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Properties

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

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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: \(1\)
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
5.47436
5.24823
4.66317
3.76410
3.63943
3.49783
2.10820
2.07637
1.28509
1.23932
−0.800729
−1.07790
−1.31629
−1.38042
−3.07757
−3.19844
−3.60350
−4.26077
−5.02970
−5.25077
−5.47436 9.74115 21.9686 −0.682625 −53.3265 21.3332 −76.4689 67.8901 3.73693
1.2 −5.24823 0.454828 19.5439 −14.1671 −2.38704 −5.88109 −60.5849 −26.7931 74.3519
1.3 −4.66317 6.42174 13.7452 17.5394 −29.9457 −35.2132 −26.7907 14.2388 −81.7893
1.4 −3.76410 −2.32365 6.16841 −9.75181 8.74645 25.5183 6.89427 −21.6006 36.7067
1.5 −3.63943 2.68307 5.24547 15.0109 −9.76487 11.3090 10.0249 −19.8011 −54.6312
1.6 −3.49783 −8.59022 4.23481 11.6779 30.0471 −4.40565 13.1700 46.7920 −40.8474
1.7 −2.10820 9.75918 −3.55551 −8.90056 −20.5743 −26.0391 24.3613 68.2417 18.7641
1.8 −2.07637 −0.230185 −3.68867 −3.63431 0.477951 −17.9695 24.2700 −26.9470 7.54618
1.9 −1.28509 5.89586 −6.34854 −2.92583 −7.57673 18.2361 18.4392 7.76112 3.75997
1.10 −1.23932 −8.38363 −6.46408 −5.00543 10.3900 7.05336 17.9256 43.2852 6.20334
1.11 0.800729 −3.23814 −7.35883 −8.57139 −2.59287 −21.6200 −12.2983 −16.5145 −6.86336
1.12 1.07790 −5.87249 −6.83814 18.2881 −6.32994 −27.1334 −15.9940 7.48617 19.7127
1.13 1.31629 4.05959 −6.26739 −20.8830 5.34359 −14.7875 −18.7800 −10.5197 −27.4880
1.14 1.38042 0.905162 −6.09444 10.0344 1.24951 19.2946 −19.4563 −26.1807 13.8517
1.15 3.07757 −6.32420 1.47146 −21.1262 −19.4632 26.8739 −20.0921 12.9955 −65.0174
1.16 3.19844 7.52269 2.23004 7.50356 24.0609 −20.3992 −18.4549 29.5909 23.9997
1.17 3.60350 −4.62610 4.98523 7.19376 −16.6702 7.67603 −10.8637 −5.59919 25.9227
1.18 4.26077 5.21042 10.1542 −9.13773 22.2004 7.27896 9.17836 0.148505 −38.9337
1.19 5.02970 −10.0219 17.2979 −11.0279 −50.4069 −23.4535 46.7655 73.4377 −55.4670
1.20 5.25077 −1.04322 19.5706 14.5658 −5.47772 −27.6715 60.7544 −25.9117 76.4814
\(n\): 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.m ✓ 20
11.b odd 2 1 2057.4.a.o yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2057.4.a.m ✓ 20 1.a even 1 1 trivial
2057.4.a.o yes 20 11.b odd 2 1

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
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