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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,4,Mod(293,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.293");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.d (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: \(17.3465615417\)
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: \( x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{8}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{2} q^{3} - 4 q^{4} - \beta_{12} q^{5} + \beta_1 q^{6} - 4 \beta_{3} q^{8} + (\beta_{14} - \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{2} q^{3} - 4 q^{4} - \beta_{12} q^{5} + \beta_1 q^{6} - 4 \beta_{3} q^{8} + (\beta_{14} - \beta_{3} - 2) q^{9} + ( - \beta_{10} - \beta_{4} + \beta_{2}) q^{10} + \beta_{7} q^{11} - 4 \beta_{2} q^{12} + (\beta_{10} - \beta_{9} + \beta_{6} + \cdots - \beta_1) q^{13}+ \cdots + ( - 24 \beta_{15} + 3 \beta_{14} + \cdots - 279) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} - 36 q^{9} + 256 q^{16} + 96 q^{18} + 24 q^{22} + 388 q^{25} - 720 q^{30} + 144 q^{36} + 1924 q^{37} - 1188 q^{39} + 1732 q^{43} - 336 q^{46} - 3276 q^{51} - 2664 q^{57} + 1560 q^{58} - 1024 q^{64} + 1412 q^{67} - 384 q^{72} + 2832 q^{78} + 5312 q^{79} - 252 q^{81} + 5232 q^{85} - 96 q^{88} - 4032 q^{93} - 4284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 359 \nu^{15} + 26957 \nu^{14} - 203453 \nu^{13} - 39814 \nu^{12} + 125514 \nu^{11} + \cdots - 328666497804 ) / 45036436104 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 212507 \nu^{15} + 5626388 \nu^{14} + 25621768 \nu^{13} + 99499205 \nu^{12} + \cdots + 21889683312741 ) / 8827141476384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 386047 \nu^{15} - 1698028 \nu^{14} - 7639460 \nu^{13} - 16796293 \nu^{12} + \cdots + 470964608523 ) / 4413570738192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 316769 \nu^{15} + 2119856 \nu^{14} - 16693244 \nu^{13} + 16032035 \nu^{12} + \cdots - 35154527200245 ) / 2942380492128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1885 \nu^{15} - 6534 \nu^{14} + 7202 \nu^{13} - 178985 \nu^{12} + 507040 \nu^{11} + \cdots + 6261969303 ) / 6368788944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7175 \nu^{15} + 52654 \nu^{14} + 10136 \nu^{13} - 31703 \nu^{12} - 462294 \nu^{11} + \cdots + 319261586427 ) / 15012145368 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2107411 \nu^{15} + 14963324 \nu^{14} + 66048316 \nu^{13} - 25091509 \nu^{12} + \cdots + 94333784924619 ) / 2942380492128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4626800 \nu^{15} + 2186215 \nu^{14} + 27869453 \nu^{13} + 191638681 \nu^{12} + \cdots + 112234457742453 ) / 4413570738192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 355015 \nu^{15} - 1366828 \nu^{14} - 1922912 \nu^{13} + 5844395 \nu^{12} + \cdots + 2323389370347 ) / 326931165792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 368573 \nu^{15} - 135865 \nu^{14} - 1693931 \nu^{13} - 5136598 \nu^{12} - 2816766 \nu^{11} + \cdots + 532076603436 ) / 315255052728 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2629535 \nu^{15} + 357328 \nu^{14} + 6200312 \nu^{13} + 73083961 \nu^{12} + \cdots - 4368118183785 ) / 2206785369096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1280285 \nu^{15} + 1318214 \nu^{14} - 7367714 \nu^{13} - 22903819 \nu^{12} + \cdots - 15169755356811 ) / 802467406944 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50759 \nu^{15} - 89566 \nu^{14} - 486710 \nu^{13} + 192305 \nu^{12} - 729666 \nu^{11} + \cdots - 489707469711 ) / 19106366832 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16023907 \nu^{15} + 4187999 \nu^{14} + 22163761 \nu^{13} + 308562482 \nu^{12} + \cdots + 147983176370214 ) / 4413570738192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 8014021 \nu^{15} + 2335208 \nu^{14} + 12518080 \nu^{13} + 103500293 \nu^{12} + \cdots + 115116186999015 ) / 2206785369096 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{13} + 5 \beta_{11} + 7 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} - 7 \beta_{4} - \beta_{3} + \cdots + 12 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 21 \beta_{12} + 14 \beta_{10} - 7 \beta_{9} - 7 \beta_{8} - 3 \beta_{5} + 7 \beta_{4} - 7 \beta_{3} + \cdots + 14 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{15} - 4\beta_{14} + 10\beta_{13} + 28\beta_{11} + 18\beta_{7} + 107\beta_{3} + 66 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21 \beta_{14} + 14 \beta_{13} + 21 \beta_{12} + 14 \beta_{11} + 35 \beta_{10} + 28 \beta_{9} + \cdots + 56 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{15} + 238 \beta_{14} + 190 \beta_{13} - 336 \beta_{12} + 8 \beta_{11} + 28 \beta_{10} + \cdots + 960 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 60\beta_{15} - 54\beta_{14} + 6\beta_{13} + 16\beta_{11} - 6\beta_{8} - 8\beta_{7} - 148\beta_{3} - 709 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2032 \beta_{15} - 1444 \beta_{14} + 1454 \beta_{13} - 7728 \beta_{12} - 1505 \beta_{11} + \cdots + 22048 ) / 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4046 \beta_{15} - 1162 \beta_{14} - 98 \beta_{13} + 6741 \beta_{12} - 1204 \beta_{11} - 1792 \beta_{10} + \cdots + 75320 ) / 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4340 \beta_{15} - 3808 \beta_{14} - 8102 \beta_{13} + 9300 \beta_{11} - 15172 \beta_{8} - 13954 \beta_{7} + \cdots + 555762 ) / 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 11508 \beta_{15} - 13839 \beta_{14} - 11984 \beta_{13} - 89733 \beta_{12} + 33824 \beta_{11} + \cdots + 11662 ) / 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 52781 \beta_{15} - 153922 \beta_{14} - 52562 \beta_{13} - 429408 \beta_{12} + 80584 \beta_{11} + \cdots + 1476812 ) / 84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 26092 \beta_{15} + 30512 \beta_{14} + 20352 \beta_{13} + 35244 \beta_{11} + 4688 \beta_{8} + \cdots + 642899 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 842744 \beta_{15} - 291088 \beta_{14} - 842210 \beta_{13} - 860832 \beta_{12} + 1358037 \beta_{11} + \cdots - 20153236 ) / 84 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 906976 \beta_{15} + 752248 \beta_{14} + 864976 \beta_{13} - 4539381 \beta_{12} + 1930740 \beta_{11} + \cdots - 29528338 ) / 42 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5604932 \beta_{15} - 13703140 \beta_{14} + 9361034 \beta_{13} - 1532188 \beta_{11} + 12716256 \beta_{8} + \cdots - 103021278 ) / 42 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.339489 − 2.98073i
−2.58777 − 1.51770i
2.99617 + 0.151487i
2.30541 − 1.91966i
−2.81518 − 1.03671i
−1.62928 + 2.51902i
−0.0204843 + 2.99993i
2.41164 + 1.78437i
0.339489 + 2.98073i
−2.58777 + 1.51770i
2.99617 − 0.151487i
2.30541 + 1.91966i
−2.81518 + 1.03671i
−1.62928 − 2.51902i
−0.0204843 − 2.99993i
2.41164 − 1.78437i
− 2.00000i −4.76510 − 2.07215i −4.00000 −8.55823 −4.14431 + 9.53020i 0 8.00000i 18.4124 + 19.7480i 17.1165i
293.2 − 2.00000i −4.51764 − 2.56729i −4.00000 10.5451 −5.13458 + 9.03527i 0 8.00000i 13.8181 + 23.1962i − 21.0903i
293.3 − 2.00000i −2.36753 + 4.62545i −4.00000 4.49068 9.25090 + 4.73506i 0 8.00000i −15.7896 − 21.9018i − 8.98135i
293.4 − 2.00000i −0.882945 + 5.12059i −4.00000 −19.8088 10.2412 + 1.76589i 0 8.00000i −25.4408 − 9.04239i 39.6177i
293.5 − 2.00000i 0.882945 − 5.12059i −4.00000 19.8088 −10.2412 − 1.76589i 0 8.00000i −25.4408 − 9.04239i − 39.6177i
293.6 − 2.00000i 2.36753 − 4.62545i −4.00000 −4.49068 −9.25090 − 4.73506i 0 8.00000i −15.7896 − 21.9018i 8.98135i
293.7 − 2.00000i 4.51764 + 2.56729i −4.00000 −10.5451 5.13458 − 9.03527i 0 8.00000i 13.8181 + 23.1962i 21.0903i
293.8 − 2.00000i 4.76510 + 2.07215i −4.00000 8.55823 4.14431 − 9.53020i 0 8.00000i 18.4124 + 19.7480i − 17.1165i
293.9 2.00000i −4.76510 + 2.07215i −4.00000 −8.55823 −4.14431 − 9.53020i 0 − 8.00000i 18.4124 − 19.7480i − 17.1165i
293.10 2.00000i −4.51764 + 2.56729i −4.00000 10.5451 −5.13458 − 9.03527i 0 − 8.00000i 13.8181 − 23.1962i 21.0903i
293.11 2.00000i −2.36753 − 4.62545i −4.00000 4.49068 9.25090 − 4.73506i 0 − 8.00000i −15.7896 + 21.9018i 8.98135i
293.12 2.00000i −0.882945 − 5.12059i −4.00000 −19.8088 10.2412 − 1.76589i 0 − 8.00000i −25.4408 + 9.04239i − 39.6177i
293.13 2.00000i 0.882945 + 5.12059i −4.00000 19.8088 −10.2412 + 1.76589i 0 − 8.00000i −25.4408 + 9.04239i 39.6177i
293.14 2.00000i 2.36753 + 4.62545i −4.00000 −4.49068 −9.25090 + 4.73506i 0 − 8.00000i −15.7896 + 21.9018i − 8.98135i
293.15 2.00000i 4.51764 − 2.56729i −4.00000 −10.5451 5.13458 + 9.03527i 0 − 8.00000i 13.8181 − 23.1962i − 21.0903i
293.16 2.00000i 4.76510 − 2.07215i −4.00000 8.55823 4.14431 + 9.53020i 0 − 8.00000i 18.4124 − 19.7480i 17.1165i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.16
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
21.c even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 597T_{5}^{6} + 92151T_{5}^{4} - 4819635T_{5}^{2} + 64448784 \) acting on \(S_{4}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{8} - 597 T^{6} + \cdots + 64448784)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 5391 T^{6} + \cdots + 49401285696)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 4551 T^{6} + \cdots + 16498888704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 22782 T^{6} + \cdots + 295892481600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 69773043356676)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 81\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 481 T^{3} + \cdots - 9036973466)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 433 T^{3} + \cdots + 966156928)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 45\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 33\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 25\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 353 T^{3} + \cdots + 1742477998)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1328 T^{3} + \cdots - 343833213419)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 42\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
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