↑

Properties

Label 2057.4.a.o
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2057,4,Mod(1,2057)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2057, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2057.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2057 = 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2057.a (trivial)

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: yes
Analytic conductor: \(121.366928882\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 112 x^{18} + 432 x^{17} + 5200 x^{16} - 19200 x^{15} - 130242 x^{14} + \cdots + 280158912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_1) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{14} - 1) q^{5} + ( - \beta_{5} + \beta_{2} + 6) q^{6} + (\beta_{8} - \beta_1 + 4) q^{7} + (\beta_{7} + \beta_{6} + 4 \beta_1 + 2) q^{8} + ( - \beta_{11} - \beta_{5} + \beta_{2} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + \beta_1) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{14} - 1) q^{5} + ( - \beta_{5} + \beta_{2} + 6) q^{6} + (\beta_{8} - \beta_1 + 4) q^{7} + (\beta_{7} + \beta_{6} + 4 \beta_1 + 2) q^{8} + ( - \beta_{11} - \beta_{5} + \beta_{2} + 9) q^{9} + ( - \beta_{15} - \beta_{7} - \beta_{5} + \cdots + 3) q^{10}+ \cdots + ( - 16 \beta_{19} - 2 \beta_{18} + \cdots - 19) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} + 2 q^{3} + 80 q^{4} - 14 q^{5} + 122 q^{6} + 80 q^{7} + 48 q^{8} + 192 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{2} + 2 q^{3} + 80 q^{4} - 14 q^{5} + 122 q^{6} + 80 q^{7} + 48 q^{8} + 192 q^{9} + 60 q^{10} + 134 q^{12} + 88 q^{13} - 210 q^{14} + 50 q^{15} + 320 q^{16} - 340 q^{17} + 166 q^{18} - 44 q^{19} + 144 q^{20} + 84 q^{21} + 34 q^{23} + 1262 q^{24} + 508 q^{25} - 310 q^{26} + 38 q^{27} + 456 q^{28} + 114 q^{29} - 368 q^{30} - 194 q^{31} + 448 q^{32} - 68 q^{34} + 1156 q^{35} + 1770 q^{36} + 366 q^{37} + 1290 q^{38} + 810 q^{39} - 732 q^{40} - 48 q^{41} + 900 q^{42} + 1472 q^{43} - 752 q^{45} + 1570 q^{46} - 200 q^{47} + 1518 q^{48} + 1412 q^{49} - 1072 q^{50} - 34 q^{51} + 2406 q^{52} + 418 q^{53} + 3402 q^{54} - 3064 q^{56} + 1960 q^{57} + 990 q^{58} + 1016 q^{59} + 3640 q^{60} + 240 q^{61} - 2588 q^{62} + 2220 q^{63} + 2444 q^{64} + 1400 q^{65} - 712 q^{67} - 1360 q^{68} + 566 q^{69} - 340 q^{70} - 746 q^{71} - 152 q^{72} - 24 q^{73} + 2486 q^{74} - 2340 q^{75} + 2556 q^{76} + 4358 q^{78} + 3778 q^{79} - 1470 q^{80} + 4396 q^{81} + 3632 q^{82} - 776 q^{83} - 4858 q^{84} + 238 q^{85} + 3182 q^{86} + 4 q^{87} - 886 q^{89} + 9304 q^{90} - 2976 q^{91} - 3274 q^{92} - 5358 q^{93} - 964 q^{94} + 1700 q^{95} + 11428 q^{96} - 1826 q^{97} - 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 112 x^{18} + 432 x^{17} + 5200 x^{16} - 19200 x^{15} - 130242 x^{14} + \cdots + 280158912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 80\!\cdots\!77 \nu^{19} + \cdots + 28\!\cdots\!16 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!79 \nu^{19} + \cdots + 14\!\cdots\!72 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!01 \nu^{19} + \cdots + 47\!\cdots\!68 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 87\!\cdots\!39 \nu^{19} + \cdots + 38\!\cdots\!52 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 87\!\cdots\!39 \nu^{19} + \cdots - 39\!\cdots\!52 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46\!\cdots\!57 \nu^{19} + \cdots + 11\!\cdots\!04 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!17 \nu^{19} + \cdots - 10\!\cdots\!36 ) / 62\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!99 \nu^{19} + \cdots + 13\!\cdots\!72 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!69 \nu^{19} + \cdots - 62\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 73\!\cdots\!99 \nu^{19} + \cdots - 44\!\cdots\!32 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 52\!\cdots\!13 \nu^{19} + \cdots + 12\!\cdots\!24 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 40\!\cdots\!17 \nu^{19} + \cdots + 86\!\cdots\!44 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26\!\cdots\!23 \nu^{19} + \cdots - 77\!\cdots\!36 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 27\!\cdots\!51 \nu^{19} + \cdots - 10\!\cdots\!68 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 77\!\cdots\!27 \nu^{19} + \cdots + 31\!\cdots\!64 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 53\!\cdots\!99 \nu^{19} + \cdots - 62\!\cdots\!32 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 61\!\cdots\!11 \nu^{19} + \cdots + 92\!\cdots\!52 ) / 34\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 20\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{17} - \beta_{16} + \beta_{15} + \beta_{7} - 2\beta_{5} + \beta_{4} + 29\beta_{2} + 2\beta _1 + 239 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} - 5 \beta_{18} - \beta_{17} - \beta_{16} + 3 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \cdots + 82 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41 \beta_{19} + 3 \beta_{18} - 36 \beta_{17} - 46 \beta_{16} + 45 \beta_{15} - 7 \beta_{14} + \cdots + 5572 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 103 \beta_{19} - 234 \beta_{18} - 59 \beta_{17} - 66 \beta_{16} + 157 \beta_{15} + 144 \beta_{14} + \cdots + 2958 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1337 \beta_{19} + 201 \beta_{18} - 1004 \beta_{17} - 1638 \beta_{16} + 1537 \beta_{15} - 543 \beta_{14} + \cdots + 139990 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3837 \beta_{19} - 8044 \beta_{18} - 2363 \beta_{17} - 2882 \beta_{16} + 5871 \beta_{15} + 4870 \beta_{14} + \cdots + 100850 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 40659 \beta_{19} + 8969 \beta_{18} - 25842 \beta_{17} - 53794 \beta_{16} + 48047 \beta_{15} + \cdots + 3669330 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 125885 \beta_{19} - 246610 \beta_{18} - 80545 \beta_{17} - 107652 \beta_{16} + 193915 \beta_{15} + \cdots + 3341988 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1200749 \beta_{19} + 338203 \beta_{18} - 643738 \beta_{17} - 1707716 \beta_{16} + 1448135 \beta_{15} + \cdots + 98667054 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3877749 \beta_{19} - 7152082 \beta_{18} - 2522285 \beta_{17} - 3735192 \beta_{16} + 6043871 \beta_{15} + \cdots + 108720008 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 34938169 \beta_{19} + 11704275 \beta_{18} - 15768734 \beta_{17} - 53277056 \beta_{16} + \cdots + 2696900330 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 115408013 \beta_{19} - 201026782 \beta_{18} - 75109729 \beta_{17} - 124645524 \beta_{16} + \cdots + 3487604352 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1007923281 \beta_{19} + 385924991 \beta_{18} - 381463370 \beta_{17} - 1644603540 \beta_{16} + \cdots + 74549477718 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3367810545 \beta_{19} - 5538909218 \beta_{18} - 2164717985 \beta_{17} - 4067674144 \beta_{16} + \cdots + 110631572448 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 28920468905 \beta_{19} + 12360035615 \beta_{18} - 9101483682 \beta_{17} - 50395646348 \beta_{16} + \cdots + 2077940866470 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 97161127289 \beta_{19} - 150455847186 \beta_{18} - 60965138209 \beta_{17} - 130924380976 \beta_{16} + \cdots + 3477581952296 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.25077
−5.02970
−4.26077
−3.60350
−3.19844
−3.07757
−1.38042
−1.31629
−1.07790
−0.800729
1.23932
1.28509
2.07637
2.10820
3.49783
3.63943
3.76410
4.66317
5.24823
5.47436
−5.25077 −1.04322 19.5706 14.5658 5.47772 27.6715 −60.7544 −25.9117 −76.4814
1.2 −5.02970 −10.0219 17.2979 −11.0279 50.4069 23.4535 −46.7655 73.4377 55.4670
1.3 −4.26077 5.21042 10.1542 −9.13773 −22.2004 −7.27896 −9.17836 0.148505 38.9337
1.4 −3.60350 −4.62610 4.98523 7.19376 16.6702 −7.67603 10.8637 −5.59919 −25.9227
1.5 −3.19844 7.52269 2.23004 7.50356 −24.0609 20.3992 18.4549 29.5909 −23.9997
1.6 −3.07757 −6.32420 1.47146 −21.1262 19.4632 −26.8739 20.0921 12.9955 65.0174
1.7 −1.38042 0.905162 −6.09444 10.0344 −1.24951 −19.2946 19.4563 −26.1807 −13.8517
1.8 −1.31629 4.05959 −6.26739 −20.8830 −5.34359 14.7875 18.7800 −10.5197 27.4880
1.9 −1.07790 −5.87249 −6.83814 18.2881 6.32994 27.1334 15.9940 7.48617 −19.7127
1.10 −0.800729 −3.23814 −7.35883 −8.57139 2.59287 21.6200 12.2983 −16.5145 6.86336
1.11 1.23932 −8.38363 −6.46408 −5.00543 −10.3900 −7.05336 −17.9256 43.2852 −6.20334
1.12 1.28509 5.89586 −6.34854 −2.92583 7.57673 −18.2361 −18.4392 7.76112 −3.75997
1.13 2.07637 −0.230185 −3.68867 −3.63431 −0.477951 17.9695 −24.2700 −26.9470 −7.54618
1.14 2.10820 9.75918 −3.55551 −8.90056 20.5743 26.0391 −24.3613 68.2417 −18.7641
1.15 3.49783 −8.59022 4.23481 11.6779 −30.0471 4.40565 −13.1700 46.7920 40.8474
1.16 3.63943 2.68307 5.24547 15.0109 9.76487 −11.3090 −10.0249 −19.8011 54.6312
1.17 3.76410 −2.32365 6.16841 −9.75181 −8.74645 −25.5183 −6.89427 −21.6006 −36.7067
1.18 4.66317 6.42174 13.7452 17.5394 29.9457 35.2132 26.7907 14.2388 81.7893
1.19 5.24823 0.454828 19.5439 −14.1671 2.38704 5.88109 60.5849 −26.7931 −74.3519
1.20 5.47436 9.74115 21.9686 −0.682625 53.3265 −21.3332 76.4689 67.8901 −3.73693
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2057.4.a.o yes 20
11.b odd 2 1 2057.4.a.m ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2057.4.a.m ✓ 20 11.b odd 2 1
2057.4.a.o yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{20} - 4 T_{2}^{19} - 112 T_{2}^{18} + 432 T_{2}^{17} + 5200 T_{2}^{16} - 19200 T_{2}^{15} + \cdots + 280158912 \) Copy content Toggle raw display
\( T_{3}^{20} - 2 T_{3}^{19} - 364 T_{3}^{18} + 682 T_{3}^{17} + 54067 T_{3}^{16} - 94104 T_{3}^{15} + \cdots + 141745147360 \) Copy content Toggle raw display
\( T_{5}^{20} + 14 T_{5}^{19} - 1406 T_{5}^{18} - 19078 T_{5}^{17} + 804035 T_{5}^{16} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 280158912 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 141745147360 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T + 17)^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 43\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 29\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 88\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 30\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 84\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 36\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 76\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 63\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 19\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 12\!\cdots\!60 \) Copy content Toggle raw display
show more
show less