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Properties

Label 425.4.c.b.424.2
Level $425$
Weight $4$
Character 425.424
Analytic conductor $25.076$
Analytic rank $0$
Dimension $2$
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] = [425,4,Mod(424,425)] 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("425.424"); 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([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,16,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 424.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.424
Dual form 425.4.c.b.424.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +8.00000 q^{3} +7.00000 q^{4} +8.00000i q^{6} +14.0000 q^{7} +15.0000i q^{8} +37.0000 q^{9} +20.0000i q^{11} +56.0000 q^{12} +58.0000i q^{13} +14.0000i q^{14} +41.0000 q^{16} +(-68.0000 - 17.0000i) q^{17} +37.0000i q^{18} -80.0000 q^{19} +112.000 q^{21} -20.0000 q^{22} +118.000 q^{23} +120.000i q^{24} -58.0000 q^{26} +80.0000 q^{27} +98.0000 q^{28} -126.000i q^{29} +70.0000i q^{31} +161.000i q^{32} +160.000i q^{33} +(17.0000 - 68.0000i) q^{34} +259.000 q^{36} +134.000 q^{37} -80.0000i q^{38} +464.000i q^{39} -100.000i q^{41} +112.000i q^{42} -272.000i q^{43} +140.000i q^{44} +118.000i q^{46} -464.000i q^{47} +328.000 q^{48} -147.000 q^{49} +(-544.000 - 136.000i) q^{51} +406.000i q^{52} -642.000i q^{53} +80.0000i q^{54} +210.000i q^{56} -640.000 q^{57} +126.000 q^{58} +180.000 q^{59} +110.000i q^{61} -70.0000 q^{62} +518.000 q^{63} +167.000 q^{64} -160.000 q^{66} -924.000i q^{67} +(-476.000 - 119.000i) q^{68} +944.000 q^{69} +90.0000i q^{71} +555.000i q^{72} +828.000 q^{73} +134.000i q^{74} -560.000 q^{76} +280.000i q^{77} -464.000 q^{78} +1334.00i q^{79} -359.000 q^{81} +100.000 q^{82} -552.000i q^{83} +784.000 q^{84} +272.000 q^{86} -1008.00i q^{87} -300.000 q^{88} -1490.00 q^{89} +812.000i q^{91} +826.000 q^{92} +560.000i q^{93} +464.000 q^{94} +1288.00i q^{96} -1376.00 q^{97} -147.000i q^{98} +740.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{3} + 14 q^{4} + 28 q^{7} + 74 q^{9} + 112 q^{12} + 82 q^{16} - 136 q^{17} - 160 q^{19} + 224 q^{21} - 40 q^{22} + 236 q^{23} - 116 q^{26} + 160 q^{27} + 196 q^{28} + 34 q^{34} + 518 q^{36} + 268 q^{37}+ \cdots - 2752 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.353553i 0.984251 + 0.176777i \(0.0565670\pi\)
−0.984251 + 0.176777i \(0.943433\pi\)
\(3\) 8.00000 1.53960 0.769800 − 0.638285i \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) 8.00000i 0.544331i
\(7\) 14.0000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 15.0000i 0.662913i
\(9\) 37.0000 1.37037
\(10\) 0 0
\(11\) 20.0000i 0.548202i 0.961701 + 0.274101i \(0.0883803\pi\)
−0.961701 + 0.274101i \(0.911620\pi\)
\(12\) 56.0000 1.34715
\(13\) 58.0000i 1.23741i 0.785624 + 0.618704i \(0.212342\pi\)
−0.785624 + 0.618704i \(0.787658\pi\)
\(14\) 14.0000i 0.267261i
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −68.0000 − 17.0000i −0.970143 − 0.242536i
\(18\) 37.0000i 0.484499i
\(19\) −80.0000 −0.965961 −0.482980 − 0.875631i \(-0.660446\pi\)
−0.482980 + 0.875631i \(0.660446\pi\)
\(20\) 0 0
\(21\) 112.000 1.16383
\(22\) −20.0000 −0.193819
\(23\) 118.000 1.06977 0.534885 − 0.844925i \(-0.320355\pi\)
0.534885 + 0.844925i \(0.320355\pi\)
\(24\) 120.000i 1.02062i
\(25\) 0 0
\(26\) −58.0000 −0.437490
\(27\) 80.0000 0.570222
\(28\) 98.0000 0.661438
\(29\) − 126.000i − 0.806814i −0.915021 − 0.403407i \(-0.867826\pi\)
0.915021 − 0.403407i \(-0.132174\pi\)
\(30\) 0 0
\(31\) 70.0000i 0.405560i 0.979224 + 0.202780i \(0.0649977\pi\)
−0.979224 + 0.202780i \(0.935002\pi\)
\(32\) 161.000i 0.889408i
\(33\) 160.000i 0.844013i
\(34\) 17.0000 − 68.0000i 0.0857493 − 0.342997i
\(35\) 0 0
\(36\) 259.000 1.19907
\(37\) 134.000 0.595391 0.297695 − 0.954661i \(-0.403782\pi\)
0.297695 + 0.954661i \(0.403782\pi\)
\(38\) − 80.0000i − 0.341519i
\(39\) 464.000i 1.90511i
\(40\) 0 0
\(41\) − 100.000i − 0.380912i −0.981696 − 0.190456i \(-0.939003\pi\)
0.981696 − 0.190456i \(-0.0609966\pi\)
\(42\) 112.000i 0.411476i
\(43\) − 272.000i − 0.964642i −0.875995 − 0.482321i \(-0.839794\pi\)
0.875995 − 0.482321i \(-0.160206\pi\)
\(44\) 140.000i 0.479677i
\(45\) 0 0
\(46\) 118.000i 0.378221i
\(47\) − 464.000i − 1.44003i −0.693959 − 0.720014i \(-0.744135\pi\)
0.693959 − 0.720014i \(-0.255865\pi\)
\(48\) 328.000 0.986307
\(49\) −147.000 −0.428571
\(50\) 0 0
\(51\) −544.000 − 136.000i −1.49363 − 0.373408i
\(52\) 406.000i 1.08273i
\(53\) − 642.000i − 1.66388i −0.554868 − 0.831939i \(-0.687231\pi\)
0.554868 − 0.831939i \(-0.312769\pi\)
\(54\) 80.0000i 0.201604i
\(55\) 0 0
\(56\) 210.000i 0.501115i
\(57\) −640.000 −1.48719
\(58\) 126.000 0.285252
\(59\) 180.000 0.397187 0.198593 − 0.980082i \(-0.436363\pi\)
0.198593 + 0.980082i \(0.436363\pi\)
\(60\) 0 0
\(61\) 110.000i 0.230886i 0.993314 + 0.115443i \(0.0368288\pi\)
−0.993314 + 0.115443i \(0.963171\pi\)
\(62\) −70.0000 −0.143387
\(63\) 518.000 1.03590
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) −160.000 −0.298404
\(67\) − 924.000i − 1.68484i −0.538818 − 0.842422i \(-0.681129\pi\)
0.538818 − 0.842422i \(-0.318871\pi\)
\(68\) −476.000 − 119.000i −0.848875 − 0.212219i
\(69\) 944.000 1.64702
\(70\) 0 0
\(71\) 90.0000i 0.150437i 0.997167 + 0.0752186i \(0.0239654\pi\)
−0.997167 + 0.0752186i \(0.976035\pi\)
\(72\) 555.000i 0.908436i
\(73\) 828.000 1.32754 0.663768 − 0.747939i \(-0.268956\pi\)
0.663768 + 0.747939i \(0.268956\pi\)
\(74\) 134.000i 0.210502i
\(75\) 0 0
\(76\) −560.000 −0.845216
\(77\) 280.000i 0.414402i
\(78\) −464.000 −0.673560
\(79\) 1334.00i 1.89983i 0.312505 + 0.949916i \(0.398832\pi\)
−0.312505 + 0.949916i \(0.601168\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 100.000 0.134673
\(83\) − 552.000i − 0.729998i −0.931008 − 0.364999i \(-0.881069\pi\)
0.931008 − 0.364999i \(-0.118931\pi\)
\(84\) 784.000 1.01835
\(85\) 0 0
\(86\) 272.000 0.341052
\(87\) − 1008.00i − 1.24217i
\(88\) −300.000 −0.363410
\(89\) −1490.00 −1.77460 −0.887302 − 0.461190i \(-0.847423\pi\)
−0.887302 + 0.461190i \(0.847423\pi\)
\(90\) 0 0
\(91\) 812.000i 0.935393i
\(92\) 826.000 0.936048
\(93\) 560.000i 0.624401i
\(94\) 464.000 0.509127
\(95\) 0 0
\(96\) 1288.00i 1.36933i
\(97\) −1376.00 −1.44033 −0.720163 − 0.693805i \(-0.755933\pi\)
−0.720163 + 0.693805i \(0.755933\pi\)
\(98\) − 147.000i − 0.151523i
\(99\) 740.000i 0.751240i
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.c.b.424.2 2
5.2 odd 4 425.4.d.a.101.1 2
5.3 odd 4 85.4.d.a.16.2 yes 2
5.4 even 2 425.4.c.a.424.1 2
15.8 even 4 765.4.g.a.271.1 2
17.16 even 2 425.4.c.a.424.2 2
85.13 odd 4 1445.4.a.d.1.1 1
85.33 odd 4 85.4.d.a.16.1 ✓ 2
85.38 odd 4 1445.4.a.e.1.1 1
85.67 odd 4 425.4.d.a.101.2 2
85.84 even 2 inner 425.4.c.b.424.1 2
255.203 even 4 765.4.g.a.271.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.d.a.16.1 ✓ 2 85.33 odd 4
85.4.d.a.16.2 yes 2 5.3 odd 4
425.4.c.a.424.1 2 5.4 even 2
425.4.c.a.424.2 2 17.16 even 2
425.4.c.b.424.1 2 85.84 even 2 inner
425.4.c.b.424.2 2 1.1 even 1 trivial
425.4.d.a.101.1 2 5.2 odd 4
425.4.d.a.101.2 2 85.67 odd 4
765.4.g.a.271.1 2 15.8 even 4
765.4.g.a.271.2 2 255.203 even 4
1445.4.a.d.1.1 1 85.13 odd 4
1445.4.a.e.1.1 1 85.38 odd 4