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Properties

Label 1617.4.a.n.1.3
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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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] = [1617,4,Mod(1,1617)] 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("1617.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1617, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-1,15,21,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.28053\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.28053 q^{2} +3.00000 q^{3} -6.36023 q^{4} -16.8824 q^{5} +3.84160 q^{6} -18.3888 q^{8} +9.00000 q^{9} -21.6185 q^{10} +11.0000 q^{11} -19.0807 q^{12} +68.0397 q^{13} -50.6472 q^{15} +27.3344 q^{16} -119.045 q^{17} +11.5248 q^{18} -16.8404 q^{19} +107.376 q^{20} +14.0859 q^{22} +199.722 q^{23} -55.1663 q^{24} +160.016 q^{25} +87.1272 q^{26} +27.0000 q^{27} +181.053 q^{29} -64.8555 q^{30} +31.5474 q^{31} +182.113 q^{32} +33.0000 q^{33} -152.441 q^{34} -57.2421 q^{36} +75.9047 q^{37} -21.5647 q^{38} +204.119 q^{39} +310.447 q^{40} -408.485 q^{41} -97.8817 q^{43} -69.9626 q^{44} -151.942 q^{45} +255.750 q^{46} -41.8437 q^{47} +82.0033 q^{48} +204.906 q^{50} -357.136 q^{51} -432.748 q^{52} -563.375 q^{53} +34.5744 q^{54} -185.707 q^{55} -50.5212 q^{57} +231.845 q^{58} +224.425 q^{59} +322.128 q^{60} +622.237 q^{61} +40.3975 q^{62} +14.5263 q^{64} -1148.67 q^{65} +42.2576 q^{66} -280.572 q^{67} +757.155 q^{68} +599.165 q^{69} -807.229 q^{71} -165.499 q^{72} -1038.49 q^{73} +97.1985 q^{74} +480.047 q^{75} +107.109 q^{76} +261.382 q^{78} +710.954 q^{79} -461.471 q^{80} +81.0000 q^{81} -523.079 q^{82} +191.854 q^{83} +2009.77 q^{85} -125.341 q^{86} +543.159 q^{87} -202.276 q^{88} -1562.10 q^{89} -194.566 q^{90} -1270.28 q^{92} +94.6421 q^{93} -53.5822 q^{94} +284.307 q^{95} +546.338 q^{96} -816.513 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 21 q^{5} - 3 q^{6} - 42 q^{8} + 45 q^{9} + 23 q^{10} + 55 q^{11} + 63 q^{12} - 101 q^{13} - 63 q^{15} - 7 q^{16} + 20 q^{17} - 9 q^{18} - 237 q^{19} - 85 q^{20} - 11 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.28053 0.452737 0.226369 − 0.974042i \(-0.427315\pi\)
0.226369 + 0.974042i \(0.427315\pi\)
\(3\) 3.00000 0.577350
\(4\) −6.36023 −0.795029
\(5\) −16.8824 −1.51001 −0.755004 − 0.655720i \(-0.772365\pi\)
−0.755004 + 0.655720i \(0.772365\pi\)
\(6\) 3.84160 0.261388
\(7\) 0 0
\(8\) −18.3888 −0.812676
\(9\) 9.00000 0.333333
\(10\) −21.6185 −0.683637
\(11\) 11.0000 0.301511
\(12\) −19.0807 −0.459010
\(13\) 68.0397 1.45160 0.725801 − 0.687905i \(-0.241469\pi\)
0.725801 + 0.687905i \(0.241469\pi\)
\(14\) 0 0
\(15\) −50.6472 −0.871804
\(16\) 27.3344 0.427100
\(17\) −119.045 −1.69839 −0.849197 − 0.528076i \(-0.822913\pi\)
−0.849197 + 0.528076i \(0.822913\pi\)
\(18\) 11.5248 0.150912
\(19\) −16.8404 −0.203340 −0.101670 − 0.994818i \(-0.532419\pi\)
−0.101670 + 0.994818i \(0.532419\pi\)
\(20\) 107.376 1.20050
\(21\) 0 0
\(22\) 14.0859 0.136505
\(23\) 199.722 1.81065 0.905323 − 0.424724i \(-0.139629\pi\)
0.905323 + 0.424724i \(0.139629\pi\)
\(24\) −55.1663 −0.469199
\(25\) 160.016 1.28013
\(26\) 87.1272 0.657194
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 181.053 1.15933 0.579667 − 0.814853i \(-0.303183\pi\)
0.579667 + 0.814853i \(0.303183\pi\)
\(30\) −64.8555 −0.394698
\(31\) 31.5474 0.182776 0.0913882 − 0.995815i \(-0.470870\pi\)
0.0913882 + 0.995815i \(0.470870\pi\)
\(32\) 182.113 1.00604
\(33\) 33.0000 0.174078
\(34\) −152.441 −0.768926
\(35\) 0 0
\(36\) −57.2421 −0.265010
\(37\) 75.9047 0.337261 0.168630 − 0.985679i \(-0.446066\pi\)
0.168630 + 0.985679i \(0.446066\pi\)
\(38\) −21.5647 −0.0920594
\(39\) 204.119 0.838083
\(40\) 310.447 1.22715
\(41\) −408.485 −1.55597 −0.777983 − 0.628285i \(-0.783757\pi\)
−0.777983 + 0.628285i \(0.783757\pi\)
\(42\) 0 0
\(43\) −97.8817 −0.347135 −0.173568 − 0.984822i \(-0.555530\pi\)
−0.173568 + 0.984822i \(0.555530\pi\)
\(44\) −69.9626 −0.239710
\(45\) −151.942 −0.503336
\(46\) 255.750 0.819747
\(47\) −41.8437 −0.129862 −0.0649311 − 0.997890i \(-0.520683\pi\)
−0.0649311 + 0.997890i \(0.520683\pi\)
\(48\) 82.0033 0.246586
\(49\) 0 0
\(50\) 204.906 0.579561
\(51\) −357.136 −0.980568
\(52\) −432.748 −1.15407
\(53\) −563.375 −1.46010 −0.730052 − 0.683392i \(-0.760504\pi\)
−0.730052 + 0.683392i \(0.760504\pi\)
\(54\) 34.5744 0.0871293
\(55\) −185.707 −0.455285
\(56\) 0 0
\(57\) −50.5212 −0.117398
\(58\) 231.845 0.524874
\(59\) 224.425 0.495213 0.247607 − 0.968861i \(-0.420356\pi\)
0.247607 + 0.968861i \(0.420356\pi\)
\(60\) 322.128 0.693109
\(61\) 622.237 1.30605 0.653027 − 0.757335i \(-0.273499\pi\)
0.653027 + 0.757335i \(0.273499\pi\)
\(62\) 40.3975 0.0827497
\(63\) 0 0
\(64\) 14.5263 0.0283716
\(65\) −1148.67 −2.19193
\(66\) 42.2576 0.0788114
\(67\) −280.572 −0.511603 −0.255801 − 0.966729i \(-0.582339\pi\)
−0.255801 + 0.966729i \(0.582339\pi\)
\(68\) 757.155 1.35027
\(69\) 599.165 1.04538
\(70\) 0 0
\(71\) −807.229 −1.34930 −0.674651 − 0.738137i \(-0.735706\pi\)
−0.674651 + 0.738137i \(0.735706\pi\)
\(72\) −165.499 −0.270892
\(73\) −1038.49 −1.66501 −0.832505 − 0.554017i \(-0.813094\pi\)
−0.832505 + 0.554017i \(0.813094\pi\)
\(74\) 97.1985 0.152691
\(75\) 480.047 0.739081
\(76\) 107.109 0.161661
\(77\) 0 0
\(78\) 261.382 0.379431
\(79\) 710.954 1.01251 0.506257 − 0.862383i \(-0.331029\pi\)
0.506257 + 0.862383i \(0.331029\pi\)
\(80\) −461.471 −0.644925
\(81\) 81.0000 0.111111
\(82\) −523.079 −0.704444
\(83\) 191.854 0.253719 0.126860 − 0.991921i \(-0.459510\pi\)
0.126860 + 0.991921i \(0.459510\pi\)
\(84\) 0 0
\(85\) 2009.77 2.56459
\(86\) −125.341 −0.157161
\(87\) 543.159 0.669342
\(88\) −202.276 −0.245031
\(89\) −1562.10 −1.86047 −0.930236 − 0.366962i \(-0.880398\pi\)
−0.930236 + 0.366962i \(0.880398\pi\)
\(90\) −194.566 −0.227879
\(91\) 0 0
\(92\) −1270.28 −1.43952
\(93\) 94.6421 0.105526
\(94\) −53.5822 −0.0587935
\(95\) 284.307 0.307045
\(96\) 546.338 0.580838
\(97\) −816.513 −0.854684 −0.427342 − 0.904090i \(-0.640550\pi\)
−0.427342 + 0.904090i \(0.640550\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
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 1617.4.a.n.1.3 5
7.6 odd 2 231.4.a.k.1.3 ✓ 5
21.20 even 2 693.4.a.p.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.4.a.k.1.3 ✓ 5 7.6 odd 2
693.4.a.p.1.3 5 21.20 even 2
1617.4.a.n.1.3 5 1.1 even 1 trivial