Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,2,Mod(107,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 107.2
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 225.107
Dual form 225.2.f.b.143.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.73205 + 1.73205i) q^{2} -4.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(3.46410 + 3.46410i) q^{8} +4.24264i q^{11} +(-3.67423 + 3.67423i) q^{13} -4.24264 q^{14} -4.00000 q^{16} +(1.73205 - 1.73205i) q^{17} +5.00000i q^{19} +(-7.34847 - 7.34847i) q^{22} +(-1.73205 - 1.73205i) q^{23} -12.7279i q^{26} +(4.89898 - 4.89898i) q^{28} -4.24264 q^{29} +1.00000 q^{31} +6.00000i q^{34} +(2.44949 + 2.44949i) q^{37} +(-8.66025 - 8.66025i) q^{38} +8.48528i q^{41} +(-1.22474 + 1.22474i) q^{43} +16.9706 q^{44} +6.00000 q^{46} +(5.19615 - 5.19615i) q^{47} -4.00000i q^{49} +(14.6969 + 14.6969i) q^{52} +(6.92820 + 6.92820i) q^{53} +8.48528i q^{56} +(7.34847 - 7.34847i) q^{58} +12.7279 q^{59} -7.00000 q^{61} +(-1.73205 + 1.73205i) q^{62} -8.00000i q^{64} +(3.67423 + 3.67423i) q^{67} +(-6.92820 - 6.92820i) q^{68} -8.48528i q^{71} +(2.44949 - 2.44949i) q^{73} -8.48528 q^{74} +20.0000 q^{76} +(-5.19615 + 5.19615i) q^{77} -2.00000i q^{79} +(-14.6969 - 14.6969i) q^{82} +(-1.73205 - 1.73205i) q^{83} -4.24264i q^{86} +(-14.6969 + 14.6969i) q^{88} -8.48528 q^{89} -9.00000 q^{91} +(-6.92820 + 6.92820i) q^{92} +18.0000i q^{94} +(-8.57321 - 8.57321i) q^{97} +(6.92820 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{16} + 8 q^{31} + 48 q^{46} - 56 q^{61} + 160 q^{76} - 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 + 1.73205i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 0 0
\(4\) − 4.00000i − 2.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.22474 + 1.22474i 0.462910 + 0.462910i 0.899608 − 0.436698i \(-0.143852\pi\)
−0.436698 + 0.899608i \(0.643852\pi\)
\(8\) 3.46410 + 3.46410i 1.22474 + 1.22474i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) −3.67423 + 3.67423i −1.01905 + 1.01905i −0.0192343 + 0.999815i \(0.506123\pi\)
−0.999815 + 0.0192343i \(0.993877\pi\)
\(14\) −4.24264 −1.13389
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 1.73205 − 1.73205i 0.420084 − 0.420084i −0.465149 − 0.885233i \(-0.653999\pi\)
0.885233 + 0.465149i \(0.153999\pi\)
\(18\) 0 0
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −7.34847 − 7.34847i −1.56670 − 1.56670i
\(23\) −1.73205 − 1.73205i −0.361158 − 0.361158i 0.503081 − 0.864239i \(-0.332200\pi\)
−0.864239 + 0.503081i \(0.832200\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 12.7279i − 2.49615i
\(27\) 0 0
\(28\) 4.89898 − 4.89898i 0.925820 − 0.925820i
\(29\) −4.24264 −0.787839 −0.393919 − 0.919145i \(-0.628881\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 6.00000i 1.02899i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.44949 + 2.44949i 0.402694 + 0.402694i 0.879181 − 0.476488i \(-0.158090\pi\)
−0.476488 + 0.879181i \(0.658090\pi\)
\(38\) −8.66025 − 8.66025i −1.40488 − 1.40488i
\(39\) 0 0
\(40\) 0 0
\(41\) 8.48528i 1.32518i 0.748983 + 0.662589i \(0.230542\pi\)
−0.748983 + 0.662589i \(0.769458\pi\)
\(42\) 0 0
\(43\) −1.22474 + 1.22474i −0.186772 + 0.186772i −0.794299 − 0.607527i \(-0.792162\pi\)
0.607527 + 0.794299i \(0.292162\pi\)
\(44\) 16.9706 2.55841
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 5.19615 − 5.19615i 0.757937 − 0.757937i −0.218010 − 0.975947i \(-0.569956\pi\)
0.975947 + 0.218010i \(0.0699565\pi\)
\(48\) 0 0
\(49\) − 4.00000i − 0.571429i
\(50\) 0 0
\(51\) 0 0
\(52\) 14.6969 + 14.6969i 2.03810 + 2.03810i
\(53\) 6.92820 + 6.92820i 0.951662 + 0.951662i 0.998884 − 0.0472225i \(-0.0150370\pi\)
−0.0472225 + 0.998884i \(0.515037\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.48528i 1.13389i
\(57\) 0 0
\(58\) 7.34847 − 7.34847i 0.964901 − 0.964901i
\(59\) 12.7279 1.65703 0.828517 − 0.559964i \(-0.189185\pi\)
0.828517 + 0.559964i \(0.189185\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −1.73205 + 1.73205i −0.219971 + 0.219971i
\(63\) 0 0
\(64\) − 8.00000i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 3.67423 + 3.67423i 0.448879 + 0.448879i 0.894982 − 0.446103i \(-0.147188\pi\)
−0.446103 + 0.894982i \(0.647188\pi\)
\(68\) −6.92820 − 6.92820i −0.840168 − 0.840168i
\(69\) 0 0
\(70\) 0 0
\(71\) − 8.48528i − 1.00702i −0.863990 − 0.503509i \(-0.832042\pi\)
0.863990 − 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) 2.44949 − 2.44949i 0.286691 − 0.286691i −0.549079 − 0.835770i \(-0.685021\pi\)
0.835770 + 0.549079i \(0.185021\pi\)
\(74\) −8.48528 −0.986394
\(75\) 0 0
\(76\) 20.0000 2.29416
\(77\) −5.19615 + 5.19615i −0.592157 + 0.592157i
\(78\) 0 0
\(79\) − 2.00000i − 0.225018i −0.993651 − 0.112509i \(-0.964111\pi\)
0.993651 − 0.112509i \(-0.0358886\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.6969 − 14.6969i −1.62301 − 1.62301i
\(83\) −1.73205 − 1.73205i −0.190117 − 0.190117i 0.605629 − 0.795747i \(-0.292921\pi\)
−0.795747 + 0.605629i \(0.792921\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 4.24264i − 0.457496i
\(87\) 0 0
\(88\) −14.6969 + 14.6969i −1.56670 + 1.56670i
\(89\) −8.48528 −0.899438 −0.449719 − 0.893170i \(-0.648476\pi\)
−0.449719 + 0.893170i \(0.648476\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) −6.92820 + 6.92820i −0.722315 + 0.722315i
\(93\) 0 0
\(94\) 18.0000i 1.85656i
\(95\) 0 0
\(96\) 0 0
\(97\) −8.57321 − 8.57321i −0.870478 − 0.870478i 0.122046 − 0.992524i \(-0.461054\pi\)
−0.992524 + 0.122046i \(0.961054\pi\)
\(98\) 6.92820 + 6.92820i 0.699854 + 0.699854i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 225.2.f.b.107.2 yes 8
3.2 odd 2 inner 225.2.f.b.107.4 yes 8
4.3 odd 2 3600.2.w.l.1457.1 8
5.2 odd 4 inner 225.2.f.b.143.1 yes 8
5.3 odd 4 inner 225.2.f.b.143.4 yes 8
5.4 even 2 inner 225.2.f.b.107.3 yes 8
12.11 even 2 3600.2.w.l.1457.2 8
15.2 even 4 inner 225.2.f.b.143.3 yes 8
15.8 even 4 inner 225.2.f.b.143.2 yes 8
15.14 odd 2 inner 225.2.f.b.107.1 ✓ 8
20.3 even 4 3600.2.w.l.593.1 8
20.7 even 4 3600.2.w.l.593.3 8
20.19 odd 2 3600.2.w.l.1457.3 8
60.23 odd 4 3600.2.w.l.593.2 8
60.47 odd 4 3600.2.w.l.593.4 8
60.59 even 2 3600.2.w.l.1457.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.f.b.107.1 ✓ 8 15.14 odd 2 inner
225.2.f.b.107.2 yes 8 1.1 even 1 trivial
225.2.f.b.107.3 yes 8 5.4 even 2 inner
225.2.f.b.107.4 yes 8 3.2 odd 2 inner
225.2.f.b.143.1 yes 8 5.2 odd 4 inner
225.2.f.b.143.2 yes 8 15.8 even 4 inner
225.2.f.b.143.3 yes 8 15.2 even 4 inner
225.2.f.b.143.4 yes 8 5.3 odd 4 inner
3600.2.w.l.593.1 8 20.3 even 4
3600.2.w.l.593.2 8 60.23 odd 4
3600.2.w.l.593.3 8 20.7 even 4
3600.2.w.l.593.4 8 60.47 odd 4
3600.2.w.l.1457.1 8 4.3 odd 2
3600.2.w.l.1457.2 8 12.11 even 2
3600.2.w.l.1457.3 8 20.19 odd 2
3600.2.w.l.1457.4 8 60.59 even 2