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Properties

Label 43.4.c.a
Level $43$
Weight $4$
Character orbit 43.c
Analytic conductor $2.537$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,4,Mod(6,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.6");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 43.c (of order \(3\), 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: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 60 x^{18} - 25 x^{17} + 2336 x^{16} - 645 x^{15} + 52478 x^{14} - 2415 x^{13} + \cdots + 589824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{6} - \beta_{4}) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{7} - 2 \beta_{4}) q^{5} + (\beta_{6} + 2 \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{18} + \beta_{17} + \beta_{16} + \cdots - 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{6} - \beta_{4}) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{7} - 2 \beta_{4}) q^{5} + (\beta_{6} + 2 \beta_{4} + \cdots + \beta_1) q^{6}+ \cdots + ( - 5 \beta_{19} + 30 \beta_{18} + \cdots - 397) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} - 5 q^{3} + 78 q^{4} - 19 q^{5} + 15 q^{6} - 51 q^{7} - 72 q^{8} - 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} - 5 q^{3} + 78 q^{4} - 19 q^{5} + 15 q^{6} - 51 q^{7} - 72 q^{8} - 117 q^{9} + 27 q^{10} + 54 q^{11} - 72 q^{12} - 15 q^{13} + 96 q^{14} + 65 q^{15} + 134 q^{16} - 82 q^{17} + 247 q^{18} + 78 q^{19} - 495 q^{20} - 18 q^{21} + 380 q^{22} - 61 q^{23} + 202 q^{24} - 151 q^{25} - 21 q^{26} - 194 q^{27} - 794 q^{28} - 53 q^{29} + 627 q^{30} + 253 q^{31} - 798 q^{32} - 424 q^{33} - 231 q^{34} + 710 q^{35} - 1092 q^{36} - 129 q^{37} - 854 q^{38} + 1382 q^{39} + 1345 q^{40} + 782 q^{41} + 62 q^{42} + 1025 q^{43} + 754 q^{44} + 1888 q^{45} - 40 q^{46} - 668 q^{47} - 2401 q^{48} - 115 q^{49} + 424 q^{50} + 1590 q^{51} - 564 q^{52} + 773 q^{53} + 364 q^{54} - 1242 q^{55} - 923 q^{56} - 765 q^{57} + 1328 q^{58} - 2966 q^{59} - 1075 q^{60} + 437 q^{61} + 1509 q^{62} - 2222 q^{63} - 1476 q^{64} - 2126 q^{65} + 1483 q^{66} - 642 q^{67} - 1052 q^{68} - 3503 q^{69} - 170 q^{70} - 1545 q^{71} + 3834 q^{72} + 1292 q^{73} - 2232 q^{74} + 164 q^{75} - 252 q^{76} + 1448 q^{77} + 5644 q^{78} - 1405 q^{79} - 3157 q^{80} + 974 q^{81} + 6608 q^{82} + 543 q^{83} + 7304 q^{84} + 1946 q^{85} + 2776 q^{86} + 2818 q^{87} - 5372 q^{88} - 2196 q^{89} - 1484 q^{90} - 3513 q^{91} + 2629 q^{92} - 983 q^{93} + 9878 q^{94} - 149 q^{95} + 3540 q^{96} - 850 q^{97} - 213 q^{98} - 3181 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 60 x^{18} - 25 x^{17} + 2336 x^{16} - 645 x^{15} + 52478 x^{14} - 2415 x^{13} + \cdots + 589824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!63 \nu^{19} + \cdots - 21\!\cdots\!60 ) / 18\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62\!\cdots\!89 \nu^{19} + \cdots + 54\!\cdots\!60 ) / 25\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!69 \nu^{19} + \cdots + 11\!\cdots\!04 ) / 11\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!99 \nu^{19} + \cdots + 42\!\cdots\!64 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!97 \nu^{19} + \cdots - 88\!\cdots\!28 ) / 25\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 91\!\cdots\!75 \nu^{19} + \cdots - 35\!\cdots\!88 ) / 12\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!61 \nu^{19} + \cdots + 26\!\cdots\!84 ) / 25\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 85\!\cdots\!99 \nu^{19} + \cdots - 35\!\cdots\!36 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 87\!\cdots\!51 \nu^{19} + \cdots + 37\!\cdots\!28 ) / 25\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\!\cdots\!73 \nu^{19} + \cdots + 75\!\cdots\!08 ) / 51\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!29 \nu^{19} + \cdots + 24\!\cdots\!48 ) / 80\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 66\!\cdots\!41 \nu^{19} + \cdots + 25\!\cdots\!56 ) / 85\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 60\!\cdots\!55 \nu^{19} + \cdots + 19\!\cdots\!92 ) / 51\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 79\!\cdots\!09 \nu^{19} + \cdots + 17\!\cdots\!20 ) / 57\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 22\!\cdots\!03 \nu^{19} + \cdots + 16\!\cdots\!80 ) / 57\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!27 \nu^{19} + \cdots + 20\!\cdots\!24 ) / 25\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 30\!\cdots\!17 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 51\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 66\!\cdots\!41 \nu^{19} + \cdots + 18\!\cdots\!52 ) / 53\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 12\beta_{4} - \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{17} + \beta_{11} - \beta_{8} + 19\beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} - \beta_{12} - \beta_{10} - 26 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 5 \beta_{6} + \cdots - 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} - 33 \beta_{18} - 31 \beta_{17} + 2 \beta_{16} + 7 \beta_{15} - 10 \beta_{14} - 2 \beta_{13} + \cdots + 175 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 39 \beta_{19} - 85 \beta_{18} - 114 \beta_{17} + 35 \beta_{16} - 29 \beta_{15} - 261 \beta_{14} + \cdots + 5220 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 89 \beta_{13} - 26 \beta_{12} - 797 \beta_{10} + 423 \beta_{9} + 941 \beta_{8} - 858 \beta_{7} + \cdots + 9604 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1214 \beta_{19} + 2809 \beta_{18} + 3510 \beta_{17} - 1071 \beta_{16} + 602 \beta_{15} + \cdots - 125541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 262 \beta_{19} + 25994 \beta_{18} + 23623 \beta_{17} - 3051 \beta_{16} - 11118 \beta_{15} + \cdots - 200385 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 31428 \beta_{13} - 35011 \beta_{12} - 13116 \beta_{10} - 439433 \beta_{9} - 85759 \beta_{8} + \cdots - 245466 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8957 \beta_{19} - 713246 \beta_{18} - 653525 \beta_{17} + 96345 \beta_{16} + 334769 \beta_{15} + \cdots + 6271036 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 976201 \beta_{19} - 2537170 \beta_{18} - 2919128 \beta_{17} + 900906 \beta_{16} - 56975 \beta_{15} + \cdots + 82182493 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2937448 \beta_{13} + 695832 \beta_{12} - 14300942 \beta_{10} + 19184426 \beta_{9} + \cdots + 155903951 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 26786708 \beta_{19} + 73994922 \beta_{18} + 83012388 \beta_{17} - 25459314 \beta_{16} + \cdots - 2177138374 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 30644196 \beta_{19} + 539337187 \beta_{18} + 505673069 \beta_{17} - 87877358 \beta_{16} + \cdots - 5786632798 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 713546967 \beta_{13} - 729712581 \beta_{12} + 192190169 \beta_{10} - 8246507944 \beta_{9} + \cdots - 8522961601 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1128933483 \beta_{19} - 14902802571 \beta_{18} - 14117320811 \beta_{17} + 2596956032 \beta_{16} + \cdots + 172553617129 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 19828506251 \beta_{19} - 61720140511 \beta_{18} - 66939618878 \beta_{17} + 19912043553 \beta_{16} + \cdots + 1583906721350 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 76069946855 \beta_{13} + 38117543562 \beta_{12} - 265039818431 \beta_{10} + 581108305805 \beta_{9} + \cdots + 2959713591578 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
2.65668 − 4.60150i
2.11736 − 3.66738i
1.77920 − 3.08166i
1.07390 − 1.86006i
0.0954556 − 0.165334i
−0.0898836 + 0.155683i
−0.961392 + 1.66518i
−1.79682 + 3.11219i
−1.91170 + 3.31117i
−2.46280 + 4.26569i
2.65668 + 4.60150i
2.11736 + 3.66738i
1.77920 + 3.08166i
1.07390 + 1.86006i
0.0954556 + 0.165334i
−0.0898836 − 0.155683i
−0.961392 − 1.66518i
−1.79682 − 3.11219i
−1.91170 − 3.31117i
−2.46280 − 4.26569i
−5.31336 −4.02821 + 6.97706i 20.2317 −7.95967 + 13.7866i 21.4033 − 37.0716i −8.31614 − 14.4040i −64.9916 −18.9529 − 32.8274i 42.2926 − 73.2529i
6.2 −4.23473 4.42929 − 7.67176i 9.93292 −2.98244 + 5.16574i −18.7568 + 32.4878i −11.2528 − 19.4903i −8.18537 −25.7372 − 44.5782i 12.6298 − 21.8755i
6.3 −3.55840 0.194544 − 0.336960i 4.66219 0.863899 − 1.49632i −0.692264 + 1.19904i 10.5931 + 18.3477i 11.8773 13.4243 + 23.2516i −3.07410 + 5.32449i
6.4 −2.14781 −3.42441 + 5.93126i −3.38692 8.56817 − 14.8405i 7.35498 − 12.7392i −11.4893 − 19.9001i 24.4569 −9.95322 − 17.2395i −18.4028 + 31.8746i
6.5 −0.190911 −0.719182 + 1.24566i −7.96355 −8.17312 + 14.1563i 0.137300 − 0.237810i −3.11997 − 5.40394i 3.04762 12.4656 + 21.5910i 1.56034 − 2.70259i
6.6 0.179767 3.01124 − 5.21562i −7.96768 5.48517 − 9.50060i 0.541322 − 0.937597i −4.39183 − 7.60688i −2.87047 −4.63513 − 8.02828i 0.986054 − 1.70790i
6.7 1.92278 −4.18916 + 7.25584i −4.30290 −0.0351129 + 0.0608173i −8.05486 + 13.9514i 11.7213 + 20.3018i −23.6558 −21.5982 − 37.4091i −0.0675145 + 0.116939i
6.8 3.59365 0.838961 − 1.45312i 4.91431 4.86091 − 8.41934i 3.01493 − 5.22201i 7.88013 + 13.6488i −11.0889 12.0923 + 20.9445i 17.4684 − 30.2561i
6.9 3.82341 3.88258 − 6.72483i 6.61843 −9.06615 + 15.7030i 14.8447 − 25.7118i −2.59898 − 4.50156i −5.28231 −16.6489 − 28.8368i −34.6635 + 60.0390i
6.10 4.92559 −2.49565 + 4.32260i 16.2615 −1.06165 + 1.83883i −12.2926 + 21.2914i −14.5255 − 25.1588i 40.6926 1.04342 + 1.80726i −5.22925 + 9.05732i
36.1 −5.31336 −4.02821 − 6.97706i 20.2317 −7.95967 − 13.7866i 21.4033 + 37.0716i −8.31614 + 14.4040i −64.9916 −18.9529 + 32.8274i 42.2926 + 73.2529i
36.2 −4.23473 4.42929 + 7.67176i 9.93292 −2.98244 − 5.16574i −18.7568 − 32.4878i −11.2528 + 19.4903i −8.18537 −25.7372 + 44.5782i 12.6298 + 21.8755i
36.3 −3.55840 0.194544 + 0.336960i 4.66219 0.863899 + 1.49632i −0.692264 − 1.19904i 10.5931 − 18.3477i 11.8773 13.4243 − 23.2516i −3.07410 − 5.32449i
36.4 −2.14781 −3.42441 − 5.93126i −3.38692 8.56817 + 14.8405i 7.35498 + 12.7392i −11.4893 + 19.9001i 24.4569 −9.95322 + 17.2395i −18.4028 − 31.8746i
36.5 −0.190911 −0.719182 − 1.24566i −7.96355 −8.17312 − 14.1563i 0.137300 + 0.237810i −3.11997 + 5.40394i 3.04762 12.4656 − 21.5910i 1.56034 + 2.70259i
36.6 0.179767 3.01124 + 5.21562i −7.96768 5.48517 + 9.50060i 0.541322 + 0.937597i −4.39183 + 7.60688i −2.87047 −4.63513 + 8.02828i 0.986054 + 1.70790i
36.7 1.92278 −4.18916 − 7.25584i −4.30290 −0.0351129 − 0.0608173i −8.05486 − 13.9514i 11.7213 − 20.3018i −23.6558 −21.5982 + 37.4091i −0.0675145 − 0.116939i
36.8 3.59365 0.838961 + 1.45312i 4.91431 4.86091 + 8.41934i 3.01493 + 5.22201i 7.88013 − 13.6488i −11.0889 12.0923 − 20.9445i 17.4684 + 30.2561i
36.9 3.82341 3.88258 + 6.72483i 6.61843 −9.06615 − 15.7030i 14.8447 + 25.7118i −2.59898 + 4.50156i −5.28231 −16.6489 + 28.8368i −34.6635 − 60.0390i
36.10 4.92559 −2.49565 − 4.32260i 16.2615 −1.06165 − 1.83883i −12.2926 − 21.2914i −14.5255 + 25.1588i 40.6926 1.04342 − 1.80726i −5.22925 − 9.05732i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.4.c.a ✓ 20
43.c even 3 1 inner 43.4.c.a ✓ 20
43.c even 3 1 1849.4.a.d 10
43.d odd 6 1 1849.4.a.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.c.a ✓ 20 1.a even 1 1 trivial
43.4.c.a ✓ 20 43.c even 3 1 inner
1849.4.a.d 10 43.c even 3 1
1849.4.a.f 10 43.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(43, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + T^{9} - 59 T^{8} + \cdots - 768)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 805779113104 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 175617200260096 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 54\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 30\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 98\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 43\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 24\!\cdots\!52)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 16\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 81\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 89\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 99\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 51\!\cdots\!44)^{2} \) Copy content Toggle raw display
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