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Properties

Label 61.4.b.a
Level $61$
Weight $4$
Character orbit 61.b
Analytic conductor $3.599$
Analytic rank $0$
Dimension $14$
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] = [61,4,Mod(60,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.60");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 61.b (of order \(2\), degree \(1\), 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: \(3.59911651035\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 75x^{12} + 2176x^{10} + 30960x^{8} + 227127x^{6} + 841453x^{4} + 1469744x^{2} + 950976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 3) q^{4} + ( - \beta_{8} - 1) q^{5} + (\beta_{5} + \beta_1) q^{6} + \beta_{4} q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{10} + \beta_{8} + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 3) q^{4} + ( - \beta_{8} - 1) q^{5} + (\beta_{5} + \beta_1) q^{6} + \beta_{4} q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{10} + \beta_{8} + 8) q^{9} + (\beta_{7} - \beta_1) q^{10} + ( - \beta_{9} - \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{11} - \beta_{10} + \cdots - 11) q^{12}+ \cdots + (7 \beta_{13} - 2 \beta_{9} + \cdots - 93 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} - 38 q^{4} - 14 q^{5} + 116 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{3} - 38 q^{4} - 14 q^{5} + 116 q^{9} - 150 q^{12} - 86 q^{13} - 8 q^{14} - 28 q^{15} - 158 q^{16} + 166 q^{19} + 54 q^{20} + 242 q^{22} + 204 q^{25} + 88 q^{27} + 824 q^{34} - 572 q^{36} + 1160 q^{39} - 64 q^{41} - 1936 q^{42} - 1310 q^{45} + 488 q^{46} - 1308 q^{47} + 230 q^{48} + 254 q^{49} - 50 q^{52} - 172 q^{56} + 1736 q^{57} - 470 q^{58} + 772 q^{60} - 630 q^{61} + 1546 q^{62} + 1098 q^{64} - 390 q^{65} - 292 q^{66} + 1390 q^{70} - 3032 q^{73} - 3806 q^{74} + 1978 q^{75} + 162 q^{76} - 82 q^{77} - 1682 q^{80} + 4238 q^{81} - 1822 q^{83} - 104 q^{86} + 3274 q^{88} - 1648 q^{95} + 3890 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 75x^{12} + 2176x^{10} + 30960x^{8} + 227127x^{6} + 841453x^{4} + 1469744x^{2} + 950976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 17\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1469 \nu^{13} - 111514 \nu^{11} - 3272518 \nu^{9} - 46700378 \nu^{7} - 332592721 \nu^{5} + \cdots - 1158216768 \nu ) / 6188544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 481 \nu^{13} + 34802 \nu^{11} + 954014 \nu^{9} + 12382978 \nu^{7} + 77892005 \nu^{5} + \cdots + 208711872 \nu ) / 2062848 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 481 \nu^{12} - 34802 \nu^{10} - 954014 \nu^{8} - 12382978 \nu^{6} - 77892005 \nu^{4} + \cdots - 210774720 ) / 2062848 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1895 \nu^{13} - 138718 \nu^{11} - 3839218 \nu^{9} - 49822382 \nu^{7} - 304206691 \nu^{5} + \cdots - 645710400 \nu ) / 3094272 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1895 \nu^{12} + 138718 \nu^{10} + 3839218 \nu^{8} + 49822382 \nu^{6} + 304206691 \nu^{4} + \cdots + 645710400 ) / 3094272 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2087 \nu^{13} + 150298 \nu^{11} + 4095994 \nu^{9} + 52557350 \nu^{7} + 319943755 \nu^{5} + \cdots + 632971296 \nu ) / 1547136 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7621 \nu^{12} - 528890 \nu^{10} - 13728086 \nu^{8} - 164917450 \nu^{6} - 917402393 \nu^{4} + \cdots - 1650838464 ) / 6188544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2741 \nu^{12} - 206362 \nu^{10} - 5968822 \nu^{8} - 83099114 \nu^{6} - 568369417 \nu^{4} + \cdots - 1763987904 ) / 2062848 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 233 \nu^{12} - 16882 \nu^{10} - 465214 \nu^{8} - 6091394 \nu^{6} - 38475325 \nu^{4} + \cdots - 94273344 ) / 91008 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28495 \nu^{13} + 2055278 \nu^{11} + 56194178 \nu^{9} + 726339550 \nu^{7} + 4504397003 \nu^{5} + \cdots + 11161382976 \nu ) / 6188544 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} - \beta_{10} - 25\beta_{2} + 189 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 3\beta_{9} - \beta_{7} - 4\beta_{5} + \beta_{4} - 31\beta_{3} + 334\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -38\beta_{12} + 40\beta_{11} + 41\beta_{10} - 6\beta_{8} - 43\beta_{6} + 581\beta_{2} - 3754 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -41\beta_{13} + 117\beta_{9} + 43\beta_{7} + 206\beta_{5} - 39\beta_{4} + 818\beta_{3} - 6996\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1097\beta_{12} - 1227\beta_{11} - 1268\beta_{10} + 222\beta_{8} + 2227\beta_{6} - 13565\beta_{2} + 79312 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1268\beta_{13} - 3462\beta_{9} - 1278\beta_{7} - 7258\beta_{5} + 1090\beta_{4} - 20660\beta_{3} + 152347\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 28872 \beta_{12} + 34104 \beta_{11} + 35516 \beta_{10} - 5344 \beta_{8} - 78940 \beta_{6} + \cdots - 1738767 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 35516 \beta_{13} + 93260 \beta_{9} + 32804 \beta_{7} + 219592 \beta_{5} - 27876 \beta_{4} + \cdots - 3408917 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 729545 \beta_{12} - 901745 \beta_{11} - 948333 \beta_{10} + 100808 \beta_{8} + 2397268 \beta_{6} + \cdots + 39113025 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 948333 \beta_{13} - 2407423 \beta_{9} - 783765 \beta_{7} - 6144012 \beta_{5} + 697109 \beta_{4} + \cdots + 77818306 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
60.1
− 4.92754i
− 4.32757i
− 3.89439i
− 2.98658i
− 1.94645i
− 1.56667i
− 1.28937i
1.28937i
1.56667i
1.94645i
2.98658i
3.89439i
4.32757i
4.92754i
− 4.92754i −2.22773 −16.2806 −4.70174 10.9772i 5.69013i 40.8032i −22.0372 23.1680i
60.2 − 4.32757i 9.22345 −10.7278 −10.7161 − 39.9151i − 11.5740i 11.8049i 58.0721 46.3747i
60.3 − 3.89439i 2.90463 −7.16627 19.6917 − 11.3117i − 3.96086i − 3.24685i −18.5632 − 76.6872i
60.4 − 2.98658i −9.52488 −0.919657 −2.16629 28.4468i 27.0414i − 21.1460i 63.7234 6.46979i
60.5 − 1.94645i −2.22484 4.21134 −18.8131 4.33054i − 26.8280i − 23.7687i −22.0501 36.6188i
60.6 − 1.56667i −4.69850 5.54555 9.67822 7.36099i − 14.1375i − 21.2214i −4.92407 − 15.1625i
60.7 − 1.28937i 5.54789 6.33752 0.0273216 − 7.15328i 21.0031i − 18.4864i 3.77903 − 0.0352277i
60.8 1.28937i 5.54789 6.33752 0.0273216 7.15328i − 21.0031i 18.4864i 3.77903 0.0352277i
60.9 1.56667i −4.69850 5.54555 9.67822 − 7.36099i 14.1375i 21.2214i −4.92407 15.1625i
60.10 1.94645i −2.22484 4.21134 −18.8131 − 4.33054i 26.8280i 23.7687i −22.0501 − 36.6188i
60.11 2.98658i −9.52488 −0.919657 −2.16629 − 28.4468i − 27.0414i 21.1460i 63.7234 − 6.46979i
60.12 3.89439i 2.90463 −7.16627 19.6917 11.3117i 3.96086i 3.24685i −18.5632 76.6872i
60.13 4.32757i 9.22345 −10.7278 −10.7161 39.9151i 11.5740i − 11.8049i 58.0721 − 46.3747i
60.14 4.92754i −2.22773 −16.2806 −4.70174 − 10.9772i − 5.69013i − 40.8032i −22.0372 − 23.1680i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 60.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.4.b.a ✓ 14
3.b odd 2 1 549.4.c.c 14
4.b odd 2 1 976.4.h.a 14
61.b even 2 1 inner 61.4.b.a ✓ 14
183.d odd 2 1 549.4.c.c 14
244.c odd 2 1 976.4.h.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.4.b.a ✓ 14 1.a even 1 1 trivial
61.4.b.a ✓ 14 61.b even 2 1 inner
549.4.c.c 14 3.b odd 2 1
549.4.c.c 14 183.d odd 2 1
976.4.h.a 14 4.b odd 2 1
976.4.h.a 14 244.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(61, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 75 T^{12} + \cdots + 950976 \) Copy content Toggle raw display
$3$ \( (T^{7} + T^{6} + \cdots - 32968)^{2} \) Copy content Toggle raw display
$5$ \( (T^{7} + 7 T^{6} + \cdots - 10692)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 31\!\cdots\!19 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{7} + 43 T^{6} + \cdots - 12415170996)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{7} - 83 T^{6} + \cdots + 8472689208)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 21\!\cdots\!91 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 23\!\cdots\!31)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 164786439356928)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 31\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 16\!\cdots\!69)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 27\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 52\!\cdots\!24)^{2} \) Copy content Toggle raw display
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