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Properties

Label 1175.4.a.c
Level $1175$
Weight $4$
Character orbit 1175.a
Self dual yes
Analytic conductor $69.327$
Analytic rank $1$
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] = [1175,4,Mod(1,1175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1175, 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("1175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1175 = 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1175.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: \(69.3272442567\)
Analytic rank: \(1\)
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: \( x^{8} - 3x^{7} - 32x^{6} + 70x^{5} + 337x^{4} - 471x^{3} - 1154x^{2} + 836x + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 235)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1 + 2) q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots - 6) q^{6}+ \cdots + ( - 2 \beta_{6} + 3 \beta_{5} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1 + 2) q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots - 6) q^{6}+ \cdots + (14 \beta_{7} + 79 \beta_{6} + \cdots - 159) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} + 10 q^{3} + 9 q^{4} - 42 q^{6} + 15 q^{7} + 57 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} + 10 q^{3} + 9 q^{4} - 42 q^{6} + 15 q^{7} + 57 q^{8} + 14 q^{9} - 89 q^{11} - 32 q^{12} + 68 q^{13} + 49 q^{14} - 147 q^{16} + 169 q^{17} - 111 q^{18} - 169 q^{19} - 454 q^{21} + 36 q^{22} + 93 q^{23} + 102 q^{24} + 5 q^{26} + 256 q^{27} - q^{28} - 583 q^{29} - 356 q^{31} + 89 q^{32} + 342 q^{33} - 680 q^{34} - 309 q^{36} + 578 q^{37} + 52 q^{38} - 590 q^{39} - 1336 q^{41} + 549 q^{42} + 280 q^{43} - 930 q^{44} - 327 q^{46} + 376 q^{47} - 236 q^{48} - 409 q^{49} + 831 q^{51} - 115 q^{52} + 1106 q^{53} - 1395 q^{54} - 521 q^{56} - 383 q^{57} + 393 q^{58} - 941 q^{59} - 1197 q^{61} - 1155 q^{62} - 2448 q^{63} + 197 q^{64} + 2823 q^{66} - 728 q^{67} - 648 q^{68} - 152 q^{69} + 608 q^{71} - 903 q^{72} - 1383 q^{73} - 499 q^{74} + 1594 q^{76} + 988 q^{77} - 3385 q^{78} - 922 q^{79} + 1100 q^{81} - 1106 q^{82} - 2979 q^{83} - 657 q^{84} + 1586 q^{86} - 3305 q^{87} - 3340 q^{88} - 3544 q^{89} + 318 q^{91} + 763 q^{92} - 2370 q^{93} + 141 q^{94} + 3830 q^{96} - 383 q^{97} - 4746 q^{98} - 1142 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 32x^{6} + 70x^{5} + 337x^{4} - 471x^{3} - 1154x^{2} + 836x + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 8\nu^{6} - 8\nu^{5} - 126\nu^{4} + 485\nu^{3} + 350\nu^{2} - 2900\nu + 704 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 8\nu^{6} + 16\nu^{5} - 198\nu^{4} - 19\nu^{3} + 1286\nu^{2} - 308\nu - 1024 ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 34\nu^{5} - 48\nu^{4} - 325\nu^{3} + 278\nu^{2} + 796\nu - 208 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{7} + 32\nu^{6} + 160\nu^{5} - 618\nu^{4} - 1141\nu^{3} + 2594\nu^{2} + 2260\nu + 512 ) / 192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 5\nu^{6} + 19\nu^{5} - 93\nu^{4} - 106\nu^{3} + 416\nu^{2} + 232\nu - 88 ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{3} + 2\beta_{2} + 15\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} + 5\beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{3} + 21\beta_{2} + 33\beta _1 + 131 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -30\beta_{7} + 36\beta_{6} - 6\beta_{5} - 2\beta_{4} + 16\beta_{3} + 66\beta_{2} + 267\beta _1 + 282 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -114\beta_{7} + 182\beta_{6} - 76\beta_{5} - 40\beta_{4} + 22\beta_{3} + 453\beta_{2} + 877\beta _1 + 2337 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -779\beta_{7} + 1023\beta_{6} - 308\beta_{5} - 52\beta_{4} + 215\beta_{3} + 1770\beta_{2} + 5447\beta _1 + 7668 \) 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
−3.47685
−3.03669
−2.44089
−0.130852
0.772616
2.72631
3.65187
4.93449
−3.47685 2.56354 4.08848 0 −8.91303 11.5394 13.5998 −20.4283 0
1.2 −3.03669 9.39936 1.22148 0 −28.5429 −32.7503 20.5843 61.3480 0
1.3 −2.44089 −2.03479 −2.04207 0 4.96669 25.3770 24.5116 −22.8596 0
1.4 −0.130852 −3.53759 −7.98288 0 0.462900 −3.43768 2.09139 −14.4855 0
1.5 0.772616 7.21457 −7.40306 0 5.57409 −6.12860 −11.9007 25.0500 0
1.6 2.72631 3.30065 −0.567225 0 8.99860 10.6176 −23.3569 −16.1057 0
1.7 3.65187 −7.43009 5.33612 0 −27.1337 16.6239 −9.72813 28.2062 0
1.8 4.93449 0.524343 16.3491 0 2.58736 −6.84139 41.1987 −26.7251 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(47\) \(-1\)

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 1175.4.a.c 8
5.b even 2 1 235.4.a.a ✓ 8
15.d odd 2 1 2115.4.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
235.4.a.a ✓ 8 5.b even 2 1
1175.4.a.c 8 1.a even 1 1 trivial
2115.4.a.d 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 3T_{2}^{7} - 32T_{2}^{6} + 70T_{2}^{5} + 337T_{2}^{4} - 471T_{2}^{3} - 1154T_{2}^{2} + 836T_{2} + 128 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + \cdots + 128 \) Copy content Toggle raw display
$3$ \( T^{8} - 10 T^{7} + \cdots - 16091 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 15 T^{7} + \cdots + 243987792 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 1199487826128 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 655358120496 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1522859929344 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 52362385785684 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11713716265344 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 15\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 712132073196336 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 64\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T - 47)^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 68\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 37\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 60\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 84\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 81\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 44\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 94\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 21\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 80\!\cdots\!68 \) Copy content Toggle raw display
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