Properties

Label 338.3.d.f
Level $338$
Weight $3$
Character orbit 338.d
Analytic conductor $9.210$
Analytic rank $0$
Dimension $8$
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] = [338,3,Mod(99,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(9.20983293538\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.612074651904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 26)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{2} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{3} - 2 \beta_{5} q^{4} + (\beta_{7} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{7} + (2 \beta_{5} + 2) q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{2} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{3} - 2 \beta_{5} q^{4} + (\beta_{7} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{7} + (2 \beta_{5} + 2) q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 + 9) q^{9} + ( - \beta_{7} - \beta_{6} + 2 \beta_{2} + \beta_1 - 2) q^{10} + ( - \beta_{6} - 5 \beta_{5} + \beta_{2} - 6) q^{11} + 2 \beta_{4} q^{12} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{14} + ( - 2 \beta_{7} + 15 \beta_{5} - \beta_{4} - \beta_{3} + 15 \beta_1 - 15) q^{15} - 4 q^{16} + ( - \beta_{7} - \beta_{6} - 11 \beta_{5} - 3 \beta_{2} - 4 \beta_1 + 3) q^{17} + ( - 2 \beta_{7} + 8 \beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_1 - 8) q^{18} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 10 \beta_1 + 1) q^{19} + (2 \beta_{6} - 4 \beta_{2} + 4) q^{20} + ( - \beta_{6} + 3 \beta_{5} - 19 \beta_{2} + 22) q^{21} + ( - \beta_{7} + \beta_{6} - \beta_{2} + 11) q^{22} + (\beta_{7} + \beta_{6} - 10 \beta_{5} - \beta_{4} + 3 \beta_{2} + 4 \beta_1 - 3) q^{23} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 2) q^{24} + ( - 22 \beta_{5} - 3 \beta_{4} - 11 \beta_{2} - 11 \beta_1 + 11) q^{25} + (2 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} + 5 \beta_{3} - 16 \beta_{2} + 23 \beta_1 + 3) q^{27} + (2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 4) q^{28} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 3 \beta_1 + 5) q^{29} + (2 \beta_{7} + 2 \beta_{6} - 32 \beta_{5} + 2 \beta_{4} - 18 \beta_{2} - 16 \beta_1 + 18) q^{30} + (5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - \beta_1 - 5) q^{31} + ( - 4 \beta_{5} + 4) q^{32} + (\beta_{6} - 20 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 17 \beta_{2} - 5 \beta_1 + 7) q^{33} + (2 \beta_{6} + 11 \beta_{5} + 6 \beta_{2} + 5) q^{34} + ( - 3 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 7 \beta_{2} + 25) q^{35} + (2 \beta_{7} + 2 \beta_{6} - 18 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 2) q^{36} + (6 \beta_{6} - 19 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 23 \beta_{2} - 5 \beta_1 + 14) q^{37} + (\beta_{7} + \beta_{6} + 6 \beta_{5} - 4 \beta_{4} + 11 \beta_{2} + 12 \beta_1 - 11) q^{38} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{2} - 2 \beta_1 - 4) q^{40} + (8 \beta_{7} + 10 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 25 \beta_1 - 10) q^{41} + ( - \beta_{7} + \beta_{6} + 19 \beta_{2} - 20 \beta_1 - 25) q^{42} + (\beta_{7} + \beta_{6} + 10 \beta_{5} - 5 \beta_{4} - 7 \beta_{2} - 6 \beta_1 + 7) q^{43} + (2 \beta_{7} + 10 \beta_{5} - 10) q^{44} + (7 \beta_{7} - 29 \beta_{5} - \beta_{4} - \beta_{3} - 28 \beta_1 + 29) q^{45} + ( - 2 \beta_{6} + 9 \beta_{5} + \beta_{4} - \beta_{3} - 6 \beta_{2} - \beta_1 + 17) q^{46} + (2 \beta_{6} - 13 \beta_{5} + \beta_{4} - \beta_{3} - 10 \beta_{2} - \beta_1 - 1) q^{47} + ( - 4 \beta_{5} - 4 \beta_{3} - 4 \beta_1 + 4) q^{48} + (2 \beta_{7} + 2 \beta_{6} - 25 \beta_{5} + 3 \beta_{4} - 24 \beta_{2} - 22 \beta_1 + 24) q^{49} + (19 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 22 \beta_{2} - 3 \beta_1 + 3) q^{50} + ( - 3 \beta_{7} - 3 \beta_{6} - 22 \beta_{5} + 3 \beta_{4} - 13 \beta_{2} + \cdots + 13) q^{51}+ \cdots + (2 \beta_{6} - 18 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 34 \beta_{2} - 4 \beta_1 - 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 6 q^{5} - 10 q^{7} + 16 q^{8} + 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 6 q^{5} - 10 q^{7} + 16 q^{8} + 84 q^{9} - 42 q^{11} + 20 q^{14} - 60 q^{15} - 32 q^{16} - 84 q^{18} - 22 q^{19} + 12 q^{20} + 102 q^{21} + 84 q^{22} + 72 q^{27} - 20 q^{28} + 12 q^{29} - 32 q^{31} + 32 q^{32} - 54 q^{33} + 60 q^{34} + 156 q^{35} - 32 q^{37} - 24 q^{40} + 12 q^{41} - 204 q^{42} - 84 q^{44} + 102 q^{45} + 108 q^{46} - 60 q^{47} + 88 q^{50} - 132 q^{53} - 72 q^{54} + 324 q^{55} + 294 q^{57} - 12 q^{58} - 234 q^{59} + 120 q^{60} - 72 q^{61} + 156 q^{63} + 108 q^{66} + 14 q^{67} - 120 q^{68} - 156 q^{70} - 162 q^{71} + 168 q^{72} - 166 q^{73} + 64 q^{74} + 44 q^{76} - 96 q^{79} + 24 q^{80} + 24 q^{81} + 240 q^{83} + 204 q^{84} + 234 q^{85} - 132 q^{86} - 720 q^{87} - 210 q^{89} - 216 q^{92} + 444 q^{93} + 120 q^{94} - 146 q^{97} + 16 q^{98} - 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{7} - 349\nu^{6} + 32\nu^{5} + 19195\nu^{4} - 8968\nu^{3} - 345859\nu^{2} + 102689\nu + 2068174 ) / 86552 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} + 349\nu^{6} + 32\nu^{5} - 19195\nu^{4} - 8968\nu^{3} + 345859\nu^{2} + 102689\nu - 1981622 ) / 86552 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 74\nu^{5} + 1718\nu^{3} - 10073\nu + 1396 ) / 2792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31 \nu^{7} + 1396 \nu^{6} + 2294 \nu^{5} - 76780 \nu^{4} - 53258 \nu^{3} + 1469988 \nu^{2} + 398815 \nu - 9700804 ) / 86552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -20\nu^{7} + 1131\nu^{5} - 22145\nu^{3} + 148063\nu ) / 21638 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 318 \nu^{7} - 698 \nu^{6} - 16901 \nu^{5} + 49209 \nu^{4} + 292601 \nu^{3} - 1135297 \nu^{2} - 1582807 \nu + 8604595 ) / 86552 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 318 \nu^{7} - 698 \nu^{6} + 16901 \nu^{5} + 49209 \nu^{4} - 292601 \nu^{3} - 1135297 \nu^{2} + 1582807 \nu + 8604595 ) / 86552 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} - 5\beta_{2} + 3\beta _1 + 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 5\beta_{5} + 20\beta_{3} + 14\beta_{2} + 14\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{7} + 8\beta_{6} - 41\beta_{5} + 82\beta_{4} - 189\beta_{2} + 107\beta _1 + 716 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 140\beta_{7} - 140\beta_{6} - 627\beta_{5} + 774\beta_{3} + 827\beta_{2} + 827\beta _1 - 1214 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 220\beta_{7} + 220\beta_{6} - 632\beta_{5} + 1264\beta_{4} - 2596\beta_{2} + 1332\beta _1 + 6663 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3488\beta_{7} - 3488\beta_{6} - 19145\beta_{5} + 14286\beta_{3} + 23167\beta_{2} + 23167\beta _1 - 30310 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−3.90972 + 0.500000i
−4.71318 + 0.500000i
3.90972 + 0.500000i
4.71318 + 0.500000i
−3.90972 0.500000i
−4.71318 0.500000i
3.90972 0.500000i
4.71318 0.500000i
−1.00000 1.00000i −4.77574 2.00000i 5.88981 + 5.88981i 4.77574 + 4.77574i −0.251956 + 0.251956i 2.00000 2.00000i 13.8077 11.7796i
99.2 −1.00000 1.00000i −3.84716 2.00000i −3.77418 3.77418i 3.84716 + 3.84716i −7.25532 + 7.25532i 2.00000 2.00000i 5.80063 7.54837i
99.3 −1.00000 1.00000i 3.04369 2.00000i −4.79174 4.79174i −3.04369 3.04369i −3.11407 + 3.11407i 2.00000 2.00000i 0.264067 9.58347i
99.4 −1.00000 1.00000i 5.57921 2.00000i −0.323893 0.323893i −5.57921 5.57921i 5.62134 5.62134i 2.00000 2.00000i 22.1276 0.647786i
239.1 −1.00000 + 1.00000i −4.77574 2.00000i 5.88981 5.88981i 4.77574 4.77574i −0.251956 0.251956i 2.00000 + 2.00000i 13.8077 11.7796i
239.2 −1.00000 + 1.00000i −3.84716 2.00000i −3.77418 + 3.77418i 3.84716 3.84716i −7.25532 7.25532i 2.00000 + 2.00000i 5.80063 7.54837i
239.3 −1.00000 + 1.00000i 3.04369 2.00000i −4.79174 + 4.79174i −3.04369 + 3.04369i −3.11407 3.11407i 2.00000 + 2.00000i 0.264067 9.58347i
239.4 −1.00000 + 1.00000i 5.57921 2.00000i −0.323893 + 0.323893i −5.57921 + 5.57921i 5.62134 + 5.62134i 2.00000 + 2.00000i 22.1276 0.647786i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.4
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 338.3.d.f 8
13.b even 2 1 338.3.d.g 8
13.c even 3 1 338.3.f.i 8
13.c even 3 1 338.3.f.j 8
13.d odd 4 1 inner 338.3.d.f 8
13.d odd 4 1 338.3.d.g 8
13.e even 6 1 26.3.f.b 8
13.e even 6 1 338.3.f.h 8
13.f odd 12 1 26.3.f.b 8
13.f odd 12 1 338.3.f.h 8
13.f odd 12 1 338.3.f.i 8
13.f odd 12 1 338.3.f.j 8
39.h odd 6 1 234.3.bb.f 8
39.k even 12 1 234.3.bb.f 8
52.i odd 6 1 208.3.bd.f 8
52.l even 12 1 208.3.bd.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.b 8 13.e even 6 1
26.3.f.b 8 13.f odd 12 1
208.3.bd.f 8 52.i odd 6 1
208.3.bd.f 8 52.l even 12 1
234.3.bb.f 8 39.h odd 6 1
234.3.bb.f 8 39.k even 12 1
338.3.d.f 8 1.a even 1 1 trivial
338.3.d.f 8 13.d odd 4 1 inner
338.3.d.g 8 13.b even 2 1
338.3.d.g 8 13.d odd 4 1
338.3.f.h 8 13.e even 6 1
338.3.f.h 8 13.f odd 12 1
338.3.f.i 8 13.c even 3 1
338.3.f.i 8 13.f odd 12 1
338.3.f.j 8 13.c even 3 1
338.3.f.j 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{4} - 39T_{3}^{2} - 12T_{3} + 312 \) Copy content Toggle raw display
\( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} + 90T_{5}^{5} + 4245T_{5}^{4} + 30312T_{5}^{3} + 109512T_{5}^{2} + 64584T_{5} + 19044 \) Copy content Toggle raw display
\( T_{7}^{8} + 10T_{7}^{7} + 50T_{7}^{6} - 146T_{7}^{5} + 5017T_{7}^{4} + 38816T_{7}^{3} + 147968T_{7}^{2} + 69632T_{7} + 16384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 39 T^{2} - 12 T + 312)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 19044 \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{7} + 50 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{8} + 42 T^{7} + 882 T^{6} + \cdots + 389376 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 624 T^{6} + 21630 T^{4} + \cdots + 471969 \) Copy content Toggle raw display
$19$ \( T^{8} + 22 T^{7} + \cdots + 1228362304 \) Copy content Toggle raw display
$23$ \( T^{8} + 1266 T^{6} + \cdots + 2508807744 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} - 930 T^{2} + \cdots + 159549)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 32 T^{7} + \cdots + 8111524096 \) Copy content Toggle raw display
$37$ \( T^{8} + 32 T^{7} + \cdots + 321419829721 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 326485389321 \) Copy content Toggle raw display
$43$ \( T^{8} + 4698 T^{6} + \cdots + 325666531584 \) Copy content Toggle raw display
$47$ \( T^{8} + 60 T^{7} + \cdots + 1853819136 \) Copy content Toggle raw display
$53$ \( (T^{4} + 66 T^{3} - 5319 T^{2} + \cdots - 5234376)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 234 T^{7} + \cdots + 70410089309184 \) Copy content Toggle raw display
$61$ \( (T^{4} + 36 T^{3} - 1710 T^{2} + \cdots + 559869)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{7} + \cdots + 2456391674944 \) Copy content Toggle raw display
$71$ \( T^{8} + 162 T^{7} + \cdots + 950999436864 \) Copy content Toggle raw display
$73$ \( T^{8} + 166 T^{7} + \cdots + 3554348548804 \) Copy content Toggle raw display
$79$ \( (T^{4} + 48 T^{3} - 14880 T^{2} + \cdots + 2312448)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 154848357540864 \) Copy content Toggle raw display
$89$ \( T^{8} + 210 T^{7} + \cdots + 14950765690884 \) Copy content Toggle raw display
$97$ \( T^{8} + 146 T^{7} + \cdots + 9988090235236 \) Copy content Toggle raw display
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