LMFDB - Embedded newform 387.4.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,4,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	387.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:	$22.8337391722$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.15653$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.15653 q^{2} +1.96369 q^{4} -1.36370 q^{5} +13.0131 q^{7} -19.0538 q^{8} +O(q^{10})$	\(q+3.15653 q^{2} +1.96369 q^{4} -1.36370 q^{5} +13.0131 q^{7} -19.0538 q^{8} -4.30455 q^{10} -64.7677 q^{11} -19.2944 q^{13} +41.0763 q^{14} -75.8534 q^{16} +54.1213 q^{17} -69.0659 q^{19} -2.67787 q^{20} -204.441 q^{22} -29.6031 q^{23} -123.140 q^{25} -60.9032 q^{26} +25.5537 q^{28} -13.1279 q^{29} +185.439 q^{31} -87.0032 q^{32} +170.836 q^{34} -17.7460 q^{35} -369.949 q^{37} -218.009 q^{38} +25.9836 q^{40} +294.860 q^{41} -43.0000 q^{43} -127.183 q^{44} -93.4431 q^{46} -367.319 q^{47} -173.659 q^{49} -388.696 q^{50} -37.8881 q^{52} -708.046 q^{53} +88.3236 q^{55} -247.950 q^{56} -41.4385 q^{58} -116.159 q^{59} +218.910 q^{61} +585.344 q^{62} +332.199 q^{64} +26.3116 q^{65} -133.114 q^{67} +106.277 q^{68} -56.0156 q^{70} +926.738 q^{71} +455.867 q^{73} -1167.75 q^{74} -135.624 q^{76} -842.831 q^{77} -620.178 q^{79} +103.441 q^{80} +930.734 q^{82} +1317.85 q^{83} -73.8050 q^{85} -135.731 q^{86} +1234.07 q^{88} -509.295 q^{89} -251.080 q^{91} -58.1312 q^{92} -1159.45 q^{94} +94.1850 q^{95} +965.870 q^{97} -548.159 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * 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 $

$2$	3.15653	1.11600	0.558001	−	0.829840i	$-0.311569\pi$
$2$	3.15653	1.11600	0.558001	+	0.829840i	$0.311569\pi$
$3$	0	0
$3$	0	0
$4$	1.96369	0.245461
$5$	−1.36370	−0.121973	−0.0609864	−	0.998139i	$-0.519425\pi$
$5$	−1.36370	−0.121973	−0.0609864	+	0.998139i	$0.519425\pi$
$6$	0	0
$7$	13.0131	0.702643	0.351321	−	0.936255i	$-0.385732\pi$
$7$	13.0131	0.702643	0.351321	+	0.936255i	$0.385732\pi$
$8$	−19.0538	−0.842067
$9$	0	0
$10$	−4.30455	−0.136122
$11$	−64.7677	−1.77529	−0.887646	−	0.460527i	$-0.847661\pi$
$11$	−64.7677	−1.77529	−0.887646	+	0.460527i	$0.847661\pi$
$12$	0	0
$13$	−19.2944	−0.411638	−0.205819	−	0.978590i	$-0.565986\pi$
$13$	−19.2944	−0.411638	−0.205819	+	0.978590i	$0.565986\pi$
$14$	41.0763	0.784151
$15$	0	0
$16$	−75.8534	−1.18521
$17$	54.1213	0.772138	0.386069	−	0.922470i	$-0.373833\pi$
$17$	54.1213	0.772138	0.386069	+	0.922470i	$0.373833\pi$
$18$	0	0
$19$	−69.0659	−0.833938	−0.416969	−	0.908921i	$-0.636908\pi$
$19$	−69.0659	−0.833938	−0.416969	+	0.908921i	$0.636908\pi$
$20$	−2.67787	−0.0299395
$21$	0	0
$22$	−204.441	−1.98123
$23$	−29.6031	−0.268377	−0.134189	−	0.990956i	$-0.542843\pi$
$23$	−29.6031	−0.268377	−0.134189	+	0.990956i	$0.542843\pi$
$24$	0	0
$25$	−123.140	−0.985123
$26$	−60.9032	−0.459389
$27$	0	0
$28$	25.5537	0.172471
$29$	−13.1279	−0.0840614	−0.0420307	−	0.999116i	$-0.513383\pi$
$29$	−13.1279	−0.0840614	−0.0420307	+	0.999116i	$0.513383\pi$
$30$	0	0
$31$	185.439	1.07438	0.537191	−	0.843461i	$-0.319486\pi$
$31$	185.439	1.07438	0.537191	+	0.843461i	$0.319486\pi$
$32$	−87.0032	−0.480629
$33$	0	0
$34$	170.836	0.861707
$35$	−17.7460	−0.0857032
$36$	0	0
$37$	−369.949	−1.64376	−0.821881	−	0.569659i	$-0.807075\pi$
$37$	−369.949	−1.64376	−0.821881	+	0.569659i	$0.807075\pi$
$38$	−218.009	−0.930676
$39$	0	0
$40$	25.9836	0.102709
$41$	294.860	1.12315	0.561577	−	0.827424i	$-0.310195\pi$
$41$	294.860	1.12315	0.561577	+	0.827424i	$0.310195\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−127.183	−0.435764
$45$	0	0
$46$	−93.4431	−0.299509
$47$	−367.319	−1.13998	−0.569989	−	0.821652i	$-0.693053\pi$
$47$	−367.319	−1.13998	−0.569989	+	0.821652i	$0.693053\pi$
$48$	0	0
$49$	−173.659	−0.506293
$50$	−388.696	−1.09940
$51$	0	0
$52$	−37.8881	−0.101041
$53$	−708.046	−1.83505	−0.917524	−	0.397680i	$-0.869815\pi$
$53$	−708.046	−1.83505	−0.917524	+	0.397680i	$0.869815\pi$
$54$	0	0
$55$	88.3236	0.216537
$56$	−247.950	−0.591672
$57$	0	0
$58$	−41.4385	−0.0938127
$59$	−116.159	−0.256316	−0.128158	−	0.991754i	$-0.540906\pi$
$59$	−116.159	−0.256316	−0.128158	+	0.991754i	$0.540906\pi$
$60$	0	0
$61$	218.910	0.459484	0.229742	−	0.973252i	$-0.426212\pi$
$61$	218.910	0.459484	0.229742	+	0.973252i	$0.426212\pi$
$62$	585.344	1.19901
$63$	0	0
$64$	332.199	0.648827
$65$	26.3116	0.0502086
$66$	0	0
$67$	−133.114	−0.242723	−0.121361	−	0.992608i	$-0.538726\pi$
$67$	−133.114	−0.242723	−0.121361	+	0.992608i	$0.538726\pi$
$68$	106.277	0.189529
$69$	0	0
$70$	−56.0156	−0.0956450
$71$	926.738	1.54906	0.774532	−	0.632535i	$-0.217985\pi$
$71$	926.738	1.54906	0.774532	+	0.632535i	$0.217985\pi$
$72$	0	0
$73$	455.867	0.730893	0.365446	−	0.930832i	$-0.380916\pi$
$73$	455.867	0.730893	0.365446	+	0.930832i	$0.380916\pi$
$74$	−1167.75	−1.83444
$75$	0	0
$76$	−135.624	−0.204699
$77$	−842.831	−1.24740
$78$	0	0
$79$	−620.178	−0.883234	−0.441617	−	0.897204i	$-0.645595\pi$
$79$	−620.178	−0.883234	−0.441617	+	0.897204i	$0.645595\pi$
$80$	103.441	0.144563
$81$	0	0
$82$	930.734	1.25344
$83$	1317.85	1.74281	0.871405	−	0.490564i	$-0.163209\pi$
$83$	1317.85	1.74281	0.871405	+	0.490564i	$0.163209\pi$
$84$	0	0
$85$	−73.8050	−0.0941797
$86$	−135.731	−0.170189
$87$	0	0
$88$	1234.07	1.49492
$89$	−509.295	−0.606575	−0.303287	−	0.952899i	$-0.598084\pi$
$89$	−509.295	−0.606575	−0.303287	+	0.952899i	$0.598084\pi$
$90$	0	0
$91$	−251.080	−0.289234
$92$	−58.1312	−0.0658760
$93$	0	0
$94$	−1159.45	−1.27222
$95$	94.1850	0.101718
$96$	0	0
$97$	965.870	1.01102	0.505511	−	0.862820i	$-0.331304\pi$
$97$	965.870	1.01102	0.505511	+	0.862820i	$0.331304\pi$
$98$	−548.159	−0.565024
$99$	0	0
$100$	−241.809	−0.241809
$101$	1501.80	1.47955	0.739776	−	0.672853i	$-0.234932\pi$
$101$	1501.80	1.47955	0.739776	+	0.672853i	$0.234932\pi$
$102$	0	0
$103$	1312.60	1.25567	0.627837	−	0.778345i	$-0.283940\pi$
$103$	1312.60	1.25567	0.627837	+	0.778345i	$0.283940\pi$
$104$	367.631	0.346627
$105$	0	0
$106$	−2234.97	−2.04792
$107$	611.656	0.552627	0.276313	−	0.961068i	$-0.410887\pi$
$107$	611.656	0.552627	0.276313	+	0.961068i	$0.410887\pi$
$108$	0	0
$109$	−946.664	−0.831871	−0.415935	−	0.909394i	$-0.636546\pi$
$109$	−946.664	−0.831871	−0.415935	+	0.909394i	$0.636546\pi$
$110$	278.796	0.241656
$111$	0	0
$112$	−987.090	−0.832779
$113$	−2109.50	−1.75615	−0.878076	−	0.478521i	$-0.841173\pi$
$113$	−2109.50	−1.75615	−0.878076	+	0.478521i	$0.841173\pi$
$114$	0	0
$115$	40.3696	0.0327347
$116$	−25.7790	−0.0206338
$117$	0	0
$118$	−366.660	−0.286049
$119$	704.287	0.542537
$120$	0	0
$121$	2863.86	2.15166
$122$	690.996	0.512785
$123$	0	0
$124$	364.144	0.263718
$125$	338.388	0.242131
$126$	0	0
$127$	788.583	0.550988	0.275494	−	0.961303i	$-0.411159\pi$
$127$	788.583	0.550988	0.275494	+	0.961303i	$0.411159\pi$
$128$	1744.62	1.20472
$129$	0	0
$130$	83.0535	0.0560329
$131$	−1873.57	−1.24957	−0.624787	−	0.780795i	$-0.714814\pi$
$131$	−1873.57	−1.24957	−0.624787	+	0.780795i	$0.714814\pi$
$132$	0	0
$133$	−898.764	−0.585960
$134$	−420.177	−0.270879
$135$	0	0
$136$	−1031.22	−0.650192
$137$	1860.27	1.16010	0.580050	−	0.814581i	$-0.303033\pi$
$137$	1860.27	1.16010	0.580050	+	0.814581i	$0.303033\pi$
$138$	0	0
$139$	−2822.52	−1.72232	−0.861161	−	0.508333i	$-0.830262\pi$
$139$	−2822.52	−1.72232	−0.861161	+	0.508333i	$0.830262\pi$
$140$	−34.8475	−0.0210368
$141$	0	0
$142$	2925.28	1.72876
$143$	1249.65	0.730777
$144$	0	0
$145$	17.9024	0.0102532
$146$	1438.96	0.815678
$147$	0	0
$148$	−726.463	−0.403479
$149$	−1615.30	−0.888126	−0.444063	−	0.895996i	$-0.646463\pi$
$149$	−1615.30	−0.888126	−0.444063	+	0.895996i	$0.646463\pi$
$150$	0	0
$151$	1399.96	0.754483	0.377241	−	0.926115i	$-0.376873\pi$
$151$	1399.96	0.754483	0.377241	+	0.926115i	$0.376873\pi$
$152$	1315.97	0.702232
$153$	0	0
$154$	−2660.42	−1.39210
$155$	−252.882	−0.131045
$156$	0	0
$157$	2933.25	1.49108	0.745539	−	0.666462i	$-0.232192\pi$
$157$	2933.25	1.49108	0.745539	+	0.666462i	$0.232192\pi$
$158$	−1957.61	−0.985691
$159$	0	0
$160$	118.646	0.0586237
$161$	−385.229	−0.188573
$162$	0	0
$163$	−166.212	−0.0798695	−0.0399347	−	0.999202i	$-0.512715\pi$
$163$	−166.212	−0.0798695	−0.0399347	+	0.999202i	$0.512715\pi$
$164$	579.012	0.275690
$165$	0	0
$166$	4159.85	1.94498
$167$	−2143.42	−0.993191	−0.496595	−	0.867982i	$-0.665417\pi$
$167$	−2143.42	−0.993191	−0.496595	+	0.867982i	$0.665417\pi$
$168$	0	0
$169$	−1824.73	−0.830554
$170$	−232.968	−0.105105
$171$	0	0
$172$	−84.4385	−0.0374324
$173$	119.018	0.0523050	0.0261525	−	0.999658i	$-0.491674\pi$
$173$	119.018	0.0523050	0.0261525	+	0.999658i	$0.491674\pi$
$174$	0	0
$175$	−1602.44	−0.692189
$176$	4912.85	2.10409
$177$	0	0
$178$	−1607.61	−0.676939
$179$	−2572.05	−1.07399	−0.536995	−	0.843585i	$-0.680441\pi$
$179$	−2572.05	−1.07399	−0.536995	+	0.843585i	$0.680441\pi$
$180$	0	0
$181$	1429.30	0.586956	0.293478	−	0.955966i	$-0.405187\pi$
$181$	1429.30	0.586956	0.293478	+	0.955966i	$0.405187\pi$
$182$	−792.541	−0.322786
$183$	0	0
$184$	564.052	0.225992
$185$	504.498	0.200494
$186$	0	0
$187$	−3505.31	−1.37077
$188$	−721.299	−0.279820
$189$	0	0
$190$	297.298	0.113517
$191$	−411.149	−0.155758	−0.0778788	−	0.996963i	$-0.524815\pi$
$191$	−411.149	−0.155758	−0.0778788	+	0.996963i	$0.524815\pi$
$192$	0	0
$193$	−3108.57	−1.15938	−0.579689	−	0.814838i	$-0.696826\pi$
$193$	−3108.57	−1.15938	−0.579689	+	0.814838i	$0.696826\pi$
$194$	3048.80	1.12830
$195$	0	0
$196$	−341.011	−0.124275
$197$	−1960.75	−0.709125	−0.354563	−	0.935032i	$-0.615370\pi$
$197$	−1960.75	−0.709125	−0.354563	+	0.935032i	$0.615370\pi$
$198$	0	0
$199$	456.187	0.162504	0.0812519	−	0.996694i	$-0.474108\pi$
$199$	456.187	0.162504	0.0812519	+	0.996694i	$0.474108\pi$
$200$	2346.29	0.829540
$201$	0	0
$202$	4740.48	1.65118
$203$	−170.834	−0.0590651
$204$	0	0
$205$	−402.099	−0.136994
$206$	4143.27	1.40133
$207$	0	0
$208$	1463.54	0.487877
$209$	4473.24	1.48048
$210$	0	0
$211$	−1742.50	−0.568524	−0.284262	−	0.958747i	$-0.591749\pi$
$211$	−1742.50	−0.568524	−0.284262	+	0.958747i	$0.591749\pi$
$212$	−1390.38	−0.450432
$213$	0	0
$214$	1930.71	0.616732
$215$	58.6390	0.0186007
$216$	0	0
$217$	2413.14	0.754906
$218$	−2988.17	−0.928370
$219$	0	0
$220$	173.440	0.0531514
$221$	−1044.24	−0.317841
$222$	0	0
$223$	3149.26	0.945697	0.472848	−	0.881144i	$-0.343226\pi$
$223$	3149.26	0.945697	0.472848	+	0.881144i	$0.343226\pi$
$224$	−1132.18	−0.337711
$225$	0	0
$226$	−6658.71	−1.95987
$227$	−2108.03	−0.616365	−0.308182	−	0.951327i	$-0.599721\pi$
$227$	−2108.03	−0.616365	−0.308182	+	0.951327i	$0.599721\pi$
$228$	0	0
$229$	1529.66	0.441409	0.220704	−	0.975341i	$-0.429164\pi$
$229$	1529.66	0.441409	0.220704	+	0.975341i	$0.429164\pi$
$230$	127.428	0.0365320
$231$	0	0
$232$	250.136	0.0707854
$233$	−2999.86	−0.843466	−0.421733	−	0.906720i	$-0.638578\pi$
$233$	−2999.86	−0.843466	−0.421733	+	0.906720i	$0.638578\pi$
$234$	0	0
$235$	500.912	0.139046
$236$	−228.100	−0.0629154
$237$	0	0
$238$	2223.10	0.605472
$239$	−3421.34	−0.925976	−0.462988	−	0.886364i	$-0.653223\pi$
$239$	−3421.34	−0.925976	−0.462988	+	0.886364i	$0.653223\pi$
$240$	0	0
$241$	−4496.86	−1.20194	−0.600971	−	0.799271i	$-0.705219\pi$
$241$	−4496.86	−1.20194	−0.600971	+	0.799271i	$0.705219\pi$
$242$	9039.86	2.40126
$243$	0	0
$244$	429.870	0.112785
$245$	236.818	0.0617540
$246$	0	0
$247$	1332.58	0.343280
$248$	−3533.32	−0.904701
$249$	0	0
$250$	1068.13	0.270219
$251$	−1478.05	−0.371688	−0.185844	−	0.982579i	$-0.559502\pi$
$251$	−1478.05	−0.371688	−0.185844	+	0.982579i	$0.559502\pi$
$252$	0	0
$253$	1917.33	0.476448
$254$	2489.19	0.614904
$255$	0	0
$256$	2849.36	0.695645
$257$	−677.467	−0.164433	−0.0822164	−	0.996614i	$-0.526200\pi$
$257$	−677.467	−0.164433	−0.0822164	+	0.996614i	$0.526200\pi$
$258$	0	0
$259$	−4814.19	−1.15498
$260$	51.6678	0.0123242
$261$	0	0
$262$	−5913.97	−1.39453
$263$	7753.56	1.81789	0.908944	−	0.416917i	$-0.136890\pi$
$263$	7753.56	1.81789	0.908944	+	0.416917i	$0.136890\pi$
$264$	0	0
$265$	965.560	0.223826
$266$	−2836.97	−0.653933
$267$	0	0
$268$	−261.393	−0.0595789
$269$	300.965	0.0682161	0.0341081	−	0.999418i	$-0.489141\pi$
$269$	300.965	0.0682161	0.0341081	+	0.999418i	$0.489141\pi$
$270$	0	0
$271$	−2176.33	−0.487833	−0.243917	−	0.969796i	$-0.578432\pi$
$271$	−2176.33	−0.487833	−0.243917	+	0.969796i	$0.578432\pi$
$272$	−4105.28	−0.915145
$273$	0	0
$274$	5872.00	1.29467
$275$	7975.52	1.74888
$276$	0	0
$277$	2947.64	0.639373	0.319687	−	0.947523i	$-0.396422\pi$
$277$	2947.64	0.639373	0.319687	+	0.947523i	$0.396422\pi$
$278$	−8909.36	−1.92211
$279$	0	0
$280$	338.128	0.0721679
$281$	−459.209	−0.0974879	−0.0487440	−	0.998811i	$-0.515522\pi$
$281$	−459.209	−0.0974879	−0.0487440	+	0.998811i	$0.515522\pi$
$282$	0	0
$283$	6190.86	1.30038	0.650192	−	0.759770i	$-0.274688\pi$
$283$	6190.86	1.30038	0.650192	+	0.759770i	$0.274688\pi$
$284$	1819.82	0.380234
$285$	0	0
$286$	3944.56	0.815549
$287$	3837.05	0.789176
$288$	0	0
$289$	−1983.89	−0.403803
$290$	56.5095	0.0114426
$291$	0	0
$292$	895.179	0.179405
$293$	−5693.95	−1.13530	−0.567652	−	0.823269i	$-0.692148\pi$
$293$	−5693.95	−1.13530	−0.567652	+	0.823269i	$0.692148\pi$
$294$	0	0
$295$	158.406	0.0312635
$296$	7048.93	1.38416
$297$	0	0
$298$	−5098.75	−0.991150
$299$	571.173	0.110474
$300$	0	0
$301$	−559.564	−0.107152
$302$	4419.01	0.842005
$303$	0	0
$304$	5238.89	0.988391
$305$	−298.527	−0.0560446
$306$	0	0
$307$	−5108.18	−0.949639	−0.474819	−	0.880083i	$-0.657487\pi$
$307$	−5108.18	−0.949639	−0.474819	+	0.880083i	$0.657487\pi$
$308$	−1655.05	−0.306187
$309$	0	0
$310$	−798.231	−0.146247
$311$	1048.11	0.191102	0.0955511	−	0.995425i	$-0.469539\pi$
$311$	1048.11	0.191102	0.0955511	+	0.995425i	$0.469539\pi$
$312$	0	0
$313$	4279.92	0.772893	0.386446	−	0.922312i	$-0.373702\pi$
$313$	4279.92	0.772893	0.386446	+	0.922312i	$0.373702\pi$
$314$	9258.91	1.66405
$315$	0	0
$316$	−1217.83	−0.216799
$317$	−5929.95	−1.05066	−0.525330	−	0.850899i	$-0.676058\pi$
$317$	−5929.95	−1.05066	−0.525330	+	0.850899i	$0.676058\pi$
$318$	0	0
$319$	850.261	0.149234
$320$	−453.019	−0.0791392
$321$	0	0
$322$	−1215.99	−0.210448
$323$	−3737.94	−0.643915
$324$	0	0
$325$	2375.91	0.405514
$326$	−524.653	−0.0891345
$327$	0	0
$328$	−5618.20	−0.945772
$329$	−4779.97	−0.800997
$330$	0	0
$331$	509.174	0.0845521	0.0422761	−	0.999106i	$-0.486539\pi$
$331$	509.174	0.0845521	0.0422761	+	0.999106i	$0.486539\pi$
$332$	2587.85	0.427792
$333$	0	0
$334$	−6765.77	−1.10840
$335$	181.527	0.0296056
$336$	0	0
$337$	−10856.3	−1.75484	−0.877419	−	0.479724i	$-0.840737\pi$
$337$	−10856.3	−1.75484	−0.877419	+	0.479724i	$0.840737\pi$
$338$	−5759.81	−0.926900
$339$	0	0
$340$	−144.930	−0.0231174
$341$	−12010.5	−1.90734
$342$	0	0
$343$	−6723.34	−1.05839
$344$	819.314	0.128414
$345$	0	0
$346$	375.684	0.0583725
$347$	−8973.74	−1.38829	−0.694143	−	0.719837i	$-0.744217\pi$
$347$	−8973.74	−1.38829	−0.694143	+	0.719837i	$0.744217\pi$
$348$	0	0
$349$	−5.70408	−0.000874877 0	−0.000437439	−	1.00000i	$-0.500139\pi$
$349$	−5.70408	−0.000874877 0	−0.000437439		1.00000i	$0.500139\pi$
$350$	−5058.15	−0.772485
$351$	0	0
$352$	5635.00	0.853257
$353$	1504.40	0.226830	0.113415	−	0.993548i	$-0.463821\pi$
$353$	1504.40	0.226830	0.113415	+	0.993548i	$0.463821\pi$
$354$	0	0
$355$	−1263.79	−0.188944
$356$	−1000.10	−0.148890
$357$	0	0
$358$	−8118.76	−1.19857
$359$	1296.86	0.190657	0.0953285	−	0.995446i	$-0.469610\pi$
$359$	1296.86	0.190657	0.0953285	+	0.995446i	$0.469610\pi$
$360$	0	0
$361$	−2088.90	−0.304548
$362$	4511.63	0.655044
$363$	0	0
$364$	−493.042	−0.0709957
$365$	−621.664	−0.0891490
$366$	0	0
$367$	12908.3	1.83599	0.917995	−	0.396592i	$-0.129807\pi$
$367$	12908.3	1.83599	0.917995	+	0.396592i	$0.129807\pi$
$368$	2245.50	0.318083
$369$	0	0
$370$	1592.46	0.223752
$371$	−9213.89	−1.28938
$372$	0	0
$373$	−1371.48	−0.190382	−0.0951908	−	0.995459i	$-0.530346\pi$
$373$	−1371.48	−0.190382	−0.0951908	+	0.995459i	$0.530346\pi$
$374$	−11064.6	−1.52978
$375$	0	0
$376$	6998.83	0.959939
$377$	253.293	0.0346029
$378$	0	0
$379$	−458.768	−0.0621776	−0.0310888	−	0.999517i	$-0.509897\pi$
$379$	−458.768	−0.0621776	−0.0310888	+	0.999517i	$0.509897\pi$
$380$	184.950	0.0249677
$381$	0	0
$382$	−1297.80	−0.173826
$383$	−9513.45	−1.26923	−0.634614	−	0.772829i	$-0.718841\pi$
$383$	−9513.45	−1.26923	−0.634614	+	0.772829i	$0.718841\pi$
$384$	0	0
$385$	1149.37	0.152148
$386$	−9812.31	−1.29387
$387$	0	0
$388$	1896.66	0.248166
$389$	−5643.25	−0.735538	−0.367769	−	0.929917i	$-0.619878\pi$
$389$	−5643.25	−0.735538	−0.367769	+	0.929917i	$0.619878\pi$
$390$	0	0
$391$	−1602.16	−0.207224
$392$	3308.86	0.426333
$393$	0	0
$394$	−6189.17	−0.791385
$395$	845.735	0.107730
$396$	0	0
$397$	−13816.5	−1.74668	−0.873340	−	0.487110i	$-0.838051\pi$
$397$	−13816.5	−1.74668	−0.873340	+	0.487110i	$0.838051\pi$
$398$	1439.97	0.181355
$399$	0	0
$400$	9340.62	1.16758
$401$	−3544.45	−0.441400	−0.220700	−	0.975342i	$-0.570834\pi$
$401$	−3544.45	−0.441400	−0.220700	+	0.975342i	$0.570834\pi$
$402$	0	0
$403$	−3577.93	−0.442256
$404$	2949.06	0.363172
$405$	0	0
$406$	−539.244	−0.0659168
$407$	23960.7	2.91816
$408$	0	0
$409$	−2736.88	−0.330880	−0.165440	−	0.986220i	$-0.552904\pi$
$409$	−2736.88	−0.330880	−0.165440	+	0.986220i	$0.552904\pi$
$410$	−1269.24	−0.152886
$411$	0	0
$412$	2577.54	0.308219
$413$	−1511.59	−0.180098
$414$	0	0
$415$	−1797.15	−0.212575
$416$	1678.67	0.197845
$417$	0	0
$418$	14119.9	1.65222
$419$	−9280.65	−1.08208	−0.541038	−	0.840998i	$-0.681968\pi$
$419$	−9280.65	−1.08208	−0.541038	+	0.840998i	$0.681968\pi$
$420$	0	0
$421$	−7308.00	−0.846010	−0.423005	−	0.906127i	$-0.639025\pi$
$421$	−7308.00	−0.846010	−0.423005	+	0.906127i	$0.639025\pi$
$422$	−5500.26	−0.634475
$423$	0	0
$424$	13491.0	1.54523
$425$	−6664.51	−0.760650
$426$	0	0
$427$	2848.70	0.322853
$428$	1201.10	0.135648
$429$	0	0
$430$	185.096	0.0207584
$431$	953.510	0.106564	0.0532818	−	0.998580i	$-0.483032\pi$
$431$	953.510	0.106564	0.0532818	+	0.998580i	$0.483032\pi$
$432$	0	0
$433$	3447.00	0.382569	0.191284	−	0.981535i	$-0.438735\pi$
$433$	3447.00	0.382569	0.191284	+	0.981535i	$0.438735\pi$
$434$	7617.15	0.842477
$435$	0	0
$436$	−1858.95	−0.204192
$437$	2044.57	0.223810
$438$	0	0
$439$	−9245.65	−1.00517	−0.502586	−	0.864527i	$-0.667618\pi$
$439$	−9245.65	−1.00517	−0.502586	+	0.864527i	$0.667618\pi$
$440$	−1682.90	−0.182339
$441$	0	0
$442$	−3296.16	−0.354711
$443$	3885.76	0.416745	0.208372	−	0.978050i	$-0.433183\pi$
$443$	3885.76	0.416745	0.208372	+	0.978050i	$0.433183\pi$
$444$	0	0
$445$	694.524	0.0739856
$446$	9940.75	1.05540
$447$	0	0
$448$	4322.95	0.455893
$449$	14413.1	1.51491	0.757455	−	0.652887i	$-0.226443\pi$
$449$	14413.1	1.51491	0.757455	+	0.652887i	$0.226443\pi$
$450$	0	0
$451$	−19097.4	−1.99393
$452$	−4142.40	−0.431066
$453$	0	0
$454$	−6654.06	−0.687864
$455$	342.397	0.0352787
$456$	0	0
$457$	−1765.21	−0.180685	−0.0903424	−	0.995911i	$-0.528796\pi$
$457$	−1765.21	−0.180685	−0.0903424	+	0.995911i	$0.528796\pi$
$458$	4828.41	0.492613
$459$	0	0
$460$	79.2733	0.00803508
$461$	−7626.64	−0.770517	−0.385258	−	0.922809i	$-0.625888\pi$
$461$	−7626.64	−0.770517	−0.385258	+	0.922809i	$0.625888\pi$
$462$	0	0
$463$	−5954.18	−0.597655	−0.298827	−	0.954307i	$-0.596595\pi$
$463$	−5954.18	−0.597655	−0.298827	+	0.954307i	$0.596595\pi$
$464$	995.792	0.0996304
$465$	0	0
$466$	−9469.16	−0.941310
$467$	−153.388	−0.0151991	−0.00759954	−	0.999971i	$-0.502419\pi$
$467$	−153.388	−0.0151991	−0.00759954	+	0.999971i	$0.502419\pi$
$468$	0	0
$469$	−1732.22	−0.170547
$470$	1581.14	0.155176
$471$	0	0
$472$	2213.27	0.215835
$473$	2785.01	0.270729
$474$	0	0
$475$	8504.80	0.821531
$476$	1383.00	0.133171
$477$	0	0
$478$	−10799.6	−1.03339
$479$	11334.3	1.08117	0.540584	−	0.841290i	$-0.318203\pi$
$479$	11334.3	1.08117	0.540584	+	0.841290i	$0.318203\pi$
$480$	0	0
$481$	7137.92	0.676635
$482$	−14194.5	−1.34137
$483$	0	0
$484$	5623.72	0.528148
$485$	−1317.15	−0.123317
$486$	0	0
$487$	−2845.89	−0.264804	−0.132402	−	0.991196i	$-0.542269\pi$
$487$	−2845.89	−0.264804	−0.132402	+	0.991196i	$0.542269\pi$
$488$	−4171.07	−0.386917
$489$	0	0
$490$	747.522	0.0689176
$491$	8345.94	0.767102	0.383551	−	0.923520i	$-0.374701\pi$
$491$	8345.94	0.767102	0.383551	+	0.923520i	$0.374701\pi$
$492$	0	0
$493$	−710.496	−0.0649070
$494$	4206.34	0.383101
$495$	0	0
$496$	−14066.2	−1.27337
$497$	12059.8	1.08844
$498$	0	0
$499$	17591.1	1.57813	0.789064	−	0.614311i	$-0.210566\pi$
$499$	17591.1	1.57813	0.789064	+	0.614311i	$0.210566\pi$
$500$	664.488	0.0594336
$501$	0	0
$502$	−4665.51	−0.414805
$503$	7975.83	0.707008	0.353504	−	0.935433i	$-0.384990\pi$
$503$	7975.83	0.707008	0.353504	+	0.935433i	$0.384990\pi$
$504$	0	0
$505$	−2048.00	−0.180465
$506$	6052.10	0.531717
$507$	0	0
$508$	1548.53	0.135246
$509$	9016.94	0.785204	0.392602	−	0.919708i	$-0.371575\pi$
$509$	9016.94	0.785204	0.392602	+	0.919708i	$0.371575\pi$
$510$	0	0
$511$	5932.25	0.513556
$512$	−4962.89	−0.428380
$513$	0	0
$514$	−2138.45	−0.183507
$515$	−1789.99	−0.153158
$516$	0	0
$517$	23790.4	2.02379
$518$	−15196.1	−1.28896
$519$	0	0
$520$	−501.337	−0.0422790
$521$	3086.13	0.259512	0.129756	−	0.991546i	$-0.458581\pi$
$521$	3086.13	0.259512	0.129756	+	0.991546i	$0.458581\pi$
$522$	0	0
$523$	8338.32	0.697149	0.348575	−	0.937281i	$-0.386666\pi$
$523$	8338.32	0.697149	0.348575	+	0.937281i	$0.386666\pi$
$524$	−3679.10	−0.306722
$525$	0	0
$526$	24474.3	2.02877
$527$	10036.2	0.829570
$528$	0	0
$529$	−11290.7	−0.927974
$530$	3047.82	0.249790
$531$	0	0
$532$	−1764.89	−0.143830
$533$	−5689.13	−0.462333
$534$	0	0
$535$	−834.114	−0.0674054
$536$	2536.32	0.204389
$537$	0	0
$538$	950.004	0.0761293
$539$	11247.5	0.898818
$540$	0	0
$541$	−14924.6	−1.18606	−0.593029	−	0.805181i	$-0.702068\pi$
$541$	−14924.6	−1.18606	−0.593029	+	0.805181i	$0.702068\pi$
$542$	−6869.66	−0.544423
$543$	0	0
$544$	−4708.72	−0.371112
$545$	1290.96	0.101466
$546$	0	0
$547$	10658.3	0.833120	0.416560	−	0.909108i	$-0.363236\pi$
$547$	10658.3	0.833120	0.416560	+	0.909108i	$0.363236\pi$
$548$	3652.99	0.284759
$549$	0	0
$550$	25175.0	1.95175
$551$	906.687	0.0701020
$552$	0	0
$553$	−8070.45	−0.620598
$554$	9304.31	0.713542
$555$	0	0
$556$	−5542.53	−0.422762
$557$	−4303.28	−0.327354	−0.163677	−	0.986514i	$-0.552335\pi$
$557$	−4303.28	−0.327354	−0.163677	+	0.986514i	$0.552335\pi$
$558$	0	0
$559$	829.657	0.0627742
$560$	1346.09	0.101576
$561$	0	0
$562$	−1449.51	−0.108797
$563$	13280.8	0.994173	0.497086	−	0.867701i	$-0.334403\pi$
$563$	13280.8	0.994173	0.497086	+	0.867701i	$0.334403\pi$
$564$	0	0
$565$	2876.72	0.214203
$566$	19541.7	1.45123
$567$	0	0
$568$	−17657.9	−1.30442
$569$	−11641.6	−0.857720	−0.428860	−	0.903371i	$-0.641085\pi$
$569$	−11641.6	−0.857720	−0.428860	+	0.903371i	$0.641085\pi$
$570$	0	0
$571$	21799.0	1.59765	0.798825	−	0.601563i	$-0.205455\pi$
$571$	21799.0	1.59765	0.798825	+	0.601563i	$0.205455\pi$
$572$	2453.92	0.179377
$573$	0	0
$574$	12111.8	0.880723
$575$	3645.34	0.264384
$576$	0	0
$577$	6854.44	0.494548	0.247274	−	0.968946i	$-0.420465\pi$
$577$	6854.44	0.494548	0.247274	+	0.968946i	$0.420465\pi$
$578$	−6262.20	−0.450645
$579$	0	0
$580$	35.1547	0.00251676
$581$	17149.4	1.22457
$582$	0	0
$583$	45858.5	3.25775
$584$	−8686.00	−0.615461
$585$	0	0
$586$	−17973.1	−1.26700
$587$	16741.7	1.17718	0.588589	−	0.808433i	$-0.299684\pi$
$587$	16741.7	1.17718	0.588589	+	0.808433i	$0.299684\pi$
$588$	0	0
$589$	−12807.5	−0.895967
$590$	500.012	0.0348902
$591$	0	0
$592$	28061.9	1.94820
$593$	−21885.0	−1.51553	−0.757764	−	0.652529i	$-0.773708\pi$
$593$	−21885.0	−1.51553	−0.757764	+	0.652529i	$0.773708\pi$
$594$	0	0
$595$	−960.434	−0.0661747
$596$	−3171.95	−0.218000
$597$	0	0
$598$	1802.92	0.123289
$599$	14440.9	0.985039	0.492519	−	0.870301i	$-0.336076\pi$
$599$	14440.9	0.985039	0.492519	+	0.870301i	$0.336076\pi$
$600$	0	0
$601$	2570.21	0.174444	0.0872221	−	0.996189i	$-0.472201\pi$
$601$	2570.21	0.174444	0.0872221	+	0.996189i	$0.472201\pi$
$602$	−1766.28	−0.119582
$603$	0	0
$604$	2749.08	0.185196
$605$	−3905.44	−0.262444
$606$	0	0
$607$	−11683.5	−0.781249	−0.390624	−	0.920550i	$-0.627741\pi$
$607$	−11683.5	−0.781249	−0.390624	+	0.920550i	$0.627741\pi$
$608$	6008.96	0.400815
$609$	0	0
$610$	−942.309	−0.0625458
$611$	7087.18	0.469258
$612$	0	0
$613$	12739.0	0.839356	0.419678	−	0.907673i	$-0.362143\pi$
$613$	12739.0	0.839356	0.419678	+	0.907673i	$0.362143\pi$
$614$	−16124.1	−1.05980
$615$	0	0
$616$	16059.1	1.05039
$617$	−4380.70	−0.285835	−0.142918	−	0.989735i	$-0.545648\pi$
$617$	−4380.70	−0.285835	−0.142918	+	0.989735i	$0.545648\pi$
$618$	0	0
$619$	−5157.68	−0.334902	−0.167451	−	0.985880i	$-0.553554\pi$
$619$	−5157.68	−0.334902	−0.167451	+	0.985880i	$0.553554\pi$
$620$	−496.582	−0.0321665
$621$	0	0
$622$	3308.39	0.213270
$623$	−6627.52	−0.426205
$624$	0	0
$625$	14931.1	0.955589
$626$	13509.7	0.862550
$627$	0	0
$628$	5759.99	0.366001
$629$	−20022.1	−1.26921
$630$	0	0
$631$	−27679.9	−1.74631	−0.873154	−	0.487444i	$-0.837929\pi$
$631$	−27679.9	−1.74631	−0.873154	+	0.487444i	$0.837929\pi$
$632$	11816.8	0.743743
$633$	0	0
$634$	−18718.1	−1.17254
$635$	−1075.39	−0.0672055
$636$	0	0
$637$	3350.63	0.208409
$638$	2683.88	0.166545
$639$	0	0
$640$	−2379.14	−0.146943
$641$	16453.5	1.01384	0.506921	−	0.861992i	$-0.330783\pi$
$641$	16453.5	1.01384	0.506921	+	0.861992i	$0.330783\pi$
$642$	0	0
$643$	−29203.6	−1.79110	−0.895551	−	0.444958i	$-0.853219\pi$
$643$	−29203.6	−1.79110	−0.895551	+	0.444958i	$0.853219\pi$
$644$	−756.468	−0.0462873
$645$	0	0
$646$	−11798.9	−0.718610
$647$	16835.6	1.02299	0.511495	−	0.859286i	$-0.329092\pi$
$647$	16835.6	1.02299	0.511495	+	0.859286i	$0.329092\pi$
$648$	0	0
$649$	7523.36	0.455035
$650$	7499.64	0.452554
$651$	0	0
$652$	−326.388	−0.0196048
$653$	13735.1	0.823117	0.411558	−	0.911383i	$-0.364985\pi$
$653$	13735.1	0.823117	0.411558	+	0.911383i	$0.364985\pi$
$654$	0	0
$655$	2554.98	0.152414
$656$	−22366.1	−1.33117
$657$	0	0
$658$	−15088.1	−0.893915
$659$	5708.76	0.337453	0.168727	−	0.985663i	$-0.446034\pi$
$659$	5708.76	0.337453	0.168727	+	0.985663i	$0.446034\pi$
$660$	0	0
$661$	18339.5	1.07916	0.539578	−	0.841936i	$-0.318584\pi$
$661$	18339.5	1.07916	0.539578	+	0.841936i	$0.318584\pi$
$662$	1607.22	0.0943604
$663$	0	0
$664$	−25110.1	−1.46756
$665$	1225.64	0.0714712
$666$	0	0
$667$	388.625	0.0225602
$668$	−4209.00	−0.243789
$669$	0	0
$670$	572.995	0.0330399
$671$	−14178.3	−0.815719
$672$	0	0
$673$	−482.234	−0.0276207	−0.0138104	−	0.999905i	$-0.504396\pi$
$673$	−482.234	−0.0276207	−0.0138104	+	0.999905i	$0.504396\pi$
$674$	−34268.3	−1.95840
$675$	0	0
$676$	−3583.19	−0.203868
$677$	−31086.0	−1.76474	−0.882372	−	0.470553i	$-0.844055\pi$
$677$	−31086.0	−1.76474	−0.882372	+	0.470553i	$0.844055\pi$
$678$	0	0
$679$	12569.0	0.710388
$680$	1406.27	0.0793057
$681$	0	0
$682$	−37911.4	−2.12860
$683$	−31381.0	−1.75807	−0.879034	−	0.476758i	$-0.841812\pi$
$683$	−31381.0	−1.75807	−0.879034	+	0.476758i	$0.841812\pi$
$684$	0	0
$685$	−2536.85	−0.141501
$686$	−21222.4	−1.18116
$687$	0	0
$688$	3261.70	0.180743
$689$	13661.3	0.755375
$690$	0	0
$691$	19856.9	1.09319	0.546593	−	0.837398i	$-0.315924\pi$
$691$	19856.9	1.09319	0.546593	+	0.837398i	$0.315924\pi$
$692$	233.714	0.0128388
$693$	0	0
$694$	−28325.9	−1.54933
$695$	3849.06	0.210076
$696$	0	0
$697$	15958.2	0.867230
$698$	−18.0051	−0.000976365 0
$699$	0	0
$700$	−3146.69	−0.169905
$701$	−2722.80	−0.146703	−0.0733514	−	0.997306i	$-0.523369\pi$
$701$	−2722.80	−0.146703	−0.0733514	+	0.997306i	$0.523369\pi$
$702$	0	0
$703$	25550.9	1.37079
$704$	−21515.8	−1.15186
$705$	0	0
$706$	4748.67	0.253143
$707$	19543.1	1.03960
$708$	0	0
$709$	−1225.30	−0.0649043	−0.0324521	−	0.999473i	$-0.510332\pi$
$709$	−1225.30	−0.0649043	−0.0324521	+	0.999473i	$0.510332\pi$
$710$	−3989.19	−0.210861
$711$	0	0
$712$	9704.01	0.510777
$713$	−5489.57	−0.288339
$714$	0	0
$715$	−1704.15	−0.0891349
$716$	−5050.70	−0.263622
$717$	0	0
$718$	4093.59	0.212774
$719$	28875.1	1.49772	0.748858	−	0.662730i	$-0.230602\pi$
$719$	28875.1	1.49772	0.748858	+	0.662730i	$0.230602\pi$
$720$	0	0
$721$	17081.0	0.882290
$722$	−6593.67	−0.339876
$723$	0	0
$724$	2806.70	0.144075
$725$	1616.57	0.0828108
$726$	0	0
$727$	−25072.0	−1.27905	−0.639525	−	0.768770i	$-0.720869\pi$
$727$	−25072.0	−1.27905	−0.639525	+	0.768770i	$0.720869\pi$
$728$	4784.03	0.243555
$729$	0	0
$730$	−1962.30	−0.0994905
$731$	−2327.22	−0.117750
$732$	0	0
$733$	1017.07	0.0512503	0.0256251	−	0.999672i	$-0.491842\pi$
$733$	1017.07	0.0512503	0.0256251	+	0.999672i	$0.491842\pi$
$734$	40745.5	2.04897
$735$	0	0
$736$	2575.56	0.128990
$737$	8621.47	0.430904
$738$	0	0
$739$	−1001.77	−0.0498658	−0.0249329	−	0.999689i	$-0.507937\pi$
$739$	−1001.77	−0.0498658	−0.0249329	+	0.999689i	$0.507937\pi$
$740$	990.675	0.0492134
$741$	0	0
$742$	−29083.9	−1.43895
$743$	2115.35	0.104448	0.0522238	−	0.998635i	$-0.483369\pi$
$743$	2115.35	0.104448	0.0522238	+	0.998635i	$0.483369\pi$
$744$	0	0
$745$	2202.78	0.108327
$746$	−4329.11	−0.212466
$747$	0	0
$748$	−6883.33	−0.336470
$749$	7959.56	0.388299
$750$	0	0
$751$	39172.1	1.90334	0.951671	−	0.307119i	$-0.0993649\pi$
$751$	39172.1	1.90334	0.951671	+	0.307119i	$0.0993649\pi$
$752$	27862.4	1.35111
$753$	0	0
$754$	799.528	0.0386169
$755$	−1909.12	−0.0920263
$756$	0	0
$757$	5944.10	0.285392	0.142696	−	0.989767i	$-0.454423\pi$
$757$	5944.10	0.285392	0.142696	+	0.989767i	$0.454423\pi$
$758$	−1448.12	−0.0693904
$759$	0	0
$760$	−1794.58	−0.0856531
$761$	1954.89	0.0931205	0.0465603	−	0.998915i	$-0.485174\pi$
$761$	1954.89	0.0931205	0.0465603	+	0.998915i	$0.485174\pi$
$762$	0	0
$763$	−12319.1	−0.584508
$764$	−807.367	−0.0382324
$765$	0	0
$766$	−30029.5	−1.41646
$767$	2241.21	0.105509
$768$	0	0
$769$	−39312.9	−1.84351	−0.921756	−	0.387770i	$-0.873245\pi$
$769$	−39312.9	−1.84351	−0.921756	+	0.387770i	$0.873245\pi$
$770$	3628.01	0.169798
$771$	0	0
$772$	−6104.26	−0.284582
$773$	5102.38	0.237413	0.118706	−	0.992929i	$-0.462125\pi$
$773$	5102.38	0.237413	0.118706	+	0.992929i	$0.462125\pi$
$774$	0	0
$775$	−22835.0	−1.05840
$776$	−18403.5	−0.851349
$777$	0	0
$778$	−17813.1	−0.820862
$779$	−20364.8	−0.936641
$780$	0	0
$781$	−60022.7	−2.75004
$782$	−5057.26	−0.231262
$783$	0	0
$784$	13172.6	0.600064
$785$	−4000.07	−0.181871
$786$	0	0
$787$	21898.3	0.991855	0.495928	−	0.868364i	$-0.334828\pi$
$787$	21898.3	0.991855	0.495928	+	0.868364i	$0.334828\pi$
$788$	−3850.30	−0.174062
$789$	0	0
$790$	2669.59	0.120227
$791$	−27451.2	−1.23395
$792$	0	0
$793$	−4223.73	−0.189141
$794$	−43612.3	−1.94930
$795$	0	0
$796$	895.809	0.0398883
$797$	−18262.4	−0.811652	−0.405826	−	0.913950i	$-0.633016\pi$
$797$	−18262.4	−0.811652	−0.405826	+	0.913950i	$0.633016\pi$
$798$	0	0
$799$	−19879.8	−0.880220
$800$	10713.6	0.473479
$801$	0	0
$802$	−11188.2	−0.492604
$803$	−29525.5	−1.29755
$804$	0	0
$805$	525.335	0.0230008
$806$	−11293.8	−0.493559
$807$	0	0
$808$	−28615.0	−1.24588
$809$	−26097.8	−1.13418	−0.567089	−	0.823656i	$-0.691931\pi$
$809$	−26097.8	−1.13418	−0.567089	+	0.823656i	$0.691931\pi$
$810$	0	0
$811$	1067.58	0.0462243	0.0231122	−	0.999733i	$-0.492643\pi$
$811$	1067.58	0.0462243	0.0231122	+	0.999733i	$0.492643\pi$
$812$	−335.465	−0.0144982
$813$	0	0
$814$	75632.8	3.25667
$815$	226.663	0.00974190
$816$	0	0
$817$	2969.84	0.127174
$818$	−8639.05	−0.369263
$819$	0	0
$820$	−789.596	−0.0336267
$821$	20127.0	0.855585	0.427793	−	0.903877i	$-0.359291\pi$
$821$	20127.0	0.855585	0.427793	+	0.903877i	$0.359291\pi$
$822$	0	0
$823$	−3011.70	−0.127559	−0.0637797	−	0.997964i	$-0.520315\pi$
$823$	−3011.70	−0.127559	−0.0637797	+	0.997964i	$0.520315\pi$
$824$	−25010.1	−1.05736
$825$	0	0
$826$	−4771.39	−0.200990
$827$	13138.4	0.552437	0.276219	−	0.961095i	$-0.410919\pi$
$827$	13138.4	0.552437	0.276219	+	0.961095i	$0.410919\pi$
$828$	0	0
$829$	19550.4	0.819075	0.409538	−	0.912293i	$-0.365690\pi$
$829$	19550.4	0.819075	0.409538	+	0.912293i	$0.365690\pi$
$830$	−5672.77	−0.237235
$831$	0	0
$832$	−6409.57	−0.267082
$833$	−9398.63	−0.390928
$834$	0	0
$835$	2922.98	0.121142
$836$	8784.05	0.363400
$837$	0	0
$838$	−29294.7	−1.20760
$839$	−3227.69	−0.132816	−0.0664078	−	0.997793i	$-0.521154\pi$
$839$	−3227.69	−0.132816	−0.0664078	+	0.997793i	$0.521154\pi$
$840$	0	0
$841$	−24216.7	−0.992934
$842$	−23067.9	−0.944149
$843$	0	0
$844$	−3421.72	−0.139550
$845$	2488.38	0.101305
$846$	0	0
$847$	37267.8	1.51185
$848$	53707.7	2.17492
$849$	0	0
$850$	−21036.7	−0.848887
$851$	10951.6	0.441148
$852$	0	0
$853$	12419.8	0.498530	0.249265	−	0.968435i	$-0.419811\pi$
$853$	12419.8	0.498530	0.249265	+	0.968435i	$0.419811\pi$
$854$	8992.01	0.360305
$855$	0	0
$856$	−11654.4	−0.465349
$857$	−18592.3	−0.741075	−0.370537	−	0.928818i	$-0.620826\pi$
$857$	−18592.3	−0.741075	−0.370537	+	0.928818i	$0.620826\pi$
$858$	0	0
$859$	6443.09	0.255920	0.127960	−	0.991779i	$-0.459157\pi$
$859$	6443.09	0.255920	0.127960	+	0.991779i	$0.459157\pi$
$860$	115.148	0.00456573
$861$	0	0
$862$	3009.78	0.118925
$863$	22085.1	0.871132	0.435566	−	0.900157i	$-0.356548\pi$
$863$	22085.1	0.871132	0.435566	+	0.900157i	$0.356548\pi$
$864$	0	0
$865$	−162.304	−0.00637979
$866$	10880.6	0.426948
$867$	0	0
$868$	4738.65	0.185300
$869$	40167.5	1.56800
$870$	0	0
$871$	2568.34	0.0999139
$872$	18037.5	0.700491
$873$	0	0
$874$	6453.73	0.249772
$875$	4403.49	0.170131
$876$	0	0
$877$	39786.3	1.53191	0.765956	−	0.642893i	$-0.222266\pi$
$877$	39786.3	1.53191	0.765956	+	0.642893i	$0.222266\pi$
$878$	−29184.2	−1.12177
$879$	0	0
$880$	−6699.64	−0.256642
$881$	−38834.0	−1.48508	−0.742538	−	0.669804i	$-0.766378\pi$
$881$	−38834.0	−1.48508	−0.742538	+	0.669804i	$0.766378\pi$
$882$	0	0
$883$	−5483.65	−0.208992	−0.104496	−	0.994525i	$-0.533323\pi$
$883$	−5483.65	−0.208992	−0.104496	+	0.994525i	$0.533323\pi$
$884$	−2050.55	−0.0780175
$885$	0	0
$886$	12265.5	0.465088
$887$	−21289.7	−0.805905	−0.402953	−	0.915221i	$-0.632016\pi$
$887$	−21289.7	−0.805905	−0.402953	+	0.915221i	$0.632016\pi$
$888$	0	0
$889$	10261.9	0.387148
$890$	2192.29	0.0825681
$891$	0	0
$892$	6184.17	0.232131
$893$	25369.2	0.950671
$894$	0	0
$895$	3507.50	0.130997
$896$	22703.0	0.846488
$897$	0	0
$898$	45495.3	1.69064
$899$	−2434.41	−0.0903140
$900$	0	0
$901$	−38320.4	−1.41691
$902$	−60281.5	−2.22523
$903$	0	0
$904$	40194.0	1.47880
$905$	−1949.13	−0.0715927
$906$	0	0
$907$	16875.4	0.617794	0.308897	−	0.951095i	$-0.400040\pi$
$907$	16875.4	0.617794	0.308897	+	0.951095i	$0.400040\pi$
$908$	−4139.51	−0.151293
$909$	0	0
$910$	1080.79	0.0393711
$911$	−17586.7	−0.639597	−0.319799	−	0.947486i	$-0.603615\pi$
$911$	−17586.7	−0.639597	−0.319799	+	0.947486i	$0.603615\pi$
$912$	0	0
$913$	−85354.4	−3.09400
$914$	−5571.94	−0.201645
$915$	0	0
$916$	3003.77	0.108349
$917$	−24381.0	−0.878005
$918$	0	0
$919$	−43891.7	−1.57547	−0.787733	−	0.616017i	$-0.788745\pi$
$919$	−43891.7	−1.57547	−0.787733	+	0.616017i	$0.788745\pi$
$920$	−769.196	−0.0275648
$921$	0	0
$922$	−24073.7	−0.859898
$923$	−17880.8	−0.637653
$924$	0	0
$925$	45555.6	1.61931
$926$	−18794.5	−0.666984
$927$	0	0
$928$	1142.16	0.0404024
$929$	41161.8	1.45369	0.726844	−	0.686803i	$-0.240987\pi$
$929$	41161.8	1.45369	0.726844	+	0.686803i	$0.240987\pi$
$930$	0	0
$931$	11993.9	0.422217
$932$	−5890.79	−0.207038
$933$	0	0
$934$	−484.175	−0.0169622
$935$	4780.18	0.167197
$936$	0	0
$937$	41754.1	1.45576	0.727880	−	0.685705i	$-0.240506\pi$
$937$	41754.1	1.45576	0.727880	+	0.685705i	$0.240506\pi$
$938$	−5467.82	−0.190331
$939$	0	0
$940$	983.633	0.0341304
$941$	37587.5	1.30214	0.651071	−	0.759016i	$-0.274320\pi$
$941$	37587.5	1.30214	0.651071	+	0.759016i	$0.274320\pi$
$942$	0	0
$943$	−8728.76	−0.301429
$944$	8811.06	0.303788
$945$	0	0
$946$	8790.98	0.302135
$947$	25689.0	0.881500	0.440750	−	0.897630i	$-0.354713\pi$
$947$	25689.0	0.881500	0.440750	+	0.897630i	$0.354713\pi$
$948$	0	0
$949$	−8795.66	−0.300863
$950$	26845.7	0.916830
$951$	0	0
$952$	−13419.4	−0.456853
$953$	10527.3	0.357831	0.178916	−	0.983864i	$-0.442741\pi$
$953$	10527.3	0.357831	0.178916	+	0.983864i	$0.442741\pi$
$954$	0	0
$955$	560.682	0.0189982
$956$	−6718.44	−0.227291
$957$	0	0
$958$	35777.2	1.20659
$959$	24207.9	0.815136
$960$	0	0
$961$	4596.60	0.154295
$962$	22531.1	0.755126
$963$	0	0
$964$	−8830.41	−0.295029
$965$	4239.15	0.141413
$966$	0	0
$967$	4292.71	0.142755	0.0713776	−	0.997449i	$-0.477260\pi$
$967$	4292.71	0.142755	0.0713776	+	0.997449i	$0.477260\pi$
$968$	−54567.5	−1.81184
$969$	0	0
$970$	−4157.63	−0.137622
$971$	41459.8	1.37025	0.685123	−	0.728427i	$-0.259748\pi$
$971$	41459.8	1.37025	0.685123	+	0.728427i	$0.259748\pi$
$972$	0	0
$973$	−36729.8	−1.21018
$974$	−8983.13	−0.295522
$975$	0	0
$976$	−16605.1	−0.544585
$977$	−3577.72	−0.117156	−0.0585779	−	0.998283i	$-0.518657\pi$
$977$	−3577.72	−0.117156	−0.0585779	+	0.998283i	$0.518657\pi$
$978$	0	0
$979$	32985.9	1.07685
$980$	465.035	0.0151582
$981$	0	0
$982$	26344.2	0.856088
$983$	−50532.3	−1.63960	−0.819802	−	0.572647i	$-0.805917\pi$
$983$	−50532.3	−1.63960	−0.819802	+	0.572647i	$0.805917\pi$
$984$	0	0
$985$	2673.87	0.0864939
$986$	−2242.70	−0.0724363
$987$	0	0
$988$	2616.77	0.0842618
$989$	1272.93	0.0409271
$990$	0	0
$991$	46280.6	1.48350	0.741752	−	0.670674i	$-0.233995\pi$
$991$	46280.6	1.48350	0.741752	+	0.670674i	$0.233995\pi$
$992$	−16133.8	−0.516379
$993$	0	0
$994$	38067.0	1.21470
$995$	−622.101	−0.0198210
$996$	0	0
$997$	53334.8	1.69421	0.847107	−	0.531423i	$-0.178342\pi$
$997$	53334.8	1.69421	0.847107	+	0.531423i	$0.178342\pi$
$998$	55526.9	1.76119
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.4.a.h.1.5		6
3.2	odd	2		43.4.a.b.1.2	✓	6
12.11	even	2		688.4.a.i.1.2		6
15.14	odd	2		1075.4.a.b.1.5		6
21.20	even	2		2107.4.a.c.1.2		6
129.128	even	2		1849.4.a.c.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.2	✓	6	3.2	odd	2
387.4.a.h.1.5		6	1.1	even	1	trivial
688.4.a.i.1.2		6	12.11	even	2
1075.4.a.b.1.5		6	15.14	odd	2
1849.4.a.c.1.5		6	129.128	even	2
2107.4.a.c.1.2		6	21.20	even	2

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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+3.15653 q^{2} +1.96369 q^{4} -1.36370 q^{5} +13.0131 q^{7} -19.0538 q^{8} +O(q^{10})\)	\(q+3.15653 q^{2} +1.96369 q^{4} -1.36370 q^{5} +13.0131 q^{7} -19.0538 q^{8} -4.30455 q^{10} -64.7677 q^{11} -19.2944 q^{13} +41.0763 q^{14} -75.8534 q^{16} +54.1213 q^{17} -69.0659 q^{19} -2.67787 q^{20} -204.441 q^{22} -29.6031 q^{23} -123.140 q^{25} -60.9032 q^{26} +25.5537 q^{28} -13.1279 q^{29} +185.439 q^{31} -87.0032 q^{32} +170.836 q^{34} -17.7460 q^{35} -369.949 q^{37} -218.009 q^{38} +25.9836 q^{40} +294.860 q^{41} -43.0000 q^{43} -127.183 q^{44} -93.4431 q^{46} -367.319 q^{47} -173.659 q^{49} -388.696 q^{50} -37.8881 q^{52} -708.046 q^{53} +88.3236 q^{55} -247.950 q^{56} -41.4385 q^{58} -116.159 q^{59} +218.910 q^{61} +585.344 q^{62} +332.199 q^{64} +26.3116 q^{65} -133.114 q^{67} +106.277 q^{68} -56.0156 q^{70} +926.738 q^{71} +455.867 q^{73} -1167.75 q^{74} -135.624 q^{76} -842.831 q^{77} -620.178 q^{79} +103.441 q^{80} +930.734 q^{82} +1317.85 q^{83} -73.8050 q^{85} -135.731 q^{86} +1234.07 q^{88} -509.295 q^{89} -251.080 q^{91} -58.1312 q^{92} -1159.45 q^{94} +94.1850 q^{95} +965.870 q^{97} -548.159 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * q^98

Introduction

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