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Properties

Label 672.2.h.c.575.3
Level $672$
Weight $2$
Character 672.575
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [672,2,Mod(575,672)] 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("672.575"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,4,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 672.575
Dual form 672.2.h.c.575.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.41421 - 1.00000i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(1.00000 - 2.82843i) q^{9} -1.41421 q^{11} +6.00000 q^{13} +(1.41421 + 2.00000i) q^{15} +1.41421i q^{17} -4.00000i q^{19} +(1.00000 + 1.41421i) q^{21} +7.07107 q^{23} +3.00000 q^{25} +(-1.41421 - 5.00000i) q^{27} +8.48528i q^{29} -2.00000i q^{31} +(-2.00000 + 1.41421i) q^{33} -1.41421 q^{35} -4.00000 q^{37} +(8.48528 - 6.00000i) q^{39} +4.24264i q^{41} +(4.00000 + 1.41421i) q^{45} -2.82843 q^{47} -1.00000 q^{49} +(1.41421 + 2.00000i) q^{51} -5.65685i q^{53} -2.00000i q^{55} +(-4.00000 - 5.65685i) q^{57} -11.3137 q^{59} -10.0000 q^{61} +(2.82843 + 1.00000i) q^{63} +8.48528i q^{65} -16.0000i q^{67} +(10.0000 - 7.07107i) q^{69} -4.24264 q^{71} +14.0000 q^{73} +(4.24264 - 3.00000i) q^{75} -1.41421i q^{77} +4.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -8.48528 q^{83} -2.00000 q^{85} +(8.48528 + 12.0000i) q^{87} +7.07107i q^{89} +6.00000i q^{91} +(-2.00000 - 2.82843i) q^{93} +5.65685 q^{95} +6.00000 q^{97} +(-1.41421 + 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + 24 q^{13} + 4 q^{21} + 12 q^{25} - 8 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 16 q^{57} - 40 q^{61} + 40 q^{69} + 56 q^{73} - 28 q^{81} - 8 q^{85} - 8 q^{93} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.41421 − 1.00000i 0.816497 − 0.577350i
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.00000 − 2.82843i 0.333333 − 0.942809i
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 − 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 1.41421 + 2.00000i 0.365148 + 0.516398i
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 1.00000 + 1.41421i 0.218218 + 0.308607i
\(22\) 0 0
\(23\) 7.07107 1.47442 0.737210 − 0.675664i \(-0.236143\pi\)
0.737210 + 0.675664i \(0.236143\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −1.41421 − 5.00000i −0.272166 − 0.962250i
\(28\) 0 0
\(29\) 8.48528i 1.57568i 0.615882 + 0.787839i \(0.288800\pi\)
−0.615882 + 0.787839i \(0.711200\pi\)
\(30\) 0 0
\(31\) − 2.00000i − 0.359211i −0.983739 − 0.179605i \(-0.942518\pi\)
0.983739 − 0.179605i \(-0.0574821\pi\)
\(32\) 0 0
\(33\) −2.00000 + 1.41421i −0.348155 + 0.246183i
\(34\) 0 0
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 8.48528 − 6.00000i 1.35873 − 0.960769i
\(40\) 0 0
\(41\) 4.24264i 0.662589i 0.943527 + 0.331295i \(0.107485\pi\)
−0.943527 + 0.331295i \(0.892515\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 4.00000 + 1.41421i 0.596285 + 0.210819i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 − 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.41421 + 2.00000i 0.198030 + 0.280056i
\(52\) 0 0
\(53\) − 5.65685i − 0.777029i −0.921443 − 0.388514i \(-0.872988\pi\)
0.921443 − 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) − 2.00000i − 0.269680i
\(56\) 0 0
\(57\) −4.00000 − 5.65685i −0.529813 − 0.749269i
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 − 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 2.82843 + 1.00000i 0.356348 + 0.125988i
\(64\) 0 0
\(65\) 8.48528i 1.05247i
\(66\) 0 0
\(67\) − 16.0000i − 1.95471i −0.211604 − 0.977356i \(-0.567869\pi\)
0.211604 − 0.977356i \(-0.432131\pi\)
\(68\) 0 0
\(69\) 10.0000 − 7.07107i 1.20386 − 0.851257i
\(70\) 0 0
\(71\) −4.24264 −0.503509 −0.251754 − 0.967791i \(-0.581008\pi\)
−0.251754 + 0.967791i \(0.581008\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 4.24264 − 3.00000i 0.489898 − 0.346410i
\(76\) 0 0
\(77\) − 1.41421i − 0.161165i
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 0 0
\(83\) −8.48528 −0.931381 −0.465690 − 0.884948i \(-0.654194\pi\)
−0.465690 + 0.884948i \(0.654194\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 8.48528 + 12.0000i 0.909718 + 1.28654i
\(88\) 0 0
\(89\) 7.07107i 0.749532i 0.927119 + 0.374766i \(0.122277\pi\)
−0.927119 + 0.374766i \(0.877723\pi\)
\(90\) 0 0
\(91\) 6.00000i 0.628971i
\(92\) 0 0
\(93\) −2.00000 − 2.82843i −0.207390 − 0.293294i
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) −1.41421 + 4.00000i −0.142134 + 0.402015i
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 672.2.h.c.575.3 yes 4
3.2 odd 2 inner 672.2.h.c.575.1 ✓ 4
4.3 odd 2 inner 672.2.h.c.575.2 yes 4
8.3 odd 2 1344.2.h.b.575.3 4
8.5 even 2 1344.2.h.b.575.2 4
12.11 even 2 inner 672.2.h.c.575.4 yes 4
24.5 odd 2 1344.2.h.b.575.4 4
24.11 even 2 1344.2.h.b.575.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.h.c.575.1 ✓ 4 3.2 odd 2 inner
672.2.h.c.575.2 yes 4 4.3 odd 2 inner
672.2.h.c.575.3 yes 4 1.1 even 1 trivial
672.2.h.c.575.4 yes 4 12.11 even 2 inner
1344.2.h.b.575.1 4 24.11 even 2
1344.2.h.b.575.2 4 8.5 even 2
1344.2.h.b.575.3 4 8.3 odd 2
1344.2.h.b.575.4 4 24.5 odd 2