LMFDB - Embedded newform 825.6.a.o.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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 $ x^7 - 2x^6 - 152x^5 + 358x^4 + 5771x^3 - 13444x^2 - 51316x + 92576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.2074$ 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)

$f(q)$	$=$	$q-9.20737 q^{2} -9.00000 q^{3} +52.7757 q^{4} +82.8663 q^{6} +97.7871 q^{7} -191.289 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-9.20737 q^{2} -9.00000 q^{3} +52.7757 q^{4} +82.8663 q^{6} +97.7871 q^{7} -191.289 q^{8} +81.0000 q^{9} +121.000 q^{11} -474.981 q^{12} +490.118 q^{13} -900.362 q^{14} +72.4504 q^{16} +881.457 q^{17} -745.797 q^{18} +34.4568 q^{19} -880.084 q^{21} -1114.09 q^{22} +2905.50 q^{23} +1721.60 q^{24} -4512.70 q^{26} -729.000 q^{27} +5160.78 q^{28} -1410.45 q^{29} -2535.92 q^{31} +5454.18 q^{32} -1089.00 q^{33} -8115.90 q^{34} +4274.83 q^{36} -6262.31 q^{37} -317.257 q^{38} -4411.06 q^{39} -18279.4 q^{41} +8103.26 q^{42} -14176.4 q^{43} +6385.86 q^{44} -26752.0 q^{46} +7815.36 q^{47} -652.054 q^{48} -7244.68 q^{49} -7933.11 q^{51} +25866.3 q^{52} +4614.88 q^{53} +6712.17 q^{54} -18705.6 q^{56} -310.111 q^{57} +12986.6 q^{58} -1123.78 q^{59} -672.594 q^{61} +23349.2 q^{62} +7920.76 q^{63} -52537.1 q^{64} +10026.8 q^{66} -29266.6 q^{67} +46519.5 q^{68} -26149.5 q^{69} -9699.67 q^{71} -15494.4 q^{72} -40302.7 q^{73} +57659.5 q^{74} +1818.48 q^{76} +11832.2 q^{77} +40614.3 q^{78} -72842.2 q^{79} +6561.00 q^{81} +168305. q^{82} -100283. q^{83} -46447.0 q^{84} +130527. q^{86} +12694.1 q^{87} -23146.0 q^{88} -30687.7 q^{89} +47927.2 q^{91} +153340. q^{92} +22823.3 q^{93} -71958.9 q^{94} -49087.6 q^{96} +34455.3 q^{97} +66704.5 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9}+O(q^{10}) $ 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9} + 847 q^{11} - 855 q^{12} - 113 q^{13} - 1212 q^{14} + 499 q^{16} + 1030 q^{17} + 729 q^{18} - 3803 q^{19} - 585 q^{21} + 1089 q^{22} + 514 q^{23} - 1863 q^{24} - 12111 q^{26} - 5103 q^{27} - 342 q^{28} - 2698 q^{29} - 17233 q^{31} + 9943 q^{32} - 7623 q^{33} + 4090 q^{34} + 7695 q^{36} + 23182 q^{37} - 11943 q^{38} + 1017 q^{39} - 16158 q^{41} + 10908 q^{42} - 4249 q^{43} + 11495 q^{44} - 28769 q^{46} + 7580 q^{47} - 4491 q^{48} + 37140 q^{49} - 9270 q^{51} - 23887 q^{52} + 20574 q^{53} - 6561 q^{54} - 73276 q^{56} + 34227 q^{57} + 8733 q^{58} - 364 q^{59} - 28127 q^{61} - 71917 q^{62} + 5265 q^{63} + 43379 q^{64} - 9801 q^{66} - 21493 q^{67} + 160660 q^{68} - 4626 q^{69} - 177084 q^{71} + 16767 q^{72} + 78670 q^{73} + 196750 q^{74} - 32701 q^{76} + 7865 q^{77} + 108999 q^{78} - 187432 q^{79} + 45927 q^{81} + 179552 q^{82} - 44592 q^{83} + 3078 q^{84} - 110433 q^{86} + 24282 q^{87} + 25047 q^{88} - 151168 q^{89} - 230153 q^{91} - 44767 q^{92} + 155097 q^{93} + 54040 q^{94} - 89487 q^{96} - 55589 q^{97} + 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9 + 847 * q^11 - 855 * q^12 - 113 * q^13 - 1212 * q^14 + 499 * q^16 + 1030 * q^17 + 729 * q^18 - 3803 * q^19 - 585 * q^21 + 1089 * q^22 + 514 * q^23 - 1863 * q^24 - 12111 * q^26 - 5103 * q^27 - 342 * q^28 - 2698 * q^29 - 17233 * q^31 + 9943 * q^32 - 7623 * q^33 + 4090 * q^34 + 7695 * q^36 + 23182 * q^37 - 11943 * q^38 + 1017 * q^39 - 16158 * q^41 + 10908 * q^42 - 4249 * q^43 + 11495 * q^44 - 28769 * q^46 + 7580 * q^47 - 4491 * q^48 + 37140 * q^49 - 9270 * q^51 - 23887 * q^52 + 20574 * q^53 - 6561 * q^54 - 73276 * q^56 + 34227 * q^57 + 8733 * q^58 - 364 * q^59 - 28127 * q^61 - 71917 * q^62 + 5265 * q^63 + 43379 * q^64 - 9801 * q^66 - 21493 * q^67 + 160660 * q^68 - 4626 * q^69 - 177084 * q^71 + 16767 * q^72 + 78670 * q^73 + 196750 * q^74 - 32701 * q^76 + 7865 * q^77 + 108999 * q^78 - 187432 * q^79 + 45927 * q^81 + 179552 * q^82 - 44592 * q^83 + 3078 * q^84 - 110433 * q^86 + 24282 * q^87 + 25047 * q^88 - 151168 * q^89 - 230153 * q^91 - 44767 * q^92 + 155097 * q^93 + 54040 * q^94 - 89487 * q^96 - 55589 * q^97 + 478341 * q^98 + 68607 * 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 $

$2$	−9.20737	−1.62765	−0.813824	−	0.581111i	$-0.802618\pi$
$2$	−9.20737	−1.62765	−0.813824	+	0.581111i	$0.802618\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	52.7757	1.64924
$5$	0	0
$5$	0	0
$6$	82.8663	0.939723
$7$	97.7871	0.754287	0.377143	−	0.926155i	$-0.376906\pi$
$7$	97.7871	0.754287	0.377143	+	0.926155i	$0.376906\pi$
$8$	−191.289	−1.05673
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−474.981	−0.952189
$13$	490.118	0.804345	0.402172	−	0.915564i	$-0.368255\pi$
$13$	490.118	0.804345	0.402172	+	0.915564i	$0.368255\pi$
$14$	−900.362	−1.22771
$15$	0	0
$16$	72.4504	0.0707524
$17$	881.457	0.739739	0.369870	−	0.929084i	$-0.379402\pi$
$17$	881.457	0.739739	0.369870	+	0.929084i	$0.379402\pi$
$18$	−745.797	−0.542550
$19$	34.4568	0.0218973	0.0109487	−	0.999940i	$-0.496515\pi$
$19$	34.4568	0.0218973	0.0109487	+	0.999940i	$0.496515\pi$
$20$	0	0
$21$	−880.084	−0.435488
$22$	−1114.09	−0.490755
$23$	2905.50	1.14525	0.572627	−	0.819816i	$-0.305924\pi$
$23$	2905.50	1.14525	0.572627	+	0.819816i	$0.305924\pi$
$24$	1721.60	0.610106
$25$	0	0
$26$	−4512.70	−1.30919
$27$	−729.000	−0.192450
$28$	5160.78	1.24400
$29$	−1410.45	−0.311432	−0.155716	−	0.987802i	$-0.549768\pi$
$29$	−1410.45	−0.311432	−0.155716	+	0.987802i	$0.549768\pi$
$30$	0	0
$31$	−2535.92	−0.473949	−0.236975	−	0.971516i	$-0.576156\pi$
$31$	−2535.92	−0.473949	−0.236975	+	0.971516i	$0.576156\pi$
$32$	5454.18	0.941574
$33$	−1089.00	−0.174078
$34$	−8115.90	−1.20404
$35$	0	0
$36$	4274.83	0.549747
$37$	−6262.31	−0.752022	−0.376011	−	0.926615i	$-0.622705\pi$
$37$	−6262.31	−0.752022	−0.376011	+	0.926615i	$0.622705\pi$
$38$	−317.257	−0.0356412
$39$	−4411.06	−0.464389
$40$	0	0
$41$	−18279.4	−1.69825	−0.849126	−	0.528190i	$-0.822871\pi$
$41$	−18279.4	−1.69825	−0.849126	+	0.528190i	$0.822871\pi$
$42$	8103.26	0.708821
$43$	−14176.4	−1.16922	−0.584608	−	0.811316i	$-0.698752\pi$
$43$	−14176.4	−1.16922	−0.584608	+	0.811316i	$0.698752\pi$
$44$	6385.86	0.497265
$45$	0	0
$46$	−26752.0	−1.86407
$47$	7815.36	0.516065	0.258032	−	0.966136i	$-0.416926\pi$
$47$	7815.36	0.516065	0.258032	+	0.966136i	$0.416926\pi$
$48$	−652.054	−0.0408489
$49$	−7244.68	−0.431051
$50$	0	0
$51$	−7933.11	−0.427089
$52$	25866.3	1.32656
$53$	4614.88	0.225668	0.112834	−	0.993614i	$-0.464007\pi$
$53$	4614.88	0.225668	0.112834	+	0.993614i	$0.464007\pi$
$54$	6712.17	0.313241
$55$	0	0
$56$	−18705.6	−0.797081
$57$	−310.111	−0.0126424
$58$	12986.6	0.506902
$59$	−1123.78	−0.0420293	−0.0210147	−	0.999779i	$-0.506690\pi$
$59$	−1123.78	−0.0420293	−0.0210147	+	0.999779i	$0.506690\pi$
$60$	0	0
$61$	−672.594	−0.0231435	−0.0115717	−	0.999933i	$-0.503683\pi$
$61$	−672.594	−0.0231435	−0.0115717	+	0.999933i	$0.503683\pi$
$62$	23349.2	0.771423
$63$	7920.76	0.251429
$64$	−52537.1	−1.60330
$65$	0	0
$66$	10026.8	0.283337
$67$	−29266.6	−0.796499	−0.398249	−	0.917277i	$-0.630382\pi$
$67$	−29266.6	−0.796499	−0.398249	+	0.917277i	$0.630382\pi$
$68$	46519.5	1.22001
$69$	−26149.5	−0.661213
$70$	0	0
$71$	−9699.67	−0.228355	−0.114178	−	0.993460i	$-0.536423\pi$
$71$	−9699.67	−0.228355	−0.114178	+	0.993460i	$0.536423\pi$
$72$	−15494.4	−0.352245
$73$	−40302.7	−0.885170	−0.442585	−	0.896726i	$-0.645939\pi$
$73$	−40302.7	−0.885170	−0.442585	+	0.896726i	$0.645939\pi$
$74$	57659.5	1.22403
$75$	0	0
$76$	1818.48	0.0361140
$77$	11832.2	0.227426
$78$	40614.3	0.755861
$79$	−72842.2	−1.31315	−0.656576	−	0.754260i	$-0.727996\pi$
$79$	−72842.2	−1.31315	−0.656576	+	0.754260i	$0.727996\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	168305.	2.76416
$83$	−100283.	−1.59784	−0.798918	−	0.601440i	$-0.794594\pi$
$83$	−100283.	−1.59784	−0.798918	+	0.601440i	$0.794594\pi$
$84$	−46447.0	−0.718224
$85$	0	0
$86$	130527.	1.90307
$87$	12694.1	0.179805
$88$	−23146.0	−0.318617
$89$	−30687.7	−0.410667	−0.205333	−	0.978692i	$-0.565828\pi$
$89$	−30687.7	−0.410667	−0.205333	+	0.978692i	$0.565828\pi$
$90$	0	0
$91$	47927.2	0.606707
$92$	153340.	1.88880
$93$	22823.3	0.273635
$94$	−71958.9	−0.839972
$95$	0	0
$96$	−49087.6	−0.543618
$97$	34455.3	0.371815	0.185907	−	0.982567i	$-0.440478\pi$
$97$	34455.3	0.371815	0.185907	+	0.982567i	$0.440478\pi$
$98$	66704.5	0.701600
$99$	9801.00	0.100504
$100$	0	0
$101$	77516.8	0.756123	0.378062	−	0.925780i	$-0.376591\pi$
$101$	77516.8	0.756123	0.378062	+	0.925780i	$0.376591\pi$
$102$	73043.1	0.695150
$103$	96973.6	0.900659	0.450330	−	0.892862i	$-0.351306\pi$
$103$	96973.6	0.900659	0.450330	+	0.892862i	$0.351306\pi$
$104$	−93754.3	−0.849979
$105$	0	0
$106$	−42490.9	−0.367309
$107$	175905.	1.48531	0.742656	−	0.669673i	$-0.233566\pi$
$107$	175905.	1.48531	0.742656	+	0.669673i	$0.233566\pi$
$108$	−38473.5	−0.317396
$109$	−124853.	−1.00654	−0.503271	−	0.864128i	$-0.667870\pi$
$109$	−124853.	−1.00654	−0.503271	+	0.864128i	$0.667870\pi$
$110$	0	0
$111$	56360.8	0.434180
$112$	7084.72	0.0533676
$113$	145912.	1.07496	0.537481	−	0.843276i	$-0.319376\pi$
$113$	145912.	1.07496	0.537481	+	0.843276i	$0.319376\pi$
$114$	2855.31	0.0205774
$115$	0	0
$116$	−74437.5	−0.513626
$117$	39699.5	0.268115
$118$	10347.1	0.0684089
$119$	86195.1	0.557976
$120$	0	0
$121$	14641.0	0.0909091
$122$	6192.83	0.0376695
$123$	164515.	0.980487
$124$	−133835.	−0.781656
$125$	0	0
$126$	−72929.3	−0.409238
$127$	38194.3	0.210131	0.105065	−	0.994465i	$-0.466495\pi$
$127$	38194.3	0.210131	0.105065	+	0.994465i	$0.466495\pi$
$128$	309195.	1.66804
$129$	127588.	0.675047
$130$	0	0
$131$	171732.	0.874327	0.437163	−	0.899382i	$-0.355983\pi$
$131$	171732.	0.874327	0.437163	+	0.899382i	$0.355983\pi$
$132$	−57472.7	−0.287096
$133$	3369.43	0.0165169
$134$	269468.	1.29642
$135$	0	0
$136$	−168613.	−0.781708
$137$	−246624.	−1.12262	−0.561312	−	0.827604i	$-0.689703\pi$
$137$	−246624.	−1.12262	−0.561312	+	0.827604i	$0.689703\pi$
$138$	240768.	1.07622
$139$	−13849.3	−0.0607982	−0.0303991	−	0.999538i	$-0.509678\pi$
$139$	−13849.3	−0.0607982	−0.0303991	+	0.999538i	$0.509678\pi$
$140$	0	0
$141$	−70338.2	−0.297950
$142$	89308.5	0.371682
$143$	59304.3	0.242519
$144$	5868.48	0.0235841
$145$	0	0
$146$	371082.	1.44075
$147$	65202.1	0.248868
$148$	−330498.	−1.24026
$149$	−195654.	−0.721975	−0.360987	−	0.932571i	$-0.617560\pi$
$149$	−195654.	−0.721975	−0.360987	+	0.932571i	$0.617560\pi$
$150$	0	0
$151$	205268.	0.732622	0.366311	−	0.930493i	$-0.380621\pi$
$151$	205268.	0.732622	0.366311	+	0.930493i	$0.380621\pi$
$152$	−6591.23	−0.0231397
$153$	71398.0	0.246580
$154$	−108944.	−0.370170
$155$	0	0
$156$	−232797.	−0.765888
$157$	89862.7	0.290958	0.145479	−	0.989361i	$-0.453528\pi$
$157$	89862.7	0.290958	0.145479	+	0.989361i	$0.453528\pi$
$158$	670685.	2.13735
$159$	−41533.9	−0.130290
$160$	0	0
$161$	284121.	0.863850
$162$	−60409.6	−0.180850
$163$	−27924.3	−0.0823214	−0.0411607	−	0.999153i	$-0.513106\pi$
$163$	−27924.3	−0.0823214	−0.0411607	+	0.999153i	$0.513106\pi$
$164$	−964708.	−2.80083
$165$	0	0
$166$	923343.	2.60071
$167$	−254410.	−0.705901	−0.352950	−	0.935642i	$-0.614822\pi$
$167$	−254410.	−0.705901	−0.352950	+	0.935642i	$0.614822\pi$
$168$	168351.	0.460195
$169$	−131077.	−0.353030
$170$	0	0
$171$	2791.00	0.00729911
$172$	−748169.	−1.92832
$173$	150277.	0.381748	0.190874	−	0.981615i	$-0.438868\pi$
$173$	150277.	0.381748	0.190874	+	0.981615i	$0.438868\pi$
$174$	−116879.	−0.292660
$175$	0	0
$176$	8766.50	0.0213326
$177$	10114.0	0.0242656
$178$	282553.	0.668421
$179$	−50602.0	−0.118042	−0.0590208	−	0.998257i	$-0.518798\pi$
$179$	−50602.0	−0.118042	−0.0590208	+	0.998257i	$0.518798\pi$
$180$	0	0
$181$	569456.	1.29200	0.646002	−	0.763336i	$-0.276440\pi$
$181$	569456.	1.29200	0.646002	+	0.763336i	$0.276440\pi$
$182$	−441284.	−0.987505
$183$	6053.35	0.0133619
$184$	−555792.	−1.21023
$185$	0	0
$186$	−210143.	−0.445381
$187$	106656.	0.223040
$188$	412461.	0.851115
$189$	−71286.8	−0.145163
$190$	0	0
$191$	379883.	0.753471	0.376735	−	0.926321i	$-0.377047\pi$
$191$	379883.	0.753471	0.376735	+	0.926321i	$0.377047\pi$
$192$	472834.	0.925668
$193$	−637445.	−1.23183	−0.615913	−	0.787814i	$-0.711213\pi$
$193$	−637445.	−1.23183	−0.615913	+	0.787814i	$0.711213\pi$
$194$	−317243.	−0.605184
$195$	0	0
$196$	−382343.	−0.710907
$197$	358863.	0.658814	0.329407	−	0.944188i	$-0.393151\pi$
$197$	358863.	0.658814	0.329407	+	0.944188i	$0.393151\pi$
$198$	−90241.4	−0.163585
$199$	−287892.	−0.515344	−0.257672	−	0.966232i	$-0.582955\pi$
$199$	−287892.	−0.515344	−0.257672	+	0.966232i	$0.582955\pi$
$200$	0	0
$201$	263399.	0.459859
$202$	−713726.	−1.23070
$203$	−137924.	−0.234909
$204$	−418675.	−0.704372
$205$	0	0
$206$	−892872.	−1.46596
$207$	235346.	0.381751
$208$	35509.2	0.0569093
$209$	4169.28	0.00660230
$210$	0	0
$211$	44672.7	0.0690774	0.0345387	−	0.999403i	$-0.489004\pi$
$211$	44672.7	0.0690774	0.0345387	+	0.999403i	$0.489004\pi$
$212$	243553.	0.372181
$213$	87297.0	0.131841
$214$	−1.61962e6	−2.41757
$215$	0	0
$216$	139450.	0.203369
$217$	−247981.	−0.357494
$218$	1.14957e6	1.63830
$219$	362724.	0.511053
$220$	0	0
$221$	432018.	0.595005
$222$	−518935.	−0.706693
$223$	−909913.	−1.22529	−0.612643	−	0.790359i	$-0.709894\pi$
$223$	−909913.	−1.22529	−0.612643	+	0.790359i	$0.709894\pi$
$224$	533349.	0.710217
$225$	0	0
$226$	−1.34346e6	−1.74966
$227$	−1.15501e6	−1.48772	−0.743859	−	0.668336i	$-0.767007\pi$
$227$	−1.15501e6	−1.48772	−0.743859	+	0.668336i	$0.767007\pi$
$228$	−16366.3	−0.0208504
$229$	−1.14608e6	−1.44420	−0.722099	−	0.691789i	$-0.756823\pi$
$229$	−1.14608e6	−1.44420	−0.722099	+	0.691789i	$0.756823\pi$
$230$	0	0
$231$	−106490.	−0.131304
$232$	269804.	0.329101
$233$	−237894.	−0.287074	−0.143537	−	0.989645i	$-0.545848\pi$
$233$	−237894.	−0.287074	−0.143537	+	0.989645i	$0.545848\pi$
$234$	−365528.	−0.436397
$235$	0	0
$236$	−59308.4	−0.0693164
$237$	655579.	0.758149
$238$	−793630.	−0.908188
$239$	38858.4	0.0440038	0.0220019	−	0.999758i	$-0.492996\pi$
$239$	38858.4	0.0440038	0.0220019	+	0.999758i	$0.492996\pi$
$240$	0	0
$241$	73826.4	0.0818784	0.0409392	−	0.999162i	$-0.486965\pi$
$241$	73826.4	0.0818784	0.0409392	+	0.999162i	$0.486965\pi$
$242$	−134805.	−0.147968
$243$	−59049.0	−0.0641500
$244$	−35496.6	−0.0381692
$245$	0	0
$246$	−1.51475e6	−1.59589
$247$	16887.9	0.0176130
$248$	485095.	0.500839
$249$	902547.	0.922511
$250$	0	0
$251$	−1.67060e6	−1.67374	−0.836870	−	0.547402i	$-0.815617\pi$
$251$	−1.67060e6	−1.67374	−0.836870	+	0.547402i	$0.815617\pi$
$252$	418023.	0.414667
$253$	351566.	0.345307
$254$	−351669.	−0.342019
$255$	0	0
$256$	−1.16568e6	−1.11168
$257$	−597714.	−0.564496	−0.282248	−	0.959342i	$-0.591080\pi$
$257$	−597714.	−0.564496	−0.282248	+	0.959342i	$0.591080\pi$
$258$	−1.17475e6	−1.09874
$259$	−612374.	−0.567240
$260$	0	0
$261$	−114247.	−0.103811
$262$	−1.58120e6	−1.42310
$263$	1.37719e6	1.22773	0.613867	−	0.789410i	$-0.289613\pi$
$263$	1.37719e6	1.22773	0.613867	+	0.789410i	$0.289613\pi$
$264$	208314.	0.183954
$265$	0	0
$266$	−31023.6	−0.0268837
$267$	276190.	0.237099
$268$	−1.54456e6	−1.31362
$269$	−161719.	−0.136264	−0.0681320	−	0.997676i	$-0.521704\pi$
$269$	−161719.	−0.136264	−0.0681320	+	0.997676i	$0.521704\pi$
$270$	0	0
$271$	241445.	0.199708	0.0998538	−	0.995002i	$-0.468162\pi$
$271$	241445.	0.199708	0.0998538	+	0.995002i	$0.468162\pi$
$272$	63861.9	0.0523383
$273$	−431345.	−0.350282
$274$	2.27076e6	1.82724
$275$	0	0
$276$	−1.38006e6	−1.09050
$277$	−513505.	−0.402110	−0.201055	−	0.979580i	$-0.564437\pi$
$277$	−513505.	−0.402110	−0.201055	+	0.979580i	$0.564437\pi$
$278$	127516.	0.0989581
$279$	−205410.	−0.157983
$280$	0	0
$281$	−39961.2	−0.0301907	−0.0150953	−	0.999886i	$-0.504805\pi$
$281$	−39961.2	−0.0301907	−0.0150953	+	0.999886i	$0.504805\pi$
$282$	647630.	0.484958
$283$	−1.66384e6	−1.23494	−0.617470	−	0.786594i	$-0.711842\pi$
$283$	−1.66384e6	−1.23494	−0.617470	+	0.786594i	$0.711842\pi$
$284$	−511907.	−0.376613
$285$	0	0
$286$	−546036.	−0.394736
$287$	−1.78749e6	−1.28097
$288$	441789.	0.313858
$289$	−642891.	−0.452786
$290$	0	0
$291$	−310098.	−0.214667
$292$	−2.12700e6	−1.45986
$293$	1.08636e6	0.739276	0.369638	−	0.929176i	$-0.379482\pi$
$293$	1.08636e6	0.739276	0.369638	+	0.929176i	$0.379482\pi$
$294$	−600340.	−0.405069
$295$	0	0
$296$	1.19791e6	0.794688
$297$	−88209.0	−0.0580259
$298$	1.80145e6	1.17512
$299$	1.42404e6	0.921179
$300$	0	0
$301$	−1.38627e6	−0.881924
$302$	−1.88998e6	−1.19245
$303$	−697652.	−0.436548
$304$	2496.41	0.00154929
$305$	0	0
$306$	−657388.	−0.401345
$307$	2.25020e6	1.36262	0.681310	−	0.731995i	$-0.261411\pi$
$307$	2.25020e6	1.36262	0.681310	+	0.731995i	$0.261411\pi$
$308$	624454.	0.375080
$309$	−872762.	−0.519996
$310$	0	0
$311$	1.57755e6	0.924871	0.462435	−	0.886653i	$-0.346976\pi$
$311$	1.57755e6	0.924871	0.462435	+	0.886653i	$0.346976\pi$
$312$	843789.	0.490735
$313$	−2.21406e6	−1.27741	−0.638703	−	0.769453i	$-0.720529\pi$
$313$	−2.21406e6	−1.27741	−0.638703	+	0.769453i	$0.720529\pi$
$314$	−827400.	−0.473578
$315$	0	0
$316$	−3.84429e6	−2.16570
$317$	2.33756e6	1.30652	0.653258	−	0.757136i	$-0.273402\pi$
$317$	2.33756e6	1.30652	0.653258	+	0.757136i	$0.273402\pi$
$318$	382418.	0.212066
$319$	−170665.	−0.0939003
$320$	0	0
$321$	−1.58314e6	−0.857545
$322$	−2.61601e6	−1.40604
$323$	30372.2	0.0161983
$324$	346261.	0.183249
$325$	0	0
$326$	257109.	0.133990
$327$	1.12368e6	0.581128
$328$	3.49665e6	1.79460
$329$	764241.	0.389261
$330$	0	0
$331$	−1.41311e6	−0.708934	−0.354467	−	0.935069i	$-0.615338\pi$
$331$	−1.41311e6	−0.708934	−0.354467	+	0.935069i	$0.615338\pi$
$332$	−5.29250e6	−2.63521
$333$	−507247.	−0.250674
$334$	2.34245e6	1.14896
$335$	0	0
$336$	−63762.5	−0.0308118
$337$	−1.96985e6	−0.944841	−0.472420	−	0.881373i	$-0.656620\pi$
$337$	−1.96985e6	−0.944841	−0.472420	+	0.881373i	$0.656620\pi$
$338$	1.20688e6	0.574608
$339$	−1.31320e6	−0.620630
$340$	0	0
$341$	−306847.	−0.142901
$342$	−25697.8	−0.0118804
$343$	−2.35194e6	−1.07942
$344$	2.71179e6	1.23555
$345$	0	0
$346$	−1.38366e6	−0.621352
$347$	−2.86899e6	−1.27910	−0.639551	−	0.768749i	$-0.720880\pi$
$347$	−2.86899e6	−1.27910	−0.639551	+	0.768749i	$0.720880\pi$
$348$	669938.	0.296542
$349$	730428.	0.321006	0.160503	−	0.987035i	$-0.448688\pi$
$349$	730428.	0.321006	0.160503	+	0.987035i	$0.448688\pi$
$350$	0	0
$351$	−357296.	−0.154796
$352$	659956.	0.283895
$353$	−641407.	−0.273966	−0.136983	−	0.990573i	$-0.543741\pi$
$353$	−641407.	−0.273966	−0.136983	+	0.990573i	$0.543741\pi$
$354$	−93123.7	−0.0394959
$355$	0	0
$356$	−1.61957e6	−0.677288
$357$	−775756.	−0.322147
$358$	465911.	0.192130
$359$	−3.57087e6	−1.46230	−0.731152	−	0.682214i	$-0.761017\pi$
$359$	−3.57087e6	−1.46230	−0.731152	+	0.682214i	$0.761017\pi$
$360$	0	0
$361$	−2.47491e6	−0.999521
$362$	−5.24319e6	−2.10293
$363$	−131769.	−0.0524864
$364$	2.52939e6	1.00060
$365$	0	0
$366$	−55735.4	−0.0217485
$367$	1.70607e6	0.661198	0.330599	−	0.943771i	$-0.392749\pi$
$367$	1.70607e6	0.661198	0.330599	+	0.943771i	$0.392749\pi$
$368$	210505.	0.0810294
$369$	−1.48063e6	−0.566084
$370$	0	0
$371$	451276.	0.170219
$372$	1.20452e6	0.451289
$373$	4.87852e6	1.81558	0.907791	−	0.419424i	$-0.137768\pi$
$373$	4.87852e6	1.81558	0.907791	+	0.419424i	$0.137768\pi$
$374$	−982024.	−0.363030
$375$	0	0
$376$	−1.49499e6	−0.545343
$377$	−691288.	−0.250499
$378$	656364.	0.236274
$379$	−2.02923e6	−0.725659	−0.362829	−	0.931856i	$-0.618189\pi$
$379$	−2.02923e6	−0.725659	−0.362829	+	0.931856i	$0.618189\pi$
$380$	0	0
$381$	−343749.	−0.121319
$382$	−3.49772e6	−1.22639
$383$	−3.39494e6	−1.18259	−0.591296	−	0.806455i	$-0.701383\pi$
$383$	−3.39494e6	−1.18259	−0.591296	+	0.806455i	$0.701383\pi$
$384$	−2.78275e6	−0.963045
$385$	0	0
$386$	5.86919e6	2.00498
$387$	−1.14829e6	−0.389739
$388$	1.81840e6	0.613211
$389$	2.90688e6	0.973986	0.486993	−	0.873406i	$-0.338094\pi$
$389$	2.90688e6	0.973986	0.486993	+	0.873406i	$0.338094\pi$
$390$	0	0
$391$	2.56108e6	0.847190
$392$	1.38583e6	0.455507
$393$	−1.54559e6	−0.504793
$394$	−3.30418e6	−1.07232
$395$	0	0
$396$	517254.	0.165755
$397$	−2.71276e6	−0.863845	−0.431923	−	0.901911i	$-0.642165\pi$
$397$	−2.71276e6	−0.863845	−0.431923	+	0.901911i	$0.642165\pi$
$398$	2.65073e6	0.838798
$399$	−30324.9	−0.00953602
$400$	0	0
$401$	4.93699e6	1.53321	0.766605	−	0.642119i	$-0.221944\pi$
$401$	4.93699e6	1.53321	0.766605	+	0.642119i	$0.221944\pi$
$402$	−2.42522e6	−0.748488
$403$	−1.24290e6	−0.381219
$404$	4.09100e6	1.24703
$405$	0	0
$406$	1.26992e6	0.382349
$407$	−757740.	−0.226743
$408$	1.51752e6	0.451319
$409$	411523.	0.121643	0.0608213	−	0.998149i	$-0.480628\pi$
$409$	411523.	0.121643	0.0608213	+	0.998149i	$0.480628\pi$
$410$	0	0
$411$	2.21962e6	0.648147
$412$	5.11785e6	1.48540
$413$	−109891.	−0.0317022
$414$	−2.16692e6	−0.621357
$415$	0	0
$416$	2.67319e6	0.757350
$417$	124644.	0.0351018
$418$	−38388.1	−0.0107462
$419$	−1.06540e6	−0.296469	−0.148235	−	0.988952i	$-0.547359\pi$
$419$	−1.06540e6	−0.296469	−0.148235	+	0.988952i	$0.547359\pi$
$420$	0	0
$421$	−5.18377e6	−1.42541	−0.712706	−	0.701462i	$-0.752531\pi$
$421$	−5.18377e6	−1.42541	−0.712706	+	0.701462i	$0.752531\pi$
$422$	−411318.	−0.112434
$423$	633044.	0.172022
$424$	−882778.	−0.238472
$425$	0	0
$426$	−803776.	−0.214591
$427$	−65771.1	−0.0174568
$428$	9.28348e6	2.44964
$429$	−533738.	−0.140018
$430$	0	0
$431$	7.59325e6	1.96895	0.984474	−	0.175530i	$-0.0561639\pi$
$431$	7.59325e6	1.96895	0.984474	+	0.175530i	$0.0561639\pi$
$432$	−52816.4	−0.0136163
$433$	6.48244e6	1.66157	0.830786	−	0.556592i	$-0.187891\pi$
$433$	6.48244e6	1.66157	0.830786	+	0.556592i	$0.187891\pi$
$434$	2.28325e6	0.581874
$435$	0	0
$436$	−6.58919e6	−1.66003
$437$	100114.	0.0250780
$438$	−3.33974e6	−0.831815
$439$	2.75964e6	0.683425	0.341713	−	0.939804i	$-0.388993\pi$
$439$	2.75964e6	0.683425	0.341713	+	0.939804i	$0.388993\pi$
$440$	0	0
$441$	−586819.	−0.143684
$442$	−3.97775e6	−0.968460
$443$	−5.64065e6	−1.36559	−0.682795	−	0.730610i	$-0.739236\pi$
$443$	−5.64065e6	−1.36559	−0.682795	+	0.730610i	$0.739236\pi$
$444$	2.97448e6	0.716067
$445$	0	0
$446$	8.37791e6	1.99434
$447$	1.76088e6	0.416832
$448$	−5.13745e6	−1.20935
$449$	1.73569e6	0.406309	0.203154	−	0.979147i	$-0.434881\pi$
$449$	1.73569e6	0.406309	0.203154	+	0.979147i	$0.434881\pi$
$450$	0	0
$451$	−2.21181e6	−0.512042
$452$	7.70058e6	1.77287
$453$	−1.84742e6	−0.422979
$454$	1.06346e7	2.42148
$455$	0	0
$456$	59321.0	0.0133597
$457$	−2.30513e6	−0.516303	−0.258151	−	0.966105i	$-0.583113\pi$
$457$	−2.30513e6	−0.516303	−0.258151	+	0.966105i	$0.583113\pi$
$458$	1.05524e7	2.35065
$459$	−642582.	−0.142363
$460$	0	0
$461$	5.17550e6	1.13423	0.567113	−	0.823640i	$-0.308060\pi$
$461$	5.17550e6	1.13423	0.567113	+	0.823640i	$0.308060\pi$
$462$	980494.	0.213718
$463$	−4.51839e6	−0.979560	−0.489780	−	0.871846i	$-0.662923\pi$
$463$	−4.51839e6	−0.979560	−0.489780	+	0.871846i	$0.662923\pi$
$464$	−102188.	−0.0220345
$465$	0	0
$466$	2.19038e6	0.467256
$467$	8.84437e6	1.87661	0.938306	−	0.345806i	$-0.112394\pi$
$467$	8.84437e6	1.87661	0.938306	+	0.345806i	$0.112394\pi$
$468$	2.09517e6	0.442186
$469$	−2.86190e6	−0.600788
$470$	0	0
$471$	−808765.	−0.167985
$472$	214968.	0.0444138
$473$	−1.71534e6	−0.352532
$474$	−6.03616e6	−1.23400
$475$	0	0
$476$	4.54901e6	0.920236
$477$	373805.	0.0752228
$478$	−357784.	−0.0716227
$479$	−7.60699e6	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$479$	−7.60699e6	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$480$	0	0
$481$	−3.06927e6	−0.604885
$482$	−679747.	−0.133269
$483$	−2.55709e6	−0.498744
$484$	772689.	0.149931
$485$	0	0
$486$	543686.	0.104414
$487$	4.53589e6	0.866642	0.433321	−	0.901240i	$-0.357342\pi$
$487$	4.53589e6	0.866642	0.433321	+	0.901240i	$0.357342\pi$
$488$	128660.	0.0244565
$489$	251318.	0.0475283
$490$	0	0
$491$	2.60325e6	0.487317	0.243658	−	0.969861i	$-0.421652\pi$
$491$	2.60325e6	0.487317	0.243658	+	0.969861i	$0.421652\pi$
$492$	8.68237e6	1.61706
$493$	−1.24325e6	−0.230379
$494$	−155493.	−0.0286678
$495$	0	0
$496$	−183729.	−0.0335330
$497$	−948503.	−0.172245
$498$	−8.31008e6	−1.50152
$499$	−5.53534e6	−0.995160	−0.497580	−	0.867418i	$-0.665778\pi$
$499$	−5.53534e6	−0.995160	−0.497580	+	0.867418i	$0.665778\pi$
$500$	0	0
$501$	2.28969e6	0.407552
$502$	1.53818e7	2.72426
$503$	1.85710e6	0.327277	0.163638	−	0.986520i	$-0.447677\pi$
$503$	1.85710e6	0.327277	0.163638	+	0.986520i	$0.447677\pi$
$504$	−1.51516e6	−0.265694
$505$	0	0
$506$	−3.23700e6	−0.562039
$507$	1.17970e6	0.203822
$508$	2.01573e6	0.346556
$509$	−3.23529e6	−0.553502	−0.276751	−	0.960942i	$-0.589258\pi$
$509$	−3.23529e6	−0.553502	−0.276751	+	0.960942i	$0.589258\pi$
$510$	0	0
$511$	−3.94108e6	−0.667672
$512$	838646.	0.141385
$513$	−25119.0	−0.00421414
$514$	5.50337e6	0.918801
$515$	0	0
$516$	6.73352e6	1.11331
$517$	945658.	0.155599
$518$	5.63835e6	0.923268
$519$	−1.35249e6	−0.220402
$520$	0	0
$521$	−1.26541e6	−0.204237	−0.102119	−	0.994772i	$-0.532562\pi$
$521$	−1.26541e6	−0.204237	−0.102119	+	0.994772i	$0.532562\pi$
$522$	1.05191e6	0.168967
$523$	2.77343e6	0.443366	0.221683	−	0.975119i	$-0.428845\pi$
$523$	2.77343e6	0.443366	0.221683	+	0.975119i	$0.428845\pi$
$524$	9.06329e6	1.44197
$525$	0	0
$526$	−1.26803e7	−1.99832
$527$	−2.23531e6	−0.350599
$528$	−78898.5	−0.0123164
$529$	2.00561e6	0.311607
$530$	0	0
$531$	−91026.4	−0.0140098
$532$	177824.	0.0272403
$533$	−8.95906e6	−1.36598
$534$	−2.54298e6	−0.385913
$535$	0	0
$536$	5.59839e6	0.841688
$537$	455418.	0.0681513
$538$	1.48901e6	0.221790
$539$	−876606.	−0.129967
$540$	0	0
$541$	1.16086e7	1.70524	0.852619	−	0.522533i	$-0.175013\pi$
$541$	1.16086e7	1.70524	0.852619	+	0.522533i	$0.175013\pi$
$542$	−2.22307e6	−0.325054
$543$	−5.12510e6	−0.745938
$544$	4.80763e6	0.696520
$545$	0	0
$546$	3.97155e6	0.570136
$547$	1.21550e7	1.73695	0.868475	−	0.495733i	$-0.165101\pi$
$547$	1.21550e7	1.73695	0.868475	+	0.495733i	$0.165101\pi$
$548$	−1.30158e7	−1.85148
$549$	−54480.2	−0.00771449
$550$	0	0
$551$	−48599.7	−0.00681953
$552$	5.00213e6	0.698726
$553$	−7.12302e6	−0.990493
$554$	4.72803e6	0.654494
$555$	0	0
$556$	−730906.	−0.100271
$557$	1.17133e7	1.59971	0.799856	−	0.600191i	$-0.204909\pi$
$557$	1.17133e7	1.59971	0.799856	+	0.600191i	$0.204909\pi$
$558$	1.89128e6	0.257141
$559$	−6.94810e6	−0.940452
$560$	0	0
$561$	−959907.	−0.128772
$562$	367938.	0.0491398
$563$	2.89057e6	0.384337	0.192168	−	0.981362i	$-0.438448\pi$
$563$	2.89057e6	0.384337	0.192168	+	0.981362i	$0.438448\pi$
$564$	−3.71215e6	−0.491391
$565$	0	0
$566$	1.53196e7	2.01005
$567$	641581.	0.0838096
$568$	1.85544e6	0.241311
$569$	−8.57114e6	−1.10983	−0.554917	−	0.831906i	$-0.687250\pi$
$569$	−8.57114e6	−1.10983	−0.554917	+	0.831906i	$0.687250\pi$
$570$	0	0
$571$	2.49555e6	0.320315	0.160157	−	0.987092i	$-0.448800\pi$
$571$	2.49555e6	0.320315	0.160157	+	0.987092i	$0.448800\pi$
$572$	3.12982e6	0.399972
$573$	−3.41895e6	−0.435016
$574$	1.64581e7	2.08497
$575$	0	0
$576$	−4.25550e6	−0.534435
$577$	4.74905e6	0.593837	0.296918	−	0.954903i	$-0.404041\pi$
$577$	4.74905e6	0.593837	0.296918	+	0.954903i	$0.404041\pi$
$578$	5.91933e6	0.736976
$579$	5.73701e6	0.711195
$580$	0	0
$581$	−9.80638e6	−1.20523
$582$	2.85518e6	0.349403
$583$	558401.	0.0680416
$584$	7.70948e6	0.935390
$585$	0	0
$586$	−1.00026e7	−1.20328
$587$	−3.99522e6	−0.478570	−0.239285	−	0.970949i	$-0.576913\pi$
$587$	−3.99522e6	−0.478570	−0.239285	+	0.970949i	$0.576913\pi$
$588$	3.44109e6	0.410442
$589$	−87379.9	−0.0103782
$590$	0	0
$591$	−3.22977e6	−0.380367
$592$	−453707.	−0.0532073
$593$	−8.72437e6	−1.01882	−0.509410	−	0.860524i	$-0.670136\pi$
$593$	−8.72437e6	−1.01882	−0.509410	+	0.860524i	$0.670136\pi$
$594$	812173.	0.0944457
$595$	0	0
$596$	−1.03257e7	−1.19071
$597$	2.59103e6	0.297534
$598$	−1.31117e7	−1.49936
$599$	−9.05896e6	−1.03160	−0.515800	−	0.856709i	$-0.672505\pi$
$599$	−9.05896e6	−1.03160	−0.515800	+	0.856709i	$0.672505\pi$
$600$	0	0
$601$	947925.	0.107050	0.0535251	−	0.998567i	$-0.482954\pi$
$601$	947925.	0.107050	0.0535251	+	0.998567i	$0.482954\pi$
$602$	1.27639e7	1.43546
$603$	−2.37059e6	−0.265500
$604$	1.08332e7	1.20827
$605$	0	0
$606$	6.42354e6	0.710547
$607$	−9.91130e6	−1.09184	−0.545920	−	0.837837i	$-0.683820\pi$
$607$	−9.91130e6	−1.09184	−0.545920	+	0.837837i	$0.683820\pi$
$608$	187934.	0.0206180
$609$	1.24132e6	0.135625
$610$	0	0
$611$	3.83045e6	0.415094
$612$	3.76808e6	0.406669
$613$	−1.74657e7	−1.87730	−0.938652	−	0.344865i	$-0.887925\pi$
$613$	−1.74657e7	−1.87730	−0.938652	+	0.344865i	$0.887925\pi$
$614$	−2.07184e7	−2.21787
$615$	0	0
$616$	−2.26338e6	−0.240329
$617$	−8.70104e6	−0.920149	−0.460075	−	0.887880i	$-0.652177\pi$
$617$	−8.70104e6	−0.920149	−0.460075	+	0.887880i	$0.652177\pi$
$618$	8.03585e6	0.846370
$619$	−5.31547e6	−0.557590	−0.278795	−	0.960351i	$-0.589935\pi$
$619$	−5.31547e6	−0.557590	−0.278795	+	0.960351i	$0.589935\pi$
$620$	0	0
$621$	−2.11811e6	−0.220404
$622$	−1.45250e7	−1.50536
$623$	−3.00086e6	−0.309761
$624$	−319583.	−0.0328566
$625$	0	0
$626$	2.03857e7	2.07917
$627$	−37523.5	−0.00381184
$628$	4.74257e6	0.479860
$629$	−5.51996e6	−0.556300
$630$	0	0
$631$	7.86646e6	0.786513	0.393257	−	0.919429i	$-0.371348\pi$
$631$	7.86646e6	0.786513	0.393257	+	0.919429i	$0.371348\pi$
$632$	1.39339e7	1.38765
$633$	−402054.	−0.0398818
$634$	−2.15228e7	−2.12655
$635$	0	0
$636$	−2.19198e6	−0.214879
$637$	−3.55075e6	−0.346714
$638$	1.57137e6	0.152837
$639$	−785673.	−0.0761184
$640$	0	0
$641$	−4.86198e6	−0.467377	−0.233689	−	0.972311i	$-0.575080\pi$
$641$	−4.86198e6	−0.467377	−0.233689	+	0.972311i	$0.575080\pi$
$642$	1.45766e7	1.39578
$643$	−2.34243e6	−0.223429	−0.111714	−	0.993740i	$-0.535634\pi$
$643$	−2.34243e6	−0.223429	−0.111714	+	0.993740i	$0.535634\pi$
$644$	1.49947e7	1.42470
$645$	0	0
$646$	−279648.	−0.0263652
$647$	−512871.	−0.0481668	−0.0240834	−	0.999710i	$-0.507667\pi$
$647$	−512871.	−0.0481668	−0.0240834	+	0.999710i	$0.507667\pi$
$648$	−1.25505e6	−0.117415
$649$	−135978.	−0.0126723
$650$	0	0
$651$	2.23183e6	0.206399
$652$	−1.47372e6	−0.135768
$653$	4.47741e6	0.410907	0.205454	−	0.978667i	$-0.434133\pi$
$653$	4.47741e6	0.410907	0.205454	+	0.978667i	$0.434133\pi$
$654$	−1.03461e7	−0.945872
$655$	0	0
$656$	−1.32435e6	−0.120155
$657$	−3.26452e6	−0.295057
$658$	−7.03665e6	−0.633580
$659$	−8.24188e6	−0.739286	−0.369643	−	0.929174i	$-0.620520\pi$
$659$	−8.24188e6	−0.739286	−0.369643	+	0.929174i	$0.620520\pi$
$660$	0	0
$661$	5.80782e6	0.517023	0.258511	−	0.966008i	$-0.416768\pi$
$661$	5.80782e6	0.517023	0.258511	+	0.966008i	$0.416768\pi$
$662$	1.30110e7	1.15389
$663$	−3.88816e6	−0.343527
$664$	1.91831e7	1.68849
$665$	0	0
$666$	4.67042e6	0.408009
$667$	−4.09807e6	−0.356669
$668$	−1.34267e7	−1.16420
$669$	8.18922e6	0.707420
$670$	0	0
$671$	−81383.9	−0.00697802
$672$	−4.80014e6	−0.410044
$673$	−1.00424e7	−0.854671	−0.427336	−	0.904093i	$-0.640548\pi$
$673$	−1.00424e7	−0.854671	−0.427336	+	0.904093i	$0.640548\pi$
$674$	1.81371e7	1.53787
$675$	0	0
$676$	−6.91770e6	−0.582231
$677$	−6.91467e6	−0.579829	−0.289914	−	0.957053i	$-0.593627\pi$
$677$	−6.91467e6	−0.579829	−0.289914	+	0.957053i	$0.593627\pi$
$678$	1.20912e7	1.01017
$679$	3.36928e6	0.280455
$680$	0	0
$681$	1.03951e7	0.858935
$682$	2.82525e6	0.232593
$683$	1.21705e7	0.998290	0.499145	−	0.866518i	$-0.333647\pi$
$683$	1.21705e7	0.998290	0.499145	+	0.866518i	$0.333647\pi$
$684$	147297.	0.0120380
$685$	0	0
$686$	2.16552e7	1.75692
$687$	1.03147e7	0.833808
$688$	−1.02709e6	−0.0827248
$689$	2.26184e6	0.181515
$690$	0	0
$691$	−1.55796e7	−1.24126	−0.620629	−	0.784104i	$-0.713123\pi$
$691$	−1.55796e7	−1.24126	−0.620629	+	0.784104i	$0.713123\pi$
$692$	7.93097e6	0.629595
$693$	958411.	0.0758087
$694$	2.64158e7	2.08193
$695$	0	0
$696$	−2.42824e6	−0.190006
$697$	−1.61125e7	−1.25626
$698$	−6.72532e6	−0.522485
$699$	2.14105e6	0.165742
$700$	0	0
$701$	−1.83816e7	−1.41282	−0.706411	−	0.707802i	$-0.749687\pi$
$701$	−1.83816e7	−1.41282	−0.706411	+	0.707802i	$0.749687\pi$
$702$	3.28976e6	0.251954
$703$	−215780.	−0.0164673
$704$	−6.35699e6	−0.483415
$705$	0	0
$706$	5.90567e6	0.445921
$707$	7.58015e6	0.570334
$708$	533775.	0.0400198
$709$	−1.54867e7	−1.15702	−0.578512	−	0.815674i	$-0.696366\pi$
$709$	−1.54867e7	−1.15702	−0.578512	+	0.815674i	$0.696366\pi$
$710$	0	0
$711$	−5.90021e6	−0.437717
$712$	5.87024e6	0.433966
$713$	−7.36814e6	−0.542792
$714$	7.14267e6	0.524343
$715$	0	0
$716$	−2.67055e6	−0.194679
$717$	−349725.	−0.0254056
$718$	3.28783e7	2.38012
$719$	−1.29862e7	−0.936829	−0.468415	−	0.883509i	$-0.655175\pi$
$719$	−1.29862e7	−0.936829	−0.468415	+	0.883509i	$0.655175\pi$
$720$	0	0
$721$	9.48277e6	0.679355
$722$	2.27874e7	1.62687
$723$	−664438.	−0.0472725
$724$	3.00534e7	2.13082
$725$	0	0
$726$	1.21325e6	0.0854294
$727$	−5.51579e6	−0.387054	−0.193527	−	0.981095i	$-0.561993\pi$
$727$	−5.51579e6	−0.387054	−0.193527	+	0.981095i	$0.561993\pi$
$728$	−9.16797e6	−0.641128
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.24959e7	−0.864915
$732$	319470.	0.0220370
$733$	563402.	0.0387310	0.0193655	−	0.999812i	$-0.493835\pi$
$733$	563402.	0.0387310	0.0193655	+	0.999812i	$0.493835\pi$
$734$	−1.57084e7	−1.07620
$735$	0	0
$736$	1.58471e7	1.07834
$737$	−3.54126e6	−0.240153
$738$	1.36327e7	0.921386
$739$	2.10879e7	1.42044	0.710220	−	0.703980i	$-0.248595\pi$
$739$	2.10879e7	1.42044	0.710220	+	0.703980i	$0.248595\pi$
$740$	0	0
$741$	−151991.	−0.0101689
$742$	−4.15506e6	−0.277056
$743$	1.32554e7	0.880891	0.440445	−	0.897779i	$-0.354821\pi$
$743$	1.32554e7	0.880891	0.440445	+	0.897779i	$0.354821\pi$
$744$	−4.36586e6	−0.289159
$745$	0	0
$746$	−4.49183e7	−2.95513
$747$	−8.12292e6	−0.532612
$748$	5.62886e6	0.367846
$749$	1.72012e7	1.12035
$750$	0	0
$751$	8.54268e6	0.552706	0.276353	−	0.961056i	$-0.410874\pi$
$751$	8.54268e6	0.552706	0.276353	+	0.961056i	$0.410874\pi$
$752$	566226.	0.0365128
$753$	1.50354e7	0.966334
$754$	6.36494e6	0.407724
$755$	0	0
$756$	−3.76221e6	−0.239408
$757$	−1.20390e7	−0.763570	−0.381785	−	0.924251i	$-0.624691\pi$
$757$	−1.20390e7	−0.763570	−0.381785	+	0.924251i	$0.624691\pi$
$758$	1.86838e7	1.18112
$759$	−3.16409e6	−0.199363
$760$	0	0
$761$	−2.02480e6	−0.126742	−0.0633708	−	0.997990i	$-0.520185\pi$
$761$	−2.02480e6	−0.126742	−0.0633708	+	0.997990i	$0.520185\pi$
$762$	3.16502e6	0.197465
$763$	−1.22090e7	−0.759222
$764$	2.00486e7	1.24265
$765$	0	0
$766$	3.12584e7	1.92484
$767$	−550786.	−0.0338060
$768$	1.04911e7	0.641830
$769$	1.97142e7	1.20216	0.601082	−	0.799187i	$-0.294737\pi$
$769$	1.97142e7	1.20216	0.601082	+	0.799187i	$0.294737\pi$
$770$	0	0
$771$	5.37943e6	0.325912
$772$	−3.36416e7	−2.03158
$773$	5.83885e6	0.351462	0.175731	−	0.984438i	$-0.443771\pi$
$773$	5.83885e6	0.351462	0.175731	+	0.984438i	$0.443771\pi$
$774$	1.05727e7	0.634357
$775$	0	0
$776$	−6.59093e6	−0.392909
$777$	5.51136e6	0.327496
$778$	−2.67647e7	−1.58531
$779$	−629850.	−0.0371872
$780$	0	0
$781$	−1.17366e6	−0.0688517
$782$	−2.35808e7	−1.37893
$783$	1.02822e6	0.0599351
$784$	−524880.	−0.0304979
$785$	0	0
$786$	1.42308e7	0.821625
$787$	−1.03958e7	−0.598301	−0.299150	−	0.954206i	$-0.596703\pi$
$787$	−1.03958e7	−0.598301	−0.299150	+	0.954206i	$0.596703\pi$
$788$	1.89392e7	1.08654
$789$	−1.23947e7	−0.708832
$790$	0	0
$791$	1.42683e7	0.810830
$792$	−1.87483e6	−0.106206
$793$	−329651.	−0.0186153
$794$	2.49774e7	1.40604
$795$	0	0
$796$	−1.51937e7	−0.849925
$797$	1.11544e7	0.622017	0.311009	−	0.950407i	$-0.399333\pi$
$797$	1.11544e7	0.622017	0.311009	+	0.950407i	$0.399333\pi$
$798$	279213.	0.0155213
$799$	6.88890e6	0.381753
$800$	0	0
$801$	−2.48571e6	−0.136889
$802$	−4.54567e7	−2.49553
$803$	−4.87663e6	−0.266889
$804$	1.39011e7	0.758417
$805$	0	0
$806$	1.14439e7	0.620490
$807$	1.45547e6	0.0786721
$808$	−1.48281e7	−0.799022
$809$	2.19701e7	1.18022	0.590108	−	0.807325i	$-0.299085\pi$
$809$	2.19701e7	1.18022	0.590108	+	0.807325i	$0.299085\pi$
$810$	0	0
$811$	1.53658e7	0.820358	0.410179	−	0.912005i	$-0.365466\pi$
$811$	1.53658e7	0.820358	0.410179	+	0.912005i	$0.365466\pi$
$812$	−7.27903e6	−0.387421
$813$	−2.17300e6	−0.115301
$814$	6.97679e6	0.369058
$815$	0	0
$816$	−574757.	−0.0302175
$817$	−488474.	−0.0256027
$818$	−3.78905e6	−0.197992
$819$	3.88210e6	0.202236
$820$	0	0
$821$	4.41770e6	0.228738	0.114369	−	0.993438i	$-0.463515\pi$
$821$	4.41770e6	0.228738	0.114369	+	0.993438i	$0.463515\pi$
$822$	−2.04368e7	−1.05496
$823$	−3.83104e7	−1.97159	−0.985796	−	0.167947i	$-0.946286\pi$
$823$	−3.83104e7	−1.97159	−0.985796	+	0.167947i	$0.946286\pi$
$824$	−1.85500e7	−0.951758
$825$	0	0
$826$	1.01181e6	0.0516000
$827$	−2.51868e7	−1.28059	−0.640293	−	0.768131i	$-0.721187\pi$
$827$	−2.51868e7	−1.28059	−0.640293	+	0.768131i	$0.721187\pi$
$828$	1.24205e7	0.629600
$829$	−3.85882e6	−0.195015	−0.0975077	−	0.995235i	$-0.531087\pi$
$829$	−3.85882e6	−0.195015	−0.0975077	+	0.995235i	$0.531087\pi$
$830$	0	0
$831$	4.62154e6	0.232158
$832$	−2.57494e7	−1.28961
$833$	−6.38587e6	−0.318866
$834$	−1.14764e6	−0.0571335
$835$	0	0
$836$	220036.	0.0108888
$837$	1.84869e6	0.0912116
$838$	9.80957e6	0.482547
$839$	2.92410e7	1.43413	0.717063	−	0.697008i	$-0.245486\pi$
$839$	2.92410e7	1.43413	0.717063	+	0.697008i	$0.245486\pi$
$840$	0	0
$841$	−1.85218e7	−0.903010
$842$	4.77289e7	2.32007
$843$	359651.	0.0174306
$844$	2.35763e6	0.113925
$845$	0	0
$846$	−5.82867e6	−0.279991
$847$	1.43170e6	0.0685715
$848$	334350.	0.0159666
$849$	1.49746e7	0.712993
$850$	0	0
$851$	−1.81952e7	−0.861256
$852$	4.60716e6	0.217437
$853$	−7.44565e6	−0.350373	−0.175186	−	0.984535i	$-0.556053\pi$
$853$	−7.44565e6	−0.350373	−0.175186	+	0.984535i	$0.556053\pi$
$854$	605579.	0.0284136
$855$	0	0
$856$	−3.36487e7	−1.56958
$857$	−2.31717e7	−1.07772	−0.538859	−	0.842396i	$-0.681145\pi$
$857$	−2.31717e7	−1.07772	−0.538859	+	0.842396i	$0.681145\pi$
$858$	4.91433e6	0.227901
$859$	2.29217e7	1.05990	0.529949	−	0.848030i	$-0.322211\pi$
$859$	2.29217e7	1.05990	0.529949	+	0.848030i	$0.322211\pi$
$860$	0	0
$861$	1.60874e7	0.739568
$862$	−6.99138e7	−3.20476
$863$	3.28455e6	0.150124	0.0750619	−	0.997179i	$-0.476085\pi$
$863$	3.28455e6	0.150124	0.0750619	+	0.997179i	$0.476085\pi$
$864$	−3.97610e6	−0.181206
$865$	0	0
$866$	−5.96863e7	−2.70445
$867$	5.78602e6	0.261416
$868$	−1.30873e7	−0.589593
$869$	−8.81390e6	−0.395930
$870$	0	0
$871$	−1.43441e7	−0.640660
$872$	2.38830e7	1.06365
$873$	2.79088e6	0.123938
$874$	−921791.	−0.0408182
$875$	0	0
$876$	1.91430e7	0.842850
$877$	6.15691e6	0.270311	0.135156	−	0.990824i	$-0.456847\pi$
$877$	6.15691e6	0.270311	0.135156	+	0.990824i	$0.456847\pi$
$878$	−2.54090e7	−1.11238
$879$	−9.77728e6	−0.426821
$880$	0	0
$881$	−2.47226e7	−1.07314	−0.536568	−	0.843857i	$-0.680280\pi$
$881$	−2.47226e7	−1.07314	−0.536568	+	0.843857i	$0.680280\pi$
$882$	5.40306e6	0.233867
$883$	1.67969e7	0.724980	0.362490	−	0.931988i	$-0.381927\pi$
$883$	1.67969e7	0.724980	0.362490	+	0.931988i	$0.381927\pi$
$884$	2.28000e7	0.981307
$885$	0	0
$886$	5.19356e7	2.22270
$887$	2.91188e7	1.24270	0.621348	−	0.783535i	$-0.286585\pi$
$887$	2.91188e7	1.24270	0.621348	+	0.783535i	$0.286585\pi$
$888$	−1.07812e7	−0.458813
$889$	3.73491e6	0.158499
$890$	0	0
$891$	793881.	0.0335013
$892$	−4.80213e7	−2.02079
$893$	269292.	0.0113004
$894$	−1.62131e7	−0.678456
$895$	0	0
$896$	3.02352e7	1.25818
$897$	−1.28164e7	−0.531843
$898$	−1.59811e7	−0.661328
$899$	3.57680e6	0.147603
$900$	0	0
$901$	4.06782e6	0.166936
$902$	2.03649e7	0.833425
$903$	1.24764e7	0.509179
$904$	−2.79113e7	−1.13595
$905$	0	0
$906$	1.70098e7	0.688462
$907$	2.39464e7	0.966544	0.483272	−	0.875470i	$-0.339448\pi$
$907$	2.39464e7	0.966544	0.483272	+	0.875470i	$0.339448\pi$
$908$	−6.09564e7	−2.45360
$909$	6.27886e6	0.252041
$910$	0	0
$911$	−4.74117e7	−1.89273	−0.946366	−	0.323096i	$-0.895276\pi$
$911$	−4.74117e7	−1.89273	−0.946366	+	0.323096i	$0.895276\pi$
$912$	−22467.7	−0.000894482 0
$913$	−1.21342e7	−0.481765
$914$	2.12242e7	0.840359
$915$	0	0
$916$	−6.04852e7	−2.38183
$917$	1.67932e7	0.659493
$918$	5.91649e6	0.231717
$919$	−4.59032e7	−1.79289	−0.896446	−	0.443152i	$-0.853860\pi$
$919$	−4.59032e7	−1.79289	−0.896446	+	0.443152i	$0.853860\pi$
$920$	0	0
$921$	−2.02518e7	−0.786709
$922$	−4.76527e7	−1.84612
$923$	−4.75398e6	−0.183676
$924$	−5.62009e6	−0.216553
$925$	0	0
$926$	4.16025e7	1.59438
$927$	7.85486e6	0.300220
$928$	−7.69286e6	−0.293236
$929$	−3.87522e7	−1.47319	−0.736593	−	0.676336i	$-0.763567\pi$
$929$	−3.87522e7	−1.47319	−0.736593	+	0.676336i	$0.763567\pi$
$930$	0	0
$931$	−249629.	−0.00943888
$932$	−1.25550e7	−0.473454
$933$	−1.41979e7	−0.533974
$934$	−8.14334e7	−3.05447
$935$	0	0
$936$	−7.59410e6	−0.283326
$937$	329903.	0.0122755	0.00613773	−	0.999981i	$-0.498046\pi$
$937$	329903.	0.0122755	0.00613773	+	0.999981i	$0.498046\pi$
$938$	2.63505e7	0.977873
$939$	1.99266e7	0.737511
$940$	0	0
$941$	−4.87766e7	−1.79572	−0.897858	−	0.440284i	$-0.854878\pi$
$941$	−4.87766e7	−1.79572	−0.897858	+	0.440284i	$0.854878\pi$
$942$	7.44660e6	0.273420
$943$	−5.31109e7	−1.94493
$944$	−81418.5	−0.00297367
$945$	0	0
$946$	1.57938e7	0.573798
$947$	−2.02123e7	−0.732387	−0.366194	−	0.930539i	$-0.619339\pi$
$947$	−2.02123e7	−0.732387	−0.366194	+	0.930539i	$0.619339\pi$
$948$	3.45986e7	1.25037
$949$	−1.97531e7	−0.711982
$950$	0	0
$951$	−2.10380e7	−0.754317
$952$	−1.64882e7	−0.589632
$953$	−3.92102e7	−1.39852	−0.699258	−	0.714870i	$-0.746486\pi$
$953$	−3.92102e7	−1.39852	−0.699258	+	0.714870i	$0.746486\pi$
$954$	−3.44176e6	−0.122436
$955$	0	0
$956$	2.05078e6	0.0725728
$957$	1.53598e6	0.0542133
$958$	7.00404e7	2.46567
$959$	−2.41167e7	−0.846780
$960$	0	0
$961$	−2.21982e7	−0.775372
$962$	2.82599e7	0.984540
$963$	1.42483e7	0.495104
$964$	3.89624e6	0.135037
$965$	0	0
$966$	2.35441e7	0.811780
$967$	4.99839e7	1.71895	0.859476	−	0.511175i	$-0.170790\pi$
$967$	4.99839e7	1.71895	0.859476	+	0.511175i	$0.170790\pi$
$968$	−2.80067e6	−0.0960668
$969$	−273350.	−0.00935211
$970$	0	0
$971$	−1.81402e7	−0.617441	−0.308720	−	0.951153i	$-0.599901\pi$
$971$	−1.81402e7	−0.617441	−0.308720	+	0.951153i	$0.599901\pi$
$972$	−3.11635e6	−0.105799
$973$	−1.35428e6	−0.0458593
$974$	−4.17636e7	−1.41059
$975$	0	0
$976$	−48729.8	−0.00163746
$977$	3.35402e7	1.12416	0.562082	−	0.827082i	$-0.310001\pi$
$977$	3.35402e7	1.12416	0.562082	+	0.827082i	$0.310001\pi$
$978$	−2.31398e6	−0.0773593
$979$	−3.71322e6	−0.123821
$980$	0	0
$981$	−1.01131e7	−0.335514
$982$	−2.39691e7	−0.793181
$983$	−1.34660e7	−0.444482	−0.222241	−	0.974992i	$-0.571337\pi$
$983$	−1.34660e7	−0.444482	−0.222241	+	0.974992i	$0.571337\pi$
$984$	−3.14699e7	−1.03611
$985$	0	0
$986$	1.14471e7	0.374975
$987$	−6.87817e6	−0.224740
$988$	891271.	0.0290481
$989$	−4.11896e7	−1.33905
$990$	0	0
$991$	−1.89814e7	−0.613965	−0.306982	−	0.951715i	$-0.599319\pi$
$991$	−1.89814e7	−0.613965	−0.306982	+	0.951715i	$0.599319\pi$
$992$	−1.38314e7	−0.446259
$993$	1.27180e7	0.409303
$994$	8.73322e6	0.280355
$995$	0	0
$996$	4.76325e7	1.52144
$997$	2.73471e7	0.871311	0.435656	−	0.900113i	$-0.356517\pi$
$997$	2.73471e7	0.871311	0.435656	+	0.900113i	$0.356517\pi$
$998$	5.09659e7	1.61977
$999$	4.56523e6	0.144727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.o.1.1	yes	7
5.4	even	2		825.6.a.m.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.m.1.7	✓	7	5.4	even	2
825.6.a.o.1.1	yes	7	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)

\(f(q)\)	\(=\)	\(q-9.20737 q^{2} -9.00000 q^{3} +52.7757 q^{4} +82.8663 q^{6} +97.7871 q^{7} -191.289 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-9.20737 q^{2} -9.00000 q^{3} +52.7757 q^{4} +82.8663 q^{6} +97.7871 q^{7} -191.289 q^{8} +81.0000 q^{9} +121.000 q^{11} -474.981 q^{12} +490.118 q^{13} -900.362 q^{14} +72.4504 q^{16} +881.457 q^{17} -745.797 q^{18} +34.4568 q^{19} -880.084 q^{21} -1114.09 q^{22} +2905.50 q^{23} +1721.60 q^{24} -4512.70 q^{26} -729.000 q^{27} +5160.78 q^{28} -1410.45 q^{29} -2535.92 q^{31} +5454.18 q^{32} -1089.00 q^{33} -8115.90 q^{34} +4274.83 q^{36} -6262.31 q^{37} -317.257 q^{38} -4411.06 q^{39} -18279.4 q^{41} +8103.26 q^{42} -14176.4 q^{43} +6385.86 q^{44} -26752.0 q^{46} +7815.36 q^{47} -652.054 q^{48} -7244.68 q^{49} -7933.11 q^{51} +25866.3 q^{52} +4614.88 q^{53} +6712.17 q^{54} -18705.6 q^{56} -310.111 q^{57} +12986.6 q^{58} -1123.78 q^{59} -672.594 q^{61} +23349.2 q^{62} +7920.76 q^{63} -52537.1 q^{64} +10026.8 q^{66} -29266.6 q^{67} +46519.5 q^{68} -26149.5 q^{69} -9699.67 q^{71} -15494.4 q^{72} -40302.7 q^{73} +57659.5 q^{74} +1818.48 q^{76} +11832.2 q^{77} +40614.3 q^{78} -72842.2 q^{79} +6561.00 q^{81} +168305. q^{82} -100283. q^{83} -46447.0 q^{84} +130527. q^{86} +12694.1 q^{87} -23146.0 q^{88} -30687.7 q^{89} +47927.2 q^{91} +153340. q^{92} +22823.3 q^{93} -71958.9 q^{94} -49087.6 q^{96} +34455.3 q^{97} +66704.5 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9}+O(q^{10}) \) 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9} + 847 q^{11} - 855 q^{12} - 113 q^{13} - 1212 q^{14} + 499 q^{16} + 1030 q^{17} + 729 q^{18} - 3803 q^{19} - 585 q^{21} + 1089 q^{22} + 514 q^{23} - 1863 q^{24} - 12111 q^{26} - 5103 q^{27} - 342 q^{28} - 2698 q^{29} - 17233 q^{31} + 9943 q^{32} - 7623 q^{33} + 4090 q^{34} + 7695 q^{36} + 23182 q^{37} - 11943 q^{38} + 1017 q^{39} - 16158 q^{41} + 10908 q^{42} - 4249 q^{43} + 11495 q^{44} - 28769 q^{46} + 7580 q^{47} - 4491 q^{48} + 37140 q^{49} - 9270 q^{51} - 23887 q^{52} + 20574 q^{53} - 6561 q^{54} - 73276 q^{56} + 34227 q^{57} + 8733 q^{58} - 364 q^{59} - 28127 q^{61} - 71917 q^{62} + 5265 q^{63} + 43379 q^{64} - 9801 q^{66} - 21493 q^{67} + 160660 q^{68} - 4626 q^{69} - 177084 q^{71} + 16767 q^{72} + 78670 q^{73} + 196750 q^{74} - 32701 q^{76} + 7865 q^{77} + 108999 q^{78} - 187432 q^{79} + 45927 q^{81} + 179552 q^{82} - 44592 q^{83} + 3078 q^{84} - 110433 q^{86} + 24282 q^{87} + 25047 q^{88} - 151168 q^{89} - 230153 q^{91} - 44767 q^{92} + 155097 q^{93} + 54040 q^{94} - 89487 q^{96} - 55589 q^{97} + 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q + 9 * q^2 - 63 * q^3 + 95 * q^4 - 81 * q^6 + 65 * q^7 + 207 * q^8 + 567 * q^9 + 847 * q^11 - 855 * q^12 - 113 * q^13 - 1212 * q^14 + 499 * q^16 + 1030 * q^17 + 729 * q^18 - 3803 * q^19 - 585 * q^21 + 1089 * q^22 + 514 * q^23 - 1863 * q^24 - 12111 * q^26 - 5103 * q^27 - 342 * q^28 - 2698 * q^29 - 17233 * q^31 + 9943 * q^32 - 7623 * q^33 + 4090 * q^34 + 7695 * q^36 + 23182 * q^37 - 11943 * q^38 + 1017 * q^39 - 16158 * q^41 + 10908 * q^42 - 4249 * q^43 + 11495 * q^44 - 28769 * q^46 + 7580 * q^47 - 4491 * q^48 + 37140 * q^49 - 9270 * q^51 - 23887 * q^52 + 20574 * q^53 - 6561 * q^54 - 73276 * q^56 + 34227 * q^57 + 8733 * q^58 - 364 * q^59 - 28127 * q^61 - 71917 * q^62 + 5265 * q^63 + 43379 * q^64 - 9801 * q^66 - 21493 * q^67 + 160660 * q^68 - 4626 * q^69 - 177084 * q^71 + 16767 * q^72 + 78670 * q^73 + 196750 * q^74 - 32701 * q^76 + 7865 * q^77 + 108999 * q^78 - 187432 * q^79 + 45927 * q^81 + 179552 * q^82 - 44592 * q^83 + 3078 * q^84 - 110433 * q^86 + 24282 * q^87 + 25047 * q^88 - 151168 * q^89 - 230153 * q^91 - 44767 * q^92 + 155097 * q^93 + 54040 * q^94 - 89487 * q^96 - 55589 * q^97 + 478341 * q^98 + 68607 * q^99

Introduction

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