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Properties

Label 625.2.e.j.374.3
Level $625$
Weight $2$
Character 625.374
Analytic conductor $4.991$
Analytic rank $0$
Dimension $32$
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] = [625,2,Mod(124,625)] 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("625.124"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(625, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,6,0,14,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 374.3
Character \(\chi\) \(=\) 625.374
Dual form 625.2.e.j.249.3

$q$-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+(-0.989484 - 1.36191i) q^{2} +(0.675574 + 0.219507i) q^{3} +(-0.257680 + 0.793058i) q^{4} +(-0.369521 - 1.13727i) q^{6} -4.59110i q^{7} +(-1.86699 + 0.606623i) q^{8} +(-2.01883 - 1.46677i) q^{9} +(-3.16713 + 2.30105i) q^{11} +(-0.348164 + 0.479206i) q^{12} +(0.336765 - 0.463518i) q^{13} +(-6.25266 + 4.54282i) q^{14} +(4.02276 + 2.92270i) q^{16} +(0.221226 - 0.0718808i) q^{17} +4.20081i q^{18} +(1.71507 + 5.27846i) q^{19} +(1.00778 - 3.10163i) q^{21} +(6.26765 + 2.03648i) q^{22} +(-2.89935 - 3.99061i) q^{23} -1.39445 q^{24} -0.964492 q^{26} +(-2.29449 - 3.15809i) q^{27} +(3.64101 + 1.18304i) q^{28} +(-1.27643 + 3.92845i) q^{29} +(-1.08011 - 3.32424i) q^{31} -4.44444i q^{32} +(-2.64473 + 0.859324i) q^{33} +(-0.316795 - 0.230165i) q^{34} +(1.68345 - 1.22310i) q^{36} +(-3.18373 + 4.38203i) q^{37} +(5.49173 - 7.55872i) q^{38} +(0.329255 - 0.239218i) q^{39} +(-8.43214 - 6.12630i) q^{41} +(-5.22132 + 1.69651i) q^{42} -1.38833i q^{43} +(-1.00876 - 3.10465i) q^{44} +(-2.56598 + 7.89729i) q^{46} +(0.875370 + 0.284425i) q^{47} +(2.07611 + 2.85753i) q^{48} -14.0782 q^{49} +0.165233 q^{51} +(0.280819 + 0.386514i) q^{52} +(-1.17092 - 0.380455i) q^{53} +(-2.03067 + 6.24976i) q^{54} +(2.78507 + 8.57157i) q^{56} +3.94246i q^{57} +(6.61320 - 2.14876i) q^{58} +(3.64689 + 2.64962i) q^{59} +(9.43214 - 6.85285i) q^{61} +(-3.45856 + 4.76029i) q^{62} +(-6.73409 + 9.26868i) q^{63} +(1.99259 - 1.44770i) q^{64} +(3.78724 + 2.75159i) q^{66} +(2.80987 - 0.912982i) q^{67} +0.193968i q^{68} +(-1.08276 - 3.33238i) q^{69} +(0.990558 - 3.04862i) q^{71} +(4.65893 + 1.51378i) q^{72} +(-6.04961 - 8.32657i) q^{73} +9.11816 q^{74} -4.62806 q^{76} +(10.5644 + 14.5406i) q^{77} +(-0.651586 - 0.211713i) q^{78} +(2.97123 - 9.14449i) q^{79} +(1.45651 + 4.48267i) q^{81} +17.5457i q^{82} +(-9.97030 + 3.23955i) q^{83} +(2.20009 + 1.59846i) q^{84} +(-1.89077 + 1.37373i) q^{86} +(-1.72465 + 2.37377i) q^{87} +(4.51714 - 6.21731i) q^{88} +(5.87207 - 4.26631i) q^{89} +(-2.12806 - 1.54612i) q^{91} +(3.91189 - 1.27105i) q^{92} -2.48286i q^{93} +(-0.478804 - 1.47361i) q^{94} +(0.975587 - 3.00255i) q^{96} +(-7.91252 - 2.57093i) q^{97} +(13.9302 + 19.1733i) q^{98} +9.76903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 6 q^{4} + 14 q^{6} + 24 q^{9} - 6 q^{11} + 2 q^{14} + 2 q^{16} - 20 q^{19} + 14 q^{21} - 20 q^{24} + 44 q^{26} - 16 q^{31} + 12 q^{34} + 2 q^{36} - 2 q^{39} - 16 q^{41} + 62 q^{44} + 84 q^{46} + 16 q^{49}+ \cdots + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.989484 − 1.36191i −0.699671 − 0.963014i −0.999958 − 0.00916528i \(-0.997083\pi\)
0.300287 − 0.953849i \(-0.402917\pi\)
\(3\) 0.675574 + 0.219507i 0.390043 + 0.126733i 0.497472 − 0.867480i \(-0.334262\pi\)
−0.107429 + 0.994213i \(0.534262\pi\)
\(4\) −0.257680 + 0.793058i −0.128840 + 0.396529i
\(5\) 0 0
\(6\) −0.369521 − 1.13727i −0.150856 − 0.464288i
\(7\) − 4.59110i − 1.73527i −0.497198 − 0.867637i \(-0.665638\pi\)
0.497198 − 0.867637i \(-0.334362\pi\)
\(8\) −1.86699 + 0.606623i −0.660082 + 0.214474i
\(9\) −2.01883 − 1.46677i −0.672945 − 0.488923i
\(10\) 0 0
\(11\) −3.16713 + 2.30105i −0.954926 + 0.693794i −0.951967 − 0.306201i \(-0.900942\pi\)
−0.00295900 + 0.999996i \(0.500942\pi\)
\(12\) −0.348164 + 0.479206i −0.100506 + 0.138335i
\(13\) 0.336765 − 0.463518i 0.0934019 − 0.128557i −0.759758 − 0.650205i \(-0.774683\pi\)
0.853160 + 0.521649i \(0.174683\pi\)
\(14\) −6.25266 + 4.54282i −1.67109 + 1.21412i
\(15\) 0 0
\(16\) 4.02276 + 2.92270i 1.00569 + 0.730676i
\(17\) 0.221226 − 0.0718808i 0.0536553 − 0.0174337i −0.282066 − 0.959395i \(-0.591020\pi\)
0.335722 + 0.941961i \(0.391020\pi\)
\(18\) 4.20081i 0.990140i
\(19\) 1.71507 + 5.27846i 0.393465 + 1.21096i 0.930151 + 0.367178i \(0.119676\pi\)
−0.536685 + 0.843782i \(0.680324\pi\)
\(20\) 0 0
\(21\) 1.00778 − 3.10163i 0.219916 − 0.676831i
\(22\) 6.26765 + 2.03648i 1.33627 + 0.434179i
\(23\) −2.89935 − 3.99061i −0.604556 − 0.832100i 0.391560 − 0.920153i \(-0.371936\pi\)
−0.996116 + 0.0880529i \(0.971936\pi\)
\(24\) −1.39445 −0.284641
\(25\) 0 0
\(26\) −0.964492 −0.189152
\(27\) −2.29449 − 3.15809i −0.441574 − 0.607775i
\(28\) 3.64101 + 1.18304i 0.688086 + 0.223573i
\(29\) −1.27643 + 3.92845i −0.237028 + 0.729496i 0.759818 + 0.650135i \(0.225288\pi\)
−0.996846 + 0.0793604i \(0.974712\pi\)
\(30\) 0 0
\(31\) −1.08011 − 3.32424i −0.193994 − 0.597051i −0.999987 − 0.00512633i \(-0.998368\pi\)
0.805993 − 0.591925i \(-0.201632\pi\)
\(32\) − 4.44444i − 0.785674i
\(33\) −2.64473 + 0.859324i −0.460388 + 0.149589i
\(34\) −0.316795 − 0.230165i −0.0543299 − 0.0394730i
\(35\) 0 0
\(36\) 1.68345 − 1.22310i 0.280574 − 0.203849i
\(37\) −3.18373 + 4.38203i −0.523402 + 0.720401i −0.986107 − 0.166112i \(-0.946879\pi\)
0.462705 + 0.886512i \(0.346879\pi\)
\(38\) 5.49173 − 7.55872i 0.890876 − 1.22619i
\(39\) 0.329255 − 0.239218i 0.0527230 − 0.0383055i
\(40\) 0 0
\(41\) −8.43214 − 6.12630i −1.31688 − 0.956768i −0.999965 − 0.00831339i \(-0.997354\pi\)
−0.316913 − 0.948455i \(-0.602646\pi\)
\(42\) −5.22132 + 1.69651i −0.805666 + 0.261777i
\(43\) − 1.38833i − 0.211718i −0.994381 − 0.105859i \(-0.966241\pi\)
0.994381 − 0.105859i \(-0.0337592\pi\)
\(44\) −1.00876 − 3.10465i −0.152077 − 0.468044i
\(45\) 0 0
\(46\) −2.56598 + 7.89729i −0.378334 + 1.16439i
\(47\) 0.875370 + 0.284425i 0.127686 + 0.0414876i 0.372163 − 0.928167i \(-0.378616\pi\)
−0.244477 + 0.969655i \(0.578616\pi\)
\(48\) 2.07611 + 2.85753i 0.299661 + 0.412448i
\(49\) −14.0782 −2.01118
\(50\) 0 0
\(51\) 0.165233 0.0231373
\(52\) 0.280819 + 0.386514i 0.0389425 + 0.0535998i
\(53\) −1.17092 − 0.380455i −0.160838 − 0.0522595i 0.227491 − 0.973780i \(-0.426948\pi\)
−0.388329 + 0.921521i \(0.626948\pi\)
\(54\) −2.03067 + 6.24976i −0.276339 + 0.850485i
\(55\) 0 0
\(56\) 2.78507 + 8.57157i 0.372171 + 1.14542i
\(57\) 3.94246i 0.522191i
\(58\) 6.61320 − 2.14876i 0.868356 − 0.282146i
\(59\) 3.64689 + 2.64962i 0.474785 + 0.344951i 0.799303 − 0.600928i \(-0.205202\pi\)
−0.324518 + 0.945879i \(0.605202\pi\)
\(60\) 0 0
\(61\) 9.43214 − 6.85285i 1.20766 − 0.877417i 0.212645 − 0.977129i \(-0.431792\pi\)
0.995016 + 0.0997121i \(0.0317922\pi\)
\(62\) −3.45856 + 4.76029i −0.439237 + 0.604558i
\(63\) −6.73409 + 9.26868i −0.848416 + 1.16774i
\(64\) 1.99259 − 1.44770i 0.249074 − 0.180963i
\(65\) 0 0
\(66\) 3.78724 + 2.75159i 0.466177 + 0.338697i
\(67\) 2.80987 − 0.912982i 0.343280 − 0.111539i −0.132303 − 0.991209i \(-0.542237\pi\)
0.475583 + 0.879671i \(0.342237\pi\)
\(68\) 0.193968i 0.0235220i
\(69\) −1.08276 − 3.33238i −0.130348 − 0.401171i
\(70\) 0 0
\(71\) 0.990558 − 3.04862i 0.117558 − 0.361805i −0.874914 − 0.484278i \(-0.839082\pi\)
0.992472 + 0.122473i \(0.0390824\pi\)
\(72\) 4.65893 + 1.51378i 0.549060 + 0.178400i
\(73\) −6.04961 − 8.32657i −0.708053 − 0.974551i −0.999837 − 0.0180667i \(-0.994249\pi\)
0.291784 − 0.956484i \(-0.405751\pi\)
\(74\) 9.11816 1.05996
\(75\) 0 0
\(76\) −4.62806 −0.530875
\(77\) 10.5644 + 14.5406i 1.20392 + 1.65706i
\(78\) −0.651586 − 0.211713i −0.0737775 − 0.0239718i
\(79\) 2.97123 − 9.14449i 0.334289 − 1.02884i −0.632782 − 0.774330i \(-0.718087\pi\)
0.967071 − 0.254506i \(-0.0819129\pi\)
\(80\) 0 0
\(81\) 1.45651 + 4.48267i 0.161834 + 0.498074i
\(82\) 17.5457i 1.93759i
\(83\) −9.97030 + 3.23955i −1.09438 + 0.355587i −0.799939 − 0.600082i \(-0.795135\pi\)
−0.294445 + 0.955669i \(0.595135\pi\)
\(84\) 2.20009 + 1.59846i 0.240049 + 0.174406i
\(85\) 0 0
\(86\) −1.89077 + 1.37373i −0.203887 + 0.148133i
\(87\) −1.72465 + 2.37377i −0.184902 + 0.254495i
\(88\) 4.51714 − 6.21731i 0.481529 − 0.662768i
\(89\) 5.87207 − 4.26631i 0.622438 − 0.452228i −0.231334 − 0.972874i \(-0.574309\pi\)
0.853772 + 0.520646i \(0.174309\pi\)
\(90\) 0 0
\(91\) −2.12806 − 1.54612i −0.223081 − 0.162078i
\(92\) 3.91189 − 1.27105i 0.407843 − 0.132516i
\(93\) − 2.48286i − 0.257461i
\(94\) −0.478804 − 1.47361i −0.0493848 − 0.151991i
\(95\) 0 0
\(96\) 0.975587 − 3.00255i 0.0995704 − 0.306446i
\(97\) −7.91252 − 2.57093i −0.803394 − 0.261039i −0.121597 − 0.992580i \(-0.538801\pi\)
−0.681797 + 0.731541i \(0.738801\pi\)
\(98\) 13.9302 + 19.1733i 1.40716 + 1.93679i
\(99\) 9.76903 0.981824
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 625.2.e.j.374.3 32
5.2 odd 4 625.2.d.m.251.4 16
5.3 odd 4 625.2.d.q.251.1 16
5.4 even 2 inner 625.2.e.j.374.6 32
25.2 odd 20 625.2.d.n.126.1 16
25.3 odd 20 625.2.a.e.1.7 ✓ 8
25.4 even 10 625.2.b.d.624.5 16
25.6 even 5 625.2.e.k.124.6 32
25.8 odd 20 625.2.d.p.501.4 16
25.9 even 10 inner 625.2.e.j.249.3 32
25.11 even 5 625.2.e.k.499.3 32
25.12 odd 20 625.2.d.m.376.4 16
25.13 odd 20 625.2.d.q.376.1 16
25.14 even 10 625.2.e.k.499.6 32
25.16 even 5 inner 625.2.e.j.249.6 32
25.17 odd 20 625.2.d.n.501.1 16
25.19 even 10 625.2.e.k.124.3 32
25.21 even 5 625.2.b.d.624.12 16
25.22 odd 20 625.2.a.g.1.2 yes 8
25.23 odd 20 625.2.d.p.126.4 16
75.47 even 20 5625.2.a.s.1.7 8
75.53 even 20 5625.2.a.be.1.2 8
100.3 even 20 10000.2.a.bn.1.4 8
100.47 even 20 10000.2.a.be.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
625.2.a.e.1.7 ✓ 8 25.3 odd 20
625.2.a.g.1.2 yes 8 25.22 odd 20
625.2.b.d.624.5 16 25.4 even 10
625.2.b.d.624.12 16 25.21 even 5
625.2.d.m.251.4 16 5.2 odd 4
625.2.d.m.376.4 16 25.12 odd 20
625.2.d.n.126.1 16 25.2 odd 20
625.2.d.n.501.1 16 25.17 odd 20
625.2.d.p.126.4 16 25.23 odd 20
625.2.d.p.501.4 16 25.8 odd 20
625.2.d.q.251.1 16 5.3 odd 4
625.2.d.q.376.1 16 25.13 odd 20
625.2.e.j.249.3 32 25.9 even 10 inner
625.2.e.j.249.6 32 25.16 even 5 inner
625.2.e.j.374.3 32 1.1 even 1 trivial
625.2.e.j.374.6 32 5.4 even 2 inner
625.2.e.k.124.3 32 25.19 even 10
625.2.e.k.124.6 32 25.6 even 5
625.2.e.k.499.3 32 25.11 even 5
625.2.e.k.499.6 32 25.14 even 10
5625.2.a.s.1.7 8 75.47 even 20
5625.2.a.be.1.2 8 75.53 even 20
10000.2.a.be.1.5 8 100.47 even 20
10000.2.a.bn.1.4 8 100.3 even 20