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Properties

Label 35.3.c.c.34.2
Level $35$
Weight $3$
Character 35.34
Analytic conductor $0.954$
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] = [35,3,Mod(34,35)] 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("35.34"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0] 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: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 34.2
Root \(1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 35.34
Dual form 35.3.c.c.34.4

$q$-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-3.00000i q^{2} +3.16228 q^{3} -5.00000 q^{4} +(-1.58114 + 4.74342i) q^{5} -9.48683i q^{6} +(-6.32456 + 3.00000i) q^{7} +3.00000i q^{8} +1.00000 q^{9} +(14.2302 + 4.74342i) q^{10} +14.0000 q^{11} -15.8114 q^{12} -3.16228 q^{13} +(9.00000 + 18.9737i) q^{14} +(-5.00000 + 15.0000i) q^{15} -11.0000 q^{16} +6.32456 q^{17} -3.00000i q^{18} -28.4605i q^{19} +(7.90569 - 23.7171i) q^{20} +(-20.0000 + 9.48683i) q^{21} -42.0000i q^{22} +12.0000i q^{23} +9.48683i q^{24} +(-20.0000 - 15.0000i) q^{25} +9.48683i q^{26} -25.2982 q^{27} +(31.6228 - 15.0000i) q^{28} +14.0000 q^{29} +(45.0000 + 15.0000i) q^{30} +37.9473i q^{31} +45.0000i q^{32} +44.2719 q^{33} -18.9737i q^{34} +(-4.23025 - 34.7434i) q^{35} -5.00000 q^{36} -18.0000i q^{37} -85.3815 q^{38} -10.0000 q^{39} +(-14.2302 - 4.74342i) q^{40} -18.9737i q^{41} +(28.4605 + 60.0000i) q^{42} +42.0000i q^{43} -70.0000 q^{44} +(-1.58114 + 4.74342i) q^{45} +36.0000 q^{46} +44.2719 q^{47} -34.7851 q^{48} +(31.0000 - 37.9473i) q^{49} +(-45.0000 + 60.0000i) q^{50} +20.0000 q^{51} +15.8114 q^{52} -54.0000i q^{53} +75.8947i q^{54} +(-22.1359 + 66.4078i) q^{55} +(-9.00000 - 18.9737i) q^{56} -90.0000i q^{57} -42.0000i q^{58} +9.48683i q^{59} +(25.0000 - 75.0000i) q^{60} -66.4078i q^{61} +113.842 q^{62} +(-6.32456 + 3.00000i) q^{63} +91.0000 q^{64} +(5.00000 - 15.0000i) q^{65} -132.816i q^{66} +102.000i q^{67} -31.6228 q^{68} +37.9473i q^{69} +(-104.230 + 12.6907i) q^{70} -16.0000 q^{71} +3.00000i q^{72} +63.2456 q^{73} -54.0000 q^{74} +(-63.2456 - 47.4342i) q^{75} +142.302i q^{76} +(-88.5438 + 42.0000i) q^{77} +30.0000i q^{78} -76.0000 q^{79} +(17.3925 - 52.1776i) q^{80} -89.0000 q^{81} -56.9210 q^{82} +72.7324 q^{83} +(100.000 - 47.4342i) q^{84} +(-10.0000 + 30.0000i) q^{85} +126.000 q^{86} +44.2719 q^{87} +42.0000i q^{88} +56.9210i q^{89} +(14.2302 + 4.74342i) q^{90} +(20.0000 - 9.48683i) q^{91} -60.0000i q^{92} +120.000i q^{93} -132.816i q^{94} +(135.000 + 45.0000i) q^{95} +142.302i q^{96} -69.5701 q^{97} +(-113.842 - 93.0000i) q^{98} +14.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{4} + 4 q^{9} + 56 q^{11} + 36 q^{14} - 20 q^{15} - 44 q^{16} - 80 q^{21} - 80 q^{25} + 56 q^{29} + 180 q^{30} + 40 q^{35} - 20 q^{36} - 40 q^{39} - 280 q^{44} + 144 q^{46} + 124 q^{49} - 180 q^{50}+ \cdots + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.50000i −0.661438 − 0.750000i \(-0.730053\pi\)
0.661438 − 0.750000i \(-0.269947\pi\)
\(3\) 3.16228 1.05409 0.527046 − 0.849837i \(-0.323299\pi\)
0.527046 + 0.849837i \(0.323299\pi\)
\(4\) −5.00000 −1.25000
\(5\) −1.58114 + 4.74342i −0.316228 + 0.948683i
\(6\) − 9.48683i − 1.58114i
\(7\) −6.32456 + 3.00000i −0.903508 + 0.428571i
\(8\) 3.00000i 0.375000i
\(9\) 1.00000 0.111111
\(10\) 14.2302 + 4.74342i 1.42302 + 0.474342i
\(11\) 14.0000 1.27273 0.636364 − 0.771389i \(-0.280438\pi\)
0.636364 + 0.771389i \(0.280438\pi\)
\(12\) −15.8114 −1.31762
\(13\) −3.16228 −0.243252 −0.121626 − 0.992576i \(-0.538811\pi\)
−0.121626 + 0.992576i \(0.538811\pi\)
\(14\) 9.00000 + 18.9737i 0.642857 + 1.35526i
\(15\) −5.00000 + 15.0000i −0.333333 + 1.00000i
\(16\) −11.0000 −0.687500
\(17\) 6.32456 0.372033 0.186016 − 0.982547i \(-0.440442\pi\)
0.186016 + 0.982547i \(0.440442\pi\)
\(18\) − 3.00000i − 0.166667i
\(19\) − 28.4605i − 1.49792i −0.662615 − 0.748960i \(-0.730553\pi\)
0.662615 − 0.748960i \(-0.269447\pi\)
\(20\) 7.90569 − 23.7171i 0.395285 − 1.18585i
\(21\) −20.0000 + 9.48683i −0.952381 + 0.451754i
\(22\) − 42.0000i − 1.90909i
\(23\) 12.0000i 0.521739i 0.965374 + 0.260870i \(0.0840093\pi\)
−0.965374 + 0.260870i \(0.915991\pi\)
\(24\) 9.48683i 0.395285i
\(25\) −20.0000 − 15.0000i −0.800000 − 0.600000i
\(26\) 9.48683i 0.364878i
\(27\) −25.2982 −0.936971
\(28\) 31.6228 − 15.0000i 1.12938 − 0.535714i
\(29\) 14.0000 0.482759 0.241379 − 0.970431i \(-0.422400\pi\)
0.241379 + 0.970431i \(0.422400\pi\)
\(30\) 45.0000 + 15.0000i 1.50000 + 0.500000i
\(31\) 37.9473i 1.22411i 0.790816 + 0.612054i \(0.209656\pi\)
−0.790816 + 0.612054i \(0.790344\pi\)
\(32\) 45.0000i 1.40625i
\(33\) 44.2719 1.34157
\(34\) − 18.9737i − 0.558049i
\(35\) −4.23025 − 34.7434i −0.120864 − 0.992669i
\(36\) −5.00000 −0.138889
\(37\) − 18.0000i − 0.486486i −0.969965 − 0.243243i \(-0.921789\pi\)
0.969965 − 0.243243i \(-0.0782113\pi\)
\(38\) −85.3815 −2.24688
\(39\) −10.0000 −0.256410
\(40\) −14.2302 − 4.74342i −0.355756 − 0.118585i
\(41\) − 18.9737i − 0.462772i −0.972862 − 0.231386i \(-0.925674\pi\)
0.972862 − 0.231386i \(-0.0743261\pi\)
\(42\) 28.4605 + 60.0000i 0.677631 + 1.42857i
\(43\) 42.0000i 0.976744i 0.872635 + 0.488372i \(0.162409\pi\)
−0.872635 + 0.488372i \(0.837591\pi\)
\(44\) −70.0000 −1.59091
\(45\) −1.58114 + 4.74342i −0.0351364 + 0.105409i
\(46\) 36.0000 0.782609
\(47\) 44.2719 0.941955 0.470978 − 0.882145i \(-0.343901\pi\)
0.470978 + 0.882145i \(0.343901\pi\)
\(48\) −34.7851 −0.724689
\(49\) 31.0000 − 37.9473i 0.632653 − 0.774435i
\(50\) −45.0000 + 60.0000i −0.900000 + 1.20000i
\(51\) 20.0000 0.392157
\(52\) 15.8114 0.304065
\(53\) − 54.0000i − 1.01887i −0.860510 − 0.509434i \(-0.829855\pi\)
0.860510 − 0.509434i \(-0.170145\pi\)
\(54\) 75.8947i 1.40546i
\(55\) −22.1359 + 66.4078i −0.402472 + 1.20742i
\(56\) −9.00000 − 18.9737i −0.160714 − 0.338815i
\(57\) − 90.0000i − 1.57895i
\(58\) − 42.0000i − 0.724138i
\(59\) 9.48683i 0.160794i 0.996763 + 0.0803969i \(0.0256188\pi\)
−0.996763 + 0.0803969i \(0.974381\pi\)
\(60\) 25.0000 − 75.0000i 0.416667 − 1.25000i
\(61\) − 66.4078i − 1.08865i −0.838873 − 0.544326i \(-0.816785\pi\)
0.838873 − 0.544326i \(-0.183215\pi\)
\(62\) 113.842 1.83616
\(63\) −6.32456 + 3.00000i −0.100390 + 0.0476190i
\(64\) 91.0000 1.42188
\(65\) 5.00000 − 15.0000i 0.0769231 − 0.230769i
\(66\) − 132.816i − 2.01236i
\(67\) 102.000i 1.52239i 0.648524 + 0.761194i \(0.275387\pi\)
−0.648524 + 0.761194i \(0.724613\pi\)
\(68\) −31.6228 −0.465041
\(69\) 37.9473i 0.549961i
\(70\) −104.230 + 12.6907i −1.48900 + 0.181296i
\(71\) −16.0000 −0.225352 −0.112676 − 0.993632i \(-0.535942\pi\)
−0.112676 + 0.993632i \(0.535942\pi\)
\(72\) 3.00000i 0.0416667i
\(73\) 63.2456 0.866377 0.433189 − 0.901303i \(-0.357388\pi\)
0.433189 + 0.901303i \(0.357388\pi\)
\(74\) −54.0000 −0.729730
\(75\) −63.2456 − 47.4342i −0.843274 − 0.632456i
\(76\) 142.302i 1.87240i
\(77\) −88.5438 + 42.0000i −1.14992 + 0.545455i
\(78\) 30.0000i 0.384615i
\(79\) −76.0000 −0.962025 −0.481013 − 0.876714i \(-0.659731\pi\)
−0.481013 + 0.876714i \(0.659731\pi\)
\(80\) 17.3925 − 52.1776i 0.217407 − 0.652220i
\(81\) −89.0000 −1.09877
\(82\) −56.9210 −0.694159
\(83\) 72.7324 0.876294 0.438147 − 0.898903i \(-0.355635\pi\)
0.438147 + 0.898903i \(0.355635\pi\)
\(84\) 100.000 − 47.4342i 1.19048 − 0.564692i
\(85\) −10.0000 + 30.0000i −0.117647 + 0.352941i
\(86\) 126.000 1.46512
\(87\) 44.2719 0.508872
\(88\) 42.0000i 0.477273i
\(89\) 56.9210i 0.639562i 0.947492 + 0.319781i \(0.103609\pi\)
−0.947492 + 0.319781i \(0.896391\pi\)
\(90\) 14.2302 + 4.74342i 0.158114 + 0.0527046i
\(91\) 20.0000 − 9.48683i 0.219780 − 0.104251i
\(92\) − 60.0000i − 0.652174i
\(93\) 120.000i 1.29032i
\(94\) − 132.816i − 1.41293i
\(95\) 135.000 + 45.0000i 1.42105 + 0.473684i
\(96\) 142.302i 1.48232i
\(97\) −69.5701 −0.717218 −0.358609 − 0.933488i \(-0.616749\pi\)
−0.358609 + 0.933488i \(0.616749\pi\)
\(98\) −113.842 − 93.0000i −1.16165 − 0.948980i
\(99\) 14.0000 0.141414
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.c.c.34.2 yes 4
3.2 odd 2 315.3.e.c.244.4 4
4.3 odd 2 560.3.p.f.209.2 4
5.2 odd 4 175.3.d.h.76.1 2
5.3 odd 4 175.3.d.b.76.2 2
5.4 even 2 inner 35.3.c.c.34.3 yes 4
7.2 even 3 245.3.i.c.129.1 8
7.3 odd 6 245.3.i.c.19.4 8
7.4 even 3 245.3.i.c.19.3 8
7.5 odd 6 245.3.i.c.129.2 8
7.6 odd 2 inner 35.3.c.c.34.1 ✓ 4
15.14 odd 2 315.3.e.c.244.1 4
20.19 odd 2 560.3.p.f.209.4 4
21.20 even 2 315.3.e.c.244.3 4
28.27 even 2 560.3.p.f.209.3 4
35.4 even 6 245.3.i.c.19.2 8
35.9 even 6 245.3.i.c.129.4 8
35.13 even 4 175.3.d.b.76.1 2
35.19 odd 6 245.3.i.c.129.3 8
35.24 odd 6 245.3.i.c.19.1 8
35.27 even 4 175.3.d.h.76.2 2
35.34 odd 2 inner 35.3.c.c.34.4 yes 4
105.104 even 2 315.3.e.c.244.2 4
140.139 even 2 560.3.p.f.209.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.c.c.34.1 ✓ 4 7.6 odd 2 inner
35.3.c.c.34.2 yes 4 1.1 even 1 trivial
35.3.c.c.34.3 yes 4 5.4 even 2 inner
35.3.c.c.34.4 yes 4 35.34 odd 2 inner
175.3.d.b.76.1 2 35.13 even 4
175.3.d.b.76.2 2 5.3 odd 4
175.3.d.h.76.1 2 5.2 odd 4
175.3.d.h.76.2 2 35.27 even 4
245.3.i.c.19.1 8 35.24 odd 6
245.3.i.c.19.2 8 35.4 even 6
245.3.i.c.19.3 8 7.4 even 3
245.3.i.c.19.4 8 7.3 odd 6
245.3.i.c.129.1 8 7.2 even 3
245.3.i.c.129.2 8 7.5 odd 6
245.3.i.c.129.3 8 35.19 odd 6
245.3.i.c.129.4 8 35.9 even 6
315.3.e.c.244.1 4 15.14 odd 2
315.3.e.c.244.2 4 105.104 even 2
315.3.e.c.244.3 4 21.20 even 2
315.3.e.c.244.4 4 3.2 odd 2
560.3.p.f.209.1 4 140.139 even 2
560.3.p.f.209.2 4 4.3 odd 2
560.3.p.f.209.3 4 28.27 even 2
560.3.p.f.209.4 4 20.19 odd 2