LMFDB - Embedded newform 825.6.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + \cdots + 1190400 $ x^9 - x^8 - 229x^7 + 267x^6 + 16434x^5 - 16568x^4 - 405504x^3 + 202288x^2 + 2184608*x + 1190400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.77674$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.77674 q^{2} -9.00000 q^{3} +1.37069 q^{4} +51.9906 q^{6} -150.201 q^{7} +176.937 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-5.77674 q^{2} -9.00000 q^{3} +1.37069 q^{4} +51.9906 q^{6} -150.201 q^{7} +176.937 q^{8} +81.0000 q^{9} -121.000 q^{11} -12.3362 q^{12} +1146.84 q^{13} +867.675 q^{14} -1065.98 q^{16} -1570.59 q^{17} -467.916 q^{18} +731.596 q^{19} +1351.81 q^{21} +698.985 q^{22} +2182.68 q^{23} -1592.44 q^{24} -6624.97 q^{26} -729.000 q^{27} -205.880 q^{28} +8603.71 q^{29} +6700.21 q^{31} +495.906 q^{32} +1089.00 q^{33} +9072.88 q^{34} +111.026 q^{36} -3056.37 q^{37} -4226.24 q^{38} -10321.5 q^{39} -440.525 q^{41} -7809.07 q^{42} -13592.0 q^{43} -165.854 q^{44} -12608.8 q^{46} +15365.3 q^{47} +9593.85 q^{48} +5753.49 q^{49} +14135.3 q^{51} +1571.96 q^{52} -25436.6 q^{53} +4211.24 q^{54} -26576.3 q^{56} -6584.36 q^{57} -49701.4 q^{58} -8795.36 q^{59} +15256.9 q^{61} -38705.3 q^{62} -12166.3 q^{63} +31246.7 q^{64} -6290.87 q^{66} -46228.1 q^{67} -2152.79 q^{68} -19644.2 q^{69} +68040.5 q^{71} +14331.9 q^{72} -12850.7 q^{73} +17655.8 q^{74} +1002.79 q^{76} +18174.4 q^{77} +59624.8 q^{78} +10936.2 q^{79} +6561.00 q^{81} +2544.80 q^{82} +66842.2 q^{83} +1852.92 q^{84} +78517.1 q^{86} -77433.4 q^{87} -21409.4 q^{88} -66587.8 q^{89} -172257. q^{91} +2991.78 q^{92} -60301.9 q^{93} -88761.4 q^{94} -4463.16 q^{96} +145903. q^{97} -33236.4 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) $ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * 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 $

$2$	−5.77674	−1.02119	−0.510596	−	0.859821i	$-0.670575\pi$
$2$	−5.77674	−1.02119	−0.510596	+	0.859821i	$0.670575\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	1.37069	0.0428341
$5$	0	0
$5$	0	0
$6$	51.9906	0.589586
$7$	−150.201	−1.15859	−0.579294	−	0.815119i	$-0.696672\pi$
$7$	−150.201	−1.15859	−0.579294	+	0.815119i	$0.696672\pi$
$8$	176.937	0.977451
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−12.3362	−0.0247303
$13$	1146.84	1.88210	0.941051	−	0.338265i	$-0.109840\pi$
$13$	1146.84	1.88210	0.941051	+	0.338265i	$0.109840\pi$
$14$	867.675	1.18314
$15$	0	0
$16$	−1065.98	−1.04100
$17$	−1570.59	−1.31808	−0.659038	−	0.752110i	$-0.729036\pi$
$17$	−1570.59	−1.31808	−0.659038	+	0.752110i	$0.729036\pi$
$18$	−467.916	−0.340397
$19$	731.596	0.464930	0.232465	−	0.972605i	$-0.425321\pi$
$19$	731.596	0.464930	0.232465	+	0.972605i	$0.425321\pi$
$20$	0	0
$21$	1351.81	0.668911
$22$	698.985	0.307901
$23$	2182.68	0.860342	0.430171	−	0.902747i	$-0.358453\pi$
$23$	2182.68	0.860342	0.430171	+	0.902747i	$0.358453\pi$
$24$	−1592.44	−0.564331
$25$	0	0
$26$	−6624.97	−1.92199
$27$	−729.000	−0.192450
$28$	−205.880	−0.0496271
$29$	8603.71	1.89973	0.949863	−	0.312667i	$-0.101222\pi$
$29$	8603.71	1.89973	0.949863	+	0.312667i	$0.101222\pi$
$30$	0	0
$31$	6700.21	1.25223	0.626115	−	0.779731i	$-0.284644\pi$
$31$	6700.21	1.25223	0.626115	+	0.779731i	$0.284644\pi$
$32$	495.906	0.0856100
$33$	1089.00	0.174078
$34$	9072.88	1.34601
$35$	0	0
$36$	111.026	0.0142780
$37$	−3056.37	−0.367029	−0.183515	−	0.983017i	$-0.558748\pi$
$37$	−3056.37	−0.367029	−0.183515	+	0.983017i	$0.558748\pi$
$38$	−4226.24	−0.474783
$39$	−10321.5	−1.08663
$40$	0	0
$41$	−440.525	−0.0409271	−0.0204636	−	0.999791i	$-0.506514\pi$
$41$	−440.525	−0.0409271	−0.0204636	+	0.999791i	$0.506514\pi$
$42$	−7809.07	−0.683087
$43$	−13592.0	−1.12101	−0.560507	−	0.828150i	$-0.689394\pi$
$43$	−13592.0	−1.12101	−0.560507	+	0.828150i	$0.689394\pi$
$44$	−165.854	−0.0129150
$45$	0	0
$46$	−12608.8	−0.878575
$47$	15365.3	1.01460	0.507302	−	0.861768i	$-0.330643\pi$
$47$	15365.3	1.01460	0.507302	+	0.861768i	$0.330643\pi$
$48$	9593.85	0.601021
$49$	5753.49	0.342327
$50$	0	0
$51$	14135.3	0.760991
$52$	1571.96	0.0806181
$53$	−25436.6	−1.24385	−0.621926	−	0.783076i	$-0.713650\pi$
$53$	−25436.6	−1.24385	−0.621926	+	0.783076i	$0.713650\pi$
$54$	4211.24	0.196529
$55$	0	0
$56$	−26576.3	−1.13246
$57$	−6584.36	−0.268427
$58$	−49701.4	−1.93999
$59$	−8795.36	−0.328945	−0.164473	−	0.986382i	$-0.552592\pi$
$59$	−8795.36	−0.328945	−0.164473	+	0.986382i	$0.552592\pi$
$60$	0	0
$61$	15256.9	0.524977	0.262489	−	0.964935i	$-0.415457\pi$
$61$	15256.9	0.524977	0.262489	+	0.964935i	$0.415457\pi$
$62$	−38705.3	−1.27877
$63$	−12166.3	−0.386196
$64$	31246.7	0.953575
$65$	0	0
$66$	−6290.87	−0.177767
$67$	−46228.1	−1.25811	−0.629056	−	0.777360i	$-0.716558\pi$
$67$	−46228.1	−1.25811	−0.629056	+	0.777360i	$0.716558\pi$
$68$	−2152.79	−0.0564585
$69$	−19644.2	−0.496719
$70$	0	0
$71$	68040.5	1.60185	0.800924	−	0.598766i	$-0.204342\pi$
$71$	68040.5	1.60185	0.800924	+	0.598766i	$0.204342\pi$
$72$	14331.9	0.325817
$73$	−12850.7	−0.282240	−0.141120	−	0.989993i	$-0.545070\pi$
$73$	−12850.7	−0.282240	−0.141120	+	0.989993i	$0.545070\pi$
$74$	17655.8	0.374808
$75$	0	0
$76$	1002.79	0.0199148
$77$	18174.4	0.349328
$78$	59624.8	1.10966
$79$	10936.2	0.197151	0.0985756	−	0.995130i	$-0.468571\pi$
$79$	10936.2	0.197151	0.0985756	+	0.995130i	$0.468571\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	2544.80	0.0417944
$83$	66842.2	1.06501	0.532507	−	0.846426i	$-0.321250\pi$
$83$	66842.2	1.06501	0.532507	+	0.846426i	$0.321250\pi$
$84$	1852.92	0.0286522
$85$	0	0
$86$	78517.1	1.14477
$87$	−77433.4	−1.09681
$88$	−21409.4	−0.294712
$89$	−66587.8	−0.891085	−0.445543	−	0.895261i	$-0.646989\pi$
$89$	−66587.8	−0.891085	−0.445543	+	0.895261i	$0.646989\pi$
$90$	0	0
$91$	−172257.	−2.18058
$92$	2991.78	0.0368520
$93$	−60301.9	−0.722975
$94$	−88761.4	−1.03611
$95$	0	0
$96$	−4463.16	−0.0494270
$97$	145903.	1.57447	0.787236	−	0.616652i	$-0.211511\pi$
$97$	145903.	1.57447	0.787236	+	0.616652i	$0.211511\pi$
$98$	−33236.4	−0.349582
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−125698.	−1.22610	−0.613050	−	0.790044i	$-0.710057\pi$
$101$	−125698.	−1.22610	−0.613050	+	0.790044i	$0.710057\pi$
$102$	−81655.9	−0.777118
$103$	−63242.3	−0.587374	−0.293687	−	0.955902i	$-0.594882\pi$
$103$	−63242.3	−0.587374	−0.293687	+	0.955902i	$0.594882\pi$
$104$	202918.	1.83966
$105$	0	0
$106$	146940.	1.27021
$107$	−103931.	−0.877582	−0.438791	−	0.898589i	$-0.644593\pi$
$107$	−103931.	−0.877582	−0.438791	+	0.898589i	$0.644593\pi$
$108$	−999.233	−0.00824342
$109$	−115234.	−0.928995	−0.464498	−	0.885574i	$-0.653765\pi$
$109$	−115234.	−0.928995	−0.464498	+	0.885574i	$0.653765\pi$
$110$	0	0
$111$	27507.3	0.211905
$112$	160112.	1.20609
$113$	−200077.	−1.47401	−0.737005	−	0.675887i	$-0.763761\pi$
$113$	−200077.	−1.47401	−0.737005	+	0.675887i	$0.763761\pi$
$114$	38036.1	0.274116
$115$	0	0
$116$	11793.0	0.0813730
$117$	92893.7	0.627367
$118$	50808.5	0.335917
$119$	235905.	1.52711
$120$	0	0
$121$	14641.0	0.0909091
$122$	−88134.9	−0.536103
$123$	3964.73	0.0236293
$124$	9183.91	0.0536381
$125$	0	0
$126$	70281.6	0.394381
$127$	−304565.	−1.67560	−0.837802	−	0.545974i	$-0.816160\pi$
$127$	−304565.	−1.67560	−0.837802	+	0.545974i	$0.816160\pi$
$128$	−196373.	−1.05939
$129$	122328.	0.647217
$130$	0	0
$131$	108252.	0.551135	0.275567	−	0.961282i	$-0.411134\pi$
$131$	108252.	0.551135	0.275567	+	0.961282i	$0.411134\pi$
$132$	1492.68	0.00745646
$133$	−109887.	−0.538662
$134$	267048.	1.28477
$135$	0	0
$136$	−277896.	−1.28835
$137$	−329333.	−1.49911	−0.749557	−	0.661940i	$-0.769733\pi$
$137$	−329333.	−1.49911	−0.749557	+	0.661940i	$0.769733\pi$
$138$	113479.	0.507245
$139$	40752.3	0.178902	0.0894510	−	0.995991i	$-0.471489\pi$
$139$	40752.3	0.178902	0.0894510	+	0.995991i	$0.471489\pi$
$140$	0	0
$141$	−138288.	−0.585782
$142$	−393052.	−1.63580
$143$	−138767.	−0.567475
$144$	−86344.6	−0.347000
$145$	0	0
$146$	74234.9	0.288221
$147$	−51781.4	−0.197643
$148$	−4189.33	−0.0157214
$149$	9127.01	0.0336793	0.0168397	−	0.999858i	$-0.494640\pi$
$149$	9127.01	0.0336793	0.0168397	+	0.999858i	$0.494640\pi$
$150$	0	0
$151$	−273166.	−0.974955	−0.487478	−	0.873135i	$-0.662083\pi$
$151$	−273166.	−0.974955	−0.487478	+	0.873135i	$0.662083\pi$
$152$	129447.	0.454446
$153$	−127218.	−0.439358
$154$	−104989.	−0.356731
$155$	0	0
$156$	−14147.6	−0.0465449
$157$	59757.7	0.193484	0.0967419	−	0.995309i	$-0.469158\pi$
$157$	59757.7	0.193484	0.0967419	+	0.995309i	$0.469158\pi$
$158$	−63175.7	−0.201329
$159$	228929.	0.718138
$160$	0	0
$161$	−327842.	−0.996782
$162$	−37901.2	−0.113466
$163$	101366.	0.298828	0.149414	−	0.988775i	$-0.452261\pi$
$163$	101366.	0.298828	0.149414	+	0.988775i	$0.452261\pi$
$164$	−603.823	−0.00175307
$165$	0	0
$166$	−386130.	−1.08758
$167$	332221.	0.921799	0.460900	−	0.887452i	$-0.347527\pi$
$167$	332221.	0.921799	0.460900	+	0.887452i	$0.347527\pi$
$168$	239186.	0.653828
$169$	943941.	2.54231
$170$	0	0
$171$	59259.3	0.154977
$172$	−18630.4	−0.0480176
$173$	−156146.	−0.396658	−0.198329	−	0.980135i	$-0.563551\pi$
$173$	−156146.	−0.396658	−0.198329	+	0.980135i	$0.563551\pi$
$174$	447313.	1.12005
$175$	0	0
$176$	128984.	0.313873
$177$	79158.3	0.189917
$178$	384660.	0.909970
$179$	798293.	1.86221	0.931107	−	0.364746i	$-0.118844\pi$
$179$	798293.	1.86221	0.931107	+	0.364746i	$0.118844\pi$
$180$	0	0
$181$	534674.	1.21309	0.606544	−	0.795050i	$-0.292555\pi$
$181$	534674.	1.21309	0.606544	+	0.795050i	$0.292555\pi$
$182$	995081.	2.22679
$183$	−137312.	−0.303096
$184$	386198.	0.840942
$185$	0	0
$186$	348348.	0.738297
$187$	190041.	0.397415
$188$	21061.1	0.0434596
$189$	109497.	0.222970
$190$	0	0
$191$	−226409.	−0.449066	−0.224533	−	0.974466i	$-0.572086\pi$
$191$	−226409.	−0.449066	−0.224533	+	0.974466i	$0.572086\pi$
$192$	−281221.	−0.550547
$193$	−78423.8	−0.151550	−0.0757748	−	0.997125i	$-0.524143\pi$
$193$	−78423.8	−0.151550	−0.0757748	+	0.997125i	$0.524143\pi$
$194$	−842843.	−1.60784
$195$	0	0
$196$	7886.25	0.0146633
$197$	79280.1	0.145545	0.0727727	−	0.997349i	$-0.476815\pi$
$197$	79280.1	0.145545	0.0727727	+	0.997349i	$0.476815\pi$
$198$	56617.8	0.102634
$199$	750688.	1.34378	0.671888	−	0.740653i	$-0.265484\pi$
$199$	750688.	1.34378	0.671888	+	0.740653i	$0.265484\pi$
$200$	0	0
$201$	416053.	0.726372
$202$	726126.	1.25208
$203$	−1.29229e6	−2.20100
$204$	19375.1	0.0325963
$205$	0	0
$206$	365334.	0.599822
$207$	176797.	0.286781
$208$	−1.22251e6	−1.95927
$209$	−88523.1	−0.140182
$210$	0	0
$211$	648787.	1.00322	0.501610	−	0.865094i	$-0.332741\pi$
$211$	648787.	1.00322	0.501610	+	0.865094i	$0.332741\pi$
$212$	−34865.7	−0.0532793
$213$	−612364.	−0.924828
$214$	600385.	0.896180
$215$	0	0
$216$	−128987.	−0.188110
$217$	−1.00638e6	−1.45082
$218$	665675.	0.948683
$219$	115656.	0.162951
$220$	0	0
$221$	−1.80121e6	−2.48075
$222$	−158902.	−0.216395
$223$	880655.	1.18589	0.592944	−	0.805244i	$-0.297966\pi$
$223$	880655.	1.18589	0.592944	+	0.805244i	$0.297966\pi$
$224$	−74485.9	−0.0991868
$225$	0	0
$226$	1.15579e6	1.50525
$227$	−22215.5	−0.0286148	−0.0143074	−	0.999898i	$-0.504554\pi$
$227$	−22215.5	−0.0286148	−0.0143074	+	0.999898i	$0.504554\pi$
$228$	−9025.12	−0.0114978
$229$	−753330.	−0.949285	−0.474642	−	0.880179i	$-0.657423\pi$
$229$	−753330.	−0.949285	−0.474642	+	0.880179i	$0.657423\pi$
$230$	0	0
$231$	−163569.	−0.201684
$232$	1.52232e6	1.85689
$233$	727015.	0.877311	0.438655	−	0.898655i	$-0.355455\pi$
$233$	727015.	0.877311	0.438655	+	0.898655i	$0.355455\pi$
$234$	−536623.	−0.640663
$235$	0	0
$236$	−12055.7	−0.0140901
$237$	−98426.0	−0.113825
$238$	−1.36276e6	−1.55947
$239$	−1.37068e6	−1.55217	−0.776086	−	0.630627i	$-0.782798\pi$
$239$	−1.37068e6	−1.55217	−0.776086	+	0.630627i	$0.782798\pi$
$240$	0	0
$241$	1.26915e6	1.40758	0.703788	−	0.710410i	$-0.251491\pi$
$241$	1.26915e6	1.40758	0.703788	+	0.710410i	$0.251491\pi$
$242$	−84577.2	−0.0928357
$243$	−59049.0	−0.0641500
$244$	20912.4	0.0224869
$245$	0	0
$246$	−22903.2	−0.0241300
$247$	839021.	0.875045
$248$	1.18552e6	1.22399
$249$	−601580.	−0.614886
$250$	0	0
$251$	−1.22546e6	−1.22777	−0.613884	−	0.789396i	$-0.710394\pi$
$251$	−1.22546e6	−1.22777	−0.613884	+	0.789396i	$0.710394\pi$
$252$	−16676.3	−0.0165424
$253$	−264105.	−0.259403
$254$	1.75939e6	1.71111
$255$	0	0
$256$	134501.	0.128270
$257$	704646.	0.665485	0.332742	−	0.943018i	$-0.392026\pi$
$257$	704646.	0.665485	0.332742	+	0.943018i	$0.392026\pi$
$258$	−706654.	−0.660933
$259$	459071.	0.425236
$260$	0	0
$261$	696901.	0.633242
$262$	−625343.	−0.562815
$263$	−225428.	−0.200964	−0.100482	−	0.994939i	$-0.532039\pi$
$263$	−225428.	−0.200964	−0.100482	+	0.994939i	$0.532039\pi$
$264$	192685.	0.170152
$265$	0	0
$266$	634787.	0.550078
$267$	599290.	0.514468
$268$	−63364.5	−0.0538901
$269$	177633.	0.149673	0.0748365	−	0.997196i	$-0.476157\pi$
$269$	177633.	0.149673	0.0748365	+	0.997196i	$0.476157\pi$
$270$	0	0
$271$	1.72248e6	1.42473	0.712363	−	0.701812i	$-0.247625\pi$
$271$	1.72248e6	1.42473	0.712363	+	0.701812i	$0.247625\pi$
$272$	1.67422e6	1.37212
$273$	1.55031e6	1.25896
$274$	1.90247e6	1.53088
$275$	0	0
$276$	−26926.0	−0.0212765
$277$	2.32275e6	1.81888	0.909438	−	0.415839i	$-0.136512\pi$
$277$	2.32275e6	1.81888	0.909438	+	0.415839i	$0.136512\pi$
$278$	−235415.	−0.182693
$279$	542717.	0.417410
$280$	0	0
$281$	993777.	0.750798	0.375399	−	0.926863i	$-0.377506\pi$
$281$	993777.	0.750798	0.375399	+	0.926863i	$0.377506\pi$
$282$	798852.	0.598196
$283$	1.99675e6	1.48203	0.741017	−	0.671486i	$-0.234344\pi$
$283$	1.99675e6	1.48203	0.741017	+	0.671486i	$0.234344\pi$
$284$	93262.4	0.0686137
$285$	0	0
$286$	801622.	0.579501
$287$	66167.5	0.0474177
$288$	40168.4	0.0285367
$289$	1.04689e6	0.737322
$290$	0	0
$291$	−1.31313e6	−0.909022
$292$	−17614.3	−0.0120895
$293$	−2.10760e6	−1.43423	−0.717116	−	0.696954i	$-0.754538\pi$
$293$	−2.10760e6	−1.43423	−0.717116	+	0.696954i	$0.754538\pi$
$294$	299127.	0.201831
$295$	0	0
$296$	−540786.	−0.358753
$297$	88209.0	0.0580259
$298$	−52724.4	−0.0343930
$299$	2.50318e6	1.61925
$300$	0	0
$301$	2.04153e6	1.29879
$302$	1.57801e6	0.995617
$303$	1.13128e6	0.707889
$304$	−779869.	−0.483991
$305$	0	0
$306$	734903.	0.448670
$307$	831340.	0.503423	0.251711	−	0.967802i	$-0.419007\pi$
$307$	831340.	0.503423	0.251711	+	0.967802i	$0.419007\pi$
$308$	24911.4	0.0149631
$309$	569181.	0.339120
$310$	0	0
$311$	−985788.	−0.577940	−0.288970	−	0.957338i	$-0.593313\pi$
$311$	−985788.	−0.577940	−0.288970	+	0.957338i	$0.593313\pi$
$312$	−1.82626e6	−1.06213
$313$	3.23805e6	1.86820	0.934098	−	0.357018i	$-0.116207\pi$
$313$	3.23805e6	1.86820	0.934098	+	0.357018i	$0.116207\pi$
$314$	−345205.	−0.197584
$315$	0	0
$316$	14990.2	0.00844479
$317$	−1.92482e6	−1.07582	−0.537912	−	0.843001i	$-0.680787\pi$
$317$	−1.92482e6	−1.07582	−0.537912	+	0.843001i	$0.680787\pi$
$318$	−1.32246e6	−0.733357
$319$	−1.04105e6	−0.572789
$320$	0	0
$321$	935383.	0.506672
$322$	1.89386e6	1.01791
$323$	−1.14904e6	−0.612812
$324$	8993.10	0.00475934
$325$	0	0
$326$	−585563.	−0.305161
$327$	1.03710e6	0.536356
$328$	−77945.4	−0.0400042
$329$	−2.30789e6	−1.17551
$330$	0	0
$331$	−481029.	−0.241324	−0.120662	−	0.992694i	$-0.538502\pi$
$331$	−481029.	−0.241324	−0.120662	+	0.992694i	$0.538502\pi$
$332$	91619.9	0.0456189
$333$	−247566.	−0.122343
$334$	−1.91916e6	−0.941334
$335$	0	0
$336$	−1.44101e6	−0.696336
$337$	−2.39601e6	−1.14925	−0.574623	−	0.818418i	$-0.694851\pi$
$337$	−2.39601e6	−1.14925	−0.574623	+	0.818418i	$0.694851\pi$
$338$	−5.45290e6	−2.59618
$339$	1.80069e6	0.851020
$340$	0	0
$341$	−810725.	−0.377562
$342$	−342325.	−0.158261
$343$	1.66025e6	0.761972
$344$	−2.40493e6	−1.09574
$345$	0	0
$346$	902016.	0.405064
$347$	2.16490e6	0.965192	0.482596	−	0.875843i	$-0.339694\pi$
$347$	2.16490e6	0.965192	0.482596	+	0.875843i	$0.339694\pi$
$348$	−106137.	−0.0469807
$349$	2.95271e6	1.29765	0.648824	−	0.760938i	$-0.275261\pi$
$349$	2.95271e6	1.29765	0.648824	+	0.760938i	$0.275261\pi$
$350$	0	0
$351$	−836044.	−0.362211
$352$	−60004.7	−0.0258124
$353$	−537762.	−0.229696	−0.114848	−	0.993383i	$-0.536638\pi$
$353$	−537762.	−0.229696	−0.114848	+	0.993383i	$0.536638\pi$
$354$	−457277.	−0.193941
$355$	0	0
$356$	−91271.2	−0.0381688
$357$	−2.12314e6	−0.881675
$358$	−4.61153e6	−1.90168
$359$	3.19303e6	1.30757	0.653787	−	0.756678i	$-0.273179\pi$
$359$	3.19303e6	1.30757	0.653787	+	0.756678i	$0.273179\pi$
$360$	0	0
$361$	−1.94087e6	−0.783840
$362$	−3.08867e6	−1.23880
$363$	−131769.	−0.0524864
$364$	−236110.	−0.0934032
$365$	0	0
$366$	793214.	0.309519
$367$	3.24705e6	1.25842	0.629208	−	0.777237i	$-0.283379\pi$
$367$	3.24705e6	1.25842	0.629208	+	0.777237i	$0.283379\pi$
$368$	−2.32670e6	−0.895616
$369$	−35682.5	−0.0136424
$370$	0	0
$371$	3.82061e6	1.44111
$372$	−82655.2	−0.0309680
$373$	−2.23197e6	−0.830647	−0.415323	−	0.909674i	$-0.636332\pi$
$373$	−2.23197e6	−0.830647	−0.415323	+	0.909674i	$0.636332\pi$
$374$	−1.09782e6	−0.405837
$375$	0	0
$376$	2.71870e6	0.991726
$377$	9.86705e6	3.57548
$378$	−632535.	−0.227696
$379$	−1.67352e6	−0.598456	−0.299228	−	0.954182i	$-0.596729\pi$
$379$	−1.67352e6	−0.598456	−0.299228	+	0.954182i	$0.596729\pi$
$380$	0	0
$381$	2.74109e6	0.967410
$382$	1.30791e6	0.458583
$383$	−1.51540e6	−0.527874	−0.263937	−	0.964540i	$-0.585021\pi$
$383$	−1.51540e6	−0.527874	−0.263937	+	0.964540i	$0.585021\pi$
$384$	1.76736e6	0.611641
$385$	0	0
$386$	453034.	0.154761
$387$	−1.10095e6	−0.373671
$388$	199988.	0.0674410
$389$	−3.15291e6	−1.05642	−0.528211	−	0.849113i	$-0.677137\pi$
$389$	−3.15291e6	−1.05642	−0.528211	+	0.849113i	$0.677137\pi$
$390$	0	0
$391$	−3.42810e6	−1.13400
$392$	1.01801e6	0.334608
$393$	−974268.	−0.318198
$394$	−457980.	−0.148630
$395$	0	0
$396$	−13434.1	−0.00430499
$397$	−5.36348e6	−1.70793	−0.853966	−	0.520330i	$-0.825809\pi$
$397$	−5.36348e6	−1.70793	−0.853966	+	0.520330i	$0.825809\pi$
$398$	−4.33653e6	−1.37225
$399$	988981.	0.310997
$400$	0	0
$401$	−2.02367e6	−0.628462	−0.314231	−	0.949347i	$-0.601747\pi$
$401$	−2.02367e6	−0.628462	−0.314231	+	0.949347i	$0.601747\pi$
$402$	−2.40343e6	−0.741765
$403$	7.68404e6	2.35682
$404$	−172293.	−0.0525188
$405$	0	0
$406$	7.46522e6	2.24764
$407$	369820.	0.110664
$408$	2.50106e6	0.743831
$409$	−6.34651e6	−1.87597	−0.937986	−	0.346672i	$-0.887312\pi$
$409$	−6.34651e6	−1.87597	−0.937986	+	0.346672i	$0.887312\pi$
$410$	0	0
$411$	2.96400e6	0.865513
$412$	−86685.6	−0.0251596
$413$	1.32108e6	0.381112
$414$	−1.02131e6	−0.292858
$415$	0	0
$416$	568723.	0.161127
$417$	−366771.	−0.103289
$418$	511375.	0.143152
$419$	5.32755e6	1.48249	0.741246	−	0.671234i	$-0.234235\pi$
$419$	5.32755e6	1.48249	0.741246	+	0.671234i	$0.234235\pi$
$420$	0	0
$421$	4.62270e6	1.27113	0.635566	−	0.772046i	$-0.280767\pi$
$421$	4.62270e6	1.27113	0.635566	+	0.772046i	$0.280767\pi$
$422$	−3.74787e6	−1.02448
$423$	1.24459e6	0.338201
$424$	−4.50068e6	−1.21580
$425$	0	0
$426$	3.53747e6	0.944427
$427$	−2.29160e6	−0.608233
$428$	−142458.	−0.0375904
$429$	1.24890e6	0.327632
$430$	0	0
$431$	−1.98012e6	−0.513450	−0.256725	−	0.966484i	$-0.582644\pi$
$431$	−1.98012e6	−0.513450	−0.256725	+	0.966484i	$0.582644\pi$
$432$	777102.	0.200340
$433$	−1.49070e6	−0.382094	−0.191047	−	0.981581i	$-0.561188\pi$
$433$	−1.49070e6	−0.382094	−0.191047	+	0.981581i	$0.561188\pi$
$434$	5.81360e6	1.48157
$435$	0	0
$436$	−157950.	−0.0397927
$437$	1.59684e6	0.399999
$438$	−668114.	−0.166405
$439$	−2.31922e6	−0.574355	−0.287178	−	0.957877i	$-0.592717\pi$
$439$	−2.31922e6	−0.574355	−0.287178	+	0.957877i	$0.592717\pi$
$440$	0	0
$441$	466033.	0.114109
$442$	1.04051e7	2.53332
$443$	1.40179e6	0.339371	0.169686	−	0.985498i	$-0.445725\pi$
$443$	1.40179e6	0.339371	0.169686	+	0.985498i	$0.445725\pi$
$444$	37704.0	0.00907674
$445$	0	0
$446$	−5.08731e6	−1.21102
$447$	−82143.1	−0.0194448
$448$	−4.69331e6	−1.10480
$449$	6.34855e6	1.48614	0.743069	−	0.669215i	$-0.233370\pi$
$449$	6.34855e6	1.48614	0.743069	+	0.669215i	$0.233370\pi$
$450$	0	0
$451$	53303.5	0.0123400
$452$	−274243.	−0.0631378
$453$	2.45850e6	0.562891
$454$	128333.	0.0292212
$455$	0	0
$456$	−1.16502e6	−0.262374
$457$	−3.44389e6	−0.771364	−0.385682	−	0.922632i	$-0.626034\pi$
$457$	−3.44389e6	−0.771364	−0.385682	+	0.922632i	$0.626034\pi$
$458$	4.35179e6	0.969402
$459$	1.14496e6	0.253664
$460$	0	0
$461$	−7.33956e6	−1.60849	−0.804244	−	0.594299i	$-0.797429\pi$
$461$	−7.33956e6	−1.60849	−0.804244	+	0.594299i	$0.797429\pi$
$462$	944898.	0.205959
$463$	4.47030e6	0.969135	0.484567	−	0.874754i	$-0.338977\pi$
$463$	4.47030e6	0.969135	0.484567	+	0.874754i	$0.338977\pi$
$464$	−9.17141e6	−1.97761
$465$	0	0
$466$	−4.19977e6	−0.895903
$467$	6.94168e6	1.47290	0.736449	−	0.676493i	$-0.236501\pi$
$467$	6.94168e6	1.47290	0.736449	+	0.676493i	$0.236501\pi$
$468$	127329.	0.0268727
$469$	6.94354e6	1.45763
$470$	0	0
$471$	−537819.	−0.111708
$472$	−1.55623e6	−0.321528
$473$	1.64463e6	0.337998
$474$	568581.	0.116238
$475$	0	0
$476$	323352.	0.0654122
$477$	−2.06036e6	−0.414617
$478$	7.91803e6	1.58507
$479$	−2.57692e6	−0.513170	−0.256585	−	0.966522i	$-0.582597\pi$
$479$	−2.57692e6	−0.513170	−0.256585	+	0.966522i	$0.582597\pi$
$480$	0	0
$481$	−3.50515e6	−0.690787
$482$	−7.33157e6	−1.43741
$483$	2.95058e6	0.575493
$484$	20068.3	0.00389401
$485$	0	0
$486$	341111.	0.0655095
$487$	−267618.	−0.0511321	−0.0255660	−	0.999673i	$-0.508139\pi$
$487$	−267618.	−0.0511321	−0.0255660	+	0.999673i	$0.508139\pi$
$488$	2.69951e6	0.513140
$489$	−912291.	−0.172529
$490$	0	0
$491$	−5.20972e6	−0.975239	−0.487619	−	0.873056i	$-0.662135\pi$
$491$	−5.20972e6	−0.975239	−0.487619	+	0.873056i	$0.662135\pi$
$492$	5434.41	0.00101214
$493$	−1.35129e7	−2.50398
$494$	−4.84680e6	−0.893589
$495$	0	0
$496$	−7.14231e6	−1.30357
$497$	−1.02198e7	−1.85588
$498$	3.47517e6	0.627917
$499$	7.98482e6	1.43553	0.717767	−	0.696283i	$-0.245164\pi$
$499$	7.98482e6	1.43553	0.717767	+	0.696283i	$0.245164\pi$
$500$	0	0
$501$	−2.98999e6	−0.532201
$502$	7.07919e6	1.25379
$503$	2.12921e6	0.375231	0.187615	−	0.982243i	$-0.439924\pi$
$503$	2.12921e6	0.375231	0.187615	+	0.982243i	$0.439924\pi$
$504$	−2.15268e6	−0.377488
$505$	0	0
$506$	1.52566e6	0.264900
$507$	−8.49547e6	−1.46780
$508$	−417465.	−0.0717729
$509$	8.45839e6	1.44708	0.723541	−	0.690281i	$-0.242513\pi$
$509$	8.45839e6	1.44708	0.723541	+	0.690281i	$0.242513\pi$
$510$	0	0
$511$	1.93019e6	0.327000
$512$	5.50697e6	0.928405
$513$	−533333.	−0.0894758
$514$	−4.07055e6	−0.679588
$515$	0	0
$516$	167673.	0.0277230
$517$	−1.85920e6	−0.305915
$518$	−2.65193e6	−0.434248
$519$	1.40532e6	0.229011
$520$	0	0
$521$	−473943.	−0.0764948	−0.0382474	−	0.999268i	$-0.512177\pi$
$521$	−473943.	−0.0764948	−0.0382474	+	0.999268i	$0.512177\pi$
$522$	−4.02581e6	−0.646662
$523$	9.60139e6	1.53490	0.767449	−	0.641110i	$-0.221526\pi$
$523$	9.60139e6	1.53490	0.767449	+	0.641110i	$0.221526\pi$
$524$	148380.	0.0236073
$525$	0	0
$526$	1.30224e6	0.205223
$527$	−1.05233e7	−1.65053
$528$	−1.16086e6	−0.181215
$529$	−1.67224e6	−0.259811
$530$	0	0
$531$	−712424.	−0.109648
$532$	−150621.	−0.0230731
$533$	−505210.	−0.0770290
$534$	−3.46194e6	−0.525371
$535$	0	0
$536$	−8.17949e6	−1.22974
$537$	−7.18464e6	−1.07515
$538$	−1.02614e6	−0.152845
$539$	−696172.	−0.103215
$540$	0	0
$541$	−5.96439e6	−0.876138	−0.438069	−	0.898941i	$-0.644338\pi$
$541$	−5.96439e6	−0.876138	−0.438069	+	0.898941i	$0.644338\pi$
$542$	−9.95032e6	−1.45492
$543$	−4.81206e6	−0.700377
$544$	−778865.	−0.112840
$545$	0	0
$546$	−8.95573e6	−1.28564
$547$	5.67751e6	0.811314	0.405657	−	0.914025i	$-0.367043\pi$
$547$	5.67751e6	0.811314	0.405657	+	0.914025i	$0.367043\pi$
$548$	−451414.	−0.0642131
$549$	1.23581e6	0.174992
$550$	0	0
$551$	6.29444e6	0.883239
$552$	−3.47579e6	−0.485518
$553$	−1.64264e6	−0.228417
$554$	−1.34179e7	−1.85742
$555$	0	0
$556$	55858.8	0.00766310
$557$	−8.02484e6	−1.09597	−0.547985	−	0.836488i	$-0.684605\pi$
$557$	−8.02484e6	−1.09597	−0.547985	+	0.836488i	$0.684605\pi$
$558$	−3.13513e6	−0.426256
$559$	−1.55877e7	−2.10986
$560$	0	0
$561$	−1.71037e6	−0.229447
$562$	−5.74079e6	−0.766709
$563$	5.79184e6	0.770098	0.385049	−	0.922896i	$-0.374185\pi$
$563$	5.79184e6	0.770098	0.385049	+	0.922896i	$0.374185\pi$
$564$	−189550.	−0.0250914
$565$	0	0
$566$	−1.15347e7	−1.51344
$567$	−985472.	−0.128732
$568$	1.20389e7	1.56573
$569$	668243.	0.0865274	0.0432637	−	0.999064i	$-0.486224\pi$
$569$	668243.	0.0865274	0.0432637	+	0.999064i	$0.486224\pi$
$570$	0	0
$571$	1.78527e6	0.229147	0.114573	−	0.993415i	$-0.463450\pi$
$571$	1.78527e6	0.229147	0.114573	+	0.993415i	$0.463450\pi$
$572$	−190207.	−0.0243073
$573$	2.03768e6	0.259268
$574$	−382232.	−0.0484226
$575$	0	0
$576$	2.53099e6	0.317858
$577$	621090.	0.0776631	0.0388316	−	0.999246i	$-0.487636\pi$
$577$	621090.	0.0776631	0.0388316	+	0.999246i	$0.487636\pi$
$578$	−6.04762e6	−0.752948
$579$	705815.	0.0874972
$580$	0	0
$581$	−1.00398e7	−1.23391
$582$	7.58559e6	0.928286
$583$	3.07782e6	0.375036
$584$	−2.27376e6	−0.275875
$585$	0	0
$586$	1.21751e7	1.46463
$587$	3.14798e6	0.377083	0.188542	−	0.982065i	$-0.439624\pi$
$587$	3.14798e6	0.377083	0.188542	+	0.982065i	$0.439624\pi$
$588$	−70976.3	−0.00846583
$589$	4.90184e6	0.582199
$590$	0	0
$591$	−713521.	−0.0840307
$592$	3.25803e6	0.382077
$593$	−1.45313e6	−0.169695	−0.0848474	−	0.996394i	$-0.527040\pi$
$593$	−1.45313e6	−0.169695	−0.0848474	+	0.996394i	$0.527040\pi$
$594$	−509560.	−0.0592556
$595$	0	0
$596$	12510.3	0.00144262
$597$	−6.75619e6	−0.775829
$598$	−1.44602e7	−1.65357
$599$	7.62041e6	0.867784	0.433892	−	0.900965i	$-0.357140\pi$
$599$	7.62041e6	0.867784	0.433892	+	0.900965i	$0.357140\pi$
$600$	0	0
$601$	5.30235e6	0.598801	0.299400	−	0.954128i	$-0.403213\pi$
$601$	5.30235e6	0.598801	0.299400	+	0.954128i	$0.403213\pi$
$602$	−1.17934e7	−1.32632
$603$	−3.74448e6	−0.419371
$604$	−374426.	−0.0417613
$605$	0	0
$606$	−6.53513e6	−0.722891
$607$	−1.09581e7	−1.20716	−0.603580	−	0.797302i	$-0.706260\pi$
$607$	−1.09581e7	−1.20716	−0.603580	+	0.797302i	$0.706260\pi$
$608$	362803.	0.0398026
$609$	1.16306e7	1.27075
$610$	0	0
$611$	1.76215e7	1.90959
$612$	−174376.	−0.0188195
$613$	7.21856e6	0.775889	0.387944	−	0.921683i	$-0.373185\pi$
$613$	7.21856e6	0.775889	0.387944	+	0.921683i	$0.373185\pi$
$614$	−4.80244e6	−0.514092
$615$	0	0
$616$	3.21573e6	0.341450
$617$	−2.11311e6	−0.223465	−0.111733	−	0.993738i	$-0.535640\pi$
$617$	−2.11311e6	−0.223465	−0.111733	+	0.993738i	$0.535640\pi$
$618$	−3.28801e6	−0.346307
$619$	−1.80296e7	−1.89130	−0.945648	−	0.325191i	$-0.894571\pi$
$619$	−1.80296e7	−1.89130	−0.945648	+	0.325191i	$0.894571\pi$
$620$	0	0
$621$	−1.59118e6	−0.165573
$622$	5.69464e6	0.590188
$623$	1.00016e7	1.03240
$624$	1.10026e7	1.13118
$625$	0	0
$626$	−1.87053e7	−1.90779
$627$	796708.	0.0809339
$628$	81909.3	0.00828770
$629$	4.80029e6	0.483772
$630$	0	0
$631$	5.34708e6	0.534618	0.267309	−	0.963611i	$-0.413866\pi$
$631$	5.34708e6	0.534618	0.267309	+	0.963611i	$0.413866\pi$
$632$	1.93503e6	0.192706
$633$	−5.83908e6	−0.579209
$634$	1.11192e7	1.09862
$635$	0	0
$636$	313791.	0.0307608
$637$	6.59831e6	0.644294
$638$	6.01387e6	0.584928
$639$	5.51128e6	0.533949
$640$	0	0
$641$	−6.87676e6	−0.661057	−0.330529	−	0.943796i	$-0.607227\pi$
$641$	−6.87676e6	−0.661057	−0.330529	+	0.943796i	$0.607227\pi$
$642$	−5.40346e6	−0.517410
$643$	−537253.	−0.0512450	−0.0256225	−	0.999672i	$-0.508157\pi$
$643$	−537253.	−0.0512450	−0.0256225	+	0.999672i	$0.508157\pi$
$644$	−449370.	−0.0426962
$645$	0	0
$646$	6.63768e6	0.625799
$647$	−1.77812e7	−1.66994	−0.834970	−	0.550295i	$-0.814515\pi$
$647$	−1.77812e7	−1.66994	−0.834970	+	0.550295i	$0.814515\pi$
$648$	1.16089e6	0.108606
$649$	1.06424e6	0.0991808
$650$	0	0
$651$	9.05743e6	0.837631
$652$	138941.	0.0128000
$653$	850319.	0.0780367	0.0390184	−	0.999238i	$-0.487577\pi$
$653$	850319.	0.0780367	0.0390184	+	0.999238i	$0.487577\pi$
$654$	−5.99108e6	−0.547722
$655$	0	0
$656$	469592.	0.0426051
$657$	−1.04090e6	−0.0940799
$658$	1.33321e7	1.20042
$659$	−9.40035e6	−0.843199	−0.421600	−	0.906782i	$-0.638531\pi$
$659$	−9.40035e6	−0.843199	−0.421600	+	0.906782i	$0.638531\pi$
$660$	0	0
$661$	−3.01606e6	−0.268495	−0.134248	−	0.990948i	$-0.542862\pi$
$661$	−3.01606e6	−0.268495	−0.134248	+	0.990948i	$0.542862\pi$
$662$	2.77878e6	0.246439
$663$	1.62109e7	1.43226
$664$	1.18269e7	1.04100
$665$	0	0
$666$	1.43012e6	0.124936
$667$	1.87792e7	1.63441
$668$	455373.	0.0394844
$669$	−7.92590e6	−0.684673
$670$	0	0
$671$	−1.84608e6	−0.158287
$672$	670373.	0.0572655
$673$	1.95542e7	1.66419	0.832095	−	0.554634i	$-0.187142\pi$
$673$	1.95542e7	1.66419	0.832095	+	0.554634i	$0.187142\pi$
$674$	1.38411e7	1.17360
$675$	0	0
$676$	1.29385e6	0.108897
$677$	−1.81798e7	−1.52446	−0.762231	−	0.647305i	$-0.775896\pi$
$677$	−1.81798e7	−1.52446	−0.762231	+	0.647305i	$0.775896\pi$
$678$	−1.04021e7	−0.869055
$679$	−2.19148e7	−1.82416
$680$	0	0
$681$	199939.	0.0165208
$682$	4.68335e6	0.385563
$683$	−9.35931e6	−0.767701	−0.383850	−	0.923395i	$-0.625402\pi$
$683$	−9.35931e6	−0.767701	−0.383850	+	0.923395i	$0.625402\pi$
$684$	81226.1	0.00663828
$685$	0	0
$686$	−9.59085e6	−0.778120
$687$	6.77997e6	0.548070
$688$	1.44888e7	1.16697
$689$	−2.91716e7	−2.34106
$690$	0	0
$691$	−6.10756e6	−0.486601	−0.243300	−	0.969951i	$-0.578230\pi$
$691$	−6.10756e6	−0.486601	−0.243300	+	0.969951i	$0.578230\pi$
$692$	−214028.	−0.0169905
$693$	1.47212e6	0.116443
$694$	−1.25060e7	−0.985647
$695$	0	0
$696$	−1.37009e7	−1.07207
$697$	691884.	0.0539450
$698$	−1.70570e7	−1.32515
$699$	−6.54313e6	−0.506516
$700$	0	0
$701$	−1.64071e7	−1.26106	−0.630531	−	0.776164i	$-0.717163\pi$
$701$	−1.64071e7	−1.26106	−0.630531	+	0.776164i	$0.717163\pi$
$702$	4.82960e6	0.369887
$703$	−2.23602e6	−0.170643
$704$	−3.78086e6	−0.287514
$705$	0	0
$706$	3.10651e6	0.234564
$707$	1.88801e7	1.42054
$708$	108501.	0.00813491
$709$	−1.77082e7	−1.32300	−0.661499	−	0.749946i	$-0.730079\pi$
$709$	−1.77082e7	−1.32300	−0.661499	+	0.749946i	$0.730079\pi$
$710$	0	0
$711$	885834.	0.0657171
$712$	−1.17819e7	−0.870992
$713$	1.46244e7	1.07735
$714$	1.22648e7	0.900360
$715$	0	0
$716$	1.09421e6	0.0797662
$717$	1.23361e7	0.896147
$718$	−1.84453e7	−1.33529
$719$	−2.63661e6	−0.190206	−0.0951029	−	0.995467i	$-0.530318\pi$
$719$	−2.63661e6	−0.190206	−0.0951029	+	0.995467i	$0.530318\pi$
$720$	0	0
$721$	9.49909e6	0.680524
$722$	1.12119e7	0.800452
$723$	−1.14224e7	−0.812664
$724$	732872.	0.0519615
$725$	0	0
$726$	761195.	0.0535987
$727$	8.88827e6	0.623708	0.311854	−	0.950130i	$-0.399050\pi$
$727$	8.88827e6	0.623708	0.311854	+	0.950130i	$0.399050\pi$
$728$	−3.04786e7	−2.13141
$729$	531441.	0.0370370
$730$	0	0
$731$	2.13474e7	1.47758
$732$	−188212.	−0.0129828
$733$	2.21059e7	1.51967	0.759834	−	0.650118i	$-0.225280\pi$
$733$	2.21059e7	1.51967	0.759834	+	0.650118i	$0.225280\pi$
$734$	−1.87574e7	−1.28509
$735$	0	0
$736$	1.08241e6	0.0736539
$737$	5.59361e6	0.379335
$738$	206129.	0.0139315
$739$	1.93561e7	1.30379	0.651893	−	0.758311i	$-0.273975\pi$
$739$	1.93561e7	1.30379	0.651893	+	0.758311i	$0.273975\pi$
$740$	0	0
$741$	−7.55119e6	−0.505207
$742$	−2.20707e7	−1.47165
$743$	−8.35581e6	−0.555286	−0.277643	−	0.960684i	$-0.589553\pi$
$743$	−8.35581e6	−0.555286	−0.277643	+	0.960684i	$0.589553\pi$
$744$	−1.06697e7	−0.706673
$745$	0	0
$746$	1.28935e7	0.848250
$747$	5.41422e6	0.355005
$748$	260488.	0.0170229
$749$	1.56107e7	1.01676
$750$	0	0
$751$	−2092.53	−0.000135385 0	−6.76926e−5	−	1.00000i	$-0.500022\pi$
$751$	−2092.53	−0.000135385 0	−6.76926e−5		1.00000i	$0.500022\pi$
$752$	−1.63792e7	−1.05620
$753$	1.10292e7	0.708852
$754$	−5.69994e7	−3.65125
$755$	0	0
$756$	150086.	0.00955073
$757$	1.32562e7	0.840773	0.420386	−	0.907345i	$-0.361895\pi$
$757$	1.32562e7	0.840773	0.420386	+	0.907345i	$0.361895\pi$
$758$	9.66746e6	0.611138
$759$	2.37694e6	0.149766
$760$	0	0
$761$	1.61517e7	1.01101	0.505507	−	0.862822i	$-0.331305\pi$
$761$	1.61517e7	1.01101	0.505507	+	0.862822i	$0.331305\pi$
$762$	−1.58346e7	−0.987912
$763$	1.73083e7	1.07632
$764$	−310337.	−0.0192353
$765$	0	0
$766$	8.75407e6	0.539061
$767$	−1.00868e7	−0.619109
$768$	−1.21051e6	−0.0740566
$769$	−7.83699e6	−0.477896	−0.238948	−	0.971032i	$-0.576803\pi$
$769$	−7.83699e6	−0.477896	−0.238948	+	0.971032i	$0.576803\pi$
$770$	0	0
$771$	−6.34181e6	−0.384218
$772$	−107495.	−0.00649149
$773$	1.22821e7	0.739306	0.369653	−	0.929170i	$-0.379477\pi$
$773$	1.22821e7	0.739306	0.369653	+	0.929170i	$0.379477\pi$
$774$	6.35989e6	0.381590
$775$	0	0
$776$	2.58157e7	1.53897
$777$	−4.13164e6	−0.245510
$778$	1.82135e7	1.07881
$779$	−322286.	−0.0190282
$780$	0	0
$781$	−8.23290e6	−0.482975
$782$	1.98032e7	1.15803
$783$	−6.27211e6	−0.365602
$784$	−6.13312e6	−0.356362
$785$	0	0
$786$	5.62809e6	0.324941
$787$	1.33913e7	0.770703	0.385351	−	0.922770i	$-0.374080\pi$
$787$	1.33913e7	0.770703	0.385351	+	0.922770i	$0.374080\pi$
$788$	108668.	0.00623431
$789$	2.02885e6	0.116027
$790$	0	0
$791$	3.00518e7	1.70777
$792$	−1.73416e6	−0.0982375
$793$	1.74971e7	0.988061
$794$	3.09834e7	1.74413
$795$	0	0
$796$	1.02896e6	0.0575594
$797$	1.58405e7	0.883332	0.441666	−	0.897180i	$-0.354388\pi$
$797$	1.58405e7	0.883332	0.441666	+	0.897180i	$0.354388\pi$
$798$	−5.71308e6	−0.317587
$799$	−2.41326e7	−1.33732
$800$	0	0
$801$	−5.39361e6	−0.297028
$802$	1.16902e7	0.641781
$803$	1.55493e6	0.0850985
$804$	570280.	0.0311135
$805$	0	0
$806$	−4.43887e7	−2.40677
$807$	−1.59870e6	−0.0864137
$808$	−2.22407e7	−1.19845
$809$	2.23526e7	1.20076	0.600380	−	0.799715i	$-0.295016\pi$
$809$	2.23526e7	1.20076	0.600380	+	0.799715i	$0.295016\pi$
$810$	0	0
$811$	3.12932e7	1.67070	0.835348	−	0.549722i	$-0.185266\pi$
$811$	3.12932e7	1.67070	0.835348	+	0.549722i	$0.185266\pi$
$812$	−1.77133e6	−0.0942778
$813$	−1.55023e7	−0.822566
$814$	−2.13635e6	−0.113009
$815$	0	0
$816$	−1.50680e7	−0.792191
$817$	−9.94382e6	−0.521192
$818$	3.66621e7	1.91573
$819$	−1.39528e7	−0.726860
$820$	0	0
$821$	−2.77034e6	−0.143442	−0.0717208	−	0.997425i	$-0.522849\pi$
$821$	−2.77034e6	−0.143442	−0.0717208	+	0.997425i	$0.522849\pi$
$822$	−1.71223e7	−0.883856
$823$	−1.42994e7	−0.735899	−0.367949	−	0.929846i	$-0.619940\pi$
$823$	−1.42994e7	−0.735899	−0.367949	+	0.929846i	$0.619940\pi$
$824$	−1.11899e7	−0.574129
$825$	0	0
$826$	−7.63151e6	−0.389189
$827$	−7.81938e6	−0.397565	−0.198783	−	0.980044i	$-0.563699\pi$
$827$	−7.81938e6	−0.397565	−0.198783	+	0.980044i	$0.563699\pi$
$828$	242334.	0.0122840
$829$	2.08778e7	1.05511	0.527557	−	0.849520i	$-0.323108\pi$
$829$	2.08778e7	1.05511	0.527557	+	0.849520i	$0.323108\pi$
$830$	0	0
$831$	−2.09048e7	−1.05013
$832$	3.58349e7	1.79473
$833$	−9.03636e6	−0.451213
$834$	2.11874e6	0.105478
$835$	0	0
$836$	−121338.	−0.00600455
$837$	−4.88445e6	−0.240992
$838$	−3.07758e7	−1.51391
$839$	−1.70951e7	−0.838429	−0.419215	−	0.907887i	$-0.637694\pi$
$839$	−1.70951e7	−0.838429	−0.419215	+	0.907887i	$0.637694\pi$
$840$	0	0
$841$	5.35127e7	2.60896
$842$	−2.67041e7	−1.29807
$843$	−8.94399e6	−0.433474
$844$	889286.	0.0429720
$845$	0	0
$846$	−7.18967e6	−0.345369
$847$	−2.19910e6	−0.105326
$848$	2.71150e7	1.29485
$849$	−1.79708e7	−0.855653
$850$	0	0
$851$	−6.67108e6	−0.315771
$852$	−839362.	−0.0396141
$853$	−1.04193e7	−0.490306	−0.245153	−	0.969484i	$-0.578838\pi$
$853$	−1.04193e7	−0.490306	−0.245153	+	0.969484i	$0.578838\pi$
$854$	1.32380e7	0.621123
$855$	0	0
$856$	−1.83894e7	−0.857793
$857$	867466.	0.0403460	0.0201730	−	0.999797i	$-0.493578\pi$
$857$	867466.	0.0403460	0.0201730	+	0.999797i	$0.493578\pi$
$858$	−7.21459e6	−0.334575
$859$	−1.41813e7	−0.655741	−0.327871	−	0.944723i	$-0.606331\pi$
$859$	−1.41813e7	−0.655741	−0.327871	+	0.944723i	$0.606331\pi$
$860$	0	0
$861$	−595508.	−0.0273766
$862$	1.14386e7	0.524332
$863$	−1.99795e7	−0.913185	−0.456592	−	0.889676i	$-0.650930\pi$
$863$	−1.99795e7	−0.913185	−0.456592	+	0.889676i	$0.650930\pi$
$864$	−361516.	−0.0164757
$865$	0	0
$866$	8.61138e6	0.390192
$867$	−9.42203e6	−0.425693
$868$	−1.37944e6	−0.0621445
$869$	−1.32328e6	−0.0594433
$870$	0	0
$871$	−5.30161e7	−2.36790
$872$	−2.03892e7	−0.908047
$873$	1.18181e7	0.524824
$874$	−9.22454e6	−0.408476
$875$	0	0
$876$	158528.	0.00697986
$877$	3.18448e7	1.39810	0.699052	−	0.715071i	$-0.253606\pi$
$877$	3.18448e7	1.39810	0.699052	+	0.715071i	$0.253606\pi$
$878$	1.33975e7	0.586527
$879$	1.89684e7	0.828054
$880$	0	0
$881$	1.62021e6	0.0703284	0.0351642	−	0.999382i	$-0.488805\pi$
$881$	1.62021e6	0.0703284	0.0351642	+	0.999382i	$0.488805\pi$
$882$	−2.69215e6	−0.116527
$883$	1.23112e7	0.531371	0.265685	−	0.964060i	$-0.414402\pi$
$883$	1.23112e7	0.531371	0.265685	+	0.964060i	$0.414402\pi$
$884$	−2.46890e6	−0.106261
$885$	0	0
$886$	−8.09780e6	−0.346563
$887$	9.83752e6	0.419833	0.209916	−	0.977719i	$-0.432681\pi$
$887$	9.83752e6	0.419833	0.209916	+	0.977719i	$0.432681\pi$
$888$	4.86707e6	0.207126
$889$	4.57462e7	1.94134
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.20711e6	0.0507964
$893$	1.12412e7	0.471720
$894$	474519.	0.0198568
$895$	0	0
$896$	2.94956e7	1.22740
$897$	−2.25286e7	−0.934875
$898$	−3.66739e7	−1.51763
$899$	5.76467e7	2.37889
$900$	0	0
$901$	3.99504e7	1.63949
$902$	−307920.	−0.0126015
$903$	−1.83738e7	−0.749858
$904$	−3.54011e7	−1.44077
$905$	0	0
$906$	−1.42021e7	−0.574820
$907$	3.26593e7	1.31822	0.659110	−	0.752046i	$-0.270933\pi$
$907$	3.26593e7	1.31822	0.659110	+	0.752046i	$0.270933\pi$
$908$	−30450.5	−0.00122569
$909$	−1.01816e7	−0.408700
$910$	0	0
$911$	7.81356e6	0.311927	0.155963	−	0.987763i	$-0.450152\pi$
$911$	7.81356e6	0.311927	0.155963	+	0.987763i	$0.450152\pi$
$912$	7.01882e6	0.279433
$913$	−8.08790e6	−0.321114
$914$	1.98945e7	0.787711
$915$	0	0
$916$	−1.03258e6	−0.0406617
$917$	−1.62596e7	−0.638538
$918$	−6.61413e6	−0.259039
$919$	2.97666e7	1.16263	0.581313	−	0.813680i	$-0.302539\pi$
$919$	2.97666e7	1.16263	0.581313	+	0.813680i	$0.302539\pi$
$920$	0	0
$921$	−7.48206e6	−0.290651
$922$	4.23987e7	1.64258
$923$	7.80313e7	3.01484
$924$	−224203.	−0.00863896
$925$	0	0
$926$	−2.58237e7	−0.989673
$927$	−5.12263e6	−0.195791
$928$	4.26664e6	0.162636
$929$	−3.15595e7	−1.19975	−0.599875	−	0.800094i	$-0.704783\pi$
$929$	−3.15595e7	−1.19975	−0.599875	+	0.800094i	$0.704783\pi$
$930$	0	0
$931$	4.20923e6	0.159158
$932$	996512.	0.0375788
$933$	8.87209e6	0.333674
$934$	−4.01003e7	−1.50411
$935$	0	0
$936$	1.64364e7	0.613221
$937$	−2.29765e7	−0.854940	−0.427470	−	0.904030i	$-0.640595\pi$
$937$	−2.29765e7	−0.854940	−0.427470	+	0.904030i	$0.640595\pi$
$938$	−4.01110e7	−1.48853
$939$	−2.91424e7	−1.07860
$940$	0	0
$941$	3.71098e7	1.36620	0.683100	−	0.730325i	$-0.260631\pi$
$941$	3.71098e7	1.36620	0.683100	+	0.730325i	$0.260631\pi$
$942$	3.10684e6	0.114075
$943$	−961527.	−0.0352113
$944$	9.37571e6	0.342432
$945$	0	0
$946$	−9.50057e6	−0.345161
$947$	−9.02929e6	−0.327174	−0.163587	−	0.986529i	$-0.552306\pi$
$947$	−9.02929e6	−0.327174	−0.163587	+	0.986529i	$0.552306\pi$
$948$	−134912.	−0.00487560
$949$	−1.47376e7	−0.531204
$950$	0	0
$951$	1.73234e7	0.621127
$952$	4.17404e7	1.49267
$953$	−3.24538e7	−1.15753	−0.578766	−	0.815494i	$-0.696466\pi$
$953$	−3.24538e7	−1.15753	−0.578766	+	0.815494i	$0.696466\pi$
$954$	1.19022e7	0.423404
$955$	0	0
$956$	−1.87877e6	−0.0664859
$957$	9.36944e6	0.330700
$958$	1.48862e7	0.524046
$959$	4.94664e7	1.73686
$960$	0	0
$961$	1.62636e7	0.568080
$962$	2.02483e7	0.705426
$963$	−8.41845e6	−0.292527
$964$	1.73962e6	0.0602922
$965$	0	0
$966$	−1.70447e7	−0.587689
$967$	1.38234e7	0.475387	0.237693	−	0.971340i	$-0.423609\pi$
$967$	1.38234e7	0.475387	0.237693	+	0.971340i	$0.423609\pi$
$968$	2.59054e6	0.0888591
$969$	1.03413e7	0.353807
$970$	0	0
$971$	2.63597e7	0.897208	0.448604	−	0.893731i	$-0.351921\pi$
$971$	2.63597e7	0.897208	0.448604	+	0.893731i	$0.351921\pi$
$972$	−80937.9	−0.00274781
$973$	−6.12106e6	−0.207274
$974$	1.54596e6	0.0522157
$975$	0	0
$976$	−1.62636e7	−0.546501
$977$	−4.42890e7	−1.48443	−0.742214	−	0.670163i	$-0.766224\pi$
$977$	−4.42890e7	−1.48443	−0.742214	+	0.670163i	$0.766224\pi$
$978$	5.27006e6	0.176185
$979$	8.05712e6	0.268672
$980$	0	0
$981$	−9.33393e6	−0.309665
$982$	3.00952e7	0.995906
$983$	−3.40185e7	−1.12288	−0.561438	−	0.827519i	$-0.689752\pi$
$983$	−3.40185e7	−1.12288	−0.561438	+	0.827519i	$0.689752\pi$
$984$	701509.	0.0230964
$985$	0	0
$986$	7.80604e7	2.55705
$987$	2.07710e7	0.678680
$988$	1.15004e6	0.0374817
$989$	−2.96669e7	−0.964455
$990$	0	0
$991$	−1.32232e7	−0.427712	−0.213856	−	0.976865i	$-0.568602\pi$
$991$	−1.32232e7	−0.427712	−0.213856	+	0.976865i	$0.568602\pi$
$992$	3.32268e6	0.107203
$993$	4.32926e6	0.139329
$994$	5.90370e7	1.89521
$995$	0	0
$996$	−824579.	−0.0263381
$997$	5.33000e7	1.69820	0.849101	−	0.528231i	$-0.177145\pi$
$997$	5.33000e7	1.69820	0.849101	+	0.528231i	$0.177145\pi$
$998$	−4.61262e7	−1.46596
$999$	2.22809e6	0.0706349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.s.1.3	yes	9
5.4	even	2		825.6.a.r.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.7	✓	9	5.4	even	2
825.6.a.s.1.3	yes	9	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-5.77674 q^{2} -9.00000 q^{3} +1.37069 q^{4} +51.9906 q^{6} -150.201 q^{7} +176.937 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-5.77674 q^{2} -9.00000 q^{3} +1.37069 q^{4} +51.9906 q^{6} -150.201 q^{7} +176.937 q^{8} +81.0000 q^{9} -121.000 q^{11} -12.3362 q^{12} +1146.84 q^{13} +867.675 q^{14} -1065.98 q^{16} -1570.59 q^{17} -467.916 q^{18} +731.596 q^{19} +1351.81 q^{21} +698.985 q^{22} +2182.68 q^{23} -1592.44 q^{24} -6624.97 q^{26} -729.000 q^{27} -205.880 q^{28} +8603.71 q^{29} +6700.21 q^{31} +495.906 q^{32} +1089.00 q^{33} +9072.88 q^{34} +111.026 q^{36} -3056.37 q^{37} -4226.24 q^{38} -10321.5 q^{39} -440.525 q^{41} -7809.07 q^{42} -13592.0 q^{43} -165.854 q^{44} -12608.8 q^{46} +15365.3 q^{47} +9593.85 q^{48} +5753.49 q^{49} +14135.3 q^{51} +1571.96 q^{52} -25436.6 q^{53} +4211.24 q^{54} -26576.3 q^{56} -6584.36 q^{57} -49701.4 q^{58} -8795.36 q^{59} +15256.9 q^{61} -38705.3 q^{62} -12166.3 q^{63} +31246.7 q^{64} -6290.87 q^{66} -46228.1 q^{67} -2152.79 q^{68} -19644.2 q^{69} +68040.5 q^{71} +14331.9 q^{72} -12850.7 q^{73} +17655.8 q^{74} +1002.79 q^{76} +18174.4 q^{77} +59624.8 q^{78} +10936.2 q^{79} +6561.00 q^{81} +2544.80 q^{82} +66842.2 q^{83} +1852.92 q^{84} +78517.1 q^{86} -77433.4 q^{87} -21409.4 q^{88} -66587.8 q^{89} -172257. q^{91} +2991.78 q^{92} -60301.9 q^{93} -88761.4 q^{94} -4463.16 q^{96} +145903. q^{97} -33236.4 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * q^99

Introduction

L-functions

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