LMFDB - Embedded newform 825.4.c.a.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.a.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} -15.0000 q^{6} -32.0000i q^{7} +45.0000i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} -15.0000 q^{6} -32.0000i q^{7} +45.0000i q^{8} -9.00000 q^{9} -11.0000 q^{11} +51.0000i q^{12} +38.0000i q^{13} -160.000 q^{14} +89.0000 q^{16} -2.00000i q^{17} +45.0000i q^{18} -72.0000 q^{19} -96.0000 q^{21} +55.0000i q^{22} -68.0000i q^{23} +135.000 q^{24} +190.000 q^{26} +27.0000i q^{27} +544.000i q^{28} +54.0000 q^{29} -152.000 q^{31} -85.0000i q^{32} +33.0000i q^{33} -10.0000 q^{34} +153.000 q^{36} +174.000i q^{37} +360.000i q^{38} +114.000 q^{39} +94.0000 q^{41} +480.000i q^{42} +528.000i q^{43} +187.000 q^{44} -340.000 q^{46} -340.000i q^{47} -267.000i q^{48} -681.000 q^{49} -6.00000 q^{51} -646.000i q^{52} +438.000i q^{53} +135.000 q^{54} +1440.00 q^{56} +216.000i q^{57} -270.000i q^{58} -20.0000 q^{59} +570.000 q^{61} +760.000i q^{62} +288.000i q^{63} +287.000 q^{64} +165.000 q^{66} -460.000i q^{67} +34.0000i q^{68} -204.000 q^{69} -1092.00 q^{71} -405.000i q^{72} -562.000i q^{73} +870.000 q^{74} +1224.00 q^{76} +352.000i q^{77} -570.000i q^{78} +16.0000 q^{79} +81.0000 q^{81} -470.000i q^{82} -372.000i q^{83} +1632.00 q^{84} +2640.00 q^{86} -162.000i q^{87} -495.000i q^{88} +966.000 q^{89} +1216.00 q^{91} +1156.00i q^{92} +456.000i q^{93} -1700.00 q^{94} -255.000 q^{96} -526.000i q^{97} +3405.00i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9}+O(q^{10}) $ 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9	$ 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9} - 22 q^{11} - 320 q^{14} + 178 q^{16} - 144 q^{19} - 192 q^{21} + 270 q^{24} + 380 q^{26} + 108 q^{29} - 304 q^{31} - 20 q^{34} + 306 q^{36} + 228 q^{39} + 188 q^{41} + 374 q^{44} - 680 q^{46} - 1362 q^{49} - 12 q^{51} + 270 q^{54} + 2880 q^{56} - 40 q^{59} + 1140 q^{61} + 574 q^{64} + 330 q^{66} - 408 q^{69} - 2184 q^{71} + 1740 q^{74} + 2448 q^{76} + 32 q^{79} + 162 q^{81} + 3264 q^{84} + 5280 q^{86} + 1932 q^{89} + 2432 q^{91} - 3400 q^{94} - 510 q^{96} + 198 q^{99}+O(q^{100}) $ 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9 - 22 * q^11 - 320 * q^14 + 178 * q^16 - 144 * q^19 - 192 * q^21 + 270 * q^24 + 380 * q^26 + 108 * q^29 - 304 * q^31 - 20 * q^34 + 306 * q^36 + 228 * q^39 + 188 * q^41 + 374 * q^44 - 680 * q^46 - 1362 * q^49 - 12 * q^51 + 270 * q^54 + 2880 * q^56 - 40 * q^59 + 1140 * q^61 + 574 * q^64 + 330 * q^66 - 408 * q^69 - 2184 * q^71 + 1740 * q^74 + 2448 * q^76 + 32 * q^79 + 162 * q^81 + 3264 * q^84 + 5280 * q^86 + 1932 * q^89 + 2432 * q^91 - 3400 * q^94 - 510 * q^96 + 198 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 5.00000i	− 1.76777i	−0.467707	−	0.883883i	$-0.654920\pi$
$2$	− 5.00000i	− 1.76777i	0.467707	−	0.883883i	$-0.345080\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−17.0000	−2.12500
$5$	0	0
$5$	0	0
$6$	−15.0000	−1.02062
$7$	− 32.0000i	− 1.72784i	−0.503631	−	0.863919i	$-0.668003\pi$
$7$	− 32.0000i	− 1.72784i	0.503631	−	0.863919i	$-0.331997\pi$
$8$	45.0000i	1.98874i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	51.0000i	1.22687i
$13$	38.0000i	0.810716i	0.914158	+	0.405358i	$0.132853\pi$
$13$	38.0000i	0.810716i	−0.914158	+	0.405358i	$0.867147\pi$
$14$	−160.000	−3.05441
$15$	0	0
$16$	89.0000	1.39062
$17$	− 2.00000i	− 0.0285336i	−0.999898	−	0.0142668i	$-0.995459\pi$
$17$	− 2.00000i	− 0.0285336i	0.999898	−	0.0142668i	$-0.00454142\pi$
$18$	45.0000i	0.589256i
$19$	−72.0000	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$19$	−72.0000	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	55.0000i	0.533002i
$23$	− 68.0000i	− 0.616477i	−0.951309	−	0.308239i	$-0.900260\pi$
$23$	− 68.0000i	− 0.616477i	0.951309	−	0.308239i	$-0.0997395\pi$
$24$	135.000	1.14820
$25$	0	0
$26$	190.000	1.43316
$27$	27.0000i	0.192450i
$28$	544.000i	3.67165i
$29$	54.0000	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$29$	54.0000	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	− 85.0000i	− 0.469563i
$33$	33.0000i	0.174078i
$34$	−10.0000	−0.0504408
$35$	0	0
$36$	153.000	0.708333
$37$	174.000i	0.773120i	0.922264	+	0.386560i	$0.126337\pi$
$37$	174.000i	0.773120i	−0.922264	+	0.386560i	$0.873663\pi$
$38$	360.000i	1.53683i
$39$	114.000	0.468067
$40$	0	0
$41$	94.0000	0.358057	0.179028	−	0.983844i	$-0.442705\pi$
$41$	94.0000	0.358057	0.179028	+	0.983844i	$0.442705\pi$
$42$	480.000i	1.76347i
$43$	528.000i	1.87254i	0.351280	+	0.936270i	$0.385746\pi$
$43$	528.000i	1.87254i	−0.351280	+	0.936270i	$0.614254\pi$
$44$	187.000	0.640712
$45$	0	0
$46$	−340.000	−1.08979
$47$	− 340.000i	− 1.05519i	−0.849495	−	0.527597i	$-0.823093\pi$
$47$	− 340.000i	− 1.05519i	0.849495	−	0.527597i	$-0.176907\pi$
$48$	− 267.000i	− 0.802878i
$49$	−681.000	−1.98542
$50$	0	0
$51$	−6.00000	−0.0164739
$52$	− 646.000i	− 1.72277i
$53$	438.000i	1.13517i	0.823315	+	0.567584i	$0.192122\pi$
$53$	438.000i	1.13517i	−0.823315	+	0.567584i	$0.807878\pi$
$54$	135.000	0.340207
$55$	0	0
$56$	1440.00	3.43622
$57$	216.000i	0.501928i
$58$	− 270.000i	− 0.611254i
$59$	−20.0000	−0.0441318	−0.0220659	−	0.999757i	$-0.507024\pi$
$59$	−20.0000	−0.0441318	−0.0220659	+	0.999757i	$0.507024\pi$
$60$	0	0
$61$	570.000	1.19641	0.598205	−	0.801343i	$-0.295881\pi$
$61$	570.000	1.19641	0.598205	+	0.801343i	$0.295881\pi$
$62$	760.000i	1.55678i
$63$	288.000i	0.575946i
$64$	287.000	0.560547
$65$	0	0
$66$	165.000	0.307729
$67$	− 460.000i	− 0.838775i	−0.907807	−	0.419388i	$-0.862245\pi$
$67$	− 460.000i	− 0.838775i	0.907807	−	0.419388i	$-0.137755\pi$
$68$	34.0000i	0.0606339i
$69$	−204.000	−0.355923
$70$	0	0
$71$	−1092.00	−1.82530	−0.912652	−	0.408738i	$-0.865969\pi$
$71$	−1092.00	−1.82530	−0.912652	+	0.408738i	$0.865969\pi$
$72$	− 405.000i	− 0.662913i
$73$	− 562.000i	− 0.901057i	−0.892762	−	0.450528i	$-0.851236\pi$
$73$	− 562.000i	− 0.901057i	0.892762	−	0.450528i	$-0.148764\pi$
$74$	870.000	1.36670
$75$	0	0
$76$	1224.00	1.84740
$77$	352.000i	0.520963i
$78$	− 570.000i	− 0.827433i
$79$	16.0000	0.0227866	0.0113933	−	0.999935i	$-0.496373\pi$
$79$	16.0000	0.0227866	0.0113933	+	0.999935i	$0.496373\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 470.000i	− 0.632961i
$83$	− 372.000i	− 0.491955i	−0.969275	−	0.245978i	$-0.920891\pi$
$83$	− 372.000i	− 0.491955i	0.969275	−	0.245978i	$-0.0791090\pi$
$84$	1632.00	2.11983
$85$	0	0
$86$	2640.00	3.31022
$87$	− 162.000i	− 0.199635i
$88$	− 495.000i	− 0.599627i
$89$	966.000	1.15051	0.575257	−	0.817973i	$-0.304902\pi$
$89$	966.000	1.15051	0.575257	+	0.817973i	$0.304902\pi$
$90$	0	0
$91$	1216.00	1.40079
$92$	1156.00i	1.31001i
$93$	456.000i	0.508441i
$94$	−1700.00	−1.86534
$95$	0	0
$96$	−255.000	−0.271102
$97$	− 526.000i	− 0.550590i	−0.961360	−	0.275295i	$-0.911225\pi$
$97$	− 526.000i	− 0.550590i	0.961360	−	0.275295i	$-0.0887755\pi$
$98$	3405.00i	3.50976i
$99$	99.0000	0.100504
$100$	0	0
$101$	50.0000	0.0492593	0.0246296	−	0.999697i	$-0.492159\pi$
$101$	50.0000	0.0492593	0.0246296	+	0.999697i	$0.492159\pi$
$102$	30.0000i	0.0291220i
$103$	− 944.000i	− 0.903059i	−0.892256	−	0.451530i	$-0.850879\pi$
$103$	− 944.000i	− 0.903059i	0.892256	−	0.451530i	$-0.149121\pi$
$104$	−1710.00	−1.61230
$105$	0	0
$106$	2190.00	2.00671
$107$	468.000i	0.422834i	0.977396	+	0.211417i	$0.0678079\pi$
$107$	468.000i	0.422834i	−0.977396	+	0.211417i	$0.932192\pi$
$108$	− 459.000i	− 0.408956i
$109$	−154.000	−0.135326	−0.0676630	−	0.997708i	$-0.521554\pi$
$109$	−154.000	−0.135326	−0.0676630	+	0.997708i	$0.521554\pi$
$110$	0	0
$111$	522.000	0.446361
$112$	− 2848.00i	− 2.40277i
$113$	54.0000i	0.0449548i	0.999747	+	0.0224774i	$0.00715538\pi$
$113$	54.0000i	0.0449548i	−0.999747	+	0.0224774i	$0.992845\pi$
$114$	1080.00	0.887292
$115$	0	0
$116$	−918.000	−0.734777
$117$	− 342.000i	− 0.270239i
$118$	100.000i	0.0780148i
$119$	−64.0000	−0.0493014
$120$	0	0
$121$	121.000	0.0909091
$122$	− 2850.00i	− 2.11497i
$123$	− 282.000i	− 0.206724i
$124$	2584.00	1.87137
$125$	0	0
$126$	1440.00	1.01814
$127$	− 2224.00i	− 1.55392i	−0.629549	−	0.776961i	$-0.716760\pi$
$127$	− 2224.00i	− 1.55392i	0.629549	−	0.776961i	$-0.283240\pi$
$128$	− 2115.00i	− 1.46048i
$129$	1584.00	1.08111
$130$	0	0
$131$	−2772.00	−1.84878	−0.924392	−	0.381443i	$-0.875427\pi$
$131$	−2772.00	−1.84878	−0.924392	+	0.381443i	$0.875427\pi$
$132$	− 561.000i	− 0.369915i
$133$	2304.00i	1.50212i
$134$	−2300.00	−1.48276
$135$	0	0
$136$	90.0000	0.0567459
$137$	1130.00i	0.704689i	0.935870	+	0.352345i	$0.114615\pi$
$137$	1130.00i	0.704689i	−0.935870	+	0.352345i	$0.885385\pi$
$138$	1020.00i	0.629190i
$139$	1616.00	0.986096	0.493048	−	0.870002i	$-0.335883\pi$
$139$	1616.00	0.986096	0.493048	+	0.870002i	$0.335883\pi$
$140$	0	0
$141$	−1020.00	−0.609216
$142$	5460.00i	3.22671i
$143$	− 418.000i	− 0.244440i
$144$	−801.000	−0.463542
$145$	0	0
$146$	−2810.00	−1.59286
$147$	2043.00i	1.14628i
$148$	− 2958.00i	− 1.64288i
$149$	−2066.00	−1.13593	−0.567964	−	0.823053i	$-0.692269\pi$
$149$	−2066.00	−1.13593	−0.567964	+	0.823053i	$0.692269\pi$
$150$	0	0
$151$	248.000	0.133655	0.0668277	−	0.997765i	$-0.478712\pi$
$151$	248.000	0.133655	0.0668277	+	0.997765i	$0.478712\pi$
$152$	− 3240.00i	− 1.72894i
$153$	18.0000i	0.00951120i
$154$	1760.00	0.920941
$155$	0	0
$156$	−1938.00	−0.994642
$157$	2366.00i	1.20272i	0.798977	+	0.601361i	$0.205375\pi$
$157$	2366.00i	1.20272i	−0.798977	+	0.601361i	$0.794625\pi$
$158$	− 80.0000i	− 0.0402814i
$159$	1314.00	0.655390
$160$	0	0
$161$	−2176.00	−1.06517
$162$	− 405.000i	− 0.196419i
$163$	284.000i	0.136470i	0.997669	+	0.0682350i	$0.0217368\pi$
$163$	284.000i	0.136470i	−0.997669	+	0.0682350i	$0.978263\pi$
$164$	−1598.00	−0.760871
$165$	0	0
$166$	−1860.00	−0.869663
$167$	600.000i	0.278020i	0.990291	+	0.139010i	$0.0443921\pi$
$167$	600.000i	0.278020i	−0.990291	+	0.139010i	$0.955608\pi$
$168$	− 4320.00i	− 1.98390i
$169$	753.000	0.342740
$170$	0	0
$171$	648.000	0.289788
$172$	− 8976.00i	− 3.97915i
$173$	− 138.000i	− 0.0606471i	−0.999540	−	0.0303235i	$-0.990346\pi$
$173$	− 138.000i	− 0.0606471i	0.999540	−	0.0303235i	$-0.00965376\pi$
$174$	−810.000	−0.352908
$175$	0	0
$176$	−979.000	−0.419289
$177$	60.0000i	0.0254795i
$178$	− 4830.00i	− 2.03384i
$179$	−3972.00	−1.65855	−0.829277	−	0.558838i	$-0.811248\pi$
$179$	−3972.00	−1.65855	−0.829277	+	0.558838i	$0.811248\pi$
$180$	0	0
$181$	2230.00	0.915771	0.457886	−	0.889011i	$-0.348607\pi$
$181$	2230.00	0.915771	0.457886	+	0.889011i	$0.348607\pi$
$182$	− 6080.00i	− 2.47626i
$183$	− 1710.00i	− 0.690748i
$184$	3060.00	1.22601
$185$	0	0
$186$	2280.00	0.898805
$187$	22.0000i	0.00860320i
$188$	5780.00i	2.24229i
$189$	864.000	0.332522
$190$	0	0
$191$	−772.000	−0.292461	−0.146230	−	0.989251i	$-0.546714\pi$
$191$	−772.000	−0.292461	−0.146230	+	0.989251i	$0.546714\pi$
$192$	− 861.000i	− 0.323632i
$193$	− 394.000i	− 0.146947i	−0.997297	−	0.0734734i	$-0.976592\pi$
$193$	− 394.000i	− 0.146947i	0.997297	−	0.0734734i	$-0.0234084\pi$
$194$	−2630.00	−0.973314
$195$	0	0
$196$	11577.0	4.21902
$197$	3058.00i	1.10596i	0.833196	+	0.552978i	$0.186509\pi$
$197$	3058.00i	1.10596i	−0.833196	+	0.552978i	$0.813491\pi$
$198$	− 495.000i	− 0.177667i
$199$	−2664.00	−0.948975	−0.474487	−	0.880262i	$-0.657367\pi$
$199$	−2664.00	−0.948975	−0.474487	+	0.880262i	$0.657367\pi$
$200$	0	0
$201$	−1380.00	−0.484267
$202$	− 250.000i	− 0.0870789i
$203$	− 1728.00i	− 0.597447i
$204$	102.000	0.0350070
$205$	0	0
$206$	−4720.00	−1.59640
$207$	612.000i	0.205492i
$208$	3382.00i	1.12740i
$209$	792.000	0.262123
$210$	0	0
$211$	−6000.00	−1.95762	−0.978808	−	0.204779i	$-0.934352\pi$
$211$	−6000.00	−1.95762	−0.978808	+	0.204779i	$0.934352\pi$
$212$	− 7446.00i	− 2.41223i
$213$	3276.00i	1.05384i
$214$	2340.00	0.747472
$215$	0	0
$216$	−1215.00	−0.382733
$217$	4864.00i	1.52161i
$218$	770.000i	0.239225i
$219$	−1686.00	−0.520225
$220$	0	0
$221$	76.0000	0.0231326
$222$	− 2610.00i	− 0.789062i
$223$	560.000i	0.168163i	0.996459	+	0.0840816i	$0.0267956\pi$
$223$	560.000i	0.168163i	−0.996459	+	0.0840816i	$0.973204\pi$
$224$	−2720.00	−0.811329
$225$	0	0
$226$	270.000	0.0794696
$227$	5292.00i	1.54732i	0.633599	+	0.773662i	$0.281577\pi$
$227$	5292.00i	1.54732i	−0.633599	+	0.773662i	$0.718423\pi$
$228$	− 3672.00i	− 1.06660i
$229$	5322.00	1.53575	0.767877	−	0.640597i	$-0.221313\pi$
$229$	5322.00	1.53575	0.767877	+	0.640597i	$0.221313\pi$
$230$	0	0
$231$	1056.00	0.300778
$232$	2430.00i	0.687661i
$233$	3954.00i	1.11174i	0.831270	+	0.555869i	$0.187615\pi$
$233$	3954.00i	1.11174i	−0.831270	+	0.555869i	$0.812385\pi$
$234$	−1710.00	−0.477719
$235$	0	0
$236$	340.000	0.0937801
$237$	− 48.0000i	− 0.0131558i
$238$	320.000i	0.0871534i
$239$	3360.00	0.909374	0.454687	−	0.890651i	$-0.349751\pi$
$239$	3360.00	0.909374	0.454687	+	0.890651i	$0.349751\pi$
$240$	0	0
$241$	−3278.00	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$241$	−3278.00	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$242$	− 605.000i	− 0.160706i
$243$	− 243.000i	− 0.0641500i
$244$	−9690.00	−2.54237
$245$	0	0
$246$	−1410.00	−0.365440
$247$	− 2736.00i	− 0.704808i
$248$	− 6840.00i	− 1.75137i
$249$	−1116.00	−0.284031
$250$	0	0
$251$	2092.00	0.526079	0.263040	−	0.964785i	$-0.415275\pi$
$251$	2092.00	0.526079	0.263040	+	0.964785i	$0.415275\pi$
$252$	− 4896.00i	− 1.22388i
$253$	748.000i	0.185875i
$254$	−11120.0	−2.74697
$255$	0	0
$256$	−8279.00	−2.02124
$257$	658.000i	0.159708i	0.996807	+	0.0798539i	$0.0254454\pi$
$257$	658.000i	0.159708i	−0.996807	+	0.0798539i	$0.974555\pi$
$258$	− 7920.00i	− 1.91115i
$259$	5568.00	1.33583
$260$	0	0
$261$	−486.000	−0.115259
$262$	13860.0i	3.26822i
$263$	5104.00i	1.19668i	0.801243	+	0.598339i	$0.204172\pi$
$263$	5104.00i	1.19668i	−0.801243	+	0.598339i	$0.795828\pi$
$264$	−1485.00	−0.346195
$265$	0	0
$266$	11520.0	2.65540
$267$	− 2898.00i	− 0.664250i
$268$	7820.00i	1.78240i
$269$	4238.00	0.960578	0.480289	−	0.877110i	$-0.340532\pi$
$269$	4238.00	0.960578	0.480289	+	0.877110i	$0.340532\pi$
$270$	0	0
$271$	−3376.00	−0.756743	−0.378372	−	0.925654i	$-0.623516\pi$
$271$	−3376.00	−0.756743	−0.378372	+	0.925654i	$0.623516\pi$
$272$	− 178.000i	− 0.0396795i
$273$	− 3648.00i	− 0.808744i
$274$	5650.00	1.24573
$275$	0	0
$276$	3468.00	0.756337
$277$	2074.00i	0.449872i	0.974374	+	0.224936i	$0.0722173\pi$
$277$	2074.00i	0.449872i	−0.974374	+	0.224936i	$0.927783\pi$
$278$	− 8080.00i	− 1.74319i
$279$	1368.00	0.293548
$280$	0	0
$281$	702.000	0.149031	0.0745157	−	0.997220i	$-0.476259\pi$
$281$	702.000	0.149031	0.0745157	+	0.997220i	$0.476259\pi$
$282$	5100.00i	1.07695i
$283$	− 4912.00i	− 1.03176i	−0.856661	−	0.515880i	$-0.827465\pi$
$283$	− 4912.00i	− 1.03176i	0.856661	−	0.515880i	$-0.172535\pi$
$284$	18564.0	3.87877
$285$	0	0
$286$	−2090.00	−0.432113
$287$	− 3008.00i	− 0.618664i
$288$	765.000i	0.156521i
$289$	4909.00	0.999186
$290$	0	0
$291$	−1578.00	−0.317883
$292$	9554.00i	1.91475i
$293$	3486.00i	0.695066i	0.937668	+	0.347533i	$0.112981\pi$
$293$	3486.00i	0.695066i	−0.937668	+	0.347533i	$0.887019\pi$
$294$	10215.0	2.02636
$295$	0	0
$296$	−7830.00	−1.53753
$297$	− 297.000i	− 0.0580259i
$298$	10330.0i	2.00806i
$299$	2584.00	0.499788
$300$	0	0
$301$	16896.0	3.23545
$302$	− 1240.00i	− 0.236271i
$303$	− 150.000i	− 0.0284399i
$304$	−6408.00	−1.20896
$305$	0	0
$306$	90.0000	0.0168136
$307$	8360.00i	1.55417i	0.629395	+	0.777085i	$0.283303\pi$
$307$	8360.00i	1.55417i	−0.629395	+	0.777085i	$0.716697\pi$
$308$	− 5984.00i	− 1.10705i
$309$	−2832.00	−0.521381
$310$	0	0
$311$	−5532.00	−1.00865	−0.504326	−	0.863513i	$-0.668259\pi$
$311$	−5532.00	−1.00865	−0.504326	+	0.863513i	$0.668259\pi$
$312$	5130.00i	0.930862i
$313$	− 4826.00i	− 0.871507i	−0.900066	−	0.435753i	$-0.856482\pi$
$313$	− 4826.00i	− 0.871507i	0.900066	−	0.435753i	$-0.143518\pi$
$314$	11830.0	2.12613
$315$	0	0
$316$	−272.000	−0.0484215
$317$	7570.00i	1.34124i	0.741800	+	0.670621i	$0.233972\pi$
$317$	7570.00i	1.34124i	−0.741800	+	0.670621i	$0.766028\pi$
$318$	− 6570.00i	− 1.15858i
$319$	−594.000	−0.104256
$320$	0	0
$321$	1404.00	0.244123
$322$	10880.0i	1.88298i
$323$	144.000i	0.0248061i
$324$	−1377.00	−0.236111
$325$	0	0
$326$	1420.00	0.241247
$327$	462.000i	0.0781305i
$328$	4230.00i	0.712081i
$329$	−10880.0	−1.82320
$330$	0	0
$331$	3676.00	0.610427	0.305213	−	0.952284i	$-0.401272\pi$
$331$	3676.00	0.610427	0.305213	+	0.952284i	$0.401272\pi$
$332$	6324.00i	1.04541i
$333$	− 1566.00i	− 0.257707i
$334$	3000.00	0.491475
$335$	0	0
$336$	−8544.00	−1.38724
$337$	− 5686.00i	− 0.919098i	−0.888152	−	0.459549i	$-0.848011\pi$
$337$	− 5686.00i	− 0.919098i	0.888152	−	0.459549i	$-0.151989\pi$
$338$	− 3765.00i	− 0.605885i
$339$	162.000	0.0259547
$340$	0	0
$341$	1672.00	0.265525
$342$	− 3240.00i	− 0.512278i
$343$	10816.0i	1.70265i
$344$	−23760.0	−3.72399
$345$	0	0
$346$	−690.000	−0.107210
$347$	− 1652.00i	− 0.255574i	−0.991802	−	0.127787i	$-0.959213\pi$
$347$	− 1652.00i	− 0.255574i	0.991802	−	0.127787i	$-0.0407873\pi$
$348$	2754.00i	0.424224i
$349$	6990.00	1.07211	0.536055	−	0.844183i	$-0.319914\pi$
$349$	6990.00	1.07211	0.536055	+	0.844183i	$0.319914\pi$
$350$	0	0
$351$	−1026.00	−0.156022
$352$	935.000i	0.141579i
$353$	8094.00i	1.22040i	0.792249	+	0.610199i	$0.208910\pi$
$353$	8094.00i	1.22040i	−0.792249	+	0.610199i	$0.791090\pi$
$354$	300.000	0.0450419
$355$	0	0
$356$	−16422.0	−2.44484
$357$	192.000i	0.0284642i
$358$	19860.0i	2.93194i
$359$	−1024.00	−0.150542	−0.0752711	−	0.997163i	$-0.523982\pi$
$359$	−1024.00	−0.150542	−0.0752711	+	0.997163i	$0.523982\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	− 11150.0i	− 1.61887i
$363$	− 363.000i	− 0.0524864i
$364$	−20672.0	−2.97667
$365$	0	0
$366$	−8550.00	−1.22108
$367$	− 13664.0i	− 1.94347i	−0.236066	−	0.971737i	$-0.575858\pi$
$367$	− 13664.0i	− 1.94347i	0.236066	−	0.971737i	$-0.424142\pi$
$368$	− 6052.00i	− 0.857289i
$369$	−846.000	−0.119352
$370$	0	0
$371$	14016.0	1.96139
$372$	− 7752.00i	− 1.08044i
$373$	1958.00i	0.271800i	0.990723	+	0.135900i	$0.0433926\pi$
$373$	1958.00i	0.271800i	−0.990723	+	0.135900i	$0.956607\pi$
$374$	110.000	0.0152085
$375$	0	0
$376$	15300.0	2.09850
$377$	2052.00i	0.280327i
$378$	− 4320.00i	− 0.587822i
$379$	−6124.00	−0.829997	−0.414998	−	0.909822i	$-0.636218\pi$
$379$	−6124.00	−0.829997	−0.414998	+	0.909822i	$0.636218\pi$
$380$	0	0
$381$	−6672.00	−0.897157
$382$	3860.00i	0.517002i
$383$	− 5612.00i	− 0.748720i	−0.927283	−	0.374360i	$-0.877862\pi$
$383$	− 5612.00i	− 0.748720i	0.927283	−	0.374360i	$-0.122138\pi$
$384$	−6345.00	−0.843208
$385$	0	0
$386$	−1970.00	−0.259768
$387$	− 4752.00i	− 0.624180i
$388$	8942.00i	1.17000i
$389$	−12450.0	−1.62273	−0.811363	−	0.584543i	$-0.801274\pi$
$389$	−12450.0	−1.62273	−0.811363	+	0.584543i	$0.801274\pi$
$390$	0	0
$391$	−136.000	−0.0175903
$392$	− 30645.0i	− 3.94849i
$393$	8316.00i	1.06740i
$394$	15290.0	1.95507
$395$	0	0
$396$	−1683.00	−0.213571
$397$	14830.0i	1.87480i	0.348252	+	0.937401i	$0.386775\pi$
$397$	14830.0i	1.87480i	−0.348252	+	0.937401i	$0.613225\pi$
$398$	13320.0i	1.67757i
$399$	6912.00	0.867250
$400$	0	0
$401$	−3358.00	−0.418181	−0.209090	−	0.977896i	$-0.567050\pi$
$401$	−3358.00	−0.418181	−0.209090	+	0.977896i	$0.567050\pi$
$402$	6900.00i	0.856071i
$403$	− 5776.00i	− 0.713953i
$404$	−850.000	−0.104676
$405$	0	0
$406$	−8640.00	−1.05615
$407$	− 1914.00i	− 0.233104i
$408$	− 270.000i	− 0.0327622i
$409$	−10698.0	−1.29335	−0.646677	−	0.762764i	$-0.723842\pi$
$409$	−10698.0	−1.29335	−0.646677	+	0.762764i	$0.723842\pi$
$410$	0	0
$411$	3390.00	0.406852
$412$	16048.0i	1.91900i
$413$	640.000i	0.0762526i
$414$	3060.00	0.363263
$415$	0	0
$416$	3230.00	0.380682
$417$	− 4848.00i	− 0.569323i
$418$	− 3960.00i	− 0.463373i
$419$	2044.00	0.238320	0.119160	−	0.992875i	$-0.461980\pi$
$419$	2044.00	0.238320	0.119160	+	0.992875i	$0.461980\pi$
$420$	0	0
$421$	3070.00	0.355398	0.177699	−	0.984085i	$-0.443135\pi$
$421$	3070.00	0.355398	0.177699	+	0.984085i	$0.443135\pi$
$422$	30000.0i	3.46061i
$423$	3060.00i	0.351731i
$424$	−19710.0	−2.25755
$425$	0	0
$426$	16380.0	1.86294
$427$	− 18240.0i	− 2.06720i
$428$	− 7956.00i	− 0.898523i
$429$	−1254.00	−0.141127
$430$	0	0
$431$	−12600.0	−1.40817	−0.704084	−	0.710116i	$-0.748642\pi$
$431$	−12600.0	−1.40817	−0.704084	+	0.710116i	$0.748642\pi$
$432$	2403.00i	0.267626i
$433$	9902.00i	1.09898i	0.835499	+	0.549492i	$0.185179\pi$
$433$	9902.00i	1.09898i	−0.835499	+	0.549492i	$0.814821\pi$
$434$	24320.0	2.68986
$435$	0	0
$436$	2618.00	0.287568
$437$	4896.00i	0.535944i
$438$	8430.00i	0.919637i
$439$	−11440.0	−1.24374	−0.621869	−	0.783121i	$-0.713627\pi$
$439$	−11440.0	−1.24374	−0.621869	+	0.783121i	$0.713627\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	− 380.000i	− 0.0408931i
$443$	5180.00i	0.555551i	0.960646	+	0.277776i	$0.0895972\pi$
$443$	5180.00i	0.555551i	−0.960646	+	0.277776i	$0.910403\pi$
$444$	−8874.00	−0.948517
$445$	0	0
$446$	2800.00	0.297273
$447$	6198.00i	0.655829i
$448$	− 9184.00i	− 0.968534i
$449$	−10826.0	−1.13789	−0.568943	−	0.822377i	$-0.692647\pi$
$449$	−10826.0	−1.13789	−0.568943	+	0.822377i	$0.692647\pi$
$450$	0	0
$451$	−1034.00	−0.107958
$452$	− 918.000i	− 0.0955290i
$453$	− 744.000i	− 0.0771659i
$454$	26460.0	2.73531
$455$	0	0
$456$	−9720.00	−0.998203
$457$	− 15798.0i	− 1.61707i	−0.588451	−	0.808533i	$-0.700262\pi$
$457$	− 15798.0i	− 1.61707i	0.588451	−	0.808533i	$-0.299738\pi$
$458$	− 26610.0i	− 2.71486i
$459$	54.0000	0.00549129
$460$	0	0
$461$	−3894.00	−0.393409	−0.196705	−	0.980463i	$-0.563024\pi$
$461$	−3894.00	−0.393409	−0.196705	+	0.980463i	$0.563024\pi$
$462$	− 5280.00i	− 0.531705i
$463$	15992.0i	1.60521i	0.596512	+	0.802604i	$0.296553\pi$
$463$	15992.0i	1.60521i	−0.596512	+	0.802604i	$0.703447\pi$
$464$	4806.00	0.480847
$465$	0	0
$466$	19770.0	1.96530
$467$	11844.0i	1.17361i	0.809729	+	0.586804i	$0.199614\pi$
$467$	11844.0i	1.17361i	−0.809729	+	0.586804i	$0.800386\pi$
$468$	5814.00i	0.574257i
$469$	−14720.0	−1.44927
$470$	0	0
$471$	7098.00	0.694392
$472$	− 900.000i	− 0.0877666i
$473$	− 5808.00i	− 0.564592i
$474$	−240.000	−0.0232565
$475$	0	0
$476$	1088.00	0.104766
$477$	− 3942.00i	− 0.378389i
$478$	− 16800.0i	− 1.60756i
$479$	−14936.0	−1.42472	−0.712362	−	0.701812i	$-0.752375\pi$
$479$	−14936.0	−1.42472	−0.712362	+	0.701812i	$0.752375\pi$
$480$	0	0
$481$	−6612.00	−0.626780
$482$	16390.0i	1.54885i
$483$	6528.00i	0.614978i
$484$	−2057.00	−0.193182
$485$	0	0
$486$	−1215.00	−0.113402
$487$	− 2056.00i	− 0.191306i	−0.995415	−	0.0956532i	$-0.969506\pi$
$487$	− 2056.00i	− 0.191306i	0.995415	−	0.0956532i	$-0.0304940\pi$
$488$	25650.0i	2.37935i
$489$	852.000	0.0787909
$490$	0	0
$491$	−17852.0	−1.64083	−0.820417	−	0.571766i	$-0.806259\pi$
$491$	−17852.0	−1.64083	−0.820417	+	0.571766i	$0.806259\pi$
$492$	4794.00i	0.439289i
$493$	− 108.000i	− 0.00986628i
$494$	−13680.0	−1.24594
$495$	0	0
$496$	−13528.0	−1.22465
$497$	34944.0i	3.15383i
$498$	5580.00i	0.502100i
$499$	−4508.00	−0.404420	−0.202210	−	0.979342i	$-0.564812\pi$
$499$	−4508.00	−0.404420	−0.202210	+	0.979342i	$0.564812\pi$
$500$	0	0
$501$	1800.00	0.160515
$502$	− 10460.0i	− 0.929985i
$503$	5912.00i	0.524062i	0.965059	+	0.262031i	$0.0843922\pi$
$503$	5912.00i	0.524062i	−0.965059	+	0.262031i	$0.915608\pi$
$504$	−12960.0	−1.14541
$505$	0	0
$506$	3740.00	0.328584
$507$	− 2259.00i	− 0.197881i
$508$	37808.0i	3.30208i
$509$	11406.0	0.993246	0.496623	−	0.867966i	$-0.334573\pi$
$509$	11406.0	0.993246	0.496623	+	0.867966i	$0.334573\pi$
$510$	0	0
$511$	−17984.0	−1.55688
$512$	24475.0i	2.11260i
$513$	− 1944.00i	− 0.167309i
$514$	3290.00	0.282326
$515$	0	0
$516$	−26928.0	−2.29736
$517$	3740.00i	0.318153i
$518$	− 27840.0i	− 2.36143i
$519$	−414.000	−0.0350146
$520$	0	0
$521$	−1542.00	−0.129667	−0.0648333	−	0.997896i	$-0.520652\pi$
$521$	−1542.00	−0.129667	−0.0648333	+	0.997896i	$0.520652\pi$
$522$	2430.00i	0.203751i
$523$	7504.00i	0.627394i	0.949523	+	0.313697i	$0.101568\pi$
$523$	7504.00i	0.627394i	−0.949523	+	0.313697i	$0.898432\pi$
$524$	47124.0	3.92867
$525$	0	0
$526$	25520.0	2.11545
$527$	304.000i	0.0251280i
$528$	2937.00i	0.242077i
$529$	7543.00	0.619956
$530$	0	0
$531$	180.000	0.0147106
$532$	− 39168.0i	− 3.19201i
$533$	3572.00i	0.290282i
$534$	−14490.0	−1.17424
$535$	0	0
$536$	20700.0	1.66810
$537$	11916.0i	0.957567i
$538$	− 21190.0i	− 1.69808i
$539$	7491.00	0.598627
$540$	0	0
$541$	1018.00	0.0809006	0.0404503	−	0.999182i	$-0.487121\pi$
$541$	1018.00	0.0809006	0.0404503	+	0.999182i	$0.487121\pi$
$542$	16880.0i	1.33775i
$543$	− 6690.00i	− 0.528721i
$544$	−170.000	−0.0133983
$545$	0	0
$546$	−18240.0	−1.42967
$547$	7904.00i	0.617826i	0.951090	+	0.308913i	$0.0999651\pi$
$547$	7904.00i	0.617826i	−0.951090	+	0.308913i	$0.900035\pi$
$548$	− 19210.0i	− 1.49746i
$549$	−5130.00	−0.398803
$550$	0	0
$551$	−3888.00	−0.300607
$552$	− 9180.00i	− 0.707838i
$553$	− 512.000i	− 0.0393715i
$554$	10370.0	0.795269
$555$	0	0
$556$	−27472.0	−2.09545
$557$	− 22934.0i	− 1.74460i	−0.488967	−	0.872302i	$-0.662626\pi$
$557$	− 22934.0i	− 1.74460i	0.488967	−	0.872302i	$-0.337374\pi$
$558$	− 6840.00i	− 0.518925i
$559$	−20064.0	−1.51810
$560$	0	0
$561$	66.0000	0.00496706
$562$	− 3510.00i	− 0.263453i
$563$	− 14020.0i	− 1.04951i	−0.851254	−	0.524754i	$-0.824157\pi$
$563$	− 14020.0i	− 1.04951i	0.851254	−	0.524754i	$-0.175843\pi$
$564$	17340.0	1.29458
$565$	0	0
$566$	−24560.0	−1.82391
$567$	− 2592.00i	− 0.191982i
$568$	− 49140.0i	− 3.63005i
$569$	−4230.00	−0.311653	−0.155827	−	0.987784i	$-0.549804\pi$
$569$	−4230.00	−0.311653	−0.155827	+	0.987784i	$0.549804\pi$
$570$	0	0
$571$	−8536.00	−0.625605	−0.312803	−	0.949818i	$-0.601268\pi$
$571$	−8536.00	−0.625605	−0.312803	+	0.949818i	$0.601268\pi$
$572$	7106.00i	0.519435i
$573$	2316.00i	0.168852i
$574$	−15040.0	−1.09365
$575$	0	0
$576$	−2583.00	−0.186849
$577$	− 11982.0i	− 0.864501i	−0.901754	−	0.432251i	$-0.857720\pi$
$577$	− 11982.0i	− 0.864501i	0.901754	−	0.432251i	$-0.142280\pi$
$578$	− 24545.0i	− 1.76633i
$579$	−1182.00	−0.0848398
$580$	0	0
$581$	−11904.0	−0.850019
$582$	7890.00i	0.561943i
$583$	− 4818.00i	− 0.342266i
$584$	25290.0	1.79197
$585$	0	0
$586$	17430.0	1.22871
$587$	− 20396.0i	− 1.43413i	−0.697007	−	0.717064i	$-0.745486\pi$
$587$	− 20396.0i	− 1.43413i	0.697007	−	0.717064i	$-0.254514\pi$
$588$	− 34731.0i	− 2.43585i
$589$	10944.0	0.765602
$590$	0	0
$591$	9174.00	0.638524
$592$	15486.0i	1.07512i
$593$	− 12518.0i	− 0.866868i	−0.901185	−	0.433434i	$-0.857302\pi$
$593$	− 12518.0i	− 0.866868i	0.901185	−	0.433434i	$-0.142698\pi$
$594$	−1485.00	−0.102576
$595$	0	0
$596$	35122.0	2.41385
$597$	7992.00i	0.547891i
$598$	− 12920.0i	− 0.883509i
$599$	25292.0	1.72521	0.862607	−	0.505875i	$-0.168830\pi$
$599$	25292.0	1.72521	0.862607	+	0.505875i	$0.168830\pi$
$600$	0	0
$601$	15962.0	1.08337	0.541683	−	0.840583i	$-0.317787\pi$
$601$	15962.0	1.08337	0.541683	+	0.840583i	$0.317787\pi$
$602$	− 84480.0i	− 5.71951i
$603$	4140.00i	0.279592i
$604$	−4216.00	−0.284018
$605$	0	0
$606$	−750.000	−0.0502750
$607$	− 1600.00i	− 0.106988i	−0.998568	−	0.0534942i	$-0.982964\pi$
$607$	− 1600.00i	− 0.106988i	0.998568	−	0.0534942i	$-0.0170359\pi$
$608$	6120.00i	0.408222i
$609$	−5184.00	−0.344936
$610$	0	0
$611$	12920.0	0.855462
$612$	− 306.000i	− 0.0202113i
$613$	− 2162.00i	− 0.142451i	−0.997460	−	0.0712254i	$-0.977309\pi$
$613$	− 2162.00i	− 0.142451i	0.997460	−	0.0712254i	$-0.0226910\pi$
$614$	41800.0	2.74741
$615$	0	0
$616$	−15840.0	−1.03606
$617$	− 18126.0i	− 1.18270i	−0.806415	−	0.591350i	$-0.798595\pi$
$617$	− 18126.0i	− 1.18270i	0.806415	−	0.591350i	$-0.201405\pi$
$618$	14160.0i	0.921681i
$619$	−17348.0	−1.12645	−0.563227	−	0.826302i	$-0.690440\pi$
$619$	−17348.0	−1.12645	−0.563227	+	0.826302i	$0.690440\pi$
$620$	0	0
$621$	1836.00	0.118641
$622$	27660.0i	1.78306i
$623$	− 30912.0i	− 1.98790i
$624$	10146.0	0.650906
$625$	0	0
$626$	−24130.0	−1.54062
$627$	− 2376.00i	− 0.151337i
$628$	− 40222.0i	− 2.55578i
$629$	348.000	0.0220599
$630$	0	0
$631$	10096.0	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	10096.0	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	720.000i	0.0453166i
$633$	18000.0i	1.13023i
$634$	37850.0	2.37100
$635$	0	0
$636$	−22338.0	−1.39270
$637$	− 25878.0i	− 1.60961i
$638$	2970.00i	0.184300i
$639$	9828.00	0.608435
$640$	0	0
$641$	8922.00	0.549763	0.274881	−	0.961478i	$-0.411361\pi$
$641$	8922.00	0.549763	0.274881	+	0.961478i	$0.411361\pi$
$642$	− 7020.00i	− 0.431553i
$643$	14644.0i	0.898138i	0.893497	+	0.449069i	$0.148244\pi$
$643$	14644.0i	0.898138i	−0.893497	+	0.449069i	$0.851756\pi$
$644$	36992.0	2.26349
$645$	0	0
$646$	720.000	0.0438514
$647$	6932.00i	0.421213i	0.977571	+	0.210607i	$0.0675439\pi$
$647$	6932.00i	0.421213i	−0.977571	+	0.210607i	$0.932456\pi$
$648$	3645.00i	0.220971i
$649$	220.000	0.0133062
$650$	0	0
$651$	14592.0	0.878503
$652$	− 4828.00i	− 0.289999i
$653$	5942.00i	0.356093i	0.984022	+	0.178046i	$0.0569777\pi$
$653$	5942.00i	0.356093i	−0.984022	+	0.178046i	$0.943022\pi$
$654$	2310.00	0.138116
$655$	0	0
$656$	8366.00	0.497923
$657$	5058.00i	0.300352i
$658$	54400.0i	3.22300i
$659$	−484.000	−0.0286100	−0.0143050	−	0.999898i	$-0.504554\pi$
$659$	−484.000	−0.0286100	−0.0143050	+	0.999898i	$0.504554\pi$
$660$	0	0
$661$	−17114.0	−1.00705	−0.503523	−	0.863982i	$-0.667963\pi$
$661$	−17114.0	−1.00705	−0.503523	+	0.863982i	$0.667963\pi$
$662$	− 18380.0i	− 1.07909i
$663$	− 228.000i	− 0.0133556i
$664$	16740.0	0.978370
$665$	0	0
$666$	−7830.00	−0.455565
$667$	− 3672.00i	− 0.213164i
$668$	− 10200.0i	− 0.590793i
$669$	1680.00	0.0970890
$670$	0	0
$671$	−6270.00	−0.360731
$672$	8160.00i	0.468421i
$673$	− 16154.0i	− 0.925247i	−0.886555	−	0.462623i	$-0.846908\pi$
$673$	− 16154.0i	− 0.925247i	0.886555	−	0.462623i	$-0.153092\pi$
$674$	−28430.0	−1.62475
$675$	0	0
$676$	−12801.0	−0.728323
$677$	− 3390.00i	− 0.192449i	−0.995360	−	0.0962247i	$-0.969323\pi$
$677$	− 3390.00i	− 0.192449i	0.995360	−	0.0962247i	$-0.0306767\pi$
$678$	− 810.000i	− 0.0458818i
$679$	−16832.0	−0.951330
$680$	0	0
$681$	15876.0	0.893347
$682$	− 8360.00i	− 0.469386i
$683$	25540.0i	1.43084i	0.698697	+	0.715418i	$0.253764\pi$
$683$	25540.0i	1.43084i	−0.698697	+	0.715418i	$0.746236\pi$
$684$	−11016.0	−0.615800
$685$	0	0
$686$	54080.0	3.00989
$687$	− 15966.0i	− 0.886668i
$688$	46992.0i	2.60400i
$689$	−16644.0	−0.920299
$690$	0	0
$691$	12476.0	0.686844	0.343422	−	0.939181i	$-0.388414\pi$
$691$	12476.0	0.686844	0.343422	+	0.939181i	$0.388414\pi$
$692$	2346.00i	0.128875i
$693$	− 3168.00i	− 0.173654i
$694$	−8260.00	−0.451794
$695$	0	0
$696$	7290.00	0.397021
$697$	− 188.000i	− 0.0102167i
$698$	− 34950.0i	− 1.89524i
$699$	11862.0	0.641863
$700$	0	0
$701$	−20806.0	−1.12102	−0.560508	−	0.828149i	$-0.689394\pi$
$701$	−20806.0	−1.12102	−0.560508	+	0.828149i	$0.689394\pi$
$702$	5130.00i	0.275811i
$703$	− 12528.0i	− 0.672123i
$704$	−3157.00	−0.169011
$705$	0	0
$706$	40470.0	2.15738
$707$	− 1600.00i	− 0.0851120i
$708$	− 1020.00i	− 0.0541440i
$709$	−14198.0	−0.752069	−0.376035	−	0.926606i	$-0.622713\pi$
$709$	−14198.0	−0.752069	−0.376035	+	0.926606i	$0.622713\pi$
$710$	0	0
$711$	−144.000	−0.00759553
$712$	43470.0i	2.28807i
$713$	10336.0i	0.542898i
$714$	960.000	0.0503181
$715$	0	0
$716$	67524.0	3.52443
$717$	− 10080.0i	− 0.525027i
$718$	5120.00i	0.266124i
$719$	−4596.00	−0.238389	−0.119195	−	0.992871i	$-0.538031\pi$
$719$	−4596.00	−0.238389	−0.119195	+	0.992871i	$0.538031\pi$
$720$	0	0
$721$	−30208.0	−1.56034
$722$	8375.00i	0.431697i
$723$	9834.00i	0.505851i
$724$	−37910.0	−1.94601
$725$	0	0
$726$	−1815.00	−0.0927837
$727$	19560.0i	0.997855i	0.866644	+	0.498927i	$0.166273\pi$
$727$	19560.0i	0.997855i	−0.866644	+	0.498927i	$0.833727\pi$
$728$	54720.0i	2.78579i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1056.00	0.0534303
$732$	29070.0i	1.46784i
$733$	1638.00i	0.0825388i	0.999148	+	0.0412694i	$0.0131402\pi$
$733$	1638.00i	0.0825388i	−0.999148	+	0.0412694i	$0.986860\pi$
$734$	−68320.0	−3.43561
$735$	0	0
$736$	−5780.00	−0.289475
$737$	5060.00i	0.252900i
$738$	4230.00i	0.210987i
$739$	15592.0	0.776131	0.388066	−	0.921632i	$-0.373143\pi$
$739$	15592.0	0.776131	0.388066	+	0.921632i	$0.373143\pi$
$740$	0	0
$741$	−8208.00	−0.406921
$742$	− 70080.0i	− 3.46727i
$743$	− 592.000i	− 0.0292307i	−0.999893	−	0.0146153i	$-0.995348\pi$
$743$	− 592.000i	− 0.0292307i	0.999893	−	0.0146153i	$-0.00465237\pi$
$744$	−20520.0	−1.01116
$745$	0	0
$746$	9790.00	0.480479
$747$	3348.00i	0.163985i
$748$	− 374.000i	− 0.0182818i
$749$	14976.0	0.730589
$750$	0	0
$751$	39832.0	1.93541	0.967703	−	0.252092i	$-0.0811186\pi$
$751$	39832.0	1.93541	0.967703	+	0.252092i	$0.0811186\pi$
$752$	− 30260.0i	− 1.46738i
$753$	− 6276.00i	− 0.303732i
$754$	10260.0	0.495553
$755$	0	0
$756$	−14688.0	−0.706610
$757$	10958.0i	0.526123i	0.964779	+	0.263062i	$0.0847323\pi$
$757$	10958.0i	0.526123i	−0.964779	+	0.263062i	$0.915268\pi$
$758$	30620.0i	1.46724i
$759$	2244.00	0.107315
$760$	0	0
$761$	−8970.00	−0.427283	−0.213641	−	0.976912i	$-0.568532\pi$
$761$	−8970.00	−0.427283	−0.213641	+	0.976912i	$0.568532\pi$
$762$	33360.0i	1.58596i
$763$	4928.00i	0.233821i
$764$	13124.0	0.621479
$765$	0	0
$766$	−28060.0	−1.32356
$767$	− 760.000i	− 0.0357784i
$768$	24837.0i	1.16696i
$769$	10054.0	0.471465	0.235732	−	0.971818i	$-0.424251\pi$
$769$	10054.0	0.471465	0.235732	+	0.971818i	$0.424251\pi$
$770$	0	0
$771$	1974.00	0.0922074
$772$	6698.00i	0.312262i
$773$	− 26346.0i	− 1.22587i	−0.790132	−	0.612936i	$-0.789988\pi$
$773$	− 26346.0i	− 1.22587i	0.790132	−	0.612936i	$-0.210012\pi$
$774$	−23760.0	−1.10341
$775$	0	0
$776$	23670.0	1.09498
$777$	− 16704.0i	− 0.771239i
$778$	62250.0i	2.86860i
$779$	−6768.00	−0.311282
$780$	0	0
$781$	12012.0	0.550350
$782$	680.000i	0.0310956i
$783$	1458.00i	0.0665449i
$784$	−60609.0	−2.76098
$785$	0	0
$786$	41580.0	1.88691
$787$	− 16040.0i	− 0.726511i	−0.931690	−	0.363256i	$-0.881665\pi$
$787$	− 16040.0i	− 0.726511i	0.931690	−	0.363256i	$-0.118335\pi$
$788$	− 51986.0i	− 2.35016i
$789$	15312.0	0.690902
$790$	0	0
$791$	1728.00	0.0776746
$792$	4455.00i	0.199876i
$793$	21660.0i	0.969948i
$794$	74150.0	3.31421
$795$	0	0
$796$	45288.0	2.01657
$797$	32810.0i	1.45821i	0.684404	+	0.729103i	$0.260062\pi$
$797$	32810.0i	1.45821i	−0.684404	+	0.729103i	$0.739938\pi$
$798$	− 34560.0i	− 1.53310i
$799$	−680.000	−0.0301085
$800$	0	0
$801$	−8694.00	−0.383505
$802$	16790.0i	0.739246i
$803$	6182.00i	0.271679i
$804$	23460.0	1.02907
$805$	0	0
$806$	−28880.0	−1.26210
$807$	− 12714.0i	− 0.554590i
$808$	2250.00i	0.0979638i
$809$	−18918.0	−0.822153	−0.411076	−	0.911601i	$-0.634847\pi$
$809$	−18918.0	−0.822153	−0.411076	+	0.911601i	$0.634847\pi$
$810$	0	0
$811$	−8552.00	−0.370285	−0.185143	−	0.982712i	$-0.559275\pi$
$811$	−8552.00	−0.370285	−0.185143	+	0.982712i	$0.559275\pi$
$812$	29376.0i	1.26958i
$813$	10128.0i	0.436906i
$814$	−9570.00	−0.412074
$815$	0	0
$816$	−534.000	−0.0229090
$817$	− 38016.0i	− 1.62792i
$818$	53490.0i	2.28635i
$819$	−10944.0	−0.466928
$820$	0	0
$821$	−46430.0	−1.97371	−0.986856	−	0.161600i	$-0.948335\pi$
$821$	−46430.0	−1.97371	−0.986856	+	0.161600i	$0.948335\pi$
$822$	− 16950.0i	− 0.719220i
$823$	− 16392.0i	− 0.694276i	−0.937814	−	0.347138i	$-0.887154\pi$
$823$	− 16392.0i	− 0.694276i	0.937814	−	0.347138i	$-0.112846\pi$
$824$	42480.0	1.79595
$825$	0	0
$826$	3200.00	0.134797
$827$	− 13876.0i	− 0.583453i	−0.956502	−	0.291727i	$-0.905770\pi$
$827$	− 13876.0i	− 0.583453i	0.956502	−	0.291727i	$-0.0942297\pi$
$828$	− 10404.0i	− 0.436671i
$829$	24554.0	1.02870	0.514352	−	0.857579i	$-0.328032\pi$
$829$	24554.0	1.02870	0.514352	+	0.857579i	$0.328032\pi$
$830$	0	0
$831$	6222.00	0.259734
$832$	10906.0i	0.454444i
$833$	1362.00i	0.0566513i
$834$	−24240.0	−1.00643
$835$	0	0
$836$	−13464.0	−0.557012
$837$	− 4104.00i	− 0.169480i
$838$	− 10220.0i	− 0.421294i
$839$	−19900.0	−0.818861	−0.409430	−	0.912341i	$-0.634273\pi$
$839$	−19900.0	−0.818861	−0.409430	+	0.912341i	$0.634273\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	− 15350.0i	− 0.628261i
$843$	− 2106.00i	− 0.0860433i
$844$	102000.	4.15993
$845$	0	0
$846$	15300.0	0.621779
$847$	− 3872.00i	− 0.157076i
$848$	38982.0i	1.57859i
$849$	−14736.0	−0.595687
$850$	0	0
$851$	11832.0	0.476611
$852$	− 55692.0i	− 2.23941i
$853$	− 41138.0i	− 1.65128i	−0.564200	−	0.825638i	$-0.690815\pi$
$853$	− 41138.0i	− 1.65128i	0.564200	−	0.825638i	$-0.309185\pi$
$854$	−91200.0	−3.65433
$855$	0	0
$856$	−21060.0	−0.840907
$857$	19910.0i	0.793597i	0.917906	+	0.396799i	$0.129879\pi$
$857$	19910.0i	0.793597i	−0.917906	+	0.396799i	$0.870121\pi$
$858$	6270.00i	0.249481i
$859$	−42924.0	−1.70495	−0.852473	−	0.522772i	$-0.824898\pi$
$859$	−42924.0	−1.70495	−0.852473	+	0.522772i	$0.824898\pi$
$860$	0	0
$861$	−9024.00	−0.357186
$862$	63000.0i	2.48931i
$863$	46236.0i	1.82374i	0.410474	+	0.911872i	$0.365363\pi$
$863$	46236.0i	1.82374i	−0.410474	+	0.911872i	$0.634637\pi$
$864$	2295.00	0.0903675
$865$	0	0
$866$	49510.0	1.94275
$867$	− 14727.0i	− 0.576880i
$868$	− 82688.0i	− 3.23343i
$869$	−176.000	−0.00687042
$870$	0	0
$871$	17480.0	0.680008
$872$	− 6930.00i	− 0.269128i
$873$	4734.00i	0.183530i
$874$	24480.0	0.947424
$875$	0	0
$876$	28662.0	1.10548
$877$	25746.0i	0.991312i	0.868519	+	0.495656i	$0.165072\pi$
$877$	25746.0i	0.991312i	−0.868519	+	0.495656i	$0.834928\pi$
$878$	57200.0i	2.19864i
$879$	10458.0	0.401296
$880$	0	0
$881$	−24550.0	−0.938831	−0.469416	−	0.882977i	$-0.655535\pi$
$881$	−24550.0	−0.938831	−0.469416	+	0.882977i	$0.655535\pi$
$882$	− 30645.0i	− 1.16992i
$883$	19436.0i	0.740740i	0.928884	+	0.370370i	$0.120769\pi$
$883$	19436.0i	0.740740i	−0.928884	+	0.370370i	$0.879231\pi$
$884$	−1292.00	−0.0491569
$885$	0	0
$886$	25900.0	0.982085
$887$	− 22912.0i	− 0.867316i	−0.901077	−	0.433658i	$-0.857223\pi$
$887$	− 22912.0i	− 0.867316i	0.901077	−	0.433658i	$-0.142777\pi$
$888$	23490.0i	0.887695i
$889$	−71168.0	−2.68492
$890$	0	0
$891$	−891.000	−0.0335013
$892$	− 9520.00i	− 0.357347i
$893$	24480.0i	0.917348i
$894$	30990.0	1.15935
$895$	0	0
$896$	−67680.0	−2.52347
$897$	− 7752.00i	− 0.288553i
$898$	54130.0i	2.01152i
$899$	−8208.00	−0.304507
$900$	0	0
$901$	876.000	0.0323904
$902$	5170.00i	0.190845i
$903$	− 50688.0i	− 1.86799i
$904$	−2430.00	−0.0894033
$905$	0	0
$906$	−3720.00	−0.136411
$907$	− 39900.0i	− 1.46070i	−0.683071	−	0.730352i	$-0.739356\pi$
$907$	− 39900.0i	− 1.46070i	0.683071	−	0.730352i	$-0.260644\pi$
$908$	− 89964.0i	− 3.28806i
$909$	−450.000	−0.0164198
$910$	0	0
$911$	29460.0	1.07141	0.535704	−	0.844406i	$-0.320046\pi$
$911$	29460.0	1.07141	0.535704	+	0.844406i	$0.320046\pi$
$912$	19224.0i	0.697994i
$913$	4092.00i	0.148330i
$914$	−78990.0	−2.85860
$915$	0	0
$916$	−90474.0	−3.26348
$917$	88704.0i	3.19440i
$918$	− 270.000i	− 0.00970733i
$919$	−29368.0	−1.05415	−0.527073	−	0.849820i	$-0.676711\pi$
$919$	−29368.0	−1.05415	−0.527073	+	0.849820i	$0.676711\pi$
$920$	0	0
$921$	25080.0	0.897301
$922$	19470.0i	0.695456i
$923$	− 41496.0i	− 1.47980i
$924$	−17952.0	−0.639153
$925$	0	0
$926$	79960.0	2.83763
$927$	8496.00i	0.301020i
$928$	− 4590.00i	− 0.162364i
$929$	−33954.0	−1.19913	−0.599567	−	0.800325i	$-0.704660\pi$
$929$	−33954.0	−1.19913	−0.599567	+	0.800325i	$0.704660\pi$
$930$	0	0
$931$	49032.0	1.72606
$932$	− 67218.0i	− 2.36245i
$933$	16596.0i	0.582346i
$934$	59220.0	2.07467
$935$	0	0
$936$	15390.0	0.537434
$937$	− 2854.00i	− 0.0995049i	−0.998762	−	0.0497525i	$-0.984157\pi$
$937$	− 2854.00i	− 0.0995049i	0.998762	−	0.0497525i	$-0.0158432\pi$
$938$	73600.0i	2.56197i
$939$	−14478.0	−0.503165
$940$	0	0
$941$	−6294.00	−0.218043	−0.109022	−	0.994039i	$-0.534772\pi$
$941$	−6294.00	−0.218043	−0.109022	+	0.994039i	$0.534772\pi$
$942$	− 35490.0i	− 1.22752i
$943$	− 6392.00i	− 0.220734i
$944$	−1780.00	−0.0613708
$945$	0	0
$946$	−29040.0	−0.998067
$947$	2268.00i	0.0778248i	0.999243	+	0.0389124i	$0.0123893\pi$
$947$	2268.00i	0.0778248i	−0.999243	+	0.0389124i	$0.987611\pi$
$948$	816.000i	0.0279562i
$949$	21356.0	0.730501
$950$	0	0
$951$	22710.0	0.774366
$952$	− 2880.00i	− 0.0980476i
$953$	− 26566.0i	− 0.902998i	−0.892272	−	0.451499i	$-0.850889\pi$
$953$	− 26566.0i	− 0.902998i	0.892272	−	0.451499i	$-0.149111\pi$
$954$	−19710.0	−0.668904
$955$	0	0
$956$	−57120.0	−1.93242
$957$	1782.00i	0.0601921i
$958$	74680.0i	2.51858i
$959$	36160.0	1.21759
$960$	0	0
$961$	−6687.00	−0.224464
$962$	33060.0i	1.10800i
$963$	− 4212.00i	− 0.140945i
$964$	55726.0	1.86184
$965$	0	0
$966$	32640.0	1.08714
$967$	11176.0i	0.371661i	0.982582	+	0.185830i	$0.0594975\pi$
$967$	11176.0i	0.371661i	−0.982582	+	0.185830i	$0.940503\pi$
$968$	5445.00i	0.180794i
$969$	432.000	0.0143218
$970$	0	0
$971$	−42316.0	−1.39854	−0.699271	−	0.714856i	$-0.746492\pi$
$971$	−42316.0	−1.39854	−0.699271	+	0.714856i	$0.746492\pi$
$972$	4131.00i	0.136319i
$973$	− 51712.0i	− 1.70381i
$974$	−10280.0	−0.338185
$975$	0	0
$976$	50730.0	1.66376
$977$	− 45054.0i	− 1.47534i	−0.675163	−	0.737669i	$-0.735927\pi$
$977$	− 45054.0i	− 1.47534i	0.675163	−	0.737669i	$-0.264073\pi$
$978$	− 4260.00i	− 0.139284i
$979$	−10626.0	−0.346893
$980$	0	0
$981$	1386.00	0.0451086
$982$	89260.0i	2.90061i
$983$	12300.0i	0.399094i	0.979888	+	0.199547i	$0.0639470\pi$
$983$	12300.0i	0.399094i	−0.979888	+	0.199547i	$0.936053\pi$
$984$	12690.0	0.411120
$985$	0	0
$986$	−540.000	−0.0174413
$987$	32640.0i	1.05263i
$988$	46512.0i	1.49772i
$989$	35904.0	1.15438
$990$	0	0
$991$	36280.0	1.16294	0.581469	−	0.813568i	$-0.302478\pi$
$991$	36280.0	1.16294	0.581469	+	0.813568i	$0.302478\pi$
$992$	12920.0i	0.413519i
$993$	− 11028.0i	− 0.352430i
$994$	174720.	5.57523
$995$	0	0
$996$	18972.0	0.603565
$997$	3290.00i	0.104509i	0.998634	+	0.0522544i	$0.0166407\pi$
$997$	3290.00i	0.104509i	−0.998634	+	0.0522544i	$0.983359\pi$
$998$	22540.0i	0.714921i
$999$	−4698.00	−0.148787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.a.199.1		2
5.2	odd	4		825.4.a.i.1.1		1
5.3	odd	4		33.4.a.a.1.1	✓	1
5.4	even	2	inner	825.4.c.a.199.2		2
15.2	even	4		2475.4.a.b.1.1		1
15.8	even	4		99.4.a.b.1.1		1
20.3	even	4		528.4.a.a.1.1		1
35.13	even	4		1617.4.a.a.1.1		1
40.3	even	4		2112.4.a.y.1.1		1
40.13	odd	4		2112.4.a.l.1.1		1
55.43	even	4		363.4.a.h.1.1		1
60.23	odd	4		1584.4.a.t.1.1		1
165.98	odd	4		1089.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.a.1.1	✓	1	5.3	odd	4
99.4.a.b.1.1		1	15.8	even	4
363.4.a.h.1.1		1	55.43	even	4
528.4.a.a.1.1		1	20.3	even	4
825.4.a.i.1.1		1	5.2	odd	4
825.4.c.a.199.1		2	1.1	even	1	trivial
825.4.c.a.199.2		2	5.4	even	2	inner
1089.4.a.a.1.1		1	165.98	odd	4
1584.4.a.t.1.1		1	60.23	odd	4
1617.4.a.a.1.1		1	35.13	even	4
2112.4.a.l.1.1		1	40.13	odd	4
2112.4.a.y.1.1		1	40.3	even	4
2475.4.a.b.1.1		1	15.2	even	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} -15.0000 q^{6} -32.0000i q^{7} +45.0000i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} -15.0000 q^{6} -32.0000i q^{7} +45.0000i q^{8} -9.00000 q^{9} -11.0000 q^{11} +51.0000i q^{12} +38.0000i q^{13} -160.000 q^{14} +89.0000 q^{16} -2.00000i q^{17} +45.0000i q^{18} -72.0000 q^{19} -96.0000 q^{21} +55.0000i q^{22} -68.0000i q^{23} +135.000 q^{24} +190.000 q^{26} +27.0000i q^{27} +544.000i q^{28} +54.0000 q^{29} -152.000 q^{31} -85.0000i q^{32} +33.0000i q^{33} -10.0000 q^{34} +153.000 q^{36} +174.000i q^{37} +360.000i q^{38} +114.000 q^{39} +94.0000 q^{41} +480.000i q^{42} +528.000i q^{43} +187.000 q^{44} -340.000 q^{46} -340.000i q^{47} -267.000i q^{48} -681.000 q^{49} -6.00000 q^{51} -646.000i q^{52} +438.000i q^{53} +135.000 q^{54} +1440.00 q^{56} +216.000i q^{57} -270.000i q^{58} -20.0000 q^{59} +570.000 q^{61} +760.000i q^{62} +288.000i q^{63} +287.000 q^{64} +165.000 q^{66} -460.000i q^{67} +34.0000i q^{68} -204.000 q^{69} -1092.00 q^{71} -405.000i q^{72} -562.000i q^{73} +870.000 q^{74} +1224.00 q^{76} +352.000i q^{77} -570.000i q^{78} +16.0000 q^{79} +81.0000 q^{81} -470.000i q^{82} -372.000i q^{83} +1632.00 q^{84} +2640.00 q^{86} -162.000i q^{87} -495.000i q^{88} +966.000 q^{89} +1216.00 q^{91} +1156.00i q^{92} +456.000i q^{93} -1700.00 q^{94} -255.000 q^{96} -526.000i q^{97} +3405.00i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9}+O(q^{10}) \) 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9	\( 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9} - 22 q^{11} - 320 q^{14} + 178 q^{16} - 144 q^{19} - 192 q^{21} + 270 q^{24} + 380 q^{26} + 108 q^{29} - 304 q^{31} - 20 q^{34} + 306 q^{36} + 228 q^{39} + 188 q^{41} + 374 q^{44} - 680 q^{46} - 1362 q^{49} - 12 q^{51} + 270 q^{54} + 2880 q^{56} - 40 q^{59} + 1140 q^{61} + 574 q^{64} + 330 q^{66} - 408 q^{69} - 2184 q^{71} + 1740 q^{74} + 2448 q^{76} + 32 q^{79} + 162 q^{81} + 3264 q^{84} + 5280 q^{86} + 1932 q^{89} + 2432 q^{91} - 3400 q^{94} - 510 q^{96} + 198 q^{99}+O(q^{100}) \) 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9 - 22 * q^11 - 320 * q^14 + 178 * q^16 - 144 * q^19 - 192 * q^21 + 270 * q^24 + 380 * q^26 + 108 * q^29 - 304 * q^31 - 20 * q^34 + 306 * q^36 + 228 * q^39 + 188 * q^41 + 374 * q^44 - 680 * q^46 - 1362 * q^49 - 12 * q^51 + 270 * q^54 + 2880 * q^56 - 40 * q^59 + 1140 * q^61 + 574 * q^64 + 330 * q^66 - 408 * q^69 - 2184 * q^71 + 1740 * q^74 + 2448 * q^76 + 32 * q^79 + 162 * q^81 + 3264 * q^84 + 5280 * q^86 + 1932 * q^89 + 2432 * q^91 - 3400 * q^94 - 510 * q^96 + 198 * q^99

Introduction

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