LMFDB - Embedded newform 833.4.a.f.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [833,4,Mod(1,833)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(833, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("833.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 833 = 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	833.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,4,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.1485910348$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 49x^{5} + 69x^{4} + 753x^{3} - 122x^{2} - 3621x - 2536 $ x^7 - 3x^6 - 49x^5 + 69x^4 + 753x^3 - 122x^2 - 3621x - 2536
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-4.37776$ of defining polynomial
Character	$\chi$	$=$	833.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.37776 q^{2} +0.191325 q^{3} +20.9203 q^{4} -3.18044 q^{5} +1.02890 q^{6} +69.4820 q^{8} -26.9634 q^{9} -17.1036 q^{10} +32.2124 q^{11} +4.00256 q^{12} +11.1233 q^{13} -0.608497 q^{15} +206.295 q^{16} +17.0000 q^{17} -145.003 q^{18} +158.983 q^{19} -66.5357 q^{20} +173.230 q^{22} +98.9354 q^{23} +13.2936 q^{24} -114.885 q^{25} +59.8183 q^{26} -10.3245 q^{27} +172.364 q^{29} -3.27235 q^{30} -55.6997 q^{31} +553.549 q^{32} +6.16302 q^{33} +91.4219 q^{34} -564.081 q^{36} -71.5966 q^{37} +854.973 q^{38} +2.12816 q^{39} -220.983 q^{40} +431.097 q^{41} -205.915 q^{43} +673.891 q^{44} +85.7555 q^{45} +532.051 q^{46} -130.638 q^{47} +39.4694 q^{48} -617.822 q^{50} +3.25252 q^{51} +232.702 q^{52} +338.832 q^{53} -55.5228 q^{54} -102.450 q^{55} +30.4174 q^{57} +926.932 q^{58} +250.468 q^{59} -12.7299 q^{60} -749.213 q^{61} -299.539 q^{62} +1326.49 q^{64} -35.3770 q^{65} +33.1432 q^{66} +124.307 q^{67} +355.644 q^{68} +18.9288 q^{69} -48.4180 q^{71} -1873.47 q^{72} -827.649 q^{73} -385.029 q^{74} -21.9803 q^{75} +3325.97 q^{76} +11.4447 q^{78} -1061.79 q^{79} -656.110 q^{80} +726.036 q^{81} +2318.33 q^{82} -961.960 q^{83} -54.0675 q^{85} -1107.36 q^{86} +32.9775 q^{87} +2238.18 q^{88} -1230.92 q^{89} +461.172 q^{90} +2069.76 q^{92} -10.6567 q^{93} -702.540 q^{94} -505.637 q^{95} +105.908 q^{96} +1017.07 q^{97} -868.554 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} - 5 q^{3} + 52 q^{4} - 35 q^{5} - 51 q^{6} - 6 q^{8} + 128 q^{9} - 18 q^{10} + 48 q^{11} + 16 q^{12} - 84 q^{13} - 54 q^{15} + 256 q^{16} + 119 q^{17} + 196 q^{18} - 156 q^{19} - 317 q^{20}+ \cdots + 7572 q^{99}+O(q^{100}) $ 7 * q + 4 * q^2 - 5 * q^3 + 52 * q^4 - 35 * q^5 - 51 * q^6 - 6 * q^8 + 128 * q^9 - 18 * q^10 + 48 * q^11 + 16 * q^12 - 84 * q^13 - 54 * q^15 + 256 * q^16 + 119 * q^17 + 196 * q^18 - 156 * q^19 - 317 * q^20 + 244 * q^22 + 290 * q^23 - 818 * q^24 + 94 * q^25 + 246 * q^26 + 10 * q^27 + 1074 * q^29 - 23 * q^30 + 291 * q^31 - 779 * q^32 - 26 * q^33 + 68 * q^34 - 679 * q^36 + 596 * q^37 + 1294 * q^38 - 824 * q^39 + 1159 * q^40 - 649 * q^41 + 257 * q^43 - 196 * q^44 - 720 * q^45 + 1044 * q^46 + 1108 * q^47 + 2655 * q^48 - 23 * q^50 - 85 * q^51 - 768 * q^52 + 321 * q^53 + 1497 * q^54 - 152 * q^55 + 398 * q^57 - 1166 * q^58 + 1176 * q^59 - 1495 * q^60 - 1345 * q^61 + 2824 * q^62 + 4888 * q^64 - 1334 * q^65 + 2280 * q^66 + 1379 * q^67 + 884 * q^68 + 1152 * q^69 + 2000 * q^71 - 3375 * q^72 - 367 * q^73 - 114 * q^74 - 888 * q^75 + 1836 * q^76 + 2362 * q^78 + 704 * q^79 - 4112 * q^80 + 4903 * q^81 + 1421 * q^82 + 16 * q^83 - 595 * q^85 - 2785 * q^86 + 1050 * q^87 + 2840 * q^88 - 1070 * q^89 + 3787 * q^90 - 2700 * q^92 - 1976 * q^93 - 4446 * q^94 + 3950 * q^95 + 1420 * q^96 + 39 * q^97 + 7572 * 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 $

$2$	5.37776	1.90132	0.950662	−	0.310229i	$-0.100406\pi$
$2$	5.37776	1.90132	0.950662	+	0.310229i	$0.100406\pi$
$3$	0.191325	0.0368205	0.0184102	−	0.999831i	$-0.494140\pi$
$3$	0.191325	0.0368205	0.0184102	+	0.999831i	$0.494140\pi$
$4$	20.9203	2.61503
$5$	−3.18044	−0.284467	−0.142234	−	0.989833i	$-0.545428\pi$
$5$	−3.18044	−0.284467	−0.142234	+	0.989833i	$0.545428\pi$
$6$	1.02890	0.0700076
$7$	0	0
$7$	0	0
$8$	69.4820	3.07070
$9$	−26.9634	−0.998644
$10$	−17.1036	−0.540865
$11$	32.2124	0.882945	0.441472	−	0.897275i	$-0.354456\pi$
$11$	32.2124	0.882945	0.441472	+	0.897275i	$0.354456\pi$
$12$	4.00256	0.0962867
$13$	11.1233	0.237311	0.118656	−	0.992935i	$-0.462142\pi$
$13$	11.1233	0.237311	0.118656	+	0.992935i	$0.462142\pi$
$14$	0	0
$15$	−0.608497	−0.0104742
$16$	206.295	3.22336
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−145.003	−1.89875
$19$	158.983	1.91964	0.959822	−	0.280610i	$-0.0905367\pi$
$19$	158.983	1.91964	0.959822	+	0.280610i	$0.0905367\pi$
$20$	−66.5357	−0.743891
$21$	0	0
$22$	173.230	1.67876
$23$	98.9354	0.896933	0.448467	−	0.893800i	$-0.351970\pi$
$23$	98.9354	0.896933	0.448467	+	0.893800i	$0.351970\pi$
$24$	13.2936	0.113065
$25$	−114.885	−0.919078
$26$	59.8183	0.451205
$27$	−10.3245	−0.0735910
$28$	0	0
$29$	172.364	1.10370	0.551848	−	0.833945i	$-0.313923\pi$
$29$	172.364	1.10370	0.551848	+	0.833945i	$0.313923\pi$
$30$	−3.27235	−0.0199149
$31$	−55.6997	−0.322709	−0.161354	−	0.986897i	$-0.551586\pi$
$31$	−55.6997	−0.322709	−0.161354	+	0.986897i	$0.551586\pi$
$32$	553.549	3.05796
$33$	6.16302	0.0325104
$34$	91.4219	0.461139
$35$	0	0
$36$	−564.081	−2.61149
$37$	−71.5966	−0.318119	−0.159060	−	0.987269i	$-0.550846\pi$
$37$	−71.5966	−0.318119	−0.159060	+	0.987269i	$0.550846\pi$
$38$	854.973	3.64986
$39$	2.12816	0.00873791
$40$	−220.983	−0.873514
$41$	431.097	1.64210	0.821049	−	0.570857i	$-0.193389\pi$
$41$	431.097	1.64210	0.821049	+	0.570857i	$0.193389\pi$
$42$	0	0
$43$	−205.915	−0.730275	−0.365137	−	0.930954i	$-0.618978\pi$
$43$	−205.915	−0.730275	−0.365137	+	0.930954i	$0.618978\pi$
$44$	673.891	2.30893
$45$	85.7555	0.284082
$46$	532.051	1.70536
$47$	−130.638	−0.405437	−0.202719	−	0.979237i	$-0.564978\pi$
$47$	−130.638	−0.405437	−0.202719	+	0.979237i	$0.564978\pi$
$48$	39.4694	0.118686
$49$	0	0
$50$	−617.822	−1.74747
$51$	3.25252	0.00893027
$52$	232.702	0.620576
$53$	338.832	0.878153	0.439076	−	0.898450i	$-0.355306\pi$
$53$	338.832	0.878153	0.439076	+	0.898450i	$0.355306\pi$
$54$	−55.5228	−0.139920
$55$	−102.450	−0.251169
$56$	0	0
$57$	30.4174	0.0706822
$58$	926.932	2.09848
$59$	250.468	0.552680	0.276340	−	0.961060i	$-0.410878\pi$
$59$	250.468	0.552680	0.276340	+	0.961060i	$0.410878\pi$
$60$	−12.7299	−0.0273904
$61$	−749.213	−1.57257	−0.786286	−	0.617863i	$-0.787999\pi$
$61$	−749.213	−1.57257	−0.786286	+	0.617863i	$0.787999\pi$
$62$	−299.539	−0.613573
$63$	0	0
$64$	1326.49	2.59080
$65$	−35.3770	−0.0675073
$66$	33.1432	0.0618129
$67$	124.307	0.226665	0.113332	−	0.993557i	$-0.463848\pi$
$67$	124.307	0.226665	0.113332	+	0.993557i	$0.463848\pi$
$68$	355.644	0.634239
$69$	18.9288	0.0330255
$70$	0	0
$71$	−48.4180	−0.0809318	−0.0404659	−	0.999181i	$-0.512884\pi$
$71$	−48.4180	−0.0809318	−0.0404659	+	0.999181i	$0.512884\pi$
$72$	−1873.47	−3.06654
$73$	−827.649	−1.32697	−0.663486	−	0.748189i	$-0.730924\pi$
$73$	−827.649	−1.32697	−0.663486	+	0.748189i	$0.730924\pi$
$74$	−385.029	−0.604847
$75$	−21.9803	−0.0338409
$76$	3325.97	5.01993
$77$	0	0
$78$	11.4447	0.0166136
$79$	−1061.79	−1.51217	−0.756083	−	0.654476i	$-0.772889\pi$
$79$	−1061.79	−1.51217	−0.756083	+	0.654476i	$0.772889\pi$
$80$	−656.110	−0.916941
$81$	726.036	0.995935
$82$	2318.33	3.12216
$83$	−961.960	−1.27215	−0.636077	−	0.771625i	$-0.719444\pi$
$83$	−961.960	−1.27215	−0.636077	+	0.771625i	$0.719444\pi$
$84$	0	0
$85$	−54.0675	−0.0689935
$86$	−1107.36	−1.38849
$87$	32.9775	0.0406386
$88$	2238.18	2.71126
$89$	−1230.92	−1.46604	−0.733020	−	0.680207i	$-0.761890\pi$
$89$	−1230.92	−1.46604	−0.733020	+	0.680207i	$0.761890\pi$
$90$	461.172	0.540131
$91$	0	0
$92$	2069.76	2.34551
$93$	−10.6567	−0.0118823
$94$	−702.540	−0.770867
$95$	−505.637	−0.546076
$96$	105.908	0.112595
$97$	1017.07	1.06462	0.532310	−	0.846550i	$-0.321324\pi$
$97$	1017.07	1.06462	0.532310	+	0.846550i	$0.321324\pi$
$98$	0	0
$99$	−868.554	−0.881748
$100$	−2403.42	−2.40342
$101$	−477.305	−0.470234	−0.235117	−	0.971967i	$-0.575547\pi$
$101$	−477.305	−0.470234	−0.235117	+	0.971967i	$0.575547\pi$
$102$	17.4913	0.0169793
$103$	−903.854	−0.864654	−0.432327	−	0.901717i	$-0.642307\pi$
$103$	−903.854	−0.864654	−0.432327	+	0.901717i	$0.642307\pi$
$104$	772.868	0.728711
$105$	0	0
$106$	1822.15	1.66965
$107$	485.514	0.438658	0.219329	−	0.975651i	$-0.429613\pi$
$107$	485.514	0.438658	0.219329	+	0.975651i	$0.429613\pi$
$108$	−215.992	−0.192443
$109$	1692.84	1.48756	0.743782	−	0.668422i	$-0.233030\pi$
$109$	1692.84	1.48756	0.743782	+	0.668422i	$0.233030\pi$
$110$	−550.948	−0.477553
$111$	−13.6982	−0.0117133
$112$	0	0
$113$	1119.14	0.931676	0.465838	−	0.884870i	$-0.345753\pi$
$113$	1119.14	0.931676	0.465838	+	0.884870i	$0.345753\pi$
$114$	163.577	0.134390
$115$	−314.658	−0.255148
$116$	3605.90	2.88620
$117$	−299.922	−0.236989
$118$	1346.95	1.05082
$119$	0	0
$120$	−42.2796	−0.0321632
$121$	−293.364	−0.220409
$122$	−4029.09	−2.98997
$123$	82.4795	0.0604628
$124$	−1165.25	−0.843893
$125$	762.940	0.545915
$126$	0	0
$127$	−1599.80	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$127$	−1599.80	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$128$	2705.15	1.86800
$129$	−39.3967	−0.0268891
$130$	−190.249	−0.128353
$131$	8.82846	0.00588814	0.00294407	−	0.999996i	$-0.499063\pi$
$131$	8.82846	0.00588814	0.00294407	+	0.999996i	$0.499063\pi$
$132$	128.932	0.0850158
$133$	0	0
$134$	668.494	0.430963
$135$	32.8366	0.0209342
$136$	1181.19	0.744754
$137$	−2606.16	−1.62525	−0.812625	−	0.582787i	$-0.801962\pi$
$137$	−2606.16	−1.62525	−0.812625	+	0.582787i	$0.801962\pi$
$138$	101.794	0.0627922
$139$	−1573.34	−0.960067	−0.480034	−	0.877250i	$-0.659376\pi$
$139$	−1573.34	−0.960067	−0.480034	+	0.877250i	$0.659376\pi$
$140$	0	0
$141$	−24.9943	−0.0149284
$142$	−260.380	−0.153878
$143$	358.307	0.209533
$144$	−5562.42	−3.21899
$145$	−548.194	−0.313966
$146$	−4450.89	−2.52300
$147$	0	0
$148$	−1497.82	−0.831892
$149$	1775.78	0.976360	0.488180	−	0.872743i	$-0.337661\pi$
$149$	1775.78	0.976360	0.488180	+	0.872743i	$0.337661\pi$
$150$	−118.205	−0.0643425
$151$	423.752	0.228374	0.114187	−	0.993459i	$-0.463574\pi$
$151$	423.752	0.228374	0.114187	+	0.993459i	$0.463574\pi$
$152$	11046.5	5.89465
$153$	−458.378	−0.242207
$154$	0	0
$155$	177.150	0.0918000
$156$	44.5217	0.0228499
$157$	1923.47	0.977769	0.488884	−	0.872349i	$-0.337404\pi$
$157$	1923.47	0.977769	0.488884	+	0.872349i	$0.337404\pi$
$158$	−5710.07	−2.87512
$159$	64.8269	0.0323340
$160$	−1760.53	−0.869889
$161$	0	0
$162$	3904.45	1.89359
$163$	−2264.08	−1.08795	−0.543977	−	0.839100i	$-0.683082\pi$
$163$	−2264.08	−1.08795	−0.543977	+	0.839100i	$0.683082\pi$
$164$	9018.66	4.29414
$165$	−19.6011	−0.00924816
$166$	−5173.19	−2.41878
$167$	306.177	0.141872	0.0709362	−	0.997481i	$-0.477401\pi$
$167$	306.177	0.141872	0.0709362	+	0.997481i	$0.477401\pi$
$168$	0	0
$169$	−2073.27	−0.943683
$170$	−290.762	−0.131179
$171$	−4286.73	−1.91704
$172$	−4307.80	−1.90969
$173$	3128.37	1.37483	0.687416	−	0.726264i	$-0.258745\pi$
$173$	3128.37	1.37483	0.687416	+	0.726264i	$0.258745\pi$
$174$	177.345	0.0772672
$175$	0	0
$176$	6645.25	2.84605
$177$	47.9206	0.0203499
$178$	−6619.60	−2.78742
$179$	326.634	0.136390	0.0681950	−	0.997672i	$-0.478276\pi$
$179$	326.634	0.136390	0.0681950	+	0.997672i	$0.478276\pi$
$180$	1794.03	0.742883
$181$	60.6331	0.0248996	0.0124498	−	0.999922i	$-0.496037\pi$
$181$	60.6331	0.0248996	0.0124498	+	0.999922i	$0.496037\pi$
$182$	0	0
$183$	−143.343	−0.0579028
$184$	6874.23	2.75421
$185$	227.709	0.0904945
$186$	−57.3093	−0.0225921
$187$	547.610	0.214146
$188$	−2732.99	−1.06023
$189$	0	0
$190$	−2719.19	−1.03827
$191$	−4232.87	−1.60356	−0.801779	−	0.597621i	$-0.796113\pi$
$191$	−4232.87	−1.60356	−0.801779	+	0.597621i	$0.796113\pi$
$192$	253.791	0.0953946
$193$	5292.96	1.97407	0.987034	−	0.160508i	$-0.0513134\pi$
$193$	5292.96	1.97407	0.987034	+	0.160508i	$0.0513134\pi$
$194$	5469.57	2.02419
$195$	−6.76849	−0.00248565
$196$	0	0
$197$	985.384	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$197$	985.384	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$198$	−4670.87	−1.67649
$199$	4175.42	1.48737	0.743687	−	0.668528i	$-0.233075\pi$
$199$	4175.42	1.48737	0.743687	+	0.668528i	$0.233075\pi$
$200$	−7982.43	−2.82221
$201$	23.7830	0.00834590
$202$	−2566.83	−0.894067
$203$	0	0
$204$	68.0436	0.0233530
$205$	−1371.08	−0.467123
$206$	−4860.71	−1.64399
$207$	−2667.64	−0.895717
$208$	2294.68	0.764940
$209$	5121.22	1.69494
$210$	0	0
$211$	−2895.15	−0.944600	−0.472300	−	0.881438i	$-0.656576\pi$
$211$	−2895.15	−0.944600	−0.472300	+	0.881438i	$0.656576\pi$
$212$	7088.45	2.29640
$213$	−9.26356	−0.00297995
$214$	2610.98	0.834032
$215$	654.902	0.207739
$216$	−717.369	−0.225976
$217$	0	0
$218$	9103.68	2.82834
$219$	−158.350	−0.0488597
$220$	−2143.27	−0.656815
$221$	189.096	0.0575564
$222$	−73.6656	−0.0222708
$223$	3695.53	1.10974	0.554868	−	0.831938i	$-0.312769\pi$
$223$	3695.53	1.10974	0.554868	+	0.831938i	$0.312769\pi$
$224$	0	0
$225$	3097.68	0.917832
$226$	6018.44	1.77142
$227$	1177.78	0.344371	0.172186	−	0.985065i	$-0.444917\pi$
$227$	1177.78	0.344371	0.172186	+	0.985065i	$0.444917\pi$
$228$	636.340	0.184836
$229$	−5123.88	−1.47858	−0.739291	−	0.673386i	$-0.764839\pi$
$229$	−5123.88	−1.47858	−0.739291	+	0.673386i	$0.764839\pi$
$230$	−1692.16	−0.485119
$231$	0	0
$232$	11976.2	3.38912
$233$	−3222.82	−0.906154	−0.453077	−	0.891471i	$-0.649674\pi$
$233$	−3222.82	−0.906154	−0.453077	+	0.891471i	$0.649674\pi$
$234$	−1612.91	−0.450594
$235$	415.487	0.115334
$236$	5239.85	1.44528
$237$	−203.147	−0.0556786
$238$	0	0
$239$	−4252.99	−1.15106	−0.575529	−	0.817781i	$-0.695204\pi$
$239$	−4252.99	−1.15106	−0.575529	+	0.817781i	$0.695204\pi$
$240$	−125.530	−0.0337622
$241$	−5026.77	−1.34358	−0.671790	−	0.740741i	$-0.734474\pi$
$241$	−5026.77	−1.34358	−0.671790	+	0.740741i	$0.734474\pi$
$242$	−1577.64	−0.419069
$243$	417.671	0.110262
$244$	−15673.7	−4.11233
$245$	0	0
$246$	443.555	0.114959
$247$	1768.42	0.455553
$248$	−3870.13	−0.990941
$249$	−184.047	−0.0468413
$250$	4102.90	1.03796
$251$	7086.11	1.78196	0.890978	−	0.454045i	$-0.150020\pi$
$251$	7086.11	1.78196	0.890978	+	0.454045i	$0.150020\pi$
$252$	0	0
$253$	3186.94	0.791942
$254$	−8603.35	−2.12529
$255$	−10.3445	−0.00254037
$256$	3935.72	0.960870
$257$	−2156.17	−0.523339	−0.261670	−	0.965157i	$-0.584273\pi$
$257$	−2156.17	−0.523339	−0.261670	+	0.965157i	$0.584273\pi$
$258$	−211.866	−0.0511248
$259$	0	0
$260$	−740.095	−0.176534
$261$	−4647.52	−1.10220
$262$	47.4773	0.0111953
$263$	1287.36	0.301833	0.150917	−	0.988546i	$-0.451777\pi$
$263$	1287.36	0.301833	0.150917	+	0.988546i	$0.451777\pi$
$264$	428.219	0.0998298
$265$	−1077.63	−0.249806
$266$	0	0
$267$	−235.506	−0.0539803
$268$	2600.54	0.592736
$269$	−1904.20	−0.431603	−0.215802	−	0.976437i	$-0.569236\pi$
$269$	−1904.20	−0.431603	−0.215802	+	0.976437i	$0.569236\pi$
$270$	176.587	0.0398028
$271$	−6568.08	−1.47226	−0.736130	−	0.676841i	$-0.763349\pi$
$271$	−6568.08	−1.47226	−0.736130	+	0.676841i	$0.763349\pi$
$272$	3507.02	0.781780
$273$	0	0
$274$	−14015.3	−3.09013
$275$	−3700.71	−0.811495
$276$	395.995	0.0863628
$277$	1228.75	0.266528	0.133264	−	0.991081i	$-0.457454\pi$
$277$	1228.75	0.266528	0.133264	+	0.991081i	$0.457454\pi$
$278$	−8461.06	−1.82540
$279$	1501.85	0.322271
$280$	0	0
$281$	−9140.17	−1.94042	−0.970208	−	0.242274i	$-0.922107\pi$
$281$	−9140.17	−1.94042	−0.970208	+	0.242274i	$0.922107\pi$
$282$	−134.413	−0.0283837
$283$	6922.90	1.45415	0.727073	−	0.686560i	$-0.240880\pi$
$283$	6922.90	1.45415	0.727073	+	0.686560i	$0.240880\pi$
$284$	−1012.92	−0.211639
$285$	−96.7408	−0.0201068
$286$	1926.89	0.398389
$287$	0	0
$288$	−14925.6	−3.05381
$289$	289.000	0.0588235
$290$	−2948.05	−0.596950
$291$	194.591	0.0391998
$292$	−17314.6	−3.47008
$293$	−6116.25	−1.21951	−0.609753	−	0.792592i	$-0.708731\pi$
$293$	−6116.25	−1.21951	−0.609753	+	0.792592i	$0.708731\pi$
$294$	0	0
$295$	−796.598	−0.157219
$296$	−4974.67	−0.976848
$297$	−332.578	−0.0649768
$298$	9549.71	1.85638
$299$	1100.49	0.212852
$300$	−459.834	−0.0884950
$301$	0	0
$302$	2278.83	0.434212
$303$	−91.3203	−0.0173142
$304$	32797.5	6.18771
$305$	2382.83	0.447345
$306$	−2465.04	−0.460514
$307$	3185.90	0.592276	0.296138	−	0.955145i	$-0.404301\pi$
$307$	3185.90	0.592276	0.296138	+	0.955145i	$0.404301\pi$
$308$	0	0
$309$	−172.930	−0.0318370
$310$	952.668	0.174542
$311$	−4546.41	−0.828949	−0.414475	−	0.910061i	$-0.636035\pi$
$311$	−4546.41	−0.828949	−0.414475	+	0.910061i	$0.636035\pi$
$312$	147.869	0.0268315
$313$	−4485.33	−0.809986	−0.404993	−	0.914320i	$-0.632726\pi$
$313$	−4485.33	−0.809986	−0.404993	+	0.914320i	$0.632726\pi$
$314$	10344.0	1.85905
$315$	0	0
$316$	−22213.0	−3.95436
$317$	−3874.33	−0.686448	−0.343224	−	0.939254i	$-0.611519\pi$
$317$	−3874.33	−0.686448	−0.343224	+	0.939254i	$0.611519\pi$
$318$	348.623	0.0614774
$319$	5552.25	0.974503
$320$	−4218.83	−0.736999
$321$	92.8909	0.0161516
$322$	0	0
$323$	2702.71	0.465582
$324$	15188.9	2.60440
$325$	−1277.90	−0.218108
$326$	−12175.7	−2.06855
$327$	323.882	0.0547728
$328$	29953.5	5.04239
$329$	0	0
$330$	−105.410	−0.0175837
$331$	751.095	0.124725	0.0623624	−	0.998054i	$-0.480137\pi$
$331$	751.095	0.124725	0.0623624	+	0.998054i	$0.480137\pi$
$332$	−20124.5	−3.32673
$333$	1930.49	0.317688
$334$	1646.55	0.269745
$335$	−395.352	−0.0644787
$336$	0	0
$337$	4660.66	0.753360	0.376680	−	0.926343i	$-0.377066\pi$
$337$	4660.66	0.753360	0.376680	+	0.926343i	$0.377066\pi$
$338$	−11149.6	−1.79425
$339$	214.118	0.0343048
$340$	−1131.11	−0.180420
$341$	−1794.22	−0.284934
$342$	−23053.0	−3.64492
$343$	0	0
$344$	−14307.4	−2.24245
$345$	−60.2019	−0.00939468
$346$	16823.6	2.61400
$347$	936.574	0.144893	0.0724465	−	0.997372i	$-0.476919\pi$
$347$	936.574	0.144893	0.0724465	+	0.997372i	$0.476919\pi$
$348$	689.898	0.106271
$349$	6679.01	1.02441	0.512205	−	0.858863i	$-0.328829\pi$
$349$	6679.01	1.02441	0.512205	+	0.858863i	$0.328829\pi$
$350$	0	0
$351$	−114.843	−0.0174640
$352$	17831.1	2.70001
$353$	−8627.22	−1.30080	−0.650398	−	0.759594i	$-0.725398\pi$
$353$	−8627.22	−1.30080	−0.650398	+	0.759594i	$0.725398\pi$
$354$	257.706	0.0386918
$355$	153.991	0.0230225
$356$	−25751.2	−3.83374
$357$	0	0
$358$	1756.56	0.259321
$359$	−4559.03	−0.670241	−0.335121	−	0.942175i	$-0.608777\pi$
$359$	−4559.03	−0.670241	−0.335121	+	0.942175i	$0.608777\pi$
$360$	5958.46	0.872330
$361$	18416.6	2.68503
$362$	326.070	0.0473422
$363$	−56.1278	−0.00811556
$364$	0	0
$365$	2632.29	0.377480
$366$	−770.864	−0.110092
$367$	−13758.6	−1.95693	−0.978463	−	0.206424i	$-0.933817\pi$
$367$	−13758.6	−1.95693	−0.978463	+	0.206424i	$0.933817\pi$
$368$	20409.9	2.89114
$369$	−11623.8	−1.63987
$370$	1224.56	0.172059
$371$	0	0
$372$	−222.942	−0.0310725
$373$	−2760.03	−0.383134	−0.191567	−	0.981480i	$-0.561357\pi$
$373$	−2760.03	−0.383134	−0.191567	+	0.981480i	$0.561357\pi$
$374$	2944.91	0.407160
$375$	145.969	0.0201008
$376$	−9077.01	−1.24498
$377$	1917.25	0.261919
$378$	0	0
$379$	2572.84	0.348701	0.174351	−	0.984684i	$-0.444217\pi$
$379$	2572.84	0.348701	0.174351	+	0.984684i	$0.444217\pi$
$380$	−10578.0	−1.42801
$381$	−306.082	−0.0411576
$382$	−22763.3	−3.04888
$383$	8731.32	1.16488	0.582441	−	0.812873i	$-0.302098\pi$
$383$	8731.32	1.16488	0.582441	+	0.812873i	$0.302098\pi$
$384$	517.563	0.0687806
$385$	0	0
$386$	28464.2	3.75334
$387$	5552.18	0.729285
$388$	21277.4	2.78402
$389$	5858.41	0.763581	0.381791	−	0.924249i	$-0.375308\pi$
$389$	5858.41	0.763581	0.381791	+	0.924249i	$0.375308\pi$
$390$	−36.3993	−0.00472602
$391$	1681.90	0.217538
$392$	0	0
$393$	1.68910	0.000216804 0
$394$	5299.15	0.677583
$395$	3376.97	0.430162
$396$	−18170.4	−2.30580
$397$	14939.8	1.88868	0.944340	−	0.328972i	$-0.106702\pi$
$397$	14939.8	1.88868	0.944340	+	0.328972i	$0.106702\pi$
$398$	22454.4	2.82798
$399$	0	0
$400$	−23700.2	−2.96252
$401$	7410.43	0.922841	0.461420	−	0.887182i	$-0.347340\pi$
$401$	7410.43	0.922841	0.461420	+	0.887182i	$0.347340\pi$
$402$	127.899	0.0158683
$403$	−619.564	−0.0765823
$404$	−9985.35	−1.22968
$405$	−2309.12	−0.283311
$406$	0	0
$407$	−2306.29	−0.280881
$408$	225.992	0.0274222
$409$	−4623.40	−0.558955	−0.279477	−	0.960152i	$-0.590161\pi$
$409$	−4623.40	−0.558955	−0.279477	+	0.960152i	$0.590161\pi$
$410$	−7373.33	−0.888153
$411$	−498.623	−0.0598425
$412$	−18908.9	−2.26110
$413$	0	0
$414$	−14345.9	−1.70305
$415$	3059.46	0.361887
$416$	6157.29	0.725687
$417$	−301.020	−0.0353501
$418$	27540.7	3.22263
$419$	12060.8	1.40622	0.703111	−	0.711080i	$-0.251794\pi$
$419$	12060.8	1.40622	0.703111	+	0.711080i	$0.251794\pi$
$420$	0	0
$421$	6681.87	0.773526	0.386763	−	0.922179i	$-0.373593\pi$
$421$	6681.87	0.773526	0.386763	+	0.922179i	$0.373593\pi$
$422$	−15569.4	−1.79599
$423$	3522.45	0.404887
$424$	23542.7	2.69654
$425$	−1953.04	−0.222909
$426$	−49.8172	−0.00566585
$427$	0	0
$428$	10157.1	1.14711
$429$	68.5531	0.00771509
$430$	3521.90	0.394980
$431$	7648.86	0.854832	0.427416	−	0.904055i	$-0.359424\pi$
$431$	7648.86	0.854832	0.427416	+	0.904055i	$0.359424\pi$
$432$	−2129.90	−0.237211
$433$	10356.1	1.14938	0.574691	−	0.818371i	$-0.305122\pi$
$433$	10356.1	1.14938	0.574691	+	0.818371i	$0.305122\pi$
$434$	0	0
$435$	−104.883	−0.0115604
$436$	35414.6	3.89003
$437$	15729.1	1.72179
$438$	−851.566	−0.0928982
$439$	−15791.9	−1.71687	−0.858436	−	0.512921i	$-0.828563\pi$
$439$	−15791.9	−1.71687	−0.858436	+	0.512921i	$0.828563\pi$
$440$	−7118.40	−0.771264
$441$	0	0
$442$	1016.91	0.109433
$443$	6901.36	0.740166	0.370083	−	0.928999i	$-0.379329\pi$
$443$	6901.36	0.740166	0.370083	+	0.928999i	$0.379329\pi$
$444$	−286.570	−0.0306306
$445$	3914.88	0.417040
$446$	19873.7	2.10997
$447$	339.751	0.0359500
$448$	0	0
$449$	−8151.58	−0.856786	−0.428393	−	0.903592i	$-0.640920\pi$
$449$	−8151.58	−0.856786	−0.428393	+	0.903592i	$0.640920\pi$
$450$	16658.6	1.74510
$451$	13886.7	1.44988
$452$	23412.6	2.43636
$453$	81.0742	0.00840882
$454$	6333.84	0.654761
$455$	0	0
$456$	2113.46	0.217044
$457$	2257.61	0.231087	0.115543	−	0.993302i	$-0.463139\pi$
$457$	2257.61	0.231087	0.115543	+	0.993302i	$0.463139\pi$
$458$	−27555.0	−2.81126
$459$	−175.517	−0.0178484
$460$	−6582.74	−0.667221
$461$	5234.07	0.528796	0.264398	−	0.964414i	$-0.414827\pi$
$461$	5234.07	0.528796	0.264398	+	0.964414i	$0.414827\pi$
$462$	0	0
$463$	−2206.42	−0.221471	−0.110736	−	0.993850i	$-0.535321\pi$
$463$	−2206.42	−0.221471	−0.110736	+	0.993850i	$0.535321\pi$
$464$	35557.9	3.55761
$465$	33.8931	0.00338012
$466$	−17331.5	−1.72289
$467$	−6744.18	−0.668272	−0.334136	−	0.942525i	$-0.608445\pi$
$467$	−6744.18	−0.668272	−0.334136	+	0.942525i	$0.608445\pi$
$468$	−6274.44	−0.619735
$469$	0	0
$470$	2234.39	0.219287
$471$	368.008	0.0360019
$472$	17403.0	1.69711
$473$	−6633.02	−0.644792
$474$	−1092.48	−0.105863
$475$	−18264.7	−1.76430
$476$	0	0
$477$	−9136.05	−0.876962
$478$	−22871.5	−2.18853
$479$	13168.8	1.25615	0.628076	−	0.778152i	$-0.283843\pi$
$479$	13168.8	1.25615	0.628076	+	0.778152i	$0.283843\pi$
$480$	−336.833	−0.0320297
$481$	−796.389	−0.0754932
$482$	−27032.8	−2.55458
$483$	0	0
$484$	−6137.26	−0.576377
$485$	−3234.74	−0.302850
$486$	2246.13	0.209643
$487$	−10213.2	−0.950312	−0.475156	−	0.879901i	$-0.657608\pi$
$487$	−10213.2	−0.950312	−0.475156	+	0.879901i	$0.657608\pi$
$488$	−52056.8	−4.82890
$489$	−433.175	−0.0400590
$490$	0	0
$491$	−3027.94	−0.278308	−0.139154	−	0.990271i	$-0.544438\pi$
$491$	−3027.94	−0.278308	−0.139154	+	0.990271i	$0.544438\pi$
$492$	1725.49	0.158112
$493$	2930.19	0.267686
$494$	9510.11	0.866154
$495$	2762.39	0.250828
$496$	−11490.6	−1.04021
$497$	0	0
$498$	−989.759	−0.0890606
$499$	−7982.48	−0.716122	−0.358061	−	0.933698i	$-0.616562\pi$
$499$	−7982.48	−0.716122	−0.358061	+	0.933698i	$0.616562\pi$
$500$	15960.9	1.42759
$501$	58.5792	0.00522381
$502$	38107.4	3.38808
$503$	−11111.2	−0.984942	−0.492471	−	0.870329i	$-0.663906\pi$
$503$	−11111.2	−0.984942	−0.492471	+	0.870329i	$0.663906\pi$
$504$	0	0
$505$	1518.04	0.133766
$506$	17138.6	1.50574
$507$	−396.668	−0.0347469
$508$	−33468.3	−2.92306
$509$	705.986	0.0614780	0.0307390	−	0.999527i	$-0.490214\pi$
$509$	705.986	0.0614780	0.0307390	+	0.999527i	$0.490214\pi$
$510$	−55.6299	−0.00483007
$511$	0	0
$512$	−475.873	−0.0410758
$513$	−1641.43	−0.141269
$514$	−11595.4	−0.995038
$515$	2874.66	0.245966
$516$	−824.190	−0.0703158
$517$	−4208.16	−0.357978
$518$	0	0
$519$	598.535	0.0506219
$520$	−2458.06	−0.207295
$521$	−13299.6	−1.11836	−0.559179	−	0.829047i	$-0.688884\pi$
$521$	−13299.6	−1.11836	−0.559179	+	0.829047i	$0.688884\pi$
$522$	−24993.2	−2.09564
$523$	−12355.4	−1.03301	−0.516503	−	0.856286i	$-0.672766\pi$
$523$	−12355.4	−1.03301	−0.516503	+	0.856286i	$0.672766\pi$
$524$	184.694	0.0153977
$525$	0	0
$526$	6923.12	0.573883
$527$	−946.895	−0.0782683
$528$	1271.40	0.104793
$529$	−2378.78	−0.195511
$530$	−5795.25	−0.474962
$531$	−6753.46	−0.551930
$532$	0	0
$533$	4795.22	0.389688
$534$	−1266.49	−0.102634
$535$	−1544.15	−0.124784
$536$	8637.11	0.696019
$537$	62.4932	0.00502194
$538$	−10240.3	−0.820618
$539$	0	0
$540$	686.950	0.0547437
$541$	15821.5	1.25733	0.628667	−	0.777675i	$-0.283601\pi$
$541$	15821.5	1.25733	0.628667	+	0.777675i	$0.283601\pi$
$542$	−35321.5	−2.79924
$543$	11.6006	0.000916814 0
$544$	9410.34	0.741664
$545$	−5383.98	−0.423164
$546$	0	0
$547$	−7269.65	−0.568241	−0.284120	−	0.958789i	$-0.591702\pi$
$547$	−7269.65	−0.568241	−0.284120	+	0.958789i	$0.591702\pi$
$548$	−54521.6	−4.25008
$549$	20201.3	1.57044
$550$	−19901.5	−1.54292
$551$	27403.0	2.11870
$552$	1315.21	0.101411
$553$	0	0
$554$	6607.91	0.506756
$555$	43.5663	0.00333205
$556$	−32914.8	−2.51061
$557$	−19405.6	−1.47620	−0.738098	−	0.674693i	$-0.764276\pi$
$557$	−19405.6	−1.47620	−0.738098	+	0.674693i	$0.764276\pi$
$558$	8076.60	0.612742
$559$	−2290.46	−0.173302
$560$	0	0
$561$	104.771	0.00788494
$562$	−49153.6	−3.68936
$563$	568.452	0.0425531	0.0212766	−	0.999774i	$-0.493227\pi$
$563$	568.452	0.0425531	0.0212766	+	0.999774i	$0.493227\pi$
$564$	−522.888	−0.0390382
$565$	−3559.35	−0.265031
$566$	37229.7	2.76480
$567$	0	0
$568$	−3364.18	−0.248517
$569$	−2503.91	−0.184480	−0.0922401	−	0.995737i	$-0.529403\pi$
$569$	−2503.91	−0.184480	−0.0922401	+	0.995737i	$0.529403\pi$
$570$	−520.248	−0.0382295
$571$	23822.0	1.74592	0.872961	−	0.487790i	$-0.162197\pi$
$571$	23822.0	1.74592	0.872961	+	0.487790i	$0.162197\pi$
$572$	7495.88	0.547935
$573$	−809.852	−0.0590437
$574$	0	0
$575$	−11366.2	−0.824352
$576$	−35766.7	−2.58729
$577$	293.252	0.0211581	0.0105791	−	0.999944i	$-0.496633\pi$
$577$	293.252	0.0211581	0.0105791	+	0.999944i	$0.496633\pi$
$578$	1554.17	0.111843
$579$	1012.67	0.0726861
$580$	−11468.4	−0.821030
$581$	0	0
$582$	1046.46	0.0745315
$583$	10914.6	0.775360
$584$	−57506.7	−4.07473
$585$	953.883	0.0674158
$586$	−32891.7	−2.31868
$587$	15857.6	1.11502	0.557508	−	0.830172i	$-0.311758\pi$
$587$	15857.6	1.11502	0.557508	+	0.830172i	$0.311758\pi$
$588$	0	0
$589$	−8855.32	−0.619485
$590$	−4283.91	−0.298925
$591$	188.528	0.0131219
$592$	−14770.0	−1.02541
$593$	−13364.7	−0.925500	−0.462750	−	0.886489i	$-0.653137\pi$
$593$	−13364.7	−0.925500	−0.462750	+	0.886489i	$0.653137\pi$
$594$	−1788.52	−0.123542
$595$	0	0
$596$	37149.8	2.55321
$597$	798.860	0.0547658
$598$	5918.15	0.404701
$599$	−11624.8	−0.792946	−0.396473	−	0.918046i	$-0.629766\pi$
$599$	−11624.8	−0.792946	−0.396473	+	0.918046i	$0.629766\pi$
$600$	−1527.24	−0.103915
$601$	−157.842	−0.0107130	−0.00535649	−	0.999986i	$-0.501705\pi$
$601$	−157.842	−0.0107130	−0.00535649	+	0.999986i	$0.501705\pi$
$602$	0	0
$603$	−3351.74	−0.226357
$604$	8864.99	0.597204
$605$	933.028	0.0626991
$606$	−491.098	−0.0329200
$607$	14178.9	0.948112	0.474056	−	0.880495i	$-0.342789\pi$
$607$	14178.9	0.948112	0.474056	+	0.880495i	$0.342789\pi$
$608$	88005.0	5.87019
$609$	0	0
$610$	12814.3	0.850549
$611$	−1453.13	−0.0962147
$612$	−9589.38	−0.633379
$613$	8990.35	0.592360	0.296180	−	0.955132i	$-0.404287\pi$
$613$	8990.35	0.592360	0.296180	+	0.955132i	$0.404287\pi$
$614$	17133.0	1.12611
$615$	−262.321	−0.0171997
$616$	0	0
$617$	23661.8	1.54390	0.771950	−	0.635683i	$-0.219281\pi$
$617$	23661.8	1.54390	0.771950	+	0.635683i	$0.219281\pi$
$618$	−929.974	−0.0605324
$619$	−5408.05	−0.351160	−0.175580	−	0.984465i	$-0.556180\pi$
$619$	−5408.05	−0.351160	−0.175580	+	0.984465i	$0.556180\pi$
$620$	3706.02	0.240060
$621$	−1021.46	−0.0660062
$622$	−24449.5	−1.57610
$623$	0	0
$624$	439.029	0.0281654
$625$	11934.1	0.763783
$626$	−24121.0	−1.54005
$627$	979.816	0.0624084
$628$	40239.5	2.55690
$629$	−1217.14	−0.0771552
$630$	0	0
$631$	21430.9	1.35206	0.676032	−	0.736873i	$-0.263698\pi$
$631$	21430.9	1.35206	0.676032	+	0.736873i	$0.263698\pi$
$632$	−73775.5	−4.64341
$633$	−553.914	−0.0347806
$634$	−20835.2	−1.30516
$635$	5088.08	0.317975
$636$	1356.20	0.0845545
$637$	0	0
$638$	29858.7	1.85285
$639$	1305.51	0.0808221
$640$	−8603.58	−0.531385
$641$	−15855.4	−0.976991	−0.488495	−	0.872566i	$-0.662454\pi$
$641$	−15855.4	−0.976991	−0.488495	+	0.872566i	$0.662454\pi$
$642$	499.545	0.0307094
$643$	22952.7	1.40773	0.703863	−	0.710336i	$-0.251457\pi$
$643$	22952.7	1.40773	0.703863	+	0.710336i	$0.251457\pi$
$644$	0	0
$645$	125.299	0.00764906
$646$	14534.5	0.885222
$647$	13160.3	0.799668	0.399834	−	0.916588i	$-0.369068\pi$
$647$	13160.3	0.799668	0.399834	+	0.916588i	$0.369068\pi$
$648$	50446.5	3.05822
$649$	8068.15	0.487986
$650$	−6872.22	−0.414693
$651$	0	0
$652$	−47365.2	−2.84503
$653$	−6948.05	−0.416383	−0.208192	−	0.978088i	$-0.566758\pi$
$653$	−6948.05	−0.416383	−0.208192	+	0.978088i	$0.566758\pi$
$654$	1741.76	0.104141
$655$	−28.0784	−0.00167498
$656$	88933.3	5.29308
$657$	22316.2	1.32517
$658$	0	0
$659$	20975.7	1.23990	0.619952	−	0.784640i	$-0.287152\pi$
$659$	20975.7	1.23990	0.619952	+	0.784640i	$0.287152\pi$
$660$	−410.061	−0.0241842
$661$	3682.21	0.216674	0.108337	−	0.994114i	$-0.465447\pi$
$661$	3682.21	0.216674	0.108337	+	0.994114i	$0.465447\pi$
$662$	4039.21	0.237142
$663$	36.1787	0.00211925
$664$	−66838.9	−3.90641
$665$	0	0
$666$	10381.7	0.604027
$667$	17052.9	0.989942
$668$	6405.30	0.371001
$669$	707.047	0.0408610
$670$	−2126.10	−0.122595
$671$	−24133.9	−1.38849
$672$	0	0
$673$	−24089.9	−1.37979	−0.689893	−	0.723911i	$-0.742342\pi$
$673$	−24089.9	−1.37979	−0.689893	+	0.723911i	$0.742342\pi$
$674$	25063.9	1.43238
$675$	1186.13	0.0676359
$676$	−43373.4	−2.46776
$677$	−2699.33	−0.153240	−0.0766200	−	0.997060i	$-0.524413\pi$
$677$	−2699.33	−0.153240	−0.0766200	+	0.997060i	$0.524413\pi$
$678$	1151.48	0.0652244
$679$	0	0
$680$	−3756.72	−0.211858
$681$	225.339	0.0126799
$682$	−9648.87	−0.541751
$683$	16698.5	0.935504	0.467752	−	0.883860i	$-0.345064\pi$
$683$	16698.5	0.935504	0.467752	+	0.883860i	$0.345064\pi$
$684$	−89679.4	−5.01313
$685$	8288.74	0.462331
$686$	0	0
$687$	−980.324	−0.0544421
$688$	−42479.4	−2.35394
$689$	3768.92	0.208395
$690$	−323.751	−0.0178623
$691$	−9234.63	−0.508396	−0.254198	−	0.967152i	$-0.581812\pi$
$691$	−9234.63	−0.508396	−0.254198	+	0.967152i	$0.581812\pi$
$692$	65446.4	3.59523
$693$	0	0
$694$	5036.66	0.275489
$695$	5003.93	0.273108
$696$	2291.34	0.124789
$697$	7328.65	0.398267
$698$	35918.1	1.94774
$699$	−616.605	−0.0333650
$700$	0	0
$701$	14091.1	0.759222	0.379611	−	0.925146i	$-0.376058\pi$
$701$	14091.1	0.759222	0.379611	+	0.925146i	$0.376058\pi$
$702$	−617.596	−0.0332047
$703$	−11382.6	−0.610675
$704$	42729.4	2.28754
$705$	79.4930	0.00424664
$706$	−46395.1	−2.47323
$707$	0	0
$708$	1002.51	0.0532157
$709$	507.249	0.0268690	0.0134345	−	0.999910i	$-0.495724\pi$
$709$	507.249	0.0268690	0.0134345	+	0.999910i	$0.495724\pi$
$710$	828.124	0.0437732
$711$	28629.6	1.51012
$712$	−85527.0	−4.50177
$713$	−5510.68	−0.289448
$714$	0	0
$715$	−1139.58	−0.0596052
$716$	6833.27	0.356664
$717$	−813.702	−0.0423825
$718$	−24517.4	−1.27435
$719$	−8366.20	−0.433945	−0.216973	−	0.976178i	$-0.569618\pi$
$719$	−8366.20	−0.433945	−0.216973	+	0.976178i	$0.569618\pi$
$720$	17691.0	0.915698
$721$	0	0
$722$	99040.2	5.10512
$723$	−961.746	−0.0494713
$724$	1268.46	0.0651132
$725$	−19802.0	−1.01438
$726$	−301.842	−0.0154303
$727$	−3236.14	−0.165092	−0.0825459	−	0.996587i	$-0.526305\pi$
$727$	−3236.14	−0.165092	−0.0825459	+	0.996587i	$0.526305\pi$
$728$	0	0
$729$	−19523.1	−0.991875
$730$	14155.8	0.717712
$731$	−3500.56	−0.177118
$732$	−2998.77	−0.151418
$733$	22562.5	1.13692	0.568461	−	0.822710i	$-0.307539\pi$
$733$	22562.5	1.13692	0.568461	+	0.822710i	$0.307539\pi$
$734$	−73990.2	−3.72075
$735$	0	0
$736$	54765.6	2.74278
$737$	4004.23	0.200132
$738$	−62510.2	−3.11793
$739$	−4400.04	−0.219023	−0.109512	−	0.993986i	$-0.534929\pi$
$739$	−4400.04	−0.219023	−0.109512	+	0.993986i	$0.534929\pi$
$740$	4763.73	0.236646
$741$	338.342	0.0167737
$742$	0	0
$743$	13767.1	0.679765	0.339882	−	0.940468i	$-0.389613\pi$
$743$	13767.1	0.679765	0.339882	+	0.940468i	$0.389613\pi$
$744$	−740.451	−0.0364869
$745$	−5647.77	−0.277742
$746$	−14842.8	−0.728462
$747$	25937.7	1.27043
$748$	11456.1	0.559997
$749$	0	0
$750$	784.987	0.0382182
$751$	−3708.46	−0.180191	−0.0900955	−	0.995933i	$-0.528717\pi$
$751$	−3708.46	−0.180191	−0.0900955	+	0.995933i	$0.528717\pi$
$752$	−26950.0	−1.30687
$753$	1355.75	0.0656125
$754$	10310.5	0.497994
$755$	−1347.72	−0.0649648
$756$	0	0
$757$	−13872.0	−0.666030	−0.333015	−	0.942921i	$-0.608066\pi$
$757$	−13872.0	−0.666030	−0.333015	+	0.942921i	$0.608066\pi$
$758$	13836.1	0.662994
$759$	609.741	0.0291597
$760$	−35132.6	−1.67684
$761$	−4245.12	−0.202215	−0.101107	−	0.994876i	$-0.532239\pi$
$761$	−4245.12	−0.202215	−0.101107	+	0.994876i	$0.532239\pi$
$762$	−1646.03	−0.0782540
$763$	0	0
$764$	−88552.7	−4.19335
$765$	1457.84	0.0688999
$766$	46954.9	2.21482
$767$	2786.02	0.131157
$768$	753.001	0.0353797
$769$	−24697.4	−1.15814	−0.579070	−	0.815278i	$-0.696584\pi$
$769$	−24697.4	−1.15814	−0.579070	+	0.815278i	$0.696584\pi$
$770$	0	0
$771$	−412.529	−0.0192696
$772$	110730.	5.16225
$773$	40131.9	1.86733	0.933665	−	0.358148i	$-0.116592\pi$
$773$	40131.9	1.86733	0.933665	+	0.358148i	$0.116592\pi$
$774$	29858.3	1.38661
$775$	6399.05	0.296594
$776$	70668.3	3.26913
$777$	0	0
$778$	31505.1	1.45181
$779$	68537.2	3.15224
$780$	−141.599	−0.00650005
$781$	−1559.66	−0.0714583
$782$	9044.86	0.413611
$783$	−1779.58	−0.0812221
$784$	0	0
$785$	−6117.49	−0.278143
$786$	9.08358	0.000412214 0
$787$	32376.1	1.46643	0.733217	−	0.679995i	$-0.238018\pi$
$787$	32376.1	1.46643	0.733217	+	0.679995i	$0.238018\pi$
$788$	20614.5	0.931930
$789$	246.304	0.0111136
$790$	18160.5	0.817877
$791$	0	0
$792$	−60348.9	−2.70758
$793$	−8333.71	−0.373189
$794$	80342.5	3.59099
$795$	−206.178	−0.00919797
$796$	87350.8	3.88953
$797$	6662.08	0.296089	0.148045	−	0.988981i	$-0.452702\pi$
$797$	6662.08	0.296089	0.148045	+	0.988981i	$0.452702\pi$
$798$	0	0
$799$	−2220.85	−0.0983329
$800$	−63594.4	−2.81050
$801$	33189.9	1.46405
$802$	39851.5	1.75462
$803$	−26660.5	−1.17164
$804$	497.547	0.0218248
$805$	0	0
$806$	−3331.86	−0.145608
$807$	−364.321	−0.0158918
$808$	−33164.1	−1.44395
$809$	13697.4	0.595271	0.297636	−	0.954680i	$-0.403802\pi$
$809$	13697.4	0.595271	0.297636	+	0.954680i	$0.403802\pi$
$810$	−12417.9	−0.538666
$811$	8326.87	0.360538	0.180269	−	0.983617i	$-0.442303\pi$
$811$	8326.87	0.360538	0.180269	+	0.983617i	$0.442303\pi$
$812$	0	0
$813$	−1256.64	−0.0542093
$814$	−12402.7	−0.534047
$815$	7200.78	0.309487
$816$	670.980	0.0287855
$817$	−32737.1	−1.40187
$818$	−24863.5	−1.06275
$819$	0	0
$820$	−28683.3	−1.22154
$821$	13762.0	0.585013	0.292507	−	0.956264i	$-0.405511\pi$
$821$	13762.0	0.585013	0.292507	+	0.956264i	$0.405511\pi$
$822$	−2681.47	−0.113780
$823$	−26685.8	−1.13027	−0.565133	−	0.825000i	$-0.691175\pi$
$823$	−26685.8	−1.13027	−0.565133	+	0.825000i	$0.691175\pi$
$824$	−62801.6	−2.65509
$825$	−708.037	−0.0298796
$826$	0	0
$827$	−36547.6	−1.53674	−0.768370	−	0.640006i	$-0.778932\pi$
$827$	−36547.6	−1.53674	−0.768370	+	0.640006i	$0.778932\pi$
$828$	−55807.6	−2.34233
$829$	−20113.2	−0.842655	−0.421327	−	0.906909i	$-0.638436\pi$
$829$	−20113.2	−0.842655	−0.421327	+	0.906909i	$0.638436\pi$
$830$	16453.0	0.688063
$831$	235.090	0.00981369
$832$	14754.9	0.614827
$833$	0	0
$834$	−1618.81	−0.0672120
$835$	−973.778	−0.0403581
$836$	107137.	4.43232
$837$	575.074	0.0237484
$838$	64859.9	2.67368
$839$	−42286.8	−1.74005	−0.870026	−	0.493006i	$-0.835898\pi$
$839$	−42286.8	−1.74005	−0.870026	+	0.493006i	$0.835898\pi$
$840$	0	0
$841$	5320.36	0.218146
$842$	35933.5	1.47072
$843$	−1748.74	−0.0714470
$844$	−60567.3	−2.47016
$845$	6593.92	0.268447
$846$	18942.9	0.769822
$847$	0	0
$848$	69899.3	2.83061
$849$	1324.52	0.0535424
$850$	−10503.0	−0.423823
$851$	−7083.44	−0.285332
$852$	−193.796	−0.00779266
$853$	18072.6	0.725434	0.362717	−	0.931899i	$-0.381849\pi$
$853$	18072.6	0.725434	0.362717	+	0.931899i	$0.381849\pi$
$854$	0	0
$855$	13633.7	0.545336
$856$	33734.5	1.34699
$857$	13258.2	0.528460	0.264230	−	0.964460i	$-0.414882\pi$
$857$	13258.2	0.528460	0.264230	+	0.964460i	$0.414882\pi$
$858$	368.662	0.0146689
$859$	−215.302	−0.00855182	−0.00427591	−	0.999991i	$-0.501361\pi$
$859$	−215.302	−0.00855182	−0.00427591	+	0.999991i	$0.501361\pi$
$860$	13700.7	0.543245
$861$	0	0
$862$	41133.7	1.62531
$863$	−8386.55	−0.330801	−0.165401	−	0.986226i	$-0.552892\pi$
$863$	−8386.55	−0.330801	−0.165401	+	0.986226i	$0.552892\pi$
$864$	−5715.14	−0.225038
$865$	−9949.61	−0.391095
$866$	55692.6	2.18535
$867$	55.2929	0.00216591
$868$	0	0
$869$	−34202.9	−1.33516
$870$	−564.035	−0.0219800
$871$	1382.70	0.0537901
$872$	117622.	4.56787
$873$	−27423.7	−1.06318
$874$	84587.1	3.27369
$875$	0	0
$876$	−3312.72	−0.127770
$877$	1040.20	0.0400512	0.0200256	−	0.999799i	$-0.493625\pi$
$877$	1040.20	0.0400512	0.0200256	+	0.999799i	$0.493625\pi$
$878$	−84925.0	−3.26433
$879$	−1170.19	−0.0449028
$880$	−21134.8	−0.809608
$881$	1555.88	0.0594993	0.0297497	−	0.999557i	$-0.490529\pi$
$881$	1555.88	0.0594993	0.0297497	+	0.999557i	$0.490529\pi$
$882$	0	0
$883$	−6509.84	−0.248101	−0.124051	−	0.992276i	$-0.539589\pi$
$883$	−6509.84	−0.248101	−0.124051	+	0.992276i	$0.539589\pi$
$884$	3955.94	0.150512
$885$	−152.409	−0.00578889
$886$	37113.8	1.40730
$887$	9387.24	0.355347	0.177673	−	0.984090i	$-0.443143\pi$
$887$	9387.24	0.355347	0.177673	+	0.984090i	$0.443143\pi$
$888$	−951.778	−0.0359680
$889$	0	0
$890$	21053.3	0.792929
$891$	23387.3	0.879355
$892$	77311.5	2.90200
$893$	−20769.3	−0.778295
$894$	1827.10	0.0683526
$895$	−1038.84	−0.0387985
$896$	0	0
$897$	210.550	0.00783732
$898$	−43837.2	−1.62903
$899$	−9600.63	−0.356172
$900$	64804.4	2.40016
$901$	5760.14	0.212983
$902$	74679.0	2.75670
$903$	0	0
$904$	77759.8	2.86090
$905$	−192.840	−0.00708312
$906$	435.997	0.0159879
$907$	−20839.3	−0.762906	−0.381453	−	0.924388i	$-0.624576\pi$
$907$	−20839.3	−0.762906	−0.381453	+	0.924388i	$0.624576\pi$
$908$	24639.6	0.900542
$909$	12869.8	0.469596
$910$	0	0
$911$	52129.2	1.89585	0.947923	−	0.318498i	$-0.103179\pi$
$911$	52129.2	1.89585	0.947923	+	0.318498i	$0.103179\pi$
$912$	6274.97	0.227834
$913$	−30987.0	−1.12324
$914$	12140.9	0.439370
$915$	455.894	0.0164715
$916$	−107193.	−3.86654
$917$	0	0
$918$	−943.888	−0.0339357
$919$	5187.05	0.186186	0.0930931	−	0.995657i	$-0.470325\pi$
$919$	5187.05	0.186186	0.0930931	+	0.995657i	$0.470325\pi$
$920$	−21863.1	−0.783484
$921$	609.541	0.0218079
$922$	28147.5	1.00541
$923$	−538.568	−0.0192060
$924$	0	0
$925$	8225.36	0.292376
$926$	−11865.6	−0.421088
$927$	24371.0	0.863482
$928$	95412.0	3.37506
$929$	45790.1	1.61714	0.808571	−	0.588399i	$-0.200241\pi$
$929$	45790.1	1.61714	0.808571	+	0.588399i	$0.200241\pi$
$930$	182.269	0.00642670
$931$	0	0
$932$	−67422.2	−2.36962
$933$	−869.840	−0.0305223
$934$	−36268.5	−1.27060
$935$	−1741.64	−0.0609174
$936$	−20839.2	−0.727723
$937$	−22497.8	−0.784387	−0.392194	−	0.919883i	$-0.628284\pi$
$937$	−22497.8	−0.784387	−0.392194	+	0.919883i	$0.628284\pi$
$938$	0	0
$939$	−858.154	−0.0298241
$940$	8692.10	0.301601
$941$	177.231	0.00613981	0.00306990	−	0.999995i	$-0.499023\pi$
$941$	177.231	0.00613981	0.00306990	+	0.999995i	$0.499023\pi$
$942$	1979.05	0.0684513
$943$	42650.8	1.47285
$944$	51670.3	1.78149
$945$	0	0
$946$	−35670.8	−1.22596
$947$	18745.0	0.643221	0.321610	−	0.946872i	$-0.395776\pi$
$947$	18745.0	0.643221	0.321610	+	0.946872i	$0.395776\pi$
$948$	−4249.90	−0.145601
$949$	−9206.18	−0.314905
$950$	−98223.3	−3.35451
$951$	−741.255	−0.0252753
$952$	0	0
$953$	12771.6	0.434117	0.217059	−	0.976159i	$-0.430354\pi$
$953$	12771.6	0.434117	0.217059	+	0.976159i	$0.430354\pi$
$954$	−49131.5	−1.66739
$955$	13462.4	0.456160
$956$	−88973.6	−3.01005
$957$	1062.28	0.0358816
$958$	70818.4	2.38835
$959$	0	0
$960$	−807.166	−0.0271366
$961$	−26688.5	−0.895859
$962$	−4282.79	−0.143537
$963$	−13091.1	−0.438064
$964$	−105161.	−3.51351
$965$	−16833.9	−0.561558
$966$	0	0
$967$	50377.0	1.67530	0.837650	−	0.546207i	$-0.183929\pi$
$967$	50377.0	1.67530	0.837650	+	0.546207i	$0.183929\pi$
$968$	−20383.5	−0.676810
$969$	517.096	0.0171429
$970$	−17395.7	−0.575815
$971$	−18650.8	−0.616407	−0.308204	−	0.951320i	$-0.599728\pi$
$971$	−18650.8	−0.616407	−0.308204	+	0.951320i	$0.599728\pi$
$972$	8737.79	0.288338
$973$	0	0
$974$	−54923.9	−1.80685
$975$	−244.493	−0.00803082
$976$	−154559.	−5.06897
$977$	29031.8	0.950675	0.475337	−	0.879804i	$-0.342326\pi$
$977$	29031.8	0.950675	0.475337	+	0.879804i	$0.342326\pi$
$978$	−2329.51	−0.0761651
$979$	−39650.9	−1.29443
$980$	0	0
$981$	−45644.7	−1.48555
$982$	−16283.5	−0.529153
$983$	−36526.4	−1.18516	−0.592580	−	0.805511i	$-0.701891\pi$
$983$	−36526.4	−1.18516	−0.592580	+	0.805511i	$0.701891\pi$
$984$	5730.84	0.185663
$985$	−3133.96	−0.101377
$986$	15757.8	0.508957
$987$	0	0
$988$	36995.7	1.19129
$989$	−20372.3	−0.655008
$990$	14855.4	0.476906
$991$	29262.0	0.937979	0.468990	−	0.883204i	$-0.344618\pi$
$991$	29262.0	0.937979	0.468990	+	0.883204i	$0.344618\pi$
$992$	−30832.5	−0.986829
$993$	143.703	0.00459243
$994$	0	0
$995$	−13279.7	−0.423109
$996$	−3850.31	−0.122492
$997$	24224.3	0.769499	0.384749	−	0.923021i	$-0.374288\pi$
$997$	24224.3	0.769499	0.384749	+	0.923021i	$0.374288\pi$
$998$	−42927.8	−1.36158
$999$	739.201	0.0234107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	833.4.a.f.1.7		7
7.6	odd	2		119.4.a.d.1.7	✓	7
21.20	even	2		1071.4.a.o.1.1		7
28.27	even	2		1904.4.a.p.1.4		7
119.118	odd	2		2023.4.a.g.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.4.a.d.1.7	✓	7	7.6	odd	2
833.4.a.f.1.7		7	1.1	even	1	trivial
1071.4.a.o.1.1		7	21.20	even	2
1904.4.a.p.1.4		7	28.27	even	2
2023.4.a.g.1.7		7	119.118	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:	\( N \)	\(=\)	\( 833 = 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	833.a (trivial)

\(f(q)\)	\(=\)	\(q+5.37776 q^{2} +0.191325 q^{3} +20.9203 q^{4} -3.18044 q^{5} +1.02890 q^{6} +69.4820 q^{8} -26.9634 q^{9} -17.1036 q^{10} +32.2124 q^{11} +4.00256 q^{12} +11.1233 q^{13} -0.608497 q^{15} +206.295 q^{16} +17.0000 q^{17} -145.003 q^{18} +158.983 q^{19} -66.5357 q^{20} +173.230 q^{22} +98.9354 q^{23} +13.2936 q^{24} -114.885 q^{25} +59.8183 q^{26} -10.3245 q^{27} +172.364 q^{29} -3.27235 q^{30} -55.6997 q^{31} +553.549 q^{32} +6.16302 q^{33} +91.4219 q^{34} -564.081 q^{36} -71.5966 q^{37} +854.973 q^{38} +2.12816 q^{39} -220.983 q^{40} +431.097 q^{41} -205.915 q^{43} +673.891 q^{44} +85.7555 q^{45} +532.051 q^{46} -130.638 q^{47} +39.4694 q^{48} -617.822 q^{50} +3.25252 q^{51} +232.702 q^{52} +338.832 q^{53} -55.5228 q^{54} -102.450 q^{55} +30.4174 q^{57} +926.932 q^{58} +250.468 q^{59} -12.7299 q^{60} -749.213 q^{61} -299.539 q^{62} +1326.49 q^{64} -35.3770 q^{65} +33.1432 q^{66} +124.307 q^{67} +355.644 q^{68} +18.9288 q^{69} -48.4180 q^{71} -1873.47 q^{72} -827.649 q^{73} -385.029 q^{74} -21.9803 q^{75} +3325.97 q^{76} +11.4447 q^{78} -1061.79 q^{79} -656.110 q^{80} +726.036 q^{81} +2318.33 q^{82} -961.960 q^{83} -54.0675 q^{85} -1107.36 q^{86} +32.9775 q^{87} +2238.18 q^{88} -1230.92 q^{89} +461.172 q^{90} +2069.76 q^{92} -10.6567 q^{93} -702.540 q^{94} -505.637 q^{95} +105.908 q^{96} +1017.07 q^{97} -868.554 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} - 5 q^{3} + 52 q^{4} - 35 q^{5} - 51 q^{6} - 6 q^{8} + 128 q^{9} - 18 q^{10} + 48 q^{11} + 16 q^{12} - 84 q^{13} - 54 q^{15} + 256 q^{16} + 119 q^{17} + 196 q^{18} - 156 q^{19} - 317 q^{20}+ \cdots + 7572 q^{99}+O(q^{100}) \) 7 * q + 4 * q^2 - 5 * q^3 + 52 * q^4 - 35 * q^5 - 51 * q^6 - 6 * q^8 + 128 * q^9 - 18 * q^10 + 48 * q^11 + 16 * q^12 - 84 * q^13 - 54 * q^15 + 256 * q^16 + 119 * q^17 + 196 * q^18 - 156 * q^19 - 317 * q^20 + 244 * q^22 + 290 * q^23 - 818 * q^24 + 94 * q^25 + 246 * q^26 + 10 * q^27 + 1074 * q^29 - 23 * q^30 + 291 * q^31 - 779 * q^32 - 26 * q^33 + 68 * q^34 - 679 * q^36 + 596 * q^37 + 1294 * q^38 - 824 * q^39 + 1159 * q^40 - 649 * q^41 + 257 * q^43 - 196 * q^44 - 720 * q^45 + 1044 * q^46 + 1108 * q^47 + 2655 * q^48 - 23 * q^50 - 85 * q^51 - 768 * q^52 + 321 * q^53 + 1497 * q^54 - 152 * q^55 + 398 * q^57 - 1166 * q^58 + 1176 * q^59 - 1495 * q^60 - 1345 * q^61 + 2824 * q^62 + 4888 * q^64 - 1334 * q^65 + 2280 * q^66 + 1379 * q^67 + 884 * q^68 + 1152 * q^69 + 2000 * q^71 - 3375 * q^72 - 367 * q^73 - 114 * q^74 - 888 * q^75 + 1836 * q^76 + 2362 * q^78 + 704 * q^79 - 4112 * q^80 + 4903 * q^81 + 1421 * q^82 + 16 * q^83 - 595 * q^85 - 2785 * q^86 + 1050 * q^87 + 2840 * q^88 - 1070 * q^89 + 3787 * q^90 - 2700 * q^92 - 1976 * q^93 - 4446 * q^94 + 3950 * q^95 + 1420 * q^96 + 39 * q^97 + 7572 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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