Properties

Label 304.10.a.d
Level $304$
Weight $10$
Character orbit 304.a
Self dual yes
Analytic conductor $156.571$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,10,Mod(1,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

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: yes
Analytic conductor: \(156.570894194\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 8016x^{2} - 155839x + 2105804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 57) q^{3} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 218) q^{5} + ( - \beta_{3} - 9 \beta_{2} - 20 \beta_1 - 672) q^{7} + ( - 7 \beta_{3} + 53 \beta_{2} - 95 \beta_1 + 12678) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 57) q^{3} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 218) q^{5} + ( - \beta_{3} - 9 \beta_{2} - 20 \beta_1 - 672) q^{7} + ( - 7 \beta_{3} + 53 \beta_{2} - 95 \beta_1 + 12678) q^{9} + (23 \beta_{3} + 121 \beta_{2} - 75 \beta_1 - 29748) q^{11} + ( - 48 \beta_{3} + 238 \beta_{2} + 275 \beta_1 - 1568) q^{13} + ( - 49 \beta_{3} + 8 \beta_{2} + 489 \beta_1 - 88972) q^{15} + (164 \beta_{3} - 2050 \beta_{2} + 20 \beta_1 + 168631) q^{17} + 130321 q^{19} + ( - 119 \beta_{3} - 4492 \beta_{2} + 1230 \beta_1 + 394438) q^{21} + (286 \beta_{3} + 7382 \beta_{2} - 885 \beta_1 - 724276) q^{23} + (1171 \beta_{3} - 297 \beta_{2} + 3659 \beta_1 + 67043) q^{25} + (249 \beta_{3} - 18149 \beta_{2} + 14725 \beta_1 - 709589) q^{27} + ( - 2317 \beta_{3} + 852 \beta_{2} + 550 \beta_1 + 2073202) q^{29} + ( - 1575 \beta_{3} - 17818 \beta_{2} + 9775 \beta_1 - 870276) q^{31} + ( - 1075 \beta_{3} + 22986 \beta_{2} + 21855 \beta_1 - 1318954) q^{33} + ( - 5483 \beta_{3} + 18741 \beta_{2} + 6443 \beta_1 - 1919394) q^{35} + ( - 2407 \beta_{3} - 20754 \beta_{2} + 59215 \beta_1 - 261758) q^{37} + (6388 \beta_{3} + 45366 \beta_{2} - 11925 \beta_1 - 8491292) q^{39} + (6677 \beta_{3} + 39302 \beta_{2} - 68935 \beta_1 + 2148182) q^{41} + (12765 \beta_{3} + 6087 \beta_{2} + 40335 \beta_1 - 3472138) q^{43} + ( - 13557 \beta_{3} + 122429 \beta_{2} - 37513 \beta_1 - 2179336) q^{45} + ( - 26833 \beta_{3} + 132675 \beta_{2} - 15365 \beta_1 + 9150076) q^{47} + ( - 2112 \beta_{3} - 67764 \beta_{2} + 35580 \beta_1 - 7756834) q^{49} + ( - 24726 \beta_{3} - 98679 \beta_{2} - 181350 \beta_1 + 50917929) q^{51} + ( - 5549 \beta_{3} + 200314 \beta_{2} - 211250 \beta_1 - 28392182) q^{53} + ( - 17917 \beta_{3} - 74041 \beta_{2} + 113927 \beta_1 + 44498004) q^{55} + ( - 130321 \beta_{2} - 7428297) q^{57} + (23037 \beta_{3} + 37063 \beta_{2} - 255595 \beta_1 + 99211973) q^{59} + ( - 40083 \beta_{3} - 215807 \beta_{2} + 201455 \beta_1 - 74655920) q^{61} + (3235 \beta_{3} - 751 \beta_{2} - 178455 \beta_1 + 114680548) q^{63} + ( - 8146 \beta_{3} - 229918 \beta_{2} - 425824 \beta_1 - 72911878) q^{65} + (58278 \beta_{3} + 770537 \beta_{2} + 305650 \beta_1 + 28773199) q^{67} + (28060 \beta_{3} + 674842 \beta_{2} + 824925 \beta_1 - 167542508) q^{69} + ( - 94124 \beta_{3} + 627986 \beta_{2} + 669390 \beta_1 - 850912) q^{71} + ( - 55158 \beta_{3} - 207296 \beta_{2} - 709870 \beta_1 + 33945965) q^{73} + ( - 55069 \beta_{3} + 1312583 \beta_{2} - 315801 \beta_1 - 6670777) q^{75} + ( - 99511 \beta_{3} + 947013 \beta_{2} + 918985 \beta_1 + 30611478) q^{77} + ( - 39373 \beta_{3} - 402324 \beta_{2} + 62905 \beta_1 - 17015018) q^{79} + (83152 \beta_{3} + 3150996 \beta_{2} - 1435640 \beta_1 + 247135793) q^{81} + (59700 \beta_{3} + 740496 \beta_{2} + 199770 \beta_1 - 343756056) q^{83} + (379483 \beta_{3} + 640819 \beta_{2} + 1503517 \beta_1 + 159920954) q^{85} + (157552 \beta_{3} - 2979110 \beta_{2} - 199125 \beta_1 - 159148436) q^{87} + (16075 \beta_{3} + 144650 \beta_{2} + 1442325 \beta_1 + 389375140) q^{89} + (397111 \beta_{3} + 1181751 \beta_{2} - 871825 \beta_1 - 355102557) q^{91} + (34724 \beta_{3} + 2380754 \beta_{2} - 2907810 \beta_1 + 510361898) q^{93} + (130321 \beta_{3} + 390963 \beta_{2} - 130321 \beta_1 + 28409978) q^{95} + ( - 183388 \beta_{3} + 756606 \beta_{2} + \cdots + 244332850) q^{97}+ \cdots + ( - 91877 \beta_{3} + 3661725 \beta_{2} + \cdots - 123874300) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 226 q^{3} + 866 q^{5} - 2670 q^{7} + 50606 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 226 q^{3} + 866 q^{5} - 2670 q^{7} + 50606 q^{9} - 119234 q^{11} - 6748 q^{13} - 355904 q^{15} + 678624 q^{17} + 521284 q^{19} + 1586736 q^{21} - 2911868 q^{23} + 268766 q^{25} - 2802058 q^{27} + 8291104 q^{29} - 3445468 q^{31} - 5321788 q^{33} - 7715058 q^{35} - 1005524 q^{37} - 34055900 q^{39} + 8514124 q^{41} - 13900726 q^{43} - 8962202 q^{45} + 36334954 q^{47} - 30891808 q^{49} + 203869074 q^{51} - 113969356 q^{53} + 178140098 q^{55} - 29452546 q^{57} + 396773766 q^{59} - 298192066 q^{61} + 458723694 q^{63} - 291187676 q^{65} + 113551722 q^{67} - 671519716 q^{69} - 4659620 q^{71} + 136198452 q^{73} - 29308274 q^{75} + 120551886 q^{77} - 67255424 q^{79} + 982241180 q^{81} - 1376505216 q^{83} + 638402178 q^{85} - 630635524 q^{87} + 1557211260 q^{89} - 1422773730 q^{91} + 2036686084 q^{93} + 112857986 q^{95} + 975818188 q^{97} - 502820650 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 8016x^{2} - 155839x + 2105804 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 47\nu^{2} - 12893\nu - 62941 ) / 693 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{2} + 57\nu + 3970 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 179\nu^{2} + 22001\nu - 470801 ) / 693 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 4\beta_{2} + \beta _1 + 14 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 19\beta_{3} - 92\beta_{2} + 19\beta _1 + 32026 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3893\beta_{3} + 9650\beta_{2} + 8051\beta _1 + 1551688 ) / 12 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.19740
−31.9926
97.8976
−73.1023
0 −266.984 0 745.613 0 3114.51 0 51597.3 0
1.2 0 −110.471 0 −1298.23 0 −6626.40 0 −7479.16 0
1.3 0 −55.3919 0 2263.93 0 −5766.09 0 −16614.7 0
1.4 0 206.847 0 −845.315 0 6607.98 0 23102.6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.10.a.d 4
4.b odd 2 1 38.10.a.e 4
12.b even 2 1 342.10.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.e 4 4.b odd 2 1
304.10.a.d 4 1.a even 1 1 trivial
342.10.a.i 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 226T_{3}^{3} - 39131T_{3}^{2} - 8791740T_{3} - 337930956 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(304))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 226 T^{3} + \cdots - 337930956 \) Copy content Toggle raw display
$5$ \( T^{4} - 866 T^{3} + \cdots + 1852446724000 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 786353549326443 \) Copy content Toggle raw display
$11$ \( T^{4} + 119234 T^{3} + \cdots + 30\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{4} + 6748 T^{3} + \cdots - 79\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{4} - 678624 T^{3} + \cdots + 16\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( (T - 130321)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 2911868 T^{3} + \cdots - 60\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{4} - 8291104 T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + 3445468 T^{3} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{4} + 1005524 T^{3} + \cdots + 88\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{4} - 8514124 T^{3} + \cdots - 18\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{4} + 13900726 T^{3} + \cdots - 70\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{4} - 36334954 T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + 113969356 T^{3} + \cdots - 13\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{4} - 396773766 T^{3} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 298192066 T^{3} + \cdots - 32\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{4} - 113551722 T^{3} + \cdots - 45\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{4} + 4659620 T^{3} + \cdots - 13\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{4} - 136198452 T^{3} + \cdots + 14\!\cdots\!57 \) Copy content Toggle raw display
$79$ \( T^{4} + 67255424 T^{3} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + 1376505216 T^{3} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{4} - 1557211260 T^{3} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} - 975818188 T^{3} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
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