LMFDB - Embedded newform 7803.2.a.cg.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,0,24,20,0,0,0,0,0,8,0,4,16,0,24,0,0,4,48,0,0,36,0,28,0, 0,0,64,0,0,0,0,0,0,0,0,0,0,0,36,0,4,16,0,0,0,0,24,0,0,16,0,0,20,80,0,0, 0,0,0,-16,0,-24,72,0,16,0,0,-48,72,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.3072686972$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	7803.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.533361 q^{2} -1.71553 q^{4} -2.01564 q^{5} -2.52795 q^{7} +1.98172 q^{8} +1.07506 q^{10} +2.43279 q^{11} +3.12835 q^{13} +1.34831 q^{14} +2.37408 q^{16} +5.61764 q^{19} +3.45788 q^{20} -1.29756 q^{22} +2.17884 q^{23} -0.937211 q^{25} -1.66854 q^{26} +4.33677 q^{28} +4.75129 q^{29} +4.46744 q^{31} -5.22968 q^{32} +5.09543 q^{35} +4.68703 q^{37} -2.99623 q^{38} -3.99442 q^{40} +9.39479 q^{41} -9.70418 q^{43} -4.17352 q^{44} -1.16211 q^{46} -10.4884 q^{47} -0.609457 q^{49} +0.499872 q^{50} -5.36677 q^{52} +2.83803 q^{53} -4.90362 q^{55} -5.00969 q^{56} -2.53415 q^{58} +3.50470 q^{59} -7.88955 q^{61} -2.38276 q^{62} -1.95885 q^{64} -6.30562 q^{65} -4.16861 q^{67} -2.71771 q^{70} -4.51044 q^{71} +12.5366 q^{73} -2.49988 q^{74} -9.63720 q^{76} -6.14998 q^{77} +4.52640 q^{79} -4.78528 q^{80} -5.01082 q^{82} -10.8255 q^{83} +5.17583 q^{86} +4.82111 q^{88} -16.2213 q^{89} -7.90833 q^{91} -3.73785 q^{92} +5.59411 q^{94} -11.3231 q^{95} +0.802837 q^{97} +0.325061 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} + 20 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{14} + 24 q^{16} + 4 q^{19} + 48 q^{20} + 36 q^{23} + 28 q^{25} + 64 q^{29} + 36 q^{41} + 4 q^{43} + 16 q^{44} + 24 q^{49} + 16 q^{52} + 20 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) $ 24 * q + 24 * q^4 + 20 * q^5 + 8 * q^11 + 4 * q^13 + 16 * q^14 + 24 * q^16 + 4 * q^19 + 48 * q^20 + 36 * q^23 + 28 * q^25 + 64 * q^29 + 36 * q^41 + 4 * q^43 + 16 * q^44 + 24 * q^49 + 16 * q^52 + 20 * q^55 + 80 * q^56 - 16 * q^62 - 24 * q^64 + 72 * q^65 + 16 * q^67 - 48 * q^70 + 72 * q^71 + 80 * q^74 - 40 * q^76 + 104 * q^80 + 144 * q^92 - 72 * q^94 - 12 * q^95

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 $

$2$	−0.533361	−0.377143	−0.188572	−	0.982059i	$-0.560386\pi$
$2$	−0.533361	−0.377143	−0.188572	+	0.982059i	$0.560386\pi$
$3$	0	0
$3$	0	0
$4$	−1.71553	−0.857763
$5$	−2.01564	−0.901420	−0.450710	−	0.892670i	$-0.648829\pi$
$5$	−2.01564	−0.901420	−0.450710	+	0.892670i	$0.648829\pi$
$6$	0	0
$7$	−2.52795	−0.955476	−0.477738	−	0.878502i	$-0.658543\pi$
$7$	−2.52795	−0.955476	−0.477738	+	0.878502i	$0.658543\pi$
$8$	1.98172	0.700643
$9$	0	0
$10$	1.07506	0.339965
$11$	2.43279	0.733515	0.366757	−	0.930317i	$-0.380468\pi$
$11$	2.43279	0.733515	0.366757	+	0.930317i	$0.380468\pi$
$12$	0	0
$13$	3.12835	0.867649	0.433824	−	0.900997i	$-0.357164\pi$
$13$	3.12835	0.867649	0.433824	+	0.900997i	$0.357164\pi$
$14$	1.34831	0.360352
$15$	0	0
$16$	2.37408	0.593520
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.61764	1.28877	0.644387	−	0.764699i	$-0.277113\pi$
$19$	5.61764	1.28877	0.644387	+	0.764699i	$0.277113\pi$
$20$	3.45788	0.773205
$21$	0	0
$22$	−1.29756	−0.276640
$23$	2.17884	0.454319	0.227160	−	0.973858i	$-0.427056\pi$
$23$	2.17884	0.454319	0.227160	+	0.973858i	$0.427056\pi$
$24$	0	0
$25$	−0.937211	−0.187442
$26$	−1.66854	−0.327228
$27$	0	0
$28$	4.33677	0.819572
$29$	4.75129	0.882293	0.441146	−	0.897435i	$-0.354572\pi$
$29$	4.75129	0.882293	0.441146	+	0.897435i	$0.354572\pi$
$30$	0	0
$31$	4.46744	0.802375	0.401188	−	0.915996i	$-0.368598\pi$
$31$	4.46744	0.802375	0.401188	+	0.915996i	$0.368598\pi$
$32$	−5.22968	−0.924485
$33$	0	0
$34$	0	0
$35$	5.09543	0.861285
$36$	0	0
$37$	4.68703	0.770544	0.385272	−	0.922803i	$-0.374108\pi$
$37$	4.68703	0.770544	0.385272	+	0.922803i	$0.374108\pi$
$38$	−2.99623	−0.486053
$39$	0	0
$40$	−3.99442	−0.631573
$41$	9.39479	1.46722	0.733610	−	0.679571i	$-0.237834\pi$
$41$	9.39479	1.46722	0.733610	+	0.679571i	$0.237834\pi$
$42$	0	0
$43$	−9.70418	−1.47987	−0.739937	−	0.672677i	$-0.765145\pi$
$43$	−9.70418	−1.47987	−0.739937	+	0.672677i	$0.765145\pi$
$44$	−4.17352	−0.629182
$45$	0	0
$46$	−1.16211	−0.171344
$47$	−10.4884	−1.52989	−0.764946	−	0.644094i	$-0.777234\pi$
$47$	−10.4884	−1.52989	−0.764946	+	0.644094i	$0.777234\pi$
$48$	0	0
$49$	−0.609457	−0.0870652
$50$	0.499872	0.0706926
$51$	0	0
$52$	−5.36677	−0.744237
$53$	2.83803	0.389834	0.194917	−	0.980820i	$-0.437556\pi$
$53$	2.83803	0.389834	0.194917	+	0.980820i	$0.437556\pi$
$54$	0	0
$55$	−4.90362	−0.661205
$56$	−5.00969	−0.669448
$57$	0	0
$58$	−2.53415	−0.332751
$59$	3.50470	0.456273	0.228137	−	0.973629i	$-0.426737\pi$
$59$	3.50470	0.456273	0.228137	+	0.973629i	$0.426737\pi$
$60$	0	0
$61$	−7.88955	−1.01015	−0.505077	−	0.863075i	$-0.668536\pi$
$61$	−7.88955	−1.01015	−0.505077	+	0.863075i	$0.668536\pi$
$62$	−2.38276	−0.302610
$63$	0	0
$64$	−1.95885	−0.244857
$65$	−6.30562	−0.782116
$66$	0	0
$67$	−4.16861	−0.509277	−0.254639	−	0.967036i	$-0.581957\pi$
$67$	−4.16861	−0.509277	−0.254639	+	0.967036i	$0.581957\pi$
$68$	0	0
$69$	0	0
$70$	−2.71771	−0.324828
$71$	−4.51044	−0.535291	−0.267645	−	0.963517i	$-0.586246\pi$
$71$	−4.51044	−0.535291	−0.267645	+	0.963517i	$0.586246\pi$
$72$	0	0
$73$	12.5366	1.46730	0.733651	−	0.679526i	$-0.237815\pi$
$73$	12.5366	1.46730	0.733651	+	0.679526i	$0.237815\pi$
$74$	−2.49988	−0.290605
$75$	0	0
$76$	−9.63720	−1.10546
$77$	−6.14998	−0.700856
$78$	0	0
$79$	4.52640	0.509260	0.254630	−	0.967039i	$-0.418046\pi$
$79$	4.52640	0.509260	0.254630	+	0.967039i	$0.418046\pi$
$80$	−4.78528	−0.535011
$81$	0	0
$82$	−5.01082	−0.553352
$83$	−10.8255	−1.18825	−0.594125	−	0.804373i	$-0.702501\pi$
$83$	−10.8255	−1.18825	−0.594125	+	0.804373i	$0.702501\pi$
$84$	0	0
$85$	0	0
$86$	5.17583	0.558124
$87$	0	0
$88$	4.82111	0.513932
$89$	−16.2213	−1.71945	−0.859726	−	0.510756i	$-0.829366\pi$
$89$	−16.2213	−1.71945	−0.859726	+	0.510756i	$0.829366\pi$
$90$	0	0
$91$	−7.90833	−0.829018
$92$	−3.73785	−0.389698
$93$	0	0
$94$	5.59411	0.576989
$95$	−11.3231	−1.16173
$96$	0	0
$97$	0.802837	0.0815158	0.0407579	−	0.999169i	$-0.487023\pi$
$97$	0.802837	0.0815158	0.0407579	+	0.999169i	$0.487023\pi$
$98$	0.325061	0.0328361
$99$	0	0
$100$	1.60781	0.160781
$101$	10.7867	1.07331	0.536657	−	0.843801i	$-0.319687\pi$
$101$	10.7867	1.07331	0.536657	+	0.843801i	$0.319687\pi$
$102$	0	0
$103$	13.2358	1.30416	0.652082	−	0.758149i	$-0.273896\pi$
$103$	13.2358	1.30416	0.652082	+	0.758149i	$0.273896\pi$
$104$	6.19951	0.607912
$105$	0	0
$106$	−1.51370	−0.147023
$107$	−8.20981	−0.793673	−0.396836	−	0.917889i	$-0.629892\pi$
$107$	−8.20981	−0.793673	−0.396836	+	0.917889i	$0.629892\pi$
$108$	0	0
$109$	0.228783	0.0219135	0.0109567	−	0.999940i	$-0.496512\pi$
$109$	0.228783	0.0219135	0.0109567	+	0.999940i	$0.496512\pi$
$110$	2.61540	0.249369
$111$	0	0
$112$	−6.00156	−0.567094
$113$	12.0126	1.13005	0.565024	−	0.825075i	$-0.308867\pi$
$113$	12.0126	1.13005	0.565024	+	0.825075i	$0.308867\pi$
$114$	0	0
$115$	−4.39175	−0.409533
$116$	−8.15096	−0.756798
$117$	0	0
$118$	−1.86927	−0.172081
$119$	0	0
$120$	0	0
$121$	−5.08152	−0.461956
$122$	4.20798	0.380973
$123$	0	0
$124$	−7.66400	−0.688248
$125$	11.9673	1.07038
$126$	0	0
$127$	−9.51499	−0.844319	−0.422160	−	0.906522i	$-0.638728\pi$
$127$	−9.51499	−0.844319	−0.422160	+	0.906522i	$0.638728\pi$
$128$	11.5041	1.01683
$129$	0	0
$130$	3.36317	0.294970
$131$	14.9234	1.30387	0.651934	−	0.758276i	$-0.273958\pi$
$131$	14.9234	1.30387	0.651934	+	0.758276i	$0.273958\pi$
$132$	0	0
$133$	−14.2011	−1.23139
$134$	2.22338	0.192070
$135$	0	0
$136$	0	0
$137$	14.1629	1.21002	0.605009	−	0.796219i	$-0.293170\pi$
$137$	14.1629	1.21002	0.605009	+	0.796219i	$0.293170\pi$
$138$	0	0
$139$	−15.4050	−1.30664	−0.653319	−	0.757083i	$-0.726624\pi$
$139$	−15.4050	−1.30664	−0.653319	+	0.757083i	$0.726624\pi$
$140$	−8.74135	−0.738779
$141$	0	0
$142$	2.40569	0.201881
$143$	7.61063	0.636433
$144$	0	0
$145$	−9.57687	−0.795316
$146$	−6.68656	−0.553383
$147$	0	0
$148$	−8.04073	−0.660944
$149$	6.62192	0.542489	0.271244	−	0.962511i	$-0.412565\pi$
$149$	6.62192	0.542489	0.271244	+	0.962511i	$0.412565\pi$
$150$	0	0
$151$	−17.0531	−1.38776	−0.693882	−	0.720089i	$-0.744101\pi$
$151$	−17.0531	−1.38776	−0.693882	+	0.720089i	$0.744101\pi$
$152$	11.1326	0.902971
$153$	0	0
$154$	3.28016	0.264323
$155$	−9.00472	−0.723277
$156$	0	0
$157$	14.5703	1.16284	0.581418	−	0.813605i	$-0.302498\pi$
$157$	14.5703	1.16284	0.581418	+	0.813605i	$0.302498\pi$
$158$	−2.41421	−0.192064
$159$	0	0
$160$	10.5411	0.833349
$161$	−5.50800	−0.434091
$162$	0	0
$163$	5.39557	0.422614	0.211307	−	0.977420i	$-0.432228\pi$
$163$	5.39557	0.422614	0.211307	+	0.977420i	$0.432228\pi$
$164$	−16.1170	−1.25853
$165$	0	0
$166$	5.77388	0.448140
$167$	12.4575	0.963993	0.481996	−	0.876173i	$-0.339912\pi$
$167$	12.4575	0.963993	0.481996	+	0.876173i	$0.339912\pi$
$168$	0	0
$169$	−3.21341	−0.247185
$170$	0	0
$171$	0	0
$172$	16.6478	1.26938
$173$	2.67379	0.203284	0.101642	−	0.994821i	$-0.467590\pi$
$173$	2.67379	0.203284	0.101642	+	0.994821i	$0.467590\pi$
$174$	0	0
$175$	2.36922	0.179097
$176$	5.77565	0.435356
$177$	0	0
$178$	8.65180	0.648480
$179$	14.0084	1.04704	0.523519	−	0.852014i	$-0.324619\pi$
$179$	14.0084	1.04704	0.523519	+	0.852014i	$0.324619\pi$
$180$	0	0
$181$	−13.3634	−0.993291	−0.496646	−	0.867953i	$-0.665435\pi$
$181$	−13.3634	−0.993291	−0.496646	+	0.867953i	$0.665435\pi$
$182$	4.21800	0.312659
$183$	0	0
$184$	4.31784	0.318316
$185$	−9.44735	−0.694583
$186$	0	0
$187$	0	0
$188$	17.9931	1.31228
$189$	0	0
$190$	6.03931	0.438138
$191$	−20.9983	−1.51938	−0.759691	−	0.650285i	$-0.774650\pi$
$191$	−20.9983	−1.51938	−0.759691	+	0.650285i	$0.774650\pi$
$192$	0	0
$193$	5.11341	0.368072	0.184036	−	0.982920i	$-0.441084\pi$
$193$	5.11341	0.368072	0.184036	+	0.982920i	$0.441084\pi$
$194$	−0.428202	−0.0307431
$195$	0	0
$196$	1.04554	0.0746813
$197$	3.83971	0.273568	0.136784	−	0.990601i	$-0.456323\pi$
$197$	3.83971	0.273568	0.136784	+	0.990601i	$0.456323\pi$
$198$	0	0
$199$	−7.22725	−0.512326	−0.256163	−	0.966634i	$-0.582458\pi$
$199$	−7.22725	−0.512326	−0.256163	+	0.966634i	$0.582458\pi$
$200$	−1.85729	−0.131330
$201$	0	0
$202$	−5.75319	−0.404793
$203$	−12.0110	−0.843010
$204$	0	0
$205$	−18.9365	−1.32258
$206$	−7.05947	−0.491857
$207$	0	0
$208$	7.42696	0.514967
$209$	13.6665	0.945335
$210$	0	0
$211$	−12.5596	−0.864636	−0.432318	−	0.901721i	$-0.642304\pi$
$211$	−12.5596	−0.864636	−0.432318	+	0.901721i	$0.642304\pi$
$212$	−4.86872	−0.334385
$213$	0	0
$214$	4.37880	0.299328
$215$	19.5601	1.33399
$216$	0	0
$217$	−11.2935	−0.766650
$218$	−0.122024	−0.00826451
$219$	0	0
$220$	8.41230	0.567157
$221$	0	0
$222$	0	0
$223$	4.79698	0.321229	0.160615	−	0.987017i	$-0.448652\pi$
$223$	4.79698	0.321229	0.160615	+	0.987017i	$0.448652\pi$
$224$	13.2204	0.883323
$225$	0	0
$226$	−6.40704	−0.426190
$227$	22.4744	1.49168	0.745839	−	0.666127i	$-0.232049\pi$
$227$	22.4744	1.49168	0.745839	+	0.666127i	$0.232049\pi$
$228$	0	0
$229$	−2.28961	−0.151302	−0.0756510	−	0.997134i	$-0.524103\pi$
$229$	−2.28961	−0.151302	−0.0756510	+	0.997134i	$0.524103\pi$
$230$	2.34239	0.154452
$231$	0	0
$232$	9.41572	0.618172
$233$	24.9870	1.63696	0.818478	−	0.574538i	$-0.194818\pi$
$233$	24.9870	1.63696	0.818478	+	0.574538i	$0.194818\pi$
$234$	0	0
$235$	21.1408	1.37908
$236$	−6.01241	−0.391374
$237$	0	0
$238$	0	0
$239$	16.8913	1.09261	0.546304	−	0.837587i	$-0.316034\pi$
$239$	16.8913	1.09261	0.546304	+	0.837587i	$0.316034\pi$
$240$	0	0
$241$	−29.9907	−1.93187	−0.965936	−	0.258783i	$-0.916679\pi$
$241$	−29.9907	−1.93187	−0.965936	+	0.258783i	$0.916679\pi$
$242$	2.71029	0.174224
$243$	0	0
$244$	13.5347	0.866472
$245$	1.22844	0.0784823
$246$	0	0
$247$	17.5740	1.11820
$248$	8.85320	0.562178
$249$	0	0
$250$	−6.38287	−0.403688
$251$	24.1719	1.52572	0.762858	−	0.646566i	$-0.223795\pi$
$251$	24.1719	1.52572	0.762858	+	0.646566i	$0.223795\pi$
$252$	0	0
$253$	5.30066	0.333250
$254$	5.07493	0.318429
$255$	0	0
$256$	−2.21815	−0.138634
$257$	−26.7412	−1.66807	−0.834034	−	0.551712i	$-0.813975\pi$
$257$	−26.7412	−1.66807	−0.834034	+	0.551712i	$0.813975\pi$
$258$	0	0
$259$	−11.8486	−0.736236
$260$	10.8175	0.670870
$261$	0	0
$262$	−7.95959	−0.491745
$263$	2.34258	0.144449	0.0722247	−	0.997388i	$-0.476990\pi$
$263$	2.34258	0.144449	0.0722247	+	0.997388i	$0.476990\pi$
$264$	0	0
$265$	−5.72044	−0.351404
$266$	7.57433	0.464412
$267$	0	0
$268$	7.15136	0.436839
$269$	−17.2220	−1.05005	−0.525023	−	0.851088i	$-0.675943\pi$
$269$	−17.2220	−1.05005	−0.525023	+	0.851088i	$0.675943\pi$
$270$	0	0
$271$	24.8158	1.50745	0.753726	−	0.657189i	$-0.228254\pi$
$271$	24.8158	1.50745	0.753726	+	0.657189i	$0.228254\pi$
$272$	0	0
$273$	0	0
$274$	−7.55394	−0.456350
$275$	−2.28004	−0.137492
$276$	0	0
$277$	−18.5075	−1.11201	−0.556005	−	0.831179i	$-0.687666\pi$
$277$	−18.5075	−1.11201	−0.556005	+	0.831179i	$0.687666\pi$
$278$	8.21645	0.492790
$279$	0	0
$280$	10.0977	0.603453
$281$	−17.2627	−1.02981	−0.514903	−	0.857249i	$-0.672172\pi$
$281$	−17.2627	−1.02981	−0.514903	+	0.857249i	$0.672172\pi$
$282$	0	0
$283$	−20.7111	−1.23115	−0.615574	−	0.788079i	$-0.711076\pi$
$283$	−20.7111	−1.23115	−0.615574	+	0.788079i	$0.711076\pi$
$284$	7.73778	0.459153
$285$	0	0
$286$	−4.05922	−0.240027
$287$	−23.7496	−1.40189
$288$	0	0
$289$	0	0
$290$	5.10793	0.299948
$291$	0	0
$292$	−21.5069	−1.25860
$293$	23.8401	1.39275	0.696375	−	0.717678i	$-0.254795\pi$
$293$	23.8401	1.39275	0.696375	+	0.717678i	$0.254795\pi$
$294$	0	0
$295$	−7.06421	−0.411294
$296$	9.28838	0.539876
$297$	0	0
$298$	−3.53187	−0.204596
$299$	6.81618	0.394190
$300$	0	0
$301$	24.5317	1.41398
$302$	9.09548	0.523386
$303$	0	0
$304$	13.3367	0.764913
$305$	15.9025	0.910572
$306$	0	0
$307$	13.9729	0.797474	0.398737	−	0.917065i	$-0.369449\pi$
$307$	13.9729	0.797474	0.398737	+	0.917065i	$0.369449\pi$
$308$	10.5505	0.601168
$309$	0	0
$310$	4.80277	0.272779
$311$	−9.75466	−0.553136	−0.276568	−	0.960994i	$-0.589197\pi$
$311$	−9.75466	−0.553136	−0.276568	+	0.960994i	$0.589197\pi$
$312$	0	0
$313$	33.2415	1.87892	0.939461	−	0.342655i	$-0.111326\pi$
$313$	33.2415	1.87892	0.939461	+	0.342655i	$0.111326\pi$
$314$	−7.77123	−0.438556
$315$	0	0
$316$	−7.76516	−0.436825
$317$	7.34140	0.412334	0.206167	−	0.978517i	$-0.433901\pi$
$317$	7.34140	0.412334	0.206167	+	0.978517i	$0.433901\pi$
$318$	0	0
$319$	11.5589	0.647175
$320$	3.94834	0.220719
$321$	0	0
$322$	2.93775	0.163715
$323$	0	0
$324$	0	0
$325$	−2.93193	−0.162634
$326$	−2.87779	−0.159386
$327$	0	0
$328$	18.6178	1.02800
$329$	26.5142	1.46178
$330$	0	0
$331$	2.05218	0.112798	0.0563990	−	0.998408i	$-0.482038\pi$
$331$	2.05218	0.112798	0.0563990	+	0.998408i	$0.482038\pi$
$332$	18.5714	1.01924
$333$	0	0
$334$	−6.64436	−0.363563
$335$	8.40241	0.459073
$336$	0	0
$337$	−6.09966	−0.332270	−0.166135	−	0.986103i	$-0.553129\pi$
$337$	−6.09966	−0.332270	−0.166135	+	0.986103i	$0.553129\pi$
$338$	1.71391	0.0932243
$339$	0	0
$340$	0	0
$341$	10.8683	0.588554
$342$	0	0
$343$	19.2363	1.03866
$344$	−19.2309	−1.03686
$345$	0	0
$346$	−1.42609	−0.0766673
$347$	34.4147	1.84748	0.923738	−	0.383026i	$-0.125118\pi$
$347$	34.4147	1.84748	0.923738	+	0.383026i	$0.125118\pi$
$348$	0	0
$349$	−17.4294	−0.932976	−0.466488	−	0.884528i	$-0.654481\pi$
$349$	−17.4294	−0.932976	−0.466488	+	0.884528i	$0.654481\pi$
$350$	−1.26365	−0.0675451
$351$	0	0
$352$	−12.7227	−0.678123
$353$	35.9886	1.91548	0.957739	−	0.287638i	$-0.0928700\pi$
$353$	35.9886	1.91548	0.957739	+	0.287638i	$0.0928700\pi$
$354$	0	0
$355$	9.09141	0.482522
$356$	27.8280	1.47488
$357$	0	0
$358$	−7.47155	−0.394884
$359$	18.7253	0.988285	0.494142	−	0.869381i	$-0.335482\pi$
$359$	18.7253	0.988285	0.494142	+	0.869381i	$0.335482\pi$
$360$	0	0
$361$	12.5578	0.660939
$362$	7.12750	0.374613
$363$	0	0
$364$	13.5669	0.711101
$365$	−25.2693	−1.32266
$366$	0	0
$367$	24.2752	1.26716	0.633578	−	0.773679i	$-0.281586\pi$
$367$	24.2752	1.26716	0.633578	+	0.773679i	$0.281586\pi$
$368$	5.17274	0.269648
$369$	0	0
$370$	5.03885	0.261957
$371$	−7.17441	−0.372477
$372$	0	0
$373$	1.52714	0.0790723	0.0395361	−	0.999218i	$-0.487412\pi$
$373$	1.52714	0.0790723	0.0395361	+	0.999218i	$0.487412\pi$
$374$	0	0
$375$	0	0
$376$	−20.7851	−1.07191
$377$	14.8637	0.765520
$378$	0	0
$379$	−5.78502	−0.297156	−0.148578	−	0.988901i	$-0.547470\pi$
$379$	−5.78502	−0.297156	−0.148578	+	0.988901i	$0.547470\pi$
$380$	19.4251	0.996486
$381$	0	0
$382$	11.1997	0.573025
$383$	−27.9186	−1.42658	−0.713288	−	0.700871i	$-0.752795\pi$
$383$	−27.9186	−1.42658	−0.713288	+	0.700871i	$0.752795\pi$
$384$	0	0
$385$	12.3961	0.631765
$386$	−2.72730	−0.138816
$387$	0	0
$388$	−1.37729	−0.0699212
$389$	−5.79580	−0.293859	−0.146929	−	0.989147i	$-0.546939\pi$
$389$	−5.79580	−0.293859	−0.146929	+	0.989147i	$0.546939\pi$
$390$	0	0
$391$	0	0
$392$	−1.20777	−0.0610016
$393$	0	0
$394$	−2.04795	−0.103174
$395$	−9.12358	−0.459057
$396$	0	0
$397$	−25.6588	−1.28778	−0.643889	−	0.765119i	$-0.722680\pi$
$397$	−25.6588	−1.28778	−0.643889	+	0.765119i	$0.722680\pi$
$398$	3.85474	0.193221
$399$	0	0
$400$	−2.22501	−0.111251
$401$	−1.94594	−0.0971755	−0.0485877	−	0.998819i	$-0.515472\pi$
$401$	−1.94594	−0.0971755	−0.0485877	+	0.998819i	$0.515472\pi$
$402$	0	0
$403$	13.9757	0.696180
$404$	−18.5048	−0.920648
$405$	0	0
$406$	6.40622	0.317935
$407$	11.4026	0.565205
$408$	0	0
$409$	9.72007	0.480627	0.240313	−	0.970695i	$-0.422750\pi$
$409$	9.72007	0.480627	0.240313	+	0.970695i	$0.422750\pi$
$410$	10.1000	0.498803
$411$	0	0
$412$	−22.7064	−1.11866
$413$	−8.85972	−0.435958
$414$	0	0
$415$	21.8202	1.07111
$416$	−16.3603	−0.802129
$417$	0	0
$418$	−7.28921	−0.356527
$419$	−4.92897	−0.240796	−0.120398	−	0.992726i	$-0.538417\pi$
$419$	−4.92897	−0.240796	−0.120398	+	0.992726i	$0.538417\pi$
$420$	0	0
$421$	−33.8369	−1.64911	−0.824554	−	0.565783i	$-0.808574\pi$
$421$	−33.8369	−1.64911	−0.824554	+	0.565783i	$0.808574\pi$
$422$	6.69878	0.326092
$423$	0	0
$424$	5.62418	0.273134
$425$	0	0
$426$	0	0
$427$	19.9444	0.965177
$428$	14.0841	0.680783
$429$	0	0
$430$	−10.4326	−0.503104
$431$	−14.9941	−0.722242	−0.361121	−	0.932519i	$-0.617606\pi$
$431$	−14.9941	−0.722242	−0.361121	+	0.932519i	$0.617606\pi$
$432$	0	0
$433$	0.570805	0.0274311	0.0137156	−	0.999906i	$-0.495634\pi$
$433$	0.570805	0.0274311	0.0137156	+	0.999906i	$0.495634\pi$
$434$	6.02350	0.289137
$435$	0	0
$436$	−0.392483	−0.0187965
$437$	12.2399	0.585515
$438$	0	0
$439$	−25.4402	−1.21419	−0.607097	−	0.794628i	$-0.707666\pi$
$439$	−25.4402	−1.21419	−0.607097	+	0.794628i	$0.707666\pi$
$440$	−9.71760	−0.463268
$441$	0	0
$442$	0	0
$443$	1.27194	0.0604317	0.0302158	−	0.999543i	$-0.490381\pi$
$443$	1.27194	0.0604317	0.0302158	+	0.999543i	$0.490381\pi$
$444$	0	0
$445$	32.6962	1.54995
$446$	−2.55852	−0.121150
$447$	0	0
$448$	4.95189	0.233955
$449$	30.7646	1.45187	0.725936	−	0.687762i	$-0.241407\pi$
$449$	30.7646	1.45187	0.725936	+	0.687762i	$0.241407\pi$
$450$	0	0
$451$	22.8556	1.07623
$452$	−20.6079	−0.969313
$453$	0	0
$454$	−11.9870	−0.562576
$455$	15.9403	0.747293
$456$	0	0
$457$	23.4935	1.09898	0.549490	−	0.835500i	$-0.314822\pi$
$457$	23.4935	1.09898	0.549490	+	0.835500i	$0.314822\pi$
$458$	1.22119	0.0570625
$459$	0	0
$460$	7.53416	0.351282
$461$	−8.51912	−0.396775	−0.198387	−	0.980124i	$-0.563570\pi$
$461$	−8.51912	−0.396775	−0.198387	+	0.980124i	$0.563570\pi$
$462$	0	0
$463$	17.6689	0.821143	0.410571	−	0.911828i	$-0.365329\pi$
$463$	17.6689	0.821143	0.410571	+	0.911828i	$0.365329\pi$
$464$	11.2799	0.523658
$465$	0	0
$466$	−13.3271	−0.617367
$467$	6.27728	0.290478	0.145239	−	0.989397i	$-0.453605\pi$
$467$	6.27728	0.290478	0.145239	+	0.989397i	$0.453605\pi$
$468$	0	0
$469$	10.5381	0.486602
$470$	−11.2757	−0.520109
$471$	0	0
$472$	6.94533	0.319685
$473$	−23.6083	−1.08551
$474$	0	0
$475$	−5.26491	−0.241571
$476$	0	0
$477$	0	0
$478$	−9.00917	−0.412070
$479$	33.8395	1.54617	0.773083	−	0.634304i	$-0.218713\pi$
$479$	33.8395	1.54617	0.773083	+	0.634304i	$0.218713\pi$
$480$	0	0
$481$	14.6627	0.668561
$482$	15.9959	0.728592
$483$	0	0
$484$	8.71748	0.396249
$485$	−1.61823	−0.0734799
$486$	0	0
$487$	−17.3297	−0.785283	−0.392641	−	0.919692i	$-0.628439\pi$
$487$	−17.3297	−0.785283	−0.392641	+	0.919692i	$0.628439\pi$
$488$	−15.6349	−0.707757
$489$	0	0
$490$	−0.655204	−0.0295991
$491$	−18.8939	−0.852668	−0.426334	−	0.904566i	$-0.640195\pi$
$491$	−18.8939	−0.852668	−0.426334	+	0.904566i	$0.640195\pi$
$492$	0	0
$493$	0	0
$494$	−9.37326	−0.421723
$495$	0	0
$496$	10.6061	0.476226
$497$	11.4022	0.511458
$498$	0	0
$499$	−6.48069	−0.290116	−0.145058	−	0.989423i	$-0.546337\pi$
$499$	−6.48069	−0.290116	−0.145058	+	0.989423i	$0.546337\pi$
$500$	−20.5301	−0.918136
$501$	0	0
$502$	−12.8924	−0.575414
$503$	16.6622	0.742930	0.371465	−	0.928447i	$-0.378856\pi$
$503$	16.6622	0.742930	0.371465	+	0.928447i	$0.378856\pi$
$504$	0	0
$505$	−21.7420	−0.967506
$506$	−2.82717	−0.125683
$507$	0	0
$508$	16.3232	0.724226
$509$	19.1265	0.847767	0.423884	−	0.905717i	$-0.360667\pi$
$509$	19.1265	0.847767	0.423884	+	0.905717i	$0.360667\pi$
$510$	0	0
$511$	−31.6920	−1.40197
$512$	−21.8252	−0.964546
$513$	0	0
$514$	14.2627	0.629101
$515$	−26.6786	−1.17560
$516$	0	0
$517$	−25.5161	−1.12220
$518$	6.31958	0.277667
$519$	0	0
$520$	−12.4960	−0.547984
$521$	−14.3864	−0.630279	−0.315139	−	0.949045i	$-0.602051\pi$
$521$	−14.3864	−0.630279	−0.315139	+	0.949045i	$0.602051\pi$
$522$	0	0
$523$	−6.11004	−0.267173	−0.133587	−	0.991037i	$-0.542649\pi$
$523$	−6.11004	−0.267173	−0.133587	+	0.991037i	$0.542649\pi$
$524$	−25.6015	−1.11841
$525$	0	0
$526$	−1.24944	−0.0544781
$527$	0	0
$528$	0	0
$529$	−18.2527	−0.793594
$530$	3.05106	0.132530
$531$	0	0
$532$	24.3624	1.05624
$533$	29.3902	1.27303
$534$	0	0
$535$	16.5480	0.715432
$536$	−8.26101	−0.356821
$537$	0	0
$538$	9.18557	0.396018
$539$	−1.48268	−0.0638636
$540$	0	0
$541$	28.6602	1.23220	0.616099	−	0.787669i	$-0.288712\pi$
$541$	28.6602	1.23220	0.616099	+	0.787669i	$0.288712\pi$
$542$	−13.2358	−0.568526
$543$	0	0
$544$	0	0
$545$	−0.461144	−0.0197532
$546$	0	0
$547$	−16.9731	−0.725716	−0.362858	−	0.931844i	$-0.618199\pi$
$547$	−16.9731	−0.725716	−0.362858	+	0.931844i	$0.618199\pi$
$548$	−24.2968	−1.03791
$549$	0	0
$550$	1.21609	0.0518540
$551$	26.6910	1.13708
$552$	0	0
$553$	−11.4425	−0.486586
$554$	9.87120	0.419387
$555$	0	0
$556$	26.4277	1.12079
$557$	11.2465	0.476530	0.238265	−	0.971200i	$-0.423421\pi$
$557$	11.2465	0.476530	0.238265	+	0.971200i	$0.423421\pi$
$558$	0	0
$559$	−30.3581	−1.28401
$560$	12.0970	0.511190
$561$	0	0
$562$	9.20724	0.388384
$563$	−1.34971	−0.0568836	−0.0284418	−	0.999595i	$-0.509055\pi$
$563$	−1.34971	−0.0568836	−0.0284418	+	0.999595i	$0.509055\pi$
$564$	0	0
$565$	−24.2130	−1.01865
$566$	11.0465	0.464319
$567$	0	0
$568$	−8.93842	−0.375048
$569$	−0.494896	−0.0207471	−0.0103736	−	0.999946i	$-0.503302\pi$
$569$	−0.494896	−0.0207471	−0.0103736	+	0.999946i	$0.503302\pi$
$570$	0	0
$571$	24.9578	1.04445	0.522225	−	0.852808i	$-0.325102\pi$
$571$	24.9578	1.04445	0.522225	+	0.852808i	$0.325102\pi$
$572$	−13.0562	−0.545909
$573$	0	0
$574$	12.6671	0.528715
$575$	−2.04203	−0.0851586
$576$	0	0
$577$	−23.0850	−0.961043	−0.480521	−	0.876983i	$-0.659553\pi$
$577$	−23.0850	−0.961043	−0.480521	+	0.876983i	$0.659553\pi$
$578$	0	0
$579$	0	0
$580$	16.4294	0.682193
$581$	27.3663	1.13534
$582$	0	0
$583$	6.90435	0.285949
$584$	24.8441	1.02806
$585$	0	0
$586$	−12.7154	−0.525267
$587$	20.9646	0.865303	0.432652	−	0.901561i	$-0.357578\pi$
$587$	20.9646	0.865303	0.432652	+	0.901561i	$0.357578\pi$
$588$	0	0
$589$	25.0964	1.03408
$590$	3.76777	0.155117
$591$	0	0
$592$	11.1274	0.457333
$593$	−27.5149	−1.12990	−0.564951	−	0.825124i	$-0.691105\pi$
$593$	−27.5149	−1.12990	−0.564951	+	0.825124i	$0.691105\pi$
$594$	0	0
$595$	0	0
$596$	−11.3601	−0.465327
$597$	0	0
$598$	−3.63549	−0.148666
$599$	−43.8786	−1.79283	−0.896415	−	0.443215i	$-0.853838\pi$
$599$	−43.8786	−1.79283	−0.896415	+	0.443215i	$0.853838\pi$
$600$	0	0
$601$	43.3759	1.76934	0.884670	−	0.466218i	$-0.154384\pi$
$601$	43.3759	1.76934	0.884670	+	0.466218i	$0.154384\pi$
$602$	−13.0843	−0.533275
$603$	0	0
$604$	29.2551	1.19037
$605$	10.2425	0.416417
$606$	0	0
$607$	21.0232	0.853307	0.426653	−	0.904415i	$-0.359692\pi$
$607$	21.0232	0.853307	0.426653	+	0.904415i	$0.359692\pi$
$608$	−29.3784	−1.19145
$609$	0	0
$610$	−8.48175	−0.343416
$611$	−32.8114	−1.32741
$612$	0	0
$613$	30.5509	1.23394	0.616969	−	0.786987i	$-0.288360\pi$
$613$	30.5509	1.23394	0.616969	+	0.786987i	$0.288360\pi$
$614$	−7.45259	−0.300762
$615$	0	0
$616$	−12.1875	−0.491050
$617$	−8.08373	−0.325439	−0.162719	−	0.986672i	$-0.552026\pi$
$617$	−8.08373	−0.325439	−0.162719	+	0.986672i	$0.552026\pi$
$618$	0	0
$619$	−7.34432	−0.295193	−0.147597	−	0.989048i	$-0.547154\pi$
$619$	−7.34432	−0.295193	−0.147597	+	0.989048i	$0.547154\pi$
$620$	15.4478	0.620400
$621$	0	0
$622$	5.20276	0.208612
$623$	41.0066	1.64289
$624$	0	0
$625$	−19.4356	−0.777423
$626$	−17.7297	−0.708623
$627$	0	0
$628$	−24.9957	−0.997438
$629$	0	0
$630$	0	0
$631$	43.2402	1.72137	0.860683	−	0.509141i	$-0.170037\pi$
$631$	43.2402	1.72137	0.860683	+	0.509141i	$0.170037\pi$
$632$	8.97005	0.356810
$633$	0	0
$634$	−3.91562	−0.155509
$635$	19.1788	0.761086
$636$	0	0
$637$	−1.90660	−0.0755421
$638$	−6.16507	−0.244078
$639$	0	0
$640$	−23.1881	−0.916592
$641$	9.44281	0.372969	0.186484	−	0.982458i	$-0.440291\pi$
$641$	9.44281	0.372969	0.186484	+	0.982458i	$0.440291\pi$
$642$	0	0
$643$	−0.346171	−0.0136516	−0.00682582	−	0.999977i	$-0.502173\pi$
$643$	−0.346171	−0.0136516	−0.00682582	+	0.999977i	$0.502173\pi$
$644$	9.44912	0.372347
$645$	0	0
$646$	0	0
$647$	2.21802	0.0871996	0.0435998	−	0.999049i	$-0.486117\pi$
$647$	2.21802	0.0871996	0.0435998	+	0.999049i	$0.486117\pi$
$648$	0	0
$649$	8.52622	0.334683
$650$	1.56378	0.0613363
$651$	0	0
$652$	−9.25625	−0.362503
$653$	−3.81435	−0.149267	−0.0746336	−	0.997211i	$-0.523779\pi$
$653$	−3.81435	−0.149267	−0.0746336	+	0.997211i	$0.523779\pi$
$654$	0	0
$655$	−30.0802	−1.17533
$656$	22.3040	0.870824
$657$	0	0
$658$	−14.1416	−0.551299
$659$	−29.3352	−1.14274	−0.571368	−	0.820694i	$-0.693587\pi$
$659$	−29.3352	−1.14274	−0.571368	+	0.820694i	$0.693587\pi$
$660$	0	0
$661$	−6.32915	−0.246175	−0.123088	−	0.992396i	$-0.539280\pi$
$661$	−6.32915	−0.246175	−0.123088	+	0.992396i	$0.539280\pi$
$662$	−1.09455	−0.0425410
$663$	0	0
$664$	−21.4530	−0.832538
$665$	28.6243	1.11000
$666$	0	0
$667$	10.3523	0.400843
$668$	−21.3712	−0.826877
$669$	0	0
$670$	−4.48152	−0.173136
$671$	−19.1936	−0.740962
$672$	0	0
$673$	−27.6869	−1.06725	−0.533627	−	0.845720i	$-0.679171\pi$
$673$	−27.6869	−1.06725	−0.533627	+	0.845720i	$0.679171\pi$
$674$	3.25332	0.125313
$675$	0	0
$676$	5.51269	0.212026
$677$	38.8370	1.49263	0.746314	−	0.665594i	$-0.231822\pi$
$677$	38.8370	1.49263	0.746314	+	0.665594i	$0.231822\pi$
$678$	0	0
$679$	−2.02953	−0.0778864
$680$	0	0
$681$	0	0
$682$	−5.79675	−0.221969
$683$	3.74165	0.143170	0.0715852	−	0.997434i	$-0.477194\pi$
$683$	3.74165	0.143170	0.0715852	+	0.997434i	$0.477194\pi$
$684$	0	0
$685$	−28.5472	−1.09073
$686$	−10.2599	−0.391726
$687$	0	0
$688$	−23.0385	−0.878334
$689$	8.87837	0.338239
$690$	0	0
$691$	33.9745	1.29245	0.646226	−	0.763146i	$-0.276346\pi$
$691$	33.9745	1.29245	0.646226	+	0.763146i	$0.276346\pi$
$692$	−4.58695	−0.174370
$693$	0	0
$694$	−18.3554	−0.696763
$695$	31.0509	1.17783
$696$	0	0
$697$	0	0
$698$	9.29618	0.351866
$699$	0	0
$700$	−4.06447	−0.153622
$701$	19.8395	0.749327	0.374663	−	0.927161i	$-0.377758\pi$
$701$	19.8395	0.749327	0.374663	+	0.927161i	$0.377758\pi$
$702$	0	0
$703$	26.3301	0.993057
$704$	−4.76548	−0.179606
$705$	0	0
$706$	−19.1949	−0.722410
$707$	−27.2682	−1.02553
$708$	0	0
$709$	48.8784	1.83567	0.917833	−	0.396967i	$-0.129937\pi$
$709$	48.8784	1.83567	0.917833	+	0.396967i	$0.129937\pi$
$710$	−4.84901	−0.181980
$711$	0	0
$712$	−32.1460	−1.20472
$713$	9.73382	0.364535
$714$	0	0
$715$	−15.3403	−0.573694
$716$	−24.0318	−0.898111
$717$	0	0
$718$	−9.98736	−0.372725
$719$	4.65960	0.173774	0.0868869	−	0.996218i	$-0.472308\pi$
$719$	4.65960	0.173774	0.0868869	+	0.996218i	$0.472308\pi$
$720$	0	0
$721$	−33.4595	−1.24610
$722$	−6.69787	−0.249269
$723$	0	0
$724$	22.9252	0.852008
$725$	−4.45296	−0.165379
$726$	0	0
$727$	34.8089	1.29099	0.645495	−	0.763765i	$-0.276651\pi$
$727$	34.8089	1.29099	0.645495	+	0.763765i	$0.276651\pi$
$728$	−15.6721	−0.580846
$729$	0	0
$730$	13.4777	0.498831
$731$	0	0
$732$	0	0
$733$	−20.2207	−0.746867	−0.373434	−	0.927657i	$-0.621820\pi$
$733$	−20.2207	−0.746867	−0.373434	+	0.927657i	$0.621820\pi$
$734$	−12.9475	−0.477899
$735$	0	0
$736$	−11.3946	−0.420011
$737$	−10.1414	−0.373562
$738$	0	0
$739$	−20.7563	−0.763533	−0.381767	−	0.924259i	$-0.624684\pi$
$739$	−20.7563	−0.763533	−0.381767	+	0.924259i	$0.624684\pi$
$740$	16.2072	0.595788
$741$	0	0
$742$	3.82655	0.140477
$743$	−14.0213	−0.514391	−0.257196	−	0.966359i	$-0.582798\pi$
$743$	−14.0213	−0.514391	−0.257196	+	0.966359i	$0.582798\pi$
$744$	0	0
$745$	−13.3474	−0.489010
$746$	−0.814517	−0.0298216
$747$	0	0
$748$	0	0
$749$	20.7540	0.758335
$750$	0	0
$751$	33.7354	1.23102	0.615512	−	0.788128i	$-0.288949\pi$
$751$	33.7354	1.23102	0.615512	+	0.788128i	$0.288949\pi$
$752$	−24.9003	−0.908022
$753$	0	0
$754$	−7.92773	−0.288711
$755$	34.3729	1.25096
$756$	0	0
$757$	37.1888	1.35165	0.675824	−	0.737063i	$-0.263788\pi$
$757$	37.1888	1.35165	0.675824	+	0.737063i	$0.263788\pi$
$758$	3.08550	0.112071
$759$	0	0
$760$	−22.4392	−0.813956
$761$	−10.1020	−0.366198	−0.183099	−	0.983094i	$-0.558613\pi$
$761$	−10.1020	−0.366198	−0.183099	+	0.983094i	$0.558613\pi$
$762$	0	0
$763$	−0.578353	−0.0209378
$764$	36.0231	1.30327
$765$	0	0
$766$	14.8907	0.538023
$767$	10.9639	0.395885
$768$	0	0
$769$	−51.1798	−1.84559	−0.922796	−	0.385288i	$-0.874102\pi$
$769$	−51.1798	−1.84559	−0.922796	+	0.385288i	$0.874102\pi$
$770$	−6.61162	−0.238266
$771$	0	0
$772$	−8.77219	−0.315718
$773$	21.0914	0.758604	0.379302	−	0.925273i	$-0.376164\pi$
$773$	21.0914	0.758604	0.379302	+	0.925273i	$0.376164\pi$
$774$	0	0
$775$	−4.18693	−0.150399
$776$	1.59100	0.0571135
$777$	0	0
$778$	3.09125	0.110827
$779$	52.7765	1.89092
$780$	0	0
$781$	−10.9730	−0.392644
$782$	0	0
$783$	0	0
$784$	−1.44690	−0.0516750
$785$	−29.3684	−1.04820
$786$	0	0
$787$	17.8351	0.635753	0.317877	−	0.948132i	$-0.397030\pi$
$787$	17.8351	0.635753	0.317877	+	0.948132i	$0.397030\pi$
$788$	−6.58713	−0.234657
$789$	0	0
$790$	4.86617	0.173130
$791$	−30.3672	−1.07973
$792$	0	0
$793$	−24.6813	−0.876458
$794$	13.6854	0.485677
$795$	0	0
$796$	12.3985	0.439455
$797$	−18.8627	−0.668150	−0.334075	−	0.942547i	$-0.608424\pi$
$797$	−18.8627	−0.668150	−0.334075	+	0.942547i	$0.608424\pi$
$798$	0	0
$799$	0	0
$800$	4.90131	0.173288
$801$	0	0
$802$	1.03789	0.0366491
$803$	30.4990	1.07629
$804$	0	0
$805$	11.1021	0.391299
$806$	−7.45410	−0.262560
$807$	0	0
$808$	21.3761	0.752009
$809$	−3.14126	−0.110441	−0.0552205	−	0.998474i	$-0.517586\pi$
$809$	−3.14126	−0.110441	−0.0552205	+	0.998474i	$0.517586\pi$
$810$	0	0
$811$	−47.4960	−1.66781	−0.833905	−	0.551907i	$-0.813900\pi$
$811$	−47.4960	−1.66781	−0.833905	+	0.551907i	$0.813900\pi$
$812$	20.6052	0.723102
$813$	0	0
$814$	−6.08169	−0.213163
$815$	−10.8755	−0.380953
$816$	0	0
$817$	−54.5145	−1.90722
$818$	−5.18431	−0.181265
$819$	0	0
$820$	32.4860	1.13446
$821$	14.7130	0.513486	0.256743	−	0.966480i	$-0.417351\pi$
$821$	14.7130	0.513486	0.256743	+	0.966480i	$0.417351\pi$
$822$	0	0
$823$	28.3774	0.989173	0.494587	−	0.869128i	$-0.335319\pi$
$823$	28.3774	0.989173	0.494587	+	0.869128i	$0.335319\pi$
$824$	26.2297	0.913753
$825$	0	0
$826$	4.72543	0.164419
$827$	−16.8952	−0.587504	−0.293752	−	0.955882i	$-0.594904\pi$
$827$	−16.8952	−0.587504	−0.293752	+	0.955882i	$0.594904\pi$
$828$	0	0
$829$	3.03136	0.105283	0.0526417	−	0.998613i	$-0.483236\pi$
$829$	3.03136	0.105283	0.0526417	+	0.998613i	$0.483236\pi$
$830$	−11.6380	−0.403963
$831$	0	0
$832$	−6.12798	−0.212450
$833$	0	0
$834$	0	0
$835$	−25.1098	−0.868962
$836$	−23.4453	−0.810873
$837$	0	0
$838$	2.62892	0.0908146
$839$	−37.4875	−1.29421	−0.647106	−	0.762400i	$-0.724021\pi$
$839$	−37.4875	−1.29421	−0.647106	+	0.762400i	$0.724021\pi$
$840$	0	0
$841$	−6.42523	−0.221560
$842$	18.0473	0.621950
$843$	0	0
$844$	21.5462	0.741652
$845$	6.47706	0.222818
$846$	0	0
$847$	12.8458	0.441388
$848$	6.73772	0.231374
$849$	0	0
$850$	0	0
$851$	10.2123	0.350073
$852$	0	0
$853$	−31.7823	−1.08821	−0.544103	−	0.839018i	$-0.683130\pi$
$853$	−31.7823	−1.08821	−0.544103	+	0.839018i	$0.683130\pi$
$854$	−10.6376	−0.364010
$855$	0	0
$856$	−16.2695	−0.556081
$857$	22.6117	0.772402	0.386201	−	0.922415i	$-0.373787\pi$
$857$	22.6117	0.772402	0.386201	+	0.922415i	$0.373787\pi$
$858$	0	0
$859$	36.9481	1.26065	0.630327	−	0.776330i	$-0.282921\pi$
$859$	36.9481	1.26065	0.630327	+	0.776330i	$0.282921\pi$
$860$	−33.5558	−1.14424
$861$	0	0
$862$	7.99729	0.272389
$863$	32.8262	1.11742	0.558709	−	0.829364i	$-0.311297\pi$
$863$	32.8262	1.11742	0.558709	+	0.829364i	$0.311297\pi$
$864$	0	0
$865$	−5.38938	−0.183245
$866$	−0.304445	−0.0103455
$867$	0	0
$868$	19.3742	0.657604
$869$	11.0118	0.373550
$870$	0	0
$871$	−13.0409	−0.441874
$872$	0.453384	0.0153535
$873$	0	0
$874$	−6.52830	−0.220823
$875$	−30.2527	−1.02273
$876$	0	0
$877$	−25.2628	−0.853065	−0.426532	−	0.904472i	$-0.640265\pi$
$877$	−25.2628	−0.853065	−0.426532	+	0.904472i	$0.640265\pi$
$878$	13.5688	0.457925
$879$	0	0
$880$	−11.6416	−0.392438
$881$	11.4521	0.385832	0.192916	−	0.981215i	$-0.438206\pi$
$881$	11.4521	0.385832	0.192916	+	0.981215i	$0.438206\pi$
$882$	0	0
$883$	−37.1590	−1.25050	−0.625251	−	0.780424i	$-0.715003\pi$
$883$	−37.1590	−1.25050	−0.625251	+	0.780424i	$0.715003\pi$
$884$	0	0
$885$	0	0
$886$	−0.678403	−0.0227914
$887$	−20.5460	−0.689868	−0.344934	−	0.938627i	$-0.612099\pi$
$887$	−20.5460	−0.689868	−0.344934	+	0.938627i	$0.612099\pi$
$888$	0	0
$889$	24.0535	0.806727
$890$	−17.4389	−0.584552
$891$	0	0
$892$	−8.22934	−0.275539
$893$	−58.9201	−1.97169
$894$	0	0
$895$	−28.2359	−0.943821
$896$	−29.0819	−0.971558
$897$	0	0
$898$	−16.4087	−0.547564
$899$	21.2261	0.707930
$900$	0	0
$901$	0	0
$902$	−12.1903	−0.405892
$903$	0	0
$904$	23.8055	0.791760
$905$	26.9357	0.895372
$906$	0	0
$907$	−32.9972	−1.09565	−0.547827	−	0.836592i	$-0.684545\pi$
$907$	−32.9972	−1.09565	−0.547827	+	0.836592i	$0.684545\pi$
$908$	−38.5554	−1.27951
$909$	0	0
$910$	−8.50194	−0.281837
$911$	−6.63185	−0.219723	−0.109861	−	0.993947i	$-0.535041\pi$
$911$	−6.63185	−0.219723	−0.109861	+	0.993947i	$0.535041\pi$
$912$	0	0
$913$	−26.3361	−0.871598
$914$	−12.5305	−0.414473
$915$	0	0
$916$	3.92789	0.129781
$917$	−37.7257	−1.24581
$918$	0	0
$919$	58.7318	1.93738	0.968692	−	0.248265i	$-0.0798604\pi$
$919$	58.7318	1.93738	0.968692	+	0.248265i	$0.0798604\pi$
$920$	−8.70320	−0.286936
$921$	0	0
$922$	4.54377	0.149641
$923$	−14.1103	−0.464445
$924$	0	0
$925$	−4.39274	−0.144432
$926$	−9.42390	−0.309689
$927$	0	0
$928$	−24.8477	−0.815666
$929$	49.7859	1.63342	0.816711	−	0.577047i	$-0.195795\pi$
$929$	49.7859	1.63342	0.816711	+	0.577047i	$0.195795\pi$
$930$	0	0
$931$	−3.42371	−0.112207
$932$	−42.8659	−1.40412
$933$	0	0
$934$	−3.34806	−0.109552
$935$	0	0
$936$	0	0
$937$	−22.6222	−0.739035	−0.369517	−	0.929224i	$-0.620477\pi$
$937$	−22.6222	−0.739035	−0.369517	+	0.929224i	$0.620477\pi$
$938$	−5.62059	−0.183519
$939$	0	0
$940$	−36.2676	−1.18292
$941$	−33.9667	−1.10728	−0.553642	−	0.832755i	$-0.686762\pi$
$941$	−33.9667	−1.10728	−0.553642	+	0.832755i	$0.686762\pi$
$942$	0	0
$943$	20.4697	0.666586
$944$	8.32045	0.270807
$945$	0	0
$946$	12.5917	0.409392
$947$	11.6328	0.378016	0.189008	−	0.981976i	$-0.439473\pi$
$947$	11.6328	0.378016	0.189008	+	0.981976i	$0.439473\pi$
$948$	0	0
$949$	39.2190	1.27310
$950$	2.80810	0.0911068
$951$	0	0
$952$	0	0
$953$	57.8467	1.87384	0.936919	−	0.349547i	$-0.113664\pi$
$953$	57.8467	1.87384	0.936919	+	0.349547i	$0.113664\pi$
$954$	0	0
$955$	42.3249	1.36960
$956$	−28.9775	−0.937199
$957$	0	0
$958$	−18.0487	−0.583127
$959$	−35.8031	−1.15614
$960$	0	0
$961$	−11.0420	−0.356194
$962$	−7.82051	−0.252143
$963$	0	0
$964$	51.4498	1.65709
$965$	−10.3068	−0.331787
$966$	0	0
$967$	27.3184	0.878501	0.439250	−	0.898365i	$-0.355244\pi$
$967$	27.3184	0.878501	0.439250	+	0.898365i	$0.355244\pi$
$968$	−10.0701	−0.323666
$969$	0	0
$970$	0.863100	0.0277125
$971$	−10.0632	−0.322943	−0.161471	−	0.986877i	$-0.551624\pi$
$971$	−10.0632	−0.322943	−0.161471	+	0.986877i	$0.551624\pi$
$972$	0	0
$973$	38.9432	1.24846
$974$	9.24298	0.296164
$975$	0	0
$976$	−18.7304	−0.599546
$977$	26.5667	0.849945	0.424973	−	0.905206i	$-0.360284\pi$
$977$	26.5667	0.849945	0.424973	+	0.905206i	$0.360284\pi$
$978$	0	0
$979$	−39.4630	−1.26124
$980$	−2.10743	−0.0673192
$981$	0	0
$982$	10.0773	0.321578
$983$	30.5443	0.974213	0.487106	−	0.873343i	$-0.338052\pi$
$983$	30.5443	0.974213	0.487106	+	0.873343i	$0.338052\pi$
$984$	0	0
$985$	−7.73947	−0.246600
$986$	0	0
$987$	0	0
$988$	−30.1486	−0.959154
$989$	−21.1438	−0.672335
$990$	0	0
$991$	−2.06918	−0.0657298	−0.0328649	−	0.999460i	$-0.510463\pi$
$991$	−2.06918	−0.0657298	−0.0328649	+	0.999460i	$0.510463\pi$
$992$	−23.3632	−0.741784
$993$	0	0
$994$	−6.08148	−0.192893
$995$	14.5675	0.461821
$996$	0	0
$997$	29.7880	0.943395	0.471697	−	0.881761i	$-0.343642\pi$
$997$	29.7880	0.943395	0.471697	+	0.881761i	$0.343642\pi$
$998$	3.45655	0.109415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.cg.1.10		24
3.2	odd	2		7803.2.a.cd.1.15		24
17.10	odd	16		459.2.l.a.406.8	yes	48
17.12	odd	16		459.2.l.a.433.8	yes	48
17.16	even	2		7803.2.a.cd.1.10		24
51.29	even	16		459.2.l.a.433.5	yes	48
51.44	even	16		459.2.l.a.406.5	✓	48
51.50	odd	2	inner	7803.2.a.cg.1.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.l.a.406.5	✓	48	51.44	even	16
459.2.l.a.406.8	yes	48	17.10	odd	16
459.2.l.a.433.5	yes	48	51.29	even	16
459.2.l.a.433.8	yes	48	17.12	odd	16
7803.2.a.cd.1.10		24	17.16	even	2
7803.2.a.cd.1.15		24	3.2	odd	2
7803.2.a.cg.1.10		24	1.1	even	1	trivial
7803.2.a.cg.1.15		24	51.50	odd	2	inner

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:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-0.533361 q^{2} -1.71553 q^{4} -2.01564 q^{5} -2.52795 q^{7} +1.98172 q^{8} +1.07506 q^{10} +2.43279 q^{11} +3.12835 q^{13} +1.34831 q^{14} +2.37408 q^{16} +5.61764 q^{19} +3.45788 q^{20} -1.29756 q^{22} +2.17884 q^{23} -0.937211 q^{25} -1.66854 q^{26} +4.33677 q^{28} +4.75129 q^{29} +4.46744 q^{31} -5.22968 q^{32} +5.09543 q^{35} +4.68703 q^{37} -2.99623 q^{38} -3.99442 q^{40} +9.39479 q^{41} -9.70418 q^{43} -4.17352 q^{44} -1.16211 q^{46} -10.4884 q^{47} -0.609457 q^{49} +0.499872 q^{50} -5.36677 q^{52} +2.83803 q^{53} -4.90362 q^{55} -5.00969 q^{56} -2.53415 q^{58} +3.50470 q^{59} -7.88955 q^{61} -2.38276 q^{62} -1.95885 q^{64} -6.30562 q^{65} -4.16861 q^{67} -2.71771 q^{70} -4.51044 q^{71} +12.5366 q^{73} -2.49988 q^{74} -9.63720 q^{76} -6.14998 q^{77} +4.52640 q^{79} -4.78528 q^{80} -5.01082 q^{82} -10.8255 q^{83} +5.17583 q^{86} +4.82111 q^{88} -16.2213 q^{89} -7.90833 q^{91} -3.73785 q^{92} +5.59411 q^{94} -11.3231 q^{95} +0.802837 q^{97} +0.325061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} + 20 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{14} + 24 q^{16} + 4 q^{19} + 48 q^{20} + 36 q^{23} + 28 q^{25} + 64 q^{29} + 36 q^{41} + 4 q^{43} + 16 q^{44} + 24 q^{49} + 16 q^{52} + 20 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) \) 24 * q + 24 * q^4 + 20 * q^5 + 8 * q^11 + 4 * q^13 + 16 * q^14 + 24 * q^16 + 4 * q^19 + 48 * q^20 + 36 * q^23 + 28 * q^25 + 64 * q^29 + 36 * q^41 + 4 * q^43 + 16 * q^44 + 24 * q^49 + 16 * q^52 + 20 * q^55 + 80 * q^56 - 16 * q^62 - 24 * q^64 + 72 * q^65 + 16 * q^67 - 48 * q^70 + 72 * q^71 + 80 * q^74 - 40 * q^76 + 104 * q^80 + 144 * q^92 - 72 * q^94 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more