Properties

Label 1344.4.p.b
Level $1344$
Weight $4$
Character orbit 1344.p
Analytic conductor $79.299$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(223,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta_1 q^{3} + \beta_{7} q^{5} + ( - \beta_{5} + \beta_1) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_1 q^{3} + \beta_{7} q^{5} + ( - \beta_{5} + \beta_1) q^{7} - 9 q^{9} + \beta_{10} q^{11} + (\beta_{13} - \beta_{7} + \beta_{4} + \beta_{2} + 3) q^{13} + 3 \beta_{6} q^{15} + ( - \beta_{11} - \beta_{8} - \beta_{4}) q^{17} + ( - \beta_{12} + \beta_{6} + 7 \beta_1) q^{19} + ( - 3 \beta_{2} + 3) q^{21} + (\beta_{12} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 19 \beta_1) q^{23} + (\beta_{13} + 5 \beta_{7} - 2 \beta_{4} - 2 \beta_{2} + 6) q^{25} + 27 \beta_1 q^{27} + ( - \beta_{14} - 3 \beta_{11}) q^{29} + ( - \beta_{15} - 3 \beta_{10} + 2 \beta_{5} - 2 \beta_{3}) q^{31} - 3 \beta_{11} q^{33} + ( - \beta_{15} - 2 \beta_{12} - 2 \beta_{10} + \beta_{9} - 6 \beta_{6} + \beta_{3} - 14 \beta_1) q^{35} + (\beta_{14} - \beta_{11} + 2 \beta_{8} - \beta_{4} + 3 \beta_{2}) q^{37} + (3 \beta_{12} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{3} - 9 \beta_1) q^{39} + (2 \beta_{14} + \beta_{11} + \beta_{8} - 5 \beta_{4} + 6 \beta_{2}) q^{41} + (4 \beta_{10} - \beta_{9} - 5 \beta_{5} + 6 \beta_{3}) q^{43} - 9 \beta_{7} q^{45} + ( - 2 \beta_{15} + 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{5} - 6 \beta_{3}) q^{47} + (\beta_{14} + 5 \beta_{13} + 5 \beta_{11} - \beta_{8} + \beta_{7} - 2 \beta_{4} + 3 \beta_{2} + \cdots + 24) q^{49}+ \cdots - 9 \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 144 q^{9} + 56 q^{13} + 36 q^{21} + 80 q^{25} + 392 q^{49} + 336 q^{57} + 184 q^{61} - 1536 q^{65} + 864 q^{69} - 240 q^{77} + 1296 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + \cdots + 22\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!32 \nu^{15} + \cdots - 27\!\cdots\!76 ) / 41\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\!\cdots\!01 \nu^{15} + \cdots - 21\!\cdots\!55 ) / 10\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 91\!\cdots\!68 \nu^{15} + \cdots - 90\!\cdots\!27 ) / 48\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!51 \nu^{15} + \cdots - 20\!\cdots\!12 ) / 10\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!51 \nu^{15} + \cdots + 20\!\cdots\!12 ) / 10\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 63\!\cdots\!46 \nu^{15} + \cdots - 77\!\cdots\!93 ) / 27\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!71 \nu^{15} + \cdots + 20\!\cdots\!98 ) / 11\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!76 \nu^{15} + \cdots + 59\!\cdots\!77 ) / 19\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!25 \nu^{15} + \cdots + 25\!\cdots\!41 ) / 92\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!09 \nu^{15} + \cdots - 25\!\cdots\!35 ) / 12\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!68 \nu^{15} + \cdots + 42\!\cdots\!29 ) / 45\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!50 \nu^{15} + \cdots - 17\!\cdots\!40 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 47\!\cdots\!55 \nu^{15} + \cdots + 54\!\cdots\!94 ) / 39\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32\!\cdots\!76 \nu^{15} + \cdots - 25\!\cdots\!08 ) / 19\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{15} + \cdots + 59\!\cdots\!81 ) / 42\!\cdots\!22 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + 5\beta_{11} - \beta_{8} + \beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + 342\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{15} - 7 \beta_{14} - 13 \beta_{13} - 13 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} - 28 \beta_{9} - 28 \beta_{8} + 157 \beta_{7} + 157 \beta_{6} + 41 \beta_{5} - 41 \beta_{4} + 553 \beta_{3} - 553 \beta_{2} - 97 \beta _1 + 97 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 163 \beta_{15} + 79 \beta_{13} - 843 \beta_{10} - 198 \beta_{9} - 64 \beta_{7} + 269 \beta_{5} - 71 \beta_{4} - 71 \beta_{3} - 71 \beta_{2} - 44560 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1496 \beta_{15} + 1496 \beta_{14} - 2047 \beta_{13} + 2047 \beta_{12} + 417 \beta_{11} + 417 \beta_{10} - 5238 \beta_{9} + 5238 \beta_{8} + 24568 \beta_{7} - 24568 \beta_{6} - 75344 \beta_{5} - 75344 \beta_{4} + \cdots + 5299 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 92451 \beta_{14} - 81136 \beta_{12} - 523953 \beta_{11} + 129075 \beta_{8} + 78256 \beta_{6} - 36503 \beta_{5} + 151104 \beta_{4} - 36503 \beta_{3} - 22029 \beta_{2} - 25601422 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1044781 \beta_{15} + 1044781 \beta_{14} + 845557 \beta_{13} + 845557 \beta_{12} - 80373 \beta_{11} + 80373 \beta_{10} + 3483834 \beta_{9} + 3483834 \beta_{8} - 13191643 \beta_{7} + \cdots + 5582015 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12704042 \beta_{15} - 16333579 \beta_{13} + 80518170 \beta_{10} + 20422200 \beta_{9} + 16195387 \beta_{7} - 22134178 \beta_{5} + 7004246 \beta_{4} + 1711978 \beta_{3} + \cdots + 3764173504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 172651005 \beta_{15} - 172651005 \beta_{14} + 52776727 \beta_{13} - 52776727 \beta_{12} + 61321509 \beta_{11} + 61321509 \beta_{10} + 567056502 \beta_{9} - 567056502 \beta_{8} + \cdots + 2274028613 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6902289991 \beta_{14} + 12082786564 \beta_{12} + 49433985051 \beta_{11} - 12879763521 \beta_{8} - 11882130856 \beta_{6} + 5669850665 \beta_{5} + \cdots + 2231044630090 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 111743533385 \beta_{15} - 111743533385 \beta_{14} + 14565283037 \beta_{13} + 14565283037 \beta_{12} + 68277298119 \beta_{11} - 68277298119 \beta_{10} + \cdots - 2315920372097 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 930489180369 \beta_{15} + 2139104560648 \beta_{13} - 7592699813517 \beta_{10} - 2032338076482 \beta_{9} - 2069856552361 \beta_{7} + \cdots - 332000524438189 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 17880852446077 \beta_{15} + 17880852446077 \beta_{14} + 9101406164066 \beta_{13} - 9101406164066 \beta_{12} - 15330070159428 \beta_{11} + \cdots - 506243338698578 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 497993622759701 \beta_{14} + \cdots - 19\!\cdots\!10 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 11\!\cdots\!03 \beta_{15} + \cdots + 40\!\cdots\!09 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
11.6544 + 11.6544i
−6.36140 6.36140i
−2.77393 2.77393i
−12.1800 12.1800i
12.6006 + 12.6006i
5.50749 + 5.50749i
4.51719 + 4.51719i
−11.9644 11.9644i
11.6544 11.6544i
−6.36140 + 6.36140i
−2.77393 + 2.77393i
−12.1800 + 12.1800i
12.6006 12.6006i
5.50749 5.50749i
4.51719 4.51719i
−11.9644 + 11.9644i
0 3.00000i 0 −10.6345 0 −18.0158 4.29303i 0 −9.00000 0
223.2 0 3.00000i 0 −10.6345 0 18.0158 4.29303i 0 −9.00000 0
223.3 0 3.00000i 0 −10.4276 0 −9.40604 + 15.9539i 0 −9.00000 0
223.4 0 3.00000i 0 −10.4276 0 9.40604 + 15.9539i 0 −9.00000 0
223.5 0 3.00000i 0 4.35176 0 −7.09314 17.1081i 0 −9.00000 0
223.6 0 3.00000i 0 4.35176 0 7.09314 17.1081i 0 −9.00000 0
223.7 0 3.00000i 0 16.7103 0 −16.4816 + 8.44725i 0 −9.00000 0
223.8 0 3.00000i 0 16.7103 0 16.4816 + 8.44725i 0 −9.00000 0
223.9 0 3.00000i 0 −10.6345 0 −18.0158 + 4.29303i 0 −9.00000 0
223.10 0 3.00000i 0 −10.6345 0 18.0158 + 4.29303i 0 −9.00000 0
223.11 0 3.00000i 0 −10.4276 0 −9.40604 15.9539i 0 −9.00000 0
223.12 0 3.00000i 0 −10.4276 0 9.40604 15.9539i 0 −9.00000 0
223.13 0 3.00000i 0 4.35176 0 −7.09314 + 17.1081i 0 −9.00000 0
223.14 0 3.00000i 0 4.35176 0 7.09314 + 17.1081i 0 −9.00000 0
223.15 0 3.00000i 0 16.7103 0 −16.4816 8.44725i 0 −9.00000 0
223.16 0 3.00000i 0 16.7103 0 16.4816 8.44725i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.p.b yes 16
4.b odd 2 1 inner 1344.4.p.b yes 16
7.b odd 2 1 1344.4.p.a 16
8.b even 2 1 1344.4.p.a 16
8.d odd 2 1 1344.4.p.a 16
28.d even 2 1 1344.4.p.a 16
56.e even 2 1 inner 1344.4.p.b yes 16
56.h odd 2 1 inner 1344.4.p.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.p.a 16 7.b odd 2 1
1344.4.p.a 16 8.b even 2 1
1344.4.p.a 16 8.d odd 2 1
1344.4.p.a 16 28.d even 2 1
1344.4.p.b yes 16 1.a even 1 1 trivial
1344.4.p.b yes 16 4.b odd 2 1 inner
1344.4.p.b yes 16 56.e even 2 1 inner
1344.4.p.b yes 16 56.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 260T_{5}^{2} - 804T_{5} + 8064 \) acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 260 T^{2} - 804 T + 8064)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} - 196 T^{14} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4620 T^{6} + \cdots + 10053320832)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 14 T^{3} - 4604 T^{2} + \cdots + 624672)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 28080 T^{6} + \cdots + 645802409513088)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 9728 T^{6} + \cdots + 1880891559936)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 18576 T^{6} + \cdots + 4988879424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 123804 T^{6} + \cdots + 18\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 142620 T^{6} + \cdots + 63\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 208512 T^{6} + \cdots + 51\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 427104 T^{6} + \cdots + 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 168372 T^{6} + \cdots + 16\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 634464 T^{6} + \cdots + 46\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 793308 T^{6} + \cdots + 93\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 683056 T^{6} + \cdots + 28\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 46 T^{3} - 394604 T^{2} + \cdots - 1820784672)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 1819044 T^{6} + \cdots + 42\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 1963872 T^{6} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 1728096 T^{6} + \cdots + 77\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 1665844 T^{6} + \cdots + 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 1273408 T^{6} + \cdots + 41\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 3975840 T^{6} + \cdots + 29\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 4368480 T^{6} + \cdots + 55\!\cdots\!28)^{2} \) Copy content Toggle raw display
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