LMFDB - Embedded newform 1521.4.a.bl.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 82x^{10} + 2137x^{8} - 22488x^{6} + 89784x^{4} - 67392x^{2} + 11664 $ x^12 - 82x^10 + 2137x^8 - 22488x^6 + 89784x^4 - 67392*x^2 + 11664
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.503193$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22886 q^{2} -6.48991 q^{4} -17.3039 q^{5} +19.1933 q^{7} -17.8060 q^{8} +O(q^{10})$	$q+1.22886 q^{2} -6.48991 q^{4} -17.3039 q^{5} +19.1933 q^{7} -17.8060 q^{8} -21.2640 q^{10} +32.6522 q^{11} +23.5859 q^{14} +30.0382 q^{16} -86.5265 q^{17} -76.0656 q^{19} +112.301 q^{20} +40.1249 q^{22} +75.3223 q^{23} +174.425 q^{25} -124.563 q^{28} +286.769 q^{29} -200.458 q^{31} +179.361 q^{32} -106.329 q^{34} -332.120 q^{35} +173.136 q^{37} -93.4739 q^{38} +308.114 q^{40} +496.146 q^{41} -187.437 q^{43} -211.910 q^{44} +92.5605 q^{46} +254.441 q^{47} +25.3846 q^{49} +214.344 q^{50} +63.6710 q^{53} -565.011 q^{55} -341.758 q^{56} +352.398 q^{58} -67.9177 q^{59} +400.870 q^{61} -246.334 q^{62} -19.8962 q^{64} -510.965 q^{67} +561.549 q^{68} -408.128 q^{70} +200.675 q^{71} -168.714 q^{73} +212.760 q^{74} +493.659 q^{76} +626.705 q^{77} -1116.83 q^{79} -519.777 q^{80} +609.693 q^{82} -724.027 q^{83} +1497.25 q^{85} -230.333 q^{86} -581.407 q^{88} -821.285 q^{89} -488.835 q^{92} +312.671 q^{94} +1316.23 q^{95} -19.5681 q^{97} +31.1940 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 32 q^{4}+O(q^{10}) $ 12 * q + 32 * q^4	$ 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) $ 12 * q + 32 * q^4 - 112 * q^10 + 184 * q^16 - 584 * q^22 + 92 * q^25 - 448 * q^40 - 1620 * q^43 - 2136 * q^49 - 920 * q^55 - 2588 * q^61 + 184 * q^64 - 6380 * q^79 - 2536 * q^82 - 10280 * q^88 - 3320 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.22886	0.434467	0.217233	−	0.976120i	$-0.430297\pi$
$2$	1.22886	0.434467	0.217233	+	0.976120i	$0.430297\pi$
$3$	0	0
$3$	0	0
$4$	−6.48991	−0.811238
$5$	−17.3039	−1.54771	−0.773854	−	0.633364i	$-0.781674\pi$
$5$	−17.3039	−1.54771	−0.773854	+	0.633364i	$0.781674\pi$
$6$	0	0
$7$	19.1933	1.03634	0.518172	−	0.855277i	$-0.326613\pi$
$7$	19.1933	1.03634	0.518172	+	0.855277i	$0.326613\pi$
$8$	−17.8060	−0.786923
$9$	0	0
$10$	−21.2640	−0.672428
$11$	32.6522	0.895001	0.447501	−	0.894284i	$-0.352314\pi$
$11$	32.6522	0.895001	0.447501	+	0.894284i	$0.352314\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	23.5859	0.450257
$15$	0	0
$16$	30.0382	0.469346
$17$	−86.5265	−1.23446	−0.617229	−	0.786784i	$-0.711745\pi$
$17$	−86.5265	−1.23446	−0.617229	+	0.786784i	$0.711745\pi$
$18$	0	0
$19$	−76.0656	−0.918455	−0.459228	−	0.888319i	$-0.651874\pi$
$19$	−76.0656	−0.918455	−0.459228	+	0.888319i	$0.651874\pi$
$20$	112.301	1.25556
$21$	0	0
$22$	40.1249	0.388849
$23$	75.3223	0.682861	0.341430	−	0.939907i	$-0.389089\pi$
$23$	75.3223	0.682861	0.341430	+	0.939907i	$0.389089\pi$
$24$	0	0
$25$	174.425	1.39540
$26$	0	0
$27$	0	0
$28$	−124.563	−0.840722
$29$	286.769	1.83626	0.918131	−	0.396277i	$-0.129698\pi$
$29$	286.769	1.83626	0.918131	+	0.396277i	$0.129698\pi$
$30$	0	0
$31$	−200.458	−1.16140	−0.580698	−	0.814119i	$-0.697220\pi$
$31$	−200.458	−1.16140	−0.580698	+	0.814119i	$0.697220\pi$
$32$	179.361	0.990839
$33$	0	0
$34$	−106.329	−0.536331
$35$	−332.120	−1.60396
$36$	0	0
$37$	173.136	0.769281	0.384641	−	0.923066i	$-0.374325\pi$
$37$	173.136	0.769281	0.384641	+	0.923066i	$0.374325\pi$
$38$	−93.4739	−0.399039
$39$	0	0
$40$	308.114	1.21793
$41$	496.146	1.88988	0.944939	−	0.327246i	$-0.106121\pi$
$41$	496.146	1.88988	0.944939	+	0.327246i	$0.106121\pi$
$42$	0	0
$43$	−187.437	−0.664741	−0.332370	−	0.943149i	$-0.607848\pi$
$43$	−187.437	−0.664741	−0.332370	+	0.943149i	$0.607848\pi$
$44$	−211.910	−0.726060
$45$	0	0
$46$	92.5605	0.296680
$47$	254.441	0.789659	0.394830	−	0.918754i	$-0.370804\pi$
$47$	254.441	0.789659	0.394830	+	0.918754i	$0.370804\pi$
$48$	0	0
$49$	25.3846	0.0740075
$50$	214.344	0.606255
$51$	0	0
$52$	0	0
$53$	63.6710	0.165017	0.0825083	−	0.996590i	$-0.473707\pi$
$53$	63.6710	0.165017	0.0825083	+	0.996590i	$0.473707\pi$
$54$	0	0
$55$	−565.011	−1.38520
$56$	−341.758	−0.815523
$57$	0	0
$58$	352.398	0.797795
$59$	−67.9177	−0.149867	−0.0749334	−	0.997189i	$-0.523874\pi$
$59$	−67.9177	−0.149867	−0.0749334	+	0.997189i	$0.523874\pi$
$60$	0	0
$61$	400.870	0.841411	0.420706	−	0.907197i	$-0.361783\pi$
$61$	400.870	0.841411	0.420706	+	0.907197i	$0.361783\pi$
$62$	−246.334	−0.504588
$63$	0	0
$64$	−19.8962	−0.0388597
$65$	0	0
$66$	0	0
$67$	−510.965	−0.931707	−0.465853	−	0.884862i	$-0.654253\pi$
$67$	−510.965	−0.931707	−0.465853	+	0.884862i	$0.654253\pi$
$68$	561.549	1.00144
$69$	0	0
$70$	−408.128	−0.696866
$71$	200.675	0.335433	0.167717	−	0.985835i	$-0.446361\pi$
$71$	200.675	0.335433	0.167717	+	0.985835i	$0.446361\pi$
$72$	0	0
$73$	−168.714	−0.270499	−0.135250	−	0.990812i	$-0.543184\pi$
$73$	−168.714	−0.270499	−0.135250	+	0.990812i	$0.543184\pi$
$74$	212.760	0.334227
$75$	0	0
$76$	493.659	0.745086
$77$	626.705	0.927529
$78$	0	0
$79$	−1116.83	−1.59055	−0.795276	−	0.606247i	$-0.792674\pi$
$79$	−1116.83	−1.59055	−0.795276	+	0.606247i	$0.792674\pi$
$80$	−519.777	−0.726411
$81$	0	0
$82$	609.693	0.821090
$83$	−724.027	−0.957497	−0.478749	−	0.877952i	$-0.658909\pi$
$83$	−724.027	−0.957497	−0.478749	+	0.877952i	$0.658909\pi$
$84$	0	0
$85$	1497.25	1.91058
$86$	−230.333	−0.288808
$87$	0	0
$88$	−581.407	−0.704297
$89$	−821.285	−0.978158	−0.489079	−	0.872240i	$-0.662667\pi$
$89$	−821.285	−0.978158	−0.489079	+	0.872240i	$0.662667\pi$
$90$	0	0
$91$	0	0
$92$	−488.835	−0.553963
$93$	0	0
$94$	312.671	0.343081
$95$	1316.23	1.42150
$96$	0	0
$97$	−19.5681	−0.0204828	−0.0102414	−	0.999948i	$-0.503260\pi$
$97$	−19.5681	−0.0204828	−0.0102414	+	0.999948i	$0.503260\pi$
$98$	31.1940	0.0321538
$99$	0	0
$100$	−1132.00	−1.13200
$101$	246.812	0.243156	0.121578	−	0.992582i	$-0.461205\pi$
$101$	246.812	0.243156	0.121578	+	0.992582i	$0.461205\pi$
$102$	0	0
$103$	910.658	0.871163	0.435581	−	0.900149i	$-0.356543\pi$
$103$	910.658	0.871163	0.435581	+	0.900149i	$0.356543\pi$
$104$	0	0
$105$	0	0
$106$	78.2426	0.0716942
$107$	−1193.53	−1.07834	−0.539172	−	0.842195i	$-0.681263\pi$
$107$	−1193.53	−1.07834	−0.539172	+	0.842195i	$0.681263\pi$
$108$	0	0
$109$	359.194	0.315638	0.157819	−	0.987468i	$-0.449554\pi$
$109$	359.194	0.315638	0.157819	+	0.987468i	$0.449554\pi$
$110$	−694.318	−0.601824
$111$	0	0
$112$	576.533	0.486404
$113$	−229.013	−0.190652	−0.0953262	−	0.995446i	$-0.530389\pi$
$113$	−229.013	−0.190652	−0.0953262	+	0.995446i	$0.530389\pi$
$114$	0	0
$115$	−1303.37	−1.05687
$116$	−1861.10	−1.48965
$117$	0	0
$118$	−83.4613	−0.0651121
$119$	−1660.73	−1.27932
$120$	0	0
$121$	−264.832	−0.198972
$122$	492.612	0.365565
$123$	0	0
$124$	1300.95	0.942169
$125$	−855.244	−0.611963
$126$	0	0
$127$	−1874.10	−1.30944	−0.654722	−	0.755870i	$-0.727214\pi$
$127$	−1874.10	−1.30944	−0.654722	+	0.755870i	$0.727214\pi$
$128$	−1459.34	−1.00772
$129$	0	0
$130$	0	0
$131$	−2763.13	−1.84287	−0.921434	−	0.388535i	$-0.872981\pi$
$131$	−2763.13	−1.84287	−0.921434	+	0.388535i	$0.872981\pi$
$132$	0	0
$133$	−1459.95	−0.951835
$134$	−627.904	−0.404796
$135$	0	0
$136$	1540.70	0.971423
$137$	−2740.44	−1.70899	−0.854496	−	0.519458i	$-0.826134\pi$
$137$	−2740.44	−1.70899	−0.854496	+	0.519458i	$0.826134\pi$
$138$	0	0
$139$	−2248.83	−1.37225	−0.686126	−	0.727483i	$-0.740690\pi$
$139$	−2248.83	−1.37225	−0.686126	+	0.727483i	$0.740690\pi$
$140$	2155.43	1.30119
$141$	0	0
$142$	246.601	0.145735
$143$	0	0
$144$	0	0
$145$	−4962.22	−2.84200
$146$	−207.325	−0.117523
$147$	0	0
$148$	−1123.64	−0.624070
$149$	892.770	0.490863	0.245431	−	0.969414i	$-0.421070\pi$
$149$	892.770	0.490863	0.245431	+	0.969414i	$0.421070\pi$
$150$	0	0
$151$	1738.22	0.936784	0.468392	−	0.883521i	$-0.344834\pi$
$151$	1738.22	0.936784	0.468392	+	0.883521i	$0.344834\pi$
$152$	1354.43	0.722754
$153$	0	0
$154$	770.132	0.402981
$155$	3468.70	1.79750
$156$	0	0
$157$	−1109.37	−0.563930	−0.281965	−	0.959425i	$-0.590986\pi$
$157$	−1109.37	−0.563930	−0.281965	+	0.959425i	$0.590986\pi$
$158$	−1372.43	−0.691042
$159$	0	0
$160$	−3103.64	−1.53353
$161$	1445.69	0.707678
$162$	0	0
$163$	−3020.17	−1.45128	−0.725639	−	0.688076i	$-0.758456\pi$
$163$	−3020.17	−1.45128	−0.725639	+	0.688076i	$0.758456\pi$
$164$	−3219.94	−1.53314
$165$	0	0
$166$	−889.726	−0.416001
$167$	1375.11	0.637179	0.318590	−	0.947893i	$-0.396791\pi$
$167$	1375.11	0.637179	0.318590	+	0.947893i	$0.396791\pi$
$168$	0	0
$169$	0	0
$170$	1839.90	0.830083
$171$	0	0
$172$	1216.45	0.539263
$173$	1772.38	0.778913	0.389456	−	0.921045i	$-0.372663\pi$
$173$	1772.38	0.778913	0.389456	+	0.921045i	$0.372663\pi$
$174$	0	0
$175$	3347.80	1.44611
$176$	980.813	0.420066
$177$	0	0
$178$	−1009.24	−0.424977
$179$	−169.777	−0.0708923	−0.0354461	−	0.999372i	$-0.511285\pi$
$179$	−169.777	−0.0708923	−0.0354461	+	0.999372i	$0.511285\pi$
$180$	0	0
$181$	−132.544	−0.0544305	−0.0272153	−	0.999630i	$-0.508664\pi$
$181$	−132.544	−0.0544305	−0.0272153	+	0.999630i	$0.508664\pi$
$182$	0	0
$183$	0	0
$184$	−1341.19	−0.537359
$185$	−2995.93	−1.19062
$186$	0	0
$187$	−2825.28	−1.10484
$188$	−1651.30	−0.640602
$189$	0	0
$190$	1617.46	0.617595
$191$	−1518.57	−0.575287	−0.287643	−	0.957738i	$-0.592872\pi$
$191$	−1518.57	−0.575287	−0.287643	+	0.957738i	$0.592872\pi$
$192$	0	0
$193$	4276.22	1.59487	0.797433	−	0.603408i	$-0.206191\pi$
$193$	4276.22	1.59487	0.797433	+	0.603408i	$0.206191\pi$
$194$	−24.0464	−0.00889912
$195$	0	0
$196$	−164.744	−0.0600378
$197$	2810.65	1.01650	0.508250	−	0.861210i	$-0.330293\pi$
$197$	2810.65	1.01650	0.508250	+	0.861210i	$0.330293\pi$
$198$	0	0
$199$	−2835.27	−1.00998	−0.504992	−	0.863124i	$-0.668504\pi$
$199$	−2835.27	−1.00998	−0.504992	+	0.863124i	$0.668504\pi$
$200$	−3105.82	−1.09807
$201$	0	0
$202$	303.298	0.105643
$203$	5504.05	1.90300
$204$	0	0
$205$	−8585.26	−2.92498
$206$	1119.07	0.378491
$207$	0	0
$208$	0	0
$209$	−2483.71	−0.822019
$210$	0	0
$211$	−3462.93	−1.12985	−0.564924	−	0.825143i	$-0.691094\pi$
$211$	−3462.93	−1.12985	−0.564924	+	0.825143i	$0.691094\pi$
$212$	−413.219	−0.133868
$213$	0	0
$214$	−1466.68	−0.468505
$215$	3243.39	1.02882
$216$	0	0
$217$	−3847.45	−1.20360
$218$	441.398	0.137134
$219$	0	0
$220$	3666.87	1.12373
$221$	0	0
$222$	0	0
$223$	−4198.64	−1.26082	−0.630408	−	0.776264i	$-0.717112\pi$
$223$	−4198.64	−1.26082	−0.630408	+	0.776264i	$0.717112\pi$
$224$	3442.54	1.02685
$225$	0	0
$226$	−281.424	−0.0828321
$227$	4790.26	1.40062	0.700310	−	0.713839i	$-0.253045\pi$
$227$	4790.26	1.40062	0.700310	+	0.713839i	$0.253045\pi$
$228$	0	0
$229$	−5308.01	−1.53172	−0.765859	−	0.643009i	$-0.777686\pi$
$229$	−5308.01	−1.53172	−0.765859	+	0.643009i	$0.777686\pi$
$230$	−1601.66	−0.459174
$231$	0	0
$232$	−5106.21	−1.44500
$233$	−1894.10	−0.532562	−0.266281	−	0.963895i	$-0.585795\pi$
$233$	−1894.10	−0.532562	−0.266281	+	0.963895i	$0.585795\pi$
$234$	0	0
$235$	−4402.81	−1.22216
$236$	440.780	0.121578
$237$	0	0
$238$	−2040.81	−0.555823
$239$	2627.50	0.711125	0.355562	−	0.934653i	$-0.384289\pi$
$239$	2627.50	0.711125	0.355562	+	0.934653i	$0.384289\pi$
$240$	0	0
$241$	5662.87	1.51360	0.756800	−	0.653646i	$-0.226761\pi$
$241$	5662.87	1.51360	0.756800	+	0.653646i	$0.226761\pi$
$242$	−325.441	−0.0864469
$243$	0	0
$244$	−2601.61	−0.682585
$245$	−439.252	−0.114542
$246$	0	0
$247$	0	0
$248$	3569.36	0.913929
$249$	0	0
$250$	−1050.97	−0.265878
$251$	−3275.99	−0.823819	−0.411909	−	0.911225i	$-0.635138\pi$
$251$	−3275.99	−0.823819	−0.411909	+	0.911225i	$0.635138\pi$
$252$	0	0
$253$	2459.44	0.611161
$254$	−2303.00	−0.568910
$255$	0	0
$256$	−1634.15	−0.398962
$257$	−6790.12	−1.64808	−0.824039	−	0.566533i	$-0.808284\pi$
$257$	−6790.12	−1.64808	−0.824039	+	0.566533i	$0.808284\pi$
$258$	0	0
$259$	3323.06	0.797239
$260$	0	0
$261$	0	0
$262$	−3395.49	−0.800665
$263$	7749.35	1.81690	0.908451	−	0.417991i	$-0.137266\pi$
$263$	7749.35	1.81690	0.908451	+	0.417991i	$0.137266\pi$
$264$	0	0
$265$	−1101.76	−0.255397
$266$	−1794.08	−0.413541
$267$	0	0
$268$	3316.12	0.755837
$269$	1610.04	0.364928	0.182464	−	0.983212i	$-0.441593\pi$
$269$	1610.04	0.364928	0.182464	+	0.983212i	$0.441593\pi$
$270$	0	0
$271$	3444.46	0.772090	0.386045	−	0.922480i	$-0.373841\pi$
$271$	3444.46	0.772090	0.386045	+	0.922480i	$0.373841\pi$
$272$	−2599.10	−0.579388
$273$	0	0
$274$	−3367.62	−0.742501
$275$	5695.36	1.24888
$276$	0	0
$277$	1811.13	0.392852	0.196426	−	0.980519i	$-0.437066\pi$
$277$	1811.13	0.392852	0.196426	+	0.980519i	$0.437066\pi$
$278$	−2763.49	−0.596198
$279$	0	0
$280$	5913.74	1.26219
$281$	1564.44	0.332122	0.166061	−	0.986115i	$-0.446895\pi$
$281$	1564.44	0.332122	0.166061	+	0.986115i	$0.446895\pi$
$282$	0	0
$283$	849.105	0.178354	0.0891768	−	0.996016i	$-0.471576\pi$
$283$	849.105	0.178354	0.0891768	+	0.996016i	$0.471576\pi$
$284$	−1302.36	−0.272116
$285$	0	0
$286$	0	0
$287$	9522.71	1.95856
$288$	0	0
$289$	2573.84	0.523884
$290$	−6097.86	−1.23475
$291$	0	0
$292$	1094.94	0.219440
$293$	3271.70	0.652338	0.326169	−	0.945311i	$-0.394242\pi$
$293$	3271.70	0.652338	0.326169	+	0.945311i	$0.394242\pi$
$294$	0	0
$295$	1175.24	0.231950
$296$	−3082.87	−0.605365
$297$	0	0
$298$	1097.09	0.213264
$299$	0	0
$300$	0	0
$301$	−3597.54	−0.688900
$302$	2136.03	0.407001
$303$	0	0
$304$	−2284.87	−0.431074
$305$	−6936.61	−1.30226
$306$	0	0
$307$	4994.44	0.928494	0.464247	−	0.885706i	$-0.346325\pi$
$307$	4994.44	0.928494	0.464247	+	0.885706i	$0.346325\pi$
$308$	−4067.26	−0.752447
$309$	0	0
$310$	4262.54	0.780955
$311$	−4677.02	−0.852764	−0.426382	−	0.904543i	$-0.640212\pi$
$311$	−4677.02	−0.852764	−0.426382	+	0.904543i	$0.640212\pi$
$312$	0	0
$313$	2098.70	0.378995	0.189498	−	0.981881i	$-0.439314\pi$
$313$	2098.70	0.378995	0.189498	+	0.981881i	$0.439314\pi$
$314$	−1363.25	−0.245009
$315$	0	0
$316$	7248.15	1.29032
$317$	−3753.15	−0.664977	−0.332489	−	0.943107i	$-0.607888\pi$
$317$	−3753.15	−0.664977	−0.332489	+	0.943107i	$0.607888\pi$
$318$	0	0
$319$	9363.63	1.64346
$320$	344.281	0.0601434
$321$	0	0
$322$	1776.55	0.307463
$323$	6581.70	1.13379
$324$	0	0
$325$	0	0
$326$	−3711.37	−0.630532
$327$	0	0
$328$	−8834.40	−1.48719
$329$	4883.57	0.818358
$330$	0	0
$331$	10048.9	1.66870	0.834349	−	0.551236i	$-0.185844\pi$
$331$	10048.9	1.66870	0.834349	+	0.551236i	$0.185844\pi$
$332$	4698.87	0.776758
$333$	0	0
$334$	1689.81	0.276833
$335$	8841.70	1.44201
$336$	0	0
$337$	−170.304	−0.0275283	−0.0137642	−	0.999905i	$-0.504381\pi$
$337$	−170.304	−0.0275283	−0.0137642	+	0.999905i	$0.504381\pi$
$338$	0	0
$339$	0	0
$340$	−9716.99	−1.54994
$341$	−6545.39	−1.03945
$342$	0	0
$343$	−6096.10	−0.959646
$344$	3337.51	0.523100
$345$	0	0
$346$	2178.01	0.338412
$347$	8636.54	1.33612	0.668060	−	0.744107i	$-0.267125\pi$
$347$	8636.54	1.33612	0.668060	+	0.744107i	$0.267125\pi$
$348$	0	0
$349$	−3169.13	−0.486073	−0.243036	−	0.970017i	$-0.578143\pi$
$349$	−3169.13	−0.486073	−0.243036	+	0.970017i	$0.578143\pi$
$350$	4113.97	0.628288
$351$	0	0
$352$	5856.53	0.886802
$353$	693.601	0.104580	0.0522899	−	0.998632i	$-0.483348\pi$
$353$	693.601	0.104580	0.0522899	+	0.998632i	$0.483348\pi$
$354$	0	0
$355$	−3472.46	−0.519153
$356$	5330.06	0.793519
$357$	0	0
$358$	−208.632	−0.0308004
$359$	1432.95	0.210664	0.105332	−	0.994437i	$-0.466409\pi$
$359$	1432.95	0.210664	0.105332	+	0.994437i	$0.466409\pi$
$360$	0	0
$361$	−1073.02	−0.156440
$362$	−162.878	−0.0236483
$363$	0	0
$364$	0	0
$365$	2919.41	0.418654
$366$	0	0
$367$	−8942.69	−1.27195	−0.635973	−	0.771711i	$-0.719401\pi$
$367$	−8942.69	−1.27195	−0.635973	+	0.771711i	$0.719401\pi$
$368$	2262.54	0.320498
$369$	0	0
$370$	−3681.57	−0.517286
$371$	1222.06	0.171014
$372$	0	0
$373$	−1402.19	−0.194645	−0.0973226	−	0.995253i	$-0.531028\pi$
$373$	−1402.19	−0.194645	−0.0973226	+	0.995253i	$0.531028\pi$
$374$	−3471.87	−0.480017
$375$	0	0
$376$	−4530.58	−0.621401
$377$	0	0
$378$	0	0
$379$	8038.56	1.08948	0.544740	−	0.838605i	$-0.316628\pi$
$379$	8038.56	1.08948	0.544740	+	0.838605i	$0.316628\pi$
$380$	−8542.22	−1.15318
$381$	0	0
$382$	−1866.11	−0.249943
$383$	5255.01	0.701093	0.350546	−	0.936545i	$-0.385996\pi$
$383$	5255.01	0.701093	0.350546	+	0.936545i	$0.385996\pi$
$384$	0	0
$385$	−10844.4	−1.43554
$386$	5254.87	0.692916
$387$	0	0
$388$	126.995	0.0166165
$389$	6096.06	0.794556	0.397278	−	0.917698i	$-0.369955\pi$
$389$	6096.06	0.794556	0.397278	+	0.917698i	$0.369955\pi$
$390$	0	0
$391$	−6517.38	−0.842962
$392$	−451.999	−0.0582382
$393$	0	0
$394$	3453.89	0.441635
$395$	19325.6	2.46171
$396$	0	0
$397$	−1939.83	−0.245233	−0.122616	−	0.992454i	$-0.539128\pi$
$397$	−1939.83	−0.245233	−0.122616	+	0.992454i	$0.539128\pi$
$398$	−3484.14	−0.438804
$399$	0	0
$400$	5239.41	0.654926
$401$	−4161.61	−0.518257	−0.259128	−	0.965843i	$-0.583435\pi$
$401$	−4161.61	−0.518257	−0.259128	+	0.965843i	$0.583435\pi$
$402$	0	0
$403$	0	0
$404$	−1601.79	−0.197258
$405$	0	0
$406$	6763.70	0.826790
$407$	5653.28	0.688508
$408$	0	0
$409$	−8420.44	−1.01801	−0.509003	−	0.860765i	$-0.669986\pi$
$409$	−8420.44	−1.01801	−0.509003	+	0.860765i	$0.669986\pi$
$410$	−10550.1	−1.27081
$411$	0	0
$412$	−5910.09	−0.706721
$413$	−1303.57	−0.155313
$414$	0	0
$415$	12528.5	1.48193
$416$	0	0
$417$	0	0
$418$	−3052.13	−0.357140
$419$	−10414.2	−1.21424	−0.607119	−	0.794611i	$-0.707675\pi$
$419$	−10414.2	−1.21424	−0.607119	+	0.794611i	$0.707675\pi$
$420$	0	0
$421$	−189.641	−0.0219538	−0.0109769	−	0.999940i	$-0.503494\pi$
$421$	−189.641	−0.0219538	−0.0109769	+	0.999940i	$0.503494\pi$
$422$	−4255.45	−0.490881
$423$	0	0
$424$	−1133.73	−0.129855
$425$	−15092.4	−1.72256
$426$	0	0
$427$	7694.03	0.871991
$428$	7745.90	0.874795
$429$	0	0
$430$	3985.66	0.446990
$431$	−8706.95	−0.973084	−0.486542	−	0.873657i	$-0.661742\pi$
$431$	−8706.95	−0.973084	−0.486542	+	0.873657i	$0.661742\pi$
$432$	0	0
$433$	−14669.5	−1.62811	−0.814056	−	0.580786i	$-0.802745\pi$
$433$	−14669.5	−1.62811	−0.814056	+	0.580786i	$0.802745\pi$
$434$	−4727.98	−0.522926
$435$	0	0
$436$	−2331.13	−0.256057
$437$	−5729.44	−0.627177
$438$	0	0
$439$	−17375.7	−1.88906	−0.944532	−	0.328420i	$-0.893484\pi$
$439$	−17375.7	−1.88906	−0.944532	+	0.328420i	$0.893484\pi$
$440$	10060.6	1.09005
$441$	0	0
$442$	0	0
$443$	−16688.1	−1.78979	−0.894894	−	0.446280i	$-0.852749\pi$
$443$	−16688.1	−1.78979	−0.894894	+	0.446280i	$0.852749\pi$
$444$	0	0
$445$	14211.4	1.51390
$446$	−5159.54	−0.547783
$447$	0	0
$448$	−381.874	−0.0402720
$449$	−5154.08	−0.541729	−0.270864	−	0.962618i	$-0.587309\pi$
$449$	−5154.08	−0.541729	−0.270864	+	0.962618i	$0.587309\pi$
$450$	0	0
$451$	16200.3	1.69144
$452$	1486.27	0.154665
$453$	0	0
$454$	5886.55	0.608523
$455$	0	0
$456$	0	0
$457$	10311.4	1.05547	0.527733	−	0.849410i	$-0.323042\pi$
$457$	10311.4	1.05547	0.527733	+	0.849410i	$0.323042\pi$
$458$	−6522.79	−0.665480
$459$	0	0
$460$	8458.75	0.857372
$461$	−12192.0	−1.23175	−0.615877	−	0.787842i	$-0.711198\pi$
$461$	−12192.0	−1.23175	−0.615877	+	0.787842i	$0.711198\pi$
$462$	0	0
$463$	1676.71	0.168301	0.0841505	−	0.996453i	$-0.473182\pi$
$463$	1676.71	0.168301	0.0841505	+	0.996453i	$0.473182\pi$
$464$	8614.00	0.861843
$465$	0	0
$466$	−2327.59	−0.231381
$467$	−13042.4	−1.29236	−0.646179	−	0.763186i	$-0.723634\pi$
$467$	−13042.4	−1.29236	−0.646179	+	0.763186i	$0.723634\pi$
$468$	0	0
$469$	−9807.14	−0.965568
$470$	−5410.43	−0.530989
$471$	0	0
$472$	1209.35	0.117934
$473$	−6120.23	−0.594944
$474$	0	0
$475$	−13267.7	−1.28161
$476$	10778.0	1.03783
$477$	0	0
$478$	3228.83	0.308960
$479$	−15333.8	−1.46267	−0.731333	−	0.682020i	$-0.761102\pi$
$479$	−15333.8	−1.46267	−0.731333	+	0.682020i	$0.761102\pi$
$480$	0	0
$481$	0	0
$482$	6958.87	0.657610
$483$	0	0
$484$	1718.74	0.161414
$485$	338.604	0.0317015
$486$	0	0
$487$	−9698.44	−0.902420	−0.451210	−	0.892418i	$-0.649007\pi$
$487$	−9698.44	−0.902420	−0.451210	+	0.892418i	$0.649007\pi$
$488$	−7137.90	−0.662126
$489$	0	0
$490$	−539.779	−0.0497647
$491$	1751.77	0.161010	0.0805051	−	0.996754i	$-0.474347\pi$
$491$	1751.77	0.161010	0.0805051	+	0.996754i	$0.474347\pi$
$492$	0	0
$493$	−24813.1	−2.26679
$494$	0	0
$495$	0	0
$496$	−6021.38	−0.545097
$497$	3851.63	0.347624
$498$	0	0
$499$	8495.40	0.762137	0.381069	−	0.924547i	$-0.375556\pi$
$499$	8495.40	0.762137	0.381069	+	0.924547i	$0.375556\pi$
$500$	5550.46	0.496448
$501$	0	0
$502$	−4025.72	−0.357922
$503$	14083.1	1.24838	0.624191	−	0.781271i	$-0.285429\pi$
$503$	14083.1	1.24838	0.624191	+	0.781271i	$0.285429\pi$
$504$	0	0
$505$	−4270.82	−0.376335
$506$	3022.30	0.265529
$507$	0	0
$508$	12162.7	1.06227
$509$	−11011.0	−0.958846	−0.479423	−	0.877584i	$-0.659154\pi$
$509$	−11011.0	−0.958846	−0.479423	+	0.877584i	$0.659154\pi$
$510$	0	0
$511$	−3238.18	−0.280330
$512$	9666.56	0.834386
$513$	0	0
$514$	−8344.09	−0.716035
$515$	−15757.9	−1.34831
$516$	0	0
$517$	8308.05	0.706746
$518$	4083.57	0.346374
$519$	0	0
$520$	0	0
$521$	16418.6	1.38064	0.690318	−	0.723506i	$-0.257471\pi$
$521$	16418.6	1.38064	0.690318	+	0.723506i	$0.257471\pi$
$522$	0	0
$523$	−864.779	−0.0723024	−0.0361512	−	0.999346i	$-0.511510\pi$
$523$	−864.779	−0.0723024	−0.0361512	+	0.999346i	$0.511510\pi$
$524$	17932.5	1.49501
$525$	0	0
$526$	9522.85	0.789384
$527$	17344.9	1.43369
$528$	0	0
$529$	−6493.55	−0.533701
$530$	−1353.90	−0.110962
$531$	0	0
$532$	9474.97	0.772165
$533$	0	0
$534$	0	0
$535$	20652.7	1.66896
$536$	9098.27	0.733182
$537$	0	0
$538$	1978.51	0.158549
$539$	828.863	0.0662368
$540$	0	0
$541$	−2776.78	−0.220671	−0.110336	−	0.993894i	$-0.535193\pi$
$541$	−2776.78	−0.220671	−0.110336	+	0.993894i	$0.535193\pi$
$542$	4232.76	0.335447
$543$	0	0
$544$	−15519.5	−1.22315
$545$	−6215.45	−0.488515
$546$	0	0
$547$	9520.68	0.744195	0.372098	−	0.928194i	$-0.378639\pi$
$547$	9520.68	0.744195	0.372098	+	0.928194i	$0.378639\pi$
$548$	17785.2	1.38640
$549$	0	0
$550$	6998.79	0.542599
$551$	−21813.2	−1.68652
$552$	0	0
$553$	−21435.8	−1.64836
$554$	2225.62	0.170681
$555$	0	0
$556$	14594.7	1.11322
$557$	−13124.5	−0.998389	−0.499194	−	0.866490i	$-0.666371\pi$
$557$	−13124.5	−0.998389	−0.499194	+	0.866490i	$0.666371\pi$
$558$	0	0
$559$	0	0
$560$	−9976.27	−0.752811
$561$	0	0
$562$	1922.47	0.144296
$563$	−14094.4	−1.05508	−0.527538	−	0.849532i	$-0.676885\pi$
$563$	−14094.4	−1.05508	−0.527538	+	0.849532i	$0.676885\pi$
$564$	0	0
$565$	3962.82	0.295074
$566$	1043.43	0.0774888
$567$	0	0
$568$	−3573.23	−0.263960
$569$	13274.7	0.978038	0.489019	−	0.872273i	$-0.337355\pi$
$569$	13274.7	0.978038	0.489019	+	0.872273i	$0.337355\pi$
$570$	0	0
$571$	−26739.7	−1.95976	−0.979879	−	0.199594i	$-0.936038\pi$
$571$	−26739.7	−1.95976	−0.979879	+	0.199594i	$0.936038\pi$
$572$	0	0
$573$	0	0
$574$	11702.1	0.850931
$575$	13138.1	0.952863
$576$	0	0
$577$	6088.48	0.439284	0.219642	−	0.975581i	$-0.429511\pi$
$577$	6088.48	0.439284	0.219642	+	0.975581i	$0.429511\pi$
$578$	3162.89	0.227610
$579$	0	0
$580$	32204.3	2.30554
$581$	−13896.5	−0.992296
$582$	0	0
$583$	2079.00	0.147690
$584$	3004.13	0.212862
$585$	0	0
$586$	4020.46	0.283419
$587$	1171.39	0.0823652	0.0411826	−	0.999152i	$-0.486887\pi$
$587$	1171.39	0.0823652	0.0411826	+	0.999152i	$0.486887\pi$
$588$	0	0
$589$	15247.9	1.06669
$590$	1444.21	0.100775
$591$	0	0
$592$	5200.69	0.361059
$593$	−1550.29	−0.107357	−0.0536787	−	0.998558i	$-0.517095\pi$
$593$	−1550.29	−0.107357	−0.0536787	+	0.998558i	$0.517095\pi$
$594$	0	0
$595$	28737.2	1.98002
$596$	−5793.99	−0.398207
$597$	0	0
$598$	0	0
$599$	−17160.2	−1.17053	−0.585263	−	0.810843i	$-0.699009\pi$
$599$	−17160.2	−1.17053	−0.585263	+	0.810843i	$0.699009\pi$
$600$	0	0
$601$	−2830.59	−0.192117	−0.0960584	−	0.995376i	$-0.530624\pi$
$601$	−2830.59	−0.192117	−0.0960584	+	0.995376i	$0.530624\pi$
$602$	−4420.87	−0.299304
$603$	0	0
$604$	−11280.9	−0.759955
$605$	4582.63	0.307951
$606$	0	0
$607$	−7458.10	−0.498707	−0.249353	−	0.968413i	$-0.580218\pi$
$607$	−7458.10	−0.498707	−0.249353	+	0.968413i	$0.580218\pi$
$608$	−13643.2	−0.910041
$609$	0	0
$610$	−8524.10	−0.565788
$611$	0	0
$612$	0	0
$613$	8848.38	0.583006	0.291503	−	0.956570i	$-0.405845\pi$
$613$	8848.38	0.583006	0.291503	+	0.956570i	$0.405845\pi$
$614$	6137.46	0.403400
$615$	0	0
$616$	−11159.1	−0.729894
$617$	−5431.36	−0.354390	−0.177195	−	0.984176i	$-0.556702\pi$
$617$	−5431.36	−0.354390	−0.177195	+	0.984176i	$0.556702\pi$
$618$	0	0
$619$	−9667.38	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$619$	−9667.38	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$620$	−22511.5	−1.45820
$621$	0	0
$622$	−5747.40	−0.370498
$623$	−15763.2	−1.01371
$624$	0	0
$625$	−7004.06	−0.448260
$626$	2579.00	0.164661
$627$	0	0
$628$	7199.68	0.457482
$629$	−14980.9	−0.949644
$630$	0	0
$631$	−15250.7	−0.962157	−0.481079	−	0.876677i	$-0.659755\pi$
$631$	−15250.7	−0.962157	−0.481079	+	0.876677i	$0.659755\pi$
$632$	19886.4	1.25164
$633$	0	0
$634$	−4612.09	−0.288911
$635$	32429.2	2.02664
$636$	0	0
$637$	0	0
$638$	11506.6	0.714028
$639$	0	0
$640$	25252.2	1.55966
$641$	212.816	0.0131134	0.00655672	−	0.999979i	$-0.497913\pi$
$641$	212.816	0.0131134	0.00655672	+	0.999979i	$0.497913\pi$
$642$	0	0
$643$	−18916.8	−1.16019	−0.580097	−	0.814548i	$-0.696985\pi$
$643$	−18916.8	−1.16019	−0.580097	+	0.814548i	$0.696985\pi$
$644$	−9382.38	−0.574096
$645$	0	0
$646$	8087.97	0.492596
$647$	−16913.2	−1.02771	−0.513853	−	0.857878i	$-0.671782\pi$
$647$	−16913.2	−1.02771	−0.513853	+	0.857878i	$0.671782\pi$
$648$	0	0
$649$	−2217.67	−0.134131
$650$	0	0
$651$	0	0
$652$	19600.7	1.17733
$653$	29432.3	1.76382	0.881911	−	0.471416i	$-0.156257\pi$
$653$	29432.3	1.76382	0.881911	+	0.471416i	$0.156257\pi$
$654$	0	0
$655$	47812.9	2.85222
$656$	14903.3	0.887007
$657$	0	0
$658$	6001.21	0.355549
$659$	−4943.12	−0.292195	−0.146098	−	0.989270i	$-0.546671\pi$
$659$	−4943.12	−0.292195	−0.146098	+	0.989270i	$0.546671\pi$
$660$	0	0
$661$	905.208	0.0532655	0.0266327	−	0.999645i	$-0.491522\pi$
$661$	905.208	0.0532655	0.0266327	+	0.999645i	$0.491522\pi$
$662$	12348.7	0.724994
$663$	0	0
$664$	12892.0	0.753477
$665$	25262.9	1.47316
$666$	0	0
$667$	21600.1	1.25391
$668$	−8924.31	−0.516904
$669$	0	0
$670$	10865.2	0.626506
$671$	13089.3	0.753064
$672$	0	0
$673$	22290.6	1.27673	0.638365	−	0.769734i	$-0.279611\pi$
$673$	22290.6	1.27673	0.638365	+	0.769734i	$0.279611\pi$
$674$	−209.279	−0.0119602
$675$	0	0
$676$	0	0
$677$	8514.65	0.483375	0.241687	−	0.970354i	$-0.422299\pi$
$677$	8514.65	0.483375	0.241687	+	0.970354i	$0.422299\pi$
$678$	0	0
$679$	−375.577	−0.0212273
$680$	−26660.0	−1.50348
$681$	0	0
$682$	−8043.36	−0.451607
$683$	−456.917	−0.0255980	−0.0127990	−	0.999918i	$-0.504074\pi$
$683$	−456.917	−0.0255980	−0.0127990	+	0.999918i	$0.504074\pi$
$684$	0	0
$685$	47420.4	2.64502
$686$	−7491.25	−0.416935
$687$	0	0
$688$	−5630.26	−0.311994
$689$	0	0
$690$	0	0
$691$	25436.2	1.40035	0.700174	−	0.713972i	$-0.253106\pi$
$691$	25436.2	1.40035	0.700174	+	0.713972i	$0.253106\pi$
$692$	−11502.6	−0.631884
$693$	0	0
$694$	10613.1	0.580500
$695$	38913.5	2.12384
$696$	0	0
$697$	−42929.8	−2.33297
$698$	−3894.41	−0.211183
$699$	0	0
$700$	−21726.9	−1.17314
$701$	23082.4	1.24367	0.621833	−	0.783150i	$-0.286388\pi$
$701$	23082.4	1.24367	0.621833	+	0.783150i	$0.286388\pi$
$702$	0	0
$703$	−13169.7	−0.706550
$704$	−649.654	−0.0347795
$705$	0	0
$706$	852.338	0.0454365
$707$	4737.16	0.251993
$708$	0	0
$709$	1870.80	0.0990962	0.0495481	−	0.998772i	$-0.484222\pi$
$709$	1870.80	0.0990962	0.0495481	+	0.998772i	$0.484222\pi$
$710$	−4267.17	−0.225555
$711$	0	0
$712$	14623.8	0.769735
$713$	−15098.9	−0.793071
$714$	0	0
$715$	0	0
$716$	1101.84	0.0575105
$717$	0	0
$718$	1760.90	0.0915266
$719$	−302.543	−0.0156926	−0.00784628	−	0.999969i	$-0.502498\pi$
$719$	−302.543	−0.0156926	−0.00784628	+	0.999969i	$0.502498\pi$
$720$	0	0
$721$	17478.6	0.902824
$722$	−1318.59	−0.0679679
$723$	0	0
$724$	860.199	0.0441561
$725$	50019.6	2.56232
$726$	0	0
$727$	9812.27	0.500574	0.250287	−	0.968172i	$-0.419475\pi$
$727$	9812.27	0.500574	0.250287	+	0.968172i	$0.419475\pi$
$728$	0	0
$729$	0	0
$730$	3587.54	0.181891
$731$	16218.3	0.820594
$732$	0	0
$733$	−6317.58	−0.318343	−0.159171	−	0.987251i	$-0.550882\pi$
$733$	−6317.58	−0.318343	−0.159171	+	0.987251i	$0.550882\pi$
$734$	−10989.3	−0.552619
$735$	0	0
$736$	13509.9	0.676605
$737$	−16684.2	−0.833879
$738$	0	0
$739$	11220.2	0.558515	0.279257	−	0.960216i	$-0.409912\pi$
$739$	11220.2	0.558515	0.279257	+	0.960216i	$0.409912\pi$
$740$	19443.3	0.965879
$741$	0	0
$742$	1501.74	0.0742999
$743$	−11901.4	−0.587645	−0.293822	−	0.955860i	$-0.594927\pi$
$743$	−11901.4	−0.587645	−0.293822	+	0.955860i	$0.594927\pi$
$744$	0	0
$745$	−15448.4	−0.759712
$746$	−1723.09	−0.0845669
$747$	0	0
$748$	18335.8	0.896289
$749$	−22907.8	−1.11754
$750$	0	0
$751$	−25602.1	−1.24399	−0.621994	−	0.783022i	$-0.713677\pi$
$751$	−25602.1	−1.24399	−0.621994	+	0.783022i	$0.713677\pi$
$752$	7642.93	0.370624
$753$	0	0
$754$	0	0
$755$	−30078.0	−1.44987
$756$	0	0
$757$	−932.976	−0.0447947	−0.0223974	−	0.999749i	$-0.507130\pi$
$757$	−932.976	−0.0447947	−0.0223974	+	0.999749i	$0.507130\pi$
$758$	9878.25	0.473343
$759$	0	0
$760$	−23436.9	−1.11861
$761$	989.705	0.0471442	0.0235721	−	0.999722i	$-0.492496\pi$
$761$	989.705	0.0471442	0.0235721	+	0.999722i	$0.492496\pi$
$762$	0	0
$763$	6894.13	0.327109
$764$	9855.37	0.466695
$765$	0	0
$766$	6457.66	0.304602
$767$	0	0
$768$	0	0
$769$	27496.0	1.28938	0.644689	−	0.764445i	$-0.276987\pi$
$769$	27496.0	1.28938	0.644689	+	0.764445i	$0.276987\pi$
$770$	−13326.3	−0.623696
$771$	0	0
$772$	−27752.3	−1.29382
$773$	−11905.1	−0.553940	−0.276970	−	0.960879i	$-0.589330\pi$
$773$	−11905.1	−0.553940	−0.276970	+	0.960879i	$0.589330\pi$
$774$	0	0
$775$	−34964.8	−1.62061
$776$	348.430	0.0161184
$777$	0	0
$778$	7491.19	0.345208
$779$	−37739.7	−1.73577
$780$	0	0
$781$	6552.49	0.300213
$782$	−8008.94	−0.366239
$783$	0	0
$784$	762.506	0.0347352
$785$	19196.4	0.872799
$786$	0	0
$787$	−41425.4	−1.87631	−0.938155	−	0.346215i	$-0.887467\pi$
$787$	−41425.4	−1.87631	−0.938155	+	0.346215i	$0.887467\pi$
$788$	−18240.8	−0.824623
$789$	0	0
$790$	23748.4	1.06953
$791$	−4395.52	−0.197581
$792$	0	0
$793$	0	0
$794$	−2383.78	−0.106546
$795$	0	0
$796$	18400.6	0.819337
$797$	26554.1	1.18017	0.590084	−	0.807342i	$-0.299095\pi$
$797$	26554.1	1.18017	0.590084	+	0.807342i	$0.299095\pi$
$798$	0	0
$799$	−22015.9	−0.974800
$800$	31285.0	1.38262
$801$	0	0
$802$	−5114.03	−0.225165
$803$	−5508.88	−0.242097
$804$	0	0
$805$	−25016.0	−1.09528
$806$	0	0
$807$	0	0
$808$	−4394.75	−0.191345
$809$	−41111.0	−1.78663	−0.893317	−	0.449428i	$-0.851628\pi$
$809$	−41111.0	−1.78663	−0.893317	+	0.449428i	$0.851628\pi$
$810$	0	0
$811$	14420.8	0.624393	0.312197	−	0.950018i	$-0.398935\pi$
$811$	14420.8	0.624393	0.312197	+	0.950018i	$0.398935\pi$
$812$	−35720.8	−1.54379
$813$	0	0
$814$	6947.08	0.299134
$815$	52260.8	2.24615
$816$	0	0
$817$	14257.5	0.610535
$818$	−10347.5	−0.442290
$819$	0	0
$820$	55717.6	2.37286
$821$	7778.12	0.330644	0.165322	−	0.986240i	$-0.447134\pi$
$821$	7778.12	0.330644	0.165322	+	0.986240i	$0.447134\pi$
$822$	0	0
$823$	14342.1	0.607452	0.303726	−	0.952759i	$-0.401769\pi$
$823$	14342.1	0.607452	0.303726	+	0.952759i	$0.401769\pi$
$824$	−16215.2	−0.685538
$825$	0	0
$826$	−1601.90	−0.0674785
$827$	6196.47	0.260547	0.130273	−	0.991478i	$-0.458414\pi$
$827$	6196.47	0.260547	0.130273	+	0.991478i	$0.458414\pi$
$828$	0	0
$829$	28302.7	1.18576	0.592879	−	0.805291i	$-0.297991\pi$
$829$	28302.7	1.18576	0.592879	+	0.805291i	$0.297991\pi$
$830$	15395.7	0.643848
$831$	0	0
$832$	0	0
$833$	−2196.44	−0.0913591
$834$	0	0
$835$	−23794.7	−0.986167
$836$	16119.1	0.666853
$837$	0	0
$838$	−12797.5	−0.527546
$839$	−17202.1	−0.707846	−0.353923	−	0.935275i	$-0.615152\pi$
$839$	−17202.1	−0.707846	−0.353923	+	0.935275i	$0.615152\pi$
$840$	0	0
$841$	57847.3	2.37186
$842$	−233.042	−0.00953819
$843$	0	0
$844$	22474.1	0.916576
$845$	0	0
$846$	0	0
$847$	−5083.02	−0.206204
$848$	1912.56	0.0774499
$849$	0	0
$850$	−18546.4	−0.748396
$851$	13041.0	0.525312
$852$	0	0
$853$	−5373.54	−0.215694	−0.107847	−	0.994168i	$-0.534396\pi$
$853$	−5373.54	−0.215694	−0.107847	+	0.994168i	$0.534396\pi$
$854$	9454.87	0.378851
$855$	0	0
$856$	21252.0	0.848574
$857$	12381.5	0.493517	0.246758	−	0.969077i	$-0.420635\pi$
$857$	12381.5	0.493517	0.246758	+	0.969077i	$0.420635\pi$
$858$	0	0
$859$	6953.37	0.276189	0.138094	−	0.990419i	$-0.455902\pi$
$859$	6953.37	0.276189	0.138094	+	0.990419i	$0.455902\pi$
$860$	−21049.3	−0.834622
$861$	0	0
$862$	−10699.6	−0.422773
$863$	1003.99	0.0396018	0.0198009	−	0.999804i	$-0.493697\pi$
$863$	1003.99	0.0396018	0.0198009	+	0.999804i	$0.493697\pi$
$864$	0	0
$865$	−30669.2	−1.20553
$866$	−18026.8	−0.707361
$867$	0	0
$868$	24969.6	0.976410
$869$	−36467.1	−1.42355
$870$	0	0
$871$	0	0
$872$	−6395.82	−0.248383
$873$	0	0
$874$	−7040.67	−0.272488
$875$	−16415.0	−0.634204
$876$	0	0
$877$	−15394.9	−0.592760	−0.296380	−	0.955070i	$-0.595779\pi$
$877$	−15394.9	−0.592760	−0.296380	+	0.955070i	$0.595779\pi$
$878$	−21352.3	−0.820736
$879$	0	0
$880$	−16971.9	−0.650139
$881$	−20107.5	−0.768942	−0.384471	−	0.923137i	$-0.625616\pi$
$881$	−20107.5	−0.768942	−0.384471	+	0.923137i	$0.625616\pi$
$882$	0	0
$883$	−756.804	−0.0288431	−0.0144216	−	0.999896i	$-0.504591\pi$
$883$	−756.804	−0.0288431	−0.0144216	+	0.999896i	$0.504591\pi$
$884$	0	0
$885$	0	0
$886$	−20507.3	−0.777603
$887$	3312.30	0.125384	0.0626922	−	0.998033i	$-0.480031\pi$
$887$	3312.30	0.125384	0.0626922	+	0.998033i	$0.480031\pi$
$888$	0	0
$889$	−35970.2	−1.35703
$890$	17463.8	0.657740
$891$	0	0
$892$	27248.8	1.02282
$893$	−19354.2	−0.725267
$894$	0	0
$895$	2937.80	0.109721
$896$	−28009.6	−1.04435
$897$	0	0
$898$	−6333.64	−0.235363
$899$	−57485.0	−2.13263
$900$	0	0
$901$	−5509.23	−0.203706
$902$	19907.8	0.734876
$903$	0	0
$904$	4077.81	0.150029
$905$	2293.53	0.0842425
$906$	0	0
$907$	−6927.91	−0.253625	−0.126812	−	0.991927i	$-0.540475\pi$
$907$	−6927.91	−0.253625	−0.126812	+	0.991927i	$0.540475\pi$
$908$	−31088.3	−1.13624
$909$	0	0
$910$	0	0
$911$	26006.9	0.945825	0.472913	−	0.881109i	$-0.343203\pi$
$911$	26006.9	0.945825	0.472913	+	0.881109i	$0.343203\pi$
$912$	0	0
$913$	−23641.1	−0.856961
$914$	12671.3	0.458565
$915$	0	0
$916$	34448.5	1.24259
$917$	−53033.7	−1.90984
$918$	0	0
$919$	−10390.7	−0.372969	−0.186484	−	0.982458i	$-0.559709\pi$
$919$	−10390.7	−0.372969	−0.186484	+	0.982458i	$0.559709\pi$
$920$	23207.9	0.831674
$921$	0	0
$922$	−14982.2	−0.535156
$923$	0	0
$924$	0	0
$925$	30199.3	1.07345
$926$	2060.44	0.0731213
$927$	0	0
$928$	51435.1	1.81944
$929$	−50627.3	−1.78797	−0.893987	−	0.448092i	$-0.852104\pi$
$929$	−50627.3	−1.78797	−0.893987	+	0.448092i	$0.852104\pi$
$930$	0	0
$931$	−1930.89	−0.0679726
$932$	12292.6	0.432035
$933$	0	0
$934$	−16027.3	−0.561486
$935$	48888.4	1.70997
$936$	0	0
$937$	10080.3	0.351452	0.175726	−	0.984439i	$-0.443773\pi$
$937$	10080.3	0.351452	0.175726	+	0.984439i	$0.443773\pi$
$938$	−12051.6	−0.419508
$939$	0	0
$940$	28573.9	0.991464
$941$	12037.5	0.417014	0.208507	−	0.978021i	$-0.433140\pi$
$941$	12037.5	0.417014	0.208507	+	0.978021i	$0.433140\pi$
$942$	0	0
$943$	37370.9	1.29052
$944$	−2040.12	−0.0703394
$945$	0	0
$946$	−7520.90	−0.258484
$947$	7240.89	0.248466	0.124233	−	0.992253i	$-0.460353\pi$
$947$	7240.89	0.248466	0.124233	+	0.992253i	$0.460353\pi$
$948$	0	0
$949$	0	0
$950$	−16304.2	−0.556818
$951$	0	0
$952$	29571.1	1.00673
$953$	5185.94	0.176274	0.0881371	−	0.996108i	$-0.471909\pi$
$953$	5185.94	0.176274	0.0881371	+	0.996108i	$0.471909\pi$
$954$	0	0
$955$	26277.2	0.890376
$956$	−17052.2	−0.576892
$957$	0	0
$958$	−18843.0	−0.635480
$959$	−52598.3	−1.77110
$960$	0	0
$961$	10392.3	0.348840
$962$	0	0
$963$	0	0
$964$	−36751.5	−1.22789
$965$	−73995.3	−2.46839
$966$	0	0
$967$	−15850.0	−0.527095	−0.263548	−	0.964646i	$-0.584893\pi$
$967$	−15850.0	−0.527095	−0.263548	+	0.964646i	$0.584893\pi$
$968$	4715.61	0.156576
$969$	0	0
$970$	416.096	0.0137732
$971$	20473.3	0.676641	0.338321	−	0.941031i	$-0.390141\pi$
$971$	20473.3	0.676641	0.338321	+	0.941031i	$0.390141\pi$
$972$	0	0
$973$	−43162.5	−1.42212
$974$	−11918.0	−0.392072
$975$	0	0
$976$	12041.4	0.394913
$977$	5520.56	0.180776	0.0903880	−	0.995907i	$-0.471189\pi$
$977$	5520.56	0.180776	0.0903880	+	0.995907i	$0.471189\pi$
$978$	0	0
$979$	−26816.8	−0.875452
$980$	2850.71	0.0929209
$981$	0	0
$982$	2152.67	0.0699536
$983$	58064.7	1.88400	0.942002	−	0.335608i	$-0.108942\pi$
$983$	58064.7	1.88400	0.942002	+	0.335608i	$0.108942\pi$
$984$	0	0
$985$	−48635.2	−1.57324
$986$	−30491.8	−0.984844
$987$	0	0
$988$	0	0
$989$	−14118.2	−0.453925
$990$	0	0
$991$	58554.8	1.87695	0.938473	−	0.345352i	$-0.112241\pi$
$991$	58554.8	1.87695	0.938473	+	0.345352i	$0.112241\pi$
$992$	−35954.3	−1.15076
$993$	0	0
$994$	4733.11	0.151031
$995$	49061.1	1.56316
$996$	0	0
$997$	10327.6	0.328063	0.164031	−	0.986455i	$-0.447550\pi$
$997$	10327.6	0.328063	0.164031	+	0.986455i	$0.447550\pi$
$998$	10439.6	0.331123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bl.1.8		12
3.2	odd	2	inner	1521.4.a.bl.1.6		12
13.6	odd	12		117.4.q.f.10.3	✓	12
13.11	odd	12		117.4.q.f.82.3	yes	12
13.12	even	2	inner	1521.4.a.bl.1.5		12
39.11	even	12		117.4.q.f.82.4	yes	12
39.32	even	12		117.4.q.f.10.4	yes	12
39.38	odd	2	inner	1521.4.a.bl.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.q.f.10.3	✓	12	13.6	odd	12
117.4.q.f.10.4	yes	12	39.32	even	12
117.4.q.f.82.3	yes	12	13.11	odd	12
117.4.q.f.82.4	yes	12	39.11	even	12
1521.4.a.bl.1.5		12	13.12	even	2	inner
1521.4.a.bl.1.6		12	3.2	odd	2	inner
1521.4.a.bl.1.7		12	39.38	odd	2	inner
1521.4.a.bl.1.8		12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.22886 q^{2} -6.48991 q^{4} -17.3039 q^{5} +19.1933 q^{7} -17.8060 q^{8} +O(q^{10})\)	\(q+1.22886 q^{2} -6.48991 q^{4} -17.3039 q^{5} +19.1933 q^{7} -17.8060 q^{8} -21.2640 q^{10} +32.6522 q^{11} +23.5859 q^{14} +30.0382 q^{16} -86.5265 q^{17} -76.0656 q^{19} +112.301 q^{20} +40.1249 q^{22} +75.3223 q^{23} +174.425 q^{25} -124.563 q^{28} +286.769 q^{29} -200.458 q^{31} +179.361 q^{32} -106.329 q^{34} -332.120 q^{35} +173.136 q^{37} -93.4739 q^{38} +308.114 q^{40} +496.146 q^{41} -187.437 q^{43} -211.910 q^{44} +92.5605 q^{46} +254.441 q^{47} +25.3846 q^{49} +214.344 q^{50} +63.6710 q^{53} -565.011 q^{55} -341.758 q^{56} +352.398 q^{58} -67.9177 q^{59} +400.870 q^{61} -246.334 q^{62} -19.8962 q^{64} -510.965 q^{67} +561.549 q^{68} -408.128 q^{70} +200.675 q^{71} -168.714 q^{73} +212.760 q^{74} +493.659 q^{76} +626.705 q^{77} -1116.83 q^{79} -519.777 q^{80} +609.693 q^{82} -724.027 q^{83} +1497.25 q^{85} -230.333 q^{86} -581.407 q^{88} -821.285 q^{89} -488.835 q^{92} +312.671 q^{94} +1316.23 q^{95} -19.5681 q^{97} +31.1940 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 32 q^{4}+O(q^{10}) \) 12 * q + 32 * q^4	\( 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) \) 12 * q + 32 * q^4 - 112 * q^10 + 184 * q^16 - 584 * q^22 + 92 * q^25 - 448 * q^40 - 1620 * q^43 - 2136 * q^49 - 920 * q^55 - 2588 * q^61 + 184 * q^64 - 6380 * q^79 - 2536 * q^82 - 10280 * q^88 - 3320 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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