LMFDB - Embedded newform 8550.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{7} -1.00000 q^{8} -1.41421 q^{11} -0.171573 q^{13} +1.58579 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} +1.41421 q^{22} +9.24264 q^{23} +0.171573 q^{26} -1.58579 q^{28} +5.82843 q^{29} -2.24264 q^{31} -1.00000 q^{32} -1.00000 q^{34} -8.48528 q^{37} +1.00000 q^{38} -4.24264 q^{41} -10.2426 q^{43} -1.41421 q^{44} -9.24264 q^{46} -4.48528 q^{49} -0.171573 q^{52} -11.4853 q^{53} +1.58579 q^{56} -5.82843 q^{58} +12.8995 q^{59} +5.75736 q^{61} +2.24264 q^{62} +1.00000 q^{64} -13.2426 q^{67} +1.00000 q^{68} +10.5858 q^{71} +5.48528 q^{73} +8.48528 q^{74} -1.00000 q^{76} +2.24264 q^{77} +10.4853 q^{79} +4.24264 q^{82} -2.48528 q^{83} +10.2426 q^{86} +1.41421 q^{88} +7.07107 q^{89} +0.272078 q^{91} +9.24264 q^{92} +11.6569 q^{97} +4.48528 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} - 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 6 q^{26} - 6 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 12 q^{43} - 10 q^{46} + 8 q^{49} - 6 q^{52} - 6 q^{53} + 6 q^{56} - 6 q^{58} + 6 q^{59} + 20 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{67} + 2 q^{68} + 24 q^{71} - 6 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} + 12 q^{83} + 12 q^{86} + 26 q^{91} + 10 q^{92} + 12 q^{97} - 8 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 - 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 6 * q^26 - 6 * q^28 + 6 * q^29 + 4 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 12 * q^43 - 10 * q^46 + 8 * q^49 - 6 * q^52 - 6 * q^53 + 6 * q^56 - 6 * q^58 + 6 * q^59 + 20 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^67 + 2 * q^68 + 24 * q^71 - 6 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^83 + 12 * q^86 + 26 * q^91 + 10 * q^92 + 12 * q^97 - 8 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.58579	−0.599371	−0.299685	−	0.954038i	$-0.596882\pi$
$7$	−1.58579	−0.599371	−0.299685	+	0.954038i	$0.596882\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−0.171573	−0.0475858	−0.0237929	−	0.999717i	$-0.507574\pi$
$13$	−0.171573	−0.0475858	−0.0237929	+	0.999717i	$0.507574\pi$
$14$	1.58579	0.423819
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	1.41421	0.301511
$23$	9.24264	1.92722	0.963612	−	0.267305i	$-0.0861332\pi$
$23$	9.24264	1.92722	0.963612	+	0.267305i	$0.0861332\pi$
$24$	0	0
$25$	0	0
$26$	0.171573	0.0336482
$27$	0	0
$28$	−1.58579	−0.299685
$29$	5.82843	1.08231	0.541156	−	0.840922i	$-0.317987\pi$
$29$	5.82843	1.08231	0.541156	+	0.840922i	$0.317987\pi$
$30$	0	0
$31$	−2.24264	−0.402790	−0.201395	−	0.979510i	$-0.564548\pi$
$31$	−2.24264	−0.402790	−0.201395	+	0.979510i	$0.564548\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	0	0
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	−1.41421	−0.213201
$45$	0	0
$46$	−9.24264	−1.36275
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−4.48528	−0.640754
$50$	0	0
$51$	0	0
$52$	−0.171573	−0.0237929
$53$	−11.4853	−1.57762	−0.788812	−	0.614634i	$-0.789304\pi$
$53$	−11.4853	−1.57762	−0.788812	+	0.614634i	$0.789304\pi$
$54$	0	0
$55$	0	0
$56$	1.58579	0.211910
$57$	0	0
$58$	−5.82843	−0.765310
$59$	12.8995	1.67937	0.839686	−	0.543073i	$-0.182739\pi$
$59$	12.8995	1.67937	0.839686	+	0.543073i	$0.182739\pi$
$60$	0	0
$61$	5.75736	0.737154	0.368577	−	0.929597i	$-0.379845\pi$
$61$	5.75736	0.737154	0.368577	+	0.929597i	$0.379845\pi$
$62$	2.24264	0.284816
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−13.2426	−1.61785	−0.808923	−	0.587915i	$-0.799949\pi$
$67$	−13.2426	−1.61785	−0.808923	+	0.587915i	$0.799949\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	10.5858	1.25630	0.628151	−	0.778092i	$-0.283812\pi$
$71$	10.5858	1.25630	0.628151	+	0.778092i	$0.283812\pi$
$72$	0	0
$73$	5.48528	0.642004	0.321002	−	0.947079i	$-0.395980\pi$
$73$	5.48528	0.642004	0.321002	+	0.947079i	$0.395980\pi$
$74$	8.48528	0.986394
$75$	0	0
$76$	−1.00000	−0.114708
$77$	2.24264	0.255573
$78$	0	0
$79$	10.4853	1.17969	0.589843	−	0.807518i	$-0.299190\pi$
$79$	10.4853	1.17969	0.589843	+	0.807518i	$0.299190\pi$
$80$	0	0
$81$	0	0
$82$	4.24264	0.468521
$83$	−2.48528	−0.272795	−0.136398	−	0.990654i	$-0.543552\pi$
$83$	−2.48528	−0.272795	−0.136398	+	0.990654i	$0.543552\pi$
$84$	0	0
$85$	0	0
$86$	10.2426	1.10449
$87$	0	0
$88$	1.41421	0.150756
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	0.272078	0.0285215
$92$	9.24264	0.963612
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	4.48528	0.453082
$99$	0	0
$100$	0	0
$101$	1.07107	0.106575	0.0532876	−	0.998579i	$-0.483030\pi$
$101$	1.07107	0.106575	0.0532876	+	0.998579i	$0.483030\pi$
$102$	0	0
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	0.171573	0.0168241
$105$	0	0
$106$	11.4853	1.11555
$107$	−5.72792	−0.553739	−0.276870	−	0.960908i	$-0.589297\pi$
$107$	−5.72792	−0.553739	−0.276870	+	0.960908i	$0.589297\pi$
$108$	0	0
$109$	15.9706	1.52970	0.764851	−	0.644207i	$-0.222812\pi$
$109$	15.9706	1.52970	0.764851	+	0.644207i	$0.222812\pi$
$110$	0	0
$111$	0	0
$112$	−1.58579	−0.149843
$113$	−1.75736	−0.165318	−0.0826592	−	0.996578i	$-0.526341\pi$
$113$	−1.75736	−0.165318	−0.0826592	+	0.996578i	$0.526341\pi$
$114$	0	0
$115$	0	0
$116$	5.82843	0.541156
$117$	0	0
$118$	−12.8995	−1.18749
$119$	−1.58579	−0.145369
$120$	0	0
$121$	−9.00000	−0.818182
$122$	−5.75736	−0.521247
$123$	0	0
$124$	−2.24264	−0.201395
$125$	0	0
$126$	0	0
$127$	−14.4853	−1.28536	−0.642680	−	0.766134i	$-0.722178\pi$
$127$	−14.4853	−1.28536	−0.642680	+	0.766134i	$0.722178\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	1.58579	0.137505
$134$	13.2426	1.14399
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	13.0000	1.11066	0.555332	−	0.831628i	$-0.312591\pi$
$137$	13.0000	1.11066	0.555332	+	0.831628i	$0.312591\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	−10.5858	−0.888339
$143$	0.242641	0.0202906
$144$	0	0
$145$	0	0
$146$	−5.48528	−0.453965
$147$	0	0
$148$	−8.48528	−0.697486
$149$	−6.34315	−0.519651	−0.259825	−	0.965656i	$-0.583665\pi$
$149$	−6.34315	−0.519651	−0.259825	+	0.965656i	$0.583665\pi$
$150$	0	0
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−2.24264	−0.180717
$155$	0	0
$156$	0	0
$157$	−0.343146	−0.0273860	−0.0136930	−	0.999906i	$-0.504359\pi$
$157$	−0.343146	−0.0273860	−0.0136930	+	0.999906i	$0.504359\pi$
$158$	−10.4853	−0.834164
$159$	0	0
$160$	0	0
$161$	−14.6569	−1.15512
$162$	0	0
$163$	1.75736	0.137647	0.0688235	−	0.997629i	$-0.478075\pi$
$163$	1.75736	0.137647	0.0688235	+	0.997629i	$0.478075\pi$
$164$	−4.24264	−0.331295
$165$	0	0
$166$	2.48528	0.192895
$167$	−9.75736	−0.755047	−0.377524	−	0.926000i	$-0.623224\pi$
$167$	−9.75736	−0.755047	−0.377524	+	0.926000i	$0.623224\pi$
$168$	0	0
$169$	−12.9706	−0.997736
$170$	0	0
$171$	0	0
$172$	−10.2426	−0.780994
$173$	−16.4853	−1.25335	−0.626676	−	0.779280i	$-0.715585\pi$
$173$	−16.4853	−1.25335	−0.626676	+	0.779280i	$0.715585\pi$
$174$	0	0
$175$	0	0
$176$	−1.41421	−0.106600
$177$	0	0
$178$	−7.07107	−0.529999
$179$	0.343146	0.0256479	0.0128240	−	0.999918i	$-0.495918\pi$
$179$	0.343146	0.0256479	0.0128240	+	0.999918i	$0.495918\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	−0.272078	−0.0201678
$183$	0	0
$184$	−9.24264	−0.681377
$185$	0	0
$186$	0	0
$187$	−1.41421	−0.103418
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.5563	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$191$	18.5563	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$192$	0	0
$193$	11.6569	0.839079	0.419539	−	0.907737i	$-0.362192\pi$
$193$	11.6569	0.839079	0.419539	+	0.907737i	$0.362192\pi$
$194$	−11.6569	−0.836913
$195$	0	0
$196$	−4.48528	−0.320377
$197$	−20.2426	−1.44223	−0.721114	−	0.692816i	$-0.756370\pi$
$197$	−20.2426	−1.44223	−0.721114	+	0.692816i	$0.756370\pi$
$198$	0	0
$199$	0.757359	0.0536878	0.0268439	−	0.999640i	$-0.491454\pi$
$199$	0.757359	0.0536878	0.0268439	+	0.999640i	$0.491454\pi$
$200$	0	0
$201$	0	0
$202$	−1.07107	−0.0753601
$203$	−9.24264	−0.648706
$204$	0	0
$205$	0	0
$206$	−4.24264	−0.295599
$207$	0	0
$208$	−0.171573	−0.0118964
$209$	1.41421	0.0978232
$210$	0	0
$211$	5.72792	0.394326	0.197163	−	0.980371i	$-0.436827\pi$
$211$	5.72792	0.394326	0.197163	+	0.980371i	$0.436827\pi$
$212$	−11.4853	−0.788812
$213$	0	0
$214$	5.72792	0.391553
$215$	0	0
$216$	0	0
$217$	3.55635	0.241421
$218$	−15.9706	−1.08166
$219$	0	0
$220$	0	0
$221$	−0.171573	−0.0115412
$222$	0	0
$223$	20.8284	1.39477	0.697387	−	0.716694i	$-0.254346\pi$
$223$	20.8284	1.39477	0.697387	+	0.716694i	$0.254346\pi$
$224$	1.58579	0.105955
$225$	0	0
$226$	1.75736	0.116898
$227$	−25.2426	−1.67541	−0.837706	−	0.546121i	$-0.816104\pi$
$227$	−25.2426	−1.67541	−0.837706	+	0.546121i	$0.816104\pi$
$228$	0	0
$229$	−18.9706	−1.25361	−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706	−1.25361	−0.626805	+	0.779176i	$0.715638\pi$
$230$	0	0
$231$	0	0
$232$	−5.82843	−0.382655
$233$	8.97056	0.587681	0.293841	−	0.955854i	$-0.405067\pi$
$233$	8.97056	0.587681	0.293841	+	0.955854i	$0.405067\pi$
$234$	0	0
$235$	0	0
$236$	12.8995	0.839686
$237$	0	0
$238$	1.58579	0.102791
$239$	−12.8995	−0.834399	−0.417199	−	0.908815i	$-0.636988\pi$
$239$	−12.8995	−0.834399	−0.417199	+	0.908815i	$0.636988\pi$
$240$	0	0
$241$	−24.9706	−1.60850	−0.804248	−	0.594294i	$-0.797431\pi$
$241$	−24.9706	−1.60850	−0.804248	+	0.594294i	$0.797431\pi$
$242$	9.00000	0.578542
$243$	0	0
$244$	5.75736	0.368577
$245$	0	0
$246$	0	0
$247$	0.171573	0.0109169
$248$	2.24264	0.142408
$249$	0	0
$250$	0	0
$251$	27.5563	1.73934	0.869671	−	0.493632i	$-0.164331\pi$
$251$	27.5563	1.73934	0.869671	+	0.493632i	$0.164331\pi$
$252$	0	0
$253$	−13.0711	−0.821771
$254$	14.4853	0.908887
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.72792	−0.294920	−0.147460	−	0.989068i	$-0.547110\pi$
$257$	−4.72792	−0.294920	−0.147460	+	0.989068i	$0.547110\pi$
$258$	0	0
$259$	13.4558	0.836105
$260$	0	0
$261$	0	0
$262$	16.9706	1.04844
$263$	6.97056	0.429823	0.214912	−	0.976633i	$-0.431054\pi$
$263$	6.97056	0.429823	0.214912	+	0.976633i	$0.431054\pi$
$264$	0	0
$265$	0	0
$266$	−1.58579	−0.0972308
$267$	0	0
$268$	−13.2426	−0.808923
$269$	−28.6274	−1.74544	−0.872722	−	0.488217i	$-0.837647\pi$
$269$	−28.6274	−1.74544	−0.872722	+	0.488217i	$0.837647\pi$
$270$	0	0
$271$	−18.7574	−1.13943	−0.569714	−	0.821843i	$-0.692946\pi$
$271$	−18.7574	−1.13943	−0.569714	+	0.821843i	$0.692946\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−13.0000	−0.785359
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	0	0
$281$	4.24264	0.253095	0.126547	−	0.991961i	$-0.459610\pi$
$281$	4.24264	0.253095	0.126547	+	0.991961i	$0.459610\pi$
$282$	0	0
$283$	−3.85786	−0.229326	−0.114663	−	0.993404i	$-0.536579\pi$
$283$	−3.85786	−0.229326	−0.114663	+	0.993404i	$0.536579\pi$
$284$	10.5858	0.628151
$285$	0	0
$286$	−0.242641	−0.0143476
$287$	6.72792	0.397137
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	5.48528	0.321002
$293$	11.4853	0.670977	0.335489	−	0.942044i	$-0.391099\pi$
$293$	11.4853	0.670977	0.335489	+	0.942044i	$0.391099\pi$
$294$	0	0
$295$	0	0
$296$	8.48528	0.493197
$297$	0	0
$298$	6.34315	0.367449
$299$	−1.58579	−0.0917084
$300$	0	0
$301$	16.2426	0.936210
$302$	−6.48528	−0.373186
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	6.34315	0.362022	0.181011	−	0.983481i	$-0.442063\pi$
$307$	6.34315	0.362022	0.181011	+	0.983481i	$0.442063\pi$
$308$	2.24264	0.127786
$309$	0	0
$310$	0	0
$311$	13.2426	0.750921	0.375461	−	0.926838i	$-0.377485\pi$
$311$	13.2426	0.750921	0.375461	+	0.926838i	$0.377485\pi$
$312$	0	0
$313$	−25.9706	−1.46794	−0.733971	−	0.679180i	$-0.762335\pi$
$313$	−25.9706	−1.46794	−0.733971	+	0.679180i	$0.762335\pi$
$314$	0.343146	0.0193648
$315$	0	0
$316$	10.4853	0.589843
$317$	7.48528	0.420415	0.210208	−	0.977657i	$-0.432586\pi$
$317$	7.48528	0.420415	0.210208	+	0.977657i	$0.432586\pi$
$318$	0	0
$319$	−8.24264	−0.461499
$320$	0	0
$321$	0	0
$322$	14.6569	0.816795
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	0	0
$326$	−1.75736	−0.0973311
$327$	0	0
$328$	4.24264	0.234261
$329$	0	0
$330$	0	0
$331$	−10.7574	−0.591278	−0.295639	−	0.955300i	$-0.595533\pi$
$331$	−10.7574	−0.591278	−0.295639	+	0.955300i	$0.595533\pi$
$332$	−2.48528	−0.136398
$333$	0	0
$334$	9.75736	0.533899
$335$	0	0
$336$	0	0
$337$	−33.8995	−1.84662	−0.923312	−	0.384052i	$-0.874528\pi$
$337$	−33.8995	−1.84662	−0.923312	+	0.384052i	$0.874528\pi$
$338$	12.9706	0.705506
$339$	0	0
$340$	0	0
$341$	3.17157	0.171750
$342$	0	0
$343$	18.2132	0.983421
$344$	10.2426	0.552246
$345$	0	0
$346$	16.4853	0.886254
$347$	−22.4853	−1.20707	−0.603537	−	0.797335i	$-0.706242\pi$
$347$	−22.4853	−1.20707	−0.603537	+	0.797335i	$0.706242\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	1.41421	0.0753778
$353$	−19.4853	−1.03710	−0.518548	−	0.855048i	$-0.673527\pi$
$353$	−19.4853	−1.03710	−0.518548	+	0.855048i	$0.673527\pi$
$354$	0	0
$355$	0	0
$356$	7.07107	0.374766
$357$	0	0
$358$	−0.343146	−0.0181358
$359$	31.2426	1.64892	0.824462	−	0.565918i	$-0.191478\pi$
$359$	31.2426	1.64892	0.824462	+	0.565918i	$0.191478\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−8.48528	−0.445976
$363$	0	0
$364$	0.272078	0.0142608
$365$	0	0
$366$	0	0
$367$	−25.4558	−1.32878	−0.664392	−	0.747384i	$-0.731309\pi$
$367$	−25.4558	−1.32878	−0.664392	+	0.747384i	$0.731309\pi$
$368$	9.24264	0.481806
$369$	0	0
$370$	0	0
$371$	18.2132	0.945582
$372$	0	0
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	1.41421	0.0731272
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	2.75736	0.141636	0.0708180	−	0.997489i	$-0.477439\pi$
$379$	2.75736	0.141636	0.0708180	+	0.997489i	$0.477439\pi$
$380$	0	0
$381$	0	0
$382$	−18.5563	−0.949425
$383$	3.75736	0.191992	0.0959960	−	0.995382i	$-0.469396\pi$
$383$	3.75736	0.191992	0.0959960	+	0.995382i	$0.469396\pi$
$384$	0	0
$385$	0	0
$386$	−11.6569	−0.593318
$387$	0	0
$388$	11.6569	0.591787
$389$	−37.0711	−1.87958	−0.939789	−	0.341756i	$-0.888979\pi$
$389$	−37.0711	−1.87958	−0.939789	+	0.341756i	$0.888979\pi$
$390$	0	0
$391$	9.24264	0.467420
$392$	4.48528	0.226541
$393$	0	0
$394$	20.2426	1.01981
$395$	0	0
$396$	0	0
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	−0.757359	−0.0379630
$399$	0	0
$400$	0	0
$401$	22.5858	1.12788	0.563940	−	0.825816i	$-0.309285\pi$
$401$	22.5858	1.12788	0.563940	+	0.825816i	$0.309285\pi$
$402$	0	0
$403$	0.384776	0.0191671
$404$	1.07107	0.0532876
$405$	0	0
$406$	9.24264	0.458705
$407$	12.0000	0.594818
$408$	0	0
$409$	−17.2132	−0.851138	−0.425569	−	0.904926i	$-0.639926\pi$
$409$	−17.2132	−0.851138	−0.425569	+	0.904926i	$0.639926\pi$
$410$	0	0
$411$	0	0
$412$	4.24264	0.209020
$413$	−20.4558	−1.00657
$414$	0	0
$415$	0	0
$416$	0.171573	0.00841205
$417$	0	0
$418$	−1.41421	−0.0691714
$419$	−16.5858	−0.810269	−0.405134	−	0.914257i	$-0.632775\pi$
$419$	−16.5858	−0.810269	−0.405134	+	0.914257i	$0.632775\pi$
$420$	0	0
$421$	2.51472	0.122560	0.0612799	−	0.998121i	$-0.480482\pi$
$421$	2.51472	0.122560	0.0612799	+	0.998121i	$0.480482\pi$
$422$	−5.72792	−0.278831
$423$	0	0
$424$	11.4853	0.557775
$425$	0	0
$426$	0	0
$427$	−9.12994	−0.441829
$428$	−5.72792	−0.276870
$429$	0	0
$430$	0	0
$431$	−30.3848	−1.46358	−0.731792	−	0.681528i	$-0.761316\pi$
$431$	−30.3848	−1.46358	−0.731792	+	0.681528i	$0.761316\pi$
$432$	0	0
$433$	−21.5563	−1.03593	−0.517966	−	0.855401i	$-0.673311\pi$
$433$	−21.5563	−1.03593	−0.517966	+	0.855401i	$0.673311\pi$
$434$	−3.55635	−0.170710
$435$	0	0
$436$	15.9706	0.764851
$437$	−9.24264	−0.442135
$438$	0	0
$439$	−14.2426	−0.679764	−0.339882	−	0.940468i	$-0.610387\pi$
$439$	−14.2426	−0.679764	−0.339882	+	0.940468i	$0.610387\pi$
$440$	0	0
$441$	0	0
$442$	0.171573	0.00816089
$443$	−4.24264	−0.201574	−0.100787	−	0.994908i	$-0.532136\pi$
$443$	−4.24264	−0.201574	−0.100787	+	0.994908i	$0.532136\pi$
$444$	0	0
$445$	0	0
$446$	−20.8284	−0.986255
$447$	0	0
$448$	−1.58579	−0.0749214
$449$	5.31371	0.250769	0.125385	−	0.992108i	$-0.459983\pi$
$449$	5.31371	0.250769	0.125385	+	0.992108i	$0.459983\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−1.75736	−0.0826592
$453$	0	0
$454$	25.2426	1.18470
$455$	0	0
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	18.9706	0.886436
$459$	0	0
$460$	0	0
$461$	−27.5563	−1.28343	−0.641714	−	0.766944i	$-0.721776\pi$
$461$	−27.5563	−1.28343	−0.641714	+	0.766944i	$0.721776\pi$
$462$	0	0
$463$	14.1421	0.657241	0.328620	−	0.944462i	$-0.393416\pi$
$463$	14.1421	0.657241	0.328620	+	0.944462i	$0.393416\pi$
$464$	5.82843	0.270578
$465$	0	0
$466$	−8.97056	−0.415553
$467$	−24.7279	−1.14427	−0.572136	−	0.820159i	$-0.693885\pi$
$467$	−24.7279	−1.14427	−0.572136	+	0.820159i	$0.693885\pi$
$468$	0	0
$469$	21.0000	0.969690
$470$	0	0
$471$	0	0
$472$	−12.8995	−0.593747
$473$	14.4853	0.666034
$474$	0	0
$475$	0	0
$476$	−1.58579	−0.0726844
$477$	0	0
$478$	12.8995	0.590009
$479$	−31.1127	−1.42158	−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127	−1.42158	−0.710788	+	0.703407i	$0.751661\pi$
$480$	0	0
$481$	1.45584	0.0663808
$482$	24.9706	1.13738
$483$	0	0
$484$	−9.00000	−0.409091
$485$	0	0
$486$	0	0
$487$	1.79899	0.0815200	0.0407600	−	0.999169i	$-0.487022\pi$
$487$	1.79899	0.0815200	0.0407600	+	0.999169i	$0.487022\pi$
$488$	−5.75736	−0.260623
$489$	0	0
$490$	0	0
$491$	2.44365	0.110280	0.0551402	−	0.998479i	$-0.482439\pi$
$491$	2.44365	0.110280	0.0551402	+	0.998479i	$0.482439\pi$
$492$	0	0
$493$	5.82843	0.262499
$494$	−0.171573	−0.00771943
$495$	0	0
$496$	−2.24264	−0.100698
$497$	−16.7868	−0.752991
$498$	0	0
$499$	−34.2426	−1.53291	−0.766456	−	0.642297i	$-0.777981\pi$
$499$	−34.2426	−1.53291	−0.766456	+	0.642297i	$0.777981\pi$
$500$	0	0
$501$	0	0
$502$	−27.5563	−1.22990
$503$	39.7279	1.77138	0.885690	−	0.464277i	$-0.153686\pi$
$503$	39.7279	1.77138	0.885690	+	0.464277i	$0.153686\pi$
$504$	0	0
$505$	0	0
$506$	13.0711	0.581080
$507$	0	0
$508$	−14.4853	−0.642680
$509$	4.97056	0.220316	0.110158	−	0.993914i	$-0.464864\pi$
$509$	4.97056	0.220316	0.110158	+	0.993914i	$0.464864\pi$
$510$	0	0
$511$	−8.69848	−0.384798
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.72792	0.208540
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−13.4558	−0.591216
$519$	0	0
$520$	0	0
$521$	−0.686292	−0.0300670	−0.0150335	−	0.999887i	$-0.504785\pi$
$521$	−0.686292	−0.0300670	−0.0150335	+	0.999887i	$0.504785\pi$
$522$	0	0
$523$	27.7279	1.21246	0.606229	−	0.795290i	$-0.292682\pi$
$523$	27.7279	1.21246	0.606229	+	0.795290i	$0.292682\pi$
$524$	−16.9706	−0.741362
$525$	0	0
$526$	−6.97056	−0.303931
$527$	−2.24264	−0.0976910
$528$	0	0
$529$	62.4264	2.71419
$530$	0	0
$531$	0	0
$532$	1.58579	0.0687526
$533$	0.727922	0.0315298
$534$	0	0
$535$	0	0
$536$	13.2426	0.571995
$537$	0	0
$538$	28.6274	1.23422
$539$	6.34315	0.273219
$540$	0	0
$541$	18.2426	0.784312	0.392156	−	0.919899i	$-0.371729\pi$
$541$	18.2426	0.784312	0.392156	+	0.919899i	$0.371729\pi$
$542$	18.7574	0.805698
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	−5.31371	−0.227198	−0.113599	−	0.993527i	$-0.536238\pi$
$547$	−5.31371	−0.227198	−0.113599	+	0.993527i	$0.536238\pi$
$548$	13.0000	0.555332
$549$	0	0
$550$	0	0
$551$	−5.82843	−0.248299
$552$	0	0
$553$	−16.6274	−0.707070
$554$	0	0
$555$	0	0
$556$	−12.0000	−0.508913
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	1.75736	0.0743284
$560$	0	0
$561$	0	0
$562$	−4.24264	−0.178965
$563$	−28.9706	−1.22096	−0.610482	−	0.792030i	$-0.709024\pi$
$563$	−28.9706	−1.22096	−0.610482	+	0.792030i	$0.709024\pi$
$564$	0	0
$565$	0	0
$566$	3.85786	0.162158
$567$	0	0
$568$	−10.5858	−0.444170
$569$	−28.2843	−1.18574	−0.592869	−	0.805299i	$-0.702005\pi$
$569$	−28.2843	−1.18574	−0.592869	+	0.805299i	$0.702005\pi$
$570$	0	0
$571$	6.24264	0.261246	0.130623	−	0.991432i	$-0.458302\pi$
$571$	6.24264	0.261246	0.130623	+	0.991432i	$0.458302\pi$
$572$	0.242641	0.0101453
$573$	0	0
$574$	−6.72792	−0.280818
$575$	0	0
$576$	0	0
$577$	−2.31371	−0.0963209	−0.0481605	−	0.998840i	$-0.515336\pi$
$577$	−2.31371	−0.0963209	−0.0481605	+	0.998840i	$0.515336\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	3.94113	0.163505
$582$	0	0
$583$	16.2426	0.672701
$584$	−5.48528	−0.226983
$585$	0	0
$586$	−11.4853	−0.474453
$587$	21.7574	0.898022	0.449011	−	0.893526i	$-0.351776\pi$
$587$	21.7574	0.898022	0.449011	+	0.893526i	$0.351776\pi$
$588$	0	0
$589$	2.24264	0.0924064
$590$	0	0
$591$	0	0
$592$	−8.48528	−0.348743
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	0	0
$596$	−6.34315	−0.259825
$597$	0	0
$598$	1.58579	0.0648476
$599$	−27.2132	−1.11190	−0.555951	−	0.831215i	$-0.687646\pi$
$599$	−27.2132	−1.11190	−0.555951	+	0.831215i	$0.687646\pi$
$600$	0	0
$601$	−11.7574	−0.479593	−0.239796	−	0.970823i	$-0.577081\pi$
$601$	−11.7574	−0.479593	−0.239796	+	0.970823i	$0.577081\pi$
$602$	−16.2426	−0.662001
$603$	0	0
$604$	6.48528	0.263882
$605$	0	0
$606$	0	0
$607$	−8.82843	−0.358335	−0.179167	−	0.983819i	$-0.557340\pi$
$607$	−8.82843	−0.358335	−0.179167	+	0.983819i	$0.557340\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−47.6985	−1.92652	−0.963262	−	0.268564i	$-0.913451\pi$
$613$	−47.6985	−1.92652	−0.963262	+	0.268564i	$0.913451\pi$
$614$	−6.34315	−0.255989
$615$	0	0
$616$	−2.24264	−0.0903586
$617$	12.4853	0.502639	0.251319	−	0.967904i	$-0.419136\pi$
$617$	12.4853	0.502639	0.251319	+	0.967904i	$0.419136\pi$
$618$	0	0
$619$	16.2426	0.652847	0.326423	−	0.945224i	$-0.394156\pi$
$619$	16.2426	0.652847	0.326423	+	0.945224i	$0.394156\pi$
$620$	0	0
$621$	0	0
$622$	−13.2426	−0.530982
$623$	−11.2132	−0.449248
$624$	0	0
$625$	0	0
$626$	25.9706	1.03799
$627$	0	0
$628$	−0.343146	−0.0136930
$629$	−8.48528	−0.338330
$630$	0	0
$631$	−5.02944	−0.200219	−0.100109	−	0.994976i	$-0.531919\pi$
$631$	−5.02944	−0.200219	−0.100109	+	0.994976i	$0.531919\pi$
$632$	−10.4853	−0.417082
$633$	0	0
$634$	−7.48528	−0.297279
$635$	0	0
$636$	0	0
$637$	0.769553	0.0304908
$638$	8.24264	0.326329
$639$	0	0
$640$	0	0
$641$	−30.0416	−1.18657	−0.593287	−	0.804991i	$-0.702170\pi$
$641$	−30.0416	−1.18657	−0.593287	+	0.804991i	$0.702170\pi$
$642$	0	0
$643$	−2.48528	−0.0980099	−0.0490050	−	0.998799i	$-0.515605\pi$
$643$	−2.48528	−0.0980099	−0.0490050	+	0.998799i	$0.515605\pi$
$644$	−14.6569	−0.577561
$645$	0	0
$646$	1.00000	0.0393445
$647$	−27.2426	−1.07102	−0.535509	−	0.844529i	$-0.679880\pi$
$647$	−27.2426	−1.07102	−0.535509	+	0.844529i	$0.679880\pi$
$648$	0	0
$649$	−18.2426	−0.716086
$650$	0	0
$651$	0	0
$652$	1.75736	0.0688235
$653$	4.97056	0.194513	0.0972566	−	0.995259i	$-0.468993\pi$
$653$	4.97056	0.194513	0.0972566	+	0.995259i	$0.468993\pi$
$654$	0	0
$655$	0	0
$656$	−4.24264	−0.165647
$657$	0	0
$658$	0	0
$659$	0.899495	0.0350393	0.0175197	−	0.999847i	$-0.494423\pi$
$659$	0.899495	0.0350393	0.0175197	+	0.999847i	$0.494423\pi$
$660$	0	0
$661$	32.4558	1.26239	0.631193	−	0.775626i	$-0.282566\pi$
$661$	32.4558	1.26239	0.631193	+	0.775626i	$0.282566\pi$
$662$	10.7574	0.418097
$663$	0	0
$664$	2.48528	0.0964476
$665$	0	0
$666$	0	0
$667$	53.8701	2.08586
$668$	−9.75736	−0.377524
$669$	0	0
$670$	0	0
$671$	−8.14214	−0.314324
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	33.8995	1.30576
$675$	0	0
$676$	−12.9706	−0.498868
$677$	26.9411	1.03543	0.517716	−	0.855553i	$-0.326782\pi$
$677$	26.9411	1.03543	0.517716	+	0.855553i	$0.326782\pi$
$678$	0	0
$679$	−18.4853	−0.709400
$680$	0	0
$681$	0	0
$682$	−3.17157	−0.121446
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	0	0
$686$	−18.2132	−0.695383
$687$	0	0
$688$	−10.2426	−0.390497
$689$	1.97056	0.0750725
$690$	0	0
$691$	5.51472	0.209790	0.104895	−	0.994483i	$-0.466549\pi$
$691$	5.51472	0.209790	0.104895	+	0.994483i	$0.466549\pi$
$692$	−16.4853	−0.626676
$693$	0	0
$694$	22.4853	0.853530
$695$	0	0
$696$	0	0
$697$	−4.24264	−0.160701
$698$	−6.00000	−0.227103
$699$	0	0
$700$	0	0
$701$	16.9706	0.640969	0.320485	−	0.947254i	$-0.396154\pi$
$701$	16.9706	0.640969	0.320485	+	0.947254i	$0.396154\pi$
$702$	0	0
$703$	8.48528	0.320028
$704$	−1.41421	−0.0533002
$705$	0	0
$706$	19.4853	0.733338
$707$	−1.69848	−0.0638781
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	0	0
$712$	−7.07107	−0.264999
$713$	−20.7279	−0.776267
$714$	0	0
$715$	0	0
$716$	0.343146	0.0128240
$717$	0	0
$718$	−31.2426	−1.16596
$719$	24.8995	0.928594	0.464297	−	0.885679i	$-0.346307\pi$
$719$	24.8995	0.928594	0.464297	+	0.885679i	$0.346307\pi$
$720$	0	0
$721$	−6.72792	−0.250561
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	8.48528	0.315353
$725$	0	0
$726$	0	0
$727$	15.7279	0.583316	0.291658	−	0.956523i	$-0.405793\pi$
$727$	15.7279	0.583316	0.291658	+	0.956523i	$0.405793\pi$
$728$	−0.272078	−0.0100839
$729$	0	0
$730$	0	0
$731$	−10.2426	−0.378838
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	25.4558	0.939592
$735$	0	0
$736$	−9.24264	−0.340688
$737$	18.7279	0.689852
$738$	0	0
$739$	26.7279	0.983203	0.491601	−	0.870820i	$-0.336412\pi$
$739$	26.7279	0.983203	0.491601	+	0.870820i	$0.336412\pi$
$740$	0	0
$741$	0	0
$742$	−18.2132	−0.668628
$743$	−18.7279	−0.687061	−0.343530	−	0.939142i	$-0.611623\pi$
$743$	−18.7279	−0.687061	−0.343530	+	0.939142i	$0.611623\pi$
$744$	0	0
$745$	0	0
$746$	9.00000	0.329513
$747$	0	0
$748$	−1.41421	−0.0517088
$749$	9.08326	0.331895
$750$	0	0
$751$	−1.27208	−0.0464188	−0.0232094	−	0.999731i	$-0.507388\pi$
$751$	−1.27208	−0.0464188	−0.0232094	+	0.999731i	$0.507388\pi$
$752$	0	0
$753$	0	0
$754$	1.00000	0.0364179
$755$	0	0
$756$	0	0
$757$	−23.6569	−0.859823	−0.429911	−	0.902871i	$-0.641455\pi$
$757$	−23.6569	−0.859823	−0.429911	+	0.902871i	$0.641455\pi$
$758$	−2.75736	−0.100152
$759$	0	0
$760$	0	0
$761$	−10.0294	−0.363567	−0.181783	−	0.983339i	$-0.558187\pi$
$761$	−10.0294	−0.363567	−0.181783	+	0.983339i	$0.558187\pi$
$762$	0	0
$763$	−25.3259	−0.916859
$764$	18.5563	0.671345
$765$	0	0
$766$	−3.75736	−0.135759
$767$	−2.21320	−0.0799141
$768$	0	0
$769$	14.4558	0.521291	0.260646	−	0.965435i	$-0.416065\pi$
$769$	14.4558	0.521291	0.260646	+	0.965435i	$0.416065\pi$
$770$	0	0
$771$	0	0
$772$	11.6569	0.419539
$773$	−19.9706	−0.718291	−0.359146	−	0.933282i	$-0.616932\pi$
$773$	−19.9706	−0.718291	−0.359146	+	0.933282i	$0.616932\pi$
$774$	0	0
$775$	0	0
$776$	−11.6569	−0.418457
$777$	0	0
$778$	37.0711	1.32906
$779$	4.24264	0.152008
$780$	0	0
$781$	−14.9706	−0.535689
$782$	−9.24264	−0.330516
$783$	0	0
$784$	−4.48528	−0.160189
$785$	0	0
$786$	0	0
$787$	−35.1838	−1.25417	−0.627083	−	0.778953i	$-0.715751\pi$
$787$	−35.1838	−1.25417	−0.627083	+	0.778953i	$0.715751\pi$
$788$	−20.2426	−0.721114
$789$	0	0
$790$	0	0
$791$	2.78680	0.0990871
$792$	0	0
$793$	−0.987807	−0.0350780
$794$	24.0000	0.851728
$795$	0	0
$796$	0.757359	0.0268439
$797$	−30.5147	−1.08089	−0.540443	−	0.841380i	$-0.681743\pi$
$797$	−30.5147	−1.08089	−0.540443	+	0.841380i	$0.681743\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−22.5858	−0.797532
$803$	−7.75736	−0.273751
$804$	0	0
$805$	0	0
$806$	−0.384776	−0.0135532
$807$	0	0
$808$	−1.07107	−0.0376800
$809$	−10.7990	−0.379672	−0.189836	−	0.981816i	$-0.560796\pi$
$809$	−10.7990	−0.379672	−0.189836	+	0.981816i	$0.560796\pi$
$810$	0	0
$811$	−6.69848	−0.235216	−0.117608	−	0.993060i	$-0.537523\pi$
$811$	−6.69848	−0.235216	−0.117608	+	0.993060i	$0.537523\pi$
$812$	−9.24264	−0.324353
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	10.2426	0.358345
$818$	17.2132	0.601846
$819$	0	0
$820$	0	0
$821$	−17.6569	−0.616228	−0.308114	−	0.951349i	$-0.599698\pi$
$821$	−17.6569	−0.616228	−0.308114	+	0.951349i	$0.599698\pi$
$822$	0	0
$823$	−9.72792	−0.339094	−0.169547	−	0.985522i	$-0.554230\pi$
$823$	−9.72792	−0.339094	−0.169547	+	0.985522i	$0.554230\pi$
$824$	−4.24264	−0.147799
$825$	0	0
$826$	20.4558	0.711750
$827$	38.6985	1.34568	0.672839	−	0.739789i	$-0.265075\pi$
$827$	38.6985	1.34568	0.672839	+	0.739789i	$0.265075\pi$
$828$	0	0
$829$	−40.4558	−1.40509	−0.702545	−	0.711640i	$-0.747953\pi$
$829$	−40.4558	−1.40509	−0.702545	+	0.711640i	$0.747953\pi$
$830$	0	0
$831$	0	0
$832$	−0.171573	−0.00594822
$833$	−4.48528	−0.155406
$834$	0	0
$835$	0	0
$836$	1.41421	0.0489116
$837$	0	0
$838$	16.5858	0.572946
$839$	22.6274	0.781185	0.390593	−	0.920564i	$-0.372270\pi$
$839$	22.6274	0.781185	0.390593	+	0.920564i	$0.372270\pi$
$840$	0	0
$841$	4.97056	0.171399
$842$	−2.51472	−0.0866629
$843$	0	0
$844$	5.72792	0.197163
$845$	0	0
$846$	0	0
$847$	14.2721	0.490394
$848$	−11.4853	−0.394406
$849$	0	0
$850$	0	0
$851$	−78.4264	−2.68842
$852$	0	0
$853$	−39.9411	−1.36756	−0.683779	−	0.729689i	$-0.739665\pi$
$853$	−39.9411	−1.36756	−0.683779	+	0.729689i	$0.739665\pi$
$854$	9.12994	0.312420
$855$	0	0
$856$	5.72792	0.195776
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−20.7279	−0.707228	−0.353614	−	0.935392i	$-0.615047\pi$
$859$	−20.7279	−0.707228	−0.353614	+	0.935392i	$0.615047\pi$
$860$	0	0
$861$	0	0
$862$	30.3848	1.03491
$863$	−24.7279	−0.841748	−0.420874	−	0.907119i	$-0.638277\pi$
$863$	−24.7279	−0.841748	−0.420874	+	0.907119i	$0.638277\pi$
$864$	0	0
$865$	0	0
$866$	21.5563	0.732515
$867$	0	0
$868$	3.55635	0.120710
$869$	−14.8284	−0.503020
$870$	0	0
$871$	2.27208	0.0769864
$872$	−15.9706	−0.540831
$873$	0	0
$874$	9.24264	0.312637
$875$	0	0
$876$	0	0
$877$	47.1421	1.59188	0.795938	−	0.605378i	$-0.206978\pi$
$877$	47.1421	1.59188	0.795938	+	0.605378i	$0.206978\pi$
$878$	14.2426	0.480666
$879$	0	0
$880$	0	0
$881$	−39.5980	−1.33409	−0.667045	−	0.745018i	$-0.732441\pi$
$881$	−39.5980	−1.33409	−0.667045	+	0.745018i	$0.732441\pi$
$882$	0	0
$883$	2.48528	0.0836364	0.0418182	−	0.999125i	$-0.486685\pi$
$883$	2.48528	0.0836364	0.0418182	+	0.999125i	$0.486685\pi$
$884$	−0.171573	−0.00577062
$885$	0	0
$886$	4.24264	0.142534
$887$	−2.78680	−0.0935715	−0.0467857	−	0.998905i	$-0.514898\pi$
$887$	−2.78680	−0.0935715	−0.0467857	+	0.998905i	$0.514898\pi$
$888$	0	0
$889$	22.9706	0.770408
$890$	0	0
$891$	0	0
$892$	20.8284	0.697387
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.58579	0.0529774
$897$	0	0
$898$	−5.31371	−0.177321
$899$	−13.0711	−0.435945
$900$	0	0
$901$	−11.4853	−0.382630
$902$	−6.00000	−0.199778
$903$	0	0
$904$	1.75736	0.0584489
$905$	0	0
$906$	0	0
$907$	20.6985	0.687282	0.343641	−	0.939101i	$-0.388340\pi$
$907$	20.6985	0.687282	0.343641	+	0.939101i	$0.388340\pi$
$908$	−25.2426	−0.837706
$909$	0	0
$910$	0	0
$911$	16.2843	0.539522	0.269761	−	0.962927i	$-0.413055\pi$
$911$	16.2843	0.539522	0.269761	+	0.962927i	$0.413055\pi$
$912$	0	0
$913$	3.51472	0.116320
$914$	−3.00000	−0.0992312
$915$	0	0
$916$	−18.9706	−0.626805
$917$	26.9117	0.888702
$918$	0	0
$919$	36.2132	1.19456	0.597282	−	0.802032i	$-0.296247\pi$
$919$	36.2132	1.19456	0.597282	+	0.802032i	$0.296247\pi$
$920$	0	0
$921$	0	0
$922$	27.5563	0.907520
$923$	−1.81623	−0.0597821
$924$	0	0
$925$	0	0
$926$	−14.1421	−0.464739
$927$	0	0
$928$	−5.82843	−0.191327
$929$	17.8284	0.584932	0.292466	−	0.956276i	$-0.405524\pi$
$929$	17.8284	0.584932	0.292466	+	0.956276i	$0.405524\pi$
$930$	0	0
$931$	4.48528	0.146999
$932$	8.97056	0.293841
$933$	0	0
$934$	24.7279	0.809122
$935$	0	0
$936$	0	0
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	−21.0000	−0.685674
$939$	0	0
$940$	0	0
$941$	−24.8579	−0.810343	−0.405172	−	0.914241i	$-0.632788\pi$
$941$	−24.8579	−0.810343	−0.405172	+	0.914241i	$0.632788\pi$
$942$	0	0
$943$	−39.2132	−1.27696
$944$	12.8995	0.419843
$945$	0	0
$946$	−14.4853	−0.470957
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−0.941125	−0.0305502
$950$	0	0
$951$	0	0
$952$	1.58579	0.0513956
$953$	−6.72792	−0.217939	−0.108969	−	0.994045i	$-0.534755\pi$
$953$	−6.72792	−0.217939	−0.108969	+	0.994045i	$0.534755\pi$
$954$	0	0
$955$	0	0
$956$	−12.8995	−0.417199
$957$	0	0
$958$	31.1127	1.00521
$959$	−20.6152	−0.665700
$960$	0	0
$961$	−25.9706	−0.837760
$962$	−1.45584	−0.0469383
$963$	0	0
$964$	−24.9706	−0.804248
$965$	0	0
$966$	0	0
$967$	19.1127	0.614623	0.307311	−	0.951609i	$-0.400571\pi$
$967$	19.1127	0.614623	0.307311	+	0.951609i	$0.400571\pi$
$968$	9.00000	0.289271
$969$	0	0
$970$	0	0
$971$	−30.3431	−0.973758	−0.486879	−	0.873469i	$-0.661865\pi$
$971$	−30.3431	−0.973758	−0.486879	+	0.873469i	$0.661865\pi$
$972$	0	0
$973$	19.0294	0.610056
$974$	−1.79899	−0.0576434
$975$	0	0
$976$	5.75736	0.184289
$977$	−8.24264	−0.263705	−0.131853	−	0.991269i	$-0.542093\pi$
$977$	−8.24264	−0.263705	−0.131853	+	0.991269i	$0.542093\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	0	0
$982$	−2.44365	−0.0779800
$983$	−0.544156	−0.0173559	−0.00867794	−	0.999962i	$-0.502762\pi$
$983$	−0.544156	−0.0173559	−0.00867794	+	0.999962i	$0.502762\pi$
$984$	0	0
$985$	0	0
$986$	−5.82843	−0.185615
$987$	0	0
$988$	0.171573	0.00545846
$989$	−94.6690	−3.01030
$990$	0	0
$991$	22.2426	0.706561	0.353280	−	0.935517i	$-0.385066\pi$
$991$	22.2426	0.706561	0.353280	+	0.935517i	$0.385066\pi$
$992$	2.24264	0.0712039
$993$	0	0
$994$	16.7868	0.532445
$995$	0	0
$996$	0	0
$997$	−23.2721	−0.737034	−0.368517	−	0.929621i	$-0.620134\pi$
$997$	−23.2721	−0.737034	−0.368517	+	0.929621i	$0.620134\pi$
$998$	34.2426	1.08393
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bn.1.2		2
3.2	odd	2		950.2.a.g.1.1		2
5.2	odd	4		1710.2.d.c.1369.2		4
5.3	odd	4		1710.2.d.c.1369.4		4
5.4	even	2		8550.2.a.cb.1.1		2
12.11	even	2		7600.2.a.bg.1.2		2
15.2	even	4		190.2.b.a.39.4	yes	4
15.8	even	4		190.2.b.a.39.1	✓	4
15.14	odd	2		950.2.a.f.1.2		2
60.23	odd	4		1520.2.d.e.609.4		4
60.47	odd	4		1520.2.d.e.609.1		4
60.59	even	2		7600.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.1	✓	4	15.8	even	4
190.2.b.a.39.4	yes	4	15.2	even	4
950.2.a.f.1.2		2	15.14	odd	2
950.2.a.g.1.1		2	3.2	odd	2
1520.2.d.e.609.1		4	60.47	odd	4
1520.2.d.e.609.4		4	60.23	odd	4
1710.2.d.c.1369.2		4	5.2	odd	4
1710.2.d.c.1369.4		4	5.3	odd	4
7600.2.a.v.1.1		2	60.59	even	2
7600.2.a.bg.1.2		2	12.11	even	2
8550.2.a.bn.1.2		2	1.1	even	1	trivial
8550.2.a.cb.1.1		2	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.58579 q^{7} -1.00000 q^{8} -1.41421 q^{11} -0.171573 q^{13} +1.58579 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} +1.41421 q^{22} +9.24264 q^{23} +0.171573 q^{26} -1.58579 q^{28} +5.82843 q^{29} -2.24264 q^{31} -1.00000 q^{32} -1.00000 q^{34} -8.48528 q^{37} +1.00000 q^{38} -4.24264 q^{41} -10.2426 q^{43} -1.41421 q^{44} -9.24264 q^{46} -4.48528 q^{49} -0.171573 q^{52} -11.4853 q^{53} +1.58579 q^{56} -5.82843 q^{58} +12.8995 q^{59} +5.75736 q^{61} +2.24264 q^{62} +1.00000 q^{64} -13.2426 q^{67} +1.00000 q^{68} +10.5858 q^{71} +5.48528 q^{73} +8.48528 q^{74} -1.00000 q^{76} +2.24264 q^{77} +10.4853 q^{79} +4.24264 q^{82} -2.48528 q^{83} +10.2426 q^{86} +1.41421 q^{88} +7.07107 q^{89} +0.272078 q^{91} +9.24264 q^{92} +11.6569 q^{97} +4.48528 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} - 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 6 q^{26} - 6 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 12 q^{43} - 10 q^{46} + 8 q^{49} - 6 q^{52} - 6 q^{53} + 6 q^{56} - 6 q^{58} + 6 q^{59} + 20 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{67} + 2 q^{68} + 24 q^{71} - 6 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} + 12 q^{83} + 12 q^{86} + 26 q^{91} + 10 q^{92} + 12 q^{97} - 8 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 - 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 6 * q^26 - 6 * q^28 + 6 * q^29 + 4 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 12 * q^43 - 10 * q^46 + 8 * q^49 - 6 * q^52 - 6 * q^53 + 6 * q^56 - 6 * q^58 + 6 * q^59 + 20 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^67 + 2 * q^68 + 24 * q^71 - 6 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^83 + 12 * q^86 + 26 * q^91 + 10 * q^92 + 12 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more