994796	0.497398	−	0.867522i	$-0.334289\pi$
$467$	4.68842e6	0.994796	0.497398	+	0.867522i	$0.334289\pi$
$468$	2.91538e6	0.615291
$469$	−1.17164e7	−2.45958
$470$	0	0
$471$	−4.24945e6	−0.882634
$472$	−244313.	−0.0504768
$473$	828318.	0.170233
$474$	−6.50683e6	−1.33022
$475$	0	0
$476$	−1.98757e6	−0.402072
$477$	−9.77043e6	−1.96615
$478$	−3.79448e6	−0.759595
$479$	2.64873e6	0.527471	0.263735	−	0.964595i	$-0.415045\pi$
$479$	2.64873e6	0.527471	0.263735	+	0.964595i	$0.415045\pi$
$480$	0	0
$481$	−5.29172e6	−1.04288
$482$	848994.	0.166451
$483$	1.57725e6	0.307634
$484$	−530260.	−0.102890
$485$	0	0
$486$	−5.37318e6	−1.03191
$487$	−2.23774e6	−0.427550	−0.213775	−	0.976883i	$-0.568576\pi$
$487$	−2.23774e6	−0.427550	−0.213775	+	0.976883i	$0.568576\pi$
$488$	9.48583e6	1.80312
$489$	2.66736e6	0.504439
$490$	0	0
$491$	3.10094e6	0.580484	0.290242	−	0.956953i	$-0.406264\pi$
$491$	3.10094e6	0.580484	0.290242	+	0.956953i	$0.406264\pi$
$492$	−4.10138e6	−0.763867
$493$	650066.	0.120459
$494$	3.72168e6	0.686153
$495$	0	0
$496$	2.95792e6	0.539861
$497$	−1.30363e7	−2.36735
$498$	9.06264e6	1.63750
$499$	5.58008e6	1.00320	0.501601	−	0.865099i	$-0.332744\pi$
$499$	5.58008e6	1.00320	0.501601	+	0.865099i	$0.332744\pi$
$500$	0	0
$501$	3.06827e6	0.546135
$502$	−3.44697e6	−0.610490
$503$	−5.10876e6	−0.900317	−0.450158	−	0.892949i	$-0.648632\pi$
$503$	−5.10876e6	−0.900317	−0.450158	+	0.892949i	$0.648632\pi$
$504$	−1.11513e7	−1.95547
$505$	0	0
$506$	−731231.	−0.126963
$507$	1.82327e6	0.315014
$508$	2.91814e6	0.501703
$509$	2.93415e6	0.501982	0.250991	−	0.967989i	$-0.419244\pi$
$509$	2.93415e6	0.501982	0.250991	+	0.967989i	$0.419244\pi$
$510$	0	0
$511$	3.15914e6	0.535200
$512$	−4.47501e6	−0.754430
$513$	−2.54100e6	−0.426296
$514$	405351.	0.0676743
$515$	0	0
$516$	589880.	0.0975303
$517$	1.19995e7	1.97440
$518$	5.96729e6	0.977131
$519$	−1.31948e7	−2.15023
$520$	0	0
$521$	−9.22452e6	−1.48885	−0.744423	−	0.667709i	$-0.767275\pi$
$521$	−9.22452e6	−1.48885	−0.744423	+	0.667709i	$0.767275\pi$
$522$	1.07526e6	0.172717
$523$	4.64359e6	0.742335	0.371168	−	0.928566i	$-0.378958\pi$
$523$	4.64359e6	0.742335	0.371168	+	0.928566i	$0.378958\pi$
$524$	−1.43360e6	−0.228087
$525$	0	0
$526$	3.87731e6	0.611035
$527$	−6.02791e6	−0.945453
$528$	4.45399e6	0.695288
$529$	−6.29327e6	−0.977771
$530$	0	0
$531$	406322.	0.0625365
$532$	3.01503e6	0.461862
$533$	−8.60244e6	−1.31161
$534$	5.18376e6	0.786669
$535$	0	0
$536$	1.31209e7	1.97266
$537$	−7.21924e6	−1.08033
$538$	−3.97823e6	−0.592562
$539$	−6.16805e6	−0.914484
$540$	0	0
$541$	3.62594e6	0.532632	0.266316	−	0.963886i	$-0.414193\pi$
$541$	3.62594e6	0.532632	0.266316	+	0.963886i	$0.414193\pi$
$542$	−982690.	−0.143687
$543$	−1.67573e7	−2.43897
$544$	3.79546e6	0.549879
$545$	0	0
$546$	−1.20406e7	−1.72849
$547$	5.56425e6	0.795131	0.397565	−	0.917574i	$-0.369855\pi$
$547$	5.56425e6	0.795131	0.397565	+	0.917574i	$0.369855\pi$
$548$	−2.25782e6	−0.321173
$549$	−1.57761e7	−2.23392
$550$	0	0
$551$	−986114.	−0.138372
$552$	−1.76633e6	−0.246732
$553$	−1.10564e7	−1.53745
$554$	−7.60271e6	−1.05243
$555$	0	0
$556$	−3.94169e6	−0.540748
$557$	−6.83963e6	−0.934103	−0.467052	−	0.884230i	$-0.654684\pi$
$557$	−6.83963e6	−0.934103	−0.467052	+	0.884230i	$0.654684\pi$
$558$	−9.97059e6	−1.35561
$559$	1.23724e6	0.167465
$560$	0	0
$561$	−9.07675e6	−1.21765
$562$	−1.16690e6	−0.155845
$563$	5.62439e6	0.747833	0.373916	−	0.927462i	$-0.378015\pi$
$563$	5.62439e6	0.747833	0.373916	+	0.927462i	$0.378015\pi$
$564$	8.54534e6	1.13118
$565$	0	0
$566$	−8.82866e6	−1.15839
$567$	−5.61726e6	−0.733782
$568$	1.45990e7	1.89869
$569$	−4.17021e6	−0.539979	−0.269990	−	0.962863i	$-0.587020\pi$
$569$	−4.17021e6	−0.539979	−0.269990	+	0.962863i	$0.587020\pi$
$570$	0	0
$571$	−8.60607e6	−1.10462	−0.552312	−	0.833637i	$-0.686254\pi$
$571$	−8.60607e6	−1.10462	−0.552312	+	0.833637i	$0.686254\pi$
$572$	−4.01027e6	−0.512487
$573$	2.51239e6	0.319669
$574$	9.70069e6	1.22892
$575$	0	0
$576$	1.06230e7	1.33410
$577$	3.23538e6	0.404563	0.202282	−	0.979327i	$-0.435164\pi$
$577$	3.23538e6	0.404563	0.202282	+	0.979327i	$0.435164\pi$
$578$	−3.01189e6	−0.374990
$579$	−268815.	−0.0333240
$580$	0	0
$581$	1.53992e7	1.89260
$582$	−1.80108e6	−0.220407
$583$	1.34398e7	1.63765
$584$	−3.53785e6	−0.429247
$585$	0	0
$586$	−4.97003e6	−0.597881
$587$	1.12970e7	1.35322	0.676609	−	0.736342i	$-0.263449\pi$
$587$	1.12970e7	1.35322	0.676609	+	0.736342i	$0.263449\pi$
$588$	−4.39253e6	−0.523928
$589$	9.14401e6	1.08605
$590$	0	0
$591$	−1.06008e6	−0.124845
$592$	−3.29714e6	−0.386663
$593$	−1.61368e7	−1.88444	−0.942218	−	0.335000i	$-0.891264\pi$
$593$	−1.61368e7	−1.88444	−0.942218	+	0.335000i	$0.891264\pi$
$594$	−3.81125e6	−0.443201
$595$	0	0
$596$	−1.63206e6	−0.188201
$597$	2.40948e7	2.76686
$598$	−1.09223e6	−0.124899
$599$	1.14121e7	1.29957	0.649785	−	0.760118i	$-0.274859\pi$
$599$	1.14121e7	1.29957	0.649785	+	0.760118i	$0.274859\pi$
$600$	0	0
$601$	1.15192e7	1.30088	0.650438	−	0.759559i	$-0.274585\pi$
$601$	1.15192e7	1.30088	0.650438	+	0.759559i	$0.274585\pi$
$602$	−1.39520e6	−0.156908
$603$	−2.18217e7	−2.44396
$604$	6.89600e6	0.769139
$605$	0	0
$606$	1.87979e7	2.07935
$607$	−1.48986e7	−1.64125	−0.820625	−	0.571466i	$-0.806375\pi$
$607$	−1.48986e7	−1.64125	−0.820625	+	0.571466i	$0.806375\pi$
$608$	−5.75751e6	−0.631648
$609$	3.19034e6	0.348573
$610$	0	0
$611$	1.79234e7	1.94230
$612$	−3.70183e6	−0.399520
$613$	5.39999e6	0.580419	0.290210	−	0.956963i	$-0.406275\pi$
$613$	5.39999e6	0.580419	0.290210	+	0.956963i	$0.406275\pi$
$614$	8.11508e6	0.868704
$615$	0	0
$616$	1.53393e7	1.62875
$617$	1.26228e7	1.33489	0.667443	−	0.744660i	$-0.267389\pi$
$617$	1.26228e7	1.33489	0.667443	+	0.744660i	$0.267389\pi$
$618$	1.13792e7	1.19851
$619$	1.43557e7	1.50591	0.752953	−	0.658074i	$-0.228629\pi$
$619$	1.43557e7	1.50591	0.752953	+	0.658074i	$0.228629\pi$
$620$	0	0
$621$	745725.	0.0775979
$622$	8.40642e6	0.871235
$623$	8.80824e6	0.909220
$624$	6.65285e6	0.683984
$625$	0	0
$626$	4.07550e6	0.415666
$627$	1.37689e7	1.39872
$628$	2.38395e6	0.241211
$629$	6.71920e6	0.677160
$630$	0	0
$631$	−6.76873e6	−0.676759	−0.338379	−	0.941010i	$-0.609879\pi$
$631$	−6.76873e6	−0.676759	−0.338379	+	0.941010i	$0.609879\pi$
$632$	1.23818e7	1.23308
$633$	1.80956e7	1.79500
$634$	−147364.	−0.0145603
$635$	0	0
$636$	9.57103e6	0.938244
$637$	−9.21310e6	−0.899616
$638$	−1.47907e6	−0.143859
$639$	−2.42799e7	−2.35232
$640$	0	0
$641$	−1.51643e6	−0.145773	−0.0728867	−	0.997340i	$-0.523221\pi$
$641$	−1.51643e6	−0.145773	−0.0728867	+	0.997340i	$0.523221\pi$
$642$	4.52124e6	0.432932
$643$	−6.58876e6	−0.628458	−0.314229	−	0.949347i	$-0.601746\pi$
$643$	−6.58876e6	−0.628458	−0.314229	+	0.949347i	$0.601746\pi$
$644$	−884842.	−0.0840719
$645$	0	0
$646$	−4.72563e6	−0.445532
$647$	−9.99129e6	−0.938341	−0.469171	−	0.883108i	$-0.655447\pi$
$647$	−9.99129e6	−0.938341	−0.469171	+	0.883108i	$0.655447\pi$
$648$	6.29065e6	0.588516
$649$	−558918.	−0.0520878
$650$	0	0
$651$	−2.95833e7	−2.73586
$652$	−1.49639e6	−0.137856
$653$	1.32714e7	1.21796	0.608980	−	0.793186i	$-0.291579\pi$
$653$	1.32714e7	1.21796	0.608980	+	0.793186i	$0.291579\pi$
$654$	−8.04425e6	−0.735430
$655$	0	0
$656$	−5.35997e6	−0.486298
$657$	5.88387e6	0.531802
$658$	−2.02116e7	−1.81985
$659$	3.69381e6	0.331330	0.165665	−	0.986182i	$-0.447023\pi$
$659$	3.69381e6	0.331330	0.165665	+	0.986182i	$0.447023\pi$
$660$	0	0
$661$	8.97800e6	0.799238	0.399619	−	0.916681i	$-0.369142\pi$
$661$	8.97800e6	0.799238	0.399619	+	0.916681i	$0.369142\pi$
$662$	7.50845e6	0.665894
$663$	−1.35578e7	−1.19786
$664$	−1.72453e7	−1.51792
$665$	0	0
$666$	1.11140e7	0.970927
$667$	289402.	0.0251876
$668$	−1.72131e6	−0.149251
$669$	1.23331e7	1.06538
$670$	0	0
$671$	2.17009e7	1.86068
$672$	1.86270e7	1.59118
$673$	−8.59992e6	−0.731909	−0.365954	−	0.930633i	$-0.619257\pi$
$673$	−8.59992e6	−0.731909	−0.365954	+	0.930633i	$0.619257\pi$
$674$	4.72322e6	0.400487
$675$	0	0
$676$	−1.02285e6	−0.0860889
$677$	8.01663e6	0.672233	0.336117	−	0.941820i	$-0.390886\pi$
$677$	8.01663e6	0.672233	0.336117	+	0.941820i	$0.390886\pi$
$678$	−2.37689e6	−0.198579
$679$	−3.06039e6	−0.254743
$680$	0	0
$681$	−1.49667e7	−1.23669
$682$	1.37151e7	1.12911
$683$	−2.81320e6	−0.230754	−0.115377	−	0.993322i	$-0.536808\pi$
$683$	−2.81320e6	−0.230754	−0.115377	+	0.993322i	$0.536808\pi$
$684$	5.61547e6	0.458930
$685$	0	0
$686$	−2.29273e6	−0.186012
$687$	1.90876e7	1.54298
$688$	770895.	0.0620904
$689$	2.00747e7	1.61102
$690$	0	0
$691$	1.04922e7	0.835935	0.417967	−	0.908462i	$-0.362743\pi$
$691$	1.04922e7	0.835935	0.417967	+	0.908462i	$0.362743\pi$
$692$	7.40231e6	0.587628
$693$	−2.55111e7	−2.01788
$694$	2.42434e6	0.191071
$695$	0	0
$696$	−3.57279e6	−0.279566
$697$	1.09230e7	0.851650
$698$	8.92521e6	0.693394
$699$	−2.44175e6	−0.189020
$700$	0	0
$701$	−2.37537e7	−1.82573	−0.912865	−	0.408263i	$-0.866135\pi$
$701$	−2.37537e7	−1.82573	−0.912865	+	0.408263i	$0.866135\pi$
$702$	−5.69279e6	−0.435996
$703$	−1.01927e7	−0.777856
$704$	−1.46125e7	−1.11120
$705$	0	0
$706$	−2.45182e6	−0.185130
$707$	3.19413e7	2.40328
$708$	−398029.	−0.0298423
$709$	−1.72000e7	−1.28503	−0.642516	−	0.766273i	$-0.722109\pi$
$709$	−1.72000e7	−1.28503	−0.642516	+	0.766273i	$0.722109\pi$
$710$	0	0
$711$	−2.05925e7	−1.52769
$712$	−9.86417e6	−0.729223
$713$	−2.68356e6	−0.197691
$714$	1.52887e7	1.12234
$715$	0	0
$716$	4.05001e6	0.295239
$717$	−2.09687e7	−1.52326
$718$	1.11100e7	0.804270
$719$	2.22034e7	1.60176	0.800878	−	0.598827i	$-0.204366\pi$
$719$	2.22034e7	1.60176	0.800878	+	0.598827i	$0.204366\pi$
$720$	0	0
$721$	1.93356e7	1.38522
$722$	−3.51660e6	−0.251062
$723$	4.69164e6	0.333794
$724$	9.40090e6	0.666535
$725$	0	0
$726$	4.07884e6	0.287207
$727$	1.95017e7	1.36847	0.684236	−	0.729261i	$-0.260136\pi$
$727$	1.95017e7	1.36847	0.684236	+	0.729261i	$0.260136\pi$
$728$	2.29120e7	1.60227
$729$	−2.18865e7	−1.52531
$730$	0	0
$731$	−1.57100e6	−0.108738
$732$	1.54541e7	1.06602
$733$	−1.07906e7	−0.741798	−0.370899	−	0.928673i	$-0.620950\pi$
$733$	−1.07906e7	−0.741798	−0.370899	+	0.928673i	$0.620950\pi$
$734$	−1.17420e6	−0.0804457
$735$	0	0
$736$	1.68970e6	0.114978
$737$	3.00169e7	2.03562
$738$	1.80675e7	1.22111
$739$	−1.44802e7	−0.975358	−0.487679	−	0.873023i	$-0.662156\pi$
$739$	−1.44802e7	−0.975358	−0.487679	+	0.873023i	$0.662156\pi$
$740$	0	0
$741$	2.05664e7	1.37598
$742$	−2.26376e7	−1.50946
$743$	−5.44908e6	−0.362119	−0.181059	−	0.983472i	$-0.557953\pi$
$743$	−5.44908e6	−0.362119	−0.181059	+	0.983472i	$0.557953\pi$
$744$	3.31297e7	2.19424
$745$	0	0
$746$	1.16067e6	0.0763594
$747$	2.86809e7	1.88058
$748$	5.09207e6	0.332767
$749$	7.68248e6	0.500376
$750$	0	0
$751$	−2.78004e6	−0.179867	−0.0899335	−	0.995948i	$-0.528665\pi$
$751$	−2.78004e6	−0.179867	−0.0899335	+	0.995948i	$0.528665\pi$
$752$	1.11676e7	0.720139
$753$	−1.90484e7	−1.22425
$754$	−2.20926e6	−0.141521
$755$	0	0
$756$	−4.61188e6	−0.293477
$757$	−2.46778e7	−1.56519	−0.782595	−	0.622531i	$-0.786105\pi$
$757$	−2.46778e7	−1.56519	−0.782595	+	0.622531i	$0.786105\pi$
$758$	−2.25281e6	−0.142414
$759$	−4.04087e6	−0.254607
$760$	0	0
$761$	−8.63006e6	−0.540197	−0.270098	−	0.962833i	$-0.587056\pi$
$761$	−8.63006e6	−0.540197	−0.270098	+	0.962833i	$0.587056\pi$
$762$	−2.24467e7	−1.40044
$763$	−1.36688e7	−0.849999
$764$	−1.40946e6	−0.0873611
$765$	0	0
$766$	−8.65345e6	−0.532865
$767$	−834845.	−0.0512410
$768$	−2.51164e7	−1.53657
$769$	−1.79104e6	−0.109217	−0.0546083	−	0.998508i	$-0.517391\pi$
$769$	−1.79104e6	−0.109217	−0.0546083	+	0.998508i	$0.517391\pi$
$770$	0	0
$771$	2.24002e6	0.135711
$772$	150806.	0.00910698
$773$	2.64220e7	1.59044	0.795219	−	0.606322i	$-0.207356\pi$
$773$	2.64220e7	1.59044	0.795219	+	0.606322i	$0.207356\pi$
$774$	−2.59855e6	−0.155911
$775$	0	0
$776$	3.42727e6	0.204312
$777$	3.29759e7	1.95950
$778$	−4.78253e6	−0.283275
$779$	−1.65696e7	−0.978293
$780$	0	0
$781$	3.33984e7	1.95929
$782$	1.38686e6	0.0810993
$783$	1.50839e6	0.0879244
$784$	−5.74045e6	−0.333546
$785$	0	0
$786$	1.10275e7	0.636677
$787$	−4.11413e6	−0.236778	−0.118389	−	0.992967i	$-0.537773\pi$
$787$	−4.11413e6	−0.236778	−0.118389	+	0.992967i	$0.537773\pi$
$788$	594708.	0.0341184
$789$	2.14265e7	1.22534
$790$	0	0
$791$	−4.03880e6	−0.229515
$792$	2.85693e7	1.61840
$793$	3.24142e7	1.83043
$794$	1.63039e7	0.917786
$795$	0	0
$796$	−1.35172e7	−0.756143
$797$	−3.23381e6	−0.180330	−0.0901651	−	0.995927i	$-0.528739\pi$
$797$	−3.23381e6	−0.180330	−0.0901651	+	0.995927i	$0.528739\pi$
$798$	−2.31920e7	−1.28923
$799$	−2.27584e7	−1.26117
$800$	0	0
$801$	1.64053e7	0.903447
$802$	7.11687e6	0.390709
$803$	−8.09359e6	−0.442948
$804$	2.13763e7	1.16625
$805$	0	0
$806$	2.04860e7	1.11076
$807$	−2.19841e7	−1.18830
$808$	−3.57704e7	−1.92750
$809$	−1.52689e7	−0.820231	−0.410115	−	0.912034i	$-0.634512\pi$
$809$	−1.52689e7	−0.820231	−0.410115	+	0.912034i	$0.634512\pi$
$810$	0	0
$811$	−3.04480e7	−1.62558	−0.812788	−	0.582560i	$-0.802051\pi$
$811$	−3.04480e7	−1.62558	−0.812788	+	0.582560i	$0.802051\pi$
$812$	−1.78978e6	−0.0952600
$813$	−5.43045e6	−0.288144
$814$	−1.52880e7	−0.808703
$815$	0	0
$816$	−8.44751e6	−0.444123
$817$	2.38312e6	0.124908
$818$	−9.24334e6	−0.482998
$819$	−3.81054e7	−1.98508
$820$	0	0
$821$	−2.25035e6	−0.116518	−0.0582590	−	0.998302i	$-0.518555\pi$
$821$	−2.25035e6	−0.116518	−0.0582590	+	0.998302i	$0.518555\pi$
$822$	1.73675e7	0.896516
$823$	1.66620e7	0.857485	0.428743	−	0.903427i	$-0.358957\pi$
$823$	1.66620e7	0.857485	0.428743	+	0.903427i	$0.358957\pi$
$824$	−2.16535e7	−1.11099
$825$	0	0
$826$	941428.	0.0480106
$827$	−3.70905e7	−1.88581	−0.942907	−	0.333056i	$-0.891920\pi$
$827$	−3.70905e7	−1.88581	−0.942907	+	0.333056i	$0.891920\pi$
$828$	−1.64801e6	−0.0835381
$829$	2.88276e7	1.45688	0.728438	−	0.685111i	$-0.240246\pi$
$829$	2.88276e7	1.45688	0.728438	+	0.685111i	$0.240246\pi$
$830$	0	0
$831$	−4.20134e7	−2.11050
$832$	−2.18264e7	−1.09313
$833$	1.16984e7	0.584137
$834$	3.03200e7	1.50943
$835$	0	0
$836$	−7.72439e6	−0.382251
$837$	−1.39870e7	−0.690096
$838$	2.80270e7	1.37869
$839$	−7.10869e6	−0.348646	−0.174323	−	0.984689i	$-0.555774\pi$
$839$	−7.10869e6	−0.348646	−0.174323	+	0.984689i	$0.555774\pi$
$840$	0	0
$841$	−1.99258e7	−0.971460
$842$	1.42560e7	0.692975
$843$	−6.44843e6	−0.312525
$844$	−1.01517e7	−0.490548
$845$	0	0
$846$	−3.76440e7	−1.80830
$847$	6.93075e6	0.331949
$848$	1.25081e7	0.597311
$849$	−4.87882e7	−2.32298
$850$	0	0
$851$	2.99131e6	0.141592
$852$	2.37844e7	1.12252
$853$	−6.01755e6	−0.283170	−0.141585	−	0.989926i	$-0.545220\pi$
$853$	−6.01755e6	−0.283170	−0.141585	+	0.989926i	$0.545220\pi$
$854$	−3.65524e7	−1.71503
$855$	0	0
$856$	−8.60345e6	−0.401317
$857$	−1.40484e7	−0.653392	−0.326696	−	0.945129i	$-0.605935\pi$
$857$	−1.40484e7	−0.653392	−0.326696	+	0.945129i	$0.605935\pi$
$858$	3.08476e7	1.43055
$859$	2.38780e7	1.10412	0.552060	−	0.833805i	$-0.313842\pi$
$859$	2.38780e7	1.10412	0.552060	+	0.833805i	$0.313842\pi$
$860$	0	0
$861$	5.36071e7	2.46442
$862$	−9.63792e6	−0.441789
$863$	−3.12051e7	−1.42626	−0.713131	−	0.701031i	$-0.752723\pi$
$863$	−3.12051e7	−1.42626	−0.713131	+	0.701031i	$0.752723\pi$
$864$	8.80686e6	0.401362
$865$	0	0
$866$	1.64748e7	0.746491
$867$	−1.66440e7	−0.751988
$868$	1.65963e7	0.747671
$869$	2.83261e7	1.27244
$870$	0	0
$871$	4.48357e7	2.00253
$872$	1.53074e7	0.681725
$873$	−5.69996e6	−0.253126
$874$	−2.10380e6	−0.0931591
$875$	0	0
$876$	−5.76379e6	−0.253775
$877$	2.97111e6	0.130443	0.0652213	−	0.997871i	$-0.479225\pi$
$877$	2.97111e6	0.130443	0.0652213	+	0.997871i	$0.479225\pi$
$878$	2.13286e7	0.933739
$879$	−2.74649e7	−1.19896
$880$	0	0
$881$	3.55211e7	1.54187	0.770934	−	0.636915i	$-0.219790\pi$
$881$	3.55211e7	1.54187	0.770934	+	0.636915i	$0.219790\pi$
$882$	1.93500e7	0.837549
$883$	−1.87105e7	−0.807576	−0.403788	−	0.914853i	$-0.632307\pi$
$883$	−1.87105e7	−0.807576	−0.403788	+	0.914853i	$0.632307\pi$
$884$	7.60593e6	0.327357
$885$	0	0
$886$	3.19347e6	0.136672
$887$	−9.33897e6	−0.398557	−0.199278	−	0.979943i	$-0.563860\pi$
$887$	−9.33897e6	−0.398557	−0.199278	+	0.979943i	$0.563860\pi$
$888$	−3.69291e7	−1.57158
$889$	−3.81415e7	−1.61861
$890$	0	0
$891$	1.43912e7	0.607300
$892$	−6.91887e6	−0.291154
$893$	3.45232e7	1.44871
$894$	1.25541e7	0.525340
$895$	0	0
$896$	−382635.	−0.0159226
$897$	−6.03577e6	−0.250468
$898$	−7.56715e6	−0.313142
$899$	−5.42807e6	−0.223999
$900$	0	0
$901$	−2.54901e7	−1.04607
$902$	−2.48528e7	−1.01709
$903$	−7.71002e6	−0.314656
$904$	4.52297e6	0.184078
$905$	0	0
$906$	−5.30451e7	−2.14696
$907$	−3.89432e7	−1.57186	−0.785930	−	0.618315i	$-0.787816\pi$
$907$	−3.89432e7	−1.57186	−0.785930	+	0.618315i	$0.787816\pi$
$908$	8.39637e6	0.337969
$909$	5.94904e7	2.38802
$910$	0	0
$911$	4.28351e7	1.71003	0.855015	−	0.518604i	$-0.173548\pi$
$911$	4.28351e7	1.71003	0.855015	+	0.518604i	$0.173548\pi$
$912$	1.28144e7	0.510166
$913$	−3.94522e7	−1.56637
$914$	−6.17692e6	−0.244572
$915$	0	0
$916$	−1.07082e7	−0.421675
$917$	1.87378e7	0.735862
$918$	7.22847e6	0.283100
$919$	881759.	0.0344398	0.0172199	−	0.999852i	$-0.494518\pi$
$919$	881759.	0.0344398	0.0172199	+	0.999852i	$0.494518\pi$
$920$	0	0
$921$	4.48448e7	1.74206
$922$	−3.46269e7	−1.34149
$923$	4.98866e7	1.92743
$924$	2.49904e7	0.962928
$925$	0	0
$926$	2.34165e7	0.897419
$927$	3.60124e7	1.37643
$928$	3.41778e6	0.130279
$929$	−1.81365e7	−0.689468	−0.344734	−	0.938700i	$-0.612031\pi$
$929$	−1.81365e7	−0.689468	−0.344734	+	0.938700i	$0.612031\pi$
$930$	0	0
$931$	−1.77459e7	−0.671000
$932$	1.36982e6	0.0516565
$933$	4.64548e7	1.74714
$934$	2.02320e7	0.758876
$935$	0	0
$936$	4.26735e7	1.59209
$937$	1.75363e7	0.652512	0.326256	−	0.945281i	$-0.394213\pi$
$937$	1.75363e7	0.652512	0.326256	+	0.945281i	$0.394213\pi$
$938$	−5.05597e7	−1.87628
$939$	2.25217e7	0.833559
$940$	0	0
$941$	7.69478e6	0.283284	0.141642	−	0.989918i	$-0.454762\pi$
$941$	7.69478e6	0.283284	0.141642	+	0.989918i	$0.454762\pi$
$942$	−1.83377e7	−0.673313
$943$	4.86281e6	0.178077
$944$	−520171.	−0.0189984
$945$	0	0
$946$	3.57444e6	0.129862
$947$	3.00889e7	1.09026	0.545131	−	0.838351i	$-0.316480\pi$
$947$	3.00889e7	1.09026	0.545131	+	0.838351i	$0.316480\pi$
$948$	2.01722e7	0.729008
$949$	−1.20892e7	−0.435746
$950$	0	0
$951$	−814352.	−0.0291985
$952$	−2.90927e7	−1.04038
$953$	−2.31755e7	−0.826601	−0.413301	−	0.910595i	$-0.635624\pi$
$953$	−2.31755e7	−0.826601	−0.413301	+	0.910595i	$0.635624\pi$
$954$	−4.21624e7	−1.49987
$955$	0	0
$956$	1.17635e7	0.416285
$957$	−8.17353e6	−0.288489
$958$	1.14301e7	0.402379
$959$	2.95109e7	1.03618
$960$	0	0
$961$	2.17041e7	0.758112
$962$	−2.28354e7	−0.795556
$963$	1.43086e7	0.497199
$964$	−2.63202e6	−0.0912212
$965$	0	0
$966$	6.80633e6	0.234677
$967$	3.96944e7	1.36510	0.682548	−	0.730841i	$-0.260872\pi$
$967$	3.96944e7	1.36510	0.682548	+	0.730841i	$0.260872\pi$
$968$	−7.76160e6	−0.266234
$969$	−2.61144e7	−0.893450
$970$	0	0
$971$	−6.90625e6	−0.235068	−0.117534	−	0.993069i	$-0.537499\pi$
$971$	−6.90625e6	−0.235068	−0.117534	+	0.993069i	$0.537499\pi$
$972$	1.66577e7	0.565522
$973$	5.15197e7	1.74458
$974$	−9.65654e6	−0.326155
$975$	0	0
$976$	2.01965e7	0.678658
$977$	1.38767e7	0.465104	0.232552	−	0.972584i	$-0.425292\pi$
$977$	1.38767e7	0.465104	0.232552	+	0.972584i	$0.425292\pi$
$978$	1.15105e7	0.384809
$979$	−2.25664e7	−0.752498
$980$	0	0
$981$	−2.54580e7	−0.844602
$982$	1.33815e7	0.442819
$983$	−1.20330e7	−0.397183	−0.198591	−	0.980082i	$-0.563637\pi$
$983$	−1.20330e7	−0.397183	−0.198591	+	0.980082i	$0.563637\pi$
$984$	−6.00335e7	−1.97654
$985$	0	0
$986$	2.80523e6	0.0918918
$987$	−1.11692e8	−3.64945
$988$	−1.15378e7	−0.376037
$989$	−699391.	−0.0227368
$990$	0	0
$991$	9.67955e6	0.313091	0.156546	−	0.987671i	$-0.449964\pi$
$991$	9.67955e6	0.313091	0.156546	+	0.987671i	$0.449964\pi$
$992$	−3.16922e7	−1.02252
$993$	4.14925e7	1.33535
$994$	−5.62554e7	−1.80592
$995$	0	0
$996$	−2.80956e7	−0.897408
$997$	273483.	0.00871351	0.00435676	−	0.999991i	$-0.498613\pi$
$997$	273483.	0.00871351	0.00435676	+	0.999991i	$0.498613\pi$
$998$	2.40797e7	0.765289
$999$	1.55910e7	0.494266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.8		10
5.4	even	2		43.6.a.b.1.3	✓	10
15.14	odd	2		387.6.a.e.1.8		10
20.19	odd	2		688.6.a.h.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.3	✓	10	5.4	even	2
387.6.a.e.1.8		10	15.14	odd	2
688.6.a.h.1.9		10	20.19	odd	2
1075.6.a.b.1.8		10	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+4.31531 q^{2} +23.8469 q^{3} -13.3781 q^{4} +102.907 q^{6} +174.859 q^{7} -195.821 q^{8} +325.673 q^{9} +O(q^{10})\)	\(q+4.31531 q^{2} +23.8469 q^{3} -13.3781 q^{4} +102.907 q^{6} +174.859 q^{7} -195.821 q^{8} +325.673 q^{9} -447.981 q^{11} -319.027 q^{12} -669.141 q^{13} +754.569 q^{14} -416.925 q^{16} +849.648 q^{17} +1405.38 q^{18} -1288.87 q^{19} +4169.83 q^{21} -1933.18 q^{22} +378.254 q^{23} -4669.71 q^{24} -2887.55 q^{26} +1971.50 q^{27} -2339.28 q^{28} +765.100 q^{29} -7094.59 q^{31} +4467.10 q^{32} -10683.0 q^{33} +3666.49 q^{34} -4356.90 q^{36} +7908.22 q^{37} -5561.87 q^{38} -15956.9 q^{39} +12855.9 q^{41} +17994.1 q^{42} -1849.00 q^{43} +5993.15 q^{44} +1632.28 q^{46} -26785.7 q^{47} -9942.37 q^{48} +13768.5 q^{49} +20261.5 q^{51} +8951.86 q^{52} -30000.7 q^{53} +8507.60 q^{54} -34240.9 q^{56} -30735.5 q^{57} +3301.64 q^{58} +1247.64 q^{59} -48441.4 q^{61} -30615.3 q^{62} +56946.8 q^{63} +32618.5 q^{64} -46100.2 q^{66} -67004.8 q^{67} -11366.7 q^{68} +9020.17 q^{69} -74553.1 q^{71} -63773.5 q^{72} +18066.8 q^{73} +34126.4 q^{74} +17242.7 q^{76} -78333.4 q^{77} -68859.0 q^{78} -63230.5 q^{79} -32124.6 q^{81} +55477.3 q^{82} +88066.7 q^{83} -55784.5 q^{84} -7979.00 q^{86} +18245.2 q^{87} +87724.0 q^{88} +50373.5 q^{89} -117005. q^{91} -5060.33 q^{92} -169184. q^{93} -115588. q^{94} +106526. q^{96} -17502.1 q^{97} +59415.5 q^{98} -145895. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * q^99

Introduction

L-functions

Modular forms

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