LMFDB - Embedded newform 1078.2.a.q.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:	$1$
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:	yes
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} -1.00000 q^{11} +1.41421 q^{12} -2.82843 q^{13} -2.00000 q^{15} +1.00000 q^{16} +5.65685 q^{17} +1.00000 q^{18} -2.82843 q^{19} -1.41421 q^{20} +1.00000 q^{22} -2.00000 q^{23} -1.41421 q^{24} -3.00000 q^{25} +2.82843 q^{26} -5.65685 q^{27} -6.00000 q^{29} +2.00000 q^{30} +1.41421 q^{31} -1.00000 q^{32} -1.41421 q^{33} -5.65685 q^{34} -1.00000 q^{36} -2.00000 q^{37} +2.82843 q^{38} -4.00000 q^{39} +1.41421 q^{40} -5.65685 q^{41} +4.00000 q^{43} -1.00000 q^{44} +1.41421 q^{45} +2.00000 q^{46} -4.24264 q^{47} +1.41421 q^{48} +3.00000 q^{50} +8.00000 q^{51} -2.82843 q^{52} -12.0000 q^{53} +5.65685 q^{54} +1.41421 q^{55} -4.00000 q^{57} +6.00000 q^{58} -4.24264 q^{59} -2.00000 q^{60} +5.65685 q^{61} -1.41421 q^{62} +1.00000 q^{64} +4.00000 q^{65} +1.41421 q^{66} -2.00000 q^{67} +5.65685 q^{68} -2.82843 q^{69} -10.0000 q^{71} +1.00000 q^{72} +14.1421 q^{73} +2.00000 q^{74} -4.24264 q^{75} -2.82843 q^{76} +4.00000 q^{78} -8.00000 q^{79} -1.41421 q^{80} -5.00000 q^{81} +5.65685 q^{82} +2.82843 q^{83} -8.00000 q^{85} -4.00000 q^{86} -8.48528 q^{87} +1.00000 q^{88} +7.07107 q^{89} -1.41421 q^{90} -2.00000 q^{92} +2.00000 q^{93} +4.24264 q^{94} +4.00000 q^{95} -1.41421 q^{96} +15.5563 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 2 q^{32} - 2 q^{36} - 4 q^{37} - 8 q^{39} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 6 q^{50} + 16 q^{51} - 24 q^{53} - 8 q^{57} + 12 q^{58} - 4 q^{60} + 2 q^{64} + 8 q^{65} - 4 q^{67} - 20 q^{71} + 2 q^{72} + 4 q^{74} + 8 q^{78} - 16 q^{79} - 10 q^{81} - 16 q^{85} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{93} + 8 q^{95} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 2 * q^11 - 4 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^22 - 4 * q^23 - 6 * q^25 - 12 * q^29 + 4 * q^30 - 2 * q^32 - 2 * q^36 - 4 * q^37 - 8 * q^39 + 8 * q^43 - 2 * q^44 + 4 * q^46 + 6 * q^50 + 16 * q^51 - 24 * q^53 - 8 * q^57 + 12 * q^58 - 4 * q^60 + 2 * q^64 + 8 * q^65 - 4 * q^67 - 20 * q^71 + 2 * q^72 + 4 * q^74 + 8 * q^78 - 16 * q^79 - 10 * q^81 - 16 * q^85 - 8 * q^86 + 2 * q^88 - 4 * q^92 + 4 * q^93 + 8 * q^95 + 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	1.41421	0.447214
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.41421	0.408248
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	1.00000	0.235702
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	−1.41421	−0.316228
$21$	0	0
$22$	1.00000	0.213201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−1.41421	−0.288675
$25$	−3.00000	−0.600000
$26$	2.82843	0.554700
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	−1.00000	−0.176777
$33$	−1.41421	−0.246183
$34$	−5.65685	−0.970143
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	2.82843	0.458831
$39$	−4.00000	−0.640513
$40$	1.41421	0.223607
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	1.41421	0.210819
$46$	2.00000	0.294884
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	3.00000	0.424264
$51$	8.00000	1.12022
$52$	−2.82843	−0.392232
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	5.65685	0.769800
$55$	1.41421	0.190693
$56$	0	0
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	−4.24264	−0.552345	−0.276172	−	0.961108i	$-0.589066\pi$
$59$	−4.24264	−0.552345	−0.276172	+	0.961108i	$0.589066\pi$
$60$	−2.00000	−0.258199
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	−1.41421	−0.179605
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	1.41421	0.174078
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	5.65685	0.685994
$69$	−2.82843	−0.340503
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000	0.117851
$73$	14.1421	1.65521	0.827606	−	0.561310i	$-0.189702\pi$
$73$	14.1421	1.65521	0.827606	+	0.561310i	$0.189702\pi$
$74$	2.00000	0.232495
$75$	−4.24264	−0.489898
$76$	−2.82843	−0.324443
$77$	0	0
$78$	4.00000	0.452911
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.41421	−0.158114
$81$	−5.00000	−0.555556
$82$	5.65685	0.624695
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	−4.00000	−0.431331
$87$	−8.48528	−0.909718
$88$	1.00000	0.106600
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	−1.41421	−0.149071
$91$	0	0
$92$	−2.00000	−0.208514
$93$	2.00000	0.207390
$94$	4.24264	0.437595
$95$	4.00000	0.410391
$96$	−1.41421	−0.144338
$97$	15.5563	1.57951	0.789754	−	0.613424i	$-0.210208\pi$
$97$	15.5563	1.57951	0.789754	+	0.613424i	$0.210208\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−3.00000	−0.300000
$101$	−2.82843	−0.281439	−0.140720	−	0.990050i	$-0.544942\pi$
$101$	−2.82843	−0.281439	−0.140720	+	0.990050i	$0.544942\pi$
$102$	−8.00000	−0.792118
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	12.0000	1.16554
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	−5.65685	−0.544331
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.41421	−0.134840
$111$	−2.82843	−0.268462
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	4.00000	0.374634
$115$	2.82843	0.263752
$116$	−6.00000	−0.557086
$117$	2.82843	0.261488
$118$	4.24264	0.390567
$119$	0	0
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	−5.65685	−0.512148
$123$	−8.00000	−0.721336
$124$	1.41421	0.127000
$125$	11.3137	1.01193
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	5.65685	0.498058
$130$	−4.00000	−0.350823
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	−1.41421	−0.123091
$133$	0	0
$134$	2.00000	0.172774
$135$	8.00000	0.688530
$136$	−5.65685	−0.485071
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	2.82843	0.240772
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	10.0000	0.839181
$143$	2.82843	0.236525
$144$	−1.00000	−0.0833333
$145$	8.48528	0.704664
$146$	−14.1421	−1.17041
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	4.24264	0.346410
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	2.82843	0.229416
$153$	−5.65685	−0.457330
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−4.00000	−0.320256
$157$	−18.3848	−1.46726	−0.733632	−	0.679546i	$-0.762177\pi$
$157$	−18.3848	−1.46726	−0.733632	+	0.679546i	$0.762177\pi$
$158$	8.00000	0.636446
$159$	−16.9706	−1.34585
$160$	1.41421	0.111803
$161$	0	0
$162$	5.00000	0.392837
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−5.65685	−0.441726
$165$	2.00000	0.155700
$166$	−2.82843	−0.219529
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	8.00000	0.613572
$171$	2.82843	0.216295
$172$	4.00000	0.304997
$173$	−14.1421	−1.07521	−0.537603	−	0.843198i	$-0.680670\pi$
$173$	−14.1421	−1.07521	−0.537603	+	0.843198i	$0.680670\pi$
$174$	8.48528	0.643268
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−6.00000	−0.450988
$178$	−7.07107	−0.529999
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	1.41421	0.105409
$181$	−12.7279	−0.946059	−0.473029	−	0.881047i	$-0.656840\pi$
$181$	−12.7279	−0.946059	−0.473029	+	0.881047i	$0.656840\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	2.00000	0.147442
$185$	2.82843	0.207950
$186$	−2.00000	−0.146647
$187$	−5.65685	−0.413670
$188$	−4.24264	−0.309426
$189$	0	0
$190$	−4.00000	−0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.41421	0.102062
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−15.5563	−1.11688
$195$	5.65685	0.405096
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−1.00000	−0.0710669
$199$	−21.2132	−1.50376	−0.751882	−	0.659298i	$-0.770854\pi$
$199$	−21.2132	−1.50376	−0.751882	+	0.659298i	$0.770854\pi$
$200$	3.00000	0.212132
$201$	−2.82843	−0.199502
$202$	2.82843	0.199007
$203$	0	0
$204$	8.00000	0.560112
$205$	8.00000	0.558744
$206$	−4.24264	−0.295599
$207$	2.00000	0.139010
$208$	−2.82843	−0.196116
$209$	2.82843	0.195646
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−12.0000	−0.824163
$213$	−14.1421	−0.969003
$214$	20.0000	1.36717
$215$	−5.65685	−0.385794
$216$	5.65685	0.384900
$217$	0	0
$218$	−10.0000	−0.677285
$219$	20.0000	1.35147
$220$	1.41421	0.0953463
$221$	−16.0000	−1.07628
$222$	2.82843	0.189832
$223$	15.5563	1.04173	0.520865	−	0.853639i	$-0.325609\pi$
$223$	15.5563	1.04173	0.520865	+	0.853639i	$0.325609\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	18.0000	1.19734
$227$	28.2843	1.87729	0.938647	−	0.344881i	$-0.112081\pi$
$227$	28.2843	1.87729	0.938647	+	0.344881i	$0.112081\pi$
$228$	−4.00000	−0.264906
$229$	4.24264	0.280362	0.140181	−	0.990126i	$-0.455232\pi$
$229$	4.24264	0.280362	0.140181	+	0.990126i	$0.455232\pi$
$230$	−2.82843	−0.186501
$231$	0	0
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−2.82843	−0.184900
$235$	6.00000	0.391397
$236$	−4.24264	−0.276172
$237$	−11.3137	−0.734904
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−2.00000	−0.129099
$241$	5.65685	0.364390	0.182195	−	0.983262i	$-0.441680\pi$
$241$	5.65685	0.364390	0.182195	+	0.983262i	$0.441680\pi$
$242$	−1.00000	−0.0642824
$243$	9.89949	0.635053
$244$	5.65685	0.362143
$245$	0	0
$246$	8.00000	0.510061
$247$	8.00000	0.509028
$248$	−1.41421	−0.0898027
$249$	4.00000	0.253490
$250$	−11.3137	−0.715542
$251$	24.0416	1.51749	0.758747	−	0.651385i	$-0.225812\pi$
$251$	24.0416	1.51749	0.758747	+	0.651385i	$0.225812\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	−12.0000	−0.752947
$255$	−11.3137	−0.708492
$256$	1.00000	0.0625000
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	−5.65685	−0.352180
$259$	0	0
$260$	4.00000	0.248069
$261$	6.00000	0.371391
$262$	11.3137	0.698963
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	1.41421	0.0870388
$265$	16.9706	1.04249
$266$	0	0
$267$	10.0000	0.611990
$268$	−2.00000	−0.122169
$269$	24.0416	1.46584	0.732922	−	0.680313i	$-0.238156\pi$
$269$	24.0416	1.46584	0.732922	+	0.680313i	$0.238156\pi$
$270$	−8.00000	−0.486864
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	2.00000	0.120824
$275$	3.00000	0.180907
$276$	−2.82843	−0.170251
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−2.82843	−0.169638
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	6.00000	0.357295
$283$	−16.9706	−1.00880	−0.504398	−	0.863472i	$-0.668285\pi$
$283$	−16.9706	−1.00880	−0.504398	+	0.863472i	$0.668285\pi$
$284$	−10.0000	−0.593391
$285$	5.65685	0.335083
$286$	−2.82843	−0.167248
$287$	0	0
$288$	1.00000	0.0589256
$289$	15.0000	0.882353
$290$	−8.48528	−0.498273
$291$	22.0000	1.28966
$292$	14.1421	0.827606
$293$	−16.9706	−0.991431	−0.495715	−	0.868485i	$-0.665094\pi$
$293$	−16.9706	−0.991431	−0.495715	+	0.868485i	$0.665094\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	2.00000	0.116248
$297$	5.65685	0.328244
$298$	10.0000	0.579284
$299$	5.65685	0.327144
$300$	−4.24264	−0.244949
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−4.00000	−0.229794
$304$	−2.82843	−0.162221
$305$	−8.00000	−0.458079
$306$	5.65685	0.323381
$307$	11.3137	0.645707	0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137	0.645707	0.322854	+	0.946449i	$0.395358\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	2.00000	0.113592
$311$	9.89949	0.561349	0.280674	−	0.959803i	$-0.409442\pi$
$311$	9.89949	0.561349	0.280674	+	0.959803i	$0.409442\pi$
$312$	4.00000	0.226455
$313$	−26.8701	−1.51879	−0.759393	−	0.650633i	$-0.774504\pi$
$313$	−26.8701	−1.51879	−0.759393	+	0.650633i	$0.774504\pi$
$314$	18.3848	1.03751
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	16.9706	0.951662
$319$	6.00000	0.335936
$320$	−1.41421	−0.0790569
$321$	−28.2843	−1.57867
$322$	0	0
$323$	−16.0000	−0.890264
$324$	−5.00000	−0.277778
$325$	8.48528	0.470679
$326$	22.0000	1.21847
$327$	14.1421	0.782062
$328$	5.65685	0.312348
$329$	0	0
$330$	−2.00000	−0.110096
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	2.82843	0.155230
$333$	2.00000	0.109599
$334$	−11.3137	−0.619059
$335$	2.82843	0.154533
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	5.00000	0.271964
$339$	−25.4558	−1.38257
$340$	−8.00000	−0.433861
$341$	−1.41421	−0.0765840
$342$	−2.82843	−0.152944
$343$	0	0
$344$	−4.00000	−0.215666
$345$	4.00000	0.215353
$346$	14.1421	0.760286
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−8.48528	−0.454859
$349$	5.65685	0.302804	0.151402	−	0.988472i	$-0.451621\pi$
$349$	5.65685	0.302804	0.151402	+	0.988472i	$0.451621\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	1.00000	0.0533002
$353$	−1.41421	−0.0752710	−0.0376355	−	0.999292i	$-0.511983\pi$
$353$	−1.41421	−0.0752710	−0.0376355	+	0.999292i	$0.511983\pi$
$354$	6.00000	0.318896
$355$	14.1421	0.750587
$356$	7.07107	0.374766
$357$	0	0
$358$	−20.0000	−1.05703
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	−1.41421	−0.0745356
$361$	−11.0000	−0.578947
$362$	12.7279	0.668965
$363$	1.41421	0.0742270
$364$	0	0
$365$	−20.0000	−1.04685
$366$	−8.00000	−0.418167
$367$	−29.6985	−1.55025	−0.775124	−	0.631809i	$-0.782313\pi$
$367$	−29.6985	−1.55025	−0.775124	+	0.631809i	$0.782313\pi$
$368$	−2.00000	−0.104257
$369$	5.65685	0.294484
$370$	−2.82843	−0.147043
$371$	0	0
$372$	2.00000	0.103695
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	5.65685	0.292509
$375$	16.0000	0.826236
$376$	4.24264	0.218797
$377$	16.9706	0.874028
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	4.00000	0.205196
$381$	16.9706	0.869428
$382$	0	0
$383$	−26.8701	−1.37300	−0.686498	−	0.727132i	$-0.740853\pi$
$383$	−26.8701	−1.37300	−0.686498	+	0.727132i	$0.740853\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−4.00000	−0.203331
$388$	15.5563	0.789754
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	−5.65685	−0.286446
$391$	−11.3137	−0.572159
$392$	0	0
$393$	−16.0000	−0.807093
$394$	−18.0000	−0.906827
$395$	11.3137	0.569254
$396$	1.00000	0.0502519
$397$	24.0416	1.20661	0.603307	−	0.797509i	$-0.293849\pi$
$397$	24.0416	1.20661	0.603307	+	0.797509i	$0.293849\pi$
$398$	21.2132	1.06332
$399$	0	0
$400$	−3.00000	−0.150000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	2.82843	0.141069
$403$	−4.00000	−0.199254
$404$	−2.82843	−0.140720
$405$	7.07107	0.351364
$406$	0	0
$407$	2.00000	0.0991363
$408$	−8.00000	−0.396059
$409$	−31.1127	−1.53842	−0.769212	−	0.638994i	$-0.779351\pi$
$409$	−31.1127	−1.53842	−0.769212	+	0.638994i	$0.779351\pi$
$410$	−8.00000	−0.395092
$411$	−2.82843	−0.139516
$412$	4.24264	0.209020
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	−4.00000	−0.196352
$416$	2.82843	0.138675
$417$	4.00000	0.195881
$418$	−2.82843	−0.138343
$419$	12.7279	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$419$	12.7279	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−28.0000	−1.36302
$423$	4.24264	0.206284
$424$	12.0000	0.582772
$425$	−16.9706	−0.823193
$426$	14.1421	0.685189
$427$	0	0
$428$	−20.0000	−0.966736
$429$	4.00000	0.193122
$430$	5.65685	0.272798
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−5.65685	−0.272166
$433$	−38.1838	−1.83499	−0.917497	−	0.397742i	$-0.869794\pi$
$433$	−38.1838	−1.83499	−0.917497	+	0.397742i	$0.869794\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	10.0000	0.478913
$437$	5.65685	0.270604
$438$	−20.0000	−0.955637
$439$	25.4558	1.21494	0.607471	−	0.794342i	$-0.292184\pi$
$439$	25.4558	1.21494	0.607471	+	0.794342i	$0.292184\pi$
$440$	−1.41421	−0.0674200
$441$	0	0
$442$	16.0000	0.761042
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−2.82843	−0.134231
$445$	−10.0000	−0.474045
$446$	−15.5563	−0.736614
$447$	−14.1421	−0.668900
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−3.00000	−0.141421
$451$	5.65685	0.266371
$452$	−18.0000	−0.846649
$453$	22.6274	1.06313
$454$	−28.2843	−1.32745
$455$	0	0
$456$	4.00000	0.187317
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−4.24264	−0.198246
$459$	−32.0000	−1.49363
$460$	2.82843	0.131876
$461$	−28.2843	−1.31733	−0.658665	−	0.752436i	$-0.728879\pi$
$461$	−28.2843	−1.31733	−0.658665	+	0.752436i	$0.728879\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	−6.00000	−0.278543
$465$	−2.82843	−0.131165
$466$	−6.00000	−0.277945
$467$	−4.24264	−0.196326	−0.0981630	−	0.995170i	$-0.531297\pi$
$467$	−4.24264	−0.196326	−0.0981630	+	0.995170i	$0.531297\pi$
$468$	2.82843	0.130744
$469$	0	0
$470$	−6.00000	−0.276759
$471$	−26.0000	−1.19802
$472$	4.24264	0.195283
$473$	−4.00000	−0.183920
$474$	11.3137	0.519656
$475$	8.48528	0.389331
$476$	0	0
$477$	12.0000	0.549442
$478$	16.0000	0.731823
$479$	−25.4558	−1.16311	−0.581554	−	0.813508i	$-0.697555\pi$
$479$	−25.4558	−1.16311	−0.581554	+	0.813508i	$0.697555\pi$
$480$	2.00000	0.0912871
$481$	5.65685	0.257930
$482$	−5.65685	−0.257663
$483$	0	0
$484$	1.00000	0.0454545
$485$	−22.0000	−0.998969
$486$	−9.89949	−0.449050
$487$	10.0000	0.453143	0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000	0.453143	0.226572	+	0.973995i	$0.427248\pi$
$488$	−5.65685	−0.256074
$489$	−31.1127	−1.40696
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	−8.00000	−0.360668
$493$	−33.9411	−1.52863
$494$	−8.00000	−0.359937
$495$	−1.41421	−0.0635642
$496$	1.41421	0.0635001
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−30.0000	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$500$	11.3137	0.505964
$501$	16.0000	0.714827
$502$	−24.0416	−1.07303
$503$	−19.7990	−0.882793	−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990	−0.882793	−0.441397	+	0.897312i	$0.645517\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	−2.00000	−0.0889108
$507$	−7.07107	−0.314037
$508$	12.0000	0.532414
$509$	−4.24264	−0.188052	−0.0940259	−	0.995570i	$-0.529974\pi$
$509$	−4.24264	−0.188052	−0.0940259	+	0.995570i	$0.529974\pi$
$510$	11.3137	0.500979
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	16.0000	0.706417
$514$	−9.89949	−0.436648
$515$	−6.00000	−0.264392
$516$	5.65685	0.249029
$517$	4.24264	0.186591
$518$	0	0
$519$	−20.0000	−0.877903
$520$	−4.00000	−0.175412
$521$	15.5563	0.681536	0.340768	−	0.940147i	$-0.389313\pi$
$521$	15.5563	0.681536	0.340768	+	0.940147i	$0.389313\pi$
$522$	−6.00000	−0.262613
$523$	16.9706	0.742071	0.371035	−	0.928619i	$-0.379003\pi$
$523$	16.9706	0.742071	0.371035	+	0.928619i	$0.379003\pi$
$524$	−11.3137	−0.494242
$525$	0	0
$526$	−24.0000	−1.04645
$527$	8.00000	0.348485
$528$	−1.41421	−0.0615457
$529$	−19.0000	−0.826087
$530$	−16.9706	−0.737154
$531$	4.24264	0.184115
$532$	0	0
$533$	16.0000	0.693037
$534$	−10.0000	−0.432742
$535$	28.2843	1.22284
$536$	2.00000	0.0863868
$537$	28.2843	1.22056
$538$	−24.0416	−1.03651
$539$	0	0
$540$	8.00000	0.344265
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	8.48528	0.364474
$543$	−18.0000	−0.772454
$544$	−5.65685	−0.242536
$545$	−14.1421	−0.605783
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−2.00000	−0.0854358
$549$	−5.65685	−0.241429
$550$	−3.00000	−0.127920
$551$	16.9706	0.722970
$552$	2.82843	0.120386
$553$	0	0
$554$	10.0000	0.424859
$555$	4.00000	0.169791
$556$	2.82843	0.119952
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	1.41421	0.0598684
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−8.00000	−0.337760
$562$	−18.0000	−0.759284
$563$	14.1421	0.596020	0.298010	−	0.954563i	$-0.403677\pi$
$563$	14.1421	0.596020	0.298010	+	0.954563i	$0.403677\pi$
$564$	−6.00000	−0.252646
$565$	25.4558	1.07094
$566$	16.9706	0.713326
$567$	0	0
$568$	10.0000	0.419591
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−5.65685	−0.236940
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	2.82843	0.118262
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	−1.00000	−0.0416667
$577$	29.6985	1.23636	0.618182	−	0.786035i	$-0.287869\pi$
$577$	29.6985	1.23636	0.618182	+	0.786035i	$0.287869\pi$
$578$	−15.0000	−0.623918
$579$	8.48528	0.352636
$580$	8.48528	0.352332
$581$	0	0
$582$	−22.0000	−0.911929
$583$	12.0000	0.496989
$584$	−14.1421	−0.585206
$585$	−4.00000	−0.165380
$586$	16.9706	0.701047
$587$	1.41421	0.0583708	0.0291854	−	0.999574i	$-0.490709\pi$
$587$	1.41421	0.0583708	0.0291854	+	0.999574i	$0.490709\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−6.00000	−0.247016
$591$	25.4558	1.04711
$592$	−2.00000	−0.0821995
$593$	−19.7990	−0.813047	−0.406524	−	0.913640i	$-0.633259\pi$
$593$	−19.7990	−0.813047	−0.406524	+	0.913640i	$0.633259\pi$
$594$	−5.65685	−0.232104
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−30.0000	−1.22782
$598$	−5.65685	−0.231326
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	4.24264	0.173205
$601$	−45.2548	−1.84598	−0.922992	−	0.384820i	$-0.874263\pi$
$601$	−45.2548	−1.84598	−0.922992	+	0.384820i	$0.874263\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	16.0000	0.651031
$605$	−1.41421	−0.0574960
$606$	4.00000	0.162489
$607$	33.9411	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$608$	2.82843	0.114708
$609$	0	0
$610$	8.00000	0.323911
$611$	12.0000	0.485468
$612$	−5.65685	−0.228665
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−11.3137	−0.456584
$615$	11.3137	0.456213
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−6.00000	−0.241355
$619$	−12.7279	−0.511578	−0.255789	−	0.966733i	$-0.582335\pi$
$619$	−12.7279	−0.511578	−0.255789	+	0.966733i	$0.582335\pi$
$620$	−2.00000	−0.0803219
$621$	11.3137	0.454003
$622$	−9.89949	−0.396934
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−1.00000	−0.0400000
$626$	26.8701	1.07394
$627$	4.00000	0.159745
$628$	−18.3848	−0.733632
$629$	−11.3137	−0.451107
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	8.00000	0.318223
$633$	39.5980	1.57388
$634$	26.0000	1.03259
$635$	−16.9706	−0.673456
$636$	−16.9706	−0.672927
$637$	0	0
$638$	−6.00000	−0.237542
$639$	10.0000	0.395594
$640$	1.41421	0.0559017
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	28.2843	1.11629
$643$	12.7279	0.501940	0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279	0.501940	0.250970	+	0.967995i	$0.419250\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	16.0000	0.629512
$647$	−41.0122	−1.61236	−0.806178	−	0.591673i	$-0.798468\pi$
$647$	−41.0122	−1.61236	−0.806178	+	0.591673i	$0.798468\pi$
$648$	5.00000	0.196419
$649$	4.24264	0.166538
$650$	−8.48528	−0.332820
$651$	0	0
$652$	−22.0000	−0.861586
$653$	−48.0000	−1.87839	−0.939193	−	0.343391i	$-0.888424\pi$
$653$	−48.0000	−1.87839	−0.939193	+	0.343391i	$0.888424\pi$
$654$	−14.1421	−0.553001
$655$	16.0000	0.625172
$656$	−5.65685	−0.220863
$657$	−14.1421	−0.551737
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	2.00000	0.0778499
$661$	−38.1838	−1.48518	−0.742588	−	0.669748i	$-0.766402\pi$
$661$	−38.1838	−1.48518	−0.742588	+	0.669748i	$0.766402\pi$
$662$	−28.0000	−1.08825
$663$	−22.6274	−0.878776
$664$	−2.82843	−0.109764
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	12.0000	0.464642
$668$	11.3137	0.437741
$669$	22.0000	0.850569
$670$	−2.82843	−0.109272
$671$	−5.65685	−0.218380
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	2.00000	0.0770371
$675$	16.9706	0.653197
$676$	−5.00000	−0.192308
$677$	−31.1127	−1.19576	−0.597879	−	0.801586i	$-0.703990\pi$
$677$	−31.1127	−1.19576	−0.597879	+	0.801586i	$0.703990\pi$
$678$	25.4558	0.977626
$679$	0	0
$680$	8.00000	0.306786
$681$	40.0000	1.53280
$682$	1.41421	0.0541530
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	2.82843	0.108148
$685$	2.82843	0.108069
$686$	0	0
$687$	6.00000	0.228914
$688$	4.00000	0.152499
$689$	33.9411	1.29305
$690$	−4.00000	−0.152277
$691$	32.5269	1.23738	0.618691	−	0.785634i	$-0.287663\pi$
$691$	32.5269	1.23738	0.618691	+	0.785634i	$0.287663\pi$
$692$	−14.1421	−0.537603
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	8.48528	0.321634
$697$	−32.0000	−1.21209
$698$	−5.65685	−0.214115
$699$	8.48528	0.320943
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	−16.0000	−0.603881
$703$	5.65685	0.213352
$704$	−1.00000	−0.0376889
$705$	8.48528	0.319574
$706$	1.41421	0.0532246
$707$	0	0
$708$	−6.00000	−0.225494
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	−14.1421	−0.530745
$711$	8.00000	0.300023
$712$	−7.07107	−0.264999
$713$	−2.82843	−0.105925
$714$	0	0
$715$	−4.00000	−0.149592
$716$	20.0000	0.747435
$717$	−22.6274	−0.845036
$718$	−32.0000	−1.19423
$719$	15.5563	0.580154	0.290077	−	0.957003i	$-0.406319\pi$
$719$	15.5563	0.580154	0.290077	+	0.957003i	$0.406319\pi$
$720$	1.41421	0.0527046
$721$	0	0
$722$	11.0000	0.409378
$723$	8.00000	0.297523
$724$	−12.7279	−0.473029
$725$	18.0000	0.668503
$726$	−1.41421	−0.0524864
$727$	32.5269	1.20636	0.603178	−	0.797606i	$-0.293901\pi$
$727$	32.5269	1.20636	0.603178	+	0.797606i	$0.293901\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	20.0000	0.740233
$731$	22.6274	0.836905
$732$	8.00000	0.295689
$733$	16.9706	0.626822	0.313411	−	0.949618i	$-0.398528\pi$
$733$	16.9706	0.626822	0.313411	+	0.949618i	$0.398528\pi$
$734$	29.6985	1.09619
$735$	0	0
$736$	2.00000	0.0737210
$737$	2.00000	0.0736709
$738$	−5.65685	−0.208232
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	2.82843	0.103975
$741$	11.3137	0.415619
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	−2.00000	−0.0733236
$745$	14.1421	0.518128
$746$	14.0000	0.512576
$747$	−2.82843	−0.103487
$748$	−5.65685	−0.206835
$749$	0	0
$750$	−16.0000	−0.584237
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−4.24264	−0.154713
$753$	34.0000	1.23903
$754$	−16.9706	−0.618031
$755$	−22.6274	−0.823496
$756$	0	0
$757$	−44.0000	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$757$	−44.0000	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$758$	−6.00000	−0.217930
$759$	2.82843	0.102665
$760$	−4.00000	−0.145095
$761$	−25.4558	−0.922774	−0.461387	−	0.887199i	$-0.652648\pi$
$761$	−25.4558	−0.922774	−0.461387	+	0.887199i	$0.652648\pi$
$762$	−16.9706	−0.614779
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	26.8701	0.970855
$767$	12.0000	0.433295
$768$	1.41421	0.0510310
$769$	−8.48528	−0.305987	−0.152994	−	0.988227i	$-0.548891\pi$
$769$	−8.48528	−0.305987	−0.152994	+	0.988227i	$0.548891\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	6.00000	0.215945
$773$	1.41421	0.0508657	0.0254329	−	0.999677i	$-0.491904\pi$
$773$	1.41421	0.0508657	0.0254329	+	0.999677i	$0.491904\pi$
$774$	4.00000	0.143777
$775$	−4.24264	−0.152400
$776$	−15.5563	−0.558440
$777$	0	0
$778$	28.0000	1.00385
$779$	16.0000	0.573259
$780$	5.65685	0.202548
$781$	10.0000	0.357828
$782$	11.3137	0.404577
$783$	33.9411	1.21296
$784$	0	0
$785$	26.0000	0.927980
$786$	16.0000	0.570701
$787$	−42.4264	−1.51234	−0.756169	−	0.654376i	$-0.772931\pi$
$787$	−42.4264	−1.51234	−0.756169	+	0.654376i	$0.772931\pi$
$788$	18.0000	0.641223
$789$	33.9411	1.20834
$790$	−11.3137	−0.402524
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−16.0000	−0.568177
$794$	−24.0416	−0.853206
$795$	24.0000	0.851192
$796$	−21.2132	−0.751882
$797$	18.3848	0.651222	0.325611	−	0.945504i	$-0.394430\pi$
$797$	18.3848	0.651222	0.325611	+	0.945504i	$0.394430\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	3.00000	0.106066
$801$	−7.07107	−0.249844
$802$	−36.0000	−1.27120
$803$	−14.1421	−0.499065
$804$	−2.82843	−0.0997509
$805$	0	0
$806$	4.00000	0.140894
$807$	34.0000	1.19686
$808$	2.82843	0.0995037
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−7.07107	−0.248452
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	−2.00000	−0.0701000
$815$	31.1127	1.08983
$816$	8.00000	0.280056
$817$	−11.3137	−0.395817
$818$	31.1127	1.08783
$819$	0	0
$820$	8.00000	0.279372
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	2.82843	0.0986527
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−4.24264	−0.147799
$825$	4.24264	0.147710
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	2.00000	0.0695048
$829$	35.3553	1.22794	0.613971	−	0.789329i	$-0.289571\pi$
$829$	35.3553	1.22794	0.613971	+	0.789329i	$0.289571\pi$
$830$	4.00000	0.138842
$831$	−14.1421	−0.490585
$832$	−2.82843	−0.0980581
$833$	0	0
$834$	−4.00000	−0.138509
$835$	−16.0000	−0.553703
$836$	2.82843	0.0978232
$837$	−8.00000	−0.276520
$838$	−12.7279	−0.439679
$839$	43.8406	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$839$	43.8406	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−8.00000	−0.275698
$843$	25.4558	0.876746
$844$	28.0000	0.963800
$845$	7.07107	0.243252
$846$	−4.24264	−0.145865
$847$	0	0
$848$	−12.0000	−0.412082
$849$	−24.0000	−0.823678
$850$	16.9706	0.582086
$851$	4.00000	0.137118
$852$	−14.1421	−0.484502
$853$	2.82843	0.0968435	0.0484218	−	0.998827i	$-0.484581\pi$
$853$	2.82843	0.0968435	0.0484218	+	0.998827i	$0.484581\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	20.0000	0.683586
$857$	8.48528	0.289852	0.144926	−	0.989443i	$-0.453706\pi$
$857$	8.48528	0.289852	0.144926	+	0.989443i	$0.453706\pi$
$858$	−4.00000	−0.136558
$859$	−38.1838	−1.30281	−0.651407	−	0.758729i	$-0.725821\pi$
$859$	−38.1838	−1.30281	−0.651407	+	0.758729i	$0.725821\pi$
$860$	−5.65685	−0.192897
$861$	0	0
$862$	20.0000	0.681203
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	5.65685	0.192450
$865$	20.0000	0.680020
$866$	38.1838	1.29754
$867$	21.2132	0.720438
$868$	0	0
$869$	8.00000	0.271381
$870$	−12.0000	−0.406838
$871$	5.65685	0.191675
$872$	−10.0000	−0.338643
$873$	−15.5563	−0.526503
$874$	−5.65685	−0.191346
$875$	0	0
$876$	20.0000	0.675737
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	−25.4558	−0.859093
$879$	−24.0000	−0.809500
$880$	1.41421	0.0476731
$881$	−4.24264	−0.142938	−0.0714691	−	0.997443i	$-0.522769\pi$
$881$	−4.24264	−0.142938	−0.0714691	+	0.997443i	$0.522769\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−16.0000	−0.538138
$885$	8.48528	0.285230
$886$	12.0000	0.403148
$887$	−48.0833	−1.61448	−0.807239	−	0.590225i	$-0.799039\pi$
$887$	−48.0833	−1.61448	−0.807239	+	0.590225i	$0.799039\pi$
$888$	2.82843	0.0949158
$889$	0	0
$890$	10.0000	0.335201
$891$	5.00000	0.167506
$892$	15.5563	0.520865
$893$	12.0000	0.401565
$894$	14.1421	0.472984
$895$	−28.2843	−0.945439
$896$	0	0
$897$	8.00000	0.267112
$898$	12.0000	0.400445
$899$	−8.48528	−0.283000
$900$	3.00000	0.100000
$901$	−67.8823	−2.26149
$902$	−5.65685	−0.188353
$903$	0	0
$904$	18.0000	0.598671
$905$	18.0000	0.598340
$906$	−22.6274	−0.751746
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	28.2843	0.938647
$909$	2.82843	0.0938130
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	−4.00000	−0.132453
$913$	−2.82843	−0.0936073
$914$	−2.00000	−0.0661541
$915$	−11.3137	−0.374020
$916$	4.24264	0.140181
$917$	0	0
$918$	32.0000	1.05616
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−2.82843	−0.0932505
$921$	16.0000	0.527218
$922$	28.2843	0.931493
$923$	28.2843	0.930988
$924$	0	0
$925$	6.00000	0.197279
$926$	14.0000	0.460069
$927$	−4.24264	−0.139347
$928$	6.00000	0.196960
$929$	15.5563	0.510387	0.255194	−	0.966890i	$-0.417861\pi$
$929$	15.5563	0.510387	0.255194	+	0.966890i	$0.417861\pi$
$930$	2.82843	0.0927478
$931$	0	0
$932$	6.00000	0.196537
$933$	14.0000	0.458339
$934$	4.24264	0.138823
$935$	8.00000	0.261628
$936$	−2.82843	−0.0924500
$937$	25.4558	0.831606	0.415803	−	0.909455i	$-0.363501\pi$
$937$	25.4558	0.831606	0.415803	+	0.909455i	$0.363501\pi$
$938$	0	0
$939$	−38.0000	−1.24008
$940$	6.00000	0.195698
$941$	−22.6274	−0.737633	−0.368816	−	0.929502i	$-0.620237\pi$
$941$	−22.6274	−0.737633	−0.368816	+	0.929502i	$0.620237\pi$
$942$	26.0000	0.847126
$943$	11.3137	0.368425
$944$	−4.24264	−0.138086
$945$	0	0
$946$	4.00000	0.130051
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−11.3137	−0.367452
$949$	−40.0000	−1.29845
$950$	−8.48528	−0.275299
$951$	−36.7696	−1.19233
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−16.0000	−0.517477
$957$	8.48528	0.274290
$958$	25.4558	0.822441
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−29.0000	−0.935484
$962$	−5.65685	−0.182384
$963$	20.0000	0.644491
$964$	5.65685	0.182195
$965$	−8.48528	−0.273151
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	−1.00000	−0.0321412
$969$	−22.6274	−0.726897
$970$	22.0000	0.706377
$971$	−26.8701	−0.862301	−0.431151	−	0.902280i	$-0.641892\pi$
$971$	−26.8701	−0.862301	−0.431151	+	0.902280i	$0.641892\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	−10.0000	−0.320421
$975$	12.0000	0.384308
$976$	5.65685	0.181071
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	31.1127	0.994874
$979$	−7.07107	−0.225992
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−4.00000	−0.127645
$983$	−57.9828	−1.84936	−0.924681	−	0.380742i	$-0.875669\pi$
$983$	−57.9828	−1.84936	−0.924681	+	0.380742i	$0.875669\pi$
$984$	8.00000	0.255031
$985$	−25.4558	−0.811091
$986$	33.9411	1.08091
$987$	0	0
$988$	8.00000	0.254514
$989$	−8.00000	−0.254385
$990$	1.41421	0.0449467
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	−1.41421	−0.0449013
$993$	39.5980	1.25660
$994$	0	0
$995$	30.0000	0.951064
$996$	4.00000	0.126745
$997$	−59.3970	−1.88112	−0.940560	−	0.339626i	$-0.889699\pi$
$997$	−59.3970	−1.88112	−0.940560	+	0.339626i	$0.889699\pi$
$998$	30.0000	0.949633
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.q.1.2	yes	2
3.2	odd	2		9702.2.a.dq.1.2		2
4.3	odd	2		8624.2.a.bw.1.1		2
7.2	even	3		1078.2.e.s.67.1		4
7.3	odd	6		1078.2.e.s.177.2		4
7.4	even	3		1078.2.e.s.177.1		4
7.5	odd	6		1078.2.e.s.67.2		4
7.6	odd	2	inner	1078.2.a.q.1.1	✓	2
21.20	even	2		9702.2.a.dq.1.1		2
28.27	even	2		8624.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.q.1.1	✓	2	7.6	odd	2	inner
1078.2.a.q.1.2	yes	2	1.1	even	1	trivial
1078.2.e.s.67.1		4	7.2	even	3
1078.2.e.s.67.2		4	7.5	odd	6
1078.2.e.s.177.1		4	7.4	even	3
1078.2.e.s.177.2		4	7.3	odd	6
8624.2.a.bw.1.1		2	4.3	odd	2
8624.2.a.bw.1.2		2	28.27	even	2
9702.2.a.dq.1.1		2	21.20	even	2
9702.2.a.dq.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} -1.00000 q^{11} +1.41421 q^{12} -2.82843 q^{13} -2.00000 q^{15} +1.00000 q^{16} +5.65685 q^{17} +1.00000 q^{18} -2.82843 q^{19} -1.41421 q^{20} +1.00000 q^{22} -2.00000 q^{23} -1.41421 q^{24} -3.00000 q^{25} +2.82843 q^{26} -5.65685 q^{27} -6.00000 q^{29} +2.00000 q^{30} +1.41421 q^{31} -1.00000 q^{32} -1.41421 q^{33} -5.65685 q^{34} -1.00000 q^{36} -2.00000 q^{37} +2.82843 q^{38} -4.00000 q^{39} +1.41421 q^{40} -5.65685 q^{41} +4.00000 q^{43} -1.00000 q^{44} +1.41421 q^{45} +2.00000 q^{46} -4.24264 q^{47} +1.41421 q^{48} +3.00000 q^{50} +8.00000 q^{51} -2.82843 q^{52} -12.0000 q^{53} +5.65685 q^{54} +1.41421 q^{55} -4.00000 q^{57} +6.00000 q^{58} -4.24264 q^{59} -2.00000 q^{60} +5.65685 q^{61} -1.41421 q^{62} +1.00000 q^{64} +4.00000 q^{65} +1.41421 q^{66} -2.00000 q^{67} +5.65685 q^{68} -2.82843 q^{69} -10.0000 q^{71} +1.00000 q^{72} +14.1421 q^{73} +2.00000 q^{74} -4.24264 q^{75} -2.82843 q^{76} +4.00000 q^{78} -8.00000 q^{79} -1.41421 q^{80} -5.00000 q^{81} +5.65685 q^{82} +2.82843 q^{83} -8.00000 q^{85} -4.00000 q^{86} -8.48528 q^{87} +1.00000 q^{88} +7.07107 q^{89} -1.41421 q^{90} -2.00000 q^{92} +2.00000 q^{93} +4.24264 q^{94} +4.00000 q^{95} -1.41421 q^{96} +15.5563 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 2 q^{32} - 2 q^{36} - 4 q^{37} - 8 q^{39} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 6 q^{50} + 16 q^{51} - 24 q^{53} - 8 q^{57} + 12 q^{58} - 4 q^{60} + 2 q^{64} + 8 q^{65} - 4 q^{67} - 20 q^{71} + 2 q^{72} + 4 q^{74} + 8 q^{78} - 16 q^{79} - 10 q^{81} - 16 q^{85} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{93} + 8 q^{95} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 2 * q^11 - 4 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^22 - 4 * q^23 - 6 * q^25 - 12 * q^29 + 4 * q^30 - 2 * q^32 - 2 * q^36 - 4 * q^37 - 8 * q^39 + 8 * q^43 - 2 * q^44 + 4 * q^46 + 6 * q^50 + 16 * q^51 - 24 * q^53 - 8 * q^57 + 12 * q^58 - 4 * q^60 + 2 * q^64 + 8 * q^65 - 4 * q^67 - 20 * q^71 + 2 * q^72 + 4 * q^74 + 8 * q^78 - 16 * q^79 - 10 * q^81 - 16 * q^85 - 8 * q^86 + 2 * q^88 - 4 * q^92 + 4 * q^93 + 8 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

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Database

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