LMFDB - Embedded newform 531.5.b.a.296.47

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			296.47
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.01293i q^{2} +11.9481 q^{4} +10.0983i q^{5} -71.1730 q^{7} +56.2577i q^{8} +O(q^{10})$	\(q+2.01293i q^{2} +11.9481 q^{4} +10.0983i q^{5} -71.1730 q^{7} +56.2577i q^{8} -20.3272 q^{10} -4.11828i q^{11} -100.854 q^{13} -143.267i q^{14} +77.9265 q^{16} -303.195i q^{17} -418.270 q^{19} +120.655i q^{20} +8.28983 q^{22} -234.681i q^{23} +523.025 q^{25} -203.013i q^{26} -850.382 q^{28} -911.107i q^{29} +734.050 q^{31} +1056.98i q^{32} +610.311 q^{34} -718.725i q^{35} +2410.10 q^{37} -841.950i q^{38} -568.106 q^{40} -281.427i q^{41} +651.599 q^{43} -49.2056i q^{44} +472.397 q^{46} -767.605i q^{47} +2664.60 q^{49} +1052.81i q^{50} -1205.02 q^{52} -2207.09i q^{53} +41.5876 q^{55} -4004.03i q^{56} +1834.00 q^{58} +453.188i q^{59} -2776.60 q^{61} +1477.59i q^{62} -880.815 q^{64} -1018.46i q^{65} -2162.76 q^{67} -3622.60i q^{68} +1446.75 q^{70} -5551.90i q^{71} +3915.04 q^{73} +4851.38i q^{74} -4997.53 q^{76} +293.111i q^{77} -7515.90 q^{79} +786.923i q^{80} +566.494 q^{82} +4650.58i q^{83} +3061.74 q^{85} +1311.63i q^{86} +231.685 q^{88} -2791.14i q^{89} +7178.12 q^{91} -2803.99i q^{92} +1545.14 q^{94} -4223.81i q^{95} +8287.06 q^{97} +5363.67i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.01293i	0.503234i	0.967827	+	0.251617i	$0.0809623\pi$
$2$	2.01293i	0.503234i	−0.967827	+	0.251617i	$0.919038\pi$
$3$	0	0
$3$	0	0
$4$	11.9481	0.746756
$5$	10.0983i	0.403931i	0.979393	+	0.201966i	$0.0647329\pi$
$5$	10.0983i	0.403931i	−0.979393	+	0.201966i	$0.935267\pi$
$6$	0	0
$7$	−71.1730	−1.45251	−0.726256	−	0.687425i	$-0.758741\pi$
$7$	−71.1730	−1.45251	−0.726256	+	0.687425i	$0.758741\pi$
$8$	56.2577i	0.879026i
$9$	0	0
$10$	−20.3272	−0.203272
$11$	− 4.11828i	− 0.0340354i	−0.999855	−	0.0170177i	$-0.994583\pi$
$11$	− 4.11828i	− 0.0340354i	0.999855	−	0.0170177i	$-0.00541716\pi$
$12$	0	0
$13$	−100.854	−0.596772	−0.298386	−	0.954445i	$-0.596448\pi$
$13$	−100.854	−0.596772	−0.298386	+	0.954445i	$0.596448\pi$
$14$	− 143.267i	− 0.730952i
$15$	0	0
$16$	77.9265	0.304400
$17$	− 303.195i	− 1.04912i	−0.851375	−	0.524558i	$-0.824231\pi$
$17$	− 303.195i	− 1.04912i	0.851375	−	0.524558i	$-0.175769\pi$
$18$	0	0
$19$	−418.270	−1.15864	−0.579321	−	0.815099i	$-0.696682\pi$
$19$	−418.270	−1.15864	−0.579321	+	0.815099i	$0.696682\pi$
$20$	120.655i	0.301638i
$21$	0	0
$22$	8.28983	0.0171278
$23$	− 234.681i	− 0.443631i	−0.975089	−	0.221816i	$-0.928802\pi$
$23$	− 234.681i	− 0.443631i	0.975089	−	0.221816i	$-0.0711983\pi$
$24$	0	0
$25$	523.025	0.836840
$26$	− 203.013i	− 0.300316i
$27$	0	0
$28$	−850.382	−1.08467
$29$	− 911.107i	− 1.08336i	−0.840584	−	0.541681i	$-0.817788\pi$
$29$	− 911.107i	− 1.08336i	0.840584	−	0.541681i	$-0.182212\pi$
$30$	0	0
$31$	734.050	0.763840	0.381920	−	0.924195i	$-0.375263\pi$
$31$	734.050	0.763840	0.381920	+	0.924195i	$0.375263\pi$
$32$	1056.98i	1.03221i
$33$	0	0
$34$	610.311	0.527951
$35$	− 718.725i	− 0.586714i
$36$	0	0
$37$	2410.10	1.76048	0.880242	−	0.474524i	$-0.157380\pi$
$37$	2410.10	1.76048	0.880242	+	0.474524i	$0.157380\pi$
$38$	− 841.950i	− 0.583068i
$39$	0	0
$40$	−568.106	−0.355066
$41$	− 281.427i	− 0.167416i	−0.996490	−	0.0837082i	$-0.973324\pi$
$41$	− 281.427i	− 0.167416i	0.996490	−	0.0837082i	$-0.0266764\pi$
$42$	0	0
$43$	651.599	0.352406	0.176203	−	0.984354i	$-0.443618\pi$
$43$	651.599	0.352406	0.176203	+	0.984354i	$0.443618\pi$
$44$	− 49.2056i	− 0.0254161i
$45$	0	0
$46$	472.397	0.223250
$47$	− 767.605i	− 0.347490i	−0.984791	−	0.173745i	$-0.944413\pi$
$47$	− 767.605i	− 0.347490i	0.984791	−	0.173745i	$-0.0555868\pi$
$48$	0	0
$49$	2664.60	1.10979
$50$	1052.81i	0.421126i
$51$	0	0
$52$	−1205.02	−0.445643
$53$	− 2207.09i	− 0.785721i	−0.919598	−	0.392860i	$-0.871486\pi$
$53$	− 2207.09i	− 0.785721i	0.919598	−	0.392860i	$-0.128514\pi$
$54$	0	0
$55$	41.5876	0.0137480
$56$	− 4004.03i	− 1.27680i
$57$	0	0
$58$	1834.00	0.545184
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	−2776.60	−0.746196	−0.373098	−	0.927792i	$-0.621704\pi$
$61$	−2776.60	−0.746196	−0.373098	+	0.927792i	$0.621704\pi$
$62$	1477.59i	0.384390i
$63$	0	0
$64$	−880.815	−0.215043
$65$	− 1018.46i	− 0.241055i
$66$	0	0
$67$	−2162.76	−0.481790	−0.240895	−	0.970551i	$-0.577441\pi$
$67$	−2162.76	−0.481790	−0.240895	+	0.970551i	$0.577441\pi$
$68$	− 3622.60i	− 0.783434i
$69$	0	0
$70$	1446.75	0.295254
$71$	− 5551.90i	− 1.10135i	−0.834720	−	0.550674i	$-0.814371\pi$
$71$	− 5551.90i	− 1.10135i	0.834720	−	0.550674i	$-0.185629\pi$
$72$	0	0
$73$	3915.04	0.734667	0.367334	−	0.930089i	$-0.380271\pi$
$73$	3915.04	0.734667	0.367334	+	0.930089i	$0.380271\pi$
$74$	4851.38i	0.885935i
$75$	0	0
$76$	−4997.53	−0.865223
$77$	293.111i	0.0494368i
$78$	0	0
$79$	−7515.90	−1.20428	−0.602139	−	0.798391i	$-0.705685\pi$
$79$	−7515.90	−1.20428	−0.602139	+	0.798391i	$0.705685\pi$
$80$	786.923i	0.122957i
$81$	0	0
$82$	566.494	0.0842496
$83$	4650.58i	0.675074i	0.941312	+	0.337537i	$0.109594\pi$
$83$	4650.58i	0.675074i	−0.941312	+	0.337537i	$0.890406\pi$
$84$	0	0
$85$	3061.74	0.423771
$86$	1311.63i	0.177343i
$87$	0	0
$88$	231.685	0.0299180
$89$	− 2791.14i	− 0.352372i	−0.984357	−	0.176186i	$-0.943624\pi$
$89$	− 2791.14i	− 0.352372i	0.984357	−	0.176186i	$-0.0563759\pi$
$90$	0	0
$91$	7178.12	0.866818
$92$	− 2803.99i	− 0.331284i
$93$	0	0
$94$	1545.14	0.174869
$95$	− 4223.81i	− 0.468012i
$96$	0	0
$97$	8287.06	0.880758	0.440379	−	0.897812i	$-0.354844\pi$
$97$	8287.06	0.880758	0.440379	+	0.897812i	$0.354844\pi$
$98$	5363.67i	0.558483i
$99$	0	0
$100$	6249.15	0.624915
$101$	− 978.165i	− 0.0958892i	−0.998850	−	0.0479446i	$-0.984733\pi$
$101$	− 978.165i	− 0.0958892i	0.998850	−	0.0479446i	$-0.0152671\pi$
$102$	0	0
$103$	722.082	0.0680632	0.0340316	−	0.999421i	$-0.489165\pi$
$103$	722.082	0.0680632	0.0340316	+	0.999421i	$0.489165\pi$
$104$	− 5673.84i	− 0.524578i
$105$	0	0
$106$	4442.73	0.395401
$107$	− 14124.9i	− 1.23373i	−0.787070	−	0.616864i	$-0.788403\pi$
$107$	− 14124.9i	− 1.23373i	0.787070	−	0.616864i	$-0.211597\pi$
$108$	0	0
$109$	−6486.68	−0.545971	−0.272985	−	0.962018i	$-0.588011\pi$
$109$	−6486.68	−0.545971	−0.272985	+	0.962018i	$0.588011\pi$
$110$	83.7130i	0.00691843i
$111$	0	0
$112$	−5546.26	−0.442145
$113$	− 12295.1i	− 0.962885i	−0.876478	−	0.481443i	$-0.840113\pi$
$113$	− 12295.1i	− 0.962885i	0.876478	−	0.481443i	$-0.159887\pi$
$114$	0	0
$115$	2369.87	0.179197
$116$	− 10886.0i	− 0.809007i
$117$	0	0
$118$	−912.237	−0.0655154
$119$	21579.3i	1.52385i
$120$	0	0
$121$	14624.0	0.998842
$122$	− 5589.10i	− 0.375511i
$123$	0	0
$124$	8770.50	0.570402
$125$	11593.1i	0.741957i
$126$	0	0
$127$	21572.0	1.33747	0.668735	−	0.743501i	$-0.266836\pi$
$127$	21572.0	1.33747	0.668735	+	0.743501i	$0.266836\pi$
$128$	15138.7i	0.923994i
$129$	0	0
$130$	2050.09	0.121307
$131$	− 25012.8i	− 1.45754i	−0.684759	−	0.728770i	$-0.740092\pi$
$131$	− 25012.8i	− 1.45754i	0.684759	−	0.728770i	$-0.259908\pi$
$132$	0	0
$133$	29769.6	1.68294
$134$	− 4353.49i	− 0.242453i
$135$	0	0
$136$	17057.0	0.922201
$137$	− 24686.5i	− 1.31528i	−0.753331	−	0.657641i	$-0.771554\pi$
$137$	− 24686.5i	− 1.31528i	0.753331	−	0.657641i	$-0.228446\pi$
$138$	0	0
$139$	−13721.8	−0.710200	−0.355100	−	0.934828i	$-0.615553\pi$
$139$	−13721.8	−0.710200	−0.355100	+	0.934828i	$0.615553\pi$
$140$	− 8587.40i	− 0.438132i
$141$	0	0
$142$	11175.6	0.554236
$143$	415.347i	0.0203114i
$144$	0	0
$145$	9200.62	0.437604
$146$	7880.72i	0.369709i
$147$	0	0
$148$	28796.1	1.31465
$149$	− 20259.0i	− 0.912527i	−0.889845	−	0.456264i	$-0.849187\pi$
$149$	− 20259.0i	− 0.912527i	0.889845	−	0.456264i	$-0.150813\pi$
$150$	0	0
$151$	26893.3	1.17948	0.589739	−	0.807594i	$-0.299230\pi$
$151$	26893.3	1.17948	0.589739	+	0.807594i	$0.299230\pi$
$152$	− 23530.9i	− 1.01848i
$153$	0	0
$154$	−590.013	−0.0248782
$155$	7412.64i	0.308539i
$156$	0	0
$157$	13400.4	0.543649	0.271825	−	0.962347i	$-0.412373\pi$
$157$	13400.4	0.543649	0.271825	+	0.962347i	$0.412373\pi$
$158$	− 15129.0i	− 0.606033i
$159$	0	0
$160$	−10673.7	−0.416942
$161$	16703.0i	0.644379i
$162$	0	0
$163$	−11104.1	−0.417936	−0.208968	−	0.977922i	$-0.567010\pi$
$163$	−11104.1	−0.417936	−0.208968	+	0.977922i	$0.567010\pi$
$164$	− 3362.52i	− 0.125019i
$165$	0	0
$166$	−9361.32	−0.339720
$167$	10686.8i	0.383189i	0.981474	+	0.191595i	$0.0613659\pi$
$167$	10686.8i	0.383189i	−0.981474	+	0.191595i	$0.938634\pi$
$168$	0	0
$169$	−18389.4	−0.643863
$170$	6163.09i	0.213256i
$171$	0	0
$172$	7785.37	0.263162
$173$	46121.4i	1.54103i	0.637423	+	0.770514i	$0.280000\pi$
$173$	46121.4i	1.54103i	−0.637423	+	0.770514i	$0.720000\pi$
$174$	0	0
$175$	−37225.3	−1.21552
$176$	− 320.923i	− 0.0103604i
$177$	0	0
$178$	5618.37	0.177325
$179$	− 20149.6i	− 0.628868i	−0.949279	−	0.314434i	$-0.898185\pi$
$179$	− 20149.6i	− 0.628868i	0.949279	−	0.314434i	$-0.101815\pi$
$180$	0	0
$181$	−36098.3	−1.10187	−0.550934	−	0.834549i	$-0.685728\pi$
$181$	−36098.3	−1.10187	−0.550934	+	0.834549i	$0.685728\pi$
$182$	14449.1i	0.436212i
$183$	0	0
$184$	13202.6	0.389964
$185$	24337.9i	0.711115i
$186$	0	0
$187$	−1248.64	−0.0357071
$188$	− 9171.41i	− 0.259490i
$189$	0	0
$190$	8502.25	0.235519
$191$	15061.1i	0.412848i	0.978463	+	0.206424i	$0.0661827\pi$
$191$	15061.1i	0.412848i	−0.978463	+	0.206424i	$0.933817\pi$
$192$	0	0
$193$	−14395.5	−0.386466	−0.193233	−	0.981153i	$-0.561897\pi$
$193$	−14395.5	−0.386466	−0.193233	+	0.981153i	$0.561897\pi$
$194$	16681.3i	0.443227i
$195$	0	0
$196$	31836.9	0.828741
$197$	− 49167.0i	− 1.26690i	−0.773785	−	0.633448i	$-0.781639\pi$
$197$	− 49167.0i	− 1.26690i	0.773785	−	0.633448i	$-0.218361\pi$
$198$	0	0
$199$	10649.4	0.268918	0.134459	−	0.990919i	$-0.457070\pi$
$199$	10649.4	0.268918	0.134459	+	0.990919i	$0.457070\pi$
$200$	29424.2i	0.735604i
$201$	0	0
$202$	1968.98	0.0482546
$203$	64846.3i	1.57360i
$204$	0	0
$205$	2841.93	0.0676247
$206$	1453.50i	0.0342517i
$207$	0	0
$208$	−7859.23	−0.181658
$209$	1722.55i	0.0394349i
$210$	0	0
$211$	−34762.9	−0.780821	−0.390410	−	0.920641i	$-0.627667\pi$
$211$	−34762.9	−0.780821	−0.390410	+	0.920641i	$0.627667\pi$
$212$	− 26370.5i	− 0.586742i
$213$	0	0
$214$	28432.6	0.620853
$215$	6580.03i	0.142348i
$216$	0	0
$217$	−52244.6	−1.10949
$218$	− 13057.3i	− 0.274751i
$219$	0	0
$220$	496.892	0.0102664
$221$	30578.5i	0.626083i
$222$	0	0
$223$	−21988.2	−0.442160	−0.221080	−	0.975256i	$-0.570958\pi$
$223$	−21988.2	−0.442160	−0.221080	+	0.975256i	$0.570958\pi$
$224$	− 75228.8i	− 1.49930i
$225$	0	0
$226$	24749.2	0.484556
$227$	58681.3i	1.13880i	0.822060	+	0.569401i	$0.192825\pi$
$227$	58681.3i	1.13880i	−0.822060	+	0.569401i	$0.807175\pi$
$228$	0	0
$229$	54030.2	1.03030	0.515152	−	0.857099i	$-0.327736\pi$
$229$	54030.2	1.03030	0.515152	+	0.857099i	$0.327736\pi$
$230$	4770.40i	0.0901777i
$231$	0	0
$232$	51256.8	0.952304
$233$	− 37117.9i	− 0.683709i	−0.939753	−	0.341854i	$-0.888945\pi$
$233$	− 37117.9i	− 0.683709i	0.939753	−	0.341854i	$-0.111055\pi$
$234$	0	0
$235$	7751.49	0.140362
$236$	5414.73i	0.0972193i
$237$	0	0
$238$	−43437.7	−0.766854
$239$	− 38508.5i	− 0.674156i	−0.941477	−	0.337078i	$-0.890561\pi$
$239$	− 38508.5i	− 0.674156i	0.941477	−	0.337078i	$-0.109439\pi$
$240$	0	0
$241$	−36821.2	−0.633963	−0.316982	−	0.948432i	$-0.602669\pi$
$241$	−36821.2	−0.633963	−0.316982	+	0.948432i	$0.602669\pi$
$242$	29437.2i	0.502651i
$243$	0	0
$244$	−33175.0	−0.557226
$245$	26907.9i	0.448278i
$246$	0	0
$247$	42184.4	0.691445
$248$	41296.0i	0.671435i
$249$	0	0
$250$	−23336.1	−0.373378
$251$	68363.7i	1.08512i	0.840017	+	0.542560i	$0.182545\pi$
$251$	68363.7i	1.08512i	−0.840017	+	0.542560i	$0.817455\pi$
$252$	0	0
$253$	−966.483	−0.0150992
$254$	43423.1i	0.673060i
$255$	0	0
$256$	−44566.3	−0.680028
$257$	− 90916.6i	− 1.37650i	−0.725472	−	0.688251i	$-0.758379\pi$
$257$	− 90916.6i	− 1.37650i	0.725472	−	0.688251i	$-0.241621\pi$
$258$	0	0
$259$	−171534.	−2.55712
$260$	− 12168.6i	− 0.180009i
$261$	0	0
$262$	50349.2	0.733483
$263$	− 102861.i	− 1.48710i	−0.668679	−	0.743552i	$-0.733140\pi$
$263$	− 102861.i	− 1.48710i	0.668679	−	0.743552i	$-0.266860\pi$
$264$	0	0
$265$	22287.8	0.317377
$266$	59924.2i	0.846913i
$267$	0	0
$268$	−25840.8	−0.359780
$269$	− 91308.7i	− 1.26185i	−0.775844	−	0.630925i	$-0.782676\pi$
$269$	− 91308.7i	− 1.26185i	0.775844	−	0.630925i	$-0.217324\pi$
$270$	0	0
$271$	−141309.	−1.92411	−0.962055	−	0.272857i	$-0.912031\pi$
$271$	−141309.	−1.92411	−0.962055	+	0.272857i	$0.912031\pi$
$272$	− 23626.9i	− 0.319351i
$273$	0	0
$274$	49692.4	0.661894
$275$	− 2153.96i	− 0.0284822i
$276$	0	0
$277$	−132664.	−1.72899	−0.864495	−	0.502642i	$-0.832362\pi$
$277$	−132664.	−1.72899	−0.864495	+	0.502642i	$0.832362\pi$
$278$	− 27621.0i	− 0.357397i
$279$	0	0
$280$	40433.8	0.515737
$281$	28055.2i	0.355304i	0.984093	+	0.177652i	$0.0568501\pi$
$281$	28055.2i	0.355304i	−0.984093	+	0.177652i	$0.943150\pi$
$282$	0	0
$283$	50386.4	0.629130	0.314565	−	0.949236i	$-0.398141\pi$
$283$	50386.4	0.629130	0.314565	+	0.949236i	$0.398141\pi$
$284$	− 66334.6i	− 0.822438i
$285$	0	0
$286$	−836.066	−0.0102214
$287$	20030.0i	0.243174i
$288$	0	0
$289$	−8405.98	−0.100645
$290$	18520.2i	0.220217i
$291$	0	0
$292$	46777.3	0.548617
$293$	− 38134.8i	− 0.444208i	−0.975023	−	0.222104i	$-0.928708\pi$
$293$	− 38134.8i	− 0.444208i	0.975023	−	0.222104i	$-0.0712925\pi$
$294$	0	0
$295$	−4576.41	−0.0525873
$296$	135587.i	1.54751i
$297$	0	0
$298$	40780.1	0.459214
$299$	23668.6i	0.264747i
$300$	0	0
$301$	−46376.3	−0.511874
$302$	54134.4i	0.593553i
$303$	0	0
$304$	−32594.3	−0.352691
$305$	− 28038.8i	− 0.301412i
$306$	0	0
$307$	−65574.8	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$307$	−65574.8	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$308$	3502.11i	0.0369172i
$309$	0	0
$310$	−14921.2	−0.155267
$311$	13905.9i	0.143773i	0.997413	+	0.0718866i	$0.0229020\pi$
$311$	13905.9i	0.143773i	−0.997413	+	0.0718866i	$0.977098\pi$
$312$	0	0
$313$	30105.3	0.307294	0.153647	−	0.988126i	$-0.450898\pi$
$313$	30105.3	0.307294	0.153647	+	0.988126i	$0.450898\pi$
$314$	26974.1i	0.273583i
$315$	0	0
$316$	−89800.7	−0.899302
$317$	− 105142.i	− 1.04630i	−0.852240	−	0.523151i	$-0.824756\pi$
$317$	− 105142.i	− 1.04630i	0.852240	−	0.523151i	$-0.175244\pi$
$318$	0	0
$319$	−3752.20	−0.0368726
$320$	− 8894.72i	− 0.0868625i
$321$	0	0
$322$	−33622.0	−0.324273
$323$	126817.i	1.21555i
$324$	0	0
$325$	−52749.4	−0.499402
$326$	− 22351.9i	− 0.210319i
$327$	0	0
$328$	15832.4	0.147163
$329$	54632.8i	0.504733i
$330$	0	0
$331$	−18384.3	−0.167800	−0.0838998	−	0.996474i	$-0.526738\pi$
$331$	−18384.3	−0.167800	−0.0838998	+	0.996474i	$0.526738\pi$
$332$	55565.6i	0.504115i
$333$	0	0
$334$	−21511.7	−0.192834
$335$	− 21840.1i	− 0.194610i
$336$	0	0
$337$	−67724.2	−0.596326	−0.298163	−	0.954515i	$-0.596374\pi$
$337$	−67724.2	−0.596326	−0.298163	+	0.954515i	$0.596374\pi$
$338$	− 37016.6i	− 0.324014i
$339$	0	0
$340$	36582.0	0.316453
$341$	− 3023.03i	− 0.0259976i
$342$	0	0
$343$	−18761.3	−0.159469
$344$	36657.5i	0.309774i
$345$	0	0
$346$	−92839.4	−0.775497
$347$	1033.68i	0.00858473i	0.999991	+	0.00429236i	$0.00136631\pi$
$347$	1033.68i	0.00858473i	−0.999991	+	0.00429236i	$0.998634\pi$
$348$	0	0
$349$	−140074.	−1.15002	−0.575012	−	0.818145i	$-0.695003\pi$
$349$	−140074.	−1.15002	−0.575012	+	0.818145i	$0.695003\pi$
$350$	− 74932.0i	− 0.611690i
$351$	0	0
$352$	4352.96	0.0351317
$353$	182705.i	1.46622i	0.680109	+	0.733111i	$0.261933\pi$
$353$	182705.i	1.46622i	−0.680109	+	0.733111i	$0.738067\pi$
$354$	0	0
$355$	56064.6	0.444869
$356$	− 33348.7i	− 0.263136i
$357$	0	0
$358$	40559.7	0.316467
$359$	49312.9i	0.382624i	0.981529	+	0.191312i	$0.0612742\pi$
$359$	49312.9i	0.382624i	−0.981529	+	0.191312i	$0.938726\pi$
$360$	0	0
$361$	44628.8	0.342453
$362$	− 72663.5i	− 0.554497i
$363$	0	0
$364$	85764.8	0.647301
$365$	39535.2i	0.296755i
$366$	0	0
$367$	25551.2	0.189705	0.0948526	−	0.995491i	$-0.469762\pi$
$367$	25551.2	0.189705	0.0948526	+	0.995491i	$0.469762\pi$
$368$	− 18287.9i	− 0.135042i
$369$	0	0
$370$	−48990.6	−0.357857
$371$	157085.i	1.14127i
$372$	0	0
$373$	160325.	1.15235	0.576173	−	0.817327i	$-0.304545\pi$
$373$	160325.	1.15235	0.576173	+	0.817327i	$0.304545\pi$
$374$	− 2513.43i	− 0.0179690i
$375$	0	0
$376$	43183.7	0.305453
$377$	91889.2i	0.646520i
$378$	0	0
$379$	−209111.	−1.45579	−0.727893	−	0.685691i	$-0.759500\pi$
$379$	−209111.	−1.45579	−0.727893	+	0.685691i	$0.759500\pi$
$380$	− 50466.4i	− 0.349491i
$381$	0	0
$382$	−30317.1	−0.207759
$383$	212147.i	1.44623i	0.690725	+	0.723117i	$0.257291\pi$
$383$	212147.i	1.44623i	−0.690725	+	0.723117i	$0.742709\pi$
$384$	0	0
$385$	−2959.91	−0.0199690
$386$	− 28977.1i	− 0.194483i
$387$	0	0
$388$	99014.5	0.657712
$389$	92564.4i	0.611709i	0.952078	+	0.305854i	$0.0989421\pi$
$389$	92564.4i	0.611709i	−0.952078	+	0.305854i	$0.901058\pi$
$390$	0	0
$391$	−71154.0	−0.465421
$392$	149904.i	0.975533i
$393$	0	0
$394$	98969.9	0.637545
$395$	− 75897.6i	− 0.486445i
$396$	0	0
$397$	81617.8	0.517850	0.258925	−	0.965897i	$-0.416632\pi$
$397$	81617.8	0.517850	0.258925	+	0.965897i	$0.416632\pi$
$398$	21436.6i	0.135329i
$399$	0	0
$400$	40757.5	0.254734
$401$	− 31534.8i	− 0.196111i	−0.995181	−	0.0980554i	$-0.968738\pi$
$401$	− 31534.8i	− 0.196111i	0.995181	−	0.0980554i	$-0.0312622\pi$
$402$	0	0
$403$	−74032.2	−0.455838
$404$	− 11687.2i	− 0.0716058i
$405$	0	0
$406$	−130531.	−0.791886
$407$	− 9925.49i	− 0.0599188i
$408$	0	0
$409$	107413.	0.642108	0.321054	−	0.947061i	$-0.395963\pi$
$409$	107413.	0.642108	0.321054	+	0.947061i	$0.395963\pi$
$410$	5720.61i	0.0340310i
$411$	0	0
$412$	8627.51	0.0508266
$413$	− 32254.7i	− 0.189101i
$414$	0	0
$415$	−46962.9	−0.272683
$416$	− 106602.i	− 0.615994i
$417$	0	0
$418$	−3467.39	−0.0198449
$419$	− 83932.3i	− 0.478081i	−0.971010	−	0.239040i	$-0.923167\pi$
$419$	− 83932.3i	− 0.478081i	0.971010	−	0.239040i	$-0.0768328\pi$
$420$	0	0
$421$	−216482.	−1.22140	−0.610699	−	0.791863i	$-0.709111\pi$
$421$	−216482.	−1.22140	−0.610699	+	0.791863i	$0.709111\pi$
$422$	− 69975.5i	− 0.392935i
$423$	0	0
$424$	124166.	0.690669
$425$	− 158578.i	− 0.877942i
$426$	0	0
$427$	197619.	1.08386
$428$	− 168766.i	− 0.921293i
$429$	0	0
$430$	−13245.2	−0.0716342
$431$	277065.i	1.49151i	0.666220	+	0.745755i	$0.267911\pi$
$431$	277065.i	1.49151i	−0.666220	+	0.745755i	$0.732089\pi$
$432$	0	0
$433$	197182.	1.05170	0.525850	−	0.850577i	$-0.323747\pi$
$433$	197182.	1.05170	0.525850	+	0.850577i	$0.323747\pi$
$434$	− 105165.i	− 0.558331i
$435$	0	0
$436$	−77503.5	−0.407707
$437$	98160.0i	0.514010i
$438$	0	0
$439$	−205163.	−1.06456	−0.532280	−	0.846568i	$-0.678665\pi$
$439$	−205163.	−1.06456	−0.532280	+	0.846568i	$0.678665\pi$
$440$	2339.62i	0.0120848i
$441$	0	0
$442$	−61552.6	−0.315066
$443$	− 239620.i	− 1.22100i	−0.792016	−	0.610500i	$-0.790969\pi$
$443$	− 239620.i	− 1.22100i	0.792016	−	0.610500i	$-0.209031\pi$
$444$	0	0
$445$	28185.7	0.142334
$446$	− 44260.7i	− 0.222510i
$447$	0	0
$448$	62690.3	0.312352
$449$	− 273606.i	− 1.35716i	−0.734524	−	0.678582i	$-0.762595\pi$
$449$	− 273606.i	− 1.35716i	0.734524	−	0.678582i	$-0.237405\pi$
$450$	0	0
$451$	−1159.00	−0.00569808
$452$	− 146903.i	− 0.719040i
$453$	0	0
$454$	−118122.	−0.573084
$455$	72486.6i	0.350135i
$456$	0	0
$457$	192676.	0.922560	0.461280	−	0.887255i	$-0.347390\pi$
$457$	192676.	0.922560	0.461280	+	0.887255i	$0.347390\pi$
$458$	108759.i	0.518483i
$459$	0	0
$460$	28315.5	0.133816
$461$	− 171132.i	− 0.805245i	−0.915366	−	0.402623i	$-0.868099\pi$
$461$	− 171132.i	− 0.805245i	0.915366	−	0.402623i	$-0.131901\pi$
$462$	0	0
$463$	−123861.	−0.577794	−0.288897	−	0.957360i	$-0.593288\pi$
$463$	−123861.	−0.577794	−0.288897	+	0.957360i	$0.593288\pi$
$464$	− 70999.4i	− 0.329776i
$465$	0	0
$466$	74715.8	0.344065
$467$	371001.i	1.70114i	0.525858	+	0.850572i	$0.323744\pi$
$467$	371001.i	1.70114i	−0.525858	+	0.850572i	$0.676256\pi$
$468$	0	0
$469$	153930.	0.699805
$470$	15603.2i	0.0706348i
$471$	0	0
$472$	−25495.3	−0.114439
$473$	− 2683.47i	− 0.0119943i
$474$	0	0
$475$	−218766.	−0.969598
$476$	257831.i	1.13795i
$477$	0	0
$478$	77515.0	0.339258
$479$	309094.i	1.34716i	0.739113	+	0.673581i	$0.235245\pi$
$479$	309094.i	1.34716i	−0.739113	+	0.673581i	$0.764755\pi$
$480$	0	0
$481$	−243070.	−1.05061
$482$	− 74118.7i	− 0.319032i
$483$	0	0
$484$	174729.	0.745891
$485$	83685.0i	0.355766i
$486$	0	0
$487$	302160.	1.27403	0.637013	−	0.770853i	$-0.280170\pi$
$487$	302160.	1.27403	0.637013	+	0.770853i	$0.280170\pi$
$488$	− 156205.i	− 0.655926i
$489$	0	0
$490$	−54163.8	−0.225589
$491$	− 38466.5i	− 0.159558i	−0.996813	−	0.0797791i	$-0.974579\pi$
$491$	− 38466.5i	− 0.159558i	0.996813	−	0.0797791i	$-0.0254215\pi$
$492$	0	0
$493$	−276243.	−1.13657
$494$	84914.4i	0.347959i
$495$	0	0
$496$	57201.9	0.232513
$497$	395145.i	1.59972i
$498$	0	0
$499$	−286908.	−1.15224	−0.576119	−	0.817366i	$-0.695433\pi$
$499$	−286908.	−1.15224	−0.576119	+	0.817366i	$0.695433\pi$
$500$	138515.i	0.554061i
$501$	0	0
$502$	−137612.	−0.546069
$503$	193369.i	0.764279i	0.924105	+	0.382139i	$0.124813\pi$
$503$	193369.i	0.764279i	−0.924105	+	0.382139i	$0.875187\pi$
$504$	0	0
$505$	9877.78	0.0387326
$506$	− 1945.47i	− 0.00759841i
$507$	0	0
$508$	257745.	0.998763
$509$	201633.i	0.778263i	0.921182	+	0.389132i	$0.127225\pi$
$509$	201633.i	0.778263i	−0.921182	+	0.389132i	$0.872775\pi$
$510$	0	0
$511$	−278645.	−1.06711
$512$	152510.i	0.581781i
$513$	0	0
$514$	183009.	0.692702
$515$	7291.79i	0.0274928i
$516$	0	0
$517$	−3161.21	−0.0118269
$518$	− 345288.i	− 1.28683i
$519$	0	0
$520$	57296.0	0.211893
$521$	− 355141.i	− 1.30836i	−0.756341	−	0.654178i	$-0.773015\pi$
$521$	− 355141.i	− 1.30836i	0.756341	−	0.654178i	$-0.226985\pi$
$522$	0	0
$523$	193142.	0.706113	0.353056	−	0.935602i	$-0.385142\pi$
$523$	193142.	0.706113	0.353056	+	0.935602i	$0.385142\pi$
$524$	− 298856.i	− 1.08843i
$525$	0	0
$526$	207053.	0.748360
$527$	− 222560.i	− 0.801357i
$528$	0	0
$529$	224766.	0.803191
$530$	44863.9i	0.159715i
$531$	0	0
$532$	355689.	1.25675
$533$	28383.2i	0.0999094i
$534$	0	0
$535$	142638.	0.498341
$536$	− 121672.i	− 0.423506i
$537$	0	0
$538$	183799.	0.635005
$539$	− 10973.6i	− 0.0377721i
$540$	0	0
$541$	−341165.	−1.16565	−0.582827	−	0.812596i	$-0.698053\pi$
$541$	−341165.	−1.16565	−0.582827	+	0.812596i	$0.698053\pi$
$542$	− 284445.i	− 0.968276i
$543$	0	0
$544$	320472.	1.08291
$545$	− 65504.3i	− 0.220535i
$546$	0	0
$547$	367378.	1.22783	0.613916	−	0.789372i	$-0.289594\pi$
$547$	367378.	1.22783	0.613916	+	0.789372i	$0.289594\pi$
$548$	− 294957.i	− 0.982195i
$549$	0	0
$550$	4335.79	0.0143332
$551$	381089.i	1.25523i
$552$	0	0
$553$	534929.	1.74923
$554$	− 267043.i	− 0.870086i
$555$	0	0
$556$	−163949.	−0.530346
$557$	30569.6i	0.0985326i	0.998786	+	0.0492663i	$0.0156883\pi$
$557$	30569.6i	0.0985326i	−0.998786	+	0.0492663i	$0.984312\pi$
$558$	0	0
$559$	−65716.7	−0.210306
$560$	− 56007.7i	− 0.178596i
$561$	0	0
$562$	−56473.2	−0.178801
$563$	− 77213.5i	− 0.243599i	−0.992555	−	0.121800i	$-0.961133\pi$
$563$	− 77213.5i	− 0.243599i	0.992555	−	0.121800i	$-0.0388666\pi$
$564$	0	0
$565$	124159.	0.388939
$566$	101425.i	0.316599i
$567$	0	0
$568$	312337.	0.968114
$569$	408704.i	1.26236i	0.775636	+	0.631181i	$0.217429\pi$
$569$	408704.i	1.26236i	−0.775636	+	0.631181i	$0.782571\pi$
$570$	0	0
$571$	−254842.	−0.781626	−0.390813	−	0.920470i	$-0.627806\pi$
$571$	−254842.	−0.781626	−0.390813	+	0.920470i	$0.627806\pi$
$572$	4962.61i	0.0151676i
$573$	0	0
$574$	−40319.1	−0.122373
$575$	− 122744.i	− 0.371248i
$576$	0	0
$577$	73724.5	0.221442	0.110721	−	0.993852i	$-0.464684\pi$
$577$	73724.5	0.221442	0.110721	+	0.993852i	$0.464684\pi$
$578$	− 16920.7i	− 0.0506480i
$579$	0	0
$580$	109930.	0.326783
$581$	− 330996.i	− 0.980552i
$582$	0	0
$583$	−9089.42	−0.0267423
$584$	220251.i	0.645792i
$585$	0	0
$586$	76762.9	0.223540
$587$	− 473911.i	− 1.37537i	−0.726008	−	0.687686i	$-0.758626\pi$
$587$	− 473911.i	− 1.37537i	0.726008	−	0.687686i	$-0.241374\pi$
$588$	0	0
$589$	−307031.	−0.885018
$590$	− 9212.02i	− 0.0264637i
$591$	0	0
$592$	187811.	0.535892
$593$	180552.i	0.513444i	0.966485	+	0.256722i	$0.0826425\pi$
$593$	180552.i	0.513444i	−0.966485	+	0.256722i	$0.917358\pi$
$594$	0	0
$595$	−217914.	−0.615532
$596$	− 242057.i	− 0.681435i
$597$	0	0
$598$	−47643.4	−0.133229
$599$	− 630100.i	− 1.75613i	−0.478545	−	0.878063i	$-0.658836\pi$
$599$	− 630100.i	− 1.75613i	0.478545	−	0.878063i	$-0.341164\pi$
$600$	0	0
$601$	−265027.	−0.733739	−0.366869	−	0.930272i	$-0.619570\pi$
$601$	−265027.	−0.733739	−0.366869	+	0.930272i	$0.619570\pi$
$602$	− 93352.5i	− 0.257592i
$603$	0	0
$604$	321324.	0.880783
$605$	147678.i	0.403463i
$606$	0	0
$607$	593043.	1.60957	0.804783	−	0.593569i	$-0.202282\pi$
$607$	593043.	1.60957	0.804783	+	0.593569i	$0.202282\pi$
$608$	− 442105.i	− 1.19596i
$609$	0	0
$610$	56440.3	0.151681
$611$	77416.3i	0.207372i
$612$	0	0
$613$	−140385.	−0.373593	−0.186797	−	0.982399i	$-0.559811\pi$
$613$	−140385.	−0.373593	−0.186797	+	0.982399i	$0.559811\pi$
$614$	− 131998.i	− 0.350131i
$615$	0	0
$616$	−16489.7	−0.0434562
$617$	− 504095.i	− 1.32416i	−0.749432	−	0.662082i	$-0.769673\pi$
$617$	− 504095.i	− 1.32416i	0.749432	−	0.662082i	$-0.230327\pi$
$618$	0	0
$619$	525814.	1.37230	0.686152	−	0.727458i	$-0.259298\pi$
$619$	525814.	1.37230	0.686152	+	0.727458i	$0.259298\pi$
$620$	88566.9i	0.230403i
$621$	0	0
$622$	−27991.7	−0.0723515
$623$	198654.i	0.511824i
$624$	0	0
$625$	209820.	0.537140
$626$	60599.9i	0.154640i
$627$	0	0
$628$	160109.	0.405973
$629$	− 730731.i	− 1.84695i
$630$	0	0
$631$	497175.	1.24868	0.624340	−	0.781153i	$-0.285368\pi$
$631$	497175.	1.24868	0.624340	+	0.781153i	$0.285368\pi$
$632$	− 422827.i	− 1.05859i
$633$	0	0
$634$	211644.	0.526534
$635$	217841.i	0.540246i
$636$	0	0
$637$	−268737.	−0.662290
$638$	− 7552.93i	− 0.0185556i
$639$	0	0
$640$	−152875.	−0.373230
$641$	682246.i	1.66045i	0.557431	+	0.830223i	$0.311787\pi$
$641$	682246.i	1.66045i	−0.557431	+	0.830223i	$0.688213\pi$
$642$	0	0
$643$	94765.6	0.229208	0.114604	−	0.993411i	$-0.463440\pi$
$643$	94765.6	0.229208	0.114604	+	0.993411i	$0.463440\pi$
$644$	199569.i	0.481194i
$645$	0	0
$646$	−255275.	−0.611706
$647$	− 572865.i	− 1.36850i	−0.729249	−	0.684249i	$-0.760130\pi$
$647$	− 572865.i	− 1.36850i	0.729249	−	0.684249i	$-0.239870\pi$
$648$	0	0
$649$	1866.35	0.00443103
$650$	− 106181.i	− 0.251316i
$651$	0	0
$652$	−132673.	−0.312096
$653$	− 144167.i	− 0.338095i	−0.985608	−	0.169047i	$-0.945931\pi$
$653$	− 144167.i	− 0.338095i	0.985608	−	0.169047i	$-0.0540691\pi$
$654$	0	0
$655$	252587.	0.588746
$656$	− 21930.6i	− 0.0509616i
$657$	0	0
$658$	−109972.	−0.253998
$659$	− 301362.i	− 0.693933i	−0.937877	−	0.346967i	$-0.887212\pi$
$659$	− 301362.i	− 0.693933i	0.937877	−	0.346967i	$-0.112788\pi$
$660$	0	0
$661$	504343.	1.15431	0.577156	−	0.816634i	$-0.304162\pi$
$661$	504343.	1.15431	0.577156	+	0.816634i	$0.304162\pi$
$662$	− 37006.4i	− 0.0844424i
$663$	0	0
$664$	−261631.	−0.593408
$665$	300621.i	0.679792i
$666$	0	0
$667$	−213820.	−0.480613
$668$	127686.i	0.286149i
$669$	0	0
$670$	43962.7	0.0979343
$671$	11434.8i	0.0253971i
$672$	0	0
$673$	−148143.	−0.327078	−0.163539	−	0.986537i	$-0.552291\pi$
$673$	−148143.	−0.327078	−0.163539	+	0.986537i	$0.552291\pi$
$674$	− 136324.i	− 0.300092i
$675$	0	0
$676$	−219718.	−0.480809
$677$	873435.i	1.90569i	0.303450	+	0.952847i	$0.401861\pi$
$677$	873435.i	1.90569i	−0.303450	+	0.952847i	$0.598139\pi$
$678$	0	0
$679$	−589815.	−1.27931
$680$	172247.i	0.372506i
$681$	0	0
$682$	6085.15	0.0130829
$683$	− 522534.i	− 1.12014i	−0.828445	−	0.560071i	$-0.810774\pi$
$683$	− 522534.i	− 1.12014i	0.828445	−	0.560071i	$-0.189226\pi$
$684$	0	0
$685$	249292.	0.531284
$686$	− 37765.3i	− 0.0802500i
$687$	0	0
$688$	50776.8	0.107273
$689$	222595.i	0.468896i
$690$	0	0
$691$	−16169.3	−0.0338638	−0.0169319	−	0.999857i	$-0.505390\pi$
$691$	−16169.3	−0.0338638	−0.0169319	+	0.999857i	$0.505390\pi$
$692$	551063.i	1.15077i
$693$	0	0
$694$	−2080.73	−0.00432012
$695$	− 138566.i	− 0.286872i
$696$	0	0
$697$	−85327.1	−0.175639
$698$	− 281960.i	− 0.578731i
$699$	0	0
$700$	−444771.	−0.907696
$701$	415990.i	0.846540i	0.906004	+	0.423270i	$0.139118\pi$
$701$	415990.i	0.846540i	−0.906004	+	0.423270i	$0.860882\pi$
$702$	0	0
$703$	−1.00807e6	−2.03977
$704$	3627.45i	0.00731907i
$705$	0	0
$706$	−367772.	−0.737852
$707$	69619.0i	0.139280i
$708$	0	0
$709$	−368967.	−0.733997	−0.366999	−	0.930221i	$-0.619615\pi$
$709$	−368967.	−0.733997	−0.366999	+	0.930221i	$0.619615\pi$
$710$	112854.i	0.223873i
$711$	0	0
$712$	157023.	0.309744
$713$	− 172268.i	− 0.338863i
$714$	0	0
$715$	−4194.29	−0.00820439
$716$	− 240749.i	− 0.469611i
$717$	0	0
$718$	−99263.7	−0.192549
$719$	− 373869.i	− 0.723205i	−0.932332	−	0.361602i	$-0.882230\pi$
$719$	− 373869.i	− 0.723205i	0.932332	−	0.361602i	$-0.117770\pi$
$720$	0	0
$721$	−51392.8	−0.0988625
$722$	89834.9i	0.172334i
$723$	0	0
$724$	−431306.	−0.822826
$725$	− 476532.i	− 0.906600i
$726$	0	0
$727$	−83762.0	−0.158481	−0.0792407	−	0.996856i	$-0.525250\pi$
$727$	−83762.0	−0.158481	−0.0792407	+	0.996856i	$0.525250\pi$
$728$	403824.i	0.761956i
$729$	0	0
$730$	−79581.7	−0.149337
$731$	− 197561.i	− 0.369715i
$732$	0	0
$733$	−889288.	−1.65514	−0.827570	−	0.561363i	$-0.810277\pi$
$733$	−889288.	−1.65514	−0.827570	+	0.561363i	$0.810277\pi$
$734$	51432.9i	0.0954660i
$735$	0	0
$736$	248054.	0.457921
$737$	8906.84i	0.0163979i
$738$	0	0
$739$	1.00495e6	1.84016	0.920079	−	0.391733i	$-0.128124\pi$
$739$	1.00495e6	1.84016	0.920079	+	0.391733i	$0.128124\pi$
$740$	290791.i	0.531029i
$741$	0	0
$742$	−316202.	−0.574324
$743$	148884.i	0.269693i	0.990867	+	0.134846i	$0.0430541\pi$
$743$	148884.i	0.269693i	−0.990867	+	0.134846i	$0.956946\pi$
$744$	0	0
$745$	204581.	0.368598
$746$	322723.i	0.579900i
$747$	0	0
$748$	−14918.9	−0.0266645
$749$	1.00532e6i	1.79200i
$750$	0	0
$751$	−352149.	−0.624376	−0.312188	−	0.950020i	$-0.601062\pi$
$751$	−352149.	−0.624376	−0.312188	+	0.950020i	$0.601062\pi$
$752$	− 59816.7i	− 0.105776i
$753$	0	0
$754$	−184967.	−0.325351
$755$	271576.i	0.476428i
$756$	0	0
$757$	−666652.	−1.16334	−0.581671	−	0.813424i	$-0.697601\pi$
$757$	−666652.	−1.16334	−0.581671	+	0.813424i	$0.697601\pi$
$758$	− 420926.i	− 0.732601i
$759$	0	0
$760$	237622.	0.411395
$761$	− 364827.i	− 0.629967i	−0.949097	−	0.314984i	$-0.898001\pi$
$761$	− 364827.i	− 0.629967i	0.949097	−	0.314984i	$-0.101999\pi$
$762$	0	0
$763$	461677.	0.793029
$764$	179952.i	0.308297i
$765$	0	0
$766$	−427037.	−0.727794
$767$	− 45706.0i	− 0.0776931i
$768$	0	0
$769$	643294.	1.08782	0.543910	−	0.839144i	$-0.316943\pi$
$769$	643294.	1.08782	0.543910	+	0.839144i	$0.316943\pi$
$770$	− 5958.11i	− 0.0100491i
$771$	0	0
$772$	−171998.	−0.288596
$773$	586359.i	0.981306i	0.871355	+	0.490653i	$0.163242\pi$
$773$	586359.i	0.981306i	−0.871355	+	0.490653i	$0.836758\pi$
$774$	0	0
$775$	383926.	0.639211
$776$	466211.i	0.774210i
$777$	0	0
$778$	−186326.	−0.307832
$779$	117712.i	0.193976i
$780$	0	0
$781$	−22864.3	−0.0374848
$782$	− 143228.i	− 0.234215i
$783$	0	0
$784$	207643.	0.337820
$785$	135321.i	0.219597i
$786$	0	0
$787$	876083.	1.41448	0.707239	−	0.706975i	$-0.249941\pi$
$787$	876083.	1.41448	0.707239	+	0.706975i	$0.249941\pi$
$788$	− 587452.i	− 0.946062i
$789$	0	0
$790$	152777.	0.244796
$791$	875078.i	1.39860i
$792$	0	0
$793$	280032.	0.445309
$794$	164291.i	0.260599i
$795$	0	0
$796$	127240.	0.200816
$797$	− 1.10906e6i	− 1.74598i	−0.487738	−	0.872990i	$-0.662178\pi$
$797$	− 1.10906e6i	− 1.74598i	0.487738	−	0.872990i	$-0.337822\pi$
$798$	0	0
$799$	−232734.	−0.364557
$800$	552829.i	0.863795i
$801$	0	0
$802$	63477.5	0.0986896
$803$	− 16123.2i	− 0.0250047i
$804$	0	0
$805$	−168671.	−0.260285
$806$	− 149022.i	− 0.229393i
$807$	0	0
$808$	55029.3	0.0842891
$809$	− 37849.4i	− 0.0578312i	−0.999582	−	0.0289156i	$-0.990795\pi$
$809$	− 37849.4i	− 0.0578312i	0.999582	−	0.0289156i	$-0.00920541\pi$
$810$	0	0
$811$	912976.	1.38809	0.694045	−	0.719932i	$-0.255827\pi$
$811$	912976.	1.38809	0.694045	+	0.719932i	$0.255827\pi$
$812$	774790.i	1.17509i
$813$	0	0
$814$	19979.4	0.0301531
$815$	− 112133.i	− 0.168817i
$816$	0	0
$817$	−272544.	−0.408313
$818$	216214.i	0.323130i
$819$	0	0
$820$	33955.6	0.0504991
$821$	− 659471.i	− 0.978384i	−0.872176	−	0.489192i	$-0.837292\pi$
$821$	− 659471.i	− 0.978384i	0.872176	−	0.489192i	$-0.162708\pi$
$822$	0	0
$823$	−1.02704e6	−1.51631	−0.758156	−	0.652073i	$-0.773899\pi$
$823$	−1.02704e6	−1.51631	−0.758156	+	0.652073i	$0.773899\pi$
$824$	40622.7i	0.0598293i
$825$	0	0
$826$	64926.7	0.0951619
$827$	866459.i	1.26688i	0.773790	+	0.633442i	$0.218359\pi$
$827$	866459.i	1.26688i	−0.773790	+	0.633442i	$0.781641\pi$
$828$	0	0
$829$	597032.	0.868737	0.434369	−	0.900735i	$-0.356972\pi$
$829$	597032.	0.868737	0.434369	+	0.900735i	$0.356972\pi$
$830$	− 94533.2i	− 0.137223i
$831$	0	0
$832$	88834.1	0.128332
$833$	− 807893.i	− 1.16430i
$834$	0	0
$835$	−107918.	−0.154782
$836$	20581.2i	0.0294482i
$837$	0	0
$838$	168950.	0.240586
$839$	519159.i	0.737525i	0.929524	+	0.368762i	$0.120218\pi$
$839$	519159.i	0.737525i	−0.929524	+	0.368762i	$0.879782\pi$
$840$	0	0
$841$	−122836.	−0.173673
$842$	− 435763.i	− 0.614648i
$843$	0	0
$844$	−415351.	−0.583083
$845$	− 185701.i	− 0.260076i
$846$	0	0
$847$	−1.04084e6	−1.45083
$848$	− 171991.i	− 0.239174i
$849$	0	0
$850$	319208.	0.441810
$851$	− 565606.i	− 0.781006i
$852$	0	0
$853$	−140457.	−0.193039	−0.0965194	−	0.995331i	$-0.530771\pi$
$853$	−140457.	−0.193039	−0.0965194	+	0.995331i	$0.530771\pi$
$854$	397794.i	0.545434i
$855$	0	0
$856$	794637.	1.08448
$857$	13314.5i	0.0181285i	0.999959	+	0.00906427i	$0.00288529\pi$
$857$	13314.5i	0.0181285i	−0.999959	+	0.00906427i	$0.997115\pi$
$858$	0	0
$859$	546504.	0.740639	0.370320	−	0.928904i	$-0.379248\pi$
$859$	546504.	0.740639	0.370320	+	0.928904i	$0.379248\pi$
$860$	78618.8i	0.106299i
$861$	0	0
$862$	−557713.	−0.750578
$863$	1.24644e6i	1.67360i	0.547512	+	0.836798i	$0.315575\pi$
$863$	1.24644e6i	1.67360i	−0.547512	+	0.836798i	$0.684425\pi$
$864$	0	0
$865$	−465747.	−0.622469
$866$	396915.i	0.529251i
$867$	0	0
$868$	−624223.	−0.828515
$869$	30952.6i	0.0409881i
$870$	0	0
$871$	218124.	0.287519
$872$	− 364926.i	− 0.479923i
$873$	0	0
$874$	−197590.	−0.258667
$875$	− 825114.i	− 1.07770i
$876$	0	0
$877$	318374.	0.413940	0.206970	−	0.978347i	$-0.433640\pi$
$877$	318374.	0.413940	0.206970	+	0.978347i	$0.433640\pi$
$878$	− 412980.i	− 0.535723i
$879$	0	0
$880$	3240.77	0.00418488
$881$	1.33695e6i	1.72252i	0.508166	+	0.861259i	$0.330324\pi$
$881$	1.33695e6i	1.72252i	−0.508166	+	0.861259i	$0.669676\pi$
$882$	0	0
$883$	−755088.	−0.968448	−0.484224	−	0.874944i	$-0.660898\pi$
$883$	−755088.	−0.968448	−0.484224	+	0.874944i	$0.660898\pi$
$884$	365355.i	0.467531i
$885$	0	0
$886$	482339.	0.614448
$887$	− 322944.i	− 0.410469i	−0.978713	−	0.205234i	$-0.934204\pi$
$887$	− 322944.i	− 0.410469i	0.978713	−	0.205234i	$-0.0657957\pi$
$888$	0	0
$889$	−1.53535e6	−1.94269
$890$	56735.9i	0.0716272i
$891$	0	0
$892$	−262717.	−0.330185
$893$	321066.i	0.402616i
$894$	0	0
$895$	203476.	0.254019
$896$	− 1.07747e6i	− 1.34211i
$897$	0	0
$898$	550750.	0.682971
$899$	− 668798.i	− 0.827515i
$900$	0	0
$901$	−669178.	−0.824312
$902$	− 2332.98i	− 0.00286747i
$903$	0	0
$904$	691693.	0.846402
$905$	− 364530.i	− 0.445078i
$906$	0	0
$907$	171494.	0.208466	0.104233	−	0.994553i	$-0.466761\pi$
$907$	171494.	0.208466	0.104233	+	0.994553i	$0.466761\pi$
$908$	701130.i	0.850407i
$909$	0	0
$910$	−145911.	−0.176199
$911$	49958.8i	0.0601970i	0.999547	+	0.0300985i	$0.00958210\pi$
$911$	49958.8i	0.0601970i	−0.999547	+	0.0300985i	$0.990418\pi$
$912$	0	0
$913$	19152.4	0.0229764
$914$	387844.i	0.464263i
$915$	0	0
$916$	645557.	0.769385
$917$	1.78024e6i	2.11709i
$918$	0	0
$919$	−99945.9	−0.118341	−0.0591703	−	0.998248i	$-0.518846\pi$
$919$	−99945.9	−0.118341	−0.0591703	+	0.998248i	$0.518846\pi$
$920$	133324.i	0.157518i
$921$	0	0
$922$	344477.	0.405226
$923$	559933.i	0.657254i
$924$	0	0
$925$	1.26054e6	1.47324
$926$	− 249324.i	− 0.290765i
$927$	0	0
$928$	963026.	1.11826
$929$	648214.i	0.751082i	0.926806	+	0.375541i	$0.122543\pi$
$929$	648214.i	0.751082i	−0.926806	+	0.375541i	$0.877457\pi$
$930$	0	0
$931$	−1.11452e6	−1.28585
$932$	− 443488.i	− 0.510564i
$933$	0	0
$934$	−746801.	−0.856073
$935$	− 12609.1i	− 0.0144232i
$936$	0	0
$937$	−142420.	−0.162215	−0.0811076	−	0.996705i	$-0.525846\pi$
$937$	−142420.	−0.162215	−0.0811076	+	0.996705i	$0.525846\pi$
$938$	309851.i	0.352166i
$939$	0	0
$940$	92615.5	0.104816
$941$	− 1.44043e6i	− 1.62672i	−0.581761	−	0.813360i	$-0.697636\pi$
$941$	− 1.44043e6i	− 1.62672i	0.581761	−	0.813360i	$-0.302364\pi$
$942$	0	0
$943$	−66045.6	−0.0742712
$944$	35315.3i	0.0396295i
$945$	0	0
$946$	5401.65	0.00603593
$947$	734836.i	0.819390i	0.912223	+	0.409695i	$0.134365\pi$
$947$	734836.i	0.819390i	−0.912223	+	0.409695i	$0.865635\pi$
$948$	0	0
$949$	−394849.	−0.438429
$950$	− 440361.i	− 0.487934i
$951$	0	0
$952$	−1.21400e6	−1.33951
$953$	145389.i	0.160083i	0.996792	+	0.0800416i	$0.0255053\pi$
$953$	145389.i	0.160083i	−0.996792	+	0.0800416i	$0.974495\pi$
$954$	0	0
$955$	−152091.	−0.166762
$956$	− 460103.i	− 0.503430i
$957$	0	0
$958$	−622187.	−0.677938
$959$	1.75702e6i	1.91046i
$960$	0	0
$961$	−384691.	−0.416549
$962$	− 489283.i	− 0.528701i
$963$	0	0
$964$	−439944.	−0.473416
$965$	− 145369.i	− 0.156106i
$966$	0	0
$967$	−451493.	−0.482834	−0.241417	−	0.970421i	$-0.577612\pi$
$967$	−451493.	−0.482834	−0.241417	+	0.970421i	$0.577612\pi$
$968$	822715.i	0.878008i
$969$	0	0
$970$	−168452.	−0.179033
$971$	− 951654.i	− 1.00935i	−0.863310	−	0.504674i	$-0.831613\pi$
$971$	− 951654.i	− 1.00935i	0.863310	−	0.504674i	$-0.168387\pi$
$972$	0	0
$973$	976620.	1.03157
$974$	608227.i	0.641133i
$975$	0	0
$976$	−216370.	−0.227142
$977$	1.69534e6i	1.77610i	0.459745	+	0.888051i	$0.347941\pi$
$977$	1.69534e6i	1.77610i	−0.459745	+	0.888051i	$0.652059\pi$
$978$	0	0
$979$	−11494.7	−0.0119931
$980$	321498.i	0.334754i
$981$	0	0
$982$	77430.5	0.0802951
$983$	859756.i	0.889751i	0.895593	+	0.444875i	$0.146752\pi$
$983$	859756.i	0.889751i	−0.895593	+	0.444875i	$0.853248\pi$
$984$	0	0
$985$	496502.	0.511739
$986$	− 556059.i	− 0.571962i
$987$	0	0
$988$	504023.	0.516341
$989$	− 152918.i	− 0.156339i
$990$	0	0
$991$	−1.45339e6	−1.47990	−0.739952	−	0.672659i	$-0.765152\pi$
$991$	−1.45339e6	−1.47990	−0.739952	+	0.672659i	$0.765152\pi$
$992$	775879.i	0.788444i
$993$	0	0
$994$	−795402.	−0.805033
$995$	107541.i	0.108624i
$996$	0	0
$997$	808314.	0.813186	0.406593	−	0.913609i	$-0.366717\pi$
$997$	808314.	0.813186	0.406593	+	0.913609i	$0.366717\pi$
$998$	− 577527.i	− 0.579844i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.47	yes	76
3.2	odd	2	inner	531.5.b.a.296.30	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.30	✓	76	3.2	odd	2	inner
531.5.b.a.296.47	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)

\(f(q)\)	\(=\)	\(q+2.01293i q^{2} +11.9481 q^{4} +10.0983i q^{5} -71.1730 q^{7} +56.2577i q^{8} +O(q^{10})\)	\(q+2.01293i q^{2} +11.9481 q^{4} +10.0983i q^{5} -71.1730 q^{7} +56.2577i q^{8} -20.3272 q^{10} -4.11828i q^{11} -100.854 q^{13} -143.267i q^{14} +77.9265 q^{16} -303.195i q^{17} -418.270 q^{19} +120.655i q^{20} +8.28983 q^{22} -234.681i q^{23} +523.025 q^{25} -203.013i q^{26} -850.382 q^{28} -911.107i q^{29} +734.050 q^{31} +1056.98i q^{32} +610.311 q^{34} -718.725i q^{35} +2410.10 q^{37} -841.950i q^{38} -568.106 q^{40} -281.427i q^{41} +651.599 q^{43} -49.2056i q^{44} +472.397 q^{46} -767.605i q^{47} +2664.60 q^{49} +1052.81i q^{50} -1205.02 q^{52} -2207.09i q^{53} +41.5876 q^{55} -4004.03i q^{56} +1834.00 q^{58} +453.188i q^{59} -2776.60 q^{61} +1477.59i q^{62} -880.815 q^{64} -1018.46i q^{65} -2162.76 q^{67} -3622.60i q^{68} +1446.75 q^{70} -5551.90i q^{71} +3915.04 q^{73} +4851.38i q^{74} -4997.53 q^{76} +293.111i q^{77} -7515.90 q^{79} +786.923i q^{80} +566.494 q^{82} +4650.58i q^{83} +3061.74 q^{85} +1311.63i q^{86} +231.685 q^{88} -2791.14i q^{89} +7178.12 q^{91} -2803.99i q^{92} +1545.14 q^{94} -4223.81i q^{95} +8287.06 q^{97} +5363.67i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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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