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Properties

Label 275.6.a.h
Level $275$
Weight $6$
Character orbit 275.a
Self dual yes
Analytic conductor $44.106$
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] = [275,6,Mod(1,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(44.1055504486\)
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: \( x^{8} - 4x^{7} - 176x^{6} + 437x^{5} + 9155x^{4} - 9342x^{3} - 165530x^{2} - 26499x + 527670 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 - 2) q^{3} + (\beta_{3} - \beta_{2} + 14) q^{4} + (\beta_{6} + \beta_{4} + 2 \beta_{3} + \cdots + 53) q^{6}+ \cdots + (3 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 101) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 - 2) q^{3} + (\beta_{3} - \beta_{2} + 14) q^{4} + (\beta_{6} + \beta_{4} + 2 \beta_{3} + \cdots + 53) q^{6}+ \cdots + ( - 363 \beta_{6} - 121 \beta_{5} + \cdots - 12221) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 9 q^{3} + 112 q^{4} + 392 q^{6} - 155 q^{7} + 21 q^{8} + 757 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 9 q^{3} + 112 q^{4} + 392 q^{6} - 155 q^{7} + 21 q^{8} + 757 q^{9} - 968 q^{11} - 2955 q^{12} - 2100 q^{13} - 2793 q^{14} + 1464 q^{16} - 1697 q^{17} - 6331 q^{18} + 2286 q^{19} - 5062 q^{21} + 484 q^{22} - 2721 q^{23} + 6489 q^{24} + 9644 q^{26} - 1296 q^{27} + 3024 q^{28} + 6685 q^{29} - 6798 q^{31} - 5572 q^{32} + 1089 q^{33} - 16502 q^{34} + 24706 q^{36} - 21848 q^{37} - 4406 q^{38} + 22645 q^{39} - 17458 q^{41} - 1495 q^{42} - 30150 q^{43} - 13552 q^{44} - 70414 q^{46} + 31094 q^{47} - 124208 q^{48} + 92141 q^{49} - 72157 q^{51} - 96720 q^{52} - 18679 q^{53} + 168485 q^{54} - 145895 q^{56} + 47919 q^{57} - 126159 q^{58} + 65726 q^{59} - 31185 q^{61} + 53512 q^{62} - 92454 q^{63} + 155907 q^{64} - 47432 q^{66} - 74368 q^{67} - 21815 q^{68} + 33584 q^{69} - 60718 q^{71} - 412182 q^{72} - 273795 q^{73} + 78453 q^{74} - 211566 q^{76} + 18755 q^{77} - 539660 q^{78} + 60613 q^{79} + 38416 q^{81} - 19849 q^{82} - 119189 q^{83} - 103007 q^{84} - 404465 q^{86} - 162565 q^{87} - 2541 q^{88} - 109189 q^{89} - 117455 q^{91} - 122595 q^{92} - 370471 q^{93} + 194347 q^{94} + 188439 q^{96} - 243923 q^{97} - 415790 q^{98} - 91597 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 176x^{6} + 437x^{5} + 9155x^{4} - 9342x^{3} - 165530x^{2} - 26499x + 527670 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 149 \nu^{7} + 1637 \nu^{6} + 17571 \nu^{5} - 202192 \nu^{4} - 349899 \nu^{3} + 5164217 \nu^{2} + \cdots - 18620574 ) / 284448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 149 \nu^{7} + 1637 \nu^{6} + 17571 \nu^{5} - 202192 \nu^{4} - 349899 \nu^{3} + 5448665 \nu^{2} + \cdots - 31420734 ) / 284448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 191 \nu^{7} + 1303 \nu^{6} + 30001 \nu^{5} - 175824 \nu^{4} - 1308713 \nu^{3} + 6156163 \nu^{2} + \cdots - 42423354 ) / 284448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83 \nu^{7} + 892 \nu^{6} + 11160 \nu^{5} - 114102 \nu^{4} - 377602 \nu^{3} + 3143640 \nu^{2} + \cdots - 12855966 ) / 35556 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1381 \nu^{7} - 11593 \nu^{6} - 184651 \nu^{5} + 1392212 \nu^{4} + 5430871 \nu^{3} + \cdots + 150760230 ) / 284448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2231 \nu^{7} - 18943 \nu^{6} - 291729 \nu^{5} + 2230584 \nu^{4} + 8178865 \nu^{3} + \cdots + 178710714 ) / 284448 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 2\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} - 2\beta_{4} - \beta_{2} + 80\beta _1 + 68 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 2\beta_{6} + 5\beta_{5} - 17\beta_{4} + 108\beta_{3} - 142\beta_{2} + 299\beta _1 + 3595 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 133\beta_{7} - 278\beta_{6} + 45\beta_{5} - 329\beta_{4} + 89\beta_{3} - 453\beta_{2} + 8032\beta _1 + 11162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 64 \beta_{7} - 660 \beta_{6} + 947 \beta_{5} - 3069 \beta_{4} + 11572 \beta_{3} - 17017 \beta_{2} + \cdots + 358583 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15396 \beta_{7} - 32624 \beta_{6} + 8926 \beta_{5} - 44750 \beta_{4} + 25736 \beta_{3} - 81906 \beta_{2} + \cdots + 1652508 \) 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
−9.56077
−5.24613
−5.02240
−2.42114
1.77463
5.99293
7.22770
11.2552
−10.5608 −28.0828 79.5298 0 296.576 162.322 −501.951 545.645 0
1.2 −6.24613 −19.7009 7.01418 0 123.054 −252.886 156.065 145.124 0
1.3 −6.02240 9.32944 4.26933 0 −56.1857 203.853 167.005 −155.962 0
1.4 −3.42114 17.7070 −20.2958 0 −60.5780 −111.912 178.911 70.5371 0
1.5 0.774628 −15.2298 −31.4000 0 −11.7975 96.7581 −49.1114 −11.0520 0
1.6 4.99293 27.7791 −7.07068 0 138.699 −208.539 −195.077 528.680 0
1.7 6.22770 7.33544 6.78423 0 45.6829 96.8075 −157.036 −189.191 0
1.8 10.2552 −8.13748 73.1689 0 −83.4514 −141.404 422.195 −176.781 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
\(5\) \( +1 \)
\(11\) \( +1 \)

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 275.6.a.h ✓ 8
5.b even 2 1 275.6.a.i yes 8
5.c odd 4 2 275.6.b.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.6.a.h ✓ 8 1.a even 1 1 trivial
275.6.a.i yes 8 5.b even 2 1
275.6.b.g 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{7} - 176T_{2}^{6} - 647T_{2}^{5} + 8630T_{2}^{4} + 28044T_{2}^{3} - 136952T_{2}^{2} - 347856T_{2} + 335712 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(275))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 335712 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 2308108500 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T + 121)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 11\!\cdots\!75 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 18\!\cdots\!51 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 59\!\cdots\!17 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 44\!\cdots\!11 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 71\!\cdots\!13 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 75\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 99\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 42\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 55\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 34\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 44\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 29\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 29\!\cdots\!91 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 28\!\cdots\!25 \) Copy content Toggle raw display
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