Properties

Label 1175.2.c.g
Level $1175$
Weight $2$
Character orbit 1175.c
Analytic conductor $9.382$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1175,2,Mod(424,1175)]
 
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([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1175.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1175 = 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1175.c (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: \(9.38242223750\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 21x^{12} + 172x^{10} + 696x^{8} + 1440x^{6} + 1385x^{4} + 429x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 235)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{8} q^{6} + (\beta_{12} + \beta_{11} - \beta_{9} + \beta_1) q^{7} + (\beta_{12} + \beta_{11} - \beta_{9} + \beta_{6}) q^{8} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{8} q^{6} + (\beta_{12} + \beta_{11} - \beta_{9} + \beta_1) q^{7} + (\beta_{12} + \beta_{11} - \beta_{9} + \beta_{6}) q^{8} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{9} + (\beta_{8} - \beta_{5} - \beta_{4}) q^{11} + (\beta_{12} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_1) q^{12} + (\beta_{7} + \beta_1) q^{13} + (\beta_{13} + 2 \beta_{8} + 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{14} + (\beta_{8} + 2 \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{16} + ( - \beta_{11} + 2 \beta_{9} + \beta_{6}) q^{17} + ( - \beta_{12} + 2 \beta_{9} - \beta_{7}) q^{18} + ( - 2 \beta_{13} - \beta_{8} - \beta_{5} + \beta_{4} - 2) q^{19} + ( - \beta_{8} + \beta_{3} - \beta_{2} + 1) q^{21} + (\beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{7} + \beta_{6}) q^{22} + ( - \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \beta_1) q^{23} + (\beta_{13} - 2 \beta_{3} - 1) q^{24} + (\beta_{8} - \beta_{5} - \beta_{4} - 2) q^{26} + ( - \beta_{12} + 2 \beta_{10} - \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_1) q^{27} + ( - 3 \beta_{12} - 2 \beta_{11} - \beta_{10} + 5 \beta_{9} - \beta_1) q^{28} + ( - 2 \beta_{3} - 4) q^{29} + (\beta_{8} + \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{31} + ( - 2 \beta_{12} - \beta_{11} - \beta_{10} + 3 \beta_{9} + \beta_{7} - \beta_{6} - \beta_1) q^{32} + (2 \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{6} + 2 \beta_1) q^{33} + ( - 2 \beta_{13} - 2 \beta_{8} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2) q^{34} + ( - 2 \beta_{8} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 6) q^{36} + ( - 3 \beta_{11} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 2 \beta_1) q^{37} + (3 \beta_{11} - \beta_{10} - \beta_{6}) q^{38} + (2 \beta_{13} - \beta_{8} - \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{39} + (2 \beta_{8} + 2) q^{41} + (\beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{7} - 2 \beta_{6} + 3 \beta_1) q^{42} + ( - \beta_{11} - \beta_{10} + 6 \beta_{9} + \beta_{7} - 3 \beta_{6} + \beta_1) q^{43} + ( - \beta_{8} + \beta_{5} - \beta_{4}) q^{44} + ( - 2 \beta_{8} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 6) q^{46} - \beta_{9} q^{47} + (\beta_{12} - \beta_{11} - \beta_{10} + 5 \beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_1) q^{48} + (2 \beta_{13} + 2 \beta_{8} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 1) q^{49} + ( - \beta_{8} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3) q^{51} + (\beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{6}) q^{52} + ( - \beta_{12} + 3 \beta_{10} - \beta_{9} + \beta_{6} - \beta_1) q^{53} + ( - \beta_{13} - 2 \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{54} + ( - 2 \beta_{8} - \beta_{5} + 4 \beta_{3} + 2 \beta_{2} - 4) q^{56} + ( - 2 \beta_{12} + 3 \beta_{11} + \beta_{10} - 4 \beta_{9} + 2 \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{57}+ \cdots + ( - 2 \beta_{13} - 2 \beta_{8} - 2 \beta_{5} + 4 \beta_{4} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4} + 2 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{4} + 2 q^{6} - 20 q^{9} + 2 q^{11} - 20 q^{14} - 2 q^{16} - 6 q^{19} + 14 q^{21} - 16 q^{24} - 26 q^{26} - 52 q^{29} - 10 q^{31} - 12 q^{34} - 68 q^{36} - 2 q^{39} + 24 q^{41} - 6 q^{44} - 72 q^{46} - 12 q^{49} - 28 q^{51} + 4 q^{54} - 54 q^{56} - 26 q^{59} + 40 q^{61} + 18 q^{64} - 82 q^{66} - 32 q^{69} + 22 q^{71} - 96 q^{74} + 22 q^{76} - 34 q^{79} + 30 q^{81} - 92 q^{84} - 60 q^{86} - 26 q^{89} - 22 q^{91} + 2 q^{94} - 58 q^{96} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 21x^{12} + 172x^{10} + 696x^{8} + 1440x^{6} + 1385x^{4} + 429x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{12} + 52\nu^{10} + 320\nu^{8} + 832\nu^{6} + 824\nu^{4} + 163\nu^{2} - 4 ) / 56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{12} + 156\nu^{10} + 960\nu^{8} + 2440\nu^{6} + 1912\nu^{4} - 855\nu^{2} - 348 ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{12} + 132\nu^{10} + 912\nu^{8} + 2776\nu^{6} + 3416\nu^{4} + 1095\nu^{2} + 156 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{13} + 156\nu^{11} + 960\nu^{9} + 2440\nu^{7} + 1912\nu^{5} - 855\nu^{3} - 348\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{13} + 84\nu^{11} + 464\nu^{9} + 744\nu^{7} - 1128\nu^{5} - 3211\nu^{3} - 468\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{12} - 108\nu^{10} - 864\nu^{8} - 3112\nu^{6} - 4808\nu^{4} - 2261\nu^{2} + 12 ) / 112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} + 24\nu^{11} + 224\nu^{9} + 1016\nu^{7} + 2272\nu^{5} + 2209\nu^{3} + 592\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{13} - 68\nu^{11} - 624\nu^{9} - 2952\nu^{7} - 7432\nu^{5} - 8963\nu^{3} - 3548\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -19\nu^{13} - 356\nu^{11} - 2496\nu^{9} - 8168\nu^{7} - 12648\nu^{5} - 8195\nu^{3} - 1564\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3\nu^{13} + 62\nu^{11} + 496\nu^{9} + 1940\nu^{7} + 3820\nu^{5} + 3395\nu^{3} + 886\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5\nu^{12} + 92\nu^{10} + 624\nu^{8} + 1912\nu^{6} + 2552\nu^{4} + 1077\nu^{2} + 4 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{9} + \beta_{6} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 2\beta_{5} - \beta_{4} - 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{12} - 9\beta_{11} - \beta_{10} + 11\beta_{9} + \beta_{7} - 9\beta_{6} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{8} - 20\beta_{5} + 8\beta_{4} + 3\beta_{3} + 46\beta_{2} - 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 79\beta_{12} + 66\beta_{11} + 10\beta_{10} - 95\beta_{9} - 10\beta_{7} + 64\beta_{6} - 100\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{13} + 81\beta_{8} + 155\beta_{5} - 54\beta_{4} - 44\beta_{3} - 299\beta_{2} + 501 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -577\beta_{12} - 456\beta_{11} - 79\beta_{10} + 739\beta_{9} + 74\beta_{7} - 425\beta_{6} + 564\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -31\beta_{13} - 613\beta_{8} - 1107\beta_{5} + 351\beta_{4} + 435\beta_{3} + 1948\beta_{2} - 3107 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4072\beta_{12} + 3086\beta_{11} + 582\beta_{10} - 5436\beta_{9} - 494\beta_{7} + 2762\beta_{6} - 3335\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 324\beta_{13} + 4484\beta_{8} + 7652\beta_{5} - 2268\beta_{4} - 3660\beta_{3} - 12761\beta_{2} + 19755 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 28233 \beta_{12} - 20737 \beta_{11} - 4160 \beta_{10} + 38721 \beta_{9} + 3168 \beta_{7} - 17873 \beta_{6} + 20380 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1175\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(851\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
2.59170i
2.24729i
1.89749i
1.68601i
1.49903i
0.730051i
0.0980806i
0.0980806i
0.730051i
1.49903i
1.68601i
1.89749i
2.24729i
2.59170i
2.59170i 0.345837i −4.71692 0 −0.896307 3.55345i 7.04146i 2.88040 0
424.2 2.24729i 2.14783i −3.05029 0 4.82679 4.71377i 2.36031i −1.61318 0
424.3 1.89749i 1.57213i −1.60046 0 −2.98310 0.327488i 0.758120i 0.528410 0
424.4 1.68601i 0.335153i −0.842624 0 −0.565071 4.20242i 1.95135i 2.88767 0
424.5 1.49903i 2.14267i −0.247101 0 3.21194 0.914748i 2.62765i −1.59105 0
424.6 0.730051i 3.14616i 1.46703 0 −2.29686 0.964297i 2.53111i −6.89835 0
424.7 0.0980806i 3.03215i 1.99038 0 −0.297395 0.786854i 0.391379i −6.19391 0
424.8 0.0980806i 3.03215i 1.99038 0 −0.297395 0.786854i 0.391379i −6.19391 0
424.9 0.730051i 3.14616i 1.46703 0 −2.29686 0.964297i 2.53111i −6.89835 0
424.10 1.49903i 2.14267i −0.247101 0 3.21194 0.914748i 2.62765i −1.59105 0
424.11 1.68601i 0.335153i −0.842624 0 −0.565071 4.20242i 1.95135i 2.88767 0
424.12 1.89749i 1.57213i −1.60046 0 −2.98310 0.327488i 0.758120i 0.528410 0
424.13 2.24729i 2.14783i −3.05029 0 4.82679 4.71377i 2.36031i −1.61318 0
424.14 2.59170i 0.345837i −4.71692 0 −0.896307 3.55345i 7.04146i 2.88040 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.14
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 1175.2.c.g 14
5.b even 2 1 inner 1175.2.c.g 14
5.c odd 4 1 235.2.a.e 7
5.c odd 4 1 1175.2.a.h 7
15.e even 4 1 2115.2.a.v 7
20.e even 4 1 3760.2.a.bi 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
235.2.a.e 7 5.c odd 4 1
1175.2.a.h 7 5.c odd 4 1
1175.2.c.g 14 1.a even 1 1 trivial
1175.2.c.g 14 5.b even 2 1 inner
2115.2.a.v 7 15.e even 4 1
3760.2.a.bi 7 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1175, [\chi])\):

\( T_{2}^{14} + 21T_{2}^{12} + 172T_{2}^{10} + 696T_{2}^{8} + 1440T_{2}^{6} + 1385T_{2}^{4} + 429T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{7} - T_{11}^{6} - 46T_{11}^{5} + 40T_{11}^{4} + 512T_{11}^{3} - 80T_{11}^{2} - 1408T_{11} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 21 T^{12} + 172 T^{10} + 696 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{14} + 31 T^{12} + 365 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 55 T^{12} + 1029 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{7} - T^{6} - 46 T^{5} + 40 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 74 T^{12} + 1625 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{14} + 114 T^{12} + \cdots + 11723776 \) Copy content Toggle raw display
$19$ \( (T^{7} + 3 T^{6} - 100 T^{5} - 384 T^{4} + \cdots - 4352)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 261 T^{12} + \cdots + 23279435776 \) Copy content Toggle raw display
$29$ \( (T^{7} + 26 T^{6} + 248 T^{5} + 1024 T^{4} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 5 T^{6} - 74 T^{5} - 632 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 402 T^{12} + \cdots + 158391264256 \) Copy content Toggle raw display
$41$ \( (T^{7} - 12 T^{6} - 36 T^{5} + 888 T^{4} + \cdots - 54784)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 489 T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$47$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$53$ \( T^{14} + 278 T^{12} + \cdots + 270536704 \) Copy content Toggle raw display
$59$ \( (T^{7} + 13 T^{6} - 197 T^{5} + \cdots + 191656)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 20 T^{6} - 18 T^{5} + 1332 T^{4} + \cdots - 6218)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 416 T^{12} + \cdots + 40282095616 \) Copy content Toggle raw display
$71$ \( (T^{7} - 11 T^{6} - 227 T^{5} + \cdots + 550688)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 482 T^{12} + \cdots + 1001595904 \) Copy content Toggle raw display
$79$ \( (T^{7} + 17 T^{6} - 157 T^{5} + \cdots - 1921952)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 405 T^{12} + 41972 T^{10} + \cdots + 4194304 \) Copy content Toggle raw display
$89$ \( (T^{7} + 13 T^{6} - 187 T^{5} + \cdots - 14252)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 1006 T^{12} + \cdots + 224660832256 \) Copy content Toggle raw display
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