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Properties

Label 175.3.g.b
Level $175$
Weight $3$
Character orbit 175.g
Analytic conductor $4.768$
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] = [175,3,Mod(43,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 175.g (of order \(4\), degree \(2\), 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: \(4.76840462631\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} + \cdots + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{10} - \beta_{4} - \beta_{3} + \beta_1) q^{4} + ( - 2 \beta_{9} - 2 \beta_{6} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{10} + 2 \beta_{9} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{10} - \beta_{4} - \beta_{3} + \beta_1) q^{4} + ( - 2 \beta_{9} - 2 \beta_{6} + \cdots - 2) q^{6}+ \cdots + ( - 6 \beta_{11} - 8 \beta_{10} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 4 q^{3} - 24 q^{6} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 4 q^{3} - 24 q^{6} - 24 q^{8} - 12 q^{11} - 16 q^{12} + 4 q^{13} + 40 q^{16} + 12 q^{17} + 56 q^{18} + 28 q^{21} + 68 q^{22} + 16 q^{23} - 56 q^{26} - 164 q^{27} - 96 q^{31} - 32 q^{32} - 124 q^{33} + 232 q^{36} + 104 q^{37} - 80 q^{38} - 208 q^{41} + 140 q^{42} - 76 q^{43} - 80 q^{46} + 164 q^{47} + 392 q^{48} + 220 q^{51} - 216 q^{52} + 204 q^{53} + 168 q^{56} + 236 q^{57} - 356 q^{58} + 280 q^{61} - 568 q^{62} - 112 q^{63} - 544 q^{66} - 324 q^{67} - 184 q^{68} + 144 q^{71} + 440 q^{72} + 248 q^{73} - 632 q^{76} + 56 q^{77} - 12 q^{78} - 260 q^{81} + 376 q^{82} + 224 q^{83} + 456 q^{86} - 244 q^{87} + 24 q^{88} + 84 q^{91} + 424 q^{92} - 236 q^{93} + 504 q^{96} - 564 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} + \cdots + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 216885938 \nu^{11} + 1060104974 \nu^{10} - 3418831332 \nu^{9} + 4776047991 \nu^{8} + \cdots + 9734414919250 ) / 2042321889134 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 650167512 \nu^{11} + 2167388474 \nu^{10} - 4094056596 \nu^{9} - 5876565360 \nu^{8} + \cdots - 412177775220 ) / 2042321889134 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20608888761 \nu^{11} - 85686392604 \nu^{10} + 175708052458 \nu^{9} + 144400827108 \nu^{8} + \cdots + 8569191768820 ) / 10211609445670 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22907692353 \nu^{11} + 65097403720 \nu^{10} - 78953995253 \nu^{9} - 352975937926 \nu^{8} + \cdots - 40619239593408 ) / 8169287556536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 62789472393 \nu^{11} + 265477896952 \nu^{10} - 559679834849 \nu^{9} + \cdots + 4818810061550 ) / 20423218891340 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 137676730578 \nu^{11} + 596855018097 \nu^{10} - 1287825684554 \nu^{9} + \cdots + 10093227434800 ) / 40846437782680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 150422124598 \nu^{11} + 616051382677 \nu^{10} - 1215756736104 \nu^{9} + \cdots - 126398845752920 ) / 40846437782680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 86748937073 \nu^{11} - 338454191902 \nu^{10} + 673792264554 \nu^{9} + 692145335679 \nu^{8} + \cdots + 70443542020170 ) / 20423218891340 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19958721249 \nu^{11} - 83519004130 \nu^{10} + 171613995862 \nu^{9} + 138524261748 \nu^{8} + \cdots + 8157013993600 ) / 2042321889134 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 409998418757 \nu^{11} + 1617251253578 \nu^{10} - 3359047754671 \nu^{9} + \cdots - 181679855170840 ) / 40846437782680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 5\beta_{4} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 2\beta_{9} - 2\beta_{4} - 9\beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + 2\beta_{8} - 2\beta_{7} - 2\beta_{6} - 13\beta_{3} + 11\beta_{2} - 13\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} - 17\beta_{10} - 4\beta_{7} - 30\beta_{6} - 2\beta_{5} + 38\beta_{4} + 17\beta_{2} - 89\beta _1 - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{11} - 123 \beta_{10} + 44 \beta_{9} - 32 \beta_{8} - 32 \beta_{7} - 44 \beta_{6} + \cdots - 159 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40 \beta_{11} - 235 \beta_{10} + 382 \beta_{9} - 96 \beta_{8} + 40 \beta_{5} + 550 \beta_{4} + \cdots + 550 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 702 \beta_{9} - 422 \beta_{8} + 422 \beta_{7} + 702 \beta_{6} + 176 \beta_{5} + 1925 \beta_{3} + \cdots + 4049 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 598 \beta_{11} + 3049 \beta_{10} + 1580 \beta_{7} + 4710 \beta_{6} + 598 \beta_{5} - 7254 \beta_{4} + \cdots + 7254 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2776 \beta_{11} + 16859 \beta_{10} - 9856 \beta_{9} + 5308 \beta_{8} + 5308 \beta_{7} + \cdots + 23323 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8084 \beta_{11} + 38487 \beta_{10} - 57726 \beta_{9} + 22488 \beta_{8} - 8084 \beta_{5} + \cdots - 92062 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.85871 + 1.85871i
−1.36503 + 1.36503i
−0.109772 + 0.109772i
0.950261 − 0.950261i
1.90845 − 1.90845i
2.47480 − 2.47480i
−1.85871 − 1.85871i
−1.36503 − 1.36503i
−0.109772 − 0.109772i
0.950261 + 0.950261i
1.90845 + 1.90845i
2.47480 + 2.47480i
−1.85871 + 1.85871i −0.163806 − 0.163806i − 2.90963i 0 0.608937 1.87083 − 1.87083i −2.02669 − 2.02669i − 8.94634i 0
43.2 −1.36503 + 1.36503i 3.78207 + 3.78207i 0.273387i 0 −10.3253 −1.87083 + 1.87083i −5.83330 − 5.83330i 19.6082i 0
43.3 −0.109772 + 0.109772i −1.67281 − 1.67281i 3.97590i 0 0.367256 −1.87083 + 1.87083i −0.875529 − 0.875529i − 3.40339i 0
43.4 0.950261 − 0.950261i −0.269488 − 0.269488i 2.19401i 0 −0.512169 1.87083 − 1.87083i 5.88593 + 5.88593i − 8.85475i 0
43.5 1.90845 − 1.90845i 3.30412 + 3.30412i − 3.28438i 0 12.6115 1.87083 − 1.87083i 1.36573 + 1.36573i 12.8345i 0
43.6 2.47480 − 2.47480i −2.98009 − 2.98009i − 8.24929i 0 −14.7503 −1.87083 + 1.87083i −10.5161 − 10.5161i 8.76185i 0
57.1 −1.85871 − 1.85871i −0.163806 + 0.163806i 2.90963i 0 0.608937 1.87083 + 1.87083i −2.02669 + 2.02669i 8.94634i 0
57.2 −1.36503 − 1.36503i 3.78207 − 3.78207i − 0.273387i 0 −10.3253 −1.87083 − 1.87083i −5.83330 + 5.83330i − 19.6082i 0
57.3 −0.109772 − 0.109772i −1.67281 + 1.67281i − 3.97590i 0 0.367256 −1.87083 − 1.87083i −0.875529 + 0.875529i 3.40339i 0
57.4 0.950261 + 0.950261i −0.269488 + 0.269488i − 2.19401i 0 −0.512169 1.87083 + 1.87083i 5.88593 − 5.88593i 8.85475i 0
57.5 1.90845 + 1.90845i 3.30412 − 3.30412i 3.28438i 0 12.6115 1.87083 + 1.87083i 1.36573 − 1.36573i − 12.8345i 0
57.6 2.47480 + 2.47480i −2.98009 + 2.98009i 8.24929i 0 −14.7503 −1.87083 − 1.87083i −10.5161 + 10.5161i − 8.76185i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.3.g.b 12
5.b even 2 1 35.3.g.a ✓ 12
5.c odd 4 1 35.3.g.a ✓ 12
5.c odd 4 1 inner 175.3.g.b 12
15.d odd 2 1 315.3.o.a 12
15.e even 4 1 315.3.o.a 12
20.d odd 2 1 560.3.bh.e 12
20.e even 4 1 560.3.bh.e 12
35.c odd 2 1 245.3.g.a 12
35.f even 4 1 245.3.g.a 12
35.i odd 6 2 245.3.m.c 24
35.j even 6 2 245.3.m.d 24
35.k even 12 2 245.3.m.c 24
35.l odd 12 2 245.3.m.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.g.a ✓ 12 5.b even 2 1
35.3.g.a ✓ 12 5.c odd 4 1
175.3.g.b 12 1.a even 1 1 trivial
175.3.g.b 12 5.c odd 4 1 inner
245.3.g.a 12 35.c odd 2 1
245.3.g.a 12 35.f even 4 1
245.3.m.c 24 35.i odd 6 2
245.3.m.c 24 35.k even 12 2
245.3.m.d 24 35.j even 6 2
245.3.m.d 24 35.l odd 12 2
315.3.o.a 12 15.d odd 2 1
315.3.o.a 12 15.e even 4 1
560.3.bh.e 12 20.d odd 2 1
560.3.bh.e 12 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 4 T_{2}^{11} + 8 T_{2}^{10} + 8 T_{2}^{9} + 70 T_{2}^{8} - 248 T_{2}^{7} + 464 T_{2}^{6} + \cdots + 100 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 484 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{3} \) Copy content Toggle raw display
$11$ \( (T^{6} + 6 T^{5} + \cdots + 255536)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 7014062500 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 70807081216 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 2061234490000 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 133532630919424 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 196000000 \) Copy content Toggle raw display
$31$ \( (T^{6} + 48 T^{5} + \cdots + 116022400)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + 104 T^{5} + \cdots + 38452576)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 159385575040000 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} - 140 T^{5} + \cdots + 35862860384)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{6} - 72 T^{5} + \cdots + 15911820320)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
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