LMFDB - Embedded newform 405.5.c.b.161.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,5,Mod(161,405)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			161.9
Character	$\chi$	$=$	405.161
Dual form			405.5.c.b.161.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.73920i q^{2} +2.01838 q^{4} -11.1803i q^{5} +39.4150 q^{7} -67.3743i q^{8} -41.8055 q^{10} +194.836i q^{11} -96.0237 q^{13} -147.381i q^{14} -219.632 q^{16} -404.005i q^{17} -559.350 q^{19} -22.5662i q^{20} +728.531 q^{22} -780.548i q^{23} -125.000 q^{25} +359.052i q^{26} +79.5545 q^{28} -478.560i q^{29} +622.668 q^{31} -256.741i q^{32} -1510.65 q^{34} -440.673i q^{35} +1404.72 q^{37} +2091.52i q^{38} -753.268 q^{40} -100.775i q^{41} -598.795 q^{43} +393.253i q^{44} -2918.63 q^{46} -2136.73i q^{47} -847.457 q^{49} +467.400i q^{50} -193.812 q^{52} -2456.58i q^{53} +2178.33 q^{55} -2655.56i q^{56} -1789.43 q^{58} -4176.14i q^{59} +691.586 q^{61} -2328.28i q^{62} -4474.12 q^{64} +1073.58i q^{65} -5439.08 q^{67} -815.435i q^{68} -1647.77 q^{70} +199.240i q^{71} -5155.25 q^{73} -5252.52i q^{74} -1128.98 q^{76} +7679.46i q^{77} -5537.57 q^{79} +2455.56i q^{80} -376.819 q^{82} +952.982i q^{83} -4516.91 q^{85} +2239.02i q^{86} +13126.9 q^{88} +8559.26i q^{89} -3784.78 q^{91} -1575.44i q^{92} -7989.67 q^{94} +6253.72i q^{95} -5110.24 q^{97} +3168.81i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.73920i	− 0.934800i	−0.884046	−	0.467400i	$-0.845191\pi$
$2$	− 3.73920i	− 0.934800i	0.884046	−	0.467400i	$-0.154809\pi$
$3$	0	0
$3$	0	0
$4$	2.01838	0.126149
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	39.4150	0.804388	0.402194	−	0.915554i	$-0.368248\pi$
$7$	39.4150	0.804388	0.402194	+	0.915554i	$0.368248\pi$
$8$	− 67.3743i	− 1.05272i
$9$	0	0
$10$	−41.8055	−0.418055
$11$	194.836i	1.61021i	0.593129	+	0.805107i	$0.297892\pi$
$11$	194.836i	1.61021i	−0.593129	+	0.805107i	$0.702108\pi$
$12$	0	0
$13$	−96.0237	−0.568188	−0.284094	−	0.958796i	$-0.591693\pi$
$13$	−96.0237	−0.568188	−0.284094	+	0.958796i	$0.591693\pi$
$14$	− 147.381i	− 0.751942i
$15$	0	0
$16$	−219.632	−0.857938
$17$	− 404.005i	− 1.39794i	−0.715151	−	0.698970i	$-0.753642\pi$
$17$	− 404.005i	− 1.39794i	0.715151	−	0.698970i	$-0.246358\pi$
$18$	0	0
$19$	−559.350	−1.54945	−0.774723	−	0.632301i	$-0.782111\pi$
$19$	−559.350	−1.54945	−0.774723	+	0.632301i	$0.782111\pi$
$20$	− 22.5662i	− 0.0564154i
$21$	0	0
$22$	728.531	1.50523
$23$	− 780.548i	− 1.47552i	−0.675065	−	0.737758i	$-0.735884\pi$
$23$	− 780.548i	− 1.47552i	0.675065	−	0.737758i	$-0.264116\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	359.052i	0.531142i
$27$	0	0
$28$	79.5545	0.101473
$29$	− 478.560i	− 0.569037i	−0.958671	−	0.284518i	$-0.908166\pi$
$29$	− 478.560i	− 0.569037i	0.958671	−	0.284518i	$-0.0918337\pi$
$30$	0	0
$31$	622.668	0.647938	0.323969	−	0.946068i	$-0.394983\pi$
$31$	622.668	0.647938	0.323969	+	0.946068i	$0.394983\pi$
$32$	− 256.741i	− 0.250724i
$33$	0	0
$34$	−1510.65	−1.30679
$35$	− 440.673i	− 0.359733i
$36$	0	0
$37$	1404.72	1.02609	0.513045	−	0.858362i	$-0.328517\pi$
$37$	1404.72	1.02609	0.513045	+	0.858362i	$0.328517\pi$
$38$	2091.52i	1.44842i
$39$	0	0
$40$	−753.268	−0.470792
$41$	− 100.775i	− 0.0599496i	−0.999551	−	0.0299748i	$-0.990457\pi$
$41$	− 100.775i	− 0.0599496i	0.999551	−	0.0299748i	$-0.00954270\pi$
$42$	0	0
$43$	−598.795	−0.323848	−0.161924	−	0.986803i	$-0.551770\pi$
$43$	−598.795	−0.323848	−0.161924	+	0.986803i	$0.551770\pi$
$44$	393.253i	0.203127i
$45$	0	0
$46$	−2918.63	−1.37931
$47$	− 2136.73i	− 0.967285i	−0.875266	−	0.483642i	$-0.839314\pi$
$47$	− 2136.73i	− 0.967285i	0.875266	−	0.483642i	$-0.160686\pi$
$48$	0	0
$49$	−847.457	−0.352960
$50$	467.400i	0.186960i
$51$	0	0
$52$	−193.812	−0.0716762
$53$	− 2456.58i	− 0.874540i	−0.899330	−	0.437270i	$-0.855945\pi$
$53$	− 2456.58i	− 0.874540i	0.899330	−	0.437270i	$-0.144055\pi$
$54$	0	0
$55$	2178.33	0.720110
$56$	− 2655.56i	− 0.846798i
$57$	0	0
$58$	−1789.43	−0.531936
$59$	− 4176.14i	− 1.19970i	−0.800114	−	0.599848i	$-0.795228\pi$
$59$	− 4176.14i	− 1.19970i	0.800114	−	0.599848i	$-0.204772\pi$
$60$	0	0
$61$	691.586	0.185860	0.0929301	−	0.995673i	$-0.470377\pi$
$61$	691.586	0.185860	0.0929301	+	0.995673i	$0.470377\pi$
$62$	− 2328.28i	− 0.605692i
$63$	0	0
$64$	−4474.12	−1.09231
$65$	1073.58i	0.254101i
$66$	0	0
$67$	−5439.08	−1.21165	−0.605823	−	0.795599i	$-0.707156\pi$
$67$	−5439.08	−1.21165	−0.605823	+	0.795599i	$0.707156\pi$
$68$	− 815.435i	− 0.176348i
$69$	0	0
$70$	−1647.77	−0.336279
$71$	199.240i	0.0395239i	0.999805	+	0.0197620i	$0.00629084\pi$
$71$	199.240i	0.0395239i	−0.999805	+	0.0197620i	$0.993709\pi$
$72$	0	0
$73$	−5155.25	−0.967395	−0.483698	−	0.875235i	$-0.660707\pi$
$73$	−5155.25	−0.967395	−0.483698	+	0.875235i	$0.660707\pi$
$74$	− 5252.52i	− 0.959189i
$75$	0	0
$76$	−1128.98	−0.195461
$77$	7679.46i	1.29524i
$78$	0	0
$79$	−5537.57	−0.887288	−0.443644	−	0.896203i	$-0.646315\pi$
$79$	−5537.57	−0.887288	−0.443644	+	0.896203i	$0.646315\pi$
$80$	2455.56i	0.383681i
$81$	0	0
$82$	−376.819	−0.0560409
$83$	952.982i	0.138334i	0.997605	+	0.0691670i	$0.0220341\pi$
$83$	952.982i	0.138334i	−0.997605	+	0.0691670i	$0.977966\pi$
$84$	0	0
$85$	−4516.91	−0.625178
$86$	2239.02i	0.302733i
$87$	0	0
$88$	13126.9	1.69511
$89$	8559.26i	1.08058i	0.841479	+	0.540289i	$0.181685\pi$
$89$	8559.26i	1.08058i	−0.841479	+	0.540289i	$0.818315\pi$
$90$	0	0
$91$	−3784.78	−0.457043
$92$	− 1575.44i	− 0.186135i
$93$	0	0
$94$	−7989.67	−0.904218
$95$	6253.72i	0.692933i
$96$	0	0
$97$	−5110.24	−0.543122	−0.271561	−	0.962421i	$-0.587540\pi$
$97$	−5110.24	−0.543122	−0.271561	+	0.962421i	$0.587540\pi$
$98$	3168.81i	0.329947i
$99$	0	0
$100$	−252.298	−0.0252298
$101$	12646.4i	1.23973i	0.784710	+	0.619863i	$0.212812\pi$
$101$	12646.4i	1.23973i	−0.784710	+	0.619863i	$0.787188\pi$
$102$	0	0
$103$	7094.84	0.668756	0.334378	−	0.942439i	$-0.391474\pi$
$103$	7094.84	0.668756	0.334378	+	0.942439i	$0.391474\pi$
$104$	6469.54i	0.598145i
$105$	0	0
$106$	−9185.66	−0.817520
$107$	− 8389.28i	− 0.732752i	−0.930467	−	0.366376i	$-0.880598\pi$
$107$	− 8389.28i	− 0.732752i	0.930467	−	0.366376i	$-0.119402\pi$
$108$	0	0
$109$	18772.4	1.58004	0.790018	−	0.613084i	$-0.210071\pi$
$109$	18772.4	1.58004	0.790018	+	0.613084i	$0.210071\pi$
$110$	− 8145.22i	− 0.673159i
$111$	0	0
$112$	−8656.80	−0.690115
$113$	− 9416.81i	− 0.737474i	−0.929534	−	0.368737i	$-0.879790\pi$
$113$	− 9416.81i	− 0.737474i	0.929534	−	0.368737i	$-0.120210\pi$
$114$	0	0
$115$	−8726.80	−0.659871
$116$	− 965.916i	− 0.0717833i
$117$	0	0
$118$	−15615.4	−1.12148
$119$	− 15923.8i	− 1.12449i
$120$	0	0
$121$	−23320.0	−1.59279
$122$	− 2585.98i	− 0.173742i
$123$	0	0
$124$	1256.78	0.0817366
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	2810.49	0.174251	0.0871253	−	0.996197i	$-0.472232\pi$
$127$	2810.49	0.174251	0.0871253	+	0.996197i	$0.472232\pi$
$128$	12621.8i	0.770372i
$129$	0	0
$130$	4014.32	0.237534
$131$	− 4025.51i	− 0.234573i	−0.993098	−	0.117287i	$-0.962580\pi$
$131$	− 4025.51i	− 0.234573i	0.993098	−	0.117287i	$-0.0374196\pi$
$132$	0	0
$133$	−22046.8	−1.24636
$134$	20337.8i	1.13265i
$135$	0	0
$136$	−27219.5	−1.47164
$137$	30008.6i	1.59884i	0.600773	+	0.799419i	$0.294859\pi$
$137$	30008.6i	1.59884i	−0.600773	+	0.799419i	$0.705141\pi$
$138$	0	0
$139$	36870.4	1.90831	0.954153	−	0.299319i	$-0.0967595\pi$
$139$	36870.4	1.90831	0.954153	+	0.299319i	$0.0967595\pi$
$140$	− 889.446i	− 0.0453799i
$141$	0	0
$142$	744.999	0.0369470
$143$	− 18708.9i	− 0.914904i
$144$	0	0
$145$	−5350.46	−0.254481
$146$	19276.5i	0.904321i
$147$	0	0
$148$	2835.25	0.129440
$149$	1198.03i	0.0539629i	0.999636	+	0.0269815i	$0.00858951\pi$
$149$	1198.03i	0.0539629i	−0.999636	+	0.0269815i	$0.991410\pi$
$150$	0	0
$151$	−4160.96	−0.182490	−0.0912452	−	0.995828i	$-0.529085\pi$
$151$	−4160.96	−0.182490	−0.0912452	+	0.995828i	$0.529085\pi$
$152$	37685.8i	1.63114i
$153$	0	0
$154$	28715.0	1.21079
$155$	− 6961.64i	− 0.289767i
$156$	0	0
$157$	34111.3	1.38388	0.691941	−	0.721954i	$-0.256756\pi$
$157$	34111.3	1.38388	0.691941	+	0.721954i	$0.256756\pi$
$158$	20706.1i	0.829437i
$159$	0	0
$160$	−2870.45	−0.112127
$161$	− 30765.3i	− 1.18689i
$162$	0	0
$163$	−44109.2	−1.66017	−0.830087	−	0.557634i	$-0.811709\pi$
$163$	−44109.2	−1.66017	−0.830087	+	0.557634i	$0.811709\pi$
$164$	− 203.403i	− 0.00756257i
$165$	0	0
$166$	3563.39	0.129315
$167$	− 1388.36i	− 0.0497816i	−0.999690	−	0.0248908i	$-0.992076\pi$
$167$	− 1388.36i	− 0.0497816i	0.999690	−	0.0248908i	$-0.00792380\pi$
$168$	0	0
$169$	−19340.4	−0.677163
$170$	16889.6i	0.584416i
$171$	0	0
$172$	−1208.60	−0.0408530
$173$	37629.2i	1.25728i	0.777695	+	0.628642i	$0.216389\pi$
$173$	37629.2i	1.25728i	−0.777695	+	0.628642i	$0.783611\pi$
$174$	0	0
$175$	−4926.88	−0.160878
$176$	− 42792.2i	− 1.38146i
$177$	0	0
$178$	32004.8	1.01012
$179$	− 36571.1i	− 1.14138i	−0.821164	−	0.570692i	$-0.806675\pi$
$179$	− 36571.1i	− 1.14138i	0.821164	−	0.570692i	$-0.193325\pi$
$180$	0	0
$181$	6564.99	0.200390	0.100195	−	0.994968i	$-0.468053\pi$
$181$	6564.99	0.200390	0.100195	+	0.994968i	$0.468053\pi$
$182$	14152.0i	0.427244i
$183$	0	0
$184$	−52588.9	−1.55331
$185$	− 15705.2i	− 0.458881i
$186$	0	0
$187$	78714.6	2.25098
$188$	− 4312.74i	− 0.122022i
$189$	0	0
$190$	23383.9	0.647754
$191$	− 37757.9i	− 1.03500i	−0.855682	−	0.517501i	$-0.826862\pi$
$191$	− 37757.9i	− 1.03500i	0.855682	−	0.517501i	$-0.173138\pi$
$192$	0	0
$193$	10782.7	0.289477	0.144738	−	0.989470i	$-0.453766\pi$
$193$	10782.7	0.289477	0.144738	+	0.989470i	$0.453766\pi$
$194$	19108.2i	0.507711i
$195$	0	0
$196$	−1710.49	−0.0445255
$197$	− 10726.4i	− 0.276390i	−0.990405	−	0.138195i	$-0.955870\pi$
$197$	− 10726.4i	− 0.276390i	0.990405	−	0.138195i	$-0.0441301\pi$
$198$	0	0
$199$	73986.5	1.86830	0.934150	−	0.356882i	$-0.116160\pi$
$199$	73986.5	1.86830	0.934150	+	0.356882i	$0.116160\pi$
$200$	8421.79i	0.210545i
$201$	0	0
$202$	47287.6	1.15890
$203$	− 18862.5i	− 0.457726i
$204$	0	0
$205$	−1126.70	−0.0268103
$206$	− 26529.0i	− 0.625153i
$207$	0	0
$208$	21089.9	0.487470
$209$	− 108981.i	− 2.49494i
$210$	0	0
$211$	1863.40	0.0418545	0.0209272	−	0.999781i	$-0.493338\pi$
$211$	1863.40	0.0418545	0.0209272	+	0.999781i	$0.493338\pi$
$212$	− 4958.32i	− 0.110322i
$213$	0	0
$214$	−31369.2	−0.684977
$215$	6694.73i	0.144829i
$216$	0	0
$217$	24542.5	0.521193
$218$	− 70193.8i	− 1.47702i
$219$	0	0
$220$	4396.70	0.0908410
$221$	38794.0i	0.794292i
$222$	0	0
$223$	58130.7	1.16895	0.584475	−	0.811411i	$-0.301300\pi$
$223$	58130.7	1.16895	0.584475	+	0.811411i	$0.301300\pi$
$224$	− 10119.5i	− 0.201679i
$225$	0	0
$226$	−35211.3	−0.689391
$227$	− 35042.0i	− 0.680045i	−0.940417	−	0.340022i	$-0.889565\pi$
$227$	− 35042.0i	− 0.680045i	0.940417	−	0.340022i	$-0.110435\pi$
$228$	0	0
$229$	−35184.8	−0.670940	−0.335470	−	0.942051i	$-0.608895\pi$
$229$	−35184.8	−0.670940	−0.335470	+	0.942051i	$0.608895\pi$
$230$	32631.2i	0.616848i
$231$	0	0
$232$	−32242.7	−0.599039
$233$	− 45567.1i	− 0.839342i	−0.907676	−	0.419671i	$-0.862145\pi$
$233$	− 45567.1i	− 0.839342i	0.907676	−	0.419671i	$-0.137855\pi$
$234$	0	0
$235$	−23889.4	−0.432583
$236$	− 8429.05i	− 0.151340i
$237$	0	0
$238$	−59542.5	−1.05117
$239$	− 33460.4i	− 0.585781i	−0.956146	−	0.292891i	$-0.905383\pi$
$239$	− 33460.4i	− 0.585781i	0.956146	−	0.292891i	$-0.0946172\pi$
$240$	0	0
$241$	15156.6	0.260957	0.130478	−	0.991451i	$-0.458349\pi$
$241$	15156.6	0.260957	0.130478	+	0.991451i	$0.458349\pi$
$242$	87198.3i	1.48894i
$243$	0	0
$244$	1395.88	0.0234460
$245$	9474.86i	0.157849i
$246$	0	0
$247$	53710.9	0.880376
$248$	− 41951.9i	− 0.682100i
$249$	0	0
$250$	5225.69	0.0836111
$251$	105439.i	1.67361i	0.547498	+	0.836807i	$0.315581\pi$
$251$	105439.i	1.67361i	−0.547498	+	0.836807i	$0.684419\pi$
$252$	0	0
$253$	152079.	2.37590
$254$	− 10509.0i	− 0.162889i
$255$	0	0
$256$	−24390.6	−0.372171
$257$	48805.9i	0.738935i	0.929244	+	0.369468i	$0.120460\pi$
$257$	48805.9i	0.738935i	−0.929244	+	0.369468i	$0.879540\pi$
$258$	0	0
$259$	55366.9	0.825374
$260$	2166.89i	0.0320546i
$261$	0	0
$262$	−15052.2	−0.219279
$263$	55153.5i	0.797373i	0.917087	+	0.398686i	$0.130534\pi$
$263$	55153.5i	0.797373i	−0.917087	+	0.398686i	$0.869466\pi$
$264$	0	0
$265$	−27465.4	−0.391106
$266$	82437.3i	1.16509i
$267$	0	0
$268$	−10978.1	−0.152848
$269$	20335.5i	0.281028i	0.990079	+	0.140514i	$0.0448755\pi$
$269$	20335.5i	0.281028i	−0.990079	+	0.140514i	$0.955124\pi$
$270$	0	0
$271$	−35462.8	−0.482874	−0.241437	−	0.970416i	$-0.577619\pi$
$271$	−35462.8	−0.482874	−0.241437	+	0.970416i	$0.577619\pi$
$272$	88732.4i	1.19935i
$273$	0	0
$274$	112208.	1.49459
$275$	− 24354.5i	− 0.322043i
$276$	0	0
$277$	131199.	1.70990	0.854948	−	0.518714i	$-0.173589\pi$
$277$	131199.	1.70990	0.854948	+	0.518714i	$0.173589\pi$
$278$	− 137866.i	− 1.78388i
$279$	0	0
$280$	−29690.1	−0.378700
$281$	− 47907.8i	− 0.606727i	−0.952875	−	0.303364i	$-0.901890\pi$
$281$	− 47907.8i	− 0.606727i	0.952875	−	0.303364i	$-0.0981097\pi$
$282$	0	0
$283$	96557.4	1.20563	0.602813	−	0.797883i	$-0.294047\pi$
$283$	96557.4	1.20563	0.602813	+	0.797883i	$0.294047\pi$
$284$	402.142i	0.00498590i
$285$	0	0
$286$	−69956.2	−0.855253
$287$	− 3972.06i	− 0.0482227i
$288$	0	0
$289$	−79698.8	−0.954236
$290$	20006.5i	0.237889i
$291$	0	0
$292$	−10405.3	−0.122036
$293$	− 54246.3i	− 0.631881i	−0.948779	−	0.315940i	$-0.897680\pi$
$293$	− 54246.3i	− 0.631881i	0.948779	−	0.315940i	$-0.102320\pi$
$294$	0	0
$295$	−46690.7	−0.536521
$296$	− 94641.9i	− 1.08019i
$297$	0	0
$298$	4479.68	0.0504445
$299$	74951.2i	0.838371i
$300$	0	0
$301$	−23601.5	−0.260500
$302$	15558.7i	0.170592i
$303$	0	0
$304$	122851.	1.32933
$305$	− 7732.16i	− 0.0831192i
$306$	0	0
$307$	94546.7	1.00316	0.501579	−	0.865112i	$-0.332753\pi$
$307$	94546.7	1.00316	0.501579	+	0.865112i	$0.332753\pi$
$308$	15500.1i	0.163393i
$309$	0	0
$310$	−26031.0	−0.270874
$311$	− 150838.i	− 1.55951i	−0.626082	−	0.779757i	$-0.715343\pi$
$311$	− 150838.i	− 1.55951i	0.626082	−	0.779757i	$-0.284657\pi$
$312$	0	0
$313$	7404.54	0.0755804	0.0377902	−	0.999286i	$-0.487968\pi$
$313$	7404.54	0.0755804	0.0377902	+	0.999286i	$0.487968\pi$
$314$	− 127549.i	− 1.29365i
$315$	0	0
$316$	−11176.9	−0.111930
$317$	34534.4i	0.343664i	0.985126	+	0.171832i	$0.0549686\pi$
$317$	34534.4i	0.343664i	−0.985126	+	0.171832i	$0.945031\pi$
$318$	0	0
$319$	93240.7	0.916272
$320$	50022.2i	0.488498i
$321$	0	0
$322$	−115038.	−1.10950
$323$	225980.i	2.16603i
$324$	0	0
$325$	12003.0	0.113638
$326$	164933.i	1.55193i
$327$	0	0
$328$	−6789.67	−0.0631104
$329$	− 84219.3i	− 0.778072i
$330$	0	0
$331$	−10569.7	−0.0964730	−0.0482365	−	0.998836i	$-0.515360\pi$
$331$	−10569.7	−0.0964730	−0.0482365	+	0.998836i	$0.515360\pi$
$332$	1923.48i	0.0174507i
$333$	0	0
$334$	−5191.35	−0.0465358
$335$	60810.8i	0.541865i
$336$	0	0
$337$	−51655.3	−0.454837	−0.227418	−	0.973797i	$-0.573028\pi$
$337$	−51655.3	−0.454837	−0.227418	+	0.973797i	$0.573028\pi$
$338$	72317.8i	0.633012i
$339$	0	0
$340$	−9116.84	−0.0788654
$341$	121318.i	1.04332i
$342$	0	0
$343$	−128038.	−1.08830
$344$	40343.4i	0.340923i
$345$	0	0
$346$	140703.	1.17531
$347$	149562.i	1.24212i	0.783764	+	0.621059i	$0.213297\pi$
$347$	149562.i	1.24212i	−0.783764	+	0.621059i	$0.786703\pi$
$348$	0	0
$349$	−33931.9	−0.278585	−0.139293	−	0.990251i	$-0.544483\pi$
$349$	−33931.9	−0.278585	−0.139293	+	0.990251i	$0.544483\pi$
$350$	18422.6i	0.150388i
$351$	0	0
$352$	50022.4	0.403719
$353$	− 194511.i	− 1.56097i	−0.625176	−	0.780484i	$-0.714973\pi$
$353$	− 194511.i	− 1.56097i	0.625176	−	0.780484i	$-0.285027\pi$
$354$	0	0
$355$	2227.57	0.0176756
$356$	17275.8i	0.136314i
$357$	0	0
$358$	−136747.	−1.06697
$359$	− 111524.i	− 0.865328i	−0.901555	−	0.432664i	$-0.857574\pi$
$359$	− 111524.i	− 0.865328i	0.901555	−	0.432664i	$-0.142426\pi$
$360$	0	0
$361$	182551.	1.40078
$362$	− 24547.8i	− 0.187325i
$363$	0	0
$364$	−7639.12	−0.0576555
$365$	57637.4i	0.432632i
$366$	0	0
$367$	−109590.	−0.813649	−0.406824	−	0.913506i	$-0.633364\pi$
$367$	−109590.	−0.813649	−0.406824	+	0.913506i	$0.633364\pi$
$368$	171433.i	1.26590i
$369$	0	0
$370$	−58724.9	−0.428962
$371$	− 96826.2i	− 0.703469i
$372$	0	0
$373$	198092.	1.42380	0.711899	−	0.702282i	$-0.247835\pi$
$373$	198092.	1.42380	0.711899	+	0.702282i	$0.247835\pi$
$374$	− 294330.i	− 2.10422i
$375$	0	0
$376$	−143961.	−1.01828
$377$	45953.1i	0.323320i
$378$	0	0
$379$	−210813.	−1.46764	−0.733820	−	0.679344i	$-0.762265\pi$
$379$	−210813.	−1.46764	−0.733820	+	0.679344i	$0.762265\pi$
$380$	12622.4i	0.0874127i
$381$	0	0
$382$	−141184.	−0.967521
$383$	27083.6i	0.184633i	0.995730	+	0.0923165i	$0.0294272\pi$
$383$	27083.6i	0.184633i	−0.995730	+	0.0923165i	$0.970573\pi$
$384$	0	0
$385$	85859.0	0.579248
$386$	− 40318.7i	− 0.270603i
$387$	0	0
$388$	−10314.4	−0.0685142
$389$	103620.i	0.684770i	0.939560	+	0.342385i	$0.111235\pi$
$389$	103620.i	0.684770i	−0.939560	+	0.342385i	$0.888765\pi$
$390$	0	0
$391$	−315345.	−2.06268
$392$	57096.9i	0.371570i
$393$	0	0
$394$	−40108.3	−0.258370
$395$	61911.9i	0.396807i
$396$	0	0
$397$	−7753.40	−0.0491939	−0.0245970	−	0.999697i	$-0.507830\pi$
$397$	−7753.40	−0.0491939	−0.0245970	+	0.999697i	$0.507830\pi$
$398$	− 276650.i	− 1.74649i
$399$	0	0
$400$	27454.0	0.171588
$401$	276889.i	1.72194i	0.508660	+	0.860968i	$0.330141\pi$
$401$	276889.i	1.72194i	−0.508660	+	0.860968i	$0.669859\pi$
$402$	0	0
$403$	−59790.9	−0.368150
$404$	25525.3i	0.156390i
$405$	0	0
$406$	−70530.5	−0.427883
$407$	273689.i	1.65222i
$408$	0	0
$409$	−75753.2	−0.452850	−0.226425	−	0.974029i	$-0.572704\pi$
$409$	−75753.2	−0.452850	−0.226425	+	0.974029i	$0.572704\pi$
$410$	4212.96i	0.0250622i
$411$	0	0
$412$	14320.1	0.0843628
$413$	− 164603.i	− 0.965022i
$414$	0	0
$415$	10654.7	0.0618648
$416$	24653.2i	0.142458i
$417$	0	0
$418$	−407504.	−2.33227
$419$	− 193009.i	− 1.09938i	−0.835368	−	0.549691i	$-0.814746\pi$
$419$	− 193009.i	− 1.09938i	0.835368	−	0.549691i	$-0.185254\pi$
$420$	0	0
$421$	249334.	1.40675	0.703375	−	0.710819i	$-0.251675\pi$
$421$	249334.	1.40675	0.703375	+	0.710819i	$0.251675\pi$
$422$	− 6967.64i	− 0.0391256i
$423$	0	0
$424$	−165511.	−0.920649
$425$	50500.6i	0.279588i
$426$	0	0
$427$	27258.9	0.149504
$428$	− 16932.8i	− 0.0924358i
$429$	0	0
$430$	25032.9	0.135386
$431$	2912.99i	0.0156814i	0.999969	+	0.00784068i	$0.00249579\pi$
$431$	2912.99i	0.0156814i	−0.999969	+	0.00784068i	$0.997504\pi$
$432$	0	0
$433$	100384.	0.535411	0.267706	−	0.963501i	$-0.413735\pi$
$433$	100384.	0.535411	0.267706	+	0.963501i	$0.413735\pi$
$434$	− 91769.2i	− 0.487212i
$435$	0	0
$436$	37889.8	0.199319
$437$	436600.i	2.28623i
$438$	0	0
$439$	324968.	1.68621	0.843105	−	0.537749i	$-0.180725\pi$
$439$	324968.	1.68621	0.843105	+	0.537749i	$0.180725\pi$
$440$	− 146764.i	− 0.758077i
$441$	0	0
$442$	145059.	0.742505
$443$	− 46397.7i	− 0.236423i	−0.992988	−	0.118211i	$-0.962284\pi$
$443$	− 46397.7i	− 0.236423i	0.992988	−	0.118211i	$-0.0377160\pi$
$444$	0	0
$445$	95695.4	0.483249
$446$	− 217363.i	− 1.09274i
$447$	0	0
$448$	−176347.	−0.878644
$449$	− 267544.i	− 1.32710i	−0.748133	−	0.663549i	$-0.769049\pi$
$449$	− 267544.i	− 1.32710i	0.748133	−	0.663549i	$-0.230951\pi$
$450$	0	0
$451$	19634.6	0.0965317
$452$	− 19006.7i	− 0.0930315i
$453$	0	0
$454$	−131029.	−0.635706
$455$	42315.1i	0.204396i
$456$	0	0
$457$	−105948.	−0.507293	−0.253647	−	0.967297i	$-0.581630\pi$
$457$	−105948.	−0.507293	−0.253647	+	0.967297i	$0.581630\pi$
$458$	131563.i	0.627195i
$459$	0	0
$460$	−17614.0	−0.0832419
$461$	− 104388.i	− 0.491187i	−0.969373	−	0.245594i	$-0.921017\pi$
$461$	− 104388.i	− 0.491187i	0.969373	−	0.245594i	$-0.0789829\pi$
$462$	0	0
$463$	−71157.0	−0.331937	−0.165968	−	0.986131i	$-0.553075\pi$
$463$	−71157.0	−0.331937	−0.165968	+	0.986131i	$0.553075\pi$
$464$	105107.i	0.488198i
$465$	0	0
$466$	−170384.	−0.784617
$467$	335261.i	1.53727i	0.639690	+	0.768633i	$0.279063\pi$
$467$	335261.i	1.53727i	−0.639690	+	0.768633i	$0.720937\pi$
$468$	0	0
$469$	−214381.	−0.974634
$470$	89327.2i	0.404378i
$471$	0	0
$472$	−281365.	−1.26295
$473$	− 116667.i	− 0.521465i
$474$	0	0
$475$	69918.7	0.309889
$476$	− 32140.4i	− 0.141853i
$477$	0	0
$478$	−125115.	−0.547589
$479$	− 187820.i	− 0.818599i	−0.912400	−	0.409299i	$-0.865773\pi$
$479$	− 187820.i	− 0.818599i	0.912400	−	0.409299i	$-0.134227\pi$
$480$	0	0
$481$	−134886.	−0.583012
$482$	− 56673.6i	− 0.243942i
$483$	0	0
$484$	−47068.7	−0.200929
$485$	57134.2i	0.242892i
$486$	0	0
$487$	−376679.	−1.58823	−0.794115	−	0.607768i	$-0.792065\pi$
$487$	−376679.	−1.58823	−0.794115	+	0.607768i	$0.792065\pi$
$488$	− 46595.1i	− 0.195659i
$489$	0	0
$490$	35428.4	0.147557
$491$	− 435214.i	− 1.80526i	−0.430416	−	0.902630i	$-0.641633\pi$
$491$	− 435214.i	− 1.80526i	0.430416	−	0.902630i	$-0.358367\pi$
$492$	0	0
$493$	−193341.	−0.795480
$494$	− 200836.i	− 0.822976i
$495$	0	0
$496$	−136758.	−0.555890
$497$	7853.05i	0.0317926i
$498$	0	0
$499$	183510.	0.736985	0.368492	−	0.929631i	$-0.379874\pi$
$499$	183510.	0.736985	0.368492	+	0.929631i	$0.379874\pi$
$500$	2820.77i	0.0112831i
$501$	0	0
$502$	394259.	1.56449
$503$	− 402594.i	− 1.59123i	−0.605806	−	0.795613i	$-0.707149\pi$
$503$	− 402594.i	− 1.59123i	0.605806	−	0.795613i	$-0.292851\pi$
$504$	0	0
$505$	141392.	0.554422
$506$	− 568653.i	− 2.22099i
$507$	0	0
$508$	5672.63	0.0219815
$509$	390783.i	1.50834i	0.656678	+	0.754171i	$0.271961\pi$
$509$	390783.i	1.50834i	−0.656678	+	0.754171i	$0.728039\pi$
$510$	0	0
$511$	−203194.	−0.778161
$512$	293150.i	1.11828i
$513$	0	0
$514$	182495.	0.690757
$515$	− 79322.7i	− 0.299077i
$516$	0	0
$517$	416312.	1.55754
$518$	− 207028.i	− 0.771560i
$519$	0	0
$520$	72331.6	0.267499
$521$	− 4492.48i	− 0.0165505i	−0.999966	−	0.00827524i	$-0.997366\pi$
$521$	− 4492.48i	− 0.0165505i	0.999966	−	0.00827524i	$-0.00263412\pi$
$522$	0	0
$523$	365385.	1.33582	0.667909	−	0.744243i	$-0.267189\pi$
$523$	365385.	1.33582	0.667909	+	0.744243i	$0.267189\pi$
$524$	− 8125.01i	− 0.0295911i
$525$	0	0
$526$	206230.	0.745384
$527$	− 251561.i	− 0.905778i
$528$	0	0
$529$	−329415.	−1.17715
$530$	102699.i	0.365606i
$531$	0	0
$532$	−44498.8	−0.157226
$533$	9676.82i	0.0340626i
$534$	0	0
$535$	−93795.0	−0.327697
$536$	366454.i	1.27553i
$537$	0	0
$538$	76038.5	0.262705
$539$	− 165115.i	− 0.568341i
$540$	0	0
$541$	325408.	1.11182	0.555909	−	0.831243i	$-0.312370\pi$
$541$	325408.	1.11182	0.555909	+	0.831243i	$0.312370\pi$
$542$	132602.i	0.451391i
$543$	0	0
$544$	−103725.	−0.350497
$545$	− 209882.i	− 0.706613i
$546$	0	0
$547$	−504866.	−1.68734	−0.843669	−	0.536864i	$-0.819609\pi$
$547$	−504866.	−1.68734	−0.843669	+	0.536864i	$0.819609\pi$
$548$	60568.8i	0.201692i
$549$	0	0
$550$	−91066.3	−0.301046
$551$	267683.i	0.881692i
$552$	0	0
$553$	−218263.	−0.713724
$554$	− 490578.i	− 1.59841i
$555$	0	0
$556$	74418.4	0.240730
$557$	− 68974.4i	− 0.222320i	−0.993803	−	0.111160i	$-0.964543\pi$
$557$	− 68974.4i	− 0.222320i	0.993803	−	0.111160i	$-0.0354565\pi$
$558$	0	0
$559$	57498.6	0.184007
$560$	96786.0i	0.308629i
$561$	0	0
$562$	−179137.	−0.567169
$563$	167958.i	0.529888i	0.964264	+	0.264944i	$0.0853535\pi$
$563$	167958.i	0.529888i	−0.964264	+	0.264944i	$0.914647\pi$
$564$	0	0
$565$	−105283.	−0.329809
$566$	− 361047.i	− 1.12702i
$567$	0	0
$568$	13423.7	0.0416078
$569$	132519.i	0.409310i	0.978834	+	0.204655i	$0.0656073\pi$
$569$	132519.i	0.409310i	−0.978834	+	0.204655i	$0.934393\pi$
$570$	0	0
$571$	448800.	1.37651	0.688257	−	0.725467i	$-0.258376\pi$
$571$	448800.	1.37651	0.688257	+	0.725467i	$0.258376\pi$
$572$	− 37761.6i	− 0.115414i
$573$	0	0
$574$	−14852.3	−0.0450786
$575$	97568.5i	0.295103i
$576$	0	0
$577$	−206080.	−0.618991	−0.309496	−	0.950901i	$-0.600160\pi$
$577$	−206080.	−0.618991	−0.309496	+	0.950901i	$0.600160\pi$
$578$	298010.i	0.892020i
$579$	0	0
$580$	−10799.3	−0.0321025
$581$	37561.8i	0.111274i
$582$	0	0
$583$	478631.	1.40820
$584$	347331.i	1.01840i
$585$	0	0
$586$	−202838.	−0.590682
$587$	147864.i	0.429126i	0.976710	+	0.214563i	$0.0688327\pi$
$587$	147864.i	0.429126i	−0.976710	+	0.214563i	$0.931167\pi$
$588$	0	0
$589$	−348289.	−1.00394
$590$	174586.i	0.501540i
$591$	0	0
$592$	−308521.	−0.880321
$593$	22222.5i	0.0631951i	0.999501	+	0.0315975i	$0.0100595\pi$
$593$	22222.5i	0.0631951i	−0.999501	+	0.0315975i	$0.989941\pi$
$594$	0	0
$595$	−178034.	−0.502885
$596$	2418.08i	0.00680735i
$597$	0	0
$598$	280257.	0.783709
$599$	− 31912.6i	− 0.0889425i	−0.999011	−	0.0444712i	$-0.985840\pi$
$599$	− 31912.6i	− 0.0889425i	0.999011	−	0.0444712i	$-0.0141603\pi$
$600$	0	0
$601$	433326.	1.19968	0.599841	−	0.800119i	$-0.295231\pi$
$601$	433326.	1.19968	0.599841	+	0.800119i	$0.295231\pi$
$602$	88250.8i	0.243515i
$603$	0	0
$604$	−8398.41	−0.0230209
$605$	260726.i	0.712318i
$606$	0	0
$607$	135910.	0.368871	0.184435	−	0.982845i	$-0.440954\pi$
$607$	135910.	0.368871	0.184435	+	0.982845i	$0.440954\pi$
$608$	143608.i	0.388483i
$609$	0	0
$610$	−28912.1	−0.0776998
$611$	205177.i	0.549599i
$612$	0	0
$613$	205158.	0.545967	0.272984	−	0.962019i	$-0.411989\pi$
$613$	205158.	0.545967	0.272984	+	0.962019i	$0.411989\pi$
$614$	− 353529.i	− 0.937752i
$615$	0	0
$616$	517399.	1.36353
$617$	− 224.386i	0 0.000589421i	−1.00000		0.000294711i	$-0.999906\pi$
$617$	− 224.386i	0 0.000589421i	1.00000		0.000294711i	$-9.38093e-5\pi$
$618$	0	0
$619$	−428963.	−1.11954	−0.559769	−	0.828649i	$-0.689110\pi$
$619$	−428963.	−1.11954	−0.559769	+	0.828649i	$0.689110\pi$
$620$	− 14051.2i	− 0.0365537i
$621$	0	0
$622$	−564013.	−1.45783
$623$	337363.i	0.869204i
$624$	0	0
$625$	15625.0	0.0400000
$626$	− 27687.0i	− 0.0706526i
$627$	0	0
$628$	68849.6	0.174575
$629$	− 567512.i	− 1.43441i
$630$	0	0
$631$	587087.	1.47450	0.737248	−	0.675623i	$-0.236125\pi$
$631$	587087.	1.47450	0.737248	+	0.675623i	$0.236125\pi$
$632$	373090.i	0.934070i
$633$	0	0
$634$	129131.	0.321257
$635$	− 31422.2i	− 0.0779272i
$636$	0	0
$637$	81376.0	0.200548
$638$	− 348646.i	− 0.856531i
$639$	0	0
$640$	141116.	0.344521
$641$	191293.i	0.465567i	0.972529	+	0.232784i	$0.0747834\pi$
$641$	191293.i	0.465567i	−0.972529	+	0.232784i	$0.925217\pi$
$642$	0	0
$643$	406807.	0.983936	0.491968	−	0.870613i	$-0.336278\pi$
$643$	406807.	0.983936	0.491968	+	0.870613i	$0.336278\pi$
$644$	− 62096.1i	− 0.149724i
$645$	0	0
$646$	844984.	2.02481
$647$	408318.i	0.975416i	0.873007	+	0.487708i	$0.162167\pi$
$647$	408318.i	0.975416i	−0.873007	+	0.487708i	$0.837833\pi$
$648$	0	0
$649$	813663.	1.93177
$650$	− 44881.5i	− 0.106228i
$651$	0	0
$652$	−89029.1	−0.209429
$653$	− 217919.i	− 0.511055i	−0.966802	−	0.255528i	$-0.917751\pi$
$653$	− 217919.i	− 0.511055i	0.966802	−	0.255528i	$-0.0822492\pi$
$654$	0	0
$655$	−45006.6	−0.104904
$656$	22133.5i	0.0514330i
$657$	0	0
$658$	−314913.	−0.727342
$659$	− 228067.i	− 0.525160i	−0.964910	−	0.262580i	$-0.915427\pi$
$659$	− 228067.i	− 0.525160i	0.964910	−	0.262580i	$-0.0845734\pi$
$660$	0	0
$661$	454570.	1.04039	0.520196	−	0.854047i	$-0.325859\pi$
$661$	454570.	1.04039	0.520196	+	0.854047i	$0.325859\pi$
$662$	39522.1i	0.0901829i
$663$	0	0
$664$	64206.5	0.145627
$665$	246490.i	0.557387i
$666$	0	0
$667$	−373539.	−0.839624
$668$	− 2802.23i	− 0.00627988i
$669$	0	0
$670$	227384.	0.506535
$671$	134746.i	0.299275i
$672$	0	0
$673$	262700.	0.580002	0.290001	−	0.957026i	$-0.406344\pi$
$673$	262700.	0.580002	0.290001	+	0.957026i	$0.406344\pi$
$674$	193150.i	0.425181i
$675$	0	0
$676$	−39036.4	−0.0854232
$677$	235310.i	0.513409i	0.966490	+	0.256705i	$0.0826367\pi$
$677$	235310.i	0.513409i	−0.966490	+	0.256705i	$0.917363\pi$
$678$	0	0
$679$	−201420.	−0.436881
$680$	304324.i	0.658140i
$681$	0	0
$682$	453633.	0.975295
$683$	− 103031.i	− 0.220864i	−0.993884	−	0.110432i	$-0.964777\pi$
$683$	− 103031.i	− 0.220864i	0.993884	−	0.110432i	$-0.0352235\pi$
$684$	0	0
$685$	335506.	0.715022
$686$	478760.i	1.01735i
$687$	0	0
$688$	131515.	0.277842
$689$	235890.i	0.496903i
$690$	0	0
$691$	183510.	0.384330	0.192165	−	0.981363i	$-0.438449\pi$
$691$	183510.	0.384330	0.192165	+	0.981363i	$0.438449\pi$
$692$	75950.1i	0.158605i
$693$	0	0
$694$	559243.	1.16113
$695$	− 412223.i	− 0.853420i
$696$	0	0
$697$	−40713.7	−0.0838059
$698$	126878.i	0.260421i
$699$	0	0
$700$	−9944.31	−0.0202945
$701$	740488.i	1.50689i	0.657510	+	0.753445i	$0.271610\pi$
$701$	740488.i	1.50689i	−0.657510	+	0.753445i	$0.728390\pi$
$702$	0	0
$703$	−785728.	−1.58987
$704$	− 871719.i	− 1.75886i
$705$	0	0
$706$	−727314.	−1.45919
$707$	498460.i	0.997221i
$708$	0	0
$709$	89664.5	0.178373	0.0891863	−	0.996015i	$-0.471573\pi$
$709$	89664.5	0.178373	0.0891863	+	0.996015i	$0.471573\pi$
$710$	− 8329.34i	− 0.0165232i
$711$	0	0
$712$	576674.	1.13755
$713$	− 486023.i	− 0.956043i
$714$	0	0
$715$	−209172.	−0.409158
$716$	− 73814.4i	− 0.143984i
$717$	0	0
$718$	−417012.	−0.808909
$719$	764844.i	1.47950i	0.672882	+	0.739750i	$0.265056\pi$
$719$	764844.i	1.47950i	−0.672882	+	0.739750i	$0.734944\pi$
$720$	0	0
$721$	279643.	0.537939
$722$	− 682596.i	− 1.30945i
$723$	0	0
$724$	13250.6	0.0252790
$725$	59820.0i	0.113807i
$726$	0	0
$727$	−48065.0	−0.0909410	−0.0454705	−	0.998966i	$-0.514479\pi$
$727$	−48065.0	−0.0909410	−0.0454705	+	0.998966i	$0.514479\pi$
$728$	254997.i	0.481141i
$729$	0	0
$730$	215518.	0.404425
$731$	241916.i	0.452720i
$732$	0	0
$733$	−339984.	−0.632776	−0.316388	−	0.948630i	$-0.602470\pi$
$733$	−339984.	−0.632776	−0.316388	+	0.948630i	$0.602470\pi$
$734$	409777.i	0.760599i
$735$	0	0
$736$	−200399.	−0.369947
$737$	− 1.05973e6i	− 1.95101i
$738$	0	0
$739$	−591893.	−1.08381	−0.541907	−	0.840439i	$-0.682297\pi$
$739$	−591893.	−1.08381	−0.541907	+	0.840439i	$0.682297\pi$
$740$	− 31699.1i	− 0.0578873i
$741$	0	0
$742$	−362053.	−0.657603
$743$	− 312219.i	− 0.565563i	−0.959184	−	0.282782i	$-0.908743\pi$
$743$	− 312219.i	− 0.565563i	0.959184	−	0.282782i	$-0.0912572\pi$
$744$	0	0
$745$	13394.4	0.0241329
$746$	− 740705.i	− 1.33097i
$747$	0	0
$748$	158876.	0.283959
$749$	− 330664.i	− 0.589417i
$750$	0	0
$751$	−623828.	−1.10608	−0.553038	−	0.833156i	$-0.686532\pi$
$751$	−623828.	−1.10608	−0.553038	+	0.833156i	$0.686532\pi$
$752$	469295.i	0.829870i
$753$	0	0
$754$	171828.	0.302239
$755$	46521.0i	0.0816122i
$756$	0	0
$757$	992275.	1.73157	0.865785	−	0.500416i	$-0.166820\pi$
$757$	992275.	1.73157	0.865785	+	0.500416i	$0.166820\pi$
$758$	788273.i	1.37195i
$759$	0	0
$760$	421340.	0.729467
$761$	8453.84i	0.0145977i	0.999973	+	0.00729885i	$0.00232332\pi$
$761$	8453.84i	0.0145977i	−0.999973	+	0.00729885i	$0.997677\pi$
$762$	0	0
$763$	739914.	1.27096
$764$	− 76209.9i	− 0.130564i
$765$	0	0
$766$	101271.	0.172595
$767$	401009.i	0.681653i
$768$	0	0
$769$	−181141.	−0.306312	−0.153156	−	0.988202i	$-0.548944\pi$
$769$	−181141.	−0.306312	−0.153156	+	0.988202i	$0.548944\pi$
$770$	− 321044.i	− 0.541481i
$771$	0	0
$772$	21763.6	0.0365171
$773$	− 384785.i	− 0.643961i	−0.946746	−	0.321980i	$-0.895651\pi$
$773$	− 384785.i	− 0.643961i	0.946746	−	0.321980i	$-0.104349\pi$
$774$	0	0
$775$	−77833.5	−0.129588
$776$	344299.i	0.571758i
$777$	0	0
$778$	387456.	0.640123
$779$	56368.6i	0.0928886i
$780$	0	0
$781$	−38819.2	−0.0636420
$782$	1.17914e6i	1.92820i
$783$	0	0
$784$	186129.	0.302818
$785$	− 381376.i	− 0.618891i
$786$	0	0
$787$	−400539.	−0.646689	−0.323345	−	0.946281i	$-0.604807\pi$
$787$	−400539.	−0.646689	−0.323345	+	0.946281i	$0.604807\pi$
$788$	− 21650.0i	− 0.0348663i
$789$	0	0
$790$	231501.	0.370936
$791$	− 371164.i	− 0.593216i
$792$	0	0
$793$	−66408.7	−0.105603
$794$	28991.5i	0.0459865i
$795$	0	0
$796$	149333.	0.235684
$797$	− 212231.i	− 0.334112i	−0.985947	−	0.167056i	$-0.946574\pi$
$797$	− 212231.i	− 0.334112i	0.985947	−	0.167056i	$-0.0534261\pi$
$798$	0	0
$799$	−863250.	−1.35221
$800$	32092.6i	0.0501447i
$801$	0	0
$802$	1.03534e6	1.60967
$803$	− 1.00443e6i	− 1.55771i
$804$	0	0
$805$	−343967.	−0.530792
$806$	223570.i	0.344147i
$807$	0	0
$808$	852046.	1.30509
$809$	804505.i	1.22923i	0.788829	+	0.614613i	$0.210688\pi$
$809$	804505.i	1.22923i	−0.788829	+	0.614613i	$0.789312\pi$
$810$	0	0
$811$	−400098.	−0.608310	−0.304155	−	0.952623i	$-0.598374\pi$
$811$	−400098.	−0.608310	−0.304155	+	0.952623i	$0.598374\pi$
$812$	− 38071.6i	− 0.0577416i
$813$	0	0
$814$	1.02338e6	1.54450
$815$	493156.i	0.742453i
$816$	0	0
$817$	334936.	0.501785
$818$	283256.i	0.423324i
$819$	0	0
$820$	−2274.11	−0.00338208
$821$	− 170567.i	− 0.253051i	−0.991963	−	0.126525i	$-0.959617\pi$
$821$	− 170567.i	− 0.253051i	0.991963	−	0.126525i	$-0.0403825\pi$
$822$	0	0
$823$	325490.	0.480550	0.240275	−	0.970705i	$-0.422762\pi$
$823$	325490.	0.480550	0.240275	+	0.970705i	$0.422762\pi$
$824$	− 478010.i	− 0.704016i
$825$	0	0
$826$	−615483.	−0.902102
$827$	− 1.18917e6i	− 1.73873i	−0.494173	−	0.869363i	$-0.664529\pi$
$827$	− 1.18917e6i	− 1.73873i	0.494173	−	0.869363i	$-0.335471\pi$
$828$	0	0
$829$	−995746.	−1.44890	−0.724452	−	0.689325i	$-0.757907\pi$
$829$	−995746.	−1.44890	−0.724452	+	0.689325i	$0.757907\pi$
$830$	− 39839.9i	− 0.0578312i
$831$	0	0
$832$	429622.	0.620640
$833$	342377.i	0.493417i
$834$	0	0
$835$	−15522.3	−0.0222630
$836$	− 219966.i	− 0.314734i
$837$	0	0
$838$	−721697.	−1.02770
$839$	428816.i	0.609182i	0.952483	+	0.304591i	$0.0985198\pi$
$839$	428816.i	0.609182i	−0.952483	+	0.304591i	$0.901480\pi$
$840$	0	0
$841$	478261.	0.676197
$842$	− 932309.i	− 1.31503i
$843$	0	0
$844$	3761.05	0.00527989
$845$	216233.i	0.302836i
$846$	0	0
$847$	−919160.	−1.28122
$848$	539544.i	0.750301i
$849$	0	0
$850$	188832.	0.261359
$851$	− 1.09645e6i	− 1.51401i
$852$	0	0
$853$	−710107.	−0.975945	−0.487973	−	0.872859i	$-0.662263\pi$
$853$	−710107.	−0.975945	−0.487973	+	0.872859i	$0.662263\pi$
$854$	− 101926.i	− 0.139756i
$855$	0	0
$856$	−565222.	−0.771386
$857$	− 737753.i	− 1.00450i	−0.864723	−	0.502250i	$-0.832506\pi$
$857$	− 737753.i	− 1.00450i	0.864723	−	0.502250i	$-0.167494\pi$
$858$	0	0
$859$	−93951.8	−0.127326	−0.0636632	−	0.997971i	$-0.520278\pi$
$859$	−93951.8	−0.127326	−0.0636632	+	0.997971i	$0.520278\pi$
$860$	13512.5i	0.0182700i
$861$	0	0
$862$	10892.2	0.0146589
$863$	− 142070.i	− 0.190757i	−0.995441	−	0.0953787i	$-0.969594\pi$
$863$	− 142070.i	− 0.190757i	0.995441	−	0.0953787i	$-0.0304062\pi$
$864$	0	0
$865$	420708.	0.562274
$866$	− 375355.i	− 0.500502i
$867$	0	0
$868$	49536.0	0.0657479
$869$	− 1.07892e6i	− 1.42872i
$870$	0	0
$871$	522281.	0.688443
$872$	− 1.26478e6i	− 1.66334i
$873$	0	0
$874$	1.63253e6	2.13717
$875$	55084.1i	0.0719466i
$876$	0	0
$877$	747229.	0.971526	0.485763	−	0.874091i	$-0.338542\pi$
$877$	747229.	0.971526	0.485763	+	0.874091i	$0.338542\pi$
$878$	− 1.21512e6i	− 1.57627i
$879$	0	0
$880$	−478432.	−0.617809
$881$	− 227197.i	− 0.292718i	−0.989232	−	0.146359i	$-0.953244\pi$
$881$	− 227197.i	− 0.292718i	0.989232	−	0.146359i	$-0.0467555\pi$
$882$	0	0
$883$	−924566.	−1.18581	−0.592907	−	0.805271i	$-0.702020\pi$
$883$	−924566.	−1.18581	−0.592907	+	0.805271i	$0.702020\pi$
$884$	78301.1i	0.100199i
$885$	0	0
$886$	−173490.	−0.221008
$887$	181751.i	0.231009i	0.993307	+	0.115504i	$0.0368485\pi$
$887$	181751.i	0.231009i	−0.993307	+	0.115504i	$0.963152\pi$
$888$	0	0
$889$	110775.	0.140165
$890$	− 357824.i	− 0.451741i
$891$	0	0
$892$	117330.	0.147462
$893$	1.19518e6i	1.49875i
$894$	0	0
$895$	−408877.	−0.510443
$896$	497487.i	0.619678i
$897$	0	0
$898$	−1.00040e6	−1.24057
$899$	− 297984.i	− 0.368701i
$900$	0	0
$901$	−992471.	−1.22255
$902$	− 73417.9i	− 0.0902378i
$903$	0	0
$904$	−634451.	−0.776357
$905$	− 73398.8i	− 0.0896172i
$906$	0	0
$907$	−679686.	−0.826216	−0.413108	−	0.910682i	$-0.635557\pi$
$907$	−679686.	−0.826216	−0.413108	+	0.910682i	$0.635557\pi$
$908$	− 70728.2i	− 0.0857868i
$909$	0	0
$910$	158225.	0.191069
$911$	− 102126.i	− 0.123055i	−0.998105	−	0.0615277i	$-0.980403\pi$
$911$	− 102126.i	− 0.123055i	0.998105	−	0.0615277i	$-0.0195972\pi$
$912$	0	0
$913$	−185675.	−0.222747
$914$	396160.i	0.474218i
$915$	0	0
$916$	−71016.2	−0.0846382
$917$	− 158665.i	− 0.188688i
$918$	0	0
$919$	−160680.	−0.190252	−0.0951262	−	0.995465i	$-0.530325\pi$
$919$	−160680.	−0.190252	−0.0951262	+	0.995465i	$0.530325\pi$
$920$	587962.i	0.694662i
$921$	0	0
$922$	−390326.	−0.459162
$923$	− 19131.8i	− 0.0224570i
$924$	0	0
$925$	−175590.	−0.205218
$926$	266070.i	0.310295i
$927$	0	0
$928$	−122866.	−0.142671
$929$	774310.i	0.897188i	0.893736	+	0.448594i	$0.148075\pi$
$929$	774310.i	0.897188i	−0.893736	+	0.448594i	$0.851925\pi$
$930$	0	0
$931$	474025.	0.546892
$932$	− 91971.6i	− 0.105882i
$933$	0	0
$934$	1.25361e6	1.43704
$935$	− 880056.i	− 1.00667i
$936$	0	0
$937$	−505619.	−0.575896	−0.287948	−	0.957646i	$-0.592973\pi$
$937$	−505619.	−0.575896	−0.287948	+	0.957646i	$0.592973\pi$
$938$	801615.i	0.911088i
$939$	0	0
$940$	−48217.9	−0.0545698
$941$	− 413188.i	− 0.466626i	−0.972402	−	0.233313i	$-0.925043\pi$
$941$	− 413188.i	− 0.466626i	0.972402	−	0.233313i	$-0.0749567\pi$
$942$	0	0
$943$	−78659.9	−0.0884566
$944$	917215.i	1.02927i
$945$	0	0
$946$	−436241.	−0.487465
$947$	111959.i	0.124841i	0.998050	+	0.0624206i	$0.0198820\pi$
$947$	111959.i	0.124841i	−0.998050	+	0.0624206i	$0.980118\pi$
$948$	0	0
$949$	495026.	0.549662
$950$	− 261440.i	− 0.289684i
$951$	0	0
$952$	−1.07286e6	−1.18377
$953$	180864.i	0.199143i	0.995030	+	0.0995717i	$0.0317473\pi$
$953$	180864.i	0.199143i	−0.995030	+	0.0995717i	$0.968253\pi$
$954$	0	0
$955$	−422147.	−0.462867
$956$	− 67535.8i	− 0.0738956i
$957$	0	0
$958$	−702297.	−0.765226
$959$	1.18279e6i	1.28609i
$960$	0	0
$961$	−535805.	−0.580177
$962$	504367.i	0.545000i
$963$	0	0
$964$	30591.8	0.0329193
$965$	− 120554.i	− 0.129458i
$966$	0	0
$967$	−768555.	−0.821906	−0.410953	−	0.911656i	$-0.634804\pi$
$967$	−768555.	−0.821906	−0.410953	+	0.911656i	$0.634804\pi$
$968$	1.57117e6i	1.67677i
$969$	0	0
$970$	213636.	0.227055
$971$	1.66534e6i	1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$971$	1.66534e6i	1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$972$	0	0
$973$	1.45325e6	1.53502
$974$	1.40848e6i	1.48468i
$975$	0	0
$976$	−151894.	−0.159456
$977$	261412.i	0.273865i	0.990580	+	0.136932i	$0.0437243\pi$
$977$	261412.i	0.273865i	−0.990580	+	0.136932i	$0.956276\pi$
$978$	0	0
$979$	−1.66765e6	−1.73996
$980$	19123.9i	0.0199124i
$981$	0	0
$982$	−1.62735e6	−1.68756
$983$	− 1.51022e6i	− 1.56291i	−0.623963	−	0.781454i	$-0.714479\pi$
$983$	− 1.51022e6i	− 1.56291i	0.623963	−	0.781454i	$-0.285521\pi$
$984$	0	0
$985$	−119925.	−0.123606
$986$	722939.i	0.743614i
$987$	0	0
$988$	108409.	0.111058
$989$	467389.i	0.477843i
$990$	0	0
$991$	1.25493e6	1.27783	0.638914	−	0.769278i	$-0.279384\pi$
$991$	1.25493e6	1.27783	0.638914	+	0.769278i	$0.279384\pi$
$992$	− 159865.i	− 0.162453i
$993$	0	0
$994$	29364.1	0.0297197
$995$	− 827194.i	− 0.835529i
$996$	0	0
$997$	1.12846e6	1.13527	0.567633	−	0.823282i	$-0.307859\pi$
$997$	1.12846e6	1.13527	0.567633	+	0.823282i	$0.307859\pi$
$998$	− 686180.i	− 0.688933i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.c.b.161.9		32
3.2	odd	2	inner	405.5.c.b.161.24		32
9.2	odd	6		135.5.i.a.71.12		32
9.4	even	3		135.5.i.a.116.12		32
9.5	odd	6		45.5.i.a.11.5	✓	32
9.7	even	3		45.5.i.a.41.5	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.i.a.11.5	✓	32	9.5	odd	6
45.5.i.a.41.5	yes	32	9.7	even	3
135.5.i.a.71.12		32	9.2	odd	6
135.5.i.a.116.12		32	9.4	even	3
405.5.c.b.161.9		32	1.1	even	1	trivial
405.5.c.b.161.24		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.73920i q^{2} +2.01838 q^{4} -11.1803i q^{5} +39.4150 q^{7} -67.3743i q^{8} -41.8055 q^{10} +194.836i q^{11} -96.0237 q^{13} -147.381i q^{14} -219.632 q^{16} -404.005i q^{17} -559.350 q^{19} -22.5662i q^{20} +728.531 q^{22} -780.548i q^{23} -125.000 q^{25} +359.052i q^{26} +79.5545 q^{28} -478.560i q^{29} +622.668 q^{31} -256.741i q^{32} -1510.65 q^{34} -440.673i q^{35} +1404.72 q^{37} +2091.52i q^{38} -753.268 q^{40} -100.775i q^{41} -598.795 q^{43} +393.253i q^{44} -2918.63 q^{46} -2136.73i q^{47} -847.457 q^{49} +467.400i q^{50} -193.812 q^{52} -2456.58i q^{53} +2178.33 q^{55} -2655.56i q^{56} -1789.43 q^{58} -4176.14i q^{59} +691.586 q^{61} -2328.28i q^{62} -4474.12 q^{64} +1073.58i q^{65} -5439.08 q^{67} -815.435i q^{68} -1647.77 q^{70} +199.240i q^{71} -5155.25 q^{73} -5252.52i q^{74} -1128.98 q^{76} +7679.46i q^{77} -5537.57 q^{79} +2455.56i q^{80} -376.819 q^{82} +952.982i q^{83} -4516.91 q^{85} +2239.02i q^{86} +13126.9 q^{88} +8559.26i q^{89} -3784.78 q^{91} -1575.44i q^{92} -7989.67 q^{94} +6253.72i q^{95} -5110.24 q^{97} +3168.81i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} + 52 q^{7} - 20 q^{13} + 2048 q^{16} + 508 q^{19} - 1344 q^{22} - 4000 q^{25} - 1664 q^{28} + 2944 q^{31} + 1188 q^{34} + 2068 q^{37} - 3300 q^{40} - 1136 q^{43} + 5724 q^{46} + 3348 q^{49}+ \cdots - 46532 q^{97}+O(q^{100}) \) 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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