LMFDB - Embedded newform 765.6.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [765,6,Mod(1,765)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("765.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	765.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,7,0,43,-125,0,-204] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$122.693622157$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.29890$ of defining polynomial
Character	$\chi$	$=$	765.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.29890 q^{2} -13.5195 q^{4} -25.0000 q^{5} +45.8012 q^{7} -195.684 q^{8} -107.472 q^{10} +504.686 q^{11} +513.142 q^{13} +196.895 q^{14} -408.601 q^{16} -289.000 q^{17} -2084.02 q^{19} +337.987 q^{20} +2169.60 q^{22} -4718.18 q^{23} +625.000 q^{25} +2205.94 q^{26} -619.207 q^{28} +3092.07 q^{29} +4157.17 q^{31} +4505.34 q^{32} -1242.38 q^{34} -1145.03 q^{35} +4514.40 q^{37} -8958.99 q^{38} +4892.09 q^{40} -7375.94 q^{41} -17592.2 q^{43} -6823.09 q^{44} -20283.0 q^{46} +14737.7 q^{47} -14709.3 q^{49} +2686.81 q^{50} -6937.40 q^{52} +1008.07 q^{53} -12617.2 q^{55} -8962.53 q^{56} +13292.5 q^{58} -496.132 q^{59} -13046.2 q^{61} +17871.2 q^{62} +32443.2 q^{64} -12828.5 q^{65} +21128.0 q^{67} +3907.12 q^{68} -4922.36 q^{70} +41206.2 q^{71} +53173.8 q^{73} +19406.9 q^{74} +28174.8 q^{76} +23115.2 q^{77} +22083.6 q^{79} +10215.0 q^{80} -31708.4 q^{82} +63967.7 q^{83} +7225.00 q^{85} -75627.1 q^{86} -98758.8 q^{88} +145034. q^{89} +23502.5 q^{91} +63787.2 q^{92} +63356.1 q^{94} +52100.5 q^{95} -131574. q^{97} -63233.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{2} + 43 q^{4} - 125 q^{5} - 204 q^{7} + 63 q^{8} - 175 q^{10} + 792 q^{11} + 88 q^{13} - 860 q^{14} - 2365 q^{16} - 1445 q^{17} - 5160 q^{19} - 1075 q^{20} - 3058 q^{22} + 6140 q^{23} + 3125 q^{25}+ \cdots - 49329 q^{98}+O(q^{100}) $ 5 * q + 7 * q^2 + 43 * q^4 - 125 * q^5 - 204 * q^7 + 63 * q^8 - 175 * q^10 + 792 * q^11 + 88 * q^13 - 860 * q^14 - 2365 * q^16 - 1445 * q^17 - 5160 * q^19 - 1075 * q^20 - 3058 * q^22 + 6140 * q^23 + 3125 * q^25 + 9643 * q^26 - 4568 * q^28 + 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 2023 * q^34 + 5100 * q^35 + 9834 * q^37 - 10625 * q^38 - 1575 * q^40 + 5302 * q^41 - 4916 * q^43 + 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 26483 * q^49 + 4375 * q^50 - 11525 * q^52 - 56492 * q^53 - 19800 * q^55 - 73228 * q^56 + 54213 * q^58 + 96200 * q^59 - 7284 * q^61 - 122335 * q^62 + 28483 * q^64 - 2200 * q^65 - 60308 * q^67 - 12427 * q^68 + 21500 * q^70 + 25304 * q^71 + 163328 * q^73 + 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 50940 * q^79 + 59125 * q^80 + 191390 * q^82 + 179060 * q^83 + 36125 * q^85 - 87086 * q^86 + 167634 * q^88 + 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 100515 * q^94 + 129000 * q^95 - 106076 * q^97 - 49329 * q^98

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 $

$2$	4.29890	0.759945	0.379973	−	0.924998i	$-0.375933\pi$
$2$	4.29890	0.759945	0.379973	+	0.924998i	$0.375933\pi$
$3$	0	0
$3$	0	0
$4$	−13.5195	−0.422483
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	45.8012	0.353290	0.176645	−	0.984275i	$-0.443476\pi$
$7$	45.8012	0.353290	0.176645	+	0.984275i	$0.443476\pi$
$8$	−195.684	−1.08101
$9$	0	0
$10$	−107.472	−0.339858
$11$	504.686	1.25759	0.628796	−	0.777570i	$-0.283548\pi$
$11$	504.686	1.25759	0.628796	+	0.777570i	$0.283548\pi$
$12$	0	0
$13$	513.142	0.842130	0.421065	−	0.907031i	$-0.361656\pi$
$13$	513.142	0.842130	0.421065	+	0.907031i	$0.361656\pi$
$14$	196.895	0.268481
$15$	0	0
$16$	−408.601	−0.399025
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	0	0
$19$	−2084.02	−1.32440	−0.662198	−	0.749329i	$-0.730376\pi$
$19$	−2084.02	−1.32440	−0.662198	+	0.749329i	$0.730376\pi$
$20$	337.987	0.188940
$21$	0	0
$22$	2169.60	0.955701
$23$	−4718.18	−1.85975	−0.929875	−	0.367876i	$-0.880085\pi$
$23$	−4718.18	−1.85975	−0.929875	+	0.367876i	$0.880085\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	2205.94	0.639972
$27$	0	0
$28$	−619.207	−0.149259
$29$	3092.07	0.682739	0.341369	−	0.939929i	$-0.389109\pi$
$29$	3092.07	0.682739	0.341369	+	0.939929i	$0.389109\pi$
$30$	0	0
$31$	4157.17	0.776950	0.388475	−	0.921459i	$-0.373002\pi$
$31$	4157.17	0.776950	0.388475	+	0.921459i	$0.373002\pi$
$32$	4505.34	0.777772
$33$	0	0
$34$	−1242.38	−0.184314
$35$	−1145.03	−0.157996
$36$	0	0
$37$	4514.40	0.542120	0.271060	−	0.962562i	$-0.412626\pi$
$37$	4514.40	0.542120	0.271060	+	0.962562i	$0.412626\pi$
$38$	−8958.99	−1.00647
$39$	0	0
$40$	4892.09	0.483442
$41$	−7375.94	−0.685264	−0.342632	−	0.939470i	$-0.611318\pi$
$41$	−7375.94	−0.685264	−0.342632	+	0.939470i	$0.611318\pi$
$42$	0	0
$43$	−17592.2	−1.45094	−0.725469	−	0.688255i	$-0.758377\pi$
$43$	−17592.2	−1.45094	−0.725469	+	0.688255i	$0.758377\pi$
$44$	−6823.09	−0.531311
$45$	0	0
$46$	−20283.0	−1.41331
$47$	14737.7	0.973164	0.486582	−	0.873635i	$-0.338243\pi$
$47$	14737.7	0.973164	0.486582	+	0.873635i	$0.338243\pi$
$48$	0	0
$49$	−14709.3	−0.875186
$50$	2686.81	0.151989
$51$	0	0
$52$	−6937.40	−0.355786
$53$	1008.07	0.0492946	0.0246473	−	0.999696i	$-0.492154\pi$
$53$	1008.07	0.0492946	0.0246473	+	0.999696i	$0.492154\pi$
$54$	0	0
$55$	−12617.2	−0.562412
$56$	−8962.53	−0.381910
$57$	0	0
$58$	13292.5	0.518844
$59$	−496.132	−0.0185553	−0.00927764	−	0.999957i	$-0.502953\pi$
$59$	−496.132	−0.0185553	−0.00927764	+	0.999957i	$0.502953\pi$
$60$	0	0
$61$	−13046.2	−0.448910	−0.224455	−	0.974484i	$-0.572060\pi$
$61$	−13046.2	−0.448910	−0.224455	+	0.974484i	$0.572060\pi$
$62$	17871.2	0.590439
$63$	0	0
$64$	32443.2	0.990089
$65$	−12828.5	−0.376612
$66$	0	0
$67$	21128.0	0.575004	0.287502	−	0.957780i	$-0.407175\pi$
$67$	21128.0	0.575004	0.287502	+	0.957780i	$0.407175\pi$
$68$	3907.12	0.102467
$69$	0	0
$70$	−4922.36	−0.120068
$71$	41206.2	0.970100	0.485050	−	0.874487i	$-0.338801\pi$
$71$	41206.2	0.970100	0.485050	+	0.874487i	$0.338801\pi$
$72$	0	0
$73$	53173.8	1.16786	0.583929	−	0.811805i	$-0.301515\pi$
$73$	53173.8	1.16786	0.583929	+	0.811805i	$0.301515\pi$
$74$	19406.9	0.411981
$75$	0	0
$76$	28174.8	0.559535
$77$	23115.2	0.444295
$78$	0	0
$79$	22083.6	0.398110	0.199055	−	0.979988i	$-0.436213\pi$
$79$	22083.6	0.398110	0.199055	+	0.979988i	$0.436213\pi$
$80$	10215.0	0.178449
$81$	0	0
$82$	−31708.4	−0.520763
$83$	63967.7	1.01922	0.509608	−	0.860407i	$-0.329791\pi$
$83$	63967.7	1.01922	0.509608	+	0.860407i	$0.329791\pi$
$84$	0	0
$85$	7225.00	0.108465
$86$	−75627.1	−1.10263
$87$	0	0
$88$	−98758.8	−1.35947
$89$	145034.	1.94087	0.970433	−	0.241369i	$-0.0775964\pi$
$89$	145034.	1.94087	0.970433	+	0.241369i	$0.0775964\pi$
$90$	0	0
$91$	23502.5	0.297516
$92$	63787.2	0.785713
$93$	0	0
$94$	63356.1	0.739552
$95$	52100.5	0.592288
$96$	0	0
$97$	−131574.	−1.41984	−0.709921	−	0.704282i	$-0.751269\pi$
$97$	−131574.	−1.41984	−0.709921	+	0.704282i	$0.751269\pi$
$98$	−63233.6	−0.665094
$99$	0	0
$100$	−8449.66	−0.0844966
$101$	101611.	0.991149	0.495575	−	0.868565i	$-0.334957\pi$
$101$	101611.	0.991149	0.495575	+	0.868565i	$0.334957\pi$
$102$	0	0
$103$	72080.6	0.669461	0.334731	−	0.942314i	$-0.391355\pi$
$103$	72080.6	0.669461	0.334731	+	0.942314i	$0.391355\pi$
$104$	−100413.	−0.910350
$105$	0	0
$106$	4333.57	0.0374612
$107$	175657.	1.48322	0.741610	−	0.670831i	$-0.234062\pi$
$107$	175657.	1.48322	0.741610	+	0.670831i	$0.234062\pi$
$108$	0	0
$109$	−16726.2	−0.134844	−0.0674220	−	0.997725i	$-0.521477\pi$
$109$	−16726.2	−0.134844	−0.0674220	+	0.997725i	$0.521477\pi$
$110$	−54239.9	−0.427402
$111$	0	0
$112$	−18714.4	−0.140971
$113$	106978.	0.788134	0.394067	−	0.919082i	$-0.371068\pi$
$113$	106978.	0.788134	0.394067	+	0.919082i	$0.371068\pi$
$114$	0	0
$115$	117954.	0.831706
$116$	−41803.2	−0.288446
$117$	0	0
$118$	−2132.82	−0.0141010
$119$	−13236.5	−0.0856854
$120$	0	0
$121$	93657.2	0.581537
$122$	−56084.3	−0.341147
$123$	0	0
$124$	−56202.6	−0.328248
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	156823.	0.862781	0.431390	−	0.902165i	$-0.358023\pi$
$127$	156823.	0.862781	0.431390	+	0.902165i	$0.358023\pi$
$128$	−4700.61	−0.0253588
$129$	0	0
$130$	−55148.6	−0.286204
$131$	−124664.	−0.634690	−0.317345	−	0.948310i	$-0.602791\pi$
$131$	−124664.	−0.634690	−0.317345	+	0.948310i	$0.602791\pi$
$132$	0	0
$133$	−95450.5	−0.467896
$134$	90827.1	0.436972
$135$	0	0
$136$	56552.6	0.262183
$137$	49378.2	0.224767	0.112384	−	0.993665i	$-0.464151\pi$
$137$	49378.2	0.224767	0.112384	+	0.993665i	$0.464151\pi$
$138$	0	0
$139$	41751.8	0.183290	0.0916449	−	0.995792i	$-0.470788\pi$
$139$	41751.8	0.183290	0.0916449	+	0.995792i	$0.470788\pi$
$140$	15480.2	0.0667507
$141$	0	0
$142$	177141.	0.737223
$143$	258976.	1.05906
$144$	0	0
$145$	−77301.8	−0.305330
$146$	228589.	0.887509
$147$	0	0
$148$	−61032.2	−0.229037
$149$	46782.8	0.172632	0.0863159	−	0.996268i	$-0.472491\pi$
$149$	46782.8	0.172632	0.0863159	+	0.996268i	$0.472491\pi$
$150$	0	0
$151$	−55545.1	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$151$	−55545.1	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$152$	407808.	1.43168
$153$	0	0
$154$	99370.0	0.337640
$155$	−103929.	−0.347463
$156$	0	0
$157$	64212.8	0.207909	0.103954	−	0.994582i	$-0.466850\pi$
$157$	64212.8	0.207909	0.103954	+	0.994582i	$0.466850\pi$
$158$	94935.4	0.302542
$159$	0	0
$160$	−112633.	−0.347830
$161$	−216098.	−0.657031
$162$	0	0
$163$	595853.	1.75659	0.878294	−	0.478121i	$-0.158682\pi$
$163$	595853.	1.75659	0.878294	+	0.478121i	$0.158682\pi$
$164$	99718.8	0.289513
$165$	0	0
$166$	274991.	0.774548
$167$	533223.	1.47951	0.739755	−	0.672876i	$-0.234941\pi$
$167$	533223.	1.47951	0.739755	+	0.672876i	$0.234941\pi$
$168$	0	0
$169$	−107979.	−0.290818
$170$	31059.5	0.0824276
$171$	0	0
$172$	237837.	0.612997
$173$	−743259.	−1.88810	−0.944050	−	0.329804i	$-0.893017\pi$
$173$	−743259.	−1.88810	−0.944050	+	0.329804i	$0.893017\pi$
$174$	0	0
$175$	28625.7	0.0706580
$176$	−206215.	−0.501810
$177$	0	0
$178$	623488.	1.47495
$179$	369567.	0.862105	0.431053	−	0.902327i	$-0.358142\pi$
$179$	369567.	0.862105	0.431053	+	0.902327i	$0.358142\pi$
$180$	0	0
$181$	490167.	1.11211	0.556055	−	0.831145i	$-0.312314\pi$
$181$	490167.	1.11211	0.556055	+	0.831145i	$0.312314\pi$
$182$	101035.	0.226096
$183$	0	0
$184$	923270.	2.01041
$185$	−112860.	−0.242443
$186$	0	0
$187$	−145854.	−0.305011
$188$	−199246.	−0.411146
$189$	0	0
$190$	223975.	0.450106
$191$	7647.56	0.0151684	0.00758420	−	0.999971i	$-0.497586\pi$
$191$	7647.56	0.0151684	0.00758420	+	0.999971i	$0.497586\pi$
$192$	0	0
$193$	−195606.	−0.377998	−0.188999	−	0.981977i	$-0.560524\pi$
$193$	−195606.	−0.377998	−0.188999	+	0.981977i	$0.560524\pi$
$194$	−565622.	−1.07900
$195$	0	0
$196$	198861.	0.369751
$197$	885254.	1.62518	0.812591	−	0.582834i	$-0.198056\pi$
$197$	885254.	1.62518	0.812591	+	0.582834i	$0.198056\pi$
$198$	0	0
$199$	−877413.	−1.57062	−0.785310	−	0.619102i	$-0.787497\pi$
$199$	−877413.	−1.57062	−0.785310	+	0.619102i	$0.787497\pi$
$200$	−122302.	−0.216202
$201$	0	0
$202$	436817.	0.753219
$203$	141620.	0.241205
$204$	0	0
$205$	184399.	0.306459
$206$	309867.	0.508754
$207$	0	0
$208$	−209670.	−0.336031
$209$	−1.05178e6	−1.66555
$210$	0	0
$211$	−325844.	−0.503852	−0.251926	−	0.967746i	$-0.581064\pi$
$211$	−325844.	−0.503852	−0.251926	+	0.967746i	$0.581064\pi$
$212$	−13628.5	−0.0208261
$213$	0	0
$214$	755131.	1.12717
$215$	439805.	0.648879
$216$	0	0
$217$	190403.	0.274489
$218$	−71904.4	−0.102474
$219$	0	0
$220$	170577.	0.237610
$221$	−148298.	−0.204246
$222$	0	0
$223$	−758258.	−1.02107	−0.510534	−	0.859858i	$-0.670552\pi$
$223$	−758258.	−1.02107	−0.510534	+	0.859858i	$0.670552\pi$
$224$	206350.	0.274779
$225$	0	0
$226$	459890.	0.598939
$227$	−487718.	−0.628209	−0.314105	−	0.949388i	$-0.601704\pi$
$227$	−487718.	−0.628209	−0.314105	+	0.949388i	$0.601704\pi$
$228$	0	0
$229$	−413467.	−0.521017	−0.260509	−	0.965472i	$-0.583890\pi$
$229$	−413467.	−0.521017	−0.260509	+	0.965472i	$0.583890\pi$
$230$	507074.	0.632051
$231$	0	0
$232$	−605068.	−0.738047
$233$	−750445.	−0.905584	−0.452792	−	0.891616i	$-0.649572\pi$
$233$	−750445.	−0.905584	−0.452792	+	0.891616i	$0.649572\pi$
$234$	0	0
$235$	−368443.	−0.435212
$236$	6707.44	0.00783929
$237$	0	0
$238$	−56902.5	−0.0651162
$239$	421206.	0.476980	0.238490	−	0.971145i	$-0.423348\pi$
$239$	421206.	0.476980	0.238490	+	0.971145i	$0.423348\pi$
$240$	0	0
$241$	212219.	0.235365	0.117682	−	0.993051i	$-0.462454\pi$
$241$	212219.	0.235365	0.117682	+	0.993051i	$0.462454\pi$
$242$	402623.	0.441937
$243$	0	0
$244$	176377.	0.189657
$245$	367731.	0.391395
$246$	0	0
$247$	−1.06940e6	−1.11531
$248$	−813489.	−0.839890
$249$	0	0
$250$	−67170.3	−0.0679716
$251$	937463.	0.939226	0.469613	−	0.882872i	$-0.344394\pi$
$251$	937463.	0.939226	0.469613	+	0.882872i	$0.344394\pi$
$252$	0	0
$253$	−2.38120e6	−2.33881
$254$	674166.	0.655666
$255$	0	0
$256$	−1.05839e6	−1.00936
$257$	1.48510e6	1.40256	0.701281	−	0.712885i	$-0.252612\pi$
$257$	1.48510e6	1.40256	0.701281	+	0.712885i	$0.252612\pi$
$258$	0	0
$259$	206765.	0.191526
$260$	173435.	0.159112
$261$	0	0
$262$	−535917.	−0.482330
$263$	420914.	0.375235	0.187618	−	0.982242i	$-0.439923\pi$
$263$	420914.	0.375235	0.187618	+	0.982242i	$0.439923\pi$
$264$	0	0
$265$	−25201.6	−0.0220452
$266$	−410332.	−0.355575
$267$	0	0
$268$	−285639.	−0.242930
$269$	1.19320e6	1.00538	0.502691	−	0.864466i	$-0.332343\pi$
$269$	1.19320e6	1.00538	0.502691	+	0.864466i	$0.332343\pi$
$270$	0	0
$271$	2.07801e6	1.71880	0.859398	−	0.511308i	$-0.170839\pi$
$271$	2.07801e6	1.71880	0.859398	+	0.511308i	$0.170839\pi$
$272$	118086.	0.0967777
$273$	0	0
$274$	212272.	0.170811
$275$	315429.	0.251518
$276$	0	0
$277$	883691.	0.691992	0.345996	−	0.938236i	$-0.387541\pi$
$277$	883691.	0.691992	0.345996	+	0.938236i	$0.387541\pi$
$278$	179487.	0.139290
$279$	0	0
$280$	224063.	0.170795
$281$	−353265.	−0.266892	−0.133446	−	0.991056i	$-0.542604\pi$
$281$	−353265.	−0.266892	−0.133446	+	0.991056i	$0.542604\pi$
$282$	0	0
$283$	−78060.8	−0.0579384	−0.0289692	−	0.999580i	$-0.509222\pi$
$283$	−78060.8	−0.0579384	−0.0289692	+	0.999580i	$0.509222\pi$
$284$	−557085.	−0.409851
$285$	0	0
$286$	1.11331e6	0.804824
$287$	−337827.	−0.242097
$288$	0	0
$289$	83521.0	0.0588235
$290$	−332313.	−0.232034
$291$	0	0
$292$	−718881.	−0.493401
$293$	1.79083e6	1.21867	0.609335	−	0.792913i	$-0.291436\pi$
$293$	1.79083e6	1.21867	0.609335	+	0.792913i	$0.291436\pi$
$294$	0	0
$295$	12403.3	0.00829817
$296$	−883393.	−0.586037
$297$	0	0
$298$	201115.	0.131191
$299$	−2.42109e6	−1.56615
$300$	0	0
$301$	−805743.	−0.512602
$302$	−238783.	−0.150656
$303$	0	0
$304$	851533.	0.528466
$305$	326155.	0.200759
$306$	0	0
$307$	−1.87896e6	−1.13781	−0.568907	−	0.822402i	$-0.692634\pi$
$307$	−1.87896e6	−1.13781	−0.568907	+	0.822402i	$0.692634\pi$
$308$	−312505.	−0.187707
$309$	0	0
$310$	−446781.	−0.264053
$311$	2.65990e6	1.55943	0.779714	−	0.626136i	$-0.215365\pi$
$311$	2.65990e6	1.55943	0.779714	+	0.626136i	$0.215365\pi$
$312$	0	0
$313$	576430.	0.332572	0.166286	−	0.986078i	$-0.446822\pi$
$313$	576430.	0.332572	0.166286	+	0.986078i	$0.446822\pi$
$314$	276044.	0.157999
$315$	0	0
$316$	−298559.	−0.168195
$317$	760504.	0.425063	0.212532	−	0.977154i	$-0.431829\pi$
$317$	760504.	0.425063	0.212532	+	0.977154i	$0.431829\pi$
$318$	0	0
$319$	1.56053e6	0.858607
$320$	−811081.	−0.442781
$321$	0	0
$322$	−928983.	−0.499308
$323$	602281.	0.321213
$324$	0	0
$325$	320714.	0.168426
$326$	2.56151e6	1.33491
$327$	0	0
$328$	1.44335e6	0.740777
$329$	675005.	0.343809
$330$	0	0
$331$	−915317.	−0.459200	−0.229600	−	0.973285i	$-0.573742\pi$
$331$	−915317.	−0.459200	−0.229600	+	0.973285i	$0.573742\pi$
$332$	−864810.	−0.430601
$333$	0	0
$334$	2.29227e6	1.12435
$335$	−528200.	−0.257150
$336$	0	0
$337$	−1.58665e6	−0.761040	−0.380520	−	0.924773i	$-0.624255\pi$
$337$	−1.58665e6	−0.761040	−0.380520	+	0.924773i	$0.624255\pi$
$338$	−464189.	−0.221006
$339$	0	0
$340$	−97678.1	−0.0458247
$341$	2.09806e6	0.977086
$342$	0	0
$343$	−1.44348e6	−0.662484
$344$	3.44250e6	1.56848
$345$	0	0
$346$	−3.19519e6	−1.43485
$347$	−3.71862e6	−1.65790	−0.828950	−	0.559322i	$-0.811061\pi$
$347$	−3.71862e6	−1.65790	−0.828950	+	0.559322i	$0.811061\pi$
$348$	0	0
$349$	2.20157e6	0.967540	0.483770	−	0.875195i	$-0.339267\pi$
$349$	2.20157e6	0.967540	0.483770	+	0.875195i	$0.339267\pi$
$350$	123059.	0.0536962
$351$	0	0
$352$	2.27378e6	0.978120
$353$	2.24412e6	0.958539	0.479269	−	0.877668i	$-0.340902\pi$
$353$	2.24412e6	0.958539	0.479269	+	0.877668i	$0.340902\pi$
$354$	0	0
$355$	−1.03015e6	−0.433842
$356$	−1.96079e6	−0.819984
$357$	0	0
$358$	1.58873e6	0.655153
$359$	2.48856e6	1.01909	0.509544	−	0.860444i	$-0.329814\pi$
$359$	2.48856e6	1.01909	0.509544	+	0.860444i	$0.329814\pi$
$360$	0	0
$361$	1.86703e6	0.754023
$362$	2.10718e6	0.845143
$363$	0	0
$364$	−317741.	−0.125695
$365$	−1.32934e6	−0.522282
$366$	0	0
$367$	−4.14031e6	−1.60460	−0.802302	−	0.596919i	$-0.796391\pi$
$367$	−4.14031e6	−1.60460	−0.802302	+	0.596919i	$0.796391\pi$
$368$	1.92785e6	0.742086
$369$	0	0
$370$	−485173.	−0.184244
$371$	46170.6	0.0174153
$372$	0	0
$373$	1.79738e6	0.668910	0.334455	−	0.942412i	$-0.391448\pi$
$373$	1.79738e6	0.668910	0.334455	+	0.942412i	$0.391448\pi$
$374$	−627013.	−0.231792
$375$	0	0
$376$	−2.88393e6	−1.05200
$377$	1.58667e6	0.574955
$378$	0	0
$379$	90472.6	0.0323533	0.0161767	−	0.999869i	$-0.494851\pi$
$379$	90472.6	0.0323533	0.0161767	+	0.999869i	$0.494851\pi$
$380$	−704370.	−0.250232
$381$	0	0
$382$	32876.1	0.0115272
$383$	−2.92745e6	−1.01975	−0.509873	−	0.860250i	$-0.670308\pi$
$383$	−2.92745e6	−1.01975	−0.509873	+	0.860250i	$0.670308\pi$
$384$	0	0
$385$	−577880.	−0.198695
$386$	−840892.	−0.287258
$387$	0	0
$388$	1.77881e6	0.599859
$389$	−1.35743e6	−0.454823	−0.227411	−	0.973799i	$-0.573026\pi$
$389$	−1.35743e6	−0.454823	−0.227411	+	0.973799i	$0.573026\pi$
$390$	0	0
$391$	1.36355e6	0.451056
$392$	2.87836e6	0.946084
$393$	0	0
$394$	3.80562e6	1.23505
$395$	−552091.	−0.178040
$396$	0	0
$397$	−223344.	−0.0711210	−0.0355605	−	0.999368i	$-0.511322\pi$
$397$	−223344.	−0.0711210	−0.0355605	+	0.999368i	$0.511322\pi$
$398$	−3.77191e6	−1.19359
$399$	0	0
$400$	−255376.	−0.0798049
$401$	−3.43768e6	−1.06759	−0.533796	−	0.845613i	$-0.679235\pi$
$401$	−3.43768e6	−1.06759	−0.533796	+	0.845613i	$0.679235\pi$
$402$	0	0
$403$	2.13322e6	0.654293
$404$	−1.37373e6	−0.418744
$405$	0	0
$406$	608812.	0.183302
$407$	2.27835e6	0.681766
$408$	0	0
$409$	−2.01964e6	−0.596987	−0.298494	−	0.954412i	$-0.596484\pi$
$409$	−2.01964e6	−0.596987	−0.298494	+	0.954412i	$0.596484\pi$
$410$	792711.	0.232892
$411$	0	0
$412$	−974491.	−0.282836
$413$	−22723.4	−0.00655539
$414$	0	0
$415$	−1.59919e6	−0.455807
$416$	2.31188e6	0.654985
$417$	0	0
$418$	−4.52148e6	−1.26573
$419$	−2.77100e6	−0.771085	−0.385542	−	0.922690i	$-0.625986\pi$
$419$	−2.77100e6	−0.771085	−0.385542	+	0.922690i	$0.625986\pi$
$420$	0	0
$421$	−1.08269e6	−0.297714	−0.148857	−	0.988859i	$-0.547559\pi$
$421$	−1.08269e6	−0.297714	−0.148857	+	0.988859i	$0.547559\pi$
$422$	−1.40077e6	−0.382900
$423$	0	0
$424$	−197262.	−0.0532879
$425$	−180625.	−0.0485071
$426$	0	0
$427$	−597531.	−0.158595
$428$	−2.37479e6	−0.626636
$429$	0	0
$430$	1.89068e6	0.493113
$431$	3.08443e6	0.799800	0.399900	−	0.916559i	$-0.369045\pi$
$431$	3.08443e6	0.799800	0.399900	+	0.916559i	$0.369045\pi$
$432$	0	0
$433$	3.83729e6	0.983569	0.491784	−	0.870717i	$-0.336345\pi$
$433$	3.83729e6	0.983569	0.491784	+	0.870717i	$0.336345\pi$
$434$	818523.	0.208596
$435$	0	0
$436$	226130.	0.0569694
$437$	9.83277e6	2.46304
$438$	0	0
$439$	−1.77903e6	−0.440577	−0.220289	−	0.975435i	$-0.570700\pi$
$439$	−1.77903e6	−0.440577	−0.220289	+	0.975435i	$0.570700\pi$
$440$	2.46897e6	0.607973
$441$	0	0
$442$	−637518.	−0.155216
$443$	−2.29093e6	−0.554629	−0.277315	−	0.960779i	$-0.589444\pi$
$443$	−2.29093e6	−0.554629	−0.277315	+	0.960779i	$0.589444\pi$
$444$	0	0
$445$	−3.62586e6	−0.867982
$446$	−3.25967e6	−0.775956
$447$	0	0
$448$	1.48594e6	0.349789
$449$	−6.34594e6	−1.48552	−0.742762	−	0.669555i	$-0.766485\pi$
$449$	−6.34594e6	−1.48552	−0.742762	+	0.669555i	$0.766485\pi$
$450$	0	0
$451$	−3.72254e6	−0.861782
$452$	−1.44629e6	−0.332973
$453$	0	0
$454$	−2.09665e6	−0.477405
$455$	−587562.	−0.133053
$456$	0	0
$457$	1.39090e6	0.311535	0.155767	−	0.987794i	$-0.450215\pi$
$457$	1.39090e6	0.311535	0.155767	+	0.987794i	$0.450215\pi$
$458$	−1.77745e6	−0.395944
$459$	0	0
$460$	−1.59468e6	−0.351382
$461$	4.36882e6	0.957441	0.478721	−	0.877967i	$-0.341101\pi$
$461$	4.36882e6	0.957441	0.478721	+	0.877967i	$0.341101\pi$
$462$	0	0
$463$	737398.	0.159864	0.0799318	−	0.996800i	$-0.474530\pi$
$463$	737398.	0.159864	0.0799318	+	0.996800i	$0.474530\pi$
$464$	−1.26342e6	−0.272430
$465$	0	0
$466$	−3.22609e6	−0.688194
$467$	−1.95716e6	−0.415273	−0.207636	−	0.978206i	$-0.566577\pi$
$467$	−1.95716e6	−0.415273	−0.207636	+	0.978206i	$0.566577\pi$
$468$	0	0
$469$	967686.	0.203143
$470$	−1.58390e6	−0.330738
$471$	0	0
$472$	97084.9	0.0200584
$473$	−8.87854e6	−1.82469
$474$	0	0
$475$	−1.30251e6	−0.264879
$476$	178951.	0.0362006
$477$	0	0
$478$	1.81072e6	0.362478
$479$	3.22574e6	0.642377	0.321189	−	0.947015i	$-0.395918\pi$
$479$	3.22574e6	0.642377	0.321189	+	0.947015i	$0.395918\pi$
$480$	0	0
$481$	2.31652e6	0.456535
$482$	912308.	0.178864
$483$	0	0
$484$	−1.26619e6	−0.245690
$485$	3.28934e6	0.634972
$486$	0	0
$487$	5.66306e6	1.08200	0.541001	−	0.841022i	$-0.318045\pi$
$487$	5.66306e6	1.08200	0.541001	+	0.841022i	$0.318045\pi$
$488$	2.55293e6	0.485276
$489$	0	0
$490$	1.58084e6	0.297439
$491$	−290042.	−0.0542947	−0.0271474	−	0.999631i	$-0.508642\pi$
$491$	−290042.	−0.0542947	−0.0271474	+	0.999631i	$0.508642\pi$
$492$	0	0
$493$	−893609.	−0.165588
$494$	−4.59723e6	−0.847576
$495$	0	0
$496$	−1.69862e6	−0.310022
$497$	1.88729e6	0.342726
$498$	0	0
$499$	7.87433e6	1.41567	0.707835	−	0.706377i	$-0.249672\pi$
$499$	7.87433e6	1.41567	0.707835	+	0.706377i	$0.249672\pi$
$500$	211242.	0.0377880
$501$	0	0
$502$	4.03006e6	0.713760
$503$	−7.91006e6	−1.39399	−0.696995	−	0.717076i	$-0.745480\pi$
$503$	−7.91006e6	−1.39399	−0.696995	+	0.717076i	$0.745480\pi$
$504$	0	0
$505$	−2.54029e6	−0.443255
$506$	−1.02365e7	−1.77736
$507$	0	0
$508$	−2.12016e6	−0.364510
$509$	4.88210e6	0.835241	0.417621	−	0.908621i	$-0.362864\pi$
$509$	4.88210e6	0.835241	0.417621	+	0.908621i	$0.362864\pi$
$510$	0	0
$511$	2.43542e6	0.412593
$512$	−4.39950e6	−0.741700
$513$	0	0
$514$	6.38428e6	1.06587
$515$	−1.80201e6	−0.299392
$516$	0	0
$517$	7.43793e6	1.22384
$518$	888860.	0.145549
$519$	0	0
$520$	2.51034e6	0.407121
$521$	4.67038e6	0.753803	0.376901	−	0.926253i	$-0.376990\pi$
$521$	4.67038e6	0.753803	0.376901	+	0.926253i	$0.376990\pi$
$522$	0	0
$523$	−8.87695e6	−1.41909	−0.709544	−	0.704661i	$-0.751099\pi$
$523$	−8.87695e6	−1.41909	−0.709544	+	0.704661i	$0.751099\pi$
$524$	1.68539e6	0.268146
$525$	0	0
$526$	1.80947e6	0.285158
$527$	−1.20142e6	−0.188438
$528$	0	0
$529$	1.58248e7	2.45867
$530$	−108339.	−0.0167531
$531$	0	0
$532$	1.29044e6	0.197678
$533$	−3.78490e6	−0.577081
$534$	0	0
$535$	−4.39142e6	−0.663316
$536$	−4.13440e6	−0.621585
$537$	0	0
$538$	5.12943e6	0.764035
$539$	−7.42356e6	−1.10063
$540$	0	0
$541$	−9.68582e6	−1.42280	−0.711399	−	0.702789i	$-0.751938\pi$
$541$	−9.68582e6	−1.42280	−0.711399	+	0.702789i	$0.751938\pi$
$542$	8.93315e6	1.30619
$543$	0	0
$544$	−1.30204e6	−0.188638
$545$	418156.	0.0603041
$546$	0	0
$547$	−1.98238e6	−0.283282	−0.141641	−	0.989918i	$-0.545238\pi$
$547$	−1.98238e6	−0.283282	−0.141641	+	0.989918i	$0.545238\pi$
$548$	−667566.	−0.0949605
$549$	0	0
$550$	1.35600e6	0.191140
$551$	−6.44394e6	−0.904216
$552$	0	0
$553$	1.01146e6	0.140648
$554$	3.79890e6	0.525876
$555$	0	0
$556$	−564462.	−0.0774369
$557$	7.73137e6	1.05589	0.527945	−	0.849279i	$-0.322963\pi$
$557$	7.73137e6	1.05589	0.527945	+	0.849279i	$0.322963\pi$
$558$	0	0
$559$	−9.02729e6	−1.22188
$560$	467860.	0.0630443
$561$	0	0
$562$	−1.51865e6	−0.202823
$563$	6.73841e6	0.895955	0.447978	−	0.894045i	$-0.352144\pi$
$563$	6.73841e6	0.895955	0.447978	+	0.894045i	$0.352144\pi$
$564$	0	0
$565$	−2.67446e6	−0.352464
$566$	−335575.	−0.0440300
$567$	0	0
$568$	−8.06337e6	−1.04869
$569$	−1.38027e6	−0.178724	−0.0893622	−	0.995999i	$-0.528483\pi$
$569$	−1.38027e6	−0.178724	−0.0893622	+	0.995999i	$0.528483\pi$
$570$	0	0
$571$	7.93779e6	1.01885	0.509424	−	0.860516i	$-0.329859\pi$
$571$	7.93779e6	1.01885	0.509424	+	0.860516i	$0.329859\pi$
$572$	−3.50121e6	−0.447433
$573$	0	0
$574$	−1.45228e6	−0.183980
$575$	−2.94886e6	−0.371950
$576$	0	0
$577$	8.26007e6	1.03287	0.516433	−	0.856327i	$-0.327259\pi$
$577$	8.26007e6	1.03287	0.516433	+	0.856327i	$0.327259\pi$
$578$	359048.	0.0447027
$579$	0	0
$580$	1.04508e6	0.128997
$581$	2.92980e6	0.360078
$582$	0	0
$583$	508757.	0.0619925
$584$	−1.04052e7	−1.26247
$585$	0	0
$586$	7.69862e6	0.926123
$587$	−1.42684e7	−1.70915	−0.854573	−	0.519331i	$-0.826181\pi$
$587$	−1.42684e7	−1.70915	−0.854573	+	0.519331i	$0.826181\pi$
$588$	0	0
$589$	−8.66361e6	−1.02899
$590$	53320.6	0.00630616
$591$	0	0
$592$	−1.84459e6	−0.216319
$593$	−1.26496e7	−1.47720	−0.738601	−	0.674143i	$-0.764513\pi$
$593$	−1.26496e7	−1.47720	−0.738601	+	0.674143i	$0.764513\pi$
$594$	0	0
$595$	330913.	0.0383197
$596$	−632478.	−0.0729340
$597$	0	0
$598$	−1.04080e7	−1.19019
$599$	4.60166e6	0.524019	0.262010	−	0.965065i	$-0.415615\pi$
$599$	4.60166e6	0.524019	0.262010	+	0.965065i	$0.415615\pi$
$600$	0	0
$601$	−8.69808e6	−0.982284	−0.491142	−	0.871080i	$-0.663420\pi$
$601$	−8.69808e6	−0.982284	−0.491142	+	0.871080i	$0.663420\pi$
$602$	−3.46381e6	−0.389549
$603$	0	0
$604$	750940.	0.0837555
$605$	−2.34143e6	−0.260071
$606$	0	0
$607$	8.58642e6	0.945889	0.472945	−	0.881092i	$-0.343191\pi$
$607$	8.58642e6	0.945889	0.472945	+	0.881092i	$0.343191\pi$
$608$	−9.38921e6	−1.03008
$609$	0	0
$610$	1.40211e6	0.152566
$611$	7.56255e6	0.819531
$612$	0	0
$613$	1.80976e6	0.194522	0.0972611	−	0.995259i	$-0.468992\pi$
$613$	1.80976e6	0.194522	0.0972611	+	0.995259i	$0.468992\pi$
$614$	−8.07745e6	−0.864676
$615$	0	0
$616$	−4.52327e6	−0.480287
$617$	−428692.	−0.0453348	−0.0226674	−	0.999743i	$-0.507216\pi$
$617$	−428692.	−0.0453348	−0.0226674	+	0.999743i	$0.507216\pi$
$618$	0	0
$619$	1.35257e7	1.41883	0.709417	−	0.704789i	$-0.248958\pi$
$619$	1.35257e7	1.41883	0.709417	+	0.704789i	$0.248958\pi$
$620$	1.40507e6	0.146797
$621$	0	0
$622$	1.14347e7	1.18508
$623$	6.64274e6	0.685689
$624$	0	0
$625$	390625.	0.0400000
$626$	2.47802e6	0.252737
$627$	0	0
$628$	−868123.	−0.0878379
$629$	−1.30466e6	−0.131483
$630$	0	0
$631$	943645.	0.0943486	0.0471743	−	0.998887i	$-0.484978\pi$
$631$	943645.	0.0943486	0.0471743	+	0.998887i	$0.484978\pi$
$632$	−4.32141e6	−0.430360
$633$	0	0
$634$	3.26933e6	0.323025
$635$	−3.92057e6	−0.385847
$636$	0	0
$637$	−7.54793e6	−0.737020
$638$	6.70855e6	0.652494
$639$	0	0
$640$	117515.	0.0113408
$641$	1.91645e7	1.84227	0.921134	−	0.389247i	$-0.127265\pi$
$641$	1.91645e7	1.84227	0.921134	+	0.389247i	$0.127265\pi$
$642$	0	0
$643$	−5.39663e6	−0.514748	−0.257374	−	0.966312i	$-0.582857\pi$
$643$	−5.39663e6	−0.514748	−0.257374	+	0.966312i	$0.582857\pi$
$644$	2.92153e6	0.277585
$645$	0	0
$646$	2.58915e6	0.244104
$647$	7.22529e6	0.678570	0.339285	−	0.940684i	$-0.389815\pi$
$647$	7.22529e6	0.678570	0.339285	+	0.940684i	$0.389815\pi$
$648$	0	0
$649$	−250391.	−0.0233350
$650$	1.37872e6	0.127994
$651$	0	0
$652$	−8.05561e6	−0.742129
$653$	−1.67534e7	−1.53752	−0.768758	−	0.639540i	$-0.779125\pi$
$653$	−1.67534e7	−1.53752	−0.768758	+	0.639540i	$0.779125\pi$
$654$	0	0
$655$	3.11659e6	0.283842
$656$	3.01382e6	0.273437
$657$	0	0
$658$	2.90178e6	0.261276
$659$	−1.70752e7	−1.53162	−0.765810	−	0.643067i	$-0.777662\pi$
$659$	−1.70752e7	−1.53162	−0.765810	+	0.643067i	$0.777662\pi$
$660$	0	0
$661$	−1.45834e7	−1.29824	−0.649122	−	0.760685i	$-0.724863\pi$
$661$	−1.45834e7	−1.29824	−0.649122	+	0.760685i	$0.724863\pi$
$662$	−3.93486e6	−0.348967
$663$	0	0
$664$	−1.25174e7	−1.10178
$665$	2.38626e6	0.209249
$666$	0	0
$667$	−1.45889e7	−1.26972
$668$	−7.20889e6	−0.625068
$669$	0	0
$670$	−2.27068e6	−0.195420
$671$	−6.58423e6	−0.564545
$672$	0	0
$673$	9.19877e6	0.782874	0.391437	−	0.920205i	$-0.371978\pi$
$673$	9.19877e6	0.782874	0.391437	+	0.920205i	$0.371978\pi$
$674$	−6.82086e6	−0.578348
$675$	0	0
$676$	1.45981e6	0.122866
$677$	1.16519e7	0.977068	0.488534	−	0.872545i	$-0.337532\pi$
$677$	1.16519e7	0.977068	0.488534	+	0.872545i	$0.337532\pi$
$678$	0	0
$679$	−6.02623e6	−0.501616
$680$	−1.41381e6	−0.117252
$681$	0	0
$682$	9.01937e6	0.742532
$683$	1.79480e7	1.47219	0.736094	−	0.676879i	$-0.236668\pi$
$683$	1.79480e7	1.47219	0.736094	+	0.676879i	$0.236668\pi$
$684$	0	0
$685$	−1.23445e6	−0.100519
$686$	−6.20538e6	−0.503452
$687$	0	0
$688$	7.18820e6	0.578960
$689$	517281.	0.0415124
$690$	0	0
$691$	−1.29794e7	−1.03410	−0.517048	−	0.855956i	$-0.672969\pi$
$691$	−1.29794e7	−1.03410	−0.517048	+	0.855956i	$0.672969\pi$
$692$	1.00485e7	0.797690
$693$	0	0
$694$	−1.59860e7	−1.25991
$695$	−1.04380e6	−0.0819697
$696$	0	0
$697$	2.13165e6	0.166201
$698$	9.46433e6	0.735278
$699$	0	0
$700$	−387004.	−0.0298518
$701$	−1.99415e7	−1.53272	−0.766358	−	0.642413i	$-0.777933\pi$
$701$	−1.99415e7	−1.53272	−0.766358	+	0.642413i	$0.777933\pi$
$702$	0	0
$703$	−9.40808e6	−0.717981
$704$	1.63737e7	1.24513
$705$	0	0
$706$	9.64726e6	0.728437
$707$	4.65392e6	0.350163
$708$	0	0
$709$	−1.53176e7	−1.14439	−0.572196	−	0.820117i	$-0.693908\pi$
$709$	−1.53176e7	−1.14439	−0.572196	+	0.820117i	$0.693908\pi$
$710$	−4.42853e6	−0.329696
$711$	0	0
$712$	−2.83808e7	−2.09810
$713$	−1.96142e7	−1.44493
$714$	0	0
$715$	−6.47439e6	−0.473624
$716$	−4.99634e6	−0.364225
$717$	0	0
$718$	1.06981e7	0.774452
$719$	1.24260e7	0.896413	0.448207	−	0.893930i	$-0.352063\pi$
$719$	1.24260e7	0.896413	0.448207	+	0.893930i	$0.352063\pi$
$720$	0	0
$721$	3.30137e6	0.236514
$722$	8.02619e6	0.573016
$723$	0	0
$724$	−6.62680e6	−0.469848
$725$	1.93255e6	0.136548
$726$	0	0
$727$	1.09417e7	0.767803	0.383902	−	0.923374i	$-0.374580\pi$
$727$	1.09417e7	0.767803	0.383902	+	0.923374i	$0.374580\pi$
$728$	−4.59905e6	−0.321618
$729$	0	0
$730$	−5.71472e6	−0.396906
$731$	5.08415e6	0.351904
$732$	0	0
$733$	−8.63822e6	−0.593833	−0.296916	−	0.954903i	$-0.595958\pi$
$733$	−8.63822e6	−0.593833	−0.296916	+	0.954903i	$0.595958\pi$
$734$	−1.77988e7	−1.21941
$735$	0	0
$736$	−2.12570e7	−1.44646
$737$	1.06630e7	0.723121
$738$	0	0
$739$	−1.28244e7	−0.863825	−0.431913	−	0.901915i	$-0.642161\pi$
$739$	−1.28244e7	−0.863825	−0.431913	+	0.901915i	$0.642161\pi$
$740$	1.52581e6	0.102428
$741$	0	0
$742$	198483.	0.0132347
$743$	−1.16623e7	−0.775020	−0.387510	−	0.921865i	$-0.626665\pi$
$743$	−1.16623e7	−0.775020	−0.387510	+	0.921865i	$0.626665\pi$
$744$	0	0
$745$	−1.16957e6	−0.0772032
$746$	7.72675e6	0.508335
$747$	0	0
$748$	1.97187e6	0.128862
$749$	8.04529e6	0.524007
$750$	0	0
$751$	2.02785e7	1.31200	0.656002	−	0.754759i	$-0.272246\pi$
$751$	2.02785e7	1.31200	0.656002	+	0.754759i	$0.272246\pi$
$752$	−6.02186e6	−0.388317
$753$	0	0
$754$	6.82094e6	0.436934
$755$	1.38863e6	0.0886582
$756$	0	0
$757$	4.98405e6	0.316114	0.158057	−	0.987430i	$-0.449477\pi$
$757$	4.98405e6	0.316114	0.158057	+	0.987430i	$0.449477\pi$
$758$	388933.	0.0245868
$759$	0	0
$760$	−1.01952e7	−0.640268
$761$	−1.63795e7	−1.02527	−0.512635	−	0.858607i	$-0.671331\pi$
$761$	−1.63795e7	−1.02527	−0.512635	+	0.858607i	$0.671331\pi$
$762$	0	0
$763$	−766081.	−0.0476391
$764$	−103391.	−0.00640839
$765$	0	0
$766$	−1.25848e7	−0.774951
$767$	−254586.	−0.0156259
$768$	0	0
$769$	−2.09430e7	−1.27709	−0.638547	−	0.769582i	$-0.720464\pi$
$769$	−2.09430e7	−1.27709	−0.638547	+	0.769582i	$0.720464\pi$
$770$	−2.48425e6	−0.150997
$771$	0	0
$772$	2.64449e6	0.159698
$773$	1.56215e7	0.940318	0.470159	−	0.882582i	$-0.344197\pi$
$773$	1.56215e7	0.940318	0.470159	+	0.882582i	$0.344197\pi$
$774$	0	0
$775$	2.59823e6	0.155390
$776$	2.57468e7	1.53486
$777$	0	0
$778$	−5.83544e6	−0.345640
$779$	1.53716e7	0.907560
$780$	0	0
$781$	2.07962e7	1.21999
$782$	5.86178e6	0.342778
$783$	0	0
$784$	6.01022e6	0.349221
$785$	−1.60532e6	−0.0929796
$786$	0	0
$787$	2.34939e7	1.35213	0.676065	−	0.736842i	$-0.263684\pi$
$787$	2.34939e7	1.35213	0.676065	+	0.736842i	$0.263684\pi$
$788$	−1.19682e7	−0.686613
$789$	0	0
$790$	−2.37338e6	−0.135301
$791$	4.89974e6	0.278440
$792$	0	0
$793$	−6.69454e6	−0.378040
$794$	−960133.	−0.0540480
$795$	0	0
$796$	1.18622e7	0.663561
$797$	2.03038e6	0.113222	0.0566111	−	0.998396i	$-0.481970\pi$
$797$	2.03038e6	0.113222	0.0566111	+	0.998396i	$0.481970\pi$
$798$	0	0
$799$	−4.25921e6	−0.236027
$800$	2.81584e6	0.155554
$801$	0	0
$802$	−1.47783e7	−0.811311
$803$	2.68361e7	1.46869
$804$	0	0
$805$	5.40245e6	0.293833
$806$	9.17048e6	0.497227
$807$	0	0
$808$	−1.98837e7	−1.07144
$809$	1.53040e7	0.822120	0.411060	−	0.911608i	$-0.365159\pi$
$809$	1.53040e7	0.822120	0.411060	+	0.911608i	$0.365159\pi$
$810$	0	0
$811$	−1.82768e6	−0.0975769	−0.0487884	−	0.998809i	$-0.515536\pi$
$811$	−1.82768e6	−0.0975769	−0.0487884	+	0.998809i	$0.515536\pi$
$812$	−1.91463e6	−0.101905
$813$	0	0
$814$	9.79441e6	0.518104
$815$	−1.48963e7	−0.785570
$816$	0	0
$817$	3.66625e7	1.92162
$818$	−8.68222e6	−0.453678
$819$	0	0
$820$	−2.49297e6	−0.129474
$821$	−3.84838e6	−0.199260	−0.0996301	−	0.995025i	$-0.531766\pi$
$821$	−3.84838e6	−0.199260	−0.0996301	+	0.995025i	$0.531766\pi$
$822$	0	0
$823$	8.89074e6	0.457550	0.228775	−	0.973479i	$-0.426528\pi$
$823$	8.89074e6	0.457550	0.228775	+	0.973479i	$0.426528\pi$
$824$	−1.41050e7	−0.723694
$825$	0	0
$826$	−97685.7	−0.00498174
$827$	−2.28605e7	−1.16231	−0.581155	−	0.813793i	$-0.697399\pi$
$827$	−2.28605e7	−1.16231	−0.581155	+	0.813793i	$0.697399\pi$
$828$	0	0
$829$	1.27813e7	0.645934	0.322967	−	0.946410i	$-0.395320\pi$
$829$	1.27813e7	0.645934	0.322967	+	0.946410i	$0.395320\pi$
$830$	−6.87477e6	−0.346388
$831$	0	0
$832$	1.66480e7	0.833783
$833$	4.25097e6	0.212264
$834$	0	0
$835$	−1.33306e7	−0.661657
$836$	1.42194e7	0.703666
$837$	0	0
$838$	−1.19123e7	−0.585982
$839$	2.44397e7	1.19865	0.599323	−	0.800508i	$-0.295437\pi$
$839$	2.44397e7	1.19865	0.599323	+	0.800508i	$0.295437\pi$
$840$	0	0
$841$	−1.09502e7	−0.533868
$842$	−4.65438e6	−0.226247
$843$	0	0
$844$	4.40523e6	0.212869
$845$	2.69946e6	0.130058
$846$	0	0
$847$	4.28961e6	0.205451
$848$	−411897.	−0.0196698
$849$	0	0
$850$	−776489.	−0.0368628
$851$	−2.12997e7	−1.00821
$852$	0	0
$853$	−3.88293e7	−1.82720	−0.913601	−	0.406612i	$-0.866710\pi$
$853$	−3.88293e7	−1.82720	−0.913601	+	0.406612i	$0.866710\pi$
$854$	−2.56872e6	−0.120524
$855$	0	0
$856$	−3.43732e7	−1.60338
$857$	2.51074e7	1.16775	0.583875	−	0.811843i	$-0.301536\pi$
$857$	2.51074e7	1.16775	0.583875	+	0.811843i	$0.301536\pi$
$858$	0	0
$859$	−1.46587e6	−0.0677817	−0.0338909	−	0.999426i	$-0.510790\pi$
$859$	−1.46587e6	−0.0677817	−0.0338909	+	0.999426i	$0.510790\pi$
$860$	−5.94593e6	−0.274141
$861$	0	0
$862$	1.32596e7	0.607804
$863$	1.85564e7	0.848138	0.424069	−	0.905630i	$-0.360601\pi$
$863$	1.85564e7	0.848138	0.424069	+	0.905630i	$0.360601\pi$
$864$	0	0
$865$	1.85815e7	0.844384
$866$	1.64961e7	0.747459
$867$	0	0
$868$	−2.57415e6	−0.115967
$869$	1.11453e7	0.500660
$870$	0	0
$871$	1.08417e7	0.484228
$872$	3.27305e6	0.145768
$873$	0	0
$874$	4.22701e7	1.87178
$875$	−715643.	−0.0315992
$876$	0	0
$877$	−4.32523e7	−1.89894	−0.949468	−	0.313864i	$-0.898377\pi$
$877$	−4.32523e7	−1.89894	−0.949468	+	0.313864i	$0.898377\pi$
$878$	−7.64787e6	−0.334814
$879$	0	0
$880$	5.15539e6	0.224416
$881$	−2.75182e7	−1.19448	−0.597241	−	0.802062i	$-0.703736\pi$
$881$	−2.75182e7	−1.19448	−0.597241	+	0.802062i	$0.703736\pi$
$882$	0	0
$883$	3.80483e7	1.64223	0.821115	−	0.570763i	$-0.193352\pi$
$883$	3.80483e7	1.64223	0.821115	+	0.570763i	$0.193352\pi$
$884$	2.00491e6	0.0862907
$885$	0	0
$886$	−9.84849e6	−0.421488
$887$	−1.82142e7	−0.777322	−0.388661	−	0.921381i	$-0.627062\pi$
$887$	−1.82142e7	−0.777322	−0.388661	+	0.921381i	$0.627062\pi$
$888$	0	0
$889$	7.18267e6	0.304812
$890$	−1.55872e7	−0.659619
$891$	0	0
$892$	1.02512e7	0.431384
$893$	−3.07137e7	−1.28885
$894$	0	0
$895$	−9.23917e6	−0.385545
$896$	−215293.	−0.00895902
$897$	0	0
$898$	−2.72805e7	−1.12892
$899$	1.28543e7	0.530454
$900$	0	0
$901$	−291331.	−0.0119557
$902$	−1.60028e7	−0.654907
$903$	0	0
$904$	−2.09339e7	−0.851980
$905$	−1.22542e7	−0.497351
$906$	0	0
$907$	3.24189e7	1.30852	0.654259	−	0.756271i	$-0.272981\pi$
$907$	3.24189e7	1.30852	0.654259	+	0.756271i	$0.272981\pi$
$908$	6.59369e6	0.265408
$909$	0	0
$910$	−2.52587e6	−0.101113
$911$	−4.17461e7	−1.66655	−0.833277	−	0.552855i	$-0.813538\pi$
$911$	−4.17461e7	−1.66655	−0.833277	+	0.552855i	$0.813538\pi$
$912$	0	0
$913$	3.22836e7	1.28176
$914$	5.97935e6	0.236749
$915$	0	0
$916$	5.58985e6	0.220121
$917$	−5.70974e6	−0.224230
$918$	0	0
$919$	−5.75989e6	−0.224970	−0.112485	−	0.993653i	$-0.535881\pi$
$919$	−5.75989e6	−0.224970	−0.112485	+	0.993653i	$0.535881\pi$
$920$	−2.30817e7	−0.899081
$921$	0	0
$922$	1.87811e7	0.727603
$923$	2.11446e7	0.816950
$924$	0	0
$925$	2.82150e6	0.108424
$926$	3.17000e6	0.121488
$927$	0	0
$928$	1.39308e7	0.531015
$929$	3.14157e6	0.119428	0.0597142	−	0.998216i	$-0.480981\pi$
$929$	3.14157e6	0.119428	0.0597142	+	0.998216i	$0.480981\pi$
$930$	0	0
$931$	3.06544e7	1.15909
$932$	1.01456e7	0.382594
$933$	0	0
$934$	−8.41362e6	−0.315585
$935$	3.64636e6	0.136405
$936$	0	0
$937$	2.19517e7	0.816808	0.408404	−	0.912801i	$-0.366085\pi$
$937$	2.19517e7	0.816808	0.408404	+	0.912801i	$0.366085\pi$
$938$	4.15999e6	0.154378
$939$	0	0
$940$	4.98116e6	0.183870
$941$	−580958.	−0.0213880	−0.0106940	−	0.999943i	$-0.503404\pi$
$941$	−580958.	−0.0213880	−0.0106940	+	0.999943i	$0.503404\pi$
$942$	0	0
$943$	3.48010e7	1.27442
$944$	202720.	0.00740401
$945$	0	0
$946$	−3.81680e7	−1.38666
$947$	−1.94273e7	−0.703945	−0.351972	−	0.936010i	$-0.614489\pi$
$947$	−1.94273e7	−0.703945	−0.351972	+	0.936010i	$0.614489\pi$
$948$	0	0
$949$	2.72857e7	0.983488
$950$	−5.59937e6	−0.201294
$951$	0	0
$952$	2.59017e6	0.0926267
$953$	5.06290e7	1.80579	0.902895	−	0.429860i	$-0.141437\pi$
$953$	5.06290e7	1.80579	0.902895	+	0.429860i	$0.141437\pi$
$954$	0	0
$955$	−191189.	−0.00678351
$956$	−5.69448e6	−0.201516
$957$	0	0
$958$	1.38671e7	0.488172
$959$	2.26158e6	0.0794081
$960$	0	0
$961$	−1.13471e7	−0.396349
$962$	9.95851e6	0.346942
$963$	0	0
$964$	−2.86909e6	−0.0994377
$965$	4.89016e6	0.169046
$966$	0	0
$967$	2.24825e7	0.773175	0.386587	−	0.922253i	$-0.373654\pi$
$967$	2.24825e7	0.773175	0.386587	+	0.922253i	$0.373654\pi$
$968$	−1.83272e7	−0.628647
$969$	0	0
$970$	1.41406e7	0.482544
$971$	2.03554e7	0.692836	0.346418	−	0.938080i	$-0.387398\pi$
$971$	2.03554e7	0.692836	0.346418	+	0.938080i	$0.387398\pi$
$972$	0	0
$973$	1.91228e6	0.0647545
$974$	2.43449e7	0.822263
$975$	0	0
$976$	5.33069e6	0.179126
$977$	1.70028e7	0.569880	0.284940	−	0.958545i	$-0.408026\pi$
$977$	1.70028e7	0.569880	0.284940	+	0.958545i	$0.408026\pi$
$978$	0	0
$979$	7.31968e7	2.44082
$980$	−4.97153e6	−0.165358
$981$	0	0
$982$	−1.24686e6	−0.0412610
$983$	−4.26423e7	−1.40753	−0.703763	−	0.710435i	$-0.748498\pi$
$983$	−4.26423e7	−1.40753	−0.703763	+	0.710435i	$0.748498\pi$
$984$	0	0
$985$	−2.21313e7	−0.726804
$986$	−3.84153e6	−0.125838
$987$	0	0
$988$	1.44577e7	0.471201
$989$	8.30031e7	2.69838
$990$	0	0
$991$	−2.07849e7	−0.672302	−0.336151	−	0.941808i	$-0.609125\pi$
$991$	−2.07849e7	−0.672302	−0.336151	+	0.941808i	$0.609125\pi$
$992$	1.87294e7	0.604290
$993$	0	0
$994$	8.11327e6	0.260453
$995$	2.19353e7	0.702403
$996$	0	0
$997$	−2.54002e6	−0.0809282	−0.0404641	−	0.999181i	$-0.512884\pi$
$997$	−2.54002e6	−0.0809282	−0.0404641	+	0.999181i	$0.512884\pi$
$998$	3.38510e7	1.07583
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.6.a.g.1.3		5
3.2	odd	2		85.6.a.a.1.3	✓	5
15.14	odd	2		425.6.a.d.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.3	✓	5	3.2	odd	2
425.6.a.d.1.3		5	15.14	odd	2
765.6.a.g.1.3		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)

\(f(q)\)	\(=\)	\(q+4.29890 q^{2} -13.5195 q^{4} -25.0000 q^{5} +45.8012 q^{7} -195.684 q^{8} -107.472 q^{10} +504.686 q^{11} +513.142 q^{13} +196.895 q^{14} -408.601 q^{16} -289.000 q^{17} -2084.02 q^{19} +337.987 q^{20} +2169.60 q^{22} -4718.18 q^{23} +625.000 q^{25} +2205.94 q^{26} -619.207 q^{28} +3092.07 q^{29} +4157.17 q^{31} +4505.34 q^{32} -1242.38 q^{34} -1145.03 q^{35} +4514.40 q^{37} -8958.99 q^{38} +4892.09 q^{40} -7375.94 q^{41} -17592.2 q^{43} -6823.09 q^{44} -20283.0 q^{46} +14737.7 q^{47} -14709.3 q^{49} +2686.81 q^{50} -6937.40 q^{52} +1008.07 q^{53} -12617.2 q^{55} -8962.53 q^{56} +13292.5 q^{58} -496.132 q^{59} -13046.2 q^{61} +17871.2 q^{62} +32443.2 q^{64} -12828.5 q^{65} +21128.0 q^{67} +3907.12 q^{68} -4922.36 q^{70} +41206.2 q^{71} +53173.8 q^{73} +19406.9 q^{74} +28174.8 q^{76} +23115.2 q^{77} +22083.6 q^{79} +10215.0 q^{80} -31708.4 q^{82} +63967.7 q^{83} +7225.00 q^{85} -75627.1 q^{86} -98758.8 q^{88} +145034. q^{89} +23502.5 q^{91} +63787.2 q^{92} +63356.1 q^{94} +52100.5 q^{95} -131574. q^{97} -63233.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{2} + 43 q^{4} - 125 q^{5} - 204 q^{7} + 63 q^{8} - 175 q^{10} + 792 q^{11} + 88 q^{13} - 860 q^{14} - 2365 q^{16} - 1445 q^{17} - 5160 q^{19} - 1075 q^{20} - 3058 q^{22} + 6140 q^{23} + 3125 q^{25}+ \cdots - 49329 q^{98}+O(q^{100}) \) 5 * q + 7 * q^2 + 43 * q^4 - 125 * q^5 - 204 * q^7 + 63 * q^8 - 175 * q^10 + 792 * q^11 + 88 * q^13 - 860 * q^14 - 2365 * q^16 - 1445 * q^17 - 5160 * q^19 - 1075 * q^20 - 3058 * q^22 + 6140 * q^23 + 3125 * q^25 + 9643 * q^26 - 4568 * q^28 + 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 2023 * q^34 + 5100 * q^35 + 9834 * q^37 - 10625 * q^38 - 1575 * q^40 + 5302 * q^41 - 4916 * q^43 + 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 26483 * q^49 + 4375 * q^50 - 11525 * q^52 - 56492 * q^53 - 19800 * q^55 - 73228 * q^56 + 54213 * q^58 + 96200 * q^59 - 7284 * q^61 - 122335 * q^62 + 28483 * q^64 - 2200 * q^65 - 60308 * q^67 - 12427 * q^68 + 21500 * q^70 + 25304 * q^71 + 163328 * q^73 + 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 50940 * q^79 + 59125 * q^80 + 191390 * q^82 + 179060 * q^83 + 36125 * q^85 - 87086 * q^86 + 167634 * q^88 + 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 100515 * q^94 + 129000 * q^95 - 106076 * q^97 - 49329 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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