LMFDB - Embedded newform 825.2.c.d.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,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, 2, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.d.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +2.41421i q^{7} +1.58579i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +2.41421i q^{7} +1.58579i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.82843i q^{12} +2.82843i q^{13} -1.00000 q^{14} +3.00000 q^{16} +0.414214i q^{17} -0.414214i q^{18} -3.58579 q^{19} -2.41421 q^{21} -0.414214i q^{22} +1.00000i q^{23} -1.58579 q^{24} -1.17157 q^{26} -1.00000i q^{27} +4.41421i q^{28} -6.82843 q^{29} +8.48528 q^{31} +4.41421i q^{32} -1.00000i q^{33} -0.171573 q^{34} -1.82843 q^{36} +5.82843i q^{37} -1.48528i q^{38} -2.82843 q^{39} +8.89949 q^{41} -1.00000i q^{42} -0.343146i q^{43} -1.82843 q^{44} -0.414214 q^{46} -9.48528i q^{47} +3.00000i q^{48} +1.17157 q^{49} -0.414214 q^{51} +5.17157i q^{52} -3.65685i q^{53} +0.414214 q^{54} -3.82843 q^{56} -3.58579i q^{57} -2.82843i q^{58} -11.0000 q^{59} +3.17157 q^{61} +3.51472i q^{62} -2.41421i q^{63} +4.17157 q^{64} +0.414214 q^{66} +11.6569i q^{67} +0.757359i q^{68} -1.00000 q^{69} +2.17157 q^{71} -1.58579i q^{72} -3.17157i q^{73} -2.41421 q^{74} -6.55635 q^{76} -2.41421i q^{77} -1.17157i q^{78} -4.75736 q^{79} +1.00000 q^{81} +3.68629i q^{82} +12.4853i q^{83} -4.41421 q^{84} +0.142136 q^{86} -6.82843i q^{87} -1.58579i q^{88} +7.65685 q^{89} -6.82843 q^{91} +1.82843i q^{92} +8.48528i q^{93} +3.92893 q^{94} -4.41421 q^{96} -0.171573i q^{97} +0.485281i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 12 q^{16} - 20 q^{19} - 4 q^{21} - 12 q^{24} - 16 q^{26} - 16 q^{29} - 12 q^{34} + 4 q^{36} - 4 q^{41} + 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{51} - 4 q^{54} - 4 q^{56} - 44 q^{59} + 24 q^{61} + 28 q^{64} - 4 q^{66} - 4 q^{69} + 20 q^{71} - 4 q^{74} + 36 q^{76} - 36 q^{79} + 4 q^{81} - 12 q^{84} - 56 q^{86} + 8 q^{89} - 16 q^{91} + 44 q^{94} - 12 q^{96} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 - 4 * q^11 - 4 * q^14 + 12 * q^16 - 20 * q^19 - 4 * q^21 - 12 * q^24 - 16 * q^26 - 16 * q^29 - 12 * q^34 + 4 * q^36 - 4 * q^41 + 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^51 - 4 * q^54 - 4 * q^56 - 44 * q^59 + 24 * q^61 + 28 * q^64 - 4 * q^66 - 4 * q^69 + 20 * q^71 - 4 * q^74 + 36 * q^76 - 36 * q^79 + 4 * q^81 - 12 * q^84 - 56 * q^86 + 8 * q^89 - 16 * q^91 + 44 * q^94 - 12 * q^96 + 4 * 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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	−0.414214	−0.169102
$7$	2.41421i	0.912487i	0.889855	+	0.456243i	$0.150805\pi$
$7$	2.41421i	0.912487i	−0.889855	+	0.456243i	$0.849195\pi$
$8$	1.58579i	0.560660i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.82843i	0.527821i
$13$	2.82843i	0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$	2.82843i	0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	3.00000	0.750000
$17$	0.414214i	0.100462i	0.998738	+	0.0502308i	$0.0159957\pi$
$17$	0.414214i	0.100462i	−0.998738	+	0.0502308i	$0.984004\pi$
$18$	− 0.414214i	− 0.0976311i
$19$	−3.58579	−0.822636	−0.411318	−	0.911492i	$-0.634931\pi$
$19$	−3.58579	−0.822636	−0.411318	+	0.911492i	$0.634931\pi$
$20$	0	0
$21$	−2.41421	−0.526825
$22$	− 0.414214i	− 0.0883106i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	−1.58579	−0.323697
$25$	0	0
$26$	−1.17157	−0.229764
$27$	− 1.00000i	− 0.192450i
$28$	4.41421i	0.834208i
$29$	−6.82843	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$29$	−6.82843	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	4.41421i	0.780330i
$33$	− 1.00000i	− 0.174078i
$34$	−0.171573	−0.0294245
$35$	0	0
$36$	−1.82843	−0.304738
$37$	5.82843i	0.958188i	0.877764	+	0.479094i	$0.159035\pi$
$37$	5.82843i	0.958188i	−0.877764	+	0.479094i	$0.840965\pi$
$38$	− 1.48528i	− 0.240944i
$39$	−2.82843	−0.452911
$40$	0	0
$41$	8.89949	1.38987	0.694934	−	0.719074i	$-0.255434\pi$
$41$	8.89949	1.38987	0.694934	+	0.719074i	$0.255434\pi$
$42$	− 1.00000i	− 0.154303i
$43$	− 0.343146i	− 0.0523292i	−0.999658	−	0.0261646i	$-0.991671\pi$
$43$	− 0.343146i	− 0.0523292i	0.999658	−	0.0261646i	$-0.00832941\pi$
$44$	−1.82843	−0.275646
$45$	0	0
$46$	−0.414214	−0.0610725
$47$	− 9.48528i	− 1.38357i	−0.722103	−	0.691785i	$-0.756824\pi$
$47$	− 9.48528i	− 1.38357i	0.722103	−	0.691785i	$-0.243176\pi$
$48$	3.00000i	0.433013i
$49$	1.17157	0.167368
$50$	0	0
$51$	−0.414214	−0.0580015
$52$	5.17157i	0.717168i
$53$	− 3.65685i	− 0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$	− 3.65685i	− 0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$	0.414214	0.0563673
$55$	0	0
$56$	−3.82843	−0.511595
$57$	− 3.58579i	− 0.474949i
$58$	− 2.82843i	− 0.371391i
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	3.51472i	0.446370i
$63$	− 2.41421i	− 0.304162i
$64$	4.17157	0.521447
$65$	0	0
$66$	0.414214	0.0509862
$67$	11.6569i	1.42411i	0.702123	+	0.712056i	$0.252236\pi$
$67$	11.6569i	1.42411i	−0.702123	+	0.712056i	$0.747764\pi$
$68$	0.757359i	0.0918433i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.17157	0.257718	0.128859	−	0.991663i	$-0.458868\pi$
$71$	2.17157	0.257718	0.128859	+	0.991663i	$0.458868\pi$
$72$	− 1.58579i	− 0.186887i
$73$	− 3.17157i	− 0.371205i	−0.982625	−	0.185602i	$-0.940576\pi$
$73$	− 3.17157i	− 0.371205i	0.982625	−	0.185602i	$-0.0594236\pi$
$74$	−2.41421	−0.280647
$75$	0	0
$76$	−6.55635	−0.752065
$77$	− 2.41421i	− 0.275125i
$78$	− 1.17157i	− 0.132655i
$79$	−4.75736	−0.535245	−0.267622	−	0.963524i	$-0.586238\pi$
$79$	−4.75736	−0.535245	−0.267622	+	0.963524i	$0.586238\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.68629i	0.407083i
$83$	12.4853i	1.37044i	0.728337	+	0.685219i	$0.240293\pi$
$83$	12.4853i	1.37044i	−0.728337	+	0.685219i	$0.759707\pi$
$84$	−4.41421	−0.481630
$85$	0	0
$86$	0.142136	0.0153269
$87$	− 6.82843i	− 0.732084i
$88$	− 1.58579i	− 0.169045i
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	−6.82843	−0.715814
$92$	1.82843i	0.190627i
$93$	8.48528i	0.879883i
$94$	3.92893	0.405238
$95$	0	0
$96$	−4.41421	−0.450524
$97$	− 0.171573i	− 0.0174206i	−0.999962	−	0.00871029i	$-0.997227\pi$
$97$	− 0.171573i	− 0.0174206i	0.999962	−	0.00871029i	$-0.00277261\pi$
$98$	0.485281i	0.0490208i
$99$	1.00000	0.100504
$100$	0	0
$101$	−4.89949	−0.487518	−0.243759	−	0.969836i	$-0.578381\pi$
$101$	−4.89949	−0.487518	−0.243759	+	0.969836i	$0.578381\pi$
$102$	− 0.171573i	− 0.0169882i
$103$	− 2.34315i	− 0.230877i	−0.993315	−	0.115439i	$-0.963173\pi$
$103$	− 2.34315i	− 0.230877i	0.993315	−	0.115439i	$-0.0368273\pi$
$104$	−4.48528	−0.439818
$105$	0	0
$106$	1.51472	0.147122
$107$	− 17.3137i	− 1.67378i	−0.547372	−	0.836890i	$-0.684372\pi$
$107$	− 17.3137i	− 1.67378i	0.547372	−	0.836890i	$-0.315628\pi$
$108$	− 1.82843i	− 0.175940i
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	−5.82843	−0.553210
$112$	7.24264i	0.684365i
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	1.48528	0.139109
$115$	0	0
$116$	−12.4853	−1.15923
$117$	− 2.82843i	− 0.261488i
$118$	− 4.55635i	− 0.419446i
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.31371i	0.118938i
$123$	8.89949i	0.802440i
$124$	15.5147	1.39326
$125$	0	0
$126$	1.00000	0.0890871
$127$	1.24264i	0.110267i	0.998479	+	0.0551333i	$0.0175584\pi$
$127$	1.24264i	0.110267i	−0.998479	+	0.0551333i	$0.982442\pi$
$128$	10.5563i	0.933058i
$129$	0.343146	0.0302123
$130$	0	0
$131$	1.17157	0.102361	0.0511804	−	0.998689i	$-0.483702\pi$
$131$	1.17157	0.102361	0.0511804	+	0.998689i	$0.483702\pi$
$132$	− 1.82843i	− 0.159144i
$133$	− 8.65685i	− 0.750644i
$134$	−4.82843	−0.417113
$135$	0	0
$136$	−0.656854	−0.0563248
$137$	− 16.1421i	− 1.37912i	−0.724231	−	0.689558i	$-0.757805\pi$
$137$	− 16.1421i	− 1.37912i	0.724231	−	0.689558i	$-0.242195\pi$
$138$	− 0.414214i	− 0.0352602i
$139$	−14.9706	−1.26979	−0.634893	−	0.772600i	$-0.718956\pi$
$139$	−14.9706	−1.26979	−0.634893	+	0.772600i	$0.718956\pi$
$140$	0	0
$141$	9.48528	0.798805
$142$	0.899495i	0.0754839i
$143$	− 2.82843i	− 0.236525i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	1.31371	0.108723
$147$	1.17157i	0.0966297i
$148$	10.6569i	0.875988i
$149$	17.7279	1.45233	0.726164	−	0.687522i	$-0.241301\pi$
$149$	17.7279	1.45233	0.726164	+	0.687522i	$0.241301\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 5.68629i	− 0.461219i
$153$	− 0.414214i	− 0.0334872i
$154$	1.00000	0.0805823
$155$	0	0
$156$	−5.17157	−0.414057
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 1.97056i	− 0.156770i
$159$	3.65685	0.290007
$160$	0	0
$161$	−2.41421	−0.190267
$162$	0.414214i	0.0325437i
$163$	− 23.7990i	− 1.86408i	−0.362354	−	0.932040i	$-0.618027\pi$
$163$	− 23.7990i	− 1.86408i	0.362354	−	0.932040i	$-0.381973\pi$
$164$	16.2721	1.27064
$165$	0	0
$166$	−5.17157	−0.401392
$167$	17.7990i	1.37733i	0.725081	+	0.688664i	$0.241802\pi$
$167$	17.7990i	1.37733i	−0.725081	+	0.688664i	$0.758198\pi$
$168$	− 3.82843i	− 0.295370i
$169$	5.00000	0.384615
$170$	0	0
$171$	3.58579	0.274212
$172$	− 0.627417i	− 0.0478401i
$173$	18.5563i	1.41081i	0.708803	+	0.705407i	$0.249236\pi$
$173$	18.5563i	1.41081i	−0.708803	+	0.705407i	$0.750764\pi$
$174$	2.82843	0.214423
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 11.0000i	− 0.826811i
$178$	3.17157i	0.237719i
$179$	22.7990	1.70408	0.852038	−	0.523480i	$-0.175366\pi$
$179$	22.7990	1.70408	0.852038	+	0.523480i	$0.175366\pi$
$180$	0	0
$181$	11.9706	0.889765	0.444882	−	0.895589i	$-0.353245\pi$
$181$	11.9706	0.889765	0.444882	+	0.895589i	$0.353245\pi$
$182$	− 2.82843i	− 0.209657i
$183$	3.17157i	0.234449i
$184$	−1.58579	−0.116906
$185$	0	0
$186$	−3.51472	−0.257712
$187$	− 0.414214i	− 0.0302903i
$188$	− 17.3431i	− 1.26488i
$189$	2.41421	0.175608
$190$	0	0
$191$	11.8284	0.855875	0.427937	−	0.903808i	$-0.359240\pi$
$191$	11.8284	0.855875	0.427937	+	0.903808i	$0.359240\pi$
$192$	4.17157i	0.301057i
$193$	− 19.3137i	− 1.39023i	−0.718898	−	0.695116i	$-0.755353\pi$
$193$	− 19.3137i	− 1.39023i	0.718898	−	0.695116i	$-0.244647\pi$
$194$	0.0710678	0.00510237
$195$	0	0
$196$	2.14214	0.153010
$197$	− 13.2426i	− 0.943499i	−0.881733	−	0.471750i	$-0.843623\pi$
$197$	− 13.2426i	− 0.943499i	0.881733	−	0.471750i	$-0.156377\pi$
$198$	0.414214i	0.0294369i
$199$	−5.17157	−0.366603	−0.183302	−	0.983057i	$-0.558678\pi$
$199$	−5.17157	−0.366603	−0.183302	+	0.983057i	$0.558678\pi$
$200$	0	0
$201$	−11.6569	−0.822211
$202$	− 2.02944i	− 0.142791i
$203$	− 16.4853i	− 1.15704i
$204$	−0.757359	−0.0530258
$205$	0	0
$206$	0.970563	0.0676223
$207$	− 1.00000i	− 0.0695048i
$208$	8.48528i	0.588348i
$209$	3.58579	0.248034
$210$	0	0
$211$	9.31371	0.641182	0.320591	−	0.947218i	$-0.396118\pi$
$211$	9.31371	0.641182	0.320591	+	0.947218i	$0.396118\pi$
$212$	− 6.68629i	− 0.459216i
$213$	2.17157i	0.148794i
$214$	7.17157	0.490239
$215$	0	0
$216$	1.58579	0.107899
$217$	20.4853i	1.39063i
$218$	7.17157i	0.485720i
$219$	3.17157	0.214315
$220$	0	0
$221$	−1.17157	−0.0788085
$222$	− 2.41421i	− 0.162031i
$223$	− 26.8284i	− 1.79656i	−0.439419	−	0.898282i	$-0.644816\pi$
$223$	− 26.8284i	− 1.79656i	0.439419	−	0.898282i	$-0.355184\pi$
$224$	−10.6569	−0.712041
$225$	0	0
$226$	4.14214	0.275531
$227$	1.51472i	0.100535i	0.998736	+	0.0502677i	$0.0160075\pi$
$227$	1.51472i	0.100535i	−0.998736	+	0.0502677i	$0.983993\pi$
$228$	− 6.55635i	− 0.434205i
$229$	19.4853	1.28762	0.643812	−	0.765184i	$-0.277352\pi$
$229$	19.4853	1.28762	0.643812	+	0.765184i	$0.277352\pi$
$230$	0	0
$231$	2.41421	0.158844
$232$	− 10.8284i	− 0.710921i
$233$	− 14.5563i	− 0.953618i	−0.879007	−	0.476809i	$-0.841793\pi$
$233$	− 14.5563i	− 0.953618i	0.879007	−	0.476809i	$-0.158207\pi$
$234$	1.17157	0.0765881
$235$	0	0
$236$	−20.1127	−1.30923
$237$	− 4.75736i	− 0.309024i
$238$	− 0.414214i	− 0.0268495i
$239$	−12.3431	−0.798412	−0.399206	−	0.916861i	$-0.630714\pi$
$239$	−12.3431	−0.798412	−0.399206	+	0.916861i	$0.630714\pi$
$240$	0	0
$241$	−14.1421	−0.910975	−0.455488	−	0.890242i	$-0.650535\pi$
$241$	−14.1421	−0.910975	−0.455488	+	0.890242i	$0.650535\pi$
$242$	0.414214i	0.0266267i
$243$	1.00000i	0.0641500i
$244$	5.79899	0.371242
$245$	0	0
$246$	−3.68629	−0.235029
$247$	− 10.1421i	− 0.645329i
$248$	13.4558i	0.854447i
$249$	−12.4853	−0.791223
$250$	0	0
$251$	−24.9706	−1.57613	−0.788064	−	0.615593i	$-0.788916\pi$
$251$	−24.9706	−1.57613	−0.788064	+	0.615593i	$0.788916\pi$
$252$	− 4.41421i	− 0.278069i
$253$	− 1.00000i	− 0.0628695i
$254$	−0.514719	−0.0322963
$255$	0	0
$256$	3.97056	0.248160
$257$	13.3137i	0.830486i	0.909710	+	0.415243i	$0.136304\pi$
$257$	13.3137i	0.830486i	−0.909710	+	0.415243i	$0.863696\pi$
$258$	0.142136i	0.00884898i
$259$	−14.0711	−0.874334
$260$	0	0
$261$	6.82843	0.422669
$262$	0.485281i	0.0299808i
$263$	− 10.9706i	− 0.676474i	−0.941061	−	0.338237i	$-0.890169\pi$
$263$	− 10.9706i	− 0.676474i	0.941061	−	0.338237i	$-0.109831\pi$
$264$	1.58579	0.0975984
$265$	0	0
$266$	3.58579	0.219859
$267$	7.65685i	0.468592i
$268$	21.3137i	1.30194i
$269$	−23.7990	−1.45105	−0.725525	−	0.688196i	$-0.758403\pi$
$269$	−23.7990	−1.45105	−0.725525	+	0.688196i	$0.758403\pi$
$270$	0	0
$271$	−10.8995	−0.662097	−0.331049	−	0.943614i	$-0.607402\pi$
$271$	−10.8995	−0.662097	−0.331049	+	0.943614i	$0.607402\pi$
$272$	1.24264i	0.0753462i
$273$	− 6.82843i	− 0.413275i
$274$	6.68629	0.403934
$275$	0	0
$276$	−1.82843	−0.110058
$277$	0.828427i	0.0497754i	0.999690	+	0.0248877i	$0.00792281\pi$
$277$	0.828427i	0.0497754i	−0.999690	+	0.0248877i	$0.992077\pi$
$278$	− 6.20101i	− 0.371912i
$279$	−8.48528	−0.508001
$280$	0	0
$281$	17.9289	1.06955	0.534775	−	0.844994i	$-0.320396\pi$
$281$	17.9289	1.06955	0.534775	+	0.844994i	$0.320396\pi$
$282$	3.92893i	0.233965i
$283$	18.8995i	1.12346i	0.827321	+	0.561729i	$0.189864\pi$
$283$	18.8995i	1.12346i	−0.827321	+	0.561729i	$0.810136\pi$
$284$	3.97056	0.235610
$285$	0	0
$286$	1.17157	0.0692766
$287$	21.4853i	1.26824i
$288$	− 4.41421i	− 0.260110i
$289$	16.8284	0.989907
$290$	0	0
$291$	0.171573	0.0100578
$292$	− 5.79899i	− 0.339360i
$293$	20.4142i	1.19261i	0.802758	+	0.596306i	$0.203365\pi$
$293$	20.4142i	1.19261i	−0.802758	+	0.596306i	$0.796635\pi$
$294$	−0.485281	−0.0283022
$295$	0	0
$296$	−9.24264	−0.537218
$297$	1.00000i	0.0580259i
$298$	7.34315i	0.425377i
$299$	−2.82843	−0.163572
$300$	0	0
$301$	0.828427	0.0477497
$302$	5.79899i	0.333694i
$303$	− 4.89949i	− 0.281469i
$304$	−10.7574	−0.616977
$305$	0	0
$306$	0.171573	0.00980817
$307$	− 29.3137i	− 1.67302i	−0.547950	−	0.836511i	$-0.684592\pi$
$307$	− 29.3137i	− 1.67302i	0.547950	−	0.836511i	$-0.315408\pi$
$308$	− 4.41421i	− 0.251523i
$309$	2.34315	0.133297
$310$	0	0
$311$	2.34315	0.132868	0.0664338	−	0.997791i	$-0.478838\pi$
$311$	2.34315	0.132868	0.0664338	+	0.997791i	$0.478838\pi$
$312$	− 4.48528i	− 0.253929i
$313$	1.14214i	0.0645573i	0.999479	+	0.0322787i	$0.0102764\pi$
$313$	1.14214i	0.0645573i	−0.999479	+	0.0322787i	$0.989724\pi$
$314$	2.48528	0.140253
$315$	0	0
$316$	−8.69848	−0.489328
$317$	− 25.1716i	− 1.41378i	−0.707325	−	0.706888i	$-0.750098\pi$
$317$	− 25.1716i	− 1.41378i	0.707325	−	0.706888i	$-0.249902\pi$
$318$	1.51472i	0.0849412i
$319$	6.82843	0.382319
$320$	0	0
$321$	17.3137	0.966357
$322$	− 1.00000i	− 0.0557278i
$323$	− 1.48528i	− 0.0826433i
$324$	1.82843	0.101579
$325$	0	0
$326$	9.85786	0.545977
$327$	17.3137i	0.957450i
$328$	14.1127i	0.779243i
$329$	22.8995	1.26249
$330$	0	0
$331$	−3.85786	−0.212047	−0.106024	−	0.994364i	$-0.533812\pi$
$331$	−3.85786	−0.212047	−0.106024	+	0.994364i	$0.533812\pi$
$332$	22.8284i	1.25287i
$333$	− 5.82843i	− 0.319396i
$334$	−7.37258	−0.403410
$335$	0	0
$336$	−7.24264	−0.395118
$337$	24.1421i	1.31511i	0.753408	+	0.657553i	$0.228408\pi$
$337$	24.1421i	1.31511i	−0.753408	+	0.657553i	$0.771592\pi$
$338$	2.07107i	0.112651i
$339$	10.0000	0.543125
$340$	0	0
$341$	−8.48528	−0.459504
$342$	1.48528i	0.0803148i
$343$	19.7279i	1.06521i
$344$	0.544156	0.0293389
$345$	0	0
$346$	−7.68629	−0.413218
$347$	26.8284i	1.44023i	0.693857	+	0.720113i	$0.255910\pi$
$347$	26.8284i	1.44023i	−0.693857	+	0.720113i	$0.744090\pi$
$348$	− 12.4853i	− 0.669281i
$349$	−14.4853	−0.775379	−0.387690	−	0.921790i	$-0.626727\pi$
$349$	−14.4853	−0.775379	−0.387690	+	0.921790i	$0.626727\pi$
$350$	0	0
$351$	2.82843	0.150970
$352$	− 4.41421i	− 0.235278i
$353$	12.4853i	0.664524i	0.943187	+	0.332262i	$0.107812\pi$
$353$	12.4853i	0.664524i	−0.943187	+	0.332262i	$0.892188\pi$
$354$	4.55635	0.242167
$355$	0	0
$356$	14.0000	0.741999
$357$	− 1.00000i	− 0.0529256i
$358$	9.44365i	0.499112i
$359$	−32.4853	−1.71451	−0.857254	−	0.514894i	$-0.827831\pi$
$359$	−32.4853	−1.71451	−0.857254	+	0.514894i	$0.827831\pi$
$360$	0	0
$361$	−6.14214	−0.323270
$362$	4.95837i	0.260606i
$363$	1.00000i	0.0524864i
$364$	−12.4853	−0.654407
$365$	0	0
$366$	−1.31371	−0.0686686
$367$	21.3137i	1.11257i	0.830993	+	0.556283i	$0.187773\pi$
$367$	21.3137i	1.11257i	−0.830993	+	0.556283i	$0.812227\pi$
$368$	3.00000i	0.156386i
$369$	−8.89949	−0.463289
$370$	0	0
$371$	8.82843	0.458349
$372$	15.5147i	0.804401i
$373$	− 12.3431i	− 0.639104i	−0.947569	−	0.319552i	$-0.896468\pi$
$373$	− 12.3431i	− 0.639104i	0.947569	−	0.319552i	$-0.103532\pi$
$374$	0.171573	0.00887182
$375$	0	0
$376$	15.0416	0.775713
$377$	− 19.3137i	− 0.994707i
$378$	1.00000i	0.0514344i
$379$	−14.8284	−0.761685	−0.380843	−	0.924640i	$-0.624366\pi$
$379$	−14.8284	−0.761685	−0.380843	+	0.924640i	$0.624366\pi$
$380$	0	0
$381$	−1.24264	−0.0636624
$382$	4.89949i	0.250680i
$383$	20.0000i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$	20.0000i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$	−10.5563	−0.538701
$385$	0	0
$386$	8.00000	0.407189
$387$	0.343146i	0.0174431i
$388$	− 0.313708i	− 0.0159261i
$389$	6.34315	0.321610	0.160805	−	0.986986i	$-0.448591\pi$
$389$	6.34315	0.321610	0.160805	+	0.986986i	$0.448591\pi$
$390$	0	0
$391$	−0.414214	−0.0209477
$392$	1.85786i	0.0938363i
$393$	1.17157i	0.0590980i
$394$	5.48528	0.276344
$395$	0	0
$396$	1.82843	0.0918819
$397$	− 31.9411i	− 1.60308i	−0.597942	−	0.801540i	$-0.704015\pi$
$397$	− 31.9411i	− 1.60308i	0.597942	−	0.801540i	$-0.295985\pi$
$398$	− 2.14214i	− 0.107376i
$399$	8.65685	0.433385
$400$	0	0
$401$	−7.79899	−0.389463	−0.194731	−	0.980857i	$-0.562384\pi$
$401$	−7.79899	−0.389463	−0.194731	+	0.980857i	$0.562384\pi$
$402$	− 4.82843i	− 0.240820i
$403$	24.0000i	1.19553i
$404$	−8.95837	−0.445696
$405$	0	0
$406$	6.82843	0.338889
$407$	− 5.82843i	− 0.288904i
$408$	− 0.656854i	− 0.0325191i
$409$	−24.1421	−1.19375	−0.596876	−	0.802334i	$-0.703592\pi$
$409$	−24.1421	−1.19375	−0.596876	+	0.802334i	$0.703592\pi$
$410$	0	0
$411$	16.1421	0.796233
$412$	− 4.28427i	− 0.211071i
$413$	− 26.5563i	− 1.30675i
$414$	0.414214	0.0203575
$415$	0	0
$416$	−12.4853	−0.612141
$417$	− 14.9706i	− 0.733112i
$418$	1.48528i	0.0726475i
$419$	8.51472	0.415971	0.207986	−	0.978132i	$-0.433309\pi$
$419$	8.51472	0.415971	0.207986	+	0.978132i	$0.433309\pi$
$420$	0	0
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	3.85786i	0.187798i
$423$	9.48528i	0.461190i
$424$	5.79899	0.281624
$425$	0	0
$426$	−0.899495	−0.0435807
$427$	7.65685i	0.370541i
$428$	− 31.6569i	− 1.53019i
$429$	2.82843	0.136558
$430$	0	0
$431$	6.82843	0.328914	0.164457	−	0.986384i	$-0.447413\pi$
$431$	6.82843	0.328914	0.164457	+	0.986384i	$0.447413\pi$
$432$	− 3.00000i	− 0.144338i
$433$	9.31371i	0.447588i	0.974636	+	0.223794i	$0.0718443\pi$
$433$	9.31371i	0.447588i	−0.974636	+	0.223794i	$0.928156\pi$
$434$	−8.48528	−0.407307
$435$	0	0
$436$	31.6569	1.51609
$437$	− 3.58579i	− 0.171531i
$438$	1.31371i	0.0627714i
$439$	−27.7279	−1.32338	−0.661691	−	0.749777i	$-0.730161\pi$
$439$	−27.7279	−1.32338	−0.661691	+	0.749777i	$0.730161\pi$
$440$	0	0
$441$	−1.17157	−0.0557892
$442$	− 0.485281i	− 0.0230825i
$443$	25.9706i	1.23390i	0.787003	+	0.616949i	$0.211632\pi$
$443$	25.9706i	1.23390i	−0.787003	+	0.616949i	$0.788368\pi$
$444$	−10.6569	−0.505752
$445$	0	0
$446$	11.1127	0.526202
$447$	17.7279i	0.838502i
$448$	10.0711i	0.475813i
$449$	−10.4853	−0.494831	−0.247416	−	0.968909i	$-0.579581\pi$
$449$	−10.4853	−0.494831	−0.247416	+	0.968909i	$0.579581\pi$
$450$	0	0
$451$	−8.89949	−0.419061
$452$	− 18.2843i	− 0.860020i
$453$	14.0000i	0.657777i
$454$	−0.627417	−0.0294461
$455$	0	0
$456$	5.68629	0.266285
$457$	− 32.1421i	− 1.50355i	−0.659422	−	0.751773i	$-0.729199\pi$
$457$	− 32.1421i	− 1.50355i	0.659422	−	0.751773i	$-0.270801\pi$
$458$	8.07107i	0.377136i
$459$	0.414214	0.0193338
$460$	0	0
$461$	40.7696	1.89883	0.949414	−	0.314028i	$-0.101679\pi$
$461$	40.7696	1.89883	0.949414	+	0.314028i	$0.101679\pi$
$462$	1.00000i	0.0465242i
$463$	1.02944i	0.0478420i	0.999714	+	0.0239210i	$0.00761502\pi$
$463$	1.02944i	0.0478420i	−0.999714	+	0.0239210i	$0.992385\pi$
$464$	−20.4853	−0.951005
$465$	0	0
$466$	6.02944	0.279308
$467$	34.6274i	1.60237i	0.598420	+	0.801183i	$0.295796\pi$
$467$	34.6274i	1.60237i	−0.598420	+	0.801183i	$0.704204\pi$
$468$	− 5.17157i	− 0.239056i
$469$	−28.1421	−1.29948
$470$	0	0
$471$	6.00000	0.276465
$472$	− 17.4437i	− 0.802909i
$473$	0.343146i	0.0157779i
$474$	1.97056	0.0905109
$475$	0	0
$476$	−1.82843	−0.0838058
$477$	3.65685i	0.167436i
$478$	− 5.11270i	− 0.233849i
$479$	7.51472	0.343356	0.171678	−	0.985153i	$-0.445081\pi$
$479$	7.51472	0.343356	0.171678	+	0.985153i	$0.445081\pi$
$480$	0	0
$481$	−16.4853	−0.751664
$482$	− 5.85786i	− 0.266818i
$483$	− 2.41421i	− 0.109851i
$484$	1.82843	0.0831103
$485$	0	0
$486$	−0.414214	−0.0187891
$487$	− 10.4853i	− 0.475133i	−0.971371	−	0.237567i	$-0.923650\pi$
$487$	− 10.4853i	− 0.475133i	0.971371	−	0.237567i	$-0.0763498\pi$
$488$	5.02944i	0.227672i
$489$	23.7990	1.07623
$490$	0	0
$491$	4.14214	0.186932	0.0934660	−	0.995622i	$-0.470205\pi$
$491$	4.14214	0.186932	0.0934660	+	0.995622i	$0.470205\pi$
$492$	16.2721i	0.733602i
$493$	− 2.82843i	− 0.127386i
$494$	4.20101	0.189012
$495$	0	0
$496$	25.4558	1.14300
$497$	5.24264i	0.235165i
$498$	− 5.17157i	− 0.231744i
$499$	−40.8284	−1.82773	−0.913866	−	0.406017i	$-0.866918\pi$
$499$	−40.8284	−1.82773	−0.913866	+	0.406017i	$0.866918\pi$
$500$	0	0
$501$	−17.7990	−0.795200
$502$	− 10.3431i	− 0.461637i
$503$	22.2843i	0.993607i	0.867863	+	0.496803i	$0.165493\pi$
$503$	22.2843i	0.993607i	−0.867863	+	0.496803i	$0.834507\pi$
$504$	3.82843	0.170532
$505$	0	0
$506$	0.414214	0.0184140
$507$	5.00000i	0.222058i
$508$	2.27208i	0.100807i
$509$	40.6274	1.80078	0.900389	−	0.435085i	$-0.143282\pi$
$509$	40.6274	1.80078	0.900389	+	0.435085i	$0.143282\pi$
$510$	0	0
$511$	7.65685	0.338719
$512$	22.7574i	1.00574i
$513$	3.58579i	0.158316i
$514$	−5.51472	−0.243244
$515$	0	0
$516$	0.627417	0.0276205
$517$	9.48528i	0.417162i
$518$	− 5.82843i	− 0.256086i
$519$	−18.5563	−0.814533
$520$	0	0
$521$	−7.85786	−0.344259	−0.172130	−	0.985074i	$-0.555065\pi$
$521$	−7.85786	−0.344259	−0.172130	+	0.985074i	$0.555065\pi$
$522$	2.82843i	0.123797i
$523$	− 0.213203i	− 0.00932274i	−0.999989	−	0.00466137i	$-0.998516\pi$
$523$	− 0.213203i	− 0.00932274i	0.999989	−	0.00466137i	$-0.00148376\pi$
$524$	2.14214	0.0935796
$525$	0	0
$526$	4.54416	0.198135
$527$	3.51472i	0.153104i
$528$	− 3.00000i	− 0.130558i
$529$	22.0000	0.956522
$530$	0	0
$531$	11.0000	0.477359
$532$	− 15.8284i	− 0.686249i
$533$	25.1716i	1.09030i
$534$	−3.17157	−0.137247
$535$	0	0
$536$	−18.4853	−0.798443
$537$	22.7990i	0.983849i
$538$	− 9.85786i	− 0.425003i
$539$	−1.17157	−0.0504632
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	− 4.51472i	− 0.193924i
$543$	11.9706i	0.513706i
$544$	−1.82843	−0.0783932
$545$	0	0
$546$	2.82843	0.121046
$547$	17.8701i	0.764068i	0.924148	+	0.382034i	$0.124776\pi$
$547$	17.8701i	0.764068i	−0.924148	+	0.382034i	$0.875224\pi$
$548$	− 29.5147i	− 1.26081i
$549$	−3.17157	−0.135359
$550$	0	0
$551$	24.4853	1.04311
$552$	− 1.58579i	− 0.0674956i
$553$	− 11.4853i	− 0.488404i
$554$	−0.343146	−0.0145789
$555$	0	0
$556$	−27.3726	−1.16086
$557$	10.8284i	0.458815i	0.973330	+	0.229408i	$0.0736789\pi$
$557$	10.8284i	0.458815i	−0.973330	+	0.229408i	$0.926321\pi$
$558$	− 3.51472i	− 0.148790i
$559$	0.970563	0.0410504
$560$	0	0
$561$	0.414214	0.0174881
$562$	7.42641i	0.313264i
$563$	− 7.31371i	− 0.308236i	−0.988052	−	0.154118i	$-0.950746\pi$
$563$	− 7.31371i	− 0.308236i	0.988052	−	0.154118i	$-0.0492536\pi$
$564$	17.3431	0.730278
$565$	0	0
$566$	−7.82843	−0.329053
$567$	2.41421i	0.101387i
$568$	3.44365i	0.144492i
$569$	−6.75736	−0.283283	−0.141642	−	0.989918i	$-0.545238\pi$
$569$	−6.75736	−0.283283	−0.141642	+	0.989918i	$0.545238\pi$
$570$	0	0
$571$	−42.9706	−1.79826	−0.899131	−	0.437680i	$-0.855800\pi$
$571$	−42.9706	−1.79826	−0.899131	+	0.437680i	$0.855800\pi$
$572$	− 5.17157i	− 0.216234i
$573$	11.8284i	0.494140i
$574$	−8.89949	−0.371458
$575$	0	0
$576$	−4.17157	−0.173816
$577$	9.97056i	0.415080i	0.978227	+	0.207540i	$0.0665457\pi$
$577$	9.97056i	0.415080i	−0.978227	+	0.207540i	$0.933454\pi$
$578$	6.97056i	0.289937i
$579$	19.3137	0.802650
$580$	0	0
$581$	−30.1421	−1.25051
$582$	0.0710678i	0.00294586i
$583$	3.65685i	0.151451i
$584$	5.02944	0.208120
$585$	0	0
$586$	−8.45584	−0.349308
$587$	25.3431i	1.04602i	0.852325	+	0.523012i	$0.175192\pi$
$587$	25.3431i	1.04602i	−0.852325	+	0.523012i	$0.824808\pi$
$588$	2.14214i	0.0883402i
$589$	−30.4264	−1.25370
$590$	0	0
$591$	13.2426	0.544729
$592$	17.4853i	0.718641i
$593$	− 35.7990i	− 1.47009i	−0.678019	−	0.735044i	$-0.737161\pi$
$593$	− 35.7990i	− 1.47009i	0.678019	−	0.735044i	$-0.262839\pi$
$594$	−0.414214	−0.0169954
$595$	0	0
$596$	32.4142	1.32774
$597$	− 5.17157i	− 0.211658i
$598$	− 1.17157i	− 0.0479092i
$599$	−13.6863	−0.559207	−0.279603	−	0.960116i	$-0.590203\pi$
$599$	−13.6863	−0.559207	−0.279603	+	0.960116i	$0.590203\pi$
$600$	0	0
$601$	−9.17157	−0.374116	−0.187058	−	0.982349i	$-0.559895\pi$
$601$	−9.17157	−0.374116	−0.187058	+	0.982349i	$0.559895\pi$
$602$	0.343146i	0.0139856i
$603$	− 11.6569i	− 0.474704i
$604$	25.5980	1.04157
$605$	0	0
$606$	2.02944	0.0824403
$607$	− 30.9706i	− 1.25706i	−0.777787	−	0.628528i	$-0.783658\pi$
$607$	− 30.9706i	− 1.25706i	0.777787	−	0.628528i	$-0.216342\pi$
$608$	− 15.8284i	− 0.641927i
$609$	16.4853	0.668017
$610$	0	0
$611$	26.8284	1.08536
$612$	− 0.757359i	− 0.0306144i
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	12.1421	0.490017
$615$	0	0
$616$	3.82843	0.154252
$617$	38.1421i	1.53554i	0.640723	+	0.767772i	$0.278635\pi$
$617$	38.1421i	1.53554i	−0.640723	+	0.767772i	$0.721365\pi$
$618$	0.970563i	0.0390418i
$619$	6.62742	0.266378	0.133189	−	0.991091i	$-0.457478\pi$
$619$	6.62742	0.266378	0.133189	+	0.991091i	$0.457478\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0.970563i	0.0389160i
$623$	18.4853i	0.740597i
$624$	−8.48528	−0.339683
$625$	0	0
$626$	−0.473088	−0.0189084
$627$	3.58579i	0.143203i
$628$	− 10.9706i	− 0.437773i
$629$	−2.41421	−0.0962610
$630$	0	0
$631$	−18.6274	−0.741546	−0.370773	−	0.928724i	$-0.620907\pi$
$631$	−18.6274	−0.741546	−0.370773	+	0.928724i	$0.620907\pi$
$632$	− 7.54416i	− 0.300090i
$633$	9.31371i	0.370187i
$634$	10.4264	0.414086
$635$	0	0
$636$	6.68629	0.265129
$637$	3.31371i	0.131294i
$638$	2.82843i	0.111979i
$639$	−2.17157	−0.0859061
$640$	0	0
$641$	−25.5147	−1.00777	−0.503885	−	0.863771i	$-0.668097\pi$
$641$	−25.5147	−1.00777	−0.503885	+	0.863771i	$0.668097\pi$
$642$	7.17157i	0.283039i
$643$	0.970563i	0.0382753i	0.999817	+	0.0191376i	$0.00609207\pi$
$643$	0.970563i	0.0382753i	−0.999817	+	0.0191376i	$0.993908\pi$
$644$	−4.41421	−0.173944
$645$	0	0
$646$	0.615224	0.0242057
$647$	− 28.6569i	− 1.12662i	−0.826247	−	0.563309i	$-0.809528\pi$
$647$	− 28.6569i	− 1.12662i	0.826247	−	0.563309i	$-0.190472\pi$
$648$	1.58579i	0.0622956i
$649$	11.0000	0.431788
$650$	0	0
$651$	−20.4853	−0.802881
$652$	− 43.5147i	− 1.70417i
$653$	− 23.5147i	− 0.920202i	−0.887867	−	0.460101i	$-0.847813\pi$
$653$	− 23.5147i	− 0.920202i	0.887867	−	0.460101i	$-0.152187\pi$
$654$	−7.17157	−0.280431
$655$	0	0
$656$	26.6985	1.04240
$657$	3.17157i	0.123735i
$658$	9.48528i	0.369775i
$659$	47.1127	1.83525	0.917625	−	0.397447i	$-0.130104\pi$
$659$	47.1127	1.83525	0.917625	+	0.397447i	$0.130104\pi$
$660$	0	0
$661$	−23.3431	−0.907943	−0.453972	−	0.891016i	$-0.649993\pi$
$661$	−23.3431	−0.907943	−0.453972	+	0.891016i	$0.649993\pi$
$662$	− 1.59798i	− 0.0621072i
$663$	− 1.17157i	− 0.0455001i
$664$	−19.7990	−0.768350
$665$	0	0
$666$	2.41421	0.0935489
$667$	− 6.82843i	− 0.264398i
$668$	32.5442i	1.25917i
$669$	26.8284	1.03725
$670$	0	0
$671$	−3.17157	−0.122437
$672$	− 10.6569i	− 0.411097i
$673$	0.343146i	0.0132273i	0.999978	+	0.00661365i	$0.00210520\pi$
$673$	0.343146i	0.0132273i	−0.999978	+	0.00661365i	$0.997895\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	9.14214	0.351621
$677$	− 23.5147i	− 0.903744i	−0.892083	−	0.451872i	$-0.850756\pi$
$677$	− 23.5147i	− 0.903744i	0.892083	−	0.451872i	$-0.149244\pi$
$678$	4.14214i	0.159078i
$679$	0.414214	0.0158961
$680$	0	0
$681$	−1.51472	−0.0580441
$682$	− 3.51472i	− 0.134586i
$683$	12.5147i	0.478862i	0.970913	+	0.239431i	$0.0769610\pi$
$683$	12.5147i	0.478862i	−0.970913	+	0.239431i	$0.923039\pi$
$684$	6.55635	0.250688
$685$	0	0
$686$	−8.17157	−0.311992
$687$	19.4853i	0.743410i
$688$	− 1.02944i	− 0.0392469i
$689$	10.3431	0.394042
$690$	0	0
$691$	−35.4558	−1.34880	−0.674402	−	0.738364i	$-0.735598\pi$
$691$	−35.4558	−1.34880	−0.674402	+	0.738364i	$0.735598\pi$
$692$	33.9289i	1.28978i
$693$	2.41421i	0.0917084i
$694$	−11.1127	−0.421832
$695$	0	0
$696$	10.8284	0.410450
$697$	3.68629i	0.139628i
$698$	− 6.00000i	− 0.227103i
$699$	14.5563	0.550572
$700$	0	0
$701$	−41.3848	−1.56308	−0.781541	−	0.623854i	$-0.785566\pi$
$701$	−41.3848	−1.56308	−0.781541	+	0.623854i	$0.785566\pi$
$702$	1.17157i	0.0442182i
$703$	− 20.8995i	− 0.788239i
$704$	−4.17157	−0.157222
$705$	0	0
$706$	−5.17157	−0.194635
$707$	− 11.8284i	− 0.444854i
$708$	− 20.1127i	− 0.755881i
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	0	0
$711$	4.75736	0.178415
$712$	12.1421i	0.455046i
$713$	8.48528i	0.317776i
$714$	0.414214	0.0155016
$715$	0	0
$716$	41.6863	1.55789
$717$	− 12.3431i	− 0.460963i
$718$	− 13.4558i	− 0.502168i
$719$	−9.65685	−0.360140	−0.180070	−	0.983654i	$-0.557632\pi$
$719$	−9.65685	−0.360140	−0.180070	+	0.983654i	$0.557632\pi$
$720$	0	0
$721$	5.65685	0.210672
$722$	− 2.54416i	− 0.0946837i
$723$	− 14.1421i	− 0.525952i
$724$	21.8873	0.813435
$725$	0	0
$726$	−0.414214	−0.0153729
$727$	− 16.9706i	− 0.629403i	−0.949191	−	0.314702i	$-0.898096\pi$
$727$	− 16.9706i	− 0.629403i	0.949191	−	0.314702i	$-0.101904\pi$
$728$	− 10.8284i	− 0.401328i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0.142136	0.00525708
$732$	5.79899i	0.214337i
$733$	32.1421i	1.18720i	0.804761	+	0.593598i	$0.202293\pi$
$733$	32.1421i	1.18720i	−0.804761	+	0.593598i	$0.797707\pi$
$734$	−8.82843	−0.325863
$735$	0	0
$736$	−4.41421	−0.162710
$737$	− 11.6569i	− 0.429386i
$738$	− 3.68629i	− 0.135694i
$739$	−20.4142	−0.750949	−0.375474	−	0.926833i	$-0.622520\pi$
$739$	−20.4142	−0.750949	−0.375474	+	0.926833i	$0.622520\pi$
$740$	0	0
$741$	10.1421	0.372581
$742$	3.65685i	0.134247i
$743$	− 31.1127i	− 1.14141i	−0.821154	−	0.570707i	$-0.806669\pi$
$743$	− 31.1127i	− 1.14141i	0.821154	−	0.570707i	$-0.193331\pi$
$744$	−13.4558	−0.493315
$745$	0	0
$746$	5.11270	0.187189
$747$	− 12.4853i	− 0.456813i
$748$	− 0.757359i	− 0.0276918i
$749$	41.7990	1.52730
$750$	0	0
$751$	31.5980	1.15303	0.576513	−	0.817088i	$-0.304413\pi$
$751$	31.5980	1.15303	0.576513	+	0.817088i	$0.304413\pi$
$752$	− 28.4558i	− 1.03768i
$753$	− 24.9706i	− 0.909978i
$754$	8.00000	0.291343
$755$	0	0
$756$	4.41421	0.160543
$757$	10.6863i	0.388400i	0.980962	+	0.194200i	$0.0622111\pi$
$757$	10.6863i	0.388400i	−0.980962	+	0.194200i	$0.937789\pi$
$758$	− 6.14214i	− 0.223092i
$759$	1.00000	0.0362977
$760$	0	0
$761$	−5.17157	−0.187469	−0.0937347	−	0.995597i	$-0.529881\pi$
$761$	−5.17157	−0.187469	−0.0937347	+	0.995597i	$0.529881\pi$
$762$	− 0.514719i	− 0.0186463i
$763$	41.7990i	1.51323i
$764$	21.6274	0.782452
$765$	0	0
$766$	−8.28427	−0.299323
$767$	− 31.1127i	− 1.12341i
$768$	3.97056i	0.143275i
$769$	7.65685	0.276113	0.138057	−	0.990424i	$-0.455914\pi$
$769$	7.65685	0.276113	0.138057	+	0.990424i	$0.455914\pi$
$770$	0	0
$771$	−13.3137	−0.479481
$772$	− 35.3137i	− 1.27097i
$773$	− 0.828427i	− 0.0297965i	−0.999889	−	0.0148982i	$-0.995258\pi$
$773$	− 0.828427i	− 0.0297965i	0.999889	−	0.0148982i	$-0.00474243\pi$
$774$	−0.142136	−0.00510896
$775$	0	0
$776$	0.272078	0.00976703
$777$	− 14.0711i	− 0.504797i
$778$	2.62742i	0.0941975i
$779$	−31.9117	−1.14335
$780$	0	0
$781$	−2.17157	−0.0777050
$782$	− 0.171573i	− 0.00613543i
$783$	6.82843i	0.244028i
$784$	3.51472	0.125526
$785$	0	0
$786$	−0.485281	−0.0173094
$787$	7.92893i	0.282636i	0.989964	+	0.141318i	$0.0451340\pi$
$787$	7.92893i	0.282636i	−0.989964	+	0.141318i	$0.954866\pi$
$788$	− 24.2132i	− 0.862560i
$789$	10.9706	0.390562
$790$	0	0
$791$	24.1421	0.858396
$792$	1.58579i	0.0563485i
$793$	8.97056i	0.318554i
$794$	13.2304	0.469531
$795$	0	0
$796$	−9.45584	−0.335154
$797$	− 40.9706i	− 1.45125i	−0.688089	−	0.725626i	$-0.741550\pi$
$797$	− 40.9706i	− 1.45125i	0.688089	−	0.725626i	$-0.258450\pi$
$798$	3.58579i	0.126935i
$799$	3.92893	0.138996
$800$	0	0
$801$	−7.65685	−0.270542
$802$	− 3.23045i	− 0.114071i
$803$	3.17157i	0.111922i
$804$	−21.3137	−0.751677
$805$	0	0
$806$	−9.94113	−0.350161
$807$	− 23.7990i	− 0.837764i
$808$	− 7.76955i	− 0.273332i
$809$	−7.72792	−0.271699	−0.135850	−	0.990729i	$-0.543376\pi$
$809$	−7.72792	−0.271699	−0.135850	+	0.990729i	$0.543376\pi$
$810$	0	0
$811$	10.2132	0.358634	0.179317	−	0.983791i	$-0.442611\pi$
$811$	10.2132	0.358634	0.179317	+	0.983791i	$0.442611\pi$
$812$	− 30.1421i	− 1.05778i
$813$	− 10.8995i	− 0.382262i
$814$	2.41421	0.0846181
$815$	0	0
$816$	−1.24264	−0.0435011
$817$	1.23045i	0.0430479i
$818$	− 10.0000i	− 0.349642i
$819$	6.82843	0.238605
$820$	0	0
$821$	15.7990	0.551389	0.275694	−	0.961245i	$-0.411092\pi$
$821$	15.7990	0.551389	0.275694	+	0.961245i	$0.411092\pi$
$822$	6.68629i	0.233211i
$823$	− 14.9706i	− 0.521841i	−0.965360	−	0.260921i	$-0.915974\pi$
$823$	− 14.9706i	− 0.521841i	0.965360	−	0.260921i	$-0.0840260\pi$
$824$	3.71573	0.129444
$825$	0	0
$826$	11.0000	0.382739
$827$	12.6863i	0.441146i	0.975371	+	0.220573i	$0.0707927\pi$
$827$	12.6863i	0.441146i	−0.975371	+	0.220573i	$0.929207\pi$
$828$	− 1.82843i	− 0.0635422i
$829$	47.9411	1.66506	0.832532	−	0.553977i	$-0.186890\pi$
$829$	47.9411	1.66506	0.832532	+	0.553977i	$0.186890\pi$
$830$	0	0
$831$	−0.828427	−0.0287378
$832$	11.7990i	0.409056i
$833$	0.485281i	0.0168140i
$834$	6.20101	0.214723
$835$	0	0
$836$	6.55635	0.226756
$837$	− 8.48528i	− 0.293294i
$838$	3.52691i	0.121835i
$839$	47.3137	1.63345	0.816725	−	0.577027i	$-0.195787\pi$
$839$	47.3137	1.63345	0.816725	+	0.577027i	$0.195787\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	− 11.1838i	− 0.385418i
$843$	17.9289i	0.617505i
$844$	17.0294	0.586177
$845$	0	0
$846$	−3.92893	−0.135079
$847$	2.41421i	0.0829534i
$848$	− 10.9706i	− 0.376731i
$849$	−18.8995	−0.648629
$850$	0	0
$851$	−5.82843	−0.199796
$852$	3.97056i	0.136029i
$853$	− 19.1716i	− 0.656422i	−0.944604	−	0.328211i	$-0.893554\pi$
$853$	− 19.1716i	− 0.656422i	0.944604	−	0.328211i	$-0.106446\pi$
$854$	−3.17157	−0.108529
$855$	0	0
$856$	27.4558	0.938421
$857$	36.6985i	1.25360i	0.779182	+	0.626798i	$0.215635\pi$
$857$	36.6985i	1.25360i	−0.779182	+	0.626798i	$0.784365\pi$
$858$	1.17157i	0.0399968i
$859$	20.4853	0.698949	0.349474	−	0.936946i	$-0.386360\pi$
$859$	20.4853	0.698949	0.349474	+	0.936946i	$0.386360\pi$
$860$	0	0
$861$	−21.4853	−0.732216
$862$	2.82843i	0.0963366i
$863$	− 28.6863i	− 0.976493i	−0.872706	−	0.488246i	$-0.837637\pi$
$863$	− 28.6863i	− 0.976493i	0.872706	−	0.488246i	$-0.162363\pi$
$864$	4.41421	0.150175
$865$	0	0
$866$	−3.85786	−0.131096
$867$	16.8284i	0.571523i
$868$	37.4558i	1.27133i
$869$	4.75736	0.161382
$870$	0	0
$871$	−32.9706	−1.11716
$872$	27.4558i	0.929772i
$873$	0.171573i	0.00580686i
$874$	1.48528	0.0502404
$875$	0	0
$876$	5.79899	0.195930
$877$	− 15.1127i	− 0.510320i	−0.966899	−	0.255160i	$-0.917872\pi$
$877$	− 15.1127i	− 0.510320i	0.966899	−	0.255160i	$-0.0821281\pi$
$878$	− 11.4853i	− 0.387609i
$879$	−20.4142	−0.688554
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	− 0.485281i	− 0.0163403i
$883$	38.6274i	1.29992i	0.759970	+	0.649958i	$0.225214\pi$
$883$	38.6274i	1.29992i	−0.759970	+	0.649958i	$0.774786\pi$
$884$	−2.14214	−0.0720478
$885$	0	0
$886$	−10.7574	−0.361401
$887$	22.1421i	0.743460i	0.928341	+	0.371730i	$0.121235\pi$
$887$	22.1421i	0.743460i	−0.928341	+	0.371730i	$0.878765\pi$
$888$	− 9.24264i	− 0.310163i
$889$	−3.00000	−0.100617
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 49.0538i	− 1.64244i
$893$	34.0122i	1.13817i
$894$	−7.34315	−0.245592
$895$	0	0
$896$	−25.4853	−0.851403
$897$	− 2.82843i	− 0.0944384i
$898$	− 4.34315i	− 0.144933i
$899$	−57.9411	−1.93244
$900$	0	0
$901$	1.51472	0.0504626
$902$	− 3.68629i	− 0.122740i
$903$	0.828427i	0.0275683i
$904$	15.8579	0.527425
$905$	0	0
$906$	−5.79899	−0.192659
$907$	2.48528i	0.0825224i	0.999148	+	0.0412612i	$0.0131376\pi$
$907$	2.48528i	0.0825224i	−0.999148	+	0.0412612i	$0.986862\pi$
$908$	2.76955i	0.0919108i
$909$	4.89949	0.162506
$910$	0	0
$911$	28.5147	0.944735	0.472367	−	0.881402i	$-0.343400\pi$
$911$	28.5147	0.944735	0.472367	+	0.881402i	$0.343400\pi$
$912$	− 10.7574i	− 0.356212i
$913$	− 12.4853i	− 0.413203i
$914$	13.3137	0.440378
$915$	0	0
$916$	35.6274	1.17716
$917$	2.82843i	0.0934029i
$918$	0.171573i	0.00566275i
$919$	−1.78680	−0.0589410	−0.0294705	−	0.999566i	$-0.509382\pi$
$919$	−1.78680	−0.0589410	−0.0294705	+	0.999566i	$0.509382\pi$
$920$	0	0
$921$	29.3137	0.965920
$922$	16.8873i	0.556154i
$923$	6.14214i	0.202171i
$924$	4.41421	0.145217
$925$	0	0
$926$	−0.426407	−0.0140126
$927$	2.34315i	0.0769590i
$928$	− 30.1421i	− 0.989464i
$929$	13.7990	0.452730	0.226365	−	0.974043i	$-0.427316\pi$
$929$	13.7990	0.452730	0.226365	+	0.974043i	$0.427316\pi$
$930$	0	0
$931$	−4.20101	−0.137683
$932$	− 26.6152i	− 0.871811i
$933$	2.34315i	0.0767111i
$934$	−14.3431	−0.469322
$935$	0	0
$936$	4.48528	0.146606
$937$	16.0000i	0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$	16.0000i	0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$	− 11.6569i	− 0.380610i
$939$	−1.14214	−0.0372722
$940$	0	0
$941$	−22.8995	−0.746502	−0.373251	−	0.927730i	$-0.621757\pi$
$941$	−22.8995	−0.746502	−0.373251	+	0.927730i	$0.621757\pi$
$942$	2.48528i	0.0809748i
$943$	8.89949i	0.289807i
$944$	−33.0000	−1.07406
$945$	0	0
$946$	−0.142136	−0.00462123
$947$	36.7990i	1.19581i	0.801568	+	0.597903i	$0.203999\pi$
$947$	36.7990i	1.19581i	−0.801568	+	0.597903i	$0.796001\pi$
$948$	− 8.69848i	− 0.282514i
$949$	8.97056	0.291197
$950$	0	0
$951$	25.1716	0.816244
$952$	− 1.58579i	− 0.0513956i
$953$	− 29.0416i	− 0.940751i	−0.882466	−	0.470375i	$-0.844119\pi$
$953$	− 29.0416i	− 0.940751i	0.882466	−	0.470375i	$-0.155881\pi$
$954$	−1.51472	−0.0490408
$955$	0	0
$956$	−22.5685	−0.729919
$957$	6.82843i	0.220732i
$958$	3.11270i	0.100567i
$959$	38.9706	1.25843
$960$	0	0
$961$	41.0000	1.32258
$962$	− 6.82843i	− 0.220157i
$963$	17.3137i	0.557926i
$964$	−25.8579	−0.832826
$965$	0	0
$966$	1.00000	0.0321745
$967$	− 38.0000i	− 1.22200i	−0.791632	−	0.610999i	$-0.790768\pi$
$967$	− 38.0000i	− 1.22200i	0.791632	−	0.610999i	$-0.209232\pi$
$968$	1.58579i	0.0509691i
$969$	1.48528	0.0477141
$970$	0	0
$971$	−50.3137	−1.61464	−0.807322	−	0.590111i	$-0.799084\pi$
$971$	−50.3137	−1.61464	−0.807322	+	0.590111i	$0.799084\pi$
$972$	1.82843i	0.0586468i
$973$	− 36.1421i	− 1.15866i
$974$	4.34315	0.139163
$975$	0	0
$976$	9.51472	0.304559
$977$	− 60.5685i	− 1.93776i	−0.247532	−	0.968880i	$-0.579620\pi$
$977$	− 60.5685i	− 1.93776i	0.247532	−	0.968880i	$-0.420380\pi$
$978$	9.85786i	0.315220i
$979$	−7.65685	−0.244714
$980$	0	0
$981$	−17.3137	−0.552784
$982$	1.71573i	0.0547511i
$983$	51.2843i	1.63571i	0.575421	+	0.817857i	$0.304838\pi$
$983$	51.2843i	1.63571i	−0.575421	+	0.817857i	$0.695162\pi$
$984$	−14.1127	−0.449896
$985$	0	0
$986$	1.17157	0.0373105
$987$	22.8995i	0.728899i
$988$	− 18.5442i	− 0.589968i
$989$	0.343146	0.0109114
$990$	0	0
$991$	−42.2843	−1.34320	−0.671602	−	0.740912i	$-0.734394\pi$
$991$	−42.2843	−1.34320	−0.671602	+	0.740912i	$0.734394\pi$
$992$	37.4558i	1.18922i
$993$	− 3.85786i	− 0.122426i
$994$	−2.17157	−0.0688781
$995$	0	0
$996$	−22.8284	−0.723346
$997$	47.2548i	1.49658i	0.663374	+	0.748288i	$0.269124\pi$
$997$	47.2548i	1.49658i	−0.663374	+	0.748288i	$0.730876\pi$
$998$	− 16.9117i	− 0.535330i
$999$	5.82843	0.184403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.d.199.3		4
3.2	odd	2		2475.2.c.o.199.2		4
5.2	odd	4		825.2.a.f.1.1	yes	2
5.3	odd	4		825.2.a.d.1.2	✓	2
5.4	even	2	inner	825.2.c.d.199.2		4
15.2	even	4		2475.2.a.l.1.2		2
15.8	even	4		2475.2.a.w.1.1		2
15.14	odd	2		2475.2.c.o.199.3		4
55.32	even	4		9075.2.a.w.1.2		2
55.43	even	4		9075.2.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.d.1.2	✓	2	5.3	odd	4
825.2.a.f.1.1	yes	2	5.2	odd	4
825.2.c.d.199.2		4	5.4	even	2	inner
825.2.c.d.199.3		4	1.1	even	1	trivial
2475.2.a.l.1.2		2	15.2	even	4
2475.2.a.w.1.1		2	15.8	even	4
2475.2.c.o.199.2		4	3.2	odd	2
2475.2.c.o.199.3		4	15.14	odd	2
9075.2.a.w.1.2		2	55.32	even	4
9075.2.a.ca.1.1		2	55.43	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +2.41421i q^{7} +1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +2.41421i q^{7} +1.58579i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.82843i q^{12} +2.82843i q^{13} -1.00000 q^{14} +3.00000 q^{16} +0.414214i q^{17} -0.414214i q^{18} -3.58579 q^{19} -2.41421 q^{21} -0.414214i q^{22} +1.00000i q^{23} -1.58579 q^{24} -1.17157 q^{26} -1.00000i q^{27} +4.41421i q^{28} -6.82843 q^{29} +8.48528 q^{31} +4.41421i q^{32} -1.00000i q^{33} -0.171573 q^{34} -1.82843 q^{36} +5.82843i q^{37} -1.48528i q^{38} -2.82843 q^{39} +8.89949 q^{41} -1.00000i q^{42} -0.343146i q^{43} -1.82843 q^{44} -0.414214 q^{46} -9.48528i q^{47} +3.00000i q^{48} +1.17157 q^{49} -0.414214 q^{51} +5.17157i q^{52} -3.65685i q^{53} +0.414214 q^{54} -3.82843 q^{56} -3.58579i q^{57} -2.82843i q^{58} -11.0000 q^{59} +3.17157 q^{61} +3.51472i q^{62} -2.41421i q^{63} +4.17157 q^{64} +0.414214 q^{66} +11.6569i q^{67} +0.757359i q^{68} -1.00000 q^{69} +2.17157 q^{71} -1.58579i q^{72} -3.17157i q^{73} -2.41421 q^{74} -6.55635 q^{76} -2.41421i q^{77} -1.17157i q^{78} -4.75736 q^{79} +1.00000 q^{81} +3.68629i q^{82} +12.4853i q^{83} -4.41421 q^{84} +0.142136 q^{86} -6.82843i q^{87} -1.58579i q^{88} +7.65685 q^{89} -6.82843 q^{91} +1.82843i q^{92} +8.48528i q^{93} +3.92893 q^{94} -4.41421 q^{96} -0.171573i q^{97} +0.485281i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 12 q^{16} - 20 q^{19} - 4 q^{21} - 12 q^{24} - 16 q^{26} - 16 q^{29} - 12 q^{34} + 4 q^{36} - 4 q^{41} + 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{51} - 4 q^{54} - 4 q^{56} - 44 q^{59} + 24 q^{61} + 28 q^{64} - 4 q^{66} - 4 q^{69} + 20 q^{71} - 4 q^{74} + 36 q^{76} - 36 q^{79} + 4 q^{81} - 12 q^{84} - 56 q^{86} + 8 q^{89} - 16 q^{91} + 44 q^{94} - 12 q^{96} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 - 4 * q^11 - 4 * q^14 + 12 * q^16 - 20 * q^19 - 4 * q^21 - 12 * q^24 - 16 * q^26 - 16 * q^29 - 12 * q^34 + 4 * q^36 - 4 * q^41 + 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^51 - 4 * q^54 - 4 * q^56 - 44 * q^59 + 24 * q^61 + 28 * q^64 - 4 * q^66 - 4 * q^69 + 20 * q^71 - 4 * q^74 + 36 * q^76 - 36 * q^79 + 4 * q^81 - 12 * q^84 - 56 * q^86 + 8 * q^89 - 16 * q^91 + 44 * q^94 - 12 * q^96 + 4 * q^99

Introduction

L-functions

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