LMFDB - Embedded newform 825.6.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$7$
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} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 $ x^7 - x^6 - 209x^5 + 137x^4 + 12724x^3 - 1040x^2 - 218208*x - 8784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-10.8333$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.8333 q^{2} -9.00000 q^{3} +85.3606 q^{4} -97.4998 q^{6} +188.476 q^{7} +578.072 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+10.8333 q^{2} -9.00000 q^{3} +85.3606 q^{4} -97.4998 q^{6} +188.476 q^{7} +578.072 q^{8} +81.0000 q^{9} +121.000 q^{11} -768.246 q^{12} -1062.44 q^{13} +2041.82 q^{14} +3530.89 q^{16} +2033.15 q^{17} +877.498 q^{18} +285.586 q^{19} -1696.29 q^{21} +1310.83 q^{22} +589.341 q^{23} -5202.65 q^{24} -11509.7 q^{26} -729.000 q^{27} +16088.4 q^{28} +5682.63 q^{29} +1897.21 q^{31} +19753.0 q^{32} -1089.00 q^{33} +22025.8 q^{34} +6914.21 q^{36} -6513.03 q^{37} +3093.84 q^{38} +9561.93 q^{39} -17560.3 q^{41} -18376.4 q^{42} -8948.19 q^{43} +10328.6 q^{44} +6384.51 q^{46} +12072.6 q^{47} -31778.1 q^{48} +18716.2 q^{49} -18298.4 q^{51} -90690.2 q^{52} +7805.32 q^{53} -7897.48 q^{54} +108953. q^{56} -2570.28 q^{57} +61561.7 q^{58} +32104.2 q^{59} +360.190 q^{61} +20553.1 q^{62} +15266.6 q^{63} +101001. q^{64} -11797.5 q^{66} -40493.0 q^{67} +173551. q^{68} -5304.07 q^{69} +72765.1 q^{71} +46823.8 q^{72} -22289.1 q^{73} -70557.7 q^{74} +24377.8 q^{76} +22805.6 q^{77} +103587. q^{78} +51779.4 q^{79} +6561.00 q^{81} -190236. q^{82} -77834.1 q^{83} -144796. q^{84} -96938.5 q^{86} -51143.6 q^{87} +69946.7 q^{88} +48019.3 q^{89} -200244. q^{91} +50306.5 q^{92} -17074.9 q^{93} +130786. q^{94} -177777. q^{96} +115985. q^{97} +202759. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} - 63 q^{3} + 195 q^{4} + 9 q^{6} - 153 q^{8} + 567 q^{9}+O(q^{10}) $ 7 * q - q^2 - 63 * q^3 + 195 * q^4 + 9 * q^6 - 153 * q^8 + 567 * q^9	\( 7 q - q^{2} - 63 q^{3} + 195 q^{4} + 9 q^{6} - 153 q^{8} + 567 q^{9} + 847 q^{11} - 1755 q^{12} - 1418 q^{13} + 2548 q^{14} + 3699 q^{16} - 630 q^{17} - 81 q^{18} + 2572 q^{19} - 121 q^{22} - 536 q^{23} + 1377 q^{24} - 7626 q^{26} - 5103 q^{27} + 11368 q^{28} - 1038 q^{29} + 1872 q^{31} + 7523 q^{32} - 7623 q^{33} + 20790 q^{34} + 15795 q^{36} - 24298 q^{37} + 18952 q^{38} + 12762 q^{39} - 17658 q^{41} - 22932 q^{42} - 7244 q^{43} + 23595 q^{44} + 31016 q^{46} - 34560 q^{47} - 33291 q^{48} + 78735 q^{49} + 5670 q^{51} - 110222 q^{52} + 10214 q^{53} + 729 q^{54} + 81124 q^{56} - 23148 q^{57} + 5718 q^{58} + 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} + 1089 q^{66} - 64908 q^{67} + 136010 q^{68} + 4824 q^{69} + 61816 q^{71} - 12393 q^{72} + 11890 q^{73} - 124050 q^{74} - 47216 q^{76} + 68634 q^{78} + 18928 q^{79} + 45927 q^{81} - 36398 q^{82} - 17492 q^{83} - 102312 q^{84} - 216688 q^{86} + 9342 q^{87} - 18513 q^{88} + 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 16848 q^{93} - 30800 q^{94} - 67707 q^{96} + 172546 q^{97} + 615271 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q - q^2 - 63 * q^3 + 195 * q^4 + 9 * q^6 - 153 * q^8 + 567 * q^9 + 847 * q^11 - 1755 * q^12 - 1418 * q^13 + 2548 * q^14 + 3699 * q^16 - 630 * q^17 - 81 * q^18 + 2572 * q^19 - 121 * q^22 - 536 * q^23 + 1377 * q^24 - 7626 * q^26 - 5103 * q^27 + 11368 * q^28 - 1038 * q^29 + 1872 * q^31 + 7523 * q^32 - 7623 * q^33 + 20790 * q^34 + 15795 * q^36 - 24298 * q^37 + 18952 * q^38 + 12762 * q^39 - 17658 * q^41 - 22932 * q^42 - 7244 * q^43 + 23595 * q^44 + 31016 * q^46 - 34560 * q^47 - 33291 * q^48 + 78735 * q^49 + 5670 * q^51 - 110222 * q^52 + 10214 * q^53 + 729 * q^54 + 81124 * q^56 - 23148 * q^57 + 5718 * q^58 + 94676 * q^59 + 69538 * q^61 + 4208 * q^62 + 112339 * q^64 + 1089 * q^66 - 64908 * q^67 + 136010 * q^68 + 4824 * q^69 + 61816 * q^71 - 12393 * q^72 + 11890 * q^73 - 124050 * q^74 - 47216 * q^76 + 68634 * q^78 + 18928 * q^79 + 45927 * q^81 - 36398 * q^82 - 17492 * q^83 - 102312 * q^84 - 216688 * q^86 + 9342 * q^87 - 18513 * q^88 + 25302 * q^89 + 3392 * q^91 + 27408 * q^92 - 16848 * q^93 - 30800 * q^94 - 67707 * q^96 + 172546 * q^97 + 615271 * 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 $

$2$	10.8333	1.91508	0.957538	−	0.288306i	$-0.0930919\pi$
$2$	10.8333	1.91508	0.957538	+	0.288306i	$0.0930919\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	85.3606	2.66752
$5$	0	0
$5$	0	0
$6$	−97.4998	−1.10567
$7$	188.476	1.45382	0.726911	−	0.686732i	$-0.240955\pi$
$7$	188.476	1.45382	0.726911	+	0.686732i	$0.240955\pi$
$8$	578.072	3.19343
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−768.246	−1.54009
$13$	−1062.44	−1.74359	−0.871795	−	0.489870i	$-0.837044\pi$
$13$	−1062.44	−1.74359	−0.871795	+	0.489870i	$0.837044\pi$
$14$	2041.82	2.78418
$15$	0	0
$16$	3530.89	3.44814
$17$	2033.15	1.70627	0.853135	−	0.521690i	$-0.174698\pi$
$17$	2033.15	1.70627	0.853135	+	0.521690i	$0.174698\pi$
$18$	877.498	0.638359
$19$	285.586	0.181490	0.0907451	−	0.995874i	$-0.471075\pi$
$19$	285.586	0.181490	0.0907451	+	0.995874i	$0.471075\pi$
$20$	0	0
$21$	−1696.29	−0.839365
$22$	1310.83	0.577417
$23$	589.341	0.232299	0.116149	−	0.993232i	$-0.462945\pi$
$23$	589.341	0.232299	0.116149	+	0.993232i	$0.462945\pi$
$24$	−5202.65	−1.84373
$25$	0	0
$26$	−11509.7	−3.33911
$27$	−729.000	−0.192450
$28$	16088.4	3.87810
$29$	5682.63	1.25474	0.627371	−	0.778721i	$-0.284131\pi$
$29$	5682.63	1.25474	0.627371	+	0.778721i	$0.284131\pi$
$30$	0	0
$31$	1897.21	0.354577	0.177289	−	0.984159i	$-0.443267\pi$
$31$	1897.21	0.354577	0.177289	+	0.984159i	$0.443267\pi$
$32$	19753.0	3.41002
$33$	−1089.00	−0.174078
$34$	22025.8	3.26764
$35$	0	0
$36$	6914.21	0.889173
$37$	−6513.03	−0.782130	−0.391065	−	0.920363i	$-0.627893\pi$
$37$	−6513.03	−0.782130	−0.391065	+	0.920363i	$0.627893\pi$
$38$	3093.84	0.347568
$39$	9561.93	1.00666
$40$	0	0
$41$	−17560.3	−1.63144	−0.815721	−	0.578445i	$-0.803660\pi$
$41$	−17560.3	−1.63144	−0.815721	+	0.578445i	$0.803660\pi$
$42$	−18376.4	−1.60745
$43$	−8948.19	−0.738013	−0.369007	−	0.929427i	$-0.620302\pi$
$43$	−8948.19	−0.738013	−0.369007	+	0.929427i	$0.620302\pi$
$44$	10328.6	0.804287
$45$	0	0
$46$	6384.51	0.444870
$47$	12072.6	0.797176	0.398588	−	0.917130i	$-0.369500\pi$
$47$	12072.6	0.797176	0.398588	+	0.917130i	$0.369500\pi$
$48$	−31778.1	−1.99078
$49$	18716.2	1.11360
$50$	0	0
$51$	−18298.4	−0.985116
$52$	−90690.2	−4.65106
$53$	7805.32	0.381681	0.190841	−	0.981621i	$-0.438879\pi$
$53$	7805.32	0.381681	0.190841	+	0.981621i	$0.438879\pi$
$54$	−7897.48	−0.368557
$55$	0	0
$56$	108953.	4.64267
$57$	−2570.28	−0.104783
$58$	61561.7	2.40293
$59$	32104.2	1.20069	0.600347	−	0.799740i	$-0.295029\pi$
$59$	32104.2	1.20069	0.600347	+	0.799740i	$0.295029\pi$
$60$	0	0
$61$	360.190	0.0123939	0.00619693	−	0.999981i	$-0.498027\pi$
$61$	360.190	0.0123939	0.00619693	+	0.999981i	$0.498027\pi$
$62$	20553.1	0.679043
$63$	15266.6	0.484607
$64$	101001.	3.08232
$65$	0	0
$66$	−11797.5	−0.333372
$67$	−40493.0	−1.10203	−0.551015	−	0.834496i	$-0.685759\pi$
$67$	−40493.0	−1.10203	−0.551015	+	0.834496i	$0.685759\pi$
$68$	173551.	4.55151
$69$	−5304.07	−0.134118
$70$	0	0
$71$	72765.1	1.71308	0.856539	−	0.516082i	$-0.172610\pi$
$71$	72765.1	1.71308	0.856539	+	0.516082i	$0.172610\pi$
$72$	46823.8	1.06448
$73$	−22289.1	−0.489536	−0.244768	−	0.969582i	$-0.578712\pi$
$73$	−22289.1	−0.489536	−0.244768	+	0.969582i	$0.578712\pi$
$74$	−70557.7	−1.49784
$75$	0	0
$76$	24377.8	0.484129
$77$	22805.6	0.438344
$78$	103587.	1.92784
$79$	51779.4	0.933446	0.466723	−	0.884403i	$-0.345434\pi$
$79$	51779.4	0.933446	0.466723	+	0.884403i	$0.345434\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−190236.	−3.12434
$83$	−77834.1	−1.24015	−0.620076	−	0.784542i	$-0.712898\pi$
$83$	−77834.1	−1.24015	−0.620076	+	0.784542i	$0.712898\pi$
$84$	−144796.	−2.23902
$85$	0	0
$86$	−96938.5	−1.41335
$87$	−51143.6	−0.724425
$88$	69946.7	0.962855
$89$	48019.3	0.642601	0.321300	−	0.946977i	$-0.395880\pi$
$89$	48019.3	0.642601	0.321300	+	0.946977i	$0.395880\pi$
$90$	0	0
$91$	−200244.	−2.53487
$92$	50306.5	0.619661
$93$	−17074.9	−0.204715
$94$	130786.	1.52665
$95$	0	0
$96$	−177777.	−1.96878
$97$	115985.	1.25161	0.625807	−	0.779978i	$-0.284770\pi$
$97$	115985.	1.25161	0.625807	+	0.779978i	$0.284770\pi$
$98$	202759.	2.13263
$99$	9801.00	0.100504
$100$	0	0
$101$	125600.	1.22514	0.612569	−	0.790417i	$-0.290136\pi$
$101$	125600.	1.22514	0.612569	+	0.790417i	$0.290136\pi$
$102$	−198232.	−1.88657
$103$	−197051.	−1.83015	−0.915073	−	0.403288i	$-0.867867\pi$
$103$	−197051.	−1.83015	−0.915073	+	0.403288i	$0.867867\pi$
$104$	−614165.	−5.56803
$105$	0	0
$106$	84557.4	0.730949
$107$	−148145.	−1.25091	−0.625456	−	0.780259i	$-0.715087\pi$
$107$	−148145.	−1.25091	−0.625456	+	0.780259i	$0.715087\pi$
$108$	−62227.9	−0.513364
$109$	−100008.	−0.806246	−0.403123	−	0.915146i	$-0.632075\pi$
$109$	−100008.	−0.806246	−0.403123	+	0.915146i	$0.632075\pi$
$110$	0	0
$111$	58617.3	0.451563
$112$	665489.	5.01298
$113$	22869.8	0.168487	0.0842435	−	0.996445i	$-0.473153\pi$
$113$	22869.8	0.168487	0.0842435	+	0.996445i	$0.473153\pi$
$114$	−27844.6	−0.200668
$115$	0	0
$116$	485073.	3.34705
$117$	−86057.3	−0.581197
$118$	347795.	2.29942
$119$	383201.	2.48061
$120$	0	0
$121$	14641.0	0.0909091
$122$	3902.05	0.0237352
$123$	158042.	0.941914
$124$	161947.	0.945842
$125$	0	0
$126$	165387.	0.928060
$127$	−10903.4	−0.0599861	−0.0299931	−	0.999550i	$-0.509549\pi$
$127$	−10903.4	−0.0599861	−0.0299931	+	0.999550i	$0.509549\pi$
$128$	462085.	2.49285
$129$	80533.7	0.426092
$130$	0	0
$131$	75423.0	0.383995	0.191997	−	0.981395i	$-0.438503\pi$
$131$	75423.0	0.383995	0.191997	+	0.981395i	$0.438503\pi$
$132$	−92957.7	−0.464355
$133$	53826.2	0.263854
$134$	−438674.	−2.11047
$135$	0	0
$136$	1.17531e6	5.44885
$137$	371462.	1.69088	0.845440	−	0.534071i	$-0.179338\pi$
$137$	371462.	1.69088	0.845440	+	0.534071i	$0.179338\pi$
$138$	−57460.6	−0.256846
$139$	−7305.52	−0.0320711	−0.0160356	−	0.999871i	$-0.505104\pi$
$139$	−7305.52	−0.0320711	−0.0160356	+	0.999871i	$0.505104\pi$
$140$	0	0
$141$	−108653.	−0.460250
$142$	788287.	3.28068
$143$	−128555.	−0.525712
$144$	286002.	1.14938
$145$	0	0
$146$	−241464.	−0.937499
$147$	−168446.	−0.642936
$148$	−555956.	−2.08635
$149$	−116850.	−0.431185	−0.215593	−	0.976483i	$-0.569168\pi$
$149$	−116850.	−0.431185	−0.215593	+	0.976483i	$0.569168\pi$
$150$	0	0
$151$	165567.	0.590925	0.295462	−	0.955354i	$-0.404526\pi$
$151$	165567.	0.590925	0.295462	+	0.955354i	$0.404526\pi$
$152$	165089.	0.579576
$153$	164685.	0.568757
$154$	247060.	0.839462
$155$	0	0
$156$	816212.	2.68529
$157$	−305132.	−0.987958	−0.493979	−	0.869474i	$-0.664458\pi$
$157$	−305132.	−0.987958	−0.493979	+	0.869474i	$0.664458\pi$
$158$	560942.	1.78762
$159$	−70247.9	−0.220364
$160$	0	0
$161$	111077.	0.337721
$162$	71077.3	0.212786
$163$	446634.	1.31669	0.658344	−	0.752718i	$-0.271257\pi$
$163$	446634.	1.31669	0.658344	+	0.752718i	$0.271257\pi$
$164$	−1.49896e6	−4.35190
$165$	0	0
$166$	−843201.	−2.37499
$167$	−134567.	−0.373377	−0.186688	−	0.982419i	$-0.559775\pi$
$167$	−134567.	−0.373377	−0.186688	+	0.982419i	$0.559775\pi$
$168$	−980575.	−2.68045
$169$	757478.	2.04011
$170$	0	0
$171$	23132.5	0.0604967
$172$	−763823.	−1.96866
$173$	−559521.	−1.42135	−0.710676	−	0.703520i	$-0.751611\pi$
$173$	−559521.	−1.42135	−0.710676	+	0.703520i	$0.751611\pi$
$174$	−554055.	−1.38733
$175$	0	0
$176$	427238.	1.03965
$177$	−288938.	−0.693221
$178$	520208.	1.23063
$179$	123724.	0.288617	0.144309	−	0.989533i	$-0.453904\pi$
$179$	123724.	0.288617	0.144309	+	0.989533i	$0.453904\pi$
$180$	0	0
$181$	665605.	1.51015	0.755075	−	0.655638i	$-0.227600\pi$
$181$	665605.	1.51015	0.755075	+	0.655638i	$0.227600\pi$
$182$	−2.16930e6	−4.85447
$183$	−3241.71	−0.00715560
$184$	340681.	0.741829
$185$	0	0
$186$	−184977.	−0.392046
$187$	246012.	0.514460
$188$	1.03052e6	2.12648
$189$	−137399.	−0.279788
$190$	0	0
$191$	310886.	0.616621	0.308311	−	0.951286i	$-0.400236\pi$
$191$	310886.	0.616621	0.308311	+	0.951286i	$0.400236\pi$
$192$	−909013.	−1.77958
$193$	−76821.1	−0.148452	−0.0742262	−	0.997241i	$-0.523649\pi$
$193$	−76821.1	−0.148452	−0.0742262	+	0.997241i	$0.523649\pi$
$194$	1.25650e6	2.39694
$195$	0	0
$196$	1.59763e6	2.97054
$197$	670364.	1.23068	0.615340	−	0.788262i	$-0.289019\pi$
$197$	670364.	1.23068	0.615340	+	0.788262i	$0.289019\pi$
$198$	106177.	0.192472
$199$	−793545.	−1.42049	−0.710247	−	0.703953i	$-0.751417\pi$
$199$	−793545.	−1.42049	−0.710247	+	0.703953i	$0.751417\pi$
$200$	0	0
$201$	364437.	0.636257
$202$	1.36066e6	2.34623
$203$	1.07104e6	1.82417
$204$	−1.56196e6	−2.62781
$205$	0	0
$206$	−2.13472e6	−3.50487
$207$	47736.6	0.0774329
$208$	−3.75135e6	−6.01214
$209$	34555.9	0.0547214
$210$	0	0
$211$	−106650.	−0.164913	−0.0824563	−	0.996595i	$-0.526276\pi$
$211$	−106650.	−0.164913	−0.0824563	+	0.996595i	$0.526276\pi$
$212$	666267.	1.01814
$213$	−654886.	−0.989046
$214$	−1.60490e6	−2.39559
$215$	0	0
$216$	−421415.	−0.614575
$217$	357579.	0.515492
$218$	−1.08342e6	−1.54402
$219$	200602.	0.282634
$220$	0	0
$221$	−2.16010e6	−2.97504
$222$	635019.	0.864778
$223$	832483.	1.12102	0.560510	−	0.828148i	$-0.310605\pi$
$223$	832483.	1.12102	0.560510	+	0.828148i	$0.310605\pi$
$224$	3.72296e6	4.95757
$225$	0	0
$226$	247756.	0.322666
$227$	−1.12488e6	−1.44891	−0.724455	−	0.689322i	$-0.757909\pi$
$227$	−1.12488e6	−1.44891	−0.724455	+	0.689322i	$0.757909\pi$
$228$	−219400.	−0.279512
$229$	−1.28251e6	−1.61611	−0.808057	−	0.589104i	$-0.799481\pi$
$229$	−1.28251e6	−1.61611	−0.808057	+	0.589104i	$0.799481\pi$
$230$	0	0
$231$	−205250.	−0.253078
$232$	3.28497e6	4.00692
$233$	−482579.	−0.582343	−0.291171	−	0.956671i	$-0.594045\pi$
$233$	−482579.	−0.582343	−0.291171	+	0.956671i	$0.594045\pi$
$234$	−932286.	−1.11304
$235$	0	0
$236$	2.74044e6	3.20287
$237$	−466015.	−0.538926
$238$	4.15133e6	4.75056
$239$	218277.	0.247180	0.123590	−	0.992333i	$-0.460559\pi$
$239$	218277.	0.247180	0.123590	+	0.992333i	$0.460559\pi$
$240$	0	0
$241$	673119.	0.746533	0.373267	−	0.927724i	$-0.378238\pi$
$241$	673119.	0.746533	0.373267	+	0.927724i	$0.378238\pi$
$242$	158610.	0.174098
$243$	−59049.0	−0.0641500
$244$	30746.0	0.0330609
$245$	0	0
$246$	1.71212e6	1.80384
$247$	−303417.	−0.316445
$248$	1.09672e6	1.13232
$249$	700507.	0.716002
$250$	0	0
$251$	−924557.	−0.926295	−0.463148	−	0.886281i	$-0.653280\pi$
$251$	−924557.	−0.926295	−0.463148	+	0.886281i	$0.653280\pi$
$252$	1.30316e6	1.29270
$253$	71310.2	0.0700407
$254$	−118119.	−0.114878
$255$	0	0
$256$	1.77386e6	1.69169
$257$	471931.	0.445704	0.222852	−	0.974852i	$-0.428463\pi$
$257$	471931.	0.445704	0.222852	+	0.974852i	$0.428463\pi$
$258$	872447.	0.815999
$259$	−1.22755e6	−1.13708
$260$	0	0
$261$	460293.	0.418247
$262$	817081.	0.735380
$263$	−869418.	−0.775067	−0.387534	−	0.921856i	$-0.626673\pi$
$263$	−869418.	−0.775067	−0.387534	+	0.921856i	$0.626673\pi$
$264$	−629521.	−0.555904
$265$	0	0
$266$	583116.	0.505302
$267$	−432174.	−0.371006
$268$	−3.45651e6	−2.93968
$269$	−460155.	−0.387725	−0.193862	−	0.981029i	$-0.562102\pi$
$269$	−460155.	−0.387725	−0.193862	+	0.981029i	$0.562102\pi$
$270$	0	0
$271$	−488213.	−0.403818	−0.201909	−	0.979404i	$-0.564715\pi$
$271$	−488213.	−0.403818	−0.201909	+	0.979404i	$0.564715\pi$
$272$	7.17885e6	5.88346
$273$	1.80219e6	1.46351
$274$	4.02416e6	3.23816
$275$	0	0
$276$	−452758.	−0.357762
$277$	1.94033e6	1.51942	0.759708	−	0.650264i	$-0.225342\pi$
$277$	1.94033e6	1.51942	0.759708	+	0.650264i	$0.225342\pi$
$278$	−79142.9	−0.0614186
$279$	153674.	0.118192
$280$	0	0
$281$	−1.76289e6	−1.33186	−0.665930	−	0.746015i	$-0.731965\pi$
$281$	−1.76289e6	−1.33186	−0.665930	+	0.746015i	$0.731965\pi$
$282$	−1.17707e6	−0.881414
$283$	461211.	0.342321	0.171160	−	0.985243i	$-0.445248\pi$
$283$	461211.	0.342321	0.171160	+	0.985243i	$0.445248\pi$
$284$	6.21127e6	4.56967
$285$	0	0
$286$	−1.39267e6	−1.00678
$287$	−3.30969e6	−2.37183
$288$	1.59999e6	1.13667
$289$	2.71386e6	1.91136
$290$	0	0
$291$	−1.04386e6	−0.722620
$292$	−1.90261e6	−1.30585
$293$	−1.72981e6	−1.17715	−0.588573	−	0.808444i	$-0.700310\pi$
$293$	−1.72981e6	−1.17715	−0.588573	+	0.808444i	$0.700310\pi$
$294$	−1.82483e6	−1.23127
$295$	0	0
$296$	−3.76500e6	−2.49768
$297$	−88209.0	−0.0580259
$298$	−1.26588e6	−0.825753
$299$	−626137.	−0.405034
$300$	0	0
$301$	−1.68652e6	−1.07294
$302$	1.79364e6	1.13167
$303$	−1.13040e6	−0.707334
$304$	1.00837e6	0.625804
$305$	0	0
$306$	1.78409e6	1.08921
$307$	−502403.	−0.304233	−0.152117	−	0.988363i	$-0.548609\pi$
$307$	−502403.	−0.304233	−0.152117	+	0.988363i	$0.548609\pi$
$308$	1.94670e6	1.16929
$309$	1.77346e6	1.05664
$310$	0	0
$311$	−1.84484e6	−1.08158	−0.540789	−	0.841158i	$-0.681874\pi$
$311$	−1.84484e6	−1.08158	−0.540789	+	0.841158i	$0.681874\pi$
$312$	5.52748e6	3.21470
$313$	−2.25237e6	−1.29951	−0.649753	−	0.760146i	$-0.725128\pi$
$313$	−2.25237e6	−1.29951	−0.649753	+	0.760146i	$0.725128\pi$
$314$	−3.30559e6	−1.89202
$315$	0	0
$316$	4.41992e6	2.48999
$317$	−1.22564e6	−0.685039	−0.342520	−	0.939511i	$-0.611280\pi$
$317$	−1.22564e6	−0.685039	−0.342520	+	0.939511i	$0.611280\pi$
$318$	−761017.	−0.422014
$319$	687598.	0.378319
$320$	0	0
$321$	1.33330e6	0.722214
$322$	1.20333e6	0.646762
$323$	580640.	0.309671
$324$	560051.	0.296391
$325$	0	0
$326$	4.83852e6	2.52156
$327$	900070.	0.465487
$328$	−1.01511e7	−5.20989
$329$	2.27539e6	1.15895
$330$	0	0
$331$	−841537.	−0.422185	−0.211093	−	0.977466i	$-0.567702\pi$
$331$	−841537.	−0.422185	−0.211093	+	0.977466i	$0.567702\pi$
$332$	−6.64397e6	−3.30813
$333$	−527556.	−0.260710
$334$	−1.45781e6	−0.715045
$335$	0	0
$336$	−5.98940e6	−2.89425
$337$	−1.88515e6	−0.904214	−0.452107	−	0.891964i	$-0.649327\pi$
$337$	−1.88515e6	−0.904214	−0.452107	+	0.891964i	$0.649327\pi$
$338$	8.20599e6	3.90696
$339$	−205828.	−0.0972760
$340$	0	0
$341$	229562.	0.106909
$342$	250601.	0.115856
$343$	359847.	0.165152
$344$	−5.17270e6	−2.35679
$345$	0	0
$346$	−6.06147e6	−2.72200
$347$	1.60117e6	0.713860	0.356930	−	0.934131i	$-0.383823\pi$
$347$	1.60117e6	0.713860	0.356930	+	0.934131i	$0.383823\pi$
$348$	−4.36565e6	−1.93242
$349$	−4.20595e6	−1.84842	−0.924209	−	0.381887i	$-0.875274\pi$
$349$	−4.20595e6	−1.84842	−0.924209	+	0.381887i	$0.875274\pi$
$350$	0	0
$351$	774516.	0.335554
$352$	2.39011e6	1.02816
$353$	3.54610e6	1.51466	0.757328	−	0.653035i	$-0.226505\pi$
$353$	3.54610e6	1.51466	0.757328	+	0.653035i	$0.226505\pi$
$354$	−3.13015e6	−1.32757
$355$	0	0
$356$	4.09896e6	1.71415
$357$	−3.44881e6	−1.43218
$358$	1.34034e6	0.552724
$359$	−3.28782e6	−1.34639	−0.673197	−	0.739463i	$-0.735080\pi$
$359$	−3.28782e6	−1.34639	−0.673197	+	0.739463i	$0.735080\pi$
$360$	0	0
$361$	−2.39454e6	−0.967061
$362$	7.21071e6	2.89205
$363$	−131769.	−0.0524864
$364$	−1.70929e7	−6.76181
$365$	0	0
$366$	−35118.4	−0.0137035
$367$	−2.05002e6	−0.794499	−0.397250	−	0.917711i	$-0.630035\pi$
$367$	−2.05002e6	−0.794499	−0.397250	+	0.917711i	$0.630035\pi$
$368$	2.08090e6	0.800998
$369$	−1.42238e6	−0.543814
$370$	0	0
$371$	1.47112e6	0.554897
$372$	−1.45752e6	−0.546082
$373$	−390699.	−0.145402	−0.0727010	−	0.997354i	$-0.523162\pi$
$373$	−390699.	−0.145402	−0.0727010	+	0.997354i	$0.523162\pi$
$374$	2.66512e6	0.985230
$375$	0	0
$376$	6.97881e6	2.54573
$377$	−6.03743e6	−2.18775
$378$	−1.48849e6	−0.535816
$379$	−1.42019e6	−0.507864	−0.253932	−	0.967222i	$-0.581724\pi$
$379$	−1.42019e6	−0.507864	−0.253932	+	0.967222i	$0.581724\pi$
$380$	0	0
$381$	98130.2	0.0346330
$382$	3.36793e6	1.18088
$383$	−2.23045e6	−0.776953	−0.388476	−	0.921459i	$-0.626998\pi$
$383$	−2.23045e6	−0.776953	−0.388476	+	0.921459i	$0.626998\pi$
$384$	−4.15876e6	−1.43925
$385$	0	0
$386$	−832226.	−0.284298
$387$	−724803.	−0.246004
$388$	9.90051e6	3.33871
$389$	−1.57882e6	−0.529005	−0.264502	−	0.964385i	$-0.585208\pi$
$389$	−1.57882e6	−0.529005	−0.264502	+	0.964385i	$0.585208\pi$
$390$	0	0
$391$	1.19822e6	0.396364
$392$	1.08193e7	3.55620
$393$	−678807.	−0.221700
$394$	7.26226e6	2.35685
$395$	0	0
$396$	836619.	0.268096
$397$	1.46256e6	0.465734	0.232867	−	0.972509i	$-0.425189\pi$
$397$	1.46256e6	0.465734	0.232867	+	0.972509i	$0.425189\pi$
$398$	−8.59672e6	−2.72035
$399$	−484436.	−0.152336
$400$	0	0
$401$	2.26635e6	0.703828	0.351914	−	0.936032i	$-0.385531\pi$
$401$	2.26635e6	0.703828	0.351914	+	0.936032i	$0.385531\pi$
$402$	3.94806e6	1.21848
$403$	−2.01566e6	−0.618238
$404$	1.07213e7	3.26808
$405$	0	0
$406$	1.16029e7	3.49343
$407$	−788077.	−0.235821
$408$	−1.05778e7	−3.14590
$409$	−5.88429e6	−1.73935	−0.869673	−	0.493628i	$-0.835670\pi$
$409$	−5.88429e6	−1.73935	−0.869673	+	0.493628i	$0.835670\pi$
$410$	0	0
$411$	−3.34316e6	−0.976230
$412$	−1.68204e7	−4.88195
$413$	6.05088e6	1.74559
$414$	517145.	0.148290
$415$	0	0
$416$	−2.09863e7	−5.94569
$417$	65749.6	0.0185163
$418$	374355.	0.104796
$419$	−3.04647e6	−0.847737	−0.423869	−	0.905724i	$-0.639328\pi$
$419$	−3.04647e6	−0.847737	−0.423869	+	0.905724i	$0.639328\pi$
$420$	0	0
$421$	2.55503e6	0.702571	0.351286	−	0.936268i	$-0.385745\pi$
$421$	2.55503e6	0.702571	0.351286	+	0.936268i	$0.385745\pi$
$422$	−1.15537e6	−0.315820
$423$	977877.	0.265725
$424$	4.51204e6	1.21887
$425$	0	0
$426$	−7.09458e6	−1.89410
$427$	67887.2	0.0180185
$428$	−1.26457e7	−3.33683
$429$	1.15699e6	0.303520
$430$	0	0
$431$	−2.99185e6	−0.775794	−0.387897	−	0.921703i	$-0.626798\pi$
$431$	−2.99185e6	−0.775794	−0.387897	+	0.921703i	$0.626798\pi$
$432$	−2.57402e6	−0.663595
$433$	−1.22139e6	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$433$	−1.22139e6	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$434$	3.87376e6	0.987207
$435$	0	0
$436$	−8.53673e6	−2.15068
$437$	168308.	0.0421599
$438$	2.17318e6	0.541265
$439$	−2.06150e6	−0.510530	−0.255265	−	0.966871i	$-0.582163\pi$
$439$	−2.06150e6	−0.510530	−0.255265	+	0.966871i	$0.582163\pi$
$440$	0	0
$441$	1.51602e6	0.371199
$442$	−2.34010e7	−5.69742
$443$	1.37260e6	0.332303	0.166151	−	0.986100i	$-0.446866\pi$
$443$	1.37260e6	0.332303	0.166151	+	0.986100i	$0.446866\pi$
$444$	5.00361e6	1.20455
$445$	0	0
$446$	9.01855e6	2.14684
$447$	1.05165e6	0.248945
$448$	1.90364e7	4.48114
$449$	1.54405e6	0.361448	0.180724	−	0.983534i	$-0.442156\pi$
$449$	1.54405e6	0.361448	0.180724	+	0.983534i	$0.442156\pi$
$450$	0	0
$451$	−2.12479e6	−0.491898
$452$	1.95218e6	0.449442
$453$	−1.49011e6	−0.341171
$454$	−1.21862e7	−2.77477
$455$	0	0
$456$	−1.48580e6	−0.334618
$457$	−5.05440e6	−1.13208	−0.566042	−	0.824376i	$-0.691526\pi$
$457$	−5.05440e6	−1.13208	−0.566042	+	0.824376i	$0.691526\pi$
$458$	−1.38938e7	−3.09498
$459$	−1.48217e6	−0.328372
$460$	0	0
$461$	−1.79144e6	−0.392601	−0.196300	−	0.980544i	$-0.562893\pi$
$461$	−1.79144e6	−0.392601	−0.196300	+	0.980544i	$0.562893\pi$
$462$	−2.22354e6	−0.484664
$463$	641130.	0.138993	0.0694966	−	0.997582i	$-0.477861\pi$
$463$	641130.	0.138993	0.0694966	+	0.997582i	$0.477861\pi$
$464$	2.00648e7	4.32652
$465$	0	0
$466$	−5.22793e6	−1.11523
$467$	−6.65471e6	−1.41201	−0.706004	−	0.708208i	$-0.749504\pi$
$467$	−6.65471e6	−1.41201	−0.706004	+	0.708208i	$0.749504\pi$
$468$	−7.34591e6	−1.55035
$469$	−7.63197e6	−1.60215
$470$	0	0
$471$	2.74619e6	0.570398
$472$	1.85586e7	3.83433
$473$	−1.08273e6	−0.222519
$474$	−5.04848e6	−1.03208
$475$	0	0
$476$	3.27103e7	6.61708
$477$	632231.	0.127227
$478$	2.36466e6	0.473368
$479$	−260876.	−0.0519511	−0.0259755	−	0.999663i	$-0.508269\pi$
$479$	−260876.	−0.0519511	−0.0259755	+	0.999663i	$0.508269\pi$
$480$	0	0
$481$	6.91968e6	1.36371
$482$	7.29210e6	1.42967
$483$	−999690.	−0.194983
$484$	1.24976e6	0.242502
$485$	0	0
$486$	−639696.	−0.122852
$487$	1.01034e6	0.193038	0.0965191	−	0.995331i	$-0.469229\pi$
$487$	1.01034e6	0.193038	0.0965191	+	0.995331i	$0.469229\pi$
$488$	208216.	0.0395789
$489$	−4.01970e6	−0.760190
$490$	0	0
$491$	259876.	0.0486477	0.0243239	−	0.999704i	$-0.492257\pi$
$491$	259876.	0.0486477	0.0243239	+	0.999704i	$0.492257\pi$
$492$	1.34906e7	2.51257
$493$	1.15537e7	2.14093
$494$	−3.28701e6	−0.606016
$495$	0	0
$496$	6.69885e6	1.22263
$497$	1.37145e7	2.49051
$498$	7.58881e6	1.37120
$499$	−454122.	−0.0816434	−0.0408217	−	0.999166i	$-0.512998\pi$
$499$	−454122.	−0.0816434	−0.0408217	+	0.999166i	$0.512998\pi$
$500$	0	0
$501$	1.21110e6	0.215569
$502$	−1.00160e7	−1.77393
$503$	−4.20300e6	−0.740695	−0.370347	−	0.928893i	$-0.620761\pi$
$503$	−4.20300e6	−0.740695	−0.370347	+	0.928893i	$0.620761\pi$
$504$	8.82518e6	1.54756
$505$	0	0
$506$	772526.	0.134133
$507$	−6.81730e6	−1.17786
$508$	−930717.	−0.160014
$509$	4.32091e6	0.739231	0.369616	−	0.929185i	$-0.379489\pi$
$509$	4.32091e6	0.739231	0.369616	+	0.929185i	$0.379489\pi$
$510$	0	0
$511$	−4.20096e6	−0.711698
$512$	4.43009e6	0.746858
$513$	−208192.	−0.0349278
$514$	5.11258e6	0.853557
$515$	0	0
$516$	6.87441e6	1.13661
$517$	1.46078e6	0.240358
$518$	−1.32984e7	−2.17759
$519$	5.03569e6	0.820618
$520$	0	0
$521$	9.81457e6	1.58408	0.792040	−	0.610469i	$-0.209019\pi$
$521$	9.81457e6	1.58408	0.792040	+	0.610469i	$0.209019\pi$
$522$	4.98649e6	0.800975
$523$	−9.06730e6	−1.44952	−0.724759	−	0.689002i	$-0.758049\pi$
$523$	−9.06730e6	−1.44952	−0.724759	+	0.689002i	$0.758049\pi$
$524$	6.43815e6	1.02431
$525$	0	0
$526$	−9.41868e6	−1.48431
$527$	3.85732e6	0.605005
$528$	−3.84514e6	−0.600244
$529$	−6.08902e6	−0.946037
$530$	0	0
$531$	2.60044e6	0.400231
$532$	4.59464e6	0.703837
$533$	1.86567e7	2.84457
$534$	−4.68188e6	−0.710504
$535$	0	0
$536$	−2.34079e7	−3.51925
$537$	−1.11352e6	−0.166633
$538$	−4.98500e6	−0.742523
$539$	2.26467e6	0.335763
$540$	0	0
$541$	−2.82663e6	−0.415217	−0.207609	−	0.978212i	$-0.566568\pi$
$541$	−2.82663e6	−0.415217	−0.207609	+	0.978212i	$0.566568\pi$
$542$	−5.28896e6	−0.773343
$543$	−5.99045e6	−0.871886
$544$	4.01608e7	5.81842
$545$	0	0
$546$	1.95237e7	2.80273
$547$	4.57486e6	0.653746	0.326873	−	0.945068i	$-0.394005\pi$
$547$	4.57486e6	0.653746	0.326873	+	0.945068i	$0.394005\pi$
$548$	3.17082e7	4.51045
$549$	29175.4	0.00413129
$550$	0	0
$551$	1.62288e6	0.227723
$552$	−3.06613e6	−0.428295
$553$	9.75918e6	1.35706
$554$	2.10202e7	2.90980
$555$	0	0
$556$	−623603.	−0.0855503
$557$	1.10502e6	0.150915	0.0754573	−	0.997149i	$-0.475958\pi$
$557$	1.10502e6	0.150915	0.0754573	+	0.997149i	$0.475958\pi$
$558$	1.66480e6	0.226348
$559$	9.50688e6	1.28679
$560$	0	0
$561$	−2.21410e6	−0.297024
$562$	−1.90979e7	−2.55061
$563$	−650286.	−0.0864637	−0.0432318	−	0.999065i	$-0.513765\pi$
$563$	−650286.	−0.0864637	−0.0432318	+	0.999065i	$0.513765\pi$
$564$	−9.27468e6	−1.22773
$565$	0	0
$566$	4.99644e6	0.655571
$567$	1.23659e6	0.161536
$568$	4.20635e7	5.47059
$569$	1.07156e7	1.38751	0.693754	−	0.720212i	$-0.255956\pi$
$569$	1.07156e7	1.38751	0.693754	+	0.720212i	$0.255956\pi$
$570$	0	0
$571$	1.51877e7	1.94941	0.974703	−	0.223506i	$-0.0717503\pi$
$571$	1.51877e7	1.94941	0.974703	+	0.223506i	$0.0717503\pi$
$572$	−1.09735e7	−1.40235
$573$	−2.79798e6	−0.356006
$574$	−3.58549e7	−4.54223
$575$	0	0
$576$	8.18112e6	1.02744
$577$	3.85869e6	0.482503	0.241251	−	0.970463i	$-0.422442\pi$
$577$	3.85869e6	0.482503	0.241251	+	0.970463i	$0.422442\pi$
$578$	2.94000e7	3.66040
$579$	691390.	0.0857090
$580$	0	0
$581$	−1.46699e7	−1.80296
$582$	−1.13085e7	−1.38387
$583$	944444.	0.115081
$584$	−1.28847e7	−1.56330
$585$	0	0
$586$	−1.87396e7	−2.25432
$587$	−1.52399e7	−1.82552	−0.912758	−	0.408501i	$-0.866052\pi$
$587$	−1.52399e7	−1.82552	−0.912758	+	0.408501i	$0.866052\pi$
$588$	−1.43787e7	−1.71504
$589$	541817.	0.0643523
$590$	0	0
$591$	−6.03328e6	−0.710533
$592$	−2.29968e7	−2.69689
$593$	7.49470e6	0.875221	0.437610	−	0.899165i	$-0.355825\pi$
$593$	7.49470e6	0.875221	0.437610	+	0.899165i	$0.355825\pi$
$594$	−955595.	−0.111124
$595$	0	0
$596$	−9.97441e6	−1.15020
$597$	7.14191e6	0.820122
$598$	−6.78313e6	−0.775671
$599$	−2.30188e6	−0.262129	−0.131064	−	0.991374i	$-0.541839\pi$
$599$	−2.30188e6	−0.262129	−0.131064	+	0.991374i	$0.541839\pi$
$600$	0	0
$601$	8.48232e6	0.957918	0.478959	−	0.877837i	$-0.341014\pi$
$601$	8.48232e6	0.957918	0.478959	+	0.877837i	$0.341014\pi$
$602$	−1.82706e7	−2.05476
$603$	−3.27994e6	−0.367343
$604$	1.41329e7	1.57630
$605$	0	0
$606$	−1.22459e7	−1.35460
$607$	8.51451e6	0.937968	0.468984	−	0.883207i	$-0.344620\pi$
$607$	8.51451e6	0.937968	0.468984	+	0.883207i	$0.344620\pi$
$608$	5.64118e6	0.618886
$609$	−9.63936e6	−1.05319
$610$	0	0
$611$	−1.28263e7	−1.38995
$612$	1.40576e7	1.51717
$613$	5.62250e6	0.604336	0.302168	−	0.953255i	$-0.402290\pi$
$613$	5.62250e6	0.604336	0.302168	+	0.953255i	$0.402290\pi$
$614$	−5.44269e6	−0.582630
$615$	0	0
$616$	1.31833e7	1.39982
$617$	−754007.	−0.0797375	−0.0398687	−	0.999205i	$-0.512694\pi$
$617$	−754007.	−0.0797375	−0.0398687	+	0.999205i	$0.512694\pi$
$618$	1.92124e7	2.02354
$619$	1.08312e7	1.13619	0.568095	−	0.822963i	$-0.307681\pi$
$619$	1.08312e7	1.13619	0.568095	+	0.822963i	$0.307681\pi$
$620$	0	0
$621$	−429629.	−0.0447059
$622$	−1.99857e7	−2.07130
$623$	9.05050e6	0.934227
$624$	3.37622e7	3.47111
$625$	0	0
$626$	−2.44006e7	−2.48865
$627$	−311003.	−0.0315934
$628$	−2.60462e7	−2.63540
$629$	−1.32420e7	−1.33453
$630$	0	0
$631$	−1.23176e7	−1.23156	−0.615778	−	0.787920i	$-0.711158\pi$
$631$	−1.23176e7	−1.23156	−0.615778	+	0.787920i	$0.711158\pi$
$632$	2.99322e7	2.98089
$633$	959848.	0.0952123
$634$	−1.32778e7	−1.31190
$635$	0	0
$636$	−5.99640e6	−0.587825
$637$	−1.98848e7	−1.94166
$638$	7.44896e6	0.724509
$639$	5.89397e6	0.571026
$640$	0	0
$641$	−1.56214e7	−1.50167	−0.750835	−	0.660490i	$-0.770348\pi$
$641$	−1.56214e7	−1.50167	−0.750835	+	0.660490i	$0.770348\pi$
$642$	1.44441e7	1.38310
$643$	9.36087e6	0.892871	0.446435	−	0.894816i	$-0.352693\pi$
$643$	9.36087e6	0.892871	0.446435	+	0.894816i	$0.352693\pi$
$644$	9.48157e6	0.900877
$645$	0	0
$646$	6.29026e6	0.593044
$647$	−3.69425e6	−0.346949	−0.173474	−	0.984838i	$-0.555499\pi$
$647$	−3.69425e6	−0.346949	−0.173474	+	0.984838i	$0.555499\pi$
$648$	3.79273e6	0.354825
$649$	3.88461e6	0.362023
$650$	0	0
$651$	−3.21821e6	−0.297620
$652$	3.81249e7	3.51229
$653$	1.75496e7	1.61059	0.805294	−	0.592875i	$-0.202007\pi$
$653$	1.75496e7	1.61059	0.805294	+	0.592875i	$0.202007\pi$
$654$	9.75074e6	0.891443
$655$	0	0
$656$	−6.20035e7	−5.62544
$657$	−1.80541e6	−0.163179
$658$	2.46500e7	2.21948
$659$	−1.36928e7	−1.22822	−0.614112	−	0.789219i	$-0.710486\pi$
$659$	−1.36928e7	−1.22822	−0.614112	+	0.789219i	$0.710486\pi$
$660$	0	0
$661$	−9.06556e6	−0.807032	−0.403516	−	0.914973i	$-0.632212\pi$
$661$	−9.06556e6	−0.807032	−0.403516	+	0.914973i	$0.632212\pi$
$662$	−9.11663e6	−0.808518
$663$	1.94409e7	1.71764
$664$	−4.49937e7	−3.96034
$665$	0	0
$666$	−5.71517e6	−0.499280
$667$	3.34900e6	0.291475
$668$	−1.14867e7	−0.995990
$669$	−7.49235e6	−0.647221
$670$	0	0
$671$	43583.0	0.00373689
$672$	−3.35067e7	−2.86225
$673$	−4.40527e6	−0.374917	−0.187458	−	0.982273i	$-0.560025\pi$
$673$	−4.40527e6	−0.374917	−0.187458	+	0.982273i	$0.560025\pi$
$674$	−2.04224e7	−1.73164
$675$	0	0
$676$	6.46588e7	5.44203
$677$	−1.84727e7	−1.54903	−0.774514	−	0.632557i	$-0.782005\pi$
$677$	−1.84727e7	−1.54903	−0.774514	+	0.632557i	$0.782005\pi$
$678$	−2.22980e6	−0.186291
$679$	2.18603e7	1.81963
$680$	0	0
$681$	1.01239e7	0.836529
$682$	2.48692e6	0.204739
$683$	−8.03048e6	−0.658703	−0.329352	−	0.944207i	$-0.606830\pi$
$683$	−8.03048e6	−0.658703	−0.329352	+	0.944207i	$0.606830\pi$
$684$	1.97460e6	0.161376
$685$	0	0
$686$	3.89834e6	0.316278
$687$	1.15426e7	0.933064
$688$	−3.15951e7	−2.54477
$689$	−8.29265e6	−0.665496
$690$	0	0
$691$	1.59119e7	1.26773	0.633864	−	0.773444i	$-0.281468\pi$
$691$	1.59119e7	1.26773	0.633864	+	0.773444i	$0.281468\pi$
$692$	−4.77611e7	−3.79148
$693$	1.84725e6	0.146115
$694$	1.73459e7	1.36710
$695$	0	0
$696$	−2.95647e7	−2.31340
$697$	−3.57027e7	−2.78368
$698$	−4.55643e7	−3.53986
$699$	4.34321e6	0.336216
$700$	0	0
$701$	−5.66506e6	−0.435421	−0.217710	−	0.976013i	$-0.569859\pi$
$701$	−5.66506e6	−0.435421	−0.217710	+	0.976013i	$0.569859\pi$
$702$	8.39057e6	0.642612
$703$	−1.86003e6	−0.141949
$704$	1.22212e7	0.929354
$705$	0	0
$706$	3.84160e7	2.90068
$707$	2.36725e7	1.78113
$708$	−2.46639e7	−1.84918
$709$	1.42504e7	1.06466	0.532332	−	0.846535i	$-0.321316\pi$
$709$	1.42504e7	1.06466	0.532332	+	0.846535i	$0.321316\pi$
$710$	0	0
$711$	4.19413e6	0.311149
$712$	2.77586e7	2.05210
$713$	1.11810e6	0.0823678
$714$	−3.73620e7	−2.74274
$715$	0	0
$716$	1.05612e7	0.769892
$717$	−1.96449e6	−0.142709
$718$	−3.56180e7	−2.57845
$719$	9.73583e6	0.702345	0.351173	−	0.936311i	$-0.385783\pi$
$719$	9.73583e6	0.702345	0.351173	+	0.936311i	$0.385783\pi$
$720$	0	0
$721$	−3.71394e7	−2.66071
$722$	−2.59408e7	−1.85200
$723$	−6.05807e6	−0.431011
$724$	5.68165e7	4.02836
$725$	0	0
$726$	−1.42749e6	−0.100515
$727$	1.67013e7	1.17196	0.585982	−	0.810324i	$-0.300709\pi$
$727$	1.67013e7	1.17196	0.585982	+	0.810324i	$0.300709\pi$
$728$	−1.15755e8	−8.09492
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.81930e7	−1.25925
$732$	−276714.	−0.0190877
$733$	−6.25234e6	−0.429816	−0.214908	−	0.976634i	$-0.568945\pi$
$733$	−6.25234e6	−0.429816	−0.214908	+	0.976634i	$0.568945\pi$
$734$	−2.22085e7	−1.52153
$735$	0	0
$736$	1.16412e7	0.792144
$737$	−4.89966e6	−0.332274
$738$	−1.54091e7	−1.04145
$739$	−5.71072e6	−0.384662	−0.192331	−	0.981330i	$-0.561605\pi$
$739$	−5.71072e6	−0.384662	−0.192331	+	0.981330i	$0.561605\pi$
$740$	0	0
$741$	2.73075e6	0.182699
$742$	1.59371e7	1.06267
$743$	−1.76929e7	−1.17578	−0.587892	−	0.808940i	$-0.700042\pi$
$743$	−1.76929e7	−1.17578	−0.587892	+	0.808940i	$0.700042\pi$
$744$	−9.87051e6	−0.653743
$745$	0	0
$746$	−4.23257e6	−0.278456
$747$	−6.30457e6	−0.413384
$748$	2.09997e7	1.37233
$749$	−2.79217e7	−1.81860
$750$	0	0
$751$	8.42479e6	0.545078	0.272539	−	0.962145i	$-0.412137\pi$
$751$	8.42479e6	0.545078	0.272539	+	0.962145i	$0.412137\pi$
$752$	4.26269e7	2.74878
$753$	8.32102e6	0.534797
$754$	−6.54053e7	−4.18972
$755$	0	0
$756$	−1.17285e7	−0.746340
$757$	1.85996e7	1.17968	0.589840	−	0.807520i	$-0.299191\pi$
$757$	1.85996e7	1.17968	0.589840	+	0.807520i	$0.299191\pi$
$758$	−1.53853e7	−0.972599
$759$	−641792.	−0.0404380
$760$	0	0
$761$	2.84836e7	1.78292	0.891462	−	0.453096i	$-0.149681\pi$
$761$	2.84836e7	1.78292	0.891462	+	0.453096i	$0.149681\pi$
$762$	1.06308e6	0.0663249
$763$	−1.88491e7	−1.17214
$764$	2.65375e7	1.64485
$765$	0	0
$766$	−2.41631e7	−1.48792
$767$	−3.41087e7	−2.09352
$768$	−1.59648e7	−0.976696
$769$	−7.45086e6	−0.454350	−0.227175	−	0.973854i	$-0.572949\pi$
$769$	−7.45086e6	−0.454350	−0.227175	+	0.973854i	$0.572949\pi$
$770$	0	0
$771$	−4.24738e6	−0.257327
$772$	−6.55749e6	−0.395999
$773$	5.32037e6	0.320253	0.160126	−	0.987097i	$-0.448810\pi$
$773$	5.32037e6	0.320253	0.160126	+	0.987097i	$0.448810\pi$
$774$	−7.85202e6	−0.471117
$775$	0	0
$776$	6.70474e7	3.99694
$777$	1.10480e7	0.656492
$778$	−1.71039e7	−1.01308
$779$	−5.01497e6	−0.296091
$780$	0	0
$781$	8.80458e6	0.516513
$782$	1.29807e7	0.759068
$783$	−4.14264e6	−0.241475
$784$	6.60851e7	3.83984
$785$	0	0
$786$	−7.35373e6	−0.424572
$787$	1.42655e7	0.821013	0.410507	−	0.911858i	$-0.365352\pi$
$787$	1.42655e7	0.821013	0.410507	+	0.911858i	$0.365352\pi$
$788$	5.72227e7	3.28286
$789$	7.82477e6	0.447485
$790$	0	0
$791$	4.31041e6	0.244950
$792$	5.66568e6	0.320952
$793$	−382679.	−0.0216098
$794$	1.58444e7	0.891916
$795$	0	0
$796$	−6.77375e7	−3.78919
$797$	3.82425e6	0.213256	0.106628	−	0.994299i	$-0.465995\pi$
$797$	3.82425e6	0.213256	0.106628	+	0.994299i	$0.465995\pi$
$798$	−5.24804e6	−0.291736
$799$	2.45453e7	1.36020
$800$	0	0
$801$	3.88957e6	0.214200
$802$	2.45521e7	1.34788
$803$	−2.69698e6	−0.147601
$804$	3.11086e7	1.69723
$805$	0	0
$806$	−2.18363e7	−1.18397
$807$	4.14140e6	0.223853
$808$	7.26056e7	3.91239
$809$	2.01604e7	1.08300	0.541499	−	0.840701i	$-0.317857\pi$
$809$	2.01604e7	1.08300	0.541499	+	0.840701i	$0.317857\pi$
$810$	0	0
$811$	−8.80372e6	−0.470017	−0.235009	−	0.971993i	$-0.575512\pi$
$811$	−8.80372e6	−0.470017	−0.235009	+	0.971993i	$0.575512\pi$
$812$	9.14246e7	4.86601
$813$	4.39391e6	0.233144
$814$	−8.53748e6	−0.451615
$815$	0	0
$816$	−6.46097e7	−3.39682
$817$	−2.55548e6	−0.133942
$818$	−6.37464e7	−3.33098
$819$	−1.62198e7	−0.844957
$820$	0	0
$821$	1.33018e7	0.688734	0.344367	−	0.938835i	$-0.388094\pi$
$821$	1.33018e7	0.688734	0.344367	+	0.938835i	$0.388094\pi$
$822$	−3.62174e7	−1.86955
$823$	−2.17096e7	−1.11726	−0.558628	−	0.829419i	$-0.688672\pi$
$823$	−2.17096e7	−1.11726	−0.558628	+	0.829419i	$0.688672\pi$
$824$	−1.13910e8	−5.84444
$825$	0	0
$826$	6.55510e7	3.34295
$827$	1.18446e7	0.602220	0.301110	−	0.953589i	$-0.402643\pi$
$827$	1.18446e7	0.602220	0.301110	+	0.953589i	$0.402643\pi$
$828$	4.07482e6	0.206554
$829$	−6.82282e6	−0.344808	−0.172404	−	0.985026i	$-0.555153\pi$
$829$	−6.82282e6	−0.344808	−0.172404	+	0.985026i	$0.555153\pi$
$830$	0	0
$831$	−1.74630e7	−0.877235
$832$	−1.07308e8	−5.37430
$833$	3.80530e7	1.90010
$834$	712286.	0.0354601
$835$	0	0
$836$	2.94972e6	0.145970
$837$	−1.38307e6	−0.0682384
$838$	−3.30033e7	−1.62348
$839$	1.51092e6	0.0741030	0.0370515	−	0.999313i	$-0.488203\pi$
$839$	1.51092e6	0.0741030	0.0370515	+	0.999313i	$0.488203\pi$
$840$	0	0
$841$	1.17811e7	0.574375
$842$	2.76794e7	1.34548
$843$	1.58660e7	0.768949
$844$	−9.10369e6	−0.439908
$845$	0	0
$846$	1.05936e7	0.508885
$847$	2.75948e6	0.132166
$848$	2.75598e7	1.31609
$849$	−4.15090e6	−0.197639
$850$	0	0
$851$	−3.83839e6	−0.181688
$852$	−5.59015e7	−2.63830
$853$	4.40414e6	0.207247	0.103624	−	0.994617i	$-0.466956\pi$
$853$	4.40414e6	0.207247	0.103624	+	0.994617i	$0.466956\pi$
$854$	735443.	0.0345068
$855$	0	0
$856$	−8.56383e7	−3.99470
$857$	−2.26455e6	−0.105324	−0.0526622	−	0.998612i	$-0.516771\pi$
$857$	−2.26455e6	−0.105324	−0.0526622	+	0.998612i	$0.516771\pi$
$858$	1.25341e7	0.581264
$859$	1.55123e7	0.717288	0.358644	−	0.933474i	$-0.383239\pi$
$859$	1.55123e7	0.717288	0.358644	+	0.933474i	$0.383239\pi$
$860$	0	0
$861$	2.97872e7	1.36937
$862$	−3.24116e7	−1.48570
$863$	2.26673e7	1.03603	0.518016	−	0.855371i	$-0.326671\pi$
$863$	2.26673e7	1.03603	0.518016	+	0.855371i	$0.326671\pi$
$864$	−1.43999e7	−0.656260
$865$	0	0
$866$	−1.32316e7	−0.599541
$867$	−2.44247e7	−1.10352
$868$	3.05231e7	1.37509
$869$	6.26531e6	0.281445
$870$	0	0
$871$	4.30213e7	1.92149
$872$	−5.78117e7	−2.57469
$873$	9.39475e6	0.417205
$874$	1.82333e6	0.0807395
$875$	0	0
$876$	1.71235e7	0.753931
$877$	−587041.	−0.0257733	−0.0128866	−	0.999917i	$-0.504102\pi$
$877$	−587041.	−0.0257733	−0.0128866	+	0.999917i	$0.504102\pi$
$878$	−2.23328e7	−0.977705
$879$	1.55683e7	0.679625
$880$	0	0
$881$	−2.61087e7	−1.13330	−0.566651	−	0.823958i	$-0.691761\pi$
$881$	−2.61087e7	−1.13330	−0.566651	+	0.823958i	$0.691761\pi$
$882$	1.64235e7	0.710875
$883$	2.97921e6	0.128588	0.0642939	−	0.997931i	$-0.479520\pi$
$883$	2.97921e6	0.128588	0.0642939	+	0.997931i	$0.479520\pi$
$884$	−1.84387e8	−7.93597
$885$	0	0
$886$	1.48698e7	0.636385
$887$	−489724.	−0.0208998	−0.0104499	−	0.999945i	$-0.503326\pi$
$887$	−489724.	−0.0208998	−0.0104499	+	0.999945i	$0.503326\pi$
$888$	3.38850e7	1.44203
$889$	−2.05502e6	−0.0872092
$890$	0	0
$891$	793881.	0.0335013
$892$	7.10613e7	2.99034
$893$	3.44775e6	0.144680
$894$	1.13929e7	0.476749
$895$	0	0
$896$	8.70920e7	3.62417
$897$	5.63523e6	0.233846
$898$	1.67272e7	0.692200
$899$	1.07811e7	0.444903
$900$	0	0
$901$	1.58694e7	0.651252
$902$	−2.30185e7	−0.942023
$903$	1.51787e7	0.619462
$904$	1.32204e7	0.538051
$905$	0	0
$906$	−1.61428e7	−0.653368
$907$	8.30654e6	0.335275	0.167638	−	0.985849i	$-0.446386\pi$
$907$	8.30654e6	0.335275	0.167638	+	0.985849i	$0.446386\pi$
$908$	−9.60204e7	−3.86500
$909$	1.01736e7	0.408379
$910$	0	0
$911$	3.30461e7	1.31924	0.659620	−	0.751599i	$-0.270717\pi$
$911$	3.30461e7	1.31924	0.659620	+	0.751599i	$0.270717\pi$
$912$	−9.07537e6	−0.361308
$913$	−9.41793e6	−0.373920
$914$	−5.47559e7	−2.16803
$915$	0	0
$916$	−1.09476e8	−4.31102
$917$	1.42154e7	0.558260
$918$	−1.60568e7	−0.628857
$919$	1.92221e7	0.750779	0.375389	−	0.926867i	$-0.377509\pi$
$919$	1.92221e7	0.750779	0.375389	+	0.926867i	$0.377509\pi$
$920$	0	0
$921$	4.52163e6	0.175649
$922$	−1.94073e7	−0.751860
$923$	−7.73083e7	−2.98691
$924$	−1.75203e7	−0.675090
$925$	0	0
$926$	6.94556e6	0.266183
$927$	−1.59611e7	−0.610049
$928$	1.12249e8	4.27870
$929$	−1.05025e7	−0.399258	−0.199629	−	0.979872i	$-0.563974\pi$
$929$	−1.05025e7	−0.399258	−0.199629	+	0.979872i	$0.563974\pi$
$930$	0	0
$931$	5.34510e6	0.202107
$932$	−4.11932e7	−1.55341
$933$	1.66036e7	0.624449
$934$	−7.20925e7	−2.70410
$935$	0	0
$936$	−4.97473e7	−1.85601
$937$	1.83991e7	0.684616	0.342308	−	0.939588i	$-0.388791\pi$
$937$	1.83991e7	0.684616	0.342308	+	0.939588i	$0.388791\pi$
$938$	−8.26795e7	−3.06825
$939$	2.02713e7	0.750270
$940$	0	0
$941$	−3.48290e7	−1.28223	−0.641117	−	0.767443i	$-0.721529\pi$
$941$	−3.48290e7	−1.28223	−0.641117	+	0.767443i	$0.721529\pi$
$942$	2.97503e7	1.09236
$943$	−1.03490e7	−0.378982
$944$	1.13357e8	4.14016
$945$	0	0
$946$	−1.17296e7	−0.426142
$947$	1.51753e7	0.549871	0.274936	−	0.961463i	$-0.411343\pi$
$947$	1.51753e7	0.549871	0.274936	+	0.961463i	$0.411343\pi$
$948$	−3.97793e7	−1.43759
$949$	2.36807e7	0.853550
$950$	0	0
$951$	1.10308e7	0.395508
$952$	2.21518e8	7.92166
$953$	4.86395e7	1.73483	0.867415	−	0.497586i	$-0.165780\pi$
$953$	4.86395e7	1.73483	0.867415	+	0.497586i	$0.165780\pi$
$954$	6.84915e6	0.243650
$955$	0	0
$956$	1.86322e7	0.659356
$957$	−6.18838e6	−0.218422
$958$	−2.82615e6	−0.0994903
$959$	7.00117e7	2.45824
$960$	0	0
$961$	−2.50297e7	−0.874275
$962$	7.49631e7	2.61162
$963$	−1.19997e7	−0.416971
$964$	5.74578e7	1.99139
$965$	0	0
$966$	−1.08299e7	−0.373408
$967$	−5.63261e7	−1.93706	−0.968531	−	0.248893i	$-0.919933\pi$
$967$	−5.63261e7	−1.93706	−0.968531	+	0.248893i	$0.919933\pi$
$968$	8.46355e6	0.290312
$969$	−5.22576e6	−0.178789
$970$	0	0
$971$	2.29679e7	0.781759	0.390880	−	0.920442i	$-0.372171\pi$
$971$	2.29679e7	0.781759	0.390880	+	0.920442i	$0.372171\pi$
$972$	−5.04046e6	−0.171121
$973$	−1.37692e6	−0.0466257
$974$	1.09453e7	0.369683
$975$	0	0
$976$	1.27179e6	0.0427358
$977$	−3.90870e7	−1.31008	−0.655038	−	0.755596i	$-0.727347\pi$
$977$	−3.90870e7	−1.31008	−0.655038	+	0.755596i	$0.727347\pi$
$978$	−4.35467e7	−1.45582
$979$	5.81034e6	0.193751
$980$	0	0
$981$	−8.10063e6	−0.268749
$982$	2.81532e6	0.0931641
$983$	3.76365e7	1.24230	0.621148	−	0.783693i	$-0.286667\pi$
$983$	3.76365e7	1.24230	0.621148	+	0.783693i	$0.286667\pi$
$984$	9.13600e7	3.00793
$985$	0	0
$986$	1.25164e8	4.10004
$987$	−2.04785e7	−0.669122
$988$	−2.58999e7	−0.844122
$989$	−5.27353e6	−0.171439
$990$	0	0
$991$	2.08100e7	0.673112	0.336556	−	0.941663i	$-0.390738\pi$
$991$	2.08100e7	0.673112	0.336556	+	0.941663i	$0.390738\pi$
$992$	3.74755e7	1.20912
$993$	7.57383e6	0.243749
$994$	1.48573e8	4.76952
$995$	0	0
$996$	5.97957e7	1.90995
$997$	−4.32288e6	−0.137732	−0.0688661	−	0.997626i	$-0.521938\pi$
$997$	−4.32288e6	−0.137732	−0.0688661	+	0.997626i	$0.521938\pi$
$998$	−4.91965e6	−0.156353
$999$	4.74800e6	0.150521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.n.1.7		7
5.4	even	2		165.6.a.h.1.1	✓	7
15.14	odd	2		495.6.a.n.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.h.1.1	✓	7	5.4	even	2
495.6.a.n.1.7		7	15.14	odd	2
825.6.a.n.1.7		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)

\(f(q)\)	\(=\)	\(q+10.8333 q^{2} -9.00000 q^{3} +85.3606 q^{4} -97.4998 q^{6} +188.476 q^{7} +578.072 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+10.8333 q^{2} -9.00000 q^{3} +85.3606 q^{4} -97.4998 q^{6} +188.476 q^{7} +578.072 q^{8} +81.0000 q^{9} +121.000 q^{11} -768.246 q^{12} -1062.44 q^{13} +2041.82 q^{14} +3530.89 q^{16} +2033.15 q^{17} +877.498 q^{18} +285.586 q^{19} -1696.29 q^{21} +1310.83 q^{22} +589.341 q^{23} -5202.65 q^{24} -11509.7 q^{26} -729.000 q^{27} +16088.4 q^{28} +5682.63 q^{29} +1897.21 q^{31} +19753.0 q^{32} -1089.00 q^{33} +22025.8 q^{34} +6914.21 q^{36} -6513.03 q^{37} +3093.84 q^{38} +9561.93 q^{39} -17560.3 q^{41} -18376.4 q^{42} -8948.19 q^{43} +10328.6 q^{44} +6384.51 q^{46} +12072.6 q^{47} -31778.1 q^{48} +18716.2 q^{49} -18298.4 q^{51} -90690.2 q^{52} +7805.32 q^{53} -7897.48 q^{54} +108953. q^{56} -2570.28 q^{57} +61561.7 q^{58} +32104.2 q^{59} +360.190 q^{61} +20553.1 q^{62} +15266.6 q^{63} +101001. q^{64} -11797.5 q^{66} -40493.0 q^{67} +173551. q^{68} -5304.07 q^{69} +72765.1 q^{71} +46823.8 q^{72} -22289.1 q^{73} -70557.7 q^{74} +24377.8 q^{76} +22805.6 q^{77} +103587. q^{78} +51779.4 q^{79} +6561.00 q^{81} -190236. q^{82} -77834.1 q^{83} -144796. q^{84} -96938.5 q^{86} -51143.6 q^{87} +69946.7 q^{88} +48019.3 q^{89} -200244. q^{91} +50306.5 q^{92} -17074.9 q^{93} +130786. q^{94} -177777. q^{96} +115985. q^{97} +202759. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} - 63 q^{3} + 195 q^{4} + 9 q^{6} - 153 q^{8} + 567 q^{9}+O(q^{10}) \) 7 * q - q^2 - 63 * q^3 + 195 * q^4 + 9 * q^6 - 153 * q^8 + 567 * q^9	\( 7 q - q^{2} - 63 q^{3} + 195 q^{4} + 9 q^{6} - 153 q^{8} + 567 q^{9} + 847 q^{11} - 1755 q^{12} - 1418 q^{13} + 2548 q^{14} + 3699 q^{16} - 630 q^{17} - 81 q^{18} + 2572 q^{19} - 121 q^{22} - 536 q^{23} + 1377 q^{24} - 7626 q^{26} - 5103 q^{27} + 11368 q^{28} - 1038 q^{29} + 1872 q^{31} + 7523 q^{32} - 7623 q^{33} + 20790 q^{34} + 15795 q^{36} - 24298 q^{37} + 18952 q^{38} + 12762 q^{39} - 17658 q^{41} - 22932 q^{42} - 7244 q^{43} + 23595 q^{44} + 31016 q^{46} - 34560 q^{47} - 33291 q^{48} + 78735 q^{49} + 5670 q^{51} - 110222 q^{52} + 10214 q^{53} + 729 q^{54} + 81124 q^{56} - 23148 q^{57} + 5718 q^{58} + 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} + 1089 q^{66} - 64908 q^{67} + 136010 q^{68} + 4824 q^{69} + 61816 q^{71} - 12393 q^{72} + 11890 q^{73} - 124050 q^{74} - 47216 q^{76} + 68634 q^{78} + 18928 q^{79} + 45927 q^{81} - 36398 q^{82} - 17492 q^{83} - 102312 q^{84} - 216688 q^{86} + 9342 q^{87} - 18513 q^{88} + 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 16848 q^{93} - 30800 q^{94} - 67707 q^{96} + 172546 q^{97} + 615271 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q - q^2 - 63 * q^3 + 195 * q^4 + 9 * q^6 - 153 * q^8 + 567 * q^9 + 847 * q^11 - 1755 * q^12 - 1418 * q^13 + 2548 * q^14 + 3699 * q^16 - 630 * q^17 - 81 * q^18 + 2572 * q^19 - 121 * q^22 - 536 * q^23 + 1377 * q^24 - 7626 * q^26 - 5103 * q^27 + 11368 * q^28 - 1038 * q^29 + 1872 * q^31 + 7523 * q^32 - 7623 * q^33 + 20790 * q^34 + 15795 * q^36 - 24298 * q^37 + 18952 * q^38 + 12762 * q^39 - 17658 * q^41 - 22932 * q^42 - 7244 * q^43 + 23595 * q^44 + 31016 * q^46 - 34560 * q^47 - 33291 * q^48 + 78735 * q^49 + 5670 * q^51 - 110222 * q^52 + 10214 * q^53 + 729 * q^54 + 81124 * q^56 - 23148 * q^57 + 5718 * q^58 + 94676 * q^59 + 69538 * q^61 + 4208 * q^62 + 112339 * q^64 + 1089 * q^66 - 64908 * q^67 + 136010 * q^68 + 4824 * q^69 + 61816 * q^71 - 12393 * q^72 + 11890 * q^73 - 124050 * q^74 - 47216 * q^76 + 68634 * q^78 + 18928 * q^79 + 45927 * q^81 - 36398 * q^82 - 17492 * q^83 - 102312 * q^84 - 216688 * q^86 + 9342 * q^87 - 18513 * q^88 + 25302 * q^89 + 3392 * q^91 + 27408 * q^92 - 16848 * q^93 - 30800 * q^94 - 67707 * q^96 + 172546 * q^97 + 615271 * q^98 + 68607 * q^99

Introduction

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