Properties

Label 338.8.b.b
Level $338$
Weight $8$
Character orbit 338.b
Analytic conductor $105.586$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,8,Mod(337,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\), degree \(1\), not 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: \(105.586138614\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 i q^{2} - 39 q^{3} - 64 q^{4} + 385 i q^{5} + 312 i q^{6} + 293 i q^{7} + 512 i q^{8} - 666 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 i q^{2} - 39 q^{3} - 64 q^{4} + 385 i q^{5} + 312 i q^{6} + 293 i q^{7} + 512 i q^{8} - 666 q^{9} + 3080 q^{10} + 5402 i q^{11} + 2496 q^{12} + 2344 q^{14} - 15015 i q^{15} + 4096 q^{16} + 21011 q^{17} + 5328 i q^{18} - 27326 i q^{19} - 24640 i q^{20} - 11427 i q^{21} + 43216 q^{22} + 63072 q^{23} - 19968 i q^{24} - 70100 q^{25} + 111267 q^{27} - 18752 i q^{28} + 122238 q^{29} - 120120 q^{30} - 208396 i q^{31} - 32768 i q^{32} - 210678 i q^{33} - 168088 i q^{34} - 112805 q^{35} + 42624 q^{36} + 442379 i q^{37} - 218608 q^{38} - 197120 q^{40} + 58000 i q^{41} - 91416 q^{42} + 202025 q^{43} - 345728 i q^{44} - 256410 i q^{45} - 504576 i q^{46} - 588511 i q^{47} - 159744 q^{48} + 737694 q^{49} + 560800 i q^{50} - 819429 q^{51} + 1684336 q^{53} - 890136 i q^{54} - 2079770 q^{55} - 150016 q^{56} + 1065714 i q^{57} - 977904 i q^{58} + 442630 i q^{59} + 960960 i q^{60} - 1083608 q^{61} - 1667168 q^{62} - 195138 i q^{63} - 262144 q^{64} - 1685424 q^{66} + 3443486 i q^{67} - 1344704 q^{68} - 2459808 q^{69} + 902440 i q^{70} + 2084705 i q^{71} - 340992 i q^{72} - 5937890 i q^{73} + 3539032 q^{74} + 2733900 q^{75} + 1748864 i q^{76} - 1582786 q^{77} - 6609256 q^{79} + 1576960 i q^{80} - 2882871 q^{81} + 464000 q^{82} - 142740 i q^{83} + 731328 i q^{84} + 8089235 i q^{85} - 1616200 i q^{86} - 4767282 q^{87} - 2765824 q^{88} + 6985286 i q^{89} - 2051280 q^{90} - 4036608 q^{92} + 8127444 i q^{93} - 4708088 q^{94} + 10520510 q^{95} + 1277952 i q^{96} - 200762 i q^{97} - 5901552 i q^{98} - 3597732 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 78 q^{3} - 128 q^{4} - 1332 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 78 q^{3} - 128 q^{4} - 1332 q^{9} + 6160 q^{10} + 4992 q^{12} + 4688 q^{14} + 8192 q^{16} + 42022 q^{17} + 86432 q^{22} + 126144 q^{23} - 140200 q^{25} + 222534 q^{27} + 244476 q^{29} - 240240 q^{30} - 225610 q^{35} + 85248 q^{36} - 437216 q^{38} - 394240 q^{40} - 182832 q^{42} + 404050 q^{43} - 319488 q^{48} + 1475388 q^{49} - 1638858 q^{51} + 3368672 q^{53} - 4159540 q^{55} - 300032 q^{56} - 2167216 q^{61} - 3334336 q^{62} - 524288 q^{64} - 3370848 q^{66} - 2689408 q^{68} - 4919616 q^{69} + 7078064 q^{74} + 5467800 q^{75} - 3165572 q^{77} - 13218512 q^{79} - 5765742 q^{81} + 928000 q^{82} - 9534564 q^{87} - 5531648 q^{88} - 4102560 q^{90} - 8073216 q^{92} - 9416176 q^{94} + 21041020 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
8.00000i −39.0000 −64.0000 385.000i 312.000i 293.000i 512.000i −666.000 3080.00
337.2 8.00000i −39.0000 −64.0000 385.000i 312.000i 293.000i 512.000i −666.000 3080.00
\(n\): e.g. 2-40 or 990-1000
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 338.8.b.b 2
13.b even 2 1 inner 338.8.b.b 2
13.d odd 4 1 26.8.a.a 1
13.d odd 4 1 338.8.a.c 1
39.f even 4 1 234.8.a.d 1
52.f even 4 1 208.8.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.a 1 13.d odd 4 1
208.8.a.c 1 52.f even 4 1
234.8.a.d 1 39.f even 4 1
338.8.a.c 1 13.d odd 4 1
338.8.b.b 2 1.a even 1 1 trivial
338.8.b.b 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 39 \) acting on \(S_{8}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 64 \) Copy content Toggle raw display
$3$ \( (T + 39)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 148225 \) Copy content Toggle raw display
$7$ \( T^{2} + 85849 \) Copy content Toggle raw display
$11$ \( T^{2} + 29181604 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 21011)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 746710276 \) Copy content Toggle raw display
$23$ \( (T - 63072)^{2} \) Copy content Toggle raw display
$29$ \( (T - 122238)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 43428892816 \) Copy content Toggle raw display
$37$ \( T^{2} + 195699179641 \) Copy content Toggle raw display
$41$ \( T^{2} + 3364000000 \) Copy content Toggle raw display
$43$ \( (T - 202025)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 346345197121 \) Copy content Toggle raw display
$53$ \( (T - 1684336)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 195921316900 \) Copy content Toggle raw display
$61$ \( (T + 1083608)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 11857595832196 \) Copy content Toggle raw display
$71$ \( T^{2} + 4345994937025 \) Copy content Toggle raw display
$73$ \( T^{2} + 35258537652100 \) Copy content Toggle raw display
$79$ \( (T + 6609256)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 20374707600 \) Copy content Toggle raw display
$89$ \( T^{2} + 48794220501796 \) Copy content Toggle raw display
$97$ \( T^{2} + 40305380644 \) Copy content Toggle raw display
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