LMFDB - Embedded newform 7800.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.76156 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.76156 q^{7} +1.00000 q^{9} +4.49084 q^{11} -1.00000 q^{13} -3.62620 q^{17} -6.14931 q^{19} -2.76156 q^{21} -5.52311 q^{23} +1.00000 q^{27} +6.25240 q^{29} +8.49084 q^{31} +4.49084 q^{33} +0.896916 q^{37} -1.00000 q^{39} +2.89692 q^{41} -0.270718 q^{43} -3.38776 q^{47} +0.626198 q^{49} -3.62620 q^{51} +3.27072 q^{53} -6.14931 q^{57} -9.53707 q^{59} +11.4200 q^{61} -2.76156 q^{63} +12.4340 q^{67} -5.52311 q^{69} +8.89692 q^{71} -3.25240 q^{73} -12.4017 q^{77} +13.6724 q^{79} +1.00000 q^{81} +3.03228 q^{83} +6.25240 q^{87} -2.20617 q^{89} +2.76156 q^{91} +8.49084 q^{93} +17.7572 q^{97} +4.49084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.76156	−1.04377	−0.521885	−	0.853016i	$-0.674771\pi$
$7$	−2.76156	−1.04377	−0.521885	+	0.853016i	$0.674771\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.49084	1.35404	0.677019	−	0.735965i	$-0.263271\pi$
$11$	4.49084	1.35404	0.677019	+	0.735965i	$0.263271\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.62620	−0.879482	−0.439741	−	0.898125i	$-0.644930\pi$
$17$	−3.62620	−0.879482	−0.439741	+	0.898125i	$0.644930\pi$
$18$	0	0
$19$	−6.14931	−1.41075	−0.705375	−	0.708835i	$-0.749221\pi$
$19$	−6.14931	−1.41075	−0.705375	+	0.708835i	$0.749221\pi$
$20$	0	0
$21$	−2.76156	−0.602621
$22$	0	0
$23$	−5.52311	−1.15165	−0.575824	−	0.817573i	$-0.695319\pi$
$23$	−5.52311	−1.15165	−0.575824	+	0.817573i	$0.695319\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.25240	1.16104	0.580520	−	0.814246i	$-0.302849\pi$
$29$	6.25240	1.16104	0.580520	+	0.814246i	$0.302849\pi$
$30$	0	0
$31$	8.49084	1.52500	0.762500	−	0.646988i	$-0.223972\pi$
$31$	8.49084	1.52500	0.762500	+	0.646988i	$0.223972\pi$
$32$	0	0
$33$	4.49084	0.781755
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.896916	0.147452	0.0737261	−	0.997279i	$-0.476511\pi$
$37$	0.896916	0.147452	0.0737261	+	0.997279i	$0.476511\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.89692	0.452422	0.226211	−	0.974078i	$-0.427366\pi$
$41$	2.89692	0.452422	0.226211	+	0.974078i	$0.427366\pi$
$42$	0	0
$43$	−0.270718	−0.0412841	−0.0206421	−	0.999787i	$-0.506571\pi$
$43$	−0.270718	−0.0412841	−0.0206421	+	0.999787i	$0.506571\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.38776	−0.494155	−0.247077	−	0.968996i	$-0.579470\pi$
$47$	−3.38776	−0.494155	−0.247077	+	0.968996i	$0.579470\pi$
$48$	0	0
$49$	0.626198	0.0894569
$50$	0	0
$51$	−3.62620	−0.507769
$52$	0	0
$53$	3.27072	0.449268	0.224634	−	0.974443i	$-0.427881\pi$
$53$	3.27072	0.449268	0.224634	+	0.974443i	$0.427881\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.14931	−0.814496
$58$	0	0
$59$	−9.53707	−1.24162	−0.620810	−	0.783961i	$-0.713196\pi$
$59$	−9.53707	−1.24162	−0.620810	+	0.783961i	$0.713196\pi$
$60$	0	0
$61$	11.4200	1.46219	0.731093	−	0.682278i	$-0.239011\pi$
$61$	11.4200	1.46219	0.731093	+	0.682278i	$0.239011\pi$
$62$	0	0
$63$	−2.76156	−0.347924
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.4340	1.51905	0.759526	−	0.650476i	$-0.225431\pi$
$67$	12.4340	1.51905	0.759526	+	0.650476i	$0.225431\pi$
$68$	0	0
$69$	−5.52311	−0.664905
$70$	0	0
$71$	8.89692	1.05587	0.527935	−	0.849285i	$-0.322967\pi$
$71$	8.89692	1.05587	0.527935	+	0.849285i	$0.322967\pi$
$72$	0	0
$73$	−3.25240	−0.380664	−0.190332	−	0.981720i	$-0.560956\pi$
$73$	−3.25240	−0.380664	−0.190332	+	0.981720i	$0.560956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.4017	−1.41331
$78$	0	0
$79$	13.6724	1.53827	0.769134	−	0.639087i	$-0.220688\pi$
$79$	13.6724	1.53827	0.769134	+	0.639087i	$0.220688\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.03228	0.332835	0.166418	−	0.986055i	$-0.446780\pi$
$83$	3.03228	0.332835	0.166418	+	0.986055i	$0.446780\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.25240	0.670327
$88$	0	0
$89$	−2.20617	−0.233853	−0.116927	−	0.993141i	$-0.537304\pi$
$89$	−2.20617	−0.233853	−0.116927	+	0.993141i	$0.537304\pi$
$90$	0	0
$91$	2.76156	0.289490
$92$	0	0
$93$	8.49084	0.880459
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.7572	1.80297	0.901485	−	0.432811i	$-0.142478\pi$
$97$	17.7572	1.80297	0.901485	+	0.432811i	$0.142478\pi$
$98$	0	0
$99$	4.49084	0.451346
$100$	0	0
$101$	5.83237	0.580342	0.290171	−	0.956975i	$-0.406288\pi$
$101$	5.83237	0.580342	0.290171	+	0.956975i	$0.406288\pi$
$102$	0	0
$103$	3.45856	0.340782	0.170391	−	0.985376i	$-0.445497\pi$
$103$	3.45856	0.340782	0.170391	+	0.985376i	$0.445497\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.832365	0.0804678	0.0402339	−	0.999190i	$-0.487190\pi$
$107$	0.832365	0.0804678	0.0402339	+	0.999190i	$0.487190\pi$
$108$	0	0
$109$	−12.1493	−1.16369	−0.581847	−	0.813299i	$-0.697670\pi$
$109$	−12.1493	−1.16369	−0.581847	+	0.813299i	$0.697670\pi$
$110$	0	0
$111$	0.896916	0.0851315
$112$	0	0
$113$	−4.50479	−0.423775	−0.211888	−	0.977294i	$-0.567961\pi$
$113$	−4.50479	−0.423775	−0.211888	+	0.977294i	$0.567961\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	10.0140	0.917978
$120$	0	0
$121$	9.16763	0.833421
$122$	0	0
$123$	2.89692	0.261206
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.62620	−0.587980	−0.293990	−	0.955808i	$-0.594983\pi$
$127$	−6.62620	−0.587980	−0.293990	+	0.955808i	$0.594983\pi$
$128$	0	0
$129$	−0.270718	−0.0238354
$130$	0	0
$131$	1.10308	0.0963769	0.0481884	−	0.998838i	$-0.484655\pi$
$131$	1.10308	0.0963769	0.0481884	+	0.998838i	$0.484655\pi$
$132$	0	0
$133$	16.9817	1.47250
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.4479	1.74699	0.873493	−	0.486837i	$-0.161850\pi$
$137$	20.4479	1.74699	0.873493	+	0.486837i	$0.161850\pi$
$138$	0	0
$139$	16.5693	1.40539	0.702697	−	0.711490i	$-0.251979\pi$
$139$	16.5693	1.40539	0.702697	+	0.711490i	$0.251979\pi$
$140$	0	0
$141$	−3.38776	−0.285300
$142$	0	0
$143$	−4.49084	−0.375543
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.626198	0.0516479
$148$	0	0
$149$	3.79383	0.310803	0.155401	−	0.987851i	$-0.450333\pi$
$149$	3.79383	0.310803	0.155401	+	0.987851i	$0.450333\pi$
$150$	0	0
$151$	−5.80779	−0.472631	−0.236315	−	0.971676i	$-0.575940\pi$
$151$	−5.80779	−0.472631	−0.236315	+	0.971676i	$0.575940\pi$
$152$	0	0
$153$	−3.62620	−0.293161
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.6541	−1.08972	−0.544858	−	0.838528i	$-0.683416\pi$
$157$	−13.6541	−1.08972	−0.544858	+	0.838528i	$0.683416\pi$
$158$	0	0
$159$	3.27072	0.259385
$160$	0	0
$161$	15.2524	1.20206
$162$	0	0
$163$	−7.25240	−0.568052	−0.284026	−	0.958817i	$-0.591670\pi$
$163$	−7.25240	−0.568052	−0.284026	+	0.958817i	$0.591670\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.14931	0.166319	0.0831594	−	0.996536i	$-0.473499\pi$
$167$	2.14931	0.166319	0.0831594	+	0.996536i	$0.473499\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.14931	−0.470250
$172$	0	0
$173$	25.5693	1.94400	0.972001	−	0.234978i	$-0.0755020\pi$
$173$	25.5693	1.94400	0.972001	+	0.234978i	$0.0755020\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.53707	−0.716850
$178$	0	0
$179$	−21.5510	−1.61080	−0.805399	−	0.592732i	$-0.798049\pi$
$179$	−21.5510	−1.61080	−0.805399	+	0.592732i	$0.798049\pi$
$180$	0	0
$181$	−0.607876	−0.0451831	−0.0225915	−	0.999745i	$-0.507192\pi$
$181$	−0.607876	−0.0451831	−0.0225915	+	0.999745i	$0.507192\pi$
$182$	0	0
$183$	11.4200	0.844193
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−16.2847	−1.19085
$188$	0	0
$189$	−2.76156	−0.200874
$190$	0	0
$191$	−21.2803	−1.53979	−0.769894	−	0.638171i	$-0.779691\pi$
$191$	−21.2803	−1.53979	−0.769894	+	0.638171i	$0.779691\pi$
$192$	0	0
$193$	−7.45856	−0.536879	−0.268440	−	0.963297i	$-0.586508\pi$
$193$	−7.45856	−0.536879	−0.268440	+	0.963297i	$0.586508\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.45856	0.531401	0.265700	−	0.964056i	$-0.414397\pi$
$197$	7.45856	0.531401	0.265700	+	0.964056i	$0.414397\pi$
$198$	0	0
$199$	19.3372	1.37077	0.685387	−	0.728179i	$-0.259633\pi$
$199$	19.3372	1.37077	0.685387	+	0.728179i	$0.259633\pi$
$200$	0	0
$201$	12.4340	0.877026
$202$	0	0
$203$	−17.2663	−1.21186
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.52311	−0.383883
$208$	0	0
$209$	−27.6156	−1.91021
$210$	0	0
$211$	−8.02791	−0.552664	−0.276332	−	0.961062i	$-0.589119\pi$
$211$	−8.02791	−0.552664	−0.276332	+	0.961062i	$0.589119\pi$
$212$	0	0
$213$	8.89692	0.609607
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−23.4479	−1.59175
$218$	0	0
$219$	−3.25240	−0.219777
$220$	0	0
$221$	3.62620	0.243924
$222$	0	0
$223$	16.5048	1.10524	0.552621	−	0.833432i	$-0.313628\pi$
$223$	16.5048	1.10524	0.552621	+	0.833432i	$0.313628\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.69701	−0.179007	−0.0895033	−	0.995987i	$-0.528528\pi$
$227$	−2.69701	−0.179007	−0.0895033	+	0.995987i	$0.528528\pi$
$228$	0	0
$229$	−10.5616	−0.697933	−0.348967	−	0.937135i	$-0.613467\pi$
$229$	−10.5616	−0.697933	−0.348967	+	0.937135i	$0.613467\pi$
$230$	0	0
$231$	−12.4017	−0.815973
$232$	0	0
$233$	25.2158	1.65194	0.825969	−	0.563715i	$-0.190628\pi$
$233$	25.2158	1.65194	0.825969	+	0.563715i	$0.190628\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.6724	0.888120
$238$	0	0
$239$	−8.51875	−0.551032	−0.275516	−	0.961297i	$-0.588849\pi$
$239$	−8.51875	−0.551032	−0.275516	+	0.961297i	$0.588849\pi$
$240$	0	0
$241$	7.45856	0.480448	0.240224	−	0.970717i	$-0.422779\pi$
$241$	7.45856	0.480448	0.240224	+	0.970717i	$0.422779\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.14931	0.391271
$248$	0	0
$249$	3.03228	0.192163
$250$	0	0
$251$	−9.64452	−0.608757	−0.304378	−	0.952551i	$-0.598449\pi$
$251$	−9.64452	−0.608757	−0.304378	+	0.952551i	$0.598449\pi$
$252$	0	0
$253$	−24.8034	−1.55938
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.3555	−0.833092	−0.416546	−	0.909115i	$-0.636760\pi$
$257$	−13.3555	−0.833092	−0.416546	+	0.909115i	$0.636760\pi$
$258$	0	0
$259$	−2.47689	−0.153906
$260$	0	0
$261$	6.25240	0.387014
$262$	0	0
$263$	9.25240	0.570527	0.285264	−	0.958449i	$-0.407919\pi$
$263$	9.25240	0.570527	0.285264	+	0.958449i	$0.407919\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.20617	−0.135015
$268$	0	0
$269$	25.2986	1.54248	0.771242	−	0.636542i	$-0.219636\pi$
$269$	25.2986	1.54248	0.771242	+	0.636542i	$0.219636\pi$
$270$	0	0
$271$	31.3309	1.90322	0.951608	−	0.307314i	$-0.0994300\pi$
$271$	31.3309	1.90322	0.951608	+	0.307314i	$0.0994300\pi$
$272$	0	0
$273$	2.76156	0.167137
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.0096	1.38251	0.691256	−	0.722610i	$-0.257058\pi$
$277$	23.0096	1.38251	0.691256	+	0.722610i	$0.257058\pi$
$278$	0	0
$279$	8.49084	0.508333
$280$	0	0
$281$	−9.19554	−0.548560	−0.274280	−	0.961650i	$-0.588440\pi$
$281$	−9.19554	−0.548560	−0.274280	+	0.961650i	$0.588440\pi$
$282$	0	0
$283$	−9.93545	−0.590601	−0.295301	−	0.955404i	$-0.595420\pi$
$283$	−9.93545	−0.590601	−0.295301	+	0.955404i	$0.595420\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−3.85069	−0.226511
$290$	0	0
$291$	17.7572	1.04094
$292$	0	0
$293$	23.7938	1.39005	0.695025	−	0.718985i	$-0.255393\pi$
$293$	23.7938	1.39005	0.695025	+	0.718985i	$0.255393\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.49084	0.260585
$298$	0	0
$299$	5.52311	0.319410
$300$	0	0
$301$	0.747604	0.0430912
$302$	0	0
$303$	5.83237	0.335061
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.0664	−1.43062	−0.715309	−	0.698809i	$-0.753714\pi$
$307$	−25.0664	−1.43062	−0.715309	+	0.698809i	$0.753714\pi$
$308$	0	0
$309$	3.45856	0.196751
$310$	0	0
$311$	18.3632	1.04128	0.520640	−	0.853776i	$-0.325693\pi$
$311$	18.3632	1.04128	0.520640	+	0.853776i	$0.325693\pi$
$312$	0	0
$313$	−10.7293	−0.606455	−0.303227	−	0.952918i	$-0.598064\pi$
$313$	−10.7293	−0.606455	−0.303227	+	0.952918i	$0.598064\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.20617	−0.460904	−0.230452	−	0.973084i	$-0.574021\pi$
$317$	−8.20617	−0.460904	−0.230452	+	0.973084i	$0.574021\pi$
$318$	0	0
$319$	28.0785	1.57209
$320$	0	0
$321$	0.832365	0.0464581
$322$	0	0
$323$	22.2986	1.24073
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.1493	−0.671859
$328$	0	0
$329$	9.35548	0.515784
$330$	0	0
$331$	30.7110	1.68803	0.844014	−	0.536322i	$-0.180187\pi$
$331$	30.7110	1.68803	0.844014	+	0.536322i	$0.180187\pi$
$332$	0	0
$333$	0.896916	0.0491507
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.13099	0.442923	0.221462	−	0.975169i	$-0.428917\pi$
$337$	8.13099	0.442923	0.221462	+	0.975169i	$0.428917\pi$
$338$	0	0
$339$	−4.50479	−0.244667
$340$	0	0
$341$	38.1310	2.06491
$342$	0	0
$343$	17.6016	0.950398
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.9248	−0.801206	−0.400603	−	0.916252i	$-0.631199\pi$
$347$	−14.9248	−0.801206	−0.400603	+	0.916252i	$0.631199\pi$
$348$	0	0
$349$	−16.5048	−0.883481	−0.441741	−	0.897143i	$-0.645639\pi$
$349$	−16.5048	−0.883481	−0.441741	+	0.897143i	$0.645639\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	23.0664	1.22770	0.613851	−	0.789422i	$-0.289619\pi$
$353$	23.0664	1.22770	0.613851	+	0.789422i	$0.289619\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.0140	0.529995
$358$	0	0
$359$	32.6681	1.72415	0.862077	−	0.506777i	$-0.169163\pi$
$359$	32.6681	1.72415	0.862077	+	0.506777i	$0.169163\pi$
$360$	0	0
$361$	18.8140	0.990213
$362$	0	0
$363$	9.16763	0.481176
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.1493	−1.67818	−0.839090	−	0.543992i	$-0.816912\pi$
$367$	−32.1493	−1.67818	−0.839090	+	0.543992i	$0.816912\pi$
$368$	0	0
$369$	2.89692	0.150807
$370$	0	0
$371$	−9.03228	−0.468932
$372$	0	0
$373$	6.81404	0.352818	0.176409	−	0.984317i	$-0.443552\pi$
$373$	6.81404	0.352818	0.176409	+	0.984317i	$0.443552\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.25240	−0.322015
$378$	0	0
$379$	1.65078	0.0847948	0.0423974	−	0.999101i	$-0.486500\pi$
$379$	1.65078	0.0847948	0.0423974	+	0.999101i	$0.486500\pi$
$380$	0	0
$381$	−6.62620	−0.339470
$382$	0	0
$383$	10.0202	0.512009	0.256004	−	0.966676i	$-0.417594\pi$
$383$	10.0202	0.512009	0.256004	+	0.966676i	$0.417594\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.270718	−0.0137614
$388$	0	0
$389$	19.4017	0.983706	0.491853	−	0.870678i	$-0.336320\pi$
$389$	19.4017	0.983706	0.491853	+	0.870678i	$0.336320\pi$
$390$	0	0
$391$	20.0279	1.01285
$392$	0	0
$393$	1.10308	0.0556432
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.56165	−0.128565	−0.0642827	−	0.997932i	$-0.520476\pi$
$397$	−2.56165	−0.128565	−0.0642827	+	0.997932i	$0.520476\pi$
$398$	0	0
$399$	16.9817	0.850147
$400$	0	0
$401$	−3.55102	−0.177330	−0.0886648	−	0.996062i	$-0.528260\pi$
$401$	−3.55102	−0.177330	−0.0886648	+	0.996062i	$0.528260\pi$
$402$	0	0
$403$	−8.49084	−0.422959
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.02791	0.199656
$408$	0	0
$409$	28.0558	1.38727	0.693635	−	0.720326i	$-0.256008\pi$
$409$	28.0558	1.38727	0.693635	+	0.720326i	$0.256008\pi$
$410$	0	0
$411$	20.4479	1.00862
$412$	0	0
$413$	26.3372	1.29597
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.5693	0.811404
$418$	0	0
$419$	36.3188	1.77429	0.887146	−	0.461490i	$-0.152685\pi$
$419$	36.3188	1.77429	0.887146	+	0.461490i	$0.152685\pi$
$420$	0	0
$421$	10.2062	0.497418	0.248709	−	0.968578i	$-0.419994\pi$
$421$	10.2062	0.497418	0.248709	+	0.968578i	$0.419994\pi$
$422$	0	0
$423$	−3.38776	−0.164718
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−31.5371	−1.52619
$428$	0	0
$429$	−4.49084	−0.216820
$430$	0	0
$431$	−11.8140	−0.569062	−0.284531	−	0.958667i	$-0.591838\pi$
$431$	−11.8140	−0.569062	−0.284531	+	0.958667i	$0.591838\pi$
$432$	0	0
$433$	−0.598291	−0.0287521	−0.0143760	−	0.999897i	$-0.504576\pi$
$433$	−0.598291	−0.0287521	−0.0143760	+	0.999897i	$0.504576\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	33.9634	1.62469
$438$	0	0
$439$	−13.6724	−0.652549	−0.326275	−	0.945275i	$-0.605793\pi$
$439$	−13.6724	−0.652549	−0.326275	+	0.945275i	$0.605793\pi$
$440$	0	0
$441$	0.626198	0.0298190
$442$	0	0
$443$	−32.1772	−1.52879	−0.764393	−	0.644751i	$-0.776961\pi$
$443$	−32.1772	−1.52879	−0.764393	+	0.644751i	$0.776961\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.79383	0.179442
$448$	0	0
$449$	−9.81404	−0.463153	−0.231577	−	0.972817i	$-0.574388\pi$
$449$	−9.81404	−0.463153	−0.231577	+	0.972817i	$0.574388\pi$
$450$	0	0
$451$	13.0096	0.612597
$452$	0	0
$453$	−5.80779	−0.272874
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.5877	0.916272	0.458136	−	0.888882i	$-0.348517\pi$
$457$	19.5877	0.916272	0.458136	+	0.888882i	$0.348517\pi$
$458$	0	0
$459$	−3.62620	−0.169256
$460$	0	0
$461$	11.7938	0.549294	0.274647	−	0.961545i	$-0.411439\pi$
$461$	11.7938	0.549294	0.274647	+	0.961545i	$0.411439\pi$
$462$	0	0
$463$	−38.8819	−1.80700	−0.903498	−	0.428592i	$-0.859010\pi$
$463$	−38.8819	−1.80700	−0.903498	+	0.428592i	$0.859010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.4017	−1.17545	−0.587725	−	0.809060i	$-0.699976\pi$
$467$	−25.4017	−1.17545	−0.587725	+	0.809060i	$0.699976\pi$
$468$	0	0
$469$	−34.3372	−1.58554
$470$	0	0
$471$	−13.6541	−0.629148
$472$	0	0
$473$	−1.21575	−0.0559003
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.27072	0.149756
$478$	0	0
$479$	34.0987	1.55801	0.779005	−	0.627018i	$-0.215725\pi$
$479$	34.0987	1.55801	0.779005	+	0.627018i	$0.215725\pi$
$480$	0	0
$481$	−0.896916	−0.0408959
$482$	0	0
$483$	15.2524	0.694008
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.7616	−0.759539	−0.379769	−	0.925081i	$-0.623997\pi$
$487$	−16.7616	−0.759539	−0.379769	+	0.925081i	$0.623997\pi$
$488$	0	0
$489$	−7.25240	−0.327965
$490$	0	0
$491$	1.65222	0.0745635	0.0372817	−	0.999305i	$-0.488130\pi$
$491$	1.65222	0.0745635	0.0372817	+	0.999305i	$0.488130\pi$
$492$	0	0
$493$	−22.6724	−1.02111
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.5693	−1.10209
$498$	0	0
$499$	−22.7326	−1.01765	−0.508826	−	0.860870i	$-0.669920\pi$
$499$	−22.7326	−1.01765	−0.508826	+	0.860870i	$0.669920\pi$
$500$	0	0
$501$	2.14931	0.0960242
$502$	0	0
$503$	−6.09246	−0.271649	−0.135825	−	0.990733i	$-0.543368\pi$
$503$	−6.09246	−0.271649	−0.135825	+	0.990733i	$0.543368\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	7.87090	0.348871	0.174436	−	0.984669i	$-0.444190\pi$
$509$	7.87090	0.348871	0.174436	+	0.984669i	$0.444190\pi$
$510$	0	0
$511$	8.98168	0.397326
$512$	0	0
$513$	−6.14931	−0.271499
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−15.2139	−0.669105
$518$	0	0
$519$	25.5693	1.12237
$520$	0	0
$521$	19.7938	0.867184	0.433592	−	0.901109i	$-0.357246\pi$
$521$	19.7938	0.867184	0.433592	+	0.901109i	$0.357246\pi$
$522$	0	0
$523$	−27.1878	−1.18884	−0.594421	−	0.804154i	$-0.702619\pi$
$523$	−27.1878	−1.18884	−0.594421	+	0.804154i	$0.702619\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30.7895	−1.34121
$528$	0	0
$529$	7.50479	0.326295
$530$	0	0
$531$	−9.53707	−0.413873
$532$	0	0
$533$	−2.89692	−0.125479
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−21.5510	−0.929995
$538$	0	0
$539$	2.81215	0.121128
$540$	0	0
$541$	−38.0558	−1.63615	−0.818074	−	0.575114i	$-0.804958\pi$
$541$	−38.0558	−1.63615	−0.818074	+	0.575114i	$0.804958\pi$
$542$	0	0
$543$	−0.607876	−0.0260865
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−10.3353	−0.441904	−0.220952	−	0.975285i	$-0.570916\pi$
$547$	−10.3353	−0.441904	−0.220952	+	0.975285i	$0.570916\pi$
$548$	0	0
$549$	11.4200	0.487395
$550$	0	0
$551$	−38.4479	−1.63794
$552$	0	0
$553$	−37.7572	−1.60560
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.25240	−0.0530657	−0.0265329	−	0.999648i	$-0.508447\pi$
$557$	−1.25240	−0.0530657	−0.0265329	+	0.999648i	$0.508447\pi$
$558$	0	0
$559$	0.270718	0.0114502
$560$	0	0
$561$	−16.2847	−0.687539
$562$	0	0
$563$	45.1868	1.90440	0.952198	−	0.305480i	$-0.0988171\pi$
$563$	45.1868	1.90440	0.952198	+	0.305480i	$0.0988171\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.76156	−0.115975
$568$	0	0
$569$	−15.8603	−0.664897	−0.332449	−	0.943121i	$-0.607875\pi$
$569$	−15.8603	−0.664897	−0.332449	+	0.943121i	$0.607875\pi$
$570$	0	0
$571$	−29.5789	−1.23784	−0.618920	−	0.785454i	$-0.712429\pi$
$571$	−29.5789	−1.23784	−0.618920	+	0.785454i	$0.712429\pi$
$572$	0	0
$573$	−21.2803	−0.888997
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−26.8401	−1.11737	−0.558683	−	0.829381i	$-0.688693\pi$
$577$	−26.8401	−1.11737	−0.558683	+	0.829381i	$0.688693\pi$
$578$	0	0
$579$	−7.45856	−0.309967
$580$	0	0
$581$	−8.37380	−0.347404
$582$	0	0
$583$	14.6883	0.608326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.4359	0.967302	0.483651	−	0.875261i	$-0.339310\pi$
$587$	23.4359	0.967302	0.483651	+	0.875261i	$0.339310\pi$
$588$	0	0
$589$	−52.2128	−2.15139
$590$	0	0
$591$	7.45856	0.306804
$592$	0	0
$593$	43.2880	1.77763	0.888813	−	0.458271i	$-0.151531\pi$
$593$	43.2880	1.77763	0.888813	+	0.458271i	$0.151531\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.3372	0.791417
$598$	0	0
$599$	3.58767	0.146588	0.0732940	−	0.997310i	$-0.476649\pi$
$599$	3.58767	0.146588	0.0732940	+	0.997310i	$0.476649\pi$
$600$	0	0
$601$	31.0646	1.26715	0.633575	−	0.773681i	$-0.281587\pi$
$601$	31.0646	1.26715	0.633575	+	0.773681i	$0.281587\pi$
$602$	0	0
$603$	12.4340	0.506351
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.2697	1.22861	0.614304	−	0.789069i	$-0.289437\pi$
$607$	30.2697	1.22861	0.614304	+	0.789069i	$0.289437\pi$
$608$	0	0
$609$	−17.2663	−0.699668
$610$	0	0
$611$	3.38776	0.137054
$612$	0	0
$613$	46.0558	1.86018	0.930088	−	0.367336i	$-0.119730\pi$
$613$	46.0558	1.86018	0.930088	+	0.367336i	$0.119730\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.43835	0.0579059	0.0289530	−	0.999581i	$-0.490783\pi$
$617$	1.43835	0.0579059	0.0289530	+	0.999581i	$0.490783\pi$
$618$	0	0
$619$	−36.2620	−1.45749	−0.728746	−	0.684784i	$-0.759897\pi$
$619$	−36.2620	−1.45749	−0.728746	+	0.684784i	$0.759897\pi$
$620$	0	0
$621$	−5.52311	−0.221635
$622$	0	0
$623$	6.09246	0.244089
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−27.6156	−1.10286
$628$	0	0
$629$	−3.25240	−0.129682
$630$	0	0
$631$	34.8401	1.38696	0.693480	−	0.720475i	$-0.256076\pi$
$631$	34.8401	1.38696	0.693480	+	0.720475i	$0.256076\pi$
$632$	0	0
$633$	−8.02791	−0.319081
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.626198	−0.0248109
$638$	0	0
$639$	8.89692	0.351957
$640$	0	0
$641$	−29.8882	−1.18051	−0.590256	−	0.807216i	$-0.700973\pi$
$641$	−29.8882	−1.18051	−0.590256	+	0.807216i	$0.700973\pi$
$642$	0	0
$643$	−18.8603	−0.743777	−0.371888	−	0.928278i	$-0.621290\pi$
$643$	−18.8603	−0.743777	−0.371888	+	0.928278i	$0.621290\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.50479	0.255730	0.127865	−	0.991792i	$-0.459188\pi$
$647$	6.50479	0.255730	0.127865	+	0.991792i	$0.459188\pi$
$648$	0	0
$649$	−42.8294	−1.68120
$650$	0	0
$651$	−23.4479	−0.918997
$652$	0	0
$653$	−28.1589	−1.10194	−0.550971	−	0.834524i	$-0.685743\pi$
$653$	−28.1589	−1.10194	−0.550971	+	0.834524i	$0.685743\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.25240	−0.126888
$658$	0	0
$659$	30.2139	1.17697	0.588483	−	0.808510i	$-0.299726\pi$
$659$	30.2139	1.17697	0.588483	+	0.808510i	$0.299726\pi$
$660$	0	0
$661$	36.0356	1.40162	0.700811	−	0.713347i	$-0.252822\pi$
$661$	36.0356	1.40162	0.700811	+	0.713347i	$0.252822\pi$
$662$	0	0
$663$	3.62620	0.140830
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34.5327	−1.33711
$668$	0	0
$669$	16.5048	0.638112
$670$	0	0
$671$	51.2855	1.97986
$672$	0	0
$673$	−1.68305	−0.0648769	−0.0324385	−	0.999474i	$-0.510327\pi$
$673$	−1.68305	−0.0648769	−0.0324385	+	0.999474i	$0.510327\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.50479	−0.173133	−0.0865666	−	0.996246i	$-0.527590\pi$
$677$	−4.50479	−0.173133	−0.0865666	+	0.996246i	$0.527590\pi$
$678$	0	0
$679$	−49.0375	−1.88189
$680$	0	0
$681$	−2.69701	−0.103350
$682$	0	0
$683$	−35.5650	−1.36086	−0.680428	−	0.732815i	$-0.738206\pi$
$683$	−35.5650	−1.36086	−0.680428	+	0.732815i	$0.738206\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.5616	−0.402952
$688$	0	0
$689$	−3.27072	−0.124604
$690$	0	0
$691$	6.29237	0.239373	0.119686	−	0.992812i	$-0.461811\pi$
$691$	6.29237	0.239373	0.119686	+	0.992812i	$0.461811\pi$
$692$	0	0
$693$	−12.4017	−0.471102
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.5048	−0.397897
$698$	0	0
$699$	25.2158	0.953747
$700$	0	0
$701$	−36.9065	−1.39394	−0.696970	−	0.717101i	$-0.745469\pi$
$701$	−36.9065	−1.39394	−0.696970	+	0.717101i	$0.745469\pi$
$702$	0	0
$703$	−5.51542	−0.208018
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.1064	−0.605744
$708$	0	0
$709$	−5.25240	−0.197258	−0.0986289	−	0.995124i	$-0.531446\pi$
$709$	−5.25240	−0.197258	−0.0986289	+	0.995124i	$0.531446\pi$
$710$	0	0
$711$	13.6724	0.512756
$712$	0	0
$713$	−46.8959	−1.75626
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.51875	−0.318138
$718$	0	0
$719$	−22.6339	−0.844102	−0.422051	−	0.906572i	$-0.638690\pi$
$719$	−22.6339	−0.844102	−0.422051	+	0.906572i	$0.638690\pi$
$720$	0	0
$721$	−9.55102	−0.355699
$722$	0	0
$723$	7.45856	0.277387
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.5606	1.28178	0.640891	−	0.767632i	$-0.278565\pi$
$727$	34.5606	1.28178	0.640891	+	0.767632i	$0.278565\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.981678	0.0363087
$732$	0	0
$733$	−11.4942	−0.424547	−0.212273	−	0.977210i	$-0.568087\pi$
$733$	−11.4942	−0.424547	−0.212273	+	0.977210i	$0.568087\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	55.8390	2.05686
$738$	0	0
$739$	20.4619	0.752703	0.376351	−	0.926477i	$-0.377179\pi$
$739$	20.4619	0.752703	0.376351	+	0.926477i	$0.377179\pi$
$740$	0	0
$741$	6.14931	0.225901
$742$	0	0
$743$	−34.2836	−1.25774	−0.628872	−	0.777509i	$-0.716483\pi$
$743$	−34.2836	−1.25774	−0.628872	+	0.777509i	$0.716483\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.03228	0.110945
$748$	0	0
$749$	−2.29862	−0.0839899
$750$	0	0
$751$	−23.6358	−0.862482	−0.431241	−	0.902237i	$-0.641924\pi$
$751$	−23.6358	−0.862482	−0.431241	+	0.902237i	$0.641924\pi$
$752$	0	0
$753$	−9.64452	−0.351466
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.54913	0.274378	0.137189	−	0.990545i	$-0.456193\pi$
$757$	7.54913	0.274378	0.137189	+	0.990545i	$0.456193\pi$
$758$	0	0
$759$	−24.8034	−0.900307
$760$	0	0
$761$	48.2418	1.74876	0.874381	−	0.485239i	$-0.161268\pi$
$761$	48.2418	1.74876	0.874381	+	0.485239i	$0.161268\pi$
$762$	0	0
$763$	33.5510	1.21463
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.53707	0.344364
$768$	0	0
$769$	24.0558	0.867475	0.433737	−	0.901039i	$-0.357195\pi$
$769$	24.0558	0.867475	0.433737	+	0.901039i	$0.357195\pi$
$770$	0	0
$771$	−13.3555	−0.480986
$772$	0	0
$773$	37.8863	1.36268	0.681338	−	0.731969i	$-0.261399\pi$
$773$	37.8863	1.36268	0.681338	+	0.731969i	$0.261399\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.47689	−0.0888578
$778$	0	0
$779$	−17.8140	−0.638254
$780$	0	0
$781$	39.9546	1.42969
$782$	0	0
$783$	6.25240	0.223442
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14.3213	−0.510500	−0.255250	−	0.966875i	$-0.582158\pi$
$787$	−14.3213	−0.510500	−0.255250	+	0.966875i	$0.582158\pi$
$788$	0	0
$789$	9.25240	0.329394
$790$	0	0
$791$	12.4402	0.442324
$792$	0	0
$793$	−11.4200	−0.405537
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.97398	0.282453	0.141226	−	0.989977i	$-0.454895\pi$
$797$	7.97398	0.282453	0.141226	+	0.989977i	$0.454895\pi$
$798$	0	0
$799$	12.2847	0.434600
$800$	0	0
$801$	−2.20617	−0.0779511
$802$	0	0
$803$	−14.6060	−0.515434
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.2986	0.890554
$808$	0	0
$809$	7.42192	0.260941	0.130470	−	0.991452i	$-0.458351\pi$
$809$	7.42192	0.260941	0.130470	+	0.991452i	$0.458351\pi$
$810$	0	0
$811$	−54.6391	−1.91864	−0.959319	−	0.282323i	$-0.908895\pi$
$811$	−54.6391	−1.91864	−0.959319	+	0.282323i	$0.908895\pi$
$812$	0	0
$813$	31.3309	1.09882
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.66473	0.0582416
$818$	0	0
$819$	2.76156	0.0964966
$820$	0	0
$821$	−2.12910	−0.0743062	−0.0371531	−	0.999310i	$-0.511829\pi$
$821$	−2.12910	−0.0743062	−0.0371531	+	0.999310i	$0.511829\pi$
$822$	0	0
$823$	−36.3188	−1.26600	−0.632998	−	0.774154i	$-0.718176\pi$
$823$	−36.3188	−1.26600	−0.632998	+	0.774154i	$0.718176\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.0419	0.766471	0.383235	−	0.923651i	$-0.374810\pi$
$827$	22.0419	0.766471	0.383235	+	0.923651i	$0.374810\pi$
$828$	0	0
$829$	48.9344	1.69956	0.849781	−	0.527136i	$-0.176734\pi$
$829$	48.9344	1.69956	0.849781	+	0.527136i	$0.176734\pi$
$830$	0	0
$831$	23.0096	0.798194
$832$	0	0
$833$	−2.27072	−0.0786757
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.49084	0.293486
$838$	0	0
$839$	10.2986	0.355548	0.177774	−	0.984071i	$-0.443110\pi$
$839$	10.2986	0.355548	0.177774	+	0.984071i	$0.443110\pi$
$840$	0	0
$841$	10.0925	0.348016
$842$	0	0
$843$	−9.19554	−0.316711
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−25.3169	−0.869901
$848$	0	0
$849$	−9.93545	−0.340984
$850$	0	0
$851$	−4.95377	−0.169813
$852$	0	0
$853$	24.0356	0.822963	0.411482	−	0.911418i	$-0.365011\pi$
$853$	24.0356	0.822963	0.411482	+	0.911418i	$0.365011\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.0096	−0.512718	−0.256359	−	0.966582i	$-0.582523\pi$
$857$	−15.0096	−0.512718	−0.256359	+	0.966582i	$0.582523\pi$
$858$	0	0
$859$	−24.2707	−0.828106	−0.414053	−	0.910253i	$-0.635887\pi$
$859$	−24.2707	−0.828106	−0.414053	+	0.910253i	$0.635887\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	42.4985	1.44667	0.723333	−	0.690499i	$-0.242609\pi$
$863$	42.4985	1.44667	0.723333	+	0.690499i	$0.242609\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.85069	−0.130776
$868$	0	0
$869$	61.4007	2.08287
$870$	0	0
$871$	−12.4340	−0.421309
$872$	0	0
$873$	17.7572	0.600990
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.56165	−0.154036	−0.0770179	−	0.997030i	$-0.524540\pi$
$877$	−4.56165	−0.154036	−0.0770179	+	0.997030i	$0.524540\pi$
$878$	0	0
$879$	23.7938	0.802546
$880$	0	0
$881$	6.10308	0.205618	0.102809	−	0.994701i	$-0.467217\pi$
$881$	6.10308	0.205618	0.102809	+	0.994701i	$0.467217\pi$
$882$	0	0
$883$	13.0096	0.437807	0.218904	−	0.975746i	$-0.429752\pi$
$883$	13.0096	0.437807	0.218904	+	0.975746i	$0.429752\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.4307	−0.383804	−0.191902	−	0.981414i	$-0.561466\pi$
$887$	−11.4307	−0.383804	−0.191902	+	0.981414i	$0.561466\pi$
$888$	0	0
$889$	18.2986	0.613716
$890$	0	0
$891$	4.49084	0.150449
$892$	0	0
$893$	20.8324	0.697129
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.52311	0.184411
$898$	0	0
$899$	53.0881	1.77059
$900$	0	0
$901$	−11.8603	−0.395123
$902$	0	0
$903$	0.747604	0.0248787
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−19.8863	−0.660313	−0.330157	−	0.943926i	$-0.607102\pi$
$907$	−19.8863	−0.660313	−0.330157	+	0.943926i	$0.607102\pi$
$908$	0	0
$909$	5.83237	0.193447
$910$	0	0
$911$	−57.8217	−1.91572	−0.957860	−	0.287236i	$-0.907264\pi$
$911$	−57.8217	−1.91572	−0.957860	+	0.287236i	$0.907264\pi$
$912$	0	0
$913$	13.6175	0.450672
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.04623	−0.100595
$918$	0	0
$919$	−8.17722	−0.269742	−0.134871	−	0.990863i	$-0.543062\pi$
$919$	−8.17722	−0.269742	−0.134871	+	0.990863i	$0.543062\pi$
$920$	0	0
$921$	−25.0664	−0.825967
$922$	0	0
$923$	−8.89692	−0.292846
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.45856	0.113594
$928$	0	0
$929$	−14.7312	−0.483314	−0.241657	−	0.970362i	$-0.577691\pi$
$929$	−14.7312	−0.483314	−0.241657	+	0.970362i	$0.577691\pi$
$930$	0	0
$931$	−3.85069	−0.126201
$932$	0	0
$933$	18.3632	0.601183
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.9431	−0.618846	−0.309423	−	0.950925i	$-0.600136\pi$
$937$	−18.9431	−0.618846	−0.309423	+	0.950925i	$0.600136\pi$
$938$	0	0
$939$	−10.7293	−0.350137
$940$	0	0
$941$	12.3353	0.402118	0.201059	−	0.979579i	$-0.435562\pi$
$941$	12.3353	0.402118	0.201059	+	0.979579i	$0.435562\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.1710	0.590477	0.295238	−	0.955424i	$-0.404601\pi$
$947$	18.1710	0.590477	0.295238	+	0.955424i	$0.404601\pi$
$948$	0	0
$949$	3.25240	0.105577
$950$	0	0
$951$	−8.20617	−0.266103
$952$	0	0
$953$	−37.4113	−1.21187	−0.605935	−	0.795514i	$-0.707201\pi$
$953$	−37.4113	−1.21187	−0.605935	+	0.795514i	$0.707201\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	28.0785	0.907649
$958$	0	0
$959$	−56.4681	−1.82345
$960$	0	0
$961$	41.0943	1.32562
$962$	0	0
$963$	0.832365	0.0268226
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−38.4821	−1.23750	−0.618750	−	0.785588i	$-0.712361\pi$
$967$	−38.4821	−1.23750	−0.618750	+	0.785588i	$0.712361\pi$
$968$	0	0
$969$	22.2986	0.716335
$970$	0	0
$971$	−44.5896	−1.43095	−0.715473	−	0.698640i	$-0.753789\pi$
$971$	−44.5896	−1.43095	−0.715473	+	0.698640i	$0.753789\pi$
$972$	0	0
$973$	−45.7572	−1.46691
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.2524	1.06384	0.531919	−	0.846795i	$-0.321471\pi$
$977$	33.2524	1.06384	0.531919	+	0.846795i	$0.321471\pi$
$978$	0	0
$979$	−9.90754	−0.316646
$980$	0	0
$981$	−12.1493	−0.387898
$982$	0	0
$983$	10.9186	0.348248	0.174124	−	0.984724i	$-0.444291\pi$
$983$	10.9186	0.348248	0.174124	+	0.984724i	$0.444291\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	9.35548	0.297788
$988$	0	0
$989$	1.49521	0.0475448
$990$	0	0
$991$	17.3738	0.551897	0.275949	−	0.961172i	$-0.411008\pi$
$991$	17.3738	0.551897	0.275949	+	0.961172i	$0.411008\pi$
$992$	0	0
$993$	30.7110	0.974583
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.0752	0.382425	0.191212	−	0.981549i	$-0.438758\pi$
$997$	12.0752	0.382425	0.191212	+	0.981549i	$0.438758\pi$
$998$	0	0
$999$	0.896916	0.0283772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bn.1.1	yes	3
5.4	even	2		7800.2.a.bm.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bm.1.3	✓	3	5.4	even	2
7800.2.a.bn.1.1	yes	3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.76156 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.76156 q^{7} +1.00000 q^{9} +4.49084 q^{11} -1.00000 q^{13} -3.62620 q^{17} -6.14931 q^{19} -2.76156 q^{21} -5.52311 q^{23} +1.00000 q^{27} +6.25240 q^{29} +8.49084 q^{31} +4.49084 q^{33} +0.896916 q^{37} -1.00000 q^{39} +2.89692 q^{41} -0.270718 q^{43} -3.38776 q^{47} +0.626198 q^{49} -3.62620 q^{51} +3.27072 q^{53} -6.14931 q^{57} -9.53707 q^{59} +11.4200 q^{61} -2.76156 q^{63} +12.4340 q^{67} -5.52311 q^{69} +8.89692 q^{71} -3.25240 q^{73} -12.4017 q^{77} +13.6724 q^{79} +1.00000 q^{81} +3.03228 q^{83} +6.25240 q^{87} -2.20617 q^{89} +2.76156 q^{91} +8.49084 q^{93} +17.7572 q^{97} +4.49084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * q^99

Introduction

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