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Properties

Label 175.5.d.i
Level $175$
Weight $5$
Character orbit 175.d
Analytic conductor $18.090$
Analytic rank $0$
Dimension $12$
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] = [175,5,Mod(76,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.76");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.d (of order \(2\), degree \(1\), 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: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{2} - \beta_1 + 10) q^{4} + ( - \beta_{4} - \beta_{3}) q^{6} + ( - \beta_{5} + 4) q^{7} + (\beta_{6} - 2 \beta_{2} - 8 \beta_1 + 12) q^{8} + (\beta_{11} + 2 \beta_1 - 35) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{2} - \beta_1 + 10) q^{4} + ( - \beta_{4} - \beta_{3}) q^{6} + ( - \beta_{5} + 4) q^{7} + (\beta_{6} - 2 \beta_{2} - 8 \beta_1 + 12) q^{8} + (\beta_{11} + 2 \beta_1 - 35) q^{9} + (\beta_{11} + 2 \beta_{8} + \beta_{6} + \cdots + 14) q^{11}+ \cdots + ( - 24 \beta_{11} - 128 \beta_{8} + \cdots + 756) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 122 q^{4} + 50 q^{7} + 186 q^{8} - 434 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 122 q^{4} + 50 q^{7} + 186 q^{8} - 434 q^{9} + 126 q^{11} + 78 q^{14} + 578 q^{16} - 734 q^{18} - 642 q^{21} - 2264 q^{22} + 756 q^{23} - 1414 q^{28} - 2190 q^{29} + 8682 q^{32} - 3582 q^{36} - 5564 q^{37} + 8634 q^{39} - 5580 q^{42} - 3944 q^{43} - 11196 q^{44} + 7844 q^{46} - 8796 q^{49} + 7206 q^{51} - 11760 q^{53} - 6606 q^{56} + 12900 q^{57} + 18496 q^{58} + 4310 q^{63} + 20146 q^{64} + 24096 q^{67} - 5664 q^{71} - 39214 q^{72} + 17604 q^{74} - 26904 q^{77} - 20100 q^{78} - 1590 q^{79} - 11912 q^{81} - 43128 q^{84} + 17604 q^{86} + 7268 q^{88} - 7182 q^{91} + 60252 q^{92} + 70980 q^{93} - 57714 q^{98} + 23084 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 545049904483 \nu^{11} + 3342865644154 \nu^{10} + 54129391887267 \nu^{9} + \cdots - 25\!\cdots\!60 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 230060779665 \nu^{11} + 4634987015638 \nu^{10} + 6235493671465 \nu^{9} + \cdots + 88\!\cdots\!20 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22050672377839 \nu^{11} - 167967216649588 \nu^{10} + \cdots + 17\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19497288667487 \nu^{11} - 197356811169334 \nu^{10} + \cdots + 11\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10105263298801 \nu^{11} + 7668759064702 \nu^{10} + \cdots - 10\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 775009347434 \nu^{11} + 4971706132333 \nu^{10} + 90922162125716 \nu^{9} + \cdots - 17\!\cdots\!20 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43753662828889 \nu^{11} - 51018613474922 \nu^{10} + \cdots - 30\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19651585016711 \nu^{11} - 155617378754012 \nu^{10} + \cdots + 53\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24193243798023 \nu^{11} - 70693370829664 \nu^{10} + \cdots + 29\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 545049904483 \nu^{11} + 3342865644154 \nu^{10} + 54129391887267 \nu^{9} + \cdots - 93\!\cdots\!00 ) / 53\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15067388428199 \nu^{11} - 153255880811244 \nu^{10} + \cdots + 54\!\cdots\!20 ) / 10\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} + 30\beta _1 + 30 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} - 2\beta_{7} + 2\beta_{5} + 6\beta_{3} - 15\beta_{2} + 15\beta _1 + 330 ) / 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 61 \beta_{10} + 18 \beta_{9} - 18 \beta_{8} - 18 \beta_{7} - 30 \beta_{6} + 18 \beta_{5} + \cdots + 1560 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 72 \beta_{10} + 36 \beta_{9} - 248 \beta_{8} - 128 \beta_{7} - 15 \beta_{6} + 128 \beta_{5} + \cdots + 5220 ) / 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 48 \beta_{11} - 659 \beta_{10} + 312 \beta_{9} - 336 \beta_{8} - 330 \beta_{7} - 60 \beta_{6} + \cdots + 3456 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 360 \beta_{11} - 5940 \beta_{10} + 3060 \beta_{9} - 11456 \beta_{8} - 6896 \beta_{7} - 615 \beta_{6} + \cdots - 313980 ) / 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5280 \beta_{11} - 151789 \beta_{10} + 82572 \beta_{9} - 65292 \beta_{8} - 87822 \beta_{7} + \cdots - 3826560 ) / 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10440 \beta_{11} - 296304 \beta_{10} + 173952 \beta_{9} - 124796 \beta_{8} - 282656 \beta_{7} + \cdots - 44467140 ) / 15 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1549440 \beta_{11} - 4258591 \beta_{10} + 2622348 \beta_{9} + 1106820 \beta_{8} - 2862090 \beta_{7} + \cdots - 450961440 ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 806616 \beta_{11} - 1477620 \beta_{10} + 1009440 \beta_{9} + 5222204 \beta_{8} - 942232 \beta_{7} + \cdots - 645718428 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 125578560 \beta_{11} + 82146923 \beta_{10} - 29772204 \beta_{9} + 381367164 \beta_{8} + \cdots - 32941939680 ) / 30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
7.50299 − 2.23607i
7.50299 + 2.23607i
5.81769 + 2.23607i
5.81769 − 2.23607i
2.17355 − 2.23607i
2.17355 + 2.23607i
−2.23331 + 2.23607i
−2.23331 − 2.23607i
−3.62973 + 2.23607i
−3.62973 − 2.23607i
−6.63119 − 2.23607i
−6.63119 + 2.23607i
−6.50299 − 5.52807i 26.2889 0 35.9490i 19.9894 + 44.7373i −66.9085 50.4404 0
76.2 −6.50299 5.52807i 26.2889 0 − 35.9490i 19.9894 − 44.7373i −66.9085 50.4404 0
76.3 −4.81769 − 14.5704i 7.21016 0 70.1957i −44.8989 − 19.6236i 42.3467 −131.296 0
76.4 −4.81769 14.5704i 7.21016 0 − 70.1957i −44.8989 + 19.6236i 42.3467 −131.296 0
76.5 −1.17355 − 9.10378i −14.6228 0 10.6837i 40.5002 − 27.5814i 35.9373 −1.87877 0
76.6 −1.17355 9.10378i −14.6228 0 − 10.6837i 40.5002 + 27.5814i 35.9373 −1.87877 0
76.7 3.23331 − 14.6970i −5.54568 0 − 47.5200i 26.6804 + 41.0994i −69.6640 −135.002 0
76.8 3.23331 14.6970i −5.54568 0 47.5200i 26.6804 − 41.0994i −69.6640 −135.002 0
76.9 4.62973 − 0.296012i 5.43443 0 − 1.37046i −15.2405 − 46.5696i −48.9158 80.9124 0
76.10 4.62973 0.296012i 5.43443 0 1.37046i −15.2405 + 46.5696i −48.9158 80.9124 0
76.11 7.63119 − 12.6955i 42.2350 0 − 96.8817i −2.03056 − 48.9579i 200.204 −80.1757 0
76.12 7.63119 12.6955i 42.2350 0 96.8817i −2.03056 + 48.9579i 200.204 −80.1757 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.d.i 12
5.b even 2 1 35.5.d.a ✓ 12
5.c odd 4 2 175.5.c.d 24
7.b odd 2 1 inner 175.5.d.i 12
15.d odd 2 1 315.5.h.a 12
20.d odd 2 1 560.5.f.b 12
35.c odd 2 1 35.5.d.a ✓ 12
35.f even 4 2 175.5.c.d 24
105.g even 2 1 315.5.h.a 12
140.c even 2 1 560.5.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.d.a ✓ 12 5.b even 2 1
35.5.d.a ✓ 12 35.c odd 2 1
175.5.c.d 24 5.c odd 4 2
175.5.c.d 24 35.f even 4 2
175.5.d.i 12 1.a even 1 1 trivial
175.5.d.i 12 7.b odd 2 1 inner
315.5.h.a 12 15.d odd 2 1
315.5.h.a 12 105.g even 2 1
560.5.f.b 12 20.d odd 2 1
560.5.f.b 12 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} - 74T_{2}^{4} + 168T_{2}^{3} + 1348T_{2}^{2} - 2340T_{2} - 4200 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 3 T^{5} + \cdots - 4200)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 1640250000 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 5924759078688)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 449297249971200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 58\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
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