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Properties

Label 425.4.b.b.324.1
Level $425$
Weight $4$
Character 425.324
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(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,-30] 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: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
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.b.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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} -5.00000i q^{3} -1.00000 q^{4} -15.0000 q^{6} +22.0000i q^{7} -21.0000i q^{8} +2.00000 q^{9} +60.0000 q^{11} +5.00000i q^{12} -31.0000i q^{13} +66.0000 q^{14} -71.0000 q^{16} -17.0000i q^{17} -6.00000i q^{18} +61.0000 q^{19} +110.000 q^{21} -180.000i q^{22} -78.0000i q^{23} -105.000 q^{24} -93.0000 q^{26} -145.000i q^{27} -22.0000i q^{28} -69.0000 q^{29} -31.0000 q^{31} +45.0000i q^{32} -300.000i q^{33} -51.0000 q^{34} -2.00000 q^{36} -56.0000i q^{37} -183.000i q^{38} -155.000 q^{39} -6.00000 q^{41} -330.000i q^{42} -538.000i q^{43} -60.0000 q^{44} -234.000 q^{46} +465.000i q^{47} +355.000i q^{48} -141.000 q^{49} -85.0000 q^{51} +31.0000i q^{52} +723.000i q^{53} -435.000 q^{54} +462.000 q^{56} -305.000i q^{57} +207.000i q^{58} +753.000 q^{59} +35.0000 q^{61} +93.0000i q^{62} +44.0000i q^{63} -433.000 q^{64} -900.000 q^{66} +322.000i q^{67} +17.0000i q^{68} -390.000 q^{69} -99.0000 q^{71} -42.0000i q^{72} -1123.00i q^{73} -168.000 q^{74} -61.0000 q^{76} +1320.00i q^{77} +465.000i q^{78} -488.000 q^{79} -671.000 q^{81} +18.0000i q^{82} -852.000i q^{83} -110.000 q^{84} -1614.00 q^{86} +345.000i q^{87} -1260.00i q^{88} -1215.00 q^{89} +682.000 q^{91} +78.0000i q^{92} +155.000i q^{93} +1395.00 q^{94} +225.000 q^{96} +601.000i q^{97} +423.000i q^{98} +120.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 30 q^{6} + 4 q^{9} + 120 q^{11} + 132 q^{14} - 142 q^{16} + 122 q^{19} + 220 q^{21} - 210 q^{24} - 186 q^{26} - 138 q^{29} - 62 q^{31} - 102 q^{34} - 4 q^{36} - 310 q^{39} - 12 q^{41} - 120 q^{44}+ \cdots + 240 q^{99}+O(q^{100}) \) 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 \(-0.177932\pi\)
\(3\) − 5.00000i − 0.962250i −0.876652 − 0.481125i \(-0.840228\pi\)
0.876652 − 0.481125i \(-0.159772\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) −15.0000 −1.02062
\(7\) 22.0000i 1.18789i 0.804506 + 0.593944i \(0.202430\pi\)
−0.804506 + 0.593944i \(0.797570\pi\)
\(8\) − 21.0000i − 0.928078i
\(9\) 2.00000 0.0740741
\(10\) 0 0
\(11\) 60.0000 1.64461 0.822304 − 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) 5.00000i 0.120281i
\(13\) − 31.0000i − 0.661373i −0.943741 − 0.330687i \(-0.892720\pi\)
0.943741 − 0.330687i \(-0.107280\pi\)
\(14\) 66.0000 1.25995
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) − 17.0000i − 0.242536i
\(18\) − 6.00000i − 0.0785674i
\(19\) 61.0000 0.736545 0.368273 − 0.929718i \(-0.379949\pi\)
0.368273 + 0.929718i \(0.379949\pi\)
\(20\) 0 0
\(21\) 110.000 1.14305
\(22\) − 180.000i − 1.74437i
\(23\) − 78.0000i − 0.707136i −0.935409 − 0.353568i \(-0.884968\pi\)
0.935409 − 0.353568i \(-0.115032\pi\)
\(24\) −105.000 −0.893043
\(25\) 0 0
\(26\) −93.0000 −0.701492
\(27\) − 145.000i − 1.03353i
\(28\) − 22.0000i − 0.148486i
\(29\) −69.0000 −0.441827 −0.220913 − 0.975293i \(-0.570904\pi\)
−0.220913 + 0.975293i \(0.570904\pi\)
\(30\) 0 0
\(31\) −31.0000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 45.0000i 0.248592i
\(33\) − 300.000i − 1.58252i
\(34\) −51.0000 −0.257248
\(35\) 0 0
\(36\) −2.00000 −0.00925926
\(37\) − 56.0000i − 0.248820i −0.992231 − 0.124410i \(-0.960296\pi\)
0.992231 − 0.124410i \(-0.0397038\pi\)
\(38\) − 183.000i − 0.781224i
\(39\) −155.000 −0.636407
\(40\) 0 0
\(41\) −6.00000 −0.0228547 −0.0114273 − 0.999935i \(-0.503638\pi\)
−0.0114273 + 0.999935i \(0.503638\pi\)
\(42\) − 330.000i − 1.21238i
\(43\) − 538.000i − 1.90801i −0.299798 − 0.954003i \(-0.596919\pi\)
0.299798 − 0.954003i \(-0.403081\pi\)
\(44\) −60.0000 −0.205576
\(45\) 0 0
\(46\) −234.000 −0.750031
\(47\) 465.000i 1.44313i 0.692345 + 0.721566i \(0.256577\pi\)
−0.692345 + 0.721566i \(0.743423\pi\)
\(48\) 355.000i 1.06750i
\(49\) −141.000 −0.411079
\(50\) 0 0
\(51\) −85.0000 −0.233380
\(52\) 31.0000i 0.0826717i
\(53\) 723.000i 1.87381i 0.349590 + 0.936903i \(0.386321\pi\)
−0.349590 + 0.936903i \(0.613679\pi\)
\(54\) −435.000 −1.09622
\(55\) 0 0
\(56\) 462.000 1.10245
\(57\) − 305.000i − 0.708741i
\(58\) 207.000i 0.468628i
\(59\) 753.000 1.66156 0.830782 − 0.556598i \(-0.187894\pi\)
0.830782 + 0.556598i \(0.187894\pi\)
\(60\) 0 0
\(61\) 35.0000 0.0734638 0.0367319 − 0.999325i \(-0.488305\pi\)
0.0367319 + 0.999325i \(0.488305\pi\)
\(62\) 93.0000i 0.190500i
\(63\) 44.0000i 0.0879917i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) −900.000 −1.67852
\(67\) 322.000i 0.587143i 0.955937 + 0.293571i \(0.0948438\pi\)
−0.955937 + 0.293571i \(0.905156\pi\)
\(68\) 17.0000i 0.0303170i
\(69\) −390.000 −0.680442
\(70\) 0 0
\(71\) −99.0000 −0.165481 −0.0827404 − 0.996571i \(-0.526367\pi\)
−0.0827404 + 0.996571i \(0.526367\pi\)
\(72\) − 42.0000i − 0.0687465i
\(73\) − 1123.00i − 1.80051i −0.435363 − 0.900255i \(-0.643380\pi\)
0.435363 − 0.900255i \(-0.356620\pi\)
\(74\) −168.000 −0.263914
\(75\) 0 0
\(76\) −61.0000 −0.0920682
\(77\) 1320.00i 1.95361i
\(78\) 465.000i 0.675011i
\(79\) −488.000 −0.694991 −0.347496 − 0.937682i \(-0.612968\pi\)
−0.347496 + 0.937682i \(0.612968\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 18.0000i 0.0242411i
\(83\) − 852.000i − 1.12674i −0.826206 − 0.563368i \(-0.809505\pi\)
0.826206 − 0.563368i \(-0.190495\pi\)
\(84\) −110.000 −0.142881
\(85\) 0 0
\(86\) −1614.00 −2.02375
\(87\) 345.000i 0.425148i
\(88\) − 1260.00i − 1.52632i
\(89\) −1215.00 −1.44708 −0.723538 − 0.690285i \(-0.757485\pi\)
−0.723538 + 0.690285i \(0.757485\pi\)
\(90\) 0 0
\(91\) 682.000 0.785638
\(92\) 78.0000i 0.0883920i
\(93\) 155.000i 0.172825i
\(94\) 1395.00 1.53067
\(95\) 0 0
\(96\) 225.000 0.239208
\(97\) 601.000i 0.629096i 0.949242 + 0.314548i \(0.101853\pi\)
−0.949242 + 0.314548i \(0.898147\pi\)
\(98\) 423.000i 0.436015i
\(99\) 120.000 0.121823
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.b.324.1 2
5.2 odd 4 85.4.a.b.1.1 ✓ 1
5.3 odd 4 425.4.a.b.1.1 1
5.4 even 2 inner 425.4.b.b.324.2 2
15.2 even 4 765.4.a.c.1.1 1
20.7 even 4 1360.4.a.g.1.1 1
85.67 odd 4 1445.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.b.1.1 ✓ 1 5.2 odd 4
425.4.a.b.1.1 1 5.3 odd 4
425.4.b.b.324.1 2 1.1 even 1 trivial
425.4.b.b.324.2 2 5.4 even 2 inner
765.4.a.c.1.1 1 15.2 even 4
1360.4.a.g.1.1 1 20.7 even 4
1445.4.a.g.1.1 1 85.67 odd 4