LMFDB - Embedded newform 495.6.a.j.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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 $ x^5 - 2x^4 - 119x^3 + 206x^2 + 1428x - 1320
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			$0.870481$ 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)

$f(q)$	$=$	$q+0.870481 q^{2} -31.2423 q^{4} +25.0000 q^{5} -236.601 q^{7} -55.0512 q^{8} +O(q^{10})$	\(q+0.870481 q^{2} -31.2423 q^{4} +25.0000 q^{5} -236.601 q^{7} -55.0512 q^{8} +21.7620 q^{10} +121.000 q^{11} -261.860 q^{13} -205.957 q^{14} +951.831 q^{16} -1586.33 q^{17} -711.422 q^{19} -781.057 q^{20} +105.328 q^{22} -2962.00 q^{23} +625.000 q^{25} -227.944 q^{26} +7391.95 q^{28} -4421.36 q^{29} -4442.31 q^{31} +2590.19 q^{32} -1380.87 q^{34} -5915.02 q^{35} -11609.3 q^{37} -619.279 q^{38} -1376.28 q^{40} -4684.06 q^{41} +20190.0 q^{43} -3780.31 q^{44} -2578.36 q^{46} +8283.84 q^{47} +39173.0 q^{49} +544.051 q^{50} +8181.09 q^{52} +24486.5 q^{53} +3025.00 q^{55} +13025.2 q^{56} -3848.71 q^{58} -44449.2 q^{59} -17286.1 q^{61} -3866.95 q^{62} -28203.9 q^{64} -6546.49 q^{65} +51551.5 q^{67} +49560.5 q^{68} -5148.91 q^{70} -16729.9 q^{71} +49070.5 q^{73} -10105.7 q^{74} +22226.4 q^{76} -28628.7 q^{77} +11585.1 q^{79} +23795.8 q^{80} -4077.38 q^{82} -20786.3 q^{83} -39658.2 q^{85} +17575.0 q^{86} -6661.19 q^{88} -28462.9 q^{89} +61956.2 q^{91} +92539.6 q^{92} +7210.93 q^{94} -17785.5 q^{95} -50359.6 q^{97} +34099.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8	$ 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8} + 50 q^{10} + 605 q^{11} + 1082 q^{13} - 432 q^{14} + 4770 q^{16} - 2174 q^{17} + 1632 q^{19} + 2050 q^{20} + 242 q^{22} - 1212 q^{23} + 3125 q^{25} - 5600 q^{26} + 16508 q^{28} - 82 q^{29} + 12120 q^{31} + 4864 q^{32} - 4524 q^{34} + 4600 q^{35} - 6530 q^{37} + 15132 q^{38} - 600 q^{40} - 6782 q^{41} + 46184 q^{43} + 9922 q^{44} + 12048 q^{46} + 11692 q^{47} + 34445 q^{49} + 1250 q^{50} + 50020 q^{52} - 10314 q^{53} + 15125 q^{55} - 54928 q^{56} + 75048 q^{58} - 92892 q^{59} + 106 q^{61} - 97160 q^{62} + 44550 q^{64} + 27050 q^{65} + 100476 q^{67} - 119928 q^{68} - 10800 q^{70} + 13772 q^{71} + 94154 q^{73} + 47924 q^{74} - 51524 q^{76} + 22264 q^{77} + 178744 q^{79} + 119250 q^{80} - 299848 q^{82} + 100116 q^{83} - 54350 q^{85} + 167704 q^{86} - 2904 q^{88} - 119410 q^{89} + 47536 q^{91} + 404560 q^{92} - 310288 q^{94} + 40800 q^{95} + 100682 q^{97} + 16434 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8 + 50 * q^10 + 605 * q^11 + 1082 * q^13 - 432 * q^14 + 4770 * q^16 - 2174 * q^17 + 1632 * q^19 + 2050 * q^20 + 242 * q^22 - 1212 * q^23 + 3125 * q^25 - 5600 * q^26 + 16508 * q^28 - 82 * q^29 + 12120 * q^31 + 4864 * q^32 - 4524 * q^34 + 4600 * q^35 - 6530 * q^37 + 15132 * q^38 - 600 * q^40 - 6782 * q^41 + 46184 * q^43 + 9922 * q^44 + 12048 * q^46 + 11692 * q^47 + 34445 * q^49 + 1250 * q^50 + 50020 * q^52 - 10314 * q^53 + 15125 * q^55 - 54928 * q^56 + 75048 * q^58 - 92892 * q^59 + 106 * q^61 - 97160 * q^62 + 44550 * q^64 + 27050 * q^65 + 100476 * q^67 - 119928 * q^68 - 10800 * q^70 + 13772 * q^71 + 94154 * q^73 + 47924 * q^74 - 51524 * q^76 + 22264 * q^77 + 178744 * q^79 + 119250 * q^80 - 299848 * q^82 + 100116 * q^83 - 54350 * q^85 + 167704 * q^86 - 2904 * q^88 - 119410 * q^89 + 47536 * q^91 + 404560 * q^92 - 310288 * q^94 + 40800 * q^95 + 100682 * q^97 + 16434 * 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 $

$2$	0.870481	0.153881	0.0769404	−	0.997036i	$-0.475485\pi$
$2$	0.870481	0.153881	0.0769404	+	0.997036i	$0.475485\pi$
$3$	0	0
$3$	0	0
$4$	−31.2423	−0.976321
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−236.601	−1.82504	−0.912518	−	0.409037i	$-0.865865\pi$
$7$	−236.601	−1.82504	−0.912518	+	0.409037i	$0.865865\pi$
$8$	−55.0512	−0.304118
$9$	0	0
$10$	21.7620	0.0688176
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−261.860	−0.429744	−0.214872	−	0.976642i	$-0.568934\pi$
$13$	−261.860	−0.429744	−0.214872	+	0.976642i	$0.568934\pi$
$14$	−205.957	−0.280838
$15$	0	0
$16$	951.831	0.929523
$17$	−1586.33	−1.33129	−0.665643	−	0.746271i	$-0.731843\pi$
$17$	−1586.33	−1.33129	−0.665643	+	0.746271i	$0.731843\pi$
$18$	0	0
$19$	−711.422	−0.452109	−0.226054	−	0.974115i	$-0.572583\pi$
$19$	−711.422	−0.452109	−0.226054	+	0.974115i	$0.572583\pi$
$20$	−781.057	−0.436624
$21$	0	0
$22$	105.328	0.0463968
$23$	−2962.00	−1.16752	−0.583761	−	0.811925i	$-0.698420\pi$
$23$	−2962.00	−1.16752	−0.583761	+	0.811925i	$0.698420\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−227.944	−0.0661294
$27$	0	0
$28$	7391.95	1.78182
$29$	−4421.36	−0.976250	−0.488125	−	0.872774i	$-0.662319\pi$
$29$	−4421.36	−0.976250	−0.488125	+	0.872774i	$0.662319\pi$
$30$	0	0
$31$	−4442.31	−0.830242	−0.415121	−	0.909766i	$-0.636261\pi$
$31$	−4442.31	−0.830242	−0.415121	+	0.909766i	$0.636261\pi$
$32$	2590.19	0.447153
$33$	0	0
$34$	−1380.87	−0.204859
$35$	−5915.02	−0.816181
$36$	0	0
$37$	−11609.3	−1.39413	−0.697063	−	0.717010i	$-0.745510\pi$
$37$	−11609.3	−1.39413	−0.697063	+	0.717010i	$0.745510\pi$
$38$	−619.279	−0.0695708
$39$	0	0
$40$	−1376.28	−0.136006
$41$	−4684.06	−0.435174	−0.217587	−	0.976041i	$-0.569819\pi$
$41$	−4684.06	−0.435174	−0.217587	+	0.976041i	$0.569819\pi$
$42$	0	0
$43$	20190.0	1.66520	0.832598	−	0.553878i	$-0.186853\pi$
$43$	20190.0	1.66520	0.832598	+	0.553878i	$0.186853\pi$
$44$	−3780.31	−0.294372
$45$	0	0
$46$	−2578.36	−0.179659
$47$	8283.84	0.547000	0.273500	−	0.961872i	$-0.411819\pi$
$47$	8283.84	0.547000	0.273500	+	0.961872i	$0.411819\pi$
$48$	0	0
$49$	39173.0	2.33075
$50$	544.051	0.0307761
$51$	0	0
$52$	8181.09	0.419568
$53$	24486.5	1.19739	0.598697	−	0.800976i	$-0.295686\pi$
$53$	24486.5	1.19739	0.598697	+	0.800976i	$0.295686\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	13025.2	0.555025
$57$	0	0
$58$	−3848.71	−0.150226
$59$	−44449.2	−1.66239	−0.831197	−	0.555979i	$-0.812344\pi$
$59$	−44449.2	−1.66239	−0.831197	+	0.555979i	$0.812344\pi$
$60$	0	0
$61$	−17286.1	−0.594802	−0.297401	−	0.954753i	$-0.596120\pi$
$61$	−17286.1	−0.594802	−0.297401	+	0.954753i	$0.596120\pi$
$62$	−3866.95	−0.127758
$63$	0	0
$64$	−28203.9	−0.860715
$65$	−6546.49	−0.192188
$66$	0	0
$67$	51551.5	1.40299	0.701494	−	0.712675i	$-0.252517\pi$
$67$	51551.5	1.40299	0.701494	+	0.712675i	$0.252517\pi$
$68$	49560.5	1.29976
$69$	0	0
$70$	−5148.91	−0.125594
$71$	−16729.9	−0.393865	−0.196932	−	0.980417i	$-0.563098\pi$
$71$	−16729.9	−0.393865	−0.196932	+	0.980417i	$0.563098\pi$
$72$	0	0
$73$	49070.5	1.07774	0.538870	−	0.842389i	$-0.318852\pi$
$73$	49070.5	1.07774	0.538870	+	0.842389i	$0.318852\pi$
$74$	−10105.7	−0.214529
$75$	0	0
$76$	22226.4	0.441403
$77$	−28628.7	−0.550269
$78$	0	0
$79$	11585.1	0.208850	0.104425	−	0.994533i	$-0.466700\pi$
$79$	11585.1	0.208850	0.104425	+	0.994533i	$0.466700\pi$
$80$	23795.8	0.415695
$81$	0	0
$82$	−4077.38	−0.0669648
$83$	−20786.3	−0.331194	−0.165597	−	0.986193i	$-0.552955\pi$
$83$	−20786.3	−0.331194	−0.165597	+	0.986193i	$0.552955\pi$
$84$	0	0
$85$	−39658.2	−0.595369
$86$	17575.0	0.256241
$87$	0	0
$88$	−6661.19	−0.0916949
$89$	−28462.9	−0.380894	−0.190447	−	0.981697i	$-0.560994\pi$
$89$	−28462.9	−0.380894	−0.190447	+	0.981697i	$0.560994\pi$
$90$	0	0
$91$	61956.2	0.784299
$92$	92539.6	1.13988
$93$	0	0
$94$	7210.93	0.0841728
$95$	−17785.5	−0.202189
$96$	0	0
$97$	−50359.6	−0.543442	−0.271721	−	0.962376i	$-0.587593\pi$
$97$	−50359.6	−0.543442	−0.271721	+	0.962376i	$0.587593\pi$
$98$	34099.3	0.358658
$99$	0	0
$100$	−19526.4	−0.195264
$101$	93581.1	0.912819	0.456409	−	0.889770i	$-0.349135\pi$
$101$	93581.1	0.912819	0.456409	+	0.889770i	$0.349135\pi$
$102$	0	0
$103$	41230.4	0.382934	0.191467	−	0.981499i	$-0.438675\pi$
$103$	41230.4	0.382934	0.191467	+	0.981499i	$0.438675\pi$
$104$	14415.7	0.130693
$105$	0	0
$106$	21315.0	0.184256
$107$	−9821.73	−0.0829332	−0.0414666	−	0.999140i	$-0.513203\pi$
$107$	−9821.73	−0.0829332	−0.0414666	+	0.999140i	$0.513203\pi$
$108$	0	0
$109$	−84192.7	−0.678748	−0.339374	−	0.940652i	$-0.610215\pi$
$109$	−84192.7	−0.678748	−0.339374	+	0.940652i	$0.610215\pi$
$110$	2633.20	0.0207493
$111$	0	0
$112$	−225204.	−1.69641
$113$	212433.	1.56504	0.782522	−	0.622623i	$-0.213933\pi$
$113$	212433.	1.56504	0.782522	+	0.622623i	$0.213933\pi$
$114$	0	0
$115$	−74050.0	−0.522132
$116$	138133.	0.953133
$117$	0	0
$118$	−38692.2	−0.255810
$119$	375327.	2.42964
$120$	0	0
$121$	14641.0	0.0909091
$122$	−15047.2	−0.0915286
$123$	0	0
$124$	138788.	0.810583
$125$	15625.0	0.0894427
$126$	0	0
$127$	282182.	1.55246	0.776228	−	0.630452i	$-0.217130\pi$
$127$	282182.	1.55246	0.776228	+	0.630452i	$0.217130\pi$
$128$	−107437.	−0.579601
$129$	0	0
$130$	−5698.60	−0.0295740
$131$	80587.8	0.410290	0.205145	−	0.978732i	$-0.434233\pi$
$131$	80587.8	0.410290	0.205145	+	0.978732i	$0.434233\pi$
$132$	0	0
$133$	168323.	0.825115
$134$	44874.6	0.215893
$135$	0	0
$136$	87329.3	0.404867
$137$	300753.	1.36901	0.684507	−	0.729006i	$-0.260017\pi$
$137$	300753.	1.36901	0.684507	+	0.729006i	$0.260017\pi$
$138$	0	0
$139$	−84700.7	−0.371835	−0.185917	−	0.982565i	$-0.559526\pi$
$139$	−84700.7	−0.371835	−0.185917	+	0.982565i	$0.559526\pi$
$140$	184799.	0.796854
$141$	0	0
$142$	−14563.0	−0.0606082
$143$	−31685.0	−0.129573
$144$	0	0
$145$	−110534.	−0.436592
$146$	42715.0	0.165843
$147$	0	0
$148$	362701.	1.36111
$149$	−300589.	−1.10919	−0.554597	−	0.832119i	$-0.687128\pi$
$149$	−300589.	−1.10919	−0.554597	+	0.832119i	$0.687128\pi$
$150$	0	0
$151$	−402879.	−1.43791	−0.718956	−	0.695055i	$-0.755380\pi$
$151$	−402879.	−1.43791	−0.718956	+	0.695055i	$0.755380\pi$
$152$	39164.6	0.137494
$153$	0	0
$154$	−24920.7	−0.0846758
$155$	−111058.	−0.371296
$156$	0	0
$157$	367370.	1.18947	0.594737	−	0.803920i	$-0.297256\pi$
$157$	367370.	1.18947	0.594737	+	0.803920i	$0.297256\pi$
$158$	10084.6	0.0321379
$159$	0	0
$160$	64754.7	0.199973
$161$	700812.	2.13077
$162$	0	0
$163$	422639.	1.24595	0.622975	−	0.782242i	$-0.285924\pi$
$163$	422639.	1.24595	0.622975	+	0.782242i	$0.285924\pi$
$164$	146341.	0.424869
$165$	0	0
$166$	−18094.1	−0.0509644
$167$	−533906.	−1.48140	−0.740702	−	0.671833i	$-0.765507\pi$
$167$	−533906.	−1.48140	−0.740702	+	0.671833i	$0.765507\pi$
$168$	0	0
$169$	−302723.	−0.815320
$170$	−34521.7	−0.0916158
$171$	0	0
$172$	−630781.	−1.62576
$173$	−280697.	−0.713054	−0.356527	−	0.934285i	$-0.616039\pi$
$173$	−280697.	−0.713054	−0.356527	+	0.934285i	$0.616039\pi$
$174$	0	0
$175$	−147876.	−0.365007
$176$	115172.	0.280262
$177$	0	0
$178$	−24776.4	−0.0586122
$179$	274428.	0.640170	0.320085	−	0.947389i	$-0.396288\pi$
$179$	274428.	0.640170	0.320085	+	0.947389i	$0.396288\pi$
$180$	0	0
$181$	126946.	0.288020	0.144010	−	0.989576i	$-0.454000\pi$
$181$	126946.	0.288020	0.144010	+	0.989576i	$0.454000\pi$
$182$	53931.7	0.120688
$183$	0	0
$184$	163062.	0.355064
$185$	−290233.	−0.623472
$186$	0	0
$187$	−191946.	−0.401398
$188$	−258806.	−0.534047
$189$	0	0
$190$	−15482.0	−0.0311130
$191$	−523390.	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$191$	−523390.	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$192$	0	0
$193$	732070.	1.41468	0.707342	−	0.706871i	$-0.249894\pi$
$193$	732070.	1.41468	0.707342	+	0.706871i	$0.249894\pi$
$194$	−43837.1	−0.0836253
$195$	0	0
$196$	−1.22385e6	−2.27556
$197$	−518770.	−0.952379	−0.476189	−	0.879343i	$-0.657982\pi$
$197$	−518770.	−0.952379	−0.476189	+	0.879343i	$0.657982\pi$
$198$	0	0
$199$	788448.	1.41137	0.705685	−	0.708526i	$-0.250640\pi$
$199$	788448.	1.41137	0.705685	+	0.708526i	$0.250640\pi$
$200$	−34407.0	−0.0608235
$201$	0	0
$202$	81460.5	0.140465
$203$	1.04610e6	1.78169
$204$	0	0
$205$	−117101.	−0.194616
$206$	35890.3	0.0589262
$207$	0	0
$208$	−249246.	−0.399457
$209$	−86082.0	−0.136316
$210$	0	0
$211$	602584.	0.931775	0.465888	−	0.884844i	$-0.345735\pi$
$211$	602584.	0.931775	0.465888	+	0.884844i	$0.345735\pi$
$212$	−765013.	−1.16904
$213$	0	0
$214$	−8549.62	−0.0127618
$215$	504750.	0.744698
$216$	0	0
$217$	1.05105e6	1.51522
$218$	−73288.1	−0.104446
$219$	0	0
$220$	−94507.8	−0.131647
$221$	415396.	0.572113
$222$	0	0
$223$	−1.19775e6	−1.61289	−0.806445	−	0.591309i	$-0.798611\pi$
$223$	−1.19775e6	−1.61289	−0.806445	+	0.591309i	$0.798611\pi$
$224$	−612841.	−0.816071
$225$	0	0
$226$	184919.	0.240830
$227$	−1.10750e6	−1.42653	−0.713263	−	0.700897i	$-0.752783\pi$
$227$	−1.10750e6	−1.42653	−0.713263	+	0.700897i	$0.752783\pi$
$228$	0	0
$229$	794404.	1.00104	0.500521	−	0.865724i	$-0.333142\pi$
$229$	794404.	1.00104	0.500521	+	0.865724i	$0.333142\pi$
$230$	−64459.1	−0.0803461
$231$	0	0
$232$	243401.	0.296895
$233$	755284.	0.911424	0.455712	−	0.890127i	$-0.349385\pi$
$233$	755284.	0.911424	0.455712	+	0.890127i	$0.349385\pi$
$234$	0	0
$235$	207096.	0.244626
$236$	1.38869e6	1.62303
$237$	0	0
$238$	326715.	0.373875
$239$	1.15093e6	1.30333	0.651664	−	0.758508i	$-0.274071\pi$
$239$	1.15093e6	1.30333	0.651664	+	0.758508i	$0.274071\pi$
$240$	0	0
$241$	−685503.	−0.760268	−0.380134	−	0.924931i	$-0.624122\pi$
$241$	−685503.	−0.760268	−0.380134	+	0.924931i	$0.624122\pi$
$242$	12744.7	0.0139892
$243$	0	0
$244$	540057.	0.580718
$245$	979324.	1.04234
$246$	0	0
$247$	186293.	0.194291
$248$	244554.	0.252491
$249$	0	0
$250$	13601.3	0.0137635
$251$	−433601.	−0.434416	−0.217208	−	0.976125i	$-0.569695\pi$
$251$	−433601.	−0.434416	−0.217208	+	0.976125i	$0.569695\pi$
$252$	0	0
$253$	−358402.	−0.352021
$254$	245634.	0.238893
$255$	0	0
$256$	809003.	0.771525
$257$	−151494.	−0.143075	−0.0715374	−	0.997438i	$-0.522791\pi$
$257$	−151494.	−0.143075	−0.0715374	+	0.997438i	$0.522791\pi$
$258$	0	0
$259$	2.74677e6	2.54433
$260$	204527.	0.187637
$261$	0	0
$262$	70150.1	0.0631357
$263$	−626051.	−0.558111	−0.279055	−	0.960275i	$-0.590021\pi$
$263$	−626051.	−0.558111	−0.279055	+	0.960275i	$0.590021\pi$
$264$	0	0
$265$	612162.	0.535490
$266$	146522.	0.126969
$267$	0	0
$268$	−1.61059e6	−1.36977
$269$	−438844.	−0.369768	−0.184884	−	0.982760i	$-0.559191\pi$
$269$	−438844.	−0.369768	−0.184884	+	0.982760i	$0.559191\pi$
$270$	0	0
$271$	−901700.	−0.745828	−0.372914	−	0.927866i	$-0.621641\pi$
$271$	−901700.	−0.745828	−0.372914	+	0.927866i	$0.621641\pi$
$272$	−1.50992e6	−1.23746
$273$	0	0
$274$	261799.	0.210665
$275$	75625.0	0.0603023
$276$	0	0
$277$	902407.	0.706648	0.353324	−	0.935501i	$-0.385051\pi$
$277$	902407.	0.706648	0.353324	+	0.935501i	$0.385051\pi$
$278$	−73730.3	−0.0572182
$279$	0	0
$280$	325629.	0.248215
$281$	−470571.	−0.355516	−0.177758	−	0.984074i	$-0.556884\pi$
$281$	−470571.	−0.355516	−0.177758	+	0.984074i	$0.556884\pi$
$282$	0	0
$283$	308437.	0.228929	0.114464	−	0.993427i	$-0.463485\pi$
$283$	308437.	0.228929	0.114464	+	0.993427i	$0.463485\pi$
$284$	522679.	0.384538
$285$	0	0
$286$	−27581.2	−0.0199388
$287$	1.10825e6	0.794207
$288$	0	0
$289$	1.09659e6	0.772321
$290$	−96217.7	−0.0671831
$291$	0	0
$292$	−1.53307e6	−1.05222
$293$	−2.27250e6	−1.54645	−0.773223	−	0.634134i	$-0.781357\pi$
$293$	−2.27250e6	−1.54645	−0.773223	+	0.634134i	$0.781357\pi$
$294$	0	0
$295$	−1.11123e6	−0.743445
$296$	639106.	0.423978
$297$	0	0
$298$	−261657.	−0.170684
$299$	775628.	0.501736
$300$	0	0
$301$	−4.77697e6	−3.03904
$302$	−350699.	−0.221267
$303$	0	0
$304$	−677154.	−0.420246
$305$	−432153.	−0.266004
$306$	0	0
$307$	−1.98871e6	−1.20428	−0.602138	−	0.798392i	$-0.705684\pi$
$307$	−1.98871e6	−1.20428	−0.602138	+	0.798392i	$0.705684\pi$
$308$	894426.	0.537239
$309$	0	0
$310$	−96673.7	−0.0571352
$311$	−50454.7	−0.0295802	−0.0147901	−	0.999891i	$-0.504708\pi$
$311$	−50454.7	−0.0295802	−0.0147901	+	0.999891i	$0.504708\pi$
$312$	0	0
$313$	2.84588e6	1.64193	0.820967	−	0.570975i	$-0.193435\pi$
$313$	2.84588e6	1.64193	0.820967	+	0.570975i	$0.193435\pi$
$314$	319789.	0.183037
$315$	0	0
$316$	−361946.	−0.203904
$317$	1.79904e6	1.00553	0.502763	−	0.864424i	$-0.332317\pi$
$317$	1.79904e6	1.00553	0.502763	+	0.864424i	$0.332317\pi$
$318$	0	0
$319$	−534985.	−0.294350
$320$	−705097.	−0.384923
$321$	0	0
$322$	610043.	0.327884
$323$	1.12855e6	0.601886
$324$	0	0
$325$	−163662.	−0.0859489
$326$	367899.	0.191728
$327$	0	0
$328$	257863.	0.132344
$329$	−1.95996e6	−0.998294
$330$	0	0
$331$	−3.48765e6	−1.74970	−0.874848	−	0.484397i	$-0.839039\pi$
$331$	−3.48765e6	−1.74970	−0.874848	+	0.484397i	$0.839039\pi$
$332$	649412.	0.323352
$333$	0	0
$334$	−464755.	−0.227960
$335$	1.28879e6	0.627436
$336$	0	0
$337$	1.53928e6	0.738319	0.369159	−	0.929366i	$-0.379646\pi$
$337$	1.53928e6	0.738319	0.369159	+	0.929366i	$0.379646\pi$
$338$	−263514.	−0.125462
$339$	0	0
$340$	1.23901e6	0.581271
$341$	−537520.	−0.250327
$342$	0	0
$343$	−5.29181e6	−2.42867
$344$	−1.11148e6	−0.506415
$345$	0	0
$346$	−244341.	−0.109725
$347$	−3.93686e6	−1.75520	−0.877600	−	0.479394i	$-0.840856\pi$
$347$	−3.93686e6	−1.75520	−0.877600	+	0.479394i	$0.840856\pi$
$348$	0	0
$349$	1.61586e6	0.710134	0.355067	−	0.934841i	$-0.384458\pi$
$349$	1.61586e6	0.710134	0.355067	+	0.934841i	$0.384458\pi$
$350$	−128723.	−0.0561675
$351$	0	0
$352$	313413.	0.134822
$353$	−2.96650e6	−1.26709	−0.633545	−	0.773706i	$-0.718401\pi$
$353$	−2.96650e6	−1.26709	−0.633545	+	0.773706i	$0.718401\pi$
$354$	0	0
$355$	−418247.	−0.176142
$356$	889245.	0.371875
$357$	0	0
$358$	238884.	0.0985099
$359$	−4.24769e6	−1.73947	−0.869735	−	0.493520i	$-0.835710\pi$
$359$	−4.24769e6	−1.73947	−0.869735	+	0.493520i	$0.835710\pi$
$360$	0	0
$361$	−1.96998e6	−0.795598
$362$	110504.	0.0443207
$363$	0	0
$364$	−1.93565e6	−0.765727
$365$	1.22676e6	0.481980
$366$	0	0
$367$	563502.	0.218389	0.109194	−	0.994020i	$-0.465173\pi$
$367$	563502.	0.218389	0.109194	+	0.994020i	$0.465173\pi$
$368$	−2.81932e6	−1.08524
$369$	0	0
$370$	−252642.	−0.0959403
$371$	−5.79352e6	−2.18528
$372$	0	0
$373$	3.70820e6	1.38004	0.690019	−	0.723791i	$-0.257602\pi$
$373$	3.70820e6	1.38004	0.690019	+	0.723791i	$0.257602\pi$
$374$	−167085.	−0.0617674
$375$	0	0
$376$	−456035.	−0.166352
$377$	1.15778e6	0.419538
$378$	0	0
$379$	−3.91792e6	−1.40106	−0.700531	−	0.713622i	$-0.747053\pi$
$379$	−3.91792e6	−1.40106	−0.700531	+	0.713622i	$0.747053\pi$
$380$	555661.	0.197402
$381$	0	0
$382$	−455601.	−0.159745
$383$	−3.45894e6	−1.20489	−0.602443	−	0.798162i	$-0.705806\pi$
$383$	−3.45894e6	−1.20489	−0.602443	+	0.798162i	$0.705806\pi$
$384$	0	0
$385$	−715718.	−0.246088
$386$	637253.	0.217693
$387$	0	0
$388$	1.57335e6	0.530574
$389$	3.65030e6	1.22308	0.611539	−	0.791214i	$-0.290551\pi$
$389$	3.65030e6	1.22308	0.611539	+	0.791214i	$0.290551\pi$
$390$	0	0
$391$	4.69871e6	1.55431
$392$	−2.15652e6	−0.708823
$393$	0	0
$394$	−451580.	−0.146553
$395$	289628.	0.0934004
$396$	0	0
$397$	−5.30688e6	−1.68991	−0.844953	−	0.534840i	$-0.820372\pi$
$397$	−5.30688e6	−1.68991	−0.844953	+	0.534840i	$0.820372\pi$
$398$	686329.	0.217182
$399$	0	0
$400$	594895.	0.185905
$401$	4.54397e6	1.41115	0.705577	−	0.708633i	$-0.250688\pi$
$401$	4.54397e6	1.41115	0.705577	+	0.708633i	$0.250688\pi$
$402$	0	0
$403$	1.16326e6	0.356792
$404$	−2.92368e6	−0.891204
$405$	0	0
$406$	910608.	0.274168
$407$	−1.40473e6	−0.420345
$408$	0	0
$409$	−658153.	−0.194544	−0.0972721	−	0.995258i	$-0.531012\pi$
$409$	−658153.	−0.194544	−0.0972721	+	0.995258i	$0.531012\pi$
$410$	−101935.	−0.0299476
$411$	0	0
$412$	−1.28813e6	−0.373867
$413$	1.05167e7	3.03393
$414$	0	0
$415$	−519658.	−0.148115
$416$	−678266.	−0.192162
$417$	0	0
$418$	−74932.7	−0.0209764
$419$	−642863.	−0.178889	−0.0894445	−	0.995992i	$-0.528509\pi$
$419$	−642863.	−0.178889	−0.0894445	+	0.995992i	$0.528509\pi$
$420$	0	0
$421$	1.59472e6	0.438509	0.219254	−	0.975668i	$-0.429638\pi$
$421$	1.59472e6	0.438509	0.219254	+	0.975668i	$0.429638\pi$
$422$	524537.	0.143382
$423$	0	0
$424$	−1.34801e6	−0.364148
$425$	−991456.	−0.266257
$426$	0	0
$427$	4.08991e6	1.08553
$428$	306853.	0.0809694
$429$	0	0
$430$	439375.	0.114595
$431$	−5.42169e6	−1.40586	−0.702929	−	0.711260i	$-0.748125\pi$
$431$	−5.42169e6	−1.40586	−0.702929	+	0.711260i	$0.748125\pi$
$432$	0	0
$433$	631386.	0.161836	0.0809180	−	0.996721i	$-0.474215\pi$
$433$	631386.	0.161836	0.0809180	+	0.996721i	$0.474215\pi$
$434$	914923.	0.233163
$435$	0	0
$436$	2.63037e6	0.662675
$437$	2.10723e6	0.527847
$438$	0	0
$439$	−3.54365e6	−0.877586	−0.438793	−	0.898588i	$-0.644594\pi$
$439$	−3.54365e6	−0.877586	−0.438793	+	0.898588i	$0.644594\pi$
$440$	−166530.	−0.0410072
$441$	0	0
$442$	361594.	0.0880371
$443$	93555.3	0.0226495	0.0113248	−	0.999936i	$-0.496395\pi$
$443$	93555.3	0.0226495	0.0113248	+	0.999936i	$0.496395\pi$
$444$	0	0
$445$	−711572.	−0.170341
$446$	−1.04262e6	−0.248193
$447$	0	0
$448$	6.67307e6	1.57083
$449$	3.09047e6	0.723450	0.361725	−	0.932285i	$-0.382188\pi$
$449$	3.09047e6	0.723450	0.361725	+	0.932285i	$0.382188\pi$
$450$	0	0
$451$	−566771.	−0.131210
$452$	−6.63689e6	−1.52798
$453$	0	0
$454$	−964059.	−0.219515
$455$	1.54891e6	0.350749
$456$	0	0
$457$	2.95911e6	0.662781	0.331391	−	0.943494i	$-0.392482\pi$
$457$	2.95911e6	0.662781	0.331391	+	0.943494i	$0.392482\pi$
$458$	691513.	0.154041
$459$	0	0
$460$	2.31349e6	0.509768
$461$	7.17368e6	1.57213	0.786067	−	0.618141i	$-0.212114\pi$
$461$	7.17368e6	1.57213	0.786067	+	0.618141i	$0.212114\pi$
$462$	0	0
$463$	5.68008e6	1.23141	0.615704	−	0.787977i	$-0.288872\pi$
$463$	5.68008e6	1.23141	0.615704	+	0.787977i	$0.288872\pi$
$464$	−4.20839e6	−0.907446
$465$	0	0
$466$	657460.	0.140251
$467$	3.21993e6	0.683210	0.341605	−	0.939844i	$-0.389030\pi$
$467$	3.21993e6	0.683210	0.341605	+	0.939844i	$0.389030\pi$
$468$	0	0
$469$	−1.21971e7	−2.56050
$470$	180273.	0.0376432
$471$	0	0
$472$	2.44698e6	0.505563
$473$	2.44299e6	0.502075
$474$	0	0
$475$	−444639.	−0.0904218
$476$	−1.17261e7	−2.37211
$477$	0	0
$478$	1.00186e6	0.200557
$479$	−8.30526e6	−1.65392	−0.826960	−	0.562261i	$-0.809932\pi$
$479$	−8.30526e6	−1.65392	−0.826960	+	0.562261i	$0.809932\pi$
$480$	0	0
$481$	3.04001e6	0.599118
$482$	−596717.	−0.116991
$483$	0	0
$484$	−457418.	−0.0887564
$485$	−1.25899e6	−0.243035
$486$	0	0
$487$	3.60963e6	0.689669	0.344834	−	0.938664i	$-0.387935\pi$
$487$	3.60963e6	0.689669	0.344834	+	0.938664i	$0.387935\pi$
$488$	951620.	0.180890
$489$	0	0
$490$	852483.	0.160397
$491$	−1.83609e6	−0.343708	−0.171854	−	0.985122i	$-0.554976\pi$
$491$	−1.83609e6	−0.343708	−0.171854	+	0.985122i	$0.554976\pi$
$492$	0	0
$493$	7.01374e6	1.29967
$494$	162164.	0.0298977
$495$	0	0
$496$	−4.22833e6	−0.771729
$497$	3.95830e6	0.718817
$498$	0	0
$499$	5.87611e6	1.05642	0.528212	−	0.849112i	$-0.322863\pi$
$499$	5.87611e6	1.05642	0.528212	+	0.849112i	$0.322863\pi$
$500$	−488160.	−0.0873248
$501$	0	0
$502$	−377441.	−0.0668483
$503$	−9.30612e6	−1.64002	−0.820009	−	0.572351i	$-0.806032\pi$
$503$	−9.30612e6	−1.64002	−0.820009	+	0.572351i	$0.806032\pi$
$504$	0	0
$505$	2.33953e6	0.408225
$506$	−311982.	−0.0541693
$507$	0	0
$508$	−8.81599e6	−1.51570
$509$	3.08301e6	0.527449	0.263725	−	0.964598i	$-0.415049\pi$
$509$	3.08301e6	0.527449	0.263725	+	0.964598i	$0.415049\pi$
$510$	0	0
$511$	−1.16101e7	−1.96691
$512$	4.14221e6	0.698324
$513$	0	0
$514$	−131873.	−0.0220164
$515$	1.03076e6	0.171253
$516$	0	0
$517$	1.00235e6	0.164927
$518$	2.39101e6	0.391523
$519$	0	0
$520$	360392.	0.0584476
$521$	−8.90068e6	−1.43658	−0.718289	−	0.695745i	$-0.755074\pi$
$521$	−8.90068e6	−1.43658	−0.718289	+	0.695745i	$0.755074\pi$
$522$	0	0
$523$	3.85500e6	0.616269	0.308135	−	0.951343i	$-0.400295\pi$
$523$	3.85500e6	0.616269	0.308135	+	0.951343i	$0.400295\pi$
$524$	−2.51775e6	−0.400575
$525$	0	0
$526$	−544966.	−0.0858825
$527$	7.04697e6	1.10529
$528$	0	0
$529$	2.33710e6	0.363109
$530$	532875.	0.0824017
$531$	0	0
$532$	−5.25879e6	−0.805577
$533$	1.22657e6	0.187013
$534$	0	0
$535$	−245543.	−0.0370888
$536$	−2.83797e6	−0.426674
$537$	0	0
$538$	−382005.	−0.0569001
$539$	4.73993e6	0.702749
$540$	0	0
$541$	−6.47766e6	−0.951535	−0.475768	−	0.879571i	$-0.657830\pi$
$541$	−6.47766e6	−0.951535	−0.475768	+	0.879571i	$0.657830\pi$
$542$	−784913.	−0.114769
$543$	0	0
$544$	−4.10889e6	−0.595289
$545$	−2.10482e6	−0.303545
$546$	0	0
$547$	1.13365e7	1.61998	0.809990	−	0.586443i	$-0.199472\pi$
$547$	1.13365e7	1.61998	0.809990	+	0.586443i	$0.199472\pi$
$548$	−9.39620e6	−1.33660
$549$	0	0
$550$	65830.1	0.00927936
$551$	3.14545e6	0.441371
$552$	0	0
$553$	−2.74105e6	−0.381158
$554$	785528.	0.108740
$555$	0	0
$556$	2.64624e6	0.363030
$557$	3.76282e6	0.513897	0.256948	−	0.966425i	$-0.417283\pi$
$557$	3.76282e6	0.513897	0.256948	+	0.966425i	$0.417283\pi$
$558$	0	0
$559$	−5.28695e6	−0.715608
$560$	−5.63010e6	−0.758658
$561$	0	0
$562$	−409623.	−0.0547071
$563$	1.31021e7	1.74208	0.871040	−	0.491211i	$-0.163446\pi$
$563$	1.31021e7	1.74208	0.871040	+	0.491211i	$0.163446\pi$
$564$	0	0
$565$	5.31083e6	0.699909
$566$	268489.	0.0352278
$567$	0	0
$568$	921000.	0.119781
$569$	8.95224e6	1.15918	0.579590	−	0.814908i	$-0.303213\pi$
$569$	8.95224e6	1.15918	0.579590	+	0.814908i	$0.303213\pi$
$570$	0	0
$571$	738601.	0.0948025	0.0474012	−	0.998876i	$-0.484906\pi$
$571$	738601.	0.0948025	0.0474012	+	0.998876i	$0.484906\pi$
$572$	989912.	0.126505
$573$	0	0
$574$	964712.	0.122213
$575$	−1.85125e6	−0.233505
$576$	0	0
$577$	−1.32997e7	−1.66304	−0.831522	−	0.555492i	$-0.812530\pi$
$577$	−1.32997e7	−1.66304	−0.831522	+	0.555492i	$0.812530\pi$
$578$	954557.	0.118845
$579$	0	0
$580$	3.45333e6	0.426254
$581$	4.91806e6	0.604441
$582$	0	0
$583$	2.96286e6	0.361028
$584$	−2.70139e6	−0.327760
$585$	0	0
$586$	−1.97817e6	−0.237968
$587$	549228.	0.0657896	0.0328948	−	0.999459i	$-0.489527\pi$
$587$	549228.	0.0657896	0.0328948	+	0.999459i	$0.489527\pi$
$588$	0	0
$589$	3.16036e6	0.375360
$590$	−967304.	−0.114402
$591$	0	0
$592$	−1.10501e7	−1.29587
$593$	6.12348e6	0.715092	0.357546	−	0.933896i	$-0.383614\pi$
$593$	6.12348e6	0.715092	0.357546	+	0.933896i	$0.383614\pi$
$594$	0	0
$595$	9.38318e6	1.08657
$596$	9.39109e6	1.08293
$597$	0	0
$598$	675169.	0.0772076
$599$	1.58398e7	1.80378	0.901888	−	0.431969i	$-0.142181\pi$
$599$	1.58398e7	1.80378	0.901888	+	0.431969i	$0.142181\pi$
$600$	0	0
$601$	435863.	0.0492226	0.0246113	−	0.999697i	$-0.492165\pi$
$601$	435863.	0.0492226	0.0246113	+	0.999697i	$0.492165\pi$
$602$	−4.15826e6	−0.467650
$603$	0	0
$604$	1.25869e7	1.40386
$605$	366025.	0.0406558
$606$	0	0
$607$	−1.14421e7	−1.26047	−0.630237	−	0.776403i	$-0.717042\pi$
$607$	−1.14421e7	−1.26047	−0.630237	+	0.776403i	$0.717042\pi$
$608$	−1.84272e6	−0.202162
$609$	0	0
$610$	−376181.	−0.0409328
$611$	−2.16921e6	−0.235070
$612$	0	0
$613$	8.57657e6	0.921855	0.460927	−	0.887438i	$-0.347517\pi$
$613$	8.57657e6	0.921855	0.460927	+	0.887438i	$0.347517\pi$
$614$	−1.73114e6	−0.185315
$615$	0	0
$616$	1.57604e6	0.167346
$617$	−5.84036e6	−0.617628	−0.308814	−	0.951122i	$-0.599932\pi$
$617$	−5.84036e6	−0.617628	−0.308814	+	0.951122i	$0.599932\pi$
$618$	0	0
$619$	−1.09233e7	−1.14585	−0.572926	−	0.819607i	$-0.694192\pi$
$619$	−1.09233e7	−1.14585	−0.572926	+	0.819607i	$0.694192\pi$
$620$	3.46970e6	0.362504
$621$	0	0
$622$	−43919.9	−0.00455182
$623$	6.73434e6	0.695145
$624$	0	0
$625$	390625.	0.0400000
$626$	2.47728e6	0.252662
$627$	0	0
$628$	−1.14775e7	−1.16131
$629$	1.84162e7	1.85598
$630$	0	0
$631$	1.30665e7	1.30643	0.653213	−	0.757174i	$-0.273421\pi$
$631$	1.30665e7	1.30643	0.653213	+	0.757174i	$0.273421\pi$
$632$	−637776.	−0.0635148
$633$	0	0
$634$	1.56603e6	0.154731
$635$	7.05454e6	0.694280
$636$	0	0
$637$	−1.02578e7	−1.00163
$638$	−465694.	−0.0452948
$639$	0	0
$640$	−2.68592e6	−0.259205
$641$	−1.14242e6	−0.109820	−0.0549100	−	0.998491i	$-0.517487\pi$
$641$	−1.14242e6	−0.109820	−0.0549100	+	0.998491i	$0.517487\pi$
$642$	0	0
$643$	−2.24945e6	−0.214560	−0.107280	−	0.994229i	$-0.534214\pi$
$643$	−2.24945e6	−0.214560	−0.107280	+	0.994229i	$0.534214\pi$
$644$	−2.18949e7	−2.08031
$645$	0	0
$646$	982381.	0.0926187
$647$	1.20553e7	1.13218	0.566092	−	0.824342i	$-0.308455\pi$
$647$	1.20553e7	1.13218	0.566092	+	0.824342i	$0.308455\pi$
$648$	0	0
$649$	−5.37835e6	−0.501230
$650$	−142465.	−0.0132259
$651$	0	0
$652$	−1.32042e7	−1.21645
$653$	9.71866e6	0.891915	0.445957	−	0.895054i	$-0.352863\pi$
$653$	9.71866e6	0.891915	0.445957	+	0.895054i	$0.352863\pi$
$654$	0	0
$655$	2.01469e6	0.183487
$656$	−4.45843e6	−0.404504
$657$	0	0
$658$	−1.70611e6	−0.153618
$659$	1.54218e6	0.138331	0.0691657	−	0.997605i	$-0.477966\pi$
$659$	1.54218e6	0.138331	0.0691657	+	0.997605i	$0.477966\pi$
$660$	0	0
$661$	−9.41171e6	−0.837847	−0.418924	−	0.908021i	$-0.637593\pi$
$661$	−9.41171e6	−0.837847	−0.418924	+	0.908021i	$0.637593\pi$
$662$	−3.03593e6	−0.269245
$663$	0	0
$664$	1.14431e6	0.100722
$665$	4.20807e6	0.369003
$666$	0	0
$667$	1.30961e7	1.13979
$668$	1.66804e7	1.44633
$669$	0	0
$670$	1.12186e6	0.0965502
$671$	−2.09162e6	−0.179340
$672$	0	0
$673$	−1803.50	−0.000153489 0	−7.67447e−5	−	1.00000i	$-0.500024\pi$
$673$	−1803.50	−0.000153489 0	−7.67447e−5		1.00000i	$0.500024\pi$
$674$	1.33992e6	0.113613
$675$	0	0
$676$	9.45774e6	0.796014
$677$	1.56916e7	1.31582	0.657909	−	0.753097i	$-0.271441\pi$
$677$	1.56916e7	1.31582	0.657909	+	0.753097i	$0.271441\pi$
$678$	0	0
$679$	1.19151e7	0.991801
$680$	2.18323e6	0.181062
$681$	0	0
$682$	−467901.	−0.0385206
$683$	7.57015e6	0.620945	0.310472	−	0.950582i	$-0.399513\pi$
$683$	7.57015e6	0.620945	0.310472	+	0.950582i	$0.399513\pi$
$684$	0	0
$685$	7.51882e6	0.612242
$686$	−4.60642e6	−0.373726
$687$	0	0
$688$	1.92175e7	1.54784
$689$	−6.41202e6	−0.514573
$690$	0	0
$691$	2.32066e6	0.184892	0.0924458	−	0.995718i	$-0.470532\pi$
$691$	2.32066e6	0.184892	0.0924458	+	0.995718i	$0.470532\pi$
$692$	8.76961e6	0.696169
$693$	0	0
$694$	−3.42696e6	−0.270091
$695$	−2.11752e6	−0.166290
$696$	0	0
$697$	7.43046e6	0.579340
$698$	1.40658e6	0.109276
$699$	0	0
$700$	4.61997e6	0.356364
$701$	−9.11425e6	−0.700529	−0.350265	−	0.936651i	$-0.613908\pi$
$701$	−9.11425e6	−0.700529	−0.350265	+	0.936651i	$0.613908\pi$
$702$	0	0
$703$	8.25911e6	0.630296
$704$	−3.41267e6	−0.259515
$705$	0	0
$706$	−2.58228e6	−0.194981
$707$	−2.21414e7	−1.66593
$708$	0	0
$709$	531982.	0.0397449	0.0198725	−	0.999803i	$-0.493674\pi$
$709$	531982.	0.0397449	0.0198725	+	0.999803i	$0.493674\pi$
$710$	−364076.	−0.0271048
$711$	0	0
$712$	1.56692e6	0.115837
$713$	1.31581e7	0.969327
$714$	0	0
$715$	−792126.	−0.0579467
$716$	−8.57375e6	−0.625012
$717$	0	0
$718$	−3.69753e6	−0.267671
$719$	1.99692e7	1.44058	0.720291	−	0.693672i	$-0.244008\pi$
$719$	1.99692e7	1.44058	0.720291	+	0.693672i	$0.244008\pi$
$720$	0	0
$721$	−9.75515e6	−0.698869
$722$	−1.71483e6	−0.122427
$723$	0	0
$724$	−3.96608e6	−0.281200
$725$	−2.76335e6	−0.195250
$726$	0	0
$727$	−4.30052e6	−0.301776	−0.150888	−	0.988551i	$-0.548213\pi$
$727$	−4.30052e6	−0.301776	−0.150888	+	0.988551i	$0.548213\pi$
$728$	−3.41076e6	−0.238519
$729$	0	0
$730$	1.06787e6	0.0741674
$731$	−3.20280e7	−2.21685
$732$	0	0
$733$	1.82364e7	1.25366	0.626828	−	0.779158i	$-0.284353\pi$
$733$	1.82364e7	1.25366	0.626828	+	0.779158i	$0.284353\pi$
$734$	490517.	0.0336058
$735$	0	0
$736$	−7.67214e6	−0.522062
$737$	6.23773e6	0.423017
$738$	0	0
$739$	5.93210e6	0.399574	0.199787	−	0.979839i	$-0.435975\pi$
$739$	5.93210e6	0.399574	0.199787	+	0.979839i	$0.435975\pi$
$740$	9.06752e6	0.608708
$741$	0	0
$742$	−5.04315e6	−0.336273
$743$	−7.32407e6	−0.486722	−0.243361	−	0.969936i	$-0.578250\pi$
$743$	−7.32407e6	−0.486722	−0.243361	+	0.969936i	$0.578250\pi$
$744$	0	0
$745$	−7.51473e6	−0.496047
$746$	3.22792e6	0.212361
$747$	0	0
$748$	5.99682e6	0.391893
$749$	2.32383e6	0.151356
$750$	0	0
$751$	1.41381e7	0.914723	0.457362	−	0.889281i	$-0.348794\pi$
$751$	1.41381e7	0.914723	0.457362	+	0.889281i	$0.348794\pi$
$752$	7.88482e6	0.508449
$753$	0	0
$754$	1.00782e6	0.0645588
$755$	−1.00720e7	−0.643054
$756$	0	0
$757$	−2.19228e7	−1.39045	−0.695225	−	0.718792i	$-0.744695\pi$
$757$	−2.19228e7	−1.39045	−0.695225	+	0.718792i	$0.744695\pi$
$758$	−3.41047e6	−0.215596
$759$	0	0
$760$	979115.	0.0614893
$761$	2.25432e7	1.41109	0.705543	−	0.708667i	$-0.250703\pi$
$761$	2.25432e7	1.41109	0.705543	+	0.708667i	$0.250703\pi$
$762$	0	0
$763$	1.99201e7	1.23874
$764$	1.63519e7	1.01352
$765$	0	0
$766$	−3.01094e6	−0.185409
$767$	1.16394e7	0.714404
$768$	0	0
$769$	2.81117e7	1.71424	0.857121	−	0.515116i	$-0.172251\pi$
$769$	2.81117e7	1.71424	0.857121	+	0.515116i	$0.172251\pi$
$770$	−623018.	−0.0378682
$771$	0	0
$772$	−2.28715e7	−1.38119
$773$	−1.67755e7	−1.00978	−0.504890	−	0.863184i	$-0.668467\pi$
$773$	−1.67755e7	−1.00978	−0.504890	+	0.863184i	$0.668467\pi$
$774$	0	0
$775$	−2.77644e6	−0.166048
$776$	2.77236e6	0.165270
$777$	0	0
$778$	3.17751e6	0.188208
$779$	3.33234e6	0.196746
$780$	0	0
$781$	−2.02432e6	−0.118755
$782$	4.09014e6	0.239178
$783$	0	0
$784$	3.72861e7	2.16649
$785$	9.18426e6	0.531949
$786$	0	0
$787$	1.15307e7	0.663622	0.331811	−	0.943346i	$-0.392340\pi$
$787$	1.15307e7	0.663622	0.331811	+	0.943346i	$0.392340\pi$
$788$	1.62076e7	0.929827
$789$	0	0
$790$	252116.	0.0143725
$791$	−5.02619e7	−2.85626
$792$	0	0
$793$	4.52653e6	0.255613
$794$	−4.61953e6	−0.260044
$795$	0	0
$796$	−2.46329e7	−1.37795
$797$	6.49963e6	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$797$	6.49963e6	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$798$	0	0
$799$	−1.31409e7	−0.728213
$800$	1.61887e6	0.0894307
$801$	0	0
$802$	3.95544e6	0.217149
$803$	5.93754e6	0.324951
$804$	0	0
$805$	1.75203e7	0.952909
$806$	1.01260e6	0.0549034
$807$	0	0
$808$	−5.15175e6	−0.277604
$809$	−2.37202e7	−1.27423	−0.637113	−	0.770770i	$-0.719872\pi$
$809$	−2.37202e7	−1.27423	−0.637113	+	0.770770i	$0.719872\pi$
$810$	0	0
$811$	1.67924e6	0.0896521	0.0448261	−	0.998995i	$-0.485727\pi$
$811$	1.67924e6	0.0896521	0.0448261	+	0.998995i	$0.485727\pi$
$812$	−3.26825e7	−1.73950
$813$	0	0
$814$	−1.22279e6	−0.0646829
$815$	1.05660e7	0.557206
$816$	0	0
$817$	−1.43636e7	−0.752850
$818$	−572909.	−0.0299366
$819$	0	0
$820$	3.65851e6	0.190007
$821$	−7.37762e6	−0.381996	−0.190998	−	0.981590i	$-0.561172\pi$
$821$	−7.37762e6	−0.381996	−0.190998	+	0.981590i	$0.561172\pi$
$822$	0	0
$823$	1.85719e7	0.955775	0.477888	−	0.878421i	$-0.341403\pi$
$823$	1.85719e7	0.955775	0.477888	+	0.878421i	$0.341403\pi$
$824$	−2.26978e6	−0.116457
$825$	0	0
$826$	9.15460e6	0.466863
$827$	−9.05431e6	−0.460353	−0.230177	−	0.973149i	$-0.573930\pi$
$827$	−9.05431e6	−0.460353	−0.230177	+	0.973149i	$0.573930\pi$
$828$	0	0
$829$	1.19762e7	0.605247	0.302623	−	0.953110i	$-0.402138\pi$
$829$	1.19762e7	0.605247	0.302623	+	0.953110i	$0.402138\pi$
$830$	−452353.	−0.0227920
$831$	0	0
$832$	7.38546e6	0.369887
$833$	−6.21413e7	−3.10290
$834$	0	0
$835$	−1.33477e7	−0.662504
$836$	2.68940e6	0.133088
$837$	0	0
$838$	−559600.	−0.0275276
$839$	2.57562e7	1.26321	0.631607	−	0.775289i	$-0.282396\pi$
$839$	2.57562e7	1.26321	0.631607	+	0.775289i	$0.282396\pi$
$840$	0	0
$841$	−962724.	−0.0469366
$842$	1.38817e6	0.0674780
$843$	0	0
$844$	−1.88261e7	−0.909711
$845$	−7.56806e6	−0.364622
$846$	0	0
$847$	−3.46407e6	−0.165912
$848$	2.33070e7	1.11300
$849$	0	0
$850$	−863044.	−0.0409718
$851$	3.43867e7	1.62767
$852$	0	0
$853$	1.82933e7	0.860835	0.430418	−	0.902630i	$-0.358366\pi$
$853$	1.82933e7	0.860835	0.430418	+	0.902630i	$0.358366\pi$
$854$	3.56019e6	0.167043
$855$	0	0
$856$	540698.	0.0252214
$857$	−87786.9	−0.00408298	−0.00204149	−	0.999998i	$-0.500650\pi$
$857$	−87786.9	−0.00408298	−0.00204149	+	0.999998i	$0.500650\pi$
$858$	0	0
$859$	2.19314e7	1.01410	0.507052	−	0.861915i	$-0.330735\pi$
$859$	2.19314e7	1.01410	0.507052	+	0.861915i	$0.330735\pi$
$860$	−1.57695e7	−0.727064
$861$	0	0
$862$	−4.71948e6	−0.216334
$863$	1.34254e7	0.613622	0.306811	−	0.951771i	$-0.400738\pi$
$863$	1.34254e7	0.613622	0.306811	+	0.951771i	$0.400738\pi$
$864$	0	0
$865$	−7.01742e6	−0.318887
$866$	549610.	0.0249035
$867$	0	0
$868$	−3.28373e7	−1.47934
$869$	1.40180e6	0.0629705
$870$	0	0
$871$	−1.34993e7	−0.602927
$872$	4.63491e6	0.206419
$873$	0	0
$874$	1.83430e6	0.0812255
$875$	−3.69689e6	−0.163236
$876$	0	0
$877$	−3.39784e7	−1.49178	−0.745889	−	0.666071i	$-0.767975\pi$
$877$	−3.39784e7	−1.49178	−0.745889	+	0.666071i	$0.767975\pi$
$878$	−3.08468e6	−0.135044
$879$	0	0
$880$	2.87929e6	0.125337
$881$	−2.05313e7	−0.891204	−0.445602	−	0.895231i	$-0.647010\pi$
$881$	−2.05313e7	−0.891204	−0.445602	+	0.895231i	$0.647010\pi$
$882$	0	0
$883$	4.34334e7	1.87466	0.937328	−	0.348448i	$-0.113291\pi$
$883$	4.34334e7	1.87466	0.937328	+	0.348448i	$0.113291\pi$
$884$	−1.29779e7	−0.558565
$885$	0	0
$886$	81438.1	0.00348532
$887$	−2.25883e7	−0.963996	−0.481998	−	0.876172i	$-0.660089\pi$
$887$	−2.25883e7	−0.963996	−0.481998	+	0.876172i	$0.660089\pi$
$888$	0	0
$889$	−6.67644e7	−2.83329
$890$	−619410.	−0.0262122
$891$	0	0
$892$	3.74205e7	1.57470
$893$	−5.89331e6	−0.247304
$894$	0	0
$895$	6.86070e6	0.286293
$896$	2.54197e7	1.05779
$897$	0	0
$898$	2.69020e6	0.111325
$899$	1.96411e7	0.810524
$900$	0	0
$901$	−3.88436e7	−1.59407
$902$	−493363.	−0.0201907
$903$	0	0
$904$	−1.16947e7	−0.475957
$905$	3.17365e6	0.128806
$906$	0	0
$907$	−2.30511e7	−0.930408	−0.465204	−	0.885204i	$-0.654019\pi$
$907$	−2.30511e7	−0.930408	−0.465204	+	0.885204i	$0.654019\pi$
$908$	3.46009e7	1.39275
$909$	0	0
$910$	1.34829e6	0.0539735
$911$	−1.47626e7	−0.589340	−0.294670	−	0.955599i	$-0.595210\pi$
$911$	−1.47626e7	−0.589340	−0.294670	+	0.955599i	$0.595210\pi$
$912$	0	0
$913$	−2.51515e6	−0.0998588
$914$	2.57585e6	0.101989
$915$	0	0
$916$	−2.48190e7	−0.977338
$917$	−1.90671e7	−0.748794
$918$	0	0
$919$	−1.78995e7	−0.699120	−0.349560	−	0.936914i	$-0.613669\pi$
$919$	−1.78995e7	−0.699120	−0.349560	+	0.936914i	$0.613669\pi$
$920$	4.07654e6	0.158790
$921$	0	0
$922$	6.24455e6	0.241921
$923$	4.38088e6	0.169261
$924$	0	0
$925$	−7.25581e6	−0.278825
$926$	4.94440e6	0.189490
$927$	0	0
$928$	−1.14522e7	−0.436533
$929$	−1.43860e6	−0.0546893	−0.0273446	−	0.999626i	$-0.508705\pi$
$929$	−1.43860e6	−0.0546893	−0.0273446	+	0.999626i	$0.508705\pi$
$930$	0	0
$931$	−2.78685e7	−1.05375
$932$	−2.35968e7	−0.889842
$933$	0	0
$934$	2.80289e6	0.105133
$935$	−4.79865e6	−0.179511
$936$	0	0
$937$	2.73893e7	1.01914	0.509568	−	0.860430i	$-0.329805\pi$
$937$	2.73893e7	1.01914	0.509568	+	0.860430i	$0.329805\pi$
$938$	−1.06174e7	−0.394012
$939$	0	0
$940$	−6.47015e6	−0.238833
$941$	4.13474e7	1.52221	0.761104	−	0.648629i	$-0.224657\pi$
$941$	4.13474e7	1.52221	0.761104	+	0.648629i	$0.224657\pi$
$942$	0	0
$943$	1.38742e7	0.508075
$944$	−4.23081e7	−1.54523
$945$	0	0
$946$	2.12658e6	0.0772597
$947$	−1.87351e7	−0.678862	−0.339431	−	0.940631i	$-0.610234\pi$
$947$	−1.87351e7	−0.678862	−0.339431	+	0.940631i	$0.610234\pi$
$948$	0	0
$949$	−1.28496e7	−0.463152
$950$	−387049.	−0.0139142
$951$	0	0
$952$	−2.06622e7	−0.738897
$953$	4.62088e6	0.164813	0.0824067	−	0.996599i	$-0.473739\pi$
$953$	4.62088e6	0.164813	0.0824067	+	0.996599i	$0.473739\pi$
$954$	0	0
$955$	−1.30847e7	−0.464255
$956$	−3.59576e7	−1.27247
$957$	0	0
$958$	−7.22957e6	−0.254506
$959$	−7.11584e7	−2.49850
$960$	0	0
$961$	−8.89502e6	−0.310698
$962$	2.64627e6	0.0921926
$963$	0	0
$964$	2.14167e7	0.742266
$965$	1.83018e7	0.632666
$966$	0	0
$967$	−4.11291e7	−1.41444	−0.707218	−	0.706996i	$-0.750050\pi$
$967$	−4.11291e7	−1.41444	−0.707218	+	0.706996i	$0.750050\pi$
$968$	−806004.	−0.0276471
$969$	0	0
$970$	−1.09593e6	−0.0373984
$971$	9.94366e6	0.338453	0.169227	−	0.985577i	$-0.445873\pi$
$971$	9.94366e6	0.338453	0.169227	+	0.985577i	$0.445873\pi$
$972$	0	0
$973$	2.00403e7	0.678611
$974$	3.14211e6	0.106127
$975$	0	0
$976$	−1.64535e7	−0.552882
$977$	−2.02974e7	−0.680304	−0.340152	−	0.940370i	$-0.610479\pi$
$977$	−2.02974e7	−0.680304	−0.340152	+	0.940370i	$0.610479\pi$
$978$	0	0
$979$	−3.44401e6	−0.114844
$980$	−3.05963e7	−1.01766
$981$	0	0
$982$	−1.59828e6	−0.0528900
$983$	−9.11502e6	−0.300866	−0.150433	−	0.988620i	$-0.548067\pi$
$983$	−9.11502e6	−0.300866	−0.150433	+	0.988620i	$0.548067\pi$
$984$	0	0
$985$	−1.29693e7	−0.425917
$986$	6.10532e6	0.199994
$987$	0	0
$988$	−5.82020e6	−0.189691
$989$	−5.98028e7	−1.94415
$990$	0	0
$991$	2.81678e7	0.911106	0.455553	−	0.890209i	$-0.349442\pi$
$991$	2.81678e7	0.911106	0.455553	+	0.890209i	$0.349442\pi$
$992$	−1.15064e7	−0.371246
$993$	0	0
$994$	3.44563e6	0.110612
$995$	1.97112e7	0.631183
$996$	0	0
$997$	−2.37470e6	−0.0756609	−0.0378304	−	0.999284i	$-0.512045\pi$
$997$	−2.37470e6	−0.0756609	−0.0378304	+	0.999284i	$0.512045\pi$
$998$	5.11504e6	0.162563
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.j.1.3		5
3.2	odd	2		165.6.a.f.1.3	✓	5
15.14	odd	2		825.6.a.l.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.f.1.3	✓	5	3.2	odd	2
495.6.a.j.1.3		5	1.1	even	1	trivial
825.6.a.l.1.3		5	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)

\(f(q)\)	\(=\)	\(q+0.870481 q^{2} -31.2423 q^{4} +25.0000 q^{5} -236.601 q^{7} -55.0512 q^{8} +O(q^{10})\)	\(q+0.870481 q^{2} -31.2423 q^{4} +25.0000 q^{5} -236.601 q^{7} -55.0512 q^{8} +21.7620 q^{10} +121.000 q^{11} -261.860 q^{13} -205.957 q^{14} +951.831 q^{16} -1586.33 q^{17} -711.422 q^{19} -781.057 q^{20} +105.328 q^{22} -2962.00 q^{23} +625.000 q^{25} -227.944 q^{26} +7391.95 q^{28} -4421.36 q^{29} -4442.31 q^{31} +2590.19 q^{32} -1380.87 q^{34} -5915.02 q^{35} -11609.3 q^{37} -619.279 q^{38} -1376.28 q^{40} -4684.06 q^{41} +20190.0 q^{43} -3780.31 q^{44} -2578.36 q^{46} +8283.84 q^{47} +39173.0 q^{49} +544.051 q^{50} +8181.09 q^{52} +24486.5 q^{53} +3025.00 q^{55} +13025.2 q^{56} -3848.71 q^{58} -44449.2 q^{59} -17286.1 q^{61} -3866.95 q^{62} -28203.9 q^{64} -6546.49 q^{65} +51551.5 q^{67} +49560.5 q^{68} -5148.91 q^{70} -16729.9 q^{71} +49070.5 q^{73} -10105.7 q^{74} +22226.4 q^{76} -28628.7 q^{77} +11585.1 q^{79} +23795.8 q^{80} -4077.38 q^{82} -20786.3 q^{83} -39658.2 q^{85} +17575.0 q^{86} -6661.19 q^{88} -28462.9 q^{89} +61956.2 q^{91} +92539.6 q^{92} +7210.93 q^{94} -17785.5 q^{95} -50359.6 q^{97} +34099.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8	\( 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8} + 50 q^{10} + 605 q^{11} + 1082 q^{13} - 432 q^{14} + 4770 q^{16} - 2174 q^{17} + 1632 q^{19} + 2050 q^{20} + 242 q^{22} - 1212 q^{23} + 3125 q^{25} - 5600 q^{26} + 16508 q^{28} - 82 q^{29} + 12120 q^{31} + 4864 q^{32} - 4524 q^{34} + 4600 q^{35} - 6530 q^{37} + 15132 q^{38} - 600 q^{40} - 6782 q^{41} + 46184 q^{43} + 9922 q^{44} + 12048 q^{46} + 11692 q^{47} + 34445 q^{49} + 1250 q^{50} + 50020 q^{52} - 10314 q^{53} + 15125 q^{55} - 54928 q^{56} + 75048 q^{58} - 92892 q^{59} + 106 q^{61} - 97160 q^{62} + 44550 q^{64} + 27050 q^{65} + 100476 q^{67} - 119928 q^{68} - 10800 q^{70} + 13772 q^{71} + 94154 q^{73} + 47924 q^{74} - 51524 q^{76} + 22264 q^{77} + 178744 q^{79} + 119250 q^{80} - 299848 q^{82} + 100116 q^{83} - 54350 q^{85} + 167704 q^{86} - 2904 q^{88} - 119410 q^{89} + 47536 q^{91} + 404560 q^{92} - 310288 q^{94} + 40800 q^{95} + 100682 q^{97} + 16434 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8 + 50 * q^10 + 605 * q^11 + 1082 * q^13 - 432 * q^14 + 4770 * q^16 - 2174 * q^17 + 1632 * q^19 + 2050 * q^20 + 242 * q^22 - 1212 * q^23 + 3125 * q^25 - 5600 * q^26 + 16508 * q^28 - 82 * q^29 + 12120 * q^31 + 4864 * q^32 - 4524 * q^34 + 4600 * q^35 - 6530 * q^37 + 15132 * q^38 - 600 * q^40 - 6782 * q^41 + 46184 * q^43 + 9922 * q^44 + 12048 * q^46 + 11692 * q^47 + 34445 * q^49 + 1250 * q^50 + 50020 * q^52 - 10314 * q^53 + 15125 * q^55 - 54928 * q^56 + 75048 * q^58 - 92892 * q^59 + 106 * q^61 - 97160 * q^62 + 44550 * q^64 + 27050 * q^65 + 100476 * q^67 - 119928 * q^68 - 10800 * q^70 + 13772 * q^71 + 94154 * q^73 + 47924 * q^74 - 51524 * q^76 + 22264 * q^77 + 178744 * q^79 + 119250 * q^80 - 299848 * q^82 + 100116 * q^83 - 54350 * q^85 + 167704 * q^86 - 2904 * q^88 - 119410 * q^89 + 47536 * q^91 + 404560 * q^92 - 310288 * q^94 + 40800 * q^95 + 100682 * q^97 + 16434 * q^98

Introduction

L-functions

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Database

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