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Properties

Label 123.4.e.a.73.6
Level $123$
Weight $4$
Character 123.73
Analytic conductor $7.257$
Analytic rank $0$
Dimension $44$
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] = [123,4,Mod(73,123)] 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("123.73"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(123, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.e (of order \(4\), 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: \(7.25723493071\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 73.6
Character \(\chi\) \(=\) 123.73
Dual form 123.4.e.a.91.17

$q$-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-3.81117i q^{2} +(-2.12132 + 2.12132i) q^{3} -6.52505 q^{4} -7.20336i q^{5} +(8.08472 + 8.08472i) q^{6} +(18.1690 - 18.1690i) q^{7} -5.62130i q^{8} -9.00000i q^{9} -27.4532 q^{10} +(-28.2436 + 28.2436i) q^{11} +(13.8417 - 13.8417i) q^{12} +(-12.8573 + 12.8573i) q^{13} +(-69.2451 - 69.2451i) q^{14} +(15.2806 + 15.2806i) q^{15} -73.6241 q^{16} +(-76.3026 - 76.3026i) q^{17} -34.3006 q^{18} +(0.815247 + 0.815247i) q^{19} +47.0022i q^{20} +77.0844i q^{21} +(107.641 + 107.641i) q^{22} -168.591 q^{23} +(11.9246 + 11.9246i) q^{24} +73.1116 q^{25} +(49.0012 + 49.0012i) q^{26} +(19.0919 + 19.0919i) q^{27} +(-118.553 + 118.553i) q^{28} +(193.344 - 193.344i) q^{29} +(58.2371 - 58.2371i) q^{30} +228.762 q^{31} +235.624i q^{32} -119.828i q^{33} +(-290.803 + 290.803i) q^{34} +(-130.878 - 130.878i) q^{35} +58.7254i q^{36} -223.382 q^{37} +(3.10705 - 3.10705i) q^{38} -54.5487i q^{39} -40.4923 q^{40} +(225.845 + 133.848i) q^{41} +293.782 q^{42} -103.858i q^{43} +(184.291 - 184.291i) q^{44} -64.8302 q^{45} +642.529i q^{46} +(-280.590 - 280.590i) q^{47} +(156.180 - 156.180i) q^{48} -317.223i q^{49} -278.641i q^{50} +323.725 q^{51} +(83.8942 - 83.8942i) q^{52} +(171.271 - 171.271i) q^{53} +(72.7625 - 72.7625i) q^{54} +(203.449 + 203.449i) q^{55} +(-102.133 - 102.133i) q^{56} -3.45880 q^{57} +(-736.869 - 736.869i) q^{58} +816.036 q^{59} +(-99.7068 - 99.7068i) q^{60} +265.021i q^{61} -871.853i q^{62} +(-163.521 - 163.521i) q^{63} +309.011 q^{64} +(92.6154 + 92.6154i) q^{65} -456.683 q^{66} +(563.392 + 563.392i) q^{67} +(497.878 + 497.878i) q^{68} +(357.635 - 357.635i) q^{69} +(-498.797 + 498.797i) q^{70} +(31.8597 - 31.8597i) q^{71} -50.5917 q^{72} -550.801i q^{73} +851.346i q^{74} +(-155.093 + 155.093i) q^{75} +(-5.31953 - 5.31953i) q^{76} +1026.31i q^{77} -207.895 q^{78} +(287.128 - 287.128i) q^{79} +530.341i q^{80} -81.0000 q^{81} +(510.116 - 860.733i) q^{82} -169.263 q^{83} -502.979i q^{84} +(-549.635 + 549.635i) q^{85} -395.820 q^{86} +820.291i q^{87} +(158.766 + 158.766i) q^{88} +(-98.6371 + 98.6371i) q^{89} +247.079i q^{90} +467.206i q^{91} +1100.06 q^{92} +(-485.278 + 485.278i) q^{93} +(-1069.38 + 1069.38i) q^{94} +(5.87252 - 5.87252i) q^{95} +(-499.834 - 499.834i) q^{96} +(-61.0351 - 61.0351i) q^{97} -1208.99 q^{98} +(254.193 + 254.193i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 192 q^{4} + 224 q^{10} + 96 q^{11} - 24 q^{12} - 196 q^{13} + 148 q^{14} + 24 q^{15} + 896 q^{16} - 308 q^{17} + 112 q^{19} - 612 q^{22} - 208 q^{23} - 180 q^{24} - 1452 q^{25} + 1172 q^{26} - 220 q^{28}+ \cdots - 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(88\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.81117i − 1.34745i −0.738981 − 0.673727i \(-0.764692\pi\)
0.738981 − 0.673727i \(-0.235308\pi\)
\(3\) −2.12132 + 2.12132i −0.408248 + 0.408248i
\(4\) −6.52505 −0.815631
\(5\) − 7.20336i − 0.644288i −0.946691 − 0.322144i \(-0.895596\pi\)
0.946691 − 0.322144i \(-0.104404\pi\)
\(6\) 8.08472 + 8.08472i 0.550096 + 0.550096i
\(7\) 18.1690 − 18.1690i 0.981032 − 0.981032i −0.0187915 − 0.999823i \(-0.505982\pi\)
0.999823 + 0.0187915i \(0.00598188\pi\)
\(8\) − 5.62130i − 0.248429i
\(9\) − 9.00000i − 0.333333i
\(10\) −27.4532 −0.868148
\(11\) −28.2436 + 28.2436i −0.774161 + 0.774161i −0.978831 − 0.204670i \(-0.934388\pi\)
0.204670 + 0.978831i \(0.434388\pi\)
\(12\) 13.8417 − 13.8417i 0.332980 − 0.332980i
\(13\) −12.8573 + 12.8573i −0.274305 + 0.274305i −0.830830 − 0.556526i \(-0.812134\pi\)
0.556526 + 0.830830i \(0.312134\pi\)
\(14\) −69.2451 − 69.2451i −1.32189 − 1.32189i
\(15\) 15.2806 + 15.2806i 0.263029 + 0.263029i
\(16\) −73.6241 −1.15038
\(17\) −76.3026 − 76.3026i −1.08859 − 1.08859i −0.995674 − 0.0929206i \(-0.970380\pi\)
−0.0929206 − 0.995674i \(-0.529620\pi\)
\(18\) −34.3006 −0.449151
\(19\) 0.815247 + 0.815247i 0.00984371 + 0.00984371i 0.712012 − 0.702168i \(-0.247784\pi\)
−0.702168 + 0.712012i \(0.747784\pi\)
\(20\) 47.0022i 0.525501i
\(21\) 77.0844i 0.801009i
\(22\) 107.641 + 107.641i 1.04315 + 1.04315i
\(23\) −168.591 −1.52842 −0.764209 − 0.644968i \(-0.776870\pi\)
−0.764209 + 0.644968i \(0.776870\pi\)
\(24\) 11.9246 + 11.9246i 0.101421 + 0.101421i
\(25\) 73.1116 0.584893
\(26\) 49.0012 + 49.0012i 0.369613 + 0.369613i
\(27\) 19.0919 + 19.0919i 0.136083 + 0.136083i
\(28\) −118.553 + 118.553i −0.800160 + 0.800160i
\(29\) 193.344 − 193.344i 1.23804 − 1.23804i 0.277239 − 0.960801i \(-0.410581\pi\)
0.960801 − 0.277239i \(-0.0894194\pi\)
\(30\) 58.2371 − 58.2371i 0.354420 − 0.354420i
\(31\) 228.762 1.32538 0.662692 − 0.748892i \(-0.269414\pi\)
0.662692 + 0.748892i \(0.269414\pi\)
\(32\) 235.624i 1.30165i
\(33\) − 119.828i − 0.632100i
\(34\) −290.803 + 290.803i −1.46683 + 1.46683i
\(35\) −130.878 − 130.878i −0.632067 − 0.632067i
\(36\) 58.7254i 0.271877i
\(37\) −223.382 −0.992533 −0.496266 − 0.868170i \(-0.665296\pi\)
−0.496266 + 0.868170i \(0.665296\pi\)
\(38\) 3.10705 − 3.10705i 0.0132639 − 0.0132639i
\(39\) − 54.5487i − 0.223969i
\(40\) −40.4923 −0.160060
\(41\) 225.845 + 133.848i 0.860269 + 0.509841i
\(42\) 293.782 1.07932
\(43\) − 103.858i − 0.368329i −0.982895 − 0.184165i \(-0.941042\pi\)
0.982895 − 0.184165i \(-0.0589580\pi\)
\(44\) 184.291 − 184.291i 0.631430 − 0.631430i
\(45\) −64.8302 −0.214763
\(46\) 642.529i 2.05947i
\(47\) −280.590 − 280.590i −0.870814 − 0.870814i 0.121747 − 0.992561i \(-0.461150\pi\)
−0.992561 + 0.121747i \(0.961150\pi\)
\(48\) 156.180 − 156.180i 0.469640 − 0.469640i
\(49\) − 317.223i − 0.924847i
\(50\) − 278.641i − 0.788116i
\(51\) 323.725 0.888833
\(52\) 83.8942 − 83.8942i 0.223731 − 0.223731i
\(53\) 171.271 − 171.271i 0.443885 − 0.443885i −0.449430 − 0.893316i \(-0.648373\pi\)
0.893316 + 0.449430i \(0.148373\pi\)
\(54\) 72.7625 − 72.7625i 0.183365 − 0.183365i
\(55\) 203.449 + 203.449i 0.498783 + 0.498783i
\(56\) −102.133 − 102.133i −0.243717 − 0.243717i
\(57\) −3.45880 −0.00803736
\(58\) −736.869 − 736.869i −1.66820 − 1.66820i
\(59\) 816.036 1.80066 0.900329 − 0.435210i \(-0.143326\pi\)
0.900329 + 0.435210i \(0.143326\pi\)
\(60\) −99.7068 − 99.7068i −0.214535 − 0.214535i
\(61\) 265.021i 0.556270i 0.960542 + 0.278135i \(0.0897163\pi\)
−0.960542 + 0.278135i \(0.910284\pi\)
\(62\) − 871.853i − 1.78589i
\(63\) −163.521 − 163.521i −0.327011 − 0.327011i
\(64\) 309.011 0.603537
\(65\) 92.6154 + 92.6154i 0.176731 + 0.176731i
\(66\) −456.683 −0.851725
\(67\) 563.392 + 563.392i 1.02730 + 1.02730i 0.999617 + 0.0276863i \(0.00881394\pi\)
0.0276863 + 0.999617i \(0.491186\pi\)
\(68\) 497.878 + 497.878i 0.887891 + 0.887891i
\(69\) 357.635 − 357.635i 0.623974 − 0.623974i
\(70\) −498.797 + 498.797i −0.851681 + 0.851681i
\(71\) 31.8597 − 31.8597i 0.0532543 − 0.0532543i −0.679978 − 0.733232i \(-0.738011\pi\)
0.733232 + 0.679978i \(0.238011\pi\)
\(72\) −50.5917 −0.0828096
\(73\) − 550.801i − 0.883101i −0.897236 − 0.441551i \(-0.854429\pi\)
0.897236 − 0.441551i \(-0.145571\pi\)
\(74\) 851.346i 1.33739i
\(75\) −155.093 + 155.093i −0.238782 + 0.238782i
\(76\) −5.31953 − 5.31953i −0.00802884 − 0.00802884i
\(77\) 1026.31i 1.51895i
\(78\) −207.895 −0.301788
\(79\) 287.128 − 287.128i 0.408917 − 0.408917i −0.472444 − 0.881361i \(-0.656628\pi\)
0.881361 + 0.472444i \(0.156628\pi\)
\(80\) 530.341i 0.741174i
\(81\) −81.0000 −0.111111
\(82\) 510.116 − 860.733i 0.686987 − 1.15917i
\(83\) −169.263 −0.223844 −0.111922 − 0.993717i \(-0.535701\pi\)
−0.111922 + 0.993717i \(0.535701\pi\)
\(84\) − 502.979i − 0.653328i
\(85\) −549.635 + 549.635i −0.701368 + 0.701368i
\(86\) −395.820 −0.496307
\(87\) 820.291i 1.01086i
\(88\) 158.766 + 158.766i 0.192324 + 0.192324i
\(89\) −98.6371 + 98.6371i −0.117478 + 0.117478i −0.763402 − 0.645924i \(-0.776472\pi\)
0.645924 + 0.763402i \(0.276472\pi\)
\(90\) 247.079i 0.289383i
\(91\) 467.206i 0.538203i
\(92\) 1100.06 1.24663
\(93\) −485.278 + 485.278i −0.541086 + 0.541086i
\(94\) −1069.38 + 1069.38i −1.17338 + 1.17338i
\(95\) 5.87252 − 5.87252i 0.00634219 − 0.00634219i
\(96\) −499.834 − 499.834i −0.531397 − 0.531397i
\(97\) −61.0351 − 61.0351i −0.0638885 − 0.0638885i 0.674441 − 0.738329i \(-0.264385\pi\)
−0.738329 + 0.674441i \(0.764385\pi\)
\(98\) −1208.99 −1.24619
\(99\) 254.193 + 254.193i 0.258054 + 0.258054i
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 123.4.e.a.73.6 ✓ 44
41.9 even 4 inner 123.4.e.a.91.17 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.4.e.a.73.6 ✓ 44 1.1 even 1 trivial
123.4.e.a.91.17 yes 44 41.9 even 4 inner