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Properties

Label 43.3.d.a
Level $43$
Weight $3$
Character orbit 43.d
Analytic conductor $1.172$
Analytic rank $0$
Dimension $12$
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] = [43,3,Mod(7,43)]
 
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([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 43.d (of order \(6\), 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: \(1.17166513675\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 37x^{10} + 483x^{8} + 2718x^{6} + 6923x^{4} + 7253x^{2} + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + \beta_{3} q^{3} + (\beta_{11} + \beta_{9} + \beta_{5} + \cdots - 2) q^{4}+ \cdots + (2 \beta_{11} + \beta_{10} - \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + \beta_{3} q^{3} + (\beta_{11} + \beta_{9} + \beta_{5} + \cdots - 2) q^{4}+ \cdots + (11 \beta_{11} + 11 \beta_{10} + \cdots - 11 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 26 q^{4} - 3 q^{5} - 9 q^{6} + 9 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 26 q^{4} - 3 q^{5} - 9 q^{6} + 9 q^{7} - 12 q^{9} - q^{10} + 28 q^{11} - 6 q^{12} + 24 q^{13} - 18 q^{14} - 13 q^{15} + 110 q^{16} - 7 q^{17} + 33 q^{18} + 66 q^{19} - 99 q^{20} - 80 q^{21} - 16 q^{23} - 2 q^{24} - 21 q^{25} + 9 q^{26} - 192 q^{28} - 111 q^{29} + 99 q^{30} - 29 q^{31} - 114 q^{33} + 213 q^{34} + 38 q^{35} + 152 q^{36} + 120 q^{37} + 172 q^{38} - 29 q^{40} + 94 q^{41} + 5 q^{43} - 174 q^{44} + 156 q^{46} - 18 q^{47} - 213 q^{48} - 99 q^{49} - 198 q^{50} - 234 q^{52} - 58 q^{53} + 128 q^{54} - 258 q^{55} + 315 q^{56} + 51 q^{57} - 196 q^{58} + 336 q^{59} - 5 q^{60} + 204 q^{61} + 261 q^{62} - 153 q^{63} - 604 q^{64} - 201 q^{66} + 115 q^{67} - 106 q^{68} + 423 q^{69} - 66 q^{71} + 294 q^{72} + 249 q^{73} - 214 q^{74} - 438 q^{76} + 117 q^{77} + 136 q^{78} + 236 q^{79} + 681 q^{80} + 110 q^{81} - 4 q^{83} + 248 q^{84} + 102 q^{86} - 408 q^{87} - 45 q^{89} - 44 q^{90} - 156 q^{91} - 483 q^{92} - 567 q^{93} - 389 q^{95} - 278 q^{96} - 370 q^{97} - 879 q^{98} + 157 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 37x^{10} + 483x^{8} + 2718x^{6} + 6923x^{4} + 7253x^{2} + 1849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{10} + 137\nu^{8} + 1563\nu^{6} + 6752\nu^{4} + 10271\nu^{2} + 124\nu + 3053 ) / 248 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{10} + 137\nu^{8} + 1563\nu^{6} + 6752\nu^{4} + 10271\nu^{2} - 124\nu + 3053 ) / 248 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1485 \nu^{11} + 26101 \nu^{10} - 54558 \nu^{9} + 904290 \nu^{8} - 695669 \nu^{7} + \cdots + 18940167 ) / 810464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1485 \nu^{11} + 25327 \nu^{10} - 54558 \nu^{9} + 861118 \nu^{8} - 695669 \nu^{7} + \cdots + 11827709 ) / 810464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1485 \nu^{11} - 26101 \nu^{10} - 54558 \nu^{9} - 904290 \nu^{8} - 695669 \nu^{7} + \cdots - 18940167 ) / 810464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1370 \nu^{11} + 12857 \nu^{10} + 48798 \nu^{9} + 441352 \nu^{8} + 595576 \nu^{7} + 5030871 \nu^{6} + \cdots + 9110281 ) / 405232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71\nu^{11} + 2455\nu^{9} + 28402\nu^{7} + 125769\nu^{5} + 201197\nu^{3} + 73310\nu + 5332 ) / 10664 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12339 \nu^{11} - 387 \nu^{10} + 422014 \nu^{9} - 21586 \nu^{8} + 4765455 \nu^{7} - 412155 \nu^{6} + \cdots - 3556229 ) / 810464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16407 \nu^{11} + 30315 \nu^{10} - 564102 \nu^{9} + 1049286 \nu^{8} - 6440307 \nu^{7} + \cdots + 27766433 ) / 810464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16407 \nu^{11} + 30315 \nu^{10} + 564102 \nu^{9} + 1049286 \nu^{8} + 6440307 \nu^{7} + \cdots + 22903649 ) / 810464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8946 \nu^{11} - 8643 \nu^{10} + 309330 \nu^{9} - 296356 \nu^{8} + 3567988 \nu^{7} - 3379585 \nu^{6} + \cdots - 6970343 ) / 405232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{5} + \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{11} - 2 \beta_{10} - \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{11} + 3\beta_{10} - 13\beta_{9} - 12\beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} - 15\beta _1 + 62 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 52 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} - 74 \beta_{8} - 46 \beta_{7} - 28 \beta_{6} + 41 \beta_{5} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 227 \beta_{11} - 61 \beta_{10} + 166 \beta_{9} + 154 \beta_{5} + 10 \beta_{4} + 63 \beta_{3} + \cdots - 774 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 789 \beta_{11} - 619 \beta_{10} - 170 \beta_{9} + 1148 \beta_{8} + 934 \beta_{7} + 340 \beta_{6} + \cdots - 637 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3164 \beta_{11} + 1023 \beta_{10} - 2141 \beta_{9} - 2061 \beta_{5} - 66 \beta_{4} - 1037 \beta_{3} + \cdots + 10235 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11644 \beta_{11} + 9675 \beta_{10} + 1969 \beta_{9} - 17120 \beta_{8} - 17240 \beta_{7} - 3938 \beta_{6} + \cdots + 10589 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 44107 \beta_{11} - 16266 \beta_{10} + 27841 \beta_{9} + 28102 \beta_{5} + 41 \beta_{4} + 15964 \beta_{3} + \cdots - 138121 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 170609 \beta_{11} - 148565 \beta_{10} - 22044 \beta_{9} + 252482 \beta_{8} + 298456 \beta_{7} + \cdots - 171272 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
− 3.53204i
− 1.61947i
− 0.604188i
1.51156i
2.15301i
3.82317i
− 3.82317i
− 2.15301i
− 1.51156i
0.604188i
1.61947i
3.53204i
− 3.53204i 0.357493 − 0.206399i −8.47532 2.82830 − 1.63292i −0.729010 − 1.26268i 0.813980 + 0.469951i 15.8070i −4.41480 + 7.64666i −5.76753 − 9.98966i
7.2 − 1.61947i 2.75697 − 1.59174i 1.37732 −6.40200 + 3.69620i −2.57777 − 4.46483i 1.18578 + 0.684612i − 8.70840i 0.567259 − 0.982521i 5.98588 + 10.3678i
7.3 − 0.604188i −1.35447 + 0.782004i 3.63496 4.60389 − 2.65806i 0.472478 + 0.818356i −0.191259 − 0.110424i − 4.61295i −3.27694 + 5.67582i −1.60597 − 2.78161i
7.4 1.51156i −3.66976 + 2.11873i 1.71518 −4.51092 + 2.60438i −3.20260 − 5.54707i 1.23619 + 0.713712i 8.63885i 4.47807 − 7.75625i −3.93669 − 6.81854i
7.5 2.15301i 2.77779 − 1.60376i −0.635471 −0.468965 + 0.270757i 3.45292 + 5.98063i −7.68536 − 4.43714i 7.24388i 0.644090 − 1.11560i −0.582944 − 1.00969i
7.6 3.82317i −0.868030 + 0.501157i −10.6167 2.44970 − 1.41434i −1.91601 − 3.31863i 9.14067 + 5.27737i − 25.2966i −3.99768 + 6.92419i 5.40725 + 9.36563i
37.1 − 3.82317i −0.868030 − 0.501157i −10.6167 2.44970 + 1.41434i −1.91601 + 3.31863i 9.14067 − 5.27737i 25.2966i −3.99768 − 6.92419i 5.40725 − 9.36563i
37.2 − 2.15301i 2.77779 + 1.60376i −0.635471 −0.468965 − 0.270757i 3.45292 − 5.98063i −7.68536 + 4.43714i − 7.24388i 0.644090 + 1.11560i −0.582944 + 1.00969i
37.3 − 1.51156i −3.66976 − 2.11873i 1.71518 −4.51092 − 2.60438i −3.20260 + 5.54707i 1.23619 − 0.713712i − 8.63885i 4.47807 + 7.75625i −3.93669 + 6.81854i
37.4 0.604188i −1.35447 − 0.782004i 3.63496 4.60389 + 2.65806i 0.472478 − 0.818356i −0.191259 + 0.110424i 4.61295i −3.27694 − 5.67582i −1.60597 + 2.78161i
37.5 1.61947i 2.75697 + 1.59174i 1.37732 −6.40200 − 3.69620i −2.57777 + 4.46483i 1.18578 − 0.684612i 8.70840i 0.567259 + 0.982521i 5.98588 − 10.3678i
37.6 3.53204i 0.357493 + 0.206399i −8.47532 2.82830 + 1.63292i −0.729010 + 1.26268i 0.813980 − 0.469951i − 15.8070i −4.41480 − 7.64666i −5.76753 + 9.98966i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.d.a ✓ 12
3.b odd 2 1 387.3.j.c 12
4.b odd 2 1 688.3.t.c 12
43.d odd 6 1 inner 43.3.d.a ✓ 12
129.h even 6 1 387.3.j.c 12
172.f even 6 1 688.3.t.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.d.a ✓ 12 1.a even 1 1 trivial
43.3.d.a ✓ 12 43.d odd 6 1 inner
387.3.j.c 12 3.b odd 2 1
387.3.j.c 12 129.h even 6 1
688.3.t.c 12 4.b odd 2 1
688.3.t.c 12 172.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(43, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 37 T^{10} + \cdots + 1849 \) Copy content Toggle raw display
$3$ \( T^{12} - 21 T^{10} + \cdots + 784 \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 1048576 \) Copy content Toggle raw display
$7$ \( T^{12} - 9 T^{11} + \cdots + 1444 \) Copy content Toggle raw display
$11$ \( (T^{6} - 14 T^{5} + \cdots - 178808)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 232783090576 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 98916610336489 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 102837203793424 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 485480586573376 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 462743427606784 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} - 47 T^{5} + \cdots - 6575768)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 39\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( (T^{6} + 9 T^{5} + \cdots + 1304641296)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 98\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{6} - 168 T^{5} + \cdots + 141472984)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 99\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 85\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 85\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( (T^{6} + 185 T^{5} + \cdots + 170062861216)^{2} \) Copy content Toggle raw display
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