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Properties

Label 36.4.b.b.35.1
Level $36$
Weight $4$
Character 36.35
Analytic conductor $2.124$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [36,4,Mod(35,36)] 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("36.35"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 36.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.12406876021\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 13x^{2} - 12x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 35.1
Root \(0.500000 - 0.522278i\) of defining polynomial
Character \(\chi\) \(=\) 36.35
Dual form 36.4.b.b.35.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.73861 - 0.707107i) q^{2} +(7.00000 + 3.87298i) q^{4} -9.89949i q^{5} -30.9839i q^{7} +(-16.4317 - 15.5563i) q^{8} +(-7.00000 + 27.1109i) q^{10} +43.8178 q^{11} -28.0000 q^{13} +(-21.9089 + 84.8528i) q^{14} +(34.0000 + 54.2218i) q^{16} +49.4975i q^{17} +(38.3406 - 69.2965i) q^{20} +(-120.000 - 30.9839i) q^{22} -43.8178 q^{23} +27.0000 q^{25} +(76.6812 + 19.7990i) q^{26} +(120.000 - 216.887i) q^{28} -137.179i q^{29} +216.887i q^{31} +(-54.7723 - 172.534i) q^{32} +(35.0000 - 135.554i) q^{34} -306.725 q^{35} +266.000 q^{37} +(-154.000 + 162.665i) q^{40} +227.688i q^{41} +61.9677i q^{43} +(306.725 + 169.706i) q^{44} +(120.000 + 30.9839i) q^{46} +306.725 q^{47} -617.000 q^{49} +(-73.9425 - 19.0919i) q^{50} +(-196.000 - 108.444i) q^{52} -120.208i q^{53} -433.774i q^{55} +(-481.996 + 509.117i) q^{56} +(-97.0000 + 375.679i) q^{58} +613.449 q^{59} +350.000 q^{61} +(153.362 - 593.970i) q^{62} +(28.0000 + 511.234i) q^{64} +277.186i q^{65} -433.774i q^{67} +(-191.703 + 346.482i) q^{68} +(840.000 + 216.887i) q^{70} -657.267 q^{71} -112.000 q^{73} +(-728.471 - 188.090i) q^{74} -1357.65i q^{77} -216.887i q^{79} +(536.768 - 336.583i) q^{80} +(161.000 - 623.550i) q^{82} +306.725 q^{83} +490.000 q^{85} +(43.8178 - 169.706i) q^{86} +(-720.000 - 681.645i) q^{88} +1296.83i q^{89} +867.548i q^{91} +(-306.725 - 169.706i) q^{92} +(-840.000 - 216.887i) q^{94} -616.000 q^{97} +(1689.72 + 436.285i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{4} - 28 q^{10} - 112 q^{13} + 136 q^{16} - 480 q^{22} + 108 q^{25} + 480 q^{28} + 140 q^{34} + 1064 q^{37} - 616 q^{40} + 480 q^{46} - 2468 q^{49} - 784 q^{52} - 388 q^{58} + 1400 q^{61} + 112 q^{64}+ \cdots - 2464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73861 − 0.707107i −0.968246 − 0.250000i
\(3\) 0 0
\(4\) 7.00000 + 3.87298i 0.875000 + 0.484123i
\(5\) − 9.89949i − 0.885438i −0.896660 − 0.442719i \(-0.854014\pi\)
0.896660 − 0.442719i \(-0.145986\pi\)
\(6\) 0 0
\(7\) − 30.9839i − 1.67297i −0.547989 − 0.836486i \(-0.684606\pi\)
0.547989 − 0.836486i \(-0.315394\pi\)
\(8\) −16.4317 − 15.5563i −0.726184 − 0.687500i
\(9\) 0 0
\(10\) −7.00000 + 27.1109i −0.221359 + 0.857321i
\(11\) 43.8178 1.20105 0.600526 − 0.799605i \(-0.294958\pi\)
0.600526 + 0.799605i \(0.294958\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 − 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) −21.9089 + 84.8528i −0.418243 + 1.61985i
\(15\) 0 0
\(16\) 34.0000 + 54.2218i 0.531250 + 0.847215i
\(17\) 49.4975i 0.706171i 0.935591 + 0.353085i \(0.114867\pi\)
−0.935591 + 0.353085i \(0.885133\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 38.3406 − 69.2965i 0.428661 − 0.774758i
\(21\) 0 0
\(22\) −120.000 − 30.9839i −1.16291 − 0.300263i
\(23\) −43.8178 −0.397245 −0.198623 − 0.980076i \(-0.563647\pi\)
−0.198623 + 0.980076i \(0.563647\pi\)
\(24\) 0 0
\(25\) 27.0000 0.216000
\(26\) 76.6812 + 19.7990i 0.578400 + 0.149342i
\(27\) 0 0
\(28\) 120.000 − 216.887i 0.809924 − 1.46385i
\(29\) − 137.179i − 0.878395i −0.898391 − 0.439197i \(-0.855263\pi\)
0.898391 − 0.439197i \(-0.144737\pi\)
\(30\) 0 0
\(31\) 216.887i 1.25658i 0.777978 + 0.628291i \(0.216245\pi\)
−0.777978 + 0.628291i \(0.783755\pi\)
\(32\) −54.7723 − 172.534i −0.302577 − 0.953125i
\(33\) 0 0
\(34\) 35.0000 − 135.554i 0.176543 − 0.683747i
\(35\) −306.725 −1.48131
\(36\) 0 0
\(37\) 266.000 1.18190 0.590948 − 0.806710i \(-0.298754\pi\)
0.590948 + 0.806710i \(0.298754\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −154.000 + 162.665i −0.608738 + 0.642991i
\(41\) 227.688i 0.867291i 0.901083 + 0.433646i \(0.142773\pi\)
−0.901083 + 0.433646i \(0.857227\pi\)
\(42\) 0 0
\(43\) 61.9677i 0.219767i 0.993944 + 0.109884i \(0.0350478\pi\)
−0.993944 + 0.109884i \(0.964952\pi\)
\(44\) 306.725 + 169.706i 1.05092 + 0.581456i
\(45\) 0 0
\(46\) 120.000 + 30.9839i 0.384631 + 0.0993113i
\(47\) 306.725 0.951923 0.475962 − 0.879466i \(-0.342100\pi\)
0.475962 + 0.879466i \(0.342100\pi\)
\(48\) 0 0
\(49\) −617.000 −1.79883
\(50\) −73.9425 − 19.0919i −0.209141 − 0.0540000i
\(51\) 0 0
\(52\) −196.000 − 108.444i −0.522698 − 0.289200i
\(53\) − 120.208i − 0.311545i −0.987793 − 0.155772i \(-0.950213\pi\)
0.987793 − 0.155772i \(-0.0497866\pi\)
\(54\) 0 0
\(55\) − 433.774i − 1.06346i
\(56\) −481.996 + 509.117i −1.15017 + 1.21489i
\(57\) 0 0
\(58\) −97.0000 + 375.679i −0.219599 + 0.850502i
\(59\) 613.449 1.35363 0.676816 − 0.736152i \(-0.263359\pi\)
0.676816 + 0.736152i \(0.263359\pi\)
\(60\) 0 0
\(61\) 350.000 0.734638 0.367319 − 0.930095i \(-0.380276\pi\)
0.367319 + 0.930095i \(0.380276\pi\)
\(62\) 153.362 − 593.970i 0.314146 − 1.21668i
\(63\) 0 0
\(64\) 28.0000 + 511.234i 0.0546875 + 0.998504i
\(65\) 277.186i 0.528933i
\(66\) 0 0
\(67\) − 433.774i − 0.790954i −0.918476 − 0.395477i \(-0.870579\pi\)
0.918476 − 0.395477i \(-0.129421\pi\)
\(68\) −191.703 + 346.482i −0.341873 + 0.617899i
\(69\) 0 0
\(70\) 840.000 + 216.887i 1.43427 + 0.370328i
\(71\) −657.267 −1.09864 −0.549319 − 0.835613i \(-0.685113\pi\)
−0.549319 + 0.835613i \(0.685113\pi\)
\(72\) 0 0
\(73\) −112.000 −0.179570 −0.0897850 − 0.995961i \(-0.528618\pi\)
−0.0897850 + 0.995961i \(0.528618\pi\)
\(74\) −728.471 − 188.090i −1.14437 − 0.295474i
\(75\) 0 0
\(76\) 0 0
\(77\) − 1357.65i − 2.00932i
\(78\) 0 0
\(79\) − 216.887i − 0.308882i −0.988002 − 0.154441i \(-0.950642\pi\)
0.988002 − 0.154441i \(-0.0493577\pi\)
\(80\) 536.768 − 336.583i 0.750156 − 0.470389i
\(81\) 0 0
\(82\) 161.000 − 623.550i 0.216823 − 0.839751i
\(83\) 306.725 0.405631 0.202816 − 0.979217i \(-0.434991\pi\)
0.202816 + 0.979217i \(0.434991\pi\)
\(84\) 0 0
\(85\) 490.000 0.625270
\(86\) 43.8178 − 169.706i 0.0549418 − 0.212789i
\(87\) 0 0
\(88\) −720.000 − 681.645i −0.872185 − 0.825723i
\(89\) 1296.83i 1.54454i 0.635294 + 0.772270i \(0.280879\pi\)
−0.635294 + 0.772270i \(0.719121\pi\)
\(90\) 0 0
\(91\) 867.548i 0.999382i
\(92\) −306.725 − 169.706i −0.347590 − 0.192316i
\(93\) 0 0
\(94\) −840.000 − 216.887i −0.921696 − 0.237981i
\(95\) 0 0
\(96\) 0 0
\(97\) −616.000 −0.644797 −0.322399 − 0.946604i \(-0.604489\pi\)
−0.322399 + 0.946604i \(0.604489\pi\)
\(98\) 1689.72 + 436.285i 1.74171 + 0.449708i
\(99\) 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 36.4.b.b.35.1 ✓ 4
3.2 odd 2 inner 36.4.b.b.35.4 yes 4
4.3 odd 2 inner 36.4.b.b.35.3 yes 4
8.3 odd 2 576.4.c.e.575.4 4
8.5 even 2 576.4.c.e.575.3 4
12.11 even 2 inner 36.4.b.b.35.2 yes 4
16.3 odd 4 2304.4.f.g.1151.1 8
16.5 even 4 2304.4.f.g.1151.8 8
16.11 odd 4 2304.4.f.g.1151.6 8
16.13 even 4 2304.4.f.g.1151.3 8
24.5 odd 2 576.4.c.e.575.1 4
24.11 even 2 576.4.c.e.575.2 4
48.5 odd 4 2304.4.f.g.1151.4 8
48.11 even 4 2304.4.f.g.1151.2 8
48.29 odd 4 2304.4.f.g.1151.7 8
48.35 even 4 2304.4.f.g.1151.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.4.b.b.35.1 ✓ 4 1.1 even 1 trivial
36.4.b.b.35.2 yes 4 12.11 even 2 inner
36.4.b.b.35.3 yes 4 4.3 odd 2 inner
36.4.b.b.35.4 yes 4 3.2 odd 2 inner
576.4.c.e.575.1 4 24.5 odd 2
576.4.c.e.575.2 4 24.11 even 2
576.4.c.e.575.3 4 8.5 even 2
576.4.c.e.575.4 4 8.3 odd 2
2304.4.f.g.1151.1 8 16.3 odd 4
2304.4.f.g.1151.2 8 48.11 even 4
2304.4.f.g.1151.3 8 16.13 even 4
2304.4.f.g.1151.4 8 48.5 odd 4
2304.4.f.g.1151.5 8 48.35 even 4
2304.4.f.g.1151.6 8 16.11 odd 4
2304.4.f.g.1151.7 8 48.29 odd 4
2304.4.f.g.1151.8 8 16.5 even 4