LMFDB - Embedded newform 8910.2.a.bw.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8910,2,Mod(1,8910)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8910, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8910.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8910.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-6,0,6,-6,0,-2,-6,0,6,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.1467082010$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1526544576.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 26x^{4} + 30x^{3} + 129x^{2} - 180x + 36 $ x^6 - 2x^5 - 26x^4 + 30x^3 + 129x^2 - 180*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.244904$ of defining polynomial
Character	$\chi$	$=$	8910.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.244904 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +5.37770 q^{13} +0.244904 q^{14} +1.00000 q^{16} -2.66909 q^{17} +3.73205 q^{19} -1.00000 q^{20} -1.00000 q^{22} +1.30292 q^{23} +1.00000 q^{25} -5.37770 q^{26} -0.244904 q^{28} -0.755096 q^{29} -7.84180 q^{31} -1.00000 q^{32} +2.66909 q^{34} +0.244904 q^{35} -11.0695 q^{37} -3.73205 q^{38} +1.00000 q^{40} -0.656874 q^{41} +12.8138 q^{43} +1.00000 q^{44} -1.30292 q^{46} -10.2564 q^{47} -6.94002 q^{49} -1.00000 q^{50} +5.37770 q^{52} -10.1727 q^{53} -1.00000 q^{55} +0.244904 q^{56} +0.755096 q^{58} -9.74358 q^{59} +3.38932 q^{61} +7.84180 q^{62} +1.00000 q^{64} -5.37770 q^{65} -5.21425 q^{67} -2.66909 q^{68} -0.244904 q^{70} +8.14698 q^{71} +10.6576 q^{73} +11.0695 q^{74} +3.73205 q^{76} -0.244904 q^{77} -4.70366 q^{79} -1.00000 q^{80} +0.656874 q^{82} -6.32596 q^{83} +2.66909 q^{85} -12.8138 q^{86} -1.00000 q^{88} -5.34244 q^{89} -1.31702 q^{91} +1.30292 q^{92} +10.2564 q^{94} -3.73205 q^{95} -5.40341 q^{97} +6.94002 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 2 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 6 q^{16} - 8 q^{17} + 12 q^{19} - 6 q^{20} - 6 q^{22} - 10 q^{23} + 6 q^{25} + 2 q^{26} - 2 q^{28} - 4 q^{29}+ \cdots - 14 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 2 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 6 * q^16 - 8 * q^17 + 12 * q^19 - 6 * q^20 - 6 * q^22 - 10 * q^23 + 6 * q^25 + 2 * q^26 - 2 * q^28 - 4 * q^29 + 8 * q^31 - 6 * q^32 + 8 * q^34 + 2 * q^35 - 4 * q^37 - 12 * q^38 + 6 * q^40 - 4 * q^41 - 4 * q^43 + 6 * q^44 + 10 * q^46 - 18 * q^47 + 14 * q^49 - 6 * q^50 - 2 * q^52 - 14 * q^53 - 6 * q^55 + 2 * q^56 + 4 * q^58 - 4 * q^59 + 16 * q^61 - 8 * q^62 + 6 * q^64 + 2 * q^65 + 6 * q^67 - 8 * q^68 - 2 * q^70 - 18 * q^71 + 2 * q^73 + 4 * q^74 + 12 * q^76 - 2 * q^77 + 14 * q^79 - 6 * q^80 + 4 * q^82 - 30 * q^83 + 8 * q^85 + 4 * q^86 - 6 * q^88 + 4 * q^89 + 20 * q^91 - 10 * q^92 + 18 * q^94 - 12 * q^95 - 20 * q^97 - 14 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.244904	−0.0925651	−0.0462825	−	0.998928i	$-0.514737\pi$
$7$	−0.244904	−0.0925651	−0.0462825	+	0.998928i	$0.514737\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.37770	1.49151	0.745753	−	0.666223i	$-0.232090\pi$
$13$	5.37770	1.49151	0.745753	+	0.666223i	$0.232090\pi$
$14$	0.244904	0.0654534
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.66909	−0.647350	−0.323675	−	0.946168i	$-0.604918\pi$
$17$	−2.66909	−0.647350	−0.323675	+	0.946168i	$0.604918\pi$
$18$	0	0
$19$	3.73205	0.856191	0.428096	−	0.903733i	$-0.359185\pi$
$19$	3.73205	0.856191	0.428096	+	0.903733i	$0.359185\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	1.30292	0.271678	0.135839	−	0.990731i	$-0.456627\pi$
$23$	1.30292	0.271678	0.135839	+	0.990731i	$0.456627\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.37770	−1.05465
$27$	0	0
$28$	−0.244904	−0.0462825
$29$	−0.755096	−0.140218	−0.0701089	−	0.997539i	$-0.522335\pi$
$29$	−0.755096	−0.140218	−0.0701089	+	0.997539i	$0.522335\pi$
$30$	0	0
$31$	−7.84180	−1.40843	−0.704214	−	0.709987i	$-0.748701\pi$
$31$	−7.84180	−1.40843	−0.704214	+	0.709987i	$0.748701\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.66909	0.457745
$35$	0.244904	0.0413964
$36$	0	0
$37$	−11.0695	−1.81982	−0.909911	−	0.414804i	$-0.863850\pi$
$37$	−11.0695	−1.81982	−0.909911	+	0.414804i	$0.863850\pi$
$38$	−3.73205	−0.605419
$39$	0	0
$40$	1.00000	0.158114
$41$	−0.656874	−0.102586	−0.0512932	−	0.998684i	$-0.516334\pi$
$41$	−0.656874	−0.102586	−0.0512932	+	0.998684i	$0.516334\pi$
$42$	0	0
$43$	12.8138	1.95409	0.977044	−	0.213038i	$-0.0683359\pi$
$43$	12.8138	1.95409	0.977044	+	0.213038i	$0.0683359\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−1.30292	−0.192105
$47$	−10.2564	−1.49605	−0.748027	−	0.663668i	$-0.768999\pi$
$47$	−10.2564	−1.49605	−0.748027	+	0.663668i	$0.768999\pi$
$48$	0	0
$49$	−6.94002	−0.991432
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.37770	0.745753
$53$	−10.1727	−1.39733	−0.698665	−	0.715449i	$-0.746222\pi$
$53$	−10.1727	−1.39733	−0.698665	+	0.715449i	$0.746222\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0.244904	0.0327267
$57$	0	0
$58$	0.755096	0.0991489
$59$	−9.74358	−1.26851	−0.634253	−	0.773126i	$-0.718692\pi$
$59$	−9.74358	−1.26851	−0.634253	+	0.773126i	$0.718692\pi$
$60$	0	0
$61$	3.38932	0.433958	0.216979	−	0.976176i	$-0.430380\pi$
$61$	3.38932	0.433958	0.216979	+	0.976176i	$0.430380\pi$
$62$	7.84180	0.995910
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.37770	−0.667021
$66$	0	0
$67$	−5.21425	−0.637022	−0.318511	−	0.947919i	$-0.603183\pi$
$67$	−5.21425	−0.637022	−0.318511	+	0.947919i	$0.603183\pi$
$68$	−2.66909	−0.323675
$69$	0	0
$70$	−0.244904	−0.0292717
$71$	8.14698	0.966869	0.483435	−	0.875381i	$-0.339389\pi$
$71$	8.14698	0.966869	0.483435	+	0.875381i	$0.339389\pi$
$72$	0	0
$73$	10.6576	1.24737	0.623687	−	0.781674i	$-0.285634\pi$
$73$	10.6576	1.24737	0.623687	+	0.781674i	$0.285634\pi$
$74$	11.0695	1.28681
$75$	0	0
$76$	3.73205	0.428096
$77$	−0.244904	−0.0279094
$78$	0	0
$79$	−4.70366	−0.529203	−0.264602	−	0.964358i	$-0.585240\pi$
$79$	−4.70366	−0.529203	−0.264602	+	0.964358i	$0.585240\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	0.656874	0.0725396
$83$	−6.32596	−0.694365	−0.347182	−	0.937798i	$-0.612862\pi$
$83$	−6.32596	−0.694365	−0.347182	+	0.937798i	$0.612862\pi$
$84$	0	0
$85$	2.66909	0.289504
$86$	−12.8138	−1.38175
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−5.34244	−0.566297	−0.283149	−	0.959076i	$-0.591379\pi$
$89$	−5.34244	−0.566297	−0.283149	+	0.959076i	$0.591379\pi$
$90$	0	0
$91$	−1.31702	−0.138061
$92$	1.30292	0.135839
$93$	0	0
$94$	10.2564	1.05787
$95$	−3.73205	−0.382900
$96$	0	0
$97$	−5.40341	−0.548633	−0.274316	−	0.961640i	$-0.588452\pi$
$97$	−5.40341	−0.548633	−0.274316	+	0.961640i	$0.588452\pi$
$98$	6.94002	0.701048
$99$	0	0
$100$	1.00000	0.100000
$101$	18.4994	1.84076	0.920378	−	0.391029i	$-0.127881\pi$
$101$	18.4994	1.84076	0.920378	+	0.391029i	$0.127881\pi$
$102$	0	0
$103$	3.73431	0.367953	0.183976	−	0.982931i	$-0.441103\pi$
$103$	3.73431	0.367953	0.183976	+	0.982931i	$0.441103\pi$
$104$	−5.37770	−0.527327
$105$	0	0
$106$	10.1727	0.988061
$107$	−11.7159	−1.13262	−0.566308	−	0.824193i	$-0.691629\pi$
$107$	−11.7159	−1.13262	−0.566308	+	0.824193i	$0.691629\pi$
$108$	0	0
$109$	1.71983	0.164730	0.0823651	−	0.996602i	$-0.473753\pi$
$109$	1.71983	0.164730	0.0823651	+	0.996602i	$0.473753\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−0.244904	−0.0231413
$113$	17.8339	1.67767	0.838835	−	0.544385i	$-0.183237\pi$
$113$	17.8339	1.67767	0.838835	+	0.544385i	$0.183237\pi$
$114$	0	0
$115$	−1.30292	−0.121498
$116$	−0.755096	−0.0701089
$117$	0	0
$118$	9.74358	0.896969
$119$	0.653672	0.0599220
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.38932	−0.306855
$123$	0	0
$124$	−7.84180	−0.704214
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.85035	0.696605	0.348303	−	0.937382i	$-0.386758\pi$
$127$	7.85035	0.696605	0.348303	+	0.937382i	$0.386758\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.37770	0.471655
$131$	−0.0464883	−0.00406170	−0.00203085	−	0.999998i	$-0.500646\pi$
$131$	−0.0464883	−0.00406170	−0.00203085	+	0.999998i	$0.500646\pi$
$132$	0	0
$133$	−0.913995	−0.0792534
$134$	5.21425	0.450443
$135$	0	0
$136$	2.66909	0.228873
$137$	5.47563	0.467815	0.233907	−	0.972259i	$-0.424849\pi$
$137$	5.47563	0.467815	0.233907	+	0.972259i	$0.424849\pi$
$138$	0	0
$139$	−4.58548	−0.388935	−0.194467	−	0.980909i	$-0.562298\pi$
$139$	−4.58548	−0.388935	−0.194467	+	0.980909i	$0.562298\pi$
$140$	0.244904	0.0206982
$141$	0	0
$142$	−8.14698	−0.683680
$143$	5.37770	0.449706
$144$	0	0
$145$	0.755096	0.0627073
$146$	−10.6576	−0.882027
$147$	0	0
$148$	−11.0695	−0.909911
$149$	−3.53012	−0.289199	−0.144599	−	0.989490i	$-0.546189\pi$
$149$	−3.53012	−0.289199	−0.144599	+	0.989490i	$0.546189\pi$
$150$	0	0
$151$	9.77296	0.795312	0.397656	−	0.917535i	$-0.369824\pi$
$151$	9.77296	0.795312	0.397656	+	0.917535i	$0.369824\pi$
$152$	−3.73205	−0.302709
$153$	0	0
$154$	0.244904	0.0197349
$155$	7.84180	0.629869
$156$	0	0
$157$	−5.18947	−0.414165	−0.207082	−	0.978323i	$-0.566397\pi$
$157$	−5.18947	−0.414165	−0.207082	+	0.978323i	$0.566397\pi$
$158$	4.70366	0.374203
$159$	0	0
$160$	1.00000	0.0790569
$161$	−0.319091	−0.0251479
$162$	0	0
$163$	−18.9321	−1.48288	−0.741438	−	0.671021i	$-0.765856\pi$
$163$	−18.9321	−1.48288	−0.741438	+	0.671021i	$0.765856\pi$
$164$	−0.656874	−0.0512932
$165$	0	0
$166$	6.32596	0.490990
$167$	−25.3359	−1.96055	−0.980276	−	0.197631i	$-0.936675\pi$
$167$	−25.3359	−1.96055	−0.980276	+	0.197631i	$0.936675\pi$
$168$	0	0
$169$	15.9196	1.22459
$170$	−2.66909	−0.204710
$171$	0	0
$172$	12.8138	0.977044
$173$	8.97666	0.682483	0.341242	−	0.939976i	$-0.389153\pi$
$173$	8.97666	0.682483	0.341242	+	0.939976i	$0.389153\pi$
$174$	0	0
$175$	−0.244904	−0.0185130
$176$	1.00000	0.0753778
$177$	0	0
$178$	5.34244	0.400433
$179$	14.1974	1.06116	0.530581	−	0.847634i	$-0.321974\pi$
$179$	14.1974	1.06116	0.530581	+	0.847634i	$0.321974\pi$
$180$	0	0
$181$	15.8646	1.17920	0.589601	−	0.807694i	$-0.299285\pi$
$181$	15.8646	1.17920	0.589601	+	0.807694i	$0.299285\pi$
$182$	1.31702	0.0976241
$183$	0	0
$184$	−1.30292	−0.0960525
$185$	11.0695	0.813849
$186$	0	0
$187$	−2.66909	−0.195183
$188$	−10.2564	−0.748027
$189$	0	0
$190$	3.73205	0.270751
$191$	−1.22803	−0.0888574	−0.0444287	−	0.999013i	$-0.514147\pi$
$191$	−1.22803	−0.0888574	−0.0444287	+	0.999013i	$0.514147\pi$
$192$	0	0
$193$	−17.4354	−1.25503	−0.627515	−	0.778605i	$-0.715928\pi$
$193$	−17.4354	−1.25503	−0.627515	+	0.778605i	$0.715928\pi$
$194$	5.40341	0.387942
$195$	0	0
$196$	−6.94002	−0.495716
$197$	5.18660	0.369530	0.184765	−	0.982783i	$-0.440848\pi$
$197$	5.18660	0.369530	0.184765	+	0.982783i	$0.440848\pi$
$198$	0	0
$199$	27.1395	1.92387	0.961935	−	0.273277i	$-0.0881076\pi$
$199$	27.1395	1.92387	0.961935	+	0.273277i	$0.0881076\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−18.4994	−1.30161
$203$	0.184926	0.0129793
$204$	0	0
$205$	0.656874	0.0458781
$206$	−3.73431	−0.260182
$207$	0	0
$208$	5.37770	0.372876
$209$	3.73205	0.258151
$210$	0	0
$211$	−14.9823	−1.03142	−0.515712	−	0.856762i	$-0.672473\pi$
$211$	−14.9823	−1.03142	−0.515712	+	0.856762i	$0.672473\pi$
$212$	−10.1727	−0.698665
$213$	0	0
$214$	11.7159	0.800881
$215$	−12.8138	−0.873895
$216$	0	0
$217$	1.92049	0.130371
$218$	−1.71983	−0.116482
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	−14.3536	−0.965525
$222$	0	0
$223$	10.9914	0.736041	0.368020	−	0.929818i	$-0.380036\pi$
$223$	10.9914	0.736041	0.368020	+	0.929818i	$0.380036\pi$
$224$	0.244904	0.0163634
$225$	0	0
$226$	−17.8339	−1.18629
$227$	1.16282	0.0771791	0.0385895	−	0.999255i	$-0.487714\pi$
$227$	1.16282	0.0771791	0.0385895	+	0.999255i	$0.487714\pi$
$228$	0	0
$229$	−21.8910	−1.44660	−0.723298	−	0.690536i	$-0.757375\pi$
$229$	−21.8910	−1.44660	−0.723298	+	0.690536i	$0.757375\pi$
$230$	1.30292	0.0859120
$231$	0	0
$232$	0.755096	0.0495745
$233$	−27.0119	−1.76961	−0.884805	−	0.465961i	$-0.845709\pi$
$233$	−27.0119	−1.76961	−0.884805	+	0.465961i	$0.845709\pi$
$234$	0	0
$235$	10.2564	0.669056
$236$	−9.74358	−0.634253
$237$	0	0
$238$	−0.653672	−0.0423712
$239$	−18.2549	−1.18081	−0.590405	−	0.807107i	$-0.701032\pi$
$239$	−18.2549	−1.18081	−0.590405	+	0.807107i	$0.701032\pi$
$240$	0	0
$241$	−8.96893	−0.577740	−0.288870	−	0.957368i	$-0.593279\pi$
$241$	−8.96893	−0.577740	−0.288870	+	0.957368i	$0.593279\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	3.38932	0.216979
$245$	6.94002	0.443382
$246$	0	0
$247$	20.0698	1.27701
$248$	7.84180	0.497955
$249$	0	0
$250$	1.00000	0.0632456
$251$	−15.8686	−1.00162	−0.500809	−	0.865558i	$-0.666964\pi$
$251$	−15.8686	−1.00162	−0.500809	+	0.865558i	$0.666964\pi$
$252$	0	0
$253$	1.30292	0.0819139
$254$	−7.85035	−0.492574
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.86558	−0.303506	−0.151753	−	0.988418i	$-0.548492\pi$
$257$	−4.86558	−0.303506	−0.151753	+	0.988418i	$0.548492\pi$
$258$	0	0
$259$	2.71098	0.168452
$260$	−5.37770	−0.333511
$261$	0	0
$262$	0.0464883	0.00287206
$263$	−6.67146	−0.411380	−0.205690	−	0.978617i	$-0.565944\pi$
$263$	−6.67146	−0.411380	−0.205690	+	0.978617i	$0.565944\pi$
$264$	0	0
$265$	10.1727	0.624905
$266$	0.913995	0.0560406
$267$	0	0
$268$	−5.21425	−0.318511
$269$	−16.6115	−1.01282	−0.506410	−	0.862293i	$-0.669028\pi$
$269$	−16.6115	−1.01282	−0.506410	+	0.862293i	$0.669028\pi$
$270$	0	0
$271$	22.4154	1.36164	0.680818	−	0.732453i	$-0.261624\pi$
$271$	22.4154	1.36164	0.680818	+	0.732453i	$0.261624\pi$
$272$	−2.66909	−0.161837
$273$	0	0
$274$	−5.47563	−0.330795
$275$	1.00000	0.0603023
$276$	0	0
$277$	22.6226	1.35926	0.679631	−	0.733554i	$-0.262140\pi$
$277$	22.6226	1.35926	0.679631	+	0.733554i	$0.262140\pi$
$278$	4.58548	0.275019
$279$	0	0
$280$	−0.244904	−0.0146358
$281$	−20.3902	−1.21638	−0.608189	−	0.793792i	$-0.708104\pi$
$281$	−20.3902	−1.21638	−0.608189	+	0.793792i	$0.708104\pi$
$282$	0	0
$283$	18.0154	1.07091	0.535453	−	0.844565i	$-0.320141\pi$
$283$	18.0154	1.07091	0.535453	+	0.844565i	$0.320141\pi$
$284$	8.14698	0.483435
$285$	0	0
$286$	−5.37770	−0.317990
$287$	0.160871	0.00949592
$288$	0	0
$289$	−9.87595	−0.580938
$290$	−0.755096	−0.0443408
$291$	0	0
$292$	10.6576	0.623687
$293$	−32.5118	−1.89936	−0.949681	−	0.313219i	$-0.898593\pi$
$293$	−32.5118	−1.89936	−0.949681	+	0.313219i	$0.898593\pi$
$294$	0	0
$295$	9.74358	0.567293
$296$	11.0695	0.643404
$297$	0	0
$298$	3.53012	0.204494
$299$	7.00671	0.405208
$300$	0	0
$301$	−3.13816	−0.180880
$302$	−9.77296	−0.562370
$303$	0	0
$304$	3.73205	0.214048
$305$	−3.38932	−0.194072
$306$	0	0
$307$	−30.1273	−1.71946	−0.859728	−	0.510751i	$-0.829367\pi$
$307$	−30.1273	−1.71946	−0.859728	+	0.510751i	$0.829367\pi$
$308$	−0.244904	−0.0139547
$309$	0	0
$310$	−7.84180	−0.445384
$311$	9.29102	0.526846	0.263423	−	0.964680i	$-0.415149\pi$
$311$	9.29102	0.526846	0.263423	+	0.964680i	$0.415149\pi$
$312$	0	0
$313$	−20.5036	−1.15893	−0.579467	−	0.814996i	$-0.696739\pi$
$313$	−20.5036	−1.15893	−0.579467	+	0.814996i	$0.696739\pi$
$314$	5.18947	0.292859
$315$	0	0
$316$	−4.70366	−0.264602
$317$	22.5858	1.26854	0.634272	−	0.773110i	$-0.281300\pi$
$317$	22.5858	1.26854	0.634272	+	0.773110i	$0.281300\pi$
$318$	0	0
$319$	−0.755096	−0.0422772
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0.319091	0.0177822
$323$	−9.96118	−0.554255
$324$	0	0
$325$	5.37770	0.298301
$326$	18.9321	1.04855
$327$	0	0
$328$	0.656874	0.0362698
$329$	2.51184	0.138482
$330$	0	0
$331$	12.1160	0.665957	0.332979	−	0.942934i	$-0.391946\pi$
$331$	12.1160	0.665957	0.332979	+	0.942934i	$0.391946\pi$
$332$	−6.32596	−0.347182
$333$	0	0
$334$	25.3359	1.38632
$335$	5.21425	0.284885
$336$	0	0
$337$	−29.4419	−1.60380	−0.801900	−	0.597458i	$-0.796177\pi$
$337$	−29.4419	−1.60380	−0.801900	+	0.597458i	$0.796177\pi$
$338$	−15.9196	−0.865914
$339$	0	0
$340$	2.66909	0.144752
$341$	−7.84180	−0.424657
$342$	0	0
$343$	3.41397	0.184337
$344$	−12.8138	−0.690874
$345$	0	0
$346$	−8.97666	−0.482588
$347$	−24.8035	−1.33152	−0.665760	−	0.746166i	$-0.731892\pi$
$347$	−24.8035	−1.33152	−0.665760	+	0.746166i	$0.731892\pi$
$348$	0	0
$349$	1.80944	0.0968571	0.0484285	−	0.998827i	$-0.484579\pi$
$349$	1.80944	0.0968571	0.0484285	+	0.998827i	$0.484579\pi$
$350$	0.244904	0.0130907
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−14.6947	−0.782121	−0.391060	−	0.920365i	$-0.627892\pi$
$353$	−14.6947	−0.782121	−0.391060	+	0.920365i	$0.627892\pi$
$354$	0	0
$355$	−8.14698	−0.432397
$356$	−5.34244	−0.283149
$357$	0	0
$358$	−14.1974	−0.750355
$359$	−32.6998	−1.72583	−0.862914	−	0.505350i	$-0.831363\pi$
$359$	−32.6998	−1.72583	−0.862914	+	0.505350i	$0.831363\pi$
$360$	0	0
$361$	−5.07180	−0.266937
$362$	−15.8646	−0.833822
$363$	0	0
$364$	−1.31702	−0.0690307
$365$	−10.6576	−0.557843
$366$	0	0
$367$	21.0799	1.10036	0.550181	−	0.835045i	$-0.314559\pi$
$367$	21.0799	1.10036	0.550181	+	0.835045i	$0.314559\pi$
$368$	1.30292	0.0679194
$369$	0	0
$370$	−11.0695	−0.575478
$371$	2.49134	0.129344
$372$	0	0
$373$	−30.5860	−1.58368	−0.791841	−	0.610728i	$-0.790877\pi$
$373$	−30.5860	−1.58368	−0.791841	+	0.610728i	$0.790877\pi$
$374$	2.66909	0.138015
$375$	0	0
$376$	10.2564	0.528935
$377$	−4.06068	−0.209136
$378$	0	0
$379$	−5.82392	−0.299155	−0.149577	−	0.988750i	$-0.547791\pi$
$379$	−5.82392	−0.299155	−0.149577	+	0.988750i	$0.547791\pi$
$380$	−3.73205	−0.191450
$381$	0	0
$382$	1.22803	0.0628317
$383$	5.97542	0.305330	0.152665	−	0.988278i	$-0.451214\pi$
$383$	5.97542	0.305330	0.152665	+	0.988278i	$0.451214\pi$
$384$	0	0
$385$	0.244904	0.0124815
$386$	17.4354	0.887440
$387$	0	0
$388$	−5.40341	−0.274316
$389$	38.1731	1.93546	0.967728	−	0.251998i	$-0.0810878\pi$
$389$	38.1731	1.93546	0.967728	+	0.251998i	$0.0810878\pi$
$390$	0	0
$391$	−3.47761	−0.175870
$392$	6.94002	0.350524
$393$	0	0
$394$	−5.18660	−0.261297
$395$	4.70366	0.236667
$396$	0	0
$397$	−30.5803	−1.53478	−0.767392	−	0.641179i	$-0.778446\pi$
$397$	−30.5803	−1.53478	−0.767392	+	0.641179i	$0.778446\pi$
$398$	−27.1395	−1.36038
$399$	0	0
$400$	1.00000	0.0500000
$401$	18.8798	0.942810	0.471405	−	0.881917i	$-0.343747\pi$
$401$	18.8798	0.942810	0.471405	+	0.881917i	$0.343747\pi$
$402$	0	0
$403$	−42.1708	−2.10068
$404$	18.4994	0.920378
$405$	0	0
$406$	−0.184926	−0.00917773
$407$	−11.0695	−0.548697
$408$	0	0
$409$	20.7108	1.02409	0.512043	−	0.858960i	$-0.328889\pi$
$409$	20.7108	1.02409	0.512043	+	0.858960i	$0.328889\pi$
$410$	−0.656874	−0.0324407
$411$	0	0
$412$	3.73431	0.183976
$413$	2.38624	0.117419
$414$	0	0
$415$	6.32596	0.310529
$416$	−5.37770	−0.263663
$417$	0	0
$418$	−3.73205	−0.182541
$419$	10.3608	0.506160	0.253080	−	0.967445i	$-0.418556\pi$
$419$	10.3608	0.506160	0.253080	+	0.967445i	$0.418556\pi$
$420$	0	0
$421$	−36.5611	−1.78188	−0.890941	−	0.454120i	$-0.849954\pi$
$421$	−36.5611	−1.78188	−0.890941	+	0.454120i	$0.849954\pi$
$422$	14.9823	0.729327
$423$	0	0
$424$	10.1727	0.494030
$425$	−2.66909	−0.129470
$426$	0	0
$427$	−0.830060	−0.0401694
$428$	−11.7159	−0.566308
$429$	0	0
$430$	12.8138	0.617937
$431$	−19.6563	−0.946813	−0.473406	−	0.880844i	$-0.656976\pi$
$431$	−19.6563	−0.946813	−0.473406	+	0.880844i	$0.656976\pi$
$432$	0	0
$433$	32.8936	1.58077	0.790384	−	0.612612i	$-0.209881\pi$
$433$	32.8936	1.58077	0.790384	+	0.612612i	$0.209881\pi$
$434$	−1.92049	−0.0921865
$435$	0	0
$436$	1.71983	0.0823651
$437$	4.86256	0.232608
$438$	0	0
$439$	14.5870	0.696202	0.348101	−	0.937457i	$-0.386827\pi$
$439$	14.5870	0.696202	0.348101	+	0.937457i	$0.386827\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	14.3536	0.682729
$443$	−3.60743	−0.171394	−0.0856971	−	0.996321i	$-0.527312\pi$
$443$	−3.60743	−0.171394	−0.0856971	+	0.996321i	$0.527312\pi$
$444$	0	0
$445$	5.34244	0.253256
$446$	−10.9914	−0.520460
$447$	0	0
$448$	−0.244904	−0.0115706
$449$	4.31769	0.203764	0.101882	−	0.994796i	$-0.467514\pi$
$449$	4.31769	0.203764	0.101882	+	0.994796i	$0.467514\pi$
$450$	0	0
$451$	−0.656874	−0.0309310
$452$	17.8339	0.838835
$453$	0	0
$454$	−1.16282	−0.0545738
$455$	1.31702	0.0617429
$456$	0	0
$457$	6.94112	0.324692	0.162346	−	0.986734i	$-0.448094\pi$
$457$	6.94112	0.324692	0.162346	+	0.986734i	$0.448094\pi$
$458$	21.8910	1.02290
$459$	0	0
$460$	−1.30292	−0.0607489
$461$	−21.6827	−1.00986	−0.504931	−	0.863159i	$-0.668482\pi$
$461$	−21.6827	−1.00986	−0.504931	+	0.863159i	$0.668482\pi$
$462$	0	0
$463$	−25.9403	−1.20555	−0.602775	−	0.797912i	$-0.705938\pi$
$463$	−25.9403	−1.20555	−0.602775	+	0.797912i	$0.705938\pi$
$464$	−0.755096	−0.0350544
$465$	0	0
$466$	27.0119	1.25130
$467$	−27.6311	−1.27861	−0.639307	−	0.768952i	$-0.720779\pi$
$467$	−27.6311	−1.27861	−0.639307	+	0.768952i	$0.720779\pi$
$468$	0	0
$469$	1.27699	0.0589660
$470$	−10.2564	−0.473094
$471$	0	0
$472$	9.74358	0.448485
$473$	12.8138	0.589180
$474$	0	0
$475$	3.73205	0.171238
$476$	0.653672	0.0299610
$477$	0	0
$478$	18.2549	0.834958
$479$	28.6571	1.30937	0.654687	−	0.755900i	$-0.272800\pi$
$479$	28.6571	1.30937	0.654687	+	0.755900i	$0.272800\pi$
$480$	0	0
$481$	−59.5287	−2.71427
$482$	8.96893	0.408524
$483$	0	0
$484$	1.00000	0.0454545
$485$	5.40341	0.245356
$486$	0	0
$487$	23.4297	1.06170	0.530851	−	0.847465i	$-0.321872\pi$
$487$	23.4297	1.06170	0.530851	+	0.847465i	$0.321872\pi$
$488$	−3.38932	−0.153427
$489$	0	0
$490$	−6.94002	−0.313518
$491$	34.9801	1.57863	0.789314	−	0.613989i	$-0.210436\pi$
$491$	34.9801	1.57863	0.789314	+	0.613989i	$0.210436\pi$
$492$	0	0
$493$	2.01542	0.0907699
$494$	−20.0698	−0.902985
$495$	0	0
$496$	−7.84180	−0.352107
$497$	−1.99523	−0.0894983
$498$	0	0
$499$	−7.37154	−0.329996	−0.164998	−	0.986294i	$-0.552762\pi$
$499$	−7.37154	−0.329996	−0.164998	+	0.986294i	$0.552762\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	15.8686	0.708251
$503$	3.53995	0.157839	0.0789193	−	0.996881i	$-0.474853\pi$
$503$	3.53995	0.157839	0.0789193	+	0.996881i	$0.474853\pi$
$504$	0	0
$505$	−18.4994	−0.823211
$506$	−1.30292	−0.0579218
$507$	0	0
$508$	7.85035	0.348303
$509$	−24.4026	−1.08163	−0.540814	−	0.841142i	$-0.681884\pi$
$509$	−24.4026	−1.08163	−0.540814	+	0.841142i	$0.681884\pi$
$510$	0	0
$511$	−2.61008	−0.115463
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.86558	0.214611
$515$	−3.73431	−0.164554
$516$	0	0
$517$	−10.2564	−0.451077
$518$	−2.71098	−0.119114
$519$	0	0
$520$	5.37770	0.235828
$521$	11.6635	0.510988	0.255494	−	0.966811i	$-0.417762\pi$
$521$	11.6635	0.510988	0.255494	+	0.966811i	$0.417762\pi$
$522$	0	0
$523$	5.70574	0.249495	0.124747	−	0.992189i	$-0.460188\pi$
$523$	5.70574	0.249495	0.124747	+	0.992189i	$0.460188\pi$
$524$	−0.0464883	−0.00203085
$525$	0	0
$526$	6.67146	0.290890
$527$	20.9305	0.911746
$528$	0	0
$529$	−21.3024	−0.926191
$530$	−10.1727	−0.441874
$531$	0	0
$532$	−0.913995	−0.0396267
$533$	−3.53247	−0.153008
$534$	0	0
$535$	11.7159	0.506522
$536$	5.21425	0.225221
$537$	0	0
$538$	16.6115	0.716172
$539$	−6.94002	−0.298928
$540$	0	0
$541$	21.6734	0.931814	0.465907	−	0.884834i	$-0.345728\pi$
$541$	21.6734	0.931814	0.465907	+	0.884834i	$0.345728\pi$
$542$	−22.4154	−0.962822
$543$	0	0
$544$	2.66909	0.114436
$545$	−1.71983	−0.0736696
$546$	0	0
$547$	17.6072	0.752829	0.376414	−	0.926451i	$-0.377157\pi$
$547$	17.6072	0.752829	0.376414	+	0.926451i	$0.377157\pi$
$548$	5.47563	0.233907
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−2.81806	−0.120053
$552$	0	0
$553$	1.15195	0.0489858
$554$	−22.6226	−0.961143
$555$	0	0
$556$	−4.58548	−0.194467
$557$	−24.1148	−1.02178	−0.510888	−	0.859647i	$-0.670683\pi$
$557$	−24.1148	−1.02178	−0.510888	+	0.859647i	$0.670683\pi$
$558$	0	0
$559$	68.9088	2.91453
$560$	0.244904	0.0103491
$561$	0	0
$562$	20.3902	0.860109
$563$	1.81089	0.0763199	0.0381600	−	0.999272i	$-0.487850\pi$
$563$	1.81089	0.0763199	0.0381600	+	0.999272i	$0.487850\pi$
$564$	0	0
$565$	−17.8339	−0.750277
$566$	−18.0154	−0.757245
$567$	0	0
$568$	−8.14698	−0.341840
$569$	19.9810	0.837649	0.418824	−	0.908067i	$-0.362442\pi$
$569$	19.9810	0.837649	0.418824	+	0.908067i	$0.362442\pi$
$570$	0	0
$571$	−17.6296	−0.737776	−0.368888	−	0.929474i	$-0.620261\pi$
$571$	−17.6296	−0.737776	−0.368888	+	0.929474i	$0.620261\pi$
$572$	5.37770	0.224853
$573$	0	0
$574$	−0.160871	−0.00671463
$575$	1.30292	0.0543355
$576$	0	0
$577$	−19.6470	−0.817915	−0.408957	−	0.912554i	$-0.634108\pi$
$577$	−19.6470	−0.817915	−0.408957	+	0.912554i	$0.634108\pi$
$578$	9.87595	0.410786
$579$	0	0
$580$	0.755096	0.0313536
$581$	1.54926	0.0642739
$582$	0	0
$583$	−10.1727	−0.421311
$584$	−10.6576	−0.441014
$585$	0	0
$586$	32.5118	1.34305
$587$	−29.8613	−1.23251	−0.616253	−	0.787548i	$-0.711350\pi$
$587$	−29.8613	−1.23251	−0.616253	+	0.787548i	$0.711350\pi$
$588$	0	0
$589$	−29.2660	−1.20588
$590$	−9.74358	−0.401137
$591$	0	0
$592$	−11.0695	−0.454955
$593$	15.0014	0.616035	0.308018	−	0.951381i	$-0.400334\pi$
$593$	15.0014	0.616035	0.308018	+	0.951381i	$0.400334\pi$
$594$	0	0
$595$	−0.653672	−0.0267979
$596$	−3.53012	−0.144599
$597$	0	0
$598$	−7.00671	−0.286526
$599$	22.9062	0.935923	0.467962	−	0.883749i	$-0.344989\pi$
$599$	22.9062	0.935923	0.467962	+	0.883749i	$0.344989\pi$
$600$	0	0
$601$	5.41954	0.221068	0.110534	−	0.993872i	$-0.464744\pi$
$601$	5.41954	0.221068	0.110534	+	0.993872i	$0.464744\pi$
$602$	3.13816	0.127902
$603$	0	0
$604$	9.77296	0.397656
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−45.6589	−1.85324	−0.926619	−	0.376003i	$-0.877298\pi$
$607$	−45.6589	−1.85324	−0.926619	+	0.376003i	$0.877298\pi$
$608$	−3.73205	−0.151355
$609$	0	0
$610$	3.38932	0.137230
$611$	−55.1560	−2.23137
$612$	0	0
$613$	3.12054	0.126037	0.0630187	−	0.998012i	$-0.479927\pi$
$613$	3.12054	0.126037	0.0630187	+	0.998012i	$0.479927\pi$
$614$	30.1273	1.21584
$615$	0	0
$616$	0.244904	0.00986747
$617$	−3.51277	−0.141419	−0.0707093	−	0.997497i	$-0.522526\pi$
$617$	−3.51277	−0.141419	−0.0707093	+	0.997497i	$0.522526\pi$
$618$	0	0
$619$	−34.3158	−1.37927	−0.689634	−	0.724159i	$-0.742228\pi$
$619$	−34.3158	−1.37927	−0.689634	+	0.724159i	$0.742228\pi$
$620$	7.84180	0.314934
$621$	0	0
$622$	−9.29102	−0.372536
$623$	1.30839	0.0524193
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.5036	0.819489
$627$	0	0
$628$	−5.18947	−0.207082
$629$	29.5456	1.17806
$630$	0	0
$631$	−22.2642	−0.886322	−0.443161	−	0.896442i	$-0.646143\pi$
$631$	−22.2642	−0.886322	−0.443161	+	0.896442i	$0.646143\pi$
$632$	4.70366	0.187102
$633$	0	0
$634$	−22.5858	−0.896997
$635$	−7.85035	−0.311531
$636$	0	0
$637$	−37.3213	−1.47873
$638$	0.755096	0.0298945
$639$	0	0
$640$	1.00000	0.0395285
$641$	−11.5678	−0.456900	−0.228450	−	0.973556i	$-0.573366\pi$
$641$	−11.5678	−0.456900	−0.228450	+	0.973556i	$0.573366\pi$
$642$	0	0
$643$	0.138516	0.00546254	0.00273127	−	0.999996i	$-0.499131\pi$
$643$	0.138516	0.00546254	0.00273127	+	0.999996i	$0.499131\pi$
$644$	−0.319091	−0.0125739
$645$	0	0
$646$	9.96118	0.391917
$647$	−6.69636	−0.263261	−0.131630	−	0.991299i	$-0.542021\pi$
$647$	−6.69636	−0.263261	−0.131630	+	0.991299i	$0.542021\pi$
$648$	0	0
$649$	−9.74358	−0.382469
$650$	−5.37770	−0.210931
$651$	0	0
$652$	−18.9321	−0.741438
$653$	−2.15144	−0.0841924	−0.0420962	−	0.999114i	$-0.513404\pi$
$653$	−2.15144	−0.0841924	−0.0420962	+	0.999114i	$0.513404\pi$
$654$	0	0
$655$	0.0464883	0.00181645
$656$	−0.656874	−0.0256466
$657$	0	0
$658$	−2.51184	−0.0979219
$659$	8.56357	0.333589	0.166795	−	0.985992i	$-0.446658\pi$
$659$	8.56357	0.333589	0.166795	+	0.985992i	$0.446658\pi$
$660$	0	0
$661$	21.0533	0.818879	0.409440	−	0.912337i	$-0.365724\pi$
$661$	21.0533	0.818879	0.409440	+	0.912337i	$0.365724\pi$
$662$	−12.1160	−0.470903
$663$	0	0
$664$	6.32596	0.245495
$665$	0.913995	0.0354432
$666$	0	0
$667$	−0.983829	−0.0380940
$668$	−25.3359	−0.980276
$669$	0	0
$670$	−5.21425	−0.201444
$671$	3.38932	0.130843
$672$	0	0
$673$	8.97030	0.345780	0.172890	−	0.984941i	$-0.444690\pi$
$673$	8.97030	0.345780	0.172890	+	0.984941i	$0.444690\pi$
$674$	29.4419	1.13406
$675$	0	0
$676$	15.9196	0.612294
$677$	4.25694	0.163607	0.0818037	−	0.996648i	$-0.473932\pi$
$677$	4.25694	0.163607	0.0818037	+	0.996648i	$0.473932\pi$
$678$	0	0
$679$	1.32332	0.0507842
$680$	−2.66909	−0.102355
$681$	0	0
$682$	7.84180	0.300278
$683$	25.1183	0.961123	0.480562	−	0.876961i	$-0.340433\pi$
$683$	25.1183	0.961123	0.480562	+	0.876961i	$0.340433\pi$
$684$	0	0
$685$	−5.47563	−0.209213
$686$	−3.41397	−0.130346
$687$	0	0
$688$	12.8138	0.488522
$689$	−54.7058	−2.08412
$690$	0	0
$691$	−1.98416	−0.0754811	−0.0377406	−	0.999288i	$-0.512016\pi$
$691$	−1.98416	−0.0754811	−0.0377406	+	0.999288i	$0.512016\pi$
$692$	8.97666	0.341242
$693$	0	0
$694$	24.8035	0.941526
$695$	4.58548	0.173937
$696$	0	0
$697$	1.75326	0.0664093
$698$	−1.80944	−0.0684883
$699$	0	0
$700$	−0.244904	−0.00925651
$701$	25.9263	0.979225	0.489612	−	0.871940i	$-0.337138\pi$
$701$	25.9263	0.979225	0.489612	+	0.871940i	$0.337138\pi$
$702$	0	0
$703$	−41.3121	−1.55812
$704$	1.00000	0.0376889
$705$	0	0
$706$	14.6947	0.553043
$707$	−4.53057	−0.170390
$708$	0	0
$709$	0.429643	0.0161356	0.00806779	−	0.999967i	$-0.497432\pi$
$709$	0.429643	0.0161356	0.00806779	+	0.999967i	$0.497432\pi$
$710$	8.14698	0.305751
$711$	0	0
$712$	5.34244	0.200216
$713$	−10.2172	−0.382638
$714$	0	0
$715$	−5.37770	−0.201115
$716$	14.1974	0.530581
$717$	0	0
$718$	32.6998	1.22034
$719$	−29.2134	−1.08948	−0.544738	−	0.838606i	$-0.683371\pi$
$719$	−29.2134	−1.08948	−0.544738	+	0.838606i	$0.683371\pi$
$720$	0	0
$721$	−0.914549	−0.0340596
$722$	5.07180	0.188753
$723$	0	0
$724$	15.8646	0.589601
$725$	−0.755096	−0.0280436
$726$	0	0
$727$	−15.9710	−0.592331	−0.296166	−	0.955137i	$-0.595708\pi$
$727$	−15.9710	−0.592331	−0.296166	+	0.955137i	$0.595708\pi$
$728$	1.31702	0.0488120
$729$	0	0
$730$	10.6576	0.394454
$731$	−34.2012	−1.26498
$732$	0	0
$733$	−35.0695	−1.29532	−0.647662	−	0.761928i	$-0.724253\pi$
$733$	−35.0695	−1.29532	−0.647662	+	0.761928i	$0.724253\pi$
$734$	−21.0799	−0.778073
$735$	0	0
$736$	−1.30292	−0.0480263
$737$	−5.21425	−0.192069
$738$	0	0
$739$	6.22948	0.229155	0.114578	−	0.993414i	$-0.463449\pi$
$739$	6.22948	0.229155	0.114578	+	0.993414i	$0.463449\pi$
$740$	11.0695	0.406924
$741$	0	0
$742$	−2.49134	−0.0914599
$743$	32.1666	1.18008	0.590040	−	0.807374i	$-0.299112\pi$
$743$	32.1666	1.18008	0.590040	+	0.807374i	$0.299112\pi$
$744$	0	0
$745$	3.53012	0.129334
$746$	30.5860	1.11983
$747$	0	0
$748$	−2.66909	−0.0975916
$749$	2.86927	0.104841
$750$	0	0
$751$	50.3650	1.83785	0.918923	−	0.394437i	$-0.129060\pi$
$751$	50.3650	1.83785	0.918923	+	0.394437i	$0.129060\pi$
$752$	−10.2564	−0.374014
$753$	0	0
$754$	4.06068	0.147881
$755$	−9.77296	−0.355674
$756$	0	0
$757$	0.485112	0.0176317	0.00881586	−	0.999961i	$-0.497194\pi$
$757$	0.485112	0.0176317	0.00881586	+	0.999961i	$0.497194\pi$
$758$	5.82392	0.211534
$759$	0	0
$760$	3.73205	0.135376
$761$	40.0985	1.45357	0.726785	−	0.686865i	$-0.241013\pi$
$761$	40.0985	1.45357	0.726785	+	0.686865i	$0.241013\pi$
$762$	0	0
$763$	−0.421195	−0.0152483
$764$	−1.22803	−0.0444287
$765$	0	0
$766$	−5.97542	−0.215901
$767$	−52.3980	−1.89198
$768$	0	0
$769$	30.2534	1.09097	0.545484	−	0.838122i	$-0.316346\pi$
$769$	30.2534	1.09097	0.545484	+	0.838122i	$0.316346\pi$
$770$	−0.244904	−0.00882574
$771$	0	0
$772$	−17.4354	−0.627515
$773$	44.5404	1.60201	0.801004	−	0.598660i	$-0.204300\pi$
$773$	44.5404	1.60201	0.801004	+	0.598660i	$0.204300\pi$
$774$	0	0
$775$	−7.84180	−0.281686
$776$	5.40341	0.193971
$777$	0	0
$778$	−38.1731	−1.36857
$779$	−2.45149	−0.0878336
$780$	0	0
$781$	8.14698	0.291522
$782$	3.47761	0.124359
$783$	0	0
$784$	−6.94002	−0.247858
$785$	5.18947	0.185220
$786$	0	0
$787$	−14.4932	−0.516628	−0.258314	−	0.966061i	$-0.583167\pi$
$787$	−14.4932	−0.516628	−0.258314	+	0.966061i	$0.583167\pi$
$788$	5.18660	0.184765
$789$	0	0
$790$	−4.70366	−0.167349
$791$	−4.36759	−0.155294
$792$	0	0
$793$	18.2268	0.647251
$794$	30.5803	1.08526
$795$	0	0
$796$	27.1395	0.961935
$797$	−16.3416	−0.578851	−0.289425	−	0.957201i	$-0.593464\pi$
$797$	−16.3416	−0.578851	−0.289425	+	0.957201i	$0.593464\pi$
$798$	0	0
$799$	27.3753	0.968470
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−18.8798	−0.666667
$803$	10.6576	0.376098
$804$	0	0
$805$	0.319091	0.0112465
$806$	42.1708	1.48540
$807$	0	0
$808$	−18.4994	−0.650806
$809$	−23.0812	−0.811493	−0.405747	−	0.913986i	$-0.632988\pi$
$809$	−23.0812	−0.811493	−0.405747	+	0.913986i	$0.632988\pi$
$810$	0	0
$811$	43.5350	1.52872	0.764360	−	0.644790i	$-0.223055\pi$
$811$	43.5350	1.52872	0.764360	+	0.644790i	$0.223055\pi$
$812$	0.184926	0.00648964
$813$	0	0
$814$	11.0695	0.387987
$815$	18.9321	0.663163
$816$	0	0
$817$	47.8218	1.67307
$818$	−20.7108	−0.724137
$819$	0	0
$820$	0.656874	0.0229390
$821$	−21.8047	−0.760988	−0.380494	−	0.924783i	$-0.624246\pi$
$821$	−21.8047	−0.760988	−0.380494	+	0.924783i	$0.624246\pi$
$822$	0	0
$823$	−25.2264	−0.879335	−0.439668	−	0.898161i	$-0.644904\pi$
$823$	−25.2264	−0.879335	−0.439668	+	0.898161i	$0.644904\pi$
$824$	−3.73431	−0.130091
$825$	0	0
$826$	−2.38624	−0.0830280
$827$	−4.61825	−0.160592	−0.0802962	−	0.996771i	$-0.525587\pi$
$827$	−4.61825	−0.160592	−0.0802962	+	0.996771i	$0.525587\pi$
$828$	0	0
$829$	28.1487	0.977646	0.488823	−	0.872383i	$-0.337426\pi$
$829$	28.1487	0.977646	0.488823	+	0.872383i	$0.337426\pi$
$830$	−6.32596	−0.219577
$831$	0	0
$832$	5.37770	0.186438
$833$	18.5235	0.641803
$834$	0	0
$835$	25.3359	0.876786
$836$	3.73205	0.129076
$837$	0	0
$838$	−10.3608	−0.357909
$839$	19.4500	0.671489	0.335744	−	0.941953i	$-0.391012\pi$
$839$	19.4500	0.671489	0.335744	+	0.941953i	$0.391012\pi$
$840$	0	0
$841$	−28.4298	−0.980339
$842$	36.5611	1.25998
$843$	0	0
$844$	−14.9823	−0.515712
$845$	−15.9196	−0.547652
$846$	0	0
$847$	−0.244904	−0.00841501
$848$	−10.1727	−0.349332
$849$	0	0
$850$	2.66909	0.0915491
$851$	−14.4227	−0.494405
$852$	0	0
$853$	−40.0697	−1.37196	−0.685980	−	0.727621i	$-0.740626\pi$
$853$	−40.0697	−1.37196	−0.685980	+	0.727621i	$0.740626\pi$
$854$	0.830060	0.0284041
$855$	0	0
$856$	11.7159	0.400441
$857$	27.3356	0.933767	0.466883	−	0.884319i	$-0.345377\pi$
$857$	27.3356	0.933767	0.466883	+	0.884319i	$0.345377\pi$
$858$	0	0
$859$	−19.9778	−0.681635	−0.340818	−	0.940129i	$-0.610704\pi$
$859$	−19.9778	−0.681635	−0.340818	+	0.940129i	$0.610704\pi$
$860$	−12.8138	−0.436947
$861$	0	0
$862$	19.6563	0.669498
$863$	43.6948	1.48739	0.743693	−	0.668521i	$-0.233072\pi$
$863$	43.6948	1.48739	0.743693	+	0.668521i	$0.233072\pi$
$864$	0	0
$865$	−8.97666	−0.305216
$866$	−32.8936	−1.11777
$867$	0	0
$868$	1.92049	0.0651857
$869$	−4.70366	−0.159561
$870$	0	0
$871$	−28.0407	−0.950122
$872$	−1.71983	−0.0582409
$873$	0	0
$874$	−4.86256	−0.164479
$875$	0.244904	0.00827927
$876$	0	0
$877$	11.3636	0.383720	0.191860	−	0.981422i	$-0.438548\pi$
$877$	11.3636	0.383720	0.191860	+	0.981422i	$0.438548\pi$
$878$	−14.5870	−0.492289
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−42.3780	−1.42775	−0.713875	−	0.700273i	$-0.753062\pi$
$881$	−42.3780	−1.42775	−0.713875	+	0.700273i	$0.753062\pi$
$882$	0	0
$883$	−48.2231	−1.62284	−0.811418	−	0.584467i	$-0.801304\pi$
$883$	−48.2231	−1.62284	−0.811418	+	0.584467i	$0.801304\pi$
$884$	−14.3536	−0.482763
$885$	0	0
$886$	3.60743	0.121194
$887$	15.7171	0.527728	0.263864	−	0.964560i	$-0.415003\pi$
$887$	15.7171	0.527728	0.263864	+	0.964560i	$0.415003\pi$
$888$	0	0
$889$	−1.92258	−0.0644814
$890$	−5.34244	−0.179079
$891$	0	0
$892$	10.9914	0.368020
$893$	−38.2775	−1.28091
$894$	0	0
$895$	−14.1974	−0.474566
$896$	0.244904	0.00818168
$897$	0	0
$898$	−4.31769	−0.144083
$899$	5.92131	0.197487
$900$	0	0
$901$	27.1519	0.904560
$902$	0.656874	0.0218715
$903$	0	0
$904$	−17.8339	−0.593146
$905$	−15.8646	−0.527356
$906$	0	0
$907$	−22.6927	−0.753498	−0.376749	−	0.926315i	$-0.622958\pi$
$907$	−22.6927	−0.753498	−0.376749	+	0.926315i	$0.622958\pi$
$908$	1.16282	0.0385895
$909$	0	0
$910$	−1.31702	−0.0436588
$911$	0.577461	0.0191321	0.00956607	−	0.999954i	$-0.496955\pi$
$911$	0.577461	0.0191321	0.00956607	+	0.999954i	$0.496955\pi$
$912$	0	0
$913$	−6.32596	−0.209359
$914$	−6.94112	−0.229592
$915$	0	0
$916$	−21.8910	−0.723298
$917$	0.0113852	0.000375972 0
$918$	0	0
$919$	−47.9577	−1.58198	−0.790989	−	0.611831i	$-0.790433\pi$
$919$	−47.9577	−1.58198	−0.790989	+	0.611831i	$0.790433\pi$
$920$	1.30292	0.0429560
$921$	0	0
$922$	21.6827	0.714081
$923$	43.8120	1.44209
$924$	0	0
$925$	−11.0695	−0.363964
$926$	25.9403	0.852452
$927$	0	0
$928$	0.755096	0.0247872
$929$	−54.2644	−1.78036	−0.890179	−	0.455611i	$-0.849421\pi$
$929$	−54.2644	−1.78036	−0.890179	+	0.455611i	$0.849421\pi$
$930$	0	0
$931$	−25.9005	−0.848855
$932$	−27.0119	−0.884805
$933$	0	0
$934$	27.6311	0.904116
$935$	2.66909	0.0872886
$936$	0	0
$937$	4.07367	0.133081	0.0665405	−	0.997784i	$-0.478804\pi$
$937$	4.07367	0.133081	0.0665405	+	0.997784i	$0.478804\pi$
$938$	−1.27699	−0.0416953
$939$	0	0
$940$	10.2564	0.334528
$941$	−0.723441	−0.0235835	−0.0117918	−	0.999930i	$-0.503754\pi$
$941$	−0.723441	−0.0235835	−0.0117918	+	0.999930i	$0.503754\pi$
$942$	0	0
$943$	−0.855854	−0.0278704
$944$	−9.74358	−0.317126
$945$	0	0
$946$	−12.8138	−0.416613
$947$	−33.7945	−1.09817	−0.549086	−	0.835766i	$-0.685024\pi$
$947$	−33.7945	−1.09817	−0.549086	+	0.835766i	$0.685024\pi$
$948$	0	0
$949$	57.3132	1.86047
$950$	−3.73205	−0.121084
$951$	0	0
$952$	−0.653672	−0.0211856
$953$	−41.4520	−1.34276	−0.671381	−	0.741112i	$-0.734299\pi$
$953$	−41.4520	−1.34276	−0.671381	+	0.741112i	$0.734299\pi$
$954$	0	0
$955$	1.22803	0.0397383
$956$	−18.2549	−0.590405
$957$	0	0
$958$	−28.6571	−0.925868
$959$	−1.34100	−0.0433033
$960$	0	0
$961$	30.4938	0.983672
$962$	59.5287	1.91928
$963$	0	0
$964$	−8.96893	−0.288870
$965$	17.4354	0.561266
$966$	0	0
$967$	−1.78603	−0.0574349	−0.0287175	−	0.999588i	$-0.509142\pi$
$967$	−1.78603	−0.0574349	−0.0287175	+	0.999588i	$0.509142\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−5.40341	−0.173493
$971$	16.2976	0.523014	0.261507	−	0.965202i	$-0.415780\pi$
$971$	16.2976	0.523014	0.261507	+	0.965202i	$0.415780\pi$
$972$	0	0
$973$	1.12300	0.0360018
$974$	−23.4297	−0.750737
$975$	0	0
$976$	3.38932	0.108490
$977$	−12.9577	−0.414552	−0.207276	−	0.978283i	$-0.566460\pi$
$977$	−12.9577	−0.414552	−0.207276	+	0.978283i	$0.566460\pi$
$978$	0	0
$979$	−5.34244	−0.170745
$980$	6.94002	0.221691
$981$	0	0
$982$	−34.9801	−1.11626
$983$	51.0597	1.62855	0.814276	−	0.580478i	$-0.197134\pi$
$983$	51.0597	1.62855	0.814276	+	0.580478i	$0.197134\pi$
$984$	0	0
$985$	−5.18660	−0.165259
$986$	−2.01542	−0.0641840
$987$	0	0
$988$	20.0698	0.638507
$989$	16.6954	0.530882
$990$	0	0
$991$	−50.2854	−1.59737	−0.798684	−	0.601751i	$-0.794470\pi$
$991$	−50.2854	−1.59737	−0.798684	+	0.601751i	$0.794470\pi$
$992$	7.84180	0.248977
$993$	0	0
$994$	1.99523	0.0632849
$995$	−27.1395	−0.860381
$996$	0	0
$997$	31.5446	0.999027	0.499514	−	0.866306i	$-0.333512\pi$
$997$	31.5446	0.999027	0.499514	+	0.866306i	$0.333512\pi$
$998$	7.37154	0.233342
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8910.2.a.bw.1.4	✓	6
3.2	odd	2		8910.2.a.bz.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8910.2.a.bw.1.4	✓	6	1.1	even	1	trivial
8910.2.a.bz.1.4	yes	6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8910.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.244904 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +5.37770 q^{13} +0.244904 q^{14} +1.00000 q^{16} -2.66909 q^{17} +3.73205 q^{19} -1.00000 q^{20} -1.00000 q^{22} +1.30292 q^{23} +1.00000 q^{25} -5.37770 q^{26} -0.244904 q^{28} -0.755096 q^{29} -7.84180 q^{31} -1.00000 q^{32} +2.66909 q^{34} +0.244904 q^{35} -11.0695 q^{37} -3.73205 q^{38} +1.00000 q^{40} -0.656874 q^{41} +12.8138 q^{43} +1.00000 q^{44} -1.30292 q^{46} -10.2564 q^{47} -6.94002 q^{49} -1.00000 q^{50} +5.37770 q^{52} -10.1727 q^{53} -1.00000 q^{55} +0.244904 q^{56} +0.755096 q^{58} -9.74358 q^{59} +3.38932 q^{61} +7.84180 q^{62} +1.00000 q^{64} -5.37770 q^{65} -5.21425 q^{67} -2.66909 q^{68} -0.244904 q^{70} +8.14698 q^{71} +10.6576 q^{73} +11.0695 q^{74} +3.73205 q^{76} -0.244904 q^{77} -4.70366 q^{79} -1.00000 q^{80} +0.656874 q^{82} -6.32596 q^{83} +2.66909 q^{85} -12.8138 q^{86} -1.00000 q^{88} -5.34244 q^{89} -1.31702 q^{91} +1.30292 q^{92} +10.2564 q^{94} -3.73205 q^{95} -5.40341 q^{97} +6.94002 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 2 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 6 q^{16} - 8 q^{17} + 12 q^{19} - 6 q^{20} - 6 q^{22} - 10 q^{23} + 6 q^{25} + 2 q^{26} - 2 q^{28} - 4 q^{29}+ \cdots - 14 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 2 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 6 * q^16 - 8 * q^17 + 12 * q^19 - 6 * q^20 - 6 * q^22 - 10 * q^23 + 6 * q^25 + 2 * q^26 - 2 * q^28 - 4 * q^29 + 8 * q^31 - 6 * q^32 + 8 * q^34 + 2 * q^35 - 4 * q^37 - 12 * q^38 + 6 * q^40 - 4 * q^41 - 4 * q^43 + 6 * q^44 + 10 * q^46 - 18 * q^47 + 14 * q^49 - 6 * q^50 - 2 * q^52 - 14 * q^53 - 6 * q^55 + 2 * q^56 + 4 * q^58 - 4 * q^59 + 16 * q^61 - 8 * q^62 + 6 * q^64 + 2 * q^65 + 6 * q^67 - 8 * q^68 - 2 * q^70 - 18 * q^71 + 2 * q^73 + 4 * q^74 + 12 * q^76 - 2 * q^77 + 14 * q^79 - 6 * q^80 + 4 * q^82 - 30 * q^83 + 8 * q^85 + 4 * q^86 - 6 * q^88 + 4 * q^89 + 20 * q^91 - 10 * q^92 + 18 * q^94 - 12 * q^95 - 20 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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