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Properties

Label 175.4.k.d
Level $175$
Weight $4$
Character orbit 175.k
Analytic conductor $10.325$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,4,Mod(74,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.74");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 75 x^{18} + 3638 x^{16} - 105775 x^{14} + 2246038 x^{12} - 30934571 x^{10} + 307864753 x^{8} + \cdots + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_1) q^{2} + (\beta_{8} + \beta_{6}) q^{3} + ( - \beta_{12} - 7 \beta_{7} + 7) q^{4} + ( - \beta_{10} + \beta_{9} + \beta_{4} + \cdots + 3) q^{6} + (\beta_{17} + 4 \beta_{15} + \cdots - \beta_1) q^{7}+ \cdots + ( - 15 \beta_{10} + 15 \beta_{9} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 70 q^{4} + 64 q^{6} + 162 q^{9} - 94 q^{11} - 14 q^{14} - 342 q^{16} - 42 q^{19} + 356 q^{21} + 1696 q^{24} + 94 q^{26} - 760 q^{29} - 776 q^{31} - 520 q^{34} + 4916 q^{36} + 28 q^{39} - 1124 q^{41}+ \cdots - 140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 75 x^{18} + 3638 x^{16} - 105775 x^{14} + 2246038 x^{12} - 30934571 x^{10} + 307864753 x^{8} + \cdots + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 47\!\cdots\!27 \nu^{18} + \cdots - 25\!\cdots\!16 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\!\cdots\!47 \nu^{18} + \cdots - 37\!\cdots\!64 ) / 82\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!31 \nu^{18} + \cdots - 20\!\cdots\!84 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3217726829 \nu^{18} - 235003656367 \nu^{16} + 11151370466878 \nu^{14} + \cdots - 82\!\cdots\!52 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 62\!\cdots\!59 \nu^{19} + \cdots - 52\!\cdots\!24 \nu ) / 62\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!93 \nu^{18} + \cdots - 77\!\cdots\!68 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!75 \nu^{19} + \cdots + 46\!\cdots\!48 \nu ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!37 \nu^{18} + \cdots + 74\!\cdots\!44 ) / 41\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 91\!\cdots\!37 \nu^{18} + \cdots - 30\!\cdots\!88 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 94\!\cdots\!07 \nu^{18} + \cdots - 27\!\cdots\!92 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!31 \nu^{18} + \cdots + 83\!\cdots\!32 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!41 \nu^{19} + \cdots + 90\!\cdots\!24 \nu ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!93 \nu^{19} + \cdots - 77\!\cdots\!68 \nu ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1570085297079 \nu^{19} - 117653430022397 \nu^{17} + \cdots - 70\!\cdots\!52 \nu ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 50\!\cdots\!73 \nu^{19} + \cdots + 56\!\cdots\!76 \nu ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10\!\cdots\!19 \nu^{19} + \cdots - 78\!\cdots\!12 \nu ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 12\!\cdots\!23 \nu^{19} + \cdots + 56\!\cdots\!00 \nu ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 82\!\cdots\!55 \nu^{19} + \cdots - 10\!\cdots\!72 \nu ) / 35\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + 15\beta_{7} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} + 21\beta_{14} + 4\beta_{13} + 4\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 29\beta_{12} - 4\beta_{11} - 8\beta_{9} + 331\beta_{7} - 2\beta_{5} - 4\beta_{4} - 327 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{19} - 2 \beta_{18} - 6 \beta_{17} - 2 \beta_{16} - 54 \beta_{15} + 491 \beta_{14} + \cdots - 491 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152\beta_{10} - 152\beta_{9} - 184\beta_{4} - 68\beta_{3} - 793\beta_{2} - 7823 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 316\beta_{19} - 316\beta_{18} - 216\beta_{17} + 216\beta_{16} + 1498\beta_{8} - 4288\beta_{6} - 12129\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 21237 \beta_{12} + 6300 \beta_{11} + 2908 \beta_{10} + 4604 \beta_{9} - 198787 \beta_{7} + \cdots - 1696 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3910 \beta_{19} - 7996 \beta_{18} + 3910 \beta_{17} + 11906 \beta_{16} + 46638 \beta_{15} + \cdots - 117870 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 563953 \beta_{12} + 193392 \beta_{11} - 63616 \beta_{10} + 259552 \beta_{9} - 5084415 \beta_{7} + \cdots + 4954639 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 257008 \beta_{19} + 137800 \beta_{18} + 394808 \beta_{17} + 137800 \beta_{16} + 1528562 \beta_{15} + \cdots + 7959597 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -3551028\beta_{10} + 3551028\beta_{9} + 5644276\beta_{4} + 2474570\beta_{3} + 14923853\beta_{2} + 130147655 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12305342 \beta_{19} + 12305342 \beta_{18} + 7737524 \beta_{17} - 7737524 \beta_{16} + \cdots + 207325763 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 394440265 \beta_{12} - 160537576 \beta_{11} - 31144744 \beta_{10} - 95841160 \beta_{9} + 3430326263 \beta_{7} + \cdots + 64696416 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 145314092 \beta_{19} + 225233992 \beta_{18} - 145314092 \beta_{17} - 370548084 \beta_{16} + \cdots + 2199176248 \beta_{6} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 10423659045 \beta_{12} - 4502384652 \beta_{11} + 1932660320 \beta_{10} - 5139448664 \beta_{9} + \cdots - 87233905479 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 6435044972 \beta_{19} - 4488836174 \beta_{18} - 10923881146 \beta_{17} - 4488836174 \beta_{16} + \cdots - 142690202287 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 68687475808 \beta_{10} - 68687475808 \beta_{9} - 125253090336 \beta_{4} - 79093800528 \beta_{3} + \cdots - 2345819043151 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 317478119920 \beta_{19} - 317478119920 \beta_{18} - 181818704864 \beta_{17} + 181818704864 \beta_{16} + \cdots - 3759177093573 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−4.50647 − 2.60181i
−4.35203 − 2.51265i
−3.33093 − 1.92311i
−2.42233 − 1.39853i
−0.197018 − 0.113749i
0.197018 + 0.113749i
2.42233 + 1.39853i
3.33093 + 1.92311i
4.35203 + 2.51265i
4.50647 + 2.60181i
−4.50647 + 2.60181i
−4.35203 + 2.51265i
−3.33093 + 1.92311i
−2.42233 + 1.39853i
−0.197018 + 0.113749i
0.197018 − 0.113749i
2.42233 − 1.39853i
3.33093 − 1.92311i
4.35203 − 2.51265i
4.50647 − 2.60181i
−4.50647 + 2.60181i −4.24917 − 2.45326i 9.53885 − 16.5218i 0 25.5317 −4.23212 + 18.0302i 57.6442i −1.46305 − 2.53407i 0
74.2 −4.35203 + 2.51265i −7.22596 − 4.17191i 8.62677 − 14.9420i 0 41.9301 7.81984 − 16.7884i 46.5017i 21.3097 + 36.9094i 0
74.3 −3.33093 + 1.92311i 7.76647 + 4.48397i 3.39672 − 5.88330i 0 −34.4927 13.8005 − 12.3509i − 4.64067i 26.7120 + 46.2666i 0
74.4 −2.42233 + 1.39853i 5.38134 + 3.10692i −0.0882227 + 0.152806i 0 −17.3805 −7.92622 − 16.7384i − 22.8700i 5.80586 + 10.0560i 0
74.5 −0.197018 + 0.113749i −1.56628 − 0.904291i −3.97412 + 6.88338i 0 0.411448 16.6030 + 8.20612i − 3.62818i −11.8645 − 20.5499i 0
74.6 0.197018 − 0.113749i 1.56628 + 0.904291i −3.97412 + 6.88338i 0 0.411448 −16.6030 − 8.20612i 3.62818i −11.8645 − 20.5499i 0
74.7 2.42233 − 1.39853i −5.38134 − 3.10692i −0.0882227 + 0.152806i 0 −17.3805 7.92622 + 16.7384i 22.8700i 5.80586 + 10.0560i 0
74.8 3.33093 − 1.92311i −7.76647 − 4.48397i 3.39672 − 5.88330i 0 −34.4927 −13.8005 + 12.3509i 4.64067i 26.7120 + 46.2666i 0
74.9 4.35203 − 2.51265i 7.22596 + 4.17191i 8.62677 − 14.9420i 0 41.9301 −7.81984 + 16.7884i − 46.5017i 21.3097 + 36.9094i 0
74.10 4.50647 − 2.60181i 4.24917 + 2.45326i 9.53885 − 16.5218i 0 25.5317 4.23212 − 18.0302i − 57.6442i −1.46305 − 2.53407i 0
149.1 −4.50647 − 2.60181i −4.24917 + 2.45326i 9.53885 + 16.5218i 0 25.5317 −4.23212 − 18.0302i − 57.6442i −1.46305 + 2.53407i 0
149.2 −4.35203 − 2.51265i −7.22596 + 4.17191i 8.62677 + 14.9420i 0 41.9301 7.81984 + 16.7884i − 46.5017i 21.3097 − 36.9094i 0
149.3 −3.33093 − 1.92311i 7.76647 − 4.48397i 3.39672 + 5.88330i 0 −34.4927 13.8005 + 12.3509i 4.64067i 26.7120 − 46.2666i 0
149.4 −2.42233 − 1.39853i 5.38134 − 3.10692i −0.0882227 − 0.152806i 0 −17.3805 −7.92622 + 16.7384i 22.8700i 5.80586 − 10.0560i 0
149.5 −0.197018 − 0.113749i −1.56628 + 0.904291i −3.97412 − 6.88338i 0 0.411448 16.6030 − 8.20612i 3.62818i −11.8645 + 20.5499i 0
149.6 0.197018 + 0.113749i 1.56628 − 0.904291i −3.97412 − 6.88338i 0 0.411448 −16.6030 + 8.20612i − 3.62818i −11.8645 + 20.5499i 0
149.7 2.42233 + 1.39853i −5.38134 + 3.10692i −0.0882227 − 0.152806i 0 −17.3805 7.92622 − 16.7384i − 22.8700i 5.80586 − 10.0560i 0
149.8 3.33093 + 1.92311i −7.76647 + 4.48397i 3.39672 + 5.88330i 0 −34.4927 −13.8005 − 12.3509i − 4.64067i 26.7120 − 46.2666i 0
149.9 4.35203 + 2.51265i 7.22596 − 4.17191i 8.62677 + 14.9420i 0 41.9301 −7.81984 − 16.7884i 46.5017i 21.3097 − 36.9094i 0
149.10 4.50647 + 2.60181i 4.24917 − 2.45326i 9.53885 + 16.5218i 0 25.5317 4.23212 + 18.0302i 57.6442i −1.46305 + 2.53407i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.k.d 20
5.b even 2 1 inner 175.4.k.d 20
5.c odd 4 1 35.4.e.c ✓ 10
5.c odd 4 1 175.4.e.d 10
7.c even 3 1 inner 175.4.k.d 20
15.e even 4 1 315.4.j.g 10
20.e even 4 1 560.4.q.n 10
35.f even 4 1 245.4.e.o 10
35.j even 6 1 inner 175.4.k.d 20
35.k even 12 1 245.4.a.n 5
35.k even 12 1 245.4.e.o 10
35.k even 12 1 1225.4.a.bf 5
35.l odd 12 1 35.4.e.c ✓ 10
35.l odd 12 1 175.4.e.d 10
35.l odd 12 1 245.4.a.m 5
35.l odd 12 1 1225.4.a.bg 5
105.w odd 12 1 2205.4.a.bt 5
105.x even 12 1 315.4.j.g 10
105.x even 12 1 2205.4.a.bu 5
140.w even 12 1 560.4.q.n 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.c ✓ 10 5.c odd 4 1
35.4.e.c ✓ 10 35.l odd 12 1
175.4.e.d 10 5.c odd 4 1
175.4.e.d 10 35.l odd 12 1
175.4.k.d 20 1.a even 1 1 trivial
175.4.k.d 20 5.b even 2 1 inner
175.4.k.d 20 7.c even 3 1 inner
175.4.k.d 20 35.j even 6 1 inner
245.4.a.m 5 35.l odd 12 1
245.4.a.n 5 35.k even 12 1
245.4.e.o 10 35.f even 4 1
245.4.e.o 10 35.k even 12 1
315.4.j.g 10 15.e even 4 1
315.4.j.g 10 105.x even 12 1
560.4.q.n 10 20.e even 4 1
560.4.q.n 10 140.w even 12 1
1225.4.a.bf 5 35.k even 12 1
1225.4.a.bg 5 35.l odd 12 1
2205.4.a.bt 5 105.w odd 12 1
2205.4.a.bu 5 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 75 T_{2}^{18} + 3638 T_{2}^{16} - 105775 T_{2}^{14} + 2246038 T_{2}^{12} - 30934571 T_{2}^{10} + \cdots + 16777216 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 75 T^{18} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 289812354063376 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 1044811065600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 94\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{5} + 190 T^{4} + \cdots + 13190815450)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 63\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + 281 T^{4} + \cdots + 77000100765)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 53\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 28\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 6439908260352)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 85\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 74\!\cdots\!44)^{2} \) Copy content Toggle raw display
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