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Properties

Label 675.2.r.a.46.17
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 46.17
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.17

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.251657 + 0.279494i) q^{2} +(0.194272 - 1.84837i) q^{4} +(0.102616 - 2.23371i) q^{5} +(1.17975 - 2.04339i) q^{7} +(1.17403 - 0.852985i) q^{8} +(0.650132 - 0.533449i) q^{10} +(-0.251022 - 0.278789i) q^{11} +(0.237736 - 0.264032i) q^{13} +(0.868006 - 0.184500i) q^{14} +(-3.10202 - 0.659354i) q^{16} +(-3.45606 + 2.51098i) q^{17} +(-3.26679 + 2.37346i) q^{19} +(-4.10879 - 0.623618i) q^{20} +(0.0147481 - 0.140318i) q^{22} +(4.49630 - 0.955717i) q^{23} +(-4.97894 - 0.458427i) q^{25} +0.133623 q^{26} +(-3.54774 - 2.57759i) q^{28} +(0.336493 - 0.149816i) q^{29} +(5.37131 + 2.39146i) q^{31} +(-2.04755 - 3.54645i) q^{32} +(-1.57154 - 0.334042i) q^{34} +(-4.44328 - 2.84490i) q^{35} +(-3.03352 - 9.33621i) q^{37} +(-1.48548 - 0.315748i) q^{38} +(-1.78485 - 2.70998i) q^{40} +(5.77601 - 6.41491i) q^{41} +(-3.86472 + 6.69390i) q^{43} +(-0.564071 + 0.409822i) q^{44} +(1.39864 + 1.01617i) q^{46} +(0.611825 - 0.272402i) q^{47} +(0.716381 + 1.24081i) q^{49} +(-1.12486 - 1.50695i) q^{50} +(-0.441844 - 0.490718i) q^{52} +(8.98419 + 6.52740i) q^{53} +(-0.648493 + 0.532104i) q^{55} +(-0.357913 - 3.40531i) q^{56} +(0.126554 + 0.0563453i) q^{58} +(7.75940 - 8.61768i) q^{59} +(4.33263 + 4.81187i) q^{61} +(0.683331 + 2.10308i) q^{62} +(-1.48405 + 4.56744i) q^{64} +(-0.565377 - 0.558127i) q^{65} +(-11.6258 - 5.17613i) q^{67} +(3.96980 + 6.87589i) q^{68} +(-0.323050 - 1.95781i) q^{70} +(1.67206 + 1.21482i) q^{71} +(1.94846 - 5.99673i) q^{73} +(1.84600 - 3.19737i) q^{74} +(3.75239 + 6.49933i) q^{76} +(-0.865817 + 0.184035i) q^{77} +(10.8685 - 4.83898i) q^{79} +(-1.79112 + 6.86136i) q^{80} +3.24650 q^{82} +(-0.919146 - 8.74509i) q^{83} +(5.25415 + 7.97751i) q^{85} +(-2.84349 + 0.604402i) q^{86} +(-0.532511 - 0.113189i) q^{88} +(0.603343 - 1.85690i) q^{89} +(-0.259051 - 0.797278i) q^{91} +(-0.893017 - 8.49649i) q^{92} +(0.230105 + 0.102449i) q^{94} +(4.96640 + 7.54061i) q^{95} +(10.2797 - 4.57681i) q^{97} +(-0.166516 + 0.512483i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.251657 + 0.279494i 0.177949 + 0.197632i 0.825520 − 0.564373i \(-0.190882\pi\)
−0.647571 + 0.762005i \(0.724215\pi\)
\(3\) 0 0
\(4\) 0.194272 − 1.84837i 0.0971358 − 0.924185i
\(5\) 0.102616 − 2.23371i 0.0458911 − 0.998946i
\(6\) 0 0
\(7\) 1.17975 − 2.04339i 0.445903 − 0.772327i −0.552211 − 0.833704i \(-0.686216\pi\)
0.998115 + 0.0613768i \(0.0195491\pi\)
\(8\) 1.17403 − 0.852985i 0.415084 − 0.301576i
\(9\) 0 0
\(10\) 0.650132 − 0.533449i 0.205590 − 0.168692i
\(11\) −0.251022 − 0.278789i −0.0756861 − 0.0840580i 0.704112 − 0.710089i \(-0.251345\pi\)
−0.779798 + 0.626031i \(0.784678\pi\)
\(12\) 0 0
\(13\) 0.237736 − 0.264032i 0.0659360 − 0.0732294i −0.709274 − 0.704933i \(-0.750977\pi\)
0.775210 + 0.631703i \(0.217644\pi\)
\(14\) 0.868006 − 0.184500i 0.231984 − 0.0493098i
\(15\) 0 0
\(16\) −3.10202 − 0.659354i −0.775505 − 0.164839i
\(17\) −3.45606 + 2.51098i −0.838218 + 0.609001i −0.921872 − 0.387494i \(-0.873341\pi\)
0.0836542 + 0.996495i \(0.473341\pi\)
\(18\) 0 0
\(19\) −3.26679 + 2.37346i −0.749452 + 0.544509i −0.895657 − 0.444746i \(-0.853294\pi\)
0.146205 + 0.989254i \(0.453294\pi\)
\(20\) −4.10879 − 0.623618i −0.918754 − 0.139445i
\(21\) 0 0
\(22\) 0.0147481 − 0.140318i 0.00314430 − 0.0299160i
\(23\) 4.49630 − 0.955717i 0.937543 − 0.199281i 0.286291 − 0.958143i \(-0.407578\pi\)
0.651252 + 0.758862i \(0.274244\pi\)
\(24\) 0 0
\(25\) −4.97894 − 0.458427i −0.995788 − 0.0916855i
\(26\) 0.133623 0.0262057
\(27\) 0 0
\(28\) −3.54774 − 2.57759i −0.670460 − 0.487118i
\(29\) 0.336493 − 0.149816i 0.0624852 − 0.0278202i −0.375256 − 0.926921i \(-0.622445\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(30\) 0 0
\(31\) 5.37131 + 2.39146i 0.964716 + 0.429519i 0.827775 − 0.561060i \(-0.189606\pi\)
0.136941 + 0.990579i \(0.456273\pi\)
\(32\) −2.04755 − 3.54645i −0.361958 − 0.626930i
\(33\) 0 0
\(34\) −1.57154 − 0.334042i −0.269518 − 0.0572878i
\(35\) −4.44328 − 2.84490i −0.751051 − 0.480877i
\(36\) 0 0
\(37\) −3.03352 − 9.33621i −0.498707 − 1.53486i −0.811099 − 0.584910i \(-0.801130\pi\)
0.312391 − 0.949953i \(-0.398870\pi\)
\(38\) −1.48548 − 0.315748i −0.240976 − 0.0512211i
\(39\) 0 0
\(40\) −1.78485 − 2.70998i −0.282209 − 0.428486i
\(41\) 5.77601 − 6.41491i 0.902061 − 1.00184i −0.0979168 − 0.995195i \(-0.531218\pi\)
0.999978 − 0.00664576i \(-0.00211543\pi\)
\(42\) 0 0
\(43\) −3.86472 + 6.69390i −0.589365 + 1.02081i 0.404951 + 0.914338i \(0.367289\pi\)
−0.994316 + 0.106471i \(0.966045\pi\)
\(44\) −0.564071 + 0.409822i −0.0850370 + 0.0617830i
\(45\) 0 0
\(46\) 1.39864 + 1.01617i 0.206219 + 0.149827i
\(47\) 0.611825 − 0.272402i 0.0892438 − 0.0397339i −0.361629 − 0.932322i \(-0.617779\pi\)
0.450873 + 0.892588i \(0.351113\pi\)
\(48\) 0 0
\(49\) 0.716381 + 1.24081i 0.102340 + 0.177258i
\(50\) −1.12486 − 1.50695i −0.159079 − 0.213115i
\(51\) 0 0
\(52\) −0.441844 − 0.490718i −0.0612728 − 0.0680503i
\(53\) 8.98419 + 6.52740i 1.23407 + 0.896607i 0.997189 − 0.0749310i \(-0.0238736\pi\)
0.236885 + 0.971538i \(0.423874\pi\)
\(54\) 0 0
\(55\) −0.648493 + 0.532104i −0.0874427 + 0.0717489i
\(56\) −0.357913 − 3.40531i −0.0478281 − 0.455054i
\(57\) 0 0
\(58\) 0.126554 + 0.0563453i 0.0166173 + 0.00739850i
\(59\) 7.75940 − 8.61768i 1.01019 − 1.12193i 0.0176673 − 0.999844i \(-0.494376\pi\)
0.992520 − 0.122083i \(-0.0389573\pi\)
\(60\) 0 0
\(61\) 4.33263 + 4.81187i 0.554736 + 0.616097i 0.953660 − 0.300887i \(-0.0972829\pi\)
−0.398924 + 0.916984i \(0.630616\pi\)
\(62\) 0.683331 + 2.10308i 0.0867831 + 0.267091i
\(63\) 0 0
\(64\) −1.48405 + 4.56744i −0.185506 + 0.570930i
\(65\) −0.565377 − 0.558127i −0.0701264 − 0.0692272i
\(66\) 0 0
\(67\) −11.6258 − 5.17613i −1.42032 − 0.632365i −0.454300 − 0.890849i \(-0.650111\pi\)
−0.966016 + 0.258484i \(0.916777\pi\)
\(68\) 3.96980 + 6.87589i 0.481409 + 0.833825i
\(69\) 0 0
\(70\) −0.323050 − 1.95781i −0.0386118 − 0.234003i
\(71\) 1.67206 + 1.21482i 0.198437 + 0.144173i 0.682567 − 0.730823i \(-0.260864\pi\)
−0.484129 + 0.874996i \(0.660864\pi\)
\(72\) 0 0
\(73\) 1.94846 − 5.99673i 0.228050 − 0.701864i −0.769918 − 0.638143i \(-0.779703\pi\)
0.997968 − 0.0637218i \(-0.0202970\pi\)
\(74\) 1.84600 − 3.19737i 0.214594 − 0.371687i
\(75\) 0 0
\(76\) 3.75239 + 6.49933i 0.430428 + 0.745524i
\(77\) −0.865817 + 0.184035i −0.0986690 + 0.0209727i
\(78\) 0 0
\(79\) 10.8685 − 4.83898i 1.22281 − 0.544428i 0.309187 − 0.951001i \(-0.399943\pi\)
0.913618 + 0.406573i \(0.133276\pi\)
\(80\) −1.79112 + 6.86136i −0.200254 + 0.767123i
\(81\) 0 0
\(82\) 3.24650 0.358516
\(83\) −0.919146 − 8.74509i −0.100889 − 0.959899i −0.921491 − 0.388399i \(-0.873028\pi\)
0.820602 − 0.571500i \(-0.193638\pi\)
\(84\) 0 0
\(85\) 5.25415 + 7.97751i 0.569893 + 0.865283i
\(86\) −2.84349 + 0.604402i −0.306621 + 0.0651743i
\(87\) 0 0
\(88\) −0.532511 − 0.113189i −0.0567659 − 0.0120660i
\(89\) 0.603343 − 1.85690i 0.0639542 − 0.196831i −0.913974 − 0.405774i \(-0.867002\pi\)
0.977928 + 0.208943i \(0.0670022\pi\)
\(90\) 0 0
\(91\) −0.259051 − 0.797278i −0.0271560 − 0.0835775i
\(92\) −0.893017 − 8.49649i −0.0931035 − 0.885821i
\(93\) 0 0
\(94\) 0.230105 + 0.102449i 0.0237335 + 0.0105668i
\(95\) 4.96640 + 7.54061i 0.509542 + 0.773650i
\(96\) 0 0
\(97\) 10.2797 − 4.57681i 1.04374 − 0.464704i 0.188034 − 0.982162i \(-0.439788\pi\)
0.855708 + 0.517458i \(0.173122\pi\)
\(98\) −0.166516 + 0.512483i −0.0168206 + 0.0517686i
\(99\) 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.46.17 224
3.2 odd 2 225.2.q.a.196.12 yes 224
9.4 even 3 inner 675.2.r.a.496.12 224
9.5 odd 6 225.2.q.a.121.17 yes 224
25.6 even 5 inner 675.2.r.a.181.12 224
75.56 odd 10 225.2.q.a.106.17 yes 224
225.31 even 15 inner 675.2.r.a.631.17 224
225.131 odd 30 225.2.q.a.31.12 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.12 ✓ 224 225.131 odd 30
225.2.q.a.106.17 yes 224 75.56 odd 10
225.2.q.a.121.17 yes 224 9.5 odd 6
225.2.q.a.196.12 yes 224 3.2 odd 2
675.2.r.a.46.17 224 1.1 even 1 trivial
675.2.r.a.181.12 224 25.6 even 5 inner
675.2.r.a.496.12 224 9.4 even 3 inner
675.2.r.a.631.17 224 225.31 even 15 inner