LMFDB - Embedded newform 1856.4.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 $ x^5 - x^4 - 34x^3 + 74x^2 + 94*x - 198
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.92646$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.03215 q^{3} -15.2584 q^{5} +33.3511 q^{7} -10.7418 q^{9} +O(q^{10})$	$q+4.03215 q^{3} -15.2584 q^{5} +33.3511 q^{7} -10.7418 q^{9} +4.44728 q^{11} +38.9073 q^{13} -61.5242 q^{15} -18.1054 q^{17} -75.1894 q^{19} +134.476 q^{21} +187.833 q^{23} +107.819 q^{25} -152.180 q^{27} -29.0000 q^{29} +68.3647 q^{31} +17.9321 q^{33} -508.885 q^{35} -44.9780 q^{37} +156.880 q^{39} +299.248 q^{41} +138.211 q^{43} +163.903 q^{45} -531.585 q^{47} +769.293 q^{49} -73.0035 q^{51} +242.701 q^{53} -67.8585 q^{55} -303.175 q^{57} -500.944 q^{59} +325.172 q^{61} -358.250 q^{63} -593.664 q^{65} +263.489 q^{67} +757.369 q^{69} +726.633 q^{71} +851.378 q^{73} +434.744 q^{75} +148.322 q^{77} +85.0429 q^{79} -323.585 q^{81} -914.025 q^{83} +276.259 q^{85} -116.932 q^{87} -1071.97 q^{89} +1297.60 q^{91} +275.657 q^{93} +1147.27 q^{95} -662.199 q^{97} -47.7718 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9	$ 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 - 36 * q^11 - 26 * q^13 + 88 * q^15 + 82 * q^17 - 156 * q^19 - 72 * q^21 + 336 * q^23 + 151 * q^25 - 352 * q^27 - 145 * q^29 + 432 * q^31 + 108 * q^33 - 600 * q^35 + 18 * q^37 + 688 * q^39 + 82 * q^41 - 340 * q^43 + 146 * q^45 + 680 * q^47 - 115 * q^49 - 608 * q^51 + 102 * q^53 + 736 * q^55 - 576 * q^57 - 924 * q^59 + 618 * q^61 + 584 * q^63 - 704 * q^65 - 44 * q^67 + 1056 * q^69 + 1032 * q^71 - 1078 * q^73 + 468 * q^75 + 888 * q^77 + 200 * q^79 - 1843 * q^81 - 452 * q^83 + 1700 * q^85 + 116 * q^87 - 1790 * q^89 + 1128 * q^91 + 1884 * q^93 + 1024 * q^95 - 2518 * q^97 + 1500 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	4.03215	0.775987	0.387993	−	0.921662i	$-0.373168\pi$
$3$	4.03215	0.775987	0.387993	+	0.921662i	$0.373168\pi$
$4$	0	0
$5$	−15.2584	−1.36475	−0.682377	−	0.731000i	$-0.739054\pi$
$5$	−15.2584	−1.36475	−0.682377	+	0.731000i	$0.739054\pi$
$6$	0	0
$7$	33.3511	1.80079	0.900394	−	0.435075i	$-0.143278\pi$
$7$	33.3511	1.80079	0.900394	+	0.435075i	$0.143278\pi$
$8$	0	0
$9$	−10.7418	−0.397844
$10$	0	0
$11$	4.44728	0.121901	0.0609503	−	0.998141i	$-0.480587\pi$
$11$	4.44728	0.121901	0.0609503	+	0.998141i	$0.480587\pi$
$12$	0	0
$13$	38.9073	0.830072	0.415036	−	0.909805i	$-0.363769\pi$
$13$	38.9073	0.830072	0.415036	+	0.909805i	$0.363769\pi$
$14$	0	0
$15$	−61.5242	−1.05903
$16$	0	0
$17$	−18.1054	−0.258306	−0.129153	−	0.991625i	$-0.541226\pi$
$17$	−18.1054	−0.258306	−0.129153	+	0.991625i	$0.541226\pi$
$18$	0	0
$19$	−75.1894	−0.907876	−0.453938	−	0.891033i	$-0.649981\pi$
$19$	−75.1894	−0.907876	−0.453938	+	0.891033i	$0.649981\pi$
$20$	0	0
$21$	134.476	1.39739
$22$	0	0
$23$	187.833	1.70286	0.851431	−	0.524466i	$-0.175735\pi$
$23$	187.833	1.70286	0.851431	+	0.524466i	$0.175735\pi$
$24$	0	0
$25$	107.819	0.862556
$26$	0	0
$27$	−152.180	−1.08471
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	68.3647	0.396086	0.198043	−	0.980193i	$-0.436541\pi$
$31$	68.3647	0.396086	0.198043	+	0.980193i	$0.436541\pi$
$32$	0	0
$33$	17.9321	0.0945932
$34$	0	0
$35$	−508.885	−2.45763
$36$	0	0
$37$	−44.9780	−0.199847	−0.0999235	−	0.994995i	$-0.531860\pi$
$37$	−44.9780	−0.199847	−0.0999235	+	0.994995i	$0.531860\pi$
$38$	0	0
$39$	156.880	0.644125
$40$	0	0
$41$	299.248	1.13987	0.569936	−	0.821689i	$-0.306968\pi$
$41$	299.248	1.13987	0.569936	+	0.821689i	$0.306968\pi$
$42$	0	0
$43$	138.211	0.490161	0.245080	−	0.969503i	$-0.421186\pi$
$43$	138.211	0.490161	0.245080	+	0.969503i	$0.421186\pi$
$44$	0	0
$45$	163.903	0.542960
$46$	0	0
$47$	−531.585	−1.64978	−0.824890	−	0.565294i	$-0.808763\pi$
$47$	−531.585	−1.64978	−0.824890	+	0.565294i	$0.808763\pi$
$48$	0	0
$49$	769.293	2.24284
$50$	0	0
$51$	−73.0035	−0.200442
$52$	0	0
$53$	242.701	0.629011	0.314505	−	0.949256i	$-0.398161\pi$
$53$	242.701	0.629011	0.314505	+	0.949256i	$0.398161\pi$
$54$	0	0
$55$	−67.8585	−0.166364
$56$	0	0
$57$	−303.175	−0.704500
$58$	0	0
$59$	−500.944	−1.10538	−0.552689	−	0.833387i	$-0.686398\pi$
$59$	−500.944	−1.10538	−0.552689	+	0.833387i	$0.686398\pi$
$60$	0	0
$61$	325.172	0.682525	0.341262	−	0.939968i	$-0.389146\pi$
$61$	325.172	0.682525	0.341262	+	0.939968i	$0.389146\pi$
$62$	0	0
$63$	−358.250	−0.716433
$64$	0	0
$65$	−593.664	−1.13285
$66$	0	0
$67$	263.489	0.480453	0.240226	−	0.970717i	$-0.422778\pi$
$67$	263.489	0.480453	0.240226	+	0.970717i	$0.422778\pi$
$68$	0	0
$69$	757.369	1.32140
$70$	0	0
$71$	726.633	1.21458	0.607292	−	0.794479i	$-0.292256\pi$
$71$	726.633	1.21458	0.607292	+	0.794479i	$0.292256\pi$
$72$	0	0
$73$	851.378	1.36502	0.682508	−	0.730878i	$-0.260889\pi$
$73$	851.378	1.36502	0.682508	+	0.730878i	$0.260889\pi$
$74$	0	0
$75$	434.744	0.669332
$76$	0	0
$77$	148.322	0.219517
$78$	0	0
$79$	85.0429	0.121115	0.0605574	−	0.998165i	$-0.480712\pi$
$79$	85.0429	0.121115	0.0605574	+	0.998165i	$0.480712\pi$
$80$	0	0
$81$	−323.585	−0.443876
$82$	0	0
$83$	−914.025	−1.20876	−0.604381	−	0.796695i	$-0.706580\pi$
$83$	−914.025	−1.20876	−0.604381	+	0.796695i	$0.706580\pi$
$84$	0	0
$85$	276.259	0.352524
$86$	0	0
$87$	−116.932	−0.144097
$88$	0	0
$89$	−1071.97	−1.27673	−0.638365	−	0.769733i	$-0.720389\pi$
$89$	−1071.97	−1.27673	−0.638365	+	0.769733i	$0.720389\pi$
$90$	0	0
$91$	1297.60	1.49478
$92$	0	0
$93$	275.657	0.307358
$94$	0	0
$95$	1147.27	1.23903
$96$	0	0
$97$	−662.199	−0.693155	−0.346578	−	0.938021i	$-0.612656\pi$
$97$	−662.199	−0.693155	−0.346578	+	0.938021i	$0.612656\pi$
$98$	0	0
$99$	−47.7718	−0.0484974
$100$	0	0
$101$	770.155	0.758746	0.379373	−	0.925244i	$-0.376140\pi$
$101$	770.155	0.758746	0.379373	+	0.925244i	$0.376140\pi$
$102$	0	0
$103$	637.552	0.609902	0.304951	−	0.952368i	$-0.401360\pi$
$103$	637.552	0.609902	0.304951	+	0.952368i	$0.401360\pi$
$104$	0	0
$105$	−2051.90	−1.90709
$106$	0	0
$107$	573.723	0.518354	0.259177	−	0.965830i	$-0.416549\pi$
$107$	573.723	0.518354	0.259177	+	0.965830i	$0.416549\pi$
$108$	0	0
$109$	−1237.73	−1.08764	−0.543821	−	0.839202i	$-0.683023\pi$
$109$	−1237.73	−1.08764	−0.543821	+	0.839202i	$0.683023\pi$
$110$	0	0
$111$	−181.358	−0.155079
$112$	0	0
$113$	1584.92	1.31944	0.659719	−	0.751512i	$-0.270675\pi$
$113$	1584.92	1.31944	0.659719	+	0.751512i	$0.270675\pi$
$114$	0	0
$115$	−2866.03	−2.32399
$116$	0	0
$117$	−417.934	−0.330240
$118$	0	0
$119$	−603.833	−0.465154
$120$	0	0
$121$	−1311.22	−0.985140
$122$	0	0
$123$	1206.61	0.884525
$124$	0	0
$125$	262.147	0.187577
$126$	0	0
$127$	−1746.60	−1.22036	−0.610180	−	0.792263i	$-0.708903\pi$
$127$	−1746.60	−1.22036	−0.610180	+	0.792263i	$0.708903\pi$
$128$	0	0
$129$	557.285	0.380358
$130$	0	0
$131$	2070.14	1.38068	0.690340	−	0.723485i	$-0.257461\pi$
$131$	2070.14	1.38068	0.690340	+	0.723485i	$0.257461\pi$
$132$	0	0
$133$	−2507.65	−1.63489
$134$	0	0
$135$	2322.03	1.48036
$136$	0	0
$137$	−1637.38	−1.02110	−0.510550	−	0.859848i	$-0.670558\pi$
$137$	−1637.38	−1.02110	−0.510550	+	0.859848i	$0.670558\pi$
$138$	0	0
$139$	3132.77	1.91164	0.955822	−	0.293947i	$-0.0949690\pi$
$139$	3132.77	1.91164	0.955822	+	0.293947i	$0.0949690\pi$
$140$	0	0
$141$	−2143.43	−1.28021
$142$	0	0
$143$	173.032	0.101186
$144$	0	0
$145$	442.494	0.253429
$146$	0	0
$147$	3101.90	1.74041
$148$	0	0
$149$	2692.17	1.48021	0.740104	−	0.672492i	$-0.234776\pi$
$149$	2692.17	1.48021	0.740104	+	0.672492i	$0.234776\pi$
$150$	0	0
$151$	1009.68	0.544149	0.272075	−	0.962276i	$-0.412290\pi$
$151$	1009.68	0.544149	0.272075	+	0.962276i	$0.412290\pi$
$152$	0	0
$153$	194.484	0.102765
$154$	0	0
$155$	−1043.14	−0.540560
$156$	0	0
$157$	3748.70	1.90560	0.952798	−	0.303606i	$-0.0981907\pi$
$157$	3748.70	1.90560	0.952798	+	0.303606i	$0.0981907\pi$
$158$	0	0
$159$	978.607	0.488104
$160$	0	0
$161$	6264.42	3.06649
$162$	0	0
$163$	3280.94	1.57658	0.788292	−	0.615302i	$-0.210966\pi$
$163$	3280.94	1.57658	0.788292	+	0.615302i	$0.210966\pi$
$164$	0	0
$165$	−273.615	−0.129097
$166$	0	0
$167$	117.766	0.0545688	0.0272844	−	0.999628i	$-0.491314\pi$
$167$	117.766	0.0545688	0.0272844	+	0.999628i	$0.491314\pi$
$168$	0	0
$169$	−683.223	−0.310980
$170$	0	0
$171$	807.670	0.361193
$172$	0	0
$173$	142.767	0.0627419	0.0313710	−	0.999508i	$-0.490013\pi$
$173$	142.767	0.0627419	0.0313710	+	0.999508i	$0.490013\pi$
$174$	0	0
$175$	3595.89	1.55328
$176$	0	0
$177$	−2019.88	−0.857760
$178$	0	0
$179$	−2572.24	−1.07407	−0.537034	−	0.843561i	$-0.680455\pi$
$179$	−2572.24	−1.07407	−0.537034	+	0.843561i	$0.680455\pi$
$180$	0	0
$181$	4026.13	1.65337	0.826686	−	0.562664i	$-0.190224\pi$
$181$	4026.13	1.65337	0.826686	+	0.562664i	$0.190224\pi$
$182$	0	0
$183$	1311.14	0.529630
$184$	0	0
$185$	686.294	0.272742
$186$	0	0
$187$	−80.5197	−0.0314876
$188$	0	0
$189$	−5075.38	−1.95333
$190$	0	0
$191$	1524.39	0.577491	0.288745	−	0.957406i	$-0.406762\pi$
$191$	1524.39	0.577491	0.288745	+	0.957406i	$0.406762\pi$
$192$	0	0
$193$	2912.51	1.08625	0.543127	−	0.839651i	$-0.317240\pi$
$193$	2912.51	1.08625	0.543127	+	0.839651i	$0.317240\pi$
$194$	0	0
$195$	−2393.74	−0.879073
$196$	0	0
$197$	−3981.79	−1.44006	−0.720028	−	0.693945i	$-0.755871\pi$
$197$	−3981.79	−1.44006	−0.720028	+	0.693945i	$0.755871\pi$
$198$	0	0
$199$	3625.36	1.29143	0.645716	−	0.763578i	$-0.276559\pi$
$199$	3625.36	1.29143	0.645716	+	0.763578i	$0.276559\pi$
$200$	0	0
$201$	1062.43	0.372825
$202$	0	0
$203$	−967.181	−0.334398
$204$	0	0
$205$	−4566.06	−1.55564
$206$	0	0
$207$	−2017.66	−0.677474
$208$	0	0
$209$	−334.389	−0.110671
$210$	0	0
$211$	3272.30	1.06765	0.533826	−	0.845594i	$-0.320754\pi$
$211$	3272.30	1.06765	0.533826	+	0.845594i	$0.320754\pi$
$212$	0	0
$213$	2929.89	0.942501
$214$	0	0
$215$	−2108.88	−0.668949
$216$	0	0
$217$	2280.04	0.713267
$218$	0	0
$219$	3432.88	1.05923
$220$	0	0
$221$	−704.431	−0.214412
$222$	0	0
$223$	−518.919	−0.155827	−0.0779134	−	0.996960i	$-0.524826\pi$
$223$	−518.919	−0.155827	−0.0779134	+	0.996960i	$0.524826\pi$
$224$	0	0
$225$	−1158.18	−0.343163
$226$	0	0
$227$	−3926.80	−1.14815	−0.574076	−	0.818802i	$-0.694639\pi$
$227$	−3926.80	−1.14815	−0.574076	+	0.818802i	$0.694639\pi$
$228$	0	0
$229$	2754.41	0.794833	0.397417	−	0.917638i	$-0.369907\pi$
$229$	2754.41	0.794833	0.397417	+	0.917638i	$0.369907\pi$
$230$	0	0
$231$	598.054	0.170342
$232$	0	0
$233$	5219.48	1.46755	0.733776	−	0.679392i	$-0.237756\pi$
$233$	5219.48	1.46755	0.733776	+	0.679392i	$0.237756\pi$
$234$	0	0
$235$	8111.15	2.25154
$236$	0	0
$237$	342.905	0.0939836
$238$	0	0
$239$	3620.68	0.979927	0.489964	−	0.871743i	$-0.337010\pi$
$239$	3620.68	0.979927	0.489964	+	0.871743i	$0.337010\pi$
$240$	0	0
$241$	−2436.44	−0.651224	−0.325612	−	0.945503i	$-0.605570\pi$
$241$	−2436.44	−0.651224	−0.325612	+	0.945503i	$0.605570\pi$
$242$	0	0
$243$	2804.13	0.740267
$244$	0	0
$245$	−11738.2	−3.06092
$246$	0	0
$247$	−2925.42	−0.753603
$248$	0	0
$249$	−3685.48	−0.937984
$250$	0	0
$251$	3292.45	0.827958	0.413979	−	0.910286i	$-0.364139\pi$
$251$	3292.45	0.827958	0.413979	+	0.910286i	$0.364139\pi$
$252$	0	0
$253$	835.345	0.207580
$254$	0	0
$255$	1113.92	0.273554
$256$	0	0
$257$	6562.16	1.59275	0.796374	−	0.604804i	$-0.206749\pi$
$257$	6562.16	1.59275	0.796374	+	0.604804i	$0.206749\pi$
$258$	0	0
$259$	−1500.06	−0.359882
$260$	0	0
$261$	311.512	0.0738778
$262$	0	0
$263$	−7429.23	−1.74185	−0.870924	−	0.491417i	$-0.836479\pi$
$263$	−7429.23	−1.74185	−0.870924	+	0.491417i	$0.836479\pi$
$264$	0	0
$265$	−3703.24	−0.858446
$266$	0	0
$267$	−4322.36	−0.990727
$268$	0	0
$269$	−2988.93	−0.677465	−0.338732	−	0.940883i	$-0.609998\pi$
$269$	−2988.93	−0.677465	−0.338732	+	0.940883i	$0.609998\pi$
$270$	0	0
$271$	−3291.47	−0.737795	−0.368897	−	0.929470i	$-0.620265\pi$
$271$	−3291.47	−0.737795	−0.368897	+	0.929470i	$0.620265\pi$
$272$	0	0
$273$	5232.11	1.15993
$274$	0	0
$275$	479.504	0.105146
$276$	0	0
$277$	3512.78	0.761958	0.380979	−	0.924584i	$-0.375587\pi$
$277$	3512.78	0.761958	0.380979	+	0.924584i	$0.375587\pi$
$278$	0	0
$279$	−734.360	−0.157581
$280$	0	0
$281$	656.844	0.139445	0.0697225	−	0.997566i	$-0.477789\pi$
$281$	656.844	0.139445	0.0697225	+	0.997566i	$0.477789\pi$
$282$	0	0
$283$	3682.78	0.773563	0.386782	−	0.922171i	$-0.373587\pi$
$283$	3682.78	0.773563	0.386782	+	0.922171i	$0.373587\pi$
$284$	0	0
$285$	4625.97	0.961470
$286$	0	0
$287$	9980.25	2.05267
$288$	0	0
$289$	−4585.20	−0.933278
$290$	0	0
$291$	−2670.08	−0.537880
$292$	0	0
$293$	−5539.43	−1.10449	−0.552247	−	0.833680i	$-0.686230\pi$
$293$	−5539.43	−1.10449	−0.552247	+	0.833680i	$0.686230\pi$
$294$	0	0
$295$	7643.62	1.50857
$296$	0	0
$297$	−676.789	−0.132227
$298$	0	0
$299$	7308.06	1.41350
$300$	0	0
$301$	4609.47	0.882676
$302$	0	0
$303$	3105.38	0.588777
$304$	0	0
$305$	−4961.61	−0.931479
$306$	0	0
$307$	−1024.53	−0.190466	−0.0952330	−	0.995455i	$-0.530360\pi$
$307$	−1024.53	−0.190466	−0.0952330	+	0.995455i	$0.530360\pi$
$308$	0	0
$309$	2570.70	0.473276
$310$	0	0
$311$	−4149.85	−0.756644	−0.378322	−	0.925674i	$-0.623499\pi$
$311$	−4149.85	−0.756644	−0.378322	+	0.925674i	$0.623499\pi$
$312$	0	0
$313$	6210.77	1.12158	0.560788	−	0.827959i	$-0.310498\pi$
$313$	6210.77	1.12158	0.560788	+	0.827959i	$0.310498\pi$
$314$	0	0
$315$	5466.34	0.977756
$316$	0	0
$317$	3373.41	0.597697	0.298848	−	0.954301i	$-0.403398\pi$
$317$	3373.41	0.597697	0.298848	+	0.954301i	$0.403398\pi$
$318$	0	0
$319$	−128.971	−0.0226364
$320$	0	0
$321$	2313.33	0.402236
$322$	0	0
$323$	1361.33	0.234509
$324$	0	0
$325$	4194.96	0.715984
$326$	0	0
$327$	−4990.70	−0.843995
$328$	0	0
$329$	−17728.9	−2.97090
$330$	0	0
$331$	1892.69	0.314295	0.157147	−	0.987575i	$-0.449770\pi$
$331$	1892.69	0.314295	0.157147	+	0.987575i	$0.449770\pi$
$332$	0	0
$333$	483.145	0.0795080
$334$	0	0
$335$	−4020.43	−0.655700
$336$	0	0
$337$	−6617.71	−1.06970	−0.534851	−	0.844946i	$-0.679632\pi$
$337$	−6617.71	−1.06970	−0.534851	+	0.844946i	$0.679632\pi$
$338$	0	0
$339$	6390.62	1.02387
$340$	0	0
$341$	304.037	0.0482831
$342$	0	0
$343$	14217.3	2.23809
$344$	0	0
$345$	−11556.3	−1.80339
$346$	0	0
$347$	−4516.16	−0.698674	−0.349337	−	0.936997i	$-0.613593\pi$
$347$	−4516.16	−0.698674	−0.349337	+	0.936997i	$0.613593\pi$
$348$	0	0
$349$	11069.8	1.69786	0.848931	−	0.528504i	$-0.177247\pi$
$349$	11069.8	1.69786	0.848931	+	0.528504i	$0.177247\pi$
$350$	0	0
$351$	−5920.93	−0.900387
$352$	0	0
$353$	−5286.28	−0.797054	−0.398527	−	0.917157i	$-0.630479\pi$
$353$	−5286.28	−0.797054	−0.398527	+	0.917157i	$0.630479\pi$
$354$	0	0
$355$	−11087.3	−1.65761
$356$	0	0
$357$	−2434.74	−0.360953
$358$	0	0
$359$	3783.59	0.556240	0.278120	−	0.960546i	$-0.410289\pi$
$359$	3783.59	0.556240	0.278120	+	0.960546i	$0.410289\pi$
$360$	0	0
$361$	−1205.55	−0.175761
$362$	0	0
$363$	−5287.04	−0.764456
$364$	0	0
$365$	−12990.7	−1.86291
$366$	0	0
$367$	5343.09	0.759964	0.379982	−	0.924994i	$-0.375930\pi$
$367$	5343.09	0.759964	0.379982	+	0.924994i	$0.375930\pi$
$368$	0	0
$369$	−3214.46	−0.453491
$370$	0	0
$371$	8094.34	1.13272
$372$	0	0
$373$	7908.34	1.09780	0.548899	−	0.835889i	$-0.315047\pi$
$373$	7908.34	1.09780	0.548899	+	0.835889i	$0.315047\pi$
$374$	0	0
$375$	1057.02	0.145558
$376$	0	0
$377$	−1128.31	−0.154141
$378$	0	0
$379$	−12720.5	−1.72403	−0.862015	−	0.506882i	$-0.830798\pi$
$379$	−12720.5	−1.72403	−0.862015	+	0.506882i	$0.830798\pi$
$380$	0	0
$381$	−7042.55	−0.946984
$382$	0	0
$383$	5000.21	0.667099	0.333549	−	0.942733i	$-0.391754\pi$
$383$	5000.21	0.667099	0.333549	+	0.942733i	$0.391754\pi$
$384$	0	0
$385$	−2263.15	−0.299587
$386$	0	0
$387$	−1484.63	−0.195008
$388$	0	0
$389$	−6.77892	−0.000883561 0	−0.000441780	−	1.00000i	$-0.500141\pi$
$389$	−6.77892	−0.000883561 0	−0.000441780		1.00000i	$0.500141\pi$
$390$	0	0
$391$	−3400.78	−0.439859
$392$	0	0
$393$	8347.11	1.07139
$394$	0	0
$395$	−1297.62	−0.165292
$396$	0	0
$397$	5653.11	0.714664	0.357332	−	0.933977i	$-0.383687\pi$
$397$	5653.11	0.714664	0.357332	+	0.933977i	$0.383687\pi$
$398$	0	0
$399$	−10111.2	−1.26865
$400$	0	0
$401$	−2403.50	−0.299314	−0.149657	−	0.988738i	$-0.547817\pi$
$401$	−2403.50	−0.299314	−0.149657	+	0.988738i	$0.547817\pi$
$402$	0	0
$403$	2659.89	0.328780
$404$	0	0
$405$	4937.40	0.605781
$406$	0	0
$407$	−200.030	−0.0243615
$408$	0	0
$409$	9718.01	1.17488	0.587439	−	0.809269i	$-0.300136\pi$
$409$	9718.01	1.17488	0.587439	+	0.809269i	$0.300136\pi$
$410$	0	0
$411$	−6602.14	−0.792359
$412$	0	0
$413$	−16707.0	−1.99055
$414$	0	0
$415$	13946.6	1.64966
$416$	0	0
$417$	12631.8	1.48341
$418$	0	0
$419$	−4134.77	−0.482093	−0.241046	−	0.970514i	$-0.577491\pi$
$419$	−4134.77	−0.482093	−0.241046	+	0.970514i	$0.577491\pi$
$420$	0	0
$421$	−14938.9	−1.72940	−0.864700	−	0.502289i	$-0.832491\pi$
$421$	−14938.9	−1.72940	−0.864700	+	0.502289i	$0.832491\pi$
$422$	0	0
$423$	5710.18	0.656355
$424$	0	0
$425$	−1952.11	−0.222803
$426$	0	0
$427$	10844.8	1.22908
$428$	0	0
$429$	697.689	0.0785192
$430$	0	0
$431$	−10759.0	−1.20242	−0.601210	−	0.799091i	$-0.705314\pi$
$431$	−10759.0	−1.20242	−0.601210	+	0.799091i	$0.705314\pi$
$432$	0	0
$433$	10624.1	1.17913	0.589565	−	0.807721i	$-0.299299\pi$
$433$	10624.1	1.17913	0.589565	+	0.807721i	$0.299299\pi$
$434$	0	0
$435$	1784.20	0.196657
$436$	0	0
$437$	−14123.0	−1.54599
$438$	0	0
$439$	−15028.5	−1.63388	−0.816938	−	0.576726i	$-0.804330\pi$
$439$	−15028.5	−1.63388	−0.816938	+	0.576726i	$0.804330\pi$
$440$	0	0
$441$	−8263.59	−0.892300
$442$	0	0
$443$	3151.82	0.338030	0.169015	−	0.985613i	$-0.445941\pi$
$443$	3151.82	0.338030	0.169015	+	0.985613i	$0.445941\pi$
$444$	0	0
$445$	16356.6	1.74242
$446$	0	0
$447$	10855.2	1.14862
$448$	0	0
$449$	−16469.1	−1.73102	−0.865509	−	0.500894i	$-0.833005\pi$
$449$	−16469.1	−1.73102	−0.865509	+	0.500894i	$0.833005\pi$
$450$	0	0
$451$	1330.84	0.138951
$452$	0	0
$453$	4071.18	0.422253
$454$	0	0
$455$	−19799.3	−2.04001
$456$	0	0
$457$	−10916.8	−1.11743	−0.558714	−	0.829360i	$-0.688705\pi$
$457$	−10916.8	−1.11743	−0.558714	+	0.829360i	$0.688705\pi$
$458$	0	0
$459$	2755.28	0.280186
$460$	0	0
$461$	−5937.01	−0.599814	−0.299907	−	0.953968i	$-0.596956\pi$
$461$	−5937.01	−0.599814	−0.299907	+	0.953968i	$0.596956\pi$
$462$	0	0
$463$	12136.5	1.21821	0.609107	−	0.793088i	$-0.291528\pi$
$463$	12136.5	1.21821	0.609107	+	0.793088i	$0.291528\pi$
$464$	0	0
$465$	−4206.08	−0.419468
$466$	0	0
$467$	−3949.99	−0.391400	−0.195700	−	0.980664i	$-0.562698\pi$
$467$	−3949.99	−0.391400	−0.195700	+	0.980664i	$0.562698\pi$
$468$	0	0
$469$	8787.65	0.865194
$470$	0	0
$471$	15115.3	1.47872
$472$	0	0
$473$	614.661	0.0597509
$474$	0	0
$475$	−8106.89	−0.783094
$476$	0	0
$477$	−2607.05	−0.250248
$478$	0	0
$479$	7792.99	0.743363	0.371681	−	0.928360i	$-0.378781\pi$
$479$	7792.99	0.743363	0.371681	+	0.928360i	$0.378781\pi$
$480$	0	0
$481$	−1749.97	−0.165888
$482$	0	0
$483$	25259.1	2.37956
$484$	0	0
$485$	10104.1	0.945987
$486$	0	0
$487$	−8860.10	−0.824414	−0.412207	−	0.911090i	$-0.635242\pi$
$487$	−8860.10	−0.824414	−0.412207	+	0.911090i	$0.635242\pi$
$488$	0	0
$489$	13229.2	1.22341
$490$	0	0
$491$	−14615.1	−1.34332	−0.671662	−	0.740857i	$-0.734419\pi$
$491$	−14615.1	−1.34332	−0.671662	+	0.740857i	$0.734419\pi$
$492$	0	0
$493$	525.056	0.0479662
$494$	0	0
$495$	728.922	0.0661871
$496$	0	0
$497$	24234.0	2.18721
$498$	0	0
$499$	8815.62	0.790864	0.395432	−	0.918495i	$-0.370595\pi$
$499$	8815.62	0.790864	0.395432	+	0.918495i	$0.370595\pi$
$500$	0	0
$501$	474.849	0.0423447
$502$	0	0
$503$	14354.9	1.27247	0.636235	−	0.771495i	$-0.280491\pi$
$503$	14354.9	1.27247	0.636235	+	0.771495i	$0.280491\pi$
$504$	0	0
$505$	−11751.4	−1.03550
$506$	0	0
$507$	−2754.85	−0.241316
$508$	0	0
$509$	11443.7	0.996530	0.498265	−	0.867025i	$-0.333971\pi$
$509$	11443.7	0.996530	0.498265	+	0.867025i	$0.333971\pi$
$510$	0	0
$511$	28394.3	2.45811
$512$	0	0
$513$	11442.4	0.984781
$514$	0	0
$515$	−9728.04	−0.832366
$516$	0	0
$517$	−2364.11	−0.201109
$518$	0	0
$519$	575.656	0.0486869
$520$	0	0
$521$	14465.0	1.21636	0.608179	−	0.793800i	$-0.291900\pi$
$521$	14465.0	1.21636	0.608179	+	0.793800i	$0.291900\pi$
$522$	0	0
$523$	−1440.84	−0.120466	−0.0602328	−	0.998184i	$-0.519184\pi$
$523$	−1440.84	−0.120466	−0.0602328	+	0.998184i	$0.519184\pi$
$524$	0	0
$525$	14499.2	1.20533
$526$	0	0
$527$	−1237.77	−0.102311
$528$	0	0
$529$	23114.2	1.89974
$530$	0	0
$531$	5381.04	0.439769
$532$	0	0
$533$	11642.9	0.946176
$534$	0	0
$535$	−8754.11	−0.707426
$536$	0	0
$537$	−10371.6	−0.833463
$538$	0	0
$539$	3421.26	0.273403
$540$	0	0
$541$	4676.44	0.371638	0.185819	−	0.982584i	$-0.440506\pi$
$541$	4676.44	0.371638	0.185819	+	0.982584i	$0.440506\pi$
$542$	0	0
$543$	16234.0	1.28299
$544$	0	0
$545$	18885.8	1.48436
$546$	0	0
$547$	−11647.5	−0.910444	−0.455222	−	0.890378i	$-0.650440\pi$
$547$	−11647.5	−0.910444	−0.455222	+	0.890378i	$0.650440\pi$
$548$	0	0
$549$	−3492.93	−0.271539
$550$	0	0
$551$	2180.49	0.168588
$552$	0	0
$553$	2836.27	0.218102
$554$	0	0
$555$	2767.24	0.211644
$556$	0	0
$557$	22051.8	1.67750	0.838748	−	0.544520i	$-0.183288\pi$
$557$	22051.8	1.67750	0.838748	+	0.544520i	$0.183288\pi$
$558$	0	0
$559$	5377.40	0.406869
$560$	0	0
$561$	−324.667	−0.0244340
$562$	0	0
$563$	−6690.91	−0.500867	−0.250434	−	0.968134i	$-0.580573\pi$
$563$	−6690.91	−0.500867	−0.250434	+	0.968134i	$0.580573\pi$
$564$	0	0
$565$	−24183.4	−1.80071
$566$	0	0
$567$	−10791.9	−0.799326
$568$	0	0
$569$	4559.72	0.335946	0.167973	−	0.985792i	$-0.446278\pi$
$569$	4559.72	0.335946	0.167973	+	0.985792i	$0.446278\pi$
$570$	0	0
$571$	20039.6	1.46871	0.734354	−	0.678767i	$-0.237485\pi$
$571$	20039.6	1.46871	0.734354	+	0.678767i	$0.237485\pi$
$572$	0	0
$573$	6146.55	0.448125
$574$	0	0
$575$	20252.0	1.46881
$576$	0	0
$577$	12342.3	0.890498	0.445249	−	0.895407i	$-0.353115\pi$
$577$	12342.3	0.890498	0.445249	+	0.895407i	$0.353115\pi$
$578$	0	0
$579$	11743.7	0.842918
$580$	0	0
$581$	−30483.7	−2.17672
$582$	0	0
$583$	1079.36	0.0766768
$584$	0	0
$585$	6377.02	0.450696
$586$	0	0
$587$	−2455.05	−0.172625	−0.0863125	−	0.996268i	$-0.527508\pi$
$587$	−2455.05	−0.172625	−0.0863125	+	0.996268i	$0.527508\pi$
$588$	0	0
$589$	−5140.30	−0.359597
$590$	0	0
$591$	−16055.2	−1.11746
$592$	0	0
$593$	−10297.5	−0.713101	−0.356550	−	0.934276i	$-0.616047\pi$
$593$	−10297.5	−0.713101	−0.356550	+	0.934276i	$0.616047\pi$
$594$	0	0
$595$	9213.54	0.634821
$596$	0	0
$597$	14618.0	1.00213
$598$	0	0
$599$	−2344.02	−0.159890	−0.0799448	−	0.996799i	$-0.525474\pi$
$599$	−2344.02	−0.159890	−0.0799448	+	0.996799i	$0.525474\pi$
$600$	0	0
$601$	−17480.0	−1.18639	−0.593197	−	0.805058i	$-0.702134\pi$
$601$	−17480.0	−1.18639	−0.593197	+	0.805058i	$0.702134\pi$
$602$	0	0
$603$	−2830.35	−0.191145
$604$	0	0
$605$	20007.2	1.34447
$606$	0	0
$607$	8368.57	0.559588	0.279794	−	0.960060i	$-0.409734\pi$
$607$	8368.57	0.559588	0.279794	+	0.960060i	$0.409734\pi$
$608$	0	0
$609$	−3899.81	−0.259488
$610$	0	0
$611$	−20682.5	−1.36944
$612$	0	0
$613$	5956.33	0.392454	0.196227	−	0.980559i	$-0.437131\pi$
$613$	5956.33	0.392454	0.196227	+	0.980559i	$0.437131\pi$
$614$	0	0
$615$	−18411.0	−1.20716
$616$	0	0
$617$	1936.64	0.126363	0.0631817	−	0.998002i	$-0.479875\pi$
$617$	1936.64	0.126363	0.0631817	+	0.998002i	$0.479875\pi$
$618$	0	0
$619$	6576.41	0.427025	0.213512	−	0.976940i	$-0.431510\pi$
$619$	6576.41	0.427025	0.213512	+	0.976940i	$0.431510\pi$
$620$	0	0
$621$	−28584.5	−1.84711
$622$	0	0
$623$	−35751.5	−2.29912
$624$	0	0
$625$	−17477.4	−1.11855
$626$	0	0
$627$	−1348.30	−0.0858789
$628$	0	0
$629$	814.344	0.0516216
$630$	0	0
$631$	−6337.99	−0.399860	−0.199930	−	0.979810i	$-0.564071\pi$
$631$	−6337.99	−0.399860	−0.199930	+	0.979810i	$0.564071\pi$
$632$	0	0
$633$	13194.4	0.828484
$634$	0	0
$635$	26650.4	1.66549
$636$	0	0
$637$	29931.1	1.86172
$638$	0	0
$639$	−7805.34	−0.483215
$640$	0	0
$641$	−24293.8	−1.49695	−0.748476	−	0.663162i	$-0.769214\pi$
$641$	−24293.8	−1.49695	−0.748476	+	0.663162i	$0.769214\pi$
$642$	0	0
$643$	7533.76	0.462056	0.231028	−	0.972947i	$-0.425791\pi$
$643$	7533.76	0.462056	0.231028	+	0.972947i	$0.425791\pi$
$644$	0	0
$645$	−8503.30	−0.519096
$646$	0	0
$647$	13097.6	0.795858	0.397929	−	0.917416i	$-0.369729\pi$
$647$	13097.6	0.795858	0.397929	+	0.917416i	$0.369729\pi$
$648$	0	0
$649$	−2227.84	−0.134746
$650$	0	0
$651$	9193.44	0.553486
$652$	0	0
$653$	−5674.78	−0.340079	−0.170039	−	0.985437i	$-0.554389\pi$
$653$	−5674.78	−0.340079	−0.170039	+	0.985437i	$0.554389\pi$
$654$	0	0
$655$	−31587.1	−1.88429
$656$	0	0
$657$	−9145.32	−0.543064
$658$	0	0
$659$	−17394.5	−1.02822	−0.514108	−	0.857725i	$-0.671877\pi$
$659$	−17394.5	−1.02822	−0.514108	+	0.857725i	$0.671877\pi$
$660$	0	0
$661$	−7891.30	−0.464351	−0.232176	−	0.972674i	$-0.574584\pi$
$661$	−7891.30	−0.464351	−0.232176	+	0.972674i	$0.574584\pi$
$662$	0	0
$663$	−2840.37	−0.166381
$664$	0	0
$665$	38262.8	2.23123
$666$	0	0
$667$	−5447.15	−0.316214
$668$	0	0
$669$	−2092.36	−0.120920
$670$	0	0
$671$	1446.13	0.0832001
$672$	0	0
$673$	−16403.4	−0.939529	−0.469764	−	0.882792i	$-0.655661\pi$
$673$	−16403.4	−0.939529	−0.469764	+	0.882792i	$0.655661\pi$
$674$	0	0
$675$	−16408.0	−0.935622
$676$	0	0
$677$	−7065.04	−0.401080	−0.200540	−	0.979685i	$-0.564270\pi$
$677$	−7065.04	−0.401080	−0.200540	+	0.979685i	$0.564270\pi$
$678$	0	0
$679$	−22085.0	−1.24823
$680$	0	0
$681$	−15833.4	−0.890951
$682$	0	0
$683$	−27873.6	−1.56157	−0.780787	−	0.624797i	$-0.785182\pi$
$683$	−27873.6	−1.56157	−0.780787	+	0.624797i	$0.785182\pi$
$684$	0	0
$685$	24983.8	1.39355
$686$	0	0
$687$	11106.2	0.616780
$688$	0	0
$689$	9442.85	0.522125
$690$	0	0
$691$	−11914.0	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$691$	−11914.0	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$692$	0	0
$693$	−1593.24	−0.0873336
$694$	0	0
$695$	−47801.2	−2.60892
$696$	0	0
$697$	−5418.00	−0.294435
$698$	0	0
$699$	21045.7	1.13880
$700$	0	0
$701$	3946.24	0.212621	0.106311	−	0.994333i	$-0.466096\pi$
$701$	3946.24	0.212621	0.106311	+	0.994333i	$0.466096\pi$
$702$	0	0
$703$	3381.87	0.181436
$704$	0	0
$705$	32705.3	1.74717
$706$	0	0
$707$	25685.5	1.36634
$708$	0	0
$709$	10512.7	0.556857	0.278429	−	0.960457i	$-0.410186\pi$
$709$	10512.7	0.556857	0.278429	+	0.960457i	$0.410186\pi$
$710$	0	0
$711$	−913.514	−0.0481849
$712$	0	0
$713$	12841.1	0.674480
$714$	0	0
$715$	−2640.19	−0.138094
$716$	0	0
$717$	14599.1	0.760411
$718$	0	0
$719$	−31708.0	−1.64466	−0.822328	−	0.569014i	$-0.807325\pi$
$719$	−31708.0	−1.64466	−0.822328	+	0.569014i	$0.807325\pi$
$720$	0	0
$721$	21263.0	1.09830
$722$	0	0
$723$	−9824.09	−0.505341
$724$	0	0
$725$	−3126.77	−0.160173
$726$	0	0
$727$	33102.6	1.68873	0.844365	−	0.535768i	$-0.179978\pi$
$727$	33102.6	1.68873	0.844365	+	0.535768i	$0.179978\pi$
$728$	0	0
$729$	20043.5	1.01831
$730$	0	0
$731$	−2502.35	−0.126611
$732$	0	0
$733$	−35146.1	−1.77101	−0.885506	−	0.464627i	$-0.846188\pi$
$733$	−35146.1	−1.77101	−0.885506	+	0.464627i	$0.846188\pi$
$734$	0	0
$735$	−47330.2	−2.37524
$736$	0	0
$737$	1171.81	0.0585674
$738$	0	0
$739$	5619.25	0.279712	0.139856	−	0.990172i	$-0.455336\pi$
$739$	5619.25	0.279712	0.139856	+	0.990172i	$0.455336\pi$
$740$	0	0
$741$	−11795.7	−0.584786
$742$	0	0
$743$	−33101.4	−1.63442	−0.817210	−	0.576340i	$-0.804480\pi$
$743$	−33101.4	−1.63442	−0.817210	+	0.576340i	$0.804480\pi$
$744$	0	0
$745$	−41078.2	−2.02012
$746$	0	0
$747$	9818.27	0.480899
$748$	0	0
$749$	19134.3	0.933446
$750$	0	0
$751$	4211.12	0.204615	0.102307	−	0.994753i	$-0.467377\pi$
$751$	4211.12	0.204615	0.102307	+	0.994753i	$0.467377\pi$
$752$	0	0
$753$	13275.6	0.642485
$754$	0	0
$755$	−15406.1	−0.742631
$756$	0	0
$757$	−11295.5	−0.542325	−0.271163	−	0.962534i	$-0.587408\pi$
$757$	−11295.5	−0.542325	−0.271163	+	0.962534i	$0.587408\pi$
$758$	0	0
$759$	3368.23	0.161079
$760$	0	0
$761$	−14621.4	−0.696486	−0.348243	−	0.937404i	$-0.613222\pi$
$761$	−14621.4	−0.696486	−0.348243	+	0.937404i	$0.613222\pi$
$762$	0	0
$763$	−41279.6	−1.95861
$764$	0	0
$765$	−2967.52	−0.140250
$766$	0	0
$767$	−19490.4	−0.917545
$768$	0	0
$769$	−27028.1	−1.26743	−0.633717	−	0.773565i	$-0.718472\pi$
$769$	−27028.1	−1.26743	−0.633717	+	0.773565i	$0.718472\pi$
$770$	0	0
$771$	26459.6	1.23595
$772$	0	0
$773$	−15328.5	−0.713233	−0.356617	−	0.934251i	$-0.616070\pi$
$773$	−15328.5	−0.713233	−0.356617	+	0.934251i	$0.616070\pi$
$774$	0	0
$775$	7371.05	0.341646
$776$	0	0
$777$	−6048.48	−0.279264
$778$	0	0
$779$	−22500.3	−1.03486
$780$	0	0
$781$	3231.54	0.148058
$782$	0	0
$783$	4413.23	0.201425
$784$	0	0
$785$	−57199.2	−2.60067
$786$	0	0
$787$	4984.32	0.225758	0.112879	−	0.993609i	$-0.463993\pi$
$787$	4984.32	0.225758	0.112879	+	0.993609i	$0.463993\pi$
$788$	0	0
$789$	−29955.8	−1.35165
$790$	0	0
$791$	52858.7	2.37603
$792$	0	0
$793$	12651.6	0.566545
$794$	0	0
$795$	−14932.0	−0.666143
$796$	0	0
$797$	4318.53	0.191933	0.0959663	−	0.995385i	$-0.469406\pi$
$797$	4318.53	0.191933	0.0959663	+	0.995385i	$0.469406\pi$
$798$	0	0
$799$	9624.54	0.426147
$800$	0	0
$801$	11514.9	0.507940
$802$	0	0
$803$	3786.32	0.166396
$804$	0	0
$805$	−95585.2	−4.18501
$806$	0	0
$807$	−12051.8	−0.525704
$808$	0	0
$809$	−23533.7	−1.02274	−0.511372	−	0.859359i	$-0.670863\pi$
$809$	−23533.7	−1.02274	−0.511372	+	0.859359i	$0.670863\pi$
$810$	0	0
$811$	−37527.0	−1.62485	−0.812423	−	0.583068i	$-0.801852\pi$
$811$	−37527.0	−1.62485	−0.812423	+	0.583068i	$0.801852\pi$
$812$	0	0
$813$	−13271.7	−0.572519
$814$	0	0
$815$	−50062.0	−2.15165
$816$	0	0
$817$	−10392.0	−0.445005
$818$	0	0
$819$	−13938.5	−0.594691
$820$	0	0
$821$	−22145.4	−0.941390	−0.470695	−	0.882296i	$-0.655997\pi$
$821$	−22145.4	−0.941390	−0.470695	+	0.882296i	$0.655997\pi$
$822$	0	0
$823$	11528.5	0.488284	0.244142	−	0.969740i	$-0.421494\pi$
$823$	11528.5	0.488284	0.244142	+	0.969740i	$0.421494\pi$
$824$	0	0
$825$	1933.43	0.0815919
$826$	0	0
$827$	−21061.9	−0.885604	−0.442802	−	0.896619i	$-0.646016\pi$
$827$	−21061.9	−0.885604	−0.442802	+	0.896619i	$0.646016\pi$
$828$	0	0
$829$	6927.86	0.290247	0.145123	−	0.989414i	$-0.453642\pi$
$829$	6927.86	0.290247	0.145123	+	0.989414i	$0.453642\pi$
$830$	0	0
$831$	14164.0	0.591269
$832$	0	0
$833$	−13928.3	−0.579338
$834$	0	0
$835$	−1796.92	−0.0744730
$836$	0	0
$837$	−10403.8	−0.429638
$838$	0	0
$839$	−22460.4	−0.924220	−0.462110	−	0.886823i	$-0.652907\pi$
$839$	−22460.4	−0.924220	−0.462110	+	0.886823i	$0.652907\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	2648.49	0.108207
$844$	0	0
$845$	10424.9	0.424411
$846$	0	0
$847$	−43730.6	−1.77403
$848$	0	0
$849$	14849.5	0.600275
$850$	0	0
$851$	−8448.35	−0.340312
$852$	0	0
$853$	−38543.5	−1.54713	−0.773566	−	0.633716i	$-0.781529\pi$
$853$	−38543.5	−1.54713	−0.773566	+	0.633716i	$0.781529\pi$
$854$	0	0
$855$	−12323.8	−0.492940
$856$	0	0
$857$	4110.65	0.163847	0.0819236	−	0.996639i	$-0.473894\pi$
$857$	4110.65	0.163847	0.0819236	+	0.996639i	$0.473894\pi$
$858$	0	0
$859$	−24522.6	−0.974038	−0.487019	−	0.873391i	$-0.661916\pi$
$859$	−24522.6	−0.974038	−0.487019	+	0.873391i	$0.661916\pi$
$860$	0	0
$861$	40241.8	1.59284
$862$	0	0
$863$	9768.83	0.385324	0.192662	−	0.981265i	$-0.438288\pi$
$863$	9768.83	0.385324	0.192662	+	0.981265i	$0.438288\pi$
$864$	0	0
$865$	−2178.40	−0.0856274
$866$	0	0
$867$	−18488.2	−0.724212
$868$	0	0
$869$	378.210	0.0147640
$870$	0	0
$871$	10251.7	0.398811
$872$	0	0
$873$	7113.20	0.275768
$874$	0	0
$875$	8742.90	0.337787
$876$	0	0
$877$	−9424.03	−0.362859	−0.181429	−	0.983404i	$-0.558072\pi$
$877$	−9424.03	−0.362859	−0.181429	+	0.983404i	$0.558072\pi$
$878$	0	0
$879$	−22335.8	−0.857074
$880$	0	0
$881$	−10791.6	−0.412689	−0.206344	−	0.978479i	$-0.566157\pi$
$881$	−10791.6	−0.412689	−0.206344	+	0.978479i	$0.566157\pi$
$882$	0	0
$883$	−16891.6	−0.643770	−0.321885	−	0.946779i	$-0.604316\pi$
$883$	−16891.6	−0.643770	−0.321885	+	0.946779i	$0.604316\pi$
$884$	0	0
$885$	30820.2	1.17063
$886$	0	0
$887$	37273.1	1.41094	0.705472	−	0.708737i	$-0.250735\pi$
$887$	37273.1	1.41094	0.705472	+	0.708737i	$0.250735\pi$
$888$	0	0
$889$	−58251.0	−2.19761
$890$	0	0
$891$	−1439.07	−0.0541087
$892$	0	0
$893$	39969.6	1.49779
$894$	0	0
$895$	39248.3	1.46584
$896$	0	0
$897$	29467.2	1.09686
$898$	0	0
$899$	−1982.58	−0.0735513
$900$	0	0
$901$	−4394.20	−0.162477
$902$	0	0
$903$	18586.1	0.684945
$904$	0	0
$905$	−61432.5	−2.25645
$906$	0	0
$907$	−11547.7	−0.422753	−0.211376	−	0.977405i	$-0.567795\pi$
$907$	−11547.7	−0.422753	−0.211376	+	0.977405i	$0.567795\pi$
$908$	0	0
$909$	−8272.85	−0.301863
$910$	0	0
$911$	10637.0	0.386849	0.193424	−	0.981115i	$-0.438041\pi$
$911$	10637.0	0.386849	0.193424	+	0.981115i	$0.438041\pi$
$912$	0	0
$913$	−4064.93	−0.147349
$914$	0	0
$915$	−20005.9	−0.722816
$916$	0	0
$917$	69041.4	2.48631
$918$	0	0
$919$	45038.0	1.61661	0.808306	−	0.588762i	$-0.200385\pi$
$919$	45038.0	1.61661	0.808306	+	0.588762i	$0.200385\pi$
$920$	0	0
$921$	−4131.06	−0.147799
$922$	0	0
$923$	28271.3	1.00819
$924$	0	0
$925$	−4849.51	−0.172379
$926$	0	0
$927$	−6848.45	−0.242646
$928$	0	0
$929$	46945.9	1.65796	0.828980	−	0.559278i	$-0.188922\pi$
$929$	46945.9	1.65796	0.828980	+	0.559278i	$0.188922\pi$
$930$	0	0
$931$	−57842.7	−2.03622
$932$	0	0
$933$	−16732.8	−0.587146
$934$	0	0
$935$	1228.60	0.0429729
$936$	0	0
$937$	50149.9	1.74848	0.874240	−	0.485494i	$-0.161360\pi$
$937$	50149.9	1.74848	0.874240	+	0.485494i	$0.161360\pi$
$938$	0	0
$939$	25042.7	0.870328
$940$	0	0
$941$	−5171.70	−0.179163	−0.0895816	−	0.995979i	$-0.528553\pi$
$941$	−5171.70	−0.179163	−0.0895816	+	0.995979i	$0.528553\pi$
$942$	0	0
$943$	56208.6	1.94104
$944$	0	0
$945$	77442.3	2.66582
$946$	0	0
$947$	43135.1	1.48015	0.740075	−	0.672524i	$-0.234790\pi$
$947$	43135.1	1.48015	0.740075	+	0.672524i	$0.234790\pi$
$948$	0	0
$949$	33124.8	1.13306
$950$	0	0
$951$	13602.1	0.463805
$952$	0	0
$953$	−17374.9	−0.590585	−0.295293	−	0.955407i	$-0.595417\pi$
$953$	−17374.9	−0.590585	−0.295293	+	0.955407i	$0.595417\pi$
$954$	0	0
$955$	−23259.7	−0.788133
$956$	0	0
$957$	−520.031	−0.0175655
$958$	0	0
$959$	−54608.3	−1.83878
$960$	0	0
$961$	−25117.3	−0.843116
$962$	0	0
$963$	−6162.81	−0.206224
$964$	0	0
$965$	−44440.3	−1.48247
$966$	0	0
$967$	−45390.0	−1.50945	−0.754727	−	0.656039i	$-0.772231\pi$
$967$	−45390.0	−1.50945	−0.754727	+	0.656039i	$0.772231\pi$
$968$	0	0
$969$	5489.09	0.181976
$970$	0	0
$971$	−509.154	−0.0168275	−0.00841377	−	0.999965i	$-0.502678\pi$
$971$	−509.154	−0.0168275	−0.00841377	+	0.999965i	$0.502678\pi$
$972$	0	0
$973$	104481.	3.44246
$974$	0	0
$975$	16914.7	0.555594
$976$	0	0
$977$	7869.73	0.257702	0.128851	−	0.991664i	$-0.458871\pi$
$977$	7869.73	0.257702	0.128851	+	0.991664i	$0.458871\pi$
$978$	0	0
$979$	−4767.37	−0.155634
$980$	0	0
$981$	13295.4	0.432712
$982$	0	0
$983$	−26799.9	−0.869567	−0.434783	−	0.900535i	$-0.643175\pi$
$983$	−26799.9	−0.869567	−0.434783	+	0.900535i	$0.643175\pi$
$984$	0	0
$985$	60755.9	1.96532
$986$	0	0
$987$	−71485.6	−2.30538
$988$	0	0
$989$	25960.5	0.834677
$990$	0	0
$991$	−53441.0	−1.71303	−0.856514	−	0.516125i	$-0.827374\pi$
$991$	−53441.0	−1.71303	−0.856514	+	0.516125i	$0.827374\pi$
$992$	0	0
$993$	7631.59	0.243888
$994$	0	0
$995$	−55317.3	−1.76249
$996$	0	0
$997$	25166.4	0.799428	0.399714	−	0.916640i	$-0.369109\pi$
$997$	25166.4	0.799428	0.399714	+	0.916640i	$0.369109\pi$
$998$	0	0
$999$	6844.78	0.216776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.z.1.4		5
4.3	odd	2		1856.4.a.ba.1.2		5
8.3	odd	2		464.4.a.m.1.4		5
8.5	even	2		232.4.a.d.1.2	✓	5
24.5	odd	2		2088.4.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.d.1.2	✓	5	8.5	even	2
464.4.a.m.1.4		5	8.3	odd	2
1856.4.a.z.1.4		5	1.1	even	1	trivial
1856.4.a.ba.1.2		5	4.3	odd	2
2088.4.a.f.1.2		5	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+4.03215 q^{3} -15.2584 q^{5} +33.3511 q^{7} -10.7418 q^{9} +O(q^{10})\)	\(q+4.03215 q^{3} -15.2584 q^{5} +33.3511 q^{7} -10.7418 q^{9} +4.44728 q^{11} +38.9073 q^{13} -61.5242 q^{15} -18.1054 q^{17} -75.1894 q^{19} +134.476 q^{21} +187.833 q^{23} +107.819 q^{25} -152.180 q^{27} -29.0000 q^{29} +68.3647 q^{31} +17.9321 q^{33} -508.885 q^{35} -44.9780 q^{37} +156.880 q^{39} +299.248 q^{41} +138.211 q^{43} +163.903 q^{45} -531.585 q^{47} +769.293 q^{49} -73.0035 q^{51} +242.701 q^{53} -67.8585 q^{55} -303.175 q^{57} -500.944 q^{59} +325.172 q^{61} -358.250 q^{63} -593.664 q^{65} +263.489 q^{67} +757.369 q^{69} +726.633 q^{71} +851.378 q^{73} +434.744 q^{75} +148.322 q^{77} +85.0429 q^{79} -323.585 q^{81} -914.025 q^{83} +276.259 q^{85} -116.932 q^{87} -1071.97 q^{89} +1297.60 q^{91} +275.657 q^{93} +1147.27 q^{95} -662.199 q^{97} -47.7718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9	\( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 - 36 * q^11 - 26 * q^13 + 88 * q^15 + 82 * q^17 - 156 * q^19 - 72 * q^21 + 336 * q^23 + 151 * q^25 - 352 * q^27 - 145 * q^29 + 432 * q^31 + 108 * q^33 - 600 * q^35 + 18 * q^37 + 688 * q^39 + 82 * q^41 - 340 * q^43 + 146 * q^45 + 680 * q^47 - 115 * q^49 - 608 * q^51 + 102 * q^53 + 736 * q^55 - 576 * q^57 - 924 * q^59 + 618 * q^61 + 584 * q^63 - 704 * q^65 - 44 * q^67 + 1056 * q^69 + 1032 * q^71 - 1078 * q^73 + 468 * q^75 + 888 * q^77 + 200 * q^79 - 1843 * q^81 - 452 * q^83 + 1700 * q^85 + 116 * q^87 - 1790 * q^89 + 1128 * q^91 + 1884 * q^93 + 1024 * q^95 - 2518 * q^97 + 1500 * q^99

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