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Properties

Label 675.2.r.a.46.18
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
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] = [675,2,Mod(46,675)] 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("675.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.r (of order \(15\), degree \(8\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 46.18
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.18

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.293899 + 0.326408i) q^{2} +(0.188891 - 1.79718i) q^{4} +(-2.06853 + 0.849220i) q^{5} +(-0.888066 + 1.53818i) q^{7} +(1.35281 - 0.982874i) q^{8} +(-0.885131 - 0.425600i) q^{10} +(2.88123 + 3.19993i) q^{11} +(4.29288 - 4.76773i) q^{13} +(-0.763074 + 0.162196i) q^{14} +(-2.81678 - 0.598725i) q^{16} +(-0.162997 + 0.118424i) q^{17} +(4.79916 - 3.48679i) q^{19} +(1.13548 + 3.87794i) q^{20} +(-0.197692 + 1.88091i) q^{22} +(2.67917 - 0.569475i) q^{23} +(3.55765 - 3.51328i) q^{25} +2.81790 q^{26} +(2.59663 + 1.88657i) q^{28} +(0.675896 - 0.300928i) q^{29} +(2.52989 + 1.12638i) q^{31} +(-2.30458 - 3.99166i) q^{32} +(-0.0865590 - 0.0183987i) q^{34} +(0.530743 - 3.93593i) q^{35} +(-1.36941 - 4.21461i) q^{37} +(2.54858 + 0.541718i) q^{38} +(-1.96365 + 3.18194i) q^{40} +(8.41459 - 9.34535i) q^{41} +(-3.91700 + 6.78444i) q^{43} +(6.29511 - 4.57366i) q^{44} +(0.973285 + 0.707133i) q^{46} +(-0.452987 + 0.201683i) q^{47} +(1.92268 + 3.33017i) q^{49} +(2.19235 + 0.128696i) q^{50} +(-7.75759 - 8.61567i) q^{52} +(-5.99398 - 4.35488i) q^{53} +(-8.67738 - 4.17236i) q^{55} +(0.310448 + 2.95372i) q^{56} +(0.296870 + 0.132175i) q^{58} +(-6.51328 + 7.23373i) q^{59} +(6.51135 + 7.23158i) q^{61} +(0.375873 + 1.15682i) q^{62} +(-1.15416 + 3.55215i) q^{64} +(-4.83111 + 13.5078i) q^{65} +(0.130253 + 0.0579926i) q^{67} +(0.182041 + 0.315304i) q^{68} +(1.44070 - 0.983526i) q^{70} +(-1.95406 - 1.41971i) q^{71} +(-0.860848 + 2.64942i) q^{73} +(0.973212 - 1.68565i) q^{74} +(-5.35988 - 9.28359i) q^{76} +(-7.48079 + 1.59009i) q^{77} +(11.7563 - 5.23422i) q^{79} +(6.33505 - 1.15358i) q^{80} +5.52343 q^{82} +(0.359051 + 3.41614i) q^{83} +(0.236596 - 0.383384i) q^{85} +(-3.36569 + 0.715400i) q^{86} +(7.04289 + 1.49701i) q^{88} +(-3.29985 + 10.1559i) q^{89} +(3.52124 + 10.8373i) q^{91} +(-0.517378 - 4.92252i) q^{92} +(-0.198963 - 0.0885841i) q^{94} +(-6.96616 + 11.2881i) q^{95} +(-14.4851 + 6.44917i) q^{97} +(-0.521922 + 1.60631i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.293899 + 0.326408i 0.207818 + 0.230805i 0.838039 − 0.545611i \(-0.183702\pi\)
−0.630221 + 0.776416i \(0.717036\pi\)
\(3\) 0 0
\(4\) 0.188891 − 1.79718i 0.0944457 − 0.898591i
\(5\) −2.06853 + 0.849220i −0.925076 + 0.379783i
\(6\) 0 0
\(7\) −0.888066 + 1.53818i −0.335657 + 0.581376i −0.983611 − 0.180304i \(-0.942292\pi\)
0.647953 + 0.761680i \(0.275625\pi\)
\(8\) 1.35281 − 0.982874i 0.478290 − 0.347498i
\(9\) 0 0
\(10\) −0.885131 − 0.425600i −0.279903 − 0.134587i
\(11\) 2.88123 + 3.19993i 0.868725 + 0.964817i 0.999647 − 0.0265760i \(-0.00846040\pi\)
−0.130922 + 0.991393i \(0.541794\pi\)
\(12\) 0 0
\(13\) 4.29288 − 4.76773i 1.19063 − 1.32233i 0.256014 − 0.966673i \(-0.417591\pi\)
0.934617 − 0.355657i \(-0.115743\pi\)
\(14\) −0.763074 + 0.162196i −0.203940 + 0.0433488i
\(15\) 0 0
\(16\) −2.81678 − 0.598725i −0.704195 − 0.149681i
\(17\) −0.162997 + 0.118424i −0.0395325 + 0.0287220i −0.607376 − 0.794414i \(-0.707778\pi\)
0.567844 + 0.823137i \(0.307778\pi\)
\(18\) 0 0
\(19\) 4.79916 − 3.48679i 1.10100 − 0.799925i 0.119779 − 0.992801i \(-0.461781\pi\)
0.981223 + 0.192875i \(0.0617813\pi\)
\(20\) 1.13548 + 3.87794i 0.253900 + 0.867134i
\(21\) 0 0
\(22\) −0.197692 + 1.88091i −0.0421481 + 0.401012i
\(23\) 2.67917 − 0.569475i 0.558645 − 0.118744i 0.0800683 − 0.996789i \(-0.474486\pi\)
0.478577 + 0.878046i \(0.341153\pi\)
\(24\) 0 0
\(25\) 3.55765 − 3.51328i 0.711530 − 0.702656i
\(26\) 2.81790 0.552635
\(27\) 0 0
\(28\) 2.59663 + 1.88657i 0.490718 + 0.356527i
\(29\) 0.675896 − 0.300928i 0.125511 − 0.0558809i −0.343020 − 0.939328i \(-0.611450\pi\)
0.468531 + 0.883447i \(0.344783\pi\)
\(30\) 0 0
\(31\) 2.52989 + 1.12638i 0.454382 + 0.202304i 0.621149 − 0.783693i \(-0.286666\pi\)
−0.166767 + 0.985996i \(0.553333\pi\)
\(32\) −2.30458 − 3.99166i −0.407397 − 0.705632i
\(33\) 0 0
\(34\) −0.0865590 − 0.0183987i −0.0148448 − 0.00315535i
\(35\) 0.530743 − 3.93593i 0.0897120 − 0.665294i
\(36\) 0 0
\(37\) −1.36941 − 4.21461i −0.225129 − 0.692877i −0.998278 − 0.0586526i \(-0.981320\pi\)
0.773149 − 0.634224i \(-0.218680\pi\)
\(38\) 2.54858 + 0.541718i 0.413435 + 0.0878783i
\(39\) 0 0
\(40\) −1.96365 + 3.18194i −0.310481 + 0.503109i
\(41\) 8.41459 − 9.34535i 1.31414 − 1.45950i 0.516607 − 0.856223i \(-0.327195\pi\)
0.797532 − 0.603276i \(-0.206138\pi\)
\(42\) 0 0
\(43\) −3.91700 + 6.78444i −0.597336 + 1.03462i 0.395876 + 0.918304i \(0.370441\pi\)
−0.993213 + 0.116313i \(0.962892\pi\)
\(44\) 6.29511 − 4.57366i 0.949023 − 0.689506i
\(45\) 0 0
\(46\) 0.973285 + 0.707133i 0.143503 + 0.104261i
\(47\) −0.452987 + 0.201683i −0.0660749 + 0.0294185i −0.439508 − 0.898239i \(-0.644847\pi\)
0.373433 + 0.927657i \(0.378181\pi\)
\(48\) 0 0
\(49\) 1.92268 + 3.33017i 0.274668 + 0.475739i
\(50\) 2.19235 + 0.128696i 0.310045 + 0.0182003i
\(51\) 0 0
\(52\) −7.75759 − 8.61567i −1.07578 − 1.19478i
\(53\) −5.99398 − 4.35488i −0.823336 − 0.598189i 0.0943302 − 0.995541i \(-0.469929\pi\)
−0.917666 + 0.397352i \(0.869929\pi\)
\(54\) 0 0
\(55\) −8.67738 − 4.17236i −1.17006 − 0.562602i
\(56\) 0.310448 + 2.95372i 0.0414854 + 0.394707i
\(57\) 0 0
\(58\) 0.296870 + 0.132175i 0.0389810 + 0.0173554i
\(59\) −6.51328 + 7.23373i −0.847957 + 0.941751i −0.998905 − 0.0467905i \(-0.985101\pi\)
0.150948 + 0.988542i \(0.451767\pi\)
\(60\) 0 0
\(61\) 6.51135 + 7.23158i 0.833692 + 0.925909i 0.998170 − 0.0604716i \(-0.0192605\pi\)
−0.164477 + 0.986381i \(0.552594\pi\)
\(62\) 0.375873 + 1.15682i 0.0477359 + 0.146916i
\(63\) 0 0
\(64\) −1.15416 + 3.55215i −0.144270 + 0.444019i
\(65\) −4.83111 + 13.5078i −0.599226 + 1.67544i
\(66\) 0 0
\(67\) 0.130253 + 0.0579926i 0.0159130 + 0.00708492i 0.414678 − 0.909968i \(-0.363894\pi\)
−0.398765 + 0.917053i \(0.630561\pi\)
\(68\) 0.182041 + 0.315304i 0.0220757 + 0.0382362i
\(69\) 0 0
\(70\) 1.44070 − 0.983526i 0.172197 − 0.117554i
\(71\) −1.95406 − 1.41971i −0.231905 − 0.168489i 0.465764 − 0.884909i \(-0.345779\pi\)
−0.697669 + 0.716420i \(0.745779\pi\)
\(72\) 0 0
\(73\) −0.860848 + 2.64942i −0.100755 + 0.310091i −0.988711 − 0.149837i \(-0.952125\pi\)
0.887956 + 0.459928i \(0.152125\pi\)
\(74\) 0.973212 − 1.68565i 0.113134 − 0.195953i
\(75\) 0 0
\(76\) −5.35988 − 9.28359i −0.614821 − 1.06490i
\(77\) −7.48079 + 1.59009i −0.852515 + 0.181208i
\(78\) 0 0
\(79\) 11.7563 − 5.23422i 1.32268 − 0.588896i 0.380744 − 0.924681i \(-0.375668\pi\)
0.941939 + 0.335784i \(0.109001\pi\)
\(80\) 6.33505 − 1.15358i 0.708280 − 0.128975i
\(81\) 0 0
\(82\) 5.52343 0.609961
\(83\) 0.359051 + 3.41614i 0.0394109 + 0.374970i 0.996395 + 0.0848308i \(0.0270350\pi\)
−0.956984 + 0.290139i \(0.906298\pi\)
\(84\) 0 0
\(85\) 0.236596 − 0.383384i 0.0256624 − 0.0415838i
\(86\) −3.36569 + 0.715400i −0.362932 + 0.0771436i
\(87\) 0 0
\(88\) 7.04289 + 1.49701i 0.750775 + 0.159582i
\(89\) −3.29985 + 10.1559i −0.349784 + 1.07652i 0.609189 + 0.793025i \(0.291495\pi\)
−0.958973 + 0.283499i \(0.908505\pi\)
\(90\) 0 0
\(91\) 3.52124 + 10.8373i 0.369126 + 1.13605i
\(92\) −0.517378 − 4.92252i −0.0539404 − 0.513208i
\(93\) 0 0
\(94\) −0.198963 − 0.0885841i −0.0205215 − 0.00913675i
\(95\) −6.96616 + 11.2881i −0.714713 + 1.15813i
\(96\) 0 0
\(97\) −14.4851 + 6.44917i −1.47074 + 0.654814i −0.976697 − 0.214625i \(-0.931147\pi\)
−0.494040 + 0.869439i \(0.664480\pi\)
\(98\) −0.521922 + 1.60631i −0.0527221 + 0.162262i
\(99\) 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 675.2.r.a.46.18 224
3.2 odd 2 225.2.q.a.196.11 yes 224
9.4 even 3 inner 675.2.r.a.496.11 224
9.5 odd 6 225.2.q.a.121.18 yes 224
25.6 even 5 inner 675.2.r.a.181.11 224
75.56 odd 10 225.2.q.a.106.18 yes 224
225.31 even 15 inner 675.2.r.a.631.18 224
225.131 odd 30 225.2.q.a.31.11 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.11 ✓ 224 225.131 odd 30
225.2.q.a.106.18 yes 224 75.56 odd 10
225.2.q.a.121.18 yes 224 9.5 odd 6
225.2.q.a.196.11 yes 224 3.2 odd 2
675.2.r.a.46.18 224 1.1 even 1 trivial
675.2.r.a.181.11 224 25.6 even 5 inner
675.2.r.a.496.11 224 9.4 even 3 inner
675.2.r.a.631.18 224 225.31 even 15 inner