LMFDB - Embedded newform 825.6.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$13$
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			1.1
Root			$11.4306$ 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)

$f(q)$	$=$	$q-10.4306 q^{2} +9.00000 q^{3} +76.7973 q^{4} -93.8753 q^{6} +21.9028 q^{7} -467.262 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-10.4306 q^{2} +9.00000 q^{3} +76.7973 q^{4} -93.8753 q^{6} +21.9028 q^{7} -467.262 q^{8} +81.0000 q^{9} +121.000 q^{11} +691.175 q^{12} +460.797 q^{13} -228.459 q^{14} +2416.31 q^{16} +2318.63 q^{17} -844.878 q^{18} +800.141 q^{19} +197.125 q^{21} -1262.10 q^{22} +4781.90 q^{23} -4205.36 q^{24} -4806.39 q^{26} +729.000 q^{27} +1682.08 q^{28} +911.582 q^{29} -944.328 q^{31} -10251.1 q^{32} +1089.00 q^{33} -24184.7 q^{34} +6220.58 q^{36} +12825.4 q^{37} -8345.95 q^{38} +4147.17 q^{39} +8712.22 q^{41} -2056.13 q^{42} -20504.8 q^{43} +9292.47 q^{44} -49878.1 q^{46} -9288.25 q^{47} +21746.8 q^{48} -16327.3 q^{49} +20867.7 q^{51} +35388.0 q^{52} +23263.1 q^{53} -7603.90 q^{54} -10234.4 q^{56} +7201.27 q^{57} -9508.34 q^{58} +20726.7 q^{59} -6625.93 q^{61} +9849.90 q^{62} +1774.13 q^{63} +29603.6 q^{64} -11358.9 q^{66} +57904.9 q^{67} +178064. q^{68} +43037.1 q^{69} +38362.5 q^{71} -37848.2 q^{72} +8431.57 q^{73} -133777. q^{74} +61448.7 q^{76} +2650.24 q^{77} -43257.5 q^{78} -1683.43 q^{79} +6561.00 q^{81} -90873.6 q^{82} +7798.26 q^{83} +15138.7 q^{84} +213877. q^{86} +8204.23 q^{87} -56538.7 q^{88} -83079.0 q^{89} +10092.8 q^{91} +367237. q^{92} -8498.95 q^{93} +96882.0 q^{94} -92260.2 q^{96} +14678.5 q^{97} +170303. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.4306	−1.84389	−0.921943	−	0.387326i	$-0.873399\pi$
$2$	−10.4306	−1.84389	−0.921943	+	0.387326i	$0.873399\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	76.7973	2.39991
$5$	0	0
$5$	0	0
$6$	−93.8753	−1.06457
$7$	21.9028	0.168949	0.0844744	−	0.996426i	$-0.473079\pi$
$7$	21.9028	0.168949	0.0844744	+	0.996426i	$0.473079\pi$
$8$	−467.262	−2.58128
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	691.175	1.38559
$13$	460.797	0.756225	0.378113	−	0.925760i	$-0.376573\pi$
$13$	460.797	0.756225	0.378113	+	0.925760i	$0.376573\pi$
$14$	−228.459	−0.311522
$15$	0	0
$16$	2416.31	2.35967
$17$	2318.63	1.94585	0.972924	−	0.231127i	$-0.0742413\pi$
$17$	2318.63	1.94585	0.972924	+	0.231127i	$0.0742413\pi$
$18$	−844.878	−0.614629
$19$	800.141	0.508490	0.254245	−	0.967140i	$-0.418173\pi$
$19$	800.141	0.508490	0.254245	+	0.967140i	$0.418173\pi$
$20$	0	0
$21$	197.125	0.0975426
$22$	−1262.10	−0.555952
$23$	4781.90	1.88487	0.942434	−	0.334391i	$-0.108531\pi$
$23$	4781.90	1.88487	0.942434	+	0.334391i	$0.108531\pi$
$24$	−4205.36	−1.49030
$25$	0	0
$26$	−4806.39	−1.39439
$27$	729.000	0.192450
$28$	1682.08	0.405462
$29$	911.582	0.201280	0.100640	−	0.994923i	$-0.467911\pi$
$29$	911.582	0.201280	0.100640	+	0.994923i	$0.467911\pi$
$30$	0	0
$31$	−944.328	−0.176489	−0.0882447	−	0.996099i	$-0.528126\pi$
$31$	−944.328	−0.176489	−0.0882447	+	0.996099i	$0.528126\pi$
$32$	−10251.1	−1.76969
$33$	1089.00	0.174078
$34$	−24184.7	−3.58792
$35$	0	0
$36$	6220.58	0.799971
$37$	12825.4	1.54017	0.770084	−	0.637942i	$-0.220214\pi$
$37$	12825.4	1.54017	0.770084	+	0.637942i	$0.220214\pi$
$38$	−8345.95	−0.937598
$39$	4147.17	0.436607
$40$	0	0
$41$	8712.22	0.809411	0.404706	−	0.914447i	$-0.367374\pi$
$41$	8712.22	0.809411	0.404706	+	0.914447i	$0.367374\pi$
$42$	−2056.13	−0.179857
$43$	−20504.8	−1.69116	−0.845578	−	0.533852i	$-0.820744\pi$
$43$	−20504.8	−1.69116	−0.845578	+	0.533852i	$0.820744\pi$
$44$	9292.47	0.723601
$45$	0	0
$46$	−49878.1	−3.47548
$47$	−9288.25	−0.613323	−0.306662	−	0.951819i	$-0.599212\pi$
$47$	−9288.25	−0.613323	−0.306662	+	0.951819i	$0.599212\pi$
$48$	21746.8	1.36236
$49$	−16327.3	−0.971456
$50$	0	0
$51$	20867.7	1.12344
$52$	35388.0	1.81488
$53$	23263.1	1.13757	0.568785	−	0.822486i	$-0.307414\pi$
$53$	23263.1	1.13757	0.568785	+	0.822486i	$0.307414\pi$
$54$	−7603.90	−0.354856
$55$	0	0
$56$	−10234.4	−0.436104
$57$	7201.27	0.293577
$58$	−9508.34	−0.371137
$59$	20726.7	0.775177	0.387588	−	0.921833i	$-0.373308\pi$
$59$	20726.7	0.775177	0.387588	+	0.921833i	$0.373308\pi$
$60$	0	0
$61$	−6625.93	−0.227993	−0.113997	−	0.993481i	$-0.536365\pi$
$61$	−6625.93	−0.227993	−0.113997	+	0.993481i	$0.536365\pi$
$62$	9849.90	0.325426
$63$	1774.13	0.0563162
$64$	29603.6	0.903429
$65$	0	0
$66$	−11358.9	−0.320979
$67$	57904.9	1.57590	0.787950	−	0.615739i	$-0.211143\pi$
$67$	57904.9	1.57590	0.787950	+	0.615739i	$0.211143\pi$
$68$	178064.	4.66987
$69$	43037.1	1.08823
$70$	0	0
$71$	38362.5	0.903153	0.451577	−	0.892232i	$-0.350862\pi$
$71$	38362.5	0.903153	0.451577	+	0.892232i	$0.350862\pi$
$72$	−37848.2	−0.860427
$73$	8431.57	0.185183	0.0925916	−	0.995704i	$-0.470485\pi$
$73$	8431.57	0.185183	0.0925916	+	0.995704i	$0.470485\pi$
$74$	−133777.	−2.83989
$75$	0	0
$76$	61448.7	1.22033
$77$	2650.24	0.0509400
$78$	−43257.5	−0.805053
$79$	−1683.43	−0.0303478	−0.0151739	−	0.999885i	$-0.504830\pi$
$79$	−1683.43	−0.0303478	−0.0151739	+	0.999885i	$0.504830\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−90873.6	−1.49246
$83$	7798.26	0.124252	0.0621259	−	0.998068i	$-0.480212\pi$
$83$	7798.26	0.124252	0.0621259	+	0.998068i	$0.480212\pi$
$84$	15138.7	0.234094
$85$	0	0
$86$	213877.	3.11830
$87$	8204.23	0.116209
$88$	−56538.7	−0.778286
$89$	−83079.0	−1.11177	−0.555887	−	0.831258i	$-0.687621\pi$
$89$	−83079.0	−1.11177	−0.555887	+	0.831258i	$0.687621\pi$
$90$	0	0
$91$	10092.8	0.127763
$92$	367237.	4.52352
$93$	−8498.95	−0.101896
$94$	96882.0	1.13090
$95$	0	0
$96$	−92260.2	−1.02173
$97$	14678.5	0.158399	0.0791997	−	0.996859i	$-0.474764\pi$
$97$	14678.5	0.158399	0.0791997	+	0.996859i	$0.474764\pi$
$98$	170303.	1.79125
$99$	9801.00	0.100504
$100$	0	0
$101$	−19178.1	−0.187069	−0.0935345	−	0.995616i	$-0.529817\pi$
$101$	−19178.1	−0.187069	−0.0935345	+	0.995616i	$0.529817\pi$
$102$	−217662.	−2.07149
$103$	−127841.	−1.18735	−0.593675	−	0.804705i	$-0.702323\pi$
$103$	−127841.	−1.18735	−0.593675	+	0.804705i	$0.702323\pi$
$104$	−215313.	−1.95203
$105$	0	0
$106$	−242648.	−2.09755
$107$	52736.6	0.445300	0.222650	−	0.974898i	$-0.428529\pi$
$107$	52736.6	0.445300	0.222650	+	0.974898i	$0.428529\pi$
$108$	55985.2	0.461864
$109$	−233661.	−1.88374	−0.941869	−	0.335979i	$-0.890933\pi$
$109$	−233661.	−1.88374	−0.941869	+	0.335979i	$0.890933\pi$
$110$	0	0
$111$	115429.	0.889216
$112$	52923.9	0.398664
$113$	−36635.5	−0.269902	−0.134951	−	0.990852i	$-0.543088\pi$
$113$	−36635.5	−0.269902	−0.134951	+	0.990852i	$0.543088\pi$
$114$	−75113.5	−0.541323
$115$	0	0
$116$	70007.0	0.483055
$117$	37324.6	0.252075
$118$	−216192.	−1.42934
$119$	50784.5	0.328748
$120$	0	0
$121$	14641.0	0.0909091
$122$	69112.4	0.420394
$123$	78410.0	0.467314
$124$	−72521.8	−0.423559
$125$	0	0
$126$	−18505.2	−0.103841
$127$	256889.	1.41330	0.706652	−	0.707561i	$-0.250205\pi$
$127$	256889.	1.41330	0.706652	+	0.707561i	$0.250205\pi$
$128$	19253.6	0.103869
$129$	−184543.	−0.976390
$130$	0	0
$131$	−109508.	−0.557531	−0.278765	−	0.960359i	$-0.589925\pi$
$131$	−109508.	−0.557531	−0.278765	+	0.960359i	$0.589925\pi$
$132$	83632.2	0.417771
$133$	17525.4	0.0859088
$134$	−603983.	−2.90578
$135$	0	0
$136$	−1.08341e6	−5.02278
$137$	149903.	0.682355	0.341177	−	0.939999i	$-0.389174\pi$
$137$	149903.	0.682355	0.341177	+	0.939999i	$0.389174\pi$
$138$	−448903.	−2.00657
$139$	−113251.	−0.497168	−0.248584	−	0.968610i	$-0.579965\pi$
$139$	−113251.	−0.497168	−0.248584	+	0.968610i	$0.579965\pi$
$140$	0	0
$141$	−83594.3	−0.354102
$142$	−400144.	−1.66531
$143$	55756.4	0.228011
$144$	195721.	0.786558
$145$	0	0
$146$	−87946.3	−0.341456
$147$	−146945.	−0.560871
$148$	984959.	3.69627
$149$	−61807.9	−0.228075	−0.114038	−	0.993476i	$-0.536378\pi$
$149$	−61807.9	−0.228075	−0.114038	+	0.993476i	$0.536378\pi$
$150$	0	0
$151$	−256778.	−0.916465	−0.458233	−	0.888832i	$-0.651517\pi$
$151$	−256778.	−0.916465	−0.458233	+	0.888832i	$0.651517\pi$
$152$	−373876.	−1.31256
$153$	187809.	0.648616
$154$	−27643.6	−0.0939275
$155$	0	0
$156$	318492.	1.04782
$157$	−116054.	−0.375761	−0.187880	−	0.982192i	$-0.560162\pi$
$157$	−116054.	−0.375761	−0.187880	+	0.982192i	$0.560162\pi$
$158$	17559.2	0.0559579
$159$	209368.	0.656776
$160$	0	0
$161$	104737.	0.318446
$162$	−68435.1	−0.204876
$163$	−322117.	−0.949609	−0.474805	−	0.880091i	$-0.657481\pi$
$163$	−322117.	−0.949609	−0.474805	+	0.880091i	$0.657481\pi$
$164$	669075.	1.94252
$165$	0	0
$166$	−81340.5	−0.229106
$167$	−435361.	−1.20797	−0.603987	−	0.796994i	$-0.706422\pi$
$167$	−435361.	−1.20797	−0.603987	+	0.796994i	$0.706422\pi$
$168$	−92109.2	−0.251785
$169$	−158959.	−0.428123
$170$	0	0
$171$	64811.5	0.169497
$172$	−1.57471e6	−4.05863
$173$	−700319.	−1.77902	−0.889510	−	0.456916i	$-0.848954\pi$
$173$	−700319.	−1.77902	−0.889510	+	0.456916i	$0.848954\pi$
$174$	−85575.0	−0.214276
$175$	0	0
$176$	292373.	0.711469
$177$	186541.	0.447548
$178$	866563.	2.04998
$179$	228192.	0.532314	0.266157	−	0.963930i	$-0.414246\pi$
$179$	228192.	0.532314	0.266157	+	0.963930i	$0.414246\pi$
$180$	0	0
$181$	511598.	1.16073	0.580366	−	0.814356i	$-0.302909\pi$
$181$	511598.	1.16073	0.580366	+	0.814356i	$0.302909\pi$
$182$	−105273.	−0.235581
$183$	−59633.4	−0.131632
$184$	−2.23440e6	−4.86538
$185$	0	0
$186$	88649.1	0.187885
$187$	280554.	0.586695
$188$	−713312.	−1.47192
$189$	15967.2	0.0325142
$190$	0	0
$191$	−803150.	−1.59299	−0.796496	−	0.604644i	$-0.793315\pi$
$191$	−803150.	−1.59299	−0.796496	+	0.604644i	$0.793315\pi$
$192$	266432.	0.521595
$193$	−578708.	−1.11832	−0.559160	−	0.829059i	$-0.688876\pi$
$193$	−578708.	−1.11832	−0.559160	+	0.829059i	$0.688876\pi$
$194$	−153106.	−0.292070
$195$	0	0
$196$	−1.25389e6	−2.33141
$197$	278554.	0.511381	0.255690	−	0.966759i	$-0.417697\pi$
$197$	278554.	0.511381	0.255690	+	0.966759i	$0.417697\pi$
$198$	−102230.	−0.185317
$199$	843631.	1.51015	0.755074	−	0.655639i	$-0.227601\pi$
$199$	843631.	1.51015	0.755074	+	0.655639i	$0.227601\pi$
$200$	0	0
$201$	521144.	0.909846
$202$	200039.	0.344934
$203$	19966.2	0.0340060
$204$	1.60258e6	2.69615
$205$	0	0
$206$	1.33346e6	2.18934
$207$	387334.	0.628290
$208$	1.11343e6	1.78445
$209$	96817.1	0.153316
$210$	0	0
$211$	48614.4	0.0751724	0.0375862	−	0.999293i	$-0.488033\pi$
$211$	48614.4	0.0751724	0.0375862	+	0.999293i	$0.488033\pi$
$212$	1.78654e6	2.73007
$213$	345263.	0.521436
$214$	−550074.	−0.821082
$215$	0	0
$216$	−340634.	−0.496768
$217$	−20683.4	−0.0298177
$218$	2.43723e6	3.47340
$219$	75884.1	0.106916
$220$	0	0
$221$	1.06842e6	1.47150
$222$	−1.20399e6	−1.63961
$223$	473625.	0.637782	0.318891	−	0.947791i	$-0.396690\pi$
$223$	473625.	0.637782	0.318891	+	0.947791i	$0.396690\pi$
$224$	−224529.	−0.298987
$225$	0	0
$226$	382130.	0.497668
$227$	767793.	0.988962	0.494481	−	0.869188i	$-0.335358\pi$
$227$	767793.	0.988962	0.494481	+	0.869188i	$0.335358\pi$
$228$	553038.	0.704560
$229$	580162.	0.731073	0.365536	−	0.930797i	$-0.380886\pi$
$229$	580162.	0.731073	0.365536	+	0.930797i	$0.380886\pi$
$230$	0	0
$231$	23852.2	0.0294102
$232$	−425947.	−0.519560
$233$	298663.	0.360406	0.180203	−	0.983629i	$-0.442325\pi$
$233$	298663.	0.360406	0.180203	+	0.983629i	$0.442325\pi$
$234$	−389317.	−0.464798
$235$	0	0
$236$	1.59176e6	1.86036
$237$	−15150.9	−0.0175213
$238$	−529712.	−0.606174
$239$	324060.	0.366970	0.183485	−	0.983023i	$-0.441262\pi$
$239$	324060.	0.366970	0.183485	+	0.983023i	$0.441262\pi$
$240$	0	0
$241$	−679917.	−0.754073	−0.377036	−	0.926198i	$-0.623057\pi$
$241$	−679917.	−0.754073	−0.377036	+	0.926198i	$0.623057\pi$
$242$	−152714.	−0.167626
$243$	59049.0	0.0641500
$244$	−508853.	−0.547165
$245$	0	0
$246$	−817863.	−0.861673
$247$	368703.	0.384533
$248$	441249.	0.455569
$249$	70184.4	0.0717368
$250$	0	0
$251$	1.78432e6	1.78767	0.893836	−	0.448393i	$-0.148004\pi$
$251$	1.78432e6	1.78767	0.893836	+	0.448393i	$0.148004\pi$
$252$	136248.	0.135154
$253$	578610.	0.568309
$254$	−2.67950e6	−2.60597
$255$	0	0
$256$	−1.14814e6	−1.09495
$257$	−891074.	−0.841552	−0.420776	−	0.907165i	$-0.638242\pi$
$257$	−891074.	−0.841552	−0.420776	+	0.907165i	$0.638242\pi$
$258$	1.92489e6	1.80035
$259$	280913.	0.260209
$260$	0	0
$261$	73838.1	0.0670933
$262$	1.14224e6	1.02802
$263$	−1.27856e6	−1.13980	−0.569902	−	0.821713i	$-0.693019\pi$
$263$	−1.27856e6	−1.13980	−0.569902	+	0.821713i	$0.693019\pi$
$264$	−508848.	−0.449344
$265$	0	0
$266$	−182800.	−0.158406
$267$	−747711.	−0.641883
$268$	4.44694e6	3.78203
$269$	454281.	0.382775	0.191388	−	0.981515i	$-0.438701\pi$
$269$	454281.	0.382775	0.191388	+	0.981515i	$0.438701\pi$
$270$	0	0
$271$	−2.02780e6	−1.67726	−0.838632	−	0.544698i	$-0.816644\pi$
$271$	−2.02780e6	−1.67726	−0.838632	+	0.544698i	$0.816644\pi$
$272$	5.60252e6	4.59157
$273$	90834.8	0.0737642
$274$	−1.56358e6	−1.25818
$275$	0	0
$276$	3.30513e6	2.61166
$277$	1.27638e6	0.999496	0.499748	−	0.866171i	$-0.333426\pi$
$277$	1.27638e6	0.999496	0.499748	+	0.866171i	$0.333426\pi$
$278$	1.18127e6	0.916722
$279$	−76490.6	−0.0588298
$280$	0	0
$281$	−1.02183e6	−0.771994	−0.385997	−	0.922500i	$-0.626143\pi$
$281$	−1.02183e6	−0.771994	−0.385997	+	0.922500i	$0.626143\pi$
$282$	871938.	0.652924
$283$	−75339.2	−0.0559184	−0.0279592	−	0.999609i	$-0.508901\pi$
$283$	−75339.2	−0.0559184	−0.0279592	+	0.999609i	$0.508901\pi$
$284$	2.94614e6	2.16749
$285$	0	0
$286$	−581573.	−0.420425
$287$	190822.	0.136749
$288$	−830342.	−0.589896
$289$	3.95618e6	2.78632
$290$	0	0
$291$	132107.	0.0914519
$292$	647522.	0.444424
$293$	661394.	0.450082	0.225041	−	0.974349i	$-0.427748\pi$
$293$	661394.	0.450082	0.225041	+	0.974349i	$0.427748\pi$
$294$	1.53273e6	1.03418
$295$	0	0
$296$	−5.99284e6	−3.97561
$297$	88209.0	0.0580259
$298$	644693.	0.420545
$299$	2.20349e6	1.42539
$300$	0	0
$301$	−449112.	−0.285719
$302$	2.67835e6	1.68986
$303$	−172603.	−0.108004
$304$	1.93339e6	1.19987
$305$	0	0
$306$	−1.95896e6	−1.19597
$307$	962329.	0.582744	0.291372	−	0.956610i	$-0.405888\pi$
$307$	962329.	0.582744	0.291372	+	0.956610i	$0.405888\pi$
$308$	203531.	0.122252
$309$	−1.15057e6	−0.685517
$310$	0	0
$311$	645247.	0.378290	0.189145	−	0.981949i	$-0.439428\pi$
$311$	645247.	0.378290	0.189145	+	0.981949i	$0.439428\pi$
$312$	−1.93782e6	−1.12701
$313$	−3.21424e6	−1.85446	−0.927229	−	0.374495i	$-0.877816\pi$
$313$	−3.21424e6	−1.85446	−0.927229	+	0.374495i	$0.877816\pi$
$314$	1.21051e6	0.692860
$315$	0	0
$316$	−129283.	−0.0728321
$317$	1.59691e6	0.892552	0.446276	−	0.894895i	$-0.352750\pi$
$317$	1.59691e6	0.892552	0.446276	+	0.894895i	$0.352750\pi$
$318$	−2.18383e6	−1.21102
$319$	110301.	0.0606882
$320$	0	0
$321$	474629.	0.257094
$322$	−1.09247e6	−0.587178
$323$	1.85523e6	0.989445
$324$	503867.	0.266657
$325$	0	0
$326$	3.35987e6	1.75097
$327$	−2.10295e6	−1.08758
$328$	−4.07089e6	−2.08932
$329$	−203439.	−0.103620
$330$	0	0
$331$	1.89235e6	0.949363	0.474682	−	0.880158i	$-0.342563\pi$
$331$	1.89235e6	0.949363	0.474682	+	0.880158i	$0.342563\pi$
$332$	598885.	0.298194
$333$	1.03886e6	0.513389
$334$	4.54107e6	2.22737
$335$	0	0
$336$	476315.	0.230169
$337$	1.58818e6	0.761772	0.380886	−	0.924622i	$-0.375619\pi$
$337$	1.58818e6	0.761772	0.380886	+	0.924622i	$0.375619\pi$
$338$	1.65804e6	0.789410
$339$	−329719.	−0.155828
$340$	0	0
$341$	−114264.	−0.0532136
$342$	−676022.	−0.312533
$343$	−725734.	−0.333075
$344$	9.58110e6	4.36535
$345$	0	0
$346$	7.30474e6	3.28031
$347$	−1.31246e6	−0.585141	−0.292571	−	0.956244i	$-0.594511\pi$
$347$	−1.31246e6	−0.585141	−0.292571	+	0.956244i	$0.594511\pi$
$348$	630063.	0.278892
$349$	−3.90078e6	−1.71430	−0.857151	−	0.515065i	$-0.827768\pi$
$349$	−3.90078e6	−1.71430	−0.857151	+	0.515065i	$0.827768\pi$
$350$	0	0
$351$	335921.	0.145536
$352$	−1.24039e6	−0.533581
$353$	3.32103e6	1.41852	0.709261	−	0.704946i	$-0.249029\pi$
$353$	3.32103e6	1.41852	0.709261	+	0.704946i	$0.249029\pi$
$354$	−1.94573e6	−0.825228
$355$	0	0
$356$	−6.38024e6	−2.66816
$357$	457060.	0.189803
$358$	−2.38018e6	−0.981527
$359$	2.37759e6	0.973647	0.486823	−	0.873500i	$-0.338155\pi$
$359$	2.37759e6	0.973647	0.486823	+	0.873500i	$0.338155\pi$
$360$	0	0
$361$	−1.83587e6	−0.741438
$362$	−5.33627e6	−2.14026
$363$	131769.	0.0524864
$364$	775096.	0.306621
$365$	0	0
$366$	622011.	0.242714
$367$	−3.52479e6	−1.36605	−0.683027	−	0.730393i	$-0.739337\pi$
$367$	−3.52479e6	−1.36605	−0.683027	+	0.730393i	$0.739337\pi$
$368$	1.15545e7	4.44768
$369$	705690.	0.269804
$370$	0	0
$371$	509528.	0.192191
$372$	−652696.	−0.244542
$373$	−3.82557e6	−1.42372	−0.711859	−	0.702322i	$-0.752147\pi$
$373$	−3.82557e6	−1.42372	−0.711859	+	0.702322i	$0.752147\pi$
$374$	−2.92634e6	−1.08180
$375$	0	0
$376$	4.34005e6	1.58316
$377$	420054.	0.152213
$378$	−166547.	−0.0599525
$379$	3.70497e6	1.32491	0.662456	−	0.749101i	$-0.269514\pi$
$379$	3.70497e6	1.32491	0.662456	+	0.749101i	$0.269514\pi$
$380$	0	0
$381$	2.31200e6	0.815972
$382$	8.37733e6	2.93729
$383$	−789569.	−0.275038	−0.137519	−	0.990499i	$-0.543913\pi$
$383$	−789569.	−0.275038	−0.137519	+	0.990499i	$0.543913\pi$
$384$	173283.	0.0599691
$385$	0	0
$386$	6.03627e6	2.06206
$387$	−1.66089e6	−0.563719
$388$	1.12727e6	0.380145
$389$	−4.73042e6	−1.58499	−0.792494	−	0.609879i	$-0.791218\pi$
$389$	−4.73042e6	−1.58499	−0.792494	+	0.609879i	$0.791218\pi$
$390$	0	0
$391$	1.10875e7	3.66767
$392$	7.62911e6	2.50760
$393$	−985575.	−0.321891
$394$	−2.90549e6	−0.942928
$395$	0	0
$396$	752690.	0.241200
$397$	−2.42880e6	−0.773421	−0.386711	−	0.922201i	$-0.626389\pi$
$397$	−2.42880e6	−0.773421	−0.386711	+	0.922201i	$0.626389\pi$
$398$	−8.79957e6	−2.78454
$399$	157728.	0.0495995
$400$	0	0
$401$	276251.	0.0857914	0.0428957	−	0.999080i	$-0.486342\pi$
$401$	276251.	0.0857914	0.0428957	+	0.999080i	$0.486342\pi$
$402$	−5.43585e6	−1.67765
$403$	−435144.	−0.133466
$404$	−1.47282e6	−0.448949
$405$	0	0
$406$	−208259.	−0.0627032
$407$	1.55188e6	0.464378
$408$	−9.75066e6	−2.89990
$409$	−563175.	−0.166470	−0.0832349	−	0.996530i	$-0.526525\pi$
$409$	−563175.	−0.166470	−0.0832349	+	0.996530i	$0.526525\pi$
$410$	0	0
$411$	1.34913e6	0.393958
$412$	−9.81787e6	−2.84954
$413$	453974.	0.130965
$414$	−4.04012e6	−1.15849
$415$	0	0
$416$	−4.72369e6	−1.33828
$417$	−1.01926e6	−0.287040
$418$	−1.00986e6	−0.282696
$419$	707010.	0.196739	0.0983695	−	0.995150i	$-0.468637\pi$
$419$	707010.	0.196739	0.0983695	+	0.995150i	$0.468637\pi$
$420$	0	0
$421$	4.56766e6	1.25600	0.627999	−	0.778214i	$-0.283874\pi$
$421$	4.56766e6	1.25600	0.627999	+	0.778214i	$0.283874\pi$
$422$	−507077.	−0.138609
$423$	−752349.	−0.204441
$424$	−1.08700e7	−2.93639
$425$	0	0
$426$	−3.60130e6	−0.961468
$427$	−145127.	−0.0385192
$428$	4.05003e6	1.06868
$429$	501808.	0.131642
$430$	0	0
$431$	6.16454e6	1.59848	0.799241	−	0.601011i	$-0.205235\pi$
$431$	6.16454e6	1.59848	0.799241	+	0.601011i	$0.205235\pi$
$432$	1.76149e6	0.454120
$433$	−2.94520e6	−0.754910	−0.377455	−	0.926028i	$-0.623201\pi$
$433$	−2.94520e6	−0.754910	−0.377455	+	0.926028i	$0.623201\pi$
$434$	215741.	0.0549804
$435$	0	0
$436$	−1.79446e7	−4.52081
$437$	3.82620e6	0.958438
$438$	−791517.	−0.197140
$439$	−5.90123e6	−1.46144	−0.730720	−	0.682677i	$-0.760816\pi$
$439$	−5.90123e6	−1.46144	−0.730720	+	0.682677i	$0.760816\pi$
$440$	0	0
$441$	−1.32251e6	−0.323819
$442$	−1.11442e7	−2.71328
$443$	4.58354e6	1.10966	0.554832	−	0.831963i	$-0.312783\pi$
$443$	4.58354e6	1.10966	0.554832	+	0.831963i	$0.312783\pi$
$444$	8.86463e6	2.13404
$445$	0	0
$446$	−4.94019e6	−1.17600
$447$	−556271.	−0.131679
$448$	648401.	0.152633
$449$	6.61995e6	1.54967	0.774834	−	0.632164i	$-0.217833\pi$
$449$	6.61995e6	1.54967	0.774834	+	0.632164i	$0.217833\pi$
$450$	0	0
$451$	1.05418e6	0.244047
$452$	−2.81350e6	−0.647741
$453$	−2.31101e6	−0.529122
$454$	−8.00854e6	−1.82353
$455$	0	0
$456$	−3.36488e6	−0.757805
$457$	−6.37429e6	−1.42771	−0.713857	−	0.700292i	$-0.753053\pi$
$457$	−6.37429e6	−1.42771	−0.713857	+	0.700292i	$0.753053\pi$
$458$	−6.05143e6	−1.34801
$459$	1.69028e6	0.374478
$460$	0	0
$461$	−5.36939e6	−1.17672	−0.588359	−	0.808600i	$-0.700226\pi$
$461$	−5.36939e6	−1.17672	−0.588359	+	0.808600i	$0.700226\pi$
$462$	−248792.	−0.0542290
$463$	7.13889e6	1.54767	0.773835	−	0.633387i	$-0.218336\pi$
$463$	7.13889e6	1.54767	0.773835	+	0.633387i	$0.218336\pi$
$464$	2.20266e6	0.474955
$465$	0	0
$466$	−3.11523e6	−0.664547
$467$	−5.11617e6	−1.08556	−0.542779	−	0.839876i	$-0.682628\pi$
$467$	−5.11617e6	−1.08556	−0.542779	+	0.839876i	$0.682628\pi$
$468$	2.86642e6	0.604959
$469$	1.26828e6	0.266246
$470$	0	0
$471$	−1.04449e6	−0.216946
$472$	−9.68481e6	−2.00095
$473$	−2.48108e6	−0.509903
$474$	158033.	0.0323073
$475$	0	0
$476$	3.90011e6	0.788968
$477$	1.88431e6	0.379190
$478$	−3.38013e6	−0.676650
$479$	−715144.	−0.142415	−0.0712074	−	0.997462i	$-0.522685\pi$
$479$	−715144.	−0.142415	−0.0712074	+	0.997462i	$0.522685\pi$
$480$	0	0
$481$	5.90993e6	1.16471
$482$	7.09194e6	1.39042
$483$	942634.	0.183855
$484$	1.12439e6	0.218174
$485$	0	0
$486$	−615916.	−0.118285
$487$	9.41305e6	1.79849	0.899245	−	0.437445i	$-0.144117\pi$
$487$	9.41305e6	1.79849	0.899245	+	0.437445i	$0.144117\pi$
$488$	3.09605e6	0.588515
$489$	−2.89906e6	−0.548257
$490$	0	0
$491$	−2.03825e6	−0.381551	−0.190776	−	0.981634i	$-0.561100\pi$
$491$	−2.03825e6	−0.381551	−0.190776	+	0.981634i	$0.561100\pi$
$492$	6.02167e6	1.12151
$493$	2.11362e6	0.391660
$494$	−3.84579e6	−0.709036
$495$	0	0
$496$	−2.28179e6	−0.416458
$497$	840248.	0.152587
$498$	−732065.	−0.132274
$499$	−2.29272e6	−0.412192	−0.206096	−	0.978532i	$-0.566076\pi$
$499$	−2.29272e6	−0.412192	−0.206096	+	0.978532i	$0.566076\pi$
$500$	0	0
$501$	−3.91825e6	−0.697425
$502$	−1.86115e7	−3.29626
$503$	6.36926e6	1.12246	0.561228	−	0.827661i	$-0.310329\pi$
$503$	6.36926e6	1.12246	0.561228	+	0.827661i	$0.310329\pi$
$504$	−828983.	−0.145368
$505$	0	0
$506$	−6.03525e6	−1.04790
$507$	−1.43063e6	−0.247177
$508$	1.97284e7	3.39181
$509$	−5.66660e6	−0.969456	−0.484728	−	0.874665i	$-0.661081\pi$
$509$	−5.66660e6	−0.969456	−0.484728	+	0.874665i	$0.661081\pi$
$510$	0	0
$511$	184675.	0.0312864
$512$	1.13597e7	1.91510
$513$	583303.	0.0978590
$514$	9.29442e6	1.55172
$515$	0	0
$516$	−1.41724e7	−2.34325
$517$	−1.12388e6	−0.184924
$518$	−2.93009e6	−0.479796
$519$	−6.30287e6	−1.02712
$520$	0	0
$521$	6.07157e6	0.979956	0.489978	−	0.871735i	$-0.337005\pi$
$521$	6.07157e6	0.979956	0.489978	+	0.871735i	$0.337005\pi$
$522$	−770175.	−0.123712
$523$	−2.50083e6	−0.399789	−0.199894	−	0.979817i	$-0.564060\pi$
$523$	−2.50083e6	−0.399789	−0.199894	+	0.979817i	$0.564060\pi$
$524$	−8.40994e6	−1.33803
$525$	0	0
$526$	1.33361e7	2.10167
$527$	−2.18954e6	−0.343421
$528$	2.63136e6	0.410767
$529$	1.64302e7	2.55273
$530$	0	0
$531$	1.67886e6	0.258392
$532$	1.34590e6	0.206174
$533$	4.01457e6	0.612098
$534$	7.79907e6	1.18356
$535$	0	0
$536$	−2.70568e7	−4.06784
$537$	2.05373e6	0.307332
$538$	−4.73842e6	−0.705794
$539$	−1.97560e6	−0.292905
$540$	0	0
$541$	2.70948e6	0.398009	0.199004	−	0.979999i	$-0.436229\pi$
$541$	2.70948e6	0.398009	0.199004	+	0.979999i	$0.436229\pi$
$542$	2.11511e7	3.09268
$543$	4.60438e6	0.670149
$544$	−2.37686e7	−3.44354
$545$	0	0
$546$	−947461.	−0.136013
$547$	9.99302e6	1.42800	0.714000	−	0.700145i	$-0.246881\pi$
$547$	9.99302e6	1.42800	0.714000	+	0.700145i	$0.246881\pi$
$548$	1.15122e7	1.63759
$549$	−536700.	−0.0759978
$550$	0	0
$551$	729394.	0.102349
$552$	−2.01096e7	−2.80903
$553$	−36871.9	−0.00512722
$554$	−1.33134e7	−1.84296
$555$	0	0
$556$	−8.69734e6	−1.19316
$557$	−2.85519e6	−0.389940	−0.194970	−	0.980809i	$-0.562461\pi$
$557$	−2.85519e6	−0.389940	−0.194970	+	0.980809i	$0.562461\pi$
$558$	797842.	0.108475
$559$	−9.44854e6	−1.27890
$560$	0	0
$561$	2.52499e6	0.338729
$562$	1.06583e7	1.42347
$563$	−8.74359e6	−1.16257	−0.581285	−	0.813700i	$-0.697450\pi$
$563$	−8.74359e6	−1.16257	−0.581285	+	0.813700i	$0.697450\pi$
$564$	−6.41981e6	−0.849815
$565$	0	0
$566$	785832.	0.103107
$567$	143704.	0.0187721
$568$	−1.79254e7	−2.33129
$569$	4.89251e6	0.633507	0.316753	−	0.948508i	$-0.397407\pi$
$569$	4.89251e6	0.633507	0.316753	+	0.948508i	$0.397407\pi$
$570$	0	0
$571$	1.65650e6	0.212619	0.106310	−	0.994333i	$-0.466097\pi$
$571$	1.65650e6	0.212619	0.106310	+	0.994333i	$0.466097\pi$
$572$	4.28194e6	0.547206
$573$	−7.22835e6	−0.919714
$574$	−1.99039e6	−0.252150
$575$	0	0
$576$	2.39789e6	0.301143
$577$	1.33417e7	1.66829	0.834144	−	0.551547i	$-0.185962\pi$
$577$	1.33417e7	1.66829	0.834144	+	0.551547i	$0.185962\pi$
$578$	−4.12653e7	−5.13766
$579$	−5.20837e6	−0.645663
$580$	0	0
$581$	170804.	0.0209922
$582$	−1.37795e6	−0.168627
$583$	2.81484e6	0.342990
$584$	−3.93975e6	−0.478010
$585$	0	0
$586$	−6.89874e6	−0.829900
$587$	3.82733e6	0.458460	0.229230	−	0.973372i	$-0.426379\pi$
$587$	3.82733e6	0.458460	0.229230	+	0.973372i	$0.426379\pi$
$588$	−1.12850e7	−1.34604
$589$	−755596.	−0.0897432
$590$	0	0
$591$	2.50699e6	0.295246
$592$	3.09902e7	3.63430
$593$	3.31931e6	0.387625	0.193812	−	0.981039i	$-0.437915\pi$
$593$	3.31931e6	0.387625	0.193812	+	0.981039i	$0.437915\pi$
$594$	−920072.	−0.106993
$595$	0	0
$596$	−4.74668e6	−0.547362
$597$	7.59268e6	0.871885
$598$	−2.29837e7	−2.62825
$599$	−7.82165e6	−0.890699	−0.445350	−	0.895357i	$-0.646921\pi$
$599$	−7.82165e6	−0.890699	−0.445350	+	0.895357i	$0.646921\pi$
$600$	0	0
$601$	1.52108e7	1.71778	0.858889	−	0.512162i	$-0.171155\pi$
$601$	1.52108e7	1.71778	0.858889	+	0.512162i	$0.171155\pi$
$602$	4.68451e6	0.526833
$603$	4.69030e6	0.525300
$604$	−1.97199e7	−2.19944
$605$	0	0
$606$	1.80035e6	0.199148
$607$	−1.34079e7	−1.47703	−0.738516	−	0.674236i	$-0.764473\pi$
$607$	−1.34079e7	−1.47703	−0.738516	+	0.674236i	$0.764473\pi$
$608$	−8.20235e6	−0.899870
$609$	179696.	0.0196334
$610$	0	0
$611$	−4.28000e6	−0.463811
$612$	1.44232e7	1.55662
$613$	1.15459e7	1.24102	0.620509	−	0.784199i	$-0.286926\pi$
$613$	1.15459e7	1.24102	0.620509	+	0.784199i	$0.286926\pi$
$614$	−1.00377e7	−1.07451
$615$	0	0
$616$	−1.23836e6	−0.131490
$617$	−1.07188e7	−1.13353	−0.566763	−	0.823881i	$-0.691805\pi$
$617$	−1.07188e7	−1.13353	−0.566763	+	0.823881i	$0.691805\pi$
$618$	1.20012e7	1.26401
$619$	8.11579e6	0.851342	0.425671	−	0.904878i	$-0.360038\pi$
$619$	8.11579e6	0.851342	0.425671	+	0.904878i	$0.360038\pi$
$620$	0	0
$621$	3.48601e6	0.362743
$622$	−6.73031e6	−0.697524
$623$	−1.81966e6	−0.187833
$624$	1.00208e7	1.03025
$625$	0	0
$626$	3.35264e7	3.41941
$627$	871354.	0.0885168
$628$	−8.91264e6	−0.901793
$629$	2.97374e7	2.99693
$630$	0	0
$631$	1.00476e7	1.00459	0.502295	−	0.864696i	$-0.332489\pi$
$631$	1.00476e7	1.00459	0.502295	+	0.864696i	$0.332489\pi$
$632$	786603.	0.0783362
$633$	437529.	0.0434008
$634$	−1.66568e7	−1.64576
$635$	0	0
$636$	1.60789e7	1.57621
$637$	−7.52356e6	−0.734640
$638$	−1.15051e6	−0.111902
$639$	3.10737e6	0.301051
$640$	0	0
$641$	−1.25041e6	−0.120200	−0.0601002	−	0.998192i	$-0.519142\pi$
$641$	−1.25041e6	−0.120200	−0.0601002	+	0.998192i	$0.519142\pi$
$642$	−4.95067e6	−0.474052
$643$	1.68108e7	1.60347	0.801734	−	0.597681i	$-0.203911\pi$
$643$	1.68108e7	1.60347	0.801734	+	0.597681i	$0.203911\pi$
$644$	8.04353e6	0.764244
$645$	0	0
$646$	−1.93511e7	−1.82442
$647$	1.96602e6	0.184640	0.0923202	−	0.995729i	$-0.470572\pi$
$647$	1.96602e6	0.184640	0.0923202	+	0.995729i	$0.470572\pi$
$648$	−3.06571e6	−0.286809
$649$	2.50793e6	0.233725
$650$	0	0
$651$	−186151.	−0.0172152
$652$	−2.47377e7	−2.27898
$653$	−2.26355e6	−0.207734	−0.103867	−	0.994591i	$-0.533122\pi$
$653$	−2.26355e6	−0.207734	−0.103867	+	0.994591i	$0.533122\pi$
$654$	2.19350e7	2.00537
$655$	0	0
$656$	2.10514e7	1.90995
$657$	682957.	0.0617277
$658$	2.12199e6	0.191064
$659$	−4.02181e6	−0.360752	−0.180376	−	0.983598i	$-0.557731\pi$
$659$	−4.02181e6	−0.360752	−0.180376	+	0.983598i	$0.557731\pi$
$660$	0	0
$661$	−1.60875e7	−1.43214	−0.716071	−	0.698027i	$-0.754061\pi$
$661$	−1.60875e7	−1.43214	−0.716071	+	0.698027i	$0.754061\pi$
$662$	−1.97384e7	−1.75052
$663$	9.61575e6	0.849570
$664$	−3.64383e6	−0.320729
$665$	0	0
$666$	−1.08359e7	−0.946631
$667$	4.35909e6	0.379386
$668$	−3.34345e7	−2.89904
$669$	4.26262e6	0.368224
$670$	0	0
$671$	−801738.	−0.0687426
$672$	−2.02076e6	−0.172620
$673$	5.69741e6	0.484886	0.242443	−	0.970166i	$-0.422051\pi$
$673$	5.69741e6	0.484886	0.242443	+	0.970166i	$0.422051\pi$
$674$	−1.65657e7	−1.40462
$675$	0	0
$676$	−1.22076e7	−1.02746
$677$	−2.57893e6	−0.216256	−0.108128	−	0.994137i	$-0.534486\pi$
$677$	−2.57893e6	−0.216256	−0.108128	+	0.994137i	$0.534486\pi$
$678$	3.43917e6	0.287329
$679$	321501.	0.0267614
$680$	0	0
$681$	6.91014e6	0.570978
$682$	1.19184e6	0.0981197
$683$	−1.28680e7	−1.05550	−0.527750	−	0.849400i	$-0.676964\pi$
$683$	−1.28680e7	−1.05550	−0.527750	+	0.849400i	$0.676964\pi$
$684$	4.97734e6	0.406778
$685$	0	0
$686$	7.56983e6	0.614152
$687$	5.22146e6	0.422085
$688$	−4.95458e7	−3.99058
$689$	1.07196e7	0.860259
$690$	0	0
$691$	−3.61306e6	−0.287859	−0.143930	−	0.989588i	$-0.545974\pi$
$691$	−3.61306e6	−0.287859	−0.143930	+	0.989588i	$0.545974\pi$
$692$	−5.37826e7	−4.26949
$693$	214670.	0.0169800
$694$	1.36897e7	1.07893
$695$	0	0
$696$	−3.83353e6	−0.299968
$697$	2.02004e7	1.57499
$698$	4.06874e7	3.16098
$699$	2.68797e6	0.208080
$700$	0	0
$701$	1.60588e7	1.23429	0.617147	−	0.786848i	$-0.288288\pi$
$701$	1.60588e7	1.23429	0.617147	+	0.786848i	$0.288288\pi$
$702$	−3.50386e6	−0.268351
$703$	1.02622e7	0.783161
$704$	3.58203e6	0.272394
$705$	0	0
$706$	−3.46403e7	−2.61559
$707$	−420054.	−0.0316051
$708$	1.43258e7	1.07408
$709$	−6.79151e6	−0.507400	−0.253700	−	0.967283i	$-0.581648\pi$
$709$	−6.79151e6	−0.507400	−0.253700	+	0.967283i	$0.581648\pi$
$710$	0	0
$711$	−136358.	−0.0101159
$712$	3.88197e7	2.86980
$713$	−4.51568e6	−0.332659
$714$	−4.76741e6	−0.349975
$715$	0	0
$716$	1.75245e7	1.27751
$717$	2.91654e6	0.211870
$718$	−2.47997e7	−1.79529
$719$	9.17382e6	0.661802	0.330901	−	0.943666i	$-0.392647\pi$
$719$	9.17382e6	0.661802	0.330901	+	0.943666i	$0.392647\pi$
$720$	0	0
$721$	−2.80009e6	−0.200601
$722$	1.91492e7	1.36713
$723$	−6.11925e6	−0.435364
$724$	3.92893e7	2.78566
$725$	0	0
$726$	−1.37443e6	−0.0967789
$727$	−1.19986e7	−0.841966	−0.420983	−	0.907069i	$-0.638315\pi$
$727$	−1.19986e7	−0.841966	−0.420983	+	0.907069i	$0.638315\pi$
$728$	−4.71596e6	−0.329793
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.75429e7	−3.29073
$732$	−4.57968e6	−0.315906
$733$	−3.87027e6	−0.266061	−0.133031	−	0.991112i	$-0.542471\pi$
$733$	−3.87027e6	−0.266061	−0.133031	+	0.991112i	$0.542471\pi$
$734$	3.67657e7	2.51885
$735$	0	0
$736$	−4.90199e7	−3.33563
$737$	7.00650e6	0.475152
$738$	−7.36077e6	−0.497487
$739$	−1.26699e7	−0.853416	−0.426708	−	0.904390i	$-0.640327\pi$
$739$	−1.26699e7	−0.853416	−0.426708	+	0.904390i	$0.640327\pi$
$740$	0	0
$741$	3.31832e6	0.222010
$742$	−5.31468e6	−0.354378
$743$	1.07430e7	0.713929	0.356965	−	0.934118i	$-0.383812\pi$
$743$	1.07430e7	0.713929	0.356965	+	0.934118i	$0.383812\pi$
$744$	3.97124e6	0.263023
$745$	0	0
$746$	3.99030e7	2.62517
$747$	631659.	0.0414173
$748$	2.15458e7	1.40802
$749$	1.15508e6	0.0752329
$750$	0	0
$751$	−1.60278e7	−1.03699	−0.518493	−	0.855082i	$-0.673507\pi$
$751$	−1.60278e7	−1.03699	−0.518493	+	0.855082i	$0.673507\pi$
$752$	−2.24433e7	−1.44724
$753$	1.60589e7	1.03211
$754$	−4.38141e6	−0.280663
$755$	0	0
$756$	1.22623e6	0.0780313
$757$	2.27246e7	1.44131	0.720654	−	0.693294i	$-0.243841\pi$
$757$	2.27246e7	1.44131	0.720654	+	0.693294i	$0.243841\pi$
$758$	−3.86451e7	−2.44299
$759$	5.20749e6	0.328114
$760$	0	0
$761$	1.62869e7	1.01948	0.509738	−	0.860330i	$-0.329742\pi$
$761$	1.62869e7	1.01948	0.509738	+	0.860330i	$0.329742\pi$
$762$	−2.41155e7	−1.50456
$763$	−5.11784e6	−0.318255
$764$	−6.16797e7	−3.82304
$765$	0	0
$766$	8.23568e6	0.507139
$767$	9.55081e6	0.586208
$768$	−1.03333e7	−0.632171
$769$	8.82491e6	0.538139	0.269069	−	0.963121i	$-0.413284\pi$
$769$	8.82491e6	0.538139	0.269069	+	0.963121i	$0.413284\pi$
$770$	0	0
$771$	−8.01966e6	−0.485870
$772$	−4.44432e7	−2.68387
$773$	−2.36592e7	−1.42413	−0.712067	−	0.702111i	$-0.752241\pi$
$773$	−2.36592e7	−1.42413	−0.712067	+	0.702111i	$0.752241\pi$
$774$	1.73240e7	1.03943
$775$	0	0
$776$	−6.85872e6	−0.408873
$777$	2.52822e6	0.150232
$778$	4.93411e7	2.92254
$779$	6.97101e6	0.411578
$780$	0	0
$781$	4.64187e6	0.272311
$782$	−1.15649e8	−6.76276
$783$	664543.	0.0387363
$784$	−3.94517e7	−2.29232
$785$	0	0
$786$	1.02801e7	0.593529
$787$	−1.96332e7	−1.12994	−0.564969	−	0.825112i	$-0.691112\pi$
$787$	−1.96332e7	−1.12994	−0.564969	+	0.825112i	$0.691112\pi$
$788$	2.13922e7	1.22727
$789$	−1.15070e7	−0.658066
$790$	0	0
$791$	−802420.	−0.0455995
$792$	−4.57963e6	−0.259429
$793$	−3.05321e6	−0.172414
$794$	2.53339e7	1.42610
$795$	0	0
$796$	6.47885e7	3.62423
$797$	2.52639e7	1.40882	0.704408	−	0.709795i	$-0.251212\pi$
$797$	2.52639e7	1.40882	0.704408	+	0.709795i	$0.251212\pi$
$798$	−1.64520e6	−0.0914557
$799$	−2.15360e7	−1.19343
$800$	0	0
$801$	−6.72940e6	−0.370591
$802$	−2.88147e6	−0.158189
$803$	1.02022e6	0.0558348
$804$	4.00225e7	2.18355
$805$	0	0
$806$	4.53880e6	0.246096
$807$	4.08853e6	0.220995
$808$	8.96119e6	0.482878
$809$	−1.60088e6	−0.0859978	−0.0429989	−	0.999075i	$-0.513691\pi$
$809$	−1.60088e6	−0.0859978	−0.0429989	+	0.999075i	$0.513691\pi$
$810$	0	0
$811$	−1.70463e7	−0.910079	−0.455040	−	0.890471i	$-0.650375\pi$
$811$	−1.70463e7	−0.910079	−0.455040	+	0.890471i	$0.650375\pi$
$812$	1.53335e6	0.0816115
$813$	−1.82502e7	−0.968369
$814$	−1.61870e7	−0.856260
$815$	0	0
$816$	5.04227e7	2.65094
$817$	−1.64067e7	−0.859937
$818$	5.87425e6	0.306951
$819$	817513.	0.0425878
$820$	0	0
$821$	1.11036e7	0.574919	0.287459	−	0.957793i	$-0.407189\pi$
$821$	1.11036e7	0.574919	0.287459	+	0.957793i	$0.407189\pi$
$822$	−1.40722e7	−0.726413
$823$	−1.15634e7	−0.595097	−0.297548	−	0.954707i	$-0.596169\pi$
$823$	−1.15634e7	−0.595097	−0.297548	+	0.954707i	$0.596169\pi$
$824$	5.97355e7	3.06488
$825$	0	0
$826$	−4.73522e6	−0.241485
$827$	3.02353e7	1.53727	0.768636	−	0.639686i	$-0.220936\pi$
$827$	3.02353e7	1.53727	0.768636	+	0.639686i	$0.220936\pi$
$828$	2.97462e7	1.50784
$829$	−1.68978e7	−0.853974	−0.426987	−	0.904258i	$-0.640425\pi$
$829$	−1.68978e7	−0.853974	−0.426987	+	0.904258i	$0.640425\pi$
$830$	0	0
$831$	1.14874e7	0.577060
$832$	1.36412e7	0.683196
$833$	−3.78569e7	−1.89031
$834$	1.06314e7	0.529270
$835$	0	0
$836$	7.43529e6	0.367944
$837$	−688415.	−0.0339654
$838$	−7.37454e6	−0.362764
$839$	−6.03413e6	−0.295944	−0.147972	−	0.988992i	$-0.547275\pi$
$839$	−6.03413e6	−0.295944	−0.147972	+	0.988992i	$0.547275\pi$
$840$	0	0
$841$	−1.96802e7	−0.959486
$842$	−4.76434e7	−2.31592
$843$	−9.19650e6	−0.445711
$844$	3.73345e6	0.180407
$845$	0	0
$846$	7.84744e6	0.376966
$847$	320679.	0.0153590
$848$	5.62108e7	2.68429
$849$	−678053.	−0.0322845
$850$	0	0
$851$	6.13300e7	2.90301
$852$	2.65152e7	1.25140
$853$	2.44547e7	1.15077	0.575386	−	0.817882i	$-0.304852\pi$
$853$	2.44547e7	1.15077	0.575386	+	0.817882i	$0.304852\pi$
$854$	1.51376e6	0.0710250
$855$	0	0
$856$	−2.46418e7	−1.14945
$857$	−9.19253e6	−0.427546	−0.213773	−	0.976883i	$-0.568575\pi$
$857$	−9.19253e6	−0.427546	−0.213773	+	0.976883i	$0.568575\pi$
$858$	−5.23415e6	−0.242733
$859$	−1.84087e7	−0.851219	−0.425610	−	0.904907i	$-0.639940\pi$
$859$	−1.84087e7	−0.851219	−0.425610	+	0.904907i	$0.639940\pi$
$860$	0	0
$861$	1.71740e6	0.0789521
$862$	−6.42999e7	−2.94742
$863$	8.55693e6	0.391103	0.195551	−	0.980693i	$-0.437350\pi$
$863$	8.55693e6	0.391103	0.195551	+	0.980693i	$0.437350\pi$
$864$	−7.47308e6	−0.340577
$865$	0	0
$866$	3.07202e7	1.39197
$867$	3.56056e7	1.60868
$868$	−1.58843e6	−0.0715598
$869$	−203695.	−0.00915021
$870$	0	0
$871$	2.66824e7	1.19174
$872$	1.09181e8	4.86246
$873$	1.18896e6	0.0527998
$874$	−3.99095e7	−1.76725
$875$	0	0
$876$	5.82769e6	0.256588
$877$	3.62307e7	1.59066	0.795330	−	0.606177i	$-0.207298\pi$
$877$	3.62307e7	1.59066	0.795330	+	0.606177i	$0.207298\pi$
$878$	6.15533e7	2.69473
$879$	5.95255e6	0.259855
$880$	0	0
$881$	−2.54635e7	−1.10530	−0.552649	−	0.833414i	$-0.686383\pi$
$881$	−2.54635e7	−1.10530	−0.552649	+	0.833414i	$0.686383\pi$
$882$	1.37945e7	0.597085
$883$	−1.25573e6	−0.0541995	−0.0270998	−	0.999633i	$-0.508627\pi$
$883$	−1.25573e6	−0.0541995	−0.0270998	+	0.999633i	$0.508627\pi$
$884$	8.20515e7	3.53147
$885$	0	0
$886$	−4.78090e7	−2.04609
$887$	−2.38712e7	−1.01874	−0.509371	−	0.860547i	$-0.670122\pi$
$887$	−2.38712e7	−1.01874	−0.509371	+	0.860547i	$0.670122\pi$
$888$	−5.39356e7	−2.29532
$889$	5.62659e6	0.238776
$890$	0	0
$891$	793881.	0.0335013
$892$	3.63731e7	1.53062
$893$	−7.43192e6	−0.311869
$894$	5.80224e6	0.242802
$895$	0	0
$896$	421709.	0.0175486
$897$	1.98314e7	0.822947
$898$	−6.90500e7	−2.85741
$899$	−860832.	−0.0355238
$900$	0	0
$901$	5.39385e7	2.21354
$902$	−1.09957e7	−0.449994
$903$	−4.04201e6	−0.164960
$904$	1.71184e7	0.696692
$905$	0	0
$906$	2.41052e7	0.975640
$907$	−1.51874e7	−0.613007	−0.306504	−	0.951870i	$-0.599159\pi$
$907$	−1.51874e7	−0.613007	−0.306504	+	0.951870i	$0.599159\pi$
$908$	5.89644e7	2.37342
$909$	−1.55342e6	−0.0623563
$910$	0	0
$911$	2.87742e6	0.114870	0.0574352	−	0.998349i	$-0.481708\pi$
$911$	2.87742e6	0.114870	0.0574352	+	0.998349i	$0.481708\pi$
$912$	1.74005e7	0.692746
$913$	943590.	0.0374633
$914$	6.64876e7	2.63254
$915$	0	0
$916$	4.45549e7	1.75451
$917$	−2.39854e6	−0.0941941
$918$	−1.76306e7	−0.690495
$919$	3.83454e7	1.49770	0.748849	−	0.662741i	$-0.230607\pi$
$919$	3.83454e7	1.49770	0.748849	+	0.662741i	$0.230607\pi$
$920$	0	0
$921$	8.66096e6	0.336447
$922$	5.60059e7	2.16973
$923$	1.76773e7	0.682987
$924$	1.83178e6	0.0705820
$925$	0	0
$926$	−7.44629e7	−2.85373
$927$	−1.03552e7	−0.395783
$928$	−9.34474e6	−0.356203
$929$	3.31484e7	1.26015	0.630077	−	0.776533i	$-0.283023\pi$
$929$	3.31484e7	1.26015	0.630077	+	0.776533i	$0.283023\pi$
$930$	0	0
$931$	−1.30641e7	−0.493976
$932$	2.29365e7	0.864943
$933$	5.80722e6	0.218406
$934$	5.33647e7	2.00164
$935$	0	0
$936$	−1.74403e7	−0.650677
$937$	2.93312e7	1.09139	0.545695	−	0.837984i	$-0.316266\pi$
$937$	2.93312e7	1.09139	0.545695	+	0.837984i	$0.316266\pi$
$938$	−1.32289e7	−0.490928
$939$	−2.89281e7	−1.07067
$940$	0	0
$941$	−9.43943e6	−0.347513	−0.173757	−	0.984789i	$-0.555591\pi$
$941$	−9.43943e6	−0.347513	−0.173757	+	0.984789i	$0.555591\pi$
$942$	1.08946e7	0.400023
$943$	4.16610e7	1.52563
$944$	5.00821e7	1.82917
$945$	0	0
$946$	2.58791e7	0.940202
$947$	2.37783e7	0.861601	0.430801	−	0.902447i	$-0.358231\pi$
$947$	2.37783e7	0.861601	0.430801	+	0.902447i	$0.358231\pi$
$948$	−1.16355e6	−0.0420496
$949$	3.88524e6	0.140040
$950$	0	0
$951$	1.43722e7	0.515315
$952$	−2.37297e7	−0.848592
$953$	−2.31442e7	−0.825488	−0.412744	−	0.910847i	$-0.635429\pi$
$953$	−2.31442e7	−0.825488	−0.412744	+	0.910847i	$0.635429\pi$
$954$	−1.96545e7	−0.699183
$955$	0	0
$956$	2.48869e7	0.880696
$957$	992712.	0.0350383
$958$	7.45938e6	0.262597
$959$	3.28331e6	0.115283
$960$	0	0
$961$	−2.77374e7	−0.968851
$962$	−6.16440e7	−2.14760
$963$	4.27167e6	0.148433
$964$	−5.22158e7	−1.80971
$965$	0	0
$966$	−9.83223e6	−0.339008
$967$	7.71333e6	0.265262	0.132631	−	0.991165i	$-0.457657\pi$
$967$	7.71333e6	0.265262	0.132631	+	0.991165i	$0.457657\pi$
$968$	−6.84118e6	−0.234662
$969$	1.66971e7	0.571256
$970$	0	0
$971$	−4.66559e7	−1.58803	−0.794014	−	0.607899i	$-0.792012\pi$
$971$	−4.66559e7	−1.58803	−0.794014	+	0.607899i	$0.792012\pi$
$972$	4.53480e6	0.153955
$973$	−2.48051e6	−0.0839960
$974$	−9.81837e7	−3.31621
$975$	0	0
$976$	−1.60103e7	−0.537990
$977$	1.28308e7	0.430050	0.215025	−	0.976609i	$-0.431017\pi$
$977$	1.28308e7	0.430050	0.215025	+	0.976609i	$0.431017\pi$
$978$	3.02389e7	1.01092
$979$	−1.00526e7	−0.335212
$980$	0	0
$981$	−1.89266e7	−0.627913
$982$	2.12601e7	0.703537
$983$	7.06277e6	0.233126	0.116563	−	0.993183i	$-0.462812\pi$
$983$	7.06277e6	0.233126	0.116563	+	0.993183i	$0.462812\pi$
$984$	−3.66380e7	−1.20627
$985$	0	0
$986$	−2.20463e7	−0.722176
$987$	−1.83095e6	−0.0598251
$988$	2.83154e7	0.922847
$989$	−9.80518e7	−3.18761
$990$	0	0
$991$	1.51956e7	0.491512	0.245756	−	0.969332i	$-0.420964\pi$
$991$	1.51956e7	0.491512	0.245756	+	0.969332i	$0.420964\pi$
$992$	9.68043e6	0.312331
$993$	1.70312e7	0.548115
$994$	−8.76428e6	−0.281352
$995$	0	0
$996$	5.38997e6	0.172162
$997$	3.19469e7	1.01787	0.508934	−	0.860806i	$-0.330040\pi$
$997$	3.19469e7	1.01787	0.508934	+	0.860806i	$0.330040\pi$
$998$	2.39144e7	0.760035
$999$	9.34975e6	0.296405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.1		13
5.2	odd	4		165.6.c.b.34.2	✓	26
5.3	odd	4		165.6.c.b.34.25	yes	26
5.4	even	2		825.6.a.v.1.13		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.2	✓	26	5.2	odd	4
165.6.c.b.34.25	yes	26	5.3	odd	4
825.6.a.v.1.13		13	5.4	even	2
825.6.a.y.1.1		13	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)

\(f(q)\)	\(=\)	\(q-10.4306 q^{2} +9.00000 q^{3} +76.7973 q^{4} -93.8753 q^{6} +21.9028 q^{7} -467.262 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-10.4306 q^{2} +9.00000 q^{3} +76.7973 q^{4} -93.8753 q^{6} +21.9028 q^{7} -467.262 q^{8} +81.0000 q^{9} +121.000 q^{11} +691.175 q^{12} +460.797 q^{13} -228.459 q^{14} +2416.31 q^{16} +2318.63 q^{17} -844.878 q^{18} +800.141 q^{19} +197.125 q^{21} -1262.10 q^{22} +4781.90 q^{23} -4205.36 q^{24} -4806.39 q^{26} +729.000 q^{27} +1682.08 q^{28} +911.582 q^{29} -944.328 q^{31} -10251.1 q^{32} +1089.00 q^{33} -24184.7 q^{34} +6220.58 q^{36} +12825.4 q^{37} -8345.95 q^{38} +4147.17 q^{39} +8712.22 q^{41} -2056.13 q^{42} -20504.8 q^{43} +9292.47 q^{44} -49878.1 q^{46} -9288.25 q^{47} +21746.8 q^{48} -16327.3 q^{49} +20867.7 q^{51} +35388.0 q^{52} +23263.1 q^{53} -7603.90 q^{54} -10234.4 q^{56} +7201.27 q^{57} -9508.34 q^{58} +20726.7 q^{59} -6625.93 q^{61} +9849.90 q^{62} +1774.13 q^{63} +29603.6 q^{64} -11358.9 q^{66} +57904.9 q^{67} +178064. q^{68} +43037.1 q^{69} +38362.5 q^{71} -37848.2 q^{72} +8431.57 q^{73} -133777. q^{74} +61448.7 q^{76} +2650.24 q^{77} -43257.5 q^{78} -1683.43 q^{79} +6561.00 q^{81} -90873.6 q^{82} +7798.26 q^{83} +15138.7 q^{84} +213877. q^{86} +8204.23 q^{87} -56538.7 q^{88} -83079.0 q^{89} +10092.8 q^{91} +367237. q^{92} -8498.95 q^{93} +96882.0 q^{94} -92260.2 q^{96} +14678.5 q^{97} +170303. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 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

Introduction

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