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Properties

Label 350.10.c.p
Level $350$
Weight $10$
Character orbit 350.c
Analytic conductor $180.263$
Analytic rank $0$
Dimension $10$
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] = [350,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.99");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(180.262542657\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 130927 x^{8} + 5651134143 x^{6} + 89949011321029 x^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16 \beta_{5} q^{2} + ( - 19 \beta_{5} + \beta_1) q^{3} - 256 q^{4} + (16 \beta_{2} + 304) q^{6} + 2401 \beta_{5} q^{7} - 4096 \beta_{5} q^{8} + (\beta_{3} - 20 \beta_{2} - 6867) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 \beta_{5} q^{2} + ( - 19 \beta_{5} + \beta_1) q^{3} - 256 q^{4} + (16 \beta_{2} + 304) q^{6} + 2401 \beta_{5} q^{7} - 4096 \beta_{5} q^{8} + (\beta_{3} - 20 \beta_{2} - 6867) q^{9} + (\beta_{6} - \beta_{3} + 5 \beta_{2} + 16916) q^{11} + (4864 \beta_{5} - 256 \beta_1) q^{12} + ( - \beta_{9} + \beta_{8} + \cdots + 189 \beta_1) q^{13}+ \cdots + ( - 2286 \beta_{6} - 12215 \beta_{4} + \cdots - 553879874) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2560 q^{4} + 3072 q^{6} - 68710 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2560 q^{4} + 3072 q^{6} - 68710 q^{9} + 169166 q^{11} - 384160 q^{14} + 655360 q^{16} - 258788 q^{19} + 460992 q^{21} - 786432 q^{24} + 731328 q^{26} - 10983534 q^{29} + 3701604 q^{31} - 15563200 q^{34} + 17589760 q^{36} - 49998656 q^{39} + 15646924 q^{41} - 43306496 q^{44} + 79090976 q^{46} - 57648010 q^{49} - 88802104 q^{51} - 46483200 q^{54} + 98344960 q^{56} - 44519492 q^{59} - 257689872 q^{61} - 167772160 q^{64} + 75092800 q^{66} + 257810336 q^{69} - 349073186 q^{71} + 46812960 q^{74} + 66249728 q^{76} - 1180709434 q^{79} + 97584250 q^{81} - 118013952 q^{84} + 966018784 q^{86} - 2475604840 q^{89} + 109744908 q^{91} + 685004224 q^{94} + 201326592 q^{96} - 5548629278 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 130927 x^{8} + 5651134143 x^{6} + 89949011321029 x^{4} + \cdots + 12\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 377747 \nu^{8} + 55358966934 \nu^{6} + \cdots - 35\!\cdots\!00 ) / 55\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 377747 \nu^{8} - 55358966934 \nu^{6} + \cdots + 81\!\cdots\!60 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 515847013 \nu^{8} + 183298915408104 \nu^{6} + \cdots + 12\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 209005112 \nu^{9} + 30142883138949 \nu^{7} + \cdots + 24\!\cdots\!00 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1137437633 \nu^{8} - 130726553238318 \nu^{6} + \cdots - 35\!\cdots\!00 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8252105718293 \nu^{9} + \cdots - 65\!\cdots\!00 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 119733687037936 \nu^{9} + \cdots + 30\!\cdots\!00 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27134871911069 \nu^{9} + \cdots - 73\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 18\beta_{2} - 26189 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{9} - 36\beta_{8} - 59\beta_{7} - 462206\beta_{5} - 44631\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -711\beta_{6} + 2770\beta_{4} - 56895\beta_{3} - 2377514\beta_{2} + 1168398535 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 330085\beta_{9} + 2355957\beta_{8} + 4626379\beta_{7} + 61710362716\beta_{5} + 2252034935\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 73742688\beta_{6} - 177557810\beta_{4} + 3086283539\beta_{3} + 188871832174\beta_{2} - 58942565702361 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16101641625 \beta_{9} - 130184389350 \beta_{8} - 317359448583 \beta_{7} + \cdots - 118947809124799 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6546807123813 \beta_{6} + 9423714645210 \beta_{4} - 167369346985327 \beta_{3} + \cdots + 31\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 661779606894365 \beta_{9} + \cdots + 64\!\cdots\!67 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 240.017i
− 92.4713i
− 17.1406i
136.563i
214.066i
− 214.066i
− 136.563i
17.1406i
92.4713i
240.017i
− 16.0000i − 221.017i −256.000 0 −3536.28 − 2401.00i 4096.00i −29165.7 0
99.2 − 16.0000i − 73.4713i −256.000 0 −1175.54 − 2401.00i 4096.00i 14285.0 0
99.3 − 16.0000i 1.85943i −256.000 0 29.7509 − 2401.00i 4096.00i 19679.5 0
99.4 − 16.0000i 155.563i −256.000 0 2489.01 − 2401.00i 4096.00i −4516.84 0
99.5 − 16.0000i 233.066i −256.000 0 3729.06 − 2401.00i 4096.00i −34636.9 0
99.6 16.0000i − 233.066i −256.000 0 3729.06 2401.00i − 4096.00i −34636.9 0
99.7 16.0000i − 155.563i −256.000 0 2489.01 2401.00i − 4096.00i −4516.84 0
99.8 16.0000i − 1.85943i −256.000 0 29.7509 2401.00i − 4096.00i 19679.5 0
99.9 16.0000i 73.4713i −256.000 0 −1175.54 2401.00i − 4096.00i 14285.0 0
99.10 16.0000i 221.017i −256.000 0 −3536.28 2401.00i − 4096.00i −29165.7 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.10
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 350.10.c.p 10
5.b even 2 1 inner 350.10.c.p 10
5.c odd 4 1 350.10.a.q ✓ 5
5.c odd 4 1 350.10.a.t yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.10.a.q ✓ 5 5.c odd 4 1
350.10.a.t yes 5 5.c odd 4 1
350.10.c.p 10 1.a even 1 1 trivial
350.10.c.p 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 132770 T_{3}^{8} + 5838123745 T_{3}^{6} + 92034037296000 T_{3}^{4} + \cdots + 11\!\cdots\!16 \) acting on \(S_{10}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 256)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5764801)^{5} \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 20\!\cdots\!43)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 49\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 37\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 41\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
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