LMFDB - Embedded newform 495.6.a.k.1.4

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 120x^{4} + 70x^{3} + 2825x^{2} - 2101x - 2690 $ x^6 - x^5 - 120x^4 + 70x^3 + 2825x^2 - 2101x - 2690
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.47168$ 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)

$f(q)$	$=$	$q+0.471685 q^{2} -31.7775 q^{4} +25.0000 q^{5} -149.335 q^{7} -30.0829 q^{8} +O(q^{10})$	\(q+0.471685 q^{2} -31.7775 q^{4} +25.0000 q^{5} -149.335 q^{7} -30.0829 q^{8} +11.7921 q^{10} +121.000 q^{11} +286.689 q^{13} -70.4389 q^{14} +1002.69 q^{16} -245.955 q^{17} +1740.24 q^{19} -794.438 q^{20} +57.0739 q^{22} -2642.52 q^{23} +625.000 q^{25} +135.227 q^{26} +4745.48 q^{28} +5497.36 q^{29} +5314.70 q^{31} +1435.61 q^{32} -116.013 q^{34} -3733.37 q^{35} -39.6138 q^{37} +820.843 q^{38} -752.072 q^{40} -6364.06 q^{41} -15127.8 q^{43} -3845.08 q^{44} -1246.44 q^{46} -541.984 q^{47} +5493.85 q^{49} +294.803 q^{50} -9110.28 q^{52} -22754.2 q^{53} +3025.00 q^{55} +4492.42 q^{56} +2593.02 q^{58} +30386.6 q^{59} +7710.80 q^{61} +2506.86 q^{62} -31409.0 q^{64} +7167.23 q^{65} +17180.5 q^{67} +7815.84 q^{68} -1760.97 q^{70} +2314.77 q^{71} -43460.8 q^{73} -18.6852 q^{74} -55300.4 q^{76} -18069.5 q^{77} -28825.6 q^{79} +25067.3 q^{80} -3001.83 q^{82} -62012.6 q^{83} -6148.87 q^{85} -7135.57 q^{86} -3640.03 q^{88} -9224.55 q^{89} -42812.7 q^{91} +83972.7 q^{92} -255.645 q^{94} +43505.9 q^{95} -27948.9 q^{97} +2591.36 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{2} + 53 q^{4} + 150 q^{5} - 80 q^{7} - 255 q^{8}+O(q^{10}) $ 6 * q - 5 * q^2 + 53 * q^4 + 150 * q^5 - 80 * q^7 - 255 * q^8	$ 6 q - 5 q^{2} + 53 q^{4} + 150 q^{5} - 80 q^{7} - 255 q^{8} - 125 q^{10} + 726 q^{11} - 420 q^{13} - 776 q^{14} + 1169 q^{16} - 2820 q^{17} - 220 q^{19} + 1325 q^{20} - 605 q^{22} - 2680 q^{23} + 3750 q^{25} - 1896 q^{26} - 11760 q^{28} - 1092 q^{29} - 7688 q^{31} - 4535 q^{32} - 354 q^{34} - 2000 q^{35} - 14020 q^{37} - 1570 q^{38} - 6375 q^{40} + 8196 q^{41} - 17340 q^{43} + 6413 q^{44} - 9982 q^{46} - 4200 q^{47} - 16890 q^{49} - 3125 q^{50} - 13440 q^{52} - 32900 q^{53} + 18150 q^{55} + 41824 q^{56} - 98010 q^{58} - 44512 q^{59} + 26636 q^{61} + 50680 q^{62} - 74607 q^{64} - 10500 q^{65} - 9920 q^{67} + 34810 q^{68} - 19400 q^{70} - 27344 q^{71} - 106620 q^{73} + 244014 q^{74} - 5638 q^{76} - 9680 q^{77} - 7168 q^{79} + 29225 q^{80} - 90250 q^{82} + 113480 q^{83} - 70500 q^{85} + 96314 q^{86} - 30855 q^{88} + 38352 q^{89} - 115232 q^{91} + 116910 q^{92} - 161846 q^{94} - 5500 q^{95} - 299100 q^{97} + 70715 q^{98}+O(q^{100}) $ 6 * q - 5 * q^2 + 53 * q^4 + 150 * q^5 - 80 * q^7 - 255 * q^8 - 125 * q^10 + 726 * q^11 - 420 * q^13 - 776 * q^14 + 1169 * q^16 - 2820 * q^17 - 220 * q^19 + 1325 * q^20 - 605 * q^22 - 2680 * q^23 + 3750 * q^25 - 1896 * q^26 - 11760 * q^28 - 1092 * q^29 - 7688 * q^31 - 4535 * q^32 - 354 * q^34 - 2000 * q^35 - 14020 * q^37 - 1570 * q^38 - 6375 * q^40 + 8196 * q^41 - 17340 * q^43 + 6413 * q^44 - 9982 * q^46 - 4200 * q^47 - 16890 * q^49 - 3125 * q^50 - 13440 * q^52 - 32900 * q^53 + 18150 * q^55 + 41824 * q^56 - 98010 * q^58 - 44512 * q^59 + 26636 * q^61 + 50680 * q^62 - 74607 * q^64 - 10500 * q^65 - 9920 * q^67 + 34810 * q^68 - 19400 * q^70 - 27344 * q^71 - 106620 * q^73 + 244014 * q^74 - 5638 * q^76 - 9680 * q^77 - 7168 * q^79 + 29225 * q^80 - 90250 * q^82 + 113480 * q^83 - 70500 * q^85 + 96314 * q^86 - 30855 * q^88 + 38352 * q^89 - 115232 * q^91 + 116910 * q^92 - 161846 * q^94 - 5500 * q^95 - 299100 * q^97 + 70715 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.471685	0.0833829	0.0416914	−	0.999131i	$-0.486725\pi$
$2$	0.471685	0.0833829	0.0416914	+	0.999131i	$0.486725\pi$
$3$	0	0
$3$	0	0
$4$	−31.7775	−0.993047
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−149.335	−1.15190	−0.575951	−	0.817484i	$-0.695368\pi$
$7$	−149.335	−1.15190	−0.575951	+	0.817484i	$0.695368\pi$
$8$	−30.0829	−0.166186
$9$	0	0
$10$	11.7921	0.0372900
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	286.689	0.470493	0.235247	−	0.971936i	$-0.424410\pi$
$13$	286.689	0.470493	0.235247	+	0.971936i	$0.424410\pi$
$14$	−70.4389	−0.0960489
$15$	0	0
$16$	1002.69	0.979190
$17$	−245.955	−0.206411	−0.103206	−	0.994660i	$-0.532910\pi$
$17$	−245.955	−0.206411	−0.103206	+	0.994660i	$0.532910\pi$
$18$	0	0
$19$	1740.24	1.10592	0.552961	−	0.833207i	$-0.313498\pi$
$19$	1740.24	1.10592	0.552961	+	0.833207i	$0.313498\pi$
$20$	−794.438	−0.444104
$21$	0	0
$22$	57.0739	0.0251409
$23$	−2642.52	−1.04159	−0.520797	−	0.853680i	$-0.674365\pi$
$23$	−2642.52	−1.04159	−0.520797	+	0.853680i	$0.674365\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	135.227	0.0392311
$27$	0	0
$28$	4745.48	1.14389
$29$	5497.36	1.21383	0.606917	−	0.794765i	$-0.292406\pi$
$29$	5497.36	1.21383	0.606917	+	0.794765i	$0.292406\pi$
$30$	0	0
$31$	5314.70	0.993287	0.496643	−	0.867955i	$-0.334566\pi$
$31$	5314.70	0.993287	0.496643	+	0.867955i	$0.334566\pi$
$32$	1435.61	0.247834
$33$	0	0
$34$	−116.013	−0.0172112
$35$	−3733.37	−0.515146
$36$	0	0
$37$	−39.6138	−0.00475710	−0.00237855	−	0.999997i	$-0.500757\pi$
$37$	−39.6138	−0.00475710	−0.00237855	+	0.999997i	$0.500757\pi$
$38$	820.843	0.0922149
$39$	0	0
$40$	−752.072	−0.0743206
$41$	−6364.06	−0.591254	−0.295627	−	0.955303i	$-0.595529\pi$
$41$	−6364.06	−0.591254	−0.295627	+	0.955303i	$0.595529\pi$
$42$	0	0
$43$	−15127.8	−1.24769	−0.623844	−	0.781549i	$-0.714430\pi$
$43$	−15127.8	−1.24769	−0.623844	+	0.781549i	$0.714430\pi$
$44$	−3845.08	−0.299415
$45$	0	0
$46$	−1246.44	−0.0868512
$47$	−541.984	−0.0357883	−0.0178942	−	0.999840i	$-0.505696\pi$
$47$	−541.984	−0.0357883	−0.0178942	+	0.999840i	$0.505696\pi$
$48$	0	0
$49$	5493.85	0.326878
$50$	294.803	0.0166766
$51$	0	0
$52$	−9110.28	−0.467222
$53$	−22754.2	−1.11269	−0.556343	−	0.830953i	$-0.687796\pi$
$53$	−22754.2	−1.11269	−0.556343	+	0.830953i	$0.687796\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	4492.42	0.191430
$57$	0	0
$58$	2593.02	0.101213
$59$	30386.6	1.13646	0.568228	−	0.822871i	$-0.307629\pi$
$59$	30386.6	1.13646	0.568228	+	0.822871i	$0.307629\pi$
$60$	0	0
$61$	7710.80	0.265323	0.132661	−	0.991161i	$-0.457648\pi$
$61$	7710.80	0.265323	0.132661	+	0.991161i	$0.457648\pi$
$62$	2506.86	0.0828231
$63$	0	0
$64$	−31409.0	−0.958525
$65$	7167.23	0.210411
$66$	0	0
$67$	17180.5	0.467571	0.233786	−	0.972288i	$-0.424889\pi$
$67$	17180.5	0.467571	0.233786	+	0.972288i	$0.424889\pi$
$68$	7815.84	0.204976
$69$	0	0
$70$	−1760.97	−0.0429544
$71$	2314.77	0.0544956	0.0272478	−	0.999629i	$-0.491326\pi$
$71$	2314.77	0.0544956	0.0272478	+	0.999629i	$0.491326\pi$
$72$	0	0
$73$	−43460.8	−0.954533	−0.477266	−	0.878759i	$-0.658372\pi$
$73$	−43460.8	−0.954533	−0.477266	+	0.878759i	$0.658372\pi$
$74$	−18.6852	−0.000396661 0
$75$	0	0
$76$	−55300.4	−1.09823
$77$	−18069.5	−0.347312
$78$	0	0
$79$	−28825.6	−0.519650	−0.259825	−	0.965656i	$-0.583665\pi$
$79$	−28825.6	−0.519650	−0.259825	+	0.965656i	$0.583665\pi$
$80$	25067.3	0.437907
$81$	0	0
$82$	−3001.83	−0.0493005
$83$	−62012.6	−0.988063	−0.494031	−	0.869444i	$-0.664477\pi$
$83$	−62012.6	−0.988063	−0.494031	+	0.869444i	$0.664477\pi$
$84$	0	0
$85$	−6148.87	−0.0923099
$86$	−7135.57	−0.104036
$87$	0	0
$88$	−3640.03	−0.0501070
$89$	−9224.55	−0.123444	−0.0617220	−	0.998093i	$-0.519659\pi$
$89$	−9224.55	−0.123444	−0.0617220	+	0.998093i	$0.519659\pi$
$90$	0	0
$91$	−42812.7	−0.541962
$92$	83972.7	1.03435
$93$	0	0
$94$	−255.645	−0.00298413
$95$	43505.9	0.494583
$96$	0	0
$97$	−27948.9	−0.301602	−0.150801	−	0.988564i	$-0.548185\pi$
$97$	−27948.9	−0.301602	−0.150801	+	0.988564i	$0.548185\pi$
$98$	2591.36	0.0272561
$99$	0	0
$100$	−19860.9	−0.198609
$101$	−40574.0	−0.395771	−0.197885	−	0.980225i	$-0.563407\pi$
$101$	−40574.0	−0.395771	−0.197885	+	0.980225i	$0.563407\pi$
$102$	0	0
$103$	−32941.9	−0.305954	−0.152977	−	0.988230i	$-0.548886\pi$
$103$	−32941.9	−0.305954	−0.152977	+	0.988230i	$0.548886\pi$
$104$	−8624.44	−0.0781894
$105$	0	0
$106$	−10732.8	−0.0927789
$107$	157921.	1.33346	0.666729	−	0.745300i	$-0.267694\pi$
$107$	157921.	1.33346	0.666729	+	0.745300i	$0.267694\pi$
$108$	0	0
$109$	5420.17	0.0436965	0.0218482	−	0.999761i	$-0.493045\pi$
$109$	5420.17	0.0436965	0.0218482	+	0.999761i	$0.493045\pi$
$110$	1426.85	0.0112433
$111$	0	0
$112$	−149737.	−1.12793
$113$	−177192.	−1.30541	−0.652707	−	0.757611i	$-0.726367\pi$
$113$	−177192.	−1.30541	−0.652707	+	0.757611i	$0.726367\pi$
$114$	0	0
$115$	−66063.0	−0.465815
$116$	−174692.	−1.20539
$117$	0	0
$118$	14332.9	0.0947610
$119$	36729.6	0.237765
$120$	0	0
$121$	14641.0	0.0909091
$122$	3637.07	0.0221234
$123$	0	0
$124$	−168888.	−0.986381
$125$	15625.0	0.0894427
$126$	0	0
$127$	−257555.	−1.41697	−0.708485	−	0.705726i	$-0.750621\pi$
$127$	−257555.	−1.41697	−0.708485	+	0.705726i	$0.750621\pi$
$128$	−60754.5	−0.327758
$129$	0	0
$130$	3380.68	0.0175447
$131$	−94371.2	−0.480464	−0.240232	−	0.970715i	$-0.577224\pi$
$131$	−94371.2	−0.480464	−0.240232	+	0.970715i	$0.577224\pi$
$132$	0	0
$133$	−259878.	−1.27391
$134$	8103.76	0.0389874
$135$	0	0
$136$	7399.03	0.0343027
$137$	−67446.7	−0.307015	−0.153507	−	0.988147i	$-0.549057\pi$
$137$	−67446.7	−0.307015	−0.153507	+	0.988147i	$0.549057\pi$
$138$	0	0
$139$	−136319.	−0.598439	−0.299219	−	0.954184i	$-0.596726\pi$
$139$	−136319.	−0.598439	−0.299219	+	0.954184i	$0.596726\pi$
$140$	118637.	0.511565
$141$	0	0
$142$	1091.84	0.00454400
$143$	34689.4	0.141859
$144$	0	0
$145$	137434.	0.542843
$146$	−20499.8	−0.0795917
$147$	0	0
$148$	1258.83	0.00472402
$149$	−82127.9	−0.303057	−0.151529	−	0.988453i	$-0.548420\pi$
$149$	−82127.9	−0.303057	−0.151529	+	0.988453i	$0.548420\pi$
$150$	0	0
$151$	−124161.	−0.443143	−0.221572	−	0.975144i	$-0.571119\pi$
$151$	−124161.	−0.443143	−0.221572	+	0.975144i	$0.571119\pi$
$152$	−52351.3	−0.183789
$153$	0	0
$154$	−8523.11	−0.0289598
$155$	132868.	0.444211
$156$	0	0
$157$	−162792.	−0.527088	−0.263544	−	0.964647i	$-0.584891\pi$
$157$	−162792.	−0.527088	−0.263544	+	0.964647i	$0.584891\pi$
$158$	−13596.6	−0.0433299
$159$	0	0
$160$	35890.2	0.110835
$161$	394620.	1.19982
$162$	0	0
$163$	1366.75	0.00402920	0.00201460	−	0.999998i	$-0.499359\pi$
$163$	1366.75	0.00402920	0.00201460	+	0.999998i	$0.499359\pi$
$164$	202234.	0.587144
$165$	0	0
$166$	−29250.4	−0.0823875
$167$	−51149.4	−0.141922	−0.0709609	−	0.997479i	$-0.522607\pi$
$167$	−51149.4	−0.141922	−0.0709609	+	0.997479i	$0.522607\pi$
$168$	0	0
$169$	−289102.	−0.778636
$170$	−2900.33	−0.00769706
$171$	0	0
$172$	480725.	1.23901
$173$	−83412.3	−0.211892	−0.105946	−	0.994372i	$-0.533787\pi$
$173$	−83412.3	−0.211892	−0.105946	+	0.994372i	$0.533787\pi$
$174$	0	0
$175$	−93334.2	−0.230380
$176$	121326.	0.295237
$177$	0	0
$178$	−4351.08	−0.0102931
$179$	−670715.	−1.56461	−0.782304	−	0.622897i	$-0.785956\pi$
$179$	−670715.	−1.56461	−0.782304	+	0.622897i	$0.785956\pi$
$180$	0	0
$181$	−583865.	−1.32470	−0.662348	−	0.749196i	$-0.730440\pi$
$181$	−583865.	−1.32470	−0.662348	+	0.749196i	$0.730440\pi$
$182$	−20194.1	−0.0451903
$183$	0	0
$184$	79494.6	0.173098
$185$	−990.345	−0.00212744
$186$	0	0
$187$	−29760.5	−0.0622353
$188$	17222.9	0.0355395
$189$	0	0
$190$	20521.1	0.0412398
$191$	400495.	0.794354	0.397177	−	0.917742i	$-0.369990\pi$
$191$	400495.	0.794354	0.397177	+	0.917742i	$0.369990\pi$
$192$	0	0
$193$	29979.9	0.0579345	0.0289672	−	0.999580i	$-0.490778\pi$
$193$	29979.9	0.0579345	0.0289672	+	0.999580i	$0.490778\pi$
$194$	−13183.0	−0.0251485
$195$	0	0
$196$	−174581.	−0.324606
$197$	965421.	1.77236	0.886178	−	0.463345i	$-0.153351\pi$
$197$	965421.	1.77236	0.886178	+	0.463345i	$0.153351\pi$
$198$	0	0
$199$	206778.	0.370145	0.185073	−	0.982725i	$-0.440748\pi$
$199$	206778.	0.370145	0.185073	+	0.982725i	$0.440748\pi$
$200$	−18801.8	−0.0332372
$201$	0	0
$202$	−19138.1	−0.0330005
$203$	−820947.	−1.39822
$204$	0	0
$205$	−159101.	−0.264417
$206$	−15538.2	−0.0255113
$207$	0	0
$208$	287461.	0.460702
$209$	210569.	0.333448
$210$	0	0
$211$	613055.	0.947967	0.473983	−	0.880534i	$-0.342816\pi$
$211$	613055.	0.947967	0.473983	+	0.880534i	$0.342816\pi$
$212$	723073.	1.10495
$213$	0	0
$214$	74488.8	0.111188
$215$	−378196.	−0.557983
$216$	0	0
$217$	−793669.	−1.14417
$218$	2556.61	0.00364354
$219$	0	0
$220$	−96127.0	−0.133902
$221$	−70512.7	−0.0971150
$222$	0	0
$223$	−760007.	−1.02342	−0.511712	−	0.859157i	$-0.670988\pi$
$223$	−760007.	−1.02342	−0.511712	+	0.859157i	$0.670988\pi$
$224$	−214386.	−0.285480
$225$	0	0
$226$	−83578.8	−0.108849
$227$	395354.	0.509239	0.254620	−	0.967041i	$-0.418050\pi$
$227$	395354.	0.509239	0.254620	+	0.967041i	$0.418050\pi$
$228$	0	0
$229$	−467856.	−0.589554	−0.294777	−	0.955566i	$-0.595245\pi$
$229$	−467856.	−0.589554	−0.294777	+	0.955566i	$0.595245\pi$
$230$	−31160.9	−0.0388410
$231$	0	0
$232$	−165376.	−0.201722
$233$	55995.3	0.0675712	0.0337856	−	0.999429i	$-0.489244\pi$
$233$	55995.3	0.0675712	0.0337856	+	0.999429i	$0.489244\pi$
$234$	0	0
$235$	−13549.6	−0.0160050
$236$	−965612.	−1.12855
$237$	0	0
$238$	17324.8	0.0198256
$239$	−1.20258e6	−1.36182	−0.680909	−	0.732368i	$-0.738415\pi$
$239$	−1.20258e6	−1.36182	−0.680909	+	0.732368i	$0.738415\pi$
$240$	0	0
$241$	−1.57692e6	−1.74890	−0.874452	−	0.485112i	$-0.838779\pi$
$241$	−1.57692e6	−1.74890	−0.874452	+	0.485112i	$0.838779\pi$
$242$	6905.94	0.00758026
$243$	0	0
$244$	−245030.	−0.263478
$245$	137346.	0.146185
$246$	0	0
$247$	498907.	0.520328
$248$	−159882.	−0.165070
$249$	0	0
$250$	7370.07	0.00745799
$251$	1.55233e6	1.55525	0.777623	−	0.628731i	$-0.216426\pi$
$251$	1.55233e6	1.55525	0.777623	+	0.628731i	$0.216426\pi$
$252$	0	0
$253$	−319745.	−0.314053
$254$	−121485.	−0.118151
$255$	0	0
$256$	976429.	0.931196
$257$	1.41677e6	1.33803	0.669015	−	0.743249i	$-0.266716\pi$
$257$	1.41677e6	1.33803	0.669015	+	0.743249i	$0.266716\pi$
$258$	0	0
$259$	5915.71	0.00547971
$260$	−227757.	−0.208948
$261$	0	0
$262$	−44513.4	−0.0400625
$263$	353340.	0.314995	0.157497	−	0.987519i	$-0.449657\pi$
$263$	353340.	0.314995	0.157497	+	0.987519i	$0.449657\pi$
$264$	0	0
$265$	−568856.	−0.497608
$266$	−122580.	−0.106223
$267$	0	0
$268$	−545952.	−0.464320
$269$	−1.93636e6	−1.63157	−0.815784	−	0.578357i	$-0.803694\pi$
$269$	−1.93636e6	−1.63157	−0.815784	+	0.578357i	$0.803694\pi$
$270$	0	0
$271$	−955799.	−0.790575	−0.395288	−	0.918557i	$-0.629355\pi$
$271$	−955799.	−0.790575	−0.395288	+	0.918557i	$0.629355\pi$
$272$	−246617.	−0.202116
$273$	0	0
$274$	−31813.6	−0.0255998
$275$	75625.0	0.0603023
$276$	0	0
$277$	−1.10202e6	−0.862957	−0.431478	−	0.902123i	$-0.642008\pi$
$277$	−1.10202e6	−0.862957	−0.431478	+	0.902123i	$0.642008\pi$
$278$	−64299.7	−0.0498996
$279$	0	0
$280$	112310.	0.0856101
$281$	−1.04062e6	−0.786191	−0.393095	−	0.919498i	$-0.628596\pi$
$281$	−1.04062e6	−0.786191	−0.393095	+	0.919498i	$0.628596\pi$
$282$	0	0
$283$	1.00251e6	0.744083	0.372042	−	0.928216i	$-0.378658\pi$
$283$	1.00251e6	0.744083	0.372042	+	0.928216i	$0.378658\pi$
$284$	−73557.5	−0.0541167
$285$	0	0
$286$	16362.5	0.0118286
$287$	950374.	0.681067
$288$	0	0
$289$	−1.35936e6	−0.957394
$290$	64825.5	0.0452638
$291$	0	0
$292$	1.38108e6	0.947896
$293$	734910.	0.500110	0.250055	−	0.968232i	$-0.419551\pi$
$293$	734910.	0.500110	0.250055	+	0.968232i	$0.419551\pi$
$294$	0	0
$295$	759666.	0.508239
$296$	1191.70	0.000790563 0
$297$	0	0
$298$	−38738.5	−0.0252698
$299$	−757583.	−0.490063
$300$	0	0
$301$	2.25911e6	1.43721
$302$	−58565.0	−0.0369505
$303$	0	0
$304$	1.74492e6	1.08291
$305$	192770.	0.118656
$306$	0	0
$307$	195697.	0.118505	0.0592527	−	0.998243i	$-0.481128\pi$
$307$	195697.	0.118505	0.0592527	+	0.998243i	$0.481128\pi$
$308$	574204.	0.344897
$309$	0	0
$310$	62671.6	0.0370396
$311$	1.29994e6	0.762119	0.381060	−	0.924551i	$-0.375559\pi$
$311$	1.29994e6	0.762119	0.381060	+	0.924551i	$0.375559\pi$
$312$	0	0
$313$	1.25859e6	0.726142	0.363071	−	0.931761i	$-0.381728\pi$
$313$	1.25859e6	0.726142	0.363071	+	0.931761i	$0.381728\pi$
$314$	−76786.3	−0.0439501
$315$	0	0
$316$	916006.	0.516037
$317$	−2.54425e6	−1.42204	−0.711018	−	0.703173i	$-0.751766\pi$
$317$	−2.54425e6	−1.42204	−0.711018	+	0.703173i	$0.751766\pi$
$318$	0	0
$319$	665181.	0.365985
$320$	−785224.	−0.428665
$321$	0	0
$322$	186136.	0.100044
$323$	−428020.	−0.228275
$324$	0	0
$325$	179181.	0.0940986
$326$	644.674	0.000335966 0
$327$	0	0
$328$	191449.	0.0982582
$329$	80937.0	0.0412247
$330$	0	0
$331$	3.18220e6	1.59646	0.798229	−	0.602354i	$-0.205770\pi$
$331$	3.18220e6	1.59646	0.798229	+	0.602354i	$0.205770\pi$
$332$	1.97061e6	0.981193
$333$	0	0
$334$	−24126.4	−0.0118339
$335$	429511.	0.209104
$336$	0	0
$337$	2.44663e6	1.17353	0.586765	−	0.809757i	$-0.300401\pi$
$337$	2.44663e6	1.17353	0.586765	+	0.809757i	$0.300401\pi$
$338$	−136365.	−0.0649249
$339$	0	0
$340$	195396.	0.0916681
$341$	643079.	0.299487
$342$	0	0
$343$	1.68945e6	0.775370
$344$	455089.	0.207348
$345$	0	0
$346$	−39344.3	−0.0176682
$347$	−4.40372e6	−1.96334	−0.981671	−	0.190586i	$-0.938961\pi$
$347$	−4.40372e6	−1.96334	−0.981671	+	0.190586i	$0.938961\pi$
$348$	0	0
$349$	2.11416e6	0.929124	0.464562	−	0.885541i	$-0.346212\pi$
$349$	2.11416e6	0.929124	0.464562	+	0.885541i	$0.346212\pi$
$350$	−44024.3	−0.0192098
$351$	0	0
$352$	173708.	0.0747247
$353$	669632.	0.286022	0.143011	−	0.989721i	$-0.454322\pi$
$353$	669632.	0.286022	0.143011	+	0.989721i	$0.454322\pi$
$354$	0	0
$355$	57869.2	0.0243712
$356$	293133.	0.122586
$357$	0	0
$358$	−316366.	−0.130462
$359$	4.30507e6	1.76297	0.881483	−	0.472216i	$-0.156546\pi$
$359$	4.30507e6	1.76297	0.881483	+	0.472216i	$0.156546\pi$
$360$	0	0
$361$	552323.	0.223062
$362$	−275400.	−0.110457
$363$	0	0
$364$	1.36048e6	0.538194
$365$	−1.08652e6	−0.426880
$366$	0	0
$367$	4.17487e6	1.61800	0.808999	−	0.587810i	$-0.200010\pi$
$367$	4.17487e6	1.61800	0.808999	+	0.587810i	$0.200010\pi$
$368$	−2.64963e6	−1.01992
$369$	0	0
$370$	−467.131	−0.000177392 0
$371$	3.39800e6	1.28171
$372$	0	0
$373$	2.35076e6	0.874856	0.437428	−	0.899254i	$-0.355890\pi$
$373$	2.35076e6	0.874856	0.437428	+	0.899254i	$0.355890\pi$
$374$	−14037.6	−0.00518936
$375$	0	0
$376$	16304.4	0.00594752
$377$	1.57604e6	0.571100
$378$	0	0
$379$	−827735.	−0.296001	−0.148001	−	0.988987i	$-0.547284\pi$
$379$	−827735.	−0.296001	−0.148001	+	0.988987i	$0.547284\pi$
$380$	−1.38251e6	−0.491144
$381$	0	0
$382$	188908.	0.0662355
$383$	−1.30481e6	−0.454517	−0.227258	−	0.973834i	$-0.572976\pi$
$383$	−1.30481e6	−0.454517	−0.227258	+	0.973834i	$0.572976\pi$
$384$	0	0
$385$	−451737.	−0.155322
$386$	14141.1	0.00483074
$387$	0	0
$388$	888145.	0.299505
$389$	316440.	0.106027	0.0530135	−	0.998594i	$-0.483117\pi$
$389$	316440.	0.106027	0.0530135	+	0.998594i	$0.483117\pi$
$390$	0	0
$391$	649941.	0.214997
$392$	−165271.	−0.0543226
$393$	0	0
$394$	455374.	0.147784
$395$	−720640.	−0.232395
$396$	0	0
$397$	−4.04587e6	−1.28836	−0.644178	−	0.764875i	$-0.722800\pi$
$397$	−4.04587e6	−1.28836	−0.644178	+	0.764875i	$0.722800\pi$
$398$	97534.1	0.0308638
$399$	0	0
$400$	626682.	0.195838
$401$	3.54667e6	1.10144	0.550719	−	0.834691i	$-0.314354\pi$
$401$	3.54667e6	1.10144	0.550719	+	0.834691i	$0.314354\pi$
$402$	0	0
$403$	1.52367e6	0.467335
$404$	1.28934e6	0.393019
$405$	0	0
$406$	−387228.	−0.116587
$407$	−4793.27	−0.00143432
$408$	0	0
$409$	2.09268e6	0.618578	0.309289	−	0.950968i	$-0.399909\pi$
$409$	2.09268e6	0.618578	0.309289	+	0.950968i	$0.399909\pi$
$410$	−75045.7	−0.0220478
$411$	0	0
$412$	1.04681e6	0.303826
$413$	−4.53778e6	−1.30909
$414$	0	0
$415$	−1.55031e6	−0.441875
$416$	411573.	0.116604
$417$	0	0
$418$	99322.0	0.0278038
$419$	5.52544e6	1.53756	0.768780	−	0.639514i	$-0.220864\pi$
$419$	5.52544e6	1.53756	0.768780	+	0.639514i	$0.220864\pi$
$420$	0	0
$421$	−3.31894e6	−0.912628	−0.456314	−	0.889819i	$-0.650831\pi$
$421$	−3.31894e6	−0.912628	−0.456314	+	0.889819i	$0.650831\pi$
$422$	289169.	0.0790442
$423$	0	0
$424$	684513.	0.184913
$425$	−153722.	−0.0412822
$426$	0	0
$427$	−1.15149e6	−0.305626
$428$	−5.01833e6	−1.32419
$429$	0	0
$430$	−178389.	−0.0465262
$431$	1.56390e6	0.405523	0.202762	−	0.979228i	$-0.435008\pi$
$431$	1.56390e6	0.405523	0.202762	+	0.979228i	$0.435008\pi$
$432$	0	0
$433$	−4.40151e6	−1.12819	−0.564095	−	0.825710i	$-0.690775\pi$
$433$	−4.40151e6	−1.12819	−0.564095	+	0.825710i	$0.690775\pi$
$434$	−374362.	−0.0954041
$435$	0	0
$436$	−172240.	−0.0433927
$437$	−4.59861e6	−1.15192
$438$	0	0
$439$	−420527.	−0.104144	−0.0520719	−	0.998643i	$-0.516582\pi$
$439$	−420527.	−0.104144	−0.0520719	+	0.998643i	$0.516582\pi$
$440$	−91000.7	−0.0224085
$441$	0	0
$442$	−33259.8	−0.00809773
$443$	−1.18107e6	−0.285934	−0.142967	−	0.989727i	$-0.545664\pi$
$443$	−1.18107e6	−0.285934	−0.142967	+	0.989727i	$0.545664\pi$
$444$	0	0
$445$	−230614.	−0.0552059
$446$	−358484.	−0.0853360
$447$	0	0
$448$	4.69045e6	1.10413
$449$	−15765.7	−0.00369060	−0.00184530	−	0.999998i	$-0.500587\pi$
$449$	−15765.7	−0.00369060	−0.00184530	+	0.999998i	$0.500587\pi$
$450$	0	0
$451$	−770051.	−0.178270
$452$	5.63072e6	1.29634
$453$	0	0
$454$	186483.	0.0424618
$455$	−1.07032e6	−0.242373
$456$	0	0
$457$	−1.97300e6	−0.441912	−0.220956	−	0.975284i	$-0.570918\pi$
$457$	−1.97300e6	−0.441912	−0.220956	+	0.975284i	$0.570918\pi$
$458$	−220681.	−0.0491587
$459$	0	0
$460$	2.09932e6	0.462577
$461$	−3.27825e6	−0.718439	−0.359220	−	0.933253i	$-0.616957\pi$
$461$	−3.27825e6	−0.718439	−0.359220	+	0.933253i	$0.616957\pi$
$462$	0	0
$463$	−1.77922e6	−0.385724	−0.192862	−	0.981226i	$-0.561777\pi$
$463$	−1.77922e6	−0.385724	−0.192862	+	0.981226i	$0.561777\pi$
$464$	5.51215e6	1.18857
$465$	0	0
$466$	26412.1	0.00563428
$467$	−3.12040e6	−0.662091	−0.331045	−	0.943615i	$-0.607401\pi$
$467$	−3.12040e6	−0.662091	−0.331045	+	0.943615i	$0.607401\pi$
$468$	0	0
$469$	−2.56564e6	−0.538596
$470$	−6391.14	−0.00133455
$471$	0	0
$472$	−914118.	−0.188863
$473$	−1.83047e6	−0.376192
$474$	0	0
$475$	1.08765e6	0.221184
$476$	−1.16718e6	−0.236112
$477$	0	0
$478$	−567239.	−0.113552
$479$	1.31784e6	0.262437	0.131218	−	0.991353i	$-0.458111\pi$
$479$	1.31784e6	0.262437	0.131218	+	0.991353i	$0.458111\pi$
$480$	0	0
$481$	−11356.9	−0.00223818
$482$	−743807.	−0.145829
$483$	0	0
$484$	−465255.	−0.0902770
$485$	−698721.	−0.134881
$486$	0	0
$487$	2.79057e6	0.533175	0.266588	−	0.963811i	$-0.414104\pi$
$487$	2.79057e6	0.533175	0.266588	+	0.963811i	$0.414104\pi$
$488$	−231963.	−0.0440930
$489$	0	0
$490$	64784.1	0.0121893
$491$	7.86286e6	1.47189	0.735947	−	0.677039i	$-0.236737\pi$
$491$	7.86286e6	1.47189	0.735947	+	0.677039i	$0.236737\pi$
$492$	0	0
$493$	−1.35210e6	−0.250549
$494$	235327.	0.0433865
$495$	0	0
$496$	5.32900e6	0.972617
$497$	−345675.	−0.0627736
$498$	0	0
$499$	−5.97700e6	−1.07456	−0.537281	−	0.843403i	$-0.680549\pi$
$499$	−5.97700e6	−1.07456	−0.537281	+	0.843403i	$0.680549\pi$
$500$	−496524.	−0.0888209
$501$	0	0
$502$	732209.	0.129681
$503$	−1.02848e7	−1.81249	−0.906243	−	0.422757i	$-0.861062\pi$
$503$	−1.02848e7	−1.81249	−0.906243	+	0.422757i	$0.861062\pi$
$504$	0	0
$505$	−1.01435e6	−0.176994
$506$	−150819.	−0.0261866
$507$	0	0
$508$	8.18446e6	1.40712
$509$	3.69976e6	0.632964	0.316482	−	0.948599i	$-0.397498\pi$
$509$	3.69976e6	0.632964	0.316482	+	0.948599i	$0.397498\pi$
$510$	0	0
$511$	6.49021e6	1.09953
$512$	2.40471e6	0.405404
$513$	0	0
$514$	668268.	0.111569
$515$	−823548.	−0.136827
$516$	0	0
$517$	−65580.0	−0.0107906
$518$	2790.35	0.000456914 0
$519$	0	0
$520$	−215611.	−0.0349673
$521$	−2.49481e6	−0.402665	−0.201332	−	0.979523i	$-0.564527\pi$
$521$	−2.49481e6	−0.402665	−0.201332	+	0.979523i	$0.564527\pi$
$522$	0	0
$523$	−1.03447e7	−1.65372	−0.826860	−	0.562407i	$-0.809875\pi$
$523$	−1.03447e7	−1.65372	−0.826860	+	0.562407i	$0.809875\pi$
$524$	2.99888e6	0.477124
$525$	0	0
$526$	166665.	0.0262652
$527$	−1.30718e6	−0.205025
$528$	0	0
$529$	546573.	0.0849198
$530$	−268321.	−0.0414920
$531$	0	0
$532$	8.25826e6	1.26506
$533$	−1.82451e6	−0.278181
$534$	0	0
$535$	3.94802e6	0.596341
$536$	−516838.	−0.0777038
$537$	0	0
$538$	−913351.	−0.136045
$539$	664755.	0.0985576
$540$	0	0
$541$	−1.19438e7	−1.75449	−0.877245	−	0.480043i	$-0.840621\pi$
$541$	−1.19438e7	−1.75449	−0.877245	+	0.480043i	$0.840621\pi$
$542$	−450836.	−0.0659204
$543$	0	0
$544$	−353094.	−0.0511556
$545$	135504.	0.0195417
$546$	0	0
$547$	−7.77394e6	−1.11089	−0.555447	−	0.831552i	$-0.687453\pi$
$547$	−7.77394e6	−1.11089	−0.555447	+	0.831552i	$0.687453\pi$
$548$	2.14329e6	0.304880
$549$	0	0
$550$	35671.2	0.00502818
$551$	9.56671e6	1.34240
$552$	0	0
$553$	4.30466e6	0.598586
$554$	−519805.	−0.0719558
$555$	0	0
$556$	4.33188e6	0.594278
$557$	7.74940e6	1.05835	0.529176	−	0.848512i	$-0.322501\pi$
$557$	7.74940e6	1.05835	0.529176	+	0.848512i	$0.322501\pi$
$558$	0	0
$559$	−4.33699e6	−0.587028
$560$	−3.74341e6	−0.504426
$561$	0	0
$562$	−490846.	−0.0655549
$563$	−4.32908e6	−0.575605	−0.287803	−	0.957690i	$-0.592925\pi$
$563$	−4.32908e6	−0.575605	−0.287803	+	0.957690i	$0.592925\pi$
$564$	0	0
$565$	−4.42980e6	−0.583799
$566$	472867.	0.0620438
$567$	0	0
$568$	−69634.9	−0.00905641
$569$	−5.41017e6	−0.700536	−0.350268	−	0.936650i	$-0.613909\pi$
$569$	−5.41017e6	−0.700536	−0.350268	+	0.936650i	$0.613909\pi$
$570$	0	0
$571$	−2.54301e6	−0.326406	−0.163203	−	0.986592i	$-0.552183\pi$
$571$	−2.54301e6	−0.326406	−0.163203	+	0.986592i	$0.552183\pi$
$572$	−1.10234e6	−0.140873
$573$	0	0
$574$	448277.	0.0567893
$575$	−1.65158e6	−0.208319
$576$	0	0
$577$	−5.20722e6	−0.651128	−0.325564	−	0.945520i	$-0.605554\pi$
$577$	−5.20722e6	−0.651128	−0.325564	+	0.945520i	$0.605554\pi$
$578$	−641191.	−0.0798303
$579$	0	0
$580$	−4.36731e6	−0.539069
$581$	9.26063e6	1.13815
$582$	0	0
$583$	−2.75326e6	−0.335487
$584$	1.30743e6	0.158630
$585$	0	0
$586$	346646.	0.0417006
$587$	−1.09767e7	−1.31485	−0.657426	−	0.753519i	$-0.728355\pi$
$587$	−1.09767e7	−1.31485	−0.657426	+	0.753519i	$0.728355\pi$
$588$	0	0
$589$	9.24884e6	1.09850
$590$	358323.	0.0423784
$591$	0	0
$592$	−39720.4	−0.00465811
$593$	−4.69137e6	−0.547851	−0.273926	−	0.961751i	$-0.588322\pi$
$593$	−4.69137e6	−0.547851	−0.273926	+	0.961751i	$0.588322\pi$
$594$	0	0
$595$	918240.	0.106332
$596$	2.60982e6	0.300950
$597$	0	0
$598$	−357340.	−0.0408629
$599$	1.43577e7	1.63500	0.817498	−	0.575932i	$-0.195361\pi$
$599$	1.43577e7	1.63500	0.817498	+	0.575932i	$0.195361\pi$
$600$	0	0
$601$	−6.72679e6	−0.759665	−0.379832	−	0.925055i	$-0.624018\pi$
$601$	−6.72679e6	−0.759665	−0.379832	+	0.925055i	$0.624018\pi$
$602$	1.06559e6	0.119839
$603$	0	0
$604$	3.94554e6	0.440062
$605$	366025.	0.0406558
$606$	0	0
$607$	−2.49688e6	−0.275059	−0.137529	−	0.990498i	$-0.543916\pi$
$607$	−2.49688e6	−0.275059	−0.137529	+	0.990498i	$0.543916\pi$
$608$	2.49829e6	0.274085
$609$	0	0
$610$	90926.7	0.00989388
$611$	−155381.	−0.0168382
$612$	0	0
$613$	−3.35968e6	−0.361116	−0.180558	−	0.983564i	$-0.557790\pi$
$613$	−3.35968e6	−0.361116	−0.180558	+	0.983564i	$0.557790\pi$
$614$	92307.2	0.00988132
$615$	0	0
$616$	543582.	0.0577183
$617$	−2.68310e6	−0.283742	−0.141871	−	0.989885i	$-0.545312\pi$
$617$	−2.68310e6	−0.283742	−0.141871	+	0.989885i	$0.545312\pi$
$618$	0	0
$619$	−1.66985e7	−1.75167	−0.875834	−	0.482612i	$-0.839688\pi$
$619$	−1.66985e7	−1.75167	−0.875834	+	0.482612i	$0.839688\pi$
$620$	−4.22220e6	−0.441123
$621$	0	0
$622$	613163.	0.0635477
$623$	1.37755e6	0.142196
$624$	0	0
$625$	390625.	0.0400000
$626$	593655.	0.0605478
$627$	0	0
$628$	5.17311e6	0.523423
$629$	9743.21	0.000981919 0
$630$	0	0
$631$	−233175.	−0.0233136	−0.0116568	−	0.999932i	$-0.503711\pi$
$631$	−233175.	−0.0233136	−0.0116568	+	0.999932i	$0.503711\pi$
$632$	867158.	0.0863585
$633$	0	0
$634$	−1.20008e6	−0.118574
$635$	−6.43887e6	−0.633688
$636$	0	0
$637$	1.57503e6	0.153794
$638$	313756.	0.0305169
$639$	0	0
$640$	−1.51886e6	−0.146578
$641$	1.58412e7	1.52280	0.761401	−	0.648281i	$-0.224512\pi$
$641$	1.58412e7	1.52280	0.761401	+	0.648281i	$0.224512\pi$
$642$	0	0
$643$	−4.16078e6	−0.396869	−0.198435	−	0.980114i	$-0.563586\pi$
$643$	−4.16078e6	−0.396869	−0.198435	+	0.980114i	$0.563586\pi$
$644$	−1.25400e7	−1.19147
$645$	0	0
$646$	−201890.	−0.0190342
$647$	1.40495e7	1.31948	0.659738	−	0.751496i	$-0.270667\pi$
$647$	1.40495e7	1.31948	0.659738	+	0.751496i	$0.270667\pi$
$648$	0	0
$649$	3.67678e6	0.342654
$650$	84516.9	0.00784621
$651$	0	0
$652$	−43431.8	−0.00400119
$653$	−1.85267e7	−1.70025	−0.850127	−	0.526577i	$-0.823475\pi$
$653$	−1.85267e7	−1.70025	−0.850127	+	0.526577i	$0.823475\pi$
$654$	0	0
$655$	−2.35928e6	−0.214870
$656$	−6.38118e6	−0.578951
$657$	0	0
$658$	38176.7	0.00343743
$659$	−4.59915e6	−0.412538	−0.206269	−	0.978495i	$-0.566132\pi$
$659$	−4.59915e6	−0.412538	−0.206269	+	0.978495i	$0.566132\pi$
$660$	0	0
$661$	−1.45949e7	−1.29927	−0.649634	−	0.760248i	$-0.725078\pi$
$661$	−1.45949e7	−1.29927	−0.649634	+	0.760248i	$0.725078\pi$
$662$	1.50100e6	0.133117
$663$	0	0
$664$	1.86552e6	0.164202
$665$	−6.49694e6	−0.569711
$666$	0	0
$667$	−1.45269e7	−1.26432
$668$	1.62540e6	0.140935
$669$	0	0
$670$	202594.	0.0174357
$671$	933007.	0.0799979
$672$	0	0
$673$	−1.37384e7	−1.16923	−0.584614	−	0.811311i	$-0.698754\pi$
$673$	−1.37384e7	−1.16923	−0.584614	+	0.811311i	$0.698754\pi$
$674$	1.15404e6	0.0978523
$675$	0	0
$676$	9.18695e6	0.773223
$677$	1.53523e7	1.28736	0.643682	−	0.765293i	$-0.277406\pi$
$677$	1.53523e7	1.28736	0.643682	+	0.765293i	$0.277406\pi$
$678$	0	0
$679$	4.17373e6	0.347416
$680$	184976.	0.0153406
$681$	0	0
$682$	303330.	0.0249721
$683$	−3.39229e6	−0.278254	−0.139127	−	0.990275i	$-0.544430\pi$
$683$	−3.39229e6	−0.278254	−0.139127	+	0.990275i	$0.544430\pi$
$684$	0	0
$685$	−1.68617e6	−0.137301
$686$	796886.	0.0646526
$687$	0	0
$688$	−1.51685e7	−1.22172
$689$	−6.52340e6	−0.523511
$690$	0	0
$691$	−2.40972e7	−1.91987	−0.959933	−	0.280229i	$-0.909590\pi$
$691$	−2.40972e7	−1.91987	−0.959933	+	0.280229i	$0.909590\pi$
$692$	2.65064e6	0.210419
$693$	0	0
$694$	−2.07717e6	−0.163709
$695$	−3.40798e6	−0.267630
$696$	0	0
$697$	1.56527e6	0.122042
$698$	997216.	0.0774730
$699$	0	0
$700$	2.96593e6	0.228779
$701$	4.86430e6	0.373874	0.186937	−	0.982372i	$-0.440144\pi$
$701$	4.86430e6	0.373874	0.186937	+	0.982372i	$0.440144\pi$
$702$	0	0
$703$	−68937.4	−0.00526098
$704$	−3.80048e6	−0.289006
$705$	0	0
$706$	315855.	0.0238493
$707$	6.05910e6	0.455889
$708$	0	0
$709$	−2.53274e7	−1.89224	−0.946118	−	0.323821i	$-0.895032\pi$
$709$	−2.53274e7	−1.89224	−0.946118	+	0.323821i	$0.895032\pi$
$710$	27296.0	0.00203214
$711$	0	0
$712$	277501.	0.0205147
$713$	−1.40442e7	−1.03460
$714$	0	0
$715$	867235.	0.0634413
$716$	2.13137e7	1.55373
$717$	0	0
$718$	2.03063e6	0.147001
$719$	−1.07333e7	−0.774302	−0.387151	−	0.922016i	$-0.626541\pi$
$719$	−1.07333e7	−0.774302	−0.387151	+	0.922016i	$0.626541\pi$
$720$	0	0
$721$	4.91937e6	0.352429
$722$	260523.	0.0185995
$723$	0	0
$724$	1.85538e7	1.31549
$725$	3.43585e6	0.242767
$726$	0	0
$727$	−6.33268e6	−0.444377	−0.222188	−	0.975004i	$-0.571320\pi$
$727$	−6.33268e6	−0.444377	−0.222188	+	0.975004i	$0.571320\pi$
$728$	1.28793e6	0.0900665
$729$	0	0
$730$	−512495.	−0.0355945
$731$	3.72077e6	0.257537
$732$	0	0
$733$	1.74869e6	0.120213	0.0601066	−	0.998192i	$-0.480856\pi$
$733$	1.74869e6	0.120213	0.0601066	+	0.998192i	$0.480856\pi$
$734$	1.96922e6	0.134913
$735$	0	0
$736$	−3.79362e6	−0.258142
$737$	2.07884e6	0.140978
$738$	0	0
$739$	−1.34607e7	−0.906686	−0.453343	−	0.891336i	$-0.649769\pi$
$739$	−1.34607e7	−0.906686	−0.453343	+	0.891336i	$0.649769\pi$
$740$	31470.7	0.00211265
$741$	0	0
$742$	1.60278e6	0.106872
$743$	−2.33346e7	−1.55070	−0.775350	−	0.631531i	$-0.782427\pi$
$743$	−2.33346e7	−1.55070	−0.775350	+	0.631531i	$0.782427\pi$
$744$	0	0
$745$	−2.05320e6	−0.135531
$746$	1.10882e6	0.0729480
$747$	0	0
$748$	945716.	0.0618026
$749$	−2.35830e7	−1.53601
$750$	0	0
$751$	1.89952e7	1.22898	0.614489	−	0.788925i	$-0.289362\pi$
$751$	1.89952e7	1.22898	0.614489	+	0.788925i	$0.289362\pi$
$752$	−543442.	−0.0350436
$753$	0	0
$754$	743392.	0.0476200
$755$	−3.10403e6	−0.198180
$756$	0	0
$757$	1.71731e7	1.08920	0.544602	−	0.838694i	$-0.316681\pi$
$757$	1.71731e7	1.08920	0.544602	+	0.838694i	$0.316681\pi$
$758$	−390430.	−0.0246814
$759$	0	0
$760$	−1.30878e6	−0.0821928
$761$	2.82842e7	1.77045	0.885224	−	0.465166i	$-0.154005\pi$
$761$	2.82842e7	1.77045	0.885224	+	0.465166i	$0.154005\pi$
$762$	0	0
$763$	−809419.	−0.0503341
$764$	−1.27267e7	−0.788831
$765$	0	0
$766$	−615458.	−0.0378989
$767$	8.71153e6	0.534695
$768$	0	0
$769$	−5.65815e6	−0.345032	−0.172516	−	0.985007i	$-0.555190\pi$
$769$	−5.65815e6	−0.345032	−0.172516	+	0.985007i	$0.555190\pi$
$770$	−213078.	−0.0129512
$771$	0	0
$772$	−952687.	−0.0575317
$773$	−1.60345e7	−0.965176	−0.482588	−	0.875847i	$-0.660303\pi$
$773$	−1.60345e7	−0.965176	−0.482588	+	0.875847i	$0.660303\pi$
$774$	0	0
$775$	3.32169e6	0.198657
$776$	840782.	0.0501221
$777$	0	0
$778$	149260.	0.00884084
$779$	−1.10750e7	−0.653881
$780$	0	0
$781$	280087.	0.0164310
$782$	306567.	0.0179271
$783$	0	0
$784$	5.50863e6	0.320076
$785$	−4.06979e6	−0.235721
$786$	0	0
$787$	−1.07093e7	−0.616344	−0.308172	−	0.951331i	$-0.599717\pi$
$787$	−1.07093e7	−0.616344	−0.308172	+	0.951331i	$0.599717\pi$
$788$	−3.06787e7	−1.76003
$789$	0	0
$790$	−339915.	−0.0193777
$791$	2.64609e7	1.50371
$792$	0	0
$793$	2.21060e6	0.124833
$794$	−1.90838e6	−0.107427
$795$	0	0
$796$	−6.57090e6	−0.367572
$797$	1.65404e7	0.922361	0.461181	−	0.887306i	$-0.347426\pi$
$797$	1.65404e7	0.922361	0.461181	+	0.887306i	$0.347426\pi$
$798$	0	0
$799$	133304.	0.00738711
$800$	897254.	0.0495667
$801$	0	0
$802$	1.67291e6	0.0918411
$803$	−5.25876e6	−0.287802
$804$	0	0
$805$	9.86550e6	0.536574
$806$	718691.	0.0389677
$807$	0	0
$808$	1.22058e6	0.0657716
$809$	−4.62480e6	−0.248440	−0.124220	−	0.992255i	$-0.539643\pi$
$809$	−4.62480e6	−0.248440	−0.124220	+	0.992255i	$0.539643\pi$
$810$	0	0
$811$	1.05606e6	0.0563814	0.0281907	−	0.999603i	$-0.491025\pi$
$811$	1.05606e6	0.0563814	0.0281907	+	0.999603i	$0.491025\pi$
$812$	2.60876e7	1.38850
$813$	0	0
$814$	−2260.91	−0.000119598 0
$815$	34168.7	0.00180191
$816$	0	0
$817$	−2.63260e7	−1.37984
$818$	987084.	0.0515788
$819$	0	0
$820$	5.05585e6	0.262579
$821$	2.57120e7	1.33130	0.665652	−	0.746262i	$-0.268153\pi$
$821$	2.57120e7	1.33130	0.665652	+	0.746262i	$0.268153\pi$
$822$	0	0
$823$	−8.76266e6	−0.450958	−0.225479	−	0.974248i	$-0.572395\pi$
$823$	−8.76266e6	−0.450958	−0.225479	+	0.974248i	$0.572395\pi$
$824$	990987.	0.0508452
$825$	0	0
$826$	−2.14040e6	−0.109155
$827$	2.08101e7	1.05806	0.529031	−	0.848603i	$-0.322556\pi$
$827$	2.08101e7	1.05806	0.529031	+	0.848603i	$0.322556\pi$
$828$	0	0
$829$	−1.48581e7	−0.750890	−0.375445	−	0.926845i	$-0.622510\pi$
$829$	−1.48581e7	−0.750890	−0.375445	+	0.926845i	$0.622510\pi$
$830$	−731260.	−0.0368448
$831$	0	0
$832$	−9.00461e6	−0.450979
$833$	−1.35124e6	−0.0674714
$834$	0	0
$835$	−1.27873e6	−0.0634694
$836$	−6.69135e6	−0.331129
$837$	0	0
$838$	2.60627e6	0.128206
$839$	2.14888e7	1.05392	0.526958	−	0.849891i	$-0.323332\pi$
$839$	2.14888e7	1.05392	0.526958	+	0.849891i	$0.323332\pi$
$840$	0	0
$841$	9.70983e6	0.473393
$842$	−1.56549e6	−0.0760976
$843$	0	0
$844$	−1.94814e7	−0.941376
$845$	−7.22755e6	−0.348217
$846$	0	0
$847$	−2.18641e6	−0.104718
$848$	−2.28155e7	−1.08953
$849$	0	0
$850$	−72508.2	−0.00344223
$851$	104680.	0.00495497
$852$	0	0
$853$	−1.20844e7	−0.568662	−0.284331	−	0.958726i	$-0.591771\pi$
$853$	−1.20844e7	−0.568662	−0.284331	+	0.958726i	$0.591771\pi$
$854$	−543140.	−0.0254840
$855$	0	0
$856$	−4.75071e6	−0.221602
$857$	1.41426e7	0.657773	0.328887	−	0.944369i	$-0.393327\pi$
$857$	1.41426e7	0.657773	0.328887	+	0.944369i	$0.393327\pi$
$858$	0	0
$859$	2.34876e7	1.08607	0.543033	−	0.839711i	$-0.317276\pi$
$859$	2.34876e7	1.08607	0.543033	+	0.839711i	$0.317276\pi$
$860$	1.20181e7	0.554103
$861$	0	0
$862$	737667.	0.0338137
$863$	−1.01191e7	−0.462502	−0.231251	−	0.972894i	$-0.574282\pi$
$863$	−1.01191e7	−0.462502	−0.231251	+	0.972894i	$0.574282\pi$
$864$	0	0
$865$	−2.08531e6	−0.0947611
$866$	−2.07613e6	−0.0940717
$867$	0	0
$868$	2.52208e7	1.13621
$869$	−3.48790e6	−0.156680
$870$	0	0
$871$	4.92546e6	0.219989
$872$	−163054.	−0.00726175
$873$	0	0
$874$	−2.16909e6	−0.0960506
$875$	−2.33335e6	−0.103029
$876$	0	0
$877$	1.09697e7	0.481611	0.240805	−	0.970573i	$-0.422588\pi$
$877$	1.09697e7	0.481611	0.240805	+	0.970573i	$0.422588\pi$
$878$	−198356.	−0.00868380
$879$	0	0
$880$	3.03314e6	0.132034
$881$	−4.44391e7	−1.92897	−0.964485	−	0.264136i	$-0.914913\pi$
$881$	−4.44391e7	−1.92897	−0.964485	+	0.264136i	$0.914913\pi$
$882$	0	0
$883$	4.03611e7	1.74205	0.871026	−	0.491237i	$-0.163455\pi$
$883$	4.03611e7	1.74205	0.871026	+	0.491237i	$0.163455\pi$
$884$	2.24072e6	0.0964398
$885$	0	0
$886$	−557091.	−0.0238420
$887$	1.97927e7	0.844689	0.422344	−	0.906436i	$-0.361207\pi$
$887$	1.97927e7	0.844689	0.422344	+	0.906436i	$0.361207\pi$
$888$	0	0
$889$	3.84619e7	1.63221
$890$	−108777.	−0.00460322
$891$	0	0
$892$	2.41511e7	1.01631
$893$	−943180.	−0.0395791
$894$	0	0
$895$	−1.67679e7	−0.699714
$896$	9.07276e6	0.377545
$897$	0	0
$898$	−7436.43	−0.000307733 0
$899$	2.92168e7	1.20569
$900$	0	0
$901$	5.59652e6	0.229671
$902$	−363221.	−0.0148647
$903$	0	0
$904$	5.33045e6	0.216941
$905$	−1.45966e7	−0.592422
$906$	0	0
$907$	1.84300e7	0.743886	0.371943	−	0.928256i	$-0.378692\pi$
$907$	1.84300e7	0.743886	0.371943	+	0.928256i	$0.378692\pi$
$908$	−1.25634e7	−0.505699
$909$	0	0
$910$	−504852.	−0.0202097
$911$	−2.70614e7	−1.08033	−0.540163	−	0.841561i	$-0.681637\pi$
$911$	−2.70614e7	−1.08033	−0.540163	+	0.841561i	$0.681637\pi$
$912$	0	0
$913$	−7.50352e6	−0.297912
$914$	−930632.	−0.0368479
$915$	0	0
$916$	1.48673e7	0.585455
$917$	1.40929e7	0.553448
$918$	0	0
$919$	4.74394e7	1.85289	0.926447	−	0.376425i	$-0.122847\pi$
$919$	4.74394e7	1.85289	0.926447	+	0.376425i	$0.122847\pi$
$920$	1.98737e6	0.0774120
$921$	0	0
$922$	−1.54630e6	−0.0599055
$923$	663619.	0.0256398
$924$	0	0
$925$	−24758.6	−0.000951420 0
$926$	−839230.	−0.0321628
$927$	0	0
$928$	7.89205e6	0.300829
$929$	−796073.	−0.0302631	−0.0151316	−	0.999886i	$-0.504817\pi$
$929$	−796073.	−0.0302631	−0.0151316	+	0.999886i	$0.504817\pi$
$930$	0	0
$931$	9.56059e6	0.361502
$932$	−1.77939e6	−0.0671014
$933$	0	0
$934$	−1.47184e6	−0.0552070
$935$	−744014.	−0.0278325
$936$	0	0
$937$	2.45501e7	0.913491	0.456746	−	0.889597i	$-0.349015\pi$
$937$	2.45501e7	0.913491	0.456746	+	0.889597i	$0.349015\pi$
$938$	−1.21017e6	−0.0449097
$939$	0	0
$940$	430572.	0.0158938
$941$	2.23594e7	0.823163	0.411581	−	0.911373i	$-0.364977\pi$
$941$	2.23594e7	0.823163	0.411581	+	0.911373i	$0.364977\pi$
$942$	0	0
$943$	1.68172e7	0.615848
$944$	3.04684e7	1.11281
$945$	0	0
$946$	−863404.	−0.0313680
$947$	−4.43364e7	−1.60652	−0.803259	−	0.595630i	$-0.796902\pi$
$947$	−4.43364e7	−1.60652	−0.803259	+	0.595630i	$0.796902\pi$
$948$	0	0
$949$	−1.24598e7	−0.449101
$950$	513027.	0.0184430
$951$	0	0
$952$	−1.10493e6	−0.0395133
$953$	1.97632e7	0.704897	0.352448	−	0.935831i	$-0.385349\pi$
$953$	1.97632e7	0.704897	0.352448	+	0.935831i	$0.385349\pi$
$954$	0	0
$955$	1.00124e7	0.355246
$956$	3.82150e7	1.35235
$957$	0	0
$958$	621606.	0.0218827
$959$	1.00721e7	0.353651
$960$	0	0
$961$	−383101.	−0.0133815
$962$	−5356.86	−0.000186626 0
$963$	0	0
$964$	5.01105e7	1.73674
$965$	749498.	0.0259091
$966$	0	0
$967$	4.38576e7	1.50827	0.754135	−	0.656720i	$-0.228057\pi$
$967$	4.38576e7	1.50827	0.754135	+	0.656720i	$0.228057\pi$
$968$	−440443.	−0.0151078
$969$	0	0
$970$	−329576.	−0.0112467
$971$	−2.44563e7	−0.832421	−0.416211	−	0.909268i	$-0.636642\pi$
$971$	−2.44563e7	−0.832421	−0.416211	+	0.909268i	$0.636642\pi$
$972$	0	0
$973$	2.03572e7	0.689343
$974$	1.31627e6	0.0444577
$975$	0	0
$976$	7.73155e6	0.259802
$977$	2.21082e7	0.740998	0.370499	−	0.928833i	$-0.379187\pi$
$977$	2.21082e7	0.740998	0.370499	+	0.928833i	$0.379187\pi$
$978$	0	0
$979$	−1.11617e6	−0.0372198
$980$	−4.36452e6	−0.145168
$981$	0	0
$982$	3.70879e6	0.122731
$983$	−4.16717e7	−1.37549	−0.687744	−	0.725953i	$-0.741399\pi$
$983$	−4.16717e7	−1.37549	−0.687744	+	0.725953i	$0.741399\pi$
$984$	0	0
$985$	2.41355e7	0.792622
$986$	−637766.	−0.0208915
$987$	0	0
$988$	−1.58540e7	−0.516711
$989$	3.99756e7	1.29958
$990$	0	0
$991$	1.47175e7	0.476047	0.238023	−	0.971259i	$-0.423501\pi$
$991$	1.47175e7	0.476047	0.238023	+	0.971259i	$0.423501\pi$
$992$	7.62982e6	0.246170
$993$	0	0
$994$	−163050.	−0.00523424
$995$	5.16946e6	0.165534
$996$	0	0
$997$	−1.80255e7	−0.574313	−0.287157	−	0.957884i	$-0.592710\pi$
$997$	−1.80255e7	−0.574313	−0.287157	+	0.957884i	$0.592710\pi$
$998$	−2.81926e6	−0.0896001
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.k.1.4	✓	6
3.2	odd	2		495.6.a.m.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.6.a.k.1.4	✓	6	1.1	even	1	trivial
495.6.a.m.1.3	yes	6	3.2	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)

\(f(q)\)	\(=\)	\(q+0.471685 q^{2} -31.7775 q^{4} +25.0000 q^{5} -149.335 q^{7} -30.0829 q^{8} +O(q^{10})\)	\(q+0.471685 q^{2} -31.7775 q^{4} +25.0000 q^{5} -149.335 q^{7} -30.0829 q^{8} +11.7921 q^{10} +121.000 q^{11} +286.689 q^{13} -70.4389 q^{14} +1002.69 q^{16} -245.955 q^{17} +1740.24 q^{19} -794.438 q^{20} +57.0739 q^{22} -2642.52 q^{23} +625.000 q^{25} +135.227 q^{26} +4745.48 q^{28} +5497.36 q^{29} +5314.70 q^{31} +1435.61 q^{32} -116.013 q^{34} -3733.37 q^{35} -39.6138 q^{37} +820.843 q^{38} -752.072 q^{40} -6364.06 q^{41} -15127.8 q^{43} -3845.08 q^{44} -1246.44 q^{46} -541.984 q^{47} +5493.85 q^{49} +294.803 q^{50} -9110.28 q^{52} -22754.2 q^{53} +3025.00 q^{55} +4492.42 q^{56} +2593.02 q^{58} +30386.6 q^{59} +7710.80 q^{61} +2506.86 q^{62} -31409.0 q^{64} +7167.23 q^{65} +17180.5 q^{67} +7815.84 q^{68} -1760.97 q^{70} +2314.77 q^{71} -43460.8 q^{73} -18.6852 q^{74} -55300.4 q^{76} -18069.5 q^{77} -28825.6 q^{79} +25067.3 q^{80} -3001.83 q^{82} -62012.6 q^{83} -6148.87 q^{85} -7135.57 q^{86} -3640.03 q^{88} -9224.55 q^{89} -42812.7 q^{91} +83972.7 q^{92} -255.645 q^{94} +43505.9 q^{95} -27948.9 q^{97} +2591.36 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{2} + 53 q^{4} + 150 q^{5} - 80 q^{7} - 255 q^{8}+O(q^{10}) \) 6 * q - 5 * q^2 + 53 * q^4 + 150 * q^5 - 80 * q^7 - 255 * q^8	\( 6 q - 5 q^{2} + 53 q^{4} + 150 q^{5} - 80 q^{7} - 255 q^{8} - 125 q^{10} + 726 q^{11} - 420 q^{13} - 776 q^{14} + 1169 q^{16} - 2820 q^{17} - 220 q^{19} + 1325 q^{20} - 605 q^{22} - 2680 q^{23} + 3750 q^{25} - 1896 q^{26} - 11760 q^{28} - 1092 q^{29} - 7688 q^{31} - 4535 q^{32} - 354 q^{34} - 2000 q^{35} - 14020 q^{37} - 1570 q^{38} - 6375 q^{40} + 8196 q^{41} - 17340 q^{43} + 6413 q^{44} - 9982 q^{46} - 4200 q^{47} - 16890 q^{49} - 3125 q^{50} - 13440 q^{52} - 32900 q^{53} + 18150 q^{55} + 41824 q^{56} - 98010 q^{58} - 44512 q^{59} + 26636 q^{61} + 50680 q^{62} - 74607 q^{64} - 10500 q^{65} - 9920 q^{67} + 34810 q^{68} - 19400 q^{70} - 27344 q^{71} - 106620 q^{73} + 244014 q^{74} - 5638 q^{76} - 9680 q^{77} - 7168 q^{79} + 29225 q^{80} - 90250 q^{82} + 113480 q^{83} - 70500 q^{85} + 96314 q^{86} - 30855 q^{88} + 38352 q^{89} - 115232 q^{91} + 116910 q^{92} - 161846 q^{94} - 5500 q^{95} - 299100 q^{97} + 70715 q^{98}+O(q^{100}) \) 6 * q - 5 * q^2 + 53 * q^4 + 150 * q^5 - 80 * q^7 - 255 * q^8 - 125 * q^10 + 726 * q^11 - 420 * q^13 - 776 * q^14 + 1169 * q^16 - 2820 * q^17 - 220 * q^19 + 1325 * q^20 - 605 * q^22 - 2680 * q^23 + 3750 * q^25 - 1896 * q^26 - 11760 * q^28 - 1092 * q^29 - 7688 * q^31 - 4535 * q^32 - 354 * q^34 - 2000 * q^35 - 14020 * q^37 - 1570 * q^38 - 6375 * q^40 + 8196 * q^41 - 17340 * q^43 + 6413 * q^44 - 9982 * q^46 - 4200 * q^47 - 16890 * q^49 - 3125 * q^50 - 13440 * q^52 - 32900 * q^53 + 18150 * q^55 + 41824 * q^56 - 98010 * q^58 - 44512 * q^59 + 26636 * q^61 + 50680 * q^62 - 74607 * q^64 - 10500 * q^65 - 9920 * q^67 + 34810 * q^68 - 19400 * q^70 - 27344 * q^71 - 106620 * q^73 + 244014 * q^74 - 5638 * q^76 - 9680 * q^77 - 7168 * q^79 + 29225 * q^80 - 90250 * q^82 + 113480 * q^83 - 70500 * q^85 + 96314 * q^86 - 30855 * q^88 + 38352 * q^89 - 115232 * q^91 + 116910 * q^92 - 161846 * q^94 - 5500 * q^95 - 299100 * q^97 + 70715 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more