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Properties

Label 1375.2.b.f
Level $1375$
Weight $2$
Character orbit 1375.b
Analytic conductor $10.979$
Analytic rank $0$
Dimension $20$
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] = [1375,2,Mod(749,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1375.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1375.b (of order \(2\), degree \(1\), not 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: \(10.9794302779\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 31 x^{18} + 401 x^{16} + 2814 x^{14} + 11674 x^{12} + 29278 x^{10} + 43479 x^{8} + 35324 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{8} + \beta_{6} - 2) q^{4} + ( - \beta_{18} - \beta_{16} - \beta_{6}) q^{6} + (\beta_{7} + \beta_{2} - \beta_1) q^{7} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{9} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{8} + \beta_{6} - 2) q^{4} + ( - \beta_{18} - \beta_{16} - \beta_{6}) q^{6} + (\beta_{7} + \beta_{2} - \beta_1) q^{7} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 22 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 22 q^{4} + 2 q^{6} - 16 q^{9} + 20 q^{11} + 28 q^{14} + 26 q^{16} - 10 q^{19} - 6 q^{21} - 48 q^{24} + 4 q^{26} + 14 q^{29} - 6 q^{31} - 2 q^{34} + 12 q^{36} + 32 q^{39} - 8 q^{41} - 22 q^{44} + 72 q^{46} - 10 q^{49} + 50 q^{51} + 14 q^{54} - 88 q^{56} - 8 q^{59} - 30 q^{61} - 28 q^{64} + 2 q^{66} - 36 q^{69} + 44 q^{71} - 28 q^{74} - 10 q^{76} + 18 q^{79} - 60 q^{81} + 84 q^{84} - 54 q^{86} + 86 q^{89} - 52 q^{91} + 40 q^{94} + 182 q^{96} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 31 x^{18} + 401 x^{16} + 2814 x^{14} + 11674 x^{12} + 29278 x^{10} + 43479 x^{8} + 35324 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12940 \nu^{19} + 184061 \nu^{17} + 11786788 \nu^{15} + 164394685 \nu^{13} + \cdots + 174682395 \nu ) / 2868647 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 402 \nu^{19} - 10367 \nu^{17} - 100273 \nu^{15} - 407100 \nu^{13} - 163868 \nu^{11} + \cdots + 1237036 \nu ) / 47027 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 335803 \nu^{18} + 9834731 \nu^{16} + 118056564 \nu^{14} + 749858571 \nu^{12} + 2719465988 \nu^{10} + \cdots - 27750138 ) / 14343235 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 355462 \nu^{18} + 10591934 \nu^{16} + 129806151 \nu^{14} + 844763109 \nu^{12} + 3147439857 \nu^{10} + \cdots + 59600978 ) / 14343235 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19374 \nu^{18} + 569568 \nu^{16} + 6862872 \nu^{14} + 43698723 \nu^{12} + 158179019 \nu^{10} + \cdots + 794126 ) / 462685 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19374 \nu^{19} + 569568 \nu^{17} + 6862872 \nu^{15} + 43698723 \nu^{13} + 158179019 \nu^{11} + \cdots + 794126 \nu ) / 462685 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19374 \nu^{18} - 569568 \nu^{16} - 6862872 \nu^{14} - 43698723 \nu^{12} - 158179019 \nu^{10} + \cdots + 1056614 ) / 462685 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 768027 \nu^{18} + 22280139 \nu^{16} + 264082731 \nu^{14} + 1648039654 \nu^{12} + 5824776337 \nu^{10} + \cdots + 40233188 ) / 14343235 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 813722 \nu^{19} + 25528554 \nu^{17} + 335051136 \nu^{15} + 2392616184 \nu^{13} + \cdots + 454533103 \nu ) / 14343235 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 833381 \nu^{19} - 26285757 \nu^{17} - 346800723 \nu^{15} - 2487520722 \nu^{13} + \cdots - 556227454 \nu ) / 14343235 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27096 \nu^{18} - 802602 \nu^{16} - 9768903 \nu^{14} - 63103112 \nu^{12} - 233457136 \nu^{10} + \cdots - 1900319 ) / 462685 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 193801 \nu^{18} + 5623822 \nu^{16} + 66601446 \nu^{14} + 414434066 \nu^{12} + 1455549470 \nu^{10} + \cdots + 12825591 ) / 2868647 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1291706 \nu^{19} - 40023227 \nu^{17} - 517216903 \nu^{15} - 3623111097 \nu^{13} + \cdots - 626955229 \nu ) / 14343235 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1872641 \nu^{19} - 56922632 \nu^{17} - 718216348 \nu^{15} - 4882866972 \nu^{13} + \cdots - 478162609 \nu ) / 14343235 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 379792 \nu^{18} + 11015118 \nu^{16} + 130234480 \nu^{14} + 806836912 \nu^{12} + 2801690233 \nu^{10} + \cdots + 434008 ) / 2868647 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 834 \nu^{19} - 25428 \nu^{17} - 322147 \nu^{15} - 2202543 \nu^{13} - 8848274 \nu^{11} + \cdots - 330951 \nu ) / 5735 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 495979 \nu^{18} - 14394999 \nu^{16} - 170434369 \nu^{14} - 1058788087 \nu^{12} - 3696805377 \nu^{10} + \cdots - 5099248 ) / 2868647 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 4979013 \nu^{19} - 150985401 \nu^{17} - 1898961199 \nu^{15} - 12854968816 \nu^{13} + \cdots - 1956411732 \nu ) / 14343235 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{11} + \beta_{10} + \beta_{7} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - \beta_{16} - 7\beta_{8} - 8\beta_{6} + \beta_{5} - \beta_{4} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{15} + 10\beta_{14} - 9\beta_{11} - 8\beta_{10} - 8\beta_{7} + \beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{18} + 12\beta_{16} + \beta_{12} + 2\beta_{9} + 46\beta_{8} + 55\beta_{6} - 7\beta_{5} + 10\beta_{4} - 133 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{19} - \beta_{17} + 70 \beta_{15} - 79 \beta_{14} + 67 \beta_{11} + 53 \beta_{10} + \cdots - 182 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 108 \beta_{18} - 108 \beta_{16} - 15 \beta_{12} - 27 \beta_{9} - 302 \beta_{8} - 369 \beta_{6} + \cdots + 843 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12 \beta_{19} + 21 \beta_{17} - 521 \beta_{15} + 584 \beta_{14} - 475 \beta_{11} - 337 \beta_{10} + \cdots + 1188 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 875 \beta_{18} + 872 \beta_{16} + 6 \beta_{13} + 160 \beta_{12} + 257 \beta_{9} + 2000 \beta_{8} + \cdots - 5492 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 94 \beta_{19} - 277 \beta_{17} + 3798 \beta_{15} - 4211 \beta_{14} + 3316 \beta_{11} + \cdots - 7932 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6737 \beta_{18} - 6674 \beta_{16} - 120 \beta_{13} - 1491 \beta_{12} - 2138 \beta_{9} - 13369 \beta_{8} + \cdots + 36451 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 584 \beta_{19} + 2960 \beta_{17} - 27342 \beta_{15} + 30008 \beta_{14} - 23034 \beta_{11} + \cdots + 53749 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 50415 \beta_{18} + 49565 \beta_{16} + 1526 \beta_{13} + 12954 \beta_{12} + 16683 \beta_{9} + \cdots - 245294 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2879 \beta_{19} - 28123 \beta_{17} + 195172 \beta_{15} - 212382 \beta_{14} + 159823 \beta_{11} + \cdots - 368023 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 370799 \beta_{18} - 361400 \beta_{16} - 15845 \beta_{13} - 107910 \beta_{12} - 125947 \beta_{9} + \cdots + 1668373 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 8638 \beta_{19} + 248405 \beta_{17} - 1384768 \beta_{15} + 1496332 \beta_{14} - 1109498 \beta_{11} + \cdots + 2539010 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2697140 \beta_{18} + 2604314 \beta_{16} + 146941 \beta_{13} + 874172 \beta_{12} + 934176 \beta_{9} + \cdots - 11442193 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 32822 \beta_{19} - 2090335 \beta_{17} + 9782010 \beta_{15} - 10508574 \beta_{14} + 7711393 \beta_{11} + \cdots - 17614795 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
− 2.66248i
− 2.59876i
− 2.31697i
− 2.13316i
− 1.50007i
− 1.42354i
− 1.33740i
− 1.05550i
− 0.240697i
− 0.0403026i
0.0403026i
0.240697i
1.05550i
1.33740i
1.42354i
1.50007i
2.13316i
2.31697i
2.59876i
2.66248i
− 2.66248i 2.56820i −5.08878 0 6.83778 4.86099i 8.22381i −3.59567 0
749.2 − 2.59876i 1.31435i −4.75353 0 3.41566 − 1.34106i 7.15575i 1.27249 0
749.3 − 2.31697i − 1.59544i −3.36836 0 −3.69660 3.55159i 3.17044i 0.454556 0
749.4 − 2.13316i − 1.48754i −2.55039 0 −3.17318 − 1.52830i 1.17407i 0.787211 0
749.5 − 1.50007i − 3.20096i −0.250195 0 −4.80165 − 1.22398i − 2.62482i −7.24615 0
749.6 − 1.42354i − 1.10147i −0.0264672 0 −1.56798 3.64589i − 2.80940i 1.78677 0
749.7 − 1.33740i 0.843654i 0.211361 0 1.12830 0.195477i − 2.95747i 2.28825 0
749.8 − 1.05550i 2.27769i 0.885918 0 2.40411 − 3.74463i − 3.04609i −2.18789 0
749.9 − 0.240697i 2.16755i 1.94206 0 0.521723 0.0324147i − 0.948843i −1.69828 0
749.10 − 0.0403026i − 1.69153i 1.99838 0 −0.0681733 − 2.40221i − 0.161145i 0.138711 0
749.11 0.0403026i 1.69153i 1.99838 0 −0.0681733 2.40221i 0.161145i 0.138711 0
749.12 0.240697i − 2.16755i 1.94206 0 0.521723 − 0.0324147i 0.948843i −1.69828 0
749.13 1.05550i − 2.27769i 0.885918 0 2.40411 3.74463i 3.04609i −2.18789 0
749.14 1.33740i − 0.843654i 0.211361 0 1.12830 − 0.195477i 2.95747i 2.28825 0
749.15 1.42354i 1.10147i −0.0264672 0 −1.56798 − 3.64589i 2.80940i 1.78677 0
749.16 1.50007i 3.20096i −0.250195 0 −4.80165 1.22398i 2.62482i −7.24615 0
749.17 2.13316i 1.48754i −2.55039 0 −3.17318 1.52830i − 1.17407i 0.787211 0
749.18 2.31697i 1.59544i −3.36836 0 −3.69660 − 3.55159i − 3.17044i 0.454556 0
749.19 2.59876i − 1.31435i −4.75353 0 3.41566 1.34106i − 7.15575i 1.27249 0
749.20 2.66248i − 2.56820i −5.08878 0 6.83778 − 4.86099i − 8.22381i −3.59567 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.2.b.f 20
5.b even 2 1 inner 1375.2.b.f 20
5.c odd 4 1 1375.2.a.e ✓ 10
5.c odd 4 1 1375.2.a.f yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.2.a.e ✓ 10 5.c odd 4 1
1375.2.a.f yes 10 5.c odd 4 1
1375.2.b.f 20 1.a even 1 1 trivial
1375.2.b.f 20 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 31 T_{2}^{18} + 401 T_{2}^{16} + 2814 T_{2}^{14} + 11674 T_{2}^{12} + 29278 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(1375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 31 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} + 38 T^{18} + \cdots + 39601 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 75 T^{18} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T - 1)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 97 T^{18} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 3862498201 \) Copy content Toggle raw display
$19$ \( (T^{10} + 5 T^{9} + \cdots + 6875)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 4264922998561 \) Copy content Toggle raw display
$29$ \( (T^{10} - 7 T^{9} + \cdots - 94275)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 3 T^{9} + \cdots - 46765975)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 277022216241 \) Copy content Toggle raw display
$41$ \( (T^{10} + 4 T^{9} + \cdots + 43846839)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 14293775665521 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 675036731397961 \) Copy content Toggle raw display
$59$ \( (T^{10} + 4 T^{9} + \cdots - 22938975)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 15 T^{9} + \cdots - 1576271)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 4180875788961 \) Copy content Toggle raw display
$71$ \( (T^{10} - 22 T^{9} + \cdots - 6126849)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 43136640486801 \) Copy content Toggle raw display
$79$ \( (T^{10} - 9 T^{9} + \cdots - 8717525)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{10} - 43 T^{9} + \cdots - 212815125)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 303003614186001 \) Copy content Toggle raw display
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