LMFDB - Embedded newform 495.6.a.b.1.1

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:	$3$
Coefficient field:	3.3.18257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.78415$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.78415 q^{2} +1.45634 q^{4} -25.0000 q^{5} +17.6498 q^{7} +176.669 q^{8} +O(q^{10})$	\(q-5.78415 q^{2} +1.45634 q^{4} -25.0000 q^{5} +17.6498 q^{7} +176.669 q^{8} +144.604 q^{10} -121.000 q^{11} +674.396 q^{13} -102.089 q^{14} -1068.48 q^{16} +2117.62 q^{17} -2307.79 q^{19} -36.4084 q^{20} +699.882 q^{22} +3072.47 q^{23} +625.000 q^{25} -3900.80 q^{26} +25.7040 q^{28} +1437.44 q^{29} -5157.66 q^{31} +526.846 q^{32} -12248.6 q^{34} -441.244 q^{35} +6928.88 q^{37} +13348.6 q^{38} -4416.72 q^{40} -2844.78 q^{41} -11665.3 q^{43} -176.217 q^{44} -17771.6 q^{46} -3451.40 q^{47} -16495.5 q^{49} -3615.09 q^{50} +982.146 q^{52} +18167.6 q^{53} +3025.00 q^{55} +3118.17 q^{56} -8314.36 q^{58} +8976.88 q^{59} -378.820 q^{61} +29832.7 q^{62} +31144.1 q^{64} -16859.9 q^{65} -12233.7 q^{67} +3083.97 q^{68} +2552.22 q^{70} -55867.0 q^{71} +19739.6 q^{73} -40077.6 q^{74} -3360.92 q^{76} -2135.62 q^{77} -13495.8 q^{79} +26712.0 q^{80} +16454.6 q^{82} -42078.3 q^{83} -52940.5 q^{85} +67474.0 q^{86} -21376.9 q^{88} +81223.2 q^{89} +11902.9 q^{91} +4474.55 q^{92} +19963.4 q^{94} +57694.8 q^{95} +152101. q^{97} +95412.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8	$ 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25} - 2132 q^{26} - 580 q^{28} + 4538 q^{29} - 14968 q^{31} + 2496 q^{32} - 13704 q^{34} + 1700 q^{35} - 6190 q^{37} + 11668 q^{38} - 600 q^{40} + 8926 q^{41} - 33592 q^{43} + 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 14693 q^{49} - 1250 q^{50} + 18780 q^{52} + 22934 q^{53} + 9075 q^{55} + 40012 q^{56} - 32304 q^{58} + 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 35474 q^{64} - 7250 q^{65} + 16868 q^{67} + 71288 q^{68} + 22900 q^{70} - 4856 q^{71} + 1910 q^{73} - 29404 q^{74} + 6116 q^{76} + 8228 q^{77} - 36844 q^{79} + 14750 q^{80} + 84000 q^{82} + 48796 q^{83} + 10850 q^{85} + 83492 q^{86} - 2904 q^{88} + 188978 q^{89} - 93208 q^{91} + 6976 q^{92} + 70472 q^{94} + 71400 q^{95} + 247526 q^{97} + 154654 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.78415	−1.02250	−0.511251	−	0.859431i	$-0.670818\pi$
$2$	−5.78415	−1.02250	−0.511251	+	0.859431i	$0.670818\pi$
$3$	0	0
$3$	0	0
$4$	1.45634	0.0455105
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	17.6498	0.136143	0.0680713	−	0.997680i	$-0.478315\pi$
$7$	17.6498	0.136143	0.0680713	+	0.997680i	$0.478315\pi$
$8$	176.669	0.975968
$9$	0	0
$10$	144.604	0.457277
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	674.396	1.10677	0.553384	−	0.832926i	$-0.313336\pi$
$13$	674.396	1.10677	0.553384	+	0.832926i	$0.313336\pi$
$14$	−102.089	−0.139206
$15$	0	0
$16$	−1068.48	−1.04344
$17$	2117.62	1.77716	0.888579	−	0.458724i	$-0.151693\pi$
$17$	2117.62	1.77716	0.888579	+	0.458724i	$0.151693\pi$
$18$	0	0
$19$	−2307.79	−1.46660	−0.733302	−	0.679903i	$-0.762022\pi$
$19$	−2307.79	−1.46660	−0.733302	+	0.679903i	$0.762022\pi$
$20$	−36.4084	−0.0203529
$21$	0	0
$22$	699.882	0.308296
$23$	3072.47	1.21107	0.605534	−	0.795820i	$-0.292960\pi$
$23$	3072.47	1.21107	0.605534	+	0.795820i	$0.292960\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−3900.80	−1.13167
$27$	0	0
$28$	25.7040	0.00619591
$29$	1437.44	0.317391	0.158696	−	0.987328i	$-0.449271\pi$
$29$	1437.44	0.317391	0.158696	+	0.987328i	$0.449271\pi$
$30$	0	0
$31$	−5157.66	−0.963937	−0.481969	−	0.876188i	$-0.660078\pi$
$31$	−5157.66	−0.963937	−0.481969	+	0.876188i	$0.660078\pi$
$32$	526.846	0.0909513
$33$	0	0
$34$	−12248.6	−1.81715
$35$	−441.244	−0.0608848
$36$	0	0
$37$	6928.88	0.832067	0.416034	−	0.909349i	$-0.363420\pi$
$37$	6928.88	0.832067	0.416034	+	0.909349i	$0.363420\pi$
$38$	13348.6	1.49961
$39$	0	0
$40$	−4416.72	−0.436466
$41$	−2844.78	−0.264295	−0.132147	−	0.991230i	$-0.542187\pi$
$41$	−2844.78	−0.264295	−0.132147	+	0.991230i	$0.542187\pi$
$42$	0	0
$43$	−11665.3	−0.962113	−0.481057	−	0.876689i	$-0.659747\pi$
$43$	−11665.3	−0.962113	−0.481057	+	0.876689i	$0.659747\pi$
$44$	−176.217	−0.0137219
$45$	0	0
$46$	−17771.6	−1.23832
$47$	−3451.40	−0.227903	−0.113952	−	0.993486i	$-0.536351\pi$
$47$	−3451.40	−0.227903	−0.113952	+	0.993486i	$0.536351\pi$
$48$	0	0
$49$	−16495.5	−0.981465
$50$	−3615.09	−0.204500
$51$	0	0
$52$	982.146	0.0503695
$53$	18167.6	0.888401	0.444200	−	0.895927i	$-0.353488\pi$
$53$	18167.6	0.888401	0.444200	+	0.895927i	$0.353488\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	3118.17	0.132871
$57$	0	0
$58$	−8314.36	−0.324533
$59$	8976.88	0.335734	0.167867	−	0.985810i	$-0.446312\pi$
$59$	8976.88	0.335734	0.167867	+	0.985810i	$0.446312\pi$
$60$	0	0
$61$	−378.820	−0.0130349	−0.00651746	−	0.999979i	$-0.502075\pi$
$61$	−378.820	−0.0130349	−0.00651746	+	0.999979i	$0.502075\pi$
$62$	29832.7	0.985628
$63$	0	0
$64$	31144.1	0.950441
$65$	−16859.9	−0.494962
$66$	0	0
$67$	−12233.7	−0.332945	−0.166473	−	0.986046i	$-0.553238\pi$
$67$	−12233.7	−0.332945	−0.166473	+	0.986046i	$0.553238\pi$
$68$	3083.97	0.0808793
$69$	0	0
$70$	2552.22	0.0622548
$71$	−55867.0	−1.31525	−0.657627	−	0.753344i	$-0.728440\pi$
$71$	−55867.0	−1.31525	−0.657627	+	0.753344i	$0.728440\pi$
$72$	0	0
$73$	19739.6	0.433541	0.216771	−	0.976223i	$-0.430448\pi$
$73$	19739.6	0.433541	0.216771	+	0.976223i	$0.430448\pi$
$74$	−40077.6	−0.850791
$75$	0	0
$76$	−3360.92	−0.0667459
$77$	−2135.62	−0.0410485
$78$	0	0
$79$	−13495.8	−0.243293	−0.121647	−	0.992573i	$-0.538817\pi$
$79$	−13495.8	−0.243293	−0.121647	+	0.992573i	$0.538817\pi$
$80$	26712.0	0.466640
$81$	0	0
$82$	16454.6	0.270242
$83$	−42078.3	−0.670444	−0.335222	−	0.942139i	$-0.608811\pi$
$83$	−42078.3	−0.670444	−0.335222	+	0.942139i	$0.608811\pi$
$84$	0	0
$85$	−52940.5	−0.794769
$86$	67474.0	0.983763
$87$	0	0
$88$	−21376.9	−0.294265
$89$	81223.2	1.08694	0.543469	−	0.839429i	$-0.317110\pi$
$89$	81223.2	1.08694	0.543469	+	0.839429i	$0.317110\pi$
$90$	0	0
$91$	11902.9	0.150678
$92$	4474.55	0.0551163
$93$	0	0
$94$	19963.4	0.233031
$95$	57694.8	0.655885
$96$	0	0
$97$	152101.	1.64136	0.820680	−	0.571388i	$-0.193595\pi$
$97$	152101.	1.64136	0.820680	+	0.571388i	$0.193595\pi$
$98$	95412.3	1.00355
$99$	0	0
$100$	910.210	0.00910210
$101$	−95207.6	−0.928684	−0.464342	−	0.885656i	$-0.653709\pi$
$101$	−95207.6	−0.928684	−0.464342	+	0.885656i	$0.653709\pi$
$102$	0	0
$103$	−150824.	−1.40081	−0.700403	−	0.713748i	$-0.746996\pi$
$103$	−150824.	−1.40081	−0.700403	+	0.713748i	$0.746996\pi$
$104$	119145.	1.08017
$105$	0	0
$106$	−105084.	−0.908392
$107$	−192242.	−1.62327	−0.811633	−	0.584168i	$-0.801421\pi$
$107$	−192242.	−1.62327	−0.811633	+	0.584168i	$0.801421\pi$
$108$	0	0
$109$	96302.8	0.776377	0.388188	−	0.921580i	$-0.373101\pi$
$109$	96302.8	0.776377	0.388188	+	0.921580i	$0.373101\pi$
$110$	−17497.0	−0.137874
$111$	0	0
$112$	−18858.5	−0.142056
$113$	210371.	1.54985	0.774926	−	0.632052i	$-0.217787\pi$
$113$	210371.	1.54985	0.774926	+	0.632052i	$0.217787\pi$
$114$	0	0
$115$	−76811.8	−0.541606
$116$	2093.39	0.0144446
$117$	0	0
$118$	−51923.6	−0.343289
$119$	37375.5	0.241947
$120$	0	0
$121$	14641.0	0.0909091
$122$	2191.15	0.0133282
$123$	0	0
$124$	−7511.29	−0.0438692
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−63529.8	−0.349517	−0.174758	−	0.984611i	$-0.555914\pi$
$127$	−63529.8	−0.349517	−0.174758	+	0.984611i	$0.555914\pi$
$128$	−197001.	−1.06278
$129$	0	0
$130$	97520.1	0.506099
$131$	88396.0	0.450043	0.225022	−	0.974354i	$-0.427755\pi$
$131$	88396.0	0.450043	0.225022	+	0.974354i	$0.427755\pi$
$132$	0	0
$133$	−40732.0	−0.199667
$134$	70761.8	0.340437
$135$	0	0
$136$	374118.	1.73445
$137$	42642.4	0.194106	0.0970532	−	0.995279i	$-0.469058\pi$
$137$	42642.4	0.194106	0.0970532	+	0.995279i	$0.469058\pi$
$138$	0	0
$139$	169503.	0.744114	0.372057	−	0.928210i	$-0.378653\pi$
$139$	169503.	0.744114	0.372057	+	0.928210i	$0.378653\pi$
$140$	−642.599	−0.00277090
$141$	0	0
$142$	323143.	1.34485
$143$	−81601.9	−0.333703
$144$	0	0
$145$	−35936.0	−0.141942
$146$	−114176.	−0.443297
$147$	0	0
$148$	10090.8	0.0378678
$149$	72462.8	0.267393	0.133696	−	0.991022i	$-0.457315\pi$
$149$	72462.8	0.267393	0.133696	+	0.991022i	$0.457315\pi$
$150$	0	0
$151$	−61316.2	−0.218843	−0.109422	−	0.993995i	$-0.534900\pi$
$151$	−61316.2	−0.218843	−0.109422	+	0.993995i	$0.534900\pi$
$152$	−407716.	−1.43136
$153$	0	0
$154$	12352.7	0.0419722
$155$	128942.	0.431086
$156$	0	0
$157$	505789.	1.63765	0.818824	−	0.574045i	$-0.194627\pi$
$157$	505789.	1.63765	0.818824	+	0.574045i	$0.194627\pi$
$158$	78061.5	0.248768
$159$	0	0
$160$	−13171.2	−0.0406747
$161$	54228.4	0.164878
$162$	0	0
$163$	387702.	1.14295	0.571477	−	0.820618i	$-0.306371\pi$
$163$	387702.	1.14295	0.571477	+	0.820618i	$0.306371\pi$
$164$	−4142.95	−0.0120282
$165$	0	0
$166$	243387.	0.685530
$167$	635422.	1.76308	0.881538	−	0.472113i	$-0.156509\pi$
$167$	635422.	1.76308	0.881538	+	0.472113i	$0.156509\pi$
$168$	0	0
$169$	83516.8	0.224935
$170$	306216.	0.812653
$171$	0	0
$172$	−16988.6	−0.0437862
$173$	679820.	1.72695	0.863473	−	0.504395i	$-0.168284\pi$
$173$	679820.	1.72695	0.863473	+	0.504395i	$0.168284\pi$
$174$	0	0
$175$	11031.1	0.0272285
$176$	129286.	0.314609
$177$	0	0
$178$	−469807.	−1.11140
$179$	63083.3	0.147157	0.0735787	−	0.997289i	$-0.476558\pi$
$179$	63083.3	0.147157	0.0735787	+	0.997289i	$0.476558\pi$
$180$	0	0
$181$	523614.	1.18800	0.593998	−	0.804466i	$-0.297549\pi$
$181$	523614.	1.18800	0.593998	+	0.804466i	$0.297549\pi$
$182$	−68848.3	−0.154069
$183$	0	0
$184$	542810.	1.18196
$185$	−173222.	−0.372112
$186$	0	0
$187$	−256232.	−0.535833
$188$	−5026.39	−0.0103720
$189$	0	0
$190$	−333715.	−0.670644
$191$	684311.	1.35728	0.678641	−	0.734470i	$-0.262569\pi$
$191$	684311.	1.35728	0.678641	+	0.734470i	$0.262569\pi$
$192$	0	0
$193$	527417.	1.01920	0.509601	−	0.860411i	$-0.329793\pi$
$193$	527417.	1.01920	0.509601	+	0.860411i	$0.329793\pi$
$194$	−879777.	−1.67829
$195$	0	0
$196$	−24023.0	−0.0446669
$197$	467139.	0.857591	0.428796	−	0.903402i	$-0.358938\pi$
$197$	467139.	0.857591	0.428796	+	0.903402i	$0.358938\pi$
$198$	0	0
$199$	−436613.	−0.781563	−0.390782	−	0.920483i	$-0.627795\pi$
$199$	−436613.	−0.781563	−0.390782	+	0.920483i	$0.627795\pi$
$200$	110418.	0.195194
$201$	0	0
$202$	550694.	0.949581
$203$	25370.5	0.0432104
$204$	0	0
$205$	71119.4	0.118196
$206$	872389.	1.43233
$207$	0	0
$208$	−720580.	−1.15484
$209$	279243.	0.442198
$210$	0	0
$211$	−747312.	−1.15557	−0.577785	−	0.816189i	$-0.696083\pi$
$211$	−747312.	−1.15557	−0.577785	+	0.816189i	$0.696083\pi$
$212$	26458.2	0.0404315
$213$	0	0
$214$	1.11196e6	1.65979
$215$	291634.	0.430270
$216$	0	0
$217$	−91031.6	−0.131233
$218$	−557029.	−0.793847
$219$	0	0
$220$	4405.41	0.00613663
$221$	1.42812e6	1.96690
$222$	0	0
$223$	870654.	1.17242	0.586211	−	0.810159i	$-0.300619\pi$
$223$	870654.	1.17242	0.586211	+	0.810159i	$0.300619\pi$
$224$	9298.71	0.0123823
$225$	0	0
$226$	−1.21682e6	−1.58473
$227$	−82323.8	−0.106038	−0.0530189	−	0.998594i	$-0.516884\pi$
$227$	−82323.8	−0.106038	−0.0530189	+	0.998594i	$0.516884\pi$
$228$	0	0
$229$	−340134.	−0.428609	−0.214305	−	0.976767i	$-0.568748\pi$
$229$	−340134.	−0.428609	−0.214305	+	0.976767i	$0.568748\pi$
$230$	444291.	0.553793
$231$	0	0
$232$	253951.	0.309763
$233$	248384.	0.299732	0.149866	−	0.988706i	$-0.452116\pi$
$233$	248384.	0.299732	0.149866	+	0.988706i	$0.452116\pi$
$234$	0	0
$235$	86284.9	0.101921
$236$	13073.4	0.0152794
$237$	0	0
$238$	−216185.	−0.247391
$239$	1.37978e6	1.56248	0.781242	−	0.624229i	$-0.214587\pi$
$239$	1.37978e6	1.56248	0.781242	+	0.624229i	$0.214587\pi$
$240$	0	0
$241$	−1.53382e6	−1.70110	−0.850552	−	0.525891i	$-0.823732\pi$
$241$	−1.53382e6	−1.70110	−0.850552	+	0.525891i	$0.823732\pi$
$242$	−84685.7	−0.0929547
$243$	0	0
$244$	−551.689	−0.000593225 0
$245$	412387.	0.438925
$246$	0	0
$247$	−1.55637e6	−1.62319
$248$	−911199.	−0.940772
$249$	0	0
$250$	90377.3	0.0914554
$251$	−338591.	−0.339227	−0.169614	−	0.985511i	$-0.554252\pi$
$251$	−338591.	−0.339227	−0.169614	+	0.985511i	$0.554252\pi$
$252$	0	0
$253$	−371769.	−0.365151
$254$	367466.	0.357382
$255$	0	0
$256$	142872.	0.136253
$257$	−1.60985e6	−1.52038	−0.760189	−	0.649702i	$-0.774894\pi$
$257$	−1.60985e6	−1.52038	−0.760189	+	0.649702i	$0.774894\pi$
$258$	0	0
$259$	122293.	0.113280
$260$	−24553.7	−0.0225259
$261$	0	0
$262$	−511295.	−0.460170
$263$	−458118.	−0.408402	−0.204201	−	0.978929i	$-0.565460\pi$
$263$	−458118.	−0.408402	−0.204201	+	0.978929i	$0.565460\pi$
$264$	0	0
$265$	−454191.	−0.397305
$266$	235600.	0.204160
$267$	0	0
$268$	−17816.4	−0.0151525
$269$	692380.	0.583397	0.291698	−	0.956510i	$-0.405780\pi$
$269$	692380.	0.583397	0.291698	+	0.956510i	$0.405780\pi$
$270$	0	0
$271$	−486690.	−0.402558	−0.201279	−	0.979534i	$-0.564510\pi$
$271$	−486690.	−0.402558	−0.201279	+	0.979534i	$0.564510\pi$
$272$	−2.26264e6	−1.85436
$273$	0	0
$274$	−246650.	−0.198474
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	2.38271e6	1.86583	0.932915	−	0.360098i	$-0.117257\pi$
$277$	2.38271e6	1.86583	0.932915	+	0.360098i	$0.117257\pi$
$278$	−980428.	−0.760858
$279$	0	0
$280$	−77954.1	−0.0594216
$281$	−1.12154e6	−0.847320	−0.423660	−	0.905821i	$-0.639255\pi$
$281$	−1.12154e6	−0.847320	−0.423660	+	0.905821i	$0.639255\pi$
$282$	0	0
$283$	965956.	0.716954	0.358477	−	0.933539i	$-0.383296\pi$
$283$	965956.	0.716954	0.358477	+	0.933539i	$0.383296\pi$
$284$	−81361.1	−0.0598578
$285$	0	0
$286$	471997.	0.341212
$287$	−50209.6	−0.0359817
$288$	0	0
$289$	3.06446e6	2.15829
$290$	207859.	0.145136
$291$	0	0
$292$	28747.4	0.0197307
$293$	−2.20819e6	−1.50269	−0.751343	−	0.659912i	$-0.770594\pi$
$293$	−2.20819e6	−1.50269	−0.751343	+	0.659912i	$0.770594\pi$
$294$	0	0
$295$	−224422.	−0.150145
$296$	1.22412e6	0.812071
$297$	0	0
$298$	−419135.	−0.273410
$299$	2.07206e6	1.34037
$300$	0	0
$301$	−205891.	−0.130985
$302$	354662.	0.223768
$303$	0	0
$304$	2.46584e6	1.53031
$305$	9470.49	0.00582939
$306$	0	0
$307$	805480.	0.487763	0.243881	−	0.969805i	$-0.421579\pi$
$307$	805480.	0.487763	0.243881	+	0.969805i	$0.421579\pi$
$308$	−3110.18	−0.00186814
$309$	0	0
$310$	−745817.	−0.440786
$311$	870766.	0.510505	0.255253	−	0.966874i	$-0.417841\pi$
$311$	870766.	0.510505	0.255253	+	0.966874i	$0.417841\pi$
$312$	0	0
$313$	−3.16630e6	−1.82680	−0.913400	−	0.407063i	$-0.866553\pi$
$313$	−3.16630e6	−1.82680	−0.913400	+	0.407063i	$0.866553\pi$
$314$	−2.92556e6	−1.67450
$315$	0	0
$316$	−19654.4	−0.0110724
$317$	−1.72288e6	−0.962955	−0.481478	−	0.876458i	$-0.659900\pi$
$317$	−1.72288e6	−0.962955	−0.481478	+	0.876458i	$0.659900\pi$
$318$	0	0
$319$	−173930.	−0.0956970
$320$	−778602.	−0.425050
$321$	0	0
$322$	−313665.	−0.168588
$323$	−4.88703e6	−2.60639
$324$	0	0
$325$	421497.	0.221354
$326$	−2.24252e6	−1.16867
$327$	0	0
$328$	−502584.	−0.257943
$329$	−60916.3	−0.0310273
$330$	0	0
$331$	−3.20493e6	−1.60786	−0.803932	−	0.594722i	$-0.797262\pi$
$331$	−3.20493e6	−1.60786	−0.803932	+	0.594722i	$0.797262\pi$
$332$	−61280.1	−0.0305122
$333$	0	0
$334$	−3.67537e6	−1.80275
$335$	305844.	0.148898
$336$	0	0
$337$	2.44638e6	1.17341	0.586703	−	0.809802i	$-0.300425\pi$
$337$	2.44638e6	1.17341	0.586703	+	0.809802i	$0.300425\pi$
$338$	−483073.	−0.229996
$339$	0	0
$340$	−77099.2	−0.0361703
$341$	624077.	0.290638
$342$	0	0
$343$	−587781.	−0.269762
$344$	−2.06090e6	−0.938992
$345$	0	0
$346$	−3.93218e6	−1.76581
$347$	3.05055e6	1.36005	0.680025	−	0.733189i	$-0.261969\pi$
$347$	3.05055e6	1.36005	0.680025	+	0.733189i	$0.261969\pi$
$348$	0	0
$349$	−3.08755e6	−1.35691	−0.678454	−	0.734643i	$-0.737350\pi$
$349$	−3.08755e6	−1.35691	−0.678454	+	0.734643i	$0.737350\pi$
$350$	−63805.5	−0.0278412
$351$	0	0
$352$	−63748.4	−0.0274228
$353$	−2.98932e6	−1.27684	−0.638418	−	0.769690i	$-0.720411\pi$
$353$	−2.98932e6	−1.27684	−0.638418	+	0.769690i	$0.720411\pi$
$354$	0	0
$355$	1.39668e6	0.588200
$356$	118288.	0.0494671
$357$	0	0
$358$	−364883.	−0.150469
$359$	2.29991e6	0.941836	0.470918	−	0.882177i	$-0.343923\pi$
$359$	2.29991e6	0.941836	0.470918	+	0.882177i	$0.343923\pi$
$360$	0	0
$361$	2.84981e6	1.15093
$362$	−3.02866e6	−1.21473
$363$	0	0
$364$	17334.7	0.00685743
$365$	−493489.	−0.193885
$366$	0	0
$367$	3.58207e6	1.38825	0.694127	−	0.719852i	$-0.255790\pi$
$367$	3.58207e6	1.38825	0.694127	+	0.719852i	$0.255790\pi$
$368$	−3.28288e6	−1.26368
$369$	0	0
$370$	1.00194e6	0.380485
$371$	320655.	0.120949
$372$	0	0
$373$	511254.	0.190268	0.0951338	−	0.995464i	$-0.469672\pi$
$373$	511254.	0.190268	0.0951338	+	0.995464i	$0.469672\pi$
$374$	1.48208e6	0.547891
$375$	0	0
$376$	−609755.	−0.222426
$377$	969404.	0.351278
$378$	0	0
$379$	2.45209e6	0.876877	0.438438	−	0.898761i	$-0.355532\pi$
$379$	2.45209e6	0.876877	0.438438	+	0.898761i	$0.355532\pi$
$380$	84023.0	0.0298497
$381$	0	0
$382$	−3.95815e6	−1.38782
$383$	1.46315e6	0.509674	0.254837	−	0.966984i	$-0.417978\pi$
$383$	1.46315e6	0.509674	0.254837	+	0.966984i	$0.417978\pi$
$384$	0	0
$385$	53390.5	0.0183575
$386$	−3.05065e6	−1.04214
$387$	0	0
$388$	221511.	0.0746991
$389$	803978.	0.269383	0.134691	−	0.990888i	$-0.456996\pi$
$389$	803978.	0.269383	0.134691	+	0.990888i	$0.456996\pi$
$390$	0	0
$391$	6.50633e6	2.15226
$392$	−2.91424e6	−0.957878
$393$	0	0
$394$	−2.70200e6	−0.876889
$395$	337394.	0.108804
$396$	0	0
$397$	−2.87304e6	−0.914883	−0.457441	−	0.889240i	$-0.651234\pi$
$397$	−2.87304e6	−0.914883	−0.457441	+	0.889240i	$0.651234\pi$
$398$	2.52543e6	0.799150
$399$	0	0
$400$	−667801.	−0.208688
$401$	3.45072e6	1.07164	0.535820	−	0.844332i	$-0.320002\pi$
$401$	3.45072e6	1.07164	0.535820	+	0.844332i	$0.320002\pi$
$402$	0	0
$403$	−3.47831e6	−1.06685
$404$	−138654.	−0.0422649
$405$	0	0
$406$	−146747.	−0.0441827
$407$	−838394.	−0.250878
$408$	0	0
$409$	−1.17776e6	−0.348136	−0.174068	−	0.984734i	$-0.555691\pi$
$409$	−1.17776e6	−0.348136	−0.174068	+	0.984734i	$0.555691\pi$
$410$	−411365.	−0.120856
$411$	0	0
$412$	−219651.	−0.0637513
$413$	158440.	0.0457077
$414$	0	0
$415$	1.05196e6	0.299832
$416$	355303.	0.100662
$417$	0	0
$418$	−1.61518e6	−0.452148
$419$	443011.	0.123276	0.0616381	−	0.998099i	$-0.480368\pi$
$419$	443011.	0.123276	0.0616381	+	0.998099i	$0.480368\pi$
$420$	0	0
$421$	−3.41894e6	−0.940127	−0.470063	−	0.882633i	$-0.655769\pi$
$421$	−3.41894e6	−0.940127	−0.470063	+	0.882633i	$0.655769\pi$
$422$	4.32256e6	1.18157
$423$	0	0
$424$	3.20966e6	0.867050
$425$	1.32351e6	0.355432
$426$	0	0
$427$	−6686.08	−0.00177461
$428$	−279969.	−0.0738756
$429$	0	0
$430$	−1.68685e6	−0.439952
$431$	4.24640e6	1.10110	0.550551	−	0.834801i	$-0.314417\pi$
$431$	4.24640e6	1.10110	0.550551	+	0.834801i	$0.314417\pi$
$432$	0	0
$433$	−1.03407e6	−0.265050	−0.132525	−	0.991180i	$-0.542309\pi$
$433$	−1.03407e6	−0.265050	−0.132525	+	0.991180i	$0.542309\pi$
$434$	526540.	0.134186
$435$	0	0
$436$	140249.	0.0353333
$437$	−7.09063e6	−1.77616
$438$	0	0
$439$	3.55155e6	0.879542	0.439771	−	0.898110i	$-0.355060\pi$
$439$	3.55155e6	0.879542	0.439771	+	0.898110i	$0.355060\pi$
$440$	534424.	0.131599
$441$	0	0
$442$	−8.26043e6	−2.01116
$443$	−2.35135e6	−0.569257	−0.284629	−	0.958638i	$-0.591870\pi$
$443$	−2.35135e6	−0.569257	−0.284629	+	0.958638i	$0.591870\pi$
$444$	0	0
$445$	−2.03058e6	−0.486094
$446$	−5.03599e6	−1.19880
$447$	0	0
$448$	549685.	0.129395
$449$	−3.01947e6	−0.706831	−0.353415	−	0.935467i	$-0.614980\pi$
$449$	−3.01947e6	−0.706831	−0.353415	+	0.935467i	$0.614980\pi$
$450$	0	0
$451$	344218.	0.0796878
$452$	306371.	0.0705345
$453$	0	0
$454$	476173.	0.108424
$455$	−297573.	−0.0673853
$456$	0	0
$457$	3.80671e6	0.852626	0.426313	−	0.904576i	$-0.359812\pi$
$457$	3.80671e6	0.852626	0.426313	+	0.904576i	$0.359812\pi$
$458$	1.96738e6	0.438254
$459$	0	0
$460$	−111864.	−0.0246487
$461$	−7.43481e6	−1.62936	−0.814680	−	0.579910i	$-0.803088\pi$
$461$	−7.43481e6	−1.62936	−0.814680	+	0.579910i	$0.803088\pi$
$462$	0	0
$463$	−108747.	−0.0235757	−0.0117879	−	0.999931i	$-0.503752\pi$
$463$	−108747.	−0.0235757	−0.0117879	+	0.999931i	$0.503752\pi$
$464$	−1.53588e6	−0.331178
$465$	0	0
$466$	−1.43669e6	−0.306477
$467$	6.85473e6	1.45445	0.727224	−	0.686400i	$-0.240810\pi$
$467$	6.85473e6	1.45445	0.727224	+	0.686400i	$0.240810\pi$
$468$	0	0
$469$	−215923.	−0.0453280
$470$	−499085.	−0.104215
$471$	0	0
$472$	1.58594e6	0.327666
$473$	1.41151e6	0.290088
$474$	0	0
$475$	−1.44237e6	−0.293321
$476$	54431.3	0.0110111
$477$	0	0
$478$	−7.98085e6	−1.59764
$479$	9.48659e6	1.88917	0.944585	−	0.328266i	$-0.106464\pi$
$479$	9.48659e6	1.88917	0.944585	+	0.328266i	$0.106464\pi$
$480$	0	0
$481$	4.67281e6	0.920905
$482$	8.87182e6	1.73938
$483$	0	0
$484$	21322.2	0.00413732
$485$	−3.80253e6	−0.734038
$486$	0	0
$487$	9.79771e6	1.87198	0.935992	−	0.352021i	$-0.114506\pi$
$487$	9.79771e6	1.87198	0.935992	+	0.352021i	$0.114506\pi$
$488$	−66925.7	−0.0127216
$489$	0	0
$490$	−2.38531e6	−0.448801
$491$	−9.17094e6	−1.71676	−0.858381	−	0.513012i	$-0.828530\pi$
$491$	−9.17094e6	−1.71676	−0.858381	+	0.513012i	$0.828530\pi$
$492$	0	0
$493$	3.04395e6	0.564054
$494$	9.00225e6	1.65972
$495$	0	0
$496$	5.51087e6	1.00581
$497$	−986040.	−0.179062
$498$	0	0
$499$	−1.91031e6	−0.343441	−0.171720	−	0.985146i	$-0.554933\pi$
$499$	−1.91031e6	−0.343441	−0.171720	+	0.985146i	$0.554933\pi$
$500$	−22755.2	−0.00407058
$501$	0	0
$502$	1.95846e6	0.346861
$503$	−7.08413e6	−1.24844	−0.624219	−	0.781250i	$-0.714583\pi$
$503$	−7.08413e6	−1.24844	−0.624219	+	0.781250i	$0.714583\pi$
$504$	0	0
$505$	2.38019e6	0.415320
$506$	2.15037e6	0.373367
$507$	0	0
$508$	−92520.7	−0.0159067
$509$	7.51321e6	1.28538	0.642689	−	0.766127i	$-0.277819\pi$
$509$	7.51321e6	1.28538	0.642689	+	0.766127i	$0.277819\pi$
$510$	0	0
$511$	348398.	0.0590234
$512$	5.47764e6	0.923461
$513$	0	0
$514$	9.31158e6	1.55459
$515$	3.77060e6	0.626459
$516$	0	0
$517$	417619.	0.0687154
$518$	−707361.	−0.115829
$519$	0	0
$520$	−2.97862e6	−0.483066
$521$	1.03994e7	1.67847	0.839235	−	0.543770i	$-0.183003\pi$
$521$	1.03994e7	1.67847	0.839235	+	0.543770i	$0.183003\pi$
$522$	0	0
$523$	−9.14037e6	−1.46120	−0.730600	−	0.682806i	$-0.760759\pi$
$523$	−9.14037e6	−1.46120	−0.730600	+	0.682806i	$0.760759\pi$
$524$	128734.	0.0204817
$525$	0	0
$526$	2.64982e6	0.417592
$527$	−1.09220e7	−1.71307
$528$	0	0
$529$	3.00374e6	0.466684
$530$	2.62711e6	0.406245
$531$	0	0
$532$	−59319.5	−0.00908695
$533$	−1.91850e6	−0.292513
$534$	0	0
$535$	4.80606e6	0.725947
$536$	−2.16132e6	−0.324944
$537$	0	0
$538$	−4.00483e6	−0.596524
$539$	1.99595e6	0.295923
$540$	0	0
$541$	6.47380e6	0.950969	0.475484	−	0.879724i	$-0.342273\pi$
$541$	6.47380e6	0.950969	0.475484	+	0.879724i	$0.342273\pi$
$542$	2.81508e6	0.411617
$543$	0	0
$544$	1.11566e6	0.161635
$545$	−2.40757e6	−0.347206
$546$	0	0
$547$	−226463.	−0.0323616	−0.0161808	−	0.999869i	$-0.505151\pi$
$547$	−226463.	−0.0323616	−0.0161808	+	0.999869i	$0.505151\pi$
$548$	62101.6	0.00883388
$549$	0	0
$550$	437426.	0.0616592
$551$	−3.31731e6	−0.465487
$552$	0	0
$553$	−238197.	−0.0331226
$554$	−1.37819e7	−1.90781
$555$	0	0
$556$	246853.	0.0338650
$557$	5.45555e6	0.745076	0.372538	−	0.928017i	$-0.378488\pi$
$557$	5.45555e6	0.745076	0.372538	+	0.928017i	$0.378488\pi$
$558$	0	0
$559$	−7.86706e6	−1.06484
$560$	471461.	0.0635296
$561$	0	0
$562$	6.48712e6	0.866386
$563$	1.30266e7	1.73205	0.866024	−	0.500002i	$-0.166667\pi$
$563$	1.30266e7	1.73205	0.866024	+	0.500002i	$0.166667\pi$
$564$	0	0
$565$	−5.25928e6	−0.693115
$566$	−5.58723e6	−0.733087
$567$	0	0
$568$	−9.86997e6	−1.28365
$569$	−3.02736e6	−0.391998	−0.195999	−	0.980604i	$-0.562795\pi$
$569$	−3.02736e6	−0.391998	−0.195999	+	0.980604i	$0.562795\pi$
$570$	0	0
$571$	9.79560e6	1.25731	0.628653	−	0.777686i	$-0.283607\pi$
$571$	9.79560e6	1.25731	0.628653	+	0.777686i	$0.283607\pi$
$572$	−118840.	−0.0151870
$573$	0	0
$574$	290420.	0.0367914
$575$	1.92029e6	0.242213
$576$	0	0
$577$	56282.7	0.00703778	0.00351889	−	0.999994i	$-0.498880\pi$
$577$	56282.7	0.00703778	0.00351889	+	0.999994i	$0.498880\pi$
$578$	−1.77253e7	−2.20686
$579$	0	0
$580$	−52334.9	−0.00645983
$581$	−742671.	−0.0912759
$582$	0	0
$583$	−2.19829e6	−0.267863
$584$	3.48737e6	0.423122
$585$	0	0
$586$	1.27725e7	1.53650
$587$	7.92842e6	0.949711	0.474855	−	0.880064i	$-0.342500\pi$
$587$	7.92842e6	0.949711	0.474855	+	0.880064i	$0.342500\pi$
$588$	0	0
$589$	1.19028e7	1.41371
$590$	1.29809e6	0.153523
$591$	0	0
$592$	−7.40338e6	−0.868212
$593$	5.11910e6	0.597801	0.298900	−	0.954284i	$-0.403380\pi$
$593$	5.11910e6	0.597801	0.298900	+	0.954284i	$0.403380\pi$
$594$	0	0
$595$	−934388.	−0.108202
$596$	105530.	0.0121692
$597$	0	0
$598$	−1.19851e7	−1.37053
$599$	1.00831e7	1.14822	0.574110	−	0.818778i	$-0.305348\pi$
$599$	1.00831e7	1.14822	0.574110	+	0.818778i	$0.305348\pi$
$600$	0	0
$601$	1.32233e7	1.49332	0.746659	−	0.665207i	$-0.231657\pi$
$601$	1.32233e7	1.49332	0.746659	+	0.665207i	$0.231657\pi$
$602$	1.19090e6	0.133932
$603$	0	0
$604$	−89296.9	−0.00995965
$605$	−366025.	−0.0406558
$606$	0	0
$607$	937160.	0.103239	0.0516193	−	0.998667i	$-0.483562\pi$
$607$	937160.	0.103239	0.0516193	+	0.998667i	$0.483562\pi$
$608$	−1.21585e6	−0.133390
$609$	0	0
$610$	−54778.7	−0.00596056
$611$	−2.32761e6	−0.252236
$612$	0	0
$613$	3.05919e6	0.328818	0.164409	−	0.986392i	$-0.447428\pi$
$613$	3.05919e6	0.328818	0.164409	+	0.986392i	$0.447428\pi$
$614$	−4.65901e6	−0.498738
$615$	0	0
$616$	−377298.	−0.0400620
$617$	−8.74474e6	−0.924771	−0.462385	−	0.886679i	$-0.653006\pi$
$617$	−8.74474e6	−0.924771	−0.462385	+	0.886679i	$0.653006\pi$
$618$	0	0
$619$	−4.14486e6	−0.434793	−0.217397	−	0.976083i	$-0.569757\pi$
$619$	−4.14486e6	−0.434793	−0.217397	+	0.976083i	$0.569757\pi$
$620$	187782.	0.0196189
$621$	0	0
$622$	−5.03664e6	−0.521993
$623$	1.43357e6	0.147979
$624$	0	0
$625$	390625.	0.0400000
$626$	1.83143e7	1.86791
$627$	0	0
$628$	736598.	0.0745301
$629$	1.46727e7	1.47872
$630$	0	0
$631$	−4.44508e6	−0.444433	−0.222216	−	0.974997i	$-0.571329\pi$
$631$	−4.44508e6	−0.444433	−0.222216	+	0.974997i	$0.571329\pi$
$632$	−2.38429e6	−0.237446
$633$	0	0
$634$	9.96537e6	0.984624
$635$	1.58825e6	0.156309
$636$	0	0
$637$	−1.11245e7	−1.08625
$638$	1.00604e6	0.0978504
$639$	0	0
$640$	4.92502e6	0.475289
$641$	−1.08278e7	−1.04086	−0.520432	−	0.853903i	$-0.674229\pi$
$641$	−1.08278e7	−1.04086	−0.520432	+	0.853903i	$0.674229\pi$
$642$	0	0
$643$	−1.23064e7	−1.17383	−0.586915	−	0.809649i	$-0.699658\pi$
$643$	−1.23064e7	−1.17383	−0.586915	+	0.809649i	$0.699658\pi$
$644$	78974.7	0.00750367
$645$	0	0
$646$	2.82673e7	2.66504
$647$	1.26477e7	1.18782	0.593912	−	0.804530i	$-0.297583\pi$
$647$	1.26477e7	1.18782	0.593912	+	0.804530i	$0.297583\pi$
$648$	0	0
$649$	−1.08620e6	−0.101228
$650$	−2.43800e6	−0.226334
$651$	0	0
$652$	564624.	0.0520164
$653$	578855.	0.0531235	0.0265618	−	0.999647i	$-0.491544\pi$
$653$	578855.	0.0531235	0.0265618	+	0.999647i	$0.491544\pi$
$654$	0	0
$655$	−2.20990e6	−0.201265
$656$	3.03959e6	0.275775
$657$	0	0
$658$	352349.	0.0317255
$659$	−1.71481e7	−1.53816	−0.769081	−	0.639151i	$-0.779286\pi$
$659$	−1.71481e7	−1.53816	−0.769081	+	0.639151i	$0.779286\pi$
$660$	0	0
$661$	6.99329e6	0.622555	0.311278	−	0.950319i	$-0.399243\pi$
$661$	6.99329e6	0.622555	0.311278	+	0.950319i	$0.399243\pi$
$662$	1.85378e7	1.64404
$663$	0	0
$664$	−7.43392e6	−0.654332
$665$	1.01830e6	0.0892939
$666$	0	0
$667$	4.41649e6	0.384382
$668$	925387.	0.0802384
$669$	0	0
$670$	−1.76904e6	−0.152248
$671$	45837.2	0.00393017
$672$	0	0
$673$	1.98109e7	1.68604	0.843018	−	0.537885i	$-0.180777\pi$
$673$	1.98109e7	1.68604	0.843018	+	0.537885i	$0.180777\pi$
$674$	−1.41502e7	−1.19981
$675$	0	0
$676$	121628.	0.0102369
$677$	2.85012e6	0.238996	0.119498	−	0.992834i	$-0.461871\pi$
$677$	2.85012e6	0.238996	0.119498	+	0.992834i	$0.461871\pi$
$678$	0	0
$679$	2.68455e6	0.223459
$680$	−9.35295e6	−0.775669
$681$	0	0
$682$	−3.60975e6	−0.297178
$683$	−4.58865e6	−0.376386	−0.188193	−	0.982132i	$-0.560263\pi$
$683$	−4.58865e6	−0.376386	−0.188193	+	0.982132i	$0.560263\pi$
$684$	0	0
$685$	−1.06606e6	−0.0868070
$686$	3.39981e6	0.275832
$687$	0	0
$688$	1.24642e7	1.00391
$689$	1.22522e7	0.983254
$690$	0	0
$691$	−2.65609e6	−0.211616	−0.105808	−	0.994387i	$-0.533743\pi$
$691$	−2.65609e6	−0.211616	−0.105808	+	0.994387i	$0.533743\pi$
$692$	990046.	0.0785942
$693$	0	0
$694$	−1.76448e7	−1.39065
$695$	−4.23757e6	−0.332778
$696$	0	0
$697$	−6.02416e6	−0.469693
$698$	1.78588e7	1.38744
$699$	0	0
$700$	16065.0	0.00123918
$701$	7.90907e6	0.607898	0.303949	−	0.952688i	$-0.401695\pi$
$701$	7.90907e6	0.607898	0.303949	+	0.952688i	$0.401695\pi$
$702$	0	0
$703$	−1.59904e7	−1.22031
$704$	−3.76843e6	−0.286569
$705$	0	0
$706$	1.72906e7	1.30557
$707$	−1.68039e6	−0.126433
$708$	0	0
$709$	−2.38057e7	−1.77855	−0.889273	−	0.457378i	$-0.848789\pi$
$709$	−2.38057e7	−1.77855	−0.889273	+	0.457378i	$0.848789\pi$
$710$	−8.07857e6	−0.601435
$711$	0	0
$712$	1.43496e7	1.06082
$713$	−1.58468e7	−1.16739
$714$	0	0
$715$	2.04005e6	0.149237
$716$	91870.5	0.00669720
$717$	0	0
$718$	−1.33030e7	−0.963029
$719$	2.00866e7	1.44905	0.724527	−	0.689246i	$-0.242058\pi$
$719$	2.00866e7	1.44905	0.724527	+	0.689246i	$0.242058\pi$
$720$	0	0
$721$	−2.66201e6	−0.190709
$722$	−1.64837e7	−1.17683
$723$	0	0
$724$	762558.	0.0540663
$725$	898400.	0.0634782
$726$	0	0
$727$	−1.00897e7	−0.708012	−0.354006	−	0.935243i	$-0.615181\pi$
$727$	−1.00897e7	−0.708012	−0.354006	+	0.935243i	$0.615181\pi$
$728$	2.10288e6	0.147057
$729$	0	0
$730$	2.85441e6	0.198248
$731$	−2.47028e7	−1.70983
$732$	0	0
$733$	−1.48230e7	−1.01901	−0.509504	−	0.860468i	$-0.670171\pi$
$733$	−1.48230e7	−1.01901	−0.509504	+	0.860468i	$0.670171\pi$
$734$	−2.07192e7	−1.41949
$735$	0	0
$736$	1.61872e6	0.110148
$737$	1.48028e6	0.100387
$738$	0	0
$739$	−1.85393e7	−1.24877	−0.624383	−	0.781118i	$-0.714650\pi$
$739$	−1.85393e7	−1.24877	−0.624383	+	0.781118i	$0.714650\pi$
$740$	−252269.	−0.0169350
$741$	0	0
$742$	−1.85471e6	−0.123671
$743$	8.68829e6	0.577381	0.288690	−	0.957423i	$-0.406780\pi$
$743$	8.68829e6	0.577381	0.288690	+	0.957423i	$0.406780\pi$
$744$	0	0
$745$	−1.81157e6	−0.119582
$746$	−2.95717e6	−0.194549
$747$	0	0
$748$	−373160.	−0.0243860
$749$	−3.39303e6	−0.220995
$750$	0	0
$751$	−1.37377e7	−0.888821	−0.444411	−	0.895823i	$-0.646587\pi$
$751$	−1.37377e7	−0.888821	−0.444411	+	0.895823i	$0.646587\pi$
$752$	3.68776e6	0.237803
$753$	0	0
$754$	−5.60717e6	−0.359183
$755$	1.53291e6	0.0978696
$756$	0	0
$757$	5.59097e6	0.354607	0.177304	−	0.984156i	$-0.443263\pi$
$757$	5.59097e6	0.354607	0.177304	+	0.984156i	$0.443263\pi$
$758$	−1.41832e7	−0.896608
$759$	0	0
$760$	1.01929e7	0.640123
$761$	2.44147e7	1.52823	0.764117	−	0.645078i	$-0.223175\pi$
$761$	2.44147e7	1.52823	0.764117	+	0.645078i	$0.223175\pi$
$762$	0	0
$763$	1.69972e6	0.105698
$764$	996586.	0.0617706
$765$	0	0
$766$	−8.46309e6	−0.521143
$767$	6.05397e6	0.371580
$768$	0	0
$769$	1.99742e7	1.21802	0.609008	−	0.793164i	$-0.291568\pi$
$769$	1.99742e7	1.21802	0.609008	+	0.793164i	$0.291568\pi$
$770$	−308819.	−0.0187705
$771$	0	0
$772$	768096.	0.0463844
$773$	2.64430e7	1.59170	0.795852	−	0.605492i	$-0.207024\pi$
$773$	2.64430e7	1.59170	0.795852	+	0.605492i	$0.207024\pi$
$774$	0	0
$775$	−3.22354e6	−0.192787
$776$	2.68716e7	1.60191
$777$	0	0
$778$	−4.65032e6	−0.275445
$779$	6.56515e6	0.387616
$780$	0	0
$781$	6.75991e6	0.396564
$782$	−3.76336e7	−2.20069
$783$	0	0
$784$	1.76251e7	1.02410
$785$	−1.26447e7	−0.732378
$786$	0	0
$787$	−2.87892e6	−0.165689	−0.0828443	−	0.996562i	$-0.526400\pi$
$787$	−2.87892e6	−0.165689	−0.0828443	+	0.996562i	$0.526400\pi$
$788$	680310.	0.0390294
$789$	0	0
$790$	−1.95154e6	−0.111252
$791$	3.71300e6	0.211001
$792$	0	0
$793$	−255474.	−0.0144266
$794$	1.66181e7	0.935469
$795$	0	0
$796$	−635855.	−0.0355693
$797$	2.55634e7	1.42552	0.712759	−	0.701409i	$-0.247446\pi$
$797$	2.55634e7	1.42552	0.712759	+	0.701409i	$0.247446\pi$
$798$	0	0
$799$	−7.30875e6	−0.405020
$800$	329279.	0.0181903
$801$	0	0
$802$	−1.99595e7	−1.09575
$803$	−2.38849e6	−0.130718
$804$	0	0
$805$	−1.35571e6	−0.0737356
$806$	2.01190e7	1.09086
$807$	0	0
$808$	−1.68202e7	−0.906365
$809$	2.32458e6	0.124874	0.0624371	−	0.998049i	$-0.480113\pi$
$809$	2.32458e6	0.124874	0.0624371	+	0.998049i	$0.480113\pi$
$810$	0	0
$811$	1.44240e7	0.770076	0.385038	−	0.922901i	$-0.374188\pi$
$811$	1.44240e7	0.770076	0.385038	+	0.922901i	$0.374188\pi$
$812$	36947.9	0.00196653
$813$	0	0
$814$	4.84939e6	0.256523
$815$	−9.69255e6	−0.511145
$816$	0	0
$817$	2.69212e7	1.41104
$818$	6.81233e6	0.355969
$819$	0	0
$820$	103574.	0.00537916
$821$	−9.95289e6	−0.515337	−0.257668	−	0.966233i	$-0.582954\pi$
$821$	−9.95289e6	−0.515337	−0.257668	+	0.966233i	$0.582954\pi$
$822$	0	0
$823$	2.46452e7	1.26833	0.634165	−	0.773198i	$-0.281344\pi$
$823$	2.46452e7	1.26833	0.634165	+	0.773198i	$0.281344\pi$
$824$	−2.66460e7	−1.36714
$825$	0	0
$826$	−916439.	−0.0467362
$827$	2.09014e7	1.06270	0.531351	−	0.847152i	$-0.321685\pi$
$827$	2.09014e7	1.06270	0.531351	+	0.847152i	$0.321685\pi$
$828$	0	0
$829$	−8.31387e6	−0.420162	−0.210081	−	0.977684i	$-0.567373\pi$
$829$	−8.31387e6	−0.420162	−0.210081	+	0.977684i	$0.567373\pi$
$830$	−6.08467e6	−0.306579
$831$	0	0
$832$	2.10034e7	1.05192
$833$	−3.49312e7	−1.74422
$834$	0	0
$835$	−1.58855e7	−0.788472
$836$	406671.	0.0201246
$837$	0	0
$838$	−2.56244e6	−0.126050
$839$	4.03176e7	1.97738	0.988690	−	0.149977i	$-0.0479198\pi$
$839$	4.03176e7	1.97738	0.988690	+	0.149977i	$0.0479198\pi$
$840$	0	0
$841$	−1.84449e7	−0.899263
$842$	1.97757e7	0.961282
$843$	0	0
$844$	−1.08834e6	−0.0525905
$845$	−2.08792e6	−0.100594
$846$	0	0
$847$	258410.	0.0123766
$848$	−1.94118e7	−0.926992
$849$	0	0
$850$	−7.65539e6	−0.363430
$851$	2.12888e7	1.00769
$852$	0	0
$853$	−2.11646e7	−0.995951	−0.497976	−	0.867191i	$-0.665923\pi$
$853$	−2.11646e7	−0.995951	−0.497976	+	0.867191i	$0.665923\pi$
$854$	38673.3	0.00181454
$855$	0	0
$856$	−3.39633e7	−1.58425
$857$	1.03622e7	0.481946	0.240973	−	0.970532i	$-0.422533\pi$
$857$	1.03622e7	0.481946	0.240973	+	0.970532i	$0.422533\pi$
$858$	0	0
$859$	−1.77409e7	−0.820336	−0.410168	−	0.912010i	$-0.634530\pi$
$859$	−1.77409e7	−0.820336	−0.410168	+	0.912010i	$0.634530\pi$
$860$	424716.	0.0195818
$861$	0	0
$862$	−2.45618e7	−1.12588
$863$	2.62577e6	0.120014	0.0600068	−	0.998198i	$-0.480888\pi$
$863$	2.62577e6	0.120014	0.0600068	+	0.998198i	$0.480888\pi$
$864$	0	0
$865$	−1.69955e7	−0.772314
$866$	5.98118e6	0.271014
$867$	0	0
$868$	−132572.	−0.00597247
$869$	1.63299e6	0.0733557
$870$	0	0
$871$	−8.25039e6	−0.368493
$872$	1.70137e7	0.757718
$873$	0	0
$874$	4.10132e7	1.81612
$875$	−275778.	−0.0121770
$876$	0	0
$877$	7.30171e6	0.320572	0.160286	−	0.987071i	$-0.448758\pi$
$877$	7.30171e6	0.320572	0.160286	+	0.987071i	$0.448758\pi$
$878$	−2.05427e7	−0.899334
$879$	0	0
$880$	−3.23216e6	−0.140697
$881$	−6.08524e6	−0.264142	−0.132071	−	0.991240i	$-0.542163\pi$
$881$	−6.08524e6	−0.264142	−0.132071	+	0.991240i	$0.542163\pi$
$882$	0	0
$883$	1.66311e7	0.717828	0.358914	−	0.933371i	$-0.383147\pi$
$883$	1.66311e7	0.717828	0.358914	+	0.933371i	$0.383147\pi$
$884$	2.07981e6	0.0895146
$885$	0	0
$886$	1.36006e7	0.582067
$887$	1.47741e7	0.630511	0.315256	−	0.949007i	$-0.397910\pi$
$887$	1.47741e7	0.630511	0.315256	+	0.949007i	$0.397910\pi$
$888$	0	0
$889$	−1.12129e6	−0.0475841
$890$	1.17452e7	0.497032
$891$	0	0
$892$	1.26796e6	0.0533574
$893$	7.96511e6	0.334244
$894$	0	0
$895$	−1.57708e6	−0.0658108
$896$	−3.47702e6	−0.144689
$897$	0	0
$898$	1.74651e7	0.722736
$899$	−7.41383e6	−0.305945
$900$	0	0
$901$	3.84722e7	1.57883
$902$	−1.99101e6	−0.0814810
$903$	0	0
$904$	3.71661e7	1.51261
$905$	−1.30904e7	−0.531288
$906$	0	0
$907$	−9.23123e6	−0.372599	−0.186299	−	0.982493i	$-0.559649\pi$
$907$	−9.23123e6	−0.372599	−0.186299	+	0.982493i	$0.559649\pi$
$908$	−119891.	−0.00482583
$909$	0	0
$910$	1.72121e6	0.0689016
$911$	1.55931e7	0.622495	0.311247	−	0.950329i	$-0.399253\pi$
$911$	1.55931e7	0.622495	0.311247	+	0.950329i	$0.399253\pi$
$912$	0	0
$913$	5.09147e6	0.202146
$914$	−2.20185e7	−0.871812
$915$	0	0
$916$	−495349.	−0.0195062
$917$	1.56017e6	0.0612700
$918$	0	0
$919$	−4.21091e6	−0.164470	−0.0822351	−	0.996613i	$-0.526206\pi$
$919$	−4.21091e6	−0.164470	−0.0822351	+	0.996613i	$0.526206\pi$
$920$	−1.35703e7	−0.528590
$921$	0	0
$922$	4.30040e7	1.66602
$923$	−3.76765e7	−1.45568
$924$	0	0
$925$	4.33055e6	0.166413
$926$	629009.	0.0241062
$927$	0	0
$928$	757310.	0.0288671
$929$	−4.10363e7	−1.56002	−0.780008	−	0.625770i	$-0.784785\pi$
$929$	−4.10363e7	−1.56002	−0.780008	+	0.625770i	$0.784785\pi$
$930$	0	0
$931$	3.80682e7	1.43942
$932$	361730.	0.0136410
$933$	0	0
$934$	−3.96488e7	−1.48718
$935$	6.40581e6	0.239632
$936$	0	0
$937$	2.41565e7	0.898844	0.449422	−	0.893320i	$-0.351630\pi$
$937$	2.41565e7	0.898844	0.449422	+	0.893320i	$0.351630\pi$
$938$	1.24893e6	0.0463479
$939$	0	0
$940$	125660.	0.00463849
$941$	−3.13080e7	−1.15261	−0.576304	−	0.817236i	$-0.695505\pi$
$941$	−3.13080e7	−1.15261	−0.576304	+	0.817236i	$0.695505\pi$
$942$	0	0
$943$	−8.74049e6	−0.320079
$944$	−9.59164e6	−0.350318
$945$	0	0
$946$	−8.16436e6	−0.296616
$947$	−5699.21	−0.000206509 0	−0.000103255	−	1.00000i	$-0.500033\pi$
$947$	−5699.21	−0.000206509 0	−0.000103255		1.00000i	$0.500033\pi$
$948$	0	0
$949$	1.33123e7	0.479829
$950$	8.34288e6	0.299921
$951$	0	0
$952$	6.60310e6	0.236132
$953$	4.24159e7	1.51285	0.756427	−	0.654078i	$-0.226943\pi$
$953$	4.24159e7	1.51285	0.756427	+	0.654078i	$0.226943\pi$
$954$	0	0
$955$	−1.71078e7	−0.606995
$956$	2.00942e6	0.0711094
$957$	0	0
$958$	−5.48718e7	−1.93168
$959$	752628.	0.0264261
$960$	0	0
$961$	−2.02765e6	−0.0708247
$962$	−2.70282e7	−0.941628
$963$	0	0
$964$	−2.23375e6	−0.0774180
$965$	−1.31854e7	−0.455801
$966$	0	0
$967$	1.43147e7	0.492284	0.246142	−	0.969234i	$-0.420837\pi$
$967$	1.43147e7	0.492284	0.246142	+	0.969234i	$0.420837\pi$
$968$	2.58661e6	0.0887243
$969$	0	0
$970$	2.19944e7	0.750556
$971$	1.06931e7	0.363960	0.181980	−	0.983302i	$-0.441749\pi$
$971$	1.06931e7	0.363960	0.181980	+	0.983302i	$0.441749\pi$
$972$	0	0
$973$	2.99168e6	0.101306
$974$	−5.66714e7	−1.91411
$975$	0	0
$976$	404762.	0.0136011
$977$	−5.44429e7	−1.82476	−0.912378	−	0.409349i	$-0.865756\pi$
$977$	−5.44429e7	−1.82476	−0.912378	+	0.409349i	$0.865756\pi$
$978$	0	0
$979$	−9.82800e6	−0.327724
$980$	600574.	0.0199757
$981$	0	0
$982$	5.30461e7	1.75539
$983$	−4.16457e7	−1.37463	−0.687316	−	0.726359i	$-0.741211\pi$
$983$	−4.16457e7	−1.37463	−0.687316	+	0.726359i	$0.741211\pi$
$984$	0	0
$985$	−1.16785e7	−0.383526
$986$	−1.76067e7	−0.576746
$987$	0	0
$988$	−2.26659e6	−0.0738722
$989$	−3.58414e7	−1.16518
$990$	0	0
$991$	−9.62055e6	−0.311183	−0.155591	−	0.987821i	$-0.549728\pi$
$991$	−9.62055e6	−0.311183	−0.155591	+	0.987821i	$0.549728\pi$
$992$	−2.71730e6	−0.0876714
$993$	0	0
$994$	5.70340e6	0.183091
$995$	1.09153e7	0.349526
$996$	0	0
$997$	3.05865e7	0.974523	0.487262	−	0.873256i	$-0.337996\pi$
$997$	3.05865e7	0.974523	0.487262	+	0.873256i	$0.337996\pi$
$998$	1.10495e7	0.351169
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.c.1.3	✓	3	3.2	odd	2
495.6.a.b.1.1		3	1.1	even	1	trivial
825.6.a.g.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-5.78415 q^{2} +1.45634 q^{4} -25.0000 q^{5} +17.6498 q^{7} +176.669 q^{8} +O(q^{10})\)	\(q-5.78415 q^{2} +1.45634 q^{4} -25.0000 q^{5} +17.6498 q^{7} +176.669 q^{8} +144.604 q^{10} -121.000 q^{11} +674.396 q^{13} -102.089 q^{14} -1068.48 q^{16} +2117.62 q^{17} -2307.79 q^{19} -36.4084 q^{20} +699.882 q^{22} +3072.47 q^{23} +625.000 q^{25} -3900.80 q^{26} +25.7040 q^{28} +1437.44 q^{29} -5157.66 q^{31} +526.846 q^{32} -12248.6 q^{34} -441.244 q^{35} +6928.88 q^{37} +13348.6 q^{38} -4416.72 q^{40} -2844.78 q^{41} -11665.3 q^{43} -176.217 q^{44} -17771.6 q^{46} -3451.40 q^{47} -16495.5 q^{49} -3615.09 q^{50} +982.146 q^{52} +18167.6 q^{53} +3025.00 q^{55} +3118.17 q^{56} -8314.36 q^{58} +8976.88 q^{59} -378.820 q^{61} +29832.7 q^{62} +31144.1 q^{64} -16859.9 q^{65} -12233.7 q^{67} +3083.97 q^{68} +2552.22 q^{70} -55867.0 q^{71} +19739.6 q^{73} -40077.6 q^{74} -3360.92 q^{76} -2135.62 q^{77} -13495.8 q^{79} +26712.0 q^{80} +16454.6 q^{82} -42078.3 q^{83} -52940.5 q^{85} +67474.0 q^{86} -21376.9 q^{88} +81223.2 q^{89} +11902.9 q^{91} +4474.55 q^{92} +19963.4 q^{94} +57694.8 q^{95} +152101. q^{97} +95412.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8	\( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25} - 2132 q^{26} - 580 q^{28} + 4538 q^{29} - 14968 q^{31} + 2496 q^{32} - 13704 q^{34} + 1700 q^{35} - 6190 q^{37} + 11668 q^{38} - 600 q^{40} + 8926 q^{41} - 33592 q^{43} + 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 14693 q^{49} - 1250 q^{50} + 18780 q^{52} + 22934 q^{53} + 9075 q^{55} + 40012 q^{56} - 32304 q^{58} + 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 35474 q^{64} - 7250 q^{65} + 16868 q^{67} + 71288 q^{68} + 22900 q^{70} - 4856 q^{71} + 1910 q^{73} - 29404 q^{74} + 6116 q^{76} + 8228 q^{77} - 36844 q^{79} + 14750 q^{80} + 84000 q^{82} + 48796 q^{83} + 10850 q^{85} + 83492 q^{86} - 2904 q^{88} + 188978 q^{89} - 93208 q^{91} + 6976 q^{92} + 70472 q^{94} + 71400 q^{95} + 247526 q^{97} + 154654 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * q^98

Introduction

L-functions

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