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Properties

Label 425.4.d.a.101.1
Level $425$
Weight $4$
Character 425.101
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(101,425)] 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("425.101"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,-14,0,0,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.101
Dual form 425.4.d.a.101.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -8.00000i q^{3} -7.00000 q^{4} +8.00000i q^{6} +14.0000i q^{7} +15.0000 q^{8} -37.0000 q^{9} +20.0000i q^{11} +56.0000i q^{12} +58.0000 q^{13} -14.0000i q^{14} +41.0000 q^{16} +(17.0000 - 68.0000i) q^{17} +37.0000 q^{18} +80.0000 q^{19} +112.000 q^{21} -20.0000i q^{22} -118.000i q^{23} -120.000i q^{24} -58.0000 q^{26} +80.0000i q^{27} -98.0000i q^{28} +126.000i q^{29} +70.0000i q^{31} -161.000 q^{32} +160.000 q^{33} +(-17.0000 + 68.0000i) q^{34} +259.000 q^{36} +134.000i q^{37} -80.0000 q^{38} -464.000i q^{39} -100.000i q^{41} -112.000 q^{42} -272.000 q^{43} -140.000i q^{44} +118.000i q^{46} +464.000 q^{47} -328.000i q^{48} +147.000 q^{49} +(-544.000 - 136.000i) q^{51} -406.000 q^{52} -642.000 q^{53} -80.0000i q^{54} +210.000i q^{56} -640.000i q^{57} -126.000i q^{58} -180.000 q^{59} +110.000i q^{61} -70.0000i q^{62} -518.000i q^{63} -167.000 q^{64} -160.000 q^{66} +924.000 q^{67} +(-119.000 + 476.000i) q^{68} -944.000 q^{69} +90.0000i q^{71} -555.000 q^{72} -828.000i q^{73} -134.000i q^{74} -560.000 q^{76} -280.000 q^{77} +464.000i q^{78} -1334.00i q^{79} -359.000 q^{81} +100.000i q^{82} -552.000 q^{83} -784.000 q^{84} +272.000 q^{86} +1008.00 q^{87} +300.000i q^{88} +1490.00 q^{89} +812.000i q^{91} +826.000i q^{92} +560.000 q^{93} -464.000 q^{94} +1288.00i q^{96} -1376.00i q^{97} -147.000 q^{98} -740.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 14 q^{4} + 30 q^{8} - 74 q^{9} + 116 q^{13} + 82 q^{16} + 34 q^{17} + 74 q^{18} + 160 q^{19} + 224 q^{21} - 116 q^{26} - 322 q^{32} + 320 q^{33} - 34 q^{34} + 518 q^{36} - 160 q^{38} - 224 q^{42}+ \cdots - 294 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 − 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) − 8.00000i − 1.53960i −0.638285 − 0.769800i \(-0.720356\pi\)
0.638285 − 0.769800i \(-0.279644\pi\)
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) 8.00000i 0.544331i
\(7\) 14.0000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 15.0000 0.662913
\(9\) −37.0000 −1.37037
\(10\) 0 0
\(11\) 20.0000i 0.548202i 0.961701 + 0.274101i \(0.0883803\pi\)
−0.961701 + 0.274101i \(0.911620\pi\)
\(12\) 56.0000i 1.34715i
\(13\) 58.0000 1.23741 0.618704 − 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) − 14.0000i − 0.267261i
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) 17.0000 − 68.0000i 0.242536 − 0.970143i
\(18\) 37.0000 0.484499
\(19\) 80.0000 0.965961 0.482980 − 0.875631i \(-0.339554\pi\)
0.482980 + 0.875631i \(0.339554\pi\)
\(20\) 0 0
\(21\) 112.000 1.16383
\(22\) − 20.0000i − 0.193819i
\(23\) − 118.000i − 1.06977i −0.844925 − 0.534885i \(-0.820355\pi\)
0.844925 − 0.534885i \(-0.179645\pi\)
\(24\) − 120.000i − 1.02062i
\(25\) 0 0
\(26\) −58.0000 −0.437490
\(27\) 80.0000i 0.570222i
\(28\) − 98.0000i − 0.661438i
\(29\) 126.000i 0.806814i 0.915021 + 0.403407i \(0.132174\pi\)
−0.915021 + 0.403407i \(0.867826\pi\)
\(30\) 0 0
\(31\) 70.0000i 0.405560i 0.979224 + 0.202780i \(0.0649977\pi\)
−0.979224 + 0.202780i \(0.935002\pi\)
\(32\) −161.000 −0.889408
\(33\) 160.000 0.844013
\(34\) −17.0000 + 68.0000i −0.0857493 + 0.342997i
\(35\) 0 0
\(36\) 259.000 1.19907
\(37\) 134.000i 0.595391i 0.954661 + 0.297695i \(0.0962180\pi\)
−0.954661 + 0.297695i \(0.903782\pi\)
\(38\) −80.0000 −0.341519
\(39\) − 464.000i − 1.90511i
\(40\) 0 0
\(41\) − 100.000i − 0.380912i −0.981696 − 0.190456i \(-0.939003\pi\)
0.981696 − 0.190456i \(-0.0609966\pi\)
\(42\) −112.000 −0.411476
\(43\) −272.000 −0.964642 −0.482321 − 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(44\) − 140.000i − 0.479677i
\(45\) 0 0
\(46\) 118.000i 0.378221i
\(47\) 464.000 1.44003 0.720014 − 0.693959i \(-0.244135\pi\)
0.720014 + 0.693959i \(0.244135\pi\)
\(48\) − 328.000i − 0.986307i
\(49\) 147.000 0.428571
\(50\) 0 0
\(51\) −544.000 − 136.000i −1.49363 − 0.373408i
\(52\) −406.000 −1.08273
\(53\) −642.000 −1.66388 −0.831939 − 0.554868i \(-0.812769\pi\)
−0.831939 + 0.554868i \(0.812769\pi\)
\(54\) − 80.0000i − 0.201604i
\(55\) 0 0
\(56\) 210.000i 0.501115i
\(57\) − 640.000i − 1.48719i
\(58\) − 126.000i − 0.285252i
\(59\) −180.000 −0.397187 −0.198593 − 0.980082i \(-0.563637\pi\)
−0.198593 + 0.980082i \(0.563637\pi\)
\(60\) 0 0
\(61\) 110.000i 0.230886i 0.993314 + 0.115443i \(0.0368288\pi\)
−0.993314 + 0.115443i \(0.963171\pi\)
\(62\) − 70.0000i − 0.143387i
\(63\) − 518.000i − 1.03590i
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) −160.000 −0.298404
\(67\) 924.000 1.68484 0.842422 − 0.538818i \(-0.181129\pi\)
0.842422 + 0.538818i \(0.181129\pi\)
\(68\) −119.000 + 476.000i −0.212219 + 0.848875i
\(69\) −944.000 −1.64702
\(70\) 0 0
\(71\) 90.0000i 0.150437i 0.997167 + 0.0752186i \(0.0239654\pi\)
−0.997167 + 0.0752186i \(0.976035\pi\)
\(72\) −555.000 −0.908436
\(73\) − 828.000i − 1.32754i −0.747939 − 0.663768i \(-0.768956\pi\)
0.747939 − 0.663768i \(-0.231044\pi\)
\(74\) − 134.000i − 0.210502i
\(75\) 0 0
\(76\) −560.000 −0.845216
\(77\) −280.000 −0.414402
\(78\) 464.000i 0.673560i
\(79\) − 1334.00i − 1.89983i −0.312505 − 0.949916i \(-0.601168\pi\)
0.312505 − 0.949916i \(-0.398832\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 100.000i 0.134673i
\(83\) −552.000 −0.729998 −0.364999 − 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) −784.000 −1.01835
\(85\) 0 0
\(86\) 272.000 0.341052
\(87\) 1008.00 1.24217
\(88\) 300.000i 0.363410i
\(89\) 1490.00 1.77460 0.887302 − 0.461190i \(-0.152577\pi\)
0.887302 + 0.461190i \(0.152577\pi\)
\(90\) 0 0
\(91\) 812.000i 0.935393i
\(92\) 826.000i 0.936048i
\(93\) 560.000 0.624401
\(94\) −464.000 −0.509127
\(95\) 0 0
\(96\) 1288.00i 1.36933i
\(97\) − 1376.00i − 1.44033i −0.693805 − 0.720163i \(-0.744067\pi\)
0.693805 − 0.720163i \(-0.255933\pi\)
\(98\) −147.000 −0.151523
\(99\) − 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.d.a.101.1 2
5.2 odd 4 425.4.c.a.424.1 2
5.3 odd 4 425.4.c.b.424.2 2
5.4 even 2 85.4.d.a.16.2 yes 2
15.14 odd 2 765.4.g.a.271.1 2
17.16 even 2 inner 425.4.d.a.101.2 2
85.4 even 4 1445.4.a.e.1.1 1
85.33 odd 4 425.4.c.a.424.2 2
85.64 even 4 1445.4.a.d.1.1 1
85.67 odd 4 425.4.c.b.424.1 2
85.84 even 2 85.4.d.a.16.1 ✓ 2
255.254 odd 2 765.4.g.a.271.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.d.a.16.1 ✓ 2 85.84 even 2
85.4.d.a.16.2 yes 2 5.4 even 2
425.4.c.a.424.1 2 5.2 odd 4
425.4.c.a.424.2 2 85.33 odd 4
425.4.c.b.424.1 2 85.67 odd 4
425.4.c.b.424.2 2 5.3 odd 4
425.4.d.a.101.1 2 1.1 even 1 trivial
425.4.d.a.101.2 2 17.16 even 2 inner
765.4.g.a.271.1 2 15.14 odd 2
765.4.g.a.271.2 2 255.254 odd 2
1445.4.a.d.1.1 1 85.64 even 4
1445.4.a.e.1.1 1 85.4 even 4