LMFDB - Embedded newform 8512.2.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51820 q^{3} +2.85952 q^{5} -1.00000 q^{7} +3.34132 q^{9} +2.34132 q^{11} +0.859523 q^{13} +7.20085 q^{15} +4.34132 q^{17} +1.00000 q^{19} -2.51820 q^{21} +4.85952 q^{23} +3.17687 q^{25} +0.859523 q^{27} -9.37772 q^{29} -7.37772 q^{31} +5.89592 q^{33} -2.85952 q^{35} -2.17687 q^{37} +2.16445 q^{39} +6.69507 q^{41} -0.353748 q^{43} +9.55460 q^{45} +5.54217 q^{47} +1.00000 q^{49} +10.9323 q^{51} -9.55460 q^{53} +6.69507 q^{55} +2.51820 q^{57} +14.5786 q^{59} +12.9323 q^{61} -3.34132 q^{63} +2.45783 q^{65} +4.34132 q^{67} +12.2372 q^{69} +7.89592 q^{71} +2.23725 q^{73} +8.00000 q^{75} -2.34132 q^{77} +7.36530 q^{79} -7.85952 q^{81} +1.83555 q^{83} +12.4141 q^{85} -23.6150 q^{87} -3.31735 q^{89} -0.859523 q^{91} -18.5786 q^{93} +2.85952 q^{95} +18.2248 q^{97} +7.82313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39}+ \cdots + 26 q^{99}+O(q^{100}) $ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.51820	1.45388	0.726941	−	0.686700i	$-0.240941\pi$
$3$	2.51820	1.45388	0.726941	+	0.686700i	$0.240941\pi$
$4$	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
$9$	3.34132	1.11377
$10$	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	$-0.461969\pi$
$13$	0.859523	0.238389	0.119194	+	0.992871i	$0.461969\pi$
$14$	0	0
$15$	7.20085	1.85925
$16$	0	0
$17$	4.34132	1.05293	0.526463	−	0.850198i	$-0.323518\pi$
$17$	4.34132	1.05293	0.526463	+	0.850198i	$0.323518\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.51820	−0.549516
$22$	0	0
$23$	4.85952	1.01328	0.506640	−	0.862158i	$-0.330887\pi$
$23$	4.85952	1.01328	0.506640	+	0.862158i	$0.330887\pi$
$24$	0	0
$25$	3.17687	0.635375
$26$	0	0
$27$	0.859523	0.165415
$28$	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
$33$	5.89592	1.02635
$34$	0	0
$35$	−2.85952	−0.483348
$36$	0	0
$37$	−2.17687	−0.357876	−0.178938	−	0.983860i	$-0.557266\pi$
$37$	−2.17687	−0.357876	−0.178938	+	0.983860i	$0.557266\pi$
$38$	0	0
$39$	2.16445	0.346589
$40$	0	0
$41$	6.69507	1.04559	0.522797	−	0.852457i	$-0.324888\pi$
$41$	6.69507	1.04559	0.522797	+	0.852457i	$0.324888\pi$
$42$	0	0
$43$	−0.353748	−0.0539461	−0.0269730	−	0.999636i	$-0.508587\pi$
$43$	−0.353748	−0.0539461	−0.0269730	+	0.999636i	$0.508587\pi$
$44$	0	0
$45$	9.55460	1.42432
$46$	0	0
$47$	5.54217	0.808409	0.404204	−	0.914669i	$-0.367548\pi$
$47$	5.54217	0.808409	0.404204	+	0.914669i	$0.367548\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.9323	1.53083
$52$	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
$57$	2.51820	0.333544
$58$	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	$-0.189532\pi$
$61$	12.9323	1.65581	0.827907	+	0.560866i	$0.189532\pi$
$62$	0	0
$63$	−3.34132	−0.420967
$64$	0	0
$65$	2.45783	0.304856
$66$	0	0
$67$	4.34132	0.530377	0.265189	−	0.964197i	$-0.414566\pi$
$67$	4.34132	0.530377	0.265189	+	0.964197i	$0.414566\pi$
$68$	0	0
$69$	12.2372	1.47319
$70$	0	0
$71$	7.89592	0.937073	0.468537	−	0.883444i	$-0.344781\pi$
$71$	7.89592	0.937073	0.468537	+	0.883444i	$0.344781\pi$
$72$	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
$75$	8.00000	0.923760
$76$	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
$81$	−7.85952	−0.873280
$82$	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
$87$	−23.6150	−2.53179
$88$	0	0
$89$	−3.31735	−0.351638	−0.175819	−	0.984422i	$-0.556257\pi$
$89$	−3.31735	−0.351638	−0.175819	+	0.984422i	$0.556257\pi$
$90$	0	0
$91$	−0.859523	−0.0901025
$92$	0	0
$93$	−18.5786	−1.92651
$94$	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
$99$	7.82313	0.786254
$100$	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
$105$	−7.20085	−0.702731
$106$	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	$-0.508821\pi$
$109$	−0.578570	−0.0554170	−0.0277085	+	0.999616i	$0.508821\pi$
$110$	0	0
$111$	−5.48180	−0.520310
$112$	0	0
$113$	4.51820	0.425036	0.212518	−	0.977157i	$-0.431834\pi$
$113$	4.51820	0.425036	0.212518	+	0.977157i	$0.431834\pi$
$114$	0	0
$115$	13.8959	1.29580
$116$	0	0
$117$	2.87195	0.265512
$118$	0	0
$119$	−4.34132	−0.397969
$120$	0	0
$121$	−5.51820	−0.501654
$122$	0	0
$123$	16.8595	1.52017
$124$	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
$129$	−0.890808	−0.0784313
$130$	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
$135$	2.45783	0.211536
$136$	0	0
$137$	−13.4381	−1.14809	−0.574047	−	0.818822i	$-0.694627\pi$
$137$	−13.4381	−1.14809	−0.574047	+	0.818822i	$0.694627\pi$
$138$	0	0
$139$	−11.6150	−0.985169	−0.492584	−	0.870265i	$-0.663948\pi$
$139$	−11.6150	−0.985169	−0.492584	+	0.870265i	$0.663948\pi$
$140$	0	0
$141$	13.9563	1.17533
$142$	0	0
$143$	2.01242	0.168287
$144$	0	0
$145$	−26.8158	−2.22693
$146$	0	0
$147$	2.51820	0.207698
$148$	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
$153$	14.5058	1.17272
$154$	0	0
$155$	−21.0968	−1.69453
$156$	0	0
$157$	−16.8158	−1.34205	−0.671024	−	0.741436i	$-0.734145\pi$
$157$	−16.8158	−1.34205	−0.671024	+	0.741436i	$0.734145\pi$
$158$	0	0
$159$	−24.0604	−1.90811
$160$	0	0
$161$	−4.85952	−0.382984
$162$	0	0
$163$	−13.2736	−1.03967	−0.519836	−	0.854266i	$-0.674007\pi$
$163$	−13.2736	−1.03967	−0.519836	+	0.854266i	$0.674007\pi$
$164$	0	0
$165$	16.8595	1.31251
$166$	0	0
$167$	14.2497	1.10267	0.551336	−	0.834283i	$-0.314118\pi$
$167$	14.2497	1.10267	0.551336	+	0.834283i	$0.314118\pi$
$168$	0	0
$169$	−12.2612	−0.943171
$170$	0	0
$171$	3.34132	0.255517
$172$	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
$177$	36.7117	2.75942
$178$	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	$-0.584067\pi$
$181$	−7.02397	−0.522088	−0.261044	+	0.965327i	$0.584067\pi$
$182$	0	0
$183$	32.5661	2.40736
$184$	0	0
$185$	−6.22482	−0.457658
$186$	0	0
$187$	10.1645	0.743298
$188$	0	0
$189$	−0.859523	−0.0625211
$190$	0	0
$191$	−15.6753	−1.13423	−0.567114	−	0.823639i	$-0.691940\pi$
$191$	−15.6753	−1.13423	−0.567114	+	0.823639i	$0.691940\pi$
$192$	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
$195$	6.18930	0.443225
$196$	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
$201$	10.9323	0.771106
$202$	0	0
$203$	9.37772	0.658187
$204$	0	0
$205$	19.1447	1.33713
$206$	0	0
$207$	16.2372	1.12857
$208$	0	0
$209$	2.34132	0.161953
$210$	0	0
$211$	−12.8471	−0.884431	−0.442215	−	0.896909i	$-0.645807\pi$
$211$	−12.8471	−0.884431	−0.442215	+	0.896909i	$0.645807\pi$
$212$	0	0
$213$	19.8835	1.36239
$214$	0	0
$215$	−1.01155	−0.0689872
$216$	0	0
$217$	7.37772	0.500832
$218$	0	0
$219$	5.63383	0.380699
$220$	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
$225$	10.6150	0.707665
$226$	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	$-0.182262\pi$
$229$	25.4381	1.68100	0.840498	+	0.541814i	$0.182262\pi$
$230$	0	0
$231$	−5.89592	−0.387923
$232$	0	0
$233$	15.6587	1.02583	0.512917	−	0.858438i	$-0.328565\pi$
$233$	15.6587	1.02583	0.512917	+	0.858438i	$0.328565\pi$
$234$	0	0
$235$	15.8480	1.03381
$236$	0	0
$237$	18.5473	1.20478
$238$	0	0
$239$	−17.0051	−1.09997	−0.549985	−	0.835175i	$-0.685366\pi$
$239$	−17.0051	−1.09997	−0.549985	+	0.835175i	$0.685366\pi$
$240$	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
$243$	−22.3704	−1.43506
$244$	0	0
$245$	2.85952	0.182688
$246$	0	0
$247$	0.859523	0.0546902
$248$	0	0
$249$	4.62228	0.292925
$250$	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
$255$	31.2612	1.95765
$256$	0	0
$257$	2.49335	0.155531	0.0777655	−	0.996972i	$-0.475221\pi$
$257$	2.49335	0.155531	0.0777655	+	0.996972i	$0.475221\pi$
$258$	0	0
$259$	2.17687	0.135264
$260$	0	0
$261$	−31.3340	−1.93953
$262$	0	0
$263$	6.72636	0.414765	0.207382	−	0.978260i	$-0.433506\pi$
$263$	6.72636	0.414765	0.207382	+	0.978260i	$0.433506\pi$
$264$	0	0
$265$	−27.3216	−1.67835
$266$	0	0
$267$	−8.35375	−0.511241
$268$	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
$273$	−2.16445	−0.130999
$274$	0	0
$275$	7.43809	0.448534
$276$	0	0
$277$	9.31735	0.559825	0.279913	−	0.960025i	$-0.409694\pi$
$277$	9.31735	0.559825	0.279913	+	0.960025i	$0.409694\pi$
$278$	0	0
$279$	−24.6514	−1.47584
$280$	0	0
$281$	−12.9323	−0.771477	−0.385739	−	0.922608i	$-0.626053\pi$
$281$	−12.9323	−0.771477	−0.385739	+	0.922608i	$0.626053\pi$
$282$	0	0
$283$	−19.9083	−1.18343	−0.591714	−	0.806148i	$-0.701549\pi$
$283$	−19.9083	−1.18343	−0.591714	+	0.806148i	$0.701549\pi$
$284$	0	0
$285$	7.20085	0.426541
$286$	0	0
$287$	−6.69507	−0.395198
$288$	0	0
$289$	1.84710	0.108653
$290$	0	0
$291$	45.8937	2.69034
$292$	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
$297$	2.01242	0.116773
$298$	0	0
$299$	4.17687	0.241555
$300$	0	0
$301$	0.353748	0.0203897
$302$	0	0
$303$	−18.1019	−1.03993
$304$	0	0
$305$	36.9803	2.11748
$306$	0	0
$307$	29.5713	1.68772	0.843860	−	0.536563i	$-0.180278\pi$
$307$	29.5713	1.68772	0.843860	+	0.536563i	$0.180278\pi$
$308$	0	0
$309$	13.1281	0.746829
$310$	0	0
$311$	−17.7794	−1.00818	−0.504089	−	0.863652i	$-0.668172\pi$
$311$	−17.7794	−1.00818	−0.504089	+	0.863652i	$0.668172\pi$
$312$	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
$315$	−9.55460	−0.538341
$316$	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
$321$	9.81070	0.547580
$322$	0	0
$323$	4.34132	0.241558
$324$	0	0
$325$	2.73060	0.151466
$326$	0	0
$327$	−1.45695	−0.0805698
$328$	0	0
$329$	−5.54217	−0.305550
$330$	0	0
$331$	−29.8835	−1.64255	−0.821273	−	0.570536i	$-0.806736\pi$
$331$	−29.8835	−1.64255	−0.821273	+	0.570536i	$0.806736\pi$
$332$	0	0
$333$	−7.27364	−0.398593
$334$	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
$339$	11.3777	0.617953
$340$	0	0
$341$	−17.2736	−0.935420
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	34.9927	1.88394
$346$	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	$-0.476343\pi$
$349$	2.77431	0.148505	0.0742526	+	0.997239i	$0.476343\pi$
$350$	0	0
$351$	0.738780	0.0394332
$352$	0	0
$353$	−19.3049	−1.02750	−0.513749	−	0.857941i	$-0.671744\pi$
$353$	−19.3049	−1.02750	−0.513749	+	0.857941i	$0.671744\pi$
$354$	0	0
$355$	22.5786	1.19835
$356$	0	0
$357$	−10.9323	−0.578600
$358$	0	0
$359$	28.5349	1.50601	0.753006	−	0.658013i	$-0.228603\pi$
$359$	28.5349	1.50601	0.753006	+	0.658013i	$0.228603\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.8959	−0.729347
$364$	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
$369$	22.3704	1.16456
$370$	0	0
$371$	9.55460	0.496050
$372$	0	0
$373$	−17.3216	−0.896878	−0.448439	−	0.893813i	$-0.648020\pi$
$373$	−17.3216	−0.896878	−0.448439	+	0.893813i	$0.648020\pi$
$374$	0	0
$375$	−13.1281	−0.677930
$376$	0	0
$377$	−8.06037	−0.415130
$378$	0	0
$379$	16.2248	0.833413	0.416707	−	0.909041i	$-0.363184\pi$
$379$	16.2248	0.833413	0.416707	+	0.909041i	$0.363184\pi$
$380$	0	0
$381$	−31.2299	−1.59996
$382$	0	0
$383$	−4.65136	−0.237674	−0.118837	−	0.992914i	$-0.537917\pi$
$383$	−4.65136	−0.237674	−0.118837	+	0.992914i	$0.537917\pi$
$384$	0	0
$385$	−6.69507	−0.341213
$386$	0	0
$387$	−1.18199	−0.0600838
$388$	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
$393$	5.00511	0.252475
$394$	0	0
$395$	21.0612	1.05971
$396$	0	0
$397$	33.9126	1.70202	0.851012	−	0.525146i	$-0.175989\pi$
$397$	33.9126	1.70202	0.851012	+	0.525146i	$0.175989\pi$
$398$	0	0
$399$	−2.51820	−0.126068
$400$	0	0
$401$	4.69507	0.234461	0.117230	−	0.993105i	$-0.462598\pi$
$401$	4.69507	0.234461	0.117230	+	0.993105i	$0.462598\pi$
$402$	0	0
$403$	−6.34132	−0.315884
$404$	0	0
$405$	−22.4745	−1.11677
$406$	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
$411$	−33.8398	−1.66919
$412$	0	0
$413$	−14.5786	−0.717365
$414$	0	0
$415$	5.24880	0.257653
$416$	0	0
$417$	−29.2488	−1.43232
$418$	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	$-0.601066\pi$
$421$	−12.8116	−0.624398	−0.312199	+	0.950017i	$0.601066\pi$
$422$	0	0
$423$	18.5182	0.900386
$424$	0	0
$425$	13.7918	0.669003
$426$	0	0
$427$	−12.9323	−0.625839
$428$	0	0
$429$	5.06768	0.244670
$430$	0	0
$431$	21.4381	1.03264	0.516318	−	0.856397i	$-0.327302\pi$
$431$	21.4381	1.03264	0.516318	+	0.856397i	$0.327302\pi$
$432$	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
$435$	−67.5276	−3.23770
$436$	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
$441$	3.34132	0.159111
$442$	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
$447$	−10.3349	−0.488823
$448$	0	0
$449$	−3.00731	−0.141924	−0.0709619	−	0.997479i	$-0.522607\pi$
$449$	−3.00731	−0.141924	−0.0709619	+	0.997479i	$0.522607\pi$
$450$	0	0
$451$	15.6753	0.738123
$452$	0	0
$453$	29.6629	1.39369
$454$	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
$459$	3.73147	0.174170
$460$	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
$465$	−53.1259	−2.46365
$466$	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
$471$	−42.3456	−1.95118
$472$	0	0
$473$	−0.828239	−0.0380825
$474$	0	0
$475$	3.17687	0.145765
$476$	0	0
$477$	−31.9250	−1.46175
$478$	0	0
$479$	0.0603715	0.00275844	0.00137922	−	0.999999i	$-0.499561\pi$
$479$	0.0603715	0.00275844	0.00137922	+	0.999999i	$0.499561\pi$
$480$	0	0
$481$	−1.87107	−0.0853136
$482$	0	0
$483$	−12.2372	−0.556814
$484$	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
$489$	−33.4257	−1.51156
$490$	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
$495$	22.3704	1.00548
$496$	0	0
$497$	−7.89592	−0.354180
$498$	0	0
$499$	39.5546	1.77071	0.885353	−	0.464919i	$-0.153916\pi$
$499$	39.5546	1.77071	0.885353	+	0.464919i	$0.153916\pi$
$500$	0	0
$501$	35.8835	1.60316
$502$	0	0
$503$	28.8531	1.28650	0.643248	−	0.765658i	$-0.277587\pi$
$503$	28.8531	1.28650	0.643248	+	0.765658i	$0.277587\pi$
$504$	0	0
$505$	−20.5555	−0.914706
$506$	0	0
$507$	−30.8762	−1.37126
$508$	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
$513$	0.859523	0.0379489
$514$	0	0
$515$	14.9075	0.656902
$516$	0	0
$517$	12.9760	0.570685
$518$	0	0
$519$	−52.0830	−2.28619
$520$	0	0
$521$	−8.88437	−0.389231	−0.194616	−	0.980880i	$-0.562346\pi$
$521$	−8.88437	−0.389231	−0.194616	+	0.980880i	$0.562346\pi$
$522$	0	0
$523$	31.0531	1.35786	0.678928	−	0.734205i	$-0.262445\pi$
$523$	31.0531	1.35786	0.678928	+	0.734205i	$0.262445\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	−32.0291	−1.39521
$528$	0	0
$529$	0.614968	0.0267377
$530$	0	0
$531$	48.7117	2.11391
$532$	0	0
$533$	5.75457	0.249258
$534$	0	0
$535$	11.1405	0.481645
$536$	0	0
$537$	−35.4068	−1.52792
$538$	0	0
$539$	2.34132	0.100848
$540$	0	0
$541$	40.1687	1.72699	0.863493	−	0.504360i	$-0.168272\pi$
$541$	40.1687	1.72699	0.863493	+	0.504360i	$0.168272\pi$
$542$	0	0
$543$	−17.6878	−0.759055
$544$	0	0
$545$	−1.65443	−0.0708682
$546$	0	0
$547$	12.6638	0.541464	0.270732	−	0.962655i	$-0.412734\pi$
$547$	12.6638	0.541464	0.270732	+	0.962655i	$0.412734\pi$
$548$	0	0
$549$	43.2111	1.84420
$550$	0	0
$551$	−9.37772	−0.399504
$552$	0	0
$553$	−7.36530	−0.313204
$554$	0	0
$555$	−15.6753	−0.665381
$556$	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
$561$	25.5961	1.08067
$562$	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
$567$	7.85952	0.330069
$568$	0	0
$569$	−32.4912	−1.36210	−0.681050	−	0.732237i	$-0.738476\pi$
$569$	−32.4912	−1.36210	−0.681050	+	0.732237i	$0.738476\pi$
$570$	0	0
$571$	−23.9687	−1.00306	−0.501530	−	0.865140i	$-0.667229\pi$
$571$	−23.9687	−1.00306	−0.501530	+	0.865140i	$0.667229\pi$
$572$	0	0
$573$	−39.4736	−1.64903
$574$	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
$579$	−1.68776	−0.0701410
$580$	0	0
$581$	−1.83555	−0.0761514
$582$	0	0
$583$	−22.3704	−0.926488
$584$	0	0
$585$	8.21240	0.339541
$586$	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
$591$	60.3893	2.48408
$592$	0	0
$593$	−12.6034	−0.517560	−0.258780	−	0.965936i	$-0.583321\pi$
$593$	−12.6034	−0.517560	−0.258780	+	0.965936i	$0.583321\pi$
$594$	0	0
$595$	−12.4141	−0.508929
$596$	0	0
$597$	42.7597	1.75004
$598$	0	0
$599$	37.7002	1.54039	0.770194	−	0.637810i	$-0.220159\pi$
$599$	37.7002	1.54039	0.770194	+	0.637810i	$0.220159\pi$
$600$	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
$603$	14.5058	0.590721
$604$	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
$609$	23.6150	0.956927
$610$	0	0
$611$	4.76363	0.192716
$612$	0	0
$613$	−30.1498	−1.21774	−0.608870	−	0.793270i	$-0.708377\pi$
$613$	−30.1498	−1.21774	−0.608870	+	0.793270i	$0.708377\pi$
$614$	0	0
$615$	48.2102	1.94402
$616$	0	0
$617$	16.6702	0.671118	0.335559	−	0.942019i	$-0.391075\pi$
$617$	16.6702	0.671118	0.335559	+	0.942019i	$0.391075\pi$
$618$	0	0
$619$	0.823999	0.0331193	0.0165596	−	0.999863i	$-0.494729\pi$
$619$	0.823999	0.0331193	0.0165596	+	0.999863i	$0.494729\pi$
$620$	0	0
$621$	4.17687	0.167612
$622$	0	0
$623$	3.31735	0.132907
$624$	0	0
$625$	−30.7918	−1.23167
$626$	0	0
$627$	5.89592	0.235460
$628$	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
$633$	−32.3516	−1.28586
$634$	0	0
$635$	−35.4629	−1.40730
$636$	0	0
$637$	0.859523	0.0340556
$638$	0	0
$639$	26.3828	1.04369
$640$	0	0
$641$	−12.8407	−0.507176	−0.253588	−	0.967312i	$-0.581611\pi$
$641$	−12.8407	−0.507176	−0.253588	+	0.967312i	$0.581611\pi$
$642$	0	0
$643$	43.3820	1.71082	0.855409	−	0.517953i	$-0.173306\pi$
$643$	43.3820	1.71082	0.855409	+	0.517953i	$0.173306\pi$
$644$	0	0
$645$	−2.54729	−0.100299
$646$	0	0
$647$	35.0531	1.37808	0.689039	−	0.724724i	$-0.258033\pi$
$647$	35.0531	1.37808	0.689039	+	0.724724i	$0.258033\pi$
$648$	0	0
$649$	34.1332	1.33984
$650$	0	0
$651$	18.5786	0.728152
$652$	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
$657$	7.47536	0.291642
$658$	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	$-0.328187\pi$
$661$	26.4265	1.02787	0.513937	+	0.857828i	$0.328187\pi$
$662$	0	0
$663$	9.39658	0.364933
$664$	0	0
$665$	−2.85952	−0.110888
$666$	0	0
$667$	−45.5713	−1.76453
$668$	0	0
$669$	21.4818	0.830534
$670$	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
$675$	2.73060	0.105101
$676$	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
$681$	−69.2466	−2.65354
$682$	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
$687$	64.0582	2.44397
$688$	0	0
$689$	−8.21240	−0.312867
$690$	0	0
$691$	−45.8895	−1.74572	−0.872859	−	0.487972i	$-0.837737\pi$
$691$	−45.8895	−1.74572	−0.872859	+	0.487972i	$0.837737\pi$
$692$	0	0
$693$	−7.82313	−0.297176
$694$	0	0
$695$	−33.2133	−1.25985
$696$	0	0
$697$	29.0655	1.10093
$698$	0	0
$699$	39.4317	1.49144
$700$	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
$705$	39.9083	1.50304
$706$	0	0
$707$	7.18842	0.270349
$708$	0	0
$709$	−43.8085	−1.64526	−0.822631	−	0.568575i	$-0.807495\pi$
$709$	−43.8085	−1.64526	−0.822631	+	0.568575i	$0.807495\pi$
$710$	0	0
$711$	24.6099	0.922942
$712$	0	0
$713$	−35.8522	−1.34268
$714$	0	0
$715$	5.75457	0.215209
$716$	0	0
$717$	−42.8223	−1.59923
$718$	0	0
$719$	−46.2184	−1.72365	−0.861827	−	0.507202i	$-0.830680\pi$
$719$	−46.2184	−1.72365	−0.861827	+	0.507202i	$0.830680\pi$
$720$	0	0
$721$	−5.21327	−0.194152
$722$	0	0
$723$	6.04795	0.224926
$724$	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
$729$	−32.7546	−1.21313
$730$	0	0
$731$	−1.53574	−0.0568012
$732$	0	0
$733$	2.30405	0.0851022	0.0425511	−	0.999094i	$-0.486451\pi$
$733$	2.30405	0.0851022	0.0425511	+	0.999094i	$0.486451\pi$
$734$	0	0
$735$	7.20085	0.265607
$736$	0	0
$737$	10.1645	0.374412
$738$	0	0
$739$	−15.1405	−0.556951	−0.278476	−	0.960443i	$-0.589829\pi$
$739$	−15.1405	−0.556951	−0.278476	+	0.960443i	$0.589829\pi$
$740$	0	0
$741$	2.16445	0.0795131
$742$	0	0
$743$	−51.2714	−1.88097	−0.940483	−	0.339839i	$-0.889627\pi$
$743$	−51.2714	−1.88097	−0.940483	+	0.339839i	$0.889627\pi$
$744$	0	0
$745$	−11.7357	−0.429963
$746$	0	0
$747$	6.13317	0.224401
$748$	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
$753$	−14.5537	−0.530367
$754$	0	0
$755$	33.6835	1.22587
$756$	0	0
$757$	−35.5109	−1.29067	−0.645333	−	0.763902i	$-0.723281\pi$
$757$	−35.5109	−1.29067	−0.645333	+	0.763902i	$0.723281\pi$
$758$	0	0
$759$	28.6514	1.03998
$760$	0	0
$761$	33.2861	1.20662	0.603309	−	0.797507i	$-0.293848\pi$
$761$	33.2861	1.20662	0.603309	+	0.797507i	$0.293848\pi$
$762$	0	0
$763$	0.578570	0.0209456
$764$	0	0
$765$	41.4796	1.49970
$766$	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
$771$	6.27876	0.226124
$772$	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
$777$	5.48180	0.196659
$778$	0	0
$779$	6.69507	0.239876
$780$	0	0
$781$	18.4869	0.661514
$782$	0	0
$783$	−8.06037	−0.288054
$784$	0	0
$785$	−48.0852	−1.71624
$786$	0	0
$787$	−7.85441	−0.279979	−0.139990	−	0.990153i	$-0.544707\pi$
$787$	−7.85441	−0.279979	−0.139990	+	0.990153i	$0.544707\pi$
$788$	0	0
$789$	16.9383	0.603020
$790$	0	0
$791$	−4.51820	−0.160649
$792$	0	0
$793$	11.1156	0.394728
$794$	0	0
$795$	−68.8012	−2.44013
$796$	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
$801$	−11.0843	−0.391646
$802$	0	0
$803$	5.23812	0.184849
$804$	0	0
$805$	−13.8959	−0.489767
$806$	0	0
$807$	−25.6587	−0.903228
$808$	0	0
$809$	21.4942	0.755697	0.377848	−	0.925867i	$-0.376664\pi$
$809$	21.4942	0.755697	0.377848	+	0.925867i	$0.376664\pi$
$810$	0	0
$811$	1.15714	0.0406327	0.0203163	−	0.999794i	$-0.493533\pi$
$811$	1.15714	0.0406327	0.0203163	+	0.999794i	$0.493533\pi$
$812$	0	0
$813$	55.6731	1.95254
$814$	0	0
$815$	−37.9563	−1.32955
$816$	0	0
$817$	−0.353748	−0.0123761
$818$	0	0
$819$	−2.87195	−0.100354
$820$	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
$825$	18.7306	0.652116
$826$	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	$-0.931113\pi$
$829$	−56.2415	−1.95335	−0.976674	+	0.214729i	$0.931113\pi$
$830$	0	0
$831$	23.4629	0.813920
$832$	0	0
$833$	4.34132	0.150418
$834$	0	0
$835$	40.7473	1.41012
$836$	0	0
$837$	−6.34132	−0.219188
$838$	0	0
$839$	−21.9272	−0.757011	−0.378506	−	0.925599i	$-0.623562\pi$
$839$	−21.9272	−0.757011	−0.378506	+	0.925599i	$0.623562\pi$
$840$	0	0
$841$	58.9417	2.03247
$842$	0	0
$843$	−32.5661	−1.12164
$844$	0	0
$845$	−35.0612	−1.20614
$846$	0	0
$847$	5.51820	0.189608
$848$	0	0
$849$	−50.1332	−1.72057
$850$	0	0
$851$	−10.5786	−0.362629
$852$	0	0
$853$	−21.2008	−0.725903	−0.362952	−	0.931808i	$-0.618231\pi$
$853$	−21.2008	−0.725903	−0.362952	+	0.931808i	$0.618231\pi$
$854$	0	0
$855$	9.55460	0.326760
$856$	0	0
$857$	35.0240	1.19640	0.598198	−	0.801348i	$-0.295884\pi$
$857$	35.0240	1.19640	0.598198	+	0.801348i	$0.295884\pi$
$858$	0	0
$859$	43.6441	1.48912	0.744558	−	0.667558i	$-0.232660\pi$
$859$	43.6441	1.48912	0.744558	+	0.667558i	$0.232660\pi$
$860$	0	0
$861$	−16.8595	−0.574571
$862$	0	0
$863$	−5.32159	−0.181149	−0.0905745	−	0.995890i	$-0.528870\pi$
$863$	−5.32159	−0.181149	−0.0905745	+	0.995890i	$0.528870\pi$
$864$	0	0
$865$	−59.1425	−2.01091
$866$	0	0
$867$	4.65136	0.157969
$868$	0	0
$869$	17.2446	0.584981
$870$	0	0
$871$	3.73147	0.126436
$872$	0	0
$873$	60.8950	2.06099
$874$	0	0
$875$	5.21327	0.176241
$876$	0	0
$877$	15.6942	0.529955	0.264978	−	0.964255i	$-0.414635\pi$
$877$	15.6942	0.529955	0.264978	+	0.964255i	$0.414635\pi$
$878$	0	0
$879$	−50.8094	−1.71376
$880$	0	0
$881$	−22.7743	−0.767286	−0.383643	−	0.923482i	$-0.625331\pi$
$881$	−22.7743	−0.767286	−0.383643	+	0.923482i	$0.625331\pi$
$882$	0	0
$883$	−41.4548	−1.39506	−0.697532	−	0.716554i	$-0.745718\pi$
$883$	−41.4548	−1.39506	−0.697532	+	0.716554i	$0.745718\pi$
$884$	0	0
$885$	104.978	3.52880
$886$	0	0
$887$	−30.0415	−1.00870	−0.504348	−	0.863501i	$-0.668267\pi$
$887$	−30.0415	−1.00870	−0.504348	+	0.863501i	$0.668267\pi$
$888$	0	0
$889$	12.4017	0.415940
$890$	0	0
$891$	−18.4017	−0.616480
$892$	0	0
$893$	5.54217	0.185462
$894$	0	0
$895$	−40.2060	−1.34394
$896$	0	0
$897$	10.5182	0.351192
$898$	0	0
$899$	69.1862	2.30749
$900$	0	0
$901$	−41.4796	−1.38189
$902$	0	0
$903$	0.890808	0.0296442
$904$	0	0
$905$	−20.0852	−0.667655
$906$	0	0
$907$	24.9323	0.827864	0.413932	−	0.910308i	$-0.364155\pi$
$907$	24.9323	0.827864	0.413932	+	0.910308i	$0.364155\pi$
$908$	0	0
$909$	−24.0189	−0.796655
$910$	0	0
$911$	−34.8034	−1.15309	−0.576544	−	0.817066i	$-0.695599\pi$
$911$	−34.8034	−1.15309	−0.576544	+	0.817066i	$0.695599\pi$
$912$	0	0
$913$	4.29762	0.142230
$914$	0	0
$915$	93.1237	3.07857
$916$	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
$921$	74.4663	2.45375
$922$	0	0
$923$	6.78673	0.223388
$924$	0	0
$925$	−6.91565	−0.227385
$926$	0	0
$927$	17.4192	0.572123
$928$	0	0
$929$	15.2985	0.501927	0.250964	−	0.967997i	$-0.419253\pi$
$929$	15.2985	0.501927	0.250964	+	0.967997i	$0.419253\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−44.7721	−1.46577
$934$	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
$939$	60.5575	1.97622
$940$	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
$945$	−2.45783	−0.0799531
$946$	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
$951$	−39.0595	−1.26659
$952$	0	0
$953$	19.5961	0.634780	0.317390	−	0.948295i	$-0.397194\pi$
$953$	19.5961	0.634780	0.317390	+	0.948295i	$0.397194\pi$
$954$	0	0
$955$	−44.8240	−1.45047
$956$	0	0
$957$	−55.2903	−1.78728
$958$	0	0
$959$	13.4381	0.433939
$960$	0	0
$961$	23.4308	0.755832
$962$	0	0
$963$	13.0175	0.419484
$964$	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
$969$	10.9323	0.351197
$970$	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
$975$	6.87619	0.220214
$976$	0	0
$977$	−40.9969	−1.31161	−0.655804	−	0.754931i	$-0.727670\pi$
$977$	−40.9969	−1.31161	−0.655804	+	0.754931i	$0.727670\pi$
$978$	0	0
$979$	−7.76699	−0.248234
$980$	0	0
$981$	−1.93319	−0.0617220
$982$	0	0
$983$	54.1126	1.72592	0.862961	−	0.505270i	$-0.168607\pi$
$983$	54.1126	1.72592	0.862961	+	0.505270i	$0.168607\pi$
$984$	0	0
$985$	68.5746	2.18497
$986$	0	0
$987$	−13.9563	−0.444234
$988$	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
$993$	−75.2526	−2.38807
$994$	0	0
$995$	48.5555	1.53931
$996$	0	0
$997$	−24.7845	−0.784934	−0.392467	−	0.919766i	$-0.628378\pi$
$997$	−24.7845	−0.784934	−0.392467	+	0.919766i	$0.628378\pi$
$998$	0	0
$999$	−1.87107	−0.0591982

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.e.1.1	✓	3	8.5	even	2
2128.2.a.r.1.3		3	8.3	odd	2
3724.2.a.i.1.3		3	56.13	odd	2
4788.2.a.o.1.3		3	24.5	odd	2
8512.2.a.bl.1.1		3	4.3	odd	2
8512.2.a.bn.1.3		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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51820 q^{3} +2.85952 q^{5} -1.00000 q^{7} +3.34132 q^{9} +2.34132 q^{11} +0.859523 q^{13} +7.20085 q^{15} +4.34132 q^{17} +1.00000 q^{19} -2.51820 q^{21} +4.85952 q^{23} +3.17687 q^{25} +0.859523 q^{27} -9.37772 q^{29} -7.37772 q^{31} +5.89592 q^{33} -2.85952 q^{35} -2.17687 q^{37} +2.16445 q^{39} +6.69507 q^{41} -0.353748 q^{43} +9.55460 q^{45} +5.54217 q^{47} +1.00000 q^{49} +10.9323 q^{51} -9.55460 q^{53} +6.69507 q^{55} +2.51820 q^{57} +14.5786 q^{59} +12.9323 q^{61} -3.34132 q^{63} +2.45783 q^{65} +4.34132 q^{67} +12.2372 q^{69} +7.89592 q^{71} +2.23725 q^{73} +8.00000 q^{75} -2.34132 q^{77} +7.36530 q^{79} -7.85952 q^{81} +1.83555 q^{83} +12.4141 q^{85} -23.6150 q^{87} -3.31735 q^{89} -0.859523 q^{91} -18.5786 q^{93} +2.85952 q^{95} +18.2248 q^{97} +7.82313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39}+ \cdots + 26 q^{99}+O(q^{100}) \) 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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