Properties

Label 175.3.i.d
Level $175$
Weight $3$
Character orbit 175.i
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(26,175)]
 
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([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 175.i (of order \(6\), 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 \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} - 3618 x^{3} + 3579 x^{2} - 1386 x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{11} - \beta_{9} + \beta_{8} + 2 \beta_{3}) q^{4} + (\beta_{10} - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{6} + (\beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{3} + 2) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2}) q^{8} + ( - 2 \beta_{11} + \beta_{10} + \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{11} - \beta_{9} + \beta_{8} + 2 \beta_{3}) q^{4} + (\beta_{10} - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{6} + (\beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{3} + 2) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2}) q^{8} + ( - 2 \beta_{11} + \beta_{10} + \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots - 1) q^{11}+ \cdots + ( - 15 \beta_{10} - 3 \beta_{8} - \beta_{7} + 17 \beta_{6} + 15 \beta_{5} + \beta_{4} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 6 q^{3} - 10 q^{4} + 2 q^{7} + 4 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 6 q^{3} - 10 q^{4} + 2 q^{7} + 4 q^{8} + 14 q^{9} - 14 q^{11} - 18 q^{12} - 2 q^{14} - 22 q^{16} - 48 q^{17} - 64 q^{18} - 30 q^{19} - 84 q^{21} + 88 q^{22} + 14 q^{23} - 36 q^{24} + 66 q^{26} - 202 q^{28} + 64 q^{29} + 132 q^{31} + 54 q^{32} + 192 q^{33} + 156 q^{36} - 44 q^{37} + 300 q^{38} - 24 q^{39} + 138 q^{42} + 4 q^{43} + 6 q^{44} - 214 q^{46} - 204 q^{47} - 24 q^{49} - 132 q^{51} - 252 q^{52} - 196 q^{53} + 168 q^{54} - 460 q^{56} + 48 q^{57} - 158 q^{58} + 72 q^{59} + 72 q^{61} - 536 q^{63} - 140 q^{64} + 744 q^{66} + 138 q^{67} + 348 q^{68} - 8 q^{71} + 196 q^{72} + 528 q^{73} + 50 q^{74} + 176 q^{77} + 312 q^{78} - 12 q^{79} - 310 q^{81} + 378 q^{82} - 276 q^{84} - 40 q^{86} - 138 q^{87} - 604 q^{88} + 204 q^{89} - 480 q^{91} - 732 q^{92} - 84 q^{93} - 42 q^{94} + 540 q^{96} - 898 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} - 3618 x^{3} + 3579 x^{2} - 1386 x + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 115287976 \nu^{11} - 155566808 \nu^{10} - 1678212789 \nu^{9} - 3282242448 \nu^{8} - 24792751384 \nu^{7} - 39686977172 \nu^{6} + \cdots + 423574271316 ) / 1182297257421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 960485876 \nu^{11} + 1805683776 \nu^{10} - 18404798452 \nu^{9} + 23294419987 \nu^{8} - 237640796192 \nu^{7} + \cdots + 279635242467 ) / 1182297257421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6740525471 \nu^{11} + 28434054878 \nu^{10} - 139462837212 \nu^{9} + 440457052261 \nu^{8} - 1761800596603 \nu^{7} + \cdots + 26654089921641 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22979355133 \nu^{11} + 77959111894 \nu^{10} - 482194495722 \nu^{9} + 1167796047734 \nu^{8} - 6080191620431 \nu^{7} + \cdots + 71882680944312 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9058064565 \nu^{11} - 1839528853 \nu^{10} - 143550656091 \nu^{9} - 113877990722 \nu^{8} - 1903870510779 \nu^{7} + \cdots - 3111541612074 ) / 2364594514842 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27689168462 \nu^{11} + 31316075951 \nu^{10} - 479491314879 \nu^{9} + 276064172287 \nu^{8} - 6134603554204 \nu^{7} + \cdots - 5553545565645 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 180695 \nu^{11} + 14981 \nu^{10} + 2869513 \nu^{9} + 1970534 \nu^{8} + 37887233 \nu^{7} + 23983213 \nu^{6} + 143464943 \nu^{5} + 206144666 \nu^{4} + \cdots + 316111698 ) / 43464414 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19955657983 \nu^{11} - 28552570644 \nu^{10} + 349387669412 \nu^{9} - 306297243081 \nu^{8} + 4462134283841 \nu^{7} + \cdots - 3994606473165 ) / 2364594514842 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 65452485346 \nu^{11} + 87736926802 \nu^{10} - 1160252919867 \nu^{9} + 889059660188 \nu^{8} - 14929298435582 \nu^{7} + \cdots - 21103413269022 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22423423930 \nu^{11} - 52060304809 \nu^{10} + 426694594745 \nu^{9} - 699708273647 \nu^{8} + 5409953327366 \nu^{7} + \cdots - 24649338083895 ) / 2364594514842 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{6} - 6\beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} - 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{11} + 4\beta_{10} - 12\beta_{9} + 14\beta_{8} + 2\beta_{5} + 57\beta_{3} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} - 16 \beta_{7} - 14 \beta_{6} + 4 \beta_{5} - 32 \beta_{4} + 6 \beta_{3} + 113 \beta_{2} - 129 \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{10} - 177\beta_{8} + 4\beta_{7} - 143\beta_{6} + 34\beta_{5} - 4\beta_{4} + 38\beta_{2} + 658 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{11} - 84 \beta_{10} + 189 \beta_{9} - 58 \beta_{8} + 422 \beta_{7} - 42 \beta_{5} + 211 \beta_{4} - 138 \beta_{3} - 211 \beta_{2} + 1543 \beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2176 \beta_{11} - 464 \beta_{10} + 1732 \beta_{9} + 100 \beta_{7} + 1732 \beta_{6} - 928 \beta_{5} + 200 \beta_{4} - 7233 \beta_{3} - 716 \beta_{2} + 816 \beta _1 - 8161 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 664 \beta_{10} + 1116 \beta_{8} - 2640 \beta_{7} + 2548 \beta_{6} - 664 \beta_{5} + 2640 \beta_{4} - 13337 \beta_{2} - 3268 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26537 \beta_{11} + 11888 \beta_{10} - 21165 \beta_{9} + 26537 \beta_{8} - 3560 \beta_{7} + 5944 \beta_{5} - 1780 \beta_{4} + 85950 \beta_{3} + 1780 \beta_{2} - 13040 \beta _1 + 5944 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18380 \beta_{11} + 9504 \beta_{10} - 34205 \beta_{9} - 32481 \beta_{7} - 34205 \beta_{6} + 19008 \beta_{5} - 64962 \beta_{4} + 44388 \beta_{3} + 193242 \beta_{2} - 225723 \beta _1 + 63396 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.68940 2.92612i
−0.925400 1.60284i
0.242987 + 0.420865i
0.410701 + 0.711354i
1.18241 + 2.04800i
1.77870 + 3.08079i
−1.68940 + 2.92612i
−0.925400 + 1.60284i
0.242987 0.420865i
0.410701 0.711354i
1.18241 2.04800i
1.77870 3.08079i
−1.68940 + 2.92612i 1.83681 1.06048i −3.70812 6.42265i 0 7.16630i −4.91879 4.98051i 11.5428 −2.25076 + 3.89842i 0
26.2 −0.925400 + 1.60284i −0.731043 + 0.422068i 0.287270 + 0.497567i 0 1.56233i 6.88972 + 1.23763i −8.46656 −4.14372 + 7.17713i 0
26.3 0.242987 0.420865i 4.74894 2.74180i 1.88192 + 3.25957i 0 2.66488i −5.87843 3.80053i 3.77301 10.5350 18.2471i 0
26.4 0.410701 0.711354i −0.507487 + 0.292998i 1.66265 + 2.87979i 0 0.481337i −1.91172 + 6.73389i 6.01701 −4.32830 + 7.49684i 0
26.5 1.18241 2.04800i −4.45439 + 2.57174i −0.796202 1.37906i 0 12.1634i 1.42520 6.85338i 5.69355 8.72772 15.1168i 0
26.6 1.77870 3.08079i 2.10717 1.21658i −4.32752 7.49548i 0 8.65567i 5.39402 4.46146i −16.5598 −1.53989 + 2.66717i 0
101.1 −1.68940 2.92612i 1.83681 + 1.06048i −3.70812 + 6.42265i 0 7.16630i −4.91879 + 4.98051i 11.5428 −2.25076 3.89842i 0
101.2 −0.925400 1.60284i −0.731043 0.422068i 0.287270 0.497567i 0 1.56233i 6.88972 1.23763i −8.46656 −4.14372 7.17713i 0
101.3 0.242987 + 0.420865i 4.74894 + 2.74180i 1.88192 3.25957i 0 2.66488i −5.87843 + 3.80053i 3.77301 10.5350 + 18.2471i 0
101.4 0.410701 + 0.711354i −0.507487 0.292998i 1.66265 2.87979i 0 0.481337i −1.91172 6.73389i 6.01701 −4.32830 7.49684i 0
101.5 1.18241 + 2.04800i −4.45439 2.57174i −0.796202 + 1.37906i 0 12.1634i 1.42520 + 6.85338i 5.69355 8.72772 + 15.1168i 0
101.6 1.77870 + 3.08079i 2.10717 + 1.21658i −4.32752 + 7.49548i 0 8.65567i 5.39402 + 4.46146i −16.5598 −1.53989 2.66717i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.3.i.d 12
5.b even 2 1 35.3.h.a 12
5.c odd 4 2 175.3.j.b 24
7.d odd 6 1 inner 175.3.i.d 12
15.d odd 2 1 315.3.w.c 12
20.d odd 2 1 560.3.bx.c 12
35.c odd 2 1 245.3.h.c 12
35.i odd 6 1 35.3.h.a 12
35.i odd 6 1 245.3.d.a 12
35.j even 6 1 245.3.d.a 12
35.j even 6 1 245.3.h.c 12
35.k even 12 2 175.3.j.b 24
105.p even 6 1 315.3.w.c 12
140.s even 6 1 560.3.bx.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.h.a 12 5.b even 2 1
35.3.h.a 12 35.i odd 6 1
175.3.i.d 12 1.a even 1 1 trivial
175.3.i.d 12 7.d odd 6 1 inner
175.3.j.b 24 5.c odd 4 2
175.3.j.b 24 35.k even 12 2
245.3.d.a 12 35.i odd 6 1
245.3.d.a 12 35.j even 6 1
245.3.h.c 12 35.c odd 2 1
245.3.h.c 12 35.j even 6 1
315.3.w.c 12 15.d odd 2 1
315.3.w.c 12 105.p even 6 1
560.3.bx.c 12 20.d odd 2 1
560.3.bx.c 12 140.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 2 T_{2}^{11} + 19 T_{2}^{10} - 26 T_{2}^{9} + 244 T_{2}^{8} - 338 T_{2}^{7} + 1249 T_{2}^{6} - 986 T_{2}^{5} + 3532 T_{2}^{4} - 3618 T_{2}^{3} + 3579 T_{2}^{2} - 1386 T_{2} + 441 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + 19 T^{10} - 26 T^{9} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} - 16 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} + 14 T^{11} + \cdots + 4134617957376 \) Copy content Toggle raw display
$13$ \( T^{12} + 546 T^{10} + \cdots + 77792400 \) Copy content Toggle raw display
$17$ \( T^{12} + 48 T^{11} + \cdots + 8707129344 \) Copy content Toggle raw display
$19$ \( T^{12} + 30 T^{11} + \cdots + 1590595171344 \) Copy content Toggle raw display
$23$ \( T^{12} - 14 T^{11} + \cdots + 44219321357121 \) Copy content Toggle raw display
$29$ \( (T^{6} - 32 T^{5} - 1494 T^{4} + \cdots + 486804)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 132 T^{11} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 110961448062976 \) Copy content Toggle raw display
$41$ \( T^{12} + 13194 T^{10} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{6} - 2 T^{5} - 3556 T^{4} + \cdots - 9258464)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 204 T^{11} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + 196 T^{11} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} - 72 T^{11} + \cdots + 92\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} - 72 T^{11} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{12} - 138 T^{11} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{6} + 4 T^{5} - 15420 T^{4} + \cdots + 46762703904)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 528 T^{11} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + 12 T^{11} + \cdots + 560965048576 \) Copy content Toggle raw display
$83$ \( T^{12} + 39468 T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} - 204 T^{11} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{12} + 48624 T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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