LMFDB - Embedded newform 1008.4.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.45017 q^{5} -7.00000 q^{7} +O(q^{10})$	$q-3.45017 q^{5} -7.00000 q^{7} -27.2990 q^{11} +58.7492 q^{13} -49.2990 q^{17} +157.148 q^{19} -82.1993 q^{23} -113.096 q^{25} +194.096 q^{29} -115.698 q^{31} +24.1512 q^{35} +327.498 q^{37} +136.103 q^{41} -311.698 q^{43} -355.698 q^{47} +49.0000 q^{49} -677.588 q^{53} +94.1861 q^{55} -197.148 q^{59} +61.0449 q^{61} -202.694 q^{65} -1017.39 q^{67} -279.807 q^{71} +629.794 q^{73} +191.093 q^{77} -20.2060 q^{79} -260.158 q^{83} +170.090 q^{85} +909.974 q^{89} -411.244 q^{91} -542.186 q^{95} +100.488 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 22 q^{5} - 14 q^{7}+O(q^{10}) $ 2 * q - 22 * q^5 - 14 * q^7	$ 2 q - 22 q^{5} - 14 q^{7} + 36 q^{11} + 42 q^{13} - 8 q^{17} + 118 q^{19} - 104 q^{23} + 106 q^{25} + 56 q^{29} - 20 q^{31} + 154 q^{35} + 504 q^{37} + 544 q^{41} - 412 q^{43} - 500 q^{47} + 98 q^{49} - 268 q^{53} - 1080 q^{55} - 198 q^{59} - 346 q^{61} + 108 q^{65} - 1008 q^{67} - 1224 q^{71} + 716 q^{73} - 252 q^{77} - 584 q^{79} - 1230 q^{83} - 596 q^{85} - 596 q^{89} - 294 q^{91} + 184 q^{95} - 856 q^{97}+O(q^{100}) $ 2 * q - 22 * q^5 - 14 * q^7 + 36 * q^11 + 42 * q^13 - 8 * q^17 + 118 * q^19 - 104 * q^23 + 106 * q^25 + 56 * q^29 - 20 * q^31 + 154 * q^35 + 504 * q^37 + 544 * q^41 - 412 * q^43 - 500 * q^47 + 98 * q^49 - 268 * q^53 - 1080 * q^55 - 198 * q^59 - 346 * q^61 + 108 * q^65 - 1008 * q^67 - 1224 * q^71 + 716 * q^73 - 252 * q^77 - 584 * q^79 - 1230 * q^83 - 596 * q^85 - 596 * q^89 - 294 * q^91 + 184 * q^95 - 856 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.45017	−0.308592	−0.154296	−	0.988025i	$-0.549311\pi$
$5$	−3.45017	−0.308592	−0.154296	+	0.988025i	$0.549311\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−27.2990	−0.748269	−0.374135	−	0.927374i	$-0.622060\pi$
$11$	−27.2990	−0.748269	−0.374135	+	0.927374i	$0.622060\pi$
$12$	0	0
$13$	58.7492	1.25339	0.626696	−	0.779264i	$-0.284407\pi$
$13$	58.7492	1.25339	0.626696	+	0.779264i	$0.284407\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−49.2990	−0.703339	−0.351670	−	0.936124i	$-0.614386\pi$
$17$	−49.2990	−0.703339	−0.351670	+	0.936124i	$0.614386\pi$
$18$	0	0
$19$	157.148	1.89748	0.948742	−	0.316052i	$-0.102357\pi$
$19$	157.148	1.89748	0.948742	+	0.316052i	$0.102357\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−82.1993	−0.745206	−0.372603	−	0.927991i	$-0.621535\pi$
$23$	−82.1993	−0.745206	−0.372603	+	0.927991i	$0.621535\pi$
$24$	0	0
$25$	−113.096	−0.904771
$26$	0	0
$27$	0	0
$28$	0	0
$29$	194.096	1.24285	0.621427	−	0.783472i	$-0.286553\pi$
$29$	194.096	1.24285	0.621427	+	0.783472i	$0.286553\pi$
$30$	0	0
$31$	−115.698	−0.670320	−0.335160	−	0.942161i	$-0.608790\pi$
$31$	−115.698	−0.670320	−0.335160	+	0.942161i	$0.608790\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	24.1512	0.116637
$36$	0	0
$37$	327.498	1.45515	0.727573	−	0.686030i	$-0.240648\pi$
$37$	327.498	1.45515	0.727573	+	0.686030i	$0.240648\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	136.103	0.518432	0.259216	−	0.965819i	$-0.416536\pi$
$41$	136.103	0.518432	0.259216	+	0.965819i	$0.416536\pi$
$42$	0	0
$43$	−311.698	−1.10543	−0.552715	−	0.833371i	$-0.686408\pi$
$43$	−311.698	−1.10543	−0.552715	+	0.833371i	$0.686408\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−355.698	−1.10391	−0.551956	−	0.833873i	$-0.686118\pi$
$47$	−355.698	−1.10391	−0.551956	+	0.833873i	$0.686118\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−677.588	−1.75611	−0.878055	−	0.478559i	$-0.841159\pi$
$53$	−677.588	−1.75611	−0.878055	+	0.478559i	$0.841159\pi$
$54$	0	0
$55$	94.1861	0.230910
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−197.148	−0.435025	−0.217512	−	0.976058i	$-0.569794\pi$
$59$	−197.148	−0.435025	−0.217512	+	0.976058i	$0.569794\pi$
$60$	0	0
$61$	61.0449	0.128131	0.0640655	−	0.997946i	$-0.479593\pi$
$61$	61.0449	0.128131	0.0640655	+	0.997946i	$0.479593\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−202.694	−0.386787
$66$	0	0
$67$	−1017.39	−1.85513	−0.927566	−	0.373660i	$-0.878103\pi$
$67$	−1017.39	−1.85513	−0.927566	+	0.373660i	$0.878103\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−279.807	−0.467704	−0.233852	−	0.972272i	$-0.575133\pi$
$71$	−279.807	−0.467704	−0.233852	+	0.972272i	$0.575133\pi$
$72$	0	0
$73$	629.794	1.00975	0.504875	−	0.863192i	$-0.331538\pi$
$73$	629.794	1.00975	0.504875	+	0.863192i	$0.331538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	191.093	0.282819
$78$	0	0
$79$	−20.2060	−0.0287766	−0.0143883	−	0.999896i	$-0.504580\pi$
$79$	−20.2060	−0.0287766	−0.0143883	+	0.999896i	$0.504580\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−260.158	−0.344049	−0.172024	−	0.985093i	$-0.555031\pi$
$83$	−260.158	−0.344049	−0.172024	+	0.985093i	$0.555031\pi$
$84$	0	0
$85$	170.090	0.217045
$86$	0	0
$87$	0	0
$88$	0	0
$89$	909.974	1.08379	0.541893	−	0.840447i	$-0.317708\pi$
$89$	909.974	1.08379	0.541893	+	0.840447i	$0.317708\pi$
$90$	0	0
$91$	−411.244	−0.473737
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−542.186	−0.585549
$96$	0	0
$97$	100.488	0.105186	0.0525931	−	0.998616i	$-0.483251\pi$
$97$	100.488	0.105186	0.0525931	+	0.998616i	$0.483251\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1868.25	−1.84057	−0.920285	−	0.391249i	$-0.872043\pi$
$101$	−1868.25	−1.84057	−0.920285	+	0.391249i	$0.872043\pi$
$102$	0	0
$103$	−7.10629	−0.00679809	−0.00339905	−	0.999994i	$-0.501082\pi$
$103$	−7.10629	−0.00679809	−0.00339905	+	0.999994i	$0.501082\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1078.01	−0.973970	−0.486985	−	0.873410i	$-0.661903\pi$
$107$	−1078.01	−0.973970	−0.486985	+	0.873410i	$0.661903\pi$
$108$	0	0
$109$	457.698	0.402197	0.201099	−	0.979571i	$-0.435549\pi$
$109$	457.698	0.402197	0.201099	+	0.979571i	$0.435549\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1931.25	−1.60776	−0.803880	−	0.594791i	$-0.797235\pi$
$113$	−1931.25	−1.60776	−0.803880	+	0.594791i	$0.797235\pi$
$114$	0	0
$115$	283.601	0.229965
$116$	0	0
$117$	0	0
$118$	0	0
$119$	345.093	0.265837
$120$	0	0
$121$	−585.764	−0.440093
$122$	0	0
$123$	0	0
$124$	0	0
$125$	821.472	0.587797
$126$	0	0
$127$	−1616.00	−1.12911	−0.564554	−	0.825396i	$-0.690952\pi$
$127$	−1616.00	−1.12911	−0.564554	+	0.825396i	$0.690952\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	507.650	0.338577	0.169288	−	0.985567i	$-0.445853\pi$
$131$	507.650	0.338577	0.169288	+	0.985567i	$0.445853\pi$
$132$	0	0
$133$	−1100.03	−0.717181
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2763.36	1.72328	0.861642	−	0.507516i	$-0.169436\pi$
$137$	2763.36	1.72328	0.861642	+	0.507516i	$0.169436\pi$
$138$	0	0
$139$	−1227.55	−0.749063	−0.374531	−	0.927214i	$-0.622196\pi$
$139$	−1227.55	−0.749063	−0.374531	+	0.927214i	$0.622196\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1603.79	−0.937874
$144$	0	0
$145$	−669.665	−0.383535
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.61126	0.00253536	0.00126768	−	0.999999i	$-0.499596\pi$
$149$	4.61126	0.00253536	0.00126768	+	0.999999i	$0.499596\pi$
$150$	0	0
$151$	−74.3656	−0.0400780	−0.0200390	−	0.999799i	$-0.506379\pi$
$151$	−74.3656	−0.0400780	−0.0200390	+	0.999799i	$0.506379\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	399.176	0.206855
$156$	0	0
$157$	−3241.65	−1.64785	−0.823923	−	0.566702i	$-0.808219\pi$
$157$	−3241.65	−1.64785	−0.823923	+	0.566702i	$0.808219\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	575.395	0.281662
$162$	0	0
$163$	943.093	0.453183	0.226591	−	0.973990i	$-0.427242\pi$
$163$	943.093	0.453183	0.226591	+	0.973990i	$0.427242\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2325.46	−1.07754	−0.538771	−	0.842452i	$-0.681111\pi$
$167$	−2325.46	−1.07754	−0.538771	+	0.842452i	$0.681111\pi$
$168$	0	0
$169$	1254.47	0.570990
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2920.14	−1.28332	−0.641659	−	0.766990i	$-0.721753\pi$
$173$	−2920.14	−1.28332	−0.641659	+	0.766990i	$0.721753\pi$
$174$	0	0
$175$	791.675	0.341971
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3049.14	1.27320	0.636602	−	0.771192i	$-0.280339\pi$
$179$	3049.14	1.27320	0.636602	+	0.771192i	$0.280339\pi$
$180$	0	0
$181$	−3046.00	−1.25087	−0.625434	−	0.780277i	$-0.715078\pi$
$181$	−3046.00	−1.25087	−0.625434	+	0.780277i	$0.715078\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1129.92	−0.449047
$186$	0	0
$187$	1345.81	0.526287
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2056.55	−0.779093	−0.389547	−	0.921007i	$-0.627368\pi$
$191$	−2056.55	−0.779093	−0.389547	+	0.921007i	$0.627368\pi$
$192$	0	0
$193$	−718.455	−0.267956	−0.133978	−	0.990984i	$-0.542775\pi$
$193$	−718.455	−0.267956	−0.133978	+	0.990984i	$0.542775\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1200.16	0.434050	0.217025	−	0.976166i	$-0.430365\pi$
$197$	1200.16	0.434050	0.217025	+	0.976166i	$0.430365\pi$
$198$	0	0
$199$	950.335	0.338530	0.169265	−	0.985571i	$-0.445861\pi$
$199$	950.335	0.338530	0.169265	+	0.985571i	$0.445861\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1358.67	−0.469755
$204$	0	0
$205$	−469.578	−0.159984
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4289.98	−1.41983
$210$	0	0
$211$	−2558.56	−0.834779	−0.417390	−	0.908728i	$-0.637055\pi$
$211$	−2558.56	−0.834779	−0.417390	+	0.908728i	$0.637055\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1075.41	0.341127
$216$	0	0
$217$	809.884	0.253357
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2896.28	−0.881559
$222$	0	0
$223$	−419.821	−0.126068	−0.0630342	−	0.998011i	$-0.520078\pi$
$223$	−419.821	−0.126068	−0.0630342	+	0.998011i	$0.520078\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4425.95	1.29410	0.647050	−	0.762448i	$-0.276003\pi$
$227$	4425.95	1.29410	0.647050	+	0.762448i	$0.276003\pi$
$228$	0	0
$229$	−682.254	−0.196876	−0.0984380	−	0.995143i	$-0.531385\pi$
$229$	−682.254	−0.196876	−0.0984380	+	0.995143i	$0.531385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4280.74	−1.20361	−0.601804	−	0.798644i	$-0.705551\pi$
$233$	−4280.74	−1.20361	−0.601804	+	0.798644i	$0.705551\pi$
$234$	0	0
$235$	1227.22	0.340658
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4828.95	1.30694	0.653470	−	0.756952i	$-0.273312\pi$
$239$	4828.95	1.30694	0.653470	+	0.756952i	$0.273312\pi$
$240$	0	0
$241$	−959.139	−0.256363	−0.128182	−	0.991751i	$-0.540914\pi$
$241$	−959.139	−0.256363	−0.128182	+	0.991751i	$0.540914\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−169.058	−0.0440846
$246$	0	0
$247$	9232.31	2.37829
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5226.44	−1.31430	−0.657151	−	0.753759i	$-0.728239\pi$
$251$	−5226.44	−1.31430	−0.657151	+	0.753759i	$0.728239\pi$
$252$	0	0
$253$	2243.96	0.557615
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3288.78	0.798242	0.399121	−	0.916898i	$-0.369315\pi$
$257$	3288.78	0.798242	0.399121	+	0.916898i	$0.369315\pi$
$258$	0	0
$259$	−2292.49	−0.549993
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6681.35	−1.56650	−0.783250	−	0.621707i	$-0.786440\pi$
$263$	−6681.35	−1.56650	−0.783250	+	0.621707i	$0.786440\pi$
$264$	0	0
$265$	2337.79	0.541922
$266$	0	0
$267$	0	0
$268$	0	0
$269$	299.875	0.0679692	0.0339846	−	0.999422i	$-0.489180\pi$
$269$	299.875	0.0679692	0.0339846	+	0.999422i	$0.489180\pi$
$270$	0	0
$271$	−126.405	−0.0283342	−0.0141671	−	0.999900i	$-0.504510\pi$
$271$	−126.405	−0.0283342	−0.0141671	+	0.999900i	$0.504510\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3087.42	0.677012
$276$	0	0
$277$	−1439.53	−0.312249	−0.156124	−	0.987737i	$-0.549900\pi$
$277$	−1439.53	−0.312249	−0.156124	+	0.987737i	$0.549900\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1024.45	0.217485	0.108742	−	0.994070i	$-0.465318\pi$
$281$	1024.45	0.217485	0.108742	+	0.994070i	$0.465318\pi$
$282$	0	0
$283$	147.168	0.0309124	0.0154562	−	0.999881i	$-0.495080\pi$
$283$	147.168	0.0309124	0.0154562	+	0.999881i	$0.495080\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−952.721	−0.195949
$288$	0	0
$289$	−2482.61	−0.505314
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4611.20	−0.919418	−0.459709	−	0.888070i	$-0.652046\pi$
$293$	−4611.20	−0.919418	−0.459709	+	0.888070i	$0.652046\pi$
$294$	0	0
$295$	680.193	0.134245
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4829.14	−0.934035
$300$	0	0
$301$	2181.88	0.417813
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−210.615	−0.0395402
$306$	0	0
$307$	1136.28	0.211242	0.105621	−	0.994406i	$-0.466317\pi$
$307$	1136.28	0.211242	0.105621	+	0.994406i	$0.466317\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8798.71	1.60427	0.802137	−	0.597140i	$-0.203696\pi$
$311$	8798.71	1.60427	0.802137	+	0.597140i	$0.203696\pi$
$312$	0	0
$313$	−8738.34	−1.57802	−0.789010	−	0.614381i	$-0.789406\pi$
$313$	−8738.34	−1.57802	−0.789010	+	0.614381i	$0.789406\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−613.475	−0.108695	−0.0543473	−	0.998522i	$-0.517308\pi$
$317$	−613.475	−0.108695	−0.0543473	+	0.998522i	$0.517308\pi$
$318$	0	0
$319$	−5298.64	−0.929990
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7747.23	−1.33457
$324$	0	0
$325$	−6644.32	−1.13403
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2489.88	0.417239
$330$	0	0
$331$	−2920.52	−0.484974	−0.242487	−	0.970155i	$-0.577963\pi$
$331$	−2920.52	−0.484974	−0.242487	+	0.970155i	$0.577963\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3510.16	0.572479
$336$	0	0
$337$	5181.86	0.837609	0.418804	−	0.908077i	$-0.362449\pi$
$337$	5181.86	0.837609	0.418804	+	0.908077i	$0.362449\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3158.43	0.501580
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6366.41	0.984919	0.492459	−	0.870335i	$-0.336098\pi$
$347$	6366.41	0.984919	0.492459	+	0.870335i	$0.336098\pi$
$348$	0	0
$349$	2114.58	0.324330	0.162165	−	0.986764i	$-0.448152\pi$
$349$	2114.58	0.324330	0.162165	+	0.986764i	$0.448152\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8424.62	1.27025	0.635124	−	0.772411i	$-0.280949\pi$
$353$	8424.62	1.27025	0.635124	+	0.772411i	$0.280949\pi$
$354$	0	0
$355$	965.381	0.144330
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−597.090	−0.0877805	−0.0438903	−	0.999036i	$-0.513975\pi$
$359$	−597.090	−0.0877805	−0.0438903	+	0.999036i	$0.513975\pi$
$360$	0	0
$361$	17836.4	2.60044
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2172.89	−0.311601
$366$	0	0
$367$	7563.58	1.07579	0.537896	−	0.843011i	$-0.319219\pi$
$367$	7563.58	1.07579	0.537896	+	0.843011i	$0.319219\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4743.12	0.663748
$372$	0	0
$373$	9760.01	1.35484	0.677419	−	0.735598i	$-0.263099\pi$
$373$	9760.01	1.35484	0.677419	+	0.735598i	$0.263099\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11403.0	1.55778
$378$	0	0
$379$	9275.91	1.25718	0.628590	−	0.777737i	$-0.283632\pi$
$379$	9275.91	1.25718	0.628590	+	0.777737i	$0.283632\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8008.24	1.06841	0.534206	−	0.845354i	$-0.320611\pi$
$383$	8008.24	1.06841	0.534206	+	0.845354i	$0.320611\pi$
$384$	0	0
$385$	−659.303	−0.0872758
$386$	0	0
$387$	0	0
$388$	0	0
$389$	208.861	0.0272228	0.0136114	−	0.999907i	$-0.495667\pi$
$389$	208.861	0.0272228	0.0136114	+	0.999907i	$0.495667\pi$
$390$	0	0
$391$	4052.35	0.524133
$392$	0	0
$393$	0	0
$394$	0	0
$395$	69.7139	0.00888022
$396$	0	0
$397$	−4004.51	−0.506249	−0.253124	−	0.967434i	$-0.581458\pi$
$397$	−4004.51	−0.506249	−0.253124	+	0.967434i	$0.581458\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5085.30	0.633286	0.316643	−	0.948545i	$-0.397444\pi$
$401$	5085.30	0.633286	0.316643	+	0.948545i	$0.397444\pi$
$402$	0	0
$403$	−6797.14	−0.840173
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8940.38	−1.08884
$408$	0	0
$409$	−15143.3	−1.83078	−0.915390	−	0.402569i	$-0.868117\pi$
$409$	−15143.3	−1.83078	−0.915390	+	0.402569i	$0.868117\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1380.03	0.164424
$414$	0	0
$415$	897.587	0.106171
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8407.07	−0.980220	−0.490110	−	0.871661i	$-0.663043\pi$
$419$	−8407.07	−0.980220	−0.490110	+	0.871661i	$0.663043\pi$
$420$	0	0
$421$	10020.9	1.16006	0.580032	−	0.814593i	$-0.303040\pi$
$421$	10020.9	1.16006	0.580032	+	0.814593i	$0.303040\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5575.54	0.636361
$426$	0	0
$427$	−427.314	−0.0484290
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15424.5	−1.72384	−0.861918	−	0.507048i	$-0.830737\pi$
$431$	−15424.5	−1.72384	−0.861918	+	0.507048i	$0.830737\pi$
$432$	0	0
$433$	10209.8	1.13314	0.566571	−	0.824013i	$-0.308270\pi$
$433$	10209.8	1.13314	0.566571	+	0.824013i	$0.308270\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12917.4	−1.41402
$438$	0	0
$439$	−13469.1	−1.46434	−0.732168	−	0.681124i	$-0.761491\pi$
$439$	−13469.1	−1.46434	−0.732168	+	0.681124i	$0.761491\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17987.5	1.92915	0.964573	−	0.263817i	$-0.0849816\pi$
$443$	17987.5	1.92915	0.964573	+	0.263817i	$0.0849816\pi$
$444$	0	0
$445$	−3139.56	−0.334448
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16413.6	1.72518	0.862591	−	0.505903i	$-0.168840\pi$
$449$	16413.6	1.72518	0.862591	+	0.505903i	$0.168840\pi$
$450$	0	0
$451$	−3715.48	−0.387927
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1418.86	0.146192
$456$	0	0
$457$	283.139	0.0289818	0.0144909	−	0.999895i	$-0.495387\pi$
$457$	283.139	0.0289818	0.0144909	+	0.999895i	$0.495387\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13769.3	−1.39111	−0.695555	−	0.718473i	$-0.744841\pi$
$461$	−13769.3	−1.39111	−0.695555	+	0.718473i	$0.744841\pi$
$462$	0	0
$463$	−12715.4	−1.27632	−0.638161	−	0.769903i	$-0.720304\pi$
$463$	−12715.4	−1.27632	−0.638161	+	0.769903i	$0.720304\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8052.37	0.797900	0.398950	−	0.916973i	$-0.369375\pi$
$467$	8052.37	0.797900	0.398950	+	0.916973i	$0.369375\pi$
$468$	0	0
$469$	7121.72	0.701174
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8509.04	0.827158
$474$	0	0
$475$	−17772.8	−1.71679
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5407.58	0.515822	0.257911	−	0.966169i	$-0.416966\pi$
$479$	5407.58	0.515822	0.257911	+	0.966169i	$0.416966\pi$
$480$	0	0
$481$	19240.3	1.82387
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−346.702	−0.0324596
$486$	0	0
$487$	−2672.17	−0.248639	−0.124320	−	0.992242i	$-0.539675\pi$
$487$	−2672.17	−0.248639	−0.124320	+	0.992242i	$0.539675\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6537.75	0.600905	0.300453	−	0.953797i	$-0.402862\pi$
$491$	6537.75	0.600905	0.300453	+	0.953797i	$0.402862\pi$
$492$	0	0
$493$	−9568.76	−0.874148
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1958.65	0.176776
$498$	0	0
$499$	14621.8	1.31175	0.655875	−	0.754870i	$-0.272300\pi$
$499$	14621.8	1.31175	0.655875	+	0.754870i	$0.272300\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2928.33	−0.259578	−0.129789	−	0.991542i	$-0.541430\pi$
$503$	−2928.33	−0.259578	−0.129789	+	0.991542i	$0.541430\pi$
$504$	0	0
$505$	6445.76	0.567986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2262.06	0.196982	0.0984910	−	0.995138i	$-0.468598\pi$
$509$	2262.06	0.196982	0.0984910	+	0.995138i	$0.468598\pi$
$510$	0	0
$511$	−4408.56	−0.381650
$512$	0	0
$513$	0	0
$514$	0	0
$515$	24.5179	0.00209784
$516$	0	0
$517$	9710.19	0.826023
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6284.94	0.528499	0.264250	−	0.964454i	$-0.414876\pi$
$521$	6284.94	0.528499	0.264250	+	0.964454i	$0.414876\pi$
$522$	0	0
$523$	23595.8	1.97279	0.986397	−	0.164378i	$-0.0525616\pi$
$523$	23595.8	1.97279	0.986397	+	0.164378i	$0.0525616\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5703.78	0.471462
$528$	0	0
$529$	−5410.27	−0.444667
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7995.94	0.649798
$534$	0	0
$535$	3719.30	0.300560
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1337.65	−0.106896
$540$	0	0
$541$	−11156.9	−0.886642	−0.443321	−	0.896363i	$-0.646200\pi$
$541$	−11156.9	−0.886642	−0.443321	+	0.896363i	$0.646200\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1579.13	−0.124115
$546$	0	0
$547$	−4578.08	−0.357851	−0.178926	−	0.983863i	$-0.557262\pi$
$547$	−4578.08	−0.357851	−0.178926	+	0.983863i	$0.557262\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	30501.8	2.35830
$552$	0	0
$553$	141.442	0.0108765
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19499.5	1.48334	0.741669	−	0.670766i	$-0.234035\pi$
$557$	19499.5	1.48334	0.741669	+	0.670766i	$0.234035\pi$
$558$	0	0
$559$	−18312.0	−1.38554
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4797.53	−0.359133	−0.179566	−	0.983746i	$-0.557469\pi$
$563$	−4797.53	−0.359133	−0.179566	+	0.983746i	$0.557469\pi$
$564$	0	0
$565$	6663.14	0.496142
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9461.07	0.697063	0.348531	−	0.937297i	$-0.386680\pi$
$569$	9461.07	0.697063	0.348531	+	0.937297i	$0.386680\pi$
$570$	0	0
$571$	−8146.88	−0.597086	−0.298543	−	0.954396i	$-0.596501\pi$
$571$	−8146.88	−0.597086	−0.298543	+	0.954396i	$0.596501\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9296.45	0.674241
$576$	0	0
$577$	231.608	0.0167105	0.00835526	−	0.999965i	$-0.497340\pi$
$577$	231.608	0.0167105	0.00835526	+	0.999965i	$0.497340\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1821.10	0.130038
$582$	0	0
$583$	18497.5	1.31404
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12579.0	−0.884481	−0.442241	−	0.896896i	$-0.645816\pi$
$587$	−12579.0	−0.884481	−0.442241	+	0.896896i	$0.645816\pi$
$588$	0	0
$589$	−18181.6	−1.27192
$590$	0	0
$591$	0	0
$592$	0	0
$593$	447.349	0.0309788	0.0154894	−	0.999880i	$-0.495069\pi$
$593$	447.349	0.0309788	0.0154894	+	0.999880i	$0.495069\pi$
$594$	0	0
$595$	−1190.63	−0.0820353
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10640.9	0.725833	0.362916	−	0.931822i	$-0.381781\pi$
$599$	10640.9	0.725833	0.362916	+	0.931822i	$0.381781\pi$
$600$	0	0
$601$	−9179.29	−0.623013	−0.311507	−	0.950244i	$-0.600834\pi$
$601$	−9179.29	−0.623013	−0.311507	+	0.950244i	$0.600834\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2020.98	0.135809
$606$	0	0
$607$	−5404.92	−0.361415	−0.180707	−	0.983537i	$-0.557839\pi$
$607$	−5404.92	−0.361415	−0.180707	+	0.983537i	$0.557839\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20896.9	−1.38363
$612$	0	0
$613$	−10971.0	−0.722864	−0.361432	−	0.932398i	$-0.617712\pi$
$613$	−10971.0	−0.722864	−0.361432	+	0.932398i	$0.617712\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5008.36	0.326789	0.163395	−	0.986561i	$-0.447756\pi$
$617$	5008.36	0.326789	0.163395	+	0.986561i	$0.447756\pi$
$618$	0	0
$619$	26422.1	1.71566	0.857830	−	0.513934i	$-0.171812\pi$
$619$	26422.1	1.71566	0.857830	+	0.513934i	$0.171812\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6369.81	−0.409633
$624$	0	0
$625$	11302.8	0.723381
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16145.3	−1.02346
$630$	0	0
$631$	−119.528	−0.00754093	−0.00377047	−	0.999993i	$-0.501200\pi$
$631$	−119.528	−0.00754093	−0.00377047	+	0.999993i	$0.501200\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5575.47	0.348434
$636$	0	0
$637$	2878.71	0.179056
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23746.6	−1.46324	−0.731618	−	0.681714i	$-0.761235\pi$
$641$	−23746.6	−1.46324	−0.731618	+	0.681714i	$0.761235\pi$
$642$	0	0
$643$	18138.5	1.11246	0.556231	−	0.831028i	$-0.312247\pi$
$643$	18138.5	1.11246	0.556231	+	0.831028i	$0.312247\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8176.52	−0.496835	−0.248417	−	0.968653i	$-0.579910\pi$
$647$	−8176.52	−0.496835	−0.248417	+	0.968653i	$0.579910\pi$
$648$	0	0
$649$	5381.94	0.325516
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24315.5	−1.45718	−0.728589	−	0.684951i	$-0.759824\pi$
$653$	−24315.5	−1.45718	−0.728589	+	0.684951i	$0.759824\pi$
$654$	0	0
$655$	−1751.47	−0.104482
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13443.8	−0.794680	−0.397340	−	0.917671i	$-0.630067\pi$
$659$	−13443.8	−0.794680	−0.397340	+	0.917671i	$0.630067\pi$
$660$	0	0
$661$	−20267.3	−1.19260	−0.596299	−	0.802762i	$-0.703363\pi$
$661$	−20267.3	−1.19260	−0.596299	+	0.802762i	$0.703363\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3795.30	0.221317
$666$	0	0
$667$	−15954.6	−0.926183
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1666.46	−0.0958765
$672$	0	0
$673$	20471.1	1.17252	0.586258	−	0.810125i	$-0.300601\pi$
$673$	20471.1	1.17252	0.586258	+	0.810125i	$0.300601\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17693.5	−1.00446	−0.502229	−	0.864735i	$-0.667486\pi$
$677$	−17693.5	−1.00446	−0.502229	+	0.864735i	$0.667486\pi$
$678$	0	0
$679$	−703.419	−0.0397566
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16613.3	−0.930734	−0.465367	−	0.885118i	$-0.654078\pi$
$683$	−16613.3	−0.930734	−0.465367	+	0.885118i	$0.654078\pi$
$684$	0	0
$685$	−9534.06	−0.531792
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39807.7	−2.20109
$690$	0	0
$691$	8207.34	0.451841	0.225920	−	0.974146i	$-0.427461\pi$
$691$	8207.34	0.451841	0.225920	+	0.974146i	$0.427461\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4235.26	0.231155
$696$	0	0
$697$	−6709.74	−0.364634
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21752.7	1.17202	0.586011	−	0.810303i	$-0.300698\pi$
$701$	21752.7	1.17202	0.586011	+	0.810303i	$0.300698\pi$
$702$	0	0
$703$	51465.7	2.76112
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13077.7	0.695670
$708$	0	0
$709$	−14636.1	−0.775277	−0.387638	−	0.921811i	$-0.626709\pi$
$709$	−14636.1	−0.775277	−0.387638	+	0.921811i	$0.626709\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9510.27	0.499527
$714$	0	0
$715$	5533.35	0.289421
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33477.7	−1.73645	−0.868224	−	0.496173i	$-0.834738\pi$
$719$	−33477.7	−1.73645	−0.868224	+	0.496173i	$0.834738\pi$
$720$	0	0
$721$	49.7440	0.00256944
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21951.6	−1.12450
$726$	0	0
$727$	−20073.4	−1.02404	−0.512022	−	0.858972i	$-0.671103\pi$
$727$	−20073.4	−1.02404	−0.512022	+	0.858972i	$0.671103\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15366.4	0.777492
$732$	0	0
$733$	−19982.3	−1.00691	−0.503455	−	0.864022i	$-0.667938\pi$
$733$	−19982.3	−1.00691	−0.503455	+	0.864022i	$0.667938\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27773.7	1.38814
$738$	0	0
$739$	−11224.8	−0.558744	−0.279372	−	0.960183i	$-0.590126\pi$
$739$	−11224.8	−0.558744	−0.279372	+	0.960183i	$0.590126\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7894.51	0.389800	0.194900	−	0.980823i	$-0.437562\pi$
$743$	7894.51	0.389800	0.194900	+	0.980823i	$0.437562\pi$
$744$	0	0
$745$	−15.9096	−0.000782393 0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7546.05	0.368126
$750$	0	0
$751$	2613.52	0.126989	0.0634945	−	0.997982i	$-0.479775\pi$
$751$	2613.52	0.126989	0.0634945	+	0.997982i	$0.479775\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	256.574	0.0123678
$756$	0	0
$757$	12055.8	0.578831	0.289415	−	0.957204i	$-0.406539\pi$
$757$	12055.8	0.578831	0.289415	+	0.957204i	$0.406539\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1196.38	−0.0569893	−0.0284946	−	0.999594i	$-0.509071\pi$
$761$	−1196.38	−0.0569893	−0.0284946	+	0.999594i	$0.509071\pi$
$762$	0	0
$763$	−3203.88	−0.152016
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11582.3	−0.545256
$768$	0	0
$769$	5216.18	0.244604	0.122302	−	0.992493i	$-0.460972\pi$
$769$	5216.18	0.244604	0.122302	+	0.992493i	$0.460972\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17623.0	−0.819994	−0.409997	−	0.912087i	$-0.634470\pi$
$773$	−17623.0	−0.819994	−0.409997	+	0.912087i	$0.634470\pi$
$774$	0	0
$775$	13085.0	0.606486
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21388.3	0.983716
$780$	0	0
$781$	7638.46	0.349969
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11184.2	0.508512
$786$	0	0
$787$	4779.24	0.216470	0.108235	−	0.994125i	$-0.465480\pi$
$787$	4779.24	0.216470	0.108235	+	0.994125i	$0.465480\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13518.8	0.607676
$792$	0	0
$793$	3586.34	0.160598
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37903.6	−1.68459	−0.842293	−	0.539020i	$-0.818795\pi$
$797$	−37903.6	−1.68459	−0.842293	+	0.539020i	$0.818795\pi$
$798$	0	0
$799$	17535.5	0.776424
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17192.8	−0.755565
$804$	0	0
$805$	−1985.21	−0.0869186
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19848.0	0.862570	0.431285	−	0.902216i	$-0.358060\pi$
$809$	19848.0	0.862570	0.431285	+	0.902216i	$0.358060\pi$
$810$	0	0
$811$	−7098.27	−0.307342	−0.153671	−	0.988122i	$-0.549110\pi$
$811$	−7098.27	−0.307342	−0.153671	+	0.988122i	$0.549110\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3253.83	−0.139849
$816$	0	0
$817$	−48982.6	−2.09753
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17294.5	−0.735179	−0.367589	−	0.929988i	$-0.619817\pi$
$821$	−17294.5	−0.735179	−0.367589	+	0.929988i	$0.619817\pi$
$822$	0	0
$823$	−17003.5	−0.720176	−0.360088	−	0.932918i	$-0.617253\pi$
$823$	−17003.5	−0.720176	−0.360088	+	0.932918i	$0.617253\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24445.1	1.02786	0.513929	−	0.857833i	$-0.328189\pi$
$827$	24445.1	1.02786	0.513929	+	0.857833i	$0.328189\pi$
$828$	0	0
$829$	19644.3	0.823011	0.411506	−	0.911407i	$-0.365003\pi$
$829$	19644.3	0.823011	0.411506	+	0.911407i	$0.365003\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2415.65	−0.100477
$834$	0	0
$835$	8023.22	0.332521
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23132.6	−0.951879	−0.475939	−	0.879478i	$-0.657892\pi$
$839$	−23132.6	−0.951879	−0.475939	+	0.879478i	$0.657892\pi$
$840$	0	0
$841$	13284.4	0.544688
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4328.11	−0.176203
$846$	0	0
$847$	4100.35	0.166340
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−26920.1	−1.08438
$852$	0	0
$853$	−45519.8	−1.82716	−0.913581	−	0.406658i	$-0.866694\pi$
$853$	−45519.8	−1.82716	−0.913581	+	0.406658i	$0.866694\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15295.3	−0.609659	−0.304829	−	0.952407i	$-0.598599\pi$
$857$	−15295.3	−0.609659	−0.304829	+	0.952407i	$0.598599\pi$
$858$	0	0
$859$	−6495.13	−0.257987	−0.128994	−	0.991645i	$-0.541175\pi$
$859$	−6495.13	−0.257987	−0.128994	+	0.991645i	$0.541175\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34002.5	1.34120	0.670601	−	0.741818i	$-0.266036\pi$
$863$	34002.5	1.34120	0.670601	+	0.741818i	$0.266036\pi$
$864$	0	0
$865$	10075.0	0.396022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	551.603	0.0215326
$870$	0	0
$871$	−59770.7	−2.32521
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5750.30	−0.222167
$876$	0	0
$877$	2063.66	0.0794581	0.0397291	−	0.999210i	$-0.487351\pi$
$877$	2063.66	0.0794581	0.0397291	+	0.999210i	$0.487351\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1681.78	0.0643141	0.0321570	−	0.999483i	$-0.489762\pi$
$881$	1681.78	0.0643141	0.0321570	+	0.999483i	$0.489762\pi$
$882$	0	0
$883$	−11207.2	−0.427125	−0.213563	−	0.976929i	$-0.568507\pi$
$883$	−11207.2	−0.427125	−0.213563	+	0.976929i	$0.568507\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8759.49	0.331584	0.165792	−	0.986161i	$-0.446982\pi$
$887$	8759.49	0.331584	0.165792	+	0.986161i	$0.446982\pi$
$888$	0	0
$889$	11312.0	0.426763
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−55897.1	−2.09465
$894$	0	0
$895$	−10520.0	−0.392901
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−22456.5	−0.833110
$900$	0	0
$901$	33404.4	1.23514
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10509.2	0.386008
$906$	0	0
$907$	−27925.3	−1.02232	−0.511161	−	0.859485i	$-0.670784\pi$
$907$	−27925.3	−1.02232	−0.511161	+	0.859485i	$0.670784\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23965.3	0.871577	0.435789	−	0.900049i	$-0.356470\pi$
$911$	23965.3	0.871577	0.435789	+	0.900049i	$0.356470\pi$
$912$	0	0
$913$	7102.05	0.257441
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3553.55	−0.127970
$918$	0	0
$919$	19192.2	0.688893	0.344446	−	0.938806i	$-0.388067\pi$
$919$	19192.2	0.688893	0.344446	+	0.938806i	$0.388067\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16438.4	−0.586217
$924$	0	0
$925$	−37038.9	−1.31657
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28616.5	1.01063	0.505317	−	0.862934i	$-0.331376\pi$
$929$	28616.5	1.01063	0.505317	+	0.862934i	$0.331376\pi$
$930$	0	0
$931$	7700.24	0.271069
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4643.28	−0.162408
$936$	0	0
$937$	−27352.7	−0.953656	−0.476828	−	0.878997i	$-0.658213\pi$
$937$	−27352.7	−0.953656	−0.476828	+	0.878997i	$0.658213\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34254.4	1.18668	0.593338	−	0.804953i	$-0.297810\pi$
$941$	34254.4	1.18668	0.593338	+	0.804953i	$0.297810\pi$
$942$	0	0
$943$	−11187.6	−0.386339
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1099.15	−0.0377166	−0.0188583	−	0.999822i	$-0.506003\pi$
$947$	−1099.15	−0.0377166	−0.0188583	+	0.999822i	$0.506003\pi$
$948$	0	0
$949$	36999.9	1.26561
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11037.5	−0.375174	−0.187587	−	0.982248i	$-0.560067\pi$
$953$	−11037.5	−0.375174	−0.187587	+	0.982248i	$0.560067\pi$
$954$	0	0
$955$	7095.44	0.240422
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19343.5	−0.651340
$960$	0	0
$961$	−16405.0	−0.550671
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2478.79	0.0826892
$966$	0	0
$967$	−11405.9	−0.379307	−0.189653	−	0.981851i	$-0.560736\pi$
$967$	−11405.9	−0.379307	−0.189653	+	0.981851i	$0.560736\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27331.4	−0.903302	−0.451651	−	0.892195i	$-0.649165\pi$
$971$	−27331.4	−0.903302	−0.451651	+	0.892195i	$0.649165\pi$
$972$	0	0
$973$	8592.87	0.283119
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4661.46	0.152644	0.0763221	−	0.997083i	$-0.475682\pi$
$977$	4661.46	0.152644	0.0763221	+	0.997083i	$0.475682\pi$
$978$	0	0
$979$	−24841.4	−0.810964
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7066.31	−0.229278	−0.114639	−	0.993407i	$-0.536571\pi$
$983$	−7066.31	−0.229278	−0.114639	+	0.993407i	$0.536571\pi$
$984$	0	0
$985$	−4140.75	−0.133944
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25621.3	0.823773
$990$	0	0
$991$	22127.8	0.709296	0.354648	−	0.935000i	$-0.384601\pi$
$991$	22127.8	0.709296	0.354648	+	0.935000i	$0.384601\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3278.81	−0.104468
$996$	0	0
$997$	−45937.0	−1.45922	−0.729609	−	0.683865i	$-0.760298\pi$
$997$	−45937.0	−1.45922	−0.729609	+	0.683865i	$0.760298\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.a.x.1.2		2
3.2	odd	2		112.4.a.h.1.1		2
4.3	odd	2		504.4.a.i.1.2		2
12.11	even	2		56.4.a.c.1.2	✓	2
21.20	even	2		784.4.a.t.1.2		2
24.5	odd	2		448.4.a.r.1.2		2
24.11	even	2		448.4.a.s.1.1		2
60.23	odd	4		1400.4.g.h.449.3		4
60.47	odd	4		1400.4.g.h.449.2		4
60.59	even	2		1400.4.a.i.1.1		2
84.11	even	6		392.4.i.l.177.1		4
84.23	even	6		392.4.i.l.361.1		4
84.47	odd	6		392.4.i.i.361.2		4
84.59	odd	6		392.4.i.i.177.2		4
84.83	odd	2		392.4.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.a.c.1.2	✓	2	12.11	even	2
112.4.a.h.1.1		2	3.2	odd	2
392.4.a.h.1.1		2	84.83	odd	2
392.4.i.i.177.2		4	84.59	odd	6
392.4.i.i.361.2		4	84.47	odd	6
392.4.i.l.177.1		4	84.11	even	6
392.4.i.l.361.1		4	84.23	even	6
448.4.a.r.1.2		2	24.5	odd	2
448.4.a.s.1.1		2	24.11	even	2
504.4.a.i.1.2		2	4.3	odd	2
784.4.a.t.1.2		2	21.20	even	2
1008.4.a.x.1.2		2	1.1	even	1	trivial
1400.4.a.i.1.1		2	60.59	even	2
1400.4.g.h.449.2		4	60.47	odd	4
1400.4.g.h.449.3		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.45017 q^{5} -7.00000 q^{7} +O(q^{10})\)	\(q-3.45017 q^{5} -7.00000 q^{7} -27.2990 q^{11} +58.7492 q^{13} -49.2990 q^{17} +157.148 q^{19} -82.1993 q^{23} -113.096 q^{25} +194.096 q^{29} -115.698 q^{31} +24.1512 q^{35} +327.498 q^{37} +136.103 q^{41} -311.698 q^{43} -355.698 q^{47} +49.0000 q^{49} -677.588 q^{53} +94.1861 q^{55} -197.148 q^{59} +61.0449 q^{61} -202.694 q^{65} -1017.39 q^{67} -279.807 q^{71} +629.794 q^{73} +191.093 q^{77} -20.2060 q^{79} -260.158 q^{83} +170.090 q^{85} +909.974 q^{89} -411.244 q^{91} -542.186 q^{95} +100.488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 22 q^{5} - 14 q^{7}+O(q^{10}) \) 2 * q - 22 * q^5 - 14 * q^7	\( 2 q - 22 q^{5} - 14 q^{7} + 36 q^{11} + 42 q^{13} - 8 q^{17} + 118 q^{19} - 104 q^{23} + 106 q^{25} + 56 q^{29} - 20 q^{31} + 154 q^{35} + 504 q^{37} + 544 q^{41} - 412 q^{43} - 500 q^{47} + 98 q^{49} - 268 q^{53} - 1080 q^{55} - 198 q^{59} - 346 q^{61} + 108 q^{65} - 1008 q^{67} - 1224 q^{71} + 716 q^{73} - 252 q^{77} - 584 q^{79} - 1230 q^{83} - 596 q^{85} - 596 q^{89} - 294 q^{91} + 184 q^{95} - 856 q^{97}+O(q^{100}) \) 2 * q - 22 * q^5 - 14 * q^7 + 36 * q^11 + 42 * q^13 - 8 * q^17 + 118 * q^19 - 104 * q^23 + 106 * q^25 + 56 * q^29 - 20 * q^31 + 154 * q^35 + 504 * q^37 + 544 * q^41 - 412 * q^43 - 500 * q^47 + 98 * q^49 - 268 * q^53 - 1080 * q^55 - 198 * q^59 - 346 * q^61 + 108 * q^65 - 1008 * q^67 - 1224 * q^71 + 716 * q^73 - 252 * q^77 - 584 * q^79 - 1230 * q^83 - 596 * q^85 - 596 * q^89 - 294 * q^91 + 184 * q^95 - 856 * q^97

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