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Properties

Label 1575.4.a.bi.1.4
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,10,0,0,28,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{42 +2 \sqrt{329}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.42371\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.42371 q^{2} +11.5692 q^{4} +7.00000 q^{7} +15.7890 q^{8} -23.3775 q^{11} +3.56918 q^{13} +30.9659 q^{14} -22.7075 q^{16} -57.5082 q^{17} -51.1384 q^{19} -103.415 q^{22} -65.6390 q^{23} +15.7890 q^{26} +80.9843 q^{28} +41.6147 q^{29} -167.538 q^{31} -226.763 q^{32} -254.399 q^{34} -224.538 q^{37} -226.221 q^{38} +196.688 q^{41} +58.9685 q^{43} -270.458 q^{44} -290.368 q^{46} -41.9282 q^{47} +49.0000 q^{49} +41.2925 q^{52} +33.3445 q^{53} +110.523 q^{56} +184.091 q^{58} -229.212 q^{59} -700.613 q^{61} -741.138 q^{62} -821.475 q^{64} -453.890 q^{67} -665.322 q^{68} +930.571 q^{71} +370.182 q^{73} -993.289 q^{74} -591.629 q^{76} -163.642 q^{77} -54.6007 q^{79} +870.091 q^{82} +430.045 q^{83} +260.859 q^{86} -369.107 q^{88} -737.202 q^{89} +24.9843 q^{91} -759.390 q^{92} -185.478 q^{94} +150.371 q^{97} +216.762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 28 q^{7} - 22 q^{13} + 18 q^{16} - 132 q^{19} - 196 q^{22} + 70 q^{28} - 126 q^{31} - 546 q^{34} - 354 q^{37} - 272 q^{43} - 182 q^{46} + 196 q^{49} + 274 q^{52} - 98 q^{58} - 1170 q^{61}+ \cdots + 1980 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42371 1.56402 0.782008 − 0.623268i \(-0.214195\pi\)
0.782008 + 0.623268i \(0.214195\pi\)
\(3\) 0 0
\(4\) 11.5692 1.44615
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 15.7890 0.697782
\(9\) 0 0
\(10\) 0 0
\(11\) −23.3775 −0.640779 −0.320390 − 0.947286i \(-0.603814\pi\)
−0.320390 + 0.947286i \(0.603814\pi\)
\(12\) 0 0
\(13\) 3.56918 0.0761471 0.0380735 − 0.999275i \(-0.487878\pi\)
0.0380735 + 0.999275i \(0.487878\pi\)
\(14\) 30.9659 0.591143
\(15\) 0 0
\(16\) −22.7075 −0.354805
\(17\) −57.5082 −0.820458 −0.410229 − 0.911983i \(-0.634551\pi\)
−0.410229 + 0.911983i \(0.634551\pi\)
\(18\) 0 0
\(19\) −51.1384 −0.617471 −0.308735 − 0.951148i \(-0.599906\pi\)
−0.308735 + 0.951148i \(0.599906\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −103.415 −1.00219
\(23\) −65.6390 −0.595073 −0.297537 − 0.954710i \(-0.596165\pi\)
−0.297537 + 0.954710i \(0.596165\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 15.7890 0.119095
\(27\) 0 0
\(28\) 80.9843 0.546592
\(29\) 41.6147 0.266471 0.133235 − 0.991084i \(-0.457463\pi\)
0.133235 + 0.991084i \(0.457463\pi\)
\(30\) 0 0
\(31\) −167.538 −0.970666 −0.485333 − 0.874329i \(-0.661302\pi\)
−0.485333 + 0.874329i \(0.661302\pi\)
\(32\) −226.763 −1.25270
\(33\) 0 0
\(34\) −254.399 −1.28321
\(35\) 0 0
\(36\) 0 0
\(37\) −224.538 −0.997669 −0.498835 − 0.866697i \(-0.666239\pi\)
−0.498835 + 0.866697i \(0.666239\pi\)
\(38\) −226.221 −0.965734
\(39\) 0 0
\(40\) 0 0
\(41\) 196.688 0.749208 0.374604 − 0.927185i \(-0.377779\pi\)
0.374604 + 0.927185i \(0.377779\pi\)
\(42\) 0 0
\(43\) 58.9685 0.209131 0.104565 − 0.994518i \(-0.466655\pi\)
0.104565 + 0.994518i \(0.466655\pi\)
\(44\) −270.458 −0.926661
\(45\) 0 0
\(46\) −290.368 −0.930704
\(47\) −41.9282 −0.130125 −0.0650623 − 0.997881i \(-0.520725\pi\)
−0.0650623 + 0.997881i \(0.520725\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 41.2925 0.110120
\(53\) 33.3445 0.0864192 0.0432096 − 0.999066i \(-0.486242\pi\)
0.0432096 + 0.999066i \(0.486242\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 110.523 0.263737
\(57\) 0 0
\(58\) 184.091 0.416765
\(59\) −229.212 −0.505777 −0.252888 − 0.967495i \(-0.581381\pi\)
−0.252888 + 0.967495i \(0.581381\pi\)
\(60\) 0 0
\(61\) −700.613 −1.47056 −0.735281 − 0.677762i \(-0.762950\pi\)
−0.735281 + 0.677762i \(0.762950\pi\)
\(62\) −741.138 −1.51814
\(63\) 0 0
\(64\) −821.475 −1.60444
\(65\) 0 0
\(66\) 0 0
\(67\) −453.890 −0.827634 −0.413817 − 0.910360i \(-0.635805\pi\)
−0.413817 + 0.910360i \(0.635805\pi\)
\(68\) −665.322 −1.18650
\(69\) 0 0
\(70\) 0 0
\(71\) 930.571 1.55547 0.777735 − 0.628592i \(-0.216368\pi\)
0.777735 + 0.628592i \(0.216368\pi\)
\(72\) 0 0
\(73\) 370.182 0.593514 0.296757 − 0.954953i \(-0.404095\pi\)
0.296757 + 0.954953i \(0.404095\pi\)
\(74\) −993.289 −1.56037
\(75\) 0 0
\(76\) −591.629 −0.892954
\(77\) −163.642 −0.242192
\(78\) 0 0
\(79\) −54.6007 −0.0777602 −0.0388801 − 0.999244i \(-0.512379\pi\)
−0.0388801 + 0.999244i \(0.512379\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 870.091 1.17177
\(83\) 430.045 0.568718 0.284359 − 0.958718i \(-0.408219\pi\)
0.284359 + 0.958718i \(0.408219\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 260.859 0.327084
\(87\) 0 0
\(88\) −369.107 −0.447124
\(89\) −737.202 −0.878014 −0.439007 − 0.898484i \(-0.644670\pi\)
−0.439007 + 0.898484i \(0.644670\pi\)
\(90\) 0 0
\(91\) 24.9843 0.0287809
\(92\) −759.390 −0.860564
\(93\) 0 0
\(94\) −185.478 −0.203517
\(95\) 0 0
\(96\) 0 0
\(97\) 150.371 0.157401 0.0787004 − 0.996898i \(-0.474923\pi\)
0.0787004 + 0.996898i \(0.474923\pi\)
\(98\) 216.762 0.223431
\(99\) 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 1575.4.a.bi.1.4 yes 4
3.2 odd 2 inner 1575.4.a.bi.1.1 yes 4
5.4 even 2 1575.4.a.bh.1.1 ✓ 4
15.14 odd 2 1575.4.a.bh.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.1 ✓ 4 5.4 even 2
1575.4.a.bh.1.4 yes 4 15.14 odd 2
1575.4.a.bi.1.1 yes 4 3.2 odd 2 inner
1575.4.a.bi.1.4 yes 4 1.1 even 1 trivial