LMFDB - Embedded newform 784.4.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{113})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 59x^{2} + 60x + 674 $ x^4 - 2x^3 - 59x^2 + 60*x + 674
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.40086$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.39533 q^{3} +20.9517 q^{5} +2.10956 q^{9} +O(q^{10})$	$q-5.39533 q^{3} +20.9517 q^{5} +2.10956 q^{9} +39.6301 q^{11} +44.3153 q^{13} -113.041 q^{15} +91.9239 q^{17} -131.319 q^{19} +105.041 q^{23} +313.973 q^{25} +134.292 q^{27} -65.4110 q^{29} +75.1083 q^{31} -213.818 q^{33} -92.9728 q^{37} -239.096 q^{39} -359.695 q^{41} -103.630 q^{43} +44.1989 q^{45} -217.576 q^{47} -495.959 q^{51} +435.343 q^{53} +830.318 q^{55} +708.507 q^{57} -876.977 q^{59} -99.5279 q^{61} +928.480 q^{65} +527.863 q^{67} -566.732 q^{69} +1048.47 q^{71} +100.080 q^{73} -1693.99 q^{75} +262.384 q^{79} -781.508 q^{81} +28.5458 q^{83} +1925.96 q^{85} +352.914 q^{87} -633.626 q^{89} -405.234 q^{93} -2751.34 q^{95} +191.016 q^{97} +83.6023 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 136 q^{9}+O(q^{10}) $ 4 * q + 136 * q^9	$ 4 q + 136 q^{9} + 116 q^{11} - 112 q^{15} + 80 q^{23} + 448 q^{25} + 36 q^{29} + 436 q^{37} - 2232 q^{39} - 372 q^{43} + 780 q^{51} + 976 q^{53} + 3812 q^{57} + 780 q^{65} + 1176 q^{67} + 2408 q^{71} - 56 q^{79} + 2104 q^{81} + 4940 q^{85} + 2376 q^{93} - 4032 q^{95} + 2588 q^{99}+O(q^{100}) $ 4 * q + 136 * q^9 + 116 * q^11 - 112 * q^15 + 80 * q^23 + 448 * q^25 + 36 * q^29 + 436 * q^37 - 2232 * q^39 - 372 * q^43 + 780 * q^51 + 976 * q^53 + 3812 * q^57 + 780 * q^65 + 1176 * q^67 + 2408 * q^71 - 56 * q^79 + 2104 * q^81 + 4940 * q^85 + 2376 * q^93 - 4032 * q^95 + 2588 * 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
$3$	−5.39533	−1.03833	−0.519166	−	0.854674i	$-0.673757\pi$
$3$	−5.39533	−1.03833	−0.519166	+	0.854674i	$0.673757\pi$
$4$	0	0
$5$	20.9517	1.87397	0.936987	−	0.349363i	$-0.113602\pi$
$5$	20.9517	1.87397	0.936987	+	0.349363i	$0.113602\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.10956	0.0781319
$10$	0	0
$11$	39.6301	1.08627	0.543134	−	0.839646i	$-0.317238\pi$
$11$	39.6301	1.08627	0.543134	+	0.839646i	$0.317238\pi$
$12$	0	0
$13$	44.3153	0.945450	0.472725	−	0.881210i	$-0.343270\pi$
$13$	44.3153	0.945450	0.472725	+	0.881210i	$0.343270\pi$
$14$	0	0
$15$	−113.041	−1.94581
$16$	0	0
$17$	91.9239	1.31146	0.655730	−	0.754996i	$-0.272361\pi$
$17$	91.9239	1.31146	0.655730	+	0.754996i	$0.272361\pi$
$18$	0	0
$19$	−131.319	−1.58561	−0.792804	−	0.609477i	$-0.791379\pi$
$19$	−131.319	−1.58561	−0.792804	+	0.609477i	$0.791379\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	105.041	0.952287	0.476143	−	0.879368i	$-0.342034\pi$
$23$	105.041	0.952287	0.476143	+	0.879368i	$0.342034\pi$
$24$	0	0
$25$	313.973	2.51178
$26$	0	0
$27$	134.292	0.957204
$28$	0	0
$29$	−65.4110	−0.418846	−0.209423	−	0.977825i	$-0.567158\pi$
$29$	−65.4110	−0.418846	−0.209423	+	0.977825i	$0.567158\pi$
$30$	0	0
$31$	75.1083	0.435156	0.217578	−	0.976043i	$-0.430184\pi$
$31$	75.1083	0.435156	0.217578	+	0.976043i	$0.430184\pi$
$32$	0	0
$33$	−213.818	−1.12791
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−92.9728	−0.413098	−0.206549	−	0.978436i	$-0.566223\pi$
$37$	−92.9728	−0.413098	−0.206549	+	0.978436i	$0.566223\pi$
$38$	0	0
$39$	−239.096	−0.981691
$40$	0	0
$41$	−359.695	−1.37012	−0.685060	−	0.728487i	$-0.740224\pi$
$41$	−359.695	−1.37012	−0.685060	+	0.728487i	$0.740224\pi$
$42$	0	0
$43$	−103.630	−0.367522	−0.183761	−	0.982971i	$-0.558827\pi$
$43$	−103.630	−0.367522	−0.183761	+	0.982971i	$0.558827\pi$
$44$	0	0
$45$	44.1989	0.146417
$46$	0	0
$47$	−217.576	−0.675249	−0.337624	−	0.941281i	$-0.609623\pi$
$47$	−217.576	−0.675249	−0.337624	+	0.941281i	$0.609623\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−495.959	−1.36173
$52$	0	0
$53$	435.343	1.12828	0.564141	−	0.825679i	$-0.309208\pi$
$53$	435.343	1.12828	0.564141	+	0.825679i	$0.309208\pi$
$54$	0	0
$55$	830.318	2.03564
$56$	0	0
$57$	708.507	1.64639
$58$	0	0
$59$	−876.977	−1.93513	−0.967565	−	0.252621i	$-0.918707\pi$
$59$	−876.977	−1.93513	−0.967565	+	0.252621i	$0.918707\pi$
$60$	0	0
$61$	−99.5279	−0.208906	−0.104453	−	0.994530i	$-0.533309\pi$
$61$	−99.5279	−0.208906	−0.104453	+	0.994530i	$0.533309\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	928.480	1.77175
$66$	0	0
$67$	527.863	0.962519	0.481259	−	0.876578i	$-0.340180\pi$
$67$	527.863	0.962519	0.481259	+	0.876578i	$0.340180\pi$
$68$	0	0
$69$	−566.732	−0.988789
$70$	0	0
$71$	1048.47	1.75254	0.876268	−	0.481824i	$-0.160026\pi$
$71$	1048.47	1.75254	0.876268	+	0.481824i	$0.160026\pi$
$72$	0	0
$73$	100.080	0.160458	0.0802290	−	0.996776i	$-0.474435\pi$
$73$	100.080	0.160458	0.0802290	+	0.996776i	$0.474435\pi$
$74$	0	0
$75$	−1693.99	−2.60806
$76$	0	0
$77$	0	0
$78$	0	0
$79$	262.384	0.373677	0.186838	−	0.982391i	$-0.440176\pi$
$79$	262.384	0.373677	0.186838	+	0.982391i	$0.440176\pi$
$80$	0	0
$81$	−781.508	−1.07203
$82$	0	0
$83$	28.5458	0.0377507	0.0188754	−	0.999822i	$-0.493991\pi$
$83$	28.5458	0.0377507	0.0188754	+	0.999822i	$0.493991\pi$
$84$	0	0
$85$	1925.96	2.45764
$86$	0	0
$87$	352.914	0.434900
$88$	0	0
$89$	−633.626	−0.754654	−0.377327	−	0.926080i	$-0.623157\pi$
$89$	−633.626	−0.754654	−0.377327	+	0.926080i	$0.623157\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−405.234	−0.451836
$94$	0	0
$95$	−2751.34	−2.97139
$96$	0	0
$97$	191.016	0.199945	0.0999726	−	0.994990i	$-0.468124\pi$
$97$	191.016	0.199945	0.0999726	+	0.994990i	$0.468124\pi$
$98$	0	0
$99$	83.6023	0.0848722
$100$	0	0
$101$	1156.14	1.13901	0.569506	−	0.821987i	$-0.307135\pi$
$101$	1156.14	1.13901	0.569506	+	0.821987i	$0.307135\pi$
$102$	0	0
$103$	1324.07	1.26665	0.633324	−	0.773887i	$-0.281690\pi$
$103$	1324.07	1.26665	0.633324	+	0.773887i	$0.281690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1591.04	1.43749	0.718747	−	0.695272i	$-0.244716\pi$
$107$	1591.04	1.43749	0.718747	+	0.695272i	$0.244716\pi$
$108$	0	0
$109$	1405.91	1.23542	0.617712	−	0.786404i	$-0.288060\pi$
$109$	1405.91	1.23542	0.617712	+	0.786404i	$0.288060\pi$
$110$	0	0
$111$	501.619	0.428933
$112$	0	0
$113$	−1329.79	−1.10705	−0.553525	−	0.832833i	$-0.686718\pi$
$113$	−1329.79	−1.10705	−0.553525	+	0.832833i	$0.686718\pi$
$114$	0	0
$115$	2200.79	1.78456
$116$	0	0
$117$	93.4859	0.0738699
$118$	0	0
$119$	0	0
$120$	0	0
$121$	239.548	0.179976
$122$	0	0
$123$	1940.67	1.42264
$124$	0	0
$125$	3959.30	2.83304
$126$	0	0
$127$	564.466	0.394396	0.197198	−	0.980364i	$-0.436816\pi$
$127$	564.466	0.394396	0.197198	+	0.980364i	$0.436816\pi$
$128$	0	0
$129$	559.119	0.381610
$130$	0	0
$131$	−714.032	−0.476223	−0.238112	−	0.971238i	$-0.576528\pi$
$131$	−714.032	−0.476223	−0.238112	+	0.971238i	$0.576528\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2813.64	1.79378
$136$	0	0
$137$	−388.301	−0.242152	−0.121076	−	0.992643i	$-0.538634\pi$
$137$	−388.301	−0.242152	−0.121076	+	0.992643i	$0.538634\pi$
$138$	0	0
$139$	119.095	0.0726726	0.0363363	−	0.999340i	$-0.488431\pi$
$139$	119.095	0.0726726	0.0363363	+	0.999340i	$0.488431\pi$
$140$	0	0
$141$	1173.89	0.701132
$142$	0	0
$143$	1756.22	1.02701
$144$	0	0
$145$	−1370.47	−0.784906
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1673.67	0.920218	0.460109	−	0.887863i	$-0.347810\pi$
$149$	1673.67	0.920218	0.460109	+	0.887863i	$0.347810\pi$
$150$	0	0
$151$	1817.18	0.979337	0.489668	−	0.871909i	$-0.337118\pi$
$151$	1817.18	0.979337	0.489668	+	0.871909i	$0.337118\pi$
$152$	0	0
$153$	193.919	0.102467
$154$	0	0
$155$	1573.64	0.815472
$156$	0	0
$157$	1272.73	0.646971	0.323486	−	0.946233i	$-0.395145\pi$
$157$	1272.73	0.646971	0.323486	+	0.946233i	$0.395145\pi$
$158$	0	0
$159$	−2348.82	−1.17153
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1750.75	−0.841286	−0.420643	−	0.907226i	$-0.638195\pi$
$163$	−1750.75	−0.841286	−0.420643	+	0.907226i	$0.638195\pi$
$164$	0	0
$165$	−4479.84	−2.11367
$166$	0	0
$167$	375.000	0.173763	0.0868814	−	0.996219i	$-0.472310\pi$
$167$	375.000	0.173763	0.0868814	+	0.996219i	$0.472310\pi$
$168$	0	0
$169$	−233.153	−0.106123
$170$	0	0
$171$	−277.025	−0.123887
$172$	0	0
$173$	−855.843	−0.376119	−0.188059	−	0.982158i	$-0.560220\pi$
$173$	−855.843	−0.376119	−0.188059	+	0.982158i	$0.560220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4731.58	2.00931
$178$	0	0
$179$	3890.27	1.62443	0.812215	−	0.583359i	$-0.198262\pi$
$179$	3890.27	1.62443	0.812215	+	0.583359i	$0.198262\pi$
$180$	0	0
$181$	3195.24	1.31216	0.656079	−	0.754692i	$-0.272214\pi$
$181$	3195.24	1.31216	0.656079	+	0.754692i	$0.272214\pi$
$182$	0	0
$183$	536.985	0.216913
$184$	0	0
$185$	−1947.94	−0.774135
$186$	0	0
$187$	3642.96	1.42460
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1545.78	−0.585597	−0.292798	−	0.956174i	$-0.594586\pi$
$191$	−1545.78	−0.585597	−0.292798	+	0.956174i	$0.594586\pi$
$192$	0	0
$193$	−989.767	−0.369145	−0.184572	−	0.982819i	$-0.559090\pi$
$193$	−989.767	−0.369145	−0.184572	+	0.982819i	$0.559090\pi$
$194$	0	0
$195$	−5009.45	−1.83966
$196$	0	0
$197$	−3316.66	−1.19950	−0.599752	−	0.800186i	$-0.704734\pi$
$197$	−3316.66	−1.19950	−0.599752	+	0.800186i	$0.704734\pi$
$198$	0	0
$199$	−575.971	−0.205173	−0.102587	−	0.994724i	$-0.532712\pi$
$199$	−575.971	−0.205173	−0.102587	+	0.994724i	$0.532712\pi$
$200$	0	0
$201$	−2848.00	−0.999413
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7536.21	−2.56757
$206$	0	0
$207$	221.591	0.0744040
$208$	0	0
$209$	−5204.17	−1.72239
$210$	0	0
$211$	4647.32	1.51628	0.758138	−	0.652094i	$-0.226109\pi$
$211$	4647.32	1.51628	0.758138	+	0.652094i	$0.226109\pi$
$212$	0	0
$213$	−5656.82	−1.81971
$214$	0	0
$215$	−2171.23	−0.688727
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−539.962	−0.166608
$220$	0	0
$221$	4073.64	1.23992
$222$	0	0
$223$	1798.07	0.539944	0.269972	−	0.962868i	$-0.412986\pi$
$223$	1798.07	0.539944	0.269972	+	0.962868i	$0.412986\pi$
$224$	0	0
$225$	662.345	0.196250
$226$	0	0
$227$	2538.47	0.742220	0.371110	−	0.928589i	$-0.378977\pi$
$227$	2538.47	0.742220	0.371110	+	0.928589i	$0.378977\pi$
$228$	0	0
$229$	−4188.31	−1.20861	−0.604304	−	0.796754i	$-0.706549\pi$
$229$	−4188.31	−1.20861	−0.604304	+	0.796754i	$0.706549\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3951.84	−1.11113	−0.555565	−	0.831473i	$-0.687498\pi$
$233$	−3951.84	−1.11113	−0.555565	+	0.831473i	$0.687498\pi$
$234$	0	0
$235$	−4558.58	−1.26540
$236$	0	0
$237$	−1415.65	−0.388001
$238$	0	0
$239$	−2491.45	−0.674304	−0.337152	−	0.941450i	$-0.609464\pi$
$239$	−2491.45	−0.674304	−0.337152	+	0.941450i	$0.609464\pi$
$240$	0	0
$241$	−1259.93	−0.336760	−0.168380	−	0.985722i	$-0.553854\pi$
$241$	−1259.93	−0.336760	−0.168380	+	0.985722i	$0.553854\pi$
$242$	0	0
$243$	590.606	0.155915
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5819.42	−1.49911
$248$	0	0
$249$	−154.014	−0.0391977
$250$	0	0
$251$	−3894.99	−0.979480	−0.489740	−	0.871868i	$-0.662908\pi$
$251$	−3894.99	−0.979480	−0.489740	+	0.871868i	$0.662908\pi$
$252$	0	0
$253$	4162.80	1.03444
$254$	0	0
$255$	−10391.2	−2.55185
$256$	0	0
$257$	−491.393	−0.119269	−0.0596347	−	0.998220i	$-0.518994\pi$
$257$	−491.393	−0.119269	−0.0596347	+	0.998220i	$0.518994\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−137.989	−0.0327252
$262$	0	0
$263$	2787.65	0.653588	0.326794	−	0.945096i	$-0.394032\pi$
$263$	2787.65	0.653588	0.326794	+	0.945096i	$0.394032\pi$
$264$	0	0
$265$	9121.16	2.11437
$266$	0	0
$267$	3418.62	0.783581
$268$	0	0
$269$	−1658.09	−0.375819	−0.187909	−	0.982186i	$-0.560171\pi$
$269$	−1658.09	−0.375819	−0.187909	+	0.982186i	$0.560171\pi$
$270$	0	0
$271$	−2393.97	−0.536618	−0.268309	−	0.963333i	$-0.586465\pi$
$271$	−2393.97	−0.536618	−0.268309	+	0.963333i	$0.586465\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12442.8	2.72847
$276$	0	0
$277$	3514.33	0.762295	0.381148	−	0.924514i	$-0.375529\pi$
$277$	3514.33	0.762295	0.381148	+	0.924514i	$0.375529\pi$
$278$	0	0
$279$	158.446	0.0339996
$280$	0	0
$281$	−750.906	−0.159414	−0.0797069	−	0.996818i	$-0.525398\pi$
$281$	−750.906	−0.159414	−0.0797069	+	0.996818i	$0.525398\pi$
$282$	0	0
$283$	1364.57	0.286627	0.143314	−	0.989677i	$-0.454224\pi$
$283$	1364.57	0.286627	0.143314	+	0.989677i	$0.454224\pi$
$284$	0	0
$285$	14844.4	3.08529
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3537.00	0.719927
$290$	0	0
$291$	−1030.59	−0.207609
$292$	0	0
$293$	−9585.96	−1.91132	−0.955661	−	0.294468i	$-0.904858\pi$
$293$	−9585.96	−1.91132	−0.955661	+	0.294468i	$0.904858\pi$
$294$	0	0
$295$	−18374.1	−3.62639
$296$	0	0
$297$	5322.01	1.03978
$298$	0	0
$299$	4654.93	0.900340
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6237.75	−1.18267
$304$	0	0
$305$	−2085.28	−0.391484
$306$	0	0
$307$	170.222	0.0316451	0.0158226	−	0.999875i	$-0.494963\pi$
$307$	170.222	0.0316451	0.0158226	+	0.999875i	$0.494963\pi$
$308$	0	0
$309$	−7143.81	−1.31520
$310$	0	0
$311$	−2084.59	−0.380084	−0.190042	−	0.981776i	$-0.560862\pi$
$311$	−2084.59	−0.380084	−0.190042	+	0.981776i	$0.560862\pi$
$312$	0	0
$313$	8112.50	1.46500	0.732501	−	0.680766i	$-0.238353\pi$
$313$	8112.50	1.46500	0.732501	+	0.680766i	$0.238353\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2621.87	0.464539	0.232269	−	0.972651i	$-0.425385\pi$
$317$	2621.87	0.464539	0.232269	+	0.972651i	$0.425385\pi$
$318$	0	0
$319$	−2592.25	−0.454978
$320$	0	0
$321$	−8584.20	−1.49260
$322$	0	0
$323$	−12071.3	−2.07946
$324$	0	0
$325$	13913.8	2.37477
$326$	0	0
$327$	−7585.32	−1.28278
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9265.11	−1.53854	−0.769270	−	0.638924i	$-0.779380\pi$
$331$	−9265.11	−1.53854	−0.769270	+	0.638924i	$0.779380\pi$
$332$	0	0
$333$	−196.132	−0.0322762
$334$	0	0
$335$	11059.6	1.80374
$336$	0	0
$337$	−6530.57	−1.05562	−0.527808	−	0.849364i	$-0.676986\pi$
$337$	−6530.57	−1.05562	−0.527808	+	0.849364i	$0.676986\pi$
$338$	0	0
$339$	7174.68	1.14948
$340$	0	0
$341$	2976.55	0.472696
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−11874.0	−1.85297
$346$	0	0
$347$	−495.580	−0.0766689	−0.0383344	−	0.999265i	$-0.512205\pi$
$347$	−495.580	−0.0766689	−0.0383344	+	0.999265i	$0.512205\pi$
$348$	0	0
$349$	1755.80	0.269301	0.134650	−	0.990893i	$-0.457009\pi$
$349$	1755.80	0.269301	0.134650	+	0.990893i	$0.457009\pi$
$350$	0	0
$351$	5951.19	0.904989
$352$	0	0
$353$	−7094.55	−1.06970	−0.534851	−	0.844947i	$-0.679632\pi$
$353$	−7094.55	−1.06970	−0.534851	+	0.844947i	$0.679632\pi$
$354$	0	0
$355$	21967.1	3.28421
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2441.18	0.358887	0.179444	−	0.983768i	$-0.442570\pi$
$359$	2441.18	0.358887	0.179444	+	0.983768i	$0.442570\pi$
$360$	0	0
$361$	10385.6	1.51415
$362$	0	0
$363$	−1292.44	−0.186875
$364$	0	0
$365$	2096.83	0.300694
$366$	0	0
$367$	−6810.86	−0.968730	−0.484365	−	0.874866i	$-0.660949\pi$
$367$	−6810.86	−0.968730	−0.484365	+	0.874866i	$0.660949\pi$
$368$	0	0
$369$	−758.799	−0.107050
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13215.6	1.83452	0.917262	−	0.398283i	$-0.130394\pi$
$373$	13215.6	1.83452	0.917262	+	0.398283i	$0.130394\pi$
$374$	0	0
$375$	−21361.7	−2.94164
$376$	0	0
$377$	−2898.71	−0.395998
$378$	0	0
$379$	4735.42	0.641800	0.320900	−	0.947113i	$-0.396015\pi$
$379$	4735.42	0.641800	0.320900	+	0.947113i	$0.396015\pi$
$380$	0	0
$381$	−3045.48	−0.409514
$382$	0	0
$383$	4075.94	0.543789	0.271894	−	0.962327i	$-0.412350\pi$
$383$	4075.94	0.543789	0.271894	+	0.962327i	$0.412350\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−218.614	−0.0287152
$388$	0	0
$389$	−4768.27	−0.621493	−0.310746	−	0.950493i	$-0.600579\pi$
$389$	−4768.27	−0.621493	−0.310746	+	0.950493i	$0.600579\pi$
$390$	0	0
$391$	9655.79	1.24889
$392$	0	0
$393$	3852.44	0.494478
$394$	0	0
$395$	5497.38	0.700261
$396$	0	0
$397$	−8257.55	−1.04392	−0.521958	−	0.852971i	$-0.674798\pi$
$397$	−8257.55	−1.04392	−0.521958	+	0.852971i	$0.674798\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4619.22	0.575244	0.287622	−	0.957744i	$-0.407135\pi$
$401$	4619.22	0.575244	0.287622	+	0.957744i	$0.407135\pi$
$402$	0	0
$403$	3328.45	0.411419
$404$	0	0
$405$	−16373.9	−2.00895
$406$	0	0
$407$	−3684.52	−0.448735
$408$	0	0
$409$	7643.11	0.924028	0.462014	−	0.886873i	$-0.347127\pi$
$409$	7643.11	0.924028	0.462014	+	0.886873i	$0.347127\pi$
$410$	0	0
$411$	2095.01	0.251434
$412$	0	0
$413$	0	0
$414$	0	0
$415$	598.082	0.0707439
$416$	0	0
$417$	−642.556	−0.0754583
$418$	0	0
$419$	1168.23	0.136209	0.0681046	−	0.997678i	$-0.478305\pi$
$419$	1168.23	0.136209	0.0681046	+	0.997678i	$0.478305\pi$
$420$	0	0
$421$	−3705.30	−0.428943	−0.214472	−	0.976730i	$-0.568803\pi$
$421$	−3705.30	−0.428943	−0.214472	+	0.976730i	$0.568803\pi$
$422$	0	0
$423$	−458.990	−0.0527585
$424$	0	0
$425$	28861.6	3.29410
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9475.39	−1.06638
$430$	0	0
$431$	−5711.40	−0.638303	−0.319151	−	0.947704i	$-0.603398\pi$
$431$	−5711.40	−0.638303	−0.319151	+	0.947704i	$0.603398\pi$
$432$	0	0
$433$	−4682.43	−0.519684	−0.259842	−	0.965651i	$-0.583670\pi$
$433$	−4682.43	−0.519684	−0.259842	+	0.965651i	$0.583670\pi$
$434$	0	0
$435$	7394.14	0.814993
$436$	0	0
$437$	−13793.9	−1.50995
$438$	0	0
$439$	4031.86	0.438338	0.219169	−	0.975687i	$-0.429665\pi$
$439$	4031.86	0.438338	0.219169	+	0.975687i	$0.429665\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2595.64	−0.278381	−0.139190	−	0.990266i	$-0.544450\pi$
$443$	−2595.64	−0.278381	−0.139190	+	0.990266i	$0.544450\pi$
$444$	0	0
$445$	−13275.5	−1.41420
$446$	0	0
$447$	−9030.00	−0.955491
$448$	0	0
$449$	206.770	0.0217329	0.0108665	−	0.999941i	$-0.496541\pi$
$449$	206.770	0.0217329	0.0108665	+	0.999941i	$0.496541\pi$
$450$	0	0
$451$	−14254.8	−1.48832
$452$	0	0
$453$	−9804.27	−1.01688
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7624.95	−0.780482	−0.390241	−	0.920713i	$-0.627608\pi$
$457$	−7624.95	−0.780482	−0.390241	+	0.920713i	$0.627608\pi$
$458$	0	0
$459$	12344.6	1.25534
$460$	0	0
$461$	15421.6	1.55804	0.779021	−	0.626997i	$-0.215716\pi$
$461$	15421.6	1.55804	0.779021	+	0.626997i	$0.215716\pi$
$462$	0	0
$463$	−17128.8	−1.71932	−0.859659	−	0.510868i	$-0.829324\pi$
$463$	−17128.8	−1.71932	−0.859659	+	0.510868i	$0.829324\pi$
$464$	0	0
$465$	−8490.33	−0.846730
$466$	0	0
$467$	−15706.7	−1.55636	−0.778179	−	0.628042i	$-0.783856\pi$
$467$	−15706.7	−1.55636	−0.778179	+	0.628042i	$0.783856\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6866.77	−0.671771
$472$	0	0
$473$	−4106.88	−0.399227
$474$	0	0
$475$	−41230.4	−3.98270
$476$	0	0
$477$	918.383	0.0881548
$478$	0	0
$479$	−20166.9	−1.92370	−0.961849	−	0.273582i	$-0.911792\pi$
$479$	−20166.9	−1.92370	−0.961849	+	0.273582i	$0.911792\pi$
$480$	0	0
$481$	−4120.12	−0.390564
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4002.10	0.374692
$486$	0	0
$487$	−1162.84	−0.108200	−0.0541001	−	0.998536i	$-0.517229\pi$
$487$	−1162.84	−0.108200	−0.0541001	+	0.998536i	$0.517229\pi$
$488$	0	0
$489$	9445.89	0.873534
$490$	0	0
$491$	17831.8	1.63897	0.819487	−	0.573098i	$-0.194259\pi$
$491$	17831.8	1.63897	0.819487	+	0.573098i	$0.194259\pi$
$492$	0	0
$493$	−6012.83	−0.549299
$494$	0	0
$495$	1751.61	0.159048
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2760.13	−0.247616	−0.123808	−	0.992306i	$-0.539511\pi$
$499$	−2760.13	−0.247616	−0.123808	+	0.992306i	$0.539511\pi$
$500$	0	0
$501$	−2023.25	−0.180423
$502$	0	0
$503$	16914.7	1.49938	0.749691	−	0.661788i	$-0.230202\pi$
$503$	16914.7	1.49938	0.749691	+	0.661788i	$0.230202\pi$
$504$	0	0
$505$	24223.1	2.13448
$506$	0	0
$507$	1257.94	0.110191
$508$	0	0
$509$	6910.50	0.601773	0.300886	−	0.953660i	$-0.402717\pi$
$509$	6910.50	0.601773	0.300886	+	0.953660i	$0.402717\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−17635.0	−1.51775
$514$	0	0
$515$	27741.5	2.37367
$516$	0	0
$517$	−8622.56	−0.733500
$518$	0	0
$519$	4617.55	0.390536
$520$	0	0
$521$	11084.4	0.932084	0.466042	−	0.884763i	$-0.345680\pi$
$521$	11084.4	0.932084	0.466042	+	0.884763i	$0.345680\pi$
$522$	0	0
$523$	−18190.0	−1.52083	−0.760416	−	0.649436i	$-0.775005\pi$
$523$	−18190.0	−1.52083	−0.760416	+	0.649436i	$0.775005\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6904.24	0.570690
$528$	0	0
$529$	−1133.35	−0.0931498
$530$	0	0
$531$	−1850.04	−0.151196
$532$	0	0
$533$	−15940.0	−1.29538
$534$	0	0
$535$	33335.0	2.69383
$536$	0	0
$537$	−20989.3	−1.68670
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5141.34	0.408583	0.204292	−	0.978910i	$-0.434511\pi$
$541$	5141.34	0.408583	0.204292	+	0.978910i	$0.434511\pi$
$542$	0	0
$543$	−17239.4	−1.36245
$544$	0	0
$545$	29456.1	2.31515
$546$	0	0
$547$	−17385.5	−1.35896	−0.679478	−	0.733695i	$-0.737794\pi$
$547$	−17385.5	−1.35896	−0.679478	+	0.733695i	$0.737794\pi$
$548$	0	0
$549$	−209.960	−0.0163222
$550$	0	0
$551$	8589.68	0.664125
$552$	0	0
$553$	0	0
$554$	0	0
$555$	10509.8	0.803809
$556$	0	0
$557$	22744.8	1.73021	0.865105	−	0.501591i	$-0.167252\pi$
$557$	22744.8	1.73021	0.865105	+	0.501591i	$0.167252\pi$
$558$	0	0
$559$	−4592.40	−0.347474
$560$	0	0
$561$	−19654.9	−1.47920
$562$	0	0
$563$	19606.0	1.46767	0.733833	−	0.679330i	$-0.237729\pi$
$563$	19606.0	1.46767	0.733833	+	0.679330i	$0.237729\pi$
$564$	0	0
$565$	−27861.4	−2.07458
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10750.9	−0.792093	−0.396046	−	0.918231i	$-0.629618\pi$
$569$	−10750.9	−0.792093	−0.396046	+	0.918231i	$0.629618\pi$
$570$	0	0
$571$	19753.9	1.44777	0.723883	−	0.689923i	$-0.242356\pi$
$571$	19753.9	1.44777	0.723883	+	0.689923i	$0.242356\pi$
$572$	0	0
$573$	8340.01	0.608043
$574$	0	0
$575$	32980.1	2.39194
$576$	0	0
$577$	−3522.99	−0.254183	−0.127092	−	0.991891i	$-0.540564\pi$
$577$	−3522.99	−0.254183	−0.127092	+	0.991891i	$0.540564\pi$
$578$	0	0
$579$	5340.12	0.383295
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17252.7	1.22561
$584$	0	0
$585$	1958.69	0.138430
$586$	0	0
$587$	−9495.13	−0.667642	−0.333821	−	0.942636i	$-0.608338\pi$
$587$	−9495.13	−0.667642	−0.333821	+	0.942636i	$0.608338\pi$
$588$	0	0
$589$	−9863.11	−0.689987
$590$	0	0
$591$	17894.5	1.24548
$592$	0	0
$593$	−11510.1	−0.797071	−0.398535	−	0.917153i	$-0.630481\pi$
$593$	−11510.1	−0.797071	−0.398535	+	0.917153i	$0.630481\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3107.55	0.213038
$598$	0	0
$599$	7559.63	0.515656	0.257828	−	0.966191i	$-0.416993\pi$
$599$	7559.63	0.515656	0.257828	+	0.966191i	$0.416993\pi$
$600$	0	0
$601$	11707.1	0.794581	0.397291	−	0.917693i	$-0.369951\pi$
$601$	11707.1	0.794581	0.397291	+	0.917693i	$0.369951\pi$
$602$	0	0
$603$	1113.56	0.0752035
$604$	0	0
$605$	5018.94	0.337271
$606$	0	0
$607$	643.054	0.0429996	0.0214998	−	0.999769i	$-0.493156\pi$
$607$	643.054	0.0429996	0.0214998	+	0.999769i	$0.493156\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9641.94	−0.638414
$612$	0	0
$613$	−11798.5	−0.777382	−0.388691	−	0.921368i	$-0.627073\pi$
$613$	−11798.5	−0.777382	−0.388691	+	0.921368i	$0.627073\pi$
$614$	0	0
$615$	40660.3	2.66599
$616$	0	0
$617$	−22419.9	−1.46287	−0.731434	−	0.681912i	$-0.761149\pi$
$617$	−22419.9	−1.46287	−0.731434	+	0.681912i	$0.761149\pi$
$618$	0	0
$619$	−19371.1	−1.25782	−0.628909	−	0.777479i	$-0.716498\pi$
$619$	−19371.1	−1.25782	−0.628909	+	0.777479i	$0.716498\pi$
$620$	0	0
$621$	14106.2	0.911533
$622$	0	0
$623$	0	0
$624$	0	0
$625$	43707.3	2.79727
$626$	0	0
$627$	28078.2	1.78841
$628$	0	0
$629$	−8546.42	−0.541762
$630$	0	0
$631$	−26001.3	−1.64041	−0.820204	−	0.572072i	$-0.806140\pi$
$631$	−26001.3	−1.64041	−0.820204	+	0.572072i	$0.806140\pi$
$632$	0	0
$633$	−25073.8	−1.57440
$634$	0	0
$635$	11826.5	0.739088
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2211.80	0.136929
$640$	0	0
$641$	14290.6	0.880572	0.440286	−	0.897858i	$-0.354877\pi$
$641$	14290.6	0.880572	0.440286	+	0.897858i	$0.354877\pi$
$642$	0	0
$643$	−12643.0	−0.775414	−0.387707	−	0.921783i	$-0.626733\pi$
$643$	−12643.0	−0.775414	−0.387707	+	0.921783i	$0.626733\pi$
$644$	0	0
$645$	11714.5	0.715127
$646$	0	0
$647$	−29350.3	−1.78343	−0.891715	−	0.452597i	$-0.850498\pi$
$647$	−29350.3	−1.78343	−0.891715	+	0.452597i	$0.850498\pi$
$648$	0	0
$649$	−34754.7	−2.10207
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20227.2	1.21217	0.606087	−	0.795398i	$-0.292738\pi$
$653$	20227.2	1.21217	0.606087	+	0.795398i	$0.292738\pi$
$654$	0	0
$655$	−14960.2	−0.892431
$656$	0	0
$657$	211.124	0.0125369
$658$	0	0
$659$	8525.05	0.503928	0.251964	−	0.967737i	$-0.418924\pi$
$659$	8525.05	0.503928	0.251964	+	0.967737i	$0.418924\pi$
$660$	0	0
$661$	20811.3	1.22461	0.612305	−	0.790622i	$-0.290242\pi$
$661$	20811.3	1.22461	0.612305	+	0.790622i	$0.290242\pi$
$662$	0	0
$663$	−21978.6	−1.28745
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6870.85	−0.398861
$668$	0	0
$669$	−9701.16	−0.560640
$670$	0	0
$671$	−3944.30	−0.226927
$672$	0	0
$673$	31955.2	1.83029	0.915143	−	0.403128i	$-0.132077\pi$
$673$	31955.2	1.83029	0.915143	+	0.403128i	$0.132077\pi$
$674$	0	0
$675$	42164.1	2.40429
$676$	0	0
$677$	13912.6	0.789816	0.394908	−	0.918721i	$-0.370777\pi$
$677$	13912.6	0.789816	0.394908	+	0.918721i	$0.370777\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−13695.9	−0.770670
$682$	0	0
$683$	13041.2	0.730610	0.365305	−	0.930888i	$-0.380965\pi$
$683$	13041.2	0.730610	0.365305	+	0.930888i	$0.380965\pi$
$684$	0	0
$685$	−8135.57	−0.453787
$686$	0	0
$687$	22597.3	1.25494
$688$	0	0
$689$	19292.3	1.06673
$690$	0	0
$691$	−17292.6	−0.952012	−0.476006	−	0.879442i	$-0.657916\pi$
$691$	−17292.6	−0.952012	−0.476006	+	0.879442i	$0.657916\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2495.24	0.136187
$696$	0	0
$697$	−33064.5	−1.79686
$698$	0	0
$699$	21321.5	1.15372
$700$	0	0
$701$	21720.2	1.17027	0.585136	−	0.810935i	$-0.301041\pi$
$701$	21720.2	1.17027	0.585136	+	0.810935i	$0.301041\pi$
$702$	0	0
$703$	12209.0	0.655011
$704$	0	0
$705$	24595.0	1.31390
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13345.8	−0.706928	−0.353464	−	0.935448i	$-0.614996\pi$
$709$	−13345.8	−0.706928	−0.353464	+	0.935448i	$0.614996\pi$
$710$	0	0
$711$	553.515	0.0291961
$712$	0	0
$713$	7889.46	0.414394
$714$	0	0
$715$	36795.8	1.92459
$716$	0	0
$717$	13442.2	0.700151
$718$	0	0
$719$	−33463.2	−1.73570	−0.867850	−	0.496827i	$-0.834498\pi$
$719$	−33463.2	−1.73570	−0.867850	+	0.496827i	$0.834498\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6797.73	0.349668
$724$	0	0
$725$	−20537.3	−1.05205
$726$	0	0
$727$	−8602.89	−0.438877	−0.219438	−	0.975626i	$-0.570423\pi$
$727$	−8602.89	−0.438877	−0.219438	+	0.975626i	$0.570423\pi$
$728$	0	0
$729$	17914.2	0.910136
$730$	0	0
$731$	−9526.09	−0.481990
$732$	0	0
$733$	−19341.5	−0.974615	−0.487307	−	0.873230i	$-0.662021\pi$
$733$	−19341.5	−0.974615	−0.487307	+	0.873230i	$0.662021\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20919.3	1.04555
$738$	0	0
$739$	−34678.1	−1.72619	−0.863094	−	0.505043i	$-0.831477\pi$
$739$	−34678.1	−1.72619	−0.863094	+	0.505043i	$0.831477\pi$
$740$	0	0
$741$	31397.7	1.55658
$742$	0	0
$743$	1394.70	0.0688650	0.0344325	−	0.999407i	$-0.489038\pi$
$743$	1394.70	0.0688650	0.0344325	+	0.999407i	$0.489038\pi$
$744$	0	0
$745$	35066.2	1.72447
$746$	0	0
$747$	60.2191	0.00294954
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17002.5	−0.826137	−0.413069	−	0.910700i	$-0.635543\pi$
$751$	−17002.5	−0.826137	−0.413069	+	0.910700i	$0.635543\pi$
$752$	0	0
$753$	21014.7	1.01703
$754$	0	0
$755$	38072.9	1.83525
$756$	0	0
$757$	28020.1	1.34532	0.672660	−	0.739952i	$-0.265152\pi$
$757$	28020.1	1.34532	0.672660	+	0.739952i	$0.265152\pi$
$758$	0	0
$759$	−22459.7	−1.07409
$760$	0	0
$761$	12467.0	0.593859	0.296930	−	0.954899i	$-0.404037\pi$
$761$	12467.0	0.593859	0.296930	+	0.954899i	$0.404037\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4062.93	0.192020
$766$	0	0
$767$	−38863.5	−1.82957
$768$	0	0
$769$	−35493.5	−1.66441	−0.832203	−	0.554472i	$-0.812920\pi$
$769$	−35493.5	−1.66441	−0.832203	+	0.554472i	$0.812920\pi$
$770$	0	0
$771$	2651.23	0.123841
$772$	0	0
$773$	−35934.5	−1.67202	−0.836012	−	0.548712i	$-0.815119\pi$
$773$	−35934.5	−1.67202	−0.836012	+	0.548712i	$0.815119\pi$
$774$	0	0
$775$	23581.9	1.09302
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47234.6	2.17247
$780$	0	0
$781$	41550.9	1.90372
$782$	0	0
$783$	−8784.18	−0.400921
$784$	0	0
$785$	26665.7	1.21241
$786$	0	0
$787$	5305.90	0.240324	0.120162	−	0.992754i	$-0.461659\pi$
$787$	5305.90	0.240324	0.120162	+	0.992754i	$0.461659\pi$
$788$	0	0
$789$	−15040.3	−0.678641
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4410.61	−0.197510
$794$	0	0
$795$	−49211.6	−2.19542
$796$	0	0
$797$	−2903.22	−0.129030	−0.0645152	−	0.997917i	$-0.520550\pi$
$797$	−2903.22	−0.129030	−0.0645152	+	0.997917i	$0.520550\pi$
$798$	0	0
$799$	−20000.4	−0.885561
$800$	0	0
$801$	−1336.67	−0.0589626
$802$	0	0
$803$	3966.17	0.174300
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8945.92	0.390225
$808$	0	0
$809$	12823.2	0.557282	0.278641	−	0.960395i	$-0.410116\pi$
$809$	12823.2	0.557282	0.278641	+	0.960395i	$0.410116\pi$
$810$	0	0
$811$	−17453.7	−0.755714	−0.377857	−	0.925864i	$-0.623339\pi$
$811$	−17453.7	−0.755714	−0.377857	+	0.925864i	$0.623339\pi$
$812$	0	0
$813$	12916.3	0.557188
$814$	0	0
$815$	−36681.2	−1.57655
$816$	0	0
$817$	13608.6	0.582746
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46985.9	−1.99734	−0.998671	−	0.0515324i	$-0.983589\pi$
$821$	−46985.9	−1.99734	−0.998671	+	0.0515324i	$0.983589\pi$
$822$	0	0
$823$	−8265.75	−0.350092	−0.175046	−	0.984560i	$-0.556007\pi$
$823$	−8265.75	−0.350092	−0.175046	+	0.984560i	$0.556007\pi$
$824$	0	0
$825$	−67132.9	−2.83305
$826$	0	0
$827$	7412.42	0.311675	0.155837	−	0.987783i	$-0.450192\pi$
$827$	7412.42	0.311675	0.155837	+	0.987783i	$0.450192\pi$
$828$	0	0
$829$	−19478.9	−0.816081	−0.408041	−	0.912964i	$-0.633788\pi$
$829$	−19478.9	−0.816081	−0.408041	+	0.912964i	$0.633788\pi$
$830$	0	0
$831$	−18961.0	−0.791515
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7856.89	0.325627
$836$	0	0
$837$	10086.4	0.416533
$838$	0	0
$839$	1592.37	0.0655241	0.0327621	−	0.999463i	$-0.489570\pi$
$839$	1592.37	0.0655241	0.0327621	+	0.999463i	$0.489570\pi$
$840$	0	0
$841$	−20110.4	−0.824568
$842$	0	0
$843$	4051.38	0.165524
$844$	0	0
$845$	−4884.95	−0.198873
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7362.32	−0.297614
$850$	0	0
$851$	−9765.97	−0.393388
$852$	0	0
$853$	33764.8	1.35532	0.677658	−	0.735377i	$-0.262995\pi$
$853$	33764.8	1.35532	0.677658	+	0.735377i	$0.262995\pi$
$854$	0	0
$855$	−5804.13	−0.232160
$856$	0	0
$857$	−26925.2	−1.07322	−0.536608	−	0.843832i	$-0.680295\pi$
$857$	−26925.2	−1.07322	−0.536608	+	0.843832i	$0.680295\pi$
$858$	0	0
$859$	12786.2	0.507870	0.253935	−	0.967221i	$-0.418275\pi$
$859$	12786.2	0.507870	0.253935	+	0.967221i	$0.418275\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15300.0	−0.603498	−0.301749	−	0.953387i	$-0.597570\pi$
$863$	−15300.0	−0.603498	−0.301749	+	0.953387i	$0.597570\pi$
$864$	0	0
$865$	−17931.3	−0.704837
$866$	0	0
$867$	−19083.3	−0.747522
$868$	0	0
$869$	10398.3	0.405913
$870$	0	0
$871$	23392.4	0.910014
$872$	0	0
$873$	402.959	0.0156221
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26846.7	1.03369	0.516846	−	0.856078i	$-0.327106\pi$
$877$	26846.7	1.03369	0.516846	+	0.856078i	$0.327106\pi$
$878$	0	0
$879$	51719.4	1.98459
$880$	0	0
$881$	5695.09	0.217789	0.108895	−	0.994053i	$-0.465269\pi$
$881$	5695.09	0.217789	0.108895	+	0.994053i	$0.465269\pi$
$882$	0	0
$883$	−8723.47	−0.332467	−0.166233	−	0.986086i	$-0.553160\pi$
$883$	−8723.47	−0.332467	−0.166233	+	0.986086i	$0.553160\pi$
$884$	0	0
$885$	99134.5	3.76539
$886$	0	0
$887$	−21734.5	−0.822741	−0.411371	−	0.911468i	$-0.634950\pi$
$887$	−21734.5	−0.822741	−0.411371	+	0.911468i	$0.634950\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−30971.3	−1.16451
$892$	0	0
$893$	28571.7	1.07068
$894$	0	0
$895$	81507.8	3.04414
$896$	0	0
$897$	−25114.9	−0.934851
$898$	0	0
$899$	−4912.91	−0.182263
$900$	0	0
$901$	40018.4	1.47970
$902$	0	0
$903$	0	0
$904$	0	0
$905$	66945.7	2.45895
$906$	0	0
$907$	−36924.9	−1.35179	−0.675893	−	0.737000i	$-0.736242\pi$
$907$	−36924.9	−1.35179	−0.675893	+	0.737000i	$0.736242\pi$
$908$	0	0
$909$	2438.95	0.0889932
$910$	0	0
$911$	−11281.1	−0.410274	−0.205137	−	0.978733i	$-0.565764\pi$
$911$	−11281.1	−0.410274	−0.205137	+	0.978733i	$0.565764\pi$
$912$	0	0
$913$	1131.27	0.0410073
$914$	0	0
$915$	11250.7	0.406490
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11131.6	0.399561	0.199780	−	0.979841i	$-0.435977\pi$
$919$	11131.6	0.399561	0.199780	+	0.979841i	$0.435977\pi$
$920$	0	0
$921$	−918.401	−0.0328581
$922$	0	0
$923$	46463.1	1.65694
$924$	0	0
$925$	−29190.9	−1.03761
$926$	0	0
$927$	2793.21	0.0989657
$928$	0	0
$929$	3237.43	0.114334	0.0571672	−	0.998365i	$-0.481793\pi$
$929$	3237.43	0.114334	0.0571672	+	0.998365i	$0.481793\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11247.0	0.394653
$934$	0	0
$935$	76326.1	2.66966
$936$	0	0
$937$	−7596.60	−0.264856	−0.132428	−	0.991193i	$-0.542277\pi$
$937$	−7596.60	−0.264856	−0.132428	+	0.991193i	$0.542277\pi$
$938$	0	0
$939$	−43769.6	−1.52116
$940$	0	0
$941$	−46173.5	−1.59959	−0.799795	−	0.600273i	$-0.795059\pi$
$941$	−46173.5	−1.59959	−0.799795	+	0.600273i	$0.795059\pi$
$942$	0	0
$943$	−37782.8	−1.30475
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37030.9	1.27069	0.635345	−	0.772229i	$-0.280858\pi$
$947$	37030.9	1.27069	0.635345	+	0.772229i	$0.280858\pi$
$948$	0	0
$949$	4435.06	0.151705
$950$	0	0
$951$	−14145.8	−0.482345
$952$	0	0
$953$	−9117.27	−0.309903	−0.154951	−	0.987922i	$-0.549522\pi$
$953$	−9117.27	−0.309903	−0.154951	+	0.987922i	$0.549522\pi$
$954$	0	0
$955$	−32386.8	−1.09739
$956$	0	0
$957$	13986.0	0.472418
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24149.7	−0.810639
$962$	0	0
$963$	3356.40	0.112314
$964$	0	0
$965$	−20737.3	−0.691768
$966$	0	0
$967$	−27287.4	−0.907448	−0.453724	−	0.891142i	$-0.649905\pi$
$967$	−27287.4	−0.907448	−0.453724	+	0.891142i	$0.649905\pi$
$968$	0	0
$969$	65128.7	2.15917
$970$	0	0
$971$	42639.8	1.40925	0.704623	−	0.709582i	$-0.251116\pi$
$971$	42639.8	1.40925	0.704623	+	0.709582i	$0.251116\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−75069.5	−2.46579
$976$	0	0
$977$	7737.38	0.253368	0.126684	−	0.991943i	$-0.459567\pi$
$977$	7737.38	0.253368	0.126684	+	0.991943i	$0.459567\pi$
$978$	0	0
$979$	−25110.7	−0.819756
$980$	0	0
$981$	2965.84	0.0965261
$982$	0	0
$983$	28837.9	0.935692	0.467846	−	0.883810i	$-0.345030\pi$
$983$	28837.9	0.935692	0.467846	+	0.883810i	$0.345030\pi$
$984$	0	0
$985$	−69489.6	−2.24784
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10885.4	−0.349986
$990$	0	0
$991$	45025.4	1.44327	0.721634	−	0.692275i	$-0.243391\pi$
$991$	45025.4	1.44327	0.721634	+	0.692275i	$0.243391\pi$
$992$	0	0
$993$	49988.3	1.59751
$994$	0	0
$995$	−12067.6	−0.384490
$996$	0	0
$997$	22268.3	0.707365	0.353682	−	0.935366i	$-0.384929\pi$
$997$	22268.3	0.707365	0.353682	+	0.935366i	$0.384929\pi$
$998$	0	0
$999$	−12485.5	−0.395419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bh.1.2		4
4.3	odd	2		392.4.a.n.1.3	yes	4
7.6	odd	2	inner	784.4.a.bh.1.3		4
28.3	even	6		392.4.i.o.177.3		8
28.11	odd	6		392.4.i.o.177.2		8
28.19	even	6		392.4.i.o.361.3		8
28.23	odd	6		392.4.i.o.361.2		8
28.27	even	2		392.4.a.n.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.n.1.2	✓	4	28.27	even	2
392.4.a.n.1.3	yes	4	4.3	odd	2
392.4.i.o.177.2		8	28.11	odd	6
392.4.i.o.177.3		8	28.3	even	6
392.4.i.o.361.2		8	28.23	odd	6
392.4.i.o.361.3		8	28.19	even	6
784.4.a.bh.1.2		4	1.1	even	1	trivial
784.4.a.bh.1.3		4	7.6	odd	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-5.39533 q^{3} +20.9517 q^{5} +2.10956 q^{9} +O(q^{10})\)	\(q-5.39533 q^{3} +20.9517 q^{5} +2.10956 q^{9} +39.6301 q^{11} +44.3153 q^{13} -113.041 q^{15} +91.9239 q^{17} -131.319 q^{19} +105.041 q^{23} +313.973 q^{25} +134.292 q^{27} -65.4110 q^{29} +75.1083 q^{31} -213.818 q^{33} -92.9728 q^{37} -239.096 q^{39} -359.695 q^{41} -103.630 q^{43} +44.1989 q^{45} -217.576 q^{47} -495.959 q^{51} +435.343 q^{53} +830.318 q^{55} +708.507 q^{57} -876.977 q^{59} -99.5279 q^{61} +928.480 q^{65} +527.863 q^{67} -566.732 q^{69} +1048.47 q^{71} +100.080 q^{73} -1693.99 q^{75} +262.384 q^{79} -781.508 q^{81} +28.5458 q^{83} +1925.96 q^{85} +352.914 q^{87} -633.626 q^{89} -405.234 q^{93} -2751.34 q^{95} +191.016 q^{97} +83.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 136 q^{9}+O(q^{10}) \) 4 * q + 136 * q^9	\( 4 q + 136 q^{9} + 116 q^{11} - 112 q^{15} + 80 q^{23} + 448 q^{25} + 36 q^{29} + 436 q^{37} - 2232 q^{39} - 372 q^{43} + 780 q^{51} + 976 q^{53} + 3812 q^{57} + 780 q^{65} + 1176 q^{67} + 2408 q^{71} - 56 q^{79} + 2104 q^{81} + 4940 q^{85} + 2376 q^{93} - 4032 q^{95} + 2588 q^{99}+O(q^{100}) \) 4 * q + 136 * q^9 + 116 * q^11 - 112 * q^15 + 80 * q^23 + 448 * q^25 + 36 * q^29 + 436 * q^37 - 2232 * q^39 - 372 * q^43 + 780 * q^51 + 976 * q^53 + 3812 * q^57 + 780 * q^65 + 1176 * q^67 + 2408 * q^71 - 56 * q^79 + 2104 * q^81 + 4940 * q^85 + 2376 * q^93 - 4032 * q^95 + 2588 * q^99

Introduction

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