LMFDB - Embedded newform 1078.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +0.585786 q^{5} +2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} +0.585786 q^{10} +1.00000 q^{11} +2.41421 q^{12} +3.82843 q^{13} +1.41421 q^{15} +1.00000 q^{16} -3.65685 q^{17} +2.82843 q^{18} +0.585786 q^{19} +0.585786 q^{20} +1.00000 q^{22} -6.24264 q^{23} +2.41421 q^{24} -4.65685 q^{25} +3.82843 q^{26} -0.414214 q^{27} +2.65685 q^{29} +1.41421 q^{30} +4.00000 q^{31} +1.00000 q^{32} +2.41421 q^{33} -3.65685 q^{34} +2.82843 q^{36} -9.41421 q^{37} +0.585786 q^{38} +9.24264 q^{39} +0.585786 q^{40} +5.41421 q^{41} -5.65685 q^{43} +1.00000 q^{44} +1.65685 q^{45} -6.24264 q^{46} +10.4853 q^{47} +2.41421 q^{48} -4.65685 q^{50} -8.82843 q^{51} +3.82843 q^{52} +7.89949 q^{53} -0.414214 q^{54} +0.585786 q^{55} +1.41421 q^{57} +2.65685 q^{58} +5.58579 q^{59} +1.41421 q^{60} -11.8284 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.24264 q^{65} +2.41421 q^{66} +2.75736 q^{67} -3.65685 q^{68} -15.0711 q^{69} -11.0711 q^{71} +2.82843 q^{72} +9.41421 q^{73} -9.41421 q^{74} -11.2426 q^{75} +0.585786 q^{76} +9.24264 q^{78} -13.2426 q^{79} +0.585786 q^{80} -9.48528 q^{81} +5.41421 q^{82} +12.1421 q^{83} -2.14214 q^{85} -5.65685 q^{86} +6.41421 q^{87} +1.00000 q^{88} -12.4853 q^{89} +1.65685 q^{90} -6.24264 q^{92} +9.65685 q^{93} +10.4853 q^{94} +0.343146 q^{95} +2.41421 q^{96} +3.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 4 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 - 6 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^33 + 4 * q^34 - 16 * q^37 + 4 * q^38 + 10 * q^39 + 4 * q^40 + 8 * q^41 + 2 * q^44 - 8 * q^45 - 4 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^58 + 14 * q^59 - 18 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 + 2 * q^66 + 14 * q^67 + 4 * q^68 - 16 * q^69 - 8 * q^71 + 16 * q^73 - 16 * q^74 - 14 * q^75 + 4 * q^76 + 10 * q^78 - 18 * q^79 + 4 * q^80 - 2 * q^81 + 8 * q^82 - 4 * q^83 + 24 * q^85 + 10 * q^87 + 2 * q^88 - 8 * q^89 - 8 * q^90 - 4 * q^92 + 8 * q^93 + 4 * q^94 + 12 * q^95 + 2 * q^96 + 2 * q^97

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	1.00000	0.500000
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	2.41421	0.985599
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	2.82843	0.942809
$10$	0.585786	0.185242
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.41421	0.696923
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	1.00000	0.250000
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	2.82843	0.666667
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	0.585786	0.130986
$21$	0	0
$22$	1.00000	0.213201
$23$	−6.24264	−1.30168	−0.650840	−	0.759215i	$-0.725583\pi$
$23$	−6.24264	−1.30168	−0.650840	+	0.759215i	$0.725583\pi$
$24$	2.41421	0.492799
$25$	−4.65685	−0.931371
$26$	3.82843	0.750816
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	2.65685	0.493365	0.246683	−	0.969096i	$-0.420659\pi$
$29$	2.65685	0.493365	0.246683	+	0.969096i	$0.420659\pi$
$30$	1.41421	0.258199
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	2.41421	0.420261
$34$	−3.65685	−0.627145
$35$	0	0
$36$	2.82843	0.471405
$37$	−9.41421	−1.54769	−0.773844	−	0.633377i	$-0.781668\pi$
$37$	−9.41421	−1.54769	−0.773844	+	0.633377i	$0.781668\pi$
$38$	0.585786	0.0950271
$39$	9.24264	1.48001
$40$	0.585786	0.0926210
$41$	5.41421	0.845558	0.422779	−	0.906233i	$-0.361055\pi$
$41$	5.41421	0.845558	0.422779	+	0.906233i	$0.361055\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	1.00000	0.150756
$45$	1.65685	0.246989
$46$	−6.24264	−0.920427
$47$	10.4853	1.52944	0.764718	−	0.644365i	$-0.222878\pi$
$47$	10.4853	1.52944	0.764718	+	0.644365i	$0.222878\pi$
$48$	2.41421	0.348462
$49$	0	0
$50$	−4.65685	−0.658579
$51$	−8.82843	−1.23623
$52$	3.82843	0.530907
$53$	7.89949	1.08508	0.542540	−	0.840030i	$-0.317463\pi$
$53$	7.89949	1.08508	0.542540	+	0.840030i	$0.317463\pi$
$54$	−0.414214	−0.0563673
$55$	0.585786	0.0789874
$56$	0	0
$57$	1.41421	0.187317
$58$	2.65685	0.348862
$59$	5.58579	0.727207	0.363604	−	0.931554i	$-0.381546\pi$
$59$	5.58579	0.727207	0.363604	+	0.931554i	$0.381546\pi$
$60$	1.41421	0.182574
$61$	−11.8284	−1.51447	−0.757237	−	0.653140i	$-0.773451\pi$
$61$	−11.8284	−1.51447	−0.757237	+	0.653140i	$0.773451\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.24264	0.278165
$66$	2.41421	0.297169
$67$	2.75736	0.336865	0.168433	−	0.985713i	$-0.446129\pi$
$67$	2.75736	0.336865	0.168433	+	0.985713i	$0.446129\pi$
$68$	−3.65685	−0.443459
$69$	−15.0711	−1.81434
$70$	0	0
$71$	−11.0711	−1.31389	−0.656947	−	0.753937i	$-0.728152\pi$
$71$	−11.0711	−1.31389	−0.656947	+	0.753937i	$0.728152\pi$
$72$	2.82843	0.333333
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	−9.41421	−1.09438
$75$	−11.2426	−1.29819
$76$	0.585786	0.0671943
$77$	0	0
$78$	9.24264	1.04652
$79$	−13.2426	−1.48991	−0.744957	−	0.667113i	$-0.767530\pi$
$79$	−13.2426	−1.48991	−0.744957	+	0.667113i	$0.767530\pi$
$80$	0.585786	0.0654929
$81$	−9.48528	−1.05392
$82$	5.41421	0.597900
$83$	12.1421	1.33277	0.666386	−	0.745607i	$-0.267840\pi$
$83$	12.1421	1.33277	0.666386	+	0.745607i	$0.267840\pi$
$84$	0	0
$85$	−2.14214	−0.232347
$86$	−5.65685	−0.609994
$87$	6.41421	0.687676
$88$	1.00000	0.106600
$89$	−12.4853	−1.32344	−0.661719	−	0.749752i	$-0.730173\pi$
$89$	−12.4853	−1.32344	−0.661719	+	0.749752i	$0.730173\pi$
$90$	1.65685	0.174648
$91$	0	0
$92$	−6.24264	−0.650840
$93$	9.65685	1.00137
$94$	10.4853	1.08147
$95$	0.343146	0.0352060
$96$	2.41421	0.246400
$97$	3.82843	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$97$	3.82843	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	−4.65685	−0.465685
$101$	−6.17157	−0.614094	−0.307047	−	0.951694i	$-0.599341\pi$
$101$	−6.17157	−0.614094	−0.307047	+	0.951694i	$0.599341\pi$
$102$	−8.82843	−0.874145
$103$	−13.4142	−1.32174	−0.660871	−	0.750500i	$-0.729813\pi$
$103$	−13.4142	−1.32174	−0.660871	+	0.750500i	$0.729813\pi$
$104$	3.82843	0.375408
$105$	0	0
$106$	7.89949	0.767267
$107$	−3.07107	−0.296891	−0.148446	−	0.988921i	$-0.547427\pi$
$107$	−3.07107	−0.296891	−0.148446	+	0.988921i	$0.547427\pi$
$108$	−0.414214	−0.0398577
$109$	16.4853	1.57900	0.789502	−	0.613748i	$-0.210339\pi$
$109$	16.4853	1.57900	0.789502	+	0.613748i	$0.210339\pi$
$110$	0.585786	0.0558525
$111$	−22.7279	−2.15724
$112$	0	0
$113$	−8.17157	−0.768717	−0.384358	−	0.923184i	$-0.625577\pi$
$113$	−8.17157	−0.768717	−0.384358	+	0.923184i	$0.625577\pi$
$114$	1.41421	0.132453
$115$	−3.65685	−0.341003
$116$	2.65685	0.246683
$117$	10.8284	1.00109
$118$	5.58579	0.514213
$119$	0	0
$120$	1.41421	0.129099
$121$	1.00000	0.0909091
$122$	−11.8284	−1.07090
$123$	13.0711	1.17858
$124$	4.00000	0.359211
$125$	−5.65685	−0.505964
$126$	0	0
$127$	15.7279	1.39563	0.697814	−	0.716279i	$-0.254156\pi$
$127$	15.7279	1.39563	0.697814	+	0.716279i	$0.254156\pi$
$128$	1.00000	0.0883883
$129$	−13.6569	−1.20242
$130$	2.24264	0.196693
$131$	0.585786	0.0511804	0.0255902	−	0.999673i	$-0.491853\pi$
$131$	0.585786	0.0511804	0.0255902	+	0.999673i	$0.491853\pi$
$132$	2.41421	0.210130
$133$	0	0
$134$	2.75736	0.238200
$135$	−0.242641	−0.0208832
$136$	−3.65685	−0.313573
$137$	−16.6569	−1.42309	−0.711546	−	0.702640i	$-0.752004\pi$
$137$	−16.6569	−1.42309	−0.711546	+	0.702640i	$0.752004\pi$
$138$	−15.0711	−1.28293
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	25.3137	2.13180
$142$	−11.0711	−0.929063
$143$	3.82843	0.320149
$144$	2.82843	0.235702
$145$	1.55635	0.129248
$146$	9.41421	0.779126
$147$	0	0
$148$	−9.41421	−0.773844
$149$	17.6569	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$149$	17.6569	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$150$	−11.2426	−0.917958
$151$	−15.7279	−1.27992	−0.639960	−	0.768408i	$-0.721049\pi$
$151$	−15.7279	−1.27992	−0.639960	+	0.768408i	$0.721049\pi$
$152$	0.585786	0.0475136
$153$	−10.3431	−0.836194
$154$	0	0
$155$	2.34315	0.188206
$156$	9.24264	0.740003
$157$	−17.6569	−1.40917	−0.704585	−	0.709619i	$-0.748867\pi$
$157$	−17.6569	−1.40917	−0.704585	+	0.709619i	$0.748867\pi$
$158$	−13.2426	−1.05353
$159$	19.0711	1.51243
$160$	0.585786	0.0463105
$161$	0	0
$162$	−9.48528	−0.745234
$163$	9.72792	0.761950	0.380975	−	0.924585i	$-0.375588\pi$
$163$	9.72792	0.761950	0.380975	+	0.924585i	$0.375588\pi$
$164$	5.41421	0.422779
$165$	1.41421	0.110096
$166$	12.1421	0.942412
$167$	−13.7279	−1.06230	−0.531149	−	0.847278i	$-0.678240\pi$
$167$	−13.7279	−1.06230	−0.531149	+	0.847278i	$0.678240\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	−2.14214	−0.164294
$171$	1.65685	0.126703
$172$	−5.65685	−0.431331
$173$	9.82843	0.747241	0.373621	−	0.927582i	$-0.378116\pi$
$173$	9.82843	0.747241	0.373621	+	0.927582i	$0.378116\pi$
$174$	6.41421	0.486260
$175$	0	0
$176$	1.00000	0.0753778
$177$	13.4853	1.01362
$178$	−12.4853	−0.935811
$179$	0.899495	0.0672314	0.0336157	−	0.999435i	$-0.489298\pi$
$179$	0.899495	0.0672314	0.0336157	+	0.999435i	$0.489298\pi$
$180$	1.65685	0.123495
$181$	7.65685	0.569129	0.284565	−	0.958657i	$-0.408151\pi$
$181$	7.65685	0.569129	0.284565	+	0.958657i	$0.408151\pi$
$182$	0	0
$183$	−28.5563	−2.11095
$184$	−6.24264	−0.460214
$185$	−5.51472	−0.405450
$186$	9.65685	0.708075
$187$	−3.65685	−0.267416
$188$	10.4853	0.764718
$189$	0	0
$190$	0.343146	0.0248944
$191$	−7.17157	−0.518917	−0.259458	−	0.965754i	$-0.583544\pi$
$191$	−7.17157	−0.518917	−0.259458	+	0.965754i	$0.583544\pi$
$192$	2.41421	0.174231
$193$	21.8995	1.57636	0.788180	−	0.615445i	$-0.211024\pi$
$193$	21.8995	1.57636	0.788180	+	0.615445i	$0.211024\pi$
$194$	3.82843	0.274865
$195$	5.41421	0.387720
$196$	0	0
$197$	−0.514719	−0.0366722	−0.0183361	−	0.999832i	$-0.505837\pi$
$197$	−0.514719	−0.0366722	−0.0183361	+	0.999832i	$0.505837\pi$
$198$	2.82843	0.201008
$199$	−0.100505	−0.00712462	−0.00356231	−	0.999994i	$-0.501134\pi$
$199$	−0.100505	−0.00712462	−0.00356231	+	0.999994i	$0.501134\pi$
$200$	−4.65685	−0.329289
$201$	6.65685	0.469538
$202$	−6.17157	−0.434230
$203$	0	0
$204$	−8.82843	−0.618114
$205$	3.17157	0.221512
$206$	−13.4142	−0.934613
$207$	−17.6569	−1.22724
$208$	3.82843	0.265454
$209$	0.585786	0.0405197
$210$	0	0
$211$	7.41421	0.510416	0.255208	−	0.966886i	$-0.417856\pi$
$211$	7.41421	0.510416	0.255208	+	0.966886i	$0.417856\pi$
$212$	7.89949	0.542540
$213$	−26.7279	−1.83137
$214$	−3.07107	−0.209934
$215$	−3.31371	−0.225993
$216$	−0.414214	−0.0281837
$217$	0	0
$218$	16.4853	1.11652
$219$	22.7279	1.53581
$220$	0.585786	0.0394937
$221$	−14.0000	−0.941742
$222$	−22.7279	−1.52540
$223$	−8.58579	−0.574947	−0.287473	−	0.957789i	$-0.592815\pi$
$223$	−8.58579	−0.574947	−0.287473	+	0.957789i	$0.592815\pi$
$224$	0	0
$225$	−13.1716	−0.878105
$226$	−8.17157	−0.543565
$227$	28.8284	1.91341	0.956705	−	0.291059i	$-0.0940078\pi$
$227$	28.8284	1.91341	0.956705	+	0.291059i	$0.0940078\pi$
$228$	1.41421	0.0936586
$229$	−23.3137	−1.54061	−0.770307	−	0.637674i	$-0.779897\pi$
$229$	−23.3137	−1.54061	−0.770307	+	0.637674i	$0.779897\pi$
$230$	−3.65685	−0.241126
$231$	0	0
$232$	2.65685	0.174431
$233$	1.41421	0.0926482	0.0463241	−	0.998926i	$-0.485249\pi$
$233$	1.41421	0.0926482	0.0463241	+	0.998926i	$0.485249\pi$
$234$	10.8284	0.707876
$235$	6.14214	0.400669
$236$	5.58579	0.363604
$237$	−31.9706	−2.07671
$238$	0	0
$239$	20.2132	1.30748	0.653742	−	0.756718i	$-0.273198\pi$
$239$	20.2132	1.30748	0.653742	+	0.756718i	$0.273198\pi$
$240$	1.41421	0.0912871
$241$	−12.2426	−0.788618	−0.394309	−	0.918978i	$-0.629016\pi$
$241$	−12.2426	−0.788618	−0.394309	+	0.918978i	$0.629016\pi$
$242$	1.00000	0.0642824
$243$	−21.6569	−1.38929
$244$	−11.8284	−0.757237
$245$	0	0
$246$	13.0711	0.833381
$247$	2.24264	0.142696
$248$	4.00000	0.254000
$249$	29.3137	1.85768
$250$	−5.65685	−0.357771
$251$	26.1421	1.65008	0.825038	−	0.565077i	$-0.191153\pi$
$251$	26.1421	1.65008	0.825038	+	0.565077i	$0.191153\pi$
$252$	0	0
$253$	−6.24264	−0.392471
$254$	15.7279	0.986858
$255$	−5.17157	−0.323856
$256$	1.00000	0.0625000
$257$	25.1421	1.56832	0.784162	−	0.620557i	$-0.213093\pi$
$257$	25.1421	1.56832	0.784162	+	0.620557i	$0.213093\pi$
$258$	−13.6569	−0.850239
$259$	0	0
$260$	2.24264	0.139083
$261$	7.51472	0.465149
$262$	0.585786	0.0361900
$263$	−17.0416	−1.05083	−0.525416	−	0.850845i	$-0.676090\pi$
$263$	−17.0416	−1.05083	−0.525416	+	0.850845i	$0.676090\pi$
$264$	2.41421	0.148585
$265$	4.62742	0.284260
$266$	0	0
$267$	−30.1421	−1.84467
$268$	2.75736	0.168433
$269$	−2.34315	−0.142864	−0.0714321	−	0.997445i	$-0.522757\pi$
$269$	−2.34315	−0.142864	−0.0714321	+	0.997445i	$0.522757\pi$
$270$	−0.242641	−0.0147666
$271$	−4.55635	−0.276779	−0.138389	−	0.990378i	$-0.544192\pi$
$271$	−4.55635	−0.276779	−0.138389	+	0.990378i	$0.544192\pi$
$272$	−3.65685	−0.221729
$273$	0	0
$274$	−16.6569	−1.00628
$275$	−4.65685	−0.280819
$276$	−15.0711	−0.907172
$277$	1.82843	0.109860	0.0549298	−	0.998490i	$-0.482507\pi$
$277$	1.82843	0.109860	0.0549298	+	0.998490i	$0.482507\pi$
$278$	0	0
$279$	11.3137	0.677334
$280$	0	0
$281$	8.72792	0.520664	0.260332	−	0.965519i	$-0.416168\pi$
$281$	8.72792	0.520664	0.260332	+	0.965519i	$0.416168\pi$
$282$	25.3137	1.50741
$283$	23.4142	1.39183	0.695915	−	0.718124i	$-0.254999\pi$
$283$	23.4142	1.39183	0.695915	+	0.718124i	$0.254999\pi$
$284$	−11.0711	−0.656947
$285$	0.828427	0.0490718
$286$	3.82843	0.226380
$287$	0	0
$288$	2.82843	0.166667
$289$	−3.62742	−0.213377
$290$	1.55635	0.0913920
$291$	9.24264	0.541813
$292$	9.41421	0.550925
$293$	10.8284	0.632603	0.316302	−	0.948659i	$-0.397559\pi$
$293$	10.8284	0.632603	0.316302	+	0.948659i	$0.397559\pi$
$294$	0	0
$295$	3.27208	0.190508
$296$	−9.41421	−0.547190
$297$	−0.414214	−0.0240351
$298$	17.6569	1.02283
$299$	−23.8995	−1.38214
$300$	−11.2426	−0.649094
$301$	0	0
$302$	−15.7279	−0.905040
$303$	−14.8995	−0.855954
$304$	0.585786	0.0335972
$305$	−6.92893	−0.396750
$306$	−10.3431	−0.591278
$307$	−9.89949	−0.564994	−0.282497	−	0.959268i	$-0.591163\pi$
$307$	−9.89949	−0.564994	−0.282497	+	0.959268i	$0.591163\pi$
$308$	0	0
$309$	−32.3848	−1.84231
$310$	2.34315	0.133082
$311$	−16.7279	−0.948553	−0.474277	−	0.880376i	$-0.657290\pi$
$311$	−16.7279	−0.948553	−0.474277	+	0.880376i	$0.657290\pi$
$312$	9.24264	0.523261
$313$	20.6569	1.16759	0.583797	−	0.811900i	$-0.301566\pi$
$313$	20.6569	1.16759	0.583797	+	0.811900i	$0.301566\pi$
$314$	−17.6569	−0.996434
$315$	0	0
$316$	−13.2426	−0.744957
$317$	8.68629	0.487871	0.243935	−	0.969791i	$-0.421562\pi$
$317$	8.68629	0.487871	0.243935	+	0.969791i	$0.421562\pi$
$318$	19.0711	1.06945
$319$	2.65685	0.148755
$320$	0.585786	0.0327465
$321$	−7.41421	−0.413821
$322$	0	0
$323$	−2.14214	−0.119192
$324$	−9.48528	−0.526960
$325$	−17.8284	−0.988943
$326$	9.72792	0.538780
$327$	39.7990	2.20089
$328$	5.41421	0.298950
$329$	0	0
$330$	1.41421	0.0778499
$331$	−24.0711	−1.32307	−0.661533	−	0.749916i	$-0.730094\pi$
$331$	−24.0711	−1.32307	−0.661533	+	0.749916i	$0.730094\pi$
$332$	12.1421	0.666386
$333$	−26.6274	−1.45917
$334$	−13.7279	−0.751158
$335$	1.61522	0.0882491
$336$	0	0
$337$	−28.2426	−1.53847	−0.769237	−	0.638963i	$-0.779364\pi$
$337$	−28.2426	−1.53847	−0.769237	+	0.638963i	$0.779364\pi$
$338$	1.65685	0.0901210
$339$	−19.7279	−1.07147
$340$	−2.14214	−0.116174
$341$	4.00000	0.216612
$342$	1.65685	0.0895924
$343$	0	0
$344$	−5.65685	−0.304997
$345$	−8.82843	−0.475307
$346$	9.82843	0.528380
$347$	17.4142	0.934844	0.467422	−	0.884034i	$-0.345183\pi$
$347$	17.4142	0.934844	0.467422	+	0.884034i	$0.345183\pi$
$348$	6.41421	0.343838
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−1.58579	−0.0846430
$352$	1.00000	0.0533002
$353$	2.68629	0.142977	0.0714884	−	0.997441i	$-0.477225\pi$
$353$	2.68629	0.142977	0.0714884	+	0.997441i	$0.477225\pi$
$354$	13.4853	0.716735
$355$	−6.48528	−0.344203
$356$	−12.4853	−0.661719
$357$	0	0
$358$	0.899495	0.0475398
$359$	19.2426	1.01559	0.507794	−	0.861479i	$-0.330461\pi$
$359$	19.2426	1.01559	0.507794	+	0.861479i	$0.330461\pi$
$360$	1.65685	0.0873239
$361$	−18.6569	−0.981940
$362$	7.65685	0.402435
$363$	2.41421	0.126713
$364$	0	0
$365$	5.51472	0.288654
$366$	−28.5563	−1.49266
$367$	10.7279	0.559993	0.279996	−	0.960001i	$-0.409667\pi$
$367$	10.7279	0.559993	0.279996	+	0.960001i	$0.409667\pi$
$368$	−6.24264	−0.325420
$369$	15.3137	0.797200
$370$	−5.51472	−0.286697
$371$	0	0
$372$	9.65685	0.500685
$373$	−19.9706	−1.03404	−0.517018	−	0.855974i	$-0.672958\pi$
$373$	−19.9706	−1.03404	−0.517018	+	0.855974i	$0.672958\pi$
$374$	−3.65685	−0.189091
$375$	−13.6569	−0.705237
$376$	10.4853	0.540737
$377$	10.1716	0.523863
$378$	0	0
$379$	27.8701	1.43159	0.715794	−	0.698311i	$-0.246065\pi$
$379$	27.8701	1.43159	0.715794	+	0.698311i	$0.246065\pi$
$380$	0.343146	0.0176030
$381$	37.9706	1.94529
$382$	−7.17157	−0.366930
$383$	6.38478	0.326247	0.163123	−	0.986606i	$-0.447843\pi$
$383$	6.38478	0.326247	0.163123	+	0.986606i	$0.447843\pi$
$384$	2.41421	0.123200
$385$	0	0
$386$	21.8995	1.11465
$387$	−16.0000	−0.813326
$388$	3.82843	0.194359
$389$	−22.7279	−1.15235	−0.576176	−	0.817326i	$-0.695456\pi$
$389$	−22.7279	−1.15235	−0.576176	+	0.817326i	$0.695456\pi$
$390$	5.41421	0.274159
$391$	22.8284	1.15448
$392$	0	0
$393$	1.41421	0.0713376
$394$	−0.514719	−0.0259311
$395$	−7.75736	−0.390315
$396$	2.82843	0.142134
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−0.100505	−0.00503786
$399$	0	0
$400$	−4.65685	−0.232843
$401$	18.3137	0.914543	0.457271	−	0.889327i	$-0.348827\pi$
$401$	18.3137	0.914543	0.457271	+	0.889327i	$0.348827\pi$
$402$	6.65685	0.332014
$403$	15.3137	0.762830
$404$	−6.17157	−0.307047
$405$	−5.55635	−0.276097
$406$	0	0
$407$	−9.41421	−0.466645
$408$	−8.82843	−0.437072
$409$	−2.72792	−0.134887	−0.0674435	−	0.997723i	$-0.521484\pi$
$409$	−2.72792	−0.134887	−0.0674435	+	0.997723i	$0.521484\pi$
$410$	3.17157	0.156633
$411$	−40.2132	−1.98357
$412$	−13.4142	−0.660871
$413$	0	0
$414$	−17.6569	−0.867787
$415$	7.11270	0.349149
$416$	3.82843	0.187704
$417$	0	0
$418$	0.585786	0.0286518
$419$	26.1421	1.27713	0.638563	−	0.769569i	$-0.279529\pi$
$419$	26.1421	1.27713	0.638563	+	0.769569i	$0.279529\pi$
$420$	0	0
$421$	0.686292	0.0334478	0.0167239	−	0.999860i	$-0.494676\pi$
$421$	0.686292	0.0334478	0.0167239	+	0.999860i	$0.494676\pi$
$422$	7.41421	0.360918
$423$	29.6569	1.44197
$424$	7.89949	0.383633
$425$	17.0294	0.826049
$426$	−26.7279	−1.29497
$427$	0	0
$428$	−3.07107	−0.148446
$429$	9.24264	0.446239
$430$	−3.31371	−0.159801
$431$	−17.5858	−0.847078	−0.423539	−	0.905878i	$-0.639212\pi$
$431$	−17.5858	−0.847078	−0.423539	+	0.905878i	$0.639212\pi$
$432$	−0.414214	−0.0199289
$433$	−26.1421	−1.25631	−0.628155	−	0.778088i	$-0.716190\pi$
$433$	−26.1421	−1.25631	−0.628155	+	0.778088i	$0.716190\pi$
$434$	0	0
$435$	3.75736	0.180152
$436$	16.4853	0.789502
$437$	−3.65685	−0.174931
$438$	22.7279	1.08598
$439$	27.3848	1.30700	0.653502	−	0.756925i	$-0.273299\pi$
$439$	27.3848	1.30700	0.653502	+	0.756925i	$0.273299\pi$
$440$	0.585786	0.0279263
$441$	0	0
$442$	−14.0000	−0.665912
$443$	12.6274	0.599947	0.299973	−	0.953948i	$-0.403022\pi$
$443$	12.6274	0.599947	0.299973	+	0.953948i	$0.403022\pi$
$444$	−22.7279	−1.07862
$445$	−7.31371	−0.346703
$446$	−8.58579	−0.406549
$447$	42.6274	2.01621
$448$	0	0
$449$	22.3431	1.05444	0.527219	−	0.849729i	$-0.323235\pi$
$449$	22.3431	1.05444	0.527219	+	0.849729i	$0.323235\pi$
$450$	−13.1716	−0.620914
$451$	5.41421	0.254945
$452$	−8.17157	−0.384358
$453$	−37.9706	−1.78401
$454$	28.8284	1.35299
$455$	0	0
$456$	1.41421	0.0662266
$457$	−11.6569	−0.545285	−0.272642	−	0.962115i	$-0.587898\pi$
$457$	−11.6569	−0.545285	−0.272642	+	0.962115i	$0.587898\pi$
$458$	−23.3137	−1.08938
$459$	1.51472	0.0707010
$460$	−3.65685	−0.170502
$461$	8.31371	0.387208	0.193604	−	0.981080i	$-0.437982\pi$
$461$	8.31371	0.387208	0.193604	+	0.981080i	$0.437982\pi$
$462$	0	0
$463$	12.8284	0.596188	0.298094	−	0.954537i	$-0.403649\pi$
$463$	12.8284	0.596188	0.298094	+	0.954537i	$0.403649\pi$
$464$	2.65685	0.123341
$465$	5.65685	0.262330
$466$	1.41421	0.0655122
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	10.8284	0.500544
$469$	0	0
$470$	6.14214	0.283316
$471$	−42.6274	−1.96417
$472$	5.58579	0.257107
$473$	−5.65685	−0.260102
$474$	−31.9706	−1.46846
$475$	−2.72792	−0.125166
$476$	0	0
$477$	22.3431	1.02302
$478$	20.2132	0.924530
$479$	11.9289	0.545047	0.272523	−	0.962149i	$-0.412142\pi$
$479$	11.9289	0.545047	0.272523	+	0.962149i	$0.412142\pi$
$480$	1.41421	0.0645497
$481$	−36.0416	−1.64336
$482$	−12.2426	−0.557637
$483$	0	0
$484$	1.00000	0.0454545
$485$	2.24264	0.101833
$486$	−21.6569	−0.982375
$487$	−9.65685	−0.437594	−0.218797	−	0.975770i	$-0.570213\pi$
$487$	−9.65685	−0.437594	−0.218797	+	0.975770i	$0.570213\pi$
$488$	−11.8284	−0.535448
$489$	23.4853	1.06204
$490$	0	0
$491$	19.1716	0.865201	0.432600	−	0.901586i	$-0.357596\pi$
$491$	19.1716	0.865201	0.432600	+	0.901586i	$0.357596\pi$
$492$	13.0711	0.589289
$493$	−9.71573	−0.437574
$494$	2.24264	0.100901
$495$	1.65685	0.0744701
$496$	4.00000	0.179605
$497$	0	0
$498$	29.3137	1.31358
$499$	22.1421	0.991218	0.495609	−	0.868546i	$-0.334945\pi$
$499$	22.1421	0.991218	0.495609	+	0.868546i	$0.334945\pi$
$500$	−5.65685	−0.252982
$501$	−33.1421	−1.48068
$502$	26.1421	1.16678
$503$	−4.21320	−0.187857	−0.0939287	−	0.995579i	$-0.529943\pi$
$503$	−4.21320	−0.187857	−0.0939287	+	0.995579i	$0.529943\pi$
$504$	0	0
$505$	−3.61522	−0.160875
$506$	−6.24264	−0.277519
$507$	4.00000	0.177646
$508$	15.7279	0.697814
$509$	−9.31371	−0.412823	−0.206411	−	0.978465i	$-0.566179\pi$
$509$	−9.31371	−0.412823	−0.206411	+	0.978465i	$0.566179\pi$
$510$	−5.17157	−0.229001
$511$	0	0
$512$	1.00000	0.0441942
$513$	−0.242641	−0.0107128
$514$	25.1421	1.10897
$515$	−7.85786	−0.346259
$516$	−13.6569	−0.601209
$517$	10.4853	0.461142
$518$	0	0
$519$	23.7279	1.04154
$520$	2.24264	0.0983463
$521$	28.2843	1.23916	0.619578	−	0.784935i	$-0.287304\pi$
$521$	28.2843	1.23916	0.619578	+	0.784935i	$0.287304\pi$
$522$	7.51472	0.328910
$523$	12.7279	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$523$	12.7279	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$524$	0.585786	0.0255902
$525$	0	0
$526$	−17.0416	−0.743050
$527$	−14.6274	−0.637180
$528$	2.41421	0.105065
$529$	15.9706	0.694372
$530$	4.62742	0.201002
$531$	15.7990	0.685618
$532$	0	0
$533$	20.7279	0.897826
$534$	−30.1421	−1.30438
$535$	−1.79899	−0.0777771
$536$	2.75736	0.119100
$537$	2.17157	0.0937103
$538$	−2.34315	−0.101020
$539$	0	0
$540$	−0.242641	−0.0104416
$541$	−12.8579	−0.552803	−0.276401	−	0.961042i	$-0.589142\pi$
$541$	−12.8579	−0.552803	−0.276401	+	0.961042i	$0.589142\pi$
$542$	−4.55635	−0.195712
$543$	18.4853	0.793279
$544$	−3.65685	−0.156786
$545$	9.65685	0.413654
$546$	0	0
$547$	34.8701	1.49094	0.745468	−	0.666541i	$-0.232226\pi$
$547$	34.8701	1.49094	0.745468	+	0.666541i	$0.232226\pi$
$548$	−16.6569	−0.711546
$549$	−33.4558	−1.42786
$550$	−4.65685	−0.198569
$551$	1.55635	0.0663027
$552$	−15.0711	−0.641467
$553$	0	0
$554$	1.82843	0.0776824
$555$	−13.3137	−0.565135
$556$	0	0
$557$	7.51472	0.318409	0.159204	−	0.987246i	$-0.449107\pi$
$557$	7.51472	0.318409	0.159204	+	0.987246i	$0.449107\pi$
$558$	11.3137	0.478947
$559$	−21.6569	−0.915987
$560$	0	0
$561$	−8.82843	−0.372736
$562$	8.72792	0.368165
$563$	−9.07107	−0.382300	−0.191150	−	0.981561i	$-0.561222\pi$
$563$	−9.07107	−0.382300	−0.191150	+	0.981561i	$0.561222\pi$
$564$	25.3137	1.06590
$565$	−4.78680	−0.201382
$566$	23.4142	0.984173
$567$	0	0
$568$	−11.0711	−0.464532
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0.828427	0.0346990
$571$	26.3848	1.10417	0.552084	−	0.833788i	$-0.313833\pi$
$571$	26.3848	1.10417	0.552084	+	0.833788i	$0.313833\pi$
$572$	3.82843	0.160075
$573$	−17.3137	−0.723291
$574$	0	0
$575$	29.0711	1.21235
$576$	2.82843	0.117851
$577$	9.68629	0.403246	0.201623	−	0.979463i	$-0.435378\pi$
$577$	9.68629	0.403246	0.201623	+	0.979463i	$0.435378\pi$
$578$	−3.62742	−0.150881
$579$	52.8701	2.19720
$580$	1.55635	0.0646239
$581$	0	0
$582$	9.24264	0.383120
$583$	7.89949	0.327164
$584$	9.41421	0.389563
$585$	6.34315	0.262257
$586$	10.8284	0.447318
$587$	25.1005	1.03601	0.518004	−	0.855378i	$-0.326675\pi$
$587$	25.1005	1.03601	0.518004	+	0.855378i	$0.326675\pi$
$588$	0	0
$589$	2.34315	0.0965476
$590$	3.27208	0.134709
$591$	−1.24264	−0.0511154
$592$	−9.41421	−0.386922
$593$	−23.6985	−0.973180	−0.486590	−	0.873630i	$-0.661759\pi$
$593$	−23.6985	−0.973180	−0.486590	+	0.873630i	$0.661759\pi$
$594$	−0.414214	−0.0169954
$595$	0	0
$596$	17.6569	0.723253
$597$	−0.242641	−0.00993062
$598$	−23.8995	−0.977323
$599$	42.6274	1.74171	0.870855	−	0.491541i	$-0.163566\pi$
$599$	42.6274	1.74171	0.870855	+	0.491541i	$0.163566\pi$
$600$	−11.2426	−0.458979
$601$	−31.9411	−1.30291	−0.651453	−	0.758689i	$-0.725840\pi$
$601$	−31.9411	−1.30291	−0.651453	+	0.758689i	$0.725840\pi$
$602$	0	0
$603$	7.79899	0.317599
$604$	−15.7279	−0.639960
$605$	0.585786	0.0238156
$606$	−14.8995	−0.605251
$607$	42.9706	1.74412	0.872061	−	0.489398i	$-0.162783\pi$
$607$	42.9706	1.74412	0.872061	+	0.489398i	$0.162783\pi$
$608$	0.585786	0.0237568
$609$	0	0
$610$	−6.92893	−0.280544
$611$	40.1421	1.62398
$612$	−10.3431	−0.418097
$613$	28.6274	1.15625	0.578125	−	0.815948i	$-0.303785\pi$
$613$	28.6274	1.15625	0.578125	+	0.815948i	$0.303785\pi$
$614$	−9.89949	−0.399511
$615$	7.65685	0.308754
$616$	0	0
$617$	−41.9706	−1.68967	−0.844836	−	0.535026i	$-0.820302\pi$
$617$	−41.9706	−1.68967	−0.844836	+	0.535026i	$0.820302\pi$
$618$	−32.3848	−1.30271
$619$	21.9411	0.881888	0.440944	−	0.897535i	$-0.354644\pi$
$619$	21.9411	0.881888	0.440944	+	0.897535i	$0.354644\pi$
$620$	2.34315	0.0941030
$621$	2.58579	0.103764
$622$	−16.7279	−0.670729
$623$	0	0
$624$	9.24264	0.370002
$625$	19.9706	0.798823
$626$	20.6569	0.825614
$627$	1.41421	0.0564782
$628$	−17.6569	−0.704585
$629$	34.4264	1.37267
$630$	0	0
$631$	23.2721	0.926447	0.463223	−	0.886242i	$-0.346693\pi$
$631$	23.2721	0.926447	0.463223	+	0.886242i	$0.346693\pi$
$632$	−13.2426	−0.526764
$633$	17.8995	0.711441
$634$	8.68629	0.344977
$635$	9.21320	0.365615
$636$	19.0711	0.756217
$637$	0	0
$638$	2.65685	0.105186
$639$	−31.3137	−1.23875
$640$	0.585786	0.0231552
$641$	15.2843	0.603692	0.301846	−	0.953357i	$-0.402397\pi$
$641$	15.2843	0.603692	0.301846	+	0.953357i	$0.402397\pi$
$642$	−7.41421	−0.292616
$643$	−1.58579	−0.0625373	−0.0312687	−	0.999511i	$-0.509955\pi$
$643$	−1.58579	−0.0625373	−0.0312687	+	0.999511i	$0.509955\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	−2.14214	−0.0842812
$647$	−30.1838	−1.18665	−0.593323	−	0.804964i	$-0.702184\pi$
$647$	−30.1838	−1.18665	−0.593323	+	0.804964i	$0.702184\pi$
$648$	−9.48528	−0.372617
$649$	5.58579	0.219261
$650$	−17.8284	−0.699288
$651$	0	0
$652$	9.72792	0.380975
$653$	−18.3848	−0.719452	−0.359726	−	0.933058i	$-0.617130\pi$
$653$	−18.3848	−0.719452	−0.359726	+	0.933058i	$0.617130\pi$
$654$	39.7990	1.55626
$655$	0.343146	0.0134078
$656$	5.41421	0.211390
$657$	26.6274	1.03883
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	1.41421	0.0550482
$661$	−18.9706	−0.737869	−0.368935	−	0.929455i	$-0.620277\pi$
$661$	−18.9706	−0.737869	−0.368935	+	0.929455i	$0.620277\pi$
$662$	−24.0711	−0.935549
$663$	−33.7990	−1.31264
$664$	12.1421	0.471206
$665$	0	0
$666$	−26.6274	−1.03179
$667$	−16.5858	−0.642204
$668$	−13.7279	−0.531149
$669$	−20.7279	−0.801388
$670$	1.61522	0.0624015
$671$	−11.8284	−0.456631
$672$	0	0
$673$	5.55635	0.214182	0.107091	−	0.994249i	$-0.465846\pi$
$673$	5.55635	0.214182	0.107091	+	0.994249i	$0.465846\pi$
$674$	−28.2426	−1.08787
$675$	1.92893	0.0742446
$676$	1.65685	0.0637252
$677$	35.3137	1.35722	0.678608	−	0.734501i	$-0.262584\pi$
$677$	35.3137	1.35722	0.678608	+	0.734501i	$0.262584\pi$
$678$	−19.7279	−0.757646
$679$	0	0
$680$	−2.14214	−0.0821472
$681$	69.5980	2.66700
$682$	4.00000	0.153168
$683$	13.5858	0.519846	0.259923	−	0.965629i	$-0.416303\pi$
$683$	13.5858	0.519846	0.259923	+	0.965629i	$0.416303\pi$
$684$	1.65685	0.0633514
$685$	−9.75736	−0.372810
$686$	0	0
$687$	−56.2843	−2.14738
$688$	−5.65685	−0.215666
$689$	30.2426	1.15215
$690$	−8.82843	−0.336092
$691$	−28.0711	−1.06787	−0.533937	−	0.845524i	$-0.679288\pi$
$691$	−28.0711	−1.06787	−0.533937	+	0.845524i	$0.679288\pi$
$692$	9.82843	0.373621
$693$	0	0
$694$	17.4142	0.661035
$695$	0	0
$696$	6.41421	0.243130
$697$	−19.7990	−0.749940
$698$	0	0
$699$	3.41421	0.129137
$700$	0	0
$701$	−26.1127	−0.986263	−0.493132	−	0.869955i	$-0.664148\pi$
$701$	−26.1127	−0.986263	−0.493132	+	0.869955i	$0.664148\pi$
$702$	−1.58579	−0.0598517
$703$	−5.51472	−0.207992
$704$	1.00000	0.0376889
$705$	14.8284	0.558471
$706$	2.68629	0.101100
$707$	0	0
$708$	13.4853	0.506808
$709$	−12.0416	−0.452233	−0.226116	−	0.974100i	$-0.572603\pi$
$709$	−12.0416	−0.452233	−0.226116	+	0.974100i	$0.572603\pi$
$710$	−6.48528	−0.243388
$711$	−37.4558	−1.40470
$712$	−12.4853	−0.467906
$713$	−24.9706	−0.935155
$714$	0	0
$715$	2.24264	0.0838700
$716$	0.899495	0.0336157
$717$	48.7990	1.82243
$718$	19.2426	0.718129
$719$	1.51472	0.0564895	0.0282447	−	0.999601i	$-0.491008\pi$
$719$	1.51472	0.0564895	0.0282447	+	0.999601i	$0.491008\pi$
$720$	1.65685	0.0617473
$721$	0	0
$722$	−18.6569	−0.694336
$723$	−29.5563	−1.09921
$724$	7.65685	0.284565
$725$	−12.3726	−0.459506
$726$	2.41421	0.0895999
$727$	−36.4264	−1.35098	−0.675490	−	0.737369i	$-0.736068\pi$
$727$	−36.4264	−1.35098	−0.675490	+	0.737369i	$0.736068\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	5.51472	0.204109
$731$	20.6863	0.765110
$732$	−28.5563	−1.05547
$733$	13.0000	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$734$	10.7279	0.395975
$735$	0	0
$736$	−6.24264	−0.230107
$737$	2.75736	0.101569
$738$	15.3137	0.563705
$739$	−52.4264	−1.92854	−0.964268	−	0.264928i	$-0.914652\pi$
$739$	−52.4264	−1.92854	−0.964268	+	0.264928i	$0.914652\pi$
$740$	−5.51472	−0.202725
$741$	5.41421	0.198896
$742$	0	0
$743$	−13.3137	−0.488433	−0.244216	−	0.969721i	$-0.578531\pi$
$743$	−13.3137	−0.488433	−0.244216	+	0.969721i	$0.578531\pi$
$744$	9.65685	0.354037
$745$	10.3431	0.378944
$746$	−19.9706	−0.731174
$747$	34.3431	1.25655
$748$	−3.65685	−0.133708
$749$	0	0
$750$	−13.6569	−0.498678
$751$	45.6569	1.66604	0.833021	−	0.553241i	$-0.186609\pi$
$751$	45.6569	1.66604	0.833021	+	0.553241i	$0.186609\pi$
$752$	10.4853	0.382359
$753$	63.1127	2.29995
$754$	10.1716	0.370427
$755$	−9.21320	−0.335303
$756$	0	0
$757$	8.34315	0.303237	0.151618	−	0.988439i	$-0.451552\pi$
$757$	8.34315	0.303237	0.151618	+	0.988439i	$0.451552\pi$
$758$	27.8701	1.01229
$759$	−15.0711	−0.547045
$760$	0.343146	0.0124472
$761$	−30.9706	−1.12268	−0.561341	−	0.827585i	$-0.689714\pi$
$761$	−30.9706	−1.12268	−0.561341	+	0.827585i	$0.689714\pi$
$762$	37.9706	1.37553
$763$	0	0
$764$	−7.17157	−0.259458
$765$	−6.05887	−0.219059
$766$	6.38478	0.230691
$767$	21.3848	0.772160
$768$	2.41421	0.0871154
$769$	−22.9706	−0.828340	−0.414170	−	0.910200i	$-0.635928\pi$
$769$	−22.9706	−0.828340	−0.414170	+	0.910200i	$0.635928\pi$
$770$	0	0
$771$	60.6985	2.18600
$772$	21.8995	0.788180
$773$	21.9411	0.789167	0.394584	−	0.918860i	$-0.370889\pi$
$773$	21.9411	0.789167	0.394584	+	0.918860i	$0.370889\pi$
$774$	−16.0000	−0.575108
$775$	−18.6274	−0.669117
$776$	3.82843	0.137433
$777$	0	0
$778$	−22.7279	−0.814835
$779$	3.17157	0.113633
$780$	5.41421	0.193860
$781$	−11.0711	−0.396154
$782$	22.8284	0.816343
$783$	−1.10051	−0.0393288
$784$	0	0
$785$	−10.3431	−0.369163
$786$	1.41421	0.0504433
$787$	−13.5563	−0.483232	−0.241616	−	0.970372i	$-0.577677\pi$
$787$	−13.5563	−0.483232	−0.241616	+	0.970372i	$0.577677\pi$
$788$	−0.514719	−0.0183361
$789$	−41.1421	−1.46470
$790$	−7.75736	−0.275994
$791$	0	0
$792$	2.82843	0.100504
$793$	−45.2843	−1.60809
$794$	−22.0000	−0.780751
$795$	11.1716	0.396215
$796$	−0.100505	−0.00356231
$797$	7.65685	0.271220	0.135610	−	0.990762i	$-0.456701\pi$
$797$	7.65685	0.271220	0.135610	+	0.990762i	$0.456701\pi$
$798$	0	0
$799$	−38.3431	−1.35648
$800$	−4.65685	−0.164645
$801$	−35.3137	−1.24775
$802$	18.3137	0.646680
$803$	9.41421	0.332220
$804$	6.65685	0.234769
$805$	0	0
$806$	15.3137	0.539402
$807$	−5.65685	−0.199131
$808$	−6.17157	−0.217115
$809$	−46.6274	−1.63933	−0.819666	−	0.572841i	$-0.805841\pi$
$809$	−46.6274	−1.63933	−0.819666	+	0.572841i	$0.805841\pi$
$810$	−5.55635	−0.195230
$811$	24.2843	0.852736	0.426368	−	0.904550i	$-0.359793\pi$
$811$	24.2843	0.852736	0.426368	+	0.904550i	$0.359793\pi$
$812$	0	0
$813$	−11.0000	−0.385787
$814$	−9.41421	−0.329968
$815$	5.69848	0.199609
$816$	−8.82843	−0.309057
$817$	−3.31371	−0.115932
$818$	−2.72792	−0.0953796
$819$	0	0
$820$	3.17157	0.110756
$821$	−29.4853	−1.02904	−0.514522	−	0.857477i	$-0.672031\pi$
$821$	−29.4853	−1.02904	−0.514522	+	0.857477i	$0.672031\pi$
$822$	−40.2132	−1.40260
$823$	−43.4558	−1.51478	−0.757388	−	0.652965i	$-0.773525\pi$
$823$	−43.4558	−1.51478	−0.757388	+	0.652965i	$0.773525\pi$
$824$	−13.4142	−0.467306
$825$	−11.2426	−0.391419
$826$	0	0
$827$	−21.3553	−0.742598	−0.371299	−	0.928513i	$-0.621087\pi$
$827$	−21.3553	−0.742598	−0.371299	+	0.928513i	$0.621087\pi$
$828$	−17.6569	−0.613618
$829$	−21.7574	−0.755664	−0.377832	−	0.925874i	$-0.623330\pi$
$829$	−21.7574	−0.755664	−0.377832	+	0.925874i	$0.623330\pi$
$830$	7.11270	0.246885
$831$	4.41421	0.153127
$832$	3.82843	0.132727
$833$	0	0
$834$	0	0
$835$	−8.04163	−0.278292
$836$	0.585786	0.0202598
$837$	−1.65685	−0.0572693
$838$	26.1421	0.903065
$839$	18.4853	0.638183	0.319091	−	0.947724i	$-0.396622\pi$
$839$	18.4853	0.638183	0.319091	+	0.947724i	$0.396622\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	0.686292	0.0236512
$843$	21.0711	0.725726
$844$	7.41421	0.255208
$845$	0.970563	0.0333884
$846$	29.6569	1.01962
$847$	0	0
$848$	7.89949	0.271270
$849$	56.5269	1.94000
$850$	17.0294	0.584105
$851$	58.7696	2.01459
$852$	−26.7279	−0.915684
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	0.970563	0.0331925
$856$	−3.07107	−0.104967
$857$	34.6274	1.18285	0.591425	−	0.806360i	$-0.298566\pi$
$857$	34.6274	1.18285	0.591425	+	0.806360i	$0.298566\pi$
$858$	9.24264	0.315539
$859$	5.72792	0.195434	0.0977171	−	0.995214i	$-0.468846\pi$
$859$	5.72792	0.195434	0.0977171	+	0.995214i	$0.468846\pi$
$860$	−3.31371	−0.112997
$861$	0	0
$862$	−17.5858	−0.598974
$863$	35.2132	1.19867	0.599336	−	0.800498i	$-0.295431\pi$
$863$	35.2132	1.19867	0.599336	+	0.800498i	$0.295431\pi$
$864$	−0.414214	−0.0140918
$865$	5.75736	0.195756
$866$	−26.1421	−0.888346
$867$	−8.75736	−0.297416
$868$	0	0
$869$	−13.2426	−0.449226
$870$	3.75736	0.127386
$871$	10.5563	0.357688
$872$	16.4853	0.558262
$873$	10.8284	0.366487
$874$	−3.65685	−0.123695
$875$	0	0
$876$	22.7279	0.767905
$877$	19.6274	0.662771	0.331385	−	0.943495i	$-0.392484\pi$
$877$	19.6274	0.662771	0.331385	+	0.943495i	$0.392484\pi$
$878$	27.3848	0.924191
$879$	26.1421	0.881752
$880$	0.585786	0.0197469
$881$	−6.45584	−0.217503	−0.108751	−	0.994069i	$-0.534685\pi$
$881$	−6.45584	−0.217503	−0.108751	+	0.994069i	$0.534685\pi$
$882$	0	0
$883$	15.7279	0.529287	0.264643	−	0.964346i	$-0.414746\pi$
$883$	15.7279	0.529287	0.264643	+	0.964346i	$0.414746\pi$
$884$	−14.0000	−0.470871
$885$	7.89949	0.265539
$886$	12.6274	0.424226
$887$	3.10051	0.104105	0.0520524	−	0.998644i	$-0.483424\pi$
$887$	3.10051	0.104105	0.0520524	+	0.998644i	$0.483424\pi$
$888$	−22.7279	−0.762699
$889$	0	0
$890$	−7.31371	−0.245156
$891$	−9.48528	−0.317769
$892$	−8.58579	−0.287473
$893$	6.14214	0.205539
$894$	42.6274	1.42567
$895$	0.526912	0.0176127
$896$	0	0
$897$	−57.6985	−1.92650
$898$	22.3431	0.745600
$899$	10.6274	0.354444
$900$	−13.1716	−0.439052
$901$	−28.8873	−0.962376
$902$	5.41421	0.180274
$903$	0	0
$904$	−8.17157	−0.271782
$905$	4.48528	0.149096
$906$	−37.9706	−1.26149
$907$	10.6863	0.354832	0.177416	−	0.984136i	$-0.443226\pi$
$907$	10.6863	0.354832	0.177416	+	0.984136i	$0.443226\pi$
$908$	28.8284	0.956705
$909$	−17.4558	−0.578974
$910$	0	0
$911$	−30.4853	−1.01002	−0.505011	−	0.863113i	$-0.668512\pi$
$911$	−30.4853	−1.01002	−0.505011	+	0.863113i	$0.668512\pi$
$912$	1.41421	0.0468293
$913$	12.1421	0.401846
$914$	−11.6569	−0.385574
$915$	−16.7279	−0.553008
$916$	−23.3137	−0.770307
$917$	0	0
$918$	1.51472	0.0499932
$919$	22.1421	0.730402	0.365201	−	0.930929i	$-0.381000\pi$
$919$	22.1421	0.730402	0.365201	+	0.930929i	$0.381000\pi$
$920$	−3.65685	−0.120563
$921$	−23.8995	−0.787515
$922$	8.31371	0.273797
$923$	−42.3848	−1.39511
$924$	0	0
$925$	43.8406	1.44147
$926$	12.8284	0.421568
$927$	−37.9411	−1.24615
$928$	2.65685	0.0872155
$929$	40.4558	1.32731	0.663657	−	0.748037i	$-0.269004\pi$
$929$	40.4558	1.32731	0.663657	+	0.748037i	$0.269004\pi$
$930$	5.65685	0.185496
$931$	0	0
$932$	1.41421	0.0463241
$933$	−40.3848	−1.32214
$934$	34.0000	1.11251
$935$	−2.14214	−0.0700553
$936$	10.8284	0.353938
$937$	19.4142	0.634235	0.317117	−	0.948386i	$-0.397285\pi$
$937$	19.4142	0.634235	0.317117	+	0.948386i	$0.397285\pi$
$938$	0	0
$939$	49.8701	1.62745
$940$	6.14214	0.200334
$941$	−24.6569	−0.803790	−0.401895	−	0.915686i	$-0.631648\pi$
$941$	−24.6569	−0.803790	−0.401895	+	0.915686i	$0.631648\pi$
$942$	−42.6274	−1.38888
$943$	−33.7990	−1.10065
$944$	5.58579	0.181802
$945$	0	0
$946$	−5.65685	−0.183920
$947$	34.8284	1.13177	0.565886	−	0.824484i	$-0.308534\pi$
$947$	34.8284	1.13177	0.565886	+	0.824484i	$0.308534\pi$
$948$	−31.9706	−1.03836
$949$	36.0416	1.16996
$950$	−2.72792	−0.0885055
$951$	20.9706	0.680017
$952$	0	0
$953$	5.55635	0.179988	0.0899939	−	0.995942i	$-0.471315\pi$
$953$	5.55635	0.179988	0.0899939	+	0.995942i	$0.471315\pi$
$954$	22.3431	0.723386
$955$	−4.20101	−0.135941
$956$	20.2132	0.653742
$957$	6.41421	0.207342
$958$	11.9289	0.385406
$959$	0	0
$960$	1.41421	0.0456435
$961$	−15.0000	−0.483871
$962$	−36.0416	−1.16203
$963$	−8.68629	−0.279912
$964$	−12.2426	−0.394309
$965$	12.8284	0.412962
$966$	0	0
$967$	−36.2843	−1.16682	−0.583412	−	0.812177i	$-0.698283\pi$
$967$	−36.2843	−1.16682	−0.583412	+	0.812177i	$0.698283\pi$
$968$	1.00000	0.0321412
$969$	−5.17157	−0.166135
$970$	2.24264	0.0720069
$971$	−22.2721	−0.714745	−0.357372	−	0.933962i	$-0.616327\pi$
$971$	−22.2721	−0.714745	−0.357372	+	0.933962i	$0.616327\pi$
$972$	−21.6569	−0.694644
$973$	0	0
$974$	−9.65685	−0.309426
$975$	−43.0416	−1.37844
$976$	−11.8284	−0.378619
$977$	40.0000	1.27971	0.639857	−	0.768494i	$-0.278994\pi$
$977$	40.0000	1.27971	0.639857	+	0.768494i	$0.278994\pi$
$978$	23.4853	0.750976
$979$	−12.4853	−0.399031
$980$	0	0
$981$	46.6274	1.48870
$982$	19.1716	0.611789
$983$	−32.4264	−1.03424	−0.517121	−	0.855912i	$-0.672996\pi$
$983$	−32.4264	−1.03424	−0.517121	+	0.855912i	$0.672996\pi$
$984$	13.0711	0.416690
$985$	−0.301515	−0.00960707
$986$	−9.71573	−0.309412
$987$	0	0
$988$	2.24264	0.0713479
$989$	35.3137	1.12291
$990$	1.65685	0.0526583
$991$	−1.61522	−0.0513093	−0.0256546	−	0.999671i	$-0.508167\pi$
$991$	−1.61522	−0.0513093	−0.0256546	+	0.999671i	$0.508167\pi$
$992$	4.00000	0.127000
$993$	−58.1127	−1.84415
$994$	0	0
$995$	−0.0588745	−0.00186645
$996$	29.3137	0.928840
$997$	−13.1716	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$997$	−13.1716	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$998$	22.1421	0.700897
$999$	3.89949	0.123375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.x.1.2		2
3.2	odd	2		9702.2.a.ch.1.2		2
4.3	odd	2		8624.2.a.bh.1.1		2
7.2	even	3		1078.2.e.m.67.1		4
7.3	odd	6		154.2.e.e.23.2	✓	4
7.4	even	3		1078.2.e.m.177.1		4
7.5	odd	6		154.2.e.e.67.2	yes	4
7.6	odd	2		1078.2.a.t.1.1		2
21.5	even	6		1386.2.k.t.991.2		4
21.17	even	6		1386.2.k.t.793.2		4
21.20	even	2		9702.2.a.cx.1.1		2
28.3	even	6		1232.2.q.f.177.1		4
28.19	even	6		1232.2.q.f.529.1		4
28.27	even	2		8624.2.a.cc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.e.23.2	✓	4	7.3	odd	6
154.2.e.e.67.2	yes	4	7.5	odd	6
1078.2.a.t.1.1		2	7.6	odd	2
1078.2.a.x.1.2		2	1.1	even	1	trivial
1078.2.e.m.67.1		4	7.2	even	3
1078.2.e.m.177.1		4	7.4	even	3
1232.2.q.f.177.1		4	28.3	even	6
1232.2.q.f.529.1		4	28.19	even	6
1386.2.k.t.793.2		4	21.17	even	6
1386.2.k.t.991.2		4	21.5	even	6
8624.2.a.bh.1.1		2	4.3	odd	2
8624.2.a.cc.1.2		2	28.27	even	2
9702.2.a.ch.1.2		2	3.2	odd	2
9702.2.a.cx.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +0.585786 q^{5} +2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} +0.585786 q^{10} +1.00000 q^{11} +2.41421 q^{12} +3.82843 q^{13} +1.41421 q^{15} +1.00000 q^{16} -3.65685 q^{17} +2.82843 q^{18} +0.585786 q^{19} +0.585786 q^{20} +1.00000 q^{22} -6.24264 q^{23} +2.41421 q^{24} -4.65685 q^{25} +3.82843 q^{26} -0.414214 q^{27} +2.65685 q^{29} +1.41421 q^{30} +4.00000 q^{31} +1.00000 q^{32} +2.41421 q^{33} -3.65685 q^{34} +2.82843 q^{36} -9.41421 q^{37} +0.585786 q^{38} +9.24264 q^{39} +0.585786 q^{40} +5.41421 q^{41} -5.65685 q^{43} +1.00000 q^{44} +1.65685 q^{45} -6.24264 q^{46} +10.4853 q^{47} +2.41421 q^{48} -4.65685 q^{50} -8.82843 q^{51} +3.82843 q^{52} +7.89949 q^{53} -0.414214 q^{54} +0.585786 q^{55} +1.41421 q^{57} +2.65685 q^{58} +5.58579 q^{59} +1.41421 q^{60} -11.8284 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.24264 q^{65} +2.41421 q^{66} +2.75736 q^{67} -3.65685 q^{68} -15.0711 q^{69} -11.0711 q^{71} +2.82843 q^{72} +9.41421 q^{73} -9.41421 q^{74} -11.2426 q^{75} +0.585786 q^{76} +9.24264 q^{78} -13.2426 q^{79} +0.585786 q^{80} -9.48528 q^{81} +5.41421 q^{82} +12.1421 q^{83} -2.14214 q^{85} -5.65685 q^{86} +6.41421 q^{87} +1.00000 q^{88} -12.4853 q^{89} +1.65685 q^{90} -6.24264 q^{92} +9.65685 q^{93} +10.4853 q^{94} +0.343146 q^{95} +2.41421 q^{96} +3.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 4 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 - 6 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^33 + 4 * q^34 - 16 * q^37 + 4 * q^38 + 10 * q^39 + 4 * q^40 + 8 * q^41 + 2 * q^44 - 8 * q^45 - 4 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^58 + 14 * q^59 - 18 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 + 2 * q^66 + 14 * q^67 + 4 * q^68 - 16 * q^69 - 8 * q^71 + 16 * q^73 - 16 * q^74 - 14 * q^75 + 4 * q^76 + 10 * q^78 - 18 * q^79 + 4 * q^80 - 2 * q^81 + 8 * q^82 - 4 * q^83 + 24 * q^85 + 10 * q^87 + 2 * q^88 - 8 * q^89 - 8 * q^90 - 4 * q^92 + 8 * q^93 + 4 * q^94 + 12 * q^95 + 2 * q^96 + 2 * q^97

Introduction

L-functions

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