LMFDB - Embedded newform 784.4.a.bd.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.1929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 13 $ x^3 - x^2 - 10*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27144$ 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)

$f(q)$	$=$	$q+0.138208 q^{3} +10.0858 q^{5} -26.9809 q^{9} +O(q^{10})$	$q+0.138208 q^{3} +10.0858 q^{5} -26.9809 q^{9} +18.6711 q^{11} -10.0850 q^{13} +1.39394 q^{15} -73.6394 q^{17} +76.0801 q^{19} -146.864 q^{23} -23.2772 q^{25} -7.46061 q^{27} -157.665 q^{29} +69.8134 q^{31} +2.58051 q^{33} +309.673 q^{37} -1.39383 q^{39} -482.865 q^{41} -17.3692 q^{43} -272.123 q^{45} -346.690 q^{47} -10.1776 q^{51} -73.4511 q^{53} +188.313 q^{55} +10.5149 q^{57} +704.382 q^{59} -841.393 q^{61} -101.715 q^{65} +891.809 q^{67} -20.2978 q^{69} +525.472 q^{71} -376.060 q^{73} -3.21710 q^{75} -1233.50 q^{79} +727.453 q^{81} -771.757 q^{83} -742.710 q^{85} -21.7907 q^{87} +175.379 q^{89} +9.64879 q^{93} +767.327 q^{95} -1313.94 q^{97} -503.764 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 13 q^{5} + 50 q^{9}+O(q^{10}) $ 3 * q + q^3 - 13 * q^5 + 50 * q^9	$ 3 q + q^{3} - 13 q^{5} + 50 q^{9} + 11 q^{11} - 70 q^{13} + 41 q^{15} - 97 q^{17} - 81 q^{19} - 191 q^{23} + 12 q^{25} + 115 q^{27} - 162 q^{29} + 597 q^{31} - 343 q^{33} + 217 q^{37} + 806 q^{39} - 698 q^{41} + 308 q^{43} - 1118 q^{45} + 139 q^{47} - 1475 q^{51} + 197 q^{53} + 147 q^{55} - 1147 q^{57} + 353 q^{59} - 449 q^{61} + 906 q^{65} + 519 q^{67} - 2557 q^{69} - 224 q^{71} - 1701 q^{73} - 1132 q^{75} - 1143 q^{79} + 251 q^{81} - 1380 q^{83} - 1025 q^{85} + 1458 q^{87} - 1749 q^{89} - 601 q^{93} + 2167 q^{95} - 2602 q^{97} - 1094 q^{99}+O(q^{100}) $ 3 * q + q^3 - 13 * q^5 + 50 * q^9 + 11 * q^11 - 70 * q^13 + 41 * q^15 - 97 * q^17 - 81 * q^19 - 191 * q^23 + 12 * q^25 + 115 * q^27 - 162 * q^29 + 597 * q^31 - 343 * q^33 + 217 * q^37 + 806 * q^39 - 698 * q^41 + 308 * q^43 - 1118 * q^45 + 139 * q^47 - 1475 * q^51 + 197 * q^53 + 147 * q^55 - 1147 * q^57 + 353 * q^59 - 449 * q^61 + 906 * q^65 + 519 * q^67 - 2557 * q^69 - 224 * q^71 - 1701 * q^73 - 1132 * q^75 - 1143 * q^79 + 251 * q^81 - 1380 * q^83 - 1025 * q^85 + 1458 * q^87 - 1749 * q^89 - 601 * q^93 + 2167 * q^95 - 2602 * q^97 - 1094 * 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
$3$	0.138208	0.0265982	0.0132991	−	0.999912i	$-0.495767\pi$
$3$	0.138208	0.0265982	0.0132991	+	0.999912i	$0.495767\pi$
$4$	0	0
$5$	10.0858	0.902099	0.451049	−	0.892499i	$-0.351050\pi$
$5$	10.0858	0.902099	0.451049	+	0.892499i	$0.351050\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−26.9809	−0.999293
$10$	0	0
$11$	18.6711	0.511778	0.255889	−	0.966706i	$-0.417632\pi$
$11$	18.6711	0.511778	0.255889	+	0.966706i	$0.417632\pi$
$12$	0	0
$13$	−10.0850	−0.215159	−0.107580	−	0.994196i	$-0.534310\pi$
$13$	−10.0850	−0.215159	−0.107580	+	0.994196i	$0.534310\pi$
$14$	0	0
$15$	1.39394	0.0239942
$16$	0	0
$17$	−73.6394	−1.05060	−0.525299	−	0.850918i	$-0.676047\pi$
$17$	−73.6394	−1.05060	−0.525299	+	0.850918i	$0.676047\pi$
$18$	0	0
$19$	76.0801	0.918630	0.459315	−	0.888273i	$-0.348095\pi$
$19$	76.0801	0.918630	0.459315	+	0.888273i	$0.348095\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−146.864	−1.33145	−0.665724	−	0.746198i	$-0.731877\pi$
$23$	−146.864	−1.33145	−0.665724	+	0.746198i	$0.731877\pi$
$24$	0	0
$25$	−23.2772	−0.186218
$26$	0	0
$27$	−7.46061	−0.0531776
$28$	0	0
$29$	−157.665	−1.00958	−0.504789	−	0.863243i	$-0.668430\pi$
$29$	−157.665	−1.00958	−0.504789	+	0.863243i	$0.668430\pi$
$30$	0	0
$31$	69.8134	0.404479	0.202240	−	0.979336i	$-0.435178\pi$
$31$	69.8134	0.404479	0.202240	+	0.979336i	$0.435178\pi$
$32$	0	0
$33$	2.58051	0.0136124
$34$	0	0
$35$	0	0
$36$	0	0
$37$	309.673	1.37595	0.687973	−	0.725737i	$-0.258501\pi$
$37$	309.673	1.37595	0.687973	+	0.725737i	$0.258501\pi$
$38$	0	0
$39$	−1.39383	−0.00572285
$40$	0	0
$41$	−482.865	−1.83929	−0.919644	−	0.392753i	$-0.871523\pi$
$41$	−482.865	−1.83929	−0.919644	+	0.392753i	$0.871523\pi$
$42$	0	0
$43$	−17.3692	−0.0615994	−0.0307997	−	0.999526i	$-0.509805\pi$
$43$	−17.3692	−0.0615994	−0.0307997	+	0.999526i	$0.509805\pi$
$44$	0	0
$45$	−272.123	−0.901461
$46$	0	0
$47$	−346.690	−1.07596	−0.537978	−	0.842959i	$-0.680812\pi$
$47$	−346.690	−1.07596	−0.537978	+	0.842959i	$0.680812\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−10.1776	−0.0279440
$52$	0	0
$53$	−73.4511	−0.190364	−0.0951820	−	0.995460i	$-0.530343\pi$
$53$	−73.4511	−0.190364	−0.0951820	+	0.995460i	$0.530343\pi$
$54$	0	0
$55$	188.313	0.461675
$56$	0	0
$57$	10.5149	0.0244339
$58$	0	0
$59$	704.382	1.55428	0.777142	−	0.629325i	$-0.216669\pi$
$59$	704.382	1.55428	0.777142	+	0.629325i	$0.216669\pi$
$60$	0	0
$61$	−841.393	−1.76606	−0.883028	−	0.469321i	$-0.844499\pi$
$61$	−841.393	−1.76606	−0.883028	+	0.469321i	$0.844499\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−101.715	−0.194095
$66$	0	0
$67$	891.809	1.62615	0.813073	−	0.582162i	$-0.197793\pi$
$67$	891.809	1.62615	0.813073	+	0.582162i	$0.197793\pi$
$68$	0	0
$69$	−20.2978	−0.0354141
$70$	0	0
$71$	525.472	0.878339	0.439170	−	0.898404i	$-0.355273\pi$
$71$	525.472	0.878339	0.439170	+	0.898404i	$0.355273\pi$
$72$	0	0
$73$	−376.060	−0.602938	−0.301469	−	0.953476i	$-0.597477\pi$
$73$	−376.060	−0.602938	−0.301469	+	0.953476i	$0.597477\pi$
$74$	0	0
$75$	−3.21710	−0.00495305
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1233.50	−1.75671	−0.878354	−	0.478010i	$-0.841358\pi$
$79$	−1233.50	−1.75671	−0.878354	+	0.478010i	$0.841358\pi$
$80$	0	0
$81$	727.453	0.997878
$82$	0	0
$83$	−771.757	−1.02062	−0.510309	−	0.859991i	$-0.670469\pi$
$83$	−771.757	−1.02062	−0.510309	+	0.859991i	$0.670469\pi$
$84$	0	0
$85$	−742.710	−0.947744
$86$	0	0
$87$	−21.7907	−0.0268529
$88$	0	0
$89$	175.379	0.208878	0.104439	−	0.994531i	$-0.466695\pi$
$89$	175.379	0.208878	0.104439	+	0.994531i	$0.466695\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.64879	0.0107584
$94$	0	0
$95$	767.327	0.828695
$96$	0	0
$97$	−1313.94	−1.37536	−0.687680	−	0.726014i	$-0.741371\pi$
$97$	−1313.94	−1.37536	−0.687680	+	0.726014i	$0.741371\pi$
$98$	0	0
$99$	−503.764	−0.511416
$100$	0	0
$101$	−431.665	−0.425270	−0.212635	−	0.977132i	$-0.568205\pi$
$101$	−431.665	−0.425270	−0.212635	+	0.977132i	$0.568205\pi$
$102$	0	0
$103$	−162.154	−0.155122	−0.0775609	−	0.996988i	$-0.524713\pi$
$103$	−162.154	−0.155122	−0.0775609	+	0.996988i	$0.524713\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	181.508	0.163991	0.0819957	−	0.996633i	$-0.473871\pi$
$107$	181.508	0.163991	0.0819957	+	0.996633i	$0.473871\pi$
$108$	0	0
$109$	1317.33	1.15759	0.578797	−	0.815472i	$-0.303522\pi$
$109$	1317.33	1.15759	0.578797	+	0.815472i	$0.303522\pi$
$110$	0	0
$111$	42.7994	0.0365977
$112$	0	0
$113$	352.404	0.293375	0.146687	−	0.989183i	$-0.453139\pi$
$113$	352.404	0.293375	0.146687	+	0.989183i	$0.453139\pi$
$114$	0	0
$115$	−1481.24	−1.20110
$116$	0	0
$117$	272.102	0.215007
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−982.388	−0.738083
$122$	0	0
$123$	−66.7359	−0.0489217
$124$	0	0
$125$	−1495.49	−1.07009
$126$	0	0
$127$	−1887.08	−1.31851	−0.659257	−	0.751918i	$-0.729129\pi$
$127$	−1887.08	−1.31851	−0.659257	+	0.751918i	$0.729129\pi$
$128$	0	0
$129$	−2.40056	−0.00163843
$130$	0	0
$131$	−1226.70	−0.818149	−0.409075	−	0.912501i	$-0.634148\pi$
$131$	−1226.70	−0.818149	−0.409075	+	0.912501i	$0.634148\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−75.2460	−0.0479714
$136$	0	0
$137$	−925.580	−0.577209	−0.288605	−	0.957448i	$-0.593191\pi$
$137$	−925.580	−0.577209	−0.288605	+	0.957448i	$0.593191\pi$
$138$	0	0
$139$	−2743.39	−1.67404	−0.837020	−	0.547172i	$-0.815704\pi$
$139$	−2743.39	−1.67404	−0.837020	+	0.547172i	$0.815704\pi$
$140$	0	0
$141$	−47.9154	−0.0286185
$142$	0	0
$143$	−188.298	−0.110114
$144$	0	0
$145$	−1590.18	−0.910739
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1118.77	−0.615122	−0.307561	−	0.951528i	$-0.599513\pi$
$149$	−1118.77	−0.615122	−0.307561	+	0.951528i	$0.599513\pi$
$150$	0	0
$151$	−1006.62	−0.542500	−0.271250	−	0.962509i	$-0.587437\pi$
$151$	−1006.62	−0.542500	−0.271250	+	0.962509i	$0.587437\pi$
$152$	0	0
$153$	1986.86	1.04986
$154$	0	0
$155$	704.122	0.364880
$156$	0	0
$157$	1877.21	0.954253	0.477126	−	0.878835i	$-0.341678\pi$
$157$	1877.21	0.954253	0.477126	+	0.878835i	$0.341678\pi$
$158$	0	0
$159$	−10.1516	−0.00506334
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2441.84	1.17337	0.586686	−	0.809815i	$-0.300432\pi$
$163$	2441.84	1.17337	0.586686	+	0.809815i	$0.300432\pi$
$164$	0	0
$165$	26.0264	0.0122797
$166$	0	0
$167$	−2456.79	−1.13840	−0.569198	−	0.822200i	$-0.692746\pi$
$167$	−2456.79	−1.13840	−0.569198	+	0.822200i	$0.692746\pi$
$168$	0	0
$169$	−2095.29	−0.953707
$170$	0	0
$171$	−2052.71	−0.917980
$172$	0	0
$173$	−318.126	−0.139807	−0.0699037	−	0.997554i	$-0.522269\pi$
$173$	−318.126	−0.139807	−0.0699037	+	0.997554i	$0.522269\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	97.3515	0.0413412
$178$	0	0
$179$	−342.653	−0.143079	−0.0715394	−	0.997438i	$-0.522791\pi$
$179$	−342.653	−0.143079	−0.0715394	+	0.997438i	$0.522791\pi$
$180$	0	0
$181$	−250.686	−0.102946	−0.0514732	−	0.998674i	$-0.516392\pi$
$181$	−250.686	−0.102946	−0.0514732	+	0.998674i	$0.516392\pi$
$182$	0	0
$183$	−116.288	−0.0469739
$184$	0	0
$185$	3123.29	1.24124
$186$	0	0
$187$	−1374.93	−0.537674
$188$	0	0
$189$	0	0
$190$	0	0
$191$	136.231	0.0516090	0.0258045	−	0.999667i	$-0.491785\pi$
$191$	136.231	0.0516090	0.0258045	+	0.999667i	$0.491785\pi$
$192$	0	0
$193$	3395.83	1.26651	0.633256	−	0.773942i	$-0.281718\pi$
$193$	3395.83	1.26651	0.633256	+	0.773942i	$0.281718\pi$
$194$	0	0
$195$	−14.0578	−0.00516257
$196$	0	0
$197$	4840.71	1.75069	0.875345	−	0.483498i	$-0.160634\pi$
$197$	4840.71	1.75069	0.875345	+	0.483498i	$0.160634\pi$
$198$	0	0
$199$	2542.60	0.905731	0.452865	−	0.891579i	$-0.350402\pi$
$199$	2542.60	0.905731	0.452865	+	0.891579i	$0.350402\pi$
$200$	0	0
$201$	123.255	0.0432526
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4870.06	−1.65922
$206$	0	0
$207$	3962.53	1.33051
$208$	0	0
$209$	1420.50	0.470135
$210$	0	0
$211$	−3205.26	−1.04578	−0.522889	−	0.852401i	$-0.675146\pi$
$211$	−3205.26	−1.04578	−0.522889	+	0.852401i	$0.675146\pi$
$212$	0	0
$213$	72.6246	0.0233622
$214$	0	0
$215$	−175.182	−0.0555687
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−51.9745	−0.0160371
$220$	0	0
$221$	742.652	0.226046
$222$	0	0
$223$	4710.02	1.41438	0.707189	−	0.707025i	$-0.249963\pi$
$223$	4710.02	1.41438	0.707189	+	0.707025i	$0.249963\pi$
$224$	0	0
$225$	628.040	0.186086
$226$	0	0
$227$	−6158.58	−1.80070	−0.900351	−	0.435165i	$-0.856690\pi$
$227$	−6158.58	−1.80070	−0.900351	+	0.435165i	$0.856690\pi$
$228$	0	0
$229$	3345.35	0.965358	0.482679	−	0.875797i	$-0.339664\pi$
$229$	3345.35	0.965358	0.482679	+	0.875797i	$0.339664\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2251.39	0.633019	0.316510	−	0.948589i	$-0.397489\pi$
$233$	2251.39	0.633019	0.316510	+	0.948589i	$0.397489\pi$
$234$	0	0
$235$	−3496.63	−0.970618
$236$	0	0
$237$	−170.480	−0.0467253
$238$	0	0
$239$	3170.94	0.858205	0.429102	−	0.903256i	$-0.358830\pi$
$239$	3170.94	0.858205	0.429102	+	0.903256i	$0.358830\pi$
$240$	0	0
$241$	−2493.32	−0.666428	−0.333214	−	0.942851i	$-0.608133\pi$
$241$	−2493.32	−0.666428	−0.333214	+	0.942851i	$0.608133\pi$
$242$	0	0
$243$	301.976	0.0797193
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−767.266	−0.197652
$248$	0	0
$249$	−106.663	−0.0271466
$250$	0	0
$251$	−1958.03	−0.492389	−0.246194	−	0.969220i	$-0.579180\pi$
$251$	−1958.03	−0.492389	−0.246194	+	0.969220i	$0.579180\pi$
$252$	0	0
$253$	−2742.12	−0.681406
$254$	0	0
$255$	−102.649	−0.0252083
$256$	0	0
$257$	−1571.86	−0.381517	−0.190758	−	0.981637i	$-0.561095\pi$
$257$	−1571.86	−0.381517	−0.190758	+	0.981637i	$0.561095\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4253.96	1.00886
$262$	0	0
$263$	23.2692	0.00545567	0.00272783	−	0.999996i	$-0.499132\pi$
$263$	23.2692	0.00545567	0.00272783	+	0.999996i	$0.499132\pi$
$264$	0	0
$265$	−740.811	−0.171727
$266$	0	0
$267$	24.2388	0.00555578
$268$	0	0
$269$	−1781.99	−0.403902	−0.201951	−	0.979396i	$-0.564728\pi$
$269$	−1781.99	−0.403902	−0.201951	+	0.979396i	$0.564728\pi$
$270$	0	0
$271$	3807.01	0.853356	0.426678	−	0.904404i	$-0.359684\pi$
$271$	3807.01	0.853356	0.426678	+	0.904404i	$0.359684\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−434.612	−0.0953022
$276$	0	0
$277$	−1938.15	−0.420405	−0.210202	−	0.977658i	$-0.567412\pi$
$277$	−1938.15	−0.420405	−0.210202	+	0.977658i	$0.567412\pi$
$278$	0	0
$279$	−1883.63	−0.404193
$280$	0	0
$281$	−383.074	−0.0813248	−0.0406624	−	0.999173i	$-0.512947\pi$
$281$	−383.074	−0.0813248	−0.0406624	+	0.999173i	$0.512947\pi$
$282$	0	0
$283$	4681.71	0.983387	0.491694	−	0.870768i	$-0.336378\pi$
$283$	4681.71	0.983387	0.491694	+	0.870768i	$0.336378\pi$
$284$	0	0
$285$	106.051	0.0220418
$286$	0	0
$287$	0	0
$288$	0	0
$289$	509.761	0.103758
$290$	0	0
$291$	−181.597	−0.0365821
$292$	0	0
$293$	5917.65	1.17991	0.589954	−	0.807437i	$-0.299146\pi$
$293$	5917.65	1.17991	0.589954	+	0.807437i	$0.299146\pi$
$294$	0	0
$295$	7104.24	1.40212
$296$	0	0
$297$	−139.298	−0.0272151
$298$	0	0
$299$	1481.12	0.286473
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−59.6597	−0.0113114
$304$	0	0
$305$	−8486.10	−1.59316
$306$	0	0
$307$	8668.63	1.61155	0.805773	−	0.592225i	$-0.201750\pi$
$307$	8668.63	1.61155	0.805773	+	0.592225i	$0.201750\pi$
$308$	0	0
$309$	−22.4111	−0.00412596
$310$	0	0
$311$	10069.2	1.83592	0.917959	−	0.396675i	$-0.129836\pi$
$311$	10069.2	1.83592	0.917959	+	0.396675i	$0.129836\pi$
$312$	0	0
$313$	27.2033	0.00491252	0.00245626	−	0.999997i	$-0.499218\pi$
$313$	27.2033	0.00491252	0.00245626	+	0.999997i	$0.499218\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3733.49	0.661493	0.330747	−	0.943720i	$-0.392699\pi$
$317$	3733.49	0.661493	0.330747	+	0.943720i	$0.392699\pi$
$318$	0	0
$319$	−2943.80	−0.516680
$320$	0	0
$321$	25.0860	0.00436187
$322$	0	0
$323$	−5602.49	−0.965112
$324$	0	0
$325$	234.750	0.0400664
$326$	0	0
$327$	182.067	0.0307899
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5156.13	0.856213	0.428106	−	0.903728i	$-0.359181\pi$
$331$	5156.13	0.856213	0.428106	+	0.903728i	$0.359181\pi$
$332$	0	0
$333$	−8355.26	−1.37497
$334$	0	0
$335$	8994.58	1.46694
$336$	0	0
$337$	2243.77	0.362689	0.181344	−	0.983420i	$-0.441955\pi$
$337$	2243.77	0.362689	0.181344	+	0.983420i	$0.441955\pi$
$338$	0	0
$339$	48.7051	0.00780324
$340$	0	0
$341$	1303.50	0.207004
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−204.719	−0.0319470
$346$	0	0
$347$	−7539.78	−1.16645	−0.583223	−	0.812312i	$-0.698209\pi$
$347$	−7539.78	−1.16645	−0.583223	+	0.812312i	$0.698209\pi$
$348$	0	0
$349$	−5466.59	−0.838452	−0.419226	−	0.907882i	$-0.637699\pi$
$349$	−5466.59	−0.838452	−0.419226	+	0.907882i	$0.637699\pi$
$350$	0	0
$351$	75.2401	0.0114416
$352$	0	0
$353$	−4991.97	−0.752680	−0.376340	−	0.926482i	$-0.622817\pi$
$353$	−4991.97	−0.752680	−0.376340	+	0.926482i	$0.622817\pi$
$354$	0	0
$355$	5299.79	0.792349
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8295.83	−1.21960	−0.609801	−	0.792555i	$-0.708751\pi$
$359$	−8295.83	−1.21960	−0.609801	+	0.792555i	$0.708751\pi$
$360$	0	0
$361$	−1070.82	−0.156118
$362$	0	0
$363$	−135.774	−0.0196317
$364$	0	0
$365$	−3792.85	−0.543909
$366$	0	0
$367$	8715.56	1.23964	0.619821	−	0.784743i	$-0.287205\pi$
$367$	8715.56	1.23964	0.619821	+	0.784743i	$0.287205\pi$
$368$	0	0
$369$	13028.1	1.83799
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10446.1	1.45008	0.725038	−	0.688708i	$-0.241822\pi$
$373$	10446.1	1.45008	0.725038	+	0.688708i	$0.241822\pi$
$374$	0	0
$375$	−206.689	−0.0284623
$376$	0	0
$377$	1590.05	0.217220
$378$	0	0
$379$	6890.55	0.933889	0.466944	−	0.884287i	$-0.345355\pi$
$379$	6890.55	0.933889	0.466944	+	0.884287i	$0.345355\pi$
$380$	0	0
$381$	−260.810	−0.0350701
$382$	0	0
$383$	1284.74	0.171403	0.0857016	−	0.996321i	$-0.472687\pi$
$383$	1284.74	0.171403	0.0857016	+	0.996321i	$0.472687\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	468.636	0.0615558
$388$	0	0
$389$	3588.88	0.467772	0.233886	−	0.972264i	$-0.424856\pi$
$389$	3588.88	0.467772	0.233886	+	0.972264i	$0.424856\pi$
$390$	0	0
$391$	10815.0	1.39882
$392$	0	0
$393$	−169.541	−0.0217613
$394$	0	0
$395$	−12440.8	−1.58472
$396$	0	0
$397$	−9080.24	−1.14792	−0.573960	−	0.818883i	$-0.694593\pi$
$397$	−9080.24	−1.14792	−0.573960	+	0.818883i	$0.694593\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4877.67	0.607430	0.303715	−	0.952763i	$-0.401773\pi$
$401$	4877.67	0.607430	0.303715	+	0.952763i	$0.401773\pi$
$402$	0	0
$403$	−704.067	−0.0870275
$404$	0	0
$405$	7336.93	0.900185
$406$	0	0
$407$	5781.96	0.704179
$408$	0	0
$409$	2707.55	0.327334	0.163667	−	0.986516i	$-0.447668\pi$
$409$	2707.55	0.327334	0.163667	+	0.986516i	$0.447668\pi$
$410$	0	0
$411$	−127.923	−0.0153527
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7783.76	−0.920699
$416$	0	0
$417$	−379.160	−0.0445265
$418$	0	0
$419$	−6892.00	−0.803571	−0.401785	−	0.915734i	$-0.631610\pi$
$419$	−6892.00	−0.803571	−0.401785	+	0.915734i	$0.631610\pi$
$420$	0	0
$421$	−3225.76	−0.373429	−0.186715	−	0.982414i	$-0.559784\pi$
$421$	−3225.76	−0.373429	−0.186715	+	0.982414i	$0.559784\pi$
$422$	0	0
$423$	9354.00	1.07519
$424$	0	0
$425$	1714.12	0.195640
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−26.0244	−0.00292883
$430$	0	0
$431$	10035.4	1.12155	0.560774	−	0.827969i	$-0.310504\pi$
$431$	10035.4	1.12155	0.560774	+	0.827969i	$0.310504\pi$
$432$	0	0
$433$	−3468.25	−0.384927	−0.192464	−	0.981304i	$-0.561648\pi$
$433$	−3468.25	−0.384927	−0.192464	+	0.981304i	$0.561648\pi$
$434$	0	0
$435$	−219.776	−0.0242240
$436$	0	0
$437$	−11173.4	−1.22311
$438$	0	0
$439$	12317.8	1.33917	0.669587	−	0.742733i	$-0.266471\pi$
$439$	12317.8	1.33917	0.669587	+	0.742733i	$0.266471\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9774.91	1.04835	0.524176	−	0.851610i	$-0.324373\pi$
$443$	9774.91	1.04835	0.524176	+	0.851610i	$0.324373\pi$
$444$	0	0
$445$	1768.83	0.188429
$446$	0	0
$447$	−154.623	−0.0163611
$448$	0	0
$449$	12437.2	1.30723	0.653616	−	0.756826i	$-0.273251\pi$
$449$	12437.2	1.30723	0.653616	+	0.756826i	$0.273251\pi$
$450$	0	0
$451$	−9015.64	−0.941308
$452$	0	0
$453$	−139.123	−0.0144295
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9135.45	−0.935095	−0.467547	−	0.883968i	$-0.654862\pi$
$457$	−9135.45	−0.935095	−0.467547	+	0.883968i	$0.654862\pi$
$458$	0	0
$459$	549.395	0.0558683
$460$	0	0
$461$	10894.4	1.10066	0.550328	−	0.834948i	$-0.314502\pi$
$461$	10894.4	1.10066	0.550328	+	0.834948i	$0.314502\pi$
$462$	0	0
$463$	−2747.26	−0.275758	−0.137879	−	0.990449i	$-0.544028\pi$
$463$	−2747.26	−0.275758	−0.137879	+	0.990449i	$0.544028\pi$
$464$	0	0
$465$	97.3155	0.00970516
$466$	0	0
$467$	−12892.2	−1.27747	−0.638735	−	0.769427i	$-0.720542\pi$
$467$	−12892.2	−1.27747	−0.638735	+	0.769427i	$0.720542\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	259.446	0.0253814
$472$	0	0
$473$	−324.302	−0.0315252
$474$	0	0
$475$	−1770.93	−0.171065
$476$	0	0
$477$	1981.78	0.190229
$478$	0	0
$479$	−1718.30	−0.163906	−0.0819529	−	0.996636i	$-0.526116\pi$
$479$	−1718.30	−0.163906	−0.0819529	+	0.996636i	$0.526116\pi$
$480$	0	0
$481$	−3123.05	−0.296047
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13252.1	−1.24071
$486$	0	0
$487$	−5884.53	−0.547543	−0.273771	−	0.961795i	$-0.588271\pi$
$487$	−5884.53	−0.547543	−0.273771	+	0.961795i	$0.588271\pi$
$488$	0	0
$489$	337.482	0.0312096
$490$	0	0
$491$	−8154.69	−0.749524	−0.374762	−	0.927121i	$-0.622276\pi$
$491$	−8154.69	−0.749524	−0.374762	+	0.927121i	$0.622276\pi$
$492$	0	0
$493$	11610.4	1.06066
$494$	0	0
$495$	−5080.85	−0.461348
$496$	0	0
$497$	0	0
$498$	0	0
$499$	18095.6	1.62338	0.811692	−	0.584085i	$-0.198547\pi$
$499$	18095.6	1.62338	0.811692	+	0.584085i	$0.198547\pi$
$500$	0	0
$501$	−339.549	−0.0302793
$502$	0	0
$503$	10122.4	0.897286	0.448643	−	0.893711i	$-0.351907\pi$
$503$	10122.4	0.897286	0.448643	+	0.893711i	$0.351907\pi$
$504$	0	0
$505$	−4353.67	−0.383636
$506$	0	0
$507$	−289.587	−0.0253669
$508$	0	0
$509$	6737.35	0.586695	0.293348	−	0.956006i	$-0.405231\pi$
$509$	6737.35	0.586695	0.293348	+	0.956006i	$0.405231\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−567.604	−0.0488505
$514$	0	0
$515$	−1635.45	−0.139935
$516$	0	0
$517$	−6473.10	−0.550651
$518$	0	0
$519$	−43.9677	−0.00371863
$520$	0	0
$521$	12606.4	1.06007	0.530037	−	0.847975i	$-0.322178\pi$
$521$	12606.4	1.06007	0.530037	+	0.847975i	$0.322178\pi$
$522$	0	0
$523$	2602.38	0.217579	0.108790	−	0.994065i	$-0.465302\pi$
$523$	2602.38	0.217579	0.108790	+	0.994065i	$0.465302\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5141.02	−0.424946
$528$	0	0
$529$	9402.08	0.772753
$530$	0	0
$531$	−19004.9	−1.55318
$532$	0	0
$533$	4869.68	0.395740
$534$	0	0
$535$	1830.65	0.147936
$536$	0	0
$537$	−47.3575	−0.00380564
$538$	0	0
$539$	0	0
$540$	0	0
$541$	772.535	0.0613934	0.0306967	−	0.999529i	$-0.490227\pi$
$541$	772.535	0.0613934	0.0306967	+	0.999529i	$0.490227\pi$
$542$	0	0
$543$	−34.6468	−0.00273819
$544$	0	0
$545$	13286.3	1.04426
$546$	0	0
$547$	1367.93	0.106926	0.0534628	−	0.998570i	$-0.482974\pi$
$547$	1367.93	0.106926	0.0534628	+	0.998570i	$0.482974\pi$
$548$	0	0
$549$	22701.6	1.76481
$550$	0	0
$551$	−11995.2	−0.927428
$552$	0	0
$553$	0	0
$554$	0	0
$555$	431.665	0.0330147
$556$	0	0
$557$	−2863.94	−0.217862	−0.108931	−	0.994049i	$-0.534743\pi$
$557$	−2863.94	−0.217862	−0.108931	+	0.994049i	$0.534743\pi$
$558$	0	0
$559$	175.168	0.0132537
$560$	0	0
$561$	−190.027	−0.0143012
$562$	0	0
$563$	−10088.3	−0.755191	−0.377595	−	0.925971i	$-0.623249\pi$
$563$	−10088.3	−0.755191	−0.377595	+	0.925971i	$0.623249\pi$
$564$	0	0
$565$	3554.26	0.264653
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5306.53	0.390969	0.195485	−	0.980707i	$-0.437372\pi$
$569$	5306.53	0.390969	0.195485	+	0.980707i	$0.437372\pi$
$570$	0	0
$571$	−11268.0	−0.825835	−0.412917	−	0.910769i	$-0.635490\pi$
$571$	−11268.0	−0.825835	−0.412917	+	0.910769i	$0.635490\pi$
$572$	0	0
$573$	18.8282	0.00137271
$574$	0	0
$575$	3418.59	0.247939
$576$	0	0
$577$	−4836.97	−0.348987	−0.174494	−	0.984658i	$-0.555829\pi$
$577$	−4836.97	−0.348987	−0.174494	+	0.984658i	$0.555829\pi$
$578$	0	0
$579$	469.331	0.0336870
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1371.42	−0.0974242
$584$	0	0
$585$	2744.36	0.193958
$586$	0	0
$587$	−25970.7	−1.82611	−0.913053	−	0.407842i	$-0.866282\pi$
$587$	−25970.7	−1.82611	−0.913053	+	0.407842i	$0.866282\pi$
$588$	0	0
$589$	5311.41	0.371567
$590$	0	0
$591$	669.026	0.0465652
$592$	0	0
$593$	−26521.5	−1.83661	−0.918303	−	0.395879i	$-0.870440\pi$
$593$	−26521.5	−1.83661	−0.918303	+	0.395879i	$0.870440\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	351.409	0.0240908
$598$	0	0
$599$	7176.69	0.489535	0.244768	−	0.969582i	$-0.421288\pi$
$599$	7176.69	0.489535	0.244768	+	0.969582i	$0.421288\pi$
$600$	0	0
$601$	−12975.9	−0.880698	−0.440349	−	0.897827i	$-0.645145\pi$
$601$	−12975.9	−0.880698	−0.440349	+	0.897827i	$0.645145\pi$
$602$	0	0
$603$	−24061.8	−1.62500
$604$	0	0
$605$	−9908.14	−0.665824
$606$	0	0
$607$	11959.2	0.799683	0.399841	−	0.916584i	$-0.369065\pi$
$607$	11959.2	0.799683	0.399841	+	0.916584i	$0.369065\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3496.36	0.231502
$612$	0	0
$613$	13742.6	0.905477	0.452738	−	0.891643i	$-0.350447\pi$
$613$	13742.6	0.905477	0.452738	+	0.891643i	$0.350447\pi$
$614$	0	0
$615$	−673.083	−0.0441322
$616$	0	0
$617$	−20188.8	−1.31729	−0.658647	−	0.752452i	$-0.728871\pi$
$617$	−20188.8	−1.31729	−0.658647	+	0.752452i	$0.728871\pi$
$618$	0	0
$619$	−25735.6	−1.67109	−0.835543	−	0.549425i	$-0.814847\pi$
$619$	−25735.6	−1.67109	−0.835543	+	0.549425i	$0.814847\pi$
$620$	0	0
$621$	1095.70	0.0708032
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−12173.5	−0.779105
$626$	0	0
$627$	196.325	0.0125047
$628$	0	0
$629$	−22804.2	−1.44557
$630$	0	0
$631$	−9962.93	−0.628554	−0.314277	−	0.949331i	$-0.601762\pi$
$631$	−9962.93	−0.628554	−0.314277	+	0.949331i	$0.601762\pi$
$632$	0	0
$633$	−442.993	−0.0278158
$634$	0	0
$635$	−19032.6	−1.18943
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−14177.7	−0.877718
$640$	0	0
$641$	−12314.3	−0.758792	−0.379396	−	0.925234i	$-0.623868\pi$
$641$	−12314.3	−0.758792	−0.379396	+	0.925234i	$0.623868\pi$
$642$	0	0
$643$	−2307.78	−0.141540	−0.0707699	−	0.997493i	$-0.522546\pi$
$643$	−2307.78	−0.141540	−0.0707699	+	0.997493i	$0.522546\pi$
$644$	0	0
$645$	−24.2115	−0.00147803
$646$	0	0
$647$	−11555.2	−0.702133	−0.351066	−	0.936351i	$-0.614181\pi$
$647$	−11555.2	−0.702133	−0.351066	+	0.936351i	$0.614181\pi$
$648$	0	0
$649$	13151.6	0.795449
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30345.3	−1.81854	−0.909268	−	0.416211i	$-0.863358\pi$
$653$	−30345.3	−1.81854	−0.909268	+	0.416211i	$0.863358\pi$
$654$	0	0
$655$	−12372.3	−0.738052
$656$	0	0
$657$	10146.4	0.602511
$658$	0	0
$659$	8359.93	0.494168	0.247084	−	0.968994i	$-0.420528\pi$
$659$	8359.93	0.494168	0.247084	+	0.968994i	$0.420528\pi$
$660$	0	0
$661$	−17471.3	−1.02807	−0.514034	−	0.857770i	$-0.671850\pi$
$661$	−17471.3	−1.02807	−0.514034	+	0.857770i	$0.671850\pi$
$662$	0	0
$663$	102.641	0.00601242
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23155.4	1.34420
$668$	0	0
$669$	650.964	0.0376199
$670$	0	0
$671$	−15709.8	−0.903829
$672$	0	0
$673$	1761.15	0.100873	0.0504364	−	0.998727i	$-0.483939\pi$
$673$	1761.15	0.100873	0.0504364	+	0.998727i	$0.483939\pi$
$674$	0	0
$675$	173.662	0.00990260
$676$	0	0
$677$	11643.3	0.660989	0.330494	−	0.943808i	$-0.392785\pi$
$677$	11643.3	0.660989	0.330494	+	0.943808i	$0.392785\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−851.167	−0.0478954
$682$	0	0
$683$	33098.4	1.85428	0.927141	−	0.374714i	$-0.122259\pi$
$683$	33098.4	1.85428	0.927141	+	0.374714i	$0.122259\pi$
$684$	0	0
$685$	−9335.19	−0.520700
$686$	0	0
$687$	462.355	0.0256768
$688$	0	0
$689$	740.753	0.0409586
$690$	0	0
$691$	9410.30	0.518067	0.259034	−	0.965868i	$-0.416596\pi$
$691$	9410.30	0.518067	0.259034	+	0.965868i	$0.416596\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27669.2	−1.51015
$696$	0	0
$697$	35557.9	1.93235
$698$	0	0
$699$	311.161	0.0168372
$700$	0	0
$701$	18474.2	0.995377	0.497689	−	0.867356i	$-0.334182\pi$
$701$	18474.2	0.995377	0.497689	+	0.867356i	$0.334182\pi$
$702$	0	0
$703$	23560.0	1.26398
$704$	0	0
$705$	−483.264	−0.0258167
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−3193.42	−0.169156	−0.0845778	−	0.996417i	$-0.526954\pi$
$709$	−3193.42	−0.169156	−0.0845778	+	0.996417i	$0.526954\pi$
$710$	0	0
$711$	33281.0	1.75547
$712$	0	0
$713$	−10253.1	−0.538543
$714$	0	0
$715$	−1899.13	−0.0993336
$716$	0	0
$717$	438.250	0.0228267
$718$	0	0
$719$	−29137.7	−1.51134	−0.755669	−	0.654954i	$-0.772688\pi$
$719$	−29137.7	−1.51134	−0.755669	+	0.654954i	$0.772688\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−344.598	−0.0177258
$724$	0	0
$725$	3670.01	0.188001
$726$	0	0
$727$	3647.99	0.186103	0.0930513	−	0.995661i	$-0.470338\pi$
$727$	3647.99	0.186103	0.0930513	+	0.995661i	$0.470338\pi$
$728$	0	0
$729$	−19599.5	−0.995758
$730$	0	0
$731$	1279.06	0.0647163
$732$	0	0
$733$	−23157.5	−1.16690	−0.583452	−	0.812148i	$-0.698298\pi$
$733$	−23157.5	−1.16690	−0.583452	+	0.812148i	$0.698298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16651.1	0.832227
$738$	0	0
$739$	6574.35	0.327255	0.163627	−	0.986522i	$-0.447681\pi$
$739$	6574.35	0.327255	0.163627	+	0.986522i	$0.447681\pi$
$740$	0	0
$741$	−106.043	−0.00525718
$742$	0	0
$743$	10914.1	0.538894	0.269447	−	0.963015i	$-0.413159\pi$
$743$	10914.1	0.538894	0.269447	+	0.963015i	$0.413159\pi$
$744$	0	0
$745$	−11283.6	−0.554900
$746$	0	0
$747$	20822.7	1.01990
$748$	0	0
$749$	0	0
$750$	0	0
$751$	3877.18	0.188389	0.0941946	−	0.995554i	$-0.469972\pi$
$751$	3877.18	0.188389	0.0941946	+	0.995554i	$0.469972\pi$
$752$	0	0
$753$	−270.616	−0.0130967
$754$	0	0
$755$	−10152.5	−0.489388
$756$	0	0
$757$	−25438.2	−1.22136	−0.610679	−	0.791878i	$-0.709104\pi$
$757$	−25438.2	−1.22136	−0.610679	+	0.791878i	$0.709104\pi$
$758$	0	0
$759$	−378.984	−0.0181242
$760$	0	0
$761$	−31475.2	−1.49931	−0.749655	−	0.661829i	$-0.769780\pi$
$761$	−31475.2	−1.49931	−0.749655	+	0.661829i	$0.769780\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	20039.0	0.947073
$766$	0	0
$767$	−7103.68	−0.334419
$768$	0	0
$769$	−9474.89	−0.444308	−0.222154	−	0.975012i	$-0.571309\pi$
$769$	−9474.89	−0.444308	−0.222154	+	0.975012i	$0.571309\pi$
$770$	0	0
$771$	−217.244	−0.0101477
$772$	0	0
$773$	3173.16	0.147647	0.0738233	−	0.997271i	$-0.476480\pi$
$773$	3173.16	0.147647	0.0738233	+	0.997271i	$0.476480\pi$
$774$	0	0
$775$	−1625.06	−0.0753212
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36736.4	−1.68963
$780$	0	0
$781$	9811.17	0.449515
$782$	0	0
$783$	1176.28	0.0536869
$784$	0	0
$785$	18933.1	0.860830
$786$	0	0
$787$	34587.5	1.56659	0.783297	−	0.621647i	$-0.213536\pi$
$787$	34587.5	1.56659	0.783297	+	0.621647i	$0.213536\pi$
$788$	0	0
$789$	3.21600	0.000145111 0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8485.44	0.379983
$794$	0	0
$795$	−102.386	−0.00456763
$796$	0	0
$797$	−11454.2	−0.509069	−0.254535	−	0.967064i	$-0.581922\pi$
$797$	−11454.2	−0.509069	−0.254535	+	0.967064i	$0.581922\pi$
$798$	0	0
$799$	25530.0	1.13040
$800$	0	0
$801$	−4731.88	−0.208730
$802$	0	0
$803$	−7021.46	−0.308570
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−246.285	−0.0107431
$808$	0	0
$809$	−22630.4	−0.983489	−0.491745	−	0.870740i	$-0.663641\pi$
$809$	−22630.4	−0.983489	−0.491745	+	0.870740i	$0.663641\pi$
$810$	0	0
$811$	39982.6	1.73117	0.865585	−	0.500763i	$-0.166947\pi$
$811$	39982.6	1.73117	0.865585	+	0.500763i	$0.166947\pi$
$812$	0	0
$813$	526.161	0.0226977
$814$	0	0
$815$	24627.8	1.05850
$816$	0	0
$817$	−1321.45	−0.0565871
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12938.7	0.550015	0.275007	−	0.961442i	$-0.411320\pi$
$821$	12938.7	0.550015	0.275007	+	0.961442i	$0.411320\pi$
$822$	0	0
$823$	17828.0	0.755098	0.377549	−	0.925990i	$-0.376767\pi$
$823$	17828.0	0.755098	0.377549	+	0.925990i	$0.376767\pi$
$824$	0	0
$825$	−60.0670	−0.00253487
$826$	0	0
$827$	−1987.48	−0.0835687	−0.0417844	−	0.999127i	$-0.513304\pi$
$827$	−1987.48	−0.0835687	−0.0417844	+	0.999127i	$0.513304\pi$
$828$	0	0
$829$	−8375.91	−0.350914	−0.175457	−	0.984487i	$-0.556140\pi$
$829$	−8375.91	−0.350914	−0.175457	+	0.984487i	$0.556140\pi$
$830$	0	0
$831$	−267.868	−0.0111820
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24778.6	−1.02695
$836$	0	0
$837$	−520.851	−0.0215092
$838$	0	0
$839$	−1002.18	−0.0412385	−0.0206192	−	0.999787i	$-0.506564\pi$
$839$	−1002.18	−0.0412385	−0.0206192	+	0.999787i	$0.506564\pi$
$840$	0	0
$841$	469.405	0.0192466
$842$	0	0
$843$	−52.9440	−0.00216309
$844$	0	0
$845$	−21132.6	−0.860338
$846$	0	0
$847$	0	0
$848$	0	0
$849$	647.051	0.0261563
$850$	0	0
$851$	−45479.9	−1.83200
$852$	0	0
$853$	−5818.51	−0.233554	−0.116777	−	0.993158i	$-0.537256\pi$
$853$	−5818.51	−0.233554	−0.116777	+	0.993158i	$0.537256\pi$
$854$	0	0
$855$	−20703.2	−0.828109
$856$	0	0
$857$	−2576.15	−0.102683	−0.0513416	−	0.998681i	$-0.516350\pi$
$857$	−2576.15	−0.102683	−0.0513416	+	0.998681i	$0.516350\pi$
$858$	0	0
$859$	−40025.9	−1.58983	−0.794915	−	0.606720i	$-0.792485\pi$
$859$	−40025.9	−1.58983	−0.794915	+	0.606720i	$0.792485\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12873.5	0.507786	0.253893	−	0.967232i	$-0.418289\pi$
$863$	12873.5	0.507786	0.253893	+	0.967232i	$0.418289\pi$
$864$	0	0
$865$	−3208.55	−0.126120
$866$	0	0
$867$	70.4532	0.00275976
$868$	0	0
$869$	−23030.9	−0.899046
$870$	0	0
$871$	−8993.87	−0.349880
$872$	0	0
$873$	35451.2	1.37439
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15972.9	0.615013	0.307507	−	0.951546i	$-0.400505\pi$
$877$	15972.9	0.615013	0.307507	+	0.951546i	$0.400505\pi$
$878$	0	0
$879$	817.868	0.0313834
$880$	0	0
$881$	−42155.2	−1.61208	−0.806042	−	0.591859i	$-0.798394\pi$
$881$	−42155.2	−1.61208	−0.806042	+	0.591859i	$0.798394\pi$
$882$	0	0
$883$	−4837.76	−0.184376	−0.0921878	−	0.995742i	$-0.529386\pi$
$883$	−4837.76	−0.184376	−0.0921878	+	0.995742i	$0.529386\pi$
$884$	0	0
$885$	981.865	0.0372938
$886$	0	0
$887$	2096.58	0.0793646	0.0396823	−	0.999212i	$-0.487365\pi$
$887$	2096.58	0.0793646	0.0396823	+	0.999212i	$0.487365\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	13582.4	0.510692
$892$	0	0
$893$	−26376.2	−0.988405
$894$	0	0
$895$	−3455.92	−0.129071
$896$	0	0
$897$	204.703	0.00761967
$898$	0	0
$899$	−11007.2	−0.408353
$900$	0	0
$901$	5408.90	0.199996
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2528.36	−0.0928679
$906$	0	0
$907$	33342.0	1.22062	0.610311	−	0.792162i	$-0.291045\pi$
$907$	33342.0	1.22062	0.610311	+	0.792162i	$0.291045\pi$
$908$	0	0
$909$	11646.7	0.424969
$910$	0	0
$911$	25012.6	0.909665	0.454832	−	0.890577i	$-0.349699\pi$
$911$	25012.6	0.909665	0.454832	+	0.890577i	$0.349699\pi$
$912$	0	0
$913$	−14409.6	−0.522330
$914$	0	0
$915$	−1172.85	−0.0423751
$916$	0	0
$917$	0	0
$918$	0	0
$919$	25789.3	0.925690	0.462845	−	0.886439i	$-0.346829\pi$
$919$	25789.3	0.925690	0.462845	+	0.886439i	$0.346829\pi$
$920$	0	0
$921$	1198.08	0.0428642
$922$	0	0
$923$	−5299.38	−0.188983
$924$	0	0
$925$	−7208.33	−0.256225
$926$	0	0
$927$	4375.07	0.155012
$928$	0	0
$929$	−5417.03	−0.191310	−0.0956550	−	0.995415i	$-0.530495\pi$
$929$	−5417.03	−0.191310	−0.0956550	+	0.995415i	$0.530495\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1391.64	0.0488321
$934$	0	0
$935$	−13867.3	−0.485035
$936$	0	0
$937$	13166.4	0.459046	0.229523	−	0.973303i	$-0.426283\pi$
$937$	13166.4	0.459046	0.229523	+	0.973303i	$0.426283\pi$
$938$	0	0
$939$	3.75972	0.000130664 0
$940$	0	0
$941$	−52187.4	−1.80793	−0.903964	−	0.427609i	$-0.859356\pi$
$941$	−52187.4	−1.80793	−0.903964	+	0.427609i	$0.859356\pi$
$942$	0	0
$943$	70915.5	2.44892
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1138.64	−0.0390717	−0.0195359	−	0.999809i	$-0.506219\pi$
$947$	−1138.64	−0.0390717	−0.0195359	+	0.999809i	$0.506219\pi$
$948$	0	0
$949$	3792.55	0.129728
$950$	0	0
$951$	515.999	0.0175945
$952$	0	0
$953$	47788.3	1.62436	0.812180	−	0.583407i	$-0.198281\pi$
$953$	47788.3	1.62436	0.812180	+	0.583407i	$0.198281\pi$
$954$	0	0
$955$	1373.99	0.0465564
$956$	0	0
$957$	−406.857	−0.0137428
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24917.1	−0.836396
$962$	0	0
$963$	−4897.26	−0.163875
$964$	0	0
$965$	34249.5	1.14252
$966$	0	0
$967$	−32462.6	−1.07955	−0.539777	−	0.841808i	$-0.681491\pi$
$967$	−32462.6	−1.07955	−0.539777	+	0.841808i	$0.681491\pi$
$968$	0	0
$969$	−774.311	−0.0256702
$970$	0	0
$971$	41221.0	1.36235	0.681176	−	0.732120i	$-0.261469\pi$
$971$	41221.0	1.36235	0.681176	+	0.732120i	$0.261469\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	32.4444	0.00106570
$976$	0	0
$977$	−20992.4	−0.687416	−0.343708	−	0.939076i	$-0.611683\pi$
$977$	−20992.4	−0.687416	−0.343708	+	0.939076i	$0.611683\pi$
$978$	0	0
$979$	3274.53	0.106899
$980$	0	0
$981$	−35542.9	−1.15678
$982$	0	0
$983$	808.930	0.0262470	0.0131235	−	0.999914i	$-0.495823\pi$
$983$	808.930	0.0262470	0.0131235	+	0.999914i	$0.495823\pi$
$984$	0	0
$985$	48822.3	1.57930
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2550.91	0.0820164
$990$	0	0
$991$	10025.1	0.321351	0.160676	−	0.987007i	$-0.448633\pi$
$991$	10025.1	0.321351	0.160676	+	0.987007i	$0.448633\pi$
$992$	0	0
$993$	712.620	0.0227737
$994$	0	0
$995$	25644.1	0.817058
$996$	0	0
$997$	53647.3	1.70414	0.852070	−	0.523428i	$-0.175347\pi$
$997$	53647.3	1.70414	0.852070	+	0.523428i	$0.175347\pi$
$998$	0	0
$999$	−2310.35	−0.0731694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bd.1.2		3
4.3	odd	2		392.4.a.j.1.2		3
7.3	odd	6		112.4.i.f.65.2		6
7.5	odd	6		112.4.i.f.81.2		6
7.6	odd	2		784.4.a.bc.1.2		3
28.3	even	6		56.4.i.a.9.2	✓	6
28.11	odd	6		392.4.i.n.177.2		6
28.19	even	6		56.4.i.a.25.2	yes	6
28.23	odd	6		392.4.i.n.361.2		6
28.27	even	2		392.4.a.k.1.2		3
56.3	even	6		448.4.i.l.65.2		6
56.5	odd	6		448.4.i.k.193.2		6
56.19	even	6		448.4.i.l.193.2		6
56.45	odd	6		448.4.i.k.65.2		6
84.47	odd	6		504.4.s.i.361.1		6
84.59	odd	6		504.4.s.i.289.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.i.a.9.2	✓	6	28.3	even	6
56.4.i.a.25.2	yes	6	28.19	even	6
112.4.i.f.65.2		6	7.3	odd	6
112.4.i.f.81.2		6	7.5	odd	6
392.4.a.j.1.2		3	4.3	odd	2
392.4.a.k.1.2		3	28.27	even	2
392.4.i.n.177.2		6	28.11	odd	6
392.4.i.n.361.2		6	28.23	odd	6
448.4.i.k.65.2		6	56.45	odd	6
448.4.i.k.193.2		6	56.5	odd	6
448.4.i.l.65.2		6	56.3	even	6
448.4.i.l.193.2		6	56.19	even	6
504.4.s.i.289.1		6	84.59	odd	6
504.4.s.i.361.1		6	84.47	odd	6
784.4.a.bc.1.2		3	7.6	odd	2
784.4.a.bd.1.2		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.138208 q^{3} +10.0858 q^{5} -26.9809 q^{9} +O(q^{10})\)	\(q+0.138208 q^{3} +10.0858 q^{5} -26.9809 q^{9} +18.6711 q^{11} -10.0850 q^{13} +1.39394 q^{15} -73.6394 q^{17} +76.0801 q^{19} -146.864 q^{23} -23.2772 q^{25} -7.46061 q^{27} -157.665 q^{29} +69.8134 q^{31} +2.58051 q^{33} +309.673 q^{37} -1.39383 q^{39} -482.865 q^{41} -17.3692 q^{43} -272.123 q^{45} -346.690 q^{47} -10.1776 q^{51} -73.4511 q^{53} +188.313 q^{55} +10.5149 q^{57} +704.382 q^{59} -841.393 q^{61} -101.715 q^{65} +891.809 q^{67} -20.2978 q^{69} +525.472 q^{71} -376.060 q^{73} -3.21710 q^{75} -1233.50 q^{79} +727.453 q^{81} -771.757 q^{83} -742.710 q^{85} -21.7907 q^{87} +175.379 q^{89} +9.64879 q^{93} +767.327 q^{95} -1313.94 q^{97} -503.764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 13 q^{5} + 50 q^{9}+O(q^{10}) \) 3 * q + q^3 - 13 * q^5 + 50 * q^9	\( 3 q + q^{3} - 13 q^{5} + 50 q^{9} + 11 q^{11} - 70 q^{13} + 41 q^{15} - 97 q^{17} - 81 q^{19} - 191 q^{23} + 12 q^{25} + 115 q^{27} - 162 q^{29} + 597 q^{31} - 343 q^{33} + 217 q^{37} + 806 q^{39} - 698 q^{41} + 308 q^{43} - 1118 q^{45} + 139 q^{47} - 1475 q^{51} + 197 q^{53} + 147 q^{55} - 1147 q^{57} + 353 q^{59} - 449 q^{61} + 906 q^{65} + 519 q^{67} - 2557 q^{69} - 224 q^{71} - 1701 q^{73} - 1132 q^{75} - 1143 q^{79} + 251 q^{81} - 1380 q^{83} - 1025 q^{85} + 1458 q^{87} - 1749 q^{89} - 601 q^{93} + 2167 q^{95} - 2602 q^{97} - 1094 q^{99}+O(q^{100}) \) 3 * q + q^3 - 13 * q^5 + 50 * q^9 + 11 * q^11 - 70 * q^13 + 41 * q^15 - 97 * q^17 - 81 * q^19 - 191 * q^23 + 12 * q^25 + 115 * q^27 - 162 * q^29 + 597 * q^31 - 343 * q^33 + 217 * q^37 + 806 * q^39 - 698 * q^41 + 308 * q^43 - 1118 * q^45 + 139 * q^47 - 1475 * q^51 + 197 * q^53 + 147 * q^55 - 1147 * q^57 + 353 * q^59 - 449 * q^61 + 906 * q^65 + 519 * q^67 - 2557 * q^69 - 224 * q^71 - 1701 * q^73 - 1132 * q^75 - 1143 * q^79 + 251 * q^81 - 1380 * q^83 - 1025 * q^85 + 1458 * q^87 - 1749 * q^89 - 601 * q^93 + 2167 * q^95 - 2602 * q^97 - 1094 * q^99

Introduction

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