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Properties

Label 115.4.a.f
Level $115$
Weight $4$
Character orbit 115.a
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $8$
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] = [115,4,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + \beta_{6} q^{3} + (\beta_{2} + 5) q^{4} + 5 q^{5} + (\beta_{7} + \beta_{5} + \beta_{3} - \beta_1 + 1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_1 + 1) q^{7} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_1 + 1) q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + \beta_{6} q^{3} + (\beta_{2} + 5) q^{4} + 5 q^{5} + (\beta_{7} + \beta_{5} + \beta_{3} - \beta_1 + 1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_1 + 1) q^{7} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_1 + 1) q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 20) q^{9} + ( - 5 \beta_1 + 5) q^{10} + (\beta_{7} + \beta_{6} - 3 \beta_{4} - \beta_{2} - \beta_1 + 5) q^{11} + (\beta_{7} + 3 \beta_{6} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{12} + ( - \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 5) q^{13} + ( - 2 \beta_{7} - 8 \beta_{6} + 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 20) q^{14} + 5 \beta_{6} q^{15} + (2 \beta_{7} - 5 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 13) q^{16} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 11 \beta_1 + 6) q^{17} + ( - 8 \beta_{6} + 2 \beta_{5} - 7 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 13) q^{18} + (4 \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 20) q^{19} + (5 \beta_{2} + 25) q^{20} + (\beta_{7} - 2 \beta_{6} + 5 \beta_{5} + 7 \beta_{4} - \beta_{3} - 5 \beta_{2} + 20 \beta_1 + 37) q^{21} + (3 \beta_{7} + 5 \beta_{6} + 2 \beta_{5} - 13 \beta_{4} - 4 \beta_{3} - \beta_{2} + \cdots + 19) q^{22}+ \cdots + ( - 40 \beta_{7} - 29 \beta_{6} - 15 \beta_{5} - 10 \beta_{4} + 27 \beta_{3} + \cdots + 184) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 42 q^{4} + 40 q^{5} + 11 q^{7} + 3 q^{8} + 158 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} + 42 q^{4} + 40 q^{5} + 11 q^{7} + 3 q^{8} + 158 q^{9} + 30 q^{10} + 41 q^{11} + 48 q^{12} + 28 q^{13} + 161 q^{14} + 98 q^{16} + 71 q^{17} + 84 q^{18} + 177 q^{19} + 210 q^{20} + 292 q^{21} + 171 q^{22} + 184 q^{23} + 177 q^{24} + 200 q^{25} + 396 q^{26} - 495 q^{27} + 179 q^{28} + 225 q^{29} - 36 q^{31} - 360 q^{32} - 53 q^{33} - 1051 q^{34} + 55 q^{35} - 396 q^{36} - 348 q^{37} - 309 q^{38} - 1077 q^{39} + 15 q^{40} + 620 q^{41} - 2288 q^{42} - 390 q^{43} - 141 q^{44} + 790 q^{45} + 138 q^{46} + 123 q^{47} - 1304 q^{48} + 881 q^{49} + 150 q^{50} + 957 q^{51} - 1890 q^{52} + 1406 q^{53} - 2949 q^{54} + 205 q^{55} + 1031 q^{56} - 1142 q^{57} - 129 q^{58} - 676 q^{59} + 240 q^{60} + 1447 q^{61} - 1556 q^{62} - 58 q^{63} - 2883 q^{64} + 140 q^{65} - 119 q^{66} - 1582 q^{67} + 953 q^{68} + 805 q^{70} + 1396 q^{71} - 1728 q^{72} + 17 q^{73} - 964 q^{74} + 2271 q^{76} + 488 q^{77} - 1919 q^{78} + 708 q^{79} + 490 q^{80} + 4316 q^{81} - 3332 q^{82} + 1486 q^{83} - 3970 q^{84} + 355 q^{85} + 3570 q^{86} - 803 q^{87} - 1031 q^{88} + 1360 q^{89} + 420 q^{90} + 2693 q^{91} + 966 q^{92} + 3833 q^{93} - 3207 q^{94} + 885 q^{95} + 4956 q^{96} - 855 q^{97} + 1817 q^{98} + 1319 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -21\nu^{7} + 372\nu^{6} - 611\nu^{5} - 10501\nu^{4} + 19940\nu^{3} + 88331\nu^{2} - 77178\nu - 104240 ) / 3680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -43\nu^{7} + 236\nu^{6} + 1027\nu^{5} - 3803\nu^{4} - 10340\nu^{3} + 373\nu^{2} + 31226\nu + 17520 ) / 3680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\nu^{7} - 484\nu^{6} - 1233\nu^{5} + 15097\nu^{4} + 6860\nu^{3} - 135927\nu^{2} - 25774\nu + 211440 ) / 3680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79\nu^{7} - 348\nu^{6} - 2871\nu^{5} + 8399\nu^{4} + 33460\nu^{3} - 36929\nu^{2} - 71618\nu + 30800 ) / 3680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 233\nu^{7} - 1236\nu^{6} - 6977\nu^{5} + 29593\nu^{4} + 63260\nu^{3} - 146183\nu^{2} - 44766\nu + 109520 ) / 3680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 3\beta_{2} + 23\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 9\beta_{6} + 5\beta_{5} - 2\beta_{4} + 6\beta_{3} + 30\beta_{2} + 87\beta _1 + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{7} - 79\beta_{6} + 45\beta_{5} - 36\beta_{4} + 54\beta_{3} + 133\beta_{2} + 652\beta _1 + 956 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 128\beta_{7} - 548\beta_{6} + 266\beta_{5} - 133\beta_{4} + 353\beta_{3} + 951\beta_{2} + 3189\beta _1 + 7538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 860 \beta_{7} - 3858 \beta_{6} + 1852 \beta_{5} - 1258 \beta_{4} + 2456 \beta_{3} + 5030 \beta_{2} + 20593 \beta _1 + 36204 \) 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
6.03341
4.85207
1.76920
0.954235
−1.16661
−3.05234
−3.57019
−3.81977
−5.03341 −4.21852 17.3352 5.00000 21.2335 −19.9088 −46.9881 −9.20409 −25.1671
1.2 −3.85207 8.45088 6.83843 5.00000 −32.5534 16.9190 4.47444 44.4174 −19.2603
1.3 −0.769202 −0.0181662 −7.40833 5.00000 0.0139735 10.0918 11.8521 −26.9997 −3.84601
1.4 0.0457645 −10.2193 −7.99791 5.00000 −0.467681 −33.8594 −0.732137 77.4338 0.228823
1.5 2.16661 8.59146 −3.30580 5.00000 18.6143 9.86011 −24.4953 46.8131 10.8331
1.6 4.05234 −8.94300 8.42146 5.00000 −36.2401 32.7645 1.70790 52.9772 20.2617
1.7 4.57019 4.96142 12.8867 5.00000 22.6746 −18.4259 22.3330 −2.38431 22.8510
1.8 4.81977 1.39521 15.2302 5.00000 6.72460 13.5587 34.8481 −25.0534 24.0989
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(23\) \(-1\)

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 115.4.a.f ✓ 8
3.b odd 2 1 1035.4.a.r 8
4.b odd 2 1 1840.4.a.v 8
5.b even 2 1 575.4.a.n 8
5.c odd 4 2 575.4.b.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.f ✓ 8 1.a even 1 1 trivial
575.4.a.n 8 5.b even 2 1
575.4.b.k 16 5.c odd 4 2
1035.4.a.r 8 3.b odd 2 1
1840.4.a.v 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{7} - 35T_{2}^{6} + 249T_{2}^{5} + 170T_{2}^{4} - 2565T_{2}^{3} + 1916T_{2}^{2} + 2802T_{2} - 132 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} - 35 T^{6} + 249 T^{5} + \cdots - 132 \) Copy content Toggle raw display
$3$ \( T^{8} - 187 T^{6} + 165 T^{5} + \cdots + 3520 \) Copy content Toggle raw display
$5$ \( (T - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 11 T^{7} + \cdots - 9289584128 \) Copy content Toggle raw display
$11$ \( T^{8} - 41 T^{7} + \cdots - 11345758080 \) Copy content Toggle raw display
$13$ \( T^{8} - 28 T^{7} + \cdots - 5251923133176 \) Copy content Toggle raw display
$17$ \( T^{8} - 71 T^{7} + \cdots + 52834929586560 \) Copy content Toggle raw display
$19$ \( T^{8} - 177 T^{7} + \cdots - 54772847726720 \) Copy content Toggle raw display
$23$ \( (T - 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 225 T^{7} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + 36 T^{7} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + 348 T^{7} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} - 620 T^{7} + \cdots - 18\!\cdots\!22 \) Copy content Toggle raw display
$43$ \( T^{8} + 390 T^{7} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} - 123 T^{7} + \cdots - 30\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{8} - 1406 T^{7} + \cdots - 62\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + 676 T^{7} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} - 1447 T^{7} + \cdots - 94\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{8} + 1582 T^{7} + \cdots - 27\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{8} - 1396 T^{7} + \cdots + 30\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{8} - 17 T^{7} + \cdots - 59\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} - 708 T^{7} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{8} - 1486 T^{7} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{8} - 1360 T^{7} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{8} + 855 T^{7} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
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