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Properties

Label 103.4.a.b
Level $103$
Weight $4$
Character orbit 103.a
Self dual yes
Analytic conductor $6.077$
Analytic rank $0$
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] = [103,4,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(6.07719673059\)
Analytic rank: \(0\)
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: \( x^{15} - 6 x^{14} - 74 x^{13} + 425 x^{12} + 2222 x^{11} - 11921 x^{10} - 33775 x^{9} + 168643 x^{8} + \cdots + 1082272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{9} + 1) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + (\beta_{8} + 2) q^{5} + ( - \beta_{14} - \beta_{13} - \beta_{10} + \cdots + 3) q^{6}+ \cdots + (3 \beta_{14} + \beta_{13} + \beta_{12} + \cdots + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{9} + 1) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + (\beta_{8} + 2) q^{5} + ( - \beta_{14} - \beta_{13} - \beta_{10} + \cdots + 3) q^{6}+ \cdots + (74 \beta_{14} + 28 \beta_{13} + \cdots + 236) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 9 q^{2} + 10 q^{3} + 67 q^{4} + 30 q^{5} + 19 q^{6} + 23 q^{7} + 132 q^{8} + 201 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 9 q^{2} + 10 q^{3} + 67 q^{4} + 30 q^{5} + 19 q^{6} + 23 q^{7} + 132 q^{8} + 201 q^{9} + 58 q^{10} + 126 q^{11} + 159 q^{12} + 133 q^{13} + 19 q^{14} - 26 q^{15} + 291 q^{16} + 415 q^{17} + 318 q^{18} - 19 q^{19} + 213 q^{20} + 100 q^{21} + 217 q^{22} + 365 q^{23} + 378 q^{24} + 885 q^{25} + 201 q^{26} - 26 q^{27} - 784 q^{28} + 173 q^{29} - 1954 q^{30} - 140 q^{31} - 100 q^{32} - 222 q^{33} - 1218 q^{34} + 378 q^{35} - 236 q^{36} - 590 q^{37} - 486 q^{38} - 640 q^{39} - 1925 q^{40} + 499 q^{41} - 2965 q^{42} + 462 q^{43} - 200 q^{44} - 636 q^{45} - 1811 q^{46} - 28 q^{47} + 321 q^{48} + 446 q^{49} + 892 q^{50} + 120 q^{51} - 1233 q^{52} + 1854 q^{53} - 3281 q^{54} - 458 q^{55} - 1709 q^{56} + 1530 q^{57} - 306 q^{58} - 143 q^{59} - 4465 q^{60} - 173 q^{61} + 893 q^{62} + 977 q^{63} - 2426 q^{64} + 2948 q^{65} - 2097 q^{66} - 732 q^{67} + 2108 q^{68} - 330 q^{69} - 564 q^{70} + 1708 q^{71} + 2718 q^{72} + 1048 q^{73} + 891 q^{74} + 678 q^{75} + 124 q^{76} + 3568 q^{77} - 952 q^{78} + 1441 q^{79} - 634 q^{80} + 4851 q^{81} - 447 q^{82} + 2315 q^{83} - 3087 q^{84} + 1828 q^{85} + 2485 q^{86} + 2026 q^{87} + 1621 q^{88} + 3320 q^{89} + 16 q^{90} + 270 q^{91} + 2052 q^{92} + 3000 q^{93} + 512 q^{94} + 4382 q^{95} + 2623 q^{96} + 2525 q^{97} + 3883 q^{98} + 3450 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 6 x^{14} - 74 x^{13} + 425 x^{12} + 2222 x^{11} - 11921 x^{10} - 33775 x^{9} + 168643 x^{8} + \cdots + 1082272 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 87738578284969 \nu^{14} - 740431703218492 \nu^{13} + \cdots + 10\!\cdots\!52 ) / 33\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 128964619181685 \nu^{14} + 335921015278060 \nu^{13} + \cdots + 17\!\cdots\!00 ) / 33\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 211101401730855 \nu^{14} - 801779285905284 \nu^{13} + \cdots - 12\!\cdots\!88 ) / 33\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 132583689186063 \nu^{14} - 617514445769060 \nu^{13} + \cdots - 11\!\cdots\!00 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 292476267723619 \nu^{14} + \cdots - 99\!\cdots\!72 ) / 33\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7640755755441 \nu^{14} - 27199900662612 \nu^{13} - 668842682814034 \nu^{12} + \cdots - 63\!\cdots\!28 ) / 46\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74112940607513 \nu^{14} - 384028833988243 \nu^{13} + \cdots - 36\!\cdots\!20 ) / 42\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 152361495350903 \nu^{14} + 600300981155038 \nu^{13} + \cdots + 10\!\cdots\!28 ) / 84\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 697316083740029 \nu^{14} + \cdots + 61\!\cdots\!32 ) / 33\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 884741429018709 \nu^{14} + \cdots + 61\!\cdots\!76 ) / 33\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42478421200561 \nu^{14} - 144887753184516 \nu^{13} + \cdots - 31\!\cdots\!76 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49030474883767 \nu^{14} + 234144803765464 \nu^{13} + \cdots + 26\!\cdots\!56 ) / 15\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + 3\beta_{2} + 20\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} + 4 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots + 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{14} - 2 \beta_{13} + 21 \beta_{12} + 29 \beta_{11} - 55 \beta_{10} - 13 \beta_{9} + \cdots + 582 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 91 \beta_{14} - 6 \beta_{13} + 230 \beta_{12} + 78 \beta_{11} - 335 \beta_{10} - 100 \beta_{9} + \cdots + 6132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 492 \beta_{14} - 118 \beta_{13} + 1295 \beta_{12} + 743 \beta_{11} - 2451 \beta_{10} - 749 \beta_{9} + \cdots + 22950 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3856 \beta_{14} - 554 \beta_{13} + 10022 \beta_{12} + 2450 \beta_{11} - 15056 \beta_{10} + \cdots + 177726 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 23203 \beta_{14} - 5710 \beta_{13} + 58716 \beta_{12} + 18510 \beta_{11} - 100909 \beta_{10} + \cdots + 831380 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 159927 \beta_{14} - 32218 \beta_{13} + 399533 \beta_{12} + 69144 \beta_{11} - 621759 \beta_{10} + \cdots + 5584289 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 991901 \beta_{14} - 254998 \beta_{13} + 2388323 \beta_{12} + 449254 \beta_{11} - 3991387 \beta_{10} + \cdots + 29294495 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6466784 \beta_{14} - 1541800 \beta_{13} + 15334546 \beta_{12} + 1782753 \beta_{11} - 24571627 \beta_{10} + \cdots + 184046659 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 40275970 \beta_{14} - 10795158 \beta_{13} + 92488853 \beta_{12} + 10339583 \beta_{11} + \cdots + 1024546612 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 255339995 \beta_{14} - 66753332 \beta_{13} + 577116091 \beta_{12} + 40921590 \beta_{11} + \cdots + 6240911171 \) 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
6.00985
5.41960
4.53738
3.48089
2.59388
1.73477
1.66595
1.02873
−0.228514
−1.84051
−3.09744
−3.19833
−3.76099
−4.12860
−4.21667
−5.00985 −1.85333 17.0986 10.9587 9.28493 −13.6104 −45.5829 −23.5652 −54.9015
1.2 −4.41960 6.98819 11.5328 13.1462 −30.8850 18.9645 −15.6137 21.8348 −58.1007
1.3 −3.53738 −4.82519 4.51308 −19.8337 17.0686 −24.5332 12.3346 −3.71750 70.1594
1.4 −2.48089 0.712803 −1.84520 −7.35709 −1.76838 5.14519 24.4248 −26.4919 18.2521
1.5 −1.59388 10.0675 −5.45955 −10.1911 −16.0464 16.4917 21.4529 74.3551 16.2433
1.6 −0.734772 −7.66179 −7.46011 4.02320 5.62966 −19.6029 11.3597 31.7030 −2.95613
1.7 −0.665955 5.86326 −7.55650 21.9948 −3.90467 −3.29090 10.3599 7.37787 −14.6475
1.8 −0.0287334 −8.19366 −7.99917 −14.0560 0.235432 19.9018 0.459711 40.1361 0.403877
1.9 1.22851 −1.72646 −6.49075 9.67020 −2.12098 32.6999 −17.8021 −24.0193 11.8800
1.10 2.84051 7.84631 0.0685111 5.52155 22.2875 3.43796 −22.5295 34.5646 15.6840
1.11 4.09744 4.69212 8.78905 1.10239 19.2257 16.4270 3.23308 −4.98399 4.51697
1.12 4.19833 −10.3334 9.62595 18.2758 −43.3829 6.67994 6.82629 79.7789 76.7276
1.13 4.76099 1.53382 14.6671 18.4077 7.30249 −29.9175 31.7419 −24.6474 87.6387
1.14 5.12860 −1.50184 18.3025 −1.20041 −7.70233 19.9856 52.8373 −24.7445 −6.15641
1.15 5.21667 8.39163 19.2137 −20.4621 43.7764 −25.7787 58.4980 43.4194 −106.744
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(103\) \(1\)

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 103.4.a.b ✓ 15
3.b odd 2 1 927.4.a.f 15
4.b odd 2 1 1648.4.a.i 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.4.a.b ✓ 15 1.a even 1 1 trivial
927.4.a.f 15 3.b odd 2 1
1648.4.a.i 15 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 9 T_{2}^{14} - 53 T_{2}^{13} + 628 T_{2}^{12} + 731 T_{2}^{11} - 16404 T_{2}^{10} + \cdots - 33296 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(103))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 9 T^{14} + \cdots - 33296 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 2095800448 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 129605596751104 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 78\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 22\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 84\!\cdots\!10 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 54\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 30\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 34\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 13\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 98\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 43\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 14\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 23\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 32\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 57\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 90\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 46\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 47\!\cdots\!24 \) Copy content Toggle raw display
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