LMFDB - Embedded newform 4620.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 6 $ x^3 - 9*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.58423$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.26245 q^{13} +1.00000 q^{15} -5.84667 q^{17} -0.678221 q^{19} -1.00000 q^{21} -3.84667 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.75268 q^{29} +3.26245 q^{31} -1.00000 q^{33} -1.00000 q^{35} -3.90600 q^{37} +3.26245 q^{39} +2.09400 q^{41} -4.58423 q^{43} -1.00000 q^{45} +3.26245 q^{47} +1.00000 q^{49} +5.84667 q^{51} +11.0151 q^{53} -1.00000 q^{55} +0.678221 q^{57} -5.94067 q^{59} +1.84667 q^{61} +1.00000 q^{63} +3.26245 q^{65} -11.6933 q^{67} +3.84667 q^{69} +16.4309 q^{71} +6.52489 q^{73} -1.00000 q^{75} +1.00000 q^{77} -5.07446 q^{79} +1.00000 q^{81} -5.84667 q^{83} +5.84667 q^{85} -9.75268 q^{87} -2.58423 q^{89} -3.26245 q^{91} -3.26245 q^{93} +0.678221 q^{95} -2.77222 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{15} - 3 q^{21} + 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{33} - 3 q^{35} - 6 q^{37} + 12 q^{41} - 6 q^{43} - 3 q^{45} + 3 q^{49} - 3 q^{55} - 6 q^{59} - 12 q^{61} + 3 q^{63} - 6 q^{69} + 24 q^{71} - 3 q^{75} + 3 q^{77} + 6 q^{79} + 3 q^{81} - 6 q^{87} - 12 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^15 - 3 * q^21 + 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^33 - 3 * q^35 - 6 * q^37 + 12 * q^41 - 6 * q^43 - 3 * q^45 + 3 * q^49 - 3 * q^55 - 6 * q^59 - 12 * q^61 + 3 * q^63 - 6 * q^69 + 24 * q^71 - 3 * q^75 + 3 * q^77 + 6 * q^79 + 3 * q^81 - 6 * q^87 - 12 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.26245	−0.904840	−0.452420	−	0.891805i	$-0.649439\pi$
$13$	−3.26245	−0.904840	−0.452420	+	0.891805i	$0.649439\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.84667	−1.41803	−0.709013	−	0.705195i	$-0.750859\pi$
$17$	−5.84667	−1.41803	−0.709013	+	0.705195i	$0.750859\pi$
$18$	0	0
$19$	−0.678221	−0.155595	−0.0777973	−	0.996969i	$-0.524789\pi$
$19$	−0.678221	−0.155595	−0.0777973	+	0.996969i	$0.524789\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−3.84667	−0.802087	−0.401043	−	0.916059i	$-0.631352\pi$
$23$	−3.84667	−0.802087	−0.401043	+	0.916059i	$0.631352\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.75268	1.81103	0.905513	−	0.424318i	$-0.139486\pi$
$29$	9.75268	1.81103	0.905513	+	0.424318i	$0.139486\pi$
$30$	0	0
$31$	3.26245	0.585953	0.292976	−	0.956120i	$-0.405354\pi$
$31$	3.26245	0.585953	0.292976	+	0.956120i	$0.405354\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−3.90600	−0.642143	−0.321072	−	0.947055i	$-0.604043\pi$
$37$	−3.90600	−0.642143	−0.321072	+	0.947055i	$0.604043\pi$
$38$	0	0
$39$	3.26245	0.522410
$40$	0	0
$41$	2.09400	0.327027	0.163514	−	0.986541i	$-0.447717\pi$
$41$	2.09400	0.327027	0.163514	+	0.986541i	$0.447717\pi$
$42$	0	0
$43$	−4.58423	−0.699088	−0.349544	−	0.936920i	$-0.613663\pi$
$43$	−4.58423	−0.699088	−0.349544	+	0.936920i	$0.613663\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.26245	0.475877	0.237938	−	0.971280i	$-0.423528\pi$
$47$	3.26245	0.475877	0.237938	+	0.971280i	$0.423528\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.84667	0.818698
$52$	0	0
$53$	11.0151	1.51304	0.756522	−	0.653969i	$-0.226897\pi$
$53$	11.0151	1.51304	0.756522	+	0.653969i	$0.226897\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0.678221	0.0898326
$58$	0	0
$59$	−5.94067	−0.773409	−0.386705	−	0.922204i	$-0.626387\pi$
$59$	−5.94067	−0.773409	−0.386705	+	0.922204i	$0.626387\pi$
$60$	0	0
$61$	1.84667	0.236442	0.118221	−	0.992987i	$-0.462281\pi$
$61$	1.84667	0.236442	0.118221	+	0.992987i	$0.462281\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.26245	0.404657
$66$	0	0
$67$	−11.6933	−1.42857	−0.714285	−	0.699855i	$-0.753248\pi$
$67$	−11.6933	−1.42857	−0.714285	+	0.699855i	$0.753248\pi$
$68$	0	0
$69$	3.84667	0.463085
$70$	0	0
$71$	16.4309	1.94999	0.974994	−	0.222229i	$-0.0713334\pi$
$71$	16.4309	1.94999	0.974994	+	0.222229i	$0.0713334\pi$
$72$	0	0
$73$	6.52489	0.763681	0.381840	−	0.924228i	$-0.375290\pi$
$73$	6.52489	0.763681	0.381840	+	0.924228i	$0.375290\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−5.07446	−0.570921	−0.285460	−	0.958391i	$-0.592147\pi$
$79$	−5.07446	−0.570921	−0.285460	+	0.958391i	$0.592147\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.84667	−0.641756	−0.320878	−	0.947121i	$-0.603978\pi$
$83$	−5.84667	−0.641756	−0.320878	+	0.947121i	$0.603978\pi$
$84$	0	0
$85$	5.84667	0.634161
$86$	0	0
$87$	−9.75268	−1.04560
$88$	0	0
$89$	−2.58423	−0.273927	−0.136964	−	0.990576i	$-0.543734\pi$
$89$	−2.58423	−0.273927	−0.136964	+	0.990576i	$0.543734\pi$
$90$	0	0
$91$	−3.26245	−0.341997
$92$	0	0
$93$	−3.26245	−0.338300
$94$	0	0
$95$	0.678221	0.0695840
$96$	0	0
$97$	−2.77222	−0.281476	−0.140738	−	0.990047i	$-0.544948\pi$
$97$	−2.77222	−0.281476	−0.140738	+	0.990047i	$0.544948\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−7.69334	−0.765516	−0.382758	−	0.923849i	$-0.625026\pi$
$101$	−7.69334	−0.765516	−0.382758	+	0.923849i	$0.625026\pi$
$102$	0	0
$103$	5.75268	0.566828	0.283414	−	0.958998i	$-0.408533\pi$
$103$	5.75268	0.566828	0.283414	+	0.958998i	$0.408533\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	18.2429	1.76361	0.881804	−	0.471616i	$-0.156329\pi$
$107$	18.2429	1.76361	0.881804	+	0.471616i	$0.156329\pi$
$108$	0	0
$109$	6.43090	0.615968	0.307984	−	0.951391i	$-0.400346\pi$
$109$	6.43090	0.615968	0.307984	+	0.951391i	$0.400346\pi$
$110$	0	0
$111$	3.90600	0.370742
$112$	0	0
$113$	−3.01512	−0.283639	−0.141819	−	0.989893i	$-0.545295\pi$
$113$	−3.01512	−0.283639	−0.141819	+	0.989893i	$0.545295\pi$
$114$	0	0
$115$	3.84667	0.358704
$116$	0	0
$117$	−3.26245	−0.301613
$118$	0	0
$119$	−5.84667	−0.535964
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.09400	−0.188809
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.584225	0.0518416	0.0259208	−	0.999664i	$-0.491748\pi$
$127$	0.584225	0.0518416	0.0259208	+	0.999664i	$0.491748\pi$
$128$	0	0
$129$	4.58423	0.403619
$130$	0	0
$131$	7.26245	0.634523	0.317261	−	0.948338i	$-0.397237\pi$
$131$	7.26245	0.634523	0.317261	+	0.948338i	$0.397237\pi$
$132$	0	0
$133$	−0.678221	−0.0588092
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−1.69334	−0.144672	−0.0723361	−	0.997380i	$-0.523045\pi$
$137$	−1.69334	−0.144672	−0.0723361	+	0.997380i	$0.523045\pi$
$138$	0	0
$139$	−12.3369	−1.04640	−0.523201	−	0.852209i	$-0.675262\pi$
$139$	−12.3369	−1.04640	−0.523201	+	0.852209i	$0.675262\pi$
$140$	0	0
$141$	−3.26245	−0.274748
$142$	0	0
$143$	−3.26245	−0.272819
$144$	0	0
$145$	−9.75268	−0.809916
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−5.16845	−0.423416	−0.211708	−	0.977333i	$-0.567903\pi$
$149$	−5.16845	−0.423416	−0.211708	+	0.977333i	$0.567903\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	0	0
$153$	−5.84667	−0.472675
$154$	0	0
$155$	−3.26245	−0.262046
$156$	0	0
$157$	2.39623	0.191240	0.0956201	−	0.995418i	$-0.469517\pi$
$157$	2.39623	0.191240	0.0956201	+	0.995418i	$0.469517\pi$
$158$	0	0
$159$	−11.0151	−0.873556
$160$	0	0
$161$	−3.84667	−0.303160
$162$	0	0
$163$	−0.549562	−0.0430450	−0.0215225	−	0.999768i	$-0.506851\pi$
$163$	−0.549562	−0.0430450	−0.0215225	+	0.999768i	$0.506851\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	19.0498	1.47412	0.737058	−	0.675829i	$-0.236214\pi$
$167$	19.0498	1.47412	0.737058	+	0.675829i	$0.236214\pi$
$168$	0	0
$169$	−2.35644	−0.181265
$170$	0	0
$171$	−0.678221	−0.0518649
$172$	0	0
$173$	−10.9805	−0.834829	−0.417414	−	0.908716i	$-0.637064\pi$
$173$	−10.9805	−0.834829	−0.417414	+	0.908716i	$0.637064\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	5.94067	0.446528
$178$	0	0
$179$	23.1438	1.72985	0.864924	−	0.501903i	$-0.167367\pi$
$179$	23.1438	1.72985	0.864924	+	0.501903i	$0.167367\pi$
$180$	0	0
$181$	−0.336902	−0.0250417	−0.0125209	−	0.999922i	$-0.503986\pi$
$181$	−0.336902	−0.0250417	−0.0125209	+	0.999922i	$0.503986\pi$
$182$	0	0
$183$	−1.84667	−0.136510
$184$	0	0
$185$	3.90600	0.287175
$186$	0	0
$187$	−5.84667	−0.427551
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−15.3867	−1.11334	−0.556671	−	0.830733i	$-0.687922\pi$
$191$	−15.3867	−1.11334	−0.556671	+	0.830733i	$0.687922\pi$
$192$	0	0
$193$	7.88134	0.567311	0.283655	−	0.958926i	$-0.408453\pi$
$193$	7.88134	0.567311	0.283655	+	0.958926i	$0.408453\pi$
$194$	0	0
$195$	−3.26245	−0.233629
$196$	0	0
$197$	24.2182	1.72548	0.862739	−	0.505650i	$-0.168747\pi$
$197$	24.2182	1.72548	0.862739	+	0.505650i	$0.168747\pi$
$198$	0	0
$199$	20.6738	1.46553	0.732764	−	0.680483i	$-0.238230\pi$
$199$	20.6738	1.46553	0.732764	+	0.680483i	$0.238230\pi$
$200$	0	0
$201$	11.6933	0.824785
$202$	0	0
$203$	9.75268	0.684504
$204$	0	0
$205$	−2.09400	−0.146251
$206$	0	0
$207$	−3.84667	−0.267362
$208$	0	0
$209$	−0.678221	−0.0469135
$210$	0	0
$211$	24.5249	1.68836	0.844182	−	0.536057i	$-0.180087\pi$
$211$	24.5249	1.68836	0.844182	+	0.536057i	$0.180087\pi$
$212$	0	0
$213$	−16.4309	−1.12583
$214$	0	0
$215$	4.58423	0.312642
$216$	0	0
$217$	3.26245	0.221469
$218$	0	0
$219$	−6.52489	−0.440911
$220$	0	0
$221$	19.0745	1.28309
$222$	0	0
$223$	16.3962	1.09797	0.548987	−	0.835831i	$-0.315014\pi$
$223$	16.3962	1.09797	0.548987	+	0.835831i	$0.315014\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.3716	0.821130	0.410565	−	0.911831i	$-0.365332\pi$
$227$	12.3716	0.821130	0.410565	+	0.911831i	$0.365332\pi$
$228$	0	0
$229$	−22.4309	−1.48228	−0.741138	−	0.671353i	$-0.765713\pi$
$229$	−22.4309	−1.48228	−0.741138	+	0.671353i	$0.765713\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	25.9363	1.69914	0.849570	−	0.527476i	$-0.176861\pi$
$233$	25.9363	1.69914	0.849570	+	0.527476i	$0.176861\pi$
$234$	0	0
$235$	−3.26245	−0.212819
$236$	0	0
$237$	5.07446	0.329621
$238$	0	0
$239$	10.2473	0.662844	0.331422	−	0.943483i	$-0.392472\pi$
$239$	10.2473	0.662844	0.331422	+	0.943483i	$0.392472\pi$
$240$	0	0
$241$	−7.16845	−0.461761	−0.230880	−	0.972982i	$-0.574161\pi$
$241$	−7.16845	−0.461761	−0.230880	+	0.972982i	$0.574161\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.21266	0.140788
$248$	0	0
$249$	5.84667	0.370518
$250$	0	0
$251$	25.0498	1.58113	0.790564	−	0.612379i	$-0.209787\pi$
$251$	25.0498	1.58113	0.790564	+	0.612379i	$0.209787\pi$
$252$	0	0
$253$	−3.84667	−0.241838
$254$	0	0
$255$	−5.84667	−0.366133
$256$	0	0
$257$	15.5993	0.973061	0.486530	−	0.873664i	$-0.338262\pi$
$257$	15.5993	0.973061	0.486530	+	0.873664i	$0.338262\pi$
$258$	0	0
$259$	−3.90600	−0.242707
$260$	0	0
$261$	9.75268	0.603676
$262$	0	0
$263$	−17.5054	−1.07943	−0.539713	−	0.841849i	$-0.681467\pi$
$263$	−17.5054	−1.07943	−0.539713	+	0.841849i	$0.681467\pi$
$264$	0	0
$265$	−11.0151	−0.676654
$266$	0	0
$267$	2.58423	0.158152
$268$	0	0
$269$	−10.0896	−0.615172	−0.307586	−	0.951520i	$-0.599521\pi$
$269$	−10.0896	−0.615172	−0.307586	+	0.951520i	$0.599521\pi$
$270$	0	0
$271$	17.6587	1.07269	0.536344	−	0.843999i	$-0.319805\pi$
$271$	17.6587	1.07269	0.536344	+	0.843999i	$0.319805\pi$
$272$	0	0
$273$	3.26245	0.197452
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	12.8618	0.772790	0.386395	−	0.922333i	$-0.373720\pi$
$277$	12.8618	0.772790	0.386395	+	0.922333i	$0.373720\pi$
$278$	0	0
$279$	3.26245	0.195318
$280$	0	0
$281$	3.13821	0.187210	0.0936048	−	0.995609i	$-0.470161\pi$
$281$	3.13821	0.187210	0.0936048	+	0.995609i	$0.470161\pi$
$282$	0	0
$283$	20.3122	1.20744	0.603718	−	0.797198i	$-0.293685\pi$
$283$	20.3122	1.20744	0.603718	+	0.797198i	$0.293685\pi$
$284$	0	0
$285$	−0.678221	−0.0401744
$286$	0	0
$287$	2.09400	0.123605
$288$	0	0
$289$	17.1836	1.01080
$290$	0	0
$291$	2.77222	0.162510
$292$	0	0
$293$	8.18357	0.478089	0.239045	−	0.971009i	$-0.423166\pi$
$293$	8.18357	0.478089	0.239045	+	0.971009i	$0.423166\pi$
$294$	0	0
$295$	5.94067	0.345879
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	12.5496	0.725760
$300$	0	0
$301$	−4.58423	−0.264230
$302$	0	0
$303$	7.69334	0.441971
$304$	0	0
$305$	−1.84667	−0.105740
$306$	0	0
$307$	0.118665	0.00677254	0.00338627	−	0.999994i	$-0.498922\pi$
$307$	0.118665	0.00677254	0.00338627	+	0.999994i	$0.498922\pi$
$308$	0	0
$309$	−5.75268	−0.327258
$310$	0	0
$311$	12.1880	0.691118	0.345559	−	0.938397i	$-0.387689\pi$
$311$	12.1880	0.691118	0.345559	+	0.938397i	$0.387689\pi$
$312$	0	0
$313$	−15.6340	−0.883687	−0.441843	−	0.897092i	$-0.645675\pi$
$313$	−15.6340	−0.883687	−0.441843	+	0.897092i	$0.645675\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−0.455566	−0.0255872	−0.0127936	−	0.999918i	$-0.504072\pi$
$317$	−0.455566	−0.0255872	−0.0127936	+	0.999918i	$0.504072\pi$
$318$	0	0
$319$	9.75268	0.546045
$320$	0	0
$321$	−18.2429	−1.01822
$322$	0	0
$323$	3.96534	0.220637
$324$	0	0
$325$	−3.26245	−0.180968
$326$	0	0
$327$	−6.43090	−0.355629
$328$	0	0
$329$	3.26245	0.179865
$330$	0	0
$331$	11.9653	0.657674	0.328837	−	0.944387i	$-0.393343\pi$
$331$	11.9653	0.657674	0.328837	+	0.944387i	$0.393343\pi$
$332$	0	0
$333$	−3.90600	−0.214048
$334$	0	0
$335$	11.6933	0.638876
$336$	0	0
$337$	−22.6145	−1.23189	−0.615944	−	0.787790i	$-0.711225\pi$
$337$	−22.6145	−1.23189	−0.615944	+	0.787790i	$0.711225\pi$
$338$	0	0
$339$	3.01512	0.163759
$340$	0	0
$341$	3.26245	0.176671
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.84667	−0.207098
$346$	0	0
$347$	2.11866	0.113736	0.0568679	−	0.998382i	$-0.481889\pi$
$347$	2.11866	0.113736	0.0568679	+	0.998382i	$0.481889\pi$
$348$	0	0
$349$	−18.5205	−0.991378	−0.495689	−	0.868500i	$-0.665084\pi$
$349$	−18.5205	−0.991378	−0.495689	+	0.868500i	$0.665084\pi$
$350$	0	0
$351$	3.26245	0.174137
$352$	0	0
$353$	31.9116	1.69848	0.849241	−	0.528005i	$-0.177060\pi$
$353$	31.9116	1.69848	0.849241	+	0.528005i	$0.177060\pi$
$354$	0	0
$355$	−16.4309	−0.872061
$356$	0	0
$357$	5.84667	0.309439
$358$	0	0
$359$	−11.2971	−0.596239	−0.298119	−	0.954529i	$-0.596359\pi$
$359$	−11.2971	−0.596239	−0.298119	+	0.954529i	$0.596359\pi$
$360$	0	0
$361$	−18.5400	−0.975790
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−6.52489	−0.341529
$366$	0	0
$367$	−19.0398	−0.993869	−0.496935	−	0.867788i	$-0.665541\pi$
$367$	−19.0398	−0.993869	−0.496935	+	0.867788i	$0.665541\pi$
$368$	0	0
$369$	2.09400	0.109009
$370$	0	0
$371$	11.0151	0.571877
$372$	0	0
$373$	−7.72243	−0.399852	−0.199926	−	0.979811i	$-0.564070\pi$
$373$	−7.72243	−0.399852	−0.199926	+	0.979811i	$0.564070\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−31.8176	−1.63869
$378$	0	0
$379$	16.0347	0.823645	0.411823	−	0.911264i	$-0.364892\pi$
$379$	16.0347	0.823645	0.411823	+	0.911264i	$0.364892\pi$
$380$	0	0
$381$	−0.584225	−0.0299308
$382$	0	0
$383$	4.43090	0.226408	0.113204	−	0.993572i	$-0.463889\pi$
$383$	4.43090	0.226408	0.113204	+	0.993572i	$0.463889\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−4.58423	−0.233029
$388$	0	0
$389$	−20.9558	−1.06250	−0.531250	−	0.847215i	$-0.678278\pi$
$389$	−20.9558	−1.06250	−0.531250	+	0.847215i	$0.678278\pi$
$390$	0	0
$391$	22.4902	1.13738
$392$	0	0
$393$	−7.26245	−0.366342
$394$	0	0
$395$	5.07446	0.255324
$396$	0	0
$397$	−25.5054	−1.28008	−0.640038	−	0.768343i	$-0.721082\pi$
$397$	−25.5054	−1.28008	−0.640038	+	0.768343i	$0.721082\pi$
$398$	0	0
$399$	0.678221	0.0339535
$400$	0	0
$401$	12.8865	0.643519	0.321760	−	0.946821i	$-0.395726\pi$
$401$	12.8865	0.643519	0.321760	+	0.946821i	$0.395726\pi$
$402$	0	0
$403$	−10.6436	−0.530193
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−3.90600	−0.193613
$408$	0	0
$409$	−3.54443	−0.175261	−0.0876305	−	0.996153i	$-0.527929\pi$
$409$	−3.54443	−0.175261	−0.0876305	+	0.996153i	$0.527929\pi$
$410$	0	0
$411$	1.69334	0.0835265
$412$	0	0
$413$	−5.94067	−0.292321
$414$	0	0
$415$	5.84667	0.287002
$416$	0	0
$417$	12.3369	0.604141
$418$	0	0
$419$	24.3469	1.18942	0.594712	−	0.803939i	$-0.297266\pi$
$419$	24.3469	1.18942	0.594712	+	0.803939i	$0.297266\pi$
$420$	0	0
$421$	−14.8965	−0.726009	−0.363004	−	0.931787i	$-0.618249\pi$
$421$	−14.8965	−0.726009	−0.363004	+	0.931787i	$0.618249\pi$
$422$	0	0
$423$	3.26245	0.158626
$424$	0	0
$425$	−5.84667	−0.283605
$426$	0	0
$427$	1.84667	0.0893667
$428$	0	0
$429$	3.26245	0.157512
$430$	0	0
$431$	15.8120	0.761638	0.380819	−	0.924650i	$-0.375642\pi$
$431$	15.8120	0.761638	0.380819	+	0.924650i	$0.375642\pi$
$432$	0	0
$433$	7.23778	0.347825	0.173913	−	0.984761i	$-0.444359\pi$
$433$	7.23778	0.347825	0.173913	+	0.984761i	$0.444359\pi$
$434$	0	0
$435$	9.75268	0.467605
$436$	0	0
$437$	2.60889	0.124800
$438$	0	0
$439$	22.8965	1.09279	0.546394	−	0.837528i	$-0.316000\pi$
$439$	22.8965	1.09279	0.546394	+	0.837528i	$0.316000\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.7431	−1.36563	−0.682814	−	0.730593i	$-0.739244\pi$
$443$	−28.7431	−1.36563	−0.682814	+	0.730593i	$0.739244\pi$
$444$	0	0
$445$	2.58423	0.122504
$446$	0	0
$447$	5.16845	0.244459
$448$	0	0
$449$	21.0745	0.994565	0.497283	−	0.867589i	$-0.334331\pi$
$449$	21.0745	0.994565	0.497283	+	0.867589i	$0.334331\pi$
$450$	0	0
$451$	2.09400	0.0986024
$452$	0	0
$453$	−14.0000	−0.657777
$454$	0	0
$455$	3.26245	0.152946
$456$	0	0
$457$	22.4958	1.05231	0.526155	−	0.850389i	$-0.323633\pi$
$457$	22.4958	1.05231	0.526155	+	0.850389i	$0.323633\pi$
$458$	0	0
$459$	5.84667	0.272899
$460$	0	0
$461$	12.7925	0.595805	0.297902	−	0.954596i	$-0.403713\pi$
$461$	12.7925	0.595805	0.297902	+	0.954596i	$0.403713\pi$
$462$	0	0
$463$	−6.21824	−0.288986	−0.144493	−	0.989506i	$-0.546155\pi$
$463$	−6.21824	−0.288986	−0.144493	+	0.989506i	$0.546155\pi$
$464$	0	0
$465$	3.26245	0.151292
$466$	0	0
$467$	18.5249	0.857230	0.428615	−	0.903487i	$-0.359002\pi$
$467$	18.5249	0.857230	0.428615	+	0.903487i	$0.359002\pi$
$468$	0	0
$469$	−11.6933	−0.539948
$470$	0	0
$471$	−2.39623	−0.110413
$472$	0	0
$473$	−4.58423	−0.210783
$474$	0	0
$475$	−0.678221	−0.0311189
$476$	0	0
$477$	11.0151	0.504348
$478$	0	0
$479$	−32.1242	−1.46779	−0.733897	−	0.679261i	$-0.762301\pi$
$479$	−32.1242	−1.46779	−0.733897	+	0.679261i	$0.762301\pi$
$480$	0	0
$481$	12.7431	0.581037
$482$	0	0
$483$	3.84667	0.175030
$484$	0	0
$485$	2.77222	0.125880
$486$	0	0
$487$	25.1685	1.14049	0.570246	−	0.821474i	$-0.306848\pi$
$487$	25.1685	1.14049	0.570246	+	0.821474i	$0.306848\pi$
$488$	0	0
$489$	0.549562	0.0248521
$490$	0	0
$491$	22.9905	1.03754	0.518772	−	0.854912i	$-0.326389\pi$
$491$	22.9905	1.03754	0.518772	+	0.854912i	$0.326389\pi$
$492$	0	0
$493$	−57.0207	−2.56808
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	16.4309	0.737026
$498$	0	0
$499$	−13.5098	−0.604780	−0.302390	−	0.953184i	$-0.597785\pi$
$499$	−13.5098	−0.604780	−0.302390	+	0.953184i	$0.597785\pi$
$500$	0	0
$501$	−19.0498	−0.851082
$502$	0	0
$503$	−10.8965	−0.485849	−0.242925	−	0.970045i	$-0.578107\pi$
$503$	−10.8965	−0.485849	−0.242925	+	0.970045i	$0.578107\pi$
$504$	0	0
$505$	7.69334	0.342349
$506$	0	0
$507$	2.35644	0.104653
$508$	0	0
$509$	24.4958	1.08576	0.542879	−	0.839811i	$-0.317334\pi$
$509$	24.4958	1.08576	0.542879	+	0.839811i	$0.317334\pi$
$510$	0	0
$511$	6.52489	0.288644
$512$	0	0
$513$	0.678221	0.0299442
$514$	0	0
$515$	−5.75268	−0.253493
$516$	0	0
$517$	3.26245	0.143482
$518$	0	0
$519$	10.9805	0.481989
$520$	0	0
$521$	2.27757	0.0997821	0.0498911	−	0.998755i	$-0.484113\pi$
$521$	2.27757	0.0997821	0.0498911	+	0.998755i	$0.484113\pi$
$522$	0	0
$523$	−15.5747	−0.681033	−0.340517	−	0.940239i	$-0.610602\pi$
$523$	−15.5747	−0.681033	−0.340517	+	0.940239i	$0.610602\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−19.0745	−0.830896
$528$	0	0
$529$	−8.20311	−0.356657
$530$	0	0
$531$	−5.94067	−0.257803
$532$	0	0
$533$	−6.83155	−0.295907
$534$	0	0
$535$	−18.2429	−0.788710
$536$	0	0
$537$	−23.1438	−0.998728
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−14.2429	−0.612350	−0.306175	−	0.951975i	$-0.599049\pi$
$541$	−14.2429	−0.612350	−0.306175	+	0.951975i	$0.599049\pi$
$542$	0	0
$543$	0.336902	0.0144578
$544$	0	0
$545$	−6.43090	−0.275469
$546$	0	0
$547$	0.890881	0.0380913	0.0190457	−	0.999819i	$-0.493937\pi$
$547$	0.890881	0.0380913	0.0190457	+	0.999819i	$0.493937\pi$
$548$	0	0
$549$	1.84667	0.0788140
$550$	0	0
$551$	−6.61447	−0.281786
$552$	0	0
$553$	−5.07446	−0.215788
$554$	0	0
$555$	−3.90600	−0.165801
$556$	0	0
$557$	−31.5356	−1.33621	−0.668103	−	0.744069i	$-0.732893\pi$
$557$	−31.5356	−1.33621	−0.668103	+	0.744069i	$0.732893\pi$
$558$	0	0
$559$	14.9558	0.632563
$560$	0	0
$561$	5.84667	0.246847
$562$	0	0
$563$	−34.4858	−1.45340	−0.726702	−	0.686953i	$-0.758948\pi$
$563$	−34.4858	−1.45340	−0.726702	+	0.686953i	$0.758948\pi$
$564$	0	0
$565$	3.01512	0.126847
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−26.6145	−1.11574	−0.557868	−	0.829929i	$-0.688381\pi$
$569$	−26.6145	−1.11574	−0.557868	+	0.829929i	$0.688381\pi$
$570$	0	0
$571$	−44.0358	−1.84284	−0.921421	−	0.388566i	$-0.872971\pi$
$571$	−44.0358	−1.84284	−0.921421	+	0.388566i	$0.872971\pi$
$572$	0	0
$573$	15.3867	0.642788
$574$	0	0
$575$	−3.84667	−0.160417
$576$	0	0
$577$	21.7627	0.905992	0.452996	−	0.891513i	$-0.350355\pi$
$577$	21.7627	0.905992	0.452996	+	0.891513i	$0.350355\pi$
$578$	0	0
$579$	−7.88134	−0.327537
$580$	0	0
$581$	−5.84667	−0.242561
$582$	0	0
$583$	11.0151	0.456200
$584$	0	0
$585$	3.26245	0.134886
$586$	0	0
$587$	−20.4309	−0.843273	−0.421637	−	0.906765i	$-0.638544\pi$
$587$	−20.4309	−0.843273	−0.421637	+	0.906765i	$0.638544\pi$
$588$	0	0
$589$	−2.21266	−0.0911711
$590$	0	0
$591$	−24.2182	−0.996205
$592$	0	0
$593$	−28.2182	−1.15878	−0.579392	−	0.815049i	$-0.696710\pi$
$593$	−28.2182	−1.15878	−0.579392	+	0.815049i	$0.696710\pi$
$594$	0	0
$595$	5.84667	0.239690
$596$	0	0
$597$	−20.6738	−0.846122
$598$	0	0
$599$	24.7925	1.01299	0.506496	−	0.862242i	$-0.330940\pi$
$599$	24.7925	1.01299	0.506496	+	0.862242i	$0.330940\pi$
$600$	0	0
$601$	44.7387	1.82493	0.912465	−	0.409155i	$-0.134176\pi$
$601$	44.7387	1.82493	0.912465	+	0.409155i	$0.134176\pi$
$602$	0	0
$603$	−11.6933	−0.476190
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	12.9311	0.524858	0.262429	−	0.964951i	$-0.415476\pi$
$607$	12.9311	0.524858	0.262429	+	0.964951i	$0.415476\pi$
$608$	0	0
$609$	−9.75268	−0.395198
$610$	0	0
$611$	−10.6436	−0.430592
$612$	0	0
$613$	24.3671	0.984180	0.492090	−	0.870544i	$-0.336233\pi$
$613$	24.3671	0.984180	0.492090	+	0.870544i	$0.336233\pi$
$614$	0	0
$615$	2.09400	0.0844381
$616$	0	0
$617$	29.5054	1.18784	0.593920	−	0.804524i	$-0.297579\pi$
$617$	29.5054	1.18784	0.593920	+	0.804524i	$0.297579\pi$
$618$	0	0
$619$	−30.0302	−1.20702	−0.603509	−	0.797356i	$-0.706231\pi$
$619$	−30.0302	−1.20702	−0.603509	+	0.797356i	$0.706231\pi$
$620$	0	0
$621$	3.84667	0.154362
$622$	0	0
$623$	−2.58423	−0.103535
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.678221	0.0270855
$628$	0	0
$629$	22.8371	0.910576
$630$	0	0
$631$	10.6089	0.422333	0.211167	−	0.977450i	$-0.432274\pi$
$631$	10.6089	0.422333	0.211167	+	0.977450i	$0.432274\pi$
$632$	0	0
$633$	−24.5249	−0.974777
$634$	0	0
$635$	−0.584225	−0.0231843
$636$	0	0
$637$	−3.26245	−0.129263
$638$	0	0
$639$	16.4309	0.649996
$640$	0	0
$641$	27.7236	1.09502	0.547508	−	0.836800i	$-0.315577\pi$
$641$	27.7236	1.09502	0.547508	+	0.836800i	$0.315577\pi$
$642$	0	0
$643$	27.7829	1.09565	0.547826	−	0.836592i	$-0.315456\pi$
$643$	27.7829	1.09565	0.547826	+	0.836592i	$0.315456\pi$
$644$	0	0
$645$	−4.58423	−0.180504
$646$	0	0
$647$	−0.187991	−0.00739070	−0.00369535	−	0.999993i	$-0.501176\pi$
$647$	−0.187991	−0.00739070	−0.00369535	+	0.999993i	$0.501176\pi$
$648$	0	0
$649$	−5.94067	−0.233192
$650$	0	0
$651$	−3.26245	−0.127865
$652$	0	0
$653$	−32.3022	−1.26408	−0.632042	−	0.774934i	$-0.717783\pi$
$653$	−32.3022	−1.26408	−0.632042	+	0.774934i	$0.717783\pi$
$654$	0	0
$655$	−7.26245	−0.283767
$656$	0	0
$657$	6.52489	0.254560
$658$	0	0
$659$	−8.58423	−0.334394	−0.167197	−	0.985924i	$-0.553472\pi$
$659$	−8.58423	−0.334394	−0.167197	+	0.985924i	$0.553472\pi$
$660$	0	0
$661$	−26.4309	−1.02804	−0.514021	−	0.857777i	$-0.671845\pi$
$661$	−26.4309	−1.02804	−0.514021	+	0.857777i	$0.671845\pi$
$662$	0	0
$663$	−19.0745	−0.740790
$664$	0	0
$665$	0.678221	0.0263003
$666$	0	0
$667$	−37.5153	−1.45260
$668$	0	0
$669$	−16.3962	−0.633915
$670$	0	0
$671$	1.84667	0.0712900
$672$	0	0
$673$	13.8713	0.534701	0.267350	−	0.963599i	$-0.413852\pi$
$673$	13.8713	0.534701	0.267350	+	0.963599i	$0.413852\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	22.2138	0.853746	0.426873	−	0.904311i	$-0.359615\pi$
$677$	22.2138	0.853746	0.426873	+	0.904311i	$0.359615\pi$
$678$	0	0
$679$	−2.77222	−0.106388
$680$	0	0
$681$	−12.3716	−0.474079
$682$	0	0
$683$	42.3369	1.61998	0.809988	−	0.586446i	$-0.199473\pi$
$683$	42.3369	1.61998	0.809988	+	0.586446i	$0.199473\pi$
$684$	0	0
$685$	1.69334	0.0646994
$686$	0	0
$687$	22.4309	0.855792
$688$	0	0
$689$	−35.9363	−1.36906
$690$	0	0
$691$	−5.35644	−0.203769	−0.101884	−	0.994796i	$-0.532487\pi$
$691$	−5.35644	−0.203769	−0.101884	+	0.994796i	$0.532487\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	12.3369	0.467965
$696$	0	0
$697$	−12.2429	−0.463733
$698$	0	0
$699$	−25.9363	−0.980999
$700$	0	0
$701$	18.2473	0.689192	0.344596	−	0.938751i	$-0.388016\pi$
$701$	18.2473	0.689192	0.344596	+	0.938751i	$0.388016\pi$
$702$	0	0
$703$	2.64913	0.0999140
$704$	0	0
$705$	3.26245	0.122871
$706$	0	0
$707$	−7.69334	−0.289338
$708$	0	0
$709$	−21.9160	−0.823073	−0.411536	−	0.911393i	$-0.635008\pi$
$709$	−21.9160	−0.823073	−0.411536	+	0.911393i	$0.635008\pi$
$710$	0	0
$711$	−5.07446	−0.190307
$712$	0	0
$713$	−12.5496	−0.469985
$714$	0	0
$715$	3.26245	0.122009
$716$	0	0
$717$	−10.2473	−0.382693
$718$	0	0
$719$	−6.61447	−0.246678	−0.123339	−	0.992365i	$-0.539360\pi$
$719$	−6.61447	−0.246678	−0.123339	+	0.992365i	$0.539360\pi$
$720$	0	0
$721$	5.75268	0.214241
$722$	0	0
$723$	7.16845	0.266598
$724$	0	0
$725$	9.75268	0.362205
$726$	0	0
$727$	45.1394	1.67413	0.837063	−	0.547106i	$-0.184271\pi$
$727$	45.1394	1.67413	0.837063	+	0.547106i	$0.184271\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	26.8025	0.991325
$732$	0	0
$733$	−43.8813	−1.62079	−0.810397	−	0.585881i	$-0.800749\pi$
$733$	−43.8813	−1.62079	−0.810397	+	0.585881i	$0.800749\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−11.6933	−0.430730
$738$	0	0
$739$	8.90088	0.327424	0.163712	−	0.986508i	$-0.447653\pi$
$739$	8.90088	0.327424	0.163712	+	0.986508i	$0.447653\pi$
$740$	0	0
$741$	−2.21266	−0.0812841
$742$	0	0
$743$	−11.4751	−0.420981	−0.210490	−	0.977596i	$-0.567506\pi$
$743$	−11.4751	−0.420981	−0.210490	+	0.977596i	$0.567506\pi$
$744$	0	0
$745$	5.16845	0.189357
$746$	0	0
$747$	−5.84667	−0.213919
$748$	0	0
$749$	18.2429	0.666581
$750$	0	0
$751$	−28.3325	−1.03387	−0.516934	−	0.856026i	$-0.672927\pi$
$751$	−28.3325	−1.03387	−0.516934	+	0.856026i	$0.672927\pi$
$752$	0	0
$753$	−25.0498	−0.912865
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	31.5356	1.14618	0.573090	−	0.819492i	$-0.305744\pi$
$757$	31.5356	1.14618	0.573090	+	0.819492i	$0.305744\pi$
$758$	0	0
$759$	3.84667	0.139625
$760$	0	0
$761$	−1.35644	−0.0491710	−0.0245855	−	0.999698i	$-0.507827\pi$
$761$	−1.35644	−0.0491710	−0.0245855	+	0.999698i	$0.507827\pi$
$762$	0	0
$763$	6.43090	0.232814
$764$	0	0
$765$	5.84667	0.211387
$766$	0	0
$767$	19.3811	0.699811
$768$	0	0
$769$	17.5400	0.632509	0.316255	−	0.948674i	$-0.397575\pi$
$769$	17.5400	0.632509	0.316255	+	0.948674i	$0.397575\pi$
$770$	0	0
$771$	−15.5993	−0.561797
$772$	0	0
$773$	−41.5747	−1.49534	−0.747669	−	0.664072i	$-0.768827\pi$
$773$	−41.5747	−1.49534	−0.747669	+	0.664072i	$0.768827\pi$
$774$	0	0
$775$	3.26245	0.117191
$776$	0	0
$777$	3.90600	0.140127
$778$	0	0
$779$	−1.42019	−0.0508837
$780$	0	0
$781$	16.4309	0.587944
$782$	0	0
$783$	−9.75268	−0.348532
$784$	0	0
$785$	−2.39623	−0.0855253
$786$	0	0
$787$	−31.7627	−1.13222	−0.566108	−	0.824331i	$-0.691552\pi$
$787$	−31.7627	−1.13222	−0.566108	+	0.824331i	$0.691552\pi$
$788$	0	0
$789$	17.5054	0.623207
$790$	0	0
$791$	−3.01512	−0.107205
$792$	0	0
$793$	−6.02467	−0.213942
$794$	0	0
$795$	11.0151	0.390666
$796$	0	0
$797$	−23.9116	−0.846992	−0.423496	−	0.905898i	$-0.639197\pi$
$797$	−23.9116	−0.846992	−0.423496	+	0.905898i	$0.639197\pi$
$798$	0	0
$799$	−19.0745	−0.674806
$800$	0	0
$801$	−2.58423	−0.0913091
$802$	0	0
$803$	6.52489	0.230258
$804$	0	0
$805$	3.84667	0.135577
$806$	0	0
$807$	10.0896	0.355170
$808$	0	0
$809$	22.8315	0.802715	0.401357	−	0.915922i	$-0.368539\pi$
$809$	22.8315	0.802715	0.401357	+	0.915922i	$0.368539\pi$
$810$	0	0
$811$	6.30666	0.221457	0.110728	−	0.993851i	$-0.464682\pi$
$811$	6.30666	0.221457	0.110728	+	0.993851i	$0.464682\pi$
$812$	0	0
$813$	−17.6587	−0.619317
$814$	0	0
$815$	0.549562	0.0192503
$816$	0	0
$817$	3.10912	0.108774
$818$	0	0
$819$	−3.26245	−0.113999
$820$	0	0
$821$	8.65355	0.302011	0.151006	−	0.988533i	$-0.451749\pi$
$821$	8.65355	0.302011	0.151006	+	0.988533i	$0.451749\pi$
$822$	0	0
$823$	−37.5500	−1.30891	−0.654456	−	0.756100i	$-0.727102\pi$
$823$	−37.5500	−1.30891	−0.654456	+	0.756100i	$0.727102\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−18.7925	−0.653478	−0.326739	−	0.945115i	$-0.605950\pi$
$827$	−18.7925	−0.653478	−0.326739	+	0.945115i	$0.605950\pi$
$828$	0	0
$829$	−17.1382	−0.595234	−0.297617	−	0.954685i	$-0.596192\pi$
$829$	−17.1382	−0.595234	−0.297617	+	0.954685i	$0.596192\pi$
$830$	0	0
$831$	−12.8618	−0.446171
$832$	0	0
$833$	−5.84667	−0.202575
$834$	0	0
$835$	−19.0498	−0.659245
$836$	0	0
$837$	−3.26245	−0.112767
$838$	0	0
$839$	37.3767	1.29039	0.645193	−	0.764019i	$-0.276777\pi$
$839$	37.3767	1.29039	0.645193	+	0.764019i	$0.276777\pi$
$840$	0	0
$841$	66.1147	2.27982
$842$	0	0
$843$	−3.13821	−0.108086
$844$	0	0
$845$	2.35644	0.0810641
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−20.3122	−0.697114
$850$	0	0
$851$	15.0251	0.515054
$852$	0	0
$853$	−23.8669	−0.817188	−0.408594	−	0.912716i	$-0.633981\pi$
$853$	−23.8669	−0.817188	−0.408594	+	0.912716i	$0.633981\pi$
$854$	0	0
$855$	0.678221	0.0231947
$856$	0	0
$857$	37.3867	1.27710	0.638552	−	0.769578i	$-0.279534\pi$
$857$	37.3867	1.27710	0.638552	+	0.769578i	$0.279534\pi$
$858$	0	0
$859$	−29.2234	−0.997088	−0.498544	−	0.866864i	$-0.666132\pi$
$859$	−29.2234	−0.997088	−0.498544	+	0.866864i	$0.666132\pi$
$860$	0	0
$861$	−2.09400	−0.0713632
$862$	0	0
$863$	−38.3022	−1.30382	−0.651912	−	0.758295i	$-0.726033\pi$
$863$	−38.3022	−1.30382	−0.651912	+	0.758295i	$0.726033\pi$
$864$	0	0
$865$	10.9805	0.373347
$866$	0	0
$867$	−17.1836	−0.583585
$868$	0	0
$869$	−5.07446	−0.172139
$870$	0	0
$871$	38.1489	1.29263
$872$	0	0
$873$	−2.77222	−0.0938253
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−4.77222	−0.161146	−0.0805732	−	0.996749i	$-0.525675\pi$
$877$	−4.77222	−0.161146	−0.0805732	+	0.996749i	$0.525675\pi$
$878$	0	0
$879$	−8.18357	−0.276025
$880$	0	0
$881$	−17.7829	−0.599122	−0.299561	−	0.954077i	$-0.596840\pi$
$881$	−17.7829	−0.599122	−0.299561	+	0.954077i	$0.596840\pi$
$882$	0	0
$883$	5.23778	0.176265	0.0881326	−	0.996109i	$-0.471910\pi$
$883$	5.23778	0.176265	0.0881326	+	0.996109i	$0.471910\pi$
$884$	0	0
$885$	−5.94067	−0.199693
$886$	0	0
$887$	−6.34132	−0.212921	−0.106460	−	0.994317i	$-0.533952\pi$
$887$	−6.34132	−0.212921	−0.106460	+	0.994317i	$0.533952\pi$
$888$	0	0
$889$	0.584225	0.0195943
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−2.21266	−0.0740439
$894$	0	0
$895$	−23.1438	−0.773611
$896$	0	0
$897$	−12.5496	−0.419018
$898$	0	0
$899$	31.8176	1.06118
$900$	0	0
$901$	−64.4018	−2.14554
$902$	0	0
$903$	4.58423	0.152553
$904$	0	0
$905$	0.336902	0.0111990
$906$	0	0
$907$	25.7873	0.856255	0.428127	−	0.903718i	$-0.359173\pi$
$907$	25.7873	0.856255	0.428127	+	0.903718i	$0.359173\pi$
$908$	0	0
$909$	−7.69334	−0.255172
$910$	0	0
$911$	46.1489	1.52898	0.764491	−	0.644635i	$-0.222990\pi$
$911$	46.1489	1.52898	0.764491	+	0.644635i	$0.222990\pi$
$912$	0	0
$913$	−5.84667	−0.193497
$914$	0	0
$915$	1.84667	0.0610491
$916$	0	0
$917$	7.26245	0.239827
$918$	0	0
$919$	24.0996	0.794972	0.397486	−	0.917608i	$-0.369883\pi$
$919$	24.0996	0.794972	0.397486	+	0.917608i	$0.369883\pi$
$920$	0	0
$921$	−0.118665	−0.00391013
$922$	0	0
$923$	−53.6049	−1.76443
$924$	0	0
$925$	−3.90600	−0.128429
$926$	0	0
$927$	5.75268	0.188943
$928$	0	0
$929$	36.2182	1.18828	0.594141	−	0.804361i	$-0.297492\pi$
$929$	36.2182	1.18828	0.594141	+	0.804361i	$0.297492\pi$
$930$	0	0
$931$	−0.678221	−0.0222278
$932$	0	0
$933$	−12.1880	−0.399017
$934$	0	0
$935$	5.84667	0.191207
$936$	0	0
$937$	−49.5412	−1.61844	−0.809220	−	0.587506i	$-0.800110\pi$
$937$	−49.5412	−1.61844	−0.809220	+	0.587506i	$0.800110\pi$
$938$	0	0
$939$	15.6340	0.510197
$940$	0	0
$941$	13.7873	0.449454	0.224727	−	0.974422i	$-0.427851\pi$
$941$	13.7873	0.449454	0.224727	+	0.974422i	$0.427851\pi$
$942$	0	0
$943$	−8.05491	−0.262304
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	5.62843	0.182900	0.0914498	−	0.995810i	$-0.470850\pi$
$947$	5.62843	0.182900	0.0914498	+	0.995810i	$0.470850\pi$
$948$	0	0
$949$	−21.2871	−0.691009
$950$	0	0
$951$	0.455566	0.0147728
$952$	0	0
$953$	40.7734	1.32078	0.660390	−	0.750923i	$-0.270391\pi$
$953$	40.7734	1.32078	0.660390	+	0.750923i	$0.270391\pi$
$954$	0	0
$955$	15.3867	0.497902
$956$	0	0
$957$	−9.75268	−0.315259
$958$	0	0
$959$	−1.69334	−0.0546809
$960$	0	0
$961$	−20.3564	−0.656659
$962$	0	0
$963$	18.2429	0.587869
$964$	0	0
$965$	−7.88134	−0.253709
$966$	0	0
$967$	−19.2971	−0.620553	−0.310277	−	0.950646i	$-0.600422\pi$
$967$	−19.2971	−0.620553	−0.310277	+	0.950646i	$0.600422\pi$
$968$	0	0
$969$	−3.96534	−0.127385
$970$	0	0
$971$	3.22778	0.103584	0.0517922	−	0.998658i	$-0.483507\pi$
$971$	3.22778	0.103584	0.0517922	+	0.998658i	$0.483507\pi$
$972$	0	0
$973$	−12.3369	−0.395503
$974$	0	0
$975$	3.26245	0.104482
$976$	0	0
$977$	−56.8080	−1.81745	−0.908725	−	0.417395i	$-0.862943\pi$
$977$	−56.8080	−1.81745	−0.908725	+	0.417395i	$0.862943\pi$
$978$	0	0
$979$	−2.58423	−0.0825922
$980$	0	0
$981$	6.43090	0.205323
$982$	0	0
$983$	−8.49465	−0.270937	−0.135469	−	0.990782i	$-0.543254\pi$
$983$	−8.49465	−0.270937	−0.135469	+	0.990782i	$0.543254\pi$
$984$	0	0
$985$	−24.2182	−0.771657
$986$	0	0
$987$	−3.26245	−0.103845
$988$	0	0
$989$	17.6340	0.560729
$990$	0	0
$991$	19.1729	0.609046	0.304523	−	0.952505i	$-0.401503\pi$
$991$	19.1729	0.609046	0.304523	+	0.952505i	$0.401503\pi$
$992$	0	0
$993$	−11.9653	−0.379708
$994$	0	0
$995$	−20.6738	−0.655404
$996$	0	0
$997$	−11.6933	−0.370332	−0.185166	−	0.982707i	$-0.559282\pi$
$997$	−11.6933	−0.370332	−0.185166	+	0.982707i	$0.559282\pi$
$998$	0	0
$999$	3.90600	0.123581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.t.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.t.1.1	✓	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.26245 q^{13} +1.00000 q^{15} -5.84667 q^{17} -0.678221 q^{19} -1.00000 q^{21} -3.84667 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.75268 q^{29} +3.26245 q^{31} -1.00000 q^{33} -1.00000 q^{35} -3.90600 q^{37} +3.26245 q^{39} +2.09400 q^{41} -4.58423 q^{43} -1.00000 q^{45} +3.26245 q^{47} +1.00000 q^{49} +5.84667 q^{51} +11.0151 q^{53} -1.00000 q^{55} +0.678221 q^{57} -5.94067 q^{59} +1.84667 q^{61} +1.00000 q^{63} +3.26245 q^{65} -11.6933 q^{67} +3.84667 q^{69} +16.4309 q^{71} +6.52489 q^{73} -1.00000 q^{75} +1.00000 q^{77} -5.07446 q^{79} +1.00000 q^{81} -5.84667 q^{83} +5.84667 q^{85} -9.75268 q^{87} -2.58423 q^{89} -3.26245 q^{91} -3.26245 q^{93} +0.678221 q^{95} -2.77222 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{15} - 3 q^{21} + 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{33} - 3 q^{35} - 6 q^{37} + 12 q^{41} - 6 q^{43} - 3 q^{45} + 3 q^{49} - 3 q^{55} - 6 q^{59} - 12 q^{61} + 3 q^{63} - 6 q^{69} + 24 q^{71} - 3 q^{75} + 3 q^{77} + 6 q^{79} + 3 q^{81} - 6 q^{87} - 12 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^15 - 3 * q^21 + 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^33 - 3 * q^35 - 6 * q^37 + 12 * q^41 - 6 * q^43 - 3 * q^45 + 3 * q^49 - 3 * q^55 - 6 * q^59 - 12 * q^61 + 3 * q^63 - 6 * q^69 + 24 * q^71 - 3 * q^75 + 3 * q^77 + 6 * q^79 + 3 * q^81 - 6 * q^87 - 12 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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