LMFDB - Embedded newform 3584.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18 $ x^10 - 2x^9 - 19x^8 + 44x^7 + 86x^6 - 236x^5 - 58x^4 + 368x^3 - 194x^2 - 12*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.791624$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.47533 q^{3} +2.59708 q^{5} -1.00000 q^{7} +3.12724 q^{9} +O(q^{10})$	$q+2.47533 q^{3} +2.59708 q^{5} -1.00000 q^{7} +3.12724 q^{9} +4.33400 q^{11} -4.86704 q^{13} +6.42863 q^{15} -5.17347 q^{17} +7.11102 q^{19} -2.47533 q^{21} -2.29943 q^{23} +1.74484 q^{25} +0.314955 q^{27} +8.20254 q^{29} +1.04427 q^{31} +10.7281 q^{33} -2.59708 q^{35} +4.77071 q^{37} -12.0475 q^{39} +7.95701 q^{41} +8.63925 q^{43} +8.12170 q^{45} -6.55587 q^{47} +1.00000 q^{49} -12.8060 q^{51} +10.3248 q^{53} +11.2558 q^{55} +17.6021 q^{57} +12.7679 q^{59} +3.05977 q^{61} -3.12724 q^{63} -12.6401 q^{65} -2.21177 q^{67} -5.69184 q^{69} -8.38367 q^{71} +2.61760 q^{73} +4.31905 q^{75} -4.33400 q^{77} -12.3837 q^{79} -8.60210 q^{81} -3.18153 q^{83} -13.4359 q^{85} +20.3039 q^{87} +8.12920 q^{89} +4.86704 q^{91} +2.58491 q^{93} +18.4679 q^{95} +1.91772 q^{97} +13.5534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{7} + 22 q^{9}+O(q^{10}) $ 10 * q - 10 * q^7 + 22 * q^9	$ 10 q - 10 q^{7} + 22 q^{9} - 16 q^{15} + 16 q^{17} - 8 q^{23} + 30 q^{25} + 8 q^{31} + 12 q^{33} + 20 q^{41} + 24 q^{47} + 10 q^{49} + 32 q^{55} + 28 q^{57} - 22 q^{63} + 32 q^{65} + 48 q^{73} - 40 q^{79} + 62 q^{81} + 8 q^{87} + 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) $ 10 * q - 10 * q^7 + 22 * q^9 - 16 * q^15 + 16 * q^17 - 8 * q^23 + 30 * q^25 + 8 * q^31 + 12 * q^33 + 20 * q^41 + 24 * q^47 + 10 * q^49 + 32 * q^55 + 28 * q^57 - 22 * q^63 + 32 * q^65 + 48 * q^73 - 40 * q^79 + 62 * q^81 + 8 * q^87 + 16 * q^89 + 32 * q^95 + 32 * 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
$3$	2.47533	1.42913	0.714565	−	0.699569i	$-0.246625\pi$
$3$	2.47533	1.42913	0.714565	+	0.699569i	$0.246625\pi$
$4$	0	0
$5$	2.59708	1.16145	0.580726	−	0.814099i	$-0.302769\pi$
$5$	2.59708	1.16145	0.580726	+	0.814099i	$0.302769\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.12724	1.04241
$10$	0	0
$11$	4.33400	1.30675	0.653375	−	0.757035i	$-0.273353\pi$
$11$	4.33400	1.30675	0.653375	+	0.757035i	$0.273353\pi$
$12$	0	0
$13$	−4.86704	−1.34987	−0.674937	−	0.737875i	$-0.735829\pi$
$13$	−4.86704	−1.34987	−0.674937	+	0.737875i	$0.735829\pi$
$14$	0	0
$15$	6.42863	1.65986
$16$	0	0
$17$	−5.17347	−1.25475	−0.627375	−	0.778717i	$-0.715871\pi$
$17$	−5.17347	−1.25475	−0.627375	+	0.778717i	$0.715871\pi$
$18$	0	0
$19$	7.11102	1.63138	0.815690	−	0.578489i	$-0.196357\pi$
$19$	7.11102	1.63138	0.815690	+	0.578489i	$0.196357\pi$
$20$	0	0
$21$	−2.47533	−0.540160
$22$	0	0
$23$	−2.29943	−0.479464	−0.239732	−	0.970839i	$-0.577060\pi$
$23$	−2.29943	−0.479464	−0.239732	+	0.970839i	$0.577060\pi$
$24$	0	0
$25$	1.74484	0.348968
$26$	0	0
$27$	0.314955	0.0606131
$28$	0	0
$29$	8.20254	1.52317	0.761586	−	0.648064i	$-0.224421\pi$
$29$	8.20254	1.52317	0.761586	+	0.648064i	$0.224421\pi$
$30$	0	0
$31$	1.04427	0.187557	0.0937783	−	0.995593i	$-0.470106\pi$
$31$	1.04427	0.187557	0.0937783	+	0.995593i	$0.470106\pi$
$32$	0	0
$33$	10.7281	1.86751
$34$	0	0
$35$	−2.59708	−0.438987
$36$	0	0
$37$	4.77071	0.784300	0.392150	−	0.919901i	$-0.371731\pi$
$37$	4.77071	0.784300	0.392150	+	0.919901i	$0.371731\pi$
$38$	0	0
$39$	−12.0475	−1.92915
$40$	0	0
$41$	7.95701	1.24268	0.621338	−	0.783543i	$-0.286589\pi$
$41$	7.95701	1.24268	0.621338	+	0.783543i	$0.286589\pi$
$42$	0	0
$43$	8.63925	1.31747	0.658736	−	0.752374i	$-0.271091\pi$
$43$	8.63925	1.31747	0.658736	+	0.752374i	$0.271091\pi$
$44$	0	0
$45$	8.12170	1.21071
$46$	0	0
$47$	−6.55587	−0.956271	−0.478136	−	0.878286i	$-0.658687\pi$
$47$	−6.55587	−0.956271	−0.478136	+	0.878286i	$0.658687\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.8060	−1.79320
$52$	0	0
$53$	10.3248	1.41821	0.709107	−	0.705100i	$-0.249098\pi$
$53$	10.3248	1.41821	0.709107	+	0.705100i	$0.249098\pi$
$54$	0	0
$55$	11.2558	1.51773
$56$	0	0
$57$	17.6021	2.33145
$58$	0	0
$59$	12.7679	1.66224	0.831118	−	0.556096i	$-0.187701\pi$
$59$	12.7679	1.66224	0.831118	+	0.556096i	$0.187701\pi$
$60$	0	0
$61$	3.05977	0.391764	0.195882	−	0.980628i	$-0.437243\pi$
$61$	3.05977	0.391764	0.195882	+	0.980628i	$0.437243\pi$
$62$	0	0
$63$	−3.12724	−0.393995
$64$	0	0
$65$	−12.6401	−1.56781
$66$	0	0
$67$	−2.21177	−0.270211	−0.135106	−	0.990831i	$-0.543137\pi$
$67$	−2.21177	−0.270211	−0.135106	+	0.990831i	$0.543137\pi$
$68$	0	0
$69$	−5.69184	−0.685217
$70$	0	0
$71$	−8.38367	−0.994959	−0.497480	−	0.867476i	$-0.665741\pi$
$71$	−8.38367	−0.994959	−0.497480	+	0.867476i	$0.665741\pi$
$72$	0	0
$73$	2.61760	0.306367	0.153184	−	0.988198i	$-0.451047\pi$
$73$	2.61760	0.306367	0.153184	+	0.988198i	$0.451047\pi$
$74$	0	0
$75$	4.31905	0.498721
$76$	0	0
$77$	−4.33400	−0.493905
$78$	0	0
$79$	−12.3837	−1.39327	−0.696636	−	0.717425i	$-0.745321\pi$
$79$	−12.3837	−1.39327	−0.696636	+	0.717425i	$0.745321\pi$
$80$	0	0
$81$	−8.60210	−0.955789
$82$	0	0
$83$	−3.18153	−0.349218	−0.174609	−	0.984638i	$-0.555866\pi$
$83$	−3.18153	−0.349218	−0.174609	+	0.984638i	$0.555866\pi$
$84$	0	0
$85$	−13.4359	−1.45733
$86$	0	0
$87$	20.3039	2.17681
$88$	0	0
$89$	8.12920	0.861693	0.430847	−	0.902425i	$-0.358215\pi$
$89$	8.12920	0.861693	0.430847	+	0.902425i	$0.358215\pi$
$90$	0	0
$91$	4.86704	0.510204
$92$	0	0
$93$	2.58491	0.268043
$94$	0	0
$95$	18.4679	1.89477
$96$	0	0
$97$	1.91772	0.194715	0.0973573	−	0.995249i	$-0.468961\pi$
$97$	1.91772	0.194715	0.0973573	+	0.995249i	$0.468961\pi$
$98$	0	0
$99$	13.5534	1.36217
$100$	0	0
$101$	14.2412	1.41706	0.708528	−	0.705683i	$-0.249360\pi$
$101$	14.2412	1.41706	0.708528	+	0.705683i	$0.249360\pi$
$102$	0	0
$103$	−4.04555	−0.398620	−0.199310	−	0.979937i	$-0.563870\pi$
$103$	−4.04555	−0.398620	−0.199310	+	0.979937i	$0.563870\pi$
$104$	0	0
$105$	−6.42863	−0.627370
$106$	0	0
$107$	0.256771	0.0248230	0.0124115	−	0.999923i	$-0.496049\pi$
$107$	0.256771	0.0248230	0.0124115	+	0.999923i	$0.496049\pi$
$108$	0	0
$109$	−9.79765	−0.938445	−0.469223	−	0.883080i	$-0.655466\pi$
$109$	−9.79765	−0.938445	−0.469223	+	0.883080i	$0.655466\pi$
$110$	0	0
$111$	11.8091	1.12087
$112$	0	0
$113$	−11.0918	−1.04343	−0.521714	−	0.853121i	$-0.674707\pi$
$113$	−11.0918	−1.04343	−0.521714	+	0.853121i	$0.674707\pi$
$114$	0	0
$115$	−5.97181	−0.556874
$116$	0	0
$117$	−15.2204	−1.40713
$118$	0	0
$119$	5.17347	0.474251
$120$	0	0
$121$	7.78354	0.707594
$122$	0	0
$123$	19.6962	1.77595
$124$	0	0
$125$	−8.45392	−0.756141
$126$	0	0
$127$	−6.55390	−0.581565	−0.290782	−	0.956789i	$-0.593916\pi$
$127$	−6.55390	−0.581565	−0.290782	+	0.956789i	$0.593916\pi$
$128$	0	0
$129$	21.3850	1.88284
$130$	0	0
$131$	−14.8032	−1.29336	−0.646679	−	0.762762i	$-0.723843\pi$
$131$	−14.8032	−1.29336	−0.646679	+	0.762762i	$0.723843\pi$
$132$	0	0
$133$	−7.11102	−0.616604
$134$	0	0
$135$	0.817964	0.0703991
$136$	0	0
$137$	3.12792	0.267236	0.133618	−	0.991033i	$-0.457340\pi$
$137$	3.12792	0.267236	0.133618	+	0.991033i	$0.457340\pi$
$138$	0	0
$139$	−7.52176	−0.637987	−0.318994	−	0.947757i	$-0.603345\pi$
$139$	−7.52176	−0.637987	−0.318994	+	0.947757i	$0.603345\pi$
$140$	0	0
$141$	−16.2279	−1.36664
$142$	0	0
$143$	−21.0937	−1.76395
$144$	0	0
$145$	21.3027	1.76909
$146$	0	0
$147$	2.47533	0.204161
$148$	0	0
$149$	−8.66522	−0.709883	−0.354941	−	0.934889i	$-0.615499\pi$
$149$	−8.66522	−0.709883	−0.354941	+	0.934889i	$0.615499\pi$
$150$	0	0
$151$	−14.2134	−1.15667	−0.578337	−	0.815798i	$-0.696298\pi$
$151$	−14.2134	−1.15667	−0.578337	+	0.815798i	$0.696298\pi$
$152$	0	0
$153$	−16.1787	−1.30797
$154$	0	0
$155$	2.71206	0.217838
$156$	0	0
$157$	−19.3326	−1.54291	−0.771455	−	0.636284i	$-0.780471\pi$
$157$	−19.3326	−1.54291	−0.771455	+	0.636284i	$0.780471\pi$
$158$	0	0
$159$	25.5571	2.02681
$160$	0	0
$161$	2.29943	0.181220
$162$	0	0
$163$	−14.4788	−1.13407	−0.567034	−	0.823694i	$-0.691909\pi$
$163$	−14.4788	−1.13407	−0.567034	+	0.823694i	$0.691909\pi$
$164$	0	0
$165$	27.8617	2.16903
$166$	0	0
$167$	16.6444	1.28798	0.643991	−	0.765033i	$-0.277277\pi$
$167$	16.6444	1.28798	0.643991	+	0.765033i	$0.277277\pi$
$168$	0	0
$169$	10.6881	0.822160
$170$	0	0
$171$	22.2379	1.70057
$172$	0	0
$173$	20.4252	1.55290	0.776450	−	0.630179i	$-0.217019\pi$
$173$	20.4252	1.55290	0.776450	+	0.630179i	$0.217019\pi$
$174$	0	0
$175$	−1.74484	−0.131898
$176$	0	0
$177$	31.6047	2.37555
$178$	0	0
$179$	−3.44508	−0.257497	−0.128749	−	0.991677i	$-0.541096\pi$
$179$	−3.44508	−0.257497	−0.128749	+	0.991677i	$0.541096\pi$
$180$	0	0
$181$	−17.7954	−1.32272	−0.661359	−	0.750069i	$-0.730020\pi$
$181$	−17.7954	−1.32272	−0.661359	+	0.750069i	$0.730020\pi$
$182$	0	0
$183$	7.57393	0.559881
$184$	0	0
$185$	12.3899	0.910926
$186$	0	0
$187$	−22.4218	−1.63964
$188$	0	0
$189$	−0.314955	−0.0229096
$190$	0	0
$191$	−8.34438	−0.603778	−0.301889	−	0.953343i	$-0.597617\pi$
$191$	−8.34438	−0.603778	−0.301889	+	0.953343i	$0.597617\pi$
$192$	0	0
$193$	24.6401	1.77363	0.886817	−	0.462121i	$-0.152911\pi$
$193$	24.6401	1.77363	0.886817	+	0.462121i	$0.152911\pi$
$194$	0	0
$195$	−31.2884	−2.24061
$196$	0	0
$197$	−13.0924	−0.932794	−0.466397	−	0.884576i	$-0.654448\pi$
$197$	−13.0924	−0.932794	−0.466397	+	0.884576i	$0.654448\pi$
$198$	0	0
$199$	−5.38865	−0.381992	−0.190996	−	0.981591i	$-0.561172\pi$
$199$	−5.38865	−0.381992	−0.190996	+	0.981591i	$0.561172\pi$
$200$	0	0
$201$	−5.47486	−0.386167
$202$	0	0
$203$	−8.20254	−0.575705
$204$	0	0
$205$	20.6650	1.44331
$206$	0	0
$207$	−7.19086	−0.499799
$208$	0	0
$209$	30.8192	2.13181
$210$	0	0
$211$	−0.555853	−0.0382665	−0.0191332	−	0.999817i	$-0.506091\pi$
$211$	−0.555853	−0.0382665	−0.0191332	+	0.999817i	$0.506091\pi$
$212$	0	0
$213$	−20.7523	−1.42193
$214$	0	0
$215$	22.4368	1.53018
$216$	0	0
$217$	−1.04427	−0.0708898
$218$	0	0
$219$	6.47942	0.437839
$220$	0	0
$221$	25.1795	1.69376
$222$	0	0
$223$	−15.8585	−1.06197	−0.530983	−	0.847382i	$-0.678177\pi$
$223$	−15.8585	−1.06197	−0.530983	+	0.847382i	$0.678177\pi$
$224$	0	0
$225$	5.45654	0.363769
$226$	0	0
$227$	14.4882	0.961617	0.480808	−	0.876826i	$-0.340343\pi$
$227$	14.4882	0.961617	0.480808	+	0.876826i	$0.340343\pi$
$228$	0	0
$229$	14.7683	0.975920	0.487960	−	0.872866i	$-0.337741\pi$
$229$	14.7683	0.975920	0.487960	+	0.872866i	$0.337741\pi$
$230$	0	0
$231$	−10.7281	−0.705854
$232$	0	0
$233$	−11.4749	−0.751743	−0.375872	−	0.926672i	$-0.622657\pi$
$233$	−11.4749	−0.751743	−0.375872	+	0.926672i	$0.622657\pi$
$234$	0	0
$235$	−17.0261	−1.11066
$236$	0	0
$237$	−30.6536	−1.99117
$238$	0	0
$239$	−17.1592	−1.10994	−0.554970	−	0.831871i	$-0.687270\pi$
$239$	−17.1592	−1.10994	−0.554970	+	0.831871i	$0.687270\pi$
$240$	0	0
$241$	−13.6863	−0.881615	−0.440807	−	0.897602i	$-0.645308\pi$
$241$	−13.6863	−0.881615	−0.440807	+	0.897602i	$0.645308\pi$
$242$	0	0
$243$	−22.2379	−1.42656
$244$	0	0
$245$	2.59708	0.165922
$246$	0	0
$247$	−34.6096	−2.20216
$248$	0	0
$249$	−7.87532	−0.499078
$250$	0	0
$251$	−6.71977	−0.424148	−0.212074	−	0.977254i	$-0.568022\pi$
$251$	−6.71977	−0.424148	−0.212074	+	0.977254i	$0.568022\pi$
$252$	0	0
$253$	−9.96572	−0.626539
$254$	0	0
$255$	−33.2583	−2.08272
$256$	0	0
$257$	−26.1130	−1.62888	−0.814442	−	0.580244i	$-0.802957\pi$
$257$	−26.1130	−1.62888	−0.814442	+	0.580244i	$0.802957\pi$
$258$	0	0
$259$	−4.77071	−0.296438
$260$	0	0
$261$	25.6513	1.58777
$262$	0	0
$263$	−13.6189	−0.839776	−0.419888	−	0.907576i	$-0.637931\pi$
$263$	−13.6189	−0.839776	−0.419888	+	0.907576i	$0.637931\pi$
$264$	0	0
$265$	26.8143	1.64719
$266$	0	0
$267$	20.1224	1.23147
$268$	0	0
$269$	−20.4252	−1.24535	−0.622673	−	0.782482i	$-0.713953\pi$
$269$	−20.4252	−1.24535	−0.622673	+	0.782482i	$0.713953\pi$
$270$	0	0
$271$	23.8585	1.44930	0.724651	−	0.689116i	$-0.242001\pi$
$271$	23.8585	1.44930	0.724651	+	0.689116i	$0.242001\pi$
$272$	0	0
$273$	12.0475	0.729149
$274$	0	0
$275$	7.56214	0.456014
$276$	0	0
$277$	−3.49540	−0.210018	−0.105009	−	0.994471i	$-0.533487\pi$
$277$	−3.49540	−0.210018	−0.105009	+	0.994471i	$0.533487\pi$
$278$	0	0
$279$	3.26568	0.195511
$280$	0	0
$281$	9.49361	0.566341	0.283171	−	0.959070i	$-0.408614\pi$
$281$	9.49361	0.566341	0.283171	+	0.959070i	$0.408614\pi$
$282$	0	0
$283$	−24.7620	−1.47195	−0.735973	−	0.677011i	$-0.763275\pi$
$283$	−24.7620	−1.47195	−0.735973	+	0.677011i	$0.763275\pi$
$284$	0	0
$285$	45.7141	2.70787
$286$	0	0
$287$	−7.95701	−0.469687
$288$	0	0
$289$	9.76479	0.574399
$290$	0	0
$291$	4.74697	0.278272
$292$	0	0
$293$	24.7968	1.44864	0.724322	−	0.689462i	$-0.242153\pi$
$293$	24.7968	1.44864	0.724322	+	0.689462i	$0.242153\pi$
$294$	0	0
$295$	33.1592	1.93061
$296$	0	0
$297$	1.36501	0.0792061
$298$	0	0
$299$	11.1914	0.647216
$300$	0	0
$301$	−8.63925	−0.497958
$302$	0	0
$303$	35.2517	2.02516
$304$	0	0
$305$	7.94648	0.455014
$306$	0	0
$307$	−1.54995	−0.0884604	−0.0442302	−	0.999021i	$-0.514084\pi$
$307$	−1.54995	−0.0884604	−0.0442302	+	0.999021i	$0.514084\pi$
$308$	0	0
$309$	−10.0141	−0.569680
$310$	0	0
$311$	4.19899	0.238103	0.119052	−	0.992888i	$-0.462015\pi$
$311$	4.19899	0.238103	0.119052	+	0.992888i	$0.462015\pi$
$312$	0	0
$313$	−28.9953	−1.63891	−0.819455	−	0.573144i	$-0.805724\pi$
$313$	−28.9953	−1.63891	−0.819455	+	0.573144i	$0.805724\pi$
$314$	0	0
$315$	−8.12170	−0.457606
$316$	0	0
$317$	−13.0280	−0.731724	−0.365862	−	0.930669i	$-0.619226\pi$
$317$	−13.0280	−0.731724	−0.365862	+	0.930669i	$0.619226\pi$
$318$	0	0
$319$	35.5498	1.99041
$320$	0	0
$321$	0.635592	0.0354753
$322$	0	0
$323$	−36.7887	−2.04698
$324$	0	0
$325$	−8.49222	−0.471064
$326$	0	0
$327$	−24.2524	−1.34116
$328$	0	0
$329$	6.55587	0.361437
$330$	0	0
$331$	−28.9289	−1.59008	−0.795038	−	0.606560i	$-0.792549\pi$
$331$	−28.9289	−1.59008	−0.795038	+	0.606560i	$0.792549\pi$
$332$	0	0
$333$	14.9191	0.817564
$334$	0	0
$335$	−5.74416	−0.313837
$336$	0	0
$337$	−27.8591	−1.51758	−0.758792	−	0.651333i	$-0.774210\pi$
$337$	−27.8591	−1.51758	−0.758792	+	0.651333i	$0.774210\pi$
$338$	0	0
$339$	−27.4558	−1.49119
$340$	0	0
$341$	4.52587	0.245090
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−14.7822	−0.795845
$346$	0	0
$347$	2.43095	0.130500	0.0652500	−	0.997869i	$-0.479216\pi$
$347$	2.43095	0.130500	0.0652500	+	0.997869i	$0.479216\pi$
$348$	0	0
$349$	19.9275	1.06670	0.533348	−	0.845896i	$-0.320934\pi$
$349$	19.9275	1.06670	0.533348	+	0.845896i	$0.320934\pi$
$350$	0	0
$351$	−1.53290	−0.0818200
$352$	0	0
$353$	−17.9445	−0.955090	−0.477545	−	0.878607i	$-0.658473\pi$
$353$	−17.9445	−0.955090	−0.477545	+	0.878607i	$0.658473\pi$
$354$	0	0
$355$	−21.7731	−1.15560
$356$	0	0
$357$	12.8060	0.677767
$358$	0	0
$359$	9.88158	0.521530	0.260765	−	0.965402i	$-0.416025\pi$
$359$	9.88158	0.521530	0.260765	+	0.965402i	$0.416025\pi$
$360$	0	0
$361$	31.5666	1.66140
$362$	0	0
$363$	19.2668	1.01124
$364$	0	0
$365$	6.79814	0.355831
$366$	0	0
$367$	20.8573	1.08874	0.544370	−	0.838845i	$-0.316769\pi$
$367$	20.8573	1.08874	0.544370	+	0.838845i	$0.316769\pi$
$368$	0	0
$369$	24.8835	1.29538
$370$	0	0
$371$	−10.3248	−0.536035
$372$	0	0
$373$	22.8110	1.18111	0.590554	−	0.806998i	$-0.298909\pi$
$373$	22.8110	1.18111	0.590554	+	0.806998i	$0.298909\pi$
$374$	0	0
$375$	−20.9262	−1.08062
$376$	0	0
$377$	−39.9221	−2.05609
$378$	0	0
$379$	22.6421	1.16305	0.581524	−	0.813529i	$-0.302457\pi$
$379$	22.6421	1.16305	0.581524	+	0.813529i	$0.302457\pi$
$380$	0	0
$381$	−16.2230	−0.831132
$382$	0	0
$383$	22.7924	1.16463	0.582317	−	0.812962i	$-0.302146\pi$
$383$	22.7924	1.16463	0.582317	+	0.812962i	$0.302146\pi$
$384$	0	0
$385$	−11.2558	−0.573646
$386$	0	0
$387$	27.0170	1.37335
$388$	0	0
$389$	30.6528	1.55416	0.777079	−	0.629403i	$-0.216701\pi$
$389$	30.6528	1.55416	0.777079	+	0.629403i	$0.216701\pi$
$390$	0	0
$391$	11.8960	0.601608
$392$	0	0
$393$	−36.6427	−1.84838
$394$	0	0
$395$	−32.1614	−1.61822
$396$	0	0
$397$	17.3327	0.869902	0.434951	−	0.900454i	$-0.356766\pi$
$397$	17.3327	0.869902	0.434951	+	0.900454i	$0.356766\pi$
$398$	0	0
$399$	−17.6021	−0.881207
$400$	0	0
$401$	12.7300	0.635707	0.317853	−	0.948140i	$-0.397038\pi$
$401$	12.7300	0.635707	0.317853	+	0.948140i	$0.397038\pi$
$402$	0	0
$403$	−5.08251	−0.253178
$404$	0	0
$405$	−22.3404	−1.11010
$406$	0	0
$407$	20.6762	1.02488
$408$	0	0
$409$	8.86582	0.438387	0.219193	−	0.975681i	$-0.429657\pi$
$409$	8.86582	0.438387	0.219193	+	0.975681i	$0.429657\pi$
$410$	0	0
$411$	7.74262	0.381915
$412$	0	0
$413$	−12.7679	−0.628266
$414$	0	0
$415$	−8.26269	−0.405600
$416$	0	0
$417$	−18.6188	−0.911767
$418$	0	0
$419$	5.48647	0.268031	0.134016	−	0.990979i	$-0.457213\pi$
$419$	5.48647	0.268031	0.134016	+	0.990979i	$0.457213\pi$
$420$	0	0
$421$	−3.49540	−0.170355	−0.0851777	−	0.996366i	$-0.527146\pi$
$421$	−3.49540	−0.170355	−0.0851777	+	0.996366i	$0.527146\pi$
$422$	0	0
$423$	−20.5017	−0.996829
$424$	0	0
$425$	−9.02689	−0.437868
$426$	0	0
$427$	−3.05977	−0.148073
$428$	0	0
$429$	−52.2139	−2.52091
$430$	0	0
$431$	20.5604	0.990359	0.495179	−	0.868791i	$-0.335102\pi$
$431$	20.5604	0.990359	0.495179	+	0.868791i	$0.335102\pi$
$432$	0	0
$433$	−7.02552	−0.337625	−0.168813	−	0.985648i	$-0.553993\pi$
$433$	−7.02552	−0.337625	−0.168813	+	0.985648i	$0.553993\pi$
$434$	0	0
$435$	52.7310	2.52826
$436$	0	0
$437$	−16.3513	−0.782188
$438$	0	0
$439$	−14.5154	−0.692784	−0.346392	−	0.938090i	$-0.612593\pi$
$439$	−14.5154	−0.692784	−0.346392	+	0.938090i	$0.612593\pi$
$440$	0	0
$441$	3.12724	0.148916
$442$	0	0
$443$	8.22851	0.390948	0.195474	−	0.980709i	$-0.437375\pi$
$443$	8.22851	0.390948	0.195474	+	0.980709i	$0.437375\pi$
$444$	0	0
$445$	21.1122	1.00081
$446$	0	0
$447$	−21.4492	−1.01451
$448$	0	0
$449$	34.2448	1.61611	0.808055	−	0.589107i	$-0.200520\pi$
$449$	34.2448	1.61611	0.808055	+	0.589107i	$0.200520\pi$
$450$	0	0
$451$	34.4857	1.62387
$452$	0	0
$453$	−35.1829	−1.65304
$454$	0	0
$455$	12.6401	0.592578
$456$	0	0
$457$	−5.67557	−0.265492	−0.132746	−	0.991150i	$-0.542379\pi$
$457$	−5.67557	−0.265492	−0.132746	+	0.991150i	$0.542379\pi$
$458$	0	0
$459$	−1.62941	−0.0760543
$460$	0	0
$461$	27.3622	1.27439	0.637193	−	0.770704i	$-0.280096\pi$
$461$	27.3622	1.27439	0.637193	+	0.770704i	$0.280096\pi$
$462$	0	0
$463$	13.4561	0.625359	0.312679	−	0.949859i	$-0.398773\pi$
$463$	13.4561	0.625359	0.312679	+	0.949859i	$0.398773\pi$
$464$	0	0
$465$	6.71323	0.311319
$466$	0	0
$467$	−16.2660	−0.752703	−0.376351	−	0.926477i	$-0.622821\pi$
$467$	−16.2660	−0.752703	−0.376351	+	0.926477i	$0.622821\pi$
$468$	0	0
$469$	2.21177	0.102130
$470$	0	0
$471$	−47.8545	−2.20502
$472$	0	0
$473$	37.4425	1.72161
$474$	0	0
$475$	12.4076	0.569300
$476$	0	0
$477$	32.2880	1.47836
$478$	0	0
$479$	1.44806	0.0661634	0.0330817	−	0.999453i	$-0.489468\pi$
$479$	1.44806	0.0661634	0.0330817	+	0.999453i	$0.489468\pi$
$480$	0	0
$481$	−23.2192	−1.05871
$482$	0	0
$483$	5.69184	0.258988
$484$	0	0
$485$	4.98047	0.226151
$486$	0	0
$487$	−28.9008	−1.30962	−0.654811	−	0.755793i	$-0.727252\pi$
$487$	−28.9008	−1.30962	−0.654811	+	0.755793i	$0.727252\pi$
$488$	0	0
$489$	−35.8398	−1.62073
$490$	0	0
$491$	−2.84168	−0.128243	−0.0641217	−	0.997942i	$-0.520425\pi$
$491$	−2.84168	−0.128243	−0.0641217	+	0.997942i	$0.520425\pi$
$492$	0	0
$493$	−42.4356	−1.91120
$494$	0	0
$495$	35.1994	1.58210
$496$	0	0
$497$	8.38367	0.376059
$498$	0	0
$499$	−5.59165	−0.250317	−0.125158	−	0.992137i	$-0.539944\pi$
$499$	−5.59165	−0.250317	−0.125158	+	0.992137i	$0.539944\pi$
$500$	0	0
$501$	41.2003	1.84070
$502$	0	0
$503$	−37.3662	−1.66608	−0.833038	−	0.553215i	$-0.813401\pi$
$503$	−37.3662	−1.66608	−0.833038	+	0.553215i	$0.813401\pi$
$504$	0	0
$505$	36.9857	1.64584
$506$	0	0
$507$	26.4565	1.17497
$508$	0	0
$509$	−6.73873	−0.298689	−0.149344	−	0.988785i	$-0.547716\pi$
$509$	−6.73873	−0.298689	−0.149344	+	0.988785i	$0.547716\pi$
$510$	0	0
$511$	−2.61760	−0.115796
$512$	0	0
$513$	2.23965	0.0988830
$514$	0	0
$515$	−10.5066	−0.462977
$516$	0	0
$517$	−28.4131	−1.24961
$518$	0	0
$519$	50.5590	2.21929
$520$	0	0
$521$	16.6470	0.729317	0.364658	−	0.931141i	$-0.381186\pi$
$521$	16.6470	0.729317	0.364658	+	0.931141i	$0.381186\pi$
$522$	0	0
$523$	11.7662	0.514500	0.257250	−	0.966345i	$-0.417184\pi$
$523$	11.7662	0.514500	0.257250	+	0.966345i	$0.417184\pi$
$524$	0	0
$525$	−4.31905	−0.188499
$526$	0	0
$527$	−5.40251	−0.235337
$528$	0	0
$529$	−17.7126	−0.770114
$530$	0	0
$531$	39.9282	1.73274
$532$	0	0
$533$	−38.7271	−1.67746
$534$	0	0
$535$	0.666856	0.0288307
$536$	0	0
$537$	−8.52770	−0.367997
$538$	0	0
$539$	4.33400	0.186679
$540$	0	0
$541$	28.2033	1.21256	0.606278	−	0.795253i	$-0.292662\pi$
$541$	28.2033	1.21256	0.606278	+	0.795253i	$0.292662\pi$
$542$	0	0
$543$	−44.0493	−1.89034
$544$	0	0
$545$	−25.4453	−1.08996
$546$	0	0
$547$	22.7117	0.971084	0.485542	−	0.874213i	$-0.338622\pi$
$547$	22.7117	0.971084	0.485542	+	0.874213i	$0.338622\pi$
$548$	0	0
$549$	9.56863	0.408379
$550$	0	0
$551$	58.3284	2.48487
$552$	0	0
$553$	12.3837	0.526607
$554$	0	0
$555$	30.6691	1.30183
$556$	0	0
$557$	37.2920	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$557$	37.2920	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$558$	0	0
$559$	−42.0476	−1.77842
$560$	0	0
$561$	−55.5013	−2.34327
$562$	0	0
$563$	−19.8636	−0.837153	−0.418576	−	0.908182i	$-0.637471\pi$
$563$	−19.8636	−0.837153	−0.418576	+	0.908182i	$0.637471\pi$
$564$	0	0
$565$	−28.8063	−1.21189
$566$	0	0
$567$	8.60210	0.361254
$568$	0	0
$569$	5.50827	0.230919	0.115459	−	0.993312i	$-0.463166\pi$
$569$	5.50827	0.230919	0.115459	+	0.993312i	$0.463166\pi$
$570$	0	0
$571$	−41.8289	−1.75048	−0.875242	−	0.483686i	$-0.839298\pi$
$571$	−41.8289	−1.75048	−0.875242	+	0.483686i	$0.839298\pi$
$572$	0	0
$573$	−20.6551	−0.862878
$574$	0	0
$575$	−4.01214	−0.167318
$576$	0	0
$577$	8.07056	0.335982	0.167991	−	0.985789i	$-0.446272\pi$
$577$	8.07056	0.335982	0.167991	+	0.985789i	$0.446272\pi$
$578$	0	0
$579$	60.9923	2.53475
$580$	0	0
$581$	3.18153	0.131992
$582$	0	0
$583$	44.7475	1.85325
$584$	0	0
$585$	−39.5286	−1.63431
$586$	0	0
$587$	−13.6933	−0.565181	−0.282590	−	0.959241i	$-0.591194\pi$
$587$	−13.6933	−0.565181	−0.282590	+	0.959241i	$0.591194\pi$
$588$	0	0
$589$	7.42584	0.305976
$590$	0	0
$591$	−32.4079	−1.33308
$592$	0	0
$593$	−25.6900	−1.05496	−0.527482	−	0.849566i	$-0.676864\pi$
$593$	−25.6900	−1.05496	−0.527482	+	0.849566i	$0.676864\pi$
$594$	0	0
$595$	13.4359	0.550820
$596$	0	0
$597$	−13.3387	−0.545916
$598$	0	0
$599$	−20.0051	−0.817387	−0.408693	−	0.912672i	$-0.634015\pi$
$599$	−20.0051	−0.817387	−0.408693	+	0.912672i	$0.634015\pi$
$600$	0	0
$601$	41.4055	1.68897	0.844483	−	0.535583i	$-0.179908\pi$
$601$	41.4055	1.68897	0.844483	+	0.535583i	$0.179908\pi$
$602$	0	0
$603$	−6.91674	−0.281671
$604$	0	0
$605$	20.2145	0.821836
$606$	0	0
$607$	33.6015	1.36384	0.681921	−	0.731425i	$-0.261145\pi$
$607$	33.6015	1.36384	0.681921	+	0.731425i	$0.261145\pi$
$608$	0	0
$609$	−20.3039	−0.822757
$610$	0	0
$611$	31.9077	1.29085
$612$	0	0
$613$	40.4668	1.63444	0.817219	−	0.576328i	$-0.195515\pi$
$613$	40.4668	1.63444	0.817219	+	0.576328i	$0.195515\pi$
$614$	0	0
$615$	51.1526	2.06267
$616$	0	0
$617$	−28.3082	−1.13965	−0.569824	−	0.821767i	$-0.692988\pi$
$617$	−28.3082	−1.13965	−0.569824	+	0.821767i	$0.692988\pi$
$618$	0	0
$619$	20.3384	0.817470	0.408735	−	0.912653i	$-0.365970\pi$
$619$	20.3384	0.817470	0.408735	+	0.912653i	$0.365970\pi$
$620$	0	0
$621$	−0.724216	−0.0290618
$622$	0	0
$623$	−8.12920	−0.325689
$624$	0	0
$625$	−30.6797	−1.22719
$626$	0	0
$627$	76.2875	3.04663
$628$	0	0
$629$	−24.6811	−0.984101
$630$	0	0
$631$	−43.1143	−1.71635	−0.858176	−	0.513355i	$-0.828402\pi$
$631$	−43.1143	−1.71635	−0.858176	+	0.513355i	$0.828402\pi$
$632$	0	0
$633$	−1.37592	−0.0546878
$634$	0	0
$635$	−17.0210	−0.675459
$636$	0	0
$637$	−4.86704	−0.192839
$638$	0	0
$639$	−26.2177	−1.03716
$640$	0	0
$641$	−26.7326	−1.05587	−0.527937	−	0.849284i	$-0.677034\pi$
$641$	−26.7326	−1.05587	−0.527937	+	0.849284i	$0.677034\pi$
$642$	0	0
$643$	30.0199	1.18387	0.591934	−	0.805987i	$-0.298365\pi$
$643$	30.0199	1.18387	0.591934	+	0.805987i	$0.298365\pi$
$644$	0	0
$645$	55.5385	2.18683
$646$	0	0
$647$	−11.3001	−0.444253	−0.222127	−	0.975018i	$-0.571300\pi$
$647$	−11.3001	−0.444253	−0.222127	+	0.975018i	$0.571300\pi$
$648$	0	0
$649$	55.3360	2.17213
$650$	0	0
$651$	−2.58491	−0.101311
$652$	0	0
$653$	−15.3418	−0.600369	−0.300185	−	0.953881i	$-0.597048\pi$
$653$	−15.3418	−0.600369	−0.300185	+	0.953881i	$0.597048\pi$
$654$	0	0
$655$	−38.4451	−1.50217
$656$	0	0
$657$	8.18587	0.319361
$658$	0	0
$659$	12.3676	0.481774	0.240887	−	0.970553i	$-0.422562\pi$
$659$	12.3676	0.481774	0.240887	+	0.970553i	$0.422562\pi$
$660$	0	0
$661$	−10.1256	−0.393841	−0.196921	−	0.980419i	$-0.563094\pi$
$661$	−10.1256	−0.393841	−0.196921	+	0.980419i	$0.563094\pi$
$662$	0	0
$663$	62.3274	2.42060
$664$	0	0
$665$	−18.4679	−0.716155
$666$	0	0
$667$	−18.8611	−0.730307
$668$	0	0
$669$	−39.2550	−1.51769
$670$	0	0
$671$	13.2610	0.511937
$672$	0	0
$673$	51.5498	1.98710	0.993549	−	0.113405i	$-0.0361759\pi$
$673$	51.5498	1.98710	0.993549	+	0.113405i	$0.0361759\pi$
$674$	0	0
$675$	0.549546	0.0211521
$676$	0	0
$677$	21.3472	0.820438	0.410219	−	0.911987i	$-0.365452\pi$
$677$	21.3472	0.820438	0.410219	+	0.911987i	$0.365452\pi$
$678$	0	0
$679$	−1.91772	−0.0735952
$680$	0	0
$681$	35.8631	1.37428
$682$	0	0
$683$	−21.1763	−0.810289	−0.405145	−	0.914253i	$-0.632779\pi$
$683$	−21.1763	−0.810289	−0.405145	+	0.914253i	$0.632779\pi$
$684$	0	0
$685$	8.12347	0.310382
$686$	0	0
$687$	36.5565	1.39472
$688$	0	0
$689$	−50.2510	−1.91441
$690$	0	0
$691$	21.7108	0.825916	0.412958	−	0.910750i	$-0.364496\pi$
$691$	21.7108	0.825916	0.412958	+	0.910750i	$0.364496\pi$
$692$	0	0
$693$	−13.5534	−0.514853
$694$	0	0
$695$	−19.5346	−0.740991
$696$	0	0
$697$	−41.1653	−1.55925
$698$	0	0
$699$	−28.4040	−1.07434
$700$	0	0
$701$	−40.7622	−1.53957	−0.769784	−	0.638304i	$-0.779636\pi$
$701$	−40.7622	−1.53957	−0.769784	+	0.638304i	$0.779636\pi$
$702$	0	0
$703$	33.9246	1.27949
$704$	0	0
$705$	−42.1452	−1.58728
$706$	0	0
$707$	−14.2412	−0.535597
$708$	0	0
$709$	31.1810	1.17103	0.585513	−	0.810663i	$-0.300893\pi$
$709$	31.1810	1.17103	0.585513	+	0.810663i	$0.300893\pi$
$710$	0	0
$711$	−38.7267	−1.45236
$712$	0	0
$713$	−2.40123	−0.0899267
$714$	0	0
$715$	−54.7822	−2.04874
$716$	0	0
$717$	−42.4747	−1.58625
$718$	0	0
$719$	26.9222	1.00403	0.502014	−	0.864860i	$-0.332593\pi$
$719$	26.9222	1.00403	0.502014	+	0.864860i	$0.332593\pi$
$720$	0	0
$721$	4.04555	0.150664
$722$	0	0
$723$	−33.8782	−1.25994
$724$	0	0
$725$	14.3121	0.531539
$726$	0	0
$727$	45.5056	1.68771	0.843854	−	0.536572i	$-0.180281\pi$
$727$	45.5056	1.68771	0.843854	+	0.536572i	$0.180281\pi$
$728$	0	0
$729$	−29.2397	−1.08295
$730$	0	0
$731$	−44.6949	−1.65310
$732$	0	0
$733$	12.7040	0.469233	0.234616	−	0.972088i	$-0.424617\pi$
$733$	12.7040	0.469233	0.234616	+	0.972088i	$0.424617\pi$
$734$	0	0
$735$	6.42863	0.237123
$736$	0	0
$737$	−9.58582	−0.353098
$738$	0	0
$739$	−41.1451	−1.51355	−0.756773	−	0.653678i	$-0.773225\pi$
$739$	−41.1451	−1.51355	−0.756773	+	0.653678i	$0.773225\pi$
$740$	0	0
$741$	−85.6701	−3.14717
$742$	0	0
$743$	−7.92011	−0.290561	−0.145280	−	0.989391i	$-0.546408\pi$
$743$	−7.92011	−0.290561	−0.145280	+	0.989391i	$0.546408\pi$
$744$	0	0
$745$	−22.5043	−0.824494
$746$	0	0
$747$	−9.94940	−0.364029
$748$	0	0
$749$	−0.256771	−0.00938221
$750$	0	0
$751$	1.86395	0.0680164	0.0340082	−	0.999422i	$-0.489173\pi$
$751$	1.86395	0.0680164	0.0340082	+	0.999422i	$0.489173\pi$
$752$	0	0
$753$	−16.6336	−0.606163
$754$	0	0
$755$	−36.9135	−1.34342
$756$	0	0
$757$	38.6105	1.40332	0.701661	−	0.712511i	$-0.252442\pi$
$757$	38.6105	1.40332	0.701661	+	0.712511i	$0.252442\pi$
$758$	0	0
$759$	−24.6684	−0.895406
$760$	0	0
$761$	18.8473	0.683215	0.341607	−	0.939843i	$-0.389029\pi$
$761$	18.8473	0.683215	0.341607	+	0.939843i	$0.389029\pi$
$762$	0	0
$763$	9.79765	0.354699
$764$	0	0
$765$	−42.0174	−1.51914
$766$	0	0
$767$	−62.1418	−2.24381
$768$	0	0
$769$	−25.0914	−0.904819	−0.452410	−	0.891810i	$-0.649435\pi$
$769$	−25.0914	−0.904819	−0.452410	+	0.891810i	$0.649435\pi$
$770$	0	0
$771$	−64.6382	−2.32789
$772$	0	0
$773$	−39.1646	−1.40865	−0.704327	−	0.709876i	$-0.748751\pi$
$773$	−39.1646	−1.40865	−0.704327	+	0.709876i	$0.748751\pi$
$774$	0	0
$775$	1.82209	0.0654514
$776$	0	0
$777$	−11.8091	−0.423648
$778$	0	0
$779$	56.5825	2.02728
$780$	0	0
$781$	−36.3348	−1.30016
$782$	0	0
$783$	2.58343	0.0923242
$784$	0	0
$785$	−50.2084	−1.79201
$786$	0	0
$787$	42.4453	1.51301	0.756505	−	0.653988i	$-0.226905\pi$
$787$	42.4453	1.51301	0.756505	+	0.653988i	$0.226905\pi$
$788$	0	0
$789$	−33.7112	−1.20015
$790$	0	0
$791$	11.0918	0.394378
$792$	0	0
$793$	−14.8920	−0.528831
$794$	0	0
$795$	66.3740	2.35404
$796$	0	0
$797$	6.15784	0.218122	0.109061	−	0.994035i	$-0.465216\pi$
$797$	6.15784	0.218122	0.109061	+	0.994035i	$0.465216\pi$
$798$	0	0
$799$	33.9166	1.19988
$800$	0	0
$801$	25.4219	0.898240
$802$	0	0
$803$	11.3447	0.400346
$804$	0	0
$805$	5.97181	0.210479
$806$	0	0
$807$	−50.5590	−1.77976
$808$	0	0
$809$	−34.1150	−1.19942	−0.599709	−	0.800218i	$-0.704717\pi$
$809$	−34.1150	−1.19942	−0.599709	+	0.800218i	$0.704717\pi$
$810$	0	0
$811$	−20.1521	−0.707636	−0.353818	−	0.935314i	$-0.615117\pi$
$811$	−20.1521	−0.707636	−0.353818	+	0.935314i	$0.615117\pi$
$812$	0	0
$813$	59.0576	2.07124
$814$	0	0
$815$	−37.6027	−1.31716
$816$	0	0
$817$	61.4339	2.14930
$818$	0	0
$819$	15.2204	0.531844
$820$	0	0
$821$	10.7385	0.374777	0.187388	−	0.982286i	$-0.439998\pi$
$821$	10.7385	0.374777	0.187388	+	0.982286i	$0.439998\pi$
$822$	0	0
$823$	−25.9633	−0.905023	−0.452511	−	0.891759i	$-0.649472\pi$
$823$	−25.9633	−0.905023	−0.452511	+	0.891759i	$0.649472\pi$
$824$	0	0
$825$	18.7188	0.651704
$826$	0	0
$827$	5.51645	0.191826	0.0959129	−	0.995390i	$-0.469423\pi$
$827$	5.51645	0.191826	0.0959129	+	0.995390i	$0.469423\pi$
$828$	0	0
$829$	2.96251	0.102892	0.0514461	−	0.998676i	$-0.483617\pi$
$829$	2.96251	0.102892	0.0514461	+	0.998676i	$0.483617\pi$
$830$	0	0
$831$	−8.65225	−0.300143
$832$	0	0
$833$	−5.17347	−0.179250
$834$	0	0
$835$	43.2269	1.49593
$836$	0	0
$837$	0.328898	0.0113684
$838$	0	0
$839$	19.9002	0.687030	0.343515	−	0.939147i	$-0.388382\pi$
$839$	19.9002	0.687030	0.343515	+	0.939147i	$0.388382\pi$
$840$	0	0
$841$	38.2816	1.32005
$842$	0	0
$843$	23.4998	0.809375
$844$	0	0
$845$	27.7578	0.954899
$846$	0	0
$847$	−7.78354	−0.267446
$848$	0	0
$849$	−61.2939	−2.10360
$850$	0	0
$851$	−10.9699	−0.376044
$852$	0	0
$853$	−6.64487	−0.227516	−0.113758	−	0.993508i	$-0.536289\pi$
$853$	−6.64487	−0.227516	−0.113758	+	0.993508i	$0.536289\pi$
$854$	0	0
$855$	57.7536	1.97513
$856$	0	0
$857$	−42.8168	−1.46259	−0.731297	−	0.682059i	$-0.761085\pi$
$857$	−42.8168	−1.46259	−0.731297	+	0.682059i	$0.761085\pi$
$858$	0	0
$859$	−25.7625	−0.879007	−0.439503	−	0.898241i	$-0.644846\pi$
$859$	−25.7625	−0.879007	−0.439503	+	0.898241i	$0.644846\pi$
$860$	0	0
$861$	−19.6962	−0.671244
$862$	0	0
$863$	3.83270	0.130467	0.0652334	−	0.997870i	$-0.479221\pi$
$863$	3.83270	0.130467	0.0652334	+	0.997870i	$0.479221\pi$
$864$	0	0
$865$	53.0459	1.80362
$866$	0	0
$867$	24.1710	0.820891
$868$	0	0
$869$	−53.6708	−1.82066
$870$	0	0
$871$	10.7648	0.364751
$872$	0	0
$873$	5.99715	0.202973
$874$	0	0
$875$	8.45392	0.285795
$876$	0	0
$877$	−35.8226	−1.20964	−0.604822	−	0.796361i	$-0.706756\pi$
$877$	−35.8226	−1.20964	−0.604822	+	0.796361i	$0.706756\pi$
$878$	0	0
$879$	61.3801	2.07030
$880$	0	0
$881$	−12.9974	−0.437892	−0.218946	−	0.975737i	$-0.570262\pi$
$881$	−12.9974	−0.437892	−0.218946	+	0.975737i	$0.570262\pi$
$882$	0	0
$883$	1.72113	0.0579205	0.0289603	−	0.999581i	$-0.490780\pi$
$883$	1.72113	0.0579205	0.0289603	+	0.999581i	$0.490780\pi$
$884$	0	0
$885$	82.0799	2.75909
$886$	0	0
$887$	−3.31810	−0.111411	−0.0557054	−	0.998447i	$-0.517741\pi$
$887$	−3.31810	−0.111411	−0.0557054	+	0.998447i	$0.517741\pi$
$888$	0	0
$889$	6.55390	0.219811
$890$	0	0
$891$	−37.2815	−1.24898
$892$	0	0
$893$	−46.6189	−1.56004
$894$	0	0
$895$	−8.94716	−0.299071
$896$	0	0
$897$	27.7024	0.924956
$898$	0	0
$899$	8.56567	0.285681
$900$	0	0
$901$	−53.4148	−1.77951
$902$	0	0
$903$	−21.3850	−0.711647
$904$	0	0
$905$	−46.2160	−1.53627
$906$	0	0
$907$	54.3809	1.80569	0.902845	−	0.429967i	$-0.141475\pi$
$907$	54.3809	1.80569	0.902845	+	0.429967i	$0.141475\pi$
$908$	0	0
$909$	44.5357	1.47716
$910$	0	0
$911$	−17.5771	−0.582355	−0.291178	−	0.956669i	$-0.594047\pi$
$911$	−17.5771	−0.582355	−0.291178	+	0.956669i	$0.594047\pi$
$912$	0	0
$913$	−13.7887	−0.456341
$914$	0	0
$915$	19.6701	0.650274
$916$	0	0
$917$	14.8032	0.488844
$918$	0	0
$919$	31.0625	1.02466	0.512328	−	0.858790i	$-0.328783\pi$
$919$	31.0625	1.02466	0.512328	+	0.858790i	$0.328783\pi$
$920$	0	0
$921$	−3.83663	−0.126421
$922$	0	0
$923$	40.8037	1.34307
$924$	0	0
$925$	8.32414	0.273696
$926$	0	0
$927$	−12.6514	−0.415526
$928$	0	0
$929$	−2.69366	−0.0883761	−0.0441880	−	0.999023i	$-0.514070\pi$
$929$	−2.69366	−0.0883761	−0.0441880	+	0.999023i	$0.514070\pi$
$930$	0	0
$931$	7.11102	0.233054
$932$	0	0
$933$	10.3939	0.340280
$934$	0	0
$935$	−58.2313	−1.90437
$936$	0	0
$937$	10.9026	0.356172	0.178086	−	0.984015i	$-0.443009\pi$
$937$	10.9026	0.356172	0.178086	+	0.984015i	$0.443009\pi$
$938$	0	0
$939$	−71.7727	−2.34221
$940$	0	0
$941$	56.8385	1.85288	0.926441	−	0.376440i	$-0.122852\pi$
$941$	56.8385	1.85288	0.926441	+	0.376440i	$0.122852\pi$
$942$	0	0
$943$	−18.2966	−0.595818
$944$	0	0
$945$	−0.817964	−0.0266084
$946$	0	0
$947$	−1.18879	−0.0386304	−0.0193152	−	0.999813i	$-0.506149\pi$
$947$	−1.18879	−0.0386304	−0.0193152	+	0.999813i	$0.506149\pi$
$948$	0	0
$949$	−12.7400	−0.413558
$950$	0	0
$951$	−32.2485	−1.04573
$952$	0	0
$953$	−32.1824	−1.04249	−0.521245	−	0.853407i	$-0.674532\pi$
$953$	−32.1824	−1.04249	−0.521245	+	0.853407i	$0.674532\pi$
$954$	0	0
$955$	−21.6711	−0.701259
$956$	0	0
$957$	87.9973	2.84455
$958$	0	0
$959$	−3.12792	−0.101006
$960$	0	0
$961$	−29.9095	−0.964822
$962$	0	0
$963$	0.802984	0.0258758
$964$	0	0
$965$	63.9924	2.05999
$966$	0	0
$967$	−25.1967	−0.810272	−0.405136	−	0.914256i	$-0.632776\pi$
$967$	−25.1967	−0.810272	−0.405136	+	0.914256i	$0.632776\pi$
$968$	0	0
$969$	−91.0639	−2.92539
$970$	0	0
$971$	−38.2008	−1.22592	−0.612961	−	0.790113i	$-0.710022\pi$
$971$	−38.2008	−1.22592	−0.612961	+	0.790113i	$0.710022\pi$
$972$	0	0
$973$	7.52176	0.241137
$974$	0	0
$975$	−21.0210	−0.673211
$976$	0	0
$977$	−15.8192	−0.506102	−0.253051	−	0.967453i	$-0.581434\pi$
$977$	−15.8192	−0.506102	−0.253051	+	0.967453i	$0.581434\pi$
$978$	0	0
$979$	35.2319	1.12602
$980$	0	0
$981$	−30.6396	−0.978247
$982$	0	0
$983$	−15.3177	−0.488560	−0.244280	−	0.969705i	$-0.578552\pi$
$983$	−15.3177	−0.488560	−0.244280	+	0.969705i	$0.578552\pi$
$984$	0	0
$985$	−34.0020	−1.08339
$986$	0	0
$987$	16.2279	0.516540
$988$	0	0
$989$	−19.8653	−0.631681
$990$	0	0
$991$	9.88980	0.314160	0.157080	−	0.987586i	$-0.449792\pi$
$991$	9.88980	0.314160	0.157080	+	0.987586i	$0.449792\pi$
$992$	0	0
$993$	−71.6084	−2.27242
$994$	0	0
$995$	−13.9948	−0.443664
$996$	0	0
$997$	10.0074	0.316938	0.158469	−	0.987364i	$-0.449344\pi$
$997$	10.0074	0.316938	0.158469	+	0.987364i	$0.449344\pi$
$998$	0	0
$999$	1.50256	0.0475388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.o.1.8	yes	10
4.3	odd	2		3584.2.a.p.1.3	yes	10
8.3	odd	2		3584.2.a.p.1.8	yes	10
8.5	even	2	inner	3584.2.a.o.1.3	✓	10
16.3	odd	4		3584.2.b.g.1793.3		10
16.5	even	4		3584.2.b.h.1793.3		10
16.11	odd	4		3584.2.b.g.1793.8		10
16.13	even	4		3584.2.b.h.1793.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.o.1.3	✓	10	8.5	even	2	inner
3584.2.a.o.1.8	yes	10	1.1	even	1	trivial
3584.2.a.p.1.3	yes	10	4.3	odd	2
3584.2.a.p.1.8	yes	10	8.3	odd	2
3584.2.b.g.1793.3		10	16.3	odd	4
3584.2.b.g.1793.8		10	16.11	odd	4
3584.2.b.h.1793.3		10	16.5	even	4
3584.2.b.h.1793.8		10	16.13	even	4

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q+2.47533 q^{3} +2.59708 q^{5} -1.00000 q^{7} +3.12724 q^{9} +O(q^{10})\)	\(q+2.47533 q^{3} +2.59708 q^{5} -1.00000 q^{7} +3.12724 q^{9} +4.33400 q^{11} -4.86704 q^{13} +6.42863 q^{15} -5.17347 q^{17} +7.11102 q^{19} -2.47533 q^{21} -2.29943 q^{23} +1.74484 q^{25} +0.314955 q^{27} +8.20254 q^{29} +1.04427 q^{31} +10.7281 q^{33} -2.59708 q^{35} +4.77071 q^{37} -12.0475 q^{39} +7.95701 q^{41} +8.63925 q^{43} +8.12170 q^{45} -6.55587 q^{47} +1.00000 q^{49} -12.8060 q^{51} +10.3248 q^{53} +11.2558 q^{55} +17.6021 q^{57} +12.7679 q^{59} +3.05977 q^{61} -3.12724 q^{63} -12.6401 q^{65} -2.21177 q^{67} -5.69184 q^{69} -8.38367 q^{71} +2.61760 q^{73} +4.31905 q^{75} -4.33400 q^{77} -12.3837 q^{79} -8.60210 q^{81} -3.18153 q^{83} -13.4359 q^{85} +20.3039 q^{87} +8.12920 q^{89} +4.86704 q^{91} +2.58491 q^{93} +18.4679 q^{95} +1.91772 q^{97} +13.5534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 10 * q - 10 * q^7 + 22 * q^9	\( 10 q - 10 q^{7} + 22 q^{9} - 16 q^{15} + 16 q^{17} - 8 q^{23} + 30 q^{25} + 8 q^{31} + 12 q^{33} + 20 q^{41} + 24 q^{47} + 10 q^{49} + 32 q^{55} + 28 q^{57} - 22 q^{63} + 32 q^{65} + 48 q^{73} - 40 q^{79} + 62 q^{81} + 8 q^{87} + 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 10 * q - 10 * q^7 + 22 * q^9 - 16 * q^15 + 16 * q^17 - 8 * q^23 + 30 * q^25 + 8 * q^31 + 12 * q^33 + 20 * q^41 + 24 * q^47 + 10 * q^49 + 32 * q^55 + 28 * q^57 - 22 * q^63 + 32 * q^65 + 48 * q^73 - 40 * q^79 + 62 * q^81 + 8 * q^87 + 16 * q^89 + 32 * q^95 + 32 * q^97

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