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Properties

Label 5.14.b.a
Level $5$
Weight $14$
Character orbit 5.b
Analytic conductor $5.362$
Analytic rank $0$
Dimension $6$
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] = [5,14,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 5.b (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: \(5.36154644760\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9276x^{4} + 17959899x^{2} + 616730624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4}\cdot 5^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 5492) q^{4} + (\beta_{4} + 5 \beta_{2} + 90 \beta_1 + 4095) q^{5} + (\beta_{5} - 5 \beta_{4} + \cdots + 19492) q^{6}+ \cdots + (3 \beta_{5} - 15 \beta_{4} + \cdots - 1182273) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 5492) q^{4} + (\beta_{4} + 5 \beta_{2} + 90 \beta_1 + 4095) q^{5} + (\beta_{5} - 5 \beta_{4} + \cdots + 19492) q^{6}+ \cdots + (51598602 \beta_{5} + \cdots - 5152581102636) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 32952 q^{4} + 24570 q^{5} + 116952 q^{6} - 7093638 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 32952 q^{4} + 24570 q^{5} + 116952 q^{6} - 7093638 q^{9} + 7569720 q^{10} + 6217992 q^{11} + 56525256 q^{14} - 98916360 q^{15} - 7721184 q^{16} - 60021000 q^{19} - 243910440 q^{20} + 1000810872 q^{21} - 3917436000 q^{24} - 1377188850 q^{25} + 10130482512 q^{26} + 2395934100 q^{29} - 27051397560 q^{30} + 19077466752 q^{31} - 33499247424 q^{34} - 31251529080 q^{35} + 135136680696 q^{36} - 91653753456 q^{39} - 106750428000 q^{40} + 165295569132 q^{41} - 58123052064 q^{44} - 136158095610 q^{45} + 175967457672 q^{46} + 153339894258 q^{49} - 188342679600 q^{50} - 143598281088 q^{51} + 61323922800 q^{54} + 657500853240 q^{55} - 584547213600 q^{56} - 371496936600 q^{59} + 1987994985120 q^{60} - 1897108304508 q^{61} + 2574945548928 q^{64} + 21533145840 q^{65} - 5935468419936 q^{66} + 2439927663864 q^{69} + 2248910887320 q^{70} + 576526386672 q^{71} - 3259347984 q^{74} + 3962457334800 q^{75} - 2243925362400 q^{76} - 6035582613600 q^{79} - 6270647352480 q^{80} + 3505417725726 q^{81} - 2765147233824 q^{84} - 1561112311680 q^{85} + 27540813895992 q^{86} - 9947300393700 q^{89} - 29614620301560 q^{90} + 11770479000432 q^{91} + 15588924740136 q^{94} - 16585499115000 q^{95} + 25919517958272 q^{96} - 30915486615816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9276x^{4} + 17959899x^{2} + 616730624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 34300\nu^{3} + 148003099\nu ) / 21420000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 34300\nu^{3} + 469303099\nu ) / 10710000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{4} - 32200\nu^{2} - 32729896 ) / 2625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -189\nu^{5} + 1088\nu^{4} - 1722700\nu^{3} + 3998400\nu^{2} - 3187265711\nu - 5815388288 ) / 476000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42529 \nu^{5} - 277440 \nu^{4} - 387744700 \nu^{3} - 1162392000 \nu^{2} - 719011997371 \nu + 1041386413440 ) / 21420000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 5\beta_{4} - 16\beta_{3} + 2\beta_{2} - 309200 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 15\beta_{4} - 3\beta_{3} - 52064\beta_{2} + 359290\beta_1 ) / 300 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -322\beta_{5} + 1610\beta_{4} + 2527\beta_{3} - 644\beta_{2} + 66832504 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -10290\beta_{5} - 51450\beta_{4} + 10290\beta_{3} + 30576421\beta_{2} - 293758502\beta_1 ) / 30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
51.9014i
− 80.9153i
5.91342i
− 5.91342i
80.9153i
− 51.9014i
− 152.321i 2014.00i −15009.6 −14163.9 + 31938.8i 306775. 69973.1i 1.03846e6i −2.46189e6 4.86494e6 + 2.15745e6i
4.2 − 127.310i − 2045.53i −8015.83 34862.0 + 2311.33i −260416. 258383.i − 22427.7i −2.58986e6 294255. − 4.43828e6i
4.3 − 40.5284i 298.988i 6549.45 −8413.14 − 33910.5i 12117.5 − 377278.i − 597447.i 1.50493e6 −1.37434e6 + 340971.i
4.4 40.5284i − 298.988i 6549.45 −8413.14 + 33910.5i 12117.5 377278.i 597447.i 1.50493e6 −1.37434e6 − 340971.i
4.5 127.310i 2045.53i −8015.83 34862.0 − 2311.33i −260416. − 258383.i 22427.7i −2.58986e6 294255. + 4.43828e6i
4.6 152.321i − 2014.00i −15009.6 −14163.9 − 31938.8i 306775. − 69973.1i − 1.03846e6i −2.46189e6 4.86494e6 − 2.15745e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.14.b.a ✓ 6
3.b odd 2 1 45.14.b.b 6
4.b odd 2 1 80.14.c.c 6
5.b even 2 1 inner 5.14.b.a ✓ 6
5.c odd 4 2 25.14.a.e 6
15.d odd 2 1 45.14.b.b 6
20.d odd 2 1 80.14.c.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.14.b.a ✓ 6 1.a even 1 1 trivial
5.14.b.a ✓ 6 5.b even 2 1 inner
25.14.a.e 6 5.c odd 4 2
45.14.b.b 6 3.b odd 2 1
45.14.b.b 6 15.d odd 2 1
80.14.c.c 6 4.b odd 2 1
80.14.c.c 6 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots + 617678127104 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{3} + \cdots - 39\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 18\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 85\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 18\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 50\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
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