Properties

Label 14.9.b.a
Level $14$
Weight $9$
Character orbit 14.b
Analytic conductor $5.703$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [14,9,Mod(13,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.13");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 14.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: \(5.70330054086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.3520512.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 120x^{2} + 3438 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{3} q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} + 128 q^{4} + (2 \beta_{2} + 7 \beta_1) q^{5} + (14 \beta_{2} + 6 \beta_1) q^{6} + (147 \beta_{3} + 49 \beta_1 - 1519) q^{7} + 256 \beta_{3} q^{8} + (1062 \beta_{3} - 3423) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{3} q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} + 128 q^{4} + (2 \beta_{2} + 7 \beta_1) q^{5} + (14 \beta_{2} + 6 \beta_1) q^{6} + (147 \beta_{3} + 49 \beta_1 - 1519) q^{7} + 256 \beta_{3} q^{8} + (1062 \beta_{3} - 3423) q^{9} + ( - 6 \beta_{2} + 98 \beta_1) q^{10} + (1308 \beta_{3} - 3390) q^{11} + ( - 128 \beta_{2} + 256 \beta_1) q^{12} + (360 \beta_{2} - 229 \beta_1) q^{13} + ( - 3038 \beta_{3} + 98 \beta_{2} + 490 \beta_1 + 9408) q^{14} + ( - 3774 \beta_{3} - 3264) q^{15} + 16384 q^{16} + ( - 622 \beta_{2} - 1658 \beta_1) q^{17} + ( - 6846 \beta_{3} + 67968) q^{18} + ( - 2391 \beta_{2} - 2888 \beta_1) q^{19} + (256 \beta_{2} + 896 \beta_1) q^{20} + ( - 7350 \beta_{3} + 2548 \beta_{2} - 2597 \beta_1 - 103488) q^{21} + ( - 6780 \beta_{3} + 83712) q^{22} + (11952 \beta_{3} - 223518) q^{23} + (1792 \beta_{2} + 768 \beta_1) q^{24} + ( - 11658 \beta_{3} + 304225) q^{25} + ( - 4058 \beta_{2} + 2750 \beta_1) q^{26} + (4296 \beta_{2} + 9462 \beta_1) q^{27} + (18816 \beta_{3} + 6272 \beta_1 - 194432) q^{28} + (135396 \beta_{3} + 79266) q^{29} + ( - 6528 \beta_{3} - 241536) q^{30} + ( - 8790 \beta_{2} + 24378 \beta_1) q^{31} + 32768 \beta_{3} q^{32} + (12546 \beta_{2} - 2856 \beta_1) q^{33} + (2904 \beta_{2} - 25288 \beta_1) q^{34} + ( - 85554 \beta_{3} - 3479 \beta_{2} - 3430 \beta_1 - 413952) q^{35} + (135936 \beta_{3} - 438144) q^{36} + ( - 89676 \beta_{3} - 623774) q^{37} + (18134 \beta_{2} - 62354 \beta_1) q^{38} + ( - 455970 \beta_{3} + 2557248) q^{39} + ( - 768 \beta_{2} + 12544 \beta_1) q^{40} + ( - 39488 \beta_{2} - 12442 \beta_1) q^{41} + ( - 206976 \beta_{3} - 30674 \beta_{2} + 9702 \beta_1 - 470400) q^{42} + (291720 \beta_{3} + 2296642) q^{43} + (167424 \beta_{3} - 433920) q^{44} + ( - 10032 \beta_{2} + 28077 \beta_1) q^{45} + ( - 447036 \beta_{3} + 764928) q^{46} + (73286 \beta_{2} - 5030 \beta_1) q^{47} + ( - 16384 \beta_{2} + 32768 \beta_1) q^{48} + ( - 893172 \beta_{3} + 14406 \beta_{2} - 76832 \beta_1 + 232897) q^{49} + (608450 \beta_{3} - 746112) q^{50} + (1095864 \beta_{3} - 81024) q^{51} + (46080 \beta_{2} - 29312 \beta_1) q^{52} + ( - 300696 \beta_{3} - 9681822) q^{53} + ( - 24036 \beta_{2} + 154764 \beta_1) q^{54} + ( - 10704 \beta_{2} + 40362 \beta_1) q^{55} + ( - 388864 \beta_{3} + 12544 \beta_{2} + 62720 \beta_1 + 1204224) q^{56} + (3689742 \beta_{3} - 7672704) q^{57} + (158532 \beta_{3} + 8665344) q^{58} + ( - 32825 \beta_{2} + 219680 \beta_1) q^{59} + ( - 483072 \beta_{3} - 417792) q^{60} + ( - 127506 \beta_{2} - 315443 \beta_1) q^{61} + (136656 \beta_{2} + 120720 \beta_1) q^{62} + ( - 2116359 \beta_{3} + 52038 \beta_{2} + \cdots + 10195185) q^{63}+ \cdots + ( - 8077464 \beta_{3} + 56055042) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 512 q^{4} - 6076 q^{7} - 13692 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 512 q^{4} - 6076 q^{7} - 13692 q^{9} - 13560 q^{11} + 37632 q^{14} - 13056 q^{15} + 65536 q^{16} + 271872 q^{18} - 413952 q^{21} + 334848 q^{22} - 894072 q^{23} + 1216900 q^{25} - 777728 q^{28} + 317064 q^{29} - 966144 q^{30} - 1655808 q^{35} - 1752576 q^{36} - 2495096 q^{37} + 10228992 q^{39} - 1881600 q^{42} + 9186568 q^{43} - 1735680 q^{44} + 3059712 q^{46} + 931588 q^{49} - 2984448 q^{50} - 324096 q^{51} - 38727288 q^{53} + 4816896 q^{56} - 30690816 q^{57} + 34661376 q^{58} - 1671168 q^{60} + 40780740 q^{63} + 8388608 q^{64} - 11891712 q^{65} - 12320248 q^{67} - 21901824 q^{70} + 62168712 q^{71} + 34799616 q^{72} - 22957056 q^{74} + 45208968 q^{77} - 116728320 q^{78} + 24889736 q^{79} - 70788348 q^{81} - 52985856 q^{84} + 89943552 q^{85} + 74680320 q^{86} + 42860544 q^{88} + 38158848 q^{91} - 114441216 q^{92} - 408466944 q^{93} + 227967744 q^{95} - 228652032 q^{98} + 224220168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 120x^{2} + 3438 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 156\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 108\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{2} + 240 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{3} - 240 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39\beta_{2} - 81\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
8.52807i
8.52807i
6.87547i
6.87547i
−11.3137 126.458i 128.000 143.012i 1430.71i −2350.56 489.571i −1448.15 −9430.58 1617.99i
13.2 −11.3137 126.458i 128.000 143.012i 1430.71i −2350.56 + 489.571i −1448.15 −9430.58 1617.99i
13.3 11.3137 63.0589i 128.000 390.317i 713.430i −687.442 2300.48i 1448.15 2584.58 4415.94i
13.4 11.3137 63.0589i 128.000 390.317i 713.430i −687.442 + 2300.48i 1448.15 2584.58 4415.94i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.9.b.a 4
3.b odd 2 1 126.9.c.a 4
4.b odd 2 1 112.9.c.c 4
5.b even 2 1 350.9.b.a 4
5.c odd 4 2 350.9.d.a 8
7.b odd 2 1 inner 14.9.b.a 4
7.c even 3 2 98.9.d.a 8
7.d odd 6 2 98.9.d.a 8
21.c even 2 1 126.9.c.a 4
28.d even 2 1 112.9.c.c 4
35.c odd 2 1 350.9.b.a 4
35.f even 4 2 350.9.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.b.a 4 1.a even 1 1 trivial
14.9.b.a 4 7.b odd 2 1 inner
98.9.d.a 8 7.c even 3 2
98.9.d.a 8 7.d odd 6 2
112.9.c.c 4 4.b odd 2 1
112.9.c.c 4 28.d even 2 1
126.9.c.a 4 3.b odd 2 1
126.9.c.a 4 21.c even 2 1
350.9.b.a 4 5.b even 2 1
350.9.b.a 4 35.c odd 2 1
350.9.d.a 8 5.c odd 4 2
350.9.d.a 8 35.f even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 19968 T^{2} + \cdots + 63589248 \) Copy content Toggle raw display
$5$ \( T^{4} + 172800 T^{2} + \cdots + 3115873152 \) Copy content Toggle raw display
$7$ \( T^{4} + 6076 T^{3} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6780 T - 43255548)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 202796647224192 \) Copy content Toggle raw display
$17$ \( T^{4} + 11879663616 T^{2} + \cdots + 23\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{4} + 94768846848 T^{2} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{2} + 447036 T + 45389086596)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 158532 T - 580343359356)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2154089189376 T^{2} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T^{2} + 1247548 T + 131756883844)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 19894697542656 T^{2} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4593284 T + 2551346607364)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 65772041361408 T^{2} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{2} + 19363644 T + 90844298538372)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 118891469721600 T^{2} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 459923108721408 T^{2} + \cdots + 38\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6160124 T - 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 31084356 T - 599142107080188)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( (T^{2} - 12444868 T - 50828841800444)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 97\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 95\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 34\!\cdots\!12 \) Copy content Toggle raw display
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