Properties

Label 13.14.a.b
Level $13$
Weight $14$
Character orbit 13.a
Self dual yes
Analytic conductor $13.940$
Analytic rank $0$
Dimension $7$
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] = [13,14,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(13.9400207637\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 35596x^{5} + 594856x^{4} + 263632368x^{3} + 252644208x^{2} - 410033371968x - 5442114981888 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 18) q^{2} + (\beta_{2} + \beta_1 + 104) q^{3} + (\beta_{3} + 2 \beta_{2} + 12 \beta_1 + 2306) q^{4} + ( - \beta_{6} + \beta_{3} + \cdots + 4979) q^{5}+ \cdots + (\beta_{6} + 36 \beta_{5} + \cdots + 351176) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 18) q^{2} + (\beta_{2} + \beta_1 + 104) q^{3} + (\beta_{3} + 2 \beta_{2} + 12 \beta_1 + 2306) q^{4} + ( - \beta_{6} + \beta_{3} + \cdots + 4979) q^{5}+ \cdots + ( - 32222537 \beta_{6} + \cdots - 2486366775433) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 127 q^{2} + 728 q^{3} + 16153 q^{4} + 34886 q^{5} + 83045 q^{6} + 131864 q^{7} + 127671 q^{8} + 2466055 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 127 q^{2} + 728 q^{3} + 16153 q^{4} + 34886 q^{5} + 83045 q^{6} + 131864 q^{7} + 127671 q^{8} + 2466055 q^{9} + 3335529 q^{10} + 8912632 q^{11} + 25360909 q^{12} + 33787663 q^{13} + 67318187 q^{14} + 78942892 q^{15} + 159240961 q^{16} + 222199706 q^{17} + 596088532 q^{18} + 652665968 q^{19} + 935513953 q^{20} + 1217880580 q^{21} + 1374511830 q^{22} + 1794723984 q^{23} + 1895354895 q^{24} + 246154813 q^{25} + 613004743 q^{26} - 1915909876 q^{27} - 1789780165 q^{28} - 4735865782 q^{29} - 9918164303 q^{30} - 9436127356 q^{31} - 17224649401 q^{32} - 25638368752 q^{33} - 29930612727 q^{34} - 20876359292 q^{35} - 31714020118 q^{36} - 14832967330 q^{37} - 7473991662 q^{38} + 3513916952 q^{39} + 10553061291 q^{40} + 59907995430 q^{41} + 69292649863 q^{42} + 72489466184 q^{43} + 82565539934 q^{44} + 60104811962 q^{45} + 118453633200 q^{46} + 195641345592 q^{47} + 69842813569 q^{48} + 55402933947 q^{49} + 87739798742 q^{50} + 142246774012 q^{51} + 77967445777 q^{52} + 409286178474 q^{53} - 46365876001 q^{54} + 418486272912 q^{55} - 217713137127 q^{56} + 243297441528 q^{57} - 1099013150598 q^{58} - 160361676600 q^{59} - 1108676787295 q^{60} - 147184998046 q^{61} - 2220407254292 q^{62} + 256048021580 q^{63} - 2276361582767 q^{64} + 168388058774 q^{65} - 2841576718138 q^{66} + 1533110763584 q^{67} - 1734599536391 q^{68} + 1341159298056 q^{69} - 1953936478521 q^{70} + 1288841132520 q^{71} - 1858742324166 q^{72} + 4340518042046 q^{73} - 2522725784507 q^{74} + 3488845434748 q^{75} + 792218810282 q^{76} + 3260905865264 q^{77} + 400842353405 q^{78} + 6986884509272 q^{79} + 2848664165317 q^{80} + 9402260691943 q^{81} + 3785376233100 q^{82} + 9485774126680 q^{83} + 2428655741171 q^{84} + 9239892384264 q^{85} - 1372999514421 q^{86} + 5364161696536 q^{87} - 2985250612134 q^{88} - 1338832432586 q^{89} - 13443400684582 q^{90} + 636482341976 q^{91} - 8786066700672 q^{92} - 12529067281768 q^{93} - 18098521583937 q^{94} - 119505100680 q^{95} - 29672968067657 q^{96} + 1532637634742 q^{97} - 14020062796872 q^{98} - 17432744537624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 35596x^{5} + 594856x^{4} + 263632368x^{3} + 252644208x^{2} - 410033371968x - 5442114981888 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 969 \nu^{6} - 153125 \nu^{5} - 37273116 \nu^{4} + 5141065076 \nu^{3} + 205885624704 \nu^{2} + \cdots - 372250932068352 ) / 215581424640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 969 \nu^{6} + 153125 \nu^{5} + 37273116 \nu^{4} - 5141065076 \nu^{3} + \cdots - 724411775075328 ) / 107790712320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2191 \nu^{6} + 13092 \nu^{5} + 104657371 \nu^{4} - 1116041580 \nu^{3} + \cdots + 22\!\cdots\!04 ) / 26947678080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 90703 \nu^{6} - 3354157 \nu^{5} - 3281844634 \nu^{4} + 139887596428 \nu^{3} + \cdots - 19\!\cdots\!80 ) / 215581424640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51142 \nu^{6} + 709161 \nu^{5} + 1685935189 \nu^{4} - 60895203816 \nu^{3} + \cdots + 92\!\cdots\!84 ) / 107790712320 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} - 24\beta _1 + 10174 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{6} - 10\beta_{5} - 8\beta_{4} - 77\beta_{3} + 4\beta_{2} + 20778\beta _1 - 242636 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 34\beta_{6} + 302\beta_{5} + 1056\beta_{4} + 28003\beta_{3} + 50428\beta_{2} - 1445466\beta _1 + 211264092 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 187182 \beta_{6} - 296962 \beta_{5} - 268640 \beta_{4} - 3095465 \beta_{3} - 3011468 \beta_{2} + \cdots - 14663379932 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3561850 \beta_{6} + 17744918 \beta_{5} + 40612416 \beta_{4} + 784047859 \beta_{3} + 1240168804 \beta_{2} + \cdots + 5319008263956 \) Copy content Toggle raw display

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
−175.134
−70.2059
−37.9195
−15.2660
48.8519
117.726
132.948
−157.134 286.918 16499.3 17641.6 −45084.7 −100427. −1.30536e6 −1.51200e6 −2.77210e6
1.2 −52.2059 −945.382 −5466.54 −30724.8 49354.5 −524401. 713057. −700576. 1.60402e6
1.3 −19.9195 1765.94 −7795.21 53134.3 −35176.8 17946.1 318458. 1.52424e6 −1.05841e6
1.4 2.73404 −42.7679 −8184.53 −45417.0 −116.929 591080. −44774.1 −1.59249e6 −124172.
1.5 66.8519 −2327.76 −3722.83 7472.85 −155615. −60830.3 −796528. 3.82413e6 499574.
1.6 135.726 2027.33 10229.5 −15679.1 275162. 295835. 276541. 2.51576e6 −2.12805e6
1.7 150.948 −36.2889 14593.4 48458.2 −5477.75 −87338.4 966274. −1.59301e6 7.31468e6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

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 13.14.a.b 7
3.b odd 2 1 117.14.a.d 7
13.b even 2 1 169.14.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.a.b 7 1.a even 1 1 trivial
117.14.a.d 7 3.b odd 2 1
169.14.a.b 7 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 127 T_{2}^{6} - 28684 T_{2}^{5} + 3589516 T_{2}^{4} + 109262496 T_{2}^{3} + \cdots + 611898900480 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + \cdots + 611898900480 \) Copy content Toggle raw display
$3$ \( T^{7} + \cdots - 35\!\cdots\!72 \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 87\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T - 4826809)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 10\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 59\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 57\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 17\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 69\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 44\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 26\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 21\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 39\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 45\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 22\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
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