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Properties

Label 546.4.i.a.235.1
Level $546$
Weight $4$
Character 546.235
Analytic conductor $32.215$
Analytic rank $0$
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] = [546,4,Mod(79,546)] 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("546.79"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 235.1
Root \(1.93649 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 546.235
Dual form 546.4.i.a.79.1

$q$-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+(-1.00000 - 1.73205i) q^{2} +(1.50000 - 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-0.436492 - 0.756026i) q^{5} -6.00000 q^{6} +(-10.8095 + 15.0385i) q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-0.872983 + 1.51205i) q^{10} +(0.682458 - 1.18205i) q^{11} +(6.00000 + 10.3923i) q^{12} +13.0000 q^{13} +(36.8569 + 3.68410i) q^{14} -2.61895 q^{15} +(-8.00000 - 13.8564i) q^{16} +(-12.5716 + 21.7746i) q^{17} +(-9.00000 + 15.5885i) q^{18} +(-6.26210 - 10.8463i) q^{19} +3.49193 q^{20} +(22.8569 + 50.6415i) q^{21} -2.72983 q^{22} +(39.5474 + 68.4981i) q^{23} +(12.0000 - 20.7846i) q^{24} +(62.1190 - 107.593i) q^{25} +(-13.0000 - 22.5167i) q^{26} -27.0000 q^{27} +(-30.4758 - 67.5220i) q^{28} +3.98387 q^{29} +(2.61895 + 4.53615i) q^{30} +(97.5706 - 168.997i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(-2.04738 - 3.54616i) q^{33} +50.2863 q^{34} +(16.0877 + 1.60808i) q^{35} +36.0000 q^{36} +(-90.7853 - 157.245i) q^{37} +(-12.5242 + 21.6926i) q^{38} +(19.5000 - 33.7750i) q^{39} +(-3.49193 - 6.04821i) q^{40} +408.554 q^{41} +(64.8569 - 90.2308i) q^{42} +295.044 q^{43} +(2.72983 + 4.72821i) q^{44} +(-3.92843 + 6.80423i) q^{45} +(79.0948 - 136.996i) q^{46} +(-178.389 - 308.979i) q^{47} -48.0000 q^{48} +(-109.310 - 325.116i) q^{49} -248.476 q^{50} +(37.7147 + 65.3238i) q^{51} +(-26.0000 + 45.0333i) q^{52} +(181.292 - 314.008i) q^{53} +(27.0000 + 46.7654i) q^{54} -1.19155 q^{55} +(-86.4758 + 120.308i) q^{56} -37.5726 q^{57} +(-3.98387 - 6.90026i) q^{58} +(-155.809 + 269.870i) q^{59} +(5.23790 - 9.07231i) q^{60} +(121.380 + 210.236i) q^{61} -390.282 q^{62} +(165.856 + 16.5785i) q^{63} +64.0000 q^{64} +(-5.67439 - 9.82833i) q^{65} +(-4.09475 + 7.09232i) q^{66} +(128.690 - 222.897i) q^{67} +(-50.2863 - 87.0984i) q^{68} +237.284 q^{69} +(-13.3024 - 29.4728i) q^{70} +75.4435 q^{71} +(-36.0000 - 62.3538i) q^{72} +(127.737 - 221.247i) q^{73} +(-181.571 + 314.489i) q^{74} +(-186.357 - 322.780i) q^{75} +50.0968 q^{76} +(10.3992 + 23.0405i) q^{77} -78.0000 q^{78} +(-27.8548 - 48.2459i) q^{79} +(-6.98387 + 12.0964i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(-408.554 - 707.637i) q^{82} +33.5201 q^{83} +(-221.141 - 22.1046i) q^{84} +21.9496 q^{85} +(-295.044 - 511.032i) q^{86} +(5.97580 - 10.3504i) q^{87} +(5.45967 - 9.45642i) q^{88} +(-551.114 - 954.557i) q^{89} +15.7137 q^{90} +(-140.523 + 195.500i) q^{91} -316.379 q^{92} +(-292.712 - 506.991i) q^{93} +(-356.778 + 617.958i) q^{94} +(-5.46671 + 9.46862i) q^{95} +(48.0000 + 83.1384i) q^{96} +1335.57 q^{97} +(-453.806 + 514.447i) q^{98} -12.2843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 6 q^{3} - 8 q^{4} + 6 q^{5} - 24 q^{6} - 20 q^{7} + 32 q^{8} - 18 q^{9} + 12 q^{10} - 36 q^{11} + 24 q^{12} + 52 q^{13} + 8 q^{14} + 36 q^{15} - 32 q^{16} - 120 q^{17} - 36 q^{18} - 118 q^{19}+ \cdots + 648 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.73205i −0.353553 − 0.612372i
\(3\) 1.50000 − 2.59808i 0.288675 − 0.500000i
\(4\) −2.00000 + 3.46410i −0.250000 + 0.433013i
\(5\) −0.436492 − 0.756026i −0.0390410 − 0.0676210i 0.845845 − 0.533429i \(-0.179097\pi\)
−0.884886 + 0.465808i \(0.845764\pi\)
\(6\) −6.00000 −0.408248
\(7\) −10.8095 + 15.0385i −0.583657 + 0.812000i
\(8\) 8.00000 0.353553
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −0.872983 + 1.51205i −0.0276062 + 0.0478153i
\(11\) 0.682458 − 1.18205i 0.0187063 − 0.0324002i −0.856521 − 0.516113i \(-0.827379\pi\)
0.875227 + 0.483712i \(0.160712\pi\)
\(12\) 6.00000 + 10.3923i 0.144338 + 0.250000i
\(13\) 13.0000 0.277350
\(14\) 36.8569 + 3.68410i 0.703601 + 0.0703298i
\(15\) −2.61895 −0.0450807
\(16\) −8.00000 − 13.8564i −0.125000 − 0.216506i
\(17\) −12.5716 + 21.7746i −0.179356 + 0.310654i −0.941660 − 0.336565i \(-0.890735\pi\)
0.762304 + 0.647219i \(0.224068\pi\)
\(18\) −9.00000 + 15.5885i −0.117851 + 0.204124i
\(19\) −6.26210 − 10.8463i −0.0756118 − 0.130963i 0.825740 − 0.564050i \(-0.190758\pi\)
−0.901352 + 0.433087i \(0.857424\pi\)
\(20\) 3.49193 0.0390410
\(21\) 22.8569 + 50.6415i 0.237513 + 0.526233i
\(22\) −2.72983 −0.0264547
\(23\) 39.5474 + 68.4981i 0.358530 + 0.620993i 0.987716 − 0.156263i \(-0.0499446\pi\)
−0.629185 + 0.777255i \(0.716611\pi\)
\(24\) 12.0000 − 20.7846i 0.102062 − 0.176777i
\(25\) 62.1190 − 107.593i 0.496952 − 0.860745i
\(26\) −13.0000 − 22.5167i −0.0980581 − 0.169842i
\(27\) −27.0000 −0.192450
\(28\) −30.4758 − 67.5220i −0.205692 − 0.455731i
\(29\) 3.98387 0.0255098 0.0127549 − 0.999919i \(-0.495940\pi\)
0.0127549 + 0.999919i \(0.495940\pi\)
\(30\) 2.61895 + 4.53615i 0.0159384 + 0.0276062i
\(31\) 97.5706 − 168.997i 0.565296 − 0.979122i −0.431726 − 0.902005i \(-0.642095\pi\)
0.997022 − 0.0771171i \(-0.0245715\pi\)
\(32\) −16.0000 + 27.7128i −0.0883883 + 0.153093i
\(33\) −2.04738 − 3.54616i −0.0108001 − 0.0187063i
\(34\) 50.2863 0.253648
\(35\) 16.0877 + 1.60808i 0.0776948 + 0.00776614i
\(36\) 36.0000 0.166667
\(37\) −90.7853 − 157.245i −0.403379 − 0.698672i 0.590753 − 0.806853i \(-0.298831\pi\)
−0.994131 + 0.108180i \(0.965498\pi\)
\(38\) −12.5242 + 21.6926i −0.0534656 + 0.0926052i
\(39\) 19.5000 − 33.7750i 0.0800641 − 0.138675i
\(40\) −3.49193 − 6.04821i −0.0138031 − 0.0239076i
\(41\) 408.554 1.55623 0.778116 − 0.628121i \(-0.216176\pi\)
0.778116 + 0.628121i \(0.216176\pi\)
\(42\) 64.8569 − 90.2308i 0.238277 − 0.331498i
\(43\) 295.044 1.04637 0.523184 − 0.852220i \(-0.324744\pi\)
0.523184 + 0.852220i \(0.324744\pi\)
\(44\) 2.72983 + 4.72821i 0.00935313 + 0.0162001i
\(45\) −3.92843 + 6.80423i −0.0130137 + 0.0225403i
\(46\) 79.0948 − 136.996i 0.253519 − 0.439108i
\(47\) −178.389 − 308.979i −0.553632 − 0.958920i −0.998009 − 0.0630792i \(-0.979908\pi\)
0.444376 − 0.895840i \(-0.353425\pi\)
\(48\) −48.0000 −0.144338
\(49\) −109.310 − 325.116i −0.318690 − 0.947859i
\(50\) −248.476 −0.702796
\(51\) 37.7147 + 65.3238i 0.103551 + 0.179356i
\(52\) −26.0000 + 45.0333i −0.0693375 + 0.120096i
\(53\) 181.292 − 314.008i 0.469857 − 0.813816i −0.529549 − 0.848279i \(-0.677639\pi\)
0.999406 + 0.0344633i \(0.0109722\pi\)
\(54\) 27.0000 + 46.7654i 0.0680414 + 0.117851i
\(55\) −1.19155 −0.00292125
\(56\) −86.4758 + 120.308i −0.206354 + 0.287086i
\(57\) −37.5726 −0.0873090
\(58\) −3.98387 − 6.90026i −0.00901909 − 0.0156215i
\(59\) −155.809 + 269.870i −0.343808 + 0.595493i −0.985136 − 0.171774i \(-0.945050\pi\)
0.641329 + 0.767266i \(0.278384\pi\)
\(60\) 5.23790 − 9.07231i 0.0112702 − 0.0195205i
\(61\) 121.380 + 210.236i 0.254772 + 0.441279i 0.964834 − 0.262861i \(-0.0846661\pi\)
−0.710061 + 0.704140i \(0.751333\pi\)
\(62\) −390.282 −0.799450
\(63\) 165.856 + 16.5785i 0.331680 + 0.0331538i
\(64\) 64.0000 0.125000
\(65\) −5.67439 − 9.82833i −0.0108280 − 0.0187547i
\(66\) −4.09475 + 7.09232i −0.00763680 + 0.0132273i
\(67\) 128.690 − 222.897i 0.234656 − 0.406435i −0.724517 − 0.689257i \(-0.757937\pi\)
0.959173 + 0.282822i \(0.0912704\pi\)
\(68\) −50.2863 − 87.0984i −0.0896781 − 0.155327i
\(69\) 237.284 0.413995
\(70\) −13.3024 − 29.4728i −0.0227135 − 0.0503239i
\(71\) 75.4435 0.126106 0.0630528 − 0.998010i \(-0.479916\pi\)
0.0630528 + 0.998010i \(0.479916\pi\)
\(72\) −36.0000 − 62.3538i −0.0589256 − 0.102062i
\(73\) 127.737 − 221.247i 0.204801 − 0.354726i −0.745268 − 0.666765i \(-0.767679\pi\)
0.950069 + 0.312039i \(0.101012\pi\)
\(74\) −181.571 + 314.489i −0.285232 + 0.494036i
\(75\) −186.357 − 322.780i −0.286915 − 0.496952i
\(76\) 50.0968 0.0756118
\(77\) 10.3992 + 23.0405i 0.0153909 + 0.0341001i
\(78\) −78.0000 −0.113228
\(79\) −27.8548 − 48.2459i −0.0396697 − 0.0687100i 0.845509 − 0.533961i \(-0.179297\pi\)
−0.885179 + 0.465251i \(0.845964\pi\)
\(80\) −6.98387 + 12.0964i −0.00976025 + 0.0169052i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) −408.554 − 707.637i −0.550211 − 0.952993i
\(83\) 33.5201 0.0443290 0.0221645 − 0.999754i \(-0.492944\pi\)
0.0221645 + 0.999754i \(0.492944\pi\)
\(84\) −221.141 − 22.1046i −0.287244 − 0.0287120i
\(85\) 21.9496 0.0280090
\(86\) −295.044 − 511.032i −0.369947 − 0.640767i
\(87\) 5.97580 − 10.3504i 0.00736406 − 0.0127549i
\(88\) 5.45967 − 9.45642i 0.00661366 − 0.0114552i
\(89\) −551.114 − 954.557i −0.656381 − 1.13689i −0.981546 − 0.191228i \(-0.938753\pi\)
0.325164 − 0.945658i \(-0.394580\pi\)
\(90\) 15.7137 0.0184041
\(91\) −140.523 + 195.500i −0.161877 + 0.225208i
\(92\) −316.379 −0.358530
\(93\) −292.712 − 506.991i −0.326374 − 0.565296i
\(94\) −356.778 + 617.958i −0.391477 + 0.678059i
\(95\) −5.46671 + 9.46862i −0.00590392 + 0.0102259i
\(96\) 48.0000 + 83.1384i 0.0510310 + 0.0883883i
\(97\) 1335.57 1.39800 0.699001 − 0.715121i \(-0.253628\pi\)
0.699001 + 0.715121i \(0.253628\pi\)
\(98\) −453.806 + 514.447i −0.467769 + 0.530275i
\(99\) −12.2843 −0.0124708
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 546.4.i.a.235.1 yes 4
7.2 even 3 inner 546.4.i.a.79.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.4.i.a.79.1 ✓ 4 7.2 even 3 inner
546.4.i.a.235.1 yes 4 1.1 even 1 trivial