Properties

Label 425.4.c.c
Level $425$
Weight $4$
Character orbit 425.c
Analytic conductor $25.076$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,4,Mod(424,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.424");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.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: \(25.0758117524\)
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: \( x^{8} + 833x^{4} + 71824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 17)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} + 2 \beta_{3}) q^{6} + (\beta_{7} - \beta_{4}) q^{7} + ( - 7 \beta_{5} - \beta_1) q^{8} + ( - 6 \beta_{2} + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} + 2 \beta_{3}) q^{6} + (\beta_{7} - \beta_{4}) q^{7} + ( - 7 \beta_{5} - \beta_1) q^{8} + ( - 6 \beta_{2} + 13) q^{9} - 7 \beta_{3} q^{11} + ( - \beta_{7} + 2 \beta_{4}) q^{12} + ( - 14 \beta_{5} + 42 \beta_1) q^{13} + 8 \beta_{3} q^{14} + (7 \beta_{2} - 63) q^{16} + (\beta_{7} - 14 \beta_{5} - 8 \beta_{4} - 21 \beta_1) q^{17} + ( - 13 \beta_{5} + 61 \beta_1) q^{18} + 28 q^{19} + ( - 34 \beta_{2} + 48) q^{21} + (7 \beta_{7} - 14 \beta_{4}) q^{22} + (7 \beta_{7} + 7 \beta_{4}) q^{23} + ( - 7 \beta_{6} + 6 \beta_{3}) q^{24} + (42 \beta_{2} - 154) q^{26} + (6 \beta_{7} + 8 \beta_{4}) q^{27} + 8 \beta_{4} q^{28} + 7 \beta_{6} q^{29} + (\beta_{6} + 7 \beta_{3}) q^{31} + (7 \beta_{5} - 127 \beta_1) q^{32} + (42 \beta_{5} - 280 \beta_1) q^{33} + ( - 7 \beta_{6} + 22 \beta_{3} - 21 \beta_{2} - 91) q^{34} + (13 \beta_{2} - 61) q^{36} + ( - 7 \beta_{7} - 28 \beta_{4}) q^{37} + ( - 28 \beta_{5} + 28 \beta_1) q^{38} + ( - 14 \beta_{6} + 56 \beta_{3}) q^{39} + ( - 14 \beta_{6} + 20 \beta_{3}) q^{41} + ( - 48 \beta_{5} + 320 \beta_1) q^{42} + ( - 76 \beta_{5} - 92 \beta_1) q^{43} + ( - 7 \beta_{6} + 14 \beta_{3}) q^{44} + (14 \beta_{6} + 28 \beta_{3}) q^{46} + ( - 28 \beta_{5} - 224 \beta_1) q^{47} + ( - 7 \beta_{7} + 70 \beta_{4}) q^{48} + ( - 14 \beta_{2} - 71) q^{49} + ( - 14 \beta_{6} - 7 \beta_{3} - 76 \beta_{2} + 328) q^{51} + (42 \beta_{5} - 154 \beta_1) q^{52} + ( - 92 \beta_{5} - 98 \beta_1) q^{53} + (14 \beta_{6} + 20 \beta_{3}) q^{54} + (8 \beta_{6} + 48 \beta_{3}) q^{56} - 28 \beta_{4} q^{57} + ( - 7 \beta_{7} - 42 \beta_{4}) q^{58} + ( - 140 \beta_{2} + 364) q^{59} + (21 \beta_{6} - 64 \beta_{3}) q^{61} + ( - 8 \beta_{7} + 8 \beta_{4}) q^{62} + (7 \beta_{7} - 55 \beta_{4}) q^{63} + ( - 71 \beta_{2} - 321) q^{64} + ( - 280 \beta_{2} + 616) q^{66} + (64 \beta_{5} + 28 \beta_1) q^{67} + ( - 7 \beta_{7} - 21 \beta_{5} + 22 \beta_{4} - 91 \beta_1) q^{68} + ( - 154 \beta_{2} - 224) q^{69} + ( - 7 \beta_{6} + 21 \beta_{3}) q^{71} + ( - 43 \beta_{5} + 323 \beta_1) q^{72} + ( - 56 \beta_{7} + 48 \beta_{4}) q^{73} + ( - 35 \beta_{6} + 14 \beta_{3}) q^{74} + (28 \beta_{2} - 28) q^{76} + (238 \beta_{5} - 336 \beta_1) q^{77} + ( - 42 \beta_{7} + 196 \beta_{4}) q^{78} + (21 \beta_{6} + 77 \beta_{3}) q^{79} + (42 \beta_{2} - 623) q^{81} + ( - 6 \beta_{7} + 124 \beta_{4}) q^{82} + (84 \beta_{5} - 756 \beta_1) q^{83} + (48 \beta_{2} - 320) q^{84} + ( - 92 \beta_{2} - 516) q^{86} + (196 \beta_{5} - 56 \beta_1) q^{87} + (49 \beta_{7} - 42 \beta_{4}) q^{88} + (42 \beta_{2} - 518) q^{89} + ( - 28 \beta_{6} + 140 \beta_{3}) q^{91} + (14 \beta_{7} + 28 \beta_{4}) q^{92} + ( - 14 \beta_{5} + 272 \beta_1) q^{93} - 224 \beta_{2} q^{94} + (7 \beta_{6} - 134 \beta_{3}) q^{96} + ( - 22 \beta_{7} + 28 \beta_{4}) q^{97} + (71 \beta_{5} + 41 \beta_1) q^{98} + (42 \beta_{6} - 133 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} + 80 q^{9} - 476 q^{16} + 224 q^{19} + 248 q^{21} - 1064 q^{26} - 812 q^{34} - 436 q^{36} - 624 q^{49} + 2320 q^{51} + 2352 q^{59} - 2852 q^{64} + 3808 q^{66} - 2408 q^{69} - 112 q^{76} - 4816 q^{81} - 2368 q^{84} - 4496 q^{86} - 3976 q^{89} - 896 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 833x^{4} + 71824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 1101\nu^{2} ) / 9916 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 472 ) / 111 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 1101\nu^{3} + 9916\nu ) / 9916 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 1101\nu^{3} + 9916\nu ) / 9916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} - 3026\nu^{2} ) / 7437 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\nu^{7} + 268\nu^{5} + 15407\nu^{3} + 156244\nu ) / 29748 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -23\nu^{7} + 268\nu^{5} - 15407\nu^{3} + 156244\nu ) / 29748 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{5} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - 23\beta_{4} + 23\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 111\beta_{2} - 472 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 111\beta_{7} + 111\beta_{6} - 583\beta_{4} - 583\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3303\beta_{5} + 12104\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3303\beta_{7} + 3303\beta_{6} + 15407\beta_{4} - 15407\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
3.68218 + 3.68218i
−3.68218 3.68218i
2.22297 2.22297i
−2.22297 + 2.22297i
2.22297 + 2.22297i
−2.22297 2.22297i
3.68218 3.68218i
−3.68218 + 3.68218i
3.37228i −7.36435 −3.37228 0 24.8347i −17.4703 15.6060i 27.2337 0
424.2 3.37228i 7.36435 −3.37228 0 24.8347i 17.4703 15.6060i 27.2337 0
424.3 2.37228i −4.44593 2.37228 0 10.5470i 14.9929 24.6060i −7.23369 0
424.4 2.37228i 4.44593 2.37228 0 10.5470i −14.9929 24.6060i −7.23369 0
424.5 2.37228i −4.44593 2.37228 0 10.5470i 14.9929 24.6060i −7.23369 0
424.6 2.37228i 4.44593 2.37228 0 10.5470i −14.9929 24.6060i −7.23369 0
424.7 3.37228i −7.36435 −3.37228 0 24.8347i −17.4703 15.6060i 27.2337 0
424.8 3.37228i 7.36435 −3.37228 0 24.8347i 17.4703 15.6060i 27.2337 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.4.c.c 8
5.b even 2 1 inner 425.4.c.c 8
5.c odd 4 1 17.4.b.a 4
5.c odd 4 1 425.4.d.c 4
15.e even 4 1 153.4.d.b 4
17.b even 2 1 inner 425.4.c.c 8
20.e even 4 1 272.4.b.d 4
85.c even 2 1 inner 425.4.c.c 8
85.f odd 4 1 289.4.a.e 4
85.g odd 4 1 17.4.b.a 4
85.g odd 4 1 425.4.d.c 4
85.i odd 4 1 289.4.a.e 4
255.o even 4 1 153.4.d.b 4
340.r even 4 1 272.4.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.b.a 4 5.c odd 4 1
17.4.b.a 4 85.g odd 4 1
153.4.d.b 4 15.e even 4 1
153.4.d.b 4 255.o even 4 1
272.4.b.d 4 20.e even 4 1
272.4.b.d 4 340.r even 4 1
289.4.a.e 4 85.f odd 4 1
289.4.a.e 4 85.i odd 4 1
425.4.c.c 8 1.a even 1 1 trivial
425.4.c.c 8 5.b even 2 1 inner
425.4.c.c 8 17.b even 2 1 inner
425.4.c.c 8 85.c even 2 1 inner
425.4.d.c 4 5.c odd 4 1
425.4.d.c 4 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{4} + 17T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{3}^{4} - 74T_{3}^{2} + 1072 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 74 T^{2} + 1072)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 530 T^{2} + 68608)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3626 T^{2} + 2573872)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5684 T^{2} + 153664)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( (T - 28)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 28322 T^{2} + 10295488)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 23520 T^{2} + 92659392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4274 T^{2} + 4390912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 86240 T^{2} + \cdots + 1245754048)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 116960 T^{2} + \cdots + 2799480832)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 129104 T^{2} + \cdots + 945685504)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 126224 T^{2} + \cdots + 2517630976)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 181128 T^{2} + \cdots + 2410024464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 588 T - 75264)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 482528 T^{2} + \cdots + 7146728128)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 74784 T^{2} + \cdots + 911556864)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 52626 T^{2} + 92659392)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 1611264 T^{2} + \cdots + 647466319872)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 689234 T^{2} + \cdots + 70925616832)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1136016 T^{2} + \cdots + 203928109056)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 994 T + 232456)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 275552 T^{2} + \cdots + 16878665728)^{2} \) Copy content Toggle raw display
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