Properties

Label 13.14.b.a
Level $13$
Weight $14$
Character orbit 13.b
Analytic conductor $13.940$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,14,Mod(12,13)]
 
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([1]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.12");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 13.b (of order \(2\), degree \(1\), 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: \(13.9400207637\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 81921 x^{12} + 2522899104 x^{10} + 37246030694192 x^{8} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{6}\cdot 13^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 104) q^{3} + ( - \beta_{3} + \beta_{2} - 3511) q^{4} + ( - \beta_{7} + 24 \beta_1) q^{5} + ( - \beta_{8} - \beta_{7} + 207 \beta_1) q^{6} + (\beta_{9} + \beta_{7} - 444 \beta_1) q^{7} + (\beta_{10} + \beta_{9} + \cdots - 3944 \beta_1) q^{8}+ \cdots + ( - \beta_{4} + 283 \beta_{3} + \cdots + 367804) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 104) q^{3} + ( - \beta_{3} + \beta_{2} - 3511) q^{4} + ( - \beta_{7} + 24 \beta_1) q^{5} + ( - \beta_{8} - \beta_{7} + 207 \beta_1) q^{6} + (\beta_{9} + \beta_{7} - 444 \beta_1) q^{7} + (\beta_{10} + \beta_{9} + \cdots - 3944 \beta_1) q^{8}+ \cdots + ( - 199232 \beta_{13} + \cdots - 63373502405 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 1456 q^{3} - 49154 q^{4} + 5149262 q^{9} - 3968358 q^{10} - 21959110 q^{12} - 14362478 q^{13} + 72843942 q^{14} + 243447170 q^{16} - 178512492 q^{17} + 1330288056 q^{22} - 1641693744 q^{23} + 1601386654 q^{25} + 138020766 q^{26} + 6360372232 q^{27} - 10348326348 q^{29} - 11230427010 q^{30} + 20242816056 q^{35} + 33181294876 q^{36} - 72234754500 q^{38} + 3102574280 q^{39} - 50528567766 q^{40} + 90123482634 q^{42} - 42288191696 q^{43} + 256465114846 q^{48} + 202681235282 q^{49} - 454251099624 q^{51} + 605927200580 q^{52} - 541269433788 q^{53} + 294791560608 q^{55} - 1027260370698 q^{56} + 1693279412900 q^{61} + 1808450592168 q^{62} - 4815047793506 q^{64} + 534977124552 q^{65} + 1417411440480 q^{66} + 65016763122 q^{68} - 4207322259360 q^{69} + 7595100503238 q^{74} + 1662071406584 q^{75} - 1391939668824 q^{77} - 361798757502 q^{78} - 5060278211056 q^{79} + 6637938210830 q^{81} + 507818992992 q^{82} - 13033010125344 q^{87} + 2586979258080 q^{88} - 4680167629284 q^{90} + 3450737986176 q^{91} + 7241299383420 q^{92} + 11219792119974 q^{94} + 11940434574336 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 81921 x^{12} + 2522899104 x^{10} + 37246030694192 x^{8} + \cdots + 56\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 171838599480073 \nu^{12} + \cdots + 42\!\cdots\!68 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 171838599480073 \nu^{12} + \cdots + 48\!\cdots\!68 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!77 \nu^{12} + \cdots - 16\!\cdots\!68 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42\!\cdots\!29 \nu^{12} + \cdots - 10\!\cdots\!44 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!89 \nu^{12} + \cdots - 55\!\cdots\!24 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!71 \nu^{13} + \cdots - 31\!\cdots\!84 \nu ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!51 \nu^{13} + \cdots + 71\!\cdots\!84 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!07 \nu^{13} + \cdots - 95\!\cdots\!28 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 66\!\cdots\!53 \nu^{13} + \cdots + 43\!\cdots\!48 \nu ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!89 \nu^{13} + \cdots + 15\!\cdots\!96 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!69 \nu^{13} + \cdots + 90\!\cdots\!28 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!69 \nu^{13} + \cdots - 90\!\cdots\!28 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 11703 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} - 4\beta_{7} - 20328\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{13} - 4\beta_{12} - 10\beta_{6} - 9\beta_{5} + 13\beta_{4} + 35530\beta_{3} - 31079\beta_{2} + 237893152 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 78 \beta_{13} - 78 \beta_{12} + 20 \beta_{11} - 40853 \beta_{10} - 53397 \beta_{9} + \cdots + 506348112 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 176572 \beta_{13} + 176572 \beta_{12} + 387202 \beta_{6} + 416277 \beta_{5} - 597001 \beta_{4} + \cdots - 5925541103548 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3975030 \beta_{13} + 3975030 \beta_{12} - 2257220 \beta_{11} + 1402215925 \beta_{10} + \cdots - 14227924191120 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6180648732 \beta_{13} - 6180648732 \beta_{12} - 12275537154 \beta_{6} - 15210322677 \beta_{5} + \cdots + 16\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 156075488310 \beta_{13} - 156075488310 \beta_{12} + 136854428100 \beta_{11} + \cdots + 42\!\cdots\!04 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 202945672691804 \beta_{13} + 202945672691804 \beta_{12} + 376255147926434 \beta_{6} + \cdots - 49\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 55\!\cdots\!74 \beta_{13} + \cdots - 13\!\cdots\!24 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 65\!\cdots\!04 \beta_{13} + \cdots + 15\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18\!\cdots\!22 \beta_{13} + \cdots + 41\!\cdots\!84 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
178.493i
131.418i
110.094i
109.869i
87.5010i
28.8440i
10.5212i
10.5212i
28.8440i
87.5010i
109.869i
110.094i
131.418i
178.493i
178.493i 850.060 −23667.6 1220.92i 151729.i 257415.i 2.76227e6i −871721. −217925.
12.2 131.418i −1774.35 −9078.57 27762.3i 233181.i 2614.44i 116510.i 1.55400e6 3.64846e6
12.3 110.094i 787.740 −3928.66 28789.1i 86725.3i 370728.i 469368.i −973789. 3.16950e6
12.4 109.869i −652.778 −3879.10 61212.2i 71719.8i 994.788i 473853.i −1.16820e6 −6.72530e6
12.5 87.5010i 2407.03 535.569 31938.7i 210617.i 191671.i 763671.i 4.19945e6 −2.79467e6
12.6 28.8440i 701.398 7360.02 35879.6i 20231.1i 318380.i 448583.i −1.10236e6 1.03491e6
12.7 10.5212i −1591.09 8081.30 9424.40i 16740.2i 484840.i 171215.i 937259. −99156.0
12.8 10.5212i −1591.09 8081.30 9424.40i 16740.2i 484840.i 171215.i 937259. −99156.0
12.9 28.8440i 701.398 7360.02 35879.6i 20231.1i 318380.i 448583.i −1.10236e6 1.03491e6
12.10 87.5010i 2407.03 535.569 31938.7i 210617.i 191671.i 763671.i 4.19945e6 −2.79467e6
12.11 109.869i −652.778 −3879.10 61212.2i 71719.8i 994.788i 473853.i −1.16820e6 −6.72530e6
12.12 110.094i 787.740 −3928.66 28789.1i 86725.3i 370728.i 469368.i −973789. 3.16950e6
12.13 131.418i −1774.35 −9078.57 27762.3i 233181.i 2614.44i 116510.i 1.55400e6 3.64846e6
12.14 178.493i 850.060 −23667.6 1220.92i 151729.i 257415.i 2.76227e6i −871721. −217925.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.14.b.a 14
3.b odd 2 1 117.14.b.c 14
13.b even 2 1 inner 13.14.b.a 14
13.d odd 4 2 169.14.a.d 14
39.d odd 2 1 117.14.b.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.b.a 14 1.a even 1 1 trivial
13.14.b.a 14 13.b even 2 1 inner
117.14.b.c 14 3.b odd 2 1
117.14.b.c 14 39.d odd 2 1
169.14.a.d 14 13.d odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( (T^{7} + \cdots + 20\!\cdots\!92)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 23\!\cdots\!37 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 74\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 65\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 50\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 20\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
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