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Properties

Label 465.2.i.f
Level $465$
Weight $2$
Character orbit 465.i
Analytic conductor $3.713$
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] = [465,2,Mod(211,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.211");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.i (of order \(3\), degree \(2\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 13 x^{12} - 8 x^{11} + 114 x^{10} - 65 x^{9} + 491 x^{8} - 152 x^{7} + 1434 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{6} + 1) q^{3} + ( - \beta_{3} + 2) q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} + (\beta_{12} - \beta_1) q^{7} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_{2}) q^{8}+ \cdots + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{6} + 1) q^{3} + ( - \beta_{3} + 2) q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} + (\beta_{12} - \beta_1) q^{7} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_{2}) q^{8}+ \cdots + (\beta_{9} - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 7 q^{3} + 22 q^{4} - 7 q^{5} + q^{6} - 2 q^{7} + 6 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} + 7 q^{3} + 22 q^{4} - 7 q^{5} + q^{6} - 2 q^{7} + 6 q^{8} - 7 q^{9} - q^{10} + 11 q^{12} + 2 q^{13} - 14 q^{14} - 14 q^{15} + 30 q^{16} + 7 q^{17} - q^{18} - 4 q^{19} - 11 q^{20} + 2 q^{21} - 2 q^{22} - 22 q^{23} + 3 q^{24} - 7 q^{25} + 9 q^{26} - 14 q^{27} + 11 q^{28} - 32 q^{29} - 2 q^{30} + 15 q^{31} + 24 q^{32} - 6 q^{34} + 4 q^{35} - 11 q^{36} - 5 q^{37} - 4 q^{38} + 4 q^{39} - 3 q^{40} - 5 q^{41} + 14 q^{42} + 17 q^{43} - 18 q^{44} - 7 q^{45} - 60 q^{46} - 6 q^{47} + 15 q^{48} - 21 q^{49} - q^{50} - 7 q^{51} + 3 q^{52} - 7 q^{53} - 2 q^{54} - 62 q^{56} + 4 q^{57} - 20 q^{58} + 6 q^{59} - 22 q^{60} + 36 q^{61} + 55 q^{62} + 4 q^{63} + 42 q^{64} + 2 q^{65} - 4 q^{66} + 7 q^{68} - 11 q^{69} + 28 q^{70} - 6 q^{71} - 3 q^{72} - 14 q^{73} - 18 q^{74} + 7 q^{75} - 8 q^{76} - 4 q^{77} + 18 q^{78} + 6 q^{79} - 15 q^{80} - 7 q^{81} + 12 q^{82} - q^{83} - 11 q^{84} - 14 q^{85} - 53 q^{86} - 16 q^{87} - 21 q^{88} + 40 q^{89} - q^{90} - 94 q^{91} + 46 q^{92} + 27 q^{93} + 104 q^{94} + 8 q^{95} + 12 q^{96} + 40 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 13 x^{12} - 8 x^{11} + 114 x^{10} - 65 x^{9} + 491 x^{8} - 152 x^{7} + 1434 x^{6} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8623861013 \nu^{13} - 156986689185 \nu^{12} + 420807649865 \nu^{11} + \cdots - 118715417454080 ) / 237084365707536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37090707043 \nu^{13} + 77174364174 \nu^{12} - 477331126105 \nu^{11} + \cdots + 236532438602704 ) / 59271091426884 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 795460726447 \nu^{13} + 1888845646844 \nu^{12} - 9774877008500 \nu^{11} + \cdots + 711117373819920 ) / 474168731415072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3327624204469 \nu^{13} - 6082734396635 \nu^{12} - 33661636308177 \nu^{11} + \cdots - 47\!\cdots\!68 ) / 18\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 463732099430 \nu^{13} - 455108238417 \nu^{12} + 5871530603405 \nu^{11} + \cdots - 170008018609664 ) / 237084365707536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6666535962649 \nu^{13} + 6629992079097 \nu^{12} - 70897350077269 \nu^{11} + \cdots + 79\!\cdots\!08 ) / 18\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1767615970043 \nu^{13} + 441948037822 \nu^{12} + 20303111636130 \nu^{11} + \cdots - 206955909546928 ) / 474168731415072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7274431172435 \nu^{13} - 5860380417935 \nu^{12} + 99483598703283 \nu^{11} + \cdots - 18\!\cdots\!24 ) / 18\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3950803585253 \nu^{13} - 7216357987317 \nu^{12} + 49998649635473 \nu^{11} + \cdots - 56\!\cdots\!80 ) / 948337462830144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 426641392387 \nu^{13} + 377933874243 \nu^{12} - 5394199477300 \nu^{11} + \cdots - 66524419993040 ) / 59271091426884 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2032308562822 \nu^{13} + 365730916455 \nu^{12} - 25135176668035 \nu^{11} + \cdots - 480197528096624 ) / 237084365707536 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4238600000522 \nu^{13} - 2504523180417 \nu^{12} + 54742837097405 \nu^{11} + \cdots + 837481459539472 ) / 474168731415072 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 4\beta_{6} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - 7\beta_{11} + \beta_{9} - \beta_{8} - 23\beta_{6} - \beta_{5} - \beta_{4} - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 10 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} - \beta_{7} + \cdots - 39 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{9} + 13\beta_{8} + 11\beta_{7} + 3\beta_{5} - 3\beta_{4} - 46\beta_{3} + 145 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{13} - 80 \beta_{12} + 57 \beta_{11} - 15 \beta_{9} - 56 \beta_{8} + 11 \beta_{6} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 91 \beta_{13} - 5 \beta_{12} + 306 \beta_{11} + 5 \beta_{10} - 42 \beta_{9} - 42 \beta_{8} + \cdots + 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 597 \beta_{10} - 380 \beta_{9} - 156 \beta_{8} + 86 \beta_{7} - 536 \beta_{5} + 536 \beta_{4} + \cdots - 102 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 683 \beta_{13} + 91 \beta_{12} - 2071 \beta_{11} + 995 \beta_{9} - 576 \beta_{8} - 6341 \beta_{6} + \cdots - 6341 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 592 \beta_{13} + 4325 \beta_{12} - 2837 \beta_{11} - 4325 \beta_{10} + 3954 \beta_{9} + \cdots - 11963 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1087 \beta_{10} - 4044 \beta_{9} + 7713 \beta_{8} + 4917 \beta_{7} + 3669 \beta_{5} - 3669 \beta_{4} + \cdots + 42935 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3830 \beta_{13} - 30882 \beta_{12} + 20240 \beta_{11} - 11596 \beta_{9} - 17165 \beta_{8} + \cdots + 8290 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
−1.29670 − 2.24596i
−0.929715 − 1.61031i
−0.529511 − 0.917140i
0.231391 + 0.400780i
0.563224 + 0.975533i
1.12108 + 1.94177i
1.34023 + 2.32135i
−1.29670 + 2.24596i
−0.929715 + 1.61031i
−0.529511 + 0.917140i
0.231391 − 0.400780i
0.563224 − 0.975533i
1.12108 − 1.94177i
1.34023 − 2.32135i
−2.59341 0.500000 + 0.866025i 4.72576 −0.500000 + 0.866025i −1.29670 − 2.24596i 2.21228 + 3.83178i −7.06901 −0.500000 + 0.866025i 1.29670 − 2.24596i
211.2 −1.85943 0.500000 + 0.866025i 1.45748 −0.500000 + 0.866025i −0.929715 − 1.61031i −0.788222 − 1.36524i 1.00878 −0.500000 + 0.866025i 0.929715 − 1.61031i
211.3 −1.05902 0.500000 + 0.866025i −0.878473 −0.500000 + 0.866025i −0.529511 − 0.917140i −0.194669 − 0.337177i 3.04837 −0.500000 + 0.866025i 0.529511 − 0.917140i
211.4 0.462781 0.500000 + 0.866025i −1.78583 −0.500000 + 0.866025i 0.231391 + 0.400780i −2.48291 − 4.30052i −1.75201 −0.500000 + 0.866025i −0.231391 + 0.400780i
211.5 1.12645 0.500000 + 0.866025i −0.731113 −0.500000 + 0.866025i 0.563224 + 0.975533i 1.38919 + 2.40614i −3.07646 −0.500000 + 0.866025i −0.563224 + 0.975533i
211.6 2.24217 0.500000 + 0.866025i 3.02731 −0.500000 + 0.866025i 1.12108 + 1.94177i 0.698473 + 1.20979i 2.30340 −0.500000 + 0.866025i −1.12108 + 1.94177i
211.7 2.68046 0.500000 + 0.866025i 5.18487 −0.500000 + 0.866025i 1.34023 + 2.32135i −1.83414 − 3.17683i 8.53693 −0.500000 + 0.866025i −1.34023 + 2.32135i
346.1 −2.59341 0.500000 − 0.866025i 4.72576 −0.500000 − 0.866025i −1.29670 + 2.24596i 2.21228 − 3.83178i −7.06901 −0.500000 − 0.866025i 1.29670 + 2.24596i
346.2 −1.85943 0.500000 − 0.866025i 1.45748 −0.500000 − 0.866025i −0.929715 + 1.61031i −0.788222 + 1.36524i 1.00878 −0.500000 − 0.866025i 0.929715 + 1.61031i
346.3 −1.05902 0.500000 − 0.866025i −0.878473 −0.500000 − 0.866025i −0.529511 + 0.917140i −0.194669 + 0.337177i 3.04837 −0.500000 − 0.866025i 0.529511 + 0.917140i
346.4 0.462781 0.500000 − 0.866025i −1.78583 −0.500000 − 0.866025i 0.231391 − 0.400780i −2.48291 + 4.30052i −1.75201 −0.500000 − 0.866025i −0.231391 − 0.400780i
346.5 1.12645 0.500000 − 0.866025i −0.731113 −0.500000 − 0.866025i 0.563224 − 0.975533i 1.38919 − 2.40614i −3.07646 −0.500000 − 0.866025i −0.563224 − 0.975533i
346.6 2.24217 0.500000 − 0.866025i 3.02731 −0.500000 − 0.866025i 1.12108 − 1.94177i 0.698473 − 1.20979i 2.30340 −0.500000 − 0.866025i −1.12108 − 1.94177i
346.7 2.68046 0.500000 − 0.866025i 5.18487 −0.500000 − 0.866025i 1.34023 − 2.32135i −1.83414 + 3.17683i 8.53693 −0.500000 − 0.866025i −1.34023 − 2.32135i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.i.f ✓ 14
31.c even 3 1 inner 465.2.i.f ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.i.f ✓ 14 1.a even 1 1 trivial
465.2.i.f ✓ 14 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\):

\( T_{2}^{7} - T_{2}^{6} - 12T_{2}^{5} + 10T_{2}^{4} + 40T_{2}^{3} - 25T_{2}^{2} - 32T_{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{14} + 2 T_{7}^{13} + 37 T_{7}^{12} + 28 T_{7}^{11} + 907 T_{7}^{10} + 655 T_{7}^{9} + \cdots + 36864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} - T^{6} - 12 T^{5} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + 2 T^{13} + \cdots + 36864 \) Copy content Toggle raw display
$11$ \( T^{14} + 42 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{14} - 2 T^{13} + \cdots + 6718464 \) Copy content Toggle raw display
$17$ \( T^{14} - 7 T^{13} + \cdots + 399424 \) Copy content Toggle raw display
$19$ \( T^{14} + 4 T^{13} + \cdots + 14047504 \) Copy content Toggle raw display
$23$ \( (T^{7} + 11 T^{6} + \cdots - 49152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 16 T^{6} + \cdots - 279624)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 27512614111 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 21437645056 \) Copy content Toggle raw display
$41$ \( T^{14} + 5 T^{13} + \cdots + 21233664 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 772395264 \) Copy content Toggle raw display
$47$ \( (T^{7} + 3 T^{6} + \cdots - 35132)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 5780865024 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 645566464 \) Copy content Toggle raw display
$61$ \( (T^{7} - 18 T^{6} + \cdots + 31462)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 2378317824 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 11627740224 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 92893286656 \) Copy content Toggle raw display
$79$ \( T^{14} - 6 T^{13} + \cdots + 20903184 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 3940323984 \) Copy content Toggle raw display
$89$ \( (T^{7} - 20 T^{6} + \cdots + 346512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} - 20 T^{6} + \cdots - 4360848)^{2} \) Copy content Toggle raw display
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