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Properties

Label 475.3.c.h
Level $475$
Weight $3$
Character orbit 475.c
Analytic conductor $12.943$
Analytic rank $0$
Dimension $14$
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] = [475,3,Mod(151,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (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: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{5} q^{6} + ( - \beta_{9} - 1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{6} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{5} q^{6} + ( - \beta_{9} - 1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{6} - 3) q^{9} - \beta_{11} q^{11} + (\beta_{7} + 2 \beta_{4}) q^{12} + \beta_{12} q^{13} + ( - \beta_{13} - \beta_{10} + \cdots - 2 \beta_1) q^{14}+ \cdots + ( - 4 \beta_{11} + \beta_{9} + \cdots - 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 28 q^{4} - 4 q^{6} - 20 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 28 q^{4} - 4 q^{6} - 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} + 22 q^{17} + 39 q^{19} + 12 q^{23} - 44 q^{24} + 30 q^{26} + 98 q^{28} + 4 q^{36} + 37 q^{38} - 32 q^{39} + 250 q^{42} + 90 q^{43} - 52 q^{44} + 148 q^{47} + 234 q^{49} + 98 q^{54} - 195 q^{57} - 274 q^{58} + 222 q^{61} + 518 q^{62} + 198 q^{63} - 218 q^{64} + 92 q^{66} + 80 q^{68} - 228 q^{73} - 92 q^{74} - 351 q^{76} - 260 q^{77} + 402 q^{81} + 58 q^{82} - 280 q^{83} - 282 q^{87} + 302 q^{92} - 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -33\nu^{13} - 1316\nu^{11} - 19576\nu^{9} - 134589\nu^{7} - 419015\nu^{5} - 459879\nu^{3} - 47160\nu ) / 1400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{12} + 79\nu^{10} + 1164\nu^{8} + 7946\nu^{6} + 24705\nu^{4} + 27471\nu^{2} + 3465 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{12} - 353\nu^{10} - 5148\nu^{8} - 34637\nu^{6} - 105680\nu^{4} - 115562\nu^{2} - 15075 ) / 160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 128\nu^{13} + 5131\nu^{11} + 76716\nu^{9} + 529424\nu^{7} + 1649415\nu^{5} + 1797789\nu^{3} + 161685\nu ) / 1400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\nu^{12} + 842\nu^{10} + 12612\nu^{8} + 87493\nu^{6} + 276205\nu^{4} + 313973\nu^{2} + 45470 ) / 200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -89\nu^{12} - 3553\nu^{10} - 52908\nu^{8} - 364237\nu^{6} - 1138320\nu^{4} - 1273082\nu^{2} - 167155 ) / 800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36 \nu^{13} - 791 \nu^{12} + 1372 \nu^{11} - 31507 \nu^{10} + 19192 \nu^{9} - 467852 \nu^{8} + \cdots - 1319745 ) / 5600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -17\nu^{12} - 684\nu^{10} - 10274\nu^{8} - 71311\nu^{6} - 224085\nu^{4} - 249771\nu^{2} - 30990 ) / 100 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 947 \nu^{13} - 37919 \nu^{11} - 566884 \nu^{9} - 3922751 \nu^{7} - 12349860 \nu^{5} + \cdots - 2133165 \nu ) / 5600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 113 \nu^{13} + 113 \nu^{12} - 4501 \nu^{11} + 4501 \nu^{10} - 66836 \nu^{9} + 66836 \nu^{8} + \cdots + 188535 ) / 800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - 14\beta_{2} + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} + 3 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{11} - 28\beta_{9} - 21\beta_{8} + 24\beta_{6} + 19\beta_{5} + 188\beta_{2} - 659 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 28 \beta_{13} + 24 \beta_{12} + 24 \beta_{11} - 76 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -77\beta_{11} + 541\beta_{9} + 358\beta_{8} - 425\beta_{6} - 288\beta_{5} - 2584\beta_{2} + 8194 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 551 \beta_{13} - 435 \beta_{12} - 425 \beta_{11} + 1401 \beta_{10} - 425 \beta_{9} - 425 \beta_{8} + \cdots - 425 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1428\beta_{11} - 9129\beta_{9} - 5677\beta_{8} + 6809\beta_{6} + 4125\beta_{5} + 36335\beta_{2} - 108047 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9425 \beta_{13} + 7105 \beta_{12} + 6809 \beta_{11} - 23043 \beta_{10} + 6809 \beta_{9} + 6809 \beta_{8} + \cdots + 6809 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 23511 \beta_{11} + 144625 \beta_{9} + 86966 \beta_{8} - 104605 \beta_{6} - 58436 \beta_{5} + \cdots + 1481391 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 150497 \beta_{13} - 110477 \beta_{12} - 104605 \beta_{11} + 359707 \beta_{10} - 104605 \beta_{9} + \cdots - 104605 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
− 3.82982i
− 3.01757i
− 2.92865i
− 2.34822i
− 1.41833i
− 1.40632i
− 0.382406i
0.382406i
1.40632i
1.41833i
2.34822i
2.92865i
3.01757i
3.82982i
− 3.82982i − 0.102062i −10.6675 0 −0.390878 −9.96826 25.5353i 8.98958 0
151.2 − 3.01757i 4.83172i −5.10575 0 14.5801 6.15539 3.33669i −14.3455 0
151.3 − 2.92865i − 2.02259i −4.57698 0 −5.92346 8.32815 1.68976i 4.90912 0
151.4 − 2.34822i − 5.73703i −1.51415 0 −13.4718 −9.56678 − 5.83734i −23.9135 0
151.5 − 1.41833i − 1.48141i 1.98835 0 −2.10112 2.50491 − 8.49344i 6.80543 0
151.6 − 1.40632i 2.91656i 2.02225 0 4.10162 −11.7208 − 8.46924i 0.493705 0
151.7 − 0.382406i 3.15259i 3.85377 0 1.20557 4.26742 − 3.00333i −0.938823 0
151.8 0.382406i − 3.15259i 3.85377 0 1.20557 4.26742 3.00333i −0.938823 0
151.9 1.40632i − 2.91656i 2.02225 0 4.10162 −11.7208 8.46924i 0.493705 0
151.10 1.41833i 1.48141i 1.98835 0 −2.10112 2.50491 8.49344i 6.80543 0
151.11 2.34822i 5.73703i −1.51415 0 −13.4718 −9.56678 5.83734i −23.9135 0
151.12 2.92865i 2.02259i −4.57698 0 −5.92346 8.32815 − 1.68976i 4.90912 0
151.13 3.01757i − 4.83172i −5.10575 0 14.5801 6.15539 − 3.33669i −14.3455 0
151.14 3.82982i 0.102062i −10.6675 0 −0.390878 −9.96826 − 25.5353i 8.98958 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.h ✓ 14
5.b even 2 1 475.3.c.i yes 14
5.c odd 4 2 475.3.d.d 28
19.b odd 2 1 inner 475.3.c.h ✓ 14
95.d odd 2 1 475.3.c.i yes 14
95.g even 4 2 475.3.d.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.3.c.h ✓ 14 1.a even 1 1 trivial
475.3.c.h ✓ 14 19.b odd 2 1 inner
475.3.c.i yes 14 5.b even 2 1
475.3.c.i yes 14 95.d odd 2 1
475.3.d.d 28 5.c odd 4 2
475.3.d.d 28 95.g even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{14} + 42T_{2}^{12} + 677T_{2}^{10} + 5313T_{2}^{8} + 21125T_{2}^{6} + 40138T_{2}^{4} + 30565T_{2}^{2} + 3675 \) Copy content Toggle raw display
\( T_{7}^{7} + 10T_{7}^{6} - 180T_{7}^{5} - 1276T_{7}^{4} + 13005T_{7}^{3} + 33200T_{7}^{2} - 383375T_{7} + 612500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 42 T^{12} + \cdots + 3675 \) Copy content Toggle raw display
$3$ \( T^{14} + 81 T^{12} + \cdots + 6075 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T^{7} + 10 T^{6} + \cdots + 612500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + 2 T^{6} + \cdots + 1993920)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 51013467967500 \) Copy content Toggle raw display
$17$ \( (T^{7} - 11 T^{6} + \cdots - 16723125)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 79\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( (T^{7} - 6 T^{6} + \cdots + 45858750)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} - 45 T^{6} + \cdots - 188721350000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} - 74 T^{6} + \cdots + 340225818750)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} - 111 T^{6} + \cdots - 121256990560)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 71\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + 114 T^{6} + \cdots - 694377415625)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 5812813755000)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
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