LMFDB - Embedded newform 336.6.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{505}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 126 $ x^2 - x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.7361$ of defining polynomial
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -28.4166 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -28.4166 q^{5} -49.0000 q^{7} +81.0000 q^{9} -424.083 q^{11} +508.500 q^{13} +255.750 q^{15} -539.916 q^{17} -2603.00 q^{19} +441.000 q^{21} -261.251 q^{23} -2317.50 q^{25} -729.000 q^{27} +6879.66 q^{29} -5687.00 q^{31} +3816.75 q^{33} +1392.41 q^{35} +4909.50 q^{37} -4576.50 q^{39} -5723.42 q^{41} +1733.99 q^{43} -2301.75 q^{45} +10147.8 q^{47} +2401.00 q^{49} +4859.24 q^{51} -31181.5 q^{53} +12051.0 q^{55} +23427.0 q^{57} +38845.5 q^{59} +13651.0 q^{61} -3969.00 q^{63} -14449.8 q^{65} +30741.5 q^{67} +2351.26 q^{69} +45627.9 q^{71} +21753.5 q^{73} +20857.5 q^{75} +20780.1 q^{77} -32295.5 q^{79} +6561.00 q^{81} +46637.3 q^{83} +15342.6 q^{85} -61917.0 q^{87} -63757.4 q^{89} -24916.5 q^{91} +51183.0 q^{93} +73968.4 q^{95} +115122. q^{97} -34350.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} + 78 q^{5} - 98 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 + 78 * q^5 - 98 * q^7 + 162 * q^9	$ 2 q - 18 q^{3} + 78 q^{5} - 98 q^{7} + 162 q^{9} - 174 q^{11} + 208 q^{13} - 702 q^{15} + 1482 q^{17} - 352 q^{19} + 882 q^{21} - 3354 q^{23} + 5882 q^{25} - 1458 q^{27} + 276 q^{29} - 6520 q^{31} + 1566 q^{33} - 3822 q^{35} + 13864 q^{37} - 1872 q^{39} - 12930 q^{41} - 12712 q^{43} + 6318 q^{45} + 28116 q^{47} + 4802 q^{49} - 13338 q^{51} - 46992 q^{53} + 38664 q^{55} + 3168 q^{57} + 65556 q^{59} - 13148 q^{61} - 7938 q^{63} - 46428 q^{65} + 75236 q^{67} + 30186 q^{69} + 66042 q^{71} + 60496 q^{73} - 52938 q^{75} + 8526 q^{77} + 34916 q^{79} + 13122 q^{81} + 82488 q^{83} + 230508 q^{85} - 2484 q^{87} + 42510 q^{89} - 10192 q^{91} + 58680 q^{93} + 313512 q^{95} + 213256 q^{97} - 14094 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 + 78 * q^5 - 98 * q^7 + 162 * q^9 - 174 * q^11 + 208 * q^13 - 702 * q^15 + 1482 * q^17 - 352 * q^19 + 882 * q^21 - 3354 * q^23 + 5882 * q^25 - 1458 * q^27 + 276 * q^29 - 6520 * q^31 + 1566 * q^33 - 3822 * q^35 + 13864 * q^37 - 1872 * q^39 - 12930 * q^41 - 12712 * q^43 + 6318 * q^45 + 28116 * q^47 + 4802 * q^49 - 13338 * q^51 - 46992 * q^53 + 38664 * q^55 + 3168 * q^57 + 65556 * q^59 - 13148 * q^61 - 7938 * q^63 - 46428 * q^65 + 75236 * q^67 + 30186 * q^69 + 66042 * q^71 + 60496 * q^73 - 52938 * q^75 + 8526 * q^77 + 34916 * q^79 + 13122 * q^81 + 82488 * q^83 + 230508 * q^85 - 2484 * q^87 + 42510 * q^89 - 10192 * q^91 + 58680 * q^93 + 313512 * q^95 + 213256 * q^97 - 14094 * q^99

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	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−28.4166	−0.508332	−0.254166	−	0.967161i	$-0.581801\pi$
$5$	−28.4166	−0.508332	−0.254166	+	0.967161i	$0.581801\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−424.083	−1.05674	−0.528371	−	0.849013i	$-0.677197\pi$
$11$	−424.083	−1.05674	−0.528371	+	0.849013i	$0.677197\pi$
$12$	0	0
$13$	508.500	0.834511	0.417256	−	0.908789i	$-0.362992\pi$
$13$	508.500	0.834511	0.417256	+	0.908789i	$0.362992\pi$
$14$	0	0
$15$	255.750	0.293486
$16$	0	0
$17$	−539.916	−0.453110	−0.226555	−	0.973998i	$-0.572746\pi$
$17$	−539.916	−0.453110	−0.226555	+	0.973998i	$0.572746\pi$
$18$	0	0
$19$	−2603.00	−1.65421	−0.827104	−	0.562050i	$-0.810013\pi$
$19$	−2603.00	−1.65421	−0.827104	+	0.562050i	$0.810013\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	−261.251	−0.102977	−0.0514883	−	0.998674i	$-0.516396\pi$
$23$	−261.251	−0.102977	−0.0514883	+	0.998674i	$0.516396\pi$
$24$	0	0
$25$	−2317.50	−0.741599
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	6879.66	1.51905	0.759525	−	0.650478i	$-0.225431\pi$
$29$	6879.66	1.51905	0.759525	+	0.650478i	$0.225431\pi$
$30$	0	0
$31$	−5687.00	−1.06287	−0.531433	−	0.847100i	$-0.678346\pi$
$31$	−5687.00	−1.06287	−0.531433	+	0.847100i	$0.678346\pi$
$32$	0	0
$33$	3816.75	0.610111
$34$	0	0
$35$	1392.41	0.192131
$36$	0	0
$37$	4909.50	0.589567	0.294783	−	0.955564i	$-0.404752\pi$
$37$	4909.50	0.589567	0.294783	+	0.955564i	$0.404752\pi$
$38$	0	0
$39$	−4576.50	−0.481805
$40$	0	0
$41$	−5723.42	−0.531736	−0.265868	−	0.964009i	$-0.585658\pi$
$41$	−5723.42	−0.531736	−0.265868	+	0.964009i	$0.585658\pi$
$42$	0	0
$43$	1733.99	0.143013	0.0715066	−	0.997440i	$-0.477219\pi$
$43$	1733.99	0.143013	0.0715066	+	0.997440i	$0.477219\pi$
$44$	0	0
$45$	−2301.75	−0.169444
$46$	0	0
$47$	10147.8	0.670083	0.335042	−	0.942203i	$-0.391250\pi$
$47$	10147.8	0.670083	0.335042	+	0.942203i	$0.391250\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	4859.24	0.261603
$52$	0	0
$53$	−31181.5	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$53$	−31181.5	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$54$	0	0
$55$	12051.0	0.537176
$56$	0	0
$57$	23427.0	0.955057
$58$	0	0
$59$	38845.5	1.45282	0.726408	−	0.687264i	$-0.241188\pi$
$59$	38845.5	1.45282	0.726408	+	0.687264i	$0.241188\pi$
$60$	0	0
$61$	13651.0	0.469720	0.234860	−	0.972029i	$-0.424537\pi$
$61$	13651.0	0.469720	0.234860	+	0.972029i	$0.424537\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	−14449.8	−0.424209
$66$	0	0
$67$	30741.5	0.836639	0.418320	−	0.908300i	$-0.362619\pi$
$67$	30741.5	0.836639	0.418320	+	0.908300i	$0.362619\pi$
$68$	0	0
$69$	2351.26	0.0594536
$70$	0	0
$71$	45627.9	1.07420	0.537099	−	0.843519i	$-0.319520\pi$
$71$	45627.9	1.07420	0.537099	+	0.843519i	$0.319520\pi$
$72$	0	0
$73$	21753.5	0.477774	0.238887	−	0.971047i	$-0.423218\pi$
$73$	21753.5	0.477774	0.238887	+	0.971047i	$0.423218\pi$
$74$	0	0
$75$	20857.5	0.428162
$76$	0	0
$77$	20780.1	0.399411
$78$	0	0
$79$	−32295.5	−0.582202	−0.291101	−	0.956692i	$-0.594022\pi$
$79$	−32295.5	−0.582202	−0.291101	+	0.956692i	$0.594022\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	46637.3	0.743085	0.371542	−	0.928416i	$-0.378829\pi$
$83$	46637.3	0.743085	0.371542	+	0.928416i	$0.378829\pi$
$84$	0	0
$85$	15342.6	0.230330
$86$	0	0
$87$	−61917.0	−0.877024
$88$	0	0
$89$	−63757.4	−0.853209	−0.426604	−	0.904438i	$-0.640290\pi$
$89$	−63757.4	−0.853209	−0.426604	+	0.904438i	$0.640290\pi$
$90$	0	0
$91$	−24916.5	−0.315416
$92$	0	0
$93$	51183.0	0.613646
$94$	0	0
$95$	73968.4	0.840886
$96$	0	0
$97$	115122.	1.24231	0.621156	−	0.783687i	$-0.286663\pi$
$97$	115122.	1.24231	0.621156	+	0.783687i	$0.286663\pi$
$98$	0	0
$99$	−34350.7	−0.352248
$100$	0	0
$101$	136803.	1.33442	0.667208	−	0.744872i	$-0.267489\pi$
$101$	136803.	1.33442	0.667208	+	0.744872i	$0.267489\pi$
$102$	0	0
$103$	30426.0	0.282587	0.141293	−	0.989968i	$-0.454874\pi$
$103$	30426.0	0.282587	0.141293	+	0.989968i	$0.454874\pi$
$104$	0	0
$105$	−12531.7	−0.110927
$106$	0	0
$107$	134612.	1.13664	0.568321	−	0.822807i	$-0.307593\pi$
$107$	134612.	1.13664	0.568321	+	0.822807i	$0.307593\pi$
$108$	0	0
$109$	7722.00	0.0622535	0.0311267	−	0.999515i	$-0.490090\pi$
$109$	7722.00	0.0622535	0.0311267	+	0.999515i	$0.490090\pi$
$110$	0	0
$111$	−44185.5	−0.340387
$112$	0	0
$113$	214596.	1.58098	0.790489	−	0.612476i	$-0.209826\pi$
$113$	214596.	1.58098	0.790489	+	0.612476i	$0.209826\pi$
$114$	0	0
$115$	7423.87	0.0523463
$116$	0	0
$117$	41188.5	0.278170
$118$	0	0
$119$	26455.9	0.171259
$120$	0	0
$121$	18795.5	0.116705
$122$	0	0
$123$	51510.8	0.306998
$124$	0	0
$125$	154657.	0.885310
$126$	0	0
$127$	−19445.6	−0.106982	−0.0534911	−	0.998568i	$-0.517035\pi$
$127$	−19445.6	−0.106982	−0.0534911	+	0.998568i	$0.517035\pi$
$128$	0	0
$129$	−15605.9	−0.0825688
$130$	0	0
$131$	63451.1	0.323043	0.161522	−	0.986869i	$-0.448360\pi$
$131$	63451.1	0.323043	0.161522	+	0.986869i	$0.448360\pi$
$132$	0	0
$133$	127547.	0.625231
$134$	0	0
$135$	20715.7	0.0978285
$136$	0	0
$137$	−231176.	−1.05231	−0.526153	−	0.850390i	$-0.676366\pi$
$137$	−231176.	−1.05231	−0.526153	+	0.850390i	$0.676366\pi$
$138$	0	0
$139$	−419126.	−1.83996	−0.919978	−	0.391971i	$-0.871793\pi$
$139$	−419126.	−1.83996	−0.919978	+	0.391971i	$0.871793\pi$
$140$	0	0
$141$	−91330.5	−0.386873
$142$	0	0
$143$	−215646.	−0.881864
$144$	0	0
$145$	−195497.	−0.772182
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−19393.6	−0.0715635	−0.0357818	−	0.999360i	$-0.511392\pi$
$149$	−19393.6	−0.0715635	−0.0357818	+	0.999360i	$0.511392\pi$
$150$	0	0
$151$	545294.	1.94620	0.973102	−	0.230376i	$-0.0739956\pi$
$151$	545294.	1.94620	0.973102	+	0.230376i	$0.0739956\pi$
$152$	0	0
$153$	−43733.2	−0.151037
$154$	0	0
$155$	161605.	0.540289
$156$	0	0
$157$	9538.03	0.0308823	0.0154411	−	0.999881i	$-0.495085\pi$
$157$	9538.03	0.0308823	0.0154411	+	0.999881i	$0.495085\pi$
$158$	0	0
$159$	280633.	0.880332
$160$	0	0
$161$	12801.3	0.0389215
$162$	0	0
$163$	566744.	1.67078	0.835388	−	0.549661i	$-0.185243\pi$
$163$	566744.	1.67078	0.835388	+	0.549661i	$0.185243\pi$
$164$	0	0
$165$	−108459.	−0.310139
$166$	0	0
$167$	−501555.	−1.39164	−0.695821	−	0.718215i	$-0.744959\pi$
$167$	−501555.	−1.39164	−0.695821	+	0.718215i	$0.744959\pi$
$168$	0	0
$169$	−112721.	−0.303591
$170$	0	0
$171$	−210843.	−0.551402
$172$	0	0
$173$	111671.	0.283677	0.141839	−	0.989890i	$-0.454699\pi$
$173$	111671.	0.283677	0.141839	+	0.989890i	$0.454699\pi$
$174$	0	0
$175$	113557.	0.280298
$176$	0	0
$177$	−349609.	−0.838784
$178$	0	0
$179$	528436.	1.23271	0.616354	−	0.787470i	$-0.288609\pi$
$179$	528436.	1.23271	0.616354	+	0.787470i	$0.288609\pi$
$180$	0	0
$181$	−290808.	−0.659797	−0.329898	−	0.944016i	$-0.607014\pi$
$181$	−290808.	−0.659797	−0.329898	+	0.944016i	$0.607014\pi$
$182$	0	0
$183$	−122859.	−0.271193
$184$	0	0
$185$	−139511.	−0.299696
$186$	0	0
$187$	228969.	0.478821
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	−51989.0	−0.103116	−0.0515582	−	0.998670i	$-0.516419\pi$
$191$	−51989.0	−0.103116	−0.0515582	+	0.998670i	$0.516419\pi$
$192$	0	0
$193$	−526004.	−1.01647	−0.508236	−	0.861218i	$-0.669702\pi$
$193$	−526004.	−1.01647	−0.508236	+	0.861218i	$0.669702\pi$
$194$	0	0
$195$	130049.	0.244917
$196$	0	0
$197$	977503.	1.79454	0.897269	−	0.441485i	$-0.145548\pi$
$197$	977503.	1.79454	0.897269	+	0.441485i	$0.145548\pi$
$198$	0	0
$199$	−1.01780e6	−1.82192	−0.910962	−	0.412491i	$-0.864659\pi$
$199$	−1.01780e6	−1.82192	−0.910962	+	0.412491i	$0.864659\pi$
$200$	0	0
$201$	−276674.	−0.483034
$202$	0	0
$203$	−337103.	−0.574147
$204$	0	0
$205$	162640.	0.270298
$206$	0	0
$207$	−21161.3	−0.0343255
$208$	0	0
$209$	1.10389e6	1.74807
$210$	0	0
$211$	−248461.	−0.384195	−0.192098	−	0.981376i	$-0.561529\pi$
$211$	−248461.	−0.384195	−0.192098	+	0.981376i	$0.561529\pi$
$212$	0	0
$213$	−410651.	−0.620189
$214$	0	0
$215$	−49274.2	−0.0726982
$216$	0	0
$217$	278663.	0.401726
$218$	0	0
$219$	−195782.	−0.275843
$220$	0	0
$221$	−274547.	−0.378125
$222$	0	0
$223$	118450.	0.159505	0.0797525	−	0.996815i	$-0.474587\pi$
$223$	118450.	0.159505	0.0797525	+	0.996815i	$0.474587\pi$
$224$	0	0
$225$	−187717.	−0.247200
$226$	0	0
$227$	−717556.	−0.924254	−0.462127	−	0.886814i	$-0.652914\pi$
$227$	−717556.	−0.924254	−0.462127	+	0.886814i	$0.652914\pi$
$228$	0	0
$229$	−1.33467e6	−1.68184	−0.840918	−	0.541162i	$-0.817984\pi$
$229$	−1.33467e6	−1.68184	−0.840918	+	0.541162i	$0.817984\pi$
$230$	0	0
$231$	−187021.	−0.230600
$232$	0	0
$233$	−68576.4	−0.0827532	−0.0413766	−	0.999144i	$-0.513174\pi$
$233$	−68576.4	−0.0827532	−0.0413766	+	0.999144i	$0.513174\pi$
$234$	0	0
$235$	−288367.	−0.340625
$236$	0	0
$237$	290659.	0.336135
$238$	0	0
$239$	−461972.	−0.523144	−0.261572	−	0.965184i	$-0.584241\pi$
$239$	−461972.	−0.523144	−0.261572	+	0.965184i	$0.584241\pi$
$240$	0	0
$241$	142028.	0.157518	0.0787592	−	0.996894i	$-0.474904\pi$
$241$	142028.	0.157518	0.0787592	+	0.996894i	$0.474904\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−68228.3	−0.0726188
$246$	0	0
$247$	−1.32362e6	−1.38045
$248$	0	0
$249$	−419736.	−0.429020
$250$	0	0
$251$	672493.	0.673757	0.336879	−	0.941548i	$-0.390629\pi$
$251$	672493.	0.673757	0.336879	+	0.941548i	$0.390629\pi$
$252$	0	0
$253$	110792.	0.108820
$254$	0	0
$255$	−138083.	−0.132981
$256$	0	0
$257$	1.16584e6	1.10104	0.550522	−	0.834821i	$-0.314429\pi$
$257$	1.16584e6	1.10104	0.550522	+	0.834821i	$0.314429\pi$
$258$	0	0
$259$	−240566.	−0.222835
$260$	0	0
$261$	557253.	0.506350
$262$	0	0
$263$	1.23336e6	1.09952	0.549758	−	0.835324i	$-0.314720\pi$
$263$	1.23336e6	1.09952	0.549758	+	0.835324i	$0.314720\pi$
$264$	0	0
$265$	886073.	0.775094
$266$	0	0
$267$	573816.	0.492600
$268$	0	0
$269$	−71305.8	−0.0600819	−0.0300410	−	0.999549i	$-0.509564\pi$
$269$	−71305.8	−0.0600819	−0.0300410	+	0.999549i	$0.509564\pi$
$270$	0	0
$271$	−1.20152e6	−0.993821	−0.496911	−	0.867802i	$-0.665532\pi$
$271$	−1.20152e6	−0.993821	−0.496911	+	0.867802i	$0.665532\pi$
$272$	0	0
$273$	224248.	0.182105
$274$	0	0
$275$	982811.	0.783679
$276$	0	0
$277$	−1.64304e6	−1.28661	−0.643307	−	0.765608i	$-0.722438\pi$
$277$	−1.64304e6	−1.28661	−0.643307	+	0.765608i	$0.722438\pi$
$278$	0	0
$279$	−460647.	−0.354289
$280$	0	0
$281$	−37163.4	−0.0280770	−0.0140385	−	0.999901i	$-0.504469\pi$
$281$	−37163.4	−0.0280770	−0.0140385	+	0.999901i	$0.504469\pi$
$282$	0	0
$283$	19110.2	0.0141841	0.00709203	−	0.999975i	$-0.497743\pi$
$283$	19110.2	0.0141841	0.00709203	+	0.999975i	$0.497743\pi$
$284$	0	0
$285$	−665716.	−0.485486
$286$	0	0
$287$	280447.	0.200977
$288$	0	0
$289$	−1.12835e6	−0.794691
$290$	0	0
$291$	−1.03610e6	−0.717249
$292$	0	0
$293$	−1.49662e6	−1.01846	−0.509230	−	0.860631i	$-0.670070\pi$
$293$	−1.49662e6	−1.01846	−0.509230	+	0.860631i	$0.670070\pi$
$294$	0	0
$295$	−1.10386e6	−0.738513
$296$	0	0
$297$	309157.	0.203370
$298$	0	0
$299$	−132846.	−0.0859351
$300$	0	0
$301$	−84965.7	−0.0540539
$302$	0	0
$303$	−1.23122e6	−0.770425
$304$	0	0
$305$	−387915.	−0.238774
$306$	0	0
$307$	1.82699e6	1.10634	0.553172	−	0.833067i	$-0.313417\pi$
$307$	1.82699e6	1.10634	0.553172	+	0.833067i	$0.313417\pi$
$308$	0	0
$309$	−273834.	−0.163152
$310$	0	0
$311$	1.65586e6	0.970784	0.485392	−	0.874297i	$-0.338677\pi$
$311$	1.65586e6	0.970784	0.485392	+	0.874297i	$0.338677\pi$
$312$	0	0
$313$	−3.03384e6	−1.75038	−0.875189	−	0.483781i	$-0.839263\pi$
$313$	−3.03384e6	−1.75038	−0.875189	+	0.483781i	$0.839263\pi$
$314$	0	0
$315$	112786.	0.0640438
$316$	0	0
$317$	−1.40939e6	−0.787741	−0.393871	−	0.919166i	$-0.628864\pi$
$317$	−1.40939e6	−0.787741	−0.393871	+	0.919166i	$0.628864\pi$
$318$	0	0
$319$	−2.91755e6	−1.60524
$320$	0	0
$321$	−1.21151e6	−0.656241
$322$	0	0
$323$	1.40540e6	0.749538
$324$	0	0
$325$	−1.17845e6	−0.618873
$326$	0	0
$327$	−69498.0	−0.0359421
$328$	0	0
$329$	−497244.	−0.253268
$330$	0	0
$331$	942703.	0.472939	0.236469	−	0.971639i	$-0.424010\pi$
$331$	942703.	0.472939	0.236469	+	0.971639i	$0.424010\pi$
$332$	0	0
$333$	397670.	0.196522
$334$	0	0
$335$	−873570.	−0.425290
$336$	0	0
$337$	1.23769e6	0.593658	0.296829	−	0.954931i	$-0.404071\pi$
$337$	1.23769e6	0.593658	0.296829	+	0.954931i	$0.404071\pi$
$338$	0	0
$339$	−1.93137e6	−0.912778
$340$	0	0
$341$	2.41176e6	1.12318
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−66814.8	−0.0302221
$346$	0	0
$347$	4.10799e6	1.83150	0.915748	−	0.401754i	$-0.131599\pi$
$347$	4.10799e6	1.83150	0.915748	+	0.401754i	$0.131599\pi$
$348$	0	0
$349$	705558.	0.310076	0.155038	−	0.987908i	$-0.450450\pi$
$349$	705558.	0.310076	0.155038	+	0.987908i	$0.450450\pi$
$350$	0	0
$351$	−370696.	−0.160602
$352$	0	0
$353$	−3.72785e6	−1.59229	−0.796143	−	0.605108i	$-0.793130\pi$
$353$	−3.72785e6	−1.59229	−0.796143	+	0.605108i	$0.793130\pi$
$354$	0	0
$355$	−1.29659e6	−0.546049
$356$	0	0
$357$	−238103.	−0.0988767
$358$	0	0
$359$	2.87917e6	1.17905	0.589524	−	0.807751i	$-0.299315\pi$
$359$	2.87917e6	1.17905	0.589524	+	0.807751i	$0.299315\pi$
$360$	0	0
$361$	4.29950e6	1.73640
$362$	0	0
$363$	−169159.	−0.0673797
$364$	0	0
$365$	−618161.	−0.242868
$366$	0	0
$367$	−1.31628e6	−0.510132	−0.255066	−	0.966924i	$-0.582097\pi$
$367$	−1.31628e6	−0.510132	−0.255066	+	0.966924i	$0.582097\pi$
$368$	0	0
$369$	−463597.	−0.177245
$370$	0	0
$371$	1.52789e6	0.576313
$372$	0	0
$373$	4.05141e6	1.50777	0.753884	−	0.657008i	$-0.228178\pi$
$373$	4.05141e6	1.50777	0.753884	+	0.657008i	$0.228178\pi$
$374$	0	0
$375$	−1.39192e6	−0.511134
$376$	0	0
$377$	3.49831e6	1.26766
$378$	0	0
$379$	4.92472e6	1.76110	0.880549	−	0.473954i	$-0.157174\pi$
$379$	4.92472e6	1.76110	0.880549	+	0.473954i	$0.157174\pi$
$380$	0	0
$381$	175010.	0.0617662
$382$	0	0
$383$	−4.55593e6	−1.58701	−0.793506	−	0.608562i	$-0.791747\pi$
$383$	−4.55593e6	−1.58701	−0.793506	+	0.608562i	$0.791747\pi$
$384$	0	0
$385$	−590499.	−0.203033
$386$	0	0
$387$	140453.	0.0476711
$388$	0	0
$389$	3.68937e6	1.23617	0.618085	−	0.786112i	$-0.287909\pi$
$389$	3.68937e6	1.23617	0.618085	+	0.786112i	$0.287909\pi$
$390$	0	0
$391$	141054.	0.0466597
$392$	0	0
$393$	−571059.	−0.186509
$394$	0	0
$395$	917728.	0.295952
$396$	0	0
$397$	1.54578e6	0.492233	0.246117	−	0.969240i	$-0.420845\pi$
$397$	1.54578e6	0.492233	0.246117	+	0.969240i	$0.420845\pi$
$398$	0	0
$399$	−1.14792e6	−0.360978
$400$	0	0
$401$	5.66295e6	1.75866	0.879331	−	0.476212i	$-0.157990\pi$
$401$	5.66295e6	1.75866	0.879331	+	0.476212i	$0.157990\pi$
$402$	0	0
$403$	−2.89184e6	−0.886975
$404$	0	0
$405$	−186441.	−0.0564813
$406$	0	0
$407$	−2.08204e6	−0.623020
$408$	0	0
$409$	727819.	0.215137	0.107568	−	0.994198i	$-0.465694\pi$
$409$	727819.	0.215137	0.107568	+	0.994198i	$0.465694\pi$
$410$	0	0
$411$	2.08059e6	0.607549
$412$	0	0
$413$	−1.90343e6	−0.549113
$414$	0	0
$415$	−1.32528e6	−0.377734
$416$	0	0
$417$	3.77213e6	1.06230
$418$	0	0
$419$	6.46306e6	1.79847	0.899235	−	0.437465i	$-0.144124\pi$
$419$	6.46306e6	1.79847	0.899235	+	0.437465i	$0.144124\pi$
$420$	0	0
$421$	−3.50166e6	−0.962874	−0.481437	−	0.876481i	$-0.659885\pi$
$421$	−3.50166e6	−0.962874	−0.481437	+	0.876481i	$0.659885\pi$
$422$	0	0
$423$	821975.	0.223361
$424$	0	0
$425$	1.25125e6	0.336026
$426$	0	0
$427$	−668898.	−0.177538
$428$	0	0
$429$	1.94082e6	0.509144
$430$	0	0
$431$	−5.35407e6	−1.38832	−0.694162	−	0.719819i	$-0.744225\pi$
$431$	−5.35407e6	−1.38832	−0.694162	+	0.719819i	$0.744225\pi$
$432$	0	0
$433$	−1.54042e6	−0.394839	−0.197420	−	0.980319i	$-0.563256\pi$
$433$	−1.54042e6	−0.394839	−0.197420	+	0.980319i	$0.563256\pi$
$434$	0	0
$435$	1.75947e6	0.445819
$436$	0	0
$437$	680036.	0.170345
$438$	0	0
$439$	3.00831e6	0.745009	0.372505	−	0.928030i	$-0.378499\pi$
$439$	3.00831e6	0.745009	0.372505	+	0.928030i	$0.378499\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	6.93545e6	1.67906	0.839528	−	0.543317i	$-0.182832\pi$
$443$	6.93545e6	1.67906	0.839528	+	0.543317i	$0.182832\pi$
$444$	0	0
$445$	1.81177e6	0.433713
$446$	0	0
$447$	174542.	0.0413172
$448$	0	0
$449$	5.41345e6	1.26724	0.633619	−	0.773646i	$-0.281569\pi$
$449$	5.41345e6	1.26724	0.633619	+	0.773646i	$0.281569\pi$
$450$	0	0
$451$	2.42720e6	0.561908
$452$	0	0
$453$	−4.90764e6	−1.12364
$454$	0	0
$455$	708042.	0.160336
$456$	0	0
$457$	551975.	0.123632	0.0618158	−	0.998088i	$-0.480311\pi$
$457$	551975.	0.123632	0.0618158	+	0.998088i	$0.480311\pi$
$458$	0	0
$459$	393599.	0.0872011
$460$	0	0
$461$	4.44254e6	0.973597	0.486799	−	0.873514i	$-0.338165\pi$
$461$	4.44254e6	0.973597	0.486799	+	0.873514i	$0.338165\pi$
$462$	0	0
$463$	4.62590e6	1.00287	0.501434	−	0.865196i	$-0.332806\pi$
$463$	4.62590e6	1.00287	0.501434	+	0.865196i	$0.332806\pi$
$464$	0	0
$465$	−1.45445e6	−0.311936
$466$	0	0
$467$	−5.50087e6	−1.16718	−0.583592	−	0.812047i	$-0.698353\pi$
$467$	−5.50087e6	−1.16718	−0.583592	+	0.812047i	$0.698353\pi$
$468$	0	0
$469$	−1.50633e6	−0.316220
$470$	0	0
$471$	−85842.2	−0.0178299
$472$	0	0
$473$	−735357.	−0.151128
$474$	0	0
$475$	6.03244e6	1.22676
$476$	0	0
$477$	−2.52570e6	−0.508260
$478$	0	0
$479$	3.56375e6	0.709690	0.354845	−	0.934925i	$-0.384534\pi$
$479$	3.56375e6	0.709690	0.354845	+	0.934925i	$0.384534\pi$
$480$	0	0
$481$	2.49648e6	0.492000
$482$	0	0
$483$	−115212.	−0.0224713
$484$	0	0
$485$	−3.27139e6	−0.631507
$486$	0	0
$487$	−3.74040e6	−0.714653	−0.357327	−	0.933979i	$-0.616312\pi$
$487$	−3.74040e6	−0.714653	−0.357327	+	0.933979i	$0.616312\pi$
$488$	0	0
$489$	−5.10070e6	−0.964623
$490$	0	0
$491$	−5.00459e6	−0.936838	−0.468419	−	0.883506i	$-0.655176\pi$
$491$	−5.00459e6	−0.936838	−0.468419	+	0.883506i	$0.655176\pi$
$492$	0	0
$493$	−3.71444e6	−0.688297
$494$	0	0
$495$	976131.	0.179059
$496$	0	0
$497$	−2.23577e6	−0.406009
$498$	0	0
$499$	−2.58167e6	−0.464141	−0.232070	−	0.972699i	$-0.574550\pi$
$499$	−2.58167e6	−0.464141	−0.232070	+	0.972699i	$0.574550\pi$
$500$	0	0
$501$	4.51400e6	0.803465
$502$	0	0
$503$	4.02089e6	0.708601	0.354301	−	0.935132i	$-0.384719\pi$
$503$	4.02089e6	0.708601	0.354301	+	0.935132i	$0.384719\pi$
$504$	0	0
$505$	−3.88747e6	−0.678326
$506$	0	0
$507$	1.01449e6	0.175278
$508$	0	0
$509$	1.13802e7	1.94696	0.973479	−	0.228776i	$-0.0734723\pi$
$509$	1.13802e7	1.94696	0.973479	+	0.228776i	$0.0734723\pi$
$510$	0	0
$511$	−1.06592e6	−0.180581
$512$	0	0
$513$	1.89759e6	0.318352
$514$	0	0
$515$	−864604.	−0.143648
$516$	0	0
$517$	−4.30353e6	−0.708106
$518$	0	0
$519$	−1.00504e6	−0.163781
$520$	0	0
$521$	−4.72031e6	−0.761862	−0.380931	−	0.924603i	$-0.624396\pi$
$521$	−4.72031e6	−0.761862	−0.380931	+	0.924603i	$0.624396\pi$
$522$	0	0
$523$	8.46281e6	1.35288	0.676441	−	0.736496i	$-0.263521\pi$
$523$	8.46281e6	1.35288	0.676441	+	0.736496i	$0.263521\pi$
$524$	0	0
$525$	−1.02202e6	−0.161830
$526$	0	0
$527$	3.07050e6	0.481596
$528$	0	0
$529$	−6.36809e6	−0.989396
$530$	0	0
$531$	3.14649e6	0.484272
$532$	0	0
$533$	−2.91036e6	−0.443739
$534$	0	0
$535$	−3.82521e6	−0.577792
$536$	0	0
$537$	−4.75592e6	−0.711704
$538$	0	0
$539$	−1.01822e6	−0.150963
$540$	0	0
$541$	1.09051e7	1.60190	0.800952	−	0.598728i	$-0.204327\pi$
$541$	1.09051e7	1.60190	0.800952	+	0.598728i	$0.204327\pi$
$542$	0	0
$543$	2.61727e6	0.380934
$544$	0	0
$545$	−219433.	−0.0316454
$546$	0	0
$547$	5.04856e6	0.721438	0.360719	−	0.932675i	$-0.382531\pi$
$547$	5.04856e6	0.721438	0.360719	+	0.932675i	$0.382531\pi$
$548$	0	0
$549$	1.10573e6	0.156573
$550$	0	0
$551$	−1.79077e7	−2.51282
$552$	0	0
$553$	1.58248e6	0.220052
$554$	0	0
$555$	1.25560e6	0.173029
$556$	0	0
$557$	−1.09175e7	−1.49103	−0.745516	−	0.666488i	$-0.767797\pi$
$557$	−1.09175e7	−1.49103	−0.745516	+	0.666488i	$0.767797\pi$
$558$	0	0
$559$	881735.	0.119346
$560$	0	0
$561$	−2.06072e6	−0.276447
$562$	0	0
$563$	2.16432e6	0.287774	0.143887	−	0.989594i	$-0.454040\pi$
$563$	2.16432e6	0.287774	0.143887	+	0.989594i	$0.454040\pi$
$564$	0	0
$565$	−6.09810e6	−0.803662
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	9.80819e6	1.27001	0.635007	−	0.772507i	$-0.280997\pi$
$569$	9.80819e6	1.27001	0.635007	+	0.772507i	$0.280997\pi$
$570$	0	0
$571$	−1.54139e7	−1.97844	−0.989221	−	0.146430i	$-0.953222\pi$
$571$	−1.54139e7	−1.97844	−0.989221	+	0.146430i	$0.953222\pi$
$572$	0	0
$573$	467901.	0.0595343
$574$	0	0
$575$	605448.	0.0763673
$576$	0	0
$577$	−7.40379e6	−0.925794	−0.462897	−	0.886412i	$-0.653190\pi$
$577$	−7.40379e6	−0.925794	−0.462897	+	0.886412i	$0.653190\pi$
$578$	0	0
$579$	4.73403e6	0.586860
$580$	0	0
$581$	−2.28523e6	−0.280860
$582$	0	0
$583$	1.32235e7	1.61130
$584$	0	0
$585$	−1.17044e6	−0.141403
$586$	0	0
$587$	−9.83732e6	−1.17837	−0.589185	−	0.807998i	$-0.700551\pi$
$587$	−9.83732e6	−1.17837	−0.589185	+	0.807998i	$0.700551\pi$
$588$	0	0
$589$	1.48032e7	1.75820
$590$	0	0
$591$	−8.79753e6	−1.03608
$592$	0	0
$593$	6.01144e6	0.702007	0.351004	−	0.936374i	$-0.385840\pi$
$593$	6.01144e6	0.702007	0.351004	+	0.936374i	$0.385840\pi$
$594$	0	0
$595$	−751786.	−0.0870567
$596$	0	0
$597$	9.16021e6	1.05189
$598$	0	0
$599$	604201.	0.0688041	0.0344020	−	0.999408i	$-0.489047\pi$
$599$	604201.	0.0688041	0.0344020	+	0.999408i	$0.489047\pi$
$600$	0	0
$601$	2.52672e6	0.285346	0.142673	−	0.989770i	$-0.454430\pi$
$601$	2.52672e6	0.285346	0.142673	+	0.989770i	$0.454430\pi$
$602$	0	0
$603$	2.49006e6	0.278880
$604$	0	0
$605$	−534103.	−0.0593249
$606$	0	0
$607$	−1.29052e7	−1.42165	−0.710823	−	0.703371i	$-0.751677\pi$
$607$	−1.29052e7	−1.42165	−0.710823	+	0.703371i	$0.751677\pi$
$608$	0	0
$609$	3.03393e6	0.331484
$610$	0	0
$611$	5.16017e6	0.559192
$612$	0	0
$613$	−3.27264e6	−0.351760	−0.175880	−	0.984412i	$-0.556277\pi$
$613$	−3.27264e6	−0.351760	−0.175880	+	0.984412i	$0.556277\pi$
$614$	0	0
$615$	−1.46376e6	−0.156057
$616$	0	0
$617$	1.31134e7	1.38676	0.693381	−	0.720571i	$-0.256120\pi$
$617$	1.31134e7	1.38676	0.693381	+	0.720571i	$0.256120\pi$
$618$	0	0
$619$	2.56630e6	0.269203	0.134602	−	0.990900i	$-0.457025\pi$
$619$	2.56630e6	0.269203	0.134602	+	0.990900i	$0.457025\pi$
$620$	0	0
$621$	190452.	0.0198179
$622$	0	0
$623$	3.12411e6	0.322483
$624$	0	0
$625$	2.84734e6	0.291567
$626$	0	0
$627$	−9.93499e6	−1.00925
$628$	0	0
$629$	−2.65072e6	−0.267139
$630$	0	0
$631$	−1.08637e7	−1.08618	−0.543092	−	0.839673i	$-0.682747\pi$
$631$	−1.08637e7	−1.08618	−0.543092	+	0.839673i	$0.682747\pi$
$632$	0	0
$633$	2.23615e6	0.221815
$634$	0	0
$635$	552577.	0.0543824
$636$	0	0
$637$	1.22091e6	0.119216
$638$	0	0
$639$	3.69586e6	0.358066
$640$	0	0
$641$	−1.55104e7	−1.49100	−0.745500	−	0.666506i	$-0.767789\pi$
$641$	−1.55104e7	−1.49100	−0.745500	+	0.666506i	$0.767789\pi$
$642$	0	0
$643$	−3.47784e6	−0.331728	−0.165864	−	0.986149i	$-0.553041\pi$
$643$	−3.47784e6	−0.331728	−0.165864	+	0.986149i	$0.553041\pi$
$644$	0	0
$645$	443468.	0.0419723
$646$	0	0
$647$	−9.45505e6	−0.887980	−0.443990	−	0.896032i	$-0.646437\pi$
$647$	−9.45505e6	−0.887980	−0.443990	+	0.896032i	$0.646437\pi$
$648$	0	0
$649$	−1.64737e7	−1.53525
$650$	0	0
$651$	−2.50797e6	−0.231937
$652$	0	0
$653$	−1.27403e7	−1.16922	−0.584610	−	0.811315i	$-0.698752\pi$
$653$	−1.27403e7	−1.16922	−0.584610	+	0.811315i	$0.698752\pi$
$654$	0	0
$655$	−1.80306e6	−0.164213
$656$	0	0
$657$	1.76203e6	0.159258
$658$	0	0
$659$	−8.02252e6	−0.719610	−0.359805	−	0.933027i	$-0.617157\pi$
$659$	−8.02252e6	−0.719610	−0.359805	+	0.933027i	$0.617157\pi$
$660$	0	0
$661$	2.12008e7	1.88734	0.943669	−	0.330892i	$-0.107350\pi$
$661$	2.12008e7	1.88734	0.943669	+	0.330892i	$0.107350\pi$
$662$	0	0
$663$	2.47092e6	0.218311
$664$	0	0
$665$	−3.62445e6	−0.317825
$666$	0	0
$667$	−1.79732e6	−0.156427
$668$	0	0
$669$	−1.06605e6	−0.0920903
$670$	0	0
$671$	−5.78915e6	−0.496374
$672$	0	0
$673$	−1.73938e7	−1.48032	−0.740162	−	0.672428i	$-0.765251\pi$
$673$	−1.73938e7	−1.48032	−0.740162	+	0.672428i	$0.765251\pi$
$674$	0	0
$675$	1.68945e6	0.142721
$676$	0	0
$677$	−2.35417e7	−1.97408	−0.987042	−	0.160464i	$-0.948701\pi$
$677$	−2.35417e7	−1.97408	−0.987042	+	0.160464i	$0.948701\pi$
$678$	0	0
$679$	−5.64100e6	−0.469550
$680$	0	0
$681$	6.45801e6	0.533618
$682$	0	0
$683$	1.31970e7	1.08249	0.541243	−	0.840866i	$-0.317954\pi$
$683$	1.31970e7	1.08249	0.541243	+	0.840866i	$0.317954\pi$
$684$	0	0
$685$	6.56925e6	0.534921
$686$	0	0
$687$	1.20120e7	0.971009
$688$	0	0
$689$	−1.58558e7	−1.27245
$690$	0	0
$691$	9.74172e6	0.776140	0.388070	−	0.921630i	$-0.373142\pi$
$691$	9.74172e6	0.776140	0.388070	+	0.921630i	$0.373142\pi$
$692$	0	0
$693$	1.68319e6	0.133137
$694$	0	0
$695$	1.19101e7	0.935308
$696$	0	0
$697$	3.09016e6	0.240935
$698$	0	0
$699$	617187.	0.0477776
$700$	0	0
$701$	−1.23931e7	−0.952542	−0.476271	−	0.879299i	$-0.658012\pi$
$701$	−1.23931e7	−0.952542	−0.476271	+	0.879299i	$0.658012\pi$
$702$	0	0
$703$	−1.27794e7	−0.975266
$704$	0	0
$705$	2.59530e6	0.196660
$706$	0	0
$707$	−6.70333e6	−0.504362
$708$	0	0
$709$	1.18843e7	0.887884	0.443942	−	0.896055i	$-0.353580\pi$
$709$	1.18843e7	0.887884	0.443942	+	0.896055i	$0.353580\pi$
$710$	0	0
$711$	−2.61593e6	−0.194067
$712$	0	0
$713$	1.48573e6	0.109450
$714$	0	0
$715$	6.12793e6	0.448279
$716$	0	0
$717$	4.15775e6	0.302037
$718$	0	0
$719$	−1.57469e7	−1.13599	−0.567993	−	0.823034i	$-0.692280\pi$
$719$	−1.57469e7	−1.13599	−0.567993	+	0.823034i	$0.692280\pi$
$720$	0	0
$721$	−1.49087e6	−0.106808
$722$	0	0
$723$	−1.27825e6	−0.0909433
$724$	0	0
$725$	−1.59436e7	−1.12653
$726$	0	0
$727$	9.34544e6	0.655788	0.327894	−	0.944714i	$-0.393661\pi$
$727$	9.34544e6	0.655788	0.327894	+	0.944714i	$0.393661\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−936210.	−0.0648008
$732$	0	0
$733$	−1.42667e7	−0.980763	−0.490381	−	0.871508i	$-0.663143\pi$
$733$	−1.42667e7	−0.980763	−0.490381	+	0.871508i	$0.663143\pi$
$734$	0	0
$735$	614055.	0.0419265
$736$	0	0
$737$	−1.30370e7	−0.884112
$738$	0	0
$739$	1.75852e7	1.18451	0.592253	−	0.805752i	$-0.298239\pi$
$739$	1.75852e7	1.18451	0.592253	+	0.805752i	$0.298239\pi$
$740$	0	0
$741$	1.19126e7	0.797006
$742$	0	0
$743$	−36857.8	−0.00244939	−0.00122469	−	0.999999i	$-0.500390\pi$
$743$	−36857.8	−0.00244939	−0.00122469	+	0.999999i	$0.500390\pi$
$744$	0	0
$745$	551099.	0.0363780
$746$	0	0
$747$	3.77762e6	0.247695
$748$	0	0
$749$	−6.59598e6	−0.429610
$750$	0	0
$751$	1.73284e7	1.12114	0.560570	−	0.828107i	$-0.310582\pi$
$751$	1.73284e7	1.12114	0.560570	+	0.828107i	$0.310582\pi$
$752$	0	0
$753$	−6.05244e6	−0.388994
$754$	0	0
$755$	−1.54954e7	−0.989317
$756$	0	0
$757$	1.04739e7	0.664308	0.332154	−	0.943225i	$-0.392225\pi$
$757$	1.04739e7	0.664308	0.332154	+	0.943225i	$0.392225\pi$
$758$	0	0
$759$	−997129.	−0.0628271
$760$	0	0
$761$	−615867.	−0.0385501	−0.0192751	−	0.999814i	$-0.506136\pi$
$761$	−615867.	−0.0385501	−0.0192751	+	0.999814i	$0.506136\pi$
$762$	0	0
$763$	−378378.	−0.0235296
$764$	0	0
$765$	1.24275e6	0.0767767
$766$	0	0
$767$	1.97529e7	1.21239
$768$	0	0
$769$	2.82564e7	1.72306	0.861531	−	0.507705i	$-0.169506\pi$
$769$	2.82564e7	1.72306	0.861531	+	0.507705i	$0.169506\pi$
$770$	0	0
$771$	−1.04925e7	−0.635688
$772$	0	0
$773$	−1.80327e7	−1.08546	−0.542729	−	0.839908i	$-0.682609\pi$
$773$	−1.80327e7	−1.08546	−0.542729	+	0.839908i	$0.682609\pi$
$774$	0	0
$775$	1.31796e7	0.788221
$776$	0	0
$777$	2.16509e6	0.128654
$778$	0	0
$779$	1.48980e7	0.879601
$780$	0	0
$781$	−1.93500e7	−1.13515
$782$	0	0
$783$	−5.01527e6	−0.292341
$784$	0	0
$785$	−271038.	−0.0156984
$786$	0	0
$787$	−3.13051e7	−1.80168	−0.900841	−	0.434149i	$-0.857049\pi$
$787$	−3.13051e7	−1.80168	−0.900841	+	0.434149i	$0.857049\pi$
$788$	0	0
$789$	−1.11003e7	−0.634806
$790$	0	0
$791$	−1.05152e7	−0.597554
$792$	0	0
$793$	6.94152e6	0.391987
$794$	0	0
$795$	−7.97465e6	−0.447501
$796$	0	0
$797$	1.38176e7	0.770525	0.385262	−	0.922807i	$-0.374111\pi$
$797$	1.38176e7	0.770525	0.385262	+	0.922807i	$0.374111\pi$
$798$	0	0
$799$	−5.47898e6	−0.303621
$800$	0	0
$801$	−5.16435e6	−0.284403
$802$	0	0
$803$	−9.22529e6	−0.504884
$804$	0	0
$805$	−363770.	−0.0197850
$806$	0	0
$807$	641752.	0.0346883
$808$	0	0
$809$	8.02702e6	0.431204	0.215602	−	0.976481i	$-0.430829\pi$
$809$	8.02702e6	0.431204	0.215602	+	0.976481i	$0.430829\pi$
$810$	0	0
$811$	−5.68487e6	−0.303507	−0.151753	−	0.988418i	$-0.548492\pi$
$811$	−5.68487e6	−0.303507	−0.151753	+	0.988418i	$0.548492\pi$
$812$	0	0
$813$	1.08137e7	0.573783
$814$	0	0
$815$	−1.61050e7	−0.849308
$816$	0	0
$817$	−4.51358e6	−0.236574
$818$	0	0
$819$	−2.01824e6	−0.105139
$820$	0	0
$821$	1.26582e7	0.655412	0.327706	−	0.944780i	$-0.393724\pi$
$821$	1.26582e7	0.655412	0.327706	+	0.944780i	$0.393724\pi$
$822$	0	0
$823$	6.69435e6	0.344516	0.172258	−	0.985052i	$-0.444894\pi$
$823$	6.69435e6	0.344516	0.172258	+	0.985052i	$0.444894\pi$
$824$	0	0
$825$	−8.84530e6	−0.452457
$826$	0	0
$827$	−6.41484e6	−0.326153	−0.163077	−	0.986613i	$-0.552142\pi$
$827$	−6.41484e6	−0.326153	−0.163077	+	0.986613i	$0.552142\pi$
$828$	0	0
$829$	−4.66483e6	−0.235749	−0.117874	−	0.993029i	$-0.537608\pi$
$829$	−4.66483e6	−0.235749	−0.117874	+	0.993029i	$0.537608\pi$
$830$	0	0
$831$	1.47873e7	0.742827
$832$	0	0
$833$	−1.29634e6	−0.0647300
$834$	0	0
$835$	1.42525e7	0.707416
$836$	0	0
$837$	4.14582e6	0.204549
$838$	0	0
$839$	7.16720e6	0.351516	0.175758	−	0.984433i	$-0.443762\pi$
$839$	7.16720e6	0.351516	0.175758	+	0.984433i	$0.443762\pi$
$840$	0	0
$841$	2.68186e7	1.30751
$842$	0	0
$843$	334471.	0.0162102
$844$	0	0
$845$	3.20315e6	0.154325
$846$	0	0
$847$	−920977.	−0.0441103
$848$	0	0
$849$	−171992.	−0.00818917
$850$	0	0
$851$	−1.28261e6	−0.0607116
$852$	0	0
$853$	−2.18869e7	−1.02994	−0.514970	−	0.857208i	$-0.672197\pi$
$853$	−2.18869e7	−1.02994	−0.514970	+	0.857208i	$0.672197\pi$
$854$	0	0
$855$	5.99144e6	0.280295
$856$	0	0
$857$	2.15656e7	1.00302	0.501509	−	0.865152i	$-0.332778\pi$
$857$	2.15656e7	1.00302	0.501509	+	0.865152i	$0.332778\pi$
$858$	0	0
$859$	−1.68364e7	−0.778514	−0.389257	−	0.921129i	$-0.627268\pi$
$859$	−1.68364e7	−0.778514	−0.389257	+	0.921129i	$0.627268\pi$
$860$	0	0
$861$	−2.52403e6	−0.116034
$862$	0	0
$863$	5.28356e6	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$863$	5.28356e6	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$864$	0	0
$865$	−3.17331e6	−0.144202
$866$	0	0
$867$	1.01551e7	0.458815
$868$	0	0
$869$	1.36960e7	0.615238
$870$	0	0
$871$	1.56320e7	0.698185
$872$	0	0
$873$	9.32492e6	0.414104
$874$	0	0
$875$	−7.57821e6	−0.334616
$876$	0	0
$877$	3.77000e7	1.65517	0.827584	−	0.561342i	$-0.189715\pi$
$877$	3.77000e7	1.65517	0.827584	+	0.561342i	$0.189715\pi$
$878$	0	0
$879$	1.34696e7	0.588008
$880$	0	0
$881$	2.59165e7	1.12496	0.562480	−	0.826811i	$-0.309847\pi$
$881$	2.59165e7	1.12496	0.562480	+	0.826811i	$0.309847\pi$
$882$	0	0
$883$	8.24975e6	0.356073	0.178036	−	0.984024i	$-0.443026\pi$
$883$	8.24975e6	0.356073	0.178036	+	0.984024i	$0.443026\pi$
$884$	0	0
$885$	9.93472e6	0.426381
$886$	0	0
$887$	−8.84834e6	−0.377618	−0.188809	−	0.982014i	$-0.560463\pi$
$887$	−8.84834e6	−0.377618	−0.188809	+	0.982014i	$0.560463\pi$
$888$	0	0
$889$	952833.	0.0404355
$890$	0	0
$891$	−2.78241e6	−0.117416
$892$	0	0
$893$	−2.64148e7	−1.10846
$894$	0	0
$895$	−1.50164e7	−0.626624
$896$	0	0
$897$	1.19561e6	0.0496147
$898$	0	0
$899$	−3.91246e7	−1.61455
$900$	0	0
$901$	1.68354e7	0.690893
$902$	0	0
$903$	764691.	0.0312081
$904$	0	0
$905$	8.26378e6	0.335396
$906$	0	0
$907$	−3.52189e6	−0.142153	−0.0710767	−	0.997471i	$-0.522644\pi$
$907$	−3.52189e6	−0.142153	−0.0710767	+	0.997471i	$0.522644\pi$
$908$	0	0
$909$	1.10810e7	0.444805
$910$	0	0
$911$	−2.67890e7	−1.06945	−0.534726	−	0.845026i	$-0.679585\pi$
$911$	−2.67890e7	−1.06945	−0.534726	+	0.845026i	$0.679585\pi$
$912$	0	0
$913$	−1.97781e7	−0.785250
$914$	0	0
$915$	3.49123e6	0.137856
$916$	0	0
$917$	−3.10910e6	−0.122099
$918$	0	0
$919$	4.21717e7	1.64715	0.823574	−	0.567209i	$-0.191977\pi$
$919$	4.21717e7	1.64715	0.823574	+	0.567209i	$0.191977\pi$
$920$	0	0
$921$	−1.64429e7	−0.638748
$922$	0	0
$923$	2.32018e7	0.896431
$924$	0	0
$925$	−1.13778e7	−0.437222
$926$	0	0
$927$	2.46451e6	0.0941956
$928$	0	0
$929$	4.24746e6	0.161469	0.0807347	−	0.996736i	$-0.474273\pi$
$929$	4.24746e6	0.161469	0.0807347	+	0.996736i	$0.474273\pi$
$930$	0	0
$931$	−6.24980e6	−0.236315
$932$	0	0
$933$	−1.49027e7	−0.560483
$934$	0	0
$935$	−6.50653e6	−0.243400
$936$	0	0
$937$	4.39330e6	0.163471	0.0817357	−	0.996654i	$-0.473954\pi$
$937$	4.39330e6	0.163471	0.0817357	+	0.996654i	$0.473954\pi$
$938$	0	0
$939$	2.73046e7	1.01058
$940$	0	0
$941$	−924321.	−0.0340290	−0.0170145	−	0.999855i	$-0.505416\pi$
$941$	−924321.	−0.0340290	−0.0170145	+	0.999855i	$0.505416\pi$
$942$	0	0
$943$	1.49525e6	0.0547563
$944$	0	0
$945$	−1.01507e6	−0.0369757
$946$	0	0
$947$	−3.87692e7	−1.40479	−0.702395	−	0.711787i	$-0.747886\pi$
$947$	−3.87692e7	−1.40479	−0.702395	+	0.711787i	$0.747886\pi$
$948$	0	0
$949$	1.10617e7	0.398708
$950$	0	0
$951$	1.26845e7	0.454803
$952$	0	0
$953$	2.28022e7	0.813289	0.406644	−	0.913587i	$-0.366699\pi$
$953$	2.28022e7	0.813289	0.406644	+	0.913587i	$0.366699\pi$
$954$	0	0
$955$	1.47735e6	0.0524174
$956$	0	0
$957$	2.62579e7	0.926789
$958$	0	0
$959$	1.13276e7	0.397734
$960$	0	0
$961$	3.71280e6	0.129686
$962$	0	0
$963$	1.09036e7	0.378881
$964$	0	0
$965$	1.49472e7	0.516705
$966$	0	0
$967$	3.66797e7	1.26142	0.630709	−	0.776019i	$-0.282764\pi$
$967$	3.66797e7	1.26142	0.630709	+	0.776019i	$0.282764\pi$
$968$	0	0
$969$	−1.26486e7	−0.432746
$970$	0	0
$971$	1.38200e7	0.470391	0.235195	−	0.971948i	$-0.424427\pi$
$971$	1.38200e7	0.470391	0.235195	+	0.971948i	$0.424427\pi$
$972$	0	0
$973$	2.05372e7	0.695438
$974$	0	0
$975$	1.06060e7	0.357306
$976$	0	0
$977$	2.05752e7	0.689616	0.344808	−	0.938673i	$-0.387944\pi$
$977$	2.05752e7	0.689616	0.344808	+	0.938673i	$0.387944\pi$
$978$	0	0
$979$	2.70384e7	0.901622
$980$	0	0
$981$	625482.	0.0207512
$982$	0	0
$983$	5.77676e7	1.90678	0.953390	−	0.301742i	$-0.0975681\pi$
$983$	5.77676e7	1.90678	0.953390	+	0.301742i	$0.0975681\pi$
$984$	0	0
$985$	−2.77773e7	−0.912221
$986$	0	0
$987$	4.47520e6	0.146224
$988$	0	0
$989$	−453008.	−0.0147270
$990$	0	0
$991$	3.65391e7	1.18188	0.590940	−	0.806715i	$-0.298757\pi$
$991$	3.65391e7	1.18188	0.590940	+	0.806715i	$0.298757\pi$
$992$	0	0
$993$	−8.48432e6	−0.273051
$994$	0	0
$995$	2.89224e7	0.926142
$996$	0	0
$997$	−3.57190e7	−1.13805	−0.569024	−	0.822321i	$-0.692679\pi$
$997$	−3.57190e7	−1.13805	−0.569024	+	0.822321i	$0.692679\pi$
$998$	0	0
$999$	−3.57903e6	−0.113462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.u.1.1		2
3.2	odd	2		1008.6.a.bf.1.2		2
4.3	odd	2		84.6.a.d.1.1	✓	2
12.11	even	2		252.6.a.e.1.2		2
28.3	even	6		588.6.i.n.373.1		4
28.11	odd	6		588.6.i.h.373.2		4
28.19	even	6		588.6.i.n.361.1		4
28.23	odd	6		588.6.i.h.361.2		4
28.27	even	2		588.6.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.d.1.1	✓	2	4.3	odd	2
252.6.a.e.1.2		2	12.11	even	2
336.6.a.u.1.1		2	1.1	even	1	trivial
588.6.a.g.1.2		2	28.27	even	2
588.6.i.h.361.2		4	28.23	odd	6
588.6.i.h.373.2		4	28.11	odd	6
588.6.i.n.361.1		4	28.19	even	6
588.6.i.n.373.1		4	28.3	even	6
1008.6.a.bf.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -28.4166 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -28.4166 q^{5} -49.0000 q^{7} +81.0000 q^{9} -424.083 q^{11} +508.500 q^{13} +255.750 q^{15} -539.916 q^{17} -2603.00 q^{19} +441.000 q^{21} -261.251 q^{23} -2317.50 q^{25} -729.000 q^{27} +6879.66 q^{29} -5687.00 q^{31} +3816.75 q^{33} +1392.41 q^{35} +4909.50 q^{37} -4576.50 q^{39} -5723.42 q^{41} +1733.99 q^{43} -2301.75 q^{45} +10147.8 q^{47} +2401.00 q^{49} +4859.24 q^{51} -31181.5 q^{53} +12051.0 q^{55} +23427.0 q^{57} +38845.5 q^{59} +13651.0 q^{61} -3969.00 q^{63} -14449.8 q^{65} +30741.5 q^{67} +2351.26 q^{69} +45627.9 q^{71} +21753.5 q^{73} +20857.5 q^{75} +20780.1 q^{77} -32295.5 q^{79} +6561.00 q^{81} +46637.3 q^{83} +15342.6 q^{85} -61917.0 q^{87} -63757.4 q^{89} -24916.5 q^{91} +51183.0 q^{93} +73968.4 q^{95} +115122. q^{97} -34350.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} + 78 q^{5} - 98 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 + 78 * q^5 - 98 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 78 q^{5} - 98 q^{7} + 162 q^{9} - 174 q^{11} + 208 q^{13} - 702 q^{15} + 1482 q^{17} - 352 q^{19} + 882 q^{21} - 3354 q^{23} + 5882 q^{25} - 1458 q^{27} + 276 q^{29} - 6520 q^{31} + 1566 q^{33} - 3822 q^{35} + 13864 q^{37} - 1872 q^{39} - 12930 q^{41} - 12712 q^{43} + 6318 q^{45} + 28116 q^{47} + 4802 q^{49} - 13338 q^{51} - 46992 q^{53} + 38664 q^{55} + 3168 q^{57} + 65556 q^{59} - 13148 q^{61} - 7938 q^{63} - 46428 q^{65} + 75236 q^{67} + 30186 q^{69} + 66042 q^{71} + 60496 q^{73} - 52938 q^{75} + 8526 q^{77} + 34916 q^{79} + 13122 q^{81} + 82488 q^{83} + 230508 q^{85} - 2484 q^{87} + 42510 q^{89} - 10192 q^{91} + 58680 q^{93} + 313512 q^{95} + 213256 q^{97} - 14094 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 78 * q^5 - 98 * q^7 + 162 * q^9 - 174 * q^11 + 208 * q^13 - 702 * q^15 + 1482 * q^17 - 352 * q^19 + 882 * q^21 - 3354 * q^23 + 5882 * q^25 - 1458 * q^27 + 276 * q^29 - 6520 * q^31 + 1566 * q^33 - 3822 * q^35 + 13864 * q^37 - 1872 * q^39 - 12930 * q^41 - 12712 * q^43 + 6318 * q^45 + 28116 * q^47 + 4802 * q^49 - 13338 * q^51 - 46992 * q^53 + 38664 * q^55 + 3168 * q^57 + 65556 * q^59 - 13148 * q^61 - 7938 * q^63 - 46428 * q^65 + 75236 * q^67 + 30186 * q^69 + 66042 * q^71 + 60496 * q^73 - 52938 * q^75 + 8526 * q^77 + 34916 * q^79 + 13122 * q^81 + 82488 * q^83 + 230508 * q^85 - 2484 * q^87 + 42510 * q^89 - 10192 * q^91 + 58680 * q^93 + 313512 * q^95 + 213256 * q^97 - 14094 * q^99

Introduction

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