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Properties

Label 1150.4.a.y
Level $1150$
Weight $4$
Character orbit 1150.a
Self dual yes
Analytic conductor $67.852$
Analytic rank $1$
Dimension $7$
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] = [1150,4,Mod(1,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, 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("1150.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(67.8521965066\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 104x^{5} + 249x^{4} + 2874x^{3} - 6310x^{2} - 21520x + 39500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 230)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + (\beta_1 + 1) q^{3} + 4 q^{4} + ( - 2 \beta_1 - 2) q^{6} + (\beta_{6} + \beta_{4} - \beta_1 + 6) q^{7} - 8 q^{8} + (\beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + (\beta_1 + 1) q^{3} + 4 q^{4} + ( - 2 \beta_1 - 2) q^{6} + (\beta_{6} + \beta_{4} - \beta_1 + 6) q^{7} - 8 q^{8} + (\beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots + 4) q^{9}+ \cdots + ( - 49 \beta_{6} + 22 \beta_{5} + \cdots - 211) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 14 q^{2} + 9 q^{3} + 28 q^{4} - 18 q^{6} + 44 q^{7} - 56 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 14 q^{2} + 9 q^{3} + 28 q^{4} - 18 q^{6} + 44 q^{7} - 56 q^{8} + 34 q^{9} - 73 q^{11} + 36 q^{12} + 45 q^{13} - 88 q^{14} + 112 q^{16} - 50 q^{17} - 68 q^{18} - 77 q^{19} - 110 q^{21} + 146 q^{22} - 161 q^{23} - 72 q^{24} - 90 q^{26} + 48 q^{27} + 176 q^{28} - 395 q^{29} - 160 q^{31} - 224 q^{32} + 153 q^{33} + 100 q^{34} + 136 q^{36} + 67 q^{37} + 154 q^{38} - 808 q^{39} - 952 q^{41} + 220 q^{42} + 506 q^{43} - 292 q^{44} + 322 q^{46} + 114 q^{47} + 144 q^{48} - 305 q^{49} - 917 q^{51} + 180 q^{52} + 307 q^{53} - 96 q^{54} - 352 q^{56} - 432 q^{57} + 790 q^{58} - 1407 q^{59} - 1871 q^{61} + 320 q^{62} + 795 q^{63} + 448 q^{64} - 306 q^{66} + 479 q^{67} - 200 q^{68} - 207 q^{69} - 1904 q^{71} - 272 q^{72} + 1890 q^{73} - 134 q^{74} - 308 q^{76} - 1622 q^{77} + 1616 q^{78} + 764 q^{79} - 2309 q^{81} + 1904 q^{82} + 1551 q^{83} - 440 q^{84} - 1012 q^{86} - 1498 q^{87} + 584 q^{88} - 1168 q^{89} - 1933 q^{91} - 644 q^{92} + 1150 q^{93} - 228 q^{94} - 288 q^{96} + 247 q^{97} + 610 q^{98} - 1671 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 104x^{5} + 249x^{4} + 2874x^{3} - 6310x^{2} - 21520x + 39500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -29\nu^{6} - 6752\nu^{5} - 6444\nu^{4} + 505959\nu^{3} - 28446\nu^{2} - 6994630\nu + 3856040 ) / 515490 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1382\nu^{6} + 4771\nu^{5} - 137298\nu^{4} - 265707\nu^{3} + 4155243\nu^{2} + 2776780\nu - 32600215 ) / 773235 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 653\nu^{6} + 352\nu^{5} - 64650\nu^{4} - 20025\nu^{3} + 1621734\nu^{2} + 1132102\nu - 8698060 ) / 309294 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -257\nu^{6} - 1770\nu^{5} + 17550\nu^{4} + 100401\nu^{3} - 340968\nu^{2} - 1109302\nu + 2747748 ) / 103098 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5444\nu^{6} - 17302\nu^{5} + 434541\nu^{4} + 797139\nu^{3} - 8352951\nu^{2} - 7431370\nu + 33980260 ) / 773235 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{2} - 2\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} + 7\beta_{5} - 11\beta_{4} + 7\beta_{3} + 52\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 72\beta_{6} - 157\beta_{5} + 88\beta_{4} - 37\beta_{3} + 73\beta_{2} - 196\beta _1 + 1389 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -393\beta_{6} + 729\beta_{5} - 942\beta_{4} + 576\beta_{3} - 171\beta_{2} + 3126\beta _1 - 3086 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4734\beta_{6} - 10755\beta_{5} + 6873\beta_{4} - 3759\beta_{3} + 4836\beta_{2} - 16262\beta _1 + 77229 \) 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
−8.41501
−4.96965
−3.36381
1.66599
3.85861
6.41782
6.80605
−2.00000 −7.41501 4.00000 0 14.8300 26.6333 −8.00000 27.9823 0
1.2 −2.00000 −3.96965 4.00000 0 7.93930 5.88989 −8.00000 −11.2419 0
1.3 −2.00000 −2.36381 4.00000 0 4.72762 −7.27979 −8.00000 −21.4124 0
1.4 −2.00000 2.66599 4.00000 0 −5.33198 1.62258 −8.00000 −19.8925 0
1.5 −2.00000 4.85861 4.00000 0 −9.71722 11.5462 −8.00000 −3.39393 0
1.6 −2.00000 7.41782 4.00000 0 −14.8356 26.7462 −8.00000 28.0240 0
1.7 −2.00000 7.80605 4.00000 0 −15.6121 −21.1583 −8.00000 33.9344 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

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 1150.4.a.y 7
5.b even 2 1 1150.4.a.z 7
5.c odd 4 2 230.4.b.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.b.a ✓ 14 5.c odd 4 2
1150.4.a.y 7 1.a even 1 1 trivial
1150.4.a.z 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):

\( T_{3}^{7} - 9T_{3}^{6} - 71T_{3}^{5} + 704T_{3}^{4} + 913T_{3}^{3} - 12449T_{3}^{2} - 1775T_{3} + 52186 \) Copy content Toggle raw display
\( T_{7}^{7} - 44T_{7}^{6} - 80T_{7}^{5} + 21943T_{7}^{4} - 179662T_{7}^{3} - 860220T_{7}^{2} + 9237692T_{7} - 12107068 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 9 T^{6} + \cdots + 52186 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 44 T^{6} + \cdots - 12107068 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 1520797760 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 10375239838 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 18410099420 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 3897418182928 \) Copy content Toggle raw display
$23$ \( (T + 23)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 43908716160740 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 108456344870975 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 44\!\cdots\!03 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 59\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 30\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 27\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 63\!\cdots\!93 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 40\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 25\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 20\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 23\!\cdots\!12 \) Copy content Toggle raw display
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