LMFDB - Embedded newform 1792.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.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:	$14.3091920422$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1792.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -1.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} -1.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} -4.00000 q^{11} +1.23607 q^{13} -1.52786 q^{15} -2.00000 q^{17} -1.23607 q^{19} +1.23607 q^{21} +6.47214 q^{23} -3.47214 q^{25} -5.52786 q^{27} -1.52786 q^{29} -4.94427 q^{33} -1.23607 q^{35} -6.47214 q^{37} +1.52786 q^{39} -2.00000 q^{41} -8.94427 q^{43} +1.81966 q^{45} -12.9443 q^{47} +1.00000 q^{49} -2.47214 q^{51} +8.94427 q^{53} +4.94427 q^{55} -1.52786 q^{57} -9.23607 q^{59} -1.23607 q^{61} -1.47214 q^{63} -1.52786 q^{65} -1.52786 q^{67} +8.00000 q^{69} -4.94427 q^{71} +14.9443 q^{73} -4.29180 q^{75} -4.00000 q^{77} +4.94427 q^{79} -2.41641 q^{81} -9.23607 q^{83} +2.47214 q^{85} -1.88854 q^{87} +2.00000 q^{89} +1.23607 q^{91} +1.52786 q^{95} +10.9443 q^{97} +5.88854 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 2 q^{13} - 12 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 20 q^{27} - 12 q^{29} + 8 q^{33} + 2 q^{35} - 4 q^{37} + 12 q^{39} - 4 q^{41} + 26 q^{45} - 8 q^{47} + 2 q^{49} + 4 q^{51} - 8 q^{55} - 12 q^{57} - 14 q^{59} + 2 q^{61} + 6 q^{63} - 12 q^{65} - 12 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} - 22 q^{75} - 8 q^{77} - 8 q^{79} + 22 q^{81} - 14 q^{83} - 4 q^{85} + 32 q^{87} + 4 q^{89} - 2 q^{91} + 12 q^{95} + 4 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 2 * q^13 - 12 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 20 * q^27 - 12 * q^29 + 8 * q^33 + 2 * q^35 - 4 * q^37 + 12 * q^39 - 4 * q^41 + 26 * q^45 - 8 * q^47 + 2 * q^49 + 4 * q^51 - 8 * q^55 - 12 * q^57 - 14 * q^59 + 2 * q^61 + 6 * q^63 - 12 * q^65 - 12 * q^67 + 16 * q^69 + 8 * q^71 + 12 * q^73 - 22 * q^75 - 8 * q^77 - 8 * q^79 + 22 * q^81 - 14 * q^83 - 4 * q^85 + 32 * q^87 + 4 * q^89 - 2 * q^91 + 12 * q^95 + 4 * q^97 - 24 * 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 $

$2$	0	0
$2$	0	0
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	−1.52786	−0.394493
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.23607	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$19$	−1.23607	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−1.52786	−0.283717	−0.141859	−	0.989887i	$-0.545308\pi$
$29$	−1.52786	−0.283717	−0.141859	+	0.989887i	$0.545308\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.94427	−0.860687
$34$	0	0
$35$	−1.23607	−0.208934
$36$	0	0
$37$	−6.47214	−1.06401	−0.532006	−	0.846740i	$-0.678562\pi$
$37$	−6.47214	−1.06401	−0.532006	+	0.846740i	$0.678562\pi$
$38$	0	0
$39$	1.52786	0.244654
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−8.94427	−1.36399	−0.681994	−	0.731357i	$-0.738887\pi$
$43$	−8.94427	−1.36399	−0.681994	+	0.731357i	$0.738887\pi$
$44$	0	0
$45$	1.81966	0.271259
$46$	0	0
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.47214	−0.346168
$52$	0	0
$53$	8.94427	1.22859	0.614295	−	0.789076i	$-0.289440\pi$
$53$	8.94427	1.22859	0.614295	+	0.789076i	$0.289440\pi$
$54$	0	0
$55$	4.94427	0.666685
$56$	0	0
$57$	−1.52786	−0.202371
$58$	0	0
$59$	−9.23607	−1.20243	−0.601217	−	0.799086i	$-0.705317\pi$
$59$	−9.23607	−1.20243	−0.601217	+	0.799086i	$0.705317\pi$
$60$	0	0
$61$	−1.23607	−0.158262	−0.0791311	−	0.996864i	$-0.525215\pi$
$61$	−1.23607	−0.158262	−0.0791311	+	0.996864i	$0.525215\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	−1.52786	−0.189508
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−4.94427	−0.586777	−0.293389	−	0.955993i	$-0.594783\pi$
$71$	−4.94427	−0.586777	−0.293389	+	0.955993i	$0.594783\pi$
$72$	0	0
$73$	14.9443	1.74909	0.874547	−	0.484940i	$-0.161159\pi$
$73$	14.9443	1.74909	0.874547	+	0.484940i	$0.161159\pi$
$74$	0	0
$75$	−4.29180	−0.495574
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−9.23607	−1.01379	−0.506895	−	0.862008i	$-0.669207\pi$
$83$	−9.23607	−1.01379	−0.506895	+	0.862008i	$0.669207\pi$
$84$	0	0
$85$	2.47214	0.268141
$86$	0	0
$87$	−1.88854	−0.202473
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	1.23607	0.129575
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.52786	0.156756
$96$	0	0
$97$	10.9443	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$97$	10.9443	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$98$	0	0
$99$	5.88854	0.591821
$100$	0	0
$101$	−14.1803	−1.41100	−0.705498	−	0.708712i	$-0.749277\pi$
$101$	−14.1803	−1.41100	−0.705498	+	0.708712i	$0.749277\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−1.52786	−0.149104
$106$	0	0
$107$	−9.52786	−0.921093	−0.460547	−	0.887635i	$-0.652347\pi$
$107$	−9.52786	−0.921093	−0.460547	+	0.887635i	$0.652347\pi$
$108$	0	0
$109$	−1.52786	−0.146343	−0.0731714	−	0.997319i	$-0.523312\pi$
$109$	−1.52786	−0.146343	−0.0731714	+	0.997319i	$0.523312\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	13.4164	1.26211	0.631055	−	0.775738i	$-0.282622\pi$
$113$	13.4164	1.26211	0.631055	+	0.775738i	$0.282622\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	−1.81966	−0.168228
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−2.47214	−0.222905
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	19.4164	1.72293	0.861464	−	0.507819i	$-0.169548\pi$
$127$	19.4164	1.72293	0.861464	+	0.507819i	$0.169548\pi$
$128$	0	0
$129$	−11.0557	−0.973403
$130$	0	0
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	0	0
$133$	−1.23607	−0.107181
$134$	0	0
$135$	6.83282	0.588075
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	17.2361	1.46194	0.730972	−	0.682407i	$-0.239067\pi$
$139$	17.2361	1.46194	0.730972	+	0.682407i	$0.239067\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	−4.94427	−0.413461
$144$	0	0
$145$	1.88854	0.156835
$146$	0	0
$147$	1.23607	0.101949
$148$	0	0
$149$	−16.9443	−1.38813	−0.694064	−	0.719913i	$-0.744182\pi$
$149$	−16.9443	−1.38813	−0.694064	+	0.719913i	$0.744182\pi$
$150$	0	0
$151$	−6.47214	−0.526695	−0.263347	−	0.964701i	$-0.584827\pi$
$151$	−6.47214	−0.526695	−0.263347	+	0.964701i	$0.584827\pi$
$152$	0	0
$153$	2.94427	0.238030
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.1803	1.77018	0.885092	−	0.465416i	$-0.154095\pi$
$157$	22.1803	1.77018	0.885092	+	0.465416i	$0.154095\pi$
$158$	0	0
$159$	11.0557	0.876776
$160$	0	0
$161$	6.47214	0.510076
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	6.11146	0.475776
$166$	0	0
$167$	4.94427	0.382599	0.191300	−	0.981532i	$-0.438730\pi$
$167$	4.94427	0.382599	0.191300	+	0.981532i	$0.438730\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	1.81966	0.139153
$172$	0	0
$173$	−3.70820	−0.281930	−0.140965	−	0.990015i	$-0.545020\pi$
$173$	−3.70820	−0.281930	−0.140965	+	0.990015i	$0.545020\pi$
$174$	0	0
$175$	−3.47214	−0.262469
$176$	0	0
$177$	−11.4164	−0.858110
$178$	0	0
$179$	−6.47214	−0.483750	−0.241875	−	0.970307i	$-0.577762\pi$
$179$	−6.47214	−0.483750	−0.241875	+	0.970307i	$0.577762\pi$
$180$	0	0
$181$	25.2361	1.87578	0.937891	−	0.346930i	$-0.112776\pi$
$181$	25.2361	1.87578	0.937891	+	0.346930i	$0.112776\pi$
$182$	0	0
$183$	−1.52786	−0.112943
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−5.52786	−0.402093
$190$	0	0
$191$	17.8885	1.29437	0.647185	−	0.762333i	$-0.275946\pi$
$191$	17.8885	1.29437	0.647185	+	0.762333i	$0.275946\pi$
$192$	0	0
$193$	12.4721	0.897764	0.448882	−	0.893591i	$-0.351822\pi$
$193$	12.4721	0.897764	0.448882	+	0.893591i	$0.351822\pi$
$194$	0	0
$195$	−1.88854	−0.135241
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	−1.88854	−0.133208
$202$	0	0
$203$	−1.52786	−0.107235
$204$	0	0
$205$	2.47214	0.172661
$206$	0	0
$207$	−9.52786	−0.662232
$208$	0	0
$209$	4.94427	0.342002
$210$	0	0
$211$	−11.4164	−0.785938	−0.392969	−	0.919552i	$-0.628552\pi$
$211$	−11.4164	−0.785938	−0.392969	+	0.919552i	$0.628552\pi$
$212$	0	0
$213$	−6.11146	−0.418750
$214$	0	0
$215$	11.0557	0.753994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	18.4721	1.24823
$220$	0	0
$221$	−2.47214	−0.166294
$222$	0	0
$223$	28.9443	1.93825	0.969126	−	0.246566i	$-0.0793023\pi$
$223$	28.9443	1.93825	0.969126	+	0.246566i	$0.0793023\pi$
$224$	0	0
$225$	5.11146	0.340764
$226$	0	0
$227$	3.70820	0.246122	0.123061	−	0.992399i	$-0.460729\pi$
$227$	3.70820	0.246122	0.123061	+	0.992399i	$0.460729\pi$
$228$	0	0
$229$	−24.6525	−1.62908	−0.814541	−	0.580106i	$-0.803011\pi$
$229$	−24.6525	−1.62908	−0.814541	+	0.580106i	$0.803011\pi$
$230$	0	0
$231$	−4.94427	−0.325309
$232$	0	0
$233$	19.8885	1.30294	0.651471	−	0.758674i	$-0.274152\pi$
$233$	19.8885	1.30294	0.651471	+	0.758674i	$0.274152\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	6.11146	0.396982
$238$	0	0
$239$	−9.52786	−0.616306	−0.308153	−	0.951337i	$-0.599711\pi$
$239$	−9.52786	−0.616306	−0.308153	+	0.951337i	$0.599711\pi$
$240$	0	0
$241$	−14.9443	−0.962645	−0.481323	−	0.876544i	$-0.659843\pi$
$241$	−14.9443	−0.962645	−0.481323	+	0.876544i	$0.659843\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	−1.23607	−0.0789695
$246$	0	0
$247$	−1.52786	−0.0972157
$248$	0	0
$249$	−11.4164	−0.723485
$250$	0	0
$251$	−16.6525	−1.05109	−0.525547	−	0.850764i	$-0.676139\pi$
$251$	−16.6525	−1.05109	−0.525547	+	0.850764i	$0.676139\pi$
$252$	0	0
$253$	−25.8885	−1.62760
$254$	0	0
$255$	3.05573	0.191357
$256$	0	0
$257$	−7.88854	−0.492074	−0.246037	−	0.969260i	$-0.579128\pi$
$257$	−7.88854	−0.492074	−0.246037	+	0.969260i	$0.579128\pi$
$258$	0	0
$259$	−6.47214	−0.402159
$260$	0	0
$261$	2.24922	0.139223
$262$	0	0
$263$	−20.9443	−1.29148	−0.645740	−	0.763558i	$-0.723451\pi$
$263$	−20.9443	−1.29148	−0.645740	+	0.763558i	$0.723451\pi$
$264$	0	0
$265$	−11.0557	−0.679148
$266$	0	0
$267$	2.47214	0.151292
$268$	0	0
$269$	27.1246	1.65382	0.826908	−	0.562337i	$-0.190097\pi$
$269$	27.1246	1.65382	0.826908	+	0.562337i	$0.190097\pi$
$270$	0	0
$271$	3.05573	0.185622	0.0928111	−	0.995684i	$-0.470415\pi$
$271$	3.05573	0.185622	0.0928111	+	0.995684i	$0.470415\pi$
$272$	0	0
$273$	1.52786	0.0924705
$274$	0	0
$275$	13.8885	0.837511
$276$	0	0
$277$	20.0000	1.20168	0.600842	−	0.799368i	$-0.294832\pi$
$277$	20.0000	1.20168	0.600842	+	0.799368i	$0.294832\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	24.6525	1.46544	0.732719	−	0.680532i	$-0.238251\pi$
$283$	24.6525	1.46544	0.732719	+	0.680532i	$0.238251\pi$
$284$	0	0
$285$	1.88854	0.111868
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	13.5279	0.793017
$292$	0	0
$293$	27.7082	1.61873	0.809365	−	0.587306i	$-0.199811\pi$
$293$	27.7082	1.61873	0.809365	+	0.587306i	$0.199811\pi$
$294$	0	0
$295$	11.4164	0.664689
$296$	0	0
$297$	22.1115	1.28304
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−8.94427	−0.515539
$302$	0	0
$303$	−17.5279	−1.00695
$304$	0	0
$305$	1.52786	0.0874852
$306$	0	0
$307$	21.5967	1.23259	0.616296	−	0.787515i	$-0.288633\pi$
$307$	21.5967	1.23259	0.616296	+	0.787515i	$0.288633\pi$
$308$	0	0
$309$	−9.88854	−0.562540
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	7.88854	0.445887	0.222943	−	0.974831i	$-0.428433\pi$
$313$	7.88854	0.445887	0.222943	+	0.974831i	$0.428433\pi$
$314$	0	0
$315$	1.81966	0.102526
$316$	0	0
$317$	−21.8885	−1.22938	−0.614692	−	0.788768i	$-0.710720\pi$
$317$	−21.8885	−1.22938	−0.614692	+	0.788768i	$0.710720\pi$
$318$	0	0
$319$	6.11146	0.342176
$320$	0	0
$321$	−11.7771	−0.657333
$322$	0	0
$323$	2.47214	0.137553
$324$	0	0
$325$	−4.29180	−0.238066
$326$	0	0
$327$	−1.88854	−0.104437
$328$	0	0
$329$	−12.9443	−0.713641
$330$	0	0
$331$	7.05573	0.387818	0.193909	−	0.981020i	$-0.437883\pi$
$331$	7.05573	0.387818	0.193909	+	0.981020i	$0.437883\pi$
$332$	0	0
$333$	9.52786	0.522124
$334$	0	0
$335$	1.88854	0.103182
$336$	0	0
$337$	−9.41641	−0.512944	−0.256472	−	0.966552i	$-0.582560\pi$
$337$	−9.41641	−0.512944	−0.256472	+	0.966552i	$0.582560\pi$
$338$	0	0
$339$	16.5836	0.900697
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−9.88854	−0.532381
$346$	0	0
$347$	−15.0557	−0.808234	−0.404117	−	0.914707i	$-0.632421\pi$
$347$	−15.0557	−0.808234	−0.404117	+	0.914707i	$0.632421\pi$
$348$	0	0
$349$	4.29180	0.229735	0.114867	−	0.993381i	$-0.463356\pi$
$349$	4.29180	0.229735	0.114867	+	0.993381i	$0.463356\pi$
$350$	0	0
$351$	−6.83282	−0.364709
$352$	0	0
$353$	−17.0557	−0.907785	−0.453892	−	0.891056i	$-0.649965\pi$
$353$	−17.0557	−0.907785	−0.453892	+	0.891056i	$0.649965\pi$
$354$	0	0
$355$	6.11146	0.324362
$356$	0	0
$357$	−2.47214	−0.130839
$358$	0	0
$359$	−3.41641	−0.180311	−0.0901556	−	0.995928i	$-0.528736\pi$
$359$	−3.41641	−0.180311	−0.0901556	+	0.995928i	$0.528736\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	0	0
$363$	6.18034	0.324384
$364$	0	0
$365$	−18.4721	−0.966876
$366$	0	0
$367$	−28.9443	−1.51088	−0.755439	−	0.655219i	$-0.772577\pi$
$367$	−28.9443	−1.51088	−0.755439	+	0.655219i	$0.772577\pi$
$368$	0	0
$369$	2.94427	0.153273
$370$	0	0
$371$	8.94427	0.464363
$372$	0	0
$373$	−32.9443	−1.70579	−0.852895	−	0.522083i	$-0.825155\pi$
$373$	−32.9443	−1.70579	−0.852895	+	0.522083i	$0.825155\pi$
$374$	0	0
$375$	12.9443	0.668439
$376$	0	0
$377$	−1.88854	−0.0972650
$378$	0	0
$379$	37.8885	1.94620	0.973102	−	0.230375i	$-0.0739953\pi$
$379$	37.8885	1.94620	0.973102	+	0.230375i	$0.0739953\pi$
$380$	0	0
$381$	24.0000	1.22956
$382$	0	0
$383$	−3.05573	−0.156140	−0.0780702	−	0.996948i	$-0.524876\pi$
$383$	−3.05573	−0.156140	−0.0780702	+	0.996948i	$0.524876\pi$
$384$	0	0
$385$	4.94427	0.251983
$386$	0	0
$387$	13.1672	0.669326
$388$	0	0
$389$	19.4164	0.984451	0.492225	−	0.870468i	$-0.336184\pi$
$389$	19.4164	0.984451	0.492225	+	0.870468i	$0.336184\pi$
$390$	0	0
$391$	−12.9443	−0.654620
$392$	0	0
$393$	−17.5279	−0.884164
$394$	0	0
$395$	−6.11146	−0.307501
$396$	0	0
$397$	3.12461	0.156820	0.0784099	−	0.996921i	$-0.475016\pi$
$397$	3.12461	0.156820	0.0784099	+	0.996921i	$0.475016\pi$
$398$	0	0
$399$	−1.52786	−0.0764889
$400$	0	0
$401$	−0.472136	−0.0235773	−0.0117887	−	0.999931i	$-0.503753\pi$
$401$	−0.472136	−0.0235773	−0.0117887	+	0.999931i	$0.503753\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.98684	0.148417
$406$	0	0
$407$	25.8885	1.28325
$408$	0	0
$409$	−14.9443	−0.738947	−0.369473	−	0.929241i	$-0.620462\pi$
$409$	−14.9443	−0.738947	−0.369473	+	0.929241i	$0.620462\pi$
$410$	0	0
$411$	−12.3607	−0.609707
$412$	0	0
$413$	−9.23607	−0.454477
$414$	0	0
$415$	11.4164	0.560409
$416$	0	0
$417$	21.3050	1.04331
$418$	0	0
$419$	6.18034	0.301929	0.150965	−	0.988539i	$-0.451762\pi$
$419$	6.18034	0.301929	0.150965	+	0.988539i	$0.451762\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	0	0
$423$	19.0557	0.926521
$424$	0	0
$425$	6.94427	0.336847
$426$	0	0
$427$	−1.23607	−0.0598175
$428$	0	0
$429$	−6.11146	−0.295064
$430$	0	0
$431$	−32.3607	−1.55876	−0.779380	−	0.626552i	$-0.784466\pi$
$431$	−32.3607	−1.55876	−0.779380	+	0.626552i	$0.784466\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	2.33437	0.111924
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−20.9443	−0.999616	−0.499808	−	0.866136i	$-0.666596\pi$
$439$	−20.9443	−0.999616	−0.499808	+	0.866136i	$0.666596\pi$
$440$	0	0
$441$	−1.47214	−0.0701017
$442$	0	0
$443$	−24.3607	−1.15741	−0.578705	−	0.815537i	$-0.696442\pi$
$443$	−24.3607	−1.15741	−0.578705	+	0.815537i	$0.696442\pi$
$444$	0	0
$445$	−2.47214	−0.117190
$446$	0	0
$447$	−20.9443	−0.990630
$448$	0	0
$449$	23.8885	1.12737	0.563685	−	0.825990i	$-0.309383\pi$
$449$	23.8885	1.12737	0.563685	+	0.825990i	$0.309383\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	−1.52786	−0.0716274
$456$	0	0
$457$	20.4721	0.957646	0.478823	−	0.877911i	$-0.341064\pi$
$457$	20.4721	0.957646	0.478823	+	0.877911i	$0.341064\pi$
$458$	0	0
$459$	11.0557	0.516037
$460$	0	0
$461$	40.0689	1.86619	0.933097	−	0.359625i	$-0.117095\pi$
$461$	40.0689	1.86619	0.933097	+	0.359625i	$0.117095\pi$
$462$	0	0
$463$	−1.88854	−0.0877681	−0.0438840	−	0.999037i	$-0.513973\pi$
$463$	−1.88854	−0.0877681	−0.0438840	+	0.999037i	$0.513973\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.6525	1.14078	0.570390	−	0.821374i	$-0.306792\pi$
$467$	24.6525	1.14078	0.570390	+	0.821374i	$0.306792\pi$
$468$	0	0
$469$	−1.52786	−0.0705502
$470$	0	0
$471$	27.4164	1.26328
$472$	0	0
$473$	35.7771	1.64503
$474$	0	0
$475$	4.29180	0.196921
$476$	0	0
$477$	−13.1672	−0.602884
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−13.5279	−0.614269
$486$	0	0
$487$	22.4721	1.01831	0.509155	−	0.860675i	$-0.329958\pi$
$487$	22.4721	1.01831	0.509155	+	0.860675i	$0.329958\pi$
$488$	0	0
$489$	−14.8328	−0.670763
$490$	0	0
$491$	−14.4721	−0.653118	−0.326559	−	0.945177i	$-0.605889\pi$
$491$	−14.4721	−0.653118	−0.326559	+	0.945177i	$0.605889\pi$
$492$	0	0
$493$	3.05573	0.137623
$494$	0	0
$495$	−7.27864	−0.327151
$496$	0	0
$497$	−4.94427	−0.221781
$498$	0	0
$499$	24.3607	1.09053	0.545267	−	0.838262i	$-0.316428\pi$
$499$	24.3607	1.09053	0.545267	+	0.838262i	$0.316428\pi$
$500$	0	0
$501$	6.11146	0.273040
$502$	0	0
$503$	11.0557	0.492951	0.246475	−	0.969149i	$-0.420728\pi$
$503$	11.0557	0.492951	0.246475	+	0.969149i	$0.420728\pi$
$504$	0	0
$505$	17.5279	0.779980
$506$	0	0
$507$	−14.1803	−0.629771
$508$	0	0
$509$	−0.652476	−0.0289205	−0.0144602	−	0.999895i	$-0.504603\pi$
$509$	−0.652476	−0.0289205	−0.0144602	+	0.999895i	$0.504603\pi$
$510$	0	0
$511$	14.9443	0.661096
$512$	0	0
$513$	6.83282	0.301676
$514$	0	0
$515$	9.88854	0.435741
$516$	0	0
$517$	51.7771	2.27715
$518$	0	0
$519$	−4.58359	−0.201197
$520$	0	0
$521$	26.9443	1.18045	0.590225	−	0.807239i	$-0.299039\pi$
$521$	26.9443	1.18045	0.590225	+	0.807239i	$0.299039\pi$
$522$	0	0
$523$	1.81966	0.0795682	0.0397841	−	0.999208i	$-0.487333\pi$
$523$	1.81966	0.0795682	0.0397841	+	0.999208i	$0.487333\pi$
$524$	0	0
$525$	−4.29180	−0.187309
$526$	0	0
$527$	0	0
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	13.5967	0.590049
$532$	0	0
$533$	−2.47214	−0.107080
$534$	0	0
$535$	11.7771	0.509168
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−0.944272	−0.0405974	−0.0202987	−	0.999794i	$-0.506462\pi$
$541$	−0.944272	−0.0405974	−0.0202987	+	0.999794i	$0.506462\pi$
$542$	0	0
$543$	31.1935	1.33864
$544$	0	0
$545$	1.88854	0.0808963
$546$	0	0
$547$	34.8328	1.48934	0.744672	−	0.667431i	$-0.232606\pi$
$547$	34.8328	1.48934	0.744672	+	0.667431i	$0.232606\pi$
$548$	0	0
$549$	1.81966	0.0776612
$550$	0	0
$551$	1.88854	0.0804547
$552$	0	0
$553$	4.94427	0.210252
$554$	0	0
$555$	9.88854	0.419745
$556$	0	0
$557$	−5.88854	−0.249506	−0.124753	−	0.992188i	$-0.539814\pi$
$557$	−5.88854	−0.249506	−0.124753	+	0.992188i	$0.539814\pi$
$558$	0	0
$559$	−11.0557	−0.467607
$560$	0	0
$561$	9.88854	0.417495
$562$	0	0
$563$	−42.5410	−1.79289	−0.896445	−	0.443155i	$-0.853859\pi$
$563$	−42.5410	−1.79289	−0.896445	+	0.443155i	$0.853859\pi$
$564$	0	0
$565$	−16.5836	−0.697677
$566$	0	0
$567$	−2.41641	−0.101480
$568$	0	0
$569$	−14.5836	−0.611376	−0.305688	−	0.952132i	$-0.598886\pi$
$569$	−14.5836	−0.611376	−0.305688	+	0.952132i	$0.598886\pi$
$570$	0	0
$571$	31.7771	1.32983	0.664915	−	0.746919i	$-0.268468\pi$
$571$	31.7771	1.32983	0.664915	+	0.746919i	$0.268468\pi$
$572$	0	0
$573$	22.1115	0.923719
$574$	0	0
$575$	−22.4721	−0.937153
$576$	0	0
$577$	−20.8328	−0.867281	−0.433641	−	0.901086i	$-0.642771\pi$
$577$	−20.8328	−0.867281	−0.433641	+	0.901086i	$0.642771\pi$
$578$	0	0
$579$	15.4164	0.640684
$580$	0	0
$581$	−9.23607	−0.383177
$582$	0	0
$583$	−35.7771	−1.48174
$584$	0	0
$585$	2.24922	0.0929940
$586$	0	0
$587$	−14.7639	−0.609373	−0.304686	−	0.952453i	$-0.598552\pi$
$587$	−14.7639	−0.609373	−0.304686	+	0.952453i	$0.598552\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	4.94427	0.203380
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	2.47214	0.101348
$596$	0	0
$597$	−29.6656	−1.21413
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−4.11146	−0.167710	−0.0838549	−	0.996478i	$-0.526723\pi$
$601$	−4.11146	−0.167710	−0.0838549	+	0.996478i	$0.526723\pi$
$602$	0	0
$603$	2.24922	0.0915955
$604$	0	0
$605$	−6.18034	−0.251267
$606$	0	0
$607$	−9.88854	−0.401364	−0.200682	−	0.979656i	$-0.564316\pi$
$607$	−9.88854	−0.401364	−0.200682	+	0.979656i	$0.564316\pi$
$608$	0	0
$609$	−1.88854	−0.0765277
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	14.4721	0.584524	0.292262	−	0.956338i	$-0.405592\pi$
$613$	14.4721	0.584524	0.292262	+	0.956338i	$0.405592\pi$
$614$	0	0
$615$	3.05573	0.123219
$616$	0	0
$617$	−23.5279	−0.947196	−0.473598	−	0.880741i	$-0.657045\pi$
$617$	−23.5279	−0.947196	−0.473598	+	0.880741i	$0.657045\pi$
$618$	0	0
$619$	9.81966	0.394685	0.197343	−	0.980335i	$-0.436769\pi$
$619$	9.81966	0.394685	0.197343	+	0.980335i	$0.436769\pi$
$620$	0	0
$621$	−35.7771	−1.43569
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	6.11146	0.244068
$628$	0	0
$629$	12.9443	0.516122
$630$	0	0
$631$	−17.8885	−0.712132	−0.356066	−	0.934461i	$-0.615882\pi$
$631$	−17.8885	−0.712132	−0.356066	+	0.934461i	$0.615882\pi$
$632$	0	0
$633$	−14.1115	−0.560880
$634$	0	0
$635$	−24.0000	−0.952411
$636$	0	0
$637$	1.23607	0.0489748
$638$	0	0
$639$	7.27864	0.287939
$640$	0	0
$641$	−10.3607	−0.409222	−0.204611	−	0.978843i	$-0.565593\pi$
$641$	−10.3607	−0.409222	−0.204611	+	0.978843i	$0.565593\pi$
$642$	0	0
$643$	−26.5410	−1.04668	−0.523338	−	0.852125i	$-0.675313\pi$
$643$	−26.5410	−1.04668	−0.523338	+	0.852125i	$0.675313\pi$
$644$	0	0
$645$	13.6656	0.538084
$646$	0	0
$647$	17.8885	0.703271	0.351636	−	0.936137i	$-0.385626\pi$
$647$	17.8885	0.703271	0.351636	+	0.936137i	$0.385626\pi$
$648$	0	0
$649$	36.9443	1.45019
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.5836	−0.492434	−0.246217	−	0.969215i	$-0.579188\pi$
$653$	−12.5836	−0.492434	−0.246217	+	0.969215i	$0.579188\pi$
$654$	0	0
$655$	17.5279	0.684870
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−42.8328	−1.66853	−0.834265	−	0.551364i	$-0.814108\pi$
$659$	−42.8328	−1.66853	−0.834265	+	0.551364i	$0.814108\pi$
$660$	0	0
$661$	22.7639	0.885414	0.442707	−	0.896666i	$-0.354018\pi$
$661$	22.7639	0.885414	0.442707	+	0.896666i	$0.354018\pi$
$662$	0	0
$663$	−3.05573	−0.118675
$664$	0	0
$665$	1.52786	0.0592480
$666$	0	0
$667$	−9.88854	−0.382886
$668$	0	0
$669$	35.7771	1.38322
$670$	0	0
$671$	4.94427	0.190872
$672$	0	0
$673$	−11.8885	−0.458270	−0.229135	−	0.973395i	$-0.573590\pi$
$673$	−11.8885	−0.458270	−0.229135	+	0.973395i	$0.573590\pi$
$674$	0	0
$675$	19.1935	0.738758
$676$	0	0
$677$	19.1246	0.735019	0.367509	−	0.930020i	$-0.380211\pi$
$677$	19.1246	0.735019	0.367509	+	0.930020i	$0.380211\pi$
$678$	0	0
$679$	10.9443	0.420003
$680$	0	0
$681$	4.58359	0.175644
$682$	0	0
$683$	−30.4721	−1.16598	−0.582992	−	0.812478i	$-0.698118\pi$
$683$	−30.4721	−1.16598	−0.582992	+	0.812478i	$0.698118\pi$
$684$	0	0
$685$	12.3607	0.472277
$686$	0	0
$687$	−30.4721	−1.16258
$688$	0	0
$689$	11.0557	0.421190
$690$	0	0
$691$	−20.2918	−0.771936	−0.385968	−	0.922512i	$-0.626133\pi$
$691$	−20.2918	−0.771936	−0.385968	+	0.922512i	$0.626133\pi$
$692$	0	0
$693$	5.88854	0.223687
$694$	0	0
$695$	−21.3050	−0.808143
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	24.5836	0.929837
$700$	0	0
$701$	−16.3607	−0.617934	−0.308967	−	0.951073i	$-0.599983\pi$
$701$	−16.3607	−0.617934	−0.308967	+	0.951073i	$0.599983\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	19.7771	0.744848
$706$	0	0
$707$	−14.1803	−0.533307
$708$	0	0
$709$	−6.47214	−0.243066	−0.121533	−	0.992587i	$-0.538781\pi$
$709$	−6.47214	−0.243066	−0.121533	+	0.992587i	$0.538781\pi$
$710$	0	0
$711$	−7.27864	−0.272970
$712$	0	0
$713$	0	0
$714$	0	0
$715$	6.11146	0.228556
$716$	0	0
$717$	−11.7771	−0.439823
$718$	0	0
$719$	19.0557	0.710659	0.355329	−	0.934741i	$-0.384369\pi$
$719$	19.0557	0.710659	0.355329	+	0.934741i	$0.384369\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−18.4721	−0.686986
$724$	0	0
$725$	5.30495	0.197021
$726$	0	0
$727$	−14.8328	−0.550119	−0.275059	−	0.961427i	$-0.588698\pi$
$727$	−14.8328	−0.550119	−0.275059	+	0.961427i	$0.588698\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	−32.0689	−1.18449	−0.592246	−	0.805757i	$-0.701758\pi$
$733$	−32.0689	−1.18449	−0.592246	+	0.805757i	$0.701758\pi$
$734$	0	0
$735$	−1.52786	−0.0563561
$736$	0	0
$737$	6.11146	0.225118
$738$	0	0
$739$	−21.8885	−0.805183	−0.402592	−	0.915380i	$-0.631890\pi$
$739$	−21.8885	−0.805183	−0.402592	+	0.915380i	$0.631890\pi$
$740$	0	0
$741$	−1.88854	−0.0693774
$742$	0	0
$743$	−25.5279	−0.936527	−0.468263	−	0.883589i	$-0.655120\pi$
$743$	−25.5279	−0.936527	−0.468263	+	0.883589i	$0.655120\pi$
$744$	0	0
$745$	20.9443	0.767339
$746$	0	0
$747$	13.5967	0.497479
$748$	0	0
$749$	−9.52786	−0.348141
$750$	0	0
$751$	−16.3607	−0.597010	−0.298505	−	0.954408i	$-0.596488\pi$
$751$	−16.3607	−0.597010	−0.298505	+	0.954408i	$0.596488\pi$
$752$	0	0
$753$	−20.5836	−0.750108
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	−43.4164	−1.57800	−0.788998	−	0.614396i	$-0.789400\pi$
$757$	−43.4164	−1.57800	−0.788998	+	0.614396i	$0.789400\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−27.8885	−1.01096	−0.505479	−	0.862839i	$-0.668684\pi$
$761$	−27.8885	−1.01096	−0.505479	+	0.862839i	$0.668684\pi$
$762$	0	0
$763$	−1.52786	−0.0553124
$764$	0	0
$765$	−3.63932	−0.131580
$766$	0	0
$767$	−11.4164	−0.412223
$768$	0	0
$769$	17.0557	0.615045	0.307523	−	0.951541i	$-0.400500\pi$
$769$	17.0557	0.615045	0.307523	+	0.951541i	$0.400500\pi$
$770$	0	0
$771$	−9.75078	−0.351166
$772$	0	0
$773$	−11.1246	−0.400124	−0.200062	−	0.979783i	$-0.564114\pi$
$773$	−11.1246	−0.400124	−0.200062	+	0.979783i	$0.564114\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	2.47214	0.0885735
$780$	0	0
$781$	19.7771	0.707680
$782$	0	0
$783$	8.44582	0.301829
$784$	0	0
$785$	−27.4164	−0.978534
$786$	0	0
$787$	4.87539	0.173789	0.0868944	−	0.996218i	$-0.472306\pi$
$787$	4.87539	0.173789	0.0868944	+	0.996218i	$0.472306\pi$
$788$	0	0
$789$	−25.8885	−0.921657
$790$	0	0
$791$	13.4164	0.477033
$792$	0	0
$793$	−1.52786	−0.0542560
$794$	0	0
$795$	−13.6656	−0.484670
$796$	0	0
$797$	33.2361	1.17728	0.588641	−	0.808395i	$-0.299663\pi$
$797$	33.2361	1.17728	0.588641	+	0.808395i	$0.299663\pi$
$798$	0	0
$799$	25.8885	0.915871
$800$	0	0
$801$	−2.94427	−0.104031
$802$	0	0
$803$	−59.7771	−2.10949
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	33.5279	1.18024
$808$	0	0
$809$	−5.41641	−0.190431	−0.0952154	−	0.995457i	$-0.530354\pi$
$809$	−5.41641	−0.190431	−0.0952154	+	0.995457i	$0.530354\pi$
$810$	0	0
$811$	−27.1246	−0.952474	−0.476237	−	0.879317i	$-0.658000\pi$
$811$	−27.1246	−0.952474	−0.476237	+	0.879317i	$0.658000\pi$
$812$	0	0
$813$	3.77709	0.132468
$814$	0	0
$815$	14.8328	0.519571
$816$	0	0
$817$	11.0557	0.386791
$818$	0	0
$819$	−1.81966	−0.0635841
$820$	0	0
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	0	0
$823$	−30.8328	−1.07476	−0.537382	−	0.843339i	$-0.680587\pi$
$823$	−30.8328	−1.07476	−0.537382	+	0.843339i	$0.680587\pi$
$824$	0	0
$825$	17.1672	0.597685
$826$	0	0
$827$	−35.4164	−1.23155	−0.615775	−	0.787922i	$-0.711157\pi$
$827$	−35.4164	−1.23155	−0.615775	+	0.787922i	$0.711157\pi$
$828$	0	0
$829$	−27.1246	−0.942077	−0.471038	−	0.882113i	$-0.656121\pi$
$829$	−27.1246	−0.942077	−0.471038	+	0.882113i	$0.656121\pi$
$830$	0	0
$831$	24.7214	0.857574
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−6.11146	−0.211496
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−26.6656	−0.919505
$842$	0	0
$843$	−7.41641	−0.255435
$844$	0	0
$845$	14.1803	0.487819
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	30.4721	1.04580
$850$	0	0
$851$	−41.8885	−1.43592
$852$	0	0
$853$	33.2361	1.13798	0.568991	−	0.822344i	$-0.307334\pi$
$853$	33.2361	1.13798	0.568991	+	0.822344i	$0.307334\pi$
$854$	0	0
$855$	−2.24922	−0.0769218
$856$	0	0
$857$	−56.8328	−1.94137	−0.970686	−	0.240351i	$-0.922737\pi$
$857$	−56.8328	−1.94137	−0.970686	+	0.240351i	$0.922737\pi$
$858$	0	0
$859$	30.1803	1.02974	0.514870	−	0.857268i	$-0.327840\pi$
$859$	30.1803	1.02974	0.514870	+	0.857268i	$0.327840\pi$
$860$	0	0
$861$	−2.47214	−0.0842502
$862$	0	0
$863$	−33.8885	−1.15358	−0.576790	−	0.816893i	$-0.695695\pi$
$863$	−33.8885	−1.15358	−0.576790	+	0.816893i	$0.695695\pi$
$864$	0	0
$865$	4.58359	0.155847
$866$	0	0
$867$	−16.0689	−0.545728
$868$	0	0
$869$	−19.7771	−0.670892
$870$	0	0
$871$	−1.88854	−0.0639909
$872$	0	0
$873$	−16.1115	−0.545290
$874$	0	0
$875$	10.4721	0.354023
$876$	0	0
$877$	−0.360680	−0.0121793	−0.00608965	−	0.999981i	$-0.501938\pi$
$877$	−0.360680	−0.0121793	−0.00608965	+	0.999981i	$0.501938\pi$
$878$	0	0
$879$	34.2492	1.15520
$880$	0	0
$881$	−20.8328	−0.701875	−0.350938	−	0.936399i	$-0.614137\pi$
$881$	−20.8328	−0.701875	−0.350938	+	0.936399i	$0.614137\pi$
$882$	0	0
$883$	−38.4721	−1.29469	−0.647345	−	0.762197i	$-0.724121\pi$
$883$	−38.4721	−1.29469	−0.647345	+	0.762197i	$0.724121\pi$
$884$	0	0
$885$	14.1115	0.474351
$886$	0	0
$887$	14.8328	0.498037	0.249019	−	0.968499i	$-0.419892\pi$
$887$	14.8328	0.498037	0.249019	+	0.968499i	$0.419892\pi$
$888$	0	0
$889$	19.4164	0.651205
$890$	0	0
$891$	9.66563	0.323811
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	9.88854	0.330169
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−17.8885	−0.595954
$902$	0	0
$903$	−11.0557	−0.367912
$904$	0	0
$905$	−31.1935	−1.03691
$906$	0	0
$907$	−35.4164	−1.17598	−0.587991	−	0.808867i	$-0.700081\pi$
$907$	−35.4164	−1.17598	−0.587991	+	0.808867i	$0.700081\pi$
$908$	0	0
$909$	20.8754	0.692393
$910$	0	0
$911$	54.4721	1.80474	0.902371	−	0.430960i	$-0.141825\pi$
$911$	54.4721	1.80474	0.902371	+	0.430960i	$0.141825\pi$
$912$	0	0
$913$	36.9443	1.22268
$914$	0	0
$915$	1.88854	0.0624333
$916$	0	0
$917$	−14.1803	−0.468276
$918$	0	0
$919$	−46.8328	−1.54487	−0.772436	−	0.635093i	$-0.780962\pi$
$919$	−46.8328	−1.54487	−0.772436	+	0.635093i	$0.780962\pi$
$920$	0	0
$921$	26.6950	0.879632
$922$	0	0
$923$	−6.11146	−0.201161
$924$	0	0
$925$	22.4721	0.738879
$926$	0	0
$927$	11.7771	0.386810
$928$	0	0
$929$	−27.8885	−0.914993	−0.457497	−	0.889211i	$-0.651254\pi$
$929$	−27.8885	−0.914993	−0.457497	+	0.889211i	$0.651254\pi$
$930$	0	0
$931$	−1.23607	−0.0405105
$932$	0	0
$933$	9.88854	0.323736
$934$	0	0
$935$	−9.88854	−0.323390
$936$	0	0
$937$	−26.9443	−0.880231	−0.440115	−	0.897941i	$-0.645063\pi$
$937$	−26.9443	−0.880231	−0.440115	+	0.897941i	$0.645063\pi$
$938$	0	0
$939$	9.75078	0.318205
$940$	0	0
$941$	0.652476	0.0212701	0.0106351	−	0.999943i	$-0.496615\pi$
$941$	0.652476	0.0212701	0.0106351	+	0.999943i	$0.496615\pi$
$942$	0	0
$943$	−12.9443	−0.421523
$944$	0	0
$945$	6.83282	0.222272
$946$	0	0
$947$	−39.0557	−1.26914	−0.634570	−	0.772865i	$-0.718823\pi$
$947$	−39.0557	−1.26914	−0.634570	+	0.772865i	$0.718823\pi$
$948$	0	0
$949$	18.4721	0.599631
$950$	0	0
$951$	−27.0557	−0.877342
$952$	0	0
$953$	31.8885	1.03297	0.516486	−	0.856296i	$-0.327240\pi$
$953$	31.8885	1.03297	0.516486	+	0.856296i	$0.327240\pi$
$954$	0	0
$955$	−22.1115	−0.715510
$956$	0	0
$957$	7.55418	0.244192
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	14.0263	0.451992
$964$	0	0
$965$	−15.4164	−0.496272
$966$	0	0
$967$	12.5836	0.404661	0.202331	−	0.979317i	$-0.435148\pi$
$967$	12.5836	0.404661	0.202331	+	0.979317i	$0.435148\pi$
$968$	0	0
$969$	3.05573	0.0981641
$970$	0	0
$971$	−15.3475	−0.492525	−0.246263	−	0.969203i	$-0.579203\pi$
$971$	−15.3475	−0.492525	−0.246263	+	0.969203i	$0.579203\pi$
$972$	0	0
$973$	17.2361	0.552563
$974$	0	0
$975$	−5.30495	−0.169894
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	2.24922	0.0718122
$982$	0	0
$983$	27.7771	0.885952	0.442976	−	0.896534i	$-0.353923\pi$
$983$	27.7771	0.885952	0.442976	+	0.896534i	$0.353923\pi$
$984$	0	0
$985$	−4.94427	−0.157538
$986$	0	0
$987$	−16.0000	−0.509286
$988$	0	0
$989$	−57.8885	−1.84075
$990$	0	0
$991$	−30.8328	−0.979437	−0.489718	−	0.871881i	$-0.662900\pi$
$991$	−30.8328	−0.979437	−0.489718	+	0.871881i	$0.662900\pi$
$992$	0	0
$993$	8.72136	0.276764
$994$	0	0
$995$	29.6656	0.940464
$996$	0	0
$997$	30.7639	0.974304	0.487152	−	0.873317i	$-0.338036\pi$
$997$	30.7639	0.974304	0.487152	+	0.873317i	$0.338036\pi$
$998$	0	0
$999$	35.7771	1.13194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.a.l.1.2		2
4.3	odd	2		1792.2.a.t.1.1		2
8.3	odd	2		1792.2.a.j.1.2		2
8.5	even	2		1792.2.a.r.1.1		2
16.3	odd	4		896.2.b.g.449.2	yes	4
16.5	even	4		896.2.b.e.449.2	✓	4
16.11	odd	4		896.2.b.g.449.3	yes	4
16.13	even	4		896.2.b.e.449.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.e.449.2	✓	4	16.5	even	4
896.2.b.e.449.3	yes	4	16.13	even	4
896.2.b.g.449.2	yes	4	16.3	odd	4
896.2.b.g.449.3	yes	4	16.11	odd	4
1792.2.a.j.1.2		2	8.3	odd	2
1792.2.a.l.1.2		2	1.1	even	1	trivial
1792.2.a.r.1.1		2	8.5	even	2
1792.2.a.t.1.1		2	4.3	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -1.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} -1.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} -4.00000 q^{11} +1.23607 q^{13} -1.52786 q^{15} -2.00000 q^{17} -1.23607 q^{19} +1.23607 q^{21} +6.47214 q^{23} -3.47214 q^{25} -5.52786 q^{27} -1.52786 q^{29} -4.94427 q^{33} -1.23607 q^{35} -6.47214 q^{37} +1.52786 q^{39} -2.00000 q^{41} -8.94427 q^{43} +1.81966 q^{45} -12.9443 q^{47} +1.00000 q^{49} -2.47214 q^{51} +8.94427 q^{53} +4.94427 q^{55} -1.52786 q^{57} -9.23607 q^{59} -1.23607 q^{61} -1.47214 q^{63} -1.52786 q^{65} -1.52786 q^{67} +8.00000 q^{69} -4.94427 q^{71} +14.9443 q^{73} -4.29180 q^{75} -4.00000 q^{77} +4.94427 q^{79} -2.41641 q^{81} -9.23607 q^{83} +2.47214 q^{85} -1.88854 q^{87} +2.00000 q^{89} +1.23607 q^{91} +1.52786 q^{95} +10.9443 q^{97} +5.88854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 2 q^{13} - 12 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 20 q^{27} - 12 q^{29} + 8 q^{33} + 2 q^{35} - 4 q^{37} + 12 q^{39} - 4 q^{41} + 26 q^{45} - 8 q^{47} + 2 q^{49} + 4 q^{51} - 8 q^{55} - 12 q^{57} - 14 q^{59} + 2 q^{61} + 6 q^{63} - 12 q^{65} - 12 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} - 22 q^{75} - 8 q^{77} - 8 q^{79} + 22 q^{81} - 14 q^{83} - 4 q^{85} + 32 q^{87} + 4 q^{89} - 2 q^{91} + 12 q^{95} + 4 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 2 * q^13 - 12 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 20 * q^27 - 12 * q^29 + 8 * q^33 + 2 * q^35 - 4 * q^37 + 12 * q^39 - 4 * q^41 + 26 * q^45 - 8 * q^47 + 2 * q^49 + 4 * q^51 - 8 * q^55 - 12 * q^57 - 14 * q^59 + 2 * q^61 + 6 * q^63 - 12 * q^65 - 12 * q^67 + 16 * q^69 + 8 * q^71 + 12 * q^73 - 22 * q^75 - 8 * q^77 - 8 * q^79 + 22 * q^81 - 14 * q^83 - 4 * q^85 + 32 * q^87 + 4 * q^89 - 2 * q^91 + 12 * q^95 + 4 * q^97 - 24 * q^99

Introduction

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