LMFDB - Embedded newform 1152.5.b.j.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1152.703");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$119.082197473$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 5x^{2} - 4x + 61 $ x^4 - 2x^3 + 5x^2 - 4*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.2
Root			$2.23205 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.5.b.j.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-30.1993i q^{5} +52.3068i q^{7} +90.0666 q^{11} +60.3987i q^{13} +338.000 q^{17} -6.92820 q^{19} -732.295i q^{23} -287.000 q^{25} +1298.57i q^{29} -1307.67i q^{31} +1579.63 q^{35} +241.595i q^{37} -578.000 q^{41} -2029.96 q^{43} +2196.89i q^{47} -335.000 q^{49} +2446.15i q^{53} -2719.95i q^{55} +1198.58 q^{59} +6402.26i q^{61} +1824.00 q^{65} -8265.35 q^{67} -4289.16i q^{71} +8734.00 q^{73} +4711.10i q^{77} +11246.0i q^{79} +13198.2 q^{83} -10207.4i q^{85} -910.000 q^{89} -3159.26 q^{91} +209.227i q^{95} +5422.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1352 q^{17} - 1148 q^{25} - 2312 q^{41} - 1340 q^{49} + 7296 q^{65} + 34936 q^{73} - 3640 q^{89} + 21688 q^{97}+O(q^{100}) $ 4 * q + 1352 * q^17 - 1148 * q^25 - 2312 * q^41 - 1340 * q^49 + 7296 * q^65 + 34936 * q^73 - 3640 * q^89 + 21688 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	− 30.1993i	− 1.20797i	−0.796994	−	0.603987i	$-0.793578\pi$
$5$	− 30.1993i	− 1.20797i	0.796994	−	0.603987i	$-0.206422\pi$
$6$	0	0
$7$	52.3068i	1.06749i	0.845647	+	0.533743i	$0.179215\pi$
$7$	52.3068i	1.06749i	−0.845647	+	0.533743i	$0.820785\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	90.0666	0.744352	0.372176	−	0.928162i	$-0.378612\pi$
$11$	90.0666	0.744352	0.372176	+	0.928162i	$0.378612\pi$
$12$	0	0
$13$	60.3987i	0.357389i	0.983905	+	0.178694i	$0.0571873\pi$
$13$	60.3987i	0.357389i	−0.983905	+	0.178694i	$0.942813\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	338.000	1.16955	0.584775	−	0.811195i	$-0.301183\pi$
$17$	338.000	1.16955	0.584775	+	0.811195i	$0.301183\pi$
$18$	0	0
$19$	−6.92820	−0.0191917	−0.00959585	−	0.999954i	$-0.503055\pi$
$19$	−6.92820	−0.0191917	−0.00959585	+	0.999954i	$0.503055\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 732.295i	− 1.38430i	−0.721753	−	0.692150i	$-0.756664\pi$
$23$	− 732.295i	− 1.38430i	0.721753	−	0.692150i	$-0.243336\pi$
$24$	0	0
$25$	−287.000	−0.459200
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1298.57i	1.54408i	0.635574	+	0.772040i	$0.280764\pi$
$29$	1298.57i	1.54408i	−0.635574	+	0.772040i	$0.719236\pi$
$30$	0	0
$31$	− 1307.67i	− 1.36074i	−0.732869	−	0.680369i	$-0.761819\pi$
$31$	− 1307.67i	− 1.36074i	0.732869	−	0.680369i	$-0.238181\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1579.63	1.28949
$36$	0	0
$37$	241.595i	0.176475i	0.996099	+	0.0882377i	$0.0281235\pi$
$37$	241.595i	0.176475i	−0.996099	+	0.0882377i	$0.971877\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−578.000	−0.343843	−0.171921	−	0.985111i	$-0.554998\pi$
$41$	−578.000	−0.343843	−0.171921	+	0.985111i	$0.554998\pi$
$42$	0	0
$43$	−2029.96	−1.09787	−0.548936	−	0.835865i	$-0.684967\pi$
$43$	−2029.96	−1.09787	−0.548936	+	0.835865i	$0.684967\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2196.89i	0.994516i	0.867603	+	0.497258i	$0.165660\pi$
$47$	2196.89i	0.994516i	−0.867603	+	0.497258i	$0.834340\pi$
$48$	0	0
$49$	−335.000	−0.139525
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2446.15i	0.870825i	0.900231	+	0.435412i	$0.143397\pi$
$53$	2446.15i	0.870825i	−0.900231	+	0.435412i	$0.856603\pi$
$54$	0	0
$55$	− 2719.95i	− 0.899158i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1198.58	0.344320	0.172160	−	0.985069i	$-0.444925\pi$
$59$	1198.58	0.344320	0.172160	+	0.985069i	$0.444925\pi$
$60$	0	0
$61$	6402.26i	1.72058i	0.509809	+	0.860288i	$0.329716\pi$
$61$	6402.26i	1.72058i	−0.509809	+	0.860288i	$0.670284\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1824.00	0.431716
$66$	0	0
$67$	−8265.35	−1.84124	−0.920622	−	0.390454i	$-0.872318\pi$
$67$	−8265.35	−1.84124	−0.920622	+	0.390454i	$0.872318\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 4289.16i	− 0.850854i	−0.904993	−	0.425427i	$-0.860124\pi$
$71$	− 4289.16i	− 0.850854i	0.904993	−	0.425427i	$-0.139876\pi$
$72$	0	0
$73$	8734.00	1.63896	0.819478	−	0.573110i	$-0.194263\pi$
$73$	8734.00	1.63896	0.819478	+	0.573110i	$0.194263\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4711.10i	0.794585i
$78$	0	0
$79$	11246.0i	1.80195i	0.433873	+	0.900974i	$0.357147\pi$
$79$	11246.0i	1.80195i	−0.433873	+	0.900974i	$0.642853\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13198.2	1.91584	0.957920	−	0.287034i	$-0.0926693\pi$
$83$	13198.2	1.91584	0.957920	+	0.287034i	$0.0926693\pi$
$84$	0	0
$85$	− 10207.4i	− 1.41279i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−910.000	−0.114884	−0.0574422	−	0.998349i	$-0.518295\pi$
$89$	−910.000	−0.114884	−0.0574422	+	0.998349i	$0.518295\pi$
$90$	0	0
$91$	−3159.26	−0.381507
$92$	0	0
$93$	0	0
$94$	0	0
$95$	209.227i	0.0231831i
$96$	0	0
$97$	5422.00	0.576257	0.288128	−	0.957592i	$-0.406967\pi$
$97$	5422.00	0.576257	0.288128	+	0.957592i	$0.406967\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10962.4i	− 1.07464i	−0.843380	−	0.537318i	$-0.819438\pi$
$101$	− 10962.4i	− 1.07464i	0.843380	−	0.537318i	$-0.180562\pi$
$102$	0	0
$103$	5387.60i	0.507833i	0.967226	+	0.253916i	$0.0817188\pi$
$103$	5387.60i	0.507833i	−0.967226	+	0.253916i	$0.918281\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6436.30	0.562171	0.281086	−	0.959683i	$-0.409305\pi$
$107$	6436.30	0.562171	0.281086	+	0.959683i	$0.409305\pi$
$108$	0	0
$109$	− 60.3987i	− 0.00508364i	−0.999997	−	0.00254182i	$-0.999191\pi$
$109$	− 60.3987i	− 0.00508364i	0.999997	−	0.00254182i	$-0.000809087\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3166.00	0.247944	0.123972	−	0.992286i	$-0.460437\pi$
$113$	3166.00	0.247944	0.123972	+	0.992286i	$0.460437\pi$
$114$	0	0
$115$	−22114.8	−1.67220
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17679.7i	1.24848i
$120$	0	0
$121$	−6529.00	−0.445939
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10207.4i	− 0.653272i
$126$	0	0
$127$	− 12919.8i	− 0.801028i	−0.916291	−	0.400514i	$-0.868832\pi$
$127$	− 12919.8i	− 0.801028i	0.916291	−	0.400514i	$-0.131168\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22676.0	1.32137	0.660684	−	0.750664i	$-0.270266\pi$
$131$	22676.0	1.32137	0.660684	+	0.750664i	$0.270266\pi$
$132$	0	0
$133$	− 362.392i	− 0.0204869i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15550.0	0.828494	0.414247	−	0.910165i	$-0.364045\pi$
$137$	15550.0	0.828494	0.414247	+	0.910165i	$0.364045\pi$
$138$	0	0
$139$	31502.5	1.63048	0.815241	−	0.579122i	$-0.196604\pi$
$139$	31502.5	1.63048	0.815241	+	0.579122i	$0.196604\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5439.91i	0.266023i
$144$	0	0
$145$	39216.0	1.86521
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 22619.3i	− 1.01884i	−0.860518	−	0.509421i	$-0.829860\pi$
$149$	− 22619.3i	− 1.01884i	0.860518	−	0.509421i	$-0.170140\pi$
$150$	0	0
$151$	− 7689.10i	− 0.337226i	−0.985682	−	0.168613i	$-0.946071\pi$
$151$	− 7689.10i	− 0.337226i	0.985682	−	0.168613i	$-0.0539289\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−39490.8	−1.64374
$156$	0	0
$157$	1207.97i	0.0490070i	0.999700	+	0.0245035i	$0.00780049\pi$
$157$	1207.97i	0.0490070i	−0.999700	+	0.0245035i	$0.992200\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38304.0	1.47772
$162$	0	0
$163$	−26749.8	−1.00680	−0.503402	−	0.864052i	$-0.667919\pi$
$163$	−26749.8	−1.00680	−0.503402	+	0.864052i	$0.667919\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 44983.8i	− 1.61296i	−0.591261	−	0.806480i	$-0.701370\pi$
$167$	− 44983.8i	− 1.61296i	0.591261	−	0.806480i	$-0.298630\pi$
$168$	0	0
$169$	24913.0	0.872273
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4197.71i	− 0.140256i	−0.997538	−	0.0701278i	$-0.977659\pi$
$173$	− 4197.71i	− 0.140256i	0.997538	−	0.0701278i	$-0.0223407\pi$
$174$	0	0
$175$	− 15012.0i	− 0.490189i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12491.6	−0.389861	−0.194931	−	0.980817i	$-0.562448\pi$
$179$	−12491.6	−0.389861	−0.194931	+	0.980817i	$0.562448\pi$
$180$	0	0
$181$	− 37870.0i	− 1.15595i	−0.816056	−	0.577973i	$-0.803844\pi$
$181$	− 37870.0i	− 1.15595i	0.816056	−	0.577973i	$-0.196156\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7296.00	0.213178
$186$	0	0
$187$	30442.5	0.870557
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 32848.7i	− 0.900432i	−0.892920	−	0.450216i	$-0.851347\pi$
$191$	− 32848.7i	− 0.900432i	0.892920	−	0.450216i	$-0.148653\pi$
$192$	0	0
$193$	44830.0	1.20352	0.601761	−	0.798676i	$-0.294466\pi$
$193$	44830.0	1.20352	0.601761	+	0.798676i	$0.294466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	42188.5i	1.08708i	0.839383	+	0.543540i	$0.182916\pi$
$197$	42188.5i	1.08708i	−0.839383	+	0.543540i	$0.817084\pi$
$198$	0	0
$199$	18778.1i	0.474183i	0.971487	+	0.237092i	$0.0761942\pi$
$199$	18778.1i	0.474183i	−0.971487	+	0.237092i	$0.923806\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−67924.1	−1.64828
$204$	0	0
$205$	17455.2i	0.415353i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−624.000	−0.0142854
$210$	0	0
$211$	63704.8	1.43089	0.715447	−	0.698667i	$-0.246223\pi$
$211$	63704.8	1.43089	0.715447	+	0.698667i	$0.246223\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	61303.6i	1.32620i
$216$	0	0
$217$	68400.0	1.45257
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20414.8i	0.417984i
$222$	0	0
$223$	− 19405.8i	− 0.390231i	−0.980780	−	0.195116i	$-0.937492\pi$
$223$	− 19405.8i	− 0.390231i	0.980780	−	0.195116i	$-0.0625082\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12089.7	0.234620	0.117310	−	0.993095i	$-0.462573\pi$
$227$	12089.7	0.234620	0.117310	+	0.993095i	$0.462573\pi$
$228$	0	0
$229$	91020.8i	1.73568i	0.496844	+	0.867840i	$0.334492\pi$
$229$	91020.8i	1.73568i	−0.496844	+	0.867840i	$0.665508\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−45166.0	−0.831955	−0.415977	−	0.909375i	$-0.636560\pi$
$233$	−45166.0	−0.831955	−0.415977	+	0.909375i	$0.636560\pi$
$234$	0	0
$235$	66344.5	1.20135
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 12135.2i	− 0.212447i	−0.994342	−	0.106223i	$-0.966124\pi$
$239$	− 12135.2i	− 0.212447i	0.994342	−	0.106223i	$-0.0338759\pi$
$240$	0	0
$241$	85822.0	1.47763	0.738813	−	0.673910i	$-0.235387\pi$
$241$	85822.0	1.47763	0.738813	+	0.673910i	$0.235387\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10116.8i	0.168543i
$246$	0	0
$247$	− 418.454i	− 0.00685889i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−58190.0	−0.923636	−0.461818	−	0.886975i	$-0.652803\pi$
$251$	−58190.0	−0.923636	−0.461818	+	0.886975i	$0.652803\pi$
$252$	0	0
$253$	− 65955.4i	− 1.03041i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	121726.	1.84297	0.921483	−	0.388420i	$-0.126979\pi$
$257$	121726.	1.84297	0.921483	+	0.388420i	$0.126979\pi$
$258$	0	0
$259$	−12637.0	−0.188385
$260$	0	0
$261$	0	0
$262$	0	0
$263$	97918.3i	1.41564i	0.706394	+	0.707819i	$0.250321\pi$
$263$	97918.3i	1.41564i	−0.706394	+	0.707819i	$0.749679\pi$
$264$	0	0
$265$	73872.0	1.05193
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14465.5i	0.199907i	0.994992	+	0.0999536i	$0.0318694\pi$
$269$	14465.5i	0.199907i	−0.994992	+	0.0999536i	$0.968131\pi$
$270$	0	0
$271$	55497.5i	0.755675i	0.925872	+	0.377837i	$0.123332\pi$
$271$	55497.5i	0.755675i	−0.925872	+	0.377837i	$0.876668\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−25849.1	−0.341807
$276$	0	0
$277$	102013.i	1.32953i	0.747053	+	0.664764i	$0.231468\pi$
$277$	102013.i	1.32953i	−0.747053	+	0.664764i	$0.768532\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	40082.0	0.507618	0.253809	−	0.967254i	$-0.418317\pi$
$281$	40082.0	0.507618	0.253809	+	0.967254i	$0.418317\pi$
$282$	0	0
$283$	48324.2	0.603381	0.301691	−	0.953406i	$-0.402449\pi$
$283$	48324.2	0.603381	0.301691	+	0.953406i	$0.402449\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 30233.3i	− 0.367047i
$288$	0	0
$289$	30723.0	0.367848
$290$	0	0
$291$	0	0
$292$	0	0
$293$	127592.i	1.48624i	0.669158	+	0.743120i	$0.266655\pi$
$293$	127592.i	1.48624i	−0.669158	+	0.743120i	$0.733345\pi$
$294$	0	0
$295$	− 36196.3i	− 0.415930i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	44229.6	0.494733
$300$	0	0
$301$	− 106181.i	− 1.17196i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	193344.	2.07841
$306$	0	0
$307$	39788.7	0.422165	0.211083	−	0.977468i	$-0.432301\pi$
$307$	39788.7	0.422165	0.211083	+	0.977468i	$0.432301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 174705.i	− 1.80627i	−0.429352	−	0.903137i	$-0.641258\pi$
$311$	− 174705.i	− 1.80627i	0.429352	−	0.903137i	$-0.358742\pi$
$312$	0	0
$313$	26930.0	0.274883	0.137441	−	0.990510i	$-0.456112\pi$
$313$	26930.0	0.274883	0.137441	+	0.990510i	$0.456112\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10841.6i	0.107888i	0.998544	+	0.0539440i	$0.0171793\pi$
$317$	10841.6i	0.107888i	−0.998544	+	0.0539440i	$0.982821\pi$
$318$	0	0
$319$	116958.i	1.14934i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2341.73	−0.0224457
$324$	0	0
$325$	− 17334.4i	− 0.164113i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−114912.	−1.06163
$330$	0	0
$331$	−127597.	−1.16462	−0.582309	−	0.812968i	$-0.697851\pi$
$331$	−127597.	−1.16462	−0.582309	+	0.812968i	$0.697851\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	249608.i	2.22417i
$336$	0	0
$337$	186482.	1.64201	0.821007	−	0.570917i	$-0.193412\pi$
$337$	186482.	1.64201	0.821007	+	0.570917i	$0.193412\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 117777.i	− 1.01287i
$342$	0	0
$343$	108066.i	0.918544i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	49224.9	0.408814	0.204407	−	0.978886i	$-0.434473\pi$
$347$	49224.9	0.408814	0.204407	+	0.978886i	$0.434473\pi$
$348$	0	0
$349$	209825.i	1.72269i	0.508023	+	0.861343i	$0.330376\pi$
$349$	209825.i	1.72269i	−0.508023	+	0.861343i	$0.669624\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	67486.0	0.541582	0.270791	−	0.962638i	$-0.412715\pi$
$353$	67486.0	0.541582	0.270791	+	0.962638i	$0.412715\pi$
$354$	0	0
$355$	−129530.	−1.02781
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 116016.i	− 0.900183i	−0.892983	−	0.450091i	$-0.851391\pi$
$359$	− 116016.i	− 0.900183i	0.892983	−	0.450091i	$-0.148609\pi$
$360$	0	0
$361$	−130273.	−0.999632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 263761.i	− 1.97982i
$366$	0	0
$367$	− 77884.8i	− 0.578257i	−0.957290	−	0.289128i	$-0.906635\pi$
$367$	− 77884.8i	− 0.578257i	0.957290	−	0.289128i	$-0.0933654\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−127950.	−0.929593
$372$	0	0
$373$	− 259835.i	− 1.86758i	−0.357816	−	0.933792i	$-0.616479\pi$
$373$	− 259835.i	− 1.86758i	0.357816	−	0.933792i	$-0.383521\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−78432.0	−0.551837
$378$	0	0
$379$	−116761.	−0.812867	−0.406433	−	0.913680i	$-0.633228\pi$
$379$	−116761.	−0.812867	−0.406433	+	0.913680i	$0.633228\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 158803.i	− 1.08259i	−0.840834	−	0.541293i	$-0.817935\pi$
$383$	− 158803.i	− 1.08259i	0.840834	−	0.541293i	$-0.182065\pi$
$384$	0	0
$385$	142272.	0.959838
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 121673.i	− 0.804073i	−0.915624	−	0.402037i	$-0.868302\pi$
$389$	− 121673.i	− 0.804073i	0.915624	−	0.402037i	$-0.131698\pi$
$390$	0	0
$391$	− 247516.i	− 1.61901i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	339621.	2.17671
$396$	0	0
$397$	− 237850.i	− 1.50911i	−0.656234	−	0.754557i	$-0.727852\pi$
$397$	− 237850.i	− 1.50911i	0.656234	−	0.754557i	$-0.272148\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	58130.0	0.361503	0.180751	−	0.983529i	$-0.442147\pi$
$401$	58130.0	0.361503	0.180751	+	0.983529i	$0.442147\pi$
$402$	0	0
$403$	78981.5	0.486312
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21759.6i	0.131360i
$408$	0	0
$409$	65186.0	0.389680	0.194840	−	0.980835i	$-0.437581\pi$
$409$	65186.0	0.389680	0.194840	+	0.980835i	$0.437581\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	62693.8i	0.367557i
$414$	0	0
$415$	− 398578.i	− 2.31428i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	267519.	1.52379	0.761897	−	0.647698i	$-0.224268\pi$
$419$	267519.	1.52379	0.761897	+	0.647698i	$0.224268\pi$
$420$	0	0
$421$	263399.i	1.48610i	0.669233	+	0.743052i	$0.266623\pi$
$421$	263399.i	1.48610i	−0.669233	+	0.743052i	$0.733377\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−97006.0	−0.537057
$426$	0	0
$427$	−334882.	−1.83669
$428$	0	0
$429$	0	0
$430$	0	0
$431$	105346.i	0.567104i	0.958957	+	0.283552i	$0.0915129\pi$
$431$	105346.i	0.567104i	−0.958957	+	0.283552i	$0.908487\pi$
$432$	0	0
$433$	83758.0	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$433$	83758.0	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5073.49i	0.0265671i
$438$	0	0
$439$	95983.0i	0.498041i	0.968498	+	0.249020i	$0.0801087\pi$
$439$	95983.0i	0.498041i	−0.968498	+	0.249020i	$0.919891\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	222679.	1.13468	0.567339	−	0.823484i	$-0.307973\pi$
$443$	222679.	1.13468	0.567339	+	0.823484i	$0.307973\pi$
$444$	0	0
$445$	27481.4i	0.138777i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	88658.0	0.439770	0.219885	−	0.975526i	$-0.429432\pi$
$449$	88658.0	0.439770	0.219885	+	0.975526i	$0.429432\pi$
$450$	0	0
$451$	−52058.5	−0.255940
$452$	0	0
$453$	0	0
$454$	0	0
$455$	95407.6i	0.460851i
$456$	0	0
$457$	−29086.0	−0.139268	−0.0696340	−	0.997573i	$-0.522183\pi$
$457$	−29086.0	−0.139268	−0.0696340	+	0.997573i	$0.522183\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	277139.i	1.30406i	0.758195	+	0.652028i	$0.226082\pi$
$461$	277139.i	1.30406i	−0.758195	+	0.652028i	$0.773918\pi$
$462$	0	0
$463$	279371.i	1.30322i	0.758553	+	0.651611i	$0.225907\pi$
$463$	279371.i	1.30322i	−0.758553	+	0.651611i	$0.774093\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−254702.	−1.16788	−0.583939	−	0.811797i	$-0.698489\pi$
$467$	−254702.	−1.16788	−0.583939	+	0.811797i	$0.698489\pi$
$468$	0	0
$469$	− 432334.i	− 1.96550i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−182832.	−0.817203
$474$	0	0
$475$	1988.39	0.00881283
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 52620.6i	− 0.229343i	−0.993403	−	0.114671i	$-0.963418\pi$
$479$	− 52620.6i	− 0.229343i	0.993403	−	0.114671i	$-0.0365815\pi$
$480$	0	0
$481$	−14592.0	−0.0630703
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 163741.i	− 0.696103i
$486$	0	0
$487$	147557.i	0.622162i	0.950383	+	0.311081i	$0.100691\pi$
$487$	147557.i	0.622162i	−0.950383	+	0.311081i	$0.899309\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	78240.2	0.324539	0.162270	−	0.986746i	$-0.448119\pi$
$491$	78240.2	0.324539	0.162270	+	0.986746i	$0.448119\pi$
$492$	0	0
$493$	438917.i	1.80588i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	224352.	0.908275
$498$	0	0
$499$	298044.	1.19696	0.598480	−	0.801138i	$-0.295771\pi$
$499$	298044.	1.19696	0.598480	+	0.801138i	$0.295771\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	132964.i	0.525530i	0.964860	+	0.262765i	$0.0846344\pi$
$503$	132964.i	0.525530i	−0.964860	+	0.262765i	$0.915366\pi$
$504$	0	0
$505$	−331056.	−1.29813
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12411.9i	0.0479075i	0.999713	+	0.0239538i	$0.00762545\pi$
$509$	12411.9i	0.0479075i	−0.999713	+	0.0239538i	$0.992375\pi$
$510$	0	0
$511$	456847.i	1.74956i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	162702.	0.613449
$516$	0	0
$517$	197866.i	0.740270i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−170018.	−0.626353	−0.313177	−	0.949695i	$-0.601393\pi$
$521$	−170018.	−0.626353	−0.313177	+	0.949695i	$0.601393\pi$
$522$	0	0
$523$	−401621.	−1.46829	−0.734147	−	0.678991i	$-0.762418\pi$
$523$	−401621.	−1.46829	−0.734147	+	0.678991i	$0.762418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 441992.i	− 1.59145i
$528$	0	0
$529$	−256415.	−0.916288
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 34910.4i	− 0.122886i
$534$	0	0
$535$	− 194372.i	− 0.679088i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−30172.3	−0.103856
$540$	0	0
$541$	138857.i	0.474430i	0.971457	+	0.237215i	$0.0762345\pi$
$541$	138857.i	0.474430i	−0.971457	+	0.237215i	$0.923765\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1824.00	−0.00614090
$546$	0	0
$547$	99094.1	0.331187	0.165593	−	0.986194i	$-0.447046\pi$
$547$	99094.1	0.331187	0.165593	+	0.986194i	$0.447046\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 8996.77i	− 0.0296335i
$552$	0	0
$553$	−588240.	−1.92355
$554$	0	0
$555$	0	0
$556$	0	0
$557$	199104.i	0.641756i	0.947120	+	0.320878i	$0.103978\pi$
$557$	199104.i	0.641756i	−0.947120	+	0.320878i	$0.896022\pi$
$558$	0	0
$559$	− 122607.i	− 0.392367i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−572457.	−1.80603	−0.903017	−	0.429605i	$-0.858653\pi$
$563$	−572457.	−1.80603	−0.903017	+	0.429605i	$0.858653\pi$
$564$	0	0
$565$	− 95611.1i	− 0.299510i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−563330.	−1.73996	−0.869978	−	0.493090i	$-0.835867\pi$
$569$	−563330.	−1.73996	−0.869978	+	0.493090i	$0.835867\pi$
$570$	0	0
$571$	113117.	0.346940	0.173470	−	0.984839i	$-0.444502\pi$
$571$	113117.	0.346940	0.173470	+	0.984839i	$0.444502\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	210169.i	0.635671i
$576$	0	0
$577$	155858.	0.468142	0.234071	−	0.972220i	$-0.424795\pi$
$577$	155858.	0.468142	0.234071	+	0.972220i	$0.424795\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	690357.i	2.04513i
$582$	0	0
$583$	220316.i	0.648200i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−270914.	−0.786239	−0.393119	−	0.919487i	$-0.628604\pi$
$587$	−270914.	−0.786239	−0.393119	+	0.919487i	$0.628604\pi$
$588$	0	0
$589$	9059.80i	0.0261149i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	418078.	1.18891	0.594454	−	0.804130i	$-0.297368\pi$
$593$	418078.	1.18891	0.594454	+	0.804130i	$0.297368\pi$
$594$	0	0
$595$	533915.	1.50813
$596$	0	0
$597$	0	0
$598$	0	0
$599$	539701.i	1.50418i	0.659060	+	0.752090i	$0.270954\pi$
$599$	539701.i	1.50418i	−0.659060	+	0.752090i	$0.729046\pi$
$600$	0	0
$601$	−439490.	−1.21675	−0.608373	−	0.793651i	$-0.708178\pi$
$601$	−439490.	−1.21675	−0.608373	+	0.793651i	$0.708178\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	197171.i	0.538683i
$606$	0	0
$607$	− 256879.i	− 0.697189i	−0.937274	−	0.348595i	$-0.886659\pi$
$607$	− 256879.i	− 0.697189i	0.937274	−	0.348595i	$-0.113341\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−132689.	−0.355429
$612$	0	0
$613$	− 222871.i	− 0.593107i	−0.955016	−	0.296553i	$-0.904163\pi$
$613$	− 222871.i	− 0.593107i	0.955016	−	0.296553i	$-0.0958373\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−588718.	−1.54645	−0.773227	−	0.634129i	$-0.781359\pi$
$617$	−588718.	−1.54645	−0.773227	+	0.634129i	$0.781359\pi$
$618$	0	0
$619$	417570.	1.08980	0.544901	−	0.838500i	$-0.316567\pi$
$619$	417570.	1.08980	0.544901	+	0.838500i	$0.316567\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 47599.2i	− 0.122638i
$624$	0	0
$625$	−487631.	−1.24834
$626$	0	0
$627$	0	0
$628$	0	0
$629$	81659.0i	0.206397i
$630$	0	0
$631$	204467.i	0.513529i	0.966474	+	0.256765i	$0.0826565\pi$
$631$	204467.i	0.513529i	−0.966474	+	0.256765i	$0.917344\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−390169.	−0.967620
$636$	0	0
$637$	− 20233.6i	− 0.0498647i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	270578.	0.658531	0.329266	−	0.944237i	$-0.393199\pi$
$641$	270578.	0.658531	0.329266	+	0.944237i	$0.393199\pi$
$642$	0	0
$643$	−584969.	−1.41485	−0.707426	−	0.706788i	$-0.750144\pi$
$643$	−584969.	−1.41485	−0.707426	+	0.706788i	$0.750144\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 96976.8i	− 0.231664i	−0.993269	−	0.115832i	$-0.963047\pi$
$647$	− 96976.8i	− 0.231664i	0.993269	−	0.115832i	$-0.0369535\pi$
$648$	0	0
$649$	107952.	0.256296
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 532082.i	− 1.24782i	−0.781496	−	0.623911i	$-0.785543\pi$
$653$	− 532082.i	− 1.24782i	0.781496	−	0.623911i	$-0.214457\pi$
$654$	0	0
$655$	− 684800.i	− 1.59618i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−253593.	−0.583938	−0.291969	−	0.956428i	$-0.594310\pi$
$659$	−253593.	−0.583938	−0.291969	+	0.956428i	$0.594310\pi$
$660$	0	0
$661$	339441.i	0.776892i	0.921471	+	0.388446i	$0.126988\pi$
$661$	339441.i	0.776892i	−0.921471	+	0.388446i	$0.873012\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10944.0	−0.0247476
$666$	0	0
$667$	950937.	2.13747
$668$	0	0
$669$	0	0
$670$	0	0
$671$	576630.i	1.28071i
$672$	0	0
$673$	−191570.	−0.422958	−0.211479	−	0.977383i	$-0.567828\pi$
$673$	−191570.	−0.422958	−0.211479	+	0.977383i	$0.567828\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 498923.i	− 1.08857i	−0.838900	−	0.544285i	$-0.816801\pi$
$677$	− 498923.i	− 1.08857i	0.838900	−	0.544285i	$-0.183199\pi$
$678$	0	0
$679$	283607.i	0.615146i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	96059.5	0.205920	0.102960	−	0.994685i	$-0.467169\pi$
$683$	96059.5	0.205920	0.102960	+	0.994685i	$0.467169\pi$
$684$	0	0
$685$	− 469600.i	− 1.00080i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−147744.	−0.311223
$690$	0	0
$691$	517474.	1.08376	0.541880	−	0.840456i	$-0.317713\pi$
$691$	517474.	1.08376	0.541880	+	0.840456i	$0.317713\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 951356.i	− 1.96958i
$696$	0	0
$697$	−195364.	−0.402142
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 214204.i	− 0.435904i	−0.975959	−	0.217952i	$-0.930062\pi$
$701$	− 214204.i	− 0.435904i	0.975959	−	0.217952i	$-0.0699377\pi$
$702$	0	0
$703$	− 1673.82i	− 0.00338686i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	573406.	1.14716
$708$	0	0
$709$	413550.i	0.822688i	0.911480	+	0.411344i	$0.134941\pi$
$709$	413550.i	0.822688i	−0.911480	+	0.411344i	$0.865059\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−957600.	−1.88367
$714$	0	0
$715$	164282.	0.321349
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 734073.i	− 1.41998i	−0.704213	−	0.709989i	$-0.748700\pi$
$719$	− 734073.i	− 1.41998i	0.704213	−	0.709989i	$-0.251300\pi$
$720$	0	0
$721$	−281808.	−0.542104
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 372690.i	− 0.709042i
$726$	0	0
$727$	− 1.03479e6i	− 1.95786i	−0.204199	−	0.978929i	$-0.565459\pi$
$727$	− 1.03479e6i	− 1.95786i	0.204199	−	0.978929i	$-0.434541\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−686128.	−1.28402
$732$	0	0
$733$	− 200222.i	− 0.372652i	−0.982488	−	0.186326i	$-0.940342\pi$
$733$	− 200222.i	− 0.372652i	0.982488	−	0.186326i	$-0.0596580\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−744432.	−1.37053
$738$	0	0
$739$	−526855.	−0.964722	−0.482361	−	0.875972i	$-0.660221\pi$
$739$	−526855.	−0.964722	−0.482361	+	0.875972i	$0.660221\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 380375.i	− 0.689024i	−0.938782	−	0.344512i	$-0.888044\pi$
$743$	− 380375.i	− 0.689024i	0.938782	−	0.344512i	$-0.111956\pi$
$744$	0	0
$745$	−683088.	−1.23073
$746$	0	0
$747$	0	0
$748$	0	0
$749$	336662.i	0.600110i
$750$	0	0
$751$	591851.i	1.04938i	0.851293	+	0.524690i	$0.175819\pi$
$751$	591851.i	1.04938i	−0.851293	+	0.524690i	$0.824181\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−232206.	−0.407360
$756$	0	0
$757$	373445.i	0.651681i	0.945425	+	0.325840i	$0.105647\pi$
$757$	373445.i	0.651681i	−0.945425	+	0.325840i	$0.894353\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37154.0	−0.0641558	−0.0320779	−	0.999485i	$-0.510212\pi$
$761$	−37154.0	−0.0641558	−0.0320779	+	0.999485i	$0.510212\pi$
$762$	0	0
$763$	3159.26	0.00542671
$764$	0	0
$765$	0	0
$766$	0	0
$767$	72392.6i	0.123056i
$768$	0	0
$769$	−455330.	−0.769970	−0.384985	−	0.922923i	$-0.625793\pi$
$769$	−455330.	−0.769970	−0.384985	+	0.922923i	$0.625793\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	57046.5i	0.0954708i	0.998860	+	0.0477354i	$0.0152004\pi$
$773$	57046.5i	0.0954708i	−0.998860	+	0.0477354i	$0.984800\pi$
$774$	0	0
$775$	375301.i	0.624851i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4004.50	0.00659893
$780$	0	0
$781$	− 386310.i	− 0.633335i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	36480.0	0.0591992
$786$	0	0
$787$	99953.2	0.161379	0.0806895	−	0.996739i	$-0.474288\pi$
$787$	99953.2	0.161379	0.0806895	+	0.996739i	$0.474288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	165603.i	0.264677i
$792$	0	0
$793$	−386688.	−0.614914
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 517345.i	− 0.814448i	−0.913328	−	0.407224i	$-0.866497\pi$
$797$	− 517345.i	− 0.814448i	0.913328	−	0.407224i	$-0.133503\pi$
$798$	0	0
$799$	742547.i	1.16314i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	786642.	1.21996
$804$	0	0
$805$	− 1.15676e6i	− 1.78505i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	570046.	0.870989	0.435495	−	0.900191i	$-0.356573\pi$
$809$	570046.	0.870989	0.435495	+	0.900191i	$0.356573\pi$
$810$	0	0
$811$	174418.	0.265185	0.132592	−	0.991171i	$-0.457670\pi$
$811$	174418.	0.265185	0.132592	+	0.991171i	$0.457670\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	807826.i	1.21619i
$816$	0	0
$817$	14064.0	0.0210700
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 438827.i	− 0.651038i	−0.945535	−	0.325519i	$-0.894461\pi$
$821$	− 438827.i	− 0.651038i	0.945535	−	0.325519i	$-0.105539\pi$
$822$	0	0
$823$	− 608380.i	− 0.898205i	−0.893480	−	0.449102i	$-0.851744\pi$
$823$	− 608380.i	− 0.898205i	0.893480	−	0.449102i	$-0.148256\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−727898.	−1.06429	−0.532144	−	0.846654i	$-0.678614\pi$
$827$	−727898.	−1.06429	−0.532144	+	0.846654i	$0.678614\pi$
$828$	0	0
$829$	− 619509.i	− 0.901444i	−0.892664	−	0.450722i	$-0.851167\pi$
$829$	− 619509.i	− 0.901444i	0.892664	−	0.450722i	$-0.148833\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−113230.	−0.163182
$834$	0	0
$835$	−1.35848e6	−1.94841
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.10064e6i	1.56358i	0.623539	+	0.781792i	$0.285694\pi$
$839$	1.10064e6i	1.56358i	−0.623539	+	0.781792i	$0.714306\pi$
$840$	0	0
$841$	−979007.	−1.38418
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 752356.i	− 1.05368i
$846$	0	0
$847$	− 341511.i	− 0.476034i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	176919.	0.244295
$852$	0	0
$853$	634911.i	0.872599i	0.899802	+	0.436299i	$0.143711\pi$
$853$	634911.i	0.872599i	−0.899802	+	0.436299i	$0.856289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	405310.	0.551856	0.275928	−	0.961178i	$-0.411015\pi$
$857$	405310.	0.551856	0.275928	+	0.961178i	$0.411015\pi$
$858$	0	0
$859$	100230.	0.135835	0.0679177	−	0.997691i	$-0.478364\pi$
$859$	100230.	0.135835	0.0679177	+	0.997691i	$0.478364\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	317816.i	0.426731i	0.976972	+	0.213366i	$0.0684425\pi$
$863$	317816.i	0.426731i	−0.976972	+	0.213366i	$0.931557\pi$
$864$	0	0
$865$	−126768.	−0.169425
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.01289e6i	1.34128i
$870$	0	0
$871$	− 499216.i	− 0.658040i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	533915.	0.697358
$876$	0	0
$877$	323133.i	0.420128i	0.977688	+	0.210064i	$0.0673673\pi$
$877$	323133.i	0.420128i	−0.977688	+	0.210064i	$0.932633\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.10419e6	1.42263	0.711315	−	0.702873i	$-0.248100\pi$
$881$	1.10419e6	1.42263	0.711315	+	0.702873i	$0.248100\pi$
$882$	0	0
$883$	−813877.	−1.04385	−0.521924	−	0.852992i	$-0.674786\pi$
$883$	−813877.	−1.04385	−0.521924	+	0.852992i	$0.674786\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	221781.i	0.281888i	0.990018	+	0.140944i	$0.0450138\pi$
$887$	221781.i	0.281888i	−0.990018	+	0.140944i	$0.954986\pi$
$888$	0	0
$889$	675792.	0.855085
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 15220.5i	− 0.0190864i
$894$	0	0
$895$	377237.i	0.470942i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.69810e6	2.10109
$900$	0	0
$901$	826797.i	1.01847i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.14365e6	−1.39635
$906$	0	0
$907$	−577071.	−0.701479	−0.350739	−	0.936473i	$-0.614070\pi$
$907$	−577071.	−0.701479	−0.350739	+	0.936473i	$0.614070\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 922692.i	− 1.11178i	−0.831255	−	0.555891i	$-0.812377\pi$
$911$	− 922692.i	− 1.11178i	0.831255	−	0.555891i	$-0.187623\pi$
$912$	0	0
$913$	1.18872e6	1.42606
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.18611e6i	1.41054i
$918$	0	0
$919$	1.47845e6i	1.75056i	0.483620	+	0.875278i	$0.339322\pi$
$919$	1.47845e6i	1.75056i	−0.483620	+	0.875278i	$0.660678\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	259059.	0.304086
$924$	0	0
$925$	− 69337.7i	− 0.0810375i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−241006.	−0.279252	−0.139626	−	0.990204i	$-0.544590\pi$
$929$	−241006.	−0.279252	−0.139626	+	0.990204i	$0.544590\pi$
$930$	0	0
$931$	2320.95	0.00267773
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 919344.i	− 1.05161i
$936$	0	0
$937$	−931058.	−1.06047	−0.530234	−	0.847851i	$-0.677896\pi$
$937$	−931058.	−1.06047	−0.530234	+	0.847851i	$0.677896\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	245490.i	0.277240i	0.990346	+	0.138620i	$0.0442666\pi$
$941$	245490.i	0.277240i	−0.990346	+	0.138620i	$0.955733\pi$
$942$	0	0
$943$	423267.i	0.475982i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	207853.	0.231770	0.115885	−	0.993263i	$-0.463030\pi$
$947$	207853.	0.231770	0.115885	+	0.993263i	$0.463030\pi$
$948$	0	0
$949$	527522.i	0.585744i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.23446e6	1.35923	0.679613	−	0.733570i	$-0.262148\pi$
$953$	1.23446e6	1.35923	0.679613	+	0.733570i	$0.262148\pi$
$954$	0	0
$955$	−992008.	−1.08770
$956$	0	0
$957$	0	0
$958$	0	0
$959$	813371.i	0.884405i
$960$	0	0
$961$	−786479.	−0.851609
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.35384e6i	− 1.45382i
$966$	0	0
$967$	− 181034.i	− 0.193601i	−0.995304	−	0.0968003i	$-0.969139\pi$
$967$	− 181034.i	− 0.193601i	0.995304	−	0.0968003i	$-0.0308608\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−97708.5	−0.103632	−0.0518160	−	0.998657i	$-0.516501\pi$
$971$	−97708.5	−0.103632	−0.0518160	+	0.998657i	$0.516501\pi$
$972$	0	0
$973$	1.64780e6i	1.74052i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.67313e6	−1.75284	−0.876419	−	0.481550i	$-0.840074\pi$
$977$	−1.67313e6	−1.75284	−0.876419	+	0.481550i	$0.840074\pi$
$978$	0	0
$979$	−81960.6	−0.0855145
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.03358e6i	− 1.06964i	−0.844966	−	0.534820i	$-0.820379\pi$
$983$	− 1.03358e6i	− 1.06964i	0.844966	−	0.534820i	$-0.179621\pi$
$984$	0	0
$985$	1.27406e6	1.31316
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.48653e6i	1.51978i
$990$	0	0
$991$	− 907889.i	− 0.924454i	−0.886762	−	0.462227i	$-0.847050\pi$
$991$	− 907889.i	− 0.924454i	0.886762	−	0.462227i	$-0.152950\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	567087.	0.572801
$996$	0	0
$997$	− 1.47844e6i	− 1.48735i	−0.668542	−	0.743675i	$-0.733081\pi$
$997$	− 1.47844e6i	− 1.48735i	0.668542	−	0.743675i	$-0.266919\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.5.b.j.703.2		4
3.2	odd	2		384.5.b.a.319.4	yes	4
4.3	odd	2	inner	1152.5.b.j.703.1		4
8.3	odd	2	inner	1152.5.b.j.703.3		4
8.5	even	2	inner	1152.5.b.j.703.4		4
12.11	even	2		384.5.b.a.319.2	yes	4
24.5	odd	2		384.5.b.a.319.1	✓	4
24.11	even	2		384.5.b.a.319.3	yes	4
48.5	odd	4		768.5.g.d.511.1		4
48.11	even	4		768.5.g.d.511.3		4
48.29	odd	4		768.5.g.d.511.4		4
48.35	even	4		768.5.g.d.511.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.a.319.1	✓	4	24.5	odd	2
384.5.b.a.319.2	yes	4	12.11	even	2
384.5.b.a.319.3	yes	4	24.11	even	2
384.5.b.a.319.4	yes	4	3.2	odd	2
768.5.g.d.511.1		4	48.5	odd	4
768.5.g.d.511.2		4	48.35	even	4
768.5.g.d.511.3		4	48.11	even	4
768.5.g.d.511.4		4	48.29	odd	4
1152.5.b.j.703.1		4	4.3	odd	2	inner
1152.5.b.j.703.2		4	1.1	even	1	trivial
1152.5.b.j.703.3		4	8.3	odd	2	inner
1152.5.b.j.703.4		4	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-30.1993i q^{5} +52.3068i q^{7} +90.0666 q^{11} +60.3987i q^{13} +338.000 q^{17} -6.92820 q^{19} -732.295i q^{23} -287.000 q^{25} +1298.57i q^{29} -1307.67i q^{31} +1579.63 q^{35} +241.595i q^{37} -578.000 q^{41} -2029.96 q^{43} +2196.89i q^{47} -335.000 q^{49} +2446.15i q^{53} -2719.95i q^{55} +1198.58 q^{59} +6402.26i q^{61} +1824.00 q^{65} -8265.35 q^{67} -4289.16i q^{71} +8734.00 q^{73} +4711.10i q^{77} +11246.0i q^{79} +13198.2 q^{83} -10207.4i q^{85} -910.000 q^{89} -3159.26 q^{91} +209.227i q^{95} +5422.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1352 q^{17} - 1148 q^{25} - 2312 q^{41} - 1340 q^{49} + 7296 q^{65} + 34936 q^{73} - 3640 q^{89} + 21688 q^{97}+O(q^{100}) \) 4 * q + 1352 * q^17 - 1148 * q^25 - 2312 * q^41 - 1340 * q^49 + 7296 * q^65 + 34936 * q^73 - 3640 * q^89 + 21688 * q^97

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