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Properties

Label 45.18.b.c
Level $45$
Weight $18$
Character orbit 45.b
Analytic conductor $82.450$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,18,Mod(19,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), 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: \(82.4499393051\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 1682025200 x^{14} - 16813165927404 x^{13} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{73}\cdot 3^{42}\cdot 5^{34} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 92073) q^{4} + ( - \beta_{3} + 188 \beta_1) q^{5} - \beta_{4} q^{7} + ( - \beta_{5} - 7 \beta_{3} - 63891 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 92073) q^{4} + ( - \beta_{3} + 188 \beta_1) q^{5} - \beta_{4} q^{7} + ( - \beta_{5} - 7 \beta_{3} - 63891 \beta_1) q^{8} + (\beta_{7} + 49 \beta_{2} - 41887458) q^{10} + ( - \beta_{8} + 134 \beta_{3} - 38 \beta_1) q^{11} + (\beta_{11} + 3 \beta_{7} + 48 \beta_{4} + \cdots - 2) q^{13}+ \cdots + ( - 147812944 \beta_{10} + \cdots + 57292151620155 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 1473168 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 1473168 q^{4} - 670199320 q^{10} + 35028482624 q^{16} + 236279498336 q^{19} + 2507484007120 q^{25} - 12501422403232 q^{31} - 36533616605360 q^{34} - 44259047245280 q^{40} - 28675167752480 q^{46} - 430197530548688 q^{49} - 16\!\cdots\!00 q^{55}+ \cdots - 20\!\cdots\!80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 1682025200 x^{14} - 16813165927404 x^{13} + \cdots + 36\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 82\!\cdots\!57 \nu^{15} + \cdots - 14\!\cdots\!92 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 43\!\cdots\!42 \nu^{15} + \cdots - 76\!\cdots\!88 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!65 \nu^{15} + \cdots + 90\!\cdots\!36 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 43\!\cdots\!07 \nu^{15} + \cdots - 23\!\cdots\!76 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36\!\cdots\!45 \nu^{15} + \cdots + 18\!\cdots\!92 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!13 \nu^{15} + \cdots - 17\!\cdots\!32 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93\!\cdots\!76 \nu^{15} + \cdots + 72\!\cdots\!84 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!83 \nu^{15} + \cdots + 21\!\cdots\!60 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!15 \nu^{15} + \cdots + 10\!\cdots\!64 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62\!\cdots\!27 \nu^{15} + \cdots - 36\!\cdots\!04 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!01 \nu^{15} + \cdots + 13\!\cdots\!12 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44\!\cdots\!77 \nu^{15} + \cdots + 67\!\cdots\!80 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!13 \nu^{15} + \cdots + 16\!\cdots\!64 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!19 \nu^{15} + \cdots - 95\!\cdots\!44 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89\!\cdots\!11 \nu^{15} + \cdots + 63\!\cdots\!48 ) / 47\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 290 \beta_{14} - 240 \beta_{13} - 2360 \beta_{12} - 480 \beta_{11} + 11168 \beta_{10} + \cdots + 388795200 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 47575 \beta_{15} - 636284 \beta_{14} - 2400075 \beta_{13} - 2680976 \beta_{12} + \cdots + 13\!\cdots\!00 ) / 622080000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 329460987 \beta_{15} - 41306178064 \beta_{14} - 38824943871 \beta_{13} + \cdots + 19\!\cdots\!20 ) / 622080000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20272791751339 \beta_{15} - 862311441852368 \beta_{14} + \cdots + 96\!\cdots\!80 ) / 622080000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20\!\cdots\!25 \beta_{15} + \cdots + 11\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 99\!\cdots\!15 \beta_{15} + \cdots + 42\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 30\!\cdots\!65 \beta_{15} + \cdots + 11\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 12\!\cdots\!35 \beta_{15} + \cdots + 38\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 41\!\cdots\!45 \beta_{15} + \cdots + 11\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15\!\cdots\!75 \beta_{15} + \cdots + 33\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 51\!\cdots\!05 \beta_{15} + \cdots + 98\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 17\!\cdots\!35 \beta_{15} + \cdots + 27\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 60\!\cdots\!25 \beta_{15} + \cdots + 77\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 20\!\cdots\!55 \beta_{15} + \cdots + 20\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 68\!\cdots\!65 \beta_{15} + \cdots + 50\!\cdots\!00 ) / 3110400000 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−3559.87 + 2204.84i
−5646.66 + 2204.84i
−7749.28 − 3526.36i
10898.0 − 3526.36i
−4797.27 + 2068.52i
−595.928 + 2068.52i
32455.2 − 2832.46i
−21003.2 − 2832.46i
32455.2 + 2832.46i
−21003.2 + 2832.46i
−4797.27 − 2068.52i
−595.928 − 2068.52i
−7749.28 + 3526.36i
10898.0 + 3526.36i
−3559.87 − 2204.84i
−5646.66 − 2204.84i
− 665.993i 0 −312475. −847574. − 211088.i 0 − 5.21087e6i 1.20813e8i 0 −1.40583e8 + 5.64478e8i
19.2 − 665.993i 0 −312475. 847574. − 211088.i 0 5.21087e6i 1.20813e8i 0 −1.40583e8 − 5.64478e8i
19.3 − 507.394i 0 −126376. −567939. + 663615.i 0 7.16981e6i − 2.38261e6i 0 3.36714e8 + 2.88169e8i
19.4 − 507.394i 0 −126376. 567939. + 663615.i 0 − 7.16981e6i − 2.38261e6i 0 3.36714e8 − 2.88169e8i
19.5 − 406.510i 0 −34178.7 −228254. − 843113.i 0 2.76153e7i − 3.93881e7i 0 −3.42734e8 + 9.27876e7i
19.6 − 406.510i 0 −34178.7 228254. − 843113.i 0 − 2.76153e7i − 3.93881e7i 0 −3.42734e8 − 9.27876e7i
19.7 − 162.279i 0 104737. −863874. − 129077.i 0 − 1.40324e7i − 3.82670e7i 0 −2.09465e7 + 1.40189e8i
19.8 − 162.279i 0 104737. 863874. − 129077.i 0 1.40324e7i − 3.82670e7i 0 −2.09465e7 − 1.40189e8i
19.9 162.279i 0 104737. −863874. + 129077.i 0 1.40324e7i 3.82670e7i 0 −2.09465e7 − 1.40189e8i
19.10 162.279i 0 104737. 863874. + 129077.i 0 − 1.40324e7i 3.82670e7i 0 −2.09465e7 + 1.40189e8i
19.11 406.510i 0 −34178.7 −228254. + 843113.i 0 − 2.76153e7i 3.93881e7i 0 −3.42734e8 − 9.27876e7i
19.12 406.510i 0 −34178.7 228254. + 843113.i 0 2.76153e7i 3.93881e7i 0 −3.42734e8 + 9.27876e7i
19.13 507.394i 0 −126376. −567939. − 663615.i 0 − 7.16981e6i 2.38261e6i 0 3.36714e8 − 2.88169e8i
19.14 507.394i 0 −126376. 567939. − 663615.i 0 7.16981e6i 2.38261e6i 0 3.36714e8 + 2.88169e8i
19.15 665.993i 0 −312475. −847574. + 211088.i 0 5.21087e6i − 1.20813e8i 0 −1.40583e8 − 5.64478e8i
19.16 665.993i 0 −312475. 847574. + 211088.i 0 − 5.21087e6i − 1.20813e8i 0 −1.40583e8 + 5.64478e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.18.b.c ✓ 16
3.b odd 2 1 inner 45.18.b.c ✓ 16
5.b even 2 1 inner 45.18.b.c ✓ 16
15.d odd 2 1 inner 45.18.b.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.18.b.c ✓ 16 1.a even 1 1 trivial
45.18.b.c ✓ 16 3.b odd 2 1 inner
45.18.b.c ✓ 16 5.b even 2 1 inner
45.18.b.c ✓ 16 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 892580T_{2}^{6} + 252842337600T_{2}^{4} + 24927770071040000T_{2}^{2} + 496934681116672000000 \) acting on \(S_{18}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 23\!\cdots\!56)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 61\!\cdots\!84)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 15\!\cdots\!24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 71\!\cdots\!84)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
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