LMFDB - Embedded newform 5292.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 756)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.42107 q^{5} +O(q^{10})$	$q+2.42107 q^{5} +4.70370 q^{11} +3.42107 q^{13} -1.70370 q^{17} -1.28263 q^{19} -1.12476 q^{23} +0.861564 q^{25} -4.70370 q^{29} +3.42107 q^{31} +8.54583 q^{37} -3.71737 q^{41} +5.54583 q^{43} +11.8285 q^{47} +10.2632 q^{53} +11.3880 q^{55} -4.12476 q^{59} +9.24953 q^{61} +8.28263 q^{65} -11.1248 q^{67} +14.9669 q^{71} -2.13844 q^{73} -14.5322 q^{79} -8.42107 q^{83} -4.12476 q^{85} -16.0917 q^{89} -3.10533 q^{95} -16.2495 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5}+O(q^{10}) $ 3 * q - q^5	$ 3 q - q^{5} + 5 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 14 q^{23} + 10 q^{25} - 5 q^{29} + 2 q^{31} - 12 q^{41} - 9 q^{43} + 9 q^{47} + 6 q^{53} - 8 q^{55} + 5 q^{59} - 7 q^{61} + 24 q^{65} - 16 q^{67} + 11 q^{71} + q^{73} - 8 q^{79} - 17 q^{83} + 5 q^{85} + 3 q^{89} + 32 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - q^5 + 5 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 14 * q^23 + 10 * q^25 - 5 * q^29 + 2 * q^31 - 12 * q^41 - 9 * q^43 + 9 * q^47 + 6 * q^53 - 8 * q^55 + 5 * q^59 - 7 * q^61 + 24 * q^65 - 16 * q^67 + 11 * q^71 + q^73 - 8 * q^79 - 17 * q^83 + 5 * q^85 + 3 * q^89 + 32 * q^95 - 14 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.42107	1.08273	0.541367	−	0.840786i	$-0.317907\pi$
$5$	2.42107	1.08273	0.541367	+	0.840786i	$0.317907\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.70370	1.41822	0.709109	−	0.705099i	$-0.249097\pi$
$11$	4.70370	1.41822	0.709109	+	0.705099i	$0.249097\pi$
$12$	0	0
$13$	3.42107	0.948833	0.474417	−	0.880300i	$-0.342659\pi$
$13$	3.42107	0.948833	0.474417	+	0.880300i	$0.342659\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.70370	−0.413207	−0.206604	−	0.978425i	$-0.566241\pi$
$17$	−1.70370	−0.413207	−0.206604	+	0.978425i	$0.566241\pi$
$18$	0	0
$19$	−1.28263	−0.294256	−0.147128	−	0.989117i	$-0.547003\pi$
$19$	−1.28263	−0.294256	−0.147128	+	0.989117i	$0.547003\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.12476	−0.234529	−0.117265	−	0.993101i	$-0.537413\pi$
$23$	−1.12476	−0.234529	−0.117265	+	0.993101i	$0.537413\pi$
$24$	0	0
$25$	0.861564	0.172313
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.70370	−0.873455	−0.436727	−	0.899594i	$-0.643863\pi$
$29$	−4.70370	−0.873455	−0.436727	+	0.899594i	$0.643863\pi$
$30$	0	0
$31$	3.42107	0.614442	0.307221	−	0.951638i	$-0.400601\pi$
$31$	3.42107	0.614442	0.307221	+	0.951638i	$0.400601\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.54583	1.40493	0.702463	−	0.711720i	$-0.252084\pi$
$37$	8.54583	1.40493	0.702463	+	0.711720i	$0.252084\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.71737	−0.580556	−0.290278	−	0.956942i	$-0.593748\pi$
$41$	−3.71737	−0.580556	−0.290278	+	0.956942i	$0.593748\pi$
$42$	0	0
$43$	5.54583	0.845731	0.422866	−	0.906192i	$-0.361024\pi$
$43$	5.54583	0.845731	0.422866	+	0.906192i	$0.361024\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.8285	1.72536	0.862679	−	0.505752i	$-0.168785\pi$
$47$	11.8285	1.72536	0.862679	+	0.505752i	$0.168785\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.2632	1.40976	0.704879	−	0.709327i	$-0.251001\pi$
$53$	10.2632	1.40976	0.704879	+	0.709327i	$0.251001\pi$
$54$	0	0
$55$	11.3880	1.53555
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.12476	−0.536998	−0.268499	−	0.963280i	$-0.586528\pi$
$59$	−4.12476	−0.536998	−0.268499	+	0.963280i	$0.586528\pi$
$60$	0	0
$61$	9.24953	1.18428	0.592140	−	0.805835i	$-0.298283\pi$
$61$	9.24953	1.18428	0.592140	+	0.805835i	$0.298283\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.28263	1.02733
$66$	0	0
$67$	−11.1248	−1.35911	−0.679553	−	0.733626i	$-0.737826\pi$
$67$	−11.1248	−1.35911	−0.679553	+	0.733626i	$0.737826\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.9669	1.77624	0.888122	−	0.459608i	$-0.152010\pi$
$71$	14.9669	1.77624	0.888122	+	0.459608i	$0.152010\pi$
$72$	0	0
$73$	−2.13844	−0.250285	−0.125143	−	0.992139i	$-0.539939\pi$
$73$	−2.13844	−0.250285	−0.125143	+	0.992139i	$0.539939\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.5322	−1.63500	−0.817498	−	0.575932i	$-0.804639\pi$
$79$	−14.5322	−1.63500	−0.817498	+	0.575932i	$0.804639\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.42107	−0.924332	−0.462166	−	0.886793i	$-0.652928\pi$
$83$	−8.42107	−0.924332	−0.462166	+	0.886793i	$0.652928\pi$
$84$	0	0
$85$	−4.12476	−0.447393
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.0917	−1.70571	−0.852856	−	0.522146i	$-0.825132\pi$
$89$	−16.0917	−1.70571	−0.852856	+	0.522146i	$0.825132\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.10533	−0.318600
$96$	0	0
$97$	−16.2495	−1.64989	−0.824945	−	0.565213i	$-0.808794\pi$
$97$	−16.2495	−1.64989	−0.824945	+	0.565213i	$0.808794\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.9863	1.29219	0.646094	−	0.763258i	$-0.276401\pi$
$101$	12.9863	1.29219	0.646094	+	0.763258i	$0.276401\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7174	1.22943	0.614717	−	0.788748i	$-0.289270\pi$
$107$	12.7174	1.22943	0.614717	+	0.788748i	$0.289270\pi$
$108$	0	0
$109$	−7.40739	−0.709500	−0.354750	−	0.934961i	$-0.615434\pi$
$109$	−7.40739	−0.709500	−0.354750	+	0.934961i	$0.615434\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.68427	−0.628803	−0.314401	−	0.949290i	$-0.601804\pi$
$113$	−6.68427	−0.628803	−0.314401	+	0.949290i	$0.601804\pi$
$114$	0	0
$115$	−2.72313	−0.253933
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.1248	1.01134
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.0194	−0.896165
$126$	0	0
$127$	−11.5322	−1.02331	−0.511657	−	0.859190i	$-0.670968\pi$
$127$	−11.5322	−1.02331	−0.511657	+	0.859190i	$0.670968\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.13844	0.798429	0.399214	−	0.916858i	$-0.369283\pi$
$131$	9.13844	0.798429	0.399214	+	0.916858i	$0.369283\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.24953	−0.192190	−0.0960950	−	0.995372i	$-0.530635\pi$
$137$	−2.24953	−0.192190	−0.0960950	+	0.995372i	$0.530635\pi$
$138$	0	0
$139$	10.5789	0.897293	0.448647	−	0.893709i	$-0.351906\pi$
$139$	10.5789	0.897293	0.448647	+	0.893709i	$0.351906\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.0917	1.34565
$144$	0	0
$145$	−11.3880	−0.945719
$146$	0	0
$147$	0	0
$148$	0	0
$149$	24.3743	1.99682	0.998410	−	0.0563721i	$-0.0179533\pi$
$149$	24.3743	1.99682	0.998410	+	0.0563721i	$0.0179533\pi$
$150$	0	0
$151$	11.1384	0.906433	0.453217	−	0.891400i	$-0.350276\pi$
$151$	11.1384	0.906433	0.453217	+	0.891400i	$0.350276\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.28263	0.665277
$156$	0	0
$157$	−3.26320	−0.260432	−0.130216	−	0.991486i	$-0.541567\pi$
$157$	−3.26320	−0.260432	−0.130216	+	0.991486i	$0.541567\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.152110	−0.0119141	−0.00595707	−	0.999982i	$-0.501896\pi$
$163$	−0.152110	−0.0119141	−0.00595707	+	0.999982i	$0.501896\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.2359	−1.87543	−0.937713	−	0.347410i	$-0.887061\pi$
$167$	−24.2359	−1.87543	−0.937713	+	0.347410i	$0.887061\pi$
$168$	0	0
$169$	−1.29630	−0.0997156
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.95322	−0.528644	−0.264322	−	0.964435i	$-0.585148\pi$
$173$	−6.95322	−0.528644	−0.264322	+	0.964435i	$0.585148\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.6979	1.09858	0.549288	−	0.835633i	$-0.314899\pi$
$179$	14.6979	1.09858	0.549288	+	0.835633i	$0.314899\pi$
$180$	0	0
$181$	−17.1053	−1.27143	−0.635715	−	0.771924i	$-0.719295\pi$
$181$	−17.1053	−1.27143	−0.635715	+	0.771924i	$0.719295\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.6900	1.52116
$186$	0	0
$187$	−8.01367	−0.586018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.9806	1.66282	0.831408	−	0.555663i	$-0.187535\pi$
$191$	22.9806	1.66282	0.831408	+	0.555663i	$0.187535\pi$
$192$	0	0
$193$	5.37429	0.386850	0.193425	−	0.981115i	$-0.438040\pi$
$193$	5.37429	0.386850	0.193425	+	0.981115i	$0.438040\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.57893	−0.254988	−0.127494	−	0.991839i	$-0.540693\pi$
$197$	−3.57893	−0.254988	−0.127494	+	0.991839i	$0.540693\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.03310	−0.417318
$210$	0	0
$211$	19.1111	1.31566	0.657831	−	0.753166i	$-0.271474\pi$
$211$	19.1111	1.31566	0.657831	+	0.753166i	$0.271474\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.4268	0.915702
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.82846	−0.392065
$222$	0	0
$223$	−24.2632	−1.62478	−0.812392	−	0.583112i	$-0.801835\pi$
$223$	−24.2632	−1.62478	−0.812392	+	0.583112i	$0.801835\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.84789	−0.255393	−0.127697	−	0.991813i	$-0.540758\pi$
$227$	−3.84789	−0.255393	−0.127697	+	0.991813i	$0.540758\pi$
$228$	0	0
$229$	−3.53216	−0.233412	−0.116706	−	0.993167i	$-0.537233\pi$
$229$	−3.53216	−0.233412	−0.116706	+	0.993167i	$0.537233\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.2495	1.52313	0.761564	−	0.648090i	$-0.224432\pi$
$233$	23.2495	1.52313	0.761564	+	0.648090i	$0.224432\pi$
$234$	0	0
$235$	28.6375	1.86810
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.28263	0.535759	0.267879	−	0.963452i	$-0.413677\pi$
$239$	8.28263	0.535759	0.267879	+	0.963452i	$0.413677\pi$
$240$	0	0
$241$	−13.5264	−0.871312	−0.435656	−	0.900113i	$-0.643484\pi$
$241$	−13.5264	−0.871312	−0.435656	+	0.900113i	$0.643484\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.38796	−0.279199
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.11109	0.322609	0.161305	−	0.986905i	$-0.448430\pi$
$251$	5.11109	0.322609	0.161305	+	0.986905i	$0.448430\pi$
$252$	0	0
$253$	−5.29055	−0.332614
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.0917	1.00377	0.501885	−	0.864934i	$-0.332640\pi$
$257$	16.0917	1.00377	0.501885	+	0.864934i	$0.332640\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.7174	0.784187	0.392093	−	0.919925i	$-0.371751\pi$
$263$	12.7174	0.784187	0.392093	+	0.919925i	$0.371751\pi$
$264$	0	0
$265$	24.8479	1.52639
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.7037	1.38427	0.692134	−	0.721769i	$-0.256671\pi$
$269$	22.7037	1.38427	0.692134	+	0.721769i	$0.256671\pi$
$270$	0	0
$271$	−15.1248	−0.918764	−0.459382	−	0.888239i	$-0.651929\pi$
$271$	−15.1248	−0.918764	−0.459382	+	0.888239i	$0.651929\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.05253	0.244377
$276$	0	0
$277$	23.5458	1.41473	0.707366	−	0.706848i	$-0.249883\pi$
$277$	23.5458	1.41473	0.707366	+	0.706848i	$0.249883\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.55950	−0.331652	−0.165826	−	0.986155i	$-0.553029\pi$
$281$	−5.55950	−0.331652	−0.165826	+	0.986155i	$0.553029\pi$
$282$	0	0
$283$	13.9532	0.829433	0.414717	−	0.909951i	$-0.363881\pi$
$283$	13.9532	0.829433	0.414717	+	0.909951i	$0.363881\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.0974	−0.829260
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.15211	−0.476251	−0.238126	−	0.971234i	$-0.576533\pi$
$293$	−8.15211	−0.476251	−0.238126	+	0.971234i	$0.576533\pi$
$294$	0	0
$295$	−9.98633	−0.581426
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.84789	−0.222529
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.3937	1.28226
$306$	0	0
$307$	−2.13844	−0.122047	−0.0610235	−	0.998136i	$-0.519436\pi$
$307$	−2.13844	−0.122047	−0.0610235	+	0.998136i	$0.519436\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.8421	−0.614801	−0.307400	−	0.951580i	$-0.599459\pi$
$311$	−10.8421	−0.614801	−0.307400	+	0.951580i	$0.599459\pi$
$312$	0	0
$313$	−32.3412	−1.82803	−0.914016	−	0.405678i	$-0.867035\pi$
$313$	−32.3412	−1.82803	−0.914016	+	0.405678i	$0.867035\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.9201	−1.23116	−0.615578	−	0.788076i	$-0.711078\pi$
$317$	−21.9201	−1.23116	−0.615578	+	0.788076i	$0.711078\pi$
$318$	0	0
$319$	−22.1248	−1.23875
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.18521	0.121588
$324$	0	0
$325$	2.94747	0.163496
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.3937	0.681220	0.340610	−	0.940205i	$-0.389366\pi$
$331$	12.3937	0.681220	0.340610	+	0.940205i	$0.389366\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−26.9338	−1.47155
$336$	0	0
$337$	7.73104	0.421137	0.210568	−	0.977579i	$-0.432469\pi$
$337$	7.73104	0.421137	0.210568	+	0.977579i	$0.432469\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.0917	0.871412
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.2222	1.51505	0.757523	−	0.652808i	$-0.226409\pi$
$347$	28.2222	1.51505	0.757523	+	0.652808i	$0.226409\pi$
$348$	0	0
$349$	1.23585	0.0661537	0.0330769	−	0.999453i	$-0.489469\pi$
$349$	1.23585	0.0661537	0.0330769	+	0.999453i	$0.489469\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.9201	−1.00702	−0.503508	−	0.863990i	$-0.667958\pi$
$353$	−18.9201	−1.00702	−0.503508	+	0.863990i	$0.667958\pi$
$354$	0	0
$355$	36.2359	1.92320
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.8616	0.626029	0.313015	−	0.949748i	$-0.398661\pi$
$359$	11.8616	0.626029	0.313015	+	0.949748i	$0.398661\pi$
$360$	0	0
$361$	−17.3549	−0.913414
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.17730	−0.270992
$366$	0	0
$367$	14.2222	0.742392	0.371196	−	0.928555i	$-0.378948\pi$
$367$	14.2222	0.742392	0.371196	+	0.928555i	$0.378948\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	23.6843	1.22632	0.613162	−	0.789957i	$-0.289897\pi$
$373$	23.6843	1.22632	0.613162	+	0.789957i	$0.289897\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.0917	−0.828763
$378$	0	0
$379$	4.42107	0.227095	0.113547	−	0.993533i	$-0.463779\pi$
$379$	4.42107	0.227095	0.113547	+	0.993533i	$0.463779\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.40164	−0.224913	−0.112457	−	0.993657i	$-0.535872\pi$
$383$	−4.40164	−0.224913	−0.112457	+	0.993657i	$0.535872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.30998	−0.167822	−0.0839112	−	0.996473i	$-0.526741\pi$
$389$	−3.30998	−0.167822	−0.0839112	+	0.996473i	$0.526741\pi$
$390$	0	0
$391$	1.91626	0.0969092
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−35.1833	−1.77026
$396$	0	0
$397$	31.1696	1.56436	0.782180	−	0.623053i	$-0.214108\pi$
$397$	31.1696	1.56436	0.782180	+	0.623053i	$0.214108\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.586849	0.0293058	0.0146529	−	0.999893i	$-0.495336\pi$
$401$	0.586849	0.0293058	0.0146529	+	0.999893i	$0.495336\pi$
$402$	0	0
$403$	11.7037	0.583003
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.1970	1.99249
$408$	0	0
$409$	−29.0312	−1.43550	−0.717750	−	0.696300i	$-0.754828\pi$
$409$	−29.0312	−1.43550	−0.717750	+	0.696300i	$0.754828\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−20.3880	−1.00081
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.1833	1.13258	0.566290	−	0.824206i	$-0.308378\pi$
$419$	23.1833	1.13258	0.566290	+	0.824206i	$0.308378\pi$
$420$	0	0
$421$	−29.0586	−1.41623	−0.708114	−	0.706098i	$-0.750454\pi$
$421$	−29.0586	−1.41623	−0.708114	+	0.706098i	$0.750454\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.46784	−0.0712008
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.98057	−0.0954007	−0.0477003	−	0.998862i	$-0.515189\pi$
$431$	−1.98057	−0.0954007	−0.0477003	+	0.998862i	$0.515189\pi$
$432$	0	0
$433$	−29.5048	−1.41791	−0.708955	−	0.705253i	$-0.750833\pi$
$433$	−29.5048	−1.41791	−0.708955	+	0.705253i	$0.750833\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.44266	0.0690116
$438$	0	0
$439$	−18.9727	−0.905515	−0.452758	−	0.891634i	$-0.649560\pi$
$439$	−18.9727	−0.905515	−0.452758	+	0.891634i	$0.649560\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.2690	1.01052	0.505259	−	0.862968i	$-0.331397\pi$
$443$	21.2690	1.01052	0.505259	+	0.862968i	$0.331397\pi$
$444$	0	0
$445$	−38.9590	−1.84683
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.2690	−1.00374	−0.501872	−	0.864942i	$-0.667355\pi$
$449$	−21.2690	−1.00374	−0.501872	+	0.864942i	$0.667355\pi$
$450$	0	0
$451$	−17.4854	−0.823354
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.4678	−0.536443	−0.268222	−	0.963357i	$-0.586436\pi$
$457$	−11.4678	−0.536443	−0.268222	+	0.963357i	$0.586436\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.40164	0.344729	0.172364	−	0.985033i	$-0.444859\pi$
$461$	7.40164	0.344729	0.172364	+	0.985033i	$0.444859\pi$
$462$	0	0
$463$	5.68427	0.264170	0.132085	−	0.991238i	$-0.457833\pi$
$463$	5.68427	0.264170	0.132085	+	0.991238i	$0.457833\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.2301	1.16751	0.583755	−	0.811930i	$-0.301583\pi$
$467$	25.2301	1.16751	0.583755	+	0.811930i	$0.301583\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26.0859	1.19943
$474$	0	0
$475$	−1.10507	−0.0507040
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.2359	0.559070	0.279535	−	0.960135i	$-0.409820\pi$
$479$	12.2359	0.559070	0.279535	+	0.960135i	$0.409820\pi$
$480$	0	0
$481$	29.2359	1.33304
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−39.3412	−1.78639
$486$	0	0
$487$	−5.98057	−0.271005	−0.135503	−	0.990777i	$-0.543265\pi$
$487$	−5.98057	−0.271005	−0.135503	+	0.990777i	$0.543265\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.20275	−0.415314	−0.207657	−	0.978202i	$-0.566584\pi$
$491$	−9.20275	−0.415314	−0.207657	+	0.978202i	$0.566584\pi$
$492$	0	0
$493$	8.01367	0.360918
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.6648	0.880319	0.440159	−	0.897920i	$-0.354922\pi$
$499$	19.6648	0.880319	0.440159	+	0.897920i	$0.354922\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.54583	−0.425628	−0.212814	−	0.977093i	$-0.568263\pi$
$503$	−9.54583	−0.425628	−0.212814	+	0.977093i	$0.568263\pi$
$504$	0	0
$505$	31.4408	1.39910
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.54583	−0.157166	−0.0785831	−	0.996908i	$-0.525040\pi$
$509$	−3.54583	−0.157166	−0.0785831	+	0.996908i	$0.525040\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.9475	0.746795
$516$	0	0
$517$	55.6375	2.44693
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.5458	0.681075	0.340538	−	0.940231i	$-0.389391\pi$
$521$	15.5458	0.681075	0.340538	+	0.940231i	$0.389391\pi$
$522$	0	0
$523$	31.3743	1.37190	0.685951	−	0.727648i	$-0.259386\pi$
$523$	31.3743	1.37190	0.685951	+	0.727648i	$0.259386\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.82846	−0.253892
$528$	0	0
$529$	−21.7349	−0.944996
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.7174	−0.550850
$534$	0	0
$535$	30.7896	1.33115
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.26896	−0.183537	−0.0917684	−	0.995780i	$-0.529252\pi$
$541$	−4.26896	−0.183537	−0.0917684	+	0.995780i	$0.529252\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.9338	−0.768199
$546$	0	0
$547$	−20.4016	−0.872311	−0.436155	−	0.899871i	$-0.643660\pi$
$547$	−20.4016	−0.872311	−0.436155	+	0.899871i	$0.643660\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.03310	0.257019
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.4933	−1.50390	−0.751950	−	0.659221i	$-0.770886\pi$
$557$	−35.4933	−1.50390	−0.751950	+	0.659221i	$0.770886\pi$
$558$	0	0
$559$	18.9727	0.802458
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.29055	−0.222970	−0.111485	−	0.993766i	$-0.535561\pi$
$563$	−5.29055	−0.222970	−0.111485	+	0.993766i	$0.535561\pi$
$564$	0	0
$565$	−16.1831	−0.680826
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.06045	−0.0444564	−0.0222282	−	0.999753i	$-0.507076\pi$
$569$	−1.06045	−0.0444564	−0.0222282	+	0.999753i	$0.507076\pi$
$570$	0	0
$571$	−29.0996	−1.21778	−0.608890	−	0.793255i	$-0.708385\pi$
$571$	−29.0996	−1.21778	−0.608890	+	0.793255i	$0.708385\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.969055	−0.0404124
$576$	0	0
$577$	1.23585	0.0514493	0.0257246	−	0.999669i	$-0.491811\pi$
$577$	1.23585	0.0514493	0.0257246	+	0.999669i	$0.491811\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	48.2750	1.99935
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.9727	1.07201	0.536003	−	0.844216i	$-0.319934\pi$
$587$	25.9727	1.07201	0.536003	+	0.844216i	$0.319934\pi$
$588$	0	0
$589$	−4.38796	−0.180803
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.3021	−0.505185	−0.252593	−	0.967573i	$-0.581283\pi$
$593$	−12.3021	−0.505185	−0.252593	+	0.967573i	$0.581283\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.7874	−0.971928	−0.485964	−	0.873979i	$-0.661531\pi$
$599$	−23.7874	−0.971928	−0.485964	+	0.873979i	$0.661531\pi$
$600$	0	0
$601$	7.92012	0.323068	0.161534	−	0.986867i	$-0.448356\pi$
$601$	7.92012	0.323068	0.161534	+	0.986867i	$0.448356\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.9338	1.09501
$606$	0	0
$607$	0.0467764	0.00189860	0.000949298	−	1.00000i	$-0.499698\pi$
$607$	0.0467764	0.00189860	0.000949298		1.00000i	$0.499698\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.4660	1.63708
$612$	0	0
$613$	19.8205	0.800544	0.400272	−	0.916396i	$-0.368916\pi$
$613$	19.8205	0.800544	0.400272	+	0.916396i	$0.368916\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.4328	1.38621	0.693107	−	0.720835i	$-0.256241\pi$
$617$	34.4328	1.38621	0.693107	+	0.720835i	$0.256241\pi$
$618$	0	0
$619$	−47.5127	−1.90970	−0.954849	−	0.297092i	$-0.903983\pi$
$619$	−47.5127	−1.90970	−0.954849	+	0.297092i	$0.903983\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.5655	−1.14262
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.5595	−0.580525
$630$	0	0
$631$	9.26320	0.368762	0.184381	−	0.982855i	$-0.440972\pi$
$631$	9.26320	0.368762	0.184381	+	0.982855i	$0.440972\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27.9201	−1.10798
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.10533	−0.122653	−0.0613266	−	0.998118i	$-0.519533\pi$
$641$	−3.10533	−0.122653	−0.0613266	+	0.998118i	$0.519533\pi$
$642$	0	0
$643$	17.7368	0.699471	0.349736	−	0.936848i	$-0.386271\pi$
$643$	17.7368	0.699471	0.349736	+	0.936848i	$0.386271\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.920120	0.0361737	0.0180868	−	0.999836i	$-0.494242\pi$
$647$	0.920120	0.0361737	0.0180868	+	0.999836i	$0.494242\pi$
$648$	0	0
$649$	−19.4016	−0.761581
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.20275	0.360131	0.180066	−	0.983655i	$-0.442369\pi$
$653$	9.20275	0.360131	0.180066	+	0.983655i	$0.442369\pi$
$654$	0	0
$655$	22.1248	0.864486
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−49.3997	−1.92434	−0.962170	−	0.272448i	$-0.912167\pi$
$659$	−49.3997	−1.92434	−0.962170	+	0.272448i	$0.912167\pi$
$660$	0	0
$661$	49.9844	1.94417	0.972085	−	0.234631i	$-0.0753881\pi$
$661$	49.9844	1.94417	0.972085	+	0.234631i	$0.0753881\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.29055	0.204851
$668$	0	0
$669$	0	0
$670$	0	0
$671$	43.5070	1.67957
$672$	0	0
$673$	9.43474	0.363682	0.181841	−	0.983328i	$-0.441794\pi$
$673$	9.43474	0.363682	0.181841	+	0.983328i	$0.441794\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.7874	0.914226	0.457113	−	0.889409i	$-0.348884\pi$
$677$	23.7874	0.914226	0.457113	+	0.889409i	$0.348884\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.6375	−1.32537	−0.662683	−	0.748900i	$-0.730582\pi$
$683$	−34.6375	−1.32537	−0.662683	+	0.748900i	$0.730582\pi$
$684$	0	0
$685$	−5.44625	−0.208091
$686$	0	0
$687$	0	0
$688$	0	0
$689$	35.1111	1.33763
$690$	0	0
$691$	−35.9201	−1.36647	−0.683233	−	0.730201i	$-0.739427\pi$
$691$	−35.9201	−1.36647	−0.683233	+	0.730201i	$0.739427\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	25.6123	0.971530
$696$	0	0
$697$	6.33327	0.239890
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.72529	−0.367319	−0.183659	−	0.982990i	$-0.558794\pi$
$701$	−9.72529	−0.367319	−0.183659	+	0.982990i	$0.558794\pi$
$702$	0	0
$703$	−10.9611	−0.413407
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.1111	−0.905511	−0.452756	−	0.891635i	$-0.649559\pi$
$709$	−24.1111	−0.905511	−0.452756	+	0.891635i	$0.649559\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.84789	−0.144105
$714$	0	0
$715$	38.9590	1.45698
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.3527	−0.945496	−0.472748	−	0.881198i	$-0.656738\pi$
$719$	−25.3527	−0.945496	−0.472748	+	0.881198i	$0.656738\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.05253	−0.150507
$726$	0	0
$727$	−47.5127	−1.76215	−0.881075	−	0.472977i	$-0.843179\pi$
$727$	−47.5127	−1.76215	−0.881075	+	0.472977i	$0.843179\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.44841	−0.349462
$732$	0	0
$733$	30.8364	1.13897	0.569484	−	0.822003i	$-0.307143\pi$
$733$	30.8364	1.13897	0.569484	+	0.822003i	$0.307143\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−52.3275	−1.92751
$738$	0	0
$739$	20.9942	0.772286	0.386143	−	0.922439i	$-0.373807\pi$
$739$	20.9942	0.772286	0.386143	+	0.922439i	$0.373807\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	45.3743	1.66462	0.832311	−	0.554309i	$-0.187018\pi$
$743$	45.3743	1.66462	0.832311	+	0.554309i	$0.187018\pi$
$744$	0	0
$745$	59.0118	2.16202
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.8421	0.468616	0.234308	−	0.972162i	$-0.424718\pi$
$751$	12.8421	0.468616	0.234308	+	0.972162i	$0.424718\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	26.9669	0.981426
$756$	0	0
$757$	12.9727	0.471499	0.235750	−	0.971814i	$-0.424245\pi$
$757$	12.9727	0.471499	0.235750	+	0.971814i	$0.424245\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41.4933	−1.50413	−0.752065	−	0.659088i	$-0.770942\pi$
$761$	−41.4933	−1.50413	−0.752065	+	0.659088i	$0.770942\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.1111	−0.509522
$768$	0	0
$769$	24.6900	0.890345	0.445173	−	0.895445i	$-0.353142\pi$
$769$	24.6900	0.890345	0.445173	+	0.895445i	$0.353142\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.6512	−1.42615	−0.713077	−	0.701086i	$-0.752699\pi$
$773$	−39.6512	−1.42615	−0.713077	+	0.701086i	$0.752699\pi$
$774$	0	0
$775$	2.94747	0.105876
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.76801	0.170832
$780$	0	0
$781$	70.3997	2.51910
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.90042	−0.281978
$786$	0	0
$787$	−18.0255	−0.642538	−0.321269	−	0.946988i	$-0.604109\pi$
$787$	−18.0255	−0.642538	−0.321269	+	0.946988i	$0.604109\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31.6432	1.12368
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.8791	−1.41259	−0.706295	−	0.707918i	$-0.749635\pi$
$797$	−39.8791	−1.41259	−0.706295	+	0.707918i	$0.749635\pi$
$798$	0	0
$799$	−20.1521	−0.712930
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.0586	−0.354959
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.1696	−1.58808	−0.794040	−	0.607865i	$-0.792026\pi$
$809$	−45.1696	−1.58808	−0.794040	+	0.607865i	$0.792026\pi$
$810$	0	0
$811$	1.70945	0.0600271	0.0300135	−	0.999549i	$-0.490445\pi$
$811$	1.70945	0.0600271	0.0300135	+	0.999549i	$0.490445\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.368267	−0.0128998
$816$	0	0
$817$	−7.11325	−0.248861
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−52.9144	−1.84672	−0.923362	−	0.383930i	$-0.874570\pi$
$821$	−52.9144	−1.84672	−0.923362	+	0.383930i	$0.874570\pi$
$822$	0	0
$823$	−31.2359	−1.08881	−0.544407	−	0.838821i	$-0.683245\pi$
$823$	−31.2359	−1.08881	−0.544407	+	0.838821i	$0.683245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.2243	1.22487	0.612435	−	0.790521i	$-0.290190\pi$
$827$	35.2243	1.22487	0.612435	+	0.790521i	$0.290190\pi$
$828$	0	0
$829$	−18.1168	−0.629224	−0.314612	−	0.949220i	$-0.601874\pi$
$829$	−18.1168	−0.629224	−0.314612	+	0.949220i	$0.601874\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−58.6766	−2.03059
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.5401	0.674598	0.337299	−	0.941398i	$-0.390487\pi$
$839$	19.5401	0.674598	0.337299	+	0.941398i	$0.390487\pi$
$840$	0	0
$841$	−6.87524	−0.237077
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.13844	−0.107965
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.61204	−0.329496
$852$	0	0
$853$	−17.3743	−0.594884	−0.297442	−	0.954740i	$-0.596134\pi$
$853$	−17.3743	−0.594884	−0.297442	+	0.954740i	$0.596134\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.5458	0.735992	0.367996	−	0.929827i	$-0.380044\pi$
$857$	21.5458	0.735992	0.367996	+	0.929827i	$0.380044\pi$
$858$	0	0
$859$	−17.3743	−0.592803	−0.296402	−	0.955063i	$-0.595787\pi$
$859$	−17.3743	−0.592803	−0.296402	+	0.955063i	$0.595787\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.4211	−0.593020	−0.296510	−	0.955030i	$-0.595823\pi$
$863$	−17.4211	−0.593020	−0.296510	+	0.955030i	$0.595823\pi$
$864$	0	0
$865$	−16.8342	−0.572381
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−68.3549	−2.31878
$870$	0	0
$871$	−38.0586	−1.28956
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.9981	1.58701	0.793507	−	0.608562i	$-0.208253\pi$
$877$	46.9981	1.58701	0.793507	+	0.608562i	$0.208253\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.1715	0.511142	0.255571	−	0.966790i	$-0.417737\pi$
$881$	15.1715	0.511142	0.255571	+	0.966790i	$0.417737\pi$
$882$	0	0
$883$	−32.4660	−1.09257	−0.546283	−	0.837601i	$-0.683958\pi$
$883$	−32.4660	−1.09257	−0.546283	+	0.837601i	$0.683958\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.86948	−0.0963477	−0.0481738	−	0.998839i	$-0.515340\pi$
$887$	−2.86948	−0.0963477	−0.0481738	+	0.998839i	$0.515340\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.1715	−0.507696
$894$	0	0
$895$	35.5847	1.18947
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.0917	−0.536687
$900$	0	0
$901$	−17.4854	−0.582522
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−41.4132	−1.37662
$906$	0	0
$907$	−32.4328	−1.07692	−0.538458	−	0.842653i	$-0.680993\pi$
$907$	−32.4328	−1.07692	−0.538458	+	0.842653i	$0.680993\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.2438	1.10142	0.550708	−	0.834698i	$-0.314358\pi$
$911$	33.2438	1.10142	0.550708	+	0.834698i	$0.314358\pi$
$912$	0	0
$913$	−39.6101	−1.31090
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−28.7505	−0.948391	−0.474195	−	0.880420i	$-0.657261\pi$
$919$	−28.7505	−0.948391	−0.474195	+	0.880420i	$0.657261\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	51.2028	1.68536
$924$	0	0
$925$	7.36278	0.242087
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−54.5458	−1.78959	−0.894795	−	0.446477i	$-0.852679\pi$
$929$	−54.5458	−1.78959	−0.894795	+	0.446477i	$0.852679\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.4016	−0.634501
$936$	0	0
$937$	−28.6979	−0.937521	−0.468760	−	0.883325i	$-0.655299\pi$
$937$	−28.6979	−0.937521	−0.468760	+	0.883325i	$0.655299\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−19.2963	−0.629042	−0.314521	−	0.949251i	$-0.601844\pi$
$941$	−19.2963	−0.629042	−0.314521	+	0.949251i	$0.601844\pi$
$942$	0	0
$943$	4.18116	0.136157
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.5322	−0.342249	−0.171125	−	0.985249i	$-0.554740\pi$
$947$	−10.5322	−0.342249	−0.171125	+	0.985249i	$0.554740\pi$
$948$	0	0
$949$	−7.31573	−0.237479
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.1970	−0.621852	−0.310926	−	0.950434i	$-0.600639\pi$
$953$	−19.1970	−0.621852	−0.310926	+	0.950434i	$0.600639\pi$
$954$	0	0
$955$	55.6375	1.80039
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.2963	−0.622461
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.0115	0.418855
$966$	0	0
$967$	−15.4854	−0.497976	−0.248988	−	0.968507i	$-0.580098\pi$
$967$	−15.4854	−0.497976	−0.248988	+	0.968507i	$0.580098\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.80903	0.0580545	0.0290273	−	0.999579i	$-0.490759\pi$
$971$	1.80903	0.0580545	0.0290273	+	0.999579i	$0.490759\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.8985	1.21248	0.606241	−	0.795281i	$-0.292677\pi$
$977$	37.8985	1.21248	0.606241	+	0.795281i	$0.292677\pi$
$978$	0	0
$979$	−75.6903	−2.41907
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38.1111	−1.21556	−0.607778	−	0.794107i	$-0.707939\pi$
$983$	−38.1111	−1.21556	−0.607778	+	0.794107i	$0.707939\pi$
$984$	0	0
$985$	−8.66484	−0.276085
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.23775	−0.198349
$990$	0	0
$991$	11.7896	0.374509	0.187254	−	0.982311i	$-0.440041\pi$
$991$	11.7896	0.374509	0.187254	+	0.982311i	$0.440041\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.9475	0.537271
$996$	0	0
$997$	−14.8069	−0.468938	−0.234469	−	0.972124i	$-0.575335\pi$
$997$	−14.8069	−0.468938	−0.234469	+	0.972124i	$0.575335\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.v.1.3		3
3.2	odd	2		5292.2.a.w.1.1		3
7.3	odd	6		756.2.k.e.541.3	yes	6
7.5	odd	6		756.2.k.e.109.3	✓	6
7.6	odd	2		5292.2.a.x.1.1		3
21.5	even	6		756.2.k.f.109.1	yes	6
21.17	even	6		756.2.k.f.541.1	yes	6
21.20	even	2		5292.2.a.u.1.3		3
63.5	even	6		2268.2.i.k.865.1		6
63.31	odd	6		2268.2.l.k.541.1		6
63.38	even	6		2268.2.i.k.2053.1		6
63.40	odd	6		2268.2.i.j.865.3		6
63.47	even	6		2268.2.l.j.109.3		6
63.52	odd	6		2268.2.i.j.2053.3		6
63.59	even	6		2268.2.l.j.541.3		6
63.61	odd	6		2268.2.l.k.109.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.e.109.3	✓	6	7.5	odd	6
756.2.k.e.541.3	yes	6	7.3	odd	6
756.2.k.f.109.1	yes	6	21.5	even	6
756.2.k.f.541.1	yes	6	21.17	even	6
2268.2.i.j.865.3		6	63.40	odd	6
2268.2.i.j.2053.3		6	63.52	odd	6
2268.2.i.k.865.1		6	63.5	even	6
2268.2.i.k.2053.1		6	63.38	even	6
2268.2.l.j.109.3		6	63.47	even	6
2268.2.l.j.541.3		6	63.59	even	6
2268.2.l.k.109.1		6	63.61	odd	6
2268.2.l.k.541.1		6	63.31	odd	6
5292.2.a.u.1.3		3	21.20	even	2
5292.2.a.v.1.3		3	1.1	even	1	trivial
5292.2.a.w.1.1		3	3.2	odd	2
5292.2.a.x.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q+2.42107 q^{5} +O(q^{10})\)	\(q+2.42107 q^{5} +4.70370 q^{11} +3.42107 q^{13} -1.70370 q^{17} -1.28263 q^{19} -1.12476 q^{23} +0.861564 q^{25} -4.70370 q^{29} +3.42107 q^{31} +8.54583 q^{37} -3.71737 q^{41} +5.54583 q^{43} +11.8285 q^{47} +10.2632 q^{53} +11.3880 q^{55} -4.12476 q^{59} +9.24953 q^{61} +8.28263 q^{65} -11.1248 q^{67} +14.9669 q^{71} -2.13844 q^{73} -14.5322 q^{79} -8.42107 q^{83} -4.12476 q^{85} -16.0917 q^{89} -3.10533 q^{95} -16.2495 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5}+O(q^{10}) \) 3 * q - q^5	\( 3 q - q^{5} + 5 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 14 q^{23} + 10 q^{25} - 5 q^{29} + 2 q^{31} - 12 q^{41} - 9 q^{43} + 9 q^{47} + 6 q^{53} - 8 q^{55} + 5 q^{59} - 7 q^{61} + 24 q^{65} - 16 q^{67} + 11 q^{71} + q^{73} - 8 q^{79} - 17 q^{83} + 5 q^{85} + 3 q^{89} + 32 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - q^5 + 5 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 14 * q^23 + 10 * q^25 - 5 * q^29 + 2 * q^31 - 12 * q^41 - 9 * q^43 + 9 * q^47 + 6 * q^53 - 8 * q^55 + 5 * q^59 - 7 * q^61 + 24 * q^65 - 16 * q^67 + 11 * q^71 + q^73 - 8 * q^79 - 17 * q^83 + 5 * q^85 + 3 * q^89 + 32 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more