LMFDB - Embedded newform 162.5.b.c.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,5,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	162.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:	$16.7459340196$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.221456830464.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 $ x^8 - 4x^7 + 38x^6 - 100x^5 + 449x^4 - 736x^3 + 1900x^2 - 1548*x + 2307
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$0.500000 + 3.61825i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.5.b.c.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.82843i q^{2} -8.00000 q^{4} +7.40592i q^{5} -60.3765 q^{7} +22.6274i q^{8} +20.9471 q^{10} -63.7457i q^{11} +289.988 q^{13} +170.770i q^{14} +64.0000 q^{16} +460.439i q^{17} +187.741 q^{19} -59.2473i q^{20} -180.300 q^{22} -41.4096i q^{23} +570.152 q^{25} -820.210i q^{26} +483.012 q^{28} +1398.90i q^{29} +636.965 q^{31} -181.019i q^{32} +1302.32 q^{34} -447.143i q^{35} +847.670 q^{37} -531.011i q^{38} -167.577 q^{40} -506.166i q^{41} +1404.86 q^{43} +509.966i q^{44} -117.124 q^{46} -2478.82i q^{47} +1244.32 q^{49} -1612.63i q^{50} -2319.90 q^{52} +773.208i q^{53} +472.095 q^{55} -1366.16i q^{56} +3956.70 q^{58} +4497.48i q^{59} -2602.95 q^{61} -1801.61i q^{62} -512.000 q^{64} +2147.63i q^{65} -3383.10 q^{67} -3683.51i q^{68} -1264.71 q^{70} +2771.07i q^{71} +3525.16 q^{73} -2397.57i q^{74} -1501.93 q^{76} +3848.74i q^{77} +5829.79 q^{79} +473.979i q^{80} -1431.65 q^{82} +5125.29i q^{83} -3409.97 q^{85} -3973.54i q^{86} +1442.40 q^{88} -1221.14i q^{89} -17508.5 q^{91} +331.277i q^{92} -7011.16 q^{94} +1390.39i q^{95} -14735.2 q^{97} -3519.46i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4} + 52 q^{7} - 20 q^{13} + 512 q^{16} + 100 q^{19} - 672 q^{22} - 1588 q^{25} - 416 q^{28} + 2956 q^{31} + 192 q^{34} - 32 q^{37} + 136 q^{43} + 2112 q^{46} - 4884 q^{49} + 160 q^{52} - 3996 q^{55}+ \cdots - 62672 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 + 52 * q^7 - 20 * q^13 + 512 * q^16 + 100 * q^19 - 672 * q^22 - 1588 * q^25 - 416 * q^28 + 2956 * q^31 + 192 * q^34 - 32 * q^37 + 136 * q^43 + 2112 * q^46 - 4884 * q^49 + 160 * q^52 - 3996 * q^55 + 4800 * q^58 + 8956 * q^61 - 4096 * q^64 - 15008 * q^67 - 12096 * q^70 + 20716 * q^73 - 800 * q^76 + 12100 * q^79 + 1152 * q^82 + 32184 * q^85 + 5376 * q^88 - 45868 * q^91 - 1344 * q^94 - 62672 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	7.40592i	0.296237i	0.988970	+	0.148118i	$0.0473216\pi$
$5$	7.40592i	0.296237i	−0.988970	+	0.148118i	$0.952678\pi$
$6$	0	0
$7$	−60.3765	−1.23217	−0.616086	−	0.787679i	$-0.711283\pi$
$7$	−60.3765	−1.23217	−0.616086	+	0.787679i	$0.711283\pi$
$8$	22.6274i	0.353553i
$9$	0	0
$10$	20.9471	0.209471
$11$	− 63.7457i	− 0.526824i	−0.964683	−	0.263412i	$-0.915152\pi$
$11$	− 63.7457i	− 0.526824i	0.964683	−	0.263412i	$-0.0848479\pi$
$12$	0	0
$13$	289.988	1.71591	0.857953	−	0.513728i	$-0.171736\pi$
$13$	289.988	1.71591	0.857953	+	0.513728i	$0.171736\pi$
$14$	170.770i	0.871278i
$15$	0	0
$16$	64.0000	0.250000
$17$	460.439i	1.59321i	0.604498	+	0.796606i	$0.293374\pi$
$17$	460.439i	1.59321i	−0.604498	+	0.796606i	$0.706626\pi$
$18$	0	0
$19$	187.741	0.520058	0.260029	−	0.965601i	$-0.416268\pi$
$19$	187.741	0.520058	0.260029	+	0.965601i	$0.416268\pi$
$20$	− 59.2473i	− 0.148118i
$21$	0	0
$22$	−180.300	−0.372521
$23$	− 41.4096i	− 0.0782790i	−0.999234	−	0.0391395i	$-0.987538\pi$
$23$	− 41.4096i	− 0.0782790i	0.999234	−	0.0391395i	$-0.0124617\pi$
$24$	0	0
$25$	570.152	0.912244
$26$	− 820.210i	− 1.21333i
$27$	0	0
$28$	483.012	0.616086
$29$	1398.90i	1.66338i	0.555239	+	0.831691i	$0.312627\pi$
$29$	1398.90i	1.66338i	−0.555239	+	0.831691i	$0.687373\pi$
$30$	0	0
$31$	636.965	0.662814	0.331407	−	0.943488i	$-0.392477\pi$
$31$	636.965	0.662814	0.331407	+	0.943488i	$0.392477\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	1302.32	1.12657
$35$	− 447.143i	− 0.365015i
$36$	0	0
$37$	847.670	0.619189	0.309594	−	0.950869i	$-0.399807\pi$
$37$	847.670	0.619189	0.309594	+	0.950869i	$0.399807\pi$
$38$	− 531.011i	− 0.367736i
$39$	0	0
$40$	−167.577	−0.104735
$41$	− 506.166i	− 0.301110i	−0.988602	−	0.150555i	$-0.951894\pi$
$41$	− 506.166i	− 0.301110i	0.988602	−	0.150555i	$-0.0481060\pi$
$42$	0	0
$43$	1404.86	0.759794	0.379897	−	0.925029i	$-0.375959\pi$
$43$	1404.86	0.759794	0.379897	+	0.925029i	$0.375959\pi$
$44$	509.966i	0.263412i
$45$	0	0
$46$	−117.124	−0.0553516
$47$	− 2478.82i	− 1.12215i	−0.827767	−	0.561073i	$-0.810389\pi$
$47$	− 2478.82i	− 1.12215i	0.827767	−	0.561073i	$-0.189611\pi$
$48$	0	0
$49$	1244.32	0.518249
$50$	− 1612.63i	− 0.645054i
$51$	0	0
$52$	−2319.90	−0.857953
$53$	773.208i	0.275261i	0.990484	+	0.137630i	$0.0439486\pi$
$53$	773.208i	0.275261i	−0.990484	+	0.137630i	$0.956051\pi$
$54$	0	0
$55$	472.095	0.156065
$56$	− 1366.16i	− 0.435639i
$57$	0	0
$58$	3956.70	1.17619
$59$	4497.48i	1.29201i	0.763334	+	0.646004i	$0.223561\pi$
$59$	4497.48i	1.29201i	−0.763334	+	0.646004i	$0.776439\pi$
$60$	0	0
$61$	−2602.95	−0.699531	−0.349765	−	0.936837i	$-0.613739\pi$
$61$	−2602.95	−0.699531	−0.349765	+	0.936837i	$0.613739\pi$
$62$	− 1801.61i	− 0.468681i
$63$	0	0
$64$	−512.000	−0.125000
$65$	2147.63i	0.508314i
$66$	0	0
$67$	−3383.10	−0.753641	−0.376821	−	0.926286i	$-0.622983\pi$
$67$	−3383.10	−0.753641	−0.376821	+	0.926286i	$0.622983\pi$
$68$	− 3683.51i	− 0.796606i
$69$	0	0
$70$	−1264.71	−0.258104
$71$	2771.07i	0.549706i	0.961486	+	0.274853i	$0.0886292\pi$
$71$	2771.07i	0.549706i	−0.961486	+	0.274853i	$0.911371\pi$
$72$	0	0
$73$	3525.16	0.661506	0.330753	−	0.943717i	$-0.392697\pi$
$73$	3525.16	0.661506	0.330753	+	0.943717i	$0.392697\pi$
$74$	− 2397.57i	− 0.437833i
$75$	0	0
$76$	−1501.93	−0.260029
$77$	3848.74i	0.649138i
$78$	0	0
$79$	5829.79	0.934112	0.467056	−	0.884228i	$-0.345315\pi$
$79$	5829.79	0.934112	0.467056	+	0.884228i	$0.345315\pi$
$80$	473.979i	0.0740592i
$81$	0	0
$82$	−1431.65	−0.212917
$83$	5125.29i	0.743982i	0.928236	+	0.371991i	$0.121325\pi$
$83$	5125.29i	0.743982i	−0.928236	+	0.371991i	$0.878675\pi$
$84$	0	0
$85$	−3409.97	−0.471968
$86$	− 3973.54i	− 0.537255i
$87$	0	0
$88$	1442.40	0.186260
$89$	− 1221.14i	− 0.154164i	−0.997025	−	0.0770822i	$-0.975440\pi$
$89$	− 1221.14i	− 0.154164i	0.997025	−	0.0770822i	$-0.0245604\pi$
$90$	0	0
$91$	−17508.5	−2.11429
$92$	331.277i	0.0391395i
$93$	0	0
$94$	−7011.16	−0.793476
$95$	1390.39i	0.154060i
$96$	0	0
$97$	−14735.2	−1.56608	−0.783038	−	0.621974i	$-0.786331\pi$
$97$	−14735.2	−1.56608	−0.783038	+	0.621974i	$0.786331\pi$
$98$	− 3519.46i	− 0.366458i
$99$	0	0
$100$	−4561.22	−0.456122
$101$	17359.8i	1.70177i	0.525351	+	0.850885i	$0.323934\pi$
$101$	17359.8i	1.70177i	−0.525351	+	0.850885i	$0.676066\pi$
$102$	0	0
$103$	−10328.7	−0.973581	−0.486791	−	0.873519i	$-0.661833\pi$
$103$	−10328.7	−0.973581	−0.486791	+	0.873519i	$0.661833\pi$
$104$	6561.68i	0.606664i
$105$	0	0
$106$	2186.96	0.194639
$107$	7260.47i	0.634157i	0.948399	+	0.317079i	$0.102702\pi$
$107$	7260.47i	0.634157i	−0.948399	+	0.317079i	$0.897298\pi$
$108$	0	0
$109$	1894.50	0.159456	0.0797280	−	0.996817i	$-0.474595\pi$
$109$	1894.50	0.159456	0.0797280	+	0.996817i	$0.474595\pi$
$110$	− 1335.29i	− 0.110354i
$111$	0	0
$112$	−3864.09	−0.308043
$113$	− 9555.08i	− 0.748303i	−0.927368	−	0.374151i	$-0.877934\pi$
$113$	− 9555.08i	− 0.748303i	0.927368	−	0.374151i	$-0.122066\pi$
$114$	0	0
$115$	306.676	0.0231891
$116$	− 11191.2i	− 0.831691i
$117$	0	0
$118$	12720.8	0.913587
$119$	− 27799.6i	− 1.96311i
$120$	0	0
$121$	10577.5	0.722456
$122$	7362.27i	0.494643i
$123$	0	0
$124$	−5095.72	−0.331407
$125$	8851.20i	0.566477i
$126$	0	0
$127$	7548.96	0.468037	0.234018	−	0.972232i	$-0.424812\pi$
$127$	7548.96	0.468037	0.234018	+	0.972232i	$0.424812\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	6074.41	0.359432
$131$	− 4822.30i	− 0.281004i	−0.990080	−	0.140502i	$-0.955128\pi$
$131$	− 4822.30i	− 0.281004i	0.990080	−	0.140502i	$-0.0448716\pi$
$132$	0	0
$133$	−11335.1	−0.640801
$134$	9568.84i	0.532905i
$135$	0	0
$136$	−10418.5	−0.563286
$137$	− 10793.0i	− 0.575041i	−0.957774	−	0.287521i	$-0.907169\pi$
$137$	− 10793.0i	− 0.575041i	0.957774	−	0.287521i	$-0.0928310\pi$
$138$	0	0
$139$	28277.2	1.46355	0.731774	−	0.681547i	$-0.238693\pi$
$139$	28277.2	1.46355	0.731774	+	0.681547i	$0.238693\pi$
$140$	3577.14i	0.182507i
$141$	0	0
$142$	7837.76	0.388701
$143$	− 18485.5i	− 0.903981i
$144$	0	0
$145$	−10360.2	−0.492755
$146$	− 9970.67i	− 0.467755i
$147$	0	0
$148$	−6781.36	−0.309594
$149$	− 9765.12i	− 0.439850i	−0.975517	−	0.219925i	$-0.929419\pi$
$149$	− 9765.12i	− 0.439850i	0.975517	−	0.219925i	$-0.0705813\pi$
$150$	0	0
$151$	766.883	0.0336337	0.0168169	−	0.999859i	$-0.494647\pi$
$151$	766.883	0.0336337	0.0168169	+	0.999859i	$0.494647\pi$
$152$	4248.09i	0.183868i
$153$	0	0
$154$	10885.9	0.459010
$155$	4717.31i	0.196350i
$156$	0	0
$157$	−5191.83	−0.210630	−0.105315	−	0.994439i	$-0.533585\pi$
$157$	−5191.83	−0.210630	−0.105315	+	0.994439i	$0.533585\pi$
$158$	− 16489.1i	− 0.660517i
$159$	0	0
$160$	1340.61	0.0523677
$161$	2500.17i	0.0964533i
$162$	0	0
$163$	24180.8	0.910113	0.455057	−	0.890462i	$-0.349619\pi$
$163$	24180.8	0.910113	0.455057	+	0.890462i	$0.349619\pi$
$164$	4049.32i	0.150555i
$165$	0	0
$166$	14496.5	0.526075
$167$	21765.5i	0.780433i	0.920723	+	0.390216i	$0.127600\pi$
$167$	21765.5i	0.780433i	−0.920723	+	0.390216i	$0.872400\pi$
$168$	0	0
$169$	55532.1	1.94433
$170$	9644.85i	0.333732i
$171$	0	0
$172$	−11238.9	−0.379897
$173$	24005.6i	0.802084i	0.916060	+	0.401042i	$0.131352\pi$
$173$	24005.6i	0.802084i	−0.916060	+	0.401042i	$0.868648\pi$
$174$	0	0
$175$	−34423.8	−1.12404
$176$	− 4079.73i	− 0.131706i
$177$	0	0
$178$	−3453.89	−0.109011
$179$	− 41131.4i	− 1.28371i	−0.766826	−	0.641855i	$-0.778165\pi$
$179$	− 41131.4i	− 1.28371i	0.766826	−	0.641855i	$-0.221835\pi$
$180$	0	0
$181$	32048.3	0.978246	0.489123	−	0.872215i	$-0.337317\pi$
$181$	32048.3	0.978246	0.489123	+	0.872215i	$0.337317\pi$
$182$	49521.4i	1.49503i
$183$	0	0
$184$	936.993	0.0276758
$185$	6277.77i	0.183426i
$186$	0	0
$187$	29351.0	0.839343
$188$	19830.5i	0.561073i
$189$	0	0
$190$	3932.63	0.108937
$191$	− 13214.7i	− 0.362235i	−0.983461	−	0.181118i	$-0.942029\pi$
$191$	− 13214.7i	− 0.362235i	0.983461	−	0.181118i	$-0.0579715\pi$
$192$	0	0
$193$	1786.92	0.0479724	0.0239862	−	0.999712i	$-0.492364\pi$
$193$	1786.92	0.0479724	0.0239862	+	0.999712i	$0.492364\pi$
$194$	41677.5i	1.10738i
$195$	0	0
$196$	−9954.53	−0.259125
$197$	− 17640.9i	− 0.454557i	−0.973830	−	0.227278i	$-0.927017\pi$
$197$	− 17640.9i	− 0.454557i	0.973830	−	0.227278i	$-0.0729828\pi$
$198$	0	0
$199$	236.938	0.00598314	0.00299157	−	0.999996i	$-0.499048\pi$
$199$	236.938	0.00598314	0.00299157	+	0.999996i	$0.499048\pi$
$200$	12901.1i	0.322527i
$201$	0	0
$202$	49100.8	1.20333
$203$	− 84460.9i	− 2.04957i
$204$	0	0
$205$	3748.62	0.0891998
$206$	29214.0i	0.688426i
$207$	0	0
$208$	18559.2	0.428976
$209$	− 11967.7i	− 0.273979i
$210$	0	0
$211$	35958.3	0.807670	0.403835	−	0.914832i	$-0.367677\pi$
$211$	35958.3	0.807670	0.403835	+	0.914832i	$0.367677\pi$
$212$	− 6185.66i	− 0.137630i
$213$	0	0
$214$	20535.7	0.448417
$215$	10404.3i	0.225079i
$216$	0	0
$217$	−38457.7	−0.816702
$218$	− 5358.44i	− 0.112752i
$219$	0	0
$220$	−3776.76	−0.0780323
$221$	133522.i	2.73380i
$222$	0	0
$223$	−54623.1	−1.09841	−0.549207	−	0.835686i	$-0.685070\pi$
$223$	−54623.1	−1.09841	−0.549207	+	0.835686i	$0.685070\pi$
$224$	10929.3i	0.217819i
$225$	0	0
$226$	−27025.8	−0.529130
$227$	− 78164.0i	− 1.51689i	−0.651735	−	0.758447i	$-0.725959\pi$
$227$	− 78164.0i	− 1.51689i	0.651735	−	0.758447i	$-0.274041\pi$
$228$	0	0
$229$	−12039.0	−0.229573	−0.114787	−	0.993390i	$-0.536618\pi$
$229$	−12039.0	−0.229573	−0.114787	+	0.993390i	$0.536618\pi$
$230$	− 867.411i	− 0.0163972i
$231$	0	0
$232$	−31653.6	−0.588094
$233$	95396.0i	1.75719i	0.477568	+	0.878595i	$0.341518\pi$
$233$	95396.0i	1.75719i	−0.477568	+	0.878595i	$0.658482\pi$
$234$	0	0
$235$	18357.9	0.332421
$236$	− 35979.8i	− 0.646004i
$237$	0	0
$238$	−78629.3	−1.38813
$239$	− 102354.i	− 1.79189i	−0.444166	−	0.895944i	$-0.646500\pi$
$239$	− 102354.i	− 1.79189i	0.444166	−	0.895944i	$-0.353500\pi$
$240$	0	0
$241$	−53667.8	−0.924016	−0.462008	−	0.886876i	$-0.652871\pi$
$241$	−53667.8	−0.924016	−0.462008	+	0.886876i	$0.652871\pi$
$242$	− 29917.6i	− 0.510854i
$243$	0	0
$244$	20823.6	0.349765
$245$	9215.30i	0.153524i
$246$	0	0
$247$	54442.6	0.892370
$248$	14412.9i	0.234340i
$249$	0	0
$250$	25035.0	0.400560
$251$	− 99789.6i	− 1.58394i	−0.610562	−	0.791969i	$-0.709056\pi$
$251$	− 99789.6i	− 1.58394i	0.610562	−	0.791969i	$-0.290944\pi$
$252$	0	0
$253$	−2639.69	−0.0412393
$254$	− 21351.7i	− 0.330952i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 2597.38i	− 0.0393250i	−0.999807	−	0.0196625i	$-0.993741\pi$
$257$	− 2597.38i	− 0.0393250i	0.999807	−	0.0196625i	$-0.00625918\pi$
$258$	0	0
$259$	−51179.3	−0.762948
$260$	− 17181.0i	− 0.254157i
$261$	0	0
$262$	−13639.5	−0.198700
$263$	126218.i	1.82478i	0.409318	+	0.912392i	$0.365767\pi$
$263$	126218.i	1.82478i	−0.409318	+	0.912392i	$0.634233\pi$
$264$	0	0
$265$	−5726.31	−0.0815424
$266$	32060.6i	0.453115i
$267$	0	0
$268$	27064.8	0.376821
$269$	14374.7i	0.198652i	0.995055	+	0.0993262i	$0.0316687\pi$
$269$	14374.7i	0.198652i	−0.995055	+	0.0993262i	$0.968331\pi$
$270$	0	0
$271$	−54121.0	−0.736931	−0.368466	−	0.929641i	$-0.620117\pi$
$271$	−54121.0	−0.736931	−0.368466	+	0.929641i	$0.620117\pi$
$272$	29468.1i	0.398303i
$273$	0	0
$274$	−30527.1	−0.406616
$275$	− 36344.8i	− 0.480592i
$276$	0	0
$277$	31611.8	0.411993	0.205996	−	0.978553i	$-0.433956\pi$
$277$	31611.8	0.411993	0.205996	+	0.978553i	$0.433956\pi$
$278$	− 79980.0i	− 1.03488i
$279$	0	0
$280$	10117.7	0.129052
$281$	− 102180.i	− 1.29406i	−0.762466	−	0.647028i	$-0.776012\pi$
$281$	− 102180.i	− 1.29406i	0.762466	−	0.647028i	$-0.223988\pi$
$282$	0	0
$283$	−145334.	−1.81466	−0.907329	−	0.420421i	$-0.861882\pi$
$283$	−145334.	−1.81466	−0.907329	+	0.420421i	$0.861882\pi$
$284$	− 22168.5i	− 0.274853i
$285$	0	0
$286$	−52284.9	−0.639211
$287$	30560.5i	0.371019i
$288$	0	0
$289$	−128483.	−1.53833
$290$	29303.0i	0.348430i
$291$	0	0
$292$	−28201.3	−0.330753
$293$	− 46243.9i	− 0.538666i	−0.963047	−	0.269333i	$-0.913197\pi$
$293$	− 46243.9i	− 0.538666i	0.963047	−	0.269333i	$-0.0868032\pi$
$294$	0	0
$295$	−33307.9	−0.382740
$296$	19180.6i	0.218916i
$297$	0	0
$298$	−27619.9	−0.311021
$299$	− 12008.3i	− 0.134319i
$300$	0	0
$301$	−84820.4	−0.936197
$302$	− 2169.07i	− 0.0237826i
$303$	0	0
$304$	12015.4	0.130014
$305$	− 19277.3i	− 0.207227i
$306$	0	0
$307$	−10736.4	−0.113915	−0.0569576	−	0.998377i	$-0.518140\pi$
$307$	−10736.4	−0.113915	−0.0569576	+	0.998377i	$0.518140\pi$
$308$	− 30789.9i	− 0.324569i
$309$	0	0
$310$	13342.6	0.138840
$311$	− 222.049i	− 0.00229577i	−0.999999	−	0.00114789i	$-0.999635\pi$
$311$	− 222.049i	− 0.00229577i	0.999999	−	0.00114789i	$-0.000365384\pi$
$312$	0	0
$313$	146676.	1.49717	0.748586	−	0.663038i	$-0.230733\pi$
$313$	146676.	1.49717	0.748586	+	0.663038i	$0.230733\pi$
$314$	14684.7i	0.148938i
$315$	0	0
$316$	−46638.3	−0.467056
$317$	176443.i	1.75584i	0.478804	+	0.877922i	$0.341070\pi$
$317$	176443.i	1.75584i	−0.478804	+	0.877922i	$0.658930\pi$
$318$	0	0
$319$	89174.2	0.876310
$320$	− 3791.83i	− 0.0370296i
$321$	0	0
$322$	7071.54	0.0682028
$323$	86443.1i	0.828563i
$324$	0	0
$325$	165337.	1.56532
$326$	− 68393.6i	− 0.643547i
$327$	0	0
$328$	11453.2	0.106458
$329$	149662.i	1.38268i
$330$	0	0
$331$	−96237.1	−0.878389	−0.439194	−	0.898392i	$-0.644736\pi$
$331$	−96237.1	−0.878389	−0.439194	+	0.898392i	$0.644736\pi$
$332$	− 41002.3i	− 0.371991i
$333$	0	0
$334$	61562.1	0.551849
$335$	− 25054.9i	− 0.223256i
$336$	0	0
$337$	118823.	1.04626	0.523130	−	0.852253i	$-0.324764\pi$
$337$	118823.	1.04626	0.523130	+	0.852253i	$0.324764\pi$
$338$	− 157068.i	− 1.37485i
$339$	0	0
$340$	27279.8	0.235984
$341$	− 40603.8i	− 0.349187i
$342$	0	0
$343$	69836.5	0.593600
$344$	31788.3i	0.268628i
$345$	0	0
$346$	67898.0	0.567159
$347$	195155.i	1.62077i	0.585901	+	0.810383i	$0.300741\pi$
$347$	195155.i	1.62077i	−0.585901	+	0.810383i	$0.699259\pi$
$348$	0	0
$349$	−78195.1	−0.641991	−0.320995	−	0.947081i	$-0.604017\pi$
$349$	−78195.1	−0.641991	−0.320995	+	0.947081i	$0.604017\pi$
$350$	97365.2i	0.794818i
$351$	0	0
$352$	−11539.2	−0.0931302
$353$	6223.18i	0.0499417i	0.999688	+	0.0249708i	$0.00794929\pi$
$353$	6223.18i	0.0499417i	−0.999688	+	0.0249708i	$0.992051\pi$
$354$	0	0
$355$	−20522.3	−0.162843
$356$	9769.08i	0.0770822i
$357$	0	0
$358$	−116337.	−0.907721
$359$	− 126328.i	− 0.980188i	−0.871670	−	0.490094i	$-0.836962\pi$
$359$	− 126328.i	− 0.980188i	0.871670	−	0.490094i	$-0.163038\pi$
$360$	0	0
$361$	−95074.4	−0.729540
$362$	− 90646.3i	− 0.691724i
$363$	0	0
$364$	140068.	1.05715
$365$	26107.1i	0.195962i
$366$	0	0
$367$	120304.	0.893202	0.446601	−	0.894733i	$-0.352634\pi$
$367$	120304.	0.893202	0.446601	+	0.894733i	$0.352634\pi$
$368$	− 2650.22i	− 0.0195698i
$369$	0	0
$370$	17756.2	0.129702
$371$	− 46683.6i	− 0.339169i
$372$	0	0
$373$	−65510.7	−0.470863	−0.235432	−	0.971891i	$-0.575650\pi$
$373$	−65510.7	−0.470863	−0.235432	+	0.971891i	$0.575650\pi$
$374$	− 83017.1i	− 0.593505i
$375$	0	0
$376$	56089.3	0.396738
$377$	405666.i	2.85421i
$378$	0	0
$379$	129148.	0.899106	0.449553	−	0.893254i	$-0.351583\pi$
$379$	129148.	0.899106	0.449553	+	0.893254i	$0.351583\pi$
$380$	− 11123.1i	− 0.0770301i
$381$	0	0
$382$	−37376.8	−0.256139
$383$	71561.3i	0.487844i	0.969795	+	0.243922i	$0.0784341\pi$
$383$	71561.3i	0.487844i	−0.969795	+	0.243922i	$0.921566\pi$
$384$	0	0
$385$	−28503.4	−0.192299
$386$	− 5054.19i	− 0.0339216i
$387$	0	0
$388$	117882.	0.783038
$389$	19856.7i	0.131222i	0.997845	+	0.0656111i	$0.0208997\pi$
$389$	19856.7i	0.131222i	−0.997845	+	0.0656111i	$0.979100\pi$
$390$	0	0
$391$	19066.6	0.124715
$392$	28155.7i	0.183229i
$393$	0	0
$394$	−49896.0	−0.321420
$395$	43174.9i	0.276718i
$396$	0	0
$397$	35293.6	0.223931	0.111966	−	0.993712i	$-0.464285\pi$
$397$	35293.6	0.223931	0.111966	+	0.993712i	$0.464285\pi$
$398$	− 670.163i	− 0.00423072i
$399$	0	0
$400$	36489.8	0.228061
$401$	− 14851.3i	− 0.0923580i	−0.998933	−	0.0461790i	$-0.985296\pi$
$401$	− 14851.3i	− 0.0923580i	0.998933	−	0.0461790i	$-0.0147045\pi$
$402$	0	0
$403$	184712.	1.13733
$404$	− 138878.i	− 0.850885i
$405$	0	0
$406$	−238891.	−1.44927
$407$	− 54035.3i	− 0.326204i
$408$	0	0
$409$	72058.4	0.430762	0.215381	−	0.976530i	$-0.430901\pi$
$409$	72058.4	0.430762	0.215381	+	0.976530i	$0.430901\pi$
$410$	− 10602.7i	− 0.0630738i
$411$	0	0
$412$	82629.8	0.486791
$413$	− 271542.i	− 1.59198i
$414$	0	0
$415$	−37957.5	−0.220395
$416$	− 52493.5i	− 0.303332i
$417$	0	0
$418$	−33849.7	−0.193732
$419$	− 289136.i	− 1.64692i	−0.567371	−	0.823462i	$-0.692040\pi$
$419$	− 289136.i	− 1.64692i	0.567371	−	0.823462i	$-0.307960\pi$
$420$	0	0
$421$	121927.	0.687914	0.343957	−	0.938985i	$-0.388233\pi$
$421$	121927.	0.687914	0.343957	+	0.938985i	$0.388233\pi$
$422$	− 101705.i	− 0.571109i
$423$	0	0
$424$	−17495.7	−0.0973195
$425$	262520.i	1.45340i
$426$	0	0
$427$	157157.	0.861943
$428$	− 58083.7i	− 0.317079i
$429$	0	0
$430$	29427.7	0.159155
$431$	− 207467.i	− 1.11685i	−0.829555	−	0.558426i	$-0.811406\pi$
$431$	− 207467.i	− 1.11685i	0.829555	−	0.558426i	$-0.188594\pi$
$432$	0	0
$433$	88879.4	0.474051	0.237026	−	0.971503i	$-0.423827\pi$
$433$	88879.4	0.474051	0.237026	+	0.971503i	$0.423827\pi$
$434$	108775.i	0.577495i
$435$	0	0
$436$	−15156.0	−0.0797280
$437$	− 7774.28i	− 0.0407096i
$438$	0	0
$439$	148284.	0.769421	0.384711	−	0.923037i	$-0.374301\pi$
$439$	148284.	0.769421	0.384711	+	0.923037i	$0.374301\pi$
$440$	10682.3i	0.0551772i
$441$	0	0
$442$	377656.	1.93309
$443$	− 205300.i	− 1.04612i	−0.852296	−	0.523060i	$-0.824790\pi$
$443$	− 205300.i	− 1.04612i	0.852296	−	0.523060i	$-0.175210\pi$
$444$	0	0
$445$	9043.63	0.0456691
$446$	154497.i	0.776697i
$447$	0	0
$448$	30912.7	0.154022
$449$	112334.i	0.557211i	0.960406	+	0.278606i	$0.0898722\pi$
$449$	112334.i	0.557211i	−0.960406	+	0.278606i	$0.910128\pi$
$450$	0	0
$451$	−32265.9	−0.158632
$452$	76440.6i	0.374151i
$453$	0	0
$454$	−221081.	−1.07261
$455$	− 129666.i	− 0.626331i
$456$	0	0
$457$	−231021.	−1.10616	−0.553082	−	0.833127i	$-0.686549\pi$
$457$	−231021.	−1.10616	−0.553082	+	0.833127i	$0.686549\pi$
$458$	34051.5i	0.162333i
$459$	0	0
$460$	−2453.41	−0.0115946
$461$	214794.i	1.01069i	0.862916	+	0.505347i	$0.168636\pi$
$461$	214794.i	1.01069i	−0.862916	+	0.505347i	$0.831364\pi$
$462$	0	0
$463$	−219326.	−1.02312	−0.511561	−	0.859247i	$-0.670932\pi$
$463$	−219326.	−1.02312	−0.511561	+	0.859247i	$0.670932\pi$
$464$	89529.9i	0.415846i
$465$	0	0
$466$	269821.	1.24252
$467$	− 62261.6i	− 0.285487i	−0.989760	−	0.142744i	$-0.954408\pi$
$467$	− 62261.6i	− 0.285487i	0.989760	−	0.142744i	$-0.0455924\pi$
$468$	0	0
$469$	204259.	0.928616
$470$	− 51924.0i	− 0.235057i
$471$	0	0
$472$	−101766.	−0.456794
$473$	− 89553.7i	− 0.400278i
$474$	0	0
$475$	107041.	0.474420
$476$	222397.i	0.981557i
$477$	0	0
$478$	−289502.	−1.26706
$479$	− 215383.i	− 0.938730i	−0.883004	−	0.469365i	$-0.844483\pi$
$479$	− 215383.i	− 0.938730i	0.883004	−	0.469365i	$-0.155517\pi$
$480$	0	0
$481$	245814.	1.06247
$482$	151795.i	0.653378i
$483$	0	0
$484$	−84619.9	−0.361228
$485$	− 109128.i	− 0.463929i
$486$	0	0
$487$	−387518.	−1.63393	−0.816966	−	0.576687i	$-0.804345\pi$
$487$	−387518.	−1.63393	−0.816966	+	0.576687i	$0.804345\pi$
$488$	− 58898.1i	− 0.247321i
$489$	0	0
$490$	26064.8	0.108558
$491$	79826.4i	0.331118i	0.986200	+	0.165559i	$0.0529429\pi$
$491$	79826.4i	0.331118i	−0.986200	+	0.165559i	$0.947057\pi$
$492$	0	0
$493$	−644110.	−2.65012
$494$	− 153987.i	− 0.631001i
$495$	0	0
$496$	40765.7	0.165704
$497$	− 167307.i	− 0.677333i
$498$	0	0
$499$	73445.5	0.294960	0.147480	−	0.989065i	$-0.452884\pi$
$499$	73445.5	0.294960	0.147480	+	0.989065i	$0.452884\pi$
$500$	− 70809.6i	− 0.283238i
$501$	0	0
$502$	−282248.	−1.12001
$503$	465077.i	1.83818i	0.394045	+	0.919091i	$0.371075\pi$
$503$	465077.i	1.83818i	−0.394045	+	0.919091i	$0.628925\pi$
$504$	0	0
$505$	−128565.	−0.504127
$506$	7466.16i	0.0291606i
$507$	0	0
$508$	−60391.7	−0.234018
$509$	− 95308.5i	− 0.367872i	−0.982938	−	0.183936i	$-0.941116\pi$
$509$	− 95308.5i	− 0.367872i	0.982938	−	0.183936i	$-0.0588839\pi$
$510$	0	0
$511$	−212837.	−0.815089
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−7346.50	−0.0278070
$515$	− 76493.7i	− 0.288410i
$516$	0	0
$517$	−158014.	−0.591173
$518$	144757.i	0.539485i
$519$	0	0
$520$	−48595.3	−0.179716
$521$	173345.i	0.638612i	0.947652	+	0.319306i	$0.103450\pi$
$521$	173345.i	0.638612i	−0.947652	+	0.319306i	$0.896550\pi$
$522$	0	0
$523$	−403535.	−1.47529	−0.737646	−	0.675187i	$-0.764063\pi$
$523$	−403535.	−1.47529	−0.737646	+	0.675187i	$0.764063\pi$
$524$	38578.4i	0.140502i
$525$	0	0
$526$	357000.	1.29032
$527$	293283.i	1.05600i
$528$	0	0
$529$	278126.	0.993872
$530$	16196.5i	0.0576592i
$531$	0	0
$532$	90681.0	0.320400
$533$	− 146782.i	− 0.516676i
$534$	0	0
$535$	−53770.4	−0.187861
$536$	− 76550.7i	− 0.266452i
$537$	0	0
$538$	40657.7	0.140468
$539$	− 79319.8i	− 0.273026i
$540$	0	0
$541$	280182.	0.957295	0.478648	−	0.878007i	$-0.341127\pi$
$541$	280182.	0.957295	0.478648	+	0.878007i	$0.341127\pi$
$542$	153077.i	0.521089i
$543$	0	0
$544$	83348.3	0.281643
$545$	14030.5i	0.0472367i
$546$	0	0
$547$	−96439.8	−0.322316	−0.161158	−	0.986929i	$-0.551523\pi$
$547$	−96439.8	−0.322316	−0.161158	+	0.986929i	$0.551523\pi$
$548$	86343.6i	0.287521i
$549$	0	0
$550$	−102799.	−0.339830
$551$	262632.i	0.865055i
$552$	0	0
$553$	−351982.	−1.15099
$554$	− 89411.7i	− 0.291323i
$555$	0	0
$556$	−226218.	−0.731774
$557$	− 403818.i	− 1.30159i	−0.759252	−	0.650797i	$-0.774435\pi$
$557$	− 403818.i	− 1.30159i	0.759252	−	0.650797i	$-0.225565\pi$
$558$	0	0
$559$	407392.	1.30373
$560$	− 28617.1i	− 0.0912537i
$561$	0	0
$562$	−289008.	−0.915035
$563$	− 14791.0i	− 0.0466639i	−0.999728	−	0.0233320i	$-0.992573\pi$
$563$	− 14791.0i	− 0.0466639i	0.999728	−	0.0233320i	$-0.00742747\pi$
$564$	0	0
$565$	70764.1	0.221675
$566$	411067.i	1.28316i
$567$	0	0
$568$	−62702.1	−0.194350
$569$	− 243297.i	− 0.751470i	−0.926727	−	0.375735i	$-0.877390\pi$
$569$	− 243297.i	− 0.751470i	0.926727	−	0.375735i	$-0.122610\pi$
$570$	0	0
$571$	−434303.	−1.33205	−0.666025	−	0.745930i	$-0.732005\pi$
$571$	−434303.	−1.33205	−0.666025	+	0.745930i	$0.732005\pi$
$572$	147884.i	0.451990i
$573$	0	0
$574$	86438.1	0.262350
$575$	− 23609.8i	− 0.0714096i
$576$	0	0
$577$	−404030.	−1.21356	−0.606781	−	0.794869i	$-0.707540\pi$
$577$	−404030.	−1.21356	−0.606781	+	0.794869i	$0.707540\pi$
$578$	363404.i	1.08776i
$579$	0	0
$580$	82881.4	0.246377
$581$	− 309447.i	− 0.916714i
$582$	0	0
$583$	49288.7	0.145014
$584$	79765.4i	0.233878i
$585$	0	0
$586$	−130798.	−0.380894
$587$	204320.i	0.592972i	0.955037	+	0.296486i	$0.0958147\pi$
$587$	204320.i	0.592972i	−0.955037	+	0.296486i	$0.904185\pi$
$588$	0	0
$589$	119584.	0.344702
$590$	94209.1i	0.270638i
$591$	0	0
$592$	54250.9	0.154797
$593$	− 569645.i	− 1.61993i	−0.586481	−	0.809963i	$-0.699487\pi$
$593$	− 569645.i	− 1.61993i	0.586481	−	0.809963i	$-0.300513\pi$
$594$	0	0
$595$	205882.	0.581546
$596$	78120.9i	0.219925i
$597$	0	0
$598$	−33964.6	−0.0949782
$599$	279594.i	0.779246i	0.920975	+	0.389623i	$0.127395\pi$
$599$	279594.i	0.779246i	−0.920975	+	0.389623i	$0.872605\pi$
$600$	0	0
$601$	228527.	0.632687	0.316344	−	0.948645i	$-0.397545\pi$
$601$	228527.	0.632687	0.316344	+	0.948645i	$0.397545\pi$
$602$	239908.i	0.661991i
$603$	0	0
$604$	−6135.06	−0.0168169
$605$	78336.0i	0.214018i
$606$	0	0
$607$	−191339.	−0.519308	−0.259654	−	0.965702i	$-0.583609\pi$
$607$	−191339.	−0.519308	−0.259654	+	0.965702i	$0.583609\pi$
$608$	− 33984.7i	− 0.0919341i
$609$	0	0
$610$	−54524.3	−0.146531
$611$	− 718828.i	− 1.92550i
$612$	0	0
$613$	−446105.	−1.18718	−0.593589	−	0.804768i	$-0.702290\pi$
$613$	−446105.	−1.18718	−0.593589	+	0.804768i	$0.702290\pi$
$614$	30367.1i	0.0805502i
$615$	0	0
$616$	−87087.1	−0.229505
$617$	− 327242.i	− 0.859604i	−0.902923	−	0.429802i	$-0.858583\pi$
$617$	− 327242.i	− 0.859604i	0.902923	−	0.429802i	$-0.141417\pi$
$618$	0	0
$619$	110326.	0.287937	0.143968	−	0.989582i	$-0.454014\pi$
$619$	110326.	0.287937	0.143968	+	0.989582i	$0.454014\pi$
$620$	− 37738.5i	− 0.0981749i
$621$	0	0
$622$	−628.051	−0.00162336
$623$	73727.8i	0.189957i
$624$	0	0
$625$	290794.	0.744433
$626$	− 414864.i	− 1.05866i
$627$	0	0
$628$	41534.6	0.105315
$629$	390300.i	0.986500i
$630$	0	0
$631$	321684.	0.807925	0.403962	−	0.914776i	$-0.367633\pi$
$631$	321684.	0.807925	0.403962	+	0.914776i	$0.367633\pi$
$632$	131913.i	0.330258i
$633$	0	0
$634$	499056.	1.24157
$635$	55907.0i	0.138650i
$636$	0	0
$637$	360837.	0.889267
$638$	− 252223.i	− 0.619645i
$639$	0	0
$640$	−10724.9	−0.0261839
$641$	− 248351.i	− 0.604436i	−0.953239	−	0.302218i	$-0.902273\pi$
$641$	− 248351.i	− 0.604436i	0.953239	−	0.302218i	$-0.0977271\pi$
$642$	0	0
$643$	437361.	1.05784	0.528918	−	0.848673i	$-0.322598\pi$
$643$	437361.	1.05784	0.528918	+	0.848673i	$0.322598\pi$
$644$	− 20001.3i	− 0.0482266i
$645$	0	0
$646$	244498.	0.585882
$647$	721599.i	1.72380i	0.507076	+	0.861901i	$0.330726\pi$
$647$	721599.i	1.72380i	−0.507076	+	0.861901i	$0.669274\pi$
$648$	0	0
$649$	286695.	0.680661
$650$	− 467645.i	− 1.10685i
$651$	0	0
$652$	−193446.	−0.455057
$653$	− 127461.i	− 0.298918i	−0.988768	−	0.149459i	$-0.952247\pi$
$653$	− 127461.i	− 0.298918i	0.988768	−	0.149459i	$-0.0477532\pi$
$654$	0	0
$655$	35713.6	0.0832436
$656$	− 32394.6i	− 0.0752775i
$657$	0	0
$658$	423309.	0.977700
$659$	− 158294.i	− 0.364496i	−0.983253	−	0.182248i	$-0.941663\pi$
$659$	− 158294.i	− 0.364496i	0.983253	−	0.182248i	$-0.0583374\pi$
$660$	0	0
$661$	−570017.	−1.30462	−0.652311	−	0.757951i	$-0.726201\pi$
$661$	−570017.	−1.30462	−0.652311	+	0.757951i	$0.726201\pi$
$662$	272200.i	0.621114i
$663$	0	0
$664$	−115972.	−0.263037
$665$	− 83947.0i	− 0.189829i
$666$	0	0
$667$	57928.1	0.130208
$668$	− 174124.i	− 0.390216i
$669$	0	0
$670$	−70866.0	−0.157866
$671$	165927.i	0.368530i
$672$	0	0
$673$	−269354.	−0.594694	−0.297347	−	0.954770i	$-0.596102\pi$
$673$	−269354.	−0.594694	−0.297347	+	0.954770i	$0.596102\pi$
$674$	− 336081.i	− 0.739817i
$675$	0	0
$676$	−444257.	−0.972166
$677$	− 454479.i	− 0.991600i	−0.868437	−	0.495800i	$-0.834875\pi$
$677$	− 454479.i	− 0.991600i	0.868437	−	0.495800i	$-0.165125\pi$
$678$	0	0
$679$	889660.	1.92968
$680$	− 77158.8i	− 0.166866i
$681$	0	0
$682$	−114845.	−0.246912
$683$	225234.i	0.482827i	0.970422	+	0.241414i	$0.0776110\pi$
$683$	225234.i	0.482827i	−0.970422	+	0.241414i	$0.922389\pi$
$684$	0	0
$685$	79931.7	0.170348
$686$	− 197527.i	− 0.419739i
$687$	0	0
$688$	89911.0	0.189948
$689$	224221.i	0.472322i
$690$	0	0
$691$	−504881.	−1.05738	−0.528692	−	0.848814i	$-0.677317\pi$
$691$	−504881.	−1.05738	−0.528692	+	0.848814i	$0.677317\pi$
$692$	− 192045.i	− 0.401042i
$693$	0	0
$694$	551981.	1.14605
$695$	209419.i	0.433557i
$696$	0	0
$697$	233058.	0.479732
$698$	221169.i	0.453956i
$699$	0	0
$700$	275390.	0.562021
$701$	19860.8i	0.0404166i	0.999796	+	0.0202083i	$0.00643294\pi$
$701$	19860.8i	0.0404166i	−0.999796	+	0.0202083i	$0.993567\pi$
$702$	0	0
$703$	159142.	0.322014
$704$	32637.8i	0.0658530i
$705$	0	0
$706$	17601.8	0.0353141
$707$	− 1.04812e6i	− 2.09688i
$708$	0	0
$709$	630841.	1.25495	0.627476	−	0.778636i	$-0.284088\pi$
$709$	630841.	1.25495	0.627476	+	0.778636i	$0.284088\pi$
$710$	58045.8i	0.115147i
$711$	0	0
$712$	27631.1	0.0545053
$713$	− 26376.5i	− 0.0518845i
$714$	0	0
$715$	136902.	0.267792
$716$	329051.i	0.641855i
$717$	0	0
$718$	−357308.	−0.693097
$719$	337555.i	0.652960i	0.945204	+	0.326480i	$0.105863\pi$
$719$	337555.i	0.652960i	−0.945204	+	0.326480i	$0.894137\pi$
$720$	0	0
$721$	623612.	1.19962
$722$	268911.i	0.515863i
$723$	0	0
$724$	−256387.	−0.489123
$725$	797589.i	1.51741i
$726$	0	0
$727$	−514962.	−0.974330	−0.487165	−	0.873310i	$-0.661969\pi$
$727$	−514962.	−0.974330	−0.487165	+	0.873310i	$0.661969\pi$
$728$	− 396171.i	− 0.747515i
$729$	0	0
$730$	73841.9	0.138566
$731$	646851.i	1.21051i
$732$	0	0
$733$	1.01614e6	1.89123	0.945617	−	0.325282i	$-0.105459\pi$
$733$	1.01614e6	1.89123	0.945617	+	0.325282i	$0.105459\pi$
$734$	− 340272.i	− 0.631589i
$735$	0	0
$736$	−7495.94	−0.0138379
$737$	215658.i	0.397036i
$738$	0	0
$739$	−663801.	−1.21548	−0.607742	−	0.794135i	$-0.707924\pi$
$739$	−663801.	−1.21548	−0.607742	+	0.794135i	$0.707924\pi$
$740$	− 50222.2i	− 0.0917132i
$741$	0	0
$742$	−132041.	−0.239829
$743$	715436.i	1.29596i	0.761656	+	0.647982i	$0.224387\pi$
$743$	715436.i	1.29596i	−0.761656	+	0.647982i	$0.775613\pi$
$744$	0	0
$745$	72319.6	0.130300
$746$	185292.i	0.332950i
$747$	0	0
$748$	−234808.	−0.419671
$749$	− 438361.i	− 0.781391i
$750$	0	0
$751$	−443186.	−0.785790	−0.392895	−	0.919583i	$-0.628526\pi$
$751$	−443186.	−0.785790	−0.392895	+	0.919583i	$0.628526\pi$
$752$	− 158644.i	− 0.280536i
$753$	0	0
$754$	1.14740e6	2.01823
$755$	5679.47i	0.00996355i
$756$	0	0
$757$	−270550.	−0.472123	−0.236061	−	0.971738i	$-0.575857\pi$
$757$	−270550.	−0.472123	−0.236061	+	0.971738i	$0.575857\pi$
$758$	− 365287.i	− 0.635764i
$759$	0	0
$760$	−31461.0	−0.0544685
$761$	− 586298.i	− 1.01239i	−0.862418	−	0.506196i	$-0.831051\pi$
$761$	− 586298.i	− 1.01239i	0.862418	−	0.506196i	$-0.168949\pi$
$762$	0	0
$763$	−114383.	−0.196477
$764$	105718.i	0.181118i
$765$	0	0
$766$	202406.	0.344958
$767$	1.30422e6i	2.21696i
$768$	0	0
$769$	−452818.	−0.765721	−0.382861	−	0.923806i	$-0.625061\pi$
$769$	−452818.	−0.765721	−0.382861	+	0.923806i	$0.625061\pi$
$770$	80619.9i	0.135976i
$771$	0	0
$772$	−14295.4	−0.0239862
$773$	217175.i	0.363455i	0.983349	+	0.181728i	$0.0581689\pi$
$773$	217175.i	0.363455i	−0.983349	+	0.181728i	$0.941831\pi$
$774$	0	0
$775$	363167.	0.604648
$776$	− 333420.i	− 0.553691i
$777$	0	0
$778$	56163.2	0.0927881
$779$	− 95028.0i	− 0.156595i
$780$	0	0
$781$	176644.	0.289598
$782$	− 53928.4i	− 0.0881870i
$783$	0	0
$784$	79636.2	0.129562
$785$	− 38450.2i	− 0.0623964i
$786$	0	0
$787$	−46838.0	−0.0756221	−0.0378110	−	0.999285i	$-0.512038\pi$
$787$	−46838.0	−0.0756221	−0.0378110	+	0.999285i	$0.512038\pi$
$788$	141127.i	0.227278i
$789$	0	0
$790$	122117.	0.195669
$791$	576902.i	0.922038i
$792$	0	0
$793$	−754826.	−1.20033
$794$	− 99825.3i	− 0.158343i
$795$	0	0
$796$	−1895.51	−0.00299157
$797$	374693.i	0.589874i	0.955517	+	0.294937i	$0.0952987\pi$
$797$	374693.i	0.589874i	−0.955517	+	0.294937i	$0.904701\pi$
$798$	0	0
$799$	1.14134e6	1.78782
$800$	− 103209.i	− 0.161263i
$801$	0	0
$802$	−42005.7	−0.0653070
$803$	− 224714.i	− 0.348497i
$804$	0	0
$805$	−18516.0	−0.0285730
$806$	− 522445.i	− 0.804212i
$807$	0	0
$808$	−392807.	−0.601667
$809$	− 1.02586e6i	− 1.56745i	−0.621109	−	0.783724i	$-0.713318\pi$
$809$	− 1.02586e6i	− 1.56745i	0.621109	−	0.783724i	$-0.286682\pi$
$810$	0	0
$811$	869028.	1.32127	0.660636	−	0.750707i	$-0.270287\pi$
$811$	869028.	1.32127	0.660636	+	0.750707i	$0.270287\pi$
$812$	675687.i	1.02479i
$813$	0	0
$814$	−152835.	−0.230661
$815$	179081.i	0.269609i
$816$	0	0
$817$	263749.	0.395137
$818$	− 203812.i	− 0.304595i
$819$	0	0
$820$	−29989.0	−0.0445999
$821$	542695.i	0.805137i	0.915390	+	0.402568i	$0.131882\pi$
$821$	542695.i	0.805137i	−0.915390	+	0.402568i	$0.868118\pi$
$822$	0	0
$823$	−571565.	−0.843851	−0.421926	−	0.906630i	$-0.638646\pi$
$823$	−571565.	−0.843851	−0.421926	+	0.906630i	$0.638646\pi$
$824$	− 233712.i	− 0.344213i
$825$	0	0
$826$	−768036.	−1.12570
$827$	− 143023.i	− 0.209120i	−0.994519	−	0.104560i	$-0.966657\pi$
$827$	− 143023.i	− 0.209120i	0.994519	−	0.104560i	$-0.0333434\pi$
$828$	0	0
$829$	−17805.6	−0.0259087	−0.0129544	−	0.999916i	$-0.504124\pi$
$829$	−17805.6	−0.0259087	−0.0129544	+	0.999916i	$0.504124\pi$
$830$	107360.i	0.155843i
$831$	0	0
$832$	−148474.	−0.214488
$833$	572931.i	0.825681i
$834$	0	0
$835$	−161193.	−0.231193
$836$	95741.4i	0.136990i
$837$	0	0
$838$	−817799.	−1.16455
$839$	963976.i	1.36944i	0.728807	+	0.684719i	$0.240075\pi$
$839$	963976.i	1.36944i	−0.728807	+	0.684719i	$0.759925\pi$
$840$	0	0
$841$	−1.24965e6	−1.76684
$842$	− 344860.i	− 0.486429i
$843$	0	0
$844$	−287666.	−0.403835
$845$	411266.i	0.575983i
$846$	0	0
$847$	−638631.	−0.890191
$848$	49485.3i	0.0688152i
$849$	0	0
$850$	742519.	1.02771
$851$	− 35101.7i	− 0.0484695i
$852$	0	0
$853$	−1.07360e6	−1.47552	−0.737760	−	0.675063i	$-0.764117\pi$
$853$	−1.07360e6	−1.47552	−0.737760	+	0.675063i	$0.764117\pi$
$854$	− 444508.i	− 0.609486i
$855$	0	0
$856$	−164286.	−0.224208
$857$	− 1.09814e6i	− 1.49519i	−0.664155	−	0.747595i	$-0.731209\pi$
$857$	− 1.09814e6i	− 1.49519i	0.664155	−	0.747595i	$-0.268791\pi$
$858$	0	0
$859$	−892.808	−0.00120996	−0.000604981	−	1.00000i	$-0.500193\pi$
$859$	−892.808	−0.00120996	−0.000604981		1.00000i	$0.500193\pi$
$860$	− 83234.1i	− 0.112539i
$861$	0	0
$862$	−586807.	−0.789733
$863$	− 395494.i	− 0.531029i	−0.964107	−	0.265514i	$-0.914458\pi$
$863$	− 395494.i	− 0.531029i	0.964107	−	0.265514i	$-0.0855418\pi$
$864$	0	0
$865$	−177783.	−0.237607
$866$	− 251389.i	− 0.335205i
$867$	0	0
$868$	307661.	0.408351
$869$	− 371624.i	− 0.492113i
$870$	0	0
$871$	−981057.	−1.29318
$872$	42867.6i	0.0563762i
$873$	0	0
$874$	−21989.0	−0.0287861
$875$	− 534404.i	− 0.697997i
$876$	0	0
$877$	−661587.	−0.860177	−0.430088	−	0.902787i	$-0.641518\pi$
$877$	−661587.	−0.860177	−0.430088	+	0.902787i	$0.641518\pi$
$878$	− 419410.i	− 0.544063i
$879$	0	0
$880$	30214.1	0.0390162
$881$	− 419496.i	− 0.540476i	−0.962794	−	0.270238i	$-0.912898\pi$
$881$	− 419496.i	− 0.540476i	0.962794	−	0.270238i	$-0.0871024\pi$
$882$	0	0
$883$	−498161.	−0.638923	−0.319461	−	0.947599i	$-0.603502\pi$
$883$	−498161.	−0.638923	−0.319461	+	0.947599i	$0.603502\pi$
$884$	− 1.06817e6i	− 1.36690i
$885$	0	0
$886$	−580676.	−0.739719
$887$	384460.i	0.488656i	0.969693	+	0.244328i	$0.0785674\pi$
$887$	384460.i	0.488656i	−0.969693	+	0.244328i	$0.921433\pi$
$888$	0	0
$889$	−455780.	−0.576702
$890$	− 25579.2i	− 0.0322929i
$891$	0	0
$892$	436985.	0.549207
$893$	− 465376.i	− 0.583580i
$894$	0	0
$895$	304616.	0.380282
$896$	− 87434.4i	− 0.108910i
$897$	0	0
$898$	317729.	0.394008
$899$	891053.i	1.10251i
$900$	0	0
$901$	−356015.	−0.438549
$902$	91261.7i	0.112170i
$903$	0	0
$904$	216207.	0.264565
$905$	237347.i	0.289792i
$906$	0	0
$907$	99712.2	0.121209	0.0606043	−	0.998162i	$-0.480697\pi$
$907$	99712.2	0.121209	0.0606043	+	0.998162i	$0.480697\pi$
$908$	625312.i	0.758447i
$909$	0	0
$910$	−366751.	−0.442883
$911$	− 1.09399e6i	− 1.31819i	−0.752060	−	0.659095i	$-0.770940\pi$
$911$	− 1.09399e6i	− 1.31819i	0.752060	−	0.659095i	$-0.229060\pi$
$912$	0	0
$913$	326715.	0.391948
$914$	653427.i	0.782177i
$915$	0	0
$916$	96312.3	0.114787
$917$	291154.i	0.346245i
$918$	0	0
$919$	102313.	0.121143	0.0605717	−	0.998164i	$-0.480708\pi$
$919$	102313.	0.121143	0.0605717	+	0.998164i	$0.480708\pi$
$920$	6939.29i	0.00819859i
$921$	0	0
$922$	607529.	0.714669
$923$	803577.i	0.943244i
$924$	0	0
$925$	483301.	0.564851
$926$	620346.i	0.723456i
$927$	0	0
$928$	253229.	0.294047
$929$	1.08787e6i	1.26051i	0.776387	+	0.630256i	$0.217050\pi$
$929$	1.08787e6i	1.26051i	−0.776387	+	0.630256i	$0.782950\pi$
$930$	0	0
$931$	233609.	0.269520
$932$	− 763168.i	− 0.878595i
$933$	0	0
$934$	−176102.	−0.201870
$935$	217371.i	0.248644i
$936$	0	0
$937$	1.30162e6	1.48253	0.741265	−	0.671213i	$-0.234226\pi$
$937$	1.30162e6	1.48253	0.741265	+	0.671213i	$0.234226\pi$
$938$	− 577733.i	− 0.656631i
$939$	0	0
$940$	−146863.	−0.166210
$941$	− 1.09614e6i	− 1.23790i	−0.785431	−	0.618950i	$-0.787558\pi$
$941$	− 1.09614e6i	− 1.23790i	0.785431	−	0.618950i	$-0.212442\pi$
$942$	0	0
$943$	−20960.1	−0.0235706
$944$	287839.i	0.323002i
$945$	0	0
$946$	−253296.	−0.283039
$947$	− 996741.i	− 1.11143i	−0.831373	−	0.555716i	$-0.812444\pi$
$947$	− 996741.i	− 1.11143i	0.831373	−	0.555716i	$-0.187556\pi$
$948$	0	0
$949$	1.02226e6	1.13508
$950$	− 302757.i	− 0.335465i
$951$	0	0
$952$	629034.	0.694065
$953$	− 501410.i	− 0.552087i	−0.961145	−	0.276044i	$-0.910977\pi$
$953$	− 501410.i	− 0.552087i	0.961145	−	0.276044i	$-0.0890234\pi$
$954$	0	0
$955$	97867.0	0.107307
$956$	818836.i	0.895944i
$957$	0	0
$958$	−609195.	−0.663782
$959$	651640.i	0.708550i
$960$	0	0
$961$	−517797.	−0.560677
$962$	− 695267.i	− 0.751280i
$963$	0	0
$964$	429342.	0.462008
$965$	13233.8i	0.0142112i
$966$	0	0
$967$	210241.	0.224835	0.112418	−	0.993661i	$-0.464141\pi$
$967$	210241.	0.224835	0.112418	+	0.993661i	$0.464141\pi$
$968$	239341.i	0.255427i
$969$	0	0
$970$	−308660.	−0.328047
$971$	− 1.41640e6i	− 1.50227i	−0.660151	−	0.751133i	$-0.729508\pi$
$971$	− 1.41640e6i	− 1.50227i	0.660151	−	0.751133i	$-0.270492\pi$
$972$	0	0
$973$	−1.70728e6	−1.80334
$974$	1.09607e6i	1.15536i
$975$	0	0
$976$	−166589.	−0.174883
$977$	− 1.80397e6i	− 1.88990i	−0.327212	−	0.944951i	$-0.606109\pi$
$977$	− 1.80397e6i	− 1.88990i	0.327212	−	0.944951i	$-0.393891\pi$
$978$	0	0
$979$	−77842.2	−0.0812175
$980$	− 73722.4i	− 0.0767622i
$981$	0	0
$982$	225783.	0.234136
$983$	1.11624e6i	1.15518i	0.816327	+	0.577590i	$0.196007\pi$
$983$	1.11624e6i	1.15518i	−0.816327	+	0.577590i	$0.803993\pi$
$984$	0	0
$985$	130647.	0.134656
$986$	1.82182e6i	1.87392i
$987$	0	0
$988$	−435541.	−0.446185
$989$	− 58174.7i	− 0.0594759i
$990$	0	0
$991$	−888178.	−0.904383	−0.452192	−	0.891921i	$-0.649358\pi$
$991$	−888178.	−0.904383	−0.452192	+	0.891921i	$0.649358\pi$
$992$	− 115303.i	− 0.117170i
$993$	0	0
$994$	−473216.	−0.478947
$995$	1754.75i	0.00177243i
$996$	0	0
$997$	809681.	0.814561	0.407280	−	0.913303i	$-0.366477\pi$
$997$	809681.	0.814561	0.407280	+	0.913303i	$0.366477\pi$
$998$	− 207735.i	− 0.208569i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.5.b.c.161.3		8
3.2	odd	2	inner	162.5.b.c.161.6		8
4.3	odd	2		1296.5.e.e.161.6		8
9.2	odd	6		18.5.d.a.5.3	✓	8
9.4	even	3		18.5.d.a.11.3	yes	8
9.5	odd	6		54.5.d.a.35.1		8
9.7	even	3		54.5.d.a.17.1		8
12.11	even	2		1296.5.e.e.161.3		8
36.7	odd	6		432.5.q.b.17.2		8
36.11	even	6		144.5.q.b.113.3		8
36.23	even	6		432.5.q.b.305.2		8
36.31	odd	6		144.5.q.b.65.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.5.d.a.5.3	✓	8	9.2	odd	6
18.5.d.a.11.3	yes	8	9.4	even	3
54.5.d.a.17.1		8	9.7	even	3
54.5.d.a.35.1		8	9.5	odd	6
144.5.q.b.65.3		8	36.31	odd	6
144.5.q.b.113.3		8	36.11	even	6
162.5.b.c.161.3		8	1.1	even	1	trivial
162.5.b.c.161.6		8	3.2	odd	2	inner
432.5.q.b.17.2		8	36.7	odd	6
432.5.q.b.305.2		8	36.23	even	6
1296.5.e.e.161.3		8	12.11	even	2
1296.5.e.e.161.6		8	4.3	odd	2

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +7.40592i q^{5} -60.3765 q^{7} +22.6274i q^{8} +20.9471 q^{10} -63.7457i q^{11} +289.988 q^{13} +170.770i q^{14} +64.0000 q^{16} +460.439i q^{17} +187.741 q^{19} -59.2473i q^{20} -180.300 q^{22} -41.4096i q^{23} +570.152 q^{25} -820.210i q^{26} +483.012 q^{28} +1398.90i q^{29} +636.965 q^{31} -181.019i q^{32} +1302.32 q^{34} -447.143i q^{35} +847.670 q^{37} -531.011i q^{38} -167.577 q^{40} -506.166i q^{41} +1404.86 q^{43} +509.966i q^{44} -117.124 q^{46} -2478.82i q^{47} +1244.32 q^{49} -1612.63i q^{50} -2319.90 q^{52} +773.208i q^{53} +472.095 q^{55} -1366.16i q^{56} +3956.70 q^{58} +4497.48i q^{59} -2602.95 q^{61} -1801.61i q^{62} -512.000 q^{64} +2147.63i q^{65} -3383.10 q^{67} -3683.51i q^{68} -1264.71 q^{70} +2771.07i q^{71} +3525.16 q^{73} -2397.57i q^{74} -1501.93 q^{76} +3848.74i q^{77} +5829.79 q^{79} +473.979i q^{80} -1431.65 q^{82} +5125.29i q^{83} -3409.97 q^{85} -3973.54i q^{86} +1442.40 q^{88} -1221.14i q^{89} -17508.5 q^{91} +331.277i q^{92} -7011.16 q^{94} +1390.39i q^{95} -14735.2 q^{97} -3519.46i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4} + 52 q^{7} - 20 q^{13} + 512 q^{16} + 100 q^{19} - 672 q^{22} - 1588 q^{25} - 416 q^{28} + 2956 q^{31} + 192 q^{34} - 32 q^{37} + 136 q^{43} + 2112 q^{46} - 4884 q^{49} + 160 q^{52} - 3996 q^{55}+ \cdots - 62672 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 + 52 * q^7 - 20 * q^13 + 512 * q^16 + 100 * q^19 - 672 * q^22 - 1588 * q^25 - 416 * q^28 + 2956 * q^31 + 192 * q^34 - 32 * q^37 + 136 * q^43 + 2112 * q^46 - 4884 * q^49 + 160 * q^52 - 3996 * q^55 + 4800 * q^58 + 8956 * q^61 - 4096 * q^64 - 15008 * q^67 - 12096 * q^70 + 20716 * q^73 - 800 * q^76 + 12100 * q^79 + 1152 * q^82 + 32184 * q^85 + 5376 * q^88 - 45868 * q^91 - 1344 * q^94 - 62672 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more