Properties

Label 175.4.f.g
Level $175$
Weight $4$
Character orbit 175.f
Analytic conductor $10.325$
Analytic rank $0$
Dimension $16$
CM no
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(118,175)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.118");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.f (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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10654x^{12} + 22102125x^{8} + 5700572500x^{4} + 44626562500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + \beta_{11} q^{3} + (\beta_{12} - \beta_{8} + \beta_{7} - 4 \beta_{4}) q^{4} + ( - \beta_{15} - \beta_{13} + \beta_{11} + \beta_{10} + \beta_{3} + \beta_1) q^{6} + (2 \beta_{8} + \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{7} + (6 \beta_{7} - \beta_{6} - 13 \beta_{4} - 12) q^{8} + ( - \beta_{12} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 14 \beta_{4} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + \beta_{11} q^{3} + (\beta_{12} - \beta_{8} + \beta_{7} - 4 \beta_{4}) q^{4} + ( - \beta_{15} - \beta_{13} + \beta_{11} + \beta_{10} + \beta_{3} + \beta_1) q^{6} + (2 \beta_{8} + \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{7} + (6 \beta_{7} - \beta_{6} - 13 \beta_{4} - 12) q^{8} + ( - \beta_{12} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 14 \beta_{4} + \cdots + 2) q^{9}+ \cdots + ( - 30 \beta_{12} + 112 \beta_{8} - 112 \beta_{7} - 32 \beta_{6} + \cdots + 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 32 q^{7} - 176 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} - 32 q^{7} - 176 q^{8} - 152 q^{11} - 504 q^{16} - 288 q^{18} + 328 q^{21} - 348 q^{22} + 72 q^{23} + 528 q^{28} - 432 q^{32} + 344 q^{36} + 256 q^{37} + 1300 q^{42} + 312 q^{43} - 1856 q^{46} + 696 q^{51} - 1768 q^{53} + 1304 q^{56} + 3920 q^{57} + 4764 q^{58} - 2544 q^{63} - 4504 q^{67} + 6368 q^{71} - 7848 q^{72} + 3016 q^{77} - 5340 q^{78} - 3088 q^{81} - 9336 q^{86} + 2048 q^{88} + 1608 q^{91} + 6328 q^{92} + 3960 q^{93} - 3308 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10654x^{12} + 22102125x^{8} + 5700572500x^{4} + 44626562500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13637\nu^{12} + 141541523\nu^{8} + 268260815950\nu^{4} + 33643813420000 ) / 2595065895000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -72908\nu^{13} - 790876257\nu^{9} - 1748160601575\nu^{5} - 620163391231250\nu ) / 50603784952500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 103709\nu^{14} + 1108253436\nu^{10} + 2329722745125\nu^{6} + 699405434108750\nu^{2} ) / 1934850601125000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 922187 \nu^{14} + 16951545 \nu^{12} + 10180112673 \nu^{10} + 177015939555 \nu^{8} + 24362198808750 \nu^{6} + \cdots - 45\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2712347 \nu^{14} + 28252575 \nu^{12} - 29527060063 \nu^{10} + 295026565925 \nu^{8} - 67007189126000 \nu^{6} + \cdots - 54\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26023741 \nu^{14} + 53408225 \nu^{12} + 277997050989 \nu^{10} + 614189471025 \nu^{8} + 580713324166500 \nu^{6} + \cdots + 41\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26023741 \nu^{14} + 53408225 \nu^{12} - 277997050989 \nu^{10} + 614189471025 \nu^{8} - 580713324166500 \nu^{6} + \cdots + 41\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21675877 \nu^{15} + 273281450 \nu^{13} + 237181688433 \nu^{11} + 2814563615175 \nu^{9} + 544908018727500 \nu^{7} + \cdots - 65\!\cdots\!50 \nu ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21675877 \nu^{15} - 273281450 \nu^{13} + 237181688433 \nu^{11} - 2814563615175 \nu^{9} + 544908018727500 \nu^{7} + \cdots + 65\!\cdots\!50 \nu ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 103709\nu^{15} + 1108253436\nu^{11} + 2329722745125\nu^{7} + 699405434108750\nu^{3} ) / 1934850601125000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11171152 \nu^{14} - 118324616783 \nu^{10} - 240090289254625 \nu^{6} - 52\!\cdots\!50 \nu^{2} ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 753130203 \nu^{15} - 2863109275 \nu^{13} - 8000341557387 \nu^{11} - 30005408823975 \nu^{9} + \cdots - 11\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 753130203 \nu^{15} + 2863109275 \nu^{13} - 8000341557387 \nu^{11} + 30005408823975 \nu^{9} + \cdots + 11\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4232737\nu^{15} + 45142420038\nu^{11} + 94008451007535\nu^{7} + 24778718008842250\nu^{3} ) / 5701359771315000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + 2\beta_{8} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 41\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 4\beta_{14} + 4\beta_{13} + 67\beta_{11} - 8\beta_{10} - 8\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 36\beta_{8} + 36\beta_{7} - 201\beta_{6} - 201\beta_{5} - 201\beta_{4} + 93\beta_{2} - 2635 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 552\beta_{14} - 552\beta_{13} + 969\beta_{10} - 969\beta_{9} + 423\beta_{3} - 5011\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9439\beta_{12} + 4672\beta_{8} - 4672\beta_{7} - 18173\beta_{6} + 18173\beta_{5} - 210644\beta_{4} + 18173 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -55129\beta_{15} - 59896\beta_{14} - 59896\beta_{13} - 401203\beta_{11} + 95537\beta_{10} + 95537\beta_{9} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 808956 \beta_{8} + 808956 \beta_{7} + 1619979 \beta_{6} + 1619979 \beta_{5} + 1619979 \beta_{4} - 919347 \beta_{2} + 16461715 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5887608 \beta_{14} + 5887608 \beta_{13} - 8908851 \beta_{10} + 8908851 \beta_{9} - 5777217 \beta_{3} + 33582919 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 86195731 \beta_{12} - 91446988 \beta_{8} + 91446988 \beta_{7} + 144279767 \beta_{6} - 144279767 \beta_{5} + 1485183926 \beta_{4} - 144279767 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 557649241 \beta_{15} + 552397984 \beta_{14} + 552397984 \beta_{13} + 2889367387 \beta_{11} - 812845823 \beta_{10} - 812845823 \beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9104513724 \beta_{8} - 9104513724 \beta_{7} - 12860150391 \beta_{6} - 12860150391 \beta_{5} - 12860150391 \beta_{4} + 7902954063 \beta_{2} + \cdots - 121492467235 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 50630722632 \beta_{14} - 50630722632 \beta_{13} + 73405265679 \beta_{10} - 73405265679 \beta_{9} + 51832282293 \beta_{3} - 252648048151 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 715809307699 \beta_{12} + 858779499652 \beta_{8} - 858779499652 \beta_{7} - 1147049021243 \beta_{6} + 1147049021243 \beta_{5} + \cdots + 1147049021243 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4727466349489 \beta_{15} - 4584496157536 \beta_{14} - 4584496157536 \beta_{13} - 22296858454723 \beta_{11} + 6594024605867 \beta_{10} + 6594024605867 \beta_{9} \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
1.19219 + 1.19219i
−1.19219 1.19219i
6.68128 + 6.68128i
−6.68128 6.68128i
2.91939 + 2.91939i
−2.91939 2.91939i
4.94129 + 4.94129i
−4.94129 4.94129i
1.19219 1.19219i
−1.19219 + 1.19219i
6.68128 6.68128i
−6.68128 + 6.68128i
2.91939 2.91939i
−2.91939 + 2.91939i
4.94129 4.94129i
−4.94129 + 4.94129i
−3.08000 + 3.08000i −1.19219 + 1.19219i 10.9728i 0 7.34390i 13.5707 + 12.6030i 9.15614 + 9.15614i 24.1574i 0
118.2 −3.08000 + 3.08000i 1.19219 1.19219i 10.9728i 0 7.34390i −12.6030 13.5707i 9.15614 + 9.15614i 24.1574i 0
118.3 −1.31781 + 1.31781i −6.68128 + 6.68128i 4.52674i 0 17.6093i 6.92319 17.1776i −16.5079 16.5079i 62.2789i 0
118.4 −1.31781 + 1.31781i 6.68128 6.68128i 4.52674i 0 17.6093i 17.1776 6.92319i −16.5079 16.5079i 62.2789i 0
118.5 −0.172516 + 0.172516i −2.91939 + 2.91939i 7.94048i 0 1.00728i −3.56610 + 18.1737i −2.74998 2.74998i 9.95432i 0
118.6 −0.172516 + 0.172516i 2.91939 2.91939i 7.94048i 0 1.00728i −18.1737 + 3.56610i −2.74998 2.74998i 9.95432i 0
118.7 3.57033 3.57033i −4.94129 + 4.94129i 17.4944i 0 35.2840i −18.5018 + 0.826888i −33.8983 33.8983i 21.8328i 0
118.8 3.57033 3.57033i 4.94129 4.94129i 17.4944i 0 35.2840i −0.826888 + 18.5018i −33.8983 33.8983i 21.8328i 0
132.1 −3.08000 3.08000i −1.19219 1.19219i 10.9728i 0 7.34390i 13.5707 12.6030i 9.15614 9.15614i 24.1574i 0
132.2 −3.08000 3.08000i 1.19219 + 1.19219i 10.9728i 0 7.34390i −12.6030 + 13.5707i 9.15614 9.15614i 24.1574i 0
132.3 −1.31781 1.31781i −6.68128 6.68128i 4.52674i 0 17.6093i 6.92319 + 17.1776i −16.5079 + 16.5079i 62.2789i 0
132.4 −1.31781 1.31781i 6.68128 + 6.68128i 4.52674i 0 17.6093i 17.1776 + 6.92319i −16.5079 + 16.5079i 62.2789i 0
132.5 −0.172516 0.172516i −2.91939 2.91939i 7.94048i 0 1.00728i −3.56610 18.1737i −2.74998 + 2.74998i 9.95432i 0
132.6 −0.172516 0.172516i 2.91939 + 2.91939i 7.94048i 0 1.00728i −18.1737 3.56610i −2.74998 + 2.74998i 9.95432i 0
132.7 3.57033 + 3.57033i −4.94129 4.94129i 17.4944i 0 35.2840i −18.5018 0.826888i −33.8983 + 33.8983i 21.8328i 0
132.8 3.57033 + 3.57033i 4.94129 + 4.94129i 17.4944i 0 35.2840i −0.826888 18.5018i −33.8983 + 33.8983i 21.8328i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.f.g 16
5.b even 2 1 35.4.f.b 16
5.c odd 4 1 35.4.f.b 16
5.c odd 4 1 inner 175.4.f.g 16
7.b odd 2 1 inner 175.4.f.g 16
35.c odd 2 1 35.4.f.b 16
35.f even 4 1 35.4.f.b 16
35.f even 4 1 inner 175.4.f.g 16
35.i odd 6 2 245.4.l.b 32
35.j even 6 2 245.4.l.b 32
35.k even 12 2 245.4.l.b 32
35.l odd 12 2 245.4.l.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.f.b 16 5.b even 2 1
35.4.f.b 16 5.c odd 4 1
35.4.f.b 16 35.c odd 2 1
35.4.f.b 16 35.f even 4 1
175.4.f.g 16 1.a even 1 1 trivial
175.4.f.g 16 5.c odd 4 1 inner
175.4.f.g 16 7.b odd 2 1 inner
175.4.f.g 16 35.f even 4 1 inner
245.4.l.b 32 35.i odd 6 2
245.4.l.b 32 35.j even 6 2
245.4.l.b 32 35.k even 12 2
245.4.l.b 32 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 2T_{2}^{6} + 20T_{2}^{5} + 549T_{2}^{4} + 1538T_{2}^{3} + 2178T_{2}^{2} + 660T_{2} + 100 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 2 T^{7} + 2 T^{6} + 20 T^{5} + \cdots + 100)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 10654 T^{12} + \cdots + 44626562500 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 32 T^{15} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + 38 T^{3} - 323 T^{2} + \cdots - 199336)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 22406254 T^{12} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + 23886114 T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 20760 T^{6} + \cdots + 132690195125000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 36 T^{7} + \cdots + 1537004857600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 82730 T^{6} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 106580 T^{6} + \cdots + 353301362000000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 128 T^{7} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 400420 T^{6} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 156 T^{7} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 112461415314 T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + 884 T^{7} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 1078040 T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 560420 T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 2252 T^{7} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 1592 T^{3} + \cdots - 101976596376)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 1359643216064 T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + 635270 T^{6} + \cdots + 89\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 4530921507244 T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} - 2894640 T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 7476920645954 T^{12} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
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