LMFDB - Embedded newform 825.6.a.s.1.2

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			1.2
Root			$-8.18565$ 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)

$f(q)$	$=$	$q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} -121.000 q^{11} -315.043 q^{12} -728.536 q^{13} -1339.68 q^{14} -918.818 q^{16} -1475.58 q^{17} -663.037 q^{18} -341.835 q^{19} -1472.95 q^{21} +990.463 q^{22} -2943.46 q^{23} +221.366 q^{24} +5963.54 q^{26} -729.000 q^{27} +5728.94 q^{28} -2823.82 q^{29} +5163.10 q^{31} +8308.19 q^{32} +1089.00 q^{33} +12078.5 q^{34} +2835.39 q^{36} -4254.93 q^{37} +2798.14 q^{38} +6556.83 q^{39} +5874.23 q^{41} +12057.1 q^{42} -1662.76 q^{43} -4235.58 q^{44} +24094.1 q^{46} +5981.04 q^{47} +8269.36 q^{48} +9978.09 q^{49} +13280.2 q^{51} -25502.3 q^{52} -27557.3 q^{53} +5967.34 q^{54} -4025.45 q^{56} +3076.51 q^{57} +23114.8 q^{58} +8248.03 q^{59} -2342.27 q^{61} -42263.3 q^{62} +13256.6 q^{63} -38605.8 q^{64} -8914.17 q^{66} -49456.4 q^{67} -51652.2 q^{68} +26491.1 q^{69} -24643.4 q^{71} -1992.29 q^{72} +23000.9 q^{73} +34829.4 q^{74} -11965.9 q^{76} -19803.0 q^{77} -53671.9 q^{78} +87214.6 q^{79} +6561.00 q^{81} -48084.4 q^{82} -90598.7 q^{83} -51560.4 q^{84} +13610.8 q^{86} +25414.4 q^{87} +2976.14 q^{88} +86239.2 q^{89} -119233. q^{91} -103035. q^{92} -46467.9 q^{93} -48958.7 q^{94} -74773.8 q^{96} -13088.6 q^{97} -81677.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.18565	−1.44703	−0.723516	−	0.690308i	$-0.757475\pi$
$2$	−8.18565	−1.44703	−0.723516	+	0.690308i	$0.757475\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	35.0048	1.09390
$5$	0	0
$5$	0	0
$6$	73.6708	0.835444
$7$	163.661	1.26241	0.631206	−	0.775615i	$-0.282560\pi$
$7$	163.661	1.26241	0.631206	+	0.775615i	$0.282560\pi$
$8$	−24.5962	−0.135876
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−315.043	−0.631563
$13$	−728.536	−1.19562	−0.597809	−	0.801638i	$-0.703962\pi$
$13$	−728.536	−1.19562	−0.597809	+	0.801638i	$0.703962\pi$
$14$	−1339.68	−1.82675
$15$	0	0
$16$	−918.818	−0.897283
$17$	−1475.58	−1.23834	−0.619169	−	0.785258i	$-0.712531\pi$
$17$	−1475.58	−1.23834	−0.619169	+	0.785258i	$0.712531\pi$
$18$	−663.037	−0.482344
$19$	−341.835	−0.217236	−0.108618	−	0.994084i	$-0.534643\pi$
$19$	−341.835	−0.217236	−0.108618	+	0.994084i	$0.534643\pi$
$20$	0	0
$21$	−1472.95	−0.728854
$22$	990.463	0.436296
$23$	−2943.46	−1.16021	−0.580107	−	0.814541i	$-0.696989\pi$
$23$	−2943.46	−1.16021	−0.580107	+	0.814541i	$0.696989\pi$
$24$	221.366	0.0784480
$25$	0	0
$26$	5963.54	1.73010
$27$	−729.000	−0.192450
$28$	5728.94	1.38095
$29$	−2823.82	−0.623508	−0.311754	−	0.950163i	$-0.600916\pi$
$29$	−2823.82	−0.623508	−0.311754	+	0.950163i	$0.600916\pi$
$30$	0	0
$31$	5163.10	0.964953	0.482476	−	0.875909i	$-0.339737\pi$
$31$	5163.10	0.964953	0.482476	+	0.875909i	$0.339737\pi$
$32$	8308.19	1.43427
$33$	1089.00	0.174078
$34$	12078.5	1.79191
$35$	0	0
$36$	2835.39	0.364633
$37$	−4254.93	−0.510962	−0.255481	−	0.966814i	$-0.582234\pi$
$37$	−4254.93	−0.510962	−0.255481	+	0.966814i	$0.582234\pi$
$38$	2798.14	0.314348
$39$	6556.83	0.690291
$40$	0	0
$41$	5874.23	0.545747	0.272873	−	0.962050i	$-0.412026\pi$
$41$	5874.23	0.545747	0.272873	+	0.962050i	$0.412026\pi$
$42$	12057.1	1.05468
$43$	−1662.76	−0.137138	−0.0685691	−	0.997646i	$-0.521843\pi$
$43$	−1662.76	−0.137138	−0.0685691	+	0.997646i	$0.521843\pi$
$44$	−4235.58	−0.329823
$45$	0	0
$46$	24094.1	1.67886
$47$	5981.04	0.394941	0.197471	−	0.980309i	$-0.436727\pi$
$47$	5981.04	0.394941	0.197471	+	0.980309i	$0.436727\pi$
$48$	8269.36	0.518047
$49$	9978.09	0.593686
$50$	0	0
$51$	13280.2	0.714955
$52$	−25502.3	−1.30789
$53$	−27557.3	−1.34755	−0.673777	−	0.738935i	$-0.735329\pi$
$53$	−27557.3	−1.34755	−0.673777	+	0.738935i	$0.735329\pi$
$54$	5967.34	0.278481
$55$	0	0
$56$	−4025.45	−0.171532
$57$	3076.51	0.125421
$58$	23114.8	0.902236
$59$	8248.03	0.308475	0.154238	−	0.988034i	$-0.450708\pi$
$59$	8248.03	0.308475	0.154238	+	0.988034i	$0.450708\pi$
$60$	0	0
$61$	−2342.27	−0.0805959	−0.0402980	−	0.999188i	$-0.512831\pi$
$61$	−2342.27	−0.0805959	−0.0402980	+	0.999188i	$0.512831\pi$
$62$	−42263.3	−1.39632
$63$	13256.6	0.420804
$64$	−38605.8	−1.17815
$65$	0	0
$66$	−8914.17	−0.251896
$67$	−49456.4	−1.34597	−0.672985	−	0.739656i	$-0.734988\pi$
$67$	−49456.4	−1.34597	−0.672985	+	0.739656i	$0.734988\pi$
$68$	−51652.2	−1.35462
$69$	26491.1	0.669849
$70$	0	0
$71$	−24643.4	−0.580169	−0.290085	−	0.957001i	$-0.593683\pi$
$71$	−24643.4	−0.580169	−0.290085	+	0.957001i	$0.593683\pi$
$72$	−1992.29	−0.0452920
$73$	23000.9	0.505170	0.252585	−	0.967575i	$-0.418719\pi$
$73$	23000.9	0.505170	0.252585	+	0.967575i	$0.418719\pi$
$74$	34829.4	0.739378
$75$	0	0
$76$	−11965.9	−0.237635
$77$	−19803.0	−0.380632
$78$	−53671.9	−0.998873
$79$	87214.6	1.57225	0.786124	−	0.618068i	$-0.212085\pi$
$79$	87214.6	1.57225	0.786124	+	0.618068i	$0.212085\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−48084.4	−0.789713
$83$	−90598.7	−1.44353	−0.721766	−	0.692137i	$-0.756669\pi$
$83$	−90598.7	−1.44353	−0.721766	+	0.692137i	$0.756669\pi$
$84$	−51560.4	−0.797294
$85$	0	0
$86$	13610.8	0.198443
$87$	25414.4	0.359983
$88$	2976.14	0.0409681
$89$	86239.2	1.15406	0.577032	−	0.816722i	$-0.304211\pi$
$89$	86239.2	1.15406	0.577032	+	0.816722i	$0.304211\pi$
$90$	0	0
$91$	−119233.	−1.50936
$92$	−103035.	−1.26916
$93$	−46467.9	−0.557116
$94$	−48958.7	−0.571492
$95$	0	0
$96$	−74773.8	−0.828078
$97$	−13088.6	−0.141242	−0.0706211	−	0.997503i	$-0.522498\pi$
$97$	−13088.6	−0.141242	−0.0706211	+	0.997503i	$0.522498\pi$
$98$	−81677.1	−0.859083
$99$	−9801.00	−0.100504
$100$	0	0
$101$	34292.9	0.334503	0.167252	−	0.985914i	$-0.446511\pi$
$101$	34292.9	0.334503	0.167252	+	0.985914i	$0.446511\pi$
$102$	−108707.	−1.03456
$103$	32773.7	0.304391	0.152196	−	0.988350i	$-0.451366\pi$
$103$	32773.7	0.304391	0.152196	+	0.988350i	$0.451366\pi$
$104$	17919.2	0.162456
$105$	0	0
$106$	225574.	1.94995
$107$	−203291.	−1.71656	−0.858278	−	0.513186i	$-0.828465\pi$
$107$	−203291.	−1.71656	−0.858278	+	0.513186i	$0.828465\pi$
$108$	−25518.5	−0.210521
$109$	203531.	1.64084	0.820418	−	0.571764i	$-0.193741\pi$
$109$	203531.	1.64084	0.820418	+	0.571764i	$0.193741\pi$
$110$	0	0
$111$	38294.4	0.295004
$112$	−150375.	−1.13274
$113$	140927.	1.03824	0.519120	−	0.854702i	$-0.326260\pi$
$113$	140927.	1.03824	0.519120	+	0.854702i	$0.326260\pi$
$114$	−25183.2	−0.181489
$115$	0	0
$116$	−98847.3	−0.682055
$117$	−59011.4	−0.398540
$118$	−67515.4	−0.446373
$119$	−241495.	−1.56329
$120$	0	0
$121$	14641.0	0.0909091
$122$	19173.0	0.116625
$123$	−52868.1	−0.315087
$124$	180733.	1.05556
$125$	0	0
$126$	−108514.	−0.608917
$127$	2686.76	0.0147815	0.00739076	−	0.999973i	$-0.497647\pi$
$127$	2686.76	0.0147815	0.00739076	+	0.999973i	$0.497647\pi$
$128$	50150.9	0.270554
$129$	14964.8	0.0791767
$130$	0	0
$131$	−280946.	−1.43036	−0.715179	−	0.698941i	$-0.753655\pi$
$131$	−280946.	−1.43036	−0.715179	+	0.698941i	$0.753655\pi$
$132$	38120.2	0.190424
$133$	−55945.2	−0.274242
$134$	404832.	1.94766
$135$	0	0
$136$	36293.5	0.168260
$137$	272454.	1.24020	0.620100	−	0.784523i	$-0.287092\pi$
$137$	272454.	1.24020	0.620100	+	0.784523i	$0.287092\pi$
$138$	−216847.	−0.969293
$139$	324210.	1.42328	0.711638	−	0.702547i	$-0.247954\pi$
$139$	324210.	1.42328	0.711638	+	0.702547i	$0.247954\pi$
$140$	0	0
$141$	−53829.4	−0.228019
$142$	201722.	0.839523
$143$	88152.9	0.360493
$144$	−74424.2	−0.299094
$145$	0	0
$146$	−188277.	−0.730997
$147$	−89802.8	−0.342765
$148$	−148943.	−0.558941
$149$	−325211.	−1.20005	−0.600024	−	0.799982i	$-0.704843\pi$
$149$	−325211.	−1.20005	−0.600024	+	0.799982i	$0.704843\pi$
$150$	0	0
$151$	−399144.	−1.42458	−0.712290	−	0.701885i	$-0.752342\pi$
$151$	−399144.	−1.42458	−0.712290	+	0.701885i	$0.752342\pi$
$152$	8407.83	0.0295172
$153$	−119522.	−0.412779
$154$	162101.	0.550786
$155$	0	0
$156$	229520.	0.755109
$157$	82871.8	0.268323	0.134161	−	0.990959i	$-0.457166\pi$
$157$	82871.8	0.268323	0.134161	+	0.990959i	$0.457166\pi$
$158$	−713907.	−2.27509
$159$	248015.	0.778011
$160$	0	0
$161$	−481730.	−1.46467
$162$	−53706.0	−0.160781
$163$	311672.	0.918818	0.459409	−	0.888225i	$-0.348061\pi$
$163$	311672.	0.918818	0.459409	+	0.888225i	$0.348061\pi$
$164$	205626.	0.596992
$165$	0	0
$166$	741609.	2.08884
$167$	129998.	0.360700	0.180350	−	0.983602i	$-0.442277\pi$
$167$	129998.	0.360700	0.180350	+	0.983602i	$0.442277\pi$
$168$	36229.0	0.0990338
$169$	159472.	0.429505
$170$	0	0
$171$	−27688.6	−0.0724121
$172$	−58204.5	−0.150015
$173$	−744520.	−1.89130	−0.945651	−	0.325183i	$-0.894574\pi$
$173$	−744520.	−1.89130	−0.945651	+	0.325183i	$0.894574\pi$
$174$	−208033.	−0.520906
$175$	0	0
$176$	111177.	0.270541
$177$	−74232.3	−0.178098
$178$	−705924.	−1.66997
$179$	309968.	0.723077	0.361539	−	0.932357i	$-0.382252\pi$
$179$	309968.	0.723077	0.361539	+	0.932357i	$0.382252\pi$
$180$	0	0
$181$	−656347.	−1.48915	−0.744573	−	0.667541i	$-0.767347\pi$
$181$	−656347.	−1.48915	−0.744573	+	0.667541i	$0.767347\pi$
$182$	976002.	2.18410
$183$	21080.5	0.0465321
$184$	72397.7	0.157645
$185$	0	0
$186$	380370.	0.806164
$187$	178545.	0.373373
$188$	209365.	0.432026
$189$	−119309.	−0.242951
$190$	0	0
$191$	−336716.	−0.667853	−0.333926	−	0.942599i	$-0.608374\pi$
$191$	−336716.	−0.667853	−0.333926	+	0.942599i	$0.608374\pi$
$192$	347452.	0.680208
$193$	599122.	1.15777	0.578884	−	0.815410i	$-0.303488\pi$
$193$	599122.	1.15777	0.578884	+	0.815410i	$0.303488\pi$
$194$	107139.	0.204382
$195$	0	0
$196$	349281.	0.649433
$197$	365021.	0.670120	0.335060	−	0.942197i	$-0.391243\pi$
$197$	365021.	0.670120	0.335060	+	0.942197i	$0.391243\pi$
$198$	80227.5	0.145432
$199$	333867.	0.597641	0.298820	−	0.954309i	$-0.403407\pi$
$199$	333867.	0.597641	0.298820	+	0.954309i	$0.403407\pi$
$200$	0	0
$201$	445108.	0.777096
$202$	−280709.	−0.484037
$203$	−462151.	−0.787125
$204$	464870.	0.782089
$205$	0	0
$206$	−268274.	−0.440463
$207$	−238420.	−0.386738
$208$	669392.	1.07281
$209$	41362.0	0.0654992
$210$	0	0
$211$	−980083.	−1.51550	−0.757751	−	0.652543i	$-0.773702\pi$
$211$	−980083.	−1.51550	−0.757751	+	0.652543i	$0.773702\pi$
$212$	−964636.	−1.47409
$213$	221791.	0.334961
$214$	1.66406e6	2.48391
$215$	0	0
$216$	17930.6	0.0261493
$217$	845000.	1.21817
$218$	−1.66604e6	−2.37434
$219$	−207008.	−0.291660
$220$	0	0
$221$	1.07501e6	1.48058
$222$	−313464.	−0.426880
$223$	80967.8	0.109031	0.0545155	−	0.998513i	$-0.482639\pi$
$223$	80967.8	0.109031	0.0545155	+	0.998513i	$0.482639\pi$
$224$	1.35973e6	1.81064
$225$	0	0
$226$	−1.15358e6	−1.50236
$227$	−524287.	−0.675312	−0.337656	−	0.941270i	$-0.609634\pi$
$227$	−524287.	−0.675312	−0.337656	+	0.941270i	$0.609634\pi$
$228$	107693.	0.137198
$229$	816517.	1.02891	0.514454	−	0.857518i	$-0.327995\pi$
$229$	816517.	1.02891	0.514454	+	0.857518i	$0.327995\pi$
$230$	0	0
$231$	178227.	0.219758
$232$	69455.2	0.0847198
$233$	338979.	0.409056	0.204528	−	0.978861i	$-0.434434\pi$
$233$	338979.	0.409056	0.204528	+	0.978861i	$0.434434\pi$
$234$	483047.	0.576699
$235$	0	0
$236$	288721.	0.337441
$237$	−784931.	−0.907738
$238$	1.97679e6	2.26214
$239$	98246.5	0.111256	0.0556278	−	0.998452i	$-0.482284\pi$
$239$	98246.5	0.111256	0.0556278	+	0.998452i	$0.482284\pi$
$240$	0	0
$241$	1.00307e6	1.11247	0.556233	−	0.831026i	$-0.312246\pi$
$241$	1.00307e6	1.11247	0.556233	+	0.831026i	$0.312246\pi$
$242$	−119846.	−0.131548
$243$	−59049.0	−0.0641500
$244$	−81990.8	−0.0881639
$245$	0	0
$246$	432759.	0.455941
$247$	249039.	0.259732
$248$	−126992.	−0.131114
$249$	815388.	0.833424
$250$	0	0
$251$	989640.	0.991501	0.495750	−	0.868465i	$-0.334893\pi$
$251$	989640.	0.991501	0.495750	+	0.868465i	$0.334893\pi$
$252$	464044.	0.460318
$253$	356158.	0.349817
$254$	−21992.8	−0.0213893
$255$	0	0
$256$	824867.	0.786655
$257$	1.15250e6	1.08845	0.544225	−	0.838939i	$-0.316824\pi$
$257$	1.15250e6	1.08845	0.544225	+	0.838939i	$0.316824\pi$
$258$	−122497.	−0.114571
$259$	−696369.	−0.645045
$260$	0	0
$261$	−228729.	−0.207836
$262$	2.29973e6	2.06977
$263$	1.03041e6	0.918589	0.459294	−	0.888284i	$-0.348102\pi$
$263$	1.03041e6	0.918589	0.459294	+	0.888284i	$0.348102\pi$
$264$	−26785.2	−0.0236530
$265$	0	0
$266$	457948.	0.396836
$267$	−776153.	−0.666299
$268$	−1.73121e6	−1.47236
$269$	−2.24263e6	−1.88963	−0.944814	−	0.327607i	$-0.893758\pi$
$269$	−2.24263e6	−1.88963	−0.944814	+	0.327607i	$0.893758\pi$
$270$	0	0
$271$	2.17227e6	1.79676	0.898382	−	0.439216i	$-0.144744\pi$
$271$	2.17227e6	1.79676	0.898382	+	0.439216i	$0.144744\pi$
$272$	1.35579e6	1.11114
$273$	1.07310e6	0.871432
$274$	−2.23021e6	−1.79461
$275$	0	0
$276$	927315.	0.732748
$277$	−1.04100e6	−0.815178	−0.407589	−	0.913165i	$-0.633630\pi$
$277$	−1.04100e6	−0.815178	−0.407589	+	0.913165i	$0.633630\pi$
$278$	−2.65387e6	−2.05952
$279$	418211.	0.321651
$280$	0	0
$281$	−1.34424e6	−1.01557	−0.507786	−	0.861483i	$-0.669536\pi$
$281$	−1.34424e6	−1.01557	−0.507786	+	0.861483i	$0.669536\pi$
$282$	440628.	0.329951
$283$	−828093.	−0.614629	−0.307314	−	0.951608i	$-0.599430\pi$
$283$	−828093.	−0.614629	−0.307314	+	0.951608i	$0.599430\pi$
$284$	−862637.	−0.634647
$285$	0	0
$286$	−721588.	−0.521644
$287$	961385.	0.688958
$288$	672964.	0.478091
$289$	757468.	0.533482
$290$	0	0
$291$	117798.	0.0815462
$292$	805142.	0.552606
$293$	1.31305e6	0.893538	0.446769	−	0.894649i	$-0.352575\pi$
$293$	1.31305e6	0.893538	0.446769	+	0.894649i	$0.352575\pi$
$294$	735094.	0.495992
$295$	0	0
$296$	104655.	0.0694274
$297$	88209.0	0.0580259
$298$	2.66206e6	1.73651
$299$	2.14441e6	1.38717
$300$	0	0
$301$	−272130.	−0.173125
$302$	3.26725e6	2.06141
$303$	−308636.	−0.193126
$304$	314084.	0.194922
$305$	0	0
$306$	978362.	0.597305
$307$	−138234.	−0.0837082	−0.0418541	−	0.999124i	$-0.513326\pi$
$307$	−138234.	−0.0837082	−0.0418541	+	0.999124i	$0.513326\pi$
$308$	−693201.	−0.416373
$309$	−294963.	−0.175740
$310$	0	0
$311$	1.17643e6	0.689710	0.344855	−	0.938656i	$-0.387928\pi$
$311$	1.17643e6	0.689710	0.344855	+	0.938656i	$0.387928\pi$
$312$	−161273.	−0.0937939
$313$	−919942.	−0.530762	−0.265381	−	0.964144i	$-0.585498\pi$
$313$	−919942.	−0.530762	−0.265381	+	0.964144i	$0.585498\pi$
$314$	−678359.	−0.388271
$315$	0	0
$316$	3.05293e6	1.71988
$317$	1.96510e6	1.09834	0.549171	−	0.835710i	$-0.314944\pi$
$317$	1.96510e6	1.09834	0.549171	+	0.835710i	$0.314944\pi$
$318$	−2.03016e6	−1.12581
$319$	341682.	0.187995
$320$	0	0
$321$	1.82962e6	0.991054
$322$	3.94327e6	2.11942
$323$	504403.	0.269012
$324$	229666.	0.121544
$325$	0	0
$326$	−2.55124e6	−1.32956
$327$	−1.83178e6	−0.947337
$328$	−144484.	−0.0741539
$329$	978866.	0.498579
$330$	0	0
$331$	1.98383e6	0.995255	0.497628	−	0.867391i	$-0.334205\pi$
$331$	1.98383e6	0.995255	0.497628	+	0.867391i	$0.334205\pi$
$332$	−3.17139e6	−1.57908
$333$	−344650.	−0.170321
$334$	−1.06412e6	−0.521945
$335$	0	0
$336$	1.35338e6	0.653989
$337$	3.28452e6	1.57543	0.787713	−	0.616043i	$-0.211265\pi$
$337$	3.28452e6	1.57543	0.787713	+	0.616043i	$0.211265\pi$
$338$	−1.30538e6	−0.621507
$339$	−1.26834e6	−0.599428
$340$	0	0
$341$	−624735.	−0.290944
$342$	226649.	0.104783
$343$	−1.11763e6	−0.512936
$344$	40897.5	0.0186338
$345$	0	0
$346$	6.09437e6	2.73677
$347$	−2.23106e6	−0.994691	−0.497345	−	0.867553i	$-0.665692\pi$
$347$	−2.23106e6	−0.994691	−0.497345	+	0.867553i	$0.665692\pi$
$348$	889625.	0.393785
$349$	1.14634e6	0.503790	0.251895	−	0.967755i	$-0.418946\pi$
$349$	1.14634e6	0.503790	0.251895	+	0.967755i	$0.418946\pi$
$350$	0	0
$351$	531103.	0.230097
$352$	−1.00529e6	−0.432449
$353$	−1.17280e6	−0.500941	−0.250470	−	0.968124i	$-0.580585\pi$
$353$	−1.17280e6	−0.500941	−0.250470	+	0.968124i	$0.580585\pi$
$354$	607639.	0.257714
$355$	0	0
$356$	3.01879e6	1.26243
$357$	2.17346e6	0.902568
$358$	−2.53729e6	−1.04632
$359$	718129.	0.294081	0.147040	−	0.989130i	$-0.453025\pi$
$359$	718129.	0.294081	0.147040	+	0.989130i	$0.453025\pi$
$360$	0	0
$361$	−2.35925e6	−0.952808
$362$	5.37263e6	2.15484
$363$	−131769.	−0.0524864
$364$	−4.17374e6	−1.65109
$365$	0	0
$366$	−172557.	−0.0673334
$367$	−933545.	−0.361801	−0.180901	−	0.983501i	$-0.557901\pi$
$367$	−933545.	−0.361801	−0.180901	+	0.983501i	$0.557901\pi$
$368$	2.70450e6	1.04104
$369$	475812.	0.181916
$370$	0	0
$371$	−4.51006e6	−1.70117
$372$	−1.62660e6	−0.609429
$373$	4.48359e6	1.66860	0.834302	−	0.551307i	$-0.185871\pi$
$373$	4.48359e6	1.66860	0.834302	+	0.551307i	$0.185871\pi$
$374$	−1.46150e6	−0.540283
$375$	0	0
$376$	−147111.	−0.0536630
$377$	2.05726e6	0.745478
$378$	976623.	0.351558
$379$	2.61145e6	0.933863	0.466931	−	0.884294i	$-0.345360\pi$
$379$	2.61145e6	0.933863	0.466931	+	0.884294i	$0.345360\pi$
$380$	0	0
$381$	−24180.8	−0.00853411
$382$	2.75624e6	0.966404
$383$	−2.87502e6	−1.00148	−0.500741	−	0.865597i	$-0.666939\pi$
$383$	−2.87502e6	−1.00148	−0.500741	+	0.865597i	$0.666939\pi$
$384$	−451358.	−0.156204
$385$	0	0
$386$	−4.90420e6	−1.67533
$387$	−134683.	−0.0457127
$388$	−458164.	−0.154505
$389$	231232.	0.0774771	0.0387386	−	0.999249i	$-0.487666\pi$
$389$	231232.	0.0774771	0.0387386	+	0.999249i	$0.487666\pi$
$390$	0	0
$391$	4.34329e6	1.43674
$392$	−245423.	−0.0806677
$393$	2.52852e6	0.825818
$394$	−2.98794e6	−0.969685
$395$	0	0
$396$	−343082.	−0.109941
$397$	2.01862e6	0.642804	0.321402	−	0.946943i	$-0.395846\pi$
$397$	2.01862e6	0.642804	0.321402	+	0.946943i	$0.395846\pi$
$398$	−2.73291e6	−0.864805
$399$	503507.	0.158334
$400$	0	0
$401$	−4.53630e6	−1.40877	−0.704386	−	0.709817i	$-0.748777\pi$
$401$	−4.53630e6	−1.40877	−0.704386	+	0.709817i	$0.748777\pi$
$402$	−3.64349e6	−1.12448
$403$	−3.76150e6	−1.15372
$404$	1.20041e6	0.365913
$405$	0	0
$406$	3.78300e6	1.13899
$407$	514847.	0.154061
$408$	−326642.	−0.0971452
$409$	629053.	0.185943	0.0929714	−	0.995669i	$-0.470363\pi$
$409$	629053.	0.185943	0.0929714	+	0.995669i	$0.470363\pi$
$410$	0	0
$411$	−2.45209e6	−0.716030
$412$	1.14724e6	0.332973
$413$	1.34988e6	0.389423
$414$	1.95162e6	0.559622
$415$	0	0
$416$	−6.05282e6	−1.71484
$417$	−2.91789e6	−0.821729
$418$	−338575.	−0.0947794
$419$	1.00216e6	0.278870	0.139435	−	0.990231i	$-0.455471\pi$
$419$	1.00216e6	0.278870	0.139435	+	0.990231i	$0.455471\pi$
$420$	0	0
$421$	−1.43349e6	−0.394175	−0.197087	−	0.980386i	$-0.563148\pi$
$421$	−1.43349e6	−0.394175	−0.197087	+	0.980386i	$0.563148\pi$
$422$	8.02261e6	2.19298
$423$	484464.	0.131647
$424$	677803.	0.183100
$425$	0	0
$426$	−1.81550e6	−0.484699
$427$	−383340.	−0.101745
$428$	−7.11614e6	−1.87774
$429$	−793376.	−0.208131
$430$	0	0
$431$	−4.80577e6	−1.24615	−0.623074	−	0.782163i	$-0.714116\pi$
$431$	−4.80577e6	−1.24615	−0.623074	+	0.782163i	$0.714116\pi$
$432$	669818.	0.172682
$433$	1.53507e6	0.393467	0.196733	−	0.980457i	$-0.436967\pi$
$433$	1.53507e6	0.393467	0.196733	+	0.980457i	$0.436967\pi$
$434$	−6.91687e6	−1.76273
$435$	0	0
$436$	7.12458e6	1.79491
$437$	1.00618e6	0.252040
$438$	1.69450e6	0.422042
$439$	7.14894e6	1.77044	0.885218	−	0.465176i	$-0.154009\pi$
$439$	7.14894e6	1.77044	0.885218	+	0.465176i	$0.154009\pi$
$440$	0	0
$441$	808225.	0.197895
$442$	−8.79966e6	−2.14245
$443$	−712048.	−0.172385	−0.0861926	−	0.996278i	$-0.527470\pi$
$443$	−712048.	−0.172385	−0.0861926	+	0.996278i	$0.527470\pi$
$444$	1.34049e6	0.322705
$445$	0	0
$446$	−662773.	−0.157771
$447$	2.92689e6	0.692849
$448$	−6.31828e6	−1.48732
$449$	−3.18417e6	−0.745385	−0.372693	−	0.927955i	$-0.621565\pi$
$449$	−3.18417e6	−0.745385	−0.372693	+	0.927955i	$0.621565\pi$
$450$	0	0
$451$	−710782.	−0.164549
$452$	4.93311e6	1.13573
$453$	3.59229e6	0.822481
$454$	4.29163e6	0.977197
$455$	0	0
$456$	−75670.4	−0.0170418
$457$	−8.26972e6	−1.85225	−0.926127	−	0.377213i	$-0.876883\pi$
$457$	−8.26972e6	−1.85225	−0.926127	+	0.377213i	$0.876883\pi$
$458$	−6.68372e6	−1.48886
$459$	1.07570e6	0.238318
$460$	0	0
$461$	7.66057e6	1.67884	0.839419	−	0.543485i	$-0.182896\pi$
$461$	7.66057e6	1.67884	0.839419	+	0.543485i	$0.182896\pi$
$462$	−1.45891e6	−0.317997
$463$	−203047.	−0.0440194	−0.0220097	−	0.999758i	$-0.507006\pi$
$463$	−203047.	−0.0440194	−0.0220097	+	0.999758i	$0.507006\pi$
$464$	2.59458e6	0.559463
$465$	0	0
$466$	−2.77476e6	−0.591917
$467$	−3.28726e6	−0.697497	−0.348748	−	0.937216i	$-0.613393\pi$
$467$	−3.28726e6	−0.697497	−0.348748	+	0.937216i	$0.613393\pi$
$468$	−2.06568e6	−0.435962
$469$	−8.09411e6	−1.69917
$470$	0	0
$471$	−745846.	−0.154916
$472$	−202870.	−0.0419144
$473$	201194.	0.0413487
$474$	6.42517e6	1.31353
$475$	0	0
$476$	−8.45348e6	−1.71009
$477$	−2.23214e6	−0.449185
$478$	−804211.	−0.160990
$479$	−3.08986e6	−0.615319	−0.307659	−	0.951497i	$-0.599546\pi$
$479$	−3.08986e6	−0.615319	−0.307659	+	0.951497i	$0.599546\pi$
$480$	0	0
$481$	3.09987e6	0.610916
$482$	−8.21074e6	−1.60977
$483$	4.33557e6	0.845627
$484$	512505.	0.0994454
$485$	0	0
$486$	483354.	0.0928271
$487$	−2.37051e6	−0.452918	−0.226459	−	0.974021i	$-0.572715\pi$
$487$	−2.37051e6	−0.452918	−0.226459	+	0.974021i	$0.572715\pi$
$488$	57611.0	0.0109510
$489$	−2.80505e6	−0.530480
$490$	0	0
$491$	−7.56399e6	−1.41595	−0.707974	−	0.706239i	$-0.750391\pi$
$491$	−7.56399e6	−1.41595	−0.707974	+	0.706239i	$0.750391\pi$
$492$	−1.85064e6	−0.344674
$493$	4.16676e6	0.772114
$494$	−2.03855e6	−0.375840
$495$	0	0
$496$	−4.74395e6	−0.865836
$497$	−4.03318e6	−0.732413
$498$	−6.67448e6	−1.20599
$499$	−1.56759e6	−0.281826	−0.140913	−	0.990022i	$-0.545004\pi$
$499$	−1.56759e6	−0.281826	−0.140913	+	0.990022i	$0.545004\pi$
$500$	0	0
$501$	−1.16999e6	−0.208251
$502$	−8.10084e6	−1.43473
$503$	2.34418e6	0.413115	0.206557	−	0.978435i	$-0.433774\pi$
$503$	2.34418e6	0.413115	0.206557	+	0.978435i	$0.433774\pi$
$504$	−326061.	−0.0571772
$505$	0	0
$506$	−2.91538e6	−0.506197
$507$	−1.43525e6	−0.247975
$508$	94049.3	0.0161695
$509$	8.79759e6	1.50511	0.752557	−	0.658527i	$-0.228820\pi$
$509$	8.79759e6	1.50511	0.752557	+	0.658527i	$0.228820\pi$
$510$	0	0
$511$	3.76436e6	0.637734
$512$	−8.35690e6	−1.40887
$513$	249198.	0.0418071
$514$	−9.43397e6	−1.57502
$515$	0	0
$516$	523841.	0.0866114
$517$	−723706.	−0.119079
$518$	5.70023e6	0.933400
$519$	6.70068e6	1.09194
$520$	0	0
$521$	−3.43077e6	−0.553729	−0.276865	−	0.960909i	$-0.589295\pi$
$521$	−3.43077e6	−0.553729	−0.276865	+	0.960909i	$0.589295\pi$
$522$	1.87230e6	0.300745
$523$	1.24453e7	1.98953	0.994763	−	0.102212i	$-0.0325920\pi$
$523$	1.24453e7	1.98953	0.994763	+	0.102212i	$0.0325920\pi$
$524$	−9.83446e6	−1.56467
$525$	0	0
$526$	−8.43458e6	−1.32923
$527$	−7.61854e6	−1.19494
$528$	−1.00059e6	−0.156197
$529$	2.22758e6	0.346095
$530$	0	0
$531$	668090.	0.102825
$532$	−1.95835e6	−0.299993
$533$	−4.27959e6	−0.652505
$534$	6.35331e6	0.964155
$535$	0	0
$536$	1.21644e6	0.182885
$537$	−2.78972e6	−0.417469
$538$	1.83574e7	2.73435
$539$	−1.20735e6	−0.179003
$540$	0	0
$541$	4.87769e6	0.716508	0.358254	−	0.933624i	$-0.383372\pi$
$541$	4.87769e6	0.716508	0.358254	+	0.933624i	$0.383372\pi$
$542$	−1.77814e7	−2.59997
$543$	5.90713e6	0.859759
$544$	−1.22594e7	−1.77611
$545$	0	0
$546$	−8.78402e6	−1.26099
$547$	3.49347e6	0.499217	0.249608	−	0.968347i	$-0.419698\pi$
$547$	3.49347e6	0.499217	0.249608	+	0.968347i	$0.419698\pi$
$548$	9.53719e6	1.35665
$549$	−189724.	−0.0268653
$550$	0	0
$551$	965280.	0.135449
$552$	−651580.	−0.0910164
$553$	1.42737e7	1.98483
$554$	8.52128e6	1.17959
$555$	0	0
$556$	1.13489e7	1.55692
$557$	1.34838e7	1.84151	0.920757	−	0.390137i	$-0.127572\pi$
$557$	1.34838e7	1.84151	0.920757	+	0.390137i	$0.127572\pi$
$558$	−3.42333e6	−0.465439
$559$	1.21138e6	0.163965
$560$	0	0
$561$	−1.60690e6	−0.215567
$562$	1.10035e7	1.46956
$563$	2.07215e6	0.275519	0.137759	−	0.990466i	$-0.456010\pi$
$563$	2.07215e6	0.275519	0.137759	+	0.990466i	$0.456010\pi$
$564$	−1.88429e6	−0.249430
$565$	0	0
$566$	6.77847e6	0.889387
$567$	1.07378e6	0.140268
$568$	606133.	0.0788310
$569$	1.11686e7	1.44617	0.723085	−	0.690759i	$-0.242723\pi$
$569$	1.11686e7	1.44617	0.723085	+	0.690759i	$0.242723\pi$
$570$	0	0
$571$	1.04065e7	1.33572	0.667861	−	0.744286i	$-0.267210\pi$
$571$	1.04065e7	1.33572	0.667861	+	0.744286i	$0.267210\pi$
$572$	3.08577e6	0.394343
$573$	3.03045e6	0.385585
$574$	−7.86956e6	−0.996944
$575$	0	0
$576$	−3.12707e6	−0.392718
$577$	4.15724e6	0.519835	0.259918	−	0.965631i	$-0.416305\pi$
$577$	4.15724e6	0.519835	0.259918	+	0.965631i	$0.416305\pi$
$578$	−6.20037e6	−0.771965
$579$	−5.39209e6	−0.668438
$580$	0	0
$581$	−1.48275e7	−1.82233
$582$	−964249.	−0.118000
$583$	3.33443e6	0.406303
$584$	−565734.	−0.0686405
$585$	0	0
$586$	−1.07482e7	−1.29298
$587$	−8.38623e6	−1.00455	−0.502275	−	0.864708i	$-0.667504\pi$
$587$	−8.38623e6	−1.00455	−0.502275	+	0.864708i	$0.667504\pi$
$588$	−3.14353e6	−0.374951
$589$	−1.76493e6	−0.209623
$590$	0	0
$591$	−3.28519e6	−0.386894
$592$	3.90951e6	0.458477
$593$	−2.10957e6	−0.246353	−0.123177	−	0.992385i	$-0.539308\pi$
$593$	−2.10957e6	−0.246353	−0.123177	+	0.992385i	$0.539308\pi$
$594$	−722048.	−0.0839653
$595$	0	0
$596$	−1.13839e7	−1.31273
$597$	−3.00480e6	−0.345048
$598$	−1.75534e7	−2.00728
$599$	3.77702e6	0.430113	0.215057	−	0.976602i	$-0.431006\pi$
$599$	3.77702e6	0.430113	0.215057	+	0.976602i	$0.431006\pi$
$600$	0	0
$601$	7.88038e6	0.889940	0.444970	−	0.895545i	$-0.353214\pi$
$601$	7.88038e6	0.889940	0.444970	+	0.895545i	$0.353214\pi$
$602$	2.22756e6	0.250517
$603$	−4.00597e6	−0.448657
$604$	−1.39719e7	−1.55835
$605$	0	0
$606$	2.52638e6	0.279459
$607$	5.43393e6	0.598608	0.299304	−	0.954158i	$-0.403246\pi$
$607$	5.43393e6	0.598608	0.299304	+	0.954158i	$0.403246\pi$
$608$	−2.84003e6	−0.311576
$609$	4.15936e6	0.454447
$610$	0	0
$611$	−4.35741e6	−0.472199
$612$	−4.18383e6	−0.451539
$613$	−1.36825e7	−1.47067	−0.735333	−	0.677706i	$-0.762974\pi$
$613$	−1.36825e7	−1.47067	−0.735333	+	0.677706i	$0.762974\pi$
$614$	1.13153e6	0.121128
$615$	0	0
$616$	487079.	0.0517187
$617$	−1.24288e7	−1.31437	−0.657184	−	0.753730i	$-0.728253\pi$
$617$	−1.24288e7	−1.31437	−0.657184	+	0.753730i	$0.728253\pi$
$618$	2.41446e6	0.254302
$619$	1.29365e7	1.35704	0.678518	−	0.734583i	$-0.262622\pi$
$619$	1.29365e7	1.35704	0.678518	+	0.734583i	$0.262622\pi$
$620$	0	0
$621$	2.14578e6	0.223283
$622$	−9.62988e6	−0.998033
$623$	1.41140e7	1.45690
$624$	−6.02453e6	−0.619386
$625$	0	0
$626$	7.53032e6	0.768029
$627$	−372258.	−0.0378160
$628$	2.90091e6	0.293518
$629$	6.27848e6	0.632744
$630$	0	0
$631$	572856.	0.0572759	0.0286379	−	0.999590i	$-0.490883\pi$
$631$	572856.	0.0572759	0.0286379	+	0.999590i	$0.490883\pi$
$632$	−2.14514e6	−0.213631
$633$	8.82075e6	0.874976
$634$	−1.60856e7	−1.58933
$635$	0	0
$636$	8.68172e6	0.851066
$637$	−7.26940e6	−0.709823
$638$	−2.79689e6	−0.272034
$639$	−1.99612e6	−0.193390
$640$	0	0
$641$	2.13431e6	0.205170	0.102585	−	0.994724i	$-0.467289\pi$
$641$	2.13431e6	0.205170	0.102585	+	0.994724i	$0.467289\pi$
$642$	−1.49766e7	−1.43409
$643$	−4.15866e6	−0.396667	−0.198333	−	0.980135i	$-0.563553\pi$
$643$	−4.15866e6	−0.396667	−0.198333	+	0.980135i	$0.563553\pi$
$644$	−1.68629e7	−1.60220
$645$	0	0
$646$	−4.12887e6	−0.389269
$647$	3.44527e6	0.323565	0.161783	−	0.986826i	$-0.448276\pi$
$647$	3.44527e6	0.323565	0.161783	+	0.986826i	$0.448276\pi$
$648$	−161375.	−0.0150973
$649$	−998012.	−0.0930088
$650$	0	0
$651$	−7.60500e6	−0.703310
$652$	1.09100e7	1.00509
$653$	2.88090e6	0.264390	0.132195	−	0.991224i	$-0.457798\pi$
$653$	2.88090e6	0.264390	0.132195	+	0.991224i	$0.457798\pi$
$654$	1.49943e7	1.37083
$655$	0	0
$656$	−5.39735e6	−0.489689
$657$	1.86307e6	0.168390
$658$	−8.01265e6	−0.721459
$659$	1.31524e7	1.17975	0.589876	−	0.807494i	$-0.299177\pi$
$659$	1.31524e7	1.17975	0.589876	+	0.807494i	$0.299177\pi$
$660$	0	0
$661$	−4.14045e6	−0.368591	−0.184295	−	0.982871i	$-0.559000\pi$
$661$	−4.14045e6	−0.368591	−0.184295	+	0.982871i	$0.559000\pi$
$662$	−1.62389e7	−1.44017
$663$	−9.67510e6	−0.854814
$664$	2.22838e6	0.196141
$665$	0	0
$666$	2.82118e6	0.246459
$667$	8.31179e6	0.723402
$668$	4.55057e6	0.394570
$669$	−728710.	−0.0629491
$670$	0	0
$671$	283415.	0.0243006
$672$	−1.22376e7	−1.04538
$673$	8.46081e6	0.720070	0.360035	−	0.932939i	$-0.382765\pi$
$673$	8.46081e6	0.720070	0.360035	+	0.932939i	$0.382765\pi$
$674$	−2.68860e7	−2.27969
$675$	0	0
$676$	5.58229e6	0.469835
$677$	7.85095e6	0.658340	0.329170	−	0.944271i	$-0.393231\pi$
$677$	7.85095e6	0.658340	0.329170	+	0.944271i	$0.393231\pi$
$678$	1.03822e7	0.867391
$679$	−2.14210e6	−0.178306
$680$	0	0
$681$	4.71858e6	0.389891
$682$	5.11386e6	0.421005
$683$	1.84419e7	1.51271	0.756354	−	0.654163i	$-0.226979\pi$
$683$	1.84419e7	1.51271	0.756354	+	0.654163i	$0.226979\pi$
$684$	−969234.	−0.0792116
$685$	0	0
$686$	9.14853e6	0.742234
$687$	−7.34866e6	−0.594040
$688$	1.52777e6	0.123052
$689$	2.00765e7	1.61116
$690$	0	0
$691$	−1.74029e7	−1.38652	−0.693259	−	0.720688i	$-0.743826\pi$
$691$	−1.74029e7	−1.38652	−0.693259	+	0.720688i	$0.743826\pi$
$692$	−2.60618e7	−2.06890
$693$	−1.60405e6	−0.126877
$694$	1.82627e7	1.43935
$695$	0	0
$696$	−625097.	−0.0489130
$697$	−8.66787e6	−0.675819
$698$	−9.38353e6	−0.729000
$699$	−3.05081e6	−0.236169
$700$	0	0
$701$	−4.85133e6	−0.372877	−0.186438	−	0.982467i	$-0.559695\pi$
$701$	−4.85133e6	−0.372877	−0.186438	+	0.982467i	$0.559695\pi$
$702$	−4.34742e6	−0.332958
$703$	1.45448e6	0.110999
$704$	4.67130e6	0.355227
$705$	0	0
$706$	9.60010e6	0.724877
$707$	5.61242e6	0.422281
$708$	−2.59848e6	−0.194822
$709$	−7.10939e6	−0.531150	−0.265575	−	0.964090i	$-0.585562\pi$
$709$	−7.10939e6	−0.531150	−0.265575	+	0.964090i	$0.585562\pi$
$710$	0	0
$711$	7.06438e6	0.524083
$712$	−2.12115e6	−0.156809
$713$	−1.51973e7	−1.11955
$714$	−1.77911e7	−1.30604
$715$	0	0
$716$	1.08504e7	0.790974
$717$	−884218.	−0.0642335
$718$	−5.87835e6	−0.425544
$719$	1.99693e6	0.144059	0.0720297	−	0.997402i	$-0.477052\pi$
$719$	1.99693e6	0.144059	0.0720297	+	0.997402i	$0.477052\pi$
$720$	0	0
$721$	5.36379e6	0.384267
$722$	1.93120e7	1.37874
$723$	−9.02760e6	−0.642283
$724$	−2.29753e7	−1.62898
$725$	0	0
$726$	1.07861e6	0.0759495
$727$	−1.47012e7	−1.03161	−0.515806	−	0.856705i	$-0.672507\pi$
$727$	−1.47012e7	−1.03161	−0.515806	+	0.856705i	$0.672507\pi$
$728$	2.93268e6	0.205086
$729$	531441.	0.0370370
$730$	0	0
$731$	2.45353e6	0.169823
$732$	737917.	0.0509014
$733$	1.03899e7	0.714249	0.357125	−	0.934057i	$-0.383757\pi$
$733$	1.03899e7	0.714249	0.357125	+	0.934057i	$0.383757\pi$
$734$	7.64166e6	0.523538
$735$	0	0
$736$	−2.44548e7	−1.66406
$737$	5.98422e6	0.405825
$738$	−3.89483e6	−0.263238
$739$	1.40217e7	0.944470	0.472235	−	0.881473i	$-0.343447\pi$
$739$	1.40217e7	0.944470	0.472235	+	0.881473i	$0.343447\pi$
$740$	0	0
$741$	−2.24135e6	−0.149956
$742$	3.69178e7	2.46165
$743$	1.62282e7	1.07845	0.539224	−	0.842162i	$-0.318717\pi$
$743$	1.62282e7	1.07845	0.539224	+	0.842162i	$0.318717\pi$
$744$	1.14293e6	0.0756986
$745$	0	0
$746$	−3.67010e7	−2.41452
$747$	−7.33849e6	−0.481178
$748$	6.24992e6	0.408433
$749$	−3.32708e7	−2.16700
$750$	0	0
$751$	−2.21572e7	−1.43356	−0.716779	−	0.697300i	$-0.754385\pi$
$751$	−2.21572e7	−1.43356	−0.716779	+	0.697300i	$0.754385\pi$
$752$	−5.49549e6	−0.354374
$753$	−8.90676e6	−0.572443
$754$	−1.68400e7	−1.07873
$755$	0	0
$756$	−4.17640e6	−0.265765
$757$	2.95113e7	1.87176	0.935878	−	0.352325i	$-0.114609\pi$
$757$	2.95113e7	1.87176	0.935878	+	0.352325i	$0.114609\pi$
$758$	−2.13764e7	−1.35133
$759$	−3.20542e6	−0.201967
$760$	0	0
$761$	1.82482e7	1.14224	0.571122	−	0.820865i	$-0.306508\pi$
$761$	1.82482e7	1.14224	0.571122	+	0.820865i	$0.306508\pi$
$762$	197936.	0.0123491
$763$	3.33103e7	2.07141
$764$	−1.17867e7	−0.730564
$765$	0	0
$766$	2.35339e7	1.44918
$767$	−6.00899e6	−0.368819
$768$	−7.42380e6	−0.454175
$769$	−1.89729e7	−1.15696	−0.578480	−	0.815697i	$-0.696354\pi$
$769$	−1.89729e7	−1.15696	−0.578480	+	0.815697i	$0.696354\pi$
$770$	0	0
$771$	−1.03725e7	−0.628417
$772$	2.09721e7	1.26648
$773$	9.39706e6	0.565644	0.282822	−	0.959172i	$-0.408729\pi$
$773$	9.39706e6	0.565644	0.282822	+	0.959172i	$0.408729\pi$
$774$	1.10247e6	0.0661477
$775$	0	0
$776$	321930.	0.0191914
$777$	6.26732e6	0.372417
$778$	−1.89278e6	−0.112112
$779$	−2.00802e6	−0.118556
$780$	0	0
$781$	2.98185e6	0.174928
$782$	−3.55527e7	−2.07900
$783$	2.05857e6	0.119994
$784$	−9.16804e6	−0.532705
$785$	0	0
$786$	−2.06975e7	−1.19498
$787$	2.29536e7	1.32103	0.660517	−	0.750811i	$-0.270337\pi$
$787$	2.29536e7	1.32103	0.660517	+	0.750811i	$0.270337\pi$
$788$	1.27775e7	0.733044
$789$	−9.27370e6	−0.530347
$790$	0	0
$791$	2.30643e7	1.31069
$792$	241067.	0.0136560
$793$	1.70643e6	0.0963620
$794$	−1.65237e7	−0.930157
$795$	0	0
$796$	1.16869e7	0.653759
$797$	1.05979e7	0.590984	0.295492	−	0.955345i	$-0.404516\pi$
$797$	1.05979e7	0.590984	0.295492	+	0.955345i	$0.404516\pi$
$798$	−4.12153e6	−0.229114
$799$	−8.82549e6	−0.489071
$800$	0	0
$801$	6.98538e6	0.384688
$802$	3.71325e7	2.03854
$803$	−2.78311e6	−0.152315
$804$	1.55809e7	0.850065
$805$	0	0
$806$	3.07903e7	1.66946
$807$	2.01836e7	1.09098
$808$	−843473.	−0.0454509
$809$	−1.95635e7	−1.05093	−0.525467	−	0.850814i	$-0.676109\pi$
$809$	−1.95635e7	−1.05093	−0.525467	+	0.850814i	$0.676109\pi$
$810$	0	0
$811$	−3.23306e7	−1.72609	−0.863043	−	0.505131i	$-0.831444\pi$
$811$	−3.23306e7	−1.72609	−0.863043	+	0.505131i	$0.831444\pi$
$812$	−1.61775e7	−0.861036
$813$	−1.95504e7	−1.03736
$814$	−4.21436e6	−0.222931
$815$	0	0
$816$	−1.22021e7	−0.641517
$817$	568389.	0.0297914
$818$	−5.14921e6	−0.269065
$819$	−9.65790e6	−0.503122
$820$	0	0
$821$	3.15924e7	1.63578	0.817890	−	0.575375i	$-0.195144\pi$
$821$	3.15924e7	1.63578	0.817890	+	0.575375i	$0.195144\pi$
$822$	2.00719e7	1.03612
$823$	−1.52405e7	−0.784333	−0.392167	−	0.919894i	$-0.628274\pi$
$823$	−1.52405e7	−0.784333	−0.392167	+	0.919894i	$0.628274\pi$
$824$	−806107.	−0.0413594
$825$	0	0
$826$	−1.10497e7	−0.563507
$827$	−2.87105e7	−1.45975	−0.729873	−	0.683583i	$-0.760421\pi$
$827$	−2.87105e7	−1.45975	−0.729873	+	0.683583i	$0.760421\pi$
$828$	−8.34584e6	−0.423052
$829$	−2.45816e7	−1.24229	−0.621146	−	0.783695i	$-0.713333\pi$
$829$	−2.45816e7	−1.24229	−0.621146	+	0.783695i	$0.713333\pi$
$830$	0	0
$831$	9.36903e6	0.470643
$832$	2.81257e7	1.40862
$833$	−1.47234e7	−0.735185
$834$	2.38848e7	1.18907
$835$	0	0
$836$	1.44787e6	0.0716495
$837$	−3.76390e6	−0.185705
$838$	−8.20334e6	−0.403534
$839$	−1.59309e6	−0.0781333	−0.0390667	−	0.999237i	$-0.512438\pi$
$839$	−1.59309e6	−0.0781333	−0.0390667	+	0.999237i	$0.512438\pi$
$840$	0	0
$841$	−1.25372e7	−0.611238
$842$	1.17340e7	0.570383
$843$	1.20981e7	0.586341
$844$	−3.43076e7	−1.65781
$845$	0	0
$846$	−3.96565e6	−0.190497
$847$	2.39617e6	0.114765
$848$	2.53201e7	1.20914
$849$	7.45283e6	0.354856
$850$	0	0
$851$	1.25242e7	0.592825
$852$	7.76373e6	0.366414
$853$	−9.71355e6	−0.457094	−0.228547	−	0.973533i	$-0.573397\pi$
$853$	−9.71355e6	−0.457094	−0.228547	+	0.973533i	$0.573397\pi$
$854$	3.13789e6	0.147229
$855$	0	0
$856$	5.00017e6	0.233239
$857$	2.81098e7	1.30739	0.653696	−	0.756758i	$-0.273218\pi$
$857$	2.81098e7	1.30739	0.653696	+	0.756758i	$0.273218\pi$
$858$	6.49429e6	0.301171
$859$	1.96952e7	0.910703	0.455352	−	0.890312i	$-0.349514\pi$
$859$	1.96952e7	0.910703	0.455352	+	0.890312i	$0.349514\pi$
$860$	0	0
$861$	−8.65247e6	−0.397770
$862$	3.93383e7	1.80322
$863$	9.07936e6	0.414981	0.207490	−	0.978237i	$-0.433470\pi$
$863$	9.07936e6	0.414981	0.207490	+	0.978237i	$0.433470\pi$
$864$	−6.05667e6	−0.276026
$865$	0	0
$866$	−1.25655e7	−0.569359
$867$	−6.81722e6	−0.308006
$868$	2.95791e7	1.33255
$869$	−1.05530e7	−0.474051
$870$	0	0
$871$	3.60308e7	1.60927
$872$	−5.00609e6	−0.222950
$873$	−1.06018e6	−0.0470807
$874$	−8.23619e6	−0.364710
$875$	0	0
$876$	−7.24628e6	−0.319047
$877$	−1.35513e7	−0.594954	−0.297477	−	0.954729i	$-0.596145\pi$
$877$	−1.35513e7	−0.594954	−0.297477	+	0.954729i	$0.596145\pi$
$878$	−5.85187e7	−2.56188
$879$	−1.18175e7	−0.515884
$880$	0	0
$881$	3.41684e7	1.48315	0.741574	−	0.670871i	$-0.234080\pi$
$881$	3.41684e7	1.48315	0.741574	+	0.670871i	$0.234080\pi$
$882$	−6.61584e6	−0.286361
$883$	4.27723e7	1.84612	0.923061	−	0.384654i	$-0.125679\pi$
$883$	4.27723e7	1.84612	0.923061	+	0.384654i	$0.125679\pi$
$884$	3.76305e7	1.61961
$885$	0	0
$886$	5.82857e6	0.249447
$887$	1.84459e7	0.787210	0.393605	−	0.919280i	$-0.371228\pi$
$887$	1.84459e7	0.787210	0.393605	+	0.919280i	$0.371228\pi$
$888$	−941896.	−0.0400839
$889$	439719.	0.0186604
$890$	0	0
$891$	−793881.	−0.0335013
$892$	2.83426e6	0.119269
$893$	−2.04453e6	−0.0857955
$894$	−2.39585e7	−1.00257
$895$	0	0
$896$	8.20777e6	0.341551
$897$	−1.92997e7	−0.800885
$898$	2.60645e7	1.07860
$899$	−1.45797e7	−0.601656
$900$	0	0
$901$	4.06628e7	1.66873
$902$	5.81821e6	0.238107
$903$	2.44917e6	0.0999537
$904$	−3.46626e6	−0.141072
$905$	0	0
$906$	−2.94052e7	−1.19016
$907$	4.12614e7	1.66543	0.832713	−	0.553704i	$-0.186786\pi$
$907$	4.12614e7	1.66543	0.832713	+	0.553704i	$0.186786\pi$
$908$	−1.83525e7	−0.738723
$909$	2.77772e6	0.111501
$910$	0	0
$911$	2.05996e7	0.822360	0.411180	−	0.911554i	$-0.365117\pi$
$911$	2.05996e7	0.822360	0.411180	+	0.911554i	$0.365117\pi$
$912$	−2.82675e6	−0.112538
$913$	1.09624e7	0.435241
$914$	6.76930e7	2.68027
$915$	0	0
$916$	2.85820e7	1.12552
$917$	−4.59801e7	−1.80570
$918$	−8.80526e6	−0.344854
$919$	5.10704e6	0.199471	0.0997356	−	0.995014i	$-0.468200\pi$
$919$	5.10704e6	0.199471	0.0997356	+	0.995014i	$0.468200\pi$
$920$	0	0
$921$	1.24410e6	0.0483290
$922$	−6.27067e7	−2.42933
$923$	1.79536e7	0.693661
$924$	6.23881e6	0.240393
$925$	0	0
$926$	1.66207e6	0.0636975
$927$	2.65467e6	0.101464
$928$	−2.34609e7	−0.894281
$929$	−213931.	−0.00813271	−0.00406636	−	0.999992i	$-0.501294\pi$
$929$	−213931.	−0.00813271	−0.00406636	+	0.999992i	$0.501294\pi$
$930$	0	0
$931$	−3.41086e6	−0.128970
$932$	1.18659e7	0.447466
$933$	−1.05879e7	−0.398205
$934$	2.69084e7	1.00930
$935$	0	0
$936$	1.45146e6	0.0541520
$937$	1.71430e7	0.637878	0.318939	−	0.947775i	$-0.396674\pi$
$937$	1.71430e7	0.637878	0.318939	+	0.947775i	$0.396674\pi$
$938$	6.62555e7	2.45875
$939$	8.27948e6	0.306436
$940$	0	0
$941$	3.27424e7	1.20542	0.602708	−	0.797962i	$-0.294088\pi$
$941$	3.27424e7	1.20542	0.602708	+	0.797962i	$0.294088\pi$
$942$	6.10523e6	0.224169
$943$	−1.72905e7	−0.633183
$944$	−7.57844e6	−0.276789
$945$	0	0
$946$	−1.64690e6	−0.0598329
$947$	−2.30829e7	−0.836403	−0.418201	−	0.908354i	$-0.637339\pi$
$947$	−2.30829e7	−0.836403	−0.418201	+	0.908354i	$0.637339\pi$
$948$	−2.74763e7	−0.992975
$949$	−1.67570e7	−0.603991
$950$	0	0
$951$	−1.76859e7	−0.634128
$952$	5.93985e6	0.212414
$953$	2.57216e7	0.917414	0.458707	−	0.888587i	$-0.348313\pi$
$953$	2.57216e7	0.917414	0.458707	+	0.888587i	$0.348313\pi$
$954$	1.82715e7	0.649984
$955$	0	0
$956$	3.43910e6	0.121703
$957$	−3.07514e6	−0.108539
$958$	2.52925e7	0.890385
$959$	4.45902e7	1.56564
$960$	0	0
$961$	−1.97158e6	−0.0688663
$962$	−2.53745e7	−0.884014
$963$	−1.64665e7	−0.572185
$964$	3.51121e7	1.21693
$965$	0	0
$966$	−3.54895e7	−1.22365
$967$	1.86464e6	0.0641253	0.0320627	−	0.999486i	$-0.489792\pi$
$967$	1.86464e6	0.0641253	0.0320627	+	0.999486i	$0.489792\pi$
$968$	−360113.	−0.0123524
$969$	−4.53963e6	−0.155314
$970$	0	0
$971$	2.90344e7	0.988245	0.494122	−	0.869392i	$-0.335489\pi$
$971$	2.90344e7	0.988245	0.494122	+	0.869392i	$0.335489\pi$
$972$	−2.06700e6	−0.0701737
$973$	5.30606e7	1.79676
$974$	1.94042e7	0.655387
$975$	0	0
$976$	2.15212e6	0.0723174
$977$	−5.86267e6	−0.196498	−0.0982492	−	0.995162i	$-0.531324\pi$
$977$	−5.86267e6	−0.196498	−0.0982492	+	0.995162i	$0.531324\pi$
$978$	2.29612e7	0.767621
$979$	−1.04349e7	−0.347963
$980$	0	0
$981$	1.64860e7	0.546945
$982$	6.19161e7	2.04892
$983$	−2.93292e7	−0.968093	−0.484047	−	0.875042i	$-0.660833\pi$
$983$	−2.93292e7	−0.968093	−0.484047	+	0.875042i	$0.660833\pi$
$984$	1.30035e6	0.0428128
$985$	0	0
$986$	−3.41076e7	−1.11727
$987$	−8.80980e6	−0.287855
$988$	8.71756e6	0.284120
$989$	4.89426e6	0.159109
$990$	0	0
$991$	2.75751e7	0.891935	0.445968	−	0.895049i	$-0.352860\pi$
$991$	2.75751e7	0.891935	0.445968	+	0.895049i	$0.352860\pi$
$992$	4.28960e7	1.38401
$993$	−1.78545e7	−0.574611
$994$	3.30141e7	1.05982
$995$	0	0
$996$	2.85425e7	0.911682
$997$	1.69502e7	0.540052	0.270026	−	0.962853i	$-0.412968\pi$
$997$	1.69502e7	0.540052	0.270026	+	0.962853i	$0.412968\pi$
$998$	1.28317e7	0.407811
$999$	3.10185e6	0.0983347

Twists

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

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

\(f(q)\)	\(=\)	\(q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} -121.000 q^{11} -315.043 q^{12} -728.536 q^{13} -1339.68 q^{14} -918.818 q^{16} -1475.58 q^{17} -663.037 q^{18} -341.835 q^{19} -1472.95 q^{21} +990.463 q^{22} -2943.46 q^{23} +221.366 q^{24} +5963.54 q^{26} -729.000 q^{27} +5728.94 q^{28} -2823.82 q^{29} +5163.10 q^{31} +8308.19 q^{32} +1089.00 q^{33} +12078.5 q^{34} +2835.39 q^{36} -4254.93 q^{37} +2798.14 q^{38} +6556.83 q^{39} +5874.23 q^{41} +12057.1 q^{42} -1662.76 q^{43} -4235.58 q^{44} +24094.1 q^{46} +5981.04 q^{47} +8269.36 q^{48} +9978.09 q^{49} +13280.2 q^{51} -25502.3 q^{52} -27557.3 q^{53} +5967.34 q^{54} -4025.45 q^{56} +3076.51 q^{57} +23114.8 q^{58} +8248.03 q^{59} -2342.27 q^{61} -42263.3 q^{62} +13256.6 q^{63} -38605.8 q^{64} -8914.17 q^{66} -49456.4 q^{67} -51652.2 q^{68} +26491.1 q^{69} -24643.4 q^{71} -1992.29 q^{72} +23000.9 q^{73} +34829.4 q^{74} -11965.9 q^{76} -19803.0 q^{77} -53671.9 q^{78} +87214.6 q^{79} +6561.00 q^{81} -48084.4 q^{82} -90598.7 q^{83} -51560.4 q^{84} +13610.8 q^{86} +25414.4 q^{87} +2976.14 q^{88} +86239.2 q^{89} -119233. q^{91} -103035. q^{92} -46467.9 q^{93} -48958.7 q^{94} -74773.8 q^{96} -13088.6 q^{97} -81677.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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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