LMFDB - Embedded newform 9300.2.g.r.3349.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] 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("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.932935936.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 32x^{4} + 256x^{2} + 324 $ x^6 + 32x^4 + 256x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.6
Root			$4.47467i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.r.3349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.47467i q^{7} -1.00000 q^{9} +2.00000 q^{11} +4.47467i q^{13} -1.07331i q^{17} -4.00000 q^{19} -4.47467 q^{21} -1.07331i q^{23} -1.00000i q^{27} +3.54798 q^{29} -1.00000 q^{31} +2.00000i q^{33} -2.47467i q^{37} -4.47467 q^{39} +4.00000 q^{41} +2.00000i q^{43} +8.02265i q^{47} -13.0226 q^{49} +1.07331 q^{51} +10.0226i q^{53} -4.00000i q^{57} -9.54798 q^{59} -0.146623 q^{61} -4.47467i q^{63} +4.32804i q^{67} +1.07331 q^{69} +12.4973 q^{71} +15.4240i q^{73} +8.94933i q^{77} -9.87602 q^{79} +1.00000 q^{81} -2.92669i q^{83} +3.54798i q^{87} -15.4014 q^{89} -20.0226 q^{91} -1.00000i q^{93} -1.85338i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 12 q^{11} - 24 q^{19} - 8 q^{29} - 6 q^{31} + 24 q^{41} - 22 q^{49} + 4 q^{51} - 28 q^{59} + 4 q^{61} + 4 q^{69} - 8 q^{71} - 8 q^{79} + 6 q^{81} - 68 q^{89} - 64 q^{91} - 12 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 12 * q^11 - 24 * q^19 - 8 * q^29 - 6 * q^31 + 24 * q^41 - 22 * q^49 + 4 * q^51 - 28 * q^59 + 4 * q^61 + 4 * q^69 - 8 * q^71 - 8 * q^79 + 6 * q^81 - 68 * q^89 - 64 * q^91 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.47467i	1.69127i	0.533765	+	0.845633i	$0.320777\pi$
$7$	4.47467i	1.69127i	−0.533765	+	0.845633i	$0.679223\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.47467i	1.24105i	0.784187	+	0.620525i	$0.213080\pi$
$13$	4.47467i	1.24105i	−0.784187	+	0.620525i	$0.786920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.07331i	− 0.260316i	−0.991493	−	0.130158i	$-0.958451\pi$
$17$	− 1.07331i	− 0.260316i	0.991493	−	0.130158i	$-0.0415485\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−4.47467	−0.976452
$22$	0	0
$23$	− 1.07331i	− 0.223801i	−0.993719	−	0.111900i	$-0.964306\pi$
$23$	− 1.07331i	− 0.223801i	0.993719	−	0.111900i	$-0.0356938\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	3.54798	0.658843	0.329422	−	0.944183i	$-0.393146\pi$
$29$	3.54798	0.658843	0.329422	+	0.944183i	$0.393146\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.47467i	− 0.406833i	−0.979092	−	0.203416i	$-0.934795\pi$
$37$	− 2.47467i	− 0.406833i	0.979092	−	0.203416i	$-0.0652045\pi$
$38$	0	0
$39$	−4.47467	−0.716520
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.02265i	1.17022i	0.810953	+	0.585112i	$0.198949\pi$
$47$	8.02265i	1.17022i	−0.810953	+	0.585112i	$0.801051\pi$
$48$	0	0
$49$	−13.0226	−1.86038
$50$	0	0
$51$	1.07331	0.150294
$52$	0	0
$53$	10.0226i	1.37672i	0.725371	+	0.688358i	$0.241668\pi$
$53$	10.0226i	1.37672i	−0.725371	+	0.688358i	$0.758332\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	−9.54798	−1.24304	−0.621520	−	0.783398i	$-0.713485\pi$
$59$	−9.54798	−1.24304	−0.621520	+	0.783398i	$0.713485\pi$
$60$	0	0
$61$	−0.146623	−0.0187731	−0.00938657	−	0.999956i	$-0.502988\pi$
$61$	−0.146623	−0.0187731	−0.00938657	+	0.999956i	$0.502988\pi$
$62$	0	0
$63$	− 4.47467i	− 0.563755i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.32804i	0.528755i	0.964419	+	0.264377i	$0.0851664\pi$
$67$	4.32804i	0.528755i	−0.964419	+	0.264377i	$0.914834\pi$
$68$	0	0
$69$	1.07331	0.129212
$70$	0	0
$71$	12.4973	1.48316	0.741579	−	0.670865i	$-0.234077\pi$
$71$	12.4973	1.48316	0.741579	+	0.670865i	$0.234077\pi$
$72$	0	0
$73$	15.4240i	1.80524i	0.430435	+	0.902621i	$0.358360\pi$
$73$	15.4240i	1.80524i	−0.430435	+	0.902621i	$0.641640\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.94933i	1.01987i
$78$	0	0
$79$	−9.87602	−1.11114	−0.555570	−	0.831470i	$-0.687500\pi$
$79$	−9.87602	−1.11114	−0.555570	+	0.831470i	$0.687500\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.92669i	− 0.321246i	−0.987016	−	0.160623i	$-0.948650\pi$
$83$	− 2.92669i	− 0.321246i	0.987016	−	0.160623i	$-0.0513503\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.54798i	0.380383i
$88$	0	0
$89$	−15.4014	−1.63254	−0.816270	−	0.577670i	$-0.803962\pi$
$89$	−15.4014	−1.63254	−0.816270	+	0.577670i	$0.803962\pi$
$90$	0	0
$91$	−20.0226	−2.09894
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.85338i	− 0.188182i	−0.995564	−	0.0940910i	$-0.970006\pi$
$97$	− 1.85338i	− 0.188182i	0.995564	−	0.0940910i	$-0.0299945\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−1.85338	−0.184418	−0.0922090	−	0.995740i	$-0.529393\pi$
$101$	−1.85338	−0.184418	−0.0922090	+	0.995740i	$0.529393\pi$
$102$	0	0
$103$	− 15.4240i	− 1.51977i	−0.650056	−	0.759886i	$-0.725255\pi$
$103$	− 15.4240i	− 1.51977i	0.650056	−	0.759886i	$-0.274745\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.21993i	0.117935i	0.998260	+	0.0589677i	$0.0187809\pi$
$107$	1.21993i	0.117935i	−0.998260	+	0.0589677i	$0.981219\pi$
$108$	0	0
$109$	−5.07331	−0.485935	−0.242968	−	0.970034i	$-0.578121\pi$
$109$	−5.07331	−0.485935	−0.242968	+	0.970034i	$0.578121\pi$
$110$	0	0
$111$	2.47467	0.234885
$112$	0	0
$113$	1.05067i	0.0988383i	0.998778	+	0.0494192i	$0.0157370\pi$
$113$	1.05067i	0.0988383i	−0.998778	+	0.0494192i	$0.984263\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.47467i	− 0.413683i
$118$	0	0
$119$	4.80271	0.440264
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.05067i	0.270703i	0.990798	+	0.135351i	$0.0432163\pi$
$127$	3.05067i	0.270703i	−0.990798	+	0.135351i	$0.956784\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	7.40136	0.646659	0.323330	−	0.946286i	$-0.395198\pi$
$131$	7.40136	0.646659	0.323330	+	0.946286i	$0.395198\pi$
$132$	0	0
$133$	− 17.8987i	− 1.55201i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.9720i	1.27914i	0.768732	+	0.639571i	$0.220888\pi$
$137$	14.9720i	1.27914i	−0.768732	+	0.639571i	$0.779112\pi$
$138$	0	0
$139$	14.8027	1.25555	0.627775	−	0.778395i	$-0.283966\pi$
$139$	14.8027	1.25555	0.627775	+	0.778395i	$0.283966\pi$
$140$	0	0
$141$	−8.02265	−0.675629
$142$	0	0
$143$	8.94933i	0.748381i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 13.0226i	− 1.07409i
$148$	0	0
$149$	9.89867	0.810931	0.405465	−	0.914110i	$-0.367110\pi$
$149$	9.89867	0.810931	0.405465	+	0.914110i	$0.367110\pi$
$150$	0	0
$151$	−4.92669	−0.400928	−0.200464	−	0.979701i	$-0.564245\pi$
$151$	−4.92669	−0.400928	−0.200464	+	0.979701i	$0.564245\pi$
$152$	0	0
$153$	1.07331i	0.0867721i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	−10.0226	−0.794848
$160$	0	0
$161$	4.80271	0.378507
$162$	0	0
$163$	− 14.4747i	− 1.13374i	−0.823806	−	0.566872i	$-0.808154\pi$
$163$	− 14.4747i	− 1.13374i	0.823806	−	0.566872i	$-0.191846\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 17.2426i	− 1.33427i	−0.744936	−	0.667135i	$-0.767520\pi$
$167$	− 17.2426i	− 1.33427i	0.744936	−	0.667135i	$-0.232480\pi$
$168$	0	0
$169$	−7.02265	−0.540204
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	− 6.94933i	− 0.528348i	−0.964475	−	0.264174i	$-0.914901\pi$
$173$	− 6.94933i	− 0.528348i	0.964475	−	0.264174i	$-0.0850993\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.54798i	− 0.717670i
$178$	0	0
$179$	−5.85338	−0.437502	−0.218751	−	0.975781i	$-0.570198\pi$
$179$	−5.85338	−0.437502	−0.218751	+	0.975781i	$0.570198\pi$
$180$	0	0
$181$	−17.7520	−1.31950	−0.659750	−	0.751486i	$-0.729338\pi$
$181$	−17.7520	−1.31950	−0.659750	+	0.751486i	$0.729338\pi$
$182$	0	0
$183$	− 0.146623i	− 0.0108387i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.14662i	− 0.156977i
$188$	0	0
$189$	4.47467	0.325484
$190$	0	0
$191$	8.35069	0.604235	0.302117	−	0.953271i	$-0.402307\pi$
$191$	8.35069	0.604235	0.302117	+	0.953271i	$0.402307\pi$
$192$	0	0
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.97198i	− 0.496733i	−0.968666	−	0.248366i	$-0.920106\pi$
$197$	− 6.97198i	− 0.496733i	0.968666	−	0.248366i	$-0.0798937\pi$
$198$	0	0
$199$	5.21993	0.370031	0.185016	−	0.982736i	$-0.440766\pi$
$199$	5.21993	0.370031	0.185016	+	0.982736i	$0.440766\pi$
$200$	0	0
$201$	−4.32804	−0.305277
$202$	0	0
$203$	15.8760i	1.11428i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.07331i	0.0746003i
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−15.0960	−1.03925	−0.519624	−	0.854395i	$-0.673928\pi$
$211$	−15.0960	−1.03925	−0.519624	+	0.854395i	$0.673928\pi$
$212$	0	0
$213$	12.4973i	0.856302i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4.47467i	− 0.303760i
$218$	0	0
$219$	−15.4240	−1.04226
$220$	0	0
$221$	4.80271	0.323065
$222$	0	0
$223$	− 18.9493i	− 1.26894i	−0.772947	−	0.634471i	$-0.781218\pi$
$223$	− 18.9493i	− 1.26894i	0.772947	−	0.634471i	$-0.218782\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.07331i	0.469472i	0.972059	+	0.234736i	$0.0754226\pi$
$227$	7.07331i	0.469472i	−0.972059	+	0.234736i	$0.924577\pi$
$228$	0	0
$229$	5.09596	0.336750	0.168375	−	0.985723i	$-0.446148\pi$
$229$	5.09596	0.336750	0.168375	+	0.985723i	$0.446148\pi$
$230$	0	0
$231$	−8.94933	−0.588823
$232$	0	0
$233$	13.8760i	0.909048i	0.890734	+	0.454524i	$0.150191\pi$
$233$	13.8760i	0.909048i	−0.890734	+	0.454524i	$0.849809\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.87602i	− 0.641517i
$238$	0	0
$239$	1.09596	0.0708916	0.0354458	−	0.999372i	$-0.488715\pi$
$239$	1.09596	0.0708916	0.0354458	+	0.999372i	$0.488715\pi$
$240$	0	0
$241$	11.2426	0.724198	0.362099	−	0.932140i	$-0.382060\pi$
$241$	11.2426	0.724198	0.362099	+	0.932140i	$0.382060\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 17.8987i	− 1.13886i
$248$	0	0
$249$	2.92669	0.185471
$250$	0	0
$251$	16.0453	1.01277	0.506385	−	0.862308i	$-0.330982\pi$
$251$	16.0453	1.01277	0.506385	+	0.862308i	$0.330982\pi$
$252$	0	0
$253$	− 2.14662i	− 0.134957i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.02265i	− 0.250926i	−0.992098	−	0.125463i	$-0.959958\pi$
$257$	− 4.02265i	− 0.250926i	0.992098	−	0.125463i	$-0.0400416\pi$
$258$	0	0
$259$	11.0733	0.688062
$260$	0	0
$261$	−3.54798	−0.219614
$262$	0	0
$263$	− 27.0960i	− 1.67081i	−0.549636	−	0.835404i	$-0.685234\pi$
$263$	− 27.0960i	− 1.67081i	0.549636	−	0.835404i	$-0.314766\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 15.4014i	− 0.942548i
$268$	0	0
$269$	−0.745267	−0.0454397	−0.0227199	−	0.999742i	$-0.507233\pi$
$269$	−0.745267	−0.0454397	−0.0227199	+	0.999742i	$0.507233\pi$
$270$	0	0
$271$	20.9946	1.27533	0.637666	−	0.770313i	$-0.279900\pi$
$271$	20.9946	1.27533	0.637666	+	0.770313i	$0.279900\pi$
$272$	0	0
$273$	− 20.0226i	− 1.21183i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.47467i	0.389025i	0.980900	+	0.194513i	$0.0623125\pi$
$277$	6.47467i	0.389025i	−0.980900	+	0.194513i	$0.937688\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	1.05067	0.0626775	0.0313387	−	0.999509i	$-0.490023\pi$
$281$	1.05067	0.0626775	0.0313387	+	0.999509i	$0.490023\pi$
$282$	0	0
$283$	7.57062i	0.450027i	0.974356	+	0.225013i	$0.0722426\pi$
$283$	7.57062i	0.450027i	−0.974356	+	0.225013i	$0.927757\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.8987i	1.05652i
$288$	0	0
$289$	15.8480	0.932235
$290$	0	0
$291$	1.85338	0.108647
$292$	0	0
$293$	− 5.87602i	− 0.343281i	−0.985160	−	0.171640i	$-0.945093\pi$
$293$	− 5.87602i	− 0.343281i	0.985160	−	0.171640i	$-0.0549067\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.00000i	− 0.116052i
$298$	0	0
$299$	4.80271	0.277748
$300$	0	0
$301$	−8.94933	−0.515831
$302$	0	0
$303$	− 1.85338i	− 0.106474i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.7172i	1.01118i	0.862775	+	0.505588i	$0.168725\pi$
$307$	17.7172i	1.01118i	−0.862775	+	0.505588i	$0.831275\pi$
$308$	0	0
$309$	15.4240	0.877441
$310$	0	0
$311$	−14.3507	−0.813753	−0.406876	−	0.913483i	$-0.633382\pi$
$311$	−14.3507	−0.813753	−0.406876	+	0.913483i	$0.633382\pi$
$312$	0	0
$313$	19.3227i	1.09218i	0.837726	+	0.546091i	$0.183885\pi$
$313$	19.3227i	1.09218i	−0.837726	+	0.546091i	$0.816115\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.0226i	1.79857i	0.437362	+	0.899285i	$0.355913\pi$
$317$	32.0226i	1.79857i	−0.437362	+	0.899285i	$0.644087\pi$
$318$	0	0
$319$	7.09596	0.397297
$320$	0	0
$321$	−1.21993	−0.0680901
$322$	0	0
$323$	4.29325i	0.238883i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.07331i	− 0.280555i
$328$	0	0
$329$	−35.8987	−1.97916
$330$	0	0
$331$	−29.8760	−1.64213	−0.821067	−	0.570831i	$-0.806621\pi$
$331$	−29.8760	−1.64213	−0.821067	+	0.570831i	$0.806621\pi$
$332$	0	0
$333$	2.47467i	0.135611i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6.47467i	− 0.352698i	−0.984328	−	0.176349i	$-0.943571\pi$
$337$	− 6.47467i	− 0.352698i	0.984328	−	0.176349i	$-0.0564287\pi$
$338$	0	0
$339$	−1.05067	−0.0570643
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	− 26.9493i	− 1.45513i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.24258i	− 0.496168i	−0.968739	−	0.248084i	$-0.920199\pi$
$347$	− 9.24258i	− 0.496168i	0.968739	−	0.248084i	$-0.0798008\pi$
$348$	0	0
$349$	−6.92669	−0.370777	−0.185389	−	0.982665i	$-0.559354\pi$
$349$	−6.92669	−0.370777	−0.185389	+	0.982665i	$0.559354\pi$
$350$	0	0
$351$	4.47467	0.238840
$352$	0	0
$353$	0.780066i	0.0415187i	0.999785	+	0.0207594i	$0.00660838\pi$
$353$	0.780066i	0.0415187i	−0.999785	+	0.0207594i	$0.993392\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.80271i	0.254186i
$358$	0	0
$359$	−32.3960	−1.70979	−0.854897	−	0.518797i	$-0.826380\pi$
$359$	−32.3960	−1.70979	−0.854897	+	0.518797i	$0.826380\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.89867i	− 0.0991097i	−0.998771	−	0.0495548i	$-0.984220\pi$
$367$	− 1.89867i	− 0.0991097i	0.998771	−	0.0495548i	$-0.0157803\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−44.8480	−2.32839
$372$	0	0
$373$	− 29.0960i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$373$	− 29.0960i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.8760i	0.817657i
$378$	0	0
$379$	29.8987	1.53579	0.767896	−	0.640575i	$-0.221304\pi$
$379$	29.8987	1.53579	0.767896	+	0.640575i	$0.221304\pi$
$380$	0	0
$381$	−3.05067	−0.156290
$382$	0	0
$383$	− 32.8254i	− 1.67730i	−0.544673	−	0.838649i	$-0.683346\pi$
$383$	− 32.8254i	− 1.67730i	0.544673	−	0.838649i	$-0.316654\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.00000i	− 0.101666i
$388$	0	0
$389$	−33.5480	−1.70095	−0.850475	−	0.526015i	$-0.823685\pi$
$389$	−33.5480	−1.70095	−0.850475	+	0.526015i	$0.823685\pi$
$390$	0	0
$391$	−1.15200	−0.0582590
$392$	0	0
$393$	7.40136i	0.373349i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.0453i	− 0.504159i	−0.967707	−	0.252079i	$-0.918886\pi$
$397$	− 10.0453i	− 0.504159i	0.967707	−	0.252079i	$-0.0811144\pi$
$398$	0	0
$399$	17.8987	0.896054
$400$	0	0
$401$	−29.5933	−1.47782	−0.738909	−	0.673806i	$-0.764659\pi$
$401$	−29.5933	−1.47782	−0.738909	+	0.673806i	$0.764659\pi$
$402$	0	0
$403$	− 4.47467i	− 0.222899i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.94933i	− 0.245329i
$408$	0	0
$409$	−5.75205	−0.284420	−0.142210	−	0.989836i	$-0.545421\pi$
$409$	−5.75205	−0.284420	−0.142210	+	0.989836i	$0.545421\pi$
$410$	0	0
$411$	−14.9720	−0.738513
$412$	0	0
$413$	− 42.7240i	− 2.10231i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.8027i	0.724892i
$418$	0	0
$419$	24.5426	1.19898	0.599492	−	0.800380i	$-0.295369\pi$
$419$	24.5426	1.19898	0.599492	+	0.800380i	$0.295369\pi$
$420$	0	0
$421$	−22.0679	−1.07553	−0.537763	−	0.843096i	$-0.680730\pi$
$421$	−22.0679	−1.07553	−0.537763	+	0.843096i	$0.680730\pi$
$422$	0	0
$423$	− 8.02265i	− 0.390074i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.656088i	− 0.0317503i
$428$	0	0
$429$	−8.94933	−0.432078
$430$	0	0
$431$	−26.3507	−1.26927	−0.634634	−	0.772813i	$-0.718849\pi$
$431$	−26.3507	−1.26927	−0.634634	+	0.772813i	$0.718849\pi$
$432$	0	0
$433$	29.6159i	1.42325i	0.702559	+	0.711625i	$0.252040\pi$
$433$	29.6159i	1.42325i	−0.702559	+	0.711625i	$0.747960\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.29325i	0.205374i
$438$	0	0
$439$	−27.7973	−1.32669	−0.663347	−	0.748312i	$-0.730865\pi$
$439$	−27.7973	−1.32669	−0.663347	+	0.748312i	$0.730865\pi$
$440$	0	0
$441$	13.0226	0.620126
$442$	0	0
$443$	− 17.2199i	− 0.818144i	−0.912502	−	0.409072i	$-0.865853\pi$
$443$	− 17.2199i	− 0.818144i	0.912502	−	0.409072i	$-0.134147\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.89867i	0.468191i
$448$	0	0
$449$	−4.30540	−0.203184	−0.101592	−	0.994826i	$-0.532394\pi$
$449$	−4.30540	−0.203184	−0.101592	+	0.994826i	$0.532394\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 4.92669i	− 0.231476i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11.3787i	− 0.532274i	−0.963935	−	0.266137i	$-0.914253\pi$
$457$	− 11.3787i	− 0.532274i	0.963935	−	0.266137i	$-0.0857473\pi$
$458$	0	0
$459$	−1.07331	−0.0500979
$460$	0	0
$461$	3.54798	0.165246	0.0826229	−	0.996581i	$-0.473670\pi$
$461$	3.54798	0.165246	0.0826229	+	0.996581i	$0.473670\pi$
$462$	0	0
$463$	− 41.7520i	− 1.94038i	−0.242341	−	0.970191i	$-0.577915\pi$
$463$	− 41.7520i	− 1.94038i	0.242341	−	0.970191i	$-0.422085\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.6281i	− 1.37102i	−0.728062	−	0.685512i	$-0.759579\pi$
$467$	− 29.6281i	− 1.37102i	0.728062	−	0.685512i	$-0.240421\pi$
$468$	0	0
$469$	−19.3666	−0.894265
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	4.00000i	0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.0226i	− 0.458905i
$478$	0	0
$479$	33.4466	1.52822	0.764108	−	0.645088i	$-0.223179\pi$
$479$	33.4466	1.52822	0.764108	+	0.645088i	$0.223179\pi$
$480$	0	0
$481$	11.0733	0.504900
$482$	0	0
$483$	4.80271i	0.218531i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 23.3892i	− 1.05987i	−0.848040	−	0.529933i	$-0.822217\pi$
$487$	− 23.3892i	− 1.05987i	0.848040	−	0.529933i	$-0.177783\pi$
$488$	0	0
$489$	14.4747	0.654567
$490$	0	0
$491$	32.7014	1.47579	0.737896	−	0.674914i	$-0.235819\pi$
$491$	32.7014	1.47579	0.737896	+	0.674914i	$0.235819\pi$
$492$	0	0
$493$	− 3.80809i	− 0.171508i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	55.9213i	2.50841i
$498$	0	0
$499$	−18.1919	−0.814382	−0.407191	−	0.913343i	$-0.633492\pi$
$499$	−18.1919	−0.814382	−0.407191	+	0.913343i	$0.633492\pi$
$500$	0	0
$501$	17.2426	0.770342
$502$	0	0
$503$	36.0679i	1.60819i	0.594501	+	0.804095i	$0.297350\pi$
$503$	36.0679i	1.60819i	−0.594501	+	0.804095i	$0.702650\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.02265i	− 0.311887i
$508$	0	0
$509$	−42.5426	−1.88567	−0.942834	−	0.333263i	$-0.891850\pi$
$509$	−42.5426	−1.88567	−0.942834	+	0.333263i	$0.891850\pi$
$510$	0	0
$511$	−69.0173	−3.05314
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0453i	0.705671i
$518$	0	0
$519$	6.94933	0.305042
$520$	0	0
$521$	8.29325	0.363334	0.181667	−	0.983360i	$-0.441851\pi$
$521$	8.29325	0.363334	0.181667	+	0.983360i	$0.441851\pi$
$522$	0	0
$523$	− 5.19729i	− 0.227262i	−0.993523	−	0.113631i	$-0.963752\pi$
$523$	− 5.19729i	− 0.227262i	0.993523	−	0.113631i	$-0.0362481\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.07331i	0.0467542i
$528$	0	0
$529$	21.8480	0.949913
$530$	0	0
$531$	9.54798	0.414347
$532$	0	0
$533$	17.8987i	0.775277i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.85338i	− 0.252592i
$538$	0	0
$539$	−26.0453	−1.12185
$540$	0	0
$541$	43.6281	1.87572	0.937859	−	0.347018i	$-0.112806\pi$
$541$	43.6281	1.87572	0.937859	+	0.347018i	$0.112806\pi$
$542$	0	0
$543$	− 17.7520i	− 0.761813i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.52533i	− 0.236246i	−0.992999	−	0.118123i	$-0.962312\pi$
$547$	− 5.52533i	− 0.236246i	0.992999	−	0.118123i	$-0.0376877\pi$
$548$	0	0
$549$	0.146623	0.00625771
$550$	0	0
$551$	−14.1919	−0.604596
$552$	0	0
$553$	− 44.1919i	− 1.87923i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.0733i	0.723419i	0.932291	+	0.361710i	$0.117807\pi$
$557$	17.0733i	0.723419i	−0.932291	+	0.361710i	$0.882193\pi$
$558$	0	0
$559$	−8.94933	−0.378517
$560$	0	0
$561$	2.14662	0.0906305
$562$	0	0
$563$	− 11.0733i	− 0.466684i	−0.972395	−	0.233342i	$-0.925034\pi$
$563$	− 11.0733i	− 0.466684i	0.972395	−	0.233342i	$-0.0749662\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.47467i	0.187918i
$568$	0	0
$569$	−35.5933	−1.49215	−0.746074	−	0.665863i	$-0.768063\pi$
$569$	−35.5933	−1.49215	−0.746074	+	0.665863i	$0.768063\pi$
$570$	0	0
$571$	34.8933	1.46024	0.730119	−	0.683320i	$-0.239464\pi$
$571$	34.8933	1.46024	0.730119	+	0.683320i	$0.239464\pi$
$572$	0	0
$573$	8.35069i	0.348855i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.8933i	1.95219i	0.217337	+	0.976097i	$0.430263\pi$
$577$	46.8933i	1.95219i	−0.217337	+	0.976097i	$0.569737\pi$
$578$	0	0
$579$	8.00000	0.332469
$580$	0	0
$581$	13.0960	0.543312
$582$	0	0
$583$	20.0453i	0.830191i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.09596i	− 0.292881i	−0.989219	−	0.146441i	$-0.953218\pi$
$587$	− 7.09596i	− 0.292881i	0.989219	−	0.146441i	$-0.0467818\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	6.97198	0.286789
$592$	0	0
$593$	18.9946i	0.780016i	0.920812	+	0.390008i	$0.127528\pi$
$593$	18.9946i	0.780016i	−0.920812	+	0.390008i	$0.872472\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.21993i	0.213638i
$598$	0	0
$599$	−21.4014	−0.874436	−0.437218	−	0.899356i	$-0.644036\pi$
$599$	−21.4014	−0.874436	−0.437218	+	0.899356i	$0.644036\pi$
$600$	0	0
$601$	17.0507	0.695511	0.347756	−	0.937585i	$-0.386944\pi$
$601$	17.0507	0.695511	0.347756	+	0.937585i	$0.386944\pi$
$602$	0	0
$603$	− 4.32804i	− 0.176252i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	47.2213i	1.91665i	0.285672	+	0.958327i	$0.407783\pi$
$607$	47.2213i	1.91665i	−0.285672	+	0.958327i	$0.592217\pi$
$608$	0	0
$609$	−15.8760	−0.643329
$610$	0	0
$611$	−35.8987	−1.45230
$612$	0	0
$613$	13.1308i	0.530346i	0.964201	+	0.265173i	$0.0854291\pi$
$613$	13.1308i	0.530346i	−0.964201	+	0.265173i	$0.914571\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.6054i	1.27239i	0.771530	+	0.636193i	$0.219492\pi$
$617$	31.6054i	1.27239i	−0.771530	+	0.636193i	$0.780508\pi$
$618$	0	0
$619$	−6.16927	−0.247964	−0.123982	−	0.992284i	$-0.539566\pi$
$619$	−6.16927	−0.247964	−0.123982	+	0.992284i	$0.539566\pi$
$620$	0	0
$621$	−1.07331	−0.0430705
$622$	0	0
$623$	− 68.9159i	− 2.76106i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8.00000i	− 0.319489i
$628$	0	0
$629$	−2.65609	−0.105905
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	− 15.0960i	− 0.600010i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 58.2720i	− 2.30882i
$638$	0	0
$639$	−12.4973	−0.494386
$640$	0	0
$641$	35.5933	1.40585	0.702925	−	0.711264i	$-0.251877\pi$
$641$	35.5933	1.40585	0.702925	+	0.711264i	$0.251877\pi$
$642$	0	0
$643$	0.904043i	0.0356520i	0.999841	+	0.0178260i	$0.00567449\pi$
$643$	0.904043i	0.0356520i	−0.999841	+	0.0178260i	$0.994326\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.24795i	0.167004i	0.996508	+	0.0835022i	$0.0266105\pi$
$647$	4.24795i	0.167004i	−0.996508	+	0.0835022i	$0.973389\pi$
$648$	0	0
$649$	−19.0960	−0.749582
$650$	0	0
$651$	4.47467	0.175376
$652$	0	0
$653$	12.0679i	0.472255i	0.971722	+	0.236127i	$0.0758783\pi$
$653$	12.0679i	0.472255i	−0.971722	+	0.236127i	$0.924122\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 15.4240i	− 0.601748i
$658$	0	0
$659$	−21.6946	−0.845102	−0.422551	−	0.906339i	$-0.638865\pi$
$659$	−21.6946	−0.845102	−0.422551	+	0.906339i	$0.638865\pi$
$660$	0	0
$661$	5.05067	0.196448	0.0982241	−	0.995164i	$-0.468684\pi$
$661$	5.05067	0.196448	0.0982241	+	0.995164i	$0.468684\pi$
$662$	0	0
$663$	4.80271i	0.186522i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.80809i	− 0.147450i
$668$	0	0
$669$	18.9493	0.732624
$670$	0	0
$671$	−0.293246	−0.0113206
$672$	0	0
$673$	− 33.7625i	− 1.30145i	−0.759313	−	0.650725i	$-0.774465\pi$
$673$	− 33.7625i	− 1.30145i	0.759313	−	0.650725i	$-0.225535\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.8760i	1.37883i	0.724368	+	0.689414i	$0.242132\pi$
$677$	35.8760i	1.37883i	−0.724368	+	0.689414i	$0.757868\pi$
$678$	0	0
$679$	8.29325	0.318266
$680$	0	0
$681$	−7.07331	−0.271050
$682$	0	0
$683$	16.6787i	0.638194i	0.947722	+	0.319097i	$0.103380\pi$
$683$	16.6787i	0.638194i	−0.947722	+	0.319097i	$0.896620\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.09596i	0.194423i
$688$	0	0
$689$	−44.8480	−1.70857
$690$	0	0
$691$	−26.8027	−1.01962	−0.509812	−	0.860286i	$-0.670285\pi$
$691$	−26.8027	−1.01962	−0.509812	+	0.860286i	$0.670285\pi$
$692$	0	0
$693$	− 8.94933i	− 0.339957i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.29325i	− 0.162618i
$698$	0	0
$699$	−13.8760	−0.524839
$700$	0	0
$701$	12.9493	0.489090	0.244545	−	0.969638i	$-0.421361\pi$
$701$	12.9493	0.489090	0.244545	+	0.969638i	$0.421361\pi$
$702$	0	0
$703$	9.89867i	0.373335i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.29325i	− 0.311900i
$708$	0	0
$709$	19.6054	0.736297	0.368149	−	0.929767i	$-0.379992\pi$
$709$	19.6054	0.736297	0.368149	+	0.929767i	$0.379992\pi$
$710$	0	0
$711$	9.87602	0.370380
$712$	0	0
$713$	1.07331i	0.0401958i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.09596i	0.0409293i
$718$	0	0
$719$	22.0906	0.823840	0.411920	−	0.911220i	$-0.364858\pi$
$719$	22.0906	0.823840	0.411920	+	0.911220i	$0.364858\pi$
$720$	0	0
$721$	69.0173	2.57034
$722$	0	0
$723$	11.2426i	0.418116i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12.3280i	0.457222i	0.973518	+	0.228611i	$0.0734184\pi$
$727$	12.3280i	0.457222i	−0.973518	+	0.228611i	$0.926582\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.14662	0.0793957
$732$	0	0
$733$	− 25.7973i	− 0.952846i	−0.879216	−	0.476423i	$-0.841933\pi$
$733$	− 25.7973i	− 0.952846i	0.879216	−	0.476423i	$-0.158067\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.65609i	0.318851i
$738$	0	0
$739$	11.1186	0.409004	0.204502	−	0.978866i	$-0.434442\pi$
$739$	11.1186	0.409004	0.204502	+	0.978866i	$0.434442\pi$
$740$	0	0
$741$	17.8987	0.657524
$742$	0	0
$743$	− 22.8027i	− 0.836550i	−0.908320	−	0.418275i	$-0.862635\pi$
$743$	− 22.8027i	− 0.836550i	0.908320	−	0.418275i	$-0.137365\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.92669i	0.107082i
$748$	0	0
$749$	−5.45880	−0.199460
$750$	0	0
$751$	−0.949334	−0.0346417	−0.0173208	−	0.999850i	$-0.505514\pi$
$751$	−0.949334	−0.0346417	−0.0173208	+	0.999850i	$0.505514\pi$
$752$	0	0
$753$	16.0453i	0.584723i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 36.8132i	− 1.33800i	−0.743263	−	0.668999i	$-0.766723\pi$
$757$	− 36.8132i	− 1.33800i	0.743263	−	0.668999i	$-0.233277\pi$
$758$	0	0
$759$	2.14662	0.0779175
$760$	0	0
$761$	35.5480	1.28861	0.644307	−	0.764767i	$-0.277146\pi$
$761$	35.5480	1.28861	0.644307	+	0.764767i	$0.277146\pi$
$762$	0	0
$763$	− 22.7014i	− 0.821845i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 42.7240i	− 1.54268i
$768$	0	0
$769$	25.4118	0.916375	0.458187	−	0.888856i	$-0.348499\pi$
$769$	25.4118	0.916375	0.458187	+	0.888856i	$0.348499\pi$
$770$	0	0
$771$	4.02265	0.144872
$772$	0	0
$773$	− 30.7240i	− 1.10507i	−0.833491	−	0.552533i	$-0.813661\pi$
$773$	− 30.7240i	− 1.10507i	0.833491	−	0.552533i	$-0.186339\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.0733i	0.397253i
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	24.9946	0.894378
$782$	0	0
$783$	− 3.54798i	− 0.126794i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.8933i	0.887350i	0.896188	+	0.443675i	$0.146326\pi$
$787$	24.8933i	0.887350i	−0.896188	+	0.443675i	$0.853674\pi$
$788$	0	0
$789$	27.0960	0.964642
$790$	0	0
$791$	−4.70138	−0.167162
$792$	0	0
$793$	− 0.656088i	− 0.0232984i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.0733i	0.604768i	0.953186	+	0.302384i	$0.0977824\pi$
$797$	17.0733i	0.604768i	−0.953186	+	0.302384i	$0.902218\pi$
$798$	0	0
$799$	8.61080	0.304628
$800$	0	0
$801$	15.4014	0.544180
$802$	0	0
$803$	30.8480i	1.08860i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 0.745267i	− 0.0262346i
$808$	0	0
$809$	−2.45202	−0.0862085	−0.0431042	−	0.999071i	$-0.513725\pi$
$809$	−2.45202	−0.0862085	−0.0431042	+	0.999071i	$0.513725\pi$
$810$	0	0
$811$	−15.1412	−0.531681	−0.265841	−	0.964017i	$-0.585649\pi$
$811$	−15.1412	−0.531681	−0.265841	+	0.964017i	$0.585649\pi$
$812$	0	0
$813$	20.9946i	0.736314i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	0	0
$819$	20.0226	0.699648
$820$	0	0
$821$	−2.45202	−0.0855761	−0.0427881	−	0.999084i	$-0.513624\pi$
$821$	−2.45202	−0.0855761	−0.0427881	+	0.999084i	$0.513624\pi$
$822$	0	0
$823$	30.0453i	1.04731i	0.851929	+	0.523657i	$0.175432\pi$
$823$	30.0453i	1.04731i	−0.851929	+	0.523657i	$0.824568\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	56.9159i	1.97916i	0.143980	+	0.989581i	$0.454010\pi$
$827$	56.9159i	1.97916i	−0.143980	+	0.989581i	$0.545990\pi$
$828$	0	0
$829$	32.8933	1.14243	0.571216	−	0.820800i	$-0.306472\pi$
$829$	32.8933	1.14243	0.571216	+	0.820800i	$0.306472\pi$
$830$	0	0
$831$	−6.47467	−0.224604
$832$	0	0
$833$	13.9774i	0.484287i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	−38.7906	−1.33920	−0.669599	−	0.742722i	$-0.733534\pi$
$839$	−38.7906	−1.33920	−0.669599	+	0.742722i	$0.733534\pi$
$840$	0	0
$841$	−16.4118	−0.565926
$842$	0	0
$843$	1.05067i	0.0361869i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 31.3227i	− 1.07626i
$848$	0	0
$849$	−7.57062	−0.259823
$850$	0	0
$851$	−2.65609	−0.0910495
$852$	0	0
$853$	48.9493i	1.67599i	0.545675	+	0.837997i	$0.316273\pi$
$853$	48.9493i	1.67599i	−0.545675	+	0.837997i	$0.683727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.6453i	− 1.93497i	−0.252932	−	0.967484i	$-0.581395\pi$
$857$	− 56.6453i	− 1.93497i	0.252932	−	0.967484i	$-0.418605\pi$
$858$	0	0
$859$	−37.8987	−1.29309	−0.646543	−	0.762878i	$-0.723786\pi$
$859$	−37.8987	−1.29309	−0.646543	+	0.762878i	$0.723786\pi$
$860$	0	0
$861$	−17.8987	−0.609985
$862$	0	0
$863$	− 10.1013i	− 0.343853i	−0.985110	−	0.171927i	$-0.945001\pi$
$863$	− 10.1013i	− 0.343853i	0.985110	−	0.171927i	$-0.0549992\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.8480i	0.538226i
$868$	0	0
$869$	−19.7520	−0.670042
$870$	0	0
$871$	−19.3666	−0.656211
$872$	0	0
$873$	1.85338i	0.0627273i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.7014i	0.428895i	0.976736	+	0.214448i	$0.0687951\pi$
$877$	12.7014i	0.428895i	−0.976736	+	0.214448i	$0.931205\pi$
$878$	0	0
$879$	5.87602	0.198193
$880$	0	0
$881$	−38.5426	−1.29853	−0.649267	−	0.760561i	$-0.724924\pi$
$881$	−38.5426	−1.29853	−0.649267	+	0.760561i	$0.724924\pi$
$882$	0	0
$883$	− 15.1973i	− 0.511429i	−0.966752	−	0.255715i	$-0.917689\pi$
$883$	− 15.1973i	− 0.511429i	0.966752	−	0.255715i	$-0.0823108\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 11.7294i	− 0.393835i	−0.980420	−	0.196917i	$-0.936907\pi$
$887$	− 11.7294i	− 0.393835i	0.980420	−	0.196917i	$-0.0630931\pi$
$888$	0	0
$889$	−13.6507	−0.457830
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	− 32.0906i	− 1.07387i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.80271i	0.160358i
$898$	0	0
$899$	−3.54798	−0.118332
$900$	0	0
$901$	10.7574	0.358382
$902$	0	0
$903$	− 8.94933i	− 0.297815i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 42.3280i	− 1.40548i	−0.711447	−	0.702740i	$-0.751960\pi$
$907$	− 42.3280i	− 1.40548i	0.711447	−	0.702740i	$-0.248040\pi$
$908$	0	0
$909$	1.85338	0.0614726
$910$	0	0
$911$	−15.8987	−0.526746	−0.263373	−	0.964694i	$-0.584835\pi$
$911$	−15.8987	−0.526746	−0.263373	+	0.964694i	$0.584835\pi$
$912$	0	0
$913$	− 5.85338i	− 0.193719i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.1186i	1.09367i
$918$	0	0
$919$	51.5494	1.70046	0.850229	−	0.526414i	$-0.176464\pi$
$919$	51.5494	1.70046	0.850229	+	0.526414i	$0.176464\pi$
$920$	0	0
$921$	−17.7172	−0.583803
$922$	0	0
$923$	55.9213i	1.84067i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.4240i	0.506591i
$928$	0	0
$929$	27.2547	0.894199	0.447099	−	0.894484i	$-0.352457\pi$
$929$	27.2547	0.894199	0.447099	+	0.894484i	$0.352457\pi$
$930$	0	0
$931$	52.0906	1.70720
$932$	0	0
$933$	− 14.3507i	− 0.469820i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.5494i	1.68404i	0.539442	+	0.842022i	$0.318635\pi$
$937$	51.5494i	1.68404i	−0.539442	+	0.842022i	$0.681365\pi$
$938$	0	0
$939$	−19.3227	−0.630571
$940$	0	0
$941$	−38.2946	−1.24837	−0.624185	−	0.781277i	$-0.714569\pi$
$941$	−38.2946	−1.24837	−0.624185	+	0.781277i	$0.714569\pi$
$942$	0	0
$943$	− 4.29325i	− 0.139807i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.0280i	− 0.813301i	−0.913584	−	0.406651i	$-0.866697\pi$
$947$	− 25.0280i	− 0.813301i	0.913584	−	0.406651i	$-0.133303\pi$
$948$	0	0
$949$	−69.0173	−2.24040
$950$	0	0
$951$	−32.0226	−1.03841
$952$	0	0
$953$	− 0.825357i	− 0.0267359i	−0.999911	−	0.0133680i	$-0.995745\pi$
$953$	− 0.825357i	− 0.0267359i	0.999911	−	0.0133680i	$-0.00425528\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.09596i	0.229380i
$958$	0	0
$959$	−66.9946	−2.16337
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 1.21993i	− 0.0393118i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10.9493i	0.352107i	0.984381	+	0.176053i	$0.0563331\pi$
$967$	10.9493i	0.352107i	−0.984381	+	0.176053i	$0.943667\pi$
$968$	0	0
$969$	−4.29325	−0.137919
$970$	0	0
$971$	35.3000	1.13283	0.566416	−	0.824120i	$-0.308330\pi$
$971$	35.3000	1.13283	0.566416	+	0.824120i	$0.308330\pi$
$972$	0	0
$973$	66.2372i	2.12347i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.6054i	− 1.26709i	−0.773706	−	0.633545i	$-0.781599\pi$
$977$	− 39.6054i	− 1.26709i	0.773706	−	0.633545i	$-0.218401\pi$
$978$	0	0
$979$	−30.8027	−0.984459
$980$	0	0
$981$	5.07331	0.161978
$982$	0	0
$983$	43.5494i	1.38901i	0.719488	+	0.694505i	$0.244376\pi$
$983$	43.5494i	1.38901i	−0.719488	+	0.694505i	$0.755624\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 35.8987i	− 1.14267i
$988$	0	0
$989$	2.14662	0.0682586
$990$	0	0
$991$	38.8933	1.23549	0.617743	−	0.786380i	$-0.288047\pi$
$991$	38.8933	1.23549	0.617743	+	0.786380i	$0.288047\pi$
$992$	0	0
$993$	− 29.8760i	− 0.948087i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 39.2879i	− 1.24426i	−0.782914	−	0.622130i	$-0.786268\pi$
$997$	− 39.2879i	− 1.24426i	0.782914	−	0.622130i	$-0.213732\pi$
$998$	0	0
$999$	−2.47467	−0.0782950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.r.3349.6		6
5.2	odd	4		1860.2.a.h.1.1	✓	3
5.3	odd	4		9300.2.a.t.1.3		3
5.4	even	2	inner	9300.2.g.r.3349.1		6
15.2	even	4		5580.2.a.i.1.1		3
20.7	even	4		7440.2.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.h.1.1	✓	3	5.2	odd	4
5580.2.a.i.1.1		3	15.2	even	4
7440.2.a.bq.1.3		3	20.7	even	4
9300.2.a.t.1.3		3	5.3	odd	4
9300.2.g.r.3349.1		6	5.4	even	2	inner
9300.2.g.r.3349.6		6	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:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.47467i q^{7} -1.00000 q^{9} +2.00000 q^{11} +4.47467i q^{13} -1.07331i q^{17} -4.00000 q^{19} -4.47467 q^{21} -1.07331i q^{23} -1.00000i q^{27} +3.54798 q^{29} -1.00000 q^{31} +2.00000i q^{33} -2.47467i q^{37} -4.47467 q^{39} +4.00000 q^{41} +2.00000i q^{43} +8.02265i q^{47} -13.0226 q^{49} +1.07331 q^{51} +10.0226i q^{53} -4.00000i q^{57} -9.54798 q^{59} -0.146623 q^{61} -4.47467i q^{63} +4.32804i q^{67} +1.07331 q^{69} +12.4973 q^{71} +15.4240i q^{73} +8.94933i q^{77} -9.87602 q^{79} +1.00000 q^{81} -2.92669i q^{83} +3.54798i q^{87} -15.4014 q^{89} -20.0226 q^{91} -1.00000i q^{93} -1.85338i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 12 q^{11} - 24 q^{19} - 8 q^{29} - 6 q^{31} + 24 q^{41} - 22 q^{49} + 4 q^{51} - 28 q^{59} + 4 q^{61} + 4 q^{69} - 8 q^{71} - 8 q^{79} + 6 q^{81} - 68 q^{89} - 64 q^{91} - 12 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 12 * q^11 - 24 * q^19 - 8 * q^29 - 6 * q^31 + 24 * q^41 - 22 * q^49 + 4 * q^51 - 28 * q^59 + 4 * q^61 + 4 * q^69 - 8 * q^71 - 8 * q^79 + 6 * q^81 - 68 * q^89 - 64 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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