LMFDB - Embedded newform 1386.4.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,4,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1386.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1386.a (trivial)

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:	yes
Analytic conductor:	$81.7766472680$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1028796.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 295x + 175 $ x^3 - x^2 - 295*x + 175
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.592735$ of defining polynomial
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +17.2278 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +17.2278 q^{5} -7.00000 q^{7} +8.00000 q^{8} +34.4555 q^{10} -11.0000 q^{11} +15.5987 q^{13} -14.0000 q^{14} +16.0000 q^{16} +131.738 q^{17} -7.96965 q^{19} +68.9111 q^{20} -22.0000 q^{22} -49.5445 q^{23} +171.796 q^{25} +31.1974 q^{26} -28.0000 q^{28} -74.7961 q^{29} -48.9350 q^{31} +32.0000 q^{32} +263.475 q^{34} -120.594 q^{35} +202.857 q^{37} -15.9393 q^{38} +137.822 q^{40} -87.9068 q^{41} +444.616 q^{43} -44.0000 q^{44} -99.0889 q^{46} -386.811 q^{47} +49.0000 q^{49} +343.592 q^{50} +62.3949 q^{52} +562.265 q^{53} -189.505 q^{55} -56.0000 q^{56} -149.592 q^{58} +322.679 q^{59} +346.790 q^{61} -97.8700 q^{62} +64.0000 q^{64} +268.731 q^{65} +984.254 q^{67} +526.950 q^{68} -241.189 q^{70} -563.185 q^{71} -132.516 q^{73} +405.714 q^{74} -31.8786 q^{76} +77.0000 q^{77} +55.2753 q^{79} +275.644 q^{80} -175.814 q^{82} -874.909 q^{83} +2269.54 q^{85} +889.232 q^{86} -88.0000 q^{88} -567.540 q^{89} -109.191 q^{91} -198.178 q^{92} -773.623 q^{94} -137.299 q^{95} +593.228 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} + 24 q^{8}+O(q^{10}) $ 3 * q + 6 * q^2 + 12 * q^4 - 7 * q^5 - 21 * q^7 + 24 * q^8	$ 3 q + 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} + 24 q^{8} - 14 q^{10} - 33 q^{11} - 15 q^{13} - 42 q^{14} + 48 q^{16} + 40 q^{17} + 41 q^{19} - 28 q^{20} - 66 q^{22} - 266 q^{23} + 330 q^{25} - 30 q^{26} - 84 q^{28} - 39 q^{29} + 332 q^{31} + 96 q^{32} + 80 q^{34} + 49 q^{35} + 553 q^{37} + 82 q^{38} - 56 q^{40} + 320 q^{41} + 290 q^{43} - 132 q^{44} - 532 q^{46} - 33 q^{47} + 147 q^{49} + 660 q^{50} - 60 q^{52} + 482 q^{53} + 77 q^{55} - 168 q^{56} - 78 q^{58} + 443 q^{59} + 546 q^{61} + 664 q^{62} + 192 q^{64} - 287 q^{65} + 661 q^{67} + 160 q^{68} + 98 q^{70} - 948 q^{71} + 1539 q^{73} + 1106 q^{74} + 164 q^{76} + 231 q^{77} + 1568 q^{79} - 112 q^{80} + 640 q^{82} - 52 q^{83} + 3028 q^{85} + 580 q^{86} - 264 q^{88} - 1042 q^{89} + 105 q^{91} - 1064 q^{92} - 66 q^{94} + 1237 q^{95} + 3670 q^{97} + 294 q^{98}+O(q^{100}) $ 3 * q + 6 * q^2 + 12 * q^4 - 7 * q^5 - 21 * q^7 + 24 * q^8 - 14 * q^10 - 33 * q^11 - 15 * q^13 - 42 * q^14 + 48 * q^16 + 40 * q^17 + 41 * q^19 - 28 * q^20 - 66 * q^22 - 266 * q^23 + 330 * q^25 - 30 * q^26 - 84 * q^28 - 39 * q^29 + 332 * q^31 + 96 * q^32 + 80 * q^34 + 49 * q^35 + 553 * q^37 + 82 * q^38 - 56 * q^40 + 320 * q^41 + 290 * q^43 - 132 * q^44 - 532 * q^46 - 33 * q^47 + 147 * q^49 + 660 * q^50 - 60 * q^52 + 482 * q^53 + 77 * q^55 - 168 * q^56 - 78 * q^58 + 443 * q^59 + 546 * q^61 + 664 * q^62 + 192 * q^64 - 287 * q^65 + 661 * q^67 + 160 * q^68 + 98 * q^70 - 948 * q^71 + 1539 * q^73 + 1106 * q^74 + 164 * q^76 + 231 * q^77 + 1568 * q^79 - 112 * q^80 + 640 * q^82 - 52 * q^83 + 3028 * q^85 + 580 * q^86 - 264 * q^88 - 1042 * q^89 + 105 * q^91 - 1064 * q^92 - 66 * q^94 + 1237 * q^95 + 3670 * q^97 + 294 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	17.2278	1.54090	0.770449	−	0.637501i	$-0.220032\pi$
$5$	17.2278	1.54090	0.770449	+	0.637501i	$0.220032\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	34.4555	1.08958
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	15.5987	0.332793	0.166396	−	0.986059i	$-0.446787\pi$
$13$	15.5987	0.332793	0.166396	+	0.986059i	$0.446787\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	131.738	1.87947	0.939737	−	0.341898i	$-0.111070\pi$
$17$	131.738	1.87947	0.939737	+	0.341898i	$0.111070\pi$
$18$	0	0
$19$	−7.96965	−0.0962297	−0.0481148	−	0.998842i	$-0.515321\pi$
$19$	−7.96965	−0.0962297	−0.0481148	+	0.998842i	$0.515321\pi$
$20$	68.9111	0.770449
$21$	0	0
$22$	−22.0000	−0.213201
$23$	−49.5445	−0.449162	−0.224581	−	0.974455i	$-0.572101\pi$
$23$	−49.5445	−0.449162	−0.224581	+	0.974455i	$0.572101\pi$
$24$	0	0
$25$	171.796	1.37437
$26$	31.1974	0.235320
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−74.7961	−0.478941	−0.239471	−	0.970904i	$-0.576974\pi$
$29$	−74.7961	−0.478941	−0.239471	+	0.970904i	$0.576974\pi$
$30$	0	0
$31$	−48.9350	−0.283516	−0.141758	−	0.989901i	$-0.545275\pi$
$31$	−48.9350	−0.283516	−0.141758	+	0.989901i	$0.545275\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	263.475	1.32899
$35$	−120.594	−0.582405
$36$	0	0
$37$	202.857	0.901337	0.450668	−	0.892691i	$-0.351186\pi$
$37$	202.857	0.901337	0.450668	+	0.892691i	$0.351186\pi$
$38$	−15.9393	−0.0680447
$39$	0	0
$40$	137.822	0.544790
$41$	−87.9068	−0.334847	−0.167424	−	0.985885i	$-0.553545\pi$
$41$	−87.9068	−0.334847	−0.167424	+	0.985885i	$0.553545\pi$
$42$	0	0
$43$	444.616	1.57682	0.788411	−	0.615149i	$-0.210904\pi$
$43$	444.616	1.57682	0.788411	+	0.615149i	$0.210904\pi$
$44$	−44.0000	−0.150756
$45$	0	0
$46$	−99.0889	−0.317606
$47$	−386.811	−1.20047	−0.600237	−	0.799822i	$-0.704927\pi$
$47$	−386.811	−1.20047	−0.600237	+	0.799822i	$0.704927\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	343.592	0.971826
$51$	0	0
$52$	62.3949	0.166396
$53$	562.265	1.45723	0.728613	−	0.684925i	$-0.240165\pi$
$53$	562.265	1.45723	0.728613	+	0.684925i	$0.240165\pi$
$54$	0	0
$55$	−189.505	−0.464598
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−149.592	−0.338663
$59$	322.679	0.712021	0.356010	−	0.934482i	$-0.384137\pi$
$59$	322.679	0.712021	0.356010	+	0.934482i	$0.384137\pi$
$60$	0	0
$61$	346.790	0.727900	0.363950	−	0.931419i	$-0.381428\pi$
$61$	346.790	0.727900	0.363950	+	0.931419i	$0.381428\pi$
$62$	−97.8700	−0.200476
$63$	0	0
$64$	64.0000	0.125000
$65$	268.731	0.512800
$66$	0	0
$67$	984.254	1.79471	0.897357	−	0.441306i	$-0.145485\pi$
$67$	984.254	1.79471	0.897357	+	0.441306i	$0.145485\pi$
$68$	526.950	0.939737
$69$	0	0
$70$	−241.189	−0.411823
$71$	−563.185	−0.941376	−0.470688	−	0.882300i	$-0.655994\pi$
$71$	−563.185	−0.941376	−0.470688	+	0.882300i	$0.655994\pi$
$72$	0	0
$73$	−132.516	−0.212464	−0.106232	−	0.994341i	$-0.533879\pi$
$73$	−132.516	−0.212464	−0.106232	+	0.994341i	$0.533879\pi$
$74$	405.714	0.637341
$75$	0	0
$76$	−31.8786	−0.0481148
$77$	77.0000	0.113961
$78$	0	0
$79$	55.2753	0.0787210	0.0393605	−	0.999225i	$-0.487468\pi$
$79$	55.2753	0.0787210	0.0393605	+	0.999225i	$0.487468\pi$
$80$	275.644	0.385225
$81$	0	0
$82$	−175.814	−0.236773
$83$	−874.909	−1.15703	−0.578517	−	0.815671i	$-0.696368\pi$
$83$	−874.909	−1.15703	−0.578517	+	0.815671i	$0.696368\pi$
$84$	0	0
$85$	2269.54	2.89608
$86$	889.232	1.11498
$87$	0	0
$88$	−88.0000	−0.106600
$89$	−567.540	−0.675945	−0.337973	−	0.941156i	$-0.609741\pi$
$89$	−567.540	−0.675945	−0.337973	+	0.941156i	$0.609741\pi$
$90$	0	0
$91$	−109.191	−0.125784
$92$	−198.178	−0.224581
$93$	0	0
$94$	−773.623	−0.848863
$95$	−137.299	−0.148280
$96$	0	0
$97$	593.228	0.620960	0.310480	−	0.950580i	$-0.399510\pi$
$97$	593.228	0.620960	0.310480	+	0.950580i	$0.399510\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	687.185	0.687185
$101$	911.792	0.898284	0.449142	−	0.893460i	$-0.351730\pi$
$101$	911.792	0.898284	0.449142	+	0.893460i	$0.351730\pi$
$102$	0	0
$103$	−718.775	−0.687602	−0.343801	−	0.939043i	$-0.611715\pi$
$103$	−718.775	−0.687602	−0.343801	+	0.939043i	$0.611715\pi$
$104$	124.790	0.117660
$105$	0	0
$106$	1124.53	1.03041
$107$	−1624.82	−1.46801	−0.734007	−	0.679142i	$-0.762352\pi$
$107$	−1624.82	−1.46801	−0.734007	+	0.679142i	$0.762352\pi$
$108$	0	0
$109$	−702.213	−0.617062	−0.308531	−	0.951214i	$-0.599837\pi$
$109$	−702.213	−0.617062	−0.308531	+	0.951214i	$0.599837\pi$
$110$	−379.011	−0.328521
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−802.338	−0.667944	−0.333972	−	0.942583i	$-0.608389\pi$
$113$	−802.338	−0.667944	−0.333972	+	0.942583i	$0.608389\pi$
$114$	0	0
$115$	−853.541	−0.692114
$116$	−299.185	−0.239471
$117$	0	0
$118$	645.358	0.503475
$119$	−922.163	−0.710374
$120$	0	0
$121$	121.000	0.0909091
$122$	693.579	0.514703
$123$	0	0
$124$	−195.740	−0.141758
$125$	806.193	0.576865
$126$	0	0
$127$	2032.57	1.42017	0.710084	−	0.704117i	$-0.248657\pi$
$127$	2032.57	1.42017	0.710084	+	0.704117i	$0.248657\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	537.462	0.362604
$131$	58.5353	0.0390401	0.0195200	−	0.999809i	$-0.493786\pi$
$131$	58.5353	0.0390401	0.0195200	+	0.999809i	$0.493786\pi$
$132$	0	0
$133$	55.7876	0.0363714
$134$	1968.51	1.26905
$135$	0	0
$136$	1053.90	0.664494
$137$	1338.19	0.834518	0.417259	−	0.908788i	$-0.362991\pi$
$137$	1338.19	0.834518	0.417259	+	0.908788i	$0.362991\pi$
$138$	0	0
$139$	−342.297	−0.208873	−0.104436	−	0.994532i	$-0.533304\pi$
$139$	−342.297	−0.208873	−0.104436	+	0.994532i	$0.533304\pi$
$140$	−482.378	−0.291203
$141$	0	0
$142$	−1126.37	−0.665654
$143$	−171.586	−0.100341
$144$	0	0
$145$	−1288.57	−0.738000
$146$	−265.033	−0.150235
$147$	0	0
$148$	811.427	0.450668
$149$	177.281	0.0974728	0.0487364	−	0.998812i	$-0.484481\pi$
$149$	177.281	0.0974728	0.0487364	+	0.998812i	$0.484481\pi$
$150$	0	0
$151$	−2329.57	−1.25548	−0.627741	−	0.778422i	$-0.716020\pi$
$151$	−2329.57	−1.25548	−0.627741	+	0.778422i	$0.716020\pi$
$152$	−63.7572	−0.0340223
$153$	0	0
$154$	154.000	0.0805823
$155$	−843.041	−0.436869
$156$	0	0
$157$	2408.40	1.22427	0.612137	−	0.790752i	$-0.290310\pi$
$157$	2408.40	1.22427	0.612137	+	0.790752i	$0.290310\pi$
$158$	110.551	0.0556642
$159$	0	0
$160$	551.289	0.272395
$161$	346.811	0.169767
$162$	0	0
$163$	3830.26	1.84055	0.920274	−	0.391274i	$-0.127966\pi$
$163$	3830.26	1.84055	0.920274	+	0.391274i	$0.127966\pi$
$164$	−351.627	−0.167424
$165$	0	0
$166$	−1749.82	−0.818146
$167$	782.568	0.362616	0.181308	−	0.983426i	$-0.441967\pi$
$167$	782.568	0.362616	0.181308	+	0.983426i	$0.441967\pi$
$168$	0	0
$169$	−1953.68	−0.889249
$170$	4539.09	2.04784
$171$	0	0
$172$	1778.46	0.788411
$173$	1854.82	0.815143	0.407571	−	0.913173i	$-0.366376\pi$
$173$	1854.82	0.815143	0.407571	+	0.913173i	$0.366376\pi$
$174$	0	0
$175$	−1202.57	−0.519463
$176$	−176.000	−0.0753778
$177$	0	0
$178$	−1135.08	−0.477965
$179$	−1722.23	−0.719136	−0.359568	−	0.933119i	$-0.617076\pi$
$179$	−1722.23	−0.719136	−0.359568	+	0.933119i	$0.617076\pi$
$180$	0	0
$181$	−2361.43	−0.969742	−0.484871	−	0.874586i	$-0.661134\pi$
$181$	−2361.43	−0.969742	−0.484871	+	0.874586i	$0.661134\pi$
$182$	−218.382	−0.0889426
$183$	0	0
$184$	−396.356	−0.158803
$185$	3494.77	1.38887
$186$	0	0
$187$	−1449.11	−0.566683
$188$	−1547.25	−0.600237
$189$	0	0
$190$	−274.599	−0.104850
$191$	587.414	0.222533	0.111266	−	0.993791i	$-0.464509\pi$
$191$	587.414	0.222533	0.111266	+	0.993791i	$0.464509\pi$
$192$	0	0
$193$	−3532.39	−1.31744	−0.658722	−	0.752386i	$-0.728903\pi$
$193$	−3532.39	−1.31744	−0.658722	+	0.752386i	$0.728903\pi$
$194$	1186.46	0.439085
$195$	0	0
$196$	196.000	0.0714286
$197$	4156.19	1.50313	0.751564	−	0.659660i	$-0.229300\pi$
$197$	4156.19	1.50313	0.751564	+	0.659660i	$0.229300\pi$
$198$	0	0
$199$	2126.74	0.757589	0.378795	−	0.925481i	$-0.376339\pi$
$199$	2126.74	0.757589	0.378795	+	0.925481i	$0.376339\pi$
$200$	1374.37	0.485913
$201$	0	0
$202$	1823.58	0.635183
$203$	523.573	0.181023
$204$	0	0
$205$	−1514.44	−0.515966
$206$	−1437.55	−0.486208
$207$	0	0
$208$	249.579	0.0831982
$209$	87.6662	0.0290143
$210$	0	0
$211$	181.344	0.0591671	0.0295836	−	0.999562i	$-0.490582\pi$
$211$	181.344	0.0591671	0.0295836	+	0.999562i	$0.490582\pi$
$212$	2249.06	0.728613
$213$	0	0
$214$	−3249.64	−1.03804
$215$	7659.75	2.42972
$216$	0	0
$217$	342.545	0.107159
$218$	−1404.43	−0.436329
$219$	0	0
$220$	−758.022	−0.232299
$221$	2054.94	0.625475
$222$	0	0
$223$	116.623	0.0350209	0.0175104	−	0.999847i	$-0.494426\pi$
$223$	116.623	0.0350209	0.0175104	+	0.999847i	$0.494426\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−1604.68	−0.472308
$227$	−2156.71	−0.630600	−0.315300	−	0.948992i	$-0.602105\pi$
$227$	−2156.71	−0.630600	−0.315300	+	0.948992i	$0.602105\pi$
$228$	0	0
$229$	5396.06	1.55713	0.778563	−	0.627567i	$-0.215949\pi$
$229$	5396.06	1.55713	0.778563	+	0.627567i	$0.215949\pi$
$230$	−1707.08	−0.489398
$231$	0	0
$232$	−598.369	−0.169331
$233$	3492.13	0.981875	0.490937	−	0.871195i	$-0.336654\pi$
$233$	3492.13	0.981875	0.490937	+	0.871195i	$0.336654\pi$
$234$	0	0
$235$	−6663.90	−1.84981
$236$	1290.72	0.356010
$237$	0	0
$238$	−1844.33	−0.502311
$239$	−1811.99	−0.490409	−0.245205	−	0.969471i	$-0.578855\pi$
$239$	−1811.99	−0.490409	−0.245205	+	0.969471i	$0.578855\pi$
$240$	0	0
$241$	546.395	0.146043	0.0730215	−	0.997330i	$-0.476736\pi$
$241$	546.395	0.146043	0.0730215	+	0.997330i	$0.476736\pi$
$242$	242.000	0.0642824
$243$	0	0
$244$	1387.16	0.363950
$245$	844.161	0.220128
$246$	0	0
$247$	−124.316	−0.0320245
$248$	−391.480	−0.100238
$249$	0	0
$250$	1612.39	0.407905
$251$	6093.48	1.53234	0.766169	−	0.642639i	$-0.222160\pi$
$251$	6093.48	1.53234	0.766169	+	0.642639i	$0.222160\pi$
$252$	0	0
$253$	544.989	0.135428
$254$	4065.14	1.00421
$255$	0	0
$256$	256.000	0.0625000
$257$	−5967.34	−1.44838	−0.724188	−	0.689603i	$-0.757785\pi$
$257$	−5967.34	−1.44838	−0.724188	+	0.689603i	$0.757785\pi$
$258$	0	0
$259$	−1420.00	−0.340673
$260$	1074.92	0.256400
$261$	0	0
$262$	117.071	0.0276055
$263$	5535.60	1.29787	0.648934	−	0.760844i	$-0.275215\pi$
$263$	5535.60	1.29787	0.648934	+	0.760844i	$0.275215\pi$
$264$	0	0
$265$	9686.57	2.24544
$266$	111.575	0.0257185
$267$	0	0
$268$	3937.02	0.897357
$269$	−1703.64	−0.386145	−0.193072	−	0.981184i	$-0.561845\pi$
$269$	−1703.64	−0.386145	−0.193072	+	0.981184i	$0.561845\pi$
$270$	0	0
$271$	−4053.71	−0.908654	−0.454327	−	0.890835i	$-0.650120\pi$
$271$	−4053.71	−0.908654	−0.454327	+	0.890835i	$0.650120\pi$
$272$	2107.80	0.469868
$273$	0	0
$274$	2676.37	0.590094
$275$	−1889.76	−0.414388
$276$	0	0
$277$	−4114.37	−0.892450	−0.446225	−	0.894921i	$-0.647232\pi$
$277$	−4114.37	−0.892450	−0.446225	+	0.894921i	$0.647232\pi$
$278$	−684.595	−0.147695
$279$	0	0
$280$	−964.755	−0.205911
$281$	−8176.45	−1.73582	−0.867911	−	0.496720i	$-0.834538\pi$
$281$	−8176.45	−1.73582	−0.867911	+	0.496720i	$0.834538\pi$
$282$	0	0
$283$	−2712.16	−0.569686	−0.284843	−	0.958574i	$-0.591941\pi$
$283$	−2712.16	−0.569686	−0.284843	+	0.958574i	$0.591941\pi$
$284$	−2252.74	−0.470688
$285$	0	0
$286$	−343.172	−0.0709516
$287$	615.347	0.126560
$288$	0	0
$289$	12441.8	2.53242
$290$	−2577.14	−0.521845
$291$	0	0
$292$	−530.066	−0.106232
$293$	4223.37	0.842089	0.421044	−	0.907040i	$-0.361664\pi$
$293$	4223.37	0.842089	0.421044	+	0.907040i	$0.361664\pi$
$294$	0	0
$295$	5559.04	1.09715
$296$	1622.85	0.318671
$297$	0	0
$298$	354.563	0.0689237
$299$	−772.830	−0.149478
$300$	0	0
$301$	−3112.31	−0.595983
$302$	−4659.14	−0.887760
$303$	0	0
$304$	−127.514	−0.0240574
$305$	5974.41	1.12162
$306$	0	0
$307$	7193.46	1.33730	0.668652	−	0.743575i	$-0.266871\pi$
$307$	7193.46	1.33730	0.668652	+	0.743575i	$0.266871\pi$
$308$	308.000	0.0569803
$309$	0	0
$310$	−1686.08	−0.308913
$311$	−3934.68	−0.717412	−0.358706	−	0.933451i	$-0.616782\pi$
$311$	−3934.68	−0.717412	−0.358706	+	0.933451i	$0.616782\pi$
$312$	0	0
$313$	1599.19	0.288791	0.144396	−	0.989520i	$-0.453876\pi$
$313$	1599.19	0.288791	0.144396	+	0.989520i	$0.453876\pi$
$314$	4816.79	0.865692
$315$	0	0
$316$	221.101	0.0393605
$317$	−1114.12	−0.197399	−0.0986994	−	0.995117i	$-0.531468\pi$
$317$	−1114.12	−0.197399	−0.0986994	+	0.995117i	$0.531468\pi$
$318$	0	0
$319$	822.758	0.144406
$320$	1102.58	0.192612
$321$	0	0
$322$	693.622	0.120044
$323$	−1049.90	−0.180861
$324$	0	0
$325$	2679.80	0.457380
$326$	7660.53	1.30146
$327$	0	0
$328$	−703.254	−0.118386
$329$	2707.68	0.453736
$330$	0	0
$331$	−7291.03	−1.21073	−0.605365	−	0.795948i	$-0.706973\pi$
$331$	−7291.03	−1.21073	−0.605365	+	0.795948i	$0.706973\pi$
$332$	−3499.64	−0.578517
$333$	0	0
$334$	1565.14	0.256409
$335$	16956.5	2.76547
$336$	0	0
$337$	3738.08	0.604233	0.302116	−	0.953271i	$-0.402307\pi$
$337$	3738.08	0.604233	0.302116	+	0.953271i	$0.402307\pi$
$338$	−3907.36	−0.628794
$339$	0	0
$340$	9078.18	1.44804
$341$	538.285	0.0854832
$342$	0	0
$343$	−343.000	−0.0539949
$344$	3556.93	0.557491
$345$	0	0
$346$	3709.65	0.576393
$347$	−7204.46	−1.11457	−0.557285	−	0.830322i	$-0.688157\pi$
$347$	−7204.46	−1.11457	−0.557285	+	0.830322i	$0.688157\pi$
$348$	0	0
$349$	−9663.02	−1.48209	−0.741045	−	0.671455i	$-0.765670\pi$
$349$	−9663.02	−1.48209	−0.741045	+	0.671455i	$0.765670\pi$
$350$	−2405.15	−0.367316
$351$	0	0
$352$	−352.000	−0.0533002
$353$	−9636.75	−1.45301	−0.726505	−	0.687161i	$-0.758857\pi$
$353$	−9636.75	−1.45301	−0.726505	+	0.687161i	$0.758857\pi$
$354$	0	0
$355$	−9702.42	−1.45057
$356$	−2270.16	−0.337973
$357$	0	0
$358$	−3444.45	−0.508506
$359$	9454.16	1.38989	0.694946	−	0.719062i	$-0.255428\pi$
$359$	9454.16	1.38989	0.694946	+	0.719062i	$0.255428\pi$
$360$	0	0
$361$	−6795.48	−0.990740
$362$	−4722.85	−0.685711
$363$	0	0
$364$	−436.764	−0.0628919
$365$	−2282.96	−0.327386
$366$	0	0
$367$	−9030.03	−1.28437	−0.642185	−	0.766549i	$-0.721972\pi$
$367$	−9030.03	−1.28437	−0.642185	+	0.766549i	$0.721972\pi$
$368$	−792.711	−0.112291
$369$	0	0
$370$	6989.54	0.982078
$371$	−3935.85	−0.550780
$372$	0	0
$373$	−2189.42	−0.303925	−0.151963	−	0.988386i	$-0.548559\pi$
$373$	−2189.42	−0.303925	−0.151963	+	0.988386i	$0.548559\pi$
$374$	−2898.23	−0.400705
$375$	0	0
$376$	−3094.49	−0.424431
$377$	−1166.72	−0.159388
$378$	0	0
$379$	−8881.76	−1.20376	−0.601881	−	0.798586i	$-0.705582\pi$
$379$	−8881.76	−1.20376	−0.601881	+	0.798586i	$0.705582\pi$
$380$	−549.198	−0.0741401
$381$	0	0
$382$	1174.83	0.157355
$383$	−9135.19	−1.21876	−0.609382	−	0.792877i	$-0.708582\pi$
$383$	−9135.19	−1.21876	−0.609382	+	0.792877i	$0.708582\pi$
$384$	0	0
$385$	1326.54	0.175602
$386$	−7064.77	−0.931574
$387$	0	0
$388$	2372.91	0.310480
$389$	−14341.4	−1.86924	−0.934622	−	0.355642i	$-0.884262\pi$
$389$	−14341.4	−1.86924	−0.934622	+	0.355642i	$0.884262\pi$
$390$	0	0
$391$	−6526.87	−0.844189
$392$	392.000	0.0505076
$393$	0	0
$394$	8312.38	1.06287
$395$	952.271	0.121301
$396$	0	0
$397$	1676.44	0.211935	0.105967	−	0.994370i	$-0.466206\pi$
$397$	1676.44	0.211935	0.105967	+	0.994370i	$0.466206\pi$
$398$	4253.47	0.535697
$399$	0	0
$400$	2748.74	0.343592
$401$	−11505.4	−1.43280	−0.716398	−	0.697692i	$-0.754211\pi$
$401$	−11505.4	−1.43280	−0.716398	+	0.697692i	$0.754211\pi$
$402$	0	0
$403$	−763.323	−0.0943519
$404$	3647.17	0.449142
$405$	0	0
$406$	1047.15	0.128002
$407$	−2231.43	−0.271763
$408$	0	0
$409$	12430.2	1.50277	0.751387	−	0.659862i	$-0.229385\pi$
$409$	12430.2	1.50277	0.751387	+	0.659862i	$0.229385\pi$
$410$	−3028.88	−0.364843
$411$	0	0
$412$	−2875.10	−0.343801
$413$	−2258.75	−0.269119
$414$	0	0
$415$	−15072.7	−1.78287
$416$	499.159	0.0588300
$417$	0	0
$418$	175.332	0.0205162
$419$	−11755.2	−1.37060	−0.685299	−	0.728261i	$-0.740329\pi$
$419$	−11755.2	−1.37060	−0.685299	+	0.728261i	$0.740329\pi$
$420$	0	0
$421$	−1881.52	−0.217814	−0.108907	−	0.994052i	$-0.534735\pi$
$421$	−1881.52	−0.217814	−0.108907	+	0.994052i	$0.534735\pi$
$422$	362.689	0.0418375
$423$	0	0
$424$	4498.12	0.515207
$425$	22632.0	2.58309
$426$	0	0
$427$	−2427.53	−0.275120
$428$	−6499.29	−0.734007
$429$	0	0
$430$	15319.5	1.71807
$431$	−16815.2	−1.87926	−0.939630	−	0.342193i	$-0.888830\pi$
$431$	−16815.2	−1.87926	−0.939630	+	0.342193i	$0.888830\pi$
$432$	0	0
$433$	−4487.62	−0.498063	−0.249031	−	0.968495i	$-0.580112\pi$
$433$	−4487.62	−0.498063	−0.249031	+	0.968495i	$0.580112\pi$
$434$	685.090	0.0757728
$435$	0	0
$436$	−2808.85	−0.308531
$437$	394.852	0.0432228
$438$	0	0
$439$	−16966.1	−1.84453	−0.922263	−	0.386563i	$-0.873662\pi$
$439$	−16966.1	−1.84453	−0.922263	+	0.386563i	$0.873662\pi$
$440$	−1516.04	−0.164260
$441$	0	0
$442$	4109.87	0.442278
$443$	−2231.40	−0.239316	−0.119658	−	0.992815i	$-0.538180\pi$
$443$	−2231.40	−0.239316	−0.119658	+	0.992815i	$0.538180\pi$
$444$	0	0
$445$	−9777.45	−1.04156
$446$	233.246	0.0247635
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−17612.7	−1.85122	−0.925608	−	0.378483i	$-0.876446\pi$
$449$	−17612.7	−1.85122	−0.925608	+	0.378483i	$0.876446\pi$
$450$	0	0
$451$	966.975	0.100960
$452$	−3209.35	−0.333972
$453$	0	0
$454$	−4313.43	−0.445902
$455$	−1881.12	−0.193820
$456$	0	0
$457$	17630.5	1.80464	0.902320	−	0.431066i	$-0.141863\pi$
$457$	17630.5	1.80464	0.902320	+	0.431066i	$0.141863\pi$
$458$	10792.1	1.10105
$459$	0	0
$460$	−3414.16	−0.346057
$461$	1705.98	0.172354	0.0861771	−	0.996280i	$-0.472535\pi$
$461$	1705.98	0.172354	0.0861771	+	0.996280i	$0.472535\pi$
$462$	0	0
$463$	7949.27	0.797913	0.398957	−	0.916970i	$-0.369372\pi$
$463$	7949.27	0.797913	0.398957	+	0.916970i	$0.369372\pi$
$464$	−1196.74	−0.119735
$465$	0	0
$466$	6984.25	0.694290
$467$	−1790.48	−0.177417	−0.0887085	−	0.996058i	$-0.528274\pi$
$467$	−1790.48	−0.177417	−0.0887085	+	0.996058i	$0.528274\pi$
$468$	0	0
$469$	−6889.78	−0.678338
$470$	−13327.8	−1.30801
$471$	0	0
$472$	2581.43	0.251737
$473$	−4890.78	−0.475430
$474$	0	0
$475$	−1369.16	−0.132255
$476$	−3688.65	−0.355187
$477$	0	0
$478$	−3623.98	−0.346772
$479$	−652.495	−0.0622407	−0.0311203	−	0.999516i	$-0.509908\pi$
$479$	−652.495	−0.0622407	−0.0311203	+	0.999516i	$0.509908\pi$
$480$	0	0
$481$	3164.31	0.299958
$482$	1092.79	0.103268
$483$	0	0
$484$	484.000	0.0454545
$485$	10220.0	0.956837
$486$	0	0
$487$	4992.01	0.464496	0.232248	−	0.972657i	$-0.425392\pi$
$487$	4992.01	0.464496	0.232248	+	0.972657i	$0.425392\pi$
$488$	2774.32	0.257351
$489$	0	0
$490$	1688.32	0.155654
$491$	18585.3	1.70824	0.854119	−	0.520077i	$-0.174097\pi$
$491$	18585.3	1.70824	0.854119	+	0.520077i	$0.174097\pi$
$492$	0	0
$493$	−9853.46	−0.900157
$494$	−248.633	−0.0226448
$495$	0	0
$496$	−782.960	−0.0708789
$497$	3942.29	0.355807
$498$	0	0
$499$	−18206.9	−1.63337	−0.816684	−	0.577085i	$-0.804190\pi$
$499$	−18206.9	−1.63337	−0.816684	+	0.577085i	$0.804190\pi$
$500$	3224.77	0.288432
$501$	0	0
$502$	12187.0	1.08353
$503$	16070.6	1.42456	0.712280	−	0.701895i	$-0.247663\pi$
$503$	16070.6	1.42456	0.712280	+	0.701895i	$0.247663\pi$
$504$	0	0
$505$	15708.1	1.38417
$506$	1089.98	0.0957617
$507$	0	0
$508$	8130.28	0.710084
$509$	14915.1	1.29883	0.649413	−	0.760436i	$-0.275015\pi$
$509$	14915.1	1.29883	0.649413	+	0.760436i	$0.275015\pi$
$510$	0	0
$511$	927.615	0.0803039
$512$	512.000	0.0441942
$513$	0	0
$514$	−11934.7	−1.02416
$515$	−12382.9	−1.05952
$516$	0	0
$517$	4254.93	0.361956
$518$	−2840.00	−0.240892
$519$	0	0
$520$	2149.85	0.181302
$521$	−5598.43	−0.470771	−0.235385	−	0.971902i	$-0.575635\pi$
$521$	−5598.43	−0.470771	−0.235385	+	0.971902i	$0.575635\pi$
$522$	0	0
$523$	−5078.21	−0.424579	−0.212289	−	0.977207i	$-0.568092\pi$
$523$	−5078.21	−0.424579	−0.212289	+	0.977207i	$0.568092\pi$
$524$	234.141	0.0195200
$525$	0	0
$526$	11071.2	0.917732
$527$	−6446.58	−0.532860
$528$	0	0
$529$	−9712.35	−0.798253
$530$	19373.1	1.58777
$531$	0	0
$532$	223.150	0.0181857
$533$	−1371.23	−0.111435
$534$	0	0
$535$	−27992.1	−2.26206
$536$	7874.03	0.634527
$537$	0	0
$538$	−3407.29	−0.273046
$539$	−539.000	−0.0430730
$540$	0	0
$541$	17562.1	1.39567	0.697833	−	0.716260i	$-0.254148\pi$
$541$	17562.1	1.39567	0.697833	+	0.716260i	$0.254148\pi$
$542$	−8107.42	−0.642515
$543$	0	0
$544$	4215.60	0.332247
$545$	−12097.6	−0.950830
$546$	0	0
$547$	15101.0	1.18039	0.590193	−	0.807262i	$-0.299052\pi$
$547$	15101.0	1.18039	0.590193	+	0.807262i	$0.299052\pi$
$548$	5352.75	0.417259
$549$	0	0
$550$	−3779.52	−0.293016
$551$	596.099	0.0460884
$552$	0	0
$553$	−386.927	−0.0297538
$554$	−8228.74	−0.631057
$555$	0	0
$556$	−1369.19	−0.104436
$557$	−8211.86	−0.624681	−0.312341	−	0.949970i	$-0.601113\pi$
$557$	−8211.86	−0.624681	−0.312341	+	0.949970i	$0.601113\pi$
$558$	0	0
$559$	6935.44	0.524755
$560$	−1929.51	−0.145601
$561$	0	0
$562$	−16352.9	−1.22741
$563$	11427.9	0.855465	0.427732	−	0.903905i	$-0.359313\pi$
$563$	11427.9	0.855465	0.427732	+	0.903905i	$0.359313\pi$
$564$	0	0
$565$	−13822.5	−1.02923
$566$	−5424.31	−0.402829
$567$	0	0
$568$	−4505.48	−0.332827
$569$	6880.75	0.506953	0.253476	−	0.967342i	$-0.418426\pi$
$569$	6880.75	0.506953	0.253476	+	0.967342i	$0.418426\pi$
$570$	0	0
$571$	8523.78	0.624709	0.312355	−	0.949966i	$-0.398882\pi$
$571$	8523.78	0.624709	0.312355	+	0.949966i	$0.398882\pi$
$572$	−686.343	−0.0501704
$573$	0	0
$574$	1230.69	0.0894917
$575$	−8511.55	−0.617315
$576$	0	0
$577$	14371.2	1.03688	0.518441	−	0.855114i	$-0.326513\pi$
$577$	14371.2	1.03688	0.518441	+	0.855114i	$0.326513\pi$
$578$	24883.6	1.79069
$579$	0	0
$580$	−5154.28	−0.369000
$581$	6124.36	0.437317
$582$	0	0
$583$	−6184.91	−0.439370
$584$	−1060.13	−0.0751174
$585$	0	0
$586$	8446.74	0.595447
$587$	−11781.0	−0.828372	−0.414186	−	0.910192i	$-0.635934\pi$
$587$	−11781.0	−0.828372	−0.414186	+	0.910192i	$0.635934\pi$
$588$	0	0
$589$	389.995	0.0272826
$590$	11118.1	0.775804
$591$	0	0
$592$	3245.71	0.225334
$593$	−28477.0	−1.97202	−0.986012	−	0.166677i	$-0.946696\pi$
$593$	−28477.0	−1.97202	−0.986012	+	0.166677i	$0.946696\pi$
$594$	0	0
$595$	−15886.8	−1.09461
$596$	709.125	0.0487364
$597$	0	0
$598$	−1545.66	−0.105697
$599$	−9800.46	−0.668507	−0.334254	−	0.942483i	$-0.608484\pi$
$599$	−9800.46	−0.668507	−0.334254	+	0.942483i	$0.608484\pi$
$600$	0	0
$601$	−9667.23	−0.656131	−0.328065	−	0.944655i	$-0.606397\pi$
$601$	−9667.23	−0.656131	−0.328065	+	0.944655i	$0.606397\pi$
$602$	−6224.63	−0.421423
$603$	0	0
$604$	−9318.28	−0.627741
$605$	2084.56	0.140082
$606$	0	0
$607$	−1908.82	−0.127639	−0.0638193	−	0.997961i	$-0.520328\pi$
$607$	−1908.82	−0.127639	−0.0638193	+	0.997961i	$0.520328\pi$
$608$	−255.029	−0.0170112
$609$	0	0
$610$	11948.8	0.793105
$611$	−6033.76	−0.399509
$612$	0	0
$613$	9970.32	0.656929	0.328465	−	0.944516i	$-0.393469\pi$
$613$	9970.32	0.656929	0.328465	+	0.944516i	$0.393469\pi$
$614$	14386.9	0.945617
$615$	0	0
$616$	616.000	0.0402911
$617$	−21216.0	−1.38432	−0.692159	−	0.721745i	$-0.743340\pi$
$617$	−21216.0	−1.38432	−0.692159	+	0.721745i	$0.743340\pi$
$618$	0	0
$619$	1144.12	0.0742911	0.0371456	−	0.999310i	$-0.488173\pi$
$619$	1144.12	0.0742911	0.0371456	+	0.999310i	$0.488173\pi$
$620$	−3372.16	−0.218434
$621$	0	0
$622$	−7869.36	−0.507287
$623$	3972.78	0.255483
$624$	0	0
$625$	−7585.60	−0.485479
$626$	3198.38	0.204206
$627$	0	0
$628$	9633.59	0.612137
$629$	26723.9	1.69404
$630$	0	0
$631$	24108.9	1.52101	0.760506	−	0.649330i	$-0.224951\pi$
$631$	24108.9	1.52101	0.760506	+	0.649330i	$0.224951\pi$
$632$	442.203	0.0278321
$633$	0	0
$634$	−2228.25	−0.139582
$635$	35016.6	2.18833
$636$	0	0
$637$	764.337	0.0475418
$638$	1645.52	0.102111
$639$	0	0
$640$	2205.15	0.136197
$641$	−19827.7	−1.22176	−0.610878	−	0.791725i	$-0.709183\pi$
$641$	−19827.7	−1.22176	−0.610878	+	0.791725i	$0.709183\pi$
$642$	0	0
$643$	−942.319	−0.0577939	−0.0288969	−	0.999582i	$-0.509199\pi$
$643$	−942.319	−0.0577939	−0.0288969	+	0.999582i	$0.509199\pi$
$644$	1387.24	0.0848837
$645$	0	0
$646$	−2099.81	−0.127888
$647$	2620.21	0.159213	0.0796066	−	0.996826i	$-0.474634\pi$
$647$	2620.21	0.159213	0.0796066	+	0.996826i	$0.474634\pi$
$648$	0	0
$649$	−3549.47	−0.214682
$650$	5359.60	0.323416
$651$	0	0
$652$	15321.1	0.920274
$653$	−5802.60	−0.347738	−0.173869	−	0.984769i	$-0.555627\pi$
$653$	−5802.60	−0.347738	−0.173869	+	0.984769i	$0.555627\pi$
$654$	0	0
$655$	1008.43	0.0601568
$656$	−1406.51	−0.0837118
$657$	0	0
$658$	5415.36	0.320840
$659$	−13941.5	−0.824103	−0.412052	−	0.911160i	$-0.635188\pi$
$659$	−13941.5	−0.824103	−0.412052	+	0.911160i	$0.635188\pi$
$660$	0	0
$661$	14780.5	0.869738	0.434869	−	0.900494i	$-0.356795\pi$
$661$	14780.5	0.869738	0.434869	+	0.900494i	$0.356795\pi$
$662$	−14582.1	−0.856115
$663$	0	0
$664$	−6999.27	−0.409073
$665$	961.096	0.0560447
$666$	0	0
$667$	3705.73	0.215122
$668$	3130.27	0.181308
$669$	0	0
$670$	33913.0	1.95548
$671$	−3814.69	−0.219470
$672$	0	0
$673$	−2994.43	−0.171511	−0.0857554	−	0.996316i	$-0.527330\pi$
$673$	−2994.43	−0.171511	−0.0857554	+	0.996316i	$0.527330\pi$
$674$	7476.17	0.427257
$675$	0	0
$676$	−7814.72	−0.444625
$677$	18613.6	1.05669	0.528344	−	0.849030i	$-0.322813\pi$
$677$	18613.6	1.05669	0.528344	+	0.849030i	$0.322813\pi$
$678$	0	0
$679$	−4152.59	−0.234701
$680$	18156.4	1.02392
$681$	0	0
$682$	1076.57	0.0604457
$683$	−9237.50	−0.517515	−0.258758	−	0.965942i	$-0.583313\pi$
$683$	−9237.50	−0.517515	−0.258758	+	0.965942i	$0.583313\pi$
$684$	0	0
$685$	23054.0	1.28591
$686$	−686.000	−0.0381802
$687$	0	0
$688$	7113.86	0.394205
$689$	8770.61	0.484954
$690$	0	0
$691$	4478.39	0.246550	0.123275	−	0.992373i	$-0.460660\pi$
$691$	4478.39	0.246550	0.123275	+	0.992373i	$0.460660\pi$
$692$	7419.30	0.407571
$693$	0	0
$694$	−14408.9	−0.788120
$695$	−5897.02	−0.321851
$696$	0	0
$697$	−11580.6	−0.629336
$698$	−19326.0	−1.04800
$699$	0	0
$700$	−4810.29	−0.259731
$701$	−18521.8	−0.997946	−0.498973	−	0.866618i	$-0.666289\pi$
$701$	−18521.8	−0.997946	−0.498973	+	0.866618i	$0.666289\pi$
$702$	0	0
$703$	−1616.70	−0.0867354
$704$	−704.000	−0.0376889
$705$	0	0
$706$	−19273.5	−1.02743
$707$	−6382.54	−0.339520
$708$	0	0
$709$	−32316.2	−1.71179	−0.855895	−	0.517150i	$-0.826993\pi$
$709$	−32316.2	−1.71179	−0.855895	+	0.517150i	$0.826993\pi$
$710$	−19404.8	−1.02570
$711$	0	0
$712$	−4540.32	−0.238983
$713$	2424.46	0.127345
$714$	0	0
$715$	−2956.04	−0.154615
$716$	−6888.91	−0.359568
$717$	0	0
$718$	18908.3	0.982802
$719$	−22295.0	−1.15641	−0.578207	−	0.815890i	$-0.696247\pi$
$719$	−22295.0	−1.15641	−0.578207	+	0.815890i	$0.696247\pi$
$720$	0	0
$721$	5031.42	0.259889
$722$	−13591.0	−0.700559
$723$	0	0
$724$	−9445.70	−0.484871
$725$	−12849.7	−0.658242
$726$	0	0
$727$	−11823.2	−0.603160	−0.301580	−	0.953441i	$-0.597514\pi$
$727$	−11823.2	−0.603160	−0.301580	+	0.953441i	$0.597514\pi$
$728$	−873.528	−0.0444713
$729$	0	0
$730$	−4565.93	−0.231497
$731$	58572.7	2.96359
$732$	0	0
$733$	7471.43	0.376485	0.188243	−	0.982123i	$-0.439721\pi$
$733$	7471.43	0.376485	0.188243	+	0.982123i	$0.439721\pi$
$734$	−18060.1	−0.908187
$735$	0	0
$736$	−1585.42	−0.0794014
$737$	−10826.8	−0.541126
$738$	0	0
$739$	334.930	0.0166720	0.00833600	−	0.999965i	$-0.497347\pi$
$739$	334.930	0.0166720	0.00833600	+	0.999965i	$0.497347\pi$
$740$	13979.1	0.694434
$741$	0	0
$742$	−7871.71	−0.389460
$743$	−31080.2	−1.53462	−0.767310	−	0.641276i	$-0.778405\pi$
$743$	−31080.2	−1.53462	−0.767310	+	0.641276i	$0.778405\pi$
$744$	0	0
$745$	3054.16	0.150196
$746$	−4378.85	−0.214907
$747$	0	0
$748$	−5796.45	−0.283341
$749$	11373.8	0.554857
$750$	0	0
$751$	−4775.62	−0.232044	−0.116022	−	0.993247i	$-0.537014\pi$
$751$	−4775.62	−0.232044	−0.116022	+	0.993247i	$0.537014\pi$
$752$	−6188.98	−0.300118
$753$	0	0
$754$	−2333.45	−0.112704
$755$	−40133.3	−1.93457
$756$	0	0
$757$	−2628.31	−0.126192	−0.0630962	−	0.998007i	$-0.520097\pi$
$757$	−2628.31	−0.126192	−0.0630962	+	0.998007i	$0.520097\pi$
$758$	−17763.5	−0.851188
$759$	0	0
$760$	−1098.40	−0.0524250
$761$	25838.8	1.23082	0.615411	−	0.788207i	$-0.288990\pi$
$761$	25838.8	1.23082	0.615411	+	0.788207i	$0.288990\pi$
$762$	0	0
$763$	4915.49	0.233228
$764$	2349.66	0.111266
$765$	0	0
$766$	−18270.4	−0.861796
$767$	5033.38	0.236955
$768$	0	0
$769$	9148.78	0.429016	0.214508	−	0.976722i	$-0.431185\pi$
$769$	9148.78	0.429016	0.214508	+	0.976722i	$0.431185\pi$
$770$	2653.08	0.124169
$771$	0	0
$772$	−14129.5	−0.658722
$773$	4945.06	0.230092	0.115046	−	0.993360i	$-0.463298\pi$
$773$	4945.06	0.230092	0.115046	+	0.993360i	$0.463298\pi$
$774$	0	0
$775$	−8406.84	−0.389655
$776$	4745.82	0.219543
$777$	0	0
$778$	−28682.7	−1.32176
$779$	700.587	0.0322222
$780$	0	0
$781$	6195.03	0.283836
$782$	−13053.7	−0.596932
$783$	0	0
$784$	784.000	0.0357143
$785$	41491.3	1.88648
$786$	0	0
$787$	22864.1	1.03560	0.517800	−	0.855502i	$-0.326751\pi$
$787$	22864.1	1.03560	0.517800	+	0.855502i	$0.326751\pi$
$788$	16624.8	0.751564
$789$	0	0
$790$	1904.54	0.0857729
$791$	5616.37	0.252459
$792$	0	0
$793$	5409.47	0.242240
$794$	3352.88	0.149860
$795$	0	0
$796$	8506.94	0.378795
$797$	−19847.5	−0.882101	−0.441050	−	0.897482i	$-0.645394\pi$
$797$	−19847.5	−0.882101	−0.441050	+	0.897482i	$0.645394\pi$
$798$	0	0
$799$	−50957.6	−2.25626
$800$	5497.48	0.242956
$801$	0	0
$802$	−23010.8	−1.01314
$803$	1457.68	0.0640603
$804$	0	0
$805$	5974.78	0.261594
$806$	−1526.65	−0.0667169
$807$	0	0
$808$	7294.34	0.317591
$809$	638.036	0.0277282	0.0138641	−	0.999904i	$-0.495587\pi$
$809$	638.036	0.0277282	0.0138641	+	0.999904i	$0.495587\pi$
$810$	0	0
$811$	−17924.7	−0.776103	−0.388052	−	0.921638i	$-0.626852\pi$
$811$	−17924.7	−0.776103	−0.388052	+	0.921638i	$0.626852\pi$
$812$	2094.29	0.0905114
$813$	0	0
$814$	−4462.85	−0.192166
$815$	65986.9	2.83610
$816$	0	0
$817$	−3543.44	−0.151737
$818$	24860.4	1.06262
$819$	0	0
$820$	−6057.75	−0.257983
$821$	−16807.2	−0.714465	−0.357233	−	0.934015i	$-0.616280\pi$
$821$	−16807.2	−0.714465	−0.357233	+	0.934015i	$0.616280\pi$
$822$	0	0
$823$	−32922.8	−1.39443	−0.697215	−	0.716862i	$-0.745578\pi$
$823$	−32922.8	−1.39443	−0.697215	+	0.716862i	$0.745578\pi$
$824$	−5750.20	−0.243104
$825$	0	0
$826$	−4517.51	−0.190296
$827$	37778.7	1.58850	0.794252	−	0.607588i	$-0.207863\pi$
$827$	37778.7	1.58850	0.794252	+	0.607588i	$0.207863\pi$
$828$	0	0
$829$	−40731.3	−1.70646	−0.853230	−	0.521534i	$-0.825360\pi$
$829$	−40731.3	−1.70646	−0.853230	+	0.521534i	$0.825360\pi$
$830$	−30145.5	−1.26068
$831$	0	0
$832$	998.318	0.0415991
$833$	6455.14	0.268496
$834$	0	0
$835$	13481.9	0.558755
$836$	350.665	0.0145072
$837$	0	0
$838$	−23510.5	−0.969160
$839$	−24335.1	−1.00136	−0.500679	−	0.865633i	$-0.666916\pi$
$839$	−24335.1	−1.00136	−0.500679	+	0.865633i	$0.666916\pi$
$840$	0	0
$841$	−18794.5	−0.770615
$842$	−3763.04	−0.154018
$843$	0	0
$844$	725.378	0.0295836
$845$	−33657.6	−1.37024
$846$	0	0
$847$	−847.000	−0.0343604
$848$	8996.24	0.364307
$849$	0	0
$850$	45264.0	1.82652
$851$	−10050.4	−0.404846
$852$	0	0
$853$	−5181.85	−0.207999	−0.104000	−	0.994577i	$-0.533164\pi$
$853$	−5181.85	−0.207999	−0.104000	+	0.994577i	$0.533164\pi$
$854$	−4855.06	−0.194539
$855$	0	0
$856$	−12998.6	−0.519021
$857$	−23157.8	−0.923051	−0.461525	−	0.887127i	$-0.652698\pi$
$857$	−23157.8	−0.923051	−0.461525	+	0.887127i	$0.652698\pi$
$858$	0	0
$859$	44044.9	1.74947	0.874734	−	0.484603i	$-0.161036\pi$
$859$	44044.9	1.74947	0.874734	+	0.484603i	$0.161036\pi$
$860$	30639.0	1.21486
$861$	0	0
$862$	−33630.4	−1.32884
$863$	9520.28	0.375520	0.187760	−	0.982215i	$-0.439877\pi$
$863$	9520.28	0.375520	0.187760	+	0.982215i	$0.439877\pi$
$864$	0	0
$865$	31954.5	1.25605
$866$	−8975.24	−0.352184
$867$	0	0
$868$	1370.18	0.0535794
$869$	−608.029	−0.0237353
$870$	0	0
$871$	15353.1	0.597267
$872$	−5617.70	−0.218164
$873$	0	0
$874$	789.704	0.0305631
$875$	−5643.35	−0.218034
$876$	0	0
$877$	13687.3	0.527011	0.263506	−	0.964658i	$-0.415121\pi$
$877$	13687.3	0.527011	0.263506	+	0.964658i	$0.415121\pi$
$878$	−33932.2	−1.30428
$879$	0	0
$880$	−3032.09	−0.116150
$881$	20450.3	0.782051	0.391026	−	0.920380i	$-0.372120\pi$
$881$	20450.3	0.782051	0.391026	+	0.920380i	$0.372120\pi$
$882$	0	0
$883$	9377.27	0.357384	0.178692	−	0.983905i	$-0.442813\pi$
$883$	9377.27	0.357384	0.178692	+	0.983905i	$0.442813\pi$
$884$	8219.75	0.312738
$885$	0	0
$886$	−4462.79	−0.169222
$887$	8610.33	0.325937	0.162969	−	0.986631i	$-0.447893\pi$
$887$	8610.33	0.325937	0.162969	+	0.986631i	$0.447893\pi$
$888$	0	0
$889$	−14228.0	−0.536773
$890$	−19554.9	−0.736496
$891$	0	0
$892$	466.492	0.0175104
$893$	3082.75	0.115521
$894$	0	0
$895$	−29670.1	−1.10812
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−35225.5	−1.30901
$899$	3660.15	0.135787
$900$	0	0
$901$	74071.4	2.73882
$902$	1933.95	0.0713897
$903$	0	0
$904$	−6418.71	−0.236154
$905$	−40682.1	−1.49427
$906$	0	0
$907$	−33607.7	−1.23035	−0.615173	−	0.788392i	$-0.710914\pi$
$907$	−33607.7	−1.23035	−0.615173	+	0.788392i	$0.710914\pi$
$908$	−8626.86	−0.315300
$909$	0	0
$910$	−3762.24	−0.137052
$911$	−43842.9	−1.59449	−0.797245	−	0.603656i	$-0.793710\pi$
$911$	−43842.9	−1.59449	−0.797245	+	0.603656i	$0.793710\pi$
$912$	0	0
$913$	9624.00	0.348859
$914$	35261.0	1.27607
$915$	0	0
$916$	21584.2	0.778563
$917$	−409.747	−0.0147558
$918$	0	0
$919$	28546.4	1.02466	0.512328	−	0.858790i	$-0.328783\pi$
$919$	28546.4	1.02466	0.512328	+	0.858790i	$0.328783\pi$
$920$	−6828.32	−0.244699
$921$	0	0
$922$	3411.96	0.121873
$923$	−8784.95	−0.313283
$924$	0	0
$925$	34850.0	1.23877
$926$	15898.5	0.564210
$927$	0	0
$928$	−2393.48	−0.0846656
$929$	21453.6	0.757664	0.378832	−	0.925465i	$-0.376326\pi$
$929$	21453.6	0.757664	0.378832	+	0.925465i	$0.376326\pi$
$930$	0	0
$931$	−390.513	−0.0137471
$932$	13968.5	0.490937
$933$	0	0
$934$	−3580.97	−0.125453
$935$	−24965.0	−0.873201
$936$	0	0
$937$	−38499.0	−1.34227	−0.671135	−	0.741336i	$-0.734193\pi$
$937$	−38499.0	−1.34227	−0.671135	+	0.741336i	$0.734193\pi$
$938$	−13779.6	−0.479657
$939$	0	0
$940$	−26655.6	−0.924904
$941$	20531.6	0.711277	0.355639	−	0.934624i	$-0.384263\pi$
$941$	20531.6	0.711277	0.355639	+	0.934624i	$0.384263\pi$
$942$	0	0
$943$	4355.29	0.150401
$944$	5162.86	0.178005
$945$	0	0
$946$	−9781.56	−0.336179
$947$	31889.6	1.09427	0.547134	−	0.837045i	$-0.315719\pi$
$947$	31889.6	1.09427	0.547134	+	0.837045i	$0.315719\pi$
$948$	0	0
$949$	−2067.09	−0.0707065
$950$	−2738.31	−0.0935185
$951$	0	0
$952$	−7377.30	−0.251155
$953$	−38924.3	−1.32307	−0.661533	−	0.749916i	$-0.730094\pi$
$953$	−38924.3	−1.32307	−0.661533	+	0.749916i	$0.730094\pi$
$954$	0	0
$955$	10119.8	0.342901
$956$	−7247.96	−0.245205
$957$	0	0
$958$	−1304.99	−0.0440108
$959$	−9367.31	−0.315418
$960$	0	0
$961$	−27396.4	−0.919619
$962$	6328.61	0.212103
$963$	0	0
$964$	2185.58	0.0730215
$965$	−60855.1	−2.03005
$966$	0	0
$967$	43233.1	1.43773	0.718863	−	0.695152i	$-0.244663\pi$
$967$	43233.1	1.43773	0.718863	+	0.695152i	$0.244663\pi$
$968$	968.000	0.0321412
$969$	0	0
$970$	20440.0	0.676586
$971$	5241.26	0.173224	0.0866118	−	0.996242i	$-0.472396\pi$
$971$	5241.26	0.173224	0.0866118	+	0.996242i	$0.472396\pi$
$972$	0	0
$973$	2396.08	0.0789464
$974$	9984.02	0.328448
$975$	0	0
$976$	5548.64	0.181975
$977$	−17573.5	−0.575461	−0.287730	−	0.957711i	$-0.592901\pi$
$977$	−17573.5	−0.575461	−0.287730	+	0.957711i	$0.592901\pi$
$978$	0	0
$979$	6242.94	0.203805
$980$	3376.64	0.110064
$981$	0	0
$982$	37170.7	1.20791
$983$	10850.5	0.352061	0.176030	−	0.984385i	$-0.443674\pi$
$983$	10850.5	0.352061	0.176030	+	0.984385i	$0.443674\pi$
$984$	0	0
$985$	71601.9	2.31617
$986$	−19706.9	−0.636507
$987$	0	0
$988$	−497.265	−0.0160123
$989$	−22028.3	−0.708249
$990$	0	0
$991$	−35028.4	−1.12282	−0.561410	−	0.827538i	$-0.689741\pi$
$991$	−35028.4	−1.12282	−0.561410	+	0.827538i	$0.689741\pi$
$992$	−1565.92	−0.0501190
$993$	0	0
$994$	7884.58	0.251593
$995$	36638.9	1.16737
$996$	0	0
$997$	51553.3	1.63762	0.818811	−	0.574064i	$-0.194634\pi$
$997$	51553.3	1.63762	0.818811	+	0.574064i	$0.194634\pi$
$998$	−36413.7	−1.15497
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.4.a.bh.1.3		3
3.2	odd	2		462.4.a.r.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.r.1.1	✓	3	3.2	odd	2
1386.4.a.bh.1.3		3	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +17.2278 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +17.2278 q^{5} -7.00000 q^{7} +8.00000 q^{8} +34.4555 q^{10} -11.0000 q^{11} +15.5987 q^{13} -14.0000 q^{14} +16.0000 q^{16} +131.738 q^{17} -7.96965 q^{19} +68.9111 q^{20} -22.0000 q^{22} -49.5445 q^{23} +171.796 q^{25} +31.1974 q^{26} -28.0000 q^{28} -74.7961 q^{29} -48.9350 q^{31} +32.0000 q^{32} +263.475 q^{34} -120.594 q^{35} +202.857 q^{37} -15.9393 q^{38} +137.822 q^{40} -87.9068 q^{41} +444.616 q^{43} -44.0000 q^{44} -99.0889 q^{46} -386.811 q^{47} +49.0000 q^{49} +343.592 q^{50} +62.3949 q^{52} +562.265 q^{53} -189.505 q^{55} -56.0000 q^{56} -149.592 q^{58} +322.679 q^{59} +346.790 q^{61} -97.8700 q^{62} +64.0000 q^{64} +268.731 q^{65} +984.254 q^{67} +526.950 q^{68} -241.189 q^{70} -563.185 q^{71} -132.516 q^{73} +405.714 q^{74} -31.8786 q^{76} +77.0000 q^{77} +55.2753 q^{79} +275.644 q^{80} -175.814 q^{82} -874.909 q^{83} +2269.54 q^{85} +889.232 q^{86} -88.0000 q^{88} -567.540 q^{89} -109.191 q^{91} -198.178 q^{92} -773.623 q^{94} -137.299 q^{95} +593.228 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} + 24 q^{8}+O(q^{10}) \) 3 * q + 6 * q^2 + 12 * q^4 - 7 * q^5 - 21 * q^7 + 24 * q^8	\( 3 q + 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} + 24 q^{8} - 14 q^{10} - 33 q^{11} - 15 q^{13} - 42 q^{14} + 48 q^{16} + 40 q^{17} + 41 q^{19} - 28 q^{20} - 66 q^{22} - 266 q^{23} + 330 q^{25} - 30 q^{26} - 84 q^{28} - 39 q^{29} + 332 q^{31} + 96 q^{32} + 80 q^{34} + 49 q^{35} + 553 q^{37} + 82 q^{38} - 56 q^{40} + 320 q^{41} + 290 q^{43} - 132 q^{44} - 532 q^{46} - 33 q^{47} + 147 q^{49} + 660 q^{50} - 60 q^{52} + 482 q^{53} + 77 q^{55} - 168 q^{56} - 78 q^{58} + 443 q^{59} + 546 q^{61} + 664 q^{62} + 192 q^{64} - 287 q^{65} + 661 q^{67} + 160 q^{68} + 98 q^{70} - 948 q^{71} + 1539 q^{73} + 1106 q^{74} + 164 q^{76} + 231 q^{77} + 1568 q^{79} - 112 q^{80} + 640 q^{82} - 52 q^{83} + 3028 q^{85} + 580 q^{86} - 264 q^{88} - 1042 q^{89} + 105 q^{91} - 1064 q^{92} - 66 q^{94} + 1237 q^{95} + 3670 q^{97} + 294 q^{98}+O(q^{100}) \) 3 * q + 6 * q^2 + 12 * q^4 - 7 * q^5 - 21 * q^7 + 24 * q^8 - 14 * q^10 - 33 * q^11 - 15 * q^13 - 42 * q^14 + 48 * q^16 + 40 * q^17 + 41 * q^19 - 28 * q^20 - 66 * q^22 - 266 * q^23 + 330 * q^25 - 30 * q^26 - 84 * q^28 - 39 * q^29 + 332 * q^31 + 96 * q^32 + 80 * q^34 + 49 * q^35 + 553 * q^37 + 82 * q^38 - 56 * q^40 + 320 * q^41 + 290 * q^43 - 132 * q^44 - 532 * q^46 - 33 * q^47 + 147 * q^49 + 660 * q^50 - 60 * q^52 + 482 * q^53 + 77 * q^55 - 168 * q^56 - 78 * q^58 + 443 * q^59 + 546 * q^61 + 664 * q^62 + 192 * q^64 - 287 * q^65 + 661 * q^67 + 160 * q^68 + 98 * q^70 - 948 * q^71 + 1539 * q^73 + 1106 * q^74 + 164 * q^76 + 231 * q^77 + 1568 * q^79 - 112 * q^80 + 640 * q^82 - 52 * q^83 + 3028 * q^85 + 580 * q^86 - 264 * q^88 - 1042 * q^89 + 105 * q^91 - 1064 * q^92 - 66 * q^94 + 1237 * q^95 + 3670 * q^97 + 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more