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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,5,Mod(15,448)] 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("448.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.3097434616\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 15.25
Character \(\chi\) \(=\) 448.15
Dual form 448.5.k.a.239.25

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.212552 - 0.212552i) q^{3} +(14.9819 + 14.9819i) q^{5} -18.5203 q^{7} -80.9096i q^{9} +(139.313 - 139.313i) q^{11} +(-186.037 + 186.037i) q^{13} -6.36887i q^{15} +167.051 q^{17} +(-146.208 - 146.208i) q^{19} +(3.93651 + 3.93651i) q^{21} -367.074 q^{23} -176.083i q^{25} +(-34.4141 + 34.4141i) q^{27} +(-1044.05 + 1044.05i) q^{29} -1417.88i q^{31} -59.2225 q^{33} +(-277.469 - 277.469i) q^{35} +(173.533 + 173.533i) q^{37} +79.0851 q^{39} +1180.37i q^{41} +(316.853 - 316.853i) q^{43} +(1212.18 - 1212.18i) q^{45} -2992.85i q^{47} +343.000 q^{49} +(-35.5070 - 35.5070i) q^{51} +(-1384.57 - 1384.57i) q^{53} +4174.37 q^{55} +62.1534i q^{57} +(3090.58 - 3090.58i) q^{59} +(1845.82 - 1845.82i) q^{61} +1498.47i q^{63} -5574.41 q^{65} +(2028.10 + 2028.10i) q^{67} +(78.0222 + 78.0222i) q^{69} -8041.72 q^{71} -5961.54i q^{73} +(-37.4267 + 37.4267i) q^{75} +(-2580.12 + 2580.12i) q^{77} +5168.83i q^{79} -6539.05 q^{81} +(-7170.62 - 7170.62i) q^{83} +(2502.75 + 2502.75i) q^{85} +443.829 q^{87} -6317.56i q^{89} +(3445.46 - 3445.46i) q^{91} +(-301.372 + 301.372i) q^{93} -4380.95i q^{95} +11147.8 q^{97} +(-11271.8 - 11271.8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −0.212552 − 0.212552i −0.0236168 − 0.0236168i 0.695200 − 0.718817i \(-0.255316\pi\)
−0.718817 + 0.695200i \(0.755316\pi\)
\(4\) 0 0
\(5\) 14.9819 + 14.9819i 0.599278 + 0.599278i 0.940120 − 0.340843i \(-0.110712\pi\)
−0.340843 + 0.940120i \(0.610712\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 0 0
\(9\) − 80.9096i − 0.998884i
\(10\) 0 0
\(11\) 139.313 − 139.313i 1.15135 − 1.15135i 0.165067 − 0.986282i \(-0.447216\pi\)
0.986282 − 0.165067i \(-0.0527842\pi\)
\(12\) 0 0
\(13\) −186.037 + 186.037i −1.10081 + 1.10081i −0.106501 + 0.994313i \(0.533965\pi\)
−0.994313 + 0.106501i \(0.966035\pi\)
\(14\) 0 0
\(15\) − 6.36887i − 0.0283061i
\(16\) 0 0
\(17\) 167.051 0.578032 0.289016 − 0.957324i \(-0.406672\pi\)
0.289016 + 0.957324i \(0.406672\pi\)
\(18\) 0 0
\(19\) −146.208 − 146.208i −0.405008 − 0.405008i 0.474986 − 0.879993i \(-0.342453\pi\)
−0.879993 + 0.474986i \(0.842453\pi\)
\(20\) 0 0
\(21\) 3.93651 + 3.93651i 0.00892633 + 0.00892633i
\(22\) 0 0
\(23\) −367.074 −0.693902 −0.346951 − 0.937883i \(-0.612783\pi\)
−0.346951 + 0.937883i \(0.612783\pi\)
\(24\) 0 0
\(25\) − 176.083i − 0.281733i
\(26\) 0 0
\(27\) −34.4141 + 34.4141i −0.0472073 + 0.0472073i
\(28\) 0 0
\(29\) −1044.05 + 1044.05i −1.24144 + 1.24144i −0.282034 + 0.959404i \(0.591009\pi\)
−0.959404 + 0.282034i \(0.908991\pi\)
\(30\) 0 0
\(31\) − 1417.88i − 1.47542i −0.675119 − 0.737709i \(-0.735908\pi\)
0.675119 − 0.737709i \(-0.264092\pi\)
\(32\) 0 0
\(33\) −59.2225 −0.0543825
\(34\) 0 0
\(35\) −277.469 − 277.469i −0.226506 − 0.226506i
\(36\) 0 0
\(37\) 173.533 + 173.533i 0.126759 + 0.126759i 0.767640 − 0.640881i \(-0.221431\pi\)
−0.640881 + 0.767640i \(0.721431\pi\)
\(38\) 0 0
\(39\) 79.0851 0.0519955
\(40\) 0 0
\(41\) 1180.37i 0.702183i 0.936341 + 0.351091i \(0.114189\pi\)
−0.936341 + 0.351091i \(0.885811\pi\)
\(42\) 0 0
\(43\) 316.853 − 316.853i 0.171364 − 0.171364i −0.616214 − 0.787579i \(-0.711334\pi\)
0.787579 + 0.616214i \(0.211334\pi\)
\(44\) 0 0
\(45\) 1212.18 − 1212.18i 0.598609 − 0.598609i
\(46\) 0 0
\(47\) − 2992.85i − 1.35484i −0.735596 − 0.677421i \(-0.763098\pi\)
0.735596 − 0.677421i \(-0.236902\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) −35.5070 − 35.5070i −0.0136513 − 0.0136513i
\(52\) 0 0
\(53\) −1384.57 − 1384.57i −0.492906 − 0.492906i 0.416315 − 0.909221i \(-0.363321\pi\)
−0.909221 + 0.416315i \(0.863321\pi\)
\(54\) 0 0
\(55\) 4174.37 1.37996
\(56\) 0 0
\(57\) 62.1534i 0.0191300i
\(58\) 0 0
\(59\) 3090.58 − 3090.58i 0.887843 − 0.887843i −0.106473 − 0.994316i \(-0.533956\pi\)
0.994316 + 0.106473i \(0.0339557\pi\)
\(60\) 0 0
\(61\) 1845.82 − 1845.82i 0.496054 − 0.496054i −0.414153 − 0.910207i \(-0.635922\pi\)
0.910207 + 0.414153i \(0.135922\pi\)
\(62\) 0 0
\(63\) 1498.47i 0.377543i
\(64\) 0 0
\(65\) −5574.41 −1.31939
\(66\) 0 0
\(67\) 2028.10 + 2028.10i 0.451793 + 0.451793i 0.895949 − 0.444156i \(-0.146497\pi\)
−0.444156 + 0.895949i \(0.646497\pi\)
\(68\) 0 0
\(69\) 78.0222 + 78.0222i 0.0163878 + 0.0163878i
\(70\) 0 0
\(71\) −8041.72 −1.59526 −0.797632 − 0.603145i \(-0.793914\pi\)
−0.797632 + 0.603145i \(0.793914\pi\)
\(72\) 0 0
\(73\) − 5961.54i − 1.11870i −0.828933 − 0.559348i \(-0.811051\pi\)
0.828933 − 0.559348i \(-0.188949\pi\)
\(74\) 0 0
\(75\) −37.4267 + 37.4267i −0.00665363 + 0.00665363i
\(76\) 0 0
\(77\) −2580.12 + 2580.12i −0.435169 + 0.435169i
\(78\) 0 0
\(79\) 5168.83i 0.828206i 0.910230 + 0.414103i \(0.135905\pi\)
−0.910230 + 0.414103i \(0.864095\pi\)
\(80\) 0 0
\(81\) −6539.05 −0.996655
\(82\) 0 0
\(83\) −7170.62 − 7170.62i −1.04088 − 1.04088i −0.999128 − 0.0417514i \(-0.986706\pi\)
−0.0417514 − 0.999128i \(-0.513294\pi\)
\(84\) 0 0
\(85\) 2502.75 + 2502.75i 0.346402 + 0.346402i
\(86\) 0 0
\(87\) 443.829 0.0586377
\(88\) 0 0
\(89\) − 6317.56i − 0.797571i −0.917044 − 0.398785i \(-0.869432\pi\)
0.917044 − 0.398785i \(-0.130568\pi\)
\(90\) 0 0
\(91\) 3445.46 − 3445.46i 0.416068 − 0.416068i
\(92\) 0 0
\(93\) −301.372 + 301.372i −0.0348447 + 0.0348447i
\(94\) 0 0
\(95\) − 4380.95i − 0.485424i
\(96\) 0 0
\(97\) 11147.8 1.18480 0.592402 − 0.805642i \(-0.298180\pi\)
0.592402 + 0.805642i \(0.298180\pi\)
\(98\) 0 0
\(99\) −11271.8 − 11271.8i −1.15007 − 1.15007i
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.k.a.15.25 96
4.3 odd 2 112.5.k.a.43.12 ✓ 96
16.3 odd 4 inner 448.5.k.a.239.25 96
16.13 even 4 112.5.k.a.99.12 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.12 ✓ 96 4.3 odd 2
112.5.k.a.99.12 yes 96 16.13 even 4
448.5.k.a.15.25 96 1.1 even 1 trivial
448.5.k.a.239.25 96 16.3 odd 4 inner