Properties

Label 675.4.a.w.1.1
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,26,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{58 +6 \sqrt{41}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 29x^{2} + 118 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.90965\) of defining polynomial
Character \(\chi\) \(=\) 675.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.90965 q^{2} +16.1047 q^{4} +2.10469 q^{7} -39.7912 q^{8} +40.3052 q^{11} +61.7328 q^{13} -10.3333 q^{14} +66.5234 q^{16} -99.2210 q^{17} +134.523 q^{19} -197.884 q^{22} -76.4985 q^{23} -303.087 q^{26} +33.8953 q^{28} +236.691 q^{29} +243.628 q^{31} -8.27738 q^{32} +487.141 q^{34} +57.4187 q^{37} -660.463 q^{38} -411.383 q^{41} +102.884 q^{43} +649.102 q^{44} +375.581 q^{46} +260.442 q^{47} -338.570 q^{49} +994.187 q^{52} -75.4706 q^{53} -83.7480 q^{56} -1162.07 q^{58} -39.2772 q^{59} -675.851 q^{61} -1196.13 q^{62} -491.548 q^{64} -601.176 q^{67} -1597.92 q^{68} +222.192 q^{71} -297.419 q^{73} -281.906 q^{74} +2166.46 q^{76} +84.8297 q^{77} -95.6999 q^{79} +2019.75 q^{82} +3.08384 q^{83} -505.126 q^{86} -1603.79 q^{88} +1342.73 q^{89} +129.928 q^{91} -1231.99 q^{92} -1278.68 q^{94} +1482.57 q^{97} +1662.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{4} - 30 q^{7} - 22 q^{13} + 74 q^{16} + 346 q^{19} - 100 q^{22} + 174 q^{28} + 744 q^{31} + 796 q^{34} + 76 q^{37} - 280 q^{43} + 1656 q^{46} - 778 q^{49} + 2440 q^{52} - 2420 q^{58} + 178 q^{61}+ \cdots + 2050 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.90965 −1.73582 −0.867912 − 0.496718i \(-0.834538\pi\)
−0.867912 + 0.496718i \(0.834538\pi\)
\(3\) 0 0
\(4\) 16.1047 2.01309
\(5\) 0 0
\(6\) 0 0
\(7\) 2.10469 0.113642 0.0568212 − 0.998384i \(-0.481904\pi\)
0.0568212 + 0.998384i \(0.481904\pi\)
\(8\) −39.7912 −1.75854
\(9\) 0 0
\(10\) 0 0
\(11\) 40.3052 1.10477 0.552385 − 0.833589i \(-0.313718\pi\)
0.552385 + 0.833589i \(0.313718\pi\)
\(12\) 0 0
\(13\) 61.7328 1.31705 0.658523 − 0.752561i \(-0.271182\pi\)
0.658523 + 0.752561i \(0.271182\pi\)
\(14\) −10.3333 −0.197263
\(15\) 0 0
\(16\) 66.5234 1.03943
\(17\) −99.2210 −1.41557 −0.707783 − 0.706430i \(-0.750305\pi\)
−0.707783 + 0.706430i \(0.750305\pi\)
\(18\) 0 0
\(19\) 134.523 1.62430 0.812152 − 0.583445i \(-0.198296\pi\)
0.812152 + 0.583445i \(0.198296\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −197.884 −1.91769
\(23\) −76.4985 −0.693524 −0.346762 − 0.937953i \(-0.612719\pi\)
−0.346762 + 0.937953i \(0.612719\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −303.087 −2.28616
\(27\) 0 0
\(28\) 33.8953 0.228772
\(29\) 236.691 1.51560 0.757801 − 0.652486i \(-0.226274\pi\)
0.757801 + 0.652486i \(0.226274\pi\)
\(30\) 0 0
\(31\) 243.628 1.41151 0.705756 − 0.708455i \(-0.250607\pi\)
0.705756 + 0.708455i \(0.250607\pi\)
\(32\) −8.27738 −0.0457265
\(33\) 0 0
\(34\) 487.141 2.45717
\(35\) 0 0
\(36\) 0 0
\(37\) 57.4187 0.255124 0.127562 − 0.991831i \(-0.459285\pi\)
0.127562 + 0.991831i \(0.459285\pi\)
\(38\) −660.463 −2.81951
\(39\) 0 0
\(40\) 0 0
\(41\) −411.383 −1.56701 −0.783503 − 0.621389i \(-0.786569\pi\)
−0.783503 + 0.621389i \(0.786569\pi\)
\(42\) 0 0
\(43\) 102.884 0.364877 0.182439 − 0.983217i \(-0.441601\pi\)
0.182439 + 0.983217i \(0.441601\pi\)
\(44\) 649.102 2.22400
\(45\) 0 0
\(46\) 375.581 1.20384
\(47\) 260.442 0.808283 0.404142 − 0.914696i \(-0.367570\pi\)
0.404142 + 0.914696i \(0.367570\pi\)
\(48\) 0 0
\(49\) −338.570 −0.987085
\(50\) 0 0
\(51\) 0 0
\(52\) 994.187 2.65133
\(53\) −75.4706 −0.195598 −0.0977989 − 0.995206i \(-0.531180\pi\)
−0.0977989 + 0.995206i \(0.531180\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −83.7480 −0.199845
\(57\) 0 0
\(58\) −1162.07 −2.63082
\(59\) −39.2772 −0.0866688 −0.0433344 − 0.999061i \(-0.513798\pi\)
−0.0433344 + 0.999061i \(0.513798\pi\)
\(60\) 0 0
\(61\) −675.851 −1.41859 −0.709294 − 0.704912i \(-0.750986\pi\)
−0.709294 + 0.704912i \(0.750986\pi\)
\(62\) −1196.13 −2.45014
\(63\) 0 0
\(64\) −491.548 −0.960055
\(65\) 0 0
\(66\) 0 0
\(67\) −601.176 −1.09620 −0.548100 − 0.836413i \(-0.684649\pi\)
−0.548100 + 0.836413i \(0.684649\pi\)
\(68\) −1597.92 −2.84966
\(69\) 0 0
\(70\) 0 0
\(71\) 222.192 0.371400 0.185700 − 0.982607i \(-0.440545\pi\)
0.185700 + 0.982607i \(0.440545\pi\)
\(72\) 0 0
\(73\) −297.419 −0.476852 −0.238426 − 0.971161i \(-0.576632\pi\)
−0.238426 + 0.971161i \(0.576632\pi\)
\(74\) −281.906 −0.442850
\(75\) 0 0
\(76\) 2166.46 3.26986
\(77\) 84.8297 0.125549
\(78\) 0 0
\(79\) −95.6999 −0.136292 −0.0681461 − 0.997675i \(-0.521708\pi\)
−0.0681461 + 0.997675i \(0.521708\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2019.75 2.72005
\(83\) 3.08384 0.00407826 0.00203913 − 0.999998i \(-0.499351\pi\)
0.00203913 + 0.999998i \(0.499351\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −505.126 −0.633363
\(87\) 0 0
\(88\) −1603.79 −1.94278
\(89\) 1342.73 1.59920 0.799601 − 0.600532i \(-0.205044\pi\)
0.799601 + 0.600532i \(0.205044\pi\)
\(90\) 0 0
\(91\) 129.928 0.149672
\(92\) −1231.99 −1.39612
\(93\) 0 0
\(94\) −1278.68 −1.40304
\(95\) 0 0
\(96\) 0 0
\(97\) 1482.57 1.55188 0.775941 − 0.630806i \(-0.217276\pi\)
0.775941 + 0.630806i \(0.217276\pi\)
\(98\) 1662.26 1.71341
\(99\) 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 675.4.a.w.1.1 ✓ 4
3.2 odd 2 inner 675.4.a.w.1.4 yes 4
5.2 odd 4 675.4.b.o.649.2 8
5.3 odd 4 675.4.b.o.649.7 8
5.4 even 2 675.4.a.x.1.4 yes 4
15.2 even 4 675.4.b.o.649.8 8
15.8 even 4 675.4.b.o.649.1 8
15.14 odd 2 675.4.a.x.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.4.a.w.1.1 ✓ 4 1.1 even 1 trivial
675.4.a.w.1.4 yes 4 3.2 odd 2 inner
675.4.a.x.1.1 yes 4 15.14 odd 2
675.4.a.x.1.4 yes 4 5.4 even 2
675.4.b.o.649.1 8 15.8 even 4
675.4.b.o.649.2 8 5.2 odd 4
675.4.b.o.649.7 8 5.3 odd 4
675.4.b.o.649.8 8 15.2 even 4