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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 / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,4,Mod(1,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: yes
Analytic conductor: \(119.360863942\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
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

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{7} - 1) q^{5} + (\beta_{6} - \beta_{4} - \beta_{2} - 2) q^{6} - 7 q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1 + 5) q^{8} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{7} - 1) q^{5} + (\beta_{6} - \beta_{4} - \beta_{2} - 2) q^{6} - 7 q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1 + 5) q^{8} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots + 18) q^{9}+ \cdots + (3 \beta_{11} + 5 \beta_{10} + \cdots + 420) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 3 q^{3} + 65 q^{4} - 12 q^{5} - 22 q^{6} - 84 q^{7} + 78 q^{8} + 233 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + 3 q^{3} + 65 q^{4} - 12 q^{5} - 22 q^{6} - 84 q^{7} + 78 q^{8} + 233 q^{9} + 36 q^{10} + 4 q^{11} + 6 q^{12} + 98 q^{13} - 21 q^{14} - 196 q^{15} + 429 q^{16} + 603 q^{18} - 37 q^{19} + 54 q^{20} - 21 q^{21} + 70 q^{22} - 369 q^{23} - 652 q^{24} + 590 q^{25} + 638 q^{26} + 501 q^{27} - 455 q^{28} + 144 q^{29} - 160 q^{30} + 67 q^{31} + 417 q^{32} + 628 q^{33} + 84 q^{35} + 999 q^{36} - 345 q^{37} - 190 q^{38} + 6 q^{39} - 466 q^{40} - 156 q^{41} + 154 q^{42} - 763 q^{43} + 1046 q^{44} + 832 q^{45} - 395 q^{46} + 561 q^{47} + 532 q^{48} + 588 q^{49} + 1923 q^{50} + 3138 q^{52} + 1156 q^{53} - 1682 q^{54} - 156 q^{55} - 546 q^{56} - 501 q^{57} - 2305 q^{58} - 473 q^{59} - 1292 q^{60} + 1850 q^{61} + 202 q^{62} - 1631 q^{63} - 712 q^{64} + 1718 q^{65} + 2142 q^{66} - 834 q^{67} + 1316 q^{69} - 252 q^{70} - 3179 q^{71} + 2046 q^{72} + 1012 q^{73} - 2695 q^{74} - 3841 q^{75} + 3272 q^{76} - 28 q^{77} + 970 q^{78} - 3897 q^{79} - 1616 q^{80} + 3448 q^{81} + 4758 q^{82} + 675 q^{83} - 42 q^{84} + 77 q^{86} + 805 q^{87} + 5398 q^{88} + 4078 q^{89} + 13384 q^{90} - 686 q^{91} - 3393 q^{92} + 3671 q^{93} + 958 q^{94} - 4126 q^{95} - 2320 q^{96} + 1474 q^{97} + 147 q^{98} + 5140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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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