LMFDB - Embedded newform 5775.2.a.ch.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.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:	$46.1136071673$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1109184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} + 7x - 2 $ x^5 - 7x^3 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.25950$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25950 q^{2} -1.00000 q^{3} -0.413654 q^{4} +1.25950 q^{6} +1.00000 q^{7} +3.04000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.25950 q^{2} -1.00000 q^{3} -0.413654 q^{4} +1.25950 q^{6} +1.00000 q^{7} +3.04000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +0.413654 q^{12} +5.74366 q^{13} -1.25950 q^{14} -3.00158 q^{16} -3.06892 q^{17} -1.25950 q^{18} +0.961580 q^{19} -1.00000 q^{21} -1.25950 q^{22} -0.702078 q^{23} -3.04000 q^{24} -7.23416 q^{26} -1.00000 q^{27} -0.413654 q^{28} +8.46116 q^{29} -7.55901 q^{31} -2.29951 q^{32} -1.00000 q^{33} +3.86532 q^{34} -0.413654 q^{36} -8.58951 q^{37} -1.21111 q^{38} -5.74366 q^{39} -4.94216 q^{41} +1.25950 q^{42} -11.4842 q^{43} -0.413654 q^{44} +0.884268 q^{46} +5.73416 q^{47} +3.00158 q^{48} +1.00000 q^{49} +3.06892 q^{51} -2.37589 q^{52} -9.65532 q^{53} +1.25950 q^{54} +3.04000 q^{56} -0.961580 q^{57} -10.6569 q^{58} +2.28842 q^{59} +6.30900 q^{61} +9.52059 q^{62} +1.00000 q^{63} +8.89940 q^{64} +1.25950 q^{66} -3.27963 q^{67} +1.26947 q^{68} +0.702078 q^{69} -11.7163 q^{71} +3.04000 q^{72} -9.25516 q^{73} +10.8185 q^{74} -0.397761 q^{76} +1.00000 q^{77} +7.23416 q^{78} +11.5575 q^{79} +1.00000 q^{81} +6.22466 q^{82} +2.28689 q^{83} +0.413654 q^{84} +14.4643 q^{86} -8.46116 q^{87} +3.04000 q^{88} +1.21516 q^{89} +5.74366 q^{91} +0.290417 q^{92} +7.55901 q^{93} -7.22219 q^{94} +2.29951 q^{96} -14.2397 q^{97} -1.25950 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 4 q^{4} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 4 * q^4 + 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 4 q^{4} + 5 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{12} - 7 q^{13} + 6 q^{16} - 13 q^{17} - q^{19} - 5 q^{21} - 4 q^{23} - 6 q^{26} - 5 q^{27} + 4 q^{28} + 4 q^{29} - 10 q^{31} + 10 q^{32} - 5 q^{33} - 18 q^{34} + 4 q^{36} - 7 q^{37} - 24 q^{38} + 7 q^{39} + q^{41} - 28 q^{43} + 4 q^{44} + 10 q^{46} - 6 q^{48} + 5 q^{49} + 13 q^{51} - 28 q^{52} + 5 q^{53} + q^{57} - 36 q^{58} + 18 q^{59} + 3 q^{61} + 14 q^{62} + 5 q^{63} - 12 q^{64} - q^{67} - 30 q^{68} + 4 q^{69} + 11 q^{71} - 15 q^{73} + 20 q^{74} - 22 q^{76} + 5 q^{77} + 6 q^{78} - 6 q^{79} + 5 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{84} - 8 q^{86} - 4 q^{87} - 10 q^{89} - 7 q^{91} + 18 q^{92} + 10 q^{93} - 26 q^{94} - 10 q^{96} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 4 * q^4 + 5 * q^7 + 5 * q^9 + 5 * q^11 - 4 * q^12 - 7 * q^13 + 6 * q^16 - 13 * q^17 - q^19 - 5 * q^21 - 4 * q^23 - 6 * q^26 - 5 * q^27 + 4 * q^28 + 4 * q^29 - 10 * q^31 + 10 * q^32 - 5 * q^33 - 18 * q^34 + 4 * q^36 - 7 * q^37 - 24 * q^38 + 7 * q^39 + q^41 - 28 * q^43 + 4 * q^44 + 10 * q^46 - 6 * q^48 + 5 * q^49 + 13 * q^51 - 28 * q^52 + 5 * q^53 + q^57 - 36 * q^58 + 18 * q^59 + 3 * q^61 + 14 * q^62 + 5 * q^63 - 12 * q^64 - q^67 - 30 * q^68 + 4 * q^69 + 11 * q^71 - 15 * q^73 + 20 * q^74 - 22 * q^76 + 5 * q^77 + 6 * q^78 - 6 * q^79 + 5 * q^81 + 8 * q^82 - 18 * q^83 - 4 * q^84 - 8 * q^86 - 4 * q^87 - 10 * q^89 - 7 * q^91 + 18 * q^92 + 10 * q^93 - 26 * q^94 - 10 * q^96 - 14 * q^97 + 5 * 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 $

$2$	−1.25950	−0.890603	−0.445301	−	0.895381i	$-0.646903\pi$
$2$	−1.25950	−0.890603	−0.445301	+	0.895381i	$0.646903\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.413654	−0.206827
$5$	0	0
$5$	0	0
$6$	1.25950	0.514190
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.04000	1.07480
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.413654	0.119412
$13$	5.74366	1.59301	0.796503	−	0.604635i	$-0.206681\pi$
$13$	5.74366	1.59301	0.796503	+	0.604635i	$0.206681\pi$
$14$	−1.25950	−0.336616
$15$	0	0
$16$	−3.00158	−0.750396
$17$	−3.06892	−0.744323	−0.372162	−	0.928168i	$-0.621383\pi$
$17$	−3.06892	−0.744323	−0.372162	+	0.928168i	$0.621383\pi$
$18$	−1.25950	−0.296868
$19$	0.961580	0.220602	0.110301	−	0.993898i	$-0.464819\pi$
$19$	0.961580	0.220602	0.110301	+	0.993898i	$0.464819\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−1.25950	−0.268527
$23$	−0.702078	−0.146393	−0.0731967	−	0.997318i	$-0.523320\pi$
$23$	−0.702078	−0.146393	−0.0731967	+	0.997318i	$0.523320\pi$
$24$	−3.04000	−0.620538
$25$	0	0
$26$	−7.23416	−1.41873
$27$	−1.00000	−0.192450
$28$	−0.413654	−0.0781732
$29$	8.46116	1.57120	0.785599	−	0.618736i	$-0.212355\pi$
$29$	8.46116	1.57120	0.785599	+	0.618736i	$0.212355\pi$
$30$	0	0
$31$	−7.55901	−1.35764	−0.678819	−	0.734306i	$-0.737508\pi$
$31$	−7.55901	−1.35764	−0.678819	+	0.734306i	$0.737508\pi$
$32$	−2.29951	−0.406499
$33$	−1.00000	−0.174078
$34$	3.86532	0.662896
$35$	0	0
$36$	−0.413654	−0.0689423
$37$	−8.58951	−1.41211	−0.706053	−	0.708159i	$-0.749526\pi$
$37$	−8.58951	−1.41211	−0.706053	+	0.708159i	$0.749526\pi$
$38$	−1.21111	−0.196468
$39$	−5.74366	−0.919722
$40$	0	0
$41$	−4.94216	−0.771835	−0.385918	−	0.922533i	$-0.626115\pi$
$41$	−4.94216	−0.771835	−0.385918	+	0.922533i	$0.626115\pi$
$42$	1.25950	0.194345
$43$	−11.4842	−1.75132	−0.875659	−	0.482930i	$-0.839573\pi$
$43$	−11.4842	−1.75132	−0.875659	+	0.482930i	$0.839573\pi$
$44$	−0.413654	−0.0623607
$45$	0	0
$46$	0.884268	0.130378
$47$	5.73416	0.836414	0.418207	−	0.908352i	$-0.362659\pi$
$47$	5.73416	0.836414	0.418207	+	0.908352i	$0.362659\pi$
$48$	3.00158	0.433241
$49$	1.00000	0.142857
$50$	0	0
$51$	3.06892	0.429735
$52$	−2.37589	−0.329476
$53$	−9.65532	−1.32626	−0.663130	−	0.748504i	$-0.730772\pi$
$53$	−9.65532	−1.32626	−0.663130	+	0.748504i	$0.730772\pi$
$54$	1.25950	0.171397
$55$	0	0
$56$	3.04000	0.406237
$57$	−0.961580	−0.127364
$58$	−10.6569	−1.39931
$59$	2.28842	0.297927	0.148964	−	0.988843i	$-0.452406\pi$
$59$	2.28842	0.297927	0.148964	+	0.988843i	$0.452406\pi$
$60$	0	0
$61$	6.30900	0.807785	0.403893	−	0.914806i	$-0.367657\pi$
$61$	6.30900	0.807785	0.403893	+	0.914806i	$0.367657\pi$
$62$	9.52059	1.20912
$63$	1.00000	0.125988
$64$	8.89940	1.11242
$65$	0	0
$66$	1.25950	0.155034
$67$	−3.27963	−0.400670	−0.200335	−	0.979727i	$-0.564203\pi$
$67$	−3.27963	−0.400670	−0.200335	+	0.979727i	$0.564203\pi$
$68$	1.26947	0.153946
$69$	0.702078	0.0845202
$70$	0	0
$71$	−11.7163	−1.39047	−0.695236	−	0.718782i	$-0.744700\pi$
$71$	−11.7163	−1.39047	−0.695236	+	0.718782i	$0.744700\pi$
$72$	3.04000	0.358268
$73$	−9.25516	−1.08323	−0.541617	−	0.840625i	$-0.682188\pi$
$73$	−9.25516	−1.08323	−0.541617	+	0.840625i	$0.682188\pi$
$74$	10.8185	1.25763
$75$	0	0
$76$	−0.397761	−0.0456264
$77$	1.00000	0.113961
$78$	7.23416	0.819107
$79$	11.5575	1.30032	0.650159	−	0.759799i	$-0.274702\pi$
$79$	11.5575	1.30032	0.650159	+	0.759799i	$0.274702\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.22466	0.687399
$83$	2.28689	0.251018	0.125509	−	0.992092i	$-0.459944\pi$
$83$	2.28689	0.251018	0.125509	+	0.992092i	$0.459944\pi$
$84$	0.413654	0.0451333
$85$	0	0
$86$	14.4643	1.55973
$87$	−8.46116	−0.907132
$88$	3.04000	0.324065
$89$	1.21516	0.128807	0.0644033	−	0.997924i	$-0.479486\pi$
$89$	1.21516	0.128807	0.0644033	+	0.997924i	$0.479486\pi$
$90$	0	0
$91$	5.74366	0.602099
$92$	0.290417	0.0302781
$93$	7.55901	0.783833
$94$	−7.22219	−0.744912
$95$	0	0
$96$	2.29951	0.234692
$97$	−14.2397	−1.44582	−0.722910	−	0.690943i	$-0.757196\pi$
$97$	−14.2397	−1.44582	−0.722910	+	0.690943i	$0.757196\pi$
$98$	−1.25950	−0.127229
$99$	1.00000	0.100504
$100$	0	0
$101$	−2.92316	−0.290865	−0.145433	−	0.989368i	$-0.546457\pi$
$101$	−2.92316	−0.290865	−0.145433	+	0.989368i	$0.546457\pi$
$102$	−3.86532	−0.382723
$103$	−3.79927	−0.374353	−0.187177	−	0.982326i	$-0.559934\pi$
$103$	−3.79927	−0.374353	−0.187177	+	0.982326i	$0.559934\pi$
$104$	17.4608	1.71217
$105$	0	0
$106$	12.1609	1.18117
$107$	5.92832	0.573112	0.286556	−	0.958063i	$-0.407490\pi$
$107$	5.92832	0.573112	0.286556	+	0.958063i	$0.407490\pi$
$108$	0.413654	0.0398039
$109$	−11.1379	−1.06682	−0.533408	−	0.845858i	$-0.679089\pi$
$109$	−11.1379	−1.06682	−0.533408	+	0.845858i	$0.679089\pi$
$110$	0	0
$111$	8.58951	0.815280
$112$	−3.00158	−0.283623
$113$	−5.49958	−0.517357	−0.258679	−	0.965963i	$-0.583287\pi$
$113$	−5.49958	−0.517357	−0.258679	+	0.965963i	$0.583287\pi$
$114$	1.21111	0.113431
$115$	0	0
$116$	−3.49999	−0.324966
$117$	5.74366	0.531002
$118$	−2.88228	−0.265335
$119$	−3.06892	−0.281328
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.94620	−0.719416
$123$	4.94216	0.445619
$124$	3.12681	0.280796
$125$	0	0
$126$	−1.25950	−0.112205
$127$	9.15643	0.812502	0.406251	−	0.913761i	$-0.366836\pi$
$127$	9.15643	0.812502	0.406251	+	0.913761i	$0.366836\pi$
$128$	−6.60980	−0.584229
$129$	11.4842	1.01112
$130$	0	0
$131$	9.84063	0.859780	0.429890	−	0.902881i	$-0.358552\pi$
$131$	9.84063	0.859780	0.429890	+	0.902881i	$0.358552\pi$
$132$	0.413654	0.0360039
$133$	0.961580	0.0833796
$134$	4.13070	0.356838
$135$	0	0
$136$	−9.32954	−0.800001
$137$	−12.4889	−1.06700	−0.533498	−	0.845801i	$-0.679123\pi$
$137$	−12.4889	−1.06700	−0.533498	+	0.845801i	$0.679123\pi$
$138$	−0.884268	−0.0752739
$139$	−3.96909	−0.336653	−0.168327	−	0.985731i	$-0.553836\pi$
$139$	−3.96909	−0.336653	−0.168327	+	0.985731i	$0.553836\pi$
$140$	0	0
$141$	−5.73416	−0.482904
$142$	14.7567	1.23836
$143$	5.74366	0.480309
$144$	−3.00158	−0.250132
$145$	0	0
$146$	11.6569	0.964732
$147$	−1.00000	−0.0824786
$148$	3.55308	0.292062
$149$	7.12681	0.583851	0.291926	−	0.956441i	$-0.405704\pi$
$149$	7.12681	0.583851	0.291926	+	0.956441i	$0.405704\pi$
$150$	0	0
$151$	15.3368	1.24809	0.624044	−	0.781390i	$-0.285489\pi$
$151$	15.3368	1.24809	0.624044	+	0.781390i	$0.285489\pi$
$152$	2.92321	0.237103
$153$	−3.06892	−0.248108
$154$	−1.25950	−0.101494
$155$	0	0
$156$	2.37589	0.190223
$157$	−12.4632	−0.994668	−0.497334	−	0.867559i	$-0.665688\pi$
$157$	−12.4632	−0.994668	−0.497334	+	0.867559i	$0.665688\pi$
$158$	−14.5567	−1.15807
$159$	9.65532	0.765716
$160$	0	0
$161$	−0.702078	−0.0553315
$162$	−1.25950	−0.0989558
$163$	−8.55085	−0.669754	−0.334877	−	0.942262i	$-0.608695\pi$
$163$	−8.55085	−0.669754	−0.334877	+	0.942262i	$0.608695\pi$
$164$	2.04434	0.159636
$165$	0	0
$166$	−2.88034	−0.223558
$167$	24.6228	1.90537	0.952683	−	0.303966i	$-0.0983108\pi$
$167$	24.6228	1.90537	0.952683	+	0.303966i	$0.0983108\pi$
$168$	−3.04000	−0.234541
$169$	19.9897	1.53767
$170$	0	0
$171$	0.961580	0.0735339
$172$	4.75047	0.362220
$173$	−7.89781	−0.600460	−0.300230	−	0.953867i	$-0.597063\pi$
$173$	−7.89781	−0.600460	−0.300230	+	0.953867i	$0.597063\pi$
$174$	10.6569	0.807894
$175$	0	0
$176$	−3.00158	−0.226253
$177$	−2.28842	−0.172008
$178$	−1.53050	−0.114716
$179$	19.8481	1.48352	0.741760	−	0.670666i	$-0.233992\pi$
$179$	19.8481	1.48352	0.741760	+	0.670666i	$0.233992\pi$
$180$	0	0
$181$	−16.8727	−1.25414	−0.627069	−	0.778963i	$-0.715746\pi$
$181$	−16.8727	−1.25414	−0.627069	+	0.778963i	$0.715746\pi$
$182$	−7.23416	−0.536231
$183$	−6.30900	−0.466375
$184$	−2.13432	−0.157344
$185$	0	0
$186$	−9.52059	−0.698083
$187$	−3.06892	−0.224422
$188$	−2.37196	−0.172993
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−19.3054	−1.39689	−0.698444	−	0.715665i	$-0.746124\pi$
$191$	−19.3054	−1.39689	−0.698444	+	0.715665i	$0.746124\pi$
$192$	−8.89940	−0.642259
$193$	19.8165	1.42642	0.713210	−	0.700951i	$-0.247241\pi$
$193$	19.8165	1.42642	0.713210	+	0.700951i	$0.247241\pi$
$194$	17.9349	1.28765
$195$	0	0
$196$	−0.413654	−0.0295467
$197$	−6.23850	−0.444474	−0.222237	−	0.974993i	$-0.571336\pi$
$197$	−6.23850	−0.444474	−0.222237	+	0.974993i	$0.571336\pi$
$198$	−1.25950	−0.0895089
$199$	−6.71470	−0.475992	−0.237996	−	0.971266i	$-0.576491\pi$
$199$	−6.71470	−0.475992	−0.237996	+	0.971266i	$0.576491\pi$
$200$	0	0
$201$	3.27963	0.231327
$202$	3.68173	0.259045
$203$	8.46116	0.593857
$204$	−1.26947	−0.0888808
$205$	0	0
$206$	4.78519	0.333400
$207$	−0.702078	−0.0487978
$208$	−17.2401	−1.19538
$209$	0.961580	0.0665139
$210$	0	0
$211$	−0.692990	−0.0477074	−0.0238537	−	0.999715i	$-0.507594\pi$
$211$	−0.692990	−0.0477074	−0.0238537	+	0.999715i	$0.507594\pi$
$212$	3.99396	0.274306
$213$	11.7163	0.802789
$214$	−7.46673	−0.510415
$215$	0	0
$216$	−3.04000	−0.206846
$217$	−7.55901	−0.513139
$218$	14.0282	0.950110
$219$	9.25516	0.625406
$220$	0	0
$221$	−17.6269	−1.18571
$222$	−10.8185	−0.726091
$223$	−1.94099	−0.129978	−0.0649890	−	0.997886i	$-0.520701\pi$
$223$	−1.94099	−0.129978	−0.0649890	+	0.997886i	$0.520701\pi$
$224$	−2.29951	−0.153642
$225$	0	0
$226$	6.92674	0.460760
$227$	−20.4471	−1.35712	−0.678560	−	0.734545i	$-0.737396\pi$
$227$	−20.4471	−1.35712	−0.678560	+	0.734545i	$0.737396\pi$
$228$	0.397761	0.0263424
$229$	−10.7814	−0.712457	−0.356228	−	0.934399i	$-0.615938\pi$
$229$	−10.7814	−0.712457	−0.356228	+	0.934399i	$0.615938\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	25.7220	1.68873
$233$	14.0724	0.921912	0.460956	−	0.887423i	$-0.347507\pi$
$233$	14.0724	0.921912	0.460956	+	0.887423i	$0.347507\pi$
$234$	−7.23416	−0.472912
$235$	0	0
$236$	−0.946615	−0.0616194
$237$	−11.5575	−0.750738
$238$	3.86532	0.250551
$239$	6.31827	0.408695	0.204348	−	0.978898i	$-0.434493\pi$
$239$	6.31827	0.408695	0.204348	+	0.978898i	$0.434493\pi$
$240$	0	0
$241$	23.0364	1.48391	0.741953	−	0.670452i	$-0.233900\pi$
$241$	23.0364	1.48391	0.741953	+	0.670452i	$0.233900\pi$
$242$	−1.25950	−0.0809639
$243$	−1.00000	−0.0641500
$244$	−2.60974	−0.167072
$245$	0	0
$246$	−6.22466	−0.396870
$247$	5.52299	0.351420
$248$	−22.9794	−1.45919
$249$	−2.28689	−0.144926
$250$	0	0
$251$	2.33400	0.147321	0.0736603	−	0.997283i	$-0.476532\pi$
$251$	2.33400	0.147321	0.0736603	+	0.997283i	$0.476532\pi$
$252$	−0.413654	−0.0260577
$253$	−0.702078	−0.0441392
$254$	−11.5326	−0.723617
$255$	0	0
$256$	−9.47373	−0.592108
$257$	−25.4344	−1.58655	−0.793276	−	0.608863i	$-0.791626\pi$
$257$	−25.4344	−1.58655	−0.793276	+	0.608863i	$0.791626\pi$
$258$	−14.4643	−0.900510
$259$	−8.58951	−0.533726
$260$	0	0
$261$	8.46116	0.523733
$262$	−12.3943	−0.765722
$263$	15.0304	0.926813	0.463407	−	0.886146i	$-0.346627\pi$
$263$	15.0304	0.926813	0.463407	+	0.886146i	$0.346627\pi$
$264$	−3.04000	−0.187099
$265$	0	0
$266$	−1.21111	−0.0742581
$267$	−1.21516	−0.0743666
$268$	1.35663	0.0828694
$269$	8.82643	0.538157	0.269078	−	0.963118i	$-0.413281\pi$
$269$	8.82643	0.538157	0.269078	+	0.963118i	$0.413281\pi$
$270$	0	0
$271$	−8.24162	−0.500643	−0.250321	−	0.968163i	$-0.580536\pi$
$271$	−8.24162	−0.500643	−0.250321	+	0.968163i	$0.580536\pi$
$272$	9.21163	0.558537
$273$	−5.74366	−0.347622
$274$	15.7298	0.950269
$275$	0	0
$276$	−0.290417	−0.0174811
$277$	−25.0010	−1.50216	−0.751082	−	0.660209i	$-0.770468\pi$
$277$	−25.0010	−1.50216	−0.751082	+	0.660209i	$0.770468\pi$
$278$	4.99907	0.299824
$279$	−7.55901	−0.452546
$280$	0	0
$281$	−20.3759	−1.21552	−0.607761	−	0.794120i	$-0.707932\pi$
$281$	−20.3759	−1.21552	−0.607761	+	0.794120i	$0.707932\pi$
$282$	7.22219	0.430075
$283$	−24.5074	−1.45682	−0.728408	−	0.685143i	$-0.759740\pi$
$283$	−24.5074	−1.45682	−0.728408	+	0.685143i	$0.759740\pi$
$284$	4.84650	0.287587
$285$	0	0
$286$	−7.23416	−0.427765
$287$	−4.94216	−0.291726
$288$	−2.29951	−0.135500
$289$	−7.58170	−0.445983
$290$	0	0
$291$	14.2397	0.834744
$292$	3.82843	0.224042
$293$	9.20044	0.537495	0.268748	−	0.963211i	$-0.413390\pi$
$293$	9.20044	0.537495	0.268748	+	0.963211i	$0.413390\pi$
$294$	1.25950	0.0734557
$295$	0	0
$296$	−26.1121	−1.51774
$297$	−1.00000	−0.0580259
$298$	−8.97624	−0.519979
$299$	−4.03250	−0.233205
$300$	0	0
$301$	−11.4842	−0.661936
$302$	−19.3167	−1.11155
$303$	2.92316	0.167931
$304$	−2.88626	−0.165538
$305$	0	0
$306$	3.86532	0.220965
$307$	11.0135	0.628572	0.314286	−	0.949328i	$-0.398235\pi$
$307$	11.0135	0.628572	0.314286	+	0.949328i	$0.398235\pi$
$308$	−0.413654	−0.0235701
$309$	3.79927	0.216133
$310$	0	0
$311$	−2.16823	−0.122949	−0.0614746	−	0.998109i	$-0.519580\pi$
$311$	−2.16823	−0.122949	−0.0614746	+	0.998109i	$0.519580\pi$
$312$	−17.4608	−0.988520
$313$	27.2482	1.54016	0.770081	−	0.637946i	$-0.220216\pi$
$313$	27.2482	1.54016	0.770081	+	0.637946i	$0.220216\pi$
$314$	15.6974	0.885854
$315$	0	0
$316$	−4.78079	−0.268941
$317$	−9.67250	−0.543262	−0.271631	−	0.962402i	$-0.587563\pi$
$317$	−9.67250	−0.543262	−0.271631	+	0.962402i	$0.587563\pi$
$318$	−12.1609	−0.681949
$319$	8.46116	0.473734
$320$	0	0
$321$	−5.92832	−0.330886
$322$	0.884268	0.0492784
$323$	−2.95102	−0.164199
$324$	−0.413654	−0.0229808
$325$	0	0
$326$	10.7698	0.596485
$327$	11.1379	0.615927
$328$	−15.0242	−0.829571
$329$	5.73416	0.316135
$330$	0	0
$331$	−21.3454	−1.17325	−0.586624	−	0.809860i	$-0.699543\pi$
$331$	−21.3454	−1.17325	−0.586624	+	0.809860i	$0.699543\pi$
$332$	−0.945980	−0.0519174
$333$	−8.58951	−0.470702
$334$	−31.0124	−1.69692
$335$	0	0
$336$	3.00158	0.163750
$337$	−9.65416	−0.525896	−0.262948	−	0.964810i	$-0.584695\pi$
$337$	−9.65416	−0.525896	−0.262948	+	0.964810i	$0.584695\pi$
$338$	−25.1770	−1.36945
$339$	5.49958	0.298696
$340$	0	0
$341$	−7.55901	−0.409343
$342$	−1.21111	−0.0654894
$343$	1.00000	0.0539949
$344$	−34.9119	−1.88232
$345$	0	0
$346$	9.94732	0.534771
$347$	−6.57377	−0.352899	−0.176449	−	0.984310i	$-0.556461\pi$
$347$	−6.57377	−0.352899	−0.176449	+	0.984310i	$0.556461\pi$
$348$	3.49999	0.187619
$349$	−32.5102	−1.74023	−0.870116	−	0.492848i	$-0.835956\pi$
$349$	−32.5102	−1.74023	−0.870116	+	0.492848i	$0.835956\pi$
$350$	0	0
$351$	−5.74366	−0.306574
$352$	−2.29951	−0.122564
$353$	12.5469	0.667805	0.333903	−	0.942608i	$-0.391634\pi$
$353$	12.5469	0.667805	0.333903	+	0.942608i	$0.391634\pi$
$354$	2.88228	0.153191
$355$	0	0
$356$	−0.502656	−0.0266407
$357$	3.06892	0.162425
$358$	−24.9988	−1.32123
$359$	20.2769	1.07017	0.535086	−	0.844798i	$-0.320279\pi$
$359$	20.2769	1.07017	0.535086	+	0.844798i	$0.320279\pi$
$360$	0	0
$361$	−18.0754	−0.951335
$362$	21.2512	1.11694
$363$	−1.00000	−0.0524864
$364$	−2.37589	−0.124530
$365$	0	0
$366$	7.94620	0.415355
$367$	−18.2028	−0.950179	−0.475090	−	0.879937i	$-0.657584\pi$
$367$	−18.2028	−0.950179	−0.475090	+	0.879937i	$0.657584\pi$
$368$	2.10734	0.109853
$369$	−4.94216	−0.257278
$370$	0	0
$371$	−9.65532	−0.501279
$372$	−3.12681	−0.162118
$373$	−16.6166	−0.860375	−0.430188	−	0.902739i	$-0.641553\pi$
$373$	−16.6166	−0.860375	−0.430188	+	0.902739i	$0.641553\pi$
$374$	3.86532	0.199871
$375$	0	0
$376$	17.4319	0.898980
$377$	48.5981	2.50293
$378$	1.25950	0.0647818
$379$	−9.48649	−0.487288	−0.243644	−	0.969865i	$-0.578343\pi$
$379$	−9.48649	−0.487288	−0.243644	+	0.969865i	$0.578343\pi$
$380$	0	0
$381$	−9.15643	−0.469098
$382$	24.3152	1.24407
$383$	1.59361	0.0814295	0.0407148	−	0.999171i	$-0.487037\pi$
$383$	1.59361	0.0814295	0.0407148	+	0.999171i	$0.487037\pi$
$384$	6.60980	0.337305
$385$	0	0
$386$	−24.9589	−1.27037
$387$	−11.4842	−0.583773
$388$	5.89029	0.299034
$389$	18.3176	0.928741	0.464370	−	0.885641i	$-0.346281\pi$
$389$	18.3176	0.928741	0.464370	+	0.885641i	$0.346281\pi$
$390$	0	0
$391$	2.15462	0.108964
$392$	3.04000	0.153543
$393$	−9.84063	−0.496394
$394$	7.85740	0.395850
$395$	0	0
$396$	−0.413654	−0.0207869
$397$	−22.3166	−1.12004	−0.560020	−	0.828479i	$-0.689206\pi$
$397$	−22.3166	−1.12004	−0.560020	+	0.828479i	$0.689206\pi$
$398$	8.45717	0.423920
$399$	−0.961580	−0.0481392
$400$	0	0
$401$	19.4530	0.971434	0.485717	−	0.874116i	$-0.338559\pi$
$401$	19.4530	0.971434	0.485717	+	0.874116i	$0.338559\pi$
$402$	−4.13070	−0.206020
$403$	−43.4164	−2.16272
$404$	1.20918	0.0601588
$405$	0	0
$406$	−10.6569	−0.528891
$407$	−8.58951	−0.425766
$408$	9.32954	0.461881
$409$	1.86889	0.0924108	0.0462054	−	0.998932i	$-0.485287\pi$
$409$	1.86889	0.0924108	0.0462054	+	0.998932i	$0.485287\pi$
$410$	0	0
$411$	12.4889	0.616030
$412$	1.57158	0.0774263
$413$	2.28842	0.112606
$414$	0.884268	0.0434594
$415$	0	0
$416$	−13.2076	−0.647555
$417$	3.96909	0.194367
$418$	−1.21111	−0.0592374
$419$	3.65191	0.178407	0.0892037	−	0.996013i	$-0.471568\pi$
$419$	3.65191	0.178407	0.0892037	+	0.996013i	$0.471568\pi$
$420$	0	0
$421$	1.60858	0.0783975	0.0391987	−	0.999231i	$-0.487519\pi$
$421$	1.60858	0.0783975	0.0391987	+	0.999231i	$0.487519\pi$
$422$	0.872822	0.0424883
$423$	5.73416	0.278805
$424$	−29.3522	−1.42547
$425$	0	0
$426$	−14.7567	−0.714966
$427$	6.30900	0.305314
$428$	−2.45227	−0.118535
$429$	−5.74366	−0.277307
$430$	0	0
$431$	−16.7365	−0.806169	−0.403085	−	0.915163i	$-0.632062\pi$
$431$	−16.7365	−0.806169	−0.403085	+	0.915163i	$0.632062\pi$
$432$	3.00158	0.144414
$433$	27.0966	1.30218	0.651089	−	0.759002i	$-0.274313\pi$
$433$	27.0966	1.30218	0.651089	+	0.759002i	$0.274313\pi$
$434$	9.52059	0.457003
$435$	0	0
$436$	4.60723	0.220646
$437$	−0.675104	−0.0322946
$438$	−11.6569	−0.556988
$439$	−5.54164	−0.264488	−0.132244	−	0.991217i	$-0.542218\pi$
$439$	−5.54164	−0.264488	−0.132244	+	0.991217i	$0.542218\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	22.2011	1.05600
$443$	22.9575	1.09074	0.545372	−	0.838194i	$-0.316388\pi$
$443$	22.9575	1.09074	0.545372	+	0.838194i	$0.316388\pi$
$444$	−3.55308	−0.168622
$445$	0	0
$446$	2.44468	0.115759
$447$	−7.12681	−0.337087
$448$	8.89940	0.420457
$449$	33.3326	1.57306	0.786531	−	0.617551i	$-0.211875\pi$
$449$	33.3326	1.57306	0.786531	+	0.617551i	$0.211875\pi$
$450$	0	0
$451$	−4.94216	−0.232717
$452$	2.27492	0.107003
$453$	−15.3368	−0.720583
$454$	25.7531	1.20865
$455$	0	0
$456$	−2.92321	−0.136892
$457$	−5.06811	−0.237076	−0.118538	−	0.992949i	$-0.537821\pi$
$457$	−5.06811	−0.237076	−0.118538	+	0.992949i	$0.537821\pi$
$458$	13.5792	0.634516
$459$	3.06892	0.143245
$460$	0	0
$461$	32.4690	1.51223	0.756115	−	0.654438i	$-0.227095\pi$
$461$	32.4690	1.51223	0.756115	+	0.654438i	$0.227095\pi$
$462$	1.25950	0.0585974
$463$	−36.6813	−1.70473	−0.852363	−	0.522950i	$-0.824831\pi$
$463$	−36.6813	−1.70473	−0.852363	+	0.522950i	$0.824831\pi$
$464$	−25.3969	−1.17902
$465$	0	0
$466$	−17.7242	−0.821057
$467$	−10.0534	−0.465217	−0.232609	−	0.972570i	$-0.574726\pi$
$467$	−10.0534	−0.465217	−0.232609	+	0.972570i	$0.574726\pi$
$468$	−2.37589	−0.109825
$469$	−3.27963	−0.151439
$470$	0	0
$471$	12.4632	0.574272
$472$	6.95681	0.320213
$473$	−11.4842	−0.528042
$474$	14.5567	0.668610
$475$	0	0
$476$	1.26947	0.0581862
$477$	−9.65532	−0.442087
$478$	−7.95788	−0.363985
$479$	7.60067	0.347283	0.173642	−	0.984809i	$-0.444447\pi$
$479$	7.60067	0.347283	0.173642	+	0.984809i	$0.444447\pi$
$480$	0	0
$481$	−49.3353	−2.24949
$482$	−29.0144	−1.32157
$483$	0.702078	0.0319456
$484$	−0.413654	−0.0188024
$485$	0	0
$486$	1.25950	0.0571322
$487$	23.5665	1.06790	0.533950	−	0.845516i	$-0.320707\pi$
$487$	23.5665	1.06790	0.533950	+	0.845516i	$0.320707\pi$
$488$	19.1794	0.868210
$489$	8.55085	0.386683
$490$	0	0
$491$	37.3108	1.68381	0.841905	−	0.539625i	$-0.181434\pi$
$491$	37.3108	1.68381	0.841905	+	0.539625i	$0.181434\pi$
$492$	−2.04434	−0.0921661
$493$	−25.9667	−1.16948
$494$	−6.95622	−0.312975
$495$	0	0
$496$	22.6890	1.01877
$497$	−11.7163	−0.525549
$498$	2.88034	0.129071
$499$	−3.00168	−0.134374	−0.0671868	−	0.997740i	$-0.521402\pi$
$499$	−3.00168	−0.134374	−0.0671868	+	0.997740i	$0.521402\pi$
$500$	0	0
$501$	−24.6228	−1.10006
$502$	−2.93967	−0.131204
$503$	−32.2669	−1.43871	−0.719356	−	0.694642i	$-0.755563\pi$
$503$	−32.2669	−1.43871	−0.719356	+	0.694642i	$0.755563\pi$
$504$	3.04000	0.135412
$505$	0	0
$506$	0.884268	0.0393105
$507$	−19.9897	−0.887772
$508$	−3.78759	−0.168047
$509$	−30.0537	−1.33211	−0.666054	−	0.745903i	$-0.732018\pi$
$509$	−30.0537	−1.33211	−0.666054	+	0.745903i	$0.732018\pi$
$510$	0	0
$511$	−9.25516	−0.409424
$512$	25.1518	1.11156
$513$	−0.961580	−0.0424548
$514$	32.0346	1.41299
$515$	0	0
$516$	−4.75047	−0.209128
$517$	5.73416	0.252188
$518$	10.8185	0.475338
$519$	7.89781	0.346676
$520$	0	0
$521$	−30.8363	−1.35096	−0.675482	−	0.737376i	$-0.736065\pi$
$521$	−30.8363	−1.35096	−0.675482	+	0.737376i	$0.736065\pi$
$522$	−10.6569	−0.466438
$523$	1.01108	0.0442115	0.0221057	−	0.999756i	$-0.492963\pi$
$523$	1.01108	0.0442115	0.0221057	+	0.999756i	$0.492963\pi$
$524$	−4.07061	−0.177826
$525$	0	0
$526$	−18.9308	−0.825422
$527$	23.1980	1.01052
$528$	3.00158	0.130627
$529$	−22.5071	−0.978569
$530$	0	0
$531$	2.28842	0.0993091
$532$	−0.397761	−0.0172451
$533$	−28.3861	−1.22954
$534$	1.53050	0.0662311
$535$	0	0
$536$	−9.97007	−0.430641
$537$	−19.8481	−0.856510
$538$	−11.1169	−0.479284
$539$	1.00000	0.0430730
$540$	0	0
$541$	23.7546	1.02129	0.510644	−	0.859792i	$-0.329407\pi$
$541$	23.7546	1.02129	0.510644	+	0.859792i	$0.329407\pi$
$542$	10.3803	0.445874
$543$	16.8727	0.724077
$544$	7.05701	0.302567
$545$	0	0
$546$	7.23416	0.309593
$547$	−40.5272	−1.73282	−0.866409	−	0.499335i	$-0.833578\pi$
$547$	−40.5272	−1.73282	−0.866409	+	0.499335i	$0.833578\pi$
$548$	5.16607	0.220683
$549$	6.30900	0.269262
$550$	0	0
$551$	8.13608	0.346609
$552$	2.13432	0.0908426
$553$	11.5575	0.491474
$554$	31.4888	1.33783
$555$	0	0
$556$	1.64183	0.0696290
$557$	25.3522	1.07421	0.537103	−	0.843517i	$-0.319519\pi$
$557$	25.3522	1.07421	0.537103	+	0.843517i	$0.319519\pi$
$558$	9.52059	0.403039
$559$	−65.9611	−2.78986
$560$	0	0
$561$	3.06892	0.129570
$562$	25.6635	1.08255
$563$	29.2630	1.23329	0.616645	−	0.787241i	$-0.288491\pi$
$563$	29.2630	1.23329	0.616645	+	0.787241i	$0.288491\pi$
$564$	2.37196	0.0998775
$565$	0	0
$566$	30.8672	1.29744
$567$	1.00000	0.0419961
$568$	−35.6177	−1.49448
$569$	−31.6036	−1.32489	−0.662446	−	0.749110i	$-0.730481\pi$
$569$	−31.6036	−1.32489	−0.662446	+	0.749110i	$0.730481\pi$
$570$	0	0
$571$	−17.0240	−0.712431	−0.356216	−	0.934404i	$-0.615933\pi$
$571$	−17.0240	−0.712431	−0.356216	+	0.934404i	$0.615933\pi$
$572$	−2.37589	−0.0993409
$573$	19.3054	0.806493
$574$	6.22466	0.259812
$575$	0	0
$576$	8.89940	0.370808
$577$	−11.0252	−0.458986	−0.229493	−	0.973310i	$-0.573707\pi$
$577$	−11.0252	−0.458986	−0.229493	+	0.973310i	$0.573707\pi$
$578$	9.54917	0.397193
$579$	−19.8165	−0.823544
$580$	0	0
$581$	2.28689	0.0948761
$582$	−17.9349	−0.743425
$583$	−9.65532	−0.399882
$584$	−28.1357	−1.16426
$585$	0	0
$586$	−11.5880	−0.478695
$587$	−27.6375	−1.14072	−0.570362	−	0.821394i	$-0.693197\pi$
$587$	−27.6375	−1.14072	−0.570362	+	0.821394i	$0.693197\pi$
$588$	0.413654	0.0170588
$589$	−7.26859	−0.299497
$590$	0	0
$591$	6.23850	0.256617
$592$	25.7821	1.05964
$593$	2.56176	0.105199	0.0525995	−	0.998616i	$-0.483249\pi$
$593$	2.56176	0.105199	0.0525995	+	0.998616i	$0.483249\pi$
$594$	1.25950	0.0516780
$595$	0	0
$596$	−2.94803	−0.120756
$597$	6.71470	0.274814
$598$	5.07894	0.207693
$599$	−17.9244	−0.732373	−0.366186	−	0.930542i	$-0.619337\pi$
$599$	−17.9244	−0.732373	−0.366186	+	0.930542i	$0.619337\pi$
$600$	0	0
$601$	11.3955	0.464833	0.232417	−	0.972616i	$-0.425337\pi$
$601$	11.3955	0.464833	0.232417	+	0.972616i	$0.425337\pi$
$602$	14.4643	0.589522
$603$	−3.27963	−0.133557
$604$	−6.34411	−0.258138
$605$	0	0
$606$	−3.68173	−0.149560
$607$	−31.2884	−1.26996	−0.634979	−	0.772529i	$-0.718991\pi$
$607$	−31.2884	−1.26996	−0.634979	+	0.772529i	$0.718991\pi$
$608$	−2.21116	−0.0896743
$609$	−8.46116	−0.342864
$610$	0	0
$611$	32.9351	1.33241
$612$	1.26947	0.0513154
$613$	21.1455	0.854058	0.427029	−	0.904238i	$-0.359560\pi$
$613$	21.1455	0.854058	0.427029	+	0.904238i	$0.359560\pi$
$614$	−13.8715	−0.559808
$615$	0	0
$616$	3.04000	0.122485
$617$	22.3393	0.899347	0.449673	−	0.893193i	$-0.351540\pi$
$617$	22.3393	0.899347	0.449673	+	0.893193i	$0.351540\pi$
$618$	−4.78519	−0.192488
$619$	−38.6140	−1.55203	−0.776014	−	0.630715i	$-0.782762\pi$
$619$	−38.6140	−1.55203	−0.776014	+	0.630715i	$0.782762\pi$
$620$	0	0
$621$	0.702078	0.0281734
$622$	2.73090	0.109499
$623$	1.21516	0.0486843
$624$	17.2401	0.690156
$625$	0	0
$626$	−34.3192	−1.37167
$627$	−0.961580	−0.0384018
$628$	5.15543	0.205724
$629$	26.3606	1.05106
$630$	0	0
$631$	5.61959	0.223712	0.111856	−	0.993724i	$-0.464320\pi$
$631$	5.61959	0.223712	0.111856	+	0.993724i	$0.464320\pi$
$632$	35.1347	1.39758
$633$	0.692990	0.0275439
$634$	12.1825	0.483830
$635$	0	0
$636$	−3.99396	−0.158371
$637$	5.74366	0.227572
$638$	−10.6569	−0.421909
$639$	−11.7163	−0.463491
$640$	0	0
$641$	−44.1559	−1.74405	−0.872026	−	0.489460i	$-0.837194\pi$
$641$	−44.1559	−1.74405	−0.872026	+	0.489460i	$0.837194\pi$
$642$	7.46673	0.294688
$643$	33.5713	1.32392	0.661961	−	0.749538i	$-0.269724\pi$
$643$	33.5713	1.32392	0.661961	+	0.749538i	$0.269724\pi$
$644$	0.290417	0.0114440
$645$	0	0
$646$	3.71681	0.146236
$647$	21.8956	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$647$	21.8956	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$648$	3.04000	0.119423
$649$	2.28842	0.0898285
$650$	0	0
$651$	7.55901	0.296261
$652$	3.53709	0.138523
$653$	−32.0519	−1.25429	−0.627144	−	0.778903i	$-0.715776\pi$
$653$	−32.0519	−1.25429	−0.627144	+	0.778903i	$0.715776\pi$
$654$	−14.0282	−0.548546
$655$	0	0
$656$	14.8343	0.579182
$657$	−9.25516	−0.361078
$658$	−7.22219	−0.281550
$659$	46.1799	1.79891	0.899457	−	0.437009i	$-0.143962\pi$
$659$	46.1799	1.79891	0.899457	+	0.437009i	$0.143962\pi$
$660$	0	0
$661$	−34.8906	−1.35709	−0.678544	−	0.734560i	$-0.737389\pi$
$661$	−34.8906	−1.35709	−0.678544	+	0.734560i	$0.737389\pi$
$662$	26.8845	1.04490
$663$	17.6269	0.684571
$664$	6.95214	0.269795
$665$	0	0
$666$	10.8185	0.419209
$667$	−5.94039	−0.230013
$668$	−10.1853	−0.394081
$669$	1.94099	0.0750428
$670$	0	0
$671$	6.30900	0.243556
$672$	2.29951	0.0887053
$673$	−29.1491	−1.12362	−0.561808	−	0.827268i	$-0.689894\pi$
$673$	−29.1491	−1.12362	−0.561808	+	0.827268i	$0.689894\pi$
$674$	12.1594	0.468364
$675$	0	0
$676$	−8.26880	−0.318031
$677$	0.779073	0.0299422	0.0149711	−	0.999888i	$-0.495234\pi$
$677$	0.779073	0.0299422	0.0149711	+	0.999888i	$0.495234\pi$
$678$	−6.92674	−0.266020
$679$	−14.2397	−0.546468
$680$	0	0
$681$	20.4471	0.783534
$682$	9.52059	0.364562
$683$	−0.522170	−0.0199803	−0.00999014	−	0.999950i	$-0.503180\pi$
$683$	−0.522170	−0.0199803	−0.00999014	+	0.999950i	$0.503180\pi$
$684$	−0.397761	−0.0152088
$685$	0	0
$686$	−1.25950	−0.0480880
$687$	10.7814	0.411337
$688$	34.4707	1.31418
$689$	−55.4569	−2.11274
$690$	0	0
$691$	−25.0233	−0.951933	−0.475967	−	0.879463i	$-0.657902\pi$
$691$	−25.0233	−0.951933	−0.475967	+	0.879463i	$0.657902\pi$
$692$	3.26696	0.124191
$693$	1.00000	0.0379869
$694$	8.27968	0.314292
$695$	0	0
$696$	−25.7220	−0.974988
$697$	15.1671	0.574495
$698$	40.9467	1.54985
$699$	−14.0724	−0.532266
$700$	0	0
$701$	−44.9790	−1.69883	−0.849416	−	0.527723i	$-0.823046\pi$
$701$	−44.9790	−1.69883	−0.849416	+	0.527723i	$0.823046\pi$
$702$	7.23416	0.273036
$703$	−8.25950	−0.311513
$704$	8.89940	0.335409
$705$	0	0
$706$	−15.8029	−0.594749
$707$	−2.92316	−0.109937
$708$	0.946615	0.0355760
$709$	24.4310	0.917525	0.458763	−	0.888559i	$-0.348293\pi$
$709$	24.4310	0.917525	0.458763	+	0.888559i	$0.348293\pi$
$710$	0	0
$711$	11.5575	0.433439
$712$	3.69409	0.138442
$713$	5.30701	0.198749
$714$	−3.86532	−0.144656
$715$	0	0
$716$	−8.21026	−0.306832
$717$	−6.31827	−0.235960
$718$	−25.5388	−0.953098
$719$	−6.30432	−0.235111	−0.117556	−	0.993066i	$-0.537506\pi$
$719$	−6.30432	−0.235111	−0.117556	+	0.993066i	$0.537506\pi$
$720$	0	0
$721$	−3.79927	−0.141492
$722$	22.7660	0.847261
$723$	−23.0364	−0.856733
$724$	6.97946	0.259390
$725$	0	0
$726$	1.25950	0.0467445
$727$	−13.4479	−0.498756	−0.249378	−	0.968406i	$-0.580226\pi$
$727$	−13.4479	−0.498756	−0.249378	+	0.968406i	$0.580226\pi$
$728$	17.4608	0.647138
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.2440	1.30355
$732$	2.60974	0.0964589
$733$	−47.1045	−1.73984	−0.869922	−	0.493189i	$-0.835831\pi$
$733$	−47.1045	−1.73984	−0.869922	+	0.493189i	$0.835831\pi$
$734$	22.9265	0.846232
$735$	0	0
$736$	1.61443	0.0595087
$737$	−3.27963	−0.120807
$738$	6.22466	0.229133
$739$	40.8905	1.50418	0.752091	−	0.659059i	$-0.229045\pi$
$739$	40.8905	1.50418	0.752091	+	0.659059i	$0.229045\pi$
$740$	0	0
$741$	−5.52299	−0.202892
$742$	12.1609	0.446440
$743$	14.1538	0.519253	0.259626	−	0.965709i	$-0.416401\pi$
$743$	14.1538	0.519253	0.259626	+	0.965709i	$0.416401\pi$
$744$	22.9794	0.842466
$745$	0	0
$746$	20.9287	0.766252
$747$	2.28689	0.0836728
$748$	1.26947	0.0464165
$749$	5.92832	0.216616
$750$	0	0
$751$	−3.26787	−0.119246	−0.0596232	−	0.998221i	$-0.518990\pi$
$751$	−3.26787	−0.119246	−0.0596232	+	0.998221i	$0.518990\pi$
$752$	−17.2116	−0.627641
$753$	−2.33400	−0.0850555
$754$	−61.2094	−2.22911
$755$	0	0
$756$	0.413654	0.0150444
$757$	36.7171	1.33450	0.667252	−	0.744832i	$-0.267470\pi$
$757$	36.7171	1.33450	0.667252	+	0.744832i	$0.267470\pi$
$758$	11.9483	0.433980
$759$	0.702078	0.0254838
$760$	0	0
$761$	46.6078	1.68953	0.844765	−	0.535138i	$-0.179740\pi$
$761$	46.6078	1.68953	0.844765	+	0.535138i	$0.179740\pi$
$762$	11.5326	0.417780
$763$	−11.1379	−0.403219
$764$	7.98574	0.288914
$765$	0	0
$766$	−2.00715	−0.0725214
$767$	13.1439	0.474600
$768$	9.47373	0.341854
$769$	−26.4744	−0.954690	−0.477345	−	0.878716i	$-0.658401\pi$
$769$	−26.4744	−0.954690	−0.477345	+	0.878716i	$0.658401\pi$
$770$	0	0
$771$	25.4344	0.915996
$772$	−8.19715	−0.295022
$773$	−44.9624	−1.61719	−0.808593	−	0.588368i	$-0.799771\pi$
$773$	−44.9624	−1.61719	−0.808593	+	0.588368i	$0.799771\pi$
$774$	14.4643	0.519910
$775$	0	0
$776$	−43.2886	−1.55397
$777$	8.58951	0.308147
$778$	−23.0711	−0.827139
$779$	−4.75228	−0.170268
$780$	0	0
$781$	−11.7163	−0.419243
$782$	−2.71375	−0.0970436
$783$	−8.46116	−0.302377
$784$	−3.00158	−0.107199
$785$	0	0
$786$	12.3943	0.442090
$787$	33.8383	1.20620	0.603102	−	0.797664i	$-0.293931\pi$
$787$	33.8383	1.20620	0.603102	+	0.797664i	$0.293931\pi$
$788$	2.58058	0.0919293
$789$	−15.0304	−0.535096
$790$	0	0
$791$	−5.49958	−0.195543
$792$	3.04000	0.108022
$793$	36.2368	1.28681
$794$	28.1078	0.997510
$795$	0	0
$796$	2.77756	0.0984480
$797$	−16.0818	−0.569648	−0.284824	−	0.958580i	$-0.591935\pi$
$797$	−16.0818	−0.569648	−0.284824	+	0.958580i	$0.591935\pi$
$798$	1.21111	0.0428729
$799$	−17.5977	−0.622562
$800$	0	0
$801$	1.21516	0.0429356
$802$	−24.5010	−0.865162
$803$	−9.25516	−0.326608
$804$	−1.35663	−0.0478446
$805$	0	0
$806$	54.6830	1.92613
$807$	−8.82643	−0.310705
$808$	−8.88641	−0.312623
$809$	14.7633	0.519049	0.259524	−	0.965737i	$-0.416434\pi$
$809$	14.7633	0.519049	0.259524	+	0.965737i	$0.416434\pi$
$810$	0	0
$811$	17.8144	0.625548	0.312774	−	0.949828i	$-0.398742\pi$
$811$	17.8144	0.625548	0.312774	+	0.949828i	$0.398742\pi$
$812$	−3.49999	−0.122826
$813$	8.24162	0.289046
$814$	10.8185	0.379189
$815$	0	0
$816$	−9.21163	−0.322472
$817$	−11.0429	−0.386344
$818$	−2.35388	−0.0823013
$819$	5.74366	0.200700
$820$	0	0
$821$	−5.54856	−0.193646	−0.0968231	−	0.995302i	$-0.530868\pi$
$821$	−5.54856	−0.193646	−0.0968231	+	0.995302i	$0.530868\pi$
$822$	−15.7298	−0.548638
$823$	39.4708	1.37586	0.687932	−	0.725775i	$-0.258519\pi$
$823$	39.4708	1.37586	0.687932	+	0.725775i	$0.258519\pi$
$824$	−11.5498	−0.402356
$825$	0	0
$826$	−2.88228	−0.100287
$827$	−48.9035	−1.70054	−0.850270	−	0.526347i	$-0.823561\pi$
$827$	−48.9035	−1.70054	−0.850270	+	0.526347i	$0.823561\pi$
$828$	0.290417	0.0100927
$829$	4.59690	0.159657	0.0798284	−	0.996809i	$-0.474563\pi$
$829$	4.59690	0.159657	0.0798284	+	0.996809i	$0.474563\pi$
$830$	0	0
$831$	25.0010	0.867275
$832$	51.1151	1.77210
$833$	−3.06892	−0.106332
$834$	−4.99907	−0.173104
$835$	0	0
$836$	−0.397761	−0.0137569
$837$	7.55901	0.261278
$838$	−4.59958	−0.158890
$839$	−10.2800	−0.354906	−0.177453	−	0.984129i	$-0.556786\pi$
$839$	−10.2800	−0.354906	−0.177453	+	0.984129i	$0.556786\pi$
$840$	0	0
$841$	42.5913	1.46866
$842$	−2.02601	−0.0698210
$843$	20.3759	0.701782
$844$	0.286658	0.00986717
$845$	0	0
$846$	−7.22219	−0.248304
$847$	1.00000	0.0343604
$848$	28.9812	0.995220
$849$	24.5074	0.841093
$850$	0	0
$851$	6.03050	0.206723
$852$	−4.84650	−0.166038
$853$	32.0457	1.09722	0.548612	−	0.836077i	$-0.315157\pi$
$853$	32.0457	1.09722	0.548612	+	0.836077i	$0.315157\pi$
$854$	−7.94620	−0.271914
$855$	0	0
$856$	18.0221	0.615983
$857$	−52.8671	−1.80590	−0.902952	−	0.429741i	$-0.858605\pi$
$857$	−52.8671	−1.80590	−0.902952	+	0.429741i	$0.858605\pi$
$858$	7.23416	0.246970
$859$	11.5954	0.395628	0.197814	−	0.980240i	$-0.436616\pi$
$859$	11.5954	0.395628	0.197814	+	0.980240i	$0.436616\pi$
$860$	0	0
$861$	4.94216	0.168428
$862$	21.0797	0.717977
$863$	31.0903	1.05832	0.529162	−	0.848521i	$-0.322506\pi$
$863$	31.0903	1.05832	0.529162	+	0.848521i	$0.322506\pi$
$864$	2.29951	0.0782307
$865$	0	0
$866$	−34.1282	−1.15972
$867$	7.58170	0.257488
$868$	3.12681	0.106131
$869$	11.5575	0.392060
$870$	0	0
$871$	−18.8371	−0.638270
$872$	−33.8592	−1.14662
$873$	−14.2397	−0.481940
$874$	0.850295	0.0287617
$875$	0	0
$876$	−3.82843	−0.129351
$877$	37.6411	1.27105	0.635525	−	0.772080i	$-0.280784\pi$
$877$	37.6411	1.27105	0.635525	+	0.772080i	$0.280784\pi$
$878$	6.97971	0.235554
$879$	−9.20044	−0.310323
$880$	0	0
$881$	35.5714	1.19843	0.599216	−	0.800587i	$-0.295479\pi$
$881$	35.5714	1.19843	0.599216	+	0.800587i	$0.295479\pi$
$882$	−1.25950	−0.0424096
$883$	−46.3469	−1.55970	−0.779849	−	0.625967i	$-0.784704\pi$
$883$	−46.3469	−1.55970	−0.779849	+	0.625967i	$0.784704\pi$
$884$	7.29142	0.245237
$885$	0	0
$886$	−28.9150	−0.971420
$887$	35.8787	1.20469	0.602345	−	0.798236i	$-0.294233\pi$
$887$	35.8787	1.20469	0.602345	+	0.798236i	$0.294233\pi$
$888$	26.1121	0.876266
$889$	9.15643	0.307097
$890$	0	0
$891$	1.00000	0.0335013
$892$	0.802896	0.0268829
$893$	5.51386	0.184514
$894$	8.97624	0.300210
$895$	0	0
$896$	−6.60980	−0.220818
$897$	4.03250	0.134641
$898$	−41.9825	−1.40097
$899$	−63.9580	−2.13312
$900$	0	0
$901$	29.6314	0.987166
$902$	6.22466	0.207258
$903$	11.4842	0.382169
$904$	−16.7187	−0.556057
$905$	0	0
$906$	19.3167	0.641753
$907$	19.2823	0.640259	0.320129	−	0.947374i	$-0.396274\pi$
$907$	19.2823	0.640259	0.320129	+	0.947374i	$0.396274\pi$
$908$	8.45801	0.280689
$909$	−2.92316	−0.0969551
$910$	0	0
$911$	−44.9754	−1.49010	−0.745051	−	0.667007i	$-0.767575\pi$
$911$	−44.9754	−1.49010	−0.745051	+	0.667007i	$0.767575\pi$
$912$	2.88626	0.0955737
$913$	2.28689	0.0756849
$914$	6.38330	0.211141
$915$	0	0
$916$	4.45978	0.147355
$917$	9.84063	0.324966
$918$	−3.86532	−0.127574
$919$	−23.9887	−0.791313	−0.395657	−	0.918398i	$-0.629483\pi$
$919$	−23.9887	−0.791313	−0.395657	+	0.918398i	$0.629483\pi$
$920$	0	0
$921$	−11.0135	−0.362906
$922$	−40.8947	−1.34680
$923$	−67.2946	−2.21503
$924$	0.413654	0.0136082
$925$	0	0
$926$	46.2002	1.51823
$927$	−3.79927	−0.124784
$928$	−19.4565	−0.638690
$929$	−20.6524	−0.677583	−0.338791	−	0.940862i	$-0.610018\pi$
$929$	−20.6524	−0.677583	−0.338791	+	0.940862i	$0.610018\pi$
$930$	0	0
$931$	0.961580	0.0315145
$932$	−5.82109	−0.190676
$933$	2.16823	0.0709848
$934$	12.6623	0.414324
$935$	0	0
$936$	17.4608	0.570723
$937$	12.9517	0.423115	0.211557	−	0.977366i	$-0.432146\pi$
$937$	12.9517	0.423115	0.211557	+	0.977366i	$0.432146\pi$
$938$	4.13070	0.134872
$939$	−27.2482	−0.889213
$940$	0	0
$941$	15.8348	0.516198	0.258099	−	0.966118i	$-0.416904\pi$
$941$	15.8348	0.516198	0.258099	+	0.966118i	$0.416904\pi$
$942$	−15.6974	−0.511448
$943$	3.46978	0.112992
$944$	−6.86889	−0.223563
$945$	0	0
$946$	14.4643	0.470276
$947$	36.9824	1.20177	0.600883	−	0.799337i	$-0.294816\pi$
$947$	36.9824	1.20177	0.600883	+	0.799337i	$0.294816\pi$
$948$	4.78079	0.155273
$949$	−53.1585	−1.72560
$950$	0	0
$951$	9.67250	0.313652
$952$	−9.32954	−0.302372
$953$	23.7461	0.769212	0.384606	−	0.923081i	$-0.374337\pi$
$953$	23.7461	0.769212	0.384606	+	0.923081i	$0.374337\pi$
$954$	12.1609	0.393723
$955$	0	0
$956$	−2.61358	−0.0845292
$957$	−8.46116	−0.273511
$958$	−9.57306	−0.309291
$959$	−12.4889	−0.403286
$960$	0	0
$961$	26.1386	0.843180
$962$	62.1379	2.00341
$963$	5.92832	0.191037
$964$	−9.52910	−0.306912
$965$	0	0
$966$	−0.884268	−0.0284509
$967$	−48.8360	−1.57046	−0.785230	−	0.619204i	$-0.787455\pi$
$967$	−48.8360	−1.57046	−0.785230	+	0.619204i	$0.787455\pi$
$968$	3.04000	0.0977094
$969$	2.95102	0.0948003
$970$	0	0
$971$	−38.0822	−1.22212	−0.611058	−	0.791586i	$-0.709256\pi$
$971$	−38.0822	−1.22212	−0.611058	+	0.791586i	$0.709256\pi$
$972$	0.413654	0.0132680
$973$	−3.96909	−0.127243
$974$	−29.6821	−0.951075
$975$	0	0
$976$	−18.9370	−0.606158
$977$	−34.5195	−1.10438	−0.552188	−	0.833719i	$-0.686207\pi$
$977$	−34.5195	−1.10438	−0.552188	+	0.833719i	$0.686207\pi$
$978$	−10.7698	−0.344381
$979$	1.21516	0.0388367
$980$	0	0
$981$	−11.1379	−0.355606
$982$	−46.9930	−1.49961
$983$	−16.8797	−0.538379	−0.269189	−	0.963087i	$-0.586756\pi$
$983$	−16.8797	−0.538379	−0.269189	+	0.963087i	$0.586756\pi$
$984$	15.0242	0.478953
$985$	0	0
$986$	32.7051	1.04154
$987$	−5.73416	−0.182520
$988$	−2.28461	−0.0726830
$989$	8.06277	0.256381
$990$	0	0
$991$	43.7980	1.39129	0.695644	−	0.718386i	$-0.255119\pi$
$991$	43.7980	1.39129	0.695644	+	0.718386i	$0.255119\pi$
$992$	17.3820	0.551878
$993$	21.3454	0.677375
$994$	14.7567	0.468055
$995$	0	0
$996$	0.945980	0.0299745
$997$	−51.3827	−1.62731	−0.813654	−	0.581350i	$-0.802525\pi$
$997$	−51.3827	−1.62731	−0.813654	+	0.581350i	$0.802525\pi$
$998$	3.78062	0.119673
$999$	8.58951	0.271760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.ch.1.2	✓	5
5.4	even	2		5775.2.a.ci.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5775.2.a.ch.1.2	✓	5	1.1	even	1	trivial
5775.2.a.ci.1.4	yes	5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25950 q^{2} -1.00000 q^{3} -0.413654 q^{4} +1.25950 q^{6} +1.00000 q^{7} +3.04000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.25950 q^{2} -1.00000 q^{3} -0.413654 q^{4} +1.25950 q^{6} +1.00000 q^{7} +3.04000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +0.413654 q^{12} +5.74366 q^{13} -1.25950 q^{14} -3.00158 q^{16} -3.06892 q^{17} -1.25950 q^{18} +0.961580 q^{19} -1.00000 q^{21} -1.25950 q^{22} -0.702078 q^{23} -3.04000 q^{24} -7.23416 q^{26} -1.00000 q^{27} -0.413654 q^{28} +8.46116 q^{29} -7.55901 q^{31} -2.29951 q^{32} -1.00000 q^{33} +3.86532 q^{34} -0.413654 q^{36} -8.58951 q^{37} -1.21111 q^{38} -5.74366 q^{39} -4.94216 q^{41} +1.25950 q^{42} -11.4842 q^{43} -0.413654 q^{44} +0.884268 q^{46} +5.73416 q^{47} +3.00158 q^{48} +1.00000 q^{49} +3.06892 q^{51} -2.37589 q^{52} -9.65532 q^{53} +1.25950 q^{54} +3.04000 q^{56} -0.961580 q^{57} -10.6569 q^{58} +2.28842 q^{59} +6.30900 q^{61} +9.52059 q^{62} +1.00000 q^{63} +8.89940 q^{64} +1.25950 q^{66} -3.27963 q^{67} +1.26947 q^{68} +0.702078 q^{69} -11.7163 q^{71} +3.04000 q^{72} -9.25516 q^{73} +10.8185 q^{74} -0.397761 q^{76} +1.00000 q^{77} +7.23416 q^{78} +11.5575 q^{79} +1.00000 q^{81} +6.22466 q^{82} +2.28689 q^{83} +0.413654 q^{84} +14.4643 q^{86} -8.46116 q^{87} +3.04000 q^{88} +1.21516 q^{89} +5.74366 q^{91} +0.290417 q^{92} +7.55901 q^{93} -7.22219 q^{94} +2.29951 q^{96} -14.2397 q^{97} -1.25950 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 4 q^{4} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 4 * q^4 + 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 4 q^{4} + 5 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{12} - 7 q^{13} + 6 q^{16} - 13 q^{17} - q^{19} - 5 q^{21} - 4 q^{23} - 6 q^{26} - 5 q^{27} + 4 q^{28} + 4 q^{29} - 10 q^{31} + 10 q^{32} - 5 q^{33} - 18 q^{34} + 4 q^{36} - 7 q^{37} - 24 q^{38} + 7 q^{39} + q^{41} - 28 q^{43} + 4 q^{44} + 10 q^{46} - 6 q^{48} + 5 q^{49} + 13 q^{51} - 28 q^{52} + 5 q^{53} + q^{57} - 36 q^{58} + 18 q^{59} + 3 q^{61} + 14 q^{62} + 5 q^{63} - 12 q^{64} - q^{67} - 30 q^{68} + 4 q^{69} + 11 q^{71} - 15 q^{73} + 20 q^{74} - 22 q^{76} + 5 q^{77} + 6 q^{78} - 6 q^{79} + 5 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{84} - 8 q^{86} - 4 q^{87} - 10 q^{89} - 7 q^{91} + 18 q^{92} + 10 q^{93} - 26 q^{94} - 10 q^{96} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 4 * q^4 + 5 * q^7 + 5 * q^9 + 5 * q^11 - 4 * q^12 - 7 * q^13 + 6 * q^16 - 13 * q^17 - q^19 - 5 * q^21 - 4 * q^23 - 6 * q^26 - 5 * q^27 + 4 * q^28 + 4 * q^29 - 10 * q^31 + 10 * q^32 - 5 * q^33 - 18 * q^34 + 4 * q^36 - 7 * q^37 - 24 * q^38 + 7 * q^39 + q^41 - 28 * q^43 + 4 * q^44 + 10 * q^46 - 6 * q^48 + 5 * q^49 + 13 * q^51 - 28 * q^52 + 5 * q^53 + q^57 - 36 * q^58 + 18 * q^59 + 3 * q^61 + 14 * q^62 + 5 * q^63 - 12 * q^64 - q^67 - 30 * q^68 + 4 * q^69 + 11 * q^71 - 15 * q^73 + 20 * q^74 - 22 * q^76 + 5 * q^77 + 6 * q^78 - 6 * q^79 + 5 * q^81 + 8 * q^82 - 18 * q^83 - 4 * q^84 - 8 * q^86 - 4 * q^87 - 10 * q^89 - 7 * q^91 + 18 * q^92 + 10 * q^93 - 26 * q^94 - 10 * q^96 - 14 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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