Properties

Label 275.4.b.d
Level $275$
Weight $4$
Character orbit 275.b
Analytic conductor $16.226$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,4,Mod(199,275)]
 
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([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(16.2255252516\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) 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 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{2} + ( - 3 \beta_{4} - \beta_{3}) q^{3} + ( - \beta_{2} + \beta_1 - 7) q^{4} + ( - 4 \beta_{2} + 3 \beta_1 + 22) q^{6} + ( - 14 \beta_{5} - 5 \beta_{4} - 3 \beta_{3}) q^{7} + (5 \beta_{5} + 5 \beta_{4} + 8 \beta_{3}) q^{8} + (9 \beta_{2} - 12 \beta_1 - 31) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{2} + ( - 3 \beta_{4} - \beta_{3}) q^{3} + ( - \beta_{2} + \beta_1 - 7) q^{4} + ( - 4 \beta_{2} + 3 \beta_1 + 22) q^{6} + ( - 14 \beta_{5} - 5 \beta_{4} - 3 \beta_{3}) q^{7} + (5 \beta_{5} + 5 \beta_{4} + 8 \beta_{3}) q^{8} + (9 \beta_{2} - 12 \beta_1 - 31) q^{9} - 11 q^{11} + (4 \beta_{5} + 15 \beta_{4} + 3 \beta_{3}) q^{12} + ( - 26 \beta_{5} - 12 \beta_{4} - 13 \beta_{3}) q^{13} + ( - 22 \beta_{2} - 19 \beta_1 - 36) q^{14} + (10 \beta_{2} - 6 \beta_1 - 55) q^{16} + (12 \beta_{5} + 25 \beta_{4} - 7 \beta_{3}) q^{17} + ( - 17 \beta_{5} - 64 \beta_{4} - 59 \beta_{3}) q^{18} + (9 \beta_{2} + 32 \beta_1 + 14) q^{19} + (\beta_{2} + 70 \beta_1 + 10) q^{21} + (11 \beta_{5} - 11 \beta_{4} + 11 \beta_{3}) q^{22} + (38 \beta_{5} + 44 \beta_{4} + 75 \beta_{3}) q^{23} + ( - 10 \beta_{2} + 23 \beta_1 + 82) q^{24} + ( - 51 \beta_{2} - 13 \beta_1 - 60) q^{26} + ( - 54 \beta_{5} + 63 \beta_{4} + 58 \beta_{3}) q^{27} + (138 \beta_{5} + 53 \beta_{4} + 87 \beta_{3}) q^{28} + ( - 11 \beta_{2} - 30 \beta_1 - 144) q^{29} + (7 \beta_{2} + 20 \beta_1 - 38) q^{31} + (27 \beta_{5} - 59 \beta_{4} + 41 \beta_{3}) q^{32} + (33 \beta_{4} + 11 \beta_{3}) q^{33} + (30 \beta_{2} + 45 \beta_1 - 154) q^{34} + ( - 68 \beta_{2} + 47 \beta_1 + 61) q^{36} + ( - 30 \beta_{5} - 9 \beta_{4} + 79 \beta_{3}) q^{37} + ( - 150 \beta_{5} + \cdots + 28 \beta_{3}) q^{38}+ \cdots + ( - 99 \beta_{2} + 132 \beta_1 + 341) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 46 q^{4} + 118 q^{6} - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 46 q^{4} + 118 q^{6} - 144 q^{9} - 66 q^{11} - 222 q^{14} - 298 q^{16} + 38 q^{19} - 78 q^{21} + 426 q^{24} - 436 q^{26} - 826 q^{29} - 254 q^{31} - 954 q^{34} + 136 q^{36} - 500 q^{39} + 116 q^{41} + 506 q^{44} + 540 q^{46} - 2584 q^{49} + 1982 q^{51} - 3250 q^{54} + 1390 q^{56} + 1260 q^{59} + 2190 q^{61} + 1906 q^{64} - 1298 q^{66} + 3996 q^{69} - 890 q^{71} + 1154 q^{74} - 438 q^{76} - 636 q^{79} + 4710 q^{81} + 1822 q^{84} - 2344 q^{86} - 5146 q^{89} - 8300 q^{91} + 620 q^{94} - 6154 q^{96} + 1584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} + 323 ) / 121 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{5} - 29\nu^{4} - 60\nu^{3} + 24\nu^{2} - 123 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{5} + 10\nu^{4} + 4\nu^{3} - 50\nu^{2} + 605\nu - 258 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 75\nu^{5} + 30\nu^{4} + 12\nu^{3} - 392\nu^{2} + 1815\nu - 774 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -45\nu^{5} - 18\nu^{4} + 17\nu^{3} + 211\nu^{2} - 968\nu + 416 ) / 121 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{5} + 10\beta_{4} + \beta_{3} - 5\beta_{2} - 5\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 6\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -50\beta_{5} - 58\beta_{4} + 7\beta_{3} - 25\beta_{2} - 33\beta _1 + 64 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.32001 + 1.32001i
−1.75233 1.75233i
0.432320 0.432320i
0.432320 + 0.432320i
−1.75233 + 1.75233i
1.32001 1.32001i
4.60975i 0.545414i −13.2498 0 −2.51422 33.3241i 24.2001i 26.7025 0
199.2 3.77801i 7.42401i −6.27334 0 28.0480 28.7933i 6.52332i −28.1159 0
199.3 3.38776i 9.87859i −3.47689 0 33.4663 19.5308i 15.3232i −70.5866 0
199.4 3.38776i 9.87859i −3.47689 0 33.4663 19.5308i 15.3232i −70.5866 0
199.5 3.77801i 7.42401i −6.27334 0 28.0480 28.7933i 6.52332i −28.1159 0
199.6 4.60975i 0.545414i −13.2498 0 −2.51422 33.3241i 24.2001i 26.7025 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.4.b.d 6
5.b even 2 1 inner 275.4.b.d 6
5.c odd 4 1 55.4.a.c 3
5.c odd 4 1 275.4.a.d 3
15.e even 4 1 495.4.a.f 3
15.e even 4 1 2475.4.a.ba 3
20.e even 4 1 880.4.a.x 3
55.e even 4 1 605.4.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.c 3 5.c odd 4 1
275.4.a.d 3 5.c odd 4 1
275.4.b.d 6 1.a even 1 1 trivial
275.4.b.d 6 5.b even 2 1 inner
495.4.a.f 3 15.e even 4 1
605.4.a.h 3 55.e even 4 1
880.4.a.x 3 20.e even 4 1
2475.4.a.ba 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 47T_{2}^{4} + 711T_{2}^{2} + 3481 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 47 T^{4} + \cdots + 3481 \) Copy content Toggle raw display
$3$ \( T^{6} + 153 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2321 T^{4} + \cdots + 351187600 \) Copy content Toggle raw display
$11$ \( (T + 11)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 2580640000 \) Copy content Toggle raw display
$17$ \( T^{6} + 12201 T^{4} + \cdots + 1577536 \) Copy content Toggle raw display
$19$ \( (T^{3} - 19 T^{2} + \cdots + 549824)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 1812319058176 \) Copy content Toggle raw display
$29$ \( (T^{3} + 413 T^{2} + \cdots + 296164)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 127 T^{2} + \cdots - 7744)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 23679785651344 \) Copy content Toggle raw display
$41$ \( (T^{3} - 58 T^{2} + \cdots + 8259848)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 688156438045696 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 822692536576 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 337667912464 \) Copy content Toggle raw display
$59$ \( (T^{3} - 630 T^{2} + \cdots + 215392)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 1095 T^{2} + \cdots - 20829740)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 400299576070144 \) Copy content Toggle raw display
$71$ \( (T^{3} + 445 T^{2} + \cdots + 69739136)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{3} + 318 T^{2} + \cdots - 502905248)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2573 T^{2} + \cdots - 1186825660)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
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