LMFDB - Embedded newform 1521.4.a.bd.1.2

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 $ x^8 - 52x^6 + 805x^4 - 4210*x^2 + 4992
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.69212$ 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)

$f(q)$	$=$	$q-3.69212 q^{2} +5.63172 q^{4} +18.7574 q^{5} +24.1383 q^{7} +8.74396 q^{8} +O(q^{10})$	$q-3.69212 q^{2} +5.63172 q^{4} +18.7574 q^{5} +24.1383 q^{7} +8.74396 q^{8} -69.2546 q^{10} +49.2741 q^{11} -89.1214 q^{14} -77.3375 q^{16} -65.3121 q^{17} -109.940 q^{19} +105.637 q^{20} -181.926 q^{22} -83.2974 q^{23} +226.841 q^{25} +135.940 q^{28} +4.99593 q^{29} +255.810 q^{31} +215.587 q^{32} +241.140 q^{34} +452.772 q^{35} -93.6356 q^{37} +405.912 q^{38} +164.014 q^{40} -67.9487 q^{41} +142.543 q^{43} +277.498 q^{44} +307.544 q^{46} +379.275 q^{47} +239.657 q^{49} -837.524 q^{50} -389.560 q^{53} +924.256 q^{55} +211.064 q^{56} -18.4456 q^{58} -133.881 q^{59} +620.792 q^{61} -944.482 q^{62} -177.273 q^{64} -119.002 q^{67} -367.819 q^{68} -1671.69 q^{70} -361.386 q^{71} +748.241 q^{73} +345.714 q^{74} -619.152 q^{76} +1189.39 q^{77} +514.165 q^{79} -1450.65 q^{80} +250.875 q^{82} -260.260 q^{83} -1225.09 q^{85} -526.286 q^{86} +430.851 q^{88} +833.631 q^{89} -469.108 q^{92} -1400.33 q^{94} -2062.19 q^{95} +1481.99 q^{97} -884.841 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{4} + 22 q^{7}+O(q^{10}) $ 8 * q + 40 * q^4 + 22 * q^7	$ 8 q + 40 q^{4} + 22 q^{7} + 36 q^{10} + 204 q^{16} - 244 q^{19} + 136 q^{22} + 354 q^{25} + 452 q^{28} - 242 q^{31} + 1292 q^{34} - 1018 q^{37} + 1700 q^{40} + 74 q^{43} + 896 q^{46} + 298 q^{49} + 1300 q^{55} - 812 q^{58} + 1148 q^{61} + 3636 q^{64} + 2198 q^{67} - 2200 q^{70} + 2176 q^{73} - 6936 q^{76} + 1862 q^{79} + 5436 q^{82} + 890 q^{85} + 3528 q^{88} - 3104 q^{94} + 4370 q^{97}+O(q^{100}) $ 8 * q + 40 * q^4 + 22 * q^7 + 36 * q^10 + 204 * q^16 - 244 * q^19 + 136 * q^22 + 354 * q^25 + 452 * q^28 - 242 * q^31 + 1292 * q^34 - 1018 * q^37 + 1700 * q^40 + 74 * q^43 + 896 * q^46 + 298 * q^49 + 1300 * q^55 - 812 * q^58 + 1148 * q^61 + 3636 * q^64 + 2198 * q^67 - 2200 * q^70 + 2176 * q^73 - 6936 * q^76 + 1862 * q^79 + 5436 * q^82 + 890 * q^85 + 3528 * q^88 - 3104 * q^94 + 4370 * q^97

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 $

$2$	−3.69212	−1.30536	−0.652680	−	0.757634i	$-0.726355\pi$
$2$	−3.69212	−1.30536	−0.652680	+	0.757634i	$0.726355\pi$
$3$	0	0
$3$	0	0
$4$	5.63172	0.703965
$5$	18.7574	1.67772	0.838858	−	0.544351i	$-0.183224\pi$
$5$	18.7574	1.67772	0.838858	+	0.544351i	$0.183224\pi$
$6$	0	0
$7$	24.1383	1.30334	0.651672	−	0.758500i	$-0.274068\pi$
$7$	24.1383	1.30334	0.651672	+	0.758500i	$0.274068\pi$
$8$	8.74396	0.386432
$9$	0	0
$10$	−69.2546	−2.19002
$11$	49.2741	1.35061	0.675305	−	0.737539i	$-0.264012\pi$
$11$	49.2741	1.35061	0.675305	+	0.737539i	$0.264012\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−89.1214	−1.70133
$15$	0	0
$16$	−77.3375	−1.20840
$17$	−65.3121	−0.931795	−0.465897	−	0.884839i	$-0.654268\pi$
$17$	−65.3121	−0.931795	−0.465897	+	0.884839i	$0.654268\pi$
$18$	0	0
$19$	−109.940	−1.32747	−0.663737	−	0.747966i	$-0.731030\pi$
$19$	−109.940	−1.32747	−0.663737	+	0.747966i	$0.731030\pi$
$20$	105.637	1.18105
$21$	0	0
$22$	−181.926	−1.76303
$23$	−83.2974	−0.755161	−0.377581	−	0.925977i	$-0.623244\pi$
$23$	−83.2974	−0.755161	−0.377581	+	0.925977i	$0.623244\pi$
$24$	0	0
$25$	226.841	1.81473
$26$	0	0
$27$	0	0
$28$	135.940	0.917509
$29$	4.99593	0.0319904	0.0159952	−	0.999872i	$-0.494908\pi$
$29$	4.99593	0.0319904	0.0159952	+	0.999872i	$0.494908\pi$
$30$	0	0
$31$	255.810	1.48209	0.741047	−	0.671453i	$-0.234330\pi$
$31$	255.810	1.48209	0.741047	+	0.671453i	$0.234330\pi$
$32$	215.587	1.19096
$33$	0	0
$34$	241.140	1.21633
$35$	452.772	2.18664
$36$	0	0
$37$	−93.6356	−0.416043	−0.208022	−	0.978124i	$-0.566702\pi$
$37$	−93.6356	−0.416043	−0.208022	+	0.978124i	$0.566702\pi$
$38$	405.912	1.73283
$39$	0	0
$40$	164.014	0.648323
$41$	−67.9487	−0.258825	−0.129412	−	0.991591i	$-0.541309\pi$
$41$	−67.9487	−0.258825	−0.129412	+	0.991591i	$0.541309\pi$
$42$	0	0
$43$	142.543	0.505527	0.252763	−	0.967528i	$-0.418661\pi$
$43$	142.543	0.505527	0.252763	+	0.967528i	$0.418661\pi$
$44$	277.498	0.950782
$45$	0	0
$46$	307.544	0.985758
$47$	379.275	1.17708	0.588541	−	0.808467i	$-0.299702\pi$
$47$	379.275	1.17708	0.588541	+	0.808467i	$0.299702\pi$
$48$	0	0
$49$	239.657	0.698708
$50$	−837.524	−2.36888
$51$	0	0
$52$	0	0
$53$	−389.560	−1.00963	−0.504813	−	0.863229i	$-0.668438\pi$
$53$	−389.560	−1.00963	−0.504813	+	0.863229i	$0.668438\pi$
$54$	0	0
$55$	924.256	2.26594
$56$	211.064	0.503654
$57$	0	0
$58$	−18.4456	−0.0417590
$59$	−133.881	−0.295420	−0.147710	−	0.989031i	$-0.547190\pi$
$59$	−133.881	−0.295420	−0.147710	+	0.989031i	$0.547190\pi$
$60$	0	0
$61$	620.792	1.30302	0.651510	−	0.758640i	$-0.274136\pi$
$61$	620.792	1.30302	0.651510	+	0.758640i	$0.274136\pi$
$62$	−944.482	−1.93467
$63$	0	0
$64$	−177.273	−0.346237
$65$	0	0
$66$	0	0
$67$	−119.002	−0.216991	−0.108496	−	0.994097i	$-0.534603\pi$
$67$	−119.002	−0.216991	−0.108496	+	0.994097i	$0.534603\pi$
$68$	−367.819	−0.655951
$69$	0	0
$70$	−1671.69	−2.85436
$71$	−361.386	−0.604066	−0.302033	−	0.953298i	$-0.597665\pi$
$71$	−361.386	−0.604066	−0.302033	+	0.953298i	$0.597665\pi$
$72$	0	0
$73$	748.241	1.19966	0.599829	−	0.800128i	$-0.295235\pi$
$73$	748.241	1.19966	0.599829	+	0.800128i	$0.295235\pi$
$74$	345.714	0.543086
$75$	0	0
$76$	−619.152	−0.934495
$77$	1189.39	1.76031
$78$	0	0
$79$	514.165	0.732254	0.366127	−	0.930565i	$-0.380684\pi$
$79$	514.165	0.732254	0.366127	+	0.930565i	$0.380684\pi$
$80$	−1450.65	−2.02735
$81$	0	0
$82$	250.875	0.337859
$83$	−260.260	−0.344183	−0.172092	−	0.985081i	$-0.555053\pi$
$83$	−260.260	−0.344183	−0.172092	+	0.985081i	$0.555053\pi$
$84$	0	0
$85$	−1225.09	−1.56329
$86$	−526.286	−0.659894
$87$	0	0
$88$	430.851	0.521919
$89$	833.631	0.992861	0.496431	−	0.868076i	$-0.334644\pi$
$89$	833.631	0.992861	0.496431	+	0.868076i	$0.334644\pi$
$90$	0	0
$91$	0	0
$92$	−469.108	−0.531607
$93$	0	0
$94$	−1400.33	−1.53652
$95$	−2062.19	−2.22712
$96$	0	0
$97$	1481.99	1.55127	0.775634	−	0.631183i	$-0.217430\pi$
$97$	1481.99	1.55127	0.775634	+	0.631183i	$0.217430\pi$
$98$	−884.841	−0.912066
$99$	0	0
$100$	1277.51	1.27751
$101$	1740.21	1.71443	0.857217	−	0.514956i	$-0.172192\pi$
$101$	1740.21	1.71443	0.857217	+	0.514956i	$0.172192\pi$
$102$	0	0
$103$	1020.16	0.975917	0.487959	−	0.872867i	$-0.337742\pi$
$103$	1020.16	0.975917	0.487959	+	0.872867i	$0.337742\pi$
$104$	0	0
$105$	0	0
$106$	1438.30	1.31792
$107$	−1507.96	−1.36243	−0.681215	−	0.732084i	$-0.738548\pi$
$107$	−1507.96	−1.36243	−0.681215	+	0.732084i	$0.738548\pi$
$108$	0	0
$109$	1074.08	0.943840	0.471920	−	0.881641i	$-0.343561\pi$
$109$	1074.08	0.943840	0.471920	+	0.881641i	$0.343561\pi$
$110$	−3412.46	−2.95787
$111$	0	0
$112$	−1866.79	−1.57496
$113$	891.337	0.742035	0.371018	−	0.928626i	$-0.379009\pi$
$113$	891.337	0.742035	0.371018	+	0.928626i	$0.379009\pi$
$114$	0	0
$115$	−1562.45	−1.26695
$116$	28.1357	0.0225201
$117$	0	0
$118$	494.304	0.385630
$119$	−1576.52	−1.21445
$120$	0	0
$121$	1096.94	0.824146
$122$	−2292.03	−1.70091
$123$	0	0
$124$	1440.65	1.04334
$125$	1910.28	1.36688
$126$	0	0
$127$	−6.55994	−0.00458347	−0.00229173	−	0.999997i	$-0.500729\pi$
$127$	−6.55994	−0.00458347	−0.00229173	+	0.999997i	$0.500729\pi$
$128$	−1070.18	−0.738999
$129$	0	0
$130$	0	0
$131$	−1267.44	−0.845320	−0.422660	−	0.906288i	$-0.638903\pi$
$131$	−1267.44	−0.845320	−0.422660	+	0.906288i	$0.638903\pi$
$132$	0	0
$133$	−2653.77	−1.73016
$134$	439.369	0.283252
$135$	0	0
$136$	−571.086	−0.360075
$137$	1350.85	0.842416	0.421208	−	0.906964i	$-0.361606\pi$
$137$	1350.85	0.842416	0.421208	+	0.906964i	$0.361606\pi$
$138$	0	0
$139$	−990.606	−0.604476	−0.302238	−	0.953233i	$-0.597734\pi$
$139$	−990.606	−0.604476	−0.302238	+	0.953233i	$0.597734\pi$
$140$	2549.89	1.53932
$141$	0	0
$142$	1334.28	0.788523
$143$	0	0
$144$	0	0
$145$	93.7108	0.0536708
$146$	−2762.59	−1.56599
$147$	0	0
$148$	−527.330	−0.292880
$149$	−2132.38	−1.17243	−0.586214	−	0.810156i	$-0.699382\pi$
$149$	−2132.38	−1.17243	−0.586214	+	0.810156i	$0.699382\pi$
$150$	0	0
$151$	251.734	0.135668	0.0678338	−	0.997697i	$-0.478391\pi$
$151$	251.734	0.135668	0.0678338	+	0.997697i	$0.478391\pi$
$152$	−961.312	−0.512978
$153$	0	0
$154$	−4391.38	−2.29784
$155$	4798.35	2.48653
$156$	0	0
$157$	−3686.27	−1.87386	−0.936931	−	0.349515i	$-0.886346\pi$
$157$	−3686.27	−1.87386	−0.936931	+	0.349515i	$0.886346\pi$
$158$	−1898.36	−0.955855
$159$	0	0
$160$	4043.86	1.99810
$161$	−2010.66	−0.984236
$162$	0	0
$163$	1267.79	0.609207	0.304604	−	0.952479i	$-0.401476\pi$
$163$	1267.79	0.609207	0.304604	+	0.952479i	$0.401476\pi$
$164$	−382.668	−0.182204
$165$	0	0
$166$	960.910	0.449283
$167$	938.997	0.435100	0.217550	−	0.976049i	$-0.430193\pi$
$167$	938.997	0.435100	0.217550	+	0.976049i	$0.430193\pi$
$168$	0	0
$169$	0	0
$170$	4523.16	2.04065
$171$	0	0
$172$	802.764	0.355873
$173$	1821.44	0.800472	0.400236	−	0.916412i	$-0.368928\pi$
$173$	1821.44	0.800472	0.400236	+	0.916412i	$0.368928\pi$
$174$	0	0
$175$	5475.56	2.36522
$176$	−3810.74	−1.63207
$177$	0	0
$178$	−3077.86	−1.29604
$179$	−645.740	−0.269636	−0.134818	−	0.990870i	$-0.543045\pi$
$179$	−645.740	−0.269636	−0.134818	+	0.990870i	$0.543045\pi$
$180$	0	0
$181$	2387.31	0.980370	0.490185	−	0.871618i	$-0.336929\pi$
$181$	2387.31	0.980370	0.490185	+	0.871618i	$0.336929\pi$
$182$	0	0
$183$	0	0
$184$	−728.349	−0.291819
$185$	−1756.36	−0.698002
$186$	0	0
$187$	−3218.19	−1.25849
$188$	2135.97	0.828625
$189$	0	0
$190$	7613.86	2.90720
$191$	−1126.36	−0.426703	−0.213351	−	0.976976i	$-0.568438\pi$
$191$	−1126.36	−0.426703	−0.213351	+	0.976976i	$0.568438\pi$
$192$	0	0
$193$	3258.63	1.21534	0.607672	−	0.794188i	$-0.292104\pi$
$193$	3258.63	1.21534	0.607672	+	0.794188i	$0.292104\pi$
$194$	−5471.67	−2.02496
$195$	0	0
$196$	1349.68	0.491866
$197$	179.494	0.0649159	0.0324580	−	0.999473i	$-0.489666\pi$
$197$	179.494	0.0649159	0.0324580	+	0.999473i	$0.489666\pi$
$198$	0	0
$199$	2953.42	1.05207	0.526036	−	0.850462i	$-0.323678\pi$
$199$	2953.42	1.05207	0.526036	+	0.850462i	$0.323678\pi$
$200$	1983.49	0.701270
$201$	0	0
$202$	−6425.07	−2.23795
$203$	120.593	0.0416945
$204$	0	0
$205$	−1274.54	−0.434234
$206$	−3766.55	−1.27392
$207$	0	0
$208$	0	0
$209$	−5417.20	−1.79290
$210$	0	0
$211$	3128.41	1.02070	0.510352	−	0.859965i	$-0.329515\pi$
$211$	3128.41	1.02070	0.510352	+	0.859965i	$0.329515\pi$
$212$	−2193.89	−0.710741
$213$	0	0
$214$	5567.56	1.77846
$215$	2673.75	0.848130
$216$	0	0
$217$	6174.83	1.93168
$218$	−3965.64	−1.23205
$219$	0	0
$220$	5205.15	1.59514
$221$	0	0
$222$	0	0
$223$	3754.12	1.12733	0.563665	−	0.826004i	$-0.309391\pi$
$223$	3754.12	1.12733	0.563665	+	0.826004i	$0.309391\pi$
$224$	5203.91	1.55224
$225$	0	0
$226$	−3290.92	−0.968623
$227$	−1549.52	−0.453062	−0.226531	−	0.974004i	$-0.572738\pi$
$227$	−1549.52	−0.453062	−0.226531	+	0.974004i	$0.572738\pi$
$228$	0	0
$229$	−1364.84	−0.393847	−0.196924	−	0.980419i	$-0.563095\pi$
$229$	−1364.84	−0.393847	−0.196924	+	0.980419i	$0.563095\pi$
$230$	5768.73	1.65382
$231$	0	0
$232$	43.6842	0.0123621
$233$	−1665.69	−0.468339	−0.234170	−	0.972196i	$-0.575237\pi$
$233$	−1665.69	−0.468339	−0.234170	+	0.972196i	$0.575237\pi$
$234$	0	0
$235$	7114.22	1.97481
$236$	−753.980	−0.207966
$237$	0	0
$238$	5820.70	1.58529
$239$	5950.06	1.61037	0.805183	−	0.593026i	$-0.202067\pi$
$239$	5950.06	1.61037	0.805183	+	0.593026i	$0.202067\pi$
$240$	0	0
$241$	1707.01	0.456257	0.228128	−	0.973631i	$-0.426739\pi$
$241$	1707.01	0.456257	0.228128	+	0.973631i	$0.426739\pi$
$242$	−4050.02	−1.07581
$243$	0	0
$244$	3496.12	0.917281
$245$	4495.35	1.17223
$246$	0	0
$247$	0	0
$248$	2236.80	0.572729
$249$	0	0
$250$	−7052.97	−1.78428
$251$	−306.390	−0.0770484	−0.0385242	−	0.999258i	$-0.512266\pi$
$251$	−306.390	−0.0770484	−0.0385242	+	0.999258i	$0.512266\pi$
$252$	0	0
$253$	−4104.41	−1.01993
$254$	24.2200	0.00598307
$255$	0	0
$256$	5369.43	1.31090
$257$	−3556.16	−0.863140	−0.431570	−	0.902079i	$-0.642040\pi$
$257$	−3556.16	−0.863140	−0.431570	+	0.902079i	$0.642040\pi$
$258$	0	0
$259$	−2260.20	−0.542248
$260$	0	0
$261$	0	0
$262$	4679.54	1.10345
$263$	7121.35	1.66966	0.834831	−	0.550506i	$-0.185565\pi$
$263$	7121.35	1.66966	0.834831	+	0.550506i	$0.185565\pi$
$264$	0	0
$265$	−7307.14	−1.69386
$266$	9798.01	2.25848
$267$	0	0
$268$	−670.186	−0.152754
$269$	−6419.06	−1.45493	−0.727467	−	0.686143i	$-0.759303\pi$
$269$	−6419.06	−1.45493	−0.727467	+	0.686143i	$0.759303\pi$
$270$	0	0
$271$	−8631.33	−1.93474	−0.967372	−	0.253360i	$-0.918464\pi$
$271$	−8631.33	−1.93474	−0.967372	+	0.253360i	$0.918464\pi$
$272$	5051.07	1.12598
$273$	0	0
$274$	−4987.50	−1.09966
$275$	11177.4	2.45099
$276$	0	0
$277$	727.474	0.157797	0.0788983	−	0.996883i	$-0.474860\pi$
$277$	727.474	0.157797	0.0788983	+	0.996883i	$0.474860\pi$
$278$	3657.43	0.789059
$279$	0	0
$280$	3959.02	0.844989
$281$	5588.39	1.18639	0.593194	−	0.805059i	$-0.297867\pi$
$281$	5588.39	1.18639	0.593194	+	0.805059i	$0.297867\pi$
$282$	0	0
$283$	−1432.76	−0.300949	−0.150474	−	0.988614i	$-0.548080\pi$
$283$	−1432.76	−0.300949	−0.150474	+	0.988614i	$0.548080\pi$
$284$	−2035.23	−0.425241
$285$	0	0
$286$	0	0
$287$	−1640.17	−0.337338
$288$	0	0
$289$	−647.332	−0.131759
$290$	−345.991	−0.0700597
$291$	0	0
$292$	4213.89	0.844517
$293$	8274.51	1.64984	0.824918	−	0.565253i	$-0.191221\pi$
$293$	8274.51	1.64984	0.824918	+	0.565253i	$0.191221\pi$
$294$	0	0
$295$	−2511.26	−0.495631
$296$	−818.746	−0.160772
$297$	0	0
$298$	7873.01	1.53044
$299$	0	0
$300$	0	0
$301$	3440.75	0.658876
$302$	−929.430	−0.177095
$303$	0	0
$304$	8502.49	1.60412
$305$	11644.5	2.18610
$306$	0	0
$307$	−628.477	−0.116837	−0.0584187	−	0.998292i	$-0.518606\pi$
$307$	−628.477	−0.116837	−0.0584187	+	0.998292i	$0.518606\pi$
$308$	6698.33	1.23920
$309$	0	0
$310$	−17716.1	−3.24582
$311$	−85.8693	−0.0156566	−0.00782830	−	0.999969i	$-0.502492\pi$
$311$	−85.8693	−0.0156566	−0.00782830	+	0.999969i	$0.502492\pi$
$312$	0	0
$313$	−2279.49	−0.411643	−0.205821	−	0.978590i	$-0.565987\pi$
$313$	−2279.49	−0.411643	−0.205821	+	0.978590i	$0.565987\pi$
$314$	13610.1	2.44606
$315$	0	0
$316$	2895.63	0.515481
$317$	−6576.19	−1.16516	−0.582580	−	0.812774i	$-0.697957\pi$
$317$	−6576.19	−1.16516	−0.582580	+	0.812774i	$0.697957\pi$
$318$	0	0
$319$	246.170	0.0432065
$320$	−3325.19	−0.580887
$321$	0	0
$322$	7423.58	1.28478
$323$	7180.42	1.23693
$324$	0	0
$325$	0	0
$326$	−4680.82	−0.795235
$327$	0	0
$328$	−594.141	−0.100018
$329$	9155.04	1.53414
$330$	0	0
$331$	−2493.35	−0.414039	−0.207020	−	0.978337i	$-0.566376\pi$
$331$	−2493.35	−0.414039	−0.207020	+	0.978337i	$0.566376\pi$
$332$	−1465.71	−0.242293
$333$	0	0
$334$	−3466.89	−0.567963
$335$	−2232.17	−0.364049
$336$	0	0
$337$	−2089.60	−0.337767	−0.168884	−	0.985636i	$-0.554016\pi$
$337$	−2089.60	−0.337767	−0.168884	+	0.985636i	$0.554016\pi$
$338$	0	0
$339$	0	0
$340$	−6899.35	−1.10050
$341$	12604.8	2.00173
$342$	0	0
$343$	−2494.53	−0.392687
$344$	1246.39	0.195352
$345$	0	0
$346$	−6724.97	−1.04490
$347$	−6468.43	−1.00070	−0.500351	−	0.865823i	$-0.666796\pi$
$347$	−6468.43	−1.00070	−0.500351	+	0.865823i	$0.666796\pi$
$348$	0	0
$349$	−5246.94	−0.804763	−0.402382	−	0.915472i	$-0.631817\pi$
$349$	−5246.94	−0.804763	−0.402382	+	0.915472i	$0.631817\pi$
$350$	−20216.4	−3.08746
$351$	0	0
$352$	10622.9	1.60853
$353$	−10269.9	−1.54847	−0.774235	−	0.632899i	$-0.781865\pi$
$353$	−10269.9	−1.54847	−0.774235	+	0.632899i	$0.781865\pi$
$354$	0	0
$355$	−6778.68	−1.01345
$356$	4694.78	0.698940
$357$	0	0
$358$	2384.15	0.351972
$359$	−3200.22	−0.470477	−0.235238	−	0.971938i	$-0.575587\pi$
$359$	−3200.22	−0.470477	−0.235238	+	0.971938i	$0.575587\pi$
$360$	0	0
$361$	5227.83	0.762185
$362$	−8814.21	−1.27974
$363$	0	0
$364$	0	0
$365$	14035.1	2.01268
$366$	0	0
$367$	22.9581	0.00326540	0.00163270	−	0.999999i	$-0.499480\pi$
$367$	22.9581	0.00326540	0.00163270	+	0.999999i	$0.499480\pi$
$368$	6442.01	0.912536
$369$	0	0
$370$	6484.70	0.911144
$371$	−9403.30	−1.31589
$372$	0	0
$373$	6194.91	0.859947	0.429973	−	0.902842i	$-0.358523\pi$
$373$	6194.91	0.859947	0.429973	+	0.902842i	$0.358523\pi$
$374$	11881.9	1.64278
$375$	0	0
$376$	3316.36	0.454863
$377$	0	0
$378$	0	0
$379$	−6955.24	−0.942656	−0.471328	−	0.881958i	$-0.656225\pi$
$379$	−6955.24	−0.942656	−0.471328	+	0.881958i	$0.656225\pi$
$380$	−11613.7	−1.56782
$381$	0	0
$382$	4158.64	0.557001
$383$	−1221.06	−0.162907	−0.0814534	−	0.996677i	$-0.525956\pi$
$383$	−1221.06	−0.162907	−0.0814534	+	0.996677i	$0.525956\pi$
$384$	0	0
$385$	22309.9	2.95330
$386$	−12031.2	−1.58646
$387$	0	0
$388$	8346.13	1.09204
$389$	−6318.58	−0.823560	−0.411780	−	0.911283i	$-0.635093\pi$
$389$	−6318.58	−0.823560	−0.411780	+	0.911283i	$0.635093\pi$
$390$	0	0
$391$	5440.33	0.703655
$392$	2095.55	0.270003
$393$	0	0
$394$	−662.714	−0.0847387
$395$	9644.41	1.22851
$396$	0	0
$397$	−3085.74	−0.390098	−0.195049	−	0.980793i	$-0.562487\pi$
$397$	−3085.74	−0.390098	−0.195049	+	0.980793i	$0.562487\pi$
$398$	−10904.4	−1.37333
$399$	0	0
$400$	−17543.3	−2.19292
$401$	10780.0	1.34246	0.671231	−	0.741248i	$-0.265766\pi$
$401$	10780.0	1.34246	0.671231	+	0.741248i	$0.265766\pi$
$402$	0	0
$403$	0	0
$404$	9800.40	1.20690
$405$	0	0
$406$	−445.244	−0.0544264
$407$	−4613.81	−0.561912
$408$	0	0
$409$	−7633.26	−0.922837	−0.461418	−	0.887183i	$-0.652659\pi$
$409$	−7633.26	−0.922837	−0.461418	+	0.887183i	$0.652659\pi$
$410$	4705.76	0.566832
$411$	0	0
$412$	5745.26	0.687012
$413$	−3231.65	−0.385035
$414$	0	0
$415$	−4881.81	−0.577442
$416$	0	0
$417$	0	0
$418$	20000.9	2.34038
$419$	3508.69	0.409095	0.204547	−	0.978857i	$-0.434428\pi$
$419$	3508.69	0.409095	0.204547	+	0.978857i	$0.434428\pi$
$420$	0	0
$421$	−12477.7	−1.44448	−0.722241	−	0.691642i	$-0.756888\pi$
$421$	−12477.7	−1.44448	−0.722241	+	0.691642i	$0.756888\pi$
$422$	−11550.5	−1.33239
$423$	0	0
$424$	−3406.29	−0.390152
$425$	−14815.5	−1.69096
$426$	0	0
$427$	14984.8	1.69828
$428$	−8492.41	−0.959103
$429$	0	0
$430$	−9871.78	−1.10711
$431$	17745.5	1.98322	0.991612	−	0.129249i	$-0.0412567\pi$
$431$	17745.5	1.98322	0.991612	+	0.129249i	$0.0412567\pi$
$432$	0	0
$433$	9696.17	1.07614	0.538069	−	0.842901i	$-0.319154\pi$
$433$	9696.17	1.07614	0.538069	+	0.842901i	$0.319154\pi$
$434$	−22798.2	−2.52154
$435$	0	0
$436$	6048.94	0.664430
$437$	9157.73	1.00246
$438$	0	0
$439$	13398.1	1.45662	0.728308	−	0.685250i	$-0.240307\pi$
$439$	13398.1	1.45662	0.728308	+	0.685250i	$0.240307\pi$
$440$	8081.66	0.875631
$441$	0	0
$442$	0	0
$443$	13630.2	1.46183	0.730913	−	0.682471i	$-0.239095\pi$
$443$	13630.2	1.46183	0.730913	+	0.682471i	$0.239095\pi$
$444$	0	0
$445$	15636.8	1.66574
$446$	−13860.6	−1.47157
$447$	0	0
$448$	−4279.08	−0.451266
$449$	−1381.79	−0.145235	−0.0726176	−	0.997360i	$-0.523135\pi$
$449$	−1381.79	−0.145235	−0.0726176	+	0.997360i	$0.523135\pi$
$450$	0	0
$451$	−3348.11	−0.349571
$452$	5019.76	0.522367
$453$	0	0
$454$	5720.99	0.591409
$455$	0	0
$456$	0	0
$457$	8261.60	0.845648	0.422824	−	0.906212i	$-0.361039\pi$
$457$	8261.60	0.845648	0.422824	+	0.906212i	$0.361039\pi$
$458$	5039.14	0.514113
$459$	0	0
$460$	−8799.26	−0.891886
$461$	−16874.2	−1.70480	−0.852399	−	0.522892i	$-0.824853\pi$
$461$	−16874.2	−1.70480	−0.852399	+	0.522892i	$0.824853\pi$
$462$	0	0
$463$	−4466.93	−0.448371	−0.224186	−	0.974546i	$-0.571972\pi$
$463$	−4466.93	−0.448371	−0.224186	+	0.974546i	$0.571972\pi$
$464$	−386.373	−0.0386571
$465$	0	0
$466$	6149.92	0.611351
$467$	−2731.37	−0.270649	−0.135324	−	0.990801i	$-0.543208\pi$
$467$	−2731.37	−0.270649	−0.135324	+	0.990801i	$0.543208\pi$
$468$	0	0
$469$	−2872.50	−0.282814
$470$	−26266.5	−2.57784
$471$	0	0
$472$	−1170.65	−0.114160
$473$	7023.69	0.682769
$474$	0	0
$475$	−24938.9	−2.40900
$476$	−8878.53	−0.854930
$477$	0	0
$478$	−21968.3	−2.10211
$479$	−5630.91	−0.537124	−0.268562	−	0.963262i	$-0.586548\pi$
$479$	−5630.91	−0.537124	−0.268562	+	0.963262i	$0.586548\pi$
$480$	0	0
$481$	0	0
$482$	−6302.46	−0.595580
$483$	0	0
$484$	6177.65	0.580170
$485$	27798.3	2.60259
$486$	0	0
$487$	12910.5	1.20129	0.600647	−	0.799514i	$-0.294910\pi$
$487$	12910.5	1.20129	0.600647	+	0.799514i	$0.294910\pi$
$488$	5428.18	0.503529
$489$	0	0
$490$	−16597.3	−1.53019
$491$	3114.67	0.286279	0.143139	−	0.989703i	$-0.454280\pi$
$491$	3114.67	0.286279	0.143139	+	0.989703i	$0.454280\pi$
$492$	0	0
$493$	−326.295	−0.0298085
$494$	0	0
$495$	0	0
$496$	−19783.7	−1.79096
$497$	−8723.24	−0.787306
$498$	0	0
$499$	11637.7	1.04404	0.522018	−	0.852934i	$-0.325179\pi$
$499$	11637.7	1.04404	0.522018	+	0.852934i	$0.325179\pi$
$500$	10758.2	0.962239
$501$	0	0
$502$	1131.23	0.100576
$503$	2832.94	0.251122	0.125561	−	0.992086i	$-0.459927\pi$
$503$	2832.94	0.251122	0.125561	+	0.992086i	$0.459927\pi$
$504$	0	0
$505$	32641.9	2.87633
$506$	15153.9	1.33137
$507$	0	0
$508$	−36.9437	−0.00322660
$509$	7699.95	0.670519	0.335260	−	0.942126i	$-0.391176\pi$
$509$	7699.95	0.670519	0.335260	+	0.942126i	$0.391176\pi$
$510$	0	0
$511$	18061.3	1.56357
$512$	−11263.1	−0.972193
$513$	0	0
$514$	13129.8	1.12671
$515$	19135.6	1.63731
$516$	0	0
$517$	18688.4	1.58978
$518$	8344.93	0.707829
$519$	0	0
$520$	0	0
$521$	−14688.2	−1.23513	−0.617565	−	0.786520i	$-0.711881\pi$
$521$	−14688.2	−1.23513	−0.617565	+	0.786520i	$0.711881\pi$
$522$	0	0
$523$	−8895.47	−0.743732	−0.371866	−	0.928286i	$-0.621282\pi$
$523$	−8895.47	−0.743732	−0.371866	+	0.928286i	$0.621282\pi$
$524$	−7137.88	−0.595076
$525$	0	0
$526$	−26292.9	−2.17951
$527$	−16707.5	−1.38101
$528$	0	0
$529$	−5228.54	−0.429731
$530$	26978.8	2.21110
$531$	0	0
$532$	−14945.3	−1.21797
$533$	0	0
$534$	0	0
$535$	−28285.4	−2.28577
$536$	−1040.55	−0.0838523
$537$	0	0
$538$	23699.9	1.89921
$539$	11808.9	0.943682
$540$	0	0
$541$	−20667.7	−1.64246	−0.821232	−	0.570594i	$-0.806713\pi$
$541$	−20667.7	−1.64246	−0.821232	+	0.570594i	$0.806713\pi$
$542$	31867.9	2.52554
$543$	0	0
$544$	−14080.5	−1.10973
$545$	20147.1	1.58350
$546$	0	0
$547$	−903.226	−0.0706018	−0.0353009	−	0.999377i	$-0.511239\pi$
$547$	−903.226	−0.0706018	−0.0353009	+	0.999377i	$0.511239\pi$
$548$	7607.61	0.593031
$549$	0	0
$550$	−41268.3	−3.19943
$551$	−549.253	−0.0424664
$552$	0	0
$553$	12411.1	0.954380
$554$	−2685.92	−0.205981
$555$	0	0
$556$	−5578.82	−0.425530
$557$	−19269.0	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$557$	−19269.0	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$558$	0	0
$559$	0	0
$560$	−35016.3	−2.64233
$561$	0	0
$562$	−20633.0	−1.54866
$563$	−22628.3	−1.69391	−0.846953	−	0.531667i	$-0.821566\pi$
$563$	−22628.3	−1.69391	−0.846953	+	0.531667i	$0.821566\pi$
$564$	0	0
$565$	16719.2	1.24492
$566$	5289.90	0.392847
$567$	0	0
$568$	−3159.95	−0.233430
$569$	−25682.6	−1.89222	−0.946109	−	0.323850i	$-0.895023\pi$
$569$	−25682.6	−1.89222	−0.946109	+	0.323850i	$0.895023\pi$
$570$	0	0
$571$	301.979	0.0221321	0.0110660	−	0.999939i	$-0.496477\pi$
$571$	301.979	0.0221321	0.0110660	+	0.999939i	$0.496477\pi$
$572$	0	0
$573$	0	0
$574$	6055.68	0.440347
$575$	−18895.3	−1.37041
$576$	0	0
$577$	17642.4	1.27290	0.636449	−	0.771319i	$-0.280402\pi$
$577$	17642.4	1.27290	0.636449	+	0.771319i	$0.280402\pi$
$578$	2390.02	0.171993
$579$	0	0
$580$	527.753	0.0377824
$581$	−6282.23	−0.448590
$582$	0	0
$583$	−19195.2	−1.36361
$584$	6542.59	0.463586
$585$	0	0
$586$	−30550.4	−2.15363
$587$	−22820.1	−1.60458	−0.802288	−	0.596937i	$-0.796384\pi$
$587$	−22820.1	−1.60458	−0.802288	+	0.596937i	$0.796384\pi$
$588$	0	0
$589$	−28123.8	−1.96744
$590$	9271.87	0.646977
$591$	0	0
$592$	7241.54	0.502746
$593$	−13688.7	−0.947940	−0.473970	−	0.880541i	$-0.657179\pi$
$593$	−13688.7	−0.947940	−0.473970	+	0.880541i	$0.657179\pi$
$594$	0	0
$595$	−29571.5	−2.03750
$596$	−12009.0	−0.825348
$597$	0	0
$598$	0	0
$599$	20276.7	1.38311	0.691556	−	0.722323i	$-0.256925\pi$
$599$	20276.7	1.38311	0.691556	+	0.722323i	$0.256925\pi$
$600$	0	0
$601$	3596.78	0.244119	0.122060	−	0.992523i	$-0.461050\pi$
$601$	3596.78	0.244119	0.122060	+	0.992523i	$0.461050\pi$
$602$	−12703.6	−0.860070
$603$	0	0
$604$	1417.69	0.0955052
$605$	20575.7	1.38268
$606$	0	0
$607$	5556.39	0.371543	0.185772	−	0.982593i	$-0.440522\pi$
$607$	5556.39	0.371543	0.185772	+	0.982593i	$0.440522\pi$
$608$	−23701.7	−1.58097
$609$	0	0
$610$	−42992.7	−2.85364
$611$	0	0
$612$	0	0
$613$	−5608.79	−0.369555	−0.184777	−	0.982780i	$-0.559156\pi$
$613$	−5608.79	−0.369555	−0.184777	+	0.982780i	$0.559156\pi$
$614$	2320.41	0.152515
$615$	0	0
$616$	10400.0	0.680240
$617$	−23125.5	−1.50891	−0.754454	−	0.656353i	$-0.772098\pi$
$617$	−23125.5	−1.50891	−0.754454	+	0.656353i	$0.772098\pi$
$618$	0	0
$619$	−11536.0	−0.749065	−0.374533	−	0.927214i	$-0.622197\pi$
$619$	−11536.0	−0.749065	−0.374533	+	0.927214i	$0.622197\pi$
$620$	27022.9	1.75043
$621$	0	0
$622$	317.039	0.0204375
$623$	20122.4	1.29404
$624$	0	0
$625$	7476.78	0.478514
$626$	8416.13	0.537342
$627$	0	0
$628$	−20760.0	−1.31913
$629$	6115.54	0.387667
$630$	0	0
$631$	−23681.2	−1.49403	−0.747016	−	0.664806i	$-0.768514\pi$
$631$	−23681.2	−1.49403	−0.747016	+	0.664806i	$0.768514\pi$
$632$	4495.84	0.282967
$633$	0	0
$634$	24280.0	1.52095
$635$	−123.048	−0.00768975
$636$	0	0
$637$	0	0
$638$	−908.888	−0.0564001
$639$	0	0
$640$	−20073.9	−1.23983
$641$	14221.5	0.876312	0.438156	−	0.898899i	$-0.355632\pi$
$641$	14221.5	0.876312	0.438156	+	0.898899i	$0.355632\pi$
$642$	0	0
$643$	−16022.4	−0.982679	−0.491340	−	0.870968i	$-0.663493\pi$
$643$	−16022.4	−0.982679	−0.491340	+	0.870968i	$0.663493\pi$
$644$	−11323.5	−0.692868
$645$	0	0
$646$	−26510.9	−1.61464
$647$	−25539.8	−1.55189	−0.775945	−	0.630801i	$-0.782726\pi$
$647$	−25539.8	−1.55189	−0.775945	+	0.630801i	$0.782726\pi$
$648$	0	0
$649$	−6596.86	−0.398998
$650$	0	0
$651$	0	0
$652$	7139.83	0.428861
$653$	11206.7	0.671597	0.335798	−	0.941934i	$-0.390994\pi$
$653$	11206.7	0.671597	0.335798	+	0.941934i	$0.390994\pi$
$654$	0	0
$655$	−23773.9	−1.41821
$656$	5254.99	0.312763
$657$	0	0
$658$	−33801.5	−2.00261
$659$	−2760.89	−0.163200	−0.0816002	−	0.996665i	$-0.526003\pi$
$659$	−2760.89	−0.163200	−0.0816002	+	0.996665i	$0.526003\pi$
$660$	0	0
$661$	29868.4	1.75756	0.878781	−	0.477226i	$-0.158358\pi$
$661$	29868.4	1.75756	0.878781	+	0.477226i	$0.158358\pi$
$662$	9205.74	0.540470
$663$	0	0
$664$	−2275.70	−0.133004
$665$	−49777.8	−2.90271
$666$	0	0
$667$	−416.148	−0.0241579
$668$	5288.17	0.306296
$669$	0	0
$670$	8241.44	0.475216
$671$	30589.0	1.75987
$672$	0	0
$673$	−2238.65	−0.128222	−0.0641110	−	0.997943i	$-0.520421\pi$
$673$	−2238.65	−0.128222	−0.0641110	+	0.997943i	$0.520421\pi$
$674$	7715.03	0.440908
$675$	0	0
$676$	0	0
$677$	−27574.0	−1.56537	−0.782685	−	0.622418i	$-0.786151\pi$
$677$	−27574.0	−1.56537	−0.782685	+	0.622418i	$0.786151\pi$
$678$	0	0
$679$	35772.6	2.02184
$680$	−10712.1	−0.604104
$681$	0	0
$682$	−46538.5	−2.61298
$683$	−1862.60	−0.104349	−0.0521745	−	0.998638i	$-0.516615\pi$
$683$	−1862.60	−0.104349	−0.0521745	+	0.998638i	$0.516615\pi$
$684$	0	0
$685$	25338.5	1.41333
$686$	9210.08	0.512598
$687$	0	0
$688$	−11023.9	−0.610877
$689$	0	0
$690$	0	0
$691$	5999.62	0.330299	0.165149	−	0.986269i	$-0.447189\pi$
$691$	5999.62	0.330299	0.165149	+	0.986269i	$0.447189\pi$
$692$	10257.8	0.563504
$693$	0	0
$694$	23882.2	1.30628
$695$	−18581.2	−1.01414
$696$	0	0
$697$	4437.87	0.241171
$698$	19372.3	1.05051
$699$	0	0
$700$	30836.8	1.66503
$701$	13737.7	0.740177	0.370088	−	0.928997i	$-0.379327\pi$
$701$	13737.7	0.740177	0.370088	+	0.928997i	$0.379327\pi$
$702$	0	0
$703$	10294.3	0.552286
$704$	−8734.99	−0.467631
$705$	0	0
$706$	37917.5	2.02131
$707$	42005.8	2.23450
$708$	0	0
$709$	1556.55	0.0824507	0.0412254	−	0.999150i	$-0.486874\pi$
$709$	1556.55	0.0824507	0.0412254	+	0.999150i	$0.486874\pi$
$710$	25027.7	1.32292
$711$	0	0
$712$	7289.23	0.383674
$713$	−21308.3	−1.11922
$714$	0	0
$715$	0	0
$716$	−3636.63	−0.189814
$717$	0	0
$718$	11815.6	0.614142
$719$	−15740.2	−0.816428	−0.408214	−	0.912886i	$-0.633848\pi$
$719$	−15740.2	−0.816428	−0.408214	+	0.912886i	$0.633848\pi$
$720$	0	0
$721$	24624.9	1.27196
$722$	−19301.7	−0.994926
$723$	0	0
$724$	13444.6	0.690146
$725$	1133.28	0.0580539
$726$	0	0
$727$	17533.2	0.894459	0.447230	−	0.894419i	$-0.352411\pi$
$727$	17533.2	0.894459	0.447230	+	0.894419i	$0.352411\pi$
$728$	0	0
$729$	0	0
$730$	−51819.2	−2.62728
$731$	−9309.80	−0.471047
$732$	0	0
$733$	−32766.4	−1.65110	−0.825549	−	0.564330i	$-0.809135\pi$
$733$	−32766.4	−1.65110	−0.825549	+	0.564330i	$0.809135\pi$
$734$	−84.7639	−0.00426252
$735$	0	0
$736$	−17957.9	−0.899369
$737$	−5863.72	−0.293070
$738$	0	0
$739$	31683.2	1.57711	0.788557	−	0.614962i	$-0.210829\pi$
$739$	31683.2	1.57711	0.788557	+	0.614962i	$0.210829\pi$
$740$	−9891.35	−0.491369
$741$	0	0
$742$	34718.1	1.71771
$743$	19757.8	0.975563	0.487781	−	0.872966i	$-0.337806\pi$
$743$	19757.8	0.975563	0.487781	+	0.872966i	$0.337806\pi$
$744$	0	0
$745$	−39998.0	−1.96700
$746$	−22872.3	−1.12254
$747$	0	0
$748$	−18124.0	−0.885933
$749$	−36399.6	−1.77572
$750$	0	0
$751$	170.877	0.00830279	0.00415140	−	0.999991i	$-0.498679\pi$
$751$	170.877	0.00830279	0.00415140	+	0.999991i	$0.498679\pi$
$752$	−29332.1	−1.42238
$753$	0	0
$754$	0	0
$755$	4721.88	0.227612
$756$	0	0
$757$	32442.0	1.55763	0.778815	−	0.627253i	$-0.215821\pi$
$757$	32442.0	1.55763	0.778815	+	0.627253i	$0.215821\pi$
$758$	25679.6	1.23051
$759$	0	0
$760$	−18031.7	−0.860632
$761$	−4594.95	−0.218879	−0.109439	−	0.993993i	$-0.534906\pi$
$761$	−4594.95	−0.218879	−0.109439	+	0.993993i	$0.534906\pi$
$762$	0	0
$763$	25926.5	1.23015
$764$	−6343.32	−0.300384
$765$	0	0
$766$	4508.30	0.212652
$767$	0	0
$768$	0	0
$769$	21213.7	0.994782	0.497391	−	0.867526i	$-0.334291\pi$
$769$	21213.7	0.994782	0.497391	+	0.867526i	$0.334291\pi$
$770$	−82370.9	−3.85512
$771$	0	0
$772$	18351.7	0.855560
$773$	2816.60	0.131056	0.0655278	−	0.997851i	$-0.479127\pi$
$773$	2816.60	0.131056	0.0655278	+	0.997851i	$0.479127\pi$
$774$	0	0
$775$	58028.3	2.68960
$776$	12958.4	0.599459
$777$	0	0
$778$	23328.9	1.07504
$779$	7470.29	0.343583
$780$	0	0
$781$	−17807.0	−0.815857
$782$	−20086.3	−0.918524
$783$	0	0
$784$	−18534.5	−0.844318
$785$	−69144.9	−3.14381
$786$	0	0
$787$	4312.13	0.195312	0.0976562	−	0.995220i	$-0.468865\pi$
$787$	4312.13	0.195312	0.0976562	+	0.995220i	$0.468865\pi$
$788$	1010.86	0.0456985
$789$	0	0
$790$	−35608.3	−1.60365
$791$	21515.4	0.967128
$792$	0	0
$793$	0	0
$794$	11392.9	0.509219
$795$	0	0
$796$	16632.8	0.740622
$797$	−1009.62	−0.0448714	−0.0224357	−	0.999748i	$-0.507142\pi$
$797$	−1009.62	−0.0448714	−0.0224357	+	0.999748i	$0.507142\pi$
$798$	0	0
$799$	−24771.2	−1.09680
$800$	48904.1	2.16128
$801$	0	0
$802$	−39801.0	−1.75240
$803$	36868.9	1.62027
$804$	0	0
$805$	−37714.8	−1.65127
$806$	0	0
$807$	0	0
$808$	15216.4	0.662512
$809$	7369.64	0.320275	0.160138	−	0.987095i	$-0.448806\pi$
$809$	7369.64	0.320275	0.160138	+	0.987095i	$0.448806\pi$
$810$	0	0
$811$	−6713.40	−0.290678	−0.145339	−	0.989382i	$-0.546427\pi$
$811$	−6713.40	−0.290678	−0.145339	+	0.989382i	$0.546427\pi$
$812$	679.147	0.0293515
$813$	0	0
$814$	17034.7	0.733497
$815$	23780.4	1.02208
$816$	0	0
$817$	−15671.2	−0.671073
$818$	28182.9	1.20463
$819$	0	0
$820$	−7177.88	−0.305686
$821$	−2409.11	−0.102410	−0.0512049	−	0.998688i	$-0.516306\pi$
$821$	−2409.11	−0.102410	−0.0512049	+	0.998688i	$0.516306\pi$
$822$	0	0
$823$	−2370.79	−0.100414	−0.0502069	−	0.998739i	$-0.515988\pi$
$823$	−2370.79	−0.100414	−0.0502069	+	0.998739i	$0.515988\pi$
$824$	8920.25	0.377126
$825$	0	0
$826$	11931.6	0.502609
$827$	10168.6	0.427567	0.213783	−	0.976881i	$-0.431421\pi$
$827$	10168.6	0.427567	0.213783	+	0.976881i	$0.431421\pi$
$828$	0	0
$829$	−19837.3	−0.831095	−0.415547	−	0.909572i	$-0.636410\pi$
$829$	−19837.3	−0.831095	−0.415547	+	0.909572i	$0.636410\pi$
$830$	18024.2	0.753770
$831$	0	0
$832$	0	0
$833$	−15652.5	−0.651052
$834$	0	0
$835$	17613.2	0.729975
$836$	−30508.2	−1.26214
$837$	0	0
$838$	−12954.5	−0.534016
$839$	−1470.08	−0.0604919	−0.0302459	−	0.999542i	$-0.509629\pi$
$839$	−1470.08	−0.0604919	−0.0302459	+	0.999542i	$0.509629\pi$
$840$	0	0
$841$	−24364.0	−0.998977
$842$	46069.2	1.88557
$843$	0	0
$844$	17618.3	0.718540
$845$	0	0
$846$	0	0
$847$	26478.2	1.07415
$848$	30127.6	1.22003
$849$	0	0
$850$	54700.4	2.20731
$851$	7799.61	0.314180
$852$	0	0
$853$	10200.0	0.409426	0.204713	−	0.978822i	$-0.434374\pi$
$853$	10200.0	0.409426	0.204713	+	0.978822i	$0.434374\pi$
$854$	−55325.8	−2.21687
$855$	0	0
$856$	−13185.5	−0.526486
$857$	2379.48	0.0948443	0.0474222	−	0.998875i	$-0.484899\pi$
$857$	2379.48	0.0948443	0.0474222	+	0.998875i	$0.484899\pi$
$858$	0	0
$859$	9651.19	0.383346	0.191673	−	0.981459i	$-0.438609\pi$
$859$	9651.19	0.383346	0.191673	+	0.981459i	$0.438609\pi$
$860$	15057.8	0.597054
$861$	0	0
$862$	−65518.4	−2.58882
$863$	−19568.7	−0.771872	−0.385936	−	0.922526i	$-0.626121\pi$
$863$	−19568.7	−0.771872	−0.385936	+	0.922526i	$0.626121\pi$
$864$	0	0
$865$	34165.5	1.34296
$866$	−35799.4	−1.40475
$867$	0	0
$868$	34774.9	1.35984
$869$	25335.0	0.988989
$870$	0	0
$871$	0	0
$872$	9391.75	0.364730
$873$	0	0
$874$	−33811.4	−1.30857
$875$	46110.9	1.78152
$876$	0	0
$877$	−1978.67	−0.0761856	−0.0380928	−	0.999274i	$-0.512128\pi$
$877$	−1978.67	−0.0761856	−0.0380928	+	0.999274i	$0.512128\pi$
$878$	−49467.2	−1.90141
$879$	0	0
$880$	−71479.6	−2.73816
$881$	4802.77	0.183665	0.0918327	−	0.995774i	$-0.470727\pi$
$881$	4802.77	0.183665	0.0918327	+	0.995774i	$0.470727\pi$
$882$	0	0
$883$	−49531.7	−1.88774	−0.943871	−	0.330315i	$-0.892845\pi$
$883$	−49531.7	−1.88774	−0.943871	+	0.330315i	$0.892845\pi$
$884$	0	0
$885$	0	0
$886$	−50324.1	−1.90821
$887$	−7718.74	−0.292187	−0.146093	−	0.989271i	$-0.546670\pi$
$887$	−7718.74	−0.292187	−0.146093	+	0.989271i	$0.546670\pi$
$888$	0	0
$889$	−158.346	−0.00597384
$890$	−57732.8	−2.17439
$891$	0	0
$892$	21142.2	0.793600
$893$	−41697.5	−1.56255
$894$	0	0
$895$	−12112.4	−0.452373
$896$	−25832.4	−0.963170
$897$	0	0
$898$	5101.72	0.189584
$899$	1278.01	0.0474128
$900$	0	0
$901$	25442.9	0.940763
$902$	12361.6	0.456316
$903$	0	0
$904$	7793.82	0.286746
$905$	44779.7	1.64478
$906$	0	0
$907$	1788.58	0.0654783	0.0327392	−	0.999464i	$-0.489577\pi$
$907$	1788.58	0.0654783	0.0327392	+	0.999464i	$0.489577\pi$
$908$	−8726.44	−0.318940
$909$	0	0
$910$	0	0
$911$	−8239.52	−0.299657	−0.149828	−	0.988712i	$-0.547872\pi$
$911$	−8239.52	−0.299657	−0.149828	+	0.988712i	$0.547872\pi$
$912$	0	0
$913$	−12824.1	−0.464857
$914$	−30502.8	−1.10388
$915$	0	0
$916$	−7686.39	−0.277255
$917$	−30593.9	−1.10174
$918$	0	0
$919$	21053.8	0.755713	0.377857	−	0.925864i	$-0.376661\pi$
$919$	21053.8	0.755713	0.377857	+	0.925864i	$0.376661\pi$
$920$	−13662.0	−0.489589
$921$	0	0
$922$	62301.7	2.22538
$923$	0	0
$924$	0	0
$925$	−21240.4	−0.755006
$926$	16492.4	0.585286
$927$	0	0
$928$	1077.06	0.0380994
$929$	−34539.0	−1.21979	−0.609896	−	0.792481i	$-0.708789\pi$
$929$	−34539.0	−1.21979	−0.609896	+	0.792481i	$0.708789\pi$
$930$	0	0
$931$	−26347.9	−0.927516
$932$	−9380.70	−0.329694
$933$	0	0
$934$	10084.6	0.353294
$935$	−60365.1	−2.11139
$936$	0	0
$937$	−6368.78	−0.222048	−0.111024	−	0.993818i	$-0.535413\pi$
$937$	−6368.78	−0.222048	−0.111024	+	0.993818i	$0.535413\pi$
$938$	10605.6	0.369175
$939$	0	0
$940$	40065.3	1.39020
$941$	−4504.64	−0.156054	−0.0780272	−	0.996951i	$-0.524862\pi$
$941$	−4504.64	−0.156054	−0.0780272	+	0.996951i	$0.524862\pi$
$942$	0	0
$943$	5659.96	0.195454
$944$	10354.0	0.356985
$945$	0	0
$946$	−25932.3	−0.891259
$947$	38765.3	1.33020	0.665101	−	0.746753i	$-0.268388\pi$
$947$	38765.3	1.33020	0.665101	+	0.746753i	$0.268388\pi$
$948$	0	0
$949$	0	0
$950$	92077.5	3.14462
$951$	0	0
$952$	−13785.0	−0.469302
$953$	−21945.3	−0.745936	−0.372968	−	0.927844i	$-0.621660\pi$
$953$	−21945.3	−0.745936	−0.372968	+	0.927844i	$0.621660\pi$
$954$	0	0
$955$	−21127.5	−0.715886
$956$	33509.1	1.13364
$957$	0	0
$958$	20790.0	0.701141
$959$	32607.2	1.09796
$960$	0	0
$961$	35648.0	1.19660
$962$	0	0
$963$	0	0
$964$	9613.38	0.321189
$965$	61123.5	2.03900
$966$	0	0
$967$	58962.4	1.96081	0.980405	−	0.196991i	$-0.0631169\pi$
$967$	58962.4	1.96081	0.980405	+	0.196991i	$0.0631169\pi$
$968$	9591.58	0.318476
$969$	0	0
$970$	−102634.	−3.39731
$971$	51664.3	1.70751	0.853753	−	0.520679i	$-0.174321\pi$
$971$	51664.3	1.70751	0.853753	+	0.520679i	$0.174321\pi$
$972$	0	0
$973$	−23911.5	−0.787841
$974$	−47667.0	−1.56812
$975$	0	0
$976$	−48010.5	−1.57457
$977$	17888.8	0.585788	0.292894	−	0.956145i	$-0.405382\pi$
$977$	17888.8	0.585788	0.292894	+	0.956145i	$0.405382\pi$
$978$	0	0
$979$	41076.4	1.34097
$980$	25316.5	0.825212
$981$	0	0
$982$	−11499.7	−0.373697
$983$	−40916.4	−1.32760	−0.663800	−	0.747910i	$-0.731058\pi$
$983$	−40916.4	−1.32760	−0.663800	+	0.747910i	$0.731058\pi$
$984$	0	0
$985$	3366.85	0.108910
$986$	1204.72	0.0389108
$987$	0	0
$988$	0	0
$989$	−11873.5	−0.381754
$990$	0	0
$991$	27804.1	0.891249	0.445624	−	0.895220i	$-0.352982\pi$
$991$	27804.1	0.891249	0.445624	+	0.895220i	$0.352982\pi$
$992$	55149.5	1.76512
$993$	0	0
$994$	32207.2	1.02772
$995$	55398.5	1.76508
$996$	0	0
$997$	41605.5	1.32162	0.660812	−	0.750552i	$-0.270212\pi$
$997$	41605.5	1.32162	0.660812	+	0.750552i	$0.270212\pi$
$998$	−42967.7	−1.36284
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bd.1.2		8
3.2	odd	2	inner	1521.4.a.bd.1.7		8
13.4	even	6		117.4.g.f.55.2	✓	16
13.10	even	6		117.4.g.f.100.2	yes	16
13.12	even	2		1521.4.a.bc.1.7		8
39.17	odd	6		117.4.g.f.55.7	yes	16
39.23	odd	6		117.4.g.f.100.7	yes	16
39.38	odd	2		1521.4.a.bc.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.2	✓	16	13.4	even	6
117.4.g.f.55.7	yes	16	39.17	odd	6
117.4.g.f.100.2	yes	16	13.10	even	6
117.4.g.f.100.7	yes	16	39.23	odd	6
1521.4.a.bc.1.2		8	39.38	odd	2
1521.4.a.bc.1.7		8	13.12	even	2
1521.4.a.bd.1.2		8	1.1	even	1	trivial
1521.4.a.bd.1.7		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.69212 q^{2} +5.63172 q^{4} +18.7574 q^{5} +24.1383 q^{7} +8.74396 q^{8} +O(q^{10})\)	\(q-3.69212 q^{2} +5.63172 q^{4} +18.7574 q^{5} +24.1383 q^{7} +8.74396 q^{8} -69.2546 q^{10} +49.2741 q^{11} -89.1214 q^{14} -77.3375 q^{16} -65.3121 q^{17} -109.940 q^{19} +105.637 q^{20} -181.926 q^{22} -83.2974 q^{23} +226.841 q^{25} +135.940 q^{28} +4.99593 q^{29} +255.810 q^{31} +215.587 q^{32} +241.140 q^{34} +452.772 q^{35} -93.6356 q^{37} +405.912 q^{38} +164.014 q^{40} -67.9487 q^{41} +142.543 q^{43} +277.498 q^{44} +307.544 q^{46} +379.275 q^{47} +239.657 q^{49} -837.524 q^{50} -389.560 q^{53} +924.256 q^{55} +211.064 q^{56} -18.4456 q^{58} -133.881 q^{59} +620.792 q^{61} -944.482 q^{62} -177.273 q^{64} -119.002 q^{67} -367.819 q^{68} -1671.69 q^{70} -361.386 q^{71} +748.241 q^{73} +345.714 q^{74} -619.152 q^{76} +1189.39 q^{77} +514.165 q^{79} -1450.65 q^{80} +250.875 q^{82} -260.260 q^{83} -1225.09 q^{85} -526.286 q^{86} +430.851 q^{88} +833.631 q^{89} -469.108 q^{92} -1400.33 q^{94} -2062.19 q^{95} +1481.99 q^{97} -884.841 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{4} + 22 q^{7}+O(q^{10}) \) 8 * q + 40 * q^4 + 22 * q^7	\( 8 q + 40 q^{4} + 22 q^{7} + 36 q^{10} + 204 q^{16} - 244 q^{19} + 136 q^{22} + 354 q^{25} + 452 q^{28} - 242 q^{31} + 1292 q^{34} - 1018 q^{37} + 1700 q^{40} + 74 q^{43} + 896 q^{46} + 298 q^{49} + 1300 q^{55} - 812 q^{58} + 1148 q^{61} + 3636 q^{64} + 2198 q^{67} - 2200 q^{70} + 2176 q^{73} - 6936 q^{76} + 1862 q^{79} + 5436 q^{82} + 890 q^{85} + 3528 q^{88} - 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 8 * q + 40 * q^4 + 22 * q^7 + 36 * q^10 + 204 * q^16 - 244 * q^19 + 136 * q^22 + 354 * q^25 + 452 * q^28 - 242 * q^31 + 1292 * q^34 - 1018 * q^37 + 1700 * q^40 + 74 * q^43 + 896 * q^46 + 298 * q^49 + 1300 * q^55 - 812 * q^58 + 1148 * q^61 + 3636 * q^64 + 2198 * q^67 - 2200 * q^70 + 2176 * q^73 - 6936 * q^76 + 1862 * q^79 + 5436 * q^82 + 890 * q^85 + 3528 * q^88 - 3104 * q^94 + 4370 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more