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Properties

Label 336.7.bh.d
Level $336$
Weight $7$
Character orbit 336.bh
Analytic conductor $77.298$
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] = [336,7,Mod(145,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), degree \(2\), 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: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 18) q^{3} + (\beta_{6} + \beta_{2} - 25 \beta_1 - 24) q^{5} + ( - 2 \beta_{6} - 2 \beta_{5} + \cdots + 76) q^{7}+ \cdots + ( - 243 \beta_1 + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_1 + 18) q^{3} + (\beta_{6} + \beta_{2} - 25 \beta_1 - 24) q^{5} + ( - 2 \beta_{6} - 2 \beta_{5} + \cdots + 76) q^{7}+ \cdots + (2187 \beta_{7} + 243 \beta_{6} + \cdots + 17496) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} - 294 q^{5} + 656 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 108 q^{3} - 294 q^{5} + 656 q^{7} + 972 q^{9} + 314 q^{11} - 5292 q^{15} - 5532 q^{17} + 18234 q^{19} + 9342 q^{21} - 3928 q^{23} - 17038 q^{25} - 8300 q^{29} + 89508 q^{31} + 8478 q^{33} + 25860 q^{35} + 64706 q^{37} - 29106 q^{39} - 45740 q^{43} - 71442 q^{45} - 483276 q^{47} - 310684 q^{49} - 49788 q^{51} - 540974 q^{53} + 328212 q^{57} + 181770 q^{59} + 418224 q^{61} + 92826 q^{63} - 414204 q^{65} + 1158902 q^{67} - 1442344 q^{71} - 378666 q^{73} - 460026 q^{75} + 1065994 q^{77} - 611452 q^{79} - 236196 q^{81} - 275112 q^{85} - 112050 q^{87} - 989196 q^{89} - 304446 q^{91} + 805572 q^{93} + 591792 q^{95} + 152604 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 405518177 \nu^{7} + 12595814413 \nu^{6} + 65484675874 \nu^{5} + 2673339327091 \nu^{4} + \cdots + 12\!\cdots\!00 ) / 11\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 405518177 \nu^{7} - 12595814413 \nu^{6} - 65484675874 \nu^{5} - 2673339327091 \nu^{4} + \cdots - 12\!\cdots\!00 ) / 57\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 364310749 \nu^{7} + 24961807811 \nu^{6} + 79053499373 \nu^{5} + 5590904486327 \nu^{4} + \cdots + 14\!\cdots\!80 ) / 157007159483580 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23765159851 \nu^{7} + 369812106601 \nu^{6} - 8515763214551 \nu^{5} + \cdots + 16\!\cdots\!96 ) / 46\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51599812183 \nu^{7} - 94536525013 \nu^{6} + 9860632241606 \nu^{5} + 16211496774869 \nu^{4} + \cdots - 35\!\cdots\!70 ) / 57\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 323868198161 \nu^{7} - 69488833736 \nu^{6} + 51830673245317 \nu^{5} + 311126494638088 \nu^{4} + \cdots + 94\!\cdots\!60 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 357421335253 \nu^{7} + 1466129677768 \nu^{6} - 71066142961631 \nu^{5} + \cdots + 32\!\cdots\!40 ) / 23\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{5} - \beta_{4} - 3\beta_{3} + 424\beta _1 - 426 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 36\beta_{7} - 52\beta_{6} + 157\beta_{5} + 20\beta_{4} - 12\beta_{3} - 177\beta_{2} + 44\beta _1 - 2094 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 217\beta_{7} + 201\beta_{6} + 217\beta_{5} - 16\beta_{4} + 434\beta_{3} - 592\beta_{2} - 72043\beta _1 + 201 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6456 \beta_{7} + 6784 \beta_{6} - 27649 \beta_{5} - 10340 \beta_{4} + 9684 \beta_{3} - 6784 \beta_{2} + \cdots + 685434 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 130791 \beta_{7} - 25869 \beta_{6} - 175899 \beta_{5} + 34733 \beta_{4} + 43597 \beta_{3} + \cdots + 13826925 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 770076 \beta_{7} + 536388 \beta_{6} - 770076 \beta_{5} + 1306464 \beta_{4} - 1540152 \beta_{3} + \cdots + 536388 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−6.30797 − 10.9257i
2.26350 + 3.92050i
7.29767 + 12.6399i
−2.75320 − 4.76869i
−6.30797 + 10.9257i
2.26350 − 3.92050i
7.29767 − 12.6399i
−2.75320 + 4.76869i
0 13.5000 − 7.79423i 0 −165.302 − 95.4373i 0 103.254 + 327.089i 0 121.500 − 210.444i 0
145.2 0 13.5000 − 7.79423i 0 −57.9943 − 33.4830i 0 240.457 + 244.600i 0 121.500 − 210.444i 0
145.3 0 13.5000 − 7.79423i 0 22.3721 + 12.9165i 0 203.121 − 276.389i 0 121.500 − 210.444i 0
145.4 0 13.5000 − 7.79423i 0 53.9244 + 31.1333i 0 −218.833 − 264.124i 0 121.500 − 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −165.302 + 95.4373i 0 103.254 − 327.089i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 −57.9943 + 33.4830i 0 240.457 − 244.600i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 22.3721 − 12.9165i 0 203.121 + 276.389i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 53.9244 − 31.1333i 0 −218.833 + 264.124i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.d 8
4.b odd 2 1 21.7.f.a ✓ 8
7.d odd 6 1 inner 336.7.bh.d 8
12.b even 2 1 63.7.m.d 8
28.d even 2 1 147.7.f.d 8
28.f even 6 1 21.7.f.a ✓ 8
28.f even 6 1 147.7.d.b 8
28.g odd 6 1 147.7.d.b 8
28.g odd 6 1 147.7.f.d 8
84.j odd 6 1 63.7.m.d 8
84.j odd 6 1 441.7.d.c 8
84.n even 6 1 441.7.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a ✓ 8 4.b odd 2 1
21.7.f.a ✓ 8 28.f even 6 1
63.7.m.d 8 12.b even 2 1
63.7.m.d 8 84.j odd 6 1
147.7.d.b 8 28.f even 6 1
147.7.d.b 8 28.g odd 6 1
147.7.f.d 8 28.d even 2 1
147.7.f.d 8 28.g odd 6 1
336.7.bh.d 8 1.a even 1 1 trivial
336.7.bh.d 8 7.d odd 6 1 inner
441.7.d.c 8 84.j odd 6 1
441.7.d.c 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 294 T_{5}^{7} + 20487 T_{5}^{6} - 2447550 T_{5}^{5} - 72000675 T_{5}^{4} + \cdots + 422734160250000 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 422734160250000 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 99\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
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