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Properties

Label 425.2.j.b
Level $425$
Weight $2$
Character orbit 425.j
Analytic conductor $3.394$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(149,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.j (of order \(4\), 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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{5} q^{3} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{5} q^{3} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + ( - 3 \beta_{10} - \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 4 q^{3} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 4 q^{3} + 12 q^{4} - 12 q^{8} - 4 q^{11} - 8 q^{12} + 4 q^{14} + 4 q^{16} - 8 q^{17} - 16 q^{21} + 20 q^{22} + 12 q^{23} - 4 q^{24} + 4 q^{27} + 4 q^{28} + 12 q^{29} + 12 q^{32} + 12 q^{34} + 12 q^{37} + 20 q^{39} - 24 q^{41} - 16 q^{42} - 16 q^{43} - 8 q^{44} - 24 q^{46} + 20 q^{48} + 32 q^{51} - 28 q^{54} + 40 q^{56} + 36 q^{58} + 40 q^{61} - 40 q^{62} - 12 q^{63} - 28 q^{64} - 60 q^{68} + 28 q^{71} + 48 q^{73} - 28 q^{74} + 48 q^{77} - 92 q^{78} + 8 q^{79} + 28 q^{81} + 40 q^{82} + 40 q^{83} - 96 q^{86} - 48 q^{87} + 72 q^{88} - 24 q^{89} - 36 q^{91} + 16 q^{92} - 72 q^{93} - 32 q^{96} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 15\nu^{9} + 34\nu^{7} - 16\nu^{5} - 75\nu^{3} - 27\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 2\nu^{9} + 15\nu^{8} - 32\nu^{7} + 35\nu^{6} - 102\nu^{5} - \nu^{4} - 100\nu^{3} - 38\nu^{2} - 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 18\nu^{8} - 84\nu^{6} - 166\nu^{4} - 133\nu^{2} - 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} + 2\nu^{9} + 15\nu^{8} + 32\nu^{7} + 35\nu^{6} + 102\nu^{5} - \nu^{4} + 100\nu^{3} - 38\nu^{2} + 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 98\nu^{6} + 188\nu^{4} + 125\nu^{2} + 13 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} + 37\nu^{9} + 182\nu^{7} + 354\nu^{5} + 258\nu^{3} + 31\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{10} + 2 \nu^{9} + 50 \nu^{8} + 32 \nu^{7} + 183 \nu^{6} + 102 \nu^{5} + 223 \nu^{4} + 100 \nu^{3} + \cdots + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{11} - 87\nu^{9} - 366\nu^{7} - 592\nu^{5} - 369\nu^{3} - 61\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{11} - \nu^{10} + 87 \nu^{9} - 17 \nu^{8} + 365 \nu^{7} - 66 \nu^{6} + 577 \nu^{5} - 86 \nu^{4} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{11} - \nu^{10} - 87 \nu^{9} - 17 \nu^{8} - 365 \nu^{7} - 66 \nu^{6} - 577 \nu^{5} - 86 \nu^{4} + \cdots + 3 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{8} + 3\beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 2\beta_{10} - 3\beta_{9} + 3\beta_{5} - 3\beta_{3} + 5\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{11} - 9\beta_{10} + 16\beta_{8} - 38\beta_{6} - 28\beta_{5} - 12\beta_{4} - 12\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -28\beta_{11} + 28\beta_{10} + 40\beta_{9} - 4\beta_{7} - 41\beta_{5} + 41\beta_{3} - 72\beta_{2} + 61\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 102\beta_{11} + 102\beta_{10} - 203\beta_{8} + 463\beta_{6} + 348\beta_{5} + 143\beta_{4} + 145\beta_{3} - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 348\beta_{11} - 348\beta_{10} - 493\beta_{9} + 60\beta_{7} + 504\beta_{5} - 504\beta_{3} + 895\beta_{2} - 718\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1222 \beta_{11} - 1222 \beta_{10} + 2482 \beta_{8} - 5618 \beta_{6} - 4240 \beta_{5} - 1726 \beta_{4} + \cdots + 225 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4240 \beta_{11} + 4240 \beta_{10} + 5998 \beta_{9} - 756 \beta_{7} - 6122 \beta_{5} + \cdots + 8667 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 14789 \beta_{11} + 14789 \beta_{10} - 30147 \beta_{8} + 68141 \beta_{6} + 51470 \beta_{5} + 20911 \beta_{4} + \cdots - 2669 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 51470 \beta_{11} - 51470 \beta_{10} - 72793 \beta_{9} + 9236 \beta_{7} + 74263 \beta_{5} + \cdots - 105040 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.254679i
1.52346i
− 3.48265i
1.35757i
− 0.455023i
− 1.19804i
− 0.254679i
− 1.52346i
3.48265i
− 1.35757i
0.455023i
1.19804i
−2.51230 0.887192 − 0.887192i 4.31167 0 −2.22889 + 2.22889i −1.14187 − 1.14187i −5.80761 1.42578i 0
149.2 −2.07061 −1.78436 + 1.78436i 2.28744 0 3.69471 − 3.69471i 0.260895 + 0.260895i −0.595174 − 3.36786i 0
149.3 −1.12708 1.75550 − 1.75550i −0.729699 0 −1.97858 + 1.97858i 1.72715 + 1.72715i 3.07658 − 3.16356i 0
149.4 0.677603 1.66705 − 1.66705i −1.54085 0 1.12960 − 1.12960i −3.02462 − 3.02462i −2.39929 − 2.55814i 0
149.5 0.783476 −0.385357 + 0.385357i −1.38617 0 −0.301918 + 0.301918i 0.840380 + 0.840380i −2.65298 2.70300i 0
149.6 2.24891 −0.140032 + 0.140032i 3.05761 0 −0.314920 + 0.314920i 1.33807 + 1.33807i 2.37848 2.96078i 0
174.1 −2.51230 0.887192 + 0.887192i 4.31167 0 −2.22889 − 2.22889i −1.14187 + 1.14187i −5.80761 − 1.42578i 0
174.2 −2.07061 −1.78436 − 1.78436i 2.28744 0 3.69471 + 3.69471i 0.260895 − 0.260895i −0.595174 3.36786i 0
174.3 −1.12708 1.75550 + 1.75550i −0.729699 0 −1.97858 − 1.97858i 1.72715 − 1.72715i 3.07658 3.16356i 0
174.4 0.677603 1.66705 + 1.66705i −1.54085 0 1.12960 + 1.12960i −3.02462 + 3.02462i −2.39929 2.55814i 0
174.5 0.783476 −0.385357 − 0.385357i −1.38617 0 −0.301918 − 0.301918i 0.840380 − 0.840380i −2.65298 − 2.70300i 0
174.6 2.24891 −0.140032 − 0.140032i 3.05761 0 −0.314920 − 0.314920i 1.33807 − 1.33807i 2.37848 − 2.96078i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.j.b 12
5.b even 2 1 425.2.j.c 12
5.c odd 4 1 85.2.e.a ✓ 12
5.c odd 4 1 425.2.e.f 12
15.e even 4 1 765.2.k.b 12
17.c even 4 1 425.2.j.c 12
20.e even 4 1 1360.2.bt.d 12
85.f odd 4 1 85.2.e.a ✓ 12
85.i odd 4 1 425.2.e.f 12
85.j even 4 1 inner 425.2.j.b 12
85.k odd 8 1 1445.2.a.n 6
85.k odd 8 1 1445.2.a.o 6
85.n odd 8 2 1445.2.d.g 12
85.n odd 8 1 7225.2.a.z 6
85.n odd 8 1 7225.2.a.bb 6
255.k even 4 1 765.2.k.b 12
340.s even 4 1 1360.2.bt.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.e.a ✓ 12 5.c odd 4 1
85.2.e.a ✓ 12 85.f odd 4 1
425.2.e.f 12 5.c odd 4 1
425.2.e.f 12 85.i odd 4 1
425.2.j.b 12 1.a even 1 1 trivial
425.2.j.b 12 85.j even 4 1 inner
425.2.j.c 12 5.b even 2 1
425.2.j.c 12 17.c even 4 1
765.2.k.b 12 15.e even 4 1
765.2.k.b 12 255.k even 4 1
1360.2.bt.d 12 20.e even 4 1
1360.2.bt.d 12 340.s even 4 1
1445.2.a.n 6 85.k odd 8 1
1445.2.a.o 6 85.k odd 8 1
1445.2.d.g 12 85.n odd 8 2
7225.2.a.z 6 85.n odd 8 1
7225.2.a.bb 6 85.n odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2T_{2}^{5} - 7T_{2}^{4} - 12T_{2}^{3} + 11T_{2}^{2} + 10T_{2} - 7 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 2 T^{5} - 7 T^{4} + \cdots - 7)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 28 T^{9} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 198916 \) Copy content Toggle raw display
$13$ \( T^{12} + 116 T^{10} + \cdots + 99856 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 136 T^{10} + \cdots + 2166784 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( T^{12} - 12 T^{11} + \cdots + 5345344 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1645275844 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} + \cdots + 3655744 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 192876544 \) Copy content Toggle raw display
$43$ \( (T^{6} + 8 T^{5} + \cdots - 15004)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 326741776 \) Copy content Toggle raw display
$53$ \( (T^{6} - 216 T^{4} + \cdots - 132112)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 240 T^{10} + \cdots + 802816 \) Copy content Toggle raw display
$61$ \( T^{12} - 40 T^{11} + \cdots + 984064 \) Copy content Toggle raw display
$67$ \( T^{12} + 108 T^{10} + \cdots + 8464 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 29133710596 \) Copy content Toggle raw display
$73$ \( T^{12} - 48 T^{11} + \cdots + 541696 \) Copy content Toggle raw display
$79$ \( T^{12} - 8 T^{11} + \cdots + 15225604 \) Copy content Toggle raw display
$83$ \( (T^{6} - 20 T^{5} + \cdots + 11204)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 12 T^{5} + \cdots - 31292)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 227086559296 \) Copy content Toggle raw display
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