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Properties

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

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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: \(0\)
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: \( x^{20} - 4 x^{19} - 112 x^{18} + 438 x^{17} + 5176 x^{16} - 19774 x^{15} - 127872 x^{14} + \cdots + 10454400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 4) q^{4} - \beta_{14} q^{5} + ( - \beta_{12} + \beta_{2} + \beta_1 + 4) q^{6} + ( - \beta_{10} + 3) q^{7} + (\beta_{3} + 4 \beta_1 + 1) q^{8} + (\beta_{8} + \beta_{2} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 4) q^{4} - \beta_{14} q^{5} + ( - \beta_{12} + \beta_{2} + \beta_1 + 4) q^{6} + ( - \beta_{10} + 3) q^{7} + (\beta_{3} + 4 \beta_1 + 1) q^{8} + (\beta_{8} + \beta_{2} + 10) q^{9} + ( - \beta_{19} + \beta_{12} - \beta_{7} + \cdots + 4) q^{10}+ \cdots + (8 \beta_{19} + 2 \beta_{18} + \cdots + 68) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} + 8 q^{3} + 80 q^{4} + 6 q^{5} + 93 q^{6} + 52 q^{7} + 30 q^{8} + 194 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{2} + 8 q^{3} + 80 q^{4} + 6 q^{5} + 93 q^{6} + 52 q^{7} + 30 q^{8} + 194 q^{9} + 66 q^{10} + 143 q^{12} + 200 q^{13} + 85 q^{14} - 70 q^{15} + 320 q^{16} + 340 q^{17} + 160 q^{18} + 188 q^{19} + 21 q^{20} - 64 q^{21} - 54 q^{23} + 664 q^{24} + 830 q^{25} + 59 q^{26} + 302 q^{27} + 227 q^{28} + 166 q^{29} + 409 q^{30} + 26 q^{31} + 534 q^{32} + 68 q^{34} + 540 q^{35} + 1813 q^{36} - 120 q^{37} + 49 q^{38} - 482 q^{39} + 1265 q^{40} + 480 q^{41} - 675 q^{42} + 1158 q^{43} - 142 q^{45} + 581 q^{46} + 372 q^{47} + 700 q^{48} - 666 q^{49} + 168 q^{50} + 136 q^{51} + 152 q^{52} - 200 q^{53} + 2051 q^{54} + 551 q^{56} + 1504 q^{57} + 2806 q^{58} + 146 q^{59} - 2549 q^{60} + 1476 q^{61} + 2844 q^{62} + 1798 q^{63} + 1532 q^{64} + 740 q^{65} - 254 q^{67} + 1360 q^{68} + 2566 q^{69} - 860 q^{70} + 394 q^{71} + 489 q^{72} + 2244 q^{73} - 3056 q^{74} + 868 q^{75} + 3345 q^{76} + 612 q^{78} + 1674 q^{79} + 3533 q^{80} - 2676 q^{81} - 3925 q^{82} - 96 q^{83} + 4468 q^{84} + 102 q^{85} - 569 q^{86} + 5498 q^{87} + 1592 q^{89} - 4486 q^{90} + 3364 q^{91} - 4854 q^{92} - 2540 q^{93} + 836 q^{94} - 136 q^{95} + 4207 q^{96} + 1802 q^{97} + 1078 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 112 x^{18} + 438 x^{17} + 5176 x^{16} - 19774 x^{15} - 127872 x^{14} + \cdots + 10454400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 20\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 63\!\cdots\!33 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!01 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!61 \nu^{19} + \cdots - 76\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\!\cdots\!07 \nu^{19} + \cdots - 49\!\cdots\!00 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 72\!\cdots\!49 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!01 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{19} + \cdots - 91\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\!\cdots\!61 \nu^{19} + \cdots + 11\!\cdots\!80 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!29 \nu^{19} + \cdots - 38\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!32 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 83\!\cdots\!41 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51\!\cdots\!40 \nu^{19} + \cdots + 33\!\cdots\!40 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 35\!\cdots\!31 \nu^{19} + \cdots - 52\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 35\!\cdots\!61 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 59\!\cdots\!99 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 70\!\cdots\!23 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 18\!\cdots\!00 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 20\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} + \beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + 2 \beta_{17} - 2 \beta_{16} - 4 \beta_{14} - 2 \beta_{13} + 3 \beta_{10} + \beta_{9} + \cdots + 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{19} + 9 \beta_{18} - 42 \beta_{16} + 36 \beta_{15} - 21 \beta_{14} - 5 \beta_{13} + \cdots + 5530 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{19} + 5 \beta_{18} + 93 \beta_{17} - 117 \beta_{16} - 12 \beta_{15} - 212 \beta_{14} + \cdots + 2782 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 316 \beta_{19} + 529 \beta_{18} + 16 \beta_{17} - 1438 \beta_{16} + 1076 \beta_{15} - 1217 \beta_{14} + \cdots + 138413 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2318 \beta_{19} + 432 \beta_{18} + 3334 \beta_{17} - 4938 \beta_{16} - 574 \beta_{15} - 8310 \beta_{14} + \cdots + 119267 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11884 \beta_{19} + 21721 \beta_{18} + 1508 \beta_{17} - 46316 \beta_{16} + 30626 \beta_{15} + \cdots + 3635969 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 79858 \beta_{19} + 24643 \beta_{18} + 109091 \beta_{17} - 183118 \beta_{16} - 17745 \beta_{15} + \cdots + 4554291 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 392407 \beta_{19} + 774803 \beta_{18} + 88050 \beta_{17} - 1459572 \beta_{16} + 857412 \beta_{15} + \cdots + 98653650 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2558404 \beta_{19} + 1155559 \beta_{18} + 3425686 \beta_{17} - 6362197 \beta_{16} - 426025 \beta_{15} + \cdots + 162618868 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 12171295 \beta_{19} + 25823618 \beta_{18} + 4129905 \beta_{17} - 45642279 \beta_{16} + \cdots + 2739473133 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 78836686 \beta_{19} + 48260249 \beta_{18} + 105462101 \beta_{17} - 213111018 \beta_{16} + \cdots + 5575902646 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 365533829 \beta_{19} + 830766088 \beta_{18} + 171163885 \beta_{17} - 1424061243 \beta_{16} + \cdots + 77421491105 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2377313144 \beta_{19} + 1869441175 \beta_{18} + 3217196736 \beta_{17} - 6983938175 \beta_{16} + \cdots + 186312172196 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 10800695069 \beta_{19} + 26227260109 \beta_{18} + 6565356094 \beta_{17} - 44424525713 \beta_{16} + \cdots + 2218977439625 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 70830467375 \beta_{19} + 68775959637 \beta_{18} + 97815013509 \beta_{17} - 225749092478 \beta_{16} + \cdots + 6118742995355 \) 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
−5.13860
−5.01027
−4.30003
−4.13940
−3.50037
−2.28396
−1.85177
−1.18708
−0.892603
−0.0851144
0.507211
0.880531
2.53888
2.73552
3.28989
3.39109
3.87013
4.40856
5.17217
5.59523
−5.13860 −8.47751 18.4052 8.94651 43.5625 2.05892 −53.4681 44.8681 −45.9725
1.2 −5.01027 6.33522 17.1028 −15.4210 −31.7412 12.1470 −45.6077 13.1350 77.2635
1.3 −4.30003 1.71484 10.4903 12.4004 −7.37385 −1.37712 −10.7083 −24.0593 −53.3220
1.4 −4.13940 −6.47640 9.13466 −16.6315 26.8084 −19.1396 −4.69683 14.9437 68.8447
1.5 −3.50037 3.21110 4.25259 −4.81532 −11.2401 −10.9403 13.1173 −16.6888 16.8554
1.6 −2.28396 −5.99751 −2.78350 18.9293 13.6981 29.6644 24.6291 8.97009 −43.2338
1.7 −1.85177 8.78109 −4.57095 9.11609 −16.2606 17.9949 23.2785 50.1076 −16.8809
1.8 −1.18708 −3.66812 −6.59083 −21.5989 4.35437 18.3464 17.3205 −13.5449 25.6397
1.9 −0.892603 −5.49427 −7.20326 5.70663 4.90421 −27.0937 13.5705 3.18703 −5.09376
1.10 −0.0851144 2.52901 −7.99276 −2.35567 −0.215256 10.0027 1.36121 −20.6041 0.200501
1.11 0.507211 7.40108 −7.74274 −2.38303 3.75391 −25.3698 −7.98489 27.7760 −1.20870
1.12 0.880531 −6.93384 −7.22467 6.48347 −6.10546 15.0809 −13.4058 21.0781 5.70890
1.13 2.53888 −0.162038 −1.55407 16.0643 −0.411396 −19.7607 −24.2567 −26.9737 40.7854
1.14 2.73552 1.14229 −0.516927 −14.9626 3.12475 −6.67864 −23.2982 −25.6952 −40.9305
1.15 3.28989 7.61827 2.82336 19.7233 25.0632 16.9779 −17.0306 31.0380 64.8876
1.16 3.39109 −8.52507 3.49951 −10.3485 −28.9093 23.5919 −15.2616 45.6768 −35.0928
1.17 3.87013 2.66577 6.97791 −2.67905 10.3169 26.3908 −3.95562 −19.8937 −10.3683
1.18 4.40856 9.44900 11.4354 −19.6923 41.6565 −12.0675 15.1450 62.2837 −86.8147
1.19 5.17217 −4.39718 18.7513 10.5778 −22.7429 −8.51519 55.6077 −7.66484 54.7101
1.20 5.59523 7.28426 23.3067 8.94016 40.7572 10.6867 85.6443 26.0605 50.0223
\(n\): 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.p yes 20
11.b odd 2 1 2057.4.a.n ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2057.4.a.n ✓ 20 11.b odd 2 1
2057.4.a.p 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{20} - 4 T_{2}^{19} - 112 T_{2}^{18} + 438 T_{2}^{17} + 5176 T_{2}^{16} - 19774 T_{2}^{15} + \cdots + 10454400 \) Copy content Toggle raw display
\( T_{3}^{20} - 8 T_{3}^{19} - 335 T_{3}^{18} + 2606 T_{3}^{17} + 47179 T_{3}^{16} - 356432 T_{3}^{15} + \cdots - 2558870782976 \) Copy content Toggle raw display
\( T_{5}^{20} - 6 T_{5}^{19} - 1647 T_{5}^{18} + 12040 T_{5}^{17} + 1101598 T_{5}^{16} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 4 T^{19} + \cdots + 10454400 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots - 2558870782976 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 68\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( (T - 17)^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 34\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 16\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 54\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 19\!\cdots\!45 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 24\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 14\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 97\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 32\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 80\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 19\!\cdots\!55 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
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