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Properties

Label 1138.2.a.e
Level $1138$
Weight $2$
Character orbit 1138.a
Self dual yes
Analytic conductor $9.087$
Analytic rank $0$
Dimension $17$
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] = [1138,2,Mod(1,1138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1138.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1138 = 2 \cdot 569 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1138.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: \(9.08697575002\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 6 x^{16} - 22 x^{15} + 189 x^{14} + 79 x^{13} - 2289 x^{12} + 1458 x^{11} + 13581 x^{10} + \cdots - 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{4} + 1) q^{5} + \beta_1 q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{4} + 1) q^{5} + \beta_1 q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{4} + 1) q^{10} + \beta_{14} q^{11} + \beta_1 q^{12} + ( - \beta_{5} + 1) q^{13} + \beta_{12} q^{14} + ( - \beta_{14} - \beta_{12} + \beta_{11} + \cdots - 1) q^{15}+ \cdots + (2 \beta_{16} + \beta_{15} + 2 \beta_{14} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 17 q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 6 q^{6} + 4 q^{7} + 17 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 17 q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 6 q^{6} + 4 q^{7} + 17 q^{8} + 29 q^{9} + 10 q^{10} + 6 q^{11} + 6 q^{12} + 18 q^{13} + 4 q^{14} - 5 q^{15} + 17 q^{16} + 22 q^{17} + 29 q^{18} + 7 q^{19} + 10 q^{20} + 7 q^{21} + 6 q^{22} + 2 q^{23} + 6 q^{24} + 29 q^{25} + 18 q^{26} + 9 q^{27} + 4 q^{28} + 13 q^{29} - 5 q^{30} - 13 q^{31} + 17 q^{32} + 22 q^{33} + 22 q^{34} + 20 q^{35} + 29 q^{36} + 3 q^{37} + 7 q^{38} - 23 q^{39} + 10 q^{40} + 27 q^{41} + 7 q^{42} + q^{43} + 6 q^{44} + 7 q^{45} + 2 q^{46} + 22 q^{47} + 6 q^{48} + 23 q^{49} + 29 q^{50} - 12 q^{51} + 18 q^{52} + 15 q^{53} + 9 q^{54} - 37 q^{55} + 4 q^{56} - 11 q^{57} + 13 q^{58} + 20 q^{59} - 5 q^{60} + 18 q^{61} - 13 q^{62} - 12 q^{63} + 17 q^{64} + 3 q^{65} + 22 q^{66} - q^{67} + 22 q^{68} - 2 q^{69} + 20 q^{70} - 16 q^{71} + 29 q^{72} + q^{73} + 3 q^{74} - 8 q^{75} + 7 q^{76} - 5 q^{77} - 23 q^{78} - 37 q^{79} + 10 q^{80} + 13 q^{81} + 27 q^{82} + 43 q^{83} + 7 q^{84} - 16 q^{85} + q^{86} - 40 q^{87} + 6 q^{88} + 26 q^{89} + 7 q^{90} - 36 q^{91} + 2 q^{92} - 30 q^{93} + 22 q^{94} - 34 q^{95} + 6 q^{96} + 23 q^{98} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 6 x^{16} - 22 x^{15} + 189 x^{14} + 79 x^{13} - 2289 x^{12} + 1458 x^{11} + 13581 x^{10} + \cdots - 3136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 56676367717119 \nu^{16} + 186048210318007 \nu^{15} + \cdots - 12\!\cdots\!46 ) / 71\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 290434274608133 \nu^{16} + 921090872613598 \nu^{15} + \cdots - 94\!\cdots\!39 ) / 24\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 874423718638805 \nu^{16} + \cdots - 14\!\cdots\!98 ) / 49\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 151170006236415 \nu^{16} - 450111373996782 \nu^{15} + \cdots + 20\!\cdots\!36 ) / 71\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 454357997602327 \nu^{16} + \cdots - 24\!\cdots\!68 ) / 14\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 267559433118609 \nu^{16} - 923780786056090 \nu^{15} + \cdots + 42\!\cdots\!10 ) / 71\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 279912998492387 \nu^{16} - 866299380496606 \nu^{15} + \cdots + 45\!\cdots\!66 ) / 71\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 159213840421028 \nu^{16} + 531106901925669 \nu^{15} + \cdots - 96\!\cdots\!12 ) / 35\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\!\cdots\!39 \nu^{16} + \cdots + 36\!\cdots\!70 ) / 49\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!85 \nu^{16} + \cdots + 32\!\cdots\!12 ) / 49\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 49\!\cdots\!83 \nu^{16} + \cdots + 80\!\cdots\!48 ) / 99\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!27 \nu^{16} + \cdots - 36\!\cdots\!00 ) / 49\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!07 \nu^{16} + \cdots + 12\!\cdots\!28 ) / 99\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!63 \nu^{16} + \cdots - 14\!\cdots\!72 ) / 99\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} - \beta_{8} + \beta_{4} + \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - 2 \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{15} + 14 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} - 12 \beta_{11} - \beta_{10} - 15 \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{16} - 31 \beta_{15} - 18 \beta_{14} + \beta_{13} + 37 \beta_{12} - 14 \beta_{11} + \cdots + 329 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7 \beta_{16} + 181 \beta_{15} + 155 \beta_{14} + 175 \beta_{13} - 40 \beta_{12} - 129 \beta_{11} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 137 \beta_{16} - 386 \beta_{15} - 246 \beta_{14} + 20 \beta_{13} + 496 \beta_{12} - 149 \beta_{11} + \cdots + 3071 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 159 \beta_{16} + 2040 \beta_{15} + 1620 \beta_{14} + 2086 \beta_{13} - 593 \beta_{12} - 1341 \beta_{11} + \cdots - 397 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1369 \beta_{16} - 4493 \beta_{15} - 3024 \beta_{14} + 295 \beta_{13} + 5962 \beta_{12} - 1463 \beta_{11} + \cdots + 30088 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2468 \beta_{16} + 22401 \beta_{15} + 16748 \beta_{14} + 24127 \beta_{13} - 7814 \beta_{12} - 13795 \beta_{11} + \cdots - 5176 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13548 \beta_{16} - 50879 \beta_{15} - 35343 \beta_{14} + 3779 \beta_{13} + 68355 \beta_{12} + \cdots + 304527 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 32904 \beta_{16} + 243601 \beta_{15} + 173942 \beta_{14} + 273504 \beta_{13} - 96722 \beta_{12} + \cdots - 66555 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 135012 \beta_{16} - 568832 \beta_{15} - 402368 \beta_{14} + 44156 \beta_{13} + 765826 \beta_{12} + \cdots + 3151393 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 406450 \beta_{16} + 2641723 \beta_{15} + 1823351 \beta_{14} + 3058659 \beta_{13} - 1153081 \beta_{12} + \cdots - 840166 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1363909 \beta_{16} - 6318777 \beta_{15} - 4515611 \beta_{14} + 481224 \beta_{13} + 8477854 \beta_{12} + \cdots + 33113159 \) 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
−3.31245
−2.73591
−2.40785
−1.80528
−1.70351
−0.958899
−0.411962
−0.268726
0.553530
1.48339
1.73491
1.84743
2.44532
2.65049
2.80333
2.81043
3.27576
1.00000 −3.31245 1.00000 2.92459 −3.31245 3.05850 1.00000 7.97235 2.92459
1.2 1.00000 −2.73591 1.00000 −0.670508 −2.73591 −4.20557 1.00000 4.48523 −0.670508
1.3 1.00000 −2.40785 1.00000 3.31769 −2.40785 −1.55152 1.00000 2.79775 3.31769
1.4 1.00000 −1.80528 1.00000 −0.723328 −1.80528 2.57817 1.00000 0.259045 −0.723328
1.5 1.00000 −1.70351 1.00000 −3.93487 −1.70351 −4.07742 1.00000 −0.0980507 −3.93487
1.6 1.00000 −0.958899 1.00000 2.18713 −0.958899 5.17669 1.00000 −2.08051 2.18713
1.7 1.00000 −0.411962 1.00000 3.45526 −0.411962 −2.34457 1.00000 −2.83029 3.45526
1.8 1.00000 −0.268726 1.00000 −2.46534 −0.268726 1.34537 1.00000 −2.92779 −2.46534
1.9 1.00000 0.553530 1.00000 2.76973 0.553530 1.32937 1.00000 −2.69360 2.76973
1.10 1.00000 1.48339 1.00000 −0.953582 1.48339 1.64975 1.00000 −0.799545 −0.953582
1.11 1.00000 1.73491 1.00000 0.539267 1.73491 3.24823 1.00000 0.00992052 0.539267
1.12 1.00000 1.84743 1.00000 4.17077 1.84743 1.83505 1.00000 0.413012 4.17077
1.13 1.00000 2.44532 1.00000 2.67980 2.44532 −3.36917 1.00000 2.97959 2.67980
1.14 1.00000 2.65049 1.00000 −3.77775 2.65049 −1.63950 1.00000 4.02510 −3.77775
1.15 1.00000 2.80333 1.00000 1.49727 2.80333 0.668565 1.00000 4.85864 1.49727
1.16 1.00000 2.81043 1.00000 1.02586 2.81043 −3.18535 1.00000 4.89854 1.02586
1.17 1.00000 3.27576 1.00000 −2.04198 3.27576 3.48340 1.00000 7.73061 −2.04198
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(569\) \(1\)

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 1138.2.a.e ✓ 17
4.b odd 2 1 9104.2.a.h 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1138.2.a.e ✓ 17 1.a even 1 1 trivial
9104.2.a.h 17 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} - 6 T_{3}^{16} - 22 T_{3}^{15} + 189 T_{3}^{14} + 79 T_{3}^{13} - 2289 T_{3}^{12} + \cdots - 3136 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 6 T^{16} + \cdots - 3136 \) Copy content Toggle raw display
$5$ \( T^{17} - 10 T^{16} + \cdots + 65072 \) Copy content Toggle raw display
$7$ \( T^{17} - 4 T^{16} + \cdots + 1835008 \) Copy content Toggle raw display
$11$ \( T^{17} - 6 T^{16} + \cdots - 3914272 \) Copy content Toggle raw display
$13$ \( T^{17} - 18 T^{16} + \cdots - 2857472 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 573114368 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 13733151880 \) Copy content Toggle raw display
$23$ \( T^{17} - 2 T^{16} + \cdots - 33444166 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 2641085440 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 52034925056 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 368137404416 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 963404756 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 4209201664 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 155414710402 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 10354921472 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 4678406039680 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 21132603005248 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 582360290816 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 12743311538176 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 192229556224 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 630847943680 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 183248643501056 \) Copy content Toggle raw display
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