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Properties

Label 448.5.c.e.321.2
Level $448$
Weight $5$
Character 448.321
Analytic conductor $46.310$
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] = [448,5,Mod(321,448)] 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("448.321"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-76,0,-252] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.3097434616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-36 +3 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 72x^{2} + 1278 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 321.2
Root \(-5.63537i\) of defining polynomial
Character \(\chi\) \(=\) 448.321
Dual form 448.5.c.e.321.3

$q$-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)
 
\(f(q)\) \(=\) \(q-11.2707i q^{3} +43.1492i q^{5} +(-44.4558 + 20.6077i) q^{7} -46.0294 q^{9} +11.8234 q^{11} +20.6077i q^{13} +486.323 q^{15} +289.172i q^{17} -104.641i q^{19} +(232.264 + 501.050i) q^{21} +73.5290 q^{23} -1236.85 q^{25} -394.144i q^{27} -950.881 q^{29} -1385.30i q^{31} -133.258i q^{33} +(-889.206 - 1918.23i) q^{35} +1279.47 q^{37} +232.264 q^{39} -1303.54i q^{41} +96.2338 q^{43} -1986.13i q^{45} -186.190i q^{47} +(1551.64 - 1832.27i) q^{49} +3259.18 q^{51} -4376.94 q^{53} +510.169i q^{55} -1179.38 q^{57} +1650.28i q^{59} -5200.50i q^{61} +(2046.28 - 948.562i) q^{63} -889.206 q^{65} -552.587 q^{67} -828.726i q^{69} -8487.61 q^{71} -317.344i q^{73} +13940.2i q^{75} +(-525.618 + 243.653i) q^{77} -624.377 q^{79} -8170.68 q^{81} -7662.33i q^{83} -12477.5 q^{85} +10717.1i q^{87} +4190.72i q^{89} +(-424.678 - 916.133i) q^{91} -15613.4 q^{93} +4515.15 q^{95} -12994.4i q^{97} -544.223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 76 q^{7} - 252 q^{9} - 360 q^{11} + 384 q^{15} - 768 q^{21} - 792 q^{23} - 2300 q^{25} - 1224 q^{29} - 4032 q^{35} + 3896 q^{37} - 768 q^{39} - 3688 q^{43} - 1532 q^{49} + 11136 q^{51} - 5832 q^{53}+ \cdots + 29592 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 11.2707i − 1.25230i −0.779701 − 0.626152i \(-0.784629\pi\)
0.779701 − 0.626152i \(-0.215371\pi\)
\(4\) 0 0
\(5\) 43.1492i 1.72597i 0.505232 + 0.862984i \(0.331407\pi\)
−0.505232 + 0.862984i \(0.668593\pi\)
\(6\) 0 0
\(7\) −44.4558 + 20.6077i −0.907262 + 0.420566i
\(8\) 0 0
\(9\) −46.0294 −0.568265
\(10\) 0 0
\(11\) 11.8234 0.0977139 0.0488569 − 0.998806i \(-0.484442\pi\)
0.0488569 + 0.998806i \(0.484442\pi\)
\(12\) 0 0
\(13\) 20.6077i 0.121939i 0.998140 + 0.0609696i \(0.0194193\pi\)
−0.998140 + 0.0609696i \(0.980581\pi\)
\(14\) 0 0
\(15\) 486.323 2.16144
\(16\) 0 0
\(17\) 289.172i 1.00059i 0.865854 + 0.500297i \(0.166776\pi\)
−0.865854 + 0.500297i \(0.833224\pi\)
\(18\) 0 0
\(19\) − 104.641i − 0.289863i −0.989442 − 0.144932i \(-0.953704\pi\)
0.989442 − 0.144932i \(-0.0462962\pi\)
\(20\) 0 0
\(21\) 232.264 + 501.050i 0.526676 + 1.13617i
\(22\) 0 0
\(23\) 73.5290 0.138996 0.0694981 − 0.997582i \(-0.477860\pi\)
0.0694981 + 0.997582i \(0.477860\pi\)
\(24\) 0 0
\(25\) −1236.85 −1.97896
\(26\) 0 0
\(27\) − 394.144i − 0.540664i
\(28\) 0 0
\(29\) −950.881 −1.13066 −0.565328 − 0.824866i \(-0.691250\pi\)
−0.565328 + 0.824866i \(0.691250\pi\)
\(30\) 0 0
\(31\) − 1385.30i − 1.44152i −0.693182 − 0.720762i \(-0.743792\pi\)
0.693182 − 0.720762i \(-0.256208\pi\)
\(32\) 0 0
\(33\) − 133.258i − 0.122367i
\(34\) 0 0
\(35\) −889.206 − 1918.23i −0.725882 − 1.56590i
\(36\) 0 0
\(37\) 1279.47 0.934602 0.467301 − 0.884098i \(-0.345226\pi\)
0.467301 + 0.884098i \(0.345226\pi\)
\(38\) 0 0
\(39\) 232.264 0.152705
\(40\) 0 0
\(41\) − 1303.54i − 0.775454i −0.921774 − 0.387727i \(-0.873260\pi\)
0.921774 − 0.387727i \(-0.126740\pi\)
\(42\) 0 0
\(43\) 96.2338 0.0520464 0.0260232 − 0.999661i \(-0.491716\pi\)
0.0260232 + 0.999661i \(0.491716\pi\)
\(44\) 0 0
\(45\) − 1986.13i − 0.980806i
\(46\) 0 0
\(47\) − 186.190i − 0.0842870i −0.999112 − 0.0421435i \(-0.986581\pi\)
0.999112 − 0.0421435i \(-0.0134187\pi\)
\(48\) 0 0
\(49\) 1551.64 − 1832.27i 0.646249 − 0.763127i
\(50\) 0 0
\(51\) 3259.18 1.25305
\(52\) 0 0
\(53\) −4376.94 −1.55818 −0.779092 − 0.626910i \(-0.784319\pi\)
−0.779092 + 0.626910i \(0.784319\pi\)
\(54\) 0 0
\(55\) 510.169i 0.168651i
\(56\) 0 0
\(57\) −1179.38 −0.362997
\(58\) 0 0
\(59\) 1650.28i 0.474081i 0.971500 + 0.237041i \(0.0761774\pi\)
−0.971500 + 0.237041i \(0.923823\pi\)
\(60\) 0 0
\(61\) − 5200.50i − 1.39761i −0.715313 − 0.698804i \(-0.753716\pi\)
0.715313 − 0.698804i \(-0.246284\pi\)
\(62\) 0 0
\(63\) 2046.28 − 948.562i 0.515565 − 0.238993i
\(64\) 0 0
\(65\) −889.206 −0.210463
\(66\) 0 0
\(67\) −552.587 −0.123098 −0.0615490 − 0.998104i \(-0.519604\pi\)
−0.0615490 + 0.998104i \(0.519604\pi\)
\(68\) 0 0
\(69\) − 828.726i − 0.174065i
\(70\) 0 0
\(71\) −8487.61 −1.68372 −0.841858 − 0.539699i \(-0.818538\pi\)
−0.841858 + 0.539699i \(0.818538\pi\)
\(72\) 0 0
\(73\) − 317.344i − 0.0595503i −0.999557 − 0.0297751i \(-0.990521\pi\)
0.999557 − 0.0297751i \(-0.00947912\pi\)
\(74\) 0 0
\(75\) 13940.2i 2.47826i
\(76\) 0 0
\(77\) −525.618 + 243.653i −0.0886521 + 0.0410951i
\(78\) 0 0
\(79\) −624.377 −0.100044 −0.0500222 − 0.998748i \(-0.515929\pi\)
−0.0500222 + 0.998748i \(0.515929\pi\)
\(80\) 0 0
\(81\) −8170.68 −1.24534
\(82\) 0 0
\(83\) − 7662.33i − 1.11226i −0.831097 − 0.556128i \(-0.812287\pi\)
0.831097 − 0.556128i \(-0.187713\pi\)
\(84\) 0 0
\(85\) −12477.5 −1.72699
\(86\) 0 0
\(87\) 10717.1i 1.41592i
\(88\) 0 0
\(89\) 4190.72i 0.529065i 0.964377 + 0.264532i \(0.0852176\pi\)
−0.964377 + 0.264532i \(0.914782\pi\)
\(90\) 0 0
\(91\) −424.678 − 916.133i −0.0512834 − 0.110631i
\(92\) 0 0
\(93\) −15613.4 −1.80523
\(94\) 0 0
\(95\) 4515.15 0.500294
\(96\) 0 0
\(97\) − 12994.4i − 1.38106i −0.723305 − 0.690529i \(-0.757378\pi\)
0.723305 − 0.690529i \(-0.242622\pi\)
\(98\) 0 0
\(99\) −544.223 −0.0555273
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 448.5.c.e.321.2 4
4.3 odd 2 448.5.c.f.321.3 4
7.6 odd 2 inner 448.5.c.e.321.3 4
8.3 odd 2 112.5.c.c.97.2 4
8.5 even 2 14.5.b.a.13.4 yes 4
24.5 odd 2 126.5.c.a.55.2 4
24.11 even 2 1008.5.f.h.433.4 4
28.27 even 2 448.5.c.f.321.2 4
40.13 odd 4 350.5.d.a.349.2 8
40.29 even 2 350.5.b.a.251.1 4
40.37 odd 4 350.5.d.a.349.7 8
56.5 odd 6 98.5.d.d.31.2 8
56.13 odd 2 14.5.b.a.13.3 ✓ 4
56.27 even 2 112.5.c.c.97.3 4
56.37 even 6 98.5.d.d.31.1 8
56.45 odd 6 98.5.d.d.19.1 8
56.53 even 6 98.5.d.d.19.2 8
168.83 odd 2 1008.5.f.h.433.1 4
168.125 even 2 126.5.c.a.55.1 4
280.13 even 4 350.5.d.a.349.3 8
280.69 odd 2 350.5.b.a.251.2 4
280.237 even 4 350.5.d.a.349.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.5.b.a.13.3 ✓ 4 56.13 odd 2
14.5.b.a.13.4 yes 4 8.5 even 2
98.5.d.d.19.1 8 56.45 odd 6
98.5.d.d.19.2 8 56.53 even 6
98.5.d.d.31.1 8 56.37 even 6
98.5.d.d.31.2 8 56.5 odd 6
112.5.c.c.97.2 4 8.3 odd 2
112.5.c.c.97.3 4 56.27 even 2
126.5.c.a.55.1 4 168.125 even 2
126.5.c.a.55.2 4 24.5 odd 2
350.5.b.a.251.1 4 40.29 even 2
350.5.b.a.251.2 4 280.69 odd 2
350.5.d.a.349.2 8 40.13 odd 4
350.5.d.a.349.3 8 280.13 even 4
350.5.d.a.349.6 8 280.237 even 4
350.5.d.a.349.7 8 40.37 odd 4
448.5.c.e.321.2 4 1.1 even 1 trivial
448.5.c.e.321.3 4 7.6 odd 2 inner
448.5.c.f.321.2 4 28.27 even 2
448.5.c.f.321.3 4 4.3 odd 2
1008.5.f.h.433.1 4 168.83 odd 2
1008.5.f.h.433.4 4 24.11 even 2