Properties

Label 775.2.bz.a
Level $775$
Weight $2$
Character orbit 775.bz
Analytic conductor $6.188$
Analytic rank $0$
Dimension $624$
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] = [775,2,Mod(58,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.58");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bz (of order \(20\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(78\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 6 q^{2} - 10 q^{4} - 16 q^{5} + 2 q^{7} - 20 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 624 q - 6 q^{2} - 10 q^{4} - 16 q^{5} + 2 q^{7} - 20 q^{8} - 20 q^{9} + 6 q^{10} - 10 q^{11} - 10 q^{12} - 10 q^{13} - 10 q^{14} + 10 q^{15} + 142 q^{16} - 10 q^{17} - 6 q^{18} - 10 q^{19} - 28 q^{20} - 10 q^{21} - 60 q^{22} - 20 q^{23} - 40 q^{24} - 44 q^{25} + 120 q^{27} - 48 q^{28} - 10 q^{29} - 6 q^{31} - 46 q^{32} - 6 q^{33} - 10 q^{34} - 42 q^{35} + 132 q^{36} + 30 q^{37} - 42 q^{38} - 10 q^{39} - 14 q^{40} - 12 q^{41} - 120 q^{42} + 10 q^{43} - 10 q^{44} - 8 q^{45} - 10 q^{46} + 96 q^{47} - 10 q^{48} + 66 q^{50} - 12 q^{51} - 80 q^{52} - 130 q^{53} - 20 q^{54} + 60 q^{55} - 4 q^{56} - 40 q^{57} - 10 q^{58} - 40 q^{59} - 200 q^{60} + 156 q^{62} + 20 q^{63} - 10 q^{64} + 70 q^{65} + 10 q^{66} - 38 q^{67} - 20 q^{68} + 50 q^{70} - 52 q^{71} + 42 q^{72} + 30 q^{73} - 110 q^{74} - 120 q^{75} - 4 q^{76} - 50 q^{77} - 84 q^{78} + 180 q^{79} - 8 q^{80} + 112 q^{81} + 66 q^{82} + 20 q^{83} - 10 q^{84} + 50 q^{85} - 10 q^{86} + 40 q^{87} + 110 q^{88} - 10 q^{89} + 118 q^{90} - 10 q^{92} - 70 q^{93} - 90 q^{94} + 34 q^{95} - 10 q^{96} + 62 q^{97} - 66 q^{98} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 −0.432453 + 2.73040i −0.813284 0.414389i −5.36597 1.74351i 1.39279 + 1.74932i 1.48316 2.04139i 0.597123 3.77009i 4.57095 8.97100i −1.27364 1.75302i −5.37868 + 3.04637i
58.2 −0.427920 + 2.70178i −2.21257 1.12736i −5.21439 1.69426i −0.783635 2.09426i 3.99268 5.49545i −0.194778 + 1.22978i 4.32511 8.48851i 1.86116 + 2.56166i 5.99356 1.22103i
58.3 −0.420856 + 2.65718i 1.36758 + 0.696816i −4.98137 1.61855i −2.22970 + 0.168664i −2.42712 + 3.34064i 0.172196 1.08720i 3.95447 7.76108i −0.378638 0.521151i 0.490211 5.99569i
58.4 −0.417323 + 2.63487i 2.25350 + 1.14822i −4.86628 1.58115i 1.91087 + 1.16128i −3.96584 + 5.45851i −0.787011 + 4.96899i 3.77471 7.40828i 1.99651 + 2.74796i −3.85727 + 4.55027i
58.5 −0.407008 + 2.56975i 0.965093 + 0.491739i −4.53583 1.47378i 0.800701 2.08779i −1.65645 + 2.27990i −0.0975664 + 0.616010i 3.27099 6.41969i −1.07376 1.47790i 5.03921 + 2.90735i
58.6 −0.385823 + 2.43599i −0.635361 0.323733i −3.88308 1.26169i −1.16565 + 1.90821i 1.03375 1.42283i −0.334318 + 2.11080i 2.33224 4.57727i −1.46447 2.01568i −4.19864 3.57574i
58.7 −0.372726 + 2.35330i 1.04430 + 0.532096i −3.49698 1.13624i 2.12831 0.685784i −1.64142 + 2.25922i 0.575570 3.63401i 1.81394 3.56005i −0.955926 1.31572i 0.820578 + 5.26416i
58.8 −0.371953 + 2.34842i −1.05680 0.538468i −3.47462 1.12897i 2.05961 0.870635i 1.65763 2.28153i −0.551526 + 3.48220i 1.78480 3.50286i −0.936471 1.28894i 1.27854 + 5.16067i
58.9 −0.359324 + 2.26869i −3.05823 1.55825i −3.11571 1.01235i −1.52232 + 1.63785i 4.63407 6.37825i 0.553964 3.49759i 1.33066 2.61157i 5.16129 + 7.10391i −3.16875 4.04219i
58.10 −0.354386 + 2.23750i 2.89940 + 1.47732i −2.97872 0.967845i −1.95864 1.07876i −4.33301 + 5.96388i −0.115111 + 0.726782i 1.16424 2.28494i 4.46070 + 6.13962i 3.10785 4.00016i
58.11 −0.337588 + 2.13145i 2.05236 + 1.04573i −2.52700 0.821071i −0.701956 + 2.12303i −2.92177 + 4.02147i 0.190417 1.20224i 0.643716 1.26336i 1.35527 + 1.86537i −4.28816 2.21289i
58.12 −0.335900 + 2.12079i −2.21885 1.13056i −2.48281 0.806715i 2.18571 0.471884i 3.14300 4.32596i 0.188721 1.19154i 0.595211 1.16817i 1.88177 + 2.59004i 0.266588 + 4.79394i
58.13 −0.333655 + 2.10662i 1.00524 + 0.512194i −2.42439 0.787732i −0.617699 2.14906i −1.41440 + 1.94675i 0.165232 1.04323i 0.531750 1.04362i −1.01520 1.39730i 4.73334 0.584210i
58.14 −0.320924 + 2.02624i −1.93450 0.985675i −2.10053 0.682503i −2.16149 + 0.572676i 2.61804 3.60342i −0.468266 + 2.95651i 0.194306 0.381347i 1.00736 + 1.38652i −0.466702 4.56348i
58.15 −0.302579 + 1.91041i −1.04392 0.531905i −1.65598 0.538061i 1.28497 + 1.82999i 1.33202 1.83337i 0.225002 1.42060i −0.227254 + 0.446011i −0.956506 1.31652i −3.88482 + 1.90110i
58.16 −0.292237 + 1.84511i 0.500677 + 0.255108i −1.41693 0.460389i −0.00183075 + 2.23607i −0.617020 + 0.849255i −0.226119 + 1.42766i −0.432661 + 0.849145i −1.57776 2.17160i −4.12527 0.656840i
58.17 −0.277660 + 1.75307i −1.36666 0.696350i −1.09406 0.355482i 0.647198 2.14036i 1.60022 2.20252i 0.563366 3.55695i −0.684635 + 1.34367i −0.380488 0.523697i 3.57251 + 1.72888i
58.18 −0.276615 + 1.74648i 1.93521 + 0.986041i −1.07156 0.348170i 1.93192 + 1.12592i −2.25741 + 3.10705i 0.0127590 0.0805573i −0.701056 + 1.37590i 1.00942 + 1.38935i −2.50079 + 3.06261i
58.19 −0.270232 + 1.70618i 0.232154 + 0.118288i −0.935906 0.304094i −1.45606 1.69703i −0.264556 + 0.364130i −0.629780 + 3.97627i −0.796737 + 1.56368i −1.72345 2.37213i 3.28891 2.02570i
58.20 −0.263622 + 1.66444i 0.760114 + 0.387297i −0.798767 0.259535i −2.20524 + 0.369994i −0.845018 + 1.16307i 0.814632 5.14338i −0.887568 + 1.74195i −1.33558 1.83827i −0.0344834 3.76805i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
775.bz even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bz.a yes 624
25.f odd 20 1 775.2.br.a 624
31.f odd 10 1 775.2.br.a 624
775.bz even 20 1 inner 775.2.bz.a yes 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.br.a 624 25.f odd 20 1
775.2.br.a 624 31.f odd 10 1
775.2.bz.a yes 624 1.a even 1 1 trivial
775.2.bz.a yes 624 775.bz even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).