LMFDB - Embedded newform 8049.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.2715885869$
Analytic rank:	$0$
Dimension:	$119$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76674 q^{2} -1.00000 q^{3} +5.65483 q^{4} -0.357353 q^{5} +2.76674 q^{6} -2.67071 q^{7} -10.1119 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.76674 q^{2} -1.00000 q^{3} +5.65483 q^{4} -0.357353 q^{5} +2.76674 q^{6} -2.67071 q^{7} -10.1119 q^{8} +1.00000 q^{9} +0.988701 q^{10} +2.49328 q^{11} -5.65483 q^{12} +0.505285 q^{13} +7.38915 q^{14} +0.357353 q^{15} +16.6674 q^{16} +3.07775 q^{17} -2.76674 q^{18} -5.33824 q^{19} -2.02077 q^{20} +2.67071 q^{21} -6.89824 q^{22} -8.10174 q^{23} +10.1119 q^{24} -4.87230 q^{25} -1.39799 q^{26} -1.00000 q^{27} -15.1024 q^{28} -7.11636 q^{29} -0.988701 q^{30} -7.62243 q^{31} -25.8905 q^{32} -2.49328 q^{33} -8.51531 q^{34} +0.954385 q^{35} +5.65483 q^{36} -10.9898 q^{37} +14.7695 q^{38} -0.505285 q^{39} +3.61353 q^{40} +2.85018 q^{41} -7.38915 q^{42} -0.938451 q^{43} +14.0991 q^{44} -0.357353 q^{45} +22.4154 q^{46} +5.39192 q^{47} -16.6674 q^{48} +0.132690 q^{49} +13.4804 q^{50} -3.07775 q^{51} +2.85730 q^{52} -2.52713 q^{53} +2.76674 q^{54} -0.890979 q^{55} +27.0061 q^{56} +5.33824 q^{57} +19.6891 q^{58} +3.30585 q^{59} +2.02077 q^{60} -7.34791 q^{61} +21.0892 q^{62} -2.67071 q^{63} +38.2973 q^{64} -0.180565 q^{65} +6.89824 q^{66} -9.63849 q^{67} +17.4041 q^{68} +8.10174 q^{69} -2.64053 q^{70} +1.81365 q^{71} -10.1119 q^{72} -10.2351 q^{73} +30.4059 q^{74} +4.87230 q^{75} -30.1869 q^{76} -6.65882 q^{77} +1.39799 q^{78} +0.770961 q^{79} -5.95615 q^{80} +1.00000 q^{81} -7.88569 q^{82} +12.0650 q^{83} +15.1024 q^{84} -1.09984 q^{85} +2.59645 q^{86} +7.11636 q^{87} -25.2119 q^{88} +6.82993 q^{89} +0.988701 q^{90} -1.34947 q^{91} -45.8140 q^{92} +7.62243 q^{93} -14.9180 q^{94} +1.90764 q^{95} +25.8905 q^{96} +2.41911 q^{97} -0.367117 q^{98} +2.49328 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9}+O(q^{10}) $ 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9	\( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9} - 10 q^{10} + 56 q^{11} - 137 q^{12} - 37 q^{13} + 31 q^{14} - 17 q^{15} + 173 q^{16} + 17 q^{17} + 11 q^{18} + 16 q^{19} + 61 q^{20} - 10 q^{21} - 3 q^{22} + 76 q^{23} - 33 q^{24} + 134 q^{25} + 47 q^{26} - 119 q^{27} - q^{28} + 47 q^{29} + 10 q^{30} + 51 q^{31} + 87 q^{32} - 56 q^{33} + 13 q^{34} + 58 q^{35} + 137 q^{36} - 67 q^{37} + 35 q^{38} + 37 q^{39} - 40 q^{40} + 47 q^{41} - 31 q^{42} + 12 q^{43} + 148 q^{44} + 17 q^{45} + 26 q^{46} + 107 q^{47} - 173 q^{48} + 163 q^{49} + 76 q^{50} - 17 q^{51} - 57 q^{52} + 64 q^{53} - 11 q^{54} + 71 q^{55} + 91 q^{56} - 16 q^{57} + 12 q^{58} + 98 q^{59} - 61 q^{60} - 50 q^{61} + 40 q^{62} + 10 q^{63} + 245 q^{64} + 40 q^{65} + 3 q^{66} + 12 q^{67} + 75 q^{68} - 76 q^{69} - 9 q^{70} + 194 q^{71} + 33 q^{72} - 79 q^{73} + 72 q^{74} - 134 q^{75} + 12 q^{76} + 71 q^{77} - 47 q^{78} + 127 q^{79} + 148 q^{80} + 119 q^{81} - 54 q^{82} + 77 q^{83} + q^{84} - 25 q^{85} + 142 q^{86} - 47 q^{87} + q^{88} + 93 q^{89} - 10 q^{90} + 61 q^{91} + 156 q^{92} - 51 q^{93} + 16 q^{94} + 138 q^{95} - 87 q^{96} - 110 q^{97} + 96 q^{98} + 56 q^{99}+O(q^{100}) \) 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9 - 10 * q^10 + 56 * q^11 - 137 * q^12 - 37 * q^13 + 31 * q^14 - 17 * q^15 + 173 * q^16 + 17 * q^17 + 11 * q^18 + 16 * q^19 + 61 * q^20 - 10 * q^21 - 3 * q^22 + 76 * q^23 - 33 * q^24 + 134 * q^25 + 47 * q^26 - 119 * q^27 - q^28 + 47 * q^29 + 10 * q^30 + 51 * q^31 + 87 * q^32 - 56 * q^33 + 13 * q^34 + 58 * q^35 + 137 * q^36 - 67 * q^37 + 35 * q^38 + 37 * q^39 - 40 * q^40 + 47 * q^41 - 31 * q^42 + 12 * q^43 + 148 * q^44 + 17 * q^45 + 26 * q^46 + 107 * q^47 - 173 * q^48 + 163 * q^49 + 76 * q^50 - 17 * q^51 - 57 * q^52 + 64 * q^53 - 11 * q^54 + 71 * q^55 + 91 * q^56 - 16 * q^57 + 12 * q^58 + 98 * q^59 - 61 * q^60 - 50 * q^61 + 40 * q^62 + 10 * q^63 + 245 * q^64 + 40 * q^65 + 3 * q^66 + 12 * q^67 + 75 * q^68 - 76 * q^69 - 9 * q^70 + 194 * q^71 + 33 * q^72 - 79 * q^73 + 72 * q^74 - 134 * q^75 + 12 * q^76 + 71 * q^77 - 47 * q^78 + 127 * q^79 + 148 * q^80 + 119 * q^81 - 54 * q^82 + 77 * q^83 + q^84 - 25 * q^85 + 142 * q^86 - 47 * q^87 + q^88 + 93 * q^89 - 10 * q^90 + 61 * q^91 + 156 * q^92 - 51 * q^93 + 16 * q^94 + 138 * q^95 - 87 * q^96 - 110 * q^97 + 96 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.76674	−1.95638	−0.978189	−	0.207717i	$-0.933397\pi$
$2$	−2.76674	−1.95638	−0.978189	+	0.207717i	$0.933397\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.65483	2.82741
$5$	−0.357353	−0.159813	−0.0799065	−	0.996802i	$-0.525462\pi$
$5$	−0.357353	−0.159813	−0.0799065	+	0.996802i	$0.525462\pi$
$6$	2.76674	1.12952
$7$	−2.67071	−1.00943	−0.504717	−	0.863285i	$-0.668403\pi$
$7$	−2.67071	−1.00943	−0.504717	+	0.863285i	$0.668403\pi$
$8$	−10.1119	−3.57511
$9$	1.00000	0.333333
$10$	0.988701	0.312655
$11$	2.49328	0.751751	0.375876	−	0.926670i	$-0.377342\pi$
$11$	2.49328	0.751751	0.375876	+	0.926670i	$0.377342\pi$
$12$	−5.65483	−1.63241
$13$	0.505285	0.140141	0.0700704	−	0.997542i	$-0.477678\pi$
$13$	0.505285	0.140141	0.0700704	+	0.997542i	$0.477678\pi$
$14$	7.38915	1.97483
$15$	0.357353	0.0922681
$16$	16.6674	4.16686
$17$	3.07775	0.746463	0.373232	−	0.927738i	$-0.378250\pi$
$17$	3.07775	0.746463	0.373232	+	0.927738i	$0.378250\pi$
$18$	−2.76674	−0.652126
$19$	−5.33824	−1.22468	−0.612339	−	0.790596i	$-0.709771\pi$
$19$	−5.33824	−1.22468	−0.612339	+	0.790596i	$0.709771\pi$
$20$	−2.02077	−0.451858
$21$	2.67071	0.582797
$22$	−6.89824	−1.47071
$23$	−8.10174	−1.68933	−0.844665	−	0.535295i	$-0.820200\pi$
$23$	−8.10174	−1.68933	−0.844665	+	0.535295i	$0.820200\pi$
$24$	10.1119	2.06409
$25$	−4.87230	−0.974460
$26$	−1.39799	−0.274169
$27$	−1.00000	−0.192450
$28$	−15.1024	−2.85409
$29$	−7.11636	−1.32148	−0.660738	−	0.750617i	$-0.729756\pi$
$29$	−7.11636	−1.32148	−0.660738	+	0.750617i	$0.729756\pi$
$30$	−0.988701	−0.180511
$31$	−7.62243	−1.36903	−0.684514	−	0.729000i	$-0.739986\pi$
$31$	−7.62243	−1.36903	−0.684514	+	0.729000i	$0.739986\pi$
$32$	−25.8905	−4.57684
$33$	−2.49328	−0.434024
$34$	−8.51531	−1.46036
$35$	0.954385	0.161321
$36$	5.65483	0.942471
$37$	−10.9898	−1.80672	−0.903358	−	0.428887i	$-0.858906\pi$
$37$	−10.9898	−1.80672	−0.903358	+	0.428887i	$0.858906\pi$
$38$	14.7695	2.39593
$39$	−0.505285	−0.0809104
$40$	3.61353	0.571350
$41$	2.85018	0.445123	0.222562	−	0.974919i	$-0.428558\pi$
$41$	2.85018	0.445123	0.222562	+	0.974919i	$0.428558\pi$
$42$	−7.38915	−1.14017
$43$	−0.938451	−0.143112	−0.0715562	−	0.997437i	$-0.522797\pi$
$43$	−0.938451	−0.143112	−0.0715562	+	0.997437i	$0.522797\pi$
$44$	14.0991	2.12551
$45$	−0.357353	−0.0532710
$46$	22.4154	3.30497
$47$	5.39192	0.786493	0.393247	−	0.919433i	$-0.371352\pi$
$47$	5.39192	0.786493	0.393247	+	0.919433i	$0.371352\pi$
$48$	−16.6674	−2.40574
$49$	0.132690	0.0189557
$50$	13.4804	1.90641
$51$	−3.07775	−0.430971
$52$	2.85730	0.396236
$53$	−2.52713	−0.347128	−0.173564	−	0.984823i	$-0.555528\pi$
$53$	−2.52713	−0.347128	−0.173564	+	0.984823i	$0.555528\pi$
$54$	2.76674	0.376505
$55$	−0.890979	−0.120140
$56$	27.0061	3.60884
$57$	5.33824	0.707068
$58$	19.6891	2.58530
$59$	3.30585	0.430385	0.215193	−	0.976572i	$-0.430962\pi$
$59$	3.30585	0.430385	0.215193	+	0.976572i	$0.430962\pi$
$60$	2.02077	0.260880
$61$	−7.34791	−0.940804	−0.470402	−	0.882452i	$-0.655891\pi$
$61$	−7.34791	−0.940804	−0.470402	+	0.882452i	$0.655891\pi$
$62$	21.0892	2.67834
$63$	−2.67071	−0.336478
$64$	38.2973	4.78716
$65$	−0.180565	−0.0223963
$66$	6.89824	0.849114
$67$	−9.63849	−1.17753	−0.588764	−	0.808305i	$-0.700385\pi$
$67$	−9.63849	−1.17753	−0.588764	+	0.808305i	$0.700385\pi$
$68$	17.4041	2.11056
$69$	8.10174	0.975335
$70$	−2.64053	−0.315604
$71$	1.81365	0.215240	0.107620	−	0.994192i	$-0.465677\pi$
$71$	1.81365	0.215240	0.107620	+	0.994192i	$0.465677\pi$
$72$	−10.1119	−1.19170
$73$	−10.2351	−1.19792	−0.598962	−	0.800777i	$-0.704420\pi$
$73$	−10.2351	−1.19792	−0.598962	+	0.800777i	$0.704420\pi$
$74$	30.4059	3.53462
$75$	4.87230	0.562605
$76$	−30.1869	−3.46267
$77$	−6.65882	−0.758843
$78$	1.39799	0.158291
$79$	0.770961	0.0867399	0.0433700	−	0.999059i	$-0.486191\pi$
$79$	0.770961	0.0867399	0.0433700	+	0.999059i	$0.486191\pi$
$80$	−5.95615	−0.665918
$81$	1.00000	0.111111
$82$	−7.88569	−0.870829
$83$	12.0650	1.32431	0.662155	−	0.749367i	$-0.269642\pi$
$83$	12.0650	1.32431	0.662155	+	0.749367i	$0.269642\pi$
$84$	15.1024	1.64781
$85$	−1.09984	−0.119295
$86$	2.59645	0.279982
$87$	7.11636	0.762954
$88$	−25.2119	−2.68760
$89$	6.82993	0.723971	0.361985	−	0.932184i	$-0.382099\pi$
$89$	6.82993	0.723971	0.361985	+	0.932184i	$0.382099\pi$
$90$	0.988701	0.104218
$91$	−1.34947	−0.141463
$92$	−45.8140	−4.77644
$93$	7.62243	0.790409
$94$	−14.9180	−1.53868
$95$	1.90764	0.195719
$96$	25.8905	2.64244
$97$	2.41911	0.245623	0.122812	−	0.992430i	$-0.460809\pi$
$97$	2.41911	0.245623	0.122812	+	0.992430i	$0.460809\pi$
$98$	−0.367117	−0.0370844
$99$	2.49328	0.250584
$100$	−27.5520	−2.75520
$101$	10.7809	1.07274	0.536371	−	0.843982i	$-0.319795\pi$
$101$	10.7809	1.07274	0.536371	+	0.843982i	$0.319795\pi$
$102$	8.51531	0.843141
$103$	19.5006	1.92145	0.960725	−	0.277502i	$-0.0895067\pi$
$103$	19.5006	1.92145	0.960725	+	0.277502i	$0.0895067\pi$
$104$	−5.10942	−0.501020
$105$	−0.954385	−0.0931385
$106$	6.99191	0.679114
$107$	−14.8789	−1.43840	−0.719198	−	0.694805i	$-0.755491\pi$
$107$	−14.8789	−1.43840	−0.719198	+	0.694805i	$0.755491\pi$
$108$	−5.65483	−0.544136
$109$	13.8714	1.32864	0.664319	−	0.747449i	$-0.268722\pi$
$109$	13.8714	1.32864	0.664319	+	0.747449i	$0.268722\pi$
$110$	2.46510	0.235038
$111$	10.9898	1.04311
$112$	−44.5139	−4.20617
$113$	−19.7381	−1.85681	−0.928404	−	0.371572i	$-0.878819\pi$
$113$	−19.7381	−1.85681	−0.928404	+	0.371572i	$0.878819\pi$
$114$	−14.7695	−1.38329
$115$	2.89518	0.269977
$116$	−40.2418	−3.73636
$117$	0.505285	0.0467136
$118$	−9.14642	−0.841996
$119$	−8.21977	−0.753505
$120$	−3.61353	−0.329869
$121$	−4.78357	−0.434870
$122$	20.3297	1.84057
$123$	−2.85018	−0.256992
$124$	−43.1035	−3.87081
$125$	3.52789	0.315544
$126$	7.38915	0.658278
$127$	−0.754581	−0.0669582	−0.0334791	−	0.999439i	$-0.510659\pi$
$127$	−0.754581	−0.0669582	−0.0334791	+	0.999439i	$0.510659\pi$
$128$	−54.1775	−4.78866
$129$	0.938451	0.0826260
$130$	0.499576	0.0438157
$131$	−3.01097	−0.263069	−0.131535	−	0.991312i	$-0.541990\pi$
$131$	−3.01097	−0.263069	−0.131535	+	0.991312i	$0.541990\pi$
$132$	−14.0991	−1.22716
$133$	14.2569	1.23623
$134$	26.6672	2.30369
$135$	0.357353	0.0307560
$136$	−31.1220	−2.66869
$137$	−16.6005	−1.41827	−0.709137	−	0.705071i	$-0.750915\pi$
$137$	−16.6005	−1.41827	−0.709137	+	0.705071i	$0.750915\pi$
$138$	−22.4154	−1.90812
$139$	−5.03384	−0.426964	−0.213482	−	0.976947i	$-0.568481\pi$
$139$	−5.03384	−0.426964	−0.213482	+	0.976947i	$0.568481\pi$
$140$	5.39689	0.456120
$141$	−5.39192	−0.454082
$142$	−5.01788	−0.421092
$143$	1.25982	0.105351
$144$	16.6674	1.38895
$145$	2.54305	0.211189
$146$	28.3177	2.34359
$147$	−0.132690	−0.0109441
$148$	−62.1456	−5.10833
$149$	10.5544	0.864654	0.432327	−	0.901717i	$-0.357693\pi$
$149$	10.5544	0.864654	0.432327	+	0.901717i	$0.357693\pi$
$150$	−13.4804	−1.10067
$151$	−11.2334	−0.914159	−0.457079	−	0.889426i	$-0.651104\pi$
$151$	−11.2334	−0.914159	−0.457079	+	0.889426i	$0.651104\pi$
$152$	53.9800	4.37836
$153$	3.07775	0.248821
$154$	18.4232	1.48458
$155$	2.72390	0.218789
$156$	−2.85730	−0.228767
$157$	0.353825	0.0282383	0.0141191	−	0.999900i	$-0.495506\pi$
$157$	0.353825	0.0282383	0.0141191	+	0.999900i	$0.495506\pi$
$158$	−2.13305	−0.169696
$159$	2.52713	0.200415
$160$	9.25204	0.731438
$161$	21.6374	1.70527
$162$	−2.76674	−0.217375
$163$	0.174862	0.0136963	0.00684814	−	0.999977i	$-0.497820\pi$
$163$	0.174862	0.0136963	0.00684814	+	0.999977i	$0.497820\pi$
$164$	16.1173	1.25855
$165$	0.890979	0.0693626
$166$	−33.3808	−2.59085
$167$	10.6463	0.823839	0.411919	−	0.911220i	$-0.364859\pi$
$167$	10.6463	0.823839	0.411919	+	0.911220i	$0.364859\pi$
$168$	−27.0061	−2.08356
$169$	−12.7447	−0.980361
$170$	3.04297	0.233385
$171$	−5.33824	−0.408226
$172$	−5.30678	−0.404638
$173$	−17.0539	−1.29659	−0.648293	−	0.761391i	$-0.724517\pi$
$173$	−17.0539	−1.29659	−0.648293	+	0.761391i	$0.724517\pi$
$174$	−19.6891	−1.49263
$175$	13.0125	0.983652
$176$	41.5565	3.13244
$177$	−3.30585	−0.248483
$178$	−18.8966	−1.41636
$179$	9.56200	0.714697	0.357349	−	0.933971i	$-0.383681\pi$
$179$	9.56200	0.714697	0.357349	+	0.933971i	$0.383681\pi$
$180$	−2.02077	−0.150619
$181$	−11.6094	−0.862919	−0.431460	−	0.902132i	$-0.642001\pi$
$181$	−11.6094	−0.862919	−0.431460	+	0.902132i	$0.642001\pi$
$182$	3.73363	0.276755
$183$	7.34791	0.543173
$184$	81.9244	6.03955
$185$	3.92724	0.288737
$186$	−21.0892	−1.54634
$187$	7.67367	0.561154
$188$	30.4904	2.22374
$189$	2.67071	0.194266
$190$	−5.27793	−0.382901
$191$	−7.42526	−0.537273	−0.268636	−	0.963242i	$-0.586573\pi$
$191$	−7.42526	−0.537273	−0.268636	+	0.963242i	$0.586573\pi$
$192$	−38.2973	−2.76387
$193$	−1.71630	−0.123542	−0.0617710	−	0.998090i	$-0.519675\pi$
$193$	−1.71630	−0.123542	−0.0617710	+	0.998090i	$0.519675\pi$
$194$	−6.69304	−0.480532
$195$	0.180565	0.0129305
$196$	0.750337	0.0535955
$197$	−4.87064	−0.347019	−0.173510	−	0.984832i	$-0.555511\pi$
$197$	−4.87064	−0.347019	−0.173510	+	0.984832i	$0.555511\pi$
$198$	−6.89824	−0.490236
$199$	19.4441	1.37835	0.689176	−	0.724594i	$-0.257973\pi$
$199$	19.4441	1.37835	0.689176	+	0.724594i	$0.257973\pi$
$200$	49.2684	3.48380
$201$	9.63849	0.679847
$202$	−29.8280	−2.09869
$203$	19.0057	1.33394
$204$	−17.4041	−1.21853
$205$	−1.01852	−0.0711365
$206$	−53.9530	−3.75908
$207$	−8.10174	−0.563110
$208$	8.42181	0.583947
$209$	−13.3097	−0.920652
$210$	2.64053	0.182214
$211$	18.1423	1.24897	0.624484	−	0.781038i	$-0.285309\pi$
$211$	18.1423	1.24897	0.624484	+	0.781038i	$0.285309\pi$
$212$	−14.2905	−0.981475
$213$	−1.81365	−0.124269
$214$	41.1660	2.81405
$215$	0.335358	0.0228712
$216$	10.1119	0.688031
$217$	20.3573	1.38194
$218$	−38.3785	−2.59932
$219$	10.2351	0.691622
$220$	−5.03834	−0.339684
$221$	1.55514	0.104610
$222$	−30.4059	−2.04071
$223$	12.0863	0.809358	0.404679	−	0.914459i	$-0.367383\pi$
$223$	12.0863	0.809358	0.404679	+	0.914459i	$0.367383\pi$
$224$	69.1460	4.62001
$225$	−4.87230	−0.324820
$226$	54.6102	3.63262
$227$	−2.84947	−0.189126	−0.0945630	−	0.995519i	$-0.530145\pi$
$227$	−2.84947	−0.189126	−0.0945630	+	0.995519i	$0.530145\pi$
$228$	30.1869	1.99917
$229$	−1.01255	−0.0669112	−0.0334556	−	0.999440i	$-0.510651\pi$
$229$	−1.01255	−0.0669112	−0.0334556	+	0.999440i	$0.510651\pi$
$230$	−8.01020	−0.528177
$231$	6.65882	0.438118
$232$	71.9603	4.72442
$233$	16.4879	1.08016	0.540080	−	0.841613i	$-0.318394\pi$
$233$	16.4879	1.08016	0.540080	+	0.841613i	$0.318394\pi$
$234$	−1.39799	−0.0913895
$235$	−1.92682	−0.125692
$236$	18.6940	1.21688
$237$	−0.770961	−0.0500793
$238$	22.7419	1.47414
$239$	4.61983	0.298832	0.149416	−	0.988774i	$-0.452261\pi$
$239$	4.61983	0.298832	0.149416	+	0.988774i	$0.452261\pi$
$240$	5.95615	0.384468
$241$	14.8956	0.959510	0.479755	−	0.877402i	$-0.340726\pi$
$241$	14.8956	0.959510	0.479755	+	0.877402i	$0.340726\pi$
$242$	13.2349	0.850770
$243$	−1.00000	−0.0641500
$244$	−41.5512	−2.66004
$245$	−0.0474170	−0.00302936
$246$	7.88569	0.502773
$247$	−2.69734	−0.171627
$248$	77.0776	4.89443
$249$	−12.0650	−0.764590
$250$	−9.76075	−0.617324
$251$	6.89289	0.435075	0.217538	−	0.976052i	$-0.430198\pi$
$251$	6.89289	0.435075	0.217538	+	0.976052i	$0.430198\pi$
$252$	−15.1024	−0.951362
$253$	−20.1999	−1.26996
$254$	2.08773	0.130996
$255$	1.09984	0.0688747
$256$	73.3003	4.58127
$257$	4.08174	0.254612	0.127306	−	0.991864i	$-0.459367\pi$
$257$	4.08174	0.254612	0.127306	+	0.991864i	$0.459367\pi$
$258$	−2.59645	−0.161648
$259$	29.3506	1.82376
$260$	−1.02106	−0.0633237
$261$	−7.11636	−0.440492
$262$	8.33055	0.514663
$263$	−14.2814	−0.880632	−0.440316	−	0.897843i	$-0.645134\pi$
$263$	−14.2814	−0.880632	−0.440316	+	0.897843i	$0.645134\pi$
$264$	25.2119	1.55168
$265$	0.903077	0.0554756
$266$	−39.4451	−2.41853
$267$	−6.82993	−0.417985
$268$	−54.5040	−3.32936
$269$	8.88497	0.541726	0.270863	−	0.962618i	$-0.412691\pi$
$269$	8.88497	0.541726	0.270863	+	0.962618i	$0.412691\pi$
$270$	−0.988701	−0.0601704
$271$	−8.84836	−0.537500	−0.268750	−	0.963210i	$-0.586611\pi$
$271$	−8.84836	−0.537500	−0.268750	+	0.963210i	$0.586611\pi$
$272$	51.2981	3.11041
$273$	1.34947	0.0816736
$274$	45.9291	2.77468
$275$	−12.1480	−0.732551
$276$	45.8140	2.75768
$277$	−11.4643	−0.688823	−0.344411	−	0.938819i	$-0.611921\pi$
$277$	−11.4643	−0.688823	−0.344411	+	0.938819i	$0.611921\pi$
$278$	13.9273	0.835304
$279$	−7.62243	−0.456343
$280$	−9.65070	−0.576739
$281$	−10.8208	−0.645516	−0.322758	−	0.946482i	$-0.604610\pi$
$281$	−10.8208	−0.645516	−0.322758	+	0.946482i	$0.604610\pi$
$282$	14.9180	0.888356
$283$	−14.2439	−0.846709	−0.423355	−	0.905964i	$-0.639148\pi$
$283$	−14.2439	−0.846709	−0.423355	+	0.905964i	$0.639148\pi$
$284$	10.2559	0.608574
$285$	−1.90764	−0.112999
$286$	−3.48558	−0.206107
$287$	−7.61200	−0.449322
$288$	−25.8905	−1.52561
$289$	−7.52748	−0.442793
$290$	−7.03595	−0.413165
$291$	−2.41911	−0.141811
$292$	−57.8776	−3.38703
$293$	−17.7127	−1.03479	−0.517393	−	0.855748i	$-0.673097\pi$
$293$	−17.7127	−1.03479	−0.517393	+	0.855748i	$0.673097\pi$
$294$	0.367117	0.0214107
$295$	−1.18136	−0.0687812
$296$	111.129	6.45921
$297$	−2.49328	−0.144675
$298$	−29.2014	−1.69159
$299$	−4.09369	−0.236744
$300$	27.5520	1.59072
$301$	2.50633	0.144462
$302$	31.0798	1.78844
$303$	−10.7809	−0.619348
$304$	−88.9748	−5.10306
$305$	2.62580	0.150353
$306$	−8.51531	−0.486788
$307$	−16.5782	−0.946168	−0.473084	−	0.881017i	$-0.656859\pi$
$307$	−16.5782	−0.946168	−0.473084	+	0.881017i	$0.656859\pi$
$308$	−37.6545	−2.14556
$309$	−19.5006	−1.10935
$310$	−7.53630	−0.428033
$311$	20.9694	1.18907	0.594534	−	0.804071i	$-0.297337\pi$
$311$	20.9694	1.18907	0.594534	+	0.804071i	$0.297337\pi$
$312$	5.10942	0.289264
$313$	−16.4549	−0.930083	−0.465042	−	0.885289i	$-0.653961\pi$
$313$	−16.4549	−0.930083	−0.465042	+	0.885289i	$0.653961\pi$
$314$	−0.978940	−0.0552448
$315$	0.954385	0.0537735
$316$	4.35965	0.245250
$317$	8.74759	0.491313	0.245657	−	0.969357i	$-0.420996\pi$
$317$	8.74759	0.491313	0.245657	+	0.969357i	$0.420996\pi$
$318$	−6.99191	−0.392087
$319$	−17.7431	−0.993420
$320$	−13.6856	−0.765051
$321$	14.8789	0.830459
$322$	−59.8650	−3.33615
$323$	−16.4298	−0.914176
$324$	5.65483	0.314157
$325$	−2.46190	−0.136562
$326$	−0.483798	−0.0267951
$327$	−13.8714	−0.767090
$328$	−28.8209	−1.59137
$329$	−14.4003	−0.793912
$330$	−2.46510	−0.135700
$331$	9.55978	0.525453	0.262727	−	0.964870i	$-0.415378\pi$
$331$	9.55978	0.525453	0.262727	+	0.964870i	$0.415378\pi$
$332$	68.2257	3.74437
$333$	−10.9898	−0.602239
$334$	−29.4556	−1.61174
$335$	3.44434	0.188184
$336$	44.5139	2.42843
$337$	−10.0321	−0.546482	−0.273241	−	0.961946i	$-0.588096\pi$
$337$	−10.0321	−0.546482	−0.273241	+	0.961946i	$0.588096\pi$
$338$	35.2612	1.91796
$339$	19.7381	1.07203
$340$	−6.21941	−0.337295
$341$	−19.0048	−1.02917
$342$	14.7695	0.798644
$343$	18.3406	0.990299
$344$	9.48957	0.511643
$345$	−2.89518	−0.155871
$346$	47.1837	2.53661
$347$	−31.7762	−1.70584	−0.852918	−	0.522044i	$-0.825170\pi$
$347$	−31.7762	−1.70584	−0.852918	+	0.522044i	$0.825170\pi$
$348$	40.2418	2.15719
$349$	17.3552	0.929003	0.464502	−	0.885572i	$-0.346233\pi$
$349$	17.3552	0.929003	0.464502	+	0.885572i	$0.346233\pi$
$350$	−36.0021	−1.92440
$351$	−0.505285	−0.0269701
$352$	−64.5522	−3.44064
$353$	19.6968	1.04835	0.524177	−	0.851609i	$-0.324373\pi$
$353$	19.6968	1.04835	0.524177	+	0.851609i	$0.324373\pi$
$354$	9.14642	0.486127
$355$	−0.648112	−0.0343982
$356$	38.6221	2.04697
$357$	8.21977	0.435036
$358$	−26.4555	−1.39822
$359$	−5.03451	−0.265711	−0.132856	−	0.991135i	$-0.542415\pi$
$359$	−5.03451	−0.265711	−0.132856	+	0.991135i	$0.542415\pi$
$360$	3.61353	0.190450
$361$	9.49684	0.499834
$362$	32.1201	1.68820
$363$	4.78357	0.251072
$364$	−7.63102	−0.399974
$365$	3.65753	0.191444
$366$	−20.3297	−1.06265
$367$	15.6351	0.816146	0.408073	−	0.912949i	$-0.366201\pi$
$367$	15.6351	0.816146	0.408073	+	0.912949i	$0.366201\pi$
$368$	−135.035	−7.03920
$369$	2.85018	0.148374
$370$	−10.8656	−0.564878
$371$	6.74923	0.350403
$372$	43.1035	2.23481
$373$	−27.0407	−1.40011	−0.700056	−	0.714088i	$-0.746842\pi$
$373$	−27.0407	−1.40011	−0.700056	+	0.714088i	$0.746842\pi$
$374$	−21.2310	−1.09783
$375$	−3.52789	−0.182180
$376$	−54.5229	−2.81180
$377$	−3.59579	−0.185193
$378$	−7.38915	−0.380057
$379$	−24.3292	−1.24971	−0.624853	−	0.780743i	$-0.714841\pi$
$379$	−24.3292	−1.24971	−0.624853	+	0.780743i	$0.714841\pi$
$380$	10.7874	0.553380
$381$	0.754581	0.0386584
$382$	20.5437	1.05111
$383$	−11.1565	−0.570072	−0.285036	−	0.958517i	$-0.592006\pi$
$383$	−11.1565	−0.570072	−0.285036	+	0.958517i	$0.592006\pi$
$384$	54.1775	2.76474
$385$	2.37955	0.121273
$386$	4.74855	0.241695
$387$	−0.938451	−0.0477042
$388$	13.6797	0.694479
$389$	−24.3326	−1.23371	−0.616857	−	0.787075i	$-0.711594\pi$
$389$	−24.3326	−1.23371	−0.616857	+	0.787075i	$0.711594\pi$
$390$	−0.499576	−0.0252970
$391$	−24.9351	−1.26102
$392$	−1.34175	−0.0677687
$393$	3.01097	0.151883
$394$	13.4758	0.678900
$395$	−0.275505	−0.0138622
$396$	14.0991	0.708504
$397$	3.19461	0.160333	0.0801664	−	0.996781i	$-0.474455\pi$
$397$	3.19461	0.160333	0.0801664	+	0.996781i	$0.474455\pi$
$398$	−53.7966	−2.69658
$399$	−14.2569	−0.713738
$400$	−81.2087	−4.06044
$401$	5.77232	0.288256	0.144128	−	0.989559i	$-0.453962\pi$
$401$	5.77232	0.288256	0.144128	+	0.989559i	$0.453962\pi$
$402$	−26.6672	−1.33004
$403$	−3.85150	−0.191857
$404$	60.9643	3.03309
$405$	−0.357353	−0.0177570
$406$	−52.5839	−2.60969
$407$	−27.4007	−1.35820
$408$	31.1220	1.54077
$409$	−1.04004	−0.0514266	−0.0257133	−	0.999669i	$-0.508186\pi$
$409$	−1.04004	−0.0514266	−0.0257133	+	0.999669i	$0.508186\pi$
$410$	2.81797	0.139170
$411$	16.6005	0.818841
$412$	110.272	5.43274
$413$	−8.82897	−0.434445
$414$	22.4154	1.10166
$415$	−4.31147	−0.211642
$416$	−13.0821	−0.641402
$417$	5.03384	0.246508
$418$	36.8245	1.80114
$419$	21.5916	1.05482	0.527409	−	0.849611i	$-0.323163\pi$
$419$	21.5916	1.05482	0.527409	+	0.849611i	$0.323163\pi$
$420$	−5.39689	−0.263341
$421$	−26.1510	−1.27452	−0.637260	−	0.770649i	$-0.719932\pi$
$421$	−26.1510	−1.27452	−0.637260	+	0.770649i	$0.719932\pi$
$422$	−50.1950	−2.44345
$423$	5.39192	0.262164
$424$	25.5542	1.24102
$425$	−14.9957	−0.727398
$426$	5.01788	0.243117
$427$	19.6241	0.949679
$428$	−84.1376	−4.06694
$429$	−1.25982	−0.0608245
$430$	−0.927847	−0.0447448
$431$	31.0868	1.49740	0.748700	−	0.662909i	$-0.230678\pi$
$431$	31.0868	1.49740	0.748700	+	0.662909i	$0.230678\pi$
$432$	−16.6674	−0.801912
$433$	−32.2543	−1.55004	−0.775021	−	0.631936i	$-0.782261\pi$
$433$	−32.2543	−1.55004	−0.775021	+	0.631936i	$0.782261\pi$
$434$	−56.3232	−2.70360
$435$	−2.54305	−0.121930
$436$	78.4404	3.75661
$437$	43.2491	2.06888
$438$	−28.3177	−1.35307
$439$	−30.1639	−1.43964	−0.719822	−	0.694158i	$-0.755777\pi$
$439$	−30.1639	−1.43964	−0.719822	+	0.694158i	$0.755777\pi$
$440$	9.00954	0.429513
$441$	0.132690	0.00631856
$442$	−4.30266	−0.204657
$443$	23.0163	1.09354	0.546769	−	0.837284i	$-0.315858\pi$
$443$	23.0163	1.09354	0.546769	+	0.837284i	$0.315858\pi$
$444$	62.1456	2.94930
$445$	−2.44069	−0.115700
$446$	−33.4396	−1.58341
$447$	−10.5544	−0.499208
$448$	−102.281	−4.83232
$449$	17.3577	0.819161	0.409581	−	0.912274i	$-0.365675\pi$
$449$	17.3577	0.819161	0.409581	+	0.912274i	$0.365675\pi$
$450$	13.4804	0.635471
$451$	7.10628	0.334622
$452$	−111.616	−5.24997
$453$	11.2334	0.527790
$454$	7.88373	0.370002
$455$	0.482237	0.0226076
$456$	−53.9800	−2.52785
$457$	−23.6898	−1.10816	−0.554081	−	0.832463i	$-0.686930\pi$
$457$	−23.6898	−1.10816	−0.554081	+	0.832463i	$0.686930\pi$
$458$	2.80146	0.130904
$459$	−3.07775	−0.143657
$460$	16.3718	0.763337
$461$	−10.1773	−0.474006	−0.237003	−	0.971509i	$-0.576165\pi$
$461$	−10.1773	−0.474006	−0.237003	+	0.971509i	$0.576165\pi$
$462$	−18.4232	−0.857124
$463$	31.1344	1.44694	0.723470	−	0.690356i	$-0.242546\pi$
$463$	31.1344	1.44694	0.723470	+	0.690356i	$0.242546\pi$
$464$	−118.611	−5.50640
$465$	−2.72390	−0.126318
$466$	−45.6178	−2.11320
$467$	30.9382	1.43165	0.715824	−	0.698281i	$-0.246051\pi$
$467$	30.9382	1.43165	0.715824	+	0.698281i	$0.246051\pi$
$468$	2.85730	0.132079
$469$	25.7416	1.18864
$470$	5.33100	0.245901
$471$	−0.353825	−0.0163034
$472$	−33.4286	−1.53868
$473$	−2.33982	−0.107585
$474$	2.13305	0.0979741
$475$	26.0095	1.19340
$476$	−46.4814	−2.13047
$477$	−2.52713	−0.115709
$478$	−12.7819	−0.584629
$479$	23.7755	1.08633	0.543165	−	0.839626i	$-0.317226\pi$
$479$	23.7755	1.08633	0.543165	+	0.839626i	$0.317226\pi$
$480$	−9.25204	−0.422296
$481$	−5.55300	−0.253195
$482$	−41.2122	−1.87716
$483$	−21.6374	−0.984536
$484$	−27.0503	−1.22956
$485$	−0.864476	−0.0392538
$486$	2.76674	0.125502
$487$	15.0640	0.682614	0.341307	−	0.939952i	$-0.389130\pi$
$487$	15.0640	0.682614	0.341307	+	0.939952i	$0.389130\pi$
$488$	74.3017	3.36348
$489$	−0.174862	−0.00790755
$490$	0.131190	0.00592658
$491$	18.3192	0.826733	0.413366	−	0.910565i	$-0.364353\pi$
$491$	18.3192	0.826733	0.413366	+	0.910565i	$0.364353\pi$
$492$	−16.1173	−0.726623
$493$	−21.9024	−0.986432
$494$	7.46282	0.335768
$495$	−0.890979	−0.0400465
$496$	−127.046	−5.70455
$497$	−4.84373	−0.217271
$498$	33.3808	1.49583
$499$	−20.2270	−0.905486	−0.452743	−	0.891641i	$-0.649555\pi$
$499$	−20.2270	−0.905486	−0.452743	+	0.891641i	$0.649555\pi$
$500$	19.9496	0.892175
$501$	−10.6463	−0.475644
$502$	−19.0708	−0.851172
$503$	14.8915	0.663981	0.331990	−	0.943283i	$-0.392280\pi$
$503$	14.8915	0.663981	0.331990	+	0.943283i	$0.392280\pi$
$504$	27.0061	1.20295
$505$	−3.85259	−0.171438
$506$	55.8878	2.48451
$507$	12.7447	0.566011
$508$	−4.26703	−0.189319
$509$	43.7102	1.93742	0.968710	−	0.248195i	$-0.0798373\pi$
$509$	43.7102	1.93742	0.968710	+	0.248195i	$0.0798373\pi$
$510$	−3.04297	−0.134745
$511$	27.3349	1.20923
$512$	−94.4477	−4.17404
$513$	5.33824	0.235689
$514$	−11.2931	−0.498117
$515$	−6.96859	−0.307073
$516$	5.30678	0.233618
$517$	13.4436	0.591247
$518$	−81.2054	−3.56796
$519$	17.0539	0.748584
$520$	1.82586	0.0800695
$521$	−15.1637	−0.664332	−0.332166	−	0.943221i	$-0.607779\pi$
$521$	−15.1637	−0.664332	−0.332166	+	0.943221i	$0.607779\pi$
$522$	19.6891	0.861768
$523$	2.34645	0.102603	0.0513016	−	0.998683i	$-0.483663\pi$
$523$	2.34645	0.102603	0.0513016	+	0.998683i	$0.483663\pi$
$524$	−17.0265	−0.743806
$525$	−13.0125	−0.567912
$526$	39.5130	1.72285
$527$	−23.4599	−1.02193
$528$	−41.5565	−1.80852
$529$	42.6383	1.85384
$530$	−2.49858	−0.108531
$531$	3.30585	0.143462
$532$	80.6203	3.49533
$533$	1.44015	0.0623799
$534$	18.8966	0.817736
$535$	5.31701	0.229875
$536$	97.4639	4.20980
$537$	−9.56200	−0.412631
$538$	−24.5824	−1.05982
$539$	0.330832	0.0142499
$540$	2.02077	0.0869600
$541$	−37.3081	−1.60400	−0.802000	−	0.597325i	$-0.796230\pi$
$541$	−37.3081	−1.60400	−0.802000	+	0.597325i	$0.796230\pi$
$542$	24.4811	1.05155
$543$	11.6094	0.498207
$544$	−79.6844	−3.41644
$545$	−4.95698	−0.212334
$546$	−3.73363	−0.159785
$547$	36.1363	1.54508	0.772539	−	0.634967i	$-0.218986\pi$
$547$	36.1363	1.54508	0.772539	+	0.634967i	$0.218986\pi$
$548$	−93.8728	−4.01005
$549$	−7.34791	−0.313601
$550$	33.6103	1.43315
$551$	37.9889	1.61838
$552$	−81.9244	−3.48693
$553$	−2.05901	−0.0875582
$554$	31.7187	1.34760
$555$	−3.92724	−0.166702
$556$	−28.4655	−1.20721
$557$	18.9730	0.803913	0.401956	−	0.915659i	$-0.368330\pi$
$557$	18.9730	0.803913	0.401956	+	0.915659i	$0.368330\pi$
$558$	21.0892	0.892779
$559$	−0.474185	−0.0200559
$560$	15.9072	0.672200
$561$	−7.67367	−0.323983
$562$	29.9383	1.26287
$563$	−37.6286	−1.58586	−0.792929	−	0.609314i	$-0.791445\pi$
$563$	−37.6286	−1.58586	−0.792929	+	0.609314i	$0.791445\pi$
$564$	−30.4904	−1.28388
$565$	7.05348	0.296742
$566$	39.4090	1.65648
$567$	−2.67071	−0.112159
$568$	−18.3395	−0.769509
$569$	−31.1946	−1.30775	−0.653874	−	0.756604i	$-0.726857\pi$
$569$	−31.1946	−1.30775	−0.653874	+	0.756604i	$0.726857\pi$
$570$	5.27793	0.221068
$571$	−10.6287	−0.444799	−0.222399	−	0.974956i	$-0.571389\pi$
$571$	−10.6287	−0.444799	−0.222399	+	0.974956i	$0.571389\pi$
$572$	7.12404	0.297871
$573$	7.42526	0.310195
$574$	21.0604	0.879044
$575$	39.4741	1.64618
$576$	38.2973	1.59572
$577$	11.0983	0.462026	0.231013	−	0.972951i	$-0.425796\pi$
$577$	11.0983	0.462026	0.231013	+	0.972951i	$0.425796\pi$
$578$	20.8265	0.866270
$579$	1.71630	0.0713270
$580$	14.3805	0.597119
$581$	−32.2222	−1.33680
$582$	6.69304	0.277435
$583$	−6.30084	−0.260954
$584$	103.497	4.28272
$585$	−0.180565	−0.00746545
$586$	49.0063	2.02443
$587$	−2.93377	−0.121090	−0.0605449	−	0.998165i	$-0.519284\pi$
$587$	−2.93377	−0.121090	−0.0605449	+	0.998165i	$0.519284\pi$
$588$	−0.750337	−0.0309434
$589$	40.6904	1.67662
$590$	3.26850	0.134562
$591$	4.87064	0.200352
$592$	−183.172	−7.52833
$593$	−29.3334	−1.20458	−0.602290	−	0.798277i	$-0.705745\pi$
$593$	−29.3334	−1.20458	−0.602290	+	0.798277i	$0.705745\pi$
$594$	6.89824	0.283038
$595$	2.93736	0.120420
$596$	59.6836	2.44473
$597$	−19.4441	−0.795792
$598$	11.3262	0.463161
$599$	13.0940	0.535005	0.267502	−	0.963557i	$-0.413802\pi$
$599$	13.0940	0.535005	0.267502	+	0.963557i	$0.413802\pi$
$600$	−49.2684	−2.01138
$601$	−1.72620	−0.0704132	−0.0352066	−	0.999380i	$-0.511209\pi$
$601$	−1.72620	−0.0704132	−0.0352066	+	0.999380i	$0.511209\pi$
$602$	−6.93436	−0.282623
$603$	−9.63849	−0.392510
$604$	−63.5228	−2.58471
$605$	1.70942	0.0694979
$606$	29.8280	1.21168
$607$	29.4759	1.19639	0.598194	−	0.801351i	$-0.295885\pi$
$607$	29.4759	1.19639	0.598194	+	0.801351i	$0.295885\pi$
$608$	138.210	5.60515
$609$	−19.0057	−0.770151
$610$	−7.26489	−0.294147
$611$	2.72446	0.110220
$612$	17.4041	0.703520
$613$	29.6584	1.19789	0.598945	−	0.800790i	$-0.295587\pi$
$613$	29.6584	1.19789	0.598945	+	0.800790i	$0.295587\pi$
$614$	45.8675	1.85106
$615$	1.01852	0.0410706
$616$	67.3336	2.71295
$617$	23.9977	0.966110	0.483055	−	0.875590i	$-0.339527\pi$
$617$	23.9977	0.966110	0.483055	+	0.875590i	$0.339527\pi$
$618$	53.9530	2.17031
$619$	45.4541	1.82695	0.913477	−	0.406890i	$-0.133387\pi$
$619$	45.4541	1.82695	0.913477	+	0.406890i	$0.133387\pi$
$620$	15.4032	0.618606
$621$	8.10174	0.325112
$622$	−58.0169	−2.32627
$623$	−18.2407	−0.730800
$624$	−8.42181	−0.337142
$625$	23.1008	0.924032
$626$	45.5262	1.81959
$627$	13.3097	0.531539
$628$	2.00082	0.0798414
$629$	−33.8239	−1.34865
$630$	−2.64053	−0.105201
$631$	18.8681	0.751129	0.375564	−	0.926796i	$-0.377449\pi$
$631$	18.8681	0.751129	0.375564	+	0.926796i	$0.377449\pi$
$632$	−7.79592	−0.310105
$633$	−18.1423	−0.721092
$634$	−24.2023	−0.961195
$635$	0.269652	0.0107008
$636$	14.2905	0.566655
$637$	0.0670461	0.00265646
$638$	49.0904	1.94351
$639$	1.81365	0.0717468
$640$	19.3605	0.765291
$641$	−43.9862	−1.73735	−0.868676	−	0.495382i	$-0.835028\pi$
$641$	−43.9862	−1.73735	−0.868676	+	0.495382i	$0.835028\pi$
$642$	−41.1660	−1.62469
$643$	8.46827	0.333956	0.166978	−	0.985961i	$-0.446599\pi$
$643$	8.46827	0.333956	0.166978	+	0.985961i	$0.446599\pi$
$644$	122.356	4.82150
$645$	−0.335358	−0.0132047
$646$	45.4568	1.78847
$647$	−24.5498	−0.965151	−0.482575	−	0.875854i	$-0.660299\pi$
$647$	−24.5498	−0.965151	−0.482575	+	0.875854i	$0.660299\pi$
$648$	−10.1119	−0.397235
$649$	8.24240	0.323543
$650$	6.81143	0.267166
$651$	−20.3573	−0.797865
$652$	0.988817	0.0387250
$653$	27.0295	1.05775	0.528873	−	0.848701i	$-0.322615\pi$
$653$	27.0295	1.05775	0.528873	+	0.848701i	$0.322615\pi$
$654$	38.3785	1.50072
$655$	1.07598	0.0420419
$656$	47.5051	1.85476
$657$	−10.2351	−0.399308
$658$	39.8417	1.55319
$659$	−21.4688	−0.836308	−0.418154	−	0.908376i	$-0.637323\pi$
$659$	−21.4688	−0.836308	−0.418154	+	0.908376i	$0.637323\pi$
$660$	5.03834	0.196117
$661$	20.6411	0.802848	0.401424	−	0.915892i	$-0.368515\pi$
$661$	20.6411	0.802848	0.401424	+	0.915892i	$0.368515\pi$
$662$	−26.4494	−1.02798
$663$	−1.55514	−0.0603966
$664$	−122.001	−4.73456
$665$	−5.09474	−0.197566
$666$	30.4059	1.17821
$667$	57.6549	2.23241
$668$	60.2033	2.32933
$669$	−12.0863	−0.467283
$670$	−9.52958	−0.368160
$671$	−18.3204	−0.707250
$672$	−69.1460	−2.66736
$673$	−36.5608	−1.40932	−0.704658	−	0.709547i	$-0.748900\pi$
$673$	−36.5608	−1.40932	−0.704658	+	0.709547i	$0.748900\pi$
$674$	27.7561	1.06912
$675$	4.87230	0.187535
$676$	−72.0690	−2.77189
$677$	8.03558	0.308832	0.154416	−	0.988006i	$-0.450650\pi$
$677$	8.03558	0.308832	0.154416	+	0.988006i	$0.450650\pi$
$678$	−54.6102	−2.09729
$679$	−6.46074	−0.247940
$680$	11.1215	0.426491
$681$	2.84947	0.109192
$682$	52.5813	2.01344
$683$	27.3412	1.04618	0.523091	−	0.852277i	$-0.324779\pi$
$683$	27.3412	1.04618	0.523091	+	0.852277i	$0.324779\pi$
$684$	−30.1869	−1.15422
$685$	5.93223	0.226659
$686$	−50.7436	−1.93740
$687$	1.01255	0.0386312
$688$	−15.6416	−0.596329
$689$	−1.27692	−0.0486469
$690$	8.01020	0.304943
$691$	−15.8126	−0.601540	−0.300770	−	0.953697i	$-0.597244\pi$
$691$	−15.8126	−0.601540	−0.300770	+	0.953697i	$0.597244\pi$
$692$	−96.4370	−3.66599
$693$	−6.65882	−0.252948
$694$	87.9164	3.33726
$695$	1.79886	0.0682345
$696$	−71.9603	−2.72765
$697$	8.77212	0.332268
$698$	−48.0173	−1.81748
$699$	−16.4879	−0.623631
$700$	73.5834	2.78119
$701$	−38.7869	−1.46496	−0.732480	−	0.680789i	$-0.761637\pi$
$701$	−38.7869	−1.46496	−0.732480	+	0.680789i	$0.761637\pi$
$702$	1.39799	0.0527638
$703$	58.6664	2.21264
$704$	95.4858	3.59875
$705$	1.92682	0.0725682
$706$	−54.4958	−2.05098
$707$	−28.7927	−1.08286
$708$	−18.6940	−0.702565
$709$	1.35327	0.0508230	0.0254115	−	0.999677i	$-0.491910\pi$
$709$	1.35327	0.0508230	0.0254115	+	0.999677i	$0.491910\pi$
$710$	1.79315	0.0672959
$711$	0.770961	0.0289133
$712$	−69.0639	−2.58828
$713$	61.7550	2.31274
$714$	−22.7419	−0.851095
$715$	−0.450199	−0.0168365
$716$	54.0715	2.02075
$717$	−4.61983	−0.172531
$718$	13.9292	0.519831
$719$	−44.8821	−1.67382	−0.836911	−	0.547340i	$-0.815641\pi$
$719$	−44.8821	−1.67382	−0.836911	+	0.547340i	$0.815641\pi$
$720$	−5.95615	−0.221973
$721$	−52.0804	−1.93958
$722$	−26.2753	−0.977864
$723$	−14.8956	−0.553973
$724$	−65.6491	−2.43983
$725$	34.6730	1.28772
$726$	−13.2349	−0.491193
$727$	23.3244	0.865054	0.432527	−	0.901621i	$-0.357622\pi$
$727$	23.3244	0.865054	0.432527	+	0.901621i	$0.357622\pi$
$728$	13.6458	0.505746
$729$	1.00000	0.0370370
$730$	−10.1194	−0.374537
$731$	−2.88831	−0.106828
$732$	41.5512	1.53578
$733$	41.9978	1.55122	0.775611	−	0.631211i	$-0.217442\pi$
$733$	41.9978	1.55122	0.775611	+	0.631211i	$0.217442\pi$
$734$	−43.2582	−1.59669
$735$	0.0474170	0.00174900
$736$	209.758	7.73179
$737$	−24.0314	−0.885209
$738$	−7.88569	−0.290276
$739$	−34.2894	−1.26136	−0.630679	−	0.776044i	$-0.717223\pi$
$739$	−34.2894	−1.26136	−0.630679	+	0.776044i	$0.717223\pi$
$740$	22.2079	0.816378
$741$	2.69734	0.0990891
$742$	−18.6733	−0.685520
$743$	−11.0369	−0.404905	−0.202452	−	0.979292i	$-0.564891\pi$
$743$	−11.0369	−0.404905	−0.202452	+	0.979292i	$0.564891\pi$
$744$	−77.0776	−2.82580
$745$	−3.77166	−0.138183
$746$	74.8144	2.73915
$747$	12.0650	0.441436
$748$	43.3933	1.58662
$749$	39.7372	1.45197
$750$	9.76075	0.356412
$751$	13.1395	0.479467	0.239733	−	0.970839i	$-0.422940\pi$
$751$	13.1395	0.479467	0.239733	+	0.970839i	$0.422940\pi$
$752$	89.8695	3.27721
$753$	−6.89289	−0.251191
$754$	9.94861	0.362307
$755$	4.01428	0.146094
$756$	15.1024	0.549269
$757$	46.5418	1.69159	0.845796	−	0.533507i	$-0.179126\pi$
$757$	46.5418	1.69159	0.845796	+	0.533507i	$0.179126\pi$
$758$	67.3124	2.44490
$759$	20.1999	0.733209
$760$	−19.2899	−0.699719
$761$	40.8261	1.47995	0.739973	−	0.672637i	$-0.234838\pi$
$761$	40.8261	1.47995	0.739973	+	0.672637i	$0.234838\pi$
$762$	−2.08773	−0.0756304
$763$	−37.0465	−1.34117
$764$	−41.9886	−1.51909
$765$	−1.09984	−0.0397648
$766$	30.8672	1.11528
$767$	1.67040	0.0603146
$768$	−73.3003	−2.64500
$769$	−36.2639	−1.30771	−0.653854	−	0.756621i	$-0.726849\pi$
$769$	−36.2639	−1.30771	−0.653854	+	0.756621i	$0.726849\pi$
$770$	−6.58358	−0.237256
$771$	−4.08174	−0.147000
$772$	−9.70539	−0.349305
$773$	1.18756	0.0427135	0.0213568	−	0.999772i	$-0.493201\pi$
$773$	1.18756	0.0427135	0.0213568	+	0.999772i	$0.493201\pi$
$774$	2.59645	0.0933274
$775$	37.1387	1.33406
$776$	−24.4619	−0.878132
$777$	−29.3506	−1.05295
$778$	67.3220	2.41361
$779$	−15.2149	−0.545132
$780$	1.02106	0.0365600
$781$	4.52193	0.161807
$782$	68.9889	2.46704
$783$	7.11636	0.254318
$784$	2.21160	0.0789856
$785$	−0.126440	−0.00451285
$786$	−8.33055	−0.297141
$787$	−19.9157	−0.709918	−0.354959	−	0.934882i	$-0.615505\pi$
$787$	−19.9157	−0.709918	−0.354959	+	0.934882i	$0.615505\pi$
$788$	−27.5427	−0.981167
$789$	14.2814	0.508433
$790$	0.762250	0.0271196
$791$	52.7148	1.87432
$792$	−25.2119	−0.895865
$793$	−3.71279	−0.131845
$794$	−8.83864	−0.313672
$795$	−0.903077	−0.0320289
$796$	109.953	3.89717
$797$	−46.4118	−1.64399	−0.821994	−	0.569496i	$-0.807139\pi$
$797$	−46.4118	−1.64399	−0.821994	+	0.569496i	$0.807139\pi$
$798$	39.4451	1.39634
$799$	16.5950	0.587088
$800$	126.146	4.45994
$801$	6.82993	0.241324
$802$	−15.9705	−0.563937
$803$	−25.5189	−0.900541
$804$	54.5040	1.92221
$805$	−7.73219	−0.272524
$806$	10.6561	0.375345
$807$	−8.88497	−0.312766
$808$	−109.016	−3.83517
$809$	−7.73978	−0.272116	−0.136058	−	0.990701i	$-0.543443\pi$
$809$	−7.73978	−0.272116	−0.136058	+	0.990701i	$0.543443\pi$
$810$	0.988701	0.0347394
$811$	−41.3835	−1.45317	−0.726586	−	0.687075i	$-0.758894\pi$
$811$	−41.3835	−1.45317	−0.726586	+	0.687075i	$0.758894\pi$
$812$	107.474	3.77160
$813$	8.84836	0.310326
$814$	75.8104	2.65715
$815$	−0.0624875	−0.00218884
$816$	−51.2981	−1.79579
$817$	5.00968	0.175267
$818$	2.87752	0.100610
$819$	−1.34947	−0.0471543
$820$	−5.75955	−0.201132
$821$	−19.1366	−0.667871	−0.333936	−	0.942596i	$-0.608377\pi$
$821$	−19.1366	−0.667871	−0.333936	+	0.942596i	$0.608377\pi$
$822$	−45.9291	−1.60196
$823$	50.4421	1.75830	0.879150	−	0.476544i	$-0.158111\pi$
$823$	50.4421	1.75830	0.879150	+	0.476544i	$0.158111\pi$
$824$	−197.189	−6.86940
$825$	12.1480	0.422939
$826$	24.4274	0.849939
$827$	−22.8181	−0.793463	−0.396731	−	0.917935i	$-0.629856\pi$
$827$	−22.8181	−0.793463	−0.396731	+	0.917935i	$0.629856\pi$
$828$	−45.8140	−1.59215
$829$	45.8778	1.59340	0.796701	−	0.604373i	$-0.206576\pi$
$829$	45.8778	1.59340	0.796701	+	0.604373i	$0.206576\pi$
$830$	11.9287	0.414051
$831$	11.4643	0.397692
$832$	19.3511	0.670877
$833$	0.408385	0.0141497
$834$	−13.9273	−0.482263
$835$	−3.80450	−0.131660
$836$	−75.2642	−2.60307
$837$	7.62243	0.263470
$838$	−59.7383	−2.06362
$839$	−20.4509	−0.706044	−0.353022	−	0.935615i	$-0.614846\pi$
$839$	−20.4509	−0.706044	−0.353022	+	0.935615i	$0.614846\pi$
$840$	9.65070	0.332981
$841$	21.6426	0.746297
$842$	72.3529	2.49344
$843$	10.8208	0.372689
$844$	102.592	3.53135
$845$	4.55435	0.156674
$846$	−14.9180	−0.512893
$847$	12.7755	0.438973
$848$	−42.1208	−1.44643
$849$	14.2439	0.488848
$850$	41.4891	1.42307
$851$	89.0367	3.05214
$852$	−10.2559	−0.351360
$853$	−39.1698	−1.34115	−0.670575	−	0.741842i	$-0.733953\pi$
$853$	−39.1698	−1.34115	−0.670575	+	0.741842i	$0.733953\pi$
$854$	−54.2948	−1.85793
$855$	1.90764	0.0652398
$856$	150.455	5.14243
$857$	29.3770	1.00350	0.501750	−	0.865013i	$-0.332690\pi$
$857$	29.3770	1.00350	0.501750	+	0.865013i	$0.332690\pi$
$858$	3.48558	0.118996
$859$	−39.8698	−1.36034	−0.680171	−	0.733054i	$-0.738094\pi$
$859$	−39.8698	−1.36034	−0.680171	+	0.733054i	$0.738094\pi$
$860$	1.89639	0.0646665
$861$	7.61200	0.259416
$862$	−86.0091	−2.92948
$863$	−21.6526	−0.737064	−0.368532	−	0.929615i	$-0.620140\pi$
$863$	−21.6526	−0.737064	−0.368532	+	0.929615i	$0.620140\pi$
$864$	25.8905	0.880812
$865$	6.09427	0.207211
$866$	89.2391	3.03247
$867$	7.52748	0.255647
$868$	115.117	3.90733
$869$	1.92222	0.0652068
$870$	7.03595	0.238541
$871$	−4.87019	−0.165020
$872$	−140.267	−4.75003
$873$	2.41911	0.0818745
$874$	−119.659	−4.04752
$875$	−9.42198	−0.318521
$876$	57.8776	1.95550
$877$	−0.306511	−0.0103501	−0.00517507	−	0.999987i	$-0.501647\pi$
$877$	−0.306511	−0.0103501	−0.00517507	+	0.999987i	$0.501647\pi$
$878$	83.4556	2.81649
$879$	17.7127	0.597434
$880$	−14.8503	−0.500605
$881$	−32.4496	−1.09325	−0.546627	−	0.837376i	$-0.684089\pi$
$881$	−32.4496	−1.09325	−0.546627	+	0.837376i	$0.684089\pi$
$882$	−0.367117	−0.0123615
$883$	−28.4334	−0.956859	−0.478430	−	0.878126i	$-0.658794\pi$
$883$	−28.4334	−0.956859	−0.478430	+	0.878126i	$0.658794\pi$
$884$	8.79405	0.295776
$885$	1.18136	0.0397108
$886$	−63.6800	−2.13937
$887$	−3.83709	−0.128837	−0.0644185	−	0.997923i	$-0.520519\pi$
$887$	−3.83709	−0.128837	−0.0644185	+	0.997923i	$0.520519\pi$
$888$	−111.129	−3.72923
$889$	2.01527	0.0675899
$890$	6.75275	0.226353
$891$	2.49328	0.0835279
$892$	68.3459	2.28839
$893$	−28.7834	−0.963200
$894$	29.2014	0.976640
$895$	−3.41701	−0.114218
$896$	144.692	4.83384
$897$	4.09369	0.136684
$898$	−48.0242	−1.60259
$899$	54.2440	1.80914
$900$	−27.5520	−0.918401
$901$	−7.77787	−0.259118
$902$	−19.6612	−0.654647
$903$	−2.50633	−0.0834055
$904$	199.591	6.63830
$905$	4.14865	0.137906
$906$	−31.0798	−1.03256
$907$	−35.2514	−1.17050	−0.585252	−	0.810852i	$-0.699004\pi$
$907$	−35.2514	−1.17050	−0.585252	+	0.810852i	$0.699004\pi$
$908$	−16.1133	−0.534738
$909$	10.7809	0.357581
$910$	−1.33422	−0.0442290
$911$	24.1992	0.801754	0.400877	−	0.916132i	$-0.368706\pi$
$911$	24.1992	0.801754	0.400877	+	0.916132i	$0.368706\pi$
$912$	88.9748	2.94625
$913$	30.0815	0.995551
$914$	65.5434	2.16798
$915$	−2.62580	−0.0868062
$916$	−5.72579	−0.189186
$917$	8.04142	0.265551
$918$	8.51531	0.281047
$919$	57.2669	1.88906	0.944531	−	0.328423i	$-0.106517\pi$
$919$	57.2669	1.88906	0.944531	+	0.328423i	$0.106517\pi$
$920$	−29.2759	−0.965198
$921$	16.5782	0.546270
$922$	28.1580	0.927334
$923$	0.916409	0.0301640
$924$	37.6545	1.23874
$925$	53.5457	1.76057
$926$	−86.1408	−2.83076
$927$	19.5006	0.640483
$928$	184.246	6.04818
$929$	−0.493235	−0.0161825	−0.00809126	−	0.999967i	$-0.502576\pi$
$929$	−0.493235	−0.0161825	−0.00809126	+	0.999967i	$0.502576\pi$
$930$	7.53630	0.247125
$931$	−0.708330	−0.0232146
$932$	93.2365	3.05406
$933$	−20.9694	−0.686509
$934$	−85.5977	−2.80084
$935$	−2.74221	−0.0896798
$936$	−5.10942	−0.167007
$937$	−15.6239	−0.510410	−0.255205	−	0.966887i	$-0.582143\pi$
$937$	−15.6239	−0.510410	−0.255205	+	0.966887i	$0.582143\pi$
$938$	−71.2202	−2.32542
$939$	16.4549	0.536984
$940$	−10.8958	−0.355383
$941$	36.1818	1.17949	0.589746	−	0.807589i	$-0.299228\pi$
$941$	36.1818	1.17949	0.589746	+	0.807589i	$0.299228\pi$
$942$	0.978940	0.0318956
$943$	−23.0914	−0.751960
$944$	55.1001	1.79335
$945$	−0.954385	−0.0310462
$946$	6.47366	0.210477
$947$	37.1073	1.20583	0.602913	−	0.797807i	$-0.294007\pi$
$947$	37.1073	1.20583	0.602913	+	0.797807i	$0.294007\pi$
$948$	−4.35965	−0.141595
$949$	−5.17163	−0.167878
$950$	−71.9615	−2.33474
$951$	−8.74759	−0.283660
$952$	83.1178	2.69386
$953$	−40.3338	−1.30654	−0.653270	−	0.757125i	$-0.726604\pi$
$953$	−40.3338	−1.30654	−0.653270	+	0.757125i	$0.726604\pi$
$954$	6.99191	0.226371
$955$	2.65344	0.0858632
$956$	26.1244	0.844922
$957$	17.7431	0.573552
$958$	−65.7805	−2.12527
$959$	44.3350	1.43165
$960$	13.6856	0.441702
$961$	27.1014	0.874239
$962$	15.3637	0.495345
$963$	−14.8789	−0.479466
$964$	84.2321	2.71293
$965$	0.613325	0.0197436
$966$	59.8650	1.92612
$967$	21.0613	0.677286	0.338643	−	0.940915i	$-0.390032\pi$
$967$	21.0613	0.677286	0.338643	+	0.940915i	$0.390032\pi$
$968$	48.3712	1.55471
$969$	16.4298	0.527800
$970$	2.39178	0.0767953
$971$	−11.5519	−0.370717	−0.185358	−	0.982671i	$-0.559345\pi$
$971$	−11.5519	−0.370717	−0.185358	+	0.982671i	$0.559345\pi$
$972$	−5.65483	−0.181379
$973$	13.4439	0.430992
$974$	−41.6780	−1.33545
$975$	2.46190	0.0788439
$976$	−122.471	−3.92020
$977$	39.2299	1.25508	0.627538	−	0.778586i	$-0.284063\pi$
$977$	39.2299	1.25508	0.627538	+	0.778586i	$0.284063\pi$
$978$	0.483798	0.0154702
$979$	17.0289	0.544246
$980$	−0.268135	−0.00856526
$981$	13.8714	0.442880
$982$	−50.6843	−1.61740
$983$	18.6145	0.593711	0.296855	−	0.954922i	$-0.404062\pi$
$983$	18.6145	0.593711	0.296855	+	0.954922i	$0.404062\pi$
$984$	28.8209	0.918775
$985$	1.74054	0.0554582
$986$	60.5980	1.92983
$987$	14.4003	0.458366
$988$	−15.2530	−0.485262
$989$	7.60309	0.241764
$990$	2.46510	0.0783462
$991$	14.9451	0.474745	0.237373	−	0.971419i	$-0.423714\pi$
$991$	14.9451	0.474745	0.237373	+	0.971419i	$0.423714\pi$
$992$	197.348	6.26582
$993$	−9.55978	−0.303371
$994$	13.4013	0.425064
$995$	−6.94839	−0.220279
$996$	−68.2257	−2.16181
$997$	61.3705	1.94362	0.971812	−	0.235756i	$-0.0757567\pi$
$997$	61.3705	1.94362	0.971812	+	0.235756i	$0.0757567\pi$
$998$	55.9629	1.77147
$999$	10.9898	0.347703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.c.1.1	✓	119

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.c.1.1	✓	119	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.76674 q^{2} -1.00000 q^{3} +5.65483 q^{4} -0.357353 q^{5} +2.76674 q^{6} -2.67071 q^{7} -10.1119 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.76674 q^{2} -1.00000 q^{3} +5.65483 q^{4} -0.357353 q^{5} +2.76674 q^{6} -2.67071 q^{7} -10.1119 q^{8} +1.00000 q^{9} +0.988701 q^{10} +2.49328 q^{11} -5.65483 q^{12} +0.505285 q^{13} +7.38915 q^{14} +0.357353 q^{15} +16.6674 q^{16} +3.07775 q^{17} -2.76674 q^{18} -5.33824 q^{19} -2.02077 q^{20} +2.67071 q^{21} -6.89824 q^{22} -8.10174 q^{23} +10.1119 q^{24} -4.87230 q^{25} -1.39799 q^{26} -1.00000 q^{27} -15.1024 q^{28} -7.11636 q^{29} -0.988701 q^{30} -7.62243 q^{31} -25.8905 q^{32} -2.49328 q^{33} -8.51531 q^{34} +0.954385 q^{35} +5.65483 q^{36} -10.9898 q^{37} +14.7695 q^{38} -0.505285 q^{39} +3.61353 q^{40} +2.85018 q^{41} -7.38915 q^{42} -0.938451 q^{43} +14.0991 q^{44} -0.357353 q^{45} +22.4154 q^{46} +5.39192 q^{47} -16.6674 q^{48} +0.132690 q^{49} +13.4804 q^{50} -3.07775 q^{51} +2.85730 q^{52} -2.52713 q^{53} +2.76674 q^{54} -0.890979 q^{55} +27.0061 q^{56} +5.33824 q^{57} +19.6891 q^{58} +3.30585 q^{59} +2.02077 q^{60} -7.34791 q^{61} +21.0892 q^{62} -2.67071 q^{63} +38.2973 q^{64} -0.180565 q^{65} +6.89824 q^{66} -9.63849 q^{67} +17.4041 q^{68} +8.10174 q^{69} -2.64053 q^{70} +1.81365 q^{71} -10.1119 q^{72} -10.2351 q^{73} +30.4059 q^{74} +4.87230 q^{75} -30.1869 q^{76} -6.65882 q^{77} +1.39799 q^{78} +0.770961 q^{79} -5.95615 q^{80} +1.00000 q^{81} -7.88569 q^{82} +12.0650 q^{83} +15.1024 q^{84} -1.09984 q^{85} +2.59645 q^{86} +7.11636 q^{87} -25.2119 q^{88} +6.82993 q^{89} +0.988701 q^{90} -1.34947 q^{91} -45.8140 q^{92} +7.62243 q^{93} -14.9180 q^{94} +1.90764 q^{95} +25.8905 q^{96} +2.41911 q^{97} -0.367117 q^{98} +2.49328 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9}+O(q^{10}) \) 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9	\( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9} - 10 q^{10} + 56 q^{11} - 137 q^{12} - 37 q^{13} + 31 q^{14} - 17 q^{15} + 173 q^{16} + 17 q^{17} + 11 q^{18} + 16 q^{19} + 61 q^{20} - 10 q^{21} - 3 q^{22} + 76 q^{23} - 33 q^{24} + 134 q^{25} + 47 q^{26} - 119 q^{27} - q^{28} + 47 q^{29} + 10 q^{30} + 51 q^{31} + 87 q^{32} - 56 q^{33} + 13 q^{34} + 58 q^{35} + 137 q^{36} - 67 q^{37} + 35 q^{38} + 37 q^{39} - 40 q^{40} + 47 q^{41} - 31 q^{42} + 12 q^{43} + 148 q^{44} + 17 q^{45} + 26 q^{46} + 107 q^{47} - 173 q^{48} + 163 q^{49} + 76 q^{50} - 17 q^{51} - 57 q^{52} + 64 q^{53} - 11 q^{54} + 71 q^{55} + 91 q^{56} - 16 q^{57} + 12 q^{58} + 98 q^{59} - 61 q^{60} - 50 q^{61} + 40 q^{62} + 10 q^{63} + 245 q^{64} + 40 q^{65} + 3 q^{66} + 12 q^{67} + 75 q^{68} - 76 q^{69} - 9 q^{70} + 194 q^{71} + 33 q^{72} - 79 q^{73} + 72 q^{74} - 134 q^{75} + 12 q^{76} + 71 q^{77} - 47 q^{78} + 127 q^{79} + 148 q^{80} + 119 q^{81} - 54 q^{82} + 77 q^{83} + q^{84} - 25 q^{85} + 142 q^{86} - 47 q^{87} + q^{88} + 93 q^{89} - 10 q^{90} + 61 q^{91} + 156 q^{92} - 51 q^{93} + 16 q^{94} + 138 q^{95} - 87 q^{96} - 110 q^{97} + 96 q^{98} + 56 q^{99}+O(q^{100}) \) 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9 - 10 * q^10 + 56 * q^11 - 137 * q^12 - 37 * q^13 + 31 * q^14 - 17 * q^15 + 173 * q^16 + 17 * q^17 + 11 * q^18 + 16 * q^19 + 61 * q^20 - 10 * q^21 - 3 * q^22 + 76 * q^23 - 33 * q^24 + 134 * q^25 + 47 * q^26 - 119 * q^27 - q^28 + 47 * q^29 + 10 * q^30 + 51 * q^31 + 87 * q^32 - 56 * q^33 + 13 * q^34 + 58 * q^35 + 137 * q^36 - 67 * q^37 + 35 * q^38 + 37 * q^39 - 40 * q^40 + 47 * q^41 - 31 * q^42 + 12 * q^43 + 148 * q^44 + 17 * q^45 + 26 * q^46 + 107 * q^47 - 173 * q^48 + 163 * q^49 + 76 * q^50 - 17 * q^51 - 57 * q^52 + 64 * q^53 - 11 * q^54 + 71 * q^55 + 91 * q^56 - 16 * q^57 + 12 * q^58 + 98 * q^59 - 61 * q^60 - 50 * q^61 + 40 * q^62 + 10 * q^63 + 245 * q^64 + 40 * q^65 + 3 * q^66 + 12 * q^67 + 75 * q^68 - 76 * q^69 - 9 * q^70 + 194 * q^71 + 33 * q^72 - 79 * q^73 + 72 * q^74 - 134 * q^75 + 12 * q^76 + 71 * q^77 - 47 * q^78 + 127 * q^79 + 148 * q^80 + 119 * q^81 - 54 * q^82 + 77 * q^83 + q^84 - 25 * q^85 + 142 * q^86 - 47 * q^87 + q^88 + 93 * q^89 - 10 * q^90 + 61 * q^91 + 156 * q^92 - 51 * q^93 + 16 * q^94 + 138 * q^95 - 87 * q^96 - 110 * q^97 + 96 * q^98 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more