Properties

Label 43.6.a.b
Level $43$
Weight $6$
Character orbit 43.a
Self dual yes
Analytic conductor $6.897$
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] = [43,6,Mod(1,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.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: \(6.89650425196\)
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: \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{6} + 3) q^{3} + (\beta_{6} + \beta_{5} - \beta_1 + 21) q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 + 13) q^{5} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_{2} - 11 \beta_1 + 9) q^{6} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 8) q^{7} + ( - 2 \beta_{9} + 5 \beta_{8} + 2 \beta_{7} + 3 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 37) q^{8}+ \cdots + (\beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} + \cdots + 133) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{6} + 3) q^{3} + (\beta_{6} + \beta_{5} - \beta_1 + 21) q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 + 13) q^{5} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_{2} - 11 \beta_1 + 9) q^{6} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 8) q^{7} + ( - 2 \beta_{9} + 5 \beta_{8} + 2 \beta_{7} + 3 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 37) q^{8}+ \cdots + (677 \beta_{9} - 740 \beta_{8} - 1481 \beta_{7} - 2577 \beta_{6} + \cdots - 26859) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 8 q^{2} + 28 q^{3} + 202 q^{4} + 138 q^{5} + 75 q^{6} + 60 q^{7} + 294 q^{8} + 1356 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 8 q^{2} + 28 q^{3} + 202 q^{4} + 138 q^{5} + 75 q^{6} + 60 q^{7} + 294 q^{8} + 1356 q^{9} - 17 q^{10} + 745 q^{11} + 4627 q^{12} + 1917 q^{13} + 1936 q^{14} + 1688 q^{15} + 5354 q^{16} + 4017 q^{17} - 2725 q^{18} - 2404 q^{19} + 1311 q^{20} - 228 q^{21} - 5836 q^{22} + 1733 q^{23} - 10711 q^{24} + 7120 q^{25} - 1484 q^{26} - 2324 q^{27} - 15028 q^{28} + 6996 q^{29} - 48420 q^{30} - 4899 q^{31} - 7554 q^{32} - 15734 q^{33} - 27033 q^{34} + 7084 q^{35} + 4433 q^{36} + 1466 q^{37} + 13905 q^{38} - 26542 q^{39} - 93211 q^{40} + 10297 q^{41} - 37642 q^{42} + 18490 q^{43} - 36140 q^{44} + 73822 q^{45} + 17991 q^{46} + 48592 q^{47} + 83607 q^{48} + 29458 q^{49} + 983 q^{50} + 92972 q^{51} + 14232 q^{52} + 127165 q^{53} - 92002 q^{54} + 106672 q^{55} - 7780 q^{56} + 34060 q^{57} - 10305 q^{58} + 99372 q^{59} + 111372 q^{60} + 17408 q^{61} + 28265 q^{62} + 2244 q^{63} + 47202 q^{64} + 54484 q^{65} - 150292 q^{66} - 2021 q^{67} + 192151 q^{68} + 1654 q^{69} - 33194 q^{70} + 11286 q^{71} - 298365 q^{72} + 49892 q^{73} - 125431 q^{74} - 44662 q^{75} - 249803 q^{76} + 98144 q^{77} - 28494 q^{78} - 91524 q^{79} + 12251 q^{80} - 26450 q^{81} - 158909 q^{82} - 105203 q^{83} - 357682 q^{84} - 87212 q^{85} + 14792 q^{86} + 181200 q^{87} - 461824 q^{88} - 62682 q^{89} - 522670 q^{90} - 295304 q^{91} + 183783 q^{92} - 238430 q^{93} + 7259 q^{94} - 305340 q^{95} - 162399 q^{96} + 108383 q^{97} + 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18033 \nu^{9} - 1188163 \nu^{8} + 3451179 \nu^{7} + 239748231 \nu^{6} - 1005840413 \nu^{5} - 14422814397 \nu^{4} + 59540816201 \nu^{3} + \cdots - 40045363612 ) / 24633272448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24107 \nu^{9} - 594245 \nu^{8} + 7832247 \nu^{7} + 116173337 \nu^{6} - 397671097 \nu^{5} - 6546797939 \nu^{4} - 20833059243 \nu^{3} + \cdots - 192111505364 ) / 24633272448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 144543 \nu^{9} - 820999 \nu^{8} + 35600547 \nu^{7} + 213763899 \nu^{6} - 2591602757 \nu^{5} - 15768164889 \nu^{4} + \cdots - 912070606876 ) / 24633272448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 474917 \nu^{9} + 1144977 \nu^{8} - 116809809 \nu^{7} - 421392509 \nu^{6} + 8979038055 \nu^{5} + 39727817087 \nu^{4} + \cdots + 2423789997252 ) / 65688726528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 474917 \nu^{9} - 1144977 \nu^{8} + 116809809 \nu^{7} + 421392509 \nu^{6} - 8979038055 \nu^{5} - 39727817087 \nu^{4} + \cdots - 5839603776708 ) / 65688726528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 808641 \nu^{9} - 7898035 \nu^{8} - 155012637 \nu^{7} + 1407118071 \nu^{6} + 9333970747 \nu^{5} - 78913681917 \nu^{4} + \cdots - 7896805510540 ) / 98533089792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5132245 \nu^{9} + 307343 \nu^{8} + 1269871233 \nu^{7} + 1411070749 \nu^{6} - 100861721975 \nu^{5} - 189530468575 \nu^{4} + \cdots - 30463422181444 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2389487 \nu^{9} - 2695583 \nu^{8} + 612520467 \nu^{7} + 1209002243 \nu^{6} - 50464104709 \nu^{5} - 122440956545 \nu^{4} + \cdots - 18517909728476 ) / 49266544896 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta _1 + 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} - 5\beta_{8} - 2\beta_{7} + 3\beta_{6} - \beta_{4} + \beta_{3} + 87\beta _1 + 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{8} - 16\beta_{7} + 126\beta_{6} + 113\beta_{5} - 7\beta_{4} + 13\beta_{3} + 20\beta_{2} + 245\beta _1 + 4542 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 278 \beta_{9} - 716 \beta_{8} - 308 \beta_{7} + 608 \beta_{6} + 92 \beta_{5} - 226 \beta_{4} + 250 \beta_{3} + 18 \beta_{2} + 8905 \beta _1 + 13392 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 202 \beta_{9} - 1946 \beta_{8} - 2948 \beta_{7} + 16013 \beta_{6} + 11839 \beta_{5} - 2292 \beta_{4} + 2520 \beta_{3} + 3422 \beta_{2} + 40767 \beta _1 + 465882 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 34356 \beta_{9} - 90155 \beta_{8} - 42446 \beta_{7} + 97339 \beta_{6} + 22646 \beta_{5} - 41129 \beta_{4} + 42101 \beta_{3} + 9022 \beta_{2} + 980257 \beta _1 + 2212914 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 68250 \beta_{9} - 383649 \beta_{8} - 450492 \beta_{7} + 2030502 \beta_{6} + 1254459 \beta_{5} - 456771 \beta_{4} + 406485 \beta_{3} + 469962 \beta_{2} + 5987031 \beta _1 + 51485248 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4138872 \beta_{9} - 11178606 \beta_{8} - 5738016 \beta_{7} + 14588868 \beta_{6} + 3971370 \beta_{5} - 6655062 \beta_{4} + 6262002 \beta_{3} + 2108976 \beta_{2} + \cdots + 324019770 \) 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
11.5305
9.86547
5.31531
3.50018
2.86024
−1.48720
−4.38824
−6.91219
−8.57770
−9.70631
−10.5305 27.4953 78.8905 86.8464 −289.538 −19.8137 −493.778 512.989 −914.532
1.2 −8.86547 1.50169 46.5966 −37.8251 −13.3132 −124.747 −129.406 −240.745 335.337
1.3 −4.31531 −23.8469 −13.3781 −52.5837 102.907 −174.859 195.821 325.673 226.915
1.4 −2.50018 16.8892 −25.7491 47.4635 −42.2260 67.4603 144.383 42.2439 −118.667
1.5 −1.86024 −14.8716 −28.5395 −42.2365 27.6647 202.971 112.618 −21.8357 78.5700
1.6 2.48720 −27.5943 −25.8138 101.308 −68.6327 15.7005 −143.795 518.448 251.974
1.7 5.38824 25.0462 −2.96684 −0.456695 134.955 166.517 −188.410 384.312 −2.46078
1.8 7.91219 12.8799 30.6028 79.5677 101.908 −172.354 −11.0549 −77.1083 629.555
1.9 9.57770 −7.84343 59.7324 28.1028 −75.1221 195.604 265.613 −181.481 269.160
1.10 10.7063 18.3440 82.6251 −72.1865 196.397 −96.4803 542.008 93.5034 −772.851
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(-1\)

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 43.6.a.b 10
3.b odd 2 1 387.6.a.e 10
4.b odd 2 1 688.6.a.h 10
5.b even 2 1 1075.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b 10 1.a even 1 1 trivial
387.6.a.e 10 3.b odd 2 1
688.6.a.h 10 4.b odd 2 1
1075.6.a.b 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 8 T_{2}^{9} - 229 T_{2}^{8} + 1734 T_{2}^{7} + 16722 T_{2}^{6} - 112716 T_{2}^{5} - 450596 T_{2}^{4} + 2163208 T_{2}^{3} + 5497616 T_{2}^{2} - 9477216 T_{2} - 20373120 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(43))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 8 T^{9} - 229 T^{8} + \cdots - 20373120 \) Copy content Toggle raw display
$3$ \( T^{10} - 28 T^{9} + \cdots + 316744901280 \) Copy content Toggle raw display
$5$ \( T^{10} - 138 T^{9} + \cdots - 25\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{10} - 60 T^{9} + \cdots - 50\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{10} - 745 T^{9} + \cdots + 46\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{10} - 1917 T^{9} + \cdots + 47\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{10} - 4017 T^{9} + \cdots - 49\!\cdots\!66 \) Copy content Toggle raw display
$19$ \( T^{10} + 2404 T^{9} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{10} - 1733 T^{9} + \cdots - 24\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{10} - 6996 T^{9} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{10} + 4899 T^{9} + \cdots - 47\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{10} - 1466 T^{9} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} - 10297 T^{9} + \cdots - 22\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( (T - 1849)^{10} \) Copy content Toggle raw display
$47$ \( T^{10} - 48592 T^{9} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} - 127165 T^{9} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} - 99372 T^{9} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} - 17408 T^{9} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{10} + 2021 T^{9} + \cdots + 89\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{10} - 11286 T^{9} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{10} - 49892 T^{9} + \cdots - 95\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + 91524 T^{9} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{10} + 105203 T^{9} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{10} + 62682 T^{9} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{10} - 108383 T^{9} + \cdots - 18\!\cdots\!58 \) Copy content Toggle raw display
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