LMFDB - Embedded newform 2352.4.a.cg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.57516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 24x + 6 $ x^3 - x^2 - 24*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.55637$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -17.8732 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -17.8732 q^{5} +9.00000 q^{9} +11.3942 q^{11} +13.0987 q^{13} +53.6196 q^{15} -53.2674 q^{17} -42.4223 q^{19} -152.085 q^{23} +194.451 q^{25} -27.0000 q^{27} +186.493 q^{29} -157.874 q^{31} -34.1825 q^{33} +3.74588 q^{37} -39.2960 q^{39} +39.3230 q^{41} -429.439 q^{43} -160.859 q^{45} +21.1869 q^{47} +159.802 q^{51} +365.904 q^{53} -203.650 q^{55} +127.267 q^{57} -226.578 q^{59} -651.973 q^{61} -234.115 q^{65} -145.433 q^{67} +456.256 q^{69} +368.962 q^{71} -608.906 q^{73} -583.354 q^{75} -910.237 q^{79} +81.0000 q^{81} -327.929 q^{83} +952.058 q^{85} -559.480 q^{87} +37.6118 q^{89} +473.621 q^{93} +758.222 q^{95} -722.013 q^{97} +102.547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9	$ 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9} - 35 q^{11} - 62 q^{13} + 33 q^{15} - 48 q^{17} - 202 q^{19} - 216 q^{23} + 130 q^{25} - 81 q^{27} + 53 q^{29} - 95 q^{31} + 105 q^{33} + 262 q^{37} + 186 q^{39} - 244 q^{41} - 360 q^{43} - 99 q^{45} - 210 q^{47} + 144 q^{51} + 393 q^{53} - 1031 q^{55} + 606 q^{57} + 1143 q^{59} + 70 q^{61} - 472 q^{65} + 628 q^{67} + 648 q^{69} - 318 q^{71} - 988 q^{73} - 390 q^{75} - 861 q^{79} + 243 q^{81} + 519 q^{83} + 1800 q^{85} - 159 q^{87} - 1766 q^{89} + 285 q^{93} + 736 q^{95} - 19 q^{97} - 315 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9 - 35 * q^11 - 62 * q^13 + 33 * q^15 - 48 * q^17 - 202 * q^19 - 216 * q^23 + 130 * q^25 - 81 * q^27 + 53 * q^29 - 95 * q^31 + 105 * q^33 + 262 * q^37 + 186 * q^39 - 244 * q^41 - 360 * q^43 - 99 * q^45 - 210 * q^47 + 144 * q^51 + 393 * q^53 - 1031 * q^55 + 606 * q^57 + 1143 * q^59 + 70 * q^61 - 472 * q^65 + 628 * q^67 + 648 * q^69 - 318 * q^71 - 988 * q^73 - 390 * q^75 - 861 * q^79 + 243 * q^81 + 519 * q^83 + 1800 * q^85 - 159 * q^87 - 1766 * q^89 + 285 * q^93 + 736 * q^95 - 19 * q^97 - 315 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−17.8732	−1.59863	−0.799314	−	0.600914i	$-0.794804\pi$
$5$	−17.8732	−1.59863	−0.799314	+	0.600914i	$0.794804\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.3942	0.312315	0.156158	−	0.987732i	$-0.450089\pi$
$11$	11.3942	0.312315	0.156158	+	0.987732i	$0.450089\pi$
$12$	0	0
$13$	13.0987	0.279455	0.139728	−	0.990190i	$-0.455377\pi$
$13$	13.0987	0.279455	0.139728	+	0.990190i	$0.455377\pi$
$14$	0	0
$15$	53.6196	0.922968
$16$	0	0
$17$	−53.2674	−0.759955	−0.379977	−	0.924996i	$-0.624068\pi$
$17$	−53.2674	−0.759955	−0.379977	+	0.924996i	$0.624068\pi$
$18$	0	0
$19$	−42.4223	−0.512228	−0.256114	−	0.966647i	$-0.582442\pi$
$19$	−42.4223	−0.512228	−0.256114	+	0.966647i	$0.582442\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−152.085	−1.37878	−0.689391	−	0.724389i	$-0.742122\pi$
$23$	−152.085	−1.37878	−0.689391	+	0.724389i	$0.742122\pi$
$24$	0	0
$25$	194.451	1.55561
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	186.493	1.19417	0.597085	−	0.802178i	$-0.296325\pi$
$29$	186.493	1.19417	0.597085	+	0.802178i	$0.296325\pi$
$30$	0	0
$31$	−157.874	−0.914676	−0.457338	−	0.889293i	$-0.651197\pi$
$31$	−157.874	−0.914676	−0.457338	+	0.889293i	$0.651197\pi$
$32$	0	0
$33$	−34.1825	−0.180315
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.74588	0.0166438	0.00832188	−	0.999965i	$-0.497351\pi$
$37$	3.74588	0.0166438	0.00832188	+	0.999965i	$0.497351\pi$
$38$	0	0
$39$	−39.2960	−0.161344
$40$	0	0
$41$	39.3230	0.149786	0.0748930	−	0.997192i	$-0.476138\pi$
$41$	39.3230	0.149786	0.0748930	+	0.997192i	$0.476138\pi$
$42$	0	0
$43$	−429.439	−1.52300	−0.761498	−	0.648168i	$-0.775536\pi$
$43$	−429.439	−1.52300	−0.761498	+	0.648168i	$0.775536\pi$
$44$	0	0
$45$	−160.859	−0.532876
$46$	0	0
$47$	21.1869	0.0657537	0.0328768	−	0.999459i	$-0.489533\pi$
$47$	21.1869	0.0657537	0.0328768	+	0.999459i	$0.489533\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	159.802	0.438760
$52$	0	0
$53$	365.904	0.948317	0.474158	−	0.880440i	$-0.342752\pi$
$53$	365.904	0.948317	0.474158	+	0.880440i	$0.342752\pi$
$54$	0	0
$55$	−203.650	−0.499276
$56$	0	0
$57$	127.267	0.295735
$58$	0	0
$59$	−226.578	−0.499964	−0.249982	−	0.968250i	$-0.580425\pi$
$59$	−226.578	−0.499964	−0.249982	+	0.968250i	$0.580425\pi$
$60$	0	0
$61$	−651.973	−1.36847	−0.684235	−	0.729262i	$-0.739864\pi$
$61$	−651.973	−1.36847	−0.684235	+	0.729262i	$0.739864\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−234.115	−0.446745
$66$	0	0
$67$	−145.433	−0.265186	−0.132593	−	0.991171i	$-0.542330\pi$
$67$	−145.433	−0.265186	−0.132593	+	0.991171i	$0.542330\pi$
$68$	0	0
$69$	456.256	0.796041
$70$	0	0
$71$	368.962	0.616728	0.308364	−	0.951268i	$-0.400218\pi$
$71$	368.962	0.616728	0.308364	+	0.951268i	$0.400218\pi$
$72$	0	0
$73$	−608.906	−0.976261	−0.488130	−	0.872771i	$-0.662321\pi$
$73$	−608.906	−0.976261	−0.488130	+	0.872771i	$0.662321\pi$
$74$	0	0
$75$	−583.354	−0.898133
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−910.237	−1.29633	−0.648163	−	0.761502i	$-0.724462\pi$
$79$	−910.237	−1.29633	−0.648163	+	0.761502i	$0.724462\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−327.929	−0.433674	−0.216837	−	0.976208i	$-0.569574\pi$
$83$	−327.929	−0.433674	−0.216837	+	0.976208i	$0.569574\pi$
$84$	0	0
$85$	952.058	1.21489
$86$	0	0
$87$	−559.480	−0.689455
$88$	0	0
$89$	37.6118	0.0447960	0.0223980	−	0.999749i	$-0.492870\pi$
$89$	37.6118	0.0447960	0.0223980	+	0.999749i	$0.492870\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	473.621	0.528089
$94$	0	0
$95$	758.222	0.818863
$96$	0	0
$97$	−722.013	−0.755766	−0.377883	−	0.925853i	$-0.623348\pi$
$97$	−722.013	−0.755766	−0.377883	+	0.925853i	$0.623348\pi$
$98$	0	0
$99$	102.547	0.104105
$100$	0	0
$101$	−1518.67	−1.49617	−0.748087	−	0.663601i	$-0.769027\pi$
$101$	−1518.67	−1.49617	−0.748087	+	0.663601i	$0.769027\pi$
$102$	0	0
$103$	−1051.88	−1.00626	−0.503132	−	0.864210i	$-0.667819\pi$
$103$	−1051.88	−1.00626	−0.503132	+	0.864210i	$0.667819\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−766.520	−0.692545	−0.346273	−	0.938134i	$-0.612553\pi$
$107$	−766.520	−0.692545	−0.346273	+	0.938134i	$0.612553\pi$
$108$	0	0
$109$	−1427.05	−1.25400	−0.627002	−	0.779018i	$-0.715718\pi$
$109$	−1427.05	−1.25400	−0.627002	+	0.779018i	$0.715718\pi$
$110$	0	0
$111$	−11.2376	−0.00960928
$112$	0	0
$113$	362.564	0.301833	0.150917	−	0.988546i	$-0.451778\pi$
$113$	362.564	0.301833	0.150917	+	0.988546i	$0.451778\pi$
$114$	0	0
$115$	2718.25	2.20416
$116$	0	0
$117$	117.888	0.0931517
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1201.17	−0.902459
$122$	0	0
$123$	−117.969	−0.0864789
$124$	0	0
$125$	−1241.32	−0.888216
$126$	0	0
$127$	−974.777	−0.681082	−0.340541	−	0.940230i	$-0.610610\pi$
$127$	−974.777	−0.681082	−0.340541	+	0.940230i	$0.610610\pi$
$128$	0	0
$129$	1288.32	0.879302
$130$	0	0
$131$	−1792.70	−1.19564	−0.597821	−	0.801629i	$-0.703967\pi$
$131$	−1792.70	−1.19564	−0.597821	+	0.801629i	$0.703967\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	482.577	0.307656
$136$	0	0
$137$	1684.42	1.05043	0.525217	−	0.850969i	$-0.323984\pi$
$137$	1684.42	1.05043	0.525217	+	0.850969i	$0.323984\pi$
$138$	0	0
$139$	315.089	0.192270	0.0961350	−	0.995368i	$-0.469352\pi$
$139$	315.089	0.192270	0.0961350	+	0.995368i	$0.469352\pi$
$140$	0	0
$141$	−63.5606	−0.0379629
$142$	0	0
$143$	149.248	0.0872781
$144$	0	0
$145$	−3333.23	−1.90903
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1893.77	1.04123	0.520617	−	0.853790i	$-0.325702\pi$
$149$	1893.77	1.04123	0.520617	+	0.853790i	$0.325702\pi$
$150$	0	0
$151$	2011.84	1.08425	0.542124	−	0.840299i	$-0.317620\pi$
$151$	2011.84	1.08425	0.542124	+	0.840299i	$0.317620\pi$
$152$	0	0
$153$	−479.406	−0.253318
$154$	0	0
$155$	2821.71	1.46223
$156$	0	0
$157$	3828.50	1.94616	0.973082	−	0.230460i	$-0.0740232\pi$
$157$	3828.50	1.94616	0.973082	+	0.230460i	$0.0740232\pi$
$158$	0	0
$159$	−1097.71	−0.547511
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3509.26	1.68630	0.843148	−	0.537682i	$-0.180700\pi$
$163$	3509.26	1.68630	0.843148	+	0.537682i	$0.180700\pi$
$164$	0	0
$165$	610.950	0.288257
$166$	0	0
$167$	−343.008	−0.158939	−0.0794694	−	0.996837i	$-0.525323\pi$
$167$	−343.008	−0.158939	−0.0794694	+	0.996837i	$0.525323\pi$
$168$	0	0
$169$	−2025.42	−0.921905
$170$	0	0
$171$	−381.800	−0.170743
$172$	0	0
$173$	4187.21	1.84016	0.920081	−	0.391729i	$-0.128123\pi$
$173$	4187.21	1.84016	0.920081	+	0.391729i	$0.128123\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	679.733	0.288654
$178$	0	0
$179$	1970.29	0.822716	0.411358	−	0.911474i	$-0.365055\pi$
$179$	1970.29	0.822716	0.411358	+	0.911474i	$0.365055\pi$
$180$	0	0
$181$	3613.10	1.48376	0.741878	−	0.670535i	$-0.233935\pi$
$181$	3613.10	1.48376	0.741878	+	0.670535i	$0.233935\pi$
$182$	0	0
$183$	1955.92	0.790086
$184$	0	0
$185$	−66.9509	−0.0266072
$186$	0	0
$187$	−606.936	−0.237345
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1907.77	−0.722729	−0.361365	−	0.932425i	$-0.617689\pi$
$191$	−1907.77	−0.722729	−0.361365	+	0.932425i	$0.617689\pi$
$192$	0	0
$193$	2399.93	0.895080	0.447540	−	0.894264i	$-0.352300\pi$
$193$	2399.93	0.895080	0.447540	+	0.894264i	$0.352300\pi$
$194$	0	0
$195$	702.346	0.257928
$196$	0	0
$197$	1514.32	0.547668	0.273834	−	0.961777i	$-0.411708\pi$
$197$	1514.32	0.547668	0.273834	+	0.961777i	$0.411708\pi$
$198$	0	0
$199$	1367.78	0.487232	0.243616	−	0.969872i	$-0.421666\pi$
$199$	1367.78	0.487232	0.243616	+	0.969872i	$0.421666\pi$
$200$	0	0
$201$	436.300	0.153105
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−702.828	−0.239452
$206$	0	0
$207$	−1368.77	−0.459594
$208$	0	0
$209$	−483.366	−0.159977
$210$	0	0
$211$	−4302.52	−1.40378	−0.701891	−	0.712285i	$-0.747661\pi$
$211$	−4302.52	−1.40378	−0.701891	+	0.712285i	$0.747661\pi$
$212$	0	0
$213$	−1106.89	−0.356068
$214$	0	0
$215$	7675.45	2.43470
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1826.72	0.563644
$220$	0	0
$221$	−697.731	−0.212373
$222$	0	0
$223$	−1497.19	−0.449592	−0.224796	−	0.974406i	$-0.572172\pi$
$223$	−1497.19	−0.449592	−0.224796	+	0.974406i	$0.572172\pi$
$224$	0	0
$225$	1750.06	0.518537
$226$	0	0
$227$	1603.32	0.468795	0.234397	−	0.972141i	$-0.424688\pi$
$227$	1603.32	0.468795	0.234397	+	0.972141i	$0.424688\pi$
$228$	0	0
$229$	−1010.52	−0.291603	−0.145802	−	0.989314i	$-0.546576\pi$
$229$	−1010.52	−0.291603	−0.145802	+	0.989314i	$0.546576\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	198.217	0.0557323	0.0278661	−	0.999612i	$-0.491129\pi$
$233$	198.217	0.0557323	0.0278661	+	0.999612i	$0.491129\pi$
$234$	0	0
$235$	−378.677	−0.105116
$236$	0	0
$237$	2730.71	0.748434
$238$	0	0
$239$	1201.19	0.325098	0.162549	−	0.986700i	$-0.448028\pi$
$239$	1201.19	0.325098	0.162549	+	0.986700i	$0.448028\pi$
$240$	0	0
$241$	2732.69	0.730407	0.365204	−	0.930928i	$-0.380999\pi$
$241$	2732.69	0.730407	0.365204	+	0.930928i	$0.380999\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−555.675	−0.143145
$248$	0	0
$249$	983.788	0.250382
$250$	0	0
$251$	7565.82	1.90259	0.951295	−	0.308281i	$-0.0997537\pi$
$251$	7565.82	1.90259	0.951295	+	0.308281i	$0.0997537\pi$
$252$	0	0
$253$	−1732.88	−0.430615
$254$	0	0
$255$	−2856.18	−0.701414
$256$	0	0
$257$	−5008.68	−1.21569	−0.607846	−	0.794055i	$-0.707966\pi$
$257$	−5008.68	−1.21569	−0.607846	+	0.794055i	$0.707966\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1678.44	0.398057
$262$	0	0
$263$	6248.81	1.46509	0.732544	−	0.680720i	$-0.238333\pi$
$263$	6248.81	1.46509	0.732544	+	0.680720i	$0.238333\pi$
$264$	0	0
$265$	−6539.88	−1.51601
$266$	0	0
$267$	−112.835	−0.0258630
$268$	0	0
$269$	3588.45	0.813351	0.406676	−	0.913573i	$-0.366688\pi$
$269$	3588.45	0.813351	0.406676	+	0.913573i	$0.366688\pi$
$270$	0	0
$271$	−1983.14	−0.444529	−0.222264	−	0.974986i	$-0.571345\pi$
$271$	−1983.14	−0.444529	−0.222264	+	0.974986i	$0.571345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2215.61	0.485841
$276$	0	0
$277$	7363.91	1.59731	0.798654	−	0.601790i	$-0.205546\pi$
$277$	7363.91	1.59731	0.798654	+	0.601790i	$0.205546\pi$
$278$	0	0
$279$	−1420.86	−0.304892
$280$	0	0
$281$	−5312.05	−1.12772	−0.563861	−	0.825869i	$-0.690685\pi$
$281$	−5312.05	−1.12772	−0.563861	+	0.825869i	$0.690685\pi$
$282$	0	0
$283$	−1091.76	−0.229324	−0.114662	−	0.993405i	$-0.536578\pi$
$283$	−1091.76	−0.229324	−0.114662	+	0.993405i	$0.536578\pi$
$284$	0	0
$285$	−2274.67	−0.472770
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2075.59	−0.422469
$290$	0	0
$291$	2166.04	0.436341
$292$	0	0
$293$	7191.86	1.43397	0.716985	−	0.697089i	$-0.245522\pi$
$293$	7191.86	1.43397	0.716985	+	0.697089i	$0.245522\pi$
$294$	0	0
$295$	4049.67	0.799257
$296$	0	0
$297$	−307.642	−0.0601051
$298$	0	0
$299$	−1992.12	−0.385308
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4556.02	0.863816
$304$	0	0
$305$	11652.9	2.18767
$306$	0	0
$307$	541.355	0.100641	0.0503204	−	0.998733i	$-0.483976\pi$
$307$	541.355	0.100641	0.0503204	+	0.998733i	$0.483976\pi$
$308$	0	0
$309$	3155.65	0.580966
$310$	0	0
$311$	54.0168	0.00984892	0.00492446	−	0.999988i	$-0.498432\pi$
$311$	54.0168	0.00984892	0.00492446	+	0.999988i	$0.498432\pi$
$312$	0	0
$313$	−3772.94	−0.681340	−0.340670	−	0.940183i	$-0.610654\pi$
$313$	−3772.94	−0.681340	−0.340670	+	0.940183i	$0.610654\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1719.24	0.304612	0.152306	−	0.988333i	$-0.451330\pi$
$317$	1719.24	0.304612	0.152306	+	0.988333i	$0.451330\pi$
$318$	0	0
$319$	2124.93	0.372957
$320$	0	0
$321$	2299.56	0.399841
$322$	0	0
$323$	2259.72	0.389270
$324$	0	0
$325$	2547.06	0.434724
$326$	0	0
$327$	4281.15	0.724000
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8408.21	−1.39625	−0.698123	−	0.715978i	$-0.745981\pi$
$331$	−8408.21	−1.39625	−0.698123	+	0.715978i	$0.745981\pi$
$332$	0	0
$333$	33.7129	0.00554792
$334$	0	0
$335$	2599.36	0.423935
$336$	0	0
$337$	2789.46	0.450894	0.225447	−	0.974255i	$-0.427616\pi$
$337$	2789.46	0.450894	0.225447	+	0.974255i	$0.427616\pi$
$338$	0	0
$339$	−1087.69	−0.174263
$340$	0	0
$341$	−1798.84	−0.285667
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8154.76	−1.27257
$346$	0	0
$347$	3471.96	0.537132	0.268566	−	0.963261i	$-0.413450\pi$
$347$	3471.96	0.537132	0.268566	+	0.963261i	$0.413450\pi$
$348$	0	0
$349$	6626.12	1.01630	0.508149	−	0.861269i	$-0.330330\pi$
$349$	6626.12	1.01630	0.508149	+	0.861269i	$0.330330\pi$
$350$	0	0
$351$	−353.664	−0.0537812
$352$	0	0
$353$	−9468.40	−1.42763	−0.713813	−	0.700337i	$-0.753033\pi$
$353$	−9468.40	−1.42763	−0.713813	+	0.700337i	$0.753033\pi$
$354$	0	0
$355$	−6594.53	−0.985919
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6279.55	0.923182	0.461591	−	0.887093i	$-0.347279\pi$
$359$	6279.55	0.923182	0.461591	+	0.887093i	$0.347279\pi$
$360$	0	0
$361$	−5059.35	−0.737622
$362$	0	0
$363$	3603.52	0.521035
$364$	0	0
$365$	10883.1	1.56068
$366$	0	0
$367$	−10827.8	−1.54008	−0.770038	−	0.637998i	$-0.779763\pi$
$367$	−10827.8	−1.54008	−0.770038	+	0.637998i	$0.779763\pi$
$368$	0	0
$369$	353.907	0.0499286
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5239.23	0.727284	0.363642	−	0.931539i	$-0.381533\pi$
$373$	5239.23	0.727284	0.363642	+	0.931539i	$0.381533\pi$
$374$	0	0
$375$	3723.96	0.512812
$376$	0	0
$377$	2442.81	0.333717
$378$	0	0
$379$	11050.4	1.49768	0.748839	−	0.662751i	$-0.230611\pi$
$379$	11050.4	1.49768	0.748839	+	0.662751i	$0.230611\pi$
$380$	0	0
$381$	2924.33	0.393223
$382$	0	0
$383$	−10468.0	−1.39658	−0.698292	−	0.715813i	$-0.746056\pi$
$383$	−10468.0	−1.39658	−0.698292	+	0.715813i	$0.746056\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3864.95	−0.507665
$388$	0	0
$389$	−11614.0	−1.51377	−0.756884	−	0.653550i	$-0.773279\pi$
$389$	−11614.0	−1.51377	−0.756884	+	0.653550i	$0.773279\pi$
$390$	0	0
$391$	8101.19	1.04781
$392$	0	0
$393$	5378.11	0.690304
$394$	0	0
$395$	16268.9	2.07234
$396$	0	0
$397$	6707.30	0.847934	0.423967	−	0.905678i	$-0.360637\pi$
$397$	6707.30	0.847934	0.423967	+	0.905678i	$0.360637\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5526.38	0.688215	0.344107	−	0.938930i	$-0.388182\pi$
$401$	5526.38	0.688215	0.344107	+	0.938930i	$0.388182\pi$
$402$	0	0
$403$	−2067.94	−0.255611
$404$	0	0
$405$	−1447.73	−0.177625
$406$	0	0
$407$	42.6811	0.00519810
$408$	0	0
$409$	1318.91	0.159452	0.0797258	−	0.996817i	$-0.474596\pi$
$409$	1318.91	0.159452	0.0797258	+	0.996817i	$0.474596\pi$
$410$	0	0
$411$	−5053.25	−0.606468
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5861.15	0.693283
$416$	0	0
$417$	−945.268	−0.111007
$418$	0	0
$419$	3656.13	0.426286	0.213143	−	0.977021i	$-0.431630\pi$
$419$	3656.13	0.426286	0.213143	+	0.977021i	$0.431630\pi$
$420$	0	0
$421$	−135.389	−0.0156733	−0.00783663	−	0.999969i	$-0.502495\pi$
$421$	−135.389	−0.0156733	−0.00783663	+	0.999969i	$0.502495\pi$
$422$	0	0
$423$	190.682	0.0219179
$424$	0	0
$425$	−10357.9	−1.18219
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−447.745	−0.0503900
$430$	0	0
$431$	−8389.16	−0.937568	−0.468784	−	0.883313i	$-0.655308\pi$
$431$	−8389.16	−0.937568	−0.468784	+	0.883313i	$0.655308\pi$
$432$	0	0
$433$	8243.02	0.914859	0.457430	−	0.889246i	$-0.348770\pi$
$433$	8243.02	0.914859	0.457430	+	0.889246i	$0.348770\pi$
$434$	0	0
$435$	9999.70	1.10218
$436$	0	0
$437$	6451.81	0.706252
$438$	0	0
$439$	−18283.2	−1.98772	−0.993859	−	0.110649i	$-0.964707\pi$
$439$	−18283.2	−1.98772	−0.993859	+	0.110649i	$0.964707\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1210.44	−0.129818	−0.0649092	−	0.997891i	$-0.520676\pi$
$443$	−1210.44	−0.129818	−0.0649092	+	0.997891i	$0.520676\pi$
$444$	0	0
$445$	−672.243	−0.0716121
$446$	0	0
$447$	−5681.32	−0.601157
$448$	0	0
$449$	−8301.16	−0.872508	−0.436254	−	0.899824i	$-0.643695\pi$
$449$	−8301.16	−0.872508	−0.436254	+	0.899824i	$0.643695\pi$
$450$	0	0
$451$	448.052	0.0467804
$452$	0	0
$453$	−6035.52	−0.625990
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12293.8	−1.25838	−0.629188	−	0.777253i	$-0.716612\pi$
$457$	−12293.8	−1.25838	−0.629188	+	0.777253i	$0.716612\pi$
$458$	0	0
$459$	1438.22	0.146253
$460$	0	0
$461$	−19434.2	−1.96343	−0.981717	−	0.190346i	$-0.939039\pi$
$461$	−19434.2	−1.96343	−0.981717	+	0.190346i	$0.939039\pi$
$462$	0	0
$463$	12491.1	1.25380	0.626902	−	0.779098i	$-0.284322\pi$
$463$	12491.1	1.25380	0.626902	+	0.779098i	$0.284322\pi$
$464$	0	0
$465$	−8465.13	−0.844217
$466$	0	0
$467$	3385.17	0.335433	0.167716	−	0.985835i	$-0.446361\pi$
$467$	3385.17	0.335433	0.167716	+	0.985835i	$0.446361\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11485.5	−1.12362
$472$	0	0
$473$	−4893.09	−0.475654
$474$	0	0
$475$	−8249.07	−0.796828
$476$	0	0
$477$	3293.14	0.316106
$478$	0	0
$479$	5979.41	0.570368	0.285184	−	0.958473i	$-0.407945\pi$
$479$	5979.41	0.570368	0.285184	+	0.958473i	$0.407945\pi$
$480$	0	0
$481$	49.0661	0.00465119
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12904.7	1.20819
$486$	0	0
$487$	1114.96	0.103745	0.0518725	−	0.998654i	$-0.483481\pi$
$487$	1114.96	0.103745	0.0518725	+	0.998654i	$0.483481\pi$
$488$	0	0
$489$	−10527.8	−0.973583
$490$	0	0
$491$	−1086.23	−0.0998387	−0.0499194	−	0.998753i	$-0.515896\pi$
$491$	−1086.23	−0.0998387	−0.0499194	+	0.998753i	$0.515896\pi$
$492$	0	0
$493$	−9934.01	−0.907516
$494$	0	0
$495$	−1832.85	−0.166425
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2213.50	0.198577	0.0992884	−	0.995059i	$-0.468343\pi$
$499$	2213.50	0.198577	0.0992884	+	0.995059i	$0.468343\pi$
$500$	0	0
$501$	1029.02	0.0917634
$502$	0	0
$503$	−2643.32	−0.234314	−0.117157	−	0.993113i	$-0.537378\pi$
$503$	−2643.32	−0.234314	−0.117157	+	0.993113i	$0.537378\pi$
$504$	0	0
$505$	27143.5	2.39183
$506$	0	0
$507$	6076.27	0.532262
$508$	0	0
$509$	665.169	0.0579236	0.0289618	−	0.999581i	$-0.490780\pi$
$509$	665.169	0.0579236	0.0289618	+	0.999581i	$0.490780\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1145.40	0.0985784
$514$	0	0
$515$	18800.5	1.60864
$516$	0	0
$517$	241.406	0.0205359
$518$	0	0
$519$	−12561.6	−1.06242
$520$	0	0
$521$	11762.0	0.989063	0.494531	−	0.869160i	$-0.335340\pi$
$521$	11762.0	0.989063	0.494531	+	0.869160i	$0.335340\pi$
$522$	0	0
$523$	10122.6	0.846330	0.423165	−	0.906053i	$-0.360919\pi$
$523$	10122.6	0.846330	0.423165	+	0.906053i	$0.360919\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8409.52	0.695112
$528$	0	0
$529$	10963.0	0.901042
$530$	0	0
$531$	−2039.20	−0.166655
$532$	0	0
$533$	515.079	0.0418585
$534$	0	0
$535$	13700.2	1.10712
$536$	0	0
$537$	−5910.86	−0.474995
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16118.0	1.28090	0.640449	−	0.768001i	$-0.278748\pi$
$541$	16118.0	1.28090	0.640449	+	0.768001i	$0.278748\pi$
$542$	0	0
$543$	−10839.3	−0.856647
$544$	0	0
$545$	25505.9	2.00469
$546$	0	0
$547$	626.100	0.0489399	0.0244699	−	0.999701i	$-0.492210\pi$
$547$	626.100	0.0489399	0.0244699	+	0.999701i	$0.492210\pi$
$548$	0	0
$549$	−5867.76	−0.456156
$550$	0	0
$551$	−7911.47	−0.611688
$552$	0	0
$553$	0	0
$554$	0	0
$555$	200.853	0.0153617
$556$	0	0
$557$	20771.2	1.58008	0.790039	−	0.613057i	$-0.210060\pi$
$557$	20771.2	1.58008	0.790039	+	0.613057i	$0.210060\pi$
$558$	0	0
$559$	−5625.08	−0.425609
$560$	0	0
$561$	1820.81	0.137031
$562$	0	0
$563$	−5521.72	−0.413344	−0.206672	−	0.978410i	$-0.566263\pi$
$563$	−5521.72	−0.413344	−0.206672	+	0.978410i	$0.566263\pi$
$564$	0	0
$565$	−6480.18	−0.482519
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7574.81	0.558089	0.279044	−	0.960278i	$-0.409982\pi$
$569$	7574.81	0.558089	0.279044	+	0.960278i	$0.409982\pi$
$570$	0	0
$571$	331.248	0.0242772	0.0121386	−	0.999926i	$-0.496136\pi$
$571$	331.248	0.0242772	0.0121386	+	0.999926i	$0.496136\pi$
$572$	0	0
$573$	5723.31	0.417268
$574$	0	0
$575$	−29573.2	−2.14485
$576$	0	0
$577$	−2038.11	−0.147050	−0.0735248	−	0.997293i	$-0.523425\pi$
$577$	−2038.11	−0.147050	−0.0735248	+	0.997293i	$0.523425\pi$
$578$	0	0
$579$	−7199.78	−0.516775
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4169.17	0.296174
$584$	0	0
$585$	−2107.04	−0.148915
$586$	0	0
$587$	5232.90	0.367947	0.183973	−	0.982931i	$-0.441104\pi$
$587$	5232.90	0.367947	0.183973	+	0.982931i	$0.441104\pi$
$588$	0	0
$589$	6697.36	0.468523
$590$	0	0
$591$	−4542.95	−0.316196
$592$	0	0
$593$	5720.24	0.396125	0.198062	−	0.980189i	$-0.436535\pi$
$593$	5720.24	0.396125	0.198062	+	0.980189i	$0.436535\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4103.33	−0.281304
$598$	0	0
$599$	−18088.4	−1.23384	−0.616922	−	0.787024i	$-0.711621\pi$
$599$	−18088.4	−1.23384	−0.616922	+	0.787024i	$0.711621\pi$
$600$	0	0
$601$	1821.43	0.123623	0.0618117	−	0.998088i	$-0.480312\pi$
$601$	1821.43	0.123623	0.0618117	+	0.998088i	$0.480312\pi$
$602$	0	0
$603$	−1308.90	−0.0883955
$604$	0	0
$605$	21468.8	1.44270
$606$	0	0
$607$	2372.20	0.158624	0.0793120	−	0.996850i	$-0.474728\pi$
$607$	2372.20	0.158624	0.0793120	+	0.996850i	$0.474728\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	277.520	0.0183752
$612$	0	0
$613$	9725.08	0.640770	0.320385	−	0.947287i	$-0.396188\pi$
$613$	9725.08	0.640770	0.320385	+	0.947287i	$0.396188\pi$
$614$	0	0
$615$	2108.48	0.138248
$616$	0	0
$617$	−5329.51	−0.347744	−0.173872	−	0.984768i	$-0.555628\pi$
$617$	−5329.51	−0.347744	−0.173872	+	0.984768i	$0.555628\pi$
$618$	0	0
$619$	−15976.6	−1.03740	−0.518702	−	0.854955i	$-0.673585\pi$
$619$	−15976.6	−1.03740	−0.518702	+	0.854955i	$0.673585\pi$
$620$	0	0
$621$	4106.31	0.265347
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−2120.06	−0.135684
$626$	0	0
$627$	1450.10	0.0923625
$628$	0	0
$629$	−199.533	−0.0126485
$630$	0	0
$631$	4199.98	0.264974	0.132487	−	0.991185i	$-0.457704\pi$
$631$	4199.98	0.264974	0.132487	+	0.991185i	$0.457704\pi$
$632$	0	0
$633$	12907.6	0.810474
$634$	0	0
$635$	17422.4	1.08880
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3320.66	0.205576
$640$	0	0
$641$	2648.51	0.163198	0.0815988	−	0.996665i	$-0.473997\pi$
$641$	2648.51	0.163198	0.0815988	+	0.996665i	$0.473997\pi$
$642$	0	0
$643$	13.4305	0.000823715 0	0.000411857	−	1.00000i	$-0.499869\pi$
$643$	13.4305	0.000823715 0	0.000411857		1.00000i	$0.499869\pi$
$644$	0	0
$645$	−23026.3	−1.40568
$646$	0	0
$647$	11624.1	0.706324	0.353162	−	0.935562i	$-0.385106\pi$
$647$	11624.1	0.706324	0.353162	+	0.935562i	$0.385106\pi$
$648$	0	0
$649$	−2581.66	−0.156146
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28516.6	−1.70894	−0.854471	−	0.519499i	$-0.826119\pi$
$653$	−28516.6	−1.70894	−0.854471	+	0.519499i	$0.826119\pi$
$654$	0	0
$655$	32041.3	1.91139
$656$	0	0
$657$	−5480.15	−0.325420
$658$	0	0
$659$	−18048.6	−1.06688	−0.533440	−	0.845838i	$-0.679101\pi$
$659$	−18048.6	−1.06688	−0.533440	+	0.845838i	$0.679101\pi$
$660$	0	0
$661$	−17841.4	−1.04985	−0.524926	−	0.851148i	$-0.675907\pi$
$661$	−17841.4	−1.04985	−0.524926	+	0.851148i	$0.675907\pi$
$662$	0	0
$663$	2093.19	0.122614
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28362.9	−1.64650
$668$	0	0
$669$	4491.56	0.259572
$670$	0	0
$671$	−7428.68	−0.427394
$672$	0	0
$673$	−6826.13	−0.390978	−0.195489	−	0.980706i	$-0.562629\pi$
$673$	−6826.13	−0.390978	−0.195489	+	0.980706i	$0.562629\pi$
$674$	0	0
$675$	−5250.19	−0.299378
$676$	0	0
$677$	21286.9	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$677$	21286.9	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4809.97	−0.270659
$682$	0	0
$683$	20696.8	1.15951	0.579753	−	0.814793i	$-0.303149\pi$
$683$	20696.8	1.15951	0.579753	+	0.814793i	$0.303149\pi$
$684$	0	0
$685$	−30105.9	−1.67925
$686$	0	0
$687$	3031.56	0.168357
$688$	0	0
$689$	4792.86	0.265012
$690$	0	0
$691$	31341.9	1.72548	0.862738	−	0.505652i	$-0.168748\pi$
$691$	31341.9	1.72548	0.862738	+	0.505652i	$0.168748\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5631.66	−0.307368
$696$	0	0
$697$	−2094.63	−0.113831
$698$	0	0
$699$	−594.651	−0.0321770
$700$	0	0
$701$	−9213.32	−0.496408	−0.248204	−	0.968708i	$-0.579840\pi$
$701$	−9213.32	−0.496408	−0.248204	+	0.968708i	$0.579840\pi$
$702$	0	0
$703$	−158.909	−0.00852541
$704$	0	0
$705$	1136.03	0.0606886
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14516.5	0.768942	0.384471	−	0.923137i	$-0.374384\pi$
$709$	14516.5	0.768942	0.384471	+	0.923137i	$0.374384\pi$
$710$	0	0
$711$	−8192.14	−0.432108
$712$	0	0
$713$	24010.3	1.26114
$714$	0	0
$715$	−2667.54	−0.139525
$716$	0	0
$717$	−3603.56	−0.187695
$718$	0	0
$719$	25882.4	1.34249	0.671246	−	0.741235i	$-0.265760\pi$
$719$	25882.4	1.34249	0.671246	+	0.741235i	$0.265760\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8198.08	−0.421701
$724$	0	0
$725$	36263.9	1.85767
$726$	0	0
$727$	32181.2	1.64172	0.820862	−	0.571127i	$-0.193494\pi$
$727$	32181.2	1.64172	0.820862	+	0.571127i	$0.193494\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	22875.1	1.15741
$732$	0	0
$733$	−20836.1	−1.04993	−0.524966	−	0.851123i	$-0.675922\pi$
$733$	−20836.1	−1.04993	−0.524966	+	0.851123i	$0.675922\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1657.09	−0.0828217
$738$	0	0
$739$	−26434.9	−1.31586	−0.657931	−	0.753078i	$-0.728568\pi$
$739$	−26434.9	−1.31586	−0.657931	+	0.753078i	$0.728568\pi$
$740$	0	0
$741$	1667.03	0.0826447
$742$	0	0
$743$	−9954.69	−0.491524	−0.245762	−	0.969330i	$-0.579038\pi$
$743$	−9954.69	−0.491524	−0.245762	+	0.969330i	$0.579038\pi$
$744$	0	0
$745$	−33847.8	−1.66455
$746$	0	0
$747$	−2951.36	−0.144558
$748$	0	0
$749$	0	0
$750$	0	0
$751$	33204.5	1.61338	0.806692	−	0.590973i	$-0.201256\pi$
$751$	33204.5	1.61338	0.806692	+	0.590973i	$0.201256\pi$
$752$	0	0
$753$	−22697.5	−1.09846
$754$	0	0
$755$	−35958.1	−1.73331
$756$	0	0
$757$	1964.06	0.0942998	0.0471499	−	0.998888i	$-0.484986\pi$
$757$	1964.06	0.0942998	0.0471499	+	0.998888i	$0.484986\pi$
$758$	0	0
$759$	5198.65	0.248615
$760$	0	0
$761$	38553.8	1.83650	0.918248	−	0.396005i	$-0.129604\pi$
$761$	38553.8	1.83650	0.918248	+	0.396005i	$0.129604\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8568.53	0.404962
$766$	0	0
$767$	−2967.87	−0.139718
$768$	0	0
$769$	19715.0	0.924501	0.462251	−	0.886749i	$-0.347042\pi$
$769$	19715.0	0.924501	0.462251	+	0.886749i	$0.347042\pi$
$770$	0	0
$771$	15026.0	0.701880
$772$	0	0
$773$	14700.7	0.684019	0.342010	−	0.939696i	$-0.388892\pi$
$773$	14700.7	0.684019	0.342010	+	0.939696i	$0.388892\pi$
$774$	0	0
$775$	−30698.8	−1.42288
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1668.17	−0.0767246
$780$	0	0
$781$	4204.01	0.192614
$782$	0	0
$783$	−5035.32	−0.229818
$784$	0	0
$785$	−68427.6	−3.11119
$786$	0	0
$787$	−23918.6	−1.08336	−0.541681	−	0.840584i	$-0.682212\pi$
$787$	−23918.6	−1.08336	−0.541681	+	0.840584i	$0.682212\pi$
$788$	0	0
$789$	−18746.4	−0.845869
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8539.98	−0.382426
$794$	0	0
$795$	19619.6	0.875266
$796$	0	0
$797$	−38252.7	−1.70010	−0.850051	−	0.526700i	$-0.823429\pi$
$797$	−38252.7	−1.70010	−0.850051	+	0.526700i	$0.823429\pi$
$798$	0	0
$799$	−1128.57	−0.0499698
$800$	0	0
$801$	338.506	0.0149320
$802$	0	0
$803$	−6937.96	−0.304901
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10765.3	−0.469589
$808$	0	0
$809$	−31435.6	−1.36615	−0.683075	−	0.730348i	$-0.739358\pi$
$809$	−31435.6	−1.36615	−0.683075	+	0.730348i	$0.739358\pi$
$810$	0	0
$811$	11467.0	0.496501	0.248250	−	0.968696i	$-0.420144\pi$
$811$	11467.0	0.496501	0.248250	+	0.968696i	$0.420144\pi$
$812$	0	0
$813$	5949.43	0.256649
$814$	0	0
$815$	−62721.7	−2.69576
$816$	0	0
$817$	18217.8	0.780121
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5030.57	−0.213847	−0.106923	−	0.994267i	$-0.534100\pi$
$821$	−5030.57	−0.213847	−0.106923	+	0.994267i	$0.534100\pi$
$822$	0	0
$823$	13985.2	0.592336	0.296168	−	0.955136i	$-0.404291\pi$
$823$	13985.2	0.592336	0.296168	+	0.955136i	$0.404291\pi$
$824$	0	0
$825$	−6646.83	−0.280500
$826$	0	0
$827$	13939.5	0.586125	0.293063	−	0.956093i	$-0.405326\pi$
$827$	13939.5	0.586125	0.293063	+	0.956093i	$0.405326\pi$
$828$	0	0
$829$	20104.4	0.842286	0.421143	−	0.906994i	$-0.361629\pi$
$829$	20104.4	0.842286	0.421143	+	0.906994i	$0.361629\pi$
$830$	0	0
$831$	−22091.7	−0.922207
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6130.66	0.254084
$836$	0	0
$837$	4262.59	0.176030
$838$	0	0
$839$	15949.5	0.656302	0.328151	−	0.944625i	$-0.393575\pi$
$839$	15949.5	0.656302	0.328151	+	0.944625i	$0.393575\pi$
$840$	0	0
$841$	10390.8	0.426043
$842$	0	0
$843$	15936.1	0.651091
$844$	0	0
$845$	36200.8	1.47378
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3275.29	0.132400
$850$	0	0
$851$	−569.694	−0.0229481
$852$	0	0
$853$	−11802.0	−0.473730	−0.236865	−	0.971543i	$-0.576120\pi$
$853$	−11802.0	−0.473730	−0.236865	+	0.971543i	$0.576120\pi$
$854$	0	0
$855$	6824.00	0.272954
$856$	0	0
$857$	−9595.28	−0.382460	−0.191230	−	0.981545i	$-0.561248\pi$
$857$	−9595.28	−0.382460	−0.191230	+	0.981545i	$0.561248\pi$
$858$	0	0
$859$	21840.9	0.867521	0.433760	−	0.901028i	$-0.357186\pi$
$859$	21840.9	0.867521	0.433760	+	0.901028i	$0.357186\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26531.6	1.04652	0.523260	−	0.852173i	$-0.324716\pi$
$863$	26531.6	1.04652	0.523260	+	0.852173i	$0.324716\pi$
$864$	0	0
$865$	−74838.9	−2.94173
$866$	0	0
$867$	6226.77	0.243912
$868$	0	0
$869$	−10371.4	−0.404862
$870$	0	0
$871$	−1904.98	−0.0741077
$872$	0	0
$873$	−6498.11	−0.251922
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6832.46	−0.263074	−0.131537	−	0.991311i	$-0.541991\pi$
$877$	−6832.46	−0.263074	−0.131537	+	0.991311i	$0.541991\pi$
$878$	0	0
$879$	−21575.6	−0.827903
$880$	0	0
$881$	3994.77	0.152766	0.0763832	−	0.997079i	$-0.475663\pi$
$881$	3994.77	0.152766	0.0763832	+	0.997079i	$0.475663\pi$
$882$	0	0
$883$	−13727.0	−0.523161	−0.261580	−	0.965182i	$-0.584244\pi$
$883$	−13727.0	−0.523161	−0.261580	+	0.965182i	$0.584244\pi$
$884$	0	0
$885$	−12149.0	−0.461451
$886$	0	0
$887$	44119.4	1.67011	0.835054	−	0.550169i	$-0.185437\pi$
$887$	44119.4	1.67011	0.835054	+	0.550169i	$0.185437\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	922.926	0.0347017
$892$	0	0
$893$	−898.796	−0.0336809
$894$	0	0
$895$	−35215.3	−1.31522
$896$	0	0
$897$	5976.35	0.222458
$898$	0	0
$899$	−29442.4	−1.09228
$900$	0	0
$901$	−19490.7	−0.720678
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−64577.7	−2.37197
$906$	0	0
$907$	−36905.8	−1.35109	−0.675545	−	0.737319i	$-0.736092\pi$
$907$	−36905.8	−1.35109	−0.675545	+	0.737319i	$0.736092\pi$
$908$	0	0
$909$	−13668.1	−0.498725
$910$	0	0
$911$	3169.56	0.115271	0.0576356	−	0.998338i	$-0.481644\pi$
$911$	3169.56	0.115271	0.0576356	+	0.998338i	$0.481644\pi$
$912$	0	0
$913$	−3736.48	−0.135443
$914$	0	0
$915$	−34958.6	−1.26305
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8727.00	−0.313250	−0.156625	−	0.987658i	$-0.550061\pi$
$919$	−8727.00	−0.313250	−0.156625	+	0.987658i	$0.550061\pi$
$920$	0	0
$921$	−1624.06	−0.0581050
$922$	0	0
$923$	4832.91	0.172348
$924$	0	0
$925$	728.392	0.0258912
$926$	0	0
$927$	−9466.95	−0.335421
$928$	0	0
$929$	−19405.1	−0.685317	−0.342659	−	0.939460i	$-0.611327\pi$
$929$	−19405.1	−0.685317	−0.342659	+	0.939460i	$0.611327\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−162.050	−0.00568628
$934$	0	0
$935$	10847.9	0.379427
$936$	0	0
$937$	615.692	0.0214662	0.0107331	−	0.999942i	$-0.496583\pi$
$937$	615.692	0.0214662	0.0107331	+	0.999942i	$0.496583\pi$
$938$	0	0
$939$	11318.8	0.393372
$940$	0	0
$941$	29602.0	1.02550	0.512751	−	0.858537i	$-0.328626\pi$
$941$	29602.0	1.02550	0.512751	+	0.858537i	$0.328626\pi$
$942$	0	0
$943$	−5980.46	−0.206522
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13537.4	−0.464527	−0.232264	−	0.972653i	$-0.574613\pi$
$947$	−13537.4	−0.464527	−0.232264	+	0.972653i	$0.574613\pi$
$948$	0	0
$949$	−7975.85	−0.272821
$950$	0	0
$951$	−5157.71	−0.175868
$952$	0	0
$953$	33468.5	1.13762	0.568810	−	0.822469i	$-0.307404\pi$
$953$	33468.5	1.13762	0.568810	+	0.822469i	$0.307404\pi$
$954$	0	0
$955$	34097.9	1.15538
$956$	0	0
$957$	−6374.80	−0.215327
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4866.88	−0.163368
$962$	0	0
$963$	−6898.68	−0.230848
$964$	0	0
$965$	−42894.4	−1.43090
$966$	0	0
$967$	55733.5	1.85343	0.926715	−	0.375764i	$-0.122620\pi$
$967$	55733.5	1.85343	0.926715	+	0.375764i	$0.122620\pi$
$968$	0	0
$969$	−6779.17	−0.224745
$970$	0	0
$971$	−19491.3	−0.644187	−0.322094	−	0.946708i	$-0.604387\pi$
$971$	−19491.3	−0.644187	−0.322094	+	0.946708i	$0.604387\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7641.17	−0.250988
$976$	0	0
$977$	−6241.56	−0.204386	−0.102193	−	0.994765i	$-0.532586\pi$
$977$	−6241.56	−0.204386	−0.102193	+	0.994765i	$0.532586\pi$
$978$	0	0
$979$	428.554	0.0139905
$980$	0	0
$981$	−12843.4	−0.418001
$982$	0	0
$983$	59694.6	1.93689	0.968444	−	0.249231i	$-0.0801778\pi$
$983$	59694.6	1.93689	0.968444	+	0.249231i	$0.0801778\pi$
$984$	0	0
$985$	−27065.7	−0.875517
$986$	0	0
$987$	0	0
$988$	0	0
$989$	65311.4	2.09988
$990$	0	0
$991$	−15561.6	−0.498821	−0.249411	−	0.968398i	$-0.580237\pi$
$991$	−15561.6	−0.498821	−0.249411	+	0.968398i	$0.580237\pi$
$992$	0	0
$993$	25224.6	0.806123
$994$	0	0
$995$	−24446.6	−0.778903
$996$	0	0
$997$	−19884.9	−0.631657	−0.315829	−	0.948816i	$-0.602282\pi$
$997$	−19884.9	−0.631657	−0.315829	+	0.948816i	$0.602282\pi$
$998$	0	0
$999$	−101.139	−0.00320309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cg.1.1		3
4.3	odd	2		147.4.a.m.1.1		3
7.3	odd	6		336.4.q.k.289.1		6
7.5	odd	6		336.4.q.k.193.1		6
7.6	odd	2		2352.4.a.ci.1.3		3
12.11	even	2		441.4.a.t.1.3		3
28.3	even	6		21.4.e.b.16.3	yes	6
28.11	odd	6		147.4.e.n.79.3		6
28.19	even	6		21.4.e.b.4.3	✓	6
28.23	odd	6		147.4.e.n.67.3		6
28.27	even	2		147.4.a.l.1.1		3
84.11	even	6		441.4.e.w.226.1		6
84.23	even	6		441.4.e.w.361.1		6
84.47	odd	6		63.4.e.c.46.1		6
84.59	odd	6		63.4.e.c.37.1		6
84.83	odd	2		441.4.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.b.4.3	✓	6	28.19	even	6
21.4.e.b.16.3	yes	6	28.3	even	6
63.4.e.c.37.1		6	84.59	odd	6
63.4.e.c.46.1		6	84.47	odd	6
147.4.a.l.1.1		3	28.27	even	2
147.4.a.m.1.1		3	4.3	odd	2
147.4.e.n.67.3		6	28.23	odd	6
147.4.e.n.79.3		6	28.11	odd	6
336.4.q.k.193.1		6	7.5	odd	6
336.4.q.k.289.1		6	7.3	odd	6
441.4.a.s.1.3		3	84.83	odd	2
441.4.a.t.1.3		3	12.11	even	2
441.4.e.w.226.1		6	84.11	even	6
441.4.e.w.361.1		6	84.23	even	6
2352.4.a.cg.1.1		3	1.1	even	1	trivial
2352.4.a.ci.1.3		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -17.8732 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -17.8732 q^{5} +9.00000 q^{9} +11.3942 q^{11} +13.0987 q^{13} +53.6196 q^{15} -53.2674 q^{17} -42.4223 q^{19} -152.085 q^{23} +194.451 q^{25} -27.0000 q^{27} +186.493 q^{29} -157.874 q^{31} -34.1825 q^{33} +3.74588 q^{37} -39.2960 q^{39} +39.3230 q^{41} -429.439 q^{43} -160.859 q^{45} +21.1869 q^{47} +159.802 q^{51} +365.904 q^{53} -203.650 q^{55} +127.267 q^{57} -226.578 q^{59} -651.973 q^{61} -234.115 q^{65} -145.433 q^{67} +456.256 q^{69} +368.962 q^{71} -608.906 q^{73} -583.354 q^{75} -910.237 q^{79} +81.0000 q^{81} -327.929 q^{83} +952.058 q^{85} -559.480 q^{87} +37.6118 q^{89} +473.621 q^{93} +758.222 q^{95} -722.013 q^{97} +102.547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9	\( 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9} - 35 q^{11} - 62 q^{13} + 33 q^{15} - 48 q^{17} - 202 q^{19} - 216 q^{23} + 130 q^{25} - 81 q^{27} + 53 q^{29} - 95 q^{31} + 105 q^{33} + 262 q^{37} + 186 q^{39} - 244 q^{41} - 360 q^{43} - 99 q^{45} - 210 q^{47} + 144 q^{51} + 393 q^{53} - 1031 q^{55} + 606 q^{57} + 1143 q^{59} + 70 q^{61} - 472 q^{65} + 628 q^{67} + 648 q^{69} - 318 q^{71} - 988 q^{73} - 390 q^{75} - 861 q^{79} + 243 q^{81} + 519 q^{83} + 1800 q^{85} - 159 q^{87} - 1766 q^{89} + 285 q^{93} + 736 q^{95} - 19 q^{97} - 315 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9 - 35 * q^11 - 62 * q^13 + 33 * q^15 - 48 * q^17 - 202 * q^19 - 216 * q^23 + 130 * q^25 - 81 * q^27 + 53 * q^29 - 95 * q^31 + 105 * q^33 + 262 * q^37 + 186 * q^39 - 244 * q^41 - 360 * q^43 - 99 * q^45 - 210 * q^47 + 144 * q^51 + 393 * q^53 - 1031 * q^55 + 606 * q^57 + 1143 * q^59 + 70 * q^61 - 472 * q^65 + 628 * q^67 + 648 * q^69 - 318 * q^71 - 988 * q^73 - 390 * q^75 - 861 * q^79 + 243 * q^81 + 519 * q^83 + 1800 * q^85 - 159 * q^87 - 1766 * q^89 + 285 * q^93 + 736 * q^95 - 19 * q^97 - 315 * q^99

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