LMFDB - Embedded newform 825.6.a.o.1.6

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			1.6
Root			$6.35371$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.35371 q^{2} -9.00000 q^{3} +22.0770 q^{4} -66.1834 q^{6} +251.987 q^{7} -72.9708 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+7.35371 q^{2} -9.00000 q^{3} +22.0770 q^{4} -66.1834 q^{6} +251.987 q^{7} -72.9708 q^{8} +81.0000 q^{9} +121.000 q^{11} -198.693 q^{12} -633.803 q^{13} +1853.04 q^{14} -1243.07 q^{16} +460.381 q^{17} +595.650 q^{18} -1693.64 q^{19} -2267.89 q^{21} +889.799 q^{22} -3139.26 q^{23} +656.737 q^{24} -4660.80 q^{26} -729.000 q^{27} +5563.13 q^{28} +223.138 q^{29} -7390.32 q^{31} -6806.11 q^{32} -1089.00 q^{33} +3385.51 q^{34} +1788.24 q^{36} +5715.38 q^{37} -12454.5 q^{38} +5704.23 q^{39} +19320.6 q^{41} -16677.4 q^{42} -2050.91 q^{43} +2671.32 q^{44} -23085.2 q^{46} -3367.25 q^{47} +11187.6 q^{48} +46690.7 q^{49} -4143.43 q^{51} -13992.5 q^{52} -20043.7 q^{53} -5360.85 q^{54} -18387.7 q^{56} +15242.7 q^{57} +1640.89 q^{58} -18420.6 q^{59} +21177.7 q^{61} -54346.3 q^{62} +20411.0 q^{63} -10271.9 q^{64} -8008.19 q^{66} +45127.5 q^{67} +10163.8 q^{68} +28253.4 q^{69} -54735.9 q^{71} -5910.64 q^{72} -64414.7 q^{73} +42029.2 q^{74} -37390.4 q^{76} +30490.5 q^{77} +41947.2 q^{78} -74351.1 q^{79} +6561.00 q^{81} +142078. q^{82} -118034. q^{83} -50068.2 q^{84} -15081.8 q^{86} -2008.24 q^{87} -8829.47 q^{88} -57632.9 q^{89} -159710. q^{91} -69305.5 q^{92} +66512.9 q^{93} -24761.8 q^{94} +61255.0 q^{96} -75016.9 q^{97} +343350. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	7.35371	1.29996	0.649982	−	0.759950i	$-0.274776\pi$
$2$	7.35371	1.29996	0.649982	+	0.759950i	$0.274776\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	22.0770	0.689906
$5$	0	0
$5$	0	0
$6$	−66.1834	−0.750535
$7$	251.987	1.94372	0.971860	−	0.235558i	$-0.0756916\pi$
$7$	251.987	1.94372	0.971860	+	0.235558i	$0.0756916\pi$
$8$	−72.9708	−0.403111
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−198.693	−0.398318
$13$	−633.803	−1.04015	−0.520075	−	0.854121i	$-0.674096\pi$
$13$	−633.803	−1.04015	−0.520075	+	0.854121i	$0.674096\pi$
$14$	1853.04	2.52677
$15$	0	0
$16$	−1243.07	−1.21394
$17$	460.381	0.386362	0.193181	−	0.981163i	$-0.438119\pi$
$17$	460.381	0.386362	0.193181	+	0.981163i	$0.438119\pi$
$18$	595.650	0.433321
$19$	−1693.64	−1.07631	−0.538154	−	0.842847i	$-0.680878\pi$
$19$	−1693.64	−1.07631	−0.538154	+	0.842847i	$0.680878\pi$
$20$	0	0
$21$	−2267.89	−1.12221
$22$	889.799	0.391954
$23$	−3139.26	−1.23739	−0.618697	−	0.785630i	$-0.712339\pi$
$23$	−3139.26	−1.23739	−0.618697	+	0.785630i	$0.712339\pi$
$24$	656.737	0.232736
$25$	0	0
$26$	−4660.80	−1.35216
$27$	−729.000	−0.192450
$28$	5563.13	1.34099
$29$	223.138	0.0492695	0.0246348	−	0.999697i	$-0.492158\pi$
$29$	223.138	0.0492695	0.0246348	+	0.999697i	$0.492158\pi$
$30$	0	0
$31$	−7390.32	−1.38121	−0.690604	−	0.723233i	$-0.742655\pi$
$31$	−7390.32	−1.38121	−0.690604	+	0.723233i	$0.742655\pi$
$32$	−6806.11	−1.17496
$33$	−1089.00	−0.174078
$34$	3385.51	0.502257
$35$	0	0
$36$	1788.24	0.229969
$37$	5715.38	0.686342	0.343171	−	0.939273i	$-0.388499\pi$
$37$	5715.38	0.686342	0.343171	+	0.939273i	$0.388499\pi$
$38$	−12454.5	−1.39916
$39$	5704.23	0.600531
$40$	0	0
$41$	19320.6	1.79498	0.897492	−	0.441031i	$-0.145387\pi$
$41$	19320.6	1.79498	0.897492	+	0.441031i	$0.145387\pi$
$42$	−16677.4	−1.45883
$43$	−2050.91	−0.169151	−0.0845755	−	0.996417i	$-0.526953\pi$
$43$	−2050.91	−0.169151	−0.0845755	+	0.996417i	$0.526953\pi$
$44$	2671.32	0.208015
$45$	0	0
$46$	−23085.2	−1.60857
$47$	−3367.25	−0.222347	−0.111173	−	0.993801i	$-0.535461\pi$
$47$	−3367.25	−0.222347	−0.111173	+	0.993801i	$0.535461\pi$
$48$	11187.6	0.700866
$49$	46690.7	2.77805
$50$	0	0
$51$	−4143.43	−0.223066
$52$	−13992.5	−0.717606
$53$	−20043.7	−0.980138	−0.490069	−	0.871684i	$-0.663028\pi$
$53$	−20043.7	−0.980138	−0.490069	+	0.871684i	$0.663028\pi$
$54$	−5360.85	−0.250178
$55$	0	0
$56$	−18387.7	−0.783534
$57$	15242.7	0.621407
$58$	1640.89	0.0640486
$59$	−18420.6	−0.688928	−0.344464	−	0.938800i	$-0.611939\pi$
$59$	−18420.6	−0.688928	−0.344464	+	0.938800i	$0.611939\pi$
$60$	0	0
$61$	21177.7	0.728710	0.364355	−	0.931260i	$-0.381290\pi$
$61$	21177.7	0.728710	0.364355	+	0.931260i	$0.381290\pi$
$62$	−54346.3	−1.79552
$63$	20411.0	0.647907
$64$	−10271.9	−0.313473
$65$	0	0
$66$	−8008.19	−0.226295
$67$	45127.5	1.22816	0.614080	−	0.789244i	$-0.289527\pi$
$67$	45127.5	1.22816	0.614080	+	0.789244i	$0.289527\pi$
$68$	10163.8	0.266554
$69$	28253.4	0.714410
$70$	0	0
$71$	−54735.9	−1.28863	−0.644313	−	0.764762i	$-0.722856\pi$
$71$	−54735.9	−1.28863	−0.644313	+	0.764762i	$0.722856\pi$
$72$	−5910.64	−0.134370
$73$	−64414.7	−1.41474	−0.707372	−	0.706842i	$-0.750119\pi$
$73$	−64414.7	−1.41474	−0.707372	+	0.706842i	$0.750119\pi$
$74$	42029.2	0.892220
$75$	0	0
$76$	−37390.4	−0.742552
$77$	30490.5	0.586054
$78$	41947.2	0.780668
$79$	−74351.1	−1.34036	−0.670178	−	0.742201i	$-0.733782\pi$
$79$	−74351.1	−1.34036	−0.670178	+	0.742201i	$0.733782\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	142078.	2.33341
$83$	−118034.	−1.88067	−0.940333	−	0.340255i	$-0.889487\pi$
$83$	−118034.	−1.88067	−0.940333	+	0.340255i	$0.889487\pi$
$84$	−50068.2	−0.774218
$85$	0	0
$86$	−15081.8	−0.219890
$87$	−2008.24	−0.0284458
$88$	−8829.47	−0.121542
$89$	−57632.9	−0.771250	−0.385625	−	0.922656i	$-0.626014\pi$
$89$	−57632.9	−0.771250	−0.385625	+	0.922656i	$0.626014\pi$
$90$	0	0
$91$	−159710.	−2.02176
$92$	−69305.5	−0.853686
$93$	66512.9	0.797441
$94$	−24761.8	−0.289043
$95$	0	0
$96$	61255.0	0.678365
$97$	−75016.9	−0.809524	−0.404762	−	0.914422i	$-0.632646\pi$
$97$	−75016.9	−0.809524	−0.404762	+	0.914422i	$0.632646\pi$
$98$	343350.	3.61137
$99$	9801.00	0.100504
$100$	0	0
$101$	32528.7	0.317295	0.158648	−	0.987335i	$-0.449287\pi$
$101$	32528.7	0.317295	0.158648	+	0.987335i	$0.449287\pi$
$102$	−30469.5	−0.289978
$103$	66562.8	0.618213	0.309107	−	0.951027i	$-0.399970\pi$
$103$	66562.8	0.618213	0.309107	+	0.951027i	$0.399970\pi$
$104$	46249.1	0.419295
$105$	0	0
$106$	−147395.	−1.27414
$107$	−64861.2	−0.547678	−0.273839	−	0.961776i	$-0.588294\pi$
$107$	−64861.2	−0.547678	−0.273839	+	0.961776i	$0.588294\pi$
$108$	−16094.1	−0.132773
$109$	−47344.4	−0.381682	−0.190841	−	0.981621i	$-0.561122\pi$
$109$	−47344.4	−0.381682	−0.190841	+	0.981621i	$0.561122\pi$
$110$	0	0
$111$	−51438.4	−0.396260
$112$	−313238.	−2.35955
$113$	114489.	0.843469	0.421734	−	0.906719i	$-0.361421\pi$
$113$	114489.	0.843469	0.421734	+	0.906719i	$0.361421\pi$
$114$	112091.	0.807806
$115$	0	0
$116$	4926.22	0.0339914
$117$	−51338.0	−0.346717
$118$	−135460.	−0.895581
$119$	116010.	0.750981
$120$	0	0
$121$	14641.0	0.0909091
$122$	155735.	0.947296
$123$	−173885.	−1.03633
$124$	−163156.	−0.952905
$125$	0	0
$126$	150096.	0.842256
$127$	−315900.	−1.73796	−0.868982	−	0.494843i	$-0.835225\pi$
$127$	−315900.	−1.73796	−0.868982	+	0.494843i	$0.835225\pi$
$128$	142259.	0.767459
$129$	18458.2	0.0976594
$130$	0	0
$131$	176456.	0.898378	0.449189	−	0.893437i	$-0.351713\pi$
$131$	176456.	0.898378	0.449189	+	0.893437i	$0.351713\pi$
$132$	−24041.9	−0.120097
$133$	−426775.	−2.09204
$134$	331855.	1.59656
$135$	0	0
$136$	−33594.4	−0.155747
$137$	−345239.	−1.57152	−0.785758	−	0.618534i	$-0.787727\pi$
$137$	−345239.	−1.57152	−0.785758	+	0.618534i	$0.787727\pi$
$138$	207767.	0.928707
$139$	−140830.	−0.618239	−0.309120	−	0.951023i	$-0.600034\pi$
$139$	−140830.	−0.618239	−0.309120	+	0.951023i	$0.600034\pi$
$140$	0	0
$141$	30305.2	0.128372
$142$	−402512.	−1.67517
$143$	−76690.2	−0.313617
$144$	−100689.	−0.404645
$145$	0	0
$146$	−473687.	−1.83912
$147$	−420216.	−1.60391
$148$	126178.	0.473512
$149$	−31764.1	−0.117212	−0.0586059	−	0.998281i	$-0.518666\pi$
$149$	−31764.1	−0.117212	−0.0586059	+	0.998281i	$0.518666\pi$
$150$	0	0
$151$	−106794.	−0.381156	−0.190578	−	0.981672i	$-0.561036\pi$
$151$	−106794.	−0.381156	−0.190578	+	0.981672i	$0.561036\pi$
$152$	123586.	0.433871
$153$	37290.8	0.128787
$154$	224218.	0.761849
$155$	0	0
$156$	125932.	0.414310
$157$	−55442.9	−0.179513	−0.0897566	−	0.995964i	$-0.528609\pi$
$157$	−55442.9	−0.179513	−0.0897566	+	0.995964i	$0.528609\pi$
$158$	−546757.	−1.74241
$159$	180393.	0.565883
$160$	0	0
$161$	−791055.	−2.40515
$162$	48247.7	0.144440
$163$	108411.	0.319598	0.159799	−	0.987150i	$-0.448915\pi$
$163$	108411.	0.319598	0.159799	+	0.987150i	$0.448915\pi$
$164$	426540.	1.23837
$165$	0	0
$166$	−867987.	−2.44480
$167$	47112.2	0.130720	0.0653600	−	0.997862i	$-0.479180\pi$
$167$	47112.2	0.130720	0.0653600	+	0.997862i	$0.479180\pi$
$168$	165490.	0.452374
$169$	30413.2	0.0819116
$170$	0	0
$171$	−137185.	−0.358769
$172$	−45277.9	−0.116698
$173$	496469.	1.26118	0.630589	−	0.776117i	$-0.282813\pi$
$173$	496469.	1.26118	0.630589	+	0.776117i	$0.282813\pi$
$174$	−14768.0	−0.0369785
$175$	0	0
$176$	−150411.	−0.366015
$177$	165785.	0.397753
$178$	−423815.	−1.00260
$179$	−564791.	−1.31751	−0.658757	−	0.752356i	$-0.728917\pi$
$179$	−564791.	−1.31751	−0.658757	+	0.752356i	$0.728917\pi$
$180$	0	0
$181$	275289.	0.624585	0.312293	−	0.949986i	$-0.398903\pi$
$181$	275289.	0.624585	0.312293	+	0.949986i	$0.398903\pi$
$182$	−1.17446e6	−2.62822
$183$	−190599.	−0.420721
$184$	229074.	0.498807
$185$	0	0
$186$	489116.	1.03664
$187$	55706.1	0.116493
$188$	−74338.7	−0.153398
$189$	−183699.	−0.374069
$190$	0	0
$191$	365202.	0.724351	0.362176	−	0.932110i	$-0.382034\pi$
$191$	365202.	0.724351	0.362176	+	0.932110i	$0.382034\pi$
$192$	92446.8	0.180983
$193$	565527.	1.09285	0.546425	−	0.837508i	$-0.315988\pi$
$193$	565527.	1.09285	0.546425	+	0.837508i	$0.315988\pi$
$194$	−551652.	−1.05235
$195$	0	0
$196$	1.03079e6	1.91659
$197$	365310.	0.670650	0.335325	−	0.942103i	$-0.391154\pi$
$197$	365310.	0.670650	0.335325	+	0.942103i	$0.391154\pi$
$198$	72073.7	0.130651
$199$	−464819.	−0.832053	−0.416026	−	0.909353i	$-0.636578\pi$
$199$	−464819.	−0.832053	−0.416026	+	0.909353i	$0.636578\pi$
$200$	0	0
$201$	−406148.	−0.709078
$202$	239207.	0.412472
$203$	56228.0	0.0957662
$204$	−91474.5	−0.153895
$205$	0	0
$206$	489483.	0.803655
$207$	−254280.	−0.412465
$208$	787861.	1.26267
$209$	−204930.	−0.324519
$210$	0	0
$211$	632574.	0.978150	0.489075	−	0.872242i	$-0.337334\pi$
$211$	632574.	0.978150	0.489075	+	0.872242i	$0.337334\pi$
$212$	−442504.	−0.676203
$213$	492623.	0.743988
$214$	−476970.	−0.711962
$215$	0	0
$216$	53195.7	0.0775787
$217$	−1.86227e6	−2.68468
$218$	−348157.	−0.496173
$219$	579732.	0.816802
$220$	0	0
$221$	−291791.	−0.401875
$222$	−378263.	−0.515124
$223$	−625787.	−0.842684	−0.421342	−	0.906902i	$-0.638441\pi$
$223$	−625787.	−0.842684	−0.421342	+	0.906902i	$0.638441\pi$
$224$	−1.71505e6	−2.28380
$225$	0	0
$226$	841921.	1.09648
$227$	1.02256e6	1.31712	0.658561	−	0.752527i	$-0.271165\pi$
$227$	1.02256e6	1.31712	0.658561	+	0.752527i	$0.271165\pi$
$228$	336514.	0.428712
$229$	−1.35027e6	−1.70150	−0.850748	−	0.525574i	$-0.823851\pi$
$229$	−1.35027e6	−1.70150	−0.850748	+	0.525574i	$0.823851\pi$
$230$	0	0
$231$	−274414.	−0.338358
$232$	−16282.6	−0.0198611
$233$	−1.19472e6	−1.44171	−0.720853	−	0.693088i	$-0.756250\pi$
$233$	−1.19472e6	−1.44171	−0.720853	+	0.693088i	$0.756250\pi$
$234$	−377525.	−0.450719
$235$	0	0
$236$	−406672.	−0.475296
$237$	669160.	0.773854
$238$	853105.	0.976248
$239$	−1.26772e6	−1.43558	−0.717789	−	0.696260i	$-0.754846\pi$
$239$	−1.26772e6	−1.43558	−0.717789	+	0.696260i	$0.754846\pi$
$240$	0	0
$241$	1.43738e6	1.59415	0.797077	−	0.603878i	$-0.206379\pi$
$241$	1.43738e6	1.59415	0.797077	+	0.603878i	$0.206379\pi$
$242$	107666.	0.118179
$243$	−59049.0	−0.0641500
$244$	467540.	0.502741
$245$	0	0
$246$	−1.27870e6	−1.34720
$247$	1.07343e6	1.11952
$248$	539278.	0.556780
$249$	1.06231e6	1.08580
$250$	0	0
$251$	−513175.	−0.514139	−0.257070	−	0.966393i	$-0.582757\pi$
$251$	−513175.	−0.514139	−0.257070	+	0.966393i	$0.582757\pi$
$252$	450613.	0.446995
$253$	−379851.	−0.373088
$254$	−2.32304e6	−2.25929
$255$	0	0
$256$	1.37483e6	1.31114
$257$	907801.	0.857350	0.428675	−	0.903459i	$-0.358981\pi$
$257$	907801.	0.857350	0.428675	+	0.903459i	$0.358981\pi$
$258$	135736.	0.126954
$259$	1.44020e6	1.33406
$260$	0	0
$261$	18074.2	0.0164232
$262$	1.29761e6	1.16786
$263$	893422.	0.796466	0.398233	−	0.917284i	$-0.369624\pi$
$263$	893422.	0.796466	0.398233	+	0.917284i	$0.369624\pi$
$264$	79465.2	0.0701725
$265$	0	0
$266$	−3.13838e6	−2.71958
$267$	518696.	0.445281
$268$	996281.	0.847315
$269$	−1.13167e6	−0.953541	−0.476771	−	0.879028i	$-0.658193\pi$
$269$	−1.13167e6	−0.953541	−0.476771	+	0.879028i	$0.658193\pi$
$270$	0	0
$271$	−2.05455e6	−1.69939	−0.849694	−	0.527276i	$-0.823213\pi$
$271$	−2.05455e6	−1.69939	−0.849694	+	0.527276i	$0.823213\pi$
$272$	−572286.	−0.469019
$273$	1.43739e6	1.16726
$274$	−2.53879e6	−2.04292
$275$	0	0
$276$	623749.	0.492876
$277$	−2.32890e6	−1.82369	−0.911845	−	0.410535i	$-0.865342\pi$
$277$	−2.32890e6	−1.82369	−0.911845	+	0.410535i	$0.865342\pi$
$278$	−1.03562e6	−0.803689
$279$	−598616.	−0.460403
$280$	0	0
$281$	−321019.	−0.242530	−0.121265	−	0.992620i	$-0.538695\pi$
$281$	−321019.	−0.242530	−0.121265	+	0.992620i	$0.538695\pi$
$282$	222856.	0.166879
$283$	1.18090e6	0.876491	0.438246	−	0.898855i	$-0.355600\pi$
$283$	1.18090e6	0.876491	0.438246	+	0.898855i	$0.355600\pi$
$284$	−1.20841e6	−0.889031
$285$	0	0
$286$	−563957.	−0.407691
$287$	4.86854e6	3.48895
$288$	−551295.	−0.391654
$289$	−1.20791e6	−0.850724
$290$	0	0
$291$	675152.	0.467379
$292$	−1.42208e6	−0.976040
$293$	423746.	0.288361	0.144181	−	0.989551i	$-0.453945\pi$
$293$	423746.	0.288361	0.144181	+	0.989551i	$0.453945\pi$
$294$	−3.09015e6	−2.08502
$295$	0	0
$296$	−417056.	−0.276672
$297$	−88209.0	−0.0580259
$298$	−233584.	−0.152371
$299$	1.98967e6	1.28707
$300$	0	0
$301$	−516803.	−0.328782
$302$	−785329.	−0.495489
$303$	−292759.	−0.183191
$304$	2.10531e6	1.30657
$305$	0	0
$306$	274226.	0.167419
$307$	247177.	0.149680	0.0748398	−	0.997196i	$-0.476155\pi$
$307$	247177.	0.149680	0.0748398	+	0.997196i	$0.476155\pi$
$308$	673139.	0.404322
$309$	−599065.	−0.356926
$310$	0	0
$311$	−1.65892e6	−0.972576	−0.486288	−	0.873799i	$-0.661649\pi$
$311$	−1.65892e6	−0.972576	−0.486288	+	0.873799i	$0.661649\pi$
$312$	−416242.	−0.242080
$313$	1.43394e6	0.827311	0.413656	−	0.910433i	$-0.364252\pi$
$313$	1.43394e6	0.827311	0.413656	+	0.910433i	$0.364252\pi$
$314$	−407711.	−0.233361
$315$	0	0
$316$	−1.64145e6	−0.924720
$317$	1.21845e6	0.681018	0.340509	−	0.940241i	$-0.389401\pi$
$317$	1.21845e6	0.681018	0.340509	+	0.940241i	$0.389401\pi$
$318$	1.32656e6	0.735627
$319$	26999.7	0.0148553
$320$	0	0
$321$	583751.	0.316202
$322$	−5.81718e6	−3.12661
$323$	−779718.	−0.415845
$324$	144847.	0.0766563
$325$	0	0
$326$	797222.	0.415466
$327$	426099.	0.220364
$328$	−1.40984e6	−0.723577
$329$	−848504.	−0.432180
$330$	0	0
$331$	2.72626e6	1.36772	0.683860	−	0.729613i	$-0.260300\pi$
$331$	2.72626e6	1.36772	0.683860	+	0.729613i	$0.260300\pi$
$332$	−2.60584e6	−1.29748
$333$	462946.	0.228781
$334$	346449.	0.169931
$335$	0	0
$336$	2.81914e6	1.36229
$337$	1.35050e6	0.647769	0.323884	−	0.946097i	$-0.395011\pi$
$337$	1.35050e6	0.647769	0.323884	+	0.946097i	$0.395011\pi$
$338$	223650.	0.106482
$339$	−1.03040e6	−0.486977
$340$	0	0
$341$	−894229.	−0.416450
$342$	−1.00882e6	−0.466387
$343$	7.53032e6	3.45603
$344$	149656.	0.0681866
$345$	0	0
$346$	3.65089e6	1.63949
$347$	4.17183e6	1.85996	0.929978	−	0.367615i	$-0.119826\pi$
$347$	4.17183e6	1.85996	0.929978	+	0.367615i	$0.119826\pi$
$348$	−44335.9	−0.0196249
$349$	1.92813e6	0.847371	0.423686	−	0.905809i	$-0.360736\pi$
$349$	1.92813e6	0.847371	0.423686	+	0.905809i	$0.360736\pi$
$350$	0	0
$351$	462042.	0.200177
$352$	−823539.	−0.354264
$353$	3.84016e6	1.64026	0.820129	−	0.572178i	$-0.193901\pi$
$353$	3.84016e6	1.64026	0.820129	+	0.572178i	$0.193901\pi$
$354$	1.21914e6	0.517064
$355$	0	0
$356$	−1.27236e6	−0.532090
$357$	−1.04409e6	−0.433579
$358$	−4.15331e6	−1.71272
$359$	−2.65540e6	−1.08741	−0.543706	−	0.839276i	$-0.682979\pi$
$359$	−2.65540e6	−1.08741	−0.543706	+	0.839276i	$0.682979\pi$
$360$	0	0
$361$	392309.	0.158438
$362$	2.02439e6	0.811939
$363$	−131769.	−0.0524864
$364$	−3.52593e6	−1.39483
$365$	0	0
$366$	−1.40161e6	−0.546922
$367$	−969753.	−0.375834	−0.187917	−	0.982185i	$-0.560174\pi$
$367$	−969753.	−0.375834	−0.187917	+	0.982185i	$0.560174\pi$
$368$	3.90232e6	1.50212
$369$	1.56497e6	0.598328
$370$	0	0
$371$	−5.05075e6	−1.90511
$372$	1.46841e6	0.550160
$373$	−838185.	−0.311938	−0.155969	−	0.987762i	$-0.549850\pi$
$373$	−838185.	−0.311938	−0.155969	+	0.987762i	$0.549850\pi$
$374$	409646.	0.151436
$375$	0	0
$376$	245711.	0.0896303
$377$	−141425.	−0.0512477
$378$	−1.35087e6	−0.486277
$379$	9083.82	0.00324841	0.00162420	−	0.999999i	$-0.499483\pi$
$379$	9083.82	0.00324841	0.00162420	+	0.999999i	$0.499483\pi$
$380$	0	0
$381$	2.84310e6	1.00341
$382$	2.68558e6	0.941630
$383$	−2.72358e6	−0.948731	−0.474365	−	0.880328i	$-0.657322\pi$
$383$	−2.72358e6	−0.948731	−0.474365	+	0.880328i	$0.657322\pi$
$384$	−1.28033e6	−0.443093
$385$	0	0
$386$	4.15872e6	1.42067
$387$	−166123.	−0.0563837
$388$	−1.65615e6	−0.558496
$389$	5.18178e6	1.73622	0.868110	−	0.496372i	$-0.165335\pi$
$389$	5.18178e6	1.73622	0.868110	+	0.496372i	$0.165335\pi$
$390$	0	0
$391$	−1.44526e6	−0.478083
$392$	−3.40706e6	−1.11986
$393$	−1.58811e6	−0.518679
$394$	2.68638e6	0.871821
$395$	0	0
$396$	216377.	0.0693382
$397$	−559564.	−0.178186	−0.0890929	−	0.996023i	$-0.528397\pi$
$397$	−559564.	−0.178186	−0.0890929	+	0.996023i	$0.528397\pi$
$398$	−3.41814e6	−1.08164
$399$	3.84098e6	1.20784
$400$	0	0
$401$	−2.20014e6	−0.683265	−0.341633	−	0.939834i	$-0.610980\pi$
$401$	−2.20014e6	−0.683265	−0.341633	+	0.939834i	$0.610980\pi$
$402$	−2.98669e6	−0.921776
$403$	4.68401e6	1.43666
$404$	718137.	0.218904
$405$	0	0
$406$	413484.	0.124493
$407$	691561.	0.206940
$408$	302349.	0.0899205
$409$	−3.41580e6	−1.00968	−0.504840	−	0.863213i	$-0.668448\pi$
$409$	−3.41580e6	−1.00968	−0.504840	+	0.863213i	$0.668448\pi$
$410$	0	0
$411$	3.10715e6	0.907316
$412$	1.46951e6	0.426509
$413$	−4.64176e6	−1.33908
$414$	−1.86990e6	−0.536189
$415$	0	0
$416$	4.31373e6	1.22214
$417$	1.26747e6	0.356941
$418$	−1.50700e6	−0.421863
$419$	1.12469e6	0.312966	0.156483	−	0.987681i	$-0.449984\pi$
$419$	1.12469e6	0.312966	0.156483	+	0.987681i	$0.449984\pi$
$420$	0	0
$421$	−5.75430e6	−1.58230	−0.791148	−	0.611625i	$-0.790516\pi$
$421$	−5.75430e6	−1.58230	−0.791148	+	0.611625i	$0.790516\pi$
$422$	4.65177e6	1.27156
$423$	−272747.	−0.0741155
$424$	1.46260e6	0.395104
$425$	0	0
$426$	3.62261e6	0.967158
$427$	5.33652e6	1.41641
$428$	−1.43194e6	−0.377847
$429$	690211.	0.181067
$430$	0	0
$431$	1.30296e6	0.337861	0.168930	−	0.985628i	$-0.445969\pi$
$431$	1.30296e6	0.337861	0.168930	+	0.985628i	$0.445969\pi$
$432$	906198.	0.233622
$433$	6.66336e6	1.70794	0.853972	−	0.520320i	$-0.174187\pi$
$433$	6.66336e6	1.70794	0.853972	+	0.520320i	$0.174187\pi$
$434$	−1.36946e7	−3.48999
$435$	0	0
$436$	−1.04522e6	−0.263325
$437$	5.31677e6	1.33182
$438$	4.26318e6	1.06181
$439$	−6.12622e6	−1.51716	−0.758580	−	0.651580i	$-0.774106\pi$
$439$	−6.12622e6	−1.51716	−0.758580	+	0.651580i	$0.774106\pi$
$440$	0	0
$441$	3.78195e6	0.926017
$442$	−2.14574e6	−0.522423
$443$	6.57224e6	1.59112	0.795562	−	0.605872i	$-0.207175\pi$
$443$	6.57224e6	1.59112	0.795562	+	0.605872i	$0.207175\pi$
$444$	−1.13561e6	−0.273382
$445$	0	0
$446$	−4.60186e6	−1.09546
$447$	285877.	0.0676722
$448$	−2.58838e6	−0.609303
$449$	−1.83721e6	−0.430075	−0.215037	−	0.976606i	$-0.568987\pi$
$449$	−1.83721e6	−0.430075	−0.215037	+	0.976606i	$0.568987\pi$
$450$	0	0
$451$	2.33779e6	0.541208
$452$	2.52758e6	0.581914
$453$	961142.	0.220061
$454$	7.51964e6	1.71221
$455$	0	0
$456$	−1.11227e6	−0.250496
$457$	−5.02595e6	−1.12571	−0.562857	−	0.826554i	$-0.690298\pi$
$457$	−5.02595e6	−1.12571	−0.562857	+	0.826554i	$0.690298\pi$
$458$	−9.92947e6	−2.21188
$459$	−335618.	−0.0743555
$460$	0	0
$461$	−3.24710e6	−0.711612	−0.355806	−	0.934560i	$-0.615794\pi$
$461$	−3.24710e6	−0.711612	−0.355806	+	0.934560i	$0.615794\pi$
$462$	−2.01796e6	−0.439854
$463$	8.24253e6	1.78693	0.893466	−	0.449131i	$-0.148266\pi$
$463$	8.24253e6	1.78693	0.893466	+	0.449131i	$0.148266\pi$
$464$	−277376.	−0.0598100
$465$	0	0
$466$	−8.78563e6	−1.87417
$467$	−1.61860e6	−0.343437	−0.171718	−	0.985146i	$-0.554932\pi$
$467$	−1.61860e6	−0.343437	−0.171718	+	0.985146i	$0.554932\pi$
$468$	−1.13339e6	−0.239202
$469$	1.13716e7	2.38720
$470$	0	0
$471$	498986.	0.103642
$472$	1.34417e6	0.277714
$473$	−248160.	−0.0510010
$474$	4.92081e6	1.00598
$475$	0	0
$476$	2.56116e6	0.518106
$477$	−1.62354e6	−0.326713
$478$	−9.32241e6	−1.86620
$479$	107588.	0.0214252	0.0107126	−	0.999943i	$-0.496590\pi$
$479$	107588.	0.0214252	0.0107126	+	0.999943i	$0.496590\pi$
$480$	0	0
$481$	−3.62243e6	−0.713899
$482$	1.05701e7	2.07234
$483$	7.11949e6	1.38861
$484$	323229.	0.0627188
$485$	0	0
$486$	−434229.	−0.0833927
$487$	−8.33266e6	−1.59207	−0.796033	−	0.605253i	$-0.793072\pi$
$487$	−8.33266e6	−1.59207	−0.796033	+	0.605253i	$0.793072\pi$
$488$	−1.54536e6	−0.293751
$489$	−975698.	−0.184520
$490$	0	0
$491$	9.57192e6	1.79182	0.895912	−	0.444232i	$-0.146523\pi$
$491$	9.57192e6	1.79182	0.895912	+	0.444232i	$0.146523\pi$
$492$	−3.83886e6	−0.714974
$493$	102728.	0.0190359
$494$	7.89371e6	1.45534
$495$	0	0
$496$	9.18669e6	1.67670
$497$	−1.37928e7	−2.50473
$498$	7.81189e6	1.41151
$499$	−815890.	−0.146683	−0.0733416	−	0.997307i	$-0.523366\pi$
$499$	−815890.	−0.146683	−0.0733416	+	0.997307i	$0.523366\pi$
$500$	0	0
$501$	−424009.	−0.0754712
$502$	−3.77374e6	−0.668363
$503$	−5.19342e6	−0.915238	−0.457619	−	0.889148i	$-0.651298\pi$
$503$	−5.19342e6	−0.915238	−0.457619	+	0.889148i	$0.651298\pi$
$504$	−1.48941e6	−0.261178
$505$	0	0
$506$	−2.79331e6	−0.485001
$507$	−273719.	−0.0472917
$508$	−6.97413e6	−1.19903
$509$	179583.	0.0307236	0.0153618	−	0.999882i	$-0.495110\pi$
$509$	179583.	0.0307236	0.0153618	+	0.999882i	$0.495110\pi$
$510$	0	0
$511$	−1.62317e7	−2.74987
$512$	5.55782e6	0.936978
$513$	1.23466e6	0.207136
$514$	6.67571e6	1.11452
$515$	0	0
$516$	407501.	0.0673758
$517$	−407437.	−0.0670400
$518$	1.05908e7	1.73423
$519$	−4.46822e6	−0.728142
$520$	0	0
$521$	7.05065e6	1.13798	0.568991	−	0.822344i	$-0.307334\pi$
$521$	7.05065e6	1.13798	0.568991	+	0.822344i	$0.307334\pi$
$522$	132912.	0.0213495
$523$	−4.20948e6	−0.672936	−0.336468	−	0.941695i	$-0.609232\pi$
$523$	−4.20948e6	−0.672936	−0.336468	+	0.941695i	$0.609232\pi$
$524$	3.89563e6	0.619797
$525$	0	0
$526$	6.56996e6	1.03538
$527$	−3.40236e6	−0.533647
$528$	1.35370e6	0.211319
$529$	3.41862e6	0.531143
$530$	0	0
$531$	−1.49207e6	−0.229643
$532$	−9.42192e6	−1.44331
$533$	−1.22454e7	−1.86705
$534$	3.81434e6	0.578850
$535$	0	0
$536$	−3.29299e6	−0.495084
$537$	5.08312e6	0.760667
$538$	−8.32197e6	−1.23957
$539$	5.64957e6	0.837614
$540$	0	0
$541$	−1.28203e7	−1.88324	−0.941618	−	0.336684i	$-0.890695\pi$
$541$	−1.28203e7	−1.88324	−0.941618	+	0.336684i	$0.890695\pi$
$542$	−1.51085e7	−2.20914
$543$	−2.47760e6	−0.360605
$544$	−3.13340e6	−0.453961
$545$	0	0
$546$	1.05702e7	1.51740
$547$	4.35983e6	0.623020	0.311510	−	0.950243i	$-0.399165\pi$
$547$	4.35983e6	0.623020	0.311510	+	0.950243i	$0.399165\pi$
$548$	−7.62185e6	−1.08420
$549$	1.71539e6	0.242903
$550$	0	0
$551$	−377915.	−0.0530292
$552$	−2.06167e6	−0.287986
$553$	−1.87356e7	−2.60528
$554$	−1.71260e7	−2.37073
$555$	0	0
$556$	−3.10909e6	−0.426527
$557$	−2.40575e6	−0.328558	−0.164279	−	0.986414i	$-0.552530\pi$
$557$	−2.40575e6	−0.328558	−0.164279	+	0.986414i	$0.552530\pi$
$558$	−4.40205e6	−0.598507
$559$	1.29987e6	0.175942
$560$	0	0
$561$	−501355.	−0.0672571
$562$	−2.36068e6	−0.315280
$563$	−2.75282e6	−0.366022	−0.183011	−	0.983111i	$-0.558584\pi$
$563$	−2.75282e6	−0.366022	−0.183011	+	0.983111i	$0.558584\pi$
$564$	669049.	0.0885646
$565$	0	0
$566$	8.68400e6	1.13941
$567$	1.65329e6	0.215969
$568$	3.99413e6	0.519459
$569$	1.50203e6	0.194491	0.0972454	−	0.995260i	$-0.468997\pi$
$569$	1.50203e6	0.194491	0.0972454	+	0.995260i	$0.468997\pi$
$570$	0	0
$571$	4.08404e6	0.524204	0.262102	−	0.965040i	$-0.415584\pi$
$571$	4.08404e6	0.524204	0.262102	+	0.965040i	$0.415584\pi$
$572$	−1.69309e6	−0.216366
$573$	−3.28681e6	−0.418204
$574$	3.58018e7	4.53551
$575$	0	0
$576$	−832021.	−0.104491
$577$	2.67916e6	0.335011	0.167506	−	0.985871i	$-0.446429\pi$
$577$	2.67916e6	0.335011	0.167506	+	0.985871i	$0.446429\pi$
$578$	−8.88259e6	−1.10591
$579$	−5.08975e6	−0.630957
$580$	0	0
$581$	−2.97431e7	−3.65549
$582$	4.96487e6	0.607575
$583$	−2.42528e6	−0.295523
$584$	4.70039e6	0.570298
$585$	0	0
$586$	3.11611e6	0.374859
$587$	3.66377e6	0.438866	0.219433	−	0.975628i	$-0.429579\pi$
$587$	3.66377e6	0.438866	0.219433	+	0.975628i	$0.429579\pi$
$588$	−9.27712e6	−1.10655
$589$	1.25165e7	1.48661
$590$	0	0
$591$	−3.28779e6	−0.387200
$592$	−7.10462e6	−0.833176
$593$	764508.	0.0892782	0.0446391	−	0.999003i	$-0.485786\pi$
$593$	764508.	0.0892782	0.0446391	+	0.999003i	$0.485786\pi$
$594$	−648663.	−0.0754316
$595$	0	0
$596$	−701256.	−0.0808651
$597$	4.18337e6	0.480386
$598$	1.46315e7	1.67315
$599$	8.57025e6	0.975947	0.487974	−	0.872858i	$-0.337736\pi$
$599$	8.57025e6	0.975947	0.487974	+	0.872858i	$0.337736\pi$
$600$	0	0
$601$	−2.52254e6	−0.284873	−0.142436	−	0.989804i	$-0.545494\pi$
$601$	−2.52254e6	−0.284873	−0.142436	+	0.989804i	$0.545494\pi$
$602$	−3.80042e6	−0.427405
$603$	3.65533e6	0.409386
$604$	−2.35768e6	−0.262962
$605$	0	0
$606$	−2.15286e6	−0.238141
$607$	9.32496e6	1.02725	0.513624	−	0.858015i	$-0.328303\pi$
$607$	9.32496e6	1.02725	0.513624	+	0.858015i	$0.328303\pi$
$608$	1.15271e7	1.26462
$609$	−506052.	−0.0552906
$610$	0	0
$611$	2.13417e6	0.231274
$612$	823270.	0.0888513
$613$	−1.07420e7	−1.15461	−0.577303	−	0.816530i	$-0.695895\pi$
$613$	−1.07420e7	−1.15461	−0.577303	+	0.816530i	$0.695895\pi$
$614$	1.81767e6	0.194578
$615$	0	0
$616$	−2.22492e6	−0.236245
$617$	4.83417e6	0.511221	0.255611	−	0.966780i	$-0.417723\pi$
$617$	4.83417e6	0.511221	0.255611	+	0.966780i	$0.417723\pi$
$618$	−4.40535e6	−0.463990
$619$	−1.37750e7	−1.44499	−0.722495	−	0.691376i	$-0.757005\pi$
$619$	−1.37750e7	−1.44499	−0.722495	+	0.691376i	$0.757005\pi$
$620$	0	0
$621$	2.28852e6	0.238137
$622$	−1.21992e7	−1.26431
$623$	−1.45228e7	−1.49909
$624$	−7.09075e6	−0.729006
$625$	0	0
$626$	1.05447e7	1.07547
$627$	1.84437e6	0.187361
$628$	−1.22401e6	−0.123847
$629$	2.63125e6	0.265177
$630$	0	0
$631$	−7.90778e6	−0.790644	−0.395322	−	0.918543i	$-0.629367\pi$
$631$	−7.90778e6	−0.790644	−0.395322	+	0.918543i	$0.629367\pi$
$632$	5.42546e6	0.540311
$633$	−5.69317e6	−0.564735
$634$	8.96010e6	0.885299
$635$	0	0
$636$	3.98253e6	0.390406
$637$	−2.95927e7	−2.88959
$638$	198548.	0.0193114
$639$	−4.43361e6	−0.429542
$640$	0	0
$641$	−1.94986e7	−1.87438	−0.937190	−	0.348818i	$-0.886583\pi$
$641$	−1.94986e7	−1.87438	−0.937190	+	0.348818i	$0.886583\pi$
$642$	4.29273e6	0.411051
$643$	−2.57069e6	−0.245201	−0.122601	−	0.992456i	$-0.539123\pi$
$643$	−2.57069e6	−0.245201	−0.122601	+	0.992456i	$0.539123\pi$
$644$	−1.74641e7	−1.65933
$645$	0	0
$646$	−5.73382e6	−0.540583
$647$	1.67753e7	1.57547	0.787735	−	0.616014i	$-0.211254\pi$
$647$	1.67753e7	1.57547	0.787735	+	0.616014i	$0.211254\pi$
$648$	−478761.	−0.0447901
$649$	−2.22889e6	−0.207720
$650$	0	0
$651$	1.67604e7	1.55000
$652$	2.39339e6	0.220493
$653$	3.43799e6	0.315517	0.157758	−	0.987478i	$-0.449573\pi$
$653$	3.43799e6	0.315517	0.157758	+	0.987478i	$0.449573\pi$
$654$	3.13341e6	0.286466
$655$	0	0
$656$	−2.40168e7	−2.17899
$657$	−5.21759e6	−0.471581
$658$	−6.23965e6	−0.561818
$659$	1.55628e7	1.39596	0.697980	−	0.716117i	$-0.254083\pi$
$659$	1.55628e7	1.39596	0.697980	+	0.716117i	$0.254083\pi$
$660$	0	0
$661$	1.05335e7	0.937714	0.468857	−	0.883274i	$-0.344666\pi$
$661$	1.05335e7	0.937714	0.468857	+	0.883274i	$0.344666\pi$
$662$	2.00481e7	1.77799
$663$	2.62612e6	0.232023
$664$	8.61304e6	0.758117
$665$	0	0
$666$	3.40437e6	0.297407
$667$	−700488.	−0.0609658
$668$	1.04010e6	0.0901845
$669$	5.63209e6	0.486524
$670$	0	0
$671$	2.56250e6	0.219714
$672$	1.54355e7	1.31855
$673$	5.82081e6	0.495389	0.247694	−	0.968838i	$-0.420327\pi$
$673$	5.82081e6	0.495389	0.247694	+	0.968838i	$0.420327\pi$
$674$	9.93118e6	0.842076
$675$	0	0
$676$	671432.	0.0565113
$677$	2.08364e7	1.74723	0.873616	−	0.486617i	$-0.161769\pi$
$677$	2.08364e7	1.74723	0.873616	+	0.486617i	$0.161769\pi$
$678$	−7.57729e6	−0.633052
$679$	−1.89033e7	−1.57349
$680$	0	0
$681$	−9.20308e6	−0.760441
$682$	−6.57590e6	−0.541370
$683$	1.49980e6	0.123022	0.0615109	−	0.998106i	$-0.480408\pi$
$683$	1.49980e6	0.123022	0.0615109	+	0.998106i	$0.480408\pi$
$684$	−3.02863e6	−0.247517
$685$	0	0
$686$	5.53757e7	4.49272
$687$	1.21524e7	0.982359
$688$	2.54942e6	0.205338
$689$	1.27037e7	1.01949
$690$	0	0
$691$	9.02927e6	0.719378	0.359689	−	0.933072i	$-0.382883\pi$
$691$	9.02927e6	0.719378	0.359689	+	0.933072i	$0.382883\pi$
$692$	1.09605e7	0.870095
$693$	2.46973e6	0.195351
$694$	3.06784e7	2.41788
$695$	0	0
$696$	146543.	0.0114668
$697$	8.89482e6	0.693514
$698$	1.41789e7	1.10155
$699$	1.07525e7	0.832369
$700$	0	0
$701$	2.19167e7	1.68454	0.842268	−	0.539058i	$-0.181220\pi$
$701$	2.19167e7	1.68454	0.842268	+	0.539058i	$0.181220\pi$
$702$	3.39772e6	0.260223
$703$	−9.67978e6	−0.738716
$704$	−1.24290e6	−0.0945155
$705$	0	0
$706$	2.82394e7	2.13228
$707$	8.19683e6	0.616733
$708$	3.66004e6	0.274412
$709$	1.77866e7	1.32885	0.664425	−	0.747355i	$-0.268676\pi$
$709$	1.77866e7	1.32885	0.664425	+	0.747355i	$0.268676\pi$
$710$	0	0
$711$	−6.02244e6	−0.446785
$712$	4.20552e6	0.310899
$713$	2.32002e7	1.70910
$714$	−7.67795e6	−0.563637
$715$	0	0
$716$	−1.24689e7	−0.908961
$717$	1.14094e7	0.828832
$718$	−1.95270e7	−1.41360
$719$	1.51719e7	1.09451	0.547254	−	0.836967i	$-0.315673\pi$
$719$	1.51719e7	1.09451	0.547254	+	0.836967i	$0.315673\pi$
$720$	0	0
$721$	1.67730e7	1.20163
$722$	2.88492e6	0.205964
$723$	−1.29365e7	−0.920385
$724$	6.07755e6	0.430905
$725$	0	0
$726$	−968991.	−0.0682304
$727$	1.83411e7	1.28703	0.643515	−	0.765434i	$-0.277476\pi$
$727$	1.83411e7	1.28703	0.643515	+	0.765434i	$0.277476\pi$
$728$	1.16542e7	0.814993
$729$	531441.	0.0370370
$730$	0	0
$731$	−944198.	−0.0653536
$732$	−4.20786e6	−0.290258
$733$	2.49198e7	1.71311	0.856555	−	0.516055i	$-0.172600\pi$
$733$	2.49198e7	1.71311	0.856555	+	0.516055i	$0.172600\pi$
$734$	−7.13128e6	−0.488571
$735$	0	0
$736$	2.13662e7	1.45389
$737$	5.46043e6	0.370304
$738$	1.15083e7	0.777805
$739$	−2.11189e7	−1.42253	−0.711264	−	0.702924i	$-0.751877\pi$
$739$	−2.11189e7	−1.42253	−0.711264	+	0.702924i	$0.751877\pi$
$740$	0	0
$741$	−9.66089e6	−0.646356
$742$	−3.71417e7	−2.47658
$743$	−1.55948e6	−0.103635	−0.0518176	−	0.998657i	$-0.516501\pi$
$743$	−1.55948e6	−0.103635	−0.0518176	+	0.998657i	$0.516501\pi$
$744$	−4.85350e6	−0.321457
$745$	0	0
$746$	−6.16377e6	−0.405508
$747$	−9.56075e6	−0.626889
$748$	1.22982e6	0.0803690
$749$	−1.63442e7	−1.06453
$750$	0	0
$751$	9.09412e6	0.588384	0.294192	−	0.955746i	$-0.404949\pi$
$751$	9.09412e6	0.588384	0.294192	+	0.955746i	$0.404949\pi$
$752$	4.18572e6	0.269914
$753$	4.61857e6	0.296838
$754$	−1.04000e6	−0.0666201
$755$	0	0
$756$	−4.05552e6	−0.258073
$757$	7.88031e6	0.499808	0.249904	−	0.968271i	$-0.419601\pi$
$757$	7.88031e6	0.499808	0.249904	+	0.968271i	$0.419601\pi$
$758$	66799.8	0.00422281
$759$	3.41866e6	0.215403
$760$	0	0
$761$	6.91680e6	0.432956	0.216478	−	0.976288i	$-0.430543\pi$
$761$	6.91680e6	0.432956	0.216478	+	0.976288i	$0.430543\pi$
$762$	2.09074e7	1.30440
$763$	−1.19302e7	−0.741884
$764$	8.06256e6	0.499734
$765$	0	0
$766$	−2.00284e7	−1.23332
$767$	1.16750e7	0.716588
$768$	−1.23735e7	−0.756988
$769$	1.04635e7	0.638057	0.319029	−	0.947745i	$-0.396643\pi$
$769$	1.04635e7	0.638057	0.319029	+	0.947745i	$0.396643\pi$
$770$	0	0
$771$	−8.17021e6	−0.494991
$772$	1.24852e7	0.753964
$773$	8.77160e6	0.527996	0.263998	−	0.964523i	$-0.414959\pi$
$773$	8.77160e6	0.527996	0.263998	+	0.964523i	$0.414959\pi$
$774$	−1.22162e6	−0.0732968
$775$	0	0
$776$	5.47404e6	0.326328
$777$	−1.29618e7	−0.770219
$778$	3.81053e7	2.25702
$779$	−3.27221e7	−1.93195
$780$	0	0
$781$	−6.62305e6	−0.388535
$782$	−1.06280e7	−0.621490
$783$	−162668.	−0.00948192
$784$	−5.80398e7	−3.37237
$785$	0	0
$786$	−1.16785e7	−0.674264
$787$	−8.13351e6	−0.468103	−0.234051	−	0.972224i	$-0.575198\pi$
$787$	−8.13351e6	−0.468103	−0.234051	+	0.972224i	$0.575198\pi$
$788$	8.06495e6	0.462686
$789$	−8.04080e6	−0.459840
$790$	0	0
$791$	2.88499e7	1.63947
$792$	−715187.	−0.0405141
$793$	−1.34225e7	−0.757967
$794$	−4.11487e6	−0.231635
$795$	0	0
$796$	−1.02618e7	−0.574038
$797$	−2.51082e7	−1.40014	−0.700068	−	0.714076i	$-0.746847\pi$
$797$	−2.51082e7	−1.40014	−0.700068	+	0.714076i	$0.746847\pi$
$798$	2.82454e7	1.57015
$799$	−1.55022e6	−0.0859064
$800$	0	0
$801$	−4.66826e6	−0.257083
$802$	−1.61792e7	−0.888220
$803$	−7.79417e6	−0.426561
$804$	−8.96653e6	−0.489198
$805$	0	0
$806$	3.44448e7	1.86761
$807$	1.01850e7	0.550527
$808$	−2.37365e6	−0.127905
$809$	6.82106e6	0.366421	0.183211	−	0.983074i	$-0.441351\pi$
$809$	6.82106e6	0.366421	0.183211	+	0.983074i	$0.441351\pi$
$810$	0	0
$811$	−1.80322e7	−0.962714	−0.481357	−	0.876525i	$-0.659856\pi$
$811$	−1.80322e7	−0.962714	−0.481357	+	0.876525i	$0.659856\pi$
$812$	1.24134e6	0.0660697
$813$	1.84909e7	0.981142
$814$	5.08554e6	0.269015
$815$	0	0
$816$	5.15057e6	0.270788
$817$	3.47349e6	0.182059
$818$	−2.51188e7	−1.31255
$819$	−1.29365e7	−0.673920
$820$	0	0
$821$	−2.73897e7	−1.41817	−0.709086	−	0.705122i	$-0.750892\pi$
$821$	−2.73897e7	−1.41817	−0.709086	+	0.705122i	$0.750892\pi$
$822$	2.28491e7	1.17948
$823$	−1.92663e7	−0.991513	−0.495757	−	0.868462i	$-0.665109\pi$
$823$	−1.92663e7	−0.991513	−0.495757	+	0.868462i	$0.665109\pi$
$824$	−4.85714e6	−0.249208
$825$	0	0
$826$	−3.41341e7	−1.74076
$827$	7.69161e6	0.391069	0.195534	−	0.980697i	$-0.437356\pi$
$827$	7.69161e6	0.391069	0.195534	+	0.980697i	$0.437356\pi$
$828$	−5.61374e6	−0.284562
$829$	3.75125e7	1.89579	0.947894	−	0.318585i	$-0.103208\pi$
$829$	3.75125e7	1.89579	0.947894	+	0.318585i	$0.103208\pi$
$830$	0	0
$831$	2.09601e7	1.05291
$832$	6.51034e6	0.326058
$833$	2.14955e7	1.07333
$834$	9.32057e6	0.464010
$835$	0	0
$836$	−4.52424e6	−0.223888
$837$	5.38755e6	0.265814
$838$	8.27064e6	0.406845
$839$	9.79411e6	0.480353	0.240176	−	0.970729i	$-0.422795\pi$
$839$	9.79411e6	0.480353	0.240176	+	0.970729i	$0.422795\pi$
$840$	0	0
$841$	−2.04614e7	−0.997573
$842$	−4.23155e7	−2.05693
$843$	2.88917e6	0.140025
$844$	1.39653e7	0.674832
$845$	0	0
$846$	−2.00570e6	−0.0963475
$847$	3.68935e6	0.176702
$848$	2.49157e7	1.18982
$849$	−1.06281e7	−0.506043
$850$	0	0
$851$	−1.79421e7	−0.849276
$852$	1.08756e7	0.513282
$853$	6.89811e6	0.324607	0.162303	−	0.986741i	$-0.448108\pi$
$853$	6.89811e6	0.324607	0.162303	+	0.986741i	$0.448108\pi$
$854$	3.92432e7	1.84128
$855$	0	0
$856$	4.73297e6	0.220775
$857$	3.52694e6	0.164039	0.0820193	−	0.996631i	$-0.473863\pi$
$857$	3.52694e6	0.164039	0.0820193	+	0.996631i	$0.473863\pi$
$858$	5.07561e6	0.235380
$859$	1.95016e6	0.0901750	0.0450875	−	0.998983i	$-0.485643\pi$
$859$	1.95016e6	0.0901750	0.0450875	+	0.998983i	$0.485643\pi$
$860$	0	0
$861$	−4.38169e7	−2.01434
$862$	9.58158e6	0.439207
$863$	−1.22897e7	−0.561712	−0.280856	−	0.959750i	$-0.590618\pi$
$863$	−1.22897e7	−0.561712	−0.280856	+	0.959750i	$0.590618\pi$
$864$	4.96165e6	0.226122
$865$	0	0
$866$	4.90004e7	2.22026
$867$	1.08712e7	0.491166
$868$	−4.11133e7	−1.85218
$869$	−8.99649e6	−0.404132
$870$	0	0
$871$	−2.86020e7	−1.27747
$872$	3.45476e6	0.153860
$873$	−6.07637e6	−0.269841
$874$	3.90980e7	1.73131
$875$	0	0
$876$	1.27987e7	0.563517
$877$	−1.46545e7	−0.643385	−0.321693	−	0.946844i	$-0.604252\pi$
$877$	−1.46545e7	−0.643385	−0.321693	+	0.946844i	$0.604252\pi$
$878$	−4.50504e7	−1.97225
$879$	−3.81372e6	−0.166485
$880$	0	0
$881$	117098.	0.00508288	0.00254144	−	0.999997i	$-0.499191\pi$
$881$	117098.	0.00508288	0.00254144	+	0.999997i	$0.499191\pi$
$882$	2.78113e7	1.20379
$883$	2.46470e7	1.06381	0.531903	−	0.846806i	$-0.321477\pi$
$883$	2.46470e7	1.06381	0.531903	+	0.846806i	$0.321477\pi$
$884$	−6.44186e6	−0.277256
$885$	0	0
$886$	4.83303e7	2.06840
$887$	−9.09068e6	−0.387961	−0.193980	−	0.981005i	$-0.562140\pi$
$887$	−9.09068e6	−0.387961	−0.193980	+	0.981005i	$0.562140\pi$
$888$	3.75350e6	0.159737
$889$	−7.96030e7	−3.37812
$890$	0	0
$891$	793881.	0.0335013
$892$	−1.38155e7	−0.581373
$893$	5.70290e6	0.239313
$894$	2.10226e6	0.0879715
$895$	0	0
$896$	3.58475e7	1.49173
$897$	−1.79071e7	−0.743093
$898$	−1.35103e7	−0.559082
$899$	−1.64906e6	−0.0680515
$900$	0	0
$901$	−9.22771e6	−0.378688
$902$	1.71914e7	0.703551
$903$	4.65123e6	0.189823
$904$	−8.35438e6	−0.340011
$905$	0	0
$906$	7.06796e6	0.286071
$907$	−9.29801e6	−0.375294	−0.187647	−	0.982237i	$-0.560086\pi$
$907$	−9.29801e6	−0.375294	−0.187647	+	0.982237i	$0.560086\pi$
$908$	2.25752e7	0.908691
$909$	2.63483e6	0.105765
$910$	0	0
$911$	2.25087e7	0.898574	0.449287	−	0.893387i	$-0.351678\pi$
$911$	2.25087e7	0.898574	0.449287	+	0.893387i	$0.351678\pi$
$912$	−1.89478e7	−0.754348
$913$	−1.42821e7	−0.567042
$914$	−3.69594e7	−1.46339
$915$	0	0
$916$	−2.98099e7	−1.17387
$917$	4.44648e7	1.74620
$918$	−2.46803e6	−0.0966595
$919$	2.05921e7	0.804288	0.402144	−	0.915576i	$-0.368265\pi$
$919$	2.05921e7	0.804288	0.402144	+	0.915576i	$0.368265\pi$
$920$	0	0
$921$	−2.22459e6	−0.0864175
$922$	−2.38782e7	−0.925070
$923$	3.46918e7	1.34036
$924$	−6.05825e6	−0.233436
$925$	0	0
$926$	6.06131e7	2.32295
$927$	5.39158e6	0.206071
$928$	−1.51870e6	−0.0578898
$929$	1.69947e7	0.646064	0.323032	−	0.946388i	$-0.395298\pi$
$929$	1.69947e7	0.646064	0.323032	+	0.946388i	$0.395298\pi$
$930$	0	0
$931$	−7.90771e7	−2.99004
$932$	−2.63759e7	−0.994642
$933$	1.49303e7	0.561517
$934$	−1.19027e7	−0.446455
$935$	0	0
$936$	3.74618e6	0.139765
$937$	1.14605e7	0.426438	0.213219	−	0.977004i	$-0.431605\pi$
$937$	1.14605e7	0.426438	0.213219	+	0.977004i	$0.431605\pi$
$938$	8.36232e7	3.10327
$939$	−1.29054e7	−0.477648
$940$	0	0
$941$	1.16402e7	0.428534	0.214267	−	0.976775i	$-0.431264\pi$
$941$	1.16402e7	0.428534	0.214267	+	0.976775i	$0.431264\pi$
$942$	3.66939e6	0.134731
$943$	−6.06523e7	−2.22110
$944$	2.28981e7	0.836314
$945$	0	0
$946$	−1.82489e6	−0.0662994
$947$	−1.62911e7	−0.590303	−0.295151	−	0.955450i	$-0.595370\pi$
$947$	−1.62911e7	−0.590303	−0.295151	+	0.955450i	$0.595370\pi$
$948$	1.47731e7	0.533887
$949$	4.08262e7	1.47154
$950$	0	0
$951$	−1.09660e7	−0.393186
$952$	−8.46536e6	−0.302728
$953$	−1.77379e7	−0.632660	−0.316330	−	0.948649i	$-0.602451\pi$
$953$	−1.77379e7	−0.632660	−0.316330	+	0.948649i	$0.602451\pi$
$954$	−1.19390e7	−0.424715
$955$	0	0
$956$	−2.79874e7	−0.990415
$957$	−242997.	−0.00857672
$958$	791172.	0.0278520
$959$	−8.69960e7	−3.05459
$960$	0	0
$961$	2.59877e7	0.907737
$962$	−2.66383e7	−0.928043
$963$	−5.25376e6	−0.182559
$964$	3.17331e7	1.09982
$965$	0	0
$966$	5.23547e7	1.80515
$967$	−4.62764e7	−1.59145	−0.795726	−	0.605657i	$-0.792911\pi$
$967$	−4.62764e7	−1.59145	−0.795726	+	0.605657i	$0.792911\pi$
$968$	−1.06837e6	−0.0366464
$969$	7.01746e6	0.240088
$970$	0	0
$971$	3.08766e7	1.05095	0.525474	−	0.850810i	$-0.323888\pi$
$971$	3.08766e7	1.05095	0.525474	+	0.850810i	$0.323888\pi$
$972$	−1.30362e6	−0.0442575
$973$	−3.54873e7	−1.20168
$974$	−6.12759e7	−2.06963
$975$	0	0
$976$	−2.63254e7	−0.884607
$977$	3.37783e7	1.13214	0.566071	−	0.824357i	$-0.308463\pi$
$977$	3.37783e7	1.13214	0.566071	+	0.824357i	$0.308463\pi$
$978$	−7.17500e6	−0.239869
$979$	−6.97358e6	−0.232541
$980$	0	0
$981$	−3.83489e6	−0.127227
$982$	7.03891e7	2.32931
$983$	4.70641e7	1.55348	0.776740	−	0.629821i	$-0.216872\pi$
$983$	4.70641e7	1.55348	0.776740	+	0.629821i	$0.216872\pi$
$984$	1.26885e7	0.417757
$985$	0	0
$986$	755435.	0.0247460
$987$	7.63654e6	0.249519
$988$	2.36982e7	0.772365
$989$	6.43833e6	0.209306
$990$	0	0
$991$	−1.74875e7	−0.565644	−0.282822	−	0.959172i	$-0.591271\pi$
$991$	−1.74875e7	−0.565644	−0.282822	+	0.959172i	$0.591271\pi$
$992$	5.02993e7	1.62287
$993$	−2.45363e7	−0.789654
$994$	−1.01428e8	−3.25606
$995$	0	0
$996$	2.34525e7	0.749103
$997$	−1.72629e7	−0.550015	−0.275008	−	0.961442i	$-0.588680\pi$
$997$	−1.72629e7	−0.550015	−0.275008	+	0.961442i	$0.588680\pi$
$998$	−5.99981e6	−0.190683
$999$	−4.16651e6	−0.132087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.o.1.6	yes	7
5.4	even	2		825.6.a.m.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.m.1.2	✓	7	5.4	even	2
825.6.a.o.1.6	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)

\(f(q)\)	\(=\)	\(q+7.35371 q^{2} -9.00000 q^{3} +22.0770 q^{4} -66.1834 q^{6} +251.987 q^{7} -72.9708 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+7.35371 q^{2} -9.00000 q^{3} +22.0770 q^{4} -66.1834 q^{6} +251.987 q^{7} -72.9708 q^{8} +81.0000 q^{9} +121.000 q^{11} -198.693 q^{12} -633.803 q^{13} +1853.04 q^{14} -1243.07 q^{16} +460.381 q^{17} +595.650 q^{18} -1693.64 q^{19} -2267.89 q^{21} +889.799 q^{22} -3139.26 q^{23} +656.737 q^{24} -4660.80 q^{26} -729.000 q^{27} +5563.13 q^{28} +223.138 q^{29} -7390.32 q^{31} -6806.11 q^{32} -1089.00 q^{33} +3385.51 q^{34} +1788.24 q^{36} +5715.38 q^{37} -12454.5 q^{38} +5704.23 q^{39} +19320.6 q^{41} -16677.4 q^{42} -2050.91 q^{43} +2671.32 q^{44} -23085.2 q^{46} -3367.25 q^{47} +11187.6 q^{48} +46690.7 q^{49} -4143.43 q^{51} -13992.5 q^{52} -20043.7 q^{53} -5360.85 q^{54} -18387.7 q^{56} +15242.7 q^{57} +1640.89 q^{58} -18420.6 q^{59} +21177.7 q^{61} -54346.3 q^{62} +20411.0 q^{63} -10271.9 q^{64} -8008.19 q^{66} +45127.5 q^{67} +10163.8 q^{68} +28253.4 q^{69} -54735.9 q^{71} -5910.64 q^{72} -64414.7 q^{73} +42029.2 q^{74} -37390.4 q^{76} +30490.5 q^{77} +41947.2 q^{78} -74351.1 q^{79} +6561.00 q^{81} +142078. q^{82} -118034. q^{83} -50068.2 q^{84} -15081.8 q^{86} -2008.24 q^{87} -8829.47 q^{88} -57632.9 q^{89} -159710. q^{91} -69305.5 q^{92} +66512.9 q^{93} -24761.8 q^{94} +61255.0 q^{96} -75016.9 q^{97} +343350. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9}+O(q^{10}) \) 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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