LMFDB - Embedded newform 1856.4.a.q.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.4481.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 8 $ x^3 - 17*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$-0.476971$ 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)

$f(q)$	$=$	$q-1.95394 q^{3} +18.5450 q^{5} +7.63711 q^{7} -23.1821 q^{9} +O(q^{10})$	$q-1.95394 q^{3} +18.5450 q^{5} +7.63711 q^{7} -23.1821 q^{9} -67.0439 q^{11} +6.82134 q^{13} -36.2359 q^{15} +105.453 q^{17} -109.724 q^{19} -14.9225 q^{21} +51.0034 q^{23} +218.917 q^{25} +98.0530 q^{27} +29.0000 q^{29} +3.96376 q^{31} +131.000 q^{33} +141.630 q^{35} +41.2588 q^{37} -13.3285 q^{39} -18.3560 q^{41} -61.0816 q^{43} -429.912 q^{45} +77.6986 q^{47} -284.675 q^{49} -206.049 q^{51} -524.666 q^{53} -1243.33 q^{55} +214.394 q^{57} -606.470 q^{59} -390.164 q^{61} -177.044 q^{63} +126.502 q^{65} +746.727 q^{67} -99.6578 q^{69} +150.049 q^{71} +707.612 q^{73} -427.751 q^{75} -512.022 q^{77} -76.5205 q^{79} +434.327 q^{81} -1173.70 q^{83} +1955.62 q^{85} -56.6643 q^{87} -1242.33 q^{89} +52.0953 q^{91} -7.74495 q^{93} -2034.82 q^{95} -1574.55 q^{97} +1554.22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9	$ 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9 - 65 * q^11 - 29 * q^13 - 85 * q^15 + 244 * q^17 - 312 * q^19 + 214 * q^21 - 24 * q^23 + 436 * q^25 - 57 * q^27 + 87 * q^29 + 249 * q^31 + 393 * q^33 + 718 * q^35 - 200 * q^37 + 749 * q^39 - 470 * q^41 - 97 * q^43 - 1562 * q^45 + 377 * q^47 + 383 * q^49 - 612 * q^51 - 1007 * q^53 - 1399 * q^55 + 1128 * q^57 + 364 * q^59 + 524 * q^61 - 2244 * q^63 + 301 * q^65 + 820 * q^67 + 824 * q^69 - 782 * q^71 + 1620 * q^73 + 1356 * q^75 - 158 * q^77 + 427 * q^79 + 227 * q^81 - 1520 * q^83 + 68 * q^85 - 87 * q^87 - 2474 * q^89 + 2002 * q^91 + 1807 * q^93 + 568 * q^95 - 170 * q^97 + 1362 * 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
$3$	−1.95394	−0.376036	−0.188018	−	0.982166i	$-0.560206\pi$
$3$	−1.95394	−0.376036	−0.188018	+	0.982166i	$0.560206\pi$
$4$	0	0
$5$	18.5450	1.65871	0.829357	−	0.558718i	$-0.188707\pi$
$5$	18.5450	1.65871	0.829357	+	0.558718i	$0.188707\pi$
$6$	0	0
$7$	7.63711	0.412365	0.206183	−	0.978514i	$-0.433896\pi$
$7$	7.63711	0.412365	0.206183	+	0.978514i	$0.433896\pi$
$8$	0	0
$9$	−23.1821	−0.858597
$10$	0	0
$11$	−67.0439	−1.83768	−0.918841	−	0.394627i	$-0.870874\pi$
$11$	−67.0439	−1.83768	−0.918841	+	0.394627i	$0.870874\pi$
$12$	0	0
$13$	6.82134	0.145531	0.0727654	−	0.997349i	$-0.476818\pi$
$13$	6.82134	0.145531	0.0727654	+	0.997349i	$0.476818\pi$
$14$	0	0
$15$	−36.2359	−0.623737
$16$	0	0
$17$	105.453	1.50448	0.752238	−	0.658892i	$-0.228975\pi$
$17$	105.453	1.50448	0.752238	+	0.658892i	$0.228975\pi$
$18$	0	0
$19$	−109.724	−1.32486	−0.662430	−	0.749124i	$-0.730475\pi$
$19$	−109.724	−1.32486	−0.662430	+	0.749124i	$0.730475\pi$
$20$	0	0
$21$	−14.9225	−0.155064
$22$	0	0
$23$	51.0034	0.462389	0.231195	−	0.972908i	$-0.425737\pi$
$23$	51.0034	0.462389	0.231195	+	0.972908i	$0.425737\pi$
$24$	0	0
$25$	218.917	1.75134
$26$	0	0
$27$	98.0530	0.698900
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	3.96376	0.0229649	0.0114824	−	0.999934i	$-0.496345\pi$
$31$	3.96376	0.0229649	0.0114824	+	0.999934i	$0.496345\pi$
$32$	0	0
$33$	131.000	0.691036
$34$	0	0
$35$	141.630	0.683996
$36$	0	0
$37$	41.2588	0.183322	0.0916610	−	0.995790i	$-0.470782\pi$
$37$	41.2588	0.183322	0.0916610	+	0.995790i	$0.470782\pi$
$38$	0	0
$39$	−13.3285	−0.0547249
$40$	0	0
$41$	−18.3560	−0.0699201	−0.0349600	−	0.999389i	$-0.511130\pi$
$41$	−18.3560	−0.0699201	−0.0349600	+	0.999389i	$0.511130\pi$
$42$	0	0
$43$	−61.0816	−0.216625	−0.108312	−	0.994117i	$-0.534545\pi$
$43$	−61.0816	−0.216625	−0.108312	+	0.994117i	$0.534545\pi$
$44$	0	0
$45$	−429.912	−1.42417
$46$	0	0
$47$	77.6986	0.241138	0.120569	−	0.992705i	$-0.461528\pi$
$47$	77.6986	0.241138	0.120569	+	0.992705i	$0.461528\pi$
$48$	0	0
$49$	−284.675	−0.829955
$50$	0	0
$51$	−206.049	−0.565738
$52$	0	0
$53$	−524.666	−1.35978	−0.679891	−	0.733313i	$-0.737973\pi$
$53$	−524.666	−1.35978	−0.679891	+	0.733313i	$0.737973\pi$
$54$	0	0
$55$	−1243.33	−3.04819
$56$	0	0
$57$	214.394	0.498195
$58$	0	0
$59$	−606.470	−1.33823	−0.669116	−	0.743158i	$-0.733327\pi$
$59$	−606.470	−1.33823	−0.669116	+	0.743158i	$0.733327\pi$
$60$	0	0
$61$	−390.164	−0.818941	−0.409471	−	0.912323i	$-0.634287\pi$
$61$	−390.164	−0.818941	−0.409471	+	0.912323i	$0.634287\pi$
$62$	0	0
$63$	−177.044	−0.354055
$64$	0	0
$65$	126.502	0.241394
$66$	0	0
$67$	746.727	1.36160	0.680800	−	0.732469i	$-0.261632\pi$
$67$	746.727	1.36160	0.680800	+	0.732469i	$0.261632\pi$
$68$	0	0
$69$	−99.6578	−0.173875
$70$	0	0
$71$	150.049	0.250810	0.125405	−	0.992106i	$-0.459977\pi$
$71$	150.049	0.250810	0.125405	+	0.992106i	$0.459977\pi$
$72$	0	0
$73$	707.612	1.13452	0.567258	−	0.823540i	$-0.308004\pi$
$73$	707.612	1.13452	0.567258	+	0.823540i	$0.308004\pi$
$74$	0	0
$75$	−427.751	−0.658566
$76$	0	0
$77$	−512.022	−0.757796
$78$	0	0
$79$	−76.5205	−0.108978	−0.0544888	−	0.998514i	$-0.517353\pi$
$79$	−76.5205	−0.108978	−0.0544888	+	0.998514i	$0.517353\pi$
$80$	0	0
$81$	434.327	0.595785
$82$	0	0
$83$	−1173.70	−1.55217	−0.776086	−	0.630627i	$-0.782798\pi$
$83$	−1173.70	−1.55217	−0.776086	+	0.630627i	$0.782798\pi$
$84$	0	0
$85$	1955.62	2.49550
$86$	0	0
$87$	−56.6643	−0.0698282
$88$	0	0
$89$	−1242.33	−1.47963	−0.739813	−	0.672812i	$-0.765086\pi$
$89$	−1242.33	−1.47963	−0.739813	+	0.672812i	$0.765086\pi$
$90$	0	0
$91$	52.0953	0.0600118
$92$	0	0
$93$	−7.74495	−0.00863564
$94$	0	0
$95$	−2034.82	−2.19756
$96$	0	0
$97$	−1574.55	−1.64816	−0.824079	−	0.566475i	$-0.808307\pi$
$97$	−1574.55	−1.64816	−0.824079	+	0.566475i	$0.808307\pi$
$98$	0	0
$99$	1554.22	1.57783
$100$	0	0
$101$	849.035	0.836457	0.418228	−	0.908342i	$-0.362651\pi$
$101$	849.035	0.836457	0.418228	+	0.908342i	$0.362651\pi$
$102$	0	0
$103$	−1707.56	−1.63350	−0.816752	−	0.576988i	$-0.804228\pi$
$103$	−1707.56	−1.63350	−0.816752	+	0.576988i	$0.804228\pi$
$104$	0	0
$105$	−276.737	−0.257208
$106$	0	0
$107$	−478.551	−0.432367	−0.216184	−	0.976353i	$-0.569361\pi$
$107$	−478.551	−0.432367	−0.216184	+	0.976353i	$0.569361\pi$
$108$	0	0
$109$	−532.818	−0.468209	−0.234104	−	0.972211i	$-0.575216\pi$
$109$	−532.818	−0.468209	−0.234104	+	0.972211i	$0.575216\pi$
$110$	0	0
$111$	−80.6174	−0.0689357
$112$	0	0
$113$	1412.19	1.17565	0.587823	−	0.808990i	$-0.299985\pi$
$113$	1412.19	1.17565	0.587823	+	0.808990i	$0.299985\pi$
$114$	0	0
$115$	945.859	0.766972
$116$	0	0
$117$	−158.133	−0.124952
$118$	0	0
$119$	805.355	0.620393
$120$	0	0
$121$	3163.89	2.37708
$122$	0	0
$123$	35.8665	0.0262925
$124$	0	0
$125$	1741.69	1.24625
$126$	0	0
$127$	−1632.35	−1.14053	−0.570266	−	0.821460i	$-0.693160\pi$
$127$	−1632.35	−1.14053	−0.570266	+	0.821460i	$0.693160\pi$
$128$	0	0
$129$	119.350	0.0814587
$130$	0	0
$131$	−1143.85	−0.762890	−0.381445	−	0.924391i	$-0.624574\pi$
$131$	−1143.85	−0.762890	−0.381445	+	0.924391i	$0.624574\pi$
$132$	0	0
$133$	−837.972	−0.546326
$134$	0	0
$135$	1818.39	1.15928
$136$	0	0
$137$	−414.515	−0.258499	−0.129250	−	0.991612i	$-0.541257\pi$
$137$	−414.515	−0.258499	−0.129250	+	0.991612i	$0.541257\pi$
$138$	0	0
$139$	−710.102	−0.433310	−0.216655	−	0.976248i	$-0.569515\pi$
$139$	−710.102	−0.433310	−0.216655	+	0.976248i	$0.569515\pi$
$140$	0	0
$141$	−151.819	−0.0906768
$142$	0	0
$143$	−457.330	−0.267439
$144$	0	0
$145$	537.805	0.308016
$146$	0	0
$147$	556.238	0.312093
$148$	0	0
$149$	2997.89	1.64830	0.824149	−	0.566373i	$-0.191654\pi$
$149$	2997.89	1.64830	0.824149	+	0.566373i	$0.191654\pi$
$150$	0	0
$151$	2508.50	1.35191	0.675955	−	0.736943i	$-0.263731\pi$
$151$	2508.50	1.35191	0.675955	+	0.736943i	$0.263731\pi$
$152$	0	0
$153$	−2444.62	−1.29174
$154$	0	0
$155$	73.5078	0.0380922
$156$	0	0
$157$	−3686.80	−1.87413	−0.937066	−	0.349152i	$-0.886470\pi$
$157$	−3686.80	−1.87413	−0.937066	+	0.349152i	$0.886470\pi$
$158$	0	0
$159$	1025.17	0.511328
$160$	0	0
$161$	389.519	0.190673
$162$	0	0
$163$	1949.12	0.936605	0.468302	−	0.883568i	$-0.344866\pi$
$163$	1949.12	0.936605	0.468302	+	0.883568i	$0.344866\pi$
$164$	0	0
$165$	2429.39	1.14623
$166$	0	0
$167$	−855.533	−0.396426	−0.198213	−	0.980159i	$-0.563514\pi$
$167$	−855.533	−0.396426	−0.198213	+	0.980159i	$0.563514\pi$
$168$	0	0
$169$	−2150.47	−0.978821
$170$	0	0
$171$	2543.63	1.13752
$172$	0	0
$173$	−2725.49	−1.19777	−0.598887	−	0.800834i	$-0.704390\pi$
$173$	−2725.49	−1.19777	−0.598887	+	0.800834i	$0.704390\pi$
$174$	0	0
$175$	1671.89	0.722190
$176$	0	0
$177$	1185.01	0.503224
$178$	0	0
$179$	−2447.11	−1.02182	−0.510910	−	0.859634i	$-0.670691\pi$
$179$	−2447.11	−1.02182	−0.510910	+	0.859634i	$0.670691\pi$
$180$	0	0
$181$	2433.45	0.999320	0.499660	−	0.866222i	$-0.333458\pi$
$181$	2433.45	0.999320	0.499660	+	0.866222i	$0.333458\pi$
$182$	0	0
$183$	762.359	0.307952
$184$	0	0
$185$	765.145	0.304079
$186$	0	0
$187$	−7069.98	−2.76475
$188$	0	0
$189$	748.841	0.288202
$190$	0	0
$191$	4530.41	1.71628	0.858138	−	0.513419i	$-0.171621\pi$
$191$	4530.41	1.71628	0.858138	+	0.513419i	$0.171621\pi$
$192$	0	0
$193$	−1819.70	−0.678678	−0.339339	−	0.940664i	$-0.610203\pi$
$193$	−1819.70	−0.678678	−0.339339	+	0.940664i	$0.610203\pi$
$194$	0	0
$195$	−247.177	−0.0907729
$196$	0	0
$197$	1325.30	0.479307	0.239653	−	0.970859i	$-0.422966\pi$
$197$	1325.30	0.479307	0.239653	+	0.970859i	$0.422966\pi$
$198$	0	0
$199$	−921.014	−0.328085	−0.164043	−	0.986453i	$-0.552453\pi$
$199$	−921.014	−0.328085	−0.164043	+	0.986453i	$0.552453\pi$
$200$	0	0
$201$	−1459.06	−0.512011
$202$	0	0
$203$	221.476	0.0765743
$204$	0	0
$205$	−340.412	−0.115978
$206$	0	0
$207$	−1182.37	−0.397006
$208$	0	0
$209$	7356.31	2.43467
$210$	0	0
$211$	−342.370	−0.111705	−0.0558525	−	0.998439i	$-0.517788\pi$
$211$	−342.370	−0.111705	−0.0558525	+	0.998439i	$0.517788\pi$
$212$	0	0
$213$	−293.186	−0.0943136
$214$	0	0
$215$	−1132.76	−0.359318
$216$	0	0
$217$	30.2717	0.00946992
$218$	0	0
$219$	−1382.63	−0.426620
$220$	0	0
$221$	719.330	0.218947
$222$	0	0
$223$	4038.47	1.21272	0.606358	−	0.795192i	$-0.292630\pi$
$223$	4038.47	1.21272	0.606358	+	0.795192i	$0.292630\pi$
$224$	0	0
$225$	−5074.96	−1.50369
$226$	0	0
$227$	−2948.69	−0.862165	−0.431083	−	0.902312i	$-0.641868\pi$
$227$	−2948.69	−0.862165	−0.431083	+	0.902312i	$0.641868\pi$
$228$	0	0
$229$	−4405.85	−1.27138	−0.635692	−	0.771943i	$-0.719285\pi$
$229$	−4405.85	−1.27138	−0.635692	+	0.771943i	$0.719285\pi$
$230$	0	0
$231$	1000.46	0.284959
$232$	0	0
$233$	−4942.11	−1.38956	−0.694781	−	0.719221i	$-0.744499\pi$
$233$	−4942.11	−1.38956	−0.694781	+	0.719221i	$0.744499\pi$
$234$	0	0
$235$	1440.92	0.399980
$236$	0	0
$237$	149.517	0.0409795
$238$	0	0
$239$	−3855.85	−1.04357	−0.521787	−	0.853076i	$-0.674734\pi$
$239$	−3855.85	−1.04357	−0.521787	+	0.853076i	$0.674734\pi$
$240$	0	0
$241$	−2496.50	−0.667275	−0.333638	−	0.942701i	$-0.608276\pi$
$241$	−2496.50	−0.667275	−0.333638	+	0.942701i	$0.608276\pi$
$242$	0	0
$243$	−3496.08	−0.922937
$244$	0	0
$245$	−5279.29	−1.37666
$246$	0	0
$247$	−748.463	−0.192808
$248$	0	0
$249$	2293.34	0.583673
$250$	0	0
$251$	−5442.61	−1.36866	−0.684332	−	0.729170i	$-0.739906\pi$
$251$	−5442.61	−1.36866	−0.684332	+	0.729170i	$0.739906\pi$
$252$	0	0
$253$	−3419.47	−0.849725
$254$	0	0
$255$	−3821.18	−0.938397
$256$	0	0
$257$	3794.51	0.920992	0.460496	−	0.887662i	$-0.347672\pi$
$257$	3794.51	0.920992	0.460496	+	0.887662i	$0.347672\pi$
$258$	0	0
$259$	315.098	0.0755956
$260$	0	0
$261$	−672.281	−0.159437
$262$	0	0
$263$	−5771.07	−1.35308	−0.676539	−	0.736407i	$-0.736521\pi$
$263$	−5771.07	−1.35308	−0.676539	+	0.736407i	$0.736521\pi$
$264$	0	0
$265$	−9729.93	−2.25549
$266$	0	0
$267$	2427.44	0.556394
$268$	0	0
$269$	−433.094	−0.0981643	−0.0490822	−	0.998795i	$-0.515630\pi$
$269$	−433.094	−0.0981643	−0.0490822	+	0.998795i	$0.515630\pi$
$270$	0	0
$271$	−4929.23	−1.10491	−0.552453	−	0.833544i	$-0.686308\pi$
$271$	−4929.23	−1.10491	−0.552453	+	0.833544i	$0.686308\pi$
$272$	0	0
$273$	−101.791	−0.0225666
$274$	0	0
$275$	−14677.1	−3.21840
$276$	0	0
$277$	2954.50	0.640862	0.320431	−	0.947272i	$-0.396172\pi$
$277$	2954.50	0.640862	0.320431	+	0.947272i	$0.396172\pi$
$278$	0	0
$279$	−91.8882	−0.0197176
$280$	0	0
$281$	3383.21	0.718241	0.359120	−	0.933291i	$-0.383077\pi$
$281$	3383.21	0.718241	0.359120	+	0.933291i	$0.383077\pi$
$282$	0	0
$283$	5168.89	1.08572	0.542859	−	0.839824i	$-0.317342\pi$
$283$	5168.89	1.08572	0.542859	+	0.839824i	$0.317342\pi$
$284$	0	0
$285$	3975.93	0.826364
$286$	0	0
$287$	−140.187	−0.0288326
$288$	0	0
$289$	6207.31	1.26345
$290$	0	0
$291$	3076.58	0.619768
$292$	0	0
$293$	3081.13	0.614340	0.307170	−	0.951655i	$-0.400618\pi$
$293$	3081.13	0.614340	0.307170	+	0.951655i	$0.400618\pi$
$294$	0	0
$295$	−11247.0	−2.21974
$296$	0	0
$297$	−6573.86	−1.28436
$298$	0	0
$299$	347.912	0.0672919
$300$	0	0
$301$	−466.487	−0.0893285
$302$	0	0
$303$	−1658.97	−0.314538
$304$	0	0
$305$	−7235.60	−1.35839
$306$	0	0
$307$	−2412.26	−0.448453	−0.224227	−	0.974537i	$-0.571986\pi$
$307$	−2412.26	−0.448453	−0.224227	+	0.974537i	$0.571986\pi$
$308$	0	0
$309$	3336.48	0.614257
$310$	0	0
$311$	−9542.56	−1.73990	−0.869950	−	0.493140i	$-0.835849\pi$
$311$	−9542.56	−1.73990	−0.869950	+	0.493140i	$0.835849\pi$
$312$	0	0
$313$	2692.28	0.486188	0.243094	−	0.970003i	$-0.421838\pi$
$313$	2692.28	0.486188	0.243094	+	0.970003i	$0.421838\pi$
$314$	0	0
$315$	−3283.29	−0.587277
$316$	0	0
$317$	−7174.82	−1.27122	−0.635612	−	0.772009i	$-0.719252\pi$
$317$	−7174.82	−1.27122	−0.635612	+	0.772009i	$0.719252\pi$
$318$	0	0
$319$	−1944.27	−0.341249
$320$	0	0
$321$	935.062	0.162586
$322$	0	0
$323$	−11570.7	−1.99322
$324$	0	0
$325$	1493.31	0.254873
$326$	0	0
$327$	1041.10	0.176064
$328$	0	0
$329$	593.393	0.0994371
$330$	0	0
$331$	9152.83	1.51989	0.759947	−	0.649985i	$-0.225225\pi$
$331$	9152.83	1.51989	0.759947	+	0.649985i	$0.225225\pi$
$332$	0	0
$333$	−956.467	−0.157400
$334$	0	0
$335$	13848.0	2.25851
$336$	0	0
$337$	−1319.39	−0.213269	−0.106635	−	0.994298i	$-0.534008\pi$
$337$	−1319.39	−0.213269	−0.106635	+	0.994298i	$0.534008\pi$
$338$	0	0
$339$	−2759.35	−0.442086
$340$	0	0
$341$	−265.746	−0.0422022
$342$	0	0
$343$	−4793.62	−0.754610
$344$	0	0
$345$	−1848.15	−0.288409
$346$	0	0
$347$	274.593	0.0424810	0.0212405	−	0.999774i	$-0.493238\pi$
$347$	274.593	0.0424810	0.0212405	+	0.999774i	$0.493238\pi$
$348$	0	0
$349$	−1680.66	−0.257775	−0.128887	−	0.991659i	$-0.541141\pi$
$349$	−1680.66	−0.257775	−0.128887	+	0.991659i	$0.541141\pi$
$350$	0	0
$351$	668.853	0.101711
$352$	0	0
$353$	8331.96	1.25628	0.628138	−	0.778102i	$-0.283817\pi$
$353$	8331.96	1.25628	0.628138	+	0.778102i	$0.283817\pi$
$354$	0	0
$355$	2782.65	0.416022
$356$	0	0
$357$	−1573.62	−0.233291
$358$	0	0
$359$	−13073.6	−1.92200	−0.960998	−	0.276554i	$-0.910807\pi$
$359$	−13073.6	−1.92200	−0.960998	+	0.276554i	$0.910807\pi$
$360$	0	0
$361$	5180.28	0.755253
$362$	0	0
$363$	−6182.06	−0.893867
$364$	0	0
$365$	13122.7	1.88184
$366$	0	0
$367$	−5042.03	−0.717145	−0.358572	−	0.933502i	$-0.616736\pi$
$367$	−5042.03	−0.717145	−0.358572	+	0.933502i	$0.616736\pi$
$368$	0	0
$369$	425.531	0.0600332
$370$	0	0
$371$	−4006.94	−0.560727
$372$	0	0
$373$	12889.1	1.78921	0.894603	−	0.446861i	$-0.147458\pi$
$373$	12889.1	1.78921	0.894603	+	0.446861i	$0.147458\pi$
$374$	0	0
$375$	−3403.16	−0.468636
$376$	0	0
$377$	197.819	0.0270244
$378$	0	0
$379$	6489.95	0.879595	0.439798	−	0.898097i	$-0.355050\pi$
$379$	6489.95	0.879595	0.439798	+	0.898097i	$0.355050\pi$
$380$	0	0
$381$	3189.52	0.428882
$382$	0	0
$383$	2537.38	0.338522	0.169261	−	0.985571i	$-0.445862\pi$
$383$	2537.38	0.338522	0.169261	+	0.985571i	$0.445862\pi$
$384$	0	0
$385$	−9495.45	−1.25697
$386$	0	0
$387$	1416.00	0.185993
$388$	0	0
$389$	10725.7	1.39798	0.698989	−	0.715132i	$-0.253634\pi$
$389$	10725.7	1.39798	0.698989	+	0.715132i	$0.253634\pi$
$390$	0	0
$391$	5378.46	0.695653
$392$	0	0
$393$	2235.02	0.286875
$394$	0	0
$395$	−1419.07	−0.180763
$396$	0	0
$397$	−14172.8	−1.79172	−0.895859	−	0.444339i	$-0.853439\pi$
$397$	−14172.8	−1.79172	−0.895859	+	0.444339i	$0.853439\pi$
$398$	0	0
$399$	1637.35	0.205439
$400$	0	0
$401$	7984.76	0.994363	0.497182	−	0.867647i	$-0.334368\pi$
$401$	7984.76	0.994363	0.497182	+	0.867647i	$0.334368\pi$
$402$	0	0
$403$	27.0381	0.00334210
$404$	0	0
$405$	8054.59	0.988237
$406$	0	0
$407$	−2766.15	−0.336887
$408$	0	0
$409$	−227.055	−0.0274503	−0.0137251	−	0.999906i	$-0.504369\pi$
$409$	−227.055	−0.0274503	−0.0137251	+	0.999906i	$0.504369\pi$
$410$	0	0
$411$	809.938	0.0972051
$412$	0	0
$413$	−4631.68	−0.551840
$414$	0	0
$415$	−21766.3	−2.57461
$416$	0	0
$417$	1387.50	0.162940
$418$	0	0
$419$	11876.4	1.38472	0.692361	−	0.721551i	$-0.256571\pi$
$419$	11876.4	1.38472	0.692361	+	0.721551i	$0.256571\pi$
$420$	0	0
$421$	−10927.3	−1.26500	−0.632499	−	0.774561i	$-0.717971\pi$
$421$	−10927.3	−1.26500	−0.632499	+	0.774561i	$0.717971\pi$
$422$	0	0
$423$	−1801.22	−0.207041
$424$	0	0
$425$	23085.4	2.63484
$426$	0	0
$427$	−2979.73	−0.337703
$428$	0	0
$429$	893.596	0.100567
$430$	0	0
$431$	1085.02	0.121262	0.0606308	−	0.998160i	$-0.480689\pi$
$431$	1085.02	0.121262	0.0606308	+	0.998160i	$0.480689\pi$
$432$	0	0
$433$	−3697.66	−0.410389	−0.205194	−	0.978721i	$-0.565783\pi$
$433$	−3697.66	−0.410389	−0.205194	+	0.978721i	$0.565783\pi$
$434$	0	0
$435$	−1050.84	−0.115825
$436$	0	0
$437$	−5596.28	−0.612601
$438$	0	0
$439$	338.575	0.0368093	0.0184047	−	0.999831i	$-0.494141\pi$
$439$	338.575	0.0368093	0.0184047	+	0.999831i	$0.494141\pi$
$440$	0	0
$441$	6599.36	0.712596
$442$	0	0
$443$	−11170.8	−1.19806	−0.599028	−	0.800728i	$-0.704446\pi$
$443$	−11170.8	−1.19806	−0.599028	+	0.800728i	$0.704446\pi$
$444$	0	0
$445$	−23039.0	−2.45428
$446$	0	0
$447$	−5857.70	−0.619820
$448$	0	0
$449$	−11355.0	−1.19348	−0.596741	−	0.802434i	$-0.703538\pi$
$449$	−11355.0	−1.19348	−0.596741	+	0.802434i	$0.703538\pi$
$450$	0	0
$451$	1230.66	0.128491
$452$	0	0
$453$	−4901.46	−0.508368
$454$	0	0
$455$	966.108	0.0995425
$456$	0	0
$457$	−2057.24	−0.210577	−0.105288	−	0.994442i	$-0.533577\pi$
$457$	−2057.24	−0.210577	−0.105288	+	0.994442i	$0.533577\pi$
$458$	0	0
$459$	10340.0	1.05148
$460$	0	0
$461$	−19378.0	−1.95775	−0.978876	−	0.204456i	$-0.934457\pi$
$461$	−19378.0	−1.95775	−0.978876	+	0.204456i	$0.934457\pi$
$462$	0	0
$463$	184.925	0.0185620	0.00928101	−	0.999957i	$-0.497046\pi$
$463$	184.925	0.0185620	0.00928101	+	0.999957i	$0.497046\pi$
$464$	0	0
$465$	−143.630	−0.0143241
$466$	0	0
$467$	14640.0	1.45066	0.725332	−	0.688400i	$-0.241686\pi$
$467$	14640.0	1.45066	0.725332	+	0.688400i	$0.241686\pi$
$468$	0	0
$469$	5702.84	0.561477
$470$	0	0
$471$	7203.80	0.704742
$472$	0	0
$473$	4095.15	0.398087
$474$	0	0
$475$	−24020.4	−2.32027
$476$	0	0
$477$	12162.9	1.16750
$478$	0	0
$479$	−9831.22	−0.937786	−0.468893	−	0.883255i	$-0.655347\pi$
$479$	−9831.22	−0.937786	−0.468893	+	0.883255i	$0.655347\pi$
$480$	0	0
$481$	281.441	0.0266790
$482$	0	0
$483$	−761.098	−0.0717001
$484$	0	0
$485$	−29200.0	−2.73382
$486$	0	0
$487$	20368.4	1.89523	0.947617	−	0.319408i	$-0.103484\pi$
$487$	20368.4	1.89523	0.947617	+	0.319408i	$0.103484\pi$
$488$	0	0
$489$	−3808.46	−0.352198
$490$	0	0
$491$	17333.5	1.59318	0.796589	−	0.604521i	$-0.206635\pi$
$491$	17333.5	1.59318	0.796589	+	0.604521i	$0.206635\pi$
$492$	0	0
$493$	3058.13	0.279374
$494$	0	0
$495$	28823.0	2.61717
$496$	0	0
$497$	1145.94	0.103425
$498$	0	0
$499$	6797.87	0.609849	0.304924	−	0.952377i	$-0.401369\pi$
$499$	6797.87	0.609849	0.304924	+	0.952377i	$0.401369\pi$
$500$	0	0
$501$	1671.66	0.149071
$502$	0	0
$503$	12318.5	1.09196	0.545978	−	0.837799i	$-0.316158\pi$
$503$	12318.5	1.09196	0.545978	+	0.837799i	$0.316158\pi$
$504$	0	0
$505$	15745.3	1.38744
$506$	0	0
$507$	4201.89	0.368072
$508$	0	0
$509$	11321.6	0.985896	0.492948	−	0.870059i	$-0.335919\pi$
$509$	11321.6	0.985896	0.492948	+	0.870059i	$0.335919\pi$
$510$	0	0
$511$	5404.11	0.467835
$512$	0	0
$513$	−10758.7	−0.925944
$514$	0	0
$515$	−31666.7	−2.70952
$516$	0	0
$517$	−5209.22	−0.443136
$518$	0	0
$519$	5325.44	0.450407
$520$	0	0
$521$	9789.88	0.823229	0.411614	−	0.911358i	$-0.364965\pi$
$521$	9789.88	0.823229	0.411614	+	0.911358i	$0.364965\pi$
$522$	0	0
$523$	3258.08	0.272401	0.136200	−	0.990681i	$-0.456511\pi$
$523$	3258.08	0.272401	0.136200	+	0.990681i	$0.456511\pi$
$524$	0	0
$525$	−3266.78	−0.271570
$526$	0	0
$527$	417.990	0.0345501
$528$	0	0
$529$	−9565.65	−0.786196
$530$	0	0
$531$	14059.3	1.14900
$532$	0	0
$533$	−125.212	−0.0101755
$534$	0	0
$535$	−8874.73	−0.717174
$536$	0	0
$537$	4781.52	0.384242
$538$	0	0
$539$	19085.7	1.52519
$540$	0	0
$541$	13740.7	1.09198	0.545989	−	0.837792i	$-0.316154\pi$
$541$	13740.7	1.09198	0.545989	+	0.837792i	$0.316154\pi$
$542$	0	0
$543$	−4754.82	−0.375781
$544$	0	0
$545$	−9881.11	−0.776625
$546$	0	0
$547$	1021.13	0.0798178	0.0399089	−	0.999203i	$-0.487293\pi$
$547$	1021.13	0.0798178	0.0399089	+	0.999203i	$0.487293\pi$
$548$	0	0
$549$	9044.83	0.703140
$550$	0	0
$551$	−3181.99	−0.246020
$552$	0	0
$553$	−584.396	−0.0449386
$554$	0	0
$555$	−1495.05	−0.114345
$556$	0	0
$557$	−4146.36	−0.315417	−0.157708	−	0.987486i	$-0.550411\pi$
$557$	−4146.36	−0.315417	−0.157708	+	0.987486i	$0.550411\pi$
$558$	0	0
$559$	−416.658	−0.0315255
$560$	0	0
$561$	13814.3	1.03965
$562$	0	0
$563$	466.827	0.0349457	0.0174728	−	0.999847i	$-0.494438\pi$
$563$	466.827	0.0349457	0.0174728	+	0.999847i	$0.494438\pi$
$564$	0	0
$565$	26189.1	1.95006
$566$	0	0
$567$	3317.00	0.245681
$568$	0	0
$569$	−23329.9	−1.71888	−0.859438	−	0.511239i	$-0.829187\pi$
$569$	−23329.9	−1.71888	−0.859438	+	0.511239i	$0.829187\pi$
$570$	0	0
$571$	10730.4	0.786431	0.393216	−	0.919446i	$-0.371363\pi$
$571$	10730.4	0.786431	0.393216	+	0.919446i	$0.371363\pi$
$572$	0	0
$573$	−8852.16	−0.645382
$574$	0	0
$575$	11165.5	0.809799
$576$	0	0
$577$	−8469.82	−0.611097	−0.305549	−	0.952176i	$-0.598840\pi$
$577$	−8469.82	−0.611097	−0.305549	+	0.952176i	$0.598840\pi$
$578$	0	0
$579$	3555.59	0.255208
$580$	0	0
$581$	−8963.67	−0.640062
$582$	0	0
$583$	35175.7	2.49885
$584$	0	0
$585$	−2932.58	−0.207260
$586$	0	0
$587$	299.884	0.0210861	0.0105431	−	0.999944i	$-0.496644\pi$
$587$	299.884	0.0210861	0.0105431	+	0.999944i	$0.496644\pi$
$588$	0	0
$589$	−434.918	−0.0304253
$590$	0	0
$591$	−2589.55	−0.180237
$592$	0	0
$593$	−10265.4	−0.710873	−0.355437	−	0.934700i	$-0.615668\pi$
$593$	−10265.4	−0.710873	−0.355437	+	0.934700i	$0.615668\pi$
$594$	0	0
$595$	14935.3	1.02906
$596$	0	0
$597$	1799.61	0.123372
$598$	0	0
$599$	15166.9	1.03456	0.517281	−	0.855816i	$-0.326944\pi$
$599$	15166.9	1.03456	0.517281	+	0.855816i	$0.326944\pi$
$600$	0	0
$601$	12237.0	0.830546	0.415273	−	0.909697i	$-0.363686\pi$
$601$	12237.0	0.830546	0.415273	+	0.909697i	$0.363686\pi$
$602$	0	0
$603$	−17310.7	−1.16907
$604$	0	0
$605$	58674.3	3.94289
$606$	0	0
$607$	−4989.45	−0.333633	−0.166817	−	0.985988i	$-0.553349\pi$
$607$	−4989.45	−0.333633	−0.166817	+	0.985988i	$0.553349\pi$
$608$	0	0
$609$	−432.752	−0.0287947
$610$	0	0
$611$	530.008	0.0350930
$612$	0	0
$613$	−13819.8	−0.910566	−0.455283	−	0.890347i	$-0.650462\pi$
$613$	−13819.8	−0.910566	−0.455283	+	0.890347i	$0.650462\pi$
$614$	0	0
$615$	665.145	0.0436118
$616$	0	0
$617$	−18400.5	−1.20061	−0.600304	−	0.799772i	$-0.704954\pi$
$617$	−18400.5	−1.20061	−0.600304	+	0.799772i	$0.704954\pi$
$618$	0	0
$619$	−10113.9	−0.656723	−0.328362	−	0.944552i	$-0.606496\pi$
$619$	−10113.9	−0.656723	−0.328362	+	0.944552i	$0.606496\pi$
$620$	0	0
$621$	5001.04	0.323164
$622$	0	0
$623$	−9487.82	−0.610147
$624$	0	0
$625$	4935.00	0.315840
$626$	0	0
$627$	−14373.8	−0.915525
$628$	0	0
$629$	4350.86	0.275803
$630$	0	0
$631$	−2969.78	−0.187361	−0.0936806	−	0.995602i	$-0.529863\pi$
$631$	−2969.78	−0.187361	−0.0936806	+	0.995602i	$0.529863\pi$
$632$	0	0
$633$	668.972	0.0420051
$634$	0	0
$635$	−30271.9	−1.89182
$636$	0	0
$637$	−1941.86	−0.120784
$638$	0	0
$639$	−3478.44	−0.215344
$640$	0	0
$641$	−150.409	−0.00926804	−0.00463402	−	0.999989i	$-0.501475\pi$
$641$	−150.409	−0.00926804	−0.00463402	+	0.999989i	$0.501475\pi$
$642$	0	0
$643$	17894.8	1.09751	0.548756	−	0.835983i	$-0.315102\pi$
$643$	17894.8	1.09751	0.548756	+	0.835983i	$0.315102\pi$
$644$	0	0
$645$	2213.34	0.135117
$646$	0	0
$647$	−17611.1	−1.07012	−0.535058	−	0.844815i	$-0.679710\pi$
$647$	−17611.1	−1.07012	−0.535058	+	0.844815i	$0.679710\pi$
$648$	0	0
$649$	40660.1	2.45924
$650$	0	0
$651$	−59.1491	−0.00356104
$652$	0	0
$653$	7388.11	0.442755	0.221378	−	0.975188i	$-0.428945\pi$
$653$	7388.11	0.442755	0.221378	+	0.975188i	$0.428945\pi$
$654$	0	0
$655$	−21212.7	−1.26542
$656$	0	0
$657$	−16403.9	−0.974092
$658$	0	0
$659$	−26209.7	−1.54929	−0.774647	−	0.632394i	$-0.782072\pi$
$659$	−26209.7	−1.54929	−0.774647	+	0.632394i	$0.782072\pi$
$660$	0	0
$661$	−2052.04	−0.120749	−0.0603745	−	0.998176i	$-0.519230\pi$
$661$	−2052.04	−0.120749	−0.0603745	+	0.998176i	$0.519230\pi$
$662$	0	0
$663$	−1405.53	−0.0823322
$664$	0	0
$665$	−15540.2	−0.906199
$666$	0	0
$667$	1479.10	0.0858635
$668$	0	0
$669$	−7890.93	−0.456026
$670$	0	0
$671$	26158.2	1.50495
$672$	0	0
$673$	13523.9	0.774605	0.387303	−	0.921953i	$-0.373407\pi$
$673$	13523.9	0.774605	0.387303	+	0.921953i	$0.373407\pi$
$674$	0	0
$675$	21465.5	1.22401
$676$	0	0
$677$	−17142.3	−0.973161	−0.486581	−	0.873636i	$-0.661756\pi$
$677$	−17142.3	−0.973161	−0.486581	+	0.873636i	$0.661756\pi$
$678$	0	0
$679$	−12025.0	−0.679643
$680$	0	0
$681$	5761.57	0.324206
$682$	0	0
$683$	1380.52	0.0773412	0.0386706	−	0.999252i	$-0.487688\pi$
$683$	1380.52	0.0773412	0.0386706	+	0.999252i	$0.487688\pi$
$684$	0	0
$685$	−7687.18	−0.428777
$686$	0	0
$687$	8608.78	0.478087
$688$	0	0
$689$	−3578.93	−0.197890
$690$	0	0
$691$	20983.5	1.15521	0.577605	−	0.816316i	$-0.303987\pi$
$691$	20983.5	1.15521	0.577605	+	0.816316i	$0.303987\pi$
$692$	0	0
$693$	11869.8	0.650641
$694$	0	0
$695$	−13168.8	−0.718737
$696$	0	0
$697$	−1935.69	−0.105193
$698$	0	0
$699$	9656.59	0.522526
$700$	0	0
$701$	21041.5	1.13370	0.566852	−	0.823820i	$-0.308161\pi$
$701$	21041.5	1.13370	0.566852	+	0.823820i	$0.308161\pi$
$702$	0	0
$703$	−4527.07	−0.242876
$704$	0	0
$705$	−2815.47	−0.150407
$706$	0	0
$707$	6484.17	0.344926
$708$	0	0
$709$	31968.5	1.69338	0.846688	−	0.532090i	$-0.178593\pi$
$709$	31968.5	1.69338	0.846688	+	0.532090i	$0.178593\pi$
$710$	0	0
$711$	1773.91	0.0935678
$712$	0	0
$713$	202.165	0.0106187
$714$	0	0
$715$	−8481.18	−0.443606
$716$	0	0
$717$	7534.10	0.392422
$718$	0	0
$719$	−19937.6	−1.03414	−0.517070	−	0.855943i	$-0.672977\pi$
$719$	−19937.6	−1.03414	−0.517070	+	0.855943i	$0.672977\pi$
$720$	0	0
$721$	−13040.8	−0.673601
$722$	0	0
$723$	4878.01	0.250920
$724$	0	0
$725$	6348.59	0.325215
$726$	0	0
$727$	3277.64	0.167209	0.0836046	−	0.996499i	$-0.473357\pi$
$727$	3277.64	0.167209	0.0836046	+	0.996499i	$0.473357\pi$
$728$	0	0
$729$	−4895.69	−0.248727
$730$	0	0
$731$	−6441.23	−0.325906
$732$	0	0
$733$	−13185.3	−0.664405	−0.332203	−	0.943208i	$-0.607792\pi$
$733$	−13185.3	−0.664405	−0.332203	+	0.943208i	$0.607792\pi$
$734$	0	0
$735$	10315.4	0.517674
$736$	0	0
$737$	−50063.5	−2.50219
$738$	0	0
$739$	877.219	0.0436658	0.0218329	−	0.999762i	$-0.493050\pi$
$739$	877.219	0.0436658	0.0218329	+	0.999762i	$0.493050\pi$
$740$	0	0
$741$	1462.45	0.0725028
$742$	0	0
$743$	22616.9	1.11674	0.558368	−	0.829593i	$-0.311428\pi$
$743$	22616.9	1.11674	0.558368	+	0.829593i	$0.311428\pi$
$744$	0	0
$745$	55595.8	2.73406
$746$	0	0
$747$	27208.8	1.33269
$748$	0	0
$749$	−3654.75	−0.178293
$750$	0	0
$751$	4283.99	0.208156	0.104078	−	0.994569i	$-0.466811\pi$
$751$	4283.99	0.208156	0.104078	+	0.994569i	$0.466811\pi$
$752$	0	0
$753$	10634.6	0.514668
$754$	0	0
$755$	46520.1	2.24243
$756$	0	0
$757$	−2313.30	−0.111068	−0.0555339	−	0.998457i	$-0.517686\pi$
$757$	−2313.30	−0.111068	−0.0555339	+	0.998457i	$0.517686\pi$
$758$	0	0
$759$	6681.45	0.319527
$760$	0	0
$761$	−11861.2	−0.565006	−0.282503	−	0.959266i	$-0.591165\pi$
$761$	−11861.2	−0.565006	−0.282503	+	0.959266i	$0.591165\pi$
$762$	0	0
$763$	−4069.19	−0.193073
$764$	0	0
$765$	−45335.5	−2.14262
$766$	0	0
$767$	−4136.94	−0.194754
$768$	0	0
$769$	−5533.00	−0.259460	−0.129730	−	0.991549i	$-0.541411\pi$
$769$	−5533.00	−0.259460	−0.129730	+	0.991549i	$0.541411\pi$
$770$	0	0
$771$	−7414.25	−0.346326
$772$	0	0
$773$	4550.04	0.211712	0.105856	−	0.994381i	$-0.466242\pi$
$773$	4550.04	0.211712	0.105856	+	0.994381i	$0.466242\pi$
$774$	0	0
$775$	867.733	0.0402192
$776$	0	0
$777$	−615.684	−0.0284267
$778$	0	0
$779$	2014.09	0.0926343
$780$	0	0
$781$	−10059.8	−0.460909
$782$	0	0
$783$	2843.54	0.129782
$784$	0	0
$785$	−68371.7	−3.10865
$786$	0	0
$787$	−30380.6	−1.37605	−0.688025	−	0.725687i	$-0.741522\pi$
$787$	−30380.6	−1.37605	−0.688025	+	0.725687i	$0.741522\pi$
$788$	0	0
$789$	11276.3	0.508806
$790$	0	0
$791$	10785.1	0.484796
$792$	0	0
$793$	−2661.44	−0.119181
$794$	0	0
$795$	19011.7	0.848147
$796$	0	0
$797$	−10663.5	−0.473926	−0.236963	−	0.971519i	$-0.576152\pi$
$797$	−10663.5	−0.473926	−0.236963	+	0.971519i	$0.576152\pi$
$798$	0	0
$799$	8193.54	0.362787
$800$	0	0
$801$	28799.8	1.27040
$802$	0	0
$803$	−47441.1	−2.08488
$804$	0	0
$805$	7223.63	0.316273
$806$	0	0
$807$	846.241	0.0369134
$808$	0	0
$809$	17366.0	0.754706	0.377353	−	0.926069i	$-0.376834\pi$
$809$	17366.0	0.754706	0.377353	+	0.926069i	$0.376834\pi$
$810$	0	0
$811$	36870.4	1.59642	0.798209	−	0.602381i	$-0.205781\pi$
$811$	36870.4	1.59642	0.798209	+	0.602381i	$0.205781\pi$
$812$	0	0
$813$	9631.44	0.415485
$814$	0	0
$815$	36146.4	1.55356
$816$	0	0
$817$	6702.10	0.286997
$818$	0	0
$819$	−1207.68	−0.0515260
$820$	0	0
$821$	18414.0	0.782769	0.391385	−	0.920227i	$-0.371996\pi$
$821$	18414.0	0.782769	0.391385	+	0.920227i	$0.371996\pi$
$822$	0	0
$823$	−10266.0	−0.434813	−0.217406	−	0.976081i	$-0.569760\pi$
$823$	−10266.0	−0.434813	−0.217406	+	0.976081i	$0.569760\pi$
$824$	0	0
$825$	28678.1	1.21023
$826$	0	0
$827$	9632.80	0.405036	0.202518	−	0.979278i	$-0.435087\pi$
$827$	9632.80	0.405036	0.202518	+	0.979278i	$0.435087\pi$
$828$	0	0
$829$	1547.06	0.0648150	0.0324075	−	0.999475i	$-0.489683\pi$
$829$	1547.06	0.0648150	0.0324075	+	0.999475i	$0.489683\pi$
$830$	0	0
$831$	−5772.93	−0.240988
$832$	0	0
$833$	−30019.7	−1.24865
$834$	0	0
$835$	−15865.9	−0.657558
$836$	0	0
$837$	388.658	0.0160502
$838$	0	0
$839$	30042.2	1.23620	0.618100	−	0.786100i	$-0.287903\pi$
$839$	30042.2	1.23620	0.618100	+	0.786100i	$0.287903\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−6610.61	−0.270085
$844$	0	0
$845$	−39880.4	−1.62358
$846$	0	0
$847$	24163.0	0.980224
$848$	0	0
$849$	−10099.7	−0.408270
$850$	0	0
$851$	2104.34	0.0847661
$852$	0	0
$853$	−20127.2	−0.807903	−0.403951	−	0.914780i	$-0.632363\pi$
$853$	−20127.2	−0.807903	−0.403951	+	0.914780i	$0.632363\pi$
$854$	0	0
$855$	47171.5	1.88682
$856$	0	0
$857$	27981.2	1.11531	0.557655	−	0.830073i	$-0.311701\pi$
$857$	27981.2	1.11531	0.557655	+	0.830073i	$0.311701\pi$
$858$	0	0
$859$	−21727.5	−0.863018	−0.431509	−	0.902109i	$-0.642019\pi$
$859$	−21727.5	−0.863018	−0.431509	+	0.902109i	$0.642019\pi$
$860$	0	0
$861$	273.917	0.0108421
$862$	0	0
$863$	22813.4	0.899858	0.449929	−	0.893064i	$-0.351449\pi$
$863$	22813.4	0.899858	0.449929	+	0.893064i	$0.351449\pi$
$864$	0	0
$865$	−50544.1	−1.98677
$866$	0	0
$867$	−12128.7	−0.475102
$868$	0	0
$869$	5130.24	0.200266
$870$	0	0
$871$	5093.68	0.198155
$872$	0	0
$873$	36501.4	1.41510
$874$	0	0
$875$	13301.5	0.513911
$876$	0	0
$877$	−13995.8	−0.538886	−0.269443	−	0.963016i	$-0.586840\pi$
$877$	−13995.8	−0.538886	−0.269443	+	0.963016i	$0.586840\pi$
$878$	0	0
$879$	−6020.35	−0.231014
$880$	0	0
$881$	−31340.0	−1.19849	−0.599247	−	0.800564i	$-0.704533\pi$
$881$	−31340.0	−1.19849	−0.599247	+	0.800564i	$0.704533\pi$
$882$	0	0
$883$	−3532.88	−0.134644	−0.0673221	−	0.997731i	$-0.521446\pi$
$883$	−3532.88	−0.134644	−0.0673221	+	0.997731i	$0.521446\pi$
$884$	0	0
$885$	21976.0	0.834705
$886$	0	0
$887$	−19969.1	−0.755916	−0.377958	−	0.925823i	$-0.623374\pi$
$887$	−19969.1	−0.755916	−0.377958	+	0.925823i	$0.623374\pi$
$888$	0	0
$889$	−12466.4	−0.470316
$890$	0	0
$891$	−29119.0	−1.09486
$892$	0	0
$893$	−8525.37	−0.319474
$894$	0	0
$895$	−45381.7	−1.69491
$896$	0	0
$897$	−679.800	−0.0253042
$898$	0	0
$899$	114.949	0.00426447
$900$	0	0
$901$	−55327.6	−2.04576
$902$	0	0
$903$	911.489	0.0335908
$904$	0	0
$905$	45128.3	1.65759
$906$	0	0
$907$	28674.1	1.04973	0.524867	−	0.851184i	$-0.324115\pi$
$907$	28674.1	1.04973	0.524867	+	0.851184i	$0.324115\pi$
$908$	0	0
$909$	−19682.4	−0.718179
$910$	0	0
$911$	−10404.6	−0.378397	−0.189199	−	0.981939i	$-0.560589\pi$
$911$	−10404.6	−0.378397	−0.189199	+	0.981939i	$0.560589\pi$
$912$	0	0
$913$	78689.4	2.85240
$914$	0	0
$915$	14137.9	0.510804
$916$	0	0
$917$	−8735.71	−0.314590
$918$	0	0
$919$	−51963.1	−1.86519	−0.932593	−	0.360930i	$-0.882459\pi$
$919$	−51963.1	−1.86519	−0.932593	+	0.360930i	$0.882459\pi$
$920$	0	0
$921$	4713.42	0.168635
$922$	0	0
$923$	1023.53	0.0365005
$924$	0	0
$925$	9032.26	0.321058
$926$	0	0
$927$	39584.9	1.40252
$928$	0	0
$929$	−19454.4	−0.687059	−0.343530	−	0.939142i	$-0.611623\pi$
$929$	−19454.4	−0.687059	−0.343530	+	0.939142i	$0.611623\pi$
$930$	0	0
$931$	31235.5	1.09957
$932$	0	0
$933$	18645.6	0.654266
$934$	0	0
$935$	−131113.	−4.58593
$936$	0	0
$937$	−40950.3	−1.42773	−0.713867	−	0.700281i	$-0.753058\pi$
$937$	−40950.3	−1.42773	−0.713867	+	0.700281i	$0.753058\pi$
$938$	0	0
$939$	−5260.56	−0.182824
$940$	0	0
$941$	−31672.6	−1.09723	−0.548617	−	0.836074i	$-0.684845\pi$
$941$	−31672.6	−1.09723	−0.548617	+	0.836074i	$0.684845\pi$
$942$	0	0
$943$	−936.219	−0.0323303
$944$	0	0
$945$	13887.3	0.478045
$946$	0	0
$947$	−15659.5	−0.537343	−0.268671	−	0.963232i	$-0.586585\pi$
$947$	−15659.5	−0.537343	−0.268671	+	0.963232i	$0.586585\pi$
$948$	0	0
$949$	4826.86	0.165107
$950$	0	0
$951$	14019.2	0.478027
$952$	0	0
$953$	32414.0	1.10178	0.550888	−	0.834579i	$-0.314289\pi$
$953$	32414.0	1.10178	0.550888	+	0.834579i	$0.314289\pi$
$954$	0	0
$955$	84016.4	2.84681
$956$	0	0
$957$	3799.00	0.128322
$958$	0	0
$959$	−3165.70	−0.106596
$960$	0	0
$961$	−29775.3	−0.999473
$962$	0	0
$963$	11093.8	0.371229
$964$	0	0
$965$	−33746.3	−1.12573
$966$	0	0
$967$	53609.9	1.78281	0.891406	−	0.453207i	$-0.149720\pi$
$967$	53609.9	1.78281	0.891406	+	0.453207i	$0.149720\pi$
$968$	0	0
$969$	22608.4	0.749523
$970$	0	0
$971$	−50535.9	−1.67021	−0.835106	−	0.550090i	$-0.814593\pi$
$971$	−50535.9	−1.67021	−0.835106	+	0.550090i	$0.814593\pi$
$972$	0	0
$973$	−5423.13	−0.178682
$974$	0	0
$975$	−2917.84	−0.0958416
$976$	0	0
$977$	46165.9	1.51175	0.755874	−	0.654718i	$-0.227212\pi$
$977$	46165.9	1.51175	0.755874	+	0.654718i	$0.227212\pi$
$978$	0	0
$979$	83290.7	2.71908
$980$	0	0
$981$	12351.9	0.402002
$982$	0	0
$983$	−11758.6	−0.381526	−0.190763	−	0.981636i	$-0.561096\pi$
$983$	−11758.6	−0.381526	−0.190763	+	0.981636i	$0.561096\pi$
$984$	0	0
$985$	24577.6	0.795033
$986$	0	0
$987$	−1159.45	−0.0373920
$988$	0	0
$989$	−3115.37	−0.100165
$990$	0	0
$991$	49542.4	1.58806	0.794030	−	0.607879i	$-0.207979\pi$
$991$	49542.4	1.58806	0.794030	+	0.607879i	$0.207979\pi$
$992$	0	0
$993$	−17884.1	−0.571536
$994$	0	0
$995$	−17080.2	−0.544200
$996$	0	0
$997$	−23095.8	−0.733651	−0.366826	−	0.930290i	$-0.619555\pi$
$997$	−23095.8	−0.733651	−0.366826	+	0.930290i	$0.619555\pi$
$998$	0	0
$999$	4045.55	0.128124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.q.1.2		3
4.3	odd	2		1856.4.a.t.1.2		3
8.3	odd	2		464.4.a.h.1.2		3
8.5	even	2		232.4.a.a.1.2	✓	3
24.5	odd	2		2088.4.a.a.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.a.1.2	✓	3	8.5	even	2
464.4.a.h.1.2		3	8.3	odd	2
1856.4.a.q.1.2		3	1.1	even	1	trivial
1856.4.a.t.1.2		3	4.3	odd	2
2088.4.a.a.1.3		3	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)

\(f(q)\)	\(=\)	\(q-1.95394 q^{3} +18.5450 q^{5} +7.63711 q^{7} -23.1821 q^{9} +O(q^{10})\)	\(q-1.95394 q^{3} +18.5450 q^{5} +7.63711 q^{7} -23.1821 q^{9} -67.0439 q^{11} +6.82134 q^{13} -36.2359 q^{15} +105.453 q^{17} -109.724 q^{19} -14.9225 q^{21} +51.0034 q^{23} +218.917 q^{25} +98.0530 q^{27} +29.0000 q^{29} +3.96376 q^{31} +131.000 q^{33} +141.630 q^{35} +41.2588 q^{37} -13.3285 q^{39} -18.3560 q^{41} -61.0816 q^{43} -429.912 q^{45} +77.6986 q^{47} -284.675 q^{49} -206.049 q^{51} -524.666 q^{53} -1243.33 q^{55} +214.394 q^{57} -606.470 q^{59} -390.164 q^{61} -177.044 q^{63} +126.502 q^{65} +746.727 q^{67} -99.6578 q^{69} +150.049 q^{71} +707.612 q^{73} -427.751 q^{75} -512.022 q^{77} -76.5205 q^{79} +434.327 q^{81} -1173.70 q^{83} +1955.62 q^{85} -56.6643 q^{87} -1242.33 q^{89} +52.0953 q^{91} -7.74495 q^{93} -2034.82 q^{95} -1574.55 q^{97} +1554.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9	\( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9 - 65 * q^11 - 29 * q^13 - 85 * q^15 + 244 * q^17 - 312 * q^19 + 214 * q^21 - 24 * q^23 + 436 * q^25 - 57 * q^27 + 87 * q^29 + 249 * q^31 + 393 * q^33 + 718 * q^35 - 200 * q^37 + 749 * q^39 - 470 * q^41 - 97 * q^43 - 1562 * q^45 + 377 * q^47 + 383 * q^49 - 612 * q^51 - 1007 * q^53 - 1399 * q^55 + 1128 * q^57 + 364 * q^59 + 524 * q^61 - 2244 * q^63 + 301 * q^65 + 820 * q^67 + 824 * q^69 - 782 * q^71 + 1620 * q^73 + 1356 * q^75 - 158 * q^77 + 427 * q^79 + 227 * q^81 - 1520 * q^83 + 68 * q^85 - 87 * q^87 - 2474 * q^89 + 2002 * q^91 + 1807 * q^93 + 568 * q^95 - 170 * q^97 + 1362 * q^99

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