LMFDB - Embedded newform 4788.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.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:	$38.2323724878$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51820$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85952 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+2.85952 q^{5} -1.00000 q^{7} +2.34132 q^{11} -0.859523 q^{13} -4.34132 q^{17} -1.00000 q^{19} -4.85952 q^{23} +3.17687 q^{25} -9.37772 q^{29} -7.37772 q^{31} -2.85952 q^{35} +2.17687 q^{37} -6.69507 q^{41} +0.353748 q^{43} -5.54217 q^{47} +1.00000 q^{49} -9.55460 q^{53} +6.69507 q^{55} +14.5786 q^{59} -12.9323 q^{61} -2.45783 q^{65} -4.34132 q^{67} -7.89592 q^{71} +2.23725 q^{73} -2.34132 q^{77} +7.36530 q^{79} +1.83555 q^{83} -12.4141 q^{85} +3.31735 q^{89} +0.859523 q^{91} -2.85952 q^{95} +18.2248 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q - 2 * q^5 - 3 * q^7	$ 3 q - 2 q^{5} - 3 q^{7} + 3 q^{11} + 8 q^{13} - 9 q^{17} - 3 q^{19} - 4 q^{23} + 7 q^{25} - 11 q^{29} - 5 q^{31} + 2 q^{35} + 4 q^{37} - 11 q^{41} - 4 q^{43} + 2 q^{47} + 3 q^{49} - 9 q^{53} + 11 q^{55} + 12 q^{59} - 2 q^{61} - 26 q^{65} - 9 q^{67} - 21 q^{73} - 3 q^{77} + 6 q^{79} + 7 q^{83} - 7 q^{85} + 18 q^{89} - 8 q^{91} + 2 q^{95} + 28 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 3 * q^7 + 3 * q^11 + 8 * q^13 - 9 * q^17 - 3 * q^19 - 4 * q^23 + 7 * q^25 - 11 * q^29 - 5 * q^31 + 2 * q^35 + 4 * q^37 - 11 * q^41 - 4 * q^43 + 2 * q^47 + 3 * q^49 - 9 * q^53 + 11 * q^55 + 12 * q^59 - 2 * q^61 - 26 * q^65 - 9 * q^67 - 21 * q^73 - 3 * q^77 + 6 * q^79 + 7 * q^83 - 7 * q^85 + 18 * q^89 - 8 * q^91 + 2 * q^95 + 28 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.85952	1.27882	0.639409	−	0.768867i	$-0.279179\pi$
$5$	2.85952	1.27882	0.639409	+	0.768867i	$0.279179\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.34132	0.705936	0.352968	−	0.935635i	$-0.385172\pi$
$11$	2.34132	0.705936	0.352968	+	0.935635i	$0.385172\pi$
$12$	0	0
$13$	−0.859523	−0.238389	−0.119194	−	0.992871i	$-0.538031\pi$
$13$	−0.859523	−0.238389	−0.119194	+	0.992871i	$0.538031\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.34132	−1.05293	−0.526463	−	0.850198i	$-0.676482\pi$
$17$	−4.34132	−1.05293	−0.526463	+	0.850198i	$0.676482\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.85952	−1.01328	−0.506640	−	0.862158i	$-0.669113\pi$
$23$	−4.85952	−1.01328	−0.506640	+	0.862158i	$0.669113\pi$
$24$	0	0
$25$	3.17687	0.635375
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.37772	−1.74140	−0.870700	−	0.491815i	$-0.836334\pi$
$29$	−9.37772	−1.74140	−0.870700	+	0.491815i	$0.836334\pi$
$30$	0	0
$31$	−7.37772	−1.32508	−0.662539	−	0.749027i	$-0.730521\pi$
$31$	−7.37772	−1.32508	−0.662539	+	0.749027i	$0.730521\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.85952	−0.483348
$36$	0	0
$37$	2.17687	0.357876	0.178938	−	0.983860i	$-0.442734\pi$
$37$	2.17687	0.357876	0.178938	+	0.983860i	$0.442734\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.69507	−1.04559	−0.522797	−	0.852457i	$-0.675112\pi$
$41$	−6.69507	−1.04559	−0.522797	+	0.852457i	$0.675112\pi$
$42$	0	0
$43$	0.353748	0.0539461	0.0269730	−	0.999636i	$-0.491413\pi$
$43$	0.353748	0.0539461	0.0269730	+	0.999636i	$0.491413\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.54217	−0.808409	−0.404204	−	0.914669i	$-0.632452\pi$
$47$	−5.54217	−0.808409	−0.404204	+	0.914669i	$0.632452\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.55460	−1.31242	−0.656212	−	0.754576i	$-0.727842\pi$
$53$	−9.55460	−1.31242	−0.656212	+	0.754576i	$0.727842\pi$
$54$	0	0
$55$	6.69507	0.902763
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.5786	1.89797	0.948984	−	0.315324i	$-0.102113\pi$
$59$	14.5786	1.89797	0.948984	+	0.315324i	$0.102113\pi$
$60$	0	0
$61$	−12.9323	−1.65581	−0.827907	−	0.560866i	$-0.810468\pi$
$61$	−12.9323	−1.65581	−0.827907	+	0.560866i	$0.810468\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.45783	−0.304856
$66$	0	0
$67$	−4.34132	−0.530377	−0.265189	−	0.964197i	$-0.585434\pi$
$67$	−4.34132	−0.530377	−0.265189	+	0.964197i	$0.585434\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.89592	−0.937073	−0.468537	−	0.883444i	$-0.655219\pi$
$71$	−7.89592	−0.937073	−0.468537	+	0.883444i	$0.655219\pi$
$72$	0	0
$73$	2.23725	0.261850	0.130925	−	0.991392i	$-0.458205\pi$
$73$	2.23725	0.261850	0.130925	+	0.991392i	$0.458205\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.34132	−0.266819
$78$	0	0
$79$	7.36530	0.828661	0.414330	−	0.910127i	$-0.364016\pi$
$79$	7.36530	0.828661	0.414330	+	0.910127i	$0.364016\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.83555	0.201478	0.100739	−	0.994913i	$-0.467879\pi$
$83$	1.83555	0.201478	0.100739	+	0.994913i	$0.467879\pi$
$84$	0	0
$85$	−12.4141	−1.34650
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.31735	0.351638	0.175819	−	0.984422i	$-0.443743\pi$
$89$	3.31735	0.351638	0.175819	+	0.984422i	$0.443743\pi$
$90$	0	0
$91$	0.859523	0.0901025
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.85952	−0.293381
$96$	0	0
$97$	18.2248	1.85045	0.925225	−	0.379418i	$-0.123876\pi$
$97$	18.2248	1.85045	0.925225	+	0.379418i	$0.123876\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.18842	−0.715275	−0.357637	−	0.933860i	$-0.616418\pi$
$101$	−7.18842	−0.715275	−0.357637	+	0.933860i	$0.616418\pi$
$102$	0	0
$103$	5.21327	0.513679	0.256839	−	0.966454i	$-0.417319\pi$
$103$	5.21327	0.513679	0.256839	+	0.966454i	$0.417319\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.89592	0.376633	0.188316	−	0.982108i	$-0.439697\pi$
$107$	3.89592	0.376633	0.188316	+	0.982108i	$0.439697\pi$
$108$	0	0
$109$	0.578570	0.0554170	0.0277085	−	0.999616i	$-0.491179\pi$
$109$	0.578570	0.0554170	0.0277085	+	0.999616i	$0.491179\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.51820	−0.425036	−0.212518	−	0.977157i	$-0.568166\pi$
$113$	−4.51820	−0.425036	−0.212518	+	0.977157i	$0.568166\pi$
$114$	0	0
$115$	−13.8959	−1.29580
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.34132	0.397969
$120$	0	0
$121$	−5.51820	−0.501654
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.21327	−0.466289
$126$	0	0
$127$	−12.4017	−1.10047	−0.550236	−	0.835009i	$-0.685462\pi$
$127$	−12.4017	−1.10047	−0.550236	+	0.835009i	$0.685462\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.98758	0.173655	0.0868277	−	0.996223i	$-0.472327\pi$
$131$	1.98758	0.173655	0.0868277	+	0.996223i	$0.472327\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.4381	1.14809	0.574047	−	0.818822i	$-0.305373\pi$
$137$	13.4381	1.14809	0.574047	+	0.818822i	$0.305373\pi$
$138$	0	0
$139$	11.6150	0.985169	0.492584	−	0.870265i	$-0.336052\pi$
$139$	11.6150	0.985169	0.492584	+	0.870265i	$0.336052\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.01242	−0.168287
$144$	0	0
$145$	−26.8158	−2.22693
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.10408	−0.336219	−0.168110	−	0.985768i	$-0.553766\pi$
$149$	−4.10408	−0.336219	−0.168110	+	0.985768i	$0.553766\pi$
$150$	0	0
$151$	11.7794	0.958595	0.479298	−	0.877652i	$-0.340891\pi$
$151$	11.7794	0.958595	0.479298	+	0.877652i	$0.340891\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−21.0968	−1.69453
$156$	0	0
$157$	16.8158	1.34205	0.671024	−	0.741436i	$-0.265855\pi$
$157$	16.8158	1.34205	0.671024	+	0.741436i	$0.265855\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.85952	0.382984
$162$	0	0
$163$	13.2736	1.03967	0.519836	−	0.854266i	$-0.325993\pi$
$163$	13.2736	1.03967	0.519836	+	0.854266i	$0.325993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.2497	−1.10267	−0.551336	−	0.834283i	$-0.685882\pi$
$167$	−14.2497	−1.10267	−0.551336	+	0.834283i	$0.685882\pi$
$168$	0	0
$169$	−12.2612	−0.943171
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.6826	−1.57247	−0.786236	−	0.617926i	$-0.787973\pi$
$173$	−20.6826	−1.57247	−0.786236	+	0.617926i	$0.787973\pi$
$174$	0	0
$175$	−3.17687	−0.240149
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.0604	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$179$	−14.0604	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$180$	0	0
$181$	7.02397	0.522088	0.261044	−	0.965327i	$-0.415933\pi$
$181$	7.02397	0.522088	0.261044	+	0.965327i	$0.415933\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.22482	0.457658
$186$	0	0
$187$	−10.1645	−0.743298
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.6753	1.13423	0.567114	−	0.823639i	$-0.308060\pi$
$191$	15.6753	1.13423	0.567114	+	0.823639i	$0.308060\pi$
$192$	0	0
$193$	−0.670226	−0.0482439	−0.0241220	−	0.999709i	$-0.507679\pi$
$193$	−0.670226	−0.0482439	−0.0241220	+	0.999709i	$0.507679\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.9811	1.70859	0.854293	−	0.519792i	$-0.173991\pi$
$197$	23.9811	1.70859	0.854293	+	0.519792i	$0.173991\pi$
$198$	0	0
$199$	16.9803	1.20370	0.601850	−	0.798609i	$-0.294431\pi$
$199$	16.9803	1.20370	0.601850	+	0.798609i	$0.294431\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.37772	0.658187
$204$	0	0
$205$	−19.1447	−1.33713
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.34132	−0.161953
$210$	0	0
$211$	12.8471	0.884431	0.442215	−	0.896909i	$-0.354193\pi$
$211$	12.8471	0.884431	0.442215	+	0.896909i	$0.354193\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.01155	0.0689872
$216$	0	0
$217$	7.37772	0.500832
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.73147	0.251006
$222$	0	0
$223$	8.53062	0.571253	0.285626	−	0.958341i	$-0.407798\pi$
$223$	8.53062	0.571253	0.285626	+	0.958341i	$0.407798\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.4985	−1.82514	−0.912569	−	0.408924i	$-0.865904\pi$
$227$	−27.4985	−1.82514	−0.912569	+	0.408924i	$0.865904\pi$
$228$	0	0
$229$	−25.4381	−1.68100	−0.840498	−	0.541814i	$-0.817738\pi$
$229$	−25.4381	−1.68100	−0.840498	+	0.541814i	$0.817738\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.6587	−1.02583	−0.512917	−	0.858438i	$-0.671435\pi$
$233$	−15.6587	−1.02583	−0.512917	+	0.858438i	$0.671435\pi$
$234$	0	0
$235$	−15.8480	−1.03381
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.0051	1.09997	0.549985	−	0.835175i	$-0.314634\pi$
$239$	17.0051	1.09997	0.549985	+	0.835175i	$0.314634\pi$
$240$	0	0
$241$	2.40170	0.154707	0.0773534	−	0.997004i	$-0.475353\pi$
$241$	2.40170	0.154707	0.0773534	+	0.997004i	$0.475353\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.85952	0.182688
$246$	0	0
$247$	0.859523	0.0546902
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.77942	−0.364794	−0.182397	−	0.983225i	$-0.558386\pi$
$251$	−5.77942	−0.364794	−0.182397	+	0.983225i	$0.558386\pi$
$252$	0	0
$253$	−11.3777	−0.715311
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.49335	−0.155531	−0.0777655	−	0.996972i	$-0.524779\pi$
$257$	−2.49335	−0.155531	−0.0777655	+	0.996972i	$0.524779\pi$
$258$	0	0
$259$	−2.17687	−0.135264
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.72636	−0.414765	−0.207382	−	0.978260i	$-0.566494\pi$
$263$	−6.72636	−0.414765	−0.207382	+	0.978260i	$0.566494\pi$
$264$	0	0
$265$	−27.3216	−1.67835
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.1893	−0.621252	−0.310626	−	0.950532i	$-0.600539\pi$
$269$	−10.1893	−0.621252	−0.310626	+	0.950532i	$0.600539\pi$
$270$	0	0
$271$	22.1083	1.34298	0.671492	−	0.741011i	$-0.265654\pi$
$271$	22.1083	1.34298	0.671492	+	0.741011i	$0.265654\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.43809	0.448534
$276$	0	0
$277$	−9.31735	−0.559825	−0.279913	−	0.960025i	$-0.590306\pi$
$277$	−9.31735	−0.559825	−0.279913	+	0.960025i	$0.590306\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.9323	0.771477	0.385739	−	0.922608i	$-0.373947\pi$
$281$	12.9323	0.771477	0.385739	+	0.922608i	$0.373947\pi$
$282$	0	0
$283$	19.9083	1.18343	0.591714	−	0.806148i	$-0.298451\pi$
$283$	19.9083	1.18343	0.591714	+	0.806148i	$0.298451\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.69507	0.395198
$288$	0	0
$289$	1.84710	0.108653
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.1769	−1.17875	−0.589373	−	0.807861i	$-0.700625\pi$
$293$	−20.1769	−1.17875	−0.589373	+	0.807861i	$0.700625\pi$
$294$	0	0
$295$	41.6878	2.42716
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.17687	0.241555
$300$	0	0
$301$	−0.353748	−0.0203897
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−36.9803	−2.11748
$306$	0	0
$307$	−29.5713	−1.68772	−0.843860	−	0.536563i	$-0.819722\pi$
$307$	−29.5713	−1.68772	−0.843860	+	0.536563i	$0.819722\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.7794	1.00818	0.504089	−	0.863652i	$-0.331828\pi$
$311$	17.7794	1.00818	0.504089	+	0.863652i	$0.331828\pi$
$312$	0	0
$313$	24.0479	1.35927	0.679635	−	0.733550i	$-0.262138\pi$
$313$	24.0479	1.35927	0.679635	+	0.733550i	$0.262138\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.5109	−0.871178	−0.435589	−	0.900146i	$-0.643460\pi$
$317$	−15.5109	−0.871178	−0.435589	+	0.900146i	$0.643460\pi$
$318$	0	0
$319$	−21.9563	−1.22932
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.34132	0.241558
$324$	0	0
$325$	−2.73060	−0.151466
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.54217	0.305550
$330$	0	0
$331$	29.8835	1.64255	0.821273	−	0.570536i	$-0.193264\pi$
$331$	29.8835	1.64255	0.821273	+	0.570536i	$0.193264\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.4141	−0.678256
$336$	0	0
$337$	−33.8771	−1.84540	−0.922701	−	0.385518i	$-0.874023\pi$
$337$	−33.8771	−1.84540	−0.922701	+	0.385518i	$0.874023\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.2736	−0.935420
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.2124	1.08506	0.542529	−	0.840037i	$-0.317467\pi$
$347$	20.2124	1.08506	0.542529	+	0.840037i	$0.317467\pi$
$348$	0	0
$349$	−2.77431	−0.148505	−0.0742526	−	0.997239i	$-0.523657\pi$
$349$	−2.77431	−0.148505	−0.0742526	+	0.997239i	$0.523657\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.3049	1.02750	0.513749	−	0.857941i	$-0.328256\pi$
$353$	19.3049	1.02750	0.513749	+	0.857941i	$0.328256\pi$
$354$	0	0
$355$	−22.5786	−1.19835
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.5349	−1.50601	−0.753006	−	0.658013i	$-0.771397\pi$
$359$	−28.5349	−1.50601	−0.753006	+	0.658013i	$0.771397\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.39746	0.334858
$366$	0	0
$367$	−3.94387	−0.205868	−0.102934	−	0.994688i	$-0.532823\pi$
$367$	−3.94387	−0.205868	−0.102934	+	0.994688i	$0.532823\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.55460	0.496050
$372$	0	0
$373$	17.3216	0.896878	0.448439	−	0.893813i	$-0.351980\pi$
$373$	17.3216	0.896878	0.448439	+	0.893813i	$0.351980\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.06037	0.415130
$378$	0	0
$379$	−16.2248	−0.833413	−0.416707	−	0.909041i	$-0.636816\pi$
$379$	−16.2248	−0.833413	−0.416707	+	0.909041i	$0.636816\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.65136	0.237674	0.118837	−	0.992914i	$-0.462083\pi$
$383$	4.65136	0.237674	0.118837	+	0.992914i	$0.462083\pi$
$384$	0	0
$385$	−6.69507	−0.341213
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.6587	−0.591118	−0.295559	−	0.955324i	$-0.595506\pi$
$389$	−11.6587	−0.591118	−0.295559	+	0.955324i	$0.595506\pi$
$390$	0	0
$391$	21.0968	1.06691
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.0612	1.05971
$396$	0	0
$397$	−33.9126	−1.70202	−0.851012	−	0.525146i	$-0.824011\pi$
$397$	−33.9126	−1.70202	−0.851012	+	0.525146i	$0.824011\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.69507	−0.234461	−0.117230	−	0.993105i	$-0.537402\pi$
$401$	−4.69507	−0.234461	−0.117230	+	0.993105i	$0.537402\pi$
$402$	0	0
$403$	6.34132	0.315884
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.09677	0.252637
$408$	0	0
$409$	−11.7628	−0.581631	−0.290815	−	0.956779i	$-0.593927\pi$
$409$	−11.7628	−0.581631	−0.290815	+	0.956779i	$0.593927\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.5786	−0.717365
$414$	0	0
$415$	5.24880	0.257653
$416$	0	0
$417$	0	0
$418$	0	0
$419$	31.7606	1.55160	0.775802	−	0.630976i	$-0.217345\pi$
$419$	31.7606	1.55160	0.775802	+	0.630976i	$0.217345\pi$
$420$	0	0
$421$	12.8116	0.624398	0.312199	−	0.950017i	$-0.398934\pi$
$421$	12.8116	0.624398	0.312199	+	0.950017i	$0.398934\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.7918	−0.669003
$426$	0	0
$427$	12.9323	0.625839
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.4381	−1.03264	−0.516318	−	0.856397i	$-0.672698\pi$
$431$	−21.4381	−1.03264	−0.516318	+	0.856397i	$0.672698\pi$
$432$	0	0
$433$	−0.176874	−0.00850002	−0.00425001	−	0.999991i	$-0.501353\pi$
$433$	−0.176874	−0.00850002	−0.00425001	+	0.999991i	$0.501353\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.85952	0.232463
$438$	0	0
$439$	−4.14559	−0.197858	−0.0989291	−	0.995094i	$-0.531542\pi$
$439$	−4.14559	−0.197858	−0.0989291	+	0.995094i	$0.531542\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.6753	−0.744758	−0.372379	−	0.928081i	$-0.621458\pi$
$443$	−15.6753	−0.744758	−0.372379	+	0.928081i	$0.621458\pi$
$444$	0	0
$445$	9.48604	0.449682
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.00731	0.141924	0.0709619	−	0.997479i	$-0.477393\pi$
$449$	3.00731	0.141924	0.0709619	+	0.997479i	$0.477393\pi$
$450$	0	0
$451$	−15.6753	−0.738123
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.45783	0.115225
$456$	0	0
$457$	−22.7117	−1.06241	−0.531205	−	0.847243i	$-0.678261\pi$
$457$	−22.7117	−1.06241	−0.531205	+	0.847243i	$0.678261\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.0406	1.81830	0.909152	−	0.416465i	$-0.136731\pi$
$461$	39.0406	1.81830	0.909152	+	0.416465i	$0.136731\pi$
$462$	0	0
$463$	−26.2497	−1.21993	−0.609963	−	0.792430i	$-0.708816\pi$
$463$	−26.2497	−1.21993	−0.609963	+	0.792430i	$0.708816\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.63894	−0.307214	−0.153607	−	0.988132i	$-0.549089\pi$
$467$	−6.63894	−0.307214	−0.153607	+	0.988132i	$0.549089\pi$
$468$	0	0
$469$	4.34132	0.200464
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.828239	0.0380825
$474$	0	0
$475$	−3.17687	−0.145765
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.0603715	−0.00275844	−0.00137922	−	0.999999i	$-0.500439\pi$
$479$	−0.0603715	−0.00275844	−0.00137922	+	0.999999i	$0.500439\pi$
$480$	0	0
$481$	−1.87107	−0.0853136
$482$	0	0
$483$	0	0
$484$	0	0
$485$	52.1143	2.36639
$486$	0	0
$487$	28.6034	1.29614	0.648072	−	0.761579i	$-0.275575\pi$
$487$	28.6034	1.29614	0.648072	+	0.761579i	$0.275575\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.73060	0.303748	0.151874	−	0.988400i	$-0.451469\pi$
$491$	6.73060	0.303748	0.151874	+	0.988400i	$0.451469\pi$
$492$	0	0
$493$	40.7117	1.83356
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.89592	0.354180
$498$	0	0
$499$	−39.5546	−1.77071	−0.885353	−	0.464919i	$-0.846084\pi$
$499$	−39.5546	−1.77071	−0.885353	+	0.464919i	$0.846084\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.8531	−1.28650	−0.643248	−	0.765658i	$-0.722413\pi$
$503$	−28.8531	−1.28650	−0.643248	+	0.765658i	$0.722413\pi$
$504$	0	0
$505$	−20.5555	−0.914706
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.4496	−1.43831	−0.719153	−	0.694852i	$-0.755470\pi$
$509$	−32.4496	−1.43831	−0.719153	+	0.694852i	$0.755470\pi$
$510$	0	0
$511$	−2.23725	−0.0989699
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.9075	0.656902
$516$	0	0
$517$	−12.9760	−0.570685
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.88437	0.389231	0.194616	−	0.980880i	$-0.437654\pi$
$521$	8.88437	0.389231	0.194616	+	0.980880i	$0.437654\pi$
$522$	0	0
$523$	−31.0531	−1.35786	−0.678928	−	0.734205i	$-0.737555\pi$
$523$	−31.0531	−1.35786	−0.678928	+	0.734205i	$0.737555\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.0291	1.39521
$528$	0	0
$529$	0.614968	0.0267377
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.75457	0.249258
$534$	0	0
$535$	11.1405	0.481645
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.34132	0.100848
$540$	0	0
$541$	−40.1687	−1.72699	−0.863493	−	0.504360i	$-0.831728\pi$
$541$	−40.1687	−1.72699	−0.863493	+	0.504360i	$0.831728\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.65443	0.0708682
$546$	0	0
$547$	−12.6638	−0.541464	−0.270732	−	0.962655i	$-0.587266\pi$
$547$	−12.6638	−0.541464	−0.270732	+	0.962655i	$0.587266\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.37772	0.399504
$552$	0	0
$553$	−7.36530	−0.313204
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.7679	−1.30368	−0.651838	−	0.758358i	$-0.726002\pi$
$557$	−30.7679	−1.30368	−0.651838	+	0.758358i	$0.726002\pi$
$558$	0	0
$559$	−0.304055	−0.0128601
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.2976	1.36118	0.680591	−	0.732663i	$-0.261723\pi$
$563$	32.2976	1.36118	0.680591	+	0.732663i	$0.261723\pi$
$564$	0	0
$565$	−12.9199	−0.543544
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.4912	1.36210	0.681050	−	0.732237i	$-0.261524\pi$
$569$	32.4912	1.36210	0.681050	+	0.732237i	$0.261524\pi$
$570$	0	0
$571$	23.9687	1.00306	0.501530	−	0.865140i	$-0.332771\pi$
$571$	23.9687	1.00306	0.501530	+	0.865140i	$0.332771\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−15.4381	−0.643813
$576$	0	0
$577$	−27.0655	−1.12675	−0.563375	−	0.826201i	$-0.690498\pi$
$577$	−27.0655	−1.12675	−0.563375	+	0.826201i	$0.690498\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.83555	−0.0761514
$582$	0	0
$583$	−22.3704	−0.926488
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.15714	−0.377956	−0.188978	−	0.981981i	$-0.560517\pi$
$587$	−9.15714	−0.377956	−0.188978	+	0.981981i	$0.560517\pi$
$588$	0	0
$589$	7.37772	0.303994
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.6034	0.517560	0.258780	−	0.965936i	$-0.416679\pi$
$593$	12.6034	0.517560	0.258780	+	0.965936i	$0.416679\pi$
$594$	0	0
$595$	12.4141	0.508929
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.7002	−1.54039	−0.770194	−	0.637810i	$-0.779841\pi$
$599$	−37.7002	−1.54039	−0.770194	+	0.637810i	$0.779841\pi$
$600$	0	0
$601$	20.0291	0.817004	0.408502	−	0.912758i	$-0.366051\pi$
$601$	20.0291	0.817004	0.408502	+	0.912758i	$0.366051\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.7794	−0.641525
$606$	0	0
$607$	−19.8480	−0.805604	−0.402802	−	0.915287i	$-0.631964\pi$
$607$	−19.8480	−0.805604	−0.402802	+	0.915287i	$0.631964\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.76363	0.192716
$612$	0	0
$613$	30.1498	1.21774	0.608870	−	0.793270i	$-0.291623\pi$
$613$	30.1498	1.21774	0.608870	+	0.793270i	$0.291623\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.6702	−0.671118	−0.335559	−	0.942019i	$-0.608925\pi$
$617$	−16.6702	−0.671118	−0.335559	+	0.942019i	$0.608925\pi$
$618$	0	0
$619$	−0.823999	−0.0331193	−0.0165596	−	0.999863i	$-0.505271\pi$
$619$	−0.823999	−0.0331193	−0.0165596	+	0.999863i	$0.505271\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.31735	−0.132907
$624$	0	0
$625$	−30.7918	−1.23167
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.45052	−0.376817
$630$	0	0
$631$	40.6762	1.61929	0.809647	−	0.586917i	$-0.199658\pi$
$631$	40.6762	1.61929	0.809647	+	0.586917i	$0.199658\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−35.4629	−1.40730
$636$	0	0
$637$	−0.859523	−0.0340556
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.8407	0.507176	0.253588	−	0.967312i	$-0.418389\pi$
$641$	12.8407	0.507176	0.253588	+	0.967312i	$0.418389\pi$
$642$	0	0
$643$	−43.3820	−1.71082	−0.855409	−	0.517953i	$-0.826694\pi$
$643$	−43.3820	−1.71082	−0.855409	+	0.517953i	$0.826694\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.0531	−1.37808	−0.689039	−	0.724724i	$-0.741967\pi$
$647$	−35.0531	−1.37808	−0.689039	+	0.724724i	$0.741967\pi$
$648$	0	0
$649$	34.1332	1.33984
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−47.8085	−1.87089	−0.935446	−	0.353470i	$-0.885001\pi$
$653$	−47.8085	−1.87089	−0.935446	+	0.353470i	$0.885001\pi$
$654$	0	0
$655$	5.68352	0.222074
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.6325	−1.58282	−0.791409	−	0.611287i	$-0.790652\pi$
$659$	−40.6325	−1.58282	−0.791409	+	0.611287i	$0.790652\pi$
$660$	0	0
$661$	−26.4265	−1.02787	−0.513937	−	0.857828i	$-0.671813\pi$
$661$	−26.4265	−1.02787	−0.513937	+	0.857828i	$0.671813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.85952	0.110888
$666$	0	0
$667$	45.5713	1.76453
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30.2788	−1.16890
$672$	0	0
$673$	−38.4077	−1.48051	−0.740254	−	0.672328i	$-0.765295\pi$
$673$	−38.4077	−1.48051	−0.740254	+	0.672328i	$0.765295\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.3944	0.822253	0.411127	−	0.911578i	$-0.365135\pi$
$677$	21.3944	0.822253	0.411127	+	0.911578i	$0.365135\pi$
$678$	0	0
$679$	−18.2248	−0.699404
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.6265	−1.32495	−0.662473	−	0.749085i	$-0.730493\pi$
$683$	−34.6265	−1.32495	−0.662473	+	0.749085i	$0.730493\pi$
$684$	0	0
$685$	38.4265	1.46820
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.21240	0.312867
$690$	0	0
$691$	45.8895	1.74572	0.872859	−	0.487972i	$-0.162263\pi$
$691$	45.8895	1.74572	0.872859	+	0.487972i	$0.162263\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.2133	1.25985
$696$	0	0
$697$	29.0655	1.10093
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.0843	0.494189	0.247094	−	0.968991i	$-0.420524\pi$
$701$	13.0843	0.494189	0.247094	+	0.968991i	$0.420524\pi$
$702$	0	0
$703$	−2.17687	−0.0821024
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.18842	0.270349
$708$	0	0
$709$	43.8085	1.64526	0.822631	−	0.568575i	$-0.192505\pi$
$709$	43.8085	1.64526	0.822631	+	0.568575i	$0.192505\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.8522	1.34268
$714$	0	0
$715$	−5.75457	−0.215209
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.2184	1.72365	0.861827	−	0.507202i	$-0.169320\pi$
$719$	46.2184	1.72365	0.861827	+	0.507202i	$0.169320\pi$
$720$	0	0
$721$	−5.21327	−0.194152
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−29.7918	−1.10644
$726$	0	0
$727$	41.8085	1.55059	0.775296	−	0.631598i	$-0.217601\pi$
$727$	41.8085	1.55059	0.775296	+	0.631598i	$0.217601\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.53574	−0.0568012
$732$	0	0
$733$	−2.30405	−0.0851022	−0.0425511	−	0.999094i	$-0.513549\pi$
$733$	−2.30405	−0.0851022	−0.0425511	+	0.999094i	$0.513549\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.1645	−0.374412
$738$	0	0
$739$	15.1405	0.556951	0.278476	−	0.960443i	$-0.410171\pi$
$739$	15.1405	0.556951	0.278476	+	0.960443i	$0.410171\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.2714	1.88097	0.940483	−	0.339839i	$-0.110373\pi$
$743$	51.2714	1.88097	0.940483	+	0.339839i	$0.110373\pi$
$744$	0	0
$745$	−11.7357	−0.429963
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.89592	−0.142354
$750$	0	0
$751$	7.57770	0.276514	0.138257	−	0.990396i	$-0.455850\pi$
$751$	7.57770	0.276514	0.138257	+	0.990396i	$0.455850\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	33.6835	1.22587
$756$	0	0
$757$	35.5109	1.29067	0.645333	−	0.763902i	$-0.276719\pi$
$757$	35.5109	1.29067	0.645333	+	0.763902i	$0.276719\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33.2861	−1.20662	−0.603309	−	0.797507i	$-0.706152\pi$
$761$	−33.2861	−1.20662	−0.603309	+	0.797507i	$0.706152\pi$
$762$	0	0
$763$	−0.578570	−0.0209456
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.5306	−0.452455
$768$	0	0
$769$	52.6680	1.89926	0.949629	−	0.313377i	$-0.101460\pi$
$769$	52.6680	1.89926	0.949629	+	0.313377i	$0.101460\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.62141	0.130253	0.0651264	−	0.997877i	$-0.479255\pi$
$773$	3.62141	0.130253	0.0651264	+	0.997877i	$0.479255\pi$
$774$	0	0
$775$	−23.4381	−0.841921
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.69507	0.239876
$780$	0	0
$781$	−18.4869	−0.661514
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.0852	1.71624
$786$	0	0
$787$	7.85441	0.279979	0.139990	−	0.990153i	$-0.455293\pi$
$787$	7.85441	0.279979	0.139990	+	0.990153i	$0.455293\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.51820	0.160649
$792$	0	0
$793$	11.1156	0.394728
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20.0355	−0.709695	−0.354847	−	0.934924i	$-0.615467\pi$
$797$	−20.0355	−0.709695	−0.354847	+	0.934924i	$0.615467\pi$
$798$	0	0
$799$	24.0604	0.851195
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.23812	0.184849
$804$	0	0
$805$	13.8959	0.489767
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.4942	−0.755697	−0.377848	−	0.925867i	$-0.623336\pi$
$809$	−21.4942	−0.755697	−0.377848	+	0.925867i	$0.623336\pi$
$810$	0	0
$811$	−1.15714	−0.0406327	−0.0203163	−	0.999794i	$-0.506467\pi$
$811$	−1.15714	−0.0406327	−0.0203163	+	0.999794i	$0.506467\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	37.9563	1.32955
$816$	0	0
$817$	−0.353748	−0.0123761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.6463	0.895060	0.447530	−	0.894269i	$-0.352304\pi$
$821$	25.6463	0.895060	0.447530	+	0.894269i	$0.352304\pi$
$822$	0	0
$823$	13.8398	0.482425	0.241212	−	0.970472i	$-0.422455\pi$
$823$	13.8398	0.482425	0.241212	+	0.970472i	$0.422455\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.23812	−0.251694	−0.125847	−	0.992050i	$-0.540165\pi$
$827$	−7.23812	−0.251694	−0.125847	+	0.992050i	$0.540165\pi$
$828$	0	0
$829$	56.2415	1.95335	0.976674	−	0.214729i	$-0.0688867\pi$
$829$	56.2415	1.95335	0.976674	+	0.214729i	$0.0688867\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.34132	−0.150418
$834$	0	0
$835$	−40.7473	−1.41012
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.9272	0.757011	0.378506	−	0.925599i	$-0.376438\pi$
$839$	21.9272	0.757011	0.378506	+	0.925599i	$0.376438\pi$
$840$	0	0
$841$	58.9417	2.03247
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−35.0612	−1.20614
$846$	0	0
$847$	5.51820	0.189608
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.5786	−0.362629
$852$	0	0
$853$	21.2008	0.725903	0.362952	−	0.931808i	$-0.381769\pi$
$853$	21.2008	0.725903	0.362952	+	0.931808i	$0.381769\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.0240	−1.19640	−0.598198	−	0.801348i	$-0.704116\pi$
$857$	−35.0240	−1.19640	−0.598198	+	0.801348i	$0.704116\pi$
$858$	0	0
$859$	−43.6441	−1.48912	−0.744558	−	0.667558i	$-0.767340\pi$
$859$	−43.6441	−1.48912	−0.744558	+	0.667558i	$0.767340\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.32159	0.181149	0.0905745	−	0.995890i	$-0.471130\pi$
$863$	5.32159	0.181149	0.0905745	+	0.995890i	$0.471130\pi$
$864$	0	0
$865$	−59.1425	−2.01091
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.2446	0.584981
$870$	0	0
$871$	3.73147	0.126436
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.21327	0.176241
$876$	0	0
$877$	−15.6942	−0.529955	−0.264978	−	0.964255i	$-0.585365\pi$
$877$	−15.6942	−0.529955	−0.264978	+	0.964255i	$0.585365\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.7743	0.767286	0.383643	−	0.923482i	$-0.374669\pi$
$881$	22.7743	0.767286	0.383643	+	0.923482i	$0.374669\pi$
$882$	0	0
$883$	41.4548	1.39506	0.697532	−	0.716554i	$-0.254282\pi$
$883$	41.4548	1.39506	0.697532	+	0.716554i	$0.254282\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.0415	1.00870	0.504348	−	0.863501i	$-0.331733\pi$
$887$	30.0415	1.00870	0.504348	+	0.863501i	$0.331733\pi$
$888$	0	0
$889$	12.4017	0.415940
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.54217	0.185462
$894$	0	0
$895$	−40.2060	−1.34394
$896$	0	0
$897$	0	0
$898$	0	0
$899$	69.1862	2.30749
$900$	0	0
$901$	41.4796	1.38189
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.0852	0.667655
$906$	0	0
$907$	−24.9323	−0.827864	−0.413932	−	0.910308i	$-0.635845\pi$
$907$	−24.9323	−0.827864	−0.413932	+	0.910308i	$0.635845\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.8034	1.15309	0.576544	−	0.817066i	$-0.304401\pi$
$911$	34.8034	1.15309	0.576544	+	0.817066i	$0.304401\pi$
$912$	0	0
$913$	4.29762	0.142230
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.98758	−0.0656356
$918$	0	0
$919$	−2.10408	−0.0694072	−0.0347036	−	0.999398i	$-0.511049\pi$
$919$	−2.10408	−0.0694072	−0.0347036	+	0.999398i	$0.511049\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.78673	0.223388
$924$	0	0
$925$	6.91565	0.227385
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.2985	−0.501927	−0.250964	−	0.967997i	$-0.580747\pi$
$929$	−15.2985	−0.501927	−0.250964	+	0.967997i	$0.580747\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.0655	−0.950543
$936$	0	0
$937$	41.8522	1.36725	0.683626	−	0.729832i	$-0.260402\pi$
$937$	41.8522	1.36725	0.683626	+	0.729832i	$0.260402\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15.7918	−0.514799	−0.257400	−	0.966305i	$-0.582866\pi$
$941$	−15.7918	−0.514799	−0.257400	+	0.966305i	$0.582866\pi$
$942$	0	0
$943$	32.5349	1.05948
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.1383	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$947$	−15.1383	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$948$	0	0
$949$	−1.92296	−0.0624221
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.5961	−0.634780	−0.317390	−	0.948295i	$-0.602806\pi$
$953$	−19.5961	−0.634780	−0.317390	+	0.948295i	$0.602806\pi$
$954$	0	0
$955$	44.8240	1.45047
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.4381	−0.433939
$960$	0	0
$961$	23.4308	0.755832
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.91653	−0.0616952
$966$	0	0
$967$	−12.0852	−0.388634	−0.194317	−	0.980939i	$-0.562249\pi$
$967$	−12.0852	−0.388634	−0.194317	+	0.980939i	$0.562249\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.3589	−0.557072	−0.278536	−	0.960426i	$-0.589849\pi$
$971$	−17.3589	−0.557072	−0.278536	+	0.960426i	$0.589849\pi$
$972$	0	0
$973$	−11.6150	−0.372359
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.9969	1.31161	0.655804	−	0.754931i	$-0.272330\pi$
$977$	40.9969	1.31161	0.655804	+	0.754931i	$0.272330\pi$
$978$	0	0
$979$	7.76699	0.248234
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.1126	−1.72592	−0.862961	−	0.505270i	$-0.831393\pi$
$983$	−54.1126	−1.72592	−0.862961	+	0.505270i	$0.831393\pi$
$984$	0	0
$985$	68.5746	2.18497
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.71905	−0.0546625
$990$	0	0
$991$	−26.9241	−0.855273	−0.427637	−	0.903951i	$-0.640654\pi$
$991$	−26.9241	−0.855273	−0.427637	+	0.903951i	$0.640654\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	48.5555	1.53931
$996$	0	0
$997$	24.7845	0.784934	0.392467	−	0.919766i	$-0.371622\pi$
$997$	24.7845	0.784934	0.392467	+	0.919766i	$0.371622\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.o.1.3		3
3.2	odd	2		532.2.a.e.1.1	✓	3
12.11	even	2		2128.2.a.r.1.3		3
21.20	even	2		3724.2.a.i.1.3		3
24.5	odd	2		8512.2.a.bn.1.3		3
24.11	even	2		8512.2.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.e.1.1	✓	3	3.2	odd	2
2128.2.a.r.1.3		3	12.11	even	2
3724.2.a.i.1.3		3	21.20	even	2
4788.2.a.o.1.3		3	1.1	even	1	trivial
8512.2.a.bl.1.1		3	24.11	even	2
8512.2.a.bn.1.3		3	24.5	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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85952 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+2.85952 q^{5} -1.00000 q^{7} +2.34132 q^{11} -0.859523 q^{13} -4.34132 q^{17} -1.00000 q^{19} -4.85952 q^{23} +3.17687 q^{25} -9.37772 q^{29} -7.37772 q^{31} -2.85952 q^{35} +2.17687 q^{37} -6.69507 q^{41} +0.353748 q^{43} -5.54217 q^{47} +1.00000 q^{49} -9.55460 q^{53} +6.69507 q^{55} +14.5786 q^{59} -12.9323 q^{61} -2.45783 q^{65} -4.34132 q^{67} -7.89592 q^{71} +2.23725 q^{73} -2.34132 q^{77} +7.36530 q^{79} +1.83555 q^{83} -12.4141 q^{85} +3.31735 q^{89} +0.859523 q^{91} -2.85952 q^{95} +18.2248 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q - 2 * q^5 - 3 * q^7	\( 3 q - 2 q^{5} - 3 q^{7} + 3 q^{11} + 8 q^{13} - 9 q^{17} - 3 q^{19} - 4 q^{23} + 7 q^{25} - 11 q^{29} - 5 q^{31} + 2 q^{35} + 4 q^{37} - 11 q^{41} - 4 q^{43} + 2 q^{47} + 3 q^{49} - 9 q^{53} + 11 q^{55} + 12 q^{59} - 2 q^{61} - 26 q^{65} - 9 q^{67} - 21 q^{73} - 3 q^{77} + 6 q^{79} + 7 q^{83} - 7 q^{85} + 18 q^{89} - 8 q^{91} + 2 q^{95} + 28 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 3 * q^7 + 3 * q^11 + 8 * q^13 - 9 * q^17 - 3 * q^19 - 4 * q^23 + 7 * q^25 - 11 * q^29 - 5 * q^31 + 2 * q^35 + 4 * q^37 - 11 * q^41 - 4 * q^43 + 2 * q^47 + 3 * q^49 - 9 * q^53 + 11 * q^55 + 12 * q^59 - 2 * q^61 - 26 * q^65 - 9 * q^67 - 21 * q^73 - 3 * q^77 + 6 * q^79 + 7 * q^83 - 7 * q^85 + 18 * q^89 - 8 * q^91 + 2 * q^95 + 28 * q^97

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