Properties

Label 740.2.e.a
Level $740$
Weight $2$
Character orbit 740.e
Analytic conductor $5.909$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(369,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.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: \(5.90892974957\)
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: \( x^{20} + 4 x^{18} - 31 x^{16} - 216 x^{14} + 470 x^{12} + 6696 x^{10} + 11750 x^{8} - 135000 x^{6} + \cdots + 9765625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{13} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{14} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{14} - 1) q^{9} + \beta_{8} q^{11} - \beta_{16} q^{13} + \beta_{11} q^{15} + \beta_{15} q^{17} - \beta_{19} q^{19} + (\beta_{18} - \beta_{2}) q^{21} + (\beta_{17} + \beta_{11} + \cdots + \beta_1) q^{23}+ \cdots + (2 \beta_{14} + \beta_{12} + \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 28 q^{9} - 8 q^{25} - 8 q^{41} - 28 q^{49} - 20 q^{65} + 56 q^{71} - 28 q^{75} + 20 q^{81} - 20 q^{85} - 20 q^{95} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 4 x^{18} - 31 x^{16} - 216 x^{14} + 470 x^{12} + 6696 x^{10} + 11750 x^{8} - 135000 x^{6} + \cdots + 9765625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{18} - 7 \nu^{16} - 50 \nu^{14} - 50 \nu^{12} + 1596 \nu^{10} + 276 \nu^{8} - 22006 \nu^{6} + \cdots + 2376875 ) / 1920000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 301 \nu^{18} - 2029 \nu^{16} + 32906 \nu^{14} + 57466 \nu^{12} - 187020 \nu^{10} + \cdots - 1427734375 ) / 400000000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59 \nu^{18} - 667 \nu^{16} + 5830 \nu^{14} + 4630 \nu^{12} - 86484 \nu^{10} - 755484 \nu^{8} + \cdots - 497640625 ) / 48000000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 375 \nu^{19} + 956 \nu^{18} - 3375 \nu^{17} + 1924 \nu^{16} - 5250 \nu^{15} + \cdots + 1192187500 ) / 1200000000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{19} - 14 \nu^{17} + 63 \nu^{15} + 93 \nu^{13} - 2919 \nu^{11} - 2237 \nu^{9} + \cdots - 2265625 \nu ) / 5000000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 108 \nu^{18} + 643 \nu^{16} + 3898 \nu^{14} - 9372 \nu^{12} - 104210 \nu^{10} + \cdots - 208203125 ) / 50000000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 481 \nu^{19} - 2551 \nu^{17} - 12186 \nu^{15} + 70454 \nu^{13} + 756420 \nu^{11} + \cdots + 1926171875 \nu ) / 1000000000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{19} + 4 \nu^{17} - 31 \nu^{15} - 216 \nu^{13} + 470 \nu^{11} + 6696 \nu^{9} + \cdots + 1562500 \nu ) / 1953125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{19} - 7 \nu^{17} - 50 \nu^{15} - 50 \nu^{13} + 1596 \nu^{11} + 276 \nu^{9} + \cdots + 2376875 \nu ) / 1920000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{18} - 4 \nu^{16} + 31 \nu^{14} + 216 \nu^{12} - 470 \nu^{10} - 6696 \nu^{8} + \cdots - 1562500 ) / 390625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3803 \nu^{19} + 413 \nu^{17} + 8518 \nu^{15} + 40198 \nu^{13} - 2568660 \nu^{11} + \cdots + 1096484375 \nu ) / 6000000000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12 \nu^{18} - 27 \nu^{16} - 522 \nu^{14} - 292 \nu^{12} + 14690 \nu^{10} + 47702 \nu^{8} + \cdots + 28828125 ) / 4000000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 871 \nu^{19} + 9 \nu^{17} - 29026 \nu^{15} - 17286 \nu^{13} + 463720 \nu^{11} + \cdots + 883984375 \nu ) / 1000000000 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2989 \nu^{19} + 7331 \nu^{17} - 200534 \nu^{15} - 187874 \nu^{13} + 3565080 \nu^{11} + \cdots + 13286328125 \nu ) / 3000000000 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 7709 \nu^{19} - 23461 \nu^{17} + 325354 \nu^{15} + 648394 \nu^{13} - 3619980 \nu^{11} + \cdots - 21180859375 \nu ) / 6000000000 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 212 \nu^{18} + 43 \nu^{16} - 5542 \nu^{14} - 13212 \nu^{12} + 104270 \nu^{10} + 648202 \nu^{8} + \cdots + 247421875 ) / 20000000 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 5041 \nu^{19} + 3911 \nu^{17} + 136946 \nu^{15} + 301906 \nu^{13} - 2688220 \nu^{11} + \cdots - 5016796875 \nu ) / 2000000000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{15} - 2\beta_{13} - \beta_{10} - \beta_{9} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{14} + \beta_{12} + 2\beta_{8} + 2\beta_{5} - 2\beta_{4} - 8\beta_{3} + \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{19} + 4 \beta_{17} + \beta_{16} - \beta_{15} + 4 \beta_{13} - 2 \beta_{11} - 4 \beta_{10} + \cdots + 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{18} - 8 \beta_{14} - 12 \beta_{13} - 23 \beta_{12} - 12 \beta_{11} + 12 \beta_{10} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{19} + 28 \beta_{17} + 16 \beta_{16} + 48 \beta_{15} - 24 \beta_{13} + 15 \beta_{10} + \cdots + 52 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16 \beta_{18} + 32 \beta_{14} + 32 \beta_{13} - 36 \beta_{12} + 32 \beta_{11} - 32 \beta_{10} + \cdots - 143 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 52 \beta_{19} + 132 \beta_{17} + 128 \beta_{16} - 64 \beta_{15} - 12 \beta_{13} - 164 \beta_{11} + \cdots - 147 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 184 \beta_{18} + 488 \beta_{14} - 252 \beta_{13} - 332 \beta_{12} - 252 \beta_{11} + 252 \beta_{10} + \cdots - 644 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 316 \beta_{19} + 604 \beta_{17} + 5 \beta_{16} + 123 \beta_{15} - 2 \beta_{13} + 624 \beta_{11} + \cdots - 52 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 288 \beta_{18} - 1468 \beta_{14} - 992 \beta_{13} - 351 \beta_{12} - 992 \beta_{11} + 992 \beta_{10} + \cdots - 5577 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1096 \beta_{19} + 344 \beta_{17} + 1613 \beta_{16} + 5683 \beta_{15} - 536 \beta_{13} + \cdots - 5347 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 752 \beta_{18} - 112 \beta_{14} + 1208 \beta_{13} - 3175 \beta_{12} + 1208 \beta_{11} - 1208 \beta_{10} + \cdots + 20184 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2824 \beta_{19} - 6584 \beta_{17} - 20240 \beta_{16} - 21232 \beta_{15} - 11264 \beta_{13} + \cdots + 13320 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 9408 \beta_{18} - 10624 \beta_{14} - 24704 \beta_{13} + 11160 \beta_{12} - 24704 \beta_{11} + \cdots - 166687 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 82024 \beta_{19} - 129064 \beta_{17} + 9872 \beta_{16} - 48784 \beta_{15} - 122648 \beta_{13} + \cdots - 208871 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 211088 \beta_{18} - 87696 \beta_{14} - 32840 \beta_{13} + 192840 \beta_{12} - 32840 \beta_{11} + \cdots + 495528 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 485144 \beta_{19} - 570296 \beta_{17} - 746583 \beta_{16} - 48169 \beta_{15} - 39826 \beta_{13} + \cdots + 143704 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
−0.419357 + 2.19639i
0.419357 2.19639i
−2.21322 0.318824i
2.21322 + 0.318824i
−0.609336 + 2.15144i
0.609336 2.15144i
−2.11814 0.716581i
2.11814 + 0.716581i
−1.25219 + 1.85257i
1.25219 1.85257i
−1.25219 1.85257i
1.25219 + 1.85257i
−2.11814 + 0.716581i
2.11814 0.716581i
−0.609336 2.15144i
0.609336 + 2.15144i
−2.21322 + 0.318824i
2.21322 0.318824i
−0.419357 2.19639i
0.419357 + 2.19639i
0 3.06614i 0 −0.419357 + 2.19639i 0 3.66077i 0 −6.40124 0
369.2 0 3.06614i 0 0.419357 2.19639i 0 3.66077i 0 −6.40124 0
369.3 0 2.69030i 0 −2.21322 0.318824i 0 3.35227i 0 −4.23773 0
369.4 0 2.69030i 0 2.21322 + 0.318824i 0 3.35227i 0 −4.23773 0
369.5 0 2.00519i 0 −0.609336 + 2.15144i 0 2.83210i 0 −1.02079 0
369.6 0 2.00519i 0 0.609336 2.15144i 0 2.83210i 0 −1.02079 0
369.7 0 0.979375i 0 −2.11814 0.716581i 0 2.22692i 0 2.04083 0
369.8 0 0.979375i 0 2.11814 + 0.716581i 0 2.22692i 0 2.04083 0
369.9 0 0.617308i 0 −1.25219 + 1.85257i 0 2.09310i 0 2.61893 0
369.10 0 0.617308i 0 1.25219 1.85257i 0 2.09310i 0 2.61893 0
369.11 0 0.617308i 0 −1.25219 1.85257i 0 2.09310i 0 2.61893 0
369.12 0 0.617308i 0 1.25219 + 1.85257i 0 2.09310i 0 2.61893 0
369.13 0 0.979375i 0 −2.11814 + 0.716581i 0 2.22692i 0 2.04083 0
369.14 0 0.979375i 0 2.11814 0.716581i 0 2.22692i 0 2.04083 0
369.15 0 2.00519i 0 −0.609336 2.15144i 0 2.83210i 0 −1.02079 0
369.16 0 2.00519i 0 0.609336 + 2.15144i 0 2.83210i 0 −1.02079 0
369.17 0 2.69030i 0 −2.21322 + 0.318824i 0 3.35227i 0 −4.23773 0
369.18 0 2.69030i 0 2.21322 0.318824i 0 3.35227i 0 −4.23773 0
369.19 0 3.06614i 0 −0.419357 2.19639i 0 3.66077i 0 −6.40124 0
369.20 0 3.06614i 0 0.419357 + 2.19639i 0 3.66077i 0 −6.40124 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.e.a 20
3.b odd 2 1 6660.2.i.b 20
5.b even 2 1 inner 740.2.e.a 20
5.c odd 4 2 3700.2.h.h 20
15.d odd 2 1 6660.2.i.b 20
37.b even 2 1 inner 740.2.e.a 20
111.d odd 2 1 6660.2.i.b 20
185.d even 2 1 inner 740.2.e.a 20
185.h odd 4 2 3700.2.h.h 20
555.b odd 2 1 6660.2.i.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.e.a 20 1.a even 1 1 trivial
740.2.e.a 20 5.b even 2 1 inner
740.2.e.a 20 37.b even 2 1 inner
740.2.e.a 20 185.d even 2 1 inner
3700.2.h.h 20 5.c odd 4 2
3700.2.h.h 20 185.h odd 4 2
6660.2.i.b 20 3.b odd 2 1
6660.2.i.b 20 15.d odd 2 1
6660.2.i.b 20 111.d odd 2 1
6660.2.i.b 20 555.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + 22 T^{8} + \cdots + 100)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} + 42 T^{8} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 37 T^{3} + \cdots + 96)^{4} \) Copy content Toggle raw display
$13$ \( (T^{10} - 76 T^{8} + \cdots - 87616)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 82 T^{8} + \cdots - 129600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 88 T^{8} + \cdots + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 142 T^{8} + \cdots - 20736)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 146 T^{8} + \cdots + 10445824)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 104 T^{8} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} + \cdots + 2040)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} - 174 T^{8} + \cdots - 473344)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 250 T^{8} + \cdots + 109202500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 272 T^{8} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 454 T^{8} + \cdots + 678498304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 282 T^{8} + \cdots + 291999744)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 394 T^{8} + \cdots + 1458017856)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 14 T^{4} + \cdots - 11808)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + 408 T^{8} + \cdots + 278784)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 702 T^{8} + \cdots + 4859205264)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 334 T^{8} + \cdots + 51076)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 688 T^{8} + \cdots + 8097120256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 390 T^{8} + \cdots - 92416)^{2} \) Copy content Toggle raw display
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