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Properties

Label 25.6.e.a.14.5
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 14.5
Character \(\chi\) \(=\) 25.14
Dual form 25.6.e.a.9.5

$q$-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+(-2.41181 - 3.31957i) q^{2} +(-18.6046 - 6.04499i) q^{3} +(4.68583 - 14.4215i) q^{4} +(9.93343 + 55.0121i) q^{5} +(24.8039 + 76.3384i) q^{6} +138.174i q^{7} +(-184.051 + 59.8017i) q^{8} +(112.997 + 82.0971i) q^{9} +(158.659 - 165.653i) q^{10} +(-21.2048 + 15.4062i) q^{11} +(-174.356 + 239.980i) q^{12} +(-108.881 + 149.862i) q^{13} +(458.679 - 333.250i) q^{14} +(147.740 - 1083.52i) q^{15} +(249.845 + 181.523i) q^{16} +(-1660.27 + 539.454i) q^{17} -573.103i q^{18} +(-84.9266 - 261.377i) q^{19} +(839.904 + 114.522i) q^{20} +(835.263 - 2570.67i) q^{21} +(102.284 + 33.2340i) q^{22} +(-988.356 - 1360.36i) q^{23} +3785.69 q^{24} +(-2927.65 + 1092.92i) q^{25} +760.077 q^{26} +(1188.09 + 1635.27i) q^{27} +(1992.68 + 647.462i) q^{28} +(-2438.59 + 7505.21i) q^{29} +(-3953.15 + 2122.81i) q^{30} +(-2147.96 - 6610.75i) q^{31} +4925.54i q^{32} +(487.637 - 158.443i) q^{33} +(5794.99 + 4210.31i) q^{34} +(-7601.26 + 1372.55i) q^{35} +(1713.45 - 1244.89i) q^{36} +(1581.13 - 2176.24i) q^{37} +(-662.833 + 912.311i) q^{38} +(2931.60 - 2129.93i) q^{39} +(-5118.07 - 9530.98i) q^{40} +(10571.3 + 7680.50i) q^{41} +(-10548.0 + 3427.26i) q^{42} -23952.6i q^{43} +(122.819 + 377.997i) q^{44} +(-3393.88 + 7031.70i) q^{45} +(-2132.07 + 6561.83i) q^{46} +(13174.4 + 4280.62i) q^{47} +(-3550.95 - 4887.46i) q^{48} -2285.15 q^{49} +(10688.9 + 7082.64i) q^{50} +34149.5 q^{51} +(1651.04 + 2272.46i) q^{52} +(-36855.7 - 11975.2i) q^{53} +(2562.94 - 7887.91i) q^{54} +(-1058.16 - 1013.48i) q^{55} +(-8263.06 - 25431.1i) q^{56} +5376.19i q^{57} +(30795.4 - 10006.0i) q^{58} +(-17724.7 - 12877.8i) q^{59} +(-14933.8 - 7207.85i) q^{60} +(-12517.5 + 9094.48i) q^{61} +(-16764.3 + 23074.1i) q^{62} +(-11343.7 + 15613.3i) q^{63} +(24345.7 - 17688.2i) q^{64} +(-9325.79 - 4501.13i) q^{65} +(-1702.05 - 1236.61i) q^{66} +(5964.47 - 1937.97i) q^{67} +26471.4i q^{68} +(10164.6 + 31283.4i) q^{69} +(22889.0 + 21922.6i) q^{70} +(-13874.0 + 42699.9i) q^{71} +(-25706.7 - 8352.62i) q^{72} +(38916.3 + 53563.7i) q^{73} -11037.5 q^{74} +(61074.4 - 2635.61i) q^{75} -4167.41 q^{76} +(-2128.74 - 2929.96i) q^{77} +(-14140.9 - 4594.66i) q^{78} +(-21150.0 + 65093.1i) q^{79} +(-7504.13 + 15547.6i) q^{80} +(-22706.9 - 69884.6i) q^{81} -53616.0i q^{82} +(45240.6 - 14699.6i) q^{83} +(-33159.1 - 24091.5i) q^{84} +(-46168.6 - 85976.1i) q^{85} +(-79512.4 + 57769.1i) q^{86} +(90737.8 - 124890. i) q^{87} +(2981.45 - 4103.61i) q^{88} +(25111.4 - 18244.5i) q^{89} +(31527.6 - 5692.88i) q^{90} +(-20707.1 - 15044.6i) q^{91} +(-24249.7 + 7879.19i) q^{92} +135974. i q^{93} +(-17564.3 - 54057.2i) q^{94} +(13535.3 - 7268.36i) q^{95} +(29774.9 - 91637.6i) q^{96} +(67073.4 + 21793.5i) q^{97} +(5511.34 + 7585.71i) q^{98} -3660.88 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41181 − 3.31957i −0.426351 − 0.586822i 0.540760 − 0.841177i \(-0.318137\pi\)
−0.967111 + 0.254355i \(0.918137\pi\)
\(3\) −18.6046 − 6.04499i −1.19348 − 0.387786i −0.356124 − 0.934439i \(-0.615902\pi\)
−0.837360 + 0.546652i \(0.815902\pi\)
\(4\) 4.68583 − 14.4215i 0.146432 − 0.450672i
\(5\) 9.93343 + 55.0121i 0.177695 + 0.984086i
\(6\) 24.8039 + 76.3384i 0.281281 + 0.865695i
\(7\) 138.174i 1.06582i 0.846173 + 0.532908i \(0.178901\pi\)
−0.846173 + 0.532908i \(0.821099\pi\)
\(8\) −184.051 + 59.8017i −1.01675 + 0.330361i
\(9\) 112.997 + 82.0971i 0.465008 + 0.337848i
\(10\) 158.659 − 165.653i 0.501723 − 0.523841i
\(11\) −21.2048 + 15.4062i −0.0528388 + 0.0383896i −0.613891 − 0.789391i \(-0.710397\pi\)
0.561052 + 0.827780i \(0.310397\pi\)
\(12\) −174.356 + 239.980i −0.349529 + 0.481086i
\(13\) −108.881 + 149.862i −0.178688 + 0.245942i −0.888960 − 0.457984i \(-0.848572\pi\)
0.710273 + 0.703927i \(0.248572\pi\)
\(14\) 458.679 − 333.250i 0.625444 − 0.454412i
\(15\) 147.740 − 1083.52i 0.169539 − 1.24340i
\(16\) 249.845 + 181.523i 0.243989 + 0.177268i
\(17\) −1660.27 + 539.454i −1.39334 + 0.452722i −0.907030 − 0.421067i \(-0.861656\pi\)
−0.486306 + 0.873789i \(0.661656\pi\)
\(18\) − 573.103i − 0.416919i
\(19\) −84.9266 − 261.377i −0.0539709 − 0.166105i 0.920438 − 0.390889i \(-0.127832\pi\)
−0.974409 + 0.224784i \(0.927832\pi\)
\(20\) 839.904 + 114.522i 0.469520 + 0.0640199i
\(21\) 835.263 − 2570.67i 0.413309 − 1.27203i
\(22\) 102.284 + 33.2340i 0.0450557 + 0.0146395i
\(23\) −988.356 − 1360.36i −0.389578 − 0.536207i 0.568513 − 0.822675i \(-0.307519\pi\)
−0.958090 + 0.286467i \(0.907519\pi\)
\(24\) 3785.69 1.34158
\(25\) −2927.65 + 1092.92i −0.936849 + 0.349733i
\(26\) 760.077 0.220508
\(27\) 1188.09 + 1635.27i 0.313647 + 0.431698i
\(28\) 1992.68 + 647.462i 0.480334 + 0.156070i
\(29\) −2438.59 + 7505.21i −0.538448 + 1.65717i 0.197630 + 0.980277i \(0.436675\pi\)
−0.736078 + 0.676896i \(0.763325\pi\)
\(30\) −3953.15 + 2122.81i −0.801936 + 0.430634i
\(31\) −2147.96 − 6610.75i −0.401441 − 1.23551i −0.923830 − 0.382802i \(-0.874959\pi\)
0.522389 − 0.852707i \(-0.325041\pi\)
\(32\) 4925.54i 0.850314i
\(33\) 487.637 − 158.443i 0.0779492 − 0.0253272i
\(34\) 5794.99 + 4210.31i 0.859717 + 0.624621i
\(35\) −7601.26 + 1372.55i −1.04885 + 0.189390i
\(36\) 1713.45 − 1244.89i 0.220351 − 0.160094i
\(37\) 1581.13 − 2176.24i 0.189873 − 0.261338i −0.703458 − 0.710737i \(-0.748362\pi\)
0.893331 + 0.449399i \(0.148362\pi\)
\(38\) −662.833 + 912.311i −0.0744637 + 0.102491i
\(39\) 2931.60 − 2129.93i 0.308634 − 0.224236i
\(40\) −5118.07 − 9530.98i −0.505774 − 0.941862i
\(41\) 10571.3 + 7680.50i 0.982130 + 0.713559i 0.958184 − 0.286154i \(-0.0923769\pi\)
0.0239463 + 0.999713i \(0.492377\pi\)
\(42\) −10548.0 + 3427.26i −0.922672 + 0.299794i
\(43\) − 23952.6i − 1.97552i −0.155969 − 0.987762i \(-0.549850\pi\)
0.155969 − 0.987762i \(-0.450150\pi\)
\(44\) 122.819 + 377.997i 0.00956384 + 0.0294345i
\(45\) −3393.88 + 7031.70i −0.249842 + 0.517641i
\(46\) −2132.07 + 6561.83i −0.148561 + 0.457225i
\(47\) 13174.4 + 4280.62i 0.869932 + 0.282658i 0.709771 − 0.704433i \(-0.248798\pi\)
0.160162 + 0.987091i \(0.448798\pi\)
\(48\) −3550.95 − 4887.46i −0.222455 − 0.306183i
\(49\) −2285.15 −0.135964
\(50\) 10688.9 + 7082.64i 0.604658 + 0.400654i
\(51\) 34149.5 1.83848
\(52\) 1651.04 + 2272.46i 0.0846738 + 0.116543i
\(53\) −36855.7 − 11975.2i −1.80225 − 0.585587i −0.802316 − 0.596900i \(-0.796399\pi\)
−0.999936 + 0.0113130i \(0.996399\pi\)
\(54\) 2562.94 − 7887.91i 0.119606 − 0.368110i
\(55\) −1058.16 − 1013.48i −0.0471679 − 0.0451763i
\(56\) −8263.06 − 25431.1i −0.352104 − 1.08366i
\(57\) 5376.19i 0.219173i
\(58\) 30795.4 − 10006.0i 1.20203 − 0.390564i
\(59\) −17724.7 − 12877.8i −0.662903 − 0.481627i 0.204739 − 0.978817i \(-0.434365\pi\)
−0.867642 + 0.497189i \(0.834365\pi\)
\(60\) −14933.8 − 7207.85i −0.535539 − 0.258480i
\(61\) −12517.5 + 9094.48i −0.430717 + 0.312934i −0.781936 − 0.623359i \(-0.785767\pi\)
0.351218 + 0.936294i \(0.385767\pi\)
\(62\) −16764.3 + 23074.1i −0.553869 + 0.762335i
\(63\) −11343.7 + 15613.3i −0.360084 + 0.495613i
\(64\) 24345.7 − 17688.2i 0.742972 − 0.539801i
\(65\) −9325.79 − 4501.13i −0.273780 − 0.132141i
\(66\) −1702.05 − 1236.61i −0.0480963 − 0.0349440i
\(67\) 5964.47 − 1937.97i 0.162325 − 0.0527425i −0.226727 − 0.973958i \(-0.572803\pi\)
0.389052 + 0.921216i \(0.372803\pi\)
\(68\) 26471.4i 0.694231i
\(69\) 10164.6 + 31283.4i 0.257020 + 0.791028i
\(70\) 22889.0 + 21922.6i 0.558318 + 0.534744i
\(71\) −13874.0 + 42699.9i −0.326630 + 1.00526i 0.644069 + 0.764967i \(0.277245\pi\)
−0.970699 + 0.240297i \(0.922755\pi\)
\(72\) −25706.7 − 8352.62i −0.584407 − 0.189885i
\(73\) 38916.3 + 53563.7i 0.854722 + 1.17642i 0.982802 + 0.184660i \(0.0591184\pi\)
−0.128081 + 0.991764i \(0.540882\pi\)
\(74\) −11037.5 −0.234311
\(75\) 61074.4 − 2635.61i 1.25374 − 0.0541039i
\(76\) −4167.41 −0.0827622
\(77\) −2128.74 − 2929.96i −0.0409163 − 0.0563164i
\(78\) −14140.9 − 4594.66i −0.263173 − 0.0855100i
\(79\) −21150.0 + 65093.1i −0.381280 + 1.17346i 0.557864 + 0.829932i \(0.311621\pi\)
−0.939143 + 0.343525i \(0.888379\pi\)
\(80\) −7504.13 + 15547.6i −0.131092 + 0.271606i
\(81\) −22706.9 − 69884.6i −0.384543 − 1.18350i
\(82\) − 53616.0i − 0.880562i
\(83\) 45240.6 − 14699.6i 0.720831 − 0.234212i 0.0744477 − 0.997225i \(-0.476281\pi\)
0.646384 + 0.763013i \(0.276281\pi\)
\(84\) −33159.1 − 24091.5i −0.512749 − 0.372534i
\(85\) −46168.6 − 85976.1i −0.693106 − 1.29072i
\(86\) −79512.4 + 57769.1i −1.15928 + 0.842267i
\(87\) 90737.8 − 124890.i 1.28526 − 1.76901i
\(88\) 2981.45 − 4103.61i 0.0410412 − 0.0564884i
\(89\) 25111.4 − 18244.5i 0.336043 − 0.244150i −0.406947 − 0.913452i \(-0.633407\pi\)
0.742991 + 0.669302i \(0.233407\pi\)
\(90\) 31527.6 − 5692.88i 0.410284 − 0.0740842i
\(91\) −20707.1 − 15044.6i −0.262129 − 0.190448i
\(92\) −24249.7 + 7879.19i −0.298701 + 0.0970537i
\(93\) 135974.i 1.63023i
\(94\) −17564.3 − 54057.2i −0.205027 − 0.631007i
\(95\) 13535.3 − 7268.36i 0.153872 − 0.0826281i
\(96\) 29774.9 − 91637.6i 0.329740 − 1.01484i
\(97\) 67073.4 + 21793.5i 0.723805 + 0.235178i 0.647672 − 0.761919i \(-0.275743\pi\)
0.0761326 + 0.997098i \(0.475743\pi\)
\(98\) 5511.34 + 7585.71i 0.0579685 + 0.0797868i
\(99\) −3660.88 −0.0375403
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.14.5 yes 48
25.3 odd 20 625.6.a.g.1.32 48
25.9 even 10 inner 25.6.e.a.9.5 ✓ 48
25.22 odd 20 625.6.a.g.1.17 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.e.a.9.5 ✓ 48 25.9 even 10 inner
25.6.e.a.14.5 yes 48 1.1 even 1 trivial
625.6.a.g.1.17 48 25.22 odd 20
625.6.a.g.1.32 48 25.3 odd 20