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Properties

Label 847.2.x.a
Level $847$
Weight $2$
Character orbit 847.x
Analytic conductor $6.763$
Analytic rank $0$
Dimension $1720$
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] = [847,2,Mod(10,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([11, 45]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.x (of order \(66\), degree \(20\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(1720\)
Relative dimension: \(86\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1720 q - 11 q^{2} - 60 q^{3} - 95 q^{4} - 33 q^{5} - 22 q^{7} - 44 q^{8} + 804 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1720 q - 11 q^{2} - 60 q^{3} - 95 q^{4} - 33 q^{5} - 22 q^{7} - 44 q^{8} + 804 q^{9} + 33 q^{10} - 18 q^{11} - 15 q^{12} - 46 q^{15} + 61 q^{16} - 33 q^{17} + 66 q^{18} - 33 q^{19} - 55 q^{21} + 48 q^{22} - 13 q^{23} - 132 q^{24} - 105 q^{25} - 51 q^{26} - 22 q^{28} - 44 q^{29} - 77 q^{30} - 39 q^{31} - 11 q^{32} - 51 q^{33} - 22 q^{35} - 150 q^{36} + 2 q^{37} - 15 q^{38} + 22 q^{39} - 53 q^{42} - 44 q^{43} - 20 q^{44} - 186 q^{45} - 11 q^{46} - 21 q^{47} - 132 q^{49} - 44 q^{50} - 165 q^{51} - 231 q^{52} - 29 q^{53} + 693 q^{54} - 34 q^{56} - 110 q^{57} - 76 q^{58} + 3 q^{59} + 34 q^{60} - 33 q^{61} - 33 q^{63} + 300 q^{64} - 11 q^{65} - 90 q^{66} - 25 q^{67} - 33 q^{68} + 36 q^{70} - 80 q^{71} - 165 q^{72} - 33 q^{73} - 11 q^{74} + 21 q^{75} - 236 q^{77} - 90 q^{78} - 55 q^{79} - 39 q^{80} - 668 q^{81} - 63 q^{82} - 55 q^{84} + 176 q^{85} + 15 q^{86} - 132 q^{87} - 29 q^{88} - 270 q^{89} - 61 q^{91} - 488 q^{92} - 106 q^{93} - 99 q^{95} - 429 q^{96} - 110 q^{98} - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.81634 − 0.134159i −0.868367 − 0.501352i 5.92282 + 0.565561i 0.682147 + 0.867421i 2.37835 + 1.52848i −2.60519 − 0.461517i −11.0231 − 1.58489i −0.997293 − 1.72736i −1.80478 − 2.53447i
10.2 −2.75726 − 0.131345i −2.59140 − 1.49614i 5.59430 + 0.534191i −1.91302 − 2.43261i 6.94865 + 4.46563i 2.39759 + 1.11874i −9.89022 − 1.42200i 2.97689 + 5.15612i 4.95520 + 6.95860i
10.3 −2.66188 − 0.126801i 2.74138 + 1.58273i 5.07858 + 0.484946i −2.17682 − 2.76805i −7.09652 − 4.56066i −2.23027 − 1.42333i −8.18154 − 1.17633i 3.51009 + 6.07966i 5.44344 + 7.64425i
10.4 −2.60759 − 0.124215i −0.156257 − 0.0902149i 4.79315 + 0.457690i 1.55659 + 1.97937i 0.396247 + 0.254653i 2.62842 − 0.302344i −7.27376 − 1.04581i −1.48372 − 2.56988i −3.81308 − 5.35472i
10.5 −2.53212 − 0.120620i 2.12284 + 1.22562i 4.40615 + 0.420736i 0.698990 + 0.888838i −5.22747 − 3.35949i 1.99423 − 1.73870i −6.08778 − 0.875291i 1.50431 + 2.60554i −1.66272 − 2.33496i
10.6 −2.51092 − 0.119610i −0.674633 − 0.389500i 4.29946 + 0.410548i −1.02626 − 1.30499i 1.64736 + 1.05869i 0.282243 − 2.63065i −5.77012 − 0.829619i −1.19658 − 2.07254i 2.42076 + 3.39948i
10.7 −2.46426 − 0.117387i 2.31172 + 1.33467i 4.06785 + 0.388432i 2.54618 + 3.23774i −5.54001 − 3.56035i −0.750402 + 2.53710i −5.09475 − 0.732514i 2.06271 + 3.57272i −5.89439 − 8.27751i
10.8 −2.46098 − 0.117231i 0.813611 + 0.469739i 4.05173 + 0.386893i −2.32023 − 2.95041i −1.94721 − 1.25140i 2.58436 + 0.566636i −5.04849 − 0.725863i −1.05869 − 1.83371i 5.36415 + 7.53290i
10.9 −2.38715 − 0.113714i −2.68556 − 1.55051i 3.69459 + 0.352791i 1.10017 + 1.39899i 6.23451 + 4.00668i −2.64121 − 0.154983i −4.04836 − 0.582067i 3.30816 + 5.72990i −2.46719 − 3.46469i
10.10 −2.32487 − 0.110747i −1.14180 − 0.659217i 3.40182 + 0.324834i 1.50586 + 1.91486i 2.58153 + 1.65905i 1.11205 + 2.40070i −3.26519 − 0.469463i −0.630865 − 1.09269i −3.28887 − 4.61857i
10.11 −2.32005 − 0.110518i 0.889184 + 0.513371i 3.37949 + 0.322702i 1.20915 + 1.53756i −2.00622 − 1.28932i −2.10097 − 1.60808i −3.20685 − 0.461075i −0.972901 − 1.68511i −2.63537 − 3.70087i
10.12 −2.27557 − 0.108399i −1.58135 − 0.912990i 3.17553 + 0.303227i −0.952228 − 1.21086i 3.49950 + 2.24899i −1.38677 + 2.25319i −2.68336 − 0.385809i 0.167102 + 0.289429i 2.03561 + 2.85861i
10.13 −2.15759 − 0.102778i 0.0221198 + 0.0127709i 2.65367 + 0.253394i −1.64064 − 2.08625i −0.0464127 − 0.0298277i −1.90987 + 1.83096i −1.42338 − 0.204651i −1.49967 − 2.59751i 3.32541 + 4.66988i
10.14 −2.14570 − 0.102212i 1.52425 + 0.880026i 2.60265 + 0.248523i −0.493336 − 0.627327i −3.18064 − 2.04407i −2.59427 + 0.519385i −1.30657 − 0.187856i 0.0488912 + 0.0846820i 0.994431 + 1.39648i
10.15 −1.96582 − 0.0936434i −1.98627 − 1.14678i 1.86472 + 0.178059i −0.904452 − 1.15010i 3.79726 + 2.44035i 0.573602 − 2.58282i 0.246999 + 0.0355131i 1.13019 + 1.95755i 1.67029 + 2.34559i
10.16 −1.94545 − 0.0926733i 2.50563 + 1.44662i 1.78525 + 0.170471i −0.415136 − 0.527888i −4.74051 − 3.04654i 1.93603 + 1.80327i 0.398343 + 0.0572730i 2.68544 + 4.65132i 0.758705 + 1.06545i
10.17 −1.86475 − 0.0888291i −0.0252474 − 0.0145766i 1.47847 + 0.141177i −0.105696 − 0.134404i 0.0457854 + 0.0294245i 1.60285 + 2.10496i 0.951292 + 0.136775i −1.49958 − 2.59734i 0.185158 + 0.260019i
10.18 −1.80792 − 0.0861219i −1.72811 − 0.997726i 1.27021 + 0.121291i −0.508859 − 0.647067i 3.03836 + 1.95264i 2.51479 − 0.822081i 1.29709 + 0.186494i 0.490914 + 0.850289i 0.864250 + 1.21367i
10.19 −1.79179 − 0.0853534i −1.73157 − 0.999725i 1.21228 + 0.115759i 1.61018 + 2.04751i 3.01729 + 1.93909i −0.898356 − 2.48857i 1.38886 + 0.199688i 0.498900 + 0.864120i −2.71034 − 3.80614i
10.20 −1.78916 − 0.0852280i 1.65195 + 0.953752i 1.20287 + 0.114861i 0.349108 + 0.443927i −2.87431 − 1.84720i −0.100171 − 2.64385i 1.40357 + 0.201802i 0.319287 + 0.553021i −0.586773 − 0.824008i
See next 80 embeddings (of 1720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
121.f odd 22 1 inner
847.x even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.x.a ✓ 1720
7.d odd 6 1 inner 847.2.x.a ✓ 1720
121.f odd 22 1 inner 847.2.x.a ✓ 1720
847.x even 66 1 inner 847.2.x.a ✓ 1720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.x.a ✓ 1720 1.a even 1 1 trivial
847.2.x.a ✓ 1720 7.d odd 6 1 inner
847.2.x.a ✓ 1720 121.f odd 22 1 inner
847.2.x.a ✓ 1720 847.x even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).