LMFDB - Embedded newform 261.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,6,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	261.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:	$41.8601769712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3257317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 34x^{2} - 27x + 10 $ x^4 - 34x^2 - 27x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.275208$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.91663 q^{2} +3.00648 q^{4} +97.7313 q^{5} +139.558 q^{7} +171.544 q^{8} +O(q^{10})$	\(q-5.91663 q^{2} +3.00648 q^{4} +97.7313 q^{5} +139.558 q^{7} +171.544 q^{8} -578.240 q^{10} -533.092 q^{11} -675.965 q^{13} -825.715 q^{14} -1111.17 q^{16} +268.994 q^{17} -2649.15 q^{19} +293.828 q^{20} +3154.11 q^{22} -794.438 q^{23} +6426.40 q^{25} +3999.43 q^{26} +419.580 q^{28} +841.000 q^{29} -4231.04 q^{31} +1084.97 q^{32} -1591.54 q^{34} +13639.2 q^{35} -2689.54 q^{37} +15674.0 q^{38} +16765.2 q^{40} -1395.36 q^{41} -23810.5 q^{43} -1602.73 q^{44} +4700.39 q^{46} -11267.5 q^{47} +2669.53 q^{49} -38022.6 q^{50} -2032.28 q^{52} +3396.67 q^{53} -52099.8 q^{55} +23940.4 q^{56} -4975.88 q^{58} +2785.38 q^{59} +41551.7 q^{61} +25033.5 q^{62} +29138.0 q^{64} -66062.9 q^{65} +8574.14 q^{67} +808.728 q^{68} -80698.2 q^{70} +6995.03 q^{71} -4994.73 q^{73} +15913.0 q^{74} -7964.62 q^{76} -74397.5 q^{77} -23856.6 q^{79} -108596. q^{80} +8255.84 q^{82} -43076.9 q^{83} +26289.2 q^{85} +140878. q^{86} -91448.7 q^{88} -13806.4 q^{89} -94336.6 q^{91} -2388.47 q^{92} +66665.5 q^{94} -258905. q^{95} -176400. q^{97} -15794.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8}+O(q^{10}) $ 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8	$ 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8} - 788 q^{10} + 124 q^{11} - 460 q^{13} - 768 q^{14} - 414 q^{16} - 184 q^{17} - 2392 q^{19} - 2822 q^{20} + 5538 q^{22} + 1192 q^{23} + 1824 q^{25} - 4724 q^{26} + 44 q^{28} + 3364 q^{29} - 19212 q^{31} - 6552 q^{32} - 7612 q^{34} + 22944 q^{35} - 10928 q^{37} + 456 q^{38} - 20 q^{40} + 1120 q^{41} - 21420 q^{43} + 1932 q^{44} - 7588 q^{46} - 23772 q^{47} + 10452 q^{49} - 43240 q^{50} - 29062 q^{52} - 8860 q^{53} - 52652 q^{55} - 34304 q^{56} + 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 20734 q^{64} - 97836 q^{65} - 7840 q^{67} - 20724 q^{68} - 77496 q^{70} + 48744 q^{71} - 74992 q^{73} + 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 106076 q^{79} - 58638 q^{80} - 234132 q^{82} - 62888 q^{83} + 23848 q^{85} + 216014 q^{86} - 39426 q^{88} - 107568 q^{89} - 268896 q^{91} + 26268 q^{92} + 30542 q^{94} - 147352 q^{95} - 49520 q^{97} - 242304 q^{98}+O(q^{100}) $ 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8 - 788 * q^10 + 124 * q^11 - 460 * q^13 - 768 * q^14 - 414 * q^16 - 184 * q^17 - 2392 * q^19 - 2822 * q^20 + 5538 * q^22 + 1192 * q^23 + 1824 * q^25 - 4724 * q^26 + 44 * q^28 + 3364 * q^29 - 19212 * q^31 - 6552 * q^32 - 7612 * q^34 + 22944 * q^35 - 10928 * q^37 + 456 * q^38 - 20 * q^40 + 1120 * q^41 - 21420 * q^43 + 1932 * q^44 - 7588 * q^46 - 23772 * q^47 + 10452 * q^49 - 43240 * q^50 - 29062 * q^52 - 8860 * q^53 - 52652 * q^55 - 34304 * q^56 + 10840 * q^59 + 49448 * q^61 - 18518 * q^62 - 20734 * q^64 - 97836 * q^65 - 7840 * q^67 - 20724 * q^68 - 77496 * q^70 + 48744 * q^71 - 74992 * q^73 + 35920 * q^74 - 140792 * q^76 - 128656 * q^77 - 106076 * q^79 - 58638 * q^80 - 234132 * q^82 - 62888 * q^83 + 23848 * q^85 + 216014 * q^86 - 39426 * q^88 - 107568 * q^89 - 268896 * q^91 + 26268 * q^92 + 30542 * q^94 - 147352 * q^95 - 49520 * q^97 - 242304 * 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 $

$2$	−5.91663	−1.04592	−0.522961	−	0.852357i	$-0.675173\pi$
$2$	−5.91663	−1.04592	−0.522961	+	0.852357i	$0.675173\pi$
$3$	0	0
$3$	0	0
$4$	3.00648	0.0939526
$5$	97.7313	1.74827	0.874135	−	0.485683i	$-0.161429\pi$
$5$	97.7313	1.74827	0.874135	+	0.485683i	$0.161429\pi$
$6$	0	0
$7$	139.558	1.07649	0.538246	−	0.842788i	$-0.319087\pi$
$7$	139.558	1.07649	0.538246	+	0.842788i	$0.319087\pi$
$8$	171.544	0.947655
$9$	0	0
$10$	−578.240	−1.82855
$11$	−533.092	−1.32838	−0.664188	−	0.747566i	$-0.731222\pi$
$11$	−533.092	−1.32838	−0.664188	+	0.747566i	$0.731222\pi$
$12$	0	0
$13$	−675.965	−1.10934	−0.554672	−	0.832069i	$-0.687156\pi$
$13$	−675.965	−1.10934	−0.554672	+	0.832069i	$0.687156\pi$
$14$	−825.715	−1.12593
$15$	0	0
$16$	−1111.17	−1.08513
$17$	268.994	0.225746	0.112873	−	0.993609i	$-0.463995\pi$
$17$	268.994	0.225746	0.112873	+	0.993609i	$0.463995\pi$
$18$	0	0
$19$	−2649.15	−1.68354	−0.841768	−	0.539840i	$-0.818485\pi$
$19$	−2649.15	−1.68354	−0.841768	+	0.539840i	$0.818485\pi$
$20$	293.828	0.164255
$21$	0	0
$22$	3154.11	1.38938
$23$	−794.438	−0.313141	−0.156571	−	0.987667i	$-0.550044\pi$
$23$	−794.438	−0.313141	−0.156571	+	0.987667i	$0.550044\pi$
$24$	0	0
$25$	6426.40	2.05645
$26$	3999.43	1.16029
$27$	0	0
$28$	419.580	0.101139
$29$	841.000	0.185695
$29$	841.000	0.185695
$30$	0	0
$31$	−4231.04	−0.790756	−0.395378	−	0.918518i	$-0.629386\pi$
$31$	−4231.04	−0.790756	−0.395378	+	0.918518i	$0.629386\pi$
$32$	1084.97	0.187302
$33$	0	0
$34$	−1591.54	−0.236113
$35$	13639.2	1.88200
$36$	0	0
$37$	−2689.54	−0.322979	−0.161489	−	0.986874i	$-0.551630\pi$
$37$	−2689.54	−0.322979	−0.161489	+	0.986874i	$0.551630\pi$
$38$	15674.0	1.76085
$39$	0	0
$40$	16765.2	1.65676
$41$	−1395.36	−0.129636	−0.0648182	−	0.997897i	$-0.520647\pi$
$41$	−1395.36	−0.129636	−0.0648182	+	0.997897i	$0.520647\pi$
$42$	0	0
$43$	−23810.5	−1.96380	−0.981901	−	0.189395i	$-0.939347\pi$
$43$	−23810.5	−1.96380	−0.981901	+	0.189395i	$0.939347\pi$
$44$	−1602.73	−0.124804
$45$	0	0
$46$	4700.39	0.327521
$47$	−11267.5	−0.744016	−0.372008	−	0.928229i	$-0.621331\pi$
$47$	−11267.5	−0.744016	−0.372008	+	0.928229i	$0.621331\pi$
$48$	0	0
$49$	2669.53	0.158834
$50$	−38022.6	−2.15089
$51$	0	0
$52$	−2032.28	−0.104226
$53$	3396.67	0.166097	0.0830487	−	0.996545i	$-0.473534\pi$
$53$	3396.67	0.166097	0.0830487	+	0.996545i	$0.473534\pi$
$54$	0	0
$55$	−52099.8	−2.32236
$56$	23940.4	1.02014
$57$	0	0
$58$	−4975.88	−0.194223
$59$	2785.38	0.104173	0.0520865	−	0.998643i	$-0.483413\pi$
$59$	2785.38	0.104173	0.0520865	+	0.998643i	$0.483413\pi$
$60$	0	0
$61$	41551.7	1.42976	0.714881	−	0.699246i	$-0.246481\pi$
$61$	41551.7	1.42976	0.714881	+	0.699246i	$0.246481\pi$
$62$	25033.5	0.827069
$63$	0	0
$64$	29138.0	0.889222
$65$	−66062.9	−1.93943
$66$	0	0
$67$	8574.14	0.233348	0.116674	−	0.993170i	$-0.462777\pi$
$67$	8574.14	0.233348	0.116674	+	0.993170i	$0.462777\pi$
$68$	808.728	0.0212095
$69$	0	0
$70$	−80698.2	−1.96842
$71$	6995.03	0.164681	0.0823405	−	0.996604i	$-0.473760\pi$
$71$	6995.03	0.164681	0.0823405	+	0.996604i	$0.473760\pi$
$72$	0	0
$73$	−4994.73	−0.109699	−0.0548497	−	0.998495i	$-0.517468\pi$
$73$	−4994.73	−0.109699	−0.0548497	+	0.998495i	$0.517468\pi$
$74$	15913.0	0.337811
$75$	0	0
$76$	−7964.62	−0.158173
$77$	−74397.5	−1.42998
$78$	0	0
$79$	−23856.6	−0.430071	−0.215036	−	0.976606i	$-0.568987\pi$
$79$	−23856.6	−0.430071	−0.215036	+	0.976606i	$0.568987\pi$
$80$	−108596.	−1.89709
$81$	0	0
$82$	8255.84	0.135590
$83$	−43076.9	−0.686356	−0.343178	−	0.939270i	$-0.611503\pi$
$83$	−43076.9	−0.686356	−0.343178	+	0.939270i	$0.611503\pi$
$84$	0	0
$85$	26289.2	0.394666
$86$	140878.	2.05398
$87$	0	0
$88$	−91448.7	−1.25884
$89$	−13806.4	−0.184759	−0.0923793	−	0.995724i	$-0.529447\pi$
$89$	−13806.4	−0.184759	−0.0923793	+	0.995724i	$0.529447\pi$
$90$	0	0
$91$	−94336.6	−1.19420
$92$	−2388.47	−0.0294205
$93$	0	0
$94$	66665.5	0.778183
$95$	−258905.	−2.94327
$96$	0	0
$97$	−176400.	−1.90357	−0.951786	−	0.306762i	$-0.900755\pi$
$97$	−176400.	−1.90357	−0.951786	+	0.306762i	$0.900755\pi$
$98$	−15794.6	−0.166128
$99$	0	0
$100$	19320.9	0.193209
$101$	112043.	1.09290	0.546452	−	0.837490i	$-0.315978\pi$
$101$	112043.	1.09290	0.546452	+	0.837490i	$0.315978\pi$
$102$	0	0
$103$	38255.6	0.355305	0.177653	−	0.984093i	$-0.443150\pi$
$103$	38255.6	0.355305	0.177653	+	0.984093i	$0.443150\pi$
$104$	−115958.	−1.05127
$105$	0	0
$106$	−20096.8	−0.173725
$107$	19410.6	0.163900	0.0819499	−	0.996636i	$-0.473885\pi$
$107$	19410.6	0.163900	0.0819499	+	0.996636i	$0.473885\pi$
$108$	0	0
$109$	−51029.2	−0.411389	−0.205694	−	0.978616i	$-0.565945\pi$
$109$	−51029.2	−0.411389	−0.205694	+	0.978616i	$0.565945\pi$
$110$	308255.	2.42901
$111$	0	0
$112$	−155073.	−1.16813
$113$	−45687.3	−0.336588	−0.168294	−	0.985737i	$-0.553826\pi$
$113$	−45687.3	−0.336588	−0.168294	+	0.985737i	$0.553826\pi$
$114$	0	0
$115$	−77641.5	−0.547456
$116$	2528.45	0.0174466
$117$	0	0
$118$	−16480.1	−0.108957
$119$	37540.4	0.243014
$120$	0	0
$121$	123136.	0.764581
$122$	−245846.	−1.49542
$123$	0	0
$124$	−12720.6	−0.0742937
$125$	322650.	1.84696
$126$	0	0
$127$	267210.	1.47009	0.735044	−	0.678019i	$-0.237161\pi$
$127$	267210.	1.47009	0.735044	+	0.678019i	$0.237161\pi$
$128$	−207118.	−1.11736
$129$	0	0
$130$	390870.	2.02849
$131$	165523.	0.842713	0.421356	−	0.906895i	$-0.361554\pi$
$131$	165523.	0.842713	0.421356	+	0.906895i	$0.361554\pi$
$132$	0	0
$133$	−369711.	−1.81231
$134$	−50730.0	−0.244063
$135$	0	0
$136$	46144.3	0.213930
$137$	−91000.4	−0.414230	−0.207115	−	0.978317i	$-0.566407\pi$
$137$	−91000.4	−0.414230	−0.207115	+	0.978317i	$0.566407\pi$
$138$	0	0
$139$	−400431.	−1.75788	−0.878942	−	0.476928i	$-0.841750\pi$
$139$	−400431.	−1.75788	−0.878942	+	0.476928i	$0.841750\pi$
$140$	41006.1	0.176819
$141$	0	0
$142$	−41387.0	−0.172244
$143$	360352.	1.47362
$144$	0	0
$145$	82192.0	0.324646
$146$	29551.9	0.114737
$147$	0	0
$148$	−8086.07	−0.0303447
$149$	18312.8	0.0675756	0.0337878	−	0.999429i	$-0.489243\pi$
$149$	18312.8	0.0675756	0.0337878	+	0.999429i	$0.489243\pi$
$150$	0	0
$151$	−17899.8	−0.0638859	−0.0319430	−	0.999490i	$-0.510169\pi$
$151$	−17899.8	−0.0638859	−0.0319430	+	0.999490i	$0.510169\pi$
$152$	−454445.	−1.59541
$153$	0	0
$154$	440182.	1.49565
$155$	−413505.	−1.38246
$156$	0	0
$157$	−413598.	−1.33915	−0.669576	−	0.742744i	$-0.733524\pi$
$157$	−413598.	−1.33915	−0.669576	+	0.742744i	$0.733524\pi$
$158$	141151.	0.449821
$159$	0	0
$160$	106035.	0.327454
$161$	−110870.	−0.337094
$162$	0	0
$163$	178276.	0.525562	0.262781	−	0.964856i	$-0.415360\pi$
$163$	178276.	0.525562	0.262781	+	0.964856i	$0.415360\pi$
$164$	−4195.13	−0.0121797
$165$	0	0
$166$	254870.	0.717875
$167$	452143.	1.25454	0.627269	−	0.778802i	$-0.284172\pi$
$167$	452143.	1.25454	0.627269	+	0.778802i	$0.284172\pi$
$168$	0	0
$169$	85635.9	0.230642
$170$	−155543.	−0.412790
$171$	0	0
$172$	−71586.0	−0.184504
$173$	−754374.	−1.91634	−0.958168	−	0.286206i	$-0.907606\pi$
$173$	−754374.	−1.91634	−0.958168	+	0.286206i	$0.907606\pi$
$174$	0	0
$175$	896858.	2.21375
$176$	592356.	1.44145
$177$	0	0
$178$	81687.2	0.193243
$179$	352813.	0.823024	0.411512	−	0.911404i	$-0.365001\pi$
$179$	352813.	0.823024	0.411512	+	0.911404i	$0.365001\pi$
$180$	0	0
$181$	227087.	0.515224	0.257612	−	0.966248i	$-0.417064\pi$
$181$	227087.	0.515224	0.257612	+	0.966248i	$0.417064\pi$
$182$	558154.	1.24904
$183$	0	0
$184$	−136281.	−0.296750
$185$	−262852.	−0.564654
$186$	0	0
$187$	−143399.	−0.299876
$188$	−33875.5	−0.0699023
$189$	0	0
$190$	1.53184e6	3.07844
$191$	−183415.	−0.363790	−0.181895	−	0.983318i	$-0.558223\pi$
$191$	−183415.	−0.363790	−0.181895	+	0.983318i	$0.558223\pi$
$192$	0	0
$193$	997005.	1.92665	0.963327	−	0.268329i	$-0.0864714\pi$
$193$	997005.	1.92665	0.963327	+	0.268329i	$0.0864714\pi$
$194$	1.04369e6	1.99099
$195$	0	0
$196$	8025.90	0.0149229
$197$	861288.	1.58119	0.790593	−	0.612342i	$-0.209773\pi$
$197$	861288.	1.58119	0.790593	+	0.612342i	$0.209773\pi$
$198$	0	0
$199$	−837743.	−1.49961	−0.749805	−	0.661659i	$-0.769853\pi$
$199$	−837743.	−1.49961	−0.749805	+	0.661659i	$0.769853\pi$
$200$	1.10241e6	1.94880
$201$	0	0
$202$	−662918.	−1.14309
$203$	117369.	0.199899
$204$	0	0
$205$	−136371.	−0.226640
$206$	−226344.	−0.371621
$207$	0	0
$208$	751111.	1.20378
$209$	1.41224e6	2.23637
$210$	0	0
$211$	−637594.	−0.985912	−0.492956	−	0.870054i	$-0.664084\pi$
$211$	−637594.	−0.985912	−0.492956	+	0.870054i	$0.664084\pi$
$212$	10212.0	0.0156053
$213$	0	0
$214$	−114845.	−0.171426
$215$	−2.32703e6	−3.43326
$216$	0	0
$217$	−590477.	−0.851243
$218$	301921.	0.430281
$219$	0	0
$220$	−156637.	−0.218192
$221$	−181831.	−0.250430
$222$	0	0
$223$	45526.9	0.0613064	0.0306532	−	0.999530i	$-0.490241\pi$
$223$	45526.9	0.0613064	0.0306532	+	0.999530i	$0.490241\pi$
$224$	151416.	0.201629
$225$	0	0
$226$	270315.	0.352045
$227$	−1.33313e6	−1.71715	−0.858575	−	0.512688i	$-0.828650\pi$
$227$	−1.33313e6	−1.71715	−0.858575	+	0.512688i	$0.828650\pi$
$228$	0	0
$229$	830883.	1.04701	0.523506	−	0.852022i	$-0.324624\pi$
$229$	830883.	1.04701	0.523506	+	0.852022i	$0.324624\pi$
$230$	459376.	0.572596
$231$	0	0
$232$	144268.	0.175975
$233$	−767627.	−0.926319	−0.463159	−	0.886275i	$-0.653284\pi$
$233$	−767627.	−0.926319	−0.463159	+	0.886275i	$0.653284\pi$
$234$	0	0
$235$	−1.10119e6	−1.30074
$236$	8374.22	0.00978733
$237$	0	0
$238$	−222113.	−0.254174
$239$	1.22858e6	1.39126	0.695631	−	0.718399i	$-0.255125\pi$
$239$	1.22858e6	1.39126	0.695631	+	0.718399i	$0.255125\pi$
$240$	0	0
$241$	304030.	0.337190	0.168595	−	0.985685i	$-0.446077\pi$
$241$	304030.	0.337190	0.168595	+	0.985685i	$0.446077\pi$
$242$	−728553.	−0.799692
$243$	0	0
$244$	124924.	0.134330
$245$	260897.	0.277685
$246$	0	0
$247$	1.79073e6	1.86762
$248$	−725809.	−0.749364
$249$	0	0
$250$	−1.90900e6	−1.93177
$251$	−850953.	−0.852553	−0.426276	−	0.904593i	$-0.640175\pi$
$251$	−850953.	−0.852553	−0.426276	+	0.904593i	$0.640175\pi$
$252$	0	0
$253$	423509.	0.415969
$254$	−1.58098e6	−1.53760
$255$	0	0
$256$	293022.	0.279448
$257$	−28252.8	−0.0266826	−0.0133413	−	0.999911i	$-0.504247\pi$
$257$	−28252.8	−0.0266826	−0.0133413	+	0.999911i	$0.504247\pi$
$258$	0	0
$259$	−375348.	−0.347684
$260$	−198617.	−0.182215
$261$	0	0
$262$	−979337.	−0.881412
$263$	−393978.	−0.351223	−0.175611	−	0.984460i	$-0.556190\pi$
$263$	−393978.	−0.351223	−0.175611	+	0.984460i	$0.556190\pi$
$264$	0	0
$265$	331960.	0.290383
$266$	2.18744e6	1.89554
$267$	0	0
$268$	25778.0	0.0219236
$269$	−453727.	−0.382309	−0.191154	−	0.981560i	$-0.561223\pi$
$269$	−453727.	−0.382309	−0.191154	+	0.981560i	$0.561223\pi$
$270$	0	0
$271$	−1.54312e6	−1.27637	−0.638186	−	0.769882i	$-0.720315\pi$
$271$	−1.54312e6	−1.27637	−0.638186	+	0.769882i	$0.720315\pi$
$272$	−298898.	−0.244963
$273$	0	0
$274$	538416.	0.433253
$275$	−3.42587e6	−2.73174
$276$	0	0
$277$	−1.13023e6	−0.885050	−0.442525	−	0.896756i	$-0.645917\pi$
$277$	−1.13023e6	−0.885050	−0.442525	+	0.896756i	$0.645917\pi$
$278$	2.36920e6	1.83861
$279$	0	0
$280$	2.33972e6	1.78348
$281$	1.17984e6	0.891371	0.445685	−	0.895190i	$-0.352960\pi$
$281$	1.17984e6	0.891371	0.445685	+	0.895190i	$0.352960\pi$
$282$	0	0
$283$	345340.	0.256319	0.128160	−	0.991754i	$-0.459093\pi$
$283$	345340.	0.256319	0.128160	+	0.991754i	$0.459093\pi$
$284$	21030.5	0.0154722
$285$	0	0
$286$	−2.13207e6	−1.54130
$287$	−194734.	−0.139553
$288$	0	0
$289$	−1.34750e6	−0.949039
$290$	−486300.	−0.339554
$291$	0	0
$292$	−15016.6	−0.0103066
$293$	−2.40825e6	−1.63883	−0.819414	−	0.573202i	$-0.805701\pi$
$293$	−2.40825e6	−1.63883	−0.819414	+	0.573202i	$0.805701\pi$
$294$	0	0
$295$	272219.	0.182123
$296$	−461374.	−0.306072
$297$	0	0
$298$	−108350.	−0.0706788
$299$	537013.	0.347381
$300$	0	0
$301$	−3.32296e6	−2.11402
$302$	105906.	0.0668197
$303$	0	0
$304$	2.94365e6	1.82685
$305$	4.06090e6	2.49961
$306$	0	0
$307$	595416.	0.360558	0.180279	−	0.983616i	$-0.442300\pi$
$307$	595416.	0.360558	0.180279	+	0.983616i	$0.442300\pi$
$308$	−223675.	−0.134351
$309$	0	0
$310$	2.44655e6	1.44594
$311$	−999900.	−0.586213	−0.293107	−	0.956080i	$-0.594689\pi$
$311$	−999900.	−0.586213	−0.293107	+	0.956080i	$0.594689\pi$
$312$	0	0
$313$	−365270.	−0.210743	−0.105371	−	0.994433i	$-0.533603\pi$
$313$	−365270.	−0.210743	−0.105371	+	0.994433i	$0.533603\pi$
$314$	2.44711e6	1.40065
$315$	0	0
$316$	−71724.5	−0.0404064
$317$	−1.60635e6	−0.897823	−0.448912	−	0.893576i	$-0.648188\pi$
$317$	−1.60635e6	−0.897823	−0.448912	+	0.893576i	$0.648188\pi$
$318$	0	0
$319$	−448331.	−0.246673
$320$	2.84770e6	1.55460
$321$	0	0
$322$	655979.	0.352574
$323$	−712606.	−0.380052
$324$	0	0
$325$	−4.34403e6	−2.28131
$326$	−1.05479e6	−0.549697
$327$	0	0
$328$	−239366.	−0.122851
$329$	−1.57247e6	−0.800928
$330$	0	0
$331$	−535124.	−0.268463	−0.134232	−	0.990950i	$-0.542857\pi$
$331$	−535124.	−0.268463	−0.134232	+	0.990950i	$0.542857\pi$
$332$	−129510.	−0.0644850
$333$	0	0
$334$	−2.67516e6	−1.31215
$335$	837961.	0.407955
$336$	0	0
$337$	1.93303e6	0.927178	0.463589	−	0.886050i	$-0.346561\pi$
$337$	1.93303e6	0.927178	0.463589	+	0.886050i	$0.346561\pi$
$338$	−506676.	−0.241234
$339$	0	0
$340$	79038.0	0.0370799
$341$	2.25553e6	1.05042
$342$	0	0
$343$	−1.97300e6	−0.905508
$344$	−4.08455e6	−1.86101
$345$	0	0
$346$	4.46335e6	2.00434
$347$	2.83701e6	1.26485	0.632423	−	0.774623i	$-0.282060\pi$
$347$	2.83701e6	1.26485	0.632423	+	0.774623i	$0.282060\pi$
$348$	0	0
$349$	1.11395e6	0.489555	0.244778	−	0.969579i	$-0.421285\pi$
$349$	1.11395e6	0.489555	0.244778	+	0.969579i	$0.421285\pi$
$350$	−5.30638e6	−2.31541
$351$	0	0
$352$	−578388.	−0.248807
$353$	3.49053e6	1.49092	0.745461	−	0.666549i	$-0.232229\pi$
$353$	3.49053e6	1.49092	0.745461	+	0.666549i	$0.232229\pi$
$354$	0	0
$355$	683633.	0.287907
$356$	−41508.7	−0.0173586
$357$	0	0
$358$	−2.08746e6	−0.860819
$359$	−469676.	−0.192337	−0.0961684	−	0.995365i	$-0.530659\pi$
$359$	−469676.	−0.192337	−0.0961684	+	0.995365i	$0.530659\pi$
$360$	0	0
$361$	4.54189e6	1.83429
$362$	−1.34359e6	−0.538884
$363$	0	0
$364$	−283621.	−0.112198
$365$	−488141.	−0.191784
$366$	0	0
$367$	2.70229e6	1.04729	0.523645	−	0.851936i	$-0.324572\pi$
$367$	2.70229e6	1.04729	0.523645	+	0.851936i	$0.324572\pi$
$368$	882755.	0.339798
$369$	0	0
$370$	1.55520e6	0.590584
$371$	474033.	0.178803
$372$	0	0
$373$	2.77666e6	1.03336	0.516678	−	0.856180i	$-0.327168\pi$
$373$	2.77666e6	1.03336	0.516678	+	0.856180i	$0.327168\pi$
$374$	848438.	0.313647
$375$	0	0
$376$	−1.93287e6	−0.705071
$377$	−568487.	−0.206000
$378$	0	0
$379$	−4.41837e6	−1.58002	−0.790012	−	0.613091i	$-0.789926\pi$
$379$	−4.41837e6	−1.58002	−0.790012	+	0.613091i	$0.789926\pi$
$380$	−778393.	−0.276528
$381$	0	0
$382$	1.08520e6	0.380496
$383$	−1.37796e6	−0.479999	−0.239999	−	0.970773i	$-0.577147\pi$
$383$	−1.37796e6	−0.479999	−0.239999	+	0.970773i	$0.577147\pi$
$384$	0	0
$385$	−7.27096e6	−2.50000
$386$	−5.89891e6	−2.01513
$387$	0	0
$388$	−530344.	−0.178846
$389$	−649334.	−0.217568	−0.108784	−	0.994065i	$-0.534696\pi$
$389$	−649334.	−0.217568	−0.108784	+	0.994065i	$0.534696\pi$
$390$	0	0
$391$	−213699.	−0.0706906
$392$	457941.	0.150520
$393$	0	0
$394$	−5.09592e6	−1.65380
$395$	−2.33154e6	−0.751881
$396$	0	0
$397$	−2.52909e6	−0.805356	−0.402678	−	0.915342i	$-0.631921\pi$
$397$	−2.52909e6	−0.805356	−0.402678	+	0.915342i	$0.631921\pi$
$398$	4.95661e6	1.56847
$399$	0	0
$400$	−7.14082e6	−2.23151
$401$	3.64231e6	1.13114	0.565569	−	0.824701i	$-0.308656\pi$
$401$	3.64231e6	1.13114	0.565569	+	0.824701i	$0.308656\pi$
$402$	0	0
$403$	2.86003e6	0.877220
$404$	336856.	0.102681
$405$	0	0
$406$	−694426.	−0.209079
$407$	1.43377e6	0.429037
$408$	0	0
$409$	2.68522e6	0.793729	0.396865	−	0.917877i	$-0.370098\pi$
$409$	2.68522e6	0.793729	0.396865	+	0.917877i	$0.370098\pi$
$410$	806854.	0.237047
$411$	0	0
$412$	115015.	0.0333819
$413$	388724.	0.112141
$414$	0	0
$415$	−4.20996e6	−1.19994
$416$	−733401.	−0.207782
$417$	0	0
$418$	−8.35570e6	−2.33906
$419$	4.33500e6	1.20630	0.603148	−	0.797629i	$-0.293913\pi$
$419$	4.33500e6	1.20630	0.603148	+	0.797629i	$0.293913\pi$
$420$	0	0
$421$	3.38962e6	0.932063	0.466032	−	0.884768i	$-0.345683\pi$
$421$	3.38962e6	0.932063	0.466032	+	0.884768i	$0.345683\pi$
$422$	3.77241e6	1.03119
$423$	0	0
$424$	582677.	0.157403
$425$	1.72867e6	0.464236
$426$	0	0
$427$	5.79888e6	1.53913
$428$	58357.6	0.0153988
$429$	0	0
$430$	1.37682e7	3.59092
$431$	−290831.	−0.0754132	−0.0377066	−	0.999289i	$-0.512005\pi$
$431$	−290831.	−0.0754132	−0.0377066	+	0.999289i	$0.512005\pi$
$432$	0	0
$433$	−2.64162e6	−0.677097	−0.338549	−	0.940949i	$-0.609936\pi$
$433$	−2.64162e6	−0.677097	−0.338549	+	0.940949i	$0.609936\pi$
$434$	3.49363e6	0.890333
$435$	0	0
$436$	−153418.	−0.0386511
$437$	2.10458e6	0.527185
$438$	0	0
$439$	5.65486e6	1.40043	0.700214	−	0.713933i	$-0.253088\pi$
$439$	5.65486e6	1.40043	0.700214	+	0.713933i	$0.253088\pi$
$440$	−8.93740e6	−2.20079
$441$	0	0
$442$	1.07583e6	0.261931
$443$	−1.31778e6	−0.319031	−0.159515	−	0.987195i	$-0.550993\pi$
$443$	−1.31778e6	−0.319031	−0.159515	+	0.987195i	$0.550993\pi$
$444$	0	0
$445$	−1.34932e6	−0.323008
$446$	−269366.	−0.0641217
$447$	0	0
$448$	4.06646e6	0.957241
$449$	6.50617e6	1.52303	0.761517	−	0.648145i	$-0.224455\pi$
$449$	6.50617e6	1.52303	0.761517	+	0.648145i	$0.224455\pi$
$450$	0	0
$451$	743857.	0.172206
$452$	−137358.	−0.0316234
$453$	0	0
$454$	7.88764e6	1.79601
$455$	−9.21963e6	−2.08778
$456$	0	0
$457$	−3.50691e6	−0.785477	−0.392739	−	0.919650i	$-0.628472\pi$
$457$	−3.50691e6	−0.785477	−0.392739	+	0.919650i	$0.628472\pi$
$458$	−4.91603e6	−1.09509
$459$	0	0
$460$	−233428.	−0.0514349
$461$	1.22247e6	0.267908	0.133954	−	0.990988i	$-0.457233\pi$
$461$	1.22247e6	0.267908	0.133954	+	0.990988i	$0.457233\pi$
$462$	0	0
$463$	382178.	0.0828540	0.0414270	−	0.999142i	$-0.486810\pi$
$463$	382178.	0.0828540	0.0414270	+	0.999142i	$0.486810\pi$
$464$	−934493.	−0.201503
$465$	0	0
$466$	4.54176e6	0.968857
$467$	1.81409e6	0.384916	0.192458	−	0.981305i	$-0.438354\pi$
$467$	1.81409e6	0.384916	0.192458	+	0.981305i	$0.438354\pi$
$468$	0	0
$469$	1.19659e6	0.251197
$470$	6.51531e6	1.36047
$471$	0	0
$472$	477816.	0.0987200
$473$	1.26932e7	2.60867
$474$	0	0
$475$	−1.70245e7	−3.46210
$476$	112865.	0.0228318
$477$	0	0
$478$	−7.26906e6	−1.45515
$479$	−4.87397e6	−0.970609	−0.485304	−	0.874345i	$-0.661291\pi$
$479$	−4.87397e6	−0.970609	−0.485304	+	0.874345i	$0.661291\pi$
$480$	0	0
$481$	1.81804e6	0.358294
$482$	−1.79883e6	−0.352674
$483$	0	0
$484$	370208.	0.0718344
$485$	−1.72398e7	−3.32796
$486$	0	0
$487$	−1.84954e6	−0.353379	−0.176690	−	0.984267i	$-0.556539\pi$
$487$	−1.84954e6	−0.353379	−0.176690	+	0.984267i	$0.556539\pi$
$488$	7.12793e6	1.35492
$489$	0	0
$490$	−1.54363e6	−0.290437
$491$	5.47088e6	1.02413	0.512063	−	0.858948i	$-0.328881\pi$
$491$	5.47088e6	1.02413	0.512063	+	0.858948i	$0.328881\pi$
$492$	0	0
$493$	226224.	0.0419201
$494$	−1.05951e7	−1.95338
$495$	0	0
$496$	4.70140e6	0.858070
$497$	976215.	0.177278
$498$	0	0
$499$	5.27760e6	0.948823	0.474411	−	0.880303i	$-0.342661\pi$
$499$	5.27760e6	0.948823	0.474411	+	0.880303i	$0.342661\pi$
$500$	970044.	0.173527
$501$	0	0
$502$	5.03477e6	0.891703
$503$	−6.07721e6	−1.07099	−0.535494	−	0.844539i	$-0.679875\pi$
$503$	−6.07721e6	−1.07099	−0.535494	+	0.844539i	$0.679875\pi$
$504$	0	0
$505$	1.09501e7	1.91069
$506$	−2.50574e6	−0.435071
$507$	0	0
$508$	803363.	0.138119
$509$	−4.28972e6	−0.733896	−0.366948	−	0.930242i	$-0.619597\pi$
$509$	−4.28972e6	−0.733896	−0.366948	+	0.930242i	$0.619597\pi$
$510$	0	0
$511$	−697056.	−0.118091
$512$	4.89407e6	0.825078
$513$	0	0
$514$	167161.	0.0279079
$515$	3.73876e6	0.621169
$516$	0	0
$517$	6.00661e6	0.988333
$518$	2.22079e6	0.363650
$519$	0	0
$520$	−1.13327e7	−1.83791
$521$	−3.12541e6	−0.504444	−0.252222	−	0.967669i	$-0.581161\pi$
$521$	−3.12541e6	−0.504444	−0.252222	+	0.967669i	$0.581161\pi$
$522$	0	0
$523$	−6.57035e6	−1.05035	−0.525176	−	0.850994i	$-0.676000\pi$
$523$	−6.57035e6	−1.05035	−0.525176	+	0.850994i	$0.676000\pi$
$524$	497642.	0.0791751
$525$	0	0
$526$	2.33102e6	0.367352
$527$	−1.13813e6	−0.178510
$528$	0	0
$529$	−5.80521e6	−0.901942
$530$	−1.96409e6	−0.303718
$531$	0	0
$532$	−1.11153e6	−0.170271
$533$	943216.	0.143811
$534$	0	0
$535$	1.89702e6	0.286541
$536$	1.47084e6	0.221133
$537$	0	0
$538$	2.68453e6	0.399865
$539$	−1.42311e6	−0.210992
$540$	0	0
$541$	−6.72381e6	−0.987694	−0.493847	−	0.869549i	$-0.664410\pi$
$541$	−6.72381e6	−0.987694	−0.493847	+	0.869549i	$0.664410\pi$
$542$	9.13008e6	1.33499
$543$	0	0
$544$	291850.	0.0422827
$545$	−4.98715e6	−0.719219
$546$	0	0
$547$	1.19150e6	0.170264	0.0851322	−	0.996370i	$-0.472869\pi$
$547$	1.19150e6	0.170264	0.0851322	+	0.996370i	$0.472869\pi$
$548$	−273591.	−0.0389180
$549$	0	0
$550$	2.02696e7	2.85718
$551$	−2.22793e6	−0.312625
$552$	0	0
$553$	−3.32939e6	−0.462968
$554$	6.68716e6	0.925693
$555$	0	0
$556$	−1.20389e6	−0.165158
$557$	−2.43064e6	−0.331958	−0.165979	−	0.986129i	$-0.553078\pi$
$557$	−2.43064e6	−0.331958	−0.165979	+	0.986129i	$0.553078\pi$
$558$	0	0
$559$	1.60951e7	2.17853
$560$	−1.51555e7	−2.04220
$561$	0	0
$562$	−6.98069e6	−0.932304
$563$	1.05702e7	1.40544	0.702721	−	0.711466i	$-0.251968\pi$
$563$	1.05702e7	1.40544	0.702721	+	0.711466i	$0.251968\pi$
$564$	0	0
$565$	−4.46508e6	−0.588447
$566$	−2.04325e6	−0.268090
$567$	0	0
$568$	1.19995e6	0.156061
$569$	−5.17877e6	−0.670573	−0.335286	−	0.942116i	$-0.608833\pi$
$569$	−5.17877e6	−0.670573	−0.335286	+	0.942116i	$0.608833\pi$
$570$	0	0
$571$	−3.92258e6	−0.503479	−0.251739	−	0.967795i	$-0.581003\pi$
$571$	−3.92258e6	−0.503479	−0.251739	+	0.967795i	$0.581003\pi$
$572$	1.08339e6	0.138451
$573$	0	0
$574$	1.15217e6	0.145961
$575$	−5.10538e6	−0.643959
$576$	0	0
$577$	1.12133e7	1.40215	0.701074	−	0.713089i	$-0.252704\pi$
$577$	1.12133e7	1.40215	0.701074	+	0.713089i	$0.252704\pi$
$578$	7.97265e6	0.992620
$579$	0	0
$580$	247109.	0.0305013
$581$	−6.01174e6	−0.738857
$582$	0	0
$583$	−1.81074e6	−0.220640
$584$	−856814.	−0.103957
$585$	0	0
$586$	1.42487e7	1.71409
$587$	−1.03337e7	−1.23783	−0.618915	−	0.785458i	$-0.712427\pi$
$587$	−1.03337e7	−1.23783	−0.618915	+	0.785458i	$0.712427\pi$
$588$	0	0
$589$	1.12086e7	1.33127
$590$	−1.61062e6	−0.190486
$591$	0	0
$592$	2.98853e6	0.350472
$593$	1.58653e7	1.85272	0.926362	−	0.376634i	$-0.122919\pi$
$593$	1.58653e7	1.85272	0.926362	+	0.376634i	$0.122919\pi$
$594$	0	0
$595$	3.66887e6	0.424855
$596$	55057.2	0.00634890
$597$	0	0
$598$	−3.17730e6	−0.363334
$599$	9.66314e6	1.10040	0.550201	−	0.835032i	$-0.314551\pi$
$599$	9.66314e6	1.10040	0.550201	+	0.835032i	$0.314551\pi$
$600$	0	0
$601$	−1.07881e7	−1.21831	−0.609155	−	0.793051i	$-0.708491\pi$
$601$	−1.07881e7	−1.21831	−0.609155	+	0.793051i	$0.708491\pi$
$602$	1.96607e7	2.21110
$603$	0	0
$604$	−53815.4	−0.00600225
$605$	1.20343e7	1.33669
$606$	0	0
$607$	6.76786e6	0.745555	0.372778	−	0.927921i	$-0.378405\pi$
$607$	6.76786e6	0.745555	0.372778	+	0.927921i	$0.378405\pi$
$608$	−2.87424e6	−0.315329
$609$	0	0
$610$	−2.40268e7	−2.61440
$611$	7.61643e6	0.825370
$612$	0	0
$613$	2.91439e6	0.313253	0.156627	−	0.987658i	$-0.449938\pi$
$613$	2.91439e6	0.313253	0.156627	+	0.987658i	$0.449938\pi$
$614$	−3.52286e6	−0.377115
$615$	0	0
$616$	−1.27624e7	−1.35513
$617$	1.01418e7	1.07251	0.536254	−	0.844056i	$-0.319839\pi$
$617$	1.01418e7	1.07251	0.536254	+	0.844056i	$0.319839\pi$
$618$	0	0
$619$	−6.89280e6	−0.723051	−0.361526	−	0.932362i	$-0.617744\pi$
$619$	−6.89280e6	−0.723051	−0.361526	+	0.932362i	$0.617744\pi$
$620$	−1.24320e6	−0.129885
$621$	0	0
$622$	5.91603e6	0.613133
$623$	−1.92680e6	−0.198891
$624$	0	0
$625$	1.14505e7	1.17253
$626$	2.16116e6	0.220420
$627$	0	0
$628$	−1.24348e6	−0.125817
$629$	−723472.	−0.0729113
$630$	0	0
$631$	−7.52515e6	−0.752388	−0.376194	−	0.926541i	$-0.622767\pi$
$631$	−7.52515e6	−0.752388	−0.376194	+	0.926541i	$0.622767\pi$
$632$	−4.09245e6	−0.407559
$633$	0	0
$634$	9.50415e6	0.939053
$635$	2.61148e7	2.57011
$636$	0	0
$637$	−1.80451e6	−0.176202
$638$	2.65261e6	0.258001
$639$	0	0
$640$	−2.02419e7	−1.95345
$641$	278549.	0.0267767	0.0133883	−	0.999910i	$-0.495738\pi$
$641$	278549.	0.0267767	0.0133883	+	0.999910i	$0.495738\pi$
$642$	0	0
$643$	−3.63623e6	−0.346836	−0.173418	−	0.984848i	$-0.555481\pi$
$643$	−3.63623e6	−0.346836	−0.173418	+	0.984848i	$0.555481\pi$
$644$	−333330.	−0.0316709
$645$	0	0
$646$	4.21622e6	0.397505
$647$	1.08234e7	1.01649	0.508243	−	0.861214i	$-0.330295\pi$
$647$	1.08234e7	1.01649	0.508243	+	0.861214i	$0.330295\pi$
$648$	0	0
$649$	−1.48487e6	−0.138381
$650$	2.57020e7	2.38607
$651$	0	0
$652$	535984.	0.0493779
$653$	−8.86232e6	−0.813326	−0.406663	−	0.913578i	$-0.633308\pi$
$653$	−8.86232e6	−0.813326	−0.406663	+	0.913578i	$0.633308\pi$
$654$	0	0
$655$	1.61768e7	1.47329
$656$	1.55048e6	0.140672
$657$	0	0
$658$	9.30373e6	0.837708
$659$	4.94619e6	0.443667	0.221833	−	0.975085i	$-0.428796\pi$
$659$	4.94619e6	0.443667	0.221833	+	0.975085i	$0.428796\pi$
$660$	0	0
$661$	−6.34440e6	−0.564790	−0.282395	−	0.959298i	$-0.591129\pi$
$661$	−6.34440e6	−0.564790	−0.282395	+	0.959298i	$0.591129\pi$
$662$	3.16613e6	0.280791
$663$	0	0
$664$	−7.38958e6	−0.650429
$665$	−3.61323e7	−3.16841
$666$	0	0
$667$	−668122.	−0.0581489
$668$	1.35936e6	0.117867
$669$	0	0
$670$	−4.95790e6	−0.426689
$671$	−2.21509e7	−1.89926
$672$	0	0
$673$	1.00239e7	0.853095	0.426548	−	0.904465i	$-0.359730\pi$
$673$	1.00239e7	0.853095	0.426548	+	0.904465i	$0.359730\pi$
$674$	−1.14370e7	−0.969756
$675$	0	0
$676$	257463.	0.0216695
$677$	−3.44231e6	−0.288654	−0.144327	−	0.989530i	$-0.546102\pi$
$677$	−3.44231e6	−0.288654	−0.144327	+	0.989530i	$0.546102\pi$
$678$	0	0
$679$	−2.46181e7	−2.04918
$680$	4.50975e6	0.374007
$681$	0	0
$682$	−1.33452e7	−1.09866
$683$	−2.51352e6	−0.206172	−0.103086	−	0.994672i	$-0.532872\pi$
$683$	−2.51352e6	−0.206172	−0.103086	+	0.994672i	$0.532872\pi$
$684$	0	0
$685$	−8.89359e6	−0.724187
$686$	1.16735e7	0.947090
$687$	0	0
$688$	2.64575e7	2.13097
$689$	−2.29603e6	−0.184259
$690$	0	0
$691$	−2.01696e7	−1.60695	−0.803474	−	0.595340i	$-0.797017\pi$
$691$	−2.01696e7	−1.60695	−0.803474	+	0.595340i	$0.797017\pi$
$692$	−2.26802e6	−0.180045
$693$	0	0
$694$	−1.67856e7	−1.32293
$695$	−3.91346e7	−3.07326
$696$	0	0
$697$	−375345.	−0.0292650
$698$	−6.59082e6	−0.512037
$699$	0	0
$700$	2.69639e6	0.207988
$701$	−1.67322e6	−0.128605	−0.0643025	−	0.997930i	$-0.520482\pi$
$701$	−1.67322e6	−0.128605	−0.0643025	+	0.997930i	$0.520482\pi$
$702$	0	0
$703$	7.12499e6	0.543746
$704$	−1.55333e7	−1.18122
$705$	0	0
$706$	−2.06522e7	−1.55939
$707$	1.56366e7	1.17650
$708$	0	0
$709$	6.28913e6	0.469867	0.234933	−	0.972011i	$-0.424513\pi$
$709$	6.28913e6	0.469867	0.234933	+	0.972011i	$0.424513\pi$
$710$	−4.04480e6	−0.301128
$711$	0	0
$712$	−2.36840e6	−0.175087
$713$	3.36130e6	0.247619
$714$	0	0
$715$	3.52177e7	2.57629
$716$	1.06073e6	0.0773253
$717$	0	0
$718$	2.77890e6	0.201169
$719$	−2.87519e6	−0.207417	−0.103709	−	0.994608i	$-0.533071\pi$
$719$	−2.87519e6	−0.207417	−0.103709	+	0.994608i	$0.533071\pi$
$720$	0	0
$721$	5.33888e6	0.382483
$722$	−2.68726e7	−1.91852
$723$	0	0
$724$	682734.	0.0484067
$725$	5.40461e6	0.381873
$726$	0	0
$727$	2.76368e6	0.193933	0.0969665	−	0.995288i	$-0.469086\pi$
$727$	2.76368e6	0.193933	0.0969665	+	0.995288i	$0.469086\pi$
$728$	−1.61829e7	−1.13169
$729$	0	0
$730$	2.88815e6	0.200591
$731$	−6.40490e6	−0.443321
$732$	0	0
$733$	−1.06643e7	−0.733115	−0.366557	−	0.930395i	$-0.619464\pi$
$733$	−1.06643e7	−0.733115	−0.366557	+	0.930395i	$0.619464\pi$
$734$	−1.59885e7	−1.09538
$735$	0	0
$736$	−861940.	−0.0586520
$737$	−4.57081e6	−0.309973
$738$	0	0
$739$	−3.98603e6	−0.268491	−0.134245	−	0.990948i	$-0.542861\pi$
$739$	−3.98603e6	−0.268491	−0.134245	+	0.990948i	$0.542861\pi$
$740$	−790262.	−0.0530508
$741$	0	0
$742$	−2.80468e6	−0.187014
$743$	2.19748e7	1.46034	0.730170	−	0.683266i	$-0.239441\pi$
$743$	2.19748e7	1.46034	0.730170	+	0.683266i	$0.239441\pi$
$744$	0	0
$745$	1.78974e6	0.118140
$746$	−1.64285e7	−1.08081
$747$	0	0
$748$	−431127.	−0.0281741
$749$	2.70891e6	0.176437
$750$	0	0
$751$	−1.84068e7	−1.19091	−0.595456	−	0.803388i	$-0.703028\pi$
$751$	−1.84068e7	−1.19091	−0.595456	+	0.803388i	$0.703028\pi$
$752$	1.25201e7	0.807351
$753$	0	0
$754$	3.36352e6	0.215460
$755$	−1.74937e6	−0.111690
$756$	0	0
$757$	2.78199e7	1.76447	0.882237	−	0.470805i	$-0.156037\pi$
$757$	2.78199e7	1.76447	0.882237	+	0.470805i	$0.156037\pi$
$758$	2.61418e7	1.65258
$759$	0	0
$760$	−4.44135e7	−2.78921
$761$	−1.96481e6	−0.122987	−0.0614935	−	0.998107i	$-0.519586\pi$
$761$	−1.96481e6	−0.122987	−0.0614935	+	0.998107i	$0.519586\pi$
$762$	0	0
$763$	−7.12155e6	−0.442857
$764$	−551434.	−0.0341791
$765$	0	0
$766$	8.15288e6	0.502041
$767$	−1.88282e6	−0.115564
$768$	0	0
$769$	−1.16064e7	−0.707754	−0.353877	−	0.935292i	$-0.615137\pi$
$769$	−1.16064e7	−0.707754	−0.353877	+	0.935292i	$0.615137\pi$
$770$	4.30196e7	2.61481
$771$	0	0
$772$	2.99748e6	0.181014
$773$	−2.10057e7	−1.26441	−0.632207	−	0.774800i	$-0.717851\pi$
$773$	−2.10057e7	−1.26441	−0.632207	+	0.774800i	$0.717851\pi$
$774$	0	0
$775$	−2.71904e7	−1.62615
$776$	−3.02604e7	−1.80393
$777$	0	0
$778$	3.84187e6	0.227559
$779$	3.69652e6	0.218248
$780$	0	0
$781$	−3.72900e6	−0.218758
$782$	1.26438e6	0.0739368
$783$	0	0
$784$	−2.96630e6	−0.172355
$785$	−4.04215e7	−2.34120
$786$	0	0
$787$	−8.11927e6	−0.467283	−0.233642	−	0.972323i	$-0.575064\pi$
$787$	−8.11927e6	−0.467283	−0.233642	+	0.972323i	$0.575064\pi$
$788$	2.58945e6	0.148557
$789$	0	0
$790$	1.37948e7	0.786409
$791$	−6.37604e6	−0.362335
$792$	0	0
$793$	−2.80875e7	−1.58610
$794$	1.49637e7	0.842340
$795$	0	0
$796$	−2.51866e6	−0.140892
$797$	1.42204e7	0.792985	0.396493	−	0.918038i	$-0.370227\pi$
$797$	1.42204e7	0.792985	0.396493	+	0.918038i	$0.370227\pi$
$798$	0	0
$799$	−3.03089e6	−0.167959
$800$	6.97244e6	0.385177
$801$	0	0
$802$	−2.15502e7	−1.18308
$803$	2.66265e6	0.145722
$804$	0	0
$805$	−1.08355e7	−0.589332
$806$	−1.69218e7	−0.917504
$807$	0	0
$808$	1.92203e7	1.03570
$809$	−2.40784e7	−1.29347	−0.646736	−	0.762714i	$-0.723866\pi$
$809$	−2.40784e7	−1.29347	−0.646736	+	0.762714i	$0.723866\pi$
$810$	0	0
$811$	2.71158e7	1.44767	0.723836	−	0.689972i	$-0.242377\pi$
$811$	2.71158e7	1.44767	0.723836	+	0.689972i	$0.242377\pi$
$812$	352867.	0.0187811
$813$	0	0
$814$	−8.48311e6	−0.448739
$815$	1.74231e7	0.918824
$816$	0	0
$817$	6.30776e7	3.30613
$818$	−1.58875e7	−0.830179
$819$	0	0
$820$	−409996.	−0.0212934
$821$	1.57023e7	0.813025	0.406513	−	0.913645i	$-0.366745\pi$
$821$	1.57023e7	0.813025	0.406513	+	0.913645i	$0.366745\pi$
$822$	0	0
$823$	−1.52411e7	−0.784364	−0.392182	−	0.919888i	$-0.628280\pi$
$823$	−1.52411e7	−0.784364	−0.392182	+	0.919888i	$0.628280\pi$
$824$	6.56250e6	0.336707
$825$	0	0
$826$	−2.29993e6	−0.117291
$827$	2.27417e7	1.15627	0.578136	−	0.815940i	$-0.303780\pi$
$827$	2.27417e7	1.15627	0.578136	+	0.815940i	$0.303780\pi$
$828$	0	0
$829$	1.10029e7	0.556060	0.278030	−	0.960572i	$-0.410319\pi$
$829$	1.10029e7	0.556060	0.278030	+	0.960572i	$0.410319\pi$
$830$	2.49088e7	1.25504
$831$	0	0
$832$	−1.96963e7	−0.986453
$833$	718089.	0.0358563
$834$	0	0
$835$	4.41885e7	2.19327
$836$	4.24588e6	0.210113
$837$	0	0
$838$	−2.56486e7	−1.26169
$839$	2.33372e7	1.14458	0.572288	−	0.820053i	$-0.306056\pi$
$839$	2.33372e7	1.14458	0.572288	+	0.820053i	$0.306056\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	−2.00551e7	−0.974866
$843$	0	0
$844$	−1.91692e6	−0.0926291
$845$	8.36931e6	0.403225
$846$	0	0
$847$	1.71847e7	0.823065
$848$	−3.77427e6	−0.180237
$849$	0	0
$850$	−1.02279e7	−0.485555
$851$	2.13667e6	0.101138
$852$	0	0
$853$	1.63119e7	0.767595	0.383797	−	0.923417i	$-0.374616\pi$
$853$	1.63119e7	0.767595	0.383797	+	0.923417i	$0.374616\pi$
$854$	−3.43098e7	−1.60981
$855$	0	0
$856$	3.32976e6	0.155321
$857$	1.11103e7	0.516743	0.258372	−	0.966046i	$-0.416814\pi$
$857$	1.11103e7	0.516743	0.258372	+	0.966046i	$0.416814\pi$
$858$	0	0
$859$	2.94971e7	1.36394	0.681971	−	0.731379i	$-0.261123\pi$
$859$	2.94971e7	1.36394	0.681971	+	0.731379i	$0.261123\pi$
$860$	−6.99619e6	−0.322564
$861$	0	0
$862$	1.72074e6	0.0788764
$863$	−2.33489e7	−1.06718	−0.533592	−	0.845742i	$-0.679158\pi$
$863$	−2.33489e7	−1.06718	−0.533592	+	0.845742i	$0.679158\pi$
$864$	0	0
$865$	−7.37260e7	−3.35027
$866$	1.56295e7	0.708191
$867$	0	0
$868$	−1.77526e6	−0.0799765
$869$	1.27178e7	0.571296
$870$	0	0
$871$	−5.79582e6	−0.258863
$872$	−8.75374e6	−0.389855
$873$	0	0
$874$	−1.24520e7	−0.551394
$875$	4.50286e7	1.98824
$876$	0	0
$877$	−3.29425e7	−1.44630	−0.723149	−	0.690692i	$-0.757306\pi$
$877$	−3.29425e7	−1.44630	−0.723149	+	0.690692i	$0.757306\pi$
$878$	−3.34577e7	−1.46474
$879$	0	0
$880$	5.78917e7	2.52005
$881$	−3.22782e7	−1.40110	−0.700550	−	0.713603i	$-0.747062\pi$
$881$	−3.22782e7	−1.40110	−0.700550	+	0.713603i	$0.747062\pi$
$882$	0	0
$883$	−1.31542e7	−0.567758	−0.283879	−	0.958860i	$-0.591621\pi$
$883$	−1.31542e7	−0.567758	−0.283879	+	0.958860i	$0.591621\pi$
$884$	−546672.	−0.0235286
$885$	0	0
$886$	7.79679e6	0.333681
$887$	5.84285e6	0.249354	0.124677	−	0.992197i	$-0.460211\pi$
$887$	5.84285e6	0.249354	0.124677	+	0.992197i	$0.460211\pi$
$888$	0	0
$889$	3.72914e7	1.58254
$890$	7.98340e6	0.337841
$891$	0	0
$892$	136876.	0.00575990
$893$	2.98492e7	1.25258
$894$	0	0
$895$	3.44809e7	1.43887
$896$	−2.89050e7	−1.20283
$897$	0	0
$898$	−3.84946e7	−1.59297
$899$	−3.55830e6	−0.146840
$900$	0	0
$901$	913684.	0.0374959
$902$	−4.40112e6	−0.180114
$903$	0	0
$904$	−7.83737e6	−0.318970
$905$	2.21935e7	0.900751
$906$	0	0
$907$	−2.27296e7	−0.917431	−0.458716	−	0.888583i	$-0.651690\pi$
$907$	−2.27296e7	−0.917431	−0.458716	+	0.888583i	$0.651690\pi$
$908$	−4.00804e6	−0.161331
$909$	0	0
$910$	5.45491e7	2.18366
$911$	−2.85916e7	−1.14141	−0.570706	−	0.821154i	$-0.693330\pi$
$911$	−2.85916e7	−1.14141	−0.570706	+	0.821154i	$0.693330\pi$
$912$	0	0
$913$	2.29640e7	0.911738
$914$	2.07491e7	0.821548
$915$	0	0
$916$	2.49804e6	0.0983695
$917$	2.31001e7	0.907173
$918$	0	0
$919$	4.07727e7	1.59250	0.796252	−	0.604966i	$-0.206813\pi$
$919$	4.07727e7	1.59250	0.796252	+	0.604966i	$0.206813\pi$
$920$	−1.33189e7	−0.518799
$921$	0	0
$922$	−7.23289e6	−0.280210
$923$	−4.72840e6	−0.182688
$924$	0	0
$925$	−1.72841e7	−0.664189
$926$	−2.26121e6	−0.0866588
$927$	0	0
$928$	912458.	0.0347811
$929$	4.12838e7	1.56943	0.784713	−	0.619860i	$-0.212810\pi$
$929$	4.12838e7	1.56943	0.784713	+	0.619860i	$0.212810\pi$
$930$	0	0
$931$	−7.07198e6	−0.267403
$932$	−2.30786e6	−0.0870301
$933$	0	0
$934$	−1.07333e7	−0.402592
$935$	−1.40146e7	−0.524264
$936$	0	0
$937$	4.54057e7	1.68951	0.844755	−	0.535153i	$-0.179746\pi$
$937$	4.54057e7	1.68951	0.844755	+	0.535153i	$0.179746\pi$
$938$	−7.07979e6	−0.262732
$939$	0	0
$940$	−3.31070e6	−0.122208
$941$	−4.70900e7	−1.73362	−0.866812	−	0.498635i	$-0.833835\pi$
$941$	−4.70900e7	−1.73362	−0.866812	+	0.498635i	$0.833835\pi$
$942$	0	0
$943$	1.10853e6	0.0405946
$944$	−3.09503e6	−0.113041
$945$	0	0
$946$	−7.51010e7	−2.72846
$947$	−2.49040e6	−0.0902390	−0.0451195	−	0.998982i	$-0.514367\pi$
$947$	−2.49040e6	−0.0902390	−0.0451195	+	0.998982i	$0.514367\pi$
$948$	0	0
$949$	3.37626e6	0.121694
$950$	1.00728e8	3.62109
$951$	0	0
$952$	6.43983e6	0.230294
$953$	1.43265e7	0.510985	0.255493	−	0.966811i	$-0.417762\pi$
$953$	1.43265e7	0.510985	0.255493	+	0.966811i	$0.417762\pi$
$954$	0	0
$955$	−1.79254e7	−0.636004
$956$	3.69371e6	0.130713
$957$	0	0
$958$	2.88375e7	1.01518
$959$	−1.26999e7	−0.445916
$960$	0	0
$961$	−1.07275e7	−0.374704
$962$	−1.07566e7	−0.374748
$963$	0	0
$964$	914063.	0.0316799
$965$	9.74385e7	3.36831
$966$	0	0
$967$	−1.92616e7	−0.662407	−0.331204	−	0.943559i	$-0.607455\pi$
$967$	−1.92616e7	−0.662407	−0.331204	+	0.943559i	$0.607455\pi$
$968$	2.11233e7	0.724559
$969$	0	0
$970$	1.02002e8	3.48079
$971$	3.31255e7	1.12749	0.563747	−	0.825948i	$-0.309359\pi$
$971$	3.31255e7	1.12749	0.563747	+	0.825948i	$0.309359\pi$
$972$	0	0
$973$	−5.58834e7	−1.89235
$974$	1.09430e7	0.369607
$975$	0	0
$976$	−4.61709e7	−1.55147
$977$	5.00959e7	1.67906	0.839529	−	0.543314i	$-0.182831\pi$
$977$	5.00959e7	1.67906	0.839529	+	0.543314i	$0.182831\pi$
$978$	0	0
$979$	7.36008e6	0.245429
$980$	784382.	0.0260893
$981$	0	0
$982$	−3.23692e7	−1.07116
$983$	6.69976e6	0.221144	0.110572	−	0.993868i	$-0.464732\pi$
$983$	6.69976e6	0.221144	0.110572	+	0.993868i	$0.464732\pi$
$984$	0	0
$985$	8.41748e7	2.76434
$986$	−1.33849e6	−0.0438451
$987$	0	0
$988$	5.38381e6	0.175468
$989$	1.89160e7	0.614948
$990$	0	0
$991$	−3.54819e7	−1.14768	−0.573842	−	0.818966i	$-0.694548\pi$
$991$	−3.54819e7	−1.14768	−0.573842	+	0.818966i	$0.694548\pi$
$992$	−4.59054e6	−0.148110
$993$	0	0
$994$	−5.77590e6	−0.185419
$995$	−8.18737e7	−2.62172
$996$	0	0
$997$	−2.99768e7	−0.955097	−0.477549	−	0.878605i	$-0.658475\pi$
$997$	−2.99768e7	−0.955097	−0.477549	+	0.878605i	$0.658475\pi$
$998$	−3.12256e7	−0.992395
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.6.a.a.1.1		4
3.2	odd	2		29.6.a.a.1.4	✓	4
12.11	even	2		464.6.a.i.1.4		4
15.14	odd	2		725.6.a.a.1.1		4
87.86	odd	2		841.6.a.a.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.4	✓	4	3.2	odd	2
261.6.a.a.1.1		4	1.1	even	1	trivial
464.6.a.i.1.4		4	12.11	even	2
725.6.a.a.1.1		4	15.14	odd	2
841.6.a.a.1.1		4	87.86	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:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q-5.91663 q^{2} +3.00648 q^{4} +97.7313 q^{5} +139.558 q^{7} +171.544 q^{8} +O(q^{10})\)	\(q-5.91663 q^{2} +3.00648 q^{4} +97.7313 q^{5} +139.558 q^{7} +171.544 q^{8} -578.240 q^{10} -533.092 q^{11} -675.965 q^{13} -825.715 q^{14} -1111.17 q^{16} +268.994 q^{17} -2649.15 q^{19} +293.828 q^{20} +3154.11 q^{22} -794.438 q^{23} +6426.40 q^{25} +3999.43 q^{26} +419.580 q^{28} +841.000 q^{29} -4231.04 q^{31} +1084.97 q^{32} -1591.54 q^{34} +13639.2 q^{35} -2689.54 q^{37} +15674.0 q^{38} +16765.2 q^{40} -1395.36 q^{41} -23810.5 q^{43} -1602.73 q^{44} +4700.39 q^{46} -11267.5 q^{47} +2669.53 q^{49} -38022.6 q^{50} -2032.28 q^{52} +3396.67 q^{53} -52099.8 q^{55} +23940.4 q^{56} -4975.88 q^{58} +2785.38 q^{59} +41551.7 q^{61} +25033.5 q^{62} +29138.0 q^{64} -66062.9 q^{65} +8574.14 q^{67} +808.728 q^{68} -80698.2 q^{70} +6995.03 q^{71} -4994.73 q^{73} +15913.0 q^{74} -7964.62 q^{76} -74397.5 q^{77} -23856.6 q^{79} -108596. q^{80} +8255.84 q^{82} -43076.9 q^{83} +26289.2 q^{85} +140878. q^{86} -91448.7 q^{88} -13806.4 q^{89} -94336.6 q^{91} -2388.47 q^{92} +66665.5 q^{94} -258905. q^{95} -176400. q^{97} -15794.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8}+O(q^{10}) \) 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8	\( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8} - 788 q^{10} + 124 q^{11} - 460 q^{13} - 768 q^{14} - 414 q^{16} - 184 q^{17} - 2392 q^{19} - 2822 q^{20} + 5538 q^{22} + 1192 q^{23} + 1824 q^{25} - 4724 q^{26} + 44 q^{28} + 3364 q^{29} - 19212 q^{31} - 6552 q^{32} - 7612 q^{34} + 22944 q^{35} - 10928 q^{37} + 456 q^{38} - 20 q^{40} + 1120 q^{41} - 21420 q^{43} + 1932 q^{44} - 7588 q^{46} - 23772 q^{47} + 10452 q^{49} - 43240 q^{50} - 29062 q^{52} - 8860 q^{53} - 52652 q^{55} - 34304 q^{56} + 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 20734 q^{64} - 97836 q^{65} - 7840 q^{67} - 20724 q^{68} - 77496 q^{70} + 48744 q^{71} - 74992 q^{73} + 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 106076 q^{79} - 58638 q^{80} - 234132 q^{82} - 62888 q^{83} + 23848 q^{85} + 216014 q^{86} - 39426 q^{88} - 107568 q^{89} - 268896 q^{91} + 26268 q^{92} + 30542 q^{94} - 147352 q^{95} - 49520 q^{97} - 242304 q^{98}+O(q^{100}) \) 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8 - 788 * q^10 + 124 * q^11 - 460 * q^13 - 768 * q^14 - 414 * q^16 - 184 * q^17 - 2392 * q^19 - 2822 * q^20 + 5538 * q^22 + 1192 * q^23 + 1824 * q^25 - 4724 * q^26 + 44 * q^28 + 3364 * q^29 - 19212 * q^31 - 6552 * q^32 - 7612 * q^34 + 22944 * q^35 - 10928 * q^37 + 456 * q^38 - 20 * q^40 + 1120 * q^41 - 21420 * q^43 + 1932 * q^44 - 7588 * q^46 - 23772 * q^47 + 10452 * q^49 - 43240 * q^50 - 29062 * q^52 - 8860 * q^53 - 52652 * q^55 - 34304 * q^56 + 10840 * q^59 + 49448 * q^61 - 18518 * q^62 - 20734 * q^64 - 97836 * q^65 - 7840 * q^67 - 20724 * q^68 - 77496 * q^70 + 48744 * q^71 - 74992 * q^73 + 35920 * q^74 - 140792 * q^76 - 128656 * q^77 - 106076 * q^79 - 58638 * q^80 - 234132 * q^82 - 62888 * q^83 + 23848 * q^85 + 216014 * q^86 - 39426 * q^88 - 107568 * q^89 - 268896 * q^91 + 26268 * q^92 + 30542 * q^94 - 147352 * q^95 - 49520 * q^97 - 242304 * q^98

Introduction

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