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Properties

Label 304.4.i.f
Level $304$
Weight $4$
Character orbit 304.i
Analytic conductor $17.937$
Analytic rank $0$
Dimension $10$
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] = [304,4,Mod(49,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), 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: \(17.9365806417\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + \cdots + 1016064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} + \beta_1) q^{3} + (\beta_{8} + \beta_{7} + \beta_{3}) q^{5} + ( - \beta_{5} + \beta_{2} + 2) q^{7} + (\beta_{9} - \beta_{6} - \beta_{4} + \cdots - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} + \beta_1) q^{3} + (\beta_{8} + \beta_{7} + \beta_{3}) q^{5} + ( - \beta_{5} + \beta_{2} + 2) q^{7} + (\beta_{9} - \beta_{6} - \beta_{4} + \cdots - 10) q^{9}+ \cdots + (23 \beta_{9} - 10 \beta_{8} + \cdots - 1056) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 7 q^{3} - 4 q^{5} + 20 q^{7} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 7 q^{3} - 4 q^{5} + 20 q^{7} - 50 q^{9} + 50 q^{11} - 56 q^{13} - 10 q^{15} + 32 q^{17} - 77 q^{19} + 126 q^{21} - 184 q^{23} - 121 q^{25} + 218 q^{27} + 352 q^{29} - 264 q^{31} + 83 q^{33} + 132 q^{35} - 640 q^{37} + 324 q^{39} - 57 q^{41} + 528 q^{43} - 232 q^{45} + 434 q^{47} + 2138 q^{49} + 242 q^{51} + 780 q^{53} - 598 q^{55} + 1482 q^{57} + 343 q^{59} + 536 q^{61} + 1568 q^{63} - 1988 q^{65} - 779 q^{67} - 1156 q^{69} - 474 q^{71} + 1453 q^{73} + 2994 q^{75} - 2956 q^{77} + 1968 q^{79} - 1097 q^{81} + 698 q^{83} - 2334 q^{85} - 8372 q^{87} + 380 q^{89} - 1348 q^{91} + 1684 q^{93} - 4312 q^{95} + 883 q^{97} - 5230 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + \cdots + 1016064 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1165047 \nu^{9} - 9250992 \nu^{8} + 143004918 \nu^{7} - 836736016 \nu^{6} + \cdots - 3174016519296 ) / 2610376990464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 393603239 \nu^{9} - 640410556 \nu^{8} + 34258666518 \nu^{7} - 65425267000 \nu^{6} + \cdots - 235285623828864 ) / 328907500798464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24604755 \nu^{9} + 228981360 \nu^{8} - 3539670190 \nu^{7} + 21170339856 \nu^{6} + \cdots - 127823859374976 ) / 7831130971392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45367449 \nu^{9} - 343433424 \nu^{8} + 5308908346 \nu^{7} - 30833262864 \nu^{6} + \cdots - 150545568650112 ) / 7831130971392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 430238075 \nu^{9} + 257384052 \nu^{8} - 36268548014 \nu^{7} + 37431389032 \nu^{6} + \cdots - 109947804333696 ) / 54817916799744 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65962847 \nu^{9} - 378416976 \nu^{8} + 5849695754 \nu^{7} - 35865870088 \nu^{6} + \cdots - 179178099194880 ) / 7831130971392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1518356977 \nu^{9} + 423607099 \nu^{8} + 133515913632 \nu^{7} - 29537922338 \nu^{6} + \cdots + 410923373422464 ) / 82226875199616 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 403452304 \nu^{9} - 295261329 \nu^{8} + 33030273168 \nu^{7} - 34821843230 \nu^{6} + \cdots - 116960641689024 ) / 13704479199936 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{6} - \beta_{5} + 36\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{5} + 2\beta_{4} + 66\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 82\beta_{9} - 24\beta_{8} - 82\beta_{6} - 82\beta_{4} - 2400\beta_{3} + 36\beta _1 - 2400 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 200 \beta_{9} - 48 \beta_{8} - 48 \beta_{7} - 316 \beta_{6} + 316 \beta_{5} + 4200 \beta_{3} + \cdots - 4836 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2160\beta_{7} + 6148\beta_{5} + 6268\beta_{4} + 5088\beta_{2} + 177264 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17864\beta_{9} + 3840\beta_{8} + 21664\beta_{6} - 17864\beta_{4} - 403536\beta_{3} + 361944\beta _1 - 403536 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 476728 \beta_{9} + 165792 \beta_{8} + 165792 \beta_{7} + 452440 \beta_{6} - 452440 \beta_{5} + \cdots - 548016 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 234432\beta_{7} - 1427536\beta_{5} + 1550048\beta_{4} + 27257712\beta_{2} + 36364896 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
4.20983 + 7.29165i
2.06508 + 3.57683i
0.576829 + 0.999096i
−1.41615 − 2.45284i
−4.43560 − 7.68268i
4.20983 − 7.29165i
2.06508 − 3.57683i
0.576829 − 0.999096i
−1.41615 + 2.45284i
−4.43560 + 7.68268i
0 −4.70983 + 8.15767i 0 −1.90769 + 3.30421i 0 −30.9131 0 −30.8651 − 53.4599i 0
49.2 0 −2.56508 + 4.44285i 0 7.98141 − 13.8242i 0 31.8733 0 0.340695 + 0.590101i 0
49.3 0 −1.07683 + 1.86512i 0 −10.6285 + 18.4091i 0 18.8370 0 11.1809 + 19.3659i 0
49.4 0 0.916150 − 1.58682i 0 2.48769 − 4.30880i 0 −19.2102 0 11.8213 + 20.4752i 0
49.5 0 3.93560 − 6.81665i 0 0.0670670 − 0.116164i 0 9.41300 0 −17.4778 − 30.2725i 0
273.1 0 −4.70983 − 8.15767i 0 −1.90769 − 3.30421i 0 −30.9131 0 −30.8651 + 53.4599i 0
273.2 0 −2.56508 − 4.44285i 0 7.98141 + 13.8242i 0 31.8733 0 0.340695 − 0.590101i 0
273.3 0 −1.07683 − 1.86512i 0 −10.6285 − 18.4091i 0 18.8370 0 11.1809 − 19.3659i 0
273.4 0 0.916150 + 1.58682i 0 2.48769 + 4.30880i 0 −19.2102 0 11.8213 − 20.4752i 0
273.5 0 3.93560 + 6.81665i 0 0.0670670 + 0.116164i 0 9.41300 0 −17.4778 + 30.2725i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.4.i.f 10
4.b odd 2 1 76.4.e.a ✓ 10
12.b even 2 1 684.4.k.c 10
19.c even 3 1 inner 304.4.i.f 10
76.f even 6 1 1444.4.a.g 5
76.g odd 6 1 76.4.e.a ✓ 10
76.g odd 6 1 1444.4.a.f 5
228.m even 6 1 684.4.k.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.e.a ✓ 10 4.b odd 2 1
76.4.e.a ✓ 10 76.g odd 6 1
304.4.i.f 10 1.a even 1 1 trivial
304.4.i.f 10 19.c even 3 1 inner
684.4.k.c 10 12.b even 2 1
684.4.k.c 10 228.m even 6 1
1444.4.a.f 5 76.g odd 6 1
1444.4.a.g 5 76.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 7 T_{3}^{9} + 117 T_{3}^{8} + 380 T_{3}^{7} + 7481 T_{3}^{6} + 28659 T_{3}^{5} + \cdots + 2253001 \) acting on \(S_{4}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 7 T^{9} + \cdots + 2253001 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots + 746496 \) Copy content Toggle raw display
$7$ \( (T^{5} - 10 T^{4} + \cdots - 3356160)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 25 T^{4} + \cdots + 14618352)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 261619974144 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{5} + 132 T^{4} + \cdots + 11017558272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 320 T^{4} + \cdots - 309775727920)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 98\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 138587802934320)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
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