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Properties

Label 294.4.f.b
Level $294$
Weight $4$
Character orbit 294.f
Analytic conductor $17.347$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,4,Mod(215,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.215");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.f (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: \(17.3465615417\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 153x^{12} + 23345x^{8} - 9792x^{4} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 42)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{2} - \beta_{12} q^{3} - 4 \beta_{2} q^{4} + (\beta_{14} + 2 \beta_{13} + 2 \beta_{11}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{6}) q^{6}+ \cdots + (\beta_{10} + 4 \beta_{5} + 8 \beta_{2} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{2} - \beta_{12} q^{3} - 4 \beta_{2} q^{4} + (\beta_{14} + 2 \beta_{13} + 2 \beta_{11}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{6}) q^{6}+ \cdots + ( - 15 \beta_{10} - 15 \beta_{9} + \cdots + 489) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 60 q^{9} + 504 q^{15} - 128 q^{16} + 120 q^{18} - 480 q^{22} - 320 q^{25} - 312 q^{30} + 480 q^{36} + 592 q^{37} + 804 q^{39} - 3392 q^{43} - 1344 q^{46} - 504 q^{51} + 5064 q^{57} + 1488 q^{58} + 1008 q^{60} - 1024 q^{64} - 496 q^{67} - 480 q^{72} + 2160 q^{78} - 2824 q^{79} - 7872 q^{85} - 960 q^{88} + 1512 q^{93} + 8064 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 153x^{12} + 23345x^{8} - 9792x^{4} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 3748753\nu^{2} ) / 1213940 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 153\nu^{12} - 23345\nu^{8} + 3571785\nu^{4} - 1498176 ) / 1494080 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -40\nu^{14} - 8979\nu^{12} + 1377355\nu^{8} - 209614755\nu^{4} - 135382840\nu^{2} + 87922368 ) / 4855760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -40\nu^{14} + 8979\nu^{12} - 1377355\nu^{8} + 209614755\nu^{4} - 135382840\nu^{2} - 87922368 ) / 4855760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -153\nu^{14} + 23345\nu^{10} - 3562505\nu^{6} + 4096\nu^{2} ) / 482560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24697 \nu^{15} + 15912 \nu^{13} + 3758545 \nu^{11} - 2427880 \nu^{9} - 573563305 \nu^{7} + \cdots - 311194624 \nu ) / 155384320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24505 \nu^{15} + 31824 \nu^{13} - 3758545 \nu^{11} - 4855760 \nu^{9} + 573563305 \nu^{7} + \cdots - 156236288 \nu ) / 155384320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 206361 \nu^{15} + 196040 \nu^{13} + 31399025 \nu^{11} - 30068360 \nu^{9} + \cdots - 3843998720 \nu ) / 621537280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 111231\nu^{14} - 384\nu^{12} - 16971815\nu^{10} + 2589217295\nu^{6} - 2977792\nu^{2} - 701445632 ) / 38846080 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 110911 \nu^{14} + 68238 \nu^{12} - 16971815 \nu^{10} - 10411870 \nu^{8} + 2589217295 \nu^{6} + \cdots - 1826816 ) / 38846080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 204633 \nu^{15} + 393616 \nu^{13} - 31399025 \nu^{11} - 60136720 \nu^{9} + 4790604105 \nu^{7} + \cdots - 1929912832 \nu ) / 621537280 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 305149 \nu^{15} - 132392 \nu^{13} - 46433205 \nu^{11} + 20356840 \nu^{9} + 7084857325 \nu^{7} + \cdots + 2599220224 \nu ) / 621537280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 302653 \nu^{15} - 266320 \nu^{13} + 46433205 \nu^{11} + 40713680 \nu^{9} + \cdots + 1304967680 \nu ) / 621537280 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 253643 \nu^{15} + 329968 \nu^{13} + 38916115 \nu^{11} - 50425200 \nu^{9} + \cdots - 1617440256 \nu ) / 310768640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 255755 \nu^{15} - 164216 \nu^{13} + 38916115 \nu^{11} + 25212600 \nu^{9} + \cdots + 3221609472 \nu ) / 310768640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - 2\beta_{12} + \beta_{11} - 2\beta_{8} + \beta_{7} - 2\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - 20\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} - 2\beta_{14} + 9\beta_{13} + 9\beta_{12} + 13\beta_{11} + 13\beta_{8} - 11\beta_{7} - 11\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{10} + 13\beta_{9} + 13\beta_{3} + 459\beta_{2} + 459 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26 \beta_{15} - 52 \beta_{14} + 322 \beta_{13} - 161 \beta_{12} + 218 \beta_{11} + \cdots - 135 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -161\beta_{10} - 161\beta_{9} - 2908\beta_{5} + 161\beta_{4} - 161\beta_{2} - 161 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 644 \beta_{15} + 322 \beta_{14} - 1345 \beta_{13} + 2690 \beta_{12} - 1989 \beta_{11} + \cdots - 3334 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -663\beta_{4} + 663\beta_{3} + 23944\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3978 \beta_{15} - 3978 \beta_{14} + 24569 \beta_{13} + 24569 \beta_{12} + 16613 \beta_{11} + \cdots + 20591 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -24569\beta_{10} - 24569\beta_{9} - 443644\beta_{5} - 24569\beta_{3} - 24569\beta_{2} + 443644\beta _1 - 24569 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 49138 \beta_{15} + 98276 \beta_{14} - 410418 \beta_{13} + 205209 \beta_{12} - 606970 \beta_{11} + \cdots - 254347 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 303485\beta_{10} - 303485\beta_{9} - 303485\beta_{4} + 303485\beta_{2} - 10656603 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1213940 \beta_{15} + 606970 \beta_{14} - 3748753 \beta_{13} + 7497506 \beta_{12} + \cdots + 6283566 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -3748753\beta_{4} - 3748753\beta_{3} + 67691420\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 7497506 \beta_{15} + 7497506 \beta_{14} - 31310897 \beta_{13} - 31310897 \beta_{12} + \cdots + 38808403 \beta_{6} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.208289 − 0.777345i
0.909643 + 3.39483i
−0.909643 − 3.39483i
0.208289 + 0.777345i
−3.39483 + 0.909643i
0.777345 − 0.208289i
−0.777345 + 0.208289i
3.39483 − 0.909643i
−0.208289 + 0.777345i
0.909643 − 3.39483i
−0.909643 + 3.39483i
0.208289 − 0.777345i
−3.39483 − 0.909643i
0.777345 + 0.208289i
−0.777345 − 0.208289i
3.39483 + 0.909643i
−1.73205 − 1.00000i −4.91112 + 1.69732i 2.00000 + 3.46410i −7.07308 + 12.2509i 10.2036 + 1.97127i 0 − 8.00000i 21.2382 − 16.6715i 24.5019 − 14.1462i
215.2 −1.73205 − 1.00000i −0.368397 + 5.18308i 2.00000 + 3.46410i 5.69837 − 9.86987i 5.82116 − 8.60895i 0 − 8.00000i −26.7286 − 3.81886i −19.7397 + 11.3967i
215.3 −1.73205 − 1.00000i 0.368397 − 5.18308i 2.00000 + 3.46410i −5.69837 + 9.86987i −5.82116 + 8.60895i 0 − 8.00000i −26.7286 − 3.81886i 19.7397 − 11.3967i
215.4 −1.73205 − 1.00000i 4.91112 − 1.69732i 2.00000 + 3.46410i 7.07308 − 12.2509i −10.2036 − 1.97127i 0 − 8.00000i 21.2382 − 16.6715i −24.5019 + 14.1462i
215.5 1.73205 + 1.00000i −4.67287 − 2.27250i 2.00000 + 3.46410i −5.69837 + 9.86987i −5.82116 − 8.60895i 0 8.00000i 16.6715 + 21.2382i −19.7397 + 11.3967i
215.6 1.73205 + 1.00000i −3.92548 + 3.40449i 2.00000 + 3.46410i 7.07308 − 12.2509i −10.2036 + 1.97127i 0 8.00000i 3.81886 − 26.7286i 24.5019 − 14.1462i
215.7 1.73205 + 1.00000i 3.92548 − 3.40449i 2.00000 + 3.46410i −7.07308 + 12.2509i 10.2036 − 1.97127i 0 8.00000i 3.81886 − 26.7286i −24.5019 + 14.1462i
215.8 1.73205 + 1.00000i 4.67287 + 2.27250i 2.00000 + 3.46410i 5.69837 − 9.86987i 5.82116 + 8.60895i 0 8.00000i 16.6715 + 21.2382i 19.7397 − 11.3967i
227.1 −1.73205 + 1.00000i −4.91112 − 1.69732i 2.00000 − 3.46410i −7.07308 − 12.2509i 10.2036 − 1.97127i 0 8.00000i 21.2382 + 16.6715i 24.5019 + 14.1462i
227.2 −1.73205 + 1.00000i −0.368397 − 5.18308i 2.00000 − 3.46410i 5.69837 + 9.86987i 5.82116 + 8.60895i 0 8.00000i −26.7286 + 3.81886i −19.7397 − 11.3967i
227.3 −1.73205 + 1.00000i 0.368397 + 5.18308i 2.00000 − 3.46410i −5.69837 − 9.86987i −5.82116 − 8.60895i 0 8.00000i −26.7286 + 3.81886i 19.7397 + 11.3967i
227.4 −1.73205 + 1.00000i 4.91112 + 1.69732i 2.00000 − 3.46410i 7.07308 + 12.2509i −10.2036 + 1.97127i 0 8.00000i 21.2382 + 16.6715i −24.5019 − 14.1462i
227.5 1.73205 − 1.00000i −4.67287 + 2.27250i 2.00000 − 3.46410i −5.69837 − 9.86987i −5.82116 + 8.60895i 0 − 8.00000i 16.6715 − 21.2382i −19.7397 − 11.3967i
227.6 1.73205 − 1.00000i −3.92548 − 3.40449i 2.00000 − 3.46410i 7.07308 + 12.2509i −10.2036 − 1.97127i 0 − 8.00000i 3.81886 + 26.7286i 24.5019 + 14.1462i
227.7 1.73205 − 1.00000i 3.92548 + 3.40449i 2.00000 − 3.46410i −7.07308 − 12.2509i 10.2036 + 1.97127i 0 − 8.00000i 3.81886 + 26.7286i −24.5019 − 14.1462i
227.8 1.73205 − 1.00000i 4.67287 − 2.27250i 2.00000 − 3.46410i 5.69837 + 9.86987i 5.82116 − 8.60895i 0 − 8.00000i 16.6715 − 21.2382i 19.7397 + 11.3967i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.4.f.b 16
3.b odd 2 1 inner 294.4.f.b 16
7.b odd 2 1 inner 294.4.f.b 16
7.c even 3 1 42.4.d.a ✓ 8
7.c even 3 1 inner 294.4.f.b 16
7.d odd 6 1 42.4.d.a ✓ 8
7.d odd 6 1 inner 294.4.f.b 16
21.c even 2 1 inner 294.4.f.b 16
21.g even 6 1 42.4.d.a ✓ 8
21.g even 6 1 inner 294.4.f.b 16
21.h odd 6 1 42.4.d.a ✓ 8
21.h odd 6 1 inner 294.4.f.b 16
28.f even 6 1 336.4.k.c 8
28.g odd 6 1 336.4.k.c 8
84.j odd 6 1 336.4.k.c 8
84.n even 6 1 336.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.d.a ✓ 8 7.c even 3 1
42.4.d.a ✓ 8 7.d odd 6 1
42.4.d.a ✓ 8 21.g even 6 1
42.4.d.a ✓ 8 21.h odd 6 1
294.4.f.b 16 1.a even 1 1 trivial
294.4.f.b 16 3.b odd 2 1 inner
294.4.f.b 16 7.b odd 2 1 inner
294.4.f.b 16 7.c even 3 1 inner
294.4.f.b 16 7.d odd 6 1 inner
294.4.f.b 16 21.c even 2 1 inner
294.4.f.b 16 21.g even 6 1 inner
294.4.f.b 16 21.h odd 6 1 inner
336.4.k.c 8 28.f even 6 1
336.4.k.c 8 28.g odd 6 1
336.4.k.c 8 84.j odd 6 1
336.4.k.c 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 330T_{5}^{6} + 82908T_{5}^{4} + 8577360T_{5}^{2} + 675584064 \) acting on \(S_{4}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{8} + 330 T^{6} + \cdots + 675584064)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 1032386052096)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2958 T^{2} + 1002528)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 20352513413376)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19764 T^{2} + 54997056)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 74 T + 5476)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 94740 T^{2} + 10071072)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 424 T - 33968)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 98\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 81\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 71\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 124 T^{3} + \cdots + 1183744)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 607716 T^{2} + 12745506816)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 706 T^{3} + \cdots + 611869696)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1502862 T^{2} + 422927723808)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1338360 T^{2} + 94643822592)^{4} \) Copy content Toggle raw display
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