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Properties

Label 825.2.n.c.676.1
Level $825$
Weight $2$
Character 825.676
Analytic conductor $6.588$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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(301,825)] 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("825.301"); 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, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,-1,-9,0,1,-1,-13,-1,0,-11] 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 676.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 825.676
Dual form 825.2.n.c.526.1

$q$-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+(0.809017 + 2.48990i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(-3.92705 + 2.85317i) q^{4} +(0.809017 - 2.48990i) q^{6} +(-0.809017 + 0.587785i) q^{7} +(-6.04508 - 4.39201i) q^{8} +(0.309017 + 0.951057i) q^{9} +(-3.30902 + 0.224514i) q^{11} +4.85410 q^{12} +(-0.0729490 - 0.224514i) q^{13} +(-2.11803 - 1.53884i) q^{14} +(3.04508 - 9.37181i) q^{16} +(0.354102 - 1.08981i) q^{17} +(-2.11803 + 1.53884i) q^{18} +(-4.73607 - 3.44095i) q^{19} +1.00000 q^{21} +(-3.23607 - 8.05748i) q^{22} -0.236068 q^{23} +(2.30902 + 7.10642i) q^{24} +(0.500000 - 0.363271i) q^{26} +(0.309017 - 0.951057i) q^{27} +(1.50000 - 4.61653i) q^{28} +(4.85410 - 3.52671i) q^{29} +(-1.88197 - 5.79210i) q^{31} +10.8541 q^{32} +(2.80902 + 1.76336i) q^{33} +3.00000 q^{34} +(-3.92705 - 2.85317i) q^{36} +(-5.04508 + 3.66547i) q^{37} +(4.73607 - 14.5761i) q^{38} +(-0.0729490 + 0.224514i) q^{39} +(-0.190983 - 0.138757i) q^{41} +(0.809017 + 2.48990i) q^{42} +6.70820 q^{43} +(12.3541 - 10.3229i) q^{44} +(-0.190983 - 0.587785i) q^{46} +(-8.16312 - 5.93085i) q^{47} +(-7.97214 + 5.79210i) q^{48} +(-1.85410 + 5.70634i) q^{49} +(-0.927051 + 0.673542i) q^{51} +(0.927051 + 0.673542i) q^{52} +(0.118034 + 0.363271i) q^{53} +2.61803 q^{54} +7.47214 q^{56} +(1.80902 + 5.56758i) q^{57} +(12.7082 + 9.23305i) q^{58} +(-5.97214 + 4.33901i) q^{59} +(-3.57295 + 10.9964i) q^{61} +(12.8992 - 9.37181i) q^{62} +(-0.809017 - 0.587785i) q^{63} +(2.69098 + 8.28199i) q^{64} +(-2.11803 + 8.42075i) q^{66} -1.85410 q^{67} +(1.71885 + 5.29007i) q^{68} +(0.190983 + 0.138757i) q^{69} +(3.19098 - 9.82084i) q^{71} +(2.30902 - 7.10642i) q^{72} +(-4.61803 + 3.35520i) q^{73} +(-13.2082 - 9.59632i) q^{74} +28.4164 q^{76} +(2.54508 - 2.12663i) q^{77} -0.618034 q^{78} +(3.39919 + 10.4616i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(0.190983 - 0.587785i) q^{82} +(-0.454915 + 1.40008i) q^{83} +(-3.92705 + 2.85317i) q^{84} +(5.42705 + 16.7027i) q^{86} -6.00000 q^{87} +(20.9894 + 13.1760i) q^{88} -8.23607 q^{89} +(0.190983 + 0.138757i) q^{91} +(0.927051 - 0.673542i) q^{92} +(-1.88197 + 5.79210i) q^{93} +(8.16312 - 25.1235i) q^{94} +(-8.78115 - 6.37988i) q^{96} +(-2.42705 - 7.46969i) q^{97} -15.7082 q^{98} +(-1.23607 - 3.07768i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - 9 q^{4} + q^{6} - q^{7} - 13 q^{8} - q^{9} - 11 q^{11} + 6 q^{12} - 7 q^{13} - 4 q^{14} + q^{16} - 12 q^{17} - 4 q^{18} - 10 q^{19} + 4 q^{21} - 4 q^{22} + 8 q^{23} + 7 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 + 2.48990i 0.572061 + 1.76062i 0.645974 + 0.763359i \(0.276451\pi\)
−0.0739128 + 0.997265i \(0.523549\pi\)
\(3\) −0.809017 − 0.587785i −0.467086 − 0.339358i
\(4\) −3.92705 + 2.85317i −1.96353 + 1.42658i
\(5\) 0 0
\(6\) 0.809017 − 2.48990i 0.330280 − 1.01650i
\(7\) −0.809017 + 0.587785i −0.305780 + 0.222162i −0.730084 − 0.683358i \(-0.760519\pi\)
0.424304 + 0.905520i \(0.360519\pi\)
\(8\) −6.04508 − 4.39201i −2.13726 − 1.55281i
\(9\) 0.309017 + 0.951057i 0.103006 + 0.317019i
\(10\) 0 0
\(11\) −3.30902 + 0.224514i −0.997706 + 0.0676935i
\(12\) 4.85410 1.40126
\(13\) −0.0729490 − 0.224514i −0.0202324 − 0.0622690i 0.940431 − 0.339986i \(-0.110422\pi\)
−0.960663 + 0.277717i \(0.910422\pi\)
\(14\) −2.11803 − 1.53884i −0.566068 − 0.411273i
\(15\) 0 0
\(16\) 3.04508 − 9.37181i 0.761271 − 2.34295i
\(17\) 0.354102 − 1.08981i 0.0858823 − 0.264319i −0.898888 − 0.438178i \(-0.855624\pi\)
0.984770 + 0.173860i \(0.0556239\pi\)
\(18\) −2.11803 + 1.53884i −0.499225 + 0.362708i
\(19\) −4.73607 − 3.44095i −1.08653 − 0.789409i −0.107719 − 0.994181i \(-0.534355\pi\)
−0.978810 + 0.204772i \(0.934355\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −3.23607 − 8.05748i −0.689932 − 1.71786i
\(23\) −0.236068 −0.0492236 −0.0246118 − 0.999697i \(-0.507835\pi\)
−0.0246118 + 0.999697i \(0.507835\pi\)
\(24\) 2.30902 + 7.10642i 0.471326 + 1.45059i
\(25\) 0 0
\(26\) 0.500000 − 0.363271i 0.0980581 − 0.0712434i
\(27\) 0.309017 − 0.951057i 0.0594703 − 0.183031i
\(28\) 1.50000 − 4.61653i 0.283473 − 0.872441i
\(29\) 4.85410 − 3.52671i 0.901384 − 0.654894i −0.0374370 − 0.999299i \(-0.511919\pi\)
0.938821 + 0.344405i \(0.111919\pi\)
\(30\) 0 0
\(31\) −1.88197 − 5.79210i −0.338011 − 1.04029i −0.965220 − 0.261440i \(-0.915803\pi\)
0.627209 − 0.778851i \(-0.284197\pi\)
\(32\) 10.8541 1.91875
\(33\) 2.80902 + 1.76336i 0.488987 + 0.306961i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −3.92705 − 2.85317i −0.654508 − 0.475528i
\(37\) −5.04508 + 3.66547i −0.829407 + 0.602599i −0.919391 − 0.393344i \(-0.871318\pi\)
0.0899846 + 0.995943i \(0.471318\pi\)
\(38\) 4.73607 − 14.5761i 0.768292 − 2.36456i
\(39\) −0.0729490 + 0.224514i −0.0116812 + 0.0359510i
\(40\) 0 0
\(41\) −0.190983 − 0.138757i −0.0298265 − 0.0216702i 0.572772 − 0.819715i \(-0.305868\pi\)
−0.602599 + 0.798044i \(0.705868\pi\)
\(42\) 0.809017 + 2.48990i 0.124834 + 0.384200i
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) 12.3541 − 10.3229i 1.86245 − 1.55623i
\(45\) 0 0
\(46\) −0.190983 − 0.587785i −0.0281589 − 0.0866642i
\(47\) −8.16312 − 5.93085i −1.19071 − 0.865104i −0.197374 − 0.980328i \(-0.563241\pi\)
−0.993339 + 0.115224i \(0.963241\pi\)
\(48\) −7.97214 + 5.79210i −1.15068 + 0.836017i
\(49\) −1.85410 + 5.70634i −0.264872 + 0.815191i
\(50\) 0 0
\(51\) −0.927051 + 0.673542i −0.129813 + 0.0943147i
\(52\) 0.927051 + 0.673542i 0.128559 + 0.0934035i
\(53\) 0.118034 + 0.363271i 0.0162132 + 0.0498991i 0.958836 − 0.283961i \(-0.0916486\pi\)
−0.942623 + 0.333860i \(0.891649\pi\)
\(54\) 2.61803 0.356269
\(55\) 0 0
\(56\) 7.47214 0.998506
\(57\) 1.80902 + 5.56758i 0.239610 + 0.737444i
\(58\) 12.7082 + 9.23305i 1.66867 + 1.21236i
\(59\) −5.97214 + 4.33901i −0.777506 + 0.564891i −0.904229 − 0.427047i \(-0.859554\pi\)
0.126724 + 0.991938i \(0.459554\pi\)
\(60\) 0 0
\(61\) −3.57295 + 10.9964i −0.457469 + 1.40795i 0.410742 + 0.911751i \(0.365270\pi\)
−0.868212 + 0.496194i \(0.834730\pi\)
\(62\) 12.8992 − 9.37181i 1.63820 − 1.19022i
\(63\) −0.809017 − 0.587785i −0.101927 − 0.0740540i
\(64\) 2.69098 + 8.28199i 0.336373 + 1.03525i
\(65\) 0 0
\(66\) −2.11803 + 8.42075i −0.260712 + 1.03652i
\(67\) −1.85410 −0.226515 −0.113257 − 0.993566i \(-0.536128\pi\)
−0.113257 + 0.993566i \(0.536128\pi\)
\(68\) 1.71885 + 5.29007i 0.208441 + 0.641515i
\(69\) 0.190983 + 0.138757i 0.0229917 + 0.0167044i
\(70\) 0 0
\(71\) 3.19098 − 9.82084i 0.378700 − 1.16552i −0.562248 − 0.826968i \(-0.690063\pi\)
0.940948 − 0.338550i \(-0.109937\pi\)
\(72\) 2.30902 − 7.10642i 0.272120 − 0.837500i
\(73\) −4.61803 + 3.35520i −0.540500 + 0.392696i −0.824271 − 0.566196i \(-0.808415\pi\)
0.283771 + 0.958892i \(0.408415\pi\)
\(74\) −13.2082 − 9.59632i −1.53542 − 1.11555i
\(75\) 0 0
\(76\) 28.4164 3.25959
\(77\) 2.54508 − 2.12663i 0.290039 − 0.242352i
\(78\) −0.618034 −0.0699786
\(79\) 3.39919 + 10.4616i 0.382438 + 1.17702i 0.938322 + 0.345764i \(0.112380\pi\)
−0.555883 + 0.831260i \(0.687620\pi\)
\(80\) 0 0
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 0.190983 − 0.587785i 0.0210905 − 0.0649100i
\(83\) −0.454915 + 1.40008i −0.0499334 + 0.153679i −0.972914 − 0.231167i \(-0.925746\pi\)
0.922981 + 0.384846i \(0.125746\pi\)
\(84\) −3.92705 + 2.85317i −0.428476 + 0.311306i
\(85\) 0 0
\(86\) 5.42705 + 16.7027i 0.585214 + 1.80110i
\(87\) −6.00000 −0.643268
\(88\) 20.9894 + 13.1760i 2.23747 + 1.40457i
\(89\) −8.23607 −0.873021 −0.436511 − 0.899699i \(-0.643786\pi\)
−0.436511 + 0.899699i \(0.643786\pi\)
\(90\) 0 0
\(91\) 0.190983 + 0.138757i 0.0200205 + 0.0145457i
\(92\) 0.927051 − 0.673542i 0.0966517 − 0.0702216i
\(93\) −1.88197 + 5.79210i −0.195151 + 0.600612i
\(94\) 8.16312 − 25.1235i 0.841961 − 2.59129i
\(95\) 0 0
\(96\) −8.78115 − 6.37988i −0.896223 − 0.651144i
\(97\) −2.42705 − 7.46969i −0.246430 − 0.758433i −0.995398 − 0.0958268i \(-0.969451\pi\)
0.748968 − 0.662606i \(-0.230549\pi\)
\(98\) −15.7082 −1.58677
\(99\) −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.n.c.676.1 4
5.2 odd 4 825.2.bx.d.49.1 8
5.3 odd 4 825.2.bx.d.49.2 8
5.4 even 2 33.2.e.b.16.1 ✓ 4
11.3 even 5 9075.2.a.cb.1.2 2
11.8 odd 10 9075.2.a.u.1.1 2
11.9 even 5 inner 825.2.n.c.526.1 4
15.14 odd 2 99.2.f.a.82.1 4
20.19 odd 2 528.2.y.b.49.1 4
45.4 even 6 891.2.n.c.379.1 8
45.14 odd 6 891.2.n.b.379.1 8
45.29 odd 6 891.2.n.b.676.1 8
45.34 even 6 891.2.n.c.676.1 8
55.4 even 10 363.2.e.k.124.1 4
55.9 even 10 33.2.e.b.31.1 yes 4
55.14 even 10 363.2.a.d.1.1 2
55.19 odd 10 363.2.a.i.1.2 2
55.24 odd 10 363.2.e.f.130.1 4
55.29 odd 10 363.2.e.b.124.1 4
55.39 odd 10 363.2.e.b.202.1 4
55.42 odd 20 825.2.bx.d.724.2 8
55.49 even 10 363.2.e.k.202.1 4
55.53 odd 20 825.2.bx.d.724.1 8
55.54 odd 2 363.2.e.f.148.1 4
165.14 odd 10 1089.2.a.t.1.2 2
165.74 even 10 1089.2.a.l.1.1 2
165.119 odd 10 99.2.f.a.64.1 4
220.19 even 10 5808.2.a.ci.1.1 2
220.119 odd 10 528.2.y.b.97.1 4
220.179 odd 10 5808.2.a.cj.1.1 2
495.119 odd 30 891.2.n.b.757.1 8
495.229 even 30 891.2.n.c.460.1 8
495.284 odd 30 891.2.n.b.460.1 8
495.394 even 30 891.2.n.c.757.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 5.4 even 2
33.2.e.b.31.1 yes 4 55.9 even 10
99.2.f.a.64.1 4 165.119 odd 10
99.2.f.a.82.1 4 15.14 odd 2
363.2.a.d.1.1 2 55.14 even 10
363.2.a.i.1.2 2 55.19 odd 10
363.2.e.b.124.1 4 55.29 odd 10
363.2.e.b.202.1 4 55.39 odd 10
363.2.e.f.130.1 4 55.24 odd 10
363.2.e.f.148.1 4 55.54 odd 2
363.2.e.k.124.1 4 55.4 even 10
363.2.e.k.202.1 4 55.49 even 10
528.2.y.b.49.1 4 20.19 odd 2
528.2.y.b.97.1 4 220.119 odd 10
825.2.n.c.526.1 4 11.9 even 5 inner
825.2.n.c.676.1 4 1.1 even 1 trivial
825.2.bx.d.49.1 8 5.2 odd 4
825.2.bx.d.49.2 8 5.3 odd 4
825.2.bx.d.724.1 8 55.53 odd 20
825.2.bx.d.724.2 8 55.42 odd 20
891.2.n.b.379.1 8 45.14 odd 6
891.2.n.b.460.1 8 495.284 odd 30
891.2.n.b.676.1 8 45.29 odd 6
891.2.n.b.757.1 8 495.119 odd 30
891.2.n.c.379.1 8 45.4 even 6
891.2.n.c.460.1 8 495.229 even 30
891.2.n.c.676.1 8 45.34 even 6
891.2.n.c.757.1 8 495.394 even 30
1089.2.a.l.1.1 2 165.74 even 10
1089.2.a.t.1.2 2 165.14 odd 10
5808.2.a.ci.1.1 2 220.19 even 10
5808.2.a.cj.1.1 2 220.179 odd 10
9075.2.a.u.1.1 2 11.8 odd 10
9075.2.a.cb.1.2 2 11.3 even 5