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Properties

Label 25.6.d.a.11.7
Level $25$
Weight $6$
Character 25.11
Analytic conductor $4.010$
Analytic rank $0$
Dimension $44$
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(6,25)] 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("25.6"); 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([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), 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: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 11.7
Character \(\chi\) \(=\) 25.11
Dual form 25.6.d.a.16.7

$q$-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+(2.06882 - 1.50308i) q^{2} +(-3.25658 + 10.0227i) q^{3} +(-7.86780 + 24.2146i) q^{4} +(-54.0584 + 14.2370i) q^{5} +(8.32772 + 25.6301i) q^{6} -36.6142 q^{7} +(45.4065 + 139.747i) q^{8} +(106.742 + 77.5524i) q^{9} +(-90.4375 + 110.708i) q^{10} +(55.3269 - 40.1974i) q^{11} +(-217.074 - 157.713i) q^{12} +(278.848 + 202.595i) q^{13} +(-75.7481 + 55.0342i) q^{14} +(33.3518 - 588.175i) q^{15} +(-355.152 - 258.033i) q^{16} +(-201.510 - 620.182i) q^{17} +337.397 q^{18} +(322.638 + 992.978i) q^{19} +(80.5772 - 1421.02i) q^{20} +(119.237 - 366.973i) q^{21} +(54.0413 - 166.322i) q^{22} +(1655.07 - 1202.48i) q^{23} -1548.51 q^{24} +(2719.62 - 1539.26i) q^{25} +881.403 q^{26} +(-3196.67 + 2322.52i) q^{27} +(288.073 - 886.597i) q^{28} +(-1240.01 + 3816.37i) q^{29} +(-815.078 - 1266.96i) q^{30} +(2940.31 + 9049.34i) q^{31} -5824.62 q^{32} +(222.710 + 685.432i) q^{33} +(-1349.07 - 980.159i) q^{34} +(1979.30 - 521.276i) q^{35} +(-2717.72 + 1974.54i) q^{36} +(-3171.57 - 2304.28i) q^{37} +(2160.01 + 1569.34i) q^{38} +(-2938.64 + 2135.04i) q^{39} +(-4444.18 - 6908.04i) q^{40} +(11804.6 + 8576.56i) q^{41} +(-304.912 - 938.424i) q^{42} +11969.7 q^{43} +(538.062 + 1655.98i) q^{44} +(-6874.40 - 2672.67i) q^{45} +(1616.61 - 4975.43i) q^{46} +(7139.63 - 21973.5i) q^{47} +(3742.77 - 2719.28i) q^{48} -15466.4 q^{49} +(3312.76 - 7272.26i) q^{50} +6872.14 q^{51} +(-7099.67 + 5158.21i) q^{52} +(7924.02 - 24387.6i) q^{53} +(-3122.39 + 9609.74i) q^{54} +(-2418.59 + 2960.69i) q^{55} +(-1662.52 - 5116.72i) q^{56} -11003.0 q^{57} +(3170.96 + 9759.23i) q^{58} +(7627.30 + 5541.56i) q^{59} +(13980.0 + 5435.24i) q^{60} +(-25439.6 + 18483.0i) q^{61} +(19684.9 + 14301.9i) q^{62} +(-3908.26 - 2839.52i) q^{63} +(-685.233 + 497.851i) q^{64} +(-17958.4 - 6981.99i) q^{65} +(1491.01 + 1083.28i) q^{66} +(1432.32 + 4408.21i) q^{67} +16602.9 q^{68} +(6662.24 + 20504.3i) q^{69} +(3311.29 - 4053.48i) q^{70} +(10186.0 - 31349.2i) q^{71} +(-5990.94 + 18438.2i) q^{72} +(-66973.7 + 48659.3i) q^{73} -10024.9 q^{74} +(6570.90 + 32270.6i) q^{75} -26583.0 q^{76} +(-2025.75 + 1471.79i) q^{77} +(-2870.35 + 8834.04i) q^{78} +(21650.0 - 66631.9i) q^{79} +(22872.5 + 8892.54i) q^{80} +(-2960.19 - 9110.54i) q^{81} +37312.9 q^{82} +(-5537.05 - 17041.3i) q^{83} +(7947.97 + 5774.54i) q^{84} +(19722.8 + 30657.2i) q^{85} +(24763.1 - 17991.4i) q^{86} +(-34212.2 - 24856.6i) q^{87} +(8129.66 + 5906.55i) q^{88} +(114662. - 83306.6i) q^{89} +(-18239.1 + 4803.52i) q^{90} +(-10209.8 - 7417.84i) q^{91} +(16095.8 + 49537.8i) q^{92} -100274. q^{93} +(-18257.5 - 56190.7i) q^{94} +(-31578.3 - 49085.4i) q^{95} +(18968.3 - 58378.5i) q^{96} +(4060.24 - 12496.1i) q^{97} +(-31997.2 + 23247.3i) q^{98} +9023.10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9} - 55 q^{10} + 718 q^{11} - 437 q^{12} - 291 q^{13} - 689 q^{14} - 1125 q^{15} + 2609 q^{16} + 718 q^{17} + 8544 q^{18}+ \cdots - 680776 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.06882 − 1.50308i 0.365719 − 0.265710i −0.389715 − 0.920936i \(-0.627426\pi\)
0.755433 + 0.655225i \(0.227426\pi\)
\(3\) −3.25658 + 10.0227i −0.208909 + 0.642957i 0.790621 + 0.612306i \(0.209758\pi\)
−0.999530 + 0.0306509i \(0.990242\pi\)
\(4\) −7.86780 + 24.2146i −0.245869 + 0.756706i
\(5\) −54.0584 + 14.2370i −0.967026 + 0.254679i
\(6\) 8.32772 + 25.6301i 0.0944382 + 0.290651i
\(7\) −36.6142 −0.282426 −0.141213 − 0.989979i \(-0.545100\pi\)
−0.141213 + 0.989979i \(0.545100\pi\)
\(8\) 45.4065 + 139.747i 0.250838 + 0.772000i
\(9\) 106.742 + 77.5524i 0.439266 + 0.319146i
\(10\) −90.4375 + 110.708i −0.285989 + 0.350090i
\(11\) 55.3269 − 40.1974i 0.137865 − 0.100165i −0.516715 − 0.856157i \(-0.672845\pi\)
0.654580 + 0.755992i \(0.272845\pi\)
\(12\) −217.074 − 157.713i −0.435165 − 0.316166i
\(13\) 278.848 + 202.595i 0.457624 + 0.332483i 0.792599 − 0.609744i \(-0.208728\pi\)
−0.334974 + 0.942227i \(0.608728\pi\)
\(14\) −75.7481 + 55.0342i −0.103288 + 0.0750434i
\(15\) 33.3518 − 588.175i 0.0382729 − 0.674961i
\(16\) −355.152 − 258.033i −0.346828 − 0.251985i
\(17\) −201.510 − 620.182i −0.169112 − 0.520472i 0.830204 − 0.557459i \(-0.188224\pi\)
−0.999316 + 0.0369876i \(0.988224\pi\)
\(18\) 337.397 0.245448
\(19\) 322.638 + 992.978i 0.205037 + 0.631038i 0.999712 + 0.0240033i \(0.00764123\pi\)
−0.794675 + 0.607035i \(0.792359\pi\)
\(20\) 80.5772 − 1421.02i 0.0450440 − 0.794372i
\(21\) 119.237 − 366.973i 0.0590014 − 0.181587i
\(22\) 54.0413 − 166.322i 0.0238051 − 0.0732645i
\(23\) 1655.07 − 1202.48i 0.652375 − 0.473978i −0.211704 − 0.977334i \(-0.567901\pi\)
0.864079 + 0.503355i \(0.167901\pi\)
\(24\) −1548.51 −0.548765
\(25\) 2719.62 − 1539.26i 0.870277 − 0.492563i
\(26\) 881.403 0.255706
\(27\) −3196.67 + 2322.52i −0.843896 + 0.613126i
\(28\) 288.073 − 886.597i 0.0694396 − 0.213713i
\(29\) −1240.01 + 3816.37i −0.273799 + 0.842666i 0.715736 + 0.698371i \(0.246091\pi\)
−0.989535 + 0.144295i \(0.953909\pi\)
\(30\) −815.078 − 1266.96i −0.165347 − 0.257015i
\(31\) 2940.31 + 9049.34i 0.549527 + 1.69127i 0.709977 + 0.704225i \(0.248705\pi\)
−0.160450 + 0.987044i \(0.551295\pi\)
\(32\) −5824.62 −1.00553
\(33\) 222.710 + 685.432i 0.0356004 + 0.109567i
\(34\) −1349.07 − 980.159i −0.200142 − 0.145412i
\(35\) 1979.30 − 521.276i 0.273113 − 0.0719279i
\(36\) −2717.72 + 1974.54i −0.349501 + 0.253928i
\(37\) −3171.57 − 2304.28i −0.380864 − 0.276714i 0.380837 − 0.924642i \(-0.375636\pi\)
−0.761702 + 0.647928i \(0.775636\pi\)
\(38\) 2160.01 + 1569.34i 0.242659 + 0.176302i
\(39\) −2938.64 + 2135.04i −0.309374 + 0.224774i
\(40\) −4444.18 − 6908.04i −0.439179 − 0.682660i
\(41\) 11804.6 + 8576.56i 1.09671 + 0.796807i 0.980520 − 0.196419i \(-0.0629314\pi\)
0.116191 + 0.993227i \(0.462931\pi\)
\(42\) −304.912 − 938.424i −0.0266718 − 0.0820872i
\(43\) 11969.7 0.987214 0.493607 − 0.869685i \(-0.335678\pi\)
0.493607 + 0.869685i \(0.335678\pi\)
\(44\) 538.062 + 1655.98i 0.0418987 + 0.128951i
\(45\) −6874.40 − 2672.67i −0.506062 − 0.196750i
\(46\) 1616.61 − 4975.43i 0.112645 − 0.346686i
\(47\) 7139.63 − 21973.5i 0.471445 − 1.45096i −0.379247 − 0.925295i \(-0.623817\pi\)
0.850692 − 0.525664i \(-0.176183\pi\)
\(48\) 3742.77 − 2719.28i 0.234471 − 0.170353i
\(49\) −15466.4 −0.920236
\(50\) 3312.76 − 7272.26i 0.187398 − 0.411381i
\(51\) 6872.14 0.369970
\(52\) −7099.67 + 5158.21i −0.364108 + 0.264540i
\(53\) 7924.02 − 24387.6i 0.387486 − 1.19256i −0.547175 − 0.837018i \(-0.684297\pi\)
0.934661 − 0.355541i \(-0.115703\pi\)
\(54\) −3122.39 + 9609.74i −0.145715 + 0.448464i
\(55\) −2418.59 + 2960.69i −0.107809 + 0.131974i
\(56\) −1662.52 − 5116.72i −0.0708431 − 0.218032i
\(57\) −11003.0 −0.448565
\(58\) 3170.96 + 9759.23i 0.123772 + 0.380930i
\(59\) 7627.30 + 5541.56i 0.285260 + 0.207253i 0.721208 − 0.692718i \(-0.243587\pi\)
−0.435949 + 0.899972i \(0.643587\pi\)
\(60\) 13980.0 + 5435.24i 0.501337 + 0.194913i
\(61\) −25439.6 + 18483.0i −0.875359 + 0.635986i −0.932020 − 0.362408i \(-0.881955\pi\)
0.0566605 + 0.998394i \(0.481955\pi\)
\(62\) 19684.9 + 14301.9i 0.650360 + 0.472514i
\(63\) −3908.26 − 2839.52i −0.124060 − 0.0901349i
\(64\) −685.233 + 497.851i −0.0209116 + 0.0151932i
\(65\) −17958.4 − 6981.99i −0.527211 − 0.204973i
\(66\) 1491.01 + 1083.28i 0.0421328 + 0.0306113i
\(67\) 1432.32 + 4408.21i 0.0389809 + 0.119971i 0.968653 − 0.248416i \(-0.0799101\pi\)
−0.929672 + 0.368387i \(0.879910\pi\)
\(68\) 16602.9 0.435423
\(69\) 6662.24 + 20504.3i 0.168460 + 0.518468i
\(70\) 3311.29 − 4053.48i 0.0807705 − 0.0988743i
\(71\) 10186.0 − 31349.2i 0.239804 − 0.738042i −0.756643 − 0.653828i \(-0.773162\pi\)
0.996448 − 0.0842140i \(-0.0268379\pi\)
\(72\) −5990.94 + 18438.2i −0.136196 + 0.419167i
\(73\) −66973.7 + 48659.3i −1.47095 + 1.06871i −0.490610 + 0.871379i \(0.663226\pi\)
−0.980338 + 0.197327i \(0.936774\pi\)
\(74\) −10024.9 −0.212815
\(75\) 6570.90 + 32270.6i 0.134888 + 0.662452i
\(76\) −26583.0 −0.527923
\(77\) −2025.75 + 1471.79i −0.0389367 + 0.0282892i
\(78\) −2870.35 + 8834.04i −0.0534194 + 0.164408i
\(79\) 21650.0 − 66631.9i 0.390293 − 1.20120i −0.542274 − 0.840202i \(-0.682437\pi\)
0.932567 − 0.360997i \(-0.117563\pi\)
\(80\) 22872.5 + 8892.54i 0.399567 + 0.155346i
\(81\) −2960.19 − 9110.54i −0.0501312 − 0.154288i
\(82\) 37312.9 0.612808
\(83\) −5537.05 − 17041.3i −0.0882232 − 0.271523i 0.897205 − 0.441614i \(-0.145594\pi\)
−0.985428 + 0.170091i \(0.945594\pi\)
\(84\) 7947.97 + 5774.54i 0.122902 + 0.0892933i
\(85\) 19722.8 + 30657.2i 0.296089 + 0.460240i
\(86\) 24763.1 − 17991.4i 0.361043 − 0.262313i
\(87\) −34212.2 − 24856.6i −0.484599 − 0.352082i
\(88\) 8129.66 + 5906.55i 0.111909 + 0.0813068i
\(89\) 114662. − 83306.6i 1.53442 − 1.11482i 0.580698 − 0.814119i \(-0.302780\pi\)
0.953719 − 0.300700i \(-0.0972203\pi\)
\(90\) −18239.1 + 4803.52i −0.237355 + 0.0625106i
\(91\) −10209.8 − 7417.84i −0.129245 − 0.0939018i
\(92\) 16095.8 + 49537.8i 0.198264 + 0.610193i
\(93\) −100274. −1.20221
\(94\) −18257.5 − 56190.7i −0.213118 − 0.655911i
\(95\) −31578.3 − 49085.4i −0.358988 − 0.558012i
\(96\) 18968.3 − 58378.5i 0.210064 − 0.646510i
\(97\) 4060.24 − 12496.1i 0.0438149 − 0.134849i −0.926756 − 0.375664i \(-0.877415\pi\)
0.970571 + 0.240815i \(0.0774148\pi\)
\(98\) −31997.2 + 23247.3i −0.336548 + 0.244516i
\(99\) 9023.10 0.0925268
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.d.a.11.7 ✓ 44
25.4 even 10 625.6.a.c.1.14 22
25.16 even 5 inner 25.6.d.a.16.7 yes 44
25.21 even 5 625.6.a.d.1.9 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.d.a.11.7 ✓ 44 1.1 even 1 trivial
25.6.d.a.16.7 yes 44 25.16 even 5 inner
625.6.a.c.1.14 22 25.4 even 10
625.6.a.d.1.9 22 25.21 even 5