LMFDB - Embedded newform 1521.4.a.bi.1.4

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.588238$ 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)

$f(q)$	$=$	$q-0.213700 q^{2} -7.95433 q^{4} -15.3391 q^{5} -32.3928 q^{7} +3.40944 q^{8} +O(q^{10})$	$q-0.213700 q^{2} -7.95433 q^{4} -15.3391 q^{5} -32.3928 q^{7} +3.40944 q^{8} +3.27797 q^{10} -29.5925 q^{11} +6.92233 q^{14} +62.9061 q^{16} +78.1958 q^{17} +10.6600 q^{19} +122.013 q^{20} +6.32392 q^{22} +26.8789 q^{23} +110.289 q^{25} +257.663 q^{28} +190.785 q^{29} -128.108 q^{31} -40.7185 q^{32} -16.7104 q^{34} +496.877 q^{35} -379.934 q^{37} -2.27805 q^{38} -52.2979 q^{40} +464.631 q^{41} +322.758 q^{43} +235.389 q^{44} -5.74401 q^{46} +248.529 q^{47} +706.292 q^{49} -23.5688 q^{50} -740.167 q^{53} +453.924 q^{55} -110.441 q^{56} -40.7706 q^{58} +340.673 q^{59} -590.834 q^{61} +27.3767 q^{62} -494.547 q^{64} -340.777 q^{67} -621.995 q^{68} -106.183 q^{70} +36.2243 q^{71} -164.572 q^{73} +81.1919 q^{74} -84.7935 q^{76} +958.584 q^{77} -327.242 q^{79} -964.925 q^{80} -99.2916 q^{82} +1404.48 q^{83} -1199.46 q^{85} -68.9734 q^{86} -100.894 q^{88} -736.986 q^{89} -213.803 q^{92} -53.1107 q^{94} -163.516 q^{95} +1494.87 q^{97} -150.935 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) $ 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8	$ 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8} - 54 q^{10} + 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} - 352 q^{19} + 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} - 940 q^{28} + 97 q^{29} - 717 q^{31} + 707 q^{32} - 632 q^{34} + 418 q^{35} - 1108 q^{37} + 660 q^{38} - 1506 q^{40} + 334 q^{41} + 242 q^{43} - 307 q^{44} - 979 q^{46} - 184 q^{47} - 38 q^{49} - 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} - 1161 q^{58} + 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} - 2308 q^{67} - 2785 q^{68} + 1420 q^{70} + 96 q^{71} - 2505 q^{73} + 1191 q^{74} - 2409 q^{76} + 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 1517 q^{82} + 1539 q^{83} - 4296 q^{85} - 3763 q^{86} - 3716 q^{88} - 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} - 1445 q^{97} + 1457 q^{98}+O(q^{100}) $ 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8 - 54 * q^10 + 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 - 352 * q^19 + 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 - 940 * q^28 + 97 * q^29 - 717 * q^31 + 707 * q^32 - 632 * q^34 + 418 * q^35 - 1108 * q^37 + 660 * q^38 - 1506 * q^40 + 334 * q^41 + 242 * q^43 - 307 * q^44 - 979 * q^46 - 184 * q^47 - 38 * q^49 - 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 - 1161 * q^58 + 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 - 2308 * q^67 - 2785 * q^68 + 1420 * q^70 + 96 * q^71 - 2505 * q^73 + 1191 * q^74 - 2409 * q^76 + 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 1517 * q^82 + 1539 * q^83 - 4296 * q^85 - 3763 * q^86 - 3716 * q^88 - 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 - 1445 * q^97 + 1457 * q^98

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 $

$2$	−0.213700	−0.0755543	−0.0377772	−	0.999286i	$-0.512028\pi$
$2$	−0.213700	−0.0755543	−0.0377772	+	0.999286i	$0.512028\pi$
$3$	0	0
$3$	0	0
$4$	−7.95433	−0.994292
$5$	−15.3391	−1.37197	−0.685987	−	0.727614i	$-0.740629\pi$
$5$	−15.3391	−1.37197	−0.685987	+	0.727614i	$0.740629\pi$
$6$	0	0
$7$	−32.3928	−1.74905	−0.874523	−	0.484984i	$-0.838825\pi$
$7$	−32.3928	−1.74905	−0.874523	+	0.484984i	$0.838825\pi$
$8$	3.40944	0.150677
$9$	0	0
$10$	3.27797	0.103659
$11$	−29.5925	−0.811135	−0.405567	−	0.914065i	$-0.632926\pi$
$11$	−29.5925	−0.811135	−0.405567	+	0.914065i	$0.632926\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	6.92233	0.132148
$15$	0	0
$16$	62.9061	0.982907
$17$	78.1958	1.11560	0.557802	−	0.829974i	$-0.311645\pi$
$17$	78.1958	1.11560	0.557802	+	0.829974i	$0.311645\pi$
$18$	0	0
$19$	10.6600	0.128715	0.0643574	−	0.997927i	$-0.479500\pi$
$19$	10.6600	0.128715	0.0643574	+	0.997927i	$0.479500\pi$
$20$	122.013	1.36414
$21$	0	0
$22$	6.32392	0.0612847
$23$	26.8789	0.243680	0.121840	−	0.992550i	$-0.461121\pi$
$23$	26.8789	0.243680	0.121840	+	0.992550i	$0.461121\pi$
$24$	0	0
$25$	110.289	0.882314
$26$	0	0
$27$	0	0
$28$	257.663	1.73906
$29$	190.785	1.22165	0.610824	−	0.791766i	$-0.290838\pi$
$29$	190.785	1.22165	0.610824	+	0.791766i	$0.290838\pi$
$30$	0	0
$31$	−128.108	−0.742222	−0.371111	−	0.928589i	$-0.621023\pi$
$31$	−128.108	−0.742222	−0.371111	+	0.928589i	$0.621023\pi$
$32$	−40.7185	−0.224940
$33$	0	0
$34$	−16.7104	−0.0842887
$35$	496.877	2.39965
$36$	0	0
$37$	−379.934	−1.68813	−0.844065	−	0.536241i	$-0.819844\pi$
$37$	−379.934	−1.68813	−0.844065	+	0.536241i	$0.819844\pi$
$38$	−2.27805	−0.00972496
$39$	0	0
$40$	−52.2979	−0.206725
$41$	464.631	1.76983	0.884917	−	0.465749i	$-0.154215\pi$
$41$	464.631	1.76983	0.884917	+	0.465749i	$0.154215\pi$
$42$	0	0
$43$	322.758	1.14466	0.572328	−	0.820025i	$-0.306041\pi$
$43$	322.758	1.14466	0.572328	+	0.820025i	$0.306041\pi$
$44$	235.389	0.806504
$45$	0	0
$46$	−5.74401	−0.0184110
$47$	248.529	0.771314	0.385657	−	0.922642i	$-0.373975\pi$
$47$	248.529	0.771314	0.385657	+	0.922642i	$0.373975\pi$
$48$	0	0
$49$	706.292	2.05916
$50$	−23.5688	−0.0666626
$51$	0	0
$52$	0	0
$53$	−740.167	−1.91830	−0.959148	−	0.282904i	$-0.908702\pi$
$53$	−740.167	−1.91830	−0.959148	+	0.282904i	$0.908702\pi$
$54$	0	0
$55$	453.924	1.11286
$56$	−110.441	−0.263542
$57$	0	0
$58$	−40.7706	−0.0923008
$59$	340.673	0.751727	0.375864	−	0.926675i	$-0.377346\pi$
$59$	340.673	0.751727	0.375864	+	0.926675i	$0.377346\pi$
$60$	0	0
$61$	−590.834	−1.24014	−0.620070	−	0.784547i	$-0.712896\pi$
$61$	−590.834	−1.24014	−0.620070	+	0.784547i	$0.712896\pi$
$62$	27.3767	0.0560781
$63$	0	0
$64$	−494.547	−0.965912
$65$	0	0
$66$	0	0
$67$	−340.777	−0.621382	−0.310691	−	0.950511i	$-0.600560\pi$
$67$	−340.777	−0.621382	−0.310691	+	0.950511i	$0.600560\pi$
$68$	−621.995	−1.10924
$69$	0	0
$70$	−106.183	−0.181304
$71$	36.2243	0.0605498	0.0302749	−	0.999542i	$-0.490362\pi$
$71$	36.2243	0.0605498	0.0302749	+	0.999542i	$0.490362\pi$
$72$	0	0
$73$	−164.572	−0.263859	−0.131929	−	0.991259i	$-0.542117\pi$
$73$	−164.572	−0.263859	−0.131929	+	0.991259i	$0.542117\pi$
$74$	81.1919	0.127546
$75$	0	0
$76$	−84.7935	−0.127980
$77$	958.584	1.41871
$78$	0	0
$79$	−327.242	−0.466046	−0.233023	−	0.972471i	$-0.574862\pi$
$79$	−327.242	−0.466046	−0.233023	+	0.972471i	$0.574862\pi$
$80$	−964.925	−1.34852
$81$	0	0
$82$	−99.2916	−0.133719
$83$	1404.48	1.85737	0.928684	−	0.370871i	$-0.120941\pi$
$83$	1404.48	1.85737	0.928684	+	0.370871i	$0.120941\pi$
$84$	0	0
$85$	−1199.46	−1.53058
$86$	−68.9734	−0.0864837
$87$	0	0
$88$	−100.894	−0.122220
$89$	−736.986	−0.877756	−0.438878	−	0.898547i	$-0.644624\pi$
$89$	−736.986	−0.877756	−0.438878	+	0.898547i	$0.644624\pi$
$90$	0	0
$91$	0	0
$92$	−213.803	−0.242289
$93$	0	0
$94$	−53.1107	−0.0582761
$95$	−163.516	−0.176593
$96$	0	0
$97$	1494.87	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$97$	1494.87	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$98$	−150.935	−0.155578
$99$	0	0
$100$	−877.277	−0.877277
$101$	484.010	0.476839	0.238420	−	0.971162i	$-0.423371\pi$
$101$	484.010	0.476839	0.238420	+	0.971162i	$0.423371\pi$
$102$	0	0
$103$	214.049	0.204766	0.102383	−	0.994745i	$-0.467353\pi$
$103$	214.049	0.204766	0.102383	+	0.994745i	$0.467353\pi$
$104$	0	0
$105$	0	0
$106$	158.174	0.144936
$107$	−148.288	−0.133977	−0.0669884	−	0.997754i	$-0.521339\pi$
$107$	−148.288	−0.133977	−0.0669884	+	0.997754i	$0.521339\pi$
$108$	0	0
$109$	−504.881	−0.443659	−0.221829	−	0.975086i	$-0.571203\pi$
$109$	−504.881	−0.443659	−0.221829	+	0.975086i	$0.571203\pi$
$110$	−97.0035	−0.0840811
$111$	0	0
$112$	−2037.70	−1.71915
$113$	1136.49	0.946120	0.473060	−	0.881030i	$-0.343149\pi$
$113$	1136.49	0.946120	0.473060	+	0.881030i	$0.343149\pi$
$114$	0	0
$115$	−412.299	−0.334322
$116$	−1517.56	−1.21467
$117$	0	0
$118$	−72.8019	−0.0567962
$119$	−2532.98	−1.95124
$120$	0	0
$121$	−455.282	−0.342060
$122$	126.261	0.0936979
$123$	0	0
$124$	1019.01	0.737985
$125$	225.651	0.161463
$126$	0	0
$127$	−1286.84	−0.899122	−0.449561	−	0.893250i	$-0.648420\pi$
$127$	−1286.84	−0.899122	−0.449561	+	0.893250i	$0.648420\pi$
$128$	431.433	0.297919
$129$	0	0
$130$	0	0
$131$	−2229.88	−1.48722	−0.743610	−	0.668614i	$-0.766888\pi$
$131$	−2229.88	−1.48722	−0.743610	+	0.668614i	$0.766888\pi$
$132$	0	0
$133$	−345.308	−0.225128
$134$	72.8241	0.0469481
$135$	0	0
$136$	266.604	0.168096
$137$	−474.745	−0.296060	−0.148030	−	0.988983i	$-0.547293\pi$
$137$	−474.745	−0.296060	−0.148030	+	0.988983i	$0.547293\pi$
$138$	0	0
$139$	1927.02	1.17588	0.587941	−	0.808904i	$-0.299939\pi$
$139$	1927.02	1.17588	0.587941	+	0.808904i	$0.299939\pi$
$140$	−3952.33	−2.38595
$141$	0	0
$142$	−7.74113	−0.00457480
$143$	0	0
$144$	0	0
$145$	−2926.47	−1.67607
$146$	35.1690	0.0199357
$147$	0	0
$148$	3022.12	1.67849
$149$	1065.73	0.585959	0.292979	−	0.956119i	$-0.405353\pi$
$149$	1065.73	0.585959	0.292979	+	0.956119i	$0.405353\pi$
$150$	0	0
$151$	−877.888	−0.473123	−0.236561	−	0.971617i	$-0.576020\pi$
$151$	−877.888	−0.473123	−0.236561	+	0.971617i	$0.576020\pi$
$152$	36.3448	0.0193944
$153$	0	0
$154$	−204.849	−0.107190
$155$	1965.07	1.01831
$156$	0	0
$157$	2314.94	1.17676	0.588382	−	0.808583i	$-0.299765\pi$
$157$	2314.94	1.17676	0.588382	+	0.808583i	$0.299765\pi$
$158$	69.9317	0.0352118
$159$	0	0
$160$	624.587	0.308612
$161$	−870.681	−0.426207
$162$	0	0
$163$	−331.399	−0.159246	−0.0796232	−	0.996825i	$-0.525372\pi$
$163$	−331.399	−0.159246	−0.0796232	+	0.996825i	$0.525372\pi$
$164$	−3695.83	−1.75973
$165$	0	0
$166$	−300.137	−0.140332
$167$	−2171.62	−1.00626	−0.503129	−	0.864212i	$-0.667818\pi$
$167$	−2171.62	−1.00626	−0.503129	+	0.864212i	$0.667818\pi$
$168$	0	0
$169$	0	0
$170$	256.324	0.115642
$171$	0	0
$172$	−2567.33	−1.13812
$173$	2936.69	1.29059	0.645295	−	0.763933i	$-0.276734\pi$
$173$	2936.69	1.29059	0.645295	+	0.763933i	$0.276734\pi$
$174$	0	0
$175$	−3572.57	−1.54321
$176$	−1861.55	−0.797270
$177$	0	0
$178$	157.494	0.0663183
$179$	−1615.57	−0.674599	−0.337300	−	0.941397i	$-0.609514\pi$
$179$	−1615.57	−0.674599	−0.337300	+	0.941397i	$0.609514\pi$
$180$	0	0
$181$	725.019	0.297736	0.148868	−	0.988857i	$-0.452437\pi$
$181$	725.019	0.297736	0.148868	+	0.988857i	$0.452437\pi$
$182$	0	0
$183$	0	0
$184$	91.6419	0.0367170
$185$	5827.87	2.31607
$186$	0	0
$187$	−2314.01	−0.904905
$188$	−1976.89	−0.766911
$189$	0	0
$190$	34.9433	0.0133424
$191$	−1717.08	−0.650490	−0.325245	−	0.945630i	$-0.605447\pi$
$191$	−1717.08	−0.650490	−0.325245	+	0.945630i	$0.605447\pi$
$192$	0	0
$193$	−435.830	−0.162548	−0.0812740	−	0.996692i	$-0.525899\pi$
$193$	−435.830	−0.162548	−0.0812740	+	0.996692i	$0.525899\pi$
$194$	−319.454	−0.118224
$195$	0	0
$196$	−5618.08	−2.04741
$197$	694.556	0.251193	0.125597	−	0.992081i	$-0.459915\pi$
$197$	694.556	0.251193	0.125597	+	0.992081i	$0.459915\pi$
$198$	0	0
$199$	2899.24	1.03277	0.516386	−	0.856356i	$-0.327277\pi$
$199$	2899.24	1.03277	0.516386	+	0.856356i	$0.327277\pi$
$200$	376.024	0.132945
$201$	0	0
$202$	−103.433	−0.0360273
$203$	−6180.04	−2.13672
$204$	0	0
$205$	−7127.04	−2.42817
$206$	−45.7423	−0.0154709
$207$	0	0
$208$	0	0
$209$	−315.457	−0.104405
$210$	0	0
$211$	5250.30	1.71301	0.856507	−	0.516136i	$-0.172630\pi$
$211$	5250.30	1.71301	0.856507	+	0.516136i	$0.172630\pi$
$212$	5887.53	1.90735
$213$	0	0
$214$	31.6891	0.0101225
$215$	−4950.84	−1.57044
$216$	0	0
$217$	4149.77	1.29818
$218$	107.893	0.0335203
$219$	0	0
$220$	−3610.66	−1.10650
$221$	0	0
$222$	0	0
$223$	1382.46	0.415141	0.207570	−	0.978220i	$-0.433444\pi$
$223$	1382.46	0.415141	0.207570	+	0.978220i	$0.433444\pi$
$224$	1318.99	0.393431
$225$	0	0
$226$	−242.867	−0.0714835
$227$	1223.57	0.357760	0.178880	−	0.983871i	$-0.442753\pi$
$227$	1223.57	0.357760	0.178880	+	0.983871i	$0.442753\pi$
$228$	0	0
$229$	4019.61	1.15993	0.579964	−	0.814642i	$-0.303067\pi$
$229$	4019.61	1.15993	0.579964	+	0.814642i	$0.303067\pi$
$230$	88.1082	0.0252595
$231$	0	0
$232$	650.468	0.184075
$233$	−1658.40	−0.466290	−0.233145	−	0.972442i	$-0.574902\pi$
$233$	−1658.40	−0.466290	−0.233145	+	0.972442i	$0.574902\pi$
$234$	0	0
$235$	−3812.23	−1.05822
$236$	−2709.83	−0.747436
$237$	0	0
$238$	541.297	0.147425
$239$	−618.554	−0.167410	−0.0837049	−	0.996491i	$-0.526675\pi$
$239$	−618.554	−0.167410	−0.0837049	+	0.996491i	$0.526675\pi$
$240$	0	0
$241$	1135.25	0.303434	0.151717	−	0.988424i	$-0.451520\pi$
$241$	1135.25	0.303434	0.151717	+	0.988424i	$0.451520\pi$
$242$	97.2938	0.0258441
$243$	0	0
$244$	4699.69	1.23306
$245$	−10833.9	−2.82512
$246$	0	0
$247$	0	0
$248$	−436.777	−0.111836
$249$	0	0
$250$	−48.2215	−0.0121992
$251$	−2287.83	−0.575326	−0.287663	−	0.957732i	$-0.592878\pi$
$251$	−2287.83	−0.575326	−0.287663	+	0.957732i	$0.592878\pi$
$252$	0	0
$253$	−795.414	−0.197657
$254$	274.997	0.0679326
$255$	0	0
$256$	3864.18	0.943403
$257$	5113.49	1.24113	0.620566	−	0.784154i	$-0.286903\pi$
$257$	5113.49	1.24113	0.620566	+	0.784154i	$0.286903\pi$
$258$	0	0
$259$	12307.1	2.95262
$260$	0	0
$261$	0	0
$262$	476.526	0.112366
$263$	−2760.17	−0.647145	−0.323573	−	0.946203i	$-0.604884\pi$
$263$	−2760.17	−0.647145	−0.323573	+	0.946203i	$0.604884\pi$
$264$	0	0
$265$	11353.5	2.63185
$266$	73.7923	0.0170094
$267$	0	0
$268$	2710.66	0.617834
$269$	3310.32	0.750312	0.375156	−	0.926962i	$-0.377589\pi$
$269$	3310.32	0.750312	0.375156	+	0.926962i	$0.377589\pi$
$270$	0	0
$271$	−2522.26	−0.565374	−0.282687	−	0.959212i	$-0.591226\pi$
$271$	−2522.26	−0.565374	−0.282687	+	0.959212i	$0.591226\pi$
$272$	4918.99	1.09654
$273$	0	0
$274$	101.453	0.0223686
$275$	−3263.74	−0.715675
$276$	0	0
$277$	−3455.48	−0.749530	−0.374765	−	0.927120i	$-0.622277\pi$
$277$	−3455.48	−0.749530	−0.374765	+	0.927120i	$0.622277\pi$
$278$	−411.804	−0.0888430
$279$	0	0
$280$	1694.07	0.361572
$281$	−981.649	−0.208400	−0.104200	−	0.994556i	$-0.533228\pi$
$281$	−981.649	−0.208400	−0.104200	+	0.994556i	$0.533228\pi$
$282$	0	0
$283$	5331.10	1.11979	0.559896	−	0.828563i	$-0.310841\pi$
$283$	5331.10	1.11979	0.559896	+	0.828563i	$0.310841\pi$
$284$	−288.140	−0.0602041
$285$	0	0
$286$	0	0
$287$	−15050.7	−3.09552
$288$	0	0
$289$	1201.58	0.244572
$290$	625.387	0.126634
$291$	0	0
$292$	1309.06	0.262353
$293$	−2420.39	−0.482597	−0.241299	−	0.970451i	$-0.577573\pi$
$293$	−2420.39	−0.482597	−0.241299	+	0.970451i	$0.577573\pi$
$294$	0	0
$295$	−5225.64	−1.03135
$296$	−1295.36	−0.254363
$297$	0	0
$298$	−227.746	−0.0442717
$299$	0	0
$300$	0	0
$301$	−10455.0	−2.00205
$302$	187.605	0.0357465
$303$	0	0
$304$	670.581	0.126515
$305$	9062.88	1.70144
$306$	0	0
$307$	875.509	0.162762	0.0813810	−	0.996683i	$-0.474067\pi$
$307$	875.509	0.162762	0.0813810	+	0.996683i	$0.474067\pi$
$308$	−7624.90	−1.41061
$309$	0	0
$310$	−419.935	−0.0769377
$311$	−6419.10	−1.17040	−0.585199	−	0.810890i	$-0.698984\pi$
$311$	−6419.10	−1.17040	−0.585199	+	0.810890i	$0.698984\pi$
$312$	0	0
$313$	−5015.58	−0.905742	−0.452871	−	0.891576i	$-0.649600\pi$
$313$	−5015.58	−0.905742	−0.452871	+	0.891576i	$0.649600\pi$
$314$	−494.701	−0.0889096
$315$	0	0
$316$	2602.99	0.463386
$317$	5228.81	0.926433	0.463217	−	0.886245i	$-0.346695\pi$
$317$	5228.81	0.926433	0.463217	+	0.886245i	$0.346695\pi$
$318$	0	0
$319$	−5645.80	−0.990922
$320$	7585.92	1.32521
$321$	0	0
$322$	186.064	0.0322018
$323$	833.570	0.143595
$324$	0	0
$325$	0	0
$326$	70.8199	0.0120318
$327$	0	0
$328$	1584.13	0.266674
$329$	−8050.56	−1.34906
$330$	0	0
$331$	3186.81	0.529193	0.264596	−	0.964359i	$-0.414761\pi$
$331$	3186.81	0.529193	0.264596	+	0.964359i	$0.414761\pi$
$332$	−11171.7	−1.84677
$333$	0	0
$334$	464.075	0.0760271
$335$	5227.23	0.852520
$336$	0	0
$337$	−5600.69	−0.905309	−0.452655	−	0.891686i	$-0.649523\pi$
$337$	−5600.69	−0.905309	−0.452655	+	0.891686i	$0.649523\pi$
$338$	0	0
$339$	0	0
$340$	9540.87	1.52184
$341$	3791.04	0.602042
$342$	0	0
$343$	−11768.0	−1.85252
$344$	1100.42	0.172474
$345$	0	0
$346$	−627.570	−0.0975097
$347$	6982.33	1.08020	0.540102	−	0.841599i	$-0.318386\pi$
$347$	6982.33	1.08020	0.540102	+	0.841599i	$0.318386\pi$
$348$	0	0
$349$	−2872.20	−0.440531	−0.220265	−	0.975440i	$-0.570692\pi$
$349$	−2872.20	−0.440531	−0.220265	+	0.975440i	$0.570692\pi$
$350$	763.459	0.116596
$351$	0	0
$352$	1204.96	0.182457
$353$	−5104.55	−0.769655	−0.384827	−	0.922989i	$-0.625739\pi$
$353$	−5104.55	−0.769655	−0.384827	+	0.922989i	$0.625739\pi$
$354$	0	0
$355$	−555.650	−0.0830728
$356$	5862.23	0.872746
$357$	0	0
$358$	345.247	0.0509689
$359$	−10771.0	−1.58348	−0.791741	−	0.610857i	$-0.790825\pi$
$359$	−10771.0	−1.58348	−0.791741	+	0.610857i	$0.790825\pi$
$360$	0	0
$361$	−6745.36	−0.983433
$362$	−154.936	−0.0224952
$363$	0	0
$364$	0	0
$365$	2524.39	0.362008
$366$	0	0
$367$	−11380.5	−1.61868	−0.809340	−	0.587340i	$-0.800175\pi$
$367$	−11380.5	−1.61868	−0.809340	+	0.587340i	$0.800175\pi$
$368$	1690.84	0.239514
$369$	0	0
$370$	−1245.41	−0.174989
$371$	23976.1	3.35519
$372$	0	0
$373$	3196.75	0.443758	0.221879	−	0.975074i	$-0.428781\pi$
$373$	3196.75	0.443758	0.221879	+	0.975074i	$0.428781\pi$
$374$	494.504	0.0683695
$375$	0	0
$376$	847.346	0.116220
$377$	0	0
$378$	0	0
$379$	−2050.75	−0.277942	−0.138971	−	0.990296i	$-0.544379\pi$
$379$	−2050.75	−0.277942	−0.138971	+	0.990296i	$0.544379\pi$
$380$	1300.66	0.175585
$381$	0	0
$382$	366.940	0.0491473
$383$	−3507.54	−0.467955	−0.233978	−	0.972242i	$-0.575174\pi$
$383$	−3507.54	−0.467955	−0.233978	+	0.972242i	$0.575174\pi$
$384$	0	0
$385$	−14703.9	−1.94644
$386$	93.1369	0.0122812
$387$	0	0
$388$	−11890.7	−1.55582
$389$	6572.66	0.856676	0.428338	−	0.903618i	$-0.359099\pi$
$389$	6572.66	0.856676	0.428338	+	0.903618i	$0.359099\pi$
$390$	0	0
$391$	2101.81	0.271850
$392$	2408.06	0.310269
$393$	0	0
$394$	−148.427	−0.0189787
$395$	5019.62	0.639403
$396$	0	0
$397$	−5285.98	−0.668251	−0.334126	−	0.942529i	$-0.608441\pi$
$397$	−5285.98	−0.668251	−0.334126	+	0.942529i	$0.608441\pi$
$398$	−619.567	−0.0780304
$399$	0	0
$400$	6937.86	0.867233
$401$	919.541	0.114513	0.0572565	−	0.998360i	$-0.481765\pi$
$401$	919.541	0.114513	0.0572565	+	0.998360i	$0.481765\pi$
$402$	0	0
$403$	0	0
$404$	−3849.97	−0.474117
$405$	0	0
$406$	1320.67	0.161438
$407$	11243.2	1.36930
$408$	0	0
$409$	−1539.17	−0.186081	−0.0930403	−	0.995662i	$-0.529659\pi$
$409$	−1539.17	−0.186081	−0.0930403	+	0.995662i	$0.529659\pi$
$410$	1523.05	0.183458
$411$	0	0
$412$	−1702.62	−0.203597
$413$	−11035.4	−1.31481
$414$	0	0
$415$	−21543.5	−2.54826
$416$	0	0
$417$	0	0
$418$	67.4132	0.00788825
$419$	−10579.9	−1.23357	−0.616783	−	0.787133i	$-0.711564\pi$
$419$	−10579.9	−1.23357	−0.616783	+	0.787133i	$0.711564\pi$
$420$	0	0
$421$	74.9351	0.00867485	0.00433743	−	0.999991i	$-0.498619\pi$
$421$	74.9351	0.00867485	0.00433743	+	0.999991i	$0.498619\pi$
$422$	−1121.99	−0.129426
$423$	0	0
$424$	−2523.55	−0.289044
$425$	8624.15	0.984313
$426$	0	0
$427$	19138.8	2.16906
$428$	1179.53	0.133212
$429$	0	0
$430$	1057.99	0.118653
$431$	12165.6	1.35962	0.679810	−	0.733388i	$-0.262062\pi$
$431$	12165.6	1.35962	0.679810	+	0.733388i	$0.262062\pi$
$432$	0	0
$433$	−2869.23	−0.318445	−0.159222	−	0.987243i	$-0.550899\pi$
$433$	−2869.23	−0.318445	−0.159222	+	0.987243i	$0.550899\pi$
$434$	−886.806	−0.0980831
$435$	0	0
$436$	4015.99	0.441126
$437$	286.530	0.0313652
$438$	0	0
$439$	−3845.12	−0.418035	−0.209018	−	0.977912i	$-0.567027\pi$
$439$	−3845.12	−0.418035	−0.209018	+	0.977912i	$0.567027\pi$
$440$	1547.63	0.167682
$441$	0	0
$442$	0	0
$443$	3858.30	0.413799	0.206900	−	0.978362i	$-0.433663\pi$
$443$	3858.30	0.413799	0.206900	+	0.978362i	$0.433663\pi$
$444$	0	0
$445$	11304.7	1.20426
$446$	−295.432	−0.0313657
$447$	0	0
$448$	16019.7	1.68942
$449$	5550.51	0.583395	0.291698	−	0.956511i	$-0.405780\pi$
$449$	5550.51	0.583395	0.291698	+	0.956511i	$0.405780\pi$
$450$	0	0
$451$	−13749.6	−1.43557
$452$	−9039.99	−0.940719
$453$	0	0
$454$	−261.478	−0.0270303
$455$	0	0
$456$	0	0
$457$	8102.42	0.829355	0.414677	−	0.909968i	$-0.363894\pi$
$457$	8102.42	0.829355	0.414677	+	0.909968i	$0.363894\pi$
$458$	−858.991	−0.0876376
$459$	0	0
$460$	3279.56	0.332414
$461$	−9230.86	−0.932590	−0.466295	−	0.884629i	$-0.654412\pi$
$461$	−9230.86	−0.932590	−0.466295	+	0.884629i	$0.654412\pi$
$462$	0	0
$463$	−13934.7	−1.39870	−0.699352	−	0.714777i	$-0.746528\pi$
$463$	−13934.7	−1.39870	−0.699352	+	0.714777i	$0.746528\pi$
$464$	12001.5	1.20077
$465$	0	0
$466$	354.400	0.0352302
$467$	−10918.0	−1.08185	−0.540926	−	0.841070i	$-0.681926\pi$
$467$	−10918.0	−1.08185	−0.540926	+	0.841070i	$0.681926\pi$
$468$	0	0
$469$	11038.7	1.08682
$470$	814.673	0.0799533
$471$	0	0
$472$	1161.51	0.113268
$473$	−9551.23	−0.928470
$474$	0	0
$475$	1175.69	0.113567
$476$	20148.2	1.94010
$477$	0	0
$478$	132.185	0.0126485
$479$	11028.4	1.05198	0.525992	−	0.850490i	$-0.323694\pi$
$479$	11028.4	1.05198	0.525992	+	0.850490i	$0.323694\pi$
$480$	0	0
$481$	0	0
$482$	−242.602	−0.0229258
$483$	0	0
$484$	3621.47	0.340108
$485$	−22930.0	−2.14680
$486$	0	0
$487$	−1078.10	−0.100315	−0.0501576	−	0.998741i	$-0.515972\pi$
$487$	−1078.10	−0.100315	−0.0501576	+	0.998741i	$0.515972\pi$
$488$	−2014.41	−0.186861
$489$	0	0
$490$	2315.21	0.213450
$491$	−6572.88	−0.604134	−0.302067	−	0.953287i	$-0.597677\pi$
$491$	−6572.88	−0.604134	−0.302067	+	0.953287i	$0.597677\pi$
$492$	0	0
$493$	14918.6	1.36288
$494$	0	0
$495$	0	0
$496$	−8058.77	−0.729535
$497$	−1173.41	−0.105904
$498$	0	0
$499$	−9956.79	−0.893241	−0.446620	−	0.894724i	$-0.647373\pi$
$499$	−9956.79	−0.893241	−0.446620	+	0.894724i	$0.647373\pi$
$500$	−1794.90	−0.160541
$501$	0	0
$502$	488.910	0.0434683
$503$	11965.0	1.06062	0.530310	−	0.847804i	$-0.322076\pi$
$503$	11965.0	1.06062	0.530310	+	0.847804i	$0.322076\pi$
$504$	0	0
$505$	−7424.29	−0.654211
$506$	169.980	0.0149338
$507$	0	0
$508$	10235.9	0.893990
$509$	−7160.44	−0.623538	−0.311769	−	0.950158i	$-0.600922\pi$
$509$	−7160.44	−0.623538	−0.311769	+	0.950158i	$0.600922\pi$
$510$	0	0
$511$	5330.94	0.461501
$512$	−4277.24	−0.369197
$513$	0	0
$514$	−1092.75	−0.0937729
$515$	−3283.33	−0.280933
$516$	0	0
$517$	−7354.61	−0.625639
$518$	−2630.03	−0.223083
$519$	0	0
$520$	0	0
$521$	17213.1	1.44745	0.723725	−	0.690089i	$-0.242428\pi$
$521$	17213.1	1.44745	0.723725	+	0.690089i	$0.242428\pi$
$522$	0	0
$523$	−10110.1	−0.845281	−0.422640	−	0.906297i	$-0.638897\pi$
$523$	−10110.1	−0.845281	−0.422640	+	0.906297i	$0.638897\pi$
$524$	17737.2	1.47873
$525$	0	0
$526$	589.847	0.0488946
$527$	−10017.5	−0.828026
$528$	0	0
$529$	−11444.5	−0.940620
$530$	−2426.25	−0.198848
$531$	0	0
$532$	2746.70	0.223843
$533$	0	0
$534$	0	0
$535$	2274.61	0.183813
$536$	−1161.86	−0.0936281
$537$	0	0
$538$	−707.416	−0.0566893
$539$	−20901.0	−1.67026
$540$	0	0
$541$	5951.54	0.472970	0.236485	−	0.971635i	$-0.424005\pi$
$541$	5951.54	0.472970	0.236485	+	0.971635i	$0.424005\pi$
$542$	539.007	0.0427165
$543$	0	0
$544$	−3184.02	−0.250944
$545$	7744.43	0.608688
$546$	0	0
$547$	5157.62	0.403152	0.201576	−	0.979473i	$-0.435394\pi$
$547$	5157.62	0.403152	0.201576	+	0.979473i	$0.435394\pi$
$548$	3776.28	0.294370
$549$	0	0
$550$	697.460	0.0540724
$551$	2033.77	0.157244
$552$	0	0
$553$	10600.3	0.815136
$554$	738.436	0.0566302
$555$	0	0
$556$	−15328.2	−1.16917
$557$	23314.4	1.77354	0.886772	−	0.462206i	$-0.152942\pi$
$557$	23314.4	1.77354	0.886772	+	0.462206i	$0.152942\pi$
$558$	0	0
$559$	0	0
$560$	31256.6	2.35863
$561$	0	0
$562$	209.778	0.0157455
$563$	−2329.12	−0.174353	−0.0871764	−	0.996193i	$-0.527784\pi$
$563$	−2329.12	−0.174353	−0.0871764	+	0.996193i	$0.527784\pi$
$564$	0	0
$565$	−17432.7	−1.29805
$566$	−1139.26	−0.0846051
$567$	0	0
$568$	123.505	0.00912348
$569$	17446.6	1.28541	0.642706	−	0.766113i	$-0.277812\pi$
$569$	17446.6	1.28541	0.642706	+	0.766113i	$0.277812\pi$
$570$	0	0
$571$	−413.526	−0.0303074	−0.0151537	−	0.999885i	$-0.504824\pi$
$571$	−413.526	−0.0303074	−0.0151537	+	0.999885i	$0.504824\pi$
$572$	0	0
$573$	0	0
$574$	3216.33	0.233880
$575$	2964.45	0.215002
$576$	0	0
$577$	−21258.8	−1.53382	−0.766911	−	0.641754i	$-0.778207\pi$
$577$	−21258.8	−1.53382	−0.766911	+	0.641754i	$0.778207\pi$
$578$	−256.778	−0.0184785
$579$	0	0
$580$	23278.1	1.66650
$581$	−45495.0	−3.24862
$582$	0	0
$583$	21903.4	1.55600
$584$	−561.098	−0.0397575
$585$	0	0
$586$	517.238	0.0364623
$587$	−2965.69	−0.208530	−0.104265	−	0.994550i	$-0.533249\pi$
$587$	−2965.69	−0.208530	−0.104265	+	0.994550i	$0.533249\pi$
$588$	0	0
$589$	−1365.64	−0.0955349
$590$	1116.72	0.0779230
$591$	0	0
$592$	−23900.2	−1.65928
$593$	17786.8	1.23173	0.615865	−	0.787852i	$-0.288807\pi$
$593$	17786.8	1.23173	0.615865	+	0.787852i	$0.288807\pi$
$594$	0	0
$595$	38853.7	2.67705
$596$	−8477.16	−0.582614
$597$	0	0
$598$	0	0
$599$	−21796.4	−1.48677	−0.743387	−	0.668861i	$-0.766782\pi$
$599$	−21796.4	−1.48677	−0.743387	+	0.668861i	$0.766782\pi$
$600$	0	0
$601$	20468.7	1.38924	0.694622	−	0.719375i	$-0.255572\pi$
$601$	20468.7	1.38924	0.694622	+	0.719375i	$0.255572\pi$
$602$	2234.24	0.151264
$603$	0	0
$604$	6983.01	0.470422
$605$	6983.64	0.469298
$606$	0	0
$607$	7852.53	0.525081	0.262541	−	0.964921i	$-0.415440\pi$
$607$	7852.53	0.525081	0.262541	+	0.964921i	$0.415440\pi$
$608$	−434.061	−0.0289531
$609$	0	0
$610$	−1936.74	−0.128551
$611$	0	0
$612$	0	0
$613$	−2460.21	−0.162099	−0.0810497	−	0.996710i	$-0.525827\pi$
$613$	−2460.21	−0.162099	−0.0810497	+	0.996710i	$0.525827\pi$
$614$	−187.096	−0.0122974
$615$	0	0
$616$	3268.23	0.213768
$617$	17829.3	1.16334	0.581670	−	0.813425i	$-0.302400\pi$
$617$	17829.3	1.16334	0.581670	+	0.813425i	$0.302400\pi$
$618$	0	0
$619$	16901.7	1.09748	0.548739	−	0.835994i	$-0.315108\pi$
$619$	16901.7	1.09748	0.548739	+	0.835994i	$0.315108\pi$
$620$	−15630.8	−1.01250
$621$	0	0
$622$	1371.76	0.0884286
$623$	23873.0	1.53524
$624$	0	0
$625$	−17247.4	−1.10384
$626$	1071.83	0.0684327
$627$	0	0
$628$	−18413.8	−1.17005
$629$	−29709.3	−1.88328
$630$	0	0
$631$	−1833.82	−0.115695	−0.0578473	−	0.998325i	$-0.518424\pi$
$631$	−1833.82	−0.115695	−0.0578473	+	0.998325i	$0.518424\pi$
$632$	−1115.71	−0.0702226
$633$	0	0
$634$	−1117.40	−0.0699960
$635$	19739.0	1.23357
$636$	0	0
$637$	0	0
$638$	1206.51	0.0748684
$639$	0	0
$640$	−6617.81	−0.408737
$641$	−29172.1	−1.79755	−0.898773	−	0.438414i	$-0.855540\pi$
$641$	−29172.1	−1.79755	−0.898773	+	0.438414i	$0.855540\pi$
$642$	0	0
$643$	−25103.8	−1.53965	−0.769826	−	0.638254i	$-0.779657\pi$
$643$	−25103.8	−1.53965	−0.769826	+	0.638254i	$0.779657\pi$
$644$	6925.69	0.423774
$645$	0	0
$646$	−178.134	−0.0108492
$647$	24471.6	1.48698	0.743492	−	0.668745i	$-0.233168\pi$
$647$	24471.6	1.48698	0.743492	+	0.668745i	$0.233168\pi$
$648$	0	0
$649$	−10081.4	−0.609752
$650$	0	0
$651$	0	0
$652$	2636.06	0.158337
$653$	−6632.68	−0.397484	−0.198742	−	0.980052i	$-0.563686\pi$
$653$	−6632.68	−0.397484	−0.198742	+	0.980052i	$0.563686\pi$
$654$	0	0
$655$	34204.5	2.04043
$656$	29228.1	1.73958
$657$	0	0
$658$	1720.40	0.101928
$659$	−26436.9	−1.56272	−0.781361	−	0.624080i	$-0.785474\pi$
$659$	−26436.9	−1.56272	−0.781361	+	0.624080i	$0.785474\pi$
$660$	0	0
$661$	−27596.7	−1.62389	−0.811943	−	0.583737i	$-0.801590\pi$
$661$	−27596.7	−1.62389	−0.811943	+	0.583737i	$0.801590\pi$
$662$	−681.020	−0.0399828
$663$	0	0
$664$	4788.49	0.279863
$665$	5296.73	0.308870
$666$	0	0
$667$	5128.07	0.297691
$668$	17273.8	1.00051
$669$	0	0
$670$	−1117.06	−0.0644115
$671$	17484.3	1.00592
$672$	0	0
$673$	18860.1	1.08024	0.540122	−	0.841587i	$-0.318378\pi$
$673$	18860.1	1.08024	0.540122	+	0.841587i	$0.318378\pi$
$674$	1196.87	0.0684000
$675$	0	0
$676$	0	0
$677$	14705.5	0.834825	0.417412	−	0.908717i	$-0.362937\pi$
$677$	14705.5	0.834825	0.417412	+	0.908717i	$0.362937\pi$
$678$	0	0
$679$	−48423.1	−2.73683
$680$	−4089.47	−0.230624
$681$	0	0
$682$	−810.145	−0.0454869
$683$	−17950.2	−1.00563	−0.502814	−	0.864395i	$-0.667702\pi$
$683$	−17950.2	−1.00563	−0.502814	+	0.864395i	$0.667702\pi$
$684$	0	0
$685$	7282.18	0.406186
$686$	2514.83	0.139966
$687$	0	0
$688$	20303.5	1.12509
$689$	0	0
$690$	0	0
$691$	8692.38	0.478544	0.239272	−	0.970953i	$-0.423091\pi$
$691$	8692.38	0.478544	0.239272	+	0.970953i	$0.423091\pi$
$692$	−23359.4	−1.28322
$693$	0	0
$694$	−1492.12	−0.0816141
$695$	−29558.8	−1.61328
$696$	0	0
$697$	36332.2	1.97443
$698$	613.789	0.0332840
$699$	0	0
$700$	28417.4	1.53440
$701$	−23275.9	−1.25409	−0.627047	−	0.778982i	$-0.715736\pi$
$701$	−23275.9	−1.25409	−0.627047	+	0.778982i	$0.715736\pi$
$702$	0	0
$703$	−4050.11	−0.217287
$704$	14634.9	0.783485
$705$	0	0
$706$	1090.84	0.0581507
$707$	−15678.4	−0.834014
$708$	0	0
$709$	34164.7	1.80971	0.904854	−	0.425722i	$-0.139980\pi$
$709$	34164.7	1.80971	0.904854	+	0.425722i	$0.139980\pi$
$710$	118.742	0.00627651
$711$	0	0
$712$	−2512.71	−0.132258
$713$	−3443.40	−0.180864
$714$	0	0
$715$	0	0
$716$	12850.8	0.670748
$717$	0	0
$718$	2301.75	0.119639
$719$	8391.74	0.435270	0.217635	−	0.976030i	$-0.430166\pi$
$719$	8391.74	0.435270	0.217635	+	0.976030i	$0.430166\pi$
$720$	0	0
$721$	−6933.64	−0.358145
$722$	1441.48	0.0743026
$723$	0	0
$724$	−5767.04	−0.296036
$725$	21041.5	1.07788
$726$	0	0
$727$	23500.4	1.19887	0.599436	−	0.800423i	$-0.295392\pi$
$727$	23500.4	1.19887	0.599436	+	0.800423i	$0.295392\pi$
$728$	0	0
$729$	0	0
$730$	−539.462	−0.0273512
$731$	25238.3	1.27698
$732$	0	0
$733$	−1289.07	−0.0649563	−0.0324781	−	0.999472i	$-0.510340\pi$
$733$	−1289.07	−0.0649563	−0.0324781	+	0.999472i	$0.510340\pi$
$734$	2432.01	0.122298
$735$	0	0
$736$	−1094.47	−0.0548133
$737$	10084.5	0.504024
$738$	0	0
$739$	18197.8	0.905844	0.452922	−	0.891550i	$-0.350382\pi$
$739$	18197.8	0.905844	0.452922	+	0.891550i	$0.350382\pi$
$740$	−46356.8	−2.30285
$741$	0	0
$742$	−5123.68	−0.253499
$743$	21277.2	1.05059	0.525294	−	0.850921i	$-0.323956\pi$
$743$	21277.2	1.05059	0.525294	+	0.850921i	$0.323956\pi$
$744$	0	0
$745$	−16347.4	−0.803920
$746$	−683.146	−0.0335278
$747$	0	0
$748$	18406.4	0.899739
$749$	4803.45	0.234331
$750$	0	0
$751$	−25653.4	−1.24648	−0.623240	−	0.782030i	$-0.714184\pi$
$751$	−25653.4	−1.24648	−0.623240	+	0.782030i	$0.714184\pi$
$752$	15634.0	0.758130
$753$	0	0
$754$	0	0
$755$	13466.0	0.649112
$756$	0	0
$757$	26521.2	1.27336	0.636678	−	0.771130i	$-0.280308\pi$
$757$	26521.2	1.27336	0.636678	+	0.771130i	$0.280308\pi$
$758$	438.245	0.0209997
$759$	0	0
$760$	−557.497	−0.0266086
$761$	21607.2	1.02925	0.514625	−	0.857415i	$-0.327931\pi$
$761$	21607.2	1.02925	0.514625	+	0.857415i	$0.327931\pi$
$762$	0	0
$763$	16354.5	0.775979
$764$	13658.2	0.646776
$765$	0	0
$766$	749.560	0.0353560
$767$	0	0
$768$	0	0
$769$	17179.2	0.805591	0.402795	−	0.915290i	$-0.368039\pi$
$769$	17179.2	0.805591	0.402795	+	0.915290i	$0.368039\pi$
$770$	3142.21	0.147062
$771$	0	0
$772$	3466.74	0.161620
$773$	−12822.1	−0.596609	−0.298304	−	0.954471i	$-0.596421\pi$
$773$	−12822.1	−0.596609	−0.298304	+	0.954471i	$0.596421\pi$
$774$	0	0
$775$	−14128.9	−0.654873
$776$	5096.67	0.235773
$777$	0	0
$778$	−1404.58	−0.0647256
$779$	4952.98	0.227804
$780$	0	0
$781$	−1071.97	−0.0491140
$782$	−449.157	−0.0205394
$783$	0	0
$784$	44430.0	2.02396
$785$	−35509.1	−1.61449
$786$	0	0
$787$	35213.4	1.59495	0.797474	−	0.603354i	$-0.206169\pi$
$787$	35213.4	1.59495	0.797474	+	0.603354i	$0.206169\pi$
$788$	−5524.73	−0.249759
$789$	0	0
$790$	−1072.69	−0.0483097
$791$	−36813.9	−1.65481
$792$	0	0
$793$	0	0
$794$	1129.61	0.0504893
$795$	0	0
$796$	−23061.5	−1.02688
$797$	−7500.15	−0.333336	−0.166668	−	0.986013i	$-0.553301\pi$
$797$	−7500.15	−0.333336	−0.166668	+	0.986013i	$0.553301\pi$
$798$	0	0
$799$	19434.0	0.860481
$800$	−4490.82	−0.198468
$801$	0	0
$802$	−196.506	−0.00865195
$803$	4870.10	0.214025
$804$	0	0
$805$	13355.5	0.584745
$806$	0	0
$807$	0	0
$808$	1650.20	0.0718489
$809$	22020.5	0.956982	0.478491	−	0.878092i	$-0.341184\pi$
$809$	22020.5	0.956982	0.478491	+	0.878092i	$0.341184\pi$
$810$	0	0
$811$	−20444.6	−0.885210	−0.442605	−	0.896717i	$-0.645946\pi$
$811$	−20444.6	−0.885210	−0.442605	+	0.896717i	$0.645946\pi$
$812$	49158.1	2.12452
$813$	0	0
$814$	−2402.67	−0.103457
$815$	5083.37	0.218482
$816$	0	0
$817$	3440.62	0.147334
$818$	328.920	0.0140592
$819$	0	0
$820$	56690.8	2.41431
$821$	40353.0	1.71538	0.857692	−	0.514164i	$-0.171898\pi$
$821$	40353.0	1.71538	0.857692	+	0.514164i	$0.171898\pi$
$822$	0	0
$823$	−33110.7	−1.40239	−0.701194	−	0.712971i	$-0.747349\pi$
$823$	−33110.7	−1.40239	−0.701194	+	0.712971i	$0.747349\pi$
$824$	729.787	0.0308536
$825$	0	0
$826$	2358.26	0.0993392
$827$	−43410.4	−1.82531	−0.912653	−	0.408735i	$-0.865970\pi$
$827$	−43410.4	−1.82531	−0.912653	+	0.408735i	$0.865970\pi$
$828$	0	0
$829$	−5502.87	−0.230546	−0.115273	−	0.993334i	$-0.536774\pi$
$829$	−5502.87	−0.230546	−0.115273	+	0.993334i	$0.536774\pi$
$830$	4603.84	0.192532
$831$	0	0
$832$	0	0
$833$	55229.1	2.29721
$834$	0	0
$835$	33310.8	1.38056
$836$	2509.25	0.103809
$837$	0	0
$838$	2260.93	0.0932013
$839$	−1698.27	−0.0698819	−0.0349409	−	0.999389i	$-0.511124\pi$
$839$	−1698.27	−0.0698819	−0.0349409	+	0.999389i	$0.511124\pi$
$840$	0	0
$841$	12009.8	0.492425
$842$	−16.0136	−0.000655423 0
$843$	0	0
$844$	−41762.7	−1.70323
$845$	0	0
$846$	0	0
$847$	14747.9	0.598279
$848$	−46561.0	−1.88551
$849$	0	0
$850$	−1842.98	−0.0743691
$851$	−10212.2	−0.411363
$852$	0	0
$853$	−18686.8	−0.750087	−0.375044	−	0.927007i	$-0.622372\pi$
$853$	−18686.8	−0.750087	−0.375044	+	0.927007i	$0.622372\pi$
$854$	−4089.95	−0.163882
$855$	0	0
$856$	−505.578	−0.0201873
$857$	9659.16	0.385007	0.192503	−	0.981296i	$-0.438339\pi$
$857$	9659.16	0.385007	0.192503	+	0.981296i	$0.438339\pi$
$858$	0	0
$859$	−1559.90	−0.0619595	−0.0309798	−	0.999520i	$-0.509863\pi$
$859$	−1559.90	−0.0619595	−0.0309798	+	0.999520i	$0.509863\pi$
$860$	39380.6	1.56147
$861$	0	0
$862$	−2599.79	−0.102725
$863$	−11358.7	−0.448034	−0.224017	−	0.974585i	$-0.571917\pi$
$863$	−11358.7	−0.448034	−0.224017	+	0.974585i	$0.571917\pi$
$864$	0	0
$865$	−45046.3	−1.77066
$866$	613.155	0.0240599
$867$	0	0
$868$	−33008.7	−1.29077
$869$	9683.93	0.378026
$870$	0	0
$871$	0	0
$872$	−1721.36	−0.0668493
$873$	0	0
$874$	−61.2314	−0.00236977
$875$	−7309.45	−0.282405
$876$	0	0
$877$	33301.6	1.28223	0.641115	−	0.767444i	$-0.278472\pi$
$877$	33301.6	1.28223	0.641115	+	0.767444i	$0.278472\pi$
$878$	821.701	0.0315844
$879$	0	0
$880$	28554.6	1.09383
$881$	−28912.3	−1.10565	−0.552826	−	0.833296i	$-0.686451\pi$
$881$	−28912.3	−1.10565	−0.552826	+	0.833296i	$0.686451\pi$
$882$	0	0
$883$	−49100.9	−1.87132	−0.935661	−	0.352901i	$-0.885195\pi$
$883$	−49100.9	−1.87132	−0.935661	+	0.352901i	$0.885195\pi$
$884$	0	0
$885$	0	0
$886$	−824.517	−0.0312643
$887$	−4566.49	−0.172861	−0.0864304	−	0.996258i	$-0.527546\pi$
$887$	−4566.49	−0.172861	−0.0864304	+	0.996258i	$0.527546\pi$
$888$	0	0
$889$	41684.3	1.57261
$890$	−2415.82	−0.0909870
$891$	0	0
$892$	−10996.5	−0.412771
$893$	2649.33	0.0992795
$894$	0	0
$895$	24781.4	0.925533
$896$	−13975.3	−0.521074
$897$	0	0
$898$	−1186.14	−0.0440780
$899$	−24441.0	−0.906734
$900$	0	0
$901$	−57877.9	−2.14006
$902$	2938.29	0.108464
$903$	0	0
$904$	3874.78	0.142559
$905$	−11121.2	−0.408486
$906$	0	0
$907$	48873.3	1.78921	0.894604	−	0.446860i	$-0.147458\pi$
$907$	48873.3	1.78921	0.894604	+	0.446860i	$0.147458\pi$
$908$	−9732.72	−0.355718
$909$	0	0
$910$	0	0
$911$	−31550.1	−1.14742	−0.573710	−	0.819058i	$-0.694497\pi$
$911$	−31550.1	−1.14742	−0.573710	+	0.819058i	$0.694497\pi$
$912$	0	0
$913$	−41562.1	−1.50658
$914$	−1731.49	−0.0626613
$915$	0	0
$916$	−31973.3	−1.15331
$917$	72232.1	2.60121
$918$	0	0
$919$	−11720.3	−0.420694	−0.210347	−	0.977627i	$-0.567459\pi$
$919$	−11720.3	−0.420694	−0.210347	+	0.977627i	$0.567459\pi$
$920$	−1405.71	−0.0503748
$921$	0	0
$922$	1972.63	0.0704612
$923$	0	0
$924$	0	0
$925$	−41902.7	−1.48946
$926$	2977.84	0.105678
$927$	0	0
$928$	−7768.47	−0.274798
$929$	−6911.23	−0.244080	−0.122040	−	0.992525i	$-0.538944\pi$
$929$	−6911.23	−0.244080	−0.122040	+	0.992525i	$0.538944\pi$
$930$	0	0
$931$	7529.10	0.265044
$932$	13191.5	0.463628
$933$	0	0
$934$	2333.18	0.0817386
$935$	35494.9	1.24151
$936$	0	0
$937$	−28673.9	−0.999717	−0.499858	−	0.866107i	$-0.666615\pi$
$937$	−28673.9	−0.999717	−0.499858	+	0.866107i	$0.666615\pi$
$938$	−2358.97	−0.0821143
$939$	0	0
$940$	30323.7	1.05218
$941$	22463.6	0.778207	0.389103	−	0.921194i	$-0.372785\pi$
$941$	22463.6	0.778207	0.389103	+	0.921194i	$0.372785\pi$
$942$	0	0
$943$	12488.8	0.431272
$944$	21430.4	0.738878
$945$	0	0
$946$	2041.10	0.0701499
$947$	36806.6	1.26299	0.631497	−	0.775379i	$-0.282441\pi$
$947$	36806.6	1.26299	0.631497	+	0.775379i	$0.282441\pi$
$948$	0	0
$949$	0	0
$950$	−251.244	−0.00858046
$951$	0	0
$952$	−8636.04	−0.294008
$953$	1781.07	0.0605400	0.0302700	−	0.999542i	$-0.490363\pi$
$953$	1781.07	0.0605400	0.0302700	+	0.999542i	$0.490363\pi$
$954$	0	0
$955$	26338.5	0.892455
$956$	4920.19	0.166454
$957$	0	0
$958$	−2356.77	−0.0794819
$959$	15378.3	0.517822
$960$	0	0
$961$	−13379.3	−0.449107
$962$	0	0
$963$	0	0
$964$	−9030.12	−0.301702
$965$	6685.26	0.223012
$966$	0	0
$967$	−54777.2	−1.82163	−0.910815	−	0.412815i	$-0.864545\pi$
$967$	−54777.2	−1.82163	−0.910815	+	0.412815i	$0.864545\pi$
$968$	−1552.26	−0.0515408
$969$	0	0
$970$	4900.15	0.162200
$971$	2391.54	0.0790403	0.0395201	−	0.999219i	$-0.487417\pi$
$971$	2391.54	0.0790403	0.0395201	+	0.999219i	$0.487417\pi$
$972$	0	0
$973$	−62421.5	−2.05667
$974$	230.391	0.00757925
$975$	0	0
$976$	−37167.0	−1.21894
$977$	−43738.5	−1.43226	−0.716130	−	0.697967i	$-0.754088\pi$
$977$	−43738.5	−1.43226	−0.716130	+	0.697967i	$0.754088\pi$
$978$	0	0
$979$	21809.3	0.711979
$980$	86176.5	2.80899
$981$	0	0
$982$	1404.62	0.0456449
$983$	27712.4	0.899174	0.449587	−	0.893236i	$-0.351571\pi$
$983$	27712.4	0.899174	0.449587	+	0.893236i	$0.351571\pi$
$984$	0	0
$985$	−10653.9	−0.344631
$986$	−3188.09	−0.102971
$987$	0	0
$988$	0	0
$989$	8675.38	0.278929
$990$	0	0
$991$	3550.23	0.113801	0.0569005	−	0.998380i	$-0.481878\pi$
$991$	3550.23	0.113801	0.0569005	+	0.998380i	$0.481878\pi$
$992$	5216.37	0.166956
$993$	0	0
$994$	250.757	0.00800153
$995$	−44471.8	−1.41694
$996$	0	0
$997$	−28338.4	−0.900188	−0.450094	−	0.892981i	$-0.648610\pi$
$997$	−28338.4	−0.900188	−0.450094	+	0.892981i	$0.648610\pi$
$998$	2127.76	0.0674882
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bi.1.4		9
3.2	odd	2		507.4.a.o.1.6	✓	9
13.12	even	2		1521.4.a.bf.1.6		9
39.5	even	4		507.4.b.k.337.9		18
39.8	even	4		507.4.b.k.337.10		18
39.38	odd	2		507.4.a.p.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.6	✓	9	3.2	odd	2
507.4.a.p.1.4	yes	9	39.38	odd	2
507.4.b.k.337.9		18	39.5	even	4
507.4.b.k.337.10		18	39.8	even	4
1521.4.a.bf.1.6		9	13.12	even	2
1521.4.a.bi.1.4		9	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)

\(f(q)\)	\(=\)	\(q-0.213700 q^{2} -7.95433 q^{4} -15.3391 q^{5} -32.3928 q^{7} +3.40944 q^{8} +O(q^{10})\)	\(q-0.213700 q^{2} -7.95433 q^{4} -15.3391 q^{5} -32.3928 q^{7} +3.40944 q^{8} +3.27797 q^{10} -29.5925 q^{11} +6.92233 q^{14} +62.9061 q^{16} +78.1958 q^{17} +10.6600 q^{19} +122.013 q^{20} +6.32392 q^{22} +26.8789 q^{23} +110.289 q^{25} +257.663 q^{28} +190.785 q^{29} -128.108 q^{31} -40.7185 q^{32} -16.7104 q^{34} +496.877 q^{35} -379.934 q^{37} -2.27805 q^{38} -52.2979 q^{40} +464.631 q^{41} +322.758 q^{43} +235.389 q^{44} -5.74401 q^{46} +248.529 q^{47} +706.292 q^{49} -23.5688 q^{50} -740.167 q^{53} +453.924 q^{55} -110.441 q^{56} -40.7706 q^{58} +340.673 q^{59} -590.834 q^{61} +27.3767 q^{62} -494.547 q^{64} -340.777 q^{67} -621.995 q^{68} -106.183 q^{70} +36.2243 q^{71} -164.572 q^{73} +81.1919 q^{74} -84.7935 q^{76} +958.584 q^{77} -327.242 q^{79} -964.925 q^{80} -99.2916 q^{82} +1404.48 q^{83} -1199.46 q^{85} -68.9734 q^{86} -100.894 q^{88} -736.986 q^{89} -213.803 q^{92} -53.1107 q^{94} -163.516 q^{95} +1494.87 q^{97} -150.935 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8	\( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8} - 54 q^{10} + 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} - 352 q^{19} + 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} - 940 q^{28} + 97 q^{29} - 717 q^{31} + 707 q^{32} - 632 q^{34} + 418 q^{35} - 1108 q^{37} + 660 q^{38} - 1506 q^{40} + 334 q^{41} + 242 q^{43} - 307 q^{44} - 979 q^{46} - 184 q^{47} - 38 q^{49} - 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} - 1161 q^{58} + 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} - 2308 q^{67} - 2785 q^{68} + 1420 q^{70} + 96 q^{71} - 2505 q^{73} + 1191 q^{74} - 2409 q^{76} + 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 1517 q^{82} + 1539 q^{83} - 4296 q^{85} - 3763 q^{86} - 3716 q^{88} - 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} - 1445 q^{97} + 1457 q^{98}+O(q^{100}) \) 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8 - 54 * q^10 + 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 - 352 * q^19 + 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 - 940 * q^28 + 97 * q^29 - 717 * q^31 + 707 * q^32 - 632 * q^34 + 418 * q^35 - 1108 * q^37 + 660 * q^38 - 1506 * q^40 + 334 * q^41 + 242 * q^43 - 307 * q^44 - 979 * q^46 - 184 * q^47 - 38 * q^49 - 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 - 1161 * q^58 + 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 - 2308 * q^67 - 2785 * q^68 + 1420 * q^70 + 96 * q^71 - 2505 * q^73 + 1191 * q^74 - 2409 * q^76 + 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 1517 * q^82 + 1539 * q^83 - 4296 * q^85 - 3763 * q^86 - 3716 * q^88 - 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 - 1445 * q^97 + 1457 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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