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Properties

Label 425.2.b.b.324.1
Level $425$
Weight $2$
Character 425.324
Analytic conductor $3.394$
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] = [425,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2,0,0] 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: \(3.39364208590\)
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 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 425.2.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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{23} +2.00000 q^{26} +4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +1.00000 q^{34} +3.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{46} -9.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +12.0000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} -4.00000i q^{62} +12.0000i q^{63} -7.00000 q^{64} +4.00000i q^{67} +1.00000i q^{68} -4.00000 q^{71} -9.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} +9.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} -4.00000i q^{92} +2.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71}+ \cdots - 16 q^{91}+O(q^{100}) \) 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 − 0.707107i −0.935414 − 0.353553i \(-0.884973\pi\)
0.935414 − 0.353553i \(-0.115027\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.00000i 0.242536i
\(18\) − 3.00000i − 0.707107i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 4.00000i 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 12.0000i 1.51186i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 1.00000i 0.121268i
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) − 9.00000i − 1.06066i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 6.00000i 0.662589i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
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 425.2.b.b.324.1 2
5.2 odd 4 425.2.a.d.1.1 1
5.3 odd 4 17.2.a.a.1.1 ✓ 1
5.4 even 2 inner 425.2.b.b.324.2 2
15.2 even 4 3825.2.a.d.1.1 1
15.8 even 4 153.2.a.c.1.1 1
20.3 even 4 272.2.a.b.1.1 1
20.7 even 4 6800.2.a.n.1.1 1
35.3 even 12 833.2.e.a.324.1 2
35.13 even 4 833.2.a.a.1.1 1
35.18 odd 12 833.2.e.b.324.1 2
35.23 odd 12 833.2.e.b.18.1 2
35.33 even 12 833.2.e.a.18.1 2
40.3 even 4 1088.2.a.h.1.1 1
40.13 odd 4 1088.2.a.i.1.1 1
55.43 even 4 2057.2.a.e.1.1 1
60.23 odd 4 2448.2.a.o.1.1 1
65.38 odd 4 2873.2.a.c.1.1 1
85.3 even 16 289.2.d.d.179.2 8
85.8 odd 8 289.2.c.a.251.1 4
85.13 odd 4 289.2.b.a.288.1 2
85.23 even 16 289.2.d.d.155.2 8
85.28 even 16 289.2.d.d.155.1 8
85.33 odd 4 289.2.a.a.1.1 1
85.38 odd 4 289.2.b.a.288.2 2
85.43 odd 8 289.2.c.a.251.2 4
85.48 even 16 289.2.d.d.179.1 8
85.53 odd 8 289.2.c.a.38.2 4
85.58 even 16 289.2.d.d.134.2 8
85.63 even 16 289.2.d.d.110.1 8
85.67 odd 4 7225.2.a.g.1.1 1
85.73 even 16 289.2.d.d.110.2 8
85.78 even 16 289.2.d.d.134.1 8
85.83 odd 8 289.2.c.a.38.1 4
95.18 even 4 6137.2.a.b.1.1 1
105.83 odd 4 7497.2.a.l.1.1 1
115.68 even 4 8993.2.a.a.1.1 1
120.53 even 4 9792.2.a.n.1.1 1
120.83 odd 4 9792.2.a.i.1.1 1
255.203 even 4 2601.2.a.g.1.1 1
340.203 even 4 4624.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 ✓ 1 5.3 odd 4
153.2.a.c.1.1 1 15.8 even 4
272.2.a.b.1.1 1 20.3 even 4
289.2.a.a.1.1 1 85.33 odd 4
289.2.b.a.288.1 2 85.13 odd 4
289.2.b.a.288.2 2 85.38 odd 4
289.2.c.a.38.1 4 85.83 odd 8
289.2.c.a.38.2 4 85.53 odd 8
289.2.c.a.251.1 4 85.8 odd 8
289.2.c.a.251.2 4 85.43 odd 8
289.2.d.d.110.1 8 85.63 even 16
289.2.d.d.110.2 8 85.73 even 16
289.2.d.d.134.1 8 85.78 even 16
289.2.d.d.134.2 8 85.58 even 16
289.2.d.d.155.1 8 85.28 even 16
289.2.d.d.155.2 8 85.23 even 16
289.2.d.d.179.1 8 85.48 even 16
289.2.d.d.179.2 8 85.3 even 16
425.2.a.d.1.1 1 5.2 odd 4
425.2.b.b.324.1 2 1.1 even 1 trivial
425.2.b.b.324.2 2 5.4 even 2 inner
833.2.a.a.1.1 1 35.13 even 4
833.2.e.a.18.1 2 35.33 even 12
833.2.e.a.324.1 2 35.3 even 12
833.2.e.b.18.1 2 35.23 odd 12
833.2.e.b.324.1 2 35.18 odd 12
1088.2.a.h.1.1 1 40.3 even 4
1088.2.a.i.1.1 1 40.13 odd 4
2057.2.a.e.1.1 1 55.43 even 4
2448.2.a.o.1.1 1 60.23 odd 4
2601.2.a.g.1.1 1 255.203 even 4
2873.2.a.c.1.1 1 65.38 odd 4
3825.2.a.d.1.1 1 15.2 even 4
4624.2.a.d.1.1 1 340.203 even 4
6137.2.a.b.1.1 1 95.18 even 4
6800.2.a.n.1.1 1 20.7 even 4
7225.2.a.g.1.1 1 85.67 odd 4
7497.2.a.l.1.1 1 105.83 odd 4
8993.2.a.a.1.1 1 115.68 even 4
9792.2.a.i.1.1 1 120.83 odd 4
9792.2.a.n.1.1 1 120.53 even 4