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Properties

Label 25.6.e.a.14.1
Level $25$
Weight $6$
Character 25.14
Analytic conductor $4.010$
Analytic rank $0$
Dimension $48$
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] = [25,6,Mod(4,25)] 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("25.4"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 14.1
Character \(\chi\) \(=\) 25.14
Dual form 25.6.e.a.9.1

$q$-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+(-6.42181 - 8.83886i) q^{2} +(13.2720 + 4.31235i) q^{3} +(-26.9973 + 83.0892i) q^{4} +(-22.7744 + 51.0522i) q^{5} +(-47.1143 - 145.003i) q^{6} +134.700i q^{7} +(575.283 - 186.921i) q^{8} +(-39.0403 - 28.3645i) q^{9} +(597.496 - 126.548i) q^{10} +(-15.5182 + 11.2746i) q^{11} +(-716.619 + 986.342i) q^{12} +(-356.764 + 491.044i) q^{13} +(1190.59 - 865.016i) q^{14} +(-522.418 + 579.356i) q^{15} +(-3084.77 - 2241.22i) q^{16} +(440.708 - 143.195i) q^{17} +527.223i q^{18} +(384.174 + 1182.37i) q^{19} +(-3627.04 - 3270.58i) q^{20} +(-580.872 + 1787.74i) q^{21} +(199.309 + 64.7595i) q^{22} +(2518.98 + 3467.08i) q^{23} +8441.25 q^{24} +(-2087.65 - 2325.37i) q^{25} +6631.35 q^{26} +(-2389.05 - 3288.25i) q^{27} +(-11192.1 - 3636.53i) q^{28} +(783.871 - 2412.51i) q^{29} +(8475.72 + 897.067i) q^{30} +(-208.877 - 642.858i) q^{31} +22302.1i q^{32} +(-254.578 + 82.7173i) q^{33} +(-4095.82 - 2975.79i) q^{34} +(-6876.71 - 3067.70i) q^{35} +(3410.77 - 2478.07i) q^{36} +(654.184 - 900.407i) q^{37} +(7983.68 - 10988.6i) q^{38} +(-6852.55 + 4978.67i) q^{39} +(-3559.02 + 33626.5i) q^{40} +(-7612.59 - 5530.87i) q^{41} +(19531.8 - 6346.28i) q^{42} +4827.27i q^{43} +(-517.849 - 1593.78i) q^{44} +(2337.19 - 1347.11i) q^{45} +(14468.6 - 44529.8i) q^{46} +(14856.2 + 4827.07i) q^{47} +(-31276.3 - 43048.2i) q^{48} -1336.99 q^{49} +(-7147.09 + 33385.5i) q^{50} +6466.60 q^{51} +(-31168.8 - 42900.2i) q^{52} +(11091.5 + 3603.83i) q^{53} +(-13722.3 + 42233.0i) q^{54} +(-222.176 - 1049.01i) q^{55} +(25178.2 + 77490.5i) q^{56} +17349.1i q^{57} +(-26357.7 + 8564.14i) q^{58} +(-14835.0 - 10778.2i) q^{59} +(-34034.3 - 59048.3i) q^{60} +(-18479.3 + 13426.0i) q^{61} +(-4340.77 + 5974.55i) q^{62} +(3820.68 - 5258.72i) q^{63} +(98412.8 - 71501.1i) q^{64} +(-16943.8 - 29396.8i) q^{65} +(2365.98 + 1718.98i) q^{66} +(50463.5 - 16396.6i) q^{67} +40483.9i q^{68} +(18480.7 + 56877.9i) q^{69} +(17045.9 + 80482.5i) q^{70} +(13347.5 - 41079.3i) q^{71} +(-27761.2 - 9020.15i) q^{72} +(35942.2 + 49470.2i) q^{73} -12159.6 q^{74} +(-17679.6 - 39865.1i) q^{75} -108614. q^{76} +(-1518.68 - 2090.29i) q^{77} +(88011.5 + 28596.7i) q^{78} +(-12364.1 + 38052.7i) q^{79} +(184673. - 106442. i) q^{80} +(-13903.9 - 42791.8i) q^{81} +102805. i q^{82} +(888.788 - 288.785i) q^{83} +(-132860. - 96528.4i) q^{84} +(-2726.46 + 25760.3i) q^{85} +(42667.6 - 30999.8i) q^{86} +(20807.2 - 28638.6i) q^{87} +(-6819.88 + 9386.76i) q^{88} +(20495.5 - 14890.8i) q^{89} +(-26915.9 - 12007.2i) q^{90} +(-66143.5 - 48056.0i) q^{91} +(-356082. + 115698. i) q^{92} -9432.80i q^{93} +(-52737.9 - 162310. i) q^{94} +(-69111.7 - 7314.76i) q^{95} +(-96174.5 + 295995. i) q^{96} +(138933. + 45142.1i) q^{97} +(8585.92 + 11817.5i) q^{98} +925.632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9} + 435 q^{10} - 724 q^{11} + 315 q^{12} - 5 q^{13} - 497 q^{14} - 105 q^{15} - 6967 q^{16} - 960 q^{17} + 4075 q^{19}+ \cdots - 451848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.42181 − 8.83886i −1.13523 − 1.56251i −0.777739 − 0.628587i \(-0.783633\pi\)
−0.357487 − 0.933918i \(-0.616367\pi\)
\(3\) 13.2720 + 4.31235i 0.851402 + 0.276637i 0.702033 − 0.712144i \(-0.252276\pi\)
0.149369 + 0.988782i \(0.452276\pi\)
\(4\) −26.9973 + 83.0892i −0.843666 + 2.59654i
\(5\) −22.7744 + 51.0522i −0.407401 + 0.913249i
\(6\) −47.1143 − 145.003i −0.534287 − 1.64437i
\(7\) 134.700i 1.03901i 0.854466 + 0.519507i \(0.173884\pi\)
−0.854466 + 0.519507i \(0.826116\pi\)
\(8\) 575.283 − 186.921i 3.17802 − 1.03260i
\(9\) −39.0403 − 28.3645i −0.160660 − 0.116726i
\(10\) 597.496 − 126.548i 1.88945 − 0.400179i
\(11\) −15.5182 + 11.2746i −0.0386686 + 0.0280944i −0.606952 − 0.794739i \(-0.707608\pi\)
0.568283 + 0.822833i \(0.307608\pi\)
\(12\) −716.619 + 986.342i −1.43660 + 1.97731i
\(13\) −356.764 + 491.044i −0.585495 + 0.805865i −0.994284 − 0.106764i \(-0.965951\pi\)
0.408789 + 0.912629i \(0.365951\pi\)
\(14\) 1190.59 − 865.016i 1.62346 − 1.17952i
\(15\) −522.418 + 579.356i −0.599501 + 0.664840i
\(16\) −3084.77 − 2241.22i −3.01247 − 2.18869i
\(17\) 440.708 − 143.195i 0.369852 − 0.120172i −0.118193 − 0.992991i \(-0.537710\pi\)
0.488045 + 0.872818i \(0.337710\pi\)
\(18\) 527.223i 0.383542i
\(19\) 384.174 + 1182.37i 0.244143 + 0.751395i 0.995776 + 0.0918134i \(0.0292663\pi\)
−0.751633 + 0.659581i \(0.770734\pi\)
\(20\) −3627.04 − 3270.58i −2.02758 − 1.82831i
\(21\) −580.872 + 1787.74i −0.287430 + 0.884619i
\(22\) 199.309 + 64.7595i 0.0877952 + 0.0285264i
\(23\) 2518.98 + 3467.08i 0.992898 + 1.36661i 0.929583 + 0.368613i \(0.120167\pi\)
0.0633154 + 0.997994i \(0.479833\pi\)
\(24\) 8441.25 2.99143
\(25\) −2087.65 − 2325.37i −0.668049 − 0.744117i
\(26\) 6631.35 1.92384
\(27\) −2389.05 − 3288.25i −0.630691 − 0.868071i
\(28\) −11192.1 − 3636.53i −2.69784 − 0.876581i
\(29\) 783.871 − 2412.51i 0.173081 − 0.532689i −0.826459 − 0.562996i \(-0.809649\pi\)
0.999541 + 0.0303071i \(0.00964854\pi\)
\(30\) 8475.72 + 897.067i 1.71939 + 0.181979i
\(31\) −208.877 − 642.858i −0.0390380 − 0.120146i 0.929638 − 0.368473i \(-0.120119\pi\)
−0.968676 + 0.248327i \(0.920119\pi\)
\(32\) 22302.1i 3.85009i
\(33\) −254.578 + 82.7173i −0.0406945 + 0.0132224i
\(34\) −4095.82 − 2975.79i −0.607636 − 0.441473i
\(35\) −6876.71 − 3067.70i −0.948879 − 0.423295i
\(36\) 3410.77 − 2478.07i 0.438627 − 0.318681i
\(37\) 654.184 − 900.407i 0.0785589 − 0.108127i −0.767929 − 0.640535i \(-0.778713\pi\)
0.846488 + 0.532408i \(0.178713\pi\)
\(38\) 7983.68 − 10988.6i 0.896900 − 1.23448i
\(39\) −6852.55 + 4978.67i −0.721424 + 0.524145i
\(40\) −3559.02 + 33626.5i −0.351706 + 3.32301i
\(41\) −7612.59 − 5530.87i −0.707249 − 0.513847i 0.175036 − 0.984562i \(-0.443996\pi\)
−0.882285 + 0.470715i \(0.843996\pi\)
\(42\) 19531.8 − 6346.28i 1.70852 − 0.555132i
\(43\) 4827.27i 0.398135i 0.979986 + 0.199068i \(0.0637913\pi\)
−0.979986 + 0.199068i \(0.936209\pi\)
\(44\) −517.849 − 1593.78i −0.0403247 − 0.124107i
\(45\) 2337.19 − 1347.11i 0.172053 − 0.0991681i
\(46\) 14468.6 − 44529.8i 1.00817 − 3.10282i
\(47\) 14856.2 + 4827.07i 0.980987 + 0.318742i 0.755243 − 0.655445i \(-0.227519\pi\)
0.225744 + 0.974187i \(0.427519\pi\)
\(48\) −31276.3 − 43048.2i −1.95935 − 2.69682i
\(49\) −1336.99 −0.0795498
\(50\) −7147.09 + 33385.5i −0.404300 + 1.88857i
\(51\) 6466.60 0.348137
\(52\) −31168.8 − 42900.2i −1.59850 − 2.20014i
\(53\) 11091.5 + 3603.83i 0.542374 + 0.176228i 0.567375 − 0.823459i \(-0.307959\pi\)
−0.0250009 + 0.999687i \(0.507959\pi\)
\(54\) −13722.3 + 42233.0i −0.640389 + 1.97091i
\(55\) −222.176 − 1049.01i −0.00990355 − 0.0467597i
\(56\) 25178.2 + 77490.5i 1.07289 + 3.30201i
\(57\) 17349.1i 0.707278i
\(58\) −26357.7 + 8564.14i −1.02882 + 0.334283i
\(59\) −14835.0 − 10778.2i −0.554826 − 0.403105i 0.274736 − 0.961520i \(-0.411410\pi\)
−0.829562 + 0.558415i \(0.811410\pi\)
\(60\) −34034.3 − 59048.3i −1.22050 − 2.11753i
\(61\) −18479.3 + 13426.0i −0.635858 + 0.461978i −0.858425 − 0.512940i \(-0.828556\pi\)
0.222567 + 0.974917i \(0.428556\pi\)
\(62\) −4340.77 + 5974.55i −0.143413 + 0.197390i
\(63\) 3820.68 − 5258.72i 0.121280 − 0.166928i
\(64\) 98412.8 − 71501.1i 3.00332 − 2.18204i
\(65\) −16943.8 − 29396.8i −0.497424 − 0.863013i
\(66\) 2365.98 + 1718.98i 0.0668576 + 0.0485749i
\(67\) 50463.5 − 16396.6i 1.37338 − 0.446238i 0.472892 − 0.881120i \(-0.343210\pi\)
0.900487 + 0.434882i \(0.143210\pi\)
\(68\) 40483.9i 1.06172i
\(69\) 18480.7 + 56877.9i 0.467301 + 1.43820i
\(70\) 17045.9 + 80482.5i 0.415791 + 1.96316i
\(71\) 13347.5 − 41079.3i 0.314234 − 0.967113i −0.661835 − 0.749650i \(-0.730222\pi\)
0.976069 − 0.217463i \(-0.0697781\pi\)
\(72\) −27761.2 − 9020.15i −0.631112 − 0.205061i
\(73\) 35942.2 + 49470.2i 0.789400 + 1.08652i 0.994183 + 0.107709i \(0.0343513\pi\)
−0.204782 + 0.978808i \(0.565649\pi\)
\(74\) −12159.6 −0.258131
\(75\) −17679.6 − 39865.1i −0.362928 − 0.818350i
\(76\) −108614. −2.15700
\(77\) −1518.68 − 2090.29i −0.0291904 − 0.0401772i
\(78\) 88011.5 + 28596.7i 1.63796 + 0.532205i
\(79\) −12364.1 + 38052.7i −0.222891 + 0.685989i 0.775607 + 0.631216i \(0.217444\pi\)
−0.998499 + 0.0547737i \(0.982556\pi\)
\(80\) 184673. − 106442.i 3.22610 − 1.85946i
\(81\) −13903.9 − 42791.8i −0.235464 − 0.724683i
\(82\) 102805.i 1.68841i
\(83\) 888.788 − 288.785i 0.0141613 − 0.00460128i −0.301928 − 0.953331i \(-0.597630\pi\)
0.316089 + 0.948730i \(0.397630\pi\)
\(84\) −132860. − 96528.4i −2.05445 − 1.49265i
\(85\) −2726.46 + 25760.3i −0.0409309 + 0.386726i
\(86\) 42667.6 − 30999.8i 0.622088 − 0.451974i
\(87\) 20807.2 − 28638.6i 0.294723 − 0.405652i
\(88\) −6819.88 + 9386.76i −0.0938793 + 0.129214i
\(89\) 20495.5 − 14890.8i 0.274273 − 0.199271i −0.442143 − 0.896945i \(-0.645782\pi\)
0.716415 + 0.697674i \(0.245782\pi\)
\(90\) −26915.9 − 12007.2i −0.350270 − 0.156256i
\(91\) −66143.5 − 48056.0i −0.837305 − 0.608337i
\(92\) −356082. + 115698.i −4.38612 + 1.42514i
\(93\) − 9432.80i − 0.113092i
\(94\) −52737.9 − 162310.i −0.615606 − 1.89464i
\(95\) −69111.7 − 7314.76i −0.785675 − 0.0831555i
\(96\) −96174.5 + 295995.i −1.06508 + 3.27798i
\(97\) 138933. + 45142.1i 1.49926 + 0.487138i 0.939801 − 0.341722i \(-0.111010\pi\)
0.559456 + 0.828860i \(0.311010\pi\)
\(98\) 8585.92 + 11817.5i 0.0903071 + 0.124297i
\(99\) 925.632 0.00949184
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 25.6.e.a.14.1 yes 48
25.3 odd 20 625.6.a.g.1.48 48
25.9 even 10 inner 25.6.e.a.9.1 ✓ 48
25.22 odd 20 625.6.a.g.1.1 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.e.a.9.1 ✓ 48 25.9 even 10 inner
25.6.e.a.14.1 yes 48 1.1 even 1 trivial
625.6.a.g.1.1 48 25.22 odd 20
625.6.a.g.1.48 48 25.3 odd 20