LMFDB - Embedded newform 387.6.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,6,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	387.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:	$62.0685382676$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.58275$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.582753 q^{2} -31.6604 q^{4} -27.7074 q^{5} -103.690 q^{7} +37.0983 q^{8} +O(q^{10})$	\(q-0.582753 q^{2} -31.6604 q^{4} -27.7074 q^{5} -103.690 q^{7} +37.0983 q^{8} +16.1466 q^{10} +158.323 q^{11} -578.882 q^{13} +60.4256 q^{14} +991.514 q^{16} -253.871 q^{17} -3092.51 q^{19} +877.227 q^{20} -92.2633 q^{22} -4163.45 q^{23} -2357.30 q^{25} +337.345 q^{26} +3282.87 q^{28} -6771.63 q^{29} +6264.06 q^{31} -1764.95 q^{32} +147.944 q^{34} +2872.98 q^{35} -3294.82 q^{37} +1802.17 q^{38} -1027.90 q^{40} +6150.84 q^{41} -1849.00 q^{43} -5012.58 q^{44} +2426.26 q^{46} -8157.17 q^{47} -6055.38 q^{49} +1373.72 q^{50} +18327.6 q^{52} +30457.8 q^{53} -4386.72 q^{55} -3846.72 q^{56} +3946.19 q^{58} +45236.1 q^{59} -7251.18 q^{61} -3650.40 q^{62} -30699.9 q^{64} +16039.3 q^{65} +19685.7 q^{67} +8037.67 q^{68} -1674.24 q^{70} +48132.2 q^{71} +42502.1 q^{73} +1920.07 q^{74} +97910.0 q^{76} -16416.5 q^{77} -65151.9 q^{79} -27472.3 q^{80} -3584.42 q^{82} -70403.6 q^{83} +7034.12 q^{85} +1077.51 q^{86} +5873.52 q^{88} -29645.1 q^{89} +60024.2 q^{91} +131816. q^{92} +4753.61 q^{94} +85685.3 q^{95} -89440.4 q^{97} +3528.79 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) $ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8	$ 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) $ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * 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 $

$2$	−0.582753	−0.103017	−0.0515085	−	0.998673i	$-0.516403\pi$
$2$	−0.582753	−0.103017	−0.0515085	+	0.998673i	$0.516403\pi$
$3$	0	0
$3$	0	0
$4$	−31.6604	−0.989387
$5$	−27.7074	−0.495645	−0.247822	−	0.968805i	$-0.579715\pi$
$5$	−27.7074	−0.495645	−0.247822	+	0.968805i	$0.579715\pi$
$6$	0	0
$7$	−103.690	−0.799819	−0.399910	−	0.916555i	$-0.630958\pi$
$7$	−103.690	−0.799819	−0.399910	+	0.916555i	$0.630958\pi$
$8$	37.0983	0.204941
$9$	0	0
$10$	16.1466	0.0510599
$11$	158.323	0.394515	0.197257	−	0.980352i	$-0.436797\pi$
$11$	158.323	0.394515	0.197257	+	0.980352i	$0.436797\pi$
$12$	0	0
$13$	−578.882	−0.950017	−0.475009	−	0.879981i	$-0.657555\pi$
$13$	−578.882	−0.950017	−0.475009	+	0.879981i	$0.657555\pi$
$14$	60.4256	0.0823950
$15$	0	0
$16$	991.514	0.968275
$17$	−253.871	−0.213055	−0.106527	−	0.994310i	$-0.533973\pi$
$17$	−253.871	−0.213055	−0.106527	+	0.994310i	$0.533973\pi$
$18$	0	0
$19$	−3092.51	−1.96529	−0.982645	−	0.185494i	$-0.940612\pi$
$19$	−3092.51	−1.96529	−0.982645	+	0.185494i	$0.940612\pi$
$20$	877.227	0.490385
$21$	0	0
$22$	−92.2633	−0.0406417
$23$	−4163.45	−1.64109	−0.820547	−	0.571579i	$-0.806331\pi$
$23$	−4163.45	−1.64109	−0.820547	+	0.571579i	$0.806331\pi$
$24$	0	0
$25$	−2357.30	−0.754336
$26$	337.345	0.0978680
$27$	0	0
$28$	3282.87	0.791331
$29$	−6771.63	−1.49520	−0.747598	−	0.664151i	$-0.768793\pi$
$29$	−6771.63	−1.49520	−0.747598	+	0.664151i	$0.768793\pi$
$30$	0	0
$31$	6264.06	1.17072	0.585358	−	0.810775i	$-0.300954\pi$
$31$	6264.06	1.17072	0.585358	+	0.810775i	$0.300954\pi$
$32$	−1764.95	−0.304690
$33$	0	0
$34$	147.944	0.0219483
$35$	2872.98	0.396426
$36$	0	0
$37$	−3294.82	−0.395665	−0.197833	−	0.980236i	$-0.563390\pi$
$37$	−3294.82	−0.395665	−0.197833	+	0.980236i	$0.563390\pi$
$38$	1802.17	0.202459
$39$	0	0
$40$	−1027.90	−0.101578
$41$	6150.84	0.571445	0.285723	−	0.958312i	$-0.407766\pi$
$41$	6150.84	0.571445	0.285723	+	0.958312i	$0.407766\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−5012.58	−0.390328
$45$	0	0
$46$	2426.26	0.169061
$47$	−8157.17	−0.538635	−0.269318	−	0.963051i	$-0.586798\pi$
$47$	−8157.17	−0.538635	−0.269318	+	0.963051i	$0.586798\pi$
$48$	0	0
$49$	−6055.38	−0.360289
$50$	1373.72	0.0777095
$51$	0	0
$52$	18327.6	0.939935
$53$	30457.8	1.48939	0.744697	−	0.667403i	$-0.232594\pi$
$53$	30457.8	1.48939	0.744697	+	0.667403i	$0.232594\pi$
$54$	0	0
$55$	−4386.72	−0.195539
$56$	−3846.72	−0.163916
$57$	0	0
$58$	3946.19	0.154031
$59$	45236.1	1.69182	0.845912	−	0.533322i	$-0.179057\pi$
$59$	45236.1	1.69182	0.845912	+	0.533322i	$0.179057\pi$
$60$	0	0
$61$	−7251.18	−0.249508	−0.124754	−	0.992188i	$-0.539814\pi$
$61$	−7251.18	−0.249508	−0.124754	+	0.992188i	$0.539814\pi$
$62$	−3650.40	−0.120604
$63$	0	0
$64$	−30699.9	−0.936887
$65$	16039.3	0.470871
$66$	0	0
$67$	19685.7	0.535751	0.267876	−	0.963454i	$-0.413678\pi$
$67$	19685.7	0.535751	0.267876	+	0.963454i	$0.413678\pi$
$68$	8037.67	0.210794
$69$	0	0
$70$	−1674.24	−0.0408387
$71$	48132.2	1.13316	0.566579	−	0.824008i	$-0.308267\pi$
$71$	48132.2	1.13316	0.566579	+	0.824008i	$0.308267\pi$
$72$	0	0
$73$	42502.1	0.933475	0.466738	−	0.884396i	$-0.345429\pi$
$73$	42502.1	0.933475	0.466738	+	0.884396i	$0.345429\pi$
$74$	1920.07	0.0407603
$75$	0	0
$76$	97910.0	1.94443
$77$	−16416.5	−0.315540
$78$	0	0
$79$	−65151.9	−1.17452	−0.587258	−	0.809400i	$-0.699793\pi$
$79$	−65151.9	−1.17452	−0.587258	+	0.809400i	$0.699793\pi$
$80$	−27472.3	−0.479921
$81$	0	0
$82$	−3584.42	−0.0588686
$83$	−70403.6	−1.12176	−0.560880	−	0.827897i	$-0.689537\pi$
$83$	−70403.6	−1.12176	−0.560880	+	0.827897i	$0.689537\pi$
$84$	0	0
$85$	7034.12	0.105600
$86$	1077.51	0.0157100
$87$	0	0
$88$	5873.52	0.0808522
$89$	−29645.1	−0.396714	−0.198357	−	0.980130i	$-0.563561\pi$
$89$	−29645.1	−0.396714	−0.198357	+	0.980130i	$0.563561\pi$
$90$	0	0
$91$	60024.2	0.759842
$92$	131816.	1.62368
$93$	0	0
$94$	4753.61	0.0554887
$95$	85685.3	0.974086
$96$	0	0
$97$	−89440.4	−0.965171	−0.482585	−	0.875849i	$-0.660302\pi$
$97$	−89440.4	−0.965171	−0.482585	+	0.875849i	$0.660302\pi$
$98$	3528.79	0.0371160
$99$	0	0
$100$	74633.1	0.746331
$101$	28633.0	0.279295	0.139647	−	0.990201i	$-0.455403\pi$
$101$	28633.0	0.279295	0.139647	+	0.990201i	$0.455403\pi$
$102$	0	0
$103$	30124.9	0.279790	0.139895	−	0.990166i	$-0.455324\pi$
$103$	30124.9	0.279790	0.139895	+	0.990166i	$0.455324\pi$
$104$	−21475.5	−0.194697
$105$	0	0
$106$	−17749.4	−0.153433
$107$	−83073.3	−0.701459	−0.350729	−	0.936477i	$-0.614066\pi$
$107$	−83073.3	−0.701459	−0.350729	+	0.936477i	$0.614066\pi$
$108$	0	0
$109$	58783.5	0.473903	0.236951	−	0.971522i	$-0.423852\pi$
$109$	58783.5	0.473903	0.236951	+	0.971522i	$0.423852\pi$
$110$	2556.38	0.0201439
$111$	0	0
$112$	−102810.	−0.774445
$113$	232647.	1.71396	0.856981	−	0.515347i	$-0.172337\pi$
$113$	232647.	1.71396	0.856981	+	0.515347i	$0.172337\pi$
$114$	0	0
$115$	115358.	0.813400
$116$	214392.	1.47933
$117$	0	0
$118$	−26361.5	−0.174287
$119$	26323.9	0.170405
$120$	0	0
$121$	−135985.	−0.844358
$122$	4225.64	0.0257036
$123$	0	0
$124$	−198323.	−1.15829
$125$	151900.	0.869528
$126$	0	0
$127$	109116.	0.600313	0.300156	−	0.953890i	$-0.402961\pi$
$127$	109116.	0.600313	0.300156	+	0.953890i	$0.402961\pi$
$128$	74368.9	0.401205
$129$	0	0
$130$	−9346.94	−0.0485078
$131$	−139424.	−0.709839	−0.354920	−	0.934897i	$-0.615492\pi$
$131$	−139424.	−0.709839	−0.354920	+	0.934897i	$0.615492\pi$
$132$	0	0
$133$	320662.	1.57188
$134$	−11471.9	−0.0551915
$135$	0	0
$136$	−9418.19	−0.0436637
$137$	64194.6	0.292211	0.146106	−	0.989269i	$-0.453326\pi$
$137$	64194.6	0.292211	0.146106	+	0.989269i	$0.453326\pi$
$138$	0	0
$139$	−281632.	−1.23636	−0.618181	−	0.786036i	$-0.712130\pi$
$139$	−281632.	−1.23636	−0.618181	+	0.786036i	$0.712130\pi$
$140$	−90959.7	−0.392219
$141$	0	0
$142$	−28049.2	−0.116735
$143$	−91650.4	−0.374796
$144$	0	0
$145$	187624.	0.741086
$146$	−24768.2	−0.0961639
$147$	0	0
$148$	104315.	0.391466
$149$	−225391.	−0.831710	−0.415855	−	0.909431i	$-0.636518\pi$
$149$	−225391.	−0.831710	−0.415855	+	0.909431i	$0.636518\pi$
$150$	0	0
$151$	388390.	1.38620	0.693099	−	0.720842i	$-0.256245\pi$
$151$	388390.	1.38620	0.693099	+	0.720842i	$0.256245\pi$
$152$	−114727.	−0.402768
$153$	0	0
$154$	9566.78	0.0325060
$155$	−173561.	−0.580259
$156$	0	0
$157$	−369612.	−1.19673	−0.598366	−	0.801223i	$-0.704183\pi$
$157$	−369612.	−1.19673	−0.598366	+	0.801223i	$0.704183\pi$
$158$	37967.4	0.120995
$159$	0	0
$160$	48902.2	0.151018
$161$	431708.	1.31258
$162$	0	0
$163$	522204.	1.53947	0.769735	−	0.638363i	$-0.220388\pi$
$163$	522204.	1.53947	0.769735	+	0.638363i	$0.220388\pi$
$164$	−194738.	−0.565381
$165$	0	0
$166$	41027.9	0.115560
$167$	17232.5	0.0478141	0.0239071	−	0.999714i	$-0.492389\pi$
$167$	17232.5	0.0478141	0.0239071	+	0.999714i	$0.492389\pi$
$168$	0	0
$169$	−36189.0	−0.0974674
$170$	−4099.15	−0.0108786
$171$	0	0
$172$	58540.1	0.150880
$173$	571643.	1.45214	0.726072	−	0.687618i	$-0.241344\pi$
$173$	571643.	1.45214	0.726072	+	0.687618i	$0.241344\pi$
$174$	0	0
$175$	244429.	0.603333
$176$	156980.	0.381999
$177$	0	0
$178$	17275.7	0.0408683
$179$	191963.	0.447802	0.223901	−	0.974612i	$-0.428121\pi$
$179$	191963.	0.447802	0.223901	+	0.974612i	$0.428121\pi$
$180$	0	0
$181$	110804.	0.251396	0.125698	−	0.992069i	$-0.459883\pi$
$181$	110804.	0.251396	0.125698	+	0.992069i	$0.459883\pi$
$182$	−34979.3	−0.0782767
$183$	0	0
$184$	−154457.	−0.336327
$185$	91290.9	0.196109
$186$	0	0
$187$	−40193.8	−0.0840533
$188$	258259.	0.532919
$189$	0	0
$190$	−49933.3	−0.100348
$191$	−295668.	−0.586437	−0.293218	−	0.956046i	$-0.594726\pi$
$191$	−295668.	−0.586437	−0.293218	+	0.956046i	$0.594726\pi$
$192$	0	0
$193$	−375944.	−0.726491	−0.363246	−	0.931693i	$-0.618331\pi$
$193$	−375944.	−0.726491	−0.363246	+	0.931693i	$0.618331\pi$
$194$	52121.6	0.0994291
$195$	0	0
$196$	191716.	0.356466
$197$	732825.	1.34535	0.672674	−	0.739939i	$-0.265146\pi$
$197$	732825.	1.34535	0.672674	+	0.739939i	$0.265146\pi$
$198$	0	0
$199$	589939.	1.05603	0.528013	−	0.849236i	$-0.322937\pi$
$199$	589939.	1.05603	0.528013	+	0.849236i	$0.322937\pi$
$200$	−87451.8	−0.154594
$201$	0	0
$202$	−16685.9	−0.0287722
$203$	702150.	1.19589
$204$	0	0
$205$	−170424.	−0.283234
$206$	−17555.3	−0.0288231
$207$	0	0
$208$	−573969.	−0.919878
$209$	−489616.	−0.775336
$210$	0	0
$211$	−371450.	−0.574374	−0.287187	−	0.957875i	$-0.592720\pi$
$211$	−371450.	−0.574374	−0.287187	+	0.957875i	$0.592720\pi$
$212$	−964307.	−1.47359
$213$	0	0
$214$	48411.2	0.0722622
$215$	51231.0	0.0755851
$216$	0	0
$217$	−649520.	−0.936361
$218$	−34256.2	−0.0488201
$219$	0	0
$220$	138885.	0.193464
$221$	146962.	0.202406
$222$	0	0
$223$	−835552.	−1.12515	−0.562576	−	0.826746i	$-0.690190\pi$
$223$	−835552.	−1.12515	−0.562576	+	0.826746i	$0.690190\pi$
$224$	183008.	0.243697
$225$	0	0
$226$	−135576.	−0.176567
$227$	363342.	0.468006	0.234003	−	0.972236i	$-0.424818\pi$
$227$	363342.	0.468006	0.234003	+	0.972236i	$0.424818\pi$
$228$	0	0
$229$	−106091.	−0.133688	−0.0668438	−	0.997763i	$-0.521293\pi$
$229$	−106091.	−0.133688	−0.0668438	+	0.997763i	$0.521293\pi$
$230$	−67225.3	−0.0837941
$231$	0	0
$232$	−251216.	−0.306427
$233$	−713760.	−0.861315	−0.430658	−	0.902515i	$-0.641718\pi$
$233$	−713760.	−0.861315	−0.430658	+	0.902515i	$0.641718\pi$
$234$	0	0
$235$	226014.	0.266972
$236$	−1.43219e6	−1.67387
$237$	0	0
$238$	−15340.3	−0.0175547
$239$	70400.0	0.0797220	0.0398610	−	0.999205i	$-0.487308\pi$
$239$	70400.0	0.0797220	0.0398610	+	0.999205i	$0.487308\pi$
$240$	0	0
$241$	44600.6	0.0494650	0.0247325	−	0.999694i	$-0.492127\pi$
$241$	44600.6	0.0494650	0.0247325	+	0.999694i	$0.492127\pi$
$242$	79245.5	0.0869833
$243$	0	0
$244$	229575.	0.246860
$245$	167779.	0.178576
$246$	0	0
$247$	1.79020e6	1.86706
$248$	232386.	0.239928
$249$	0	0
$250$	−88520.3	−0.0895762
$251$	−1.05989e6	−1.06188	−0.530942	−	0.847408i	$-0.678162\pi$
$251$	−1.05989e6	−1.06188	−0.530942	+	0.847408i	$0.678162\pi$
$252$	0	0
$253$	−659170.	−0.647435
$254$	−63587.4	−0.0618425
$255$	0	0
$256$	939058.	0.895556
$257$	966091.	0.912400	0.456200	−	0.889877i	$-0.349210\pi$
$257$	966091.	0.912400	0.456200	+	0.889877i	$0.349210\pi$
$258$	0	0
$259$	341640.	0.316460
$260$	−507811.	−0.465874
$261$	0	0
$262$	81249.8	0.0731255
$263$	−855270.	−0.762454	−0.381227	−	0.924481i	$-0.624498\pi$
$263$	−855270.	−0.762454	−0.381227	+	0.924481i	$0.624498\pi$
$264$	0	0
$265$	−843907.	−0.738210
$266$	−186867.	−0.161930
$267$	0	0
$268$	−623256.	−0.530065
$269$	−1.55224e6	−1.30791	−0.653955	−	0.756533i	$-0.726892\pi$
$269$	−1.55224e6	−1.30791	−0.653955	+	0.756533i	$0.726892\pi$
$270$	0	0
$271$	−4177.31	−0.00345520	−0.00172760	−	0.999999i	$-0.500550\pi$
$271$	−4177.31	−0.00345520	−0.00172760	+	0.999999i	$0.500550\pi$
$272$	−251717.	−0.206296
$273$	0	0
$274$	−37409.5	−0.0301027
$275$	−373216.	−0.297597
$276$	0	0
$277$	−1.59749e6	−1.25095	−0.625475	−	0.780244i	$-0.715095\pi$
$277$	−1.59749e6	−1.25095	−0.625475	+	0.780244i	$0.715095\pi$
$278$	164122.	0.127366
$279$	0	0
$280$	106583.	0.0812439
$281$	−1.60304e6	−1.21110	−0.605549	−	0.795808i	$-0.707046\pi$
$281$	−1.60304e6	−1.21110	−0.605549	+	0.795808i	$0.707046\pi$
$282$	0	0
$283$	2.04664e6	1.51906	0.759530	−	0.650472i	$-0.225429\pi$
$283$	2.04664e6	1.51906	0.759530	+	0.650472i	$0.225429\pi$
$284$	−1.52389e6	−1.12113
$285$	0	0
$286$	53409.5	0.0386103
$287$	−637780.	−0.457053
$288$	0	0
$289$	−1.35541e6	−0.954608
$290$	−109338.	−0.0763446
$291$	0	0
$292$	−1.34563e6	−0.923569
$293$	−1.35028e6	−0.918869	−0.459434	−	0.888212i	$-0.651948\pi$
$293$	−1.35028e6	−0.918869	−0.459434	+	0.888212i	$0.651948\pi$
$294$	0	0
$295$	−1.25337e6	−0.838544
$296$	−122232.	−0.0810880
$297$	0	0
$298$	131347.	0.0856803
$299$	2.41014e6	1.55907
$300$	0	0
$301$	191723.	0.121971
$302$	−226335.	−0.142802
$303$	0	0
$304$	−3.06626e6	−1.90294
$305$	200911.	0.123667
$306$	0	0
$307$	−361103.	−0.218668	−0.109334	−	0.994005i	$-0.534872\pi$
$307$	−361103.	−0.218668	−0.109334	+	0.994005i	$0.534872\pi$
$308$	519754.	0.312192
$309$	0	0
$310$	101143.	0.0597766
$311$	−2.12425e6	−1.24539	−0.622694	−	0.782466i	$-0.713962\pi$
$311$	−2.12425e6	−1.24539	−0.622694	+	0.782466i	$0.713962\pi$
$312$	0	0
$313$	−541717.	−0.312545	−0.156272	−	0.987714i	$-0.549948\pi$
$313$	−541717.	−0.312545	−0.156272	+	0.987714i	$0.549948\pi$
$314$	215392.	0.123284
$315$	0	0
$316$	2.06273e6	1.16205
$317$	3.06921e6	1.71545	0.857725	−	0.514110i	$-0.171878\pi$
$317$	3.06921e6	1.71545	0.857725	+	0.514110i	$0.171878\pi$
$318$	0	0
$319$	−1.07211e6	−0.589877
$320$	850614.	0.464363
$321$	0	0
$322$	−251579.	−0.135218
$323$	785100.	0.418715
$324$	0	0
$325$	1.36460e6	0.716632
$326$	−304316.	−0.158592
$327$	0	0
$328$	228185.	0.117112
$329$	845817.	0.430811
$330$	0	0
$331$	−282184.	−0.141567	−0.0707836	−	0.997492i	$-0.522550\pi$
$331$	−282184.	−0.141567	−0.0707836	+	0.997492i	$0.522550\pi$
$332$	2.22901e6	1.10986
$333$	0	0
$334$	−10042.3	−0.00492567
$335$	−545438.	−0.265542
$336$	0	0
$337$	1.04584e6	0.501640	0.250820	−	0.968034i	$-0.419300\pi$
$337$	1.04584e6	0.501640	0.250820	+	0.968034i	$0.419300\pi$
$338$	21089.2	0.0100408
$339$	0	0
$340$	−222703.	−0.104479
$341$	991746.	0.461864
$342$	0	0
$343$	2.37060e6	1.08799
$344$	−68594.7	−0.0312532
$345$	0	0
$346$	−333127.	−0.149596
$347$	−1.90831e6	−0.850796	−0.425398	−	0.905006i	$-0.639866\pi$
$347$	−1.90831e6	−0.850796	−0.425398	+	0.905006i	$0.639866\pi$
$348$	0	0
$349$	1.46350e6	0.643175	0.321588	−	0.946880i	$-0.395784\pi$
$349$	1.46350e6	0.643175	0.321588	+	0.946880i	$0.395784\pi$
$350$	−142441.	−0.0621536
$351$	0	0
$352$	−279433.	−0.120205
$353$	−867374.	−0.370484	−0.185242	−	0.982693i	$-0.559307\pi$
$353$	−867374.	−0.370484	−0.185242	+	0.982693i	$0.559307\pi$
$354$	0	0
$355$	−1.33362e6	−0.561643
$356$	938575.	0.392504
$357$	0	0
$358$	−111867.	−0.0461313
$359$	−3.45466e6	−1.41472	−0.707359	−	0.706855i	$-0.750113\pi$
$359$	−3.45466e6	−1.41472	−0.707359	+	0.706855i	$0.750113\pi$
$360$	0	0
$361$	7.08751e6	2.86237
$362$	−64571.2	−0.0258981
$363$	0	0
$364$	−1.90039e6	−0.751778
$365$	−1.17762e6	−0.462672
$366$	0	0
$367$	−2.78797e6	−1.08050	−0.540248	−	0.841506i	$-0.681670\pi$
$367$	−2.78797e6	−1.08050	−0.540248	+	0.841506i	$0.681670\pi$
$368$	−4.12811e6	−1.58903
$369$	0	0
$370$	−53200.0	−0.0202026
$371$	−3.15817e6	−1.19125
$372$	0	0
$373$	−1.65578e6	−0.616212	−0.308106	−	0.951352i	$-0.599695\pi$
$373$	−1.65578e6	−0.616212	−0.308106	+	0.951352i	$0.599695\pi$
$374$	23423.0	0.00865892
$375$	0	0
$376$	−302617.	−0.110388
$377$	3.91997e6	1.42046
$378$	0	0
$379$	−2.35538e6	−0.842291	−0.421146	−	0.906993i	$-0.638372\pi$
$379$	−2.35538e6	−0.842291	−0.421146	+	0.906993i	$0.638372\pi$
$380$	−2.71283e6	−0.963749
$381$	0	0
$382$	172301.	0.0604130
$383$	4.13864e6	1.44165	0.720827	−	0.693115i	$-0.243762\pi$
$383$	4.13864e6	1.44165	0.720827	+	0.693115i	$0.243762\pi$
$384$	0	0
$385$	454859.	0.156396
$386$	219083.	0.0748410
$387$	0	0
$388$	2.83172e6	0.954928
$389$	5.27781e6	1.76840	0.884198	−	0.467112i	$-0.154706\pi$
$389$	5.27781e6	1.76840	0.884198	+	0.467112i	$0.154706\pi$
$390$	0	0
$391$	1.05698e6	0.349643
$392$	−224644.	−0.0738380
$393$	0	0
$394$	−427056.	−0.138594
$395$	1.80519e6	0.582143
$396$	0	0
$397$	3.71439e6	1.18280	0.591401	−	0.806378i	$-0.298575\pi$
$397$	3.71439e6	1.18280	0.591401	+	0.806378i	$0.298575\pi$
$398$	−343789.	−0.108789
$399$	0	0
$400$	−2.33730e6	−0.730405
$401$	−3.60034e6	−1.11810	−0.559052	−	0.829132i	$-0.688835\pi$
$401$	−3.60034e6	−1.11810	−0.559052	+	0.829132i	$0.688835\pi$
$402$	0	0
$403$	−3.62615e6	−1.11220
$404$	−906532.	−0.276331
$405$	0	0
$406$	−409180.	−0.123197
$407$	−521647.	−0.156096
$408$	0	0
$409$	1.65305e6	0.488627	0.244314	−	0.969696i	$-0.421437\pi$
$409$	1.65305e6	0.488627	0.244314	+	0.969696i	$0.421437\pi$
$410$	99314.8	0.0291779
$411$	0	0
$412$	−953765.	−0.276821
$413$	−4.69053e6	−1.35315
$414$	0	0
$415$	1.95070e6	0.555994
$416$	1.02170e6	0.289461
$417$	0	0
$418$	285325.	0.0798728
$419$	−1.25123e6	−0.348180	−0.174090	−	0.984730i	$-0.555698\pi$
$419$	−1.25123e6	−0.348180	−0.174090	+	0.984730i	$0.555698\pi$
$420$	0	0
$421$	3.12108e6	0.858221	0.429111	−	0.903252i	$-0.358827\pi$
$421$	3.12108e6	0.858221	0.429111	+	0.903252i	$0.358827\pi$
$422$	216464.	0.0591703
$423$	0	0
$424$	1.12993e6	0.305238
$425$	598451.	0.160715
$426$	0	0
$427$	751875.	0.199561
$428$	2.63013e6	0.694015
$429$	0	0
$430$	−29855.0	−0.00778656
$431$	−246469.	−0.0639102	−0.0319551	−	0.999489i	$-0.510173\pi$
$431$	−246469.	−0.0639102	−0.0319551	+	0.999489i	$0.510173\pi$
$432$	0	0
$433$	−7.56031e6	−1.93785	−0.968924	−	0.247357i	$-0.920438\pi$
$433$	−7.56031e6	−1.93785	−0.968924	+	0.247357i	$0.920438\pi$
$434$	378510.	0.0964612
$435$	0	0
$436$	−1.86111e6	−0.468873
$437$	1.28755e7	3.22523
$438$	0	0
$439$	−3.76145e6	−0.931525	−0.465763	−	0.884910i	$-0.654220\pi$
$439$	−3.76145e6	−0.931525	−0.465763	+	0.884910i	$0.654220\pi$
$440$	−162740.	−0.0400740
$441$	0	0
$442$	−85642.2	−0.0208513
$443$	1.02986e6	0.249327	0.124664	−	0.992199i	$-0.460215\pi$
$443$	1.02986e6	0.249327	0.124664	+	0.992199i	$0.460215\pi$
$444$	0	0
$445$	821387.	0.196629
$446$	486920.	0.115910
$447$	0	0
$448$	3.18327e6	0.749340
$449$	4.29961e6	1.00650	0.503249	−	0.864141i	$-0.332138\pi$
$449$	4.29961e6	1.00650	0.503249	+	0.864141i	$0.332138\pi$
$450$	0	0
$451$	973820.	0.225443
$452$	−7.36570e6	−1.69577
$453$	0	0
$454$	−211739.	−0.0482126
$455$	−1.66312e6	−0.376612
$456$	0	0
$457$	−1.42831e6	−0.319913	−0.159957	−	0.987124i	$-0.551135\pi$
$457$	−1.42831e6	−0.319913	−0.159957	+	0.987124i	$0.551135\pi$
$458$	61825.0	0.0137721
$459$	0	0
$460$	−3.65229e6	−0.804768
$461$	3.06394e6	0.671472	0.335736	−	0.941956i	$-0.391015\pi$
$461$	3.06394e6	0.671472	0.335736	+	0.941956i	$0.391015\pi$
$462$	0	0
$463$	−3.51890e6	−0.762877	−0.381438	−	0.924394i	$-0.624571\pi$
$463$	−3.51890e6	−0.762877	−0.381438	+	0.924394i	$0.624571\pi$
$464$	−6.71416e6	−1.44776
$465$	0	0
$466$	415945.	0.0887302
$467$	6.90153e6	1.46438	0.732188	−	0.681102i	$-0.238499\pi$
$467$	6.90153e6	1.46438	0.732188	+	0.681102i	$0.238499\pi$
$468$	0	0
$469$	−2.04121e6	−0.428504
$470$	−131710.	−0.0275027
$471$	0	0
$472$	1.67818e6	0.346724
$473$	−292740.	−0.0601629
$474$	0	0
$475$	7.28997e6	1.48249
$476$	−833426.	−0.168597
$477$	0	0
$478$	−41025.8	−0.00821272
$479$	4.37684e6	0.871609	0.435805	−	0.900041i	$-0.356464\pi$
$479$	4.37684e6	0.871609	0.435805	+	0.900041i	$0.356464\pi$
$480$	0	0
$481$	1.90731e6	0.375889
$482$	−25991.1	−0.00509574
$483$	0	0
$484$	4.30533e6	0.835398
$485$	2.47816e6	0.478382
$486$	0	0
$487$	4.08443e6	0.780385	0.390192	−	0.920733i	$-0.372409\pi$
$487$	4.08443e6	0.780385	0.390192	+	0.920733i	$0.372409\pi$
$488$	−269006.	−0.0511343
$489$	0	0
$490$	−97773.6	−0.0183963
$491$	4.67807e6	0.875716	0.437858	−	0.899044i	$-0.355737\pi$
$491$	4.67807e6	0.875716	0.437858	+	0.899044i	$0.355737\pi$
$492$	0	0
$493$	1.71912e6	0.318559
$494$	−1.04324e6	−0.192339
$495$	0	0
$496$	6.21090e6	1.13357
$497$	−4.99083e6	−0.906321
$498$	0	0
$499$	9.66466e6	1.73754	0.868771	−	0.495214i	$-0.164910\pi$
$499$	9.66466e6	1.73754	0.868771	+	0.495214i	$0.164910\pi$
$500$	−4.80922e6	−0.860300
$501$	0	0
$502$	617655.	0.109392
$503$	1.03319e7	1.82079	0.910396	−	0.413738i	$-0.135777\pi$
$503$	1.03319e7	1.82079	0.910396	+	0.413738i	$0.135777\pi$
$504$	0	0
$505$	−793345.	−0.138431
$506$	384133.	0.0666969
$507$	0	0
$508$	−3.45464e6	−0.593942
$509$	4.20006e6	0.718557	0.359278	−	0.933230i	$-0.383023\pi$
$509$	4.20006e6	0.718557	0.359278	+	0.933230i	$0.383023\pi$
$510$	0	0
$511$	−4.40704e6	−0.746611
$512$	−2.92704e6	−0.493463
$513$	0	0
$514$	−562992.	−0.0939928
$515$	−834681.	−0.138676
$516$	0	0
$517$	−1.29147e6	−0.212500
$518$	−199092.	−0.0326008
$519$	0	0
$520$	595030.	0.0965007
$521$	−6.46853e6	−1.04403	−0.522013	−	0.852938i	$-0.674819\pi$
$521$	−6.46853e6	−1.04403	−0.522013	+	0.852938i	$0.674819\pi$
$522$	0	0
$523$	−3.00248e6	−0.479983	−0.239991	−	0.970775i	$-0.577145\pi$
$523$	−3.00248e6	−0.479983	−0.239991	+	0.970775i	$0.577145\pi$
$524$	4.41423e6	0.702306
$525$	0	0
$526$	498411.	0.0785458
$527$	−1.59027e6	−0.249427
$528$	0	0
$529$	1.08979e7	1.69319
$530$	491789.	0.0760482
$531$	0	0
$532$	−1.01523e7	−1.55520
$533$	−3.56061e6	−0.542883
$534$	0	0
$535$	2.30175e6	0.347674
$536$	730304.	0.109797
$537$	0	0
$538$	904572.	0.134737
$539$	−958708.	−0.142139
$540$	0	0
$541$	−5.21767e6	−0.766449	−0.383224	−	0.923655i	$-0.625186\pi$
$541$	−5.21767e6	−0.766449	−0.383224	+	0.923655i	$0.625186\pi$
$542$	2434.34	0.000355945 0
$543$	0	0
$544$	448071.	0.0649157
$545$	−1.62874e6	−0.234887
$546$	0	0
$547$	−7.86187e6	−1.12346	−0.561730	−	0.827321i	$-0.689864\pi$
$547$	−7.86187e6	−1.12346	−0.561730	+	0.827321i	$0.689864\pi$
$548$	−2.03243e6	−0.289110
$549$	0	0
$550$	217492.	0.0306575
$551$	2.09413e7	2.93850
$552$	0	0
$553$	6.75560e6	0.939401
$554$	930944.	0.128869
$555$	0	0
$556$	8.91659e6	1.22324
$557$	−1.21531e7	−1.65978	−0.829889	−	0.557929i	$-0.811596\pi$
$557$	−1.21531e7	−1.65978	−0.829889	+	0.557929i	$0.811596\pi$
$558$	0	0
$559$	1.07035e6	0.144876
$560$	2.84860e6	0.383850
$561$	0	0
$562$	934177.	0.124764
$563$	−5.15518e6	−0.685446	−0.342723	−	0.939437i	$-0.611349\pi$
$563$	−5.15518e6	−0.685446	−0.342723	+	0.939437i	$0.611349\pi$
$564$	0	0
$565$	−6.44604e6	−0.849517
$566$	−1.19268e6	−0.156489
$567$	0	0
$568$	1.78562e6	0.232230
$569$	1.42079e6	0.183972	0.0919858	−	0.995760i	$-0.470679\pi$
$569$	1.42079e6	0.183972	0.0919858	+	0.995760i	$0.470679\pi$
$570$	0	0
$571$	−1.14786e7	−1.47332	−0.736661	−	0.676262i	$-0.763599\pi$
$571$	−1.14786e7	−1.47332	−0.736661	+	0.676262i	$0.763599\pi$
$572$	2.90169e6	0.370818
$573$	0	0
$574$	371668.	0.0470842
$575$	9.81450e6	1.23794
$576$	0	0
$577$	1.00683e7	1.25897	0.629484	−	0.777014i	$-0.283266\pi$
$577$	1.00683e7	1.25897	0.629484	+	0.777014i	$0.283266\pi$
$578$	789867.	0.0983409
$579$	0	0
$580$	−5.94026e6	−0.733221
$581$	7.30015e6	0.897205
$582$	0	0
$583$	4.82218e6	0.587587
$584$	1.57675e6	0.191307
$585$	0	0
$586$	786877.	0.0946592
$587$	1.33560e7	1.59985	0.799927	−	0.600097i	$-0.204871\pi$
$587$	1.33560e7	1.59985	0.799927	+	0.600097i	$0.204871\pi$
$588$	0	0
$589$	−1.93716e7	−2.30080
$590$	730407.	0.0863843
$591$	0	0
$592$	−3.26686e6	−0.383113
$593$	−1.20725e7	−1.40981	−0.704906	−	0.709301i	$-0.749011\pi$
$593$	−1.20725e7	−1.40981	−0.704906	+	0.709301i	$0.749011\pi$
$594$	0	0
$595$	−729367.	−0.0844606
$596$	7.13598e6	0.822883
$597$	0	0
$598$	−1.40452e6	−0.160611
$599$	−1.59735e7	−1.81900	−0.909502	−	0.415700i	$-0.863537\pi$
$599$	−1.59735e7	−1.81900	−0.909502	+	0.415700i	$0.863537\pi$
$600$	0	0
$601$	7.82358e6	0.883526	0.441763	−	0.897132i	$-0.354353\pi$
$601$	7.82358e6	0.883526	0.441763	+	0.897132i	$0.354353\pi$
$602$	−111727.	−0.0125651
$603$	0	0
$604$	−1.22966e7	−1.37149
$605$	3.76778e6	0.418502
$606$	0	0
$607$	3.67967e6	0.405356	0.202678	−	0.979245i	$-0.435036\pi$
$607$	3.67967e6	0.405356	0.202678	+	0.979245i	$0.435036\pi$
$608$	5.45813e6	0.598804
$609$	0	0
$610$	−117082.	−0.0127398
$611$	4.72204e6	0.511713
$612$	0	0
$613$	4.33193e6	0.465619	0.232810	−	0.972522i	$-0.425208\pi$
$613$	4.33193e6	0.465619	0.232810	+	0.972522i	$0.425208\pi$
$614$	210434.	0.0225265
$615$	0	0
$616$	−609025.	−0.0646671
$617$	6.45241e6	0.682353	0.341176	−	0.939999i	$-0.389175\pi$
$617$	6.45241e6	0.682353	0.341176	+	0.939999i	$0.389175\pi$
$618$	0	0
$619$	7.05387e6	0.739947	0.369974	−	0.929042i	$-0.379367\pi$
$619$	7.05387e6	0.739947	0.369974	+	0.929042i	$0.379367\pi$
$620$	5.49500e6	0.574101
$621$	0	0
$622$	1.23791e6	0.128296
$623$	3.07390e6	0.317299
$624$	0	0
$625$	3.15781e6	0.323359
$626$	315687.	0.0321974
$627$	0	0
$628$	1.17021e7	1.18403
$629$	836461.	0.0842984
$630$	0	0
$631$	1.46818e7	1.46793	0.733966	−	0.679186i	$-0.237667\pi$
$631$	1.46818e7	1.46793	0.733966	+	0.679186i	$0.237667\pi$
$632$	−2.41702e6	−0.240706
$633$	0	0
$634$	−1.78859e6	−0.176721
$635$	−3.02331e6	−0.297542
$636$	0	0
$637$	3.50535e6	0.342281
$638$	624773.	0.0607674
$639$	0	0
$640$	−2.06057e6	−0.198855
$641$	−3.24675e6	−0.312107	−0.156054	−	0.987749i	$-0.549877\pi$
$641$	−3.24675e6	−0.312107	−0.156054	+	0.987749i	$0.549877\pi$
$642$	0	0
$643$	−7.57964e6	−0.722971	−0.361486	−	0.932378i	$-0.617730\pi$
$643$	−7.57964e6	−0.722971	−0.361486	+	0.932378i	$0.617730\pi$
$644$	−1.36680e7	−1.29865
$645$	0	0
$646$	−457519.	−0.0431348
$647$	−6.13290e6	−0.575977	−0.287988	−	0.957634i	$-0.592986\pi$
$647$	−6.13290e6	−0.575977	−0.287988	+	0.957634i	$0.592986\pi$
$648$	0	0
$649$	7.16193e6	0.667449
$650$	−795223.	−0.0738254
$651$	0	0
$652$	−1.65332e7	−1.52313
$653$	1.13174e6	0.103864	0.0519318	−	0.998651i	$-0.483462\pi$
$653$	1.13174e6	0.103864	0.0519318	+	0.998651i	$0.483462\pi$
$654$	0	0
$655$	3.86308e6	0.351828
$656$	6.09864e6	0.553316
$657$	0	0
$658$	−492902.	−0.0443809
$659$	−1.03411e7	−0.927586	−0.463793	−	0.885944i	$-0.653512\pi$
$659$	−1.03411e7	−0.927586	−0.463793	+	0.885944i	$0.653512\pi$
$660$	0	0
$661$	−8.30708e6	−0.739511	−0.369756	−	0.929129i	$-0.620559\pi$
$661$	−8.30708e6	−0.739511	−0.369756	+	0.929129i	$0.620559\pi$
$662$	164444.	0.0145838
$663$	0	0
$664$	−2.61185e6	−0.229894
$665$	−8.88471e6	−0.779093
$666$	0	0
$667$	2.81933e7	2.45376
$668$	−545587.	−0.0473067
$669$	0	0
$670$	317856.	0.0273554
$671$	−1.14803e6	−0.0984344
$672$	0	0
$673$	−1.13225e7	−0.963617	−0.481809	−	0.876276i	$-0.660020\pi$
$673$	−1.13225e7	−0.963617	−0.481809	+	0.876276i	$0.660020\pi$
$674$	−609468.	−0.0516775
$675$	0	0
$676$	1.14576e6	0.0964331
$677$	−2.12894e7	−1.78522	−0.892609	−	0.450831i	$-0.851128\pi$
$677$	−2.12894e7	−1.78522	−0.892609	+	0.450831i	$0.851128\pi$
$678$	0	0
$679$	9.27407e6	0.771962
$680$	260954.	0.0216417
$681$	0	0
$682$	−577943.	−0.0475799
$683$	1.18724e7	0.973841	0.486920	−	0.873446i	$-0.338120\pi$
$683$	1.18724e7	0.973841	0.486920	+	0.873446i	$0.338120\pi$
$684$	0	0
$685$	−1.77866e6	−0.144833
$686$	−1.38147e6	−0.112081
$687$	0	0
$688$	−1.83331e6	−0.147661
$689$	−1.76315e7	−1.41495
$690$	0	0
$691$	−1.96927e7	−1.56896	−0.784478	−	0.620157i	$-0.787069\pi$
$691$	−1.96927e7	−1.56896	−0.784478	+	0.620157i	$0.787069\pi$
$692$	−1.80985e7	−1.43673
$693$	0	0
$694$	1.11207e6	0.0876465
$695$	7.80329e6	0.612796
$696$	0	0
$697$	−1.56152e6	−0.121749
$698$	−852859.	−0.0662581
$699$	0	0
$700$	−7.73870e6	−0.596930
$701$	1.36097e7	1.04605	0.523027	−	0.852316i	$-0.324803\pi$
$701$	1.36097e7	1.04605	0.523027	+	0.852316i	$0.324803\pi$
$702$	0	0
$703$	1.01893e7	0.777597
$704$	−4.86051e6	−0.369615
$705$	0	0
$706$	505464.	0.0381662
$707$	−2.96895e6	−0.223385
$708$	0	0
$709$	1.30274e7	0.973290	0.486645	−	0.873600i	$-0.338220\pi$
$709$	1.30274e7	0.973290	0.486645	+	0.873600i	$0.338220\pi$
$710$	777170.	0.0578589
$711$	0	0
$712$	−1.09978e6	−0.0813029
$713$	−2.60801e7	−1.92125
$714$	0	0
$715$	2.53939e6	0.185765
$716$	−6.07764e6	−0.443050
$717$	0	0
$718$	2.01321e6	0.145740
$719$	9.11723e6	0.657719	0.328860	−	0.944379i	$-0.393336\pi$
$719$	9.11723e6	0.657719	0.328860	+	0.944379i	$0.393336\pi$
$720$	0	0
$721$	−3.12365e6	−0.223781
$722$	−4.13026e6	−0.294873
$723$	0	0
$724$	−3.50809e6	−0.248728
$725$	1.59628e7	1.12788
$726$	0	0
$727$	−1.52424e7	−1.06959	−0.534797	−	0.844981i	$-0.679612\pi$
$727$	−1.52424e7	−1.06959	−0.534797	+	0.844981i	$0.679612\pi$
$728$	2.22680e6	0.155723
$729$	0	0
$730$	686262.	0.0476631
$731$	469408.	0.0324906
$732$	0	0
$733$	−1.61394e7	−1.10950	−0.554751	−	0.832016i	$-0.687187\pi$
$733$	−1.61394e7	−1.10950	−0.554751	+	0.832016i	$0.687187\pi$
$734$	1.62470e6	0.111310
$735$	0	0
$736$	7.34828e6	0.500025
$737$	3.11670e6	0.211362
$738$	0	0
$739$	−9.24471e6	−0.622704	−0.311352	−	0.950295i	$-0.600782\pi$
$739$	−9.24471e6	−0.622704	−0.311352	+	0.950295i	$0.600782\pi$
$740$	−2.89031e6	−0.194028
$741$	0	0
$742$	1.84043e6	0.122719
$743$	3.89612e6	0.258917	0.129459	−	0.991585i	$-0.458676\pi$
$743$	3.89612e6	0.258917	0.129459	+	0.991585i	$0.458676\pi$
$744$	0	0
$745$	6.24501e6	0.412233
$746$	964909.	0.0634804
$747$	0	0
$748$	1.27255e6	0.0831613
$749$	8.61387e6	0.561040
$750$	0	0
$751$	−2.36303e7	−1.52887	−0.764433	−	0.644704i	$-0.776981\pi$
$751$	−2.36303e7	−1.52887	−0.764433	+	0.644704i	$0.776981\pi$
$752$	−8.08795e6	−0.521547
$753$	0	0
$754$	−2.28437e6	−0.146332
$755$	−1.07613e7	−0.687062
$756$	0	0
$757$	2.86009e6	0.181401	0.0907005	−	0.995878i	$-0.471089\pi$
$757$	2.86009e6	0.181401	0.0907005	+	0.995878i	$0.471089\pi$
$758$	1.37260e6	0.0867704
$759$	0	0
$760$	3.17878e6	0.199630
$761$	2.90585e7	1.81891	0.909456	−	0.415800i	$-0.136498\pi$
$761$	2.90585e7	1.81891	0.909456	+	0.415800i	$0.136498\pi$
$762$	0	0
$763$	−6.09526e6	−0.379036
$764$	9.36097e6	0.580213
$765$	0	0
$766$	−2.41180e6	−0.148515
$767$	−2.61864e7	−1.60726
$768$	0	0
$769$	−1.85731e7	−1.13258	−0.566290	−	0.824206i	$-0.691622\pi$
$769$	−1.85731e7	−1.13258	−0.566290	+	0.824206i	$0.691622\pi$
$770$	−265071.	−0.0161115
$771$	0	0
$772$	1.19026e7	0.718781
$773$	−1.38698e7	−0.834872	−0.417436	−	0.908706i	$-0.637071\pi$
$773$	−1.38698e7	−0.834872	−0.417436	+	0.908706i	$0.637071\pi$
$774$	0	0
$775$	−1.47663e7	−0.883113
$776$	−3.31808e6	−0.197803
$777$	0	0
$778$	−3.07566e6	−0.182175
$779$	−1.90215e7	−1.12306
$780$	0	0
$781$	7.62045e6	0.447047
$782$	−615958.	−0.0360192
$783$	0	0
$784$	−6.00400e6	−0.348859
$785$	1.02410e7	0.593154
$786$	0	0
$787$	−1.95803e7	−1.12689	−0.563445	−	0.826154i	$-0.690524\pi$
$787$	−1.95803e7	−1.12689	−0.563445	+	0.826154i	$0.690524\pi$
$788$	−2.32015e7	−1.33107
$789$	0	0
$790$	−1.05198e6	−0.0599707
$791$	−2.41232e7	−1.37086
$792$	0	0
$793$	4.19757e6	0.237037
$794$	−2.16457e6	−0.121849
$795$	0	0
$796$	−1.86777e7	−1.04482
$797$	2.66479e7	1.48600	0.742998	−	0.669293i	$-0.233403\pi$
$797$	2.66479e7	1.48600	0.742998	+	0.669293i	$0.233403\pi$
$798$	0	0
$799$	2.07087e6	0.114759
$800$	4.16052e6	0.229839
$801$	0	0
$802$	2.09811e6	0.115184
$803$	6.72907e6	0.368270
$804$	0	0
$805$	−1.19615e7	−0.650573
$806$	2.11315e6	0.114576
$807$	0	0
$808$	1.06223e6	0.0572390
$809$	−4.31569e6	−0.231835	−0.115917	−	0.993259i	$-0.536981\pi$
$809$	−4.31569e6	−0.231835	−0.115917	+	0.993259i	$0.536981\pi$
$810$	0	0
$811$	1.07890e7	0.576007	0.288003	−	0.957629i	$-0.407009\pi$
$811$	1.07890e7	0.576007	0.288003	+	0.957629i	$0.407009\pi$
$812$	−2.22304e7	−1.18320
$813$	0	0
$814$	303991.	0.0160805
$815$	−1.44689e7	−0.763031
$816$	0	0
$817$	5.71805e6	0.299704
$818$	−963319.	−0.0503369
$819$	0	0
$820$	5.39568e6	0.280228
$821$	2.12759e7	1.10162	0.550808	−	0.834632i	$-0.314320\pi$
$821$	2.12759e7	1.10162	0.550808	+	0.834632i	$0.314320\pi$
$822$	0	0
$823$	7.04803e6	0.362717	0.181359	−	0.983417i	$-0.441950\pi$
$823$	7.04803e6	0.362717	0.181359	+	0.983417i	$0.441950\pi$
$824$	1.11758e6	0.0573404
$825$	0	0
$826$	2.73342e6	0.139398
$827$	−2.00838e7	−1.02113	−0.510566	−	0.859839i	$-0.670564\pi$
$827$	−2.00838e7	−1.02113	−0.510566	+	0.859839i	$0.670564\pi$
$828$	0	0
$829$	2.04193e7	1.03194	0.515970	−	0.856606i	$-0.327431\pi$
$829$	2.04193e7	1.03194	0.515970	+	0.856606i	$0.327431\pi$
$830$	−1.13678e6	−0.0572769
$831$	0	0
$832$	1.77716e7	0.890059
$833$	1.53729e6	0.0767614
$834$	0	0
$835$	−477467.	−0.0236988
$836$	1.55014e7	0.767108
$837$	0	0
$838$	729160.	0.0358685
$839$	4.92445e6	0.241520	0.120760	−	0.992682i	$-0.461467\pi$
$839$	4.92445e6	0.241520	0.120760	+	0.992682i	$0.461467\pi$
$840$	0	0
$841$	2.53438e7	1.23561
$842$	−1.81882e6	−0.0884114
$843$	0	0
$844$	1.17603e7	0.568278
$845$	1.00270e6	0.0483092
$846$	0	0
$847$	1.41003e7	0.675334
$848$	3.01994e7	1.44214
$849$	0	0
$850$	−348749.	−0.0165564
$851$	1.37178e7	0.649324
$852$	0	0
$853$	−1.25088e7	−0.588632	−0.294316	−	0.955708i	$-0.595092\pi$
$853$	−1.25088e7	−0.588632	−0.294316	+	0.955708i	$0.595092\pi$
$854$	−438157.	−0.0205582
$855$	0	0
$856$	−3.08188e6	−0.143758
$857$	−2.21879e7	−1.03196	−0.515982	−	0.856600i	$-0.672573\pi$
$857$	−2.21879e7	−1.03196	−0.515982	+	0.856600i	$0.672573\pi$
$858$	0	0
$859$	−1.40835e7	−0.651218	−0.325609	−	0.945505i	$-0.605569\pi$
$859$	−1.40835e7	−0.651218	−0.325609	+	0.945505i	$0.605569\pi$
$860$	−1.62199e6	−0.0747830
$861$	0	0
$862$	143631.	0.00658384
$863$	7.94491e6	0.363130	0.181565	−	0.983379i	$-0.441884\pi$
$863$	7.94491e6	0.363130	0.181565	+	0.983379i	$0.441884\pi$
$864$	0	0
$865$	−1.58387e7	−0.719748
$866$	4.40579e6	0.199632
$867$	0	0
$868$	2.05641e7	0.926424
$869$	−1.03151e7	−0.463364
$870$	0	0
$871$	−1.13957e7	−0.508973
$872$	2.18077e6	0.0971221
$873$	0	0
$874$	−7.50323e6	−0.332253
$875$	−1.57505e7	−0.695465
$876$	0	0
$877$	−4.03970e6	−0.177358	−0.0886788	−	0.996060i	$-0.528264\pi$
$877$	−4.03970e6	−0.177358	−0.0886788	+	0.996060i	$0.528264\pi$
$878$	2.19200e6	0.0959630
$879$	0	0
$880$	−4.34950e6	−0.189336
$881$	1.99840e7	0.867447	0.433724	−	0.901046i	$-0.357199\pi$
$881$	1.99840e7	0.867447	0.433724	+	0.901046i	$0.357199\pi$
$882$	0	0
$883$	1.66868e7	0.720230	0.360115	−	0.932908i	$-0.382737\pi$
$883$	1.66868e7	0.720230	0.360115	+	0.932908i	$0.382737\pi$
$884$	−4.65286e6	−0.200258
$885$	0	0
$886$	−600155.	−0.0256850
$887$	1.79964e6	0.0768027	0.0384014	−	0.999262i	$-0.487773\pi$
$887$	1.79964e6	0.0768027	0.0384014	+	0.999262i	$0.487773\pi$
$888$	0	0
$889$	−1.13142e7	−0.480141
$890$	−478666.	−0.0202562
$891$	0	0
$892$	2.64539e7	1.11321
$893$	2.52261e7	1.05858
$894$	0	0
$895$	−5.31881e6	−0.221951
$896$	−7.71131e6	−0.320892
$897$	0	0
$898$	−2.50561e6	−0.103687
$899$	−4.24179e7	−1.75045
$900$	0	0
$901$	−7.73238e6	−0.317323
$902$	−567496.	−0.0232245
$903$	0	0
$904$	8.63080e6	0.351261
$905$	−3.07008e6	−0.124603
$906$	0	0
$907$	−3.64748e6	−0.147223	−0.0736113	−	0.997287i	$-0.523452\pi$
$907$	−3.64748e6	−0.147223	−0.0736113	+	0.997287i	$0.523452\pi$
$908$	−1.15036e7	−0.463039
$909$	0	0
$910$	969185.	0.0387974
$911$	−4.92223e6	−0.196502	−0.0982508	−	0.995162i	$-0.531325\pi$
$911$	−4.92223e6	−0.196502	−0.0982508	+	0.995162i	$0.531325\pi$
$912$	0	0
$913$	−1.11465e7	−0.442551
$914$	832352.	0.0329565
$915$	0	0
$916$	3.35890e6	0.132269
$917$	1.44569e7	0.567743
$918$	0	0
$919$	1.52960e7	0.597432	0.298716	−	0.954342i	$-0.403442\pi$
$919$	1.52960e7	0.597432	0.298716	+	0.954342i	$0.403442\pi$
$920$	4.27959e6	0.166699
$921$	0	0
$922$	−1.78552e6	−0.0691731
$923$	−2.78629e7	−1.07652
$924$	0	0
$925$	7.76689e6	0.298464
$926$	2.05065e6	0.0785893
$927$	0	0
$928$	1.19516e7	0.455571
$929$	−3.10755e7	−1.18135	−0.590674	−	0.806910i	$-0.701138\pi$
$929$	−3.10755e7	−1.18135	−0.590674	+	0.806910i	$0.701138\pi$
$930$	0	0
$931$	1.87263e7	0.708073
$932$	2.25979e7	0.852175
$933$	0	0
$934$	−4.02188e6	−0.150856
$935$	1.11366e6	0.0416606
$936$	0	0
$937$	−3.19233e7	−1.18784	−0.593922	−	0.804523i	$-0.702421\pi$
$937$	−3.19233e7	−1.18784	−0.593922	+	0.804523i	$0.702421\pi$
$938$	1.18952e6	0.0441432
$939$	0	0
$940$	−7.15569e6	−0.264139
$941$	4.89415e7	1.80179	0.900894	−	0.434040i	$-0.142912\pi$
$941$	4.89415e7	1.80179	0.900894	+	0.434040i	$0.142912\pi$
$942$	0	0
$943$	−2.56087e7	−0.937795
$944$	4.48522e7	1.63815
$945$	0	0
$946$	170595.	0.00619781
$947$	1.45580e7	0.527505	0.263753	−	0.964590i	$-0.415040\pi$
$947$	1.45580e7	0.527505	0.263753	+	0.964590i	$0.415040\pi$
$948$	0	0
$949$	−2.46037e7	−0.886818
$950$	−4.24825e6	−0.152722
$951$	0	0
$952$	976572.	0.0349230
$953$	2.71191e7	0.967261	0.483631	−	0.875272i	$-0.339318\pi$
$953$	2.71191e7	0.967261	0.483631	+	0.875272i	$0.339318\pi$
$954$	0	0
$955$	8.19219e6	0.290664
$956$	−2.22889e6	−0.0788759
$957$	0	0
$958$	−2.55062e6	−0.0897907
$959$	−6.65633e6	−0.233716
$960$	0	0
$961$	1.06093e7	0.370576
$962$	−1.11149e6	−0.0387229
$963$	0	0
$964$	−1.41207e6	−0.0489401
$965$	1.04164e7	0.360082
$966$	0	0
$967$	1.16883e7	0.401963	0.200982	−	0.979595i	$-0.435587\pi$
$967$	1.16883e7	0.401963	0.200982	+	0.979595i	$0.435587\pi$
$968$	−5.04480e6	−0.173044
$969$	0	0
$970$	−1.44415e6	−0.0492815
$971$	−3.07622e6	−0.104706	−0.0523528	−	0.998629i	$-0.516672\pi$
$971$	−3.07622e6	−0.104706	−0.0523528	+	0.998629i	$0.516672\pi$
$972$	0	0
$973$	2.92024e7	0.988865
$974$	−2.38021e6	−0.0803930
$975$	0	0
$976$	−7.18964e6	−0.241592
$977$	−3.81186e7	−1.27762	−0.638809	−	0.769365i	$-0.720573\pi$
$977$	−3.81186e7	−1.27762	−0.638809	+	0.769365i	$0.720573\pi$
$978$	0	0
$979$	−4.69350e6	−0.156509
$980$	−5.31195e6	−0.176680
$981$	0	0
$982$	−2.72616e6	−0.0902137
$983$	3.14497e7	1.03809	0.519043	−	0.854748i	$-0.326289\pi$
$983$	3.14497e7	1.03809	0.519043	+	0.854748i	$0.326289\pi$
$984$	0	0
$985$	−2.03047e7	−0.666815
$986$	−1.00182e6	−0.0328170
$987$	0	0
$988$	−5.66783e7	−1.84725
$989$	7.69821e6	0.250264
$990$	0	0
$991$	1.08378e7	0.350555	0.175277	−	0.984519i	$-0.443918\pi$
$991$	1.08378e7	0.350555	0.175277	+	0.984519i	$0.443918\pi$
$992$	−1.10558e7	−0.356705
$993$	0	0
$994$	2.90842e6	0.0933665
$995$	−1.63457e7	−0.523414
$996$	0	0
$997$	−1.37214e7	−0.437179	−0.218590	−	0.975817i	$-0.570146\pi$
$997$	−1.37214e7	−0.437179	−0.218590	+	0.975817i	$0.570146\pi$
$998$	−5.63211e6	−0.178997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.c.1.4		8
3.2	odd	2		43.6.a.a.1.5	✓	8
12.11	even	2		688.6.a.e.1.4		8
15.14	odd	2		1075.6.a.a.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.5	✓	8	3.2	odd	2
387.6.a.c.1.4		8	1.1	even	1	trivial
688.6.a.e.1.4		8	12.11	even	2
1075.6.a.a.1.4		8	15.14	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-0.582753 q^{2} -31.6604 q^{4} -27.7074 q^{5} -103.690 q^{7} +37.0983 q^{8} +O(q^{10})\)	\(q-0.582753 q^{2} -31.6604 q^{4} -27.7074 q^{5} -103.690 q^{7} +37.0983 q^{8} +16.1466 q^{10} +158.323 q^{11} -578.882 q^{13} +60.4256 q^{14} +991.514 q^{16} -253.871 q^{17} -3092.51 q^{19} +877.227 q^{20} -92.2633 q^{22} -4163.45 q^{23} -2357.30 q^{25} +337.345 q^{26} +3282.87 q^{28} -6771.63 q^{29} +6264.06 q^{31} -1764.95 q^{32} +147.944 q^{34} +2872.98 q^{35} -3294.82 q^{37} +1802.17 q^{38} -1027.90 q^{40} +6150.84 q^{41} -1849.00 q^{43} -5012.58 q^{44} +2426.26 q^{46} -8157.17 q^{47} -6055.38 q^{49} +1373.72 q^{50} +18327.6 q^{52} +30457.8 q^{53} -4386.72 q^{55} -3846.72 q^{56} +3946.19 q^{58} +45236.1 q^{59} -7251.18 q^{61} -3650.40 q^{62} -30699.9 q^{64} +16039.3 q^{65} +19685.7 q^{67} +8037.67 q^{68} -1674.24 q^{70} +48132.2 q^{71} +42502.1 q^{73} +1920.07 q^{74} +97910.0 q^{76} -16416.5 q^{77} -65151.9 q^{79} -27472.3 q^{80} -3584.42 q^{82} -70403.6 q^{83} +7034.12 q^{85} +1077.51 q^{86} +5873.52 q^{88} -29645.1 q^{89} +60024.2 q^{91} +131816. q^{92} +4753.61 q^{94} +85685.3 q^{95} -89440.4 q^{97} +3528.79 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) \) 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8	\( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) \) 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * q^98

Introduction

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