LMFDB - Embedded newform 495.6.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.476452$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.64018 q^{2} -0.188384 q^{4} -25.0000 q^{5} +145.021 q^{7} -181.548 q^{8} +O(q^{10})$	\(q+5.64018 q^{2} -0.188384 q^{4} -25.0000 q^{5} +145.021 q^{7} -181.548 q^{8} -141.004 q^{10} +121.000 q^{11} -69.9067 q^{13} +817.946 q^{14} -1017.94 q^{16} +500.495 q^{17} -670.685 q^{19} +4.70959 q^{20} +682.462 q^{22} -791.025 q^{23} +625.000 q^{25} -394.287 q^{26} -27.3196 q^{28} -1545.91 q^{29} +2703.09 q^{31} +68.2013 q^{32} +2822.88 q^{34} -3625.53 q^{35} -2667.13 q^{37} -3782.78 q^{38} +4538.71 q^{40} -9622.09 q^{41} -6681.57 q^{43} -22.7944 q^{44} -4461.52 q^{46} +1167.06 q^{47} +4224.17 q^{49} +3525.11 q^{50} +13.1693 q^{52} -28872.9 q^{53} -3025.00 q^{55} -26328.4 q^{56} -8719.22 q^{58} -23599.3 q^{59} +17601.3 q^{61} +15245.9 q^{62} +32958.6 q^{64} +1747.67 q^{65} +16501.5 q^{67} -94.2851 q^{68} -20448.6 q^{70} -72059.4 q^{71} -45480.8 q^{73} -15043.1 q^{74} +126.346 q^{76} +17547.6 q^{77} -21688.7 q^{79} +25448.4 q^{80} -54270.3 q^{82} -6934.34 q^{83} -12512.4 q^{85} -37685.2 q^{86} -21967.3 q^{88} -42779.8 q^{89} -10138.0 q^{91} +149.016 q^{92} +6582.41 q^{94} +16767.1 q^{95} -20992.6 q^{97} +23825.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8	$ 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8 + 50 * q^10 + 363 * q^11 - 546 * q^13 + 8 * q^14 - 1360 * q^16 + 314 * q^17 + 1808 * q^19 + 500 * q^20 - 242 * q^22 - 4288 * q^23 + 1875 * q^25 - 812 * q^26 + 5888 * q^28 - 5582 * q^29 + 6328 * q^31 + 736 * q^32 + 11596 * q^34 - 3800 * q^35 + 16866 * q^37 - 9584 * q^38 - 600 * q^40 - 23282 * q^41 + 20572 * q^43 - 2420 * q^44 + 16592 * q^46 - 3432 * q^47 + 11531 * q^49 - 1250 * q^50 + 21816 * q^52 - 16138 * q^53 - 9075 * q^55 - 15648 * q^56 + 17460 * q^58 - 21972 * q^59 + 8322 * q^61 - 5056 * q^62 + 22208 * q^64 + 13650 * q^65 - 84332 * q^67 - 59832 * q^68 - 200 * q^70 - 50528 * q^71 - 53838 * q^73 - 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 6364 * q^79 + 34000 * q^80 - 68020 * q^82 - 96272 * q^83 - 7850 * q^85 - 143152 * q^86 + 2904 * q^88 + 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 49088 * q^94 - 45200 * q^95 - 103242 * q^97 - 9554 * 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 $

$2$	5.64018	0.997052	0.498526	−	0.866875i	$-0.333875\pi$
$2$	5.64018	0.997052	0.498526	+	0.866875i	$0.333875\pi$
$3$	0	0
$3$	0	0
$4$	−0.188384	−0.00588699
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	145.021	1.11863	0.559315	−	0.828955i	$-0.311064\pi$
$7$	145.021	1.11863	0.559315	+	0.828955i	$0.311064\pi$
$8$	−181.548	−1.00292
$9$	0	0
$10$	−141.004	−0.445895
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−69.9067	−0.114726	−0.0573628	−	0.998353i	$-0.518269\pi$
$13$	−69.9067	−0.114726	−0.0573628	+	0.998353i	$0.518269\pi$
$14$	817.946	1.11533
$15$	0	0
$16$	−1017.94	−0.994078
$17$	500.495	0.420027	0.210014	−	0.977698i	$-0.432649\pi$
$17$	500.495	0.420027	0.210014	+	0.977698i	$0.432649\pi$
$18$	0	0
$19$	−670.685	−0.426221	−0.213110	−	0.977028i	$-0.568359\pi$
$19$	−670.685	−0.426221	−0.213110	+	0.977028i	$0.568359\pi$
$20$	4.70959	0.00263274
$21$	0	0
$22$	682.462	0.300623
$23$	−791.025	−0.311796	−0.155898	−	0.987773i	$-0.549827\pi$
$23$	−791.025	−0.311796	−0.155898	+	0.987773i	$0.549827\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−394.287	−0.114388
$27$	0	0
$28$	−27.3196	−0.00658537
$29$	−1545.91	−0.341342	−0.170671	−	0.985328i	$-0.554594\pi$
$29$	−1545.91	−0.341342	−0.170671	+	0.985328i	$0.554594\pi$
$30$	0	0
$31$	2703.09	0.505191	0.252596	−	0.967572i	$-0.418716\pi$
$31$	2703.09	0.505191	0.252596	+	0.967572i	$0.418716\pi$
$32$	68.2013	0.0117738
$33$	0	0
$34$	2822.88	0.418789
$35$	−3625.53	−0.500267
$36$	0	0
$37$	−2667.13	−0.320287	−0.160144	−	0.987094i	$-0.551196\pi$
$37$	−2667.13	−0.320287	−0.160144	+	0.987094i	$0.551196\pi$
$38$	−3782.78	−0.424964
$39$	0	0
$40$	4538.71	0.448520
$41$	−9622.09	−0.893942	−0.446971	−	0.894548i	$-0.647497\pi$
$41$	−9622.09	−0.893942	−0.446971	+	0.894548i	$0.647497\pi$
$42$	0	0
$43$	−6681.57	−0.551070	−0.275535	−	0.961291i	$-0.588855\pi$
$43$	−6681.57	−0.551070	−0.275535	+	0.961291i	$0.588855\pi$
$44$	−22.7944	−0.00177499
$45$	0	0
$46$	−4461.52	−0.310877
$47$	1167.06	0.0770632	0.0385316	−	0.999257i	$-0.487732\pi$
$47$	1167.06	0.0770632	0.0385316	+	0.999257i	$0.487732\pi$
$48$	0	0
$49$	4224.17	0.251334
$50$	3525.11	0.199410
$51$	0	0
$52$	13.1693	0.000675389 0
$53$	−28872.9	−1.41189	−0.705944	−	0.708268i	$-0.749477\pi$
$53$	−28872.9	−1.41189	−0.705944	+	0.708268i	$0.749477\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−26328.4	−1.12190
$57$	0	0
$58$	−8719.22	−0.340336
$59$	−23599.3	−0.882610	−0.441305	−	0.897357i	$-0.645484\pi$
$59$	−23599.3	−0.882610	−0.441305	+	0.897357i	$0.645484\pi$
$60$	0	0
$61$	17601.3	0.605648	0.302824	−	0.953046i	$-0.402070\pi$
$61$	17601.3	0.605648	0.302824	+	0.953046i	$0.402070\pi$
$62$	15245.9	0.503702
$63$	0	0
$64$	32958.6	1.00582
$65$	1747.67	0.0513069
$66$	0	0
$67$	16501.5	0.449092	0.224546	−	0.974463i	$-0.427910\pi$
$67$	16501.5	0.449092	0.224546	+	0.974463i	$0.427910\pi$
$68$	−94.2851	−0.00247270
$69$	0	0
$70$	−20448.6	−0.498792
$71$	−72059.4	−1.69646	−0.848232	−	0.529625i	$-0.822333\pi$
$71$	−72059.4	−1.69646	−0.848232	+	0.529625i	$0.822333\pi$
$72$	0	0
$73$	−45480.8	−0.998898	−0.499449	−	0.866343i	$-0.666464\pi$
$73$	−45480.8	−0.998898	−0.499449	+	0.866343i	$0.666464\pi$
$74$	−15043.1	−0.319343
$75$	0	0
$76$	126.346	0.00250916
$77$	17547.6	0.337280
$78$	0	0
$79$	−21688.7	−0.390990	−0.195495	−	0.980705i	$-0.562631\pi$
$79$	−21688.7	−0.390990	−0.195495	+	0.980705i	$0.562631\pi$
$80$	25448.4	0.444565
$81$	0	0
$82$	−54270.3	−0.891307
$83$	−6934.34	−0.110487	−0.0552434	−	0.998473i	$-0.517593\pi$
$83$	−6934.34	−0.110487	−0.0552434	+	0.998473i	$0.517593\pi$
$84$	0	0
$85$	−12512.4	−0.187842
$86$	−37685.2	−0.549446
$87$	0	0
$88$	−21967.3	−0.302392
$89$	−42779.8	−0.572484	−0.286242	−	0.958157i	$-0.592406\pi$
$89$	−42779.8	−0.572484	−0.286242	+	0.958157i	$0.592406\pi$
$90$	0	0
$91$	−10138.0	−0.128336
$92$	149.016	0.00183554
$93$	0	0
$94$	6582.41	0.0768361
$95$	16767.1	0.190612
$96$	0	0
$97$	−20992.6	−0.226535	−0.113268	−	0.993565i	$-0.536132\pi$
$97$	−20992.6	−0.226535	−0.113268	+	0.993565i	$0.536132\pi$
$98$	23825.1	0.250593
$99$	0	0
$100$	−117.740	−0.00117740
$101$	−189654.	−1.84995	−0.924973	−	0.380033i	$-0.875913\pi$
$101$	−189654.	−1.84995	−0.924973	+	0.380033i	$0.875913\pi$
$102$	0	0
$103$	−160899.	−1.49438	−0.747190	−	0.664611i	$-0.768597\pi$
$103$	−160899.	−1.49438	−0.747190	+	0.664611i	$0.768597\pi$
$104$	12691.4	0.115061
$105$	0	0
$106$	−162848.	−1.40773
$107$	−113244.	−0.956216	−0.478108	−	0.878301i	$-0.658677\pi$
$107$	−113244.	−0.956216	−0.478108	+	0.878301i	$0.658677\pi$
$108$	0	0
$109$	−1768.32	−0.0142559	−0.00712797	−	0.999975i	$-0.502269\pi$
$109$	−1768.32	−0.0142559	−0.00712797	+	0.999975i	$0.502269\pi$
$110$	−17061.5	−0.134442
$111$	0	0
$112$	−147622.	−1.11201
$113$	77273.2	0.569289	0.284645	−	0.958633i	$-0.408124\pi$
$113$	77273.2	0.569289	0.284645	+	0.958633i	$0.408124\pi$
$114$	0	0
$115$	19775.6	0.139439
$116$	291.224	0.00200948
$117$	0	0
$118$	−133104.	−0.880008
$119$	72582.5	0.469855
$120$	0	0
$121$	14641.0	0.0909091
$122$	99274.6	0.603863
$123$	0	0
$124$	−509.218	−0.00297406
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	104707.	0.576058	0.288029	−	0.957622i	$-0.407000\pi$
$127$	104707.	0.576058	0.288029	+	0.957622i	$0.407000\pi$
$128$	183710.	0.991079
$129$	0	0
$130$	9857.16	0.0511556
$131$	−101248.	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$131$	−101248.	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$132$	0	0
$133$	−97263.6	−0.476783
$134$	93071.2	0.447768
$135$	0	0
$136$	−90864.0	−0.421255
$137$	−128311.	−0.584068	−0.292034	−	0.956408i	$-0.594332\pi$
$137$	−128311.	−0.584068	−0.292034	+	0.956408i	$0.594332\pi$
$138$	0	0
$139$	161754.	0.710100	0.355050	−	0.934847i	$-0.384464\pi$
$139$	161754.	0.710100	0.355050	+	0.934847i	$0.384464\pi$
$140$	682.991	0.00294507
$141$	0	0
$142$	−406428.	−1.69146
$143$	−8458.72	−0.0345911
$144$	0	0
$145$	38647.8	0.152653
$146$	−256520.	−0.995954
$147$	0	0
$148$	502.443	0.00188553
$149$	404421.	1.49234	0.746170	−	0.665755i	$-0.231890\pi$
$149$	404421.	1.49234	0.746170	+	0.665755i	$0.231890\pi$
$150$	0	0
$151$	416180.	1.48538	0.742692	−	0.669633i	$-0.233549\pi$
$151$	416180.	1.48538	0.742692	+	0.669633i	$0.233549\pi$
$152$	121762.	0.427466
$153$	0	0
$154$	98971.5	0.336286
$155$	−67577.2	−0.225929
$156$	0	0
$157$	332834.	1.07765	0.538827	−	0.842417i	$-0.318868\pi$
$157$	332834.	1.07765	0.538827	+	0.842417i	$0.318868\pi$
$158$	−122328.	−0.389837
$159$	0	0
$160$	−1705.03	−0.00526542
$161$	−114715.	−0.348785
$162$	0	0
$163$	−14764.9	−0.0435272	−0.0217636	−	0.999763i	$-0.506928\pi$
$163$	−14764.9	−0.0435272	−0.0217636	+	0.999763i	$0.506928\pi$
$164$	1812.64	0.00526263
$165$	0	0
$166$	−39110.9	−0.110161
$167$	−169960.	−0.471579	−0.235790	−	0.971804i	$-0.575768\pi$
$167$	−169960.	−0.471579	−0.235790	+	0.971804i	$0.575768\pi$
$168$	0	0
$169$	−366406.	−0.986838
$170$	−70572.1	−0.187288
$171$	0	0
$172$	1258.70	0.00324415
$173$	−725869.	−1.84392	−0.921962	−	0.387281i	$-0.873414\pi$
$173$	−725869.	−1.84392	−0.921962	+	0.387281i	$0.873414\pi$
$174$	0	0
$175$	90638.3	0.223726
$176$	−123170.	−0.299726
$177$	0	0
$178$	−241286.	−0.570797
$179$	−69311.4	−0.161686	−0.0808429	−	0.996727i	$-0.525761\pi$
$179$	−69311.4	−0.161686	−0.0808429	+	0.996727i	$0.525761\pi$
$180$	0	0
$181$	790396.	1.79328	0.896640	−	0.442761i	$-0.146001\pi$
$181$	790396.	1.79328	0.896640	+	0.442761i	$0.146001\pi$
$182$	−57179.9	−0.127957
$183$	0	0
$184$	143609.	0.312707
$185$	66678.2	0.143237
$186$	0	0
$187$	60559.9	0.126643
$188$	−219.854	−0.000453670 0
$189$	0	0
$190$	94569.6	0.190050
$191$	−418979.	−0.831015	−0.415507	−	0.909590i	$-0.636396\pi$
$191$	−418979.	−0.831015	−0.415507	+	0.909590i	$0.636396\pi$
$192$	0	0
$193$	265524.	0.513110	0.256555	−	0.966530i	$-0.417413\pi$
$193$	265524.	0.513110	0.256555	+	0.966530i	$0.417413\pi$
$194$	−118402.	−0.225868
$195$	0	0
$196$	−795.765	−0.00147960
$197$	87754.1	0.161102	0.0805512	−	0.996750i	$-0.474332\pi$
$197$	87754.1	0.161102	0.0805512	+	0.996750i	$0.474332\pi$
$198$	0	0
$199$	380988.	0.681992	0.340996	−	0.940065i	$-0.389236\pi$
$199$	380988.	0.681992	0.340996	+	0.940065i	$0.389236\pi$
$200$	−113468.	−0.200584
$201$	0	0
$202$	−1.06968e6	−1.84449
$203$	−224190.	−0.381835
$204$	0	0
$205$	240552.	0.399783
$206$	−907501.	−1.48997
$207$	0	0
$208$	71160.6	0.114046
$209$	−81152.9	−0.128510
$210$	0	0
$211$	624404.	0.965516	0.482758	−	0.875754i	$-0.339635\pi$
$211$	624404.	0.965516	0.482758	+	0.875754i	$0.339635\pi$
$212$	5439.18	0.00831177
$213$	0	0
$214$	−638717.	−0.953398
$215$	167039.	0.246446
$216$	0	0
$217$	392005.	0.565123
$218$	−9973.67	−0.0142139
$219$	0	0
$220$	569.861	0.000793802 0
$221$	−34988.0	−0.0481879
$222$	0	0
$223$	−658298.	−0.886462	−0.443231	−	0.896407i	$-0.646168\pi$
$223$	−658298.	−0.886462	−0.443231	+	0.896407i	$0.646168\pi$
$224$	9890.64	0.0131706
$225$	0	0
$226$	435835.	0.567611
$227$	−219954.	−0.283313	−0.141657	−	0.989916i	$-0.545243\pi$
$227$	−219954.	−0.283313	−0.141657	+	0.989916i	$0.545243\pi$
$228$	0	0
$229$	−313324.	−0.394825	−0.197412	−	0.980321i	$-0.563254\pi$
$229$	−313324.	−0.394825	−0.197412	+	0.980321i	$0.563254\pi$
$230$	111538.	0.139028
$231$	0	0
$232$	280657.	0.342339
$233$	1.01608e6	1.22614	0.613069	−	0.790030i	$-0.289935\pi$
$233$	1.01608e6	1.22614	0.613069	+	0.790030i	$0.289935\pi$
$234$	0	0
$235$	−29176.4	−0.0344637
$236$	4445.72	0.00519592
$237$	0	0
$238$	409378.	0.468470
$239$	−762420.	−0.863375	−0.431688	−	0.902023i	$-0.642082\pi$
$239$	−762420.	−0.863375	−0.431688	+	0.902023i	$0.642082\pi$
$240$	0	0
$241$	−22909.2	−0.0254078	−0.0127039	−	0.999919i	$-0.504044\pi$
$241$	−22909.2	−0.0254078	−0.0127039	+	0.999919i	$0.504044\pi$
$242$	82577.9	0.0906411
$243$	0	0
$244$	−3315.80	−0.00356545
$245$	−105604.	−0.112400
$246$	0	0
$247$	46885.4	0.0488985
$248$	−490741.	−0.506668
$249$	0	0
$250$	−88127.8	−0.0891791
$251$	691159.	0.692458	0.346229	−	0.938150i	$-0.387462\pi$
$251$	691159.	0.692458	0.346229	+	0.938150i	$0.387462\pi$
$252$	0	0
$253$	−95714.0	−0.0940100
$254$	590566.	0.574360
$255$	0	0
$256$	−18518.2	−0.0176604
$257$	1.34845e6	1.27351	0.636753	−	0.771068i	$-0.280277\pi$
$257$	1.34845e6	1.27351	0.636753	+	0.771068i	$0.280277\pi$
$258$	0	0
$259$	−386790.	−0.358283
$260$	−329.232	−0.000302043 0
$261$	0	0
$262$	−571057.	−0.513957
$263$	−1.34541e6	−1.19940	−0.599700	−	0.800225i	$-0.704713\pi$
$263$	−1.34541e6	−1.19940	−0.599700	+	0.800225i	$0.704713\pi$
$264$	0	0
$265$	721821.	0.631415
$266$	−548584.	−0.475378
$267$	0	0
$268$	−3108.61	−0.00264380
$269$	2.03803e6	1.71724	0.858620	−	0.512613i	$-0.171322\pi$
$269$	2.03803e6	1.71724	0.858620	+	0.512613i	$0.171322\pi$
$270$	0	0
$271$	−218053.	−0.180360	−0.0901799	−	0.995925i	$-0.528744\pi$
$271$	−218053.	−0.180360	−0.0901799	+	0.995925i	$0.528744\pi$
$272$	−509472.	−0.417540
$273$	0	0
$274$	−723698.	−0.582346
$275$	75625.0	0.0603023
$276$	0	0
$277$	−410179.	−0.321199	−0.160599	−	0.987020i	$-0.551343\pi$
$277$	−410179.	−0.321199	−0.160599	+	0.987020i	$0.551343\pi$
$278$	912324.	0.708006
$279$	0	0
$280$	658209.	0.501728
$281$	406589.	0.307178	0.153589	−	0.988135i	$-0.450917\pi$
$281$	406589.	0.307178	0.153589	+	0.988135i	$0.450917\pi$
$282$	0	0
$283$	1.26416e6	0.938291	0.469146	−	0.883121i	$-0.344562\pi$
$283$	1.26416e6	0.938291	0.469146	+	0.883121i	$0.344562\pi$
$284$	13574.8	0.00998707
$285$	0	0
$286$	−47708.7	−0.0344891
$287$	−1.39541e6	−0.999991
$288$	0	0
$289$	−1.16936e6	−0.823577
$290$	217980.	0.152203
$291$	0	0
$292$	8567.85	0.00588050
$293$	−351099.	−0.238924	−0.119462	−	0.992839i	$-0.538117\pi$
$293$	−351099.	−0.238924	−0.119462	+	0.992839i	$0.538117\pi$
$294$	0	0
$295$	589982.	0.394715
$296$	484212.	0.321223
$297$	0	0
$298$	2.28101e6	1.48794
$299$	55298.0	0.0357710
$300$	0	0
$301$	−968969.	−0.616444
$302$	2.34733e6	1.48101
$303$	0	0
$304$	682714.	0.423697
$305$	−440033.	−0.270854
$306$	0	0
$307$	−1.46228e6	−0.885493	−0.442747	−	0.896647i	$-0.645996\pi$
$307$	−1.46228e6	−0.885493	−0.442747	+	0.896647i	$0.645996\pi$
$308$	−3305.68	−0.00198556
$309$	0	0
$310$	−381148.	−0.225263
$311$	−2.20292e6	−1.29151	−0.645755	−	0.763545i	$-0.723457\pi$
$311$	−2.20292e6	−1.29151	−0.645755	+	0.763545i	$0.723457\pi$
$312$	0	0
$313$	647848.	0.373777	0.186888	−	0.982381i	$-0.440160\pi$
$313$	647848.	0.373777	0.186888	+	0.982381i	$0.440160\pi$
$314$	1.87725e6	1.07448
$315$	0	0
$316$	4085.80	0.00230175
$317$	1.62933e6	0.910670	0.455335	−	0.890320i	$-0.349519\pi$
$317$	1.62933e6	0.910670	0.455335	+	0.890320i	$0.349519\pi$
$318$	0	0
$319$	−187055.	−0.102918
$320$	−823966.	−0.449815
$321$	0	0
$322$	−647016.	−0.347756
$323$	−335675.	−0.179024
$324$	0	0
$325$	−43691.7	−0.0229451
$326$	−83276.6	−0.0433989
$327$	0	0
$328$	1.74687e6	0.896554
$329$	169248.	0.0862053
$330$	0	0
$331$	799339.	0.401016	0.200508	−	0.979692i	$-0.435741\pi$
$331$	799339.	0.401016	0.200508	+	0.979692i	$0.435741\pi$
$332$	1306.32	0.000650434 0
$333$	0	0
$334$	−958603.	−0.470189
$335$	−412537.	−0.200840
$336$	0	0
$337$	−3.76736e6	−1.80702	−0.903508	−	0.428570i	$-0.859017\pi$
$337$	−3.76736e6	−1.80702	−0.903508	+	0.428570i	$0.859017\pi$
$338$	−2.06660e6	−0.983929
$339$	0	0
$340$	2357.13	0.00110582
$341$	327074.	0.152321
$342$	0	0
$343$	−1.82478e6	−0.837481
$344$	1.21303e6	0.552681
$345$	0	0
$346$	−4.09403e6	−1.83849
$347$	−923519.	−0.411739	−0.205869	−	0.978579i	$-0.566002\pi$
$347$	−923519.	−0.411739	−0.205869	+	0.978579i	$0.566002\pi$
$348$	0	0
$349$	2.61022e6	1.14713	0.573567	−	0.819159i	$-0.305559\pi$
$349$	2.61022e6	1.14713	0.573567	+	0.819159i	$0.305559\pi$
$350$	511216.	0.223067
$351$	0	0
$352$	8252.36	0.00354994
$353$	−1.25644e6	−0.536669	−0.268335	−	0.963326i	$-0.586473\pi$
$353$	−1.25644e6	−0.536669	−0.268335	+	0.963326i	$0.586473\pi$
$354$	0	0
$355$	1.80148e6	0.758682
$356$	8059.01	0.00337021
$357$	0	0
$358$	−390928.	−0.161209
$359$	2.38200e6	0.975451	0.487726	−	0.872997i	$-0.337827\pi$
$359$	2.38200e6	0.975451	0.487726	+	0.872997i	$0.337827\pi$
$360$	0	0
$361$	−2.02628e6	−0.818336
$362$	4.45797e6	1.78799
$363$	0	0
$364$	1909.83	0.000755511 0
$365$	1.13702e6	0.446721
$366$	0	0
$367$	−3.46320e6	−1.34219	−0.671093	−	0.741373i	$-0.734175\pi$
$367$	−3.46320e6	−1.34219	−0.671093	+	0.741373i	$0.734175\pi$
$368$	805213.	0.309950
$369$	0	0
$370$	376077.	0.142815
$371$	−4.18718e6	−1.57938
$372$	0	0
$373$	811517.	0.302013	0.151006	−	0.988533i	$-0.451749\pi$
$373$	811517.	0.302013	0.151006	+	0.988533i	$0.451749\pi$
$374$	341569.	0.126270
$375$	0	0
$376$	−211877.	−0.0772884
$377$	108070.	0.0391607
$378$	0	0
$379$	−522481.	−0.186841	−0.0934206	−	0.995627i	$-0.529780\pi$
$379$	−522481.	−0.186841	−0.0934206	+	0.995627i	$0.529780\pi$
$380$	−3158.65	−0.00112213
$381$	0	0
$382$	−2.36312e6	−0.828565
$383$	2.24960e6	0.783625	0.391813	−	0.920045i	$-0.371848\pi$
$383$	2.24960e6	0.783625	0.391813	+	0.920045i	$0.371848\pi$
$384$	0	0
$385$	−438689.	−0.150836
$386$	1.49760e6	0.511598
$387$	0	0
$388$	3954.65	0.00133361
$389$	2.42055e6	0.811035	0.405518	−	0.914087i	$-0.367091\pi$
$389$	2.42055e6	0.811035	0.405518	+	0.914087i	$0.367091\pi$
$390$	0	0
$391$	−395904.	−0.130963
$392$	−766891.	−0.252068
$393$	0	0
$394$	494949.	0.160627
$395$	542217.	0.174856
$396$	0	0
$397$	1.96075e6	0.624375	0.312187	−	0.950021i	$-0.398938\pi$
$397$	1.96075e6	0.624375	0.312187	+	0.950021i	$0.398938\pi$
$398$	2.14884e6	0.679981
$399$	0	0
$400$	−636210.	−0.198816
$401$	3.86313e6	1.19972	0.599858	−	0.800107i	$-0.295224\pi$
$401$	3.86313e6	1.19972	0.599858	+	0.800107i	$0.295224\pi$
$402$	0	0
$403$	−188964.	−0.0579584
$404$	35727.8	0.0108906
$405$	0	0
$406$	−1.26447e6	−0.380710
$407$	−322722.	−0.0965702
$408$	0	0
$409$	1.75292e6	0.518149	0.259075	−	0.965857i	$-0.416582\pi$
$409$	1.75292e6	0.518149	0.259075	+	0.965857i	$0.416582\pi$
$410$	1.35676e6	0.398605
$411$	0	0
$412$	30310.8	0.00879740
$413$	−3.42240e6	−0.987314
$414$	0	0
$415$	173359.	0.0494112
$416$	−4767.73	−0.00135076
$417$	0	0
$418$	−457717.	−0.128132
$419$	4.28414e6	1.19214	0.596071	−	0.802932i	$-0.296728\pi$
$419$	4.28414e6	1.19214	0.596071	+	0.802932i	$0.296728\pi$
$420$	0	0
$421$	5.74050e6	1.57850	0.789250	−	0.614072i	$-0.210470\pi$
$421$	5.74050e6	1.57850	0.789250	+	0.614072i	$0.210470\pi$
$422$	3.52175e6	0.962670
$423$	0	0
$424$	5.24182e6	1.41601
$425$	312810.	0.0840055
$426$	0	0
$427$	2.55257e6	0.677497
$428$	21333.3	0.00562924
$429$	0	0
$430$	942131.	0.245720
$431$	−5.29228e6	−1.37230	−0.686150	−	0.727460i	$-0.740701\pi$
$431$	−5.29228e6	−1.37230	−0.686150	+	0.727460i	$0.740701\pi$
$432$	0	0
$433$	−2.37840e6	−0.609628	−0.304814	−	0.952412i	$-0.598594\pi$
$433$	−2.37840e6	−0.609628	−0.304814	+	0.952412i	$0.598594\pi$
$434$	2.21098e6	0.563457
$435$	0	0
$436$	333.123	8.39245e−5 0
$437$	530528.	0.132894
$438$	0	0
$439$	6.44393e6	1.59584	0.797921	−	0.602762i	$-0.205933\pi$
$439$	6.44393e6	1.59584	0.797921	+	0.602762i	$0.205933\pi$
$440$	549183.	0.135234
$441$	0	0
$442$	−197339.	−0.0480459
$443$	−2501.14	−0.000605520 0	−0.000302760	−	1.00000i	$-0.500096\pi$
$443$	−2501.14	−0.000605520 0	−0.000302760		1.00000i	$0.500096\pi$
$444$	0	0
$445$	1.06949e6	0.256023
$446$	−3.71292e6	−0.883849
$447$	0	0
$448$	4.77970e6	1.12514
$449$	5.04731e6	1.18153	0.590764	−	0.806844i	$-0.298826\pi$
$449$	5.04731e6	1.18153	0.590764	+	0.806844i	$0.298826\pi$
$450$	0	0
$451$	−1.16427e6	−0.269534
$452$	−14557.0	−0.00335140
$453$	0	0
$454$	−1.24058e6	−0.282478
$455$	253449.	0.0573934
$456$	0	0
$457$	4.62692e6	1.03634	0.518169	−	0.855278i	$-0.326614\pi$
$457$	4.62692e6	1.03634	0.518169	+	0.855278i	$0.326614\pi$
$458$	−1.76720e6	−0.393661
$459$	0	0
$460$	−3725.40	−0.000820879 0
$461$	−1.43209e6	−0.313846	−0.156923	−	0.987611i	$-0.550158\pi$
$461$	−1.43209e6	−0.313846	−0.156923	+	0.987611i	$0.550158\pi$
$462$	0	0
$463$	1.58115e6	0.342785	0.171393	−	0.985203i	$-0.445173\pi$
$463$	1.58115e6	0.342785	0.171393	+	0.985203i	$0.445173\pi$
$464$	1.57364e6	0.339321
$465$	0	0
$466$	5.73089e6	1.22252
$467$	−8724.92	−0.00185127	−0.000925634	−	1.00000i	$-0.500295\pi$
$467$	−8724.92	−0.00185127	−0.000925634		1.00000i	$0.500295\pi$
$468$	0	0
$469$	2.39306e6	0.502368
$470$	−164560.	−0.0343621
$471$	0	0
$472$	4.28441e6	0.885189
$473$	−808470.	−0.166154
$474$	0	0
$475$	−419178.	−0.0852441
$476$	−13673.4	−0.00276603
$477$	0	0
$478$	−4.30018e6	−0.860830
$479$	−8.15482e6	−1.62396	−0.811981	−	0.583684i	$-0.801611\pi$
$479$	−8.15482e6	−1.62396	−0.811981	+	0.583684i	$0.801611\pi$
$480$	0	0
$481$	186450.	0.0367452
$482$	−129212.	−0.0253329
$483$	0	0
$484$	−2758.13	−0.000535181 0
$485$	524814.	0.101310
$486$	0	0
$487$	9.00123e6	1.71981	0.859903	−	0.510457i	$-0.170524\pi$
$487$	9.00123e6	1.71981	0.859903	+	0.510457i	$0.170524\pi$
$488$	−3.19549e6	−0.607418
$489$	0	0
$490$	−595627.	−0.112069
$491$	−6.03644e6	−1.13000	−0.564999	−	0.825092i	$-0.691123\pi$
$491$	−6.03644e6	−1.13000	−0.564999	+	0.825092i	$0.691123\pi$
$492$	0	0
$493$	−773721.	−0.143373
$494$	264442.	0.0487543
$495$	0	0
$496$	−2.75157e6	−0.502200
$497$	−1.04501e7	−1.89772
$498$	0	0
$499$	233452.	0.0419707	0.0209853	−	0.999780i	$-0.493320\pi$
$499$	233452.	0.0419707	0.0209853	+	0.999780i	$0.493320\pi$
$500$	2943.50	0.000526548 0
$501$	0	0
$502$	3.89826e6	0.690417
$503$	7.79728e6	1.37412	0.687058	−	0.726603i	$-0.258902\pi$
$503$	7.79728e6	1.37412	0.687058	+	0.726603i	$0.258902\pi$
$504$	0	0
$505$	4.74135e6	0.827321
$506$	−539844.	−0.0937329
$507$	0	0
$508$	−19725.1	−0.00339125
$509$	−5.32415e6	−0.910869	−0.455434	−	0.890269i	$-0.650516\pi$
$509$	−5.32415e6	−0.910869	−0.455434	+	0.890269i	$0.650516\pi$
$510$	0	0
$511$	−6.59569e6	−1.11740
$512$	−5.98317e6	−1.00869
$513$	0	0
$514$	7.60548e6	1.26975
$515$	4.02248e6	0.668307
$516$	0	0
$517$	141214.	0.0232354
$518$	−2.18157e6	−0.357227
$519$	0	0
$520$	−317286.	−0.0514568
$521$	8.42076e6	1.35912	0.679559	−	0.733621i	$-0.262171\pi$
$521$	8.42076e6	1.35912	0.679559	+	0.733621i	$0.262171\pi$
$522$	0	0
$523$	−6.81759e6	−1.08988	−0.544938	−	0.838476i	$-0.683447\pi$
$523$	−6.81759e6	−1.08988	−0.544938	+	0.838476i	$0.683447\pi$
$524$	19073.5	0.00303460
$525$	0	0
$526$	−7.58833e6	−1.19586
$527$	1.35288e6	0.212194
$528$	0	0
$529$	−5.81062e6	−0.902783
$530$	4.07120e6	0.629554
$531$	0	0
$532$	18322.9	0.00280682
$533$	672649.	0.102558
$534$	0	0
$535$	2.83110e6	0.427633
$536$	−2.99581e6	−0.450404
$537$	0	0
$538$	1.14949e7	1.71218
$539$	511125.	0.0757801
$540$	0	0
$541$	−1.00313e7	−1.47354	−0.736770	−	0.676143i	$-0.763650\pi$
$541$	−1.00313e7	−1.47354	−0.736770	+	0.676143i	$0.763650\pi$
$542$	−1.22986e6	−0.179828
$543$	0	0
$544$	34134.4	0.00494533
$545$	44208.1	0.00637545
$546$	0	0
$547$	341056.	0.0487368	0.0243684	−	0.999703i	$-0.492243\pi$
$547$	341056.	0.0487368	0.0243684	+	0.999703i	$0.492243\pi$
$548$	24171.7	0.00343840
$549$	0	0
$550$	426539.	0.0601245
$551$	1.03682e6	0.145487
$552$	0	0
$553$	−3.14532e6	−0.437373
$554$	−2.31348e6	−0.320252
$555$	0	0
$556$	−30471.9	−0.00418035
$557$	−8.70989e6	−1.18953	−0.594764	−	0.803901i	$-0.702754\pi$
$557$	−8.70989e6	−1.18953	−0.594764	+	0.803901i	$0.702754\pi$
$558$	0	0
$559$	467087.	0.0632219
$560$	3.69056e6	0.497304
$561$	0	0
$562$	2.29324e6	0.306272
$563$	1.05570e7	1.40369	0.701845	−	0.712330i	$-0.252360\pi$
$563$	1.05570e7	1.40369	0.701845	+	0.712330i	$0.252360\pi$
$564$	0	0
$565$	−1.93183e6	−0.254594
$566$	7.13011e6	0.935525
$567$	0	0
$568$	1.30823e7	1.70142
$569$	2.88092e6	0.373035	0.186518	−	0.982452i	$-0.440280\pi$
$569$	2.88092e6	0.373035	0.186518	+	0.982452i	$0.440280\pi$
$570$	0	0
$571$	−8.00484e6	−1.02745	−0.513727	−	0.857954i	$-0.671735\pi$
$571$	−8.00484e6	−1.02745	−0.513727	+	0.857954i	$0.671735\pi$
$572$	1593.48	0.000203637 0
$573$	0	0
$574$	−7.87035e6	−0.997043
$575$	−494391.	−0.0623592
$576$	0	0
$577$	−1.42967e7	−1.78770	−0.893852	−	0.448361i	$-0.852008\pi$
$577$	−1.42967e7	−1.78770	−0.893852	+	0.448361i	$0.852008\pi$
$578$	−6.59541e6	−0.821149
$579$	0	0
$580$	−7280.61	−0.000898665 0
$581$	−1.00563e6	−0.123594
$582$	0	0
$583$	−3.49362e6	−0.425700
$584$	8.25697e6	1.00182
$585$	0	0
$586$	−1.98026e6	−0.238220
$587$	−9.83619e6	−1.17823	−0.589117	−	0.808047i	$-0.700524\pi$
$587$	−9.83619e6	−1.17823	−0.589117	+	0.808047i	$0.700524\pi$
$588$	0	0
$589$	−1.81292e6	−0.215323
$590$	3.32760e6	0.393552
$591$	0	0
$592$	2.71497e6	0.318390
$593$	9.84429e6	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$593$	9.84429e6	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$594$	0	0
$595$	−1.81456e6	−0.210126
$596$	−76186.3	−0.00878540
$597$	0	0
$598$	311890.	0.0356656
$599$	3.94699e6	0.449468	0.224734	−	0.974420i	$-0.427849\pi$
$599$	3.94699e6	0.449468	0.224734	+	0.974420i	$0.427849\pi$
$600$	0	0
$601$	−7.63150e6	−0.861834	−0.430917	−	0.902392i	$-0.641810\pi$
$601$	−7.63150e6	−0.861834	−0.430917	+	0.902392i	$0.641810\pi$
$602$	−5.46516e6	−0.614627
$603$	0	0
$604$	−78401.5	−0.00874444
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−1.12292e7	−1.23703	−0.618513	−	0.785775i	$-0.712264\pi$
$607$	−1.12292e7	−1.23703	−0.618513	+	0.785775i	$0.712264\pi$
$608$	−45741.6	−0.00501825
$609$	0	0
$610$	−2.48186e6	−0.270056
$611$	−81585.1	−0.00884113
$612$	0	0
$613$	2.62390e6	0.282031	0.141016	−	0.990007i	$-0.454963\pi$
$613$	2.62390e6	0.282031	0.141016	+	0.990007i	$0.454963\pi$
$614$	−8.24753e6	−0.882883
$615$	0	0
$616$	−3.18573e6	−0.338265
$617$	3.59228e6	0.379889	0.189945	−	0.981795i	$-0.439169\pi$
$617$	3.59228e6	0.379889	0.189945	+	0.981795i	$0.439169\pi$
$618$	0	0
$619$	−1.19777e7	−1.25646	−0.628228	−	0.778029i	$-0.716219\pi$
$619$	−1.19777e7	−1.25646	−0.628228	+	0.778029i	$0.716219\pi$
$620$	12730.4	0.00133004
$621$	0	0
$622$	−1.24249e7	−1.28770
$623$	−6.20398e6	−0.640398
$624$	0	0
$625$	390625.	0.0400000
$626$	3.65398e6	0.372675
$627$	0	0
$628$	−62700.6	−0.00634413
$629$	−1.33488e6	−0.134529
$630$	0	0
$631$	2.50844e6	0.250801	0.125401	−	0.992106i	$-0.459978\pi$
$631$	2.50844e6	0.250801	0.125401	+	0.992106i	$0.459978\pi$
$632$	3.93754e6	0.392132
$633$	0	0
$634$	9.18972e6	0.907986
$635$	−2.61767e6	−0.257621
$636$	0	0
$637$	−295298.	−0.0288345
$638$	−1.05503e6	−0.102615
$639$	0	0
$640$	−4.59275e6	−0.443224
$641$	−1.12431e7	−1.08079	−0.540395	−	0.841411i	$-0.681725\pi$
$641$	−1.12431e7	−1.08079	−0.540395	+	0.841411i	$0.681725\pi$
$642$	0	0
$643$	5.61271e6	0.535360	0.267680	−	0.963508i	$-0.413743\pi$
$643$	5.61271e6	0.535360	0.267680	+	0.963508i	$0.413743\pi$
$644$	21610.5	0.00205329
$645$	0	0
$646$	−1.89326e6	−0.178497
$647$	−1.10739e7	−1.04002	−0.520008	−	0.854162i	$-0.674071\pi$
$647$	−1.10739e7	−1.04002	−0.520008	+	0.854162i	$0.674071\pi$
$648$	0	0
$649$	−2.85551e6	−0.266117
$650$	−246429.	−0.0228775
$651$	0	0
$652$	2781.46	0.000256244 0
$653$	7.83486e6	0.719032	0.359516	−	0.933139i	$-0.382942\pi$
$653$	7.83486e6	0.719032	0.359516	+	0.933139i	$0.382942\pi$
$654$	0	0
$655$	2.53120e6	0.230528
$656$	9.79467e6	0.888649
$657$	0	0
$658$	954589.	0.0859511
$659$	−1.02774e7	−0.921868	−0.460934	−	0.887434i	$-0.652486\pi$
$659$	−1.02774e7	−0.921868	−0.460934	+	0.887434i	$0.652486\pi$
$660$	0	0
$661$	−1.42467e7	−1.26827	−0.634134	−	0.773223i	$-0.718643\pi$
$661$	−1.42467e7	−1.26827	−0.634134	+	0.773223i	$0.718643\pi$
$662$	4.50842e6	0.399833
$663$	0	0
$664$	1.25892e6	0.110810
$665$	2.43159e6	0.213224
$666$	0	0
$667$	1.22285e6	0.106429
$668$	32017.6	0.00277618
$669$	0	0
$670$	−2.32678e6	−0.200248
$671$	2.12976e6	0.182610
$672$	0	0
$673$	−3.85183e6	−0.327815	−0.163908	−	0.986476i	$-0.552410\pi$
$673$	−3.85183e6	−0.327815	−0.163908	+	0.986476i	$0.552410\pi$
$674$	−2.12486e7	−1.80169
$675$	0	0
$676$	69024.9	0.00580951
$677$	−1.39430e7	−1.16919	−0.584594	−	0.811326i	$-0.698746\pi$
$677$	−1.39430e7	−1.16919	−0.584594	+	0.811326i	$0.698746\pi$
$678$	0	0
$679$	−3.04437e6	−0.253409
$680$	2.27160e6	0.188391
$681$	0	0
$682$	1.84475e6	0.151872
$683$	1.24329e7	1.01981	0.509905	−	0.860231i	$-0.329681\pi$
$683$	1.24329e7	1.01981	0.509905	+	0.860231i	$0.329681\pi$
$684$	0	0
$685$	3.20778e6	0.261203
$686$	−1.02921e7	−0.835012
$687$	0	0
$688$	6.80141e6	0.547807
$689$	2.01841e6	0.161980
$690$	0	0
$691$	1.46905e7	1.17042	0.585208	−	0.810883i	$-0.301013\pi$
$691$	1.46905e7	1.17042	0.585208	+	0.810883i	$0.301013\pi$
$692$	136742.	0.0108552
$693$	0	0
$694$	−5.20881e6	−0.410525
$695$	−4.04386e6	−0.317566
$696$	0	0
$697$	−4.81581e6	−0.375480
$698$	1.47221e7	1.14375
$699$	0	0
$700$	−17074.8	−0.00131707
$701$	3.55096e6	0.272930	0.136465	−	0.990645i	$-0.456426\pi$
$701$	3.55096e6	0.272930	0.136465	+	0.990645i	$0.456426\pi$
$702$	0	0
$703$	1.78880e6	0.136513
$704$	3.98799e6	0.303265
$705$	0	0
$706$	−7.08657e6	−0.535087
$707$	−2.75039e7	−2.06941
$708$	0	0
$709$	−8.56408e6	−0.639831	−0.319915	−	0.947446i	$-0.603654\pi$
$709$	−8.56408e6	−0.639831	−0.319915	+	0.947446i	$0.603654\pi$
$710$	1.01607e7	0.756445
$711$	0	0
$712$	7.76659e6	0.574157
$713$	−2.13821e6	−0.157517
$714$	0	0
$715$	211468.	0.0154696
$716$	13057.1	0.000951843 0
$717$	0	0
$718$	1.34349e7	0.972576
$719$	−1.07318e7	−0.774195	−0.387097	−	0.922039i	$-0.626522\pi$
$719$	−1.07318e7	−0.774195	−0.387097	+	0.922039i	$0.626522\pi$
$720$	0	0
$721$	−2.33338e7	−1.67166
$722$	−1.14286e7	−0.815924
$723$	0	0
$724$	−148898.	−0.0105570
$725$	−966195.	−0.0682684
$726$	0	0
$727$	1.49237e7	1.04722	0.523611	−	0.851957i	$-0.324584\pi$
$727$	1.49237e7	1.04722	0.523611	+	0.851957i	$0.324584\pi$
$728$	1.84053e6	0.128711
$729$	0	0
$730$	6.41300e6	0.445404
$731$	−3.34409e6	−0.231465
$732$	0	0
$733$	1.21777e7	0.837155	0.418577	−	0.908181i	$-0.362529\pi$
$733$	1.21777e7	0.837155	0.418577	+	0.908181i	$0.362529\pi$
$734$	−1.95331e7	−1.33823
$735$	0	0
$736$	−53948.9	−0.00367103
$737$	1.99668e6	0.135406
$738$	0	0
$739$	4.10708e6	0.276644	0.138322	−	0.990387i	$-0.455829\pi$
$739$	4.10708e6	0.276644	0.138322	+	0.990387i	$0.455829\pi$
$740$	−12561.1	−0.000843233 0
$741$	0	0
$742$	−2.36164e7	−1.57472
$743$	−2.73077e7	−1.81474	−0.907368	−	0.420337i	$-0.861912\pi$
$743$	−2.73077e7	−1.81474	−0.907368	+	0.420337i	$0.861912\pi$
$744$	0	0
$745$	−1.01105e7	−0.667395
$746$	4.57710e6	0.301123
$747$	0	0
$748$	−11408.5	−0.000745546 0
$749$	−1.64228e7	−1.06965
$750$	0	0
$751$	−1.54004e7	−0.996398	−0.498199	−	0.867063i	$-0.666005\pi$
$751$	−1.54004e7	−0.996398	−0.498199	+	0.867063i	$0.666005\pi$
$752$	−1.18799e6	−0.0766069
$753$	0	0
$754$	609532.	0.0390452
$755$	−1.04045e7	−0.664284
$756$	0	0
$757$	8.23327e6	0.522195	0.261097	−	0.965312i	$-0.415916\pi$
$757$	8.23327e6	0.522195	0.261097	+	0.965312i	$0.415916\pi$
$758$	−2.94689e6	−0.186290
$759$	0	0
$760$	−3.04404e6	−0.191169
$761$	−1.69020e7	−1.05798	−0.528989	−	0.848629i	$-0.677429\pi$
$761$	−1.69020e7	−1.05798	−0.528989	+	0.848629i	$0.677429\pi$
$762$	0	0
$763$	−256445.	−0.0159471
$764$	78928.8	0.00489217
$765$	0	0
$766$	1.26881e7	0.781315
$767$	1.64975e6	0.101258
$768$	0	0
$769$	−1.72204e7	−1.05009	−0.525045	−	0.851075i	$-0.675951\pi$
$769$	−1.72204e7	−1.05009	−0.525045	+	0.851075i	$0.675951\pi$
$770$	−2.47429e6	−0.150391
$771$	0	0
$772$	−50020.4	−0.00302067
$773$	−1.86902e7	−1.12503	−0.562517	−	0.826786i	$-0.690167\pi$
$773$	−1.86902e7	−1.12503	−0.562517	+	0.826786i	$0.690167\pi$
$774$	0	0
$775$	1.68943e6	0.101038
$776$	3.81116e6	0.227197
$777$	0	0
$778$	1.36523e7	0.808644
$779$	6.45339e6	0.381017
$780$	0	0
$781$	−8.71919e6	−0.511503
$782$	−2.23297e6	−0.130577
$783$	0	0
$784$	−4.29994e6	−0.249846
$785$	−8.32086e6	−0.481941
$786$	0	0
$787$	−8.17563e6	−0.470527	−0.235264	−	0.971932i	$-0.575595\pi$
$787$	−8.17563e6	−0.470527	−0.235264	+	0.971932i	$0.575595\pi$
$788$	−16531.4	−0.000948408 0
$789$	0	0
$790$	3.05820e6	0.174341
$791$	1.12063e7	0.636824
$792$	0	0
$793$	−1.23045e6	−0.0694834
$794$	1.10590e7	0.622534
$795$	0	0
$796$	−71772.0	−0.00401488
$797$	3.39844e7	1.89511	0.947554	−	0.319595i	$-0.103547\pi$
$797$	3.39844e7	1.89511	0.947554	+	0.319595i	$0.103547\pi$
$798$	0	0
$799$	584106.	0.0323687
$800$	42625.8	0.00235477
$801$	0	0
$802$	2.17887e7	1.19618
$803$	−5.50318e6	−0.301179
$804$	0	0
$805$	2.86789e6	0.155981
$806$	−1.06579e6	−0.0577876
$807$	0	0
$808$	3.44314e7	1.85535
$809$	−2.50619e7	−1.34630	−0.673151	−	0.739505i	$-0.735060\pi$
$809$	−2.50619e7	−1.34630	−0.673151	+	0.739505i	$0.735060\pi$
$810$	0	0
$811$	−1.73791e7	−0.927844	−0.463922	−	0.885876i	$-0.653558\pi$
$811$	−1.73791e7	−0.927844	−0.463922	+	0.885876i	$0.653558\pi$
$812$	42233.7	0.00224786
$813$	0	0
$814$	−1.82021e6	−0.0962855
$815$	369122.	0.0194660
$816$	0	0
$817$	4.48123e6	0.234878
$818$	9.88681e6	0.516622
$819$	0	0
$820$	−45316.1	−0.00235352
$821$	1.21448e7	0.628827	0.314413	−	0.949286i	$-0.398192\pi$
$821$	1.21448e7	0.628827	0.314413	+	0.949286i	$0.398192\pi$
$822$	0	0
$823$	9.33503e6	0.480415	0.240207	−	0.970722i	$-0.422785\pi$
$823$	9.33503e6	0.480415	0.240207	+	0.970722i	$0.422785\pi$
$824$	2.92110e7	1.49875
$825$	0	0
$826$	−1.93029e7	−0.984404
$827$	1.15378e7	0.586622	0.293311	−	0.956017i	$-0.405243\pi$
$827$	1.15378e7	0.586622	0.293311	+	0.956017i	$0.405243\pi$
$828$	0	0
$829$	−1.25927e7	−0.636404	−0.318202	−	0.948023i	$-0.603079\pi$
$829$	−1.25927e7	−0.636404	−0.318202	+	0.948023i	$0.603079\pi$
$830$	977773.	0.0492655
$831$	0	0
$832$	−2.30403e6	−0.115393
$833$	2.11418e6	0.105567
$834$	0	0
$835$	4.24899e6	0.210897
$836$	15287.9	0.000756539 0
$837$	0	0
$838$	2.41633e7	1.18863
$839$	−3.69853e7	−1.81395	−0.906973	−	0.421189i	$-0.861613\pi$
$839$	−3.69853e7	−1.81395	−0.906973	+	0.421189i	$0.861613\pi$
$840$	0	0
$841$	−1.81213e7	−0.883486
$842$	3.23775e7	1.57385
$843$	0	0
$844$	−117627.	−0.00568398
$845$	9.16015e6	0.441327
$846$	0	0
$847$	2.12326e6	0.101694
$848$	2.93907e7	1.40353
$849$	0	0
$850$	1.76430e6	0.0837579
$851$	2.10976e6	0.0998642
$852$	0	0
$853$	3.49513e7	1.64471	0.822357	−	0.568972i	$-0.192659\pi$
$853$	3.49513e7	1.64471	0.822357	+	0.568972i	$0.192659\pi$
$854$	1.43969e7	0.675500
$855$	0	0
$856$	2.05593e7	0.959010
$857$	2.24410e7	1.04373	0.521867	−	0.853027i	$-0.325236\pi$
$857$	2.24410e7	1.04373	0.521867	+	0.853027i	$0.325236\pi$
$858$	0	0
$859$	3.30887e7	1.53002	0.765010	−	0.644019i	$-0.222734\pi$
$859$	3.30887e7	1.53002	0.765010	+	0.644019i	$0.222734\pi$
$860$	−31467.5	−0.00145083
$861$	0	0
$862$	−2.98494e7	−1.36826
$863$	−1.58950e6	−0.0726495	−0.0363248	−	0.999340i	$-0.511565\pi$
$863$	−1.58950e6	−0.0726495	−0.0363248	+	0.999340i	$0.511565\pi$
$864$	0	0
$865$	1.81467e7	0.824628
$866$	−1.34146e7	−0.607830
$867$	0	0
$868$	−73847.4	−0.00332687
$869$	−2.62433e6	−0.117888
$870$	0	0
$871$	−1.15356e6	−0.0515224
$872$	321036.	0.0142976
$873$	0	0
$874$	2.99228e6	0.132502
$875$	−2.26596e6	−0.100053
$876$	0	0
$877$	−1.78401e7	−0.783246	−0.391623	−	0.920126i	$-0.628086\pi$
$877$	−1.78401e7	−0.783246	−0.391623	+	0.920126i	$0.628086\pi$
$878$	3.63449e7	1.59114
$879$	0	0
$880$	3.07926e6	0.134041
$881$	1.53213e7	0.665054	0.332527	−	0.943094i	$-0.392099\pi$
$881$	1.53213e7	0.665054	0.332527	+	0.943094i	$0.392099\pi$
$882$	0	0
$883$	1.92226e7	0.829680	0.414840	−	0.909894i	$-0.363838\pi$
$883$	1.92226e7	0.829680	0.414840	+	0.909894i	$0.363838\pi$
$884$	6591.17	0.000283682 0
$885$	0	0
$886$	−14106.9	−0.000603735 0
$887$	1.78729e7	0.762756	0.381378	−	0.924419i	$-0.375450\pi$
$887$	1.78729e7	0.762756	0.381378	+	0.924419i	$0.375450\pi$
$888$	0	0
$889$	1.51847e7	0.644396
$890$	6.03214e6	0.255268
$891$	0	0
$892$	124013.	0.00521860
$893$	−782727.	−0.0328459
$894$	0	0
$895$	1.73278e6	0.0723081
$896$	2.66419e7	1.10865
$897$	0	0
$898$	2.84677e7	1.17804
$899$	−4.17874e6	−0.172443
$900$	0	0
$901$	−1.44507e7	−0.593032
$902$	−6.56670e6	−0.268739
$903$	0	0
$904$	−1.40288e7	−0.570953
$905$	−1.97599e7	−0.801979
$906$	0	0
$907$	2.36602e7	0.954993	0.477497	−	0.878634i	$-0.341544\pi$
$907$	2.36602e7	0.954993	0.477497	+	0.878634i	$0.341544\pi$
$908$	41435.7	0.00166786
$909$	0	0
$910$	1.42950e6	0.0572243
$911$	−2.62399e7	−1.04753	−0.523766	−	0.851862i	$-0.675473\pi$
$911$	−2.62399e7	−1.04753	−0.523766	+	0.851862i	$0.675473\pi$
$912$	0	0
$913$	−839056.	−0.0333130
$914$	2.60966e7	1.03328
$915$	0	0
$916$	59025.0	0.00232433
$917$	−1.46831e7	−0.576627
$918$	0	0
$919$	1.05336e7	0.411422	0.205711	−	0.978613i	$-0.434049\pi$
$919$	1.05336e7	0.411422	0.205711	+	0.978613i	$0.434049\pi$
$920$	−3.59023e6	−0.139847
$921$	0	0
$922$	−8.07723e6	−0.312921
$923$	5.03744e6	0.194628
$924$	0	0
$925$	−1.66695e6	−0.0640574
$926$	8.91800e6	0.341775
$927$	0	0
$928$	−105433.	−0.00401890
$929$	−4.93218e7	−1.87499	−0.937497	−	0.347993i	$-0.886863\pi$
$929$	−4.93218e7	−1.87499	−0.937497	+	0.347993i	$0.886863\pi$
$930$	0	0
$931$	−2.83309e6	−0.107124
$932$	−191413.	−0.00721826
$933$	0	0
$934$	−49210.1	−0.00184581
$935$	−1.51400e6	−0.0566365
$936$	0	0
$937$	3.96100e6	0.147386	0.0736929	−	0.997281i	$-0.476522\pi$
$937$	3.96100e6	0.147386	0.0736929	+	0.997281i	$0.476522\pi$
$938$	1.34973e7	0.500887
$939$	0	0
$940$	5496.36	0.000202888 0
$941$	−3.82868e7	−1.40953	−0.704766	−	0.709440i	$-0.748948\pi$
$941$	−3.82868e7	−1.40953	−0.704766	+	0.709440i	$0.748948\pi$
$942$	0	0
$943$	7.61131e6	0.278728
$944$	2.40226e7	0.877383
$945$	0	0
$946$	−4.55991e6	−0.165664
$947$	1.34832e7	0.488562	0.244281	−	0.969705i	$-0.421448\pi$
$947$	1.34832e7	0.488562	0.244281	+	0.969705i	$0.421448\pi$
$948$	0	0
$949$	3.17942e6	0.114599
$950$	−2.36424e6	−0.0849928
$951$	0	0
$952$	−1.31772e7	−0.471228
$953$	9.20696e6	0.328385	0.164193	−	0.986428i	$-0.447498\pi$
$953$	9.20696e6	0.328385	0.164193	+	0.986428i	$0.447498\pi$
$954$	0	0
$955$	1.04745e7	0.371641
$956$	143627.	0.00508268
$957$	0	0
$958$	−4.59947e7	−1.61917
$959$	−1.86079e7	−0.653356
$960$	0	0
$961$	−2.13225e7	−0.744782
$962$	1.05161e6	0.0366368
$963$	0	0
$964$	4315.71	0.000149575 0
$965$	−6.63810e6	−0.229470
$966$	0	0
$967$	2.95285e6	0.101549	0.0507745	−	0.998710i	$-0.483831\pi$
$967$	2.95285e6	0.101549	0.0507745	+	0.998710i	$0.483831\pi$
$968$	−2.65805e6	−0.0911747
$969$	0	0
$970$	2.96004e6	0.101011
$971$	3.09624e7	1.05387	0.526935	−	0.849906i	$-0.323341\pi$
$971$	3.09624e7	1.05387	0.526935	+	0.849906i	$0.323341\pi$
$972$	0	0
$973$	2.34578e7	0.794339
$974$	5.07686e7	1.71474
$975$	0	0
$976$	−1.79170e7	−0.602062
$977$	2.85400e7	0.956573	0.478287	−	0.878204i	$-0.341258\pi$
$977$	2.85400e7	0.956573	0.478287	+	0.878204i	$0.341258\pi$
$978$	0	0
$979$	−5.17635e6	−0.172610
$980$	19894.1	0.000661698 0
$981$	0	0
$982$	−3.40466e7	−1.12667
$983$	4.46869e7	1.47501	0.737507	−	0.675340i	$-0.236003\pi$
$983$	4.46869e7	1.47501	0.737507	+	0.675340i	$0.236003\pi$
$984$	0	0
$985$	−2.19385e6	−0.0720472
$986$	−4.36393e6	−0.142950
$987$	0	0
$988$	−8832.44	−0.000287865 0
$989$	5.28529e6	0.171822
$990$	0	0
$991$	5.85115e7	1.89259	0.946296	−	0.323302i	$-0.104793\pi$
$991$	5.85115e7	1.89259	0.946296	+	0.323302i	$0.104793\pi$
$992$	184354.	0.00594804
$993$	0	0
$994$	−5.89407e7	−1.89212
$995$	−9.52471e6	−0.304996
$996$	0	0
$997$	−4.32684e7	−1.37858	−0.689292	−	0.724483i	$-0.742078\pi$
$997$	−4.32684e7	−1.37858	−0.689292	+	0.724483i	$0.742078\pi$
$998$	1.31671e6	0.0418469
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.c.1.3		3
3.2	odd	2		165.6.a.d.1.1	✓	3
15.14	odd	2		825.6.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.d.1.1	✓	3	3.2	odd	2
495.6.a.c.1.3		3	1.1	even	1	trivial
825.6.a.h.1.3		3	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+5.64018 q^{2} -0.188384 q^{4} -25.0000 q^{5} +145.021 q^{7} -181.548 q^{8} +O(q^{10})\)	\(q+5.64018 q^{2} -0.188384 q^{4} -25.0000 q^{5} +145.021 q^{7} -181.548 q^{8} -141.004 q^{10} +121.000 q^{11} -69.9067 q^{13} +817.946 q^{14} -1017.94 q^{16} +500.495 q^{17} -670.685 q^{19} +4.70959 q^{20} +682.462 q^{22} -791.025 q^{23} +625.000 q^{25} -394.287 q^{26} -27.3196 q^{28} -1545.91 q^{29} +2703.09 q^{31} +68.2013 q^{32} +2822.88 q^{34} -3625.53 q^{35} -2667.13 q^{37} -3782.78 q^{38} +4538.71 q^{40} -9622.09 q^{41} -6681.57 q^{43} -22.7944 q^{44} -4461.52 q^{46} +1167.06 q^{47} +4224.17 q^{49} +3525.11 q^{50} +13.1693 q^{52} -28872.9 q^{53} -3025.00 q^{55} -26328.4 q^{56} -8719.22 q^{58} -23599.3 q^{59} +17601.3 q^{61} +15245.9 q^{62} +32958.6 q^{64} +1747.67 q^{65} +16501.5 q^{67} -94.2851 q^{68} -20448.6 q^{70} -72059.4 q^{71} -45480.8 q^{73} -15043.1 q^{74} +126.346 q^{76} +17547.6 q^{77} -21688.7 q^{79} +25448.4 q^{80} -54270.3 q^{82} -6934.34 q^{83} -12512.4 q^{85} -37685.2 q^{86} -21967.3 q^{88} -42779.8 q^{89} -10138.0 q^{91} +149.016 q^{92} +6582.41 q^{94} +16767.1 q^{95} -20992.6 q^{97} +23825.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8	\( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8 + 50 * q^10 + 363 * q^11 - 546 * q^13 + 8 * q^14 - 1360 * q^16 + 314 * q^17 + 1808 * q^19 + 500 * q^20 - 242 * q^22 - 4288 * q^23 + 1875 * q^25 - 812 * q^26 + 5888 * q^28 - 5582 * q^29 + 6328 * q^31 + 736 * q^32 + 11596 * q^34 - 3800 * q^35 + 16866 * q^37 - 9584 * q^38 - 600 * q^40 - 23282 * q^41 + 20572 * q^43 - 2420 * q^44 + 16592 * q^46 - 3432 * q^47 + 11531 * q^49 - 1250 * q^50 + 21816 * q^52 - 16138 * q^53 - 9075 * q^55 - 15648 * q^56 + 17460 * q^58 - 21972 * q^59 + 8322 * q^61 - 5056 * q^62 + 22208 * q^64 + 13650 * q^65 - 84332 * q^67 - 59832 * q^68 - 200 * q^70 - 50528 * q^71 - 53838 * q^73 - 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 6364 * q^79 + 34000 * q^80 - 68020 * q^82 - 96272 * q^83 - 7850 * q^85 - 143152 * q^86 + 2904 * q^88 + 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 49088 * q^94 - 45200 * q^95 - 103242 * q^97 - 9554 * q^98

Introduction

L-functions

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Database

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