LMFDB - Embedded newform 3584.2.a.p.1.3

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			1.3
Root			$-0.269488$ 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	$-0.753375\pi$
$3$	−2.47533	−1.42913	−0.714565	+	0.699569i	$0.753375\pi$
$4$	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	$-0.726647\pi$
$11$	−4.33400	−1.30675	−0.653375	+	0.757035i	$0.726647\pi$
$12$	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	$-0.803643\pi$
$19$	−7.11102	−1.63138	−0.815690	+	0.578489i	$0.803643\pi$
$20$	0	0
$21$	−2.47533	−0.540160
$22$	0	0
$23$	2.29943	0.479464	0.239732	−	0.970839i	$-0.422940\pi$
$23$	2.29943	0.479464	0.239732	+	0.970839i	$0.422940\pi$
$24$	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	$-0.529894\pi$
$31$	−1.04427	−0.187557	−0.0937783	+	0.995593i	$0.529894\pi$
$32$	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	$-0.728909\pi$
$43$	−8.63925	−1.31747	−0.658736	+	0.752374i	$0.728909\pi$
$44$	0	0
$45$	8.12170	1.21071
$46$	0	0
$47$	6.55587	0.956271	0.478136	−	0.878286i	$-0.341313\pi$
$47$	6.55587	0.956271	0.478136	+	0.878286i	$0.341313\pi$
$48$	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	$-0.812299\pi$
$59$	−12.7679	−1.66224	−0.831118	+	0.556096i	$0.812299\pi$
$60$	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	$-0.456863\pi$
$67$	2.21177	0.270211	0.135106	+	0.990831i	$0.456863\pi$
$68$	0	0
$69$	−5.69184	−0.685217
$70$	0	0
$71$	8.38367	0.994959	0.497480	−	0.867476i	$-0.334259\pi$
$71$	8.38367	0.994959	0.497480	+	0.867476i	$0.334259\pi$
$72$	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	$-0.254679\pi$
$79$	12.3837	1.39327	0.696636	+	0.717425i	$0.254679\pi$
$80$	0	0
$81$	−8.60210	−0.955789
$82$	0	0
$83$	3.18153	0.349218	0.174609	−	0.984638i	$-0.444134\pi$
$83$	3.18153	0.349218	0.174609	+	0.984638i	$0.444134\pi$
$84$	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	$-0.436130\pi$
$103$	4.04555	0.398620	0.199310	+	0.979937i	$0.436130\pi$
$104$	0	0
$105$	−6.42863	−0.627370
$106$	0	0
$107$	−0.256771	−0.0248230	−0.0124115	−	0.999923i	$-0.503951\pi$
$107$	−0.256771	−0.0248230	−0.0124115	+	0.999923i	$0.503951\pi$
$108$	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	$-0.406084\pi$
$127$	6.55390	0.581565	0.290782	+	0.956789i	$0.406084\pi$
$128$	0	0
$129$	21.3850	1.88284
$130$	0	0
$131$	14.8032	1.29336	0.646679	−	0.762762i	$-0.276157\pi$
$131$	14.8032	1.29336	0.646679	+	0.762762i	$0.276157\pi$
$132$	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	$-0.396655\pi$
$139$	7.52176	0.637987	0.318994	+	0.947757i	$0.396655\pi$
$140$	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	$-0.303702\pi$
$151$	14.2134	1.15667	0.578337	+	0.815798i	$0.303702\pi$
$152$	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	$-0.308091\pi$
$163$	14.4788	1.13407	0.567034	+	0.823694i	$0.308091\pi$
$164$	0	0
$165$	27.8617	2.16903
$166$	0	0
$167$	−16.6444	−1.28798	−0.643991	−	0.765033i	$-0.722723\pi$
$167$	−16.6444	−1.28798	−0.643991	+	0.765033i	$0.722723\pi$
$168$	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	$-0.458904\pi$
$179$	3.44508	0.257497	0.128749	+	0.991677i	$0.458904\pi$
$180$	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	$-0.402383\pi$
$191$	8.34438	0.603778	0.301889	+	0.953343i	$0.402383\pi$
$192$	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	$-0.438828\pi$
$199$	5.38865	0.381992	0.190996	+	0.981591i	$0.438828\pi$
$200$	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	$-0.493909\pi$
$211$	0.555853	0.0382665	0.0191332	+	0.999817i	$0.493909\pi$
$212$	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	$-0.321823\pi$
$223$	15.8585	1.06197	0.530983	+	0.847382i	$0.321823\pi$
$224$	0	0
$225$	5.45654	0.363769
$226$	0	0
$227$	−14.4882	−0.961617	−0.480808	−	0.876826i	$-0.659657\pi$
$227$	−14.4882	−0.961617	−0.480808	+	0.876826i	$0.659657\pi$
$228$	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	$-0.312730\pi$
$239$	17.1592	1.10994	0.554970	+	0.831871i	$0.312730\pi$
$240$	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	$-0.431978\pi$
$251$	6.71977	0.424148	0.212074	+	0.977254i	$0.431978\pi$
$252$	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	$-0.362069\pi$
$263$	13.6189	0.839776	0.419888	+	0.907576i	$0.362069\pi$
$264$	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	$-0.757999\pi$
$271$	−23.8585	−1.44930	−0.724651	+	0.689116i	$0.757999\pi$
$272$	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	$-0.236725\pi$
$283$	24.7620	1.47195	0.735973	+	0.677011i	$0.236725\pi$
$284$	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	$-0.485916\pi$
$307$	1.54995	0.0884604	0.0442302	+	0.999021i	$0.485916\pi$
$308$	0	0
$309$	−10.0141	−0.569680
$310$	0	0
$311$	−4.19899	−0.238103	−0.119052	−	0.992888i	$-0.537985\pi$
$311$	−4.19899	−0.238103	−0.119052	+	0.992888i	$0.537985\pi$
$312$	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	$-0.207451\pi$
$331$	28.9289	1.59008	0.795038	+	0.606560i	$0.207451\pi$
$332$	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	$-0.520784\pi$
$347$	−2.43095	−0.130500	−0.0652500	+	0.997869i	$0.520784\pi$
$348$	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	$-0.583975\pi$
$359$	−9.88158	−0.521530	−0.260765	+	0.965402i	$0.583975\pi$
$360$	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	$-0.683231\pi$
$367$	−20.8573	−1.08874	−0.544370	+	0.838845i	$0.683231\pi$
$368$	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	$-0.697543\pi$
$379$	−22.6421	−1.16305	−0.581524	+	0.813529i	$0.697543\pi$
$380$	0	0
$381$	−16.2230	−0.831132
$382$	0	0
$383$	−22.7924	−1.16463	−0.582317	−	0.812962i	$-0.697854\pi$
$383$	−22.7924	−1.16463	−0.582317	+	0.812962i	$0.697854\pi$
$384$	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	$-0.542787\pi$
$419$	−5.48647	−0.268031	−0.134016	+	0.990979i	$0.542787\pi$
$420$	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	$-0.664898\pi$
$431$	−20.5604	−0.990359	−0.495179	+	0.868791i	$0.664898\pi$
$432$	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	$-0.387407\pi$
$439$	14.5154	0.692784	0.346392	+	0.938090i	$0.387407\pi$
$440$	0	0
$441$	3.12724	0.148916
$442$	0	0
$443$	−8.22851	−0.390948	−0.195474	−	0.980709i	$-0.562625\pi$
$443$	−8.22851	−0.390948	−0.195474	+	0.980709i	$0.562625\pi$
$444$	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	$-0.601227\pi$
$463$	−13.4561	−0.625359	−0.312679	+	0.949859i	$0.601227\pi$
$464$	0	0
$465$	6.71323	0.311319
$466$	0	0
$467$	16.2660	0.752703	0.376351	−	0.926477i	$-0.377179\pi$
$467$	16.2660	0.752703	0.376351	+	0.926477i	$0.377179\pi$
$468$	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	$-0.510532\pi$
$479$	−1.44806	−0.0661634	−0.0330817	+	0.999453i	$0.510532\pi$
$480$	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	$-0.272748\pi$
$487$	28.9008	1.30962	0.654811	+	0.755793i	$0.272748\pi$
$488$	0	0
$489$	−35.8398	−1.62073
$490$	0	0
$491$	2.84168	0.128243	0.0641217	−	0.997942i	$-0.479575\pi$
$491$	2.84168	0.128243	0.0641217	+	0.997942i	$0.479575\pi$
$492$	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	$-0.460056\pi$
$499$	5.59165	0.250317	0.125158	+	0.992137i	$0.460056\pi$
$500$	0	0
$501$	41.2003	1.84070
$502$	0	0
$503$	37.3662	1.66608	0.833038	−	0.553215i	$-0.186599\pi$
$503$	37.3662	1.66608	0.833038	+	0.553215i	$0.186599\pi$
$504$	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	$-0.582816\pi$
$523$	−11.7662	−0.514500	−0.257250	+	0.966345i	$0.582816\pi$
$524$	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	$-0.661378\pi$
$547$	−22.7117	−0.971084	−0.485542	+	0.874213i	$0.661378\pi$
$548$	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	$-0.362529\pi$
$563$	19.8636	0.837153	0.418576	+	0.908182i	$0.362529\pi$
$564$	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	$-0.160702\pi$
$571$	41.8289	1.75048	0.875242	+	0.483686i	$0.160702\pi$
$572$	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	$-0.408806\pi$
$587$	13.6933	0.565181	0.282590	+	0.959241i	$0.408806\pi$
$588$	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	$-0.365985\pi$
$599$	20.0051	0.817387	0.408693	+	0.912672i	$0.365985\pi$
$600$	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	$-0.738855\pi$
$607$	−33.6015	−1.36384	−0.681921	+	0.731425i	$0.738855\pi$
$608$	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	$-0.634030\pi$
$619$	−20.3384	−0.817470	−0.408735	+	0.912653i	$0.634030\pi$
$620$	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	$-0.171598\pi$
$631$	43.1143	1.71635	0.858176	+	0.513355i	$0.171598\pi$
$632$	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	$-0.701635\pi$
$643$	−30.0199	−1.18387	−0.591934	+	0.805987i	$0.701635\pi$
$644$	0	0
$645$	55.5385	2.18683
$646$	0	0
$647$	11.3001	0.444253	0.222127	−	0.975018i	$-0.428700\pi$
$647$	11.3001	0.444253	0.222127	+	0.975018i	$0.428700\pi$
$648$	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	$-0.577438\pi$
$659$	−12.3676	−0.481774	−0.240887	+	0.970553i	$0.577438\pi$
$660$	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	$-0.367221\pi$
$683$	21.1763	0.810289	0.405145	+	0.914253i	$0.367221\pi$
$684$	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	$-0.635504\pi$
$691$	−21.7108	−0.825916	−0.412958	+	0.910750i	$0.635504\pi$
$692$	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	$-0.667407\pi$
$719$	−26.9222	−1.00403	−0.502014	+	0.864860i	$0.667407\pi$
$720$	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	$-0.819719\pi$
$727$	−45.5056	−1.68771	−0.843854	+	0.536572i	$0.819719\pi$
$728$	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	$-0.226775\pi$
$739$	41.1451	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$740$	0	0
$741$	−85.6701	−3.14717
$742$	0	0
$743$	7.92011	0.290561	0.145280	−	0.989391i	$-0.453592\pi$
$743$	7.92011	0.290561	0.145280	+	0.989391i	$0.453592\pi$
$744$	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	$-0.510827\pi$
$751$	−1.86395	−0.0680164	−0.0340082	+	0.999422i	$0.510827\pi$
$752$	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	$-0.773095\pi$
$787$	−42.4453	−1.51301	−0.756505	+	0.653988i	$0.773095\pi$
$788$	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	$-0.384883\pi$
$811$	20.1521	0.707636	0.353818	+	0.935314i	$0.384883\pi$
$812$	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	$-0.350528\pi$
$823$	25.9633	0.905023	0.452511	+	0.891759i	$0.350528\pi$
$824$	0	0
$825$	18.7188	0.651704
$826$	0	0
$827$	−5.51645	−0.191826	−0.0959129	−	0.995390i	$-0.530577\pi$
$827$	−5.51645	−0.191826	−0.0959129	+	0.995390i	$0.530577\pi$
$828$	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	$-0.611618\pi$
$839$	−19.9002	−0.687030	−0.343515	+	0.939147i	$0.611618\pi$
$840$	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	$-0.355154\pi$
$859$	25.7625	0.879007	0.439503	+	0.898241i	$0.355154\pi$
$860$	0	0
$861$	−19.6962	−0.671244
$862$	0	0
$863$	−3.83270	−0.130467	−0.0652334	−	0.997870i	$-0.520779\pi$
$863$	−3.83270	−0.130467	−0.0652334	+	0.997870i	$0.520779\pi$
$864$	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	$-0.509220\pi$
$883$	−1.72113	−0.0579205	−0.0289603	+	0.999581i	$0.509220\pi$
$884$	0	0
$885$	82.0799	2.75909
$886$	0	0
$887$	3.31810	0.111411	0.0557054	−	0.998447i	$-0.482259\pi$
$887$	3.31810	0.111411	0.0557054	+	0.998447i	$0.482259\pi$
$888$	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	$-0.858525\pi$
$907$	−54.3809	−1.80569	−0.902845	+	0.429967i	$0.858525\pi$
$908$	0	0
$909$	44.5357	1.47716
$910$	0	0
$911$	17.5771	0.582355	0.291178	−	0.956669i	$-0.405953\pi$
$911$	17.5771	0.582355	0.291178	+	0.956669i	$0.405953\pi$
$912$	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	$-0.671217\pi$
$919$	−31.0625	−1.02466	−0.512328	+	0.858790i	$0.671217\pi$
$920$	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	$-0.493851\pi$
$947$	1.18879	0.0386304	0.0193152	+	0.999813i	$0.493851\pi$
$948$	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	$-0.367224\pi$
$967$	25.1967	0.810272	0.405136	+	0.914256i	$0.367224\pi$
$968$	0	0
$969$	−91.0639	−2.92539
$970$	0	0
$971$	38.2008	1.22592	0.612961	−	0.790113i	$-0.289978\pi$
$971$	38.2008	1.22592	0.612961	+	0.790113i	$0.289978\pi$
$972$	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	$-0.421448\pi$
$983$	15.3177	0.488560	0.244280	+	0.969705i	$0.421448\pi$
$984$	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	$-0.550208\pi$
$991$	−9.88980	−0.314160	−0.157080	+	0.987586i	$0.550208\pi$
$992$	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.p.1.3	yes	10
4.3	odd	2		3584.2.a.o.1.8	yes	10
8.3	odd	2		3584.2.a.o.1.3	✓	10
8.5	even	2	inner	3584.2.a.p.1.8	yes	10
16.3	odd	4		3584.2.b.h.1793.8		10
16.5	even	4		3584.2.b.g.1793.8		10
16.11	odd	4		3584.2.b.h.1793.3		10
16.13	even	4		3584.2.b.g.1793.3		10

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