LMFDB - Embedded newform 825.4.c.i.199.4

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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
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			199.4
Root			$3.37228i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.i.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228i q^{2} -3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} -4.74456i q^{7} +15.6060i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+3.37228i q^{2} -3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} -4.74456i q^{7} +15.6060i q^{8} -9.00000 q^{9} +11.0000 q^{11} +10.1168i q^{12} +15.0217i q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684i q^{17} -30.3505i q^{18} +78.7011 q^{19} -14.2337 q^{21} +37.0951i q^{22} -112.000i q^{23} +46.8179 q^{24} -50.6576 q^{26} +27.0000i q^{27} +16.0000i q^{28} -243.125 q^{29} +278.717 q^{31} -143.606i q^{32} -33.0000i q^{33} -246.745 q^{34} +30.3505 q^{36} +102.380i q^{37} +265.402i q^{38} +45.0652 q^{39} -241.255 q^{41} -48.0000i q^{42} +280.016i q^{43} -37.0951 q^{44} +377.696 q^{46} -169.870i q^{47} +238.818i q^{48} +320.489 q^{49} +219.505 q^{51} -50.6576i q^{52} +409.652i q^{53} -91.0516 q^{54} +74.0435 q^{56} -236.103i q^{57} -819.886i q^{58} -196.000 q^{59} -701.359 q^{61} +939.913i q^{62} +42.7011i q^{63} -152.568 q^{64} +111.285 q^{66} +900.587i q^{67} -246.745i q^{68} -336.000 q^{69} +756.500 q^{71} -140.454i q^{72} +1019.81i q^{73} -345.255 q^{74} -265.402 q^{76} -52.1902i q^{77} +151.973i q^{78} +327.549 q^{79} +81.0000 q^{81} -813.581i q^{82} +756.619i q^{83} +48.0000 q^{84} -944.293 q^{86} +729.375i q^{87} +171.666i q^{88} -508.978 q^{89} +71.2716 q^{91} +377.696i q^{92} -836.152i q^{93} +572.848 q^{94} -430.818 q^{96} +614.358i q^{97} +1080.78i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9	$ 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44} + 224 q^{46} + 1236 q^{49} - 156 q^{51} - 54 q^{54} + 480 q^{56} - 784 q^{59} - 2208 q^{61} - 1426 q^{64} + 66 q^{66} - 1344 q^{69} + 912 q^{71} - 1404 q^{74} - 648 q^{76} + 460 q^{79} + 324 q^{81} + 192 q^{84} - 2904 q^{86} - 1944 q^{89} - 680 q^{91} + 1648 q^{94} - 1482 q^{96} - 396 q^{99}+O(q^{100}) $ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 + 44 * q^11 + 64 * q^14 - 238 * q^16 + 108 * q^19 + 12 * q^21 - 54 * q^24 + 188 * q^26 - 444 * q^29 - 80 * q^31 - 964 * q^34 + 18 * q^36 + 456 * q^39 - 988 * q^41 - 22 * q^44 + 224 * q^46 + 1236 * q^49 - 156 * q^51 - 54 * q^54 + 480 * q^56 - 784 * q^59 - 2208 * q^61 - 1426 * q^64 + 66 * q^66 - 1344 * q^69 + 912 * q^71 - 1404 * q^74 - 648 * q^76 + 460 * q^79 + 324 * q^81 + 192 * q^84 - 2904 * q^86 - 1944 * q^89 - 680 * q^91 + 1648 * q^94 - 1482 * q^96 - 396 * 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 $

$2$	3.37228i	1.19228i	0.802880	+	0.596141i	$0.203300\pi$
$2$	3.37228i	1.19228i	−0.802880	+	0.596141i	$0.796700\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−3.37228	−0.421535
$5$	0	0
$5$	0	0
$6$	10.1168	0.688364
$7$	− 4.74456i	− 0.256182i	−0.991762	−	0.128091i	$-0.959115\pi$
$7$	− 4.74456i	− 0.256182i	0.991762	−	0.128091i	$-0.0408850\pi$
$8$	15.6060i	0.689693i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	10.1168i	0.243373i
$13$	15.0217i	0.320483i	0.987078	+	0.160242i	$0.0512274\pi$
$13$	15.0217i	0.320483i	−0.987078	+	0.160242i	$0.948773\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	−79.6060	−1.24384
$17$	73.1684i	1.04388i	0.852982	+	0.521940i	$0.174791\pi$
$17$	73.1684i	1.04388i	−0.852982	+	0.521940i	$0.825209\pi$
$18$	− 30.3505i	− 0.397427i
$19$	78.7011	0.950277	0.475138	−	0.879911i	$-0.342398\pi$
$19$	78.7011	0.950277	0.475138	+	0.879911i	$0.342398\pi$
$20$	0	0
$21$	−14.2337	−0.147907
$22$	37.0951i	0.359486i
$23$	− 112.000i	− 1.01537i	−0.861541	−	0.507687i	$-0.830501\pi$
$23$	− 112.000i	− 1.01537i	0.861541	−	0.507687i	$-0.169499\pi$
$24$	46.8179	0.398194
$25$	0	0
$26$	−50.6576	−0.382106
$27$	27.0000i	0.192450i
$28$	16.0000i	0.107990i
$29$	−243.125	−1.55680	−0.778399	−	0.627769i	$-0.783968\pi$
$29$	−243.125	−1.55680	−0.778399	+	0.627769i	$0.783968\pi$
$30$	0	0
$31$	278.717	1.61481	0.807405	−	0.589998i	$-0.200871\pi$
$31$	278.717	1.61481	0.807405	+	0.589998i	$0.200871\pi$
$32$	− 143.606i	− 0.793318i
$33$	− 33.0000i	− 0.174078i
$34$	−246.745	−1.24460
$35$	0	0
$36$	30.3505	0.140512
$37$	102.380i	0.454898i	0.973790	+	0.227449i	$0.0730385\pi$
$37$	102.380i	0.454898i	−0.973790	+	0.227449i	$0.926961\pi$
$38$	265.402i	1.13300i
$39$	45.0652	0.185031
$40$	0	0
$41$	−241.255	−0.918970	−0.459485	−	0.888186i	$-0.651966\pi$
$41$	−241.255	−0.918970	−0.459485	+	0.888186i	$0.651966\pi$
$42$	− 48.0000i	− 0.176347i
$43$	280.016i	0.993071i	0.868016	+	0.496536i	$0.165395\pi$
$43$	280.016i	0.993071i	−0.868016	+	0.496536i	$0.834605\pi$
$44$	−37.0951	−0.127098
$45$	0	0
$46$	377.696	1.21061
$47$	− 169.870i	− 0.527192i	−0.964633	−	0.263596i	$-0.915091\pi$
$47$	− 169.870i	− 0.527192i	0.964633	−	0.263596i	$-0.0849085\pi$
$48$	238.818i	0.718133i
$49$	320.489	0.934371
$50$	0	0
$51$	219.505	0.602684
$52$	− 50.6576i	− 0.135095i
$53$	409.652i	1.06170i	0.847466	+	0.530849i	$0.178127\pi$
$53$	409.652i	1.06170i	−0.847466	+	0.530849i	$0.821873\pi$
$54$	−91.0516	−0.229455
$55$	0	0
$56$	74.0435	0.176687
$57$	− 236.103i	− 0.548643i
$58$	− 819.886i	− 1.85614i
$59$	−196.000	−0.432492	−0.216246	−	0.976339i	$-0.569381\pi$
$59$	−196.000	−0.432492	−0.216246	+	0.976339i	$0.569381\pi$
$60$	0	0
$61$	−701.359	−1.47213	−0.736064	−	0.676912i	$-0.763318\pi$
$61$	−701.359	−1.47213	−0.736064	+	0.676912i	$0.763318\pi$
$62$	939.913i	1.92531i
$63$	42.7011i	0.0853941i
$64$	−152.568	−0.297984
$65$	0	0
$66$	111.285	0.207550
$67$	900.587i	1.64215i	0.570819	+	0.821076i	$0.306626\pi$
$67$	900.587i	1.64215i	−0.570819	+	0.821076i	$0.693374\pi$
$68$	− 246.745i	− 0.440032i
$69$	−336.000	−0.586227
$70$	0	0
$71$	756.500	1.26451	0.632254	−	0.774762i	$-0.282130\pi$
$71$	756.500	1.26451	0.632254	+	0.774762i	$0.282130\pi$
$72$	− 140.454i	− 0.229898i
$73$	1019.81i	1.63507i	0.575877	+	0.817536i	$0.304661\pi$
$73$	1019.81i	1.63507i	−0.575877	+	0.817536i	$0.695339\pi$
$74$	−345.255	−0.542367
$75$	0	0
$76$	−265.402	−0.400575
$77$	− 52.1902i	− 0.0772419i
$78$	151.973i	0.220609i
$79$	327.549	0.466483	0.233241	−	0.972419i	$-0.425067\pi$
$79$	327.549	0.466483	0.233241	+	0.972419i	$0.425067\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 813.581i	− 1.09567i
$83$	756.619i	1.00060i	0.865852	+	0.500300i	$0.166777\pi$
$83$	756.619i	1.00060i	−0.865852	+	0.500300i	$0.833223\pi$
$84$	48.0000	0.0623480
$85$	0	0
$86$	−944.293	−1.18402
$87$	729.375i	0.898818i
$88$	171.666i	0.207950i
$89$	−508.978	−0.606198	−0.303099	−	0.952959i	$-0.598021\pi$
$89$	−508.978	−0.606198	−0.303099	+	0.952959i	$0.598021\pi$
$90$	0	0
$91$	71.2716	0.0821022
$92$	377.696i	0.428016i
$93$	− 836.152i	− 0.932311i
$94$	572.848	0.628561
$95$	0	0
$96$	−430.818	−0.458023
$97$	614.358i	0.643079i	0.946896	+	0.321539i	$0.104200\pi$
$97$	614.358i	0.643079i	−0.946896	+	0.321539i	$0.895800\pi$
$98$	1080.78i	1.11403i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−1015.92	−1.00087	−0.500434	−	0.865775i	$-0.666826\pi$
$101$	−1015.92	−1.00087	−0.500434	+	0.865775i	$0.666826\pi$
$102$	740.234i	0.718569i
$103$	− 1102.16i	− 1.05436i	−0.849753	−	0.527181i	$-0.823249\pi$
$103$	− 1102.16i	− 1.05436i	0.849753	−	0.527181i	$-0.176751\pi$
$104$	−234.429	−0.221035
$105$	0	0
$106$	−1381.46	−1.26584
$107$	1377.58i	1.24463i	0.782767	+	0.622315i	$0.213808\pi$
$107$	1377.58i	1.24463i	−0.782767	+	0.622315i	$0.786192\pi$
$108$	− 91.0516i	− 0.0811245i
$109$	−320.217	−0.281388	−0.140694	−	0.990053i	$-0.544933\pi$
$109$	−320.217	−0.281388	−0.140694	+	0.990053i	$0.544933\pi$
$110$	0	0
$111$	307.141	0.262636
$112$	377.696i	0.318651i
$113$	1629.45i	1.35651i	0.734828	+	0.678254i	$0.237263\pi$
$113$	1629.45i	1.35651i	−0.734828	+	0.678254i	$0.762737\pi$
$114$	796.206	0.654136
$115$	0	0
$116$	819.886	0.656245
$117$	− 135.196i	− 0.106828i
$118$	− 660.967i	− 0.515652i
$119$	347.152	0.267423
$120$	0	0
$121$	121.000	0.0909091
$122$	− 2365.18i	− 1.75519i
$123$	723.766i	0.530568i
$124$	−939.913	−0.680699
$125$	0	0
$126$	−144.000	−0.101814
$127$	2291.26i	1.60091i	0.599390	+	0.800457i	$0.295410\pi$
$127$	2291.26i	1.60091i	−0.599390	+	0.800457i	$0.704590\pi$
$128$	− 1663.35i	− 1.14860i
$129$	840.049	0.573350
$130$	0	0
$131$	−1147.41	−0.765267	−0.382633	−	0.923900i	$-0.624983\pi$
$131$	−1147.41	−0.765267	−0.382633	+	0.923900i	$0.624983\pi$
$132$	111.285i	0.0733799i
$133$	− 373.402i	− 0.243444i
$134$	−3037.03	−1.95791
$135$	0	0
$136$	−1141.86	−0.719956
$137$	1268.60i	0.791121i	0.918440	+	0.395561i	$0.129450\pi$
$137$	1268.60i	0.791121i	−0.918440	+	0.395561i	$0.870550\pi$
$138$	− 1133.09i	− 0.698947i
$139$	486.288	0.296737	0.148368	−	0.988932i	$-0.452598\pi$
$139$	486.288	0.296737	0.148368	+	0.988932i	$0.452598\pi$
$140$	0	0
$141$	−509.609	−0.304374
$142$	2551.13i	1.50765i
$143$	165.239i	0.0966294i
$144$	716.454	0.414614
$145$	0	0
$146$	−3439.10	−1.94947
$147$	− 961.467i	− 0.539459i
$148$	− 345.255i	− 0.191756i
$149$	−2354.11	−1.29434	−0.647169	−	0.762346i	$-0.724047\pi$
$149$	−2354.11	−1.29434	−0.647169	+	0.762346i	$0.724047\pi$
$150$	0	0
$151$	−570.070	−0.307229	−0.153615	−	0.988131i	$-0.549091\pi$
$151$	−570.070	−0.307229	−0.153615	+	0.988131i	$0.549091\pi$
$152$	1228.21i	0.655399i
$153$	− 658.516i	− 0.347960i
$154$	176.000	0.0920941
$155$	0	0
$156$	−151.973	−0.0779971
$157$	− 2072.67i	− 1.05361i	−0.849985	−	0.526807i	$-0.823389\pi$
$157$	− 2072.67i	− 1.05361i	0.849985	−	0.526807i	$-0.176611\pi$
$158$	1104.59i	0.556179i
$159$	1228.96	0.612972
$160$	0	0
$161$	−531.391	−0.260121
$162$	273.155i	0.132476i
$163$	− 2676.51i	− 1.28614i	−0.765808	−	0.643069i	$-0.777661\pi$
$163$	− 2676.51i	− 1.28614i	0.765808	−	0.643069i	$-0.222339\pi$
$164$	813.581	0.387378
$165$	0	0
$166$	−2551.53	−1.19300
$167$	− 1188.12i	− 0.550536i	−0.961368	−	0.275268i	$-0.911233\pi$
$167$	− 1188.12i	− 0.550536i	0.961368	−	0.275268i	$-0.0887665\pi$
$168$	− 222.130i	− 0.102010i
$169$	1971.35	0.897290
$170$	0	0
$171$	−708.310	−0.316759
$172$	− 944.293i	− 0.418615i
$173$	− 807.147i	− 0.354718i	−0.984146	−	0.177359i	$-0.943245\pi$
$173$	− 807.147i	− 0.354718i	0.984146	−	0.177359i	$-0.0567554\pi$
$174$	−2459.66	−1.07164
$175$	0	0
$176$	−875.666	−0.375033
$177$	588.000i	0.249699i
$178$	− 1716.42i	− 0.722758i
$179$	1950.39	0.814408	0.407204	−	0.913337i	$-0.366504\pi$
$179$	1950.39	0.814408	0.407204	+	0.913337i	$0.366504\pi$
$180$	0	0
$181$	1061.61	0.435959	0.217980	−	0.975953i	$-0.430053\pi$
$181$	1061.61	0.435959	0.217980	+	0.975953i	$0.430053\pi$
$182$	240.348i	0.0978889i
$183$	2104.08i	0.849933i
$184$	1747.87	0.700297
$185$	0	0
$186$	2819.74	1.11158
$187$	804.853i	0.314742i
$188$	572.848i	0.222230i
$189$	128.103	0.0493023
$190$	0	0
$191$	2136.41	0.809348	0.404674	−	0.914461i	$-0.367385\pi$
$191$	2136.41	0.809348	0.404674	+	0.914461i	$0.367385\pi$
$192$	457.704i	0.172041i
$193$	− 3947.76i	− 1.47236i	−0.676784	−	0.736181i	$-0.736627\pi$
$193$	− 3947.76i	− 1.47236i	0.676784	−	0.736181i	$-0.263373\pi$
$194$	−2071.79	−0.766731
$195$	0	0
$196$	−1080.78	−0.393870
$197$	923.886i	0.334133i	0.985946	+	0.167066i	$0.0534294\pi$
$197$	923.886i	0.334133i	−0.985946	+	0.167066i	$0.946571\pi$
$198$	− 333.856i	− 0.119829i
$199$	476.152	0.169616	0.0848078	−	0.996397i	$-0.472972\pi$
$199$	476.152	0.169616	0.0848078	+	0.996397i	$0.472972\pi$
$200$	0	0
$201$	2701.76	0.948097
$202$	− 3425.96i	− 1.19332i
$203$	1153.52i	0.398824i
$204$	−740.234	−0.254053
$205$	0	0
$206$	3716.80	1.25710
$207$	1008.00i	0.338458i
$208$	− 1195.82i	− 0.398631i
$209$	865.712	0.286519
$210$	0	0
$211$	−4918.24	−1.60467	−0.802336	−	0.596872i	$-0.796410\pi$
$211$	−4918.24	−1.60467	−0.802336	+	0.596872i	$0.796410\pi$
$212$	− 1381.46i	− 0.447543i
$213$	− 2269.50i	− 0.730064i
$214$	−4645.57	−1.48395
$215$	0	0
$216$	−421.361	−0.132731
$217$	− 1322.39i	− 0.413686i
$218$	− 1079.86i	− 0.335494i
$219$	3059.44	0.944010
$220$	0	0
$221$	−1099.12	−0.334546
$222$	1035.77i	0.313136i
$223$	− 2100.29i	− 0.630700i	−0.948975	−	0.315350i	$-0.897878\pi$
$223$	− 2100.29i	− 0.630700i	0.948975	−	0.315350i	$-0.102122\pi$
$224$	−681.348	−0.203234
$225$	0	0
$226$	−5494.95	−1.61734
$227$	− 2257.16i	− 0.659970i	−0.943986	−	0.329985i	$-0.892956\pi$
$227$	− 2257.16i	− 0.659970i	0.943986	−	0.329985i	$-0.107044\pi$
$228$	796.206i	0.231272i
$229$	5311.07	1.53260	0.766301	−	0.642482i	$-0.222095\pi$
$229$	5311.07	1.53260	0.766301	+	0.642482i	$0.222095\pi$
$230$	0	0
$231$	−156.571	−0.0445956
$232$	− 3794.20i	− 1.07371i
$233$	− 2466.27i	− 0.693435i	−0.937970	−	0.346718i	$-0.887296\pi$
$233$	− 2466.27i	− 0.693435i	0.937970	−	0.346718i	$-0.112704\pi$
$234$	455.918	0.127369
$235$	0	0
$236$	660.967	0.182311
$237$	− 982.646i	− 0.269324i
$238$	1170.70i	0.318844i
$239$	−1429.40	−0.386863	−0.193432	−	0.981114i	$-0.561962\pi$
$239$	−1429.40	−0.386863	−0.193432	+	0.981114i	$0.561962\pi$
$240$	0	0
$241$	−978.989	−0.261669	−0.130835	−	0.991404i	$-0.541766\pi$
$241$	−978.989	−0.261669	−0.130835	+	0.991404i	$0.541766\pi$
$242$	408.046i	0.108389i
$243$	− 243.000i	− 0.0641500i
$244$	2365.18	0.620553
$245$	0	0
$246$	−2440.74	−0.632586
$247$	1182.23i	0.304548i
$248$	4349.65i	1.11372i
$249$	2269.86	0.577696
$250$	0	0
$251$	−6530.63	−1.64227	−0.821135	−	0.570734i	$-0.806659\pi$
$251$	−6530.63	−1.64227	−0.821135	+	0.570734i	$0.806659\pi$
$252$	− 144.000i	− 0.0359966i
$253$	− 1232.00i	− 0.306147i
$254$	−7726.76	−1.90874
$255$	0	0
$256$	4388.74	1.07147
$257$	8130.26i	1.97335i	0.162696	+	0.986676i	$0.447981\pi$
$257$	8130.26i	1.97335i	−0.162696	+	0.986676i	$0.552019\pi$
$258$	2832.88i	0.683595i
$259$	485.750	0.116537
$260$	0	0
$261$	2188.12	0.518933
$262$	− 3869.40i	− 0.912414i
$263$	4549.42i	1.06665i	0.845910	+	0.533326i	$0.179058\pi$
$263$	4549.42i	1.06665i	−0.845910	+	0.533326i	$0.820942\pi$
$264$	514.997	0.120060
$265$	0	0
$266$	1259.22	0.290254
$267$	1526.93i	0.349988i
$268$	− 3037.03i	− 0.692225i
$269$	29.1522	0.00660760	0.00330380	−	0.999995i	$-0.498948\pi$
$269$	29.1522	0.00660760	0.00330380	+	0.999995i	$0.498948\pi$
$270$	0	0
$271$	7711.22	1.72850	0.864250	−	0.503063i	$-0.167794\pi$
$271$	7711.22	1.72850	0.864250	+	0.503063i	$0.167794\pi$
$272$	− 5824.64i	− 1.29842i
$273$	− 213.815i	− 0.0474017i
$274$	−4278.07	−0.943239
$275$	0	0
$276$	1133.09	0.247115
$277$	1127.52i	0.244571i	0.992495	+	0.122286i	$0.0390224\pi$
$277$	1127.52i	0.244571i	−0.992495	+	0.122286i	$0.960978\pi$
$278$	1639.90i	0.353794i
$279$	−2508.46	−0.538270
$280$	0	0
$281$	−1872.47	−0.397517	−0.198758	−	0.980049i	$-0.563691\pi$
$281$	−1872.47	−0.397517	−0.198758	+	0.980049i	$0.563691\pi$
$282$	− 1718.54i	− 0.362900i
$283$	− 2124.48i	− 0.446245i	−0.974790	−	0.223123i	$-0.928375\pi$
$283$	− 2124.48i	− 0.446245i	0.974790	−	0.223123i	$-0.0716251\pi$
$284$	−2551.13	−0.533034
$285$	0	0
$286$	−557.233	−0.115209
$287$	1144.65i	0.235424i
$288$	1292.45i	0.264439i
$289$	−440.621	−0.0896846
$290$	0	0
$291$	1843.07	0.371282
$292$	− 3439.10i	− 0.689241i
$293$	3324.19i	0.662802i	0.943490	+	0.331401i	$0.107521\pi$
$293$	3324.19i	0.662802i	−0.943490	+	0.331401i	$0.892479\pi$
$294$	3242.34	0.643187
$295$	0	0
$296$	−1597.75	−0.313740
$297$	297.000i	0.0580259i
$298$	− 7938.73i	− 1.54322i
$299$	1682.44	0.325411
$300$	0	0
$301$	1328.55	0.254407
$302$	− 1922.44i	− 0.366304i
$303$	3047.75i	0.577851i
$304$	−6265.07	−1.18200
$305$	0	0
$306$	2220.70	0.414866
$307$	− 1698.94i	− 0.315843i	−0.987452	−	0.157921i	$-0.949521\pi$
$307$	− 1698.94i	− 0.315843i	0.987452	−	0.157921i	$-0.0504793\pi$
$308$	176.000i	0.0325602i
$309$	−3306.49	−0.608736
$310$	0	0
$311$	6928.83	1.26334	0.631668	−	0.775239i	$-0.282370\pi$
$311$	6928.83	1.26334	0.631668	+	0.775239i	$0.282370\pi$
$312$	703.287i	0.127615i
$313$	3560.75i	0.643020i	0.946906	+	0.321510i	$0.104190\pi$
$313$	3560.75i	0.643020i	−0.946906	+	0.321510i	$0.895810\pi$
$314$	6989.64	1.25620
$315$	0	0
$316$	−1104.59	−0.196639
$317$	332.750i	0.0589561i	0.999565	+	0.0294780i	$0.00938451\pi$
$317$	332.750i	0.0589561i	−0.999565	+	0.0294780i	$0.990615\pi$
$318$	4144.39i	0.730835i
$319$	−2674.37	−0.469393
$320$	0	0
$321$	4132.73	0.718587
$322$	− 1792.00i	− 0.310137i
$323$	5758.43i	0.991975i
$324$	−273.155	−0.0468372
$325$	0	0
$326$	9025.94	1.53344
$327$	960.652i	0.162459i
$328$	− 3765.02i	− 0.633807i
$329$	−805.957	−0.135057
$330$	0	0
$331$	−541.445	−0.0899108	−0.0449554	−	0.998989i	$-0.514315\pi$
$331$	−541.445	−0.0899108	−0.0449554	+	0.998989i	$0.514315\pi$
$332$	− 2551.53i	− 0.421788i
$333$	− 921.423i	− 0.151633i
$334$	4006.67	0.656393
$335$	0	0
$336$	1133.09	0.183973
$337$	816.531i	0.131986i	0.997820	+	0.0659930i	$0.0210215\pi$
$337$	816.531i	0.131986i	−0.997820	+	0.0659930i	$0.978978\pi$
$338$	6647.94i	1.06982i
$339$	4888.34	0.783180
$340$	0	0
$341$	3065.89	0.486883
$342$	− 2388.62i	− 0.377666i
$343$	− 3147.97i	− 0.495552i
$344$	−4369.92	−0.684914
$345$	0	0
$346$	2721.93	0.422924
$347$	6260.53i	0.968539i	0.874919	+	0.484269i	$0.160914\pi$
$347$	6260.53i	0.968539i	−0.874919	+	0.484269i	$0.839086\pi$
$348$	− 2459.66i	− 0.378884i
$349$	12768.5	1.95840	0.979198	−	0.202906i	$-0.0650386\pi$
$349$	12768.5	1.95840	0.979198	+	0.202906i	$0.0650386\pi$
$350$	0	0
$351$	−405.587	−0.0616771
$352$	− 1579.67i	− 0.239194i
$353$	2649.28i	0.399453i	0.979852	+	0.199727i	$0.0640054\pi$
$353$	2649.28i	0.399453i	−0.979852	+	0.199727i	$0.935995\pi$
$354$	−1982.90	−0.297712
$355$	0	0
$356$	1716.42	0.255534
$357$	− 1041.46i	− 0.154397i
$358$	6577.27i	0.971004i
$359$	3203.91	0.471020	0.235510	−	0.971872i	$-0.424324\pi$
$359$	3203.91	0.471020	0.235510	+	0.971872i	$0.424324\pi$
$360$	0	0
$361$	−665.143	−0.0969737
$362$	3580.04i	0.519786i
$363$	− 363.000i	− 0.0524864i
$364$	−240.348	−0.0346089
$365$	0	0
$366$	−7095.54	−1.01336
$367$	− 8429.40i	− 1.19894i	−0.800397	−	0.599470i	$-0.795378\pi$
$367$	− 8429.40i	− 1.19894i	0.800397	−	0.599470i	$-0.204622\pi$
$368$	8915.87i	1.26297i
$369$	2171.30	0.306323
$370$	0	0
$371$	1943.62	0.271988
$372$	2819.74i	0.393002i
$373$	9388.53i	1.30327i	0.758533	+	0.651635i	$0.225917\pi$
$373$	9388.53i	1.30327i	−0.758533	+	0.651635i	$0.774083\pi$
$374$	−2714.19	−0.375261
$375$	0	0
$376$	2650.98	0.363600
$377$	− 3652.16i	− 0.498928i
$378$	432.000i	0.0587822i
$379$	14264.5	1.93329	0.966647	−	0.256112i	$-0.0824415\pi$
$379$	14264.5	1.93329	0.966647	+	0.256112i	$0.0824415\pi$
$380$	0	0
$381$	6873.77	0.924288
$382$	7204.58i	0.964970i
$383$	− 13462.2i	− 1.79605i	−0.439942	−	0.898026i	$-0.645001\pi$
$383$	− 13462.2i	− 1.79605i	0.439942	−	0.898026i	$-0.354999\pi$
$384$	−4990.05	−0.663144
$385$	0	0
$386$	13313.0	1.75547
$387$	− 2520.15i	− 0.331024i
$388$	− 2071.79i	− 0.271080i
$389$	941.881	0.122764	0.0613821	−	0.998114i	$-0.480449\pi$
$389$	941.881	0.122764	0.0613821	+	0.998114i	$0.480449\pi$
$390$	0	0
$391$	8194.87	1.05993
$392$	5001.54i	0.644429i
$393$	3442.24i	0.441827i
$394$	−3115.60	−0.398380
$395$	0	0
$396$	333.856	0.0423659
$397$	− 847.839i	− 0.107183i	−0.998563	−	0.0535917i	$-0.982933\pi$
$397$	− 847.839i	− 0.107183i	0.998563	−	0.0535917i	$-0.0170669\pi$
$398$	1605.72i	0.202230i
$399$	−1120.21	−0.140553
$400$	0	0
$401$	12203.6	1.51975	0.759875	−	0.650069i	$-0.225260\pi$
$401$	12203.6	1.51975	0.759875	+	0.650069i	$0.225260\pi$
$402$	9111.10i	1.13040i
$403$	4186.82i	0.517520i
$404$	3425.96	0.421901
$405$	0	0
$406$	−3890.00	−0.475511
$407$	1126.18i	0.137157i
$408$	3425.59i	0.415667i
$409$	−8759.53	−1.05900	−0.529500	−	0.848310i	$-0.677620\pi$
$409$	−8759.53	−1.05900	−0.529500	+	0.848310i	$0.677620\pi$
$410$	0	0
$411$	3805.79	0.456754
$412$	3716.80i	0.444451i
$413$	929.934i	0.110797i
$414$	−3399.26	−0.403537
$415$	0	0
$416$	2157.21	0.254245
$417$	− 1458.86i	− 0.171321i
$418$	2919.42i	0.341612i
$419$	11188.4	1.30451	0.652256	−	0.757999i	$-0.273823\pi$
$419$	11188.4	1.30451	0.652256	+	0.757999i	$0.273823\pi$
$420$	0	0
$421$	−14082.3	−1.63023	−0.815116	−	0.579298i	$-0.803327\pi$
$421$	−14082.3	−1.63023	−0.815116	+	0.579298i	$0.803327\pi$
$422$	− 16585.7i	− 1.91322i
$423$	1528.83i	0.175731i
$424$	−6393.02	−0.732246
$425$	0	0
$426$	7653.39	0.870441
$427$	3327.64i	0.377133i
$428$	− 4645.57i	− 0.524655i
$429$	495.718	0.0557890
$430$	0	0
$431$	−5616.05	−0.627647	−0.313823	−	0.949481i	$-0.601610\pi$
$431$	−5616.05	−0.627647	−0.313823	+	0.949481i	$0.601610\pi$
$432$	− 2149.36i	− 0.239378i
$433$	− 7195.75i	− 0.798627i	−0.916814	−	0.399314i	$-0.869248\pi$
$433$	− 7195.75i	− 0.798627i	0.916814	−	0.399314i	$-0.130752\pi$
$434$	4459.48	0.493230
$435$	0	0
$436$	1079.86	0.118615
$437$	− 8814.52i	− 0.964887i
$438$	10317.3i	1.12553i
$439$	−101.959	−0.0110848	−0.00554240	−	0.999985i	$-0.501764\pi$
$439$	−101.959	−0.0110848	−0.00554240	+	0.999985i	$0.501764\pi$
$440$	0	0
$441$	−2884.40	−0.311457
$442$	− 3706.53i	− 0.398873i
$443$	− 4953.74i	− 0.531285i	−0.964072	−	0.265642i	$-0.914416\pi$
$443$	− 4953.74i	− 0.531285i	0.964072	−	0.265642i	$-0.0855841\pi$
$444$	−1035.77	−0.110710
$445$	0	0
$446$	7082.78	0.751972
$447$	7062.34i	0.747287i
$448$	723.869i	0.0763383i
$449$	11602.0	1.21945	0.609723	−	0.792615i	$-0.291281\pi$
$449$	11602.0	1.21945	0.609723	+	0.792615i	$0.291281\pi$
$450$	0	0
$451$	−2653.81	−0.277080
$452$	− 5494.95i	− 0.571816i
$453$	1710.21i	0.177379i
$454$	7611.79	0.786870
$455$	0	0
$456$	3684.62	0.378395
$457$	− 3530.68i	− 0.361397i	−0.983539	−	0.180698i	$-0.942164\pi$
$457$	− 3530.68i	− 0.361397i	0.983539	−	0.180698i	$-0.0578358\pi$
$458$	17910.4i	1.82729i
$459$	−1975.55	−0.200895
$460$	0	0
$461$	11566.3	1.16854	0.584271	−	0.811559i	$-0.301381\pi$
$461$	11566.3	1.16854	0.584271	+	0.811559i	$0.301381\pi$
$462$	− 528.000i	− 0.0531705i
$463$	− 10888.5i	− 1.09294i	−0.837479	−	0.546470i	$-0.815971\pi$
$463$	− 10888.5i	− 1.09294i	0.837479	−	0.546470i	$-0.184029\pi$
$464$	19354.2	1.93641
$465$	0	0
$466$	8316.94	0.826770
$467$	10688.0i	1.05906i	0.848292	+	0.529529i	$0.177631\pi$
$467$	10688.0i	1.05906i	−0.848292	+	0.529529i	$0.822369\pi$
$468$	455.918i	0.0450317i
$469$	4272.89	0.420690
$470$	0	0
$471$	−6218.02	−0.608304
$472$	− 3058.77i	− 0.298287i
$473$	3080.18i	0.299422i
$474$	3313.76	0.321110
$475$	0	0
$476$	−1170.70	−0.112728
$477$	− 3686.87i	− 0.353900i
$478$	− 4820.35i	− 0.461250i
$479$	−2341.90	−0.223391	−0.111696	−	0.993742i	$-0.535628\pi$
$479$	−2341.90	−0.223391	−0.111696	+	0.993742i	$0.535628\pi$
$480$	0	0
$481$	−1537.93	−0.145787
$482$	− 3301.43i	− 0.311983i
$483$	1594.17i	0.150181i
$484$	−408.046	−0.0383214
$485$	0	0
$486$	819.464	0.0764849
$487$	6748.91i	0.627972i	0.949428	+	0.313986i	$0.101665\pi$
$487$	6748.91i	0.627972i	−0.949428	+	0.313986i	$0.898335\pi$
$488$	− 10945.4i	− 1.01532i
$489$	−8029.53	−0.742552
$490$	0	0
$491$	7361.40	0.676609	0.338305	−	0.941037i	$-0.390147\pi$
$491$	7361.40	0.676609	0.338305	+	0.941037i	$0.390147\pi$
$492$	− 2440.74i	− 0.223653i
$493$	− 17789.1i	− 1.62511i
$494$	−3986.80	−0.363107
$495$	0	0
$496$	−22187.6	−2.00857
$497$	− 3589.26i	− 0.323944i
$498$	7654.60i	0.688777i
$499$	−10381.7	−0.931359	−0.465680	−	0.884953i	$-0.654190\pi$
$499$	−10381.7	−0.931359	−0.465680	+	0.884953i	$0.654190\pi$
$500$	0	0
$501$	−3564.36	−0.317852
$502$	− 22023.1i	− 1.95805i
$503$	− 19149.0i	− 1.69744i	−0.528840	−	0.848721i	$-0.677373\pi$
$503$	− 19149.0i	− 1.69744i	0.528840	−	0.848721i	$-0.322627\pi$
$504$	−666.391	−0.0588957
$505$	0	0
$506$	4154.65	0.365013
$507$	− 5914.04i	− 0.518051i
$508$	− 7726.76i	− 0.674841i
$509$	−16073.2	−1.39967	−0.699836	−	0.714303i	$-0.746744\pi$
$509$	−16073.2	−1.39967	−0.699836	+	0.714303i	$0.746744\pi$
$510$	0	0
$511$	4838.58	0.418877
$512$	1493.27i	0.128894i
$513$	2124.93i	0.182881i
$514$	−27417.5	−2.35279
$515$	0	0
$516$	−2832.88	−0.241687
$517$	− 1868.56i	− 0.158954i
$518$	1638.09i	0.138945i
$519$	−2421.44	−0.204797
$520$	0	0
$521$	−18955.3	−1.59395	−0.796975	−	0.604012i	$-0.793568\pi$
$521$	−18955.3	−1.59395	−0.796975	+	0.604012i	$0.793568\pi$
$522$	7378.97i	0.618714i
$523$	4442.19i	0.371402i	0.982606	+	0.185701i	$0.0594556\pi$
$523$	4442.19i	0.371402i	−0.982606	+	0.185701i	$0.940544\pi$
$524$	3869.40	0.322587
$525$	0	0
$526$	−15341.9	−1.27175
$527$	20393.3i	1.68567i
$528$	2627.00i	0.216525i
$529$	−377.000	−0.0309855
$530$	0	0
$531$	1764.00	0.144164
$532$	1259.22i	0.102620i
$533$	− 3624.08i	− 0.294515i
$534$	−5149.25	−0.417285
$535$	0	0
$536$	−14054.5	−1.13258
$537$	− 5851.17i	− 0.470199i
$538$	98.3096i	0.00787812i
$539$	3525.38	0.281723
$540$	0	0
$541$	2180.90	0.173316	0.0866580	−	0.996238i	$-0.472381\pi$
$541$	2180.90	0.173316	0.0866580	+	0.996238i	$0.472381\pi$
$542$	26004.4i	2.06086i
$543$	− 3184.82i	− 0.251701i
$544$	10507.4	0.828129
$545$	0	0
$546$	721.044	0.0565162
$547$	8225.04i	0.642920i	0.946923	+	0.321460i	$0.104174\pi$
$547$	8225.04i	0.642920i	−0.946923	+	0.321460i	$0.895826\pi$
$548$	− 4278.07i	− 0.333485i
$549$	6312.23	0.490709
$550$	0	0
$551$	−19134.2	−1.47939
$552$	− 5243.61i	− 0.404316i
$553$	− 1554.08i	− 0.119505i
$554$	−3802.32	−0.291598
$555$	0	0
$556$	−1639.90	−0.125085
$557$	− 25181.9i	− 1.91561i	−0.287423	−	0.957804i	$-0.592799\pi$
$557$	− 25181.9i	− 1.91561i	0.287423	−	0.957804i	$-0.407201\pi$
$558$	− 8459.22i	− 0.641769i
$559$	−4206.33	−0.318263
$560$	0	0
$561$	2414.56	0.181716
$562$	− 6314.50i	− 0.473952i
$563$	4504.50i	0.337197i	0.985685	+	0.168599i	$0.0539242\pi$
$563$	4504.50i	0.337197i	−0.985685	+	0.168599i	$0.946076\pi$
$564$	1718.54	0.128304
$565$	0	0
$566$	7164.36	0.532050
$567$	− 384.310i	− 0.0284647i
$568$	11805.9i	0.872122i
$569$	13447.0	0.990732	0.495366	−	0.868684i	$-0.335034\pi$
$569$	13447.0	0.990732	0.495366	+	0.868684i	$0.335034\pi$
$570$	0	0
$571$	−2605.52	−0.190959	−0.0954795	−	0.995431i	$-0.530438\pi$
$571$	−2605.52	−0.190959	−0.0954795	+	0.995431i	$0.530438\pi$
$572$	− 557.233i	− 0.0407327i
$573$	− 6409.24i	− 0.467277i
$574$	−3860.09	−0.280691
$575$	0	0
$576$	1373.11	0.0993281
$577$	6339.65i	0.457406i	0.973496	+	0.228703i	$0.0734484\pi$
$577$	6339.65i	0.457406i	−0.973496	+	0.228703i	$0.926552\pi$
$578$	− 1485.90i	− 0.106929i
$579$	−11843.3	−0.850069
$580$	0	0
$581$	3589.83	0.256336
$582$	6215.37i	0.442672i
$583$	4506.17i	0.320114i
$584$	−15915.2	−1.12770
$585$	0	0
$586$	−11210.1	−0.790247
$587$	− 13370.6i	− 0.940140i	−0.882629	−	0.470070i	$-0.844229\pi$
$587$	− 13370.6i	− 0.940140i	0.882629	−	0.470070i	$-0.155771\pi$
$588$	3242.34i	0.227401i
$589$	21935.3	1.53452
$590$	0	0
$591$	2771.66	0.192912
$592$	− 8150.09i	− 0.565822i
$593$	− 14319.3i	− 0.991608i	−0.868434	−	0.495804i	$-0.834873\pi$
$593$	− 14319.3i	− 0.991608i	0.868434	−	0.495804i	$-0.165127\pi$
$594$	−1001.57	−0.0691832
$595$	0	0
$596$	7938.73	0.545609
$597$	− 1428.46i	− 0.0979276i
$598$	5673.65i	0.387981i
$599$	5788.63	0.394853	0.197427	−	0.980318i	$-0.436742\pi$
$599$	5788.63	0.394853	0.197427	+	0.980318i	$0.436742\pi$
$600$	0	0
$601$	23968.1	1.62675	0.813375	−	0.581739i	$-0.197628\pi$
$601$	23968.1	1.62675	0.813375	+	0.581739i	$0.197628\pi$
$602$	4480.26i	0.303325i
$603$	− 8105.28i	− 0.547384i
$604$	1922.44	0.129508
$605$	0	0
$606$	−10277.9	−0.688961
$607$	− 23526.6i	− 1.57317i	−0.617482	−	0.786585i	$-0.711847\pi$
$607$	− 23526.6i	− 1.57317i	0.617482	−	0.786585i	$-0.288153\pi$
$608$	− 11301.9i	− 0.753872i
$609$	3460.56	0.230261
$610$	0	0
$611$	2551.74	0.168956
$612$	2220.70i	0.146677i
$613$	− 1228.07i	− 0.0809159i	−0.999181	−	0.0404579i	$-0.987118\pi$
$613$	− 1228.07i	− 0.0809159i	0.999181	−	0.0404579i	$-0.0128817\pi$
$614$	5729.31	0.376573
$615$	0	0
$616$	814.478	0.0532732
$617$	− 9844.90i	− 0.642368i	−0.947017	−	0.321184i	$-0.895919\pi$
$617$	− 9844.90i	− 0.642368i	0.947017	−	0.321184i	$-0.104081\pi$
$618$	− 11150.4i	− 0.725785i
$619$	6551.68	0.425419	0.212709	−	0.977115i	$-0.431771\pi$
$619$	6551.68	0.425419	0.212709	+	0.977115i	$0.431771\pi$
$620$	0	0
$621$	3024.00	0.195409
$622$	23365.9i	1.50625i
$623$	2414.88i	0.155297i
$624$	−3587.46	−0.230150
$625$	0	0
$626$	−12007.8	−0.766661
$627$	− 2597.14i	− 0.165422i
$628$	6989.64i	0.444135i
$629$	−7491.01	−0.474859
$630$	0	0
$631$	−26440.5	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$631$	−26440.5	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$632$	5111.72i	0.321730i
$633$	14754.7i	0.926458i
$634$	−1122.13	−0.0702923
$635$	0	0
$636$	−4144.39	−0.258389
$637$	4814.31i	0.299450i
$638$	− 9018.74i	− 0.559648i
$639$	−6808.50	−0.421502
$640$	0	0
$641$	−27927.2	−1.72084	−0.860421	−	0.509584i	$-0.829799\pi$
$641$	−27927.2	−1.72084	−0.860421	+	0.509584i	$0.829799\pi$
$642$	13936.7i	0.856758i
$643$	16737.7i	1.02655i	0.858225	+	0.513274i	$0.171568\pi$
$643$	16737.7i	1.02655i	−0.858225	+	0.513274i	$0.828432\pi$
$644$	1792.00	0.109650
$645$	0	0
$646$	−19419.1	−1.18271
$647$	7818.70i	0.475092i	0.971376	+	0.237546i	$0.0763431\pi$
$647$	7818.70i	0.475092i	−0.971376	+	0.237546i	$0.923657\pi$
$648$	1264.08i	0.0766325i
$649$	−2156.00	−0.130401
$650$	0	0
$651$	−3967.17	−0.238842
$652$	9025.94i	0.542152i
$653$	− 19747.6i	− 1.18344i	−0.806144	−	0.591719i	$-0.798450\pi$
$653$	− 19747.6i	− 1.18344i	0.806144	−	0.591719i	$-0.201550\pi$
$654$	−3239.59	−0.193697
$655$	0	0
$656$	19205.4	1.14305
$657$	− 9178.33i	− 0.545024i
$658$	− 2717.91i	− 0.161026i
$659$	−7867.72	−0.465072	−0.232536	−	0.972588i	$-0.574702\pi$
$659$	−7867.72	−0.465072	−0.232536	+	0.972588i	$0.574702\pi$
$660$	0	0
$661$	4227.41	0.248755	0.124378	−	0.992235i	$-0.460307\pi$
$661$	4227.41	0.248755	0.124378	+	0.992235i	$0.460307\pi$
$662$	− 1825.90i	− 0.107199i
$663$	3297.35i	0.193150i
$664$	−11807.8	−0.690106
$665$	0	0
$666$	3107.30	0.180789
$667$	27230.0i	1.58073i
$668$	4006.67i	0.232070i
$669$	−6300.88	−0.364135
$670$	0	0
$671$	−7714.94	−0.443863
$672$	2044.04i	0.117337i
$673$	− 29397.6i	− 1.68379i	−0.539638	−	0.841897i	$-0.681439\pi$
$673$	− 29397.6i	− 1.68379i	0.539638	−	0.841897i	$-0.318561\pi$
$674$	−2753.57	−0.157364
$675$	0	0
$676$	−6647.94	−0.378239
$677$	5737.14i	0.325696i	0.986651	+	0.162848i	$0.0520680\pi$
$677$	5737.14i	0.325696i	−0.986651	+	0.162848i	$0.947932\pi$
$678$	16484.8i	0.933771i
$679$	2914.86	0.164745
$680$	0	0
$681$	−6771.49	−0.381034
$682$	10339.0i	0.580502i
$683$	− 32097.6i	− 1.79821i	−0.437729	−	0.899107i	$-0.644217\pi$
$683$	− 32097.6i	− 1.79821i	0.437729	−	0.899107i	$-0.355783\pi$
$684$	2388.62	0.133525
$685$	0	0
$686$	10615.8	0.590837
$687$	− 15933.2i	− 0.884848i
$688$	− 22291.0i	− 1.23523i
$689$	−6153.69	−0.340257
$690$	0	0
$691$	−16456.2	−0.905965	−0.452983	−	0.891519i	$-0.649640\pi$
$691$	−16456.2	−0.905965	−0.452983	+	0.891519i	$0.649640\pi$
$692$	2721.93i	0.149526i
$693$	469.712i	0.0257473i
$694$	−21112.3	−1.15477
$695$	0	0
$696$	−11382.6	−0.619909
$697$	− 17652.3i	− 0.959294i
$698$	43058.9i	2.33496i
$699$	−7398.80	−0.400355
$700$	0	0
$701$	27238.1	1.46758	0.733788	−	0.679379i	$-0.237751\pi$
$701$	27238.1	1.46758	0.733788	+	0.679379i	$0.237751\pi$
$702$	− 1367.75i	− 0.0735364i
$703$	8057.44i	0.432279i
$704$	−1678.25	−0.0898457
$705$	0	0
$706$	−8934.12	−0.476261
$707$	4820.09i	0.256405i
$708$	− 1982.90i	− 0.105257i
$709$	−28761.4	−1.52349	−0.761747	−	0.647875i	$-0.775658\pi$
$709$	−28761.4	−1.52349	−0.761747	+	0.647875i	$0.775658\pi$
$710$	0	0
$711$	−2947.94	−0.155494
$712$	− 7943.10i	− 0.418090i
$713$	− 31216.3i	− 1.63964i
$714$	3512.09	0.184085
$715$	0	0
$716$	−6577.27	−0.343302
$717$	4288.21i	0.223356i
$718$	10804.5i	0.561588i
$719$	27272.0	1.41456	0.707282	−	0.706931i	$-0.249921\pi$
$719$	27272.0	1.41456	0.707282	+	0.706931i	$0.249921\pi$
$720$	0	0
$721$	−5229.28	−0.270109
$722$	− 2243.05i	− 0.115620i
$723$	2936.97i	0.151075i
$724$	−3580.04	−0.183772
$725$	0	0
$726$	1224.14	0.0625785
$727$	3979.75i	0.203027i	0.994834	+	0.101514i	$0.0323685\pi$
$727$	3979.75i	0.203027i	−0.994834	+	0.101514i	$0.967631\pi$
$728$	1112.26i	0.0566253i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−20488.3	−1.03665
$732$	− 7095.54i	− 0.358277i
$733$	− 9342.48i	− 0.470767i	−0.971903	−	0.235384i	$-0.924365\pi$
$733$	− 9342.48i	− 0.470767i	0.971903	−	0.235384i	$-0.0756346\pi$
$734$	28426.3	1.42947
$735$	0	0
$736$	−16083.9	−0.805515
$737$	9906.45i	0.495127i
$738$	7322.23i	0.365224i
$739$	28928.0	1.43997	0.719983	−	0.693992i	$-0.244150\pi$
$739$	28928.0	1.43997	0.719983	+	0.693992i	$0.244150\pi$
$740$	0	0
$741$	3546.68	0.175831
$742$	6554.43i	0.324287i
$743$	− 4857.04i	− 0.239822i	−0.992785	−	0.119911i	$-0.961739\pi$
$743$	− 4857.04i	− 0.239822i	0.992785	−	0.119911i	$-0.0382609\pi$
$744$	13049.0	0.643008
$745$	0	0
$746$	−31660.8	−1.55386
$747$	− 6809.57i	− 0.333533i
$748$	− 2714.19i	− 0.132675i
$749$	6536.00	0.318852
$750$	0	0
$751$	14355.4	0.697517	0.348759	−	0.937213i	$-0.386603\pi$
$751$	14355.4	0.697517	0.348759	+	0.937213i	$0.386603\pi$
$752$	13522.6i	0.655744i
$753$	19591.9i	0.948165i
$754$	12316.1	0.594863
$755$	0	0
$756$	−432.000	−0.0207827
$757$	− 17714.9i	− 0.850538i	−0.905067	−	0.425269i	$-0.860179\pi$
$757$	− 17714.9i	− 0.850538i	0.905067	−	0.425269i	$-0.139821\pi$
$758$	48103.9i	2.30503i
$759$	−3696.00	−0.176754
$760$	0	0
$761$	−7945.82	−0.378497	−0.189248	−	0.981929i	$-0.560605\pi$
$761$	−7945.82	−0.378497	−0.189248	+	0.981929i	$0.560605\pi$
$762$	23180.3i	1.10201i
$763$	1519.29i	0.0720866i
$764$	−7204.58	−0.341168
$765$	0	0
$766$	45398.4	2.14140
$767$	− 2944.26i	− 0.138606i
$768$	− 13166.2i	− 0.618613i
$769$	−27308.1	−1.28057	−0.640284	−	0.768139i	$-0.721183\pi$
$769$	−27308.1	−1.28057	−0.640284	+	0.768139i	$0.721183\pi$
$770$	0	0
$771$	24390.8	1.13932
$772$	13313.0i	0.620653i
$773$	18872.6i	0.878136i	0.898454	+	0.439068i	$0.144691\pi$
$773$	18872.6i	0.878136i	−0.898454	+	0.439068i	$0.855309\pi$
$774$	8498.64	0.394674
$775$	0	0
$776$	−9587.65	−0.443527
$777$	− 1457.25i	− 0.0672826i
$778$	3176.29i	0.146369i
$779$	−18987.1	−0.873276
$780$	0	0
$781$	8321.50	0.381263
$782$	27635.4i	1.26373i
$783$	− 6564.37i	− 0.299606i
$784$	−25512.8	−1.16221
$785$	0	0
$786$	−11608.2	−0.526782
$787$	− 14512.1i	− 0.657307i	−0.944450	−	0.328654i	$-0.893405\pi$
$787$	− 14512.1i	− 0.657307i	0.944450	−	0.328654i	$-0.106595\pi$
$788$	− 3115.60i	− 0.140849i
$789$	13648.3	0.615832
$790$	0	0
$791$	7731.01	0.347513
$792$	− 1544.99i	− 0.0693167i
$793$	− 10535.6i	− 0.471792i
$794$	2859.15	0.127793
$795$	0	0
$796$	−1605.72	−0.0714989
$797$	29108.9i	1.29371i	0.762612	+	0.646856i	$0.223917\pi$
$797$	29108.9i	1.29371i	−0.762612	+	0.646856i	$0.776083\pi$
$798$	− 3777.65i	− 0.167578i
$799$	12429.1	0.550325
$800$	0	0
$801$	4580.80	0.202066
$802$	41154.0i	1.81197i
$803$	11218.0i	0.492993i
$804$	−9111.10	−0.399656
$805$	0	0
$806$	−14119.1	−0.617029
$807$	− 87.4567i	− 0.00381490i
$808$	− 15854.4i	− 0.690291i
$809$	3000.83	0.130413	0.0652063	−	0.997872i	$-0.479229\pi$
$809$	3000.83	0.130413	0.0652063	+	0.997872i	$0.479229\pi$
$810$	0	0
$811$	6239.39	0.270154	0.135077	−	0.990835i	$-0.456872\pi$
$811$	6239.39	0.270154	0.135077	+	0.990835i	$0.456872\pi$
$812$	− 3890.00i	− 0.168118i
$813$	− 23133.7i	− 0.997950i
$814$	−3797.81	−0.163530
$815$	0	0
$816$	−17473.9	−0.749645
$817$	22037.6i	0.943693i
$818$	− 29539.6i	− 1.26263i
$819$	−641.445	−0.0273674
$820$	0	0
$821$	14922.4	0.634342	0.317171	−	0.948368i	$-0.397267\pi$
$821$	14922.4	0.634342	0.317171	+	0.948368i	$0.397267\pi$
$822$	12834.2i	0.544580i
$823$	25737.8i	1.09011i	0.838399	+	0.545057i	$0.183492\pi$
$823$	25737.8i	1.09011i	−0.838399	+	0.545057i	$0.816508\pi$
$824$	17200.3	0.727186
$825$	0	0
$826$	−3136.00	−0.132101
$827$	27043.4i	1.13711i	0.822645	+	0.568555i	$0.192497\pi$
$827$	27043.4i	1.13711i	−0.822645	+	0.568555i	$0.807503\pi$
$828$	− 3399.26i	− 0.142672i
$829$	9795.41	0.410384	0.205192	−	0.978722i	$-0.434218\pi$
$829$	9795.41	0.410384	0.205192	+	0.978722i	$0.434218\pi$
$830$	0	0
$831$	3382.56	0.141203
$832$	− 2291.84i	− 0.0954990i
$833$	23449.7i	0.975370i
$834$	4919.70	0.204263
$835$	0	0
$836$	−2919.42	−0.120778
$837$	7525.37i	0.310770i
$838$	37730.5i	1.55535i
$839$	−28875.5	−1.18819	−0.594095	−	0.804395i	$-0.702490\pi$
$839$	−28875.5	−1.18819	−0.594095	+	0.804395i	$0.702490\pi$
$840$	0	0
$841$	34720.7	1.42362
$842$	− 47489.3i	− 1.94369i
$843$	5617.41i	0.229506i
$844$	16585.7	0.676426
$845$	0	0
$846$	−5155.63	−0.209520
$847$	− 574.092i	− 0.0232893i
$848$	− 32610.7i	− 1.32059i
$849$	−6373.45	−0.257640
$850$	0	0
$851$	11466.6	0.461892
$852$	7653.39i	0.307747i
$853$	47157.1i	1.89288i	0.322878	+	0.946441i	$0.395350\pi$
$853$	47157.1i	1.89288i	−0.322878	+	0.946441i	$0.604650\pi$
$854$	−11221.7	−0.449649
$855$	0	0
$856$	−21498.4	−0.858412
$857$	5021.31i	0.200145i	0.994980	+	0.100073i	$0.0319075\pi$
$857$	5021.31i	0.200145i	−0.994980	+	0.100073i	$0.968092\pi$
$858$	1671.70i	0.0665162i
$859$	22921.1	0.910428	0.455214	−	0.890382i	$-0.349563\pi$
$859$	22921.1	0.910428	0.455214	+	0.890382i	$0.349563\pi$
$860$	0	0
$861$	3433.95	0.135922
$862$	− 18938.9i	− 0.748332i
$863$	19488.1i	0.768693i	0.923189	+	0.384347i	$0.125573\pi$
$863$	19488.1i	0.768693i	−0.923189	+	0.384347i	$0.874427\pi$
$864$	3877.36	0.152674
$865$	0	0
$866$	24266.1	0.952188
$867$	1321.86i	0.0517794i
$868$	4459.48i	0.174383i
$869$	3603.04	0.140650
$870$	0	0
$871$	−13528.4	−0.526282
$872$	− 4997.30i	− 0.194071i
$873$	− 5529.22i	− 0.214360i
$874$	29725.0	1.15042
$875$	0	0
$876$	−10317.3	−0.397933
$877$	− 8455.67i	− 0.325573i	−0.986661	−	0.162787i	$-0.947952\pi$
$877$	− 8455.67i	− 0.325573i	0.986661	−	0.162787i	$-0.0520482\pi$
$878$	− 343.834i	− 0.0132162i
$879$	9972.56	0.382669
$880$	0	0
$881$	−11291.2	−0.431794	−0.215897	−	0.976416i	$-0.569268\pi$
$881$	−11291.2	−0.431794	−0.215897	+	0.976416i	$0.569268\pi$
$882$	− 9727.02i	− 0.371344i
$883$	− 31818.1i	− 1.21264i	−0.795219	−	0.606322i	$-0.792644\pi$
$883$	− 31818.1i	− 1.21264i	0.795219	−	0.606322i	$-0.207356\pi$
$884$	3706.53	0.141023
$885$	0	0
$886$	16705.4	0.633441
$887$	17481.1i	0.661732i	0.943678	+	0.330866i	$0.107341\pi$
$887$	17481.1i	0.661732i	−0.943678	+	0.330866i	$0.892659\pi$
$888$	4793.24i	0.181138i
$889$	10871.0	0.410126
$890$	0	0
$891$	891.000	0.0335013
$892$	7082.78i	0.265862i
$893$	− 13368.9i	− 0.500978i
$894$	−23816.2	−0.890976
$895$	0	0
$896$	−7891.87	−0.294251
$897$	− 5047.31i	− 0.187876i
$898$	39125.1i	1.45392i
$899$	−67763.1	−2.51393
$900$	0	0
$901$	−29973.6	−1.10829
$902$	− 8949.39i	− 0.330357i
$903$	− 3985.66i	− 0.146882i
$904$	−25429.1	−0.935574
$905$	0	0
$906$	−5767.31	−0.211486
$907$	10607.4i	0.388326i	0.980969	+	0.194163i	$0.0621990\pi$
$907$	10607.4i	0.388326i	−0.980969	+	0.194163i	$0.937801\pi$
$908$	7611.79i	0.278201i
$909$	9143.26	0.333623
$910$	0	0
$911$	−41249.2	−1.50016	−0.750080	−	0.661347i	$-0.769985\pi$
$911$	−41249.2	−1.50016	−0.750080	+	0.661347i	$0.769985\pi$
$912$	18795.2i	0.682425i
$913$	8322.81i	0.301692i
$914$	11906.5	0.430887
$915$	0	0
$916$	−17910.4	−0.646045
$917$	5443.97i	0.196048i
$918$	− 6662.10i	− 0.239523i
$919$	13858.1	0.497429	0.248714	−	0.968577i	$-0.419992\pi$
$919$	13858.1	0.497429	0.248714	+	0.968577i	$0.419992\pi$
$920$	0	0
$921$	−5096.83	−0.182352
$922$	39004.9i	1.39323i
$923$	11363.9i	0.405253i
$924$	528.000	0.0187986
$925$	0	0
$926$	36719.0	1.30309
$927$	9919.47i	0.351454i
$928$	34914.2i	1.23504i
$929$	−20893.7	−0.737890	−0.368945	−	0.929451i	$-0.620281\pi$
$929$	−20893.7	−0.737890	−0.368945	+	0.929451i	$0.620281\pi$
$930$	0	0
$931$	25222.8	0.887911
$932$	8316.94i	0.292307i
$933$	− 20786.5i	− 0.729388i
$934$	−36042.9	−1.26270
$935$	0	0
$936$	2109.86	0.0736784
$937$	3203.52i	0.111691i	0.998439	+	0.0558454i	$0.0177854\pi$
$937$	3203.52i	0.111691i	−0.998439	+	0.0558454i	$0.982215\pi$
$938$	14409.4i	0.501581i
$939$	10682.2	0.371248
$940$	0	0
$941$	19951.6	0.691182	0.345591	−	0.938385i	$-0.387678\pi$
$941$	19951.6	0.691182	0.345591	+	0.938385i	$0.387678\pi$
$942$	− 20968.9i	− 0.725270i
$943$	27020.6i	0.933099i
$944$	15602.8	0.537952
$945$	0	0
$946$	−10387.2	−0.356996
$947$	− 38216.7i	− 1.31138i	−0.755031	−	0.655689i	$-0.772378\pi$
$947$	− 38216.7i	− 1.31138i	0.755031	−	0.655689i	$-0.227622\pi$
$948$	3313.76i	0.113529i
$949$	−15319.4	−0.524014
$950$	0	0
$951$	998.249	0.0340383
$952$	5417.65i	0.184440i
$953$	47661.4i	1.62004i	0.586399	+	0.810022i	$0.300545\pi$
$953$	47661.4i	1.62004i	−0.586399	+	0.810022i	$0.699455\pi$
$954$	12433.2	0.421948
$955$	0	0
$956$	4820.35	0.163077
$957$	8023.12i	0.271004i
$958$	− 7897.56i	− 0.266345i
$959$	6018.94	0.202671
$960$	0	0
$961$	47892.3	1.60761
$962$	− 5186.34i	− 0.173819i
$963$	− 12398.2i	− 0.414876i
$964$	3301.43	0.110303
$965$	0	0
$966$	−5376.00	−0.179058
$967$	18933.2i	0.629628i	0.949153	+	0.314814i	$0.101942\pi$
$967$	18933.2i	0.629628i	−0.949153	+	0.314814i	$0.898058\pi$
$968$	1888.32i	0.0626994i
$969$	17275.3	0.572717
$970$	0	0
$971$	−40660.3	−1.34382	−0.671911	−	0.740632i	$-0.734526\pi$
$971$	−40660.3	−1.34382	−0.671911	+	0.740632i	$0.734526\pi$
$972$	819.464i	0.0270415i
$973$	− 2307.23i	− 0.0760188i
$974$	−22759.2	−0.748720
$975$	0	0
$976$	55832.3	1.83110
$977$	− 22502.8i	− 0.736876i	−0.929652	−	0.368438i	$-0.879893\pi$
$977$	− 22502.8i	− 0.736876i	0.929652	−	0.368438i	$-0.120107\pi$
$978$	− 27077.8i	− 0.885331i
$979$	−5598.76	−0.182775
$980$	0	0
$981$	2881.96	0.0937959
$982$	24824.7i	0.806709i
$983$	4435.20i	0.143907i	0.997408	+	0.0719536i	$0.0229234\pi$
$983$	4435.20i	0.143907i	−0.997408	+	0.0719536i	$0.977077\pi$
$984$	−11295.1	−0.365929
$985$	0	0
$986$	59989.8	1.93759
$987$	2417.87i	0.0779753i
$988$	− 3986.80i	− 0.128378i
$989$	31361.8	1.00834
$990$	0	0
$991$	7362.76	0.236010	0.118005	−	0.993013i	$-0.462350\pi$
$991$	7362.76	0.236010	0.118005	+	0.993013i	$0.462350\pi$
$992$	− 40025.5i	− 1.28106i
$993$	1624.33i	0.0519101i
$994$	12104.0	0.386233
$995$	0	0
$996$	−7654.60	−0.243519
$997$	− 53480.1i	− 1.69883i	−0.527728	−	0.849413i	$-0.676956\pi$
$997$	− 53480.1i	− 1.69883i	0.527728	−	0.849413i	$-0.323044\pi$
$998$	− 35010.0i	− 1.11044i
$999$	−2764.27	−0.0875452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.i.199.4		4
5.2	odd	4		825.4.a.k.1.1		2
5.3	odd	4		33.4.a.d.1.2	✓	2
5.4	even	2	inner	825.4.c.i.199.1		4
15.2	even	4		2475.4.a.o.1.2		2
15.8	even	4		99.4.a.e.1.1		2
20.3	even	4		528.4.a.o.1.1		2
35.13	even	4		1617.4.a.j.1.2		2
40.3	even	4		2112.4.a.bh.1.2		2
40.13	odd	4		2112.4.a.ba.1.2		2
55.43	even	4		363.4.a.j.1.1		2
60.23	odd	4		1584.4.a.x.1.2		2
165.98	odd	4		1089.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.d.1.2	✓	2	5.3	odd	4
99.4.a.e.1.1		2	15.8	even	4
363.4.a.j.1.1		2	55.43	even	4
528.4.a.o.1.1		2	20.3	even	4
825.4.a.k.1.1		2	5.2	odd	4
825.4.c.i.199.1		4	5.4	even	2	inner
825.4.c.i.199.4		4	1.1	even	1	trivial
1089.4.a.t.1.2		2	165.98	odd	4
1584.4.a.x.1.2		2	60.23	odd	4
1617.4.a.j.1.2		2	35.13	even	4
2112.4.a.ba.1.2		2	40.13	odd	4
2112.4.a.bh.1.2		2	40.3	even	4
2475.4.a.o.1.2		2	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)

\(f(q)\)	\(=\)	\(q+3.37228i q^{2} -3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} -4.74456i q^{7} +15.6060i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+3.37228i q^{2} -3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} -4.74456i q^{7} +15.6060i q^{8} -9.00000 q^{9} +11.0000 q^{11} +10.1168i q^{12} +15.0217i q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684i q^{17} -30.3505i q^{18} +78.7011 q^{19} -14.2337 q^{21} +37.0951i q^{22} -112.000i q^{23} +46.8179 q^{24} -50.6576 q^{26} +27.0000i q^{27} +16.0000i q^{28} -243.125 q^{29} +278.717 q^{31} -143.606i q^{32} -33.0000i q^{33} -246.745 q^{34} +30.3505 q^{36} +102.380i q^{37} +265.402i q^{38} +45.0652 q^{39} -241.255 q^{41} -48.0000i q^{42} +280.016i q^{43} -37.0951 q^{44} +377.696 q^{46} -169.870i q^{47} +238.818i q^{48} +320.489 q^{49} +219.505 q^{51} -50.6576i q^{52} +409.652i q^{53} -91.0516 q^{54} +74.0435 q^{56} -236.103i q^{57} -819.886i q^{58} -196.000 q^{59} -701.359 q^{61} +939.913i q^{62} +42.7011i q^{63} -152.568 q^{64} +111.285 q^{66} +900.587i q^{67} -246.745i q^{68} -336.000 q^{69} +756.500 q^{71} -140.454i q^{72} +1019.81i q^{73} -345.255 q^{74} -265.402 q^{76} -52.1902i q^{77} +151.973i q^{78} +327.549 q^{79} +81.0000 q^{81} -813.581i q^{82} +756.619i q^{83} +48.0000 q^{84} -944.293 q^{86} +729.375i q^{87} +171.666i q^{88} -508.978 q^{89} +71.2716 q^{91} +377.696i q^{92} -836.152i q^{93} +572.848 q^{94} -430.818 q^{96} +614.358i q^{97} +1080.78i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9	\( 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44} + 224 q^{46} + 1236 q^{49} - 156 q^{51} - 54 q^{54} + 480 q^{56} - 784 q^{59} - 2208 q^{61} - 1426 q^{64} + 66 q^{66} - 1344 q^{69} + 912 q^{71} - 1404 q^{74} - 648 q^{76} + 460 q^{79} + 324 q^{81} + 192 q^{84} - 2904 q^{86} - 1944 q^{89} - 680 q^{91} + 1648 q^{94} - 1482 q^{96} - 396 q^{99}+O(q^{100}) \) 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 + 44 * q^11 + 64 * q^14 - 238 * q^16 + 108 * q^19 + 12 * q^21 - 54 * q^24 + 188 * q^26 - 444 * q^29 - 80 * q^31 - 964 * q^34 + 18 * q^36 + 456 * q^39 - 988 * q^41 - 22 * q^44 + 224 * q^46 + 1236 * q^49 - 156 * q^51 - 54 * q^54 + 480 * q^56 - 784 * q^59 - 2208 * q^61 - 1426 * q^64 + 66 * q^66 - 1344 * q^69 + 912 * q^71 - 1404 * q^74 - 648 * q^76 + 460 * q^79 + 324 * q^81 + 192 * q^84 - 2904 * q^86 - 1944 * q^89 - 680 * q^91 + 1648 * q^94 - 1482 * q^96 - 396 * q^99

Introduction

L-functions

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