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Properties

Label 1900.3.e.h
Level $1900$
Weight $3$
Character orbit 1900.e
Analytic conductor $51.771$
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] = [1900,3,Mod(1101,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1900.1101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.e (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: \(51.7712502285\)
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: \( x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 5 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{2} - 4) q^{9} - \beta_{3} q^{11} - \beta_{12} q^{13} + \beta_{4} q^{17} + (\beta_{6} - 2) q^{19} + (\beta_{10} + \beta_{8} - \beta_1) q^{21} + (\beta_{9} - 2) q^{23} + ( - \beta_{11} + \beta_{10} - 3 \beta_1) q^{27} + ( - \beta_{12} + \beta_{11}) q^{29} + (\beta_{11} - \beta_{8} + \cdots - \beta_{5}) q^{31}+ \cdots + ( - 3 \beta_{9} + \beta_{7} + \beta_{6} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{7} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{7} - 52 q^{9} + 4 q^{11} - 6 q^{17} - 29 q^{19} - 28 q^{23} - 56 q^{39} + 22 q^{43} + 84 q^{47} + 138 q^{49} - 5 q^{57} - 50 q^{61} + 234 q^{63} - 204 q^{73} + 68 q^{77} + 66 q^{81} + 256 q^{83} + 18 q^{87} + 118 q^{93} + 92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 107 \nu^{12} + 4422 \nu^{10} + 631973 \nu^{8} + 17246726 \nu^{6} + 168176486 \nu^{4} + \cdots + 88760925 ) / 22723050 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 648 \nu^{12} - 34907 \nu^{10} - 499913 \nu^{8} + 407849 \nu^{6} + 49911024 \nu^{4} + \cdots + 843924750 ) / 37871750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1462 \nu^{13} + 23670 \nu^{12} + 110483 \nu^{11} + 2161755 \nu^{10} + 2732597 \nu^{9} + \cdots - 2922880500 ) / 340845750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1462 \nu^{13} + 23670 \nu^{12} - 110483 \nu^{11} + 2161755 \nu^{10} - 2732597 \nu^{9} + \cdots - 2922880500 ) / 340845750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13561 \nu^{12} - 1135449 \nu^{10} - 35328191 \nu^{8} - 496542482 \nu^{6} - 3026541407 \nu^{4} + \cdots - 743885250 ) / 113615250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14264 \nu^{13} - 1102351 \nu^{11} - 29091859 \nu^{9} - 259998543 \nu^{7} + \cdots + 40788043750 \nu ) / 340845750 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1401 \nu^{12} + 117049 \nu^{10} + 3662221 \nu^{8} + 52791402 \nu^{6} + 343822347 \nu^{4} + \cdots + 146643050 ) / 7574350 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26419 \nu^{13} - 2303996 \nu^{11} - 76892714 \nu^{9} - 1229628903 \nu^{7} + \cdots - 42678853750 \nu ) / 340845750 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26419 \nu^{13} - 2303996 \nu^{11} - 76892714 \nu^{9} - 1229628903 \nu^{7} + \cdots - 49836614500 \nu ) / 340845750 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5608 \nu^{13} + 473087 \nu^{11} + 14768858 \nu^{9} + 206445486 \nu^{7} + 1219089951 \nu^{5} + \cdots - 1252190975 \nu ) / 68169150 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9986 \nu^{13} + 902959 \nu^{11} + 30950281 \nu^{9} + 498348567 \nu^{7} + 3816325902 \nu^{5} + \cdots + 15865169450 \nu ) / 68169150 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 3\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - 26\beta_{2} + 276 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 6\beta_{12} + 33\beta_{11} - 35\beta_{10} - 5\beta_{8} + 2\beta_{6} - 2\beta_{5} + 496\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58\beta_{9} + 144\beta_{7} + 39\beta_{6} + 39\beta_{5} - 56\beta_{4} - 18\beta_{3} + 645\beta_{2} - 6578 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{13} + 328 \beta_{12} - 927 \beta_{11} + 1039 \beta_{10} + 260 \beta_{8} + \cdots - 12208 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2269 \beta_{9} - 5267 \beta_{7} - 1314 \beta_{6} - 1314 \beta_{5} + 1228 \beta_{4} + 1160 \beta_{3} + \cdots + 163233 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2029 \beta_{13} - 12864 \beta_{12} + 25040 \beta_{11} - 29678 \beta_{10} - 9805 \beta_{8} + \cdots + 306617 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 77045 \beta_{9} + 173705 \beta_{7} + 42099 \beta_{6} + 42099 \beta_{5} - 23710 \beta_{4} + \cdots - 4131880 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 65339 \beta_{13} + 441160 \beta_{12} - 668272 \beta_{11} + 837126 \beta_{10} + 327795 \beta_{8} + \cdots - 7809830 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2440375 \beta_{9} - 5434385 \beta_{7} - 1306588 \beta_{6} - 1306588 \beta_{5} + 390220 \beta_{4} + \cdots + 106018551 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1939013 \beta_{13} - 14096270 \beta_{12} + 17768071 \beta_{11} - 23486399 \beta_{10} + \cdots + 201243047 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
− 5.26531i
− 4.84257i
− 4.58743i
− 2.93221i
− 2.16913i
− 1.84332i
− 0.284180i
0.284180i
1.84332i
2.16913i
2.93221i
4.58743i
4.84257i
5.26531i
0 − 5.26531i 0 0 0 −12.1735 0 −18.7235 0
1101.2 0 − 4.84257i 0 0 0 6.85668 0 −14.4504 0
1101.3 0 − 4.58743i 0 0 0 1.52935 0 −12.0445 0
1101.4 0 − 2.93221i 0 0 0 5.62705 0 0.402160 0
1101.5 0 − 2.16913i 0 0 0 −8.18099 0 4.29489 0
1101.6 0 − 1.84332i 0 0 0 10.5375 0 5.60216 0
1101.7 0 − 0.284180i 0 0 0 −2.19606 0 8.91924 0
1101.8 0 0.284180i 0 0 0 −2.19606 0 8.91924 0
1101.9 0 1.84332i 0 0 0 10.5375 0 5.60216 0
1101.10 0 2.16913i 0 0 0 −8.18099 0 4.29489 0
1101.11 0 2.93221i 0 0 0 5.62705 0 0.402160 0
1101.12 0 4.58743i 0 0 0 1.52935 0 −12.0445 0
1101.13 0 4.84257i 0 0 0 6.85668 0 −14.4504 0
1101.14 0 5.26531i 0 0 0 −12.1735 0 −18.7235 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1101.14
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 1900.3.e.h yes 14
5.b even 2 1 1900.3.e.g ✓ 14
5.c odd 4 2 1900.3.g.d 28
19.b odd 2 1 inner 1900.3.e.h yes 14
95.d odd 2 1 1900.3.e.g ✓ 14
95.g even 4 2 1900.3.g.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.3.e.g ✓ 14 5.b even 2 1
1900.3.e.g ✓ 14 95.d odd 2 1
1900.3.e.h yes 14 1.a even 1 1 trivial
1900.3.e.h yes 14 19.b odd 2 1 inner
1900.3.g.d 28 5.c odd 4 2
1900.3.g.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 \(S_{3}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{14} + 89T_{3}^{12} + 3026T_{3}^{10} + 49092T_{3}^{8} + 390297T_{3}^{6} + 1440495T_{3}^{4} + 1994425T_{3}^{2} + 151875 \) Copy content Toggle raw display
\( T_{7}^{7} - 2T_{7}^{6} - 204T_{7}^{5} + 640T_{7}^{4} + 9845T_{7}^{3} - 37276T_{7}^{2} - 56107T_{7} + 135988 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 89 T^{12} + \cdots + 151875 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T^{7} - 2 T^{6} + \cdots + 135988)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} - 2 T^{6} + \cdots - 1375296)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 835543129687500 \) Copy content Toggle raw display
$17$ \( (T^{7} + 3 T^{6} + \cdots - 48317883)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 79\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( (T^{7} + 14 T^{6} + \cdots - 60378750)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} - 11 T^{6} + \cdots + 5955376816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} - 42 T^{6} + \cdots + 588156954)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 1184496469600)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 14\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + 102 T^{6} + \cdots + 322844910273)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 10116036761400)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
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