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Properties

Label 2023.4.a.m
Level $2023$
Weight $4$
Character orbit 2023.a
Self dual yes
Analytic conductor $119.361$
Analytic rank $1$
Dimension $13$
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: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 82 x^{11} + 2489 x^{9} - 39 x^{8} - 34308 x^{7} + 2394 x^{6} + 212624 x^{5} - 40975 x^{4} + \cdots - 135664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 119)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{3} - \beta_1 - 2) q^{5} + ( - \beta_{5} - 2 \beta_1 - 1) q^{6} + 7 q^{7} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{9} + \beta_{6} + \beta_{3} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{3} - \beta_1 - 2) q^{5} + ( - \beta_{5} - 2 \beta_1 - 1) q^{6} + 7 q^{7} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{8}+ \cdots + (2 \beta_{12} + 12 \beta_{11} + \cdots - 284) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 14 q^{3} + 60 q^{4} - 20 q^{5} - 7 q^{6} + 91 q^{7} + 113 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 14 q^{3} + 60 q^{4} - 20 q^{5} - 7 q^{6} + 91 q^{7} + 113 q^{9} - 92 q^{10} - 28 q^{11} - 270 q^{12} - 116 q^{13} - 92 q^{15} + 388 q^{16} - 56 q^{18} - 34 q^{19} + 197 q^{20} - 98 q^{21} - 88 q^{22} - 56 q^{23} - 104 q^{24} + 15 q^{25} + 36 q^{26} - 452 q^{27} + 420 q^{28} - 312 q^{29} - 289 q^{30} - 548 q^{31} + 195 q^{32} + 1000 q^{33} - 140 q^{35} + 617 q^{36} - 4 q^{37} + 600 q^{38} + 428 q^{39} - 1185 q^{40} - 626 q^{41} - 49 q^{42} - 226 q^{43} - 880 q^{44} + 922 q^{45} - 1288 q^{46} - 280 q^{47} - 2117 q^{48} + 637 q^{49} + 1007 q^{50} - 1194 q^{52} - 766 q^{53} - 487 q^{54} - 12 q^{55} - 2044 q^{57} - 2456 q^{58} + 326 q^{59} - 2831 q^{60} + 272 q^{61} + 2028 q^{62} + 791 q^{63} + 636 q^{64} - 2392 q^{65} + 2128 q^{66} - 1162 q^{67} + 260 q^{69} - 644 q^{70} - 804 q^{71} + 275 q^{72} - 802 q^{73} - 4010 q^{74} - 766 q^{75} + 1244 q^{76} - 196 q^{77} + 624 q^{78} + 1160 q^{79} - 774 q^{80} + 2145 q^{81} - 289 q^{82} - 26 q^{83} - 1890 q^{84} - 267 q^{86} - 1720 q^{87} - 1050 q^{88} - 2026 q^{89} - 1699 q^{90} - 812 q^{91} - 2152 q^{92} + 768 q^{93} - 3350 q^{94} + 2228 q^{95} + 3742 q^{96} - 1942 q^{97} - 3732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 82 x^{11} + 2489 x^{9} - 39 x^{8} - 34308 x^{7} + 2394 x^{6} + 212624 x^{5} - 40975 x^{4} + \cdots - 135664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1166765 \nu^{12} + 8644973 \nu^{11} + 78198029 \nu^{10} - 673903165 \nu^{9} + \cdots + 505962055344 ) / 14552776448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2102755 \nu^{12} + 29941613 \nu^{11} - 199974931 \nu^{10} - 2313408541 \nu^{9} + \cdots + 1462832423856 ) / 14552776448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2375517 \nu^{12} - 23672157 \nu^{11} - 161047917 \nu^{10} + 1800553293 \nu^{9} + \cdots - 780381076016 ) / 14552776448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5645037 \nu^{12} + 2375517 \nu^{11} + 439220877 \nu^{10} - 161047917 \nu^{9} + \cdots - 191042068944 ) / 14552776448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2456359 \nu^{12} - 6276519 \nu^{11} - 186731271 \nu^{10} + 474216447 \nu^{9} + \cdots - 26314086832 ) / 3638194112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11914493 \nu^{12} + 3819939 \nu^{11} - 952076125 \nu^{10} - 322286451 \nu^{9} + \cdots + 725818479312 ) / 14552776448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8669995 \nu^{12} - 33739259 \nu^{11} - 648170507 \nu^{10} + 2582395291 \nu^{9} + \cdots - 604871037968 ) / 7276388224 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13970 \nu^{12} - 88537 \nu^{11} - 980789 \nu^{10} + 6763309 \nu^{9} + 22400815 \nu^{8} + \cdots - 3516506904 ) / 10958416 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26871649 \nu^{12} + 50175905 \nu^{11} + 2052826753 \nu^{10} - 3774161905 \nu^{9} + \cdots + 1407953133680 ) / 14552776448 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25616123 \nu^{12} - 17435011 \nu^{11} - 2001851651 \nu^{10} + 1244925939 \nu^{9} + \cdots + 868844783856 ) / 7276388224 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 21\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + 2\beta_{11} + \beta_{10} + 2\beta_{6} - 2\beta_{5} + 27\beta_{2} + \beta _1 + 278 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} - 3 \beta_{10} - 2 \beta_{8} - \beta_{7} + 27 \beta_{6} + 41 \beta_{5} + 32 \beta_{4} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45 \beta_{12} + 72 \beta_{11} + 42 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} - 6 \beta_{7} + 131 \beta_{6} + \cdots + 6678 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9 \beta_{12} - 48 \beta_{11} - 156 \beta_{10} + 15 \beta_{9} - 96 \beta_{8} - 78 \beta_{7} + \cdots + 1897 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1547 \beta_{12} + 2120 \beta_{11} + 1400 \beta_{10} - 125 \beta_{9} - 376 \beta_{8} - 306 \beta_{7} + \cdots + 168113 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 646 \beta_{12} - 1762 \beta_{11} - 5969 \beta_{10} + 1011 \beta_{9} - 3332 \beta_{8} - 3454 \beta_{7} + \cdots + 52780 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 48175 \beta_{12} + 59153 \beta_{11} + 43327 \beta_{10} - 3797 \beta_{9} - 16358 \beta_{8} + \cdots + 4346280 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 31014 \beta_{12} - 58668 \beta_{11} - 202850 \beta_{10} + 45750 \beta_{9} - 102270 \beta_{8} + \cdots + 1324756 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1431408 \beta_{12} + 1620444 \beta_{11} + 1296936 \beta_{10} - 101320 \beta_{9} - 610176 \beta_{8} + \cdots + 114437893 \) 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
−5.37724
−4.69593
−4.26708
−2.52189
−2.49437
−0.908633
0.413674
1.24177
1.41178
2.77752
4.14261
5.04013
5.23767
−5.37724 −7.74209 20.9147 12.7656 41.6310 7.00000 −69.4453 32.9399 −68.6434
1.2 −4.69593 −0.0727600 14.0518 −10.8167 0.341676 7.00000 −28.4188 −26.9947 50.7944
1.3 −4.26708 6.42887 10.2080 9.85858 −27.4325 7.00000 −9.42160 14.3303 −42.0674
1.4 −2.52189 −9.22530 −1.64007 −0.867945 23.2652 7.00000 24.3112 58.1061 2.18886
1.5 −2.49437 −1.25019 −1.77811 −9.59796 3.11844 7.00000 24.3902 −25.4370 23.9409
1.6 −0.908633 6.84482 −7.17439 4.85745 −6.21943 7.00000 13.7880 19.8515 −4.41364
1.7 0.413674 −1.23220 −7.82887 3.83096 −0.509729 7.00000 −6.54799 −25.4817 1.58477
1.8 1.24177 −6.23411 −6.45801 −20.0505 −7.74131 7.00000 −17.9535 11.8641 −24.8981
1.9 1.41178 8.42678 −6.00687 −14.6552 11.8968 7.00000 −19.7747 44.0106 −20.6900
1.10 2.77752 −3.52385 −0.285404 12.5979 −9.78756 7.00000 −23.0128 −14.5825 34.9909
1.11 4.14261 3.50851 9.16125 1.63125 14.5344 7.00000 4.81060 −14.6904 6.75763
1.12 5.04013 −9.64379 17.4029 7.63102 −48.6059 7.00000 47.3916 66.0027 38.4613
1.13 5.23767 −0.284683 19.4332 −17.1844 −1.49108 7.00000 59.8831 −26.9190 −90.0062
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.m 13
17.b even 2 1 2023.4.a.n 13
17.c even 4 2 119.4.b.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.4.b.a ✓ 26 17.c even 4 2
2023.4.a.m 13 1.a even 1 1 trivial
2023.4.a.n 13 17.b even 2 1

Hecke kernels

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

\( T_{2}^{13} - 82 T_{2}^{11} + 2489 T_{2}^{9} - 39 T_{2}^{8} - 34308 T_{2}^{7} + 2394 T_{2}^{6} + \cdots - 135664 \) Copy content Toggle raw display
\( T_{3}^{13} + 14 T_{3}^{12} - 134 T_{3}^{11} - 2388 T_{3}^{10} + 3851 T_{3}^{9} + 139106 T_{3}^{8} + \cdots + 628160 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 82 T^{11} + \cdots - 135664 \) Copy content Toggle raw display
$3$ \( T^{13} + 14 T^{12} + \cdots + 628160 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 167104173888 \) Copy content Toggle raw display
$7$ \( (T - 7)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots - 20\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 61\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{13} \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 16\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 81\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 17\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 45\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 95\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 72\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 40\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 50\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 38\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 84\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 49\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 56\!\cdots\!64 \) Copy content Toggle raw display
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