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Properties

Label 1175.4.a.d
Level $1175$
Weight $4$
Character orbit 1175.a
Self dual yes
Analytic conductor $69.327$
Analytic rank $1$
Dimension $10$
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] = [1175,4,Mod(1,1175)]
 
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([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("1175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1175 = 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1175.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: \(69.3272442567\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 50 x^{8} + 108 x^{7} + 815 x^{6} - 1205 x^{5} - 4566 x^{4} + 5476 x^{3} + 6592 x^{2} + \cdots - 2208 \) 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 235)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{6} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{8} - 2 \beta_{6} + \cdots - 2 \beta_1) q^{6}+ \cdots + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{6} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{8} - 2 \beta_{6} + \cdots - 2 \beta_1) q^{6}+ \cdots + ( - 22 \beta_{9} - 5 \beta_{8} + \cdots - 127) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 7 q^{2} - 8 q^{3} + 33 q^{4} - 2 q^{6} - 9 q^{7} - 63 q^{8} + 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 7 q^{2} - 8 q^{3} + 33 q^{4} - 2 q^{6} - 9 q^{7} - 63 q^{8} + 88 q^{9} - 47 q^{11} - 150 q^{12} - 106 q^{13} + 397 q^{14} + 293 q^{16} - 363 q^{17} - 163 q^{18} - 27 q^{19} + 146 q^{21} - 374 q^{22} - 133 q^{23} + 234 q^{24} + 53 q^{26} - 308 q^{27} - 257 q^{28} + 635 q^{29} + 116 q^{31} - 491 q^{32} - 198 q^{33} + 132 q^{34} + 483 q^{36} - 264 q^{37} - 662 q^{38} + 314 q^{39} + 640 q^{41} - 523 q^{42} + 374 q^{43} - 1282 q^{44} - 987 q^{46} + 470 q^{47} - 1650 q^{48} + 209 q^{49} + 1123 q^{51} - 273 q^{52} - 1308 q^{53} - 227 q^{54} + 2935 q^{56} + 113 q^{57} - 407 q^{58} + 321 q^{59} + 397 q^{61} - 589 q^{62} - 1356 q^{63} - 283 q^{64} - 1355 q^{66} - 16 q^{67} - 2884 q^{68} + 416 q^{69} + 910 q^{71} + 1023 q^{72} - 965 q^{73} - 2783 q^{74} - 2526 q^{76} - 1776 q^{77} + 2649 q^{78} - 2132 q^{79} - 1874 q^{81} + 4030 q^{82} - 3345 q^{83} - 4889 q^{84} + 656 q^{86} - 1193 q^{87} + 3394 q^{88} - 1018 q^{89} - 2006 q^{91} + 2999 q^{92} - 858 q^{93} - 329 q^{94} - 686 q^{96} + 2381 q^{97} + 2588 q^{98} - 1454 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 50 x^{8} + 108 x^{7} + 815 x^{6} - 1205 x^{5} - 4566 x^{4} + 5476 x^{3} + 6592 x^{2} + \cdots - 2208 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 127 \nu^{9} - 2543 \nu^{8} + 23320 \nu^{7} + 105920 \nu^{6} - 749605 \nu^{5} - 1354101 \nu^{4} + \cdots + 1634304 ) / 1300464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 263 \nu^{9} - 4547 \nu^{8} + 11440 \nu^{7} + 148436 \nu^{6} - 619375 \nu^{5} - 1402293 \nu^{4} + \cdots - 14938416 ) / 866976 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 655 \nu^{9} + 2671 \nu^{8} + 47740 \nu^{7} - 186724 \nu^{6} - 1050961 \nu^{5} + 3778785 \nu^{4} + \cdots + 16235088 ) / 866976 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7891 \nu^{9} - 53191 \nu^{8} - 276496 \nu^{7} + 2012596 \nu^{6} + 2833669 \nu^{5} + \cdots - 25883184 ) / 5201856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24487 \nu^{9} + 43435 \nu^{8} + 1332232 \nu^{7} - 1390948 \nu^{6} - 23130721 \nu^{5} + \cdots + 68279664 ) / 5201856 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26065 \nu^{9} - 70717 \nu^{8} - 1263592 \nu^{7} + 2281564 \nu^{6} + 19414471 \nu^{5} + \cdots - 100689744 ) / 5201856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22057 \nu^{9} - 43453 \nu^{8} - 1083148 \nu^{7} + 1000996 \nu^{6} + 16442191 \nu^{5} + \cdots + 55088304 ) / 2600928 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{4} + 2\beta_{2} + 20\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 2\beta_{8} + 2\beta_{6} + 3\beta_{5} - 6\beta_{4} + 5\beta_{3} + 30\beta_{2} + 49\beta _1 + 234 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{9} + 47 \beta_{8} + 23 \beta_{7} + 4 \beta_{6} + 20 \beta_{5} - 49 \beta_{4} + 16 \beta_{3} + \cdots + 567 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 77 \beta_{9} + 158 \beta_{8} + 66 \beta_{6} + 163 \beta_{5} - 296 \beta_{4} + 265 \beta_{3} + \cdots + 6146 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 638 \beta_{9} + 1797 \beta_{8} + 497 \beta_{7} + 172 \beta_{6} + 1082 \beta_{5} - 1945 \beta_{4} + \cdots + 22229 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3685 \beta_{9} + 7722 \beta_{8} + 56 \beta_{7} + 1778 \beta_{6} + 6983 \beta_{5} - 11726 \beta_{4} + \cdots + 179998 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 26092 \beta_{9} + 64655 \beta_{8} + 10655 \beta_{7} + 6092 \beta_{6} + 44764 \beta_{5} + \cdots + 801959 \) 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
−4.48623
−3.45621
−3.19432
−1.14390
−0.337963
1.02380
1.76358
2.36767
4.60153
5.86206
−5.48623 −3.85426 22.0987 0 21.1453 −15.0341 −77.3487 −12.1447 0
1.2 −4.45621 5.99426 11.8578 0 −26.7117 −27.6211 −17.1912 8.93116 0
1.3 −4.19432 −8.80839 9.59231 0 36.9452 −6.89424 −6.67865 50.5877 0
1.4 −2.14390 8.20380 −3.40367 0 −17.5882 4.97585 24.4484 40.3023 0
1.5 −1.33796 −4.09547 −6.20986 0 5.47959 −14.8711 19.0123 −10.2271 0
1.6 0.0237969 2.09726 −7.99943 0 0.0499082 30.9022 −0.380737 −22.6015 0
1.7 0.763576 1.95391 −7.41695 0 1.49196 0.232785 −11.7720 −23.1822 0
1.8 1.36767 −8.54544 −6.12949 0 −11.6873 −19.8081 −19.3244 46.0245 0
1.9 3.60153 5.17795 4.97100 0 18.6485 8.51030 −10.9090 −0.188807 0
1.10 4.86206 −6.12362 15.6396 0 −29.7734 30.6076 37.1441 10.4987 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(47\) \(-1\)

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 1175.4.a.d 10
5.b even 2 1 235.4.a.b ✓ 10
15.d odd 2 1 2115.4.a.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
235.4.a.b ✓ 10 5.b even 2 1
1175.4.a.d 10 1.a even 1 1 trivial
2115.4.a.g 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 7 T_{2}^{9} - 32 T_{2}^{8} - 280 T_{2}^{7} + 129 T_{2}^{6} + 3027 T_{2}^{5} + 1746 T_{2}^{4} + \cdots - 128 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 7 T^{9} + \cdots - 128 \) Copy content Toggle raw display
$3$ \( T^{10} + 8 T^{9} + \cdots - 7591868 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 7862790336 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 6166281917952 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 3752713915248 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 62\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 34\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 90\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 34\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 70\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 93\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T - 47)^{10} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 75\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 61\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 61\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 27\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
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