LMFDB - Embedded newform 1216.4.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.17107$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43763 q^{3} +9.42708 q^{5} +9.58919 q^{7} -21.0580 q^{9} +O(q^{10})$	$q-2.43763 q^{3} +9.42708 q^{5} +9.58919 q^{7} -21.0580 q^{9} -37.8637 q^{11} +30.7777 q^{13} -22.9797 q^{15} +35.3347 q^{17} -19.0000 q^{19} -23.3749 q^{21} +88.7405 q^{23} -36.1302 q^{25} +117.148 q^{27} +226.228 q^{29} -320.623 q^{31} +92.2976 q^{33} +90.3981 q^{35} -346.942 q^{37} -75.0247 q^{39} -150.854 q^{41} -284.273 q^{43} -198.515 q^{45} -240.447 q^{47} -251.047 q^{49} -86.1330 q^{51} +539.492 q^{53} -356.944 q^{55} +46.3150 q^{57} +889.117 q^{59} +377.014 q^{61} -201.929 q^{63} +290.144 q^{65} -78.4735 q^{67} -216.317 q^{69} -924.711 q^{71} +576.682 q^{73} +88.0720 q^{75} -363.082 q^{77} -601.388 q^{79} +283.003 q^{81} -1020.13 q^{83} +333.103 q^{85} -551.459 q^{87} +828.104 q^{89} +295.134 q^{91} +781.559 q^{93} -179.114 q^{95} -101.923 q^{97} +797.332 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9	$ 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9} - 33 q^{11} + 35 q^{13} - 120 q^{15} + 66 q^{17} - 133 q^{19} - 33 q^{21} - 389 q^{23} + 44 q^{25} - 39 q^{27} - 233 q^{29} - 158 q^{31} - 206 q^{33} + 123 q^{35} + 436 q^{37} - 807 q^{39} - 94 q^{41} + 645 q^{43} - 103 q^{45} - 1451 q^{47} + 93 q^{49} + 1741 q^{51} - 3 q^{53} - 1971 q^{55} - 57 q^{57} + 297 q^{59} - 93 q^{61} - 2999 q^{63} - 788 q^{65} + 1641 q^{67} - 945 q^{69} - 2392 q^{71} + 324 q^{73} + 1909 q^{75} + 711 q^{77} - 2492 q^{79} + 143 q^{81} + 310 q^{83} - 2353 q^{85} - 4795 q^{87} - 440 q^{89} - 107 q^{91} - 900 q^{93} - 323 q^{95} - 532 q^{97} - 1591 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9 - 33 * q^11 + 35 * q^13 - 120 * q^15 + 66 * q^17 - 133 * q^19 - 33 * q^21 - 389 * q^23 + 44 * q^25 - 39 * q^27 - 233 * q^29 - 158 * q^31 - 206 * q^33 + 123 * q^35 + 436 * q^37 - 807 * q^39 - 94 * q^41 + 645 * q^43 - 103 * q^45 - 1451 * q^47 + 93 * q^49 + 1741 * q^51 - 3 * q^53 - 1971 * q^55 - 57 * q^57 + 297 * q^59 - 93 * q^61 - 2999 * q^63 - 788 * q^65 + 1641 * q^67 - 945 * q^69 - 2392 * q^71 + 324 * q^73 + 1909 * q^75 + 711 * q^77 - 2492 * q^79 + 143 * q^81 + 310 * q^83 - 2353 * q^85 - 4795 * q^87 - 440 * q^89 - 107 * q^91 - 900 * q^93 - 323 * q^95 - 532 * q^97 - 1591 * 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
$3$	−2.43763	−0.469122	−0.234561	−	0.972101i	$-0.575365\pi$
$3$	−2.43763	−0.469122	−0.234561	+	0.972101i	$0.575365\pi$
$4$	0	0
$5$	9.42708	0.843184	0.421592	−	0.906786i	$-0.361472\pi$
$5$	9.42708	0.843184	0.421592	+	0.906786i	$0.361472\pi$
$6$	0	0
$7$	9.58919	0.517768	0.258884	−	0.965908i	$-0.416645\pi$
$7$	9.58919	0.517768	0.258884	+	0.965908i	$0.416645\pi$
$8$	0	0
$9$	−21.0580	−0.779925
$10$	0	0
$11$	−37.8637	−1.03785	−0.518924	−	0.854820i	$-0.673667\pi$
$11$	−37.8637	−1.03785	−0.518924	+	0.854820i	$0.673667\pi$
$12$	0	0
$13$	30.7777	0.656631	0.328316	−	0.944568i	$-0.393519\pi$
$13$	30.7777	0.656631	0.328316	+	0.944568i	$0.393519\pi$
$14$	0	0
$15$	−22.9797	−0.395556
$16$	0	0
$17$	35.3347	0.504114	0.252057	−	0.967712i	$-0.418893\pi$
$17$	35.3347	0.504114	0.252057	+	0.967712i	$0.418893\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−23.3749	−0.242896
$22$	0	0
$23$	88.7405	0.804508	0.402254	−	0.915528i	$-0.368227\pi$
$23$	88.7405	0.804508	0.402254	+	0.915528i	$0.368227\pi$
$24$	0	0
$25$	−36.1302	−0.289041
$26$	0	0
$27$	117.148	0.835002
$28$	0	0
$29$	226.228	1.44860	0.724300	−	0.689485i	$-0.242163\pi$
$29$	226.228	1.44860	0.724300	+	0.689485i	$0.242163\pi$
$30$	0	0
$31$	−320.623	−1.85760	−0.928799	−	0.370585i	$-0.879157\pi$
$31$	−320.623	−1.85760	−0.928799	+	0.370585i	$0.879157\pi$
$32$	0	0
$33$	92.2976	0.486877
$34$	0	0
$35$	90.3981	0.436573
$36$	0	0
$37$	−346.942	−1.54154	−0.770768	−	0.637116i	$-0.780127\pi$
$37$	−346.942	−1.54154	−0.770768	+	0.637116i	$0.780127\pi$
$38$	0	0
$39$	−75.0247	−0.308040
$40$	0	0
$41$	−150.854	−0.574620	−0.287310	−	0.957838i	$-0.592761\pi$
$41$	−150.854	−0.574620	−0.287310	+	0.957838i	$0.592761\pi$
$42$	0	0
$43$	−284.273	−1.00817	−0.504084	−	0.863654i	$-0.668170\pi$
$43$	−284.273	−1.00817	−0.504084	+	0.863654i	$0.668170\pi$
$44$	0	0
$45$	−198.515	−0.657620
$46$	0	0
$47$	−240.447	−0.746230	−0.373115	−	0.927785i	$-0.621710\pi$
$47$	−240.447	−0.746230	−0.373115	+	0.927785i	$0.621710\pi$
$48$	0	0
$49$	−251.047	−0.731917
$50$	0	0
$51$	−86.1330	−0.236491
$52$	0	0
$53$	539.492	1.39821	0.699103	−	0.715021i	$-0.253583\pi$
$53$	539.492	1.39821	0.699103	+	0.715021i	$0.253583\pi$
$54$	0	0
$55$	−356.944	−0.875097
$56$	0	0
$57$	46.3150	0.107624
$58$	0	0
$59$	889.117	1.96192	0.980959	−	0.194215i	$-0.0622160\pi$
$59$	889.117	1.96192	0.980959	+	0.194215i	$0.0622160\pi$
$60$	0	0
$61$	377.014	0.791340	0.395670	−	0.918393i	$-0.370512\pi$
$61$	377.014	0.791340	0.395670	+	0.918393i	$0.370512\pi$
$62$	0	0
$63$	−201.929	−0.403820
$64$	0	0
$65$	290.144	0.553661
$66$	0	0
$67$	−78.4735	−0.143091	−0.0715453	−	0.997437i	$-0.522793\pi$
$67$	−78.4735	−0.143091	−0.0715453	+	0.997437i	$0.522793\pi$
$68$	0	0
$69$	−216.317	−0.377412
$70$	0	0
$71$	−924.711	−1.54568	−0.772838	−	0.634603i	$-0.781164\pi$
$71$	−924.711	−1.54568	−0.772838	+	0.634603i	$0.781164\pi$
$72$	0	0
$73$	576.682	0.924597	0.462298	−	0.886724i	$-0.347025\pi$
$73$	576.682	0.924597	0.462298	+	0.886724i	$0.347025\pi$
$74$	0	0
$75$	88.0720	0.135596
$76$	0	0
$77$	−363.082	−0.537364
$78$	0	0
$79$	−601.388	−0.856473	−0.428237	−	0.903667i	$-0.640865\pi$
$79$	−601.388	−0.856473	−0.428237	+	0.903667i	$0.640865\pi$
$80$	0	0
$81$	283.003	0.388207
$82$	0	0
$83$	−1020.13	−1.34908	−0.674538	−	0.738240i	$-0.735657\pi$
$83$	−1020.13	−1.34908	−0.674538	+	0.738240i	$0.735657\pi$
$84$	0	0
$85$	333.103	0.425060
$86$	0	0
$87$	−551.459	−0.679570
$88$	0	0
$89$	828.104	0.986279	0.493140	−	0.869950i	$-0.335849\pi$
$89$	828.104	0.986279	0.493140	+	0.869950i	$0.335849\pi$
$90$	0	0
$91$	295.134	0.339982
$92$	0	0
$93$	781.559	0.871440
$94$	0	0
$95$	−179.114	−0.193440
$96$	0	0
$97$	−101.923	−0.106688	−0.0533438	−	0.998576i	$-0.516988\pi$
$97$	−101.923	−0.106688	−0.0533438	+	0.998576i	$0.516988\pi$
$98$	0	0
$99$	797.332	0.809443
$100$	0	0
$101$	−735.677	−0.724779	−0.362389	−	0.932027i	$-0.618039\pi$
$101$	−735.677	−0.724779	−0.362389	+	0.932027i	$0.618039\pi$
$102$	0	0
$103$	−185.090	−0.177063	−0.0885313	−	0.996073i	$-0.528217\pi$
$103$	−185.090	−0.177063	−0.0885313	+	0.996073i	$0.528217\pi$
$104$	0	0
$105$	−220.357	−0.204806
$106$	0	0
$107$	717.679	0.648417	0.324209	−	0.945986i	$-0.394902\pi$
$107$	717.679	0.648417	0.324209	+	0.945986i	$0.394902\pi$
$108$	0	0
$109$	776.115	0.682003	0.341001	−	0.940063i	$-0.389234\pi$
$109$	776.115	0.682003	0.341001	+	0.940063i	$0.389234\pi$
$110$	0	0
$111$	845.715	0.723169
$112$	0	0
$113$	−1062.61	−0.884616	−0.442308	−	0.896863i	$-0.645840\pi$
$113$	−1062.61	−0.884616	−0.442308	+	0.896863i	$0.645840\pi$
$114$	0	0
$115$	836.564	0.678348
$116$	0	0
$117$	−648.116	−0.512123
$118$	0	0
$119$	338.832	0.261014
$120$	0	0
$121$	102.659	0.0771289
$122$	0	0
$123$	367.726	0.269567
$124$	0	0
$125$	−1518.99	−1.08690
$126$	0	0
$127$	−1355.40	−0.947026	−0.473513	−	0.880787i	$-0.657014\pi$
$127$	−1355.40	−0.947026	−0.473513	+	0.880787i	$0.657014\pi$
$128$	0	0
$129$	692.953	0.472954
$130$	0	0
$131$	−2095.87	−1.39784	−0.698920	−	0.715200i	$-0.746336\pi$
$131$	−2095.87	−1.39784	−0.698920	+	0.715200i	$0.746336\pi$
$132$	0	0
$133$	−182.195	−0.118784
$134$	0	0
$135$	1104.36	0.704060
$136$	0	0
$137$	−2256.24	−1.40703	−0.703516	−	0.710680i	$-0.748387\pi$
$137$	−2256.24	−1.40703	−0.703516	+	0.710680i	$0.748387\pi$
$138$	0	0
$139$	1276.76	0.779091	0.389545	−	0.921007i	$-0.372632\pi$
$139$	1276.76	0.779091	0.389545	+	0.921007i	$0.372632\pi$
$140$	0	0
$141$	586.121	0.350073
$142$	0	0
$143$	−1165.36	−0.681484
$144$	0	0
$145$	2132.67	1.22144
$146$	0	0
$147$	611.961	0.343358
$148$	0	0
$149$	682.904	0.375474	0.187737	−	0.982219i	$-0.439885\pi$
$149$	682.904	0.375474	0.187737	+	0.982219i	$0.439885\pi$
$150$	0	0
$151$	−3418.01	−1.84208	−0.921039	−	0.389472i	$-0.872658\pi$
$151$	−3418.01	−1.84208	−0.921039	+	0.389472i	$0.872658\pi$
$152$	0	0
$153$	−744.078	−0.393171
$154$	0	0
$155$	−3022.53	−1.56630
$156$	0	0
$157$	−703.314	−0.357520	−0.178760	−	0.983893i	$-0.557209\pi$
$157$	−703.314	−0.357520	−0.178760	+	0.983893i	$0.557209\pi$
$158$	0	0
$159$	−1315.08	−0.655929
$160$	0	0
$161$	850.950	0.416548
$162$	0	0
$163$	−821.522	−0.394764	−0.197382	−	0.980327i	$-0.563244\pi$
$163$	−821.522	−0.394764	−0.197382	+	0.980327i	$0.563244\pi$
$164$	0	0
$165$	870.097	0.410527
$166$	0	0
$167$	−1232.97	−0.571320	−0.285660	−	0.958331i	$-0.592213\pi$
$167$	−1232.97	−0.571320	−0.285660	+	0.958331i	$0.592213\pi$
$168$	0	0
$169$	−1249.73	−0.568835
$170$	0	0
$171$	400.101	0.178927
$172$	0	0
$173$	−3706.69	−1.62899	−0.814493	−	0.580174i	$-0.802985\pi$
$173$	−3706.69	−1.62899	−0.814493	+	0.580174i	$0.802985\pi$
$174$	0	0
$175$	−346.459	−0.149656
$176$	0	0
$177$	−2167.34	−0.920379
$178$	0	0
$179$	−3709.73	−1.54904	−0.774521	−	0.632549i	$-0.782009\pi$
$179$	−3709.73	−1.54904	−0.774521	+	0.632549i	$0.782009\pi$
$180$	0	0
$181$	1006.39	0.413283	0.206642	−	0.978417i	$-0.433747\pi$
$181$	1006.39	0.413283	0.206642	+	0.978417i	$0.433747\pi$
$182$	0	0
$183$	−919.022	−0.371235
$184$	0	0
$185$	−3270.65	−1.29980
$186$	0	0
$187$	−1337.90	−0.523193
$188$	0	0
$189$	1123.35	0.432337
$190$	0	0
$191$	1493.75	0.565886	0.282943	−	0.959137i	$-0.408689\pi$
$191$	1493.75	0.565886	0.282943	+	0.959137i	$0.408689\pi$
$192$	0	0
$193$	−69.3171	−0.0258526	−0.0129263	−	0.999916i	$-0.504115\pi$
$193$	−69.3171	−0.0258526	−0.0129263	+	0.999916i	$0.504115\pi$
$194$	0	0
$195$	−707.264	−0.259734
$196$	0	0
$197$	−528.461	−0.191123	−0.0955617	−	0.995424i	$-0.530465\pi$
$197$	−528.461	−0.191123	−0.0955617	+	0.995424i	$0.530465\pi$
$198$	0	0
$199$	−2827.61	−1.00726	−0.503629	−	0.863920i	$-0.668002\pi$
$199$	−2827.61	−1.00726	−0.503629	+	0.863920i	$0.668002\pi$
$200$	0	0
$201$	191.289	0.0671269
$202$	0	0
$203$	2169.34	0.750039
$204$	0	0
$205$	−1422.11	−0.484510
$206$	0	0
$207$	−1868.69	−0.627455
$208$	0	0
$209$	719.410	0.238099
$210$	0	0
$211$	−2566.80	−0.837468	−0.418734	−	0.908109i	$-0.637526\pi$
$211$	−2566.80	−0.837468	−0.418734	+	0.908109i	$0.637526\pi$
$212$	0	0
$213$	2254.10	0.725111
$214$	0	0
$215$	−2679.87	−0.850072
$216$	0	0
$217$	−3074.51	−0.961804
$218$	0	0
$219$	−1405.74	−0.433749
$220$	0	0
$221$	1087.52	0.331017
$222$	0	0
$223$	1293.99	0.388573	0.194286	−	0.980945i	$-0.437761\pi$
$223$	1293.99	0.388573	0.194286	+	0.980945i	$0.437761\pi$
$224$	0	0
$225$	760.828	0.225431
$226$	0	0
$227$	−4680.99	−1.36867	−0.684335	−	0.729168i	$-0.739907\pi$
$227$	−4680.99	−1.36867	−0.684335	+	0.729168i	$0.739907\pi$
$228$	0	0
$229$	4590.15	1.32457	0.662283	−	0.749254i	$-0.269588\pi$
$229$	4590.15	1.32457	0.662283	+	0.749254i	$0.269588\pi$
$230$	0	0
$231$	885.060	0.252089
$232$	0	0
$233$	4091.75	1.15047	0.575235	−	0.817988i	$-0.304911\pi$
$233$	4091.75	1.15047	0.575235	+	0.817988i	$0.304911\pi$
$234$	0	0
$235$	−2266.71	−0.629209
$236$	0	0
$237$	1465.96	0.401790
$238$	0	0
$239$	4506.64	1.21971	0.609855	−	0.792513i	$-0.291228\pi$
$239$	4506.64	1.21971	0.609855	+	0.792513i	$0.291228\pi$
$240$	0	0
$241$	4831.08	1.29127	0.645637	−	0.763644i	$-0.276592\pi$
$241$	4831.08	1.29127	0.645637	+	0.763644i	$0.276592\pi$
$242$	0	0
$243$	−3852.84	−1.01712
$244$	0	0
$245$	−2366.64	−0.617140
$246$	0	0
$247$	−584.777	−0.150642
$248$	0	0
$249$	2486.69	0.632881
$250$	0	0
$251$	−3766.28	−0.947113	−0.473556	−	0.880764i	$-0.657030\pi$
$251$	−3766.28	−0.947113	−0.473556	+	0.880764i	$0.657030\pi$
$252$	0	0
$253$	−3360.04	−0.834957
$254$	0	0
$255$	−811.983	−0.199405
$256$	0	0
$257$	−3263.14	−0.792021	−0.396010	−	0.918246i	$-0.629606\pi$
$257$	−3263.14	−0.792021	−0.396010	+	0.918246i	$0.629606\pi$
$258$	0	0
$259$	−3326.89	−0.798158
$260$	0	0
$261$	−4763.89	−1.12980
$262$	0	0
$263$	4864.93	1.14062	0.570312	−	0.821428i	$-0.306822\pi$
$263$	4864.93	1.14062	0.570312	+	0.821428i	$0.306822\pi$
$264$	0	0
$265$	5085.83	1.17894
$266$	0	0
$267$	−2018.61	−0.462685
$268$	0	0
$269$	−240.732	−0.0545638	−0.0272819	−	0.999628i	$-0.508685\pi$
$269$	−240.732	−0.0545638	−0.0272819	+	0.999628i	$0.508685\pi$
$270$	0	0
$271$	−1603.13	−0.359347	−0.179674	−	0.983726i	$-0.557504\pi$
$271$	−1603.13	−0.359347	−0.179674	+	0.983726i	$0.557504\pi$
$272$	0	0
$273$	−719.426	−0.159493
$274$	0	0
$275$	1368.02	0.299981
$276$	0	0
$277$	6141.93	1.33225	0.666124	−	0.745841i	$-0.267952\pi$
$277$	6141.93	1.33225	0.666124	+	0.745841i	$0.267952\pi$
$278$	0	0
$279$	6751.66	1.44879
$280$	0	0
$281$	3784.11	0.803350	0.401675	−	0.915782i	$-0.368428\pi$
$281$	3784.11	0.803350	0.401675	+	0.915782i	$0.368428\pi$
$282$	0	0
$283$	−4591.13	−0.964362	−0.482181	−	0.876072i	$-0.660155\pi$
$283$	−4591.13	−0.964362	−0.482181	+	0.876072i	$0.660155\pi$
$284$	0	0
$285$	436.615	0.0907468
$286$	0	0
$287$	−1446.57	−0.297520
$288$	0	0
$289$	−3664.46	−0.745869
$290$	0	0
$291$	248.450	0.0500495
$292$	0	0
$293$	−4699.81	−0.937084	−0.468542	−	0.883441i	$-0.655221\pi$
$293$	−4699.81	−0.937084	−0.468542	+	0.883441i	$0.655221\pi$
$294$	0	0
$295$	8381.77	1.65426
$296$	0	0
$297$	−4435.64	−0.866605
$298$	0	0
$299$	2731.23	0.528265
$300$	0	0
$301$	−2725.95	−0.521997
$302$	0	0
$303$	1793.31	0.340010
$304$	0	0
$305$	3554.14	0.667245
$306$	0	0
$307$	5219.15	0.970269	0.485135	−	0.874440i	$-0.338771\pi$
$307$	5219.15	0.970269	0.485135	+	0.874440i	$0.338771\pi$
$308$	0	0
$309$	451.181	0.0830640
$310$	0	0
$311$	6801.34	1.24009	0.620046	−	0.784566i	$-0.287114\pi$
$311$	6801.34	1.24009	0.620046	+	0.784566i	$0.287114\pi$
$312$	0	0
$313$	−6207.21	−1.12093	−0.560467	−	0.828177i	$-0.689378\pi$
$313$	−6207.21	−1.12093	−0.560467	+	0.828177i	$0.689378\pi$
$314$	0	0
$315$	−1903.60	−0.340494
$316$	0	0
$317$	−5047.46	−0.894301	−0.447151	−	0.894459i	$-0.647561\pi$
$317$	−5047.46	−0.894301	−0.447151	+	0.894459i	$0.647561\pi$
$318$	0	0
$319$	−8565.81	−1.50343
$320$	0	0
$321$	−1749.44	−0.304187
$322$	0	0
$323$	−671.360	−0.115652
$324$	0	0
$325$	−1112.01	−0.189794
$326$	0	0
$327$	−1891.88	−0.319943
$328$	0	0
$329$	−2305.69	−0.386374
$330$	0	0
$331$	7678.32	1.27504	0.637521	−	0.770433i	$-0.279960\pi$
$331$	7678.32	1.27504	0.637521	+	0.770433i	$0.279960\pi$
$332$	0	0
$333$	7305.88	1.20228
$334$	0	0
$335$	−739.776	−0.120652
$336$	0	0
$337$	7014.71	1.13387	0.566937	−	0.823761i	$-0.308128\pi$
$337$	7014.71	1.13387	0.566937	+	0.823761i	$0.308128\pi$
$338$	0	0
$339$	2590.24	0.414993
$340$	0	0
$341$	12140.0	1.92790
$342$	0	0
$343$	−5696.43	−0.896730
$344$	0	0
$345$	−2039.23	−0.318228
$346$	0	0
$347$	−2411.56	−0.373082	−0.186541	−	0.982447i	$-0.559728\pi$
$347$	−2411.56	−0.373082	−0.186541	+	0.982447i	$0.559728\pi$
$348$	0	0
$349$	−1768.55	−0.271256	−0.135628	−	0.990760i	$-0.543305\pi$
$349$	−1768.55	−0.271256	−0.135628	+	0.990760i	$0.543305\pi$
$350$	0	0
$351$	3605.53	0.548288
$352$	0	0
$353$	−725.636	−0.109410	−0.0547050	−	0.998503i	$-0.517422\pi$
$353$	−725.636	−0.109410	−0.0547050	+	0.998503i	$0.517422\pi$
$354$	0	0
$355$	−8717.32	−1.30329
$356$	0	0
$357$	−825.946	−0.122447
$358$	0	0
$359$	−3393.30	−0.498863	−0.249431	−	0.968392i	$-0.580244\pi$
$359$	−3393.30	−0.498863	−0.249431	+	0.968392i	$0.580244\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−250.243	−0.0361828
$364$	0	0
$365$	5436.43	0.779605
$366$	0	0
$367$	4902.25	0.697262	0.348631	−	0.937260i	$-0.386647\pi$
$367$	4902.25	0.697262	0.348631	+	0.937260i	$0.386647\pi$
$368$	0	0
$369$	3176.67	0.448160
$370$	0	0
$371$	5173.29	0.723946
$372$	0	0
$373$	−11121.7	−1.54386	−0.771932	−	0.635705i	$-0.780709\pi$
$373$	−11121.7	−1.54386	−0.771932	+	0.635705i	$0.780709\pi$
$374$	0	0
$375$	3702.73	0.509888
$376$	0	0
$377$	6962.77	0.951197
$378$	0	0
$379$	6826.18	0.925164	0.462582	−	0.886576i	$-0.346923\pi$
$379$	6826.18	0.925164	0.462582	+	0.886576i	$0.346923\pi$
$380$	0	0
$381$	3303.96	0.444271
$382$	0	0
$383$	−7637.04	−1.01889	−0.509445	−	0.860503i	$-0.670149\pi$
$383$	−7637.04	−1.01889	−0.509445	+	0.860503i	$0.670149\pi$
$384$	0	0
$385$	−3422.80	−0.453097
$386$	0	0
$387$	5986.22	0.786296
$388$	0	0
$389$	−198.410	−0.0258607	−0.0129303	−	0.999916i	$-0.504116\pi$
$389$	−198.410	−0.0258607	−0.0129303	+	0.999916i	$0.504116\pi$
$390$	0	0
$391$	3135.62	0.405563
$392$	0	0
$393$	5108.95	0.655757
$394$	0	0
$395$	−5669.33	−0.722164
$396$	0	0
$397$	−13398.8	−1.69387	−0.846933	−	0.531700i	$-0.821553\pi$
$397$	−13398.8	−1.69387	−0.846933	+	0.531700i	$0.821553\pi$
$398$	0	0
$399$	444.123	0.0557242
$400$	0	0
$401$	−13426.2	−1.67200	−0.835999	−	0.548731i	$-0.815111\pi$
$401$	−13426.2	−1.67200	−0.835999	+	0.548731i	$0.815111\pi$
$402$	0	0
$403$	−9868.04	−1.21976
$404$	0	0
$405$	2667.89	0.327330
$406$	0	0
$407$	13136.5	1.59988
$408$	0	0
$409$	−945.230	−0.114275	−0.0571377	−	0.998366i	$-0.518197\pi$
$409$	−945.230	−0.114275	−0.0571377	+	0.998366i	$0.518197\pi$
$410$	0	0
$411$	5499.87	0.660069
$412$	0	0
$413$	8525.91	1.01582
$414$	0	0
$415$	−9616.80	−1.13752
$416$	0	0
$417$	−3112.27	−0.365489
$418$	0	0
$419$	5013.06	0.584497	0.292248	−	0.956342i	$-0.405597\pi$
$419$	5013.06	0.584497	0.292248	+	0.956342i	$0.405597\pi$
$420$	0	0
$421$	−1574.62	−0.182285	−0.0911426	−	0.995838i	$-0.529052\pi$
$421$	−1574.62	−0.182285	−0.0911426	+	0.995838i	$0.529052\pi$
$422$	0	0
$423$	5063.32	0.582003
$424$	0	0
$425$	−1276.65	−0.145710
$426$	0	0
$427$	3615.26	0.409730
$428$	0	0
$429$	2840.71	0.319699
$430$	0	0
$431$	9535.61	1.06569	0.532847	−	0.846212i	$-0.321122\pi$
$431$	9535.61	1.06569	0.532847	+	0.846212i	$0.321122\pi$
$432$	0	0
$433$	9182.73	1.01915	0.509577	−	0.860425i	$-0.329802\pi$
$433$	9182.73	1.01915	0.509577	+	0.860425i	$0.329802\pi$
$434$	0	0
$435$	−5198.65	−0.573003
$436$	0	0
$437$	−1686.07	−0.184567
$438$	0	0
$439$	898.925	0.0977297	0.0488648	−	0.998805i	$-0.484440\pi$
$439$	898.925	0.0977297	0.0488648	+	0.998805i	$0.484440\pi$
$440$	0	0
$441$	5286.55	0.570840
$442$	0	0
$443$	6638.61	0.711987	0.355993	−	0.934488i	$-0.384143\pi$
$443$	6638.61	0.711987	0.355993	+	0.934488i	$0.384143\pi$
$444$	0	0
$445$	7806.60	0.831615
$446$	0	0
$447$	−1664.67	−0.176143
$448$	0	0
$449$	1509.78	0.158688	0.0793442	−	0.996847i	$-0.474717\pi$
$449$	1509.78	0.158688	0.0793442	+	0.996847i	$0.474717\pi$
$450$	0	0
$451$	5711.88	0.596368
$452$	0	0
$453$	8331.84	0.864159
$454$	0	0
$455$	2782.25	0.286668
$456$	0	0
$457$	−2910.97	−0.297964	−0.148982	−	0.988840i	$-0.547600\pi$
$457$	−2910.97	−0.297964	−0.148982	+	0.988840i	$0.547600\pi$
$458$	0	0
$459$	4139.38	0.420936
$460$	0	0
$461$	2610.80	0.263768	0.131884	−	0.991265i	$-0.457897\pi$
$461$	2610.80	0.263768	0.131884	+	0.991265i	$0.457897\pi$
$462$	0	0
$463$	−19238.6	−1.93109	−0.965545	−	0.260237i	$-0.916199\pi$
$463$	−19238.6	−1.93109	−0.965545	+	0.260237i	$0.916199\pi$
$464$	0	0
$465$	7367.82	0.734784
$466$	0	0
$467$	−1300.33	−0.128848	−0.0644241	−	0.997923i	$-0.520521\pi$
$467$	−1300.33	−0.128848	−0.0644241	+	0.997923i	$0.520521\pi$
$468$	0	0
$469$	−752.497	−0.0740876
$470$	0	0
$471$	1714.42	0.167720
$472$	0	0
$473$	10763.6	1.04633
$474$	0	0
$475$	686.474	0.0663107
$476$	0	0
$477$	−11360.6	−1.09050
$478$	0	0
$479$	−14734.4	−1.40550	−0.702749	−	0.711438i	$-0.748044\pi$
$479$	−14734.4	−1.40550	−0.702749	+	0.711438i	$0.748044\pi$
$480$	0	0
$481$	−10678.1	−1.01222
$482$	0	0
$483$	−2074.30	−0.195412
$484$	0	0
$485$	−960.834	−0.0899572
$486$	0	0
$487$	9192.13	0.855308	0.427654	−	0.903943i	$-0.359340\pi$
$487$	9192.13	0.855308	0.427654	+	0.903943i	$0.359340\pi$
$488$	0	0
$489$	2002.57	0.185193
$490$	0	0
$491$	17240.8	1.58466	0.792330	−	0.610093i	$-0.208868\pi$
$491$	17240.8	1.58466	0.792330	+	0.610093i	$0.208868\pi$
$492$	0	0
$493$	7993.70	0.730259
$494$	0	0
$495$	7516.51	0.682509
$496$	0	0
$497$	−8867.23	−0.800301
$498$	0	0
$499$	17911.8	1.60690	0.803451	−	0.595371i	$-0.202995\pi$
$499$	17911.8	1.60690	0.803451	+	0.595371i	$0.202995\pi$
$500$	0	0
$501$	3005.54	0.268019
$502$	0	0
$503$	−1995.19	−0.176861	−0.0884307	−	0.996082i	$-0.528185\pi$
$503$	−1995.19	−0.176861	−0.0884307	+	0.996082i	$0.528185\pi$
$504$	0	0
$505$	−6935.29	−0.611121
$506$	0	0
$507$	3046.38	0.266853
$508$	0	0
$509$	−4595.38	−0.400170	−0.200085	−	0.979779i	$-0.564122\pi$
$509$	−4595.38	−0.400170	−0.200085	+	0.979779i	$0.564122\pi$
$510$	0	0
$511$	5529.92	0.478726
$512$	0	0
$513$	−2225.80	−0.191563
$514$	0	0
$515$	−1744.86	−0.149296
$516$	0	0
$517$	9104.21	0.774473
$518$	0	0
$519$	9035.54	0.764193
$520$	0	0
$521$	−15124.4	−1.27181	−0.635905	−	0.771768i	$-0.719373\pi$
$521$	−15124.4	−1.27181	−0.635905	+	0.771768i	$0.719373\pi$
$522$	0	0
$523$	11770.3	0.984092	0.492046	−	0.870569i	$-0.336249\pi$
$523$	11770.3	0.984092	0.492046	+	0.870569i	$0.336249\pi$
$524$	0	0
$525$	844.539	0.0702071
$526$	0	0
$527$	−11329.1	−0.936440
$528$	0	0
$529$	−4292.12	−0.352767
$530$	0	0
$531$	−18723.0	−1.53015
$532$	0	0
$533$	−4642.94	−0.377313
$534$	0	0
$535$	6765.62	0.546735
$536$	0	0
$537$	9042.95	0.726689
$538$	0	0
$539$	9505.58	0.759618
$540$	0	0
$541$	−7134.26	−0.566961	−0.283481	−	0.958978i	$-0.591489\pi$
$541$	−7134.26	−0.566961	−0.283481	+	0.958978i	$0.591489\pi$
$542$	0	0
$543$	−2453.20	−0.193880
$544$	0	0
$545$	7316.50	0.575054
$546$	0	0
$547$	23355.9	1.82565	0.912823	−	0.408356i	$-0.133898\pi$
$547$	23355.9	1.82565	0.912823	+	0.408356i	$0.133898\pi$
$548$	0	0
$549$	−7939.16	−0.617186
$550$	0	0
$551$	−4298.33	−0.332332
$552$	0	0
$553$	−5766.82	−0.443454
$554$	0	0
$555$	7972.62	0.609764
$556$	0	0
$557$	−7717.97	−0.587111	−0.293556	−	0.955942i	$-0.594839\pi$
$557$	−7717.97	−0.587111	−0.293556	+	0.955942i	$0.594839\pi$
$558$	0	0
$559$	−8749.29	−0.661995
$560$	0	0
$561$	3261.31	0.245442
$562$	0	0
$563$	−10920.7	−0.817502	−0.408751	−	0.912646i	$-0.634035\pi$
$563$	−10920.7	−0.817502	−0.408751	+	0.912646i	$0.634035\pi$
$564$	0	0
$565$	−10017.3	−0.745894
$566$	0	0
$567$	2713.77	0.201001
$568$	0	0
$569$	−9060.49	−0.667549	−0.333775	−	0.942653i	$-0.608322\pi$
$569$	−9060.49	−0.667549	−0.333775	+	0.942653i	$0.608322\pi$
$570$	0	0
$571$	13551.5	0.993189	0.496595	−	0.867983i	$-0.334584\pi$
$571$	13551.5	0.993189	0.496595	+	0.867983i	$0.334584\pi$
$572$	0	0
$573$	−3641.22	−0.265470
$574$	0	0
$575$	−3206.21	−0.232536
$576$	0	0
$577$	23214.2	1.67491	0.837453	−	0.546510i	$-0.184044\pi$
$577$	23214.2	1.67491	0.837453	+	0.546510i	$0.184044\pi$
$578$	0	0
$579$	168.969	0.0121280
$580$	0	0
$581$	−9782.18	−0.698508
$582$	0	0
$583$	−20427.2	−1.45113
$584$	0	0
$585$	−6109.84	−0.431814
$586$	0	0
$587$	11562.3	0.812993	0.406496	−	0.913652i	$-0.366750\pi$
$587$	11562.3	0.812993	0.406496	+	0.913652i	$0.366750\pi$
$588$	0	0
$589$	6091.83	0.426162
$590$	0	0
$591$	1288.19	0.0896601
$592$	0	0
$593$	−4831.74	−0.334597	−0.167298	−	0.985906i	$-0.553504\pi$
$593$	−4831.74	−0.334597	−0.167298	+	0.985906i	$0.553504\pi$
$594$	0	0
$595$	3194.19	0.220083
$596$	0	0
$597$	6892.68	0.472527
$598$	0	0
$599$	7303.29	0.498171	0.249085	−	0.968482i	$-0.419870\pi$
$599$	7303.29	0.498171	0.249085	+	0.968482i	$0.419870\pi$
$600$	0	0
$601$	8753.05	0.594083	0.297042	−	0.954864i	$-0.404000\pi$
$601$	8753.05	0.594083	0.297042	+	0.954864i	$0.404000\pi$
$602$	0	0
$603$	1652.49	0.111600
$604$	0	0
$605$	967.770	0.0650338
$606$	0	0
$607$	−10886.6	−0.727965	−0.363982	−	0.931406i	$-0.618583\pi$
$607$	−10886.6	−0.727965	−0.363982	+	0.931406i	$0.618583\pi$
$608$	0	0
$609$	−5288.05	−0.351860
$610$	0	0
$611$	−7400.41	−0.489998
$612$	0	0
$613$	−16505.2	−1.08750	−0.543750	−	0.839247i	$-0.682996\pi$
$613$	−16505.2	−1.08750	−0.543750	+	0.839247i	$0.682996\pi$
$614$	0	0
$615$	3466.58	0.227294
$616$	0	0
$617$	16403.5	1.07031	0.535155	−	0.844754i	$-0.320253\pi$
$617$	16403.5	1.07031	0.535155	+	0.844754i	$0.320253\pi$
$618$	0	0
$619$	5618.63	0.364833	0.182417	−	0.983221i	$-0.441608\pi$
$619$	5618.63	0.364833	0.182417	+	0.983221i	$0.441608\pi$
$620$	0	0
$621$	10395.7	0.671765
$622$	0	0
$623$	7940.85	0.510664
$624$	0	0
$625$	−9803.34	−0.627414
$626$	0	0
$627$	−1753.65	−0.111697
$628$	0	0
$629$	−12259.1	−0.777110
$630$	0	0
$631$	25840.5	1.63026	0.815131	−	0.579276i	$-0.196665\pi$
$631$	25840.5	1.63026	0.815131	+	0.579276i	$0.196665\pi$
$632$	0	0
$633$	6256.91	0.392875
$634$	0	0
$635$	−12777.5	−0.798517
$636$	0	0
$637$	−7726.67	−0.480599
$638$	0	0
$639$	19472.5	1.20551
$640$	0	0
$641$	17293.1	1.06558	0.532790	−	0.846248i	$-0.321144\pi$
$641$	17293.1	1.06558	0.532790	+	0.846248i	$0.321144\pi$
$642$	0	0
$643$	25338.3	1.55404	0.777019	−	0.629478i	$-0.216731\pi$
$643$	25338.3	1.55404	0.777019	+	0.629478i	$0.216731\pi$
$644$	0	0
$645$	6532.52	0.398787
$646$	0	0
$647$	−5761.08	−0.350064	−0.175032	−	0.984563i	$-0.556003\pi$
$647$	−5761.08	−0.350064	−0.175032	+	0.984563i	$0.556003\pi$
$648$	0	0
$649$	−33665.2	−2.03617
$650$	0	0
$651$	7494.52	0.451203
$652$	0	0
$653$	25471.5	1.52646	0.763229	−	0.646129i	$-0.223613\pi$
$653$	25471.5	1.52646	0.763229	+	0.646129i	$0.223613\pi$
$654$	0	0
$655$	−19757.9	−1.17864
$656$	0	0
$657$	−12143.8	−0.721116
$658$	0	0
$659$	−11251.4	−0.665089	−0.332545	−	0.943088i	$-0.607907\pi$
$659$	−11251.4	−0.665089	−0.332545	+	0.943088i	$0.607907\pi$
$660$	0	0
$661$	11094.9	0.652859	0.326430	−	0.945222i	$-0.394154\pi$
$661$	11094.9	0.652859	0.326430	+	0.945222i	$0.394154\pi$
$662$	0	0
$663$	−2650.98	−0.155287
$664$	0	0
$665$	−1717.56	−0.100157
$666$	0	0
$667$	20075.6	1.16541
$668$	0	0
$669$	−3154.26	−0.182288
$670$	0	0
$671$	−14275.2	−0.821291
$672$	0	0
$673$	108.246	0.00619994	0.00309997	−	0.999995i	$-0.499013\pi$
$673$	108.246	0.00619994	0.00309997	+	0.999995i	$0.499013\pi$
$674$	0	0
$675$	−4232.56	−0.241350
$676$	0	0
$677$	−8045.86	−0.456761	−0.228381	−	0.973572i	$-0.573343\pi$
$677$	−8045.86	−0.456761	−0.228381	+	0.973572i	$0.573343\pi$
$678$	0	0
$679$	−977.357	−0.0552394
$680$	0	0
$681$	11410.5	0.642073
$682$	0	0
$683$	26682.8	1.49486	0.747429	−	0.664341i	$-0.231288\pi$
$683$	26682.8	1.49486	0.747429	+	0.664341i	$0.231288\pi$
$684$	0	0
$685$	−21269.7	−1.18639
$686$	0	0
$687$	−11189.1	−0.621383
$688$	0	0
$689$	16604.3	0.918106
$690$	0	0
$691$	872.171	0.0480159	0.0240079	−	0.999712i	$-0.492357\pi$
$691$	872.171	0.0480159	0.0240079	+	0.999712i	$0.492357\pi$
$692$	0	0
$693$	7645.77	0.419104
$694$	0	0
$695$	12036.1	0.656916
$696$	0	0
$697$	−5330.38	−0.289674
$698$	0	0
$699$	−9974.17	−0.539710
$700$	0	0
$701$	−4411.58	−0.237693	−0.118847	−	0.992913i	$-0.537920\pi$
$701$	−4411.58	−0.237693	−0.118847	+	0.992913i	$0.537920\pi$
$702$	0	0
$703$	6591.89	0.353653
$704$	0	0
$705$	5525.40	0.295176
$706$	0	0
$707$	−7054.55	−0.375267
$708$	0	0
$709$	12916.0	0.684161	0.342081	−	0.939671i	$-0.388868\pi$
$709$	12916.0	0.684161	0.342081	+	0.939671i	$0.388868\pi$
$710$	0	0
$711$	12664.0	0.667985
$712$	0	0
$713$	−28452.2	−1.49445
$714$	0	0
$715$	−10985.9	−0.574616
$716$	0	0
$717$	−10985.5	−0.572192
$718$	0	0
$719$	20954.9	1.08691	0.543453	−	0.839439i	$-0.317117\pi$
$719$	20954.9	1.08691	0.543453	+	0.839439i	$0.317117\pi$
$720$	0	0
$721$	−1774.86	−0.0916773
$722$	0	0
$723$	−11776.4	−0.605765
$724$	0	0
$725$	−8173.65	−0.418706
$726$	0	0
$727$	32170.7	1.64119	0.820596	−	0.571509i	$-0.193642\pi$
$727$	32170.7	1.64119	0.820596	+	0.571509i	$0.193642\pi$
$728$	0	0
$729$	1750.72	0.0889457
$730$	0	0
$731$	−10044.7	−0.508232
$732$	0	0
$733$	14716.9	0.741584	0.370792	−	0.928716i	$-0.379086\pi$
$733$	14716.9	0.741584	0.370792	+	0.928716i	$0.379086\pi$
$734$	0	0
$735$	5769.00	0.289514
$736$	0	0
$737$	2971.30	0.148506
$738$	0	0
$739$	5677.64	0.282619	0.141309	−	0.989965i	$-0.454869\pi$
$739$	5677.64	0.282619	0.141309	+	0.989965i	$0.454869\pi$
$740$	0	0
$741$	1425.47	0.0706693
$742$	0	0
$743$	7330.81	0.361967	0.180983	−	0.983486i	$-0.442072\pi$
$743$	7330.81	0.361967	0.180983	+	0.983486i	$0.442072\pi$
$744$	0	0
$745$	6437.79	0.316594
$746$	0	0
$747$	21481.8	1.05218
$748$	0	0
$749$	6881.96	0.335729
$750$	0	0
$751$	−8139.92	−0.395512	−0.197756	−	0.980251i	$-0.563365\pi$
$751$	−8139.92	−0.395512	−0.197756	+	0.980251i	$0.563365\pi$
$752$	0	0
$753$	9180.79	0.444311
$754$	0	0
$755$	−32221.8	−1.55321
$756$	0	0
$757$	8241.25	0.395685	0.197842	−	0.980234i	$-0.436607\pi$
$757$	8241.25	0.395685	0.197842	+	0.980234i	$0.436607\pi$
$758$	0	0
$759$	8190.54	0.391697
$760$	0	0
$761$	13502.0	0.643165	0.321583	−	0.946881i	$-0.395785\pi$
$761$	13502.0	0.643165	0.321583	+	0.946881i	$0.395785\pi$
$762$	0	0
$763$	7442.31	0.353119
$764$	0	0
$765$	−7014.48	−0.331515
$766$	0	0
$767$	27365.0	1.28826
$768$	0	0
$769$	−35922.0	−1.68450	−0.842249	−	0.539089i	$-0.818769\pi$
$769$	−35922.0	−1.68450	−0.842249	+	0.539089i	$0.818769\pi$
$770$	0	0
$771$	7954.34	0.371554
$772$	0	0
$773$	18570.5	0.864079	0.432040	−	0.901855i	$-0.357794\pi$
$773$	18570.5	0.864079	0.432040	+	0.901855i	$0.357794\pi$
$774$	0	0
$775$	11584.2	0.536923
$776$	0	0
$777$	8109.72	0.374433
$778$	0	0
$779$	2866.22	0.131827
$780$	0	0
$781$	35013.0	1.60418
$782$	0	0
$783$	26502.0	1.20958
$784$	0	0
$785$	−6630.20	−0.301455
$786$	0	0
$787$	1029.83	0.0466448	0.0233224	−	0.999728i	$-0.492576\pi$
$787$	1029.83	0.0466448	0.0233224	+	0.999728i	$0.492576\pi$
$788$	0	0
$789$	−11858.9	−0.535092
$790$	0	0
$791$	−10189.5	−0.458026
$792$	0	0
$793$	11603.7	0.519619
$794$	0	0
$795$	−12397.4	−0.553069
$796$	0	0
$797$	−7192.39	−0.319658	−0.159829	−	0.987145i	$-0.551094\pi$
$797$	−7192.39	−0.319658	−0.159829	+	0.987145i	$0.551094\pi$
$798$	0	0
$799$	−8496.13	−0.376185
$800$	0	0
$801$	−17438.2	−0.769223
$802$	0	0
$803$	−21835.3	−0.959591
$804$	0	0
$805$	8021.97	0.351227
$806$	0	0
$807$	586.814	0.0255971
$808$	0	0
$809$	41160.0	1.78876	0.894382	−	0.447304i	$-0.147616\pi$
$809$	41160.0	1.78876	0.894382	+	0.447304i	$0.147616\pi$
$810$	0	0
$811$	−24315.9	−1.05283	−0.526415	−	0.850228i	$-0.676464\pi$
$811$	−24315.9	−1.05283	−0.526415	+	0.850228i	$0.676464\pi$
$812$	0	0
$813$	3907.83	0.168578
$814$	0	0
$815$	−7744.56	−0.332859
$816$	0	0
$817$	5401.19	0.231290
$818$	0	0
$819$	−6214.91	−0.265161
$820$	0	0
$821$	13863.8	0.589343	0.294671	−	0.955599i	$-0.404790\pi$
$821$	13863.8	0.589343	0.294671	+	0.955599i	$0.404790\pi$
$822$	0	0
$823$	−18436.9	−0.780886	−0.390443	−	0.920627i	$-0.627678\pi$
$823$	−18436.9	−0.780886	−0.390443	+	0.920627i	$0.627678\pi$
$824$	0	0
$825$	−3334.73	−0.140728
$826$	0	0
$827$	−13852.5	−0.582466	−0.291233	−	0.956652i	$-0.594065\pi$
$827$	−13852.5	−0.582466	−0.291233	+	0.956652i	$0.594065\pi$
$828$	0	0
$829$	−2643.89	−0.110768	−0.0553838	−	0.998465i	$-0.517638\pi$
$829$	−2643.89	−0.110768	−0.0553838	+	0.998465i	$0.517638\pi$
$830$	0	0
$831$	−14971.7	−0.624987
$832$	0	0
$833$	−8870.70	−0.368969
$834$	0	0
$835$	−11623.3	−0.481728
$836$	0	0
$837$	−37560.1	−1.55110
$838$	0	0
$839$	−4288.75	−0.176477	−0.0882384	−	0.996099i	$-0.528124\pi$
$839$	−4288.75	−0.176477	−0.0882384	+	0.996099i	$0.528124\pi$
$840$	0	0
$841$	26790.0	1.09844
$842$	0	0
$843$	−9224.26	−0.376869
$844$	0	0
$845$	−11781.3	−0.479633
$846$	0	0
$847$	984.412	0.0399348
$848$	0	0
$849$	11191.5	0.452403
$850$	0	0
$851$	−30787.8	−1.24018
$852$	0	0
$853$	−23014.8	−0.923814	−0.461907	−	0.886928i	$-0.652835\pi$
$853$	−23014.8	−0.923814	−0.461907	+	0.886928i	$0.652835\pi$
$854$	0	0
$855$	3771.79	0.150868
$856$	0	0
$857$	−47786.3	−1.90472	−0.952362	−	0.304971i	$-0.901353\pi$
$857$	−47786.3	−1.90472	−0.952362	+	0.304971i	$0.901353\pi$
$858$	0	0
$859$	17545.4	0.696905	0.348453	−	0.937326i	$-0.386707\pi$
$859$	17545.4	0.696905	0.348453	+	0.937326i	$0.386707\pi$
$860$	0	0
$861$	3526.19	0.139573
$862$	0	0
$863$	26675.6	1.05220	0.526100	−	0.850423i	$-0.323654\pi$
$863$	26675.6	1.05220	0.526100	+	0.850423i	$0.323654\pi$
$864$	0	0
$865$	−34943.3	−1.37353
$866$	0	0
$867$	8932.59	0.349904
$868$	0	0
$869$	22770.7	0.888889
$870$	0	0
$871$	−2415.24	−0.0939577
$872$	0	0
$873$	2146.29	0.0832082
$874$	0	0
$875$	−14565.9	−0.562761
$876$	0	0
$877$	12172.5	0.468684	0.234342	−	0.972154i	$-0.424706\pi$
$877$	12172.5	0.468684	0.234342	+	0.972154i	$0.424706\pi$
$878$	0	0
$879$	11456.4	0.439607
$880$	0	0
$881$	−23201.8	−0.887273	−0.443636	−	0.896207i	$-0.646312\pi$
$881$	−23201.8	−0.887273	−0.443636	+	0.896207i	$0.646312\pi$
$882$	0	0
$883$	26967.8	1.02779	0.513895	−	0.857853i	$-0.328202\pi$
$883$	26967.8	1.02779	0.513895	+	0.857853i	$0.328202\pi$
$884$	0	0
$885$	−20431.7	−0.776048
$886$	0	0
$887$	−22005.0	−0.832984	−0.416492	−	0.909139i	$-0.636741\pi$
$887$	−22005.0	−0.832984	−0.416492	+	0.909139i	$0.636741\pi$
$888$	0	0
$889$	−12997.2	−0.490339
$890$	0	0
$891$	−10715.5	−0.402900
$892$	0	0
$893$	4568.49	0.171197
$894$	0	0
$895$	−34971.9	−1.30613
$896$	0	0
$897$	−6657.73	−0.247821
$898$	0	0
$899$	−72533.7	−2.69092
$900$	0	0
$901$	19062.8	0.704855
$902$	0	0
$903$	6644.86	0.244880
$904$	0	0
$905$	9487.30	0.348474
$906$	0	0
$907$	31155.8	1.14059	0.570293	−	0.821441i	$-0.306830\pi$
$907$	31155.8	1.14059	0.570293	+	0.821441i	$0.306830\pi$
$908$	0	0
$909$	15491.9	0.565273
$910$	0	0
$911$	−37698.9	−1.37104	−0.685522	−	0.728052i	$-0.740426\pi$
$911$	−37698.9	−1.37104	−0.685522	+	0.728052i	$0.740426\pi$
$912$	0	0
$913$	38625.7	1.40014
$914$	0	0
$915$	−8663.69	−0.313019
$916$	0	0
$917$	−20097.7	−0.723756
$918$	0	0
$919$	18251.9	0.655142	0.327571	−	0.944827i	$-0.393770\pi$
$919$	18251.9	0.655142	0.327571	+	0.944827i	$0.393770\pi$
$920$	0	0
$921$	−12722.4	−0.455175
$922$	0	0
$923$	−28460.5	−1.01494
$924$	0	0
$925$	12535.1	0.445568
$926$	0	0
$927$	3897.62	0.138095
$928$	0	0
$929$	38440.4	1.35758	0.678788	−	0.734334i	$-0.262505\pi$
$929$	38440.4	1.35758	0.678788	+	0.734334i	$0.262505\pi$
$930$	0	0
$931$	4769.90	0.167913
$932$	0	0
$933$	−16579.1	−0.581754
$934$	0	0
$935$	−12612.5	−0.441148
$936$	0	0
$937$	44092.7	1.53730	0.768649	−	0.639671i	$-0.220930\pi$
$937$	44092.7	1.53730	0.768649	+	0.639671i	$0.220930\pi$
$938$	0	0
$939$	15130.9	0.525855
$940$	0	0
$941$	13844.9	0.479630	0.239815	−	0.970819i	$-0.422913\pi$
$941$	13844.9	0.479630	0.239815	+	0.970819i	$0.422913\pi$
$942$	0	0
$943$	−13386.9	−0.462286
$944$	0	0
$945$	10589.9	0.364539
$946$	0	0
$947$	−22422.6	−0.769416	−0.384708	−	0.923038i	$-0.625698\pi$
$947$	−22422.6	−0.769416	−0.384708	+	0.923038i	$0.625698\pi$
$948$	0	0
$949$	17749.0	0.607119
$950$	0	0
$951$	12303.8	0.419536
$952$	0	0
$953$	31743.4	1.07898	0.539491	−	0.841991i	$-0.318617\pi$
$953$	31743.4	1.07898	0.539491	+	0.841991i	$0.318617\pi$
$954$	0	0
$955$	14081.7	0.477146
$956$	0	0
$957$	20880.3	0.705291
$958$	0	0
$959$	−21635.5	−0.728515
$960$	0	0
$961$	73007.8	2.45067
$962$	0	0
$963$	−15112.9	−0.505717
$964$	0	0
$965$	−653.458	−0.0217985
$966$	0	0
$967$	−9446.38	−0.314142	−0.157071	−	0.987587i	$-0.550205\pi$
$967$	−9446.38	−0.314142	−0.157071	+	0.987587i	$0.550205\pi$
$968$	0	0
$969$	1636.53	0.0542547
$970$	0	0
$971$	−9714.03	−0.321048	−0.160524	−	0.987032i	$-0.551318\pi$
$971$	−9714.03	−0.321048	−0.160524	+	0.987032i	$0.551318\pi$
$972$	0	0
$973$	12243.1	0.403388
$974$	0	0
$975$	2710.66	0.0890364
$976$	0	0
$977$	−53402.2	−1.74871	−0.874354	−	0.485288i	$-0.838715\pi$
$977$	−53402.2	−1.74871	−0.874354	+	0.485288i	$0.838715\pi$
$978$	0	0
$979$	−31355.1	−1.02361
$980$	0	0
$981$	−16343.4	−0.531911
$982$	0	0
$983$	22085.7	0.716608	0.358304	−	0.933605i	$-0.383355\pi$
$983$	22085.7	0.716608	0.358304	+	0.933605i	$0.383355\pi$
$984$	0	0
$985$	−4981.84	−0.161152
$986$	0	0
$987$	5620.42	0.181256
$988$	0	0
$989$	−25226.6	−0.811080
$990$	0	0
$991$	56382.3	1.80731	0.903654	−	0.428263i	$-0.140874\pi$
$991$	56382.3	1.80731	0.903654	+	0.428263i	$0.140874\pi$
$992$	0	0
$993$	−18716.9	−0.598150
$994$	0	0
$995$	−26656.1	−0.849303
$996$	0	0
$997$	−3946.90	−0.125376	−0.0626878	−	0.998033i	$-0.519967\pi$
$997$	−3946.90	−0.125376	−0.0626878	+	0.998033i	$0.519967\pi$
$998$	0	0
$999$	−40643.3	−1.28719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bg.1.3		7
4.3	odd	2		1216.4.a.bf.1.5		7
8.3	odd	2		608.4.a.k.1.3	yes	7
8.5	even	2		608.4.a.j.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.5	✓	7	8.5	even	2
608.4.a.k.1.3	yes	7	8.3	odd	2
1216.4.a.bf.1.5		7	4.3	odd	2
1216.4.a.bg.1.3		7	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43763 q^{3} +9.42708 q^{5} +9.58919 q^{7} -21.0580 q^{9} +O(q^{10})\)	\(q-2.43763 q^{3} +9.42708 q^{5} +9.58919 q^{7} -21.0580 q^{9} -37.8637 q^{11} +30.7777 q^{13} -22.9797 q^{15} +35.3347 q^{17} -19.0000 q^{19} -23.3749 q^{21} +88.7405 q^{23} -36.1302 q^{25} +117.148 q^{27} +226.228 q^{29} -320.623 q^{31} +92.2976 q^{33} +90.3981 q^{35} -346.942 q^{37} -75.0247 q^{39} -150.854 q^{41} -284.273 q^{43} -198.515 q^{45} -240.447 q^{47} -251.047 q^{49} -86.1330 q^{51} +539.492 q^{53} -356.944 q^{55} +46.3150 q^{57} +889.117 q^{59} +377.014 q^{61} -201.929 q^{63} +290.144 q^{65} -78.4735 q^{67} -216.317 q^{69} -924.711 q^{71} +576.682 q^{73} +88.0720 q^{75} -363.082 q^{77} -601.388 q^{79} +283.003 q^{81} -1020.13 q^{83} +333.103 q^{85} -551.459 q^{87} +828.104 q^{89} +295.134 q^{91} +781.559 q^{93} -179.114 q^{95} -101.923 q^{97} +797.332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9	\( 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9} - 33 q^{11} + 35 q^{13} - 120 q^{15} + 66 q^{17} - 133 q^{19} - 33 q^{21} - 389 q^{23} + 44 q^{25} - 39 q^{27} - 233 q^{29} - 158 q^{31} - 206 q^{33} + 123 q^{35} + 436 q^{37} - 807 q^{39} - 94 q^{41} + 645 q^{43} - 103 q^{45} - 1451 q^{47} + 93 q^{49} + 1741 q^{51} - 3 q^{53} - 1971 q^{55} - 57 q^{57} + 297 q^{59} - 93 q^{61} - 2999 q^{63} - 788 q^{65} + 1641 q^{67} - 945 q^{69} - 2392 q^{71} + 324 q^{73} + 1909 q^{75} + 711 q^{77} - 2492 q^{79} + 143 q^{81} + 310 q^{83} - 2353 q^{85} - 4795 q^{87} - 440 q^{89} - 107 q^{91} - 900 q^{93} - 323 q^{95} - 532 q^{97} - 1591 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9 - 33 * q^11 + 35 * q^13 - 120 * q^15 + 66 * q^17 - 133 * q^19 - 33 * q^21 - 389 * q^23 + 44 * q^25 - 39 * q^27 - 233 * q^29 - 158 * q^31 - 206 * q^33 + 123 * q^35 + 436 * q^37 - 807 * q^39 - 94 * q^41 + 645 * q^43 - 103 * q^45 - 1451 * q^47 + 93 * q^49 + 1741 * q^51 - 3 * q^53 - 1971 * q^55 - 57 * q^57 + 297 * q^59 - 93 * q^61 - 2999 * q^63 - 788 * q^65 + 1641 * q^67 - 945 * q^69 - 2392 * q^71 + 324 * q^73 + 1909 * q^75 + 711 * q^77 - 2492 * q^79 + 143 * q^81 + 310 * q^83 - 2353 * q^85 - 4795 * q^87 - 440 * q^89 - 107 * q^91 - 900 * q^93 - 323 * q^95 - 532 * q^97 - 1591 * q^99

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