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Properties

Label 2023.4.a.l
Level $2023$
Weight $4$
Character orbit 2023.a
Self dual yes
Analytic conductor $119.361$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2023,4,Mod(1,2023)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2023, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2023.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2023.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,3,3,65,-12,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(119.360863942\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 76 x^{10} + 195 x^{9} + 2126 x^{8} - 4299 x^{7} - 26508 x^{6} + 35641 x^{5} + \cdots + 17280 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{7} - 1) q^{5} + (\beta_{6} - \beta_{4} - \beta_{2} - 2) q^{6} - 7 q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1 + 5) q^{8}+ \cdots + (3 \beta_{11} + 5 \beta_{10} + \cdots + 420) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 3 q^{3} + 65 q^{4} - 12 q^{5} - 22 q^{6} - 84 q^{7} + 78 q^{8} + 233 q^{9} + 36 q^{10} + 4 q^{11} + 6 q^{12} + 98 q^{13} - 21 q^{14} - 196 q^{15} + 429 q^{16} + 603 q^{18} - 37 q^{19} + 54 q^{20}+ \cdots + 5140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 76 x^{10} + 195 x^{9} + 2126 x^{8} - 4299 x^{7} - 26508 x^{6} + 35641 x^{5} + \cdots + 17280 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 675589 \nu^{11} + 2609670 \nu^{10} + 50245000 \nu^{9} - 175457383 \nu^{8} - 1354717065 \nu^{7} + \cdots - 8208000768 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1788170 \nu^{11} + 6605469 \nu^{10} + 131276648 \nu^{9} - 439763862 \nu^{8} + \cdots - 41407277152 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1788170 \nu^{11} - 6605469 \nu^{10} - 131276648 \nu^{9} + 439763862 \nu^{8} + 3494372727 \nu^{7} + \cdots + 39537989792 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3029129 \nu^{11} + 11229741 \nu^{10} + 222347360 \nu^{9} - 747040555 \nu^{8} + \cdots - 76419286944 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3537994 \nu^{11} - 12982101 \nu^{10} - 259611120 \nu^{9} + 864854350 \nu^{8} + \cdots + 87970608864 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3951219 \nu^{11} - 14795746 \nu^{10} - 290298752 \nu^{9} + 988820617 \nu^{8} + \cdots + 109921616768 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5010072 \nu^{11} - 17938655 \nu^{10} - 369208056 \nu^{9} + 1189752992 \nu^{8} + \cdots + 92839718816 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5300448 \nu^{11} - 19810559 \nu^{10} - 388579480 \nu^{9} + 1319798520 \nu^{8} + \cdots + 133912342560 ) / 373857472 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3630602 \nu^{11} - 13714597 \nu^{10} - 265927372 \nu^{9} + 913370370 \nu^{8} + \cdots + 99677300480 ) / 186928736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 22\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 3\beta_{10} + \beta_{8} + \beta_{5} - 2\beta_{4} + \beta_{3} + 26\beta_{2} + 30\beta _1 + 280 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + 4 \beta_{7} + 9 \beta_{6} + 33 \beta_{5} + 39 \beta_{4} + \cdots + 190 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 44 \beta_{11} - 134 \beta_{10} - 3 \beta_{9} + 36 \beta_{8} + 11 \beta_{6} + 51 \beta_{5} - 129 \beta_{4} + \cdots + 6629 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 202 \beta_{11} - 175 \beta_{10} + 97 \beta_{9} + 16 \beta_{8} + 158 \beta_{7} + 476 \beta_{6} + \cdots + 6179 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1501 \beta_{11} - 4676 \beta_{10} - 188 \beta_{9} + 1131 \beta_{8} + 38 \beta_{7} + 755 \beta_{6} + \cdots + 164648 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7534 \beta_{11} - 7199 \beta_{10} + 3335 \beta_{9} + 802 \beta_{8} + 5012 \beta_{7} + 18508 \beta_{6} + \cdots + 191250 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 47060 \beta_{11} - 149737 \beta_{10} - 7834 \beta_{9} + 34226 \beta_{8} + 3950 \beta_{7} + \cdots + 4221847 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 250828 \beta_{11} - 261196 \beta_{10} + 100755 \beta_{9} + 28176 \beta_{8} + 154708 \beta_{7} + \cdots + 5803107 \) Copy content Toggle raw display

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.17875
−4.53864
−3.89012
−2.47421
−0.715652
−0.695388
0.205031
1.66864
3.61052
4.56003
5.01382
5.43473
−5.17875 7.17785 18.8195 −4.39960 −37.1723 −7.00000 −56.0315 24.5215 22.7844
1.2 −4.53864 −1.55300 12.5993 14.8181 7.04850 −7.00000 −20.8745 −24.5882 −67.2541
1.3 −3.89012 −5.36521 7.13303 −15.2848 20.8713 −7.00000 3.37263 1.78551 59.4598
1.4 −2.47421 2.31409 −1.87830 −4.14536 −5.72554 −7.00000 24.4410 −21.6450 10.2565
1.5 −0.715652 −5.91749 −7.48784 16.6210 4.23487 −7.00000 11.0839 8.01672 −11.8949
1.6 −0.695388 9.96181 −7.51644 −15.3331 −6.92732 −7.00000 10.7899 72.2376 10.6624
1.7 0.205031 −7.78304 −7.95796 −7.51065 −1.59576 −7.00000 −3.27187 33.5757 −1.53991
1.8 1.66864 4.24887 −5.21566 −4.69551 7.08982 −7.00000 −22.0521 −8.94707 −7.83509
1.9 3.61052 10.0777 5.03583 15.3130 36.3856 −7.00000 −10.7022 74.5595 55.2878
1.10 4.56003 −5.59626 12.7939 −16.5864 −25.5191 −7.00000 21.8602 4.31818 −75.6344
1.11 5.01382 −9.79079 17.1384 19.7425 −49.0893 −7.00000 45.8182 68.8596 98.9853
1.12 5.43473 5.22551 21.5363 −10.5392 28.3992 −7.00000 73.5664 0.305909 −57.2779
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(17\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2023.4.a.l yes 12
17.b even 2 1 2023.4.a.k ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2023.4.a.k ✓ 12 17.b even 2 1
2023.4.a.l yes 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2023))\):

\( T_{2}^{12} - 3 T_{2}^{11} - 76 T_{2}^{10} + 195 T_{2}^{9} + 2126 T_{2}^{8} - 4299 T_{2}^{7} + \cdots + 17280 \) Copy content Toggle raw display
\( T_{3}^{12} - 3 T_{3}^{11} - 274 T_{3}^{10} + 610 T_{3}^{9} + 28240 T_{3}^{8} - 42820 T_{3}^{7} + \cdots + 778459096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 17280 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 778459096 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1961997396480 \) Copy content Toggle raw display
$7$ \( (T + 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 81\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 34\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 46\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 21\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 53\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 43\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 83\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 34\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 17\!\cdots\!16 \) Copy content Toggle raw display
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