Properties

Label 400.10.c.k
Level $400$
Weight $10$
Character orbit 400.c
Analytic conductor $206.014$
Analytic rank $0$
Dimension $4$
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] = [400,10,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(206.014334466\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 29 \beta_1) q^{3} + (35 \beta_{2} + 2821 \beta_1) q^{7} + ( - 58 \beta_{3} - 18881) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 29 \beta_1) q^{3} + (35 \beta_{2} + 2821 \beta_1) q^{7} + ( - 58 \beta_{3} - 18881) q^{9} + ( - 53 \beta_{3} + 50704) q^{11} + (60 \beta_{2} - 5343 \beta_1) q^{13} + (1804 \beta_{2} + 74195 \beta_1) q^{17} + (608 \beta_{3} + 137916) q^{19} + ( - 3836 \beta_{3} - 1559236) q^{21} + ( - 2705 \beta_{2} + 146321 \beta_1) q^{23} + ( - 5926 \beta_{2} - 2018342 \beta_1) q^{27} + ( - 4444 \beta_{3} + 4964378) q^{29} + ( - 12701 \beta_{3} - 2565740) q^{31} + (44556 \beta_{2} - 395184 \beta_1) q^{33} + (37592 \beta_{2} + 2751983 \beta_1) q^{37} + (3603 \beta_{3} - 1492212) q^{39} + ( - 7538 \beta_{3} - 20917978) q^{41} + (78353 \beta_{2} + 5848513 \beta_1) q^{43} + ( - 292213 \beta_{2} + 2927937 \beta_1) q^{47} + ( - 197470 \beta_{3} - 34598557) q^{49} + ( - 126511 \beta_{3} - 72107420) q^{51} + ( - 95548 \beta_{2} - 11596067 \beta_1) q^{53} + (208444 \beta_{2} + 25401164 \beta_1) q^{57} + (213402 \beta_{3} + 89119788) q^{59} + ( - 215888 \beta_{3} + 15912610) q^{61} + ( - 1315307 \beta_{2} - 124719301 \beta_1) q^{63} + ( - 638803 \beta_{2} + 22370157 \beta_1) q^{67} + ( - 67876 \beta_{3} + 78242764) q^{69} + (298987 \beta_{3} - 56159588) q^{71} + ( - 531916 \beta_{2} - 23323631 \beta_1) q^{73} + (1176588 \beta_{2} + 77739984 \beta_1) q^{77} + ( - 1432146 \beta_{3} - 95800664) q^{79} + (1048582 \beta_{3} + 71088149) q^{81} + ( - 203907 \beta_{2} + 4317609 \beta_1) q^{83} + (4448874 \beta_{2} - 12461838 \beta_1) q^{87} + ( - 1832076 \beta_{3} + 307533574) q^{89} + (17745 \beta_{3} - 13629588) q^{91} + ( - 4039056 \beta_{2} - 521481660 \beta_1) q^{93} + ( - 4842204 \beta_{2} + 249136367 \beta_1) q^{97} + ( - 1940139 \beta_{3} - 524523024) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 75524 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 75524 q^{9} + 202816 q^{11} + 551664 q^{19} - 6236944 q^{21} + 19857512 q^{29} - 10262960 q^{31} - 5968848 q^{39} - 83671912 q^{41} - 138394228 q^{49} - 288429680 q^{51} + 356479152 q^{59} + 63650440 q^{61} + 312971056 q^{69} - 224638352 q^{71} - 383202656 q^{79} + 284352596 q^{81} + 1230134296 q^{89} - 54518352 q^{91} - 2098092096 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40\nu^{3} + 440\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -80\nu^{3} + 880\nu ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} ) / 160 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{3} + 22\beta_{2} ) / 160 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
2.34521 2.34521i
−2.34521 2.34521i
−2.34521 + 2.34521i
2.34521 + 2.34521i
0 245.617i 0 0 0 12208.6i 0 −40644.5 0
49.2 0 129.617i 0 0 0 924.582i 0 2882.53 0
49.3 0 129.617i 0 0 0 924.582i 0 2882.53 0
49.4 0 245.617i 0 0 0 12208.6i 0 −40644.5 0
\(n\): e.g. 2-40 or 990-1000
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 400.10.c.k 4
4.b odd 2 1 200.10.c.c 4
5.b even 2 1 inner 400.10.c.k 4
5.c odd 4 1 80.10.a.i 2
5.c odd 4 1 400.10.a.n 2
20.d odd 2 1 200.10.c.c 4
20.e even 4 1 40.10.a.a 2
20.e even 4 1 200.10.a.e 2
40.i odd 4 1 320.10.a.m 2
40.k even 4 1 320.10.a.r 2
60.l odd 4 1 360.10.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.a.a 2 20.e even 4 1
80.10.a.i 2 5.c odd 4 1
200.10.a.e 2 20.e even 4 1
200.10.c.c 4 4.b odd 2 1
200.10.c.c 4 20.d odd 2 1
320.10.a.m 2 40.i odd 4 1
320.10.a.r 2 40.k even 4 1
360.10.a.i 2 60.l odd 4 1
400.10.a.n 2 5.c odd 4 1
400.10.c.k 4 1.a even 1 1 trivial
400.10.c.k 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 77128T_{3}^{2} + 1013530896 \) acting on \(S_{10}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 77128 T^{2} + \cdots + 1013530896 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 127415241562896 \) Copy content Toggle raw display
$11$ \( (T^{2} - 101408 T + 2175388416)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 156985964595216 \) Copy content Toggle raw display
$17$ \( T^{4} + 273150070600 T^{2} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} - 275832 T - 33027868144)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 686397240328 T^{2} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{2} - 9928756 T + 21864370578084)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5131480 T - 16130186713200)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 160073639323912 T^{2} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} + 41835956 T + 429561344293284)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 705839994162952 T^{2} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{2} - 178239576 T + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 31825220 T - 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 112319176 T - 94\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{2} + 191601328 T - 27\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{2} - 615067148 T - 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
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