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Properties

Label 675.2.r.a.46.7
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 46.7
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.956880 - 1.06272i) q^{2} +(-0.00470356 + 0.0447514i) q^{4} +(2.07212 - 0.840433i) q^{5} +(1.92019 - 3.32586i) q^{7} +(-2.26178 + 1.64328i) q^{8} +(-2.87591 - 1.39789i) q^{10} +(-3.49787 - 3.88478i) q^{11} +(-2.25011 + 2.49900i) q^{13} +(-5.37185 + 1.14182i) q^{14} +(3.99864 + 0.849937i) q^{16} +(3.90998 - 2.84077i) q^{17} +(-1.90903 + 1.38699i) q^{19} +(0.0278642 + 0.0966831i) q^{20} +(-0.781401 + 7.43453i) q^{22} +(-4.56147 + 0.969570i) q^{23} +(3.58734 - 3.48295i) q^{25} +4.80884 q^{26} +(0.139805 + 0.101574i) q^{28} +(6.71254 - 2.98862i) q^{29} +(0.593962 + 0.264449i) q^{31} +(-0.127249 - 0.220402i) q^{32} +(-6.76033 - 1.43695i) q^{34} +(1.18369 - 8.50536i) q^{35} +(0.315262 + 0.970276i) q^{37} +(3.30069 + 0.701584i) q^{38} +(-3.30561 + 5.30596i) q^{40} +(2.50903 - 2.78656i) q^{41} +(-2.34851 + 4.06773i) q^{43} +(0.190302 - 0.138262i) q^{44} +(5.39516 + 3.91981i) q^{46} +(-10.9118 + 4.85827i) q^{47} +(-3.87423 - 6.71036i) q^{49} +(-7.13407 - 0.479583i) q^{50} +(-0.101250 - 0.112450i) q^{52} +(-2.92011 - 2.12158i) q^{53} +(-10.5129 - 5.10999i) q^{55} +(1.12228 + 10.6778i) q^{56} +(-9.59917 - 4.27383i) q^{58} +(5.20768 - 5.78372i) q^{59} +(-4.40843 - 4.89605i) q^{61} +(-0.287314 - 0.884263i) q^{62} +(2.41404 - 7.42965i) q^{64} +(-2.56225 + 7.06930i) q^{65} +(-0.930565 - 0.414314i) q^{67} +(0.108737 + 0.188339i) q^{68} +(-10.1715 + 6.88068i) q^{70} +(2.82082 + 2.04945i) q^{71} +(-1.38140 + 4.25150i) q^{73} +(0.729467 - 1.26347i) q^{74} +(-0.0530904 - 0.0919553i) q^{76} +(-19.6368 + 4.17393i) q^{77} +(1.49826 - 0.667070i) q^{79} +(8.99997 - 1.59942i) q^{80} -5.36219 q^{82} +(1.18152 + 11.2414i) q^{83} +(5.71446 - 9.17248i) q^{85} +(6.57011 - 1.39652i) q^{86} +(14.2952 + 3.03854i) q^{88} +(0.935178 - 2.87818i) q^{89} +(3.99070 + 12.2821i) q^{91} +(-0.0219345 - 0.208692i) q^{92} +(15.6043 + 6.94749i) q^{94} +(-2.79006 + 4.47841i) q^{95} +(14.8057 - 6.59193i) q^{97} +(-3.42408 + 10.5382i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.956880 − 1.06272i −0.676616 − 0.751458i 0.302856 − 0.953036i \(-0.402060\pi\)
−0.979472 + 0.201578i \(0.935393\pi\)
\(3\) 0 0
\(4\) −0.00470356 + 0.0447514i −0.00235178 + 0.0223757i
\(5\) 2.07212 − 0.840433i 0.926679 − 0.375853i
\(6\) 0 0
\(7\) 1.92019 − 3.32586i 0.725762 − 1.25706i −0.232897 − 0.972501i \(-0.574821\pi\)
0.958660 − 0.284556i \(-0.0918460\pi\)
\(8\) −2.26178 + 1.64328i −0.799662 + 0.580988i
\(9\) 0 0
\(10\) −2.87591 − 1.39789i −0.909444 − 0.442052i
\(11\) −3.49787 − 3.88478i −1.05465 − 1.17131i −0.984790 − 0.173748i \(-0.944412\pi\)
−0.0698578 − 0.997557i \(-0.522255\pi\)
\(12\) 0 0
\(13\) −2.25011 + 2.49900i −0.624069 + 0.693099i −0.969429 − 0.245371i \(-0.921090\pi\)
0.345360 + 0.938470i \(0.387757\pi\)
\(14\) −5.37185 + 1.14182i −1.43569 + 0.305165i
\(15\) 0 0
\(16\) 3.99864 + 0.849937i 0.999660 + 0.212484i
\(17\) 3.90998 − 2.84077i 0.948309 − 0.688987i −0.00209712 − 0.999998i \(-0.500668\pi\)
0.950406 + 0.311011i \(0.100668\pi\)
\(18\) 0 0
\(19\) −1.90903 + 1.38699i −0.437961 + 0.318197i −0.784824 − 0.619719i \(-0.787247\pi\)
0.346863 + 0.937916i \(0.387247\pi\)
\(20\) 0.0278642 + 0.0966831i 0.00623063 + 0.0216190i
\(21\) 0 0
\(22\) −0.781401 + 7.43453i −0.166595 + 1.58505i
\(23\) −4.56147 + 0.969570i −0.951132 + 0.202169i −0.657250 − 0.753672i \(-0.728281\pi\)
−0.293882 + 0.955842i \(0.594947\pi\)
\(24\) 0 0
\(25\) 3.58734 − 3.48295i 0.717469 − 0.696591i
\(26\) 4.80884 0.943090
\(27\) 0 0
\(28\) 0.139805 + 0.101574i 0.0264207 + 0.0191957i
\(29\) 6.71254 − 2.98862i 1.24649 − 0.554972i 0.325862 − 0.945417i \(-0.394346\pi\)
0.920626 + 0.390445i \(0.127679\pi\)
\(30\) 0 0
\(31\) 0.593962 + 0.264449i 0.106679 + 0.0474964i 0.459382 − 0.888239i \(-0.348071\pi\)
−0.352703 + 0.935735i \(0.614738\pi\)
\(32\) −0.127249 − 0.220402i −0.0224947 − 0.0389619i
\(33\) 0 0
\(34\) −6.76033 − 1.43695i −1.15939 − 0.246435i
\(35\) 1.18369 − 8.50536i 0.200080 − 1.43767i
\(36\) 0 0
\(37\) 0.315262 + 0.970276i 0.0518287 + 0.159512i 0.973621 − 0.228172i \(-0.0732749\pi\)
−0.921792 + 0.387685i \(0.873275\pi\)
\(38\) 3.30069 + 0.701584i 0.535443 + 0.113812i
\(39\) 0 0
\(40\) −3.30561 + 5.30596i −0.522664 + 0.838945i
\(41\) 2.50903 − 2.78656i 0.391845 − 0.435188i −0.514653 − 0.857399i \(-0.672079\pi\)
0.906498 + 0.422211i \(0.138746\pi\)
\(42\) 0 0
\(43\) −2.34851 + 4.06773i −0.358144 + 0.620324i −0.987651 − 0.156671i \(-0.949924\pi\)
0.629507 + 0.776995i \(0.283257\pi\)
\(44\) 0.190302 − 0.138262i 0.0286890 − 0.0208438i
\(45\) 0 0
\(46\) 5.39516 + 3.91981i 0.795473 + 0.577945i
\(47\) −10.9118 + 4.85827i −1.59166 + 0.708651i −0.995548 − 0.0942531i \(-0.969954\pi\)
−0.596108 + 0.802904i \(0.703287\pi\)
\(48\) 0 0
\(49\) −3.87423 − 6.71036i −0.553461 − 0.958623i
\(50\) −7.13407 − 0.479583i −1.00891 − 0.0678233i
\(51\) 0 0
\(52\) −0.101250 − 0.112450i −0.0140409 − 0.0155940i
\(53\) −2.92011 − 2.12158i −0.401108 − 0.291422i 0.368884 − 0.929475i \(-0.379740\pi\)
−0.769992 + 0.638054i \(0.779740\pi\)
\(54\) 0 0
\(55\) −10.5129 − 5.10999i −1.41756 − 0.689031i
\(56\) 1.12228 + 10.6778i 0.149971 + 1.42688i
\(57\) 0 0
\(58\) −9.59917 − 4.27383i −1.26043 − 0.561181i
\(59\) 5.20768 − 5.78372i 0.677982 − 0.752976i −0.301728 − 0.953394i \(-0.597563\pi\)
0.979710 + 0.200418i \(0.0642302\pi\)
\(60\) 0 0
\(61\) −4.40843 − 4.89605i −0.564441 − 0.626875i 0.391590 − 0.920140i \(-0.371925\pi\)
−0.956031 + 0.293264i \(0.905258\pi\)
\(62\) −0.287314 − 0.884263i −0.0364890 − 0.112301i
\(63\) 0 0
\(64\) 2.41404 − 7.42965i 0.301755 − 0.928707i
\(65\) −2.56225 + 7.06930i −0.317808 + 0.876839i
\(66\) 0 0
\(67\) −0.930565 − 0.414314i −0.113687 − 0.0506165i 0.349105 − 0.937084i \(-0.386486\pi\)
−0.462792 + 0.886467i \(0.653152\pi\)
\(68\) 0.108737 + 0.188339i 0.0131863 + 0.0228394i
\(69\) 0 0
\(70\) −10.1715 + 6.88068i −1.21573 + 0.822398i
\(71\) 2.82082 + 2.04945i 0.334770 + 0.243225i 0.742452 − 0.669899i \(-0.233663\pi\)
−0.407682 + 0.913124i \(0.633663\pi\)
\(72\) 0 0
\(73\) −1.38140 + 4.25150i −0.161680 + 0.497601i −0.998776 − 0.0494551i \(-0.984252\pi\)
0.837096 + 0.547056i \(0.184252\pi\)
\(74\) 0.729467 − 1.26347i 0.0847988 − 0.146876i
\(75\) 0 0
\(76\) −0.0530904 − 0.0919553i −0.00608989 − 0.0105480i
\(77\) −19.6368 + 4.17393i −2.23782 + 0.475663i
\(78\) 0 0
\(79\) 1.49826 − 0.667070i 0.168568 − 0.0750513i −0.320718 − 0.947175i \(-0.603924\pi\)
0.489286 + 0.872123i \(0.337257\pi\)
\(80\) 8.99997 − 1.59942i 1.00623 − 0.178821i
\(81\) 0 0
\(82\) −5.36219 −0.592154
\(83\) 1.18152 + 11.2414i 0.129688 + 1.23390i 0.844873 + 0.534967i \(0.179676\pi\)
−0.715185 + 0.698936i \(0.753657\pi\)
\(84\) 0 0
\(85\) 5.71446 − 9.17248i 0.619820 − 0.994895i
\(86\) 6.57011 − 1.39652i 0.708473 − 0.150591i
\(87\) 0 0
\(88\) 14.2952 + 3.03854i 1.52388 + 0.323910i
\(89\) 0.935178 − 2.87818i 0.0991286 − 0.305087i −0.889179 − 0.457559i \(-0.848724\pi\)
0.988308 + 0.152473i \(0.0487237\pi\)
\(90\) 0 0
\(91\) 3.99070 + 12.2821i 0.418339 + 1.28752i
\(92\) −0.0219345 − 0.208692i −0.00228682 − 0.0217577i
\(93\) 0 0
\(94\) 15.6043 + 6.94749i 1.60946 + 0.716579i
\(95\) −2.79006 + 4.47841i −0.286254 + 0.459475i
\(96\) 0 0
\(97\) 14.8057 − 6.59193i 1.50329 − 0.669309i 0.520474 − 0.853878i \(-0.325755\pi\)
0.982820 + 0.184569i \(0.0590888\pi\)
\(98\) −3.42408 + 10.5382i −0.345885 + 1.06452i
\(99\) 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.46.7 224
3.2 odd 2 225.2.q.a.196.22 yes 224
9.4 even 3 inner 675.2.r.a.496.22 224
9.5 odd 6 225.2.q.a.121.7 yes 224
25.6 even 5 inner 675.2.r.a.181.22 224
75.56 odd 10 225.2.q.a.106.7 yes 224
225.31 even 15 inner 675.2.r.a.631.7 224
225.131 odd 30 225.2.q.a.31.22 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.22 ✓ 224 225.131 odd 30
225.2.q.a.106.7 yes 224 75.56 odd 10
225.2.q.a.121.7 yes 224 9.5 odd 6
225.2.q.a.196.22 yes 224 3.2 odd 2
675.2.r.a.46.7 224 1.1 even 1 trivial
675.2.r.a.181.22 224 25.6 even 5 inner
675.2.r.a.496.22 224 9.4 even 3 inner
675.2.r.a.631.7 224 225.31 even 15 inner