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Properties

Label 405.4.e.q.136.1
Level $405$
Weight $4$
Character 405.136
Analytic conductor $23.896$
Analytic rank $0$
Dimension $6$
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] = [405,4,Mod(136,405)] 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("405.136"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-5,0,-17,-15,0,4] 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: no
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.84779568.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 136.1
Root \(0.327167 - 0.566669i\) of defining polynomial
Character \(\chi\) \(=\) 405.136
Dual form 405.4.e.q.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.72938 + 4.72742i) q^{2} +(-10.8990 - 18.8776i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(5.90326 - 10.2247i) q^{7} +75.3201 q^{8} +27.2938 q^{10} +(28.1188 - 48.7032i) q^{11} +(-17.2980 - 29.9611i) q^{13} +(32.2245 + 55.8144i) q^{14} +(-118.385 + 205.049i) q^{16} +39.2675 q^{17} -146.561 q^{19} +(-54.4951 + 94.3882i) q^{20} +(153.494 + 265.859i) q^{22} +(11.7889 + 20.4189i) q^{23} +(-12.5000 + 21.6506i) q^{25} +188.851 q^{26} -257.359 q^{28} +(-80.5013 + 139.432i) q^{29} +(14.7733 + 25.5880i) q^{31} +(-344.954 - 597.478i) q^{32} +(-107.176 + 185.634i) q^{34} -59.0326 q^{35} -217.688 q^{37} +(400.020 - 692.855i) q^{38} +(-188.300 - 326.146i) q^{40} +(-71.1449 - 123.227i) q^{41} +(234.015 - 405.326i) q^{43} -1225.87 q^{44} -128.705 q^{46} +(-197.159 + 341.490i) q^{47} +(101.803 + 176.328i) q^{49} +(-68.2345 - 118.186i) q^{50} +(-377.063 + 653.092i) q^{52} -134.780 q^{53} -281.188 q^{55} +(444.634 - 770.129i) q^{56} +(-439.437 - 761.128i) q^{58} +(65.5977 + 113.619i) q^{59} +(-129.901 + 224.994i) q^{61} -161.287 q^{62} +1871.88 q^{64} +(-86.4901 + 149.805i) q^{65} +(-222.622 - 385.593i) q^{67} +(-427.977 - 741.278i) q^{68} +(161.122 - 279.072i) q^{70} +560.841 q^{71} -88.6681 q^{73} +(594.152 - 1029.10i) q^{74} +(1597.37 + 2766.72i) q^{76} +(-331.985 - 575.015i) q^{77} +(-225.171 + 390.008i) q^{79} +1183.85 q^{80} +776.726 q^{82} +(-142.148 + 246.207i) q^{83} +(-98.1687 - 170.033i) q^{85} +(1277.43 + 2212.57i) q^{86} +(2117.91 - 3668.33i) q^{88} +625.305 q^{89} -408.459 q^{91} +(256.974 - 445.092i) q^{92} +(-1076.24 - 1864.11i) q^{94} +(366.402 + 634.627i) q^{95} +(96.6307 - 167.369i) q^{97} -1111.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} - 17 q^{4} - 15 q^{5} + 4 q^{7} + 150 q^{8} + 50 q^{10} - 5 q^{11} - 7 q^{13} - 60 q^{14} - 161 q^{16} + 310 q^{17} - 100 q^{19} - 85 q^{20} + 229 q^{22} - 285 q^{23} - 75 q^{25} + 370 q^{26}+ \cdots - 610 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.72938 + 4.72742i −0.964981 + 1.67140i −0.255315 + 0.966858i \(0.582179\pi\)
−0.709666 + 0.704538i \(0.751154\pi\)
\(3\) 0 0
\(4\) −10.8990 − 18.8776i −1.36238 − 2.35971i
\(5\) −2.50000 − 4.33013i −0.223607 − 0.387298i
\(6\) 0 0
\(7\) 5.90326 − 10.2247i 0.318746 − 0.552084i −0.661481 − 0.749962i \(-0.730072\pi\)
0.980227 + 0.197878i \(0.0634050\pi\)
\(8\) 75.3201 3.32871
\(9\) 0 0
\(10\) 27.2938 0.863105
\(11\) 28.1188 − 48.7032i 0.770740 − 1.33496i −0.166418 − 0.986055i \(-0.553220\pi\)
0.937158 − 0.348905i \(-0.113446\pi\)
\(12\) 0 0
\(13\) −17.2980 − 29.9611i −0.369047 − 0.639208i 0.620370 − 0.784309i \(-0.286983\pi\)
−0.989417 + 0.145101i \(0.953649\pi\)
\(14\) 32.2245 + 55.8144i 0.615168 + 1.06550i
\(15\) 0 0
\(16\) −118.385 + 205.049i −1.84976 + 3.20389i
\(17\) 39.2675 0.560221 0.280111 − 0.959968i \(-0.409629\pi\)
0.280111 + 0.959968i \(0.409629\pi\)
\(18\) 0 0
\(19\) −146.561 −1.76965 −0.884825 − 0.465924i \(-0.845722\pi\)
−0.884825 + 0.465924i \(0.845722\pi\)
\(20\) −54.4951 + 94.3882i −0.609273 + 1.05529i
\(21\) 0 0
\(22\) 153.494 + 265.859i 1.48750 + 2.57642i
\(23\) 11.7889 + 20.4189i 0.106876 + 0.185115i 0.914503 − 0.404579i \(-0.132582\pi\)
−0.807627 + 0.589693i \(0.799249\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 188.851 1.42449
\(27\) 0 0
\(28\) −257.359 −1.73701
\(29\) −80.5013 + 139.432i −0.515473 + 0.892826i 0.484365 + 0.874866i \(0.339051\pi\)
−0.999839 + 0.0179601i \(0.994283\pi\)
\(30\) 0 0
\(31\) 14.7733 + 25.5880i 0.0855921 + 0.148250i 0.905643 − 0.424040i \(-0.139388\pi\)
−0.820051 + 0.572290i \(0.806055\pi\)
\(32\) −344.954 − 597.478i −1.90562 − 3.30063i
\(33\) 0 0
\(34\) −107.176 + 185.634i −0.540603 + 0.936352i
\(35\) −59.0326 −0.285095
\(36\) 0 0
\(37\) −217.688 −0.967233 −0.483617 − 0.875280i \(-0.660677\pi\)
−0.483617 + 0.875280i \(0.660677\pi\)
\(38\) 400.020 − 692.855i 1.70768 − 2.95779i
\(39\) 0 0
\(40\) −188.300 − 326.146i −0.744322 − 1.28920i
\(41\) −71.1449 − 123.227i −0.270999 − 0.469385i 0.698119 − 0.715982i \(-0.254021\pi\)
−0.969118 + 0.246597i \(0.920687\pi\)
\(42\) 0 0
\(43\) 234.015 − 405.326i 0.829929 − 1.43748i −0.0681645 − 0.997674i \(-0.521714\pi\)
0.898093 − 0.439805i \(-0.144952\pi\)
\(44\) −1225.87 −4.20015
\(45\) 0 0
\(46\) −128.705 −0.412533
\(47\) −197.159 + 341.490i −0.611886 + 1.05982i 0.379037 + 0.925382i \(0.376255\pi\)
−0.990922 + 0.134435i \(0.957078\pi\)
\(48\) 0 0
\(49\) 101.803 + 176.328i 0.296802 + 0.514076i
\(50\) −68.2345 − 118.186i −0.192996 − 0.334279i
\(51\) 0 0
\(52\) −377.063 + 653.092i −1.00556 + 1.74168i
\(53\) −134.780 −0.349311 −0.174655 − 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(54\) 0 0
\(55\) −281.188 −0.689371
\(56\) 444.634 − 770.129i 1.06101 − 1.83773i
\(57\) 0 0
\(58\) −439.437 − 761.128i −0.994844 − 1.72312i
\(59\) 65.5977 + 113.619i 0.144747 + 0.250710i 0.929279 − 0.369379i \(-0.120430\pi\)
−0.784531 + 0.620089i \(0.787096\pi\)
\(60\) 0 0
\(61\) −129.901 + 224.994i −0.272657 + 0.472255i −0.969541 − 0.244928i \(-0.921236\pi\)
0.696885 + 0.717183i \(0.254569\pi\)
\(62\) −161.287 −0.330379
\(63\) 0 0
\(64\) 1871.88 3.65602
\(65\) −86.4901 + 149.805i −0.165043 + 0.285863i
\(66\) 0 0
\(67\) −222.622 − 385.593i −0.405935 − 0.703100i 0.588495 − 0.808501i \(-0.299721\pi\)
−0.994430 + 0.105401i \(0.966387\pi\)
\(68\) −427.977 − 741.278i −0.763233 − 1.32196i
\(69\) 0 0
\(70\) 161.122 − 279.072i 0.275111 − 0.476507i
\(71\) 560.841 0.937459 0.468729 − 0.883342i \(-0.344712\pi\)
0.468729 + 0.883342i \(0.344712\pi\)
\(72\) 0 0
\(73\) −88.6681 −0.142162 −0.0710809 − 0.997471i \(-0.522645\pi\)
−0.0710809 + 0.997471i \(0.522645\pi\)
\(74\) 594.152 − 1029.10i 0.933362 − 1.61663i
\(75\) 0 0
\(76\) 1597.37 + 2766.72i 2.41093 + 4.17585i
\(77\) −331.985 − 575.015i −0.491340 − 0.851027i
\(78\) 0 0
\(79\) −225.171 + 390.008i −0.320680 + 0.555434i −0.980629 − 0.195877i \(-0.937245\pi\)
0.659948 + 0.751311i \(0.270578\pi\)
\(80\) 1183.85 1.65448
\(81\) 0 0
\(82\) 776.726 1.04604
\(83\) −142.148 + 246.207i −0.187985 + 0.325599i −0.944578 − 0.328286i \(-0.893529\pi\)
0.756594 + 0.653886i \(0.226862\pi\)
\(84\) 0 0
\(85\) −98.1687 − 170.033i −0.125269 − 0.216973i
\(86\) 1277.43 + 2212.57i 1.60173 + 2.77428i
\(87\) 0 0
\(88\) 2117.91 − 3668.33i 2.56557 − 4.44369i
\(89\) 625.305 0.744744 0.372372 − 0.928083i \(-0.378545\pi\)
0.372372 + 0.928083i \(0.378545\pi\)
\(90\) 0 0
\(91\) −408.459 −0.470529
\(92\) 256.974 − 445.092i 0.291211 − 0.504392i
\(93\) 0 0
\(94\) −1076.24 − 1864.11i −1.18092 − 2.04541i
\(95\) 366.402 + 634.627i 0.395706 + 0.685382i
\(96\) 0 0
\(97\) 96.6307 − 167.369i 0.101148 − 0.175193i −0.811010 − 0.585032i \(-0.801082\pi\)
0.912158 + 0.409839i \(0.134415\pi\)
\(98\) −1111.44 −1.14563
\(99\) 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 405.4.e.q.136.1 6
3.2 odd 2 405.4.e.v.136.3 6
9.2 odd 6 135.4.a.e.1.1 ✓ 3
9.4 even 3 inner 405.4.e.q.271.1 6
9.5 odd 6 405.4.e.v.271.3 6
9.7 even 3 135.4.a.h.1.3 yes 3
36.7 odd 6 2160.4.a.bq.1.2 3
36.11 even 6 2160.4.a.bi.1.2 3
45.2 even 12 675.4.b.m.649.1 6
45.7 odd 12 675.4.b.n.649.6 6
45.29 odd 6 675.4.a.s.1.3 3
45.34 even 6 675.4.a.p.1.1 3
45.38 even 12 675.4.b.m.649.6 6
45.43 odd 12 675.4.b.n.649.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.e.1.1 ✓ 3 9.2 odd 6
135.4.a.h.1.3 yes 3 9.7 even 3
405.4.e.q.136.1 6 1.1 even 1 trivial
405.4.e.q.271.1 6 9.4 even 3 inner
405.4.e.v.136.3 6 3.2 odd 2
405.4.e.v.271.3 6 9.5 odd 6
675.4.a.p.1.1 3 45.34 even 6
675.4.a.s.1.3 3 45.29 odd 6
675.4.b.m.649.1 6 45.2 even 12
675.4.b.m.649.6 6 45.38 even 12
675.4.b.n.649.1 6 45.43 odd 12
675.4.b.n.649.6 6 45.7 odd 12
2160.4.a.bi.1.2 3 36.11 even 6
2160.4.a.bq.1.2 3 36.7 odd 6