LMFDB - Embedded newform 819.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,4,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("819.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	819.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:	$48.3225642947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5364412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 27x^{2} - 24x + 76 $ x^4 - 27x^2 - 24x + 76
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.63459$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.63459 q^{2} +5.21021 q^{4} -11.0031 q^{5} +7.00000 q^{7} -10.1397 q^{8} +O(q^{10})$	\(q+3.63459 q^{2} +5.21021 q^{4} -11.0031 q^{5} +7.00000 q^{7} -10.1397 q^{8} -39.9917 q^{10} +11.7038 q^{11} -13.0000 q^{13} +25.4421 q^{14} -78.5354 q^{16} +102.636 q^{17} +43.2825 q^{19} -57.3285 q^{20} +42.5383 q^{22} +25.9576 q^{23} -3.93182 q^{25} -47.2496 q^{26} +36.4715 q^{28} +272.081 q^{29} -121.190 q^{31} -204.326 q^{32} +373.039 q^{34} -77.0217 q^{35} +168.192 q^{37} +157.314 q^{38} +111.568 q^{40} +451.218 q^{41} +94.6309 q^{43} +60.9790 q^{44} +94.3450 q^{46} -50.6431 q^{47} +49.0000 q^{49} -14.2905 q^{50} -67.7327 q^{52} +398.509 q^{53} -128.778 q^{55} -70.9781 q^{56} +988.901 q^{58} +686.474 q^{59} -75.3794 q^{61} -440.476 q^{62} -114.356 q^{64} +143.040 q^{65} -336.720 q^{67} +534.754 q^{68} -279.942 q^{70} -427.161 q^{71} -134.191 q^{73} +611.308 q^{74} +225.511 q^{76} +81.9263 q^{77} +253.005 q^{79} +864.133 q^{80} +1639.99 q^{82} +193.536 q^{83} -1129.31 q^{85} +343.944 q^{86} -118.673 q^{88} +996.000 q^{89} -91.0000 q^{91} +135.244 q^{92} -184.067 q^{94} -476.242 q^{95} +761.982 q^{97} +178.095 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 26 q^{4} + 36 q^{5} + 28 q^{7} + 30 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 26 * q^4 + 36 * q^5 + 28 * q^7 + 30 * q^8	$ 4 q + 4 q^{2} + 26 q^{4} + 36 q^{5} + 28 q^{7} + 30 q^{8} - 44 q^{10} + 95 q^{11} - 52 q^{13} + 28 q^{14} + 58 q^{16} + 146 q^{17} - 48 q^{19} + 474 q^{20} - 143 q^{22} + 121 q^{23} + 506 q^{25} - 52 q^{26} + 182 q^{28} + 440 q^{29} - 283 q^{31} + 114 q^{32} + 1234 q^{34} + 252 q^{35} - 209 q^{37} - 440 q^{38} + 754 q^{40} + 93 q^{41} + 526 q^{43} - 217 q^{44} - 841 q^{46} + 783 q^{47} + 196 q^{49} - 446 q^{50} - 338 q^{52} + 340 q^{53} + 756 q^{55} + 210 q^{56} + 1916 q^{58} + 922 q^{59} - 141 q^{61} - 1745 q^{62} - 1510 q^{64} - 468 q^{65} - 523 q^{67} + 1710 q^{68} - 308 q^{70} - 1468 q^{71} - 47 q^{73} + 2249 q^{74} - 1382 q^{76} + 665 q^{77} + 1025 q^{79} + 2538 q^{80} - 1561 q^{82} + 1190 q^{83} - 568 q^{85} - 738 q^{86} - 555 q^{88} + 2962 q^{89} - 364 q^{91} + 599 q^{92} + 317 q^{94} - 2082 q^{95} + 2715 q^{97} + 196 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 26 * q^4 + 36 * q^5 + 28 * q^7 + 30 * q^8 - 44 * q^10 + 95 * q^11 - 52 * q^13 + 28 * q^14 + 58 * q^16 + 146 * q^17 - 48 * q^19 + 474 * q^20 - 143 * q^22 + 121 * q^23 + 506 * q^25 - 52 * q^26 + 182 * q^28 + 440 * q^29 - 283 * q^31 + 114 * q^32 + 1234 * q^34 + 252 * q^35 - 209 * q^37 - 440 * q^38 + 754 * q^40 + 93 * q^41 + 526 * q^43 - 217 * q^44 - 841 * q^46 + 783 * q^47 + 196 * q^49 - 446 * q^50 - 338 * q^52 + 340 * q^53 + 756 * q^55 + 210 * q^56 + 1916 * q^58 + 922 * q^59 - 141 * q^61 - 1745 * q^62 - 1510 * q^64 - 468 * q^65 - 523 * q^67 + 1710 * q^68 - 308 * q^70 - 1468 * q^71 - 47 * q^73 + 2249 * q^74 - 1382 * q^76 + 665 * q^77 + 1025 * q^79 + 2538 * q^80 - 1561 * q^82 + 1190 * q^83 - 568 * q^85 - 738 * q^86 - 555 * q^88 + 2962 * q^89 - 364 * q^91 + 599 * q^92 + 317 * q^94 - 2082 * q^95 + 2715 * q^97 + 196 * q^98

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 $

$2$	3.63459	1.28502	0.642510	−	0.766277i	$-0.277893\pi$
$2$	3.63459	1.28502	0.642510	+	0.766277i	$0.277893\pi$
$3$	0	0
$3$	0	0
$4$	5.21021	0.651276
$5$	−11.0031	−0.984147	−0.492074	−	0.870554i	$-0.663761\pi$
$5$	−11.0031	−0.984147	−0.492074	+	0.870554i	$0.663761\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−10.1397	−0.448117
$9$	0	0
$10$	−39.9917	−1.26465
$11$	11.7038	0.320801	0.160401	−	0.987052i	$-0.448721\pi$
$11$	11.7038	0.320801	0.160401	+	0.987052i	$0.448721\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	25.4421	0.485692
$15$	0	0
$16$	−78.5354	−1.22712
$17$	102.636	1.46429	0.732143	−	0.681151i	$-0.238520\pi$
$17$	102.636	1.46429	0.732143	+	0.681151i	$0.238520\pi$
$18$	0	0
$19$	43.2825	0.522616	0.261308	−	0.965256i	$-0.415846\pi$
$19$	43.2825	0.522616	0.261308	+	0.965256i	$0.415846\pi$
$20$	−57.3285	−0.640952
$21$	0	0
$22$	42.5383	0.412236
$23$	25.9576	0.235327	0.117664	−	0.993054i	$-0.462460\pi$
$23$	25.9576	0.235327	0.117664	+	0.993054i	$0.462460\pi$
$24$	0	0
$25$	−3.93182	−0.0314546
$26$	−47.2496	−0.356400
$27$	0	0
$28$	36.4715	0.246159
$29$	272.081	1.74221	0.871106	−	0.491095i	$-0.163403\pi$
$29$	272.081	1.74221	0.871106	+	0.491095i	$0.163403\pi$
$30$	0	0
$31$	−121.190	−0.702142	−0.351071	−	0.936349i	$-0.614182\pi$
$31$	−121.190	−0.702142	−0.351071	+	0.936349i	$0.614182\pi$
$32$	−204.326	−1.12875
$33$	0	0
$34$	373.039	1.88164
$35$	−77.0217	−0.371973
$36$	0	0
$37$	168.192	0.747314	0.373657	−	0.927567i	$-0.378104\pi$
$37$	168.192	0.747314	0.373657	+	0.927567i	$0.378104\pi$
$38$	157.314	0.671572
$39$	0	0
$40$	111.568	0.441013
$41$	451.218	1.71874	0.859372	−	0.511351i	$-0.170855\pi$
$41$	451.218	1.71874	0.859372	+	0.511351i	$0.170855\pi$
$42$	0	0
$43$	94.6309	0.335606	0.167803	−	0.985821i	$-0.446333\pi$
$43$	94.6309	0.335606	0.167803	+	0.985821i	$0.446333\pi$
$44$	60.9790	0.208930
$45$	0	0
$46$	94.3450	0.302400
$47$	−50.6431	−0.157171	−0.0785857	−	0.996907i	$-0.525040\pi$
$47$	−50.6431	−0.157171	−0.0785857	+	0.996907i	$0.525040\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−14.2905	−0.0404198
$51$	0	0
$52$	−67.7327	−0.180632
$53$	398.509	1.03282	0.516410	−	0.856341i	$-0.327268\pi$
$53$	398.509	1.03282	0.516410	+	0.856341i	$0.327268\pi$
$54$	0	0
$55$	−128.778	−0.315716
$56$	−70.9781	−0.169372
$57$	0	0
$58$	988.901	2.23878
$59$	686.474	1.51477	0.757384	−	0.652970i	$-0.226477\pi$
$59$	686.474	1.51477	0.757384	+	0.652970i	$0.226477\pi$
$60$	0	0
$61$	−75.3794	−0.158219	−0.0791094	−	0.996866i	$-0.525208\pi$
$61$	−75.3794	−0.158219	−0.0791094	+	0.996866i	$0.525208\pi$
$62$	−440.476	−0.902267
$63$	0	0
$64$	−114.356	−0.223352
$65$	143.040	0.272953
$66$	0	0
$67$	−336.720	−0.613983	−0.306991	−	0.951712i	$-0.599322\pi$
$67$	−336.720	−0.613983	−0.306991	+	0.951712i	$0.599322\pi$
$68$	534.754	0.953654
$69$	0	0
$70$	−279.942	−0.477992
$71$	−427.161	−0.714009	−0.357005	−	0.934103i	$-0.616202\pi$
$71$	−427.161	−0.714009	−0.357005	+	0.934103i	$0.616202\pi$
$72$	0	0
$73$	−134.191	−0.215148	−0.107574	−	0.994197i	$-0.534308\pi$
$73$	−134.191	−0.215148	−0.107574	+	0.994197i	$0.534308\pi$
$74$	611.308	0.960313
$75$	0	0
$76$	225.511	0.340367
$77$	81.9263	0.121251
$78$	0	0
$79$	253.005	0.360319	0.180160	−	0.983637i	$-0.442339\pi$
$79$	253.005	0.360319	0.180160	+	0.983637i	$0.442339\pi$
$80$	864.133	1.20766
$81$	0	0
$82$	1639.99	2.20862
$83$	193.536	0.255944	0.127972	−	0.991778i	$-0.459153\pi$
$83$	193.536	0.255944	0.127972	+	0.991778i	$0.459153\pi$
$84$	0	0
$85$	−1129.31	−1.44107
$86$	343.944	0.431261
$87$	0	0
$88$	−118.673	−0.143756
$89$	996.000	1.18624	0.593122	−	0.805112i	$-0.297895\pi$
$89$	996.000	1.18624	0.593122	+	0.805112i	$0.297895\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	135.244	0.153263
$93$	0	0
$94$	−184.067	−0.201968
$95$	−476.242	−0.514331
$96$	0	0
$97$	761.982	0.797604	0.398802	−	0.917037i	$-0.369426\pi$
$97$	761.982	0.797604	0.398802	+	0.917037i	$0.369426\pi$
$98$	178.095	0.183574
$99$	0	0
$100$	−20.4856	−0.0204856
$101$	−822.058	−0.809879	−0.404940	−	0.914343i	$-0.632707\pi$
$101$	−822.058	−0.809879	−0.404940	+	0.914343i	$0.632707\pi$
$102$	0	0
$103$	794.608	0.760146	0.380073	−	0.924956i	$-0.375899\pi$
$103$	794.608	0.760146	0.380073	+	0.924956i	$0.375899\pi$
$104$	131.816	0.124285
$105$	0	0
$106$	1448.42	1.32719
$107$	−2115.28	−1.91114	−0.955571	−	0.294761i	$-0.904760\pi$
$107$	−2115.28	−1.91114	−0.955571	+	0.294761i	$0.904760\pi$
$108$	0	0
$109$	1364.17	1.19875	0.599377	−	0.800467i	$-0.295415\pi$
$109$	1364.17	1.19875	0.599377	+	0.800467i	$0.295415\pi$
$110$	−468.053	−0.405701
$111$	0	0
$112$	−549.748	−0.463806
$113$	−1277.27	−1.06332	−0.531660	−	0.846958i	$-0.678432\pi$
$113$	−1277.27	−1.06332	−0.531660	+	0.846958i	$0.678432\pi$
$114$	0	0
$115$	−285.614	−0.231597
$116$	1417.60	1.13466
$117$	0	0
$118$	2495.05	1.94651
$119$	718.451	0.553448
$120$	0	0
$121$	−1194.02	−0.897087
$122$	−273.973	−0.203314
$123$	0	0
$124$	−631.427	−0.457289
$125$	1418.65	1.01510
$126$	0	0
$127$	1278.83	0.893526	0.446763	−	0.894652i	$-0.352577\pi$
$127$	1278.83	0.893526	0.446763	+	0.894652i	$0.352577\pi$
$128$	1218.97	0.841739
$129$	0	0
$130$	519.892	0.350750
$131$	2865.00	1.91081	0.955405	−	0.295297i	$-0.0954187\pi$
$131$	2865.00	1.91081	0.955405	+	0.295297i	$0.0954187\pi$
$132$	0	0
$133$	302.978	0.197530
$134$	−1223.84	−0.788980
$135$	0	0
$136$	−1040.70	−0.656171
$137$	−1494.96	−0.932283	−0.466141	−	0.884710i	$-0.654356\pi$
$137$	−1494.96	−0.932283	−0.466141	+	0.884710i	$0.654356\pi$
$138$	0	0
$139$	−1783.85	−1.08852	−0.544260	−	0.838917i	$-0.683189\pi$
$139$	−1783.85	−1.08852	−0.544260	+	0.838917i	$0.683189\pi$
$140$	−401.299	−0.242257
$141$	0	0
$142$	−1552.55	−0.917516
$143$	−152.149	−0.0889743
$144$	0	0
$145$	−2993.73	−1.71459
$146$	−487.728	−0.276470
$147$	0	0
$148$	876.316	0.486708
$149$	−1727.01	−0.949547	−0.474774	−	0.880108i	$-0.657470\pi$
$149$	−1727.01	−0.949547	−0.474774	+	0.880108i	$0.657470\pi$
$150$	0	0
$151$	1499.77	0.808277	0.404139	−	0.914698i	$-0.367571\pi$
$151$	1499.77	0.808277	0.404139	+	0.914698i	$0.367571\pi$
$152$	−438.873	−0.234193
$153$	0	0
$154$	297.768	0.155811
$155$	1333.47	0.691011
$156$	0	0
$157$	−1021.82	−0.519430	−0.259715	−	0.965685i	$-0.583629\pi$
$157$	−1021.82	−0.519430	−0.259715	+	0.965685i	$0.583629\pi$
$158$	919.567	0.463018
$159$	0	0
$160$	2248.22	1.11086
$161$	181.703	0.0889454
$162$	0	0
$163$	−3847.87	−1.84901	−0.924504	−	0.381172i	$-0.875520\pi$
$163$	−3847.87	−1.84901	−0.924504	+	0.381172i	$0.875520\pi$
$164$	2350.94	1.11938
$165$	0	0
$166$	703.423	0.328893
$167$	424.130	0.196528	0.0982639	−	0.995160i	$-0.468671\pi$
$167$	424.130	0.196528	0.0982639	+	0.995160i	$0.468671\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−4104.58	−1.85181
$171$	0	0
$172$	493.047	0.218573
$173$	3117.51	1.37006	0.685029	−	0.728516i	$-0.259790\pi$
$173$	3117.51	1.37006	0.685029	+	0.728516i	$0.259790\pi$
$174$	0	0
$175$	−27.5228	−0.0118887
$176$	−919.159	−0.393660
$177$	0	0
$178$	3620.05	1.52435
$179$	−2962.21	−1.23690	−0.618452	−	0.785823i	$-0.712240\pi$
$179$	−2962.21	−1.23690	−0.618452	+	0.785823i	$0.712240\pi$
$180$	0	0
$181$	−1198.27	−0.492081	−0.246041	−	0.969259i	$-0.579130\pi$
$181$	−1198.27	−0.492081	−0.246041	+	0.969259i	$0.579130\pi$
$182$	−330.747	−0.134707
$183$	0	0
$184$	−263.203	−0.105454
$185$	−1850.63	−0.735466
$186$	0	0
$187$	1201.22	0.469745
$188$	−263.861	−0.102362
$189$	0	0
$190$	−1730.94	−0.660925
$191$	3940.99	1.49298	0.746492	−	0.665394i	$-0.231737\pi$
$191$	3940.99	1.49298	0.746492	+	0.665394i	$0.231737\pi$
$192$	0	0
$193$	−1974.13	−0.736276	−0.368138	−	0.929771i	$-0.620005\pi$
$193$	−1974.13	−0.736276	−0.368138	+	0.929771i	$0.620005\pi$
$194$	2769.49	1.02494
$195$	0	0
$196$	255.300	0.0930395
$197$	−2772.61	−1.00274	−0.501372	−	0.865232i	$-0.667171\pi$
$197$	−2772.61	−1.00274	−0.501372	+	0.865232i	$0.667171\pi$
$198$	0	0
$199$	−919.870	−0.327678	−0.163839	−	0.986487i	$-0.552388\pi$
$199$	−919.870	−0.327678	−0.163839	+	0.986487i	$0.552388\pi$
$200$	39.8676	0.0140953
$201$	0	0
$202$	−2987.84	−1.04071
$203$	1904.57	0.658494
$204$	0	0
$205$	−4964.80	−1.69150
$206$	2888.07	0.976803
$207$	0	0
$208$	1020.96	0.340341
$209$	506.568	0.167656
$210$	0	0
$211$	936.550	0.305568	0.152784	−	0.988260i	$-0.451176\pi$
$211$	936.550	0.305568	0.152784	+	0.988260i	$0.451176\pi$
$212$	2076.32	0.672651
$213$	0	0
$214$	−7688.18	−2.45586
$215$	−1041.23	−0.330286
$216$	0	0
$217$	−848.332	−0.265385
$218$	4958.20	1.54042
$219$	0	0
$220$	−670.958	−0.205618
$221$	−1334.27	−0.406120
$222$	0	0
$223$	5243.34	1.57453	0.787264	−	0.616616i	$-0.211497\pi$
$223$	5243.34	1.57453	0.787264	+	0.616616i	$0.211497\pi$
$224$	−1430.28	−0.426628
$225$	0	0
$226$	−4642.34	−1.36639
$227$	3821.36	1.11732	0.558662	−	0.829396i	$-0.311315\pi$
$227$	3821.36	1.11732	0.558662	+	0.829396i	$0.311315\pi$
$228$	0	0
$229$	2144.86	0.618936	0.309468	−	0.950910i	$-0.399849\pi$
$229$	2144.86	0.618936	0.309468	+	0.950910i	$0.399849\pi$
$230$	−1038.09	−0.297606
$231$	0	0
$232$	−2758.83	−0.780714
$233$	3473.06	0.976513	0.488257	−	0.872700i	$-0.337633\pi$
$233$	3473.06	0.976513	0.488257	+	0.872700i	$0.337633\pi$
$234$	0	0
$235$	557.231	0.154680
$236$	3576.68	0.986533
$237$	0	0
$238$	2611.27	0.711192
$239$	−3691.00	−0.998959	−0.499479	−	0.866326i	$-0.666475\pi$
$239$	−3691.00	−0.998959	−0.499479	+	0.866326i	$0.666475\pi$
$240$	0	0
$241$	−914.807	−0.244514	−0.122257	−	0.992498i	$-0.539013\pi$
$241$	−914.807	−0.244514	−0.122257	+	0.992498i	$0.539013\pi$
$242$	−4339.78	−1.15277
$243$	0	0
$244$	−392.743	−0.103044
$245$	−539.152	−0.140592
$246$	0	0
$247$	−562.673	−0.144948
$248$	1228.84	0.314642
$249$	0	0
$250$	5156.20	1.30443
$251$	−5817.09	−1.46283	−0.731417	−	0.681930i	$-0.761141\pi$
$251$	−5817.09	−1.46283	−0.731417	+	0.681930i	$0.761141\pi$
$252$	0	0
$253$	303.801	0.0754933
$254$	4648.02	1.14820
$255$	0	0
$256$	5345.30	1.30500
$257$	−3066.16	−0.744210	−0.372105	−	0.928191i	$-0.621364\pi$
$257$	−3066.16	−0.744210	−0.372105	+	0.928191i	$0.621364\pi$
$258$	0	0
$259$	1177.34	0.282458
$260$	745.270	0.177768
$261$	0	0
$262$	10413.1	2.45543
$263$	−1215.01	−0.284869	−0.142435	−	0.989804i	$-0.545493\pi$
$263$	−1215.01	−0.284869	−0.142435	+	0.989804i	$0.545493\pi$
$264$	0	0
$265$	−4384.84	−1.01645
$266$	1101.20	0.253830
$267$	0	0
$268$	−1754.38	−0.399873
$269$	−3689.75	−0.836311	−0.418156	−	0.908375i	$-0.637323\pi$
$269$	−3689.75	−0.836311	−0.418156	+	0.908375i	$0.637323\pi$
$270$	0	0
$271$	5672.09	1.27142	0.635710	−	0.771928i	$-0.280708\pi$
$271$	5672.09	1.27142	0.635710	+	0.771928i	$0.280708\pi$
$272$	−8060.55	−1.79685
$273$	0	0
$274$	−5433.55	−1.19800
$275$	−46.0171	−0.0100907
$276$	0	0
$277$	6266.61	1.35929	0.679647	−	0.733539i	$-0.262133\pi$
$277$	6266.61	1.35929	0.679647	+	0.733539i	$0.262133\pi$
$278$	−6483.56	−1.39877
$279$	0	0
$280$	780.979	0.166687
$281$	7122.92	1.51216	0.756082	−	0.654477i	$-0.227111\pi$
$281$	7122.92	1.51216	0.756082	+	0.654477i	$0.227111\pi$
$282$	0	0
$283$	−2576.18	−0.541123	−0.270561	−	0.962703i	$-0.587209\pi$
$283$	−2576.18	−0.541123	−0.270561	+	0.962703i	$0.587209\pi$
$284$	−2225.60	−0.465017
$285$	0	0
$286$	−552.998	−0.114334
$287$	3158.53	0.649624
$288$	0	0
$289$	5621.12	1.14413
$290$	−10881.0	−2.20329
$291$	0	0
$292$	−699.162	−0.140121
$293$	6955.17	1.38678	0.693388	−	0.720565i	$-0.256117\pi$
$293$	6955.17	1.38678	0.693388	+	0.720565i	$0.256117\pi$
$294$	0	0
$295$	−7553.34	−1.49075
$296$	−1705.42	−0.334884
$297$	0	0
$298$	−6276.98	−1.22019
$299$	−337.448	−0.0652681
$300$	0	0
$301$	662.416	0.126847
$302$	5451.06	1.03865
$303$	0	0
$304$	−3399.21	−0.641310
$305$	829.407	0.155710
$306$	0	0
$307$	1690.34	0.314244	0.157122	−	0.987579i	$-0.449778\pi$
$307$	1690.34	0.314244	0.157122	+	0.987579i	$0.449778\pi$
$308$	426.853	0.0789682
$309$	0	0
$310$	4846.61	0.887963
$311$	−7291.22	−1.32941	−0.664706	−	0.747105i	$-0.731443\pi$
$311$	−7291.22	−1.32941	−0.664706	+	0.747105i	$0.731443\pi$
$312$	0	0
$313$	−4730.37	−0.854237	−0.427118	−	0.904196i	$-0.640471\pi$
$313$	−4730.37	−0.854237	−0.427118	+	0.904196i	$0.640471\pi$
$314$	−3713.91	−0.667477
$315$	0	0
$316$	1318.21	0.234668
$317$	−8225.90	−1.45745	−0.728726	−	0.684805i	$-0.759887\pi$
$317$	−8225.90	−1.45745	−0.728726	+	0.684805i	$0.759887\pi$
$318$	0	0
$319$	3184.37	0.558904
$320$	1258.27	0.219812
$321$	0	0
$322$	660.415	0.114297
$323$	4442.34	0.765258
$324$	0	0
$325$	51.1137	0.00872393
$326$	−13985.4	−2.37601
$327$	0	0
$328$	−4575.23	−0.770198
$329$	−354.502	−0.0594052
$330$	0	0
$331$	−7168.03	−1.19030	−0.595152	−	0.803613i	$-0.702908\pi$
$331$	−7168.03	−1.19030	−0.595152	+	0.803613i	$0.702908\pi$
$332$	1008.36	0.166690
$333$	0	0
$334$	1541.54	0.252542
$335$	3704.96	0.604249
$336$	0	0
$337$	4566.95	0.738213	0.369107	−	0.929387i	$-0.379664\pi$
$337$	4566.95	0.738213	0.369107	+	0.929387i	$0.379664\pi$
$338$	614.245	0.0988477
$339$	0	0
$340$	−5883.96	−0.938536
$341$	−1418.38	−0.225248
$342$	0	0
$343$	343.000	0.0539949
$344$	−959.531	−0.150391
$345$	0	0
$346$	11330.9	1.76055
$347$	4616.95	0.714267	0.357134	−	0.934053i	$-0.383754\pi$
$347$	4616.95	0.714267	0.357134	+	0.934053i	$0.383754\pi$
$348$	0	0
$349$	−7688.68	−1.17927	−0.589636	−	0.807669i	$-0.700729\pi$
$349$	−7688.68	−1.17927	−0.589636	+	0.807669i	$0.700729\pi$
$350$	−100.034	−0.0152772
$351$	0	0
$352$	−2391.38	−0.362105
$353$	8037.67	1.21190	0.605952	−	0.795501i	$-0.292792\pi$
$353$	8037.67	1.21190	0.605952	+	0.795501i	$0.292792\pi$
$354$	0	0
$355$	4700.09	0.702690
$356$	5189.37	0.772573
$357$	0	0
$358$	−10766.4	−1.58945
$359$	−170.112	−0.0250089	−0.0125044	−	0.999922i	$-0.503980\pi$
$359$	−170.112	−0.0250089	−0.0125044	+	0.999922i	$0.503980\pi$
$360$	0	0
$361$	−4985.62	−0.726873
$362$	−4355.22	−0.632335
$363$	0	0
$364$	−474.129	−0.0682723
$365$	1476.51	0.211738
$366$	0	0
$367$	6547.73	0.931304	0.465652	−	0.884968i	$-0.345820\pi$
$367$	6547.73	0.931304	0.465652	+	0.884968i	$0.345820\pi$
$368$	−2038.59	−0.288774
$369$	0	0
$370$	−6726.29	−0.945089
$371$	2789.57	0.390369
$372$	0	0
$373$	−12845.2	−1.78311	−0.891554	−	0.452915i	$-0.850384\pi$
$373$	−12845.2	−1.78311	−0.891554	+	0.452915i	$0.850384\pi$
$374$	4365.95	0.603631
$375$	0	0
$376$	513.507	0.0704312
$377$	−3537.05	−0.483203
$378$	0	0
$379$	9494.88	1.28686	0.643429	−	0.765505i	$-0.277511\pi$
$379$	9494.88	1.28686	0.643429	+	0.765505i	$0.277511\pi$
$380$	−2481.32	−0.334971
$381$	0	0
$382$	14323.9	1.91851
$383$	−6344.95	−0.846506	−0.423253	−	0.906012i	$-0.639112\pi$
$383$	−6344.95	−0.846506	−0.423253	+	0.906012i	$0.639112\pi$
$384$	0	0
$385$	−901.443	−0.119329
$386$	−7175.16	−0.946130
$387$	0	0
$388$	3970.09	0.519461
$389$	13196.7	1.72005	0.860026	−	0.510250i	$-0.170447\pi$
$389$	13196.7	1.72005	0.860026	+	0.510250i	$0.170447\pi$
$390$	0	0
$391$	2664.18	0.344586
$392$	−496.847	−0.0640167
$393$	0	0
$394$	−10077.3	−1.28855
$395$	−2783.83	−0.354607
$396$	0	0
$397$	−32.4042	−0.00409653	−0.00204826	−	0.999998i	$-0.500652\pi$
$397$	−32.4042	−0.00409653	−0.00204826	+	0.999998i	$0.500652\pi$
$398$	−3343.35	−0.421072
$399$	0	0
$400$	308.787	0.0385984
$401$	−3853.07	−0.479834	−0.239917	−	0.970793i	$-0.577120\pi$
$401$	−3853.07	−0.479834	−0.239917	+	0.970793i	$0.577120\pi$
$402$	0	0
$403$	1575.47	0.194739
$404$	−4283.10	−0.527455
$405$	0	0
$406$	6922.31	0.846178
$407$	1968.48	0.239739
$408$	0	0
$409$	10221.7	1.23577	0.617886	−	0.786268i	$-0.287989\pi$
$409$	10221.7	1.23577	0.617886	+	0.786268i	$0.287989\pi$
$410$	−18045.0	−2.17361
$411$	0	0
$412$	4140.08	0.495065
$413$	4805.32	0.572529
$414$	0	0
$415$	−2129.49	−0.251886
$416$	2656.24	0.313059
$417$	0	0
$418$	1841.17	0.215441
$419$	12153.6	1.41704	0.708521	−	0.705689i	$-0.249363\pi$
$419$	12153.6	1.41704	0.708521	+	0.705689i	$0.249363\pi$
$420$	0	0
$421$	13630.9	1.57798	0.788990	−	0.614405i	$-0.210604\pi$
$421$	13630.9	1.57798	0.788990	+	0.614405i	$0.210604\pi$
$422$	3403.97	0.392660
$423$	0	0
$424$	−4040.77	−0.462824
$425$	−403.546	−0.0460585
$426$	0	0
$427$	−527.656	−0.0598011
$428$	−11021.1	−1.24468
$429$	0	0
$430$	−3784.45	−0.424424
$431$	5959.31	0.666009	0.333004	−	0.942925i	$-0.391938\pi$
$431$	5959.31	0.666009	0.333004	+	0.942925i	$0.391938\pi$
$432$	0	0
$433$	13019.1	1.44494	0.722470	−	0.691402i	$-0.243007\pi$
$433$	13019.1	1.44494	0.722470	+	0.691402i	$0.243007\pi$
$434$	−3083.33	−0.341025
$435$	0	0
$436$	7107.63	0.780720
$437$	1123.51	0.122986
$438$	0	0
$439$	−892.739	−0.0970572	−0.0485286	−	0.998822i	$-0.515453\pi$
$439$	−892.739	−0.0970572	−0.0485286	+	0.998822i	$0.515453\pi$
$440$	1305.77	0.141477
$441$	0	0
$442$	−4849.50	−0.521872
$443$	8779.74	0.941621	0.470810	−	0.882234i	$-0.343962\pi$
$443$	8779.74	0.941621	0.470810	+	0.882234i	$0.343962\pi$
$444$	0	0
$445$	−10959.1	−1.16744
$446$	19057.4	2.02330
$447$	0	0
$448$	−800.495	−0.0844193
$449$	−13122.2	−1.37923	−0.689617	−	0.724174i	$-0.742221\pi$
$449$	−13122.2	−1.37923	−0.689617	+	0.724174i	$0.742221\pi$
$450$	0	0
$451$	5280.95	0.551375
$452$	−6654.83	−0.692515
$453$	0	0
$454$	13889.1	1.43578
$455$	1001.28	0.103167
$456$	0	0
$457$	−9100.56	−0.931524	−0.465762	−	0.884910i	$-0.654220\pi$
$457$	−9100.56	−0.931524	−0.465762	+	0.884910i	$0.654220\pi$
$458$	7795.67	0.795345
$459$	0	0
$460$	−1488.11	−0.150833
$461$	11153.6	1.12685	0.563423	−	0.826169i	$-0.309484\pi$
$461$	11153.6	1.12685	0.563423	+	0.826169i	$0.309484\pi$
$462$	0	0
$463$	−10006.3	−1.00439	−0.502197	−	0.864753i	$-0.667475\pi$
$463$	−10006.3	−1.00439	−0.502197	+	0.864753i	$0.667475\pi$
$464$	−21368.0	−2.13790
$465$	0	0
$466$	12623.1	1.25484
$467$	−10455.5	−1.03603	−0.518014	−	0.855372i	$-0.673328\pi$
$467$	−10455.5	−1.03603	−0.518014	+	0.855372i	$0.673328\pi$
$468$	0	0
$469$	−2357.04	−0.232064
$470$	2025.30	0.198767
$471$	0	0
$472$	−6960.66	−0.678793
$473$	1107.54	0.107663
$474$	0	0
$475$	−170.179	−0.0164387
$476$	3743.28	0.360448
$477$	0	0
$478$	−13415.3	−1.28368
$479$	11161.0	1.06463	0.532314	−	0.846547i	$-0.321322\pi$
$479$	11161.0	1.06463	0.532314	+	0.846547i	$0.321322\pi$
$480$	0	0
$481$	−2186.50	−0.207267
$482$	−3324.94	−0.314205
$483$	0	0
$484$	−6221.11	−0.584251
$485$	−8384.17	−0.784959
$486$	0	0
$487$	3941.17	0.366717	0.183359	−	0.983046i	$-0.441303\pi$
$487$	3941.17	0.366717	0.183359	+	0.983046i	$0.441303\pi$
$488$	764.326	0.0709005
$489$	0	0
$490$	−1959.59	−0.180664
$491$	−11636.5	−1.06955	−0.534775	−	0.844995i	$-0.679603\pi$
$491$	−11636.5	−1.06955	−0.534775	+	0.844995i	$0.679603\pi$
$492$	0	0
$493$	27925.3	2.55110
$494$	−2045.08	−0.186260
$495$	0	0
$496$	9517.72	0.861610
$497$	−2990.13	−0.269870
$498$	0	0
$499$	2370.55	0.212666	0.106333	−	0.994331i	$-0.466089\pi$
$499$	2370.55	0.212666	0.106333	+	0.994331i	$0.466089\pi$
$500$	7391.46	0.661113
$501$	0	0
$502$	−21142.7	−1.87977
$503$	−16697.1	−1.48010	−0.740048	−	0.672554i	$-0.765197\pi$
$503$	−16697.1	−1.48010	−0.740048	+	0.672554i	$0.765197\pi$
$504$	0	0
$505$	9045.18	0.797040
$506$	1104.19	0.0970104
$507$	0	0
$508$	6662.98	0.581933
$509$	12297.2	1.07085	0.535426	−	0.844582i	$-0.320151\pi$
$509$	12297.2	1.07085	0.535426	+	0.844582i	$0.320151\pi$
$510$	0	0
$511$	−939.335	−0.0813185
$512$	9676.19	0.835217
$513$	0	0
$514$	−11144.2	−0.956324
$515$	−8743.15	−0.748096
$516$	0	0
$517$	−592.714	−0.0504208
$518$	4279.16	0.362964
$519$	0	0
$520$	−1450.39	−0.122315
$521$	−18304.4	−1.53922	−0.769609	−	0.638516i	$-0.779549\pi$
$521$	−18304.4	−1.53922	−0.769609	+	0.638516i	$0.779549\pi$
$522$	0	0
$523$	−12673.2	−1.05958	−0.529788	−	0.848130i	$-0.677729\pi$
$523$	−12673.2	−1.05958	−0.529788	+	0.848130i	$0.677729\pi$
$524$	14927.3	1.24447
$525$	0	0
$526$	−4416.05	−0.366063
$527$	−12438.5	−1.02814
$528$	0	0
$529$	−11493.2	−0.944621
$530$	−15937.1	−1.30615
$531$	0	0
$532$	1578.58	0.128647
$533$	−5865.84	−0.476694
$534$	0	0
$535$	23274.7	1.88084
$536$	3414.24	0.275136
$537$	0	0
$538$	−13410.7	−1.07468
$539$	573.484	0.0458287
$540$	0	0
$541$	−21283.1	−1.69137	−0.845684	−	0.533684i	$-0.820807\pi$
$541$	−21283.1	−1.69137	−0.845684	+	0.533684i	$0.820807\pi$
$542$	20615.7	1.63380
$543$	0	0
$544$	−20971.1	−1.65281
$545$	−15010.1	−1.17975
$546$	0	0
$547$	−19081.2	−1.49150	−0.745751	−	0.666225i	$-0.767909\pi$
$547$	−19081.2	−1.49150	−0.745751	+	0.666225i	$0.767909\pi$
$548$	−7789.04	−0.607174
$549$	0	0
$550$	−167.253	−0.0129667
$551$	11776.4	0.910507
$552$	0	0
$553$	1771.03	0.136188
$554$	22776.5	1.74672
$555$	0	0
$556$	−9294.24	−0.708927
$557$	13693.3	1.04166	0.520828	−	0.853661i	$-0.325623\pi$
$557$	13693.3	1.04166	0.520828	+	0.853661i	$0.325623\pi$
$558$	0	0
$559$	−1230.20	−0.0930805
$560$	6048.93	0.456453
$561$	0	0
$562$	25888.9	1.94316
$563$	15762.2	1.17993	0.589964	−	0.807430i	$-0.299142\pi$
$563$	15762.2	1.17993	0.589964	+	0.807430i	$0.299142\pi$
$564$	0	0
$565$	14053.9	1.04646
$566$	−9363.33	−0.695354
$567$	0	0
$568$	4331.29	0.319960
$569$	−22202.1	−1.63578	−0.817892	−	0.575372i	$-0.804857\pi$
$569$	−22202.1	−1.63578	−0.817892	+	0.575372i	$0.804857\pi$
$570$	0	0
$571$	21989.8	1.61164	0.805819	−	0.592162i	$-0.201725\pi$
$571$	21989.8	1.61164	0.805819	+	0.592162i	$0.201725\pi$
$572$	−792.727	−0.0579468
$573$	0	0
$574$	11479.9	0.834780
$575$	−102.061	−0.00740212
$576$	0	0
$577$	−18405.1	−1.32793	−0.663965	−	0.747764i	$-0.731128\pi$
$577$	−18405.1	−1.32793	−0.663965	+	0.747764i	$0.731128\pi$
$578$	20430.4	1.47023
$579$	0	0
$580$	−15598.0	−1.11667
$581$	1354.75	0.0967376
$582$	0	0
$583$	4664.05	0.331330
$584$	1360.66	0.0964116
$585$	0	0
$586$	25279.1	1.78203
$587$	13059.4	0.918260	0.459130	−	0.888369i	$-0.348161\pi$
$587$	13059.4	0.918260	0.459130	+	0.888369i	$0.348161\pi$
$588$	0	0
$589$	−5245.42	−0.366951
$590$	−27453.3	−1.91565
$591$	0	0
$592$	−13209.0	−0.917040
$593$	−11511.8	−0.797189	−0.398595	−	0.917127i	$-0.630502\pi$
$593$	−11511.8	−0.797189	−0.398595	+	0.917127i	$0.630502\pi$
$594$	0	0
$595$	−7905.19	−0.544674
$596$	−8998.11	−0.618418
$597$	0	0
$598$	−1226.49	−0.0838708
$599$	−7113.35	−0.485215	−0.242607	−	0.970125i	$-0.578003\pi$
$599$	−7113.35	−0.485215	−0.242607	+	0.970125i	$0.578003\pi$
$600$	0	0
$601$	−7802.91	−0.529596	−0.264798	−	0.964304i	$-0.585305\pi$
$601$	−7802.91	−0.529596	−0.264798	+	0.964304i	$0.585305\pi$
$602$	2407.61	0.163001
$603$	0	0
$604$	7814.14	0.526412
$605$	13137.9	0.882865
$606$	0	0
$607$	−13290.9	−0.888732	−0.444366	−	0.895845i	$-0.646571\pi$
$607$	−13290.9	−0.888732	−0.444366	+	0.895845i	$0.646571\pi$
$608$	−8843.74	−0.589903
$609$	0	0
$610$	3014.55	0.200091
$611$	658.360	0.0435915
$612$	0	0
$613$	−3133.55	−0.206465	−0.103232	−	0.994657i	$-0.532919\pi$
$613$	−3133.55	−0.206465	−0.103232	+	0.994657i	$0.532919\pi$
$614$	6143.69	0.403810
$615$	0	0
$616$	−830.710	−0.0543348
$617$	126.313	0.00824175	0.00412088	−	0.999992i	$-0.498688\pi$
$617$	126.313	0.00824175	0.00412088	+	0.999992i	$0.498688\pi$
$618$	0	0
$619$	5888.40	0.382350	0.191175	−	0.981556i	$-0.438770\pi$
$619$	5888.40	0.382350	0.191175	+	0.981556i	$0.438770\pi$
$620$	6947.65	0.450039
$621$	0	0
$622$	−26500.6	−1.70832
$623$	6972.00	0.448358
$624$	0	0
$625$	−15118.1	−0.967556
$626$	−17192.9	−1.09771
$627$	0	0
$628$	−5323.92	−0.338292
$629$	17262.5	1.09428
$630$	0	0
$631$	−21316.2	−1.34483	−0.672413	−	0.740176i	$-0.734742\pi$
$631$	−21316.2	−1.34483	−0.672413	+	0.740176i	$0.734742\pi$
$632$	−2565.40	−0.161465
$633$	0	0
$634$	−29897.7	−1.87286
$635$	−14071.1	−0.879361
$636$	0	0
$637$	−637.000	−0.0396214
$638$	11573.9	0.718203
$639$	0	0
$640$	−13412.4	−0.828395
$641$	11687.9	0.720193	0.360097	−	0.932915i	$-0.382744\pi$
$641$	11687.9	0.720193	0.360097	+	0.932915i	$0.382744\pi$
$642$	0	0
$643$	−20771.5	−1.27395	−0.636974	−	0.770885i	$-0.719814\pi$
$643$	−20771.5	−1.27395	−0.636974	+	0.770885i	$0.719814\pi$
$644$	946.711	0.0579280
$645$	0	0
$646$	16146.1	0.983372
$647$	15685.1	0.953086	0.476543	−	0.879151i	$-0.341890\pi$
$647$	15685.1	0.953086	0.476543	+	0.879151i	$0.341890\pi$
$648$	0	0
$649$	8034.32	0.485940
$650$	185.777	0.0112104
$651$	0	0
$652$	−20048.2	−1.20422
$653$	−5161.91	−0.309343	−0.154672	−	0.987966i	$-0.549432\pi$
$653$	−5161.91	−0.309343	−0.154672	+	0.987966i	$0.549432\pi$
$654$	0	0
$655$	−31523.9	−1.88052
$656$	−35436.6	−2.10910
$657$	0	0
$658$	−1288.47	−0.0763369
$659$	19647.1	1.16137	0.580683	−	0.814130i	$-0.302786\pi$
$659$	19647.1	1.16137	0.580683	+	0.814130i	$0.302786\pi$
$660$	0	0
$661$	25197.0	1.48268	0.741340	−	0.671130i	$-0.234191\pi$
$661$	25197.0	1.48268	0.741340	+	0.671130i	$0.234191\pi$
$662$	−26052.8	−1.52956
$663$	0	0
$664$	−1962.40	−0.114693
$665$	−3333.69	−0.194399
$666$	0	0
$667$	7062.56	0.409990
$668$	2209.81	0.127994
$669$	0	0
$670$	13466.0	0.776473
$671$	−882.222	−0.0507568
$672$	0	0
$673$	31227.3	1.78860	0.894298	−	0.447472i	$-0.147676\pi$
$673$	31227.3	1.78860	0.894298	+	0.447472i	$0.147676\pi$
$674$	16599.0	0.948619
$675$	0	0
$676$	880.526	0.0500982
$677$	1846.13	0.104804	0.0524022	−	0.998626i	$-0.483312\pi$
$677$	1846.13	0.104804	0.0524022	+	0.998626i	$0.483312\pi$
$678$	0	0
$679$	5333.88	0.301466
$680$	11450.9	0.645769
$681$	0	0
$682$	−5155.23	−0.289448
$683$	5829.61	0.326594	0.163297	−	0.986577i	$-0.447787\pi$
$683$	5829.61	0.326594	0.163297	+	0.986577i	$0.447787\pi$
$684$	0	0
$685$	16449.2	0.917503
$686$	1246.66	0.0693846
$687$	0	0
$688$	−7431.87	−0.411828
$689$	−5180.62	−0.286453
$690$	0	0
$691$	−12829.3	−0.706294	−0.353147	−	0.935568i	$-0.614888\pi$
$691$	−12829.3	−0.706294	−0.353147	+	0.935568i	$0.614888\pi$
$692$	16242.9	0.892286
$693$	0	0
$694$	16780.7	0.917847
$695$	19627.9	1.07126
$696$	0	0
$697$	46311.2	2.51673
$698$	−27945.2	−1.51539
$699$	0	0
$700$	−143.399	−0.00774284
$701$	30321.8	1.63372	0.816862	−	0.576834i	$-0.195712\pi$
$701$	30321.8	1.63372	0.816862	+	0.576834i	$0.195712\pi$
$702$	0	0
$703$	7279.78	0.390558
$704$	−1338.40	−0.0716517
$705$	0	0
$706$	29213.6	1.55732
$707$	−5754.41	−0.306106
$708$	0	0
$709$	13767.1	0.729246	0.364623	−	0.931155i	$-0.381198\pi$
$709$	13767.1	0.729246	0.364623	+	0.931155i	$0.381198\pi$
$710$	17082.9	0.902971
$711$	0	0
$712$	−10099.2	−0.531576
$713$	−3145.81	−0.165233
$714$	0	0
$715$	1674.11	0.0875638
$716$	−15433.7	−0.805566
$717$	0	0
$718$	−618.288	−0.0321369
$719$	25769.7	1.33664	0.668322	−	0.743872i	$-0.267013\pi$
$719$	25769.7	1.33664	0.668322	+	0.743872i	$0.267013\pi$
$720$	0	0
$721$	5562.26	0.287308
$722$	−18120.7	−0.934046
$723$	0	0
$724$	−6243.24	−0.320481
$725$	−1069.77	−0.0548006
$726$	0	0
$727$	20584.7	1.05013	0.525064	−	0.851063i	$-0.324041\pi$
$727$	20584.7	1.05013	0.525064	+	0.851063i	$0.324041\pi$
$728$	922.715	0.0469754
$729$	0	0
$730$	5366.52	0.272087
$731$	9712.52	0.491424
$732$	0	0
$733$	12560.0	0.632900	0.316450	−	0.948609i	$-0.397509\pi$
$733$	12560.0	0.632900	0.316450	+	0.948609i	$0.397509\pi$
$734$	23798.3	1.19674
$735$	0	0
$736$	−5303.80	−0.265626
$737$	−3940.88	−0.196966
$738$	0	0
$739$	2247.01	0.111850	0.0559252	−	0.998435i	$-0.482189\pi$
$739$	2247.01	0.111850	0.0559252	+	0.998435i	$0.482189\pi$
$740$	−9642.19	−0.478992
$741$	0	0
$742$	10138.9	0.501632
$743$	17437.7	0.861006	0.430503	−	0.902589i	$-0.358336\pi$
$743$	17437.7	0.861006	0.430503	+	0.902589i	$0.358336\pi$
$744$	0	0
$745$	19002.5	0.934494
$746$	−46687.0	−2.29133
$747$	0	0
$748$	6258.63	0.305934
$749$	−14807.0	−0.722344
$750$	0	0
$751$	−13096.0	−0.636323	−0.318162	−	0.948036i	$-0.603065\pi$
$751$	−13096.0	−0.636323	−0.318162	+	0.948036i	$0.603065\pi$
$752$	3977.28	0.192867
$753$	0	0
$754$	−12855.7	−0.620925
$755$	−16502.2	−0.795464
$756$	0	0
$757$	29343.6	1.40887	0.704433	−	0.709771i	$-0.251202\pi$
$757$	29343.6	1.40887	0.704433	+	0.709771i	$0.251202\pi$
$758$	34510.0	1.65364
$759$	0	0
$760$	4828.96	0.230480
$761$	23788.7	1.13316	0.566582	−	0.824005i	$-0.308265\pi$
$761$	23788.7	1.13316	0.566582	+	0.824005i	$0.308265\pi$
$762$	0	0
$763$	9549.21	0.453086
$764$	20533.4	0.972345
$765$	0	0
$766$	−23061.3	−1.08778
$767$	−8924.16	−0.420121
$768$	0	0
$769$	11235.6	0.526875	0.263437	−	0.964676i	$-0.415144\pi$
$769$	11235.6	0.526875	0.263437	+	0.964676i	$0.415144\pi$
$770$	−3276.37	−0.153341
$771$	0	0
$772$	−10285.7	−0.479519
$773$	7742.41	0.360252	0.180126	−	0.983644i	$-0.442349\pi$
$773$	7742.41	0.360252	0.180126	+	0.983644i	$0.442349\pi$
$774$	0	0
$775$	476.499	0.0220856
$776$	−7726.29	−0.357420
$777$	0	0
$778$	47964.6	2.21030
$779$	19529.9	0.898242
$780$	0	0
$781$	−4999.38	−0.229055
$782$	9683.18	0.442800
$783$	0	0
$784$	−3848.23	−0.175302
$785$	11243.2	0.511195
$786$	0	0
$787$	73.6706	0.00333681	0.00166841	−	0.999999i	$-0.499469\pi$
$787$	73.6706	0.00333681	0.00166841	+	0.999999i	$0.499469\pi$
$788$	−14445.9	−0.653063
$789$	0	0
$790$	−10118.1	−0.455678
$791$	−8940.87	−0.401897
$792$	0	0
$793$	979.932	0.0438820
$794$	−117.776	−0.00526412
$795$	0	0
$796$	−4792.72	−0.213409
$797$	−5805.43	−0.258016	−0.129008	−	0.991644i	$-0.541179\pi$
$797$	−5805.43	−0.258016	−0.129008	+	0.991644i	$0.541179\pi$
$798$	0	0
$799$	−5197.80	−0.230144
$800$	803.373	0.0355044
$801$	0	0
$802$	−14004.3	−0.616596
$803$	−1570.53	−0.0690199
$804$	0	0
$805$	−1999.30	−0.0875353
$806$	5726.19	0.250244
$807$	0	0
$808$	8335.44	0.362921
$809$	−27095.9	−1.17755	−0.588776	−	0.808296i	$-0.700390\pi$
$809$	−27095.9	−1.17755	−0.588776	+	0.808296i	$0.700390\pi$
$810$	0	0
$811$	−28006.3	−1.21262	−0.606309	−	0.795229i	$-0.707350\pi$
$811$	−28006.3	−1.21262	−0.606309	+	0.795229i	$0.707350\pi$
$812$	9923.19	0.428862
$813$	0	0
$814$	7154.60	0.308070
$815$	42338.5	1.81970
$816$	0	0
$817$	4095.87	0.175393
$818$	37151.7	1.58799
$819$	0	0
$820$	−25867.7	−1.10163
$821$	16408.0	0.697493	0.348747	−	0.937217i	$-0.386607\pi$
$821$	16408.0	0.697493	0.348747	+	0.937217i	$0.386607\pi$
$822$	0	0
$823$	7614.47	0.322507	0.161254	−	0.986913i	$-0.448446\pi$
$823$	7614.47	0.322507	0.161254	+	0.986913i	$0.448446\pi$
$824$	−8057.11	−0.340634
$825$	0	0
$826$	17465.3	0.735711
$827$	17850.7	0.750581	0.375290	−	0.926907i	$-0.377543\pi$
$827$	17850.7	0.750581	0.375290	+	0.926907i	$0.377543\pi$
$828$	0	0
$829$	−3802.11	−0.159292	−0.0796459	−	0.996823i	$-0.525379\pi$
$829$	−3802.11	−0.159292	−0.0796459	+	0.996823i	$0.525379\pi$
$830$	−7739.83	−0.323679
$831$	0	0
$832$	1486.63	0.0619468
$833$	5029.16	0.209184
$834$	0	0
$835$	−4666.74	−0.193412
$836$	2639.33	0.109190
$837$	0	0
$838$	44173.2	1.82093
$839$	3538.68	0.145613	0.0728063	−	0.997346i	$-0.476805\pi$
$839$	3538.68	0.145613	0.0728063	+	0.997346i	$0.476805\pi$
$840$	0	0
$841$	49639.0	2.03530
$842$	49542.7	2.02774
$843$	0	0
$844$	4879.62	0.199009
$845$	−1859.52	−0.0757036
$846$	0	0
$847$	−8358.16	−0.339067
$848$	−31297.1	−1.26739
$849$	0	0
$850$	−1466.72	−0.0591861
$851$	4365.86	0.175863
$852$	0	0
$853$	−21190.1	−0.850570	−0.425285	−	0.905059i	$-0.639826\pi$
$853$	−21190.1	−0.850570	−0.425285	+	0.905059i	$0.639826\pi$
$854$	−1917.81	−0.0768456
$855$	0	0
$856$	21448.4	0.856415
$857$	19184.4	0.764673	0.382337	−	0.924023i	$-0.375119\pi$
$857$	19184.4	0.764673	0.382337	+	0.924023i	$0.375119\pi$
$858$	0	0
$859$	−37876.3	−1.50445	−0.752226	−	0.658905i	$-0.771020\pi$
$859$	−37876.3	−1.50445	−0.752226	+	0.658905i	$0.771020\pi$
$860$	−5425.04	−0.215108
$861$	0	0
$862$	21659.6	0.855834
$863$	−9543.13	−0.376422	−0.188211	−	0.982129i	$-0.560269\pi$
$863$	−9543.13	−0.376422	−0.188211	+	0.982129i	$0.560269\pi$
$864$	0	0
$865$	−34302.3	−1.34834
$866$	47319.1	1.85678
$867$	0	0
$868$	−4419.99	−0.172839
$869$	2961.10	0.115591
$870$	0	0
$871$	4377.35	0.170288
$872$	−13832.3	−0.537181
$873$	0	0
$874$	4083.49	0.158039
$875$	9930.55	0.383673
$876$	0	0
$877$	27017.4	1.04027	0.520133	−	0.854085i	$-0.325882\pi$
$877$	27017.4	1.04027	0.520133	+	0.854085i	$0.325882\pi$
$878$	−3244.74	−0.124720
$879$	0	0
$880$	10113.6	0.387419
$881$	31954.5	1.22199	0.610996	−	0.791634i	$-0.290769\pi$
$881$	31954.5	1.22199	0.610996	+	0.791634i	$0.290769\pi$
$882$	0	0
$883$	42055.4	1.60280	0.801402	−	0.598126i	$-0.204088\pi$
$883$	42055.4	1.60280	0.801402	+	0.598126i	$0.204088\pi$
$884$	−6951.81	−0.264496
$885$	0	0
$886$	31910.7	1.21000
$887$	−34830.5	−1.31848	−0.659242	−	0.751931i	$-0.729123\pi$
$887$	−34830.5	−1.31848	−0.659242	+	0.751931i	$0.729123\pi$
$888$	0	0
$889$	8951.81	0.337721
$890$	−39831.7	−1.50018
$891$	0	0
$892$	27318.9	1.02545
$893$	−2191.96	−0.0821403
$894$	0	0
$895$	32593.4	1.21729
$896$	8532.78	0.318147
$897$	0	0
$898$	−47693.9	−1.77234
$899$	−32973.6	−1.22328
$900$	0	0
$901$	40901.3	1.51234
$902$	19194.1	0.708528
$903$	0	0
$904$	12951.1	0.476492
$905$	13184.7	0.484281
$906$	0	0
$907$	28035.7	1.02636	0.513180	−	0.858281i	$-0.328467\pi$
$907$	28035.7	1.02636	0.513180	+	0.858281i	$0.328467\pi$
$908$	19910.1	0.727686
$909$	0	0
$910$	3639.24	0.132571
$911$	−19163.8	−0.696954	−0.348477	−	0.937317i	$-0.613301\pi$
$911$	−19163.8	−0.696954	−0.348477	+	0.937317i	$0.613301\pi$
$912$	0	0
$913$	2265.10	0.0821070
$914$	−33076.8	−1.19703
$915$	0	0
$916$	11175.2	0.403098
$917$	20055.0	0.722219
$918$	0	0
$919$	−34654.1	−1.24389	−0.621945	−	0.783061i	$-0.713657\pi$
$919$	−34654.1	−1.24389	−0.621945	+	0.783061i	$0.713657\pi$
$920$	2896.04	0.103782
$921$	0	0
$922$	40538.8	1.44802
$923$	5553.09	0.198031
$924$	0	0
$925$	−661.301	−0.0235064
$926$	−36368.9	−1.29067
$927$	0	0
$928$	−55593.1	−1.96652
$929$	−42110.2	−1.48718	−0.743591	−	0.668635i	$-0.766879\pi$
$929$	−42110.2	−1.48718	−0.743591	+	0.668635i	$0.766879\pi$
$930$	0	0
$931$	2120.84	0.0746594
$932$	18095.4	0.635980
$933$	0	0
$934$	−38001.6	−1.33132
$935$	−13217.2	−0.462298
$936$	0	0
$937$	−18514.6	−0.645514	−0.322757	−	0.946482i	$-0.604610\pi$
$937$	−18514.6	−0.645514	−0.322757	+	0.946482i	$0.604610\pi$
$938$	−8566.85	−0.298206
$939$	0	0
$940$	2903.29	0.100739
$941$	8119.45	0.281282	0.140641	−	0.990061i	$-0.455084\pi$
$941$	8119.45	0.281282	0.140641	+	0.990061i	$0.455084\pi$
$942$	0	0
$943$	11712.5	0.404467
$944$	−53912.5	−1.85880
$945$	0	0
$946$	4025.44	0.138349
$947$	−49354.7	−1.69357	−0.846786	−	0.531934i	$-0.821466\pi$
$947$	−49354.7	−1.69357	−0.846786	+	0.531934i	$0.821466\pi$
$948$	0	0
$949$	1744.48	0.0596714
$950$	−618.531	−0.0211240
$951$	0	0
$952$	−7284.89	−0.248009
$953$	300.006	0.0101974	0.00509872	−	0.999987i	$-0.498377\pi$
$953$	300.006	0.0101974	0.00509872	+	0.999987i	$0.498377\pi$
$954$	0	0
$955$	−43363.1	−1.46932
$956$	−19230.9	−0.650598
$957$	0	0
$958$	40565.4	1.36807
$959$	−10464.7	−0.352370
$960$	0	0
$961$	−15103.9	−0.506996
$962$	−7947.01	−0.266343
$963$	0	0
$964$	−4766.34	−0.159246
$965$	21721.6	0.724604
$966$	0	0
$967$	−20103.3	−0.668539	−0.334270	−	0.942477i	$-0.608490\pi$
$967$	−20103.3	−0.668539	−0.334270	+	0.942477i	$0.608490\pi$
$968$	12107.1	0.402000
$969$	0	0
$970$	−30473.0	−1.00869
$971$	−39115.5	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$971$	−39115.5	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$972$	0	0
$973$	−12487.0	−0.411422
$974$	14324.5	0.471239
$975$	0	0
$976$	5919.95	0.194153
$977$	−21590.6	−0.707005	−0.353502	−	0.935434i	$-0.615009\pi$
$977$	−21590.6	−0.707005	−0.353502	+	0.935434i	$0.615009\pi$
$978$	0	0
$979$	11656.9	0.380549
$980$	−2809.10	−0.0915645
$981$	0	0
$982$	−42293.9	−1.37439
$983$	7689.51	0.249499	0.124749	−	0.992188i	$-0.460187\pi$
$983$	7689.51	0.249499	0.124749	+	0.992188i	$0.460187\pi$
$984$	0	0
$985$	30507.3	0.986847
$986$	101497.	3.27821
$987$	0	0
$988$	−2931.65	−0.0944009
$989$	2456.39	0.0789774
$990$	0	0
$991$	23913.5	0.766537	0.383269	−	0.923637i	$-0.374798\pi$
$991$	23913.5	0.766537	0.383269	+	0.923637i	$0.374798\pi$
$992$	24762.3	0.792544
$993$	0	0
$994$	−10867.9	−0.346789
$995$	10121.4	0.322483
$996$	0	0
$997$	−33.8007	−0.00107370	−0.000536851	−	1.00000i	$-0.500171\pi$
$997$	−33.8007	−0.00107370	−0.000536851		1.00000i	$0.500171\pi$
$998$	8615.97	0.273280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.4.a.h.1.3		4
3.2	odd	2		91.4.a.b.1.2	✓	4
12.11	even	2		1456.4.a.s.1.3		4
15.14	odd	2		2275.4.a.h.1.3		4
21.20	even	2		637.4.a.d.1.2		4
39.38	odd	2		1183.4.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.4.a.b.1.2	✓	4	3.2	odd	2
637.4.a.d.1.2		4	21.20	even	2
819.4.a.h.1.3		4	1.1	even	1	trivial
1183.4.a.e.1.3		4	39.38	odd	2
1456.4.a.s.1.3		4	12.11	even	2
2275.4.a.h.1.3		4	15.14	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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+3.63459 q^{2} +5.21021 q^{4} -11.0031 q^{5} +7.00000 q^{7} -10.1397 q^{8} +O(q^{10})\)	\(q+3.63459 q^{2} +5.21021 q^{4} -11.0031 q^{5} +7.00000 q^{7} -10.1397 q^{8} -39.9917 q^{10} +11.7038 q^{11} -13.0000 q^{13} +25.4421 q^{14} -78.5354 q^{16} +102.636 q^{17} +43.2825 q^{19} -57.3285 q^{20} +42.5383 q^{22} +25.9576 q^{23} -3.93182 q^{25} -47.2496 q^{26} +36.4715 q^{28} +272.081 q^{29} -121.190 q^{31} -204.326 q^{32} +373.039 q^{34} -77.0217 q^{35} +168.192 q^{37} +157.314 q^{38} +111.568 q^{40} +451.218 q^{41} +94.6309 q^{43} +60.9790 q^{44} +94.3450 q^{46} -50.6431 q^{47} +49.0000 q^{49} -14.2905 q^{50} -67.7327 q^{52} +398.509 q^{53} -128.778 q^{55} -70.9781 q^{56} +988.901 q^{58} +686.474 q^{59} -75.3794 q^{61} -440.476 q^{62} -114.356 q^{64} +143.040 q^{65} -336.720 q^{67} +534.754 q^{68} -279.942 q^{70} -427.161 q^{71} -134.191 q^{73} +611.308 q^{74} +225.511 q^{76} +81.9263 q^{77} +253.005 q^{79} +864.133 q^{80} +1639.99 q^{82} +193.536 q^{83} -1129.31 q^{85} +343.944 q^{86} -118.673 q^{88} +996.000 q^{89} -91.0000 q^{91} +135.244 q^{92} -184.067 q^{94} -476.242 q^{95} +761.982 q^{97} +178.095 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 26 q^{4} + 36 q^{5} + 28 q^{7} + 30 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 26 * q^4 + 36 * q^5 + 28 * q^7 + 30 * q^8	\( 4 q + 4 q^{2} + 26 q^{4} + 36 q^{5} + 28 q^{7} + 30 q^{8} - 44 q^{10} + 95 q^{11} - 52 q^{13} + 28 q^{14} + 58 q^{16} + 146 q^{17} - 48 q^{19} + 474 q^{20} - 143 q^{22} + 121 q^{23} + 506 q^{25} - 52 q^{26} + 182 q^{28} + 440 q^{29} - 283 q^{31} + 114 q^{32} + 1234 q^{34} + 252 q^{35} - 209 q^{37} - 440 q^{38} + 754 q^{40} + 93 q^{41} + 526 q^{43} - 217 q^{44} - 841 q^{46} + 783 q^{47} + 196 q^{49} - 446 q^{50} - 338 q^{52} + 340 q^{53} + 756 q^{55} + 210 q^{56} + 1916 q^{58} + 922 q^{59} - 141 q^{61} - 1745 q^{62} - 1510 q^{64} - 468 q^{65} - 523 q^{67} + 1710 q^{68} - 308 q^{70} - 1468 q^{71} - 47 q^{73} + 2249 q^{74} - 1382 q^{76} + 665 q^{77} + 1025 q^{79} + 2538 q^{80} - 1561 q^{82} + 1190 q^{83} - 568 q^{85} - 738 q^{86} - 555 q^{88} + 2962 q^{89} - 364 q^{91} + 599 q^{92} + 317 q^{94} - 2082 q^{95} + 2715 q^{97} + 196 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 26 * q^4 + 36 * q^5 + 28 * q^7 + 30 * q^8 - 44 * q^10 + 95 * q^11 - 52 * q^13 + 28 * q^14 + 58 * q^16 + 146 * q^17 - 48 * q^19 + 474 * q^20 - 143 * q^22 + 121 * q^23 + 506 * q^25 - 52 * q^26 + 182 * q^28 + 440 * q^29 - 283 * q^31 + 114 * q^32 + 1234 * q^34 + 252 * q^35 - 209 * q^37 - 440 * q^38 + 754 * q^40 + 93 * q^41 + 526 * q^43 - 217 * q^44 - 841 * q^46 + 783 * q^47 + 196 * q^49 - 446 * q^50 - 338 * q^52 + 340 * q^53 + 756 * q^55 + 210 * q^56 + 1916 * q^58 + 922 * q^59 - 141 * q^61 - 1745 * q^62 - 1510 * q^64 - 468 * q^65 - 523 * q^67 + 1710 * q^68 - 308 * q^70 - 1468 * q^71 - 47 * q^73 + 2249 * q^74 - 1382 * q^76 + 665 * q^77 + 1025 * q^79 + 2538 * q^80 - 1561 * q^82 + 1190 * q^83 - 568 * q^85 - 738 * q^86 - 555 * q^88 + 2962 * q^89 - 364 * q^91 + 599 * q^92 + 317 * q^94 - 2082 * q^95 + 2715 * q^97 + 196 * q^98

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