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Properties

Label 656.2.u.h
Level $656$
Weight $2$
Character orbit 656.u
Analytic conductor $5.238$
Analytic rank $0$
Dimension $20$
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] = [656,2,Mod(305,656)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(656, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("656.305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 656.u (of order \(5\), degree \(4\), 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: \(5.23818637260\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 7 x^{18} - 6 x^{17} + 60 x^{16} - 92 x^{15} + 603 x^{14} - 690 x^{13} + 2935 x^{12} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 328)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} + \beta_{2} q^{5} + (\beta_{14} - \beta_{6}) q^{7} + ( - \beta_{13} - \beta_{12} - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} + \beta_{2} q^{5} + (\beta_{14} - \beta_{6}) q^{7} + ( - \beta_{13} - \beta_{12} - \beta_{7} + 1) q^{9} + ( - \beta_{14} - \beta_{12} + \cdots - \beta_{5}) q^{11}+ \cdots + (2 \beta_{19} + \beta_{18} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3} + 2 q^{5} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} + 2 q^{5} + 10 q^{9} - 9 q^{11} + 7 q^{13} - q^{15} - 8 q^{17} - q^{19} - 6 q^{21} + 11 q^{23} + 15 q^{25} - 2 q^{27} + 21 q^{29} + 5 q^{31} + 19 q^{33} - 4 q^{37} - 4 q^{39} + 9 q^{41} + 17 q^{43} + 11 q^{45} - 15 q^{47} - 25 q^{49} + 22 q^{51} + 10 q^{53} + 28 q^{55} - 20 q^{57} + 24 q^{59} + 15 q^{61} + 65 q^{63} - 29 q^{65} + 26 q^{67} - 47 q^{69} - 16 q^{71} + 14 q^{73} - 11 q^{75} + 12 q^{77} + 26 q^{79} - 60 q^{81} - 20 q^{83} - 94 q^{85} - 57 q^{87} + 5 q^{89} + 46 q^{91} + 43 q^{93} - 71 q^{95} - 22 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 7 x^{18} - 6 x^{17} + 60 x^{16} - 92 x^{15} + 603 x^{14} - 690 x^{13} + 2935 x^{12} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!35 \nu^{19} + \cdots + 22\!\cdots\!16 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\!\cdots\!34 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 77\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16\!\cdots\!43 \nu^{19} + \cdots - 41\!\cdots\!36 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\!\cdots\!25 \nu^{19} + \cdots - 23\!\cdots\!16 ) / 77\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!35 \nu^{19} + \cdots - 71\!\cdots\!92 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\!\cdots\!46 \nu^{19} + \cdots + 43\!\cdots\!80 ) / 77\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83\!\cdots\!21 \nu^{19} + \cdots + 32\!\cdots\!36 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{19} + \cdots - 15\!\cdots\!04 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!41 \nu^{19} + \cdots + 24\!\cdots\!80 ) / 15\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 81\!\cdots\!99 \nu^{19} + \cdots + 10\!\cdots\!60 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!23 \nu^{19} + \cdots - 28\!\cdots\!96 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!79 \nu^{19} + \cdots + 21\!\cdots\!28 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 35\!\cdots\!46 \nu^{19} + \cdots - 54\!\cdots\!52 ) / 77\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 72\!\cdots\!57 \nu^{19} + \cdots + 48\!\cdots\!28 ) / 15\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 40\!\cdots\!67 \nu^{19} + \cdots - 16\!\cdots\!52 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!89 \nu^{19} + \cdots - 41\!\cdots\!92 ) / 15\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 47\!\cdots\!31 \nu^{19} + \cdots - 93\!\cdots\!00 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 54\!\cdots\!61 \nu^{19} + \cdots - 33\!\cdots\!92 ) / 62\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} - \beta_{14} - \beta_{8} - 4\beta_{7} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{17} - \beta_{16} - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} - 2\beta_{14} + 8\beta_{13} + 10\beta_{12} - \beta_{11} - 8\beta_{8} - 13\beta_{6} - \beta_{5} - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{19} + \beta_{18} + 11 \beta_{17} - 11 \beta_{16} + 11 \beta_{13} + 3 \beta_{12} - 42 \beta_{11} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 62 \beta_{19} + 10 \beta_{18} + \beta_{16} + \beta_{15} + 25 \beta_{14} + 25 \beta_{13} + 92 \beta_{12} + \cdots - 183 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 102 \beta_{19} + \beta_{18} + \beta_{15} + 102 \beta_{14} - 71 \beta_{10} + 26 \beta_{9} + 177 \beta_{8} + \cdots - 43 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 244 \beta_{19} + 17 \beta_{18} - 6 \beta_{17} - 11 \beta_{16} + 81 \beta_{15} + 493 \beta_{14} + \cdots - 138 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 809 \beta_{17} + 786 \beta_{16} + 78 \beta_{15} + 625 \beta_{14} - 908 \beta_{13} - 478 \beta_{12} + \cdots + 478 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2193 \beta_{19} - 200 \beta_{18} - 271 \beta_{17} - 361 \beta_{16} - 4001 \beta_{13} - 6599 \beta_{12} + \cdots + 11311 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7956 \beta_{19} - 322 \beta_{18} - 4362 \beta_{17} + 3781 \beta_{16} - 581 \beta_{15} - 5253 \beta_{14} + \cdots + 7617 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 32883 \beta_{19} - 4943 \beta_{18} - 4943 \beta_{15} - 32883 \beta_{14} + 1645 \beta_{10} - 1472 \beta_{9} + \cdots + 54632 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 44467 \beta_{19} - 4403 \beta_{18} + 34709 \beta_{17} - 30306 \beta_{16} - 3657 \beta_{15} + \cdots + 127746 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 32843 \beta_{17} + 6269 \beta_{16} - 18640 \beta_{15} - 162157 \beta_{14} + 272306 \beta_{13} + \cdots - 449496 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 378134 \beta_{19} + 34633 \beta_{18} + 440690 \beta_{17} - 403368 \beta_{16} + 597238 \beta_{13} + \cdots - 713337 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2265492 \beta_{19} + 313208 \beta_{18} + 193870 \beta_{17} - 29066 \beta_{16} + 164804 \beta_{15} + \cdots - 6024408 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 5140943 \beta_{19} + 358674 \beta_{18} + 358674 \beta_{15} + 5140943 \beta_{14} - 2250638 \beta_{10} + \cdots - 4124897 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 11544451 \beta_{19} + 1418018 \beta_{18} - 1907039 \beta_{17} + 489021 \beta_{16} + \cdots - 10510866 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 29331148 \beta_{17} + 26006091 \beta_{16} + 2370660 \beta_{15} + 27508794 \beta_{14} + \cdots + 37609778 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/656\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(165\) \(575\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
0.688630 + 2.11938i
0.596104 + 1.83462i
0.165728 + 0.510060i
−0.330131 − 1.01604i
−0.620331 − 1.90918i
0.688630 − 2.11938i
0.596104 − 1.83462i
0.165728 − 0.510060i
−0.330131 + 1.01604i
−0.620331 + 1.90918i
−2.23116 + 1.62103i
−1.29183 + 0.938568i
0.645787 − 0.469192i
1.02434 − 0.744226i
2.35286 − 1.70946i
−2.23116 − 1.62103i
−1.29183 − 0.938568i
0.645787 + 0.469192i
1.02434 + 0.744226i
2.35286 + 1.70946i
0 −2.22845 0 2.81973 − 2.04866i 0 1.29202 − 3.97642i 0 1.96600 0
305.2 0 −1.92903 0 −3.33100 + 2.42011i 0 0.669598 − 2.06081i 0 0.721162 0
305.3 0 −0.536309 0 1.18977 − 0.864417i 0 −1.04717 + 3.22286i 0 −2.71237 0
305.4 0 1.06833 0 −1.23890 + 0.900114i 0 −0.620323 + 1.90916i 0 −1.85868 0
305.5 0 2.00743 0 1.06040 − 0.770427i 0 0.823910 − 2.53573i 0 1.02979 0
385.1 0 −2.22845 0 2.81973 + 2.04866i 0 1.29202 + 3.97642i 0 1.96600 0
385.2 0 −1.92903 0 −3.33100 − 2.42011i 0 0.669598 + 2.06081i 0 0.721162 0
385.3 0 −0.536309 0 1.18977 + 0.864417i 0 −1.04717 − 3.22286i 0 −2.71237 0
385.4 0 1.06833 0 −1.23890 − 0.900114i 0 −0.620323 − 1.90916i 0 −1.85868 0
385.5 0 2.00743 0 1.06040 + 0.770427i 0 0.823910 + 2.53573i 0 1.02979 0
529.1 0 −2.75787 0 0.675164 − 2.07794i 0 1.49390 + 1.08538i 0 4.60584 0
529.2 0 −1.59679 0 −0.571528 + 1.75898i 0 −1.03415 − 0.751356i 0 −0.450271 0
529.3 0 0.798236 0 0.897909 − 2.76348i 0 −1.99043 − 1.44613i 0 −2.36282 0
529.4 0 1.26615 0 −0.537278 + 1.65357i 0 −1.50744 − 1.09522i 0 −1.39685 0
529.5 0 2.90830 0 0.0357330 − 0.109975i 0 1.92009 + 1.39502i 0 5.45821 0
625.1 0 −2.75787 0 0.675164 + 2.07794i 0 1.49390 − 1.08538i 0 4.60584 0
625.2 0 −1.59679 0 −0.571528 − 1.75898i 0 −1.03415 + 0.751356i 0 −0.450271 0
625.3 0 0.798236 0 0.897909 + 2.76348i 0 −1.99043 + 1.44613i 0 −2.36282 0
625.4 0 1.26615 0 −0.537278 − 1.65357i 0 −1.50744 + 1.09522i 0 −1.39685 0
625.5 0 2.90830 0 0.0357330 + 0.109975i 0 1.92009 − 1.39502i 0 5.45821 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 656.2.u.h 20
4.b odd 2 1 328.2.m.c ✓ 20
41.d even 5 1 inner 656.2.u.h 20
164.j odd 10 1 328.2.m.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
328.2.m.c ✓ 20 4.b odd 2 1
328.2.m.c ✓ 20 164.j odd 10 1
656.2.u.h 20 1.a even 1 1 trivial
656.2.u.h 20 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + T_{3}^{9} - 17T_{3}^{8} - 15T_{3}^{7} + 98T_{3}^{6} + 63T_{3}^{5} - 243T_{3}^{4} - 76T_{3}^{3} + 253T_{3}^{2} - 64 \) acting on \(S_{2}^{\mathrm{new}}(656, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + T^{9} - 17 T^{8} + \cdots - 64)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots + 10000 \) Copy content Toggle raw display
$7$ \( T^{20} + 30 T^{18} + \cdots + 17808400 \) Copy content Toggle raw display
$11$ \( T^{20} + 9 T^{19} + \cdots + 6869641 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 3237610000 \) Copy content Toggle raw display
$17$ \( T^{20} + 8 T^{19} + \cdots + 1860496 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6643717081 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 111597335393296 \) Copy content Toggle raw display
$29$ \( T^{20} - 21 T^{19} + \cdots + 20394256 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 273511265406976 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 80812844176 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 11787988356496 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 477403611136 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 266099744254096 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 38831067136 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 2249088092416 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 163523184400 \) Copy content Toggle raw display
$73$ \( (T^{10} - 7 T^{9} + \cdots - 1905616)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 13 T^{9} + \cdots - 56320)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 10 T^{9} + \cdots - 106195216)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 68\!\cdots\!01 \) Copy content Toggle raw display
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