LMFDB - Embedded newform 1911.4.a.v.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1911,4,Mod(1,1911)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1911.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1911, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1911.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,-1,-33,31,17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$112.752650021$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 59 x^{9} + 36 x^{8} + 1220 x^{7} - 339 x^{6} - 10807 x^{5} + 58 x^{4} + 40509 x^{3} + \cdots - 27208 $ x^11 - x^10 - 59x^9 + 36x^8 + 1220x^7 - 339x^6 - 10807x^5 + 58x^4 + 40509x^3 + 10990x^2 - 51360*x - 27208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-4.83732$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.83732 q^{2} -3.00000 q^{3} +15.3997 q^{4} +16.3780 q^{5} -14.5120 q^{6} +35.7947 q^{8} +9.00000 q^{9} +79.2258 q^{10} +5.16661 q^{11} -46.1991 q^{12} -13.0000 q^{13} -49.1341 q^{15} +49.9531 q^{16} +6.88424 q^{17} +43.5359 q^{18} +122.280 q^{19} +252.217 q^{20} +24.9926 q^{22} -70.6956 q^{23} -107.384 q^{24} +143.240 q^{25} -62.8852 q^{26} -27.0000 q^{27} +205.007 q^{29} -237.678 q^{30} +173.826 q^{31} -44.7186 q^{32} -15.4998 q^{33} +33.3013 q^{34} +138.597 q^{36} -165.139 q^{37} +591.509 q^{38} +39.0000 q^{39} +586.247 q^{40} -116.581 q^{41} +98.2962 q^{43} +79.5642 q^{44} +147.402 q^{45} -341.977 q^{46} +559.104 q^{47} -149.859 q^{48} +692.898 q^{50} -20.6527 q^{51} -200.196 q^{52} -190.585 q^{53} -130.608 q^{54} +84.6189 q^{55} -366.841 q^{57} +991.684 q^{58} +394.926 q^{59} -756.650 q^{60} -1.15246 q^{61} +840.851 q^{62} -615.943 q^{64} -212.914 q^{65} -74.9777 q^{66} -656.911 q^{67} +106.015 q^{68} +212.087 q^{69} +260.791 q^{71} +322.153 q^{72} +1065.83 q^{73} -798.833 q^{74} -429.720 q^{75} +1883.08 q^{76} +188.656 q^{78} +356.419 q^{79} +818.133 q^{80} +81.0000 q^{81} -563.942 q^{82} +804.148 q^{83} +112.750 q^{85} +475.490 q^{86} -615.020 q^{87} +184.937 q^{88} -645.066 q^{89} +713.033 q^{90} -1088.69 q^{92} -521.477 q^{93} +2704.57 q^{94} +2002.71 q^{95} +134.156 q^{96} -1002.09 q^{97} +46.4995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - q^{2} - 33 q^{3} + 31 q^{4} + 17 q^{5} + 3 q^{6} - 54 q^{8} + 99 q^{9} + 75 q^{10} + 7 q^{11} - 93 q^{12} - 143 q^{13} - 51 q^{15} + 23 q^{16} - 20 q^{17} - 9 q^{18} + 242 q^{19} + 254 q^{20} - 290 q^{22}+ \cdots + 63 q^{99}+O(q^{100}) $ 11 * q - q^2 - 33 * q^3 + 31 * q^4 + 17 * q^5 + 3 * q^6 - 54 * q^8 + 99 * q^9 + 75 * q^10 + 7 * q^11 - 93 * q^12 - 143 * q^13 - 51 * q^15 + 23 * q^16 - 20 * q^17 - 9 * q^18 + 242 * q^19 + 254 * q^20 - 290 * q^22 - 122 * q^23 + 162 * q^24 + 208 * q^25 + 13 * q^26 - 297 * q^27 - 177 * q^29 - 225 * q^30 + 493 * q^31 - 448 * q^32 - 21 * q^33 + 855 * q^34 + 279 * q^36 - 660 * q^37 + 458 * q^38 + 429 * q^39 - 530 * q^40 + 854 * q^41 - 1018 * q^43 + 1122 * q^44 + 153 * q^45 - 757 * q^46 + 414 * q^47 - 69 * q^48 + 851 * q^50 + 60 * q^51 - 403 * q^52 - 961 * q^53 + 27 * q^54 + 1003 * q^55 - 726 * q^57 + 801 * q^58 + 2053 * q^59 - 762 * q^60 + 916 * q^61 - 653 * q^62 + 1366 * q^64 - 221 * q^65 + 870 * q^66 - 2788 * q^67 - 2458 * q^68 + 366 * q^69 + 886 * q^71 - 486 * q^72 + 1916 * q^73 - 1557 * q^74 - 624 * q^75 + 2292 * q^76 - 39 * q^78 + 651 * q^79 + 4438 * q^80 + 891 * q^81 - 2148 * q^82 + 185 * q^83 - 2052 * q^85 + 4025 * q^86 + 531 * q^87 - 4503 * q^88 + 1490 * q^89 + 675 * q^90 + 586 * q^92 - 1479 * q^93 + 1796 * q^94 - 2496 * q^95 + 1344 * q^96 + 2705 * q^97 + 63 * q^99

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 $

$2$	4.83732	1.71025	0.855126	−	0.518420i	$-0.173480\pi$
$2$	4.83732	1.71025	0.855126	+	0.518420i	$0.173480\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	15.3997	1.92496
$5$	16.3780	1.46490	0.732448	−	0.680823i	$-0.238378\pi$
$5$	16.3780	1.46490	0.732448	+	0.680823i	$0.238378\pi$
$6$	−14.5120	−0.987414
$7$	0	0
$7$	0	0
$8$	35.7947	1.58192
$9$	9.00000	0.333333
$10$	79.2258	2.50534
$11$	5.16661	0.141617	0.0708087	−	0.997490i	$-0.477442\pi$
$11$	5.16661	0.141617	0.0708087	+	0.997490i	$0.477442\pi$
$12$	−46.1991	−1.11138
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−49.1341	−0.845758
$16$	49.9531	0.780517
$17$	6.88424	0.0982160	0.0491080	−	0.998793i	$-0.484362\pi$
$17$	6.88424	0.0982160	0.0491080	+	0.998793i	$0.484362\pi$
$18$	43.5359	0.570084
$19$	122.280	1.47647	0.738237	−	0.674541i	$-0.235659\pi$
$19$	122.280	1.47647	0.738237	+	0.674541i	$0.235659\pi$
$20$	252.217	2.81987
$21$	0	0
$22$	24.9926	0.242202
$23$	−70.6956	−0.640915	−0.320458	−	0.947263i	$-0.603837\pi$
$23$	−70.6956	−0.640915	−0.320458	+	0.947263i	$0.603837\pi$
$24$	−107.384	−0.913321
$25$	143.240	1.14592
$26$	−62.8852	−0.474339
$27$	−27.0000	−0.192450
$28$	0	0
$29$	205.007	1.31272	0.656359	−	0.754449i	$-0.272096\pi$
$29$	205.007	1.31272	0.656359	+	0.754449i	$0.272096\pi$
$30$	−237.678	−1.44646
$31$	173.826	1.00710	0.503549	−	0.863967i	$-0.332027\pi$
$31$	173.826	1.00710	0.503549	+	0.863967i	$0.332027\pi$
$32$	−44.7186	−0.247038
$33$	−15.4998	−0.0817629
$34$	33.3013	0.167974
$35$	0	0
$36$	138.597	0.641654
$37$	−165.139	−0.733750	−0.366875	−	0.930270i	$-0.619572\pi$
$37$	−165.139	−0.733750	−0.366875	+	0.930270i	$0.619572\pi$
$38$	591.509	2.52514
$39$	39.0000	0.160128
$40$	586.247	2.31735
$41$	−116.581	−0.444072	−0.222036	−	0.975038i	$-0.571270\pi$
$41$	−116.581	−0.444072	−0.222036	+	0.975038i	$0.571270\pi$
$42$	0	0
$43$	98.2962	0.348605	0.174303	−	0.984692i	$-0.444233\pi$
$43$	98.2962	0.348605	0.174303	+	0.984692i	$0.444233\pi$
$44$	79.5642	0.272608
$45$	147.402	0.488299
$46$	−341.977	−1.09613
$47$	559.104	1.73519	0.867593	−	0.497275i	$-0.165666\pi$
$47$	559.104	1.73519	0.867593	+	0.497275i	$0.165666\pi$
$48$	−149.859	−0.450632
$49$	0	0
$50$	692.898	1.95981
$51$	−20.6527	−0.0567051
$52$	−200.196	−0.533888
$53$	−190.585	−0.493942	−0.246971	−	0.969023i	$-0.579435\pi$
$53$	−190.585	−0.493942	−0.246971	+	0.969023i	$0.579435\pi$
$54$	−130.608	−0.329138
$55$	84.6189	0.207455
$56$	0	0
$57$	−366.841	−0.852443
$58$	991.684	2.24508
$59$	394.926	0.871440	0.435720	−	0.900082i	$-0.356494\pi$
$59$	394.926	0.871440	0.435720	+	0.900082i	$0.356494\pi$
$60$	−756.650	−1.62805
$61$	−1.15246	−0.00241896	−0.00120948	−	0.999999i	$-0.500385\pi$
$61$	−1.15246	−0.00241896	−0.00120948	+	0.999999i	$0.500385\pi$
$62$	840.851	1.72239
$63$	0	0
$64$	−615.943	−1.20301
$65$	−212.914	−0.406289
$66$	−74.9777	−0.139835
$67$	−656.911	−1.19783	−0.598914	−	0.800814i	$-0.704401\pi$
$67$	−656.911	−1.19783	−0.598914	+	0.800814i	$0.704401\pi$
$68$	106.015	0.189062
$69$	212.087	0.370033
$70$	0	0
$71$	260.791	0.435919	0.217960	−	0.975958i	$-0.430060\pi$
$71$	260.791	0.435919	0.217960	+	0.975958i	$0.430060\pi$
$72$	322.153	0.527306
$73$	1065.83	1.70885	0.854427	−	0.519572i	$-0.173909\pi$
$73$	1065.83	1.70885	0.854427	+	0.519572i	$0.173909\pi$
$74$	−798.833	−1.25490
$75$	−429.720	−0.661597
$76$	1883.08	2.84216
$77$	0	0
$78$	188.656	0.273860
$79$	356.419	0.507598	0.253799	−	0.967257i	$-0.418320\pi$
$79$	356.419	0.507598	0.253799	+	0.967257i	$0.418320\pi$
$80$	818.133	1.14338
$81$	81.0000	0.111111
$82$	−563.942	−0.759475
$83$	804.148	1.06345	0.531727	−	0.846916i	$-0.321543\pi$
$83$	804.148	1.06345	0.531727	+	0.846916i	$0.321543\pi$
$84$	0	0
$85$	112.750	0.143876
$86$	475.490	0.596203
$87$	−615.020	−0.757898
$88$	184.937	0.224027
$89$	−645.066	−0.768279	−0.384139	−	0.923275i	$-0.625502\pi$
$89$	−645.066	−0.768279	−0.384139	+	0.923275i	$0.625502\pi$
$90$	713.033	0.835114
$91$	0	0
$92$	−1088.69	−1.23374
$93$	−521.477	−0.581448
$94$	2704.57	2.96761
$95$	2002.71	2.16288
$96$	134.156	0.142627
$97$	−1002.09	−1.04894	−0.524468	−	0.851430i	$-0.675736\pi$
$97$	−1002.09	−1.04894	−0.524468	+	0.851430i	$0.675736\pi$
$98$	0	0
$99$	46.4995	0.0472058
$100$	2205.85	2.20585
$101$	1642.29	1.61796	0.808979	−	0.587838i	$-0.200021\pi$
$101$	1642.29	1.61796	0.808979	+	0.587838i	$0.200021\pi$
$102$	−99.9038	−0.0969799
$103$	−1446.23	−1.38351	−0.691753	−	0.722134i	$-0.743161\pi$
$103$	−1446.23	−1.38351	−0.691753	+	0.722134i	$0.743161\pi$
$104$	−465.331	−0.438745
$105$	0	0
$106$	−921.923	−0.844765
$107$	−567.920	−0.513112	−0.256556	−	0.966529i	$-0.582588\pi$
$107$	−567.920	−0.513112	−0.256556	+	0.966529i	$0.582588\pi$
$108$	−415.792	−0.370459
$109$	1808.43	1.58914	0.794568	−	0.607175i	$-0.207697\pi$
$109$	1808.43	1.58914	0.794568	+	0.607175i	$0.207697\pi$
$110$	409.329	0.354800
$111$	495.418	0.423631
$112$	0	0
$113$	−1598.23	−1.33052	−0.665260	−	0.746612i	$-0.731679\pi$
$113$	−1598.23	−1.33052	−0.665260	+	0.746612i	$0.731679\pi$
$114$	−1774.53	−1.45789
$115$	−1157.85	−0.938874
$116$	3157.04	2.52693
$117$	−117.000	−0.0924500
$118$	1910.38	1.49038
$119$	0	0
$120$	−1758.74	−1.33792
$121$	−1304.31	−0.979945
$122$	−5.57480	−0.00413704
$123$	349.744	0.256385
$124$	2676.86	1.93862
$125$	298.734	0.213757
$126$	0	0
$127$	1555.80	1.08705	0.543524	−	0.839393i	$-0.317090\pi$
$127$	1555.80	1.08705	0.543524	+	0.839393i	$0.317090\pi$
$128$	−2621.77	−1.81042
$129$	−294.888	−0.201267
$130$	−1029.94	−0.694857
$131$	−724.411	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$131$	−724.411	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$132$	−238.693	−0.157390
$133$	0	0
$134$	−3177.69	−2.04859
$135$	−442.207	−0.281919
$136$	246.419	0.155370
$137$	−1706.15	−1.06399	−0.531995	−	0.846748i	$-0.678557\pi$
$137$	−1706.15	−1.06399	−0.531995	+	0.846748i	$0.678557\pi$
$138$	1025.93	0.632849
$139$	−2998.76	−1.82987	−0.914934	−	0.403605i	$-0.867757\pi$
$139$	−2998.76	−1.82987	−0.914934	+	0.403605i	$0.867757\pi$
$140$	0	0
$141$	−1677.31	−1.00181
$142$	1261.53	0.745532
$143$	−67.1659	−0.0392776
$144$	449.578	0.260172
$145$	3357.61	1.92299
$146$	5155.78	2.92257
$147$	0	0
$148$	−2543.10	−1.41244
$149$	−1404.51	−0.772228	−0.386114	−	0.922451i	$-0.626183\pi$
$149$	−1404.51	−0.772228	−0.386114	+	0.922451i	$0.626183\pi$
$150$	−2078.69	−1.13150
$151$	−918.641	−0.495086	−0.247543	−	0.968877i	$-0.579623\pi$
$151$	−918.641	−0.495086	−0.247543	+	0.968877i	$0.579623\pi$
$152$	4376.99	2.33566
$153$	61.9581	0.0327387
$154$	0	0
$155$	2846.92	1.47529
$156$	600.588	0.308241
$157$	1089.66	0.553913	0.276957	−	0.960882i	$-0.410674\pi$
$157$	1089.66	0.553913	0.276957	+	0.960882i	$0.410674\pi$
$158$	1724.11	0.868121
$159$	571.756	0.285177
$160$	−732.402	−0.361884
$161$	0	0
$162$	391.823	0.190028
$163$	−1777.50	−0.854136	−0.427068	−	0.904219i	$-0.640454\pi$
$163$	−1777.50	−0.854136	−0.427068	+	0.904219i	$0.640454\pi$
$164$	−1795.32	−0.854822
$165$	−253.857	−0.119774
$166$	3889.92	1.81877
$167$	−1165.78	−0.540182	−0.270091	−	0.962835i	$-0.587054\pi$
$167$	−1165.78	−0.540182	−0.270091	+	0.962835i	$0.587054\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	545.409	0.246065
$171$	1100.52	0.492158
$172$	1513.73	0.671052
$173$	716.703	0.314971	0.157485	−	0.987521i	$-0.449661\pi$
$173$	716.703	0.314971	0.157485	+	0.987521i	$0.449661\pi$
$174$	−2975.05	−1.29620
$175$	0	0
$176$	258.088	0.110535
$177$	−1184.78	−0.503126
$178$	−3120.39	−1.31395
$179$	1614.59	0.674189	0.337094	−	0.941471i	$-0.390556\pi$
$179$	1614.59	0.674189	0.337094	+	0.941471i	$0.390556\pi$
$180$	2269.95	0.939956
$181$	525.461	0.215786	0.107893	−	0.994163i	$-0.465590\pi$
$181$	525.461	0.215786	0.107893	+	0.994163i	$0.465590\pi$
$182$	0	0
$183$	3.45737	0.00139659
$184$	−2530.53	−1.01388
$185$	−2704.66	−1.07487
$186$	−2522.55	−0.994422
$187$	35.5682	0.0139091
$188$	8610.04	3.34017
$189$	0	0
$190$	9687.75	3.69907
$191$	2689.87	1.01902	0.509509	−	0.860465i	$-0.329827\pi$
$191$	2689.87	1.01902	0.509509	+	0.860465i	$0.329827\pi$
$192$	1847.83	0.694560
$193$	4306.51	1.60616	0.803081	−	0.595870i	$-0.203193\pi$
$193$	4306.51	1.60616	0.803081	+	0.595870i	$0.203193\pi$
$194$	−4847.43	−1.79394
$195$	638.743	0.234571
$196$	0	0
$197$	4743.03	1.71536	0.857682	−	0.514180i	$-0.171904\pi$
$197$	4743.03	1.71536	0.857682	+	0.514180i	$0.171904\pi$
$198$	224.933	0.0807338
$199$	4212.91	1.50073	0.750366	−	0.661023i	$-0.229877\pi$
$199$	4212.91	1.50073	0.750366	+	0.661023i	$0.229877\pi$
$200$	5127.23	1.81275
$201$	1970.73	0.691566
$202$	7944.28	2.76712
$203$	0	0
$204$	−318.045	−0.109155
$205$	−1909.37	−0.650519
$206$	−6995.88	−2.36614
$207$	−636.260	−0.213638
$208$	−649.390	−0.216476
$209$	631.774	0.209094
$210$	0	0
$211$	−2657.69	−0.867122	−0.433561	−	0.901124i	$-0.642743\pi$
$211$	−2657.69	−0.867122	−0.433561	+	0.901124i	$0.642743\pi$
$212$	−2934.96	−0.950819
$213$	−782.374	−0.251678
$214$	−2747.21	−0.877550
$215$	1609.90	0.510670
$216$	−966.458	−0.304440
$217$	0	0
$218$	8747.94	2.71782
$219$	−3197.50	−0.986607
$220$	1303.11	0.399343
$221$	−89.4951	−0.0272402
$222$	2396.50	0.724515
$223$	1982.11	0.595211	0.297605	−	0.954689i	$-0.403812\pi$
$223$	1982.11	0.595211	0.297605	+	0.954689i	$0.403812\pi$
$224$	0	0
$225$	1289.16	0.381973
$226$	−7731.15	−2.27553
$227$	−4680.70	−1.36859	−0.684293	−	0.729207i	$-0.739889\pi$
$227$	−4680.70	−1.36859	−0.684293	+	0.729207i	$0.739889\pi$
$228$	−5649.24	−1.64092
$229$	6347.83	1.83177	0.915887	−	0.401435i	$-0.131489\pi$
$229$	6347.83	1.83177	0.915887	+	0.401435i	$0.131489\pi$
$230$	−5600.92	−1.60571
$231$	0	0
$232$	7338.16	2.07661
$233$	5132.05	1.44297	0.721484	−	0.692431i	$-0.243460\pi$
$233$	5132.05	1.44297	0.721484	+	0.692431i	$0.243460\pi$
$234$	−565.967	−0.158113
$235$	9157.03	2.54187
$236$	6081.74	1.67749
$237$	−1069.26	−0.293062
$238$	0	0
$239$	−4937.46	−1.33631	−0.668155	−	0.744022i	$-0.732916\pi$
$239$	−4937.46	−1.33631	−0.668155	+	0.744022i	$0.732916\pi$
$240$	−2454.40	−0.660128
$241$	5595.12	1.49549	0.747745	−	0.663986i	$-0.231137\pi$
$241$	5595.12	1.49549	0.747745	+	0.663986i	$0.231137\pi$
$242$	−6309.35	−1.67595
$243$	−243.000	−0.0641500
$244$	−17.7475	−0.00465641
$245$	0	0
$246$	1691.82	0.438483
$247$	−1589.64	−0.409500
$248$	6222.04	1.59315
$249$	−2412.44	−0.613986
$250$	1445.07	0.365578
$251$	−6375.31	−1.60321	−0.801605	−	0.597853i	$-0.796020\pi$
$251$	−6375.31	−1.60321	−0.801605	+	0.597853i	$0.796020\pi$
$252$	0	0
$253$	−365.257	−0.0907648
$254$	7525.92	1.85913
$255$	−338.251	−0.0830670
$256$	−7754.79	−1.89326
$257$	−4821.50	−1.17026	−0.585130	−	0.810940i	$-0.698956\pi$
$257$	−4821.50	−1.17026	−0.585130	+	0.810940i	$0.698956\pi$
$258$	−1426.47	−0.344218
$259$	0	0
$260$	−3278.82	−0.782091
$261$	1845.06	0.437573
$262$	−3504.21	−0.826300
$263$	−2068.59	−0.484998	−0.242499	−	0.970152i	$-0.577967\pi$
$263$	−2068.59	−0.484998	−0.242499	+	0.970152i	$0.577967\pi$
$264$	−554.812	−0.129342
$265$	−3121.41	−0.723573
$266$	0	0
$267$	1935.20	0.443566
$268$	−10116.2	−2.30577
$269$	752.070	0.170463	0.0852315	−	0.996361i	$-0.472837\pi$
$269$	752.070	0.170463	0.0852315	+	0.996361i	$0.472837\pi$
$270$	−2139.10	−0.482153
$271$	4367.60	0.979015	0.489507	−	0.871999i	$-0.337177\pi$
$271$	4367.60	0.979015	0.489507	+	0.871999i	$0.337177\pi$
$272$	343.889	0.0766593
$273$	0	0
$274$	−8253.22	−1.81969
$275$	740.065	0.162282
$276$	3266.07	0.712299
$277$	713.426	0.154750	0.0773748	−	0.997002i	$-0.475346\pi$
$277$	713.426	0.154750	0.0773748	+	0.997002i	$0.475346\pi$
$278$	−14506.0	−3.12953
$279$	1564.43	0.335699
$280$	0	0
$281$	858.912	0.182343	0.0911715	−	0.995835i	$-0.470939\pi$
$281$	858.912	0.182343	0.0911715	+	0.995835i	$0.470939\pi$
$282$	−8113.71	−1.71335
$283$	−9354.01	−1.96480	−0.982400	−	0.186792i	$-0.940191\pi$
$283$	−9354.01	−1.96480	−0.982400	+	0.186792i	$0.940191\pi$
$284$	4016.11	0.839128
$285$	−6008.13	−1.24874
$286$	−324.903	−0.0671746
$287$	0	0
$288$	−402.467	−0.0823459
$289$	−4865.61	−0.990354
$290$	16241.8	3.28880
$291$	3006.27	0.605603
$292$	16413.5	3.28948
$293$	7360.81	1.46766	0.733828	−	0.679335i	$-0.237732\pi$
$293$	7360.81	1.46766	0.733828	+	0.679335i	$0.237732\pi$
$294$	0	0
$295$	6468.11	1.27657
$296$	−5911.12	−1.16073
$297$	−139.498	−0.0272543
$298$	−6794.07	−1.32070
$299$	919.043	0.177758
$300$	−6617.55	−1.27355
$301$	0	0
$302$	−4443.76	−0.846721
$303$	−4926.86	−0.934128
$304$	6108.27	1.15241
$305$	−18.8749	−0.00354353
$306$	299.711	0.0559914
$307$	−8873.13	−1.64956	−0.824782	−	0.565451i	$-0.808702\pi$
$307$	−8873.13	−1.64956	−0.824782	+	0.565451i	$0.808702\pi$
$308$	0	0
$309$	4338.69	0.798768
$310$	13771.5	2.52312
$311$	−9942.02	−1.81273	−0.906367	−	0.422491i	$-0.861156\pi$
$311$	−9942.02	−1.81273	−0.906367	+	0.422491i	$0.861156\pi$
$312$	1395.99	0.253310
$313$	1301.41	0.235016	0.117508	−	0.993072i	$-0.462509\pi$
$313$	1301.41	0.235016	0.117508	+	0.993072i	$0.462509\pi$
$314$	5271.04	0.947331
$315$	0	0
$316$	5488.74	0.977108
$317$	3168.01	0.561303	0.280652	−	0.959810i	$-0.409449\pi$
$317$	3168.01	0.561303	0.280652	+	0.959810i	$0.409449\pi$
$318$	2765.77	0.487725
$319$	1059.19	0.185904
$320$	−10087.9	−1.76229
$321$	1703.76	0.296245
$322$	0	0
$323$	841.806	0.145013
$324$	1247.38	0.213885
$325$	−1862.12	−0.317821
$326$	−8598.32	−1.46079
$327$	−5425.28	−0.917488
$328$	−4173.00	−0.702485
$329$	0	0
$330$	−1227.99	−0.204844
$331$	−595.878	−0.0989499	−0.0494750	−	0.998775i	$-0.515755\pi$
$331$	−595.878	−0.0989499	−0.0494750	+	0.998775i	$0.515755\pi$
$332$	12383.6	2.04711
$333$	−1486.25	−0.244583
$334$	−5639.24	−0.923848
$335$	−10758.9	−1.75469
$336$	0	0
$337$	−9176.25	−1.48327	−0.741636	−	0.670803i	$-0.765950\pi$
$337$	−9176.25	−1.48327	−0.741636	+	0.670803i	$0.765950\pi$
$338$	817.508	0.131558
$339$	4794.69	0.768176
$340$	1736.32	0.276956
$341$	898.090	0.142623
$342$	5323.58	0.841714
$343$	0	0
$344$	3518.48	0.551465
$345$	3473.56	0.542059
$346$	3466.93	0.538679
$347$	2898.12	0.448355	0.224178	−	0.974548i	$-0.428030\pi$
$347$	2898.12	0.448355	0.224178	+	0.974548i	$0.428030\pi$
$348$	−9471.13	−1.45892
$349$	783.587	0.120185	0.0600923	−	0.998193i	$-0.480860\pi$
$349$	783.587	0.120185	0.0600923	+	0.998193i	$0.480860\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	−231.044	−0.0349848
$353$	−5240.19	−0.790106	−0.395053	−	0.918658i	$-0.629274\pi$
$353$	−5240.19	−0.790106	−0.395053	+	0.918658i	$0.629274\pi$
$354$	−5731.15	−0.860473
$355$	4271.25	0.638576
$356$	−9933.82	−1.47891
$357$	0	0
$358$	7810.27	1.15303
$359$	3490.98	0.513223	0.256611	−	0.966515i	$-0.417394\pi$
$359$	3490.98	0.513223	0.256611	+	0.966515i	$0.417394\pi$
$360$	5276.22	0.772448
$361$	8093.46	1.17998
$362$	2541.82	0.369048
$363$	3912.92	0.565771
$364$	0	0
$365$	17456.2	2.50329
$366$	16.7244	0.00238852
$367$	−3436.90	−0.488842	−0.244421	−	0.969669i	$-0.578598\pi$
$367$	−3436.90	−0.488842	−0.244421	+	0.969669i	$0.578598\pi$
$368$	−3531.46	−0.500245
$369$	−1049.23	−0.148024
$370$	−13083.3	−1.83829
$371$	0	0
$372$	−8030.59	−1.11927
$373$	−1700.44	−0.236047	−0.118024	−	0.993011i	$-0.537656\pi$
$373$	−1700.44	−0.236047	−0.118024	+	0.993011i	$0.537656\pi$
$374$	172.055	0.0237881
$375$	−896.203	−0.123413
$376$	20013.0	2.74492
$377$	−2665.09	−0.364082
$378$	0	0
$379$	−6419.93	−0.870105	−0.435052	−	0.900405i	$-0.643270\pi$
$379$	−6419.93	−0.870105	−0.435052	+	0.900405i	$0.643270\pi$
$380$	30841.1	4.16346
$381$	−4667.41	−0.627608
$382$	13011.8	1.74278
$383$	−8897.78	−1.18709	−0.593545	−	0.804801i	$-0.702272\pi$
$383$	−8897.78	−1.18709	−0.593545	+	0.804801i	$0.702272\pi$
$384$	7865.30	1.04525
$385$	0	0
$386$	20832.0	2.74694
$387$	884.665	0.116202
$388$	−15431.9	−2.01916
$389$	−2696.07	−0.351404	−0.175702	−	0.984443i	$-0.556219\pi$
$389$	−2696.07	−0.351404	−0.175702	+	0.984443i	$0.556219\pi$
$390$	3089.81	0.401176
$391$	−486.685	−0.0629481
$392$	0	0
$393$	2173.23	0.278944
$394$	22943.6	2.93371
$395$	5837.44	0.743579
$396$	716.078	0.0908694
$397$	−6204.46	−0.784365	−0.392183	−	0.919887i	$-0.628280\pi$
$397$	−6204.46	−0.784365	−0.392183	+	0.919887i	$0.628280\pi$
$398$	20379.2	2.56663
$399$	0	0
$400$	7155.28	0.894410
$401$	6215.81	0.774072	0.387036	−	0.922065i	$-0.373499\pi$
$401$	6215.81	0.774072	0.387036	+	0.922065i	$0.373499\pi$
$402$	9533.07	1.18275
$403$	−2259.73	−0.279319
$404$	25290.7	3.11451
$405$	1326.62	0.162766
$406$	0	0
$407$	−853.211	−0.103912
$408$	−739.258	−0.0897028
$409$	909.851	0.109998	0.0549991	−	0.998486i	$-0.482484\pi$
$409$	909.851	0.109998	0.0549991	+	0.998486i	$0.482484\pi$
$410$	−9236.25	−1.11255
$411$	5118.46	0.614295
$412$	−22271.5	−2.66320
$413$	0	0
$414$	−3077.80	−0.365376
$415$	13170.4	1.55785
$416$	581.342	0.0685159
$417$	8996.28	1.05647
$418$	3056.10	0.357604
$419$	−5345.25	−0.623228	−0.311614	−	0.950209i	$-0.600870\pi$
$419$	−5345.25	−0.623228	−0.311614	+	0.950209i	$0.600870\pi$
$420$	0	0
$421$	10597.2	1.22679	0.613393	−	0.789778i	$-0.289804\pi$
$421$	10597.2	1.22679	0.613393	+	0.789778i	$0.289804\pi$
$422$	−12856.1	−1.48300
$423$	5031.94	0.578396
$424$	−6821.95	−0.781375
$425$	986.098	0.112548
$426$	−3784.60	−0.430433
$427$	0	0
$428$	−8745.80	−0.987720
$429$	201.498	0.0226769
$430$	7787.60	0.873375
$431$	547.482	0.0611863	0.0305931	−	0.999532i	$-0.490260\pi$
$431$	547.482	0.0611863	0.0305931	+	0.999532i	$0.490260\pi$
$432$	−1348.73	−0.150211
$433$	−2168.27	−0.240648	−0.120324	−	0.992735i	$-0.538393\pi$
$433$	−2168.27	−0.240648	−0.120324	+	0.992735i	$0.538393\pi$
$434$	0	0
$435$	−10072.8	−1.11024
$436$	27849.2	3.05903
$437$	−8644.67	−0.946295
$438$	−15467.3	−1.68735
$439$	10369.3	1.12734	0.563668	−	0.826001i	$-0.309390\pi$
$439$	10369.3	1.12734	0.563668	+	0.826001i	$0.309390\pi$
$440$	3028.91	0.328176
$441$	0	0
$442$	−432.917	−0.0465877
$443$	−8216.42	−0.881205	−0.440603	−	0.897702i	$-0.645235\pi$
$443$	−8216.42	−0.881205	−0.440603	+	0.897702i	$0.645235\pi$
$444$	7629.29	0.815473
$445$	−10564.9	−1.12545
$446$	9588.12	1.01796
$447$	4213.53	0.445846
$448$	0	0
$449$	−12142.2	−1.27623	−0.638113	−	0.769942i	$-0.720285\pi$
$449$	−12142.2	−1.27623	−0.638113	+	0.769942i	$0.720285\pi$
$450$	6236.08	0.653270
$451$	−602.330	−0.0628883
$452$	−24612.3	−2.56120
$453$	2755.92	0.285838
$454$	−22642.1	−2.34063
$455$	0	0
$456$	−13131.0	−1.34849
$457$	−8647.67	−0.885166	−0.442583	−	0.896728i	$-0.645938\pi$
$457$	−8647.67	−0.885166	−0.442583	+	0.896728i	$0.645938\pi$
$458$	30706.5	3.13280
$459$	−185.874	−0.0189017
$460$	−17830.6	−1.80730
$461$	−3096.97	−0.312885	−0.156443	−	0.987687i	$-0.550003\pi$
$461$	−3096.97	−0.312885	−0.156443	+	0.987687i	$0.550003\pi$
$462$	0	0
$463$	−2933.61	−0.294463	−0.147231	−	0.989102i	$-0.547036\pi$
$463$	−2933.61	−0.294463	−0.147231	+	0.989102i	$0.547036\pi$
$464$	10240.7	1.02460
$465$	−8540.77	−0.851760
$466$	24825.4	2.46784
$467$	−2017.58	−0.199919	−0.0999597	−	0.994991i	$-0.531871\pi$
$467$	−2017.58	−0.199919	−0.0999597	+	0.994991i	$0.531871\pi$
$468$	−1801.76	−0.177963
$469$	0	0
$470$	44295.5	4.34723
$471$	−3268.98	−0.319802
$472$	14136.3	1.37855
$473$	507.858	0.0493686
$474$	−5172.34	−0.501210
$475$	17515.4	1.69192
$476$	0	0
$477$	−1715.27	−0.164647
$478$	−23884.1	−2.28543
$479$	−10221.6	−0.975025	−0.487512	−	0.873116i	$-0.662096\pi$
$479$	−10221.6	−0.975025	−0.487512	+	0.873116i	$0.662096\pi$
$480$	2197.21	0.208934
$481$	2146.81	0.203506
$482$	27065.4	2.55766
$483$	0	0
$484$	−20085.9	−1.88636
$485$	−16412.2	−1.53658
$486$	−1175.47	−0.109713
$487$	−8120.41	−0.755587	−0.377794	−	0.925890i	$-0.623317\pi$
$487$	−8120.41	−0.755587	−0.377794	+	0.925890i	$0.623317\pi$
$488$	−41.2518	−0.00382660
$489$	5332.49	0.493136
$490$	0	0
$491$	−10516.9	−0.966646	−0.483323	−	0.875442i	$-0.660570\pi$
$491$	−10516.9	−0.966646	−0.483323	+	0.875442i	$0.660570\pi$
$492$	5385.95	0.493531
$493$	1411.32	0.128930
$494$	−7689.62	−0.700349
$495$	761.570	0.0691516
$496$	8683.13	0.786056
$497$	0	0
$498$	−11669.8	−1.05007
$499$	−14223.9	−1.27605	−0.638023	−	0.770017i	$-0.720248\pi$
$499$	−14223.9	−1.27605	−0.638023	+	0.770017i	$0.720248\pi$
$500$	4600.42	0.411474
$501$	3497.33	0.311874
$502$	−30839.4	−2.74189
$503$	−19730.7	−1.74901	−0.874503	−	0.485020i	$-0.838812\pi$
$503$	−19730.7	−1.74901	−0.874503	+	0.485020i	$0.838812\pi$
$504$	0	0
$505$	26897.4	2.37014
$506$	−1766.86	−0.155231
$507$	−507.000	−0.0444116
$508$	23958.9	2.09253
$509$	−20325.9	−1.77000	−0.884998	−	0.465595i	$-0.845840\pi$
$509$	−20325.9	−1.77000	−0.884998	+	0.465595i	$0.845840\pi$
$510$	−1636.23	−0.142065
$511$	0	0
$512$	−16538.3	−1.42753
$513$	−3301.57	−0.284148
$514$	−23323.1	−2.00144
$515$	−23686.4	−2.02669
$516$	−4541.19	−0.387432
$517$	2888.68	0.245733
$518$	0	0
$519$	−2150.11	−0.181848
$520$	−7621.21	−0.642716
$521$	16050.6	1.34969	0.674845	−	0.737960i	$-0.264211\pi$
$521$	16050.6	1.34969	0.674845	+	0.737960i	$0.264211\pi$
$522$	8925.16	0.748359
$523$	2211.82	0.184926	0.0924629	−	0.995716i	$-0.470526\pi$
$523$	2211.82	0.184926	0.0924629	+	0.995716i	$0.470526\pi$
$524$	−11155.7	−0.930037
$525$	0	0
$526$	−10006.4	−0.829470
$527$	1196.66	0.0989131
$528$	−774.264	−0.0638173
$529$	−7169.13	−0.589228
$530$	−15099.3	−1.23749
$531$	3554.33	0.290480
$532$	0	0
$533$	1515.56	0.123163
$534$	9361.17	0.758610
$535$	−9301.42	−0.751655
$536$	−23513.9	−1.89487
$537$	−4843.76	−0.389243
$538$	3638.01	0.291535
$539$	0	0
$540$	−6809.85	−0.542684
$541$	−14763.8	−1.17328	−0.586642	−	0.809847i	$-0.699550\pi$
$541$	−14763.8	−1.17328	−0.586642	+	0.809847i	$0.699550\pi$
$542$	21127.5	1.67436
$543$	−1576.38	−0.124584
$544$	−307.853	−0.0242631
$545$	29618.5	2.32792
$546$	0	0
$547$	8994.09	0.703034	0.351517	−	0.936181i	$-0.385666\pi$
$547$	8994.09	0.703034	0.351517	+	0.936181i	$0.385666\pi$
$548$	−26274.3	−2.04814
$549$	−10.3721	−0.000806321 0
$550$	3579.93	0.277543
$551$	25068.3	1.93819
$552$	7591.59	0.585361
$553$	0	0
$554$	3451.07	0.264661
$555$	8113.97	0.620575
$556$	−46180.0	−3.52242
$557$	−3557.63	−0.270631	−0.135316	−	0.990803i	$-0.543205\pi$
$557$	−3557.63	−0.270631	−0.135316	+	0.990803i	$0.543205\pi$
$558$	7567.66	0.574130
$559$	−1277.85	−0.0966857
$560$	0	0
$561$	−106.705	−0.00803042
$562$	4154.83	0.311852
$563$	10820.2	0.809979	0.404990	−	0.914321i	$-0.367275\pi$
$563$	10820.2	0.809979	0.404990	+	0.914321i	$0.367275\pi$
$564$	−25830.1	−1.92845
$565$	−26175.9	−1.94907
$566$	−45248.4	−3.36030
$567$	0	0
$568$	9334.96	0.689588
$569$	16487.5	1.21475	0.607374	−	0.794416i	$-0.292223\pi$
$569$	16487.5	1.21475	0.607374	+	0.794416i	$0.292223\pi$
$570$	−29063.3	−2.13566
$571$	−3472.27	−0.254484	−0.127242	−	0.991872i	$-0.540612\pi$
$571$	−3472.27	−0.254484	−0.127242	+	0.991872i	$0.540612\pi$
$572$	−1034.34	−0.0756079
$573$	−8069.62	−0.588330
$574$	0	0
$575$	−10126.4	−0.734437
$576$	−5543.49	−0.401005
$577$	−13174.6	−0.950545	−0.475273	−	0.879839i	$-0.657651\pi$
$577$	−13174.6	−0.950545	−0.475273	+	0.879839i	$0.657651\pi$
$578$	−23536.5	−1.69375
$579$	−12919.5	−0.927318
$580$	51706.1	3.70169
$581$	0	0
$582$	14542.3	1.03573
$583$	−984.680	−0.0699508
$584$	38151.2	2.70327
$585$	−1916.23	−0.135430
$586$	35606.6	2.51006
$587$	−14194.6	−0.998085	−0.499042	−	0.866578i	$-0.666315\pi$
$587$	−14194.6	−0.998085	−0.499042	+	0.866578i	$0.666315\pi$
$588$	0	0
$589$	21255.4	1.48695
$590$	31288.3	2.18325
$591$	−14229.1	−0.990366
$592$	−8249.22	−0.572704
$593$	−9446.52	−0.654169	−0.327084	−	0.944995i	$-0.606066\pi$
$593$	−9446.52	−0.654169	−0.327084	+	0.944995i	$0.606066\pi$
$594$	−674.799	−0.0466117
$595$	0	0
$596$	−21629.0	−1.48651
$597$	−12638.7	−0.866448
$598$	4445.71	0.304011
$599$	7467.14	0.509347	0.254674	−	0.967027i	$-0.418032\pi$
$599$	7467.14	0.509347	0.254674	+	0.967027i	$0.418032\pi$
$600$	−15381.7	−1.04659
$601$	1115.01	0.0756779	0.0378389	−	0.999284i	$-0.487953\pi$
$601$	1115.01	0.0756779	0.0378389	+	0.999284i	$0.487953\pi$
$602$	0	0
$603$	−5912.20	−0.399276
$604$	−14146.8	−0.953021
$605$	−21362.0	−1.43552
$606$	−23832.8	−1.59759
$607$	14706.2	0.983374	0.491687	−	0.870772i	$-0.336380\pi$
$607$	14706.2	0.983374	0.491687	+	0.870772i	$0.336380\pi$
$608$	−5468.20	−0.364745
$609$	0	0
$610$	−91.3042	−0.00606033
$611$	−7268.36	−0.481254
$612$	954.136	0.0630207
$613$	23507.6	1.54888	0.774438	−	0.632650i	$-0.218033\pi$
$613$	23507.6	1.54888	0.774438	+	0.632650i	$0.218033\pi$
$614$	−42922.2	−2.82117
$615$	5728.12	0.375577
$616$	0	0
$617$	−18523.1	−1.20861	−0.604306	−	0.796753i	$-0.706549\pi$
$617$	−18523.1	−1.20861	−0.604306	+	0.796753i	$0.706549\pi$
$618$	20987.6	1.36609
$619$	15470.8	1.00456	0.502280	−	0.864705i	$-0.332495\pi$
$619$	15470.8	1.00456	0.502280	+	0.864705i	$0.332495\pi$
$620$	43841.7	2.83988
$621$	1908.78	0.123344
$622$	−48092.8	−3.10023
$623$	0	0
$624$	1948.17	0.124983
$625$	−13012.3	−0.832788
$626$	6295.33	0.401936
$627$	−1895.32	−0.120721
$628$	16780.4	1.06626
$629$	−1136.86	−0.0720660
$630$	0	0
$631$	25835.8	1.62996	0.814982	−	0.579486i	$-0.196747\pi$
$631$	25835.8	1.62996	0.814982	+	0.579486i	$0.196747\pi$
$632$	12757.9	0.802979
$633$	7973.07	0.500633
$634$	15324.7	0.959970
$635$	25481.0	1.59241
$636$	8804.87	0.548956
$637$	0	0
$638$	5123.65	0.317942
$639$	2347.12	0.145306
$640$	−42939.4	−2.65207
$641$	−14716.1	−0.906789	−0.453394	−	0.891310i	$-0.649787\pi$
$641$	−14716.1	−0.906789	−0.453394	+	0.891310i	$0.649787\pi$
$642$	8241.64	0.506654
$643$	15531.5	0.952573	0.476286	−	0.879290i	$-0.341983\pi$
$643$	15531.5	0.952573	0.476286	+	0.879290i	$0.341983\pi$
$644$	0	0
$645$	−4829.69	−0.294836
$646$	4072.09	0.248010
$647$	−31042.5	−1.88626	−0.943128	−	0.332431i	$-0.892131\pi$
$647$	−31042.5	−1.88626	−0.943128	+	0.332431i	$0.892131\pi$
$648$	2899.37	0.175769
$649$	2040.43	0.123411
$650$	−9007.67	−0.543554
$651$	0	0
$652$	−27372.9	−1.64418
$653$	−21968.5	−1.31653	−0.658265	−	0.752786i	$-0.728709\pi$
$653$	−21968.5	−1.31653	−0.658265	+	0.752786i	$0.728709\pi$
$654$	−26243.8	−1.56914
$655$	−11864.4	−0.707758
$656$	−5823.60	−0.346606
$657$	9592.49	0.569618
$658$	0	0
$659$	−3283.83	−0.194112	−0.0970560	−	0.995279i	$-0.530943\pi$
$659$	−3283.83	−0.194112	−0.0970560	+	0.995279i	$0.530943\pi$
$660$	−3909.32	−0.230561
$661$	15962.2	0.939268	0.469634	−	0.882861i	$-0.344386\pi$
$661$	15962.2	0.939268	0.469634	+	0.882861i	$0.344386\pi$
$662$	−2882.46	−0.169229
$663$	268.485	0.0157272
$664$	28784.3	1.68230
$665$	0	0
$666$	−7189.49	−0.418299
$667$	−14493.1	−0.841341
$668$	−17952.6	−1.03983
$669$	−5946.34	−0.343645
$670$	−52044.3	−3.00097
$671$	−5.95429	−0.000342567 0
$672$	0	0
$673$	−26354.1	−1.50948	−0.754738	−	0.656027i	$-0.772236\pi$
$673$	−26354.1	−1.50948	−0.754738	+	0.656027i	$0.772236\pi$
$674$	−44388.5	−2.53677
$675$	−3867.48	−0.220532
$676$	2602.55	0.148074
$677$	29401.7	1.66913	0.834564	−	0.550912i	$-0.185720\pi$
$677$	29401.7	1.66913	0.834564	+	0.550912i	$0.185720\pi$
$678$	23193.5	1.31378
$679$	0	0
$680$	4035.86	0.227600
$681$	14042.1	0.790154
$682$	4344.35	0.243920
$683$	2437.56	0.136560	0.0682801	−	0.997666i	$-0.478249\pi$
$683$	2437.56	0.136560	0.0682801	+	0.997666i	$0.478249\pi$
$684$	16947.7	0.947386
$685$	−27943.4	−1.55863
$686$	0	0
$687$	−19043.5	−1.05758
$688$	4910.20	0.272092
$689$	2477.61	0.136995
$690$	16802.8	0.927058
$691$	20802.5	1.14524	0.572622	−	0.819819i	$-0.305926\pi$
$691$	20802.5	1.14524	0.572622	+	0.819819i	$0.305926\pi$
$692$	11037.0	0.606307
$693$	0	0
$694$	14019.1	0.766800
$695$	−49113.8	−2.68056
$696$	−22014.5	−1.19893
$697$	−802.574	−0.0436150
$698$	3790.46	0.205546
$699$	−15396.1	−0.833098
$700$	0	0
$701$	−11644.1	−0.627376	−0.313688	−	0.949526i	$-0.601565\pi$
$701$	−11644.1	−0.627376	−0.313688	+	0.949526i	$0.601565\pi$
$702$	1697.90	0.0912865
$703$	−20193.3	−1.08336
$704$	−3182.34	−0.170368
$705$	−27471.1	−1.46755
$706$	−25348.5	−1.35128
$707$	0	0
$708$	−18245.2	−0.968499
$709$	−7851.47	−0.415893	−0.207947	−	0.978140i	$-0.566678\pi$
$709$	−7851.47	−0.415893	−0.207947	+	0.978140i	$0.566678\pi$
$710$	20661.4	1.09213
$711$	3207.77	0.169199
$712$	−23090.0	−1.21535
$713$	−12288.7	−0.645464
$714$	0	0
$715$	−1100.05	−0.0575376
$716$	24864.1	1.29779
$717$	14812.4	0.771519
$718$	16887.0	0.877740
$719$	−8769.51	−0.454864	−0.227432	−	0.973794i	$-0.573033\pi$
$719$	−8769.51	−0.454864	−0.227432	+	0.973794i	$0.573033\pi$
$720$	7363.20	0.381125
$721$	0	0
$722$	39150.7	2.01806
$723$	−16785.3	−0.863421
$724$	8091.93	0.415379
$725$	29365.2	1.50427
$726$	18928.1	0.967611
$727$	−1063.85	−0.0542726	−0.0271363	−	0.999632i	$-0.508639\pi$
$727$	−1063.85	−0.0542726	−0.0271363	+	0.999632i	$0.508639\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	84441.5	4.28126
$731$	676.694	0.0342386
$732$	53.2424	0.00268838
$733$	−10750.5	−0.541716	−0.270858	−	0.962619i	$-0.587307\pi$
$733$	−10750.5	−0.541716	−0.270858	+	0.962619i	$0.587307\pi$
$734$	−16625.4	−0.836042
$735$	0	0
$736$	3161.41	0.158330
$737$	−3394.00	−0.169633
$738$	−5075.47	−0.253158
$739$	17099.4	0.851165	0.425583	−	0.904920i	$-0.360069\pi$
$739$	17099.4	0.851165	0.425583	+	0.904920i	$0.360069\pi$
$740$	−41650.9	−2.06908
$741$	4768.93	0.236425
$742$	0	0
$743$	−35309.8	−1.74346	−0.871730	−	0.489986i	$-0.837002\pi$
$743$	−35309.8	−1.74346	−0.871730	+	0.489986i	$0.837002\pi$
$744$	−18666.1	−0.919803
$745$	−23003.1	−1.13123
$746$	−8225.60	−0.403700
$747$	7237.33	0.354485
$748$	547.739	0.0267745
$749$	0	0
$750$	−4335.22	−0.211067
$751$	−31626.2	−1.53669	−0.768346	−	0.640035i	$-0.778920\pi$
$751$	−31626.2	−1.53669	−0.768346	+	0.640035i	$0.778920\pi$
$752$	27929.0	1.35434
$753$	19125.9	0.925614
$754$	−12891.9	−0.622673
$755$	−15045.5	−0.725249
$756$	0	0
$757$	8789.00	0.421984	0.210992	−	0.977488i	$-0.432331\pi$
$757$	8789.00	0.421984	0.210992	+	0.977488i	$0.432331\pi$
$758$	−31055.3	−1.48810
$759$	1095.77	0.0524031
$760$	71686.4	3.42150
$761$	−8178.79	−0.389594	−0.194797	−	0.980844i	$-0.562405\pi$
$761$	−8178.79	−0.389594	−0.194797	+	0.980844i	$0.562405\pi$
$762$	−22577.8	−1.07337
$763$	0	0
$764$	41423.2	1.96157
$765$	1014.75	0.0479587
$766$	−43041.4	−2.03022
$767$	−5134.04	−0.241694
$768$	23264.4	1.09307
$769$	−13503.3	−0.633216	−0.316608	−	0.948556i	$-0.602544\pi$
$769$	−13503.3	−0.633216	−0.316608	+	0.948556i	$0.602544\pi$
$770$	0	0
$771$	14464.5	0.675650
$772$	66318.9	3.09180
$773$	4110.06	0.191240	0.0956200	−	0.995418i	$-0.469517\pi$
$773$	4110.06	0.191240	0.0956200	+	0.995418i	$0.469517\pi$
$774$	4279.41	0.198734
$775$	24898.8	1.15405
$776$	−35869.5	−1.65933
$777$	0	0
$778$	−13041.8	−0.600989
$779$	−14255.6	−0.655661
$780$	9836.45	0.451540
$781$	1347.41	0.0617337
$782$	−2354.25	−0.107657
$783$	−5535.18	−0.252633
$784$	0	0
$785$	17846.5	0.811425
$786$	10512.6	0.477065
$787$	27340.0	1.23833	0.619166	−	0.785260i	$-0.287471\pi$
$787$	27340.0	1.23833	0.619166	+	0.785260i	$0.287471\pi$
$788$	73041.2	3.30201
$789$	6205.76	0.280014
$790$	28237.6	1.27171
$791$	0	0
$792$	1664.44	0.0746757
$793$	14.9819	0.000670900 0
$794$	−30013.0	−1.34146
$795$	9364.24	0.417755
$796$	64877.6	2.88885
$797$	4514.37	0.200637	0.100318	−	0.994955i	$-0.468014\pi$
$797$	4514.37	0.200637	0.100318	+	0.994955i	$0.468014\pi$
$798$	0	0
$799$	3849.01	0.170423
$800$	−6405.49	−0.283085
$801$	−5805.59	−0.256093
$802$	30067.9	1.32386
$803$	5506.74	0.242003
$804$	30348.7	1.33124
$805$	0	0
$806$	−10931.1	−0.477705
$807$	−2256.21	−0.0984168
$808$	58785.2	2.55948
$809$	22472.8	0.976641	0.488320	−	0.872664i	$-0.337610\pi$
$809$	22472.8	0.976641	0.488320	+	0.872664i	$0.337610\pi$
$810$	6417.29	0.278371
$811$	−26153.6	−1.13240	−0.566199	−	0.824268i	$-0.691587\pi$
$811$	−26153.6	−1.13240	−0.566199	+	0.824268i	$0.691587\pi$
$812$	0	0
$813$	−13102.8	−0.565235
$814$	−4127.26	−0.177715
$815$	−29111.9	−1.25122
$816$	−1031.67	−0.0442593
$817$	12019.7	0.514707
$818$	4401.25	0.188125
$819$	0	0
$820$	−29403.8	−1.25222
$821$	31826.7	1.35294	0.676468	−	0.736472i	$-0.263510\pi$
$821$	31826.7	1.35294	0.676468	+	0.736472i	$0.263510\pi$
$822$	24759.7	1.05060
$823$	−22129.9	−0.937303	−0.468652	−	0.883383i	$-0.655260\pi$
$823$	−22129.9	−0.937303	−0.468652	+	0.883383i	$0.655260\pi$
$824$	−51767.4	−2.18859
$825$	−2220.19	−0.0936937
$826$	0	0
$827$	−14089.5	−0.592431	−0.296216	−	0.955121i	$-0.595725\pi$
$827$	−14089.5	−0.592431	−0.296216	+	0.955121i	$0.595725\pi$
$828$	−9798.22	−0.411246
$829$	−19178.8	−0.803505	−0.401753	−	0.915748i	$-0.631599\pi$
$829$	−19178.8	−0.803505	−0.401753	+	0.915748i	$0.631599\pi$
$830$	63709.3	2.66432
$831$	−2140.28	−0.0893447
$832$	8007.26	0.333656
$833$	0	0
$834$	43517.9	1.80684
$835$	−19093.1	−0.791311
$836$	9729.13	0.402499
$837$	−4693.29	−0.193816
$838$	−25856.7	−1.06588
$839$	−12029.5	−0.494998	−0.247499	−	0.968888i	$-0.579609\pi$
$839$	−12029.5	−0.494998	−0.247499	+	0.968888i	$0.579609\pi$
$840$	0	0
$841$	17638.8	0.723227
$842$	51262.2	2.09811
$843$	−2576.73	−0.105276
$844$	−40927.6	−1.66918
$845$	2767.89	0.112684
$846$	24341.1	0.989202
$847$	0	0
$848$	−9520.33	−0.385530
$849$	28062.0	1.13438
$850$	4770.07	0.192485
$851$	11674.6	0.470271
$852$	−12048.3	−0.484471
$853$	31976.7	1.28354	0.641770	−	0.766897i	$-0.278200\pi$
$853$	31976.7	1.28354	0.641770	+	0.766897i	$0.278200\pi$
$854$	0	0
$855$	18024.4	0.720960
$856$	−20328.6	−0.811701
$857$	22460.8	0.895270	0.447635	−	0.894216i	$-0.352266\pi$
$857$	22460.8	0.895270	0.447635	+	0.894216i	$0.352266\pi$
$858$	974.710	0.0387833
$859$	−7553.74	−0.300035	−0.150018	−	0.988683i	$-0.547933\pi$
$859$	−7553.74	−0.300035	−0.150018	+	0.988683i	$0.547933\pi$
$860$	24791.9	0.983021
$861$	0	0
$862$	2648.35	0.104644
$863$	45888.0	1.81002	0.905008	−	0.425394i	$-0.139864\pi$
$863$	45888.0	1.81002	0.905008	+	0.425394i	$0.139864\pi$
$864$	1207.40	0.0475424
$865$	11738.2	0.461399
$866$	−10488.6	−0.411569
$867$	14596.8	0.571781
$868$	0	0
$869$	1841.48	0.0718848
$870$	−48725.5	−1.89879
$871$	8539.84	0.332218
$872$	64732.1	2.51388
$873$	−9018.80	−0.349645
$874$	−41817.1	−1.61840
$875$	0	0
$876$	−49240.5	−1.89918
$877$	14712.3	0.566476	0.283238	−	0.959050i	$-0.408591\pi$
$877$	14712.3	0.566476	0.283238	+	0.959050i	$0.408591\pi$
$878$	50159.8	1.92803
$879$	−22082.4	−0.847352
$880$	4226.98	0.161922
$881$	40585.8	1.55207	0.776033	−	0.630692i	$-0.217229\pi$
$881$	40585.8	1.55207	0.776033	+	0.630692i	$0.217229\pi$
$882$	0	0
$883$	−44467.7	−1.69474	−0.847370	−	0.531002i	$-0.821816\pi$
$883$	−44467.7	−1.69474	−0.847370	+	0.531002i	$0.821816\pi$
$884$	−1378.20	−0.0524364
$885$	−19404.3	−0.737027
$886$	−39745.5	−1.50708
$887$	−17573.0	−0.665212	−0.332606	−	0.943066i	$-0.607928\pi$
$887$	−17573.0	−0.665212	−0.332606	+	0.943066i	$0.607928\pi$
$888$	17733.4	0.670149
$889$	0	0
$890$	−51105.9	−1.92480
$891$	418.495	0.0157353
$892$	30523.9	1.14576
$893$	68367.4	2.56196
$894$	20382.2	0.762509
$895$	26443.7	0.987617
$896$	0	0
$897$	−2757.13	−0.102629
$898$	−58735.7	−2.18267
$899$	35635.4	1.32203
$900$	19852.7	0.735284
$901$	−1312.03	−0.0485130
$902$	−2913.67	−0.107555
$903$	0	0
$904$	−57208.2	−2.10477
$905$	8606.01	0.316103
$906$	13331.3	0.488855
$907$	−48566.0	−1.77796	−0.888979	−	0.457948i	$-0.848584\pi$
$907$	−48566.0	−1.77796	−0.888979	+	0.457948i	$0.848584\pi$
$908$	−72081.4	−2.63448
$909$	14780.6	0.539319
$910$	0	0
$911$	11129.7	0.404767	0.202384	−	0.979306i	$-0.435131\pi$
$911$	11129.7	0.404767	0.202384	+	0.979306i	$0.435131\pi$
$912$	−18324.8	−0.665346
$913$	4154.72	0.150604
$914$	−41831.6	−1.51386
$915$	56.6248	0.00204586
$916$	97754.7	3.52610
$917$	0	0
$918$	−899.134	−0.0323266
$919$	−15164.2	−0.544309	−0.272155	−	0.962254i	$-0.587736\pi$
$919$	−15164.2	−0.544309	−0.272155	+	0.962254i	$0.587736\pi$
$920$	−41445.1	−1.48522
$921$	26619.4	0.952376
$922$	−14981.0	−0.535113
$923$	−3390.29	−0.120902
$924$	0	0
$925$	−23654.6	−0.840818
$926$	−14190.8	−0.503606
$927$	−13016.1	−0.461169
$928$	−9167.61	−0.324291
$929$	−10914.1	−0.385446	−0.192723	−	0.981253i	$-0.561732\pi$
$929$	−10914.1	−0.385446	−0.192723	+	0.981253i	$0.561732\pi$
$930$	−41314.5	−1.45673
$931$	0	0
$932$	79032.0	2.77766
$933$	29826.1	1.04658
$934$	−9759.67	−0.341913
$935$	582.537	0.0203754
$936$	−4187.98	−0.146248
$937$	32662.9	1.13879	0.569397	−	0.822063i	$-0.307177\pi$
$937$	32662.9	1.13879	0.569397	+	0.822063i	$0.307177\pi$
$938$	0	0
$939$	−3904.22	−0.135686
$940$	141015.	4.89300
$941$	40450.2	1.40132	0.700658	−	0.713497i	$-0.252890\pi$
$941$	40450.2	1.40132	0.700658	+	0.713497i	$0.252890\pi$
$942$	−15813.1	−0.546942
$943$	8241.79	0.284612
$944$	19727.8	0.680174
$945$	0	0
$946$	2456.67	0.0844327
$947$	24891.8	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$947$	24891.8	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$948$	−16466.2	−0.564133
$949$	−13855.8	−0.473951
$950$	84727.7	2.89361
$951$	−9504.03	−0.324069
$952$	0	0
$953$	−18293.1	−0.621796	−0.310898	−	0.950443i	$-0.600630\pi$
$953$	−18293.1	−0.621796	−0.310898	+	0.950443i	$0.600630\pi$
$954$	−8297.31	−0.281588
$955$	44054.8	1.49275
$956$	−76035.5	−2.57235
$957$	−3177.57	−0.107332
$958$	−49445.2	−1.66754
$959$	0	0
$960$	30263.8	1.01746
$961$	424.366	0.0142448
$962$	10384.8	0.348046
$963$	−5111.28	−0.171037
$964$	86163.1	2.87876
$965$	70532.1	2.35286
$966$	0	0
$967$	−1620.93	−0.0539043	−0.0269521	−	0.999637i	$-0.508580\pi$
$967$	−1620.93	−0.0539043	−0.0269521	+	0.999637i	$0.508580\pi$
$968$	−46687.3	−1.55019
$969$	−2525.42	−0.0837235
$970$	−79391.3	−2.62794
$971$	6450.05	0.213174	0.106587	−	0.994303i	$-0.466008\pi$
$971$	6450.05	0.213174	0.106587	+	0.994303i	$0.466008\pi$
$972$	−3742.13	−0.123486
$973$	0	0
$974$	−39281.0	−1.29224
$975$	5586.36	0.183494
$976$	−57.5687	−0.00188804
$977$	7083.99	0.231972	0.115986	−	0.993251i	$-0.462997\pi$
$977$	7083.99	0.231972	0.115986	+	0.993251i	$0.462997\pi$
$978$	25795.0	0.843386
$979$	−3332.80	−0.108802
$980$	0	0
$981$	16275.8	0.529712
$982$	−50873.9	−1.65321
$983$	4587.68	0.148855	0.0744273	−	0.997226i	$-0.476287\pi$
$983$	4587.68	0.148855	0.0744273	+	0.997226i	$0.476287\pi$
$984$	12519.0	0.405580
$985$	77681.5	2.51283
$986$	6826.99	0.220503
$987$	0	0
$988$	−24480.0	−0.788272
$989$	−6949.11	−0.223426
$990$	3683.96	0.118267
$991$	38143.2	1.22266	0.611332	−	0.791374i	$-0.290634\pi$
$991$	38143.2	1.22266	0.611332	+	0.791374i	$0.290634\pi$
$992$	−7773.24	−0.248791
$993$	1787.63	0.0571288
$994$	0	0
$995$	68999.2	2.19841
$996$	−37150.9	−1.18190
$997$	14700.2	0.466962	0.233481	−	0.972361i	$-0.424988\pi$
$997$	14700.2	0.466962	0.233481	+	0.972361i	$0.424988\pi$
$998$	−68805.4	−2.18236
$999$	4458.76	0.141210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.4.a.v.1.11		11
7.2	even	3		273.4.i.d.235.1	yes	22
7.4	even	3		273.4.i.d.79.1	✓	22
7.6	odd	2		1911.4.a.w.1.11		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.4.i.d.79.1	✓	22	7.4	even	3
273.4.i.d.235.1	yes	22	7.2	even	3
1911.4.a.v.1.11		11	1.1	even	1	trivial
1911.4.a.w.1.11		11	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q+4.83732 q^{2} -3.00000 q^{3} +15.3997 q^{4} +16.3780 q^{5} -14.5120 q^{6} +35.7947 q^{8} +9.00000 q^{9} +79.2258 q^{10} +5.16661 q^{11} -46.1991 q^{12} -13.0000 q^{13} -49.1341 q^{15} +49.9531 q^{16} +6.88424 q^{17} +43.5359 q^{18} +122.280 q^{19} +252.217 q^{20} +24.9926 q^{22} -70.6956 q^{23} -107.384 q^{24} +143.240 q^{25} -62.8852 q^{26} -27.0000 q^{27} +205.007 q^{29} -237.678 q^{30} +173.826 q^{31} -44.7186 q^{32} -15.4998 q^{33} +33.3013 q^{34} +138.597 q^{36} -165.139 q^{37} +591.509 q^{38} +39.0000 q^{39} +586.247 q^{40} -116.581 q^{41} +98.2962 q^{43} +79.5642 q^{44} +147.402 q^{45} -341.977 q^{46} +559.104 q^{47} -149.859 q^{48} +692.898 q^{50} -20.6527 q^{51} -200.196 q^{52} -190.585 q^{53} -130.608 q^{54} +84.6189 q^{55} -366.841 q^{57} +991.684 q^{58} +394.926 q^{59} -756.650 q^{60} -1.15246 q^{61} +840.851 q^{62} -615.943 q^{64} -212.914 q^{65} -74.9777 q^{66} -656.911 q^{67} +106.015 q^{68} +212.087 q^{69} +260.791 q^{71} +322.153 q^{72} +1065.83 q^{73} -798.833 q^{74} -429.720 q^{75} +1883.08 q^{76} +188.656 q^{78} +356.419 q^{79} +818.133 q^{80} +81.0000 q^{81} -563.942 q^{82} +804.148 q^{83} +112.750 q^{85} +475.490 q^{86} -615.020 q^{87} +184.937 q^{88} -645.066 q^{89} +713.033 q^{90} -1088.69 q^{92} -521.477 q^{93} +2704.57 q^{94} +2002.71 q^{95} +134.156 q^{96} -1002.09 q^{97} +46.4995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - q^{2} - 33 q^{3} + 31 q^{4} + 17 q^{5} + 3 q^{6} - 54 q^{8} + 99 q^{9} + 75 q^{10} + 7 q^{11} - 93 q^{12} - 143 q^{13} - 51 q^{15} + 23 q^{16} - 20 q^{17} - 9 q^{18} + 242 q^{19} + 254 q^{20} - 290 q^{22}+ \cdots + 63 q^{99}+O(q^{100}) \) 11 * q - q^2 - 33 * q^3 + 31 * q^4 + 17 * q^5 + 3 * q^6 - 54 * q^8 + 99 * q^9 + 75 * q^10 + 7 * q^11 - 93 * q^12 - 143 * q^13 - 51 * q^15 + 23 * q^16 - 20 * q^17 - 9 * q^18 + 242 * q^19 + 254 * q^20 - 290 * q^22 - 122 * q^23 + 162 * q^24 + 208 * q^25 + 13 * q^26 - 297 * q^27 - 177 * q^29 - 225 * q^30 + 493 * q^31 - 448 * q^32 - 21 * q^33 + 855 * q^34 + 279 * q^36 - 660 * q^37 + 458 * q^38 + 429 * q^39 - 530 * q^40 + 854 * q^41 - 1018 * q^43 + 1122 * q^44 + 153 * q^45 - 757 * q^46 + 414 * q^47 - 69 * q^48 + 851 * q^50 + 60 * q^51 - 403 * q^52 - 961 * q^53 + 27 * q^54 + 1003 * q^55 - 726 * q^57 + 801 * q^58 + 2053 * q^59 - 762 * q^60 + 916 * q^61 - 653 * q^62 + 1366 * q^64 - 221 * q^65 + 870 * q^66 - 2788 * q^67 - 2458 * q^68 + 366 * q^69 + 886 * q^71 - 486 * q^72 + 1916 * q^73 - 1557 * q^74 - 624 * q^75 + 2292 * q^76 - 39 * q^78 + 651 * q^79 + 4438 * q^80 + 891 * q^81 - 2148 * q^82 + 185 * q^83 - 2052 * q^85 + 4025 * q^86 + 531 * q^87 - 4503 * q^88 + 1490 * q^89 + 675 * q^90 + 586 * q^92 - 1479 * q^93 + 1796 * q^94 - 2496 * q^95 + 1344 * q^96 + 2705 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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