LMFDB - Embedded newform 495.6.a.n.1.2

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 $ x^7 - x^6 - 209x^5 + 137x^4 + 12724x^3 - 1040x^2 - 218208*x - 8784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.12102$ 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)

$f(q)$	$=$	$q-9.12102 q^{2} +51.1931 q^{4} -25.0000 q^{5} +202.409 q^{7} -175.061 q^{8} +O(q^{10})$	\(q-9.12102 q^{2} +51.1931 q^{4} -25.0000 q^{5} +202.409 q^{7} -175.061 q^{8} +228.026 q^{10} -121.000 q^{11} -622.776 q^{13} -1846.18 q^{14} -41.4462 q^{16} +1589.54 q^{17} +1443.78 q^{19} -1279.83 q^{20} +1103.64 q^{22} +1559.87 q^{23} +625.000 q^{25} +5680.35 q^{26} +10362.0 q^{28} -4450.09 q^{29} +5066.67 q^{31} +5979.97 q^{32} -14498.2 q^{34} -5060.23 q^{35} +11660.1 q^{37} -13168.7 q^{38} +4376.52 q^{40} +4841.13 q^{41} -19248.8 q^{43} -6194.36 q^{44} -14227.7 q^{46} +13367.3 q^{47} +24162.5 q^{49} -5700.64 q^{50} -31881.8 q^{52} +4242.51 q^{53} +3025.00 q^{55} -35433.9 q^{56} +40589.4 q^{58} -13224.1 q^{59} +3994.05 q^{61} -46213.2 q^{62} -53217.2 q^{64} +15569.4 q^{65} -71148.0 q^{67} +81373.3 q^{68} +46154.5 q^{70} +72167.1 q^{71} -49113.3 q^{73} -106352. q^{74} +73911.5 q^{76} -24491.5 q^{77} +25276.5 q^{79} +1036.15 q^{80} -44156.1 q^{82} -5181.19 q^{83} -39738.4 q^{85} +175568. q^{86} +21182.3 q^{88} -74142.5 q^{89} -126056. q^{91} +79854.8 q^{92} -121924. q^{94} -36094.5 q^{95} +15523.6 q^{97} -220387. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8}+O(q^{10}) $ 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8	$ 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8} + 25 q^{10} - 847 q^{11} + 1418 q^{13} - 2548 q^{14} + 3699 q^{16} - 630 q^{17} + 2572 q^{19} - 4875 q^{20} + 121 q^{22} - 536 q^{23} + 4375 q^{25} + 7626 q^{26} - 11368 q^{28} + 1038 q^{29} + 1872 q^{31} + 7523 q^{32} + 20790 q^{34} + 24298 q^{37} + 18952 q^{38} + 3825 q^{40} + 17658 q^{41} + 7244 q^{43} - 23595 q^{44} + 31016 q^{46} - 34560 q^{47} + 78735 q^{49} - 625 q^{50} + 110222 q^{52} + 10214 q^{53} + 21175 q^{55} - 81124 q^{56} - 5718 q^{58} - 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} - 35450 q^{65} + 64908 q^{67} + 136010 q^{68} + 63700 q^{70} - 61816 q^{71} - 11890 q^{73} + 124050 q^{74} - 47216 q^{76} + 18928 q^{79} - 92475 q^{80} + 36398 q^{82} - 17492 q^{83} + 15750 q^{85} + 216688 q^{86} + 18513 q^{88} - 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 30800 q^{94} - 64300 q^{95} - 172546 q^{97} + 615271 q^{98}+O(q^{100}) $ 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8 + 25 * q^10 - 847 * q^11 + 1418 * q^13 - 2548 * q^14 + 3699 * q^16 - 630 * q^17 + 2572 * q^19 - 4875 * q^20 + 121 * q^22 - 536 * q^23 + 4375 * q^25 + 7626 * q^26 - 11368 * q^28 + 1038 * q^29 + 1872 * q^31 + 7523 * q^32 + 20790 * q^34 + 24298 * q^37 + 18952 * q^38 + 3825 * q^40 + 17658 * q^41 + 7244 * q^43 - 23595 * q^44 + 31016 * q^46 - 34560 * q^47 + 78735 * q^49 - 625 * q^50 + 110222 * q^52 + 10214 * q^53 + 21175 * q^55 - 81124 * q^56 - 5718 * q^58 - 94676 * q^59 + 69538 * q^61 + 4208 * q^62 + 112339 * q^64 - 35450 * q^65 + 64908 * q^67 + 136010 * q^68 + 63700 * q^70 - 61816 * q^71 - 11890 * q^73 + 124050 * q^74 - 47216 * q^76 + 18928 * q^79 - 92475 * q^80 + 36398 * q^82 - 17492 * q^83 + 15750 * q^85 + 216688 * q^86 + 18513 * q^88 - 25302 * q^89 + 3392 * q^91 + 27408 * q^92 - 30800 * q^94 - 64300 * q^95 - 172546 * q^97 + 615271 * 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 $

$2$	−9.12102	−1.61238	−0.806192	−	0.591654i	$-0.798475\pi$
$2$	−9.12102	−1.61238	−0.806192	+	0.591654i	$0.798475\pi$
$3$	0	0
$3$	0	0
$4$	51.1931	1.59978
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	202.409	1.56130	0.780648	−	0.624970i	$-0.214889\pi$
$7$	202.409	1.56130	0.780648	+	0.624970i	$0.214889\pi$
$8$	−175.061	−0.967083
$9$	0	0
$10$	228.026	0.721080
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−622.776	−1.02205	−0.511026	−	0.859565i	$-0.670735\pi$
$13$	−622.776	−1.02205	−0.511026	+	0.859565i	$0.670735\pi$
$14$	−1846.18	−2.51741
$15$	0	0
$16$	−41.4462	−0.0404748
$17$	1589.54	1.33398	0.666989	−	0.745068i	$-0.267583\pi$
$17$	1589.54	1.33398	0.666989	+	0.745068i	$0.267583\pi$
$18$	0	0
$19$	1443.78	0.917522	0.458761	−	0.888560i	$-0.348293\pi$
$19$	1443.78	0.917522	0.458761	+	0.888560i	$0.348293\pi$
$20$	−1279.83	−0.715445
$21$	0	0
$22$	1103.64	0.486152
$23$	1559.87	0.614851	0.307426	−	0.951572i	$-0.400532\pi$
$23$	1559.87	0.614851	0.307426	+	0.951572i	$0.400532\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	5680.35	1.64794
$27$	0	0
$28$	10362.0	2.49774
$29$	−4450.09	−0.982594	−0.491297	−	0.870992i	$-0.663477\pi$
$29$	−4450.09	−0.982594	−0.491297	+	0.870992i	$0.663477\pi$
$30$	0	0
$31$	5066.67	0.946930	0.473465	−	0.880813i	$-0.343003\pi$
$31$	5066.67	0.946930	0.473465	+	0.880813i	$0.343003\pi$
$32$	5979.97	1.03234
$33$	0	0
$34$	−14498.2	−2.15088
$35$	−5060.23	−0.698233
$36$	0	0
$37$	11660.1	1.40023	0.700113	−	0.714032i	$-0.253133\pi$
$37$	11660.1	1.40023	0.700113	+	0.714032i	$0.253133\pi$
$38$	−13168.7	−1.47940
$39$	0	0
$40$	4376.52	0.432493
$41$	4841.13	0.449767	0.224883	−	0.974386i	$-0.427800\pi$
$41$	4841.13	0.449767	0.224883	+	0.974386i	$0.427800\pi$
$42$	0	0
$43$	−19248.8	−1.58757	−0.793783	−	0.608201i	$-0.791891\pi$
$43$	−19248.8	−1.58757	−0.793783	+	0.608201i	$0.791891\pi$
$44$	−6194.36	−0.482353
$45$	0	0
$46$	−14227.7	−0.991377
$47$	13367.3	0.882674	0.441337	−	0.897341i	$-0.354504\pi$
$47$	13367.3	0.882674	0.441337	+	0.897341i	$0.354504\pi$
$48$	0	0
$49$	24162.5	1.43765
$50$	−5700.64	−0.322477
$51$	0	0
$52$	−31881.8	−1.63506
$53$	4242.51	0.207459	0.103730	−	0.994606i	$-0.466922\pi$
$53$	4242.51	0.207459	0.103730	+	0.994606i	$0.466922\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	−35433.9	−1.50990
$57$	0	0
$58$	40589.4	1.58432
$59$	−13224.1	−0.494581	−0.247291	−	0.968941i	$-0.579540\pi$
$59$	−13224.1	−0.494581	−0.247291	+	0.968941i	$0.579540\pi$
$60$	0	0
$61$	3994.05	0.137432	0.0687161	−	0.997636i	$-0.478110\pi$
$61$	3994.05	0.137432	0.0687161	+	0.997636i	$0.478110\pi$
$62$	−46213.2	−1.52682
$63$	0	0
$64$	−53217.2	−1.62406
$65$	15569.4	0.457076
$66$	0	0
$67$	−71148.0	−1.93631	−0.968157	−	0.250345i	$-0.919456\pi$
$67$	−71148.0	−1.93631	−0.968157	+	0.250345i	$0.919456\pi$
$68$	81373.3	2.13408
$69$	0	0
$70$	46154.5	1.12582
$71$	72167.1	1.69900	0.849500	−	0.527588i	$-0.176903\pi$
$71$	72167.1	1.69900	0.849500	+	0.527588i	$0.176903\pi$
$72$	0	0
$73$	−49113.3	−1.07868	−0.539340	−	0.842088i	$-0.681326\pi$
$73$	−49113.3	−1.07868	−0.539340	+	0.842088i	$0.681326\pi$
$74$	−106352.	−2.25770
$75$	0	0
$76$	73911.5	1.46784
$77$	−24491.5	−0.470749
$78$	0	0
$79$	25276.5	0.455669	0.227835	−	0.973700i	$-0.426835\pi$
$79$	25276.5	0.455669	0.227835	+	0.973700i	$0.426835\pi$
$80$	1036.15	0.0181009
$81$	0	0
$82$	−44156.1	−0.725197
$83$	−5181.19	−0.0825532	−0.0412766	−	0.999148i	$-0.513142\pi$
$83$	−5181.19	−0.0825532	−0.0412766	+	0.999148i	$0.513142\pi$
$84$	0	0
$85$	−39738.4	−0.596573
$86$	175568.	2.55977
$87$	0	0
$88$	21182.3	0.291586
$89$	−74142.5	−0.992184	−0.496092	−	0.868270i	$-0.665232\pi$
$89$	−74142.5	−0.992184	−0.496092	+	0.868270i	$0.665232\pi$
$90$	0	0
$91$	−126056.	−1.59573
$92$	79854.8	0.983629
$93$	0	0
$94$	−121924.	−1.42321
$95$	−36094.5	−0.410329
$96$	0	0
$97$	15523.6	0.167519	0.0837595	−	0.996486i	$-0.473307\pi$
$97$	15523.6	0.167519	0.0837595	+	0.996486i	$0.473307\pi$
$98$	−220387.	−2.31804
$99$	0	0
$100$	31995.7	0.319957
$101$	−53438.1	−0.521251	−0.260626	−	0.965440i	$-0.583929\pi$
$101$	−53438.1	−0.521251	−0.260626	+	0.965440i	$0.583929\pi$
$102$	0	0
$103$	49861.8	0.463100	0.231550	−	0.972823i	$-0.425620\pi$
$103$	49861.8	0.463100	0.231550	+	0.972823i	$0.425620\pi$
$104$	109024.	0.988410
$105$	0	0
$106$	−38696.0	−0.334504
$107$	−90332.4	−0.762754	−0.381377	−	0.924420i	$-0.624550\pi$
$107$	−90332.4	−0.762754	−0.381377	+	0.924420i	$0.624550\pi$
$108$	0	0
$109$	−142859.	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$109$	−142859.	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$110$	−27591.1	−0.217414
$111$	0	0
$112$	−8389.09	−0.0631931
$113$	−148783.	−1.09611	−0.548057	−	0.836441i	$-0.684632\pi$
$113$	−148783.	−1.09611	−0.548057	+	0.836441i	$0.684632\pi$
$114$	0	0
$115$	−38996.9	−0.274970
$116$	−227814.	−1.57194
$117$	0	0
$118$	120618.	0.797455
$119$	321737.	2.08273
$120$	0	0
$121$	14641.0	0.0909091
$122$	−36429.8	−0.221594
$123$	0	0
$124$	259378.	1.51488
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	285818.	1.57246	0.786230	−	0.617934i	$-0.212030\pi$
$127$	285818.	1.57246	0.786230	+	0.617934i	$0.212030\pi$
$128$	294036.	1.58627
$129$	0	0
$130$	−142009.	−0.736982
$131$	318082.	1.61942	0.809712	−	0.586827i	$-0.199623\pi$
$131$	318082.	1.61942	0.809712	+	0.586827i	$0.199623\pi$
$132$	0	0
$133$	292234.	1.43252
$134$	648943.	3.12208
$135$	0	0
$136$	−278266.	−1.29007
$137$	79864.1	0.363538	0.181769	−	0.983341i	$-0.441818\pi$
$137$	79864.1	0.363538	0.181769	+	0.983341i	$0.441818\pi$
$138$	0	0
$139$	−17479.4	−0.0767341	−0.0383671	−	0.999264i	$-0.512216\pi$
$139$	−17479.4	−0.0767341	−0.0383671	+	0.999264i	$0.512216\pi$
$140$	−259049.	−1.11702
$141$	0	0
$142$	−658238.	−2.73944
$143$	75355.9	0.308161
$144$	0	0
$145$	111252.	0.439429
$146$	447964.	1.73925
$147$	0	0
$148$	596917.	2.24006
$149$	398862.	1.47183	0.735914	−	0.677075i	$-0.236753\pi$
$149$	398862.	1.47183	0.735914	+	0.677075i	$0.236753\pi$
$150$	0	0
$151$	425340.	1.51808	0.759039	−	0.651046i	$-0.225669\pi$
$151$	425340.	1.51808	0.759039	+	0.651046i	$0.225669\pi$
$152$	−252749.	−0.887320
$153$	0	0
$154$	223388.	0.759028
$155$	−126667.	−0.423480
$156$	0	0
$157$	−466635.	−1.51087	−0.755437	−	0.655221i	$-0.772575\pi$
$157$	−466635.	−1.51087	−0.755437	+	0.655221i	$0.772575\pi$
$158$	−230548.	−0.734714
$159$	0	0
$160$	−149499.	−0.461678
$161$	315733.	0.959965
$162$	0	0
$163$	−53525.3	−0.157794	−0.0788969	−	0.996883i	$-0.525140\pi$
$163$	−53525.3	−0.157794	−0.0788969	+	0.996883i	$0.525140\pi$
$164$	247833.	0.719530
$165$	0	0
$166$	47257.7	0.133108
$167$	557685.	1.54738	0.773692	−	0.633562i	$-0.218408\pi$
$167$	557685.	1.54738	0.773692	+	0.633562i	$0.218408\pi$
$168$	0	0
$169$	16556.7	0.0445920
$170$	362455.	0.961905
$171$	0	0
$172$	−985404.	−2.53976
$173$	−471147.	−1.19685	−0.598427	−	0.801177i	$-0.704207\pi$
$173$	−471147.	−1.19685	−0.598427	+	0.801177i	$0.704207\pi$
$174$	0	0
$175$	126506.	0.312259
$176$	5014.98	0.0122036
$177$	0	0
$178$	676256.	1.59978
$179$	520092.	1.21324	0.606621	−	0.794991i	$-0.292524\pi$
$179$	520092.	1.21324	0.606621	+	0.794991i	$0.292524\pi$
$180$	0	0
$181$	390293.	0.885511	0.442756	−	0.896642i	$-0.354001\pi$
$181$	390293.	0.885511	0.442756	+	0.896642i	$0.354001\pi$
$182$	1.14976e6	2.57293
$183$	0	0
$184$	−273073.	−0.594612
$185$	−291502.	−0.626200
$186$	0	0
$187$	−192334.	−0.402209
$188$	684315.	1.41209
$189$	0	0
$190$	329219.	0.661607
$191$	−565539.	−1.12171	−0.560853	−	0.827916i	$-0.689527\pi$
$191$	−565539.	−1.12171	−0.560853	+	0.827916i	$0.689527\pi$
$192$	0	0
$193$	987933.	1.90912	0.954562	−	0.298012i	$-0.0963234\pi$
$193$	987933.	1.90912	0.954562	+	0.298012i	$0.0963234\pi$
$194$	−141592.	−0.270105
$195$	0	0
$196$	1.23696e6	2.29993
$197$	135081.	0.247987	0.123994	−	0.992283i	$-0.460430\pi$
$197$	135081.	0.247987	0.123994	+	0.992283i	$0.460430\pi$
$198$	0	0
$199$	1.03181e6	1.84700	0.923502	−	0.383593i	$-0.125313\pi$
$199$	1.03181e6	1.84700	0.923502	+	0.383593i	$0.125313\pi$
$200$	−109413.	−0.193417
$201$	0	0
$202$	487410.	0.840458
$203$	−900740.	−1.53412
$204$	0	0
$205$	−121028.	−0.201142
$206$	−454791.	−0.746695
$207$	0	0
$208$	25811.7	0.0413673
$209$	−174697.	−0.276643
$210$	0	0
$211$	48982.6	0.0757418	0.0378709	−	0.999283i	$-0.487942\pi$
$211$	48982.6	0.0757418	0.0378709	+	0.999283i	$0.487942\pi$
$212$	217187.	0.331890
$213$	0	0
$214$	823924.	1.22985
$215$	481219.	0.709981
$216$	0	0
$217$	1.02554e6	1.47844
$218$	1.30302e6	1.85699
$219$	0	0
$220$	154859.	0.215715
$221$	−989926.	−1.36340
$222$	0	0
$223$	−505459.	−0.680650	−0.340325	−	0.940308i	$-0.610537\pi$
$223$	−505459.	−0.680650	−0.340325	+	0.940308i	$0.610537\pi$
$224$	1.21040e6	1.61179
$225$	0	0
$226$	1.35705e6	1.76736
$227$	16245.1	0.0209247	0.0104623	−	0.999945i	$-0.496670\pi$
$227$	16245.1	0.0209247	0.0104623	+	0.999945i	$0.496670\pi$
$228$	0	0
$229$	1.27523e6	1.60694	0.803470	−	0.595346i	$-0.202985\pi$
$229$	1.27523e6	1.60694	0.803470	+	0.595346i	$0.202985\pi$
$230$	355691.	0.443357
$231$	0	0
$232$	779036.	0.950250
$233$	−1.06906e6	−1.29007	−0.645035	−	0.764153i	$-0.723157\pi$
$233$	−1.06906e6	−1.29007	−0.645035	+	0.764153i	$0.723157\pi$
$234$	0	0
$235$	−334183.	−0.394744
$236$	−676985.	−0.791223
$237$	0	0
$238$	−2.93457e6	−3.35817
$239$	416694.	0.471870	0.235935	−	0.971769i	$-0.424185\pi$
$239$	416694.	0.471870	0.235935	+	0.971769i	$0.424185\pi$
$240$	0	0
$241$	−393327.	−0.436225	−0.218113	−	0.975924i	$-0.569990\pi$
$241$	−393327.	−0.436225	−0.218113	+	0.975924i	$0.569990\pi$
$242$	−133541.	−0.146580
$243$	0	0
$244$	204468.	0.219862
$245$	−604064.	−0.642936
$246$	0	0
$247$	−899151.	−0.937756
$248$	−886974.	−0.915760
$249$	0	0
$250$	142516.	0.144216
$251$	−1.93167e6	−1.93530	−0.967649	−	0.252299i	$-0.918813\pi$
$251$	−1.93167e6	−1.93530	−0.967649	+	0.252299i	$0.918813\pi$
$252$	0	0
$253$	−188745.	−0.185385
$254$	−2.60695e6	−2.53541
$255$	0	0
$256$	−978962.	−0.933611
$257$	232562.	0.219637	0.109818	−	0.993952i	$-0.464973\pi$
$257$	232562.	0.219637	0.109818	+	0.993952i	$0.464973\pi$
$258$	0	0
$259$	2.36011e6	2.18617
$260$	797046.	0.731223
$261$	0	0
$262$	−2.90123e6	−2.61114
$263$	−707314.	−0.630555	−0.315277	−	0.949000i	$-0.602098\pi$
$263$	−707314.	−0.630555	−0.315277	+	0.949000i	$0.602098\pi$
$264$	0	0
$265$	−106063.	−0.0927787
$266$	−2.66548e6	−2.30978
$267$	0	0
$268$	−3.64229e6	−3.09768
$269$	−209541.	−0.176558	−0.0882791	−	0.996096i	$-0.528137\pi$
$269$	−209541.	−0.176558	−0.0882791	+	0.996096i	$0.528137\pi$
$270$	0	0
$271$	1.73625e6	1.43611	0.718057	−	0.695985i	$-0.245032\pi$
$271$	1.73625e6	1.43611	0.718057	+	0.695985i	$0.245032\pi$
$272$	−65880.2	−0.0539924
$273$	0	0
$274$	−728442.	−0.586163
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	804390.	0.629894	0.314947	−	0.949109i	$-0.398013\pi$
$277$	804390.	0.629894	0.314947	+	0.949109i	$0.398013\pi$
$278$	159430.	0.123725
$279$	0	0
$280$	885848.	0.675249
$281$	1.60563e6	1.21306	0.606528	−	0.795062i	$-0.292562\pi$
$281$	1.60563e6	1.21306	0.606528	+	0.795062i	$0.292562\pi$
$282$	0	0
$283$	−1.00408e6	−0.745247	−0.372623	−	0.927983i	$-0.621542\pi$
$283$	−1.00408e6	−0.745247	−0.372623	+	0.927983i	$0.621542\pi$
$284$	3.69446e6	2.71803
$285$	0	0
$286$	−687323.	−0.496873
$287$	979891.	0.702220
$288$	0	0
$289$	1.10677e6	0.779496
$290$	−1.01474e6	−0.708529
$291$	0	0
$292$	−2.51426e6	−1.72565
$293$	−78617.3	−0.0534994	−0.0267497	−	0.999642i	$-0.508516\pi$
$293$	−78617.3	−0.0534994	−0.0267497	+	0.999642i	$0.508516\pi$
$294$	0	0
$295$	330604.	0.221183
$296$	−2.04123e6	−1.35413
$297$	0	0
$298$	−3.63803e6	−2.37315
$299$	−971452.	−0.628410
$300$	0	0
$301$	−3.89613e6	−2.47866
$302$	−3.87954e6	−2.44772
$303$	0	0
$304$	−59839.1	−0.0371365
$305$	−99851.1	−0.0614615
$306$	0	0
$307$	2.87890e6	1.74334	0.871668	−	0.490097i	$-0.163039\pi$
$307$	2.87890e6	1.74334	0.871668	+	0.490097i	$0.163039\pi$
$308$	−1.25380e6	−0.753096
$309$	0	0
$310$	1.15533e6	0.682813
$311$	1.60084e6	0.938526	0.469263	−	0.883058i	$-0.344520\pi$
$311$	1.60084e6	0.938526	0.469263	+	0.883058i	$0.344520\pi$
$312$	0	0
$313$	−290927.	−0.167851	−0.0839253	−	0.996472i	$-0.526746\pi$
$313$	−290927.	−0.167851	−0.0839253	+	0.996472i	$0.526746\pi$
$314$	4.25619e6	2.43611
$315$	0	0
$316$	1.29398e6	0.728973
$317$	−826642.	−0.462029	−0.231015	−	0.972950i	$-0.574205\pi$
$317$	−826642.	−0.462029	−0.231015	+	0.972950i	$0.574205\pi$
$318$	0	0
$319$	538461.	0.296263
$320$	1.33043e6	0.726302
$321$	0	0
$322$	−2.87981e6	−1.54783
$323$	2.29494e6	1.22395
$324$	0	0
$325$	−389235.	−0.204411
$326$	488205.	0.254424
$327$	0	0
$328$	−847492.	−0.434962
$329$	2.70567e6	1.37812
$330$	0	0
$331$	962245.	0.482743	0.241371	−	0.970433i	$-0.422403\pi$
$331$	962245.	0.482743	0.241371	+	0.970433i	$0.422403\pi$
$332$	−265241.	−0.132067
$333$	0	0
$334$	−5.08666e6	−2.49498
$335$	1.77870e6	0.865946
$336$	0	0
$337$	−3.35494e6	−1.60920	−0.804601	−	0.593816i	$-0.797621\pi$
$337$	−3.35494e6	−1.60920	−0.804601	+	0.593816i	$0.797621\pi$
$338$	−151014.	−0.0718995
$339$	0	0
$340$	−2.03433e6	−0.954388
$341$	−613067.	−0.285510
$342$	0	0
$343$	1.48883e6	0.683298
$344$	3.36970e6	1.53531
$345$	0	0
$346$	4.29734e6	1.92979
$347$	3.13872e6	1.39936	0.699680	−	0.714457i	$-0.253326\pi$
$347$	3.13872e6	1.39936	0.699680	+	0.714457i	$0.253326\pi$
$348$	0	0
$349$	2.60178e6	1.14342	0.571711	−	0.820455i	$-0.306280\pi$
$349$	2.60178e6	1.14342	0.571711	+	0.820455i	$0.306280\pi$
$350$	−1.15386e6	−0.503482
$351$	0	0
$352$	−723577.	−0.311263
$353$	−984180.	−0.420376	−0.210188	−	0.977661i	$-0.567408\pi$
$353$	−984180.	−0.420376	−0.210188	+	0.977661i	$0.567408\pi$
$354$	0	0
$355$	−1.80418e6	−0.759816
$356$	−3.79559e6	−1.58728
$357$	0	0
$358$	−4.74377e6	−1.95621
$359$	466191.	0.190910	0.0954549	−	0.995434i	$-0.469569\pi$
$359$	466191.	0.190910	0.0954549	+	0.995434i	$0.469569\pi$
$360$	0	0
$361$	−391601.	−0.158153
$362$	−3.55987e6	−1.42778
$363$	0	0
$364$	−6.45318e6	−2.55282
$365$	1.22783e6	0.482400
$366$	0	0
$367$	−2033.56	−0.000788121 0	−0.000394060	−	1.00000i	$-0.500125\pi$
$367$	−2033.56	−0.000788121 0	−0.000394060		1.00000i	$0.500125\pi$
$368$	−64650.8	−0.0248860
$369$	0	0
$370$	2.65880e6	1.00968
$371$	858724.	0.323906
$372$	0	0
$373$	360261.	0.134074	0.0670372	−	0.997750i	$-0.478645\pi$
$373$	360261.	0.134074	0.0670372	+	0.997750i	$0.478645\pi$
$374$	1.75428e6	0.648516
$375$	0	0
$376$	−2.34010e6	−0.853619
$377$	2.77141e6	1.00426
$378$	0	0
$379$	5.16278e6	1.84623	0.923114	−	0.384526i	$-0.125635\pi$
$379$	5.16278e6	1.84623	0.923114	+	0.384526i	$0.125635\pi$
$380$	−1.84779e6	−0.656437
$381$	0	0
$382$	5.15829e6	1.80862
$383$	1.61501e6	0.562573	0.281287	−	0.959624i	$-0.409239\pi$
$383$	1.61501e6	0.562573	0.281287	+	0.959624i	$0.409239\pi$
$384$	0	0
$385$	612288.	0.210525
$386$	−9.01096e6	−3.07824
$387$	0	0
$388$	794703.	0.267994
$389$	283391.	0.0949538	0.0474769	−	0.998872i	$-0.484882\pi$
$389$	283391.	0.0949538	0.0474769	+	0.998872i	$0.484882\pi$
$390$	0	0
$391$	2.47948e6	0.820198
$392$	−4.22991e6	−1.39032
$393$	0	0
$394$	−1.23208e6	−0.399851
$395$	−631913.	−0.203782
$396$	0	0
$397$	1.75781e6	0.559754	0.279877	−	0.960036i	$-0.409706\pi$
$397$	1.75781e6	0.559754	0.279877	+	0.960036i	$0.409706\pi$
$398$	−9.41119e6	−2.97808
$399$	0	0
$400$	−25903.8	−0.00809495
$401$	2.62611e6	0.815552	0.407776	−	0.913082i	$-0.366304\pi$
$401$	2.62611e6	0.815552	0.407776	+	0.913082i	$0.366304\pi$
$402$	0	0
$403$	−3.15540e6	−0.967813
$404$	−2.73566e6	−0.833890
$405$	0	0
$406$	8.21568e6	2.47359
$407$	−1.41087e6	−0.422184
$408$	0	0
$409$	−785411.	−0.232161	−0.116080	−	0.993240i	$-0.537033\pi$
$409$	−785411.	−0.232161	−0.116080	+	0.993240i	$0.537033\pi$
$410$	1.10390e6	0.324318
$411$	0	0
$412$	2.55258e6	0.740860
$413$	−2.67669e6	−0.772188
$414$	0	0
$415$	129530.	0.0369189
$416$	−3.72418e6	−1.05511
$417$	0	0
$418$	1.59342e6	0.446056
$419$	4.44234e6	1.23617	0.618083	−	0.786113i	$-0.287910\pi$
$419$	4.44234e6	1.23617	0.618083	+	0.786113i	$0.287910\pi$
$420$	0	0
$421$	516137.	0.141925	0.0709626	−	0.997479i	$-0.477393\pi$
$421$	516137.	0.141925	0.0709626	+	0.997479i	$0.477393\pi$
$422$	−446772.	−0.122125
$423$	0	0
$424$	−742697.	−0.200630
$425$	993461.	0.266795
$426$	0	0
$427$	808432.	0.214572
$428$	−4.62440e6	−1.22024
$429$	0	0
$430$	−4.38921e6	−1.14476
$431$	366944.	0.0951494	0.0475747	−	0.998868i	$-0.484851\pi$
$431$	366944.	0.0951494	0.0475747	+	0.998868i	$0.484851\pi$
$432$	0	0
$433$	4.37319e6	1.12093	0.560466	−	0.828178i	$-0.310622\pi$
$433$	4.37319e6	1.12093	0.560466	+	0.828178i	$0.310622\pi$
$434$	−9.35398e6	−2.38381
$435$	0	0
$436$	−7.31339e6	−1.84248
$437$	2.25211e6	0.564140
$438$	0	0
$439$	3.65714e6	0.905692	0.452846	−	0.891589i	$-0.350409\pi$
$439$	3.65714e6	0.905692	0.452846	+	0.891589i	$0.350409\pi$
$440$	−529559.	−0.130401
$441$	0	0
$442$	9.02914e6	2.19832
$443$	134454.	0.0325509	0.0162755	−	0.999868i	$-0.494819\pi$
$443$	134454.	0.0325509	0.0162755	+	0.999868i	$0.494819\pi$
$444$	0	0
$445$	1.85356e6	0.443718
$446$	4.61030e6	1.09747
$447$	0	0
$448$	−1.07717e7	−2.53564
$449$	−7.14060e6	−1.67155	−0.835774	−	0.549073i	$-0.814981\pi$
$449$	−7.14060e6	−1.67155	−0.835774	+	0.549073i	$0.814981\pi$
$450$	0	0
$451$	−585777.	−0.135610
$452$	−7.61664e6	−1.75355
$453$	0	0
$454$	−148172.	−0.0337386
$455$	3.15139e6	0.713631
$456$	0	0
$457$	−1.01463e6	−0.227257	−0.113628	−	0.993523i	$-0.536247\pi$
$457$	−1.01463e6	−0.227257	−0.113628	+	0.993523i	$0.536247\pi$
$458$	−1.16314e7	−2.59100
$459$	0	0
$460$	−1.99637e6	−0.439892
$461$	−6.63887e6	−1.45493	−0.727464	−	0.686146i	$-0.759301\pi$
$461$	−6.63887e6	−1.45493	−0.727464	+	0.686146i	$0.759301\pi$
$462$	0	0
$463$	945627.	0.205006	0.102503	−	0.994733i	$-0.467315\pi$
$463$	945627.	0.205006	0.102503	+	0.994733i	$0.467315\pi$
$464$	184439.	0.0397703
$465$	0	0
$466$	9.75094e6	2.08009
$467$	4.61328e6	0.978852	0.489426	−	0.872045i	$-0.337206\pi$
$467$	4.61328e6	0.978852	0.489426	+	0.872045i	$0.337206\pi$
$468$	0	0
$469$	−1.44010e7	−3.02316
$470$	3.04810e6	0.636479
$471$	0	0
$472$	2.31503e6	0.478301
$473$	2.32910e6	0.478669
$474$	0	0
$475$	902362.	0.183504
$476$	1.64707e7	3.33193
$477$	0	0
$478$	−3.80067e6	−0.760836
$479$	1.58475e6	0.315589	0.157795	−	0.987472i	$-0.449562\pi$
$479$	1.58475e6	0.315589	0.157795	+	0.987472i	$0.449562\pi$
$480$	0	0
$481$	−7.26163e6	−1.43110
$482$	3.58754e6	0.703363
$483$	0	0
$484$	749518.	0.145435
$485$	−388091.	−0.0749168
$486$	0	0
$487$	−7.97563e6	−1.52385	−0.761925	−	0.647665i	$-0.775746\pi$
$487$	−7.97563e6	−1.52385	−0.761925	+	0.647665i	$0.775746\pi$
$488$	−699200.	−0.132908
$489$	0	0
$490$	5.50968e6	1.03666
$491$	−1.01117e6	−0.189288	−0.0946438	−	0.995511i	$-0.530171\pi$
$491$	−1.01117e6	−0.189288	−0.0946438	+	0.995511i	$0.530171\pi$
$492$	0	0
$493$	−7.07359e6	−1.31076
$494$	8.20118e6	1.51202
$495$	0	0
$496$	−209994.	−0.0383268
$497$	1.46073e7	2.65264
$498$	0	0
$499$	−735640.	−0.132256	−0.0661278	−	0.997811i	$-0.521065\pi$
$499$	−735640.	−0.132256	−0.0661278	+	0.997811i	$0.521065\pi$
$500$	−799892.	−0.143089
$501$	0	0
$502$	1.76188e7	3.12045
$503$	892146.	0.157223	0.0786114	−	0.996905i	$-0.474951\pi$
$503$	892146.	0.157223	0.0786114	+	0.996905i	$0.474951\pi$
$504$	0	0
$505$	1.33595e6	0.233111
$506$	1.72155e6	0.298911
$507$	0	0
$508$	1.46319e7	2.51560
$509$	−2.21531e6	−0.379000	−0.189500	−	0.981881i	$-0.560687\pi$
$509$	−2.21531e6	−0.379000	−0.189500	+	0.981881i	$0.560687\pi$
$510$	0	0
$511$	−9.94100e6	−1.68414
$512$	−480026.	−0.0809263
$513$	0	0
$514$	−2.12120e6	−0.354139
$515$	−1.24654e6	−0.207105
$516$	0	0
$517$	−1.61745e6	−0.266136
$518$	−2.15267e7	−3.52494
$519$	0	0
$520$	−2.72559e6	−0.442030
$521$	1.01241e7	1.63404	0.817022	−	0.576607i	$-0.195624\pi$
$521$	1.01241e7	1.63404	0.817022	+	0.576607i	$0.195624\pi$
$522$	0	0
$523$	7.94591e6	1.27025	0.635125	−	0.772409i	$-0.280948\pi$
$523$	7.94591e6	1.27025	0.635125	+	0.772409i	$0.280948\pi$
$524$	1.62836e7	2.59073
$525$	0	0
$526$	6.45143e6	1.01670
$527$	8.05366e6	1.26318
$528$	0	0
$529$	−4.00314e6	−0.621958
$530$	967401.	0.149595
$531$	0	0
$532$	1.49604e7	2.29173
$533$	−3.01494e6	−0.459686
$534$	0	0
$535$	2.25831e6	0.341114
$536$	1.24552e7	1.87258
$537$	0	0
$538$	1.91123e6	0.284680
$539$	−2.92367e6	−0.433467
$540$	0	0
$541$	1.12440e7	1.65168	0.825842	−	0.563902i	$-0.190700\pi$
$541$	1.12440e7	1.65168	0.825842	+	0.563902i	$0.190700\pi$
$542$	−1.58364e7	−2.31557
$543$	0	0
$544$	9.50539e6	1.37712
$545$	3.57147e6	0.515058
$546$	0	0
$547$	2.85517e6	0.408003	0.204001	−	0.978971i	$-0.434605\pi$
$547$	2.85517e6	0.408003	0.204001	+	0.978971i	$0.434605\pi$
$548$	4.08849e6	0.581583
$549$	0	0
$550$	689778.	0.0972305
$551$	−6.42495e6	−0.901552
$552$	0	0
$553$	5.11621e6	0.711435
$554$	−7.33687e6	−1.01563
$555$	0	0
$556$	−894823.	−0.122758
$557$	1.02655e7	1.40198	0.700991	−	0.713170i	$-0.252741\pi$
$557$	1.02655e7	1.40198	0.700991	+	0.713170i	$0.252741\pi$
$558$	0	0
$559$	1.19877e7	1.62258
$560$	209727.	0.0282608
$561$	0	0
$562$	−1.46450e7	−1.95591
$563$	5.28621e6	0.702868	0.351434	−	0.936213i	$-0.385694\pi$
$563$	5.28621e6	0.702868	0.351434	+	0.936213i	$0.385694\pi$
$564$	0	0
$565$	3.71956e6	0.490197
$566$	9.15820e6	1.20162
$567$	0	0
$568$	−1.26336e7	−1.64307
$569$	−7.85682e6	−1.01734	−0.508670	−	0.860962i	$-0.669863\pi$
$569$	−7.85682e6	−1.01734	−0.508670	+	0.860962i	$0.669863\pi$
$570$	0	0
$571$	−1.45219e7	−1.86395	−0.931973	−	0.362528i	$-0.881914\pi$
$571$	−1.45219e7	−1.86395	−0.931973	+	0.362528i	$0.881914\pi$
$572$	3.85770e6	0.492990
$573$	0	0
$574$	−8.93761e6	−1.13225
$575$	974921.	0.122970
$576$	0	0
$577$	1.19454e7	1.49370	0.746848	−	0.664995i	$-0.231566\pi$
$577$	1.19454e7	1.49370	0.746848	+	0.664995i	$0.231566\pi$
$578$	−1.00949e7	−1.25685
$579$	0	0
$580$	5.69535e6	0.702992
$581$	−1.04872e6	−0.128890
$582$	0	0
$583$	−513344.	−0.0625514
$584$	8.59781e6	1.04317
$585$	0	0
$586$	717070.	0.0862616
$587$	−1.10016e7	−1.31784	−0.658918	−	0.752215i	$-0.728986\pi$
$587$	−1.10016e7	−1.31784	−0.658918	+	0.752215i	$0.728986\pi$
$588$	0	0
$589$	7.31515e6	0.868830
$590$	−3.01544e6	−0.356633
$591$	0	0
$592$	−483266.	−0.0566738
$593$	1.63434e7	1.90856	0.954279	−	0.298918i	$-0.0966257\pi$
$593$	1.63434e7	1.90856	0.954279	+	0.298918i	$0.0966257\pi$
$594$	0	0
$595$	−8.04343e6	−0.931427
$596$	2.04190e7	2.35461
$597$	0	0
$598$	8.86064e6	1.01324
$599$	−1.17731e7	−1.34068	−0.670338	−	0.742055i	$-0.733851\pi$
$599$	−1.17731e7	−1.34068	−0.670338	+	0.742055i	$0.733851\pi$
$600$	0	0
$601$	−8.69665e6	−0.982122	−0.491061	−	0.871125i	$-0.663391\pi$
$601$	−8.69665e6	−0.982122	−0.491061	+	0.871125i	$0.663391\pi$
$602$	3.55367e7	3.99655
$603$	0	0
$604$	2.17745e7	2.42860
$605$	−366025.	−0.0406558
$606$	0	0
$607$	6.30094e6	0.694118	0.347059	−	0.937843i	$-0.387180\pi$
$607$	6.30094e6	0.694118	0.347059	+	0.937843i	$0.387180\pi$
$608$	8.63376e6	0.947199
$609$	0	0
$610$	910745.	0.0990996
$611$	−8.32486e6	−0.902139
$612$	0	0
$613$	−4.22263e6	−0.453870	−0.226935	−	0.973910i	$-0.572871\pi$
$613$	−4.22263e6	−0.453870	−0.226935	+	0.973910i	$0.572871\pi$
$614$	−2.62585e7	−2.81093
$615$	0	0
$616$	4.28750e6	0.455253
$617$	−9.34435e6	−0.988180	−0.494090	−	0.869411i	$-0.664499\pi$
$617$	−9.34435e6	−0.988180	−0.494090	+	0.869411i	$0.664499\pi$
$618$	0	0
$619$	1.74673e7	1.83231	0.916157	−	0.400819i	$-0.131274\pi$
$619$	1.74673e7	1.83231	0.916157	+	0.400819i	$0.131274\pi$
$620$	−6.48446e6	−0.677477
$621$	0	0
$622$	−1.46013e7	−1.51327
$623$	−1.50071e7	−1.54909
$624$	0	0
$625$	390625.	0.0400000
$626$	2.65355e6	0.270640
$627$	0	0
$628$	−2.38885e7	−2.41707
$629$	1.85342e7	1.86787
$630$	0	0
$631$	−1.88200e6	−0.188168	−0.0940842	−	0.995564i	$-0.529992\pi$
$631$	−1.88200e6	−0.188168	−0.0940842	+	0.995564i	$0.529992\pi$
$632$	−4.42493e6	−0.440670
$633$	0	0
$634$	7.53983e6	0.744969
$635$	−7.14544e6	−0.703225
$636$	0	0
$637$	−1.50478e7	−1.46935
$638$	−4.91132e6	−0.477690
$639$	0	0
$640$	−7.35091e6	−0.709400
$641$	4.66560e6	0.448500	0.224250	−	0.974532i	$-0.428007\pi$
$641$	4.66560e6	0.448500	0.224250	+	0.974532i	$0.428007\pi$
$642$	0	0
$643$	−6.38952e6	−0.609454	−0.304727	−	0.952440i	$-0.598565\pi$
$643$	−6.38952e6	−0.609454	−0.304727	+	0.952440i	$0.598565\pi$
$644$	1.61634e7	1.53574
$645$	0	0
$646$	−2.09322e7	−1.97349
$647$	−3.77808e6	−0.354822	−0.177411	−	0.984137i	$-0.556772\pi$
$647$	−3.77808e6	−0.354822	−0.177411	+	0.984137i	$0.556772\pi$
$648$	0	0
$649$	1.60012e6	0.149122
$650$	3.55022e6	0.329588
$651$	0	0
$652$	−2.74012e6	−0.252436
$653$	−1.75441e7	−1.61008	−0.805040	−	0.593221i	$-0.797856\pi$
$653$	−1.75441e7	−1.61008	−0.805040	+	0.593221i	$0.797856\pi$
$654$	0	0
$655$	−7.95205e6	−0.724229
$656$	−200646.	−0.0182042
$657$	0	0
$658$	−2.46785e7	−2.22205
$659$	1.84387e7	1.65393	0.826964	−	0.562255i	$-0.190066\pi$
$659$	1.84387e7	1.65393	0.826964	+	0.562255i	$0.190066\pi$
$660$	0	0
$661$	−1.01588e7	−0.904354	−0.452177	−	0.891928i	$-0.649352\pi$
$661$	−1.01588e7	−0.904354	−0.452177	+	0.891928i	$0.649352\pi$
$662$	−8.77666e6	−0.778367
$663$	0	0
$664$	907022.	0.0798358
$665$	−7.30586e6	−0.640645
$666$	0	0
$667$	−6.94159e6	−0.604149
$668$	2.85496e7	2.47548
$669$	0	0
$670$	−1.62236e7	−1.39624
$671$	−483280.	−0.0414374
$672$	0	0
$673$	−1.00569e6	−0.0855911	−0.0427955	−	0.999084i	$-0.513626\pi$
$673$	−1.00569e6	−0.0855911	−0.0427955	+	0.999084i	$0.513626\pi$
$674$	3.06005e7	2.59465
$675$	0	0
$676$	847589.	0.0713376
$677$	−761267.	−0.0638360	−0.0319180	−	0.999490i	$-0.510162\pi$
$677$	−761267.	−0.0638360	−0.0319180	+	0.999490i	$0.510162\pi$
$678$	0	0
$679$	3.14213e6	0.261547
$680$	6.95664e6	0.576935
$681$	0	0
$682$	5.59180e6	0.460352
$683$	2.20421e7	1.80802	0.904008	−	0.427516i	$-0.140611\pi$
$683$	2.20421e7	1.80802	0.904008	+	0.427516i	$0.140611\pi$
$684$	0	0
$685$	−1.99660e6	−0.162579
$686$	−1.35797e7	−1.10174
$687$	0	0
$688$	797787.	0.0642563
$689$	−2.64213e6	−0.212035
$690$	0	0
$691$	1.86615e7	1.48680	0.743399	−	0.668849i	$-0.233213\pi$
$691$	1.86615e7	1.48680	0.743399	+	0.668849i	$0.233213\pi$
$692$	−2.41195e7	−1.91471
$693$	0	0
$694$	−2.86284e7	−2.25631
$695$	436984.	0.0343166
$696$	0	0
$697$	7.69516e6	0.599979
$698$	−2.37309e7	−1.84364
$699$	0	0
$700$	6.47623e6	0.499548
$701$	2.18785e7	1.68160	0.840799	−	0.541347i	$-0.182085\pi$
$701$	2.18785e7	1.68160	0.840799	+	0.541347i	$0.182085\pi$
$702$	0	0
$703$	1.68346e7	1.28474
$704$	6.43928e6	0.489673
$705$	0	0
$706$	8.97673e6	0.677808
$707$	−1.08164e7	−0.813828
$708$	0	0
$709$	−4.96055e6	−0.370608	−0.185304	−	0.982681i	$-0.559327\pi$
$709$	−4.96055e6	−0.370608	−0.185304	+	0.982681i	$0.559327\pi$
$710$	1.64560e7	1.22512
$711$	0	0
$712$	1.29794e7	0.959524
$713$	7.90336e6	0.582221
$714$	0	0
$715$	−1.88390e6	−0.137814
$716$	2.66251e7	1.94093
$717$	0	0
$718$	−4.25214e6	−0.307820
$719$	−1.38634e7	−1.00011	−0.500056	−	0.865993i	$-0.666687\pi$
$719$	−1.38634e7	−1.00011	−0.500056	+	0.865993i	$0.666687\pi$
$720$	0	0
$721$	1.00925e7	0.723037
$722$	3.57181e6	0.255003
$723$	0	0
$724$	1.99803e7	1.41663
$725$	−2.78131e6	−0.196519
$726$	0	0
$727$	2.24625e7	1.57624	0.788120	−	0.615522i	$-0.211055\pi$
$727$	2.24625e7	1.57624	0.788120	+	0.615522i	$0.211055\pi$
$728$	2.20674e7	1.54320
$729$	0	0
$730$	−1.11991e7	−0.777814
$731$	−3.05966e7	−2.11778
$732$	0	0
$733$	1.39744e6	0.0960668	0.0480334	−	0.998846i	$-0.484705\pi$
$733$	1.39744e6	0.0960668	0.0480334	+	0.998846i	$0.484705\pi$
$734$	18548.2	0.00127075
$735$	0	0
$736$	9.32801e6	0.634738
$737$	8.60891e6	0.583820
$738$	0	0
$739$	−4.73034e6	−0.318626	−0.159313	−	0.987228i	$-0.550928\pi$
$739$	−4.73034e6	−0.318626	−0.159313	+	0.987228i	$0.550928\pi$
$740$	−1.49229e7	−1.00178
$741$	0	0
$742$	−7.83244e6	−0.522261
$743$	1.86274e6	0.123788	0.0618942	−	0.998083i	$-0.480286\pi$
$743$	1.86274e6	0.123788	0.0618942	+	0.998083i	$0.480286\pi$
$744$	0	0
$745$	−9.97155e6	−0.658221
$746$	−3.28595e6	−0.216179
$747$	0	0
$748$	−9.84617e6	−0.643448
$749$	−1.82841e7	−1.19088
$750$	0	0
$751$	1.33251e7	0.862129	0.431065	−	0.902321i	$-0.358138\pi$
$751$	1.33251e7	0.862129	0.431065	+	0.902321i	$0.358138\pi$
$752$	−554025.	−0.0357260
$753$	0	0
$754$	−2.52781e7	−1.61926
$755$	−1.06335e7	−0.678905
$756$	0	0
$757$	−2.58320e7	−1.63839	−0.819196	−	0.573514i	$-0.805580\pi$
$757$	−2.58320e7	−1.63839	−0.819196	+	0.573514i	$0.805580\pi$
$758$	−4.70898e7	−2.97683
$759$	0	0
$760$	6.31872e6	0.396822
$761$	656825.	0.0411138	0.0205569	−	0.999789i	$-0.493456\pi$
$761$	656825.	0.0411138	0.0205569	+	0.999789i	$0.493456\pi$
$762$	0	0
$763$	−2.89160e7	−1.79815
$764$	−2.89517e7	−1.79449
$765$	0	0
$766$	−1.47306e7	−0.907085
$767$	8.23568e6	0.505488
$768$	0	0
$769$	−1.46980e7	−0.896278	−0.448139	−	0.893964i	$-0.647913\pi$
$769$	−1.46980e7	−0.896278	−0.448139	+	0.893964i	$0.647913\pi$
$770$	−5.58470e6	−0.339448
$771$	0	0
$772$	5.05753e7	3.05419
$773$	1.21689e7	0.732489	0.366244	−	0.930519i	$-0.380643\pi$
$773$	1.21689e7	0.732489	0.366244	+	0.930519i	$0.380643\pi$
$774$	0	0
$775$	3.16667e6	0.189386
$776$	−2.71758e6	−0.162005
$777$	0	0
$778$	−2.58482e6	−0.153102
$779$	6.98953e6	0.412671
$780$	0	0
$781$	−8.73222e6	−0.512268
$782$	−2.26154e7	−1.32247
$783$	0	0
$784$	−1.00144e6	−0.0581885
$785$	1.16659e7	0.675683
$786$	0	0
$787$	2.21794e7	1.27648	0.638238	−	0.769839i	$-0.279664\pi$
$787$	2.21794e7	1.27648	0.638238	+	0.769839i	$0.279664\pi$
$788$	6.91523e6	0.396726
$789$	0	0
$790$	5.76370e6	0.328574
$791$	−3.01150e7	−1.71136
$792$	0	0
$793$	−2.48740e6	−0.140463
$794$	−1.60331e7	−0.902538
$795$	0	0
$796$	5.28217e7	2.95481
$797$	−1.98231e7	−1.10542	−0.552708	−	0.833375i	$-0.686405\pi$
$797$	−1.98231e7	−1.10542	−0.552708	+	0.833375i	$0.686405\pi$
$798$	0	0
$799$	2.12479e7	1.17747
$800$	3.73748e6	0.206469
$801$	0	0
$802$	−2.39528e7	−1.31498
$803$	5.94271e6	0.325234
$804$	0	0
$805$	−7.89333e6	−0.429309
$806$	2.87805e7	1.56049
$807$	0	0
$808$	9.35490e6	0.504093
$809$	2.68062e7	1.44000	0.720001	−	0.693973i	$-0.244141\pi$
$809$	2.68062e7	1.44000	0.720001	+	0.693973i	$0.244141\pi$
$810$	0	0
$811$	−9.50847e6	−0.507643	−0.253821	−	0.967251i	$-0.581688\pi$
$811$	−9.50847e6	−0.507643	−0.253821	+	0.967251i	$0.581688\pi$
$812$	−4.61117e7	−2.45426
$813$	0	0
$814$	1.28686e7	0.680723
$815$	1.33813e6	0.0705675
$816$	0	0
$817$	−2.77910e7	−1.45663
$818$	7.16375e6	0.374332
$819$	0	0
$820$	−6.19582e6	−0.321784
$821$	1.64798e7	0.853282	0.426641	−	0.904421i	$-0.359697\pi$
$821$	1.64798e7	0.853282	0.426641	+	0.904421i	$0.359697\pi$
$822$	0	0
$823$	−1.81437e7	−0.933740	−0.466870	−	0.884326i	$-0.654618\pi$
$823$	−1.81437e7	−0.933740	−0.466870	+	0.884326i	$0.654618\pi$
$824$	−8.72884e6	−0.447856
$825$	0	0
$826$	2.44142e7	1.24506
$827$	1.02430e7	0.520791	0.260395	−	0.965502i	$-0.416147\pi$
$827$	1.02430e7	0.520791	0.260395	+	0.965502i	$0.416147\pi$
$828$	0	0
$829$	5.34596e6	0.270171	0.135086	−	0.990834i	$-0.456869\pi$
$829$	5.34596e6	0.270171	0.135086	+	0.990834i	$0.456869\pi$
$830$	−1.18144e6	−0.0595275
$831$	0	0
$832$	3.31424e7	1.65988
$833$	3.84073e7	1.91779
$834$	0	0
$835$	−1.39421e7	−0.692011
$836$	−8.94329e6	−0.442570
$837$	0	0
$838$	−4.05187e7	−1.99317
$839$	−1.55327e7	−0.761804	−0.380902	−	0.924615i	$-0.624386\pi$
$839$	−1.55327e7	−0.761804	−0.380902	+	0.924615i	$0.624386\pi$
$840$	0	0
$841$	−707822.	−0.0345092
$842$	−4.70769e6	−0.228838
$843$	0	0
$844$	2.50757e6	0.121171
$845$	−413918.	−0.0199422
$846$	0	0
$847$	2.96348e6	0.141936
$848$	−175836.	−0.00839687
$849$	0	0
$850$	−9.06138e6	−0.430177
$851$	1.81883e7	0.860930
$852$	0	0
$853$	−3.48487e7	−1.63989	−0.819943	−	0.572446i	$-0.805995\pi$
$853$	−3.48487e7	−1.63989	−0.819943	+	0.572446i	$0.805995\pi$
$854$	−7.37373e6	−0.345973
$855$	0	0
$856$	1.58137e7	0.737646
$857$	2.41325e7	1.12241	0.561203	−	0.827678i	$-0.310339\pi$
$857$	2.41325e7	1.12241	0.561203	+	0.827678i	$0.310339\pi$
$858$	0	0
$859$	1.85877e6	0.0859494	0.0429747	−	0.999076i	$-0.486317\pi$
$859$	1.85877e6	0.0859494	0.0429747	+	0.999076i	$0.486317\pi$
$860$	2.46351e7	1.13582
$861$	0	0
$862$	−3.34690e6	−0.153417
$863$	−3.69085e7	−1.68694	−0.843470	−	0.537177i	$-0.819491\pi$
$863$	−3.69085e7	−1.68694	−0.843470	+	0.537177i	$0.819491\pi$
$864$	0	0
$865$	1.17787e7	0.535249
$866$	−3.98880e7	−1.80737
$867$	0	0
$868$	5.25006e7	2.36518
$869$	−3.05846e6	−0.137389
$870$	0	0
$871$	4.43092e7	1.97901
$872$	2.50090e7	1.11379
$873$	0	0
$874$	−2.05416e7	−0.909610
$875$	−3.16265e6	−0.139647
$876$	0	0
$877$	−3.57125e6	−0.156791	−0.0783955	−	0.996922i	$-0.524980\pi$
$877$	−3.57125e6	−0.156791	−0.0783955	+	0.996922i	$0.524980\pi$
$878$	−3.33569e7	−1.46032
$879$	0	0
$880$	−125375.	−0.00545762
$881$	5.31806e6	0.230841	0.115421	−	0.993317i	$-0.463178\pi$
$881$	5.31806e6	0.230841	0.115421	+	0.993317i	$0.463178\pi$
$882$	0	0
$883$	3.96429e7	1.71105	0.855526	−	0.517759i	$-0.173234\pi$
$883$	3.96429e7	1.71105	0.855526	+	0.517759i	$0.173234\pi$
$884$	−5.06773e7	−2.18114
$885$	0	0
$886$	−1.22636e6	−0.0524846
$887$	2.14528e7	0.915536	0.457768	−	0.889072i	$-0.348649\pi$
$887$	2.14528e7	0.915536	0.457768	+	0.889072i	$0.348649\pi$
$888$	0	0
$889$	5.78521e7	2.45508
$890$	−1.69064e7	−0.715445
$891$	0	0
$892$	−2.58760e7	−1.08889
$893$	1.92995e7	0.809873
$894$	0	0
$895$	−1.30023e7	−0.542579
$896$	5.95157e7	2.47663
$897$	0	0
$898$	6.51296e7	2.69518
$899$	−2.25471e7	−0.930448
$900$	0	0
$901$	6.74363e6	0.276746
$902$	5.34289e6	0.218655
$903$	0	0
$904$	2.60460e7	1.06003
$905$	−9.75732e6	−0.396013
$906$	0	0
$907$	−3.09753e7	−1.25025	−0.625126	−	0.780524i	$-0.714952\pi$
$907$	−3.09753e7	−1.25025	−0.625126	+	0.780524i	$0.714952\pi$
$908$	831638.	0.0334749
$909$	0	0
$910$	−2.87439e7	−1.15065
$911$	1.40369e6	0.0560369	0.0280184	−	0.999607i	$-0.491080\pi$
$911$	1.40369e6	0.0560369	0.0280184	+	0.999607i	$0.491080\pi$
$912$	0	0
$913$	626924.	0.0248907
$914$	9.25445e6	0.366425
$915$	0	0
$916$	6.52829e7	2.57076
$917$	6.43828e7	2.52840
$918$	0	0
$919$	−2.79914e7	−1.09329	−0.546646	−	0.837364i	$-0.684096\pi$
$919$	−2.79914e7	−1.09329	−0.546646	+	0.837364i	$0.684096\pi$
$920$	6.82682e6	0.265919
$921$	0	0
$922$	6.05533e7	2.34590
$923$	−4.49439e7	−1.73647
$924$	0	0
$925$	7.28756e6	0.280045
$926$	−8.62509e6	−0.330549
$927$	0	0
$928$	−2.66114e7	−1.01437
$929$	−1.87384e7	−0.712348	−0.356174	−	0.934420i	$-0.615919\pi$
$929$	−1.87384e7	−0.712348	−0.356174	+	0.934420i	$0.615919\pi$
$930$	0	0
$931$	3.48854e7	1.31907
$932$	−5.47286e7	−2.06383
$933$	0	0
$934$	−4.20778e7	−1.57829
$935$	4.80835e6	0.179873
$936$	0	0
$937$	9.99011e6	0.371725	0.185862	−	0.982576i	$-0.440492\pi$
$937$	9.99011e6	0.371725	0.185862	+	0.982576i	$0.440492\pi$
$938$	1.31352e8	4.87450
$939$	0	0
$940$	−1.71079e7	−0.631505
$941$	2.85997e7	1.05290	0.526450	−	0.850206i	$-0.323523\pi$
$941$	2.85997e7	1.05290	0.526450	+	0.850206i	$0.323523\pi$
$942$	0	0
$943$	7.55156e6	0.276540
$944$	548090.	0.0200181
$945$	0	0
$946$	−2.12438e7	−0.771798
$947$	−3.69010e7	−1.33710	−0.668549	−	0.743668i	$-0.733084\pi$
$947$	−3.69010e7	−1.33710	−0.668549	+	0.743668i	$0.733084\pi$
$948$	0	0
$949$	3.05866e7	1.10247
$950$	−8.23046e6	−0.295880
$951$	0	0
$952$	−5.63235e7	−2.01418
$953$	2.24465e7	0.800602	0.400301	−	0.916384i	$-0.368906\pi$
$953$	2.24465e7	0.800602	0.400301	+	0.916384i	$0.368906\pi$
$954$	0	0
$955$	1.41385e7	0.501642
$956$	2.13318e7	0.754890
$957$	0	0
$958$	−1.44546e7	−0.508852
$959$	1.61652e7	0.567591
$960$	0	0
$961$	−2.95804e6	−0.103323
$962$	6.62335e7	2.30749
$963$	0	0
$964$	−2.01356e7	−0.697866
$965$	−2.46983e7	−0.853786
$966$	0	0
$967$	−2.95230e7	−1.01530	−0.507650	−	0.861563i	$-0.669486\pi$
$967$	−2.95230e7	−1.01530	−0.507650	+	0.861563i	$0.669486\pi$
$968$	−2.56306e6	−0.0879166
$969$	0	0
$970$	3.53979e6	0.120795
$971$	−3.24650e7	−1.10501	−0.552506	−	0.833509i	$-0.686328\pi$
$971$	−3.24650e7	−1.10501	−0.552506	+	0.833509i	$0.686328\pi$
$972$	0	0
$973$	−3.53799e6	−0.119805
$974$	7.27459e7	2.45703
$975$	0	0
$976$	−165538.	−0.00556253
$977$	−4.72262e7	−1.58287	−0.791437	−	0.611250i	$-0.790667\pi$
$977$	−4.72262e7	−1.58287	−0.791437	+	0.611250i	$0.790667\pi$
$978$	0	0
$979$	8.97125e6	0.299155
$980$	−3.09239e7	−1.02856
$981$	0	0
$982$	9.22294e6	0.305204
$983$	4.61760e7	1.52417	0.762084	−	0.647478i	$-0.224176\pi$
$983$	4.61760e7	1.52417	0.762084	+	0.647478i	$0.224176\pi$
$984$	0	0
$985$	−3.37703e6	−0.110903
$986$	6.45184e7	2.11345
$987$	0	0
$988$	−4.60303e7	−1.50021
$989$	−3.00256e7	−0.976116
$990$	0	0
$991$	−5.16438e7	−1.67045	−0.835226	−	0.549906i	$-0.814663\pi$
$991$	−5.16438e7	−1.67045	−0.835226	+	0.549906i	$0.814663\pi$
$992$	3.02985e7	0.977558
$993$	0	0
$994$	−1.33234e8	−4.27708
$995$	−2.57953e7	−0.826006
$996$	0	0
$997$	−2.37613e7	−0.757065	−0.378532	−	0.925588i	$-0.623571\pi$
$997$	−2.37613e7	−0.757065	−0.378532	+	0.925588i	$0.623571\pi$
$998$	6.70979e6	0.213247
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.n.1.2		7
3.2	odd	2		165.6.a.h.1.6	✓	7
15.14	odd	2		825.6.a.n.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.h.1.6	✓	7	3.2	odd	2
495.6.a.n.1.2		7	1.1	even	1	trivial
825.6.a.n.1.2		7	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)

\(f(q)\)	\(=\)	\(q-9.12102 q^{2} +51.1931 q^{4} -25.0000 q^{5} +202.409 q^{7} -175.061 q^{8} +O(q^{10})\)	\(q-9.12102 q^{2} +51.1931 q^{4} -25.0000 q^{5} +202.409 q^{7} -175.061 q^{8} +228.026 q^{10} -121.000 q^{11} -622.776 q^{13} -1846.18 q^{14} -41.4462 q^{16} +1589.54 q^{17} +1443.78 q^{19} -1279.83 q^{20} +1103.64 q^{22} +1559.87 q^{23} +625.000 q^{25} +5680.35 q^{26} +10362.0 q^{28} -4450.09 q^{29} +5066.67 q^{31} +5979.97 q^{32} -14498.2 q^{34} -5060.23 q^{35} +11660.1 q^{37} -13168.7 q^{38} +4376.52 q^{40} +4841.13 q^{41} -19248.8 q^{43} -6194.36 q^{44} -14227.7 q^{46} +13367.3 q^{47} +24162.5 q^{49} -5700.64 q^{50} -31881.8 q^{52} +4242.51 q^{53} +3025.00 q^{55} -35433.9 q^{56} +40589.4 q^{58} -13224.1 q^{59} +3994.05 q^{61} -46213.2 q^{62} -53217.2 q^{64} +15569.4 q^{65} -71148.0 q^{67} +81373.3 q^{68} +46154.5 q^{70} +72167.1 q^{71} -49113.3 q^{73} -106352. q^{74} +73911.5 q^{76} -24491.5 q^{77} +25276.5 q^{79} +1036.15 q^{80} -44156.1 q^{82} -5181.19 q^{83} -39738.4 q^{85} +175568. q^{86} +21182.3 q^{88} -74142.5 q^{89} -126056. q^{91} +79854.8 q^{92} -121924. q^{94} -36094.5 q^{95} +15523.6 q^{97} -220387. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8}+O(q^{10}) \) 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8	\( 7 q - q^{2} + 195 q^{4} - 175 q^{5} - 153 q^{8} + 25 q^{10} - 847 q^{11} + 1418 q^{13} - 2548 q^{14} + 3699 q^{16} - 630 q^{17} + 2572 q^{19} - 4875 q^{20} + 121 q^{22} - 536 q^{23} + 4375 q^{25} + 7626 q^{26} - 11368 q^{28} + 1038 q^{29} + 1872 q^{31} + 7523 q^{32} + 20790 q^{34} + 24298 q^{37} + 18952 q^{38} + 3825 q^{40} + 17658 q^{41} + 7244 q^{43} - 23595 q^{44} + 31016 q^{46} - 34560 q^{47} + 78735 q^{49} - 625 q^{50} + 110222 q^{52} + 10214 q^{53} + 21175 q^{55} - 81124 q^{56} - 5718 q^{58} - 94676 q^{59} + 69538 q^{61} + 4208 q^{62} + 112339 q^{64} - 35450 q^{65} + 64908 q^{67} + 136010 q^{68} + 63700 q^{70} - 61816 q^{71} - 11890 q^{73} + 124050 q^{74} - 47216 q^{76} + 18928 q^{79} - 92475 q^{80} + 36398 q^{82} - 17492 q^{83} + 15750 q^{85} + 216688 q^{86} + 18513 q^{88} - 25302 q^{89} + 3392 q^{91} + 27408 q^{92} - 30800 q^{94} - 64300 q^{95} - 172546 q^{97} + 615271 q^{98}+O(q^{100}) \) 7 * q - q^2 + 195 * q^4 - 175 * q^5 - 153 * q^8 + 25 * q^10 - 847 * q^11 + 1418 * q^13 - 2548 * q^14 + 3699 * q^16 - 630 * q^17 + 2572 * q^19 - 4875 * q^20 + 121 * q^22 - 536 * q^23 + 4375 * q^25 + 7626 * q^26 - 11368 * q^28 + 1038 * q^29 + 1872 * q^31 + 7523 * q^32 + 20790 * q^34 + 24298 * q^37 + 18952 * q^38 + 3825 * q^40 + 17658 * q^41 + 7244 * q^43 - 23595 * q^44 + 31016 * q^46 - 34560 * q^47 + 78735 * q^49 - 625 * q^50 + 110222 * q^52 + 10214 * q^53 + 21175 * q^55 - 81124 * q^56 - 5718 * q^58 - 94676 * q^59 + 69538 * q^61 + 4208 * q^62 + 112339 * q^64 - 35450 * q^65 + 64908 * q^67 + 136010 * q^68 + 63700 * q^70 - 61816 * q^71 - 11890 * q^73 + 124050 * q^74 - 47216 * q^76 + 18928 * q^79 - 92475 * q^80 + 36398 * q^82 - 17492 * q^83 + 15750 * q^85 + 216688 * q^86 + 18513 * q^88 - 25302 * q^89 + 3392 * q^91 + 27408 * q^92 - 30800 * q^94 - 64300 * q^95 - 172546 * q^97 + 615271 * q^98

Introduction

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