LMFDB - Embedded newform 2888.2.a.x.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,-3,0,3,0,9,0,6,0,3,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 12x^{7} + 35x^{6} + 45x^{5} - 117x^{4} - 55x^{3} + 96x^{2} - 6x - 1 $ x^9 - 3x^8 - 12x^7 + 35x^6 + 45x^5 - 117x^4 - 55x^3 + 96x^2 - 6x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.95251$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95251 q^{3} +1.73942 q^{5} +1.92522 q^{7} +0.812302 q^{9} +4.55495 q^{11} +2.74488 q^{13} +3.39623 q^{15} +5.01391 q^{17} +3.75901 q^{21} +1.14055 q^{23} -1.97443 q^{25} -4.27151 q^{27} -5.00912 q^{29} -1.99987 q^{31} +8.89359 q^{33} +3.34876 q^{35} +7.06608 q^{37} +5.35941 q^{39} -11.0249 q^{41} +6.10421 q^{43} +1.41293 q^{45} -8.77261 q^{47} -3.29353 q^{49} +9.78972 q^{51} -0.0537475 q^{53} +7.92296 q^{55} -3.70019 q^{59} -14.9905 q^{61} +1.56386 q^{63} +4.77449 q^{65} -12.6944 q^{67} +2.22694 q^{69} +2.06940 q^{71} -0.713001 q^{73} -3.85509 q^{75} +8.76928 q^{77} +11.2592 q^{79} -10.7771 q^{81} +9.98525 q^{83} +8.72129 q^{85} -9.78037 q^{87} +16.1999 q^{89} +5.28449 q^{91} -3.90477 q^{93} -0.372404 q^{97} +3.69999 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 3 q^{11} + 6 q^{13} - 3 q^{17} - 15 q^{21} + 24 q^{23} + 30 q^{25} - 12 q^{27} + 15 q^{29} - 6 q^{31} + 18 q^{33} + 15 q^{35} - 24 q^{37} + 6 q^{39} + 12 q^{41}+ \cdots + 48 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9 + 3 * q^11 + 6 * q^13 - 3 * q^17 - 15 * q^21 + 24 * q^23 + 30 * q^25 - 12 * q^27 + 15 * q^29 - 6 * q^31 + 18 * q^33 + 15 * q^35 - 24 * q^37 + 6 * q^39 + 12 * q^41 + 9 * q^43 + 42 * q^45 - 12 * q^47 + 18 * q^49 - 12 * q^51 + 18 * q^53 + 21 * q^55 - 57 * q^59 + 15 * q^61 + 30 * q^63 + 27 * q^65 - 6 * q^67 - 3 * q^69 + 36 * q^73 - 45 * q^75 + 30 * q^77 + 3 * q^79 - 15 * q^81 + 27 * q^83 + 18 * q^85 - 18 * q^87 + 30 * q^89 + 21 * q^91 - 24 * q^93 - 3 * q^97 + 48 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.95251	1.12728	0.563642	−	0.826019i	$-0.309400\pi$
$3$	1.95251	1.12728	0.563642	+	0.826019i	$0.309400\pi$
$4$	0	0
$5$	1.73942	0.777891	0.388945	−	0.921261i	$-0.372839\pi$
$5$	1.73942	0.777891	0.388945	+	0.921261i	$0.372839\pi$
$6$	0	0
$7$	1.92522	0.727664	0.363832	−	0.931465i	$-0.381468\pi$
$7$	1.92522	0.727664	0.363832	+	0.931465i	$0.381468\pi$
$8$	0	0
$9$	0.812302	0.270767
$10$	0	0
$11$	4.55495	1.37337	0.686685	−	0.726956i	$-0.259065\pi$
$11$	4.55495	1.37337	0.686685	+	0.726956i	$0.259065\pi$
$12$	0	0
$13$	2.74488	0.761292	0.380646	−	0.924721i	$-0.375702\pi$
$13$	2.74488	0.761292	0.380646	+	0.924721i	$0.375702\pi$
$14$	0	0
$15$	3.39623	0.876903
$16$	0	0
$17$	5.01391	1.21605	0.608026	−	0.793917i	$-0.291962\pi$
$17$	5.01391	1.21605	0.608026	+	0.793917i	$0.291962\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.75901	0.820284
$22$	0	0
$23$	1.14055	0.237821	0.118911	−	0.992905i	$-0.462060\pi$
$23$	1.14055	0.237821	0.118911	+	0.992905i	$0.462060\pi$
$24$	0	0
$25$	−1.97443	−0.394886
$26$	0	0
$27$	−4.27151	−0.822052
$28$	0	0
$29$	−5.00912	−0.930170	−0.465085	−	0.885266i	$-0.653976\pi$
$29$	−5.00912	−0.930170	−0.465085	+	0.885266i	$0.653976\pi$
$30$	0	0
$31$	−1.99987	−0.359188	−0.179594	−	0.983741i	$-0.557478\pi$
$31$	−1.99987	−0.359188	−0.179594	+	0.983741i	$0.557478\pi$
$32$	0	0
$33$	8.89359	1.54818
$34$	0	0
$35$	3.34876	0.566043
$36$	0	0
$37$	7.06608	1.16166	0.580828	−	0.814026i	$-0.302729\pi$
$37$	7.06608	1.16166	0.580828	+	0.814026i	$0.302729\pi$
$38$	0	0
$39$	5.35941	0.858192
$40$	0	0
$41$	−11.0249	−1.72180	−0.860902	−	0.508770i	$-0.830100\pi$
$41$	−11.0249	−1.72180	−0.860902	+	0.508770i	$0.830100\pi$
$42$	0	0
$43$	6.10421	0.930883	0.465442	−	0.885079i	$-0.345896\pi$
$43$	6.10421	0.930883	0.465442	+	0.885079i	$0.345896\pi$
$44$	0	0
$45$	1.41293	0.210627
$46$	0	0
$47$	−8.77261	−1.27962	−0.639808	−	0.768535i	$-0.720986\pi$
$47$	−8.77261	−1.27962	−0.639808	+	0.768535i	$0.720986\pi$
$48$	0	0
$49$	−3.29353	−0.470505
$50$	0	0
$51$	9.78972	1.37084
$52$	0	0
$53$	−0.0537475	−0.00738278	−0.00369139	−	0.999993i	$-0.501175\pi$
$53$	−0.0537475	−0.00738278	−0.00369139	+	0.999993i	$0.501175\pi$
$54$	0	0
$55$	7.92296	1.06833
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.70019	−0.481724	−0.240862	−	0.970559i	$-0.577430\pi$
$59$	−3.70019	−0.481724	−0.240862	+	0.970559i	$0.577430\pi$
$60$	0	0
$61$	−14.9905	−1.91933	−0.959665	−	0.281145i	$-0.909286\pi$
$61$	−14.9905	−1.91933	−0.959665	+	0.281145i	$0.909286\pi$
$62$	0	0
$63$	1.56386	0.197028
$64$	0	0
$65$	4.77449	0.592202
$66$	0	0
$67$	−12.6944	−1.55087	−0.775435	−	0.631427i	$-0.782469\pi$
$67$	−12.6944	−1.55087	−0.775435	+	0.631427i	$0.782469\pi$
$68$	0	0
$69$	2.22694	0.268092
$70$	0	0
$71$	2.06940	0.245592	0.122796	−	0.992432i	$-0.460814\pi$
$71$	2.06940	0.245592	0.122796	+	0.992432i	$0.460814\pi$
$72$	0	0
$73$	−0.713001	−0.0834505	−0.0417253	−	0.999129i	$-0.513285\pi$
$73$	−0.713001	−0.0834505	−0.0417253	+	0.999129i	$0.513285\pi$
$74$	0	0
$75$	−3.85509	−0.445148
$76$	0	0
$77$	8.76928	0.999352
$78$	0	0
$79$	11.2592	1.26676	0.633378	−	0.773843i	$-0.281668\pi$
$79$	11.2592	1.26676	0.633378	+	0.773843i	$0.281668\pi$
$80$	0	0
$81$	−10.7771	−1.19745
$82$	0	0
$83$	9.98525	1.09602	0.548012	−	0.836471i	$-0.315385\pi$
$83$	9.98525	1.09602	0.548012	+	0.836471i	$0.315385\pi$
$84$	0	0
$85$	8.72129	0.945956
$86$	0	0
$87$	−9.78037	−1.04857
$88$	0	0
$89$	16.1999	1.71718	0.858592	−	0.512660i	$-0.171340\pi$
$89$	16.1999	1.71718	0.858592	+	0.512660i	$0.171340\pi$
$90$	0	0
$91$	5.28449	0.553965
$92$	0	0
$93$	−3.90477	−0.404906
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.372404	−0.0378119	−0.0189060	−	0.999821i	$-0.506018\pi$
$97$	−0.372404	−0.0378119	−0.0189060	+	0.999821i	$0.506018\pi$
$98$	0	0
$99$	3.69999	0.371863
$100$	0	0
$101$	18.5966	1.85043	0.925213	−	0.379447i	$-0.123886\pi$
$101$	18.5966	1.85043	0.925213	+	0.379447i	$0.123886\pi$
$102$	0	0
$103$	−8.97947	−0.884773	−0.442387	−	0.896824i	$-0.645868\pi$
$103$	−8.97947	−0.884773	−0.442387	+	0.896824i	$0.645868\pi$
$104$	0	0
$105$	6.53849	0.638091
$106$	0	0
$107$	−7.48664	−0.723761	−0.361881	−	0.932224i	$-0.617865\pi$
$107$	−7.48664	−0.723761	−0.361881	+	0.932224i	$0.617865\pi$
$108$	0	0
$109$	16.7733	1.60659	0.803297	−	0.595578i	$-0.203077\pi$
$109$	16.7733	1.60659	0.803297	+	0.595578i	$0.203077\pi$
$110$	0	0
$111$	13.7966	1.30952
$112$	0	0
$113$	−10.6659	−1.00336	−0.501682	−	0.865052i	$-0.667285\pi$
$113$	−10.6659	−1.00336	−0.501682	+	0.865052i	$0.667285\pi$
$114$	0	0
$115$	1.98389	0.184999
$116$	0	0
$117$	2.22967	0.206133
$118$	0	0
$119$	9.65288	0.884878
$120$	0	0
$121$	9.74757	0.886143
$122$	0	0
$123$	−21.5263	−1.94096
$124$	0	0
$125$	−12.1314	−1.08507
$126$	0	0
$127$	−11.0279	−0.978565	−0.489282	−	0.872125i	$-0.662741\pi$
$127$	−11.0279	−0.978565	−0.489282	+	0.872125i	$0.662741\pi$
$128$	0	0
$129$	11.9185	1.04937
$130$	0	0
$131$	14.0249	1.22536	0.612682	−	0.790330i	$-0.290091\pi$
$131$	14.0249	1.22536	0.612682	+	0.790330i	$0.290091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.42993	−0.639467
$136$	0	0
$137$	8.81909	0.753465	0.376733	−	0.926322i	$-0.377048\pi$
$137$	8.81909	0.753465	0.376733	+	0.926322i	$0.377048\pi$
$138$	0	0
$139$	6.12544	0.519553	0.259776	−	0.965669i	$-0.416351\pi$
$139$	6.12544	0.519553	0.259776	+	0.965669i	$0.416351\pi$
$140$	0	0
$141$	−17.1286	−1.44249
$142$	0	0
$143$	12.5028	1.04554
$144$	0	0
$145$	−8.71295	−0.723571
$146$	0	0
$147$	−6.43066	−0.530392
$148$	0	0
$149$	−2.43152	−0.199198	−0.0995991	−	0.995028i	$-0.531756\pi$
$149$	−2.43152	−0.199198	−0.0995991	+	0.995028i	$0.531756\pi$
$150$	0	0
$151$	−11.4850	−0.934636	−0.467318	−	0.884089i	$-0.654780\pi$
$151$	−11.4850	−0.934636	−0.467318	+	0.884089i	$0.654780\pi$
$152$	0	0
$153$	4.07281	0.329267
$154$	0	0
$155$	−3.47861	−0.279409
$156$	0	0
$157$	−2.66113	−0.212382	−0.106191	−	0.994346i	$-0.533865\pi$
$157$	−2.66113	−0.212382	−0.106191	+	0.994346i	$0.533865\pi$
$158$	0	0
$159$	−0.104943	−0.00832249
$160$	0	0
$161$	2.19581	0.173054
$162$	0	0
$163$	−4.11223	−0.322094	−0.161047	−	0.986947i	$-0.551487\pi$
$163$	−4.11223	−0.322094	−0.161047	+	0.986947i	$0.551487\pi$
$164$	0	0
$165$	15.4697	1.20431
$166$	0	0
$167$	−14.9628	−1.15786	−0.578929	−	0.815378i	$-0.696529\pi$
$167$	−14.9628	−1.15786	−0.578929	+	0.815378i	$0.696529\pi$
$168$	0	0
$169$	−5.46565	−0.420434
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.37340	0.484561	0.242280	−	0.970206i	$-0.422105\pi$
$173$	6.37340	0.484561	0.242280	+	0.970206i	$0.422105\pi$
$174$	0	0
$175$	−3.80121	−0.287344
$176$	0	0
$177$	−7.22467	−0.543039
$178$	0	0
$179$	−3.18125	−0.237778	−0.118889	−	0.992908i	$-0.537933\pi$
$179$	−3.18125	−0.237778	−0.118889	+	0.992908i	$0.537933\pi$
$180$	0	0
$181$	4.54805	0.338054	0.169027	−	0.985611i	$-0.445938\pi$
$181$	4.54805	0.338054	0.169027	+	0.985611i	$0.445938\pi$
$182$	0	0
$183$	−29.2690	−2.16363
$184$	0	0
$185$	12.2909	0.903642
$186$	0	0
$187$	22.8381	1.67009
$188$	0	0
$189$	−8.22358	−0.598178
$190$	0	0
$191$	−0.968712	−0.0700935	−0.0350468	−	0.999386i	$-0.511158\pi$
$191$	−0.968712	−0.0700935	−0.0350468	+	0.999386i	$0.511158\pi$
$192$	0	0
$193$	−15.2065	−1.09459	−0.547295	−	0.836940i	$-0.684343\pi$
$193$	−15.2065	−1.09459	−0.547295	+	0.836940i	$0.684343\pi$
$194$	0	0
$195$	9.32224	0.667580
$196$	0	0
$197$	16.4588	1.17264	0.586320	−	0.810079i	$-0.300576\pi$
$197$	16.4588	1.17264	0.586320	+	0.810079i	$0.300576\pi$
$198$	0	0
$199$	21.9412	1.55537	0.777687	−	0.628652i	$-0.216393\pi$
$199$	21.9412	1.55537	0.777687	+	0.628652i	$0.216393\pi$
$200$	0	0
$201$	−24.7860	−1.74827
$202$	0	0
$203$	−9.64365	−0.676852
$204$	0	0
$205$	−19.1770	−1.33938
$206$	0	0
$207$	0.926472	0.0643942
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−18.3925	−1.26620	−0.633098	−	0.774072i	$-0.718217\pi$
$211$	−18.3925	−1.26620	−0.633098	+	0.774072i	$0.718217\pi$
$212$	0	0
$213$	4.04052	0.276852
$214$	0	0
$215$	10.6178	0.724126
$216$	0	0
$217$	−3.85019	−0.261368
$218$	0	0
$219$	−1.39214	−0.0940723
$220$	0	0
$221$	13.7626	0.925771
$222$	0	0
$223$	19.5162	1.30690	0.653451	−	0.756969i	$-0.273321\pi$
$223$	19.5162	1.30690	0.653451	+	0.756969i	$0.273321\pi$
$224$	0	0
$225$	−1.60383	−0.106922
$226$	0	0
$227$	−7.30867	−0.485094	−0.242547	−	0.970140i	$-0.577983\pi$
$227$	−7.30867	−0.485094	−0.242547	+	0.970140i	$0.577983\pi$
$228$	0	0
$229$	21.1319	1.39644	0.698218	−	0.715885i	$-0.253977\pi$
$229$	21.1319	1.39644	0.698218	+	0.715885i	$0.253977\pi$
$230$	0	0
$231$	17.1221	1.12655
$232$	0	0
$233$	4.76072	0.311885	0.155943	−	0.987766i	$-0.450159\pi$
$233$	4.76072	0.311885	0.155943	+	0.987766i	$0.450159\pi$
$234$	0	0
$235$	−15.2592	−0.995402
$236$	0	0
$237$	21.9837	1.42799
$238$	0	0
$239$	−20.8740	−1.35023	−0.675114	−	0.737714i	$-0.735905\pi$
$239$	−20.8740	−1.35023	−0.675114	+	0.737714i	$0.735905\pi$
$240$	0	0
$241$	−7.06714	−0.455235	−0.227617	−	0.973751i	$-0.573094\pi$
$241$	−7.06714	−0.455235	−0.227617	+	0.973751i	$0.573094\pi$
$242$	0	0
$243$	−8.22784	−0.527816
$244$	0	0
$245$	−5.72883	−0.366001
$246$	0	0
$247$	0	0
$248$	0	0
$249$	19.4963	1.23553
$250$	0	0
$251$	−12.1581	−0.767415	−0.383707	−	0.923455i	$-0.625353\pi$
$251$	−12.1581	−0.767415	−0.383707	+	0.923455i	$0.625353\pi$
$252$	0	0
$253$	5.19515	0.326616
$254$	0	0
$255$	17.0284	1.06636
$256$	0	0
$257$	0.329327	0.0205429	0.0102714	−	0.999947i	$-0.496730\pi$
$257$	0.329327	0.0205429	0.0102714	+	0.999947i	$0.496730\pi$
$258$	0	0
$259$	13.6037	0.845295
$260$	0	0
$261$	−4.06892	−0.251860
$262$	0	0
$263$	−8.86600	−0.546701	−0.273350	−	0.961915i	$-0.588132\pi$
$263$	−8.86600	−0.546701	−0.273350	+	0.961915i	$0.588132\pi$
$264$	0	0
$265$	−0.0934893	−0.00574300
$266$	0	0
$267$	31.6304	1.93575
$268$	0	0
$269$	−11.5241	−0.702635	−0.351317	−	0.936256i	$-0.614266\pi$
$269$	−11.5241	−0.702635	−0.351317	+	0.936256i	$0.614266\pi$
$270$	0	0
$271$	−2.97613	−0.180787	−0.0903936	−	0.995906i	$-0.528813\pi$
$271$	−2.97613	−0.180787	−0.0903936	+	0.995906i	$0.528813\pi$
$272$	0	0
$273$	10.3180	0.624475
$274$	0	0
$275$	−8.99342	−0.542324
$276$	0	0
$277$	31.6704	1.90289	0.951445	−	0.307818i	$-0.0995989\pi$
$277$	31.6704	1.90289	0.951445	+	0.307818i	$0.0995989\pi$
$278$	0	0
$279$	−1.62450	−0.0972562
$280$	0	0
$281$	14.9322	0.890778	0.445389	−	0.895337i	$-0.353065\pi$
$281$	14.9322	0.890778	0.445389	+	0.895337i	$0.353065\pi$
$282$	0	0
$283$	−18.6129	−1.10642	−0.553212	−	0.833040i	$-0.686598\pi$
$283$	−18.6129	−1.10642	−0.553212	+	0.833040i	$0.686598\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.2254	−1.25290
$288$	0	0
$289$	8.13932	0.478783
$290$	0	0
$291$	−0.727124	−0.0426248
$292$	0	0
$293$	−11.3675	−0.664097	−0.332048	−	0.943262i	$-0.607740\pi$
$293$	−11.3675	−0.664097	−0.332048	+	0.943262i	$0.607740\pi$
$294$	0	0
$295$	−6.43618	−0.374729
$296$	0	0
$297$	−19.4565	−1.12898
$298$	0	0
$299$	3.13067	0.181051
$300$	0	0
$301$	11.7519	0.677370
$302$	0	0
$303$	36.3100	2.08595
$304$	0	0
$305$	−26.0747	−1.49303
$306$	0	0
$307$	10.6593	0.608359	0.304179	−	0.952615i	$-0.401618\pi$
$307$	10.6593	0.608359	0.304179	+	0.952615i	$0.401618\pi$
$308$	0	0
$309$	−17.5325	−0.997390
$310$	0	0
$311$	−25.4182	−1.44134	−0.720668	−	0.693281i	$-0.756165\pi$
$311$	−25.4182	−1.44134	−0.720668	+	0.693281i	$0.756165\pi$
$312$	0	0
$313$	−1.68895	−0.0954651	−0.0477325	−	0.998860i	$-0.515200\pi$
$313$	−1.68895	−0.0954651	−0.0477325	+	0.998860i	$0.515200\pi$
$314$	0	0
$315$	2.72020	0.153266
$316$	0	0
$317$	−19.6811	−1.10540	−0.552701	−	0.833379i	$-0.686403\pi$
$317$	−19.6811	−1.10540	−0.552701	+	0.833379i	$0.686403\pi$
$318$	0	0
$319$	−22.8163	−1.27747
$320$	0	0
$321$	−14.6178	−0.815884
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5.41956	−0.300623
$326$	0	0
$327$	32.7501	1.81109
$328$	0	0
$329$	−16.8892	−0.931131
$330$	0	0
$331$	−9.00708	−0.495074	−0.247537	−	0.968878i	$-0.579621\pi$
$331$	−9.00708	−0.495074	−0.247537	+	0.968878i	$0.579621\pi$
$332$	0	0
$333$	5.73979	0.314538
$334$	0	0
$335$	−22.0809	−1.20641
$336$	0	0
$337$	10.9783	0.598027	0.299014	−	0.954249i	$-0.403342\pi$
$337$	10.9783	0.598027	0.299014	+	0.954249i	$0.403342\pi$
$338$	0	0
$339$	−20.8253	−1.13107
$340$	0	0
$341$	−9.10931	−0.493297
$342$	0	0
$343$	−19.8173	−1.07003
$344$	0	0
$345$	3.87358	0.208546
$346$	0	0
$347$	2.80115	0.150374	0.0751868	−	0.997169i	$-0.476045\pi$
$347$	2.80115	0.150374	0.0751868	+	0.997169i	$0.476045\pi$
$348$	0	0
$349$	33.7780	1.80810	0.904048	−	0.427431i	$-0.140581\pi$
$349$	33.7780	1.80810	0.904048	+	0.427431i	$0.140581\pi$
$350$	0	0
$351$	−11.7248	−0.625821
$352$	0	0
$353$	2.50634	0.133399	0.0666996	−	0.997773i	$-0.478753\pi$
$353$	2.50634	0.133399	0.0666996	+	0.997773i	$0.478753\pi$
$354$	0	0
$355$	3.59954	0.191044
$356$	0	0
$357$	18.8474	0.997508
$358$	0	0
$359$	21.3669	1.12770	0.563850	−	0.825877i	$-0.309320\pi$
$359$	21.3669	1.12770	0.563850	+	0.825877i	$0.309320\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	19.0322	0.998934
$364$	0	0
$365$	−1.24021	−0.0649154
$366$	0	0
$367$	3.58067	0.186910	0.0934548	−	0.995624i	$-0.470209\pi$
$367$	3.58067	0.186910	0.0934548	+	0.995624i	$0.470209\pi$
$368$	0	0
$369$	−8.95557	−0.466208
$370$	0	0
$371$	−0.103476	−0.00537219
$372$	0	0
$373$	35.5217	1.83924	0.919622	−	0.392804i	$-0.128495\pi$
$373$	35.5217	1.83924	0.919622	+	0.392804i	$0.128495\pi$
$374$	0	0
$375$	−23.6868	−1.22318
$376$	0	0
$377$	−13.7494	−0.708131
$378$	0	0
$379$	7.31674	0.375836	0.187918	−	0.982185i	$-0.439826\pi$
$379$	7.31674	0.375836	0.187918	+	0.982185i	$0.439826\pi$
$380$	0	0
$381$	−21.5320	−1.10312
$382$	0	0
$383$	−7.37459	−0.376824	−0.188412	−	0.982090i	$-0.560334\pi$
$383$	−7.37459	−0.376824	−0.188412	+	0.982090i	$0.560334\pi$
$384$	0	0
$385$	15.2534	0.777387
$386$	0	0
$387$	4.95846	0.252053
$388$	0	0
$389$	7.83370	0.397184	0.198592	−	0.980082i	$-0.436363\pi$
$389$	7.83370	0.397184	0.198592	+	0.980082i	$0.436363\pi$
$390$	0	0
$391$	5.71862	0.289203
$392$	0	0
$393$	27.3838	1.38133
$394$	0	0
$395$	19.5844	0.985398
$396$	0	0
$397$	20.0680	1.00718	0.503591	−	0.863942i	$-0.332012\pi$
$397$	20.0680	1.00718	0.503591	+	0.863942i	$0.332012\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.33346	−0.316278	−0.158139	−	0.987417i	$-0.550549\pi$
$401$	−6.33346	−0.316278	−0.158139	+	0.987417i	$0.550549\pi$
$402$	0	0
$403$	−5.48940	−0.273447
$404$	0	0
$405$	−18.7458	−0.931487
$406$	0	0
$407$	32.1856	1.59538
$408$	0	0
$409$	−19.5731	−0.967827	−0.483913	−	0.875116i	$-0.660785\pi$
$409$	−19.5731	−0.967827	−0.483913	+	0.875116i	$0.660785\pi$
$410$	0	0
$411$	17.2194	0.849369
$412$	0	0
$413$	−7.12368	−0.350533
$414$	0	0
$415$	17.3685	0.852587
$416$	0	0
$417$	11.9600	0.585683
$418$	0	0
$419$	−9.06493	−0.442851	−0.221426	−	0.975177i	$-0.571071\pi$
$419$	−9.06493	−0.442851	−0.221426	+	0.975177i	$0.571071\pi$
$420$	0	0
$421$	25.0706	1.22186	0.610932	−	0.791683i	$-0.290795\pi$
$421$	25.0706	1.22186	0.610932	+	0.791683i	$0.290795\pi$
$422$	0	0
$423$	−7.12601	−0.346478
$424$	0	0
$425$	−9.89961	−0.480202
$426$	0	0
$427$	−28.8599	−1.39663
$428$	0	0
$429$	24.4118	1.17861
$430$	0	0
$431$	−6.58203	−0.317045	−0.158523	−	0.987355i	$-0.550673\pi$
$431$	−6.58203	−0.317045	−0.158523	+	0.987355i	$0.550673\pi$
$432$	0	0
$433$	19.6621	0.944899	0.472450	−	0.881358i	$-0.343370\pi$
$433$	19.6621	0.944899	0.472450	+	0.881358i	$0.343370\pi$
$434$	0	0
$435$	−17.0121	−0.815669
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.8664	0.804988	0.402494	−	0.915423i	$-0.368143\pi$
$439$	16.8664	0.804988	0.402494	+	0.915423i	$0.368143\pi$
$440$	0	0
$441$	−2.67534	−0.127397
$442$	0	0
$443$	−37.7997	−1.79592	−0.897959	−	0.440079i	$-0.854950\pi$
$443$	−37.7997	−1.79592	−0.897959	+	0.440079i	$0.854950\pi$
$444$	0	0
$445$	28.1783	1.33578
$446$	0	0
$447$	−4.74758	−0.224553
$448$	0	0
$449$	14.0452	0.662836	0.331418	−	0.943484i	$-0.392473\pi$
$449$	14.0452	0.662836	0.331418	+	0.943484i	$0.392473\pi$
$450$	0	0
$451$	−50.2180	−2.36467
$452$	0	0
$453$	−22.4246	−1.05360
$454$	0	0
$455$	9.19193	0.430924
$456$	0	0
$457$	−10.9491	−0.512177	−0.256089	−	0.966653i	$-0.582434\pi$
$457$	−10.9491	−0.512177	−0.256089	+	0.966653i	$0.582434\pi$
$458$	0	0
$459$	−21.4170	−0.999658
$460$	0	0
$461$	7.85327	0.365763	0.182882	−	0.983135i	$-0.441458\pi$
$461$	7.85327	0.365763	0.182882	+	0.983135i	$0.441458\pi$
$462$	0	0
$463$	−19.7742	−0.918985	−0.459492	−	0.888182i	$-0.651969\pi$
$463$	−19.7742	−0.918985	−0.459492	+	0.888182i	$0.651969\pi$
$464$	0	0
$465$	−6.79203	−0.314973
$466$	0	0
$467$	9.12248	0.422138	0.211069	−	0.977471i	$-0.432305\pi$
$467$	9.12248	0.422138	0.211069	+	0.977471i	$0.432305\pi$
$468$	0	0
$469$	−24.4395	−1.12851
$470$	0	0
$471$	−5.19590	−0.239414
$472$	0	0
$473$	27.8044	1.27845
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.0436592	−0.00199902
$478$	0	0
$479$	−34.9470	−1.59677	−0.798385	−	0.602147i	$-0.794312\pi$
$479$	−34.9470	−1.59677	−0.798385	+	0.602147i	$0.794312\pi$
$480$	0	0
$481$	19.3955	0.884359
$482$	0	0
$483$	4.28734	0.195081
$484$	0	0
$485$	−0.647767	−0.0294136
$486$	0	0
$487$	29.8364	1.35202	0.676009	−	0.736893i	$-0.263708\pi$
$487$	29.8364	1.35202	0.676009	+	0.736893i	$0.263708\pi$
$488$	0	0
$489$	−8.02917	−0.363092
$490$	0	0
$491$	−7.17461	−0.323785	−0.161893	−	0.986808i	$-0.551760\pi$
$491$	−7.17461	−0.323785	−0.161893	+	0.986808i	$0.551760\pi$
$492$	0	0
$493$	−25.1153	−1.13114
$494$	0	0
$495$	6.43583	0.289269
$496$	0	0
$497$	3.98404	0.178709
$498$	0	0
$499$	−16.2309	−0.726594	−0.363297	−	0.931673i	$-0.618349\pi$
$499$	−16.2309	−0.726594	−0.363297	+	0.931673i	$0.618349\pi$
$500$	0	0
$501$	−29.2151	−1.30523
$502$	0	0
$503$	18.6165	0.830069	0.415034	−	0.909806i	$-0.363770\pi$
$503$	18.6165	0.830069	0.415034	+	0.909806i	$0.363770\pi$
$504$	0	0
$505$	32.3472	1.43943
$506$	0	0
$507$	−10.6717	−0.473949
$508$	0	0
$509$	4.23235	0.187595	0.0937977	−	0.995591i	$-0.470099\pi$
$509$	4.23235	0.187595	0.0937977	+	0.995591i	$0.470099\pi$
$510$	0	0
$511$	−1.37268	−0.0607239
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.6190	−0.688257
$516$	0	0
$517$	−39.9588	−1.75739
$518$	0	0
$519$	12.4441	0.546237
$520$	0	0
$521$	−12.8106	−0.561242	−0.280621	−	0.959819i	$-0.590540\pi$
$521$	−12.8106	−0.561242	−0.280621	+	0.959819i	$0.590540\pi$
$522$	0	0
$523$	−32.3625	−1.41511	−0.707557	−	0.706656i	$-0.750203\pi$
$523$	−32.3625	−1.41511	−0.707557	+	0.706656i	$0.750203\pi$
$524$	0	0
$525$	−7.42190	−0.323918
$526$	0	0
$527$	−10.0272	−0.436791
$528$	0	0
$529$	−21.6991	−0.943441
$530$	0	0
$531$	−3.00567	−0.130435
$532$	0	0
$533$	−30.2621	−1.31080
$534$	0	0
$535$	−13.0224	−0.563007
$536$	0	0
$537$	−6.21143	−0.268043
$538$	0	0
$539$	−15.0019	−0.646177
$540$	0	0
$541$	18.4412	0.792850	0.396425	−	0.918067i	$-0.370251\pi$
$541$	18.4412	0.792850	0.396425	+	0.918067i	$0.370251\pi$
$542$	0	0
$543$	8.88011	0.381082
$544$	0	0
$545$	29.1758	1.24976
$546$	0	0
$547$	−32.6193	−1.39470	−0.697351	−	0.716730i	$-0.745638\pi$
$547$	−32.6193	−1.39470	−0.697351	+	0.716730i	$0.745638\pi$
$548$	0	0
$549$	−12.1768	−0.519692
$550$	0	0
$551$	0	0
$552$	0	0
$553$	21.6764	0.921773
$554$	0	0
$555$	23.9980	1.01866
$556$	0	0
$557$	13.0507	0.552974	0.276487	−	0.961018i	$-0.410830\pi$
$557$	13.0507	0.552974	0.276487	+	0.961018i	$0.410830\pi$
$558$	0	0
$559$	16.7553	0.708674
$560$	0	0
$561$	44.5917	1.88266
$562$	0	0
$563$	18.2500	0.769145	0.384573	−	0.923095i	$-0.374349\pi$
$563$	18.2500	0.769145	0.384573	+	0.923095i	$0.374349\pi$
$564$	0	0
$565$	−18.5524	−0.780507
$566$	0	0
$567$	−20.7482	−0.871343
$568$	0	0
$569$	34.4274	1.44327	0.721636	−	0.692273i	$-0.243390\pi$
$569$	34.4274	1.44327	0.721636	+	0.692273i	$0.243390\pi$
$570$	0	0
$571$	16.6525	0.696884	0.348442	−	0.937330i	$-0.386711\pi$
$571$	16.6525	0.696884	0.348442	+	0.937330i	$0.386711\pi$
$572$	0	0
$573$	−1.89142	−0.0790153
$574$	0	0
$575$	−2.25194	−0.0939122
$576$	0	0
$577$	−37.4788	−1.56026	−0.780130	−	0.625617i	$-0.784847\pi$
$577$	−37.4788	−1.56026	−0.780130	+	0.625617i	$0.784847\pi$
$578$	0	0
$579$	−29.6909	−1.23391
$580$	0	0
$581$	19.2238	0.797537
$582$	0	0
$583$	−0.244817	−0.0101393
$584$	0	0
$585$	3.87833	0.160349
$586$	0	0
$587$	−33.9793	−1.40248	−0.701238	−	0.712927i	$-0.747369\pi$
$587$	−33.9793	−1.40248	−0.701238	+	0.712927i	$0.747369\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	32.1360	1.32190
$592$	0	0
$593$	18.9644	0.778773	0.389386	−	0.921074i	$-0.372687\pi$
$593$	18.9644	0.778773	0.389386	+	0.921074i	$0.372687\pi$
$594$	0	0
$595$	16.7904	0.688339
$596$	0	0
$597$	42.8405	1.75335
$598$	0	0
$599$	33.4965	1.36863	0.684314	−	0.729187i	$-0.260102\pi$
$599$	33.4965	1.36863	0.684314	+	0.729187i	$0.260102\pi$
$600$	0	0
$601$	15.0011	0.611908	0.305954	−	0.952046i	$-0.401025\pi$
$601$	15.0011	0.611908	0.305954	+	0.952046i	$0.401025\pi$
$602$	0	0
$603$	−10.3117	−0.419925
$604$	0	0
$605$	16.9551	0.689322
$606$	0	0
$607$	7.06547	0.286779	0.143389	−	0.989666i	$-0.454200\pi$
$607$	7.06547	0.286779	0.143389	+	0.989666i	$0.454200\pi$
$608$	0	0
$609$	−18.8293	−0.763003
$610$	0	0
$611$	−24.0797	−0.974162
$612$	0	0
$613$	−26.1833	−1.05753	−0.528767	−	0.848767i	$-0.677345\pi$
$613$	−26.1833	−1.05753	−0.528767	+	0.848767i	$0.677345\pi$
$614$	0	0
$615$	−37.4432	−1.50986
$616$	0	0
$617$	−43.5538	−1.75341	−0.876705	−	0.481029i	$-0.840263\pi$
$617$	−43.5538	−1.75341	−0.876705	+	0.481029i	$0.840263\pi$
$618$	0	0
$619$	24.9837	1.00418	0.502090	−	0.864815i	$-0.332565\pi$
$619$	24.9837	1.00418	0.502090	+	0.864815i	$0.332565\pi$
$620$	0	0
$621$	−4.87187	−0.195501
$622$	0	0
$623$	31.1883	1.24953
$624$	0	0
$625$	−11.2295	−0.449180
$626$	0	0
$627$	0	0
$628$	0	0
$629$	35.4287	1.41263
$630$	0	0
$631$	−46.5146	−1.85172	−0.925858	−	0.377873i	$-0.876656\pi$
$631$	−46.5146	−1.85172	−0.925858	+	0.377873i	$0.876656\pi$
$632$	0	0
$633$	−35.9117	−1.42736
$634$	0	0
$635$	−19.1821	−0.761217
$636$	0	0
$637$	−9.04034	−0.358192
$638$	0	0
$639$	1.68098	0.0664983
$640$	0	0
$641$	−7.08499	−0.279840	−0.139920	−	0.990163i	$-0.544685\pi$
$641$	−7.08499	−0.279840	−0.139920	+	0.990163i	$0.544685\pi$
$642$	0	0
$643$	0.159804	0.00630206	0.00315103	−	0.999995i	$-0.498997\pi$
$643$	0.159804	0.00630206	0.00315103	+	0.999995i	$0.498997\pi$
$644$	0	0
$645$	20.7313	0.816295
$646$	0	0
$647$	−18.8979	−0.742955	−0.371478	−	0.928442i	$-0.621149\pi$
$647$	−18.8979	−0.742955	−0.371478	+	0.928442i	$0.621149\pi$
$648$	0	0
$649$	−16.8542	−0.661585
$650$	0	0
$651$	−7.51754	−0.294636
$652$	0	0
$653$	−11.2493	−0.440218	−0.220109	−	0.975475i	$-0.570641\pi$
$653$	−11.2493	−0.440218	−0.220109	+	0.975475i	$0.570641\pi$
$654$	0	0
$655$	24.3952	0.953199
$656$	0	0
$657$	−0.579172	−0.0225957
$658$	0	0
$659$	12.6573	0.493058	0.246529	−	0.969135i	$-0.420710\pi$
$659$	12.6573	0.493058	0.246529	+	0.969135i	$0.420710\pi$
$660$	0	0
$661$	3.69864	0.143861	0.0719303	−	0.997410i	$-0.477084\pi$
$661$	3.69864	0.143861	0.0719303	+	0.997410i	$0.477084\pi$
$662$	0	0
$663$	26.8716	1.04361
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.71316	−0.221214
$668$	0	0
$669$	38.1056	1.47325
$670$	0	0
$671$	−68.2808	−2.63595
$672$	0	0
$673$	22.2546	0.857851	0.428926	−	0.903340i	$-0.358892\pi$
$673$	22.2546	0.857851	0.428926	+	0.903340i	$0.358892\pi$
$674$	0	0
$675$	8.43378	0.324616
$676$	0	0
$677$	−23.8967	−0.918425	−0.459213	−	0.888326i	$-0.651868\pi$
$677$	−23.8967	−0.918425	−0.459213	+	0.888326i	$0.651868\pi$
$678$	0	0
$679$	−0.716960	−0.0275144
$680$	0	0
$681$	−14.2703	−0.546838
$682$	0	0
$683$	−26.9375	−1.03073	−0.515367	−	0.856970i	$-0.672344\pi$
$683$	−26.9375	−1.03073	−0.515367	+	0.856970i	$0.672344\pi$
$684$	0	0
$685$	15.3401	0.586114
$686$	0	0
$687$	41.2603	1.57418
$688$	0	0
$689$	−0.147530	−0.00562046
$690$	0	0
$691$	27.3737	1.04135	0.520673	−	0.853756i	$-0.325681\pi$
$691$	27.3737	1.04135	0.520673	+	0.853756i	$0.325681\pi$
$692$	0	0
$693$	7.12330	0.270592
$694$	0	0
$695$	10.6547	0.404156
$696$	0	0
$697$	−55.2780	−2.09380
$698$	0	0
$699$	9.29537	0.351583
$700$	0	0
$701$	−23.1852	−0.875695	−0.437847	−	0.899049i	$-0.644259\pi$
$701$	−23.1852	−0.875695	−0.437847	+	0.899049i	$0.644259\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−29.7938	−1.12210
$706$	0	0
$707$	35.8024	1.34649
$708$	0	0
$709$	−0.189146	−0.00710354	−0.00355177	−	0.999994i	$-0.501131\pi$
$709$	−0.189146	−0.00710354	−0.00355177	+	0.999994i	$0.501131\pi$
$710$	0	0
$711$	9.14585	0.342996
$712$	0	0
$713$	−2.28095	−0.0854224
$714$	0	0
$715$	21.7476	0.813312
$716$	0	0
$717$	−40.7568	−1.52209
$718$	0	0
$719$	−11.4260	−0.426119	−0.213059	−	0.977039i	$-0.568343\pi$
$719$	−11.4260	−0.426119	−0.213059	+	0.977039i	$0.568343\pi$
$720$	0	0
$721$	−17.2874	−0.643818
$722$	0	0
$723$	−13.7987	−0.513178
$724$	0	0
$725$	9.89015	0.367311
$726$	0	0
$727$	30.0799	1.11560	0.557801	−	0.829974i	$-0.311645\pi$
$727$	30.0799	1.11560	0.557801	+	0.829974i	$0.311645\pi$
$728$	0	0
$729$	16.2663	0.602454
$730$	0	0
$731$	30.6060	1.13200
$732$	0	0
$733$	−32.6737	−1.20683	−0.603414	−	0.797428i	$-0.706194\pi$
$733$	−32.6737	−1.20683	−0.603414	+	0.797428i	$0.706194\pi$
$734$	0	0
$735$	−11.1856	−0.412587
$736$	0	0
$737$	−57.8225	−2.12992
$738$	0	0
$739$	−25.5448	−0.939682	−0.469841	−	0.882751i	$-0.655689\pi$
$739$	−25.5448	−0.939682	−0.469841	+	0.882751i	$0.655689\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.0945	−0.847255	−0.423628	−	0.905836i	$-0.639244\pi$
$743$	−23.0945	−0.847255	−0.423628	+	0.905836i	$0.639244\pi$
$744$	0	0
$745$	−4.22943	−0.154954
$746$	0	0
$747$	8.11104	0.296767
$748$	0	0
$749$	−14.4134	−0.526655
$750$	0	0
$751$	12.9465	0.472426	0.236213	−	0.971701i	$-0.424094\pi$
$751$	12.9465	0.472426	0.236213	+	0.971701i	$0.424094\pi$
$752$	0	0
$753$	−23.7389	−0.865093
$754$	0	0
$755$	−19.9772	−0.727045
$756$	0	0
$757$	17.4823	0.635404	0.317702	−	0.948191i	$-0.397089\pi$
$757$	17.4823	0.635404	0.317702	+	0.948191i	$0.397089\pi$
$758$	0	0
$759$	10.1436	0.368189
$760$	0	0
$761$	−4.43066	−0.160611	−0.0803057	−	0.996770i	$-0.525590\pi$
$761$	−4.43066	−0.160611	−0.0803057	+	0.996770i	$0.525590\pi$
$762$	0	0
$763$	32.2924	1.16906
$764$	0	0
$765$	7.08432	0.256134
$766$	0	0
$767$	−10.1566	−0.366733
$768$	0	0
$769$	40.9324	1.47606	0.738031	−	0.674767i	$-0.235756\pi$
$769$	40.9324	1.47606	0.738031	+	0.674767i	$0.235756\pi$
$770$	0	0
$771$	0.643015	0.0231576
$772$	0	0
$773$	−7.58321	−0.272749	−0.136375	−	0.990657i	$-0.543545\pi$
$773$	−7.58321	−0.272749	−0.136375	+	0.990657i	$0.543545\pi$
$774$	0	0
$775$	3.94860	0.141838
$776$	0	0
$777$	26.5615	0.952887
$778$	0	0
$779$	0	0
$780$	0	0
$781$	9.42600	0.337289
$782$	0	0
$783$	21.3965	0.764648
$784$	0	0
$785$	−4.62882	−0.165210
$786$	0	0
$787$	−30.9049	−1.10164	−0.550821	−	0.834623i	$-0.685685\pi$
$787$	−30.9049	−1.10164	−0.550821	+	0.834623i	$0.685685\pi$
$788$	0	0
$789$	−17.3110	−0.616287
$790$	0	0
$791$	−20.5342	−0.730111
$792$	0	0
$793$	−41.1470	−1.46117
$794$	0	0
$795$	−0.182539	−0.00647399
$796$	0	0
$797$	17.9708	0.636559	0.318279	−	0.947997i	$-0.396895\pi$
$797$	17.9708	0.636559	0.318279	+	0.947997i	$0.396895\pi$
$798$	0	0
$799$	−43.9851	−1.55608
$800$	0	0
$801$	13.1592	0.464957
$802$	0	0
$803$	−3.24769	−0.114608
$804$	0	0
$805$	3.81943	0.134617
$806$	0	0
$807$	−22.5009	−0.792068
$808$	0	0
$809$	5.04923	0.177522	0.0887608	−	0.996053i	$-0.471709\pi$
$809$	5.04923	0.177522	0.0887608	+	0.996053i	$0.471709\pi$
$810$	0	0
$811$	−19.7394	−0.693144	−0.346572	−	0.938023i	$-0.612654\pi$
$811$	−19.7394	−0.693144	−0.346572	+	0.938023i	$0.612654\pi$
$812$	0	0
$813$	−5.81093	−0.203798
$814$	0	0
$815$	−7.15288	−0.250554
$816$	0	0
$817$	0	0
$818$	0	0
$819$	4.29260	0.149996
$820$	0	0
$821$	2.01298	0.0702535	0.0351267	−	0.999383i	$-0.488817\pi$
$821$	2.01298	0.0702535	0.0351267	+	0.999383i	$0.488817\pi$
$822$	0	0
$823$	43.0060	1.49909	0.749547	−	0.661951i	$-0.230271\pi$
$823$	43.0060	1.49909	0.749547	+	0.661951i	$0.230271\pi$
$824$	0	0
$825$	−17.5598	−0.611352
$826$	0	0
$827$	−33.6192	−1.16905	−0.584527	−	0.811375i	$-0.698720\pi$
$827$	−33.6192	−1.16905	−0.584527	+	0.811375i	$0.698720\pi$
$828$	0	0
$829$	34.7597	1.20725	0.603627	−	0.797267i	$-0.293721\pi$
$829$	34.7597	1.20725	0.603627	+	0.797267i	$0.293721\pi$
$830$	0	0
$831$	61.8368	2.14510
$832$	0	0
$833$	−16.5135	−0.572158
$834$	0	0
$835$	−26.0266	−0.900687
$836$	0	0
$837$	8.54246	0.295271
$838$	0	0
$839$	−18.5759	−0.641311	−0.320656	−	0.947196i	$-0.603903\pi$
$839$	−18.5759	−0.641311	−0.320656	+	0.947196i	$0.603903\pi$
$840$	0	0
$841$	−3.90872	−0.134783
$842$	0	0
$843$	29.1552	1.00416
$844$	0	0
$845$	−9.50704	−0.327052
$846$	0	0
$847$	18.7662	0.644814
$848$	0	0
$849$	−36.3420	−1.24725
$850$	0	0
$851$	8.05922	0.276266
$852$	0	0
$853$	−2.05122	−0.0702325	−0.0351162	−	0.999383i	$-0.511180\pi$
$853$	−2.05122	−0.0702325	−0.0351162	+	0.999383i	$0.511180\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.1261	−0.687495	−0.343747	−	0.939062i	$-0.611696\pi$
$857$	−20.1261	−0.687495	−0.343747	+	0.939062i	$0.611696\pi$
$858$	0	0
$859$	−9.55321	−0.325951	−0.162976	−	0.986630i	$-0.552109\pi$
$859$	−9.55321	−0.325951	−0.162976	+	0.986630i	$0.552109\pi$
$860$	0	0
$861$	−41.4428	−1.41237
$862$	0	0
$863$	34.7287	1.18218	0.591089	−	0.806606i	$-0.298698\pi$
$863$	34.7287	1.18218	0.591089	+	0.806606i	$0.298698\pi$
$864$	0	0
$865$	11.0860	0.376935
$866$	0	0
$867$	15.8921	0.539725
$868$	0	0
$869$	51.2850	1.73972
$870$	0	0
$871$	−34.8446	−1.18067
$872$	0	0
$873$	−0.302505	−0.0102382
$874$	0	0
$875$	−23.3557	−0.789566
$876$	0	0
$877$	26.7932	0.904743	0.452371	−	0.891830i	$-0.350578\pi$
$877$	26.7932	0.904743	0.452371	+	0.891830i	$0.350578\pi$
$878$	0	0
$879$	−22.1952	−0.748625
$880$	0	0
$881$	−15.7588	−0.530928	−0.265464	−	0.964121i	$-0.585525\pi$
$881$	−15.7588	−0.530928	−0.265464	+	0.964121i	$0.585525\pi$
$882$	0	0
$883$	41.3896	1.39287	0.696435	−	0.717620i	$-0.254768\pi$
$883$	41.3896	1.39287	0.696435	+	0.717620i	$0.254768\pi$
$884$	0	0
$885$	−12.5667	−0.422425
$886$	0	0
$887$	−46.5948	−1.56450	−0.782249	−	0.622965i	$-0.785928\pi$
$887$	−46.5948	−1.56450	−0.782249	+	0.622965i	$0.785928\pi$
$888$	0	0
$889$	−21.2311	−0.712067
$890$	0	0
$891$	−49.0890	−1.64454
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−5.53352	−0.184965
$896$	0	0
$897$	6.11267	0.204096
$898$	0	0
$899$	10.0176	0.334106
$900$	0	0
$901$	−0.269485	−0.00897785
$902$	0	0
$903$	22.9458	0.763588
$904$	0	0
$905$	7.91095	0.262969
$906$	0	0
$907$	−2.43901	−0.0809861	−0.0404930	−	0.999180i	$-0.512893\pi$
$907$	−2.43901	−0.0809861	−0.0404930	+	0.999180i	$0.512893\pi$
$908$	0	0
$909$	15.1060	0.501035
$910$	0	0
$911$	29.8844	0.990115	0.495057	−	0.868860i	$-0.335147\pi$
$911$	29.8844	0.990115	0.495057	+	0.868860i	$0.335147\pi$
$912$	0	0
$913$	45.4823	1.50524
$914$	0	0
$915$	−50.9111	−1.68307
$916$	0	0
$917$	27.0010	0.891653
$918$	0	0
$919$	40.3474	1.33094	0.665469	−	0.746426i	$-0.268232\pi$
$919$	40.3474	1.33094	0.665469	+	0.746426i	$0.268232\pi$
$920$	0	0
$921$	20.8124	0.685793
$922$	0	0
$923$	5.68024	0.186967
$924$	0	0
$925$	−13.9515	−0.458721
$926$	0	0
$927$	−7.29404	−0.239568
$928$	0	0
$929$	−52.6807	−1.72840	−0.864199	−	0.503150i	$-0.832174\pi$
$929$	−52.6807	−1.72840	−0.864199	+	0.503150i	$0.832174\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−49.6294	−1.62479
$934$	0	0
$935$	39.7250	1.29915
$936$	0	0
$937$	31.9362	1.04331	0.521655	−	0.853156i	$-0.325315\pi$
$937$	31.9362	1.04331	0.521655	+	0.853156i	$0.325315\pi$
$938$	0	0
$939$	−3.29769	−0.107616
$940$	0	0
$941$	1.68453	0.0549141	0.0274570	−	0.999623i	$-0.491259\pi$
$941$	1.68453	0.0549141	0.0274570	+	0.999623i	$0.491259\pi$
$942$	0	0
$943$	−12.5745	−0.409482
$944$	0	0
$945$	−14.3042	−0.465317
$946$	0	0
$947$	31.1218	1.01132	0.505661	−	0.862732i	$-0.331249\pi$
$947$	31.1218	1.01132	0.505661	+	0.862732i	$0.331249\pi$
$948$	0	0
$949$	−1.95710	−0.0635302
$950$	0	0
$951$	−38.4277	−1.24610
$952$	0	0
$953$	−23.3357	−0.755917	−0.377959	−	0.925822i	$-0.623374\pi$
$953$	−23.3357	−0.755917	−0.377959	+	0.925822i	$0.623374\pi$
$954$	0	0
$955$	−1.68499	−0.0545251
$956$	0	0
$957$	−44.5491	−1.44007
$958$	0	0
$959$	16.9787	0.548270
$960$	0	0
$961$	−27.0005	−0.870984
$962$	0	0
$963$	−6.08142	−0.195971
$964$	0	0
$965$	−26.4505	−0.851472
$966$	0	0
$967$	54.4984	1.75255	0.876276	−	0.481810i	$-0.160020\pi$
$967$	54.4984	1.75255	0.876276	+	0.481810i	$0.160020\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.2286	−0.937991	−0.468995	−	0.883201i	$-0.655384\pi$
$971$	−29.2286	−0.937991	−0.468995	+	0.883201i	$0.655384\pi$
$972$	0	0
$973$	11.7928	0.378060
$974$	0	0
$975$	−10.5818	−0.338888
$976$	0	0
$977$	34.0062	1.08795	0.543977	−	0.839100i	$-0.316918\pi$
$977$	34.0062	1.08795	0.543977	+	0.839100i	$0.316918\pi$
$978$	0	0
$979$	73.7896	2.35833
$980$	0	0
$981$	13.6250	0.435013
$982$	0	0
$983$	34.2563	1.09261	0.546304	−	0.837587i	$-0.316034\pi$
$983$	34.2563	1.09261	0.546304	+	0.837587i	$0.316034\pi$
$984$	0	0
$985$	28.6287	0.912186
$986$	0	0
$987$	−32.9763	−1.04965
$988$	0	0
$989$	6.96216	0.221384
$990$	0	0
$991$	43.9047	1.39468	0.697339	−	0.716742i	$-0.254367\pi$
$991$	43.9047	1.39468	0.697339	+	0.716742i	$0.254367\pi$
$992$	0	0
$993$	−17.5864	−0.558089
$994$	0	0
$995$	38.1650	1.20991
$996$	0	0
$997$	−57.6104	−1.82454	−0.912270	−	0.409590i	$-0.865672\pi$
$997$	−57.6104	−1.82454	−0.912270	+	0.409590i	$0.865672\pi$
$998$	0	0
$999$	−30.1828	−0.954941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.x.1.8		9
4.3	odd	2		5776.2.a.ce.1.2		9
19.14	odd	18		152.2.q.c.25.3	✓	18
19.15	odd	18		152.2.q.c.73.3	yes	18
19.18	odd	2		2888.2.a.y.1.2		9
76.15	even	18		304.2.u.f.225.1		18
76.71	even	18		304.2.u.f.177.1		18
76.75	even	2		5776.2.a.cd.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.c.25.3	✓	18	19.14	odd	18
152.2.q.c.73.3	yes	18	19.15	odd	18
304.2.u.f.177.1		18	76.71	even	18
304.2.u.f.225.1		18	76.15	even	18
2888.2.a.x.1.8		9	1.1	even	1	trivial
2888.2.a.y.1.2		9	19.18	odd	2
5776.2.a.cd.1.8		9	76.75	even	2
5776.2.a.ce.1.2		9	4.3	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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q+1.95251 q^{3} +1.73942 q^{5} +1.92522 q^{7} +0.812302 q^{9} +4.55495 q^{11} +2.74488 q^{13} +3.39623 q^{15} +5.01391 q^{17} +3.75901 q^{21} +1.14055 q^{23} -1.97443 q^{25} -4.27151 q^{27} -5.00912 q^{29} -1.99987 q^{31} +8.89359 q^{33} +3.34876 q^{35} +7.06608 q^{37} +5.35941 q^{39} -11.0249 q^{41} +6.10421 q^{43} +1.41293 q^{45} -8.77261 q^{47} -3.29353 q^{49} +9.78972 q^{51} -0.0537475 q^{53} +7.92296 q^{55} -3.70019 q^{59} -14.9905 q^{61} +1.56386 q^{63} +4.77449 q^{65} -12.6944 q^{67} +2.22694 q^{69} +2.06940 q^{71} -0.713001 q^{73} -3.85509 q^{75} +8.76928 q^{77} +11.2592 q^{79} -10.7771 q^{81} +9.98525 q^{83} +8.72129 q^{85} -9.78037 q^{87} +16.1999 q^{89} +5.28449 q^{91} -3.90477 q^{93} -0.372404 q^{97} +3.69999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 3 q^{11} + 6 q^{13} - 3 q^{17} - 15 q^{21} + 24 q^{23} + 30 q^{25} - 12 q^{27} + 15 q^{29} - 6 q^{31} + 18 q^{33} + 15 q^{35} - 24 q^{37} + 6 q^{39} + 12 q^{41}+ \cdots + 48 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9 + 3 * q^11 + 6 * q^13 - 3 * q^17 - 15 * q^21 + 24 * q^23 + 30 * q^25 - 12 * q^27 + 15 * q^29 - 6 * q^31 + 18 * q^33 + 15 * q^35 - 24 * q^37 + 6 * q^39 + 12 * q^41 + 9 * q^43 + 42 * q^45 - 12 * q^47 + 18 * q^49 - 12 * q^51 + 18 * q^53 + 21 * q^55 - 57 * q^59 + 15 * q^61 + 30 * q^63 + 27 * q^65 - 6 * q^67 - 3 * q^69 + 36 * q^73 - 45 * q^75 + 30 * q^77 + 3 * q^79 - 15 * q^81 + 27 * q^83 + 18 * q^85 - 18 * q^87 + 30 * q^89 + 21 * q^91 - 24 * q^93 - 3 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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