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Properties

Label 825.2.bx.d.724.1
Level $825$
Weight $2$
Character 825.724
Analytic conductor $6.588$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [825,2,Mod(49,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bx (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,18,0,2,0,0,2,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 724.1
Root \(-0.951057 - 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 825.724
Dual form 825.2.bx.d.49.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.48990 - 0.809017i) q^{2} +(-0.587785 - 0.809017i) q^{3} +(3.92705 + 2.85317i) q^{4} +(0.809017 + 2.48990i) q^{6} +(-0.587785 + 0.809017i) q^{7} +(-4.39201 - 6.04508i) q^{8} +(-0.309017 + 0.951057i) q^{9} +(-3.30902 - 0.224514i) q^{11} -4.85410i q^{12} +(-0.224514 - 0.0729490i) q^{13} +(2.11803 - 1.53884i) q^{14} +(3.04508 + 9.37181i) q^{16} +(1.08981 - 0.354102i) q^{17} +(1.53884 - 2.11803i) q^{18} +(4.73607 - 3.44095i) q^{19} +1.00000 q^{21} +(8.05748 + 3.23607i) q^{22} -0.236068i q^{23} +(-2.30902 + 7.10642i) q^{24} +(0.500000 + 0.363271i) q^{26} +(0.951057 - 0.309017i) q^{27} +(-4.61653 + 1.50000i) q^{28} +(-4.85410 - 3.52671i) q^{29} +(-1.88197 + 5.79210i) q^{31} -10.8541i q^{32} +(1.76336 + 2.80902i) q^{33} -3.00000 q^{34} +(-3.92705 + 2.85317i) q^{36} +(-3.66547 + 5.04508i) q^{37} +(-14.5761 + 4.73607i) q^{38} +(0.0729490 + 0.224514i) q^{39} +(-0.190983 + 0.138757i) q^{41} +(-2.48990 - 0.809017i) q^{42} +6.70820i q^{43} +(-12.3541 - 10.3229i) q^{44} +(-0.190983 + 0.587785i) q^{46} +(5.93085 + 8.16312i) q^{47} +(5.79210 - 7.97214i) q^{48} +(1.85410 + 5.70634i) q^{49} +(-0.927051 - 0.673542i) q^{51} +(-0.673542 - 0.927051i) q^{52} +(0.363271 + 0.118034i) q^{53} -2.61803 q^{54} +7.47214 q^{56} +(-5.56758 - 1.80902i) q^{57} +(9.23305 + 12.7082i) q^{58} +(5.97214 + 4.33901i) q^{59} +(-3.57295 - 10.9964i) q^{61} +(9.37181 - 12.8992i) q^{62} +(-0.587785 - 0.809017i) q^{63} +(-2.69098 + 8.28199i) q^{64} +(-2.11803 - 8.42075i) q^{66} +1.85410i q^{67} +(5.29007 + 1.71885i) q^{68} +(-0.190983 + 0.138757i) q^{69} +(3.19098 + 9.82084i) q^{71} +(7.10642 - 2.30902i) q^{72} +(3.35520 - 4.61803i) q^{73} +(13.2082 - 9.59632i) q^{74} +28.4164 q^{76} +(2.12663 - 2.54508i) q^{77} -0.618034i q^{78} +(-3.39919 + 10.4616i) q^{79} +(-0.809017 - 0.587785i) q^{81} +(0.587785 - 0.190983i) q^{82} +(1.40008 - 0.454915i) q^{83} +(3.92705 + 2.85317i) q^{84} +(5.42705 - 16.7027i) q^{86} +6.00000i q^{87} +(13.1760 + 20.9894i) q^{88} +8.23607 q^{89} +(0.190983 - 0.138757i) q^{91} +(0.673542 - 0.927051i) q^{92} +(5.79210 - 1.88197i) q^{93} +(-8.16312 - 25.1235i) q^{94} +(-8.78115 + 6.37988i) q^{96} +(7.46969 + 2.42705i) q^{97} -15.7082i q^{98} +(1.23607 - 3.07768i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 18 q^{4} + 2 q^{6} + 2 q^{9} - 22 q^{11} + 8 q^{14} + 2 q^{16} + 20 q^{19} + 8 q^{21} - 14 q^{24} + 4 q^{26} - 12 q^{29} - 24 q^{31} - 24 q^{34} - 18 q^{36} + 14 q^{39} - 6 q^{41} - 72 q^{44} - 6 q^{46}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.48990 − 0.809017i −1.76062 − 0.572061i −0.763359 − 0.645974i \(-0.776451\pi\)
−0.997265 + 0.0739128i \(0.976451\pi\)
\(3\) −0.587785 − 0.809017i −0.339358 − 0.467086i
\(4\) 3.92705 + 2.85317i 1.96353 + 1.42658i
\(5\) 0 0
\(6\) 0.809017 + 2.48990i 0.330280 + 1.01650i
\(7\) −0.587785 + 0.809017i −0.222162 + 0.305780i −0.905520 − 0.424304i \(-0.860519\pi\)
0.683358 + 0.730084i \(0.260519\pi\)
\(8\) −4.39201 − 6.04508i −1.55281 − 2.13726i
\(9\) −0.309017 + 0.951057i −0.103006 + 0.317019i
\(10\) 0 0
\(11\) −3.30902 − 0.224514i −0.997706 − 0.0676935i
\(12\) − 4.85410i − 1.40126i
\(13\) −0.224514 − 0.0729490i −0.0622690 − 0.0202324i 0.277717 − 0.960663i \(-0.410422\pi\)
−0.339986 + 0.940431i \(0.610422\pi\)
\(14\) 2.11803 − 1.53884i 0.566068 − 0.411273i
\(15\) 0 0
\(16\) 3.04508 + 9.37181i 0.761271 + 2.34295i
\(17\) 1.08981 − 0.354102i 0.264319 − 0.0858823i −0.173860 − 0.984770i \(-0.555624\pi\)
0.438178 + 0.898888i \(0.355624\pi\)
\(18\) 1.53884 − 2.11803i 0.362708 − 0.499225i
\(19\) 4.73607 − 3.44095i 1.08653 − 0.789409i 0.107719 − 0.994181i \(-0.465645\pi\)
0.978810 + 0.204772i \(0.0656454\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 8.05748 + 3.23607i 1.71786 + 0.689932i
\(23\) − 0.236068i − 0.0492236i −0.999697 − 0.0246118i \(-0.992165\pi\)
0.999697 − 0.0246118i \(-0.00783497\pi\)
\(24\) −2.30902 + 7.10642i −0.471326 + 1.45059i
\(25\) 0 0
\(26\) 0.500000 + 0.363271i 0.0980581 + 0.0712434i
\(27\) 0.951057 − 0.309017i 0.183031 − 0.0594703i
\(28\) −4.61653 + 1.50000i −0.872441 + 0.283473i
\(29\) −4.85410 − 3.52671i −0.901384 − 0.654894i 0.0374370 − 0.999299i \(-0.488081\pi\)
−0.938821 + 0.344405i \(0.888081\pi\)
\(30\) 0 0
\(31\) −1.88197 + 5.79210i −0.338011 + 1.04029i 0.627209 + 0.778851i \(0.284197\pi\)
−0.965220 + 0.261440i \(0.915803\pi\)
\(32\) − 10.8541i − 1.91875i
\(33\) 1.76336 + 2.80902i 0.306961 + 0.488987i
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −3.92705 + 2.85317i −0.654508 + 0.475528i
\(37\) −3.66547 + 5.04508i −0.602599 + 0.829407i −0.995943 − 0.0899846i \(-0.971318\pi\)
0.393344 + 0.919391i \(0.371318\pi\)
\(38\) −14.5761 + 4.73607i −2.36456 + 0.768292i
\(39\) 0.0729490 + 0.224514i 0.0116812 + 0.0359510i
\(40\) 0 0
\(41\) −0.190983 + 0.138757i −0.0298265 + 0.0216702i −0.602599 − 0.798044i \(-0.705868\pi\)
0.572772 + 0.819715i \(0.305868\pi\)
\(42\) −2.48990 − 0.809017i −0.384200 − 0.124834i
\(43\) 6.70820i 1.02299i 0.859286 + 0.511496i \(0.170908\pi\)
−0.859286 + 0.511496i \(0.829092\pi\)
\(44\) −12.3541 − 10.3229i −1.86245 − 1.55623i
\(45\) 0 0
\(46\) −0.190983 + 0.587785i −0.0281589 + 0.0866642i
\(47\) 5.93085 + 8.16312i 0.865104 + 1.19071i 0.980328 + 0.197374i \(0.0632413\pi\)
−0.115224 + 0.993339i \(0.536759\pi\)
\(48\) 5.79210 − 7.97214i 0.836017 − 1.15068i
\(49\) 1.85410 + 5.70634i 0.264872 + 0.815191i
\(50\) 0 0
\(51\) −0.927051 − 0.673542i −0.129813 − 0.0943147i
\(52\) −0.673542 − 0.927051i −0.0934035 − 0.128559i
\(53\) 0.363271 + 0.118034i 0.0498991 + 0.0162132i 0.333860 − 0.942623i \(-0.391649\pi\)
−0.283961 + 0.958836i \(0.591649\pi\)
\(54\) −2.61803 −0.356269
\(55\) 0 0
\(56\) 7.47214 0.998506
\(57\) −5.56758 − 1.80902i −0.737444 − 0.239610i
\(58\) 9.23305 + 12.7082i 1.21236 + 1.66867i
\(59\) 5.97214 + 4.33901i 0.777506 + 0.564891i 0.904229 − 0.427047i \(-0.140446\pi\)
−0.126724 + 0.991938i \(0.540446\pi\)
\(60\) 0 0
\(61\) −3.57295 − 10.9964i −0.457469 − 1.40795i −0.868212 − 0.496194i \(-0.834730\pi\)
0.410742 − 0.911751i \(-0.365270\pi\)
\(62\) 9.37181 − 12.8992i 1.19022 − 1.63820i
\(63\) −0.587785 − 0.809017i −0.0740540 − 0.101927i
\(64\) −2.69098 + 8.28199i −0.336373 + 1.03525i
\(65\) 0 0
\(66\) −2.11803 − 8.42075i −0.260712 − 1.03652i
\(67\) 1.85410i 0.226515i 0.993566 + 0.113257i \(0.0361284\pi\)
−0.993566 + 0.113257i \(0.963872\pi\)
\(68\) 5.29007 + 1.71885i 0.641515 + 0.208441i
\(69\) −0.190983 + 0.138757i −0.0229917 + 0.0167044i
\(70\) 0 0
\(71\) 3.19098 + 9.82084i 0.378700 + 1.16552i 0.940948 + 0.338550i \(0.109937\pi\)
−0.562248 + 0.826968i \(0.690063\pi\)
\(72\) 7.10642 − 2.30902i 0.837500 − 0.272120i
\(73\) 3.35520 − 4.61803i 0.392696 − 0.540500i −0.566196 − 0.824271i \(-0.691585\pi\)
0.958892 + 0.283771i \(0.0915854\pi\)
\(74\) 13.2082 − 9.59632i 1.53542 − 1.11555i
\(75\) 0 0
\(76\) 28.4164 3.25959
\(77\) 2.12663 − 2.54508i 0.242352 − 0.290039i
\(78\) − 0.618034i − 0.0699786i
\(79\) −3.39919 + 10.4616i −0.382438 + 1.17702i 0.555883 + 0.831260i \(0.312380\pi\)
−0.938322 + 0.345764i \(0.887620\pi\)
\(80\) 0 0
\(81\) −0.809017 − 0.587785i −0.0898908 − 0.0653095i
\(82\) 0.587785 − 0.190983i 0.0649100 − 0.0210905i
\(83\) 1.40008 − 0.454915i 0.153679 − 0.0499334i −0.231167 − 0.972914i \(-0.574254\pi\)
0.384846 + 0.922981i \(0.374254\pi\)
\(84\) 3.92705 + 2.85317i 0.428476 + 0.311306i
\(85\) 0 0
\(86\) 5.42705 − 16.7027i 0.585214 − 1.80110i
\(87\) 6.00000i 0.643268i
\(88\) 13.1760 + 20.9894i 1.40457 + 2.23747i
\(89\) 8.23607 0.873021 0.436511 − 0.899699i \(-0.356214\pi\)
0.436511 + 0.899699i \(0.356214\pi\)
\(90\) 0 0
\(91\) 0.190983 − 0.138757i 0.0200205 − 0.0145457i
\(92\) 0.673542 − 0.927051i 0.0702216 − 0.0966517i
\(93\) 5.79210 − 1.88197i 0.600612 − 0.195151i
\(94\) −8.16312 − 25.1235i −0.841961 − 2.59129i
\(95\) 0 0
\(96\) −8.78115 + 6.37988i −0.896223 + 0.651144i
\(97\) 7.46969 + 2.42705i 0.758433 + 0.246430i 0.662606 − 0.748968i \(-0.269451\pi\)
0.0958268 + 0.995398i \(0.469451\pi\)
\(98\) − 15.7082i − 1.58677i
\(99\) 1.23607 − 3.07768i 0.124230 − 0.309319i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.2.bx.d.724.1 8
5.2 odd 4 825.2.n.c.526.1 4
5.3 odd 4 33.2.e.b.31.1 yes 4
5.4 even 2 inner 825.2.bx.d.724.2 8
11.5 even 5 inner 825.2.bx.d.49.2 8
15.8 even 4 99.2.f.a.64.1 4
20.3 even 4 528.2.y.b.97.1 4
45.13 odd 12 891.2.n.c.460.1 8
45.23 even 12 891.2.n.b.460.1 8
45.38 even 12 891.2.n.b.757.1 8
45.43 odd 12 891.2.n.c.757.1 8
55.3 odd 20 363.2.e.k.202.1 4
55.7 even 20 9075.2.a.u.1.1 2
55.8 even 20 363.2.e.b.202.1 4
55.13 even 20 363.2.e.b.124.1 4
55.18 even 20 363.2.a.i.1.2 2
55.27 odd 20 825.2.n.c.676.1 4
55.28 even 20 363.2.e.f.148.1 4
55.37 odd 20 9075.2.a.cb.1.2 2
55.38 odd 20 33.2.e.b.16.1 ✓ 4
55.43 even 4 363.2.e.f.130.1 4
55.48 odd 20 363.2.a.d.1.1 2
55.49 even 10 inner 825.2.bx.d.49.1 8
55.53 odd 20 363.2.e.k.124.1 4
165.38 even 20 99.2.f.a.82.1 4
165.128 odd 20 1089.2.a.l.1.1 2
165.158 even 20 1089.2.a.t.1.2 2
220.103 even 20 5808.2.a.cj.1.1 2
220.183 odd 20 5808.2.a.ci.1.1 2
220.203 even 20 528.2.y.b.49.1 4
495.38 even 60 891.2.n.b.676.1 8
495.148 odd 60 891.2.n.c.379.1 8
495.203 even 60 891.2.n.b.379.1 8
495.313 odd 60 891.2.n.c.676.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 55.38 odd 20
33.2.e.b.31.1 yes 4 5.3 odd 4
99.2.f.a.64.1 4 15.8 even 4
99.2.f.a.82.1 4 165.38 even 20
363.2.a.d.1.1 2 55.48 odd 20
363.2.a.i.1.2 2 55.18 even 20
363.2.e.b.124.1 4 55.13 even 20
363.2.e.b.202.1 4 55.8 even 20
363.2.e.f.130.1 4 55.43 even 4
363.2.e.f.148.1 4 55.28 even 20
363.2.e.k.124.1 4 55.53 odd 20
363.2.e.k.202.1 4 55.3 odd 20
528.2.y.b.49.1 4 220.203 even 20
528.2.y.b.97.1 4 20.3 even 4
825.2.n.c.526.1 4 5.2 odd 4
825.2.n.c.676.1 4 55.27 odd 20
825.2.bx.d.49.1 8 55.49 even 10 inner
825.2.bx.d.49.2 8 11.5 even 5 inner
825.2.bx.d.724.1 8 1.1 even 1 trivial
825.2.bx.d.724.2 8 5.4 even 2 inner
891.2.n.b.379.1 8 495.203 even 60
891.2.n.b.460.1 8 45.23 even 12
891.2.n.b.676.1 8 495.38 even 60
891.2.n.b.757.1 8 45.38 even 12
891.2.n.c.379.1 8 495.148 odd 60
891.2.n.c.460.1 8 45.13 odd 12
891.2.n.c.676.1 8 495.313 odd 60
891.2.n.c.757.1 8 45.43 odd 12
1089.2.a.l.1.1 2 165.128 odd 20
1089.2.a.t.1.2 2 165.158 even 20
5808.2.a.ci.1.1 2 220.183 odd 20
5808.2.a.cj.1.1 2 220.103 even 20
9075.2.a.u.1.1 2 55.7 even 20
9075.2.a.cb.1.2 2 55.37 odd 20