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Properties

Label 6300.2.d.a.3401.4
Level $6300$
Weight $2$
Character 6300.3401
Analytic conductor $50.306$
Analytic rank $0$
Dimension $4$
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] = [6300,2,Mod(3401,6300)] 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("6300.3401"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-24,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3401.4
Root \(2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 6300.3401
Dual form 6300.2.d.a.3401.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.23607 + 1.41421i) q^{7} +1.41421i q^{11} -4.57649i q^{13} +4.47214 q^{17} +4.57649i q^{19} +3.16228i q^{23} +10.5672i q^{29} +1.74806i q^{31} -10.4721 q^{37} -8.47214 q^{41} -8.47214 q^{43} +6.47214 q^{47} +(3.00000 + 6.32456i) q^{49} -2.49458i q^{53} +10.4721 q^{59} +9.15298i q^{61} -4.91034i q^{71} -2.41577i q^{73} +(-2.00000 + 3.16228i) q^{77} -8.94427 q^{79} +14.4721 q^{83} -6.94427 q^{89} +(6.47214 - 10.2333i) q^{91} -19.3863i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{37} - 16 q^{41} - 16 q^{43} + 8 q^{47} + 12 q^{49} + 24 q^{59} - 8 q^{77} + 40 q^{83} + 8 q^{89} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.23607 + 1.41421i 0.845154 + 0.534522i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) − 4.57649i − 1.26929i −0.772804 − 0.634645i \(-0.781146\pi\)
0.772804 − 0.634645i \(-0.218854\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 4.57649i 1.04992i 0.851127 + 0.524960i \(0.175920\pi\)
−0.851127 + 0.524960i \(0.824080\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.16228i 0.659380i 0.944089 + 0.329690i \(0.106944\pi\)
−0.944089 + 0.329690i \(0.893056\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.5672i 1.96228i 0.193301 + 0.981140i \(0.438081\pi\)
−0.193301 + 0.981140i \(0.561919\pi\)
\(30\) 0 0
\(31\) 1.74806i 0.313962i 0.987602 + 0.156981i \(0.0501761\pi\)
−0.987602 + 0.156981i \(0.949824\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.4721 −1.72161 −0.860804 − 0.508936i \(-0.830039\pi\)
−0.860804 + 0.508936i \(0.830039\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.47214 −1.32313 −0.661563 − 0.749890i \(-0.730106\pi\)
−0.661563 + 0.749890i \(0.730106\pi\)
\(42\) 0 0
\(43\) −8.47214 −1.29199 −0.645994 − 0.763342i \(-0.723557\pi\)
−0.645994 + 0.763342i \(0.723557\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.47214 0.944058 0.472029 − 0.881583i \(-0.343522\pi\)
0.472029 + 0.881583i \(0.343522\pi\)
\(48\) 0 0
\(49\) 3.00000 + 6.32456i 0.428571 + 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.49458i − 0.342656i −0.985214 − 0.171328i \(-0.945194\pi\)
0.985214 − 0.171328i \(-0.0548059\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.4721 1.36336 0.681678 − 0.731652i \(-0.261251\pi\)
0.681678 + 0.731652i \(0.261251\pi\)
\(60\) 0 0
\(61\) 9.15298i 1.17192i 0.810340 + 0.585960i \(0.199282\pi\)
−0.810340 + 0.585960i \(0.800718\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 4.91034i − 0.582750i −0.956609 − 0.291375i \(-0.905887\pi\)
0.956609 − 0.291375i \(-0.0941128\pi\)
\(72\) 0 0
\(73\) − 2.41577i − 0.282744i −0.989957 − 0.141372i \(-0.954849\pi\)
0.989957 − 0.141372i \(-0.0451514\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 + 3.16228i −0.227921 + 0.360375i
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 − 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.4721 1.58852 0.794262 − 0.607576i \(-0.207858\pi\)
0.794262 + 0.607576i \(0.207858\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.94427 −0.736091 −0.368046 − 0.929808i \(-0.619973\pi\)
−0.368046 + 0.929808i \(0.619973\pi\)
\(90\) 0 0
\(91\) 6.47214 − 10.2333i 0.678464 − 1.07275i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 19.3863i − 1.96838i −0.177107 − 0.984192i \(-0.556674\pi\)
0.177107 − 0.984192i \(-0.443326\pi\)
\(98\) 0 0
\(99\) 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 6300.2.d.a.3401.4 4
3.2 odd 2 6300.2.d.b.3401.4 4
5.2 odd 4 6300.2.f.a.3149.4 8
5.3 odd 4 6300.2.f.a.3149.6 8
5.4 even 2 1260.2.d.b.881.1 yes 4
7.6 odd 2 6300.2.d.b.3401.3 4
15.2 even 4 6300.2.f.c.3149.3 8
15.8 even 4 6300.2.f.c.3149.5 8
15.14 odd 2 1260.2.d.a.881.1 ✓ 4
20.19 odd 2 5040.2.f.d.881.4 4
21.20 even 2 inner 6300.2.d.a.3401.3 4
35.13 even 4 6300.2.f.c.3149.2 8
35.27 even 4 6300.2.f.c.3149.8 8
35.34 odd 2 1260.2.d.a.881.2 yes 4
60.59 even 2 5040.2.f.b.881.4 4
105.62 odd 4 6300.2.f.a.3149.7 8
105.83 odd 4 6300.2.f.a.3149.1 8
105.104 even 2 1260.2.d.b.881.2 yes 4
140.139 even 2 5040.2.f.b.881.3 4
420.419 odd 2 5040.2.f.d.881.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.d.a.881.1 ✓ 4 15.14 odd 2
1260.2.d.a.881.2 yes 4 35.34 odd 2
1260.2.d.b.881.1 yes 4 5.4 even 2
1260.2.d.b.881.2 yes 4 105.104 even 2
5040.2.f.b.881.3 4 140.139 even 2
5040.2.f.b.881.4 4 60.59 even 2
5040.2.f.d.881.3 4 420.419 odd 2
5040.2.f.d.881.4 4 20.19 odd 2
6300.2.d.a.3401.3 4 21.20 even 2 inner
6300.2.d.a.3401.4 4 1.1 even 1 trivial
6300.2.d.b.3401.3 4 7.6 odd 2
6300.2.d.b.3401.4 4 3.2 odd 2
6300.2.f.a.3149.1 8 105.83 odd 4
6300.2.f.a.3149.4 8 5.2 odd 4
6300.2.f.a.3149.6 8 5.3 odd 4
6300.2.f.a.3149.7 8 105.62 odd 4
6300.2.f.c.3149.2 8 35.13 even 4
6300.2.f.c.3149.3 8 15.2 even 4
6300.2.f.c.3149.5 8 15.8 even 4
6300.2.f.c.3149.8 8 35.27 even 4