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Properties

Label 1150.4.a.ba
Level $1150$
Weight $4$
Character orbit 1150.a
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $9$
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 179x^{7} + 380x^{6} + 10197x^{5} - 8259x^{4} - 205207x^{3} - 105750x^{2} + 525560x + 178000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - \beta_1 q^{3} + 4 q^{4} + 2 \beta_1 q^{6} + ( - \beta_{7} - 5) q^{7} - 8 q^{8} + (\beta_{7} + \beta_{6} + \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - \beta_1 q^{3} + 4 q^{4} + 2 \beta_1 q^{6} + ( - \beta_{7} - 5) q^{7} - 8 q^{8} + (\beta_{7} + \beta_{6} + \beta_1 + 14) q^{9} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 8) q^{11}+ \cdots + ( - 37 \beta_{8} - 57 \beta_{7} + \cdots - 249) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 18 q^{2} - 3 q^{3} + 36 q^{4} + 6 q^{6} - 44 q^{7} - 72 q^{8} + 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 18 q^{2} - 3 q^{3} + 36 q^{4} + 6 q^{6} - 44 q^{7} - 72 q^{8} + 124 q^{9} + 81 q^{11} - 12 q^{12} - 59 q^{13} + 88 q^{14} + 144 q^{16} - 110 q^{17} - 248 q^{18} + 221 q^{19} + 142 q^{21} - 162 q^{22} + 207 q^{23} + 24 q^{24} + 118 q^{26} - 336 q^{27} - 176 q^{28} + 205 q^{29} + 336 q^{31} - 288 q^{32} - 437 q^{33} + 220 q^{34} + 496 q^{36} + 5 q^{37} - 442 q^{38} - 44 q^{39} + 360 q^{41} - 284 q^{42} - 366 q^{43} + 324 q^{44} - 414 q^{46} + 122 q^{47} - 48 q^{48} + 457 q^{49} + 1025 q^{51} - 236 q^{52} - 631 q^{53} + 672 q^{54} + 352 q^{56} + 384 q^{57} - 410 q^{58} + 797 q^{59} + 211 q^{61} - 672 q^{62} - 2447 q^{63} + 576 q^{64} + 874 q^{66} - 111 q^{67} - 440 q^{68} - 69 q^{69} + 2912 q^{71} - 992 q^{72} - 98 q^{73} - 10 q^{74} + 884 q^{76} - 942 q^{77} + 88 q^{78} + 1184 q^{79} + 2093 q^{81} - 720 q^{82} - 2375 q^{83} + 568 q^{84} + 732 q^{86} + 1534 q^{87} - 648 q^{88} + 2588 q^{89} + 2677 q^{91} + 828 q^{92} - 1402 q^{93} - 244 q^{94} + 96 q^{96} + 593 q^{97} - 914 q^{98} - 1753 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 179x^{7} + 380x^{6} + 10197x^{5} - 8259x^{4} - 205207x^{3} - 105750x^{2} + 525560x + 178000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1574 \nu^{8} + 18599 \nu^{7} + 167459 \nu^{6} - 2461233 \nu^{5} - 1018734 \nu^{4} + \cdots + 414540898 ) / 17971794 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49721 \nu^{8} - 363041 \nu^{7} - 8099153 \nu^{6} + 46094694 \nu^{5} + 402531573 \nu^{4} + \cdots + 1362472940 ) / 395379468 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4060 \nu^{8} + 27817 \nu^{7} + 529351 \nu^{6} - 3473625 \nu^{5} - 15637956 \nu^{4} + \cdots - 245779618 ) / 17971794 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 36121 \nu^{8} + 339447 \nu^{7} + 4826031 \nu^{6} - 44672032 \nu^{5} - 160036865 \nu^{4} + \cdots - 5750099420 ) / 131793156 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 114481 \nu^{8} - 793627 \nu^{7} - 16480393 \nu^{6} + 104406708 \nu^{5} + 640534911 \nu^{4} + \cdots - 2385585032 ) / 395379468 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 114481 \nu^{8} + 793627 \nu^{7} + 16480393 \nu^{6} - 104406708 \nu^{5} - 640534911 \nu^{4} + \cdots - 13824973156 ) / 395379468 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20228 \nu^{8} + 216703 \nu^{7} + 2276212 \nu^{6} - 27827869 \nu^{5} - 32517437 \nu^{4} + \cdots + 2097641682 ) / 65896578 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{8} + 4\beta_{7} + \beta_{6} - 2\beta_{5} - 3\beta_{4} + \beta_{3} - 5\beta_{2} + 69\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} + 108\beta_{7} + 99\beta_{6} - 20\beta_{5} - 6\beta_{4} - 11\beta_{3} + 22\beta_{2} + 55\beta _1 + 2854 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 390 \beta_{8} + 454 \beta_{7} + 142 \beta_{6} - 273 \beta_{5} - 327 \beta_{4} + 45 \beta_{3} + \cdots + 2449 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 510 \beta_{8} + 10592 \beta_{7} + 9314 \beta_{6} - 2696 \beta_{5} - 1005 \beta_{4} - 1646 \beta_{3} + \cdots + 223434 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40779 \beta_{8} + 44169 \beta_{7} + 14676 \beta_{6} - 28807 \beta_{5} - 33141 \beta_{4} + 440 \beta_{3} + \cdots + 103410 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 62097 \beta_{8} + 988266 \beta_{7} + 859737 \beta_{6} - 279231 \beta_{5} - 121077 \beta_{4} + \cdots + 18500635 \) 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
9.32355
8.38995
6.93776
1.56723
−0.330540
−2.10900
−5.52373
−5.67028
−9.58492
−2.00000 −9.32355 4.00000 0 18.6471 −14.4532 −8.00000 59.9285 0
1.2 −2.00000 −8.38995 4.00000 0 16.7799 −23.6795 −8.00000 43.3912 0
1.3 −2.00000 −6.93776 4.00000 0 13.8755 7.91891 −8.00000 21.1325 0
1.4 −2.00000 −1.56723 4.00000 0 3.13447 3.11755 −8.00000 −24.5438 0
1.5 −2.00000 0.330540 4.00000 0 −0.661080 18.7014 −8.00000 −26.8907 0
1.6 −2.00000 2.10900 4.00000 0 −4.21800 −35.7485 −8.00000 −22.5521 0
1.7 −2.00000 5.52373 4.00000 0 −11.0475 26.7233 −8.00000 3.51163 0
1.8 −2.00000 5.67028 4.00000 0 −11.3406 −11.4356 −8.00000 5.15209 0
1.9 −2.00000 9.58492 4.00000 0 −19.1698 −15.1443 −8.00000 64.8708 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.ba 9
5.b even 2 1 1150.4.a.bb 9
5.c odd 4 2 230.4.b.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.b.b ✓ 18 5.c odd 4 2
1150.4.a.ba 9 1.a even 1 1 trivial
1150.4.a.bb 9 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):

\( T_{3}^{9} + 3 T_{3}^{8} - 179 T_{3}^{7} - 380 T_{3}^{6} + 10197 T_{3}^{5} + 8259 T_{3}^{4} + \cdots - 178000 \) Copy content Toggle raw display
\( T_{7}^{9} + 44 T_{7}^{8} - 804 T_{7}^{7} - 50171 T_{7}^{6} - 65062 T_{7}^{5} + 14993076 T_{7}^{4} + \cdots + 26142461280 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots - 178000 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 26142461280 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 1738279352448 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 27230930068712 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 21\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 213356057254112 \) Copy content Toggle raw display
$23$ \( (T - 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 21\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 39\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 13\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 28\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 63\!\cdots\!28 \) Copy content Toggle raw display
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