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Properties

Label 35.3.i.a.24.4
Level $35$
Weight $3$
Character 35.24
Analytic conductor $0.954$
Analytic rank $0$
Dimension $12$
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] = [35,3,Mod(19,35)] 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("35.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 180x^{8} - 669x^{6} + 1980x^{4} - 135x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 24.4
Root \(0.226181 - 0.130586i\) of defining polynomial
Character \(\chi\) \(=\) 35.24
Dual form 35.3.i.a.19.4

$q$-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+(0.226181 - 0.130586i) q^{2} +(1.88157 - 3.25898i) q^{3} +(-1.96589 + 3.40503i) q^{4} +(2.11218 - 4.53196i) q^{5} -0.982826i q^{6} +(-1.66053 + 6.80019i) q^{7} +2.07156i q^{8} +(-2.58064 - 4.46979i) q^{9} +(-0.114075 - 1.30086i) q^{10} +(-3.04653 + 5.27675i) q^{11} +(7.39795 + 12.8136i) q^{12} -13.0886 q^{13} +(0.512426 + 1.75492i) q^{14} +(-10.7954 - 15.4108i) q^{15} +(-7.59306 - 13.1516i) q^{16} +(-5.05373 + 8.75332i) q^{17} +(-1.16738 - 0.673989i) q^{18} +(19.5373 - 11.2799i) q^{19} +(11.2792 + 16.1014i) q^{20} +(19.0373 + 18.2067i) q^{21} +1.59133i q^{22} +(30.2957 - 17.4912i) q^{23} +(6.75116 + 3.89779i) q^{24} +(-16.0774 - 19.1446i) q^{25} +(-2.96039 + 1.70918i) q^{26} +14.4457 q^{27} +(-19.8904 - 19.0226i) q^{28} -16.3474 q^{29} +(-4.45413 - 2.07590i) q^{30} +(-23.4907 - 13.5624i) q^{31} +(-10.6109 - 6.12620i) q^{32} +(11.4645 + 19.8572i) q^{33} +2.63978i q^{34} +(27.3109 + 21.8887i) q^{35} +20.2930 q^{36} +(-14.4349 + 8.33401i) q^{37} +(2.94598 - 5.10258i) q^{38} +(-24.6272 + 42.6555i) q^{39} +(9.38822 + 4.37550i) q^{40} -19.7267i q^{41} +(6.68341 + 1.63202i) q^{42} +53.4739i q^{43} +(-11.9783 - 20.7471i) q^{44} +(-25.7077 + 2.25436i) q^{45} +(4.56821 - 7.91237i) q^{46} +(-18.3385 - 31.7632i) q^{47} -57.1476 q^{48} +(-43.4852 - 22.5839i) q^{49} +(-6.13642 - 2.23067i) q^{50} +(19.0179 + 32.9400i) q^{51} +(25.7308 - 44.5671i) q^{52} +(31.9730 + 18.4596i) q^{53} +(3.26734 - 1.88640i) q^{54} +(17.4792 + 24.9522i) q^{55} +(-14.0870 - 3.43989i) q^{56} -84.8955i q^{57} +(-3.69747 + 2.13474i) q^{58} +(92.4861 + 53.3969i) q^{59} +(73.6967 - 6.46260i) q^{60} +(-18.4211 + 10.6354i) q^{61} -7.08422 q^{62} +(34.6807 - 10.1266i) q^{63} +57.5445 q^{64} +(-27.6455 + 59.3171i) q^{65} +(5.18613 + 2.99421i) q^{66} +(5.33016 + 3.07737i) q^{67} +(-19.8702 - 34.4162i) q^{68} -131.644i q^{69} +(9.03556 + 1.38440i) q^{70} -48.1133 q^{71} +(9.25943 - 5.34594i) q^{72} +(-2.82277 + 4.88919i) q^{73} +(-2.17660 + 3.76999i) q^{74} +(-92.6428 + 16.3739i) q^{75} +88.7000i q^{76} +(-30.8240 - 29.4792i) q^{77} +12.8638i q^{78} +(19.1607 + 33.1873i) q^{79} +(-75.6404 + 6.63304i) q^{80} +(50.4064 - 87.3064i) q^{81} +(-2.57602 - 4.46180i) q^{82} +102.359 q^{83} +(-99.4197 + 29.0300i) q^{84} +(28.9953 + 41.3919i) q^{85} +(6.98292 + 12.0948i) q^{86} +(-30.7588 + 53.2759i) q^{87} +(-10.9311 - 6.31106i) q^{88} +(-34.1003 + 19.6878i) q^{89} +(-5.52021 + 3.86695i) q^{90} +(21.7341 - 89.0050i) q^{91} +137.544i q^{92} +(-88.3991 + 51.0373i) q^{93} +(-8.29564 - 4.78949i) q^{94} +(-9.85369 - 112.367i) q^{95} +(-39.9304 + 23.0538i) q^{96} +33.5609 q^{97} +(-12.7847 + 0.570497i) q^{98} +31.4480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 18 q^{9} - 54 q^{10} + 6 q^{11} - 66 q^{14} + 48 q^{15} - 6 q^{16} + 18 q^{19} + 12 q^{21} + 216 q^{24} + 18 q^{25} + 18 q^{26} + 48 q^{30} - 108 q^{31} + 222 q^{35} - 204 q^{36} - 240 q^{39}+ \cdots + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.226181 − 0.130586i 0.113091 − 0.0652929i −0.442388 − 0.896824i \(-0.645869\pi\)
0.555478 + 0.831531i \(0.312535\pi\)
\(3\) 1.88157 − 3.25898i 0.627191 − 1.08633i −0.360922 − 0.932596i \(-0.617538\pi\)
0.988113 − 0.153731i \(-0.0491288\pi\)
\(4\) −1.96589 + 3.40503i −0.491474 + 0.851257i
\(5\) 2.11218 − 4.53196i 0.422436 − 0.906393i
\(6\) − 0.982826i − 0.163804i
\(7\) −1.66053 + 6.80019i −0.237219 + 0.971456i
\(8\) 2.07156i 0.258945i
\(9\) −2.58064 − 4.46979i −0.286737 − 0.496644i
\(10\) −0.114075 − 1.30086i −0.0114075 − 0.130086i
\(11\) −3.04653 + 5.27675i −0.276957 + 0.479704i −0.970627 − 0.240589i \(-0.922659\pi\)
0.693670 + 0.720293i \(0.255993\pi\)
\(12\) 7.39795 + 12.8136i 0.616496 + 1.06780i
\(13\) −13.0886 −1.00682 −0.503408 − 0.864049i \(-0.667921\pi\)
−0.503408 + 0.864049i \(0.667921\pi\)
\(14\) 0.512426 + 1.75492i 0.0366019 + 0.125351i
\(15\) −10.7954 − 15.4108i −0.719691 − 1.02738i
\(16\) −7.59306 − 13.1516i −0.474566 − 0.821973i
\(17\) −5.05373 + 8.75332i −0.297278 + 0.514901i −0.975512 − 0.219945i \(-0.929412\pi\)
0.678234 + 0.734846i \(0.262746\pi\)
\(18\) −1.16738 − 0.673989i −0.0648546 − 0.0374438i
\(19\) 19.5373 − 11.2799i 1.02828 − 0.593676i 0.111787 − 0.993732i \(-0.464343\pi\)
0.916491 + 0.400056i \(0.131009\pi\)
\(20\) 11.2792 + 16.1014i 0.563958 + 0.805070i
\(21\) 19.0373 + 18.2067i 0.906537 + 0.866986i
\(22\) 1.59133i 0.0723334i
\(23\) 30.2957 − 17.4912i 1.31720 − 0.760488i 0.333927 − 0.942599i \(-0.391626\pi\)
0.983278 + 0.182111i \(0.0582929\pi\)
\(24\) 6.75116 + 3.89779i 0.281298 + 0.162408i
\(25\) −16.0774 − 19.1446i −0.643096 − 0.765785i
\(26\) −2.96039 + 1.70918i −0.113861 + 0.0657379i
\(27\) 14.4457 0.535026
\(28\) −19.8904 − 19.0226i −0.710372 − 0.679380i
\(29\) −16.3474 −0.563703 −0.281852 − 0.959458i \(-0.590949\pi\)
−0.281852 + 0.959458i \(0.590949\pi\)
\(30\) −4.45413 − 2.07590i −0.148471 − 0.0691968i
\(31\) −23.4907 − 13.5624i −0.757766 − 0.437496i 0.0707270 − 0.997496i \(-0.477468\pi\)
−0.828493 + 0.559999i \(0.810801\pi\)
\(32\) −10.6109 − 6.12620i −0.331591 − 0.191444i
\(33\) 11.4645 + 19.8572i 0.347410 + 0.601733i
\(34\) 2.63978i 0.0776406i
\(35\) 27.3109 + 21.8887i 0.780311 + 0.625391i
\(36\) 20.2930 0.563696
\(37\) −14.4349 + 8.33401i −0.390133 + 0.225243i −0.682218 − 0.731149i \(-0.738984\pi\)
0.292085 + 0.956392i \(0.405651\pi\)
\(38\) 2.94598 − 5.10258i 0.0775257 − 0.134278i
\(39\) −24.6272 + 42.6555i −0.631466 + 1.09373i
\(40\) 9.38822 + 4.37550i 0.234706 + 0.109387i
\(41\) − 19.7267i − 0.481139i −0.970632 − 0.240569i \(-0.922666\pi\)
0.970632 − 0.240569i \(-0.0773341\pi\)
\(42\) 6.68341 + 1.63202i 0.159129 + 0.0388576i
\(43\) 53.4739i 1.24358i 0.783185 + 0.621789i \(0.213594\pi\)
−0.783185 + 0.621789i \(0.786406\pi\)
\(44\) −11.9783 − 20.7471i −0.272235 − 0.471524i
\(45\) −25.7077 + 2.25436i −0.571283 + 0.0500968i
\(46\) 4.56821 − 7.91237i 0.0993089 − 0.172008i
\(47\) −18.3385 − 31.7632i −0.390181 − 0.675813i 0.602293 − 0.798275i \(-0.294254\pi\)
−0.992473 + 0.122463i \(0.960921\pi\)
\(48\) −57.1476 −1.19058
\(49\) −43.4852 − 22.5839i −0.887454 − 0.460896i
\(50\) −6.13642 − 2.23067i −0.122728 − 0.0446135i
\(51\) 19.0179 + 32.9400i 0.372901 + 0.645883i
\(52\) 25.7308 − 44.5671i 0.494823 − 0.857059i
\(53\) 31.9730 + 18.4596i 0.603264 + 0.348294i 0.770324 − 0.637652i \(-0.220094\pi\)
−0.167061 + 0.985947i \(0.553428\pi\)
\(54\) 3.26734 − 1.88640i 0.0605063 − 0.0349333i
\(55\) 17.4792 + 24.9522i 0.317804 + 0.453676i
\(56\) −14.0870 − 3.43989i −0.251553 − 0.0614266i
\(57\) − 84.8955i − 1.48939i
\(58\) −3.69747 + 2.13474i −0.0637495 + 0.0368058i
\(59\) 92.4861 + 53.3969i 1.56756 + 0.905032i 0.996452 + 0.0841571i \(0.0268198\pi\)
0.571108 + 0.820875i \(0.306514\pi\)
\(60\) 73.6967 − 6.46260i 1.22828 − 0.107710i
\(61\) −18.4211 + 10.6354i −0.301985 + 0.174351i −0.643334 − 0.765585i \(-0.722450\pi\)
0.341349 + 0.939937i \(0.389116\pi\)
\(62\) −7.08422 −0.114262
\(63\) 34.6807 − 10.1266i 0.550487 − 0.160739i
\(64\) 57.5445 0.899133
\(65\) −27.6455 + 59.3171i −0.425315 + 0.912571i
\(66\) 5.18613 + 2.99421i 0.0785777 + 0.0453668i
\(67\) 5.33016 + 3.07737i 0.0795546 + 0.0459309i 0.539250 − 0.842146i \(-0.318708\pi\)
−0.459695 + 0.888077i \(0.652041\pi\)
\(68\) −19.8702 − 34.4162i −0.292209 − 0.506121i
\(69\) − 131.644i − 1.90789i
\(70\) 9.03556 + 1.38440i 0.129079 + 0.0197771i
\(71\) −48.1133 −0.677652 −0.338826 − 0.940849i \(-0.610030\pi\)
−0.338826 + 0.940849i \(0.610030\pi\)
\(72\) 9.25943 − 5.34594i 0.128603 − 0.0742491i
\(73\) −2.82277 + 4.88919i −0.0386681 + 0.0669752i −0.884712 − 0.466138i \(-0.845645\pi\)
0.846044 + 0.533114i \(0.178978\pi\)
\(74\) −2.17660 + 3.76999i −0.0294136 + 0.0509458i
\(75\) −92.6428 + 16.3739i −1.23524 + 0.218319i
\(76\) 88.7000i 1.16711i
\(77\) −30.8240 − 29.4792i −0.400312 − 0.382847i
\(78\) 12.8638i 0.164921i
\(79\) 19.1607 + 33.1873i 0.242540 + 0.420092i 0.961437 − 0.275024i \(-0.0886860\pi\)
−0.718897 + 0.695117i \(0.755353\pi\)
\(80\) −75.6404 + 6.63304i −0.945504 + 0.0829130i
\(81\) 50.4064 − 87.3064i 0.622301 − 1.07786i
\(82\) −2.57602 − 4.46180i −0.0314149 − 0.0544122i
\(83\) 102.359 1.23325 0.616623 − 0.787259i \(-0.288500\pi\)
0.616623 + 0.787259i \(0.288500\pi\)
\(84\) −99.4197 + 29.0300i −1.18357 + 0.345595i
\(85\) 28.9953 + 41.3919i 0.341122 + 0.486963i
\(86\) 6.98292 + 12.0948i 0.0811968 + 0.140637i
\(87\) −30.7588 + 53.2759i −0.353550 + 0.612366i
\(88\) −10.9311 − 6.31106i −0.124217 − 0.0717166i
\(89\) −34.1003 + 19.6878i −0.383149 + 0.221211i −0.679187 − 0.733965i \(-0.737668\pi\)
0.296038 + 0.955176i \(0.404334\pi\)
\(90\) −5.52021 + 3.86695i −0.0613357 + 0.0429661i
\(91\) 21.7341 − 89.0050i 0.238836 − 0.978077i
\(92\) 137.544i 1.49504i
\(93\) −88.3991 + 51.0373i −0.950528 + 0.548788i
\(94\) −8.29564 − 4.78949i −0.0882515 − 0.0509520i
\(95\) −9.85369 − 112.367i −0.103723 − 1.18281i
\(96\) −39.9304 + 23.0538i −0.415941 + 0.240144i
\(97\) 33.5609 0.345989 0.172994 − 0.984923i \(-0.444656\pi\)
0.172994 + 0.984923i \(0.444656\pi\)
\(98\) −12.7847 + 0.570497i −0.130456 + 0.00582140i
\(99\) 31.4480 0.317656
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 35.3.i.a.24.4 yes 12
3.2 odd 2 315.3.bi.c.199.3 12
4.3 odd 2 560.3.br.a.129.2 12
5.2 odd 4 175.3.i.c.101.4 12
5.3 odd 4 175.3.i.c.101.3 12
5.4 even 2 inner 35.3.i.a.24.3 yes 12
7.2 even 3 245.3.i.d.19.3 12
7.3 odd 6 245.3.c.a.244.8 12
7.4 even 3 245.3.c.a.244.7 12
7.5 odd 6 inner 35.3.i.a.19.3 ✓ 12
7.6 odd 2 245.3.i.d.129.4 12
15.14 odd 2 315.3.bi.c.199.4 12
20.19 odd 2 560.3.br.a.129.5 12
21.5 even 6 315.3.bi.c.19.4 12
28.19 even 6 560.3.br.a.369.5 12
35.4 even 6 245.3.c.a.244.6 12
35.9 even 6 245.3.i.d.19.4 12
35.12 even 12 175.3.i.c.26.4 12
35.19 odd 6 inner 35.3.i.a.19.4 yes 12
35.24 odd 6 245.3.c.a.244.5 12
35.33 even 12 175.3.i.c.26.3 12
35.34 odd 2 245.3.i.d.129.3 12
105.89 even 6 315.3.bi.c.19.3 12
140.19 even 6 560.3.br.a.369.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.i.a.19.3 ✓ 12 7.5 odd 6 inner
35.3.i.a.19.4 yes 12 35.19 odd 6 inner
35.3.i.a.24.3 yes 12 5.4 even 2 inner
35.3.i.a.24.4 yes 12 1.1 even 1 trivial
175.3.i.c.26.3 12 35.33 even 12
175.3.i.c.26.4 12 35.12 even 12
175.3.i.c.101.3 12 5.3 odd 4
175.3.i.c.101.4 12 5.2 odd 4
245.3.c.a.244.5 12 35.24 odd 6
245.3.c.a.244.6 12 35.4 even 6
245.3.c.a.244.7 12 7.4 even 3
245.3.c.a.244.8 12 7.3 odd 6
245.3.i.d.19.3 12 7.2 even 3
245.3.i.d.19.4 12 35.9 even 6
245.3.i.d.129.3 12 35.34 odd 2
245.3.i.d.129.4 12 7.6 odd 2
315.3.bi.c.19.3 12 105.89 even 6
315.3.bi.c.19.4 12 21.5 even 6
315.3.bi.c.199.3 12 3.2 odd 2
315.3.bi.c.199.4 12 15.14 odd 2
560.3.br.a.129.2 12 4.3 odd 2
560.3.br.a.129.5 12 20.19 odd 2
560.3.br.a.369.2 12 140.19 even 6
560.3.br.a.369.5 12 28.19 even 6