LMFDB - Embedded newform 312.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.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:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{55}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 55 $ x^2 - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.41620$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +12.8324 q^{5} +24.8324 q^{7} +9.00000 q^{9} +19.6648 q^{11} -13.0000 q^{13} -38.4972 q^{15} -63.6648 q^{17} -0.832397 q^{19} -74.4972 q^{21} +119.330 q^{23} +39.6704 q^{25} -27.0000 q^{27} -6.00000 q^{29} +185.503 q^{31} -58.9944 q^{33} +318.659 q^{35} +143.665 q^{37} +39.0000 q^{39} -117.156 q^{41} +67.6760 q^{43} +115.492 q^{45} +476.659 q^{47} +273.648 q^{49} +190.994 q^{51} +59.3184 q^{53} +252.346 q^{55} +2.49719 q^{57} +78.0000 q^{59} +609.955 q^{61} +223.492 q^{63} -166.821 q^{65} -654.452 q^{67} -357.989 q^{69} -390.994 q^{71} -84.3127 q^{73} -119.011 q^{75} +488.324 q^{77} +1176.65 q^{79} +81.0000 q^{81} -430.648 q^{83} -816.972 q^{85} +18.0000 q^{87} -1341.41 q^{89} -322.821 q^{91} -556.508 q^{93} -10.6816 q^{95} +802.949 q^{97} +176.983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9} - 20 q^{11} - 26 q^{13} + 12 q^{15} - 68 q^{17} + 28 q^{19} - 60 q^{21} + 120 q^{23} + 198 q^{25} - 54 q^{27} - 12 q^{29} + 460 q^{31} + 60 q^{33} + 400 q^{35}+ \cdots - 180 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9 - 20 * q^11 - 26 * q^13 + 12 * q^15 - 68 * q^17 + 28 * q^19 - 60 * q^21 + 120 * q^23 + 198 * q^25 - 54 * q^27 - 12 * q^29 + 460 * q^31 + 60 * q^33 + 400 * q^35 + 228 * q^37 + 78 * q^39 + 92 * q^41 + 432 * q^43 - 36 * q^45 + 716 * q^47 - 46 * q^49 + 204 * q^51 - 356 * q^53 + 920 * q^55 - 84 * q^57 + 156 * q^59 - 204 * q^61 + 180 * q^63 + 52 * q^65 + 204 * q^67 - 360 * q^69 - 604 * q^71 + 484 * q^73 - 594 * q^75 + 680 * q^77 + 1760 * q^79 + 162 * q^81 - 268 * q^83 - 744 * q^85 + 36 * q^87 + 76 * q^89 - 260 * q^91 - 1380 * q^93 - 496 * q^95 + 4 * q^97 - 180 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	12.8324	1.14776	0.573882	−	0.818938i	$-0.305437\pi$
$5$	12.8324	1.14776	0.573882	+	0.818938i	$0.305437\pi$
$6$	0	0
$7$	24.8324	1.34082	0.670412	−	0.741989i	$-0.266118\pi$
$7$	24.8324	1.34082	0.670412	+	0.741989i	$0.266118\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	19.6648	0.539014	0.269507	−	0.962998i	$-0.413139\pi$
$11$	19.6648	0.539014	0.269507	+	0.962998i	$0.413139\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−38.4972	−0.662662
$16$	0	0
$17$	−63.6648	−0.908293	−0.454146	−	0.890927i	$-0.650056\pi$
$17$	−63.6648	−0.908293	−0.454146	+	0.890927i	$0.650056\pi$
$18$	0	0
$19$	−0.832397	−0.0100508	−0.00502539	−	0.999987i	$-0.501600\pi$
$19$	−0.832397	−0.0100508	−0.00502539	+	0.999987i	$0.501600\pi$
$20$	0	0
$21$	−74.4972	−0.774125
$22$	0	0
$23$	119.330	1.08182	0.540912	−	0.841079i	$-0.318079\pi$
$23$	119.330	1.08182	0.540912	+	0.841079i	$0.318079\pi$
$24$	0	0
$25$	39.6704	0.317363
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	185.503	1.07475	0.537376	−	0.843343i	$-0.319416\pi$
$31$	185.503	1.07475	0.537376	+	0.843343i	$0.319416\pi$
$32$	0	0
$33$	−58.9944	−0.311200
$34$	0	0
$35$	318.659	1.53895
$36$	0	0
$37$	143.665	0.638334	0.319167	−	0.947699i	$-0.396597\pi$
$37$	143.665	0.638334	0.319167	+	0.947699i	$0.396597\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−117.156	−0.446262	−0.223131	−	0.974788i	$-0.571628\pi$
$41$	−117.156	−0.446262	−0.223131	+	0.974788i	$0.571628\pi$
$42$	0	0
$43$	67.6760	0.240012	0.120006	−	0.992773i	$-0.461709\pi$
$43$	67.6760	0.240012	0.120006	+	0.992773i	$0.461709\pi$
$44$	0	0
$45$	115.492	0.382588
$46$	0	0
$47$	476.659	1.47932	0.739658	−	0.672983i	$-0.234987\pi$
$47$	476.659	1.47932	0.739658	+	0.672983i	$0.234987\pi$
$48$	0	0
$49$	273.648	0.797807
$50$	0	0
$51$	190.994	0.524403
$52$	0	0
$53$	59.3184	0.153736	0.0768679	−	0.997041i	$-0.475508\pi$
$53$	59.3184	0.153736	0.0768679	+	0.997041i	$0.475508\pi$
$54$	0	0
$55$	252.346	0.618662
$56$	0	0
$57$	2.49719	0.00580282
$58$	0	0
$59$	78.0000	0.172114	0.0860571	−	0.996290i	$-0.472573\pi$
$59$	78.0000	0.172114	0.0860571	+	0.996290i	$0.472573\pi$
$60$	0	0
$61$	609.955	1.28027	0.640137	−	0.768261i	$-0.278877\pi$
$61$	609.955	1.28027	0.640137	+	0.768261i	$0.278877\pi$
$62$	0	0
$63$	223.492	0.446941
$64$	0	0
$65$	−166.821	−0.318333
$66$	0	0
$67$	−654.452	−1.19334	−0.596672	−	0.802485i	$-0.703511\pi$
$67$	−654.452	−1.19334	−0.596672	+	0.802485i	$0.703511\pi$
$68$	0	0
$69$	−357.989	−0.624591
$70$	0	0
$71$	−390.994	−0.653556	−0.326778	−	0.945101i	$-0.605963\pi$
$71$	−390.994	−0.653556	−0.326778	+	0.945101i	$0.605963\pi$
$72$	0	0
$73$	−84.3127	−0.135179	−0.0675894	−	0.997713i	$-0.521531\pi$
$73$	−84.3127	−0.135179	−0.0675894	+	0.997713i	$0.521531\pi$
$74$	0	0
$75$	−119.011	−0.183230
$76$	0	0
$77$	488.324	0.722723
$78$	0	0
$79$	1176.65	1.67574	0.837869	−	0.545872i	$-0.183802\pi$
$79$	1176.65	1.67574	0.837869	+	0.545872i	$0.183802\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−430.648	−0.569515	−0.284758	−	0.958600i	$-0.591913\pi$
$83$	−430.648	−0.569515	−0.284758	+	0.958600i	$0.591913\pi$
$84$	0	0
$85$	−816.972	−1.04251
$86$	0	0
$87$	18.0000	0.0221816
$88$	0	0
$89$	−1341.41	−1.59763	−0.798817	−	0.601574i	$-0.794541\pi$
$89$	−1341.41	−1.59763	−0.798817	+	0.601574i	$0.794541\pi$
$90$	0	0
$91$	−322.821	−0.371878
$92$	0	0
$93$	−556.508	−0.620508
$94$	0	0
$95$	−10.6816	−0.0115359
$96$	0	0
$97$	802.949	0.840486	0.420243	−	0.907412i	$-0.361945\pi$
$97$	802.949	0.840486	0.420243	+	0.907412i	$0.361945\pi$
$98$	0	0
$99$	176.983	0.179671
$100$	0	0
$101$	−502.369	−0.494926	−0.247463	−	0.968897i	$-0.579597\pi$
$101$	−502.369	−0.494926	−0.247463	+	0.968897i	$0.579597\pi$
$102$	0	0
$103$	1928.93	1.84527	0.922635	−	0.385674i	$-0.126031\pi$
$103$	1928.93	1.84527	0.922635	+	0.385674i	$0.126031\pi$
$104$	0	0
$105$	−955.978	−0.888513
$106$	0	0
$107$	−1759.96	−1.59011	−0.795053	−	0.606540i	$-0.792557\pi$
$107$	−1759.96	−1.59011	−0.795053	+	0.606540i	$0.792557\pi$
$108$	0	0
$109$	−53.3071	−0.0468431	−0.0234215	−	0.999726i	$-0.507456\pi$
$109$	−53.3071	−0.0468431	−0.0234215	+	0.999726i	$0.507456\pi$
$110$	0	0
$111$	−430.994	−0.368542
$112$	0	0
$113$	254.648	0.211993	0.105997	−	0.994366i	$-0.466197\pi$
$113$	254.648	0.211993	0.105997	+	0.994366i	$0.466197\pi$
$114$	0	0
$115$	1531.28	1.24168
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−1580.95	−1.21786
$120$	0	0
$121$	−944.296	−0.709463
$122$	0	0
$123$	351.469	0.257650
$124$	0	0
$125$	−1094.98	−0.783506
$126$	0	0
$127$	805.597	0.562876	0.281438	−	0.959579i	$-0.409189\pi$
$127$	805.597	0.562876	0.281438	+	0.959579i	$0.409189\pi$
$128$	0	0
$129$	−203.028	−0.138571
$130$	0	0
$131$	365.966	0.244081	0.122041	−	0.992525i	$-0.461056\pi$
$131$	365.966	0.244081	0.122041	+	0.992525i	$0.461056\pi$
$132$	0	0
$133$	−20.6704	−0.0134763
$134$	0	0
$135$	−346.475	−0.220887
$136$	0	0
$137$	−2319.79	−1.44667	−0.723333	−	0.690499i	$-0.757391\pi$
$137$	−2319.79	−1.44667	−0.723333	+	0.690499i	$0.757391\pi$
$138$	0	0
$139$	1816.67	1.10855	0.554273	−	0.832335i	$-0.312996\pi$
$139$	1816.67	1.10855	0.554273	+	0.832335i	$0.312996\pi$
$140$	0	0
$141$	−1429.98	−0.854084
$142$	0	0
$143$	−255.642	−0.149496
$144$	0	0
$145$	−76.9944	−0.0440968
$146$	0	0
$147$	−820.944	−0.460614
$148$	0	0
$149$	1752.37	0.963490	0.481745	−	0.876311i	$-0.340003\pi$
$149$	1752.37	0.963490	0.481745	+	0.876311i	$0.340003\pi$
$150$	0	0
$151$	−590.128	−0.318039	−0.159020	−	0.987275i	$-0.550833\pi$
$151$	−590.128	−0.318039	−0.159020	+	0.987275i	$0.550833\pi$
$152$	0	0
$153$	−572.983	−0.302764
$154$	0	0
$155$	2380.45	1.23356
$156$	0	0
$157$	−2338.56	−1.18877	−0.594386	−	0.804180i	$-0.702605\pi$
$157$	−2338.56	−1.18877	−0.594386	+	0.804180i	$0.702605\pi$
$158$	0	0
$159$	−177.955	−0.0887595
$160$	0	0
$161$	2963.24	1.45053
$162$	0	0
$163$	12.8549	0.00617712	0.00308856	−	0.999995i	$-0.499017\pi$
$163$	12.8549	0.00617712	0.00308856	+	0.999995i	$0.499017\pi$
$164$	0	0
$165$	−757.039	−0.357184
$166$	0	0
$167$	−3236.50	−1.49969	−0.749844	−	0.661614i	$-0.769872\pi$
$167$	−3236.50	−1.49969	−0.749844	+	0.661614i	$0.769872\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−7.49157	−0.00335026
$172$	0	0
$173$	3704.85	1.62818	0.814088	−	0.580742i	$-0.197237\pi$
$173$	3704.85	1.62818	0.814088	+	0.580742i	$0.197237\pi$
$174$	0	0
$175$	985.111	0.425528
$176$	0	0
$177$	−234.000	−0.0993702
$178$	0	0
$179$	−2487.37	−1.03863	−0.519316	−	0.854582i	$-0.673813\pi$
$179$	−2487.37	−1.03863	−0.519316	+	0.854582i	$0.673813\pi$
$180$	0	0
$181$	−4362.47	−1.79149	−0.895745	−	0.444568i	$-0.853357\pi$
$181$	−4362.47	−1.79149	−0.895745	+	0.444568i	$0.853357\pi$
$182$	0	0
$183$	−1829.87	−0.739167
$184$	0	0
$185$	1843.56	0.732657
$186$	0	0
$187$	−1251.96	−0.489583
$188$	0	0
$189$	−670.475	−0.258042
$190$	0	0
$191$	−2816.51	−1.06699	−0.533497	−	0.845802i	$-0.679122\pi$
$191$	−2816.51	−1.06699	−0.533497	+	0.845802i	$0.679122\pi$
$192$	0	0
$193$	4253.33	1.58633	0.793164	−	0.609008i	$-0.208432\pi$
$193$	4253.33	1.58633	0.793164	+	0.609008i	$0.208432\pi$
$194$	0	0
$195$	500.463	0.183789
$196$	0	0
$197$	−4164.98	−1.50631	−0.753153	−	0.657845i	$-0.771468\pi$
$197$	−4164.98	−1.50631	−0.753153	+	0.657845i	$0.771468\pi$
$198$	0	0
$199$	2252.44	0.802367	0.401183	−	0.915998i	$-0.368599\pi$
$199$	2252.44	0.802367	0.401183	+	0.915998i	$0.368599\pi$
$200$	0	0
$201$	1963.36	0.688978
$202$	0	0
$203$	−148.994	−0.0515141
$204$	0	0
$205$	−1503.40	−0.512204
$206$	0	0
$207$	1073.97	0.360608
$208$	0	0
$209$	−16.3689	−0.00541752
$210$	0	0
$211$	−4507.73	−1.47073	−0.735367	−	0.677669i	$-0.762990\pi$
$211$	−4507.73	−1.47073	−0.735367	+	0.677669i	$0.762990\pi$
$212$	0	0
$213$	1172.98	0.377331
$214$	0	0
$215$	868.446	0.275477
$216$	0	0
$217$	4606.48	1.44105
$218$	0	0
$219$	252.938	0.0780456
$220$	0	0
$221$	827.642	0.251915
$222$	0	0
$223$	−863.280	−0.259235	−0.129618	−	0.991564i	$-0.541375\pi$
$223$	−863.280	−0.259235	−0.129618	+	0.991564i	$0.541375\pi$
$224$	0	0
$225$	357.034	0.105788
$226$	0	0
$227$	−4662.68	−1.36332	−0.681658	−	0.731671i	$-0.738741\pi$
$227$	−4662.68	−1.36332	−0.681658	+	0.731671i	$0.738741\pi$
$228$	0	0
$229$	−3661.26	−1.05652	−0.528260	−	0.849083i	$-0.677155\pi$
$229$	−3661.26	−1.05652	−0.528260	+	0.849083i	$0.677155\pi$
$230$	0	0
$231$	−1464.97	−0.417264
$232$	0	0
$233$	783.845	0.220392	0.110196	−	0.993910i	$-0.464852\pi$
$233$	783.845	0.220392	0.110196	+	0.993910i	$0.464852\pi$
$234$	0	0
$235$	6116.68	1.69791
$236$	0	0
$237$	−3529.94	−0.967487
$238$	0	0
$239$	2279.44	0.616924	0.308462	−	0.951237i	$-0.400186\pi$
$239$	2279.44	0.616924	0.308462	+	0.951237i	$0.400186\pi$
$240$	0	0
$241$	4352.87	1.16346	0.581728	−	0.813383i	$-0.302377\pi$
$241$	4352.87	1.16346	0.581728	+	0.813383i	$0.302377\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3511.56	0.915695
$246$	0	0
$247$	10.8212	0.00278759
$248$	0	0
$249$	1291.94	0.328810
$250$	0	0
$251$	3851.24	0.968478	0.484239	−	0.874936i	$-0.339096\pi$
$251$	3851.24	0.968478	0.484239	+	0.874936i	$0.339096\pi$
$252$	0	0
$253$	2346.59	0.583118
$254$	0	0
$255$	2450.92	0.601891
$256$	0	0
$257$	3090.99	0.750237	0.375119	−	0.926977i	$-0.377602\pi$
$257$	3090.99	0.750237	0.375119	+	0.926977i	$0.377602\pi$
$258$	0	0
$259$	3567.54	0.855893
$260$	0	0
$261$	−54.0000	−0.0128066
$262$	0	0
$263$	−8.67041	−0.00203285	−0.00101643	−	0.999999i	$-0.500324\pi$
$263$	−8.67041	−0.00203285	−0.00101643	+	0.999999i	$0.500324\pi$
$264$	0	0
$265$	761.197	0.176453
$266$	0	0
$267$	4024.24	0.922395
$268$	0	0
$269$	−7485.51	−1.69665	−0.848326	−	0.529474i	$-0.822389\pi$
$269$	−7485.51	−1.69665	−0.848326	+	0.529474i	$0.822389\pi$
$270$	0	0
$271$	6357.84	1.42513	0.712566	−	0.701605i	$-0.247533\pi$
$271$	6357.84	1.42513	0.712566	+	0.701605i	$0.247533\pi$
$272$	0	0
$273$	968.463	0.214704
$274$	0	0
$275$	780.110	0.171063
$276$	0	0
$277$	7232.46	1.56880	0.784398	−	0.620258i	$-0.212972\pi$
$277$	7232.46	1.56880	0.784398	+	0.620258i	$0.212972\pi$
$278$	0	0
$279$	1669.53	0.358250
$280$	0	0
$281$	2886.87	0.612868	0.306434	−	0.951892i	$-0.400864\pi$
$281$	2886.87	0.612868	0.306434	+	0.951892i	$0.400864\pi$
$282$	0	0
$283$	−6413.26	−1.34710	−0.673549	−	0.739143i	$-0.735231\pi$
$283$	−6413.26	−1.34710	−0.673549	+	0.739143i	$0.735231\pi$
$284$	0	0
$285$	32.0449	0.00666028
$286$	0	0
$287$	−2909.27	−0.598359
$288$	0	0
$289$	−859.794	−0.175004
$290$	0	0
$291$	−2408.85	−0.485255
$292$	0	0
$293$	−1223.82	−0.244016	−0.122008	−	0.992529i	$-0.538933\pi$
$293$	−1223.82	−0.244016	−0.122008	+	0.992529i	$0.538933\pi$
$294$	0	0
$295$	1000.93	0.197547
$296$	0	0
$297$	−530.949	−0.103733
$298$	0	0
$299$	−1551.28	−0.300044
$300$	0	0
$301$	1680.56	0.321813
$302$	0	0
$303$	1507.11	0.285746
$304$	0	0
$305$	7827.19	1.46945
$306$	0	0
$307$	−3853.21	−0.716333	−0.358167	−	0.933658i	$-0.616598\pi$
$307$	−3853.21	−0.716333	−0.358167	+	0.933658i	$0.616598\pi$
$308$	0	0
$309$	−5786.78	−1.06537
$310$	0	0
$311$	−7694.88	−1.40301	−0.701506	−	0.712664i	$-0.747489\pi$
$311$	−7694.88	−1.40301	−0.701506	+	0.712664i	$0.747489\pi$
$312$	0	0
$313$	2743.30	0.495400	0.247700	−	0.968837i	$-0.420325\pi$
$313$	2743.30	0.495400	0.247700	+	0.968837i	$0.420325\pi$
$314$	0	0
$315$	2867.93	0.512983
$316$	0	0
$317$	4706.66	0.833920	0.416960	−	0.908925i	$-0.363096\pi$
$317$	4706.66	0.833920	0.416960	+	0.908925i	$0.363096\pi$
$318$	0	0
$319$	−117.989	−0.0207088
$320$	0	0
$321$	5279.87	0.918048
$322$	0	0
$323$	52.9944	0.00912906
$324$	0	0
$325$	−515.715	−0.0880207
$326$	0	0
$327$	159.921	0.0270449
$328$	0	0
$329$	11836.6	1.98350
$330$	0	0
$331$	−9270.83	−1.53949	−0.769745	−	0.638352i	$-0.779616\pi$
$331$	−9270.83	−1.53949	−0.769745	+	0.638352i	$0.779616\pi$
$332$	0	0
$333$	1292.98	0.212778
$334$	0	0
$335$	−8398.19	−1.36968
$336$	0	0
$337$	−6888.86	−1.11353	−0.556766	−	0.830670i	$-0.687958\pi$
$337$	−6888.86	−1.11353	−0.556766	+	0.830670i	$0.687958\pi$
$338$	0	0
$339$	−763.944	−0.122394
$340$	0	0
$341$	3647.87	0.579306
$342$	0	0
$343$	−1722.18	−0.271105
$344$	0	0
$345$	−4593.85	−0.716883
$346$	0	0
$347$	−2864.49	−0.443152	−0.221576	−	0.975143i	$-0.571120\pi$
$347$	−2864.49	−0.443152	−0.221576	+	0.975143i	$0.571120\pi$
$348$	0	0
$349$	7865.20	1.20635	0.603173	−	0.797610i	$-0.293903\pi$
$349$	7865.20	1.20635	0.603173	+	0.797610i	$0.293903\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	6071.98	0.915521	0.457761	−	0.889075i	$-0.348652\pi$
$353$	6071.98	0.915521	0.457761	+	0.889075i	$0.348652\pi$
$354$	0	0
$355$	−5017.40	−0.750129
$356$	0	0
$357$	4742.85	0.703132
$358$	0	0
$359$	4951.16	0.727889	0.363945	−	0.931421i	$-0.381430\pi$
$359$	4951.16	0.727889	0.363945	+	0.931421i	$0.381430\pi$
$360$	0	0
$361$	−6858.31	−0.999899
$362$	0	0
$363$	2832.89	0.409609
$364$	0	0
$365$	−1081.93	−0.155154
$366$	0	0
$367$	2475.62	0.352115	0.176058	−	0.984380i	$-0.443666\pi$
$367$	2475.62	0.352115	0.176058	+	0.984380i	$0.443666\pi$
$368$	0	0
$369$	−1054.41	−0.148754
$370$	0	0
$371$	1473.02	0.206133
$372$	0	0
$373$	−7342.00	−1.01918	−0.509590	−	0.860417i	$-0.670203\pi$
$373$	−7342.00	−1.01918	−0.509590	+	0.860417i	$0.670203\pi$
$374$	0	0
$375$	3284.95	0.452357
$376$	0	0
$377$	78.0000	0.0106557
$378$	0	0
$379$	−6188.34	−0.838717	−0.419359	−	0.907821i	$-0.637745\pi$
$379$	−6188.34	−0.838717	−0.419359	+	0.907821i	$0.637745\pi$
$380$	0	0
$381$	−2416.79	−0.324976
$382$	0	0
$383$	−3583.67	−0.478113	−0.239056	−	0.971006i	$-0.576838\pi$
$383$	−3583.67	−0.478113	−0.239056	+	0.971006i	$0.576838\pi$
$384$	0	0
$385$	6266.37	0.829516
$386$	0	0
$387$	609.084	0.0800039
$388$	0	0
$389$	−2377.34	−0.309861	−0.154931	−	0.987925i	$-0.549515\pi$
$389$	−2377.34	−0.309861	−0.154931	+	0.987925i	$0.549515\pi$
$390$	0	0
$391$	−7597.09	−0.982613
$392$	0	0
$393$	−1097.90	−0.140920
$394$	0	0
$395$	15099.2	1.92335
$396$	0	0
$397$	−8775.17	−1.10935	−0.554677	−	0.832066i	$-0.687158\pi$
$397$	−8775.17	−1.10935	−0.554677	+	0.832066i	$0.687158\pi$
$398$	0	0
$399$	62.0112	0.00778056
$400$	0	0
$401$	11512.5	1.43368	0.716842	−	0.697236i	$-0.245587\pi$
$401$	11512.5	1.43368	0.716842	+	0.697236i	$0.245587\pi$
$402$	0	0
$403$	−2411.54	−0.298082
$404$	0	0
$405$	1039.42	0.127529
$406$	0	0
$407$	2825.14	0.344071
$408$	0	0
$409$	−13259.2	−1.60300	−0.801499	−	0.597996i	$-0.795964\pi$
$409$	−13259.2	−1.60300	−0.801499	+	0.597996i	$0.795964\pi$
$410$	0	0
$411$	6959.38	0.835233
$412$	0	0
$413$	1936.93	0.230775
$414$	0	0
$415$	−5526.25	−0.653669
$416$	0	0
$417$	−5450.01	−0.640020
$418$	0	0
$419$	−2101.74	−0.245052	−0.122526	−	0.992465i	$-0.539099\pi$
$419$	−2101.74	−0.245052	−0.122526	+	0.992465i	$0.539099\pi$
$420$	0	0
$421$	−15777.9	−1.82653	−0.913264	−	0.407369i	$-0.866446\pi$
$421$	−15777.9	−1.82653	−0.913264	+	0.407369i	$0.866446\pi$
$422$	0	0
$423$	4289.93	0.493106
$424$	0	0
$425$	−2525.61	−0.288259
$426$	0	0
$427$	15146.6	1.71662
$428$	0	0
$429$	766.927	0.0863114
$430$	0	0
$431$	−4091.03	−0.457211	−0.228606	−	0.973519i	$-0.573417\pi$
$431$	−4091.03	−0.457211	−0.228606	+	0.973519i	$0.573417\pi$
$432$	0	0
$433$	−419.071	−0.0465110	−0.0232555	−	0.999730i	$-0.507403\pi$
$433$	−419.071	−0.0465110	−0.0232555	+	0.999730i	$0.507403\pi$
$434$	0	0
$435$	230.983	0.0254593
$436$	0	0
$437$	−99.3296	−0.0108732
$438$	0	0
$439$	−15041.1	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$439$	−15041.1	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$440$	0	0
$441$	2462.83	0.265936
$442$	0	0
$443$	−11829.0	−1.26865	−0.634326	−	0.773066i	$-0.718722\pi$
$443$	−11829.0	−1.26865	−0.634326	+	0.773066i	$0.718722\pi$
$444$	0	0
$445$	−17213.5	−1.83371
$446$	0	0
$447$	−5257.12	−0.556271
$448$	0	0
$449$	−9655.47	−1.01485	−0.507427	−	0.861695i	$-0.669403\pi$
$449$	−9655.47	−1.01485	−0.507427	+	0.861695i	$0.669403\pi$
$450$	0	0
$451$	−2303.86	−0.240542
$452$	0	0
$453$	1770.38	0.183620
$454$	0	0
$455$	−4142.57	−0.426828
$456$	0	0
$457$	−2304.65	−0.235901	−0.117951	−	0.993019i	$-0.537632\pi$
$457$	−2304.65	−0.235901	−0.117951	+	0.993019i	$0.537632\pi$
$458$	0	0
$459$	1718.95	0.174801
$460$	0	0
$461$	−16181.3	−1.63479	−0.817393	−	0.576080i	$-0.804582\pi$
$461$	−16181.3	−1.63479	−0.817393	+	0.576080i	$0.804582\pi$
$462$	0	0
$463$	13304.0	1.33540	0.667699	−	0.744431i	$-0.267279\pi$
$463$	13304.0	1.33540	0.667699	+	0.744431i	$0.267279\pi$
$464$	0	0
$465$	−7141.34	−0.712197
$466$	0	0
$467$	6280.78	0.622355	0.311178	−	0.950352i	$-0.399277\pi$
$467$	6280.78	0.622355	0.311178	+	0.950352i	$0.399277\pi$
$468$	0	0
$469$	−16251.6	−1.60006
$470$	0	0
$471$	7015.67	0.686338
$472$	0	0
$473$	1330.84	0.129370
$474$	0	0
$475$	−33.0215	−0.00318975
$476$	0	0
$477$	533.865	0.0512453
$478$	0	0
$479$	−12938.7	−1.23420	−0.617102	−	0.786883i	$-0.711693\pi$
$479$	−12938.7	−1.23420	−0.617102	+	0.786883i	$0.711693\pi$
$480$	0	0
$481$	−1867.64	−0.177042
$482$	0	0
$483$	−8889.72	−0.837466
$484$	0	0
$485$	10303.8	0.964680
$486$	0	0
$487$	17162.2	1.59690	0.798452	−	0.602059i	$-0.205653\pi$
$487$	17162.2	1.59690	0.798452	+	0.602059i	$0.205653\pi$
$488$	0	0
$489$	−38.5646	−0.00356636
$490$	0	0
$491$	−1295.35	−0.119060	−0.0595300	−	0.998227i	$-0.518960\pi$
$491$	−1295.35	−0.119060	−0.0595300	+	0.998227i	$0.518960\pi$
$492$	0	0
$493$	381.989	0.0348964
$494$	0	0
$495$	2271.12	0.206221
$496$	0	0
$497$	−9709.33	−0.876304
$498$	0	0
$499$	50.9972	0.00457504	0.00228752	−	0.999997i	$-0.499272\pi$
$499$	50.9972	0.00457504	0.00228752	+	0.999997i	$0.499272\pi$
$500$	0	0
$501$	9709.51	0.865846
$502$	0	0
$503$	18949.3	1.67973	0.839866	−	0.542794i	$-0.182633\pi$
$503$	18949.3	1.67973	0.839866	+	0.542794i	$0.182633\pi$
$504$	0	0
$505$	−6446.60	−0.568059
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−19440.3	−1.69288	−0.846439	−	0.532486i	$-0.821258\pi$
$509$	−19440.3	−1.69288	−0.846439	+	0.532486i	$0.821258\pi$
$510$	0	0
$511$	−2093.69	−0.181251
$512$	0	0
$513$	22.4747	0.00193427
$514$	0	0
$515$	24752.8	2.11794
$516$	0	0
$517$	9373.40	0.797373
$518$	0	0
$519$	−11114.5	−0.940028
$520$	0	0
$521$	2480.08	0.208549	0.104275	−	0.994549i	$-0.466748\pi$
$521$	2480.08	0.208549	0.104275	+	0.994549i	$0.466748\pi$
$522$	0	0
$523$	14540.2	1.21568	0.607840	−	0.794060i	$-0.292036\pi$
$523$	14540.2	1.21568	0.607840	+	0.794060i	$0.292036\pi$
$524$	0	0
$525$	−2955.33	−0.245679
$526$	0	0
$527$	−11810.0	−0.976189
$528$	0	0
$529$	2072.55	0.170342
$530$	0	0
$531$	702.000	0.0573714
$532$	0	0
$533$	1523.03	0.123771
$534$	0	0
$535$	−22584.4	−1.82507
$536$	0	0
$537$	7462.12	0.599654
$538$	0	0
$539$	5381.23	0.430030
$540$	0	0
$541$	14166.6	1.12582	0.562909	−	0.826519i	$-0.309682\pi$
$541$	14166.6	1.12582	0.562909	+	0.826519i	$0.309682\pi$
$542$	0	0
$543$	13087.4	1.03432
$544$	0	0
$545$	−684.058	−0.0537648
$546$	0	0
$547$	−3102.57	−0.242516	−0.121258	−	0.992621i	$-0.538693\pi$
$547$	−3102.57	−0.242516	−0.121258	+	0.992621i	$0.538693\pi$
$548$	0	0
$549$	5489.60	0.426758
$550$	0	0
$551$	4.99438	0.000386148 0
$552$	0	0
$553$	29219.0	2.24687
$554$	0	0
$555$	−5530.69	−0.423000
$556$	0	0
$557$	6829.69	0.519539	0.259770	−	0.965671i	$-0.416353\pi$
$557$	6829.69	0.519539	0.259770	+	0.965671i	$0.416353\pi$
$558$	0	0
$559$	−879.788	−0.0665672
$560$	0	0
$561$	3755.87	0.282661
$562$	0	0
$563$	15516.1	1.16150	0.580752	−	0.814080i	$-0.302758\pi$
$563$	15516.1	1.16150	0.580752	+	0.814080i	$0.302758\pi$
$564$	0	0
$565$	3267.74	0.243319
$566$	0	0
$567$	2011.42	0.148980
$568$	0	0
$569$	−9960.00	−0.733822	−0.366911	−	0.930256i	$-0.619585\pi$
$569$	−9960.00	−0.733822	−0.366911	+	0.930256i	$0.619585\pi$
$570$	0	0
$571$	22117.9	1.62102	0.810512	−	0.585722i	$-0.199189\pi$
$571$	22117.9	1.62102	0.810512	+	0.585722i	$0.199189\pi$
$572$	0	0
$573$	8449.54	0.616029
$574$	0	0
$575$	4733.85	0.343331
$576$	0	0
$577$	8171.54	0.589576	0.294788	−	0.955563i	$-0.404751\pi$
$577$	8171.54	0.589576	0.294788	+	0.955563i	$0.404751\pi$
$578$	0	0
$579$	−12760.0	−0.915867
$580$	0	0
$581$	−10694.0	−0.763619
$582$	0	0
$583$	1166.48	0.0828659
$584$	0	0
$585$	−1501.39	−0.106111
$586$	0	0
$587$	−6436.54	−0.452580	−0.226290	−	0.974060i	$-0.572660\pi$
$587$	−6436.54	−0.452580	−0.226290	+	0.974060i	$0.572660\pi$
$588$	0	0
$589$	−154.412	−0.0108021
$590$	0	0
$591$	12494.9	0.869666
$592$	0	0
$593$	16421.7	1.13720	0.568599	−	0.822615i	$-0.307486\pi$
$593$	16421.7	1.13720	0.568599	+	0.822615i	$0.307486\pi$
$594$	0	0
$595$	−20287.4	−1.39782
$596$	0	0
$597$	−6757.31	−0.463247
$598$	0	0
$599$	−914.614	−0.0623875	−0.0311938	−	0.999513i	$-0.509931\pi$
$599$	−914.614	−0.0623875	−0.0311938	+	0.999513i	$0.509931\pi$
$600$	0	0
$601$	−28107.4	−1.90770	−0.953848	−	0.300289i	$-0.902917\pi$
$601$	−28107.4	−1.90770	−0.953848	+	0.300289i	$0.902917\pi$
$602$	0	0
$603$	−5890.07	−0.397781
$604$	0	0
$605$	−12117.6	−0.814297
$606$	0	0
$607$	9581.92	0.640722	0.320361	−	0.947296i	$-0.396196\pi$
$607$	9581.92	0.640722	0.320361	+	0.947296i	$0.396196\pi$
$608$	0	0
$609$	446.983	0.0297417
$610$	0	0
$611$	−6196.57	−0.410289
$612$	0	0
$613$	−1118.28	−0.0736814	−0.0368407	−	0.999321i	$-0.511729\pi$
$613$	−1118.28	−0.0736814	−0.0368407	+	0.999321i	$0.511729\pi$
$614$	0	0
$615$	4510.19	0.295721
$616$	0	0
$617$	8230.82	0.537051	0.268525	−	0.963273i	$-0.413464\pi$
$617$	8230.82	0.537051	0.268525	+	0.963273i	$0.413464\pi$
$618$	0	0
$619$	5312.60	0.344962	0.172481	−	0.985013i	$-0.444822\pi$
$619$	5312.60	0.344962	0.172481	+	0.985013i	$0.444822\pi$
$620$	0	0
$621$	−3221.90	−0.208197
$622$	0	0
$623$	−33310.5	−2.14215
$624$	0	0
$625$	−19010.1	−1.21664
$626$	0	0
$627$	49.1067	0.00312781
$628$	0	0
$629$	−9146.39	−0.579794
$630$	0	0
$631$	−6797.18	−0.428830	−0.214415	−	0.976743i	$-0.568784\pi$
$631$	−6797.18	−0.428830	−0.214415	+	0.976743i	$0.568784\pi$
$632$	0	0
$633$	13523.2	0.849129
$634$	0	0
$635$	10337.7	0.646049
$636$	0	0
$637$	−3557.42	−0.221272
$638$	0	0
$639$	−3518.95	−0.217852
$640$	0	0
$641$	7436.63	0.458236	0.229118	−	0.973399i	$-0.426416\pi$
$641$	7436.63	0.458236	0.229118	+	0.973399i	$0.426416\pi$
$642$	0	0
$643$	3976.51	0.243886	0.121943	−	0.992537i	$-0.461088\pi$
$643$	3976.51	0.243886	0.121943	+	0.992537i	$0.461088\pi$
$644$	0	0
$645$	−2605.34	−0.159047
$646$	0	0
$647$	29726.1	1.80626	0.903132	−	0.429362i	$-0.141262\pi$
$647$	29726.1	1.80626	0.903132	+	0.429362i	$0.141262\pi$
$648$	0	0
$649$	1533.85	0.0927720
$650$	0	0
$651$	−13819.4	−0.831992
$652$	0	0
$653$	−18774.1	−1.12510	−0.562548	−	0.826765i	$-0.690179\pi$
$653$	−18774.1	−1.12510	−0.562548	+	0.826765i	$0.690179\pi$
$654$	0	0
$655$	4696.22	0.280148
$656$	0	0
$657$	−758.815	−0.0450596
$658$	0	0
$659$	7601.47	0.449334	0.224667	−	0.974436i	$-0.427871\pi$
$659$	7601.47	0.449334	0.224667	+	0.974436i	$0.427871\pi$
$660$	0	0
$661$	14701.5	0.865089	0.432544	−	0.901613i	$-0.357616\pi$
$661$	14701.5	0.865089	0.432544	+	0.901613i	$0.357616\pi$
$662$	0	0
$663$	−2482.93	−0.145443
$664$	0	0
$665$	−265.251	−0.0154677
$666$	0	0
$667$	−715.978	−0.0415634
$668$	0	0
$669$	2589.84	0.149670
$670$	0	0
$671$	11994.6	0.690086
$672$	0	0
$673$	2577.04	0.147604	0.0738020	−	0.997273i	$-0.476487\pi$
$673$	2577.04	0.147604	0.0738020	+	0.997273i	$0.476487\pi$
$674$	0	0
$675$	−1071.10	−0.0610766
$676$	0	0
$677$	10227.0	0.580583	0.290291	−	0.956938i	$-0.406248\pi$
$677$	10227.0	0.580583	0.290291	+	0.956938i	$0.406248\pi$
$678$	0	0
$679$	19939.2	1.12694
$680$	0	0
$681$	13988.0	0.787111
$682$	0	0
$683$	11239.2	0.629656	0.314828	−	0.949149i	$-0.398053\pi$
$683$	11239.2	0.629656	0.314828	+	0.949149i	$0.398053\pi$
$684$	0	0
$685$	−29768.5	−1.66043
$686$	0	0
$687$	10983.8	0.609982
$688$	0	0
$689$	−771.139	−0.0426387
$690$	0	0
$691$	−4388.42	−0.241597	−0.120798	−	0.992677i	$-0.538545\pi$
$691$	−4388.42	−0.241597	−0.120798	+	0.992677i	$0.538545\pi$
$692$	0	0
$693$	4394.92	0.240908
$694$	0	0
$695$	23312.2	1.27235
$696$	0	0
$697$	7458.74	0.405337
$698$	0	0
$699$	−2351.53	−0.127243
$700$	0	0
$701$	15247.0	0.821500	0.410750	−	0.911748i	$-0.365267\pi$
$701$	15247.0	0.821500	0.410750	+	0.911748i	$0.365267\pi$
$702$	0	0
$703$	−119.586	−0.00641576
$704$	0	0
$705$	−18350.0	−0.980287
$706$	0	0
$707$	−12475.0	−0.663609
$708$	0	0
$709$	22073.9	1.16926	0.584628	−	0.811301i	$-0.301240\pi$
$709$	22073.9	1.16926	0.584628	+	0.811301i	$0.301240\pi$
$710$	0	0
$711$	10589.8	0.558579
$712$	0	0
$713$	22136.0	1.16269
$714$	0	0
$715$	−3280.50	−0.171586
$716$	0	0
$717$	−6838.32	−0.356181
$718$	0	0
$719$	20988.2	1.08863	0.544317	−	0.838879i	$-0.316789\pi$
$719$	20988.2	1.08863	0.544317	+	0.838879i	$0.316789\pi$
$720$	0	0
$721$	47899.9	2.47418
$722$	0	0
$723$	−13058.6	−0.671722
$724$	0	0
$725$	−238.022	−0.0121930
$726$	0	0
$727$	11692.8	0.596509	0.298254	−	0.954486i	$-0.403596\pi$
$727$	11692.8	0.596509	0.298254	+	0.954486i	$0.403596\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4308.58	−0.218001
$732$	0	0
$733$	21190.2	1.06777	0.533887	−	0.845556i	$-0.320731\pi$
$733$	21190.2	1.06777	0.533887	+	0.845556i	$0.320731\pi$
$734$	0	0
$735$	−10534.7	−0.528677
$736$	0	0
$737$	−12869.7	−0.643230
$738$	0	0
$739$	6905.92	0.343759	0.171880	−	0.985118i	$-0.445016\pi$
$739$	6905.92	0.343759	0.171880	+	0.985118i	$0.445016\pi$
$740$	0	0
$741$	−32.4635	−0.00160941
$742$	0	0
$743$	−2539.96	−0.125413	−0.0627067	−	0.998032i	$-0.519973\pi$
$743$	−2539.96	−0.125413	−0.0627067	+	0.998032i	$0.519973\pi$
$744$	0	0
$745$	22487.2	1.10586
$746$	0	0
$747$	−3875.83	−0.189838
$748$	0	0
$749$	−43703.9	−2.13205
$750$	0	0
$751$	3805.24	0.184894	0.0924468	−	0.995718i	$-0.470531\pi$
$751$	3805.24	0.184894	0.0924468	+	0.995718i	$0.470531\pi$
$752$	0	0
$753$	−11553.7	−0.559151
$754$	0	0
$755$	−7572.76	−0.365034
$756$	0	0
$757$	−20266.6	−0.973052	−0.486526	−	0.873666i	$-0.661736\pi$
$757$	−20266.6	−0.973052	−0.486526	+	0.873666i	$0.661736\pi$
$758$	0	0
$759$	−7039.78	−0.336664
$760$	0	0
$761$	28056.2	1.33645	0.668224	−	0.743960i	$-0.267055\pi$
$761$	28056.2	1.33645	0.668224	+	0.743960i	$0.267055\pi$
$762$	0	0
$763$	−1323.74	−0.0628083
$764$	0	0
$765$	−7352.75	−0.347502
$766$	0	0
$767$	−1014.00	−0.0477359
$768$	0	0
$769$	16703.0	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$769$	16703.0	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$770$	0	0
$771$	−9272.98	−0.433150
$772$	0	0
$773$	25155.9	1.17050	0.585250	−	0.810853i	$-0.300997\pi$
$773$	25155.9	1.17050	0.585250	+	0.810853i	$0.300997\pi$
$774$	0	0
$775$	7358.97	0.341087
$776$	0	0
$777$	−10702.6	−0.494150
$778$	0	0
$779$	97.5206	0.00448529
$780$	0	0
$781$	−7688.82	−0.352276
$782$	0	0
$783$	162.000	0.00739388
$784$	0	0
$785$	−30009.3	−1.36443
$786$	0	0
$787$	−8287.15	−0.375356	−0.187678	−	0.982231i	$-0.560096\pi$
$787$	−8287.15	−0.375356	−0.187678	+	0.982231i	$0.560096\pi$
$788$	0	0
$789$	26.0112	0.00117367
$790$	0	0
$791$	6323.52	0.284246
$792$	0	0
$793$	−7929.42	−0.355084
$794$	0	0
$795$	−2283.59	−0.101875
$796$	0	0
$797$	28506.5	1.26694	0.633470	−	0.773767i	$-0.281630\pi$
$797$	28506.5	1.26694	0.633470	+	0.773767i	$0.281630\pi$
$798$	0	0
$799$	−30346.4	−1.34365
$800$	0	0
$801$	−12072.7	−0.532545
$802$	0	0
$803$	−1657.99	−0.0728634
$804$	0	0
$805$	38025.5	1.66487
$806$	0	0
$807$	22456.5	0.979562
$808$	0	0
$809$	−31026.7	−1.34838	−0.674191	−	0.738557i	$-0.735508\pi$
$809$	−31026.7	−1.34838	−0.674191	+	0.738557i	$0.735508\pi$
$810$	0	0
$811$	11304.9	0.489480	0.244740	−	0.969589i	$-0.421297\pi$
$811$	11304.9	0.489480	0.244740	+	0.969589i	$0.421297\pi$
$812$	0	0
$813$	−19073.5	−0.822801
$814$	0	0
$815$	164.959	0.00708988
$816$	0	0
$817$	−56.3333	−0.00241231
$818$	0	0
$819$	−2905.39	−0.123959
$820$	0	0
$821$	−32811.2	−1.39479	−0.697393	−	0.716688i	$-0.745657\pi$
$821$	−32811.2	−1.39479	−0.697393	+	0.716688i	$0.745657\pi$
$822$	0	0
$823$	39056.0	1.65420	0.827099	−	0.562056i	$-0.189990\pi$
$823$	39056.0	1.65420	0.827099	+	0.562056i	$0.189990\pi$
$824$	0	0
$825$	−2340.33	−0.0987635
$826$	0	0
$827$	8221.77	0.345706	0.172853	−	0.984948i	$-0.444701\pi$
$827$	8221.77	0.345706	0.172853	+	0.984948i	$0.444701\pi$
$828$	0	0
$829$	17201.9	0.720686	0.360343	−	0.932820i	$-0.382660\pi$
$829$	17201.9	0.720686	0.360343	+	0.932820i	$0.382660\pi$
$830$	0	0
$831$	−21697.4	−0.905744
$832$	0	0
$833$	−17421.7	−0.724643
$834$	0	0
$835$	−41532.1	−1.72129
$836$	0	0
$837$	−5008.58	−0.206836
$838$	0	0
$839$	−37656.5	−1.54952	−0.774759	−	0.632257i	$-0.782129\pi$
$839$	−37656.5	−1.54952	−0.774759	+	0.632257i	$0.782129\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−8660.60	−0.353840
$844$	0	0
$845$	2168.68	0.0882896
$846$	0	0
$847$	−23449.1	−0.951265
$848$	0	0
$849$	19239.8	0.777747
$850$	0	0
$851$	17143.5	0.690564
$852$	0	0
$853$	19882.5	0.798082	0.399041	−	0.916933i	$-0.369343\pi$
$853$	19882.5	0.798082	0.399041	+	0.916933i	$0.369343\pi$
$854$	0	0
$855$	−96.1348	−0.00384531
$856$	0	0
$857$	−45416.6	−1.81027	−0.905135	−	0.425125i	$-0.860230\pi$
$857$	−45416.6	−1.81027	−0.905135	+	0.425125i	$0.860230\pi$
$858$	0	0
$859$	−18572.2	−0.737689	−0.368845	−	0.929491i	$-0.620247\pi$
$859$	−18572.2	−0.737689	−0.368845	+	0.929491i	$0.620247\pi$
$860$	0	0
$861$	8727.82	0.345463
$862$	0	0
$863$	−43854.1	−1.72979	−0.864896	−	0.501952i	$-0.832616\pi$
$863$	−43854.1	−1.72979	−0.864896	+	0.501952i	$0.832616\pi$
$864$	0	0
$865$	47542.1	1.86876
$866$	0	0
$867$	2579.38	0.101039
$868$	0	0
$869$	23138.5	0.903246
$870$	0	0
$871$	8507.88	0.330974
$872$	0	0
$873$	7226.54	0.280162
$874$	0	0
$875$	−27191.1	−1.05054
$876$	0	0
$877$	8076.73	0.310983	0.155491	−	0.987837i	$-0.450304\pi$
$877$	8076.73	0.310983	0.155491	+	0.987837i	$0.450304\pi$
$878$	0	0
$879$	3671.47	0.140883
$880$	0	0
$881$	−28647.3	−1.09552	−0.547759	−	0.836636i	$-0.684519\pi$
$881$	−28647.3	−1.09552	−0.547759	+	0.836636i	$0.684519\pi$
$882$	0	0
$883$	25901.8	0.987164	0.493582	−	0.869699i	$-0.335687\pi$
$883$	25901.8	0.987164	0.493582	+	0.869699i	$0.335687\pi$
$884$	0	0
$885$	−3002.78	−0.114054
$886$	0	0
$887$	−9023.06	−0.341561	−0.170780	−	0.985309i	$-0.554629\pi$
$887$	−9023.06	−0.341561	−0.170780	+	0.985309i	$0.554629\pi$
$888$	0	0
$889$	20004.9	0.754717
$890$	0	0
$891$	1592.85	0.0598905
$892$	0	0
$893$	−396.770	−0.0148683
$894$	0	0
$895$	−31919.0	−1.19210
$896$	0	0
$897$	4653.85	0.173230
$898$	0	0
$899$	−1113.02	−0.0412916
$900$	0	0
$901$	−3776.49	−0.139637
$902$	0	0
$903$	−5041.67	−0.185799
$904$	0	0
$905$	−55980.9	−2.05621
$906$	0	0
$907$	−9526.00	−0.348738	−0.174369	−	0.984680i	$-0.555789\pi$
$907$	−9526.00	−0.348738	−0.174369	+	0.984680i	$0.555789\pi$
$908$	0	0
$909$	−4521.32	−0.164975
$910$	0	0
$911$	−8695.33	−0.316234	−0.158117	−	0.987420i	$-0.550542\pi$
$911$	−8695.33	−0.316234	−0.158117	+	0.987420i	$0.550542\pi$
$912$	0	0
$913$	−8468.60	−0.306977
$914$	0	0
$915$	−23481.6	−0.848389
$916$	0	0
$917$	9087.82	0.327270
$918$	0	0
$919$	−3367.18	−0.120863	−0.0604315	−	0.998172i	$-0.519248\pi$
$919$	−3367.18	−0.120863	−0.0604315	+	0.998172i	$0.519248\pi$
$920$	0	0
$921$	11559.6	0.413575
$922$	0	0
$923$	5082.93	0.181264
$924$	0	0
$925$	5699.24	0.202584
$926$	0	0
$927$	17360.3	0.615090
$928$	0	0
$929$	−25124.3	−0.887300	−0.443650	−	0.896200i	$-0.646317\pi$
$929$	−25124.3	−0.887300	−0.443650	+	0.896200i	$0.646317\pi$
$930$	0	0
$931$	−227.784	−0.00801859
$932$	0	0
$933$	23084.6	0.810029
$934$	0	0
$935$	−16065.6	−0.561926
$936$	0	0
$937$	1387.27	0.0483674	0.0241837	−	0.999708i	$-0.492301\pi$
$937$	1387.27	0.0483674	0.0241837	+	0.999708i	$0.492301\pi$
$938$	0	0
$939$	−8229.89	−0.286019
$940$	0	0
$941$	579.261	0.0200674	0.0100337	−	0.999950i	$-0.496806\pi$
$941$	579.261	0.0200674	0.0100337	+	0.999950i	$0.496806\pi$
$942$	0	0
$943$	−13980.2	−0.482777
$944$	0	0
$945$	−8603.80	−0.296171
$946$	0	0
$947$	−23162.3	−0.794798	−0.397399	−	0.917646i	$-0.630087\pi$
$947$	−23162.3	−0.794798	−0.397399	+	0.917646i	$0.630087\pi$
$948$	0	0
$949$	1096.07	0.0374919
$950$	0	0
$951$	−14120.0	−0.481464
$952$	0	0
$953$	4896.52	0.166436	0.0832182	−	0.996531i	$-0.473480\pi$
$953$	4896.52	0.166436	0.0832182	+	0.996531i	$0.473480\pi$
$954$	0	0
$955$	−36142.6	−1.22466
$956$	0	0
$957$	353.966	0.0119562
$958$	0	0
$959$	−57606.0	−1.93972
$960$	0	0
$961$	4620.29	0.155090
$962$	0	0
$963$	−15839.6	−0.530035
$964$	0	0
$965$	54580.4	1.82073
$966$	0	0
$967$	48707.2	1.61977	0.809885	−	0.586589i	$-0.199530\pi$
$967$	48707.2	1.61977	0.809885	+	0.586589i	$0.199530\pi$
$968$	0	0
$969$	−158.983	−0.00527067
$970$	0	0
$971$	55808.4	1.84446	0.922232	−	0.386636i	$-0.126363\pi$
$971$	55808.4	1.84446	0.922232	+	0.386636i	$0.126363\pi$
$972$	0	0
$973$	45112.3	1.48637
$974$	0	0
$975$	1547.15	0.0508188
$976$	0	0
$977$	12136.2	0.397413	0.198707	−	0.980059i	$-0.436326\pi$
$977$	12136.2	0.397413	0.198707	+	0.980059i	$0.436326\pi$
$978$	0	0
$979$	−26378.6	−0.861148
$980$	0	0
$981$	−479.764	−0.0156144
$982$	0	0
$983$	43305.9	1.40513	0.702566	−	0.711619i	$-0.252038\pi$
$983$	43305.9	1.40513	0.702566	+	0.711619i	$0.252038\pi$
$984$	0	0
$985$	−53446.6	−1.72888
$986$	0	0
$987$	−35509.8	−1.14518
$988$	0	0
$989$	8075.75	0.259650
$990$	0	0
$991$	−36327.3	−1.16446	−0.582228	−	0.813025i	$-0.697819\pi$
$991$	−36327.3	−1.16446	−0.582228	+	0.813025i	$0.697819\pi$
$992$	0	0
$993$	27812.5	0.888825
$994$	0	0
$995$	28904.2	0.920928
$996$	0	0
$997$	−57063.1	−1.81264	−0.906322	−	0.422588i	$-0.861122\pi$
$997$	−57063.1	−1.81264	−0.906322	+	0.422588i	$0.861122\pi$
$998$	0	0
$999$	−3878.95	−0.122847

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.b.1.2	✓	2
3.2	odd	2		936.4.a.h.1.1		2
4.3	odd	2		624.4.a.n.1.2		2
8.3	odd	2		2496.4.a.x.1.1		2
8.5	even	2		2496.4.a.bi.1.1		2
12.11	even	2		1872.4.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.b.1.2	✓	2	1.1	even	1	trivial
624.4.a.n.1.2		2	4.3	odd	2
936.4.a.h.1.1		2	3.2	odd	2
1872.4.a.bd.1.1		2	12.11	even	2
2496.4.a.x.1.1		2	8.3	odd	2
2496.4.a.bi.1.1		2	8.5	even	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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +12.8324 q^{5} +24.8324 q^{7} +9.00000 q^{9} +19.6648 q^{11} -13.0000 q^{13} -38.4972 q^{15} -63.6648 q^{17} -0.832397 q^{19} -74.4972 q^{21} +119.330 q^{23} +39.6704 q^{25} -27.0000 q^{27} -6.00000 q^{29} +185.503 q^{31} -58.9944 q^{33} +318.659 q^{35} +143.665 q^{37} +39.0000 q^{39} -117.156 q^{41} +67.6760 q^{43} +115.492 q^{45} +476.659 q^{47} +273.648 q^{49} +190.994 q^{51} +59.3184 q^{53} +252.346 q^{55} +2.49719 q^{57} +78.0000 q^{59} +609.955 q^{61} +223.492 q^{63} -166.821 q^{65} -654.452 q^{67} -357.989 q^{69} -390.994 q^{71} -84.3127 q^{73} -119.011 q^{75} +488.324 q^{77} +1176.65 q^{79} +81.0000 q^{81} -430.648 q^{83} -816.972 q^{85} +18.0000 q^{87} -1341.41 q^{89} -322.821 q^{91} -556.508 q^{93} -10.6816 q^{95} +802.949 q^{97} +176.983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9} - 20 q^{11} - 26 q^{13} + 12 q^{15} - 68 q^{17} + 28 q^{19} - 60 q^{21} + 120 q^{23} + 198 q^{25} - 54 q^{27} - 12 q^{29} + 460 q^{31} + 60 q^{33} + 400 q^{35}+ \cdots - 180 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9 - 20 * q^11 - 26 * q^13 + 12 * q^15 - 68 * q^17 + 28 * q^19 - 60 * q^21 + 120 * q^23 + 198 * q^25 - 54 * q^27 - 12 * q^29 + 460 * q^31 + 60 * q^33 + 400 * q^35 + 228 * q^37 + 78 * q^39 + 92 * q^41 + 432 * q^43 - 36 * q^45 + 716 * q^47 - 46 * q^49 + 204 * q^51 - 356 * q^53 + 920 * q^55 - 84 * q^57 + 156 * q^59 - 204 * q^61 + 180 * q^63 + 52 * q^65 + 204 * q^67 - 360 * q^69 - 604 * q^71 + 484 * q^73 - 594 * q^75 + 680 * q^77 + 1760 * q^79 + 162 * q^81 - 268 * q^83 - 744 * q^85 + 36 * q^87 + 76 * q^89 - 260 * q^91 - 1380 * q^93 - 496 * q^95 + 4 * q^97 - 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more