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Properties

Label 448.2.m.c.337.3
Level $448$
Weight $2$
Character 448.337
Analytic conductor $3.577$
Analytic rank $0$
Dimension $8$
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] = [448,2,Mod(113,448)] 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("448.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.214798336.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 337.3
Root \(1.41216 - 0.0762223i\) of defining polynomial
Character \(\chi\) \(=\) 448.337
Dual form 448.2.m.c.113.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.715276 + 0.715276i) q^{3} +(0.867721 - 0.867721i) q^{5} -1.00000i q^{7} -1.97676i q^{9} +(2.97676 - 2.97676i) q^{11} +(-2.02017 - 2.02017i) q^{13} +1.24132 q^{15} +0.264559 q^{17} +(4.53959 + 4.53959i) q^{19} +(0.715276 - 0.715276i) q^{21} +1.54621i q^{23} +3.49412i q^{25} +(3.55976 - 3.55976i) q^{27} +(-0.328129 - 0.328129i) q^{29} -6.04033 q^{31} +4.25841 q^{33} +(-0.867721 - 0.867721i) q^{35} +(6.64863 - 6.64863i) q^{37} -2.88995i q^{39} +11.0327i q^{41} +(-3.38407 + 3.38407i) q^{43} +(-1.71528 - 1.71528i) q^{45} -3.12566 q^{47} -1.00000 q^{49} +(0.189233 + 0.189233i) q^{51} +(0.430552 - 0.430552i) q^{53} -5.16599i q^{55} +6.49412i q^{57} +(4.62640 - 4.62640i) q^{59} +(4.86772 + 4.86772i) q^{61} -1.97676 q^{63} -3.50588 q^{65} +(-3.34374 - 3.34374i) q^{67} +(-1.10597 + 1.10597i) q^{69} +9.03885i q^{71} -14.8146i q^{73} +(-2.49926 + 2.49926i) q^{75} +(-2.97676 - 2.97676i) q^{77} -12.5904 q^{79} -0.837864 q^{81} +(0.715276 + 0.715276i) q^{83} +(0.229563 - 0.229563i) q^{85} -0.469405i q^{87} +10.9924i q^{89} +(-2.02017 + 2.02017i) q^{91} +(-4.32050 - 4.32050i) q^{93} +7.87820 q^{95} -14.2452 q^{97} +(-5.88434 - 5.88434i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 8 q^{15} + 24 q^{17} + 12 q^{19} - 12 q^{27} - 16 q^{29} - 16 q^{31} - 24 q^{33} + 4 q^{35} + 16 q^{37} + 32 q^{43} - 8 q^{45} - 24 q^{47} - 8 q^{49} - 8 q^{51} - 8 q^{53} + 28 q^{59} + 28 q^{61}+ \cdots - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 0.715276 + 0.715276i 0.412965 + 0.412965i 0.882770 − 0.469805i \(-0.155676\pi\)
−0.469805 + 0.882770i \(0.655676\pi\)
\(4\) 0 0
\(5\) 0.867721 − 0.867721i 0.388056 − 0.388056i −0.485937 − 0.873994i \(-0.661522\pi\)
0.873994 + 0.485937i \(0.161522\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) − 1.97676i − 0.658920i
\(10\) 0 0
\(11\) 2.97676 − 2.97676i 0.897527 − 0.897527i −0.0976898 − 0.995217i \(-0.531145\pi\)
0.995217 + 0.0976898i \(0.0311453\pi\)
\(12\) 0 0
\(13\) −2.02017 − 2.02017i −0.560293 − 0.560293i 0.369098 − 0.929391i \(-0.379667\pi\)
−0.929391 + 0.369098i \(0.879667\pi\)
\(14\) 0 0
\(15\) 1.24132 0.320507
\(16\) 0 0
\(17\) 0.264559 0.0641649 0.0320825 − 0.999485i \(-0.489786\pi\)
0.0320825 + 0.999485i \(0.489786\pi\)
\(18\) 0 0
\(19\) 4.53959 + 4.53959i 1.04145 + 1.04145i 0.999103 + 0.0423510i \(0.0134848\pi\)
0.0423510 + 0.999103i \(0.486515\pi\)
\(20\) 0 0
\(21\) 0.715276 − 0.715276i 0.156086 − 0.156086i
\(22\) 0 0
\(23\) 1.54621i 0.322407i 0.986921 + 0.161203i \(0.0515375\pi\)
−0.986921 + 0.161203i \(0.948462\pi\)
\(24\) 0 0
\(25\) 3.49412i 0.698824i
\(26\) 0 0
\(27\) 3.55976 − 3.55976i 0.685076 − 0.685076i
\(28\) 0 0
\(29\) −0.328129 − 0.328129i −0.0609320 − 0.0609320i 0.675984 − 0.736916i \(-0.263719\pi\)
−0.736916 + 0.675984i \(0.763719\pi\)
\(30\) 0 0
\(31\) −6.04033 −1.08488 −0.542438 − 0.840096i \(-0.682499\pi\)
−0.542438 + 0.840096i \(0.682499\pi\)
\(32\) 0 0
\(33\) 4.25841 0.741294
\(34\) 0 0
\(35\) −0.867721 − 0.867721i −0.146672 − 0.146672i
\(36\) 0 0
\(37\) 6.64863 − 6.64863i 1.09303 − 1.09303i 0.0978247 − 0.995204i \(-0.468812\pi\)
0.995204 − 0.0978247i \(-0.0311884\pi\)
\(38\) 0 0
\(39\) − 2.88995i − 0.462763i
\(40\) 0 0
\(41\) 11.0327i 1.72302i 0.507741 + 0.861510i \(0.330481\pi\)
−0.507741 + 0.861510i \(0.669519\pi\)
\(42\) 0 0
\(43\) −3.38407 + 3.38407i −0.516066 + 0.516066i −0.916379 − 0.400312i \(-0.868902\pi\)
0.400312 + 0.916379i \(0.368902\pi\)
\(44\) 0 0
\(45\) −1.71528 − 1.71528i −0.255698 − 0.255698i
\(46\) 0 0
\(47\) −3.12566 −0.455925 −0.227962 − 0.973670i \(-0.573206\pi\)
−0.227962 + 0.973670i \(0.573206\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0.189233 + 0.189233i 0.0264979 + 0.0264979i
\(52\) 0 0
\(53\) 0.430552 − 0.430552i 0.0591409 − 0.0591409i −0.676918 − 0.736059i \(-0.736685\pi\)
0.736059 + 0.676918i \(0.236685\pi\)
\(54\) 0 0
\(55\) − 5.16599i − 0.696582i
\(56\) 0 0
\(57\) 6.49412i 0.860167i
\(58\) 0 0
\(59\) 4.62640 − 4.62640i 0.602306 − 0.602306i −0.338618 − 0.940924i \(-0.609959\pi\)
0.940924 + 0.338618i \(0.109959\pi\)
\(60\) 0 0
\(61\) 4.86772 + 4.86772i 0.623248 + 0.623248i 0.946360 − 0.323113i \(-0.104729\pi\)
−0.323113 + 0.946360i \(0.604729\pi\)
\(62\) 0 0
\(63\) −1.97676 −0.249048
\(64\) 0 0
\(65\) −3.50588 −0.434851
\(66\) 0 0
\(67\) −3.34374 − 3.34374i −0.408503 − 0.408503i 0.472713 − 0.881216i \(-0.343275\pi\)
−0.881216 + 0.472713i \(0.843275\pi\)
\(68\) 0 0
\(69\) −1.10597 + 1.10597i −0.133143 + 0.133143i
\(70\) 0 0
\(71\) 9.03885i 1.07271i 0.843991 + 0.536357i \(0.180200\pi\)
−0.843991 + 0.536357i \(0.819800\pi\)
\(72\) 0 0
\(73\) − 14.8146i − 1.73392i −0.498377 − 0.866960i \(-0.666070\pi\)
0.498377 − 0.866960i \(-0.333930\pi\)
\(74\) 0 0
\(75\) −2.49926 + 2.49926i −0.288590 + 0.288590i
\(76\) 0 0
\(77\) −2.97676 − 2.97676i −0.339233 − 0.339233i
\(78\) 0 0
\(79\) −12.5904 −1.41653 −0.708265 − 0.705947i \(-0.750522\pi\)
−0.708265 + 0.705947i \(0.750522\pi\)
\(80\) 0 0
\(81\) −0.837864 −0.0930960
\(82\) 0 0
\(83\) 0.715276 + 0.715276i 0.0785117 + 0.0785117i 0.745272 − 0.666760i \(-0.232320\pi\)
−0.666760 + 0.745272i \(0.732320\pi\)
\(84\) 0 0
\(85\) 0.229563 − 0.229563i 0.0248996 − 0.0248996i
\(86\) 0 0
\(87\) − 0.469405i − 0.0503255i
\(88\) 0 0
\(89\) 10.9924i 1.16519i 0.812763 + 0.582595i \(0.197962\pi\)
−0.812763 + 0.582595i \(0.802038\pi\)
\(90\) 0 0
\(91\) −2.02017 + 2.02017i −0.211771 + 0.211771i
\(92\) 0 0
\(93\) −4.32050 − 4.32050i −0.448015 − 0.448015i
\(94\) 0 0
\(95\) 7.87820 0.808286
\(96\) 0 0
\(97\) −14.2452 −1.44638 −0.723189 − 0.690650i \(-0.757325\pi\)
−0.723189 + 0.690650i \(0.757325\pi\)
\(98\) 0 0
\(99\) −5.88434 − 5.88434i −0.591399 − 0.591399i
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 448.2.m.c.337.3 8
4.3 odd 2 112.2.m.c.29.4 ✓ 8
8.3 odd 2 896.2.m.e.673.3 8
8.5 even 2 896.2.m.f.673.2 8
16.3 odd 4 896.2.m.e.225.3 8
16.5 even 4 inner 448.2.m.c.113.3 8
16.11 odd 4 112.2.m.c.85.4 yes 8
16.13 even 4 896.2.m.f.225.2 8
28.3 even 6 784.2.x.j.765.2 16
28.11 odd 6 784.2.x.k.765.2 16
28.19 even 6 784.2.x.j.557.2 16
28.23 odd 6 784.2.x.k.557.2 16
28.27 even 2 784.2.m.g.589.4 8
32.5 even 8 7168.2.a.bd.1.4 8
32.11 odd 8 7168.2.a.bc.1.4 8
32.21 even 8 7168.2.a.bd.1.5 8
32.27 odd 8 7168.2.a.bc.1.5 8
112.11 odd 12 784.2.x.k.373.2 16
112.27 even 4 784.2.m.g.197.4 8
112.59 even 12 784.2.x.j.373.2 16
112.75 even 12 784.2.x.j.165.2 16
112.107 odd 12 784.2.x.k.165.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.4 ✓ 8 4.3 odd 2
112.2.m.c.85.4 yes 8 16.11 odd 4
448.2.m.c.113.3 8 16.5 even 4 inner
448.2.m.c.337.3 8 1.1 even 1 trivial
784.2.m.g.197.4 8 112.27 even 4
784.2.m.g.589.4 8 28.27 even 2
784.2.x.j.165.2 16 112.75 even 12
784.2.x.j.373.2 16 112.59 even 12
784.2.x.j.557.2 16 28.19 even 6
784.2.x.j.765.2 16 28.3 even 6
784.2.x.k.165.2 16 112.107 odd 12
784.2.x.k.373.2 16 112.11 odd 12
784.2.x.k.557.2 16 28.23 odd 6
784.2.x.k.765.2 16 28.11 odd 6
896.2.m.e.225.3 8 16.3 odd 4
896.2.m.e.673.3 8 8.3 odd 2
896.2.m.f.225.2 8 16.13 even 4
896.2.m.f.673.2 8 8.5 even 2
7168.2.a.bc.1.4 8 32.11 odd 8
7168.2.a.bc.1.5 8 32.27 odd 8
7168.2.a.bd.1.4 8 32.5 even 8
7168.2.a.bd.1.5 8 32.21 even 8