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Properties

Label 350.8.c.o
Level $350$
Weight $8$
Character orbit 350.c
Analytic conductor $109.335$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,8,Mod(99,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.99");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), 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: \(109.334758919\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4947x^{6} + 6833025x^{4} + 2898935968x^{2} + 151950276864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta_{2} q^{2} + (\beta_{6} - 10 \beta_{2}) q^{3} - 64 q^{4} + (8 \beta_1 + 80) q^{6} - 343 \beta_{2} q^{7} - 512 \beta_{2} q^{8} + (2 \beta_{4} + 3 \beta_{3} + \cdots - 545) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_{2} q^{2} + (\beta_{6} - 10 \beta_{2}) q^{3} - 64 q^{4} + (8 \beta_1 + 80) q^{6} - 343 \beta_{2} q^{7} - 512 \beta_{2} q^{8} + (2 \beta_{4} + 3 \beta_{3} + \cdots - 545) q^{9}+ \cdots + ( - 3875 \beta_{4} - 11136 \beta_{3} + \cdots + 5794491) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 512 q^{4} + 672 q^{6} - 4420 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 512 q^{4} + 672 q^{6} - 4420 q^{9} + 2648 q^{11} + 21952 q^{14} + 32768 q^{16} - 23968 q^{19} - 28812 q^{21} - 43008 q^{24} - 279328 q^{26} + 111156 q^{29} + 34412 q^{31} + 290528 q^{34} + 282880 q^{36} + 264688 q^{39} - 537320 q^{41} - 169472 q^{44} - 2324800 q^{46} - 941192 q^{49} - 2695912 q^{51} - 995904 q^{54} - 1404928 q^{56} + 2831668 q^{59} - 8667820 q^{61} - 2097152 q^{64} - 10949120 q^{66} + 21549500 q^{69} - 6741632 q^{71} + 8126368 q^{74} + 1533952 q^{76} + 18073992 q^{79} - 6560048 q^{81} + 1843968 q^{84} + 5799360 q^{86} - 614656 q^{89} + 11976188 q^{91} + 13695808 q^{94} + 2752512 q^{96} + 47422456 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4947x^{6} + 6833025x^{4} + 2898935968x^{2} + 151950276864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1513\nu^{6} - 4706363\nu^{4} + 5312751623\nu^{2} + 8623352313648 ) / 125807290560 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3737\nu^{7} - 18443627\nu^{5} - 23971148041\nu^{3} - 7215339947632\nu ) / 45408044739456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8659\nu^{6} - 38071109\nu^{4} - 38287051111\nu^{2} - 5880321819756 ) / 7862955660 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 75037\nu^{6} + 263108087\nu^{4} + 129373148173\nu^{2} - 18918085702032 ) / 62903645280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -976531\nu^{7} - 962141081\nu^{5} + 9225597170621\nu^{3} + 10805676864153456\nu ) / 170280167772960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7933991\nu^{7} + 36505508341\nu^{5} + 43791333461399\nu^{3} + 14036321718447024\nu ) / 1362241342183680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1530263\nu^{7} + 6527842813\nu^{5} + 6334996598447\nu^{3} + 1642340356378152\nu ) / 24325738253280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 10\beta_{6} + \beta_{5} - 13\beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{4} + \beta_{3} + 206\beta _1 - 12470 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1049\beta_{7} + 16250\beta_{6} - 554\beta_{5} + 386777\beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -18505\beta_{4} - 13229\beta_{3} - 624138\beta _1 + 27322352 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5134501\beta_{7} - 97603714\beta_{6} + 2483443\beta_{5} - 3155727361\beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 34048084\beta_{4} + 22330875\beta_{3} + 916647032\beta _1 - 35890745581 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13813939693\beta_{7} + 292549527154\beta_{6} - 7080279445\beta_{5} + 10273384796737\beta_{2} ) / 30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
54.4803i
− 35.1621i
− 26.1112i
7.79308i
− 7.79308i
26.1112i
35.1621i
− 54.4803i
− 8.00000i − 61.8960i −64.0000 0 −495.168 343.000i 512.000i −1644.11 0
99.2 − 8.00000i − 8.12245i −64.0000 0 −64.9796 343.000i 512.000i 2121.03 0
99.3 − 8.00000i 36.1742i −64.0000 0 289.394 343.000i 512.000i 878.424 0
99.4 − 8.00000i 75.8442i −64.0000 0 606.753 343.000i 512.000i −3565.34 0
99.5 8.00000i − 75.8442i −64.0000 0 606.753 − 343.000i − 512.000i −3565.34 0
99.6 8.00000i − 36.1742i −64.0000 0 289.394 − 343.000i − 512.000i 878.424 0
99.7 8.00000i 8.12245i −64.0000 0 −64.9796 − 343.000i − 512.000i 2121.03 0
99.8 8.00000i 61.8960i −64.0000 0 −495.168 − 343.000i − 512.000i −1644.11 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
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 350.8.c.o 8
5.b even 2 1 inner 350.8.c.o 8
5.c odd 4 1 350.8.a.v ✓ 4
5.c odd 4 1 350.8.a.y yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.8.a.v ✓ 4 5.c odd 4 1
350.8.a.y yes 4 5.c odd 4 1
350.8.c.o 8 1.a even 1 1 trivial
350.8.c.o 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 10958T_{3}^{6} + 35297113T_{3}^{4} + 31119489756T_{3}^{2} + 1902578835600 \) acting on \(S_{8}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 1902578835600 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 117649)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 395322363347625)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 79\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 81\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 22\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 28\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 44\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 76\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
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