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Properties

Label 304.4.i.g
Level $304$
Weight $4$
Character orbit 304.i
Analytic conductor $17.937$
Analytic rank $0$
Dimension $14$
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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 118 x^{12} - 188 x^{11} + 10300 x^{10} - 13520 x^{9} + 358384 x^{8} + \cdots + 11943936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{3} + (\beta_{8} + \beta_{7}) q^{5} + (\beta_{4} + \beta_{2} + 2) q^{7} + (\beta_{12} + 7 \beta_{7} - 2 \beta_{2} + \cdots - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1) q^{3} + (\beta_{8} + \beta_{7}) q^{5} + (\beta_{4} + \beta_{2} + 2) q^{7} + (\beta_{12} + 7 \beta_{7} - 2 \beta_{2} + \cdots - 7) q^{9}+ \cdots + (19 \beta_{13} - 25 \beta_{12} + \cdots + 272) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 5 q^{3} + 5 q^{5} + 28 q^{7} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 5 q^{3} + 5 q^{5} + 28 q^{7} - 46 q^{9} + 24 q^{11} - 65 q^{13} - 89 q^{15} - 37 q^{17} + 72 q^{19} - 134 q^{21} - 165 q^{23} - 64 q^{25} + 718 q^{27} - 175 q^{29} - 76 q^{31} + 100 q^{33} - 60 q^{35} + 672 q^{37} + 1078 q^{39} + 115 q^{41} - 289 q^{43} + 956 q^{45} - 421 q^{47} + 542 q^{49} + 125 q^{51} - 101 q^{53} - 66 q^{55} - 951 q^{57} - 513 q^{59} + 193 q^{61} - 56 q^{63} - 802 q^{65} - 107 q^{67} + 1222 q^{69} + 119 q^{71} + 295 q^{73} - 1576 q^{75} - 436 q^{77} + 1271 q^{79} - 907 q^{81} - 2120 q^{83} + 1405 q^{85} + 3034 q^{87} - 741 q^{89} - 2668 q^{91} + 2846 q^{93} + 17 q^{95} + 367 q^{97} + 1752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 118 x^{12} - 188 x^{11} + 10300 x^{10} - 13520 x^{9} + 358384 x^{8} + \cdots + 11943936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!93 \nu^{13} + \cdots + 54\!\cdots\!12 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 75\!\cdots\!21 \nu^{13} + \cdots + 45\!\cdots\!36 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\!\cdots\!73 \nu^{13} + \cdots + 65\!\cdots\!96 ) / 82\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!85 \nu^{13} + \cdots - 15\!\cdots\!20 ) / 45\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 75\!\cdots\!67 \nu^{13} + \cdots - 48\!\cdots\!56 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!43 \nu^{13} + \cdots - 30\!\cdots\!04 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 97\!\cdots\!95 \nu^{13} + \cdots + 17\!\cdots\!00 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!27 \nu^{13} + \cdots + 10\!\cdots\!00 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27\!\cdots\!95 \nu^{13} + \cdots - 13\!\cdots\!96 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61\!\cdots\!91 \nu^{13} + \cdots + 13\!\cdots\!68 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!73 \nu^{13} + \cdots + 12\!\cdots\!40 ) / 36\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38\!\cdots\!11 \nu^{13} + \cdots - 22\!\cdots\!64 ) / 33\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + 33\beta_{7} - 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 6\beta_{4} + 62\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{13} - 78 \beta_{12} + 6 \beta_{11} - 12 \beta_{10} - 6 \beta_{9} + 6 \beta_{8} + \cdots + 52 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 232\beta_{12} + 540\beta_{10} - 144\beta_{8} + 2112\beta_{7} + 144\beta_{6} - 4188\beta_{2} - 4188\beta _1 - 2112 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 540\beta_{9} - 252\beta_{6} + 5876\beta_{5} + 1536\beta_{4} + 684\beta_{3} + 8192\beta_{2} + 132432 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 168 \beta_{13} - 22440 \beta_{12} - 120 \beta_{11} - 42312 \beta_{10} - 168 \beta_{9} + \cdots + 296488 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 42552 \beta_{13} + 443560 \beta_{12} - 59544 \beta_{11} + 151920 \beta_{10} - 5688 \beta_{8} + \cdots - 9443328 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 32832 \beta_{9} - 1535616 \beta_{6} + 2026976 \beta_{5} + 3237072 \beta_{4} - 4032 \beta_{3} + \cdots + 30289344 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3245136 \beta_{13} - 33733296 \beta_{12} + 4747920 \beta_{11} - 13837440 \beta_{10} + \cdots + 87552256 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4341600 \beta_{13} + 176585824 \beta_{12} - 1239264 \beta_{11} + 247255776 \beta_{10} + \cdots - 2872325760 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 244777248 \beta_{9} - 34210080 \beta_{6} + 2585479520 \beta_{5} + 1213154880 \beta_{4} + \cdots + 51836084736 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 483290880 \beta_{13} - 15041648256 \beta_{12} + 272723712 \beta_{11} - 18960565440 \beta_{10} + \cdots + 121950190528 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(1\) \(1\)

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
−4.47208 + 7.74586i
−2.37870 + 4.12003i
−1.14086 + 1.97603i
−0.101755 + 0.176245i
1.53996 − 2.66730i
3.51998 − 6.09679i
4.03344 − 6.98613i
−4.47208 − 7.74586i
−2.37870 − 4.12003i
−1.14086 − 1.97603i
−0.101755 − 0.176245i
1.53996 + 2.66730i
3.51998 + 6.09679i
4.03344 + 6.98613i
0 −4.97208 + 8.61189i 0 −3.93649 + 6.81820i 0 21.1075 0 −35.9431 − 62.2552i 0
49.2 0 −2.87870 + 4.98605i 0 6.27866 − 10.8750i 0 −22.0011 0 −3.07380 − 5.32398i 0
49.3 0 −1.64086 + 2.84205i 0 −6.59079 + 11.4156i 0 −9.05159 0 8.11516 + 14.0559i 0
49.4 0 −0.601755 + 1.04227i 0 −0.973522 + 1.68619i 0 10.0882 0 12.7758 + 22.1283i 0
49.5 0 1.03996 − 1.80127i 0 8.43134 − 14.6035i 0 32.7147 0 11.3370 + 19.6362i 0
49.6 0 3.01998 − 5.23076i 0 −6.01619 + 10.4203i 0 3.03455 0 −4.74059 − 8.21095i 0
49.7 0 3.53344 − 6.12010i 0 5.30699 − 9.19198i 0 −21.8922 0 −11.4704 − 19.8673i 0
273.1 0 −4.97208 − 8.61189i 0 −3.93649 − 6.81820i 0 21.1075 0 −35.9431 + 62.2552i 0
273.2 0 −2.87870 − 4.98605i 0 6.27866 + 10.8750i 0 −22.0011 0 −3.07380 + 5.32398i 0
273.3 0 −1.64086 − 2.84205i 0 −6.59079 − 11.4156i 0 −9.05159 0 8.11516 − 14.0559i 0
273.4 0 −0.601755 − 1.04227i 0 −0.973522 − 1.68619i 0 10.0882 0 12.7758 − 22.1283i 0
273.5 0 1.03996 + 1.80127i 0 8.43134 + 14.6035i 0 32.7147 0 11.3370 − 19.6362i 0
273.6 0 3.01998 + 5.23076i 0 −6.01619 − 10.4203i 0 3.03455 0 −4.74059 + 8.21095i 0
273.7 0 3.53344 + 6.12010i 0 5.30699 + 9.19198i 0 −21.8922 0 −11.4704 + 19.8673i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.7
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.g 14
4.b odd 2 1 152.4.i.a ✓ 14
19.c even 3 1 inner 304.4.i.g 14
76.g odd 6 1 152.4.i.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.i.a ✓ 14 4.b odd 2 1
152.4.i.a ✓ 14 76.g odd 6 1
304.4.i.g 14 1.a even 1 1 trivial
304.4.i.g 14 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 5 T_{3}^{13} + 130 T_{3}^{12} + 189 T_{3}^{11} + 10011 T_{3}^{10} + 16674 T_{3}^{9} + \cdots + 403005625 \) acting on \(S_{4}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 403005625 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 29858793062400 \) Copy content Toggle raw display
$7$ \( (T^{7} - 14 T^{6} + \cdots + 92160000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} - 12 T^{6} + \cdots + 11349013440)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 811990100793600 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 71\!\cdots\!19 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 89858297474560)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 24203746164544)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 49\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 73\!\cdots\!49 \) Copy content Toggle raw display
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