LMFDB - Embedded newform 531.5.b.a.296.42

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.42
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.643121i q^{2} +15.5864 q^{4} -12.3392i q^{5} +75.5465 q^{7} +20.3139i q^{8} +O(q^{10})$	\(q+0.643121i q^{2} +15.5864 q^{4} -12.3392i q^{5} +75.5465 q^{7} +20.3139i q^{8} +7.93561 q^{10} +34.0841i q^{11} -129.794 q^{13} +48.5855i q^{14} +236.318 q^{16} +332.771i q^{17} +320.012 q^{19} -192.324i q^{20} -21.9202 q^{22} +629.038i q^{23} +472.744 q^{25} -83.4731i q^{26} +1177.50 q^{28} +1273.41i q^{29} -163.543 q^{31} +477.003i q^{32} -214.012 q^{34} -932.184i q^{35} -2022.99 q^{37} +205.807i q^{38} +250.657 q^{40} -2331.57i q^{41} +1678.35 q^{43} +531.248i q^{44} -404.548 q^{46} +1988.87i q^{47} +3306.27 q^{49} +304.032i q^{50} -2023.02 q^{52} -4947.26i q^{53} +420.571 q^{55} +1534.64i q^{56} -818.956 q^{58} -453.188i q^{59} -6089.15 q^{61} -105.178i q^{62} +3474.32 q^{64} +1601.55i q^{65} +3110.24 q^{67} +5186.71i q^{68} +599.507 q^{70} -4169.67i q^{71} +7581.35 q^{73} -1301.03i q^{74} +4987.84 q^{76} +2574.93i q^{77} +10567.7 q^{79} -2915.98i q^{80} +1499.48 q^{82} +4104.67i q^{83} +4106.14 q^{85} +1079.38i q^{86} -692.380 q^{88} +6066.41i q^{89} -9805.46 q^{91} +9804.43i q^{92} -1279.08 q^{94} -3948.70i q^{95} -794.678 q^{97} +2126.33i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

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 $

$2$	0.643121i	0.160780i	0.996763	+	0.0803902i	$0.0256166\pi$
$2$	0.643121i	0.160780i	−0.996763	+	0.0803902i	$0.974383\pi$
$3$	0	0
$3$	0	0
$4$	15.5864	0.974150
$5$	− 12.3392i	− 0.493568i	−0.969070	−	0.246784i	$-0.920626\pi$
$5$	− 12.3392i	− 0.493568i	0.969070	−	0.246784i	$-0.0793739\pi$
$6$	0	0
$7$	75.5465	1.54176	0.770882	−	0.636978i	$-0.219816\pi$
$7$	75.5465	1.54176	0.770882	+	0.636978i	$0.219816\pi$
$8$	20.3139i	0.317404i
$9$	0	0
$10$	7.93561	0.0793561
$11$	34.0841i	0.281687i	0.990032	+	0.140843i	$0.0449814\pi$
$11$	34.0841i	0.281687i	−0.990032	+	0.140843i	$0.955019\pi$
$12$	0	0
$13$	−129.794	−0.768010	−0.384005	−	0.923331i	$-0.625456\pi$
$13$	−129.794	−0.768010	−0.384005	+	0.923331i	$0.625456\pi$
$14$	48.5855i	0.247885i
$15$	0	0
$16$	236.318	0.923117
$17$	332.771i	1.15146i	0.817640	+	0.575729i	$0.195282\pi$
$17$	332.771i	1.15146i	−0.817640	+	0.575729i	$0.804718\pi$
$18$	0	0
$19$	320.012	0.886461	0.443231	−	0.896408i	$-0.353832\pi$
$19$	320.012	0.886461	0.443231	+	0.896408i	$0.353832\pi$
$20$	− 192.324i	− 0.480810i
$21$	0	0
$22$	−21.9202	−0.0452897
$23$	629.038i	1.18911i	0.804056	+	0.594554i	$0.202671\pi$
$23$	629.038i	1.18911i	−0.804056	+	0.594554i	$0.797329\pi$
$24$	0	0
$25$	472.744	0.756390
$26$	− 83.4731i	− 0.123481i
$27$	0	0
$28$	1177.50	1.50191
$29$	1273.41i	1.51416i	0.653322	+	0.757080i	$0.273375\pi$
$29$	1273.41i	1.51416i	−0.653322	+	0.757080i	$0.726625\pi$
$30$	0	0
$31$	−163.543	−0.170180	−0.0850902	−	0.996373i	$-0.527118\pi$
$31$	−163.543	−0.170180	−0.0850902	+	0.996373i	$0.527118\pi$
$32$	477.003i	0.465824i
$33$	0	0
$34$	−214.012	−0.185132
$35$	− 932.184i	− 0.760966i
$36$	0	0
$37$	−2022.99	−1.47771	−0.738855	−	0.673864i	$-0.764633\pi$
$37$	−2022.99	−1.47771	−0.738855	+	0.673864i	$0.764633\pi$
$38$	205.807i	0.142526i
$39$	0	0
$40$	250.657	0.156661
$41$	− 2331.57i	− 1.38701i	−0.720451	−	0.693506i	$-0.756065\pi$
$41$	− 2331.57i	− 1.38701i	0.720451	−	0.693506i	$-0.243935\pi$
$42$	0	0
$43$	1678.35	0.907706	0.453853	−	0.891076i	$-0.350049\pi$
$43$	1678.35	0.907706	0.453853	+	0.891076i	$0.350049\pi$
$44$	531.248i	0.274405i
$45$	0	0
$46$	−404.548	−0.191185
$47$	1988.87i	0.900349i	0.892941	+	0.450174i	$0.148638\pi$
$47$	1988.87i	0.900349i	−0.892941	+	0.450174i	$0.851362\pi$
$48$	0	0
$49$	3306.27	1.37704
$50$	304.032i	0.121613i
$51$	0	0
$52$	−2023.02	−0.748157
$53$	− 4947.26i	− 1.76122i	−0.473843	−	0.880610i	$-0.657133\pi$
$53$	− 4947.26i	− 1.76122i	0.473843	−	0.880610i	$-0.342867\pi$
$54$	0	0
$55$	420.571	0.139032
$56$	1534.64i	0.489363i
$57$	0	0
$58$	−818.956	−0.243447
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	−6089.15	−1.63643	−0.818215	−	0.574913i	$-0.805036\pi$
$61$	−6089.15	−1.63643	−0.818215	+	0.574913i	$0.805036\pi$
$62$	− 105.178i	− 0.0273617i
$63$	0	0
$64$	3474.32	0.848222
$65$	1601.55i	0.379066i
$66$	0	0
$67$	3110.24	0.692859	0.346429	−	0.938076i	$-0.387394\pi$
$67$	3110.24	0.692859	0.346429	+	0.938076i	$0.387394\pi$
$68$	5186.71i	1.12169i
$69$	0	0
$70$	599.507	0.122348
$71$	− 4169.67i	− 0.827152i	−0.910470	−	0.413576i	$-0.864280\pi$
$71$	− 4169.67i	− 0.827152i	0.910470	−	0.413576i	$-0.135720\pi$
$72$	0	0
$73$	7581.35	1.42266	0.711330	−	0.702858i	$-0.248093\pi$
$73$	7581.35	1.42266	0.711330	+	0.702858i	$0.248093\pi$
$74$	− 1301.03i	− 0.237587i
$75$	0	0
$76$	4987.84	0.863546
$77$	2574.93i	0.434295i
$78$	0	0
$79$	10567.7	1.69326	0.846632	−	0.532180i	$-0.178627\pi$
$79$	10567.7	1.69326	0.846632	+	0.532180i	$0.178627\pi$
$80$	− 2915.98i	− 0.455622i
$81$	0	0
$82$	1499.48	0.223004
$83$	4104.67i	0.595829i	0.954593	+	0.297915i	$0.0962911\pi$
$83$	4104.67i	0.595829i	−0.954593	+	0.297915i	$0.903709\pi$
$84$	0	0
$85$	4106.14	0.568323
$86$	1079.38i	0.145941i
$87$	0	0
$88$	−692.380	−0.0894086
$89$	6066.41i	0.765865i	0.923776	+	0.382932i	$0.125086\pi$
$89$	6066.41i	0.765865i	−0.923776	+	0.382932i	$0.874914\pi$
$90$	0	0
$91$	−9805.46	−1.18409
$92$	9804.43i	1.15837i
$93$	0	0
$94$	−1279.08	−0.144758
$95$	− 3948.70i	− 0.437529i
$96$	0	0
$97$	−794.678	−0.0844593	−0.0422297	−	0.999108i	$-0.513446\pi$
$97$	−794.678	−0.0844593	−0.0422297	+	0.999108i	$0.513446\pi$
$98$	2126.33i	0.221401i
$99$	0	0
$100$	7368.37	0.736837
$101$	− 9870.18i	− 0.967570i	−0.875187	−	0.483785i	$-0.839262\pi$
$101$	− 9870.18i	− 0.967570i	0.875187	−	0.483785i	$-0.160738\pi$
$102$	0	0
$103$	−7551.44	−0.711795	−0.355898	−	0.934525i	$-0.615825\pi$
$103$	−7551.44	−0.711795	−0.355898	+	0.934525i	$0.615825\pi$
$104$	− 2636.61i	− 0.243770i
$105$	0	0
$106$	3181.69	0.283169
$107$	13962.2i	1.21951i	0.792589	+	0.609756i	$0.208733\pi$
$107$	13962.2i	1.21951i	−0.792589	+	0.609756i	$0.791267\pi$
$108$	0	0
$109$	−18631.8	−1.56820	−0.784101	−	0.620634i	$-0.786875\pi$
$109$	−18631.8	−1.56820	−0.784101	+	0.620634i	$0.786875\pi$
$110$	270.478i	0.0223536i
$111$	0	0
$112$	17853.0	1.42323
$113$	− 15468.4i	− 1.21140i	−0.795691	−	0.605702i	$-0.792892\pi$
$113$	− 15468.4i	− 1.21140i	0.795691	−	0.605702i	$-0.207108\pi$
$114$	0	0
$115$	7761.83	0.586906
$116$	19847.8i	1.47502i
$117$	0	0
$118$	291.455	0.0209318
$119$	25139.7i	1.77528i
$120$	0	0
$121$	13479.3	0.920653
$122$	− 3916.06i	− 0.263106i
$123$	0	0
$124$	−2549.05	−0.165781
$125$	− 13545.3i	− 0.866899i
$126$	0	0
$127$	−5334.90	−0.330764	−0.165382	−	0.986230i	$-0.552886\pi$
$127$	−5334.90	−0.330764	−0.165382	+	0.986230i	$0.552886\pi$
$128$	9866.46i	0.602201i
$129$	0	0
$130$	−1029.99	−0.0609463
$131$	14696.0i	0.856362i	0.903693	+	0.428181i	$0.140845\pi$
$131$	14696.0i	0.856362i	−0.903693	+	0.428181i	$0.859155\pi$
$132$	0	0
$133$	24175.8	1.36671
$134$	2000.26i	0.111398i
$135$	0	0
$136$	−6759.88	−0.365478
$137$	29016.4i	1.54597i	0.634421	+	0.772987i	$0.281238\pi$
$137$	29016.4i	1.54597i	−0.634421	+	0.772987i	$0.718762\pi$
$138$	0	0
$139$	27949.9	1.44661	0.723305	−	0.690529i	$-0.242622\pi$
$139$	27949.9	1.44661	0.723305	+	0.690529i	$0.242622\pi$
$140$	− 14529.4i	− 0.741295i
$141$	0	0
$142$	2681.61	0.132990
$143$	− 4423.90i	− 0.216338i
$144$	0	0
$145$	15712.9	0.747341
$146$	4875.73i	0.228736i
$147$	0	0
$148$	−31531.1	−1.43951
$149$	− 29161.4i	− 1.31352i	−0.754100	−	0.656760i	$-0.771927\pi$
$149$	− 29161.4i	− 1.31352i	0.754100	−	0.656760i	$-0.228073\pi$
$150$	0	0
$151$	43867.3	1.92392	0.961959	−	0.273193i	$-0.0880799\pi$
$151$	43867.3	1.92392	0.961959	+	0.273193i	$0.0880799\pi$
$152$	6500.70i	0.281367i
$153$	0	0
$154$	−1655.99	−0.0698260
$155$	2018.00i	0.0839956i
$156$	0	0
$157$	−28477.1	−1.15531	−0.577653	−	0.816282i	$-0.696031\pi$
$157$	−28477.1	−1.15531	−0.577653	+	0.816282i	$0.696031\pi$
$158$	6796.28i	0.272243i
$159$	0	0
$160$	5885.84	0.229916
$161$	47521.6i	1.83332i
$162$	0	0
$163$	33769.2	1.27100	0.635500	−	0.772101i	$-0.280794\pi$
$163$	33769.2	1.27100	0.635500	+	0.772101i	$0.280794\pi$
$164$	− 36340.7i	− 1.35116i
$165$	0	0
$166$	−2639.80	−0.0957976
$167$	13408.7i	0.480787i	0.970675	+	0.240394i	$0.0772765\pi$
$167$	13408.7i	0.480787i	−0.970675	+	0.240394i	$0.922723\pi$
$168$	0	0
$169$	−11714.6	−0.410160
$170$	2640.74i	0.0913752i
$171$	0	0
$172$	26159.4	0.884242
$173$	33188.6i	1.10891i	0.832213	+	0.554456i	$0.187073\pi$
$173$	33188.6i	1.10891i	−0.832213	+	0.554456i	$0.812927\pi$
$174$	0	0
$175$	35714.1	1.16618
$176$	8054.68i	0.260030i
$177$	0	0
$178$	−3901.44	−0.123136
$179$	− 28165.3i	− 0.879038i	−0.898233	−	0.439519i	$-0.855149\pi$
$179$	− 28165.3i	− 0.879038i	0.898233	−	0.439519i	$-0.144851\pi$
$180$	0	0
$181$	−27696.7	−0.845417	−0.422709	−	0.906266i	$-0.638921\pi$
$181$	−27696.7	−0.845417	−0.422709	+	0.906266i	$0.638921\pi$
$182$	− 6306.10i	− 0.190379i
$183$	0	0
$184$	−12778.2	−0.377428
$185$	24962.0i	0.729351i
$186$	0	0
$187$	−11342.2	−0.324350
$188$	30999.3i	0.877074i
$189$	0	0
$190$	2539.49	0.0703461
$191$	− 61800.5i	− 1.69405i	−0.531556	−	0.847023i	$-0.678393\pi$
$191$	− 61800.5i	− 1.69405i	0.531556	−	0.847023i	$-0.321607\pi$
$192$	0	0
$193$	15221.0	0.408628	0.204314	−	0.978905i	$-0.434504\pi$
$193$	15221.0	0.408628	0.204314	+	0.978905i	$0.434504\pi$
$194$	− 511.074i	− 0.0135794i
$195$	0	0
$196$	51532.8	1.34144
$197$	− 23560.3i	− 0.607083i	−0.952818	−	0.303542i	$-0.901831\pi$
$197$	− 23560.3i	− 0.607083i	0.952818	−	0.303542i	$-0.0981691\pi$
$198$	0	0
$199$	19277.0	0.486780	0.243390	−	0.969929i	$-0.421741\pi$
$199$	19277.0	0.486780	0.243390	+	0.969929i	$0.421741\pi$
$200$	9603.26i	0.240082i
$201$	0	0
$202$	6347.72	0.155566
$203$	96201.5i	2.33448i
$204$	0	0
$205$	−28769.7	−0.684585
$206$	− 4856.49i	− 0.114443i
$207$	0	0
$208$	−30672.6	−0.708964
$209$	10907.3i	0.249704i
$210$	0	0
$211$	−1222.80	−0.0274656	−0.0137328	−	0.999906i	$-0.504371\pi$
$211$	−1222.80	−0.0274656	−0.0137328	+	0.999906i	$0.504371\pi$
$212$	− 77110.0i	− 1.71569i
$213$	0	0
$214$	−8979.39	−0.196074
$215$	− 20709.5i	− 0.448015i
$216$	0	0
$217$	−12355.1	−0.262378
$218$	− 11982.5i	− 0.252136i
$219$	0	0
$220$	6555.18	0.135438
$221$	− 43191.7i	− 0.884332i
$222$	0	0
$223$	18592.7	0.373879	0.186940	−	0.982371i	$-0.440143\pi$
$223$	18592.7	0.373879	0.186940	+	0.982371i	$0.440143\pi$
$224$	36035.9i	0.718190i
$225$	0	0
$226$	9948.07	0.194770
$227$	− 29659.8i	− 0.575594i	−0.957691	−	0.287797i	$-0.907077\pi$
$227$	− 29659.8i	− 0.575594i	0.957691	−	0.287797i	$-0.0929228\pi$
$228$	0	0
$229$	−88867.5	−1.69462	−0.847309	−	0.531100i	$-0.821779\pi$
$229$	−88867.5	−1.69462	−0.847309	+	0.531100i	$0.821779\pi$
$230$	4991.80i	0.0943629i
$231$	0	0
$232$	−25867.9	−0.480601
$233$	− 40439.4i	− 0.744892i	−0.928054	−	0.372446i	$-0.878519\pi$
$233$	− 40439.4i	− 0.744892i	0.928054	−	0.372446i	$-0.121481\pi$
$234$	0	0
$235$	24541.1	0.444384
$236$	− 7063.56i	− 0.126823i
$237$	0	0
$238$	−16167.9	−0.285430
$239$	− 62265.6i	− 1.09007i	−0.838415	−	0.545033i	$-0.816517\pi$
$239$	− 62265.6i	− 1.09007i	0.838415	−	0.545033i	$-0.183483\pi$
$240$	0	0
$241$	−24738.2	−0.425925	−0.212963	−	0.977060i	$-0.568311\pi$
$241$	−24738.2	−0.425925	−0.212963	+	0.977060i	$0.568311\pi$
$242$	8668.81i	0.148023i
$243$	0	0
$244$	−94907.9	−1.59413
$245$	− 40796.7i	− 0.679663i
$246$	0	0
$247$	−41535.6	−0.680811
$248$	− 3322.20i	− 0.0540160i
$249$	0	0
$250$	8711.27	0.139380
$251$	100727.i	1.59881i	0.600791	+	0.799406i	$0.294852\pi$
$251$	100727.i	1.59881i	−0.600791	+	0.799406i	$0.705148\pi$
$252$	0	0
$253$	−21440.2	−0.334956
$254$	− 3430.99i	− 0.0531804i
$255$	0	0
$256$	49243.7	0.751400
$257$	− 53236.4i	− 0.806014i	−0.915197	−	0.403007i	$-0.867965\pi$
$257$	− 53236.4i	− 0.806014i	0.915197	−	0.403007i	$-0.132035\pi$
$258$	0	0
$259$	−152829.	−2.27828
$260$	24962.4i	0.369267i
$261$	0	0
$262$	−9451.33	−0.137686
$263$	− 8400.28i	− 0.121446i	−0.998155	−	0.0607228i	$-0.980659\pi$
$263$	− 8400.28i	− 0.121446i	0.998155	−	0.0607228i	$-0.0193406\pi$
$264$	0	0
$265$	−61045.3	−0.869282
$266$	15548.0i	0.219741i
$267$	0	0
$268$	48477.5	0.674948
$269$	29840.1i	0.412378i	0.978512	+	0.206189i	$0.0661061\pi$
$269$	29840.1i	0.412378i	−0.978512	+	0.206189i	$0.933894\pi$
$270$	0	0
$271$	−117104.	−1.59453	−0.797266	−	0.603628i	$-0.793721\pi$
$271$	−117104.	−1.59453	−0.797266	+	0.603628i	$0.793721\pi$
$272$	78639.9i	1.06293i
$273$	0	0
$274$	−18661.1	−0.248562
$275$	16113.0i	0.213065i
$276$	0	0
$277$	−81805.8	−1.06616	−0.533082	−	0.846063i	$-0.678966\pi$
$277$	−81805.8	−1.06616	−0.533082	+	0.846063i	$0.678966\pi$
$278$	17975.2i	0.232586i
$279$	0	0
$280$	18936.3	0.241534
$281$	− 117212.i	− 1.48442i	−0.670165	−	0.742212i	$-0.733777\pi$
$281$	− 117212.i	− 1.48442i	0.670165	−	0.742212i	$-0.266223\pi$
$282$	0	0
$283$	100852.	1.25925	0.629626	−	0.776898i	$-0.283208\pi$
$283$	100852.	1.25925	0.629626	+	0.776898i	$0.283208\pi$
$284$	− 64990.2i	− 0.805770i
$285$	0	0
$286$	2845.10	0.0347829
$287$	− 176142.i	− 2.13844i
$288$	0	0
$289$	−27215.8	−0.325856
$290$	10105.3i	0.120158i
$291$	0	0
$292$	118166.	1.38588
$293$	− 132415.i	− 1.54241i	−0.636584	−	0.771207i	$-0.719653\pi$
$293$	− 132415.i	− 1.54241i	0.636584	−	0.771207i	$-0.280347\pi$
$294$	0	0
$295$	−5591.98	−0.0642571
$296$	− 41094.7i	− 0.469032i
$297$	0	0
$298$	18754.3	0.211188
$299$	− 81645.2i	− 0.913247i
$300$	0	0
$301$	126793.	1.39947
$302$	28212.0i	0.309328i
$303$	0	0
$304$	75624.7	0.818308
$305$	75135.3i	0.807690i
$306$	0	0
$307$	−92823.5	−0.984875	−0.492438	−	0.870348i	$-0.663894\pi$
$307$	−92823.5	−0.984875	−0.492438	+	0.870348i	$0.663894\pi$
$308$	40133.9i	0.423068i
$309$	0	0
$310$	−1297.82	−0.0135048
$311$	102477.i	1.05951i	0.848152	+	0.529753i	$0.177715\pi$
$311$	102477.i	1.05951i	−0.848152	+	0.529753i	$0.822285\pi$
$312$	0	0
$313$	−103704.	−1.05854	−0.529270	−	0.848453i	$-0.677534\pi$
$313$	−103704.	−1.05854	−0.529270	+	0.848453i	$0.677534\pi$
$314$	− 18314.3i	− 0.185751i
$315$	0	0
$316$	164712.	1.64949
$317$	− 26685.7i	− 0.265558i	−0.991146	−	0.132779i	$-0.957610\pi$
$317$	− 26685.7i	− 0.265558i	0.991146	−	0.132779i	$-0.0423901\pi$
$318$	0	0
$319$	−43403.0	−0.426519
$320$	− 42870.3i	− 0.418656i
$321$	0	0
$322$	−30562.2	−0.294762
$323$	106491.i	1.02072i
$324$	0	0
$325$	−61359.2	−0.580915
$326$	21717.7i	0.204352i
$327$	0	0
$328$	47363.2	0.440244
$329$	150252.i	1.38813i
$330$	0	0
$331$	108176.	0.987363	0.493682	−	0.869643i	$-0.335651\pi$
$331$	108176.	0.987363	0.493682	+	0.869643i	$0.335651\pi$
$332$	63977.0i	0.580427i
$333$	0	0
$334$	−8623.41	−0.0773012
$335$	− 38377.9i	− 0.341973i
$336$	0	0
$337$	−66744.5	−0.587700	−0.293850	−	0.955852i	$-0.594937\pi$
$337$	−66744.5	−0.587700	−0.293850	+	0.955852i	$0.594937\pi$
$338$	− 7533.90i	− 0.0659457i
$339$	0	0
$340$	63999.9	0.553632
$341$	− 5574.22i	− 0.0479375i
$342$	0	0
$343$	68389.8	0.581304
$344$	34093.8i	0.288110i
$345$	0	0
$346$	−21344.3	−0.178291
$347$	105067.i	0.872583i	0.899805	+	0.436291i	$0.143708\pi$
$347$	105067.i	0.872583i	−0.899805	+	0.436291i	$0.856292\pi$
$348$	0	0
$349$	43953.0	0.360859	0.180430	−	0.983588i	$-0.442251\pi$
$349$	43953.0	0.360859	0.180430	+	0.983588i	$0.442251\pi$
$350$	22968.5i	0.187498i
$351$	0	0
$352$	−16258.2	−0.131216
$353$	− 122498.i	− 0.983062i	−0.870860	−	0.491531i	$-0.836437\pi$
$353$	− 122498.i	− 0.983062i	0.870860	−	0.491531i	$-0.163563\pi$
$354$	0	0
$355$	−51450.5	−0.408256
$356$	94553.5i	0.746067i
$357$	0	0
$358$	18113.7	0.141332
$359$	− 84271.4i	− 0.653870i	−0.945047	−	0.326935i	$-0.893984\pi$
$359$	− 84271.4i	− 0.653870i	0.945047	−	0.326935i	$-0.106016\pi$
$360$	0	0
$361$	−27913.0	−0.214187
$362$	− 17812.4i	− 0.135926i
$363$	0	0
$364$	−152832.	−1.15348
$365$	− 93547.9i	− 0.702180i
$366$	0	0
$367$	41804.8	0.310380	0.155190	−	0.987885i	$-0.450401\pi$
$367$	41804.8	0.310380	0.155190	+	0.987885i	$0.450401\pi$
$368$	148653.i	1.09769i
$369$	0	0
$370$	−16053.6	−0.117265
$371$	− 373748.i	− 2.71539i
$372$	0	0
$373$	−215184.	−1.54665	−0.773326	−	0.634009i	$-0.781408\pi$
$373$	−215184.	−1.54665	−0.773326	+	0.634009i	$0.781408\pi$
$374$	− 7294.42i	− 0.0521492i
$375$	0	0
$376$	−40401.7	−0.285775
$377$	− 165280.i	− 1.16289i
$378$	0	0
$379$	−125870.	−0.876281	−0.438141	−	0.898906i	$-0.644363\pi$
$379$	−125870.	−0.876281	−0.438141	+	0.898906i	$0.644363\pi$
$380$	− 61546.0i	− 0.426219i
$381$	0	0
$382$	39745.2	0.272369
$383$	− 116714.i	− 0.795656i	−0.917460	−	0.397828i	$-0.869764\pi$
$383$	− 116714.i	− 0.795656i	0.917460	−	0.397828i	$-0.130236\pi$
$384$	0	0
$385$	31772.6	0.214354
$386$	9788.94i	0.0656993i
$387$	0	0
$388$	−12386.2	−0.0822760
$389$	76016.5i	0.502353i	0.967941	+	0.251176i	$0.0808174\pi$
$389$	76016.5i	0.502353i	−0.967941	+	0.251176i	$0.919183\pi$
$390$	0	0
$391$	−209326.	−1.36921
$392$	67163.2i	0.437078i
$393$	0	0
$394$	15152.1	0.0976070
$395$	− 130397.i	− 0.835741i
$396$	0	0
$397$	6318.83	0.0400918	0.0200459	−	0.999799i	$-0.493619\pi$
$397$	6318.83	0.0400918	0.0200459	+	0.999799i	$0.493619\pi$
$398$	12397.4i	0.0782646i
$399$	0	0
$400$	111718.	0.698237
$401$	212859.i	1.32374i	0.749619	+	0.661870i	$0.230237\pi$
$401$	212859.i	1.32374i	−0.749619	+	0.661870i	$0.769763\pi$
$402$	0	0
$403$	21226.9	0.130700
$404$	− 153841.i	− 0.942558i
$405$	0	0
$406$	−61869.2	−0.375338
$407$	− 68951.6i	− 0.416251i
$408$	0	0
$409$	177883.	1.06338	0.531690	−	0.846939i	$-0.321557\pi$
$409$	177883.	1.06338	0.531690	+	0.846939i	$0.321557\pi$
$410$	− 18502.4i	− 0.110068i
$411$	0	0
$412$	−117700.	−0.693395
$413$	− 34236.7i	− 0.200721i
$414$	0	0
$415$	50648.4	0.294082
$416$	− 61912.0i	− 0.357757i
$417$	0	0
$418$	−7014.74	−0.0401475
$419$	− 302591.i	− 1.72356i	−0.507278	−	0.861782i	$-0.669348\pi$
$419$	− 302591.i	− 1.72356i	0.507278	−	0.861782i	$-0.330652\pi$
$420$	0	0
$421$	−124662.	−0.703346	−0.351673	−	0.936123i	$-0.614387\pi$
$421$	−124662.	−0.703346	−0.351673	+	0.936123i	$0.614387\pi$
$422$	− 786.407i	− 0.00441593i
$423$	0	0
$424$	100498.	0.559019
$425$	157316.i	0.870952i
$426$	0	0
$427$	−460014.	−2.52299
$428$	217620.i	1.18799i
$429$	0	0
$430$	13318.7	0.0720320
$431$	216552.i	1.16576i	0.812559	+	0.582879i	$0.198074\pi$
$431$	216552.i	1.16576i	−0.812559	+	0.582879i	$0.801926\pi$
$432$	0	0
$433$	−264385.	−1.41014	−0.705068	−	0.709140i	$-0.749083\pi$
$433$	−264385.	−1.41014	−0.705068	+	0.709140i	$0.749083\pi$
$434$	− 7945.84i	− 0.0421852i
$435$	0	0
$436$	−290403.	−1.52766
$437$	201300.i	1.05410i
$438$	0	0
$439$	−75375.9	−0.391114	−0.195557	−	0.980692i	$-0.562651\pi$
$439$	−75375.9	−0.391114	−0.195557	+	0.980692i	$0.562651\pi$
$440$	8543.42i	0.0441293i
$441$	0	0
$442$	27777.5	0.142183
$443$	− 202332.i	− 1.03100i	−0.856890	−	0.515499i	$-0.827606\pi$
$443$	− 202332.i	− 1.03100i	0.856890	−	0.515499i	$-0.172394\pi$
$444$	0	0
$445$	74854.8	0.378007
$446$	11957.3i	0.0601125i
$447$	0	0
$448$	262472.	1.30776
$449$	− 185185.i	− 0.918572i	−0.888289	−	0.459286i	$-0.848105\pi$
$449$	− 185185.i	− 0.918572i	0.888289	−	0.459286i	$-0.151895\pi$
$450$	0	0
$451$	79469.3	0.390703
$452$	− 241097.i	− 1.18009i
$453$	0	0
$454$	19074.8	0.0925442
$455$	120992.i	0.584430i
$456$	0	0
$457$	−99841.6	−0.478056	−0.239028	−	0.971013i	$-0.576829\pi$
$457$	−99841.6	−0.478056	−0.239028	+	0.971013i	$0.576829\pi$
$458$	− 57152.6i	− 0.272461i
$459$	0	0
$460$	120979.	0.571734
$461$	266443.i	1.25372i	0.779130	+	0.626862i	$0.215661\pi$
$461$	266443.i	1.25372i	−0.779130	+	0.626862i	$0.784339\pi$
$462$	0	0
$463$	33222.2	0.154977	0.0774883	−	0.996993i	$-0.475310\pi$
$463$	33222.2	0.154977	0.0774883	+	0.996993i	$0.475310\pi$
$464$	300929.i	1.39775i
$465$	0	0
$466$	26007.5	0.119764
$467$	− 177317.i	− 0.813048i	−0.913640	−	0.406524i	$-0.866741\pi$
$467$	− 177317.i	− 0.813048i	0.913640	−	0.406524i	$-0.133259\pi$
$468$	0	0
$469$	234968.	1.06823
$470$	15782.9i	0.0714482i
$471$	0	0
$472$	9206.00	0.0413225
$473$	57205.0i	0.255689i
$474$	0	0
$475$	151284.	0.670510
$476$	391837.i	1.72939i
$477$	0	0
$478$	40044.3	0.175261
$479$	143771.i	0.626615i	0.949652	+	0.313308i	$0.101437\pi$
$479$	143771.i	0.626615i	−0.949652	+	0.313308i	$0.898563\pi$
$480$	0	0
$481$	262571.	1.13490
$482$	− 15909.6i	− 0.0684804i
$483$	0	0
$484$	210093.	0.896853
$485$	9805.69i	0.0416864i
$486$	0	0
$487$	95227.6	0.401518	0.200759	−	0.979641i	$-0.435659\pi$
$487$	95227.6	0.401518	0.200759	+	0.979641i	$0.435659\pi$
$488$	− 123694.i	− 0.519410i
$489$	0	0
$490$	26237.3	0.109276
$491$	317749.i	1.31802i	0.752134	+	0.659010i	$0.229024\pi$
$491$	317749.i	1.31802i	−0.752134	+	0.659010i	$0.770976\pi$
$492$	0	0
$493$	−423754.	−1.74349
$494$	− 26712.4i	− 0.109461i
$495$	0	0
$496$	−38648.2	−0.157096
$497$	− 315004.i	− 1.27527i
$498$	0	0
$499$	287419.	1.15429	0.577145	−	0.816642i	$-0.304167\pi$
$499$	287419.	1.15429	0.577145	+	0.816642i	$0.304167\pi$
$500$	− 211122.i	− 0.844489i
$501$	0	0
$502$	−64779.5	−0.257058
$503$	− 85572.8i	− 0.338220i	−0.985597	−	0.169110i	$-0.945911\pi$
$503$	− 85572.8i	− 0.338220i	0.985597	−	0.169110i	$-0.0540894\pi$
$504$	0	0
$505$	−121790.	−0.477562
$506$	− 13788.6i	− 0.0538543i
$507$	0	0
$508$	−83151.8	−0.322214
$509$	194747.i	0.751685i	0.926684	+	0.375843i	$0.122647\pi$
$509$	194747.i	0.751685i	−0.926684	+	0.375843i	$0.877353\pi$
$510$	0	0
$511$	572744.	2.19341
$512$	189533.i	0.723011i
$513$	0	0
$514$	34237.5	0.129591
$515$	93178.8i	0.351320i
$516$	0	0
$517$	−67788.8	−0.253616
$518$	− 98287.9i	− 0.366303i
$519$	0	0
$520$	−32533.7	−0.120317
$521$	− 280899.i	− 1.03484i	−0.855730	−	0.517422i	$-0.826892\pi$
$521$	− 280899.i	− 1.03484i	0.855730	−	0.517422i	$-0.173108\pi$
$522$	0	0
$523$	−13227.5	−0.0483588	−0.0241794	−	0.999708i	$-0.507697\pi$
$523$	−13227.5	−0.0483588	−0.0241794	+	0.999708i	$0.507697\pi$
$524$	229058.i	0.834225i
$525$	0	0
$526$	5402.40	0.0195261
$527$	− 54422.5i	− 0.195956i
$528$	0	0
$529$	−115848.	−0.413977
$530$	− 39259.6i	− 0.139763i
$531$	0	0
$532$	376814.	1.33138
$533$	302623.i	1.06524i
$534$	0	0
$535$	172283.	0.601913
$536$	63181.1i	0.219916i
$537$	0	0
$538$	−19190.8	−0.0663022
$539$	112691.i	0.387893i
$540$	0	0
$541$	61027.3	0.208511	0.104256	−	0.994551i	$-0.466754\pi$
$541$	61027.3	0.208511	0.104256	+	0.994551i	$0.466754\pi$
$542$	− 75312.1i	− 0.256369i
$543$	0	0
$544$	−158733.	−0.536376
$545$	229902.i	0.774015i
$546$	0	0
$547$	−54405.7	−0.181832	−0.0909159	−	0.995859i	$-0.528979\pi$
$547$	−54405.7	−0.181832	−0.0909159	+	0.995859i	$0.528979\pi$
$548$	452261.i	1.50601i
$549$	0	0
$550$	−10362.6	−0.0342567
$551$	407507.i	1.34224i
$552$	0	0
$553$	798349.	2.61061
$554$	− 52611.0i	− 0.171418i
$555$	0	0
$556$	435639.	1.40921
$557$	− 228591.i	− 0.736799i	−0.929668	−	0.368399i	$-0.879906\pi$
$557$	− 228591.i	− 0.736799i	0.929668	−	0.368399i	$-0.120094\pi$
$558$	0	0
$559$	−217839.	−0.697128
$560$	− 220292.i	− 0.702461i
$561$	0	0
$562$	75381.3	0.238666
$563$	234874.i	0.740999i	0.928833	+	0.370500i	$0.120814\pi$
$563$	234874.i	0.740999i	−0.928833	+	0.370500i	$0.879186\pi$
$564$	0	0
$565$	−190868.	−0.597911
$566$	64860.2i	0.202463i
$567$	0	0
$568$	84702.3	0.262542
$569$	4791.25i	0.0147987i	0.999973	+	0.00739937i	$0.00235531\pi$
$569$	4791.25i	0.0147987i	−0.999973	+	0.00739937i	$0.997645\pi$
$570$	0	0
$571$	−408393.	−1.25258	−0.626291	−	0.779590i	$-0.715428\pi$
$571$	−408393.	−1.25258	−0.626291	+	0.779590i	$0.715428\pi$
$572$	− 68952.7i	− 0.210746i
$573$	0	0
$574$	113280.	0.343820
$575$	297374.i	0.899429i
$576$	0	0
$577$	247045.	0.742034	0.371017	−	0.928626i	$-0.379009\pi$
$577$	247045.	0.742034	0.371017	+	0.928626i	$0.379009\pi$
$578$	− 17503.1i	− 0.0523913i
$579$	0	0
$580$	244907.	0.728022
$581$	310093.i	0.918628i
$582$	0	0
$583$	168623.	0.496112
$584$	154007.i	0.451558i
$585$	0	0
$586$	85158.8	0.247990
$587$	65177.1i	0.189155i	0.995517	+	0.0945777i	$0.0301501\pi$
$587$	65177.1i	0.189155i	−0.995517	+	0.0945777i	$0.969850\pi$
$588$	0	0
$589$	−52335.9	−0.150858
$590$	− 3596.32i	− 0.0103313i
$591$	0	0
$592$	−478068.	−1.36410
$593$	− 480461.i	− 1.36631i	−0.730274	−	0.683154i	$-0.760608\pi$
$593$	− 480461.i	− 1.36631i	0.730274	−	0.683154i	$-0.239392\pi$
$594$	0	0
$595$	310204.	0.876221
$596$	− 454522.i	− 1.27956i
$597$	0	0
$598$	52507.8	0.146832
$599$	55489.7i	0.154653i	0.997006	+	0.0773266i	$0.0246384\pi$
$599$	55489.7i	0.154653i	−0.997006	+	0.0773266i	$0.975362\pi$
$600$	0	0
$601$	446206.	1.23534	0.617669	−	0.786438i	$-0.288077\pi$
$601$	446206.	1.23534	0.617669	+	0.786438i	$0.288077\pi$
$602$	81543.5i	0.225007i
$603$	0	0
$604$	683732.	1.87418
$605$	− 166324.i	− 0.454405i
$606$	0	0
$607$	−315420.	−0.856074	−0.428037	−	0.903761i	$-0.640795\pi$
$607$	−315420.	−0.856074	−0.428037	+	0.903761i	$0.640795\pi$
$608$	152647.i	0.412934i
$609$	0	0
$610$	−48321.1	−0.129861
$611$	− 258143.i	− 0.691477i
$612$	0	0
$613$	723129.	1.92440	0.962200	−	0.272346i	$-0.0877994\pi$
$613$	723129.	1.92440	0.962200	+	0.272346i	$0.0877994\pi$
$614$	− 59696.8i	− 0.158349i
$615$	0	0
$616$	−52306.9	−0.137847
$617$	− 231664.i	− 0.608539i	−0.952586	−	0.304269i	$-0.901588\pi$
$617$	− 231664.i	− 0.608539i	0.952586	−	0.304269i	$-0.0984123\pi$
$618$	0	0
$619$	−62934.1	−0.164250	−0.0821249	−	0.996622i	$-0.526171\pi$
$619$	−62934.1	−0.164250	−0.0821249	+	0.996622i	$0.526171\pi$
$620$	31453.3i	0.0818243i
$621$	0	0
$622$	−65904.8	−0.170348
$623$	458296.i	1.18078i
$624$	0	0
$625$	128327.	0.328516
$626$	− 66694.4i	− 0.170193i
$627$	0	0
$628$	−443856.	−1.12544
$629$	− 673192.i	− 1.70152i
$630$	0	0
$631$	−645496.	−1.62119	−0.810597	−	0.585604i	$-0.800857\pi$
$631$	−645496.	−1.62119	−0.810597	+	0.585604i	$0.800857\pi$
$632$	214670.i	0.537449i
$633$	0	0
$634$	17162.1	0.0426965
$635$	65828.4i	0.163255i
$636$	0	0
$637$	−429133.	−1.05758
$638$	− 27913.4i	− 0.0685758i
$639$	0	0
$640$	121744.	0.297227
$641$	− 79990.0i	− 0.194679i	−0.995251	−	0.0973396i	$-0.968967\pi$
$641$	− 79990.0i	− 0.194679i	0.995251	−	0.0973396i	$-0.0310333\pi$
$642$	0	0
$643$	87848.0	0.212476	0.106238	−	0.994341i	$-0.466119\pi$
$643$	87848.0	0.212476	0.106238	+	0.994341i	$0.466119\pi$
$644$	740690.i	1.78593i
$645$	0	0
$646$	−68486.6	−0.164112
$647$	− 382715.i	− 0.914253i	−0.889402	−	0.457127i	$-0.848879\pi$
$647$	− 382715.i	− 0.914253i	0.889402	−	0.457127i	$-0.151121\pi$
$648$	0	0
$649$	15446.5	0.0366725
$650$	− 39461.4i	− 0.0933998i
$651$	0	0
$652$	526340.	1.23814
$653$	268193.i	0.628957i	0.949265	+	0.314479i	$0.101830\pi$
$653$	268193.i	0.628957i	−0.949265	+	0.314479i	$0.898170\pi$
$654$	0	0
$655$	181337.	0.422673
$656$	− 550991.i	− 1.28037i
$657$	0	0
$658$	−96630.3	−0.223183
$659$	− 501694.i	− 1.15523i	−0.816309	−	0.577615i	$-0.803984\pi$
$659$	− 501694.i	− 1.15523i	0.816309	−	0.577615i	$-0.196016\pi$
$660$	0	0
$661$	−752149.	−1.72148	−0.860738	−	0.509048i	$-0.829998\pi$
$661$	−752149.	−1.72148	−0.860738	+	0.509048i	$0.829998\pi$
$662$	69570.6i	0.158749i
$663$	0	0
$664$	−83381.7	−0.189119
$665$	− 298310.i	− 0.674567i
$666$	0	0
$667$	−801022.	−1.80050
$668$	208993.i	0.468359i
$669$	0	0
$670$	24681.7	0.0549826
$671$	− 207543.i	− 0.460960i
$672$	0	0
$673$	637182.	1.40680	0.703402	−	0.710792i	$-0.251663\pi$
$673$	637182.	1.40680	0.703402	+	0.710792i	$0.251663\pi$
$674$	− 42924.8i	− 0.0944906i
$675$	0	0
$676$	−182588.	−0.399557
$677$	766040.i	1.67137i	0.549205	+	0.835687i	$0.314931\pi$
$677$	766040.i	1.67137i	−0.549205	+	0.835687i	$0.685069\pi$
$678$	0	0
$679$	−60035.1	−0.130216
$680$	83411.6i	0.180388i
$681$	0	0
$682$	3584.90	0.00770741
$683$	401690.i	0.861093i	0.902568	+	0.430546i	$0.141679\pi$
$683$	401690.i	0.861093i	−0.902568	+	0.430546i	$0.858321\pi$
$684$	0	0
$685$	358039.	0.763044
$686$	43983.0i	0.0934623i
$687$	0	0
$688$	396624.	0.837919
$689$	642124.i	1.35263i
$690$	0	0
$691$	617328.	1.29288	0.646442	−	0.762963i	$-0.276256\pi$
$691$	617328.	1.29288	0.646442	+	0.762963i	$0.276256\pi$
$692$	517291.i	1.08025i
$693$	0	0
$694$	−67570.7	−0.140294
$695$	− 344880.i	− 0.714001i
$696$	0	0
$697$	775879.	1.59709
$698$	28267.1i	0.0580191i
$699$	0	0
$700$	556655.	1.13603
$701$	72395.7i	0.147325i	0.997283	+	0.0736626i	$0.0234688\pi$
$701$	72395.7i	0.147325i	−0.997283	+	0.0736626i	$0.976531\pi$
$702$	0	0
$703$	−647381.	−1.30993
$704$	118419.i	0.238933i
$705$	0	0
$706$	78781.3	0.158057
$707$	− 745657.i	− 1.49177i
$708$	0	0
$709$	−16238.0	−0.0323027	−0.0161514	−	0.999870i	$-0.505141\pi$
$709$	−16238.0	−0.0323027	−0.0161514	+	0.999870i	$0.505141\pi$
$710$	− 33088.9i	− 0.0656396i
$711$	0	0
$712$	−123232.	−0.243089
$713$	− 102875.i	− 0.202363i
$714$	0	0
$715$	−54587.4	−0.106778
$716$	− 438995.i	− 0.856314i
$717$	0	0
$718$	54196.7	0.105129
$719$	− 567160.i	− 1.09710i	−0.836117	−	0.548552i	$-0.815179\pi$
$719$	− 567160.i	− 1.09710i	0.836117	−	0.548552i	$-0.184821\pi$
$720$	0	0
$721$	−570484.	−1.09742
$722$	− 17951.5i	− 0.0344370i
$723$	0	0
$724$	−431692.	−0.823563
$725$	601996.i	1.14530i
$726$	0	0
$727$	48225.0	0.0912438	0.0456219	−	0.998959i	$-0.485473\pi$
$727$	48225.0	0.0912438	0.0456219	+	0.998959i	$0.485473\pi$
$728$	− 199187.i	− 0.375836i
$729$	0	0
$730$	60162.6	0.112897
$731$	558507.i	1.04519i
$732$	0	0
$733$	314553.	0.585444	0.292722	−	0.956198i	$-0.405439\pi$
$733$	314553.	0.585444	0.292722	+	0.956198i	$0.405439\pi$
$734$	26885.6i	0.0499031i
$735$	0	0
$736$	−300053.	−0.553914
$737$	106010.i	0.195169i
$738$	0	0
$739$	−333876.	−0.611358	−0.305679	−	0.952135i	$-0.598883\pi$
$739$	−333876.	−0.611358	−0.305679	+	0.952135i	$0.598883\pi$
$740$	389068.i	0.710497i
$741$	0	0
$742$	240366.	0.436581
$743$	908465.i	1.64562i	0.568313	+	0.822812i	$0.307596\pi$
$743$	908465.i	1.64562i	−0.568313	+	0.822812i	$0.692404\pi$
$744$	0	0
$745$	−359829.	−0.648312
$746$	− 138389.i	− 0.248671i
$747$	0	0
$748$	−176784.	−0.315966
$749$	1.05479e6i	1.88020i
$750$	0	0
$751$	−108334.	−0.192081	−0.0960404	−	0.995377i	$-0.530618\pi$
$751$	−108334.	−0.192081	−0.0960404	+	0.995377i	$0.530618\pi$
$752$	470006.i	0.831127i
$753$	0	0
$754$	106295.	0.186970
$755$	− 541287.i	− 0.949585i
$756$	0	0
$757$	−564745.	−0.985510	−0.492755	−	0.870168i	$-0.664010\pi$
$757$	−564745.	−0.985510	−0.492755	+	0.870168i	$0.664010\pi$
$758$	− 80949.6i	− 0.140889i
$759$	0	0
$760$	80213.5	0.138874
$761$	− 19242.9i	− 0.0332278i	−0.999862	−	0.0166139i	$-0.994711\pi$
$761$	− 19242.9i	− 0.0332278i	0.999862	−	0.0166139i	$-0.00528862\pi$
$762$	0	0
$763$	−1.40757e6	−2.41780
$764$	− 963247.i	− 1.65025i
$765$	0	0
$766$	75061.2	0.127926
$767$	58820.9i	0.0999864i
$768$	0	0
$769$	−57105.4	−0.0965661	−0.0482831	−	0.998834i	$-0.515375\pi$
$769$	−57105.4	−0.0965661	−0.0482831	+	0.998834i	$0.515375\pi$
$770$	20433.7i	0.0344639i
$771$	0	0
$772$	237240.	0.398065
$773$	− 835391.i	− 1.39808i	−0.715084	−	0.699038i	$-0.753612\pi$
$773$	− 835391.i	− 1.39808i	0.715084	−	0.699038i	$-0.246388\pi$
$774$	0	0
$775$	−77314.1	−0.128723
$776$	− 16143.0i	− 0.0268078i
$777$	0	0
$778$	−48887.8	−0.0807684
$779$	− 746130.i	− 1.22953i
$780$	0	0
$781$	142120.	0.232998
$782$	− 134622.i	− 0.220142i
$783$	0	0
$784$	781331.	1.27117
$785$	351385.i	0.570223i
$786$	0	0
$787$	−433583.	−0.700039	−0.350020	−	0.936742i	$-0.613825\pi$
$787$	−433583.	−0.700039	−0.350020	+	0.936742i	$0.613825\pi$
$788$	− 367220.i	− 0.591390i
$789$	0	0
$790$	83860.8	0.134371
$791$	− 1.16858e6i	− 1.86770i
$792$	0	0
$793$	790334.	1.25679
$794$	4063.78i	0.00644598i
$795$	0	0
$796$	300458.	0.474196
$797$	− 42073.2i	− 0.0662353i	−0.999451	−	0.0331176i	$-0.989456\pi$
$797$	− 42073.2i	− 0.0662353i	0.999451	−	0.0331176i	$-0.0105436\pi$
$798$	0	0
$799$	−661839.	−1.03671
$800$	225500.i	0.352344i
$801$	0	0
$802$	−136894.	−0.212831
$803$	258403.i	0.400744i
$804$	0	0
$805$	586379.	0.904871
$806$	13651.5i	0.0210140i
$807$	0	0
$808$	200502.	0.307111
$809$	− 1.16353e6i	− 1.77779i	−0.458109	−	0.888896i	$-0.651473\pi$
$809$	− 1.16353e6i	− 1.77779i	0.458109	−	0.888896i	$-0.348527\pi$
$810$	0	0
$811$	−589350.	−0.896048	−0.448024	−	0.894021i	$-0.647872\pi$
$811$	−589350.	−0.896048	−0.448024	+	0.894021i	$0.647872\pi$
$812$	1.49943e6i	2.27413i
$813$	0	0
$814$	44344.3	0.0669250
$815$	− 416685.i	− 0.627325i
$816$	0	0
$817$	537093.	0.804646
$818$	114401.i	0.170971i
$819$	0	0
$820$	−448416.	−0.666888
$821$	− 1.27971e6i	− 1.89856i	−0.314428	−	0.949281i	$-0.601813\pi$
$821$	− 1.27971e6i	− 1.89856i	0.314428	−	0.949281i	$-0.398187\pi$
$822$	0	0
$823$	−619404.	−0.914481	−0.457240	−	0.889343i	$-0.651162\pi$
$823$	−619404.	−0.914481	−0.457240	+	0.889343i	$0.651162\pi$
$824$	− 153399.i	− 0.225927i
$825$	0	0
$826$	22018.4	0.0322719
$827$	− 577166.i	− 0.843898i	−0.906620	−	0.421949i	$-0.861346\pi$
$827$	− 577166.i	− 0.843898i	0.906620	−	0.421949i	$-0.138654\pi$
$828$	0	0
$829$	−883675.	−1.28583	−0.642915	−	0.765937i	$-0.722275\pi$
$829$	−883675.	−1.28583	−0.642915	+	0.765937i	$0.722275\pi$
$830$	32573.0i	0.0472827i
$831$	0	0
$832$	−450945.	−0.651443
$833$	1.10023e6i	1.58560i
$834$	0	0
$835$	165452.	0.237301
$836$	170006.i	0.243249i
$837$	0	0
$838$	194603.	0.277115
$839$	− 224906.i	− 0.319505i	−0.987157	−	0.159753i	$-0.948930\pi$
$839$	− 224906.i	− 0.319505i	0.987157	−	0.159753i	$-0.0510696\pi$
$840$	0	0
$841$	−914288.	−1.29268
$842$	− 80172.6i	− 0.113084i
$843$	0	0
$844$	−19059.0	−0.0267556
$845$	144549.i	0.202442i
$846$	0	0
$847$	1.01831e6	1.41943
$848$	− 1.16913e6i	− 1.62581i
$849$	0	0
$850$	−101173.	−0.140032
$851$	− 1.27253e6i	− 1.75716i
$852$	0	0
$853$	179495.	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$853$	179495.	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$854$	− 295845.i	− 0.405647i
$855$	0	0
$856$	−283627.	−0.387079
$857$	1.30534e6i	1.77730i	0.458584	+	0.888651i	$0.348357\pi$
$857$	1.30534e6i	1.77730i	−0.458584	+	0.888651i	$0.651643\pi$
$858$	0	0
$859$	−1.14905e6	−1.55723	−0.778613	−	0.627505i	$-0.784076\pi$
$859$	−1.14905e6	−1.55723	−0.778613	+	0.627505i	$0.784076\pi$
$860$	− 322786.i	− 0.436434i
$861$	0	0
$862$	−139269.	−0.187431
$863$	1.17032e6i	1.57139i	0.618615	+	0.785694i	$0.287694\pi$
$863$	1.17032e6i	1.57139i	−0.618615	+	0.785694i	$0.712306\pi$
$864$	0	0
$865$	409521.	0.547323
$866$	− 170032.i	− 0.226722i
$867$	0	0
$868$	−192572.	−0.255595
$869$	360189.i	0.476970i
$870$	0	0
$871$	−403690.	−0.532123
$872$	− 378484.i	− 0.497754i
$873$	0	0
$874$	−129460.	−0.169478
$875$	− 1.02330e6i	− 1.33655i
$876$	0	0
$877$	−592073.	−0.769797	−0.384898	−	0.922959i	$-0.625764\pi$
$877$	−592073.	−0.769797	−0.384898	+	0.922959i	$0.625764\pi$
$878$	− 48475.8i	− 0.0628835i
$879$	0	0
$880$	99388.4	0.128343
$881$	465835.i	0.600179i	0.953911	+	0.300089i	$0.0970165\pi$
$881$	465835.i	0.600179i	−0.953911	+	0.300089i	$0.902984\pi$
$882$	0	0
$883$	77645.3	0.0995850	0.0497925	−	0.998760i	$-0.484144\pi$
$883$	77645.3	0.0995850	0.0497925	+	0.998760i	$0.484144\pi$
$884$	− 673202.i	− 0.861472i
$885$	0	0
$886$	130124.	0.165764
$887$	87742.0i	0.111522i	0.998444	+	0.0557610i	$0.0177585\pi$
$887$	87742.0i	0.111522i	−0.998444	+	0.0557610i	$0.982242\pi$
$888$	0	0
$889$	−403033.	−0.509961
$890$	48140.7i	0.0607760i
$891$	0	0
$892$	289792.	0.364215
$893$	636463.i	0.798124i
$894$	0	0
$895$	−347537.	−0.433865
$896$	745376.i	0.928452i
$897$	0	0
$898$	119096.	0.147688
$899$	− 208257.i	− 0.257680i
$900$	0	0
$901$	1.64631e6	2.02797
$902$	51108.4i	0.0628173i
$903$	0	0
$904$	314224.	0.384505
$905$	341756.i	0.417271i
$906$	0	0
$907$	−11256.2	−0.0136829	−0.00684143	−	0.999977i	$-0.502178\pi$
$907$	−11256.2	−0.0136829	−0.00684143	+	0.999977i	$0.502178\pi$
$908$	− 462289.i	− 0.560715i
$909$	0	0
$910$	−77812.3	−0.0939648
$911$	− 148491.i	− 0.178922i	−0.995990	−	0.0894608i	$-0.971486\pi$
$911$	− 148491.i	− 0.178922i	0.995990	−	0.0894608i	$-0.0285144\pi$
$912$	0	0
$913$	−139904.	−0.167837
$914$	− 64210.2i	− 0.0768620i
$915$	0	0
$916$	−1.38512e6	−1.65081
$917$	1.11023e6i	1.32031i
$918$	0	0
$919$	905313.	1.07193	0.535966	−	0.844239i	$-0.319947\pi$
$919$	905313.	1.07193	0.535966	+	0.844239i	$0.319947\pi$
$920$	157673.i	0.186287i
$921$	0	0
$922$	−171355.	−0.201574
$923$	541198.i	0.635261i
$924$	0	0
$925$	−956354.	−1.11773
$926$	21365.9i	0.0249172i
$927$	0	0
$928$	−607420.	−0.705331
$929$	− 339694.i	− 0.393601i	−0.980444	−	0.196801i	$-0.936945\pi$
$929$	− 339694.i	− 0.393601i	0.980444	−	0.196801i	$-0.0630552\pi$
$930$	0	0
$931$	1.05805e6	1.22069
$932$	− 630305.i	− 0.725636i
$933$	0	0
$934$	114036.	0.130722
$935$	139954.i	0.160089i
$936$	0	0
$937$	−897557.	−1.02231	−0.511155	−	0.859488i	$-0.670782\pi$
$937$	−897557.	−1.02231	−0.511155	+	0.859488i	$0.670782\pi$
$938$	151113.i	0.171750i
$939$	0	0
$940$	382507.	0.432896
$941$	− 1.33773e6i	− 1.51074i	−0.655301	−	0.755368i	$-0.727458\pi$
$941$	− 1.33773e6i	− 1.51074i	0.655301	−	0.755368i	$-0.272542\pi$
$942$	0	0
$943$	1.46664e6	1.64931
$944$	− 107096.i	− 0.120180i
$945$	0	0
$946$	−36789.7	−0.0411097
$947$	90505.5i	0.100920i	0.998726	+	0.0504598i	$0.0160687\pi$
$947$	90505.5i	0.100920i	−0.998726	+	0.0504598i	$0.983931\pi$
$948$	0	0
$949$	−984012.	−1.09262
$950$	97293.9i	0.107805i
$951$	0	0
$952$	−510685.	−0.563481
$953$	− 1.19834e6i	− 1.31945i	−0.751505	−	0.659727i	$-0.770672\pi$
$953$	− 1.19834e6i	− 1.31945i	0.751505	−	0.659727i	$-0.229328\pi$
$954$	0	0
$955$	−762569.	−0.836128
$956$	− 970496.i	− 1.06189i
$957$	0	0
$958$	−92462.3	−0.100747
$959$	2.19209e6i	2.38353i
$960$	0	0
$961$	−896775.	−0.971039
$962$	168865.i	0.182469i
$963$	0	0
$964$	−385579.	−0.414915
$965$	− 187815.i	− 0.201686i
$966$	0	0
$967$	1.48346e6	1.58643	0.793217	−	0.608939i	$-0.208405\pi$
$967$	1.48346e6	1.58643	0.793217	+	0.608939i	$0.208405\pi$
$968$	273816.i	0.292219i
$969$	0	0
$970$	−6306.25	−0.00670236
$971$	− 22083.2i	− 0.0234220i	−0.999931	−	0.0117110i	$-0.996272\pi$
$971$	− 22083.2i	− 0.0234220i	0.999931	−	0.0117110i	$-0.00372781\pi$
$972$	0	0
$973$	2.11152e6	2.23033
$974$	61242.9i	0.0645562i
$975$	0	0
$976$	−1.43898e6	−1.51062
$977$	1.02672e6i	1.07563i	0.843064	+	0.537814i	$0.180750\pi$
$977$	1.02672e6i	1.07563i	−0.843064	+	0.537814i	$0.819250\pi$
$978$	0	0
$979$	−206768.	−0.215734
$980$	− 635874.i	− 0.662093i
$981$	0	0
$982$	−204351.	−0.211912
$983$	− 376350.i	− 0.389480i	−0.980855	−	0.194740i	$-0.937614\pi$
$983$	− 376350.i	− 0.389480i	0.980855	−	0.194740i	$-0.0623863\pi$
$984$	0	0
$985$	−290715.	−0.299637
$986$	− 272525.i	− 0.280319i
$987$	0	0
$988$	−647390.	−0.663212
$989$	1.05575e6i	1.07936i
$990$	0	0
$991$	615927.	0.627165	0.313583	−	0.949561i	$-0.398471\pi$
$991$	615927.	0.627165	0.313583	+	0.949561i	$0.398471\pi$
$992$	− 78010.7i	− 0.0792740i
$993$	0	0
$994$	202586.	0.205039
$995$	− 237863.i	− 0.240259i
$996$	0	0
$997$	744737.	0.749226	0.374613	−	0.927181i	$-0.377776\pi$
$997$	744737.	0.749226	0.374613	+	0.927181i	$0.377776\pi$
$998$	184846.i	0.185587i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.42	yes	76
3.2	odd	2	inner	531.5.b.a.296.35	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.35	✓	76	3.2	odd	2	inner
531.5.b.a.296.42	yes	76	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.643121i q^{2} +15.5864 q^{4} -12.3392i q^{5} +75.5465 q^{7} +20.3139i q^{8} +O(q^{10})\)	\(q+0.643121i q^{2} +15.5864 q^{4} -12.3392i q^{5} +75.5465 q^{7} +20.3139i q^{8} +7.93561 q^{10} +34.0841i q^{11} -129.794 q^{13} +48.5855i q^{14} +236.318 q^{16} +332.771i q^{17} +320.012 q^{19} -192.324i q^{20} -21.9202 q^{22} +629.038i q^{23} +472.744 q^{25} -83.4731i q^{26} +1177.50 q^{28} +1273.41i q^{29} -163.543 q^{31} +477.003i q^{32} -214.012 q^{34} -932.184i q^{35} -2022.99 q^{37} +205.807i q^{38} +250.657 q^{40} -2331.57i q^{41} +1678.35 q^{43} +531.248i q^{44} -404.548 q^{46} +1988.87i q^{47} +3306.27 q^{49} +304.032i q^{50} -2023.02 q^{52} -4947.26i q^{53} +420.571 q^{55} +1534.64i q^{56} -818.956 q^{58} -453.188i q^{59} -6089.15 q^{61} -105.178i q^{62} +3474.32 q^{64} +1601.55i q^{65} +3110.24 q^{67} +5186.71i q^{68} +599.507 q^{70} -4169.67i q^{71} +7581.35 q^{73} -1301.03i q^{74} +4987.84 q^{76} +2574.93i q^{77} +10567.7 q^{79} -2915.98i q^{80} +1499.48 q^{82} +4104.67i q^{83} +4106.14 q^{85} +1079.38i q^{86} -692.380 q^{88} +6066.41i q^{89} -9805.46 q^{91} +9804.43i q^{92} -1279.08 q^{94} -3948.70i q^{95} -794.678 q^{97} +2126.33i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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