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Properties

Label 43.10.a.a
Level $43$
Weight $10$
Character orbit 43.a
Self dual yes
Analytic conductor $22.147$
Analytic rank $1$
Dimension $15$
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] = [43,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(22.1465409550\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{2} - 21) q^{3} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 216) q^{4} + (\beta_{8} - \beta_{2} + 4 \beta_1 - 315) q^{5} + (\beta_{9} - 2 \beta_{8} - 3 \beta_{3} + \cdots + 39) q^{6}+ \cdots + ( - \beta_{14} - \beta_{12} + \cdots + 4618) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{2} - 21) q^{3} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 216) q^{4} + (\beta_{8} - \beta_{2} + 4 \beta_1 - 315) q^{5} + (\beta_{9} - 2 \beta_{8} - 3 \beta_{3} + \cdots + 39) q^{6}+ \cdots + ( - 113291 \beta_{14} + \cdots - 438998692) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 32 q^{2} - 317 q^{3} + 3242 q^{4} - 4717 q^{5} + 687 q^{6} - 9680 q^{7} - 20394 q^{8} + 69516 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 32 q^{2} - 317 q^{3} + 3242 q^{4} - 4717 q^{5} + 687 q^{6} - 9680 q^{7} - 20394 q^{8} + 69516 q^{9} - 36237 q^{10} - 104484 q^{11} - 266395 q^{12} - 116174 q^{13} + 416064 q^{14} + 415388 q^{15} + 996762 q^{16} - 884265 q^{17} - 588735 q^{18} - 689535 q^{19} - 3077879 q^{20} - 2070198 q^{21} - 7276218 q^{22} - 2504077 q^{23} - 11534895 q^{24} + 1315350 q^{25} - 13343414 q^{26} - 12546986 q^{27} - 28059568 q^{28} - 18406221 q^{29} - 39503820 q^{30} - 12033699 q^{31} - 18952630 q^{32} - 14197716 q^{33} - 30383125 q^{34} - 27855546 q^{35} - 18372959 q^{36} - 8722847 q^{37} - 63941843 q^{38} - 30955510 q^{39} - 39665611 q^{40} - 18689389 q^{41} - 73185310 q^{42} - 51282015 q^{43} - 68723220 q^{44} - 216992888 q^{45} - 2067521 q^{46} - 104960741 q^{47} - 145362479 q^{48} + 92663095 q^{49} - 42446347 q^{50} + 37433407 q^{51} + 149226080 q^{52} - 215907800 q^{53} + 419158122 q^{54} + 384379852 q^{55} + 430441344 q^{56} + 258744488 q^{57} + 295963139 q^{58} + 185924544 q^{59} + 973236172 q^{60} + 247538102 q^{61} + 139798853 q^{62} + 405429926 q^{63} + 848556290 q^{64} + 94294394 q^{65} + 667230492 q^{66} + 467904656 q^{67} - 88234341 q^{68} + 163914994 q^{69} + 647526126 q^{70} - 8252944 q^{71} + 889796745 q^{72} - 715627902 q^{73} + 725122989 q^{74} - 18301762 q^{75} + 346300359 q^{76} - 1236779964 q^{77} + 2058642146 q^{78} + 560681783 q^{79} - 1157214179 q^{80} - 752010645 q^{81} + 941346367 q^{82} - 1442854698 q^{83} + 1895248718 q^{84} + 699302088 q^{85} + 109401632 q^{86} - 2094576907 q^{87} - 1464507256 q^{88} - 396710008 q^{89} + 1411356270 q^{90} - 3278076852 q^{91} + 155864647 q^{92} - 1424759183 q^{93} + 4666638949 q^{94} - 3854114395 q^{95} - 952489551 q^{96} - 3063837815 q^{97} - 6161086984 q^{98} - 6576160348 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} + \cdots + 52\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\!\cdots\!31 \nu^{14} + \cdots + 11\!\cdots\!12 ) / 77\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!31 \nu^{14} + \cdots - 67\!\cdots\!40 ) / 77\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!81 \nu^{14} + \cdots + 54\!\cdots\!28 ) / 24\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!29 \nu^{14} + \cdots + 27\!\cdots\!80 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{14} + \cdots + 39\!\cdots\!60 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!19 \nu^{14} + \cdots + 13\!\cdots\!20 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!49 \nu^{14} + \cdots + 47\!\cdots\!80 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!51 \nu^{14} + \cdots + 74\!\cdots\!00 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!61 \nu^{14} + \cdots - 97\!\cdots\!40 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40\!\cdots\!87 \nu^{14} + \cdots + 50\!\cdots\!00 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!99 \nu^{14} + \cdots - 52\!\cdots\!04 ) / 77\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 86\!\cdots\!09 \nu^{14} + \cdots - 15\!\cdots\!80 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20\!\cdots\!97 \nu^{14} + \cdots + 74\!\cdots\!08 ) / 77\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - \beta _1 + 724 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 2\beta_{9} - \beta_{8} + 3\beta_{4} - \beta_{3} - 37\beta_{2} + 1260\beta _1 - 971 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 25 \beta_{14} - 51 \beta_{13} + 3 \beta_{12} - 5 \beta_{11} - 65 \beta_{10} - 45 \beta_{9} + \cdots + 912638 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 451 \beta_{14} - 129 \beta_{13} + 361 \beta_{12} - 2211 \beta_{11} + 17 \beta_{10} + 4281 \beta_{9} + \cdots - 2545422 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 56261 \beta_{14} - 137047 \beta_{13} + 29647 \beta_{12} - 13355 \beta_{11} - 166145 \beta_{10} + \cdots + 1382593092 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1476377 \beta_{14} - 101683 \beta_{13} + 624443 \beta_{12} - 3891795 \beta_{11} + 499011 \beta_{10} + \cdots - 7998183896 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 99010825 \beta_{14} - 289235299 \beta_{13} + 92490507 \beta_{12} - 24459623 \beta_{11} + \cdots + 2294173232028 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3505734621 \beta_{14} + 504553057 \beta_{13} + 400597703 \beta_{12} - 6498434995 \beta_{11} + \cdots - 22538241354596 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 159606282881 \beta_{14} - 567035896587 \beta_{13} + 222487096275 \beta_{12} - 36914590119 \beta_{11} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7366410638861 \beta_{14} + 2712863734673 \beta_{13} - 1119132304553 \beta_{12} + \cdots - 57\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 245248710611449 \beta_{14} + \cdots + 72\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 14\!\cdots\!65 \beta_{14} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 36\!\cdots\!09 \beta_{14} + \cdots + 13\!\cdots\!40 \) 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
40.4171
39.2075
32.6728
21.7142
16.4876
15.9205
7.62988
2.38595
−0.220103
−17.0644
−22.4329
−25.8680
−28.3075
−35.4490
−45.0936
−42.4171 107.942 1287.21 267.050 −4578.58 1084.23 −32882.2 −8031.56 −11327.5
1.2 −41.2075 −186.815 1186.05 −1998.59 7698.15 −10690.1 −27776.1 15216.7 82356.9
1.3 −34.6728 96.4953 690.205 1855.36 −3345.76 −11539.9 −6178.86 −10371.7 −64330.7
1.4 −23.7142 2.00552 50.3616 −764.097 −47.5592 4079.63 10947.4 −19679.0 18119.9
1.5 −18.4876 −262.112 −170.210 −2363.43 4845.81 7025.62 12612.4 49019.5 43694.1
1.6 −17.9205 −172.986 −190.855 238.484 3099.99 −4524.82 12595.5 10241.0 −4273.75
1.7 −9.62988 189.920 −419.265 636.384 −1828.91 −3288.51 8967.97 16386.6 −6128.30
1.8 −4.38595 117.920 −492.763 −1237.15 −517.190 12549.3 4406.84 −5777.96 5426.06
1.9 −1.77990 −203.939 −508.832 1139.29 362.990 4324.96 1816.98 21908.1 −2027.82
1.10 15.0644 34.0074 −285.063 1876.98 512.302 −7041.18 −12007.3 −18526.5 28275.7
1.11 20.4329 231.971 −94.4979 −2583.58 4739.83 −1769.99 −12392.5 34127.5 −52790.0
1.12 23.8680 64.8300 57.6838 −578.837 1547.37 −413.035 −10843.6 −15480.1 −13815.7
1.13 26.3075 −164.924 180.084 879.044 −4338.73 8431.97 −8731.88 7516.88 23125.4
1.14 33.4490 8.30450 606.833 −752.925 277.777 −7164.62 3172.07 −19614.0 −25184.6
1.15 43.0936 −179.621 1345.05 −1330.99 −7740.49 −743.545 35899.3 12580.5 −57356.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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 43.10.a.a ✓ 15
3.b odd 2 1 387.10.a.c 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.a.a ✓ 15 1.a even 1 1 trivial
387.10.a.c 15 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 32 T_{2}^{14} - 4949 T_{2}^{13} - 151570 T_{2}^{12} + 9265782 T_{2}^{11} + \cdots + 99\!\cdots\!40 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(43))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots + 99\!\cdots\!40 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 94\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 24\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 24\!\cdots\!14 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 49\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 25\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 40\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( (T + 3418801)^{15} \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 82\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 22\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 26\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 23\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 15\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 52\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 17\!\cdots\!82 \) Copy content Toggle raw display
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