Properties

Label 13.9.d.a
Level $13$
Weight $9$
Character orbit 13.d
Analytic conductor $5.296$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,9,Mod(5,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 13.d (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(5.29592193079\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} + 2 x^{16} + 3828 x^{15} + 1204401 x^{14} + 47082 x^{13} + 4823826 x^{12} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2}\cdot 13^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{6} q^{3} + ( - \beta_{9} + \beta_{3} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{16} + 4 \beta_{6} + \cdots + 2177) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{6} q^{3} + ( - \beta_{9} + \beta_{3} + \cdots - \beta_1) q^{4}+ \cdots + ( - 5063 \beta_{17} - 5063 \beta_{16} + \cdots - 62717875) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} - 4 q^{3} + 166 q^{5} + 2720 q^{6} + 5308 q^{7} + 10464 q^{8} + 39362 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{2} - 4 q^{3} + 166 q^{5} + 2720 q^{6} + 5308 q^{7} + 10464 q^{8} + 39362 q^{9} - 31556 q^{11} + 71300 q^{13} - 110260 q^{14} + 121124 q^{15} - 522860 q^{16} - 308962 q^{18} + 100288 q^{19} + 736268 q^{20} + 279416 q^{21} - 977312 q^{22} - 986232 q^{24} + 2952238 q^{26} + 364916 q^{27} + 4497084 q^{28} - 2479024 q^{29} - 1892664 q^{31} + 947212 q^{32} + 1368644 q^{33} - 531576 q^{34} - 2918284 q^{35} - 8343978 q^{37} - 1101268 q^{39} - 12691908 q^{40} + 1140178 q^{41} + 15137212 q^{42} - 3867188 q^{44} + 6565370 q^{45} + 2006148 q^{46} - 13368572 q^{47} - 10106716 q^{48} + 37369598 q^{50} - 14821220 q^{52} + 50561348 q^{53} - 4995928 q^{54} + 76994128 q^{55} - 34755760 q^{57} + 22505716 q^{58} + 2127976 q^{59} - 161134196 q^{60} - 52016516 q^{61} - 15210112 q^{63} + 10413082 q^{65} - 51097328 q^{66} + 960292 q^{67} + 283187508 q^{68} - 166635032 q^{70} + 67412140 q^{71} + 96846900 q^{72} - 145213226 q^{73} - 233620024 q^{74} - 150533640 q^{76} + 212071160 q^{78} - 76829120 q^{79} + 524889520 q^{80} + 293280674 q^{81} - 241951556 q^{83} + 345496184 q^{84} + 260737764 q^{85} - 579480384 q^{86} - 618300656 q^{87} - 89187110 q^{89} + 232660948 q^{91} - 122690376 q^{92} + 669553400 q^{93} + 1069637380 q^{94} - 1324092400 q^{96} + 331183146 q^{97} + 588677614 q^{98} - 1125076336 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} + 2 x^{16} + 3828 x^{15} + 1204401 x^{14} + 47082 x^{13} + 4823826 x^{12} + \cdots + 18\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!28 \nu^{17} + \cdots - 58\!\cdots\!16 ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!89 \nu^{17} + \cdots - 15\!\cdots\!88 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!85 \nu^{17} + \cdots + 39\!\cdots\!40 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61\!\cdots\!93 \nu^{17} + \cdots - 10\!\cdots\!00 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53\!\cdots\!63 \nu^{17} + \cdots - 20\!\cdots\!92 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25\!\cdots\!23 \nu^{17} + \cdots - 63\!\cdots\!12 ) / 62\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41\!\cdots\!54 \nu^{17} + \cdots - 11\!\cdots\!92 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 66\!\cdots\!37 \nu^{17} + \cdots + 22\!\cdots\!24 ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36\!\cdots\!51 \nu^{17} + \cdots - 20\!\cdots\!68 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!66 \nu^{17} + \cdots + 57\!\cdots\!20 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\!\cdots\!01 \nu^{17} + \cdots + 15\!\cdots\!44 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!23 \nu^{17} + \cdots + 18\!\cdots\!60 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!38 \nu^{17} + \cdots - 73\!\cdots\!08 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!62 \nu^{17} + \cdots + 84\!\cdots\!28 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 39\!\cdots\!73 \nu^{17} + \cdots - 23\!\cdots\!76 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!55 \nu^{17} + \cdots + 34\!\cdots\!96 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{3} + 398\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} + \beta_{12} - \beta_{9} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots - 562 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 12 \beta_{14} - \beta_{13} - 12 \beta_{12} - 52 \beta_{11} - 21 \beta_{10} + 21 \beta_{8} + \cdots - 269110 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16 \beta_{17} + 16 \beta_{16} + 1121 \beta_{15} + 1415 \beta_{14} - 2518 \beta_{11} + 1273 \beta_{9} + \cdots - 621958 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 288 \beta_{17} + 795 \beta_{15} - 13570 \beta_{14} + 795 \beta_{13} + 13570 \beta_{12} + \cdots - 1111268 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 29072 \beta_{17} - 29072 \beta_{16} + 1084387 \beta_{13} - 1524229 \beta_{12} + 494750 \beta_{10} + \cdots + 574172222 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 645472 \beta_{16} - 757891 \beta_{15} + 12123572 \beta_{14} + 757891 \beta_{13} + 12123572 \beta_{12} + \cdots + 183724767422 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 36722608 \beta_{17} - 36722608 \beta_{16} - 1014336061 \beta_{15} - 1509472399 \beta_{14} + \cdots + 500428370502 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 909816416 \beta_{17} - 881351515 \beta_{15} + 10148277070 \beta_{14} - 881351515 \beta_{13} + \cdots + 780253792164 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 40426150480 \beta_{17} + 40426150480 \beta_{16} - 939651432275 \beta_{13} + 1444295467061 \beta_{12} + \cdots - 437822922203350 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1069772963680 \beta_{16} + 1057831398055 \beta_{15} - 8330281836160 \beta_{14} - 1057831398055 \beta_{13} + \cdots - 14\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 41635658726576 \beta_{17} + 41635658726576 \beta_{16} + 867847175486293 \beta_{15} + \cdots - 39\!\cdots\!10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11\!\cdots\!76 \beta_{17} + \cdots - 62\!\cdots\!16 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 41\!\cdots\!72 \beta_{17} + \cdots + 36\!\cdots\!30 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 11\!\cdots\!40 \beta_{16} + \cdots + 11\!\cdots\!42 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 40\!\cdots\!80 \beta_{17} + \cdots + 35\!\cdots\!26 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
21.6276 + 21.6276i
18.0289 + 18.0289i
9.74776 + 9.74776i
7.83093 + 7.83093i
−3.32100 3.32100i
−5.06366 5.06366i
−13.2448 13.2448i
−13.3036 13.3036i
−21.3021 21.3021i
21.6276 21.6276i
18.0289 18.0289i
9.74776 9.74776i
7.83093 7.83093i
−3.32100 + 3.32100i
−5.06366 + 5.06366i
−13.2448 + 13.2448i
−13.3036 + 13.3036i
−21.3021 + 21.3021i
−21.6276 + 21.6276i −83.2917 679.508i −706.219 + 706.219i 1801.40 1801.40i 966.123 + 966.123i 9159.48 + 9159.48i 376.504 30547.7i
5.2 −18.0289 + 18.0289i 107.999 394.083i 706.041 706.041i −1947.10 + 1947.10i 1701.26 + 1701.26i 2489.49 + 2489.49i 5102.69 25458.3i
5.3 −9.74776 + 9.74776i 19.1757 65.9624i −198.672 + 198.672i −186.920 + 186.920i −1520.85 1520.85i −3138.41 3138.41i −6193.29 3873.22i
5.4 −7.83093 + 7.83093i −141.994 133.353i 581.000 581.000i 1111.95 1111.95i 586.412 + 586.412i −3049.00 3049.00i 13601.3 9099.54i
5.5 3.32100 3.32100i −9.51767 233.942i −136.029 + 136.029i −31.6082 + 31.6082i 2562.74 + 2562.74i 1627.10 + 1627.10i −6470.41 903.504i
5.6 5.06366 5.06366i 152.779 204.719i −186.065 + 186.065i 773.622 773.622i −798.941 798.941i 2332.92 + 2332.92i 16780.5 1884.34i
5.7 13.2448 13.2448i 7.30083 94.8495i 788.030 788.030i 96.6980 96.6980i −1965.56 1965.56i 2134.41 + 2134.41i −6507.70 20874.6i
5.8 13.3036 13.3036i −112.756 97.9733i −454.715 + 454.715i −1500.06 + 1500.06i −1427.64 1427.64i 2102.33 + 2102.33i 6152.90 12098.7i
5.9 21.3021 21.3021i 58.3050 651.562i −310.371 + 310.371i 1242.02 1242.02i 2550.45 + 2550.45i −8426.31 8426.31i −3161.53 13223.1i
8.1 −21.6276 21.6276i −83.2917 679.508i −706.219 706.219i 1801.40 + 1801.40i 966.123 966.123i 9159.48 9159.48i 376.504 30547.7i
8.2 −18.0289 18.0289i 107.999 394.083i 706.041 + 706.041i −1947.10 1947.10i 1701.26 1701.26i 2489.49 2489.49i 5102.69 25458.3i
8.3 −9.74776 9.74776i 19.1757 65.9624i −198.672 198.672i −186.920 186.920i −1520.85 + 1520.85i −3138.41 + 3138.41i −6193.29 3873.22i
8.4 −7.83093 7.83093i −141.994 133.353i 581.000 + 581.000i 1111.95 + 1111.95i 586.412 586.412i −3049.00 + 3049.00i 13601.3 9099.54i
8.5 3.32100 + 3.32100i −9.51767 233.942i −136.029 136.029i −31.6082 31.6082i 2562.74 2562.74i 1627.10 1627.10i −6470.41 903.504i
8.6 5.06366 + 5.06366i 152.779 204.719i −186.065 186.065i 773.622 + 773.622i −798.941 + 798.941i 2332.92 2332.92i 16780.5 1884.34i
8.7 13.2448 + 13.2448i 7.30083 94.8495i 788.030 + 788.030i 96.6980 + 96.6980i −1965.56 + 1965.56i 2134.41 2134.41i −6507.70 20874.6i
8.8 13.3036 + 13.3036i −112.756 97.9733i −454.715 454.715i −1500.06 1500.06i −1427.64 + 1427.64i 2102.33 2102.33i 6152.90 12098.7i
8.9 21.3021 + 21.3021i 58.3050 651.562i −310.371 310.371i 1242.02 + 1242.02i 2550.45 2550.45i −8426.31 + 8426.31i −3161.53 13223.1i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.9.d.a 18
3.b odd 2 1 117.9.j.a 18
13.d odd 4 1 inner 13.9.d.a 18
39.f even 4 1 117.9.j.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.9.d.a 18 1.a even 1 1 trivial
13.9.d.a 18 13.d odd 4 1 inner
117.9.j.a 18 3.b odd 2 1
117.9.j.a 18 39.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$3$ \( (T^{9} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 23\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 84\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 54\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 40\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 20\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 44\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 40\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 87\!\cdots\!72 \) Copy content Toggle raw display
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