LMFDB - Embedded newform 825.6.a.r.1.5

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			1.5
Root			$-0.624305$ 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)

$f(q)$	$=$	$q+0.624305 q^{2} +9.00000 q^{3} -31.6102 q^{4} +5.61875 q^{6} +106.290 q^{7} -39.7122 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+0.624305 q^{2} +9.00000 q^{3} -31.6102 q^{4} +5.61875 q^{6} +106.290 q^{7} -39.7122 q^{8} +81.0000 q^{9} -121.000 q^{11} -284.492 q^{12} +264.362 q^{13} +66.3575 q^{14} +986.735 q^{16} +509.491 q^{17} +50.5687 q^{18} +1662.79 q^{19} +956.611 q^{21} -75.5409 q^{22} +2212.06 q^{23} -357.410 q^{24} +165.043 q^{26} +729.000 q^{27} -3359.86 q^{28} -5424.15 q^{29} -8799.95 q^{31} +1886.81 q^{32} -1089.00 q^{33} +318.078 q^{34} -2560.43 q^{36} -2600.13 q^{37} +1038.09 q^{38} +2379.26 q^{39} -9444.87 q^{41} +597.217 q^{42} +9572.69 q^{43} +3824.84 q^{44} +1381.00 q^{46} +5521.13 q^{47} +8880.62 q^{48} -5509.41 q^{49} +4585.42 q^{51} -8356.55 q^{52} +24674.9 q^{53} +455.119 q^{54} -4221.01 q^{56} +14965.1 q^{57} -3386.32 q^{58} +31867.9 q^{59} +13479.2 q^{61} -5493.85 q^{62} +8609.50 q^{63} -30397.6 q^{64} -679.868 q^{66} -29599.5 q^{67} -16105.1 q^{68} +19908.6 q^{69} -20097.8 q^{71} -3216.69 q^{72} +63873.3 q^{73} -1623.27 q^{74} -52561.3 q^{76} -12861.1 q^{77} +1485.38 q^{78} -84623.1 q^{79} +6561.00 q^{81} -5896.48 q^{82} +81972.5 q^{83} -30238.7 q^{84} +5976.28 q^{86} -48817.3 q^{87} +4805.18 q^{88} +1898.12 q^{89} +28099.1 q^{91} -69923.8 q^{92} -79199.5 q^{93} +3446.87 q^{94} +16981.3 q^{96} +26096.5 q^{97} -3439.56 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.624305	0.110363	0.0551813	−	0.998476i	$-0.482426\pi$
$2$	0.624305	0.110363	0.0551813	+	0.998476i	$0.482426\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.6102	−0.987820
$5$	0	0
$5$	0	0
$6$	5.61875	0.0637179
$7$	106.290	0.819875	0.409938	−	0.912114i	$-0.365550\pi$
$7$	106.290	0.819875	0.409938	+	0.912114i	$0.365550\pi$
$8$	−39.7122	−0.219381
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−284.492	−0.570318
$13$	264.362	0.433851	0.216926	−	0.976188i	$-0.430397\pi$
$13$	264.362	0.433851	0.216926	+	0.976188i	$0.430397\pi$
$14$	66.3575	0.0904836
$15$	0	0
$16$	986.735	0.963609
$17$	509.491	0.427577	0.213788	−	0.976880i	$-0.431420\pi$
$17$	509.491	0.427577	0.213788	+	0.976880i	$0.431420\pi$
$18$	50.5687	0.0367875
$19$	1662.79	1.05671	0.528353	−	0.849025i	$-0.322810\pi$
$19$	1662.79	1.05671	0.528353	+	0.849025i	$0.322810\pi$
$20$	0	0
$21$	956.611	0.473355
$22$	−75.5409	−0.0332756
$23$	2212.06	0.871922	0.435961	−	0.899966i	$-0.356409\pi$
$23$	2212.06	0.871922	0.435961	+	0.899966i	$0.356409\pi$
$24$	−357.410	−0.126660
$25$	0	0
$26$	165.043	0.0478809
$27$	729.000	0.192450
$28$	−3359.86	−0.809889
$29$	−5424.15	−1.19767	−0.598834	−	0.800873i	$-0.704369\pi$
$29$	−5424.15	−1.19767	−0.598834	+	0.800873i	$0.704369\pi$
$30$	0	0
$31$	−8799.95	−1.64466	−0.822329	−	0.569012i	$-0.807326\pi$
$31$	−8799.95	−1.64466	−0.822329	+	0.569012i	$0.807326\pi$
$32$	1886.81	0.325727
$33$	−1089.00	−0.174078
$34$	318.078	0.0471885
$35$	0	0
$36$	−2560.43	−0.329273
$37$	−2600.13	−0.312242	−0.156121	−	0.987738i	$-0.549899\pi$
$37$	−2600.13	−0.312242	−0.156121	+	0.987738i	$0.549899\pi$
$38$	1038.09	0.116621
$39$	2379.26	0.250484
$40$	0	0
$41$	−9444.87	−0.877478	−0.438739	−	0.898615i	$-0.644575\pi$
$41$	−9444.87	−0.877478	−0.438739	+	0.898615i	$0.644575\pi$
$42$	597.217	0.0522407
$43$	9572.69	0.789519	0.394760	−	0.918784i	$-0.370828\pi$
$43$	9572.69	0.789519	0.394760	+	0.918784i	$0.370828\pi$
$44$	3824.84	0.297839
$45$	0	0
$46$	1381.00	0.0962276
$47$	5521.13	0.364572	0.182286	−	0.983246i	$-0.441650\pi$
$47$	5521.13	0.364572	0.182286	+	0.983246i	$0.441650\pi$
$48$	8880.62	0.556340
$49$	−5509.41	−0.327805
$50$	0	0
$51$	4585.42	0.246862
$52$	−8356.55	−0.428567
$53$	24674.9	1.20661	0.603303	−	0.797512i	$-0.293851\pi$
$53$	24674.9	1.20661	0.603303	+	0.797512i	$0.293851\pi$
$54$	455.119	0.0212393
$55$	0	0
$56$	−4221.01	−0.179865
$57$	14965.1	0.610089
$58$	−3386.32	−0.132178
$59$	31867.9	1.19186	0.595928	−	0.803038i	$-0.296784\pi$
$59$	31867.9	1.19186	0.595928	+	0.803038i	$0.296784\pi$
$60$	0	0
$61$	13479.2	0.463810	0.231905	−	0.972738i	$-0.425504\pi$
$61$	13479.2	0.463810	0.231905	+	0.972738i	$0.425504\pi$
$62$	−5493.85	−0.181509
$63$	8609.50	0.273292
$64$	−30397.6	−0.927661
$65$	0	0
$66$	−679.868	−0.0192117
$67$	−29599.5	−0.805560	−0.402780	−	0.915297i	$-0.631956\pi$
$67$	−29599.5	−0.805560	−0.402780	+	0.915297i	$0.631956\pi$
$68$	−16105.1	−0.422369
$69$	19908.6	0.503404
$70$	0	0
$71$	−20097.8	−0.473155	−0.236578	−	0.971613i	$-0.576026\pi$
$71$	−20097.8	−0.473155	−0.236578	+	0.971613i	$0.576026\pi$
$72$	−3216.69	−0.0731270
$73$	63873.3	1.40285	0.701427	−	0.712742i	$-0.252547\pi$
$73$	63873.3	1.40285	0.701427	+	0.712742i	$0.252547\pi$
$74$	−1623.27	−0.0344598
$75$	0	0
$76$	−52561.3	−1.04384
$77$	−12861.1	−0.247202
$78$	1485.38	0.0276441
$79$	−84623.1	−1.52553	−0.762766	−	0.646675i	$-0.776159\pi$
$79$	−84623.1	−1.52553	−0.762766	+	0.646675i	$0.776159\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−5896.48	−0.0968408
$83$	81972.5	1.30609	0.653045	−	0.757319i	$-0.273491\pi$
$83$	81972.5	1.30609	0.653045	+	0.757319i	$0.273491\pi$
$84$	−30238.7	−0.467590
$85$	0	0
$86$	5976.28	0.0871334
$87$	−48817.3	−0.691474
$88$	4805.18	0.0661459
$89$	1898.12	0.0254009	0.0127004	−	0.999919i	$-0.495957\pi$
$89$	1898.12	0.0254009	0.0127004	+	0.999919i	$0.495957\pi$
$90$	0	0
$91$	28099.1	0.355704
$92$	−69923.8	−0.861302
$93$	−79199.5	−0.949544
$94$	3446.87	0.0402352
$95$	0	0
$96$	16981.3	0.188059
$97$	26096.5	0.281613	0.140806	−	0.990037i	$-0.455030\pi$
$97$	26096.5	0.281613	0.140806	+	0.990037i	$0.455030\pi$
$98$	−3439.56	−0.0361774
$99$	−9801.00	−0.100504
$100$	0	0
$101$	114822.	1.12001	0.560003	−	0.828491i	$-0.310800\pi$
$101$	114822.	1.12001	0.560003	+	0.828491i	$0.310800\pi$
$102$	2862.70	0.0272443
$103$	194550.	1.80691	0.903457	−	0.428679i	$-0.141021\pi$
$103$	194550.	1.80691	0.903457	+	0.428679i	$0.141021\pi$
$104$	−10498.4	−0.0951787
$105$	0	0
$106$	15404.7	0.133164
$107$	6495.07	0.0548434	0.0274217	−	0.999624i	$-0.491270\pi$
$107$	6495.07	0.0548434	0.0274217	+	0.999624i	$0.491270\pi$
$108$	−23043.9	−0.190106
$109$	−55792.2	−0.449787	−0.224894	−	0.974383i	$-0.572203\pi$
$109$	−55792.2	−0.449787	−0.224894	+	0.974383i	$0.572203\pi$
$110$	0	0
$111$	−23401.2	−0.180273
$112$	104880.	0.790039
$113$	−105074.	−0.774107	−0.387053	−	0.922057i	$-0.626507\pi$
$113$	−105074.	−0.774107	−0.387053	+	0.922057i	$0.626507\pi$
$114$	9342.81	0.0673311
$115$	0	0
$116$	171459.	1.18308
$117$	21413.3	0.144617
$118$	19895.3	0.131536
$119$	54153.9	0.350560
$120$	0	0
$121$	14641.0	0.0909091
$122$	8415.15	0.0511873
$123$	−85003.8	−0.506612
$124$	278168.	1.62463
$125$	0	0
$126$	5374.95	0.0301612
$127$	−119862.	−0.659437	−0.329718	−	0.944079i	$-0.606954\pi$
$127$	−119862.	−0.659437	−0.329718	+	0.944079i	$0.606954\pi$
$128$	−79355.4	−0.428106
$129$	86154.2	0.455829
$130$	0	0
$131$	−213844.	−1.08873	−0.544364	−	0.838849i	$-0.683229\pi$
$131$	−213844.	−1.08873	−0.544364	+	0.838849i	$0.683229\pi$
$132$	34423.6	0.171957
$133$	176738.	0.866367
$134$	−18479.1	−0.0889037
$135$	0	0
$136$	−20233.0	−0.0938023
$137$	269753.	1.22791	0.613953	−	0.789343i	$-0.289578\pi$
$137$	269753.	1.22791	0.613953	+	0.789343i	$0.289578\pi$
$138$	12429.0	0.0555570
$139$	242482.	1.06449	0.532247	−	0.846589i	$-0.321348\pi$
$139$	242482.	1.06449	0.532247	+	0.846589i	$0.321348\pi$
$140$	0	0
$141$	49690.2	0.210486
$142$	−12547.2	−0.0522186
$143$	−31987.8	−0.130811
$144$	79925.6	0.321203
$145$	0	0
$146$	39876.4	0.154823
$147$	−49584.7	−0.189258
$148$	82190.7	0.308439
$149$	366868.	1.35377	0.676884	−	0.736090i	$-0.263330\pi$
$149$	366868.	1.35377	0.676884	+	0.736090i	$0.263330\pi$
$150$	0	0
$151$	180382.	0.643800	0.321900	−	0.946774i	$-0.395678\pi$
$151$	180382.	0.643800	0.321900	+	0.946774i	$0.395678\pi$
$152$	−66033.2	−0.231821
$153$	41268.8	0.142526
$154$	−8029.25	−0.0272818
$155$	0	0
$156$	−75208.9	−0.247433
$157$	145886.	0.472349	0.236175	−	0.971711i	$-0.424106\pi$
$157$	145886.	0.472349	0.236175	+	0.971711i	$0.424106\pi$
$158$	−52830.6	−0.168362
$159$	222074.	0.696634
$160$	0	0
$161$	235120.	0.714867
$162$	4096.07	0.0122625
$163$	147546.	0.434969	0.217484	−	0.976064i	$-0.430215\pi$
$163$	147546.	0.434969	0.217484	+	0.976064i	$0.430215\pi$
$164$	298555.	0.866790
$165$	0	0
$166$	51175.9	0.144143
$167$	−366460.	−1.01680	−0.508400	−	0.861121i	$-0.669763\pi$
$167$	−366460.	−1.01680	−0.508400	+	0.861121i	$0.669763\pi$
$168$	−37989.1	−0.103845
$169$	−301406.	−0.811773
$170$	0	0
$171$	134686.	0.352235
$172$	−302595.	−0.779903
$173$	501960.	1.27513	0.637564	−	0.770398i	$-0.279942\pi$
$173$	501960.	1.27513	0.637564	+	0.770398i	$0.279942\pi$
$174$	−30476.9	−0.0763129
$175$	0	0
$176$	−119395.	−0.290539
$177$	286811.	0.688118
$178$	1185.01	0.00280331
$179$	475285.	1.10872	0.554360	−	0.832277i	$-0.312963\pi$
$179$	475285.	1.10872	0.554360	+	0.832277i	$0.312963\pi$
$180$	0	0
$181$	492648.	1.11774	0.558869	−	0.829256i	$-0.311235\pi$
$181$	492648.	1.11774	0.558869	+	0.829256i	$0.311235\pi$
$182$	17542.4	0.0392564
$183$	121313.	0.267781
$184$	−87845.8	−0.191283
$185$	0	0
$186$	−49444.7	−0.104794
$187$	−61648.4	−0.128919
$188$	−174524.	−0.360132
$189$	77485.5	0.157785
$190$	0	0
$191$	−297473.	−0.590016	−0.295008	−	0.955495i	$-0.595322\pi$
$191$	−297473.	−0.590016	−0.295008	+	0.955495i	$0.595322\pi$
$192$	−273578.	−0.535585
$193$	−397116.	−0.767404	−0.383702	−	0.923457i	$-0.625351\pi$
$193$	−397116.	−0.767404	−0.383702	+	0.923457i	$0.625351\pi$
$194$	16292.2	0.0310795
$195$	0	0
$196$	174154.	0.323812
$197$	283439.	0.520348	0.260174	−	0.965562i	$-0.416220\pi$
$197$	283439.	0.520348	0.260174	+	0.965562i	$0.416220\pi$
$198$	−6118.82	−0.0110919
$199$	442088.	0.791364	0.395682	−	0.918388i	$-0.370508\pi$
$199$	442088.	0.791364	0.395682	+	0.918388i	$0.370508\pi$
$200$	0	0
$201$	−266396.	−0.465090
$202$	71683.8	0.123607
$203$	−576533.	−0.981938
$204$	−144946.	−0.243855
$205$	0	0
$206$	121458.	0.199416
$207$	179177.	0.290641
$208$	260855.	0.418063
$209$	−201198.	−0.318609
$210$	0	0
$211$	424099.	0.655785	0.327892	−	0.944715i	$-0.393662\pi$
$211$	424099.	0.655785	0.327892	+	0.944715i	$0.393662\pi$
$212$	−779979.	−1.19191
$213$	−180881.	−0.273176
$214$	4054.91	0.00605266
$215$	0	0
$216$	−28950.2	−0.0422199
$217$	−935347.	−1.34841
$218$	−34831.4	−0.0496397
$219$	574860.	0.809938
$220$	0	0
$221$	134690.	0.185505
$222$	−14609.5	−0.0198954
$223$	1.02158e6	1.37566	0.687830	−	0.725872i	$-0.258563\pi$
$223$	1.02158e6	1.37566	0.687830	+	0.725872i	$0.258563\pi$
$224$	200550.	0.267056
$225$	0	0
$226$	−65598.5	−0.0854324
$227$	−1.15245e6	−1.48442	−0.742211	−	0.670166i	$-0.766223\pi$
$227$	−1.15245e6	−1.48442	−0.742211	+	0.670166i	$0.766223\pi$
$228$	−473052.	−0.602659
$229$	1.17552e6	1.48130	0.740650	−	0.671891i	$-0.234518\pi$
$229$	1.17552e6	1.48130	0.740650	+	0.671891i	$0.234518\pi$
$230$	0	0
$231$	−115750.	−0.142722
$232$	215405.	0.262746
$233$	613235.	0.740009	0.370004	−	0.929030i	$-0.379356\pi$
$233$	613235.	0.740009	0.370004	+	0.929030i	$0.379356\pi$
$234$	13368.5	0.0159603
$235$	0	0
$236$	−1.00735e6	−1.17734
$237$	−761608.	−0.880766
$238$	33808.5	0.0386887
$239$	1.11292e6	1.26029	0.630146	−	0.776477i	$-0.282995\pi$
$239$	1.11292e6	1.26029	0.630146	+	0.776477i	$0.282995\pi$
$240$	0	0
$241$	277378.	0.307630	0.153815	−	0.988100i	$-0.450844\pi$
$241$	277378.	0.307630	0.153815	+	0.988100i	$0.450844\pi$
$242$	9140.45	0.0100330
$243$	59049.0	0.0641500
$244$	−426082.	−0.458161
$245$	0	0
$246$	−53068.3	−0.0559110
$247$	439579.	0.458453
$248$	349465.	0.360807
$249$	737753.	0.754071
$250$	0	0
$251$	826834.	0.828388	0.414194	−	0.910189i	$-0.364063\pi$
$251$	826834.	0.828388	0.414194	+	0.910189i	$0.364063\pi$
$252$	−272148.	−0.269963
$253$	−267659.	−0.262894
$254$	−74830.6	−0.0727772
$255$	0	0
$256$	923181.	0.880414
$257$	−591626.	−0.558746	−0.279373	−	0.960183i	$-0.590127\pi$
$257$	−591626.	−0.558746	−0.279373	+	0.960183i	$0.590127\pi$
$258$	53786.5	0.0503065
$259$	−276368.	−0.255999
$260$	0	0
$261$	−439356.	−0.399223
$262$	−133504.	−0.120155
$263$	1.48336e6	1.32238	0.661192	−	0.750216i	$-0.270051\pi$
$263$	1.48336e6	1.32238	0.661192	+	0.750216i	$0.270051\pi$
$264$	43246.6	0.0381893
$265$	0	0
$266$	110339.	0.0956145
$267$	17083.1	0.0146652
$268$	935648.	0.795748
$269$	676733.	0.570213	0.285106	−	0.958496i	$-0.407971\pi$
$269$	676733.	0.570213	0.285106	+	0.958496i	$0.407971\pi$
$270$	0	0
$271$	33731.6	0.0279006	0.0139503	−	0.999903i	$-0.495559\pi$
$271$	33731.6	0.0279006	0.0139503	+	0.999903i	$0.495559\pi$
$272$	502733.	0.412017
$273$	252892.	0.205366
$274$	168408.	0.135515
$275$	0	0
$276$	−629314.	−0.497273
$277$	−1.51556e6	−1.18679	−0.593394	−	0.804912i	$-0.702212\pi$
$277$	−1.51556e6	−1.18679	−0.593394	+	0.804912i	$0.702212\pi$
$278$	151383.	0.117480
$279$	−712796.	−0.548220
$280$	0	0
$281$	−1.97949e6	−1.49551	−0.747753	−	0.663978i	$-0.768867\pi$
$281$	−1.97949e6	−1.49551	−0.747753	+	0.663978i	$0.768867\pi$
$282$	31021.9	0.0232298
$283$	691882.	0.513530	0.256765	−	0.966474i	$-0.417343\pi$
$283$	691882.	0.513530	0.256765	+	0.966474i	$0.417343\pi$
$284$	635298.	0.467392
$285$	0	0
$286$	−19970.2	−0.0144366
$287$	−1.00390e6	−0.719422
$288$	152832.	0.108576
$289$	−1.16028e6	−0.817178
$290$	0	0
$291$	234868.	0.162589
$292$	−2.01905e6	−1.38577
$293$	−1.72110e6	−1.17122	−0.585608	−	0.810594i	$-0.699144\pi$
$293$	−1.72110e6	−1.17122	−0.585608	+	0.810594i	$0.699144\pi$
$294$	−30956.0	−0.0208870
$295$	0	0
$296$	103257.	0.0684999
$297$	−88209.0	−0.0580259
$298$	229038.	0.149405
$299$	584785.	0.378284
$300$	0	0
$301$	1.01748e6	0.647307
$302$	112614.	0.0710515
$303$	1.03340e6	0.646636
$304$	1.64074e6	1.01825
$305$	0	0
$306$	25764.3	0.0157295
$307$	929998.	0.563165	0.281583	−	0.959537i	$-0.409141\pi$
$307$	929998.	0.563165	0.281583	+	0.959537i	$0.409141\pi$
$308$	406543.	0.244191
$309$	1.75095e6	1.04322
$310$	0	0
$311$	−293753.	−0.172219	−0.0861096	−	0.996286i	$-0.527444\pi$
$311$	−293753.	−0.172219	−0.0861096	+	0.996286i	$0.527444\pi$
$312$	−94485.6	−0.0549515
$313$	665923.	0.384205	0.192102	−	0.981375i	$-0.438469\pi$
$313$	665923.	0.384205	0.192102	+	0.981375i	$0.438469\pi$
$314$	91077.1	0.0521297
$315$	0	0
$316$	2.67496e6	1.50695
$317$	2.04535e6	1.14320	0.571598	−	0.820534i	$-0.306324\pi$
$317$	2.04535e6	1.14320	0.571598	+	0.820534i	$0.306324\pi$
$318$	138642.	0.0768824
$319$	656322.	0.361110
$320$	0	0
$321$	58455.7	0.0316639
$322$	146787.	0.0788946
$323$	847178.	0.451823
$324$	−207395.	−0.109758
$325$	0	0
$326$	92113.6	0.0480043
$327$	−502130.	−0.259685
$328$	375076.	0.192502
$329$	586842.	0.298904
$330$	0	0
$331$	−2.62633e6	−1.31759	−0.658793	−	0.752324i	$-0.728933\pi$
$331$	−2.62633e6	−1.31759	−0.658793	+	0.752324i	$0.728933\pi$
$332$	−2.59117e6	−1.29018
$333$	−210611.	−0.104081
$334$	−228783.	−0.112217
$335$	0	0
$336$	943922.	0.456129
$337$	966186.	0.463432	0.231716	−	0.972783i	$-0.425566\pi$
$337$	966186.	0.463432	0.231716	+	0.972783i	$0.425566\pi$
$338$	−188169.	−0.0895894
$339$	−945670.	−0.446931
$340$	0	0
$341$	1.06479e6	0.495883
$342$	84085.3	0.0388736
$343$	−2.37201e6	−1.08863
$344$	−380152.	−0.173206
$345$	0	0
$346$	313376.	0.140726
$347$	1.94573e6	0.867479	0.433740	−	0.901038i	$-0.357194\pi$
$347$	1.94573e6	0.867479	0.433740	+	0.901038i	$0.357194\pi$
$348$	1.54313e6	0.683052
$349$	1.24936e6	0.549065	0.274532	−	0.961578i	$-0.411477\pi$
$349$	1.24936e6	0.549065	0.274532	+	0.961578i	$0.411477\pi$
$350$	0	0
$351$	192720.	0.0834947
$352$	−228305.	−0.0982105
$353$	4.13513e6	1.76625	0.883126	−	0.469137i	$-0.155435\pi$
$353$	4.13513e6	1.76625	0.883126	+	0.469137i	$0.155435\pi$
$354$	179058.	0.0759425
$355$	0	0
$356$	−60000.1	−0.0250915
$357$	487385.	0.202396
$358$	296723.	0.122361
$359$	−2.34776e6	−0.961431	−0.480716	−	0.876877i	$-0.659623\pi$
$359$	−2.34776e6	−0.961431	−0.480716	+	0.876877i	$0.659623\pi$
$360$	0	0
$361$	288781.	0.116627
$362$	307563.	0.123356
$363$	131769.	0.0524864
$364$	−888218.	−0.351371
$365$	0	0
$366$	75736.4	0.0295530
$367$	−1.61235e6	−0.624878	−0.312439	−	0.949938i	$-0.601146\pi$
$367$	−1.61235e6	−0.624878	−0.312439	+	0.949938i	$0.601146\pi$
$368$	2.18272e6	0.840192
$369$	−765034.	−0.292493
$370$	0	0
$371$	2.62270e6	0.989266
$372$	2.50352e6	0.937979
$373$	541048.	0.201355	0.100678	−	0.994919i	$-0.467899\pi$
$373$	541048.	0.201355	0.100678	+	0.994919i	$0.467899\pi$
$374$	−38487.4	−0.0142279
$375$	0	0
$376$	−219256.	−0.0799803
$377$	−1.43394e6	−0.519610
$378$	48374.6	0.0174136
$379$	3.10147e6	1.10910	0.554549	−	0.832151i	$-0.312891\pi$
$379$	3.10147e6	1.10910	0.554549	+	0.832151i	$0.312891\pi$
$380$	0	0
$381$	−1.07876e6	−0.380726
$382$	−185714.	−0.0651157
$383$	−729397.	−0.254078	−0.127039	−	0.991898i	$-0.540547\pi$
$383$	−729397.	−0.254078	−0.127039	+	0.991898i	$0.540547\pi$
$384$	−714199.	−0.247167
$385$	0	0
$386$	−247922.	−0.0846927
$387$	775388.	0.263173
$388$	−824916.	−0.278183
$389$	5.35650e6	1.79476	0.897382	−	0.441255i	$-0.145467\pi$
$389$	5.35650e6	1.79476	0.897382	+	0.441255i	$0.145467\pi$
$390$	0	0
$391$	1.12703e6	0.372814
$392$	218791.	0.0719141
$393$	−1.92460e6	−0.628578
$394$	176952.	0.0574269
$395$	0	0
$396$	309812.	0.0992797
$397$	−1.13420e6	−0.361170	−0.180585	−	0.983559i	$-0.557799\pi$
$397$	−1.13420e6	−0.361170	−0.180585	+	0.983559i	$0.557799\pi$
$398$	275998.	0.0873370
$399$	1.59065e6	0.500197
$400$	0	0
$401$	−626701.	−0.194625	−0.0973127	−	0.995254i	$-0.531025\pi$
$401$	−626701.	−0.194625	−0.0973127	+	0.995254i	$0.531025\pi$
$402$	−166312.	−0.0513286
$403$	−2.32637e6	−0.713537
$404$	−3.62954e6	−1.10636
$405$	0	0
$406$	−359933.	−0.108369
$407$	314616.	0.0941444
$408$	−182097.	−0.0541568
$409$	−471719.	−0.139436	−0.0697180	−	0.997567i	$-0.522210\pi$
$409$	−471719.	−0.139436	−0.0697180	+	0.997567i	$0.522210\pi$
$410$	0	0
$411$	2.42778e6	0.708932
$412$	−6.14976e6	−1.78491
$413$	3.38725e6	0.977173
$414$	111861.	0.0320759
$415$	0	0
$416$	498802.	0.141317
$417$	2.18234e6	0.614586
$418$	−125609.	−0.0351625
$419$	3.13616e6	0.872696	0.436348	−	0.899778i	$-0.356272\pi$
$419$	3.13616e6	0.872696	0.436348	+	0.899778i	$0.356272\pi$
$420$	0	0
$421$	2.61937e6	0.720265	0.360132	−	0.932901i	$-0.382732\pi$
$421$	2.61937e6	0.720265	0.360132	+	0.932901i	$0.382732\pi$
$422$	264767.	0.0723741
$423$	447212.	0.121524
$424$	−979894.	−0.264706
$425$	0	0
$426$	−112925.	−0.0301484
$427$	1.43271e6	0.380267
$428$	−205311.	−0.0541754
$429$	−287890.	−0.0755238
$430$	0	0
$431$	6.05506e6	1.57009	0.785046	−	0.619438i	$-0.212639\pi$
$431$	6.05506e6	1.57009	0.785046	+	0.619438i	$0.212639\pi$
$432$	719330.	0.185447
$433$	−1.93261e6	−0.495365	−0.247683	−	0.968841i	$-0.579669\pi$
$433$	−1.93261e6	−0.495365	−0.247683	+	0.968841i	$0.579669\pi$
$434$	−583942.	−0.148815
$435$	0	0
$436$	1.76360e6	0.444309
$437$	3.67820e6	0.921365
$438$	358888.	0.0893868
$439$	−3.88023e6	−0.960939	−0.480469	−	0.877011i	$-0.659534\pi$
$439$	−3.88023e6	−0.960939	−0.480469	+	0.877011i	$0.659534\pi$
$440$	0	0
$441$	−446263.	−0.109268
$442$	84087.7	0.0204728
$443$	−6.92252e6	−1.67593	−0.837963	−	0.545727i	$-0.816254\pi$
$443$	−6.92252e6	−1.67593	−0.837963	+	0.545727i	$0.816254\pi$
$444$	739717.	0.178077
$445$	0	0
$446$	637779.	0.151821
$447$	3.30181e6	0.781598
$448$	−3.23096e6	−0.760566
$449$	3.31516e6	0.776048	0.388024	−	0.921649i	$-0.373158\pi$
$449$	3.31516e6	0.776048	0.388024	+	0.921649i	$0.373158\pi$
$450$	0	0
$451$	1.14283e6	0.264570
$452$	3.32143e6	0.764678
$453$	1.62344e6	0.371698
$454$	−719480.	−0.163825
$455$	0	0
$456$	−594299.	−0.133842
$457$	−3.40904e6	−0.763557	−0.381778	−	0.924254i	$-0.624688\pi$
$457$	−3.40904e6	−0.763557	−0.381778	+	0.924254i	$0.624688\pi$
$458$	733886.	0.163480
$459$	371419.	0.0822872
$460$	0	0
$461$	−2.54110e6	−0.556891	−0.278445	−	0.960452i	$-0.589819\pi$
$461$	−2.54110e6	−0.556891	−0.278445	+	0.960452i	$0.589819\pi$
$462$	−72263.3	−0.0157512
$463$	−5.00975e6	−1.08608	−0.543042	−	0.839705i	$-0.682728\pi$
$463$	−5.00975e6	−1.08608	−0.543042	+	0.839705i	$0.682728\pi$
$464$	−5.35220e6	−1.15408
$465$	0	0
$466$	382846.	0.0816693
$467$	4.45513e6	0.945297	0.472649	−	0.881251i	$-0.343298\pi$
$467$	4.45513e6	0.945297	0.472649	+	0.881251i	$0.343298\pi$
$468$	−676880.	−0.142856
$469$	−3.14614e6	−0.660459
$470$	0	0
$471$	1.31297e6	0.272711
$472$	−1.26555e6	−0.261471
$473$	−1.15830e6	−0.238049
$474$	−475476.	−0.0972036
$475$	0	0
$476$	−1.71182e6	−0.346290
$477$	1.99867e6	0.402202
$478$	694804.	0.139089
$479$	−1.72313e6	−0.343147	−0.171573	−	0.985171i	$-0.554885\pi$
$479$	−1.72313e6	−0.343147	−0.171573	+	0.985171i	$0.554885\pi$
$480$	0	0
$481$	−687376.	−0.135466
$482$	173168.	0.0339509
$483$	2.11608e6	0.412729
$484$	−462806.	−0.0898018
$485$	0	0
$486$	36864.6	0.00707976
$487$	−7.34645e6	−1.40364	−0.701819	−	0.712356i	$-0.747628\pi$
$487$	−7.34645e6	−1.40364	−0.701819	+	0.712356i	$0.747628\pi$
$488$	−535290.	−0.101751
$489$	1.32791e6	0.251129
$490$	0	0
$491$	−1.57312e6	−0.294481	−0.147240	−	0.989101i	$-0.547039\pi$
$491$	−1.57312e6	−0.294481	−0.147240	+	0.989101i	$0.547039\pi$
$492$	2.68699e6	0.500442
$493$	−2.76355e6	−0.512095
$494$	274432.	0.0505961
$495$	0	0
$496$	−8.68322e6	−1.58481
$497$	−2.13620e6	−0.387928
$498$	460583.	0.0832213
$499$	−5.80283e6	−1.04325	−0.521625	−	0.853175i	$-0.674674\pi$
$499$	−5.80283e6	−1.04325	−0.521625	+	0.853175i	$0.674674\pi$
$500$	0	0
$501$	−3.29814e6	−0.587050
$502$	516197.	0.0914231
$503$	4.44523e6	0.783384	0.391692	−	0.920096i	$-0.371890\pi$
$503$	4.44523e6	0.783384	0.391692	+	0.920096i	$0.371890\pi$
$504$	−341902.	−0.0599550
$505$	0	0
$506$	−167101.	−0.0290137
$507$	−2.71265e6	−0.468677
$508$	3.78888e6	0.651405
$509$	−5.34223e6	−0.913962	−0.456981	−	0.889476i	$-0.651069\pi$
$509$	−5.34223e6	−0.913962	−0.456981	+	0.889476i	$0.651069\pi$
$510$	0	0
$511$	6.78910e6	1.15016
$512$	3.11572e6	0.525271
$513$	1.21218e6	0.203363
$514$	−369355.	−0.0616647
$515$	0	0
$516$	−2.72335e6	−0.450277
$517$	−668057.	−0.109923
$518$	−172538.	−0.0282527
$519$	4.51764e6	0.736195
$520$	0	0
$521$	−1.08913e7	−1.75786	−0.878930	−	0.476951i	$-0.841742\pi$
$521$	−1.08913e7	−1.75786	−0.878930	+	0.476951i	$0.841742\pi$
$522$	−274292.	−0.0440592
$523$	−2.27063e6	−0.362988	−0.181494	−	0.983392i	$-0.558093\pi$
$523$	−2.27063e6	−0.362988	−0.181494	+	0.983392i	$0.558093\pi$
$524$	6.75967e6	1.07547
$525$	0	0
$526$	926071.	0.145942
$527$	−4.48349e6	−0.703218
$528$	−1.07455e6	−0.167743
$529$	−1.54313e6	−0.239752
$530$	0	0
$531$	2.58130e6	0.397285
$532$	−5.58674e6	−0.855815
$533$	−2.49686e6	−0.380695
$534$	10665.1	0.00161849
$535$	0	0
$536$	1.17546e6	0.176725
$537$	4.27757e6	0.640120
$538$	422488.	0.0629302
$539$	666639.	0.0988369
$540$	0	0
$541$	−7.70067e6	−1.13119	−0.565595	−	0.824683i	$-0.691353\pi$
$541$	−7.70067e6	−1.13119	−0.565595	+	0.824683i	$0.691353\pi$
$542$	21058.8	0.00307919
$543$	4.43383e6	0.645326
$544$	961315.	0.139274
$545$	0	0
$546$	157882.	0.0226647
$547$	4.26005e6	0.608760	0.304380	−	0.952551i	$-0.401551\pi$
$547$	4.26005e6	0.608760	0.304380	+	0.952551i	$0.401551\pi$
$548$	−8.52696e6	−1.21295
$549$	1.09182e6	0.154603
$550$	0	0
$551$	−9.01923e6	−1.26558
$552$	−790613.	−0.110437
$553$	−8.99460e6	−1.25075
$554$	−946170.	−0.130977
$555$	0	0
$556$	−7.66493e6	−1.05153
$557$	2.72296e6	0.371881	0.185940	−	0.982561i	$-0.440467\pi$
$557$	2.72296e6	0.371881	0.185940	+	0.982561i	$0.440467\pi$
$558$	−445002.	−0.0605029
$559$	2.53066e6	0.342534
$560$	0	0
$561$	−554836.	−0.0744316
$562$	−1.23581e6	−0.165048
$563$	−3.37881e6	−0.449255	−0.224627	−	0.974445i	$-0.572116\pi$
$563$	−3.37881e6	−0.449255	−0.224627	+	0.974445i	$0.572116\pi$
$564$	−1.57072e6	−0.207922
$565$	0	0
$566$	431946.	0.0566746
$567$	697369.	0.0910972
$568$	798130.	0.103801
$569$	−1.98049e6	−0.256444	−0.128222	−	0.991746i	$-0.540927\pi$
$569$	−1.98049e6	−0.256444	−0.128222	+	0.991746i	$0.540927\pi$
$570$	0	0
$571$	−1.02513e7	−1.31580	−0.657898	−	0.753107i	$-0.728554\pi$
$571$	−1.02513e7	−1.31580	−0.657898	+	0.753107i	$0.728554\pi$
$572$	1.01114e6	0.129218
$573$	−2.67725e6	−0.340646
$574$	−626737.	−0.0793973
$575$	0	0
$576$	−2.46220e6	−0.309220
$577$	1.56178e7	1.95291	0.976453	−	0.215729i	$-0.0692127\pi$
$577$	1.56178e7	1.95291	0.976453	+	0.215729i	$0.0692127\pi$
$578$	−724366.	−0.0901859
$579$	−3.57404e6	−0.443061
$580$	0	0
$581$	8.71287e6	1.07083
$582$	146630.	0.0179438
$583$	−2.98566e6	−0.363805
$584$	−2.53655e6	−0.307759
$585$	0	0
$586$	−1.07449e6	−0.129259
$587$	9.39734e6	1.12567	0.562833	−	0.826570i	$-0.309711\pi$
$587$	9.39734e6	1.12567	0.562833	+	0.826570i	$0.309711\pi$
$588$	1.56739e6	0.186953
$589$	−1.46325e7	−1.73792
$590$	0	0
$591$	2.55095e6	0.300423
$592$	−2.56564e6	−0.300879
$593$	1.28737e7	1.50337	0.751684	−	0.659523i	$-0.229242\pi$
$593$	1.28737e7	1.50337	0.751684	+	0.659523i	$0.229242\pi$
$594$	−55069.3	−0.00640389
$595$	0	0
$596$	−1.15968e7	−1.33728
$597$	3.97880e6	0.456894
$598$	365084.	0.0417484
$599$	−4.74896e6	−0.540793	−0.270397	−	0.962749i	$-0.587155\pi$
$599$	−4.74896e6	−0.540793	−0.270397	+	0.962749i	$0.587155\pi$
$600$	0	0
$601$	1.93066e6	0.218032	0.109016	−	0.994040i	$-0.465230\pi$
$601$	1.93066e6	0.218032	0.109016	+	0.994040i	$0.465230\pi$
$602$	635219.	0.0714385
$603$	−2.39756e6	−0.268520
$604$	−5.70192e6	−0.635959
$605$	0	0
$606$	645154.	0.0713644
$607$	−1.21875e7	−1.34259	−0.671293	−	0.741192i	$-0.734261\pi$
$607$	−1.21875e7	−1.34259	−0.671293	+	0.741192i	$0.734261\pi$
$608$	3.13738e6	0.344198
$609$	−5.18880e6	−0.566922
$610$	0	0
$611$	1.45958e6	0.158170
$612$	−1.30452e6	−0.140790
$613$	−1.32187e7	−1.42082	−0.710410	−	0.703788i	$-0.751490\pi$
$613$	−1.32187e7	−1.42082	−0.710410	+	0.703788i	$0.751490\pi$
$614$	580602.	0.0621524
$615$	0	0
$616$	510743.	0.0542314
$617$	1.20057e7	1.26962	0.634811	−	0.772667i	$-0.281078\pi$
$617$	1.20057e7	1.26962	0.634811	+	0.772667i	$0.281078\pi$
$618$	1.09313e6	0.115133
$619$	−39871.4	−0.00418249	−0.00209124	−	0.999998i	$-0.500666\pi$
$619$	−39871.4	−0.00418249	−0.00209124	+	0.999998i	$0.500666\pi$
$620$	0	0
$621$	1.61259e6	0.167801
$622$	−183392.	−0.0190066
$623$	201752.	0.0208256
$624$	2.34770e6	0.241369
$625$	0	0
$626$	415739.	0.0424019
$627$	−1.81078e6	−0.183949
$628$	−4.61148e6	−0.466596
$629$	−1.32474e6	−0.133507
$630$	0	0
$631$	4.92201e6	0.492118	0.246059	−	0.969255i	$-0.420864\pi$
$631$	4.92201e6	0.492118	0.246059	+	0.969255i	$0.420864\pi$
$632$	3.36057e6	0.334673
$633$	3.81689e6	0.378617
$634$	1.27693e6	0.126166
$635$	0	0
$636$	−7.01981e6	−0.688149
$637$	−1.45648e6	−0.142218
$638$	409745.	0.0398531
$639$	−1.62793e6	−0.157718
$640$	0	0
$641$	−7.42514e6	−0.713772	−0.356886	−	0.934148i	$-0.616162\pi$
$641$	−7.42514e6	−0.713772	−0.356886	+	0.934148i	$0.616162\pi$
$642$	36494.2	0.00349451
$643$	−9.55282e6	−0.911180	−0.455590	−	0.890190i	$-0.650572\pi$
$643$	−9.55282e6	−0.911180	−0.455590	+	0.890190i	$0.650572\pi$
$644$	−7.43221e6	−0.706160
$645$	0	0
$646$	528898.	0.0498644
$647$	8.28391e6	0.777992	0.388996	−	0.921240i	$-0.372822\pi$
$647$	8.28391e6	0.777992	0.388996	+	0.921240i	$0.372822\pi$
$648$	−260552.	−0.0243757
$649$	−3.85602e6	−0.359358
$650$	0	0
$651$	−8.41812e6	−0.778508
$652$	−4.66396e6	−0.429671
$653$	2.04353e6	0.187542	0.0937711	−	0.995594i	$-0.470108\pi$
$653$	2.04353e6	0.187542	0.0937711	+	0.995594i	$0.470108\pi$
$654$	−313482.	−0.0286595
$655$	0	0
$656$	−9.31958e6	−0.845545
$657$	5.17374e6	0.467618
$658$	366369.	0.0329878
$659$	−1.43938e7	−1.29110	−0.645552	−	0.763716i	$-0.723373\pi$
$659$	−1.43938e7	−1.29110	−0.645552	+	0.763716i	$0.723373\pi$
$660$	0	0
$661$	1.39764e7	1.24421	0.622103	−	0.782936i	$-0.286279\pi$
$661$	1.39764e7	1.24421	0.622103	+	0.782936i	$0.286279\pi$
$662$	−1.63963e6	−0.145412
$663$	1.21221e6	0.107101
$664$	−3.25531e6	−0.286531
$665$	0	0
$666$	−131485.	−0.0114866
$667$	−1.19985e7	−1.04427
$668$	1.15839e7	1.00442
$669$	9.19424e6	0.794238
$670$	0	0
$671$	−1.63099e6	−0.139844
$672$	1.80495e6	0.154185
$673$	4.36237e6	0.371266	0.185633	−	0.982619i	$-0.440567\pi$
$673$	4.36237e6	0.371266	0.185633	+	0.982619i	$0.440567\pi$
$674$	603195.	0.0511456
$675$	0	0
$676$	9.52751e6	0.801886
$677$	2.77129e6	0.232386	0.116193	−	0.993227i	$-0.462931\pi$
$677$	2.77129e6	0.232386	0.116193	+	0.993227i	$0.462931\pi$
$678$	−590386.	−0.0493244
$679$	2.77380e6	0.230887
$680$	0	0
$681$	−1.03720e7	−0.857031
$682$	664756.	0.0547270
$683$	−1.27204e7	−1.04340	−0.521698	−	0.853130i	$-0.674701\pi$
$683$	−1.27204e7	−1.04340	−0.521698	+	0.853130i	$0.674701\pi$
$684$	−4.25746e6	−0.347945
$685$	0	0
$686$	−1.48086e6	−0.120145
$687$	1.05797e7	0.855229
$688$	9.44571e6	0.760788
$689$	6.52310e6	0.523487
$690$	0	0
$691$	2.06818e7	1.64776	0.823879	−	0.566766i	$-0.191806\pi$
$691$	2.06818e7	1.64776	0.823879	+	0.566766i	$0.191806\pi$
$692$	−1.58671e7	−1.25960
$693$	−1.04175e6	−0.0824006
$694$	1.21473e6	0.0957373
$695$	0	0
$696$	1.93864e6	0.151696
$697$	−4.81207e6	−0.375189
$698$	779981.	0.0605962
$699$	5.51911e6	0.427244
$700$	0	0
$701$	1.40652e7	1.08106	0.540531	−	0.841324i	$-0.318224\pi$
$701$	1.40652e7	1.08106	0.540531	+	0.841324i	$0.318224\pi$
$702$	120316.	0.00921469
$703$	−4.32348e6	−0.329948
$704$	3.67811e6	0.279700
$705$	0	0
$706$	2.58158e6	0.194928
$707$	1.22044e7	0.918265
$708$	−9.06618e6	−0.679737
$709$	−2.36738e7	−1.76869	−0.884345	−	0.466833i	$-0.845395\pi$
$709$	−2.36738e7	−1.76869	−0.884345	+	0.466833i	$0.845395\pi$
$710$	0	0
$711$	−6.85447e6	−0.508510
$712$	−75378.6	−0.00557247
$713$	−1.94660e7	−1.43401
$714$	304277.	0.0223369
$715$	0	0
$716$	−1.50239e7	−1.09522
$717$	1.00163e7	0.727629
$718$	−1.46572e6	−0.106106
$719$	1.03736e7	0.748354	0.374177	−	0.927357i	$-0.377925\pi$
$719$	1.03736e7	0.748354	0.374177	+	0.927357i	$0.377925\pi$
$720$	0	0
$721$	2.06787e7	1.48144
$722$	180287.	0.0128713
$723$	2.49640e6	0.177610
$724$	−1.55727e7	−1.10412
$725$	0	0
$726$	82264.1	0.00579253
$727$	−2.09455e7	−1.46979	−0.734894	−	0.678182i	$-0.762768\pi$
$727$	−2.09455e7	−1.46979	−0.734894	+	0.678182i	$0.762768\pi$
$728$	−1.11588e6	−0.0780347
$729$	531441.	0.0370370
$730$	0	0
$731$	4.87720e6	0.337580
$732$	−3.83474e6	−0.264520
$733$	−1.15758e7	−0.795780	−0.397890	−	0.917433i	$-0.630257\pi$
$733$	−1.15758e7	−0.795780	−0.397890	+	0.917433i	$0.630257\pi$
$734$	−1.00660e6	−0.0689631
$735$	0	0
$736$	4.17375e6	0.284009
$737$	3.58154e6	0.242885
$738$	−477615.	−0.0322803
$739$	−1.84303e7	−1.24143	−0.620713	−	0.784038i	$-0.713157\pi$
$739$	−1.84303e7	−1.24143	−0.620713	+	0.784038i	$0.713157\pi$
$740$	0	0
$741$	3.95621e6	0.264688
$742$	1.63736e6	0.109178
$743$	−1.27772e7	−0.849108	−0.424554	−	0.905403i	$-0.639569\pi$
$743$	−1.27772e7	−0.849108	−0.424554	+	0.905403i	$0.639569\pi$
$744$	3.14519e6	0.208312
$745$	0	0
$746$	337779.	0.0222221
$747$	6.63977e6	0.435363
$748$	1.94872e6	0.127349
$749$	690362.	0.0449648
$750$	0	0
$751$	1.85955e7	1.20311	0.601557	−	0.798830i	$-0.294547\pi$
$751$	1.85955e7	1.20311	0.601557	+	0.798830i	$0.294547\pi$
$752$	5.44790e6	0.351305
$753$	7.44151e6	0.478270
$754$	−895215.	−0.0573455
$755$	0	0
$756$	−2.44933e6	−0.155863
$757$	1.07465e7	0.681595	0.340797	−	0.940137i	$-0.389303\pi$
$757$	1.07465e7	0.681595	0.340797	+	0.940137i	$0.389303\pi$
$758$	1.93627e6	0.122403
$759$	−2.40893e6	−0.151782
$760$	0	0
$761$	−1.04966e7	−0.657031	−0.328515	−	0.944499i	$-0.606548\pi$
$761$	−1.04966e7	−0.657031	−0.328515	+	0.944499i	$0.606548\pi$
$762$	−673476.	−0.0420179
$763$	−5.93016e6	−0.368769
$764$	9.40318e6	0.582829
$765$	0	0
$766$	−455367.	−0.0280407
$767$	8.42467e6	0.517088
$768$	8.30862e6	0.508307
$769$	−1.94421e7	−1.18557	−0.592787	−	0.805360i	$-0.701972\pi$
$769$	−1.94421e7	−1.18557	−0.592787	+	0.805360i	$0.701972\pi$
$770$	0	0
$771$	−5.32463e6	−0.322592
$772$	1.25529e7	0.758057
$773$	1.32960e7	0.800338	0.400169	−	0.916441i	$-0.368951\pi$
$773$	1.32960e7	0.800338	0.400169	+	0.916441i	$0.368951\pi$
$774$	484079.	0.0290445
$775$	0	0
$776$	−1.03635e6	−0.0617805
$777$	−2.48731e6	−0.147801
$778$	3.34409e6	0.198075
$779$	−1.57049e7	−0.927236
$780$	0	0
$781$	2.43184e6	0.142662
$782$	703608.	0.0411447
$783$	−3.95420e6	−0.230491
$784$	−5.43633e6	−0.315876
$785$	0	0
$786$	−1.20154e6	−0.0693715
$787$	−7.22388e6	−0.415752	−0.207876	−	0.978155i	$-0.566655\pi$
$787$	−7.22388e6	−0.415752	−0.207876	+	0.978155i	$0.566655\pi$
$788$	−8.95957e6	−0.514010
$789$	1.33503e7	0.763479
$790$	0	0
$791$	−1.11684e7	−0.634671
$792$	389219.	0.0220486
$793$	3.56340e6	0.201225
$794$	−708085.	−0.0398597
$795$	0	0
$796$	−1.39745e7	−0.781726
$797$	−1.01501e7	−0.566012	−0.283006	−	0.959118i	$-0.591332\pi$
$797$	−1.01501e7	−0.566012	−0.283006	+	0.959118i	$0.591332\pi$
$798$	993048.	0.0552031
$799$	2.81297e6	0.155883
$800$	0	0
$801$	153748.	0.00846696
$802$	−391253.	−0.0214794
$803$	−7.72867e6	−0.422976
$804$	8.42084e6	0.459425
$805$	0	0
$806$	−1.45237e6	−0.0787478
$807$	6.09060e6	0.329213
$808$	−4.55982e6	−0.245708
$809$	1.07717e7	0.578644	0.289322	−	0.957232i	$-0.406570\pi$
$809$	1.07717e7	0.578644	0.289322	+	0.957232i	$0.406570\pi$
$810$	0	0
$811$	2.92706e7	1.56271	0.781357	−	0.624084i	$-0.214528\pi$
$811$	2.92706e7	1.56271	0.781357	+	0.624084i	$0.214528\pi$
$812$	1.82243e7	0.969978
$813$	303585.	0.0161084
$814$	196416.	0.0103900
$815$	0	0
$816$	4.52459e6	0.237878
$817$	1.59174e7	0.834290
$818$	−294496.	−0.0153885
$819$	2.27602e6	0.118568
$820$	0	0
$821$	1.75303e7	0.907677	0.453838	−	0.891084i	$-0.350054\pi$
$821$	1.75303e7	0.907677	0.453838	+	0.891084i	$0.350054\pi$
$822$	1.51567e6	0.0782396
$823$	3.94210e6	0.202875	0.101437	−	0.994842i	$-0.467656\pi$
$823$	3.94210e6	0.202875	0.101437	+	0.994842i	$0.467656\pi$
$824$	−7.72600e6	−0.396403
$825$	0	0
$826$	2.11467e6	0.107843
$827$	1.55928e7	0.792794	0.396397	−	0.918079i	$-0.370260\pi$
$827$	1.55928e7	0.792794	0.396397	+	0.918079i	$0.370260\pi$
$828$	−5.66383e6	−0.287101
$829$	1.82374e7	0.921675	0.460837	−	0.887485i	$-0.347549\pi$
$829$	1.82374e7	0.921675	0.460837	+	0.887485i	$0.347549\pi$
$830$	0	0
$831$	−1.36400e7	−0.685192
$832$	−8.03597e6	−0.402467
$833$	−2.80700e6	−0.140162
$834$	1.36245e6	0.0678273
$835$	0	0
$836$	6.35992e6	0.314728
$837$	−6.41516e6	−0.316515
$838$	1.95792e6	0.0963130
$839$	8.42568e6	0.413238	0.206619	−	0.978421i	$-0.433754\pi$
$839$	8.42568e6	0.413238	0.206619	+	0.978421i	$0.433754\pi$
$840$	0	0
$841$	8.91021e6	0.434408
$842$	1.63529e6	0.0794903
$843$	−1.78154e7	−0.863430
$844$	−1.34059e7	−0.647797
$845$	0	0
$846$	279197.	0.0134117
$847$	1.55619e6	0.0745341
$848$	2.43476e7	1.16270
$849$	6.22694e6	0.296487
$850$	0	0
$851$	−5.75165e6	−0.272250
$852$	5.71768e6	0.269849
$853$	4.15991e7	1.95755	0.978773	−	0.204949i	$-0.0657030\pi$
$853$	4.15991e7	1.95755	0.978773	+	0.204949i	$0.0657030\pi$
$854$	894448.	0.0419672
$855$	0	0
$856$	−257934.	−0.0120316
$857$	1.04420e7	0.485659	0.242829	−	0.970069i	$-0.421924\pi$
$857$	1.04420e7	0.485659	0.242829	+	0.970069i	$0.421924\pi$
$858$	−179731.	−0.00833500
$859$	−6.29331e6	−0.291002	−0.145501	−	0.989358i	$-0.546479\pi$
$859$	−6.29331e6	−0.291002	−0.145501	+	0.989358i	$0.546479\pi$
$860$	0	0
$861$	−9.03506e6	−0.415359
$862$	3.78020e6	0.173279
$863$	−9.45460e6	−0.432132	−0.216066	−	0.976379i	$-0.569323\pi$
$863$	−9.45460e6	−0.432132	−0.216066	+	0.976379i	$0.569323\pi$
$864$	1.37549e6	0.0626863
$865$	0	0
$866$	−1.20654e6	−0.0546698
$867$	−1.04425e7	−0.471798
$868$	2.95666e7	1.33199
$869$	1.02394e7	0.459965
$870$	0	0
$871$	−7.82499e6	−0.349493
$872$	2.21563e6	0.0986748
$873$	2.11381e6	0.0938710
$874$	2.29632e6	0.101684
$875$	0	0
$876$	−1.81715e7	−0.800073
$877$	432267.	0.0189781	0.00948905	−	0.999955i	$-0.496979\pi$
$877$	432267.	0.0189781	0.00948905	+	0.999955i	$0.496979\pi$
$878$	−2.42245e6	−0.106052
$879$	−1.54899e7	−0.676202
$880$	0	0
$881$	−1.03415e7	−0.448894	−0.224447	−	0.974486i	$-0.572058\pi$
$881$	−1.03415e7	−0.448894	−0.224447	+	0.974486i	$0.572058\pi$
$882$	−278604.	−0.0120591
$883$	−1.89956e7	−0.819881	−0.409941	−	0.912112i	$-0.634451\pi$
$883$	−1.89956e7	−0.819881	−0.409941	+	0.912112i	$0.634451\pi$
$884$	−4.25759e6	−0.183245
$885$	0	0
$886$	−4.32176e6	−0.184960
$887$	−190605.	−0.00813438	−0.00406719	−	0.999992i	$-0.501295\pi$
$887$	−190605.	−0.00813438	−0.00406719	+	0.999992i	$0.501295\pi$
$888$	929312.	0.0395484
$889$	−1.27402e7	−0.540656
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−3.22925e7	−1.35890
$893$	9.18050e6	0.385246
$894$	2.06134e6	0.0862592
$895$	0	0
$896$	−8.43470e6	−0.350994
$897$	5.26307e6	0.218403
$898$	2.06967e6	0.0856467
$899$	4.77322e7	1.96975
$900$	0	0
$901$	1.25716e7	0.515917
$902$	713474.	0.0291986
$903$	9.15734e6	0.373723
$904$	4.17274e6	0.169824
$905$	0	0
$906$	1.01352e6	0.0410216
$907$	−2.99780e7	−1.21000	−0.604999	−	0.796226i	$-0.706827\pi$
$907$	−2.99780e7	−1.21000	−0.604999	+	0.796226i	$0.706827\pi$
$908$	3.64292e7	1.46634
$909$	9.30056e6	0.373335
$910$	0	0
$911$	4.55816e7	1.81967	0.909837	−	0.414966i	$-0.136206\pi$
$911$	4.55816e7	1.81967	0.909837	+	0.414966i	$0.136206\pi$
$912$	1.47666e7	0.587887
$913$	−9.91867e6	−0.393801
$914$	−2.12828e6	−0.0842681
$915$	0	0
$916$	−3.71586e7	−1.46326
$917$	−2.27295e7	−0.892622
$918$	231879.	0.00908143
$919$	−3.05489e6	−0.119318	−0.0596591	−	0.998219i	$-0.519001\pi$
$919$	−3.05489e6	−0.119318	−0.0596591	+	0.998219i	$0.519001\pi$
$920$	0	0
$921$	8.36998e6	0.325144
$922$	−1.58642e6	−0.0614599
$923$	−5.31311e6	−0.205279
$924$	3.65888e6	0.140984
$925$	0	0
$926$	−3.12761e6	−0.119863
$927$	1.57585e7	0.602305
$928$	−1.02344e7	−0.390113
$929$	−1.75975e7	−0.668978	−0.334489	−	0.942400i	$-0.608564\pi$
$929$	−1.75975e7	−0.668978	−0.334489	+	0.942400i	$0.608564\pi$
$930$	0	0
$931$	−9.16102e6	−0.346393
$932$	−1.93845e7	−0.730995
$933$	−2.64378e6	−0.0994309
$934$	2.78136e6	0.104326
$935$	0	0
$936$	−850370.	−0.0317262
$937$	−2.79612e7	−1.04042	−0.520208	−	0.854040i	$-0.674146\pi$
$937$	−2.79612e7	−1.04042	−0.520208	+	0.854040i	$0.674146\pi$
$938$	−1.96415e6	−0.0728899
$939$	5.99330e6	0.221821
$940$	0	0
$941$	3.56145e7	1.31115	0.655576	−	0.755129i	$-0.272426\pi$
$941$	3.56145e7	1.31115	0.655576	+	0.755129i	$0.272426\pi$
$942$	819694.	0.0300971
$943$	−2.08926e7	−0.765092
$944$	3.14452e7	1.14848
$945$	0	0
$946$	−723130.	−0.0262717
$947$	−3.24704e7	−1.17656	−0.588278	−	0.808659i	$-0.700194\pi$
$947$	−3.24704e7	−1.17656	−0.588278	+	0.808659i	$0.700194\pi$
$948$	2.40746e7	0.870038
$949$	1.68857e7	0.608629
$950$	0	0
$951$	1.84082e7	0.660024
$952$	−2.15057e6	−0.0769061
$953$	4.13052e6	0.147324	0.0736619	−	0.997283i	$-0.476531\pi$
$953$	4.13052e6	0.147324	0.0736619	+	0.997283i	$0.476531\pi$
$954$	1.24778e6	0.0443880
$955$	0	0
$956$	−3.51798e7	−1.24494
$957$	5.90689e6	0.208487
$958$	−1.07576e6	−0.0378705
$959$	2.86721e7	1.00673
$960$	0	0
$961$	4.88099e7	1.70490
$962$	−429132.	−0.0149504
$963$	526101.	0.0182811
$964$	−8.76798e6	−0.303883
$965$	0	0
$966$	1.32108e6	0.0455498
$967$	−4.10935e7	−1.41321	−0.706605	−	0.707608i	$-0.749774\pi$
$967$	−4.10935e7	−1.41321	−0.706605	+	0.707608i	$0.749774\pi$
$968$	−581426.	−0.0199437
$969$	7.62460e6	0.260860
$970$	0	0
$971$	−2.37238e7	−0.807487	−0.403744	−	0.914872i	$-0.632291\pi$
$971$	−2.37238e7	−0.807487	−0.403744	+	0.914872i	$0.632291\pi$
$972$	−1.86655e6	−0.0633687
$973$	2.57735e7	0.872752
$974$	−4.58642e6	−0.154909
$975$	0	0
$976$	1.33004e7	0.446932
$977$	1.39204e7	0.466568	0.233284	−	0.972409i	$-0.425053\pi$
$977$	1.39204e7	0.466568	0.233284	+	0.972409i	$0.425053\pi$
$978$	829023.	0.0277153
$979$	−229673.	−0.00765866
$980$	0	0
$981$	−4.51917e6	−0.149929
$982$	−982104.	−0.0324997
$983$	4.80216e6	0.158509	0.0792543	−	0.996854i	$-0.474746\pi$
$983$	4.80216e6	0.158509	0.0792543	+	0.996854i	$0.474746\pi$
$984$	3.37569e6	0.111141
$985$	0	0
$986$	−1.72530e6	−0.0565162
$987$	5.28158e6	0.172572
$988$	−1.38952e7	−0.452869
$989$	2.11754e7	0.688399
$990$	0	0
$991$	−4.05874e7	−1.31283	−0.656413	−	0.754401i	$-0.727927\pi$
$991$	−4.05874e7	−1.31283	−0.656413	+	0.754401i	$0.727927\pi$
$992$	−1.66039e7	−0.535710
$993$	−2.36369e7	−0.760708
$994$	−1.33364e6	−0.0428128
$995$	0	0
$996$	−2.33205e7	−0.744887
$997$	−4.33820e7	−1.38220	−0.691101	−	0.722758i	$-0.742874\pi$
$997$	−4.33820e7	−1.38220	−0.691101	+	0.722758i	$0.742874\pi$
$998$	−3.62273e6	−0.115136
$999$	−1.89549e6	−0.0600909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.r.1.5	✓	9
5.4	even	2		825.6.a.s.1.5	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.5	✓	9	1.1	even	1	trivial
825.6.a.s.1.5	yes	9	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+0.624305 q^{2} +9.00000 q^{3} -31.6102 q^{4} +5.61875 q^{6} +106.290 q^{7} -39.7122 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+0.624305 q^{2} +9.00000 q^{3} -31.6102 q^{4} +5.61875 q^{6} +106.290 q^{7} -39.7122 q^{8} +81.0000 q^{9} -121.000 q^{11} -284.492 q^{12} +264.362 q^{13} +66.3575 q^{14} +986.735 q^{16} +509.491 q^{17} +50.5687 q^{18} +1662.79 q^{19} +956.611 q^{21} -75.5409 q^{22} +2212.06 q^{23} -357.410 q^{24} +165.043 q^{26} +729.000 q^{27} -3359.86 q^{28} -5424.15 q^{29} -8799.95 q^{31} +1886.81 q^{32} -1089.00 q^{33} +318.078 q^{34} -2560.43 q^{36} -2600.13 q^{37} +1038.09 q^{38} +2379.26 q^{39} -9444.87 q^{41} +597.217 q^{42} +9572.69 q^{43} +3824.84 q^{44} +1381.00 q^{46} +5521.13 q^{47} +8880.62 q^{48} -5509.41 q^{49} +4585.42 q^{51} -8356.55 q^{52} +24674.9 q^{53} +455.119 q^{54} -4221.01 q^{56} +14965.1 q^{57} -3386.32 q^{58} +31867.9 q^{59} +13479.2 q^{61} -5493.85 q^{62} +8609.50 q^{63} -30397.6 q^{64} -679.868 q^{66} -29599.5 q^{67} -16105.1 q^{68} +19908.6 q^{69} -20097.8 q^{71} -3216.69 q^{72} +63873.3 q^{73} -1623.27 q^{74} -52561.3 q^{76} -12861.1 q^{77} +1485.38 q^{78} -84623.1 q^{79} +6561.00 q^{81} -5896.48 q^{82} +81972.5 q^{83} -30238.7 q^{84} +5976.28 q^{86} -48817.3 q^{87} +4805.18 q^{88} +1898.12 q^{89} +28099.1 q^{91} -69923.8 q^{92} -79199.5 q^{93} +3446.87 q^{94} +16981.3 q^{96} +26096.5 q^{97} -3439.56 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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

Modular forms

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