LMFDB - Embedded newform 825.6.a.s.1.4

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			1.4
Root			$-1.92452$ 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)

$f(q)$	$=$	$q-1.92452 q^{2} -9.00000 q^{3} -28.2962 q^{4} +17.3207 q^{6} +59.1120 q^{7} +116.041 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-1.92452 q^{2} -9.00000 q^{3} -28.2962 q^{4} +17.3207 q^{6} +59.1120 q^{7} +116.041 q^{8} +81.0000 q^{9} -121.000 q^{11} +254.666 q^{12} -599.621 q^{13} -113.762 q^{14} +682.155 q^{16} +2271.97 q^{17} -155.886 q^{18} -2998.77 q^{19} -532.008 q^{21} +232.867 q^{22} +1571.78 q^{23} -1044.37 q^{24} +1153.98 q^{26} -729.000 q^{27} -1672.65 q^{28} +9020.65 q^{29} +1838.01 q^{31} -5026.14 q^{32} +1089.00 q^{33} -4372.45 q^{34} -2291.99 q^{36} -3881.95 q^{37} +5771.20 q^{38} +5396.59 q^{39} -2147.54 q^{41} +1023.86 q^{42} -2613.13 q^{43} +3423.84 q^{44} -3024.92 q^{46} +8379.51 q^{47} -6139.40 q^{48} -13312.8 q^{49} -20447.7 q^{51} +16967.0 q^{52} -11261.5 q^{53} +1402.98 q^{54} +6859.43 q^{56} +26988.9 q^{57} -17360.4 q^{58} -2505.76 q^{59} -42835.6 q^{61} -3537.29 q^{62} +4788.07 q^{63} -12156.1 q^{64} -2095.80 q^{66} -8325.99 q^{67} -64288.2 q^{68} -14146.0 q^{69} +13440.4 q^{71} +9399.34 q^{72} -19484.1 q^{73} +7470.89 q^{74} +84853.9 q^{76} -7152.55 q^{77} -10385.8 q^{78} -20416.2 q^{79} +6561.00 q^{81} +4132.99 q^{82} -82162.1 q^{83} +15053.8 q^{84} +5029.03 q^{86} -81185.9 q^{87} -14041.0 q^{88} +38458.1 q^{89} -35444.8 q^{91} -44475.4 q^{92} -16542.1 q^{93} -16126.5 q^{94} +45235.3 q^{96} -139725. q^{97} +25620.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.92452	−0.340210	−0.170105	−	0.985426i	$-0.554411\pi$
$2$	−1.92452	−0.340210	−0.170105	+	0.985426i	$0.554411\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−28.2962	−0.884257
$5$	0	0
$5$	0	0
$6$	17.3207	0.196420
$7$	59.1120	0.455964	0.227982	−	0.973665i	$-0.426787\pi$
$7$	59.1120	0.455964	0.227982	+	0.973665i	$0.426787\pi$
$8$	116.041	0.641044
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	254.666	0.510526
$13$	−599.621	−0.984053	−0.492027	−	0.870580i	$-0.663744\pi$
$13$	−599.621	−0.984053	−0.492027	+	0.870580i	$0.663744\pi$
$14$	−113.762	−0.155124
$15$	0	0
$16$	682.155	0.666167
$17$	2271.97	1.90669	0.953346	−	0.301880i	$-0.0976143\pi$
$17$	2271.97	1.90669	0.953346	+	0.301880i	$0.0976143\pi$
$18$	−155.886	−0.113403
$19$	−2998.77	−1.90572	−0.952861	−	0.303408i	$-0.901875\pi$
$19$	−2998.77	−1.90572	−0.952861	+	0.303408i	$0.901875\pi$
$20$	0	0
$21$	−532.008	−0.263251
$22$	232.867	0.102577
$23$	1571.78	0.619544	0.309772	−	0.950811i	$-0.399747\pi$
$23$	1571.78	0.619544	0.309772	+	0.950811i	$0.399747\pi$
$24$	−1044.37	−0.370107
$25$	0	0
$26$	1153.98	0.334785
$27$	−729.000	−0.192450
$28$	−1672.65	−0.403189
$29$	9020.65	1.99179	0.995894	−	0.0905310i	$-0.0288564\pi$
$29$	9020.65	1.99179	0.995894	+	0.0905310i	$0.0288564\pi$
$30$	0	0
$31$	1838.01	0.343514	0.171757	−	0.985139i	$-0.445056\pi$
$31$	1838.01	0.343514	0.171757	+	0.985139i	$0.445056\pi$
$32$	−5026.14	−0.867681
$33$	1089.00	0.174078
$34$	−4372.45	−0.648676
$35$	0	0
$36$	−2291.99	−0.294752
$37$	−3881.95	−0.466171	−0.233086	−	0.972456i	$-0.574882\pi$
$37$	−3881.95	−0.466171	−0.233086	+	0.972456i	$0.574882\pi$
$38$	5771.20	0.648346
$39$	5396.59	0.568143
$40$	0	0
$41$	−2147.54	−0.199518	−0.0997590	−	0.995012i	$-0.531807\pi$
$41$	−2147.54	−0.199518	−0.0997590	+	0.995012i	$0.531807\pi$
$42$	1023.86	0.0895606
$43$	−2613.13	−0.215521	−0.107761	−	0.994177i	$-0.534368\pi$
$43$	−2613.13	−0.215521	−0.107761	+	0.994177i	$0.534368\pi$
$44$	3423.84	0.266614
$45$	0	0
$46$	−3024.92	−0.210775
$47$	8379.51	0.553317	0.276659	−	0.960968i	$-0.410773\pi$
$47$	8379.51	0.553317	0.276659	+	0.960968i	$0.410773\pi$
$48$	−6139.40	−0.384612
$49$	−13312.8	−0.792097
$50$	0	0
$51$	−20447.7	−1.10083
$52$	16967.0	0.870156
$53$	−11261.5	−0.550690	−0.275345	−	0.961345i	$-0.588792\pi$
$53$	−11261.5	−0.550690	−0.275345	+	0.961345i	$0.588792\pi$
$54$	1402.98	0.0654735
$55$	0	0
$56$	6859.43	0.292293
$57$	26988.9	1.10027
$58$	−17360.4	−0.677626
$59$	−2505.76	−0.0937152	−0.0468576	−	0.998902i	$-0.514921\pi$
$59$	−2505.76	−0.0937152	−0.0468576	+	0.998902i	$0.514921\pi$
$60$	0	0
$61$	−42835.6	−1.47394	−0.736970	−	0.675925i	$-0.763744\pi$
$61$	−42835.6	−1.47394	−0.736970	+	0.675925i	$0.763744\pi$
$62$	−3537.29	−0.116867
$63$	4788.07	0.151988
$64$	−12156.1	−0.370974
$65$	0	0
$66$	−2095.80	−0.0592230
$67$	−8325.99	−0.226594	−0.113297	−	0.993561i	$-0.536141\pi$
$67$	−8325.99	−0.226594	−0.113297	+	0.993561i	$0.536141\pi$
$68$	−64288.2	−1.68601
$69$	−14146.0	−0.357694
$70$	0	0
$71$	13440.4	0.316422	0.158211	−	0.987405i	$-0.449427\pi$
$71$	13440.4	0.316422	0.158211	+	0.987405i	$0.449427\pi$
$72$	9399.34	0.213681
$73$	−19484.1	−0.427931	−0.213966	−	0.976841i	$-0.568638\pi$
$73$	−19484.1	−0.427931	−0.213966	+	0.976841i	$0.568638\pi$
$74$	7470.89	0.158596
$75$	0	0
$76$	84853.9	1.68515
$77$	−7152.55	−0.137478
$78$	−10385.8	−0.193288
$79$	−20416.2	−0.368050	−0.184025	−	0.982922i	$-0.558913\pi$
$79$	−20416.2	−0.368050	−0.184025	+	0.982922i	$0.558913\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	4132.99	0.0678781
$83$	−82162.1	−1.30911	−0.654555	−	0.756014i	$-0.727144\pi$
$83$	−82162.1	−1.30911	−0.654555	+	0.756014i	$0.727144\pi$
$84$	15053.8	0.232781
$85$	0	0
$86$	5029.03	0.0733226
$87$	−81185.9	−1.14996
$88$	−14041.0	−0.193282
$89$	38458.1	0.514651	0.257325	−	0.966325i	$-0.417159\pi$
$89$	38458.1	0.514651	0.257325	+	0.966325i	$0.417159\pi$
$90$	0	0
$91$	−35444.8	−0.448693
$92$	−44475.4	−0.547836
$93$	−16542.1	−0.198328
$94$	−16126.5	−0.188244
$95$	0	0
$96$	45235.3	0.500956
$97$	−139725.	−1.50780	−0.753901	−	0.656988i	$-0.771830\pi$
$97$	−139725.	−1.50780	−0.753901	+	0.656988i	$0.771830\pi$
$98$	25620.7	0.269480
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−129073.	−1.25902	−0.629511	−	0.776991i	$-0.716745\pi$
$101$	−129073.	−1.25902	−0.629511	+	0.776991i	$0.716745\pi$
$102$	39352.1	0.374513
$103$	26675.0	0.247749	0.123875	−	0.992298i	$-0.460468\pi$
$103$	26675.0	0.247749	0.123875	+	0.992298i	$0.460468\pi$
$104$	−69580.8	−0.630821
$105$	0	0
$106$	21673.0	0.187350
$107$	116793.	0.986180	0.493090	−	0.869978i	$-0.335867\pi$
$107$	116793.	0.986180	0.493090	+	0.869978i	$0.335867\pi$
$108$	20627.9	0.170175
$109$	148976.	1.20102	0.600509	−	0.799618i	$-0.294965\pi$
$109$	148976.	1.20102	0.600509	+	0.799618i	$0.294965\pi$
$110$	0	0
$111$	34937.5	0.269144
$112$	40323.5	0.303748
$113$	−155580.	−1.14619	−0.573095	−	0.819489i	$-0.694258\pi$
$113$	−155580.	−1.14619	−0.573095	+	0.819489i	$0.694258\pi$
$114$	−51940.8	−0.374323
$115$	0	0
$116$	−255250.	−1.76125
$117$	−48569.3	−0.328018
$118$	4822.39	0.0318829
$119$	134301.	0.869382
$120$	0	0
$121$	14641.0	0.0909091
$122$	82437.9	0.501450
$123$	19327.9	0.115192
$124$	−52008.8	−0.303754
$125$	0	0
$126$	−9214.74	−0.0517079
$127$	−193163.	−1.06271	−0.531354	−	0.847150i	$-0.678316\pi$
$127$	−193163.	−1.06271	−0.531354	+	0.847150i	$0.678316\pi$
$128$	184231.	0.993890
$129$	23518.2	0.124431
$130$	0	0
$131$	337105.	1.71628	0.858138	−	0.513419i	$-0.171621\pi$
$131$	337105.	1.71628	0.858138	+	0.513419i	$0.171621\pi$
$132$	−30814.6	−0.153929
$133$	−177263.	−0.868940
$134$	16023.5	0.0770897
$135$	0	0
$136$	263642.	1.22227
$137$	349973.	1.59306	0.796531	−	0.604598i	$-0.206666\pi$
$137$	349973.	1.59306	0.796531	+	0.604598i	$0.206666\pi$
$138$	27224.3	0.121691
$139$	343788.	1.50923	0.754613	−	0.656171i	$-0.227825\pi$
$139$	343788.	1.50923	0.754613	+	0.656171i	$0.227825\pi$
$140$	0	0
$141$	−75415.6	−0.319458
$142$	−25866.3	−0.107650
$143$	72554.2	0.296703
$144$	55254.6	0.222056
$145$	0	0
$146$	37497.6	0.145587
$147$	119815.	0.457317
$148$	109844.	0.412215
$149$	447504.	1.65132	0.825661	−	0.564167i	$-0.190803\pi$
$149$	447504.	1.65132	0.825661	+	0.564167i	$0.190803\pi$
$150$	0	0
$151$	496536.	1.77218	0.886091	−	0.463510i	$-0.153410\pi$
$151$	496536.	1.77218	0.886091	+	0.463510i	$0.153410\pi$
$152$	−347981.	−1.22165
$153$	184030.	0.635564
$154$	13765.2	0.0467715
$155$	0	0
$156$	−152703.	−0.502385
$157$	365502.	1.18342	0.591712	−	0.806150i	$-0.298452\pi$
$157$	365502.	1.18342	0.591712	+	0.806150i	$0.298452\pi$
$158$	39291.3	0.125214
$159$	101354.	0.317941
$160$	0	0
$161$	92911.0	0.282490
$162$	−12626.8	−0.0378011
$163$	−536870.	−1.58271	−0.791353	−	0.611359i	$-0.790623\pi$
$163$	−536870.	−1.58271	−0.791353	+	0.611359i	$0.790623\pi$
$164$	60767.3	0.176425
$165$	0	0
$166$	158123.	0.445373
$167$	−158899.	−0.440890	−0.220445	−	0.975399i	$-0.570751\pi$
$167$	−158899.	−0.440890	−0.220445	+	0.975399i	$0.570751\pi$
$168$	−61734.9	−0.168755
$169$	−11747.5	−0.0316395
$170$	0	0
$171$	−242901.	−0.635240
$172$	73941.8	0.190576
$173$	19966.6	0.0507211	0.0253606	−	0.999678i	$-0.491927\pi$
$173$	19966.6	0.0507211	0.0253606	+	0.999678i	$0.491927\pi$
$174$	156244.	0.391228
$175$	0	0
$176$	−82540.8	−0.200857
$177$	22551.9	0.0541065
$178$	−74013.3	−0.175089
$179$	153145.	0.357247	0.178624	−	0.983917i	$-0.442836\pi$
$179$	153145.	0.357247	0.178624	+	0.983917i	$0.442836\pi$
$180$	0	0
$181$	−790865.	−1.79435	−0.897173	−	0.441680i	$-0.854383\pi$
$181$	−790865.	−1.79435	−0.897173	+	0.441680i	$0.854383\pi$
$182$	68214.2	0.152650
$183$	385520.	0.850980
$184$	182391.	0.397155
$185$	0	0
$186$	31835.6	0.0674731
$187$	−274908.	−0.574889
$188$	−237109.	−0.489274
$189$	−43092.6	−0.0877503
$190$	0	0
$191$	612916.	1.21568	0.607838	−	0.794061i	$-0.292037\pi$
$191$	612916.	1.21568	0.607838	+	0.794061i	$0.292037\pi$
$192$	109405.	0.214182
$193$	−884585.	−1.70941	−0.854705	−	0.519114i	$-0.826262\pi$
$193$	−884585.	−1.70941	−0.854705	+	0.519114i	$0.826262\pi$
$194$	268903.	0.512970
$195$	0	0
$196$	376701.	0.700417
$197$	14105.8	0.0258960	0.0129480	−	0.999916i	$-0.495878\pi$
$197$	14105.8	0.0258960	0.0129480	+	0.999916i	$0.495878\pi$
$198$	18862.2	0.0341924
$199$	−104501.	−0.187063	−0.0935316	−	0.995616i	$-0.529816\pi$
$199$	−104501.	−0.187063	−0.0935316	+	0.995616i	$0.529816\pi$
$200$	0	0
$201$	74933.9	0.130824
$202$	248404.	0.428332
$203$	533229.	0.908183
$204$	578594.	0.973416
$205$	0	0
$206$	−51336.7	−0.0842868
$207$	127314.	0.206515
$208$	−409035.	−0.655544
$209$	362851.	0.574597
$210$	0	0
$211$	1.08804e6	1.68244	0.841221	−	0.540692i	$-0.181837\pi$
$211$	1.08804e6	1.68244	0.841221	+	0.540692i	$0.181837\pi$
$212$	318658.	0.486952
$213$	−120964.	−0.182686
$214$	−224770.	−0.335509
$215$	0	0
$216$	−84594.1	−0.123369
$217$	108648.	0.156630
$218$	−286707.	−0.408599
$219$	175357.	0.247066
$220$	0	0
$221$	−1.36232e6	−1.87629
$222$	−67238.0	−0.0915656
$223$	818867.	1.10268	0.551342	−	0.834279i	$-0.314116\pi$
$223$	818867.	1.10268	0.551342	+	0.834279i	$0.314116\pi$
$224$	−297105.	−0.395631
$225$	0	0
$226$	299416.	0.389946
$227$	409868.	0.527934	0.263967	−	0.964532i	$-0.414969\pi$
$227$	409868.	0.527934	0.263967	+	0.964532i	$0.414969\pi$
$228$	−763685.	−0.972920
$229$	−29456.0	−0.0371180	−0.0185590	−	0.999828i	$-0.505908\pi$
$229$	−29456.0	−0.0371180	−0.0185590	+	0.999828i	$0.505908\pi$
$230$	0	0
$231$	64372.9	0.0793731
$232$	1.04677e6	1.27682
$233$	211115.	0.254758	0.127379	−	0.991854i	$-0.459344\pi$
$233$	211115.	0.254758	0.127379	+	0.991854i	$0.459344\pi$
$234$	93472.6	0.111595
$235$	0	0
$236$	70903.6	0.0828683
$237$	183745.	0.212494
$238$	−258464.	−0.295773
$239$	879085.	0.995488	0.497744	−	0.867324i	$-0.334162\pi$
$239$	879085.	0.995488	0.497744	+	0.867324i	$0.334162\pi$
$240$	0	0
$241$	−830172.	−0.920715	−0.460358	−	0.887734i	$-0.652279\pi$
$241$	−830172.	−0.920715	−0.460358	+	0.887734i	$0.652279\pi$
$242$	−28176.9	−0.0309282
$243$	−59049.0	−0.0641500
$244$	1.21208e6	1.30334
$245$	0	0
$246$	−37196.9	−0.0391894
$247$	1.79813e6	1.87533
$248$	213285.	0.220207
$249$	739458.	0.755815
$250$	0	0
$251$	−96982.6	−0.0971649	−0.0485825	−	0.998819i	$-0.515470\pi$
$251$	−96982.6	−0.0971649	−0.0485825	+	0.998819i	$0.515470\pi$
$252$	−135484.	−0.134396
$253$	−190185.	−0.186800
$254$	371745.	0.361544
$255$	0	0
$256$	34437.4	0.0328421
$257$	369343.	0.348817	0.174408	−	0.984673i	$-0.444199\pi$
$257$	369343.	0.348817	0.174408	+	0.984673i	$0.444199\pi$
$258$	−45261.3	−0.0423328
$259$	−229470.	−0.212557
$260$	0	0
$261$	730673.	0.663929
$262$	−648765.	−0.583895
$263$	−988642.	−0.881353	−0.440676	−	0.897666i	$-0.645261\pi$
$263$	−988642.	−0.881353	−0.440676	+	0.897666i	$0.645261\pi$
$264$	126369.	0.111591
$265$	0	0
$266$	341147.	0.295622
$267$	−346123.	−0.297134
$268$	235594.	0.200367
$269$	1.36005e6	1.14598	0.572988	−	0.819564i	$-0.305784\pi$
$269$	1.36005e6	1.14598	0.572988	+	0.819564i	$0.305784\pi$
$270$	0	0
$271$	−219383.	−0.181460	−0.0907298	−	0.995876i	$-0.528920\pi$
$271$	−219383.	−0.181460	−0.0907298	+	0.995876i	$0.528920\pi$
$272$	1.54984e6	1.27018
$273$	319003.	0.259053
$274$	−673529.	−0.541976
$275$	0	0
$276$	400279.	0.316293
$277$	−1.78416e6	−1.39713	−0.698563	−	0.715549i	$-0.746177\pi$
$277$	−1.78416e6	−1.39713	−0.698563	+	0.715549i	$0.746177\pi$
$278$	−661627.	−0.513454
$279$	148879.	0.114505
$280$	0	0
$281$	−1.01028e6	−0.763264	−0.381632	−	0.924314i	$-0.624638\pi$
$281$	−1.01028e6	−0.763264	−0.381632	+	0.924314i	$0.624638\pi$
$282$	145139.	0.108683
$283$	270967.	0.201118	0.100559	−	0.994931i	$-0.467937\pi$
$283$	270967.	0.201118	0.100559	+	0.994931i	$0.467937\pi$
$284$	−380313.	−0.279798
$285$	0	0
$286$	−139632.	−0.100941
$287$	−126945.	−0.0909730
$288$	−407118.	−0.289227
$289$	3.74199e6	2.63547
$290$	0	0
$291$	1.25752e6	0.870530
$292$	551327.	0.378401
$293$	−874428.	−0.595052	−0.297526	−	0.954714i	$-0.596162\pi$
$293$	−874428.	−0.595052	−0.297526	+	0.954714i	$0.596162\pi$
$294$	−230586.	−0.155584
$295$	0	0
$296$	−450466.	−0.298836
$297$	88209.0	0.0580259
$298$	−861231.	−0.561796
$299$	−942472.	−0.609664
$300$	0	0
$301$	−154467.	−0.0982700
$302$	−955594.	−0.602915
$303$	1.16166e6	0.726897
$304$	−2.04563e6	−1.26953
$305$	0	0
$306$	−354169.	−0.216225
$307$	2.20038e6	1.33245	0.666225	−	0.745750i	$-0.267909\pi$
$307$	2.20038e6	1.33245	0.666225	+	0.745750i	$0.267909\pi$
$308$	202390.	0.121566
$309$	−240075.	−0.143038
$310$	0	0
$311$	420029.	0.246251	0.123126	−	0.992391i	$-0.460708\pi$
$311$	420029.	0.246251	0.123126	+	0.992391i	$0.460708\pi$
$312$	626227.	0.364205
$313$	342934.	0.197856	0.0989282	−	0.995095i	$-0.468459\pi$
$313$	342934.	0.197856	0.0989282	+	0.995095i	$0.468459\pi$
$314$	−703415.	−0.402613
$315$	0	0
$316$	577700.	0.325450
$317$	1.83293e6	1.02446	0.512232	−	0.858847i	$-0.328819\pi$
$317$	1.83293e6	1.02446	0.512232	+	0.858847i	$0.328819\pi$
$318$	−195057.	−0.108167
$319$	−1.09150e6	−0.600546
$320$	0	0
$321$	−1.05113e6	−0.569371
$322$	−178809.	−0.0961059
$323$	−6.81312e6	−3.63362
$324$	−185652.	−0.0982508
$325$	0	0
$326$	1.03322e6	0.538453
$327$	−1.34078e6	−0.693408
$328$	−249204.	−0.127900
$329$	495329.	0.252293
$330$	0	0
$331$	1.42543e6	0.715114	0.357557	−	0.933891i	$-0.383610\pi$
$331$	1.42543e6	0.715114	0.357557	+	0.933891i	$0.383610\pi$
$332$	2.32488e6	1.15759
$333$	−314438.	−0.155390
$334$	305805.	0.149995
$335$	0	0
$336$	−362912.	−0.175369
$337$	3.93186e6	1.88592	0.942961	−	0.332903i	$-0.108028\pi$
$337$	3.93186e6	1.88592	0.942961	+	0.332903i	$0.108028\pi$
$338$	22608.3	0.0107641
$339$	1.40022e6	0.661754
$340$	0	0
$341$	−222399.	−0.103573
$342$	467467.	0.216115
$343$	−1.78044e6	−0.817131
$344$	−303231.	−0.138159
$345$	0	0
$346$	−38426.1	−0.0172558
$347$	338208.	0.150786	0.0753929	−	0.997154i	$-0.475979\pi$
$347$	338208.	0.150786	0.0753929	+	0.997154i	$0.475979\pi$
$348$	2.29725e6	1.01686
$349$	−2.82994e6	−1.24369	−0.621846	−	0.783139i	$-0.713617\pi$
$349$	−2.82994e6	−1.24369	−0.621846	+	0.783139i	$0.713617\pi$
$350$	0	0
$351$	437124.	0.189381
$352$	608163.	0.261616
$353$	4.21459e6	1.80019	0.900096	−	0.435691i	$-0.143496\pi$
$353$	4.21459e6	1.80019	0.900096	+	0.435691i	$0.143496\pi$
$354$	−43401.5	−0.0184076
$355$	0	0
$356$	−1.08822e6	−0.455083
$357$	−1.20871e6	−0.501938
$358$	−294730.	−0.121539
$359$	1.88776e6	0.773057	0.386529	−	0.922277i	$-0.373674\pi$
$359$	1.88776e6	0.773057	0.386529	+	0.922277i	$0.373674\pi$
$360$	0	0
$361$	6.51653e6	2.63177
$362$	1.52204e6	0.610455
$363$	−131769.	−0.0524864
$364$	1.00295e6	0.396760
$365$	0	0
$366$	−741941.	−0.289512
$367$	376896.	0.146068	0.0730342	−	0.997329i	$-0.476732\pi$
$367$	376896.	0.146068	0.0730342	+	0.997329i	$0.476732\pi$
$368$	1.07220e6	0.412720
$369$	−173951.	−0.0665060
$370$	0	0
$371$	−665691.	−0.251095
$372$	468079.	0.175373
$373$	−2.55080e6	−0.949300	−0.474650	−	0.880175i	$-0.657425\pi$
$373$	−2.55080e6	−0.949300	−0.474650	+	0.880175i	$0.657425\pi$
$374$	529067.	0.195583
$375$	0	0
$376$	972369.	0.354700
$377$	−5.40897e6	−1.96002
$378$	82932.6	0.0298535
$379$	3.97187e6	1.42035	0.710177	−	0.704023i	$-0.248615\pi$
$379$	3.97187e6	1.42035	0.710177	+	0.704023i	$0.248615\pi$
$380$	0	0
$381$	1.73846e6	0.613554
$382$	−1.17957e6	−0.413585
$383$	1.85461e6	0.646035	0.323018	−	0.946393i	$-0.395303\pi$
$383$	1.85461e6	0.646035	0.323018	+	0.946393i	$0.395303\pi$
$384$	−1.65808e6	−0.573822
$385$	0	0
$386$	1.70240e6	0.581559
$387$	−211664.	−0.0718405
$388$	3.95369e6	1.33329
$389$	−992340.	−0.332496	−0.166248	−	0.986084i	$-0.553165\pi$
$389$	−992340.	−0.332496	−0.166248	+	0.986084i	$0.553165\pi$
$390$	0	0
$391$	3.57104e6	1.18128
$392$	−1.54483e6	−0.507769
$393$	−3.03395e6	−0.990892
$394$	−27146.9	−0.00881009
$395$	0	0
$396$	277331.	0.0888712
$397$	148359.	0.0472430	0.0236215	−	0.999721i	$-0.492480\pi$
$397$	148359.	0.0472430	0.0236215	+	0.999721i	$0.492480\pi$
$398$	201115.	0.0636408
$399$	1.59537e6	0.501683
$400$	0	0
$401$	3.81078e6	1.18346	0.591729	−	0.806137i	$-0.298445\pi$
$401$	3.81078e6	1.18346	0.591729	+	0.806137i	$0.298445\pi$
$402$	−144212.	−0.0445077
$403$	−1.10211e6	−0.338036
$404$	3.65229e6	1.11330
$405$	0	0
$406$	−1.02621e6	−0.308973
$407$	469716.	0.140556
$408$	−2.37278e6	−0.705679
$409$	−74545.8	−0.0220351	−0.0110175	−	0.999939i	$-0.503507\pi$
$409$	−74545.8	−0.0220351	−0.0110175	+	0.999939i	$0.503507\pi$
$410$	0	0
$411$	−3.14975e6	−0.919754
$412$	−754803.	−0.219074
$413$	−148121.	−0.0427307
$414$	−245019.	−0.0702584
$415$	0	0
$416$	3.01378e6	0.853844
$417$	−3.09409e6	−0.871351
$418$	−698315.	−0.195484
$419$	1.48992e6	0.414599	0.207299	−	0.978278i	$-0.433533\pi$
$419$	1.48992e6	0.414599	0.207299	+	0.978278i	$0.433533\pi$
$420$	0	0
$421$	−2.60498e6	−0.716307	−0.358154	−	0.933663i	$-0.616594\pi$
$421$	−2.60498e6	−0.716307	−0.358154	+	0.933663i	$0.616594\pi$
$422$	−2.09396e6	−0.572384
$423$	678740.	0.184439
$424$	−1.30680e6	−0.353016
$425$	0	0
$426$	232797.	0.0621517
$427$	−2.53210e6	−0.672064
$428$	−3.30479e6	−0.872037
$429$	−652987.	−0.171302
$430$	0	0
$431$	−6.30280e6	−1.63433	−0.817166	−	0.576402i	$-0.804456\pi$
$431$	−6.30280e6	−1.63433	−0.817166	+	0.576402i	$0.804456\pi$
$432$	−497291.	−0.128204
$433$	4.19051e6	1.07411	0.537053	−	0.843549i	$-0.319538\pi$
$433$	4.19051e6	1.07411	0.537053	+	0.843549i	$0.319538\pi$
$434$	−209096.	−0.0532871
$435$	0	0
$436$	−4.21545e6	−1.06201
$437$	−4.71341e6	−1.18068
$438$	−337478.	−0.0840545
$439$	−2.10556e6	−0.521442	−0.260721	−	0.965414i	$-0.583960\pi$
$439$	−2.10556e6	−0.521442	−0.260721	+	0.965414i	$0.583960\pi$
$440$	0	0
$441$	−1.07833e6	−0.264032
$442$	2.62182e6	0.638332
$443$	−5.79037e6	−1.40184	−0.700918	−	0.713242i	$-0.747226\pi$
$443$	−5.79037e6	−1.40184	−0.700918	+	0.713242i	$0.747226\pi$
$444$	−988600.	−0.237993
$445$	0	0
$446$	−1.57593e6	−0.375145
$447$	−4.02754e6	−0.953391
$448$	−718569.	−0.169150
$449$	−2.51876e6	−0.589618	−0.294809	−	0.955556i	$-0.595256\pi$
$449$	−2.51876e6	−0.589618	−0.294809	+	0.955556i	$0.595256\pi$
$450$	0	0
$451$	259853.	0.0601569
$452$	4.40232e6	1.01353
$453$	−4.46883e6	−1.02317
$454$	−788800.	−0.179609
$455$	0	0
$456$	3.13183e6	0.705320
$457$	6.44679e6	1.44395	0.721977	−	0.691917i	$-0.243234\pi$
$457$	6.44679e6	1.44395	0.721977	+	0.691917i	$0.243234\pi$
$458$	56688.6	0.0126279
$459$	−1.65627e6	−0.366943
$460$	0	0
$461$	2.26707e6	0.496836	0.248418	−	0.968653i	$-0.420089\pi$
$461$	2.26707e6	0.496836	0.248418	+	0.968653i	$0.420089\pi$
$462$	−123887.	−0.0270035
$463$	8.82820e6	1.91390	0.956951	−	0.290251i	$-0.0937386\pi$
$463$	8.82820e6	1.91390	0.956951	+	0.290251i	$0.0937386\pi$
$464$	6.15349e6	1.32686
$465$	0	0
$466$	−406294.	−0.0866714
$467$	−3.46435e6	−0.735072	−0.367536	−	0.930009i	$-0.619799\pi$
$467$	−3.46435e6	−0.735072	−0.367536	+	0.930009i	$0.619799\pi$
$468$	1.37433e6	0.290052
$469$	−492165.	−0.103319
$470$	0	0
$471$	−3.28951e6	−0.683250
$472$	−290772.	−0.0600755
$473$	316189.	0.0649822
$474$	−353622.	−0.0722925
$475$	0	0
$476$	−3.80020e6	−0.768757
$477$	−912183.	−0.183563
$478$	−1.69182e6	−0.338675
$479$	−5.50564e6	−1.09640	−0.548200	−	0.836347i	$-0.684687\pi$
$479$	−5.50564e6	−1.09640	−0.548200	+	0.836347i	$0.684687\pi$
$480$	0	0
$481$	2.32770e6	0.458737
$482$	1.59768e6	0.313237
$483$	−836199.	−0.163095
$484$	−414285.	−0.0803870
$485$	0	0
$486$	113641.	0.0218245
$487$	1.34130e6	0.256272	0.128136	−	0.991757i	$-0.459101\pi$
$487$	1.34130e6	0.256272	0.128136	+	0.991757i	$0.459101\pi$
$488$	−4.97069e6	−0.944860
$489$	4.83183e6	0.913776
$490$	0	0
$491$	6.76450e6	1.26629	0.633144	−	0.774034i	$-0.281764\pi$
$491$	6.76450e6	1.26629	0.633144	+	0.774034i	$0.281764\pi$
$492$	−546906.	−0.101859
$493$	2.04947e7	3.79772
$494$	−3.46053e6	−0.638007
$495$	0	0
$496$	1.25381e6	0.228838
$497$	794489.	0.144277
$498$	−1.42310e6	−0.257136
$499$	6.79946e6	1.22243	0.611214	−	0.791466i	$-0.290682\pi$
$499$	6.79946e6	1.22243	0.611214	+	0.791466i	$0.290682\pi$
$500$	0	0
$501$	1.43009e6	0.254548
$502$	186645.	0.0330565
$503$	5.08998e6	0.897007	0.448503	−	0.893781i	$-0.351957\pi$
$503$	5.08998e6	0.897007	0.448503	+	0.893781i	$0.351957\pi$
$504$	555614.	0.0974309
$505$	0	0
$506$	366016.	0.0635511
$507$	105728.	0.0182670
$508$	5.46577e6	0.939706
$509$	−3.53565e6	−0.604889	−0.302444	−	0.953167i	$-0.597803\pi$
$509$	−3.53565e6	−0.604889	−0.302444	+	0.953167i	$0.597803\pi$
$510$	0	0
$511$	−1.15175e6	−0.195121
$512$	−5.96167e6	−1.00506
$513$	2.18610e6	0.366756
$514$	−710808.	−0.118671
$515$	0	0
$516$	−665476.	−0.110029
$517$	−1.01392e6	−0.166831
$518$	441619.	0.0723141
$519$	−179699.	−0.0292839
$520$	0	0
$521$	−3.72572e6	−0.601334	−0.300667	−	0.953729i	$-0.597209\pi$
$521$	−3.72572e6	−0.601334	−0.300667	+	0.953729i	$0.597209\pi$
$522$	−1.40619e6	−0.225875
$523$	1.46844e6	0.234748	0.117374	−	0.993088i	$-0.462552\pi$
$523$	1.46844e6	0.234748	0.117374	+	0.993088i	$0.462552\pi$
$524$	−9.53880e6	−1.51763
$525$	0	0
$526$	1.90266e6	0.299845
$527$	4.17591e6	0.654975
$528$	742867.	0.115965
$529$	−3.96585e6	−0.616165
$530$	0	0
$531$	−202967.	−0.0312384
$532$	5.01588e6	0.768366
$533$	1.28771e6	0.196336
$534$	666120.	0.101088
$535$	0	0
$536$	−966158.	−0.145257
$537$	−1.37830e6	−0.206257
$538$	−2.61745e6	−0.389873
$539$	1.61085e6	0.238826
$540$	0	0
$541$	−158155.	−0.0232322	−0.0116161	−	0.999933i	$-0.503698\pi$
$541$	−158155.	−0.0232322	−0.0116161	+	0.999933i	$0.503698\pi$
$542$	422207.	0.0617344
$543$	7.11779e6	1.03597
$544$	−1.14193e7	−1.65440
$545$	0	0
$546$	−613928.	−0.0881324
$547$	−7.82586e6	−1.11831	−0.559157	−	0.829062i	$-0.688875\pi$
$547$	−7.82586e6	−1.11831	−0.559157	+	0.829062i	$0.688875\pi$
$548$	−9.90290e6	−1.40868
$549$	−3.46968e6	−0.491314
$550$	0	0
$551$	−2.70509e7	−3.79579
$552$	−1.64152e6	−0.229297
$553$	−1.20684e6	−0.167817
$554$	3.43366e6	0.475316
$555$	0	0
$556$	−9.72791e6	−1.33454
$557$	−71452.4	−0.00975841	−0.00487920	−	0.999988i	$-0.501553\pi$
$557$	−71452.4	−0.00975841	−0.00487920	+	0.999988i	$0.501553\pi$
$558$	−286520.	−0.0389556
$559$	1.56689e6	0.212085
$560$	0	0
$561$	2.47418e6	0.331912
$562$	1.94430e6	0.259670
$563$	3.87924e6	0.515793	0.257897	−	0.966173i	$-0.416971\pi$
$563$	3.87924e6	0.515793	0.257897	+	0.966173i	$0.416971\pi$
$564$	2.13398e6	0.282483
$565$	0	0
$566$	−521481.	−0.0684223
$567$	387834.	0.0506626
$568$	1.55964e6	0.202840
$569$	8.73077e6	1.13050	0.565252	−	0.824918i	$-0.308779\pi$
$569$	8.73077e6	1.13050	0.565252	+	0.824918i	$0.308779\pi$
$570$	0	0
$571$	441497.	0.0566680	0.0283340	−	0.999599i	$-0.490980\pi$
$571$	441497.	0.0566680	0.0283340	+	0.999599i	$0.490980\pi$
$572$	−2.05301e6	−0.262362
$573$	−5.51625e6	−0.701871
$574$	244309.	0.0309499
$575$	0	0
$576$	−984641.	−0.123658
$577$	1.12074e7	1.40141	0.700703	−	0.713453i	$-0.252870\pi$
$577$	1.12074e7	1.40141	0.700703	+	0.713453i	$0.252870\pi$
$578$	−7.20154e6	−0.896615
$579$	7.96126e6	0.986928
$580$	0	0
$581$	−4.85676e6	−0.596907
$582$	−2.42013e6	−0.296163
$583$	1.36264e6	0.166039
$584$	−2.26096e6	−0.274323
$585$	0	0
$586$	1.68285e6	0.202443
$587$	−9.06478e6	−1.08583	−0.542915	−	0.839788i	$-0.682679\pi$
$587$	−9.06478e6	−1.08583	−0.542915	+	0.839788i	$0.682679\pi$
$588$	−3.39031e6	−0.404386
$589$	−5.51178e6	−0.654641
$590$	0	0
$591$	−126952.	−0.0149511
$592$	−2.64809e6	−0.310548
$593$	−3789.68	−0.000442554 0	−0.000221277	−	1.00000i	$-0.500070\pi$
$593$	−3789.68	−0.000442554 0	−0.000221277		1.00000i	$0.500070\pi$
$594$	−169760.	−0.0197410
$595$	0	0
$596$	−1.26627e7	−1.46019
$597$	940510.	0.108001
$598$	1.81381e6	0.207414
$599$	−6.84765e6	−0.779785	−0.389892	−	0.920860i	$-0.627488\pi$
$599$	−6.84765e6	−0.779785	−0.389892	+	0.920860i	$0.627488\pi$
$600$	0	0
$601$	−8.82312e6	−0.996405	−0.498203	−	0.867061i	$-0.666006\pi$
$601$	−8.82312e6	−0.996405	−0.498203	+	0.867061i	$0.666006\pi$
$602$	297276.	0.0334325
$603$	−674405.	−0.0755314
$604$	−1.40501e7	−1.56707
$605$	0	0
$606$	−2.23564e6	−0.247298
$607$	−1.19124e7	−1.31228	−0.656142	−	0.754638i	$-0.727813\pi$
$607$	−1.19124e7	−1.31228	−0.656142	+	0.754638i	$0.727813\pi$
$608$	1.50723e7	1.65356
$609$	−4.79906e6	−0.524340
$610$	0	0
$611$	−5.02453e6	−0.544493
$612$	−5.20734e6	−0.562002
$613$	5.84838e6	0.628615	0.314307	−	0.949321i	$-0.398228\pi$
$613$	5.84838e6	0.628615	0.314307	+	0.949321i	$0.398228\pi$
$614$	−4.23467e6	−0.453313
$615$	0	0
$616$	−829991.	−0.0881295
$617$	1.65932e6	0.175476	0.0877381	−	0.996144i	$-0.472036\pi$
$617$	1.65932e6	0.175476	0.0877381	+	0.996144i	$0.472036\pi$
$618$	462030.	0.0486630
$619$	−6.13215e6	−0.643259	−0.321630	−	0.946866i	$-0.604231\pi$
$619$	−6.13215e6	−0.643259	−0.321630	+	0.946866i	$0.604231\pi$
$620$	0	0
$621$	−1.14583e6	−0.119231
$622$	−808355.	−0.0837772
$623$	2.27333e6	0.234662
$624$	3.68131e6	0.378479
$625$	0	0
$626$	−659984.	−0.0673128
$627$	−3.26566e6	−0.331744
$628$	−1.03423e7	−1.04645
$629$	−8.81967e6	−0.888845
$630$	0	0
$631$	1.82127e7	1.82097	0.910483	−	0.413547i	$-0.135710\pi$
$631$	1.82127e7	1.82097	0.910483	+	0.413547i	$0.135710\pi$
$632$	−2.36912e6	−0.235936
$633$	−9.79239e6	−0.971358
$634$	−3.52750e6	−0.348533
$635$	0	0
$636$	−2.86793e6	−0.281142
$637$	7.98262e6	0.779466
$638$	2.10061e6	0.204312
$639$	1.08867e6	0.105474
$640$	0	0
$641$	1.32331e7	1.27209	0.636044	−	0.771653i	$-0.280570\pi$
$641$	1.32331e7	1.27209	0.636044	+	0.771653i	$0.280570\pi$
$642$	2.02293e6	0.193706
$643$	−9.98593e6	−0.952491	−0.476246	−	0.879312i	$-0.658003\pi$
$643$	−9.98593e6	−0.952491	−0.476246	+	0.879312i	$0.658003\pi$
$644$	−2.62903e6	−0.249793
$645$	0	0
$646$	1.31120e7	1.23620
$647$	−1.83105e7	−1.71965	−0.859823	−	0.510593i	$-0.829426\pi$
$647$	−1.83105e7	−1.71965	−0.859823	+	0.510593i	$0.829426\pi$
$648$	761347.	0.0712271
$649$	303197.	0.0282562
$650$	0	0
$651$	−977836.	−0.0904302
$652$	1.51914e7	1.39952
$653$	1.10272e7	1.01201	0.506004	−	0.862531i	$-0.331122\pi$
$653$	1.10272e7	1.01201	0.506004	+	0.862531i	$0.331122\pi$
$654$	2.58036e6	0.235905
$655$	0	0
$656$	−1.46496e6	−0.132912
$657$	−1.57821e6	−0.142644
$658$	−953271.	−0.0858325
$659$	8.75231e6	0.785072	0.392536	−	0.919737i	$-0.371598\pi$
$659$	8.75231e6	0.785072	0.392536	+	0.919737i	$0.371598\pi$
$660$	0	0
$661$	−1.04678e6	−0.0931866	−0.0465933	−	0.998914i	$-0.514836\pi$
$661$	−1.04678e6	−0.0931866	−0.0465933	+	0.998914i	$0.514836\pi$
$662$	−2.74326e6	−0.243289
$663$	1.22609e7	1.08327
$664$	−9.53419e6	−0.839196
$665$	0	0
$666$	605142.	0.0528654
$667$	1.41785e7	1.23400
$668$	4.49625e6	0.389860
$669$	−7.36980e6	−0.636635
$670$	0	0
$671$	5.18310e6	0.444410
$672$	2.67395e6	0.228418
$673$	−5.21480e6	−0.443813	−0.221906	−	0.975068i	$-0.571228\pi$
$673$	−5.21480e6	−0.443813	−0.221906	+	0.975068i	$0.571228\pi$
$674$	−7.56695e6	−0.641610
$675$	0	0
$676$	332410.	0.0279774
$677$	−1.81772e7	−1.52425	−0.762125	−	0.647430i	$-0.775844\pi$
$677$	−1.81772e7	−1.52425	−0.762125	+	0.647430i	$0.775844\pi$
$678$	−2.69475e6	−0.225135
$679$	−8.25942e6	−0.687503
$680$	0	0
$681$	−3.68881e6	−0.304803
$682$	428012.	0.0352367
$683$	7.22749e6	0.592837	0.296419	−	0.955058i	$-0.404208\pi$
$683$	7.22749e6	0.592837	0.296419	+	0.955058i	$0.404208\pi$
$684$	6.87317e6	0.561716
$685$	0	0
$686$	3.42649e6	0.277996
$687$	265104.	0.0214301
$688$	−1.78256e6	−0.143573
$689$	6.75264e6	0.541908
$690$	0	0
$691$	1.05510e7	0.840619	0.420309	−	0.907381i	$-0.361922\pi$
$691$	1.05510e7	0.840619	0.420309	+	0.907381i	$0.361922\pi$
$692$	−564979.	−0.0448505
$693$	−579356.	−0.0458261
$694$	−650888.	−0.0512989
$695$	0	0
$696$	−9.42091e6	−0.737174
$697$	−4.87915e6	−0.380419
$698$	5.44627e6	0.423117
$699$	−1.90003e6	−0.147085
$700$	0	0
$701$	2.26639e7	1.74197	0.870985	−	0.491310i	$-0.163482\pi$
$701$	2.26639e7	1.74197	0.870985	+	0.491310i	$0.163482\pi$
$702$	−841253.	−0.0644294
$703$	1.16411e7	0.888392
$704$	1.47088e6	0.111853
$705$	0	0
$706$	−8.11107e6	−0.612444
$707$	−7.62978e6	−0.574069
$708$	−638132.	−0.0478440
$709$	−1.63112e7	−1.21863	−0.609313	−	0.792930i	$-0.708555\pi$
$709$	−1.63112e7	−1.21863	−0.609313	+	0.792930i	$0.708555\pi$
$710$	0	0
$711$	−1.65371e6	−0.122683
$712$	4.46272e6	0.329913
$713$	2.88895e6	0.212822
$714$	2.32618e6	0.170765
$715$	0	0
$716$	−4.33341e6	−0.315898
$717$	−7.91176e6	−0.574745
$718$	−3.63304e6	−0.263002
$719$	9.77281e6	0.705013	0.352507	−	0.935809i	$-0.385329\pi$
$719$	9.77281e6	0.705013	0.352507	+	0.935809i	$0.385329\pi$
$720$	0	0
$721$	1.57681e6	0.112965
$722$	−1.25412e7	−0.895357
$723$	7.47155e6	0.531575
$724$	2.23785e7	1.58666
$725$	0	0
$726$	253592.	0.0178564
$727$	1.05257e7	0.738611	0.369306	−	0.929308i	$-0.379596\pi$
$727$	1.05257e7	0.738611	0.369306	+	0.929308i	$0.379596\pi$
$728$	−4.11306e6	−0.287631
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.93696e6	−0.410933
$732$	−1.09088e7	−0.752485
$733$	1.34590e6	0.0925238	0.0462619	−	0.998929i	$-0.485269\pi$
$733$	1.34590e6	0.0925238	0.0462619	+	0.998929i	$0.485269\pi$
$734$	−725343.	−0.0496939
$735$	0	0
$736$	−7.89999e6	−0.537566
$737$	1.00744e6	0.0683207
$738$	334772.	0.0226260
$739$	−1.89035e7	−1.27330	−0.636651	−	0.771152i	$-0.719681\pi$
$739$	−1.89035e7	−1.27330	−0.636651	+	0.771152i	$0.719681\pi$
$740$	0	0
$741$	−1.61831e7	−1.08272
$742$	1.28113e6	0.0854250
$743$	−1.35078e7	−0.897659	−0.448830	−	0.893617i	$-0.648159\pi$
$743$	−1.35078e7	−0.897659	−0.448830	+	0.893617i	$0.648159\pi$
$744$	−1.91957e6	−0.127137
$745$	0	0
$746$	4.90906e6	0.322962
$747$	−6.65513e6	−0.436370
$748$	7.77887e6	0.508350
$749$	6.90385e6	0.449662
$750$	0	0
$751$	6.35210e6	0.410977	0.205488	−	0.978660i	$-0.434122\pi$
$751$	6.35210e6	0.410977	0.205488	+	0.978660i	$0.434122\pi$
$752$	5.71613e6	0.368602
$753$	872843.	0.0560982
$754$	1.04097e7	0.666820
$755$	0	0
$756$	1.21936e6	0.0775938
$757$	−8.58261e6	−0.544352	−0.272176	−	0.962248i	$-0.587743\pi$
$757$	−8.58261e6	−0.544352	−0.272176	+	0.962248i	$0.587743\pi$
$758$	−7.64394e6	−0.483219
$759$	1.71167e6	0.107849
$760$	0	0
$761$	1.15085e7	0.720372	0.360186	−	0.932880i	$-0.382713\pi$
$761$	1.15085e7	0.720372	0.360186	+	0.932880i	$0.382713\pi$
$762$	−3.34571e6	−0.208737
$763$	8.80626e6	0.547621
$764$	−1.73432e7	−1.07497
$765$	0	0
$766$	−3.56924e6	−0.219788
$767$	1.50251e6	0.0922207
$768$	−309937.	−0.0189614
$769$	−2.72489e7	−1.66162	−0.830812	−	0.556554i	$-0.812123\pi$
$769$	−2.72489e7	−1.66162	−0.830812	+	0.556554i	$0.812123\pi$
$770$	0	0
$771$	−3.32409e6	−0.201389
$772$	2.50304e7	1.51156
$773$	1.35064e7	0.813003	0.406502	−	0.913650i	$-0.366749\pi$
$773$	1.35064e7	0.813003	0.406502	+	0.913650i	$0.366749\pi$
$774$	407351.	0.0244409
$775$	0	0
$776$	−1.62139e7	−0.966567
$777$	2.06523e6	0.122720
$778$	1.90978e6	0.113119
$779$	6.43999e6	0.380226
$780$	0	0
$781$	−1.62629e6	−0.0954047
$782$	−6.87253e6	−0.401883
$783$	−6.57606e6	−0.383320
$784$	−9.08138e6	−0.527669
$785$	0	0
$786$	5.83889e6	0.337112
$787$	7.81643e6	0.449854	0.224927	−	0.974376i	$-0.427786\pi$
$787$	7.81643e6	0.449854	0.224927	+	0.974376i	$0.427786\pi$
$788$	−399141.	−0.0228987
$789$	8.89778e6	0.508849
$790$	0	0
$791$	−9.19662e6	−0.522621
$792$	−1.13732e6	−0.0644273
$793$	2.56851e7	1.45044
$794$	−285520.	−0.0160726
$795$	0	0
$796$	2.95699e6	0.165412
$797$	−2.29316e7	−1.27876	−0.639379	−	0.768891i	$-0.720809\pi$
$797$	−2.29316e7	−1.27876	−0.639379	+	0.768891i	$0.720809\pi$
$798$	−3.07032e6	−0.170678
$799$	1.90380e7	1.05501
$800$	0	0
$801$	3.11510e6	0.171550
$802$	−7.33393e6	−0.402625
$803$	2.35758e6	0.129026
$804$	−2.12035e6	−0.115682
$805$	0	0
$806$	2.12103e6	0.115003
$807$	−1.22405e7	−0.661629
$808$	−1.49778e7	−0.807088
$809$	−1.38109e7	−0.741908	−0.370954	−	0.928651i	$-0.620969\pi$
$809$	−1.38109e7	−0.741908	−0.370954	+	0.928651i	$0.620969\pi$
$810$	0	0
$811$	1.89793e7	1.01328	0.506638	−	0.862159i	$-0.330888\pi$
$811$	1.89793e7	1.01328	0.506638	+	0.862159i	$0.330888\pi$
$812$	−1.50884e7	−0.803067
$813$	1.97445e6	0.104766
$814$	−903977.	−0.0478186
$815$	0	0
$816$	−1.39485e7	−0.733336
$817$	7.83619e6	0.410724
$818$	143465.	0.00749656
$819$	−2.87103e6	−0.149564
$820$	0	0
$821$	−3.65447e6	−0.189220	−0.0946098	−	0.995514i	$-0.530160\pi$
$821$	−3.65447e6	−0.189220	−0.0946098	+	0.995514i	$0.530160\pi$
$822$	6.06176e6	0.312910
$823$	1.37132e7	0.705731	0.352865	−	0.935674i	$-0.385207\pi$
$823$	1.37132e7	0.705731	0.352865	+	0.935674i	$0.385207\pi$
$824$	3.09541e6	0.158818
$825$	0	0
$826$	285061.	0.0145374
$827$	1.96311e7	0.998116	0.499058	−	0.866569i	$-0.333679\pi$
$827$	1.96311e7	0.998116	0.499058	+	0.866569i	$0.333679\pi$
$828$	−3.60251e6	−0.182612
$829$	−1.78813e7	−0.903676	−0.451838	−	0.892100i	$-0.649232\pi$
$829$	−1.78813e7	−0.903676	−0.451838	+	0.892100i	$0.649232\pi$
$830$	0	0
$831$	1.60575e7	0.806631
$832$	7.28903e6	0.365058
$833$	−3.02462e7	−1.51028
$834$	5.95465e6	0.296443
$835$	0	0
$836$	−1.02673e7	−0.508091
$837$	−1.33991e6	−0.0661092
$838$	−2.86738e6	−0.141051
$839$	2.96975e6	0.145651	0.0728257	−	0.997345i	$-0.476798\pi$
$839$	2.96975e6	0.145651	0.0728257	+	0.997345i	$0.476798\pi$
$840$	0	0
$841$	6.08610e7	2.96722
$842$	5.01334e6	0.243695
$843$	9.09249e6	0.440670
$844$	−3.07875e7	−1.48771
$845$	0	0
$846$	−1.30625e6	−0.0627481
$847$	865458.	0.0414513
$848$	−7.68211e6	−0.366852
$849$	−2.43870e6	−0.116115
$850$	0	0
$851$	−6.10157e6	−0.288814
$852$	3.42281e6	0.161542
$853$	−7.98055e6	−0.375544	−0.187772	−	0.982213i	$-0.560127\pi$
$853$	−7.98055e6	−0.375544	−0.187772	+	0.982213i	$0.560127\pi$
$854$	4.87307e6	0.228643
$855$	0	0
$856$	1.35528e7	0.632184
$857$	1.41979e7	0.660347	0.330173	−	0.943920i	$-0.392893\pi$
$857$	1.41979e7	0.660347	0.330173	+	0.943920i	$0.392893\pi$
$858$	1.25669e6	0.0582786
$859$	−1.26770e7	−0.586182	−0.293091	−	0.956084i	$-0.594684\pi$
$859$	−1.26770e7	−0.586182	−0.293091	+	0.956084i	$0.594684\pi$
$860$	0	0
$861$	1.14251e6	0.0525233
$862$	1.21299e7	0.556017
$863$	3.18697e7	1.45664	0.728318	−	0.685239i	$-0.240302\pi$
$863$	3.18697e7	1.45664	0.728318	+	0.685239i	$0.240302\pi$
$864$	3.66406e6	0.166985
$865$	0	0
$866$	−8.06472e6	−0.365422
$867$	−3.36780e7	−1.52159
$868$	−3.07434e6	−0.138501
$869$	2.47036e6	0.110971
$870$	0	0
$871$	4.99244e6	0.222981
$872$	1.72873e7	0.769905
$873$	−1.13177e7	−0.502601
$874$	9.07105e6	0.401679
$875$	0	0
$876$	−4.96195e6	−0.218470
$877$	−452817.	−0.0198804	−0.00994018	−	0.999951i	$-0.503164\pi$
$877$	−452817.	−0.0198804	−0.00994018	+	0.999951i	$0.503164\pi$
$878$	4.05219e6	0.177400
$879$	7.86985e6	0.343554
$880$	0	0
$881$	3.26083e7	1.41543	0.707715	−	0.706498i	$-0.249726\pi$
$881$	3.26083e7	1.41543	0.707715	+	0.706498i	$0.249726\pi$
$882$	2.07528e6	0.0898265
$883$	−2.36123e7	−1.01915	−0.509573	−	0.860427i	$-0.670197\pi$
$883$	−2.36123e7	−1.01915	−0.509573	+	0.860427i	$0.670197\pi$
$884$	3.85486e7	1.65912
$885$	0	0
$886$	1.11437e7	0.476919
$887$	3.31071e7	1.41290	0.706451	−	0.707762i	$-0.250295\pi$
$887$	3.31071e7	1.41290	0.706451	+	0.707762i	$0.250295\pi$
$888$	4.05420e6	0.172533
$889$	−1.14182e7	−0.484556
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−2.31708e7	−0.975056
$893$	−2.51282e7	−1.05447
$894$	7.75108e6	0.324353
$895$	0	0
$896$	1.08903e7	0.453178
$897$	8.48225e6	0.351990
$898$	4.84740e6	0.200594
$899$	1.65801e7	0.684206
$900$	0	0
$901$	−2.55858e7	−1.05000
$902$	−500091.	−0.0204660
$903$	1.39021e6	0.0567362
$904$	−1.80537e7	−0.734758
$905$	0	0
$906$	8.60034e6	0.348093
$907$	8.76937e6	0.353957	0.176978	−	0.984215i	$-0.443368\pi$
$907$	8.76937e6	0.353957	0.176978	+	0.984215i	$0.443368\pi$
$908$	−1.15977e7	−0.466829
$909$	−1.04549e7	−0.419674
$910$	0	0
$911$	1.15460e7	0.460932	0.230466	−	0.973080i	$-0.425975\pi$
$911$	1.15460e7	0.460932	0.230466	+	0.973080i	$0.425975\pi$
$912$	1.84107e7	0.732963
$913$	9.94161e6	0.394711
$914$	−1.24070e7	−0.491248
$915$	0	0
$916$	833493.	0.0328219
$917$	1.99269e7	0.782560
$918$	3.18752e6	0.124838
$919$	−2.95602e7	−1.15456	−0.577282	−	0.816545i	$-0.695887\pi$
$919$	−2.95602e7	−1.15456	−0.577282	+	0.816545i	$0.695887\pi$
$920$	0	0
$921$	−1.98034e7	−0.769291
$922$	−4.36302e6	−0.169029
$923$	−8.05915e6	−0.311376
$924$	−1.82151e6	−0.0701862
$925$	0	0
$926$	−1.69900e7	−0.651129
$927$	2.16068e6	0.0825830
$928$	−4.53391e7	−1.72824
$929$	4.03347e7	1.53334	0.766672	−	0.642039i	$-0.221911\pi$
$929$	4.03347e7	1.53334	0.766672	+	0.642039i	$0.221911\pi$
$930$	0	0
$931$	3.99220e7	1.50952
$932$	−5.97374e6	−0.225272
$933$	−3.78026e6	−0.142173
$934$	6.66722e6	0.250079
$935$	0	0
$936$	−5.63605e6	−0.210274
$937$	2.52134e6	0.0938172	0.0469086	−	0.998899i	$-0.485063\pi$
$937$	2.52134e6	0.0938172	0.0469086	+	0.998899i	$0.485063\pi$
$938$	947182.	0.0351501
$939$	−3.08641e6	−0.114232
$940$	0	0
$941$	3.52049e7	1.29607	0.648036	−	0.761609i	$-0.275590\pi$
$941$	3.52049e7	1.29607	0.648036	+	0.761609i	$0.275590\pi$
$942$	6.33074e6	0.232449
$943$	−3.37546e6	−0.123610
$944$	−1.70932e6	−0.0624300
$945$	0	0
$946$	−608512.	−0.0221076
$947$	−3.15673e7	−1.14383	−0.571917	−	0.820312i	$-0.693800\pi$
$947$	−3.15673e7	−1.14383	−0.571917	+	0.820312i	$0.693800\pi$
$948$	−5.19930e6	−0.187899
$949$	1.16831e7	0.421107
$950$	0	0
$951$	−1.64963e7	−0.591475
$952$	1.55844e7	0.557312
$953$	2.68995e7	0.959427	0.479713	−	0.877425i	$-0.340741\pi$
$953$	2.68995e7	0.959427	0.479713	+	0.877425i	$0.340741\pi$
$954$	1.75551e6	0.0624501
$955$	0	0
$956$	−2.48748e7	−0.880267
$957$	9.82349e6	0.346726
$958$	1.05957e7	0.373007
$959$	2.06876e7	0.726378
$960$	0	0
$961$	−2.52509e7	−0.881998
$962$	−4.47970e6	−0.156067
$963$	9.46021e6	0.328727
$964$	2.34907e7	0.814149
$965$	0	0
$966$	1.60928e6	0.0554867
$967$	2.96256e7	1.01883	0.509414	−	0.860522i	$-0.329862\pi$
$967$	2.96256e7	1.01883	0.509414	+	0.860522i	$0.329862\pi$
$968$	1.69896e6	0.0582767
$969$	6.13181e7	2.09787
$970$	0	0
$971$	−3.86746e7	−1.31637	−0.658184	−	0.752857i	$-0.728675\pi$
$971$	−3.86746e7	−1.31637	−0.658184	+	0.752857i	$0.728675\pi$
$972$	1.67086e6	0.0567251
$973$	2.03220e7	0.688152
$974$	−2.58135e6	−0.0871865
$975$	0	0
$976$	−2.92205e7	−0.981891
$977$	9.61017e6	0.322103	0.161051	−	0.986946i	$-0.448511\pi$
$977$	9.61017e6	0.322103	0.161051	+	0.986946i	$0.448511\pi$
$978$	−9.29896e6	−0.310876
$979$	−4.65343e6	−0.155173
$980$	0	0
$981$	1.20670e7	0.400339
$982$	−1.30184e7	−0.430804
$983$	2.35665e7	0.777877	0.388939	−	0.921264i	$-0.372842\pi$
$983$	2.35665e7	0.777877	0.388939	+	0.921264i	$0.372842\pi$
$984$	2.24283e6	0.0738429
$985$	0	0
$986$	−3.94424e7	−1.29202
$987$	−4.45797e6	−0.145661
$988$	−5.08802e7	−1.65827
$989$	−4.10727e6	−0.133525
$990$	0	0
$991$	−2.88338e7	−0.932649	−0.466325	−	0.884614i	$-0.654422\pi$
$991$	−2.88338e7	−0.932649	−0.466325	+	0.884614i	$0.654422\pi$
$992$	−9.23811e6	−0.298060
$993$	−1.28289e7	−0.412871
$994$	−1.52901e6	−0.0490845
$995$	0	0
$996$	−2.09239e7	−0.668335
$997$	4.93391e7	1.57200	0.786002	−	0.618225i	$-0.212148\pi$
$997$	4.93391e7	1.57200	0.786002	+	0.618225i	$0.212148\pi$
$998$	−1.30857e7	−0.415882
$999$	2.82994e6	0.0897147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.s.1.4	yes	9
5.4	even	2		825.6.a.r.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.6	✓	9	5.4	even	2
825.6.a.s.1.4	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)

\(f(q)\)	\(=\)	\(q-1.92452 q^{2} -9.00000 q^{3} -28.2962 q^{4} +17.3207 q^{6} +59.1120 q^{7} +116.041 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-1.92452 q^{2} -9.00000 q^{3} -28.2962 q^{4} +17.3207 q^{6} +59.1120 q^{7} +116.041 q^{8} +81.0000 q^{9} -121.000 q^{11} +254.666 q^{12} -599.621 q^{13} -113.762 q^{14} +682.155 q^{16} +2271.97 q^{17} -155.886 q^{18} -2998.77 q^{19} -532.008 q^{21} +232.867 q^{22} +1571.78 q^{23} -1044.37 q^{24} +1153.98 q^{26} -729.000 q^{27} -1672.65 q^{28} +9020.65 q^{29} +1838.01 q^{31} -5026.14 q^{32} +1089.00 q^{33} -4372.45 q^{34} -2291.99 q^{36} -3881.95 q^{37} +5771.20 q^{38} +5396.59 q^{39} -2147.54 q^{41} +1023.86 q^{42} -2613.13 q^{43} +3423.84 q^{44} -3024.92 q^{46} +8379.51 q^{47} -6139.40 q^{48} -13312.8 q^{49} -20447.7 q^{51} +16967.0 q^{52} -11261.5 q^{53} +1402.98 q^{54} +6859.43 q^{56} +26988.9 q^{57} -17360.4 q^{58} -2505.76 q^{59} -42835.6 q^{61} -3537.29 q^{62} +4788.07 q^{63} -12156.1 q^{64} -2095.80 q^{66} -8325.99 q^{67} -64288.2 q^{68} -14146.0 q^{69} +13440.4 q^{71} +9399.34 q^{72} -19484.1 q^{73} +7470.89 q^{74} +84853.9 q^{76} -7152.55 q^{77} -10385.8 q^{78} -20416.2 q^{79} +6561.00 q^{81} +4132.99 q^{82} -82162.1 q^{83} +15053.8 q^{84} +5029.03 q^{86} -81185.9 q^{87} -14041.0 q^{88} +38458.1 q^{89} -35444.8 q^{91} -44475.4 q^{92} -16542.1 q^{93} -16126.5 q^{94} +45235.3 q^{96} -139725. q^{97} +25620.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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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