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Properties

Label 2023.4.a.n
Level $2023$
Weight $4$
Character orbit 2023.a
Self dual yes
Analytic conductor $119.361$
Analytic rank $0$
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: \(0\)
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.n 13
17.b even 2 1 2023.4.a.m 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 17.b even 2 1
2023.4.a.n 13 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2023))\):

\( T_{2}^{13} - 82 T_{2}^{11} + 2489 T_{2}^{9} - 39 T_{2}^{8} - 34308 T_{2}^{7} + 2394 T_{2}^{6} + \cdots - 135664 \) 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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