LMFDB - Embedded newform 8880.2.a.br.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8880,2,Mod(1,8880)] 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("8880.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,2,0,1,0,2,0,-3,0,-3] 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:	$70.9071569949$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	8880.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{9} +1.37228 q^{11} +1.37228 q^{13} +1.00000 q^{15} -1.37228 q^{17} +1.37228 q^{19} +3.37228 q^{21} -3.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -2.74456 q^{31} +1.37228 q^{33} +3.37228 q^{35} -1.00000 q^{37} +1.37228 q^{39} +8.74456 q^{41} +4.00000 q^{43} +1.00000 q^{45} +4.74456 q^{47} +4.37228 q^{49} -1.37228 q^{51} +5.37228 q^{53} +1.37228 q^{55} +1.37228 q^{57} -14.7446 q^{59} -2.74456 q^{61} +3.37228 q^{63} +1.37228 q^{65} -2.74456 q^{67} -3.37228 q^{69} -1.25544 q^{71} -4.11684 q^{73} +1.00000 q^{75} +4.62772 q^{77} -4.00000 q^{79} +1.00000 q^{81} -0.627719 q^{83} -1.37228 q^{85} +6.00000 q^{87} +13.3723 q^{89} +4.62772 q^{91} -2.74456 q^{93} +1.37228 q^{95} +13.4891 q^{97} +1.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} - 3 q^{11} - 3 q^{13} + 2 q^{15} + 3 q^{17} - 3 q^{19} + q^{21} - q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + q^{35} - 2 q^{37} - 3 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 - 3 * q^11 - 3 * q^13 + 2 * q^15 + 3 * q^17 - 3 * q^19 + q^21 - q^23 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 6 * q^31 - 3 * q^33 + q^35 - 2 * q^37 - 3 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 2 * q^47 + 3 * q^49 + 3 * q^51 + 5 * q^53 - 3 * q^55 - 3 * q^57 - 18 * q^59 + 6 * q^61 + q^63 - 3 * q^65 + 6 * q^67 - q^69 - 14 * q^71 + 9 * q^73 + 2 * q^75 + 15 * q^77 - 8 * q^79 + 2 * q^81 - 7 * q^83 + 3 * q^85 + 12 * q^87 + 21 * q^89 + 15 * q^91 + 6 * q^93 - 3 * q^95 + 4 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	1.37228	0.380602	0.190301	−	0.981726i	$-0.439054\pi$
$13$	1.37228	0.380602	0.190301	+	0.981726i	$0.439054\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.37228	−0.332827	−0.166414	−	0.986056i	$-0.553219\pi$
$17$	−1.37228	−0.332827	−0.166414	+	0.986056i	$0.553219\pi$
$18$	0	0
$19$	1.37228	0.314823	0.157411	−	0.987533i	$-0.449685\pi$
$19$	1.37228	0.314823	0.157411	+	0.987533i	$0.449685\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	0	0
$23$	−3.37228	−0.703169	−0.351585	−	0.936156i	$-0.614357\pi$
$23$	−3.37228	−0.703169	−0.351585	+	0.936156i	$0.614357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.74456	−0.492938	−0.246469	−	0.969151i	$-0.579270\pi$
$31$	−2.74456	−0.492938	−0.246469	+	0.969151i	$0.579270\pi$
$32$	0	0
$33$	1.37228	0.238884
$34$	0	0
$35$	3.37228	0.570020
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	1.37228	0.219741
$40$	0	0
$41$	8.74456	1.36567	0.682836	−	0.730572i	$-0.260747\pi$
$41$	8.74456	1.36567	0.682836	+	0.730572i	$0.260747\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.74456	0.692066	0.346033	−	0.938222i	$-0.387529\pi$
$47$	4.74456	0.692066	0.346033	+	0.938222i	$0.387529\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	−1.37228	−0.192158
$52$	0	0
$53$	5.37228	0.737940	0.368970	−	0.929441i	$-0.379711\pi$
$53$	5.37228	0.737940	0.368970	+	0.929441i	$0.379711\pi$
$54$	0	0
$55$	1.37228	0.185038
$56$	0	0
$57$	1.37228	0.181763
$58$	0	0
$59$	−14.7446	−1.91958	−0.959789	−	0.280721i	$-0.909426\pi$
$59$	−14.7446	−1.91958	−0.959789	+	0.280721i	$0.909426\pi$
$60$	0	0
$61$	−2.74456	−0.351405	−0.175703	−	0.984443i	$-0.556220\pi$
$61$	−2.74456	−0.351405	−0.175703	+	0.984443i	$0.556220\pi$
$62$	0	0
$63$	3.37228	0.424868
$64$	0	0
$65$	1.37228	0.170211
$66$	0	0
$67$	−2.74456	−0.335302	−0.167651	−	0.985846i	$-0.553618\pi$
$67$	−2.74456	−0.335302	−0.167651	+	0.985846i	$0.553618\pi$
$68$	0	0
$69$	−3.37228	−0.405975
$70$	0	0
$71$	−1.25544	−0.148993	−0.0744965	−	0.997221i	$-0.523735\pi$
$71$	−1.25544	−0.148993	−0.0744965	+	0.997221i	$0.523735\pi$
$72$	0	0
$73$	−4.11684	−0.481840	−0.240920	−	0.970545i	$-0.577449\pi$
$73$	−4.11684	−0.481840	−0.240920	+	0.970545i	$0.577449\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	4.62772	0.527377
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.627719	−0.0689011	−0.0344505	−	0.999406i	$-0.510968\pi$
$83$	−0.627719	−0.0689011	−0.0344505	+	0.999406i	$0.510968\pi$
$84$	0	0
$85$	−1.37228	−0.148845
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	13.3723	1.41746	0.708729	−	0.705480i	$-0.249269\pi$
$89$	13.3723	1.41746	0.708729	+	0.705480i	$0.249269\pi$
$90$	0	0
$91$	4.62772	0.485117
$92$	0	0
$93$	−2.74456	−0.284598
$94$	0	0
$95$	1.37228	0.140793
$96$	0	0
$97$	13.4891	1.36961	0.684807	−	0.728725i	$-0.259887\pi$
$97$	13.4891	1.36961	0.684807	+	0.728725i	$0.259887\pi$
$98$	0	0
$99$	1.37228	0.137919
$100$	0	0
$101$	10.7446	1.06912	0.534562	−	0.845129i	$-0.320477\pi$
$101$	10.7446	1.06912	0.534562	+	0.845129i	$0.320477\pi$
$102$	0	0
$103$	0.744563	0.0733639	0.0366820	−	0.999327i	$-0.488321\pi$
$103$	0.744563	0.0733639	0.0366820	+	0.999327i	$0.488321\pi$
$104$	0	0
$105$	3.37228	0.329101
$106$	0	0
$107$	−3.37228	−0.326011	−0.163005	−	0.986625i	$-0.552119\pi$
$107$	−3.37228	−0.326011	−0.163005	+	0.986625i	$0.552119\pi$
$108$	0	0
$109$	4.62772	0.443255	0.221628	−	0.975131i	$-0.428863\pi$
$109$	4.62772	0.443255	0.221628	+	0.975131i	$0.428863\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	19.4891	1.83338	0.916691	−	0.399596i	$-0.130850\pi$
$113$	19.4891	1.83338	0.916691	+	0.399596i	$0.130850\pi$
$114$	0	0
$115$	−3.37228	−0.314467
$116$	0	0
$117$	1.37228	0.126867
$118$	0	0
$119$	−4.62772	−0.424222
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	8.74456	0.788471
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.37228	−0.299242	−0.149621	−	0.988743i	$-0.547805\pi$
$127$	−3.37228	−0.299242	−0.149621	+	0.988743i	$0.547805\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	4.62772	0.401274
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.7446	0.917970	0.458985	−	0.888444i	$-0.348213\pi$
$137$	10.7446	0.917970	0.458985	+	0.888444i	$0.348213\pi$
$138$	0	0
$139$	−2.74456	−0.232791	−0.116395	−	0.993203i	$-0.537134\pi$
$139$	−2.74456	−0.232791	−0.116395	+	0.993203i	$0.537134\pi$
$140$	0	0
$141$	4.74456	0.399564
$142$	0	0
$143$	1.88316	0.157477
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	4.37228	0.360620
$148$	0	0
$149$	9.25544	0.758235	0.379117	−	0.925349i	$-0.376228\pi$
$149$	9.25544	0.758235	0.379117	+	0.925349i	$0.376228\pi$
$150$	0	0
$151$	3.37228	0.274432	0.137216	−	0.990541i	$-0.456184\pi$
$151$	3.37228	0.274432	0.137216	+	0.990541i	$0.456184\pi$
$152$	0	0
$153$	−1.37228	−0.110942
$154$	0	0
$155$	−2.74456	−0.220449
$156$	0	0
$157$	3.25544	0.259812	0.129906	−	0.991526i	$-0.458532\pi$
$157$	3.25544	0.259812	0.129906	+	0.991526i	$0.458532\pi$
$158$	0	0
$159$	5.37228	0.426050
$160$	0	0
$161$	−11.3723	−0.896261
$162$	0	0
$163$	−4.86141	−0.380775	−0.190387	−	0.981709i	$-0.560974\pi$
$163$	−4.86141	−0.380775	−0.190387	+	0.981709i	$0.560974\pi$
$164$	0	0
$165$	1.37228	0.106832
$166$	0	0
$167$	1.88316	0.145723	0.0728615	−	0.997342i	$-0.476787\pi$
$167$	1.88316	0.145723	0.0728615	+	0.997342i	$0.476787\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	0	0
$171$	1.37228	0.104941
$172$	0	0
$173$	−6.86141	−0.521663	−0.260832	−	0.965384i	$-0.583997\pi$
$173$	−6.86141	−0.521663	−0.260832	+	0.965384i	$0.583997\pi$
$174$	0	0
$175$	3.37228	0.254921
$176$	0	0
$177$	−14.7446	−1.10827
$178$	0	0
$179$	−5.48913	−0.410276	−0.205138	−	0.978733i	$-0.565764\pi$
$179$	−5.48913	−0.410276	−0.205138	+	0.978733i	$0.565764\pi$
$180$	0	0
$181$	−20.9783	−1.55930	−0.779651	−	0.626215i	$-0.784603\pi$
$181$	−20.9783	−1.55930	−0.779651	+	0.626215i	$0.784603\pi$
$182$	0	0
$183$	−2.74456	−0.202884
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−1.88316	−0.137710
$188$	0	0
$189$	3.37228	0.245297
$190$	0	0
$191$	19.3723	1.40173	0.700865	−	0.713294i	$-0.252798\pi$
$191$	19.3723	1.40173	0.700865	+	0.713294i	$0.252798\pi$
$192$	0	0
$193$	−1.25544	−0.0903684	−0.0451842	−	0.998979i	$-0.514387\pi$
$193$	−1.25544	−0.0903684	−0.0451842	+	0.998979i	$0.514387\pi$
$194$	0	0
$195$	1.37228	0.0982711
$196$	0	0
$197$	−18.8614	−1.34382	−0.671910	−	0.740633i	$-0.734526\pi$
$197$	−18.8614	−1.34382	−0.671910	+	0.740633i	$0.734526\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−2.74456	−0.193587
$202$	0	0
$203$	20.2337	1.42013
$204$	0	0
$205$	8.74456	0.610747
$206$	0	0
$207$	−3.37228	−0.234390
$208$	0	0
$209$	1.88316	0.130261
$210$	0	0
$211$	9.25544	0.637171	0.318585	−	0.947894i	$-0.396792\pi$
$211$	9.25544	0.637171	0.318585	+	0.947894i	$0.396792\pi$
$212$	0	0
$213$	−1.25544	−0.0860212
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−9.25544	−0.628300
$218$	0	0
$219$	−4.11684	−0.278191
$220$	0	0
$221$	−1.88316	−0.126675
$222$	0	0
$223$	−18.9783	−1.27088	−0.635439	−	0.772151i	$-0.719181\pi$
$223$	−18.9783	−1.27088	−0.635439	+	0.772151i	$0.719181\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	4.62772	0.304482
$232$	0	0
$233$	−22.7446	−1.49005	−0.745023	−	0.667039i	$-0.767561\pi$
$233$	−22.7446	−1.49005	−0.745023	+	0.667039i	$0.767561\pi$
$234$	0	0
$235$	4.74456	0.309501
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	5.48913	0.355062	0.177531	−	0.984115i	$-0.443189\pi$
$239$	5.48913	0.355062	0.177531	+	0.984115i	$0.443189\pi$
$240$	0	0
$241$	−0.510875	−0.0329083	−0.0164542	−	0.999865i	$-0.505238\pi$
$241$	−0.510875	−0.0329083	−0.0164542	+	0.999865i	$0.505238\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.37228	0.279335
$246$	0	0
$247$	1.88316	0.119822
$248$	0	0
$249$	−0.627719	−0.0397801
$250$	0	0
$251$	26.7446	1.68810	0.844051	−	0.536263i	$-0.180164\pi$
$251$	26.7446	1.68810	0.844051	+	0.536263i	$0.180164\pi$
$252$	0	0
$253$	−4.62772	−0.290942
$254$	0	0
$255$	−1.37228	−0.0859356
$256$	0	0
$257$	−5.37228	−0.335114	−0.167557	−	0.985862i	$-0.553588\pi$
$257$	−5.37228	−0.335114	−0.167557	+	0.985862i	$0.553588\pi$
$258$	0	0
$259$	−3.37228	−0.209543
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−10.2337	−0.631036	−0.315518	−	0.948920i	$-0.602178\pi$
$263$	−10.2337	−0.631036	−0.315518	+	0.948920i	$0.602178\pi$
$264$	0	0
$265$	5.37228	0.330017
$266$	0	0
$267$	13.3723	0.818370
$268$	0	0
$269$	0.627719	0.0382727	0.0191363	−	0.999817i	$-0.493908\pi$
$269$	0.627719	0.0382727	0.0191363	+	0.999817i	$0.493908\pi$
$270$	0	0
$271$	−13.4891	−0.819406	−0.409703	−	0.912219i	$-0.634368\pi$
$271$	−13.4891	−0.819406	−0.409703	+	0.912219i	$0.634368\pi$
$272$	0	0
$273$	4.62772	0.280082
$274$	0	0
$275$	1.37228	0.0827517
$276$	0	0
$277$	25.6060	1.53851	0.769257	−	0.638940i	$-0.220627\pi$
$277$	25.6060	1.53851	0.769257	+	0.638940i	$0.220627\pi$
$278$	0	0
$279$	−2.74456	−0.164313
$280$	0	0
$281$	1.37228	0.0818634	0.0409317	−	0.999162i	$-0.486967\pi$
$281$	1.37228	0.0818634	0.0409317	+	0.999162i	$0.486967\pi$
$282$	0	0
$283$	11.6060	0.689903	0.344952	−	0.938620i	$-0.387895\pi$
$283$	11.6060	0.689903	0.344952	+	0.938620i	$0.387895\pi$
$284$	0	0
$285$	1.37228	0.0812869
$286$	0	0
$287$	29.4891	1.74069
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	13.4891	0.790747
$292$	0	0
$293$	−4.11684	−0.240509	−0.120254	−	0.992743i	$-0.538371\pi$
$293$	−4.11684	−0.240509	−0.120254	+	0.992743i	$0.538371\pi$
$294$	0	0
$295$	−14.7446	−0.858462
$296$	0	0
$297$	1.37228	0.0796278
$298$	0	0
$299$	−4.62772	−0.267628
$300$	0	0
$301$	13.4891	0.777500
$302$	0	0
$303$	10.7446	0.617259
$304$	0	0
$305$	−2.74456	−0.157153
$306$	0	0
$307$	10.7446	0.613225	0.306612	−	0.951834i	$-0.400805\pi$
$307$	10.7446	0.613225	0.306612	+	0.951834i	$0.400805\pi$
$308$	0	0
$309$	0.744563	0.0423567
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	3.37228	0.190007
$316$	0	0
$317$	−16.9783	−0.953594	−0.476797	−	0.879014i	$-0.658202\pi$
$317$	−16.9783	−0.953594	−0.476797	+	0.879014i	$0.658202\pi$
$318$	0	0
$319$	8.23369	0.460998
$320$	0	0
$321$	−3.37228	−0.188222
$322$	0	0
$323$	−1.88316	−0.104782
$324$	0	0
$325$	1.37228	0.0761205
$326$	0	0
$327$	4.62772	0.255913
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	8.74456	0.480645	0.240322	−	0.970693i	$-0.422747\pi$
$331$	8.74456	0.480645	0.240322	+	0.970693i	$0.422747\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−2.74456	−0.149951
$336$	0	0
$337$	−4.11684	−0.224259	−0.112129	−	0.993694i	$-0.535767\pi$
$337$	−4.11684	−0.224259	−0.112129	+	0.993694i	$0.535767\pi$
$338$	0	0
$339$	19.4891	1.05850
$340$	0	0
$341$	−3.76631	−0.203957
$342$	0	0
$343$	−8.86141	−0.478471
$344$	0	0
$345$	−3.37228	−0.181558
$346$	0	0
$347$	−1.48913	−0.0799404	−0.0399702	−	0.999201i	$-0.512726\pi$
$347$	−1.48913	−0.0799404	−0.0399702	+	0.999201i	$0.512726\pi$
$348$	0	0
$349$	16.7446	0.896316	0.448158	−	0.893954i	$-0.352080\pi$
$349$	16.7446	0.896316	0.448158	+	0.893954i	$0.352080\pi$
$350$	0	0
$351$	1.37228	0.0732470
$352$	0	0
$353$	3.48913	0.185707	0.0928537	−	0.995680i	$-0.470401\pi$
$353$	3.48913	0.185707	0.0928537	+	0.995680i	$0.470401\pi$
$354$	0	0
$355$	−1.25544	−0.0666317
$356$	0	0
$357$	−4.62772	−0.244925
$358$	0	0
$359$	−32.4674	−1.71356	−0.856781	−	0.515680i	$-0.827539\pi$
$359$	−32.4674	−1.71356	−0.856781	+	0.515680i	$0.827539\pi$
$360$	0	0
$361$	−17.1168	−0.900887
$362$	0	0
$363$	−9.11684	−0.478510
$364$	0	0
$365$	−4.11684	−0.215485
$366$	0	0
$367$	−20.6277	−1.07676	−0.538379	−	0.842703i	$-0.680963\pi$
$367$	−20.6277	−1.07676	−0.538379	+	0.842703i	$0.680963\pi$
$368$	0	0
$369$	8.74456	0.455224
$370$	0	0
$371$	18.1168	0.940580
$372$	0	0
$373$	3.48913	0.180660	0.0903300	−	0.995912i	$-0.471208\pi$
$373$	3.48913	0.180660	0.0903300	+	0.995912i	$0.471208\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	8.23369	0.424057
$378$	0	0
$379$	−24.2337	−1.24480	−0.622400	−	0.782699i	$-0.713842\pi$
$379$	−24.2337	−1.24480	−0.622400	+	0.782699i	$0.713842\pi$
$380$	0	0
$381$	−3.37228	−0.172767
$382$	0	0
$383$	−27.6060	−1.41060	−0.705300	−	0.708909i	$-0.749188\pi$
$383$	−27.6060	−1.41060	−0.705300	+	0.708909i	$0.749188\pi$
$384$	0	0
$385$	4.62772	0.235850
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	23.4891	1.19095	0.595473	−	0.803375i	$-0.296965\pi$
$389$	23.4891	1.19095	0.595473	+	0.803375i	$0.296965\pi$
$390$	0	0
$391$	4.62772	0.234034
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	20.7446	1.04114	0.520570	−	0.853819i	$-0.325720\pi$
$397$	20.7446	1.04114	0.520570	+	0.853819i	$0.325720\pi$
$398$	0	0
$399$	4.62772	0.231676
$400$	0	0
$401$	−36.1168	−1.80359	−0.901795	−	0.432165i	$-0.857750\pi$
$401$	−36.1168	−1.80359	−0.901795	+	0.432165i	$0.857750\pi$
$402$	0	0
$403$	−3.76631	−0.187613
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−1.37228	−0.0680215
$408$	0	0
$409$	−8.74456	−0.432391	−0.216195	−	0.976350i	$-0.569365\pi$
$409$	−8.74456	−0.432391	−0.216195	+	0.976350i	$0.569365\pi$
$410$	0	0
$411$	10.7446	0.529990
$412$	0	0
$413$	−49.7228	−2.44670
$414$	0	0
$415$	−0.627719	−0.0308135
$416$	0	0
$417$	−2.74456	−0.134402
$418$	0	0
$419$	−7.88316	−0.385117	−0.192559	−	0.981285i	$-0.561679\pi$
$419$	−7.88316	−0.385117	−0.192559	+	0.981285i	$0.561679\pi$
$420$	0	0
$421$	24.2337	1.18108	0.590539	−	0.807009i	$-0.298915\pi$
$421$	24.2337	1.18108	0.590539	+	0.807009i	$0.298915\pi$
$422$	0	0
$423$	4.74456	0.230689
$424$	0	0
$425$	−1.37228	−0.0665654
$426$	0	0
$427$	−9.25544	−0.447902
$428$	0	0
$429$	1.88316	0.0909196
$430$	0	0
$431$	31.6060	1.52241	0.761203	−	0.648514i	$-0.224609\pi$
$431$	31.6060	1.52241	0.761203	+	0.648514i	$0.224609\pi$
$432$	0	0
$433$	−10.8614	−0.521966	−0.260983	−	0.965343i	$-0.584047\pi$
$433$	−10.8614	−0.521966	−0.260983	+	0.965343i	$0.584047\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	0	0
$437$	−4.62772	−0.221374
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	13.3723	0.633907
$446$	0	0
$447$	9.25544	0.437767
$448$	0	0
$449$	36.9783	1.74511	0.872556	−	0.488515i	$-0.162461\pi$
$449$	36.9783	1.74511	0.872556	+	0.488515i	$0.162461\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	3.37228	0.158444
$454$	0	0
$455$	4.62772	0.216951
$456$	0	0
$457$	−40.4674	−1.89298	−0.946492	−	0.322727i	$-0.895400\pi$
$457$	−40.4674	−1.89298	−0.946492	+	0.322727i	$0.895400\pi$
$458$	0	0
$459$	−1.37228	−0.0640526
$460$	0	0
$461$	7.48913	0.348803	0.174402	−	0.984675i	$-0.444201\pi$
$461$	7.48913	0.348803	0.174402	+	0.984675i	$0.444201\pi$
$462$	0	0
$463$	19.7228	0.916597	0.458298	−	0.888798i	$-0.348459\pi$
$463$	19.7228	0.916597	0.458298	+	0.888798i	$0.348459\pi$
$464$	0	0
$465$	−2.74456	−0.127276
$466$	0	0
$467$	−1.48913	−0.0689085	−0.0344543	−	0.999406i	$-0.510969\pi$
$467$	−1.48913	−0.0689085	−0.0344543	+	0.999406i	$0.510969\pi$
$468$	0	0
$469$	−9.25544	−0.427376
$470$	0	0
$471$	3.25544	0.150003
$472$	0	0
$473$	5.48913	0.252390
$474$	0	0
$475$	1.37228	0.0629646
$476$	0	0
$477$	5.37228	0.245980
$478$	0	0
$479$	18.1168	0.827780	0.413890	−	0.910327i	$-0.364170\pi$
$479$	18.1168	0.827780	0.413890	+	0.910327i	$0.364170\pi$
$480$	0	0
$481$	−1.37228	−0.0625706
$482$	0	0
$483$	−11.3723	−0.517457
$484$	0	0
$485$	13.4891	0.612510
$486$	0	0
$487$	30.4674	1.38061	0.690304	−	0.723519i	$-0.257477\pi$
$487$	30.4674	1.38061	0.690304	+	0.723519i	$0.257477\pi$
$488$	0	0
$489$	−4.86141	−0.219840
$490$	0	0
$491$	−2.62772	−0.118587	−0.0592936	−	0.998241i	$-0.518885\pi$
$491$	−2.62772	−0.118587	−0.0592936	+	0.998241i	$0.518885\pi$
$492$	0	0
$493$	−8.23369	−0.370827
$494$	0	0
$495$	1.37228	0.0616795
$496$	0	0
$497$	−4.23369	−0.189907
$498$	0	0
$499$	−20.3505	−0.911015	−0.455507	−	0.890232i	$-0.650542\pi$
$499$	−20.3505	−0.911015	−0.455507	+	0.890232i	$0.650542\pi$
$500$	0	0
$501$	1.88316	0.0841332
$502$	0	0
$503$	−5.48913	−0.244748	−0.122374	−	0.992484i	$-0.539051\pi$
$503$	−5.48913	−0.244748	−0.122374	+	0.992484i	$0.539051\pi$
$504$	0	0
$505$	10.7446	0.478127
$506$	0	0
$507$	−11.1168	−0.493716
$508$	0	0
$509$	18.3505	0.813373	0.406687	−	0.913568i	$-0.366684\pi$
$509$	18.3505	0.813373	0.406687	+	0.913568i	$0.366684\pi$
$510$	0	0
$511$	−13.8832	−0.614155
$512$	0	0
$513$	1.37228	0.0605877
$514$	0	0
$515$	0.744563	0.0328094
$516$	0	0
$517$	6.51087	0.286348
$518$	0	0
$519$	−6.86141	−0.301182
$520$	0	0
$521$	−5.76631	−0.252627	−0.126313	−	0.991990i	$-0.540314\pi$
$521$	−5.76631	−0.252627	−0.126313	+	0.991990i	$0.540314\pi$
$522$	0	0
$523$	6.97825	0.305138	0.152569	−	0.988293i	$-0.451245\pi$
$523$	6.97825	0.305138	0.152569	+	0.988293i	$0.451245\pi$
$524$	0	0
$525$	3.37228	0.147178
$526$	0	0
$527$	3.76631	0.164063
$528$	0	0
$529$	−11.6277	−0.505553
$530$	0	0
$531$	−14.7446	−0.639860
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−3.37228	−0.145796
$536$	0	0
$537$	−5.48913	−0.236873
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	4.86141	0.209008	0.104504	−	0.994524i	$-0.466674\pi$
$541$	4.86141	0.209008	0.104504	+	0.994524i	$0.466674\pi$
$542$	0	0
$543$	−20.9783	−0.900263
$544$	0	0
$545$	4.62772	0.198230
$546$	0	0
$547$	2.11684	0.0905097	0.0452549	−	0.998975i	$-0.485590\pi$
$547$	2.11684	0.0905097	0.0452549	+	0.998975i	$0.485590\pi$
$548$	0	0
$549$	−2.74456	−0.117135
$550$	0	0
$551$	8.23369	0.350767
$552$	0	0
$553$	−13.4891	−0.573616
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	40.7446	1.72640	0.863201	−	0.504860i	$-0.168456\pi$
$557$	40.7446	1.72640	0.863201	+	0.504860i	$0.168456\pi$
$558$	0	0
$559$	5.48913	0.232165
$560$	0	0
$561$	−1.88316	−0.0795069
$562$	0	0
$563$	−13.2554	−0.558650	−0.279325	−	0.960197i	$-0.590111\pi$
$563$	−13.2554	−0.558650	−0.279325	+	0.960197i	$0.590111\pi$
$564$	0	0
$565$	19.4891	0.819914
$566$	0	0
$567$	3.37228	0.141623
$568$	0	0
$569$	−32.3505	−1.35620	−0.678102	−	0.734967i	$-0.737197\pi$
$569$	−32.3505	−1.35620	−0.678102	+	0.734967i	$0.737197\pi$
$570$	0	0
$571$	1.25544	0.0525384	0.0262692	−	0.999655i	$-0.491637\pi$
$571$	1.25544	0.0525384	0.0262692	+	0.999655i	$0.491637\pi$
$572$	0	0
$573$	19.3723	0.809289
$574$	0	0
$575$	−3.37228	−0.140634
$576$	0	0
$577$	26.7446	1.11339	0.556695	−	0.830717i	$-0.312069\pi$
$577$	26.7446	1.11339	0.556695	+	0.830717i	$0.312069\pi$
$578$	0	0
$579$	−1.25544	−0.0521742
$580$	0	0
$581$	−2.11684	−0.0878215
$582$	0	0
$583$	7.37228	0.305329
$584$	0	0
$585$	1.37228	0.0567368
$586$	0	0
$587$	−1.48913	−0.0614628	−0.0307314	−	0.999528i	$-0.509784\pi$
$587$	−1.48913	−0.0614628	−0.0307314	+	0.999528i	$0.509784\pi$
$588$	0	0
$589$	−3.76631	−0.155188
$590$	0	0
$591$	−18.8614	−0.775855
$592$	0	0
$593$	−22.9783	−0.943604	−0.471802	−	0.881705i	$-0.656396\pi$
$593$	−22.9783	−0.943604	−0.471802	+	0.881705i	$0.656396\pi$
$594$	0	0
$595$	−4.62772	−0.189718
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	22.9783	0.938866	0.469433	−	0.882968i	$-0.344458\pi$
$599$	22.9783	0.938866	0.469433	+	0.882968i	$0.344458\pi$
$600$	0	0
$601$	30.8614	1.25886	0.629432	−	0.777056i	$-0.283288\pi$
$601$	30.8614	1.25886	0.629432	+	0.777056i	$0.283288\pi$
$602$	0	0
$603$	−2.74456	−0.111767
$604$	0	0
$605$	−9.11684	−0.370652
$606$	0	0
$607$	−38.4674	−1.56134	−0.780671	−	0.624942i	$-0.785123\pi$
$607$	−38.4674	−1.56134	−0.780671	+	0.624942i	$0.785123\pi$
$608$	0	0
$609$	20.2337	0.819910
$610$	0	0
$611$	6.51087	0.263402
$612$	0	0
$613$	30.4674	1.23057	0.615283	−	0.788306i	$-0.289042\pi$
$613$	30.4674	1.23057	0.615283	+	0.788306i	$0.289042\pi$
$614$	0	0
$615$	8.74456	0.352615
$616$	0	0
$617$	32.2337	1.29768	0.648840	−	0.760925i	$-0.275255\pi$
$617$	32.2337	1.29768	0.648840	+	0.760925i	$0.275255\pi$
$618$	0	0
$619$	26.9783	1.08435	0.542174	−	0.840266i	$-0.317601\pi$
$619$	26.9783	1.08435	0.542174	+	0.840266i	$0.317601\pi$
$620$	0	0
$621$	−3.37228	−0.135325
$622$	0	0
$623$	45.0951	1.80670
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.88316	0.0752060
$628$	0	0
$629$	1.37228	0.0547164
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	9.25544	0.367871
$634$	0	0
$635$	−3.37228	−0.133825
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−1.25544	−0.0496643
$640$	0	0
$641$	43.7228	1.72695	0.863474	−	0.504394i	$-0.168284\pi$
$641$	43.7228	1.72695	0.863474	+	0.504394i	$0.168284\pi$
$642$	0	0
$643$	−23.6060	−0.930929	−0.465464	−	0.885067i	$-0.654113\pi$
$643$	−23.6060	−0.930929	−0.465464	+	0.885067i	$0.654113\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	39.6060	1.55707	0.778536	−	0.627600i	$-0.215963\pi$
$647$	39.6060	1.55707	0.778536	+	0.627600i	$0.215963\pi$
$648$	0	0
$649$	−20.2337	−0.794242
$650$	0	0
$651$	−9.25544	−0.362749
$652$	0	0
$653$	19.7228	0.771813	0.385907	−	0.922538i	$-0.373889\pi$
$653$	19.7228	0.771813	0.385907	+	0.922538i	$0.373889\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.11684	−0.160613
$658$	0	0
$659$	2.23369	0.0870121	0.0435061	−	0.999053i	$-0.486147\pi$
$659$	2.23369	0.0870121	0.0435061	+	0.999053i	$0.486147\pi$
$660$	0	0
$661$	−26.1168	−1.01583	−0.507914	−	0.861408i	$-0.669583\pi$
$661$	−26.1168	−1.01583	−0.507914	+	0.861408i	$0.669583\pi$
$662$	0	0
$663$	−1.88316	−0.0731357
$664$	0	0
$665$	4.62772	0.179455
$666$	0	0
$667$	−20.2337	−0.783452
$668$	0	0
$669$	−18.9783	−0.733742
$670$	0	0
$671$	−3.76631	−0.145397
$672$	0	0
$673$	2.86141	0.110299	0.0551496	−	0.998478i	$-0.482436\pi$
$673$	2.86141	0.110299	0.0551496	+	0.998478i	$0.482436\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−2.86141	−0.109973	−0.0549864	−	0.998487i	$-0.517512\pi$
$677$	−2.86141	−0.109973	−0.0549864	+	0.998487i	$0.517512\pi$
$678$	0	0
$679$	45.4891	1.74571
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−30.9783	−1.18535	−0.592675	−	0.805442i	$-0.701928\pi$
$683$	−30.9783	−1.18535	−0.592675	+	0.805442i	$0.701928\pi$
$684$	0	0
$685$	10.7446	0.410529
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	7.37228	0.280862
$690$	0	0
$691$	−22.7446	−0.865244	−0.432622	−	0.901575i	$-0.642412\pi$
$691$	−22.7446	−0.865244	−0.432622	+	0.901575i	$0.642412\pi$
$692$	0	0
$693$	4.62772	0.175792
$694$	0	0
$695$	−2.74456	−0.104107
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	−22.7446	−0.860278
$700$	0	0
$701$	−47.7228	−1.80247	−0.901233	−	0.433335i	$-0.857337\pi$
$701$	−47.7228	−1.80247	−0.901233	+	0.433335i	$0.857337\pi$
$702$	0	0
$703$	−1.37228	−0.0517566
$704$	0	0
$705$	4.74456	0.178691
$706$	0	0
$707$	36.2337	1.36271
$708$	0	0
$709$	−20.6277	−0.774690	−0.387345	−	0.921935i	$-0.626608\pi$
$709$	−20.6277	−0.774690	−0.387345	+	0.921935i	$0.626608\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	9.25544	0.346619
$714$	0	0
$715$	1.88316	0.0704260
$716$	0	0
$717$	5.48913	0.204995
$718$	0	0
$719$	13.7228	0.511775	0.255887	−	0.966707i	$-0.417632\pi$
$719$	13.7228	0.511775	0.255887	+	0.966707i	$0.417632\pi$
$720$	0	0
$721$	2.51087	0.0935099
$722$	0	0
$723$	−0.510875	−0.0189996
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−8.97825	−0.332985	−0.166492	−	0.986043i	$-0.553244\pi$
$727$	−8.97825	−0.332985	−0.166492	+	0.986043i	$0.553244\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.48913	−0.203023
$732$	0	0
$733$	3.48913	0.128874	0.0644369	−	0.997922i	$-0.479475\pi$
$733$	3.48913	0.128874	0.0644369	+	0.997922i	$0.479475\pi$
$734$	0	0
$735$	4.37228	0.161274
$736$	0	0
$737$	−3.76631	−0.138734
$738$	0	0
$739$	−25.7228	−0.946229	−0.473114	−	0.881001i	$-0.656870\pi$
$739$	−25.7228	−0.946229	−0.473114	+	0.881001i	$0.656870\pi$
$740$	0	0
$741$	1.88316	0.0691795
$742$	0	0
$743$	−22.4674	−0.824248	−0.412124	−	0.911128i	$-0.635213\pi$
$743$	−22.4674	−0.824248	−0.412124	+	0.911128i	$0.635213\pi$
$744$	0	0
$745$	9.25544	0.339093
$746$	0	0
$747$	−0.627719	−0.0229670
$748$	0	0
$749$	−11.3723	−0.415534
$750$	0	0
$751$	10.5109	0.383547	0.191774	−	0.981439i	$-0.438576\pi$
$751$	10.5109	0.383547	0.191774	+	0.981439i	$0.438576\pi$
$752$	0	0
$753$	26.7446	0.974626
$754$	0	0
$755$	3.37228	0.122730
$756$	0	0
$757$	44.1168	1.60345	0.801727	−	0.597690i	$-0.203915\pi$
$757$	44.1168	1.60345	0.801727	+	0.597690i	$0.203915\pi$
$758$	0	0
$759$	−4.62772	−0.167976
$760$	0	0
$761$	−12.5109	−0.453519	−0.226759	−	0.973951i	$-0.572813\pi$
$761$	−12.5109	−0.453519	−0.226759	+	0.973951i	$0.572813\pi$
$762$	0	0
$763$	15.6060	0.564974
$764$	0	0
$765$	−1.37228	−0.0496149
$766$	0	0
$767$	−20.2337	−0.730596
$768$	0	0
$769$	−51.4891	−1.85675	−0.928373	−	0.371651i	$-0.878792\pi$
$769$	−51.4891	−1.85675	−0.928373	+	0.371651i	$0.878792\pi$
$770$	0	0
$771$	−5.37228	−0.193478
$772$	0	0
$773$	20.1168	0.723553	0.361776	−	0.932265i	$-0.382170\pi$
$773$	20.1168	0.723553	0.361776	+	0.932265i	$0.382170\pi$
$774$	0	0
$775$	−2.74456	−0.0985876
$776$	0	0
$777$	−3.37228	−0.120980
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	−1.72281	−0.0616471
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	3.25544	0.116192
$786$	0	0
$787$	−10.7446	−0.383002	−0.191501	−	0.981492i	$-0.561336\pi$
$787$	−10.7446	−0.383002	−0.191501	+	0.981492i	$0.561336\pi$
$788$	0	0
$789$	−10.2337	−0.364329
$790$	0	0
$791$	65.7228	2.33683
$792$	0	0
$793$	−3.76631	−0.133746
$794$	0	0
$795$	5.37228	0.190535
$796$	0	0
$797$	4.51087	0.159783	0.0798917	−	0.996804i	$-0.474543\pi$
$797$	4.51087	0.159783	0.0798917	+	0.996804i	$0.474543\pi$
$798$	0	0
$799$	−6.51087	−0.230338
$800$	0	0
$801$	13.3723	0.472486
$802$	0	0
$803$	−5.64947	−0.199365
$804$	0	0
$805$	−11.3723	−0.400820
$806$	0	0
$807$	0.627719	0.0220967
$808$	0	0
$809$	−24.1168	−0.847903	−0.423952	−	0.905685i	$-0.639357\pi$
$809$	−24.1168	−0.847903	−0.423952	+	0.905685i	$0.639357\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−13.4891	−0.473084
$814$	0	0
$815$	−4.86141	−0.170288
$816$	0	0
$817$	5.48913	0.192040
$818$	0	0
$819$	4.62772	0.161706
$820$	0	0
$821$	26.3505	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$821$	26.3505	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$822$	0	0
$823$	−25.8832	−0.902230	−0.451115	−	0.892466i	$-0.648974\pi$
$823$	−25.8832	−0.902230	−0.451115	+	0.892466i	$0.648974\pi$
$824$	0	0
$825$	1.37228	0.0477767
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−35.8397	−1.24476	−0.622381	−	0.782714i	$-0.713835\pi$
$829$	−35.8397	−1.24476	−0.622381	+	0.782714i	$0.713835\pi$
$830$	0	0
$831$	25.6060	0.888261
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	1.88316	0.0651693
$836$	0	0
$837$	−2.74456	−0.0948660
$838$	0	0
$839$	9.25544	0.319533	0.159767	−	0.987155i	$-0.448926\pi$
$839$	9.25544	0.319533	0.159767	+	0.987155i	$0.448926\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	1.37228	0.0472639
$844$	0	0
$845$	−11.1168	−0.382431
$846$	0	0
$847$	−30.7446	−1.05640
$848$	0	0
$849$	11.6060	0.398316
$850$	0	0
$851$	3.37228	0.115600
$852$	0	0
$853$	40.3505	1.38158	0.690788	−	0.723057i	$-0.257264\pi$
$853$	40.3505	1.38158	0.690788	+	0.723057i	$0.257264\pi$
$854$	0	0
$855$	1.37228	0.0469310
$856$	0	0
$857$	17.8397	0.609391	0.304696	−	0.952450i	$-0.401445\pi$
$857$	17.8397	0.609391	0.304696	+	0.952450i	$0.401445\pi$
$858$	0	0
$859$	−17.8397	−0.608681	−0.304341	−	0.952563i	$-0.598436\pi$
$859$	−17.8397	−0.608681	−0.304341	+	0.952563i	$0.598436\pi$
$860$	0	0
$861$	29.4891	1.00499
$862$	0	0
$863$	−3.25544	−0.110816	−0.0554082	−	0.998464i	$-0.517646\pi$
$863$	−3.25544	−0.110816	−0.0554082	+	0.998464i	$0.517646\pi$
$864$	0	0
$865$	−6.86141	−0.233295
$866$	0	0
$867$	−15.1168	−0.513395
$868$	0	0
$869$	−5.48913	−0.186206
$870$	0	0
$871$	−3.76631	−0.127617
$872$	0	0
$873$	13.4891	0.456538
$874$	0	0
$875$	3.37228	0.114004
$876$	0	0
$877$	−54.2337	−1.83134	−0.915671	−	0.401929i	$-0.868340\pi$
$877$	−54.2337	−1.83134	−0.915671	+	0.401929i	$0.868340\pi$
$878$	0	0
$879$	−4.11684	−0.138858
$880$	0	0
$881$	−14.2337	−0.479545	−0.239773	−	0.970829i	$-0.577073\pi$
$881$	−14.2337	−0.479545	−0.239773	+	0.970829i	$0.577073\pi$
$882$	0	0
$883$	50.1168	1.68657	0.843283	−	0.537470i	$-0.180620\pi$
$883$	50.1168	1.68657	0.843283	+	0.537470i	$0.180620\pi$
$884$	0	0
$885$	−14.7446	−0.495633
$886$	0	0
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	−11.3723	−0.381414
$890$	0	0
$891$	1.37228	0.0459732
$892$	0	0
$893$	6.51087	0.217878
$894$	0	0
$895$	−5.48913	−0.183481
$896$	0	0
$897$	−4.62772	−0.154515
$898$	0	0
$899$	−16.4674	−0.549218
$900$	0	0
$901$	−7.37228	−0.245606
$902$	0	0
$903$	13.4891	0.448890
$904$	0	0
$905$	−20.9783	−0.697341
$906$	0	0
$907$	3.60597	0.119734	0.0598671	−	0.998206i	$-0.480932\pi$
$907$	3.60597	0.119734	0.0598671	+	0.998206i	$0.480932\pi$
$908$	0	0
$909$	10.7446	0.356375
$910$	0	0
$911$	−34.9783	−1.15888	−0.579441	−	0.815014i	$-0.696729\pi$
$911$	−34.9783	−1.15888	−0.579441	+	0.815014i	$0.696729\pi$
$912$	0	0
$913$	−0.861407	−0.0285084
$914$	0	0
$915$	−2.74456	−0.0907324
$916$	0	0
$917$	0	0
$918$	0	0
$919$	34.5109	1.13841	0.569204	−	0.822196i	$-0.307251\pi$
$919$	34.5109	1.13841	0.569204	+	0.822196i	$0.307251\pi$
$920$	0	0
$921$	10.7446	0.354045
$922$	0	0
$923$	−1.72281	−0.0567071
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	0.744563	0.0244546
$928$	0	0
$929$	−12.5109	−0.410468	−0.205234	−	0.978713i	$-0.565796\pi$
$929$	−12.5109	−0.410468	−0.205234	+	0.978713i	$0.565796\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−1.88316	−0.0615858
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	0	0
$941$	15.7663	0.513967	0.256984	−	0.966416i	$-0.417271\pi$
$941$	15.7663	0.513967	0.256984	+	0.966416i	$0.417271\pi$
$942$	0	0
$943$	−29.4891	−0.960298
$944$	0	0
$945$	3.37228	0.109700
$946$	0	0
$947$	28.4674	0.925065	0.462533	−	0.886602i	$-0.346941\pi$
$947$	28.4674	0.925065	0.462533	+	0.886602i	$0.346941\pi$
$948$	0	0
$949$	−5.64947	−0.183389
$950$	0	0
$951$	−16.9783	−0.550557
$952$	0	0
$953$	−50.7446	−1.64378	−0.821889	−	0.569648i	$-0.807080\pi$
$953$	−50.7446	−1.64378	−0.821889	+	0.569648i	$0.807080\pi$
$954$	0	0
$955$	19.3723	0.626872
$956$	0	0
$957$	8.23369	0.266157
$958$	0	0
$959$	36.2337	1.17005
$960$	0	0
$961$	−23.4674	−0.757012
$962$	0	0
$963$	−3.37228	−0.108670
$964$	0	0
$965$	−1.25544	−0.0404140
$966$	0	0
$967$	16.9783	0.545984	0.272992	−	0.962016i	$-0.411987\pi$
$967$	16.9783	0.545984	0.272992	+	0.962016i	$0.411987\pi$
$968$	0	0
$969$	−1.88316	−0.0604957
$970$	0	0
$971$	−50.2337	−1.61208	−0.806038	−	0.591864i	$-0.798392\pi$
$971$	−50.2337	−1.61208	−0.806038	+	0.591864i	$0.798392\pi$
$972$	0	0
$973$	−9.25544	−0.296716
$974$	0	0
$975$	1.37228	0.0439482
$976$	0	0
$977$	25.1386	0.804255	0.402127	−	0.915584i	$-0.368271\pi$
$977$	25.1386	0.804255	0.402127	+	0.915584i	$0.368271\pi$
$978$	0	0
$979$	18.3505	0.586486
$980$	0	0
$981$	4.62772	0.147752
$982$	0	0
$983$	12.5109	0.399035	0.199517	−	0.979894i	$-0.436063\pi$
$983$	12.5109	0.399035	0.199517	+	0.979894i	$0.436063\pi$
$984$	0	0
$985$	−18.8614	−0.600974
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	−13.4891	−0.428929
$990$	0	0
$991$	−22.9783	−0.729928	−0.364964	−	0.931022i	$-0.618919\pi$
$991$	−22.9783	−0.729928	−0.364964	+	0.931022i	$0.618919\pi$
$992$	0	0
$993$	8.74456	0.277500
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	16.1168	0.510426	0.255213	−	0.966885i	$-0.417854\pi$
$997$	16.1168	0.510426	0.255213	+	0.966885i	$0.417854\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.br.1.2		2
4.3	odd	2		1110.2.a.p.1.1	✓	2
12.11	even	2		3330.2.a.be.1.1		2
20.19	odd	2		5550.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.p.1.1	✓	2	4.3	odd	2
3330.2.a.be.1.1		2	12.11	even	2
5550.2.a.ca.1.2		2	20.19	odd	2
8880.2.a.br.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{9} +1.37228 q^{11} +1.37228 q^{13} +1.00000 q^{15} -1.37228 q^{17} +1.37228 q^{19} +3.37228 q^{21} -3.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -2.74456 q^{31} +1.37228 q^{33} +3.37228 q^{35} -1.00000 q^{37} +1.37228 q^{39} +8.74456 q^{41} +4.00000 q^{43} +1.00000 q^{45} +4.74456 q^{47} +4.37228 q^{49} -1.37228 q^{51} +5.37228 q^{53} +1.37228 q^{55} +1.37228 q^{57} -14.7446 q^{59} -2.74456 q^{61} +3.37228 q^{63} +1.37228 q^{65} -2.74456 q^{67} -3.37228 q^{69} -1.25544 q^{71} -4.11684 q^{73} +1.00000 q^{75} +4.62772 q^{77} -4.00000 q^{79} +1.00000 q^{81} -0.627719 q^{83} -1.37228 q^{85} +6.00000 q^{87} +13.3723 q^{89} +4.62772 q^{91} -2.74456 q^{93} +1.37228 q^{95} +13.4891 q^{97} +1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} - 3 q^{11} - 3 q^{13} + 2 q^{15} + 3 q^{17} - 3 q^{19} + q^{21} - q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + q^{35} - 2 q^{37} - 3 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 - 3 * q^11 - 3 * q^13 + 2 * q^15 + 3 * q^17 - 3 * q^19 + q^21 - q^23 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 6 * q^31 - 3 * q^33 + q^35 - 2 * q^37 - 3 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 2 * q^47 + 3 * q^49 + 3 * q^51 + 5 * q^53 - 3 * q^55 - 3 * q^57 - 18 * q^59 + 6 * q^61 + q^63 - 3 * q^65 + 6 * q^67 - q^69 - 14 * q^71 + 9 * q^73 + 2 * q^75 + 15 * q^77 - 8 * q^79 + 2 * q^81 - 7 * q^83 + 3 * q^85 + 12 * q^87 + 21 * q^89 + 15 * q^91 + 6 * q^93 - 3 * q^95 + 4 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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