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Properties

Label 3150.2.b.c.251.1
Level $3150$
Weight $2$
Character 3150.251
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3150,2,Mod(251,3150)] 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("3150.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(46)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.1
Root \(-1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 3150.251
Dual form 3150.2.b.c.251.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.12132 - 1.58114i) q^{7} +1.00000i q^{8} -1.41421i q^{11} +3.16228i q^{13} +(-1.58114 + 2.12132i) q^{14} +1.00000 q^{16} +4.47214 q^{17} -1.41421 q^{22} +6.00000i q^{23} +3.16228 q^{26} +(2.12132 + 1.58114i) q^{28} -2.82843i q^{29} -1.00000i q^{32} -4.47214i q^{34} +4.24264 q^{37} -9.48683 q^{41} +8.48528 q^{43} +1.41421i q^{44} +6.00000 q^{46} +4.47214 q^{47} +(2.00000 + 6.70820i) q^{49} -3.16228i q^{52} +6.00000i q^{53} +(1.58114 - 2.12132i) q^{56} -2.82843 q^{58} +9.48683 q^{59} -13.4164i q^{61} -1.00000 q^{64} -4.47214 q^{68} -5.65685i q^{71} -6.32456i q^{73} -4.24264i q^{74} +(-2.23607 + 3.00000i) q^{77} +4.00000 q^{79} +9.48683i q^{82} +8.94427 q^{83} -8.48528i q^{86} +1.41421 q^{88} -9.48683 q^{89} +(5.00000 - 6.70820i) q^{91} -6.00000i q^{92} -4.47214i q^{94} -12.6491i q^{97} +(6.70820 - 2.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{16} + 48 q^{46} + 16 q^{49} - 8 q^{64} + 32 q^{79} + 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.12132 − 1.58114i −0.801784 − 0.597614i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.41421i − 0.426401i −0.977008 − 0.213201i \(-0.931611\pi\)
0.977008 − 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 3.16228i 0.877058i 0.898717 + 0.438529i \(0.144500\pi\)
−0.898717 + 0.438529i \(0.855500\pi\)
\(14\) −1.58114 + 2.12132i −0.422577 + 0.566947i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.41421 −0.301511
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.16228 0.620174
\(27\) 0 0
\(28\) 2.12132 + 1.58114i 0.400892 + 0.298807i
\(29\) − 2.82843i − 0.525226i −0.964901 − 0.262613i \(-0.915416\pi\)
0.964901 − 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 4.47214i − 0.766965i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.24264 0.697486 0.348743 − 0.937218i \(-0.386609\pi\)
0.348743 + 0.937218i \(0.386609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.48683 −1.48159 −0.740797 − 0.671729i \(-0.765552\pi\)
−0.740797 + 0.671729i \(0.765552\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 − 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 4.47214 0.652328 0.326164 − 0.945313i \(-0.394244\pi\)
0.326164 + 0.945313i \(0.394244\pi\)
\(48\) 0 0
\(49\) 2.00000 + 6.70820i 0.285714 + 0.958315i
\(50\) 0 0
\(51\) 0 0
\(52\) − 3.16228i − 0.438529i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.58114 − 2.12132i 0.211289 − 0.283473i
\(57\) 0 0
\(58\) −2.82843 −0.371391
\(59\) 9.48683 1.23508 0.617540 − 0.786539i \(-0.288129\pi\)
0.617540 + 0.786539i \(0.288129\pi\)
\(60\) 0 0
\(61\) − 13.4164i − 1.71780i −0.512148 − 0.858898i \(-0.671150\pi\)
0.512148 − 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −4.47214 −0.542326
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.65685i − 0.671345i −0.941979 − 0.335673i \(-0.891036\pi\)
0.941979 − 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) − 6.32456i − 0.740233i −0.928985 − 0.370117i \(-0.879318\pi\)
0.928985 − 0.370117i \(-0.120682\pi\)
\(74\) − 4.24264i − 0.493197i
\(75\) 0 0
\(76\) 0 0
\(77\) −2.23607 + 3.00000i −0.254824 + 0.341882i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.48683i 1.04765i
\(83\) 8.94427 0.981761 0.490881 − 0.871227i \(-0.336675\pi\)
0.490881 + 0.871227i \(0.336675\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 8.48528i − 0.914991i
\(87\) 0 0
\(88\) 1.41421 0.150756
\(89\) −9.48683 −1.00560 −0.502801 − 0.864402i \(-0.667697\pi\)
−0.502801 + 0.864402i \(0.667697\pi\)
\(90\) 0 0
\(91\) 5.00000 − 6.70820i 0.524142 − 0.703211i
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) − 4.47214i − 0.461266i
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.6491i − 1.28432i −0.766570 − 0.642161i \(-0.778038\pi\)
0.766570 − 0.642161i \(-0.221962\pi\)
\(98\) 6.70820 − 2.00000i 0.677631 − 0.202031i
\(99\) 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 3150.2.b.c.251.1 8
3.2 odd 2 inner 3150.2.b.c.251.5 8
5.2 odd 4 630.2.d.d.629.4 yes 4
5.3 odd 4 630.2.d.a.629.1 ✓ 4
5.4 even 2 inner 3150.2.b.c.251.8 8
7.6 odd 2 inner 3150.2.b.c.251.2 8
15.2 even 4 630.2.d.a.629.2 yes 4
15.8 even 4 630.2.d.d.629.3 yes 4
15.14 odd 2 inner 3150.2.b.c.251.4 8
20.3 even 4 5040.2.k.d.1889.2 4
20.7 even 4 5040.2.k.a.1889.3 4
21.20 even 2 inner 3150.2.b.c.251.6 8
35.13 even 4 630.2.d.a.629.4 yes 4
35.27 even 4 630.2.d.d.629.1 yes 4
35.34 odd 2 inner 3150.2.b.c.251.7 8
60.23 odd 4 5040.2.k.a.1889.4 4
60.47 odd 4 5040.2.k.d.1889.1 4
105.62 odd 4 630.2.d.a.629.3 yes 4
105.83 odd 4 630.2.d.d.629.2 yes 4
105.104 even 2 inner 3150.2.b.c.251.3 8
140.27 odd 4 5040.2.k.a.1889.2 4
140.83 odd 4 5040.2.k.d.1889.3 4
420.83 even 4 5040.2.k.a.1889.1 4
420.167 even 4 5040.2.k.d.1889.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.d.a.629.1 ✓ 4 5.3 odd 4
630.2.d.a.629.2 yes 4 15.2 even 4
630.2.d.a.629.3 yes 4 105.62 odd 4
630.2.d.a.629.4 yes 4 35.13 even 4
630.2.d.d.629.1 yes 4 35.27 even 4
630.2.d.d.629.2 yes 4 105.83 odd 4
630.2.d.d.629.3 yes 4 15.8 even 4
630.2.d.d.629.4 yes 4 5.2 odd 4
3150.2.b.c.251.1 8 1.1 even 1 trivial
3150.2.b.c.251.2 8 7.6 odd 2 inner
3150.2.b.c.251.3 8 105.104 even 2 inner
3150.2.b.c.251.4 8 15.14 odd 2 inner
3150.2.b.c.251.5 8 3.2 odd 2 inner
3150.2.b.c.251.6 8 21.20 even 2 inner
3150.2.b.c.251.7 8 35.34 odd 2 inner
3150.2.b.c.251.8 8 5.4 even 2 inner
5040.2.k.a.1889.1 4 420.83 even 4
5040.2.k.a.1889.2 4 140.27 odd 4
5040.2.k.a.1889.3 4 20.7 even 4
5040.2.k.a.1889.4 4 60.23 odd 4
5040.2.k.d.1889.1 4 60.47 odd 4
5040.2.k.d.1889.2 4 20.3 even 4
5040.2.k.d.1889.3 4 140.83 odd 4
5040.2.k.d.1889.4 4 420.167 even 4