Properties

Label 2057.4.a.i
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 55 x^{8} + 72 x^{7} + 1037 x^{6} - 812 x^{5} - 7851 x^{4} + 2526 x^{3} + 20108 x^{2} + 4072 x - 5760 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 187)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{7} + \beta_{4} - 4) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 3) q^{6} + (\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 5) q^{7} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 9) q^{8}+ \cdots + (\beta_{9} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{7} + \beta_{4} - 4) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1 - 3) q^{6} + (\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 5) q^{7} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 9) q^{8}+ \cdots + ( - 48 \beta_{9} - 79 \beta_{8} - 24 \beta_{7} + 56 \beta_{6} - 8 \beta_{5} + \cdots + 61) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 8 q^{2} - 9 q^{3} + 40 q^{4} - 41 q^{5} - 31 q^{6} + 63 q^{7} + 96 q^{8} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 8 q^{2} - 9 q^{3} + 40 q^{4} - 41 q^{5} - 31 q^{6} + 63 q^{7} + 96 q^{8} + 61 q^{9} + 47 q^{10} - 171 q^{12} + 99 q^{13} - 95 q^{14} - 150 q^{15} + 180 q^{16} + 170 q^{17} + 207 q^{18} + 73 q^{19} - 343 q^{20} + 152 q^{21} - 58 q^{23} - 157 q^{24} + 529 q^{25} - 285 q^{26} - 246 q^{27} + 1031 q^{28} + 754 q^{29} + 754 q^{30} - 228 q^{31} + 466 q^{32} + 136 q^{34} - 20 q^{35} - 535 q^{36} - 1177 q^{37} - 747 q^{38} + 791 q^{39} - 537 q^{40} + 327 q^{41} - 632 q^{42} + 487 q^{43} - 834 q^{45} + 854 q^{46} - 573 q^{47} - 1857 q^{48} + 2215 q^{49} + 855 q^{50} - 153 q^{51} - 1673 q^{52} - 2022 q^{53} - 3572 q^{54} + 1997 q^{56} + 1310 q^{57} + 4514 q^{58} - 602 q^{59} + 4936 q^{60} + 570 q^{61} - 166 q^{62} + 1903 q^{63} + 968 q^{64} + 2018 q^{65} - 12 q^{67} + 680 q^{68} - 1034 q^{69} + 5460 q^{70} + 636 q^{71} + 2989 q^{72} + 201 q^{73} - 959 q^{74} + 3213 q^{75} - 6999 q^{76} + 3669 q^{78} + 1097 q^{79} - 1409 q^{80} + 3554 q^{81} + 2249 q^{82} + 477 q^{83} - 8476 q^{84} - 697 q^{85} - 1271 q^{86} - 1782 q^{87} - 195 q^{89} - 3292 q^{90} - 1128 q^{91} - 2064 q^{92} - 2632 q^{93} - 465 q^{94} + 1644 q^{95} - 8367 q^{96} - 994 q^{97} - 225 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 55 x^{8} + 72 x^{7} + 1037 x^{6} - 812 x^{5} - 7851 x^{4} + 2526 x^{3} + 20108 x^{2} + 4072 x - 5760 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8581 \nu^{9} + 87924 \nu^{8} - 477211 \nu^{7} - 4373978 \nu^{6} + 3699365 \nu^{5} + 63749218 \nu^{4} + 63143541 \nu^{3} - 262851596 \nu^{2} + \cdots - 66046272 ) / 38091008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3389 \nu^{9} - 51454 \nu^{8} + 53385 \nu^{7} + 1596748 \nu^{6} - 4145275 \nu^{5} - 10681936 \nu^{4} + 42656301 \nu^{3} - 21239998 \nu^{2} + \cdots + 85931328 ) / 9522752 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 65857 \nu^{9} - 19492 \nu^{8} - 3591727 \nu^{7} - 1423226 \nu^{6} + 66918273 \nu^{5} + 43172786 \nu^{4} - 500621839 \nu^{3} - 300571300 \nu^{2} + \cdots + 447588544 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 100245 \nu^{9} + 308020 \nu^{8} + 3973787 \nu^{7} - 9256382 \nu^{6} - 41124309 \nu^{5} + 72105766 \nu^{4} + 53847931 \nu^{3} - 38832332 \nu^{2} + \cdots - 190803392 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 114545 \nu^{9} + 219204 \nu^{8} + 6247231 \nu^{7} - 8101958 \nu^{6} - 112129585 \nu^{5} + 92980046 \nu^{4} + 741971295 \nu^{3} + \cdots - 77756096 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 131631 \nu^{9} - 771020 \nu^{8} - 6038849 \nu^{7} + 33447594 \nu^{6} + 97500335 \nu^{5} - 458116482 \nu^{4} - 633224833 \nu^{3} + 1991611828 \nu^{2} + \cdots - 829705408 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 107255 \nu^{9} + 351052 \nu^{8} + 5474489 \nu^{7} - 14597242 \nu^{6} - 95190391 \nu^{5} + 204902354 \nu^{4} + 647740985 \nu^{3} + \cdots + 724671168 ) / 38091008 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{3} + \beta_{2} + 20\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} + 7\beta_{8} - 2\beta_{6} + 7\beta_{5} - 3\beta_{4} + 26\beta_{2} + 45\beta _1 + 210 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 42 \beta_{9} + 74 \beta_{8} + 42 \beta_{7} + 22 \beta_{6} + 86 \beta_{5} - 10 \beta_{4} + 52 \beta_{3} + 58 \beta_{2} + 485 \beta _1 + 418 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 296 \beta_{9} + 342 \beta_{8} + 86 \beta_{7} - 82 \beta_{6} + 358 \beta_{5} - 152 \beta_{4} + 38 \beta_{3} + 699 \beta_{2} + 1677 \beta _1 + 4999 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1531 \beta_{9} + 2488 \beta_{8} + 1517 \beta_{7} + 313 \beta_{6} + 3028 \beta_{5} - 588 \beta_{4} + 1316 \beta_{3} + 2365 \beta_{2} + 13442 \beta _1 + 16115 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10275 \beta_{9} + 12859 \beta_{8} + 5048 \beta_{7} - 2678 \beta_{6} + 14211 \beta_{5} - 5725 \beta_{4} + 2362 \beta_{3} + 20172 \beta_{2} + 58807 \beta _1 + 137170 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 53272 \beta_{9} + 82310 \beta_{8} + 51012 \beta_{7} + 1064 \beta_{6} + 101470 \beta_{5} - 24920 \beta_{4} + 35658 \beta_{3} + 86246 \beta_{2} + 404569 \beta _1 + 576936 \) Copy content Toggle raw display

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
5.76153
3.99555
3.25015
2.44876
0.447769
−0.813593
−1.54722
−3.27779
−3.88370
−4.38145
−4.76153 −2.80312 14.6722 −15.6524 13.3472 31.0159 −31.7697 −19.1425 74.5294
1.2 −2.99555 −2.94594 0.973314 8.57150 8.82472 19.5556 21.0488 −18.3214 −25.6764
1.3 −2.25015 1.42575 −2.93684 −11.7202 −3.20814 −23.4792 24.6095 −24.9672 26.3722
1.4 −1.44876 9.43069 −5.90109 −19.3257 −13.6628 33.2851 20.1394 61.9379 27.9984
1.5 0.552231 −9.39468 −7.69504 9.49824 −5.18803 −15.9512 −8.66728 61.2599 5.24522
1.6 1.81359 4.59414 −4.71088 2.19824 8.33191 8.16262 −23.0524 −5.89384 3.98672
1.7 2.54722 −5.06531 −1.51166 −0.212294 −12.9025 −8.35551 −24.2283 −1.34267 −0.540760
1.8 4.27779 2.36616 10.2995 −19.2230 10.1219 −31.3553 9.83664 −21.4013 −82.2318
1.9 4.88370 2.22066 15.8506 17.8151 10.8451 27.4352 38.3399 −22.0687 87.0038
1.10 5.38145 −8.82835 20.9600 −12.9495 −47.5093 22.6868 69.7435 50.9398 −69.6868
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.i 10
11.b odd 2 1 187.4.a.d 10
33.d even 2 1 1683.4.a.m 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.a.d 10 11.b odd 2 1
1683.4.a.m 10 33.d even 2 1
2057.4.a.i 10 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{10} - 8 T_{2}^{9} - 28 T_{2}^{8} + 320 T_{2}^{7} + 43 T_{2}^{6} - 3842 T_{2}^{5} + 2272 T_{2}^{4} + 16866 T_{2}^{3} - 12040 T_{2}^{2} - 22680 T_{2} + 13336 \) Copy content Toggle raw display
\( T_{3}^{10} + 9 T_{3}^{9} - 125 T_{3}^{8} - 1124 T_{3}^{7} + 3693 T_{3}^{6} + 31974 T_{3}^{5} - 46891 T_{3}^{4} - 295260 T_{3}^{3} + 350102 T_{3}^{2} + 883601 T_{3} - 1126032 \) Copy content Toggle raw display
\( T_{5}^{10} + 41 T_{5}^{9} - 49 T_{5}^{8} - 20054 T_{5}^{7} - 156539 T_{5}^{6} + 2613964 T_{5}^{5} + 26309445 T_{5}^{4} - 128550290 T_{5}^{3} - 1161812718 T_{5}^{2} + 2573173715 T_{5} + 597348544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 8 T^{9} - 28 T^{8} + \cdots + 13336 \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{9} - 125 T^{8} + \cdots - 1126032 \) Copy content Toggle raw display
$5$ \( T^{10} + 41 T^{9} + \cdots + 597348544 \) Copy content Toggle raw display
$7$ \( T^{10} - 63 T^{9} + \cdots + 10064173681152 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 99 T^{9} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T - 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} - 73 T^{9} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + 58 T^{9} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{10} - 754 T^{9} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + 228 T^{9} + \cdots - 55\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{10} + 1177 T^{9} + \cdots - 88\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{10} - 327 T^{9} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{10} - 487 T^{9} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + 573 T^{9} + \cdots + 26\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{10} + 2022 T^{9} + \cdots - 41\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + 602 T^{9} + \cdots + 90\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{10} - 570 T^{9} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{10} + 12 T^{9} + \cdots - 24\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{10} - 636 T^{9} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} - 201 T^{9} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{10} - 1097 T^{9} + \cdots + 74\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{10} - 477 T^{9} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + 195 T^{9} + \cdots - 35\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T^{10} + 994 T^{9} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
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