Properties

Label 980.4.i.w
Level $980$
Weight $4$
Character orbit 980.i
Analytic conductor $57.822$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,4,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), degree \(2\), 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: \(57.8218718056\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 22x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (5 \beta_{2} + \beta_1 + 5) q^{3} - 5 \beta_{2} q^{5} + (10 \beta_{3} + 20 \beta_{2} + 10 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta_{2} + \beta_1 + 5) q^{3} - 5 \beta_{2} q^{5} + (10 \beta_{3} + 20 \beta_{2} + 10 \beta_1) q^{9} + ( - 26 \beta_{2} - 2 \beta_1 - 26) q^{11} + (14 \beta_{3} + 26) q^{13} + ( - 5 \beta_{3} + 25) q^{15} + ( - 38 \beta_{2} - 14 \beta_1 - 38) q^{17} + ( - 2 \beta_{3} - 88 \beta_{2} - 2 \beta_1) q^{19} + (3 \beta_{3} + 51 \beta_{2} + 3 \beta_1) q^{23} + ( - 25 \beta_{2} - 25) q^{25} + (43 \beta_{3} - 185) q^{27} + ( - 12 \beta_{3} - 153) q^{29} + (122 \beta_{2} - 12 \beta_1 + 122) q^{31} + ( - 36 \beta_{3} - 174 \beta_{2} - 36 \beta_1) q^{33} + ( - 48 \beta_{3} - 88 \beta_{2} - 48 \beta_1) q^{37} + ( - 178 \beta_{2} - 44 \beta_1 - 178) q^{39} + ( - 62 \beta_{3} + 65) q^{41} + (95 \beta_{3} + 57) q^{43} + (100 \beta_{2} + 50 \beta_1 + 100) q^{45} + (20 \beta_{3} - 358 \beta_{2} + 20 \beta_1) q^{47} + ( - 108 \beta_{3} - 498 \beta_{2} - 108 \beta_1) q^{51} + ( - 404 \beta_{2} + 22 \beta_1 - 404) q^{53} + (10 \beta_{3} - 130) q^{55} + ( - 98 \beta_{3} + 484) q^{57} + ( - 508 \beta_{2} + 38 \beta_1 - 508) q^{59} + ( - 94 \beta_{3} + 111 \beta_{2} - 94 \beta_1) q^{61} + (70 \beta_{3} - 130 \beta_{2} + 70 \beta_1) q^{65} + ( - 67 \beta_{2} - 53 \beta_1 - 67) q^{67} + (66 \beta_{3} - 321) q^{69} + (146 \beta_{3} + 148) q^{71} + (362 \beta_{2} + 150 \beta_1 + 362) q^{73} + ( - 25 \beta_{3} - 125 \beta_{2} - 25 \beta_1) q^{75} + (82 \beta_{3} + 564 \beta_{2} + 82 \beta_1) q^{79} + ( - 1331 \beta_{2} - 130 \beta_1 - 1331) q^{81} + (187 \beta_{3} - 169) q^{83} + (70 \beta_{3} - 190) q^{85} + ( - 501 \beta_{2} - 93 \beta_1 - 501) q^{87} + (220 \beta_{3} + 163 \beta_{2} + 220 \beta_1) q^{89} + (62 \beta_{3} + 346 \beta_{2} + 62 \beta_1) q^{93} + ( - 440 \beta_{2} - 10 \beta_1 - 440) q^{95} + ( - 36 \beta_{3} + 970) q^{97} + ( - 300 \beta_{3} + 960) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 10 q^{5} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} + 10 q^{5} - 40 q^{9} - 52 q^{11} + 104 q^{13} + 100 q^{15} - 76 q^{17} + 176 q^{19} - 102 q^{23} - 50 q^{25} - 740 q^{27} - 612 q^{29} + 244 q^{31} + 348 q^{33} + 176 q^{37} - 356 q^{39} + 260 q^{41} + 228 q^{43} + 200 q^{45} + 716 q^{47} + 996 q^{51} - 808 q^{53} - 520 q^{55} + 1936 q^{57} - 1016 q^{59} - 222 q^{61} + 260 q^{65} - 134 q^{67} - 1284 q^{69} + 592 q^{71} + 724 q^{73} + 250 q^{75} - 1128 q^{79} - 2662 q^{81} - 676 q^{83} - 760 q^{85} - 1002 q^{87} - 326 q^{89} - 692 q^{93} - 880 q^{95} + 3880 q^{97} + 3840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 22x^{2} + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 22 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 22\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−2.34521 + 4.06202i
2.34521 4.06202i
−2.34521 4.06202i
2.34521 + 4.06202i
0 0.154792 0.268108i 0 2.50000 + 4.33013i 0 0 0 13.4521 + 23.2997i 0
361.2 0 4.84521 8.39215i 0 2.50000 + 4.33013i 0 0 0 −33.4521 57.9407i 0
961.1 0 0.154792 + 0.268108i 0 2.50000 4.33013i 0 0 0 13.4521 23.2997i 0
961.2 0 4.84521 + 8.39215i 0 2.50000 4.33013i 0 0 0 −33.4521 + 57.9407i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.4.i.w 4
7.b odd 2 1 140.4.i.c 4
7.c even 3 1 980.4.a.n 2
7.c even 3 1 inner 980.4.i.w 4
7.d odd 6 1 140.4.i.c 4
7.d odd 6 1 980.4.a.u 2
21.c even 2 1 1260.4.s.e 4
21.g even 6 1 1260.4.s.e 4
28.d even 2 1 560.4.q.l 4
28.f even 6 1 560.4.q.l 4
35.c odd 2 1 700.4.i.h 4
35.f even 4 2 700.4.r.f 8
35.i odd 6 1 700.4.i.h 4
35.k even 12 2 700.4.r.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.4.i.c 4 7.b odd 2 1
140.4.i.c 4 7.d odd 6 1
560.4.q.l 4 28.d even 2 1
560.4.q.l 4 28.f even 6 1
700.4.i.h 4 35.c odd 2 1
700.4.i.h 4 35.i odd 6 1
700.4.r.f 8 35.f even 4 2
700.4.r.f 8 35.k even 12 2
980.4.a.n 2 7.c even 3 1
980.4.a.u 2 7.d odd 6 1
980.4.i.w 4 1.a even 1 1 trivial
980.4.i.w 4 7.c even 3 1 inner
1260.4.s.e 4 21.c even 2 1
1260.4.s.e 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{3} + 97T_{3}^{2} - 30T_{3} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} + 52T_{11}^{3} + 2116T_{11}^{2} + 30576T_{11} + 345744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10 T^{3} + 97 T^{2} - 30 T + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 52 T^{3} + 2116 T^{2} + \cdots + 345744 \) Copy content Toggle raw display
$13$ \( (T^{2} - 52 T - 3636)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 76 T^{3} + 8644 T^{2} + \cdots + 8225424 \) Copy content Toggle raw display
$19$ \( T^{4} - 176 T^{3} + \cdots + 58614336 \) Copy content Toggle raw display
$23$ \( T^{4} + 102 T^{3} + 8001 T^{2} + \cdots + 5774409 \) Copy content Toggle raw display
$29$ \( (T^{2} + 306 T + 20241)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 244 T^{3} + \cdots + 137264656 \) Copy content Toggle raw display
$37$ \( T^{4} - 176 T^{3} + \cdots + 1844187136 \) Copy content Toggle raw display
$41$ \( (T^{2} - 130 T - 80343)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 114 T - 195301)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 716 T^{3} + \cdots + 14247764496 \) Copy content Toggle raw display
$53$ \( T^{4} + 808 T^{3} + \cdots + 23276994624 \) Copy content Toggle raw display
$59$ \( T^{4} + 1016 T^{3} + \cdots + 51209879616 \) Copy content Toggle raw display
$61$ \( T^{4} + 222 T^{3} + \cdots + 33149849041 \) Copy content Toggle raw display
$67$ \( T^{4} + 134 T^{3} + \cdots + 3284321481 \) Copy content Toggle raw display
$71$ \( (T^{2} - 296 T - 447048)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 724 T^{3} + \cdots + 132463969936 \) Copy content Toggle raw display
$79$ \( T^{4} + 1128 T^{3} + \cdots + 28957148224 \) Copy content Toggle raw display
$83$ \( (T^{2} + 338 T - 740757)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 326 T^{3} + \cdots + 1077923609361 \) Copy content Toggle raw display
$97$ \( (T^{2} - 1940 T + 912388)^{2} \) Copy content Toggle raw display
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