LMFDB - Embedded newform 8512.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.25857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 9x + 3 $ x^4 - x^3 - 8x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.52280$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{3} +2.82557 q^{5} -1.00000 q^{7} -1.30278 q^{9} +O(q^{10})$	$q-1.30278 q^{3} +2.82557 q^{5} -1.00000 q^{7} -1.30278 q^{9} -4.50666 q^{11} +0.158290 q^{13} -3.68109 q^{15} +2.68109 q^{17} +1.00000 q^{19} +1.30278 q^{21} -2.15829 q^{23} +2.98386 q^{25} +5.60555 q^{27} -0.742819 q^{29} -3.63549 q^{31} +5.87117 q^{33} -2.82557 q^{35} +6.18775 q^{37} -0.206217 q^{39} +1.25718 q^{41} +10.3184 q^{43} -3.68109 q^{45} -3.66728 q^{47} +1.00000 q^{49} -3.49286 q^{51} +8.34837 q^{53} -12.7339 q^{55} -1.30278 q^{57} -1.58220 q^{59} -4.71288 q^{61} +1.30278 q^{63} +0.447261 q^{65} +4.99767 q^{67} +2.81177 q^{69} -11.5438 q^{71} +10.7795 q^{73} -3.88730 q^{75} +4.50666 q^{77} -5.10733 q^{79} -3.39445 q^{81} +15.3850 q^{83} +7.57561 q^{85} +0.967727 q^{87} -11.4767 q^{89} -0.158290 q^{91} +4.73623 q^{93} +2.82557 q^{95} +7.83279 q^{97} +5.87117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - q^5 - 4 * q^7 + 2 * q^9	$ 4 q + 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 7 q^{15} + 3 q^{17} + 4 q^{19} - 2 q^{21} - 6 q^{23} - 3 q^{25} + 8 q^{27} - 17 q^{31} + q^{33} + q^{35} - 3 q^{37} - q^{39} + 8 q^{41} + 7 q^{43} - 7 q^{45} - 5 q^{47} + 4 q^{49} + 8 q^{51} + 16 q^{53} - 17 q^{55} + 2 q^{57} + 7 q^{59} + q^{61} - 2 q^{63} - 10 q^{65} + 7 q^{67} - 3 q^{69} - 27 q^{71} - q^{73} - 8 q^{75} - 2 q^{77} - 15 q^{79} - 28 q^{81} + 3 q^{83} - q^{85} - 26 q^{87} - 9 q^{89} + 2 q^{91} - 2 q^{93} - q^{95} + 3 q^{97} + q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - q^5 - 4 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 - 7 * q^15 + 3 * q^17 + 4 * q^19 - 2 * q^21 - 6 * q^23 - 3 * q^25 + 8 * q^27 - 17 * q^31 + q^33 + q^35 - 3 * q^37 - q^39 + 8 * q^41 + 7 * q^43 - 7 * q^45 - 5 * q^47 + 4 * q^49 + 8 * q^51 + 16 * q^53 - 17 * q^55 + 2 * q^57 + 7 * q^59 + q^61 - 2 * q^63 - 10 * q^65 + 7 * q^67 - 3 * q^69 - 27 * q^71 - q^73 - 8 * q^75 - 2 * q^77 - 15 * q^79 - 28 * q^81 + 3 * q^83 - q^85 - 26 * q^87 - 9 * q^89 + 2 * q^91 - 2 * q^93 - q^95 + 3 * q^97 + q^99

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
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	0	0
$5$	2.82557	1.26363	0.631817	−	0.775117i	$-0.282309\pi$
$5$	2.82557	1.26363	0.631817	+	0.775117i	$0.282309\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	−4.50666	−1.35881	−0.679405	−	0.733764i	$-0.737762\pi$
$11$	−4.50666	−1.35881	−0.679405	+	0.733764i	$0.737762\pi$
$12$	0	0
$13$	0.158290	0.0439018	0.0219509	−	0.999759i	$-0.493012\pi$
$13$	0.158290	0.0439018	0.0219509	+	0.999759i	$0.493012\pi$
$14$	0	0
$15$	−3.68109	−0.950453
$16$	0	0
$17$	2.68109	0.650259	0.325130	−	0.945669i	$-0.394592\pi$
$17$	2.68109	0.650259	0.325130	+	0.945669i	$0.394592\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.30278	0.284289
$22$	0	0
$23$	−2.15829	−0.450035	−0.225017	−	0.974355i	$-0.572244\pi$
$23$	−2.15829	−0.450035	−0.225017	+	0.974355i	$0.572244\pi$
$24$	0	0
$25$	2.98386	0.596773
$26$	0	0
$27$	5.60555	1.07879
$28$	0	0
$29$	−0.742819	−0.137938	−0.0689690	−	0.997619i	$-0.521971\pi$
$29$	−0.742819	−0.137938	−0.0689690	+	0.997619i	$0.521971\pi$
$30$	0	0
$31$	−3.63549	−0.652954	−0.326477	−	0.945205i	$-0.605862\pi$
$31$	−3.63549	−0.652954	−0.326477	+	0.945205i	$0.605862\pi$
$32$	0	0
$33$	5.87117	1.02204
$34$	0	0
$35$	−2.82557	−0.477609
$36$	0	0
$37$	6.18775	1.01726	0.508630	−	0.860985i	$-0.330152\pi$
$37$	6.18775	1.01726	0.508630	+	0.860985i	$0.330152\pi$
$38$	0	0
$39$	−0.206217	−0.0330211
$40$	0	0
$41$	1.25718	0.196339	0.0981693	−	0.995170i	$-0.468701\pi$
$41$	1.25718	0.196339	0.0981693	+	0.995170i	$0.468701\pi$
$42$	0	0
$43$	10.3184	1.57355	0.786773	−	0.617243i	$-0.211750\pi$
$43$	10.3184	1.57355	0.786773	+	0.617243i	$0.211750\pi$
$44$	0	0
$45$	−3.68109	−0.548744
$46$	0	0
$47$	−3.66728	−0.534928	−0.267464	−	0.963568i	$-0.586186\pi$
$47$	−3.66728	−0.534928	−0.267464	+	0.963568i	$0.586186\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.49286	−0.489098
$52$	0	0
$53$	8.34837	1.14674	0.573368	−	0.819298i	$-0.305636\pi$
$53$	8.34837	1.14674	0.573368	+	0.819298i	$0.305636\pi$
$54$	0	0
$55$	−12.7339	−1.71704
$56$	0	0
$57$	−1.30278	−0.172557
$58$	0	0
$59$	−1.58220	−0.205985	−0.102992	−	0.994682i	$-0.532842\pi$
$59$	−1.58220	−0.205985	−0.102992	+	0.994682i	$0.532842\pi$
$60$	0	0
$61$	−4.71288	−0.603422	−0.301711	−	0.953399i	$-0.597558\pi$
$61$	−4.71288	−0.603422	−0.301711	+	0.953399i	$0.597558\pi$
$62$	0	0
$63$	1.30278	0.164134
$64$	0	0
$65$	0.447261	0.0554759
$66$	0	0
$67$	4.99767	0.610562	0.305281	−	0.952262i	$-0.401250\pi$
$67$	4.99767	0.610562	0.305281	+	0.952262i	$0.401250\pi$
$68$	0	0
$69$	2.81177	0.338497
$70$	0	0
$71$	−11.5438	−1.37000	−0.685000	−	0.728543i	$-0.740198\pi$
$71$	−11.5438	−1.37000	−0.685000	+	0.728543i	$0.740198\pi$
$72$	0	0
$73$	10.7795	1.26164	0.630822	−	0.775927i	$-0.282718\pi$
$73$	10.7795	1.26164	0.630822	+	0.775927i	$0.282718\pi$
$74$	0	0
$75$	−3.88730	−0.448867
$76$	0	0
$77$	4.50666	0.513582
$78$	0	0
$79$	−5.10733	−0.574619	−0.287310	−	0.957838i	$-0.592761\pi$
$79$	−5.10733	−0.574619	−0.287310	+	0.957838i	$0.592761\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	15.3850	1.68873	0.844364	−	0.535770i	$-0.179979\pi$
$83$	15.3850	1.68873	0.844364	+	0.535770i	$0.179979\pi$
$84$	0	0
$85$	7.57561	0.821690
$86$	0	0
$87$	0.967727	0.103751
$88$	0	0
$89$	−11.4767	−1.21653	−0.608265	−	0.793734i	$-0.708134\pi$
$89$	−11.4767	−1.21653	−0.608265	+	0.793734i	$0.708134\pi$
$90$	0	0
$91$	−0.158290	−0.0165933
$92$	0	0
$93$	4.73623	0.491124
$94$	0	0
$95$	2.82557	0.289898
$96$	0	0
$97$	7.83279	0.795299	0.397650	−	0.917537i	$-0.369826\pi$
$97$	7.83279	0.795299	0.397650	+	0.917537i	$0.369826\pi$
$98$	0	0
$99$	5.87117	0.590075
$100$	0	0
$101$	−0.763842	−0.0760051	−0.0380025	−	0.999278i	$-0.512099\pi$
$101$	−0.763842	−0.0760051	−0.0380025	+	0.999278i	$0.512099\pi$
$102$	0	0
$103$	−17.7772	−1.75164	−0.875818	−	0.482642i	$-0.839677\pi$
$103$	−17.7772	−1.75164	−0.875818	+	0.482642i	$0.839677\pi$
$104$	0	0
$105$	3.68109	0.359237
$106$	0	0
$107$	−6.00722	−0.580740	−0.290370	−	0.956915i	$-0.593778\pi$
$107$	−6.00722	−0.580740	−0.290370	+	0.956915i	$0.593778\pi$
$108$	0	0
$109$	−2.49822	−0.239287	−0.119643	−	0.992817i	$-0.538175\pi$
$109$	−2.49822	−0.239287	−0.119643	+	0.992817i	$0.538175\pi$
$110$	0	0
$111$	−8.06125	−0.765140
$112$	0	0
$113$	−16.2651	−1.53010	−0.765048	−	0.643974i	$-0.777285\pi$
$113$	−16.2651	−1.53010	−0.765048	+	0.643974i	$0.777285\pi$
$114$	0	0
$115$	−6.09841	−0.568679
$116$	0	0
$117$	−0.206217	−0.0190647
$118$	0	0
$119$	−2.68109	−0.245775
$120$	0	0
$121$	9.30999	0.846363
$122$	0	0
$123$	−1.63782	−0.147678
$124$	0	0
$125$	−5.69674	−0.509532
$126$	0	0
$127$	1.79330	0.159130	0.0795648	−	0.996830i	$-0.474647\pi$
$127$	1.79330	0.159130	0.0795648	+	0.996830i	$0.474647\pi$
$128$	0	0
$129$	−13.4426	−1.18355
$130$	0	0
$131$	−13.6032	−1.18852	−0.594259	−	0.804273i	$-0.702555\pi$
$131$	−13.6032	−1.18852	−0.594259	+	0.804273i	$0.702555\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	15.8389	1.36320
$136$	0	0
$137$	−15.4239	−1.31775	−0.658877	−	0.752251i	$-0.728968\pi$
$137$	−15.4239	−1.31775	−0.658877	+	0.752251i	$0.728968\pi$
$138$	0	0
$139$	−14.6296	−1.24087	−0.620435	−	0.784258i	$-0.713044\pi$
$139$	−14.6296	−1.24087	−0.620435	+	0.784258i	$0.713044\pi$
$140$	0	0
$141$	4.77765	0.402350
$142$	0	0
$143$	−0.713361	−0.0596542
$144$	0	0
$145$	−2.09889	−0.174303
$146$	0	0
$147$	−1.30278	−0.107451
$148$	0	0
$149$	4.35607	0.356863	0.178432	−	0.983952i	$-0.442898\pi$
$149$	4.35607	0.356863	0.178432	+	0.983952i	$0.442898\pi$
$150$	0	0
$151$	6.68109	0.543699	0.271850	−	0.962340i	$-0.412365\pi$
$151$	6.68109	0.543699	0.271850	+	0.962340i	$0.412365\pi$
$152$	0	0
$153$	−3.49286	−0.282381
$154$	0	0
$155$	−10.2724	−0.825095
$156$	0	0
$157$	−3.75004	−0.299286	−0.149643	−	0.988740i	$-0.547812\pi$
$157$	−3.75004	−0.299286	−0.149643	+	0.988740i	$0.547812\pi$
$158$	0	0
$159$	−10.8761	−0.862527
$160$	0	0
$161$	2.15829	0.170097
$162$	0	0
$163$	−21.3761	−1.67431	−0.837154	−	0.546968i	$-0.815782\pi$
$163$	−21.3761	−1.67431	−0.837154	+	0.546968i	$0.815782\pi$
$164$	0	0
$165$	16.5894	1.29148
$166$	0	0
$167$	−5.96051	−0.461238	−0.230619	−	0.973044i	$-0.574075\pi$
$167$	−5.96051	−0.461238	−0.230619	+	0.973044i	$0.574075\pi$
$168$	0	0
$169$	−12.9749	−0.998073
$170$	0	0
$171$	−1.30278	−0.0996257
$172$	0	0
$173$	−7.32990	−0.557282	−0.278641	−	0.960395i	$-0.589884\pi$
$173$	−7.32990	−0.557282	−0.278641	+	0.960395i	$0.589884\pi$
$174$	0	0
$175$	−2.98386	−0.225559
$176$	0	0
$177$	2.06125	0.154933
$178$	0	0
$179$	5.06125	0.378295	0.189148	−	0.981949i	$-0.439427\pi$
$179$	5.06125	0.378295	0.189148	+	0.981949i	$0.439427\pi$
$180$	0	0
$181$	−21.6944	−1.61253	−0.806266	−	0.591553i	$-0.798515\pi$
$181$	−21.6944	−1.61253	−0.806266	+	0.591553i	$0.798515\pi$
$182$	0	0
$183$	6.13982	0.453869
$184$	0	0
$185$	17.4839	1.28544
$186$	0	0
$187$	−12.0828	−0.883578
$188$	0	0
$189$	−5.60555	−0.407744
$190$	0	0
$191$	−5.88497	−0.425822	−0.212911	−	0.977072i	$-0.568294\pi$
$191$	−5.88497	−0.425822	−0.212911	+	0.977072i	$0.568294\pi$
$192$	0	0
$193$	11.6345	0.837472	0.418736	−	0.908108i	$-0.362473\pi$
$193$	11.6345	0.837472	0.418736	+	0.908108i	$0.362473\pi$
$194$	0	0
$195$	−0.582681	−0.0417266
$196$	0	0
$197$	−5.21954	−0.371877	−0.185938	−	0.982561i	$-0.559532\pi$
$197$	−5.21954	−0.371877	−0.185938	+	0.982561i	$0.559532\pi$
$198$	0	0
$199$	−2.44541	−0.173351	−0.0866754	−	0.996237i	$-0.527624\pi$
$199$	−2.44541	−0.173351	−0.0866754	+	0.996237i	$0.527624\pi$
$200$	0	0
$201$	−6.51084	−0.459239
$202$	0	0
$203$	0.742819	0.0521357
$204$	0	0
$205$	3.55226	0.248100
$206$	0	0
$207$	2.81177	0.195431
$208$	0	0
$209$	−4.50666	−0.311732
$210$	0	0
$211$	11.8527	0.815974	0.407987	−	0.912988i	$-0.366231\pi$
$211$	11.8527	0.815974	0.407987	+	0.912988i	$0.366231\pi$
$212$	0	0
$213$	15.0390	1.03046
$214$	0	0
$215$	29.1555	1.98839
$216$	0	0
$217$	3.63549	0.246793
$218$	0	0
$219$	−14.0433	−0.948956
$220$	0	0
$221$	0.424390	0.0285476
$222$	0	0
$223$	−2.79611	−0.187242	−0.0936208	−	0.995608i	$-0.529844\pi$
$223$	−2.79611	−0.187242	−0.0936208	+	0.995608i	$0.529844\pi$
$224$	0	0
$225$	−3.88730	−0.259154
$226$	0	0
$227$	10.1345	0.672647	0.336324	−	0.941746i	$-0.390816\pi$
$227$	10.1345	0.672647	0.336324	+	0.941746i	$0.390816\pi$
$228$	0	0
$229$	−5.86395	−0.387501	−0.193750	−	0.981051i	$-0.562065\pi$
$229$	−5.86395	−0.387501	−0.193750	+	0.981051i	$0.562065\pi$
$230$	0	0
$231$	−5.87117	−0.386294
$232$	0	0
$233$	19.4278	1.27276	0.636380	−	0.771376i	$-0.280431\pi$
$233$	19.4278	1.27276	0.636380	+	0.771376i	$0.280431\pi$
$234$	0	0
$235$	−10.3622	−0.675954
$236$	0	0
$237$	6.65370	0.432204
$238$	0	0
$239$	−18.2495	−1.18046	−0.590230	−	0.807235i	$-0.700963\pi$
$239$	−18.2495	−1.18046	−0.590230	+	0.807235i	$0.700963\pi$
$240$	0	0
$241$	20.8917	1.34575	0.672877	−	0.739755i	$-0.265058\pi$
$241$	20.8917	1.34575	0.672877	+	0.739755i	$0.265058\pi$
$242$	0	0
$243$	−12.3944	−0.795104
$244$	0	0
$245$	2.82557	0.180519
$246$	0	0
$247$	0.158290	0.0100718
$248$	0	0
$249$	−20.0433	−1.27019
$250$	0	0
$251$	−6.31780	−0.398776	−0.199388	−	0.979921i	$-0.563895\pi$
$251$	−6.31780	−0.398776	−0.199388	+	0.979921i	$0.563895\pi$
$252$	0	0
$253$	9.72668	0.611511
$254$	0	0
$255$	−9.86932	−0.618041
$256$	0	0
$257$	−25.6076	−1.59736	−0.798680	−	0.601756i	$-0.794468\pi$
$257$	−25.6076	−1.59736	−0.798680	+	0.601756i	$0.794468\pi$
$258$	0	0
$259$	−6.18775	−0.384488
$260$	0	0
$261$	0.967727	0.0599008
$262$	0	0
$263$	26.8707	1.65692	0.828459	−	0.560050i	$-0.189218\pi$
$263$	26.8707	1.65692	0.828459	+	0.560050i	$0.189218\pi$
$264$	0	0
$265$	23.5889	1.44906
$266$	0	0
$267$	14.9516	0.915022
$268$	0	0
$269$	−11.2939	−0.688599	−0.344299	−	0.938860i	$-0.611883\pi$
$269$	−11.2939	−0.688599	−0.344299	+	0.938860i	$0.611883\pi$
$270$	0	0
$271$	7.90721	0.480329	0.240165	−	0.970732i	$-0.422799\pi$
$271$	7.90721	0.480329	0.240165	+	0.970732i	$0.422799\pi$
$272$	0	0
$273$	0.206217	0.0124808
$274$	0	0
$275$	−13.4473	−0.810900
$276$	0	0
$277$	15.0133	0.902063	0.451032	−	0.892508i	$-0.351056\pi$
$277$	15.0133	0.902063	0.451032	+	0.892508i	$0.351056\pi$
$278$	0	0
$279$	4.73623	0.283551
$280$	0	0
$281$	−0.768505	−0.0458451	−0.0229226	−	0.999737i	$-0.507297\pi$
$281$	−0.768505	−0.0458451	−0.0229226	+	0.999737i	$0.507297\pi$
$282$	0	0
$283$	0.495187	0.0294358	0.0147179	−	0.999892i	$-0.495315\pi$
$283$	0.495187	0.0294358	0.0147179	+	0.999892i	$0.495315\pi$
$284$	0	0
$285$	−3.68109	−0.218049
$286$	0	0
$287$	−1.25718	−0.0742090
$288$	0	0
$289$	−9.81177	−0.577163
$290$	0	0
$291$	−10.2044	−0.598191
$292$	0	0
$293$	16.5062	0.964301	0.482151	−	0.876088i	$-0.339856\pi$
$293$	16.5062	0.964301	0.482151	+	0.876088i	$0.339856\pi$
$294$	0	0
$295$	−4.47061	−0.260289
$296$	0	0
$297$	−25.2623	−1.46587
$298$	0	0
$299$	−0.341637	−0.0197573
$300$	0	0
$301$	−10.3184	−0.594744
$302$	0	0
$303$	0.995114	0.0571678
$304$	0	0
$305$	−13.3166	−0.762505
$306$	0	0
$307$	−15.4816	−0.883582	−0.441791	−	0.897118i	$-0.645657\pi$
$307$	−15.4816	−0.883582	−0.441791	+	0.897118i	$0.645657\pi$
$308$	0	0
$309$	23.1597	1.31751
$310$	0	0
$311$	−32.4145	−1.83806	−0.919029	−	0.394190i	$-0.871025\pi$
$311$	−32.4145	−1.83806	−0.919029	+	0.394190i	$0.871025\pi$
$312$	0	0
$313$	−29.4944	−1.66712	−0.833562	−	0.552426i	$-0.813702\pi$
$313$	−29.4944	−1.66712	−0.833562	+	0.552426i	$0.813702\pi$
$314$	0	0
$315$	3.68109	0.207406
$316$	0	0
$317$	−20.6832	−1.16168	−0.580841	−	0.814017i	$-0.697276\pi$
$317$	−20.6832	−1.16168	−0.580841	+	0.814017i	$0.697276\pi$
$318$	0	0
$319$	3.34763	0.187432
$320$	0	0
$321$	7.82606	0.436808
$322$	0	0
$323$	2.68109	0.149180
$324$	0	0
$325$	0.472317	0.0261994
$326$	0	0
$327$	3.25463	0.179981
$328$	0	0
$329$	3.66728	0.202184
$330$	0	0
$331$	10.9773	0.603366	0.301683	−	0.953408i	$-0.402452\pi$
$331$	10.9773	0.603366	0.301683	+	0.953408i	$0.402452\pi$
$332$	0	0
$333$	−8.06125	−0.441754
$334$	0	0
$335$	14.1213	0.771528
$336$	0	0
$337$	30.2677	1.64879	0.824393	−	0.566018i	$-0.191517\pi$
$337$	30.2677	1.64879	0.824393	+	0.566018i	$0.191517\pi$
$338$	0	0
$339$	21.1898	1.15087
$340$	0	0
$341$	16.3839	0.887240
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.94486	0.427737
$346$	0	0
$347$	29.7283	1.59590	0.797949	−	0.602726i	$-0.205919\pi$
$347$	29.7283	1.59590	0.797949	+	0.602726i	$0.205919\pi$
$348$	0	0
$349$	−27.4449	−1.46909	−0.734547	−	0.678558i	$-0.762605\pi$
$349$	−27.4449	−1.46909	−0.734547	+	0.678558i	$0.762605\pi$
$350$	0	0
$351$	0.887305	0.0473608
$352$	0	0
$353$	4.45570	0.237153	0.118576	−	0.992945i	$-0.462167\pi$
$353$	4.45570	0.237153	0.118576	+	0.992945i	$0.462167\pi$
$354$	0	0
$355$	−32.6179	−1.73118
$356$	0	0
$357$	3.49286	0.184862
$358$	0	0
$359$	−27.9655	−1.47596	−0.737982	−	0.674820i	$-0.764221\pi$
$359$	−27.9655	−1.47596	−0.737982	+	0.674820i	$0.764221\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.1288	−0.636599
$364$	0	0
$365$	30.4583	1.59426
$366$	0	0
$367$	10.6038	0.553516	0.276758	−	0.960940i	$-0.410740\pi$
$367$	10.6038	0.553516	0.276758	+	0.960940i	$0.410740\pi$
$368$	0	0
$369$	−1.63782	−0.0852617
$370$	0	0
$371$	−8.34837	−0.433426
$372$	0	0
$373$	8.57987	0.444249	0.222124	−	0.975018i	$-0.428701\pi$
$373$	8.57987	0.444249	0.222124	+	0.975018i	$0.428701\pi$
$374$	0	0
$375$	7.42158	0.383249
$376$	0	0
$377$	−0.117581	−0.00605574
$378$	0	0
$379$	15.1440	0.777895	0.388948	−	0.921260i	$-0.372839\pi$
$379$	15.1440	0.777895	0.388948	+	0.921260i	$0.372839\pi$
$380$	0	0
$381$	−2.33627	−0.119691
$382$	0	0
$383$	−2.15829	−0.110283	−0.0551417	−	0.998479i	$-0.517561\pi$
$383$	−2.15829	−0.110283	−0.0551417	+	0.998479i	$0.517561\pi$
$384$	0	0
$385$	12.7339	0.648980
$386$	0	0
$387$	−13.4426	−0.683326
$388$	0	0
$389$	6.78671	0.344100	0.172050	−	0.985088i	$-0.444961\pi$
$389$	6.78671	0.344100	0.172050	+	0.985088i	$0.444961\pi$
$390$	0	0
$391$	−5.78657	−0.292639
$392$	0	0
$393$	17.7219	0.893954
$394$	0	0
$395$	−14.4311	−0.726109
$396$	0	0
$397$	10.4302	0.523475	0.261737	−	0.965139i	$-0.415705\pi$
$397$	10.4302	0.523475	0.261737	+	0.965139i	$0.415705\pi$
$398$	0	0
$399$	1.30278	0.0652204
$400$	0	0
$401$	21.9978	1.09852	0.549259	−	0.835652i	$-0.314910\pi$
$401$	21.9978	1.09852	0.549259	+	0.835652i	$0.314910\pi$
$402$	0	0
$403$	−0.575463	−0.0286659
$404$	0	0
$405$	−9.59126	−0.476594
$406$	0	0
$407$	−27.8861	−1.38226
$408$	0	0
$409$	8.24275	0.407578	0.203789	−	0.979015i	$-0.434674\pi$
$409$	8.24275	0.407578	0.203789	+	0.979015i	$0.434674\pi$
$410$	0	0
$411$	20.0939	0.991159
$412$	0	0
$413$	1.58220	0.0778548
$414$	0	0
$415$	43.4716	2.13394
$416$	0	0
$417$	19.0591	0.933330
$418$	0	0
$419$	2.22187	0.108545	0.0542727	−	0.998526i	$-0.482716\pi$
$419$	2.22187	0.108545	0.0542727	+	0.998526i	$0.482716\pi$
$420$	0	0
$421$	−39.0518	−1.90327	−0.951635	−	0.307230i	$-0.900598\pi$
$421$	−39.0518	−1.90327	−0.951635	+	0.307230i	$0.900598\pi$
$422$	0	0
$423$	4.77765	0.232297
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	4.71288	0.228072
$428$	0	0
$429$	0.929349	0.0448694
$430$	0	0
$431$	−7.53660	−0.363025	−0.181513	−	0.983389i	$-0.558099\pi$
$431$	−7.53660	−0.363025	−0.181513	+	0.983389i	$0.558099\pi$
$432$	0	0
$433$	−13.5367	−0.650535	−0.325267	−	0.945622i	$-0.605454\pi$
$433$	−13.5367	−0.650535	−0.325267	+	0.945622i	$0.605454\pi$
$434$	0	0
$435$	2.73438	0.131104
$436$	0	0
$437$	−2.15829	−0.103245
$438$	0	0
$439$	−16.3432	−0.780020	−0.390010	−	0.920811i	$-0.627528\pi$
$439$	−16.3432	−0.780020	−0.390010	+	0.920811i	$0.627528\pi$
$440$	0	0
$441$	−1.30278	−0.0620369
$442$	0	0
$443$	0.0523291	0.00248623	0.00124311	−	0.999999i	$-0.499604\pi$
$443$	0.0523291	0.00248623	0.00124311	+	0.999999i	$0.499604\pi$
$444$	0	0
$445$	−32.4283	−1.53725
$446$	0	0
$447$	−5.67498	−0.268417
$448$	0	0
$449$	−29.1105	−1.37381	−0.686905	−	0.726747i	$-0.741031\pi$
$449$	−29.1105	−1.37381	−0.686905	+	0.726747i	$0.741031\pi$
$450$	0	0
$451$	−5.66569	−0.266787
$452$	0	0
$453$	−8.70396	−0.408948
$454$	0	0
$455$	−0.447261	−0.0209679
$456$	0	0
$457$	25.3448	1.18558	0.592790	−	0.805357i	$-0.298026\pi$
$457$	25.3448	1.18558	0.592790	+	0.805357i	$0.298026\pi$
$458$	0	0
$459$	15.0290	0.701492
$460$	0	0
$461$	−3.84475	−0.179068	−0.0895339	−	0.995984i	$-0.528538\pi$
$461$	−3.84475	−0.179068	−0.0895339	+	0.995984i	$0.528538\pi$
$462$	0	0
$463$	−6.79611	−0.315842	−0.157921	−	0.987452i	$-0.550479\pi$
$463$	−6.79611	−0.315842	−0.157921	+	0.987452i	$0.550479\pi$
$464$	0	0
$465$	13.3826	0.620602
$466$	0	0
$467$	−2.44604	−0.113189	−0.0565947	−	0.998397i	$-0.518024\pi$
$467$	−2.44604	−0.113189	−0.0565947	+	0.998397i	$0.518024\pi$
$468$	0	0
$469$	−4.99767	−0.230771
$470$	0	0
$471$	4.88546	0.225110
$472$	0	0
$473$	−46.5017	−2.13815
$474$	0	0
$475$	2.98386	0.136909
$476$	0	0
$477$	−10.8761	−0.497980
$478$	0	0
$479$	9.02246	0.412247	0.206123	−	0.978526i	$-0.433915\pi$
$479$	9.02246	0.412247	0.206123	+	0.978526i	$0.433915\pi$
$480$	0	0
$481$	0.979461	0.0446596
$482$	0	0
$483$	−2.81177	−0.127940
$484$	0	0
$485$	22.1321	1.00497
$486$	0	0
$487$	−6.34348	−0.287451	−0.143725	−	0.989618i	$-0.545908\pi$
$487$	−6.34348	−0.287451	−0.143725	+	0.989618i	$0.545908\pi$
$488$	0	0
$489$	27.8483	1.25934
$490$	0	0
$491$	8.18238	0.369266	0.184633	−	0.982808i	$-0.440890\pi$
$491$	8.18238	0.369266	0.184633	+	0.982808i	$0.440890\pi$
$492$	0	0
$493$	−1.99156	−0.0896955
$494$	0	0
$495$	16.5894	0.745639
$496$	0	0
$497$	11.5438	0.517811
$498$	0	0
$499$	31.7264	1.42027	0.710135	−	0.704066i	$-0.248634\pi$
$499$	31.7264	1.42027	0.710135	+	0.704066i	$0.248634\pi$
$500$	0	0
$501$	7.76521	0.346924
$502$	0	0
$503$	10.9193	0.486868	0.243434	−	0.969917i	$-0.421726\pi$
$503$	10.9193	0.486868	0.243434	+	0.969917i	$0.421726\pi$
$504$	0	0
$505$	−2.15829	−0.0960427
$506$	0	0
$507$	16.9034	0.750708
$508$	0	0
$509$	−25.1358	−1.11413	−0.557063	−	0.830470i	$-0.688072\pi$
$509$	−25.1358	−1.11413	−0.557063	+	0.830470i	$0.688072\pi$
$510$	0	0
$511$	−10.7795	−0.476857
$512$	0	0
$513$	5.60555	0.247491
$514$	0	0
$515$	−50.2307	−2.21343
$516$	0	0
$517$	16.5272	0.726865
$518$	0	0
$519$	9.54922	0.419164
$520$	0	0
$521$	9.90529	0.433959	0.216979	−	0.976176i	$-0.430380\pi$
$521$	9.90529	0.433959	0.216979	+	0.976176i	$0.430380\pi$
$522$	0	0
$523$	14.6849	0.642124	0.321062	−	0.947058i	$-0.395960\pi$
$523$	14.6849	0.642124	0.321062	+	0.947058i	$0.395960\pi$
$524$	0	0
$525$	3.88730	0.169656
$526$	0	0
$527$	−9.74708	−0.424589
$528$	0	0
$529$	−18.3418	−0.797469
$530$	0	0
$531$	2.06125	0.0894506
$532$	0	0
$533$	0.199000	0.00861963
$534$	0	0
$535$	−16.9738	−0.733843
$536$	0	0
$537$	−6.59367	−0.284538
$538$	0	0
$539$	−4.50666	−0.194116
$540$	0	0
$541$	−39.7423	−1.70866	−0.854328	−	0.519734i	$-0.826031\pi$
$541$	−39.7423	−1.70866	−0.854328	+	0.519734i	$0.826031\pi$
$542$	0	0
$543$	28.2629	1.21288
$544$	0	0
$545$	−7.05892	−0.302371
$546$	0	0
$547$	−43.3794	−1.85477	−0.927385	−	0.374108i	$-0.877949\pi$
$547$	−43.3794	−1.85477	−0.927385	+	0.374108i	$0.877949\pi$
$548$	0	0
$549$	6.13982	0.262041
$550$	0	0
$551$	−0.742819	−0.0316452
$552$	0	0
$553$	5.10733	0.217186
$554$	0	0
$555$	−22.7776	−0.966857
$556$	0	0
$557$	−30.7666	−1.30362	−0.651812	−	0.758380i	$-0.725991\pi$
$557$	−30.7666	−1.30362	−0.651812	+	0.758380i	$0.725991\pi$
$558$	0	0
$559$	1.63331	0.0690816
$560$	0	0
$561$	15.7411	0.664590
$562$	0	0
$563$	37.1617	1.56618	0.783090	−	0.621908i	$-0.213642\pi$
$563$	37.1617	1.56618	0.783090	+	0.621908i	$0.213642\pi$
$564$	0	0
$565$	−45.9583	−1.93348
$566$	0	0
$567$	3.39445	0.142553
$568$	0	0
$569$	−2.06244	−0.0864619	−0.0432309	−	0.999065i	$-0.513765\pi$
$569$	−2.06244	−0.0864619	−0.0432309	+	0.999065i	$0.513765\pi$
$570$	0	0
$571$	−31.9553	−1.33729	−0.668644	−	0.743583i	$-0.733125\pi$
$571$	−31.9553	−1.33729	−0.668644	+	0.743583i	$0.733125\pi$
$572$	0	0
$573$	7.66680	0.320285
$574$	0	0
$575$	−6.44004	−0.268568
$576$	0	0
$577$	−21.0039	−0.874405	−0.437202	−	0.899363i	$-0.644031\pi$
$577$	−21.0039	−0.874405	−0.437202	+	0.899363i	$0.644031\pi$
$578$	0	0
$579$	−15.1572	−0.629911
$580$	0	0
$581$	−15.3850	−0.638279
$582$	0	0
$583$	−37.6233	−1.55820
$584$	0	0
$585$	−0.582681	−0.0240909
$586$	0	0
$587$	35.0998	1.44873	0.724363	−	0.689419i	$-0.242134\pi$
$587$	35.0998	1.44873	0.724363	+	0.689419i	$0.242134\pi$
$588$	0	0
$589$	−3.63549	−0.149798
$590$	0	0
$591$	6.79989	0.279710
$592$	0	0
$593$	−25.0575	−1.02899	−0.514494	−	0.857494i	$-0.672020\pi$
$593$	−25.0575	−1.02899	−0.514494	+	0.857494i	$0.672020\pi$
$594$	0	0
$595$	−7.57561	−0.310570
$596$	0	0
$597$	3.18582	0.130387
$598$	0	0
$599$	6.78745	0.277328	0.138664	−	0.990340i	$-0.455719\pi$
$599$	6.78745	0.277328	0.138664	+	0.990340i	$0.455719\pi$
$600$	0	0
$601$	36.8240	1.50208	0.751041	−	0.660256i	$-0.229552\pi$
$601$	36.8240	1.50208	0.751041	+	0.660256i	$0.229552\pi$
$602$	0	0
$603$	−6.51084	−0.265142
$604$	0	0
$605$	26.3061	1.06949
$606$	0	0
$607$	−11.3909	−0.462344	−0.231172	−	0.972913i	$-0.574256\pi$
$607$	−11.3909	−0.462344	−0.231172	+	0.972913i	$0.574256\pi$
$608$	0	0
$609$	−0.967727	−0.0392143
$610$	0	0
$611$	−0.580495	−0.0234843
$612$	0	0
$613$	34.0135	1.37379	0.686897	−	0.726755i	$-0.258972\pi$
$613$	34.0135	1.37379	0.686897	+	0.726755i	$0.258972\pi$
$614$	0	0
$615$	−4.62779	−0.186611
$616$	0	0
$617$	−12.6507	−0.509297	−0.254648	−	0.967034i	$-0.581960\pi$
$617$	−12.6507	−0.509297	−0.254648	+	0.967034i	$0.581960\pi$
$618$	0	0
$619$	7.31169	0.293882	0.146941	−	0.989145i	$-0.453057\pi$
$619$	7.31169	0.293882	0.146941	+	0.989145i	$0.453057\pi$
$620$	0	0
$621$	−12.0984	−0.485492
$622$	0	0
$623$	11.4767	0.459805
$624$	0	0
$625$	−31.0159	−1.24064
$626$	0	0
$627$	5.87117	0.234472
$628$	0	0
$629$	16.5899	0.661483
$630$	0	0
$631$	−26.3182	−1.04771	−0.523855	−	0.851808i	$-0.675507\pi$
$631$	−26.3182	−1.04771	−0.523855	+	0.851808i	$0.675507\pi$
$632$	0	0
$633$	−15.4414	−0.613741
$634$	0	0
$635$	5.06710	0.201082
$636$	0	0
$637$	0.158290	0.00627169
$638$	0	0
$639$	15.0390	0.594934
$640$	0	0
$641$	46.1526	1.82292	0.911459	−	0.411391i	$-0.134957\pi$
$641$	46.1526	1.82292	0.911459	+	0.411391i	$0.134957\pi$
$642$	0	0
$643$	−35.6897	−1.40746	−0.703732	−	0.710466i	$-0.748484\pi$
$643$	−35.6897	−1.40746	−0.703732	+	0.710466i	$0.748484\pi$
$644$	0	0
$645$	−37.9830	−1.49558
$646$	0	0
$647$	40.4256	1.58930	0.794648	−	0.607070i	$-0.207655\pi$
$647$	40.4256	1.58930	0.794648	+	0.607070i	$0.207655\pi$
$648$	0	0
$649$	7.13043	0.279894
$650$	0	0
$651$	−4.73623	−0.185628
$652$	0	0
$653$	−1.50359	−0.0588402	−0.0294201	−	0.999567i	$-0.509366\pi$
$653$	−1.50359	−0.0588402	−0.0294201	+	0.999567i	$0.509366\pi$
$654$	0	0
$655$	−38.4369	−1.50185
$656$	0	0
$657$	−14.0433	−0.547880
$658$	0	0
$659$	−15.1299	−0.589379	−0.294689	−	0.955593i	$-0.595216\pi$
$659$	−15.1299	−0.589379	−0.294689	+	0.955593i	$0.595216\pi$
$660$	0	0
$661$	2.82487	0.109875	0.0549373	−	0.998490i	$-0.482504\pi$
$661$	2.82487	0.109875	0.0549373	+	0.998490i	$0.482504\pi$
$662$	0	0
$663$	−0.552885	−0.0214723
$664$	0	0
$665$	−2.82557	−0.109571
$666$	0	0
$667$	1.60322	0.0620769
$668$	0	0
$669$	3.64271	0.140835
$670$	0	0
$671$	21.2393	0.819936
$672$	0	0
$673$	−31.0362	−1.19636	−0.598179	−	0.801363i	$-0.704109\pi$
$673$	−31.0362	−1.19636	−0.598179	+	0.801363i	$0.704109\pi$
$674$	0	0
$675$	16.7262	0.643792
$676$	0	0
$677$	−43.3471	−1.66597	−0.832983	−	0.553299i	$-0.813369\pi$
$677$	−43.3471	−1.66597	−0.832983	+	0.553299i	$0.813369\pi$
$678$	0	0
$679$	−7.83279	−0.300595
$680$	0	0
$681$	−13.2029	−0.505937
$682$	0	0
$683$	−46.9729	−1.79737	−0.898683	−	0.438598i	$-0.855475\pi$
$683$	−46.9729	−1.79737	−0.898683	+	0.438598i	$0.855475\pi$
$684$	0	0
$685$	−43.5814	−1.66516
$686$	0	0
$687$	7.63941	0.291462
$688$	0	0
$689$	1.32147	0.0503439
$690$	0	0
$691$	−6.52880	−0.248367	−0.124184	−	0.992259i	$-0.539631\pi$
$691$	−6.52880	−0.248367	−0.124184	+	0.992259i	$0.539631\pi$
$692$	0	0
$693$	−5.87117	−0.223027
$694$	0	0
$695$	−41.3371	−1.56801
$696$	0	0
$697$	3.37061	0.127671
$698$	0	0
$699$	−25.3101	−0.957316
$700$	0	0
$701$	−32.5723	−1.23024	−0.615120	−	0.788434i	$-0.710892\pi$
$701$	−32.5723	−1.23024	−0.615120	+	0.788434i	$0.710892\pi$
$702$	0	0
$703$	6.18775	0.233375
$704$	0	0
$705$	13.4996	0.508424
$706$	0	0
$707$	0.763842	0.0287272
$708$	0	0
$709$	−12.8059	−0.480935	−0.240468	−	0.970657i	$-0.577301\pi$
$709$	−12.8059	−0.480935	−0.240468	+	0.970657i	$0.577301\pi$
$710$	0	0
$711$	6.65370	0.249533
$712$	0	0
$713$	7.84645	0.293852
$714$	0	0
$715$	−2.01565	−0.0753812
$716$	0	0
$717$	23.7750	0.887893
$718$	0	0
$719$	30.2736	1.12901	0.564507	−	0.825428i	$-0.309066\pi$
$719$	30.2736	1.12901	0.564507	+	0.825428i	$0.309066\pi$
$720$	0	0
$721$	17.7772	0.662056
$722$	0	0
$723$	−27.2172	−1.01222
$724$	0	0
$725$	−2.21647	−0.0823177
$726$	0	0
$727$	36.2349	1.34388	0.671940	−	0.740606i	$-0.265461\pi$
$727$	36.2349	1.34388	0.671940	+	0.740606i	$0.265461\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	27.6646	1.02321
$732$	0	0
$733$	−32.8785	−1.21440	−0.607198	−	0.794550i	$-0.707707\pi$
$733$	−32.8785	−1.21440	−0.607198	+	0.794550i	$0.707707\pi$
$734$	0	0
$735$	−3.68109	−0.135779
$736$	0	0
$737$	−22.5228	−0.829638
$738$	0	0
$739$	−10.1868	−0.374727	−0.187363	−	0.982291i	$-0.559994\pi$
$739$	−10.1868	−0.374727	−0.187363	+	0.982291i	$0.559994\pi$
$740$	0	0
$741$	−0.206217	−0.00757556
$742$	0	0
$743$	−49.3380	−1.81004	−0.905018	−	0.425374i	$-0.860143\pi$
$743$	−49.3380	−1.81004	−0.905018	+	0.425374i	$0.860143\pi$
$744$	0	0
$745$	12.3084	0.450945
$746$	0	0
$747$	−20.0433	−0.733345
$748$	0	0
$749$	6.00722	0.219499
$750$	0	0
$751$	−30.9422	−1.12910	−0.564548	−	0.825400i	$-0.690950\pi$
$751$	−30.9422	−1.12910	−0.564548	+	0.825400i	$0.690950\pi$
$752$	0	0
$753$	8.23068	0.299942
$754$	0	0
$755$	18.8779	0.687037
$756$	0	0
$757$	41.4725	1.50734	0.753671	−	0.657252i	$-0.228281\pi$
$757$	41.4725	1.50734	0.753671	+	0.657252i	$0.228281\pi$
$758$	0	0
$759$	−12.6717	−0.459953
$760$	0	0
$761$	26.9416	0.976631	0.488315	−	0.872667i	$-0.337612\pi$
$761$	26.9416	0.976631	0.488315	+	0.872667i	$0.337612\pi$
$762$	0	0
$763$	2.49822	0.0904418
$764$	0	0
$765$	−9.86932	−0.356826
$766$	0	0
$767$	−0.250447	−0.00904310
$768$	0	0
$769$	36.4545	1.31458	0.657291	−	0.753637i	$-0.271702\pi$
$769$	36.4545	1.31458	0.657291	+	0.753637i	$0.271702\pi$
$770$	0	0
$771$	33.3610	1.20147
$772$	0	0
$773$	6.56447	0.236108	0.118054	−	0.993007i	$-0.462334\pi$
$773$	6.56447	0.236108	0.118054	+	0.993007i	$0.462334\pi$
$774$	0	0
$775$	−10.8478	−0.389665
$776$	0	0
$777$	8.06125	0.289196
$778$	0	0
$779$	1.25718	0.0450432
$780$	0	0
$781$	52.0241	1.86157
$782$	0	0
$783$	−4.16391	−0.148806
$784$	0	0
$785$	−10.5960	−0.378188
$786$	0	0
$787$	−10.4003	−0.370730	−0.185365	−	0.982670i	$-0.559347\pi$
$787$	−10.4003	−0.370730	−0.185365	+	0.982670i	$0.559347\pi$
$788$	0	0
$789$	−35.0065	−1.24626
$790$	0	0
$791$	16.2651	0.578322
$792$	0	0
$793$	−0.746003	−0.0264913
$794$	0	0
$795$	−30.7311	−1.08992
$796$	0	0
$797$	9.29130	0.329115	0.164557	−	0.986368i	$-0.447380\pi$
$797$	9.29130	0.329115	0.164557	+	0.986368i	$0.447380\pi$
$798$	0	0
$799$	−9.83231	−0.347842
$800$	0	0
$801$	14.9516	0.528288
$802$	0	0
$803$	−48.5795	−1.71433
$804$	0	0
$805$	6.09841	0.214941
$806$	0	0
$807$	14.7134	0.517935
$808$	0	0
$809$	−40.3102	−1.41723	−0.708616	−	0.705594i	$-0.750680\pi$
$809$	−40.3102	−1.41723	−0.708616	+	0.705594i	$0.750680\pi$
$810$	0	0
$811$	23.8246	0.836595	0.418298	−	0.908310i	$-0.362627\pi$
$811$	23.8246	0.836595	0.418298	+	0.908310i	$0.362627\pi$
$812$	0	0
$813$	−10.3013	−0.361283
$814$	0	0
$815$	−60.3998	−2.11571
$816$	0	0
$817$	10.3184	0.360996
$818$	0	0
$819$	0.206217	0.00720580
$820$	0	0
$821$	31.5277	1.10032	0.550162	−	0.835058i	$-0.314566\pi$
$821$	31.5277	1.10032	0.550162	+	0.835058i	$0.314566\pi$
$822$	0	0
$823$	−30.7823	−1.07300	−0.536502	−	0.843899i	$-0.680255\pi$
$823$	−30.7823	−1.07300	−0.536502	+	0.843899i	$0.680255\pi$
$824$	0	0
$825$	17.5188	0.609925
$826$	0	0
$827$	4.79611	0.166777	0.0833886	−	0.996517i	$-0.473426\pi$
$827$	4.79611	0.166777	0.0833886	+	0.996517i	$0.473426\pi$
$828$	0	0
$829$	53.9621	1.87418	0.937091	−	0.349086i	$-0.113508\pi$
$829$	53.9621	1.87418	0.937091	+	0.349086i	$0.113508\pi$
$830$	0	0
$831$	−19.5590	−0.678494
$832$	0	0
$833$	2.68109	0.0928942
$834$	0	0
$835$	−16.8419	−0.582836
$836$	0	0
$837$	−20.3789	−0.704399
$838$	0	0
$839$	28.4919	0.983648	0.491824	−	0.870695i	$-0.336330\pi$
$839$	28.4919	0.983648	0.491824	+	0.870695i	$0.336330\pi$
$840$	0	0
$841$	−28.4482	−0.980973
$842$	0	0
$843$	1.00119	0.0344828
$844$	0	0
$845$	−36.6617	−1.26120
$846$	0	0
$847$	−9.30999	−0.319895
$848$	0	0
$849$	−0.645118	−0.0221404
$850$	0	0
$851$	−13.3550	−0.457802
$852$	0	0
$853$	−35.7947	−1.22559	−0.612793	−	0.790243i	$-0.709954\pi$
$853$	−35.7947	−1.22559	−0.612793	+	0.790243i	$0.709954\pi$
$854$	0	0
$855$	−3.68109	−0.125891
$856$	0	0
$857$	33.4634	1.14309	0.571544	−	0.820572i	$-0.306345\pi$
$857$	33.4634	1.14309	0.571544	+	0.820572i	$0.306345\pi$
$858$	0	0
$859$	19.9511	0.680723	0.340361	−	0.940295i	$-0.389451\pi$
$859$	19.9511	0.680723	0.340361	+	0.940295i	$0.389451\pi$
$860$	0	0
$861$	1.63782	0.0558169
$862$	0	0
$863$	2.44197	0.0831256	0.0415628	−	0.999136i	$-0.486766\pi$
$863$	2.44197	0.0831256	0.0415628	+	0.999136i	$0.486766\pi$
$864$	0	0
$865$	−20.7112	−0.704201
$866$	0	0
$867$	12.7825	0.434118
$868$	0	0
$869$	23.0170	0.780798
$870$	0	0
$871$	0.791083	0.0268048
$872$	0	0
$873$	−10.2044	−0.345366
$874$	0	0
$875$	5.69674	0.192585
$876$	0	0
$877$	20.7289	0.699965	0.349983	−	0.936756i	$-0.386187\pi$
$877$	20.7289	0.699965	0.349983	+	0.936756i	$0.386187\pi$
$878$	0	0
$879$	−21.5038	−0.725307
$880$	0	0
$881$	−10.2207	−0.344342	−0.172171	−	0.985067i	$-0.555078\pi$
$881$	−10.2207	−0.344342	−0.172171	+	0.985067i	$0.555078\pi$
$882$	0	0
$883$	1.75381	0.0590204	0.0295102	−	0.999564i	$-0.490605\pi$
$883$	1.75381	0.0590204	0.0295102	+	0.999564i	$0.490605\pi$
$884$	0	0
$885$	5.82421	0.195779
$886$	0	0
$887$	−42.4187	−1.42428	−0.712140	−	0.702037i	$-0.752274\pi$
$887$	−42.4187	−1.42428	−0.712140	+	0.702037i	$0.752274\pi$
$888$	0	0
$889$	−1.79330	−0.0601454
$890$	0	0
$891$	15.2976	0.512490
$892$	0	0
$893$	−3.66728	−0.122721
$894$	0	0
$895$	14.3009	0.478027
$896$	0	0
$897$	0.445076	0.0148606
$898$	0	0
$899$	2.70051	0.0900672
$900$	0	0
$901$	22.3827	0.745676
$902$	0	0
$903$	13.4426	0.447342
$904$	0	0
$905$	−61.2991	−2.03765
$906$	0	0
$907$	39.0795	1.29761	0.648806	−	0.760954i	$-0.275269\pi$
$907$	39.0795	1.29761	0.648806	+	0.760954i	$0.275269\pi$
$908$	0	0
$909$	0.995114	0.0330059
$910$	0	0
$911$	7.71191	0.255507	0.127754	−	0.991806i	$-0.459223\pi$
$911$	7.71191	0.255507	0.127754	+	0.991806i	$0.459223\pi$
$912$	0	0
$913$	−69.3352	−2.29466
$914$	0	0
$915$	17.3485	0.573524
$916$	0	0
$917$	13.6032	0.449218
$918$	0	0
$919$	−32.9284	−1.08621	−0.543104	−	0.839666i	$-0.682751\pi$
$919$	−32.9284	−1.08621	−0.543104	+	0.839666i	$0.682751\pi$
$920$	0	0
$921$	20.1691	0.664593
$922$	0	0
$923$	−1.82728	−0.0601455
$924$	0	0
$925$	18.4634	0.607073
$926$	0	0
$927$	23.1597	0.760663
$928$	0	0
$929$	−26.9810	−0.885217	−0.442608	−	0.896715i	$-0.645947\pi$
$929$	−26.9810	−0.885217	−0.442608	+	0.896715i	$0.645947\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	42.2288	1.38251
$934$	0	0
$935$	−34.1407	−1.11652
$936$	0	0
$937$	−4.92335	−0.160839	−0.0804194	−	0.996761i	$-0.525626\pi$
$937$	−4.92335	−0.160839	−0.0804194	+	0.996761i	$0.525626\pi$
$938$	0	0
$939$	38.4246	1.25394
$940$	0	0
$941$	44.9945	1.46678	0.733390	−	0.679808i	$-0.237937\pi$
$941$	44.9945	1.46678	0.733390	+	0.679808i	$0.237937\pi$
$942$	0	0
$943$	−2.71336	−0.0883592
$944$	0	0
$945$	−15.8389	−0.515239
$946$	0	0
$947$	−47.1881	−1.53341	−0.766704	−	0.642001i	$-0.778105\pi$
$947$	−47.1881	−1.53341	−0.766704	+	0.642001i	$0.778105\pi$
$948$	0	0
$949$	1.70629	0.0553885
$950$	0	0
$951$	26.9455	0.873768
$952$	0	0
$953$	−37.0747	−1.20097	−0.600484	−	0.799637i	$-0.705025\pi$
$953$	−37.0747	−1.20097	−0.600484	+	0.799637i	$0.705025\pi$
$954$	0	0
$955$	−16.6284	−0.538083
$956$	0	0
$957$	−4.36122	−0.140978
$958$	0	0
$959$	15.4239	0.498064
$960$	0	0
$961$	−17.7832	−0.573651
$962$	0	0
$963$	7.82606	0.252191
$964$	0	0
$965$	32.8742	1.05826
$966$	0	0
$967$	−9.13460	−0.293749	−0.146874	−	0.989155i	$-0.546921\pi$
$967$	−9.13460	−0.293749	−0.146874	+	0.989155i	$0.546921\pi$
$968$	0	0
$969$	−3.49286	−0.112207
$970$	0	0
$971$	−44.9938	−1.44392	−0.721960	−	0.691935i	$-0.756758\pi$
$971$	−44.9938	−1.44392	−0.721960	+	0.691935i	$0.756758\pi$
$972$	0	0
$973$	14.6296	0.469005
$974$	0	0
$975$	−0.615323	−0.0197061
$976$	0	0
$977$	−12.2890	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$977$	−12.2890	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$978$	0	0
$979$	51.7217	1.65303
$980$	0	0
$981$	3.25463	0.103912
$982$	0	0
$983$	−35.0472	−1.11783	−0.558916	−	0.829224i	$-0.688782\pi$
$983$	−35.0472	−1.11783	−0.558916	+	0.829224i	$0.688782\pi$
$984$	0	0
$985$	−14.7482	−0.469916
$986$	0	0
$987$	−4.77765	−0.152074
$988$	0	0
$989$	−22.2702	−0.708150
$990$	0	0
$991$	14.4840	0.460098	0.230049	−	0.973179i	$-0.426111\pi$
$991$	14.4840	0.460098	0.230049	+	0.973179i	$0.426111\pi$
$992$	0	0
$993$	−14.3009	−0.453826
$994$	0	0
$995$	−6.90969	−0.219052
$996$	0	0
$997$	−39.7301	−1.25827	−0.629133	−	0.777298i	$-0.716590\pi$
$997$	−39.7301	−1.25827	−0.629133	+	0.777298i	$0.716590\pi$
$998$	0	0
$999$	34.6857	1.09741

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bu.1.2		4
4.3	odd	2		8512.2.a.bq.1.4		4
8.3	odd	2		1064.2.a.h.1.1	✓	4
8.5	even	2		2128.2.a.t.1.3		4
24.11	even	2		9576.2.a.ci.1.4		4
56.27	even	2		7448.2.a.bj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.h.1.1	✓	4	8.3	odd	2
2128.2.a.t.1.3		4	8.5	even	2
7448.2.a.bj.1.4		4	56.27	even	2
8512.2.a.bq.1.4		4	4.3	odd	2
8512.2.a.bu.1.2		4	1.1	even	1	trivial
9576.2.a.ci.1.4		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{3} +2.82557 q^{5} -1.00000 q^{7} -1.30278 q^{9} +O(q^{10})\)	\(q-1.30278 q^{3} +2.82557 q^{5} -1.00000 q^{7} -1.30278 q^{9} -4.50666 q^{11} +0.158290 q^{13} -3.68109 q^{15} +2.68109 q^{17} +1.00000 q^{19} +1.30278 q^{21} -2.15829 q^{23} +2.98386 q^{25} +5.60555 q^{27} -0.742819 q^{29} -3.63549 q^{31} +5.87117 q^{33} -2.82557 q^{35} +6.18775 q^{37} -0.206217 q^{39} +1.25718 q^{41} +10.3184 q^{43} -3.68109 q^{45} -3.66728 q^{47} +1.00000 q^{49} -3.49286 q^{51} +8.34837 q^{53} -12.7339 q^{55} -1.30278 q^{57} -1.58220 q^{59} -4.71288 q^{61} +1.30278 q^{63} +0.447261 q^{65} +4.99767 q^{67} +2.81177 q^{69} -11.5438 q^{71} +10.7795 q^{73} -3.88730 q^{75} +4.50666 q^{77} -5.10733 q^{79} -3.39445 q^{81} +15.3850 q^{83} +7.57561 q^{85} +0.967727 q^{87} -11.4767 q^{89} -0.158290 q^{91} +4.73623 q^{93} +2.82557 q^{95} +7.83279 q^{97} +5.87117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - q^5 - 4 * q^7 + 2 * q^9	\( 4 q + 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 7 q^{15} + 3 q^{17} + 4 q^{19} - 2 q^{21} - 6 q^{23} - 3 q^{25} + 8 q^{27} - 17 q^{31} + q^{33} + q^{35} - 3 q^{37} - q^{39} + 8 q^{41} + 7 q^{43} - 7 q^{45} - 5 q^{47} + 4 q^{49} + 8 q^{51} + 16 q^{53} - 17 q^{55} + 2 q^{57} + 7 q^{59} + q^{61} - 2 q^{63} - 10 q^{65} + 7 q^{67} - 3 q^{69} - 27 q^{71} - q^{73} - 8 q^{75} - 2 q^{77} - 15 q^{79} - 28 q^{81} + 3 q^{83} - q^{85} - 26 q^{87} - 9 q^{89} + 2 q^{91} - 2 q^{93} - q^{95} + 3 q^{97} + q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - q^5 - 4 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 - 7 * q^15 + 3 * q^17 + 4 * q^19 - 2 * q^21 - 6 * q^23 - 3 * q^25 + 8 * q^27 - 17 * q^31 + q^33 + q^35 - 3 * q^37 - q^39 + 8 * q^41 + 7 * q^43 - 7 * q^45 - 5 * q^47 + 4 * q^49 + 8 * q^51 + 16 * q^53 - 17 * q^55 + 2 * q^57 + 7 * q^59 + q^61 - 2 * q^63 - 10 * q^65 + 7 * q^67 - 3 * q^69 - 27 * q^71 - q^73 - 8 * q^75 - 2 * q^77 - 15 * q^79 - 28 * q^81 + 3 * q^83 - q^85 - 26 * q^87 - 9 * q^89 + 2 * q^91 - 2 * q^93 - q^95 + 3 * q^97 + q^99

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