Properties

Label 671.2.ck.a
Level $671$
Weight $2$
Character orbit 671.ck
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [671,2,Mod(14,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([24, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.ck (of order \(30\), degree \(8\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 9 q^{2} - 12 q^{3} - 61 q^{4} - 5 q^{5} + 3 q^{6} - 18 q^{7} - 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 9 q^{2} - 12 q^{3} - 61 q^{4} - 5 q^{5} + 3 q^{6} - 18 q^{7} - 124 q^{9} - 48 q^{10} - 2 q^{12} + q^{13} + q^{14} + 3 q^{15} + 35 q^{16} - 18 q^{17} + 21 q^{18} - 11 q^{19} - 56 q^{20} - 6 q^{21} - 6 q^{22} + 51 q^{25} + 33 q^{26} + 36 q^{27} + 36 q^{29} - 72 q^{30} - 9 q^{31} - 18 q^{32} - 64 q^{34} + 27 q^{35} + 2 q^{36} - 42 q^{39} - 75 q^{40} + 2 q^{41} + 32 q^{42} - 42 q^{43} - 63 q^{44} - 8 q^{45} + 11 q^{46} - 17 q^{47} - 63 q^{48} - 58 q^{49} + 39 q^{51} - 22 q^{52} + 234 q^{54} - 147 q^{55} - 30 q^{56} + 59 q^{57} - 72 q^{58} - 15 q^{59} + 34 q^{60} - 7 q^{61} - 42 q^{62} - 69 q^{63} + 28 q^{64} + 32 q^{65} + 126 q^{66} - 72 q^{67} - 96 q^{68} - 82 q^{70} + 75 q^{71} - 16 q^{73} - 44 q^{74} - 95 q^{75} - 44 q^{76} - 10 q^{77} + 120 q^{78} + 15 q^{79} + 64 q^{80} - 60 q^{81} + 9 q^{82} + 53 q^{83} + 51 q^{86} - 78 q^{87} - 112 q^{88} + 120 q^{90} + 33 q^{91} + 117 q^{92} + 27 q^{93} - 14 q^{95} + 204 q^{96} - 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.09795 + 2.46603i −0.791281 2.43531i −3.53757 3.92887i −0.357447 + 3.40089i 6.87435 + 0.722523i −0.902579 4.24630i 8.43822 2.74174i −2.87757 + 2.09068i −7.99424 4.61548i
14.2 −1.09088 + 2.45015i 0.874736 + 2.69216i −3.47496 3.85933i 0.162454 1.54565i −7.55043 0.793582i 0.643881 + 3.02922i 8.14518 2.64653i −4.05552 + 2.94651i 3.60985 + 2.08415i
14.3 −1.07778 + 2.42074i 0.0630027 + 0.193902i −3.36011 3.73178i −0.187024 + 1.77942i −0.537291 0.0564716i 0.849246 + 3.99539i 7.61488 2.47422i 2.39342 1.73892i −4.10594 2.37056i
14.4 −1.04401 + 2.34488i −0.583396 1.79551i −3.07026 3.40986i 0.320382 3.04823i 4.81932 + 0.506531i −0.266419 1.25340i 6.31877 2.05309i −0.456447 + 0.331628i 6.81326 + 3.93364i
14.5 −1.01763 + 2.28564i 0.541493 + 1.66655i −2.85032 3.16560i 0.0850514 0.809210i −4.36017 0.458272i −0.423656 1.99315i 5.37704 1.74711i −0.0571068 + 0.0414905i 1.76301 + 1.01788i
14.6 −1.01486 + 2.27942i −0.379729 1.16869i −2.82754 3.14030i −0.118877 + 1.13104i 3.04929 + 0.320494i 0.262125 + 1.23320i 5.28160 1.71609i 1.20542 0.875789i −2.45746 1.41881i
14.7 −0.944063 + 2.12040i 0.949585 + 2.92252i −2.26658 2.51729i −0.349328 + 3.32363i −7.09338 0.745545i −0.530768 2.49707i 3.06252 0.995074i −5.21238 + 3.78701i −6.71764 3.87843i
14.8 −0.914585 + 2.05419i −1.05536 3.24806i −2.04497 2.27117i 0.186406 1.77354i 7.63736 + 0.802719i 0.492311 + 2.31614i 2.25865 0.733881i −7.00909 + 5.09240i 3.47270 + 2.00497i
14.9 −0.841616 + 1.89030i 0.392484 + 1.20794i −1.52666 1.69552i 0.322713 3.07041i −2.61369 0.274710i 0.450217 + 2.11810i 0.554068 0.180027i 1.12197 0.815161i 5.53239 + 3.19413i
14.10 −0.840089 + 1.88687i 0.422744 + 1.30107i −1.51627 1.68399i −0.407298 + 3.87518i −2.81010 0.295353i −0.114564 0.538981i 0.522572 0.169794i 0.912972 0.663313i −6.96980 4.02402i
14.11 −0.831389 + 1.86733i −0.191751 0.590149i −1.45745 1.61867i −0.146061 + 1.38968i 1.26142 + 0.132581i −0.394481 1.85589i 0.346285 0.112515i 2.11554 1.53703i −2.47356 1.42811i
14.12 −0.791444 + 1.77761i −0.629136 1.93628i −1.19526 1.32747i −0.0926998 + 0.881979i 3.93989 + 0.414099i 0.135633 + 0.638103i −0.395488 + 0.128502i −0.926325 + 0.673015i −1.49445 0.862822i
14.13 −0.785203 + 1.76360i 0.203004 + 0.624781i −1.15546 1.28327i 0.119183 1.13395i −1.26126 0.132564i −0.590630 2.77870i −0.501574 + 0.162971i 2.07791 1.50969i 1.90624 + 1.10057i
14.14 −0.759127 + 1.70503i −0.0220173 0.0677622i −0.992581 1.10237i 0.430383 4.09482i 0.132250 + 0.0139001i 0.867496 + 4.08125i −0.917003 + 0.297952i 2.42294 1.76037i 6.65506 + 3.84230i
14.15 −0.641762 + 1.44142i 0.766175 + 2.35804i −0.327577 0.363811i 0.308577 2.93592i −3.89064 0.408923i −1.02482 4.82142i −2.26658 + 0.736458i −2.54630 + 1.85000i 4.03386 + 2.32895i
14.16 −0.616002 + 1.38356i −0.492021 1.51429i −0.196528 0.218267i −0.0864920 + 0.822917i 2.39820 + 0.252061i 0.972024 + 4.57301i −2.45770 + 0.798555i 0.376076 0.273235i −1.08528 0.626586i
14.17 −0.615500 + 1.38244i 0.695450 + 2.14037i −0.194028 0.215490i −0.177745 + 1.69113i −3.38698 0.355986i 0.875518 + 4.11899i −2.46108 + 0.799652i −1.67050 + 1.21369i −2.22847 1.28661i
14.18 −0.584331 + 1.31243i −0.155559 0.478760i −0.0427671 0.0474976i −0.236278 + 2.24803i 0.719236 + 0.0755948i −0.890049 4.18735i −2.64531 + 0.859514i 2.22204 1.61441i −2.81232 1.62369i
14.19 −0.534745 + 1.20106i 0.955432 + 2.94052i 0.181677 + 0.201773i 0.132663 1.26221i −4.04264 0.424899i 0.245631 + 1.15560i −2.84024 + 0.922849i −5.30675 + 3.85558i 1.44504 + 0.834296i
14.20 −0.480268 + 1.07870i −0.530935 1.63405i 0.405327 + 0.450161i 0.369451 3.51509i 2.01764 + 0.212062i −0.518425 2.43900i −2.92624 + 0.950792i 0.0388275 0.0282099i 3.61429 + 2.08671i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
61.f even 6 1 inner
671.ck even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.ck.a 480
11.c even 5 1 inner 671.2.ck.a 480
61.f even 6 1 inner 671.2.ck.a 480
671.ck even 30 1 inner 671.2.ck.a 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.ck.a 480 1.a even 1 1 trivial
671.2.ck.a 480 11.c even 5 1 inner
671.2.ck.a 480 61.f even 6 1 inner
671.2.ck.a 480 671.ck even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).