LMFDB - Embedded newform 531.5.b.a.296.66

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.66
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.11

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.38419i q^{2} -24.7578 q^{4} +16.4296i q^{5} +84.4308 q^{7} -55.9116i q^{8} +O(q^{10})$	\(q+6.38419i q^{2} -24.7578 q^{4} +16.4296i q^{5} +84.4308 q^{7} -55.9116i q^{8} -104.890 q^{10} -31.3181i q^{11} +43.8032 q^{13} +539.022i q^{14} -39.1754 q^{16} -243.216i q^{17} +602.103 q^{19} -406.761i q^{20} +199.940 q^{22} -401.054i q^{23} +355.068 q^{25} +279.648i q^{26} -2090.32 q^{28} +1577.03i q^{29} +306.821 q^{31} -1144.69i q^{32} +1552.74 q^{34} +1387.16i q^{35} +2509.88 q^{37} +3843.94i q^{38} +918.606 q^{40} -1680.84i q^{41} +2973.39 q^{43} +775.368i q^{44} +2560.40 q^{46} -890.845i q^{47} +4727.55 q^{49} +2266.82i q^{50} -1084.47 q^{52} +4496.31i q^{53} +514.544 q^{55} -4720.66i q^{56} -10068.0 q^{58} -453.188i q^{59} -4996.33 q^{61} +1958.80i q^{62} +6681.09 q^{64} +719.670i q^{65} -5524.56 q^{67} +6021.50i q^{68} -8855.92 q^{70} -6512.64i q^{71} +1300.13 q^{73} +16023.5i q^{74} -14906.8 q^{76} -2644.21i q^{77} -7102.80 q^{79} -643.636i q^{80} +10730.8 q^{82} +7734.10i q^{83} +3995.95 q^{85} +18982.7i q^{86} -1751.04 q^{88} -9381.04i q^{89} +3698.34 q^{91} +9929.21i q^{92} +5687.32 q^{94} +9892.33i q^{95} -16857.7 q^{97} +30181.6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	6.38419i	1.59605i	0.602627	+	0.798023i	$0.294121\pi$
$2$	6.38419i	1.59605i	−0.602627	+	0.798023i	$0.705879\pi$
$3$	0	0
$3$	0	0
$4$	−24.7578	−1.54736
$5$	16.4296i	0.657185i	0.944472	+	0.328592i	$0.106574\pi$
$5$	16.4296i	0.657185i	−0.944472	+	0.328592i	$0.893426\pi$
$6$	0	0
$7$	84.4308	1.72308	0.861538	−	0.507693i	$-0.169501\pi$
$7$	84.4308	1.72308	0.861538	+	0.507693i	$0.169501\pi$
$8$	− 55.9116i	− 0.873618i
$9$	0	0
$10$	−104.890	−1.04890
$11$	− 31.3181i	− 0.258827i	−0.991591	−	0.129414i	$-0.958691\pi$
$11$	− 31.3181i	− 0.258827i	0.991591	−	0.129414i	$-0.0413095\pi$
$12$	0	0
$13$	43.8032	0.259191	0.129595	−	0.991567i	$-0.458632\pi$
$13$	43.8032	0.259191	0.129595	+	0.991567i	$0.458632\pi$
$14$	539.022i	2.75011i
$15$	0	0
$16$	−39.1754	−0.153029
$17$	− 243.216i	− 0.841578i	−0.907158	−	0.420789i	$-0.861753\pi$
$17$	− 243.216i	− 0.841578i	0.907158	−	0.420789i	$-0.138247\pi$
$18$	0	0
$19$	602.103	1.66788	0.833938	−	0.551858i	$-0.186081\pi$
$19$	602.103	1.66788	0.833938	+	0.551858i	$0.186081\pi$
$20$	− 406.761i	− 1.01690i
$21$	0	0
$22$	199.940	0.413100
$23$	− 401.054i	− 0.758135i	−0.925369	−	0.379068i	$-0.876245\pi$
$23$	− 401.054i	− 0.758135i	0.925369	−	0.379068i	$-0.123755\pi$
$24$	0	0
$25$	355.068	0.568108
$26$	279.648i	0.413680i
$27$	0	0
$28$	−2090.32	−2.66623
$29$	1577.03i	1.87518i	0.347742	+	0.937590i	$0.386949\pi$
$29$	1577.03i	1.87518i	−0.347742	+	0.937590i	$0.613051\pi$
$30$	0	0
$31$	306.821	0.319273	0.159636	−	0.987176i	$-0.448968\pi$
$31$	306.821	0.319273	0.159636	+	0.987176i	$0.448968\pi$
$32$	− 1144.69i	− 1.11786i
$33$	0	0
$34$	1552.74	1.34320
$35$	1387.16i	1.13238i
$36$	0	0
$37$	2509.88	1.83337	0.916683	−	0.399614i	$-0.130856\pi$
$37$	2509.88	1.83337	0.916683	+	0.399614i	$0.130856\pi$
$38$	3843.94i	2.66201i
$39$	0	0
$40$	918.606	0.574128
$41$	− 1680.84i	− 0.999907i	−0.866052	−	0.499953i	$-0.833350\pi$
$41$	− 1680.84i	− 0.999907i	0.866052	−	0.499953i	$-0.166650\pi$
$42$	0	0
$43$	2973.39	1.60811	0.804055	−	0.594555i	$-0.202672\pi$
$43$	2973.39	1.60811	0.804055	+	0.594555i	$0.202672\pi$
$44$	775.368i	0.400500i
$45$	0	0
$46$	2560.40	1.21002
$47$	− 890.845i	− 0.403280i	−0.979460	−	0.201640i	$-0.935373\pi$
$47$	− 890.845i	− 0.403280i	0.979460	−	0.201640i	$-0.0646271\pi$
$48$	0	0
$49$	4727.55	1.96899
$50$	2266.82i	0.906727i
$51$	0	0
$52$	−1084.47	−0.401062
$53$	4496.31i	1.60068i	0.599546	+	0.800340i	$0.295348\pi$
$53$	4496.31i	1.60068i	−0.599546	+	0.800340i	$0.704652\pi$
$54$	0	0
$55$	514.544	0.170097
$56$	− 4720.66i	− 1.50531i
$57$	0	0
$58$	−10068.0	−2.99288
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	−4996.33	−1.34274	−0.671369	−	0.741124i	$-0.734293\pi$
$61$	−4996.33	−1.34274	−0.671369	+	0.741124i	$0.734293\pi$
$62$	1958.80i	0.509574i
$63$	0	0
$64$	6681.09	1.63113
$65$	719.670i	0.170336i
$66$	0	0
$67$	−5524.56	−1.23069	−0.615344	−	0.788259i	$-0.710983\pi$
$67$	−5524.56	−1.23069	−0.615344	+	0.788259i	$0.710983\pi$
$68$	6021.50i	1.30223i
$69$	0	0
$70$	−8855.92	−1.80733
$71$	− 6512.64i	− 1.29193i	−0.763365	−	0.645967i	$-0.776454\pi$
$71$	− 6512.64i	− 1.29193i	0.763365	−	0.645967i	$-0.223546\pi$
$72$	0	0
$73$	1300.13	0.243973	0.121986	−	0.992532i	$-0.461074\pi$
$73$	1300.13	0.243973	0.121986	+	0.992532i	$0.461074\pi$
$74$	16023.5i	2.92614i
$75$	0	0
$76$	−14906.8	−2.58081
$77$	− 2644.21i	− 0.445979i
$78$	0	0
$79$	−7102.80	−1.13809	−0.569043	−	0.822308i	$-0.692686\pi$
$79$	−7102.80	−1.13809	−0.569043	+	0.822308i	$0.692686\pi$
$80$	− 643.636i	− 0.100568i
$81$	0	0
$82$	10730.8	1.59590
$83$	7734.10i	1.12267i	0.827587	+	0.561337i	$0.189713\pi$
$83$	7734.10i	1.12267i	−0.827587	+	0.561337i	$0.810287\pi$
$84$	0	0
$85$	3995.95	0.553072
$86$	18982.7i	2.56662i
$87$	0	0
$88$	−1751.04	−0.226116
$89$	− 9381.04i	− 1.18433i	−0.805819	−	0.592163i	$-0.798274\pi$
$89$	− 9381.04i	− 1.18433i	0.805819	−	0.592163i	$-0.201726\pi$
$90$	0	0
$91$	3698.34	0.446606
$92$	9929.21i	1.17311i
$93$	0	0
$94$	5687.32	0.643653
$95$	9892.33i	1.09610i
$96$	0	0
$97$	−16857.7	−1.79165	−0.895827	−	0.444403i	$-0.853416\pi$
$97$	−16857.7	−1.79165	−0.895827	+	0.444403i	$0.853416\pi$
$98$	30181.6i	3.14260i
$99$	0	0
$100$	−8790.70	−0.879070
$101$	− 1394.85i	− 0.136736i	−0.997660	−	0.0683681i	$-0.978221\pi$
$101$	− 1394.85i	− 0.136736i	0.997660	−	0.0683681i	$-0.0217792\pi$
$102$	0	0
$103$	−6438.86	−0.606924	−0.303462	−	0.952843i	$-0.598143\pi$
$103$	−6438.86	−0.606924	−0.303462	+	0.952843i	$0.598143\pi$
$104$	− 2449.11i	− 0.226434i
$105$	0	0
$106$	−28705.3	−2.55476
$107$	15170.7i	1.32507i	0.749032	+	0.662533i	$0.230519\pi$
$107$	15170.7i	1.32507i	−0.749032	+	0.662533i	$0.769481\pi$
$108$	0	0
$109$	13352.8	1.12388	0.561938	−	0.827179i	$-0.310056\pi$
$109$	13352.8	1.12388	0.561938	+	0.827179i	$0.310056\pi$
$110$	3284.94i	0.271483i
$111$	0	0
$112$	−3307.60	−0.263680
$113$	− 12491.9i	− 0.978299i	−0.872200	−	0.489150i	$-0.837307\pi$
$113$	− 12491.9i	− 0.978299i	0.872200	−	0.489150i	$-0.162693\pi$
$114$	0	0
$115$	6589.16	0.498235
$116$	− 39043.8i	− 2.90159i
$117$	0	0
$118$	2893.23	0.207788
$119$	− 20534.9i	− 1.45010i
$120$	0	0
$121$	13660.2	0.933009
$122$	− 31897.5i	− 2.14307i
$123$	0	0
$124$	−7596.22	−0.494031
$125$	16102.1i	1.03054i
$126$	0	0
$127$	6207.75	0.384881	0.192441	−	0.981309i	$-0.438360\pi$
$127$	6207.75	0.384881	0.192441	+	0.981309i	$0.438360\pi$
$128$	24338.3i	1.48549i
$129$	0	0
$130$	−4594.51	−0.271864
$131$	6173.95i	0.359766i	0.983688	+	0.179883i	$0.0575720\pi$
$131$	6173.95i	0.359766i	−0.983688	+	0.179883i	$0.942428\pi$
$132$	0	0
$133$	50836.0	2.87388
$134$	− 35269.8i	− 1.96424i
$135$	0	0
$136$	−13598.6	−0.735218
$137$	17557.9i	0.935473i	0.883868	+	0.467736i	$0.154930\pi$
$137$	17557.9i	0.935473i	−0.883868	+	0.467736i	$0.845070\pi$
$138$	0	0
$139$	−34957.5	−1.80930	−0.904650	−	0.426155i	$-0.859868\pi$
$139$	−34957.5	−1.80930	−0.904650	+	0.426155i	$0.859868\pi$
$140$	− 34343.2i	− 1.75220i
$141$	0	0
$142$	41577.9	2.06199
$143$	− 1371.83i	− 0.0670856i
$144$	0	0
$145$	−25909.9	−1.23234
$146$	8300.28i	0.389392i
$147$	0	0
$148$	−62139.1	−2.83689
$149$	− 5530.20i	− 0.249097i	−0.992214	−	0.124549i	$-0.960252\pi$
$149$	− 5530.20i	− 0.249097i	0.992214	−	0.124549i	$-0.0397483\pi$
$150$	0	0
$151$	−351.339	−0.0154089	−0.00770446	−	0.999970i	$-0.502452\pi$
$151$	−351.339	−0.0154089	−0.00770446	+	0.999970i	$0.502452\pi$
$152$	− 33664.5i	− 1.45709i
$153$	0	0
$154$	16881.1	0.711803
$155$	5040.95i	0.209821i
$156$	0	0
$157$	14649.0	0.594304	0.297152	−	0.954830i	$-0.403963\pi$
$157$	14649.0	0.594304	0.297152	+	0.954830i	$0.403963\pi$
$158$	− 45345.6i	− 1.81644i
$159$	0	0
$160$	18806.8	0.734640
$161$	− 33861.3i	− 1.30633i
$162$	0	0
$163$	−14721.4	−0.554082	−0.277041	−	0.960858i	$-0.589354\pi$
$163$	−14721.4	−0.554082	−0.277041	+	0.960858i	$0.589354\pi$
$164$	41614.0i	1.54722i
$165$	0	0
$166$	−49375.9	−1.79184
$167$	1511.22i	0.0541870i	0.999633	+	0.0270935i	$0.00862518\pi$
$167$	1511.22i	0.0541870i	−0.999633	+	0.0270935i	$0.991375\pi$
$168$	0	0
$169$	−26642.3	−0.932820
$170$	25510.9i	0.882729i
$171$	0	0
$172$	−73614.8	−2.48833
$173$	33955.4i	1.13453i	0.823535	+	0.567266i	$0.191999\pi$
$173$	33955.4i	1.13453i	−0.823535	+	0.567266i	$0.808001\pi$
$174$	0	0
$175$	29978.6	0.978894
$176$	1226.90i	0.0396080i
$177$	0	0
$178$	59890.3	1.89024
$179$	− 21231.1i	− 0.662623i	−0.943522	−	0.331311i	$-0.892509\pi$
$179$	− 21231.1i	− 0.662623i	0.943522	−	0.331311i	$-0.107491\pi$
$180$	0	0
$181$	1207.76	0.0368657	0.0184328	−	0.999830i	$-0.494132\pi$
$181$	1207.76	0.0368657	0.0184328	+	0.999830i	$0.494132\pi$
$182$	23610.9i	0.712803i
$183$	0	0
$184$	−22423.5	−0.662321
$185$	41236.3i	1.20486i
$186$	0	0
$187$	−7617.06	−0.217823
$188$	22055.4i	0.624021i
$189$	0	0
$190$	−63154.5	−1.74943
$191$	− 12946.8i	− 0.354892i	−0.984131	−	0.177446i	$-0.943216\pi$
$191$	− 12946.8i	− 0.354892i	0.984131	−	0.177446i	$-0.0567836\pi$
$192$	0	0
$193$	7860.82	0.211034	0.105517	−	0.994417i	$-0.466350\pi$
$193$	7860.82	0.211034	0.105517	+	0.994417i	$0.466350\pi$
$194$	− 107622.i	− 2.85956i
$195$	0	0
$196$	−117044.	−3.04675
$197$	− 36685.4i	− 0.945281i	−0.881255	−	0.472640i	$-0.843301\pi$
$197$	− 36685.4i	− 0.945281i	0.881255	−	0.472640i	$-0.156699\pi$
$198$	0	0
$199$	−70653.6	−1.78414	−0.892069	−	0.451900i	$-0.850746\pi$
$199$	−70653.6	−1.78414	−0.892069	+	0.451900i	$0.850746\pi$
$200$	− 19852.4i	− 0.496310i
$201$	0	0
$202$	8904.96	0.218237
$203$	133150.i	3.23108i
$204$	0	0
$205$	27615.6	0.657123
$206$	− 41106.9i	− 0.968679i
$207$	0	0
$208$	−1716.01	−0.0396636
$209$	− 18856.7i	− 0.431692i
$210$	0	0
$211$	14850.2	0.333555	0.166777	−	0.985995i	$-0.446664\pi$
$211$	14850.2	0.333555	0.166777	+	0.985995i	$0.446664\pi$
$212$	− 111319.i	− 2.47683i
$213$	0	0
$214$	−96852.5	−2.11487
$215$	48851.7i	1.05682i
$216$	0	0
$217$	25905.1	0.550132
$218$	85246.5i	1.79376i
$219$	0	0
$220$	−12739.0	−0.263202
$221$	− 10653.7i	− 0.218129i
$222$	0	0
$223$	34364.0	0.691026	0.345513	−	0.938414i	$-0.387705\pi$
$223$	34364.0	0.691026	0.345513	+	0.938414i	$0.387705\pi$
$224$	− 96646.9i	− 1.92616i
$225$	0	0
$226$	79750.6	1.56141
$227$	10950.2i	0.212506i	0.994339	+	0.106253i	$0.0338853\pi$
$227$	10950.2i	0.212506i	−0.994339	+	0.106253i	$0.966115\pi$
$228$	0	0
$229$	29201.6	0.556846	0.278423	−	0.960459i	$-0.410188\pi$
$229$	29201.6	0.556846	0.278423	+	0.960459i	$0.410188\pi$
$230$	42066.4i	0.795206i
$231$	0	0
$232$	88174.1	1.63819
$233$	− 8886.56i	− 0.163690i	−0.996645	−	0.0818449i	$-0.973919\pi$
$233$	− 8886.56i	− 0.163690i	0.996645	−	0.0818449i	$-0.0260812\pi$
$234$	0	0
$235$	14636.2	0.265029
$236$	11219.9i	0.201450i
$237$	0	0
$238$	131099.	2.31443
$239$	90627.6i	1.58659i	0.608837	+	0.793295i	$0.291636\pi$
$239$	90627.6i	1.58659i	−0.608837	+	0.793295i	$0.708364\pi$
$240$	0	0
$241$	53499.6	0.921120	0.460560	−	0.887629i	$-0.347649\pi$
$241$	53499.6	0.921120	0.460560	+	0.887629i	$0.347649\pi$
$242$	87209.1i	1.48912i
$243$	0	0
$244$	123698.	2.07770
$245$	77671.9i	1.29399i
$246$	0	0
$247$	26374.1	0.432298
$248$	− 17154.9i	− 0.278923i
$249$	0	0
$250$	−102799.	−1.64478
$251$	− 39389.1i	− 0.625214i	−0.949883	−	0.312607i	$-0.898798\pi$
$251$	− 39389.1i	− 0.625214i	0.949883	−	0.312607i	$-0.101202\pi$
$252$	0	0
$253$	−12560.2	−0.196226
$254$	39631.4i	0.614288i
$255$	0	0
$256$	−48482.9	−0.739791
$257$	− 81869.3i	− 1.23952i	−0.784790	−	0.619762i	$-0.787229\pi$
$257$	− 81869.3i	− 1.23952i	0.784790	−	0.619762i	$-0.212771\pi$
$258$	0	0
$259$	211911.	3.15903
$260$	− 17817.5i	− 0.263572i
$261$	0	0
$262$	−39415.6	−0.574204
$263$	− 71572.4i	− 1.03475i	−0.855760	−	0.517373i	$-0.826910\pi$
$263$	− 71572.4i	− 1.03475i	0.855760	−	0.517373i	$-0.173090\pi$
$264$	0	0
$265$	−73872.7	−1.05194
$266$	324547.i	4.58684i
$267$	0	0
$268$	136776.	1.90432
$269$	98180.3i	1.35681i	0.734687	+	0.678406i	$0.237329\pi$
$269$	98180.3i	1.35681i	−0.734687	+	0.678406i	$0.762671\pi$
$270$	0	0
$271$	−2432.69	−0.0331245	−0.0165622	−	0.999863i	$-0.505272\pi$
$271$	−2432.69	−0.0331245	−0.0165622	+	0.999863i	$0.505272\pi$
$272$	9528.08i	0.128786i
$273$	0	0
$274$	−112093.	−1.49306
$275$	− 11120.0i	− 0.147042i
$276$	0	0
$277$	−23579.7	−0.307312	−0.153656	−	0.988124i	$-0.549105\pi$
$277$	−23579.7	−0.307312	−0.153656	+	0.988124i	$0.549105\pi$
$278$	− 223175.i	− 2.88773i
$279$	0	0
$280$	77558.6	0.989267
$281$	− 45120.5i	− 0.571428i	−0.958315	−	0.285714i	$-0.907769\pi$
$281$	− 45120.5i	− 0.571428i	0.958315	−	0.285714i	$-0.0922307\pi$
$282$	0	0
$283$	−32901.3	−0.410809	−0.205404	−	0.978677i	$-0.565851\pi$
$283$	−32901.3	−0.410809	−0.205404	+	0.978677i	$0.565851\pi$
$284$	161239.i	1.99909i
$285$	0	0
$286$	8758.04	0.107072
$287$	− 141915.i	− 1.72292i
$288$	0	0
$289$	24366.9	0.291746
$290$	− 165414.i	− 1.96687i
$291$	0	0
$292$	−32188.4	−0.377515
$293$	156217.i	1.81968i	0.414965	+	0.909838i	$0.363794\pi$
$293$	156217.i	1.81968i	−0.414965	+	0.909838i	$0.636206\pi$
$294$	0	0
$295$	7445.70	0.0855581
$296$	− 140331.i	− 1.60166i
$297$	0	0
$298$	35305.8	0.397570
$299$	− 17567.4i	− 0.196502i
$300$	0	0
$301$	251046.	2.77090
$302$	− 2243.01i	− 0.0245933i
$303$	0	0
$304$	−23587.6	−0.255233
$305$	− 82087.7i	− 0.882426i
$306$	0	0
$307$	−144566.	−1.53387	−0.766936	−	0.641723i	$-0.778220\pi$
$307$	−144566.	−1.53387	−0.766936	+	0.641723i	$0.778220\pi$
$308$	65464.9i	0.690092i
$309$	0	0
$310$	−32182.4	−0.334884
$311$	41579.4i	0.429890i	0.976626	+	0.214945i	$0.0689573\pi$
$311$	41579.4i	0.429890i	−0.976626	+	0.214945i	$0.931043\pi$
$312$	0	0
$313$	185290.	1.89132	0.945658	−	0.325164i	$-0.105420\pi$
$313$	185290.	1.89132	0.945658	+	0.325164i	$0.105420\pi$
$314$	93521.9i	0.948537i
$315$	0	0
$316$	175850.	1.76103
$317$	125291.i	1.24681i	0.781900	+	0.623404i	$0.214251\pi$
$317$	125291.i	1.24681i	−0.781900	+	0.623404i	$0.785749\pi$
$318$	0	0
$319$	49389.5	0.485348
$320$	109768.i	1.07195i
$321$	0	0
$322$	216177.	2.08496
$323$	− 146441.i	− 1.40365i
$324$	0	0
$325$	15553.1	0.147248
$326$	− 93984.1i	− 0.884340i
$327$	0	0
$328$	−93978.6	−0.873537
$329$	− 75214.7i	− 0.694882i
$330$	0	0
$331$	−157297.	−1.43570	−0.717851	−	0.696197i	$-0.754874\pi$
$331$	−157297.	−1.43570	−0.717851	+	0.696197i	$0.754874\pi$
$332$	− 191479.i	− 1.73718i
$333$	0	0
$334$	−9647.92	−0.0864850
$335$	− 90766.4i	− 0.808789i
$336$	0	0
$337$	186500.	1.64217	0.821085	−	0.570806i	$-0.193369\pi$
$337$	186500.	1.64217	0.821085	+	0.570806i	$0.193369\pi$
$338$	− 170089.i	− 1.48882i
$339$	0	0
$340$	−98931.0	−0.855804
$341$	− 9609.05i	− 0.0826365i
$342$	0	0
$343$	196433.	1.66965
$344$	− 166247.i	− 1.40487i
$345$	0	0
$346$	−216778.	−1.81077
$347$	137926.i	1.14548i	0.819738	+	0.572738i	$0.194119\pi$
$347$	137926.i	1.14548i	−0.819738	+	0.572738i	$0.805881\pi$
$348$	0	0
$349$	57501.5	0.472094	0.236047	−	0.971742i	$-0.424148\pi$
$349$	57501.5	0.472094	0.236047	+	0.971742i	$0.424148\pi$
$350$	191389.i	1.56236i
$351$	0	0
$352$	−35849.4	−0.289332
$353$	130353.i	1.04609i	0.852304	+	0.523046i	$0.175204\pi$
$353$	130353.i	1.04609i	−0.852304	+	0.523046i	$0.824796\pi$
$354$	0	0
$355$	107000.	0.849039
$356$	232254.i	1.83258i
$357$	0	0
$358$	135543.	1.05758
$359$	− 45021.1i	− 0.349323i	−0.984629	−	0.174662i	$-0.944117\pi$
$359$	− 45021.1i	− 0.349323i	0.984629	−	0.174662i	$-0.0558831\pi$
$360$	0	0
$361$	232208.	1.78181
$362$	7710.54i	0.0588393i
$363$	0	0
$364$	−91562.9	−0.691061
$365$	21360.7i	0.160335i
$366$	0	0
$367$	−11975.5	−0.0889120	−0.0444560	−	0.999011i	$-0.514155\pi$
$367$	−11975.5	−0.0889120	−0.0444560	+	0.999011i	$0.514155\pi$
$368$	15711.4i	0.116016i
$369$	0	0
$370$	−263260.	−1.92301
$371$	379627.i	2.75809i
$372$	0	0
$373$	−22936.6	−0.164858	−0.0824292	−	0.996597i	$-0.526268\pi$
$373$	−22936.6	−0.164858	−0.0824292	+	0.996597i	$0.526268\pi$
$374$	− 48628.7i	− 0.347656i
$375$	0	0
$376$	−49808.6	−0.352313
$377$	69078.9i	0.486030i
$378$	0	0
$379$	45087.7	0.313892	0.156946	−	0.987607i	$-0.449835\pi$
$379$	45087.7	0.313892	0.156946	+	0.987607i	$0.449835\pi$
$380$	− 244912.i	− 1.69607i
$381$	0	0
$382$	82654.9	0.566425
$383$	− 73353.7i	− 0.500063i	−0.968238	−	0.250031i	$-0.919559\pi$
$383$	− 73353.7i	− 0.500063i	0.968238	−	0.250031i	$-0.0804409\pi$
$384$	0	0
$385$	43443.3	0.293091
$386$	50184.9i	0.336821i
$387$	0	0
$388$	417359.	2.77234
$389$	− 82728.6i	− 0.546709i	−0.961913	−	0.273354i	$-0.911867\pi$
$389$	− 82728.6i	− 0.546709i	0.961913	−	0.273354i	$-0.0881332\pi$
$390$	0	0
$391$	−97542.7	−0.638030
$392$	− 264325.i	− 1.72015i
$393$	0	0
$394$	234206.	1.50871
$395$	− 116696.i	− 0.747933i
$396$	0	0
$397$	222421.	1.41122	0.705609	−	0.708602i	$-0.250674\pi$
$397$	222421.	1.41122	0.705609	+	0.708602i	$0.250674\pi$
$398$	− 451066.i	− 2.84757i
$399$	0	0
$400$	−13909.9	−0.0869369
$401$	− 76062.6i	− 0.473023i	−0.971629	−	0.236512i	$-0.923996\pi$
$401$	− 76062.6i	− 0.473023i	0.971629	−	0.236512i	$-0.0760041\pi$
$402$	0	0
$403$	13439.8	0.0827526
$404$	34533.4i	0.211581i
$405$	0	0
$406$	−850052.	−5.15695
$407$	− 78604.6i	− 0.474525i
$408$	0	0
$409$	−106082.	−0.634153	−0.317077	−	0.948400i	$-0.602701\pi$
$409$	−106082.	−0.634153	−0.317077	+	0.948400i	$0.602701\pi$
$410$	176303.i	1.04880i
$411$	0	0
$412$	159412.	0.939133
$413$	− 38263.0i	− 0.224325i
$414$	0	0
$415$	−127068.	−0.737804
$416$	− 50141.0i	− 0.289739i
$417$	0	0
$418$	120385.	0.689000
$419$	− 182880.i	− 1.04169i	−0.853652	−	0.520844i	$-0.825617\pi$
$419$	− 182880.i	− 1.04169i	0.853652	−	0.520844i	$-0.174383\pi$
$420$	0	0
$421$	−260085.	−1.46741	−0.733704	−	0.679469i	$-0.762210\pi$
$421$	−260085.	−1.46741	−0.733704	+	0.679469i	$0.762210\pi$
$422$	94806.4i	0.532369i
$423$	0	0
$424$	251396.	1.39838
$425$	− 86358.2i	− 0.478108i
$426$	0	0
$427$	−421844.	−2.31364
$428$	− 375593.i	− 2.05036i
$429$	0	0
$430$	−311878.	−1.68674
$431$	− 13132.3i	− 0.0706946i	−0.999375	−	0.0353473i	$-0.988746\pi$
$431$	− 13132.3i	− 0.0706946i	0.999375	−	0.0353473i	$-0.0112537\pi$
$432$	0	0
$433$	−250937.	−1.33841	−0.669204	−	0.743079i	$-0.733365\pi$
$433$	−250937.	−1.33841	−0.669204	+	0.743079i	$0.733365\pi$
$434$	165383.i	0.878035i
$435$	0	0
$436$	−330585.	−1.73904
$437$	− 241476.i	− 1.26448i
$438$	0	0
$439$	−151603.	−0.786644	−0.393322	−	0.919401i	$-0.628674\pi$
$439$	−151603.	−0.786644	−0.393322	+	0.919401i	$0.628674\pi$
$440$	− 28769.0i	− 0.148600i
$441$	0	0
$442$	68014.9	0.348145
$443$	− 254620.i	− 1.29743i	−0.761030	−	0.648716i	$-0.775306\pi$
$443$	− 254620.i	− 1.29743i	0.761030	−	0.648716i	$-0.224694\pi$
$444$	0	0
$445$	154127.	0.778320
$446$	219386.i	1.10291i
$447$	0	0
$448$	564090.	2.81056
$449$	211564.i	1.04942i	0.851281	+	0.524710i	$0.175826\pi$
$449$	211564.i	1.04942i	−0.851281	+	0.524710i	$0.824174\pi$
$450$	0	0
$451$	−52640.8	−0.258803
$452$	309272.i	1.51379i
$453$	0	0
$454$	−69908.2	−0.339169
$455$	60762.3i	0.293502i
$456$	0	0
$457$	−108413.	−0.519098	−0.259549	−	0.965730i	$-0.583574\pi$
$457$	−108413.	−0.519098	−0.259549	+	0.965730i	$0.583574\pi$
$458$	186428.i	0.888752i
$459$	0	0
$460$	−163133.	−0.770951
$461$	225878.i	1.06285i	0.847105	+	0.531426i	$0.178344\pi$
$461$	225878.i	1.06285i	−0.847105	+	0.531426i	$0.821656\pi$
$462$	0	0
$463$	−146905.	−0.685290	−0.342645	−	0.939465i	$-0.611323\pi$
$463$	−146905.	−0.685290	−0.342645	+	0.939465i	$0.611323\pi$
$464$	− 61780.6i	− 0.286957i
$465$	0	0
$466$	56733.4	0.261257
$467$	− 170884.i	− 0.783551i	−0.920061	−	0.391776i	$-0.871861\pi$
$467$	− 170884.i	− 0.783551i	0.920061	−	0.391776i	$-0.128139\pi$
$468$	0	0
$469$	−466443.	−2.12057
$470$	93440.5i	0.422999i
$471$	0	0
$472$	−25338.4	−0.113735
$473$	− 93121.0i	− 0.416222i
$474$	0	0
$475$	213788.	0.947535
$476$	508400.i	2.24384i
$477$	0	0
$478$	−578584.	−2.53227
$479$	− 338877.i	− 1.47697i	−0.674272	−	0.738483i	$-0.735543\pi$
$479$	− 338877.i	− 1.47697i	0.674272	−	0.738483i	$-0.264457\pi$
$480$	0	0
$481$	109941.	0.475192
$482$	341551.i	1.47015i
$483$	0	0
$484$	−338196.	−1.44370
$485$	− 276965.i	− 1.17745i
$486$	0	0
$487$	222038.	0.936202	0.468101	−	0.883675i	$-0.344938\pi$
$487$	222038.	0.936202	0.468101	+	0.883675i	$0.344938\pi$
$488$	279352.i	1.17304i
$489$	0	0
$490$	−495872.	−2.06527
$491$	129659.i	0.537825i	0.963165	+	0.268912i	$0.0866642\pi$
$491$	129659.i	0.537825i	−0.963165	+	0.268912i	$0.913336\pi$
$492$	0	0
$493$	383558.	1.57811
$494$	168377.i	0.689968i
$495$	0	0
$496$	−12019.8	−0.0488579
$497$	− 549867.i	− 2.22610i
$498$	0	0
$499$	255872.	1.02759	0.513797	−	0.857912i	$-0.328239\pi$
$499$	255872.	1.02759	0.513797	+	0.857912i	$0.328239\pi$
$500$	− 398654.i	− 1.59462i
$501$	0	0
$502$	251467.	0.997870
$503$	331218.i	1.30912i	0.756012	+	0.654558i	$0.227145\pi$
$503$	331218.i	1.30912i	−0.756012	+	0.654558i	$0.772855\pi$
$504$	0	0
$505$	22916.8	0.0898610
$506$	− 80186.8i	− 0.313186i
$507$	0	0
$508$	−153690.	−0.595551
$509$	− 320646.i	− 1.23763i	−0.785538	−	0.618814i	$-0.787614\pi$
$509$	− 320646.i	− 1.23763i	0.785538	−	0.618814i	$-0.212386\pi$
$510$	0	0
$511$	109771.	0.420384
$512$	79889.2i	0.304753i
$513$	0	0
$514$	522669.	1.97834
$515$	− 105788.i	− 0.398861i
$516$	0	0
$517$	−27899.6	−0.104380
$518$	1.35288e6i	5.04196i
$519$	0	0
$520$	40237.9	0.148809
$521$	− 99104.2i	− 0.365104i	−0.983196	−	0.182552i	$-0.941564\pi$
$521$	− 99104.2i	− 0.365104i	0.983196	−	0.182552i	$-0.0584358\pi$
$522$	0	0
$523$	318725.	1.16523	0.582617	−	0.812747i	$-0.302029\pi$
$523$	318725.	1.16523	0.582617	+	0.812747i	$0.302029\pi$
$524$	− 152854.i	− 0.556689i
$525$	0	0
$526$	456931.	1.65150
$527$	− 74623.9i	− 0.268693i
$528$	0	0
$529$	118997.	0.425231
$530$	− 471617.i	− 1.67895i
$531$	0	0
$532$	−1.25859e6	−4.44694
$533$	− 73626.4i	− 0.259167i
$534$	0	0
$535$	−249249.	−0.870813
$536$	308887.i	1.07515i
$537$	0	0
$538$	−626801.	−2.16553
$539$	− 148058.i	− 0.509629i
$540$	0	0
$541$	−507671.	−1.73455	−0.867277	−	0.497826i	$-0.834132\pi$
$541$	−507671.	−1.73455	−0.867277	+	0.497826i	$0.834132\pi$
$542$	− 15530.8i	− 0.0528682i
$543$	0	0
$544$	−278407.	−0.940766
$545$	219381.i	0.738594i
$546$	0	0
$547$	−216288.	−0.722865	−0.361432	−	0.932398i	$-0.617712\pi$
$547$	−216288.	−0.722865	−0.361432	+	0.932398i	$0.617712\pi$
$548$	− 434695.i	− 1.44752i
$549$	0	0
$550$	70992.4	0.234686
$551$	949533.i	3.12757i
$552$	0	0
$553$	−599694.	−1.96101
$554$	− 150537.i	− 0.490484i
$555$	0	0
$556$	865471.	2.79965
$557$	− 500929.i	− 1.61460i	−0.590139	−	0.807302i	$-0.700927\pi$
$557$	− 500929.i	− 1.61460i	0.590139	−	0.807302i	$-0.299073\pi$
$558$	0	0
$559$	130244.	0.416807
$560$	− 54342.7i	− 0.173287i
$561$	0	0
$562$	288058.	0.912025
$563$	− 290220.i	− 0.915610i	−0.889053	−	0.457805i	$-0.848636\pi$
$563$	− 290220.i	− 0.915610i	0.889053	−	0.457805i	$-0.151364\pi$
$564$	0	0
$565$	205237.	0.642923
$566$	− 210048.i	− 0.655670i
$567$	0	0
$568$	−364132.	−1.12866
$569$	− 394021.i	− 1.21701i	−0.793549	−	0.608506i	$-0.791769\pi$
$569$	− 394021.i	− 1.21701i	0.793549	−	0.608506i	$-0.208231\pi$
$570$	0	0
$571$	24021.1	0.0736750	0.0368375	−	0.999321i	$-0.488272\pi$
$571$	24021.1	0.0736750	0.0368375	+	0.999321i	$0.488272\pi$
$572$	33963.6i	0.103806i
$573$	0	0
$574$	906011.	2.74985
$575$	− 142401.i	− 0.430703i
$576$	0	0
$577$	21255.5	0.0638441	0.0319220	−	0.999490i	$-0.489837\pi$
$577$	21255.5	0.0638441	0.0319220	+	0.999490i	$0.489837\pi$
$578$	155563.i	0.465640i
$579$	0	0
$580$	641474.	1.90688
$581$	652996.i	1.93445i
$582$	0	0
$583$	140816.	0.414299
$584$	− 72692.4i	− 0.213139i
$585$	0	0
$586$	−997320.	−2.90429
$587$	− 67625.0i	− 0.196260i	−0.995174	−	0.0981298i	$-0.968714\pi$
$587$	− 67625.0i	− 0.196260i	0.995174	−	0.0981298i	$-0.0312860\pi$
$588$	0	0
$589$	184738.	0.532508
$590$	47534.7i	0.136555i
$591$	0	0
$592$	−98325.4	−0.280558
$593$	2627.19i	0.00747105i	0.999993	+	0.00373552i	$0.00118906\pi$
$593$	2627.19i	0.00747105i	−0.999993	+	0.00373552i	$0.998811\pi$
$594$	0	0
$595$	337381.	0.952986
$596$	136916.i	0.385444i
$597$	0	0
$598$	112154.	0.313626
$599$	− 114040.i	− 0.317836i	−0.987292	−	0.158918i	$-0.949199\pi$
$599$	− 114040.i	− 0.317836i	0.987292	−	0.158918i	$-0.0508006\pi$
$600$	0	0
$601$	−159026.	−0.440270	−0.220135	−	0.975469i	$-0.570650\pi$
$601$	−159026.	−0.440270	−0.220135	+	0.975469i	$0.570650\pi$
$602$	1.60272e6i	4.42248i
$603$	0	0
$604$	8698.38	0.0238432
$605$	224431.i	0.613159i
$606$	0	0
$607$	225627.	0.612369	0.306185	−	0.951972i	$-0.400948\pi$
$607$	225627.	0.612369	0.306185	+	0.951972i	$0.400948\pi$
$608$	− 689220.i	− 1.86445i
$609$	0	0
$610$	524063.	1.40839
$611$	− 39021.9i	− 0.104526i
$612$	0	0
$613$	273471.	0.727764	0.363882	−	0.931445i	$-0.381451\pi$
$613$	273471.	0.727764	0.363882	+	0.931445i	$0.381451\pi$
$614$	− 922936.i	− 2.44813i
$615$	0	0
$616$	−147842.	−0.389615
$617$	493931.i	1.29747i	0.761016	+	0.648733i	$0.224701\pi$
$617$	493931.i	1.29747i	−0.761016	+	0.648733i	$0.775299\pi$
$618$	0	0
$619$	126364.	0.329793	0.164897	−	0.986311i	$-0.447271\pi$
$619$	126364.	0.329793	0.164897	+	0.986311i	$0.447271\pi$
$620$	− 124803.i	− 0.324670i
$621$	0	0
$622$	−265451.	−0.686125
$623$	− 792048.i	− 2.04068i
$624$	0	0
$625$	−42634.5	−0.109144
$626$	1.18293e6i	3.01863i
$627$	0	0
$628$	−362677.	−0.919604
$629$	− 610443.i	− 1.54292i
$630$	0	0
$631$	60850.8	0.152830	0.0764148	−	0.997076i	$-0.475653\pi$
$631$	60850.8	0.152830	0.0764148	+	0.997076i	$0.475653\pi$
$632$	397128.i	0.994253i
$633$	0	0
$634$	−799878.	−1.98996
$635$	101991.i	0.252938i
$636$	0	0
$637$	207082.	0.510345
$638$	315311.i	0.774637i
$639$	0	0
$640$	−399869.	−0.976244
$641$	463957.i	1.12918i	0.825373	+	0.564588i	$0.190965\pi$
$641$	463957.i	1.12918i	−0.825373	+	0.564588i	$0.809035\pi$
$642$	0	0
$643$	−118531.	−0.286689	−0.143345	−	0.989673i	$-0.545786\pi$
$643$	−118531.	−0.286689	−0.143345	+	0.989673i	$0.545786\pi$
$644$	838331.i	2.02136i
$645$	0	0
$646$	934908.	2.24029
$647$	814030.i	1.94461i	0.233722	+	0.972304i	$0.424910\pi$
$647$	814030.i	1.94461i	−0.233722	+	0.972304i	$0.575090\pi$
$648$	0	0
$649$	−14193.0	−0.0336964
$650$	99294.0i	0.235015i
$651$	0	0
$652$	364470.	0.857366
$653$	− 671405.i	− 1.57456i	−0.616599	−	0.787278i	$-0.711490\pi$
$653$	− 671405.i	− 1.57456i	0.616599	−	0.787278i	$-0.288510\pi$
$654$	0	0
$655$	−101436.	−0.236433
$656$	65847.6i	0.153014i
$657$	0	0
$658$	480185.	1.10906
$659$	− 97060.7i	− 0.223497i	−0.993737	−	0.111749i	$-0.964355\pi$
$659$	− 97060.7i	− 0.223497i	0.993737	−	0.111749i	$-0.0356452\pi$
$660$	0	0
$661$	−332725.	−0.761522	−0.380761	−	0.924673i	$-0.624338\pi$
$661$	−332725.	−0.761522	−0.380761	+	0.924673i	$0.624338\pi$
$662$	− 1.00421e6i	− 2.29145i
$663$	0	0
$664$	432426.	0.980788
$665$	835217.i	1.88867i
$666$	0	0
$667$	632472.	1.42164
$668$	− 37414.5i	− 0.0838470i
$669$	0	0
$670$	579470.	1.29087
$671$	156475.i	0.347537i
$672$	0	0
$673$	−416464.	−0.919491	−0.459745	−	0.888051i	$-0.652059\pi$
$673$	−416464.	−0.919491	−0.459745	+	0.888051i	$0.652059\pi$
$674$	1.19065e6i	2.62098i
$675$	0	0
$676$	659605.	1.44341
$677$	261128.i	0.569739i	0.958566	+	0.284869i	$0.0919502\pi$
$677$	261128.i	0.569739i	−0.958566	+	0.284869i	$0.908050\pi$
$678$	0	0
$679$	−1.42331e6	−3.08716
$680$	− 223420.i	− 0.483174i
$681$	0	0
$682$	61346.0	0.131892
$683$	− 783856.i	− 1.68033i	−0.542331	−	0.840165i	$-0.682458\pi$
$683$	− 783856.i	− 1.68033i	0.542331	−	0.840165i	$-0.317542\pi$
$684$	0	0
$685$	−288469.	−0.614778
$686$	1.25406e6i	2.66484i
$687$	0	0
$688$	−116484.	−0.246087
$689$	196953.i	0.414882i
$690$	0	0
$691$	560121.	1.17307	0.586537	−	0.809922i	$-0.300491\pi$
$691$	560121.	1.17307	0.586537	+	0.809922i	$0.300491\pi$
$692$	− 840662.i	− 1.75553i
$693$	0	0
$694$	−880543.	−1.82823
$695$	− 574338.i	− 1.18904i
$696$	0	0
$697$	−408808.	−0.841500
$698$	367100.i	0.753484i
$699$	0	0
$700$	−742206.	−1.51471
$701$	6423.80i	0.0130724i	0.999979	+	0.00653621i	$0.00208056\pi$
$701$	6423.80i	0.0130724i	−0.999979	+	0.00653621i	$0.997919\pi$
$702$	0	0
$703$	1.51121e6	3.05783
$704$	− 209239.i	− 0.422180i
$705$	0	0
$706$	−832195.	−1.66961
$707$	− 117768.i	− 0.235607i
$708$	0	0
$709$	9597.63	0.0190929	0.00954644	−	0.999954i	$-0.496961\pi$
$709$	9597.63	0.0190929	0.00954644	+	0.999954i	$0.496961\pi$
$710$	683109.i	1.35511i
$711$	0	0
$712$	−524509.	−1.03465
$713$	− 123052.i	− 0.242052i
$714$	0	0
$715$	22538.7	0.0440876
$716$	525636.i	1.02532i
$717$	0	0
$718$	287423.	0.557536
$719$	244867.i	0.473666i	0.971550	+	0.236833i	$0.0761095\pi$
$719$	244867.i	0.473666i	−0.971550	+	0.236833i	$0.923891\pi$
$720$	0	0
$721$	−543638.	−1.04578
$722$	1.48246e6i	2.84385i
$723$	0	0
$724$	−29901.4	−0.0570446
$725$	559951.i	1.06531i
$726$	0	0
$727$	398236.	0.753481	0.376740	−	0.926319i	$-0.377045\pi$
$727$	398236.	0.753481	0.376740	+	0.926319i	$0.377045\pi$
$728$	− 206780.i	− 0.390163i
$729$	0	0
$730$	−136370.	−0.255902
$731$	− 723178.i	− 1.35335i
$732$	0	0
$733$	301885.	0.561866	0.280933	−	0.959727i	$-0.409356\pi$
$733$	301885.	0.561866	0.280933	+	0.959727i	$0.409356\pi$
$734$	− 76453.6i	− 0.141908i
$735$	0	0
$736$	−459081.	−0.847489
$737$	173019.i	0.318536i
$738$	0	0
$739$	211672.	0.387592	0.193796	−	0.981042i	$-0.437920\pi$
$739$	211672.	0.387592	0.193796	+	0.981042i	$0.437920\pi$
$740$	− 1.02092e6i	− 1.86436i
$741$	0	0
$742$	−2.42361e6	−4.40205
$743$	− 497892.i	− 0.901899i	−0.892549	−	0.450950i	$-0.851085\pi$
$743$	− 497892.i	− 0.901899i	0.892549	−	0.450950i	$-0.148915\pi$
$744$	0	0
$745$	90859.1	0.163703
$746$	− 146431.i	− 0.263122i
$747$	0	0
$748$	188582.	0.337052
$749$	1.28087e6i	2.28319i
$750$	0	0
$751$	−105180.	−0.186489	−0.0932446	−	0.995643i	$-0.529724\pi$
$751$	−105180.	−0.186489	−0.0932446	+	0.995643i	$0.529724\pi$
$752$	34899.2i	0.0617134i
$753$	0	0
$754$	−441012.	−0.775726
$755$	− 5772.36i	− 0.0101265i
$756$	0	0
$757$	−389680.	−0.680012	−0.340006	−	0.940423i	$-0.610429\pi$
$757$	−389680.	−0.680012	−0.340006	+	0.940423i	$0.610429\pi$
$758$	287848.i	0.500986i
$759$	0	0
$760$	553096.	0.957575
$761$	− 471821.i	− 0.814720i	−0.913268	−	0.407360i	$-0.866449\pi$
$761$	− 471821.i	− 0.814720i	0.913268	−	0.407360i	$-0.133551\pi$
$762$	0	0
$763$	1.12738e6	1.93652
$764$	320535.i	0.549148i
$765$	0	0
$766$	468304.	0.798123
$767$	− 19851.1i	− 0.0337438i
$768$	0	0
$769$	−90005.1	−0.152200	−0.0761000	−	0.997100i	$-0.524247\pi$
$769$	−90005.1	−0.152200	−0.0761000	+	0.997100i	$0.524247\pi$
$770$	277350.i	0.467786i
$771$	0	0
$772$	−194617.	−0.326547
$773$	344592.i	0.576695i	0.957526	+	0.288347i	$0.0931058\pi$
$773$	344592.i	0.576695i	−0.957526	+	0.288347i	$0.906894\pi$
$774$	0	0
$775$	108942.	0.181382
$776$	942539.i	1.56522i
$777$	0	0
$778$	528154.	0.872573
$779$	− 1.01204e6i	− 1.66772i
$780$	0	0
$781$	−203963.	−0.334388
$782$	− 622731.i	− 1.01833i
$783$	0	0
$784$	−185204.	−0.301313
$785$	240677.i	0.390567i
$786$	0	0
$787$	−572667.	−0.924597	−0.462298	−	0.886724i	$-0.652975\pi$
$787$	−572667.	−0.924597	−0.462298	+	0.886724i	$0.652975\pi$
$788$	908251.i	1.46269i
$789$	0	0
$790$	745010.	1.19374
$791$	− 1.05470e6i	− 1.68568i
$792$	0	0
$793$	−218855.	−0.348025
$794$	1.41997e6i	2.25237i
$795$	0	0
$796$	1.74923e6	2.76071
$797$	996132.i	1.56820i	0.620637	+	0.784098i	$0.286874\pi$
$797$	996132.i	1.56820i	−0.620637	+	0.784098i	$0.713126\pi$
$798$	0	0
$799$	−216668.	−0.339392
$800$	− 406442.i	− 0.635065i
$801$	0	0
$802$	485598.	0.754967
$803$	− 40717.6i	− 0.0631468i
$804$	0	0
$805$	556327.	0.858497
$806$	85801.9i	0.132077i
$807$	0	0
$808$	−77988.1	−0.119455
$809$	480734.i	0.734527i	0.930117	+	0.367263i	$0.119705\pi$
$809$	480734.i	0.734527i	−0.930117	+	0.367263i	$0.880295\pi$
$810$	0	0
$811$	−1.03911e6	−1.57986	−0.789930	−	0.613197i	$-0.789883\pi$
$811$	−1.03911e6	−1.57986	−0.789930	+	0.613197i	$0.789883\pi$
$812$	− 3.29649e6i	− 4.99966i
$813$	0	0
$814$	501826.	0.757364
$815$	− 241867.i	− 0.364134i
$816$	0	0
$817$	1.79029e6	2.68213
$818$	− 677246.i	− 1.01214i
$819$	0	0
$820$	−683702.	−1.01681
$821$	− 1.16284e6i	− 1.72518i	−0.505904	−	0.862590i	$-0.668841\pi$
$821$	− 1.16284e6i	− 1.72518i	0.505904	−	0.862590i	$-0.331159\pi$
$822$	0	0
$823$	−198095.	−0.292465	−0.146232	−	0.989250i	$-0.546715\pi$
$823$	−198095.	−0.292465	−0.146232	+	0.989250i	$0.546715\pi$
$824$	360007.i	0.530220i
$825$	0	0
$826$	244278.	0.358034
$827$	− 741023.i	− 1.08348i	−0.840546	−	0.541740i	$-0.817766\pi$
$827$	− 741023.i	− 1.08348i	0.840546	−	0.541740i	$-0.182234\pi$
$828$	0	0
$829$	−286940.	−0.417524	−0.208762	−	0.977966i	$-0.566943\pi$
$829$	−286940.	−0.417524	−0.208762	+	0.977966i	$0.566943\pi$
$830$	− 811227.i	− 1.17757i
$831$	0	0
$832$	292654.	0.422773
$833$	− 1.14982e6i	− 1.65706i
$834$	0	0
$835$	−24828.8	−0.0356109
$836$	466851.i	0.667984i
$837$	0	0
$838$	1.16754e6	1.66258
$839$	− 184330.i	− 0.261862i	−0.991391	−	0.130931i	$-0.958203\pi$
$839$	− 184330.i	− 0.261862i	0.991391	−	0.130931i	$-0.0417967\pi$
$840$	0	0
$841$	−1.77973e6	−2.51630
$842$	− 1.66043e6i	− 2.34205i
$843$	0	0
$844$	−367659.	−0.516131
$845$	− 437722.i	− 0.613035i
$846$	0	0
$847$	1.15334e6	1.60765
$848$	− 176145.i	− 0.244950i
$849$	0	0
$850$	551327.	0.763082
$851$	− 1.00660e6i	− 1.38994i
$852$	0	0
$853$	−1.38558e6	−1.90430	−0.952149	−	0.305635i	$-0.901131\pi$
$853$	−1.38558e6	−1.90430	−0.952149	+	0.305635i	$0.901131\pi$
$854$	− 2.69313e6i	− 3.69268i
$855$	0	0
$856$	848217.	1.15760
$857$	− 871483.i	− 1.18658i	−0.804989	−	0.593290i	$-0.797829\pi$
$857$	− 871483.i	− 1.18658i	0.804989	−	0.593290i	$-0.202171\pi$
$858$	0	0
$859$	−137699.	−0.186614	−0.0933071	−	0.995637i	$-0.529744\pi$
$859$	−137699.	−0.186614	−0.0933071	+	0.995637i	$0.529744\pi$
$860$	− 1.20946e6i	− 1.63529i
$861$	0	0
$862$	83839.0	0.112832
$863$	− 42360.1i	− 0.0568768i	−0.999596	−	0.0284384i	$-0.990947\pi$
$863$	− 42360.1i	− 0.0568768i	0.999596	−	0.0284384i	$-0.00905345\pi$
$864$	0	0
$865$	−557874.	−0.745597
$866$	− 1.60203e6i	− 2.13616i
$867$	0	0
$868$	−641355.	−0.851254
$869$	222446.i	0.294568i
$870$	0	0
$871$	−241994.	−0.318983
$872$	− 746574.i	− 0.981838i
$873$	0	0
$874$	1.54163e6	2.01816
$875$	1.35952e6i	1.77569i
$876$	0	0
$877$	−745544.	−0.969336	−0.484668	−	0.874698i	$-0.661060\pi$
$877$	−745544.	−0.969336	−0.484668	+	0.874698i	$0.661060\pi$
$878$	− 967861.i	− 1.25552i
$879$	0	0
$880$	−20157.4	−0.0260298
$881$	− 157416.i	− 0.202814i	−0.994845	−	0.101407i	$-0.967666\pi$
$881$	− 157416.i	− 0.202814i	0.994845	−	0.101407i	$-0.0323344\pi$
$882$	0	0
$883$	427912.	0.548824	0.274412	−	0.961612i	$-0.411517\pi$
$883$	427912.	0.548824	0.274412	+	0.961612i	$0.411517\pi$
$884$	263761.i	0.337526i
$885$	0	0
$886$	1.62554e6	2.07076
$887$	839984.i	1.06764i	0.845599	+	0.533819i	$0.179244\pi$
$887$	839984.i	1.06764i	−0.845599	+	0.533819i	$0.820756\pi$
$888$	0	0
$889$	524125.	0.663180
$890$	983974.i	1.24224i
$891$	0	0
$892$	−850779.	−1.06927
$893$	− 536381.i	− 0.672621i
$894$	0	0
$895$	348819.	0.435465
$896$	2.05490e6i	2.55962i
$897$	0	0
$898$	−1.35066e6	−1.67492
$899$	483865.i	0.598694i
$900$	0	0
$901$	1.09358e6	1.34710
$902$	− 336069.i	− 0.413062i
$903$	0	0
$904$	−698442.	−0.854660
$905$	19843.0i	0.0242275i
$906$	0	0
$907$	−719458.	−0.874562	−0.437281	−	0.899325i	$-0.644059\pi$
$907$	−719458.	−0.874562	−0.437281	+	0.899325i	$0.644059\pi$
$908$	− 271103.i	− 0.328824i
$909$	0	0
$910$	−387918.	−0.468443
$911$	− 24750.9i	− 0.0298232i	−0.999889	−	0.0149116i	$-0.995253\pi$
$911$	− 24750.9i	− 0.0298232i	0.999889	−	0.0149116i	$-0.00474669\pi$
$912$	0	0
$913$	242217.	0.290578
$914$	− 692130.i	− 0.828505i
$915$	0	0
$916$	−722967.	−0.861644
$917$	521271.i	0.619905i
$918$	0	0
$919$	655862.	0.776571	0.388286	−	0.921539i	$-0.373067\pi$
$919$	655862.	0.776571	0.388286	+	0.921539i	$0.373067\pi$
$920$	− 368410.i	− 0.435267i
$921$	0	0
$922$	−1.44205e6	−1.69636
$923$	− 285275.i	− 0.334857i
$924$	0	0
$925$	891177.	1.04155
$926$	− 937868.i	− 1.09375i
$927$	0	0
$928$	1.80520e6	2.09619
$929$	113201.i	0.131166i	0.997847	+	0.0655828i	$0.0208907\pi$
$929$	113201.i	0.131166i	−0.997847	+	0.0655828i	$0.979109\pi$
$930$	0	0
$931$	2.84648e6	3.28404
$932$	220012.i	0.253288i
$933$	0	0
$934$	1.09095e6	1.25058
$935$	− 125145.i	− 0.143150i
$936$	0	0
$937$	490926.	0.559161	0.279580	−	0.960122i	$-0.409805\pi$
$937$	490926.	0.559161	0.279580	+	0.960122i	$0.409805\pi$
$938$	− 2.97786e6i	− 3.38453i
$939$	0	0
$940$	−362362.	−0.410097
$941$	1.30667e6i	1.47566i	0.674988	+	0.737829i	$0.264149\pi$
$941$	1.30667e6i	1.47566i	−0.674988	+	0.737829i	$0.735851\pi$
$942$	0	0
$943$	−674108.	−0.758065
$944$	17753.8i	0.0199226i
$945$	0	0
$946$	594502.	0.664310
$947$	− 358264.i	− 0.399487i	−0.979848	−	0.199744i	$-0.935989\pi$
$947$	− 358264.i	− 0.399487i	0.979848	−	0.199744i	$-0.0640109\pi$
$948$	0	0
$949$	56950.0	0.0632355
$950$	1.36486e6i	1.51231i
$951$	0	0
$952$	−1.14814e6	−1.26684
$953$	− 1.20395e6i	− 1.32563i	−0.748782	−	0.662817i	$-0.769361\pi$
$953$	− 1.20395e6i	− 1.32563i	0.748782	−	0.662817i	$-0.230639\pi$
$954$	0	0
$955$	212711.	0.233230
$956$	− 2.24374e6i	− 2.45503i
$957$	0	0
$958$	2.16345e6	2.35731
$959$	1.48243e6i	1.61189i
$960$	0	0
$961$	−829382.	−0.898065
$962$	701883.i	0.758428i
$963$	0	0
$964$	−1.32453e6	−1.42531
$965$	129150.i	0.138689i
$966$	0	0
$967$	−1.14500e6	−1.22448	−0.612241	−	0.790671i	$-0.709732\pi$
$967$	−1.14500e6	−1.22448	−0.612241	+	0.790671i	$0.709732\pi$
$968$	− 763762.i	− 0.815093i
$969$	0	0
$970$	1.76820e6	1.87926
$971$	− 208959.i	− 0.221627i	−0.993841	−	0.110813i	$-0.964654\pi$
$971$	− 208959.i	− 0.221627i	0.993841	−	0.110813i	$-0.0353456\pi$
$972$	0	0
$973$	−2.95149e6	−3.11756
$974$	1.41753e6i	1.49422i
$975$	0	0
$976$	195733.	0.205477
$977$	245588.i	0.257287i	0.991691	+	0.128644i	$0.0410623\pi$
$977$	245588.i	0.257287i	−0.991691	+	0.128644i	$0.958938\pi$
$978$	0	0
$979$	−293796.	−0.306535
$980$	− 1.92299e6i	− 2.00228i
$981$	0	0
$982$	−827770.	−0.858394
$983$	1.24959e6i	1.29318i	0.762838	+	0.646590i	$0.223805\pi$
$983$	1.24959e6i	1.29318i	−0.762838	+	0.646590i	$0.776195\pi$
$984$	0	0
$985$	602727.	0.621224
$986$	2.44871e6i	2.51874i
$987$	0	0
$988$	−652965.	−0.668923
$989$	− 1.19249e6i	− 1.21916i
$990$	0	0
$991$	648193.	0.660020	0.330010	−	0.943977i	$-0.392948\pi$
$991$	648193.	0.660020	0.330010	+	0.943977i	$0.392948\pi$
$992$	− 351214.i	− 0.356902i
$993$	0	0
$994$	3.51045e6	3.55296
$995$	− 1.16081e6i	− 1.17251i
$996$	0	0
$997$	−719032.	−0.723366	−0.361683	−	0.932301i	$-0.617798\pi$
$997$	−719032.	−0.723366	−0.361683	+	0.932301i	$0.617798\pi$
$998$	1.63353e6i	1.64009i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.66	yes	76
3.2	odd	2	inner	531.5.b.a.296.11	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.11	✓	76	3.2	odd	2	inner
531.5.b.a.296.66	yes	76	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.38419i q^{2} -24.7578 q^{4} +16.4296i q^{5} +84.4308 q^{7} -55.9116i q^{8} +O(q^{10})\)	\(q+6.38419i q^{2} -24.7578 q^{4} +16.4296i q^{5} +84.4308 q^{7} -55.9116i q^{8} -104.890 q^{10} -31.3181i q^{11} +43.8032 q^{13} +539.022i q^{14} -39.1754 q^{16} -243.216i q^{17} +602.103 q^{19} -406.761i q^{20} +199.940 q^{22} -401.054i q^{23} +355.068 q^{25} +279.648i q^{26} -2090.32 q^{28} +1577.03i q^{29} +306.821 q^{31} -1144.69i q^{32} +1552.74 q^{34} +1387.16i q^{35} +2509.88 q^{37} +3843.94i q^{38} +918.606 q^{40} -1680.84i q^{41} +2973.39 q^{43} +775.368i q^{44} +2560.40 q^{46} -890.845i q^{47} +4727.55 q^{49} +2266.82i q^{50} -1084.47 q^{52} +4496.31i q^{53} +514.544 q^{55} -4720.66i q^{56} -10068.0 q^{58} -453.188i q^{59} -4996.33 q^{61} +1958.80i q^{62} +6681.09 q^{64} +719.670i q^{65} -5524.56 q^{67} +6021.50i q^{68} -8855.92 q^{70} -6512.64i q^{71} +1300.13 q^{73} +16023.5i q^{74} -14906.8 q^{76} -2644.21i q^{77} -7102.80 q^{79} -643.636i q^{80} +10730.8 q^{82} +7734.10i q^{83} +3995.95 q^{85} +18982.7i q^{86} -1751.04 q^{88} -9381.04i q^{89} +3698.34 q^{91} +9929.21i q^{92} +5687.32 q^{94} +9892.33i q^{95} -16857.7 q^{97} +30181.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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