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Properties

Label 425.2.c.c
Level $425$
Weight $2$
Character orbit 425.c
Analytic conductor $3.394$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 16 x^{10} - 4 x^{9} + 111 x^{8} + 4 x^{7} + 394 x^{6} + 236 x^{5} + 581 x^{4} + 1000 x^{3} + \cdots + 845 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{2} q^{3} + (\beta_{4} - \beta_1) q^{4} + (\beta_{11} - \beta_{9}) q^{6} - \beta_{3} q^{7} + ( - \beta_{7} + \beta_{5}) q^{8} + (\beta_{4} + 2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{2} q^{3} + (\beta_{4} - \beta_1) q^{4} + (\beta_{11} - \beta_{9}) q^{6} - \beta_{3} q^{7} + ( - \beta_{7} + \beta_{5}) q^{8} + (\beta_{4} + 2 \beta_1 + 3) q^{9} + (\beta_{11} - \beta_{10} + \beta_{9}) q^{11} + (\beta_{6} + \beta_{2}) q^{12} + ( - 2 \beta_{8} - \beta_{7} - \beta_{5}) q^{13} - \beta_{10} q^{14} + ( - 2 \beta_1 - 1) q^{16} + (\beta_{8} - \beta_{7} - \beta_{6}) q^{17} + (4 \beta_{8} - \beta_{7} + 4 \beta_{5}) q^{18} + ( - 3 \beta_{4} + \beta_1 - 3) q^{19} + ( - 2 \beta_{4} - 2 \beta_1 - 1) q^{21} + (2 \beta_{6} + \beta_{3} + \beta_{2}) q^{22} + (\beta_{6} + \beta_{3} + \beta_{2}) q^{23} + (2 \beta_{10} - \beta_{9}) q^{24} + ( - 2 \beta_{4} + 2 \beta_1 + 3) q^{26} + (\beta_{6} + 3 \beta_{3} - 2 \beta_{2}) q^{27} + (\beta_{6} - \beta_{3}) q^{28} + ( - \beta_{11} - 2 \beta_{10}) q^{29} + (\beta_{10} + 2 \beta_{9}) q^{31} + ( - 3 \beta_{8} - 2 \beta_{7}) q^{32} + (2 \beta_{8} + 5 \beta_{7} - 4 \beta_{5}) q^{33} + (\beta_{11} - 2 \beta_{10} - \beta_1 - 3) q^{34} + (\beta_{4} - 3) q^{36} + ( - \beta_{6} - 2 \beta_{3} - 2 \beta_{2}) q^{37} + (\beta_{8} + 3 \beta_{7} - 5 \beta_{5}) q^{38} + ( - 2 \beta_{11} + \beta_{9}) q^{39} + (\beta_{11} - 2 \beta_{9}) q^{41} + ( - \beta_{8} + 2 \beta_{7} - 6 \beta_{5}) q^{42} + ( - \beta_{8} - 5 \beta_{7} + \beta_{5}) q^{43} + ( - \beta_{11} + 3 \beta_{10} + 3 \beta_{9}) q^{44} + ( - 2 \beta_{11} + 3 \beta_{10} + \beta_{9}) q^{46} + ( - 2 \beta_{8} + 6 \beta_{5}) q^{47} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{48} + (2 \beta_{4} - 3 \beta_1) q^{49} + (\beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots - 3) q^{51}+ \cdots + ( - \beta_{11} - 6 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{4} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{4} + 32 q^{9} - 12 q^{16} - 24 q^{19} - 4 q^{21} + 44 q^{26} - 36 q^{34} - 40 q^{36} - 8 q^{49} - 16 q^{51} + 8 q^{59} + 44 q^{64} - 32 q^{66} - 48 q^{69} - 72 q^{76} + 68 q^{81} + 12 q^{84} - 8 q^{86} + 16 q^{89} + 80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16 x^{10} - 4 x^{9} + 111 x^{8} + 4 x^{7} + 394 x^{6} + 236 x^{5} + 581 x^{4} + 1000 x^{3} + \cdots + 845 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29989233626 \nu^{11} + 33559794709 \nu^{10} + 547851009554 \nu^{9} + 389434883123 \nu^{8} + \cdots + 72820295016374 ) / 41848335113881 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 33559794709 \nu^{11} - 68023271538 \nu^{10} - 509391817627 \nu^{9} + \cdots - 142052438041554 ) / 41848335113881 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61233139525 \nu^{11} - 40858076915 \nu^{10} + 1030800575596 \nu^{9} - 477371158089 \nu^{8} + \cdots + 83799065734757 ) / 41848335113881 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 73861008490 \nu^{11} - 59627477099 \nu^{10} + 1170290320882 \nu^{9} - 1237529786622 \nu^{8} + \cdots + 33899457572842 ) / 41848335113881 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5090528 \nu^{11} - 8003933 \nu^{10} + 89292082 \nu^{9} - 148858063 \nu^{8} + 717257948 \nu^{7} + \cdots + 5809264629 ) / 2104342573 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 163097602378 \nu^{11} - 197183765517 \nu^{10} + 2378203191190 \nu^{9} + \cdots + 39771430531216 ) / 41848335113881 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6644083201706 \nu^{11} + 5057919025962 \nu^{10} - 113181097539194 \nu^{9} + \cdots - 60\!\cdots\!63 ) / 12\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7412735159152 \nu^{11} - 3347729341311 \nu^{10} + 118240390871884 \nu^{9} + \cdots + 66\!\cdots\!79 ) / 12\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10105670799750 \nu^{11} + 5889350856365 \nu^{10} - 165256429174211 \nu^{9} + \cdots - 85\!\cdots\!89 ) / 12\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11208701366936 \nu^{11} - 11837852530417 \nu^{10} + 190547163399362 \nu^{9} + \cdots + 11\!\cdots\!43 ) / 12\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11218196608770 \nu^{11} - 6990467506979 \nu^{10} + 185623974635416 \nu^{9} + \cdots + 12\!\cdots\!02 ) / 12\!\cdots\!49 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{8} - \beta_{5} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - 2\beta_{2} - \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} + \beta_{10} + 3\beta_{9} + 7\beta_{8} + \beta_{7} + 3\beta_{5} - 3\beta_{3} + 3\beta_{2} + 8\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 7 \beta_{8} + 6 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{11} - 18 \beta_{10} - 18 \beta_{9} - 22 \beta_{8} - \beta_{7} + 32 \beta_{5} - 13 \beta_{4} + \cdots - 61 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 48 \beta_{11} - 38 \beta_{10} + 52 \beta_{9} + 125 \beta_{8} - 45 \beta_{7} - 46 \beta_{6} + \cdots + 59 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} + 195 \beta_{10} + 11 \beta_{9} - 29 \beta_{8} + 73 \beta_{7} - 7 \beta_{6} - 457 \beta_{5} + \cdots + 500 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 164 \beta_{11} + 60 \beta_{10} - 84 \beta_{9} - 390 \beta_{8} - 36 \beta_{7} + 168 \beta_{6} + \cdots - 424 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 205 \beta_{11} - 1598 \beta_{10} + 666 \beta_{9} + 1007 \beta_{8} - 1334 \beta_{7} - 156 \beta_{6} + \cdots - 2192 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1426 \beta_{11} + 1252 \beta_{10} + 38 \beta_{9} + 2875 \beta_{8} + 2263 \beta_{7} - 1942 \beta_{6} + \cdots + 6891 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1865 \beta_{11} + 10069 \beta_{10} - 6567 \beta_{9} - 8595 \beta_{8} + 11201 \beta_{7} + 3894 \beta_{6} + \cdots - 780 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
−0.269594 + 2.47846i
−0.269594 − 1.30838i
−0.837565 + 2.56702i
−0.837565 − 0.0858277i
1.10716 − 1.97403i
1.10716 + 1.28514i
1.10716 + 1.97403i
1.10716 − 1.28514i
−0.837565 − 2.56702i
−0.837565 + 0.0858277i
−0.269594 − 2.47846i
−0.269594 + 1.30838i
− 2.17009i −2.21547 −2.70928 0 4.80775i −1.02091 1.53919i 1.90829 0
424.2 − 2.17009i 2.21547 −2.70928 0 − 4.80775i 1.02091 1.53919i 1.90829 0
424.3 − 1.48119i −3.29112 −0.193937 0 4.87478i 2.22194 − 2.67513i 7.83146 0
424.4 − 1.48119i 3.29112 −0.193937 0 − 4.87478i −2.22194 − 2.67513i 7.83146 0
424.5 − 0.311108i −1.12261 1.90321 0 0.349253i −3.60843 − 1.21432i −1.73975 0
424.6 − 0.311108i 1.12261 1.90321 0 − 0.349253i 3.60843 − 1.21432i −1.73975 0
424.7 0.311108i −1.12261 1.90321 0 − 0.349253i −3.60843 1.21432i −1.73975 0
424.8 0.311108i 1.12261 1.90321 0 0.349253i 3.60843 1.21432i −1.73975 0
424.9 1.48119i −3.29112 −0.193937 0 − 4.87478i 2.22194 2.67513i 7.83146 0
424.10 1.48119i 3.29112 −0.193937 0 4.87478i −2.22194 2.67513i 7.83146 0
424.11 2.17009i −2.21547 −2.70928 0 − 4.80775i −1.02091 − 1.53919i 1.90829 0
424.12 2.17009i 2.21547 −2.70928 0 4.80775i 1.02091 − 1.53919i 1.90829 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.12
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 425.2.c.c 12
5.b even 2 1 inner 425.2.c.c 12
5.c odd 4 1 425.2.d.a ✓ 6
5.c odd 4 1 425.2.d.b yes 6
17.b even 2 1 inner 425.2.c.c 12
85.c even 2 1 inner 425.2.c.c 12
85.f odd 4 1 7225.2.a.ba 6
85.f odd 4 1 7225.2.a.bg 6
85.g odd 4 1 425.2.d.a ✓ 6
85.g odd 4 1 425.2.d.b yes 6
85.i odd 4 1 7225.2.a.ba 6
85.i odd 4 1 7225.2.a.bg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.c.c 12 1.a even 1 1 trivial
425.2.c.c 12 5.b even 2 1 inner
425.2.c.c 12 17.b even 2 1 inner
425.2.c.c 12 85.c even 2 1 inner
425.2.d.a ✓ 6 5.c odd 4 1
425.2.d.a ✓ 6 85.g odd 4 1
425.2.d.b yes 6 5.c odd 4 1
425.2.d.b yes 6 85.g odd 4 1
7225.2.a.ba 6 85.f odd 4 1
7225.2.a.ba 6 85.i odd 4 1
7225.2.a.bg 6 85.f odd 4 1
7225.2.a.bg 6 85.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 17T_{3}^{4} + 73T_{3}^{2} - 67 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 7 T^{4} + 11 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 17 T^{4} + \cdots - 67)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 19 T^{4} + \cdots - 67)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 66 T^{4} + \cdots + 6700)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 27 T^{4} + \cdots + 529)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 26 T^{10} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} + \cdots - 100)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 54 T^{4} + \cdots - 268)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 116 T^{4} + \cdots + 1072)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 89 T^{4} + \cdots + 1675)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 144 T^{4} + \cdots - 1072)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 100 T^{4} + \cdots + 26800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 192 T^{4} + \cdots + 204304)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 160 T^{4} + \cdots + 43264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 163 T^{4} + \cdots + 1849)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 2 T^{2} - 44 T + 20)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} + 228 T^{4} + \cdots + 107200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 80 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 91 T^{4} + \cdots + 1675)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 32 T^{4} + \cdots - 1072)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 135 T^{4} + \cdots + 67)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 300 T^{4} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} - 4 T + 20)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 332 T^{4} + \cdots - 1239232)^{2} \) Copy content Toggle raw display
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