Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(251,425)] 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("425.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 119x^{8} + 364x^{6} + 519x^{4} + 278x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 251.4
Root \(1.21647i\) of defining polynomial
Character \(\chi\) \(=\) 425.251
Dual form 425.2.e.c.276.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.21647i q^{2} +(-2.23861 - 2.23861i) q^{3} +0.520205 q^{4} +(2.72320 - 2.72320i) q^{6} +(-0.679950 + 0.679950i) q^{7} +3.06575i q^{8} +7.02277i q^{9} +(2.22126 - 2.22126i) q^{11} +(-1.16454 - 1.16454i) q^{12} +2.02534 q^{13} +(-0.827137 - 0.827137i) q^{14} -2.68898 q^{16} +(3.56346 - 2.07407i) q^{17} -8.54297 q^{18} -5.28609i q^{19} +3.04429 q^{21} +(2.70209 + 2.70209i) q^{22} +(6.01797 - 6.01797i) q^{23} +(6.86302 - 6.86302i) q^{24} +2.46376i q^{26} +(9.00542 - 9.00542i) q^{27} +(-0.353714 + 0.353714i) q^{28} +(-0.857606 - 0.857606i) q^{29} +(3.97529 + 3.97529i) q^{31} +2.86045i q^{32} -9.94509 q^{33} +(2.52305 + 4.33483i) q^{34} +3.65328i q^{36} +(-5.84955 - 5.84955i) q^{37} +6.43037 q^{38} +(-4.53395 - 4.53395i) q^{39} +(-1.04325 + 1.04325i) q^{41} +3.70328i q^{42} +7.01089i q^{43} +(1.15551 - 1.15551i) q^{44} +(7.32067 + 7.32067i) q^{46} +10.9275 q^{47} +(6.01957 + 6.01957i) q^{48} +6.07534i q^{49} +(-12.6202 - 3.33415i) q^{51} +1.05359 q^{52} +5.24568i q^{53} +(10.9548 + 10.9548i) q^{54} +(-2.08456 - 2.08456i) q^{56} +(-11.8335 + 11.8335i) q^{57} +(1.04325 - 1.04325i) q^{58} -13.8346i q^{59} +(2.70557 - 2.70557i) q^{61} +(-4.83581 + 4.83581i) q^{62} +(-4.77513 - 4.77513i) q^{63} -8.85759 q^{64} -12.0979i q^{66} -2.37336 q^{67} +(1.85373 - 1.07894i) q^{68} -26.9438 q^{69} +(2.82261 + 2.82261i) q^{71} -21.5300 q^{72} +(5.51272 + 5.51272i) q^{73} +(7.11579 - 7.11579i) q^{74} -2.74985i q^{76} +3.02069i q^{77} +(5.51540 - 5.51540i) q^{78} +(-4.74215 + 4.74215i) q^{79} -19.2510 q^{81} +(-1.26908 - 1.26908i) q^{82} +0.171341i q^{83} +1.58365 q^{84} -8.52853 q^{86} +3.83969i q^{87} +(6.80983 + 6.80983i) q^{88} +1.32080 q^{89} +(-1.37713 + 1.37713i) q^{91} +(3.13058 - 3.13058i) q^{92} -17.7982i q^{93} +13.2930i q^{94} +(6.40343 - 6.40343i) q^{96} +(1.33838 + 1.33838i) q^{97} -7.39045 q^{98} +(15.5994 + 15.5994i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 12 q^{4} + 6 q^{6} - 4 q^{11} - 4 q^{12} + 12 q^{13} + 14 q^{14} + 4 q^{16} + 6 q^{17} - 4 q^{18} + 8 q^{21} + 10 q^{22} + 12 q^{23} - 8 q^{24} + 22 q^{27} - 34 q^{28} - 6 q^{29} - 6 q^{31}+ \cdots + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21647i 0.860173i 0.902788 + 0.430086i \(0.141517\pi\)
−0.902788 + 0.430086i \(0.858483\pi\)
\(3\) −2.23861 − 2.23861i −1.29246 − 1.29246i −0.933259 − 0.359204i \(-0.883048\pi\)
−0.359204 − 0.933259i \(-0.616952\pi\)
\(4\) 0.520205 0.260103
\(5\) 0 0
\(6\) 2.72320 − 2.72320i 1.11174 − 1.11174i
\(7\) −0.679950 + 0.679950i −0.256997 + 0.256997i −0.823832 − 0.566835i \(-0.808168\pi\)
0.566835 + 0.823832i \(0.308168\pi\)
\(8\) 3.06575i 1.08391i
\(9\) 7.02277i 2.34092i
\(10\) 0 0
\(11\) 2.22126 − 2.22126i 0.669736 − 0.669736i −0.287919 − 0.957655i \(-0.592963\pi\)
0.957655 + 0.287919i \(0.0929634\pi\)
\(12\) −1.16454 − 1.16454i −0.336173 − 0.336173i
\(13\) 2.02534 0.561728 0.280864 − 0.959748i \(-0.409379\pi\)
0.280864 + 0.959748i \(0.409379\pi\)
\(14\) −0.827137 − 0.827137i −0.221062 − 0.221062i
\(15\) 0 0
\(16\) −2.68898 −0.672244
\(17\) 3.56346 − 2.07407i 0.864265 − 0.503037i
\(18\) −8.54297 −2.01360
\(19\) − 5.28609i − 1.21271i −0.795193 − 0.606357i \(-0.792630\pi\)
0.795193 − 0.606357i \(-0.207370\pi\)
\(20\) 0 0
\(21\) 3.04429 0.664318
\(22\) 2.70209 + 2.70209i 0.576088 + 0.576088i
\(23\) 6.01797 − 6.01797i 1.25483 − 1.25483i 0.301307 − 0.953527i \(-0.402577\pi\)
0.953527 − 0.301307i \(-0.0974230\pi\)
\(24\) 6.86302 − 6.86302i 1.40091 − 1.40091i
\(25\) 0 0
\(26\) 2.46376i 0.483183i
\(27\) 9.00542 − 9.00542i 1.73309 − 1.73309i
\(28\) −0.353714 + 0.353714i −0.0668456 + 0.0668456i
\(29\) −0.857606 − 0.857606i −0.159253 − 0.159253i 0.622982 − 0.782236i \(-0.285921\pi\)
−0.782236 + 0.622982i \(0.785921\pi\)
\(30\) 0 0
\(31\) 3.97529 + 3.97529i 0.713982 + 0.713982i 0.967366 − 0.253384i \(-0.0815435\pi\)
−0.253384 + 0.967366i \(0.581543\pi\)
\(32\) 2.86045i 0.505660i
\(33\) −9.94509 −1.73122
\(34\) 2.52305 + 4.33483i 0.432699 + 0.743417i
\(35\) 0 0
\(36\) 3.65328i 0.608880i
\(37\) −5.84955 − 5.84955i −0.961660 − 0.961660i 0.0376315 − 0.999292i \(-0.488019\pi\)
−0.999292 + 0.0376315i \(0.988019\pi\)
\(38\) 6.43037 1.04314
\(39\) −4.53395 − 4.53395i −0.726013 − 0.726013i
\(40\) 0 0
\(41\) −1.04325 + 1.04325i −0.162928 + 0.162928i −0.783863 − 0.620934i \(-0.786753\pi\)
0.620934 + 0.783863i \(0.286753\pi\)
\(42\) 3.70328i 0.571428i
\(43\) 7.01089i 1.06915i 0.845121 + 0.534576i \(0.179529\pi\)
−0.845121 + 0.534576i \(0.820471\pi\)
\(44\) 1.15551 − 1.15551i 0.174200 − 0.174200i
\(45\) 0 0
\(46\) 7.32067 + 7.32067i 1.07937 + 1.07937i
\(47\) 10.9275 1.59394 0.796970 − 0.604019i \(-0.206435\pi\)
0.796970 + 0.604019i \(0.206435\pi\)
\(48\) 6.01957 + 6.01957i 0.868851 + 0.868851i
\(49\) 6.07534i 0.867905i
\(50\) 0 0
\(51\) −12.6202 − 3.33415i −1.76719 − 0.466874i
\(52\) 1.05359 0.146107
\(53\) 5.24568i 0.720550i 0.932846 + 0.360275i \(0.117317\pi\)
−0.932846 + 0.360275i \(0.882683\pi\)
\(54\) 10.9548 + 10.9548i 1.49076 + 1.49076i
\(55\) 0 0
\(56\) −2.08456 − 2.08456i −0.278561 − 0.278561i
\(57\) −11.8335 + 11.8335i −1.56739 + 1.56739i
\(58\) 1.04325 − 1.04325i 0.136985 − 0.136985i
\(59\) − 13.8346i − 1.80111i −0.434747 − 0.900553i \(-0.643162\pi\)
0.434747 − 0.900553i \(-0.356838\pi\)
\(60\) 0 0
\(61\) 2.70557 − 2.70557i 0.346412 − 0.346412i −0.512359 − 0.858771i \(-0.671228\pi\)
0.858771 + 0.512359i \(0.171228\pi\)
\(62\) −4.83581 + 4.83581i −0.614148 + 0.614148i
\(63\) −4.77513 − 4.77513i −0.601610 − 0.601610i
\(64\) −8.85759 −1.10720
\(65\) 0 0
\(66\) − 12.0979i − 1.48915i
\(67\) −2.37336 −0.289953 −0.144976 − 0.989435i \(-0.546311\pi\)
−0.144976 + 0.989435i \(0.546311\pi\)
\(68\) 1.85373 − 1.07894i 0.224798 − 0.130841i
\(69\) −26.9438 −3.24366
\(70\) 0 0
\(71\) 2.82261 + 2.82261i 0.334983 + 0.334983i 0.854475 − 0.519492i \(-0.173879\pi\)
−0.519492 + 0.854475i \(0.673879\pi\)
\(72\) −21.5300 −2.53734
\(73\) 5.51272 + 5.51272i 0.645215 + 0.645215i 0.951833 − 0.306618i \(-0.0991973\pi\)
−0.306618 + 0.951833i \(0.599197\pi\)
\(74\) 7.11579 − 7.11579i 0.827194 − 0.827194i
\(75\) 0 0
\(76\) − 2.74985i − 0.315430i
\(77\) 3.02069i 0.344240i
\(78\) 5.51540 − 5.51540i 0.624496 − 0.624496i
\(79\) −4.74215 + 4.74215i −0.533534 + 0.533534i −0.921622 − 0.388089i \(-0.873135\pi\)
0.388089 + 0.921622i \(0.373135\pi\)
\(80\) 0 0
\(81\) −19.2510 −2.13900
\(82\) −1.26908 − 1.26908i −0.140147 − 0.140147i
\(83\) 0.171341i 0.0188071i 0.999956 + 0.00940355i \(0.00299329\pi\)
−0.999956 + 0.00940355i \(0.997007\pi\)
\(84\) 1.58365 0.172791
\(85\) 0 0
\(86\) −8.52853 −0.919655
\(87\) 3.83969i 0.411658i
\(88\) 6.80983 + 6.80983i 0.725930 + 0.725930i
\(89\) 1.32080 0.140004 0.0700020 − 0.997547i \(-0.477699\pi\)
0.0700020 + 0.997547i \(0.477699\pi\)
\(90\) 0 0
\(91\) −1.37713 + 1.37713i −0.144362 + 0.144362i
\(92\) 3.13058 − 3.13058i 0.326386 − 0.326386i
\(93\) − 17.7982i − 1.84559i
\(94\) 13.2930i 1.37106i
\(95\) 0 0
\(96\) 6.40343 − 6.40343i 0.653547 − 0.653547i
\(97\) 1.33838 + 1.33838i 0.135892 + 0.135892i 0.771781 − 0.635889i \(-0.219366\pi\)
−0.635889 + 0.771781i \(0.719366\pi\)
\(98\) −7.39045 −0.746548
\(99\) 15.5994 + 15.5994i 1.56780 + 1.56780i
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 425.2.e.c.251.4 ✓ 12
5.2 odd 4 425.2.j.a.149.3 12
5.3 odd 4 425.2.j.d.149.4 12
5.4 even 2 425.2.e.e.251.3 yes 12
17.2 even 8 7225.2.a.bm.1.5 12
17.4 even 4 inner 425.2.e.c.276.3 yes 12
17.15 even 8 7225.2.a.bm.1.6 12
85.4 even 4 425.2.e.e.276.4 yes 12
85.19 even 8 7225.2.a.br.1.8 12
85.38 odd 4 425.2.j.a.174.3 12
85.49 even 8 7225.2.a.br.1.7 12
85.72 odd 4 425.2.j.d.174.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.e.c.251.4 ✓ 12 1.1 even 1 trivial
425.2.e.c.276.3 yes 12 17.4 even 4 inner
425.2.e.e.251.3 yes 12 5.4 even 2
425.2.e.e.276.4 yes 12 85.4 even 4
425.2.j.a.149.3 12 5.2 odd 4
425.2.j.a.174.3 12 85.38 odd 4
425.2.j.d.149.4 12 5.3 odd 4
425.2.j.d.174.4 12 85.72 odd 4
7225.2.a.bm.1.5 12 17.2 even 8
7225.2.a.bm.1.6 12 17.15 even 8
7225.2.a.br.1.7 12 85.49 even 8
7225.2.a.br.1.8 12 85.19 even 8