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,-6,0,-6,0,0,2,0,18] 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 49.2
Root \(0.587785 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 825.49
Dual form 825.2.bx.b.724.2

$q$-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+(0.587785 - 0.190983i) q^{2} +(-0.587785 + 0.809017i) q^{3} +(-1.30902 + 0.951057i) q^{4} +(-0.190983 + 0.587785i) q^{6} +(-1.76336 - 2.42705i) q^{7} +(-1.31433 + 1.80902i) q^{8} +(-0.309017 - 0.951057i) q^{9} +(1.69098 - 2.85317i) q^{11} -1.61803i q^{12} +(1.67760 - 0.545085i) q^{13} +(-1.50000 - 1.08981i) q^{14} +(0.572949 - 1.76336i) q^{16} +(1.53884 + 0.500000i) q^{17} +(-0.363271 - 0.500000i) q^{18} +(4.73607 + 3.44095i) q^{19} +3.00000 q^{21} +(0.449028 - 2.00000i) q^{22} -3.47214i q^{23} +(-0.690983 - 2.12663i) q^{24} +(0.881966 - 0.640786i) q^{26} +(0.951057 + 0.309017i) q^{27} +(4.61653 + 1.50000i) q^{28} +(3.61803 - 2.62866i) q^{29} +(0.881966 + 2.71441i) q^{31} -5.61803i q^{32} +(1.31433 + 3.04508i) q^{33} +1.00000 q^{34} +(1.30902 + 0.951057i) q^{36} +(-0.138757 - 0.190983i) q^{37} +(3.44095 + 1.11803i) q^{38} +(-0.545085 + 1.67760i) q^{39} +(9.66312 + 7.02067i) q^{41} +(1.76336 - 0.572949i) q^{42} -6.23607i q^{43} +(0.500000 + 5.34307i) q^{44} +(-0.663119 - 2.04087i) q^{46} +(0.951057 - 1.30902i) q^{47} +(1.08981 + 1.50000i) q^{48} +(-0.618034 + 1.90211i) q^{49} +(-1.30902 + 0.951057i) q^{51} +(-1.67760 + 2.30902i) q^{52} +(-9.14729 + 2.97214i) q^{53} +0.618034 q^{54} +6.70820 q^{56} +(-5.56758 + 1.80902i) q^{57} +(1.62460 - 2.23607i) q^{58} +(8.35410 - 6.06961i) q^{59} +(2.42705 - 7.46969i) q^{61} +(1.03681 + 1.42705i) q^{62} +(-1.76336 + 2.42705i) q^{63} +(0.0729490 + 0.224514i) q^{64} +(1.35410 + 1.53884i) q^{66} -9.56231i q^{67} +(-2.48990 + 0.809017i) q^{68} +(2.80902 + 2.04087i) q^{69} +(-1.71885 + 5.29007i) q^{71} +(2.12663 + 0.690983i) q^{72} +(-1.90211 - 2.61803i) q^{73} +(-0.118034 - 0.0857567i) q^{74} -9.47214 q^{76} +(-9.90659 + 0.927051i) q^{77} +1.09017i q^{78} +(-2.92705 - 9.00854i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(7.02067 + 2.28115i) q^{82} +(-0.673542 - 0.218847i) q^{83} +(-3.92705 + 2.85317i) q^{84} +(-1.19098 - 3.66547i) q^{86} +4.47214i q^{87} +(2.93893 + 6.80902i) q^{88} -0.527864 q^{89} +(-4.28115 - 3.11044i) q^{91} +(3.30220 + 4.54508i) q^{92} +(-2.71441 - 0.881966i) q^{93} +(0.309017 - 0.951057i) q^{94} +(4.54508 + 3.30220i) q^{96} +(13.3475 - 4.33688i) q^{97} +1.23607i q^{98} +(-3.23607 - 0.726543i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} - 6 q^{6} + 2 q^{9} + 18 q^{11} - 12 q^{14} + 18 q^{16} + 20 q^{19} + 24 q^{21} - 10 q^{24} + 16 q^{26} + 20 q^{29} + 16 q^{31} + 8 q^{34} + 6 q^{36} + 18 q^{39} + 46 q^{41} + 4 q^{44} + 26 q^{46}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.587785 − 0.190983i 0.415627 − 0.135045i −0.0937362 − 0.995597i \(-0.529881\pi\)
0.509363 + 0.860552i \(0.329881\pi\)
\(3\) −0.587785 + 0.809017i −0.339358 + 0.467086i
\(4\) −1.30902 + 0.951057i −0.654508 + 0.475528i
\(5\) 0 0
\(6\) −0.190983 + 0.587785i −0.0779685 + 0.239962i
\(7\) −1.76336 − 2.42705i −0.666486 − 0.917339i 0.333188 − 0.942860i \(-0.391875\pi\)
−0.999674 + 0.0255212i \(0.991875\pi\)
\(8\) −1.31433 + 1.80902i −0.464685 + 0.639584i
\(9\) −0.309017 − 0.951057i −0.103006 − 0.317019i
\(10\) 0 0
\(11\) 1.69098 − 2.85317i 0.509851 − 0.860263i
\(12\) − 1.61803i − 0.467086i
\(13\) 1.67760 − 0.545085i 0.465282 − 0.151179i −0.0669881 − 0.997754i \(-0.521339\pi\)
0.532270 + 0.846574i \(0.321339\pi\)
\(14\) −1.50000 − 1.08981i −0.400892 − 0.291265i
\(15\) 0 0
\(16\) 0.572949 − 1.76336i 0.143237 − 0.440839i
\(17\) 1.53884 + 0.500000i 0.373224 + 0.121268i 0.489622 − 0.871935i \(-0.337135\pi\)
−0.116398 + 0.993203i \(0.537135\pi\)
\(18\) −0.363271 − 0.500000i −0.0856239 − 0.117851i
\(19\) 4.73607 + 3.44095i 1.08653 + 0.789409i 0.978810 − 0.204772i \(-0.0656454\pi\)
0.107719 + 0.994181i \(0.465645\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0.449028 − 2.00000i 0.0957331 − 0.426401i
\(23\) − 3.47214i − 0.723990i −0.932180 − 0.361995i \(-0.882096\pi\)
0.932180 − 0.361995i \(-0.117904\pi\)
\(24\) −0.690983 − 2.12663i −0.141046 − 0.434096i
\(25\) 0 0
\(26\) 0.881966 − 0.640786i 0.172968 − 0.125668i
\(27\) 0.951057 + 0.309017i 0.183031 + 0.0594703i
\(28\) 4.61653 + 1.50000i 0.872441 + 0.283473i
\(29\) 3.61803 − 2.62866i 0.671852 − 0.488129i −0.198793 − 0.980042i \(-0.563702\pi\)
0.870645 + 0.491912i \(0.163702\pi\)
\(30\) 0 0
\(31\) 0.881966 + 2.71441i 0.158406 + 0.487523i 0.998490 − 0.0549331i \(-0.0174946\pi\)
−0.840084 + 0.542456i \(0.817495\pi\)
\(32\) − 5.61803i − 0.993137i
\(33\) 1.31433 + 3.04508i 0.228795 + 0.530081i
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.30902 + 0.951057i 0.218169 + 0.158509i
\(37\) −0.138757 − 0.190983i −0.0228116 − 0.0313974i 0.797459 − 0.603373i \(-0.206177\pi\)
−0.820270 + 0.571976i \(0.806177\pi\)
\(38\) 3.44095 + 1.11803i 0.558197 + 0.181369i
\(39\) −0.545085 + 1.67760i −0.0872835 + 0.268631i
\(40\) 0 0
\(41\) 9.66312 + 7.02067i 1.50913 + 1.09644i 0.966563 + 0.256428i \(0.0825458\pi\)
0.542562 + 0.840015i \(0.317454\pi\)
\(42\) 1.76336 − 0.572949i 0.272092 − 0.0884080i
\(43\) − 6.23607i − 0.950991i −0.879718 − 0.475496i \(-0.842269\pi\)
0.879718 − 0.475496i \(-0.157731\pi\)
\(44\) 0.500000 + 5.34307i 0.0753778 + 0.805498i
\(45\) 0 0
\(46\) −0.663119 − 2.04087i −0.0977716 − 0.300910i
\(47\) 0.951057 − 1.30902i 0.138726 − 0.190940i −0.734001 − 0.679148i \(-0.762349\pi\)
0.872727 + 0.488208i \(0.162349\pi\)
\(48\) 1.08981 + 1.50000i 0.157301 + 0.216506i
\(49\) −0.618034 + 1.90211i −0.0882906 + 0.271730i
\(50\) 0 0
\(51\) −1.30902 + 0.951057i −0.183299 + 0.133175i
\(52\) −1.67760 + 2.30902i −0.232641 + 0.320203i
\(53\) −9.14729 + 2.97214i −1.25648 + 0.408254i −0.860238 − 0.509893i \(-0.829685\pi\)
−0.396240 + 0.918147i \(0.629685\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) 6.70820 0.896421
\(57\) −5.56758 + 1.80902i −0.737444 + 0.239610i
\(58\) 1.62460 − 2.23607i 0.213320 − 0.293610i
\(59\) 8.35410 − 6.06961i 1.08761 − 0.790196i 0.108617 − 0.994084i \(-0.465358\pi\)
0.978994 + 0.203888i \(0.0653577\pi\)
\(60\) 0 0
\(61\) 2.42705 − 7.46969i 0.310752 − 0.956396i −0.666716 − 0.745312i \(-0.732301\pi\)
0.977468 − 0.211084i \(-0.0676995\pi\)
\(62\) 1.03681 + 1.42705i 0.131675 + 0.181236i
\(63\) −1.76336 + 2.42705i −0.222162 + 0.305780i
\(64\) 0.0729490 + 0.224514i 0.00911863 + 0.0280642i
\(65\) 0 0
\(66\) 1.35410 + 1.53884i 0.166678 + 0.189418i
\(67\) − 9.56231i − 1.16822i −0.811674 − 0.584111i \(-0.801443\pi\)
0.811674 − 0.584111i \(-0.198557\pi\)
\(68\) −2.48990 + 0.809017i −0.301945 + 0.0981077i
\(69\) 2.80902 + 2.04087i 0.338166 + 0.245692i
\(70\) 0 0
\(71\) −1.71885 + 5.29007i −0.203990 + 0.627815i 0.795764 + 0.605607i \(0.207070\pi\)
−0.999753 + 0.0222083i \(0.992930\pi\)
\(72\) 2.12663 + 0.690983i 0.250625 + 0.0814331i
\(73\) −1.90211 − 2.61803i −0.222625 − 0.306418i 0.683065 − 0.730358i \(-0.260647\pi\)
−0.905690 + 0.423940i \(0.860647\pi\)
\(74\) −0.118034 − 0.0857567i −0.0137212 − 0.00996902i
\(75\) 0 0
\(76\) −9.47214 −1.08653
\(77\) −9.90659 + 0.927051i −1.12896 + 0.105647i
\(78\) 1.09017i 0.123437i
\(79\) −2.92705 − 9.00854i −0.329319 − 1.01354i −0.969453 − 0.245276i \(-0.921121\pi\)
0.640134 − 0.768263i \(-0.278879\pi\)
\(80\) 0 0
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 7.02067 + 2.28115i 0.775303 + 0.251911i
\(83\) −0.673542 − 0.218847i −0.0739308 − 0.0240216i 0.271818 − 0.962349i \(-0.412375\pi\)
−0.345749 + 0.938327i \(0.612375\pi\)
\(84\) −3.92705 + 2.85317i −0.428476 + 0.311306i
\(85\) 0 0
\(86\) −1.19098 − 3.66547i −0.128427 − 0.395258i
\(87\) 4.47214i 0.479463i
\(88\) 2.93893 + 6.80902i 0.313291 + 0.725844i
\(89\) −0.527864 −0.0559535 −0.0279767 − 0.999609i \(-0.508906\pi\)
−0.0279767 + 0.999609i \(0.508906\pi\)
\(90\) 0 0
\(91\) −4.28115 − 3.11044i −0.448787 − 0.326063i
\(92\) 3.30220 + 4.54508i 0.344278 + 0.473858i
\(93\) −2.71441 − 0.881966i −0.281471 − 0.0914556i
\(94\) 0.309017 − 0.951057i 0.0318727 − 0.0980940i
\(95\) 0 0
\(96\) 4.54508 + 3.30220i 0.463881 + 0.337029i
\(97\) 13.3475 − 4.33688i 1.35524 − 0.440344i 0.460788 − 0.887510i \(-0.347567\pi\)
0.894451 + 0.447167i \(0.147567\pi\)
\(98\) 1.23607i 0.124862i
\(99\) −3.23607 − 0.726543i −0.325237 − 0.0730203i
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.b.49.2 8
5.2 odd 4 825.2.n.f.676.1 4
5.3 odd 4 33.2.e.a.16.1 ✓ 4
5.4 even 2 inner 825.2.bx.b.49.1 8
11.9 even 5 inner 825.2.bx.b.724.1 8
15.8 even 4 99.2.f.b.82.1 4
20.3 even 4 528.2.y.f.49.1 4
45.13 odd 12 891.2.n.d.379.1 8
45.23 even 12 891.2.n.a.379.1 8
45.38 even 12 891.2.n.a.676.1 8
45.43 odd 12 891.2.n.d.676.1 8
55.3 odd 20 363.2.a.h.1.1 2
55.8 even 20 363.2.a.e.1.2 2
55.9 even 10 inner 825.2.bx.b.724.2 8
55.13 even 20 363.2.e.j.130.1 4
55.18 even 20 363.2.e.c.124.1 4
55.28 even 20 363.2.e.c.202.1 4
55.38 odd 20 363.2.e.h.202.1 4
55.42 odd 20 825.2.n.f.526.1 4
55.43 even 4 363.2.e.j.148.1 4
55.47 odd 20 9075.2.a.x.1.2 2
55.48 odd 20 363.2.e.h.124.1 4
55.52 even 20 9075.2.a.bv.1.1 2
55.53 odd 20 33.2.e.a.31.1 yes 4
165.8 odd 20 1089.2.a.s.1.1 2
165.53 even 20 99.2.f.b.64.1 4
165.113 even 20 1089.2.a.m.1.2 2
220.3 even 20 5808.2.a.bl.1.1 2
220.63 odd 20 5808.2.a.bm.1.1 2
220.163 even 20 528.2.y.f.97.1 4
495.218 even 60 891.2.n.a.757.1 8
495.328 odd 60 891.2.n.d.460.1 8
495.383 even 60 891.2.n.a.460.1 8
495.493 odd 60 891.2.n.d.757.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.16.1 ✓ 4 5.3 odd 4
33.2.e.a.31.1 yes 4 55.53 odd 20
99.2.f.b.64.1 4 165.53 even 20
99.2.f.b.82.1 4 15.8 even 4
363.2.a.e.1.2 2 55.8 even 20
363.2.a.h.1.1 2 55.3 odd 20
363.2.e.c.124.1 4 55.18 even 20
363.2.e.c.202.1 4 55.28 even 20
363.2.e.h.124.1 4 55.48 odd 20
363.2.e.h.202.1 4 55.38 odd 20
363.2.e.j.130.1 4 55.13 even 20
363.2.e.j.148.1 4 55.43 even 4
528.2.y.f.49.1 4 20.3 even 4
528.2.y.f.97.1 4 220.163 even 20
825.2.n.f.526.1 4 55.42 odd 20
825.2.n.f.676.1 4 5.2 odd 4
825.2.bx.b.49.1 8 5.4 even 2 inner
825.2.bx.b.49.2 8 1.1 even 1 trivial
825.2.bx.b.724.1 8 11.9 even 5 inner
825.2.bx.b.724.2 8 55.9 even 10 inner
891.2.n.a.379.1 8 45.23 even 12
891.2.n.a.460.1 8 495.383 even 60
891.2.n.a.676.1 8 45.38 even 12
891.2.n.a.757.1 8 495.218 even 60
891.2.n.d.379.1 8 45.13 odd 12
891.2.n.d.460.1 8 495.328 odd 60
891.2.n.d.676.1 8 45.43 odd 12
891.2.n.d.757.1 8 495.493 odd 60
1089.2.a.m.1.2 2 165.113 even 20
1089.2.a.s.1.1 2 165.8 odd 20
5808.2.a.bl.1.1 2 220.3 even 20
5808.2.a.bm.1.1 2 220.63 odd 20
9075.2.a.x.1.2 2 55.47 odd 20
9075.2.a.bv.1.1 2 55.52 even 20