Properties

Label 2175.4.a.k
Level $2175$
Weight $4$
Character orbit 2175.a
Self dual yes
Analytic conductor $128.329$
Analytic rank $1$
Dimension $6$
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] = [2175,4,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(128.329154262\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 435)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 8) q^{4} + 3 \beta_1 q^{6} + (\beta_{4} - 8) q^{7} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 - 7) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 8) q^{4} + 3 \beta_1 q^{6} + (\beta_{4} - 8) q^{7} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 - 7) q^{8} + 9 q^{9} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 14) q^{11} + ( - 3 \beta_{2} - 3 \beta_1 - 24) q^{12} + ( - \beta_{5} - \beta_{2} - 3 \beta_1 - 27) q^{13} + ( - 3 \beta_{5} + \beta_{4} - \beta_{2} + \cdots - 5) q^{14}+ \cdots + (9 \beta_{5} - 9 \beta_{4} + \cdots + 126) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23} + 153 q^{24} + 294 q^{26} - 162 q^{27} - 344 q^{28} + 174 q^{29} + 340 q^{31} - 499 q^{32} - 243 q^{33} + 920 q^{34} + 459 q^{36} - 332 q^{37} + 378 q^{38} + 507 q^{39} - 616 q^{41} + 90 q^{42} - 334 q^{43} - 52 q^{44} - 158 q^{46} + 85 q^{47} - 393 q^{48} + 879 q^{49} - 3 q^{51} - 2220 q^{52} + 850 q^{53} + 27 q^{54} - 624 q^{56} - 348 q^{57} - 29 q^{58} - 758 q^{59} - 36 q^{61} + 152 q^{62} - 423 q^{63} + 1795 q^{64} + 270 q^{66} - 939 q^{67} + 186 q^{68} - 156 q^{69} - 1388 q^{71} - 459 q^{72} - 1708 q^{73} - 814 q^{74} + 566 q^{76} - 2585 q^{77} - 882 q^{78} + 1250 q^{79} + 486 q^{81} - 1372 q^{82} + 748 q^{83} + 1032 q^{84} - 800 q^{86} - 522 q^{87} - 536 q^{88} + 1099 q^{89} + 539 q^{91} - 1698 q^{92} - 1020 q^{93} - 4542 q^{94} + 1497 q^{96} + 22 q^{97} + 1433 q^{98} + 729 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 21\nu + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 35\nu^{3} + 62\nu^{2} + 242\nu - 372 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} - 33\nu^{2} - 4\nu + 192 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 22\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 33\beta_{2} + 37\beta _1 + 336 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} + 8\beta_{4} + 35\beta_{3} + 39\beta_{2} + 540\beta _1 + 297 \) Copy content Toggle raw display

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
5.42717
4.39070
2.35472
−2.57352
−3.54861
−5.05047
−5.42717 −3.00000 21.4542 0 16.2815 10.2400 −73.0180 9.00000 0
1.2 −4.39070 −3.00000 11.2783 0 13.1721 −31.5336 −14.3939 9.00000 0
1.3 −2.35472 −3.00000 −2.45528 0 7.06417 3.94392 24.6193 9.00000 0
1.4 2.57352 −3.00000 −1.37700 0 −7.72056 −31.5281 −24.1319 9.00000 0
1.5 3.54861 −3.00000 4.59262 0 −10.6458 21.2667 −12.0915 9.00000 0
1.6 5.05047 −3.00000 17.5072 0 −15.1514 −19.3889 48.0160 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(29\) \( -1 \)

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 2175.4.a.k 6
5.b even 2 1 435.4.a.h 6
15.d odd 2 1 1305.4.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.h 6 5.b even 2 1
1305.4.a.h 6 15.d odd 2 1
2175.4.a.k 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\):

\( T_{2}^{6} + T_{2}^{5} - 49T_{2}^{4} - 27T_{2}^{3} + 692T_{2}^{2} + 82T_{2} - 2588 \) Copy content Toggle raw display
\( T_{7}^{6} + 47T_{7}^{5} - 364T_{7}^{4} - 31972T_{7}^{3} + 4072T_{7}^{2} + 4689056T_{7} - 16555808 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + \cdots - 2588 \) Copy content Toggle raw display
$3$ \( (T + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 47 T^{5} + \cdots - 16555808 \) Copy content Toggle raw display
$11$ \( T^{6} - 81 T^{5} + \cdots + 51862040 \) Copy content Toggle raw display
$13$ \( T^{6} + 169 T^{5} + \cdots + 158715904 \) Copy content Toggle raw display
$17$ \( T^{6} - T^{5} + \cdots - 178244224 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 105108863360 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 670601037824 \) Copy content Toggle raw display
$29$ \( (T - 29)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 1225416704 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 128139819712 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 4772989110016 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 142035778816 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 11309241395200 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 89482466320 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 452631511470080 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 205098218864640 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 22\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 756565599059968 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 226457216857600 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 72\!\cdots\!32 \) Copy content Toggle raw display
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