↑

Properties

Label 670.2.w.a
Level $670$
Weight $2$
Character orbit 670.w
Analytic conductor $5.350$
Analytic rank $0$
Dimension $1360$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [670,2,Mod(7,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(132))
 
chi = DirichletCharacter(H, H._module([33, 46]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.w (of order \(132\), degree \(40\), 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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(1360\)
Relative dimension: \(34\) over \(\Q(\zeta_{132})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{132}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1360 q - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1360 q - 4 q^{6} + 8 q^{15} - 68 q^{16} + 12 q^{20} - 24 q^{21} - 36 q^{23} - 24 q^{25} + 88 q^{31} - 44 q^{33} + 12 q^{35} + 64 q^{36} + 24 q^{37} - 24 q^{38} + 12 q^{41} + 88 q^{43} + 12 q^{46} - 108 q^{47} + 44 q^{48} - 72 q^{50} + 48 q^{51} - 288 q^{55} + 4 q^{56} + 36 q^{57} - 164 q^{60} + 20 q^{61} - 32 q^{62} - 352 q^{65} + 68 q^{67} - 484 q^{70} + 88 q^{71} - 16 q^{73} - 704 q^{75} - 32 q^{76} + 456 q^{77} - 60 q^{78} - 56 q^{81} + 48 q^{82} - 16 q^{83} - 96 q^{85} + 4 q^{86} + 96 q^{87} - 32 q^{90} - 128 q^{91} + 16 q^{92} + 80 q^{93} + 60 q^{95} + 4 q^{96} + 120 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.304486 + 0.952517i −1.62534 − 2.97659i −0.814576 − 0.580057i 1.73842 − 1.40638i 3.33015 − 0.641834i 1.95508 + 2.15066i 0.800541 − 0.599278i −4.59645 + 7.15222i 0.810272 + 2.08410i
7.2 −0.304486 + 0.952517i −1.56362 − 2.86356i −0.814576 − 0.580057i −0.902584 + 2.04581i 3.20369 − 0.617460i −3.33052 − 3.66369i 0.800541 − 0.599278i −4.13313 + 6.43127i −1.67384 − 1.48265i
7.3 −0.304486 + 0.952517i −1.12569 − 2.06154i −0.814576 − 0.580057i −1.46896 − 1.68587i 2.30641 − 0.444523i 0.520683 + 0.572770i 0.800541 − 0.599278i −1.36085 + 2.11753i 2.05310 − 0.885882i
7.4 −0.304486 + 0.952517i −0.968327 − 1.77336i −0.814576 − 0.580057i −2.10295 + 0.760013i 1.98400 − 0.382384i 1.18491 + 1.30345i 0.800541 − 0.599278i −0.585222 + 0.910624i −0.0836064 − 2.23450i
7.5 −0.304486 + 0.952517i −0.814424 − 1.49151i −0.814576 − 0.580057i 2.10590 + 0.751800i 1.66867 − 0.321609i −0.209460 − 0.230413i 0.800541 − 0.599278i 0.0606165 − 0.0943212i −1.35732 + 1.77699i
7.6 −0.304486 + 0.952517i −0.808548 − 1.48075i −0.814576 − 0.580057i 0.790679 − 2.09161i 1.65663 − 0.319289i −3.05675 − 3.36254i 0.800541 − 0.599278i 0.0830664 − 0.129254i 1.75154 + 1.39000i
7.7 −0.304486 + 0.952517i −0.467863 − 0.856828i −0.814576 − 0.580057i 0.475616 − 2.18490i 0.958601 − 0.184755i −1.63348 − 1.79689i 0.800541 − 0.599278i 1.10666 − 1.72200i 1.93634 + 1.11830i
7.8 −0.304486 + 0.952517i −0.0909985 − 0.166651i −0.814576 − 0.580057i −0.579530 + 2.15966i 0.186446 − 0.0359345i 3.19533 + 3.51498i 0.800541 − 0.599278i 1.60243 − 2.49343i −1.88066 − 1.20960i
7.9 −0.304486 + 0.952517i 0.0481924 + 0.0882578i −0.814576 − 0.580057i −0.507349 + 2.17775i −0.0987409 + 0.0190308i −0.918621 − 1.01052i 0.800541 − 0.599278i 1.61646 − 2.51525i −1.91986 − 1.14635i
7.10 −0.304486 + 0.952517i 0.0592154 + 0.108445i −0.814576 − 0.580057i 2.18858 − 0.458367i −0.121326 + 0.0233836i 2.74768 + 3.02255i 0.800541 − 0.599278i 1.61367 − 2.51092i −0.229792 + 2.22423i
7.11 −0.304486 + 0.952517i 0.330524 + 0.605309i −0.814576 − 0.580057i 1.39768 + 1.74542i −0.677207 + 0.130521i −3.33442 − 3.66799i 0.800541 − 0.599278i 1.36477 − 2.12362i −2.08812 + 0.799854i
7.12 −0.304486 + 0.952517i 0.411670 + 0.753917i −0.814576 − 0.580057i −2.01241 − 0.974782i −0.843466 + 0.162565i −0.968582 − 1.06548i 0.800541 − 0.599278i 1.22300 − 1.90303i 1.54125 − 1.62005i
7.13 −0.304486 + 0.952517i 0.623082 + 1.14109i −0.814576 − 0.580057i −2.21635 + 0.296332i −1.27663 + 0.246050i −0.0733986 − 0.0807412i 0.800541 − 0.599278i 0.708069 − 1.10178i 0.392586 − 2.20134i
7.14 −0.304486 + 0.952517i 0.697010 + 1.27648i −0.814576 − 0.580057i −0.362196 − 2.20654i −1.42810 + 0.275243i 0.103392 + 0.113735i 0.800541 − 0.599278i 0.478347 − 0.744322i 2.21205 + 0.326863i
7.15 −0.304486 + 0.952517i 1.05670 + 1.93520i −0.814576 − 0.580057i 2.23204 + 0.134090i −2.16506 + 0.417281i −0.406316 − 0.446962i 0.800541 − 0.599278i −1.00646 + 1.56608i −0.807350 + 2.08523i
7.16 −0.304486 + 0.952517i 1.41547 + 2.59224i −0.814576 − 0.580057i −0.200367 + 2.22707i −2.90015 + 0.558958i 0.493182 + 0.542519i 0.800541 − 0.599278i −3.09424 + 4.81474i −2.06032 − 0.868966i
7.17 −0.304486 + 0.952517i 1.53411 + 2.80952i −0.814576 − 0.580057i 0.233387 − 2.22385i −3.14323 + 0.605809i 2.75764 + 3.03350i 0.800541 − 0.599278i −3.91798 + 6.09650i 2.04720 + 0.899439i
7.18 0.304486 − 0.952517i −1.53754 − 2.81580i −0.814576 − 0.580057i −0.766691 − 2.10052i −3.15025 + 0.607162i 0.443261 + 0.487604i −0.800541 + 0.599278i −3.94275 + 6.13504i −2.23423 + 0.0907058i
7.19 0.304486 − 0.952517i −1.23592 − 2.26342i −0.814576 − 0.580057i −1.26114 + 1.84649i −2.53226 + 0.488054i 1.49660 + 1.64632i −0.800541 + 0.599278i −1.97364 + 3.07104i 1.37482 + 1.76348i
7.20 0.304486 − 0.952517i −1.19524 − 2.18892i −0.814576 − 0.580057i 2.15273 + 0.604783i −2.44892 + 0.471991i 2.66378 + 2.93025i −0.800541 + 0.599278i −1.74086 + 2.70883i 1.23154 − 1.86636i
See next 80 embeddings (of 1360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
67.h odd 66 1 inner
335.w even 132 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.w.a ✓ 1360
5.c odd 4 1 inner 670.2.w.a ✓ 1360
67.h odd 66 1 inner 670.2.w.a ✓ 1360
335.w even 132 1 inner 670.2.w.a ✓ 1360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.w.a ✓ 1360 1.a even 1 1 trivial
670.2.w.a ✓ 1360 5.c odd 4 1 inner
670.2.w.a ✓ 1360 67.h odd 66 1 inner
670.2.w.a ✓ 1360 335.w even 132 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(670, [\chi])\).