↑

Properties

Label 675.2.r.a.46.5
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
Inner twists $4$

Related objects

Downloads

Learn more

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 46.5
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.53828 - 1.70844i) q^{2} +(-0.343384 + 3.26708i) q^{4} +(-0.322490 + 2.21269i) q^{5} +(-1.62043 + 2.80667i) q^{7} +(2.39008 - 1.73649i) q^{8} +(4.27633 - 2.85279i) q^{10} +(-1.30655 - 1.45107i) q^{11} +(2.38463 - 2.64840i) q^{13} +(7.28770 - 1.54905i) q^{14} +(-0.216735 - 0.0460685i) q^{16} +(-0.726639 + 0.527934i) q^{17} +(0.733140 - 0.532657i) q^{19} +(-7.11830 - 1.81340i) q^{20} +(-0.469217 + 4.46430i) q^{22} +(-8.31379 + 1.76715i) q^{23} +(-4.79200 - 1.42714i) q^{25} -8.19288 q^{26} +(-8.61318 - 6.25784i) q^{28} +(6.82216 - 3.03742i) q^{29} +(-7.07738 - 3.15105i) q^{31} +(-2.69961 - 4.67586i) q^{32} +(2.01972 + 0.429305i) q^{34} +(-5.68771 - 4.49063i) q^{35} +(-1.33249 - 4.10099i) q^{37} +(-2.03779 - 0.433145i) q^{38} +(3.07155 + 5.84851i) q^{40} +(-3.03574 + 3.37153i) q^{41} +(-0.885445 + 1.53364i) q^{43} +(5.18939 - 3.77032i) q^{44} +(15.8080 + 11.4852i) q^{46} +(-7.06685 + 3.14637i) q^{47} +(-1.75159 - 3.03384i) q^{49} +(4.93328 + 10.3822i) q^{50} +(7.83371 + 8.70021i) q^{52} +(-5.94096 - 4.31636i) q^{53} +(3.63211 - 2.42303i) q^{55} +(1.00081 + 9.52203i) q^{56} +(-15.6837 - 6.98281i) q^{58} +(9.23740 - 10.2592i) q^{59} +(-4.22089 - 4.68777i) q^{61} +(5.50365 + 16.9385i) q^{62} +(-3.97259 + 12.2264i) q^{64} +(5.09108 + 6.13054i) q^{65} +(-11.0747 - 4.93079i) q^{67} +(-1.47529 - 2.55527i) q^{68} +(1.07736 + 16.6250i) q^{70} +(8.02747 + 5.83230i) q^{71} +(0.246766 - 0.759468i) q^{73} +(-4.95654 + 8.58498i) q^{74} +(1.48849 + 2.57813i) q^{76} +(6.18982 - 1.31569i) q^{77} +(-0.222600 + 0.0991078i) q^{79} +(0.171830 - 0.464711i) q^{80} +10.4299 q^{82} +(0.483482 + 4.60003i) q^{83} +(-0.933822 - 1.77808i) q^{85} +(3.98219 - 0.846440i) q^{86} +(-5.64252 - 1.19935i) q^{88} +(-2.11267 + 6.50212i) q^{89} +(3.56906 + 10.9844i) q^{91} +(-2.91860 - 27.7686i) q^{92} +(16.2462 + 7.23327i) q^{94} +(0.942175 + 1.79399i) q^{95} +(-4.33055 + 1.92809i) q^{97} +(-2.48869 + 7.65939i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53828 − 1.70844i −1.08773 − 1.20805i −0.976785 − 0.214221i \(-0.931279\pi\)
−0.110946 − 0.993826i \(-0.535388\pi\)
\(3\) 0 0
\(4\) −0.343384 + 3.26708i −0.171692 + 1.63354i
\(5\) −0.322490 + 2.21269i −0.144222 + 0.989545i
\(6\) 0 0
\(7\) −1.62043 + 2.80667i −0.612465 + 1.06082i 0.378359 + 0.925659i \(0.376489\pi\)
−0.990824 + 0.135161i \(0.956845\pi\)
\(8\) 2.39008 − 1.73649i 0.845021 − 0.613944i
\(9\) 0 0
\(10\) 4.27633 − 2.85279i 1.35229 − 0.902132i
\(11\) −1.30655 − 1.45107i −0.393938 − 0.437513i 0.513249 − 0.858240i \(-0.328442\pi\)
−0.907187 + 0.420727i \(0.861775\pi\)
\(12\) 0 0
\(13\) 2.38463 − 2.64840i 0.661379 − 0.734535i −0.315359 − 0.948972i \(-0.602125\pi\)
0.976738 + 0.214437i \(0.0687917\pi\)
\(14\) 7.28770 − 1.54905i 1.94772 − 0.414001i
\(15\) 0 0
\(16\) −0.216735 − 0.0460685i −0.0541838 − 0.0115171i
\(17\) −0.726639 + 0.527934i −0.176236 + 0.128043i −0.672406 − 0.740182i \(-0.734739\pi\)
0.496170 + 0.868225i \(0.334739\pi\)
\(18\) 0 0
\(19\) 0.733140 − 0.532657i 0.168194 − 0.122200i −0.500504 − 0.865734i \(-0.666852\pi\)
0.668698 + 0.743534i \(0.266852\pi\)
\(20\) −7.11830 − 1.81340i −1.59170 − 0.405489i
\(21\) 0 0
\(22\) −0.469217 + 4.46430i −0.100037 + 0.951792i
\(23\) −8.31379 + 1.76715i −1.73354 + 0.368476i −0.963124 − 0.269059i \(-0.913287\pi\)
−0.770421 + 0.637535i \(0.779954\pi\)
\(24\) 0 0
\(25\) −4.79200 − 1.42714i −0.958400 − 0.285428i
\(26\) −8.19288 −1.60676
\(27\) 0 0
\(28\) −8.61318 − 6.25784i −1.62774 − 1.18262i
\(29\) 6.82216 − 3.03742i 1.26684 − 0.564035i 0.340333 − 0.940305i \(-0.389460\pi\)
0.926510 + 0.376270i \(0.122794\pi\)
\(30\) 0 0
\(31\) −7.07738 − 3.15105i −1.27114 − 0.565946i −0.343402 − 0.939189i \(-0.611579\pi\)
−0.927734 + 0.373243i \(0.878246\pi\)
\(32\) −2.69961 − 4.67586i −0.477228 − 0.826582i
\(33\) 0 0
\(34\) 2.01972 + 0.429305i 0.346379 + 0.0736252i
\(35\) −5.68771 − 4.49063i −0.961399 − 0.759055i
\(36\) 0 0
\(37\) −1.33249 − 4.10099i −0.219061 − 0.674199i −0.998840 − 0.0481453i \(-0.984669\pi\)
0.779780 − 0.626054i \(-0.215331\pi\)
\(38\) −2.03779 − 0.433145i −0.330573 − 0.0702655i
\(39\) 0 0
\(40\) 3.07155 + 5.84851i 0.485655 + 0.924731i
\(41\) −3.03574 + 3.37153i −0.474103 + 0.526545i −0.931999 − 0.362461i \(-0.881937\pi\)
0.457896 + 0.889006i \(0.348603\pi\)
\(42\) 0 0
\(43\) −0.885445 + 1.53364i −0.135029 + 0.233877i −0.925609 − 0.378482i \(-0.876446\pi\)
0.790579 + 0.612359i \(0.209779\pi\)
\(44\) 5.18939 − 3.77032i 0.782331 − 0.568397i
\(45\) 0 0
\(46\) 15.8080 + 11.4852i 2.33077 + 1.69340i
\(47\) −7.06685 + 3.14637i −1.03081 + 0.458945i −0.851226 − 0.524799i \(-0.824141\pi\)
−0.179580 + 0.983743i \(0.557474\pi\)
\(48\) 0 0
\(49\) −1.75159 − 3.03384i −0.250227 − 0.433406i
\(50\) 4.93328 + 10.3822i 0.697671 + 1.46826i
\(51\) 0 0
\(52\) 7.83371 + 8.70021i 1.08634 + 1.20650i
\(53\) −5.94096 − 4.31636i −0.816053 − 0.592897i 0.0995259 − 0.995035i \(-0.468267\pi\)
−0.915579 + 0.402137i \(0.868267\pi\)
\(54\) 0 0
\(55\) 3.63211 − 2.42303i 0.489753 − 0.326721i
\(56\) 1.00081 + 9.52203i 0.133738 + 1.27243i
\(57\) 0 0
\(58\) −15.6837 − 6.98281i −2.05937 − 0.916888i
\(59\) 9.23740 − 10.2592i 1.20261 − 1.33563i 0.275286 − 0.961362i \(-0.411228\pi\)
0.927321 − 0.374267i \(-0.122106\pi\)
\(60\) 0 0
\(61\) −4.22089 − 4.68777i −0.540429 − 0.600207i 0.409640 − 0.912247i \(-0.365654\pi\)
−0.950069 + 0.312040i \(0.898988\pi\)
\(62\) 5.50365 + 16.9385i 0.698964 + 2.15119i
\(63\) 0 0
\(64\) −3.97259 + 12.2264i −0.496573 + 1.52830i
\(65\) 5.09108 + 6.13054i 0.631471 + 0.760400i
\(66\) 0 0
\(67\) −11.0747 − 4.93079i −1.35300 − 0.602392i −0.403156 − 0.915131i \(-0.632087\pi\)
−0.949839 + 0.312739i \(0.898754\pi\)
\(68\) −1.47529 − 2.55527i −0.178905 − 0.309872i
\(69\) 0 0
\(70\) 1.07736 + 16.6250i 0.128769 + 1.98706i
\(71\) 8.02747 + 5.83230i 0.952686 + 0.692167i 0.951441 − 0.307833i \(-0.0996037\pi\)
0.00124504 + 0.999999i \(0.499604\pi\)
\(72\) 0 0
\(73\) 0.246766 − 0.759468i 0.0288818 − 0.0888891i −0.935577 − 0.353124i \(-0.885119\pi\)
0.964458 + 0.264235i \(0.0851194\pi\)
\(74\) −4.95654 + 8.58498i −0.576186 + 0.997983i
\(75\) 0 0
\(76\) 1.48849 + 2.57813i 0.170741 + 0.295732i
\(77\) 6.18982 − 1.31569i 0.705396 − 0.149937i
\(78\) 0 0
\(79\) −0.222600 + 0.0991078i −0.0250444 + 0.0111505i −0.419221 − 0.907884i \(-0.637697\pi\)
0.394176 + 0.919035i \(0.371030\pi\)
\(80\) 0.171830 − 0.464711i 0.0192112 − 0.0519563i
\(81\) 0 0
\(82\) 10.4299 1.15179
\(83\) 0.483482 + 4.60003i 0.0530691 + 0.504919i 0.988479 + 0.151355i \(0.0483636\pi\)
−0.935410 + 0.353564i \(0.884970\pi\)
\(84\) 0 0
\(85\) −0.933822 − 1.77808i −0.101287 − 0.192860i
\(86\) 3.98219 − 0.846440i 0.429410 − 0.0912740i
\(87\) 0 0
\(88\) −5.64252 − 1.19935i −0.601494 − 0.127852i
\(89\) −2.11267 + 6.50212i −0.223942 + 0.689224i 0.774455 + 0.632629i \(0.218024\pi\)
−0.998397 + 0.0565946i \(0.981976\pi\)
\(90\) 0 0
\(91\) 3.56906 + 10.9844i 0.374139 + 1.15148i
\(92\) −2.91860 − 27.7686i −0.304285 − 2.89508i
\(93\) 0 0
\(94\) 16.2462 + 7.23327i 1.67567 + 0.746055i
\(95\) 0.942175 + 1.79399i 0.0966652 + 0.184059i
\(96\) 0 0
\(97\) −4.33055 + 1.92809i −0.439701 + 0.195767i −0.614633 − 0.788813i \(-0.710696\pi\)
0.174932 + 0.984580i \(0.444029\pi\)
\(98\) −2.48869 + 7.65939i −0.251395 + 0.773715i
\(99\) 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.46.5 224
3.2 odd 2 225.2.q.a.196.24 yes 224
9.4 even 3 inner 675.2.r.a.496.24 224
9.5 odd 6 225.2.q.a.121.5 yes 224
25.6 even 5 inner 675.2.r.a.181.24 224
75.56 odd 10 225.2.q.a.106.5 yes 224
225.31 even 15 inner 675.2.r.a.631.5 224
225.131 odd 30 225.2.q.a.31.24 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.24 ✓ 224 225.131 odd 30
225.2.q.a.106.5 yes 224 75.56 odd 10
225.2.q.a.121.5 yes 224 9.5 odd 6
225.2.q.a.196.24 yes 224 3.2 odd 2
675.2.r.a.46.5 224 1.1 even 1 trivial
675.2.r.a.181.24 224 25.6 even 5 inner
675.2.r.a.496.24 224 9.4 even 3 inner
675.2.r.a.631.5 224 225.31 even 15 inner