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Properties

Label 65.4.b.b
Level $65$
Weight $4$
Character orbit 65.b
Analytic conductor $3.835$
Analytic rank $0$
Dimension $16$
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] = [65,4,Mod(14,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.14");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.b (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: \(3.83512415037\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 83x^{14} + 2611x^{12} + 39621x^{10} + 299324x^{8} + 997532x^{6} + 882116x^{4} + 265856x^{2} + 25600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{4} q^{3} + (\beta_1 - 4) q^{4} + ( - \beta_{11} - 1) q^{5} + (\beta_{8} + \beta_{6} - \beta_{3}) q^{6} + ( - \beta_{14} + \beta_{13} + \cdots + 2 \beta_{2}) q^{7}+ \cdots + (\beta_{12} - \beta_{11} + \beta_{10} + \cdots - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{4} q^{3} + (\beta_1 - 4) q^{4} + ( - \beta_{11} - 1) q^{5} + (\beta_{8} + \beta_{6} - \beta_{3}) q^{6} + ( - \beta_{14} + \beta_{13} + \cdots + 2 \beta_{2}) q^{7}+ \cdots + (33 \beta_{12} - 33 \beta_{11} + \cdots - 171) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 58 q^{4} - 20 q^{5} + 4 q^{6} - 148 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 58 q^{4} - 20 q^{5} + 4 q^{6} - 148 q^{9} - 72 q^{10} + 28 q^{11} - 124 q^{14} + 28 q^{15} + 394 q^{16} - 316 q^{19} + 240 q^{20} + 312 q^{21} - 656 q^{24} + 414 q^{25} + 234 q^{26} + 248 q^{29} - 390 q^{30} + 268 q^{31} - 116 q^{34} + 830 q^{35} - 10 q^{36} - 416 q^{39} + 104 q^{40} + 152 q^{41} + 520 q^{44} + 1096 q^{45} - 1240 q^{46} + 1204 q^{49} - 952 q^{50} - 1492 q^{51} - 164 q^{54} + 460 q^{55} - 2660 q^{56} - 684 q^{59} - 820 q^{60} + 272 q^{61} + 2126 q^{64} - 494 q^{65} - 2336 q^{66} + 1088 q^{69} - 1392 q^{70} - 2780 q^{71} + 2272 q^{74} + 142 q^{75} + 6268 q^{76} - 384 q^{79} - 5484 q^{80} - 928 q^{81} + 1916 q^{84} + 1556 q^{85} + 3304 q^{86} + 1400 q^{89} - 2526 q^{90} + 312 q^{91} + 5764 q^{94} + 496 q^{95} + 8132 q^{96} - 3468 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 83x^{14} + 2611x^{12} + 39621x^{10} + 299324x^{8} + 997532x^{6} + 882116x^{4} + 265856x^{2} + 25600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15079 \nu^{14} + 1336609 \nu^{12} + 45369757 \nu^{10} + 743763647 \nu^{8} + 5991596708 \nu^{6} + \cdots + 3985190256 ) / 157759376 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7030741 \nu^{15} - 582345183 \nu^{13} - 18250336031 \nu^{11} - 274935408601 \nu^{9} + \cdots - 719567179456 \nu ) / 25241500160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7514047 \nu^{14} - 628680005 \nu^{12} - 19953613869 \nu^{10} - 304947895795 \nu^{8} + \cdots - 854790335040 ) / 6310375040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18058185 \nu^{15} - 1503044219 \nu^{13} - 47417127899 \nu^{11} - 720506347613 \nu^{9} + \cdots - 1683299862336 \nu ) / 50483000320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15745821 \nu^{15} - 127066832 \nu^{14} + 1242740759 \nu^{13} - 10313375024 \nu^{12} + \cdots - 11814853222400 ) / 100966000640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19727999 \nu^{14} + 1645459973 \nu^{12} + 52031864653 \nu^{10} + 792141197811 \nu^{8} + \cdots + 1896764943040 ) / 12620750080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13935789 \nu^{15} + 1146735383 \nu^{13} + 35585048935 \nu^{11} + 527917013121 \nu^{9} + \cdots - 373858991552 \nu ) / 25241500160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13581099 \nu^{14} - 1124646093 \nu^{12} - 35229070713 \nu^{10} - 530201347251 \nu^{8} + \cdots - 1319170658880 ) / 6310375040 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7941677 \nu^{14} + 644585939 \nu^{12} + 19670809359 \nu^{10} + 286991784573 \nu^{8} + \cdots + 738428326400 ) / 3155187520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42449209 \nu^{14} + 3493207363 \nu^{12} + 108592143003 \nu^{10} + 1621664354181 \nu^{8} + \cdots + 5178992664640 ) / 12620750080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30614715 \nu^{15} + 22150807 \nu^{14} + 2529731633 \nu^{13} + 1829036149 \nu^{12} + \cdots + 1894630402240 ) / 25241500160 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30614715 \nu^{15} - 22150807 \nu^{14} + 2529731633 \nu^{13} - 1829036149 \nu^{12} + \cdots - 1894630402240 ) / 25241500160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 46908435 \nu^{15} - 3871421689 \nu^{13} - 120749934873 \nu^{11} - 1808091041103 \nu^{9} + \cdots - 4131435010112 \nu ) / 25241500160 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 114499471 \nu^{15} - 9477470733 \nu^{13} - 296689646317 \nu^{11} - 4460781342171 \nu^{9} + \cdots - 9770334211520 \nu ) / 50483000320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 186795969 \nu^{15} + 15479328611 \nu^{13} + 485425192643 \nu^{11} + 7318956154037 \nu^{9} + \cdots + 19609191173440 \nu ) / 50483000320 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{4} + 12\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} + 2\beta_{11} + 2\beta_{9} + 6\beta_{8} + 2\beta_{6} + 13\beta _1 - 143 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 35 \beta_{15} - 31 \beta_{14} + 17 \beta_{13} + 12 \beta_{12} + 12 \beta_{11} + 3 \beta_{9} + \cdots - 208 \beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 102 \beta_{12} - 102 \beta_{11} - 8 \beta_{10} - 47 \beta_{9} - 212 \beta_{8} - 96 \beta_{6} + \cdots + 3006 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1081 \beta_{15} + 1167 \beta_{14} - 619 \beta_{13} - 466 \beta_{12} - 466 \beta_{11} + \cdots + 4488 \beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3860 \beta_{12} + 3860 \beta_{11} + 384 \beta_{10} + 1259 \beta_{9} + 7250 \beta_{8} + 3338 \beta_{6} + \cdots - 75518 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32855 \beta_{15} - 37809 \beta_{14} + 18659 \beta_{13} + 14702 \beta_{12} + 14702 \beta_{11} + \cdots - 113118 \beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 131984 \beta_{12} - 131984 \beta_{11} - 13852 \beta_{10} - 36875 \beta_{9} - 240302 \beta_{8} + \cdots + 2095094 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1002019 \beta_{15} + 1181965 \beta_{14} - 543859 \beta_{13} - 444534 \beta_{12} + \cdots + 3130690 \beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4311684 \beta_{12} + 4311684 \beta_{11} + 458200 \beta_{10} + 1123131 \beta_{9} + 7779586 \beta_{8} + \cdots - 61238090 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 30735199 \beta_{15} - 36634701 \beta_{14} + 15940295 \beta_{13} + 13402774 \beta_{12} + \cdots - 91183734 \beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 137697248 \beta_{12} - 137697248 \beta_{11} - 14655348 \beta_{10} - 34709351 \beta_{9} + \cdots + 1838809218 ) / 13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 946961211 \beta_{15} + 1134074357 \beta_{14} - 473942131 \beta_{13} - 406721614 \beta_{12} + \cdots + 2729719394 \beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 4344986900 \beta_{12} + 4344986900 \beta_{11} + 461911808 \beta_{10} + 1077681507 \beta_{9} + \cdots - 56009345146 ) / 13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 29262602031 \beta_{15} - 35125253637 \beta_{14} + 14282721567 \beta_{13} + \cdots - 82964711846 \beta_{2} ) / 13 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
− 4.36654i
− 5.57225i
− 3.31046i
− 3.04206i
− 3.39640i
− 0.517915i
0.458230i
0.810091i
− 0.810091i
− 0.458230i
0.517915i
3.39640i
3.04206i
3.31046i
5.57225i
4.36654i
− 5.36654i 6.45581i −20.7998 −11.0785 − 1.50576i 34.6454 12.0424i 68.6905i −14.6775 −8.08071 + 59.4531i
14.2 − 4.57225i − 4.87122i −12.9055 3.65995 − 10.5643i −22.2724 16.3896i 22.4293i 3.27124 −48.3027 − 16.7342i
14.3 − 4.31046i − 6.56332i −10.5801 −5.85275 + 9.52604i −28.2909 − 13.5304i 11.1213i −16.0772 41.0616 + 25.2280i
14.4 − 4.04206i 5.52038i −8.33826 11.1220 − 1.14090i 22.3137 − 26.2226i 1.36727i −3.47464 −4.61158 − 44.9557i
14.5 − 2.39640i 0.780000i 2.25725 −9.70429 − 5.55219i 1.86919 − 21.1733i − 24.5805i 26.3916 −13.3053 + 23.2554i
14.6 − 1.51792i − 6.38427i 5.69593 11.0855 + 1.45281i −9.69078 4.09127i − 20.7893i −13.7589 2.20524 − 16.8269i
14.7 − 0.541770i 2.78461i 7.70649 0.371108 − 11.1742i 1.50862 17.0055i − 8.50930i 19.2460 −6.05383 − 0.201055i
14.8 − 0.189909i 10.0956i 7.96393 −9.60307 + 5.72548i 1.91724 − 10.1698i − 3.03170i −74.9206 1.08732 + 1.82371i
14.9 0.189909i − 10.0956i 7.96393 −9.60307 − 5.72548i 1.91724 10.1698i 3.03170i −74.9206 1.08732 − 1.82371i
14.10 0.541770i − 2.78461i 7.70649 0.371108 + 11.1742i 1.50862 − 17.0055i 8.50930i 19.2460 −6.05383 + 0.201055i
14.11 1.51792i 6.38427i 5.69593 11.0855 − 1.45281i −9.69078 − 4.09127i 20.7893i −13.7589 2.20524 + 16.8269i
14.12 2.39640i − 0.780000i 2.25725 −9.70429 + 5.55219i 1.86919 21.1733i 24.5805i 26.3916 −13.3053 − 23.2554i
14.13 4.04206i − 5.52038i −8.33826 11.1220 + 1.14090i 22.3137 26.2226i − 1.36727i −3.47464 −4.61158 + 44.9557i
14.14 4.31046i 6.56332i −10.5801 −5.85275 − 9.52604i −28.2909 13.5304i − 11.1213i −16.0772 41.0616 − 25.2280i
14.15 4.57225i 4.87122i −12.9055 3.65995 + 10.5643i −22.2724 − 16.3896i − 22.4293i 3.27124 −48.3027 + 16.7342i
14.16 5.36654i − 6.45581i −20.7998 −11.0785 + 1.50576i 34.6454 − 12.0424i − 68.6905i −14.6775 −8.08071 − 59.4531i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.16
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 65.4.b.b ✓ 16
3.b odd 2 1 585.4.c.d 16
5.b even 2 1 inner 65.4.b.b ✓ 16
5.c odd 4 1 325.4.a.n 8
5.c odd 4 1 325.4.a.o 8
15.d odd 2 1 585.4.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.b.b ✓ 16 1.a even 1 1 trivial
65.4.b.b ✓ 16 5.b even 2 1 inner
325.4.a.n 8 5.c odd 4 1
325.4.a.o 8 5.c odd 4 1
585.4.c.d 16 3.b odd 2 1
585.4.c.d 16 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 93 T_{2}^{14} + 3366 T_{2}^{12} + 59586 T_{2}^{10} + 527617 T_{2}^{8} + 2116681 T_{2}^{6} + \cdots + 25600 \) acting on \(S_{4}^{\mathrm{new}}(65, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 93 T^{14} + \cdots + 25600 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 25442802064 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots - 1440249377552)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 169)^{8} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 4352926624240)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 53\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 25\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 26\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 58\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
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