↑

Properties

Label 35.5.l.a.2.12
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 2.12
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.12

$q$-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+(5.58177 - 1.49563i) q^{2} +(11.2270 + 3.00826i) q^{3} +(15.0628 - 8.69652i) q^{4} +(-22.0427 - 11.7949i) q^{5} +67.1656 q^{6} +(-38.6780 + 30.0834i) q^{7} +(5.69217 - 5.69217i) q^{8} +(46.8474 + 27.0474i) q^{9} +(-140.678 - 32.8685i) q^{10} +(-7.11215 - 12.3186i) q^{11} +(195.271 - 52.3228i) q^{12} +(195.546 - 195.546i) q^{13} +(-170.898 + 225.766i) q^{14} +(-211.991 - 198.731i) q^{15} +(-115.885 + 200.719i) q^{16} +(-10.4712 + 39.0789i) q^{17} +(301.944 + 80.9057i) q^{18} +(222.686 + 128.568i) q^{19} +(-434.600 + 14.0308i) q^{20} +(-524.736 + 221.392i) q^{21} +(-58.1225 - 58.1225i) q^{22} +(-22.4812 - 83.9010i) q^{23} +(81.0294 - 46.7823i) q^{24} +(346.762 + 519.982i) q^{25} +(799.027 - 1383.96i) q^{26} +(-221.127 - 221.127i) q^{27} +(-320.980 + 789.505i) q^{28} +1116.77i q^{29} +(-1480.51 - 792.210i) q^{30} +(-890.180 - 1541.84i) q^{31} +(-379.979 + 1418.10i) q^{32} +(-42.7904 - 159.696i) q^{33} +233.790i q^{34} +(1207.40 - 206.916i) q^{35} +940.872 q^{36} +(805.400 - 215.806i) q^{37} +(1435.27 + 384.580i) q^{38} +(2783.64 - 1607.14i) q^{39} +(-192.609 + 58.3324i) q^{40} +2737.91 q^{41} +(-2597.84 + 2020.57i) q^{42} +(-209.828 + 209.828i) q^{43} +(-214.258 - 123.702i) q^{44} +(-713.623 - 1148.76i) q^{45} +(-250.970 - 434.692i) q^{46} +(-523.197 + 140.190i) q^{47} +(-1904.86 + 1904.86i) q^{48} +(590.982 - 2327.13i) q^{49} +(2713.24 + 2383.79i) q^{50} +(-235.119 + 407.238i) q^{51} +(1244.90 - 4646.04i) q^{52} +(-3253.32 - 871.725i) q^{53} +(-1565.00 - 903.555i) q^{54} +(11.4746 + 355.422i) q^{55} +(-48.9223 + 391.401i) q^{56} +(2113.33 + 2113.33i) q^{57} +(1670.27 + 6233.55i) q^{58} +(-1429.00 + 825.035i) q^{59} +(-4921.45 - 1149.86i) q^{60} +(-438.372 + 759.283i) q^{61} +(-7274.80 - 7274.80i) q^{62} +(-2625.64 + 363.189i) q^{63} +4775.49i q^{64} +(-6616.80 + 2003.92i) q^{65} +(-477.692 - 827.387i) q^{66} +(-365.546 + 1364.24i) q^{67} +(182.125 + 679.701i) q^{68} -1009.58i q^{69} +(6429.95 - 2960.78i) q^{70} +1621.68 q^{71} +(420.621 - 112.705i) q^{72} +(5006.69 + 1341.54i) q^{73} +(4172.79 - 2409.16i) q^{74} +(2328.85 + 6880.98i) q^{75} +4472.38 q^{76} +(645.669 + 262.502i) q^{77} +(13134.0 - 13134.0i) q^{78} +(-890.409 - 514.078i) q^{79} +(4921.89 - 3057.54i) q^{80} +(-4008.22 - 6942.44i) q^{81} +(15282.4 - 4094.90i) q^{82} +(-6269.86 + 6269.86i) q^{83} +(-5978.67 + 7898.16i) q^{84} +(691.744 - 737.899i) q^{85} +(-857.387 + 1485.04i) q^{86} +(-3359.53 + 12537.9i) q^{87} +(-110.603 - 29.6360i) q^{88} +(-5777.55 - 3335.67i) q^{89} +(-5701.39 - 5344.77i) q^{90} +(-1680.65 + 13446.0i) q^{91} +(-1068.28 - 1068.28i) q^{92} +(-5355.79 - 19988.1i) q^{93} +(-2710.69 + 1565.02i) q^{94} +(-3392.16 - 5460.54i) q^{95} +(-8532.03 + 14777.9i) q^{96} +(12264.2 + 12264.2i) q^{97} +(-181.805 - 13873.4i) q^{98} -769.460i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.58177 − 1.49563i 1.39544 − 0.373908i 0.518736 − 0.854934i \(-0.326403\pi\)
0.876706 + 0.481027i \(0.159736\pi\)
\(3\) 11.2270 + 3.00826i 1.24744 + 0.334251i 0.821348 − 0.570428i \(-0.193222\pi\)
0.426094 + 0.904679i \(0.359889\pi\)
\(4\) 15.0628 − 8.69652i 0.941426 − 0.543533i
\(5\) −22.0427 − 11.7949i −0.881708 − 0.471795i
\(6\) 67.1656 1.86571
\(7\) −38.6780 + 30.0834i −0.789348 + 0.613946i
\(8\) 5.69217 − 5.69217i 0.0889401 − 0.0889401i
\(9\) 46.8474 + 27.0474i 0.578363 + 0.333918i
\(10\) −140.678 − 32.8685i −1.40678 − 0.328685i
\(11\) −7.11215 − 12.3186i −0.0587781 − 0.101807i 0.835139 − 0.550039i \(-0.185387\pi\)
−0.893917 + 0.448232i \(0.852054\pi\)
\(12\) 195.271 − 52.3228i 1.35605 − 0.363353i
\(13\) 195.546 − 195.546i 1.15708 − 1.15708i 0.171975 − 0.985101i \(-0.444985\pi\)
0.985101 − 0.171975i \(-0.0550148\pi\)
\(14\) −170.898 + 225.766i −0.871930 + 1.15187i
\(15\) −211.991 − 198.731i −0.942182 − 0.883249i
\(16\) −115.885 + 200.719i −0.452677 + 0.784060i
\(17\) −10.4712 + 39.0789i −0.0362324 + 0.135221i −0.981673 − 0.190574i \(-0.938965\pi\)
0.945440 + 0.325795i \(0.105632\pi\)
\(18\) 301.944 + 80.9057i 0.931926 + 0.249709i
\(19\) 222.686 + 128.568i 0.616859 + 0.356144i 0.775645 − 0.631169i \(-0.217425\pi\)
−0.158786 + 0.987313i \(0.550758\pi\)
\(20\) −434.600 + 14.0308i −1.08650 + 0.0350771i
\(21\) −524.736 + 221.392i −1.18988 + 0.502022i
\(22\) −58.1225 − 58.1225i −0.120088 − 0.120088i
\(23\) −22.4812 − 83.9010i −0.0424975 − 0.158603i 0.941416 − 0.337247i \(-0.109496\pi\)
−0.983914 + 0.178644i \(0.942829\pi\)
\(24\) 81.0294 − 46.7823i 0.140676 − 0.0812193i
\(25\) 346.762 + 519.982i 0.554819 + 0.831971i
\(26\) 799.027 − 1383.96i 1.18199 − 2.04727i
\(27\) −221.127 − 221.127i −0.303329 − 0.303329i
\(28\) −320.980 + 789.505i −0.409413 + 1.00702i
\(29\) 1116.77i 1.32791i 0.747774 + 0.663953i \(0.231123\pi\)
−0.747774 + 0.663953i \(0.768877\pi\)
\(30\) −1480.51 − 792.210i −1.64501 − 0.880234i
\(31\) −890.180 − 1541.84i −0.926306 − 1.60441i −0.789447 − 0.613819i \(-0.789632\pi\)
−0.136860 − 0.990590i \(-0.543701\pi\)
\(32\) −379.979 + 1418.10i −0.371073 + 1.38486i
\(33\) −42.7904 − 159.696i −0.0392933 − 0.146645i
\(34\) 233.790i 0.202241i
\(35\) 1207.40 − 206.916i 0.985631 − 0.168911i
\(36\) 940.872 0.725981
\(37\) 805.400 − 215.806i 0.588313 − 0.157638i 0.0476309 − 0.998865i \(-0.484833\pi\)
0.540682 + 0.841227i \(0.318166\pi\)
\(38\) 1435.27 + 384.580i 0.993956 + 0.266330i
\(39\) 2783.64 − 1607.14i 1.83014 − 1.05663i
\(40\) −192.609 + 58.3324i −0.120381 + 0.0364577i
\(41\) 2737.91 1.62874 0.814370 − 0.580346i \(-0.197083\pi\)
0.814370 + 0.580346i \(0.197083\pi\)
\(42\) −2597.84 + 2020.57i −1.47270 + 1.14545i
\(43\) −209.828 + 209.828i −0.113482 + 0.113482i −0.761568 − 0.648086i \(-0.775570\pi\)
0.648086 + 0.761568i \(0.275570\pi\)
\(44\) −214.258 − 123.702i −0.110670 − 0.0638956i
\(45\) −713.623 − 1148.76i −0.352407 − 0.567287i
\(46\) −250.970 − 434.692i −0.118606 − 0.205431i
\(47\) −523.197 + 140.190i −0.236848 + 0.0634632i −0.375290 − 0.926907i \(-0.622457\pi\)
0.138443 + 0.990370i \(0.455790\pi\)
\(48\) −1904.86 + 1904.86i −0.826762 + 0.826762i
\(49\) 590.982 − 2327.13i 0.246140 − 0.969234i
\(50\) 2713.24 + 2383.79i 1.08530 + 0.953516i
\(51\) −235.119 + 407.238i −0.0903957 + 0.156570i
\(52\) 1244.90 − 4646.04i 0.460393 − 1.71821i
\(53\) −3253.32 − 871.725i −1.15818 − 0.310333i −0.371940 − 0.928257i \(-0.621307\pi\)
−0.786238 + 0.617924i \(0.787974\pi\)
\(54\) −1565.00 − 903.555i −0.536695 − 0.309861i
\(55\) 11.4746 + 355.422i 0.00379327 + 0.117495i
\(56\) −48.9223 + 391.401i −0.0156002 + 0.124809i
\(57\) 2113.33 + 2113.33i 0.650455 + 0.650455i
\(58\) 1670.27 + 6233.55i 0.496514 + 1.85302i
\(59\) −1429.00 + 825.035i −0.410515 + 0.237011i −0.691011 − 0.722844i \(-0.742834\pi\)
0.280496 + 0.959855i \(0.409501\pi\)
\(60\) −4921.45 − 1149.86i −1.36707 − 0.319407i
\(61\) −438.372 + 759.283i −0.117810 + 0.204053i −0.918900 − 0.394491i \(-0.870921\pi\)
0.801089 + 0.598545i \(0.204254\pi\)
\(62\) −7274.80 − 7274.80i −1.89251 − 1.89251i
\(63\) −2625.64 + 363.189i −0.661537 + 0.0915064i
\(64\) 4775.49i 1.16589i
\(65\) −6616.80 + 2003.92i −1.56611 + 0.474301i
\(66\) −477.692 − 827.387i −0.109663 − 0.189942i
\(67\) −365.546 + 1364.24i −0.0814316 + 0.303907i −0.994615 − 0.103643i \(-0.966950\pi\)
0.913183 + 0.407550i \(0.133617\pi\)
\(68\) 182.125 + 679.701i 0.0393870 + 0.146994i
\(69\) − 1009.58i − 0.212053i
\(70\) 6429.95 − 2960.78i 1.31223 − 0.604241i
\(71\) 1621.68 0.321698 0.160849 − 0.986979i \(-0.448577\pi\)
0.160849 + 0.986979i \(0.448577\pi\)
\(72\) 420.621 − 112.705i 0.0811384 − 0.0217410i
\(73\) 5006.69 + 1341.54i 0.939517 + 0.251743i 0.695908 − 0.718131i \(-0.255002\pi\)
0.243609 + 0.969874i \(0.421669\pi\)
\(74\) 4172.79 − 2409.16i 0.762015 − 0.439949i
\(75\) 2328.85 + 6880.98i 0.414017 + 1.22328i
\(76\) 4472.38 0.774303
\(77\) 645.669 + 262.502i 0.108900 + 0.0442743i
\(78\) 13134.0 − 13134.0i 2.15877 − 2.15877i
\(79\) −890.409 − 514.078i −0.142671 − 0.0823711i 0.426965 − 0.904268i \(-0.359583\pi\)
−0.569636 + 0.821897i \(0.692916\pi\)
\(80\) 4921.89 − 3057.54i 0.769045 − 0.477741i
\(81\) −4008.22 − 6942.44i −0.610916 − 1.05814i
\(82\) 15282.4 − 4094.90i 2.27281 − 0.608998i
\(83\) −6269.86 + 6269.86i −0.910127 + 0.910127i −0.996282 − 0.0861548i \(-0.972542\pi\)
0.0861548 + 0.996282i \(0.472542\pi\)
\(84\) −5978.67 + 7898.16i −0.847317 + 1.11935i
\(85\) 691.744 − 737.899i 0.0957431 − 0.102131i
\(86\) −857.387 + 1485.04i −0.115926 + 0.200789i
\(87\) −3359.53 + 12537.9i −0.443854 + 1.65649i
\(88\) −110.603 − 29.6360i −0.0142824 − 0.00382697i
\(89\) −5777.55 − 3335.67i −0.729396 − 0.421117i 0.0888049 − 0.996049i \(-0.471695\pi\)
−0.818201 + 0.574932i \(0.805029\pi\)
\(90\) −5701.39 − 5344.77i −0.703876 − 0.659849i
\(91\) −1680.65 + 13446.0i −0.202953 + 1.62372i
\(92\) −1068.28 − 1068.28i −0.126214 − 0.126214i
\(93\) −5355.79 − 19988.1i −0.619238 − 2.31103i
\(94\) −2710.69 + 1565.02i −0.306778 + 0.177118i
\(95\) −3392.16 − 5460.54i −0.375863 − 0.605046i
\(96\) −8532.03 + 14777.9i −0.925785 + 1.60351i
\(97\) 12264.2 + 12264.2i 1.30345 + 1.30345i 0.926051 + 0.377399i \(0.123181\pi\)
0.377399 + 0.926051i \(0.376819\pi\)
\(98\) −181.805 − 13873.4i −0.0189302 − 1.44454i
\(99\) − 769.460i − 0.0785083i
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 35.5.l.a.2.12 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.3 yes 56
7.4 even 3 inner 35.5.l.a.32.3 yes 56
35.18 odd 12 inner 35.5.l.a.18.12 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.12 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.12 yes 56 35.18 odd 12 inner
35.5.l.a.23.3 yes 56 5.3 odd 4 inner
35.5.l.a.32.3 yes 56 7.4 even 3 inner