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Properties

Label 103.4.a.a
Level $103$
Weight $4$
Character orbit 103.a
Self dual yes
Analytic conductor $6.077$
Analytic rank $1$
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] = [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: \(1\)
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: \( x^{10} - x^{9} - 47x^{8} + 52x^{7} + 757x^{6} - 827x^{5} - 5127x^{4} + 5248x^{3} + 13320x^{2} - 11920x - 6768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 3) q^{4} + ( - \beta_{8} + \beta_{6} + \beta_{3} + \cdots - 3) q^{5}+ \cdots + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 3) q^{4} + ( - \beta_{8} + \beta_{6} + \beta_{3} + \cdots - 3) q^{5}+ \cdots + ( - 12 \beta_{9} + 58 \beta_{8} + \cdots - 118) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 11 q^{2} - 8 q^{3} + 27 q^{4} - 30 q^{5} - 17 q^{6} - 33 q^{7} - 108 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 11 q^{2} - 8 q^{3} + 27 q^{4} - 30 q^{5} - 17 q^{6} - 33 q^{7} - 108 q^{8} - 24 q^{9} - 2 q^{10} - 28 q^{11} - 57 q^{12} - 75 q^{13} - 149 q^{14} - 206 q^{15} - 29 q^{16} - 401 q^{17} - 222 q^{18} - 171 q^{19} - 387 q^{20} - 236 q^{21} - 179 q^{22} - 463 q^{23} - 54 q^{24} + 10 q^{25} + 66 q^{26} - 218 q^{27} + 419 q^{28} - 395 q^{29} + 710 q^{30} + 188 q^{31} + 145 q^{32} - 142 q^{33} + 1270 q^{34} - 342 q^{35} + 912 q^{36} - 142 q^{37} + 573 q^{38} + 600 q^{39} + 1891 q^{40} - 561 q^{41} + 1303 q^{42} - 108 q^{43} + 902 q^{44} + 296 q^{45} + 2298 q^{46} - 412 q^{47} + 1275 q^{48} + 459 q^{49} + 88 q^{50} + 252 q^{51} + 767 q^{52} - 1050 q^{53} + 1989 q^{54} + 362 q^{55} - 670 q^{56} - 918 q^{57} + 822 q^{58} - 503 q^{59} - 89 q^{60} - 629 q^{61} - 611 q^{62} + 57 q^{63} + 630 q^{64} - 2348 q^{65} + 1055 q^{66} - 1054 q^{67} - 2145 q^{68} - 286 q^{69} + 2144 q^{70} + 168 q^{71} + 82 q^{72} - 1642 q^{73} - 83 q^{74} + 500 q^{75} - 404 q^{76} - 2376 q^{77} - 660 q^{78} + 629 q^{79} - 3522 q^{80} - 1330 q^{81} + 1260 q^{82} - 2013 q^{83} - 1849 q^{84} - 708 q^{85} - 141 q^{86} - 1286 q^{87} - 1991 q^{88} - 926 q^{89} - 266 q^{90} - 466 q^{91} - 7224 q^{92} - 2632 q^{93} - 592 q^{94} - 886 q^{95} - 275 q^{96} - 291 q^{97} - 2552 q^{98} - 492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 47x^{8} + 52x^{7} + 757x^{6} - 827x^{5} - 5127x^{4} + 5248x^{3} + 13320x^{2} - 11920x - 6768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{9} + 3 \nu^{8} + 41 \nu^{7} - 134 \nu^{6} - 489 \nu^{5} + 1805 \nu^{4} + 1517 \nu^{3} + \cdots + 6968 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{9} + \nu^{8} - 221 \nu^{7} - 18 \nu^{6} + 3213 \nu^{5} + 335 \nu^{4} - 18001 \nu^{3} + \cdots + 11432 ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13 \nu^{9} - 7 \nu^{8} - 565 \nu^{7} + 302 \nu^{6} + 8021 \nu^{5} - 2505 \nu^{4} - 44585 \nu^{3} + \cdots + 25640 ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{9} + 5 \nu^{8} - 241 \nu^{7} - 206 \nu^{6} + 3997 \nu^{5} + 3091 \nu^{4} - 26701 \nu^{3} + \cdots + 22712 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39 \nu^{9} - 19 \nu^{8} + 1767 \nu^{7} + 718 \nu^{6} - 26911 \nu^{5} - 12237 \nu^{4} + \cdots - 137496 ) / 1024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41 \nu^{9} - 45 \nu^{8} + 1881 \nu^{7} + 1698 \nu^{6} - 29425 \nu^{5} - 23123 \nu^{4} + \cdots - 153896 ) / 1024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77 \nu^{9} + 57 \nu^{8} - 3509 \nu^{7} - 2130 \nu^{6} + 54101 \nu^{5} + 31671 \nu^{4} - 332777 \nu^{3} + \cdots + 289576 ) / 1024 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{3} - \beta_{2} + 15\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + 4\beta_{8} + 4\beta_{7} + 2\beta_{4} + 4\beta_{3} + 24\beta_{2} - 5\beta _1 + 148 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{9} - 8 \beta_{8} - 24 \beta_{7} - 24 \beta_{6} - 16 \beta_{5} - 62 \beta_{4} - 56 \beta_{3} + \cdots - 112 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 132 \beta_{9} + 124 \beta_{8} + 155 \beta_{7} + 15 \beta_{6} + 15 \beta_{5} + 88 \beta_{4} + 130 \beta_{3} + \cdots + 2635 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 176 \beta_{9} - 316 \beta_{8} - 558 \beta_{7} - 554 \beta_{6} - 226 \beta_{5} - 1528 \beta_{4} + \cdots - 3516 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3352 \beta_{9} + 3060 \beta_{8} + 4268 \beta_{7} + 592 \beta_{6} + 536 \beta_{5} + 2792 \beta_{4} + \cdots + 51930 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5672 \beta_{9} - 9260 \beta_{8} - 13405 \beta_{7} - 12729 \beta_{6} - 3361 \beta_{5} - 35170 \beta_{4} + \cdots - 99951 \) 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
4.29021
3.60584
3.10947
1.83045
1.79480
−0.417130
−2.36943
−2.74781
−3.26174
−4.83465
−5.29021 3.32131 19.9863 −17.0584 −17.5704 19.1616 −63.4102 −15.9689 90.2427
1.2 −4.60584 −8.88411 13.2137 −1.63470 40.9187 14.4844 −24.0135 51.9273 7.52915
1.3 −4.10947 6.94318 8.88772 −4.21255 −28.5328 −31.5191 −3.64805 21.2077 17.3113
1.4 −2.83045 −1.09834 0.0114397 1.75744 3.10879 24.1206 22.6112 −25.7937 −4.97435
1.5 −2.79480 −5.01212 −0.189110 17.8067 14.0079 −12.4806 22.8869 −1.87865 −49.7660
1.6 −0.582870 4.03293 −7.66026 −1.29086 −2.35067 −16.7333 9.12789 −10.7355 0.752402
1.7 1.36943 3.27954 −6.12466 −19.3768 4.49110 10.1428 −19.3428 −16.2446 −26.5352
1.8 1.74781 −4.34788 −4.94514 10.0313 −7.59929 0.0926531 −22.6257 −8.09592 17.5329
1.9 2.26174 −0.275808 −2.88455 −4.67047 −0.623806 −34.0508 −24.6180 −26.9239 −10.5634
1.10 3.83465 −5.95870 6.70451 −11.3516 −22.8495 −6.21822 −4.96773 8.50611 −43.5295
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.a ✓ 10
3.b odd 2 1 927.4.a.c 10
4.b odd 2 1 1648.4.a.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.4.a.a ✓ 10 1.a even 1 1 trivial
927.4.a.c 10 3.b odd 2 1
1648.4.a.f 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 11 T_{2}^{9} + 7 T_{2}^{8} - 272 T_{2}^{7} - 629 T_{2}^{6} + 2023 T_{2}^{5} + 5589 T_{2}^{4} + \cdots + 9584 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(103))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 11 T^{9} + \cdots + 9584 \) Copy content Toggle raw display
$3$ \( T^{10} + 8 T^{9} + \cdots + 106588 \) Copy content Toggle raw display
$5$ \( T^{10} + 30 T^{9} + \cdots - 48901564 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 8768419621 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 36137122121152 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 56855183137904 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 28\!\cdots\!79 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 82\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 46\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 59\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 24\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 92\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 53\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 46\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 38\!\cdots\!71 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 55\!\cdots\!52 \) Copy content Toggle raw display
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