↑

Properties

Label 61.3.d.a
Level $61$
Weight $3$
Character orbit 61.d
Analytic conductor $1.662$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [61,3,Mod(11,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 61.d (of order \(4\), degree \(2\), 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: \(1.66212961260\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} + 6 x^{17} + 229 x^{16} - 892 x^{15} + 1754 x^{14} + 1268 x^{13} + \cdots + 68121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{15} + 2 \beta_{11}) q^{4} - \beta_{14} q^{5} + ( - \beta_{19} - \beta_{18} - \beta_{3}) q^{6} - \beta_{4} q^{7} + ( - \beta_{11} + \beta_{5} + \beta_{3} + 1) q^{8} + ( - \beta_{19} + \beta_{16} + \beta_{9} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{15} + 2 \beta_{11}) q^{4} - \beta_{14} q^{5} + ( - \beta_{19} - \beta_{18} - \beta_{3}) q^{6} - \beta_{4} q^{7} + ( - \beta_{11} + \beta_{5} + \beta_{3} + 1) q^{8} + ( - \beta_{19} + \beta_{16} + \beta_{9} + \cdots - 3) q^{9}+ \cdots + (2 \beta_{19} - 7 \beta_{18} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{2} - 14 q^{6} + 2 q^{7} + 18 q^{8} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{2} - 14 q^{6} + 2 q^{7} + 18 q^{8} - 64 q^{9} + 14 q^{10} - 12 q^{11} + 32 q^{12} - 40 q^{13} + 64 q^{15} + 28 q^{16} - 26 q^{17} - 44 q^{18} + 84 q^{20} + 30 q^{21} - 4 q^{22} + 8 q^{23} - 6 q^{24} - 224 q^{25} - 104 q^{26} + 68 q^{28} + 130 q^{29} + 254 q^{30} - 162 q^{31} + 8 q^{32} + 26 q^{33} - 96 q^{34} + 104 q^{35} - 106 q^{37} - 90 q^{38} + 88 q^{40} - 188 q^{42} + 202 q^{43} - 338 q^{44} - 40 q^{47} + 352 q^{50} + 148 q^{51} + 88 q^{53} + 424 q^{54} + 42 q^{55} + 376 q^{56} - 200 q^{58} - 58 q^{59} - 2 q^{61} - 184 q^{62} - 654 q^{63} + 30 q^{67} - 310 q^{68} + 414 q^{69} - 192 q^{70} + 70 q^{71} - 254 q^{72} + 444 q^{73} - 660 q^{74} + 652 q^{76} - 340 q^{77} - 654 q^{78} - 8 q^{79} + 228 q^{81} + 144 q^{82} + 216 q^{83} + 654 q^{84} - 30 q^{85} - 1344 q^{86} - 256 q^{87} + 584 q^{89} + 54 q^{90} + 178 q^{91} - 286 q^{92} - 96 q^{93} + 180 q^{94} - 444 q^{95} + 130 q^{96} + 694 q^{98} - 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 8 x^{18} + 6 x^{17} + 229 x^{16} - 892 x^{15} + 1754 x^{14} + 1268 x^{13} + \cdots + 68121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 49\!\cdots\!11 \nu^{19} + \cdots - 26\!\cdots\!73 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92\!\cdots\!87 \nu^{19} + \cdots - 64\!\cdots\!41 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!97 \nu^{19} + \cdots + 18\!\cdots\!54 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46\!\cdots\!03 \nu^{19} + \cdots - 11\!\cdots\!29 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!73 \nu^{19} + \cdots + 37\!\cdots\!86 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!33 \nu^{19} + \cdots + 92\!\cdots\!44 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52\!\cdots\!57 \nu^{19} + \cdots - 37\!\cdots\!24 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!19 \nu^{19} + \cdots - 38\!\cdots\!02 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!03 \nu^{19} + \cdots + 18\!\cdots\!21 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 81\!\cdots\!27 \nu^{19} + \cdots - 34\!\cdots\!61 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 34\!\cdots\!33 \nu^{19} + \cdots + 89\!\cdots\!69 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!78 \nu^{19} + \cdots - 86\!\cdots\!29 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!16 \nu^{19} + \cdots - 51\!\cdots\!13 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 81\!\cdots\!27 \nu^{19} + \cdots + 34\!\cdots\!61 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 34\!\cdots\!81 \nu^{19} + \cdots + 15\!\cdots\!92 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 69\!\cdots\!98 \nu^{19} + \cdots - 33\!\cdots\!41 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 37\!\cdots\!94 \nu^{19} + \cdots + 32\!\cdots\!92 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!99 \nu^{19} + \cdots - 32\!\cdots\!82 ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + 6\beta_{11} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{5} - 9\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - 2\beta_{9} - \beta_{6} - 15\beta_{2} - 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{14} + \beta_{12} - 17 \beta_{11} + 16 \beta_{10} - \beta_{9} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17 \beta_{19} + 19 \beta_{18} + 19 \beta_{17} - 17 \beta_{16} - 198 \beta_{15} + 44 \beta_{14} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{19} + 21 \beta_{18} - 2 \beta_{15} + 27 \beta_{14} + 23 \beta_{13} + 13 \beta_{12} + \cdots + 226 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 245 \beta_{19} + 275 \beta_{18} - 275 \beta_{17} + 245 \beta_{16} + 9 \beta_{13} + 18 \beta_{10} + \cdots + 6543 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 316 \beta_{17} - 148 \beta_{16} + 57 \beta_{15} - 502 \beta_{14} - 86 \beta_{12} + 2797 \beta_{11} + \cdots + 2797 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3367 \beta_{19} - 3631 \beta_{18} - 3631 \beta_{17} + 3367 \beta_{16} + 31895 \beta_{15} + \cdots - 1344 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 976 \beta_{19} - 4200 \beta_{18} + 1123 \beta_{15} - 8028 \beta_{14} - 6650 \beta_{13} + 258 \beta_{12} + \cdots - 34038 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 45282 \beta_{19} - 46144 \beta_{18} + 46144 \beta_{17} - 45282 \beta_{16} - 7271 \beta_{13} + \cdots - 950622 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 52739 \beta_{17} - 389 \beta_{16} - 19355 \beta_{15} + 118997 \beta_{14} - 20867 \beta_{12} + \cdots - 415810 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 601322 \beta_{19} + 576608 \beta_{18} + 576608 \beta_{17} - 601322 \beta_{16} - 5021234 \beta_{15} + \cdots + 333258 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 165659 \beta_{19} + 644831 \beta_{18} - 314021 \beta_{15} + 1690173 \beta_{14} + 1502619 \beta_{13} + \cdots + 5136257 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 7916884 \beta_{19} + 7154072 \beta_{18} - 7154072 \beta_{17} + 7916884 \beta_{16} + 2226120 \beta_{13} + \cdots + 145211309 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7801160 \beta_{17} + 4031344 \beta_{16} + 4929676 \beta_{15} - 23393692 \beta_{14} + 8461456 \beta_{12} + \cdots + 64249088 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 103568896 \beta_{19} - 88551092 \beta_{18} - 88551092 \beta_{17} + 103568896 \beta_{16} + \cdots - 62466628 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 73848716 \beta_{19} - 94254428 \beta_{18} + 75705100 \beta_{15} - 318440800 \beta_{14} + \cdots - 813126325 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
2.52951 + 2.52951i
2.33998 + 2.33998i
1.86632 + 1.86632i
1.30769 + 1.30769i
0.964421 + 0.964421i
−0.0765757 − 0.0765757i
−1.10354 − 1.10354i
−1.64958 − 1.64958i
−1.68344 − 1.68344i
−2.49479 − 2.49479i
2.52951 − 2.52951i
2.33998 − 2.33998i
1.86632 − 1.86632i
1.30769 − 1.30769i
0.964421 − 0.964421i
−0.0765757 + 0.0765757i
−1.10354 + 1.10354i
−1.64958 + 1.64958i
−1.68344 + 1.68344i
−2.49479 + 2.49479i
−2.52951 − 2.52951i 0.490886i 8.79688i 8.10124i 1.24170 − 1.24170i −2.60873 − 2.60873i 12.1338 − 12.1338i 8.75903 20.4922 − 20.4922i
11.2 −2.33998 − 2.33998i − 4.54916i 6.95102i − 7.88002i −10.6449 + 10.6449i 7.18203 + 7.18203i 6.90532 − 6.90532i −11.6948 −18.4391 + 18.4391i
11.3 −1.86632 − 1.86632i 2.61741i 2.96631i − 6.79801i 4.88492 − 4.88492i −8.35328 − 8.35328i −1.92920 + 1.92920i 2.14918 −12.6873 + 12.6873i
11.4 −1.30769 − 1.30769i 3.56312i − 0.579893i 2.41698i 4.65946 − 4.65946i 9.44859 + 9.44859i −5.98908 + 5.98908i −3.69583 3.16067 − 3.16067i
11.5 −0.964421 − 0.964421i − 4.33156i − 2.13978i 5.54482i −4.17745 + 4.17745i −4.37358 − 4.37358i −5.92134 + 5.92134i −9.76240 5.34754 − 5.34754i
11.6 0.0765757 + 0.0765757i − 0.636780i − 3.98827i − 3.49072i 0.0487618 − 0.0487618i 0.570654 + 0.570654i 0.611707 − 0.611707i 8.59451 0.267305 − 0.267305i
11.7 1.10354 + 1.10354i 2.59770i − 1.56441i 7.09440i −2.86666 + 2.86666i −2.87678 − 2.87678i 6.14054 − 6.14054i 2.25197 −7.82894 + 7.82894i
11.8 1.64958 + 1.64958i − 4.85092i 1.44220i 1.90255i 8.00195 − 8.00195i 5.31875 + 5.31875i 4.21928 − 4.21928i −14.5314 −3.13840 + 3.13840i
11.9 1.68344 + 1.68344i 5.63743i 1.66795i − 9.05328i −9.49029 + 9.49029i 3.24878 + 3.24878i 3.92587 − 3.92587i −22.7807 15.2407 − 15.2407i
11.10 2.49479 + 2.49479i − 0.538135i 8.44800i − 1.83795i 1.34254 − 1.34254i −6.55642 − 6.55642i −11.0969 + 11.0969i 8.71041 4.58532 − 4.58532i
50.1 −2.52951 + 2.52951i − 0.490886i − 8.79688i − 8.10124i 1.24170 + 1.24170i −2.60873 + 2.60873i 12.1338 + 12.1338i 8.75903 20.4922 + 20.4922i
50.2 −2.33998 + 2.33998i 4.54916i − 6.95102i 7.88002i −10.6449 − 10.6449i 7.18203 − 7.18203i 6.90532 + 6.90532i −11.6948 −18.4391 − 18.4391i
50.3 −1.86632 + 1.86632i − 2.61741i − 2.96631i 6.79801i 4.88492 + 4.88492i −8.35328 + 8.35328i −1.92920 − 1.92920i 2.14918 −12.6873 − 12.6873i
50.4 −1.30769 + 1.30769i − 3.56312i 0.579893i − 2.41698i 4.65946 + 4.65946i 9.44859 − 9.44859i −5.98908 − 5.98908i −3.69583 3.16067 + 3.16067i
50.5 −0.964421 + 0.964421i 4.33156i 2.13978i − 5.54482i −4.17745 − 4.17745i −4.37358 + 4.37358i −5.92134 − 5.92134i −9.76240 5.34754 + 5.34754i
50.6 0.0765757 − 0.0765757i 0.636780i 3.98827i 3.49072i 0.0487618 + 0.0487618i 0.570654 − 0.570654i 0.611707 + 0.611707i 8.59451 0.267305 + 0.267305i
50.7 1.10354 − 1.10354i − 2.59770i 1.56441i − 7.09440i −2.86666 − 2.86666i −2.87678 + 2.87678i 6.14054 + 6.14054i 2.25197 −7.82894 − 7.82894i
50.8 1.64958 − 1.64958i 4.85092i − 1.44220i − 1.90255i 8.00195 + 8.00195i 5.31875 − 5.31875i 4.21928 + 4.21928i −14.5314 −3.13840 − 3.13840i
50.9 1.68344 − 1.68344i − 5.63743i − 1.66795i 9.05328i −9.49029 − 9.49029i 3.24878 − 3.24878i 3.92587 + 3.92587i −22.7807 15.2407 + 15.2407i
50.10 2.49479 − 2.49479i 0.538135i − 8.44800i 1.83795i 1.34254 + 1.34254i −6.55642 + 6.55642i −11.0969 − 11.0969i 8.71041 4.58532 + 4.58532i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.3.d.a ✓ 20
3.b odd 2 1 549.3.i.a 20
61.d odd 4 1 inner 61.3.d.a ✓ 20
183.g even 4 1 549.3.i.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.3.d.a ✓ 20 1.a even 1 1 trivial
61.3.d.a ✓ 20 61.d odd 4 1 inner
549.3.i.a 20 3.b odd 2 1
549.3.i.a 20 183.g even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(61, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 4 T^{19} + \cdots + 68121 \) Copy content Toggle raw display
$3$ \( T^{20} + 122 T^{18} + \cdots + 4822416 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 20790207498384 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{10} + 20 T^{9} + \cdots - 255741300)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 278449828217856 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 21\!\cdots\!49 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 155619032836800)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 83\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 50\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 36\!\cdots\!15)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 51\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 24\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
show more
show less