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Properties

Label 45.6.f.a
Level $45$
Weight $6$
Character orbit 45.f
Analytic conductor $7.217$
Analytic rank $0$
Dimension $20$
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] = [45,6,Mod(8,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.8");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.f (of order \(4\), degree \(2\), 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: \(7.21727189158\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 19978x^{16} + 11248353x^{12} + 1386043201x^{8} + 1477627450x^{4} + 332150625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{24}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + (\beta_{4} - 14 \beta_{2}) q^{4} + ( - \beta_{14} + 2 \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + 8 \beta_{2} + \cdots + 8) q^{7}+ \cdots + (\beta_{19} - \beta_{17} + \cdots + 15 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + (\beta_{4} - 14 \beta_{2}) q^{4} + ( - \beta_{14} + 2 \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + 8 \beta_{2} + \cdots + 8) q^{7}+ \cdots + (16 \beta_{19} - 300 \beta_{18} + \cdots - 2413 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 152 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 152 q^{7} - 2128 q^{10} + 1844 q^{13} - 6280 q^{16} - 6512 q^{22} - 5104 q^{25} + 29728 q^{28} + 43120 q^{31} - 37516 q^{37} - 32472 q^{40} + 5408 q^{43} + 118720 q^{46} - 285256 q^{52} - 98072 q^{55} + 274752 q^{58} + 163360 q^{61} - 302704 q^{67} - 457776 q^{70} + 146324 q^{73} + 760800 q^{76} - 305744 q^{82} - 365152 q^{85} + 355296 q^{88} + 499120 q^{91} - 362524 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 19978x^{16} + 11248353x^{12} + 1386043201x^{8} + 1477627450x^{4} + 332150625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4854966023630 \nu^{16} + \cdots - 62\!\cdots\!75 ) / 59\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 159785243 \nu^{18} + 3192258575999 \nu^{14} + \cdots + 34\!\cdots\!75 \nu^{2} ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 208235594721671 \nu^{17} + \cdots - 16\!\cdots\!25 \nu ) / 53\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79\!\cdots\!07 \nu^{18} + \cdots - 93\!\cdots\!25 \nu^{2} ) / 48\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!63 \nu^{18} + \cdots - 91\!\cdots\!25 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!63 \nu^{18} + \cdots - 91\!\cdots\!25 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\!\cdots\!93 \nu^{18} + \cdots + 52\!\cdots\!00 ) / 14\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!49 \nu^{18} - 412855054046793 \nu^{16} + \cdots - 29\!\cdots\!25 ) / 16\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 67\!\cdots\!53 \nu^{18} + \cdots + 31\!\cdots\!50 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67\!\cdots\!53 \nu^{18} + \cdots + 31\!\cdots\!50 ) / 72\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 72\!\cdots\!76 \nu^{19} + \cdots - 59\!\cdots\!75 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72\!\cdots\!76 \nu^{19} + \cdots - 59\!\cdots\!75 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58\!\cdots\!56 \nu^{19} + \cdots - 50\!\cdots\!50 \nu^{3} ) / 97\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!65 \nu^{19} + \cdots + 31\!\cdots\!75 \nu ) / 39\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!84 \nu^{19} + \cdots - 24\!\cdots\!75 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 62\!\cdots\!67 \nu^{19} + \cdots - 50\!\cdots\!00 \nu ) / 43\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 64\!\cdots\!68 \nu^{19} + \cdots + 99\!\cdots\!75 \nu ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10\!\cdots\!52 \nu^{19} + \cdots - 42\!\cdots\!25 \nu ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 14\!\cdots\!77 \nu^{19} + \cdots - 60\!\cdots\!50 \nu ) / 39\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 15 \beta_{19} - 18 \beta_{18} - 15 \beta_{17} + 36 \beta_{16} + 75 \beta_{15} - 15 \beta_{14} + \cdots - 438 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 71 \beta_{10} - 107 \beta_{9} + 36 \beta_{8} + 72 \beta_{7} + 270 \beta_{6} - 270 \beta_{5} + \cdots + 36 \beta_1 ) / 1080 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 555 \beta_{19} - 2043 \beta_{18} - 555 \beta_{17} - 1134 \beta_{16} - 330 \beta_{15} + \cdots - 2043 \beta_{3} ) / 540 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18097 \beta_{10} + 11851 \beta_{9} + 12492 \beta_{8} - 6246 \beta_{7} + 39960 \beta_{6} + \cdots - 4330240 ) / 1080 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 136365 \beta_{19} + 319698 \beta_{18} + 136365 \beta_{17} - 554076 \beta_{16} + \cdots + 6676998 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 842593 \beta_{10} + 1293106 \beta_{9} - 450513 \beta_{8} - 901026 \beta_{7} - 2800035 \beta_{6} + \cdots - 450513 \beta_1 ) / 540 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18699735 \beta_{19} + 76894416 \beta_{18} + 18699735 \beta_{17} + 44695908 \beta_{16} + \cdots + 76894416 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 361550707 \beta_{10} - 235339081 \beta_{9} - 252423252 \beta_{8} + 126211626 \beta_{7} + \cdots + 81635467840 ) / 1080 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1299893745 \beta_{19} - 3115094769 \beta_{18} - 1299893745 \beta_{17} + 5352619203 \beta_{16} + \cdots - 64424204769 \beta_{3} ) / 540 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 32791620941 \beta_{10} - 50387014997 \beta_{9} + 17595394056 \beta_{8} + 35190788112 \beta_{7} + \cdots + 17595394056 \beta_1 ) / 1080 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 362041420485 \beta_{19} - 1491035363496 \beta_{18} - 362041420485 \beta_{17} + \cdots - 1491035363496 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1754743437823 \beta_{10} + 1141944263059 \beta_{9} + 1225598349528 \beta_{8} - 612799174764 \beta_{7} + \cdots - 395776207244110 ) / 270 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 50427516797115 \beta_{19} + 120894535915578 \beta_{18} + 50427516797115 \beta_{17} + \cdots + 24\!\cdots\!78 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 636253236999971 \beta_{10} + 977688978308207 \beta_{9} - 341435741308236 \beta_{8} + \cdots - 341435741308236 \beta_1 ) / 1080 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 35\!\cdots\!05 \beta_{19} + \cdots + 14\!\cdots\!13 \beta_{3} ) / 540 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 13\!\cdots\!77 \beta_{10} + \cdots + 30\!\cdots\!40 ) / 1080 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 97\!\cdots\!65 \beta_{19} + \cdots - 48\!\cdots\!18 \beta_{3} ) / 1080 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 61\!\cdots\!63 \beta_{10} + \cdots + 33\!\cdots\!33 \beta_1 ) / 540 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 13\!\cdots\!35 \beta_{19} + \cdots - 56\!\cdots\!56 \beta_{3} ) / 1080 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−0.532441 + 0.532441i
0.658471 − 0.658471i
−3.15495 + 3.15495i
−8.34538 + 8.34538i
−2.58530 + 2.58530i
2.58530 − 2.58530i
8.34538 − 8.34538i
3.15495 − 3.15495i
−0.658471 + 0.658471i
0.532441 − 0.532441i
−0.532441 − 0.532441i
0.658471 + 0.658471i
−3.15495 − 3.15495i
−8.34538 − 8.34538i
−2.58530 − 2.58530i
2.58530 + 2.58530i
8.34538 + 8.34538i
3.15495 + 3.15495i
−0.658471 − 0.658471i
0.532441 + 0.532441i
−7.55546 − 7.55546i 0 82.1700i 46.0804 − 31.6480i 0 97.1320 − 97.1320i 379.058 − 379.058i 0 −587.274 − 109.044i
8.2 −5.73598 − 5.73598i 0 33.8030i −24.5731 − 50.2112i 0 −92.1304 + 92.1304i 10.3420 − 10.3420i 0 −147.059 + 428.961i
8.3 −3.49760 − 3.49760i 0 − 7.53358i −33.7137 + 44.5913i 0 24.1207 − 24.1207i −138.273 + 138.273i 0 273.880 − 38.0455i
8.4 −3.44473 − 3.44473i 0 − 8.26762i 38.9682 + 40.0809i 0 137.623 − 137.623i −138.711 + 138.711i 0 3.83296 − 272.303i
8.5 −0.956082 − 0.956082i 0 − 30.1718i 42.3334 − 36.5086i 0 −128.745 + 128.745i −59.4413 + 59.4413i 0 −75.3794 − 5.56899i
8.6 0.956082 + 0.956082i 0 − 30.1718i −42.3334 + 36.5086i 0 −128.745 + 128.745i 59.4413 − 59.4413i 0 −75.3794 − 5.56899i
8.7 3.44473 + 3.44473i 0 − 8.26762i −38.9682 − 40.0809i 0 137.623 − 137.623i 138.711 − 138.711i 0 3.83296 − 272.303i
8.8 3.49760 + 3.49760i 0 − 7.53358i 33.7137 − 44.5913i 0 24.1207 − 24.1207i 138.273 − 138.273i 0 273.880 − 38.0455i
8.9 5.73598 + 5.73598i 0 33.8030i 24.5731 + 50.2112i 0 −92.1304 + 92.1304i −10.3420 + 10.3420i 0 −147.059 + 428.961i
8.10 7.55546 + 7.55546i 0 82.1700i −46.0804 + 31.6480i 0 97.1320 − 97.1320i −379.058 + 379.058i 0 −587.274 − 109.044i
17.1 −7.55546 + 7.55546i 0 − 82.1700i 46.0804 + 31.6480i 0 97.1320 + 97.1320i 379.058 + 379.058i 0 −587.274 + 109.044i
17.2 −5.73598 + 5.73598i 0 − 33.8030i −24.5731 + 50.2112i 0 −92.1304 − 92.1304i 10.3420 + 10.3420i 0 −147.059 − 428.961i
17.3 −3.49760 + 3.49760i 0 7.53358i −33.7137 − 44.5913i 0 24.1207 + 24.1207i −138.273 − 138.273i 0 273.880 + 38.0455i
17.4 −3.44473 + 3.44473i 0 8.26762i 38.9682 − 40.0809i 0 137.623 + 137.623i −138.711 − 138.711i 0 3.83296 + 272.303i
17.5 −0.956082 + 0.956082i 0 30.1718i 42.3334 + 36.5086i 0 −128.745 − 128.745i −59.4413 − 59.4413i 0 −75.3794 + 5.56899i
17.6 0.956082 − 0.956082i 0 30.1718i −42.3334 − 36.5086i 0 −128.745 − 128.745i 59.4413 + 59.4413i 0 −75.3794 + 5.56899i
17.7 3.44473 − 3.44473i 0 8.26762i −38.9682 + 40.0809i 0 137.623 + 137.623i 138.711 + 138.711i 0 3.83296 + 272.303i
17.8 3.49760 − 3.49760i 0 7.53358i 33.7137 + 44.5913i 0 24.1207 + 24.1207i 138.273 + 138.273i 0 273.880 + 38.0455i
17.9 5.73598 − 5.73598i 0 − 33.8030i 24.5731 − 50.2112i 0 −92.1304 − 92.1304i −10.3420 − 10.3420i 0 −147.059 − 428.961i
17.10 7.55546 − 7.55546i 0 − 82.1700i −46.0804 − 31.6480i 0 97.1320 + 97.1320i −379.058 − 379.058i 0 −587.274 + 109.044i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.6.f.a ✓ 20
3.b odd 2 1 inner 45.6.f.a ✓ 20
5.b even 2 1 225.6.f.b 20
5.c odd 4 1 inner 45.6.f.a ✓ 20
5.c odd 4 1 225.6.f.b 20
15.d odd 2 1 225.6.f.b 20
15.e even 4 1 inner 45.6.f.a ✓ 20
15.e even 4 1 225.6.f.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.f.a ✓ 20 1.a even 1 1 trivial
45.6.f.a ✓ 20 3.b odd 2 1 inner
45.6.f.a ✓ 20 5.c odd 4 1 inner
45.6.f.a ✓ 20 15.e even 4 1 inner
225.6.f.b 20 5.b even 2 1
225.6.f.b 20 5.c odd 4 1
225.6.f.b 20 15.d odd 2 1
225.6.f.b 20 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 63600242200576 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 46\!\cdots\!68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 27\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 91\!\cdots\!24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 66\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 44\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 53\!\cdots\!32)^{2} \) Copy content Toggle raw display
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