Properties

Label 425.4.a.d.1.1
Level $425$
Weight $4$
Character 425.1
Self dual yes
Analytic conductor $25.076$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [425,4,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,3,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 425.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +8.00000 q^{3} +1.00000 q^{4} +24.0000 q^{6} +28.0000 q^{7} -21.0000 q^{8} +37.0000 q^{9} -24.0000 q^{11} +8.00000 q^{12} +58.0000 q^{13} +84.0000 q^{14} -71.0000 q^{16} -17.0000 q^{17} +111.000 q^{18} +116.000 q^{19} +224.000 q^{21} -72.0000 q^{22} +60.0000 q^{23} -168.000 q^{24} +174.000 q^{26} +80.0000 q^{27} +28.0000 q^{28} +30.0000 q^{29} -172.000 q^{31} -45.0000 q^{32} -192.000 q^{33} -51.0000 q^{34} +37.0000 q^{36} +58.0000 q^{37} +348.000 q^{38} +464.000 q^{39} -342.000 q^{41} +672.000 q^{42} +148.000 q^{43} -24.0000 q^{44} +180.000 q^{46} -288.000 q^{47} -568.000 q^{48} +441.000 q^{49} -136.000 q^{51} +58.0000 q^{52} -318.000 q^{53} +240.000 q^{54} -588.000 q^{56} +928.000 q^{57} +90.0000 q^{58} +252.000 q^{59} +110.000 q^{61} -516.000 q^{62} +1036.00 q^{63} +433.000 q^{64} -576.000 q^{66} +484.000 q^{67} -17.0000 q^{68} +480.000 q^{69} -708.000 q^{71} -777.000 q^{72} -362.000 q^{73} +174.000 q^{74} +116.000 q^{76} -672.000 q^{77} +1392.00 q^{78} -484.000 q^{79} -359.000 q^{81} -1026.00 q^{82} -756.000 q^{83} +224.000 q^{84} +444.000 q^{86} +240.000 q^{87} +504.000 q^{88} -774.000 q^{89} +1624.00 q^{91} +60.0000 q^{92} -1376.00 q^{93} -864.000 q^{94} -360.000 q^{96} +382.000 q^{97} +1323.00 q^{98} -888.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 − 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 8.00000 1.53960 0.769800 − 0.638285i \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) 1.00000 0.125000
\(5\) 0 0
\(6\) 24.0000 1.63299
\(7\) 28.0000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −21.0000 −0.928078
\(9\) 37.0000 1.37037
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 − 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 8.00000 0.192450
\(13\) 58.0000 1.23741 0.618704 − 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 84.0000 1.60357
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −17.0000 −0.242536
\(18\) 111.000 1.45350
\(19\) 116.000 1.40064 0.700322 − 0.713827i \(-0.253040\pi\)
0.700322 + 0.713827i \(0.253040\pi\)
\(20\) 0 0
\(21\) 224.000 2.32766
\(22\) −72.0000 −0.697748
\(23\) 60.0000 0.543951 0.271975 − 0.962304i \(-0.412323\pi\)
0.271975 + 0.962304i \(0.412323\pi\)
\(24\) −168.000 −1.42887
\(25\) 0 0
\(26\) 174.000 1.31247
\(27\) 80.0000 0.570222
\(28\) 28.0000 0.188982
\(29\) 30.0000 0.192099 0.0960493 − 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) −172.000 −0.996520 −0.498260 − 0.867028i \(-0.666027\pi\)
−0.498260 + 0.867028i \(0.666027\pi\)
\(32\) −45.0000 −0.248592
\(33\) −192.000 −1.01282
\(34\) −51.0000 −0.257248
\(35\) 0 0
\(36\) 37.0000 0.171296
\(37\) 58.0000 0.257707 0.128853 − 0.991664i \(-0.458870\pi\)
0.128853 + 0.991664i \(0.458870\pi\)
\(38\) 348.000 1.48561
\(39\) 464.000 1.90511
\(40\) 0 0
\(41\) −342.000 −1.30272 −0.651359 − 0.758770i \(-0.725801\pi\)
−0.651359 + 0.758770i \(0.725801\pi\)
\(42\) 672.000 2.46885
\(43\) 148.000 0.524879 0.262439 − 0.964948i \(-0.415473\pi\)
0.262439 + 0.964948i \(0.415473\pi\)
\(44\) −24.0000 −0.0822304
\(45\) 0 0
\(46\) 180.000 0.576947
\(47\) −288.000 −0.893811 −0.446906 − 0.894581i \(-0.647474\pi\)
−0.446906 + 0.894581i \(0.647474\pi\)
\(48\) −568.000 −1.70799
\(49\) 441.000 1.28571
\(50\) 0 0
\(51\) −136.000 −0.373408
\(52\) 58.0000 0.154676
\(53\) −318.000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 240.000 0.604812
\(55\) 0 0
\(56\) −588.000 −1.40312
\(57\) 928.000 2.15643
\(58\) 90.0000 0.203751
\(59\) 252.000 0.556061 0.278031 − 0.960572i \(-0.410318\pi\)
0.278031 + 0.960572i \(0.410318\pi\)
\(60\) 0 0
\(61\) 110.000 0.230886 0.115443 − 0.993314i \(-0.463171\pi\)
0.115443 + 0.993314i \(0.463171\pi\)
\(62\) −516.000 −1.05697
\(63\) 1036.00 2.07181
\(64\) 433.000 0.845703
\(65\) 0 0
\(66\) −576.000 −1.07425
\(67\) 484.000 0.882537 0.441269 − 0.897375i \(-0.354529\pi\)
0.441269 + 0.897375i \(0.354529\pi\)
\(68\) −17.0000 −0.0303170
\(69\) 480.000 0.837467
\(70\) 0 0
\(71\) −708.000 −1.18344 −0.591719 − 0.806144i \(-0.701551\pi\)
−0.591719 + 0.806144i \(0.701551\pi\)
\(72\) −777.000 −1.27181
\(73\) −362.000 −0.580396 −0.290198 − 0.956967i \(-0.593721\pi\)
−0.290198 + 0.956967i \(0.593721\pi\)
\(74\) 174.000 0.273339
\(75\) 0 0
\(76\) 116.000 0.175080
\(77\) −672.000 −0.994565
\(78\) 1392.00 2.02068
\(79\) −484.000 −0.689294 −0.344647 − 0.938732i \(-0.612001\pi\)
−0.344647 + 0.938732i \(0.612001\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) −1026.00 −1.38174
\(83\) −756.000 −0.999780 −0.499890 − 0.866089i \(-0.666626\pi\)
−0.499890 + 0.866089i \(0.666626\pi\)
\(84\) 224.000 0.290957
\(85\) 0 0
\(86\) 444.000 0.556718
\(87\) 240.000 0.295755
\(88\) 504.000 0.610529
\(89\) −774.000 −0.921841 −0.460920 − 0.887441i \(-0.652481\pi\)
−0.460920 + 0.887441i \(0.652481\pi\)
\(90\) 0 0
\(91\) 1624.00 1.87079
\(92\) 60.0000 0.0679938
\(93\) −1376.00 −1.53424
\(94\) −864.000 −0.948030
\(95\) 0 0
\(96\) −360.000 −0.382733
\(97\) 382.000 0.399858 0.199929 − 0.979810i \(-0.435929\pi\)
0.199929 + 0.979810i \(0.435929\pi\)
\(98\) 1323.00 1.36371
\(99\) −888.000 −0.901488
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 425.4.a.d.1.1 1
5.2 odd 4 425.4.b.c.324.2 2
5.3 odd 4 425.4.b.c.324.1 2
5.4 even 2 17.4.a.a.1.1 ✓ 1
15.14 odd 2 153.4.a.d.1.1 1
20.19 odd 2 272.4.a.d.1.1 1
35.34 odd 2 833.4.a.a.1.1 1
40.19 odd 2 1088.4.a.a.1.1 1
40.29 even 2 1088.4.a.l.1.1 1
55.54 odd 2 2057.4.a.d.1.1 1
60.59 even 2 2448.4.a.f.1.1 1
85.4 even 4 289.4.b.a.288.2 2
85.64 even 4 289.4.b.a.288.1 2
85.84 even 2 289.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.a.1.1 ✓ 1 5.4 even 2
153.4.a.d.1.1 1 15.14 odd 2
272.4.a.d.1.1 1 20.19 odd 2
289.4.a.a.1.1 1 85.84 even 2
289.4.b.a.288.1 2 85.64 even 4
289.4.b.a.288.2 2 85.4 even 4
425.4.a.d.1.1 1 1.1 even 1 trivial
425.4.b.c.324.1 2 5.3 odd 4
425.4.b.c.324.2 2 5.2 odd 4
833.4.a.a.1.1 1 35.34 odd 2
1088.4.a.a.1.1 1 40.19 odd 2
1088.4.a.l.1.1 1 40.29 even 2
2057.4.a.d.1.1 1 55.54 odd 2
2448.4.a.f.1.1 1 60.59 even 2