LMFDB - Embedded newform 825.6.a.s.1.8

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			1.8
Root			$8.36252$ 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)

$f(q)$	$=$	$q+8.36252 q^{2} -9.00000 q^{3} +37.9318 q^{4} -75.2627 q^{6} -15.4419 q^{7} +49.6048 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+8.36252 q^{2} -9.00000 q^{3} +37.9318 q^{4} -75.2627 q^{6} -15.4419 q^{7} +49.6048 q^{8} +81.0000 q^{9} -121.000 q^{11} -341.386 q^{12} +624.673 q^{13} -129.133 q^{14} -798.996 q^{16} +1254.35 q^{17} +677.364 q^{18} -440.239 q^{19} +138.977 q^{21} -1011.87 q^{22} +2534.04 q^{23} -446.443 q^{24} +5223.84 q^{26} -729.000 q^{27} -585.737 q^{28} -3653.61 q^{29} -8001.02 q^{31} -8268.98 q^{32} +1089.00 q^{33} +10489.5 q^{34} +3072.48 q^{36} -12096.5 q^{37} -3681.51 q^{38} -5622.06 q^{39} +13260.6 q^{41} +1162.20 q^{42} +4651.51 q^{43} -4589.75 q^{44} +21191.0 q^{46} +22955.5 q^{47} +7190.97 q^{48} -16568.5 q^{49} -11289.2 q^{51} +23695.0 q^{52} +13857.7 q^{53} -6096.28 q^{54} -765.990 q^{56} +3962.15 q^{57} -30553.4 q^{58} +46429.1 q^{59} +55964.9 q^{61} -66908.7 q^{62} -1250.79 q^{63} -43581.6 q^{64} +9106.79 q^{66} +3063.93 q^{67} +47579.8 q^{68} -22806.4 q^{69} +62515.4 q^{71} +4017.99 q^{72} +60834.6 q^{73} -101158. q^{74} -16699.1 q^{76} +1868.46 q^{77} -47014.6 q^{78} +53833.8 q^{79} +6561.00 q^{81} +110892. q^{82} -53585.5 q^{83} +5271.64 q^{84} +38898.4 q^{86} +32882.5 q^{87} -6002.18 q^{88} +3381.86 q^{89} -9646.11 q^{91} +96120.8 q^{92} +72009.2 q^{93} +191966. q^{94} +74420.8 q^{96} +30289.8 q^{97} -138555. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	8.36252	1.47830	0.739150	−	0.673541i	$-0.235228\pi$
$2$	8.36252	1.47830	0.739150	+	0.673541i	$0.235228\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	37.9318	1.18537
$5$	0	0
$5$	0	0
$6$	−75.2627	−0.853496
$7$	−15.4419	−0.119112	−0.0595558	−	0.998225i	$-0.518968\pi$
$7$	−15.4419	−0.119112	−0.0595558	+	0.998225i	$0.518968\pi$
$8$	49.6048	0.274030
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−341.386	−0.684373
$13$	624.673	1.02517	0.512583	−	0.858638i	$-0.328689\pi$
$13$	624.673	1.02517	0.512583	+	0.858638i	$0.328689\pi$
$14$	−129.133	−0.176083
$15$	0	0
$16$	−798.996	−0.780270
$17$	1254.35	1.05268	0.526341	−	0.850274i	$-0.323564\pi$
$17$	1254.35	1.05268	0.526341	+	0.850274i	$0.323564\pi$
$18$	677.364	0.492766
$19$	−440.239	−0.279772	−0.139886	−	0.990168i	$-0.544674\pi$
$19$	−440.239	−0.279772	−0.139886	+	0.990168i	$0.544674\pi$
$20$	0	0
$21$	138.977	0.0687692
$22$	−1011.87	−0.445724
$23$	2534.04	0.998836	0.499418	−	0.866361i	$-0.333547\pi$
$23$	2534.04	0.998836	0.499418	+	0.866361i	$0.333547\pi$
$24$	−446.443	−0.158212
$25$	0	0
$26$	5223.84	1.51550
$27$	−729.000	−0.192450
$28$	−585.737	−0.141191
$29$	−3653.61	−0.806729	−0.403364	−	0.915039i	$-0.632159\pi$
$29$	−3653.61	−0.806729	−0.403364	+	0.915039i	$0.632159\pi$
$30$	0	0
$31$	−8001.02	−1.49534	−0.747672	−	0.664068i	$-0.768828\pi$
$31$	−8001.02	−1.49534	−0.747672	+	0.664068i	$0.768828\pi$
$32$	−8268.98	−1.42750
$33$	1089.00	0.174078
$34$	10489.5	1.55618
$35$	0	0
$36$	3072.48	0.395123
$37$	−12096.5	−1.45264	−0.726318	−	0.687359i	$-0.758770\pi$
$37$	−12096.5	−1.45264	−0.726318	+	0.687359i	$0.758770\pi$
$38$	−3681.51	−0.413587
$39$	−5622.06	−0.591880
$40$	0	0
$41$	13260.6	1.23198	0.615990	−	0.787754i	$-0.288756\pi$
$41$	13260.6	1.23198	0.615990	+	0.787754i	$0.288756\pi$
$42$	1162.20	0.101661
$43$	4651.51	0.383639	0.191820	−	0.981430i	$-0.438561\pi$
$43$	4651.51	0.383639	0.191820	+	0.981430i	$0.438561\pi$
$44$	−4589.75	−0.357402
$45$	0	0
$46$	21191.0	1.47658
$47$	22955.5	1.51580	0.757901	−	0.652370i	$-0.226225\pi$
$47$	22955.5	1.51580	0.757901	+	0.652370i	$0.226225\pi$
$48$	7190.97	0.450489
$49$	−16568.5	−0.985812
$50$	0	0
$51$	−11289.2	−0.607766
$52$	23695.0	1.21520
$53$	13857.7	0.677645	0.338822	−	0.940850i	$-0.389971\pi$
$53$	13857.7	0.677645	0.338822	+	0.940850i	$0.389971\pi$
$54$	−6096.28	−0.284499
$55$	0	0
$56$	−765.990	−0.0326402
$57$	3962.15	0.161527
$58$	−30553.4	−1.19259
$59$	46429.1	1.73644	0.868220	−	0.496179i	$-0.165264\pi$
$59$	46429.1	1.73644	0.868220	+	0.496179i	$0.165264\pi$
$60$	0	0
$61$	55964.9	1.92571	0.962855	−	0.270018i	$-0.0870297\pi$
$61$	55964.9	1.92571	0.962855	+	0.270018i	$0.0870297\pi$
$62$	−66908.7	−2.21057
$63$	−1250.79	−0.0397039
$64$	−43581.6	−1.33001
$65$	0	0
$66$	9106.79	0.257339
$67$	3063.93	0.0833858	0.0416929	−	0.999130i	$-0.486725\pi$
$67$	3063.93	0.0833858	0.0416929	+	0.999130i	$0.486725\pi$
$68$	47579.8	1.24782
$69$	−22806.4	−0.576678
$70$	0	0
$71$	62515.4	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$71$	62515.4	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$72$	4017.99	0.0913435
$73$	60834.6	1.33612	0.668058	−	0.744110i	$-0.267126\pi$
$73$	60834.6	1.33612	0.668058	+	0.744110i	$0.267126\pi$
$74$	−101158.	−2.14743
$75$	0	0
$76$	−16699.1	−0.331633
$77$	1868.46	0.0359135
$78$	−47014.6	−0.874976
$79$	53833.8	0.970481	0.485241	−	0.874381i	$-0.338732\pi$
$79$	53833.8	0.970481	0.485241	+	0.874381i	$0.338732\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	110892.	1.82124
$83$	−53585.5	−0.853792	−0.426896	−	0.904301i	$-0.640393\pi$
$83$	−53585.5	−0.853792	−0.426896	+	0.904301i	$0.640393\pi$
$84$	5271.64	0.0815168
$85$	0	0
$86$	38898.4	0.567134
$87$	32882.5	0.465765
$88$	−6002.18	−0.0826233
$89$	3381.86	0.0452564	0.0226282	−	0.999744i	$-0.492797\pi$
$89$	3381.86	0.0452564	0.0226282	+	0.999744i	$0.492797\pi$
$90$	0	0
$91$	−9646.11	−0.122109
$92$	96120.8	1.18399
$93$	72009.2	0.863337
$94$	191966.	2.24081
$95$	0	0
$96$	74420.8	0.824169
$97$	30289.8	0.326865	0.163432	−	0.986555i	$-0.447743\pi$
$97$	30289.8	0.326865	0.163432	+	0.986555i	$0.447743\pi$
$98$	−138555.	−1.45733
$99$	−9801.00	−0.100504
$100$	0	0
$101$	97950.2	0.955437	0.477718	−	0.878513i	$-0.341464\pi$
$101$	97950.2	0.955437	0.477718	+	0.878513i	$0.341464\pi$
$102$	−94405.9	−0.898460
$103$	−87201.9	−0.809903	−0.404951	−	0.914338i	$-0.632712\pi$
$103$	−87201.9	−0.809903	−0.404951	+	0.914338i	$0.632712\pi$
$104$	30986.8	0.280927
$105$	0	0
$106$	115885.	1.00176
$107$	12956.6	0.109403	0.0547017	−	0.998503i	$-0.482579\pi$
$107$	12956.6	0.109403	0.0547017	+	0.998503i	$0.482579\pi$
$108$	−27652.3	−0.228124
$109$	−130438.	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$109$	−130438.	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$110$	0	0
$111$	108869.	0.838679
$112$	12338.0	0.0929393
$113$	9112.49	0.0671338	0.0335669	−	0.999436i	$-0.489313\pi$
$113$	9112.49	0.0671338	0.0335669	+	0.999436i	$0.489313\pi$
$114$	33133.6	0.238785
$115$	0	0
$116$	−138588.	−0.956271
$117$	50598.5	0.341722
$118$	388264.	2.56698
$119$	−19369.5	−0.125387
$120$	0	0
$121$	14641.0	0.0909091
$122$	468008.	2.84678
$123$	−119346.	−0.711284
$124$	−303493.	−1.77253
$125$	0	0
$126$	−10459.8	−0.0586942
$127$	67808.2	0.373055	0.186528	−	0.982450i	$-0.440277\pi$
$127$	67808.2	0.373055	0.186528	+	0.982450i	$0.440277\pi$
$128$	−99845.2	−0.538645
$129$	−41863.6	−0.221494
$130$	0	0
$131$	270464.	1.37699	0.688497	−	0.725239i	$-0.258271\pi$
$131$	270464.	1.37699	0.688497	+	0.725239i	$0.258271\pi$
$132$	41307.7	0.206346
$133$	6798.11	0.0333241
$134$	25622.2	0.123269
$135$	0	0
$136$	62221.9	0.288467
$137$	27960.6	0.127276	0.0636378	−	0.997973i	$-0.479730\pi$
$137$	27960.6	0.127276	0.0636378	+	0.997973i	$0.479730\pi$
$138$	−190719.	−0.852503
$139$	31442.3	0.138031	0.0690157	−	0.997616i	$-0.478014\pi$
$139$	31442.3	0.138031	0.0690157	+	0.997616i	$0.478014\pi$
$140$	0	0
$141$	−206600.	−0.875148
$142$	522787.	2.17572
$143$	−75585.4	−0.309099
$144$	−64718.7	−0.260090
$145$	0	0
$146$	508731.	1.97518
$147$	149117.	0.569159
$148$	−458843.	−1.72191
$149$	−120047.	−0.442982	−0.221491	−	0.975162i	$-0.571092\pi$
$149$	−120047.	−0.442982	−0.221491	+	0.975162i	$0.571092\pi$
$150$	0	0
$151$	−110901.	−0.395816	−0.197908	−	0.980221i	$-0.563415\pi$
$151$	−110901.	−0.395816	−0.197908	+	0.980221i	$0.563415\pi$
$152$	−21838.0	−0.0766661
$153$	101603.	0.350894
$154$	15625.1	0.0530909
$155$	0	0
$156$	−213255.	−0.701596
$157$	531986.	1.72247	0.861234	−	0.508208i	$-0.169692\pi$
$157$	531986.	1.72247	0.861234	+	0.508208i	$0.169692\pi$
$158$	450186.	1.43466
$159$	−124719.	−0.391238
$160$	0	0
$161$	−39130.3	−0.118973
$162$	54866.5	0.164255
$163$	459073.	1.35336	0.676680	−	0.736278i	$-0.263418\pi$
$163$	459073.	1.35336	0.676680	+	0.736278i	$0.263418\pi$
$164$	502999.	1.46035
$165$	0	0
$166$	−448110.	−1.26216
$167$	286383.	0.794613	0.397306	−	0.917686i	$-0.369945\pi$
$167$	286383.	0.794613	0.397306	+	0.917686i	$0.369945\pi$
$168$	6893.91	0.0188448
$169$	18923.3	0.0509659
$170$	0	0
$171$	−35659.4	−0.0932574
$172$	176440.	0.454754
$173$	−656155.	−1.66683	−0.833415	−	0.552647i	$-0.813618\pi$
$173$	−656155.	−1.66683	−0.833415	+	0.552647i	$0.813618\pi$
$174$	274981.	0.688540
$175$	0	0
$176$	96678.5	0.235260
$177$	−417862.	−1.00253
$178$	28280.8	0.0669025
$179$	−220000.	−0.513203	−0.256601	−	0.966517i	$-0.582603\pi$
$179$	−220000.	−0.513203	−0.256601	+	0.966517i	$0.582603\pi$
$180$	0	0
$181$	−426138.	−0.966837	−0.483419	−	0.875389i	$-0.660605\pi$
$181$	−426138.	−0.966837	−0.483419	+	0.875389i	$0.660605\pi$
$182$	−80665.8	−0.180514
$183$	−503684.	−1.11181
$184$	125701.	0.273711
$185$	0	0
$186$	602179.	1.27627
$187$	−151777.	−0.317396
$188$	870744.	1.79678
$189$	11257.1	0.0229231
$190$	0	0
$191$	163125.	0.323546	0.161773	−	0.986828i	$-0.448279\pi$
$191$	163125.	0.323546	0.161773	+	0.986828i	$0.448279\pi$
$192$	392235.	0.767879
$193$	212660.	0.410953	0.205476	−	0.978662i	$-0.434126\pi$
$193$	212660.	0.410953	0.205476	+	0.978662i	$0.434126\pi$
$194$	253300.	0.483204
$195$	0	0
$196$	−628475.	−1.16855
$197$	−729671.	−1.33956	−0.669779	−	0.742561i	$-0.733611\pi$
$197$	−729671.	−1.33956	−0.669779	+	0.742561i	$0.733611\pi$
$198$	−81961.1	−0.148575
$199$	642534.	1.15017	0.575087	−	0.818092i	$-0.304968\pi$
$199$	642534.	1.15017	0.575087	+	0.818092i	$0.304968\pi$
$200$	0	0
$201$	−27575.4	−0.0481428
$202$	819111.	1.41242
$203$	56418.6	0.0960908
$204$	−428218.	−0.720427
$205$	0	0
$206$	−729228.	−1.19728
$207$	205257.	0.332945
$208$	−499111.	−0.799906
$209$	53268.9	0.0843545
$210$	0	0
$211$	460769.	0.712487	0.356244	−	0.934393i	$-0.384057\pi$
$211$	460769.	0.712487	0.356244	+	0.934393i	$0.384057\pi$
$212$	525648.	0.803259
$213$	−562639.	−0.849729
$214$	108350.	0.161731
$215$	0	0
$216$	−36161.9	−0.0527372
$217$	123551.	0.178113
$218$	−1.09079e6	−1.55454
$219$	−547512.	−0.771406
$220$	0	0
$221$	783560.	1.07917
$222$	910418.	1.23982
$223$	841893.	1.13369	0.566846	−	0.823824i	$-0.308164\pi$
$223$	841893.	1.13369	0.566846	+	0.823824i	$0.308164\pi$
$224$	127688.	0.170032
$225$	0	0
$226$	76203.4	0.0992438
$227$	−415368.	−0.535018	−0.267509	−	0.963555i	$-0.586200\pi$
$227$	−415368.	−0.535018	−0.267509	+	0.963555i	$0.586200\pi$
$228$	150291.	0.191468
$229$	1.33133e6	1.67763	0.838815	−	0.544417i	$-0.183249\pi$
$229$	1.33133e6	1.67763	0.838815	+	0.544417i	$0.183249\pi$
$230$	0	0
$231$	−16816.2	−0.0207347
$232$	−181237.	−0.221068
$233$	−53370.5	−0.0644039	−0.0322019	−	0.999481i	$-0.510252\pi$
$233$	−53370.5	−0.0644039	−0.0322019	+	0.999481i	$0.510252\pi$
$234$	423131.	0.505168
$235$	0	0
$236$	1.76114e6	2.05832
$237$	−484504.	−0.560308
$238$	−161978.	−0.185359
$239$	160686.	0.181963	0.0909815	−	0.995853i	$-0.471000\pi$
$239$	160686.	0.181963	0.0909815	+	0.995853i	$0.471000\pi$
$240$	0	0
$241$	496394.	0.550534	0.275267	−	0.961368i	$-0.411234\pi$
$241$	496394.	0.550534	0.275267	+	0.961368i	$0.411234\pi$
$242$	122436.	0.134391
$243$	−59049.0	−0.0641500
$244$	2.12285e6	2.28268
$245$	0	0
$246$	−998030.	−1.05149
$247$	−275005.	−0.286813
$248$	−396889.	−0.409770
$249$	482270.	0.492937
$250$	0	0
$251$	1.78702e6	1.79038	0.895189	−	0.445686i	$-0.147040\pi$
$251$	1.78702e6	1.79038	0.895189	+	0.445686i	$0.147040\pi$
$252$	−47444.7	−0.0470638
$253$	−306619.	−0.301160
$254$	567048.	0.551487
$255$	0	0
$256$	559655.	0.533728
$257$	−1.37898e6	−1.30234	−0.651171	−	0.758931i	$-0.725722\pi$
$257$	−1.37898e6	−1.30234	−0.651171	+	0.758931i	$0.725722\pi$
$258$	−350086.	−0.327435
$259$	186793.	0.173026
$260$	0	0
$261$	−295943.	−0.268910
$262$	2.26177e6	2.03561
$263$	1.92969e6	1.72027	0.860136	−	0.510065i	$-0.170379\pi$
$263$	1.92969e6	1.72027	0.860136	+	0.510065i	$0.170379\pi$
$264$	54019.6	0.0477026
$265$	0	0
$266$	56849.3	0.0492630
$267$	−30436.7	−0.0261288
$268$	116220.	0.0988429
$269$	1.14161e6	0.961916	0.480958	−	0.876744i	$-0.340289\pi$
$269$	1.14161e6	0.961916	0.480958	+	0.876744i	$0.340289\pi$
$270$	0	0
$271$	−584178.	−0.483195	−0.241597	−	0.970377i	$-0.577671\pi$
$271$	−584178.	−0.483195	−0.241597	+	0.970377i	$0.577671\pi$
$272$	−1.00222e6	−0.821376
$273$	86815.0	0.0704998
$274$	233821.	0.188151
$275$	0	0
$276$	−865087.	−0.683577
$277$	−1.95194e6	−1.52850	−0.764252	−	0.644918i	$-0.776892\pi$
$277$	−1.95194e6	−1.52850	−0.764252	+	0.644918i	$0.776892\pi$
$278$	262937.	0.204052
$279$	−648083.	−0.498448
$280$	0	0
$281$	130490.	0.0985848	0.0492924	−	0.998784i	$-0.484303\pi$
$281$	130490.	0.0985848	0.0492924	+	0.998784i	$0.484303\pi$
$282$	−1.72769e6	−1.29373
$283$	−1.53762e6	−1.14125	−0.570626	−	0.821210i	$-0.693300\pi$
$283$	−1.53762e6	−1.14125	−0.570626	+	0.821210i	$0.693300\pi$
$284$	2.37132e6	1.74460
$285$	0	0
$286$	−632085.	−0.456941
$287$	−204768.	−0.146743
$288$	−669787.	−0.475834
$289$	153542.	0.108139
$290$	0	0
$291$	−272609.	−0.188715
$292$	2.30757e6	1.58379
$293$	−2.19926e6	−1.49661	−0.748303	−	0.663357i	$-0.769131\pi$
$293$	−2.19926e6	−1.49661	−0.748303	+	0.663357i	$0.769131\pi$
$294$	1.24699e6	0.841387
$295$	0	0
$296$	−600046.	−0.398066
$297$	88209.0	0.0580259
$298$	−1.00390e6	−0.654861
$299$	1.58295e6	1.02397
$300$	0	0
$301$	−71828.0	−0.0456959
$302$	−927413.	−0.585135
$303$	−881552.	−0.551622
$304$	351749.	0.218298
$305$	0	0
$306$	849653.	0.518726
$307$	136632.	0.0827385	0.0413693	−	0.999144i	$-0.486828\pi$
$307$	136632.	0.0827385	0.0413693	+	0.999144i	$0.486828\pi$
$308$	70874.2	0.0425708
$309$	784817.	0.467597
$310$	0	0
$311$	−2.03310e6	−1.19195	−0.595975	−	0.803003i	$-0.703234\pi$
$311$	−2.03310e6	−1.19195	−0.595975	+	0.803003i	$0.703234\pi$
$312$	−278881.	−0.162193
$313$	−814182.	−0.469743	−0.234872	−	0.972026i	$-0.575467\pi$
$313$	−814182.	−0.469743	−0.234872	+	0.972026i	$0.575467\pi$
$314$	4.44875e6	2.54632
$315$	0	0
$316$	2.04201e6	1.15038
$317$	−498935.	−0.278866	−0.139433	−	0.990231i	$-0.544528\pi$
$317$	−498935.	−0.278866	−0.139433	+	0.990231i	$0.544528\pi$
$318$	−1.04297e6	−0.578367
$319$	442087.	0.243238
$320$	0	0
$321$	−116609.	−0.0631641
$322$	−327228.	−0.175878
$323$	−552215.	−0.294511
$324$	248871.	0.131708
$325$	0	0
$326$	3.83901e6	2.00067
$327$	1.17395e6	0.607126
$328$	657790.	0.337600
$329$	−354476.	−0.180550
$330$	0	0
$331$	166599.	0.0835799	0.0417900	−	0.999126i	$-0.486694\pi$
$331$	166599.	0.0835799	0.0417900	+	0.999126i	$0.486694\pi$
$332$	−2.03259e6	−1.01206
$333$	−979819.	−0.484212
$334$	2.39488e6	1.17468
$335$	0	0
$336$	−111042.	−0.0536585
$337$	200668.	0.0962504	0.0481252	−	0.998841i	$-0.484675\pi$
$337$	200668.	0.0962504	0.0481252	+	0.998841i	$0.484675\pi$
$338$	158246.	0.0753428
$339$	−82012.4	−0.0387597
$340$	0	0
$341$	968124.	0.450863
$342$	−298202.	−0.137862
$343$	515380.	0.236533
$344$	230737.	0.105129
$345$	0	0
$346$	−5.48711e6	−2.46407
$347$	−1.38809e6	−0.618862	−0.309431	−	0.950922i	$-0.600139\pi$
$347$	−1.38809e6	−0.618862	−0.309431	+	0.950922i	$0.600139\pi$
$348$	1.24729e6	0.552103
$349$	−4.11254e6	−1.80737	−0.903684	−	0.428200i	$-0.859148\pi$
$349$	−4.11254e6	−1.80737	−0.903684	+	0.428200i	$0.859148\pi$
$350$	0	0
$351$	−455387.	−0.197293
$352$	1.00055e6	0.430408
$353$	1.14899e6	0.490770	0.245385	−	0.969426i	$-0.421086\pi$
$353$	1.14899e6	0.490770	0.245385	+	0.969426i	$0.421086\pi$
$354$	−3.49438e6	−1.48205
$355$	0	0
$356$	128280.	0.0536455
$357$	174326.	0.0723921
$358$	−1.83975e6	−0.758668
$359$	1.96193e6	0.803429	0.401714	−	0.915765i	$-0.368415\pi$
$359$	1.96193e6	0.803429	0.401714	+	0.915765i	$0.368415\pi$
$360$	0	0
$361$	−2.28229e6	−0.921728
$362$	−3.56359e6	−1.42927
$363$	−131769.	−0.0524864
$364$	−365894.	−0.144745
$365$	0	0
$366$	−4.21207e6	−1.64359
$367$	−2.98170e6	−1.15558	−0.577788	−	0.816187i	$-0.696084\pi$
$367$	−2.98170e6	−1.15558	−0.577788	+	0.816187i	$0.696084\pi$
$368$	−2.02469e6	−0.779362
$369$	1.07411e6	0.410660
$370$	0	0
$371$	−213989.	−0.0807154
$372$	2.73144e6	1.02337
$373$	−4.00552e6	−1.49069	−0.745344	−	0.666680i	$-0.767715\pi$
$373$	−4.00552e6	−1.49069	−0.745344	+	0.666680i	$0.767715\pi$
$374$	−1.26924e6	−0.469206
$375$	0	0
$376$	1.13870e6	0.415376
$377$	−2.28231e6	−0.827031
$378$	94137.9	0.0338871
$379$	−2.50361e6	−0.895299	−0.447650	−	0.894209i	$-0.647739\pi$
$379$	−2.50361e6	−0.895299	−0.447650	+	0.894209i	$0.647739\pi$
$380$	0	0
$381$	−610274.	−0.215384
$382$	1.36414e6	0.478299
$383$	−2.05384e6	−0.715434	−0.357717	−	0.933830i	$-0.616445\pi$
$383$	−2.05384e6	−0.715434	−0.357717	+	0.933830i	$0.616445\pi$
$384$	898607.	0.310987
$385$	0	0
$386$	1.77837e6	0.607511
$387$	376773.	0.127880
$388$	1.14895e6	0.387455
$389$	−3.29367e6	−1.10359	−0.551794	−	0.833981i	$-0.686056\pi$
$389$	−3.29367e6	−1.10359	−0.551794	+	0.833981i	$0.686056\pi$
$390$	0	0
$391$	3.17858e6	1.05146
$392$	−821880.	−0.270143
$393$	−2.43418e6	−0.795008
$394$	−6.10189e6	−1.98027
$395$	0	0
$396$	−371770.	−0.119134
$397$	2.90741e6	0.925827	0.462914	−	0.886403i	$-0.346804\pi$
$397$	2.90741e6	0.925827	0.462914	+	0.886403i	$0.346804\pi$
$398$	5.37320e6	1.70030
$399$	−61183.0	−0.0192397
$400$	0	0
$401$	−72188.2	−0.0224184	−0.0112092	−	0.999937i	$-0.503568\pi$
$401$	−72188.2	−0.0224184	−0.0112092	+	0.999937i	$0.503568\pi$
$402$	−230600.	−0.0711695
$403$	−4.99802e6	−1.53298
$404$	3.71543e6	1.13255
$405$	0	0
$406$	471802.	0.142051
$407$	1.46368e6	0.437986
$408$	−559997.	−0.166546
$409$	−3.53120e6	−1.04379	−0.521895	−	0.853009i	$-0.674775\pi$
$409$	−3.53120e6	−1.04379	−0.521895	+	0.853009i	$0.674775\pi$
$410$	0	0
$411$	−251645.	−0.0734826
$412$	−3.30772e6	−0.960033
$413$	−716951.	−0.206830
$414$	1.71647e6	0.492193
$415$	0	0
$416$	−5.16541e6	−1.46343
$417$	−282981.	−0.0796925
$418$	445463.	0.124701
$419$	2.48255e6	0.690817	0.345409	−	0.938452i	$-0.387740\pi$
$419$	2.48255e6	0.690817	0.345409	+	0.938452i	$0.387740\pi$
$420$	0	0
$421$	3.38758e6	0.931504	0.465752	−	0.884915i	$-0.345784\pi$
$421$	3.38758e6	0.931504	0.465752	+	0.884915i	$0.345784\pi$
$422$	3.85319e6	1.05327
$423$	1.85940e6	0.505267
$424$	687409.	0.185695
$425$	0	0
$426$	−4.70508e6	−1.25615
$427$	−864202.	−0.229375
$428$	491466.	0.129683
$429$	680269.	0.178459
$430$	0	0
$431$	5.13279e6	1.33095	0.665473	−	0.746422i	$-0.268230\pi$
$431$	5.13279e6	1.33095	0.665473	+	0.746422i	$0.268230\pi$
$432$	582468.	0.150163
$433$	−4.52419e6	−1.15963	−0.579817	−	0.814747i	$-0.696876\pi$
$433$	−4.52419e6	−1.15963	−0.579817	+	0.814747i	$0.696876\pi$
$434$	1.03319e6	0.263304
$435$	0	0
$436$	−4.94776e6	−1.24650
$437$	−1.11558e6	−0.279447
$438$	−4.57858e6	−1.14037
$439$	2.94510e6	0.729355	0.364677	−	0.931134i	$-0.381179\pi$
$439$	2.94510e6	0.729355	0.364677	+	0.931134i	$0.381179\pi$
$440$	0	0
$441$	−1.34205e6	−0.328604
$442$	6.55254e6	1.59534
$443$	1.33910e6	0.324192	0.162096	−	0.986775i	$-0.448175\pi$
$443$	1.33910e6	0.324192	0.162096	+	0.986775i	$0.448175\pi$
$444$	4.12959e6	0.994144
$445$	0	0
$446$	7.04035e6	1.67594
$447$	1.08043e6	0.255756
$448$	672981.	0.158419
$449$	−4.92158e6	−1.15210	−0.576048	−	0.817416i	$-0.695406\pi$
$449$	−4.92158e6	−1.15210	−0.576048	+	0.817416i	$0.695406\pi$
$450$	0	0
$451$	−1.60453e6	−0.371456
$452$	345653.	0.0795783
$453$	998110.	0.228525
$454$	−3.47352e6	−0.790916
$455$	0	0
$456$	196542.	0.0442632
$457$	4.92247e6	1.10253	0.551267	−	0.834329i	$-0.314144\pi$
$457$	4.92247e6	1.10253	0.551267	+	0.834329i	$0.314144\pi$
$458$	1.11333e7	2.48004
$459$	−914423.	−0.202589
$460$	0	0
$461$	8.40393e6	1.84175	0.920874	−	0.389861i	$-0.127477\pi$
$461$	8.40393e6	1.84175	0.920874	+	0.389861i	$0.127477\pi$
$462$	−140626.	−0.0306521
$463$	2.90332e6	0.629422	0.314711	−	0.949187i	$-0.398092\pi$
$463$	2.90332e6	0.629422	0.314711	+	0.949187i	$0.398092\pi$
$464$	2.91922e6	0.629466
$465$	0	0
$466$	−446312.	−0.0952082
$467$	−3.02045e6	−0.640883	−0.320442	−	0.947268i	$-0.603831\pi$
$467$	−3.02045e6	−0.640883	−0.320442	+	0.947268i	$0.603831\pi$
$468$	1.91929e6	0.405067
$469$	−47312.8	−0.00993222
$470$	0	0
$471$	−4.78788e6	−0.994468
$472$	2.30310e6	0.475837
$473$	−562833.	−0.115672
$474$	−4.05168e6	−0.828302
$475$	0	0
$476$	−734721.	−0.148629
$477$	1.12248e6	0.225882
$478$	1.34374e6	0.268996
$479$	−731965.	−0.145764	−0.0728822	−	0.997341i	$-0.523220\pi$
$479$	−731965.	−0.145764	−0.0728822	+	0.997341i	$0.523220\pi$
$480$	0	0
$481$	−7.55638e6	−1.48919
$482$	4.15111e6	0.813854
$483$	352173.	0.0686891
$484$	555359.	0.107761
$485$	0	0
$486$	−493799.	−0.0948329
$487$	−2.93463e6	−0.560700	−0.280350	−	0.959898i	$-0.590451\pi$
$487$	−2.93463e6	−0.560700	−0.280350	+	0.959898i	$0.590451\pi$
$488$	2.77613e6	0.527703
$489$	−4.13166e6	−0.781362
$490$	0	0
$491$	−6.20616e6	−1.16177	−0.580884	−	0.813987i	$-0.697293\pi$
$491$	−6.20616e6	−1.16177	−0.580884	+	0.813987i	$0.697293\pi$
$492$	−4.52699e6	−0.843134
$493$	−4.58292e6	−0.849229
$494$	−2.29974e6	−0.423995
$495$	0	0
$496$	6.39279e6	1.16677
$497$	−965354.	−0.175306
$498$	4.03299e6	0.728708
$499$	−6.62174e6	−1.19048	−0.595239	−	0.803549i	$-0.702942\pi$
$499$	−6.62174e6	−1.19048	−0.595239	+	0.803549i	$0.702942\pi$
$500$	0	0
$501$	−2.57744e6	−0.458770
$502$	1.49440e7	2.64672
$503$	2.50211e6	0.440946	0.220473	−	0.975393i	$-0.429240\pi$
$503$	2.50211e6	0.440946	0.220473	+	0.975393i	$0.429240\pi$
$504$	−62045.2	−0.0108801
$505$	0	0
$506$	−2.56411e6	−0.445205
$507$	−170309.	−0.0294252
$508$	2.57209e6	0.442208
$509$	4.26240e6	0.729222	0.364611	−	0.931160i	$-0.381202\pi$
$509$	4.26240e6	0.729222	0.364611	+	0.931160i	$0.381202\pi$
$510$	0	0
$511$	−939400.	−0.159147
$512$	7.87517e6	1.32765
$513$	320934.	0.0538422
$514$	−1.15317e7	−1.92525
$515$	0	0
$516$	−1.58796e6	−0.262552
$517$	−2.77762e6	−0.457031
$518$	1.56206e6	0.255784
$519$	5.90540e6	0.962345
$520$	0	0
$521$	5.90104e6	0.952433	0.476216	−	0.879328i	$-0.342008\pi$
$521$	5.90104e6	0.952433	0.476216	+	0.879328i	$0.342008\pi$
$522$	−2.47483e6	−0.397529
$523$	1.27490e6	0.203809	0.101904	−	0.994794i	$-0.467506\pi$
$523$	1.27490e6	0.203809	0.101904	+	0.994794i	$0.467506\pi$
$524$	1.02592e7	1.63225
$525$	0	0
$526$	1.61370e7	2.54308
$527$	−1.00361e7	−1.57412
$528$	−870107.	−0.135828
$529$	−14971.4	−0.00232607
$530$	0	0
$531$	3.76075e6	0.578813
$532$	257864.	0.0395014
$533$	8.28355e6	1.26299
$534$	−254528.	−0.0386262
$535$	0	0
$536$	151986.	0.0228502
$537$	1.98000e6	0.296298
$538$	9.54674e6	1.42200
$539$	2.00479e6	0.297234
$540$	0	0
$541$	1.18799e7	1.74510	0.872551	−	0.488523i	$-0.162464\pi$
$541$	1.18799e7	1.74510	0.872551	+	0.488523i	$0.162464\pi$
$542$	−4.88521e6	−0.714307
$543$	3.83524e6	0.558204
$544$	−1.03722e7	−1.50271
$545$	0	0
$546$	725992.	0.104220
$547$	1.02980e7	1.47158	0.735788	−	0.677212i	$-0.236812\pi$
$547$	1.02980e7	1.47158	0.735788	+	0.677212i	$0.236812\pi$
$548$	1.06060e6	0.150868
$549$	4.53316e6	0.641904
$550$	0	0
$551$	1.60846e6	0.225700
$552$	−1.13131e6	−0.158027
$553$	−831294.	−0.115596
$554$	−1.63231e7	−2.25959
$555$	0	0
$556$	1.19266e6	0.163618
$557$	−9.12350e6	−1.24602	−0.623008	−	0.782215i	$-0.714090\pi$
$557$	−9.12350e6	−1.24602	−0.623008	+	0.782215i	$0.714090\pi$
$558$	−5.41961e6	−0.736855
$559$	2.90567e6	0.393294
$560$	0	0
$561$	1.36599e6	0.183248
$562$	1.09122e6	0.145738
$563$	−1.08869e7	−1.44754	−0.723771	−	0.690040i	$-0.757593\pi$
$563$	−1.08869e7	−1.44754	−0.723771	+	0.690040i	$0.757593\pi$
$564$	−7.83669e6	−1.03737
$565$	0	0
$566$	−1.28584e7	−1.68711
$567$	−101314.	−0.0132346
$568$	3.10106e6	0.403311
$569$	−8.89191e6	−1.15137	−0.575684	−	0.817672i	$-0.695264\pi$
$569$	−8.89191e6	−1.15137	−0.575684	+	0.817672i	$0.695264\pi$
$570$	0	0
$571$	1.47453e7	1.89262	0.946311	−	0.323258i	$-0.104778\pi$
$571$	1.47453e7	1.89262	0.946311	+	0.323258i	$0.104778\pi$
$572$	−2.86709e6	−0.366397
$573$	−1.46812e6	−0.186800
$574$	−1.71238e6	−0.216931
$575$	0	0
$576$	−3.53011e6	−0.443335
$577$	6.49447e6	0.812090	0.406045	−	0.913853i	$-0.366908\pi$
$577$	6.49447e6	0.812090	0.406045	+	0.913853i	$0.366908\pi$
$578$	1.28400e6	0.159862
$579$	−1.91394e6	−0.237264
$580$	0	0
$581$	827460.	0.101697
$582$	−2.27970e6	−0.278978
$583$	−1.67678e6	−0.204318
$584$	3.01769e6	0.366136
$585$	0	0
$586$	−1.83914e7	−2.21243
$587$	2.10642e6	0.252319	0.126160	−	0.992010i	$-0.459735\pi$
$587$	2.10642e6	0.252319	0.126160	+	0.992010i	$0.459735\pi$
$588$	5.65627e6	0.674663
$589$	3.52236e6	0.418356
$590$	0	0
$591$	6.56704e6	0.773394
$592$	9.66508e6	1.13345
$593$	9.24728e6	1.07988	0.539942	−	0.841702i	$-0.318446\pi$
$593$	9.24728e6	1.07988	0.539942	+	0.841702i	$0.318446\pi$
$594$	737650.	0.0857796
$595$	0	0
$596$	−4.55361e6	−0.525098
$597$	−5.78280e6	−0.664053
$598$	1.32374e7	1.51374
$599$	1.30180e6	0.148245	0.0741223	−	0.997249i	$-0.476384\pi$
$599$	1.30180e6	0.148245	0.0741223	+	0.997249i	$0.476384\pi$
$600$	0	0
$601$	2.74661e6	0.310178	0.155089	−	0.987901i	$-0.450434\pi$
$601$	2.74661e6	0.310178	0.155089	+	0.987901i	$0.450434\pi$
$602$	−600663.	−0.0675523
$603$	248178.	0.0277953
$604$	−4.20668e6	−0.469188
$605$	0	0
$606$	−7.37200e6	−0.815462
$607$	2.68332e6	0.295598	0.147799	−	0.989017i	$-0.452781\pi$
$607$	2.68332e6	0.295598	0.147799	+	0.989017i	$0.452781\pi$
$608$	3.64033e6	0.399375
$609$	−507767.	−0.0554781
$610$	0	0
$611$	1.43397e7	1.55395
$612$	3.85397e6	0.415939
$613$	−2.54616e6	−0.273675	−0.136837	−	0.990594i	$-0.543694\pi$
$613$	−2.54616e6	−0.273675	−0.136837	+	0.990594i	$0.543694\pi$
$614$	1.14259e6	0.122312
$615$	0	0
$616$	92684.8	0.00984140
$617$	−7.90825e6	−0.836310	−0.418155	−	0.908376i	$-0.637323\pi$
$617$	−7.90825e6	−0.836310	−0.418155	+	0.908376i	$0.637323\pi$
$618$	6.56305e6	0.691249
$619$	−1.80718e7	−1.89573	−0.947864	−	0.318675i	$-0.896762\pi$
$619$	−1.80718e7	−1.89573	−0.947864	+	0.318675i	$0.896762\pi$
$620$	0	0
$621$	−1.84732e6	−0.192226
$622$	−1.70018e7	−1.76206
$623$	−52222.1	−0.00539057
$624$	4.49200e6	0.461826
$625$	0	0
$626$	−6.80862e6	−0.694421
$627$	−479420.	−0.0487021
$628$	2.01792e7	2.04176
$629$	−1.51733e7	−1.52916
$630$	0	0
$631$	8.68542e6	0.868395	0.434198	−	0.900818i	$-0.357032\pi$
$631$	8.68542e6	0.868395	0.434198	+	0.900818i	$0.357032\pi$
$632$	2.67041e6	0.265941
$633$	−4.14692e6	−0.411355
$634$	−4.17236e6	−0.412248
$635$	0	0
$636$	−4.73083e6	−0.463762
$637$	−1.03499e7	−1.01062
$638$	3.69696e6	0.359578
$639$	5.06375e6	0.490591
$640$	0	0
$641$	1.20048e7	1.15402	0.577008	−	0.816739i	$-0.304220\pi$
$641$	1.20048e7	1.15402	0.577008	+	0.816739i	$0.304220\pi$
$642$	−975147.	−0.0933754
$643$	−2.64892e6	−0.252663	−0.126331	−	0.991988i	$-0.540320\pi$
$643$	−2.64892e6	−0.252663	−0.126331	+	0.991988i	$0.540320\pi$
$644$	−1.48428e6	−0.141027
$645$	0	0
$646$	−4.61791e6	−0.435375
$647$	1.54484e7	1.45085	0.725425	−	0.688301i	$-0.241643\pi$
$647$	1.54484e7	1.45085	0.725425	+	0.688301i	$0.241643\pi$
$648$	325457.	0.0304478
$649$	−5.61792e6	−0.523557
$650$	0	0
$651$	−1.11196e6	−0.102834
$652$	1.74135e7	1.60423
$653$	2.03688e7	1.86932	0.934658	−	0.355548i	$-0.115706\pi$
$653$	2.03688e7	1.86932	0.934658	+	0.355548i	$0.115706\pi$
$654$	9.81715e6	0.897514
$655$	0	0
$656$	−1.05952e7	−0.961277
$657$	4.92761e6	0.445372
$658$	−2.96431e6	−0.266906
$659$	−6.76191e6	−0.606535	−0.303268	−	0.952905i	$-0.598078\pi$
$659$	−6.76191e6	−0.606535	−0.303268	+	0.952905i	$0.598078\pi$
$660$	0	0
$661$	1.49341e7	1.32946	0.664728	−	0.747085i	$-0.268547\pi$
$661$	1.49341e7	1.32946	0.664728	+	0.747085i	$0.268547\pi$
$662$	1.39319e6	0.123556
$663$	−7.05204e6	−0.623061
$664$	−2.65810e6	−0.233965
$665$	0	0
$666$	−8.19376e6	−0.715810
$667$	−9.25841e6	−0.805790
$668$	1.08630e7	0.941909
$669$	−7.57704e6	−0.654537
$670$	0	0
$671$	−6.77175e6	−0.580624
$672$	−1.14920e6	−0.0981682
$673$	6.00776e6	0.511299	0.255649	−	0.966770i	$-0.417711\pi$
$673$	6.00776e6	0.511299	0.255649	+	0.966770i	$0.417711\pi$
$674$	1.67809e6	0.142287
$675$	0	0
$676$	717794.	0.0604133
$677$	−2.14516e7	−1.79882	−0.899409	−	0.437108i	$-0.856003\pi$
$677$	−2.14516e7	−1.79882	−0.899409	+	0.437108i	$0.856003\pi$
$678$	−685831.	−0.0572984
$679$	−467731.	−0.0389334
$680$	0	0
$681$	3.73831e6	0.308893
$682$	8.09596e6	0.666511
$683$	5.40024e6	0.442956	0.221478	−	0.975165i	$-0.428912\pi$
$683$	5.40024e6	0.442956	0.221478	+	0.975165i	$0.428912\pi$
$684$	−1.35262e6	−0.110544
$685$	0	0
$686$	4.30988e6	0.349667
$687$	−1.19819e7	−0.968580
$688$	−3.71654e6	−0.299342
$689$	8.65654e6	0.694698
$690$	0	0
$691$	−1.55798e7	−1.24127	−0.620635	−	0.784100i	$-0.713125\pi$
$691$	−1.55798e7	−1.24127	−0.620635	+	0.784100i	$0.713125\pi$
$692$	−2.48892e7	−1.97581
$693$	151346.	0.0119712
$694$	−1.16079e7	−0.914864
$695$	0	0
$696$	1.63113e6	0.127634
$697$	1.66335e7	1.29688
$698$	−3.43912e7	−2.67183
$699$	480335.	0.0371836
$700$	0	0
$701$	−1.49222e7	−1.14693	−0.573465	−	0.819230i	$-0.694401\pi$
$701$	−1.49222e7	−1.14693	−0.573465	+	0.819230i	$0.694401\pi$
$702$	−3.80818e6	−0.291659
$703$	5.32536e6	0.406407
$704$	5.27338e6	0.401012
$705$	0	0
$706$	9.60842e6	0.725504
$707$	−1.51253e6	−0.113804
$708$	−1.58502e7	−1.18837
$709$	7.29943e6	0.545347	0.272674	−	0.962107i	$-0.412092\pi$
$709$	7.29943e6	0.545347	0.272674	+	0.962107i	$0.412092\pi$
$710$	0	0
$711$	4.36054e6	0.323494
$712$	167756.	0.0124016
$713$	−2.02749e7	−1.49360
$714$	1.45780e6	0.107017
$715$	0	0
$716$	−8.34498e6	−0.608335
$717$	−1.44617e6	−0.105056
$718$	1.64067e7	1.18771
$719$	−80423.4	−0.00580176	−0.00290088	−	0.999996i	$-0.500923\pi$
$719$	−80423.4	−0.00580176	−0.00290088	+	0.999996i	$0.500923\pi$
$720$	0	0
$721$	1.34656e6	0.0964689
$722$	−1.90857e7	−1.36259
$723$	−4.46755e6	−0.317851
$724$	−1.61642e7	−1.14606
$725$	0	0
$726$	−1.10192e6	−0.0775906
$727$	−1.75871e7	−1.23412	−0.617061	−	0.786916i	$-0.711677\pi$
$727$	−1.75871e7	−1.23412	−0.617061	+	0.786916i	$0.711677\pi$
$728$	−478493.	−0.0334617
$729$	531441.	0.0370370
$730$	0	0
$731$	5.83464e6	0.403850
$732$	−1.91056e7	−1.31790
$733$	1.17001e7	0.804319	0.402160	−	0.915570i	$-0.368260\pi$
$733$	1.17001e7	0.804319	0.402160	+	0.915570i	$0.368260\pi$
$734$	−2.49345e7	−1.70829
$735$	0	0
$736$	−2.09539e7	−1.42584
$737$	−370736.	−0.0251418
$738$	8.98227e6	0.607079
$739$	−1.55729e7	−1.04896	−0.524480	−	0.851422i	$-0.675740\pi$
$739$	−1.55729e7	−1.04896	−0.524480	+	0.851422i	$0.675740\pi$
$740$	0	0
$741$	2.47505e6	0.165592
$742$	−1.78949e6	−0.119322
$743$	320800.	0.0213188	0.0106594	−	0.999943i	$-0.496607\pi$
$743$	320800.	0.0213188	0.0106594	+	0.999943i	$0.496607\pi$
$744$	3.57200e6	0.236581
$745$	0	0
$746$	−3.34963e7	−2.20368
$747$	−4.34043e6	−0.284597
$748$	−5.75716e6	−0.376231
$749$	−200074.	−0.0130312
$750$	0	0
$751$	−1.56465e7	−1.01232	−0.506159	−	0.862440i	$-0.668935\pi$
$751$	−1.56465e7	−1.01232	−0.506159	+	0.862440i	$0.668935\pi$
$752$	−1.83414e7	−1.18273
$753$	−1.60832e7	−1.03368
$754$	−1.90859e7	−1.22260
$755$	0	0
$756$	427003.	0.0271723
$757$	1.55874e7	0.988629	0.494314	−	0.869283i	$-0.335419\pi$
$757$	1.55874e7	0.988629	0.494314	+	0.869283i	$0.335419\pi$
$758$	−2.09365e7	−1.32352
$759$	2.75957e6	0.173875
$760$	0	0
$761$	9.31514e6	0.583079	0.291540	−	0.956559i	$-0.405832\pi$
$761$	9.31514e6	0.583079	0.291540	+	0.956559i	$0.405832\pi$
$762$	−5.10343e6	−0.318401
$763$	2.01421e6	0.125255
$764$	6.18762e6	0.383522
$765$	0	0
$766$	−1.71753e7	−1.05763
$767$	2.90030e7	1.78014
$768$	−5.03689e6	−0.308148
$769$	1.43630e7	0.875848	0.437924	−	0.899012i	$-0.355714\pi$
$769$	1.43630e7	0.875848	0.437924	+	0.899012i	$0.355714\pi$
$770$	0	0
$771$	1.24108e7	0.751907
$772$	8.06656e6	0.487131
$773$	−4.14465e6	−0.249482	−0.124741	−	0.992189i	$-0.539810\pi$
$773$	−4.14465e6	−0.249482	−0.124741	+	0.992189i	$0.539810\pi$
$774$	3.15077e6	0.189045
$775$	0	0
$776$	1.50252e6	0.0895708
$777$	−1.68114e6	−0.0998965
$778$	−2.75434e7	−1.63143
$779$	−5.83784e6	−0.344674
$780$	0	0
$781$	−7.56437e6	−0.443757
$782$	2.65810e7	1.55437
$783$	2.66348e6	0.155255
$784$	1.32382e7	0.769200
$785$	0	0
$786$	−2.03559e7	−1.17526
$787$	−1.44908e7	−0.833980	−0.416990	−	0.908911i	$-0.636915\pi$
$787$	−1.44908e7	−0.833980	−0.416990	+	0.908911i	$0.636915\pi$
$788$	−2.76777e7	−1.58787
$789$	−1.73672e7	−0.993199
$790$	0	0
$791$	−140714.	−0.00799642
$792$	−486177.	−0.0275411
$793$	3.49597e7	1.97417
$794$	2.43133e7	1.36865
$795$	0	0
$796$	2.43725e7	1.36338
$797$	4.19488e6	0.233923	0.116962	−	0.993136i	$-0.462685\pi$
$797$	4.19488e6	0.233923	0.116962	+	0.993136i	$0.462685\pi$
$798$	−511644.	−0.0284420
$799$	2.87943e7	1.59566
$800$	0	0
$801$	273930.	0.0150855
$802$	−603676.	−0.0331412
$803$	−7.36099e6	−0.402854
$804$	−1.04598e6	−0.0570670
$805$	0	0
$806$	−4.17961e7	−2.26620
$807$	−1.02745e7	−0.555363
$808$	4.85880e6	0.261819
$809$	2.08215e7	1.11851	0.559257	−	0.828994i	$-0.311086\pi$
$809$	2.08215e7	1.11851	0.559257	+	0.828994i	$0.311086\pi$
$810$	0	0
$811$	3.14240e7	1.67768	0.838839	−	0.544379i	$-0.183235\pi$
$811$	3.14240e7	1.67768	0.838839	+	0.544379i	$0.183235\pi$
$812$	2.14006e6	0.113903
$813$	5.25761e6	0.278973
$814$	1.22401e7	0.647474
$815$	0	0
$816$	9.02000e6	0.474222
$817$	−2.04778e6	−0.107332
$818$	−2.95297e7	−1.54304
$819$	−781335.	−0.0407031
$820$	0	0
$821$	−1.81350e7	−0.938990	−0.469495	−	0.882935i	$-0.655564\pi$
$821$	−1.81350e7	−0.938990	−0.469495	+	0.882935i	$0.655564\pi$
$822$	−2.10439e6	−0.108629
$823$	1.05426e7	0.542561	0.271281	−	0.962500i	$-0.412553\pi$
$823$	1.05426e7	0.542561	0.271281	+	0.962500i	$0.412553\pi$
$824$	−4.32563e6	−0.221938
$825$	0	0
$826$	−5.99552e6	−0.305757
$827$	−2.74783e7	−1.39710	−0.698548	−	0.715563i	$-0.746170\pi$
$827$	−2.74783e7	−1.39710	−0.698548	+	0.715563i	$0.746170\pi$
$828$	7.78578e6	0.394663
$829$	2.35793e7	1.19164	0.595819	−	0.803119i	$-0.296828\pi$
$829$	2.35793e7	1.19164	0.595819	+	0.803119i	$0.296828\pi$
$830$	0	0
$831$	1.75674e7	0.882482
$832$	−2.72243e7	−1.36348
$833$	−2.07828e7	−1.03775
$834$	−2.36644e6	−0.117809
$835$	0	0
$836$	2.02059e6	0.0999912
$837$	5.83274e6	0.287779
$838$	2.07604e7	1.02123
$839$	−3.78658e7	−1.85713	−0.928565	−	0.371170i	$-0.878957\pi$
$839$	−3.78658e7	−1.85713	−0.928565	+	0.371170i	$0.878957\pi$
$840$	0	0
$841$	−7.16226e6	−0.349188
$842$	2.83288e7	1.37704
$843$	−1.17441e6	−0.0569180
$844$	1.74778e7	0.844560
$845$	0	0
$846$	1.55492e7	0.746936
$847$	−226084.	−0.0108283
$848$	−1.10723e7	−0.528746
$849$	1.38385e7	0.658903
$850$	0	0
$851$	−3.06531e7	−1.45094
$852$	−2.13419e7	−1.00724
$853$	−2.10629e7	−0.991166	−0.495583	−	0.868561i	$-0.665046\pi$
$853$	−2.10629e7	−0.991166	−0.495583	+	0.868561i	$0.665046\pi$
$854$	−7.22691e6	−0.339084
$855$	0	0
$856$	642708.	0.0299799
$857$	−6.20639e6	−0.288660	−0.144330	−	0.989530i	$-0.546103\pi$
$857$	−6.20639e6	−0.288660	−0.144330	+	0.989530i	$0.546103\pi$
$858$	5.68876e6	0.263815
$859$	−1.19855e7	−0.554206	−0.277103	−	0.960840i	$-0.589374\pi$
$859$	−1.19855e7	−0.554206	−0.277103	+	0.960840i	$0.589374\pi$
$860$	0	0
$861$	1.84292e6	0.0847223
$862$	4.29231e7	1.96754
$863$	−3.35712e7	−1.53440	−0.767202	−	0.641406i	$-0.778352\pi$
$863$	−3.35712e7	−1.53440	−0.767202	+	0.641406i	$0.778352\pi$
$864$	6.02809e6	0.274723
$865$	0	0
$866$	−3.78336e7	−1.71429
$867$	−1.38188e6	−0.0624341
$868$	4.68650e6	0.211130
$869$	−6.51389e6	−0.292611
$870$	0	0
$871$	1.91396e6	0.0854843
$872$	−6.47037e6	−0.288163
$873$	2.45348e6	0.108955
$874$	−9.32910e6	−0.413106
$875$	0	0
$876$	−2.07681e7	−0.914401
$877$	−2.91315e7	−1.27898	−0.639490	−	0.768799i	$-0.720854\pi$
$877$	−2.91315e7	−1.27898	−0.639490	+	0.768799i	$0.720854\pi$
$878$	2.46285e7	1.07820
$879$	1.97933e7	0.864065
$880$	0	0
$881$	1.56250e7	0.678237	0.339119	−	0.940744i	$-0.389871\pi$
$881$	1.56250e7	0.678237	0.339119	+	0.940744i	$0.389871\pi$
$882$	−1.12229e7	−0.485775
$883$	3.67889e6	0.158787	0.0793935	−	0.996843i	$-0.474702\pi$
$883$	3.67889e6	0.158787	0.0793935	+	0.996843i	$0.474702\pi$
$884$	2.97218e7	1.27922
$885$	0	0
$886$	1.11982e7	0.479253
$887$	−2.49631e7	−1.06534	−0.532672	−	0.846322i	$-0.678812\pi$
$887$	−2.49631e7	−1.06534	−0.532672	+	0.846322i	$0.678812\pi$
$888$	5.40041e6	0.229824
$889$	−1.04708e6	−0.0444352
$890$	0	0
$891$	−793881.	−0.0335013
$892$	3.19345e7	1.34384
$893$	−1.01059e7	−0.424079
$894$	9.03508e6	0.378084
$895$	0	0
$896$	1.54180e6	0.0641589
$897$	−1.42465e7	−0.591191
$898$	−4.11568e7	−1.70314
$899$	2.92326e7	1.20634
$900$	0	0
$901$	1.73825e7	0.713344
$902$	−1.34180e7	−0.549123
$903$	646452.	0.0263826
$904$	452023.	0.0183967
$905$	0	0
$906$	8.34672e6	0.337828
$907$	−1.41749e7	−0.572139	−0.286070	−	0.958209i	$-0.592349\pi$
$907$	−1.41749e7	−0.572139	−0.286070	+	0.958209i	$0.592349\pi$
$908$	−1.57556e7	−0.634193
$909$	7.93397e6	0.318479
$910$	0	0
$911$	3.03213e6	0.121047	0.0605233	−	0.998167i	$-0.480723\pi$
$911$	3.03213e6	0.121047	0.0605233	+	0.998167i	$0.480723\pi$
$912$	−3.16574e6	−0.126034
$913$	6.48385e6	0.257428
$914$	4.11642e7	1.62988
$915$	0	0
$916$	5.04996e7	1.98861
$917$	−4.17647e6	−0.164016
$918$	−7.64688e6	−0.299487
$919$	1.47606e6	0.0576521	0.0288260	−	0.999584i	$-0.490823\pi$
$919$	1.47606e6	0.0576521	0.0288260	+	0.999584i	$0.490823\pi$
$920$	0	0
$921$	−1.22969e6	−0.0477691
$922$	7.02780e7	2.72265
$923$	3.90517e7	1.50881
$924$	−637868.	−0.0245782
$925$	0	0
$926$	2.42791e7	0.930475
$927$	−7.06335e6	−0.269968
$928$	3.02117e7	1.15161
$929$	−2.51584e7	−0.956409	−0.478205	−	0.878248i	$-0.658712\pi$
$929$	−2.51584e7	−0.956409	−0.478205	+	0.878248i	$0.658712\pi$
$930$	0	0
$931$	7.29412e6	0.275803
$932$	−2.02444e6	−0.0763423
$933$	1.82979e7	0.688172
$934$	−2.52585e7	−0.947417
$935$	0	0
$936$	2.50993e6	0.0936422
$937$	3.23202e7	1.20261	0.601305	−	0.799019i	$-0.294648\pi$
$937$	3.23202e7	1.20261	0.601305	+	0.799019i	$0.294648\pi$
$938$	−395654.	−0.0146828
$939$	7.32764e6	0.271207
$940$	0	0
$941$	−2.88348e7	−1.06156	−0.530778	−	0.847511i	$-0.678100\pi$
$941$	−2.88348e7	−1.06156	−0.530778	+	0.847511i	$0.678100\pi$
$942$	−4.00387e7	−1.47012
$943$	3.36030e7	1.23055
$944$	−3.70966e7	−1.35489
$945$	0	0
$946$	−4.70671e6	−0.170997
$947$	1.94005e7	0.702973	0.351487	−	0.936193i	$-0.385676\pi$
$947$	1.94005e7	0.702973	0.351487	+	0.936193i	$0.385676\pi$
$948$	−1.83781e7	−0.664171
$949$	3.80018e7	1.36974
$950$	0	0
$951$	4.49042e6	0.161004
$952$	−960821.	−0.0343598
$953$	1.51638e6	0.0540848	0.0270424	−	0.999634i	$-0.491391\pi$
$953$	1.51638e6	0.0540848	0.0270424	+	0.999634i	$0.491391\pi$
$954$	9.38672e6	0.333921
$955$	0	0
$956$	6.09510e6	0.215693
$957$	−3.97879e6	−0.140433
$958$	−6.12108e6	−0.215484
$959$	−431763.	−0.0151600
$960$	0	0
$961$	3.53872e7	1.23605
$962$	−6.31904e7	−2.20147
$963$	1.04948e6	0.0364678
$964$	1.88291e7	0.652586
$965$	0	0
$966$	2.94505e6	0.101543
$967$	4.41852e6	0.151954	0.0759768	−	0.997110i	$-0.475793\pi$
$967$	4.41852e6	0.151954	0.0759768	+	0.997110i	$0.475793\pi$
$968$	726264.	0.0249119
$969$	4.96993e6	0.170036
$970$	0	0
$971$	−2.45791e6	−0.0836600	−0.0418300	−	0.999125i	$-0.513319\pi$
$971$	−2.45791e6	−0.0836600	−0.0418300	+	0.999125i	$0.513319\pi$
$972$	−2.23983e6	−0.0760414
$973$	−485528.	−0.0164412
$974$	−2.45409e7	−0.828883
$975$	0	0
$976$	−4.47157e7	−1.50257
$977$	−1.93483e7	−0.648495	−0.324248	−	0.945972i	$-0.605111\pi$
$977$	−1.93483e7	−0.648495	−0.324248	+	0.945972i	$0.605111\pi$
$978$	−3.45511e7	−1.15509
$979$	−409205.	−0.0136453
$980$	0	0
$981$	−1.05655e7	−0.350524
$982$	−5.18992e7	−1.71744
$983$	5.63062e6	0.185854	0.0929271	−	0.995673i	$-0.470378\pi$
$983$	5.63062e6	0.185854	0.0929271	+	0.995673i	$0.470378\pi$
$984$	−5.92011e6	−0.194914
$985$	0	0
$986$	−3.83248e7	−1.25541
$987$	3.19028e6	0.104240
$988$	−1.04314e7	−0.339979
$989$	1.17871e7	0.383193
$990$	0	0
$991$	1.17881e7	0.381293	0.190647	−	0.981659i	$-0.438942\pi$
$991$	1.17881e7	0.381293	0.190647	+	0.981659i	$0.438942\pi$
$992$	6.61603e7	2.13461
$993$	−1.49939e6	−0.0482549
$994$	−8.07280e6	−0.259154
$995$	0	0
$996$	1.82934e7	0.584312
$997$	−2.55288e7	−0.813379	−0.406689	−	0.913567i	$-0.633317\pi$
$997$	−2.55288e7	−0.813379	−0.406689	+	0.913567i	$0.633317\pi$
$998$	−5.53745e7	−1.75988
$999$	8.81837e6	0.279560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.s.1.8	yes	9
5.4	even	2		825.6.a.r.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.2	✓	9	5.4	even	2
825.6.a.s.1.8	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)

\(f(q)\)	\(=\)	\(q+8.36252 q^{2} -9.00000 q^{3} +37.9318 q^{4} -75.2627 q^{6} -15.4419 q^{7} +49.6048 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+8.36252 q^{2} -9.00000 q^{3} +37.9318 q^{4} -75.2627 q^{6} -15.4419 q^{7} +49.6048 q^{8} +81.0000 q^{9} -121.000 q^{11} -341.386 q^{12} +624.673 q^{13} -129.133 q^{14} -798.996 q^{16} +1254.35 q^{17} +677.364 q^{18} -440.239 q^{19} +138.977 q^{21} -1011.87 q^{22} +2534.04 q^{23} -446.443 q^{24} +5223.84 q^{26} -729.000 q^{27} -585.737 q^{28} -3653.61 q^{29} -8001.02 q^{31} -8268.98 q^{32} +1089.00 q^{33} +10489.5 q^{34} +3072.48 q^{36} -12096.5 q^{37} -3681.51 q^{38} -5622.06 q^{39} +13260.6 q^{41} +1162.20 q^{42} +4651.51 q^{43} -4589.75 q^{44} +21191.0 q^{46} +22955.5 q^{47} +7190.97 q^{48} -16568.5 q^{49} -11289.2 q^{51} +23695.0 q^{52} +13857.7 q^{53} -6096.28 q^{54} -765.990 q^{56} +3962.15 q^{57} -30553.4 q^{58} +46429.1 q^{59} +55964.9 q^{61} -66908.7 q^{62} -1250.79 q^{63} -43581.6 q^{64} +9106.79 q^{66} +3063.93 q^{67} +47579.8 q^{68} -22806.4 q^{69} +62515.4 q^{71} +4017.99 q^{72} +60834.6 q^{73} -101158. q^{74} -16699.1 q^{76} +1868.46 q^{77} -47014.6 q^{78} +53833.8 q^{79} +6561.00 q^{81} +110892. q^{82} -53585.5 q^{83} +5271.64 q^{84} +38898.4 q^{86} +32882.5 q^{87} -6002.18 q^{88} +3381.86 q^{89} -9646.11 q^{91} +96120.8 q^{92} +72009.2 q^{93} +191966. q^{94} +74420.8 q^{96} +30289.8 q^{97} -138555. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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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