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Properties

Label 425.4.b.c.324.1
Level $425$
Weight $4$
Character 425.324
Analytic conductor $25.076$
Analytic rank $1$
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(324,425)] 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("425.324"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,48] 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: \(25.0758117524\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.c.324.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} +8.00000i q^{3} -1.00000 q^{4} +24.0000 q^{6} -28.0000i q^{7} -21.0000i q^{8} -37.0000 q^{9} -24.0000 q^{11} -8.00000i q^{12} +58.0000i q^{13} -84.0000 q^{14} -71.0000 q^{16} +17.0000i q^{17} +111.000i q^{18} -116.000 q^{19} +224.000 q^{21} +72.0000i q^{22} +60.0000i q^{23} +168.000 q^{24} +174.000 q^{26} -80.0000i q^{27} +28.0000i q^{28} -30.0000 q^{29} -172.000 q^{31} +45.0000i q^{32} -192.000i q^{33} +51.0000 q^{34} +37.0000 q^{36} -58.0000i q^{37} +348.000i q^{38} -464.000 q^{39} -342.000 q^{41} -672.000i q^{42} +148.000i q^{43} +24.0000 q^{44} +180.000 q^{46} +288.000i q^{47} -568.000i q^{48} -441.000 q^{49} -136.000 q^{51} -58.0000i q^{52} -318.000i q^{53} -240.000 q^{54} -588.000 q^{56} -928.000i q^{57} +90.0000i q^{58} -252.000 q^{59} +110.000 q^{61} +516.000i q^{62} +1036.00i q^{63} -433.000 q^{64} -576.000 q^{66} -484.000i q^{67} -17.0000i q^{68} -480.000 q^{69} -708.000 q^{71} +777.000i q^{72} -362.000i q^{73} -174.000 q^{74} +116.000 q^{76} +672.000i q^{77} +1392.00i q^{78} +484.000 q^{79} -359.000 q^{81} +1026.00i q^{82} -756.000i q^{83} -224.000 q^{84} +444.000 q^{86} -240.000i q^{87} +504.000i q^{88} +774.000 q^{89} +1624.00 q^{91} -60.0000i q^{92} -1376.00i q^{93} +864.000 q^{94} -360.000 q^{96} -382.000i q^{97} +1323.00i q^{98} +888.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 48 q^{6} - 74 q^{9} - 48 q^{11} - 168 q^{14} - 142 q^{16} - 232 q^{19} + 448 q^{21} + 336 q^{24} + 348 q^{26} - 60 q^{29} - 344 q^{31} + 102 q^{34} + 74 q^{36} - 928 q^{39} - 684 q^{41}+ \cdots + 1776 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.06066i −0.847791 − 0.530330i \(-0.822068\pi\)
0.847791 − 0.530330i \(-0.177932\pi\)
\(3\) 8.00000i 1.53960i 0.638285 + 0.769800i \(0.279644\pi\)
−0.638285 + 0.769800i \(0.720356\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 24.0000 1.63299
\(7\) − 28.0000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) − 21.0000i − 0.928078i
\(9\) −37.0000 −1.37037
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 − 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) − 8.00000i − 0.192450i
\(13\) 58.0000i 1.23741i 0.785624 + 0.618704i \(0.212342\pi\)
−0.785624 + 0.618704i \(0.787658\pi\)
\(14\) −84.0000 −1.60357
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 17.0000i 0.242536i
\(18\) 111.000i 1.45350i
\(19\) −116.000 −1.40064 −0.700322 − 0.713827i \(-0.746960\pi\)
−0.700322 + 0.713827i \(0.746960\pi\)
\(20\) 0 0
\(21\) 224.000 2.32766
\(22\) 72.0000i 0.697748i
\(23\) 60.0000i 0.543951i 0.962304 + 0.271975i \(0.0876769\pi\)
−0.962304 + 0.271975i \(0.912323\pi\)
\(24\) 168.000 1.42887
\(25\) 0 0
\(26\) 174.000 1.31247
\(27\) − 80.0000i − 0.570222i
\(28\) 28.0000i 0.188982i
\(29\) −30.0000 −0.192099 −0.0960493 − 0.995377i \(-0.530621\pi\)
−0.0960493 + 0.995377i \(0.530621\pi\)
\(30\) 0 0
\(31\) −172.000 −0.996520 −0.498260 − 0.867028i \(-0.666027\pi\)
−0.498260 + 0.867028i \(0.666027\pi\)
\(32\) 45.0000i 0.248592i
\(33\) − 192.000i − 1.01282i
\(34\) 51.0000 0.257248
\(35\) 0 0
\(36\) 37.0000 0.171296
\(37\) − 58.0000i − 0.257707i −0.991664 − 0.128853i \(-0.958870\pi\)
0.991664 − 0.128853i \(-0.0411296\pi\)
\(38\) 348.000i 1.48561i
\(39\) −464.000 −1.90511
\(40\) 0 0
\(41\) −342.000 −1.30272 −0.651359 − 0.758770i \(-0.725801\pi\)
−0.651359 + 0.758770i \(0.725801\pi\)
\(42\) − 672.000i − 2.46885i
\(43\) 148.000i 0.524879i 0.964948 + 0.262439i \(0.0845270\pi\)
−0.964948 + 0.262439i \(0.915473\pi\)
\(44\) 24.0000 0.0822304
\(45\) 0 0
\(46\) 180.000 0.576947
\(47\) 288.000i 0.893811i 0.894581 + 0.446906i \(0.147474\pi\)
−0.894581 + 0.446906i \(0.852526\pi\)
\(48\) − 568.000i − 1.70799i
\(49\) −441.000 −1.28571
\(50\) 0 0
\(51\) −136.000 −0.373408
\(52\) − 58.0000i − 0.154676i
\(53\) − 318.000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) −240.000 −0.604812
\(55\) 0 0
\(56\) −588.000 −1.40312
\(57\) − 928.000i − 2.15643i
\(58\) 90.0000i 0.203751i
\(59\) −252.000 −0.556061 −0.278031 − 0.960572i \(-0.589682\pi\)
−0.278031 + 0.960572i \(0.589682\pi\)
\(60\) 0 0
\(61\) 110.000 0.230886 0.115443 − 0.993314i \(-0.463171\pi\)
0.115443 + 0.993314i \(0.463171\pi\)
\(62\) 516.000i 1.05697i
\(63\) 1036.00i 2.07181i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) −576.000 −1.07425
\(67\) − 484.000i − 0.882537i −0.897375 − 0.441269i \(-0.854529\pi\)
0.897375 − 0.441269i \(-0.145471\pi\)
\(68\) − 17.0000i − 0.0303170i
\(69\) −480.000 −0.837467
\(70\) 0 0
\(71\) −708.000 −1.18344 −0.591719 − 0.806144i \(-0.701551\pi\)
−0.591719 + 0.806144i \(0.701551\pi\)
\(72\) 777.000i 1.27181i
\(73\) − 362.000i − 0.580396i −0.956967 − 0.290198i \(-0.906279\pi\)
0.956967 − 0.290198i \(-0.0937211\pi\)
\(74\) −174.000 −0.273339
\(75\) 0 0
\(76\) 116.000 0.175080
\(77\) 672.000i 0.994565i
\(78\) 1392.00i 2.02068i
\(79\) 484.000 0.689294 0.344647 − 0.938732i \(-0.387999\pi\)
0.344647 + 0.938732i \(0.387999\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 1026.00i 1.38174i
\(83\) − 756.000i − 0.999780i −0.866089 − 0.499890i \(-0.833374\pi\)
0.866089 − 0.499890i \(-0.166626\pi\)
\(84\) −224.000 −0.290957
\(85\) 0 0
\(86\) 444.000 0.556718
\(87\) − 240.000i − 0.295755i
\(88\) 504.000i 0.610529i
\(89\) 774.000 0.921841 0.460920 − 0.887441i \(-0.347519\pi\)
0.460920 + 0.887441i \(0.347519\pi\)
\(90\) 0 0
\(91\) 1624.00 1.87079
\(92\) − 60.0000i − 0.0679938i
\(93\) − 1376.00i − 1.53424i
\(94\) 864.000 0.948030
\(95\) 0 0
\(96\) −360.000 −0.382733
\(97\) − 382.000i − 0.399858i −0.979810 − 0.199929i \(-0.935929\pi\)
0.979810 − 0.199929i \(-0.0640711\pi\)
\(98\) 1323.00i 1.36371i
\(99\) 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.b.c.324.1 2
5.2 odd 4 425.4.a.d.1.1 1
5.3 odd 4 17.4.a.a.1.1 ✓ 1
5.4 even 2 inner 425.4.b.c.324.2 2
15.8 even 4 153.4.a.d.1.1 1
20.3 even 4 272.4.a.d.1.1 1
35.13 even 4 833.4.a.a.1.1 1
40.3 even 4 1088.4.a.a.1.1 1
40.13 odd 4 1088.4.a.l.1.1 1
55.43 even 4 2057.4.a.d.1.1 1
60.23 odd 4 2448.4.a.f.1.1 1
85.13 odd 4 289.4.b.a.288.1 2
85.33 odd 4 289.4.a.a.1.1 1
85.38 odd 4 289.4.b.a.288.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.a.1.1 ✓ 1 5.3 odd 4
153.4.a.d.1.1 1 15.8 even 4
272.4.a.d.1.1 1 20.3 even 4
289.4.a.a.1.1 1 85.33 odd 4
289.4.b.a.288.1 2 85.13 odd 4
289.4.b.a.288.2 2 85.38 odd 4
425.4.a.d.1.1 1 5.2 odd 4
425.4.b.c.324.1 2 1.1 even 1 trivial
425.4.b.c.324.2 2 5.4 even 2 inner
833.4.a.a.1.1 1 35.13 even 4
1088.4.a.a.1.1 1 40.3 even 4
1088.4.a.l.1.1 1 40.13 odd 4
2057.4.a.d.1.1 1 55.43 even 4
2448.4.a.f.1.1 1 60.23 odd 4