Properties

Label 345.4.a.i.1.5
Level $345$
Weight $4$
Character 345.1
Self dual yes
Analytic conductor $20.356$
Analytic rank $1$
Dimension $6$
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] = [345,4,Mod(1,345)] 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("345.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(345, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 345.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-4,-18] 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: yes
Analytic conductor: \(20.3556589520\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 34x^{4} + 38x^{3} + 293x^{2} - 12x - 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.30064\) of defining polynomial
Character \(\chi\) \(=\) 345.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.30064 q^{2} -3.00000 q^{3} +2.89421 q^{4} +5.00000 q^{5} -9.90191 q^{6} +4.06396 q^{7} -16.8524 q^{8} +9.00000 q^{9} +16.5032 q^{10} -66.3207 q^{11} -8.68262 q^{12} +36.1993 q^{13} +13.4137 q^{14} -15.0000 q^{15} -78.7772 q^{16} -123.114 q^{17} +29.7057 q^{18} +102.288 q^{19} +14.4710 q^{20} -12.1919 q^{21} -218.901 q^{22} -23.0000 q^{23} +50.5571 q^{24} +25.0000 q^{25} +119.481 q^{26} -27.0000 q^{27} +11.7619 q^{28} -303.542 q^{29} -49.5096 q^{30} -128.869 q^{31} -125.196 q^{32} +198.962 q^{33} -406.354 q^{34} +20.3198 q^{35} +26.0479 q^{36} +155.747 q^{37} +337.615 q^{38} -108.598 q^{39} -84.2619 q^{40} -84.5879 q^{41} -40.2410 q^{42} -360.550 q^{43} -191.946 q^{44} +45.0000 q^{45} -75.9147 q^{46} +141.232 q^{47} +236.332 q^{48} -326.484 q^{49} +82.5159 q^{50} +369.342 q^{51} +104.768 q^{52} -235.209 q^{53} -89.1172 q^{54} -331.604 q^{55} -68.4874 q^{56} -306.863 q^{57} -1001.88 q^{58} -71.5654 q^{59} -43.4131 q^{60} +746.119 q^{61} -425.349 q^{62} +36.5756 q^{63} +216.991 q^{64} +180.997 q^{65} +656.702 q^{66} -288.611 q^{67} -356.317 q^{68} +69.0000 q^{69} +67.0683 q^{70} +107.869 q^{71} -151.671 q^{72} +798.988 q^{73} +514.065 q^{74} -75.0000 q^{75} +296.042 q^{76} -269.525 q^{77} -358.443 q^{78} -652.342 q^{79} -393.886 q^{80} +81.0000 q^{81} -279.194 q^{82} -18.7696 q^{83} -35.2858 q^{84} -615.570 q^{85} -1190.05 q^{86} +910.627 q^{87} +1117.66 q^{88} +1109.13 q^{89} +148.529 q^{90} +147.113 q^{91} -66.5667 q^{92} +386.606 q^{93} +466.156 q^{94} +511.439 q^{95} +375.588 q^{96} +352.382 q^{97} -1077.61 q^{98} -596.887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} - 18 q^{3} + 26 q^{4} + 30 q^{5} + 12 q^{6} - 40 q^{7} - 54 q^{8} + 54 q^{9} - 20 q^{10} - 48 q^{11} - 78 q^{12} - 56 q^{13} - 50 q^{14} - 90 q^{15} + 42 q^{16} - 84 q^{17} - 36 q^{18} + 80 q^{19}+ \cdots - 432 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.30064 1.16695 0.583476 − 0.812131i \(-0.301692\pi\)
0.583476 + 0.812131i \(0.301692\pi\)
\(3\) −3.00000 −0.577350
\(4\) 2.89421 0.361776
\(5\) 5.00000 0.447214
\(6\) −9.90191 −0.673740
\(7\) 4.06396 0.219433 0.109717 − 0.993963i \(-0.465006\pi\)
0.109717 + 0.993963i \(0.465006\pi\)
\(8\) −16.8524 −0.744777
\(9\) 9.00000 0.333333
\(10\) 16.5032 0.521877
\(11\) −66.3207 −1.81786 −0.908930 − 0.416949i \(-0.863099\pi\)
−0.908930 + 0.416949i \(0.863099\pi\)
\(12\) −8.68262 −0.208871
\(13\) 36.1993 0.772299 0.386149 − 0.922436i \(-0.373805\pi\)
0.386149 + 0.922436i \(0.373805\pi\)
\(14\) 13.4137 0.256068
\(15\) −15.0000 −0.258199
\(16\) −78.7772 −1.23089
\(17\) −123.114 −1.75644 −0.878221 − 0.478255i \(-0.841269\pi\)
−0.878221 + 0.478255i \(0.841269\pi\)
\(18\) 29.7057 0.388984
\(19\) 102.288 1.23508 0.617538 − 0.786541i \(-0.288130\pi\)
0.617538 + 0.786541i \(0.288130\pi\)
\(20\) 14.4710 0.161791
\(21\) −12.1919 −0.126690
\(22\) −218.901 −2.12135
\(23\) −23.0000 −0.208514
\(24\) 50.5571 0.429997
\(25\) 25.0000 0.200000
\(26\) 119.481 0.901235
\(27\) −27.0000 −0.192450
\(28\) 11.7619 0.0793856
\(29\) −303.542 −1.94367 −0.971834 − 0.235667i \(-0.924273\pi\)
−0.971834 + 0.235667i \(0.924273\pi\)
\(30\) −49.5096 −0.301306
\(31\) −128.869 −0.746629 −0.373315 − 0.927705i \(-0.621779\pi\)
−0.373315 + 0.927705i \(0.621779\pi\)
\(32\) −125.196 −0.691617
\(33\) 198.962 1.04954
\(34\) −406.354 −2.04968
\(35\) 20.3198 0.0981335
\(36\) 26.0479 0.120592
\(37\) 155.747 0.692019 0.346009 − 0.938231i \(-0.387537\pi\)
0.346009 + 0.938231i \(0.387537\pi\)
\(38\) 337.615 1.44127
\(39\) −108.598 −0.445887
\(40\) −84.2619 −0.333074
\(41\) −84.5879 −0.322205 −0.161103 − 0.986938i \(-0.551505\pi\)
−0.161103 + 0.986938i \(0.551505\pi\)
\(42\) −40.2410 −0.147841
\(43\) −360.550 −1.27868 −0.639342 − 0.768923i \(-0.720793\pi\)
−0.639342 + 0.768923i \(0.720793\pi\)
\(44\) −191.946 −0.657658
\(45\) 45.0000 0.149071
\(46\) −75.9147 −0.243326
\(47\) 141.232 0.438315 0.219158 − 0.975689i \(-0.429669\pi\)
0.219158 + 0.975689i \(0.429669\pi\)
\(48\) 236.332 0.710657
\(49\) −326.484 −0.951849
\(50\) 82.5159 0.233390
\(51\) 369.342 1.01408
\(52\) 104.768 0.279399
\(53\) −235.209 −0.609592 −0.304796 − 0.952418i \(-0.598588\pi\)
−0.304796 + 0.952418i \(0.598588\pi\)
\(54\) −89.1172 −0.224580
\(55\) −331.604 −0.812971
\(56\) −68.4874 −0.163429
\(57\) −306.863 −0.713071
\(58\) −1001.88 −2.26817
\(59\) −71.5654 −0.157916 −0.0789578 − 0.996878i \(-0.525159\pi\)
−0.0789578 + 0.996878i \(0.525159\pi\)
\(60\) −43.4131 −0.0934101
\(61\) 746.119 1.56608 0.783039 − 0.621972i \(-0.213668\pi\)
0.783039 + 0.621972i \(0.213668\pi\)
\(62\) −425.349 −0.871280
\(63\) 36.5756 0.0731444
\(64\) 216.991 0.423811
\(65\) 180.997 0.345383
\(66\) 656.702 1.22476
\(67\) −288.611 −0.526260 −0.263130 − 0.964760i \(-0.584755\pi\)
−0.263130 + 0.964760i \(0.584755\pi\)
\(68\) −356.317 −0.635438
\(69\) 69.0000 0.120386
\(70\) 67.0683 0.114517
\(71\) 107.869 0.180305 0.0901525 − 0.995928i \(-0.471265\pi\)
0.0901525 + 0.995928i \(0.471265\pi\)
\(72\) −151.671 −0.248259
\(73\) 798.988 1.28102 0.640510 − 0.767950i \(-0.278723\pi\)
0.640510 + 0.767950i \(0.278723\pi\)
\(74\) 514.065 0.807552
\(75\) −75.0000 −0.115470
\(76\) 296.042 0.446820
\(77\) −269.525 −0.398899
\(78\) −358.443 −0.520328
\(79\) −652.342 −0.929041 −0.464521 − 0.885562i \(-0.653773\pi\)
−0.464521 + 0.885562i \(0.653773\pi\)
\(80\) −393.886 −0.550473
\(81\) 81.0000 0.111111
\(82\) −279.194 −0.375998
\(83\) −18.7696 −0.0248221 −0.0124110 − 0.999923i \(-0.503951\pi\)
−0.0124110 + 0.999923i \(0.503951\pi\)
\(84\) −35.2858 −0.0458333
\(85\) −615.570 −0.785505
\(86\) −1190.05 −1.49216
\(87\) 910.627 1.12218
\(88\) 1117.66 1.35390
\(89\) 1109.13 1.32098 0.660491 − 0.750834i \(-0.270348\pi\)
0.660491 + 0.750834i \(0.270348\pi\)
\(90\) 148.529 0.173959
\(91\) 147.113 0.169468
\(92\) −66.5667 −0.0754355
\(93\) 386.606 0.431067
\(94\) 466.156 0.511493
\(95\) 511.439 0.552343
\(96\) 375.588 0.399305
\(97\) 352.382 0.368855 0.184428 − 0.982846i \(-0.440957\pi\)
0.184428 + 0.982846i \(0.440957\pi\)
\(98\) −1077.61 −1.11076
\(99\) −596.887 −0.605953
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 345.4.a.i.1.5 ✓ 6
3.2 odd 2 1035.4.a.q.1.2 6
5.4 even 2 1725.4.a.x.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.4.a.i.1.5 ✓ 6 1.1 even 1 trivial
1035.4.a.q.1.2 6 3.2 odd 2
1725.4.a.x.1.2 6 5.4 even 2