Properties

Label 304.10.a.e
Level $304$
Weight $10$
Character orbit 304.a
Self dual yes
Analytic conductor $156.571$
Analytic rank $0$
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: \(0\)
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: \( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 21) q^{3} + ( - 4 \beta_{3} - \beta_{2} - 5 \beta_1 - 350) q^{5} + (9 \beta_{3} - 2 \beta_{2} - 27 \beta_1 - 3075) q^{7} + (16 \beta_{3} - 18 \beta_{2} + 135 \beta_1 + 4134) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 21) q^{3} + ( - 4 \beta_{3} - \beta_{2} - 5 \beta_1 - 350) q^{5} + (9 \beta_{3} - 2 \beta_{2} - 27 \beta_1 - 3075) q^{7} + (16 \beta_{3} - 18 \beta_{2} + 135 \beta_1 + 4134) q^{9} + ( - 25 \beta_{2} + 3 \beta_1 + 26056) q^{11} + ( - 169 \beta_{3} - 33 \beta_{2} + 766 \beta_1 + 30071) q^{13} + (54 \beta_{3} - 264 \beta_{2} + 1380 \beta_1 + 147720) q^{15} + (234 \beta_{3} - 587 \beta_{2} - 276 \beta_1 - 103123) q^{17} - 130321 q^{19} + (125 \beta_{3} - 375 \beta_{2} + 2976 \beta_1 + 609189) q^{21} + ( - 93 \beta_{3} - 1531 \beta_{2} + 678 \beta_1 - 752981) q^{23} + (1350 \beta_{3} + 1845 \beta_{2} + 6795 \beta_1 + 2440945) q^{25} + ( - 3948 \beta_{3} + 1674 \beta_{2} - 12769 \beta_1 - 3097563) q^{27} + ( - 3481 \beta_{3} - 1027 \beta_{2} - 1436 \beta_1 + 1537183) q^{29} + (2150 \beta_{3} + 2050 \beta_{2} - 9482 \beta_1 - 3192456) q^{31} + ( - 2198 \beta_{3} - 1596 \beta_{2} - 38998 \beta_1 - 815454) q^{33} + (20156 \beta_{3} - 10051 \beta_{2} + 20875 \beta_1 - 2453330) q^{35} + (14570 \beta_{3} - 6770 \beta_{2} - 40262 \beta_1 + 5128462) q^{37} + ( - 12559 \beta_{3} + 7047 \beta_{2} - 93298 \beta_1 - 17471739) q^{39} + ( - 28480 \beta_{3} - 19020 \beta_{2} - 11750 \beta_1 + 2893200) q^{41} + ( - 25660 \beta_{3} - 24965 \beta_{2} + 59233 \beta_1 - 1937238) q^{43} + (33138 \beta_{3} + 28557 \beta_{2} - 352695 \beta_1 - 30988530) q^{45} + ( - 32602 \beta_{3} + 31111 \beta_{2} - 45347 \beta_1 + 7894898) q^{47} + ( - 100318 \beta_{3} + 12239 \beta_{2} + 110482 \beta_1 + 4720576) q^{49} + ( - 49576 \beta_{3} - 37392 \beta_{2} - 217655 \beta_1 + 2126961) q^{51} + ( - 78901 \beta_{3} - 35357 \beta_{2} + 259234 \beta_1 + 18108791) q^{53} + ( - 51834 \beta_{3} - 18501 \beta_{2} - 321855 \beta_1 - 5350710) q^{55} + (130321 \beta_1 + 2736741) q^{57} + (8236 \beta_{3} + 66072 \beta_{2} + 25343 \beta_1 + 37327107) q^{59} + (44408 \beta_{3} - 115309 \beta_{2} - 7739 \beta_1 + 32233368) q^{61} + ( - 258888 \beta_{3} + 71559 \beta_{2} - 636137 \beta_1 - 25788720) q^{63} + ( - 112430 \beta_{3} + 309130 \beta_{2} + \cdots + 31222100) q^{65}+ \cdots + (519682 \beta_{3} - 374571 \beta_{2} + 4927511 \beta_1 + 420494730) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9} + 104249 q^{11} + 120486 q^{13} + 591090 q^{15} - 412139 q^{17} - 521284 q^{19} + 2437006 q^{21} - 3010300 q^{23} + 9760585 q^{25} - 12387978 q^{27} + 6153240 q^{29} - 12774024 q^{31} - 3258022 q^{33} - 9823425 q^{35} + 20506048 q^{37} - 69881444 q^{39} + 11620300 q^{41} - 7698327 q^{43} - 124015815 q^{45} + 31581083 q^{47} + 18970383 q^{49} + 8594812 q^{51} + 72549422 q^{53} - 21332505 q^{55} + 10946964 q^{57} + 149234120 q^{59} + 129004373 q^{61} - 102967551 q^{63} + 124691700 q^{65} - 132595266 q^{67} - 45529972 q^{69} + 47138482 q^{71} - 39332795 q^{73} - 824627010 q^{75} - 165933719 q^{77} + 307010840 q^{79} + 1305551744 q^{81} + 746568232 q^{83} - 105005985 q^{85} + 82148208 q^{87} + 286943482 q^{89} - 3155781114 q^{91} + 1151901596 q^{93} + 181797795 q^{95} + 793519958 q^{97} + 1681833809 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{3} - 736\nu^{2} - 252737\nu - 5880008 ) / 20632 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -17\nu^{3} + 244\nu^{2} + 698613\nu + 30780440 ) / 20632 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{3} - 490\nu^{2} - 413779\nu - 18350856 ) / 10316 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 193\beta_{3} + 85\beta_{2} - 397\beta _1 + 103370 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 43865\beta_{3} + 35033\beta_{2} - 46793\beta _1 + 12429538 ) / 6 \) 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
−67.1081
−26.2676
219.264
−124.888
0 −265.578 0 −2367.11 0 −5859.40 0 50848.5 0
1.2 0 −25.2570 0 2126.71 0 −11469.0 0 −19045.1 0
1.3 0 66.6053 0 −2418.56 0 1533.95 0 −15246.7 0
1.4 0 140.229 0 1263.95 0 3487.42 0 −18.7042 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.e 4
4.b odd 2 1 38.10.a.d 4
12.b even 2 1 342.10.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.d 4 4.b odd 2 1
304.10.a.e 4 1.a even 1 1 trivial
342.10.a.l 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 84T_{3}^{3} - 44107T_{3}^{2} + 1329018T_{3} + 62650008 \) 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
$3$ \( T^{4} + 84 T^{3} - 44107 T^{2} + \cdots + 62650008 \) Copy content Toggle raw display
$5$ \( T^{4} + 1395 T^{3} + \cdots + 15389064288000 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 359494671206216 \) Copy content Toggle raw display
$11$ \( T^{4} - 104249 T^{3} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{4} - 120486 T^{3} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{4} + 412139 T^{3} + \cdots + 27\!\cdots\!62 \) Copy content Toggle raw display
$19$ \( (T + 130321)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 3010300 T^{3} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{4} - 6153240 T^{3} + \cdots - 34\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{4} + 12774024 T^{3} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{4} - 20506048 T^{3} + \cdots - 58\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{4} - 11620300 T^{3} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + 7698327 T^{3} + \cdots - 19\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{4} - 31581083 T^{3} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} - 72549422 T^{3} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{4} - 149234120 T^{3} + \cdots - 27\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{4} - 129004373 T^{3} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + 132595266 T^{3} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{4} - 47138482 T^{3} + \cdots - 32\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{4} + 39332795 T^{3} + \cdots + 35\!\cdots\!34 \) Copy content Toggle raw display
$79$ \( T^{4} - 307010840 T^{3} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} - 746568232 T^{3} + \cdots - 94\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{4} - 286943482 T^{3} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{4} - 793519958 T^{3} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
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