LMFDB - Embedded newform 825.6.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 $ x^8 - x^7 - 176x^6 + 272x^5 + 9055x^4 - 15851x^3 - 118840x^2 + 149572x - 33248
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.292443$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29244 q^{2} -9.00000 q^{3} -30.3296 q^{4} -11.6320 q^{6} +91.7671 q^{7} -80.5574 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+1.29244 q^{2} -9.00000 q^{3} -30.3296 q^{4} -11.6320 q^{6} +91.7671 q^{7} -80.5574 q^{8} +81.0000 q^{9} -121.000 q^{11} +272.966 q^{12} +70.3337 q^{13} +118.604 q^{14} +866.431 q^{16} -1175.46 q^{17} +104.688 q^{18} -961.502 q^{19} -825.904 q^{21} -156.386 q^{22} +1303.06 q^{23} +725.017 q^{24} +90.9024 q^{26} -729.000 q^{27} -2783.26 q^{28} -3346.41 q^{29} +6451.89 q^{31} +3697.65 q^{32} +1089.00 q^{33} -1519.21 q^{34} -2456.70 q^{36} -13216.4 q^{37} -1242.69 q^{38} -633.004 q^{39} +18467.9 q^{41} -1067.43 q^{42} +19255.1 q^{43} +3669.88 q^{44} +1684.14 q^{46} +17064.0 q^{47} -7797.88 q^{48} -8385.80 q^{49} +10579.1 q^{51} -2133.19 q^{52} -31237.8 q^{53} -942.191 q^{54} -7392.52 q^{56} +8653.52 q^{57} -4325.05 q^{58} +43403.2 q^{59} -3568.78 q^{61} +8338.70 q^{62} +7433.14 q^{63} -22946.8 q^{64} +1407.47 q^{66} +70764.0 q^{67} +35651.1 q^{68} -11727.6 q^{69} +41227.7 q^{71} -6525.15 q^{72} -41255.5 q^{73} -17081.4 q^{74} +29162.0 q^{76} -11103.8 q^{77} -818.121 q^{78} -57458.9 q^{79} +6561.00 q^{81} +23868.7 q^{82} -17233.5 q^{83} +25049.3 q^{84} +24886.1 q^{86} +30117.7 q^{87} +9747.45 q^{88} -24409.7 q^{89} +6454.32 q^{91} -39521.4 q^{92} -58067.0 q^{93} +22054.3 q^{94} -33278.9 q^{96} -126547. q^{97} -10838.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) $ 8 * q + 9 * q^2 - 72 * q^3 + 107 * q^4 - 81 * q^6 - 66 * q^7 + 207 * q^8 + 648 * q^9	\( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 8 * q + 9 * q^2 - 72 * q^3 + 107 * q^4 - 81 * q^6 - 66 * q^7 + 207 * q^8 + 648 * q^9 - 968 * q^11 - 963 * q^12 + 382 * q^13 + 1048 * q^14 - 325 * q^16 + 288 * q^17 + 729 * q^18 - 988 * q^19 + 594 * q^21 - 1089 * q^22 + 5972 * q^23 - 1863 * q^24 - 14579 * q^26 - 5832 * q^27 + 5942 * q^28 + 1032 * q^29 - 4682 * q^31 + 3863 * q^32 + 8712 * q^33 + 2206 * q^34 + 8667 * q^36 - 17200 * q^37 + 11011 * q^38 - 3438 * q^39 - 13220 * q^41 - 9432 * q^42 + 22872 * q^43 - 12947 * q^44 + 9101 * q^46 + 6700 * q^47 + 2925 * q^48 - 43466 * q^49 - 2592 * q^51 + 5009 * q^52 + 6224 * q^53 - 6561 * q^54 + 20992 * q^56 + 8892 * q^57 + 33015 * q^58 - 77556 * q^59 + 11554 * q^61 + 12135 * q^62 - 5346 * q^63 - 149917 * q^64 + 9801 * q^66 + 20894 * q^67 + 91776 * q^68 - 53748 * q^69 - 21648 * q^71 + 16767 * q^72 + 64660 * q^73 - 179522 * q^74 + 24401 * q^76 + 7986 * q^77 + 131211 * q^78 - 22660 * q^79 + 52488 * q^81 - 56080 * q^82 + 100390 * q^83 - 53478 * q^84 + 47271 * q^86 - 9288 * q^87 - 25047 * q^88 - 25578 * q^89 + 73250 * q^91 + 95311 * q^92 + 42138 * q^93 - 170120 * q^94 - 34767 * q^96 + 142828 * q^97 + 303397 * q^98 - 78408 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.29244	0.228474	0.114237	−	0.993454i	$-0.463558\pi$
$2$	1.29244	0.228474	0.114237	+	0.993454i	$0.463558\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−30.3296	−0.947800
$5$	0	0
$5$	0	0
$6$	−11.6320	−0.131909
$7$	91.7671	0.707851	0.353926	−	0.935274i	$-0.384847\pi$
$7$	91.7671	0.707851	0.353926	+	0.935274i	$0.384847\pi$
$8$	−80.5574	−0.445021
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	272.966	0.547212
$13$	70.3337	0.115426	0.0577132	−	0.998333i	$-0.481619\pi$
$13$	70.3337	0.115426	0.0577132	+	0.998333i	$0.481619\pi$
$14$	118.604	0.161725
$15$	0	0
$16$	866.431	0.846124
$17$	−1175.46	−0.986471	−0.493235	−	0.869896i	$-0.664186\pi$
$17$	−1175.46	−0.986471	−0.493235	+	0.869896i	$0.664186\pi$
$18$	104.688	0.0761579
$19$	−961.502	−0.611035	−0.305517	−	0.952186i	$-0.598829\pi$
$19$	−961.502	−0.611035	−0.305517	+	0.952186i	$0.598829\pi$
$20$	0	0
$21$	−825.904	−0.408678
$22$	−156.386	−0.0688874
$23$	1303.06	0.513625	0.256813	−	0.966461i	$-0.417328\pi$
$23$	1303.06	0.513625	0.256813	+	0.966461i	$0.417328\pi$
$24$	725.017	0.256933
$25$	0	0
$26$	90.9024	0.0263719
$27$	−729.000	−0.192450
$28$	−2783.26	−0.670901
$29$	−3346.41	−0.738898	−0.369449	−	0.929251i	$-0.620454\pi$
$29$	−3346.41	−0.738898	−0.369449	+	0.929251i	$0.620454\pi$
$30$	0	0
$31$	6451.89	1.20582	0.602911	−	0.797809i	$-0.294008\pi$
$31$	6451.89	1.20582	0.602911	+	0.797809i	$0.294008\pi$
$32$	3697.65	0.638338
$33$	1089.00	0.174078
$34$	−1519.21	−0.225383
$35$	0	0
$36$	−2456.70	−0.315933
$37$	−13216.4	−1.58711	−0.793557	−	0.608496i	$-0.791773\pi$
$37$	−13216.4	−1.58711	−0.793557	+	0.608496i	$0.791773\pi$
$38$	−1242.69	−0.139605
$39$	−633.004	−0.0666415
$40$	0	0
$41$	18467.9	1.71577	0.857883	−	0.513844i	$-0.171779\pi$
$41$	18467.9	1.71577	0.857883	+	0.513844i	$0.171779\pi$
$42$	−1067.43	−0.0933722
$43$	19255.1	1.58809	0.794044	−	0.607861i	$-0.207972\pi$
$43$	19255.1	1.58809	0.794044	+	0.607861i	$0.207972\pi$
$44$	3669.88	0.285772
$45$	0	0
$46$	1684.14	0.117350
$47$	17064.0	1.12677	0.563387	−	0.826193i	$-0.309498\pi$
$47$	17064.0	1.12677	0.563387	+	0.826193i	$0.309498\pi$
$48$	−7797.88	−0.488510
$49$	−8385.80	−0.498947
$50$	0	0
$51$	10579.1	0.569539
$52$	−2133.19	−0.109401
$53$	−31237.8	−1.52753	−0.763767	−	0.645491i	$-0.776653\pi$
$53$	−31237.8	−1.52753	−0.763767	+	0.645491i	$0.776653\pi$
$54$	−942.191	−0.0439698
$55$	0	0
$56$	−7392.52	−0.315009
$57$	8653.52	0.352781
$58$	−4325.05	−0.168819
$59$	43403.2	1.62327	0.811637	−	0.584162i	$-0.198577\pi$
$59$	43403.2	1.62327	0.811637	+	0.584162i	$0.198577\pi$
$60$	0	0
$61$	−3568.78	−0.122799	−0.0613996	−	0.998113i	$-0.519556\pi$
$61$	−3568.78	−0.122799	−0.0613996	+	0.998113i	$0.519556\pi$
$62$	8338.70	0.275499
$63$	7433.14	0.235950
$64$	−22946.8	−0.700280
$65$	0	0
$66$	1407.47	0.0397722
$67$	70764.0	1.92586	0.962932	−	0.269745i	$-0.0869394\pi$
$67$	70764.0	1.92586	0.962932	+	0.269745i	$0.0869394\pi$
$68$	35651.1	0.934977
$69$	−11727.6	−0.296542
$70$	0	0
$71$	41227.7	0.970607	0.485304	−	0.874346i	$-0.338709\pi$
$71$	41227.7	0.970607	0.485304	+	0.874346i	$0.338709\pi$
$72$	−6525.15	−0.148340
$73$	−41255.5	−0.906097	−0.453048	−	0.891486i	$-0.649664\pi$
$73$	−41255.5	−0.906097	−0.453048	+	0.891486i	$0.649664\pi$
$74$	−17081.4	−0.362614
$75$	0	0
$76$	29162.0	0.579139
$77$	−11103.8	−0.213425
$78$	−818.121	−0.0152258
$79$	−57458.9	−1.03583	−0.517916	−	0.855431i	$-0.673292\pi$
$79$	−57458.9	−1.03583	−0.517916	+	0.855431i	$0.673292\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	23868.7	0.392008
$83$	−17233.5	−0.274587	−0.137293	−	0.990530i	$-0.543840\pi$
$83$	−17233.5	−0.274587	−0.137293	+	0.990530i	$0.543840\pi$
$84$	25049.3	0.387345
$85$	0	0
$86$	24886.1	0.362836
$87$	30117.7	0.426603
$88$	9747.45	0.134179
$89$	−24409.7	−0.326654	−0.163327	−	0.986572i	$-0.552223\pi$
$89$	−24409.7	−0.326654	−0.163327	+	0.986572i	$0.552223\pi$
$90$	0	0
$91$	6454.32	0.0817047
$92$	−39521.4	−0.486814
$93$	−58067.0	−0.696181
$94$	22054.3	0.257438
$95$	0	0
$96$	−33278.9	−0.368545
$97$	−126547.	−1.36560	−0.682799	−	0.730606i	$-0.739238\pi$
$97$	−126547.	−1.36560	−0.682799	+	0.730606i	$0.739238\pi$
$98$	−10838.2	−0.113996
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−89974.7	−0.877641	−0.438820	−	0.898575i	$-0.644604\pi$
$101$	−89974.7	−0.877641	−0.438820	+	0.898575i	$0.644604\pi$
$102$	13672.9	0.130125
$103$	106257.	0.986876	0.493438	−	0.869781i	$-0.335740\pi$
$103$	106257.	0.986876	0.493438	+	0.869781i	$0.335740\pi$
$104$	−5665.91	−0.0513672
$105$	0	0
$106$	−40373.1	−0.349002
$107$	90929.4	0.767794	0.383897	−	0.923376i	$-0.374582\pi$
$107$	90929.4	0.767794	0.383897	+	0.923376i	$0.374582\pi$
$108$	22110.3	0.182404
$109$	46968.1	0.378649	0.189324	−	0.981915i	$-0.439370\pi$
$109$	46968.1	0.378649	0.189324	+	0.981915i	$0.439370\pi$
$110$	0	0
$111$	118947.	0.916320
$112$	79509.9	0.598930
$113$	137709.	1.01453	0.507267	−	0.861789i	$-0.330656\pi$
$113$	137709.	1.01453	0.507267	+	0.861789i	$0.330656\pi$
$114$	11184.2	0.0806013
$115$	0	0
$116$	101495.	0.700327
$117$	5697.03	0.0384755
$118$	56096.2	0.370876
$119$	−107868.	−0.698274
$120$	0	0
$121$	14641.0	0.0909091
$122$	−4612.45	−0.0280564
$123$	−166211.	−0.990598
$124$	−195683.	−1.14288
$125$	0	0
$126$	9606.90	0.0539085
$127$	−25160.6	−0.138424	−0.0692120	−	0.997602i	$-0.522048\pi$
$127$	−25160.6	−0.138424	−0.0692120	+	0.997602i	$0.522048\pi$
$128$	−147982.	−0.798334
$129$	−173296.	−0.916883
$130$	0	0
$131$	−217042.	−1.10501	−0.552505	−	0.833510i	$-0.686328\pi$
$131$	−217042.	−1.10501	−0.552505	+	0.833510i	$0.686328\pi$
$132$	−33028.9	−0.164991
$133$	−88234.2	−0.432522
$134$	91458.5	0.440009
$135$	0	0
$136$	94691.8	0.439000
$137$	−24093.9	−0.109675	−0.0548373	−	0.998495i	$-0.517464\pi$
$137$	−24093.9	−0.109675	−0.0548373	+	0.998495i	$0.517464\pi$
$138$	−15157.2	−0.0677520
$139$	−359804.	−1.57953	−0.789766	−	0.613408i	$-0.789798\pi$
$139$	−359804.	−1.57953	−0.789766	+	0.613408i	$0.789798\pi$
$140$	0	0
$141$	−153576.	−0.650543
$142$	53284.5	0.221758
$143$	−8510.38	−0.0348024
$144$	70180.9	0.282041
$145$	0	0
$146$	−53320.4	−0.207019
$147$	75472.2	0.288067
$148$	400847.	1.50427
$149$	−307960.	−1.13639	−0.568197	−	0.822892i	$-0.692359\pi$
$149$	−307960.	−1.13639	−0.568197	+	0.822892i	$0.692359\pi$
$150$	0	0
$151$	−531554.	−1.89716	−0.948582	−	0.316531i	$-0.897482\pi$
$151$	−531554.	−1.89716	−0.948582	+	0.316531i	$0.897482\pi$
$152$	77456.1	0.271924
$153$	−95212.0	−0.328824
$154$	−14351.1	−0.0487621
$155$	0	0
$156$	19198.7	0.0631628
$157$	428858.	1.38856	0.694280	−	0.719705i	$-0.255723\pi$
$157$	428858.	1.38856	0.694280	+	0.719705i	$0.255723\pi$
$158$	−74262.3	−0.236660
$159$	281140.	0.881923
$160$	0	0
$161$	119578.	0.363570
$162$	8479.72	0.0253860
$163$	−20926.3	−0.0616912	−0.0308456	−	0.999524i	$-0.509820\pi$
$163$	−20926.3	−0.0616912	−0.0308456	+	0.999524i	$0.509820\pi$
$164$	−560124.	−1.62620
$165$	0	0
$166$	−22273.4	−0.0627358
$167$	−550264.	−1.52679	−0.763396	−	0.645931i	$-0.776469\pi$
$167$	−550264.	−1.52679	−0.763396	+	0.645931i	$0.776469\pi$
$168$	66532.7	0.181870
$169$	−366346.	−0.986677
$170$	0	0
$171$	−77881.6	−0.203678
$172$	−583999.	−1.50519
$173$	−586340.	−1.48948	−0.744740	−	0.667355i	$-0.767426\pi$
$173$	−586340.	−1.48948	−0.744740	+	0.667355i	$0.767426\pi$
$174$	38925.4	0.0974676
$175$	0	0
$176$	−104838.	−0.255116
$177$	−390629.	−0.937198
$178$	−31548.2	−0.0746318
$179$	556686.	1.29861	0.649303	−	0.760530i	$-0.275061\pi$
$179$	556686.	1.29861	0.649303	+	0.760530i	$0.275061\pi$
$180$	0	0
$181$	399427.	0.906235	0.453117	−	0.891451i	$-0.350312\pi$
$181$	399427.	0.906235	0.453117	+	0.891451i	$0.350312\pi$
$182$	8341.85	0.0186674
$183$	32119.0	0.0708981
$184$	−104972.	−0.228574
$185$	0	0
$186$	−75048.3	−0.159059
$187$	142230.	0.297432
$188$	−517545.	−1.06796
$189$	−66898.2	−0.136226
$190$	0	0
$191$	−615748.	−1.22129	−0.610646	−	0.791903i	$-0.709090\pi$
$191$	−615748.	−1.22129	−0.610646	+	0.791903i	$0.709090\pi$
$192$	206521.	0.404307
$193$	−422687.	−0.816819	−0.408409	−	0.912799i	$-0.633916\pi$
$193$	−422687.	−0.816819	−0.408409	+	0.912799i	$0.633916\pi$
$194$	−163555.	−0.312003
$195$	0	0
$196$	254338.	0.472902
$197$	−116768.	−0.214367	−0.107184	−	0.994239i	$-0.534183\pi$
$197$	−116768.	−0.214367	−0.107184	+	0.994239i	$0.534183\pi$
$198$	−12667.2	−0.0229625
$199$	−674634.	−1.20764	−0.603818	−	0.797123i	$-0.706354\pi$
$199$	−674634.	−1.20764	−0.603818	+	0.797123i	$0.706354\pi$
$200$	0	0
$201$	−636876.	−1.11190
$202$	−116287.	−0.200518
$203$	−307090.	−0.523030
$204$	−320860.	−0.539809
$205$	0	0
$206$	137331.	0.225475
$207$	105548.	0.171208
$208$	60939.3	0.0976651
$209$	116342.	0.184234
$210$	0	0
$211$	347243.	0.536941	0.268471	−	0.963288i	$-0.413482\pi$
$211$	347243.	0.536941	0.268471	+	0.963288i	$0.413482\pi$
$212$	947431.	1.44780
$213$	−371050.	−0.560380
$214$	117521.	0.175421
$215$	0	0
$216$	58726.4	0.0856444
$217$	592072.	0.853542
$218$	60703.6	0.0865113
$219$	371299.	0.523135
$220$	0	0
$221$	−82674.2	−0.113865
$222$	153733.	0.209355
$223$	−314163.	−0.423051	−0.211525	−	0.977373i	$-0.567843\pi$
$223$	−314163.	−0.423051	−0.211525	+	0.977373i	$0.567843\pi$
$224$	339323.	0.451849
$225$	0	0
$226$	177981.	0.231795
$227$	1.03995e6	1.33952	0.669759	−	0.742579i	$-0.266398\pi$
$227$	1.03995e6	1.33952	0.669759	+	0.742579i	$0.266398\pi$
$228$	−262458.	−0.334366
$229$	−192732.	−0.242866	−0.121433	−	0.992600i	$-0.538749\pi$
$229$	−192732.	−0.242866	−0.121433	+	0.992600i	$0.538749\pi$
$230$	0	0
$231$	99934.4	0.123221
$232$	269578.	0.328825
$233$	644411.	0.777631	0.388815	−	0.921316i	$-0.372884\pi$
$233$	644411.	0.777631	0.388815	+	0.921316i	$0.372884\pi$
$234$	7363.09	0.00879064
$235$	0	0
$236$	−1.31640e6	−1.53854
$237$	517130.	0.598038
$238$	−139414.	−0.159537
$239$	−605351.	−0.685508	−0.342754	−	0.939425i	$-0.611360\pi$
$239$	−605351.	−0.685508	−0.342754	+	0.939425i	$0.611360\pi$
$240$	0	0
$241$	−432527.	−0.479701	−0.239851	−	0.970810i	$-0.577098\pi$
$241$	−432527.	−0.479701	−0.239851	+	0.970810i	$0.577098\pi$
$242$	18922.7	0.0207703
$243$	−59049.0	−0.0641500
$244$	108240.	0.116389
$245$	0	0
$246$	−214819.	−0.226326
$247$	−67626.0	−0.0705296
$248$	−519748.	−0.536616
$249$	155102.	0.158533
$250$	0	0
$251$	−1.02207e6	−1.02400	−0.511998	−	0.858987i	$-0.671094\pi$
$251$	−1.02207e6	−1.02400	−0.511998	+	0.858987i	$0.671094\pi$
$252$	−225444.	−0.223634
$253$	−157671.	−0.154864
$254$	−32518.6	−0.0316263
$255$	0	0
$256$	543039.	0.517882
$257$	909641.	0.859087	0.429544	−	0.903046i	$-0.358674\pi$
$257$	909641.	0.859087	0.429544	+	0.903046i	$0.358674\pi$
$258$	−223975.	−0.209484
$259$	−1.21283e6	−1.12344
$260$	0	0
$261$	−271059.	−0.246299
$262$	−280515.	−0.252466
$263$	−999592.	−0.891115	−0.445557	−	0.895253i	$-0.646994\pi$
$263$	−999592.	−0.891115	−0.445557	+	0.895253i	$0.646994\pi$
$264$	−87727.1	−0.0774682
$265$	0	0
$266$	−114038.	−0.0988199
$267$	219687.	0.188594
$268$	−2.14624e6	−1.82533
$269$	−2.14720e6	−1.80922	−0.904611	−	0.426238i	$-0.859839\pi$
$269$	−2.14720e6	−1.80922	−0.904611	+	0.426238i	$0.859839\pi$
$270$	0	0
$271$	−743716.	−0.615154	−0.307577	−	0.951523i	$-0.599518\pi$
$271$	−743716.	−0.615154	−0.307577	+	0.951523i	$0.599518\pi$
$272$	−1.01845e6	−0.834677
$273$	−58088.9	−0.0471723
$274$	−31140.0	−0.0250578
$275$	0	0
$276$	355693.	0.281062
$277$	2.30153e6	1.80226	0.901131	−	0.433547i	$-0.142738\pi$
$277$	2.30153e6	1.80226	0.901131	+	0.433547i	$0.142738\pi$
$278$	−465026.	−0.360882
$279$	522603.	0.401940
$280$	0	0
$281$	974755.	0.736427	0.368214	−	0.929741i	$-0.379970\pi$
$281$	974755.	0.736427	0.368214	+	0.929741i	$0.379970\pi$
$282$	−198488.	−0.148632
$283$	1.23221e6	0.914577	0.457289	−	0.889318i	$-0.348821\pi$
$283$	1.23221e6	0.914577	0.457289	+	0.889318i	$0.348821\pi$
$284$	−1.25042e6	−0.919941
$285$	0	0
$286$	−10999.2	−0.00795143
$287$	1.69475e6	1.21451
$288$	299510.	0.212779
$289$	−38159.6	−0.0268757
$290$	0	0
$291$	1.13892e6	0.788429
$292$	1.25126e6	0.858798
$293$	−367972.	−0.250407	−0.125203	−	0.992131i	$-0.539958\pi$
$293$	−367972.	−0.250407	−0.125203	+	0.992131i	$0.539958\pi$
$294$	97543.5	0.0658158
$295$	0	0
$296$	1.06468e6	0.706299
$297$	88209.0	0.0580259
$298$	−398021.	−0.259636
$299$	91649.4	0.0592859
$300$	0	0
$301$	1.76698e6	1.12413
$302$	−687003.	−0.433452
$303$	809772.	0.506706
$304$	−833075.	−0.517011
$305$	0	0
$306$	−123056.	−0.0751276
$307$	760172.	0.460327	0.230163	−	0.973152i	$-0.426074\pi$
$307$	760172.	0.460327	0.230163	+	0.973152i	$0.426074\pi$
$308$	336774.	0.202284
$309$	−956309.	−0.569773
$310$	0	0
$311$	−2.20267e6	−1.29136	−0.645681	−	0.763607i	$-0.723426\pi$
$311$	−2.20267e6	−1.29136	−0.645681	+	0.763607i	$0.723426\pi$
$312$	50993.2	0.0296569
$313$	685921.	0.395743	0.197872	−	0.980228i	$-0.436597\pi$
$313$	685921.	0.395743	0.197872	+	0.980228i	$0.436597\pi$
$314$	554275.	0.317250
$315$	0	0
$316$	1.74270e6	0.981761
$317$	−1.87909e6	−1.05027	−0.525134	−	0.851019i	$-0.675985\pi$
$317$	−1.87909e6	−1.05027	−0.525134	+	0.851019i	$0.675985\pi$
$318$	363358.	0.201496
$319$	404916.	0.222786
$320$	0	0
$321$	−818364.	−0.443286
$322$	154548.	0.0830663
$323$	1.13020e6	0.602768
$324$	−198992.	−0.105311
$325$	0	0
$326$	−27046.0	−0.0140948
$327$	−422713.	−0.218613
$328$	−1.48773e6	−0.763553
$329$	1.56592e6	0.797588
$330$	0	0
$331$	1.94540e6	0.975974	0.487987	−	0.872851i	$-0.337731\pi$
$331$	1.94540e6	0.975974	0.487987	+	0.872851i	$0.337731\pi$
$332$	522686.	0.260253
$333$	−1.07053e6	−0.529038
$334$	−711185.	−0.348832
$335$	0	0
$336$	−715589.	−0.345792
$337$	−1.39092e6	−0.667154	−0.333577	−	0.942723i	$-0.608256\pi$
$337$	−1.39092e6	−0.667154	−0.333577	+	0.942723i	$0.608256\pi$
$338$	−473482.	−0.225430
$339$	−1.23938e6	−0.585742
$340$	0	0
$341$	−780679.	−0.363569
$342$	−100658.	−0.0465352
$343$	−2.31187e6	−1.06103
$344$	−1.55114e6	−0.706733
$345$	0	0
$346$	−757812.	−0.340307
$347$	3.23565e6	1.44257	0.721286	−	0.692637i	$-0.243551\pi$
$347$	3.23565e6	1.44257	0.721286	+	0.692637i	$0.243551\pi$
$348$	−913458.	−0.404334
$349$	−1.54020e6	−0.676883	−0.338442	−	0.940987i	$-0.609900\pi$
$349$	−1.54020e6	−0.676883	−0.338442	+	0.940987i	$0.609900\pi$
$350$	0	0
$351$	−51273.3	−0.0222138
$352$	−447416.	−0.192466
$353$	696423.	0.297465	0.148733	−	0.988877i	$-0.452481\pi$
$353$	696423.	0.297465	0.148733	+	0.988877i	$0.452481\pi$
$354$	−504866.	−0.214125
$355$	0	0
$356$	740337.	0.309602
$357$	970814.	0.403149
$358$	719485.	0.296698
$359$	−2.60283e6	−1.06588	−0.532942	−	0.846152i	$-0.678914\pi$
$359$	−2.60283e6	−1.06588	−0.532942	+	0.846152i	$0.678914\pi$
$360$	0	0
$361$	−1.55161e6	−0.626636
$362$	516236.	0.207051
$363$	−131769.	−0.0524864
$364$	−195757.	−0.0774397
$365$	0	0
$366$	41512.0	0.0161984
$367$	−801086.	−0.310466	−0.155233	−	0.987878i	$-0.549613\pi$
$367$	−801086.	−0.310466	−0.155233	+	0.987878i	$0.549613\pi$
$368$	1.12902e6	0.434591
$369$	1.49590e6	0.571922
$370$	0	0
$371$	−2.86660e6	−1.08127
$372$	1.76115e6	0.659840
$373$	−1.94463e6	−0.723712	−0.361856	−	0.932234i	$-0.617857\pi$
$373$	−1.94463e6	−0.723712	−0.361856	+	0.932234i	$0.617857\pi$
$374$	183824.	0.0679554
$375$	0	0
$376$	−1.37463e6	−0.501438
$377$	−235366.	−0.0852884
$378$	−86462.1	−0.0311241
$379$	−3.33496e6	−1.19260	−0.596298	−	0.802763i	$-0.703362\pi$
$379$	−3.33496e6	−1.19260	−0.596298	+	0.802763i	$0.703362\pi$
$380$	0	0
$381$	226445.	0.0799191
$382$	−795820.	−0.279033
$383$	2.16994e6	0.755875	0.377938	−	0.925831i	$-0.376633\pi$
$383$	2.16994e6	0.755875	0.377938	+	0.925831i	$0.376633\pi$
$384$	1.33184e6	0.460918
$385$	0	0
$386$	−546299.	−0.186622
$387$	1.55966e6	0.529362
$388$	3.83812e6	1.29431
$389$	−561944.	−0.188286	−0.0941432	−	0.995559i	$-0.530011\pi$
$389$	−561944.	−0.188286	−0.0941432	+	0.995559i	$0.530011\pi$
$390$	0	0
$391$	−1.53170e6	−0.506676
$392$	675539.	0.222042
$393$	1.95338e6	0.637978
$394$	−150916.	−0.0489773
$395$	0	0
$396$	297260.	0.0952575
$397$	−3.98781e6	−1.26987	−0.634934	−	0.772566i	$-0.718973\pi$
$397$	−3.98781e6	−1.26987	−0.634934	+	0.772566i	$0.718973\pi$
$398$	−871926.	−0.275913
$399$	794108.	0.249717
$400$	0	0
$401$	3.07030e6	0.953499	0.476750	−	0.879039i	$-0.341815\pi$
$401$	3.07030e6	0.953499	0.476750	+	0.879039i	$0.341815\pi$
$402$	−823126.	−0.254040
$403$	453786.	0.139184
$404$	2.72890e6	0.831828
$405$	0	0
$406$	−396897.	−0.119499
$407$	1.59918e6	0.478533
$408$	−852226.	−0.253457
$409$	−34399.2	−0.0101681	−0.00508405	−	0.999987i	$-0.501618\pi$
$409$	−34399.2	−0.0101681	−0.00508405	+	0.999987i	$0.501618\pi$
$410$	0	0
$411$	216845.	0.0633206
$412$	−3.22272e6	−0.935361
$413$	3.98299e6	1.14904
$414$	136415.	0.0391166
$415$	0	0
$416$	260070.	0.0736811
$417$	3.23823e6	0.911944
$418$	150365.	0.0420926
$419$	3.35733e6	0.934241	0.467121	−	0.884194i	$-0.345291\pi$
$419$	3.35733e6	0.934241	0.467121	+	0.884194i	$0.345291\pi$
$420$	0	0
$421$	−693515.	−0.190700	−0.0953499	−	0.995444i	$-0.530397\pi$
$421$	−693515.	−0.190700	−0.0953499	+	0.995444i	$0.530397\pi$
$422$	448791.	0.122677
$423$	1.38219e6	0.375591
$424$	2.51644e6	0.679785
$425$	0	0
$426$	−479560.	−0.128032
$427$	−327497.	−0.0869235
$428$	−2.75785e6	−0.727715
$429$	76593.5	0.0200932
$430$	0	0
$431$	−3.60068e6	−0.933666	−0.466833	−	0.884346i	$-0.654605\pi$
$431$	−3.60068e6	−0.933666	−0.466833	+	0.884346i	$0.654605\pi$
$432$	−631628.	−0.162837
$433$	−2.17347e6	−0.557100	−0.278550	−	0.960422i	$-0.589854\pi$
$433$	−2.17347e6	−0.557100	−0.278550	+	0.960422i	$0.589854\pi$
$434$	765219.	0.195012
$435$	0	0
$436$	−1.42452e6	−0.358883
$437$	−1.25290e6	−0.313843
$438$	479883.	0.119523
$439$	−6.87194e6	−1.70184	−0.850919	−	0.525296i	$-0.823954\pi$
$439$	−6.87194e6	−1.70184	−0.850919	+	0.525296i	$0.823954\pi$
$440$	0	0
$441$	−679250.	−0.166316
$442$	−106852.	−0.0260151
$443$	4.97594e6	1.20466	0.602332	−	0.798246i	$-0.294238\pi$
$443$	4.97594e6	1.20466	0.602332	+	0.798246i	$0.294238\pi$
$444$	−3.60762e6	−0.868488
$445$	0	0
$446$	−406037.	−0.0966560
$447$	2.77164e6	0.656098
$448$	−2.10576e6	−0.495694
$449$	−1.82131e6	−0.426352	−0.213176	−	0.977014i	$-0.568381\pi$
$449$	−1.82131e6	−0.426352	−0.213176	+	0.977014i	$0.568381\pi$
$450$	0	0
$451$	−2.23462e6	−0.517323
$452$	−4.17666e6	−0.961576
$453$	4.78398e6	1.09533
$454$	1.34408e6	0.306045
$455$	0	0
$456$	−697105.	−0.156995
$457$	−1.52586e6	−0.341763	−0.170881	−	0.985292i	$-0.554661\pi$
$457$	−1.52586e6	−0.341763	−0.170881	+	0.985292i	$0.554661\pi$
$458$	−249096.	−0.0554885
$459$	856908.	0.189846
$460$	0	0
$461$	4.99516e6	1.09470	0.547352	−	0.836902i	$-0.315636\pi$
$461$	4.99516e6	1.09470	0.547352	+	0.836902i	$0.315636\pi$
$462$	129159.	0.0281528
$463$	−5.20396e6	−1.12819	−0.564094	−	0.825711i	$-0.690774\pi$
$463$	−5.20396e6	−1.12819	−0.564094	+	0.825711i	$0.690774\pi$
$464$	−2.89943e6	−0.625199
$465$	0	0
$466$	832865.	0.177668
$467$	2.28807e6	0.485487	0.242744	−	0.970090i	$-0.421953\pi$
$467$	2.28807e6	0.485487	0.242744	+	0.970090i	$0.421953\pi$
$468$	−172789.	−0.0364671
$469$	6.49381e6	1.36322
$470$	0	0
$471$	−3.85973e6	−0.801686
$472$	−3.49645e6	−0.722391
$473$	−2.32987e6	−0.478826
$474$	668361.	0.136636
$475$	0	0
$476$	3.27160e6	0.661824
$477$	−2.53026e6	−0.509178
$478$	−782382.	−0.156621
$479$	1.91319e6	0.380996	0.190498	−	0.981688i	$-0.438990\pi$
$479$	1.91319e6	0.380996	0.190498	+	0.981688i	$0.438990\pi$
$480$	0	0
$481$	−929557.	−0.183195
$482$	−559017.	−0.109599
$483$	−1.07621e6	−0.209907
$484$	−444056.	−0.0861636
$485$	0	0
$486$	−76317.5	−0.0146566
$487$	−5.63300e6	−1.07626	−0.538130	−	0.842862i	$-0.680869\pi$
$487$	−5.63300e6	−1.07626	−0.538130	+	0.842862i	$0.680869\pi$
$488$	287492.	0.0546482
$489$	188336.	0.0356174
$490$	0	0
$491$	−2.42388e6	−0.453741	−0.226871	−	0.973925i	$-0.572849\pi$
$491$	−2.42388e6	−0.453741	−0.226871	+	0.973925i	$0.572849\pi$
$492$	5.04112e6	0.938889
$493$	3.93356e6	0.728901
$494$	−87402.8	−0.0161142
$495$	0	0
$496$	5.59012e6	1.02027
$497$	3.78335e6	0.687045
$498$	200460.	0.0362206
$499$	6.53921e6	1.17564	0.587819	−	0.808992i	$-0.299987\pi$
$499$	6.53921e6	1.17564	0.587819	+	0.808992i	$0.299987\pi$
$500$	0	0
$501$	4.95238e6	0.881494
$502$	−1.32097e6	−0.233956
$503$	7.09673e6	1.25066	0.625329	−	0.780361i	$-0.284965\pi$
$503$	7.09673e6	1.25066	0.625329	+	0.780361i	$0.284965\pi$
$504$	−598794.	−0.105003
$505$	0	0
$506$	−203781.	−0.0353823
$507$	3.29712e6	0.569658
$508$	763110.	0.131198
$509$	1.40802e6	0.240887	0.120443	−	0.992720i	$-0.461568\pi$
$509$	1.40802e6	0.240887	0.120443	+	0.992720i	$0.461568\pi$
$510$	0	0
$511$	−3.78590e6	−0.641382
$512$	5.43728e6	0.916657
$513$	700935.	0.117594
$514$	1.17566e6	0.196279
$515$	0	0
$516$	5.25599e6	0.869021
$517$	−2.06475e6	−0.339735
$518$	−1.56751e6	−0.256677
$519$	5.27706e6	0.859951
$520$	0	0
$521$	−4.33371e6	−0.699465	−0.349732	−	0.936850i	$-0.613728\pi$
$521$	−4.33371e6	−0.699465	−0.349732	+	0.936850i	$0.613728\pi$
$522$	−350329.	−0.0562729
$523$	−5.99430e6	−0.958262	−0.479131	−	0.877743i	$-0.659048\pi$
$523$	−5.99430e6	−0.958262	−0.479131	+	0.877743i	$0.659048\pi$
$524$	6.58280e6	1.04733
$525$	0	0
$526$	−1.29192e6	−0.203596
$527$	−7.58392e6	−1.18951
$528$	943543.	0.147291
$529$	−4.73837e6	−0.736189
$530$	0	0
$531$	3.51566e6	0.541091
$532$	2.67611e6	0.409944
$533$	1.29892e6	0.198045
$534$	283933.	0.0430887
$535$	0	0
$536$	−5.70057e6	−0.857050
$537$	−5.01017e6	−0.749751
$538$	−2.77513e6	−0.413360
$539$	1.01468e6	0.150438
$540$	0	0
$541$	9.71649e6	1.42730	0.713652	−	0.700501i	$-0.247040\pi$
$541$	9.71649e6	1.42730	0.713652	+	0.700501i	$0.247040\pi$
$542$	−961211.	−0.140547
$543$	−3.59484e6	−0.523215
$544$	−4.34643e6	−0.629702
$545$	0	0
$546$	−75076.6	−0.0107776
$547$	−1.18605e7	−1.69486	−0.847430	−	0.530908i	$-0.821851\pi$
$547$	−1.18605e7	−1.69486	−0.847430	+	0.530908i	$0.821851\pi$
$548$	730758.	0.103950
$549$	−289071.	−0.0409331
$550$	0	0
$551$	3.21758e6	0.451492
$552$	944744.	0.131967
$553$	−5.27283e6	−0.733215
$554$	2.97460e6	0.411770
$555$	0	0
$556$	1.09127e7	1.49708
$557$	−4.65955e6	−0.636364	−0.318182	−	0.948030i	$-0.603072\pi$
$557$	−4.65955e6	−0.636364	−0.318182	+	0.948030i	$0.603072\pi$
$558$	675435.	0.0918329
$559$	1.35428e6	0.183307
$560$	0	0
$561$	−1.28007e6	−0.171722
$562$	1.25982e6	0.168254
$563$	−7.30425e6	−0.971191	−0.485595	−	0.874184i	$-0.661397\pi$
$563$	−7.30425e6	−0.971191	−0.485595	+	0.874184i	$0.661397\pi$
$564$	4.65790e6	0.616584
$565$	0	0
$566$	1.59257e6	0.208957
$567$	602084.	0.0786501
$568$	−3.32120e6	−0.431941
$569$	−3.49414e6	−0.452439	−0.226219	−	0.974076i	$-0.572637\pi$
$569$	−3.49414e6	−0.452439	−0.226219	+	0.974076i	$0.572637\pi$
$570$	0	0
$571$	1.42919e7	1.83443	0.917213	−	0.398396i	$-0.130433\pi$
$571$	1.42919e7	1.83443	0.917213	+	0.398396i	$0.130433\pi$
$572$	258116.	0.0329857
$573$	5.54174e6	0.705114
$574$	2.19036e6	0.277483
$575$	0	0
$576$	−1.85869e6	−0.233427
$577$	−6.93782e6	−0.867528	−0.433764	−	0.901027i	$-0.642815\pi$
$577$	−6.93782e6	−0.867528	−0.433764	+	0.901027i	$0.642815\pi$
$578$	−49319.1	−0.00614039
$579$	3.80418e6	0.471591
$580$	0	0
$581$	−1.58147e6	−0.194366
$582$	1.47199e6	0.180135
$583$	3.77978e6	0.460569
$584$	3.32344e6	0.403232
$585$	0	0
$586$	−475583.	−0.0572114
$587$	2.56707e6	0.307498	0.153749	−	0.988110i	$-0.450865\pi$
$587$	2.56707e6	0.307498	0.153749	+	0.988110i	$0.450865\pi$
$588$	−2.28904e6	−0.273030
$589$	−6.20351e6	−0.736799
$590$	0	0
$591$	1.05091e6	0.123765
$592$	−1.14511e7	−1.34289
$593$	8.76559e6	1.02363	0.511817	−	0.859095i	$-0.328973\pi$
$593$	8.76559e6	1.02363	0.511817	+	0.859095i	$0.328973\pi$
$594$	114005.	0.0132574
$595$	0	0
$596$	9.34031e6	1.07707
$597$	6.07171e6	0.697229
$598$	118452.	0.0135453
$599$	6.17238e6	0.702887	0.351444	−	0.936209i	$-0.385691\pi$
$599$	6.17238e6	0.702887	0.351444	+	0.936209i	$0.385691\pi$
$600$	0	0
$601$	−1.19370e7	−1.34806	−0.674031	−	0.738703i	$-0.735439\pi$
$601$	−1.19370e7	−1.34806	−0.674031	+	0.738703i	$0.735439\pi$
$602$	2.28373e6	0.256834
$603$	5.73189e6	0.641955
$604$	1.61218e7	1.79813
$605$	0	0
$606$	1.04658e6	0.115769
$607$	−2.00634e6	−0.221021	−0.110510	−	0.993875i	$-0.535249\pi$
$607$	−2.00634e6	−0.221021	−0.110510	+	0.993875i	$0.535249\pi$
$608$	−3.55530e6	−0.390047
$609$	2.76381e6	0.301971
$610$	0	0
$611$	1.20018e6	0.130059
$612$	2.88774e6	0.311659
$613$	−1.13241e6	−0.121718	−0.0608589	−	0.998146i	$-0.519384\pi$
$613$	−1.13241e6	−0.121718	−0.0608589	+	0.998146i	$0.519384\pi$
$614$	982479.	0.105173
$615$	0	0
$616$	894495.	0.0949787
$617$	−1.20807e6	−0.127755	−0.0638775	−	0.997958i	$-0.520347\pi$
$617$	−1.20807e6	−0.127755	−0.0638775	+	0.997958i	$0.520347\pi$
$618$	−1.23598e6	−0.130178
$619$	1.14027e7	1.19614	0.598071	−	0.801443i	$-0.295934\pi$
$619$	1.14027e7	1.19614	0.598071	+	0.801443i	$0.295934\pi$
$620$	0	0
$621$	−949934.	−0.0988472
$622$	−2.84682e6	−0.295042
$623$	−2.24001e6	−0.231222
$624$	−548454.	−0.0563870
$625$	0	0
$626$	886514.	0.0904170
$627$	−1.04708e6	−0.106368
$628$	−1.30071e7	−1.31608
$629$	1.55353e7	1.56564
$630$	0	0
$631$	1.32708e6	0.132685	0.0663426	−	0.997797i	$-0.478867\pi$
$631$	1.32708e6	0.132685	0.0663426	+	0.997797i	$0.478867\pi$
$632$	4.62874e6	0.460967
$633$	−3.12518e6	−0.310003
$634$	−2.42862e6	−0.239959
$635$	0	0
$636$	−8.52688e6	−0.835886
$637$	−589805.	−0.0575917
$638$	523331.	0.0509008
$639$	3.33945e6	0.323536
$640$	0	0
$641$	−9.61062e6	−0.923860	−0.461930	−	0.886916i	$-0.652843\pi$
$641$	−9.61062e6	−0.923860	−0.461930	+	0.886916i	$0.652843\pi$
$642$	−1.05769e6	−0.101279
$643$	1.26822e7	1.20967	0.604835	−	0.796351i	$-0.293239\pi$
$643$	1.26822e7	1.20967	0.604835	+	0.796351i	$0.293239\pi$
$644$	−3.62677e6	−0.344592
$645$	0	0
$646$	1.46072e6	0.137717
$647$	−1.27269e7	−1.19526	−0.597629	−	0.801773i	$-0.703890\pi$
$647$	−1.27269e7	−1.19526	−0.597629	+	0.801773i	$0.703890\pi$
$648$	−528537.	−0.0494468
$649$	−5.25179e6	−0.489435
$650$	0	0
$651$	−5.32864e6	−0.492793
$652$	634685.	0.0584709
$653$	7.54035e6	0.692004	0.346002	−	0.938234i	$-0.387539\pi$
$653$	7.54035e6	0.692004	0.346002	+	0.938234i	$0.387539\pi$
$654$	−546332.	−0.0499473
$655$	0	0
$656$	1.60012e7	1.45175
$657$	−3.34170e6	−0.302032
$658$	2.02386e6	0.182228
$659$	1.20388e7	1.07987	0.539933	−	0.841708i	$-0.318450\pi$
$659$	1.20388e7	1.07987	0.539933	+	0.841708i	$0.318450\pi$
$660$	0	0
$661$	6.36439e6	0.566569	0.283285	−	0.959036i	$-0.408576\pi$
$661$	6.36439e6	0.566569	0.283285	+	0.959036i	$0.408576\pi$
$662$	2.51432e6	0.222985
$663$	744068.	0.0657399
$664$	1.38829e6	0.122197
$665$	0	0
$666$	−1.38359e6	−0.120871
$667$	−4.36059e6	−0.379517
$668$	1.66893e7	1.44709
$669$	2.82746e6	0.244248
$670$	0	0
$671$	431823.	0.0370254
$672$	−3.05390e6	−0.260875
$673$	−741014.	−0.0630650	−0.0315325	−	0.999503i	$-0.510039\pi$
$673$	−741014.	−0.0630650	−0.0315325	+	0.999503i	$0.510039\pi$
$674$	−1.79768e6	−0.152427
$675$	0	0
$676$	1.11111e7	0.935172
$677$	1.43961e7	1.20718	0.603590	−	0.797295i	$-0.293736\pi$
$677$	1.43961e7	1.20718	0.603590	+	0.797295i	$0.293736\pi$
$678$	−1.60183e6	−0.133827
$679$	−1.16129e7	−0.966640
$680$	0	0
$681$	−9.35956e6	−0.773371
$682$	−1.00898e6	−0.0830659
$683$	−7.32070e6	−0.600484	−0.300242	−	0.953863i	$-0.597067\pi$
$683$	−7.32070e6	−0.600484	−0.300242	+	0.953863i	$0.597067\pi$
$684$	2.36212e6	0.193046
$685$	0	0
$686$	−2.98796e6	−0.242418
$687$	1.73459e6	0.140219
$688$	1.66832e7	1.34372
$689$	−2.19707e6	−0.176318
$690$	0	0
$691$	1.51860e7	1.20989	0.604947	−	0.796266i	$-0.293194\pi$
$691$	1.51860e7	1.20989	0.604947	+	0.796266i	$0.293194\pi$
$692$	1.77835e7	1.41173
$693$	−899409.	−0.0711417
$694$	4.18189e6	0.329590
$695$	0	0
$696$	−2.42621e6	−0.189847
$697$	−2.17082e7	−1.69255
$698$	−1.99062e6	−0.154650
$699$	−5.79970e6	−0.448965
$700$	0	0
$701$	−1.17236e7	−0.901089	−0.450544	−	0.892754i	$-0.648770\pi$
$701$	−1.17236e7	−0.901089	−0.450544	+	0.892754i	$0.648770\pi$
$702$	−66267.8	−0.00507528
$703$	1.27076e7	0.969782
$704$	2.77656e6	0.211142
$705$	0	0
$706$	900087.	0.0679630
$707$	−8.25672e6	−0.621239
$708$	1.18476e7	0.888276
$709$	6.09002e6	0.454991	0.227495	−	0.973779i	$-0.426946\pi$
$709$	6.09002e6	0.454991	0.227495	+	0.973779i	$0.426946\pi$
$710$	0	0
$711$	−4.65417e6	−0.345277
$712$	1.96638e6	0.145368
$713$	8.40723e6	0.619340
$714$	1.25472e6	0.0921090
$715$	0	0
$716$	−1.68841e7	−1.23082
$717$	5.44816e6	0.395778
$718$	−3.36401e6	−0.243527
$719$	3.50864e6	0.253114	0.126557	−	0.991959i	$-0.459607\pi$
$719$	3.50864e6	0.253114	0.126557	+	0.991959i	$0.459607\pi$
$720$	0	0
$721$	9.75086e6	0.698562
$722$	−2.00537e6	−0.143170
$723$	3.89274e6	0.276956
$724$	−1.21145e7	−0.858929
$725$	0	0
$726$	−170304.	−0.0119918
$727$	−222222.	−0.0155937	−0.00779687	−	0.999970i	$-0.502482\pi$
$727$	−222222.	−0.0155937	−0.00779687	+	0.999970i	$0.502482\pi$
$728$	−519944.	−0.0363603
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.26335e7	−1.56660
$732$	−974157.	−0.0671972
$733$	−1.95614e6	−0.134475	−0.0672373	−	0.997737i	$-0.521418\pi$
$733$	−1.95614e6	−0.134475	−0.0672373	+	0.997737i	$0.521418\pi$
$734$	−1.03536e6	−0.0709333
$735$	0	0
$736$	4.81828e6	0.327867
$737$	−8.56245e6	−0.580670
$738$	1.93337e6	0.130669
$739$	−3.61998e6	−0.243834	−0.121917	−	0.992540i	$-0.538904\pi$
$739$	−3.61998e6	−0.243834	−0.121917	+	0.992540i	$0.538904\pi$
$740$	0	0
$741$	608634.	0.0407203
$742$	−3.70492e6	−0.247041
$743$	4.37060e6	0.290449	0.145224	−	0.989399i	$-0.453610\pi$
$743$	4.37060e6	0.290449	0.145224	+	0.989399i	$0.453610\pi$
$744$	4.67773e6	0.309815
$745$	0	0
$746$	−2.51333e6	−0.165349
$747$	−1.39592e6	−0.0915289
$748$	−4.31378e6	−0.281906
$749$	8.34432e6	0.543484
$750$	0	0
$751$	8.89727e6	0.575648	0.287824	−	0.957683i	$-0.407068\pi$
$751$	8.89727e6	0.575648	0.287824	+	0.957683i	$0.407068\pi$
$752$	1.47848e7	0.953390
$753$	9.19867e6	0.591204
$754$	−304197.	−0.0194862
$755$	0	0
$756$	2.02900e6	0.129115
$757$	−1.36934e7	−0.868506	−0.434253	−	0.900791i	$-0.642988\pi$
$757$	−1.36934e7	−0.868506	−0.434253	+	0.900791i	$0.642988\pi$
$758$	−4.31025e6	−0.272477
$759$	1.41904e6	0.0894107
$760$	0	0
$761$	1.75245e7	1.09694	0.548472	−	0.836169i	$-0.315210\pi$
$761$	1.75245e7	1.09694	0.548472	+	0.836169i	$0.315210\pi$
$762$	292668.	0.0182594
$763$	4.31012e6	0.268027
$764$	1.86754e7	1.15754
$765$	0	0
$766$	2.80452e6	0.172698
$767$	3.05271e6	0.187369
$768$	−4.88735e6	−0.298999
$769$	−1.68641e7	−1.02837	−0.514183	−	0.857681i	$-0.671905\pi$
$769$	−1.68641e7	−1.02837	−0.514183	+	0.857681i	$0.671905\pi$
$770$	0	0
$771$	−8.18677e6	−0.495994
$772$	1.28199e7	0.774181
$773$	1.24437e7	0.749033	0.374516	−	0.927220i	$-0.377809\pi$
$773$	1.24437e7	0.749033	0.374516	+	0.927220i	$0.377809\pi$
$774$	2.01577e6	0.120945
$775$	0	0
$776$	1.01943e7	0.607720
$777$	1.09155e7	0.648618
$778$	−726281.	−0.0430185
$779$	−1.77569e7	−1.04839
$780$	0	0
$781$	−4.98856e6	−0.292649
$782$	−1.97963e6	−0.115762
$783$	2.43953e6	0.142201
$784$	−7.26572e6	−0.422171
$785$	0	0
$786$	2.52463e6	0.145761
$787$	−2.62304e7	−1.50962	−0.754811	−	0.655943i	$-0.772271\pi$
$787$	−2.62304e7	−1.50962	−0.754811	+	0.655943i	$0.772271\pi$
$788$	3.54152e6	0.203177
$789$	8.99633e6	0.514485
$790$	0	0
$791$	1.26372e7	0.718139
$792$	789544.	0.0447263
$793$	−251006.	−0.0141743
$794$	−5.15402e6	−0.290132
$795$	0	0
$796$	2.04614e7	1.14460
$797$	−4.99841e6	−0.278731	−0.139366	−	0.990241i	$-0.544506\pi$
$797$	−4.99841e6	−0.278731	−0.139366	+	0.990241i	$0.544506\pi$
$798$	1.02634e6	0.0570537
$799$	−2.00580e7	−1.11153
$800$	0	0
$801$	−1.97719e6	−0.108885
$802$	3.96819e6	0.217850
$803$	4.99191e6	0.273199
$804$	1.93162e7	1.05386
$805$	0	0
$806$	586492.	0.0317998
$807$	1.93248e7	1.04456
$808$	7.24813e6	0.390569
$809$	2.76591e7	1.48582	0.742912	−	0.669389i	$-0.233444\pi$
$809$	2.76591e7	1.48582	0.742912	+	0.669389i	$0.233444\pi$
$810$	0	0
$811$	−1.72731e7	−0.922184	−0.461092	−	0.887352i	$-0.652542\pi$
$811$	−1.72731e7	−0.922184	−0.461092	+	0.887352i	$0.652542\pi$
$812$	9.31393e6	0.495727
$813$	6.69344e6	0.355159
$814$	2.06685e6	0.109332
$815$	0	0
$816$	9.16607e6	0.481901
$817$	−1.85138e7	−0.970377
$818$	−44459.0	−0.00232315
$819$	522800.	0.0272349
$820$	0	0
$821$	−1.70950e7	−0.885139	−0.442570	−	0.896734i	$-0.645933\pi$
$821$	−1.70950e7	−0.885139	−0.442570	+	0.896734i	$0.645933\pi$
$822$	280260.	0.0144671
$823$	1.39800e7	0.719464	0.359732	−	0.933056i	$-0.382868\pi$
$823$	1.39800e7	0.719464	0.359732	+	0.933056i	$0.382868\pi$
$824$	−8.55976e6	−0.439181
$825$	0	0
$826$	5.14778e6	0.262525
$827$	−2.33121e7	−1.18527	−0.592636	−	0.805471i	$-0.701913\pi$
$827$	−2.33121e7	−1.18527	−0.592636	+	0.805471i	$0.701913\pi$
$828$	−3.20123e6	−0.162271
$829$	−2.71774e7	−1.37348	−0.686738	−	0.726905i	$-0.740958\pi$
$829$	−2.71774e7	−1.37348	−0.686738	+	0.726905i	$0.740958\pi$
$830$	0	0
$831$	−2.07138e7	−1.04054
$832$	−1.61393e6	−0.0808309
$833$	9.85714e6	0.492196
$834$	4.18523e6	0.208355
$835$	0	0
$836$	−3.52860e6	−0.174617
$837$	−4.70343e6	−0.232060
$838$	4.33916e6	0.213450
$839$	6.44081e6	0.315890	0.157945	−	0.987448i	$-0.449513\pi$
$839$	6.44081e6	0.315890	0.157945	+	0.987448i	$0.449513\pi$
$840$	0	0
$841$	−9.31268e6	−0.454030
$842$	−896328.	−0.0435699
$843$	−8.77280e6	−0.425176
$844$	−1.05317e7	−0.508913
$845$	0	0
$846$	1.78640e6	0.0858127
$847$	1.34356e6	0.0643501
$848$	−2.70654e7	−1.29248
$849$	−1.10899e7	−0.528031
$850$	0	0
$851$	−1.72218e7	−0.815182
$852$	1.12538e7	0.531128
$853$	3.24050e6	0.152490	0.0762448	−	0.997089i	$-0.475707\pi$
$853$	3.24050e6	0.152490	0.0762448	+	0.997089i	$0.475707\pi$
$854$	−423271.	−0.0198598
$855$	0	0
$856$	−7.32504e6	−0.341685
$857$	2.15579e7	1.00266	0.501331	−	0.865255i	$-0.332844\pi$
$857$	2.15579e7	1.00266	0.501331	+	0.865255i	$0.332844\pi$
$858$	98992.7	0.00459076
$859$	3.00808e7	1.39093	0.695467	−	0.718558i	$-0.255197\pi$
$859$	3.00808e7	1.39093	0.695467	+	0.718558i	$0.255197\pi$
$860$	0	0
$861$	−1.52527e7	−0.701196
$862$	−4.65367e6	−0.213318
$863$	3.68726e7	1.68530	0.842650	−	0.538462i	$-0.180995\pi$
$863$	3.68726e7	1.68530	0.842650	+	0.538462i	$0.180995\pi$
$864$	−2.69559e6	−0.122848
$865$	0	0
$866$	−2.80908e6	−0.127283
$867$	343436.	0.0155167
$868$	−1.79573e7	−0.808987
$869$	6.95252e6	0.312315
$870$	0	0
$871$	4.97710e6	0.222296
$872$	−3.78363e6	−0.168507
$873$	−1.02503e7	−0.455200
$874$	−1.61930e6	−0.0717049
$875$	0	0
$876$	−1.12614e7	−0.495827
$877$	1.99151e7	0.874348	0.437174	−	0.899377i	$-0.355979\pi$
$877$	1.99151e7	0.874348	0.437174	+	0.899377i	$0.355979\pi$
$878$	−8.88160e6	−0.388826
$879$	3.31175e6	0.144572
$880$	0	0
$881$	2.96980e7	1.28910	0.644551	−	0.764561i	$-0.277044\pi$
$881$	2.96980e7	1.28910	0.644551	+	0.764561i	$0.277044\pi$
$882$	−877892.	−0.0379988
$883$	−1.61839e7	−0.698526	−0.349263	−	0.937025i	$-0.613568\pi$
$883$	−1.61839e7	−0.698526	−0.349263	+	0.937025i	$0.613568\pi$
$884$	2.50748e6	0.107921
$885$	0	0
$886$	6.43112e6	0.275234
$887$	3.22208e7	1.37508	0.687540	−	0.726147i	$-0.258691\pi$
$887$	3.22208e7	1.37508	0.687540	+	0.726147i	$0.258691\pi$
$888$	−9.58209e6	−0.407782
$889$	−2.30891e6	−0.0979836
$890$	0	0
$891$	−793881.	−0.0335013
$892$	9.52842e6	0.400967
$893$	−1.64071e7	−0.688498
$894$	3.58219e6	0.149901
$895$	0	0
$896$	−1.35799e7	−0.565102
$897$	−824845.	−0.0342288
$898$	−2.35394e6	−0.0974102
$899$	−2.15907e7	−0.890979
$900$	0	0
$901$	3.67187e7	1.50687
$902$	−2.88812e6	−0.118195
$903$	−1.59029e7	−0.649016
$904$	−1.10935e7	−0.451489
$905$	0	0
$906$	6.18303e6	0.250254
$907$	−6.59225e6	−0.266082	−0.133041	−	0.991111i	$-0.542474\pi$
$907$	−6.59225e6	−0.266082	−0.133041	+	0.991111i	$0.542474\pi$
$908$	−3.15413e7	−1.26959
$909$	−7.28795e6	−0.292547
$910$	0	0
$911$	−4.51432e7	−1.80217	−0.901086	−	0.433640i	$-0.857229\pi$
$911$	−4.51432e7	−1.80217	−0.901086	+	0.433640i	$0.857229\pi$
$912$	7.49767e6	0.298497
$913$	2.08526e6	0.0827910
$914$	−1.97209e6	−0.0780838
$915$	0	0
$916$	5.84550e6	0.230188
$917$	−1.99173e7	−0.782182
$918$	1.10750e6	0.0433749
$919$	−4.40836e7	−1.72182	−0.860911	−	0.508756i	$-0.830106\pi$
$919$	−4.40836e7	−1.72182	−0.860911	+	0.508756i	$0.830106\pi$
$920$	0	0
$921$	−6.84155e6	−0.265770
$922$	6.45596e6	0.250111
$923$	2.89970e6	0.112034
$924$	−3.03097e6	−0.116789
$925$	0	0
$926$	−6.72582e6	−0.257761
$927$	8.60678e6	0.328959
$928$	−1.23739e7	−0.471667
$929$	−3.78094e7	−1.43734	−0.718672	−	0.695349i	$-0.755250\pi$
$929$	−3.78094e7	−1.43734	−0.718672	+	0.695349i	$0.755250\pi$
$930$	0	0
$931$	8.06296e6	0.304874
$932$	−1.95447e7	−0.737038
$933$	1.98240e7	0.745568
$934$	2.95721e6	0.110921
$935$	0	0
$936$	−458938.	−0.0171224
$937$	−4.50928e7	−1.67787	−0.838935	−	0.544231i	$-0.816821\pi$
$937$	−4.50928e7	−1.67787	−0.838935	+	0.544231i	$0.816821\pi$
$938$	8.39288e6	0.311461
$939$	−6.17329e6	−0.228482
$940$	0	0
$941$	4.32466e7	1.59213	0.796063	−	0.605213i	$-0.206912\pi$
$941$	4.32466e7	1.59213	0.796063	+	0.605213i	$0.206912\pi$
$942$	−4.98847e6	−0.183164
$943$	2.40649e7	0.881261
$944$	3.76059e7	1.37349
$945$	0	0
$946$	−3.01122e6	−0.109399
$947$	3.10917e7	1.12660	0.563300	−	0.826253i	$-0.309532\pi$
$947$	3.10917e7	1.12660	0.563300	+	0.826253i	$0.309532\pi$
$948$	−1.56843e7	−0.566820
$949$	−2.90165e6	−0.104588
$950$	0	0
$951$	1.69118e7	0.606373
$952$	8.68959e6	0.310747
$953$	−4.53118e7	−1.61614	−0.808071	−	0.589085i	$-0.799488\pi$
$953$	−4.53118e7	−1.61614	−0.808071	+	0.589085i	$0.799488\pi$
$954$	−3.27022e6	−0.116334
$955$	0	0
$956$	1.83601e7	0.649725
$957$	−3.64424e6	−0.128626
$958$	2.47269e6	0.0870476
$959$	−2.21103e6	−0.0776333
$960$	0	0
$961$	1.29978e7	0.454005
$962$	−1.20140e6	−0.0418552
$963$	7.36528e6	0.255931
$964$	1.31184e7	0.454661
$965$	0	0
$966$	−1.39094e6	−0.0479583
$967$	3.52998e7	1.21396	0.606982	−	0.794715i	$-0.292380\pi$
$967$	3.52998e7	1.21396	0.606982	+	0.794715i	$0.292380\pi$
$968$	−1.17944e6	−0.0404565
$969$	−1.01718e7	−0.348008
$970$	0	0
$971$	−2.90825e7	−0.989883	−0.494941	−	0.868926i	$-0.664810\pi$
$971$	−2.90825e7	−0.989883	−0.494941	+	0.868926i	$0.664810\pi$
$972$	1.79093e6	0.0608014
$973$	−3.30181e7	−1.11807
$974$	−7.28033e6	−0.245897
$975$	0	0
$976$	−3.09210e6	−0.103903
$977$	2.59538e7	0.869890	0.434945	−	0.900457i	$-0.356768\pi$
$977$	2.59538e7	0.869890	0.434945	+	0.900457i	$0.356768\pi$
$978$	243414.	0.00813764
$979$	2.95357e6	0.0984898
$980$	0	0
$981$	3.80441e6	0.126216
$982$	−3.13273e6	−0.103668
$983$	−2.55539e7	−0.843477	−0.421738	−	0.906718i	$-0.638580\pi$
$983$	−2.55539e7	−0.843477	−0.421738	+	0.906718i	$0.638580\pi$
$984$	1.33896e7	0.440837
$985$	0	0
$986$	5.08390e6	0.166535
$987$	−1.40932e7	−0.460488
$988$	2.05107e6	0.0668479
$989$	2.50906e7	0.815682
$990$	0	0
$991$	−5.48265e7	−1.77340	−0.886699	−	0.462346i	$-0.847008\pi$
$991$	−5.48265e7	−1.77340	−0.886699	+	0.462346i	$0.847008\pi$
$992$	2.38568e7	0.769722
$993$	−1.75086e7	−0.563479
$994$	4.88976e6	0.156972
$995$	0	0
$996$	−4.70418e6	−0.150257
$997$	−2.30540e7	−0.734527	−0.367264	−	0.930117i	$-0.619705\pi$
$997$	−2.30540e7	−0.734527	−0.367264	+	0.930117i	$0.619705\pi$
$998$	8.45155e6	0.268603
$999$	9.63474e6	0.305440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.q.1.4	yes	8
5.4	even	2		825.6.a.p.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.5	✓	8	5.4	even	2
825.6.a.q.1.4	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.29244 q^{2} -9.00000 q^{3} -30.3296 q^{4} -11.6320 q^{6} +91.7671 q^{7} -80.5574 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+1.29244 q^{2} -9.00000 q^{3} -30.3296 q^{4} -11.6320 q^{6} +91.7671 q^{7} -80.5574 q^{8} +81.0000 q^{9} -121.000 q^{11} +272.966 q^{12} +70.3337 q^{13} +118.604 q^{14} +866.431 q^{16} -1175.46 q^{17} +104.688 q^{18} -961.502 q^{19} -825.904 q^{21} -156.386 q^{22} +1303.06 q^{23} +725.017 q^{24} +90.9024 q^{26} -729.000 q^{27} -2783.26 q^{28} -3346.41 q^{29} +6451.89 q^{31} +3697.65 q^{32} +1089.00 q^{33} -1519.21 q^{34} -2456.70 q^{36} -13216.4 q^{37} -1242.69 q^{38} -633.004 q^{39} +18467.9 q^{41} -1067.43 q^{42} +19255.1 q^{43} +3669.88 q^{44} +1684.14 q^{46} +17064.0 q^{47} -7797.88 q^{48} -8385.80 q^{49} +10579.1 q^{51} -2133.19 q^{52} -31237.8 q^{53} -942.191 q^{54} -7392.52 q^{56} +8653.52 q^{57} -4325.05 q^{58} +43403.2 q^{59} -3568.78 q^{61} +8338.70 q^{62} +7433.14 q^{63} -22946.8 q^{64} +1407.47 q^{66} +70764.0 q^{67} +35651.1 q^{68} -11727.6 q^{69} +41227.7 q^{71} -6525.15 q^{72} -41255.5 q^{73} -17081.4 q^{74} +29162.0 q^{76} -11103.8 q^{77} -818.121 q^{78} -57458.9 q^{79} +6561.00 q^{81} +23868.7 q^{82} -17233.5 q^{83} +25049.3 q^{84} +24886.1 q^{86} +30117.7 q^{87} +9747.45 q^{88} -24409.7 q^{89} +6454.32 q^{91} -39521.4 q^{92} -58067.0 q^{93} +22054.3 q^{94} -33278.9 q^{96} -126547. q^{97} -10838.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) 8 * q + 9 * q^2 - 72 * q^3 + 107 * q^4 - 81 * q^6 - 66 * q^7 + 207 * q^8 + 648 * q^9	\( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 8 * q + 9 * q^2 - 72 * q^3 + 107 * q^4 - 81 * q^6 - 66 * q^7 + 207 * q^8 + 648 * q^9 - 968 * q^11 - 963 * q^12 + 382 * q^13 + 1048 * q^14 - 325 * q^16 + 288 * q^17 + 729 * q^18 - 988 * q^19 + 594 * q^21 - 1089 * q^22 + 5972 * q^23 - 1863 * q^24 - 14579 * q^26 - 5832 * q^27 + 5942 * q^28 + 1032 * q^29 - 4682 * q^31 + 3863 * q^32 + 8712 * q^33 + 2206 * q^34 + 8667 * q^36 - 17200 * q^37 + 11011 * q^38 - 3438 * q^39 - 13220 * q^41 - 9432 * q^42 + 22872 * q^43 - 12947 * q^44 + 9101 * q^46 + 6700 * q^47 + 2925 * q^48 - 43466 * q^49 - 2592 * q^51 + 5009 * q^52 + 6224 * q^53 - 6561 * q^54 + 20992 * q^56 + 8892 * q^57 + 33015 * q^58 - 77556 * q^59 + 11554 * q^61 + 12135 * q^62 - 5346 * q^63 - 149917 * q^64 + 9801 * q^66 + 20894 * q^67 + 91776 * q^68 - 53748 * q^69 - 21648 * q^71 + 16767 * q^72 + 64660 * q^73 - 179522 * q^74 + 24401 * q^76 + 7986 * q^77 + 131211 * q^78 - 22660 * q^79 + 52488 * q^81 - 56080 * q^82 + 100390 * q^83 - 53478 * q^84 + 47271 * q^86 - 9288 * q^87 - 25047 * q^88 - 25578 * q^89 + 73250 * q^91 + 95311 * q^92 + 42138 * q^93 - 170120 * q^94 - 34767 * q^96 + 142828 * q^97 + 303397 * q^98 - 78408 * q^99

Introduction

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