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Properties

Label 625.2.d.l.376.1
Level $625$
Weight $2$
Character 625.376
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
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(126,625)] 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("625.126"); 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([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,6,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 125)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 376.1
Root \(0.476925 + 1.46782i\) of defining polynomial
Character \(\chi\) \(=\) 625.376
Dual form 625.2.d.l.251.1

$q$-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+(-2.02029 - 1.46782i) q^{2} +(-0.476925 - 1.46782i) q^{3} +(1.30902 + 4.02874i) q^{4} +(-1.19098 + 3.66547i) q^{6} -0.953850 q^{7} +(1.72553 - 5.31064i) q^{8} +(0.500000 - 0.363271i) q^{9} +(-1.61803 - 1.17557i) q^{11} +(5.28918 - 3.84281i) q^{12} +(4.04057 - 2.93565i) q^{13} +(1.92705 + 1.40008i) q^{14} +(-4.42705 + 3.21644i) q^{16} +(-0.953850 + 2.93565i) q^{17} -1.54336 q^{18} +(0.854102 - 2.62866i) q^{19} +(0.454915 + 1.40008i) q^{21} +(1.54336 + 4.74998i) q^{22} +(-3.26889 - 2.37499i) q^{23} -8.61803 q^{24} -12.4721 q^{26} +(-4.51750 - 3.28216i) q^{27} +(-1.24861 - 3.84281i) q^{28} +(-1.80902 - 5.56758i) q^{29} +(0.618034 - 1.90211i) q^{31} +2.49721 q^{32} +(-0.953850 + 2.93565i) q^{33} +(6.23607 - 4.53077i) q^{34} +(2.11803 + 1.53884i) q^{36} +(-6.53779 + 4.74998i) q^{37} +(-5.58394 + 4.05697i) q^{38} +(-6.23607 - 4.53077i) q^{39} +(-4.11803 + 2.99193i) q^{41} +(1.13602 - 3.49631i) q^{42} +9.62451 q^{43} +(2.61803 - 8.05748i) q^{44} +(3.11803 + 9.59632i) q^{46} +(-2.02029 - 6.21780i) q^{47} +(6.83254 + 4.96413i) q^{48} -6.09017 q^{49} +4.76393 q^{51} +(17.1161 + 12.4356i) q^{52} +(1.90770 + 5.87130i) q^{53} +(4.30902 + 13.2618i) q^{54} +(-1.64590 + 5.06555i) q^{56} -4.26575 q^{57} +(-4.51750 + 13.9034i) q^{58} +(-3.61803 + 2.62866i) q^{59} +(4.92705 + 3.57971i) q^{61} +(-4.04057 + 2.93565i) q^{62} +(-0.476925 + 0.346506i) q^{63} +(3.80902 + 2.76741i) q^{64} +(6.23607 - 4.53077i) q^{66} +(-0.953850 + 2.93565i) q^{67} -13.0756 q^{68} +(-1.92705 + 5.93085i) q^{69} +(-4.38197 - 13.4863i) q^{71} +(-1.06644 - 3.28216i) q^{72} +(-7.12730 - 5.17828i) q^{73} +20.1803 q^{74} +11.7082 q^{76} +(1.54336 + 1.12132i) q^{77} +(5.94827 + 18.3069i) q^{78} +(2.23607 + 6.88191i) q^{79} +(-2.09017 + 6.43288i) q^{81} +12.7112 q^{82} +(-2.20246 + 6.77846i) q^{83} +(-5.04508 + 3.66547i) q^{84} +(-19.4443 - 14.1271i) q^{86} +(-7.30947 + 5.31064i) q^{87} +(-9.03500 + 6.56431i) q^{88} +(5.16312 + 3.75123i) q^{89} +(-3.85410 + 2.80017i) q^{91} +(5.28918 - 16.2784i) q^{92} -3.08672 q^{93} +(-5.04508 + 15.5272i) q^{94} +(-1.19098 - 3.66547i) q^{96} +(0.364338 + 1.12132i) q^{97} +(12.3039 + 8.93930i) q^{98} -1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} - 14 q^{6} + 4 q^{9} - 4 q^{11} + 2 q^{14} - 22 q^{16} - 20 q^{19} + 26 q^{21} - 60 q^{24} - 64 q^{26} - 10 q^{29} - 4 q^{31} + 32 q^{34} + 8 q^{36} - 32 q^{39} - 24 q^{41} + 12 q^{44} + 16 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.02029 − 1.46782i −1.42856 − 1.03791i −0.990283 − 0.139068i \(-0.955589\pi\)
−0.438276 − 0.898841i \(-0.644411\pi\)
\(3\) −0.476925 − 1.46782i −0.275353 − 0.847449i −0.989126 − 0.147072i \(-0.953015\pi\)
0.713773 − 0.700377i \(-0.246985\pi\)
\(4\) 1.30902 + 4.02874i 0.654508 + 2.01437i
\(5\) 0 0
\(6\) −1.19098 + 3.66547i −0.486217 + 1.49642i
\(7\) −0.953850 −0.360521 −0.180261 − 0.983619i \(-0.557694\pi\)
−0.180261 + 0.983619i \(0.557694\pi\)
\(8\) 1.72553 − 5.31064i 0.610067 − 1.87759i
\(9\) 0.500000 − 0.363271i 0.166667 − 0.121090i
\(10\) 0 0
\(11\) −1.61803 − 1.17557i −0.487856 − 0.354448i 0.316503 − 0.948591i \(-0.397491\pi\)
−0.804359 + 0.594144i \(0.797491\pi\)
\(12\) 5.28918 − 3.84281i 1.52685 − 1.10932i
\(13\) 4.04057 − 2.93565i 1.12065 − 0.814202i 0.136346 − 0.990661i \(-0.456464\pi\)
0.984308 + 0.176459i \(0.0564643\pi\)
\(14\) 1.92705 + 1.40008i 0.515026 + 0.374188i
\(15\) 0 0
\(16\) −4.42705 + 3.21644i −1.10676 + 0.804110i
\(17\) −0.953850 + 2.93565i −0.231343 + 0.711999i 0.766243 + 0.642551i \(0.222124\pi\)
−0.997586 + 0.0694484i \(0.977876\pi\)
\(18\) −1.54336 −0.363774
\(19\) 0.854102 − 2.62866i 0.195944 − 0.603055i −0.804020 − 0.594602i \(-0.797309\pi\)
0.999964 − 0.00845249i \(-0.00269054\pi\)
\(20\) 0 0
\(21\) 0.454915 + 1.40008i 0.0992706 + 0.305523i
\(22\) 1.54336 + 4.74998i 0.329046 + 1.01270i
\(23\) −3.26889 − 2.37499i −0.681611 − 0.495220i 0.192281 − 0.981340i \(-0.438412\pi\)
−0.873892 + 0.486120i \(0.838412\pi\)
\(24\) −8.61803 −1.75915
\(25\) 0 0
\(26\) −12.4721 −2.44599
\(27\) −4.51750 − 3.28216i −0.869393 − 0.631651i
\(28\) −1.24861 − 3.84281i −0.235964 − 0.726224i
\(29\) −1.80902 − 5.56758i −0.335926 − 1.03387i −0.966264 − 0.257553i \(-0.917084\pi\)
0.630338 − 0.776321i \(-0.282916\pi\)
\(30\) 0 0
\(31\) 0.618034 − 1.90211i 0.111002 − 0.341630i −0.880090 − 0.474807i \(-0.842518\pi\)
0.991092 + 0.133177i \(0.0425179\pi\)
\(32\) 2.49721 0.441449
\(33\) −0.953850 + 2.93565i −0.166044 + 0.511031i
\(34\) 6.23607 − 4.53077i 1.06948 − 0.777020i
\(35\) 0 0
\(36\) 2.11803 + 1.53884i 0.353006 + 0.256474i
\(37\) −6.53779 + 4.74998i −1.07481 + 0.780892i −0.976770 − 0.214291i \(-0.931256\pi\)
−0.0980356 + 0.995183i \(0.531256\pi\)
\(38\) −5.58394 + 4.05697i −0.905834 + 0.658127i
\(39\) −6.23607 − 4.53077i −0.998570 − 0.725504i
\(40\) 0 0
\(41\) −4.11803 + 2.99193i −0.643129 + 0.467260i −0.860924 − 0.508734i \(-0.830114\pi\)
0.217795 + 0.975995i \(0.430114\pi\)
\(42\) 1.13602 − 3.49631i 0.175292 − 0.539492i
\(43\) 9.62451 1.46772 0.733862 − 0.679299i \(-0.237716\pi\)
0.733862 + 0.679299i \(0.237716\pi\)
\(44\) 2.61803 − 8.05748i 0.394683 − 1.21471i
\(45\) 0 0
\(46\) 3.11803 + 9.59632i 0.459729 + 1.41490i
\(47\) −2.02029 − 6.21780i −0.294689 − 0.906960i −0.983326 − 0.181853i \(-0.941790\pi\)
0.688637 − 0.725107i \(-0.258210\pi\)
\(48\) 6.83254 + 4.96413i 0.986192 + 0.716511i
\(49\) −6.09017 −0.870024
\(50\) 0 0
\(51\) 4.76393 0.667084
\(52\) 17.1161 + 12.4356i 2.37358 + 1.72451i
\(53\) 1.90770 + 5.87130i 0.262043 + 0.806485i 0.992360 + 0.123377i \(0.0393723\pi\)
−0.730317 + 0.683108i \(0.760628\pi\)
\(54\) 4.30902 + 13.2618i 0.586383 + 1.80470i
\(55\) 0 0
\(56\) −1.64590 + 5.06555i −0.219942 + 0.676913i
\(57\) −4.26575 −0.565012
\(58\) −4.51750 + 13.9034i −0.593177 + 1.82561i
\(59\) −3.61803 + 2.62866i −0.471028 + 0.342222i −0.797842 − 0.602867i \(-0.794025\pi\)
0.326814 + 0.945089i \(0.394025\pi\)
\(60\) 0 0
\(61\) 4.92705 + 3.57971i 0.630844 + 0.458335i 0.856693 − 0.515827i \(-0.172515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) −4.04057 + 2.93565i −0.513153 + 0.372828i
\(63\) −0.476925 + 0.346506i −0.0600869 + 0.0436557i
\(64\) 3.80902 + 2.76741i 0.476127 + 0.345927i
\(65\) 0 0
\(66\) 6.23607 − 4.53077i 0.767607 − 0.557699i
\(67\) −0.953850 + 2.93565i −0.116531 + 0.358647i −0.992263 − 0.124151i \(-0.960379\pi\)
0.875732 + 0.482798i \(0.160379\pi\)
\(68\) −13.0756 −1.58565
\(69\) −1.92705 + 5.93085i −0.231990 + 0.713991i
\(70\) 0 0
\(71\) −4.38197 − 13.4863i −0.520044 − 1.60053i −0.773913 − 0.633291i \(-0.781703\pi\)
0.253870 − 0.967238i \(-0.418297\pi\)
\(72\) −1.06644 − 3.28216i −0.125681 − 0.386806i
\(73\) −7.12730 − 5.17828i −0.834187 − 0.606072i 0.0865537 − 0.996247i \(-0.472415\pi\)
−0.920741 + 0.390175i \(0.872415\pi\)
\(74\) 20.1803 2.34592
\(75\) 0 0
\(76\) 11.7082 1.34302
\(77\) 1.54336 + 1.12132i 0.175882 + 0.127786i
\(78\) 5.94827 + 18.3069i 0.673509 + 2.07285i
\(79\) 2.23607 + 6.88191i 0.251577 + 0.774275i 0.994485 + 0.104881i \(0.0334462\pi\)
−0.742907 + 0.669394i \(0.766554\pi\)
\(80\) 0 0
\(81\) −2.09017 + 6.43288i −0.232241 + 0.714765i
\(82\) 12.7112 1.40372
\(83\) −2.20246 + 6.77846i −0.241751 + 0.744033i 0.754403 + 0.656412i \(0.227927\pi\)
−0.996154 + 0.0876213i \(0.972073\pi\)
\(84\) −5.04508 + 3.66547i −0.550464 + 0.399935i
\(85\) 0 0
\(86\) −19.4443 − 14.1271i −2.09673 − 1.52336i
\(87\) −7.30947 + 5.31064i −0.783657 + 0.569360i
\(88\) −9.03500 + 6.56431i −0.963134 + 0.699758i
\(89\) 5.16312 + 3.75123i 0.547290 + 0.397629i 0.826785 − 0.562518i \(-0.190167\pi\)
−0.279496 + 0.960147i \(0.590167\pi\)
\(90\) 0 0
\(91\) −3.85410 + 2.80017i −0.404020 + 0.293537i
\(92\) 5.28918 − 16.2784i 0.551435 − 1.69714i
\(93\) −3.08672 −0.320078
\(94\) −5.04508 + 15.5272i −0.520361 + 1.60151i
\(95\) 0 0
\(96\) −1.19098 − 3.66547i −0.121554 − 0.374105i
\(97\) 0.364338 + 1.12132i 0.0369930 + 0.113853i 0.967848 − 0.251537i \(-0.0809359\pi\)
−0.930855 + 0.365389i \(0.880936\pi\)
\(98\) 12.3039 + 8.93930i 1.24288 + 0.903006i
\(99\) −1.23607 −0.124230
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.d.l.376.1 8
5.2 odd 4 625.2.e.b.249.2 8
5.3 odd 4 625.2.e.b.249.1 8
5.4 even 2 inner 625.2.d.l.376.2 8
25.2 odd 20 625.2.e.b.374.1 8
25.3 odd 20 625.2.e.h.499.2 8
25.4 even 10 625.2.d.k.126.1 8
25.6 even 5 125.2.a.c.1.4 yes 4
25.8 odd 20 125.2.b.a.124.1 4
25.9 even 10 625.2.d.k.501.1 8
25.11 even 5 inner 625.2.d.l.251.1 8
25.12 odd 20 625.2.e.h.124.2 8
25.13 odd 20 625.2.e.h.124.1 8
25.14 even 10 inner 625.2.d.l.251.2 8
25.16 even 5 625.2.d.k.501.2 8
25.17 odd 20 125.2.b.a.124.4 4
25.19 even 10 125.2.a.c.1.1 ✓ 4
25.21 even 5 625.2.d.k.126.2 8
25.22 odd 20 625.2.e.h.499.1 8
25.23 odd 20 625.2.e.b.374.2 8
75.8 even 20 1125.2.b.a.874.4 4
75.17 even 20 1125.2.b.a.874.1 4
75.44 odd 10 1125.2.a.k.1.4 4
75.56 odd 10 1125.2.a.k.1.1 4
100.19 odd 10 2000.2.a.o.1.2 4
100.31 odd 10 2000.2.a.o.1.3 4
100.67 even 20 2000.2.c.c.1249.2 4
100.83 even 20 2000.2.c.c.1249.3 4
175.6 odd 10 6125.2.a.o.1.4 4
175.69 odd 10 6125.2.a.o.1.1 4
200.19 odd 10 8000.2.a.bk.1.3 4
200.69 even 10 8000.2.a.bj.1.2 4
200.131 odd 10 8000.2.a.bk.1.2 4
200.181 even 10 8000.2.a.bj.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
125.2.a.c.1.1 ✓ 4 25.19 even 10
125.2.a.c.1.4 yes 4 25.6 even 5
125.2.b.a.124.1 4 25.8 odd 20
125.2.b.a.124.4 4 25.17 odd 20
625.2.d.k.126.1 8 25.4 even 10
625.2.d.k.126.2 8 25.21 even 5
625.2.d.k.501.1 8 25.9 even 10
625.2.d.k.501.2 8 25.16 even 5
625.2.d.l.251.1 8 25.11 even 5 inner
625.2.d.l.251.2 8 25.14 even 10 inner
625.2.d.l.376.1 8 1.1 even 1 trivial
625.2.d.l.376.2 8 5.4 even 2 inner
625.2.e.b.249.1 8 5.3 odd 4
625.2.e.b.249.2 8 5.2 odd 4
625.2.e.b.374.1 8 25.2 odd 20
625.2.e.b.374.2 8 25.23 odd 20
625.2.e.h.124.1 8 25.13 odd 20
625.2.e.h.124.2 8 25.12 odd 20
625.2.e.h.499.1 8 25.22 odd 20
625.2.e.h.499.2 8 25.3 odd 20
1125.2.a.k.1.1 4 75.56 odd 10
1125.2.a.k.1.4 4 75.44 odd 10
1125.2.b.a.874.1 4 75.17 even 20
1125.2.b.a.874.4 4 75.8 even 20
2000.2.a.o.1.2 4 100.19 odd 10
2000.2.a.o.1.3 4 100.31 odd 10
2000.2.c.c.1249.2 4 100.67 even 20
2000.2.c.c.1249.3 4 100.83 even 20
6125.2.a.o.1.1 4 175.69 odd 10
6125.2.a.o.1.4 4 175.6 odd 10
8000.2.a.bj.1.2 4 200.69 even 10
8000.2.a.bj.1.3 4 200.181 even 10
8000.2.a.bk.1.2 4 200.131 odd 10
8000.2.a.bk.1.3 4 200.19 odd 10