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Properties

Label 405.4.e.q.271.1
Level $405$
Weight $4$
Character 405.271
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 271.1
Root \(0.327167 + 0.566669i\) of defining polynomial
Character \(\chi\) \(=\) 405.271
Dual form 405.4.e.q.136.1

$q$-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 \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

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

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 −0.709666 − 0.704538i \(-0.751154\pi\)
−0.255315 − 0.966858i \(-0.582179\pi\)
\(3\) 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 0.980227 − 0.197878i \(-0.0634050\pi\)
−0.661481 + 0.749962i \(0.730072\pi\)
\(8\) 75.3201 3.32871
\(9\) 0 0
\(10\) 27.2938 0.863105
\(11\) 28.1188 + 48.7032i 0.770740 + 1.33496i 0.937158 + 0.348905i \(0.113446\pi\)
−0.166418 + 0.986055i \(0.553220\pi\)
\(12\) 0 0
\(13\) −17.2980 + 29.9611i −0.369047 + 0.639208i −0.989417 − 0.145101i \(-0.953649\pi\)
0.620370 + 0.784309i \(0.286983\pi\)
\(14\) 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 −0.807627 − 0.589693i \(-0.799249\pi\)
0.914503 + 0.404579i \(0.132582\pi\)
\(24\) 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 −0.999839 − 0.0179601i \(-0.994283\pi\)
0.484365 − 0.874866i \(-0.339051\pi\)
\(30\) 0 0
\(31\) 14.7733 − 25.5880i 0.0855921 − 0.148250i −0.820051 − 0.572290i \(-0.806055\pi\)
0.905643 + 0.424040i \(0.139388\pi\)
\(32\) −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 −0.969118 − 0.246597i \(-0.920687\pi\)
0.698119 + 0.715982i \(0.254021\pi\)
\(42\) 0 0
\(43\) 234.015 + 405.326i 0.829929 + 1.43748i 0.898093 + 0.439805i \(0.144952\pi\)
−0.0681645 + 0.997674i \(0.521714\pi\)
\(44\) −1225.87 −4.20015
\(45\) 0 0
\(46\) −128.705 −0.412533
\(47\) −197.159 − 341.490i −0.611886 − 1.05982i −0.990922 − 0.134435i \(-0.957078\pi\)
0.379037 − 0.925382i \(-0.376255\pi\)
\(48\) 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 −0.784531 − 0.620089i \(-0.787096\pi\)
0.929279 + 0.369379i \(0.120430\pi\)
\(60\) 0 0
\(61\) −129.901 − 224.994i −0.272657 − 0.472255i 0.696885 − 0.717183i \(-0.254569\pi\)
−0.969541 + 0.244928i \(0.921236\pi\)
\(62\) −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 −0.994430 − 0.105401i \(-0.966387\pi\)
0.588495 + 0.808501i \(0.299721\pi\)
\(68\) −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 0.659948 − 0.751311i \(-0.270578\pi\)
−0.980629 + 0.195877i \(0.937245\pi\)
\(80\) 1183.85 1.65448
\(81\) 0 0
\(82\) 776.726 1.04604
\(83\) −142.148 − 246.207i −0.187985 − 0.325599i 0.756594 − 0.653886i \(-0.226862\pi\)
−0.944578 + 0.328286i \(0.893529\pi\)
\(84\) 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 0.912158 − 0.409839i \(-0.134415\pi\)
−0.811010 + 0.585032i \(0.801082\pi\)
\(98\) −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.271.1 6
3.2 odd 2 405.4.e.v.271.3 6
9.2 odd 6 405.4.e.v.136.3 6
9.4 even 3 135.4.a.h.1.3 yes 3
9.5 odd 6 135.4.a.e.1.1 ✓ 3
9.7 even 3 inner 405.4.e.q.136.1 6
36.23 even 6 2160.4.a.bi.1.2 3
36.31 odd 6 2160.4.a.bq.1.2 3
45.4 even 6 675.4.a.p.1.1 3
45.13 odd 12 675.4.b.n.649.1 6
45.14 odd 6 675.4.a.s.1.3 3
45.22 odd 12 675.4.b.n.649.6 6
45.23 even 12 675.4.b.m.649.6 6
45.32 even 12 675.4.b.m.649.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.e.1.1 ✓ 3 9.5 odd 6
135.4.a.h.1.3 yes 3 9.4 even 3
405.4.e.q.136.1 6 9.7 even 3 inner
405.4.e.q.271.1 6 1.1 even 1 trivial
405.4.e.v.136.3 6 9.2 odd 6
405.4.e.v.271.3 6 3.2 odd 2
675.4.a.p.1.1 3 45.4 even 6
675.4.a.s.1.3 3 45.14 odd 6
675.4.b.m.649.1 6 45.32 even 12
675.4.b.m.649.6 6 45.23 even 12
675.4.b.n.649.1 6 45.13 odd 12
675.4.b.n.649.6 6 45.22 odd 12
2160.4.a.bi.1.2 3 36.23 even 6
2160.4.a.bq.1.2 3 36.31 odd 6