LMFDB - Embedded newform 825.6.a.v.1.11

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$8.39855$ 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)

$f(q)$	$=$	$q+7.39855 q^{2} -9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} -150.852 q^{7} -68.5217 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+7.39855 q^{2} -9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} -150.852 q^{7} -68.5217 q^{8} +81.0000 q^{9} +121.000 q^{11} -204.646 q^{12} +868.426 q^{13} -1116.08 q^{14} -1234.59 q^{16} +2317.87 q^{17} +599.282 q^{18} -2655.24 q^{19} +1357.67 q^{21} +895.224 q^{22} +2537.58 q^{23} +616.695 q^{24} +6425.09 q^{26} -729.000 q^{27} -3430.14 q^{28} -819.917 q^{29} +7303.36 q^{31} -6941.50 q^{32} -1089.00 q^{33} +17148.9 q^{34} +1841.82 q^{36} -2993.84 q^{37} -19644.9 q^{38} -7815.84 q^{39} -4567.72 q^{41} +10044.8 q^{42} +1022.10 q^{43} +2751.36 q^{44} +18774.4 q^{46} -24499.3 q^{47} +11111.3 q^{48} +5949.25 q^{49} -20860.8 q^{51} +19746.7 q^{52} -13318.9 q^{53} -5393.54 q^{54} +10336.6 q^{56} +23897.2 q^{57} -6066.20 q^{58} -29760.6 q^{59} -17257.1 q^{61} +54034.2 q^{62} -12219.0 q^{63} -11850.0 q^{64} -8057.02 q^{66} -18645.0 q^{67} +52704.9 q^{68} -22838.2 q^{69} +47722.0 q^{71} -5550.26 q^{72} -19156.3 q^{73} -22150.1 q^{74} -60376.2 q^{76} -18253.1 q^{77} -57825.8 q^{78} -3713.15 q^{79} +6561.00 q^{81} -33794.5 q^{82} -51919.0 q^{83} +30871.3 q^{84} +7562.07 q^{86} +7379.26 q^{87} -8291.13 q^{88} -52966.5 q^{89} -131004. q^{91} +57700.8 q^{92} -65730.2 q^{93} -181259. q^{94} +62473.5 q^{96} -114526. q^{97} +44015.8 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * 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 $

$2$	7.39855	1.30789	0.653945	−	0.756542i	$-0.273113\pi$
$2$	7.39855	1.30789	0.653945	+	0.756542i	$0.273113\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	22.7385	0.710578
$5$	0	0
$5$	0	0
$6$	−66.5869	−0.755111
$7$	−150.852	−1.16360	−0.581802	−	0.813330i	$-0.697652\pi$
$7$	−150.852	−1.16360	−0.581802	+	0.813330i	$0.697652\pi$
$8$	−68.5217	−0.378533
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−204.646	−0.410252
$13$	868.426	1.42520	0.712598	−	0.701573i	$-0.247518\pi$
$13$	868.426	1.42520	0.712598	+	0.701573i	$0.247518\pi$
$14$	−1116.08	−1.52187
$15$	0	0
$16$	−1234.59	−1.20566
$17$	2317.87	1.94521	0.972606	−	0.232458i	$-0.0746769\pi$
$17$	2317.87	1.94521	0.972606	+	0.232458i	$0.0746769\pi$
$18$	599.282	0.435964
$19$	−2655.24	−1.68741	−0.843705	−	0.536808i	$-0.819630\pi$
$19$	−2655.24	−1.68741	−0.843705	+	0.536808i	$0.819630\pi$
$20$	0	0
$21$	1357.67	0.671807
$22$	895.224	0.394344
$23$	2537.58	1.00023	0.500116	−	0.865959i	$-0.333291\pi$
$23$	2537.58	1.00023	0.500116	+	0.865959i	$0.333291\pi$
$24$	616.695	0.218546
$25$	0	0
$26$	6425.09	1.86400
$27$	−729.000	−0.192450
$28$	−3430.14	−0.826831
$29$	−819.917	−0.181040	−0.0905201	−	0.995895i	$-0.528853\pi$
$29$	−819.917	−0.181040	−0.0905201	+	0.995895i	$0.528853\pi$
$30$	0	0
$31$	7303.36	1.36495	0.682477	−	0.730907i	$-0.260903\pi$
$31$	7303.36	1.36495	0.682477	+	0.730907i	$0.260903\pi$
$32$	−6941.50	−1.19833
$33$	−1089.00	−0.174078
$34$	17148.9	2.54413
$35$	0	0
$36$	1841.82	0.236859
$37$	−2993.84	−0.359521	−0.179761	−	0.983710i	$-0.557532\pi$
$37$	−2993.84	−0.359521	−0.179761	+	0.983710i	$0.557532\pi$
$38$	−19644.9	−2.20695
$39$	−7815.84	−0.822837
$40$	0	0
$41$	−4567.72	−0.424365	−0.212182	−	0.977230i	$-0.568057\pi$
$41$	−4567.72	−0.424365	−0.212182	+	0.977230i	$0.568057\pi$
$42$	10044.8	0.878650
$43$	1022.10	0.0842992	0.0421496	−	0.999111i	$-0.486579\pi$
$43$	1022.10	0.0842992	0.0421496	+	0.999111i	$0.486579\pi$
$44$	2751.36	0.214247
$45$	0	0
$46$	18774.4	1.30819
$47$	−24499.3	−1.61774	−0.808871	−	0.587987i	$-0.799921\pi$
$47$	−24499.3	−1.61774	−0.808871	+	0.587987i	$0.799921\pi$
$48$	11111.3	0.696086
$49$	5949.25	0.353974
$50$	0	0
$51$	−20860.8	−1.12307
$52$	19746.7	1.01271
$53$	−13318.9	−0.651297	−0.325649	−	0.945491i	$-0.605583\pi$
$53$	−13318.9	−0.651297	−0.325649	+	0.945491i	$0.605583\pi$
$54$	−5393.54	−0.251704
$55$	0	0
$56$	10336.6	0.440462
$57$	23897.2	0.974226
$58$	−6066.20	−0.236781
$59$	−29760.6	−1.11304	−0.556522	−	0.830833i	$-0.687864\pi$
$59$	−29760.6	−1.11304	−0.556522	+	0.830833i	$0.687864\pi$
$60$	0	0
$61$	−17257.1	−0.593805	−0.296903	−	0.954908i	$-0.595954\pi$
$61$	−17257.1	−0.593805	−0.296903	+	0.954908i	$0.595954\pi$
$62$	54034.2	1.78521
$63$	−12219.0	−0.387868
$64$	−11850.0	−0.361634
$65$	0	0
$66$	−8057.02	−0.227675
$67$	−18645.0	−0.507429	−0.253715	−	0.967279i	$-0.581652\pi$
$67$	−18645.0	−0.507429	−0.253715	+	0.967279i	$0.581652\pi$
$68$	52704.9	1.38223
$69$	−22838.2	−0.577484
$70$	0	0
$71$	47722.0	1.12350	0.561750	−	0.827307i	$-0.310128\pi$
$71$	47722.0	1.12350	0.561750	+	0.827307i	$0.310128\pi$
$72$	−5550.26	−0.126178
$73$	−19156.3	−0.420730	−0.210365	−	0.977623i	$-0.567465\pi$
$73$	−19156.3	−0.420730	−0.210365	+	0.977623i	$0.567465\pi$
$74$	−22150.1	−0.470214
$75$	0	0
$76$	−60376.2	−1.19904
$77$	−18253.1	−0.350840
$78$	−57825.8	−1.07618
$79$	−3713.15	−0.0669383	−0.0334692	−	0.999440i	$-0.510656\pi$
$79$	−3713.15	−0.0669383	−0.0334692	+	0.999440i	$0.510656\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−33794.5	−0.555023
$83$	−51919.0	−0.827239	−0.413619	−	0.910450i	$-0.635736\pi$
$83$	−51919.0	−0.827239	−0.413619	+	0.910450i	$0.635736\pi$
$84$	30871.3	0.477371
$85$	0	0
$86$	7562.07	0.110254
$87$	7379.26	0.104524
$88$	−8291.13	−0.114132
$89$	−52966.5	−0.708803	−0.354402	−	0.935093i	$-0.615315\pi$
$89$	−52966.5	−0.708803	−0.354402	+	0.935093i	$0.615315\pi$
$90$	0	0
$91$	−131004.	−1.65836
$92$	57700.8	0.710742
$93$	−65730.2	−0.788057
$94$	−181259.	−2.11583
$95$	0	0
$96$	62473.5	0.691859
$97$	−114526.	−1.23588	−0.617940	−	0.786225i	$-0.712032\pi$
$97$	−114526.	−1.23588	−0.617940	+	0.786225i	$0.712032\pi$
$98$	44015.8	0.462960
$99$	9801.00	0.100504
$100$	0	0
$101$	154646.	1.50846	0.754232	−	0.656608i	$-0.228009\pi$
$101$	154646.	1.50846	0.754232	+	0.656608i	$0.228009\pi$
$102$	−154340.	−1.46885
$103$	117955.	1.09552	0.547762	−	0.836634i	$-0.315480\pi$
$103$	117955.	1.09552	0.547762	+	0.836634i	$0.315480\pi$
$104$	−59506.1	−0.539483
$105$	0	0
$106$	−98540.6	−0.851826
$107$	−233086.	−1.96815	−0.984073	−	0.177763i	$-0.943114\pi$
$107$	−233086.	−1.96815	−0.984073	+	0.177763i	$0.943114\pi$
$108$	−16576.4	−0.136751
$109$	−18774.3	−0.151355	−0.0756775	−	0.997132i	$-0.524112\pi$
$109$	−18774.3	−0.151355	−0.0756775	+	0.997132i	$0.524112\pi$
$110$	0	0
$111$	26944.6	0.207570
$112$	186240.	1.40291
$113$	−165056.	−1.21600	−0.608001	−	0.793936i	$-0.708029\pi$
$113$	−165056.	−1.21600	−0.608001	+	0.793936i	$0.708029\pi$
$114$	176804.	1.27418
$115$	0	0
$116$	−18643.7	−0.128643
$117$	70342.5	0.475065
$118$	−220186.	−1.45574
$119$	−349655.	−2.26346
$120$	0	0
$121$	14641.0	0.0909091
$122$	−127678.	−0.776632
$123$	41109.4	0.245007
$124$	166067.	0.969907
$125$	0	0
$126$	−90402.8	−0.507289
$127$	−203866.	−1.12159	−0.560797	−	0.827953i	$-0.689505\pi$
$127$	−203866.	−1.12159	−0.560797	+	0.827953i	$0.689505\pi$
$128$	134455.	0.725357
$129$	−9198.92	−0.0486701
$130$	0	0
$131$	82311.6	0.419067	0.209533	−	0.977802i	$-0.432806\pi$
$131$	82311.6	0.419067	0.209533	+	0.977802i	$0.432806\pi$
$132$	−24762.2	−0.123696
$133$	400548.	1.96348
$134$	−137946.	−0.663662
$135$	0	0
$136$	−158825.	−0.736326
$137$	−202042.	−0.919687	−0.459844	−	0.888000i	$-0.652095\pi$
$137$	−202042.	−0.919687	−0.459844	+	0.888000i	$0.652095\pi$
$138$	−168970.	−0.755286
$139$	−200312.	−0.879367	−0.439683	−	0.898153i	$-0.644909\pi$
$139$	−200312.	−0.879367	−0.439683	+	0.898153i	$0.644909\pi$
$140$	0	0
$141$	220494.	0.934003
$142$	353074.	1.46942
$143$	105080.	0.429713
$144$	−100002.	−0.401886
$145$	0	0
$146$	−141728.	−0.550269
$147$	−53543.2	−0.204367
$148$	−68075.4	−0.255468
$149$	80425.4	0.296775	0.148388	−	0.988929i	$-0.452592\pi$
$149$	80425.4	0.296775	0.148388	+	0.988929i	$0.452592\pi$
$150$	0	0
$151$	395216.	1.41056	0.705280	−	0.708928i	$-0.250821\pi$
$151$	395216.	1.41056	0.705280	+	0.708928i	$0.250821\pi$
$152$	181942.	0.638739
$153$	187748.	0.648404
$154$	−135046.	−0.458860
$155$	0	0
$156$	−177720.	−0.584690
$157$	510056.	1.65146	0.825732	−	0.564063i	$-0.190763\pi$
$157$	510056.	1.65146	0.825732	+	0.564063i	$0.190763\pi$
$158$	−27471.9	−0.0875480
$159$	119870.	0.376027
$160$	0	0
$161$	−382799.	−1.16387
$162$	48541.9	0.145321
$163$	−307324.	−0.905998	−0.452999	−	0.891511i	$-0.649646\pi$
$163$	−307324.	−0.905998	−0.452999	+	0.891511i	$0.649646\pi$
$164$	−103863.	−0.301544
$165$	0	0
$166$	−384125.	−1.08194
$167$	−214345.	−0.594733	−0.297367	−	0.954763i	$-0.596108\pi$
$167$	−214345.	−0.594733	−0.297367	+	0.954763i	$0.596108\pi$
$168$	−93029.6	−0.254301
$169$	382871.	1.03118
$170$	0	0
$171$	−215075.	−0.562470
$172$	23241.1	0.0599011
$173$	−366833.	−0.931865	−0.465933	−	0.884820i	$-0.654281\pi$
$173$	−366833.	−0.931865	−0.465933	+	0.884820i	$0.654281\pi$
$174$	54595.8	0.136705
$175$	0	0
$176$	−149386.	−0.363519
$177$	267846.	0.642616
$178$	−391875.	−0.927037
$179$	678162.	1.58198	0.790990	−	0.611829i	$-0.209566\pi$
$179$	678162.	1.58198	0.790990	+	0.611829i	$0.209566\pi$
$180$	0	0
$181$	−270856.	−0.614529	−0.307265	−	0.951624i	$-0.599414\pi$
$181$	−270856.	−0.614529	−0.307265	+	0.951624i	$0.599414\pi$
$182$	−969236.	−2.16896
$183$	155314.	0.342834
$184$	−173879.	−0.378620
$185$	0	0
$186$	−486308.	−1.03069
$187$	280462.	0.586504
$188$	−557077.	−1.14953
$189$	109971.	0.223936
$190$	0	0
$191$	81645.4	0.161938	0.0809689	−	0.996717i	$-0.474199\pi$
$191$	81645.4	0.161938	0.0809689	+	0.996717i	$0.474199\pi$
$192$	106650.	0.208789
$193$	170728.	0.329922	0.164961	−	0.986300i	$-0.447250\pi$
$193$	170728.	0.329922	0.164961	+	0.986300i	$0.447250\pi$
$194$	−847329.	−1.61640
$195$	0	0
$196$	135277.	0.251526
$197$	173263.	0.318082	0.159041	−	0.987272i	$-0.449160\pi$
$197$	173263.	0.318082	0.159041	+	0.987272i	$0.449160\pi$
$198$	72513.2	0.131448
$199$	42534.0	0.0761384	0.0380692	−	0.999275i	$-0.487879\pi$
$199$	42534.0	0.0761384	0.0380692	+	0.999275i	$0.487879\pi$
$200$	0	0
$201$	167805.	0.292964
$202$	1.14416e6	1.97291
$203$	123686.	0.210659
$204$	−474344.	−0.798028
$205$	0	0
$206$	872693.	1.43283
$207$	205544.	0.333410
$208$	−1.07215e6	−1.71830
$209$	−321284.	−0.508773
$210$	0	0
$211$	−625694.	−0.967511	−0.483755	−	0.875203i	$-0.660728\pi$
$211$	−625694.	−0.967511	−0.483755	+	0.875203i	$0.660728\pi$
$212$	−302852.	−0.462797
$213$	−429498.	−0.648653
$214$	−1.72450e6	−2.57412
$215$	0	0
$216$	49952.3	0.0728486
$217$	−1.10172e6	−1.58827
$218$	−138902.	−0.197956
$219$	172406.	0.242909
$220$	0	0
$221$	2.01290e6	2.77231
$222$	199351.	0.271478
$223$	42469.9	0.0571899	0.0285950	−	0.999591i	$-0.490897\pi$
$223$	42469.9	0.0571899	0.0285950	+	0.999591i	$0.490897\pi$
$224$	1.04714e6	1.39439
$225$	0	0
$226$	−1.22117e6	−1.59040
$227$	−21815.5	−0.0280996	−0.0140498	−	0.999901i	$-0.504472\pi$
$227$	−21815.5	−0.0280996	−0.0140498	+	0.999901i	$0.504472\pi$
$228$	543386.	0.692263
$229$	58861.1	0.0741720	0.0370860	−	0.999312i	$-0.488192\pi$
$229$	58861.1	0.0741720	0.0370860	+	0.999312i	$0.488192\pi$
$230$	0	0
$231$	164278.	0.202557
$232$	56182.1	0.0685296
$233$	−418005.	−0.504419	−0.252210	−	0.967673i	$-0.581157\pi$
$233$	−418005.	−0.504419	−0.252210	+	0.967673i	$0.581157\pi$
$234$	520432.	0.621333
$235$	0	0
$236$	−676712.	−0.790904
$237$	33418.3	0.0386468
$238$	−2.58694e6	−2.96035
$239$	−1.29252e6	−1.46367	−0.731834	−	0.681483i	$-0.761335\pi$
$239$	−1.29252e6	−1.46367	−0.731834	+	0.681483i	$0.761335\pi$
$240$	0	0
$241$	496148.	0.550261	0.275131	−	0.961407i	$-0.411279\pi$
$241$	496148.	0.550261	0.275131	+	0.961407i	$0.411279\pi$
$242$	108322.	0.118899
$243$	−59049.0	−0.0641500
$244$	−392401.	−0.421945
$245$	0	0
$246$	304150.	0.320443
$247$	−2.30588e6	−2.40489
$248$	−500439.	−0.516680
$249$	467271.	0.477606
$250$	0	0
$251$	−1.58588e6	−1.58886	−0.794431	−	0.607354i	$-0.792231\pi$
$251$	−1.58588e6	−1.58886	−0.794431	+	0.607354i	$0.792231\pi$
$252$	−277841.	−0.275610
$253$	307047.	0.301581
$254$	−1.50831e6	−1.46692
$255$	0	0
$256$	1.37397e6	1.31032
$257$	−1.41881e6	−1.33995	−0.669977	−	0.742381i	$-0.733696\pi$
$257$	−1.41881e6	−1.33995	−0.669977	+	0.742381i	$0.733696\pi$
$258$	−68058.6	−0.0636552
$259$	451626.	0.418340
$260$	0	0
$261$	−66413.3	−0.0603467
$262$	608987.	0.548093
$263$	−1.62046e6	−1.44461	−0.722304	−	0.691575i	$-0.756917\pi$
$263$	−1.62046e6	−1.44461	−0.722304	+	0.691575i	$0.756917\pi$
$264$	74620.1	0.0658941
$265$	0	0
$266$	2.96347e6	2.56801
$267$	476698.	0.409228
$268$	−423959.	−0.360568
$269$	1.25821e6	1.06017	0.530083	−	0.847946i	$-0.322161\pi$
$269$	1.25821e6	1.06017	0.530083	+	0.847946i	$0.322161\pi$
$270$	0	0
$271$	−2.33387e6	−1.93043	−0.965214	−	0.261462i	$-0.915796\pi$
$271$	−2.33387e6	−1.93043	−0.965214	+	0.261462i	$0.915796\pi$
$272$	−2.86163e6	−2.34526
$273$	1.17903e6	0.957457
$274$	−1.49482e6	−1.20285
$275$	0	0
$276$	−519307.	−0.410347
$277$	952978.	0.746248	0.373124	−	0.927781i	$-0.378287\pi$
$277$	952978.	0.746248	0.373124	+	0.927781i	$0.378287\pi$
$278$	−1.48202e6	−1.15012
$279$	591572.	0.454985
$280$	0	0
$281$	−194035.	−0.146593	−0.0732966	−	0.997310i	$-0.523352\pi$
$281$	−194035.	−0.146593	−0.0732966	+	0.997310i	$0.523352\pi$
$282$	1.63133e6	1.22157
$283$	589829.	0.437784	0.218892	−	0.975749i	$-0.429756\pi$
$283$	589829.	0.437784	0.218892	+	0.975749i	$0.429756\pi$
$284$	1.08513e6	0.798334
$285$	0	0
$286$	777436.	0.562017
$287$	689048.	0.493793
$288$	−562261.	−0.399445
$289$	3.95267e6	2.78385
$290$	0	0
$291$	1.03074e6	0.713535
$292$	−435584.	−0.298961
$293$	2.18674e6	1.48809	0.744045	−	0.668130i	$-0.232905\pi$
$293$	2.18674e6	1.48809	0.744045	+	0.668130i	$0.232905\pi$
$294$	−396142.	−0.267290
$295$	0	0
$296$	205143.	0.136090
$297$	−88209.0	−0.0580259
$298$	595031.	0.388150
$299$	2.20370e6	1.42553
$300$	0	0
$301$	−154186.	−0.0980908
$302$	2.92402e6	1.84486
$303$	−1.39181e6	−0.870913
$304$	3.27814e6	2.03444
$305$	0	0
$306$	1.38906e6	0.848042
$307$	−2.25874e6	−1.36780	−0.683898	−	0.729578i	$-0.739717\pi$
$307$	−2.25874e6	−1.36780	−0.683898	+	0.729578i	$0.739717\pi$
$308$	−415047.	−0.249299
$309$	−1.06159e6	−0.632501
$310$	0	0
$311$	−3.00436e6	−1.76137	−0.880687	−	0.473699i	$-0.842918\pi$
$311$	−3.00436e6	−1.76137	−0.880687	+	0.473699i	$0.842918\pi$
$312$	535554.	0.311471
$313$	−2.76965e6	−1.59795	−0.798976	−	0.601363i	$-0.794624\pi$
$313$	−2.76965e6	−1.59795	−0.798976	+	0.601363i	$0.794624\pi$
$314$	3.77367e6	2.15993
$315$	0	0
$316$	−84431.4	−0.0475649
$317$	−2.00320e6	−1.11963	−0.559817	−	0.828616i	$-0.689129\pi$
$317$	−2.00320e6	−1.11963	−0.559817	+	0.828616i	$0.689129\pi$
$318$	886866.	0.491802
$319$	−99210.0	−0.0545857
$320$	0	0
$321$	2.09778e6	1.13631
$322$	−2.83215e6	−1.52222
$323$	−6.15451e6	−3.28237
$324$	149187.	0.0789531
$325$	0	0
$326$	−2.27375e6	−1.18495
$327$	168968.	0.0873849
$328$	312988.	0.160636
$329$	3.69576e6	1.88241
$330$	0	0
$331$	1.09643e6	0.550059	0.275029	−	0.961436i	$-0.411312\pi$
$331$	1.09643e6	0.550059	0.275029	+	0.961436i	$0.411312\pi$
$332$	−1.18056e6	−0.587817
$333$	−242501.	−0.119840
$334$	−1.58584e6	−0.777846
$335$	0	0
$336$	−1.67616e6	−0.809969
$337$	−594378.	−0.285094	−0.142547	−	0.989788i	$-0.545529\pi$
$337$	−594378.	−0.285094	−0.142547	+	0.989788i	$0.545529\pi$
$338$	2.83269e6	1.34868
$339$	1.48550e6	0.702060
$340$	0	0
$341$	883706.	0.411549
$342$	−1.59124e6	−0.735649
$343$	1.63791e6	0.751718
$344$	−70036.2	−0.0319100
$345$	0	0
$346$	−2.71403e6	−1.21878
$347$	822560.	0.366728	0.183364	−	0.983045i	$-0.441301\pi$
$347$	822560.	0.366728	0.183364	+	0.983045i	$0.441301\pi$
$348$	167793.	0.0742722
$349$	956928.	0.420548	0.210274	−	0.977642i	$-0.432564\pi$
$349$	956928.	0.420548	0.210274	+	0.977642i	$0.432564\pi$
$350$	0	0
$351$	−633083.	−0.274279
$352$	−839921.	−0.361312
$353$	−218804.	−0.0934583	−0.0467291	−	0.998908i	$-0.514880\pi$
$353$	−218804.	−0.0934583	−0.0467291	+	0.998908i	$0.514880\pi$
$354$	1.98167e6	0.840472
$355$	0	0
$356$	−1.20438e6	−0.503660
$357$	3.14689e6	1.30681
$358$	5.01741e6	2.06906
$359$	−2.72858e6	−1.11738	−0.558689	−	0.829378i	$-0.688695\pi$
$359$	−2.72858e6	−1.11738	−0.558689	+	0.829378i	$0.688695\pi$
$360$	0	0
$361$	4.57422e6	1.84735
$362$	−2.00394e6	−0.803737
$363$	−131769.	−0.0524864
$364$	−2.97882e6	−1.17840
$365$	0	0
$366$	1.14910e6	0.448389
$367$	1.94374e6	0.753307	0.376654	−	0.926354i	$-0.377075\pi$
$367$	1.94374e6	0.753307	0.376654	+	0.926354i	$0.377075\pi$
$368$	−3.13288e6	−1.20594
$369$	−369985.	−0.141455
$370$	0	0
$371$	2.00918e6	0.757852
$372$	−1.49461e6	−0.559976
$373$	4.88061e6	1.81636	0.908180	−	0.418580i	$-0.137472\pi$
$373$	4.88061e6	1.81636	0.908180	+	0.418580i	$0.137472\pi$
$374$	2.07501e6	0.767083
$375$	0	0
$376$	1.67873e6	0.612368
$377$	−712038.	−0.258018
$378$	813625.	0.292883
$379$	−4.08830e6	−1.46199	−0.730995	−	0.682383i	$-0.760944\pi$
$379$	−4.08830e6	−1.46199	−0.730995	+	0.682383i	$0.760944\pi$
$380$	0	0
$381$	1.83479e6	0.647553
$382$	604057.	0.211797
$383$	−3.04558e6	−1.06090	−0.530448	−	0.847717i	$-0.677976\pi$
$383$	−3.04558e6	−1.06090	−0.530448	+	0.847717i	$0.677976\pi$
$384$	−1.21009e6	−0.418785
$385$	0	0
$386$	1.26314e6	0.431502
$387$	82790.3	0.0280997
$388$	−2.60416e6	−0.878189
$389$	2.13646e6	0.715847	0.357923	−	0.933751i	$-0.383485\pi$
$389$	2.13646e6	0.715847	0.357923	+	0.933751i	$0.383485\pi$
$390$	0	0
$391$	5.88179e6	1.94566
$392$	−407653.	−0.133991
$393$	−740805.	−0.241948
$394$	1.28189e6	0.416017
$395$	0	0
$396$	222860.	0.0714158
$397$	2.27058e6	0.723038	0.361519	−	0.932365i	$-0.382258\pi$
$397$	2.27058e6	0.723038	0.361519	+	0.932365i	$0.382258\pi$
$398$	314690.	0.0995807
$399$	−3.60493e6	−1.13361
$400$	0	0
$401$	981140.	0.304698	0.152349	−	0.988327i	$-0.451316\pi$
$401$	981140.	0.304698	0.152349	+	0.988327i	$0.451316\pi$
$402$	1.24151e6	0.383165
$403$	6.34243e6	1.94533
$404$	3.51642e6	1.07188
$405$	0	0
$406$	915096.	0.275519
$407$	−362255.	−0.108400
$408$	1.42942e6	0.425118
$409$	−1.98079e6	−0.585505	−0.292752	−	0.956188i	$-0.594571\pi$
$409$	−1.98079e6	−0.585505	−0.292752	+	0.956188i	$0.594571\pi$
$410$	0	0
$411$	1.81838e6	0.530982
$412$	2.68211e6	0.778456
$413$	4.48945e6	1.29514
$414$	1.52073e6	0.436064
$415$	0	0
$416$	−6.02818e6	−1.70786
$417$	1.80281e6	0.507703
$418$	−2.37704e6	−0.665419
$419$	153079.	0.0425972	0.0212986	−	0.999773i	$-0.493220\pi$
$419$	153079.	0.0425972	0.0212986	+	0.999773i	$0.493220\pi$
$420$	0	0
$421$	848256.	0.233250	0.116625	−	0.993176i	$-0.462792\pi$
$421$	848256.	0.233250	0.116625	+	0.993176i	$0.462792\pi$
$422$	−4.62923e6	−1.26540
$423$	−1.98444e6	−0.539247
$424$	912635.	0.246537
$425$	0	0
$426$	−3.17766e6	−0.848367
$427$	2.60327e6	0.690954
$428$	−5.30003e6	−1.39852
$429$	−945716.	−0.248095
$430$	0	0
$431$	−492966.	−0.127827	−0.0639137	−	0.997955i	$-0.520358\pi$
$431$	−492966.	−0.127827	−0.0639137	+	0.997955i	$0.520358\pi$
$432$	900018.	0.232029
$433$	−5.12774e6	−1.31434	−0.657168	−	0.753744i	$-0.728246\pi$
$433$	−5.12774e6	−1.31434	−0.657168	+	0.753744i	$0.728246\pi$
$434$	−8.15116e6	−2.07728
$435$	0	0
$436$	−426899.	−0.107550
$437$	−6.73790e6	−1.68780
$438$	1.27556e6	0.317698
$439$	−3.98100e6	−0.985895	−0.492947	−	0.870059i	$-0.664081\pi$
$439$	−3.98100e6	−0.985895	−0.492947	+	0.870059i	$0.664081\pi$
$440$	0	0
$441$	481889.	0.117991
$442$	1.48925e7	3.62588
$443$	3.09112e6	0.748353	0.374177	−	0.927357i	$-0.377925\pi$
$443$	3.09112e6	0.748353	0.374177	+	0.927357i	$0.377925\pi$
$444$	612679.	0.147494
$445$	0	0
$446$	314216.	0.0747981
$447$	−723829.	−0.171343
$448$	1.78760e6	0.420799
$449$	−1.51168e6	−0.353870	−0.176935	−	0.984223i	$-0.556618\pi$
$449$	−1.51168e6	−0.353870	−0.176935	+	0.984223i	$0.556618\pi$
$450$	0	0
$451$	−552694.	−0.127951
$452$	−3.75312e6	−0.864065
$453$	−3.55694e6	−0.814388
$454$	−161403.	−0.0367512
$455$	0	0
$456$	−1.63748e6	−0.368776
$457$	5.58801e6	1.25160	0.625802	−	0.779982i	$-0.284772\pi$
$457$	5.58801e6	1.25160	0.625802	+	0.779982i	$0.284772\pi$
$458$	435487.	0.0970088
$459$	−1.68973e6	−0.374356
$460$	0	0
$461$	−819486.	−0.179593	−0.0897964	−	0.995960i	$-0.528622\pi$
$461$	−819486.	−0.179593	−0.0897964	+	0.995960i	$0.528622\pi$
$462$	1.21542e6	0.264923
$463$	987740.	0.214136	0.107068	−	0.994252i	$-0.465854\pi$
$463$	987740.	0.214136	0.107068	+	0.994252i	$0.465854\pi$
$464$	1.01226e6	0.218272
$465$	0	0
$466$	−3.09263e6	−0.659725
$467$	3.26318e6	0.692387	0.346193	−	0.938163i	$-0.387474\pi$
$467$	3.26318e6	0.692387	0.346193	+	0.938163i	$0.387474\pi$
$468$	1.59948e6	0.337571
$469$	2.81263e6	0.590447
$470$	0	0
$471$	−4.59051e6	−0.953473
$472$	2.03925e6	0.421323
$473$	123674.	0.0254172
$474$	247247.	0.0505459
$475$	0	0
$476$	−7.95063e6	−1.60836
$477$	−1.07883e6	−0.217099
$478$	−9.56277e6	−1.91432
$479$	5.61817e6	1.11881	0.559405	−	0.828895i	$-0.311030\pi$
$479$	5.61817e6	1.11881	0.559405	+	0.828895i	$0.311030\pi$
$480$	0	0
$481$	−2.59993e6	−0.512388
$482$	3.67078e6	0.719681
$483$	3.44519e6	0.671962
$484$	332914.	0.0645980
$485$	0	0
$486$	−436877.	−0.0839012
$487$	4.31761e6	0.824938	0.412469	−	0.910972i	$-0.364666\pi$
$487$	4.31761e6	0.824938	0.412469	+	0.910972i	$0.364666\pi$
$488$	1.18249e6	0.224775
$489$	2.76591e6	0.523078
$490$	0	0
$491$	8.77622e6	1.64287	0.821436	−	0.570301i	$-0.193173\pi$
$491$	8.77622e6	1.64287	0.821436	+	0.570301i	$0.193173\pi$
$492$	934767.	0.174097
$493$	−1.90046e6	−0.352162
$494$	−1.70602e7	−3.14533
$495$	0	0
$496$	−9.01667e6	−1.64567
$497$	−7.19895e6	−1.30731
$498$	3.45712e6	0.624657
$499$	−9.08110e6	−1.63263	−0.816313	−	0.577609i	$-0.803986\pi$
$499$	−9.08110e6	−1.63263	−0.816313	+	0.577609i	$0.803986\pi$
$500$	0	0
$501$	1.92911e6	0.343370
$502$	−1.17332e7	−2.07806
$503$	−4.35992e6	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$503$	−4.35992e6	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$504$	837266.	0.146821
$505$	0	0
$506$	2.27170e6	0.394435
$507$	−3.44584e6	−0.595354
$508$	−4.63561e6	−0.796980
$509$	5.30215e6	0.907105	0.453552	−	0.891230i	$-0.350156\pi$
$509$	5.30215e6	0.907105	0.453552	+	0.891230i	$0.350156\pi$
$510$	0	0
$511$	2.88975e6	0.489563
$512$	5.86284e6	0.988400
$513$	1.93567e6	0.324742
$514$	−1.04971e7	−1.75251
$515$	0	0
$516$	−209170.	−0.0345839
$517$	−2.96442e6	−0.487767
$518$	3.34138e6	0.547143
$519$	3.30150e6	0.538013
$520$	0	0
$521$	1.80284e6	0.290981	0.145490	−	0.989360i	$-0.453524\pi$
$521$	1.80284e6	0.290981	0.145490	+	0.989360i	$0.453524\pi$
$522$	−491362.	−0.0789269
$523$	−1.48512e6	−0.237414	−0.118707	−	0.992929i	$-0.537875\pi$
$523$	−1.48512e6	−0.237414	−0.118707	+	0.992929i	$0.537875\pi$
$524$	1.87164e6	0.297779
$525$	0	0
$526$	−1.19891e7	−1.88939
$527$	1.69282e7	2.65513
$528$	1.34447e6	0.209878
$529$	2976.47	0.000462448 0
$530$	0	0
$531$	−2.41061e6	−0.371015
$532$	9.10786e6	1.39520
$533$	−3.96672e6	−0.604803
$534$	3.52687e6	0.535225
$535$	0	0
$536$	1.27759e6	0.192078
$537$	−6.10346e6	−0.913356
$538$	9.30896e6	1.38658
$539$	719859.	0.106727
$540$	0	0
$541$	−2.18504e6	−0.320972	−0.160486	−	0.987038i	$-0.551306\pi$
$541$	−2.18504e6	−0.320972	−0.160486	+	0.987038i	$0.551306\pi$
$542$	−1.72673e7	−2.52479
$543$	2.43771e6	0.354799
$544$	−1.60895e7	−2.33102
$545$	0	0
$546$	8.72313e6	1.25225
$547$	−6.00853e6	−0.858618	−0.429309	−	0.903158i	$-0.641243\pi$
$547$	−6.00853e6	−0.858618	−0.429309	+	0.903158i	$0.641243\pi$
$548$	−4.59413e6	−0.653509
$549$	−1.39783e6	−0.197935
$550$	0	0
$551$	2.17708e6	0.305489
$552$	1.56491e6	0.218596
$553$	560135.	0.0778897
$554$	7.05065e6	0.976011
$555$	0	0
$556$	−4.55479e6	−0.624858
$557$	−8.40252e6	−1.14755	−0.573775	−	0.819013i	$-0.694522\pi$
$557$	−8.40252e6	−1.14755	−0.573775	+	0.819013i	$0.694522\pi$
$558$	4.37677e6	0.595071
$559$	887620.	0.120143
$560$	0	0
$561$	−2.52416e6	−0.338618
$562$	−1.43558e6	−0.191728
$563$	1.68040e6	0.223430	0.111715	−	0.993740i	$-0.464366\pi$
$563$	1.68040e6	0.223430	0.111715	+	0.993740i	$0.464366\pi$
$564$	5.01369e6	0.663682
$565$	0	0
$566$	4.36387e6	0.572573
$567$	−989738.	−0.129289
$568$	−3.27000e6	−0.425281
$569$	1.43427e7	1.85716	0.928580	−	0.371131i	$-0.121030\pi$
$569$	1.43427e7	1.85716	0.928580	+	0.371131i	$0.121030\pi$
$570$	0	0
$571$	9.83718e6	1.26264	0.631321	−	0.775521i	$-0.282513\pi$
$571$	9.83718e6	1.26264	0.631321	+	0.775521i	$0.282513\pi$
$572$	2.38935e6	0.305344
$573$	−734809.	−0.0934949
$574$	5.09795e6	0.645827
$575$	0	0
$576$	−959852.	−0.120545
$577$	−4.88454e6	−0.610779	−0.305390	−	0.952228i	$-0.598787\pi$
$577$	−4.88454e6	−0.610779	−0.305390	+	0.952228i	$0.598787\pi$
$578$	2.92440e7	3.64097
$579$	−1.53655e6	−0.190481
$580$	0	0
$581$	7.83207e6	0.962578
$582$	7.62596e6	0.933226
$583$	−1.61159e6	−0.196374
$584$	1.31262e6	0.159260
$585$	0	0
$586$	1.61787e7	1.94626
$587$	−3.75376e6	−0.449646	−0.224823	−	0.974400i	$-0.572180\pi$
$587$	−3.75376e6	−0.449646	−0.224823	+	0.974400i	$0.572180\pi$
$588$	−1.21749e6	−0.145219
$589$	−1.93922e7	−2.30324
$590$	0	0
$591$	−1.55936e6	−0.183645
$592$	3.69617e6	0.433459
$593$	−7.63599e6	−0.891720	−0.445860	−	0.895103i	$-0.647102\pi$
$593$	−7.63599e6	−0.891720	−0.445860	+	0.895103i	$0.647102\pi$
$594$	−652618.	−0.0758915
$595$	0	0
$596$	1.82875e6	0.210882
$597$	−382806.	−0.0439585
$598$	1.63042e7	1.86443
$599$	−1.06753e6	−0.121567	−0.0607834	−	0.998151i	$-0.519360\pi$
$599$	−1.06753e6	−0.121567	−0.0607834	+	0.998151i	$0.519360\pi$
$600$	0	0
$601$	−4.83197e6	−0.545680	−0.272840	−	0.962059i	$-0.587963\pi$
$601$	−4.83197e6	−0.545680	−0.272840	+	0.962059i	$0.587963\pi$
$602$	−1.14075e6	−0.128292
$603$	−1.51025e6	−0.169143
$604$	8.98661e6	1.00231
$605$	0	0
$606$	−1.02974e7	−1.13906
$607$	8.95712e6	0.986726	0.493363	−	0.869824i	$-0.335767\pi$
$607$	8.95712e6	0.986726	0.493363	+	0.869824i	$0.335767\pi$
$608$	1.84314e7	2.02208
$609$	−1.11317e6	−0.121624
$610$	0	0
$611$	−2.12758e7	−2.30560
$612$	4.26910e6	0.460742
$613$	8.42569e6	0.905637	0.452818	−	0.891603i	$-0.350419\pi$
$613$	8.42569e6	0.905637	0.452818	+	0.891603i	$0.350419\pi$
$614$	−1.67114e7	−1.78893
$615$	0	0
$616$	1.25073e6	0.132804
$617$	4.84242e6	0.512094	0.256047	−	0.966664i	$-0.417580\pi$
$617$	4.84242e6	0.512094	0.256047	+	0.966664i	$0.417580\pi$
$618$	−7.85424e6	−0.827243
$619$	−4.22402e6	−0.443097	−0.221549	−	0.975149i	$-0.571111\pi$
$619$	−4.22402e6	−0.443097	−0.221549	+	0.975149i	$0.571111\pi$
$620$	0	0
$621$	−1.84990e6	−0.192495
$622$	−2.22279e7	−2.30368
$623$	7.99008e6	0.824767
$624$	9.64938e6	0.992059
$625$	0	0
$626$	−2.04914e7	−2.08995
$627$	2.89156e6	0.293740
$628$	1.15979e7	1.17349
$629$	−6.93934e6	−0.699345
$630$	0	0
$631$	1.18298e7	1.18278	0.591392	−	0.806384i	$-0.298579\pi$
$631$	1.18298e7	1.18278	0.591392	+	0.806384i	$0.298579\pi$
$632$	254431.	0.0253383
$633$	5.63125e6	0.558593
$634$	−1.48208e7	−1.46436
$635$	0	0
$636$	2.72567e6	0.267196
$637$	5.16648e6	0.504483
$638$	−734010.	−0.0713921
$639$	3.86549e6	0.374500
$640$	0	0
$641$	−2.19467e6	−0.210972	−0.105486	−	0.994421i	$-0.533640\pi$
$641$	−2.19467e6	−0.210972	−0.105486	+	0.994421i	$0.533640\pi$
$642$	1.55205e7	1.48617
$643$	−467202.	−0.0445633	−0.0222816	−	0.999752i	$-0.507093\pi$
$643$	−467202.	−0.0445633	−0.0222816	+	0.999752i	$0.507093\pi$
$644$	−8.70426e6	−0.827022
$645$	0	0
$646$	−4.55345e7	−4.29298
$647$	−944645.	−0.0887172	−0.0443586	−	0.999016i	$-0.514124\pi$
$647$	−944645.	−0.0887172	−0.0443586	+	0.999016i	$0.514124\pi$
$648$	−449571.	−0.0420592
$649$	−3.60104e6	−0.335595
$650$	0	0
$651$	9.91552e6	0.916986
$652$	−6.98808e6	−0.643782
$653$	−917543.	−0.0842061	−0.0421030	−	0.999113i	$-0.513406\pi$
$653$	−917543.	−0.0842061	−0.0421030	+	0.999113i	$0.513406\pi$
$654$	1.25012e6	0.114290
$655$	0	0
$656$	5.63927e6	0.511638
$657$	−1.55166e6	−0.140243
$658$	2.73433e7	2.46199
$659$	−5.17061e6	−0.463798	−0.231899	−	0.972740i	$-0.574494\pi$
$659$	−5.17061e6	−0.463798	−0.231899	+	0.972740i	$0.574494\pi$
$660$	0	0
$661$	−1.78425e6	−0.158837	−0.0794185	−	0.996841i	$-0.525306\pi$
$661$	−1.78425e6	−0.158837	−0.0794185	+	0.996841i	$0.525306\pi$
$662$	8.11195e6	0.719417
$663$	−1.81161e7	−1.60059
$664$	3.55758e6	0.313137
$665$	0	0
$666$	−1.79416e6	−0.156738
$667$	−2.08061e6	−0.181082
$668$	−4.87388e6	−0.422604
$669$	−382229.	−0.0330186
$670$	0	0
$671$	−2.08811e6	−0.179039
$672$	−9.42423e6	−0.805050
$673$	−9.10656e6	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$673$	−9.10656e6	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$674$	−4.39753e6	−0.372872
$675$	0	0
$676$	8.70591e6	0.732736
$677$	−657297.	−0.0551175	−0.0275588	−	0.999620i	$-0.508773\pi$
$677$	−657297.	−0.0551175	−0.0275588	+	0.999620i	$0.508773\pi$
$678$	1.09906e7	0.918217
$679$	1.72765e7	1.43807
$680$	0	0
$681$	196339.	0.0162233
$682$	6.53814e6	0.538262
$683$	−1.92866e7	−1.58199	−0.790997	−	0.611821i	$-0.790437\pi$
$683$	−1.92866e7	−1.58199	−0.790997	+	0.611821i	$0.790437\pi$
$684$	−4.89047e6	−0.399678
$685$	0	0
$686$	1.21182e7	0.983165
$687$	−529750.	−0.0428232
$688$	−1.26188e6	−0.101636
$689$	−1.15665e7	−0.928226
$690$	0	0
$691$	7.09653e6	0.565394	0.282697	−	0.959209i	$-0.408771\pi$
$691$	7.09653e6	0.565394	0.282697	+	0.959209i	$0.408771\pi$
$692$	−8.34123e6	−0.662163
$693$	−1.47850e6	−0.116947
$694$	6.08575e6	0.479640
$695$	0	0
$696$	−505639.	−0.0395656
$697$	−1.05874e7	−0.825480
$698$	7.07987e6	0.550031
$699$	3.76204e6	0.291227
$700$	0	0
$701$	1.57310e7	1.20910	0.604550	−	0.796567i	$-0.293353\pi$
$701$	1.57310e7	1.20910	0.604550	+	0.796567i	$0.293353\pi$
$702$	−4.68389e6	−0.358727
$703$	7.94938e6	0.606659
$704$	−1.43385e6	−0.109037
$705$	0	0
$706$	−1.61883e6	−0.122233
$707$	−2.33286e7	−1.75526
$708$	6.09041e6	0.456629
$709$	2.46071e7	1.83842	0.919210	−	0.393767i	$-0.128828\pi$
$709$	2.46071e7	1.83842	0.919210	+	0.393767i	$0.128828\pi$
$710$	0	0
$711$	−300765.	−0.0223128
$712$	3.62935e6	0.268305
$713$	1.85329e7	1.36527
$714$	2.32824e7	1.70916
$715$	0	0
$716$	1.54204e7	1.12412
$717$	1.16327e7	0.845049
$718$	−2.01875e7	−1.46141
$719$	−8.76856e6	−0.632566	−0.316283	−	0.948665i	$-0.602435\pi$
$719$	−8.76856e6	−0.632566	−0.316283	+	0.948665i	$0.602435\pi$
$720$	0	0
$721$	−1.77937e7	−1.27476
$722$	3.38426e7	2.41613
$723$	−4.46533e6	−0.317693
$724$	−6.15887e6	−0.436671
$725$	0	0
$726$	−974899.	−0.0686465
$727$	1.62173e7	1.13800	0.568999	−	0.822338i	$-0.307331\pi$
$727$	1.62173e7	1.13800	0.568999	+	0.822338i	$0.307331\pi$
$728$	8.97659e6	0.627745
$729$	531441.	0.0370370
$730$	0	0
$731$	2.36910e6	0.163980
$732$	3.53161e6	0.243610
$733$	1.12288e7	0.771919	0.385960	−	0.922516i	$-0.373870\pi$
$733$	1.12288e7	0.771919	0.385960	+	0.922516i	$0.373870\pi$
$734$	1.43808e7	0.985243
$735$	0	0
$736$	−1.76146e7	−1.19861
$737$	−2.25605e6	−0.152996
$738$	−2.73735e6	−0.185008
$739$	8.22290e6	0.553878	0.276939	−	0.960888i	$-0.410680\pi$
$739$	8.22290e6	0.553878	0.276939	+	0.960888i	$0.410680\pi$
$740$	0	0
$741$	2.07529e7	1.38846
$742$	1.48650e7	0.991188
$743$	2.24984e6	0.149513	0.0747566	−	0.997202i	$-0.476182\pi$
$743$	2.24984e6	0.149513	0.0747566	+	0.997202i	$0.476182\pi$
$744$	4.50395e6	0.298305
$745$	0	0
$746$	3.61094e7	2.37560
$747$	−4.20544e6	−0.275746
$748$	6.37729e6	0.416757
$749$	3.51615e7	2.29014
$750$	0	0
$751$	−171752.	−0.0111122	−0.00555612	−	0.999985i	$-0.501769\pi$
$751$	−171752.	−0.0111122	−0.00555612	+	0.999985i	$0.501769\pi$
$752$	3.02467e7	1.95044
$753$	1.42729e7	0.917330
$754$	−5.26804e6	−0.337459
$755$	0	0
$756$	2.50057e6	0.159124
$757$	−2.62239e7	−1.66325	−0.831624	−	0.555339i	$-0.812589\pi$
$757$	−2.62239e7	−1.66325	−0.831624	+	0.555339i	$0.812589\pi$
$758$	−3.02475e7	−1.91212
$759$	−2.76343e6	−0.174118
$760$	0	0
$761$	−2.29774e7	−1.43827	−0.719134	−	0.694871i	$-0.755461\pi$
$761$	−2.29774e7	−1.43827	−0.719134	+	0.694871i	$0.755461\pi$
$762$	1.35748e7	0.846928
$763$	2.83213e6	0.176117
$764$	1.85649e6	0.115069
$765$	0	0
$766$	−2.25329e7	−1.38754
$767$	−2.58449e7	−1.58631
$768$	−1.23657e7	−0.756515
$769$	−2.94748e6	−0.179736	−0.0898680	−	0.995954i	$-0.528645\pi$
$769$	−2.94748e6	−0.179736	−0.0898680	+	0.995954i	$0.528645\pi$
$770$	0	0
$771$	1.27693e7	0.773623
$772$	3.88209e6	0.234435
$773$	1.27928e7	0.770049	0.385024	−	0.922906i	$-0.374193\pi$
$773$	1.27928e7	0.770049	0.385024	+	0.922906i	$0.374193\pi$
$774$	612528.	0.0367514
$775$	0	0
$776$	7.84754e6	0.467821
$777$	−4.06463e6	−0.241529
$778$	1.58067e7	0.936249
$779$	1.21284e7	0.716077
$780$	0	0
$781$	5.77437e6	0.338748
$782$	4.35167e7	2.54471
$783$	597720.	0.0348412
$784$	−7.34490e6	−0.426772
$785$	0	0
$786$	−5.48088e6	−0.316442
$787$	1.94779e6	0.112100	0.0560499	−	0.998428i	$-0.482149\pi$
$787$	1.94779e6	0.112100	0.0560499	+	0.998428i	$0.482149\pi$
$788$	3.93973e6	0.226022
$789$	1.45842e7	0.834045
$790$	0	0
$791$	2.48989e7	1.41495
$792$	−671581.	−0.0380440
$793$	−1.49865e7	−0.846289
$794$	1.67990e7	0.945655
$795$	0	0
$796$	967160.	0.0541023
$797$	2.72445e7	1.51926	0.759630	−	0.650355i	$-0.225380\pi$
$797$	2.72445e7	1.51926	0.759630	+	0.650355i	$0.225380\pi$
$798$	−2.66713e7	−1.48264
$799$	−5.67862e7	−3.14685
$800$	0	0
$801$	−4.29028e6	−0.236268
$802$	7.25901e6	0.398512
$803$	−2.31791e6	−0.126855
$804$	3.81563e6	0.208174
$805$	0	0
$806$	4.69247e7	2.54428
$807$	−1.13239e7	−0.612087
$808$	−1.05966e7	−0.571003
$809$	1.56700e7	0.841780	0.420890	−	0.907112i	$-0.361718\pi$
$809$	1.56700e7	0.841780	0.420890	+	0.907112i	$0.361718\pi$
$810$	0	0
$811$	1.41672e7	0.756365	0.378182	−	0.925731i	$-0.376549\pi$
$811$	1.41672e7	0.756365	0.378182	+	0.925731i	$0.376549\pi$
$812$	2.81243e6	0.149690
$813$	2.10048e7	1.11453
$814$	−2.68016e6	−0.141775
$815$	0	0
$816$	2.57547e7	1.35404
$817$	−2.71393e6	−0.142247
$818$	−1.46550e7	−0.765776
$819$	−1.06113e7	−0.552788
$820$	0	0
$821$	6.51336e6	0.337246	0.168623	−	0.985681i	$-0.446068\pi$
$821$	6.51336e6	0.337246	0.168623	+	0.985681i	$0.446068\pi$
$822$	1.34533e7	0.694466
$823$	−2.92352e7	−1.50455	−0.752275	−	0.658849i	$-0.771044\pi$
$823$	−2.92352e7	−1.50455	−0.752275	+	0.658849i	$0.771044\pi$
$824$	−8.08246e6	−0.414692
$825$	0	0
$826$	3.32154e7	1.69390
$827$	2.56735e7	1.30533	0.652666	−	0.757646i	$-0.273651\pi$
$827$	2.56735e7	1.30533	0.652666	+	0.757646i	$0.273651\pi$
$828$	4.67376e6	0.236914
$829$	−9.02846e6	−0.456276	−0.228138	−	0.973629i	$-0.573264\pi$
$829$	−9.02846e6	−0.456276	−0.228138	+	0.973629i	$0.573264\pi$
$830$	0	0
$831$	−8.57680e6	−0.430847
$832$	−1.02909e7	−0.515399
$833$	1.37896e7	0.688555
$834$	1.33382e7	0.664019
$835$	0	0
$836$	−7.30552e6	−0.361523
$837$	−5.32415e6	−0.262686
$838$	1.13256e6	0.0557124
$839$	3.00826e7	1.47540	0.737701	−	0.675128i	$-0.235912\pi$
$839$	3.00826e7	1.47540	0.737701	+	0.675128i	$0.235912\pi$
$840$	0	0
$841$	−1.98389e7	−0.967224
$842$	6.27586e6	0.305065
$843$	1.74631e6	0.0846356
$844$	−1.42273e7	−0.687492
$845$	0	0
$846$	−1.46820e7	−0.705276
$847$	−2.20862e6	−0.105782
$848$	1.64434e7	0.785241
$849$	−5.30846e6	−0.252755
$850$	0	0
$851$	−7.59711e6	−0.359604
$852$	−9.76614e6	−0.460919
$853$	−2.48619e7	−1.16994	−0.584968	−	0.811057i	$-0.698893\pi$
$853$	−2.48619e7	−1.16994	−0.584968	+	0.811057i	$0.698893\pi$
$854$	1.92604e7	0.903692
$855$	0	0
$856$	1.59715e7	0.745008
$857$	−8.27535e6	−0.384888	−0.192444	−	0.981308i	$-0.561641\pi$
$857$	−8.27535e6	−0.384888	−0.192444	+	0.981308i	$0.561641\pi$
$858$	−6.99693e6	−0.324481
$859$	1.89682e7	0.877088	0.438544	−	0.898710i	$-0.355494\pi$
$859$	1.89682e7	0.877088	0.438544	+	0.898710i	$0.355494\pi$
$860$	0	0
$861$	−6.20143e6	−0.285091
$862$	−3.64723e6	−0.167184
$863$	−1.70264e7	−0.778209	−0.389104	−	0.921194i	$-0.627215\pi$
$863$	−1.70264e7	−0.778209	−0.389104	+	0.921194i	$0.627215\pi$
$864$	5.06035e6	0.230620
$865$	0	0
$866$	−3.79378e7	−1.71901
$867$	−3.55741e7	−1.60726
$868$	−2.50515e7	−1.12859
$869$	−449291.	−0.0201827
$870$	0	0
$871$	−1.61918e7	−0.723186
$872$	1.28645e6	0.0572928
$873$	−9.27664e6	−0.411960
$874$	−4.98506e7	−2.20746
$875$	0	0
$876$	3.92026e6	0.172605
$877$	−2.69746e7	−1.18428	−0.592142	−	0.805834i	$-0.701717\pi$
$877$	−2.69746e7	−1.18428	−0.592142	+	0.805834i	$0.701717\pi$
$878$	−2.94536e7	−1.28944
$879$	−1.96807e7	−0.859149
$880$	0	0
$881$	2.13740e7	0.927781	0.463890	−	0.885893i	$-0.346453\pi$
$881$	2.13740e7	0.927781	0.463890	+	0.885893i	$0.346453\pi$
$882$	3.56528e6	0.154320
$883$	3.26501e6	0.140923	0.0704616	−	0.997514i	$-0.477553\pi$
$883$	3.26501e6	0.140923	0.0704616	+	0.997514i	$0.477553\pi$
$884$	4.57703e7	1.96994
$885$	0	0
$886$	2.28698e7	0.978764
$887$	6.68667e6	0.285365	0.142682	−	0.989769i	$-0.454427\pi$
$887$	6.68667e6	0.285365	0.142682	+	0.989769i	$0.454427\pi$
$888$	−1.84629e6	−0.0785718
$889$	3.07536e7	1.30509
$890$	0	0
$891$	793881.	0.0335013
$892$	965702.	0.0406379
$893$	6.50516e7	2.72979
$894$	−5.35528e6	−0.224098
$895$	0	0
$896$	−2.02828e7	−0.844029
$897$	−1.98333e7	−0.823027
$898$	−1.11842e7	−0.462824
$899$	−5.98815e6	−0.247112
$900$	0	0
$901$	−3.08715e7	−1.26691
$902$	−4.08913e6	−0.167346
$903$	1.38767e6	0.0566328
$904$	1.13099e7	0.460297
$905$	0	0
$906$	−2.63162e7	−1.06513
$907$	−2.82586e7	−1.14060	−0.570299	−	0.821437i	$-0.693173\pi$
$907$	−2.82586e7	−1.14060	−0.570299	+	0.821437i	$0.693173\pi$
$908$	−496050.	−0.0199669
$909$	1.25263e7	0.502822
$910$	0	0
$911$	−9.75713e6	−0.389517	−0.194758	−	0.980851i	$-0.562392\pi$
$911$	−9.75713e6	−0.389517	−0.194758	+	0.980851i	$0.562392\pi$
$912$	−2.95033e7	−1.17458
$913$	−6.28219e6	−0.249422
$914$	4.13432e7	1.63696
$915$	0	0
$916$	1.33841e6	0.0527050
$917$	−1.24169e7	−0.487628
$918$	−1.25015e7	−0.489617
$919$	−3.69755e7	−1.44419	−0.722097	−	0.691792i	$-0.756821\pi$
$919$	−3.69755e7	−1.44419	−0.722097	+	0.691792i	$0.756821\pi$
$920$	0	0
$921$	2.03287e7	0.789697
$922$	−6.06300e6	−0.234888
$923$	4.14431e7	1.60121
$924$	3.73542e6	0.143933
$925$	0	0
$926$	7.30784e6	0.280067
$927$	9.55433e6	0.365175
$928$	5.69145e6	0.216947
$929$	−5.05419e7	−1.92137	−0.960687	−	0.277633i	$-0.910450\pi$
$929$	−5.05419e7	−1.92137	−0.960687	+	0.277633i	$0.910450\pi$
$930$	0	0
$931$	−1.57967e7	−0.597299
$932$	−9.50480e6	−0.358429
$933$	2.70393e7	1.01693
$934$	2.41428e7	0.905566
$935$	0	0
$936$	−4.81999e6	−0.179828
$937$	−1.08317e7	−0.403040	−0.201520	−	0.979484i	$-0.564588\pi$
$937$	−1.08317e7	−0.403040	−0.201520	+	0.979484i	$0.564588\pi$
$938$	2.08094e7	0.772240
$939$	2.49268e7	0.922578
$940$	0	0
$941$	1.45659e6	0.0536246	0.0268123	−	0.999640i	$-0.491464\pi$
$941$	1.45659e6	0.0536246	0.0268123	+	0.999640i	$0.491464\pi$
$942$	−3.39631e7	−1.24704
$943$	−1.15909e7	−0.424463
$944$	3.67423e7	1.34195
$945$	0	0
$946$	915011.	0.0332429
$947$	4.32433e7	1.56691	0.783455	−	0.621449i	$-0.213456\pi$
$947$	4.32433e7	1.56691	0.783455	+	0.621449i	$0.213456\pi$
$948$	759883.	0.0274616
$949$	−1.66358e7	−0.599623
$950$	0	0
$951$	1.80288e7	0.646421
$952$	2.39590e7	0.856792
$953$	−1.69154e7	−0.603322	−0.301661	−	0.953415i	$-0.597541\pi$
$953$	−1.69154e7	−0.603322	−0.301661	+	0.953415i	$0.597541\pi$
$954$	−7.98179e6	−0.283942
$955$	0	0
$956$	−2.93900e7	−1.04005
$957$	892890.	0.0315151
$958$	4.15663e7	1.46328
$959$	3.04784e7	1.07015
$960$	0	0
$961$	2.47099e7	0.863102
$962$	−1.92357e7	−0.670147
$963$	−1.88800e7	−0.656049
$964$	1.12817e7	0.391003
$965$	0	0
$966$	2.54894e7	0.878853
$967$	−2.51066e7	−0.863421	−0.431710	−	0.902012i	$-0.642090\pi$
$967$	−2.51066e7	−0.863421	−0.431710	+	0.902012i	$0.642090\pi$
$968$	−1.00323e6	−0.0344121
$969$	5.53906e7	1.89508
$970$	0	0
$971$	−3.10806e7	−1.05789	−0.528947	−	0.848655i	$-0.677413\pi$
$971$	−3.10806e7	−1.05789	−0.528947	+	0.848655i	$0.677413\pi$
$972$	−1.34269e6	−0.0455836
$973$	3.02174e7	1.02323
$974$	3.19441e7	1.07893
$975$	0	0
$976$	2.13055e7	0.715925
$977$	1.72964e7	0.579722	0.289861	−	0.957069i	$-0.406391\pi$
$977$	1.72964e7	0.579722	0.289861	+	0.957069i	$0.406391\pi$
$978$	2.04637e7	0.684129
$979$	−6.40894e6	−0.213712
$980$	0	0
$981$	−1.52072e6	−0.0504517
$982$	6.49313e7	2.14870
$983$	−2.02617e7	−0.668793	−0.334396	−	0.942433i	$-0.608532\pi$
$983$	−2.02617e7	−0.668793	−0.334396	+	0.942433i	$0.608532\pi$
$984$	−2.81689e6	−0.0927432
$985$	0	0
$986$	−1.40607e7	−0.460589
$987$	−3.32619e7	−1.08681
$988$	−5.24323e7	−1.70886
$989$	2.59367e6	0.0843186
$990$	0	0
$991$	1.64795e7	0.533041	0.266520	−	0.963829i	$-0.414126\pi$
$991$	1.64795e7	0.533041	0.266520	+	0.963829i	$0.414126\pi$
$992$	−5.06962e7	−1.63567
$993$	−9.86783e6	−0.317577
$994$	−5.32618e7	−1.70982
$995$	0	0
$996$	1.06250e7	0.339377
$997$	−364567.	−0.0116155	−0.00580777	−	0.999983i	$-0.501849\pi$
$997$	−364567.	−0.0116155	−0.00580777	+	0.999983i	$0.501849\pi$
$998$	−6.71869e7	−2.13530
$999$	2.18251e6	0.0691899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.v.1.11		13
5.2	odd	4		165.6.c.b.34.21	yes	26
5.3	odd	4		165.6.c.b.34.6	✓	26
5.4	even	2		825.6.a.y.1.3		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.6	✓	26	5.3	odd	4
165.6.c.b.34.21	yes	26	5.2	odd	4
825.6.a.v.1.11		13	1.1	even	1	trivial
825.6.a.y.1.3		13	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+7.39855 q^{2} -9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} -150.852 q^{7} -68.5217 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+7.39855 q^{2} -9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} -150.852 q^{7} -68.5217 q^{8} +81.0000 q^{9} +121.000 q^{11} -204.646 q^{12} +868.426 q^{13} -1116.08 q^{14} -1234.59 q^{16} +2317.87 q^{17} +599.282 q^{18} -2655.24 q^{19} +1357.67 q^{21} +895.224 q^{22} +2537.58 q^{23} +616.695 q^{24} +6425.09 q^{26} -729.000 q^{27} -3430.14 q^{28} -819.917 q^{29} +7303.36 q^{31} -6941.50 q^{32} -1089.00 q^{33} +17148.9 q^{34} +1841.82 q^{36} -2993.84 q^{37} -19644.9 q^{38} -7815.84 q^{39} -4567.72 q^{41} +10044.8 q^{42} +1022.10 q^{43} +2751.36 q^{44} +18774.4 q^{46} -24499.3 q^{47} +11111.3 q^{48} +5949.25 q^{49} -20860.8 q^{51} +19746.7 q^{52} -13318.9 q^{53} -5393.54 q^{54} +10336.6 q^{56} +23897.2 q^{57} -6066.20 q^{58} -29760.6 q^{59} -17257.1 q^{61} +54034.2 q^{62} -12219.0 q^{63} -11850.0 q^{64} -8057.02 q^{66} -18645.0 q^{67} +52704.9 q^{68} -22838.2 q^{69} +47722.0 q^{71} -5550.26 q^{72} -19156.3 q^{73} -22150.1 q^{74} -60376.2 q^{76} -18253.1 q^{77} -57825.8 q^{78} -3713.15 q^{79} +6561.00 q^{81} -33794.5 q^{82} -51919.0 q^{83} +30871.3 q^{84} +7562.07 q^{86} +7379.26 q^{87} -8291.13 q^{88} -52966.5 q^{89} -131004. q^{91} +57700.8 q^{92} -65730.2 q^{93} -181259. q^{94} +62473.5 q^{96} -114526. q^{97} +44015.8 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * q^99

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