Properties

Label 4.23.b.a
Level $4$
Weight $23$
Character orbit 4.b
Analytic conductor $12.268$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,23,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 23 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(12.2682973937\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} - 63342 x^{8} - 45742928 x^{7} + 34835133568 x^{6} + 12622768560288 x^{5} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{11} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 154) q^{2} + ( - \beta_{2} - 19 \beta_1) q^{3} + (\beta_{3} + 2 \beta_{2} - 146 \beta_1 + 226446) q^{4} + (\beta_{5} + \beta_{3} + 8 \beta_{2} - 3885 \beta_1 - 1709110) q^{5} + (\beta_{7} + 20 \beta_{3} - 65 \beta_{2} - 2915 \beta_1 - 79193586) q^{6} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{4} - 9 \beta_{3} - 503 \beta_{2} - 141333 \beta_1 + 4) q^{7} + (\beta_{9} - 2 \beta_{7} + 3 \beta_{6} - 37 \beta_{5} - 9 \beta_{4} + \cdots + 980443087) q^{8}+ \cdots + ( - 4 \beta_{9} - \beta_{8} + 57 \beta_{7} + 4 \beta_{6} + \cdots - 10430961350) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 154) q^{2} + ( - \beta_{2} - 19 \beta_1) q^{3} + (\beta_{3} + 2 \beta_{2} - 146 \beta_1 + 226446) q^{4} + (\beta_{5} + \beta_{3} + 8 \beta_{2} - 3885 \beta_1 - 1709110) q^{5} + (\beta_{7} + 20 \beta_{3} - 65 \beta_{2} - 2915 \beta_1 - 79193586) q^{6} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{4} - 9 \beta_{3} - 503 \beta_{2} - 141333 \beta_1 + 4) q^{7} + (\beta_{9} - 2 \beta_{7} + 3 \beta_{6} - 37 \beta_{5} - 9 \beta_{4} + \cdots + 980443087) q^{8}+ \cdots + (167656656384 \beta_{9} + \cdots - 449886184122528) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 1540 q^{2} + 2264464 q^{4} - 17091100 q^{5} - 791935776 q^{6} + 9804431680 q^{8} - 104309613702 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 1540 q^{2} + 2264464 q^{4} - 17091100 q^{5} - 791935776 q^{6} + 9804431680 q^{8} - 104309613702 q^{9} + 159414035240 q^{10} + 519021175680 q^{12} - 531230356540 q^{13} - 5894008940736 q^{14} - 27717620084480 q^{16} + 14058178115540 q^{17} + 16283956279140 q^{18} + 233643631625120 q^{20} - 313135665760512 q^{21} + 120589650366240 q^{22} - 20\!\cdots\!36 q^{24}+ \cdots - 23\!\cdots\!20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} - 63342 x^{8} - 45742928 x^{7} + 34835133568 x^{6} + 12622768560288 x^{5} + \cdots + 11\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 43 \nu^{8} - 61708 \nu^{7} - 43398024 \nu^{6} + 36484258480 \nu^{5} + 11236366738048 \nu^{4} + \cdots - 16\!\cdots\!72 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12193477 \nu^{9} - 979139351 \nu^{8} - 4293302429244 \nu^{7} + \cdots - 55\!\cdots\!48 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6915365 \nu^{9} - 91005641 \nu^{8} + 2247044895420 \nu^{7} + \cdots + 28\!\cdots\!44 ) / 69\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1567851625 \nu^{9} + 133933514147 \nu^{8} + 556463209123308 \nu^{7} + \cdots + 71\!\cdots\!80 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2954795719 \nu^{9} + 1592279695507 \nu^{8} + 438857576441772 \nu^{7} + \cdots - 41\!\cdots\!84 ) / 92\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12059814031 \nu^{9} - 15503143391237 \nu^{8} + \cdots - 12\!\cdots\!88 ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10985599895 \nu^{9} - 2735389283747 \nu^{8} + 963042238239252 \nu^{7} + \cdots + 31\!\cdots\!84 ) / 69\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 46997419903 \nu^{9} - 22896956550571 \nu^{8} + \cdots + 78\!\cdots\!92 ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8681772593 \nu^{9} + 10047012439493 \nu^{8} + \cdots + 15\!\cdots\!88 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 3\beta_{3} + 136\beta_{2} - 3183\beta _1 + 131071 ) / 262144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16 \beta_{8} - 16 \beta_{7} + 112 \beta_{5} - 27 \beta_{4} - 2993 \beta_{3} - 25336 \beta_{2} - 3474299 \beta _1 + 3321601659 ) / 262144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1024 \beta_{9} - 2064 \beta_{8} - 32752 \beta_{7} - 15360 \beta_{6} - 140400 \beta_{5} + 31799 \beta_{4} - 1385467 \beta_{3} + 16322008 \beta_{2} + 3279655767 \beta _1 + 3622281117033 ) / 262144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 173056 \beta_{9} + 143824 \beta_{8} + 29099568 \beta_{7} + 287744 \beta_{6} + 86675632 \beta_{5} + 163017 \beta_{4} - 2279565189 \beta_{3} + \cdots - 34\!\cdots\!53 ) / 262144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 662658048 \beta_{9} - 50467920 \beta_{8} - 8113890224 \beta_{7} + 957193216 \beta_{6} + 14157888976 \beta_{5} - 91723649 \beta_{4} + \cdots - 12\!\cdots\!95 ) / 262144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 588186385408 \beta_{9} + 39909369808 \beta_{8} - 1521215942608 \beta_{7} - 116501527552 \beta_{6} - 31944420709712 \beta_{5} + \cdots - 22\!\cdots\!85 ) / 262144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 213620913249280 \beta_{9} - 43366557800016 \beta_{8} + \cdots - 34\!\cdots\!55 ) / 262144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 65\!\cdots\!48 \beta_{9} + \cdots - 15\!\cdots\!13 ) / 262144 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 51\!\cdots\!60 \beta_{9} + \cdots + 34\!\cdots\!33 ) / 262144 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−501.982 + 216.483i
−501.982 216.483i
−313.209 + 431.746i
−313.209 431.746i
1.30158 + 510.489i
1.30158 510.489i
407.912 + 251.607i
407.912 251.607i
408.476 + 250.605i
408.476 250.605i
−1855.93 865.934i 142172.i 2.69462e6 + 3.21422e6i 1.40852e7 1.23111e8 2.63860e8i 7.81741e8i −2.21772e9 8.29872e9i 1.11682e10 −2.61411e10 1.21969e10i
3.2 −1855.93 + 865.934i 142172.i 2.69462e6 3.21422e6i 1.40852e7 1.23111e8 + 2.63860e8i 7.81741e8i −2.21772e9 + 8.29872e9i 1.11682e10 −2.61411e10 + 1.21969e10i
3.3 −1100.83 1726.98i 308212.i −1.77063e6 + 3.80224e6i −6.05072e7 −5.32276e8 + 3.39290e8i 1.97415e9i 8.51558e9 1.12779e9i −6.36133e10 6.66084e10 + 1.04495e11i
3.4 −1100.83 + 1726.98i 308212.i −1.77063e6 3.80224e6i −6.05072e7 −5.32276e8 3.39290e8i 1.97415e9i 8.51558e9 + 1.12779e9i −6.36133e10 6.66084e10 1.04495e11i
3.5 157.206 2041.96i 75737.5i −4.14488e6 642017.i 1.73209e7 1.54653e8 + 1.19064e7i 2.02930e9i −1.96257e9 + 8.36273e9i 2.56449e10 2.72295e9 3.53685e10i
3.6 157.206 + 2041.96i 75737.5i −4.14488e6 + 642017.i 1.73209e7 1.54653e8 1.19064e7i 2.02930e9i −1.96257e9 8.36273e9i 2.56449e10 2.72295e9 + 3.53685e10i
3.7 1783.65 1006.43i 267899.i 2.16851e6 3.59023e6i 8.57934e7 −2.69621e8 4.77838e8i 2.80689e8i 2.54549e8 8.58616e9i −4.03887e10 1.53025e11 8.63449e10i
3.8 1783.65 + 1006.43i 267899.i 2.16851e6 + 3.59023e6i 8.57934e7 −2.69621e8 + 4.77838e8i 2.80689e8i 2.54549e8 + 8.58616e9i −4.03887e10 1.53025e11 + 8.63449e10i
3.9 1785.90 1002.42i 127855.i 2.18461e6 3.58046e6i −6.52379e7 1.28165e8 + 2.28337e8i 3.27904e9i 3.12381e8 8.58425e9i 1.50341e10 −1.16509e11 + 6.53958e10i
3.10 1785.90 + 1002.42i 127855.i 2.18461e6 + 3.58046e6i −6.52379e7 1.28165e8 2.28337e8i 3.27904e9i 3.12381e8 + 8.58425e9i 1.50341e10 −1.16509e11 6.53958e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.23.b.a 10
3.b odd 2 1 36.23.d.c 10
4.b odd 2 1 inner 4.23.b.a 10
8.b even 2 1 64.23.c.e 10
8.d odd 2 1 64.23.c.e 10
12.b even 2 1 36.23.d.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.23.b.a 10 1.a even 1 1 trivial
4.23.b.a 10 4.b odd 2 1 inner
36.23.d.c 10 3.b odd 2 1
36.23.d.c 10 12.b even 2 1
64.23.c.e 10 8.b even 2 1
64.23.c.e 10 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{23}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 1540 T^{9} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{10} + 209060104896 T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{5} + 8545550 T^{4} + \cdots - 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + 265615178270 T^{4} + \cdots + 41\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 7029089057770 T^{4} + \cdots + 27\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 79\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 22\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 80\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 46\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 16\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 22\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
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