LMFDB - Embedded newform 8464.2.a.cg.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-1,0,0,0,-10,0,16,0,-23,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.01779$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.01779 q^{3} -0.684021 q^{5} +2.16510 q^{7} +1.07147 q^{9} +1.89186 q^{11} -2.22346 q^{13} -1.38021 q^{15} -2.80474 q^{17} -2.23273 q^{19} +4.36872 q^{21} -4.53211 q^{25} -3.89136 q^{27} -8.59972 q^{29} +2.31109 q^{31} +3.81737 q^{33} -1.48098 q^{35} -5.55335 q^{37} -4.48647 q^{39} +3.44767 q^{41} -1.37494 q^{43} -0.732910 q^{45} +11.4999 q^{47} -2.31234 q^{49} -5.65938 q^{51} -8.37106 q^{53} -1.29407 q^{55} -4.50517 q^{57} +8.40725 q^{59} -5.71825 q^{61} +2.31985 q^{63} +1.52089 q^{65} -15.4601 q^{67} +1.24569 q^{71} +13.4084 q^{73} -9.14485 q^{75} +4.09607 q^{77} -8.14191 q^{79} -11.0664 q^{81} -9.82785 q^{83} +1.91850 q^{85} -17.3524 q^{87} -11.0489 q^{89} -4.81402 q^{91} +4.66329 q^{93} +1.52723 q^{95} -16.8611 q^{97} +2.02707 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} + q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} + 18 q^{33} + 18 q^{35} + 24 q^{37} + 19 q^{39} + 9 q^{41} - 48 q^{43} + 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) $ 15 * q - q^3 - 10 * q^7 + 16 * q^9 - 23 * q^11 - 10 * q^15 - 29 * q^19 + q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 + 18 * q^33 + 18 * q^35 + 24 * q^37 + 19 * q^39 + 9 * q^41 - 48 * q^43 + 4 * q^45 + 36 * q^47 + 25 * q^49 - 35 * q^51 - 5 * q^53 + 10 * q^55 + 23 * q^57 + 22 * q^59 + 12 * q^61 - 35 * q^63 - 26 * q^65 - 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 - 26 * q^79 - 21 * q^81 - 68 * q^83 - 72 * q^85 - 19 * q^87 - 6 * q^89 - 71 * q^91 - 55 * q^93 + 12 * q^95 + 40 * q^97 - 63 * 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
$3$	2.01779	1.16497	0.582485	−	0.812841i	$-0.302080\pi$
$3$	2.01779	1.16497	0.582485	+	0.812841i	$0.302080\pi$
$4$	0	0
$5$	−0.684021	−0.305904	−0.152952	−	0.988234i	$-0.548878\pi$
$5$	−0.684021	−0.305904	−0.152952	+	0.988234i	$0.548878\pi$
$6$	0	0
$7$	2.16510	0.818331	0.409166	−	0.912460i	$-0.365820\pi$
$7$	2.16510	0.818331	0.409166	+	0.912460i	$0.365820\pi$
$8$	0	0
$9$	1.07147	0.357157
$10$	0	0
$11$	1.89186	0.570417	0.285208	−	0.958466i	$-0.407937\pi$
$11$	1.89186	0.570417	0.285208	+	0.958466i	$0.407937\pi$
$12$	0	0
$13$	−2.22346	−0.616677	−0.308339	−	0.951277i	$-0.599773\pi$
$13$	−2.22346	−0.616677	−0.308339	+	0.951277i	$0.599773\pi$
$14$	0	0
$15$	−1.38021	−0.356369
$16$	0	0
$17$	−2.80474	−0.680250	−0.340125	−	0.940380i	$-0.610469\pi$
$17$	−2.80474	−0.680250	−0.340125	+	0.940380i	$0.610469\pi$
$18$	0	0
$19$	−2.23273	−0.512222	−0.256111	−	0.966647i	$-0.582441\pi$
$19$	−2.23273	−0.512222	−0.256111	+	0.966647i	$0.582441\pi$
$20$	0	0
$21$	4.36872	0.953332
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.53211	−0.906423
$26$	0	0
$27$	−3.89136	−0.748893
$28$	0	0
$29$	−8.59972	−1.59693	−0.798464	−	0.602042i	$-0.794354\pi$
$29$	−8.59972	−1.59693	−0.798464	+	0.602042i	$0.794354\pi$
$30$	0	0
$31$	2.31109	0.415084	0.207542	−	0.978226i	$-0.433454\pi$
$31$	2.31109	0.415084	0.207542	+	0.978226i	$0.433454\pi$
$32$	0	0
$33$	3.81737	0.664519
$34$	0	0
$35$	−1.48098	−0.250331
$36$	0	0
$37$	−5.55335	−0.912966	−0.456483	−	0.889732i	$-0.650891\pi$
$37$	−5.55335	−0.912966	−0.456483	+	0.889732i	$0.650891\pi$
$38$	0	0
$39$	−4.48647	−0.718411
$40$	0	0
$41$	3.44767	0.538436	0.269218	−	0.963079i	$-0.413235\pi$
$41$	3.44767	0.538436	0.269218	+	0.963079i	$0.413235\pi$
$42$	0	0
$43$	−1.37494	−0.209676	−0.104838	−	0.994489i	$-0.533432\pi$
$43$	−1.37494	−0.209676	−0.104838	+	0.994489i	$0.533432\pi$
$44$	0	0
$45$	−0.732910	−0.109256
$46$	0	0
$47$	11.4999	1.67743	0.838717	−	0.544568i	$-0.183306\pi$
$47$	11.4999	1.67743	0.838717	+	0.544568i	$0.183306\pi$
$48$	0	0
$49$	−2.31234	−0.330334
$50$	0	0
$51$	−5.65938	−0.792472
$52$	0	0
$53$	−8.37106	−1.14985	−0.574927	−	0.818205i	$-0.694970\pi$
$53$	−8.37106	−1.14985	−0.574927	+	0.818205i	$0.694970\pi$
$54$	0	0
$55$	−1.29407	−0.174493
$56$	0	0
$57$	−4.50517	−0.596724
$58$	0	0
$59$	8.40725	1.09453	0.547265	−	0.836959i	$-0.315669\pi$
$59$	8.40725	1.09453	0.547265	+	0.836959i	$0.315669\pi$
$60$	0	0
$61$	−5.71825	−0.732147	−0.366073	−	0.930586i	$-0.619298\pi$
$61$	−5.71825	−0.732147	−0.366073	+	0.930586i	$0.619298\pi$
$62$	0	0
$63$	2.31985	0.292273
$64$	0	0
$65$	1.52089	0.188644
$66$	0	0
$67$	−15.4601	−1.88875	−0.944376	−	0.328868i	$-0.893333\pi$
$67$	−15.4601	−1.88875	−0.944376	+	0.328868i	$0.893333\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.24569	0.147836	0.0739180	−	0.997264i	$-0.476450\pi$
$71$	1.24569	0.147836	0.0739180	+	0.997264i	$0.476450\pi$
$72$	0	0
$73$	13.4084	1.56934	0.784668	−	0.619916i	$-0.212833\pi$
$73$	13.4084	1.56934	0.784668	+	0.619916i	$0.212833\pi$
$74$	0	0
$75$	−9.14485	−1.05596
$76$	0	0
$77$	4.09607	0.466790
$78$	0	0
$79$	−8.14191	−0.916036	−0.458018	−	0.888943i	$-0.651440\pi$
$79$	−8.14191	−0.916036	−0.458018	+	0.888943i	$0.651440\pi$
$80$	0	0
$81$	−11.0664	−1.22960
$82$	0	0
$83$	−9.82785	−1.07875	−0.539374	−	0.842067i	$-0.681339\pi$
$83$	−9.82785	−1.07875	−0.539374	+	0.842067i	$0.681339\pi$
$84$	0	0
$85$	1.91850	0.208091
$86$	0	0
$87$	−17.3524	−1.86038
$88$	0	0
$89$	−11.0489	−1.17118	−0.585590	−	0.810607i	$-0.699137\pi$
$89$	−11.0489	−1.17118	−0.585590	+	0.810607i	$0.699137\pi$
$90$	0	0
$91$	−4.81402	−0.504646
$92$	0	0
$93$	4.66329	0.483560
$94$	0	0
$95$	1.52723	0.156691
$96$	0	0
$97$	−16.8611	−1.71199	−0.855993	−	0.516988i	$-0.827053\pi$
$97$	−16.8611	−1.71199	−0.855993	+	0.516988i	$0.827053\pi$
$98$	0	0
$99$	2.02707	0.203729
$100$	0	0
$101$	6.43369	0.640176	0.320088	−	0.947388i	$-0.396288\pi$
$101$	6.43369	0.640176	0.320088	+	0.947388i	$0.396288\pi$
$102$	0	0
$103$	12.5667	1.23824	0.619118	−	0.785298i	$-0.287490\pi$
$103$	12.5667	1.23824	0.619118	+	0.785298i	$0.287490\pi$
$104$	0	0
$105$	−2.98830	−0.291628
$106$	0	0
$107$	−7.32102	−0.707750	−0.353875	−	0.935293i	$-0.615136\pi$
$107$	−7.32102	−0.707750	−0.353875	+	0.935293i	$0.615136\pi$
$108$	0	0
$109$	−8.59291	−0.823051	−0.411526	−	0.911398i	$-0.635004\pi$
$109$	−8.59291	−0.823051	−0.411526	+	0.911398i	$0.635004\pi$
$110$	0	0
$111$	−11.2055	−1.06358
$112$	0	0
$113$	11.8763	1.11723	0.558615	−	0.829427i	$-0.311333\pi$
$113$	11.8763	1.11723	0.558615	+	0.829427i	$0.311333\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.38238	−0.220251
$118$	0	0
$119$	−6.07255	−0.556670
$120$	0	0
$121$	−7.42087	−0.674625
$122$	0	0
$123$	6.95668	0.627263
$124$	0	0
$125$	6.52017	0.583182
$126$	0	0
$127$	−7.96058	−0.706387	−0.353194	−	0.935550i	$-0.614904\pi$
$127$	−7.96058	−0.706387	−0.353194	+	0.935550i	$0.614904\pi$
$128$	0	0
$129$	−2.77433	−0.244267
$130$	0	0
$131$	2.26451	0.197851	0.0989256	−	0.995095i	$-0.468459\pi$
$131$	2.26451	0.197851	0.0989256	+	0.995095i	$0.468459\pi$
$132$	0	0
$133$	−4.83408	−0.419168
$134$	0	0
$135$	2.66177	0.229089
$136$	0	0
$137$	−4.75279	−0.406058	−0.203029	−	0.979173i	$-0.565079\pi$
$137$	−4.75279	−0.406058	−0.203029	+	0.979173i	$0.565079\pi$
$138$	0	0
$139$	8.47555	0.718887	0.359444	−	0.933167i	$-0.382967\pi$
$139$	8.47555	0.718887	0.359444	+	0.933167i	$0.382967\pi$
$140$	0	0
$141$	23.2044	1.95416
$142$	0	0
$143$	−4.20647	−0.351763
$144$	0	0
$145$	5.88239	0.488506
$146$	0	0
$147$	−4.66581	−0.384829
$148$	0	0
$149$	13.6174	1.11558	0.557790	−	0.829982i	$-0.311649\pi$
$149$	13.6174	1.11558	0.557790	+	0.829982i	$0.311649\pi$
$150$	0	0
$151$	1.51335	0.123155	0.0615775	−	0.998102i	$-0.480387\pi$
$151$	1.51335	0.123155	0.0615775	+	0.998102i	$0.480387\pi$
$152$	0	0
$153$	−3.00520	−0.242956
$154$	0	0
$155$	−1.58083	−0.126976
$156$	0	0
$157$	20.3036	1.62041	0.810203	−	0.586149i	$-0.199357\pi$
$157$	20.3036	1.62041	0.810203	+	0.586149i	$0.199357\pi$
$158$	0	0
$159$	−16.8910	−1.33955
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.88736	0.304482	0.152241	−	0.988343i	$-0.451351\pi$
$163$	3.88736	0.304482	0.152241	+	0.988343i	$0.451351\pi$
$164$	0	0
$165$	−2.61116	−0.203279
$166$	0	0
$167$	−5.52957	−0.427891	−0.213945	−	0.976846i	$-0.568631\pi$
$167$	−5.52957	−0.427891	−0.213945	+	0.976846i	$0.568631\pi$
$168$	0	0
$169$	−8.05622	−0.619709
$170$	0	0
$171$	−2.39230	−0.182944
$172$	0	0
$173$	−16.1608	−1.22868	−0.614342	−	0.789040i	$-0.710578\pi$
$173$	−16.1608	−1.22868	−0.614342	+	0.789040i	$0.710578\pi$
$174$	0	0
$175$	−9.81249	−0.741754
$176$	0	0
$177$	16.9640	1.27510
$178$	0	0
$179$	−12.6453	−0.945151	−0.472576	−	0.881290i	$-0.656676\pi$
$179$	−12.6453	−0.945151	−0.472576	+	0.881290i	$0.656676\pi$
$180$	0	0
$181$	11.4188	0.848749	0.424375	−	0.905487i	$-0.360494\pi$
$181$	11.4188	0.848749	0.424375	+	0.905487i	$0.360494\pi$
$182$	0	0
$183$	−11.5382	−0.852930
$184$	0	0
$185$	3.79861	0.279279
$186$	0	0
$187$	−5.30618	−0.388026
$188$	0	0
$189$	−8.42519	−0.612843
$190$	0	0
$191$	−3.15177	−0.228054	−0.114027	−	0.993478i	$-0.536375\pi$
$191$	−3.15177	−0.228054	−0.114027	+	0.993478i	$0.536375\pi$
$192$	0	0
$193$	16.5055	1.18809	0.594044	−	0.804432i	$-0.297530\pi$
$193$	16.5055	1.18809	0.594044	+	0.804432i	$0.297530\pi$
$194$	0	0
$195$	3.06884	0.219764
$196$	0	0
$197$	−17.5007	−1.24688	−0.623438	−	0.781873i	$-0.714265\pi$
$197$	−17.5007	−1.24688	−0.623438	+	0.781873i	$0.714265\pi$
$198$	0	0
$199$	3.94636	0.279750	0.139875	−	0.990169i	$-0.455330\pi$
$199$	3.94636	0.279750	0.139875	+	0.990169i	$0.455330\pi$
$200$	0	0
$201$	−31.1952	−2.20034
$202$	0	0
$203$	−18.6193	−1.30682
$204$	0	0
$205$	−2.35828	−0.164710
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.22400	−0.292180
$210$	0	0
$211$	13.9925	0.963284	0.481642	−	0.876368i	$-0.340041\pi$
$211$	13.9925	0.963284	0.481642	+	0.876368i	$0.340041\pi$
$212$	0	0
$213$	2.51353	0.172225
$214$	0	0
$215$	0.940487	0.0641407
$216$	0	0
$217$	5.00374	0.339676
$218$	0	0
$219$	27.0553	1.82823
$220$	0	0
$221$	6.23624	0.419495
$222$	0	0
$223$	26.8055	1.79503	0.897515	−	0.440984i	$-0.145370\pi$
$223$	26.8055	1.79503	0.897515	+	0.440984i	$0.145370\pi$
$224$	0	0
$225$	−4.85604	−0.323736
$226$	0	0
$227$	−24.1452	−1.60257	−0.801286	−	0.598281i	$-0.795851\pi$
$227$	−24.1452	−1.60257	−0.801286	+	0.598281i	$0.795851\pi$
$228$	0	0
$229$	12.7876	0.845030	0.422515	−	0.906356i	$-0.361147\pi$
$229$	12.7876	0.845030	0.422515	+	0.906356i	$0.361147\pi$
$230$	0	0
$231$	8.26500	0.543797
$232$	0	0
$233$	7.91413	0.518472	0.259236	−	0.965814i	$-0.416529\pi$
$233$	7.91413	0.518472	0.259236	+	0.965814i	$0.416529\pi$
$234$	0	0
$235$	−7.86618	−0.513133
$236$	0	0
$237$	−16.4286	−1.06716
$238$	0	0
$239$	−11.0775	−0.716546	−0.358273	−	0.933617i	$-0.616634\pi$
$239$	−11.0775	−0.716546	−0.358273	+	0.933617i	$0.616634\pi$
$240$	0	0
$241$	26.1980	1.68756	0.843780	−	0.536689i	$-0.180325\pi$
$241$	26.1980	1.68756	0.843780	+	0.536689i	$0.180325\pi$
$242$	0	0
$243$	−10.6555	−0.683551
$244$	0	0
$245$	1.58169	0.101050
$246$	0	0
$247$	4.96438	0.315876
$248$	0	0
$249$	−19.8305	−1.25671
$250$	0	0
$251$	17.4314	1.10026	0.550129	−	0.835080i	$-0.314578\pi$
$251$	17.4314	1.10026	0.550129	+	0.835080i	$0.314578\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.87114	0.242420
$256$	0	0
$257$	12.3472	0.770195	0.385098	−	0.922876i	$-0.374168\pi$
$257$	12.3472	0.770195	0.385098	+	0.922876i	$0.374168\pi$
$258$	0	0
$259$	−12.0236	−0.747108
$260$	0	0
$261$	−9.21436	−0.570355
$262$	0	0
$263$	−14.7587	−0.910060	−0.455030	−	0.890476i	$-0.650371\pi$
$263$	−14.7587	−0.910060	−0.455030	+	0.890476i	$0.650371\pi$
$264$	0	0
$265$	5.72598	0.351744
$266$	0	0
$267$	−22.2943	−1.36439
$268$	0	0
$269$	2.59624	0.158295	0.0791476	−	0.996863i	$-0.474780\pi$
$269$	2.59624	0.158295	0.0791476	+	0.996863i	$0.474780\pi$
$270$	0	0
$271$	−14.5342	−0.882888	−0.441444	−	0.897289i	$-0.645534\pi$
$271$	−14.5342	−0.882888	−0.441444	+	0.897289i	$0.645534\pi$
$272$	0	0
$273$	−9.71367	−0.587898
$274$	0	0
$275$	−8.57412	−0.517039
$276$	0	0
$277$	28.2684	1.69848	0.849241	−	0.528006i	$-0.177060\pi$
$277$	28.2684	1.69848	0.849241	+	0.528006i	$0.177060\pi$
$278$	0	0
$279$	2.47627	0.148250
$280$	0	0
$281$	−10.9980	−0.656083	−0.328042	−	0.944663i	$-0.606389\pi$
$281$	−10.9980	−0.656083	−0.328042	+	0.944663i	$0.606389\pi$
$282$	0	0
$283$	−29.9745	−1.78180	−0.890898	−	0.454203i	$-0.849924\pi$
$283$	−29.9745	−1.78180	−0.890898	+	0.454203i	$0.849924\pi$
$284$	0	0
$285$	3.08163	0.182540
$286$	0	0
$287$	7.46457	0.440619
$288$	0	0
$289$	−9.13342	−0.537260
$290$	0	0
$291$	−34.0221	−1.99441
$292$	0	0
$293$	15.8721	0.927258	0.463629	−	0.886029i	$-0.346547\pi$
$293$	15.8721	0.927258	0.463629	+	0.886029i	$0.346547\pi$
$294$	0	0
$295$	−5.75073	−0.334821
$296$	0	0
$297$	−7.36191	−0.427181
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2.97688	−0.171585
$302$	0	0
$303$	12.9818	0.745787
$304$	0	0
$305$	3.91140	0.223966
$306$	0	0
$307$	5.79066	0.330490	0.165245	−	0.986253i	$-0.447158\pi$
$307$	5.79066	0.330490	0.165245	+	0.986253i	$0.447158\pi$
$308$	0	0
$309$	25.3570	1.44251
$310$	0	0
$311$	20.1064	1.14013	0.570065	−	0.821600i	$-0.306918\pi$
$311$	20.1064	1.14013	0.570065	+	0.821600i	$0.306918\pi$
$312$	0	0
$313$	34.0183	1.92283	0.961415	−	0.275101i	$-0.0887113\pi$
$313$	34.0183	1.92283	0.961415	+	0.275101i	$0.0887113\pi$
$314$	0	0
$315$	−1.58682	−0.0894074
$316$	0	0
$317$	−16.2032	−0.910061	−0.455031	−	0.890476i	$-0.650372\pi$
$317$	−16.2032	−0.910061	−0.455031	+	0.890476i	$0.650372\pi$
$318$	0	0
$319$	−16.2695	−0.910915
$320$	0	0
$321$	−14.7723	−0.824508
$322$	0	0
$323$	6.26222	0.348439
$324$	0	0
$325$	10.0770	0.558970
$326$	0	0
$327$	−17.3387	−0.958831
$328$	0	0
$329$	24.8985	1.37270
$330$	0	0
$331$	−20.7611	−1.14113	−0.570567	−	0.821251i	$-0.693277\pi$
$331$	−20.7611	−1.14113	−0.570567	+	0.821251i	$0.693277\pi$
$332$	0	0
$333$	−5.95026	−0.326072
$334$	0	0
$335$	10.5750	0.577776
$336$	0	0
$337$	−6.76630	−0.368584	−0.184292	−	0.982872i	$-0.558999\pi$
$337$	−6.76630	−0.368584	−0.184292	+	0.982872i	$0.558999\pi$
$338$	0	0
$339$	23.9639	1.30154
$340$	0	0
$341$	4.37225	0.236771
$342$	0	0
$343$	−20.1622	−1.08865
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.35663	0.126511	0.0632553	−	0.997997i	$-0.479852\pi$
$347$	2.35663	0.126511	0.0632553	+	0.997997i	$0.479852\pi$
$348$	0	0
$349$	−0.910787	−0.0487533	−0.0243766	−	0.999703i	$-0.507760\pi$
$349$	−0.910787	−0.0487533	−0.0243766	+	0.999703i	$0.507760\pi$
$350$	0	0
$351$	8.65229	0.461825
$352$	0	0
$353$	−21.8083	−1.16074	−0.580370	−	0.814353i	$-0.697092\pi$
$353$	−21.8083	−1.16074	−0.580370	+	0.814353i	$0.697092\pi$
$354$	0	0
$355$	−0.852077	−0.0452235
$356$	0	0
$357$	−12.2531	−0.648504
$358$	0	0
$359$	22.0442	1.16345	0.581725	−	0.813386i	$-0.302378\pi$
$359$	22.0442	1.16345	0.581725	+	0.813386i	$0.302378\pi$
$360$	0	0
$361$	−14.0149	−0.737628
$362$	0	0
$363$	−14.9737	−0.785918
$364$	0	0
$365$	−9.17164	−0.480065
$366$	0	0
$367$	9.06460	0.473168	0.236584	−	0.971611i	$-0.423972\pi$
$367$	9.06460	0.473168	0.236584	+	0.971611i	$0.423972\pi$
$368$	0	0
$369$	3.69409	0.192307
$370$	0	0
$371$	−18.1242	−0.940961
$372$	0	0
$373$	29.6202	1.53367	0.766837	−	0.641842i	$-0.221829\pi$
$373$	29.6202	1.53367	0.766837	+	0.641842i	$0.221829\pi$
$374$	0	0
$375$	13.1563	0.679390
$376$	0	0
$377$	19.1211	0.984789
$378$	0	0
$379$	−24.7843	−1.27308	−0.636541	−	0.771243i	$-0.719636\pi$
$379$	−24.7843	−1.27308	−0.636541	+	0.771243i	$0.719636\pi$
$380$	0	0
$381$	−16.0628	−0.822921
$382$	0	0
$383$	−9.26313	−0.473324	−0.236662	−	0.971592i	$-0.576053\pi$
$383$	−9.26313	−0.473324	−0.236662	+	0.971592i	$0.576053\pi$
$384$	0	0
$385$	−2.80180	−0.142793
$386$	0	0
$387$	−1.47321	−0.0748874
$388$	0	0
$389$	−12.2531	−0.621259	−0.310629	−	0.950531i	$-0.600540\pi$
$389$	−12.2531	−0.621259	−0.310629	+	0.950531i	$0.600540\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.56930	0.230491
$394$	0	0
$395$	5.56924	0.280219
$396$	0	0
$397$	−5.37752	−0.269890	−0.134945	−	0.990853i	$-0.543086\pi$
$397$	−5.37752	−0.269890	−0.134945	+	0.990853i	$0.543086\pi$
$398$	0	0
$399$	−9.75415	−0.488318
$400$	0	0
$401$	−37.9387	−1.89457	−0.947284	−	0.320396i	$-0.896184\pi$
$401$	−37.9387	−1.89457	−0.947284	+	0.320396i	$0.896184\pi$
$402$	0	0
$403$	−5.13861	−0.255973
$404$	0	0
$405$	7.56963	0.376138
$406$	0	0
$407$	−10.5062	−0.520771
$408$	0	0
$409$	−11.0040	−0.544111	−0.272056	−	0.962282i	$-0.587703\pi$
$409$	−11.0040	−0.544111	−0.272056	+	0.962282i	$0.587703\pi$
$410$	0	0
$411$	−9.59013	−0.473046
$412$	0	0
$413$	18.2025	0.895688
$414$	0	0
$415$	6.72246	0.329993
$416$	0	0
$417$	17.1019	0.837483
$418$	0	0
$419$	10.4458	0.510313	0.255156	−	0.966900i	$-0.417873\pi$
$419$	10.4458	0.510313	0.255156	+	0.966900i	$0.417873\pi$
$420$	0	0
$421$	−9.49732	−0.462871	−0.231436	−	0.972850i	$-0.574342\pi$
$421$	−9.49732	−0.462871	−0.231436	+	0.972850i	$0.574342\pi$
$422$	0	0
$423$	12.3218	0.599108
$424$	0	0
$425$	12.7114	0.616594
$426$	0	0
$427$	−12.3806	−0.599139
$428$	0	0
$429$	−8.48778	−0.409794
$430$	0	0
$431$	−9.98527	−0.480974	−0.240487	−	0.970652i	$-0.577307\pi$
$431$	−9.98527	−0.480974	−0.240487	+	0.970652i	$0.577307\pi$
$432$	0	0
$433$	−21.7332	−1.04443	−0.522215	−	0.852814i	$-0.674894\pi$
$433$	−21.7332	−1.04443	−0.522215	+	0.852814i	$0.674894\pi$
$434$	0	0
$435$	11.8694	0.569096
$436$	0	0
$437$	0	0
$438$	0	0
$439$	17.5206	0.836215	0.418107	−	0.908398i	$-0.362694\pi$
$439$	17.5206	0.836215	0.418107	+	0.908398i	$0.362694\pi$
$440$	0	0
$441$	−2.47760	−0.117981
$442$	0	0
$443$	−19.8304	−0.942169	−0.471085	−	0.882088i	$-0.656137\pi$
$443$	−19.8304	−0.942169	−0.471085	+	0.882088i	$0.656137\pi$
$444$	0	0
$445$	7.55768	0.358268
$446$	0	0
$447$	27.4770	1.29962
$448$	0	0
$449$	−11.6734	−0.550904	−0.275452	−	0.961315i	$-0.588827\pi$
$449$	−11.6734	−0.550904	−0.275452	+	0.961315i	$0.588827\pi$
$450$	0	0
$451$	6.52251	0.307133
$452$	0	0
$453$	3.05363	0.143472
$454$	0	0
$455$	3.29289	0.154373
$456$	0	0
$457$	−9.51241	−0.444972	−0.222486	−	0.974936i	$-0.571417\pi$
$457$	−9.51241	−0.444972	−0.222486	+	0.974936i	$0.571417\pi$
$458$	0	0
$459$	10.9143	0.509434
$460$	0	0
$461$	−3.83581	−0.178652	−0.0893258	−	0.996002i	$-0.528471\pi$
$461$	−3.83581	−0.178652	−0.0893258	+	0.996002i	$0.528471\pi$
$462$	0	0
$463$	38.6758	1.79741	0.898707	−	0.438549i	$-0.144507\pi$
$463$	38.6758	1.79741	0.898707	+	0.438549i	$0.144507\pi$
$464$	0	0
$465$	−3.18979	−0.147923
$466$	0	0
$467$	−4.49328	−0.207924	−0.103962	−	0.994581i	$-0.533152\pi$
$467$	−4.49328	−0.207924	−0.103962	+	0.994581i	$0.533152\pi$
$468$	0	0
$469$	−33.4727	−1.54562
$470$	0	0
$471$	40.9684	1.88773
$472$	0	0
$473$	−2.60119	−0.119603
$474$	0	0
$475$	10.1190	0.464290
$476$	0	0
$477$	−8.96936	−0.410679
$478$	0	0
$479$	−17.7988	−0.813247	−0.406623	−	0.913596i	$-0.633294\pi$
$479$	−17.7988	−0.813247	−0.406623	+	0.913596i	$0.633294\pi$
$480$	0	0
$481$	12.3477	0.563005
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.5333	0.523702
$486$	0	0
$487$	−25.4702	−1.15416	−0.577082	−	0.816686i	$-0.695809\pi$
$487$	−25.4702	−1.15416	−0.577082	+	0.816686i	$0.695809\pi$
$488$	0	0
$489$	7.84388	0.354712
$490$	0	0
$491$	−40.9002	−1.84580	−0.922899	−	0.385042i	$-0.874187\pi$
$491$	−40.9002	−1.84580	−0.922899	+	0.385042i	$0.874187\pi$
$492$	0	0
$493$	24.1200	1.08631
$494$	0	0
$495$	−1.38656	−0.0623213
$496$	0	0
$497$	2.69704	0.120979
$498$	0	0
$499$	17.3557	0.776949	0.388474	−	0.921460i	$-0.373002\pi$
$499$	17.3557	0.776949	0.388474	+	0.921460i	$0.373002\pi$
$500$	0	0
$501$	−11.1575	−0.498480
$502$	0	0
$503$	−7.32866	−0.326769	−0.163385	−	0.986562i	$-0.552241\pi$
$503$	−7.32866	−0.326769	−0.163385	+	0.986562i	$0.552241\pi$
$504$	0	0
$505$	−4.40078	−0.195832
$506$	0	0
$507$	−16.2558	−0.721943
$508$	0	0
$509$	15.1177	0.670082	0.335041	−	0.942204i	$-0.391250\pi$
$509$	15.1177	0.670082	0.335041	+	0.942204i	$0.391250\pi$
$510$	0	0
$511$	29.0306	1.28424
$512$	0	0
$513$	8.68834	0.383600
$514$	0	0
$515$	−8.59590	−0.378781
$516$	0	0
$517$	21.7562	0.956836
$518$	0	0
$519$	−32.6091	−1.43138
$520$	0	0
$521$	−21.7848	−0.954411	−0.477206	−	0.878792i	$-0.658350\pi$
$521$	−21.7848	−0.954411	−0.477206	+	0.878792i	$0.658350\pi$
$522$	0	0
$523$	−12.8603	−0.562343	−0.281172	−	0.959657i	$-0.590723\pi$
$523$	−12.8603	−0.562343	−0.281172	+	0.959657i	$0.590723\pi$
$524$	0	0
$525$	−19.7995	−0.864122
$526$	0	0
$527$	−6.48201	−0.282361
$528$	0	0
$529$	0	0
$530$	0	0
$531$	9.00813	0.390920
$532$	0	0
$533$	−7.66577	−0.332041
$534$	0	0
$535$	5.00774	0.216503
$536$	0	0
$537$	−25.5155	−1.10107
$538$	0	0
$539$	−4.37461	−0.188428
$540$	0	0
$541$	26.3283	1.13194	0.565971	−	0.824425i	$-0.308502\pi$
$541$	26.3283	1.13194	0.565971	+	0.824425i	$0.308502\pi$
$542$	0	0
$543$	23.0406	0.988768
$544$	0	0
$545$	5.87773	0.251774
$546$	0	0
$547$	33.1023	1.41535	0.707676	−	0.706537i	$-0.249743\pi$
$547$	33.1023	1.41535	0.707676	+	0.706537i	$0.249743\pi$
$548$	0	0
$549$	−6.12694	−0.261492
$550$	0	0
$551$	19.2008	0.817983
$552$	0	0
$553$	−17.6281	−0.749621
$554$	0	0
$555$	7.66480	0.325352
$556$	0	0
$557$	−32.0125	−1.35641	−0.678207	−	0.734871i	$-0.737243\pi$
$557$	−32.0125	−1.35641	−0.678207	+	0.734871i	$0.737243\pi$
$558$	0	0
$559$	3.05712	0.129302
$560$	0	0
$561$	−10.7067	−0.452039
$562$	0	0
$563$	11.9340	0.502959	0.251480	−	0.967863i	$-0.419083\pi$
$563$	11.9340	0.502959	0.251480	+	0.967863i	$0.419083\pi$
$564$	0	0
$565$	−8.12365	−0.341765
$566$	0	0
$567$	−23.9598	−1.00622
$568$	0	0
$569$	28.2753	1.18536	0.592682	−	0.805436i	$-0.298069\pi$
$569$	28.2753	1.18536	0.592682	+	0.805436i	$0.298069\pi$
$570$	0	0
$571$	−38.2981	−1.60272	−0.801362	−	0.598179i	$-0.795891\pi$
$571$	−38.2981	−1.60272	−0.801362	+	0.598179i	$0.795891\pi$
$572$	0	0
$573$	−6.35961	−0.265677
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.835083	−0.0347650	−0.0173825	−	0.999849i	$-0.505533\pi$
$577$	−0.835083	−0.0347650	−0.0173825	+	0.999849i	$0.505533\pi$
$578$	0	0
$579$	33.3045	1.38409
$580$	0	0
$581$	−21.2783	−0.882773
$582$	0	0
$583$	−15.8369	−0.655896
$584$	0	0
$585$	1.62960	0.0673755
$586$	0	0
$587$	−18.2288	−0.752382	−0.376191	−	0.926542i	$-0.622766\pi$
$587$	−18.2288	−0.752382	−0.376191	+	0.926542i	$0.622766\pi$
$588$	0	0
$589$	−5.16003	−0.212615
$590$	0	0
$591$	−35.3128	−1.45257
$592$	0	0
$593$	35.7660	1.46873	0.734366	−	0.678754i	$-0.237480\pi$
$593$	35.7660	1.46873	0.734366	+	0.678754i	$0.237480\pi$
$594$	0	0
$595$	4.15375	0.170287
$596$	0	0
$597$	7.96293	0.325901
$598$	0	0
$599$	5.11167	0.208857	0.104429	−	0.994532i	$-0.466699\pi$
$599$	5.11167	0.208857	0.104429	+	0.994532i	$0.466699\pi$
$600$	0	0
$601$	−41.4661	−1.69144	−0.845720	−	0.533627i	$-0.820829\pi$
$601$	−41.4661	−1.69144	−0.845720	+	0.533627i	$0.820829\pi$
$602$	0	0
$603$	−16.5651	−0.674582
$604$	0	0
$605$	5.07603	0.206370
$606$	0	0
$607$	9.78706	0.397245	0.198622	−	0.980076i	$-0.436353\pi$
$607$	9.78706	0.397245	0.198622	+	0.980076i	$0.436353\pi$
$608$	0	0
$609$	−37.5698	−1.52240
$610$	0	0
$611$	−25.5696	−1.03443
$612$	0	0
$613$	−21.0624	−0.850703	−0.425351	−	0.905028i	$-0.639850\pi$
$613$	−21.0624	−0.850703	−0.425351	+	0.905028i	$0.639850\pi$
$614$	0	0
$615$	−4.75852	−0.191882
$616$	0	0
$617$	−33.8273	−1.36183	−0.680917	−	0.732361i	$-0.738419\pi$
$617$	−33.8273	−1.36183	−0.680917	+	0.732361i	$0.738419\pi$
$618$	0	0
$619$	1.46841	0.0590204	0.0295102	−	0.999564i	$-0.490605\pi$
$619$	1.46841	0.0590204	0.0295102	+	0.999564i	$0.490605\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−23.9220	−0.958414
$624$	0	0
$625$	18.2006	0.728026
$626$	0	0
$627$	−8.52314	−0.340382
$628$	0	0
$629$	15.5757	0.621045
$630$	0	0
$631$	1.13363	0.0451290	0.0225645	−	0.999745i	$-0.492817\pi$
$631$	1.13363	0.0451290	0.0225645	+	0.999745i	$0.492817\pi$
$632$	0	0
$633$	28.2339	1.12220
$634$	0	0
$635$	5.44521	0.216086
$636$	0	0
$637$	5.14139	0.203709
$638$	0	0
$639$	1.33472	0.0528007
$640$	0	0
$641$	−33.9836	−1.34227	−0.671136	−	0.741335i	$-0.734193\pi$
$641$	−33.9836	−1.34227	−0.671136	+	0.741335i	$0.734193\pi$
$642$	0	0
$643$	1.17625	0.0463866	0.0231933	−	0.999731i	$-0.492617\pi$
$643$	1.17625	0.0463866	0.0231933	+	0.999731i	$0.492617\pi$
$644$	0	0
$645$	1.89770	0.0747220
$646$	0	0
$647$	10.2359	0.402416	0.201208	−	0.979549i	$-0.435513\pi$
$647$	10.2359	0.402416	0.201208	+	0.979549i	$0.435513\pi$
$648$	0	0
$649$	15.9053	0.624338
$650$	0	0
$651$	10.0965	0.395713
$652$	0	0
$653$	−44.6392	−1.74687	−0.873433	−	0.486945i	$-0.838111\pi$
$653$	−44.6392	−1.74687	−0.873433	+	0.486945i	$0.838111\pi$
$654$	0	0
$655$	−1.54897	−0.0605234
$656$	0	0
$657$	14.3667	0.560500
$658$	0	0
$659$	15.7671	0.614199	0.307099	−	0.951677i	$-0.400642\pi$
$659$	15.7671	0.614199	0.307099	+	0.951677i	$0.400642\pi$
$660$	0	0
$661$	17.0071	0.661501	0.330751	−	0.943718i	$-0.392698\pi$
$661$	17.0071	0.661501	0.330751	+	0.943718i	$0.392698\pi$
$662$	0	0
$663$	12.5834	0.488699
$664$	0	0
$665$	3.30661	0.128225
$666$	0	0
$667$	0	0
$668$	0	0
$669$	54.0879	2.09116
$670$	0	0
$671$	−10.8181	−0.417629
$672$	0	0
$673$	4.81165	0.185476	0.0927378	−	0.995691i	$-0.470438\pi$
$673$	4.81165	0.185476	0.0927378	+	0.995691i	$0.470438\pi$
$674$	0	0
$675$	17.6361	0.678814
$676$	0	0
$677$	−17.4412	−0.670322	−0.335161	−	0.942161i	$-0.608791\pi$
$677$	−17.4412	−0.670322	−0.335161	+	0.942161i	$0.608791\pi$
$678$	0	0
$679$	−36.5060	−1.40097
$680$	0	0
$681$	−48.7199	−1.86695
$682$	0	0
$683$	31.4732	1.20429	0.602145	−	0.798387i	$-0.294313\pi$
$683$	31.4732	1.20429	0.602145	+	0.798387i	$0.294313\pi$
$684$	0	0
$685$	3.25101	0.124215
$686$	0	0
$687$	25.8027	0.984435
$688$	0	0
$689$	18.6127	0.709088
$690$	0	0
$691$	−24.1989	−0.920570	−0.460285	−	0.887771i	$-0.652253\pi$
$691$	−24.1989	−0.920570	−0.460285	+	0.887771i	$0.652253\pi$
$692$	0	0
$693$	4.38882	0.166718
$694$	0	0
$695$	−5.79746	−0.219910
$696$	0	0
$697$	−9.66984	−0.366271
$698$	0	0
$699$	15.9691	0.604005
$700$	0	0
$701$	−15.1778	−0.573257	−0.286629	−	0.958042i	$-0.592535\pi$
$701$	−15.1778	−0.573257	−0.286629	+	0.958042i	$0.592535\pi$
$702$	0	0
$703$	12.3991	0.467641
$704$	0	0
$705$	−15.8723	−0.597785
$706$	0	0
$707$	13.9296	0.523876
$708$	0	0
$709$	−31.2973	−1.17540	−0.587698	−	0.809080i	$-0.699966\pi$
$709$	−31.2973	−1.17540	−0.587698	+	0.809080i	$0.699966\pi$
$710$	0	0
$711$	−8.72383	−0.327169
$712$	0	0
$713$	0	0
$714$	0	0
$715$	2.87732	0.107606
$716$	0	0
$717$	−22.3521	−0.834755
$718$	0	0
$719$	46.8276	1.74638	0.873188	−	0.487383i	$-0.162049\pi$
$719$	46.8276	1.74638	0.873188	+	0.487383i	$0.162049\pi$
$720$	0	0
$721$	27.2082	1.01329
$722$	0	0
$723$	52.8620	1.96596
$724$	0	0
$725$	38.9749	1.44749
$726$	0	0
$727$	40.4833	1.50144	0.750721	−	0.660619i	$-0.229706\pi$
$727$	40.4833	1.50144	0.750721	+	0.660619i	$0.229706\pi$
$728$	0	0
$729$	11.6985	0.433279
$730$	0	0
$731$	3.85635	0.142632
$732$	0	0
$733$	25.0451	0.925061	0.462530	−	0.886603i	$-0.346942\pi$
$733$	25.0451	0.925061	0.462530	+	0.886603i	$0.346942\pi$
$734$	0	0
$735$	3.19151	0.117721
$736$	0	0
$737$	−29.2483	−1.07738
$738$	0	0
$739$	−1.01275	−0.0372546	−0.0186273	−	0.999826i	$-0.505930\pi$
$739$	−1.01275	−0.0372546	−0.0186273	+	0.999826i	$0.505930\pi$
$740$	0	0
$741$	10.0171	0.367986
$742$	0	0
$743$	49.1693	1.80385	0.901923	−	0.431896i	$-0.142155\pi$
$743$	49.1693	1.80385	0.901923	+	0.431896i	$0.142155\pi$
$744$	0	0
$745$	−9.31458	−0.341260
$746$	0	0
$747$	−10.5303	−0.385283
$748$	0	0
$749$	−15.8508	−0.579174
$750$	0	0
$751$	14.2429	0.519731	0.259866	−	0.965645i	$-0.416322\pi$
$751$	14.2429	0.519731	0.259866	+	0.965645i	$0.416322\pi$
$752$	0	0
$753$	35.1728	1.28177
$754$	0	0
$755$	−1.03517	−0.0376735
$756$	0	0
$757$	−3.22786	−0.117318	−0.0586592	−	0.998278i	$-0.518683\pi$
$757$	−3.22786	−0.117318	−0.0586592	+	0.998278i	$0.518683\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17.9756	−0.651615	−0.325808	−	0.945436i	$-0.605636\pi$
$761$	−17.9756	−0.651615	−0.325808	+	0.945436i	$0.605636\pi$
$762$	0	0
$763$	−18.6045	−0.673529
$764$	0	0
$765$	2.05562	0.0743212
$766$	0	0
$767$	−18.6932	−0.674972
$768$	0	0
$769$	−1.73961	−0.0627320	−0.0313660	−	0.999508i	$-0.509986\pi$
$769$	−1.73961	−0.0627320	−0.0313660	+	0.999508i	$0.509986\pi$
$770$	0	0
$771$	24.9140	0.897255
$772$	0	0
$773$	9.05709	0.325761	0.162880	−	0.986646i	$-0.447922\pi$
$773$	9.05709	0.325761	0.162880	+	0.986646i	$0.447922\pi$
$774$	0	0
$775$	−10.4741	−0.376241
$776$	0	0
$777$	−24.2610	−0.870360
$778$	0	0
$779$	−7.69771	−0.275799
$780$	0	0
$781$	2.35667	0.0843281
$782$	0	0
$783$	33.4646	1.19593
$784$	0	0
$785$	−13.8881	−0.495688
$786$	0	0
$787$	−19.9164	−0.709943	−0.354972	−	0.934877i	$-0.615509\pi$
$787$	−19.9164	−0.709943	−0.354972	+	0.934877i	$0.615509\pi$
$788$	0	0
$789$	−29.7799	−1.06019
$790$	0	0
$791$	25.7134	0.914264
$792$	0	0
$793$	12.7143	0.451498
$794$	0	0
$795$	11.5538	0.409772
$796$	0	0
$797$	38.5349	1.36498	0.682488	−	0.730897i	$-0.260898\pi$
$797$	38.5349	1.36498	0.682488	+	0.730897i	$0.260898\pi$
$798$	0	0
$799$	−32.2543	−1.14107
$800$	0	0
$801$	−11.8386	−0.418296
$802$	0	0
$803$	25.3668	0.895176
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.23865	0.184409
$808$	0	0
$809$	32.6633	1.14838	0.574189	−	0.818722i	$-0.305317\pi$
$809$	32.6633	1.14838	0.574189	+	0.818722i	$0.305317\pi$
$810$	0	0
$811$	19.0132	0.667645	0.333822	−	0.942636i	$-0.391661\pi$
$811$	19.0132	0.667645	0.333822	+	0.942636i	$0.391661\pi$
$812$	0	0
$813$	−29.3269	−1.02854
$814$	0	0
$815$	−2.65904	−0.0931420
$816$	0	0
$817$	3.06986	0.107401
$818$	0	0
$819$	−5.15809	−0.180238
$820$	0	0
$821$	−12.5871	−0.439291	−0.219646	−	0.975580i	$-0.570490\pi$
$821$	−12.5871	−0.439291	−0.219646	+	0.975580i	$0.570490\pi$
$822$	0	0
$823$	−22.7562	−0.793231	−0.396616	−	0.917985i	$-0.629815\pi$
$823$	−22.7562	−0.793231	−0.396616	+	0.917985i	$0.629815\pi$
$824$	0	0
$825$	−17.3008	−0.602335
$826$	0	0
$827$	26.5016	0.921550	0.460775	−	0.887517i	$-0.347572\pi$
$827$	26.5016	0.921550	0.460775	+	0.887517i	$0.347572\pi$
$828$	0	0
$829$	36.3904	1.26389	0.631946	−	0.775012i	$-0.282256\pi$
$829$	36.3904	1.26389	0.631946	+	0.775012i	$0.282256\pi$
$830$	0	0
$831$	57.0396	1.97868
$832$	0	0
$833$	6.48551	0.224709
$834$	0	0
$835$	3.78234	0.130893
$836$	0	0
$837$	−8.99328	−0.310853
$838$	0	0
$839$	1.17356	0.0405158	0.0202579	−	0.999795i	$-0.493551\pi$
$839$	1.17356	0.0405158	0.0202579	+	0.999795i	$0.493551\pi$
$840$	0	0
$841$	44.9552	1.55018
$842$	0	0
$843$	−22.1916	−0.764318
$844$	0	0
$845$	5.51063	0.189571
$846$	0	0
$847$	−16.0669	−0.552066
$848$	0	0
$849$	−60.4821	−2.07574
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0.163542	0.00559959	0.00279979	−	0.999996i	$-0.499109\pi$
$853$	0.163542	0.00559959	0.00279979	+	0.999996i	$0.499109\pi$
$854$	0	0
$855$	1.63639	0.0559632
$856$	0	0
$857$	10.5196	0.359342	0.179671	−	0.983727i	$-0.442497\pi$
$857$	10.5196	0.359342	0.179671	+	0.983727i	$0.442497\pi$
$858$	0	0
$859$	−0.716641	−0.0244515	−0.0122257	−	0.999925i	$-0.503892\pi$
$859$	−0.716641	−0.0244515	−0.0122257	+	0.999925i	$0.503892\pi$
$860$	0	0
$861$	15.0619	0.513309
$862$	0	0
$863$	−53.5759	−1.82374	−0.911872	−	0.410474i	$-0.865363\pi$
$863$	−53.5759	−1.82374	−0.911872	+	0.410474i	$0.865363\pi$
$864$	0	0
$865$	11.0543	0.375859
$866$	0	0
$867$	−18.4293	−0.625892
$868$	0	0
$869$	−15.4033	−0.522522
$870$	0	0
$871$	34.3749	1.16475
$872$	0	0
$873$	−18.0662	−0.611448
$874$	0	0
$875$	14.1168	0.477236
$876$	0	0
$877$	−29.6430	−1.00097	−0.500486	−	0.865745i	$-0.666845\pi$
$877$	−29.6430	−1.00097	−0.500486	+	0.865745i	$0.666845\pi$
$878$	0	0
$879$	32.0266	1.08023
$880$	0	0
$881$	−4.91347	−0.165539	−0.0827696	−	0.996569i	$-0.526377\pi$
$881$	−4.91347	−0.165539	−0.0827696	+	0.996569i	$0.526377\pi$
$882$	0	0
$883$	11.4104	0.383989	0.191995	−	0.981396i	$-0.438504\pi$
$883$	11.4104	0.383989	0.191995	+	0.981396i	$0.438504\pi$
$884$	0	0
$885$	−11.6038	−0.390056
$886$	0	0
$887$	−27.2006	−0.913305	−0.456653	−	0.889645i	$-0.650952\pi$
$887$	−27.2006	−0.913305	−0.456653	+	0.889645i	$0.650952\pi$
$888$	0	0
$889$	−17.2355	−0.578059
$890$	0	0
$891$	−20.9360	−0.701382
$892$	0	0
$893$	−25.6761	−0.859219
$894$	0	0
$895$	8.64963	0.289125
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.8747	−0.662859
$900$	0	0
$901$	23.4787	0.782188
$902$	0	0
$903$	−6.00672	−0.199891
$904$	0	0
$905$	−7.81067	−0.259636
$906$	0	0
$907$	−18.4093	−0.611272	−0.305636	−	0.952148i	$-0.598869\pi$
$907$	−18.4093	−0.611272	−0.305636	+	0.952148i	$0.598869\pi$
$908$	0	0
$909$	6.89352	0.228644
$910$	0	0
$911$	49.7510	1.64833	0.824163	−	0.566353i	$-0.191646\pi$
$911$	49.7510	1.64833	0.824163	+	0.566353i	$0.191646\pi$
$912$	0	0
$913$	−18.5929	−0.615336
$914$	0	0
$915$	7.89239	0.260914
$916$	0	0
$917$	4.90289	0.161908
$918$	0	0
$919$	36.6449	1.20880	0.604402	−	0.796680i	$-0.293412\pi$
$919$	36.6449	1.20880	0.604402	+	0.796680i	$0.293412\pi$
$920$	0	0
$921$	11.6843	0.385012
$922$	0	0
$923$	−2.76974	−0.0911670
$924$	0	0
$925$	25.1684	0.827533
$926$	0	0
$927$	13.4649	0.442245
$928$	0	0
$929$	30.1338	0.988658	0.494329	−	0.869275i	$-0.335414\pi$
$929$	30.1338	0.988658	0.494329	+	0.869275i	$0.335414\pi$
$930$	0	0
$931$	5.16281	0.169204
$932$	0	0
$933$	40.5705	1.32822
$934$	0	0
$935$	3.62954	0.118699
$936$	0	0
$937$	24.7552	0.808719	0.404359	−	0.914600i	$-0.367495\pi$
$937$	24.7552	0.808719	0.404359	+	0.914600i	$0.367495\pi$
$938$	0	0
$939$	68.6418	2.24004
$940$	0	0
$941$	−6.15351	−0.200599	−0.100299	−	0.994957i	$-0.531980\pi$
$941$	−6.15351	−0.200599	−0.100299	+	0.994957i	$0.531980\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.76301	0.187471
$946$	0	0
$947$	59.4254	1.93107	0.965533	−	0.260279i	$-0.0838145\pi$
$947$	59.4254	1.93107	0.965533	+	0.260279i	$0.0838145\pi$
$948$	0	0
$949$	−29.8131	−0.967773
$950$	0	0
$951$	−32.6946	−1.06019
$952$	0	0
$953$	−42.8168	−1.38697	−0.693487	−	0.720469i	$-0.743926\pi$
$953$	−42.8168	−1.38697	−0.693487	+	0.720469i	$0.743926\pi$
$954$	0	0
$955$	2.15588	0.0697626
$956$	0	0
$957$	−32.8283	−1.06119
$958$	0	0
$959$	−10.2903	−0.332290
$960$	0	0
$961$	−25.6589	−0.827706
$962$	0	0
$963$	−7.84427	−0.252778
$964$	0	0
$965$	−11.2901	−0.363441
$966$	0	0
$967$	−12.8748	−0.414026	−0.207013	−	0.978338i	$-0.566374\pi$
$967$	−12.8748	−0.414026	−0.207013	+	0.978338i	$0.566374\pi$
$968$	0	0
$969$	12.6358	0.405922
$970$	0	0
$971$	40.6407	1.30422	0.652111	−	0.758123i	$-0.273884\pi$
$971$	40.6407	1.30422	0.652111	+	0.758123i	$0.273884\pi$
$972$	0	0
$973$	18.3504	0.588288
$974$	0	0
$975$	20.3332	0.651184
$976$	0	0
$977$	19.9540	0.638384	0.319192	−	0.947690i	$-0.396589\pi$
$977$	19.9540	0.638384	0.319192	+	0.947690i	$0.396589\pi$
$978$	0	0
$979$	−20.9030	−0.668061
$980$	0	0
$981$	−9.20706	−0.293959
$982$	0	0
$983$	5.51066	0.175763	0.0878813	−	0.996131i	$-0.471990\pi$
$983$	5.51066	0.175763	0.0878813	+	0.996131i	$0.471990\pi$
$984$	0	0
$985$	11.9709	0.381424
$986$	0	0
$987$	50.2398	1.59915
$988$	0	0
$989$	0	0
$990$	0	0
$991$	26.4831	0.841263	0.420632	−	0.907232i	$-0.361809\pi$
$991$	26.4831	0.841263	0.420632	+	0.907232i	$0.361809\pi$
$992$	0	0
$993$	−41.8916	−1.32939
$994$	0	0
$995$	−2.69940	−0.0855767
$996$	0	0
$997$	42.8330	1.35654	0.678268	−	0.734815i	$-0.262731\pi$
$997$	42.8330	1.35654	0.678268	+	0.734815i	$0.262731\pi$
$998$	0	0
$999$	21.6101	0.683713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.cg.1.12		15
4.3	odd	2		4232.2.a.bb.1.4		15
23.3	even	11		368.2.m.e.193.1		30
23.8	even	11		368.2.m.e.225.1		30
23.22	odd	2		8464.2.a.ch.1.12		15
92.3	odd	22		184.2.i.b.9.3	✓	30
92.31	odd	22		184.2.i.b.41.3	yes	30
92.91	even	2		4232.2.a.ba.1.4		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.9.3	✓	30	92.3	odd	22
184.2.i.b.41.3	yes	30	92.31	odd	22
368.2.m.e.193.1		30	23.3	even	11
368.2.m.e.225.1		30	23.8	even	11
4232.2.a.ba.1.4		15	92.91	even	2
4232.2.a.bb.1.4		15	4.3	odd	2
8464.2.a.cg.1.12		15	1.1	even	1	trivial
8464.2.a.ch.1.12		15	23.22	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.01779 q^{3} -0.684021 q^{5} +2.16510 q^{7} +1.07147 q^{9} +1.89186 q^{11} -2.22346 q^{13} -1.38021 q^{15} -2.80474 q^{17} -2.23273 q^{19} +4.36872 q^{21} -4.53211 q^{25} -3.89136 q^{27} -8.59972 q^{29} +2.31109 q^{31} +3.81737 q^{33} -1.48098 q^{35} -5.55335 q^{37} -4.48647 q^{39} +3.44767 q^{41} -1.37494 q^{43} -0.732910 q^{45} +11.4999 q^{47} -2.31234 q^{49} -5.65938 q^{51} -8.37106 q^{53} -1.29407 q^{55} -4.50517 q^{57} +8.40725 q^{59} -5.71825 q^{61} +2.31985 q^{63} +1.52089 q^{65} -15.4601 q^{67} +1.24569 q^{71} +13.4084 q^{73} -9.14485 q^{75} +4.09607 q^{77} -8.14191 q^{79} -11.0664 q^{81} -9.82785 q^{83} +1.91850 q^{85} -17.3524 q^{87} -11.0489 q^{89} -4.81402 q^{91} +4.66329 q^{93} +1.52723 q^{95} -16.8611 q^{97} +2.02707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} + q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} + 18 q^{33} + 18 q^{35} + 24 q^{37} + 19 q^{39} + 9 q^{41} - 48 q^{43} + 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) \) 15 * q - q^3 - 10 * q^7 + 16 * q^9 - 23 * q^11 - 10 * q^15 - 29 * q^19 + q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 + 18 * q^33 + 18 * q^35 + 24 * q^37 + 19 * q^39 + 9 * q^41 - 48 * q^43 + 4 * q^45 + 36 * q^47 + 25 * q^49 - 35 * q^51 - 5 * q^53 + 10 * q^55 + 23 * q^57 + 22 * q^59 + 12 * q^61 - 35 * q^63 - 26 * q^65 - 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 - 26 * q^79 - 21 * q^81 - 68 * q^83 - 72 * q^85 - 19 * q^87 - 6 * q^89 - 71 * q^91 - 55 * q^93 + 12 * q^95 + 40 * q^97 - 63 * q^99

Introduction

L-functions

Modular forms

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