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Properties

Label 1075.6.a.d
Level $1075$
Weight $6$
Character orbit 1075.a
Self dual yes
Analytic conductor $172.413$
Analytic rank $0$
Dimension $15$
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] = [1075,6,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, 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("1075.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(172.412606299\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 329 x^{13} + 1535 x^{12} + 41612 x^{11} - 177314 x^{10} - 2538260 x^{9} + \cdots + 297216000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 215)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} + 14) q^{4} + ( - \beta_{7} - \beta_{6} - 9) q^{6} + ( - \beta_{9} + \beta_{2} + \beta_1 + 8) q^{7} + (\beta_{11} + \beta_{9} - \beta_{7} + \cdots + 5) q^{8}+ \cdots + (\beta_{11} - 2 \beta_{6} + \beta_{4} + \cdots + 58) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} + 14) q^{4} + ( - \beta_{7} - \beta_{6} - 9) q^{6} + ( - \beta_{9} + \beta_{2} + \beta_1 + 8) q^{7} + (\beta_{11} + \beta_{9} - \beta_{7} + \cdots + 5) q^{8}+ \cdots + ( - 87 \beta_{14} + 135 \beta_{13} + \cdots - 10165) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 20 q^{3} + 203 q^{4} - 128 q^{6} + 118 q^{7} + 135 q^{8} + 917 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 20 q^{3} + 203 q^{4} - 128 q^{6} + 118 q^{7} + 135 q^{8} + 917 q^{9} + 162 q^{11} - 636 q^{12} + 474 q^{13} + 1095 q^{14} + 2651 q^{16} + 2714 q^{17} + 2609 q^{18} + 1258 q^{19} - 2900 q^{21} + 10022 q^{22} + 5490 q^{23} - 6144 q^{24} - 7471 q^{26} + 6722 q^{27} + 24385 q^{28} + 6982 q^{29} - 14994 q^{31} + 1215 q^{32} + 17178 q^{33} - 17248 q^{34} - 41877 q^{36} + 32958 q^{37} + 5649 q^{38} + 360 q^{39} - 6460 q^{41} + 25014 q^{42} + 27735 q^{43} + 32804 q^{44} - 52608 q^{46} - 10278 q^{47} + 53552 q^{48} + 3339 q^{49} - 17712 q^{51} + 43283 q^{52} + 52324 q^{53} - 37022 q^{54} - 118159 q^{56} + 95140 q^{57} + 148067 q^{58} - 19796 q^{59} - 64588 q^{61} + 142531 q^{62} + 69020 q^{63} - 144813 q^{64} - 183722 q^{66} + 151538 q^{67} + 340500 q^{68} - 49378 q^{69} - 73628 q^{71} + 268071 q^{72} + 182556 q^{73} - 48202 q^{74} - 91529 q^{76} + 34536 q^{77} + 358900 q^{78} - 272076 q^{79} - 61417 q^{81} + 38057 q^{82} + 45760 q^{83} - 148524 q^{84} + 9245 q^{86} + 165122 q^{87} + 229688 q^{88} + 147998 q^{89} - 239260 q^{91} + 297076 q^{92} + 144362 q^{93} + 34182 q^{94} - 54176 q^{96} + 548194 q^{97} - 334804 q^{98} - 157186 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 329 x^{13} + 1535 x^{12} + 41612 x^{11} - 177314 x^{10} - 2538260 x^{9} + \cdots + 297216000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 84\!\cdots\!81 \nu^{14} + \cdots + 15\!\cdots\!00 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97\!\cdots\!41 \nu^{14} + \cdots - 35\!\cdots\!20 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 56\!\cdots\!45 \nu^{14} + \cdots - 15\!\cdots\!40 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83\!\cdots\!31 \nu^{14} + \cdots - 58\!\cdots\!80 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\!\cdots\!43 \nu^{14} + \cdots + 23\!\cdots\!40 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49\!\cdots\!17 \nu^{14} + \cdots + 48\!\cdots\!60 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61\!\cdots\!81 \nu^{14} + \cdots - 14\!\cdots\!40 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67\!\cdots\!83 \nu^{14} + \cdots - 31\!\cdots\!60 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!61 \nu^{14} + \cdots + 69\!\cdots\!60 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!91 \nu^{14} + \cdots - 20\!\cdots\!00 ) / 66\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29\!\cdots\!81 \nu^{14} + \cdots - 77\!\cdots\!20 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{14} + \cdots + 10\!\cdots\!60 ) / 13\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{2} + 77\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} - 2 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 4 \beta_{7} - 10 \beta_{6} + \cdots + 3566 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{14} + 8 \beta_{13} - 2 \beta_{12} + 141 \beta_{11} - 4 \beta_{10} + 145 \beta_{9} + 10 \beta_{8} + \cdots + 719 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{14} - 238 \beta_{13} + 88 \beta_{12} - 522 \beta_{11} - 448 \beta_{10} - 426 \beta_{9} + \cdots + 311690 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1472 \beta_{14} + 1712 \beta_{13} - 1054 \beta_{12} + 16757 \beta_{11} - 1196 \beta_{10} + 18017 \beta_{9} + \cdots + 9379 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4056 \beta_{14} - 23606 \beta_{13} + 18392 \beta_{12} - 84986 \beta_{11} - 39208 \beta_{10} + \cdots + 29148914 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 199360 \beta_{14} + 252112 \beta_{13} - 216726 \beta_{12} + 1903877 \beta_{11} - 214396 \beta_{10} + \cdots - 10619829 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 859048 \beta_{14} - 2286742 \beta_{13} + 2740264 \beta_{12} - 11671770 \beta_{11} - 3112792 \beta_{10} + \cdots + 2839390418 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 24265904 \beta_{14} + 32326592 \beta_{13} - 33531638 \beta_{12} + 212988261 \beta_{11} + \cdots - 2549788341 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 136641656 \beta_{14} - 225881094 \beta_{13} + 358324760 \beta_{12} - 1478846106 \beta_{11} + \cdots + 283948425986 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2817090624 \beta_{14} + 3882122896 \beta_{13} - 4543300422 \beta_{12} + 23677804965 \beta_{11} + \cdots - 427776934181 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 18954684744 \beta_{14} - 23041937110 \beta_{13} + 43975662536 \beta_{12} - 179320906138 \beta_{11} + \cdots + 28924666539026 \) 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
−10.5579
−8.82079
−7.08317
−6.43239
−6.13228
−0.782522
−0.628687
0.197169
1.26269
2.75414
5.61972
6.87899
9.17438
9.49114
10.0595
−10.5579 7.72084 79.4693 0 −81.5159 195.394 −501.176 −183.389 0
1.2 −8.82079 4.03416 45.8064 0 −35.5845 −67.1168 −121.783 −226.726 0
1.3 −7.08317 −17.5586 18.1713 0 124.371 177.355 97.9508 65.3051 0
1.4 −6.43239 −11.6878 9.37562 0 75.1807 −96.2170 145.529 −106.394 0
1.5 −6.13228 23.7714 5.60482 0 −145.773 −218.344 161.863 322.079 0
1.6 −0.782522 −25.4538 −31.3877 0 19.9181 −58.8352 49.6023 404.895 0
1.7 −0.628687 30.0987 −31.6048 0 −18.9227 49.9325 39.9875 662.934 0
1.8 0.197169 16.7450 −31.9611 0 3.30160 −87.8670 −12.6111 37.3960 0
1.9 1.26269 −10.5408 −30.4056 0 −13.3097 −101.599 −78.7991 −131.892 0
1.10 2.75414 14.3418 −24.4147 0 39.4995 157.095 −155.374 −37.3113 0
1.11 5.61972 6.83614 −0.418707 0 38.4172 −71.3849 −182.184 −196.267 0
1.12 6.87899 −17.6851 15.3206 0 −121.656 69.5263 −114.738 69.7627 0
1.13 9.17438 21.0591 52.1693 0 193.204 160.882 185.041 200.486 0
1.14 9.49114 −22.8215 58.0817 0 −216.602 145.733 247.545 277.820 0
1.15 10.0595 1.14031 69.1935 0 11.4709 −136.554 374.147 −241.700 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(43\) \(-1\)

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 1075.6.a.d 15
5.b even 2 1 215.6.a.b ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.6.a.b ✓ 15 5.b even 2 1
1075.6.a.d 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 5 T_{2}^{14} - 329 T_{2}^{13} + 1535 T_{2}^{12} + 41612 T_{2}^{11} - 177314 T_{2}^{10} + \cdots + 297216000 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1075))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots + 297216000 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 19\!\cdots\!60 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 31\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 42\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 48\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 52\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 31\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 89\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 28\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( (T - 1849)^{15} \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 67\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 13\!\cdots\!92 \) Copy content Toggle raw display
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