LMFDB - Embedded newform 405.5.c.b.161.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,5,Mod(161,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	405.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,-256,0,0,52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$41.8648350490$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.13
Character	$\chi$	$=$	405.161
Dual form			405.5.c.b.161.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.48322i q^{2} +9.83359 q^{4} -11.1803i q^{5} +31.5696 q^{7} -64.1506i q^{8} -27.7633 q^{10} -211.258i q^{11} -287.174 q^{13} -78.3944i q^{14} -1.96292 q^{16} +195.177i q^{17} -139.924 q^{19} -109.943i q^{20} -524.600 q^{22} +473.275i q^{23} -125.000 q^{25} +713.117i q^{26} +310.443 q^{28} -1166.22i q^{29} +320.949 q^{31} -1021.54i q^{32} +484.670 q^{34} -352.959i q^{35} -2151.28 q^{37} +347.463i q^{38} -717.226 q^{40} +600.555i q^{41} +2772.71 q^{43} -2077.42i q^{44} +1175.25 q^{46} -799.712i q^{47} -1404.36 q^{49} +310.403i q^{50} -2823.95 q^{52} +78.0197i q^{53} -2361.93 q^{55} -2025.21i q^{56} -2895.99 q^{58} -1031.67i q^{59} -5163.04 q^{61} -796.988i q^{62} -2568.11 q^{64} +3210.70i q^{65} +4066.21 q^{67} +1919.30i q^{68} -876.476 q^{70} +2637.50i q^{71} +5276.42 q^{73} +5342.12i q^{74} -1375.96 q^{76} -6669.32i q^{77} -4908.41 q^{79} +21.9461i q^{80} +1491.31 q^{82} -3614.96i q^{83} +2182.15 q^{85} -6885.26i q^{86} -13552.3 q^{88} -8334.74i q^{89} -9065.96 q^{91} +4653.99i q^{92} -1985.87 q^{94} +1564.40i q^{95} +639.836 q^{97} +3487.34i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 256 q^{4} + 52 q^{7} - 20 q^{13} + 2048 q^{16} + 508 q^{19} - 1344 q^{22} - 4000 q^{25} - 1664 q^{28} + 2944 q^{31} + 1188 q^{34} + 2068 q^{37} - 3300 q^{40} - 1136 q^{43} + 5724 q^{46} + 3348 q^{49}+ \cdots - 46532 q^{97}+O(q^{100}) $ 32 * q - 256 * q^4 + 52 * q^7 - 20 * q^13 + 2048 * q^16 + 508 * q^19 - 1344 * q^22 - 4000 * q^25 - 1664 * q^28 + 2944 * q^31 + 1188 * q^34 + 2068 * q^37 - 3300 * q^40 - 1136 * q^43 + 5724 * q^46 + 3348 * q^49 - 3068 * q^52 + 16380 * q^58 + 8956 * q^61 - 37216 * q^64 - 25676 * q^67 - 4800 * q^70 + 29236 * q^73 - 12380 * q^76 + 7024 * q^79 + 34692 * q^82 + 8700 * q^85 + 32256 * q^88 - 48904 * q^91 - 10236 * q^94 - 46532 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.48322i	− 0.620806i	−0.950605	−	0.310403i	$-0.899536\pi$
$2$	− 2.48322i	− 0.620806i	0.950605	−	0.310403i	$-0.100464\pi$
$3$	0	0
$3$	0	0
$4$	9.83359	0.614600
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	31.5696	0.644278	0.322139	−	0.946692i	$-0.395598\pi$
$7$	31.5696	0.644278	0.322139	+	0.946692i	$0.395598\pi$
$8$	− 64.1506i	− 1.00235i
$9$	0	0
$10$	−27.7633	−0.277633
$11$	− 211.258i	− 1.74593i	−0.487783	−	0.872965i	$-0.662194\pi$
$11$	− 211.258i	− 1.74593i	0.487783	−	0.872965i	$-0.337806\pi$
$12$	0	0
$13$	−287.174	−1.69925	−0.849627	−	0.527385i	$-0.823173\pi$
$13$	−287.174	−1.69925	−0.849627	+	0.527385i	$0.823173\pi$
$14$	− 78.3944i	− 0.399972i
$15$	0	0
$16$	−1.96292	−0.00766764
$17$	195.177i	0.675355i	0.941262	+	0.337677i	$0.109641\pi$
$17$	195.177i	0.675355i	−0.941262	+	0.337677i	$0.890359\pi$
$18$	0	0
$19$	−139.924	−0.387601	−0.193801	−	0.981041i	$-0.562081\pi$
$19$	−139.924	−0.387601	−0.193801	+	0.981041i	$0.562081\pi$
$20$	− 109.943i	− 0.274857i
$21$	0	0
$22$	−524.600	−1.08388
$23$	473.275i	0.894659i	0.894369	+	0.447330i	$0.147625\pi$
$23$	473.275i	0.894659i	−0.894369	+	0.447330i	$0.852375\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	713.117i	1.05491i
$27$	0	0
$28$	310.443	0.395973
$29$	− 1166.22i	− 1.38671i	−0.720597	−	0.693354i	$-0.756132\pi$
$29$	− 1166.22i	− 1.38671i	0.720597	−	0.693354i	$-0.243868\pi$
$30$	0	0
$31$	320.949	0.333974	0.166987	−	0.985959i	$-0.446596\pi$
$31$	320.949	0.333974	0.166987	+	0.985959i	$0.446596\pi$
$32$	− 1021.54i	− 0.997593i
$33$	0	0
$34$	484.670	0.419264
$35$	− 352.959i	− 0.288130i
$36$	0	0
$37$	−2151.28	−1.57143	−0.785713	−	0.618591i	$-0.787704\pi$
$37$	−2151.28	−1.57143	−0.785713	+	0.618591i	$0.787704\pi$
$38$	347.463i	0.240625i
$39$	0	0
$40$	−717.226	−0.448266
$41$	600.555i	0.357261i	0.983916	+	0.178630i	$0.0571666\pi$
$41$	600.555i	0.357261i	−0.983916	+	0.178630i	$0.942833\pi$
$42$	0	0
$43$	2772.71	1.49957	0.749786	−	0.661681i	$-0.230157\pi$
$43$	2772.71	1.49957	0.749786	+	0.661681i	$0.230157\pi$
$44$	− 2077.42i	− 1.07305i
$45$	0	0
$46$	1175.25	0.555410
$47$	− 799.712i	− 0.362025i	−0.983481	−	0.181012i	$-0.942063\pi$
$47$	− 799.712i	− 0.362025i	0.983481	−	0.181012i	$-0.0579374\pi$
$48$	0	0
$49$	−1404.36	−0.584906
$50$	310.403i	0.124161i
$51$	0	0
$52$	−2823.95	−1.04436
$53$	78.0197i	0.0277749i	0.999904	+	0.0138875i	$0.00442066\pi$
$53$	78.0197i	0.0277749i	−0.999904	+	0.0138875i	$0.995579\pi$
$54$	0	0
$55$	−2361.93	−0.780804
$56$	− 2025.21i	− 0.645794i
$57$	0	0
$58$	−2895.99	−0.860877
$59$	− 1031.67i	− 0.296371i	−0.988960	−	0.148185i	$-0.952657\pi$
$59$	− 1031.67i	− 0.296371i	0.988960	−	0.148185i	$-0.0473432\pi$
$60$	0	0
$61$	−5163.04	−1.38754	−0.693771	−	0.720196i	$-0.744052\pi$
$61$	−5163.04	−1.38754	−0.693771	+	0.720196i	$0.744052\pi$
$62$	− 796.988i	− 0.207333i
$63$	0	0
$64$	−2568.11	−0.626980
$65$	3210.70i	0.759929i
$66$	0	0
$67$	4066.21	0.905817	0.452909	−	0.891557i	$-0.350386\pi$
$67$	4066.21	0.905817	0.452909	+	0.891557i	$0.350386\pi$
$68$	1919.30i	0.415073i
$69$	0	0
$70$	−876.476	−0.178873
$71$	2637.50i	0.523209i	0.965175	+	0.261604i	$0.0842516\pi$
$71$	2637.50i	0.523209i	−0.965175	+	0.261604i	$0.915748\pi$
$72$	0	0
$73$	5276.42	0.990133	0.495067	−	0.868855i	$-0.335144\pi$
$73$	5276.42	0.990133	0.495067	+	0.868855i	$0.335144\pi$
$74$	5342.12i	0.975551i
$75$	0	0
$76$	−1375.96	−0.238219
$77$	− 6669.32i	− 1.12486i
$78$	0	0
$79$	−4908.41	−0.786479	−0.393239	−	0.919436i	$-0.628646\pi$
$79$	−4908.41	−0.786479	−0.393239	+	0.919436i	$0.628646\pi$
$80$	21.9461i	0.00342907i
$81$	0	0
$82$	1491.31	0.221790
$83$	− 3614.96i	− 0.524744i	−0.964967	−	0.262372i	$-0.915495\pi$
$83$	− 3614.96i	− 0.524744i	0.964967	−	0.262372i	$-0.0845047\pi$
$84$	0	0
$85$	2182.15	0.302028
$86$	− 6885.26i	− 0.930943i
$87$	0	0
$88$	−13552.3	−1.75004
$89$	− 8334.74i	− 1.05223i	−0.850412	−	0.526117i	$-0.823647\pi$
$89$	− 8334.74i	− 1.05223i	0.850412	−	0.526117i	$-0.176353\pi$
$90$	0	0
$91$	−9065.96	−1.09479
$92$	4653.99i	0.549857i
$93$	0	0
$94$	−1985.87	−0.224747
$95$	1564.40i	0.173340i
$96$	0	0
$97$	639.836	0.0680026	0.0340013	−	0.999422i	$-0.489175\pi$
$97$	639.836	0.0680026	0.0340013	+	0.999422i	$0.489175\pi$
$98$	3487.34i	0.363113i
$99$	0	0
$100$	−1229.20	−0.122920
$101$	1787.44i	0.175222i	0.996155	+	0.0876112i	$0.0279233\pi$
$101$	1787.44i	0.175222i	−0.996155	+	0.0876112i	$0.972077\pi$
$102$	0	0
$103$	−6451.66	−0.608131	−0.304065	−	0.952651i	$-0.598344\pi$
$103$	−6451.66	−0.608131	−0.304065	+	0.952651i	$0.598344\pi$
$104$	18422.4i	1.70325i
$105$	0	0
$106$	193.741	0.0172428
$107$	4943.00i	0.431741i	0.976422	+	0.215870i	$0.0692589\pi$
$107$	4943.00i	0.431741i	−0.976422	+	0.215870i	$0.930741\pi$
$108$	0	0
$109$	−19023.5	−1.60117	−0.800586	−	0.599217i	$-0.795479\pi$
$109$	−19023.5	−1.60117	−0.800586	+	0.599217i	$0.795479\pi$
$110$	5865.21i	0.484728i
$111$	0	0
$112$	−61.9685	−0.00494009
$113$	− 4934.25i	− 0.386424i	−0.981157	−	0.193212i	$-0.938109\pi$
$113$	− 4934.25i	− 0.386424i	0.981157	−	0.193212i	$-0.0618905\pi$
$114$	0	0
$115$	5291.37	0.400104
$116$	− 11468.1i	− 0.852270i
$117$	0	0
$118$	−2561.86	−0.183989
$119$	6161.68i	0.435116i
$120$	0	0
$121$	−29988.8	−2.04827
$122$	12821.0i	0.861395i
$123$	0	0
$124$	3156.08	0.205260
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	70.4302	0.00436668	0.00218334	−	0.999998i	$-0.499305\pi$
$127$	70.4302	0.00436668	0.00218334	+	0.999998i	$0.499305\pi$
$128$	− 9967.38i	− 0.608360i
$129$	0	0
$130$	7972.89	0.471769
$131$	7673.53i	0.447150i	0.974687	+	0.223575i	$0.0717727\pi$
$131$	7673.53i	0.447150i	−0.974687	+	0.223575i	$0.928227\pi$
$132$	0	0
$133$	−4417.35	−0.249723
$134$	− 10097.3i	− 0.562337i
$135$	0	0
$136$	12520.8	0.676944
$137$	− 7910.18i	− 0.421449i	−0.977545	−	0.210725i	$-0.932418\pi$
$137$	− 7910.18i	− 0.421449i	0.977545	−	0.210725i	$-0.0675823\pi$
$138$	0	0
$139$	21356.5	1.10535	0.552676	−	0.833396i	$-0.313607\pi$
$139$	21356.5	1.10535	0.552676	+	0.833396i	$0.313607\pi$
$140$	− 3470.86i	− 0.177084i
$141$	0	0
$142$	6549.50	0.324811
$143$	60667.6i	2.96678i
$144$	0	0
$145$	−13038.8	−0.620155
$146$	− 13102.5i	− 0.614681i
$147$	0	0
$148$	−21154.8	−0.965798
$149$	− 34609.9i	− 1.55894i	−0.626442	−	0.779468i	$-0.715490\pi$
$149$	− 34609.9i	− 1.55894i	0.626442	−	0.779468i	$-0.284510\pi$
$150$	0	0
$151$	−4580.10	−0.200873	−0.100436	−	0.994943i	$-0.532024\pi$
$151$	−4580.10	−0.200873	−0.100436	+	0.994943i	$0.532024\pi$
$152$	8976.21i	0.388513i
$153$	0	0
$154$	−16561.4	−0.698323
$155$	− 3588.32i	− 0.149358i
$156$	0	0
$157$	−14695.0	−0.596168	−0.298084	−	0.954540i	$-0.596348\pi$
$157$	−14695.0	−0.596168	−0.298084	+	0.954540i	$0.596348\pi$
$158$	12188.7i	0.488251i
$159$	0	0
$160$	−11421.1	−0.446137
$161$	14941.1i	0.576409i
$162$	0	0
$163$	30100.9	1.13293	0.566466	−	0.824085i	$-0.308310\pi$
$163$	30100.9	1.13293	0.566466	+	0.824085i	$0.308310\pi$
$164$	5905.62i	0.219572i
$165$	0	0
$166$	−8976.76	−0.325764
$167$	21055.5i	0.754975i	0.926015	+	0.377487i	$0.123212\pi$
$167$	21055.5i	0.754975i	−0.926015	+	0.377487i	$0.876788\pi$
$168$	0	0
$169$	53907.8	1.88746
$170$	− 5418.77i	− 0.187501i
$171$	0	0
$172$	27265.7	0.921636
$173$	− 29421.9i	− 0.983058i	−0.870861	−	0.491529i	$-0.836438\pi$
$173$	− 29421.9i	− 0.983058i	0.870861	−	0.491529i	$-0.163562\pi$
$174$	0	0
$175$	−3946.20	−0.128856
$176$	414.681i	0.0133872i
$177$	0	0
$178$	−20697.0	−0.653233
$179$	− 53541.9i	− 1.67104i	−0.549458	−	0.835522i	$-0.685166\pi$
$179$	− 53541.9i	− 1.67104i	0.549458	−	0.835522i	$-0.314834\pi$
$180$	0	0
$181$	−4168.77	−0.127248	−0.0636239	−	0.997974i	$-0.520266\pi$
$181$	−4168.77	−0.127248	−0.0636239	+	0.997974i	$0.520266\pi$
$182$	22512.8i	0.679653i
$183$	0	0
$184$	30360.9	0.896765
$185$	24052.1i	0.702763i
$186$	0	0
$187$	41232.7	1.17912
$188$	− 7864.05i	− 0.222500i
$189$	0	0
$190$	3884.75	0.107611
$191$	− 33845.5i	− 0.927756i	−0.885899	−	0.463878i	$-0.846458\pi$
$191$	− 33845.5i	− 0.927756i	0.885899	−	0.463878i	$-0.153542\pi$
$192$	0	0
$193$	26676.7	0.716173	0.358087	−	0.933688i	$-0.383429\pi$
$193$	26676.7	0.716173	0.358087	+	0.933688i	$0.383429\pi$
$194$	− 1588.86i	− 0.0422164i
$195$	0	0
$196$	−13809.9	−0.359483
$197$	− 7023.20i	− 0.180968i	−0.995898	−	0.0904842i	$-0.971159\pi$
$197$	− 7023.20i	− 0.180968i	0.995898	−	0.0904842i	$-0.0288415\pi$
$198$	0	0
$199$	25748.9	0.650208	0.325104	−	0.945678i	$-0.394601\pi$
$199$	25748.9	0.650208	0.325104	+	0.945678i	$0.394601\pi$
$200$	8018.83i	0.200471i
$201$	0	0
$202$	4438.63	0.108779
$203$	− 36817.2i	− 0.893425i
$204$	0	0
$205$	6714.41	0.159772
$206$	16020.9i	0.377531i
$207$	0	0
$208$	563.698	0.0130293
$209$	29560.0i	0.676724i
$210$	0	0
$211$	53711.2	1.20642	0.603212	−	0.797581i	$-0.293887\pi$
$211$	53711.2	1.20642	0.603212	+	0.797581i	$0.293887\pi$
$212$	767.214i	0.0170705i
$213$	0	0
$214$	12274.6	0.268027
$215$	− 30999.8i	− 0.670629i
$216$	0	0
$217$	10132.2	0.215172
$218$	47239.7i	0.994018i
$219$	0	0
$220$	−23226.3	−0.479882
$221$	− 56049.9i	− 1.14760i
$222$	0	0
$223$	55373.6	1.11351	0.556754	−	0.830678i	$-0.312047\pi$
$223$	55373.6	1.11351	0.556754	+	0.830678i	$0.312047\pi$
$224$	− 32249.5i	− 0.642727i
$225$	0	0
$226$	−12252.9	−0.239894
$227$	11848.1i	0.229931i	0.993369	+	0.114966i	$0.0366758\pi$
$227$	11848.1i	0.229931i	−0.993369	+	0.114966i	$0.963324\pi$
$228$	0	0
$229$	91890.5	1.75226	0.876132	−	0.482071i	$-0.160115\pi$
$229$	91890.5	1.75226	0.876132	+	0.482071i	$0.160115\pi$
$230$	− 13139.7i	− 0.248387i
$231$	0	0
$232$	−74813.8	−1.38997
$233$	− 87547.4i	− 1.61262i	−0.591495	−	0.806309i	$-0.701462\pi$
$233$	− 87547.4i	− 1.61262i	0.591495	−	0.806309i	$-0.298538\pi$
$234$	0	0
$235$	−8941.06	−0.161902
$236$	− 10145.0i	− 0.182149i
$237$	0	0
$238$	15300.8	0.270123
$239$	− 49057.2i	− 0.858829i	−0.903108	−	0.429414i	$-0.858720\pi$
$239$	− 49057.2i	− 0.858829i	0.903108	−	0.429414i	$-0.141280\pi$
$240$	0	0
$241$	−96597.1	−1.66315	−0.831573	−	0.555416i	$-0.812559\pi$
$241$	−96597.1	−1.66315	−0.831573	+	0.555416i	$0.812559\pi$
$242$	74468.8i	1.27158i
$243$	0	0
$244$	−50771.3	−0.852783
$245$	15701.2i	0.261578i
$246$	0	0
$247$	40182.5	0.658632
$248$	− 20589.1i	− 0.334760i
$249$	0	0
$250$	3470.41	0.0555266
$251$	− 52332.9i	− 0.830668i	−0.909669	−	0.415334i	$-0.863665\pi$
$251$	− 52332.9i	− 0.830668i	0.909669	−	0.415334i	$-0.136335\pi$
$252$	0	0
$253$	99982.8	1.56201
$254$	− 174.894i	− 0.00271086i
$255$	0	0
$256$	−65841.0	−1.00465
$257$	66847.7i	1.01209i	0.862506	+	0.506046i	$0.168893\pi$
$257$	66847.7i	1.01209i	−0.862506	+	0.506046i	$0.831107\pi$
$258$	0	0
$259$	−67915.1	−1.01243
$260$	31572.7i	0.467052i
$261$	0	0
$262$	19055.1	0.277593
$263$	− 78933.4i	− 1.14117i	−0.821240	−	0.570583i	$-0.806717\pi$
$263$	− 78933.4i	− 1.14117i	0.821240	−	0.570583i	$-0.193283\pi$
$264$	0	0
$265$	872.287	0.0124213
$266$	10969.3i	0.155029i
$267$	0	0
$268$	39985.5	0.556715
$269$	52284.6i	0.722553i	0.932459	+	0.361276i	$0.117659\pi$
$269$	52284.6i	0.722553i	−0.932459	+	0.361276i	$0.882341\pi$
$270$	0	0
$271$	34857.3	0.474631	0.237315	−	0.971433i	$-0.423733\pi$
$271$	34857.3	0.474631	0.237315	+	0.971433i	$0.423733\pi$
$272$	− 383.117i	− 0.00517838i
$273$	0	0
$274$	−19642.8	−0.261638
$275$	26407.2i	0.349186i
$276$	0	0
$277$	104584.	1.36303	0.681513	−	0.731806i	$-0.261322\pi$
$277$	104584.	1.36303	0.681513	+	0.731806i	$0.261322\pi$
$278$	− 53033.0i	− 0.686210i
$279$	0	0
$280$	−22642.5	−0.288808
$281$	132643.i	1.67985i	0.542699	+	0.839927i	$0.317403\pi$
$281$	132643.i	1.67985i	−0.542699	+	0.839927i	$0.682597\pi$
$282$	0	0
$283$	−26871.7	−0.335523	−0.167761	−	0.985828i	$-0.553654\pi$
$283$	−26871.7	−0.335523	−0.167761	+	0.985828i	$0.553654\pi$
$284$	25936.1i	0.321564i
$285$	0	0
$286$	150651.	1.84179
$287$	18959.3i	0.230175i
$288$	0	0
$289$	45426.8	0.543896
$290$	32378.2i	0.384996i
$291$	0	0
$292$	51886.2	0.608536
$293$	2807.88i	0.0327072i	0.999866	+	0.0163536i	$0.00520575\pi$
$293$	2807.88i	0.0327072i	−0.999866	+	0.0163536i	$0.994794\pi$
$294$	0	0
$295$	−11534.4	−0.132541
$296$	138006.i	1.57512i
$297$	0	0
$298$	−85944.2	−0.967797
$299$	− 135912.i	− 1.52025i
$300$	0	0
$301$	87533.3	0.966141
$302$	11373.4i	0.124703i
$303$	0	0
$304$	274.659	0.00297199
$305$	57724.6i	0.620528i
$306$	0	0
$307$	168172.	1.78434	0.892170	−	0.451700i	$-0.149182\pi$
$307$	168172.	1.78434	0.892170	+	0.451700i	$0.149182\pi$
$308$	− 65583.4i	− 0.691341i
$309$	0	0
$310$	−8910.60	−0.0927221
$311$	120530.i	1.24616i	0.782157	+	0.623081i	$0.214119\pi$
$311$	120530.i	1.24616i	−0.782157	+	0.623081i	$0.785881\pi$
$312$	0	0
$313$	68971.9	0.704018	0.352009	−	0.935997i	$-0.385499\pi$
$313$	68971.9	0.704018	0.352009	+	0.935997i	$0.385499\pi$
$314$	36490.9i	0.370105i
$315$	0	0
$316$	−48267.3	−0.483369
$317$	92846.1i	0.923943i	0.886895	+	0.461971i	$0.152858\pi$
$317$	92846.1i	0.923943i	−0.886895	+	0.461971i	$0.847142\pi$
$318$	0	0
$319$	−246373.	−2.42110
$320$	28712.3i	0.280394i
$321$	0	0
$322$	37102.1	0.357838
$323$	− 27310.0i	− 0.261768i
$324$	0	0
$325$	35896.7	0.339851
$326$	− 74747.3i	− 0.703332i
$327$	0	0
$328$	38526.0	0.358101
$329$	− 25246.6i	− 0.233244i
$330$	0	0
$331$	3208.25	0.0292828	0.0146414	−	0.999893i	$-0.495339\pi$
$331$	3208.25	0.0292828	0.0146414	+	0.999893i	$0.495339\pi$
$332$	− 35548.0i	− 0.322507i
$333$	0	0
$334$	52285.5	0.468693
$335$	− 45461.6i	− 0.405094i
$336$	0	0
$337$	111346.	0.980423	0.490212	−	0.871603i	$-0.336919\pi$
$337$	111346.	0.980423	0.490212	+	0.871603i	$0.336919\pi$
$338$	− 133865.i	− 1.17175i
$339$	0	0
$340$	21458.4	0.185626
$341$	− 67802.9i	− 0.583095i
$342$	0	0
$343$	−120134.	−1.02112
$344$	− 177871.i	− 1.50310i
$345$	0	0
$346$	−73061.3	−0.610289
$347$	57203.0i	0.475073i	0.971379	+	0.237536i	$0.0763399\pi$
$347$	57203.0i	0.475073i	−0.971379	+	0.237536i	$0.923660\pi$
$348$	0	0
$349$	16501.6	0.135480	0.0677402	−	0.997703i	$-0.478421\pi$
$349$	16501.6	0.135480	0.0677402	+	0.997703i	$0.478421\pi$
$350$	9799.30i	0.0799943i
$351$	0	0
$352$	−215807.	−1.74173
$353$	139822.i	1.12209i	0.827787	+	0.561043i	$0.189600\pi$
$353$	139822.i	1.12209i	−0.827787	+	0.561043i	$0.810400\pi$
$354$	0	0
$355$	29488.1	0.233986
$356$	− 81960.5i	− 0.646702i
$357$	0	0
$358$	−132957.	−1.03739
$359$	− 122208.i	− 0.948222i	−0.880465	−	0.474111i	$-0.842770\pi$
$359$	− 122208.i	− 0.948222i	0.880465	−	0.474111i	$-0.157230\pi$
$360$	0	0
$361$	−110742.	−0.849765
$362$	10352.0i	0.0789963i
$363$	0	0
$364$	−89151.0	−0.672858
$365$	− 58992.2i	− 0.442801i
$366$	0	0
$367$	48313.8	0.358707	0.179353	−	0.983785i	$-0.442599\pi$
$367$	48313.8	0.358707	0.179353	+	0.983785i	$0.442599\pi$
$368$	− 928.998i	− 0.00685992i
$369$	0	0
$370$	59726.7	0.436280
$371$	2463.05i	0.0178948i
$372$	0	0
$373$	67714.9	0.486706	0.243353	−	0.969938i	$-0.421753\pi$
$373$	67714.9	0.486706	0.243353	+	0.969938i	$0.421753\pi$
$374$	− 102390.i	− 0.732006i
$375$	0	0
$376$	−51302.0	−0.362877
$377$	334908.i	2.35637i
$378$	0	0
$379$	241316.	1.67999	0.839997	−	0.542591i	$-0.182557\pi$
$379$	241316.	1.67999	0.839997	+	0.542591i	$0.182557\pi$
$380$	15383.7i	0.106535i
$381$	0	0
$382$	−84045.9	−0.575957
$383$	219647.i	1.49737i	0.662928	+	0.748683i	$0.269314\pi$
$383$	219647.i	1.49737i	−0.662928	+	0.748683i	$0.730686\pi$
$384$	0	0
$385$	−74565.3	−0.503054
$386$	− 66244.4i	− 0.444605i
$387$	0	0
$388$	6291.89	0.0417944
$389$	− 127127.i	− 0.840115i	−0.907498	−	0.420057i	$-0.862010\pi$
$389$	− 127127.i	− 0.840115i	0.907498	−	0.420057i	$-0.137990\pi$
$390$	0	0
$391$	−92372.5	−0.604212
$392$	90090.6i	0.586283i
$393$	0	0
$394$	−17440.2	−0.112346
$395$	54877.7i	0.351724i
$396$	0	0
$397$	−110607.	−0.701783	−0.350891	−	0.936416i	$-0.614121\pi$
$397$	−110607.	−0.701783	−0.350891	+	0.936416i	$0.614121\pi$
$398$	− 63940.3i	− 0.403653i
$399$	0	0
$400$	245.365	0.00153353
$401$	211742.i	1.31679i	0.752670	+	0.658397i	$0.228765\pi$
$401$	211742.i	1.31679i	−0.752670	+	0.658397i	$0.771235\pi$
$402$	0	0
$403$	−92168.1	−0.567506
$404$	17577.0i	0.107692i
$405$	0	0
$406$	−91425.3	−0.554644
$407$	454475.i	2.74360i
$408$	0	0
$409$	129566.	0.774539	0.387270	−	0.921967i	$-0.373418\pi$
$409$	129566.	0.774539	0.387270	+	0.921967i	$0.373418\pi$
$410$	− 16673.4i	− 0.0991873i
$411$	0	0
$412$	−63443.0	−0.373757
$413$	− 32569.3i	− 0.190945i
$414$	0	0
$415$	−40416.5	−0.234673
$416$	293358.i	1.69516i
$417$	0	0
$418$	73404.1	0.420115
$419$	− 154082.i	− 0.877654i	−0.898572	−	0.438827i	$-0.855394\pi$
$419$	− 154082.i	− 0.877654i	0.898572	−	0.438827i	$-0.144606\pi$
$420$	0	0
$421$	−15732.9	−0.0887655	−0.0443827	−	0.999015i	$-0.514132\pi$
$421$	−15732.9	−0.0887655	−0.0443827	+	0.999015i	$0.514132\pi$
$422$	− 133377.i	− 0.748956i
$423$	0	0
$424$	5005.01	0.0278403
$425$	− 24397.2i	− 0.135071i
$426$	0	0
$427$	−162995.	−0.893963
$428$	48607.4i	0.265348i
$429$	0	0
$430$	−76979.5	−0.416331
$431$	− 285514.i	− 1.53699i	−0.639853	−	0.768497i	$-0.721005\pi$
$431$	− 285514.i	− 1.53699i	0.639853	−	0.768497i	$-0.278995\pi$
$432$	0	0
$433$	−266643.	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$433$	−266643.	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$434$	− 25160.6i	− 0.133580i
$435$	0	0
$436$	−187070.	−0.984080
$437$	− 66222.5i	− 0.346771i
$438$	0	0
$439$	19440.0	0.100871	0.0504355	−	0.998727i	$-0.483939\pi$
$439$	19440.0	0.100871	0.0504355	+	0.998727i	$0.483939\pi$
$440$	151519.i	0.782641i
$441$	0	0
$442$	−139184.	−0.712436
$443$	− 30760.3i	− 0.156741i	−0.996924	−	0.0783705i	$-0.975028\pi$
$443$	− 30760.3i	− 0.156741i	0.996924	−	0.0783705i	$-0.0249717\pi$
$444$	0	0
$445$	−93185.3	−0.470573
$446$	− 137505.i	− 0.691272i
$447$	0	0
$448$	−81074.2	−0.403949
$449$	67901.9i	0.336813i	0.985718	+	0.168407i	$0.0538622\pi$
$449$	67901.9i	0.336813i	−0.985718	+	0.168407i	$0.946138\pi$
$450$	0	0
$451$	126872.	0.623752
$452$	− 48521.4i	− 0.237496i
$453$	0	0
$454$	29421.6	0.142743
$455$	101361.i	0.489605i
$456$	0	0
$457$	−343613.	−1.64527	−0.822635	−	0.568570i	$-0.807497\pi$
$457$	−343613.	−1.64527	−0.822635	+	0.568570i	$0.807497\pi$
$458$	− 228185.i	− 1.08782i
$459$	0	0
$460$	52033.2	0.245904
$461$	− 267166.i	− 1.25713i	−0.777757	−	0.628565i	$-0.783643\pi$
$461$	− 267166.i	− 1.25713i	0.777757	−	0.628565i	$-0.216357\pi$
$462$	0	0
$463$	−66193.0	−0.308780	−0.154390	−	0.988010i	$-0.549341\pi$
$463$	−66193.0	−0.308780	−0.154390	+	0.988010i	$0.549341\pi$
$464$	2289.20i	0.0106328i
$465$	0	0
$466$	−217400.	−1.00112
$467$	65412.7i	0.299936i	0.988691	+	0.149968i	$0.0479171\pi$
$467$	65412.7i	0.299936i	−0.988691	+	0.149968i	$0.952083\pi$
$468$	0	0
$469$	128369.	0.583598
$470$	22202.7i	0.100510i
$471$	0	0
$472$	−66182.1	−0.297068
$473$	− 585756.i	− 2.61815i
$474$	0	0
$475$	17490.5	0.0775202
$476$	60591.4i	0.267422i
$477$	0	0
$478$	−121820.	−0.533166
$479$	− 323660.i	− 1.41065i	−0.708886	−	0.705323i	$-0.750802\pi$
$479$	− 323660.i	− 1.41065i	0.708886	−	0.705323i	$-0.249198\pi$
$480$	0	0
$481$	617792.	2.67025
$482$	239872.i	1.03249i
$483$	0	0
$484$	−294897.	−1.25887
$485$	− 7153.59i	− 0.0304117i
$486$	0	0
$487$	15920.9	0.0671287	0.0335644	−	0.999437i	$-0.489314\pi$
$487$	15920.9	0.0671287	0.0335644	+	0.999437i	$0.489314\pi$
$488$	331213.i	1.39081i
$489$	0	0
$490$	38989.7	0.162389
$491$	− 185776.i	− 0.770595i	−0.922792	−	0.385298i	$-0.874099\pi$
$491$	− 185776.i	− 0.770595i	0.922792	−	0.385298i	$-0.125901\pi$
$492$	0	0
$493$	227620.	0.936520
$494$	− 99782.2i	− 0.408883i
$495$	0	0
$496$	−629.996	−0.00256079
$497$	83264.7i	0.337092i
$498$	0	0
$499$	−336466.	−1.35127	−0.675633	−	0.737238i	$-0.736129\pi$
$499$	−336466.	−1.35127	−0.675633	+	0.737238i	$0.736129\pi$
$500$	13742.9i	0.0549715i
$501$	0	0
$502$	−129954.	−0.515684
$503$	− 248667.i	− 0.982838i	−0.870923	−	0.491419i	$-0.836478\pi$
$503$	− 248667.i	− 0.982838i	0.870923	−	0.491419i	$-0.163522\pi$
$504$	0	0
$505$	19984.2	0.0783619
$506$	− 248280.i	− 0.969707i
$507$	0	0
$508$	692.582	0.00268376
$509$	− 51470.3i	− 0.198665i	−0.995054	−	0.0993325i	$-0.968329\pi$
$509$	− 51470.3i	− 0.198665i	0.995054	−	0.0993325i	$-0.0316707\pi$
$510$	0	0
$511$	166575.	0.637921
$512$	4019.95i	0.0153349i
$513$	0	0
$514$	165998.	0.628313
$515$	72131.8i	0.271964i
$516$	0	0
$517$	−168945.	−0.632070
$518$	168649.i	0.628526i
$519$	0	0
$520$	205968.	0.761718
$521$	− 58808.5i	− 0.216653i	−0.994115	−	0.108326i	$-0.965451\pi$
$521$	− 58808.5i	− 0.216653i	0.994115	−	0.108326i	$-0.0345492\pi$
$522$	0	0
$523$	79752.3	0.291568	0.145784	−	0.989316i	$-0.453430\pi$
$523$	79752.3	0.291568	0.145784	+	0.989316i	$0.453430\pi$
$524$	75458.4i	0.274818i
$525$	0	0
$526$	−196009.	−0.708444
$527$	62642.0i	0.225551i
$528$	0	0
$529$	55852.1	0.199585
$530$	− 2166.08i	− 0.00771123i
$531$	0	0
$532$	−43438.4	−0.153480
$533$	− 172464.i	− 0.607076i
$534$	0	0
$535$	55264.4	0.193080
$536$	− 260850.i	− 0.907949i
$537$	0	0
$538$	129835.	0.448565
$539$	296682.i	1.02121i
$540$	0	0
$541$	−164771.	−0.562970	−0.281485	−	0.959566i	$-0.590827\pi$
$541$	−164771.	−0.562970	−0.281485	+	0.959566i	$0.590827\pi$
$542$	− 86558.6i	− 0.294654i
$543$	0	0
$544$	199381.	0.673729
$545$	212690.i	0.716066i
$546$	0	0
$547$	223754.	0.747818	0.373909	−	0.927465i	$-0.378017\pi$
$547$	223754.	0.747818	0.373909	+	0.927465i	$0.378017\pi$
$548$	− 77785.5i	− 0.259023i
$549$	0	0
$550$	65575.0	0.216777
$551$	163182.i	0.537490i
$552$	0	0
$553$	−154957.	−0.506711
$554$	− 259705.i	− 0.846175i
$555$	0	0
$556$	210011.	0.679349
$557$	435479.i	1.40364i	0.712352	+	0.701822i	$0.247630\pi$
$557$	435479.i	1.40364i	−0.712352	+	0.701822i	$0.752370\pi$
$558$	0	0
$559$	−796249.	−2.54815
$560$	692.829i	0.00220928i
$561$	0	0
$562$	329382.	1.04286
$563$	452468.i	1.42748i	0.700409	+	0.713742i	$0.253001\pi$
$563$	452468.i	1.42748i	−0.700409	+	0.713742i	$0.746999\pi$
$564$	0	0
$565$	−55166.6	−0.172814
$566$	66728.4i	0.208295i
$567$	0	0
$568$	169197.	0.524440
$569$	221247.i	0.683367i	0.939815	+	0.341683i	$0.110997\pi$
$569$	221247.i	0.683367i	−0.939815	+	0.341683i	$0.889003\pi$
$570$	0	0
$571$	198050.	0.607438	0.303719	−	0.952762i	$-0.401772\pi$
$571$	198050.	0.607438	0.303719	+	0.952762i	$0.401772\pi$
$572$	596581.i	1.82338i
$573$	0	0
$574$	47080.2	0.142894
$575$	− 59159.3i	− 0.178932i
$576$	0	0
$577$	148915.	0.447288	0.223644	−	0.974671i	$-0.428205\pi$
$577$	148915.	0.447288	0.223644	+	0.974671i	$0.428205\pi$
$578$	− 112805.i	− 0.337654i
$579$	0	0
$580$	−128218.	−0.381147
$581$	− 114123.i	− 0.338081i
$582$	0	0
$583$	16482.3	0.0484931
$584$	− 338486.i	− 0.992463i
$585$	0	0
$586$	6972.60	0.0203048
$587$	− 416126.i	− 1.20767i	−0.797109	−	0.603836i	$-0.793638\pi$
$587$	− 416126.i	− 1.20767i	0.797109	−	0.603836i	$-0.206362\pi$
$588$	0	0
$589$	−44908.4	−0.129449
$590$	28642.5i	0.0822823i
$591$	0	0
$592$	4222.79	0.0120491
$593$	140902.i	0.400690i	0.979725	+	0.200345i	$0.0642062\pi$
$593$	140902.i	0.400690i	−0.979725	+	0.200345i	$0.935794\pi$
$594$	0	0
$595$	68889.6	0.194590
$596$	− 340340.i	− 0.958121i
$597$	0	0
$598$	−337500.	−0.943782
$599$	− 207373.i	− 0.577962i	−0.957335	−	0.288981i	$-0.906684\pi$
$599$	− 207373.i	− 0.577962i	0.957335	−	0.288981i	$-0.0933164\pi$
$600$	0	0
$601$	370713.	1.02633	0.513167	−	0.858289i	$-0.328472\pi$
$601$	370713.	1.02633	0.513167	+	0.858289i	$0.328472\pi$
$602$	− 217365.i	− 0.599786i
$603$	0	0
$604$	−45038.9	−0.123456
$605$	335285.i	0.916015i
$606$	0	0
$607$	−410172.	−1.11324	−0.556619	−	0.830768i	$-0.687902\pi$
$607$	−410172.	−1.11324	−0.556619	+	0.830768i	$0.687902\pi$
$608$	142937.i	0.386668i
$609$	0	0
$610$	143343.	0.385227
$611$	229656.i	0.615171i
$612$	0	0
$613$	207476.	0.552138	0.276069	−	0.961138i	$-0.410968\pi$
$613$	207476.	0.552138	0.276069	+	0.961138i	$0.410968\pi$
$614$	− 417610.i	− 1.10773i
$615$	0	0
$616$	−427841.	−1.12751
$617$	− 172995.i	− 0.454427i	−0.973845	−	0.227213i	$-0.927039\pi$
$617$	− 172995.i	− 0.454427i	0.973845	−	0.227213i	$-0.0729615\pi$
$618$	0	0
$619$	52137.2	0.136071	0.0680357	−	0.997683i	$-0.478327\pi$
$619$	52137.2	0.136071	0.0680357	+	0.997683i	$0.478327\pi$
$620$	− 35286.1i	− 0.0917951i
$621$	0	0
$622$	299303.	0.773626
$623$	− 263125.i	− 0.677931i
$624$	0	0
$625$	15625.0	0.0400000
$626$	− 171273.i	− 0.437058i
$627$	0	0
$628$	−144504.	−0.366405
$629$	− 419882.i	− 1.06127i
$630$	0	0
$631$	629939.	1.58212	0.791061	−	0.611737i	$-0.209529\pi$
$631$	629939.	1.58212	0.791061	+	0.611737i	$0.209529\pi$
$632$	314878.i	0.788330i
$633$	0	0
$634$	230558.	0.573589
$635$	− 787.434i	− 0.00195284i
$636$	0	0
$637$	403295.	0.993904
$638$	611800.i	1.50303i
$639$	0	0
$640$	−111439.	−0.272067
$641$	− 72867.4i	− 0.177344i	−0.996061	−	0.0886721i	$-0.971738\pi$
$641$	− 72867.4i	− 0.177344i	0.996061	−	0.0886721i	$-0.0282623\pi$
$642$	0	0
$643$	498338.	1.20532	0.602659	−	0.797999i	$-0.294108\pi$
$643$	498338.	1.20532	0.602659	+	0.797999i	$0.294108\pi$
$644$	146925.i	0.354261i
$645$	0	0
$646$	−67816.9	−0.162507
$647$	720075.i	1.72016i	0.510158	+	0.860081i	$0.329587\pi$
$647$	720075.i	1.72016i	−0.510158	+	0.860081i	$0.670413\pi$
$648$	0	0
$649$	−217947.	−0.517443
$650$	− 89139.6i	− 0.210981i
$651$	0	0
$652$	296000.	0.696300
$653$	− 640195.i	− 1.50136i	−0.660664	−	0.750681i	$-0.729725\pi$
$653$	− 640195.i	− 1.50136i	0.660664	−	0.750681i	$-0.270275\pi$
$654$	0	0
$655$	85792.7	0.199971
$656$	− 1178.84i	− 0.00273935i
$657$	0	0
$658$	−62693.0	−0.144800
$659$	176509.i	0.406440i	0.979133	+	0.203220i	$0.0651407\pi$
$659$	176509.i	0.406440i	−0.979133	+	0.203220i	$0.934859\pi$
$660$	0	0
$661$	770922.	1.76444	0.882221	−	0.470836i	$-0.156048\pi$
$661$	770922.	1.76444	0.882221	+	0.470836i	$0.156048\pi$
$662$	− 7966.81i	− 0.0181789i
$663$	0	0
$664$	−231902.	−0.525979
$665$	49387.4i	0.111679i
$666$	0	0
$667$	551943.	1.24063
$668$	207051.i	0.464007i
$669$	0	0
$670$	−112891.	−0.251485
$671$	1.09073e6i	2.42255i
$672$	0	0
$673$	226533.	0.500150	0.250075	−	0.968226i	$-0.419545\pi$
$673$	226533.	0.500150	0.250075	+	0.968226i	$0.419545\pi$
$674$	− 276496.i	− 0.608653i
$675$	0	0
$676$	530107.	1.16003
$677$	326316.i	0.711970i	0.934492	+	0.355985i	$0.115855\pi$
$677$	326316.i	0.711970i	−0.934492	+	0.355985i	$0.884145\pi$
$678$	0	0
$679$	20199.4	0.0438126
$680$	− 139986.i	− 0.302739i
$681$	0	0
$682$	−168370.	−0.361989
$683$	358820.i	0.769194i	0.923085	+	0.384597i	$0.125660\pi$
$683$	358820.i	0.769194i	−0.923085	+	0.384597i	$0.874340\pi$
$684$	0	0
$685$	−88438.5	−0.188478
$686$	298319.i	0.633918i
$687$	0	0
$688$	−5442.59	−0.0114982
$689$	− 22405.2i	− 0.0471966i
$690$	0	0
$691$	−526194.	−1.10202	−0.551011	−	0.834498i	$-0.685758\pi$
$691$	−526194.	−1.10202	−0.551011	+	0.834498i	$0.685758\pi$
$692$	− 289323.i	− 0.604187i
$693$	0	0
$694$	142048.	0.294928
$695$	− 238773.i	− 0.494329i
$696$	0	0
$697$	−117215.	−0.241278
$698$	− 40977.3i	− 0.0841071i
$699$	0	0
$700$	−38805.3	−0.0791946
$701$	270484.i	0.550434i	0.961382	+	0.275217i	$0.0887497\pi$
$701$	270484.i	0.550434i	−0.961382	+	0.275217i	$0.911250\pi$
$702$	0	0
$703$	301016.	0.609086
$704$	542533.i	1.09466i
$705$	0	0
$706$	347209.	0.696597
$707$	56428.9i	0.112892i
$708$	0	0
$709$	45089.7	0.0896985	0.0448492	−	0.998994i	$-0.485719\pi$
$709$	45089.7	0.0896985	0.0448492	+	0.998994i	$0.485719\pi$
$710$	− 73225.6i	− 0.145260i
$711$	0	0
$712$	−534679.	−1.05471
$713$	151897.i	0.298793i
$714$	0	0
$715$	678285.	1.32678
$716$	− 526509.i	− 1.02702i
$717$	0	0
$718$	−303469.	−0.588662
$719$	− 782866.i	− 1.51436i	−0.653206	−	0.757181i	$-0.726576\pi$
$719$	− 782866.i	− 1.51436i	0.653206	−	0.757181i	$-0.273424\pi$
$720$	0	0
$721$	−203676.	−0.391805
$722$	274998.i	0.527540i
$723$	0	0
$724$	−40994.0	−0.0782065
$725$	145778.i	0.277342i
$726$	0	0
$727$	−261925.	−0.495573	−0.247787	−	0.968815i	$-0.579703\pi$
$727$	−261925.	−0.495573	−0.247787	+	0.968815i	$0.579703\pi$
$728$	581587.i	1.09737i
$729$	0	0
$730$	−146491.	−0.274894
$731$	541170.i	1.01274i
$732$	0	0
$733$	320367.	0.596266	0.298133	−	0.954524i	$-0.403636\pi$
$733$	320367.	0.596266	0.298133	+	0.954524i	$0.403636\pi$
$734$	− 119974.i	− 0.222687i
$735$	0	0
$736$	483467.	0.892506
$737$	− 859018.i	− 1.58149i
$738$	0	0
$739$	−796278.	−1.45806	−0.729031	−	0.684481i	$-0.760029\pi$
$739$	−796278.	−1.45806	−0.729031	+	0.684481i	$0.760029\pi$
$740$	236518.i	0.431918i
$741$	0	0
$742$	6116.31	0.0111092
$743$	42572.3i	0.0771169i	0.999256	+	0.0385585i	$0.0122766\pi$
$743$	42572.3i	0.0771169i	−0.999256	+	0.0385585i	$0.987723\pi$
$744$	0	0
$745$	−386951.	−0.697177
$746$	− 168151.i	− 0.302150i
$747$	0	0
$748$	405466.	0.724688
$749$	156049.i	0.278161i
$750$	0	0
$751$	−953913.	−1.69133	−0.845666	−	0.533712i	$-0.820797\pi$
$751$	−953913.	−1.69133	−0.845666	+	0.533712i	$0.820797\pi$
$752$	1569.77i	0.00277587i
$753$	0	0
$754$	831652.	1.46285
$755$	51207.1i	0.0898331i
$756$	0	0
$757$	−854962.	−1.49195	−0.745976	−	0.665973i	$-0.768017\pi$
$757$	−854962.	−1.49195	−0.745976	+	0.665973i	$0.768017\pi$
$758$	− 599242.i	− 1.04295i
$759$	0	0
$760$	100357.	0.173748
$761$	546283.i	0.943297i	0.881787	+	0.471649i	$0.156341\pi$
$761$	546283.i	0.943297i	−0.881787	+	0.471649i	$0.843659\pi$
$762$	0	0
$763$	−600566.	−1.03160
$764$	− 332823.i	− 0.570199i
$765$	0	0
$766$	545433.	0.929574
$767$	296268.i	0.503609i
$768$	0	0
$769$	819785.	1.38627	0.693134	−	0.720809i	$-0.256229\pi$
$769$	819785.	1.38627	0.693134	+	0.720809i	$0.256229\pi$
$770$	185162.i	0.312299i
$771$	0	0
$772$	262328.	0.440160
$773$	− 793360.i	− 1.32773i	−0.747851	−	0.663867i	$-0.768914\pi$
$773$	− 793360.i	− 1.32773i	0.747851	−	0.663867i	$-0.231086\pi$
$774$	0	0
$775$	−40118.6	−0.0667948
$776$	− 41045.9i	− 0.0681626i
$777$	0	0
$778$	−315685.	−0.521548
$779$	− 84032.1i	− 0.138475i
$780$	0	0
$781$	557191.	0.913486
$782$	229382.i	0.375099i
$783$	0	0
$784$	2756.64	0.00448485
$785$	164295.i	0.266615i
$786$	0	0
$787$	−313788.	−0.506625	−0.253313	−	0.967384i	$-0.581520\pi$
$787$	−313788.	−0.506625	−0.253313	+	0.967384i	$0.581520\pi$
$788$	− 69063.3i	− 0.111223i
$789$	0	0
$790$	136274.	0.218352
$791$	− 155772.i	− 0.248964i
$792$	0	0
$793$	1.48269e6	2.35779
$794$	274663.i	0.435671i
$795$	0	0
$796$	253204.	0.399618
$797$	− 1.01463e6i	− 1.59732i	−0.601782	−	0.798660i	$-0.705542\pi$
$797$	− 1.01463e6i	− 1.59732i	0.601782	−	0.798660i	$-0.294458\pi$
$798$	0	0
$799$	156086.	0.244495
$800$	127692.i	0.199519i
$801$	0	0
$802$	525803.	0.817474
$803$	− 1.11468e6i	− 1.72870i
$804$	0	0
$805$	167047.	0.257778
$806$	228874.i	0.352311i
$807$	0	0
$808$	114666.	0.175635
$809$	581506.i	0.888500i	0.895903	+	0.444250i	$0.146530\pi$
$809$	581506.i	0.888500i	−0.895903	+	0.444250i	$0.853470\pi$
$810$	0	0
$811$	−470704.	−0.715660	−0.357830	−	0.933787i	$-0.616483\pi$
$811$	−470704.	−0.715660	−0.357830	+	0.933787i	$0.616483\pi$
$812$	− 362045.i	− 0.549099i
$813$	0	0
$814$	1.12856e6	1.70324
$815$	− 336538.i	− 0.506663i
$816$	0	0
$817$	−387968.	−0.581236
$818$	− 321741.i	− 0.480839i
$819$	0	0
$820$	66026.8	0.0981957
$821$	− 931729.i	− 1.38230i	−0.722710	−	0.691151i	$-0.757104\pi$
$821$	− 931729.i	− 1.38230i	0.722710	−	0.691151i	$-0.242896\pi$
$822$	0	0
$823$	−485058.	−0.716134	−0.358067	−	0.933696i	$-0.616564\pi$
$823$	−485058.	−0.716134	−0.358067	+	0.933696i	$0.616564\pi$
$824$	413878.i	0.609562i
$825$	0	0
$826$	−80876.9	−0.118540
$827$	− 73886.9i	− 0.108033i	−0.998540	−	0.0540165i	$-0.982798\pi$
$827$	− 73886.9i	− 0.108033i	0.998540	−	0.0540165i	$-0.0172024\pi$
$828$	0	0
$829$	1.14364e6	1.66410	0.832050	−	0.554701i	$-0.187167\pi$
$829$	1.14364e6	1.66410	0.832050	+	0.554701i	$0.187167\pi$
$830$	100363.i	0.145686i
$831$	0	0
$832$	737494.	1.06540
$833$	− 274099.i	− 0.395019i
$834$	0	0
$835$	235408.	0.337635
$836$	290681.i	0.415915i
$837$	0	0
$838$	−382620.	−0.544853
$839$	245686.i	0.349025i	0.984655	+	0.174512i	$0.0558349\pi$
$839$	245686.i	0.349025i	−0.984655	+	0.174512i	$0.944165\pi$
$840$	0	0
$841$	−652791.	−0.922959
$842$	39068.3i	0.0551062i
$843$	0	0
$844$	528174.	0.741468
$845$	− 602707.i	− 0.844098i
$846$	0	0
$847$	−946733.	−1.31966
$848$	− 153.146i	0 0.000212968i
$849$	0	0
$850$	−60583.7	−0.0838529
$851$	− 1.01815e6i	− 1.40589i
$852$	0	0
$853$	165938.	0.228059	0.114030	−	0.993477i	$-0.463624\pi$
$853$	165938.	0.228059	0.114030	+	0.993477i	$0.463624\pi$
$854$	404754.i	0.554977i
$855$	0	0
$856$	317096.	0.432757
$857$	− 682428.i	− 0.929171i	−0.885528	−	0.464585i	$-0.846203\pi$
$857$	− 682428.i	− 0.929171i	0.885528	−	0.464585i	$-0.153797\pi$
$858$	0	0
$859$	−995813.	−1.34956	−0.674779	−	0.738020i	$-0.735761\pi$
$859$	−995813.	−1.34956	−0.674779	+	0.738020i	$0.735761\pi$
$860$	− 304840.i	− 0.412168i
$861$	0	0
$862$	−708994.	−0.954176
$863$	935842.i	1.25655i	0.777990	+	0.628277i	$0.216240\pi$
$863$	935842.i	1.25655i	−0.777990	+	0.628277i	$0.783760\pi$
$864$	0	0
$865$	−328947.	−0.439637
$866$	662135.i	0.882898i
$867$	0	0
$868$	99636.2	0.132245
$869$	1.03694e6i	1.37314i
$870$	0	0
$871$	−1.16771e6	−1.53921
$872$	1.22037e6i	1.60494i
$873$	0	0
$874$	−164445.	−0.215277
$875$	44119.9i	0.0576260i
$876$	0	0
$877$	−341461.	−0.443957	−0.221979	−	0.975052i	$-0.571252\pi$
$877$	−341461.	−0.443957	−0.221979	+	0.975052i	$0.571252\pi$
$878$	− 48273.8i	− 0.0626214i
$879$	0	0
$880$	4636.27	0.00598692
$881$	96624.3i	0.124490i	0.998061	+	0.0622450i	$0.0198260\pi$
$881$	96624.3i	0.124490i	−0.998061	+	0.0622450i	$0.980174\pi$
$882$	0	0
$883$	−1.17097e6	−1.50184	−0.750921	−	0.660392i	$-0.770390\pi$
$883$	−1.17097e6	−1.50184	−0.750921	+	0.660392i	$0.770390\pi$
$884$	− 551172.i	− 0.705314i
$885$	0	0
$886$	−76384.7	−0.0973058
$887$	741294.i	0.942201i	0.882080	+	0.471100i	$0.156143\pi$
$887$	741294.i	0.942201i	−0.882080	+	0.471100i	$0.843857\pi$
$888$	0	0
$889$	2223.45	0.00281336
$890$	231400.i	0.292135i
$891$	0	0
$892$	544522.	0.684361
$893$	111899.i	0.140321i
$894$	0	0
$895$	−598617.	−0.747313
$896$	− 314666.i	− 0.391953i
$897$	0	0
$898$	168616.	0.209096
$899$	− 374297.i	− 0.463124i
$900$	0	0
$901$	−15227.7	−0.0187579
$902$	− 315051.i	− 0.387229i
$903$	0	0
$904$	−316535.	−0.387334
$905$	46608.2i	0.0569070i
$906$	0	0
$907$	566509.	0.688640	0.344320	−	0.938852i	$-0.388109\pi$
$907$	566509.	0.688640	0.344320	+	0.938852i	$0.388109\pi$
$908$	116510.i	0.141316i
$909$	0	0
$910$	251701.	0.303950
$911$	− 979262.i	− 1.17995i	−0.807423	−	0.589973i	$-0.799138\pi$
$911$	− 979262.i	− 1.17995i	0.807423	−	0.589973i	$-0.200862\pi$
$912$	0	0
$913$	−763688.	−0.916166
$914$	853268.i	1.02139i
$915$	0	0
$916$	903614.	1.07694
$917$	242250.i	0.288089i
$918$	0	0
$919$	440425.	0.521483	0.260742	−	0.965409i	$-0.416033\pi$
$919$	440425.	0.521483	0.260742	+	0.965409i	$0.416033\pi$
$920$	− 339445.i	− 0.401045i
$921$	0	0
$922$	−663434.	−0.780434
$923$	− 757420.i	− 0.889064i
$924$	0	0
$925$	268910.	0.314285
$926$	164372.i	0.191693i
$927$	0	0
$928$	−1.19134e6	−1.38337
$929$	202277.i	0.234377i	0.993110	+	0.117189i	$0.0373882\pi$
$929$	202277.i	0.234377i	−0.993110	+	0.117189i	$0.962612\pi$
$930$	0	0
$931$	196504.	0.226710
$932$	− 860906.i	− 0.991114i
$933$	0	0
$934$	162435.	0.186202
$935$	− 460996.i	− 0.527319i
$936$	0	0
$937$	567578.	0.646468	0.323234	−	0.946319i	$-0.395230\pi$
$937$	567578.	0.646468	0.323234	+	0.946319i	$0.395230\pi$
$938$	− 318769.i	− 0.362301i
$939$	0	0
$940$	−87922.7	−0.0995051
$941$	788530.i	0.890510i	0.895404	+	0.445255i	$0.146887\pi$
$941$	788530.i	0.890510i	−0.895404	+	0.445255i	$0.853113\pi$
$942$	0	0
$943$	−284227.	−0.319626
$944$	2025.08i	0.00227247i
$945$	0	0
$946$	−1.45456e6	−1.62536
$947$	455348.i	0.507742i	0.967238	+	0.253871i	$0.0817039\pi$
$947$	455348.i	0.507742i	−0.967238	+	0.253871i	$0.918296\pi$
$948$	0	0
$949$	−1.51525e6	−1.68249
$950$	− 43432.8i	− 0.0481250i
$951$	0	0
$952$	395275.	0.436140
$953$	− 1.71277e6i	− 1.88587i	−0.332971	−	0.942937i	$-0.608051\pi$
$953$	− 1.71277e6i	− 1.88587i	0.332971	−	0.942937i	$-0.391949\pi$
$954$	0	0
$955$	−378404.	−0.414905
$956$	− 482408.i	− 0.527836i
$957$	0	0
$958$	−803721.	−0.875738
$959$	− 249721.i	− 0.271530i
$960$	0	0
$961$	−820513.	−0.888461
$962$	− 1.53412e6i	− 1.65771i
$963$	0	0
$964$	−949897.	−1.02217
$965$	− 298255.i	− 0.320283i
$966$	0	0
$967$	789798.	0.844623	0.422312	−	0.906451i	$-0.361219\pi$
$967$	789798.	0.844623	0.422312	+	0.906451i	$0.361219\pi$
$968$	1.92380e6i	2.05309i
$969$	0	0
$970$	−17764.0	−0.0188798
$971$	34673.8i	0.0367759i	0.999831	+	0.0183880i	$0.00585340\pi$
$971$	34673.8i	0.0367759i	−0.999831	+	0.0183880i	$0.994147\pi$
$972$	0	0
$973$	674217.	0.712154
$974$	− 39535.1i	− 0.0416739i
$975$	0	0
$976$	10134.6	0.0106392
$977$	191652.i	0.200782i	0.994948	+	0.100391i	$0.0320094\pi$
$977$	191652.i	0.200782i	−0.994948	+	0.100391i	$0.967991\pi$
$978$	0	0
$979$	−1.76078e6	−1.83713
$980$	154399.i	0.160766i
$981$	0	0
$982$	−461323.	−0.478390
$983$	894753.i	0.925968i	0.886367	+	0.462984i	$0.153221\pi$
$983$	894753.i	0.925968i	−0.886367	+	0.462984i	$0.846779\pi$
$984$	0	0
$985$	−78521.8	−0.0809315
$986$	− 565232.i	− 0.581397i
$987$	0	0
$988$	395138.	0.404795
$989$	1.31225e6i	1.34161i
$990$	0	0
$991$	−619450.	−0.630752	−0.315376	−	0.948967i	$-0.602131\pi$
$991$	−619450.	−0.630752	−0.315376	+	0.948967i	$0.602131\pi$
$992$	− 327861.i	− 0.333170i
$993$	0	0
$994$	206765.	0.209269
$995$	− 287881.i	− 0.290782i
$996$	0	0
$997$	369211.	0.371437	0.185718	−	0.982603i	$-0.440539\pi$
$997$	369211.	0.371437	0.185718	+	0.982603i	$0.440539\pi$
$998$	835522.i	0.838874i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.c.b.161.13		32
3.2	odd	2	inner	405.5.c.b.161.20		32
9.2	odd	6		135.5.i.a.71.10		32
9.4	even	3		135.5.i.a.116.10		32
9.5	odd	6		45.5.i.a.11.7	✓	32
9.7	even	3		45.5.i.a.41.7	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.i.a.11.7	✓	32	9.5	odd	6
45.5.i.a.41.7	yes	32	9.7	even	3
135.5.i.a.71.10		32	9.2	odd	6
135.5.i.a.116.10		32	9.4	even	3
405.5.c.b.161.13		32	1.1	even	1	trivial
405.5.c.b.161.20		32	3.2	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	405.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.48322i q^{2} +9.83359 q^{4} -11.1803i q^{5} +31.5696 q^{7} -64.1506i q^{8} -27.7633 q^{10} -211.258i q^{11} -287.174 q^{13} -78.3944i q^{14} -1.96292 q^{16} +195.177i q^{17} -139.924 q^{19} -109.943i q^{20} -524.600 q^{22} +473.275i q^{23} -125.000 q^{25} +713.117i q^{26} +310.443 q^{28} -1166.22i q^{29} +320.949 q^{31} -1021.54i q^{32} +484.670 q^{34} -352.959i q^{35} -2151.28 q^{37} +347.463i q^{38} -717.226 q^{40} +600.555i q^{41} +2772.71 q^{43} -2077.42i q^{44} +1175.25 q^{46} -799.712i q^{47} -1404.36 q^{49} +310.403i q^{50} -2823.95 q^{52} +78.0197i q^{53} -2361.93 q^{55} -2025.21i q^{56} -2895.99 q^{58} -1031.67i q^{59} -5163.04 q^{61} -796.988i q^{62} -2568.11 q^{64} +3210.70i q^{65} +4066.21 q^{67} +1919.30i q^{68} -876.476 q^{70} +2637.50i q^{71} +5276.42 q^{73} +5342.12i q^{74} -1375.96 q^{76} -6669.32i q^{77} -4908.41 q^{79} +21.9461i q^{80} +1491.31 q^{82} -3614.96i q^{83} +2182.15 q^{85} -6885.26i q^{86} -13552.3 q^{88} -8334.74i q^{89} -9065.96 q^{91} +4653.99i q^{92} -1985.87 q^{94} +1564.40i q^{95} +639.836 q^{97} +3487.34i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} + 52 q^{7} - 20 q^{13} + 2048 q^{16} + 508 q^{19} - 1344 q^{22} - 4000 q^{25} - 1664 q^{28} + 2944 q^{31} + 1188 q^{34} + 2068 q^{37} - 3300 q^{40} - 1136 q^{43} + 5724 q^{46} + 3348 q^{49}+ \cdots - 46532 q^{97}+O(q^{100}) \) 32 * q - 256 * q^4 + 52 * q^7 - 20 * q^13 + 2048 * q^16 + 508 * q^19 - 1344 * q^22 - 4000 * q^25 - 1664 * q^28 + 2944 * q^31 + 1188 * q^34 + 2068 * q^37 - 3300 * q^40 - 1136 * q^43 + 5724 * q^46 + 3348 * q^49 - 3068 * q^52 + 16380 * q^58 + 8956 * q^61 - 37216 * q^64 - 25676 * q^67 - 4800 * q^70 + 29236 * q^73 - 12380 * q^76 + 7024 * q^79 + 34692 * q^82 + 8700 * q^85 + 32256 * q^88 - 48904 * q^91 - 10236 * q^94 - 46532 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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