LMFDB - Embedded newform 7168.2.a.be.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, 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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.9433055232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 $ x^8 - 4x^7 - 6x^6 + 32x^5 + 9x^4 - 76x^3 - 4x^2 + 48*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.38887$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.09974 q^{3} +2.59647 q^{5} -1.00000 q^{7} +1.40890 q^{9} +O(q^{10})$	$q+2.09974 q^{3} +2.59647 q^{5} -1.00000 q^{7} +1.40890 q^{9} +0.454591 q^{11} +6.53068 q^{13} +5.45192 q^{15} +1.84172 q^{17} +5.50056 q^{19} -2.09974 q^{21} +5.88497 q^{23} +1.74168 q^{25} -3.34090 q^{27} -8.69134 q^{29} -5.69821 q^{31} +0.954522 q^{33} -2.59647 q^{35} +2.35999 q^{37} +13.7127 q^{39} +10.7333 q^{41} -0.753925 q^{43} +3.65817 q^{45} -0.465401 q^{47} +1.00000 q^{49} +3.86712 q^{51} +0.881386 q^{53} +1.18033 q^{55} +11.5497 q^{57} -10.3546 q^{59} +10.7092 q^{61} -1.40890 q^{63} +16.9567 q^{65} -8.71619 q^{67} +12.3569 q^{69} +0.162532 q^{71} -3.49118 q^{73} +3.65707 q^{75} -0.454591 q^{77} +8.28703 q^{79} -11.2417 q^{81} +3.55356 q^{83} +4.78197 q^{85} -18.2495 q^{87} +1.60040 q^{89} -6.53068 q^{91} -11.9647 q^{93} +14.2821 q^{95} -8.88621 q^{97} +0.640472 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q + 8 * q^5 - 8 * q^7 + 12 * q^9	$ 8 q + 8 q^{5} - 8 q^{7} + 12 q^{9} - 12 q^{11} + 20 q^{13} + 4 q^{17} - 4 q^{19} - 8 q^{23} + 12 q^{25} + 12 q^{27} + 8 q^{29} + 4 q^{31} + 8 q^{33} - 8 q^{35} + 8 q^{37} + 16 q^{39} - 12 q^{41} + 4 q^{43} + 52 q^{45} - 20 q^{47} + 8 q^{49} - 32 q^{51} + 40 q^{53} - 24 q^{55} - 4 q^{57} - 4 q^{59} - 8 q^{61} - 12 q^{63} + 36 q^{65} - 28 q^{67} + 4 q^{69} + 16 q^{71} + 16 q^{73} + 28 q^{75} + 12 q^{77} + 20 q^{81} + 8 q^{83} + 16 q^{85} - 20 q^{87} + 16 q^{89} - 20 q^{91} + 16 q^{93} + 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q + 8 * q^5 - 8 * q^7 + 12 * q^9 - 12 * q^11 + 20 * q^13 + 4 * q^17 - 4 * q^19 - 8 * q^23 + 12 * q^25 + 12 * q^27 + 8 * q^29 + 4 * q^31 + 8 * q^33 - 8 * q^35 + 8 * q^37 + 16 * q^39 - 12 * q^41 + 4 * q^43 + 52 * q^45 - 20 * q^47 + 8 * q^49 - 32 * q^51 + 40 * q^53 - 24 * q^55 - 4 * q^57 - 4 * q^59 - 8 * q^61 - 12 * q^63 + 36 * q^65 - 28 * q^67 + 4 * q^69 + 16 * q^71 + 16 * q^73 + 28 * q^75 + 12 * q^77 + 20 * q^81 + 8 * q^83 + 16 * q^85 - 20 * q^87 + 16 * q^89 - 20 * q^91 + 16 * q^93 + 40 * q^95 - 36 * q^97 + 4 * 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
$3$	2.09974	1.21228	0.606142	−	0.795356i	$-0.292716\pi$
$3$	2.09974	1.21228	0.606142	+	0.795356i	$0.292716\pi$
$4$	0	0
$5$	2.59647	1.16118	0.580589	−	0.814196i	$-0.302822\pi$
$5$	2.59647	1.16118	0.580589	+	0.814196i	$0.302822\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.40890	0.469633
$10$	0	0
$11$	0.454591	0.137064	0.0685322	−	0.997649i	$-0.478168\pi$
$11$	0.454591	0.137064	0.0685322	+	0.997649i	$0.478168\pi$
$12$	0	0
$13$	6.53068	1.81128	0.905642	−	0.424044i	$-0.139390\pi$
$13$	6.53068	1.81128	0.905642	+	0.424044i	$0.139390\pi$
$14$	0	0
$15$	5.45192	1.40768
$16$	0	0
$17$	1.84172	0.446682	0.223341	−	0.974740i	$-0.428304\pi$
$17$	1.84172	0.446682	0.223341	+	0.974740i	$0.428304\pi$
$18$	0	0
$19$	5.50056	1.26191	0.630957	−	0.775818i	$-0.282662\pi$
$19$	5.50056	1.26191	0.630957	+	0.775818i	$0.282662\pi$
$20$	0	0
$21$	−2.09974	−0.458200
$22$	0	0
$23$	5.88497	1.22710	0.613550	−	0.789656i	$-0.289741\pi$
$23$	5.88497	1.22710	0.613550	+	0.789656i	$0.289741\pi$
$24$	0	0
$25$	1.74168	0.348336
$26$	0	0
$27$	−3.34090	−0.642956
$28$	0	0
$29$	−8.69134	−1.61394	−0.806970	−	0.590592i	$-0.798894\pi$
$29$	−8.69134	−1.61394	−0.806970	+	0.590592i	$0.798894\pi$
$30$	0	0
$31$	−5.69821	−1.02343	−0.511714	−	0.859156i	$-0.670989\pi$
$31$	−5.69821	−1.02343	−0.511714	+	0.859156i	$0.670989\pi$
$32$	0	0
$33$	0.954522	0.166161
$34$	0	0
$35$	−2.59647	−0.438884
$36$	0	0
$37$	2.35999	0.387980	0.193990	−	0.981003i	$-0.437857\pi$
$37$	2.35999	0.387980	0.193990	+	0.981003i	$0.437857\pi$
$38$	0	0
$39$	13.7127	2.19579
$40$	0	0
$41$	10.7333	1.67626	0.838132	−	0.545468i	$-0.183648\pi$
$41$	10.7333	1.67626	0.838132	+	0.545468i	$0.183648\pi$
$42$	0	0
$43$	−0.753925	−0.114972	−0.0574862	−	0.998346i	$-0.518309\pi$
$43$	−0.753925	−0.114972	−0.0574862	+	0.998346i	$0.518309\pi$
$44$	0	0
$45$	3.65817	0.545327
$46$	0	0
$47$	−0.465401	−0.0678856	−0.0339428	−	0.999424i	$-0.510806\pi$
$47$	−0.465401	−0.0678856	−0.0339428	+	0.999424i	$0.510806\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.86712	0.541505
$52$	0	0
$53$	0.881386	0.121068	0.0605338	−	0.998166i	$-0.480720\pi$
$53$	0.881386	0.121068	0.0605338	+	0.998166i	$0.480720\pi$
$54$	0	0
$55$	1.18033	0.159156
$56$	0	0
$57$	11.5497	1.52980
$58$	0	0
$59$	−10.3546	−1.34806	−0.674030	−	0.738704i	$-0.735438\pi$
$59$	−10.3546	−1.34806	−0.674030	+	0.738704i	$0.735438\pi$
$60$	0	0
$61$	10.7092	1.37118	0.685589	−	0.727989i	$-0.259545\pi$
$61$	10.7092	1.37118	0.685589	+	0.727989i	$0.259545\pi$
$62$	0	0
$63$	−1.40890	−0.177504
$64$	0	0
$65$	16.9567	2.10322
$66$	0	0
$67$	−8.71619	−1.06485	−0.532426	−	0.846477i	$-0.678719\pi$
$67$	−8.71619	−1.06485	−0.532426	+	0.846477i	$0.678719\pi$
$68$	0	0
$69$	12.3569	1.48759
$70$	0	0
$71$	0.162532	0.0192890	0.00964452	−	0.999953i	$-0.496930\pi$
$71$	0.162532	0.0192890	0.00964452	+	0.999953i	$0.496930\pi$
$72$	0	0
$73$	−3.49118	−0.408612	−0.204306	−	0.978907i	$-0.565494\pi$
$73$	−3.49118	−0.408612	−0.204306	+	0.978907i	$0.565494\pi$
$74$	0	0
$75$	3.65707	0.422283
$76$	0	0
$77$	−0.454591	−0.0518054
$78$	0	0
$79$	8.28703	0.932363	0.466182	−	0.884689i	$-0.345629\pi$
$79$	8.28703	0.932363	0.466182	+	0.884689i	$0.345629\pi$
$80$	0	0
$81$	−11.2417	−1.24908
$82$	0	0
$83$	3.55356	0.390054	0.195027	−	0.980798i	$-0.437521\pi$
$83$	3.55356	0.390054	0.195027	+	0.980798i	$0.437521\pi$
$84$	0	0
$85$	4.78197	0.518677
$86$	0	0
$87$	−18.2495	−1.95655
$88$	0	0
$89$	1.60040	0.169642	0.0848209	−	0.996396i	$-0.472968\pi$
$89$	1.60040	0.169642	0.0848209	+	0.996396i	$0.472968\pi$
$90$	0	0
$91$	−6.53068	−0.684601
$92$	0	0
$93$	−11.9647	−1.24069
$94$	0	0
$95$	14.2821	1.46531
$96$	0	0
$97$	−8.88621	−0.902258	−0.451129	−	0.892459i	$-0.648979\pi$
$97$	−8.88621	−0.902258	−0.451129	+	0.892459i	$0.648979\pi$
$98$	0	0
$99$	0.640472	0.0643699
$100$	0	0
$101$	13.8668	1.37980	0.689901	−	0.723904i	$-0.257654\pi$
$101$	13.8668	1.37980	0.689901	+	0.723904i	$0.257654\pi$
$102$	0	0
$103$	14.3236	1.41135	0.705674	−	0.708537i	$-0.250644\pi$
$103$	14.3236	1.41135	0.705674	+	0.708537i	$0.250644\pi$
$104$	0	0
$105$	−5.45192	−0.532052
$106$	0	0
$107$	−15.1314	−1.46280	−0.731401	−	0.681947i	$-0.761133\pi$
$107$	−15.1314	−1.46280	−0.731401	+	0.681947i	$0.761133\pi$
$108$	0	0
$109$	−10.4465	−1.00059	−0.500296	−	0.865854i	$-0.666776\pi$
$109$	−10.4465	−1.00059	−0.500296	+	0.865854i	$0.666776\pi$
$110$	0	0
$111$	4.95537	0.470343
$112$	0	0
$113$	−8.05995	−0.758217	−0.379108	−	0.925352i	$-0.623769\pi$
$113$	−8.05995	−0.758217	−0.379108	+	0.925352i	$0.623769\pi$
$114$	0	0
$115$	15.2802	1.42488
$116$	0	0
$117$	9.20105	0.850638
$118$	0	0
$119$	−1.84172	−0.168830
$120$	0	0
$121$	−10.7933	−0.981213
$122$	0	0
$123$	22.5372	2.03211
$124$	0	0
$125$	−8.46014	−0.756698
$126$	0	0
$127$	−0.367471	−0.0326078	−0.0163039	−	0.999867i	$-0.505190\pi$
$127$	−0.367471	−0.0326078	−0.0163039	+	0.999867i	$0.505190\pi$
$128$	0	0
$129$	−1.58304	−0.139379
$130$	0	0
$131$	−13.1642	−1.15016	−0.575081	−	0.818096i	$-0.695030\pi$
$131$	−13.1642	−1.15016	−0.575081	+	0.818096i	$0.695030\pi$
$132$	0	0
$133$	−5.50056	−0.476959
$134$	0	0
$135$	−8.67456	−0.746587
$136$	0	0
$137$	6.85872	0.585980	0.292990	−	0.956115i	$-0.405350\pi$
$137$	6.85872	0.585980	0.292990	+	0.956115i	$0.405350\pi$
$138$	0	0
$139$	−22.8721	−1.93998	−0.969991	−	0.243141i	$-0.921822\pi$
$139$	−22.8721	−1.93998	−0.969991	+	0.243141i	$0.921822\pi$
$140$	0	0
$141$	−0.977219	−0.0822967
$142$	0	0
$143$	2.96879	0.248262
$144$	0	0
$145$	−22.5668	−1.87407
$146$	0	0
$147$	2.09974	0.173183
$148$	0	0
$149$	21.4973	1.76112	0.880562	−	0.473931i	$-0.157165\pi$
$149$	21.4973	1.76112	0.880562	+	0.473931i	$0.157165\pi$
$150$	0	0
$151$	−4.76306	−0.387612	−0.193806	−	0.981040i	$-0.562083\pi$
$151$	−4.76306	−0.387612	−0.193806	+	0.981040i	$0.562083\pi$
$152$	0	0
$153$	2.59479	0.209776
$154$	0	0
$155$	−14.7952	−1.18838
$156$	0	0
$157$	16.6718	1.33055	0.665277	−	0.746596i	$-0.268313\pi$
$157$	16.6718	1.33055	0.665277	+	0.746596i	$0.268313\pi$
$158$	0	0
$159$	1.85068	0.146768
$160$	0	0
$161$	−5.88497	−0.463800
$162$	0	0
$163$	13.6908	1.07234	0.536171	−	0.844109i	$-0.319870\pi$
$163$	13.6908	1.07234	0.536171	+	0.844109i	$0.319870\pi$
$164$	0	0
$165$	2.47839	0.192942
$166$	0	0
$167$	13.1916	1.02079	0.510397	−	0.859939i	$-0.329499\pi$
$167$	13.1916	1.02079	0.510397	+	0.859939i	$0.329499\pi$
$168$	0	0
$169$	29.6497	2.28075
$170$	0	0
$171$	7.74972	0.592636
$172$	0	0
$173$	9.06309	0.689054	0.344527	−	0.938776i	$-0.388039\pi$
$173$	9.06309	0.689054	0.344527	+	0.938776i	$0.388039\pi$
$174$	0	0
$175$	−1.74168	−0.131659
$176$	0	0
$177$	−21.7420	−1.63423
$178$	0	0
$179$	−12.8125	−0.957653	−0.478826	−	0.877910i	$-0.658938\pi$
$179$	−12.8125	−0.957653	−0.478826	+	0.877910i	$0.658938\pi$
$180$	0	0
$181$	−19.3980	−1.44184	−0.720920	−	0.693019i	$-0.756280\pi$
$181$	−19.3980	−1.44184	−0.720920	+	0.693019i	$0.756280\pi$
$182$	0	0
$183$	22.4866	1.66226
$184$	0	0
$185$	6.12766	0.450515
$186$	0	0
$187$	0.837227	0.0612241
$188$	0	0
$189$	3.34090	0.243015
$190$	0	0
$191$	19.2622	1.39376	0.696882	−	0.717186i	$-0.254570\pi$
$191$	19.2622	1.39376	0.696882	+	0.717186i	$0.254570\pi$
$192$	0	0
$193$	−2.38323	−0.171549	−0.0857744	−	0.996315i	$-0.527336\pi$
$193$	−2.38323	−0.171549	−0.0857744	+	0.996315i	$0.527336\pi$
$194$	0	0
$195$	35.6047	2.54971
$196$	0	0
$197$	8.71532	0.620941	0.310470	−	0.950583i	$-0.399513\pi$
$197$	8.71532	0.620941	0.310470	+	0.950583i	$0.399513\pi$
$198$	0	0
$199$	−9.33136	−0.661483	−0.330741	−	0.943721i	$-0.607299\pi$
$199$	−9.33136	−0.661483	−0.330741	+	0.943721i	$0.607299\pi$
$200$	0	0
$201$	−18.3017	−1.29090
$202$	0	0
$203$	8.69134	0.610012
$204$	0	0
$205$	27.8688	1.94644
$206$	0	0
$207$	8.29132	0.576286
$208$	0	0
$209$	2.50050	0.172963
$210$	0	0
$211$	27.3470	1.88264	0.941321	−	0.337513i	$-0.109586\pi$
$211$	27.3470	1.88264	0.941321	+	0.337513i	$0.109586\pi$
$212$	0	0
$213$	0.341275	0.0233838
$214$	0	0
$215$	−1.95755	−0.133504
$216$	0	0
$217$	5.69821	0.386819
$218$	0	0
$219$	−7.33057	−0.495354
$220$	0	0
$221$	12.0277	0.809067
$222$	0	0
$223$	−19.2604	−1.28977	−0.644886	−	0.764279i	$-0.723095\pi$
$223$	−19.2604	−1.28977	−0.644886	+	0.764279i	$0.723095\pi$
$224$	0	0
$225$	2.45385	0.163590
$226$	0	0
$227$	−0.530703	−0.0352240	−0.0176120	−	0.999845i	$-0.505606\pi$
$227$	−0.530703	−0.0352240	−0.0176120	+	0.999845i	$0.505606\pi$
$228$	0	0
$229$	−13.2167	−0.873382	−0.436691	−	0.899612i	$-0.643850\pi$
$229$	−13.2167	−0.873382	−0.436691	+	0.899612i	$0.643850\pi$
$230$	0	0
$231$	−0.954522	−0.0628029
$232$	0	0
$233$	13.8761	0.909055	0.454527	−	0.890733i	$-0.349808\pi$
$233$	13.8761	0.909055	0.454527	+	0.890733i	$0.349808\pi$
$234$	0	0
$235$	−1.20840	−0.0788274
$236$	0	0
$237$	17.4006	1.13029
$238$	0	0
$239$	−15.2148	−0.984165	−0.492082	−	0.870549i	$-0.663764\pi$
$239$	−15.2148	−0.984165	−0.492082	+	0.870549i	$0.663764\pi$
$240$	0	0
$241$	28.2964	1.82273	0.911364	−	0.411601i	$-0.135030\pi$
$241$	28.2964	1.82273	0.911364	+	0.411601i	$0.135030\pi$
$242$	0	0
$243$	−13.5819	−0.871281
$244$	0	0
$245$	2.59647	0.165883
$246$	0	0
$247$	35.9223	2.28568
$248$	0	0
$249$	7.46154	0.472856
$250$	0	0
$251$	12.6157	0.796295	0.398147	−	0.917321i	$-0.369653\pi$
$251$	12.6157	0.796295	0.398147	+	0.917321i	$0.369653\pi$
$252$	0	0
$253$	2.67525	0.168192
$254$	0	0
$255$	10.0409	0.628784
$256$	0	0
$257$	−27.4810	−1.71422	−0.857110	−	0.515133i	$-0.827742\pi$
$257$	−27.4810	−1.71422	−0.857110	+	0.515133i	$0.827742\pi$
$258$	0	0
$259$	−2.35999	−0.146643
$260$	0	0
$261$	−12.2452	−0.757959
$262$	0	0
$263$	14.9308	0.920674	0.460337	−	0.887744i	$-0.347729\pi$
$263$	14.9308	0.920674	0.460337	+	0.887744i	$0.347729\pi$
$264$	0	0
$265$	2.28850	0.140581
$266$	0	0
$267$	3.36042	0.205654
$268$	0	0
$269$	−0.00645316	−0.000393456 0	−0.000196728	−	1.00000i	$-0.500063\pi$
$269$	−0.00645316	−0.000393456 0	−0.000196728		1.00000i	$0.500063\pi$
$270$	0	0
$271$	−5.74567	−0.349025	−0.174512	−	0.984655i	$-0.555835\pi$
$271$	−5.74567	−0.349025	−0.174512	+	0.984655i	$0.555835\pi$
$272$	0	0
$273$	−13.7127	−0.829931
$274$	0	0
$275$	0.791752	0.0477445
$276$	0	0
$277$	−15.9643	−0.959203	−0.479602	−	0.877486i	$-0.659219\pi$
$277$	−15.9643	−0.959203	−0.479602	+	0.877486i	$0.659219\pi$
$278$	0	0
$279$	−8.02819	−0.480635
$280$	0	0
$281$	9.71094	0.579306	0.289653	−	0.957132i	$-0.406460\pi$
$281$	9.71094	0.579306	0.289653	+	0.957132i	$0.406460\pi$
$282$	0	0
$283$	17.8953	1.06377	0.531884	−	0.846817i	$-0.321484\pi$
$283$	17.8953	1.06377	0.531884	+	0.846817i	$0.321484\pi$
$284$	0	0
$285$	29.9886	1.77637
$286$	0	0
$287$	−10.7333	−0.633568
$288$	0	0
$289$	−13.6081	−0.800475
$290$	0	0
$291$	−18.6587	−1.09379
$292$	0	0
$293$	1.28175	0.0748807	0.0374403	−	0.999299i	$-0.488080\pi$
$293$	1.28175	0.0748807	0.0374403	+	0.999299i	$0.488080\pi$
$294$	0	0
$295$	−26.8856	−1.56534
$296$	0	0
$297$	−1.51874	−0.0881263
$298$	0	0
$299$	38.4328	2.22263
$300$	0	0
$301$	0.753925	0.0434555
$302$	0	0
$303$	29.1167	1.67271
$304$	0	0
$305$	27.8063	1.59218
$306$	0	0
$307$	−20.9201	−1.19398	−0.596988	−	0.802251i	$-0.703636\pi$
$307$	−20.9201	−1.19398	−0.596988	+	0.802251i	$0.703636\pi$
$308$	0	0
$309$	30.0758	1.71095
$310$	0	0
$311$	−11.6043	−0.658022	−0.329011	−	0.944326i	$-0.606715\pi$
$311$	−11.6043	−0.658022	−0.329011	+	0.944326i	$0.606715\pi$
$312$	0	0
$313$	18.5621	1.04919	0.524596	−	0.851351i	$-0.324216\pi$
$313$	18.5621	1.04919	0.524596	+	0.851351i	$0.324216\pi$
$314$	0	0
$315$	−3.65817	−0.206114
$316$	0	0
$317$	22.4666	1.26185	0.630924	−	0.775845i	$-0.282676\pi$
$317$	22.4666	1.26185	0.630924	+	0.775845i	$0.282676\pi$
$318$	0	0
$319$	−3.95100	−0.221214
$320$	0	0
$321$	−31.7719	−1.77333
$322$	0	0
$323$	10.1305	0.563674
$324$	0	0
$325$	11.3744	0.630936
$326$	0	0
$327$	−21.9349	−1.21300
$328$	0	0
$329$	0.465401	0.0256584
$330$	0	0
$331$	7.77317	0.427252	0.213626	−	0.976915i	$-0.431473\pi$
$331$	7.77317	0.427252	0.213626	+	0.976915i	$0.431473\pi$
$332$	0	0
$333$	3.32499	0.182208
$334$	0	0
$335$	−22.6314	−1.23648
$336$	0	0
$337$	2.73515	0.148993	0.0744966	−	0.997221i	$-0.476265\pi$
$337$	2.73515	0.148993	0.0744966	+	0.997221i	$0.476265\pi$
$338$	0	0
$339$	−16.9238	−0.919174
$340$	0	0
$341$	−2.59035	−0.140275
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	32.0843	1.72736
$346$	0	0
$347$	−35.3848	−1.89956	−0.949779	−	0.312922i	$-0.898692\pi$
$347$	−35.3848	−1.89956	−0.949779	+	0.312922i	$0.898692\pi$
$348$	0	0
$349$	12.3409	0.660596	0.330298	−	0.943877i	$-0.392851\pi$
$349$	12.3409	0.660596	0.330298	+	0.943877i	$0.392851\pi$
$350$	0	0
$351$	−21.8183	−1.16458
$352$	0	0
$353$	−22.2643	−1.18501	−0.592504	−	0.805567i	$-0.701861\pi$
$353$	−22.2643	−1.18501	−0.592504	+	0.805567i	$0.701861\pi$
$354$	0	0
$355$	0.422011	0.0223980
$356$	0	0
$357$	−3.86712	−0.204670
$358$	0	0
$359$	−18.3142	−0.966588	−0.483294	−	0.875458i	$-0.660560\pi$
$359$	−18.3142	−0.966588	−0.483294	+	0.875458i	$0.660560\pi$
$360$	0	0
$361$	11.2561	0.592427
$362$	0	0
$363$	−22.6632	−1.18951
$364$	0	0
$365$	−9.06477	−0.474472
$366$	0	0
$367$	−19.3398	−1.00953	−0.504764	−	0.863257i	$-0.668420\pi$
$367$	−19.3398	−1.00953	−0.504764	+	0.863257i	$0.668420\pi$
$368$	0	0
$369$	15.1222	0.787228
$370$	0	0
$371$	−0.881386	−0.0457593
$372$	0	0
$373$	7.97146	0.412747	0.206373	−	0.978473i	$-0.433834\pi$
$373$	7.97146	0.412747	0.206373	+	0.978473i	$0.433834\pi$
$374$	0	0
$375$	−17.7641	−0.917333
$376$	0	0
$377$	−56.7603	−2.92330
$378$	0	0
$379$	−4.58476	−0.235503	−0.117752	−	0.993043i	$-0.537569\pi$
$379$	−4.58476	−0.235503	−0.117752	+	0.993043i	$0.537569\pi$
$380$	0	0
$381$	−0.771592	−0.0395299
$382$	0	0
$383$	−5.39804	−0.275827	−0.137913	−	0.990444i	$-0.544040\pi$
$383$	−5.39804	−0.275827	−0.137913	+	0.990444i	$0.544040\pi$
$384$	0	0
$385$	−1.18033	−0.0601554
$386$	0	0
$387$	−1.06220	−0.0539948
$388$	0	0
$389$	22.1427	1.12268	0.561340	−	0.827586i	$-0.310286\pi$
$389$	22.1427	1.12268	0.561340	+	0.827586i	$0.310286\pi$
$390$	0	0
$391$	10.8384	0.548123
$392$	0	0
$393$	−27.6414	−1.39432
$394$	0	0
$395$	21.5171	1.08264
$396$	0	0
$397$	3.17580	0.159389	0.0796945	−	0.996819i	$-0.474606\pi$
$397$	3.17580	0.159389	0.0796945	+	0.996819i	$0.474606\pi$
$398$	0	0
$399$	−11.5497	−0.578209
$400$	0	0
$401$	−19.3307	−0.965331	−0.482665	−	0.875805i	$-0.660331\pi$
$401$	−19.3307	−0.965331	−0.482665	+	0.875805i	$0.660331\pi$
$402$	0	0
$403$	−37.2131	−1.85372
$404$	0	0
$405$	−29.1888	−1.45040
$406$	0	0
$407$	1.07283	0.0531783
$408$	0	0
$409$	0.497591	0.0246043	0.0123021	−	0.999924i	$-0.496084\pi$
$409$	0.497591	0.0246043	0.0123021	+	0.999924i	$0.496084\pi$
$410$	0	0
$411$	14.4015	0.710375
$412$	0	0
$413$	10.3546	0.509519
$414$	0	0
$415$	9.22673	0.452922
$416$	0	0
$417$	−48.0253	−2.35181
$418$	0	0
$419$	−4.96496	−0.242554	−0.121277	−	0.992619i	$-0.538699\pi$
$419$	−4.96496	−0.242554	−0.121277	+	0.992619i	$0.538699\pi$
$420$	0	0
$421$	1.91040	0.0931073	0.0465537	−	0.998916i	$-0.485176\pi$
$421$	1.91040	0.0931073	0.0465537	+	0.998916i	$0.485176\pi$
$422$	0	0
$423$	−0.655702	−0.0318813
$424$	0	0
$425$	3.20768	0.155595
$426$	0	0
$427$	−10.7092	−0.518256
$428$	0	0
$429$	6.23367	0.300964
$430$	0	0
$431$	−19.6166	−0.944896	−0.472448	−	0.881358i	$-0.656630\pi$
$431$	−19.6166	−0.944896	−0.472448	+	0.881358i	$0.656630\pi$
$432$	0	0
$433$	2.76217	0.132741	0.0663706	−	0.997795i	$-0.478858\pi$
$433$	2.76217	0.132741	0.0663706	+	0.997795i	$0.478858\pi$
$434$	0	0
$435$	−47.3844	−2.27191
$436$	0	0
$437$	32.3706	1.54850
$438$	0	0
$439$	3.05296	0.145710	0.0728550	−	0.997343i	$-0.476789\pi$
$439$	3.05296	0.145710	0.0728550	+	0.997343i	$0.476789\pi$
$440$	0	0
$441$	1.40890	0.0670904
$442$	0	0
$443$	−1.93814	−0.0920836	−0.0460418	−	0.998940i	$-0.514661\pi$
$443$	−1.93814	−0.0920836	−0.0460418	+	0.998940i	$0.514661\pi$
$444$	0	0
$445$	4.15539	0.196985
$446$	0	0
$447$	45.1386	2.13498
$448$	0	0
$449$	38.6173	1.82246	0.911232	−	0.411894i	$-0.135133\pi$
$449$	38.6173	1.82246	0.911232	+	0.411894i	$0.135133\pi$
$450$	0	0
$451$	4.87927	0.229756
$452$	0	0
$453$	−10.0012	−0.469896
$454$	0	0
$455$	−16.9567	−0.794944
$456$	0	0
$457$	−36.0959	−1.68849	−0.844247	−	0.535955i	$-0.819952\pi$
$457$	−36.0959	−1.68849	−0.844247	+	0.535955i	$0.819952\pi$
$458$	0	0
$459$	−6.15298	−0.287197
$460$	0	0
$461$	−6.07115	−0.282762	−0.141381	−	0.989955i	$-0.545154\pi$
$461$	−6.07115	−0.282762	−0.141381	+	0.989955i	$0.545154\pi$
$462$	0	0
$463$	3.38348	0.157243	0.0786217	−	0.996905i	$-0.474948\pi$
$463$	3.38348	0.157243	0.0786217	+	0.996905i	$0.474948\pi$
$464$	0	0
$465$	−31.0661	−1.44066
$466$	0	0
$467$	30.7940	1.42498	0.712489	−	0.701683i	$-0.247568\pi$
$467$	30.7940	1.42498	0.712489	+	0.701683i	$0.247568\pi$
$468$	0	0
$469$	8.71619	0.402476
$470$	0	0
$471$	35.0064	1.61301
$472$	0	0
$473$	−0.342727	−0.0157586
$474$	0	0
$475$	9.58022	0.439570
$476$	0	0
$477$	1.24178	0.0568573
$478$	0	0
$479$	6.74054	0.307983	0.153992	−	0.988072i	$-0.450787\pi$
$479$	6.74054	0.307983	0.153992	+	0.988072i	$0.450787\pi$
$480$	0	0
$481$	15.4123	0.702743
$482$	0	0
$483$	−12.3569	−0.562258
$484$	0	0
$485$	−23.0728	−1.04768
$486$	0	0
$487$	−30.4472	−1.37970	−0.689848	−	0.723954i	$-0.742323\pi$
$487$	−30.4472	−1.37970	−0.689848	+	0.723954i	$0.742323\pi$
$488$	0	0
$489$	28.7470	1.29998
$490$	0	0
$491$	−15.7806	−0.712166	−0.356083	−	0.934454i	$-0.615888\pi$
$491$	−15.7806	−0.712166	−0.356083	+	0.934454i	$0.615888\pi$
$492$	0	0
$493$	−16.0070	−0.720918
$494$	0	0
$495$	1.66297	0.0747449
$496$	0	0
$497$	−0.162532	−0.00729057
$498$	0	0
$499$	24.6859	1.10509	0.552547	−	0.833482i	$-0.313656\pi$
$499$	24.6859	1.10509	0.552547	+	0.833482i	$0.313656\pi$
$500$	0	0
$501$	27.6988	1.23749
$502$	0	0
$503$	−26.7354	−1.19207	−0.596035	−	0.802958i	$-0.703258\pi$
$503$	−26.7354	−1.19207	−0.596035	+	0.802958i	$0.703258\pi$
$504$	0	0
$505$	36.0049	1.60220
$506$	0	0
$507$	62.2566	2.76491
$508$	0	0
$509$	−19.4609	−0.862588	−0.431294	−	0.902212i	$-0.641943\pi$
$509$	−19.4609	−0.862588	−0.431294	+	0.902212i	$0.641943\pi$
$510$	0	0
$511$	3.49118	0.154441
$512$	0	0
$513$	−18.3768	−0.811355
$514$	0	0
$515$	37.1909	1.63883
$516$	0	0
$517$	−0.211567	−0.00930470
$518$	0	0
$519$	19.0301	0.835330
$520$	0	0
$521$	36.5859	1.60286	0.801428	−	0.598091i	$-0.204074\pi$
$521$	36.5859	1.60286	0.801428	+	0.598091i	$0.204074\pi$
$522$	0	0
$523$	5.68929	0.248775	0.124388	−	0.992234i	$-0.460303\pi$
$523$	5.68929	0.248775	0.124388	+	0.992234i	$0.460303\pi$
$524$	0	0
$525$	−3.65707	−0.159608
$526$	0	0
$527$	−10.4945	−0.457147
$528$	0	0
$529$	11.6328	0.505776
$530$	0	0
$531$	−14.5886	−0.633093
$532$	0	0
$533$	70.0958	3.03619
$534$	0	0
$535$	−39.2882	−1.69858
$536$	0	0
$537$	−26.9029	−1.16095
$538$	0	0
$539$	0.454591	0.0195806
$540$	0	0
$541$	25.3799	1.09117	0.545584	−	0.838056i	$-0.316308\pi$
$541$	25.3799	1.09117	0.545584	+	0.838056i	$0.316308\pi$
$542$	0	0
$543$	−40.7306	−1.74792
$544$	0	0
$545$	−27.1241	−1.16187
$546$	0	0
$547$	−26.7752	−1.14482	−0.572412	−	0.819966i	$-0.693992\pi$
$547$	−26.7752	−1.14482	−0.572412	+	0.819966i	$0.693992\pi$
$548$	0	0
$549$	15.0882	0.643950
$550$	0	0
$551$	−47.8072	−2.03665
$552$	0	0
$553$	−8.28703	−0.352400
$554$	0	0
$555$	12.8665	0.546152
$556$	0	0
$557$	42.9236	1.81873	0.909366	−	0.415998i	$-0.136568\pi$
$557$	42.9236	1.81873	0.909366	+	0.415998i	$0.136568\pi$
$558$	0	0
$559$	−4.92364	−0.208248
$560$	0	0
$561$	1.75796	0.0742210
$562$	0	0
$563$	−25.7662	−1.08592	−0.542958	−	0.839760i	$-0.682696\pi$
$563$	−25.7662	−1.08592	−0.542958	+	0.839760i	$0.682696\pi$
$564$	0	0
$565$	−20.9275	−0.880425
$566$	0	0
$567$	11.2417	0.472107
$568$	0	0
$569$	−40.3261	−1.69056	−0.845279	−	0.534325i	$-0.820566\pi$
$569$	−40.3261	−1.69056	−0.845279	+	0.534325i	$0.820566\pi$
$570$	0	0
$571$	25.6720	1.07434	0.537169	−	0.843475i	$-0.319494\pi$
$571$	25.6720	1.07434	0.537169	+	0.843475i	$0.319494\pi$
$572$	0	0
$573$	40.4456	1.68964
$574$	0	0
$575$	10.2497	0.427444
$576$	0	0
$577$	28.8535	1.20119	0.600594	−	0.799554i	$-0.294931\pi$
$577$	28.8535	1.20119	0.600594	+	0.799554i	$0.294931\pi$
$578$	0	0
$579$	−5.00416	−0.207966
$580$	0	0
$581$	−3.55356	−0.147427
$582$	0	0
$583$	0.400670	0.0165941
$584$	0	0
$585$	23.8903	0.987743
$586$	0	0
$587$	−31.1230	−1.28458	−0.642292	−	0.766460i	$-0.722016\pi$
$587$	−31.1230	−1.28458	−0.642292	+	0.766460i	$0.722016\pi$
$588$	0	0
$589$	−31.3433	−1.29148
$590$	0	0
$591$	18.2999	0.752757
$592$	0	0
$593$	−24.6606	−1.01269	−0.506344	−	0.862331i	$-0.669004\pi$
$593$	−24.6606	−1.01269	−0.506344	+	0.862331i	$0.669004\pi$
$594$	0	0
$595$	−4.78197	−0.196042
$596$	0	0
$597$	−19.5934	−0.801905
$598$	0	0
$599$	10.8236	0.442239	0.221120	−	0.975247i	$-0.429029\pi$
$599$	10.8236	0.442239	0.221120	+	0.975247i	$0.429029\pi$
$600$	0	0
$601$	16.6417	0.678828	0.339414	−	0.940637i	$-0.389771\pi$
$601$	16.6417	0.678828	0.339414	+	0.940637i	$0.389771\pi$
$602$	0	0
$603$	−12.2802	−0.500089
$604$	0	0
$605$	−28.0247	−1.13936
$606$	0	0
$607$	−22.3716	−0.908037	−0.454019	−	0.890992i	$-0.650010\pi$
$607$	−22.3716	−0.908037	−0.454019	+	0.890992i	$0.650010\pi$
$608$	0	0
$609$	18.2495	0.739508
$610$	0	0
$611$	−3.03938	−0.122960
$612$	0	0
$613$	46.2891	1.86960	0.934800	−	0.355175i	$-0.115579\pi$
$613$	46.2891	1.86960	0.934800	+	0.355175i	$0.115579\pi$
$614$	0	0
$615$	58.5172	2.35964
$616$	0	0
$617$	3.37531	0.135885	0.0679424	−	0.997689i	$-0.478357\pi$
$617$	3.37531	0.135885	0.0679424	+	0.997689i	$0.478357\pi$
$618$	0	0
$619$	−5.05532	−0.203190	−0.101595	−	0.994826i	$-0.532395\pi$
$619$	−5.05532	−0.203190	−0.101595	+	0.994826i	$0.532395\pi$
$620$	0	0
$621$	−19.6611	−0.788972
$622$	0	0
$623$	−1.60040	−0.0641186
$624$	0	0
$625$	−30.6750	−1.22700
$626$	0	0
$627$	5.25040	0.209681
$628$	0	0
$629$	4.34644	0.173304
$630$	0	0
$631$	31.2089	1.24241	0.621204	−	0.783649i	$-0.286644\pi$
$631$	31.2089	1.24241	0.621204	+	0.783649i	$0.286644\pi$
$632$	0	0
$633$	57.4214	2.28230
$634$	0	0
$635$	−0.954129	−0.0378634
$636$	0	0
$637$	6.53068	0.258755
$638$	0	0
$639$	0.228991	0.00905876
$640$	0	0
$641$	25.7829	1.01836	0.509182	−	0.860659i	$-0.329948\pi$
$641$	25.7829	1.01836	0.509182	+	0.860659i	$0.329948\pi$
$642$	0	0
$643$	2.57885	0.101700	0.0508500	−	0.998706i	$-0.483807\pi$
$643$	2.57885	0.101700	0.0508500	+	0.998706i	$0.483807\pi$
$644$	0	0
$645$	−4.11033	−0.161844
$646$	0	0
$647$	−10.0909	−0.396713	−0.198356	−	0.980130i	$-0.563560\pi$
$647$	−10.0909	−0.396713	−0.198356	+	0.980130i	$0.563560\pi$
$648$	0	0
$649$	−4.70713	−0.184771
$650$	0	0
$651$	11.9647	0.468935
$652$	0	0
$653$	−19.6009	−0.767041	−0.383520	−	0.923532i	$-0.625288\pi$
$653$	−19.6009	−0.767041	−0.383520	+	0.923532i	$0.625288\pi$
$654$	0	0
$655$	−34.1805	−1.33554
$656$	0	0
$657$	−4.91872	−0.191898
$658$	0	0
$659$	19.6681	0.766162	0.383081	−	0.923715i	$-0.374863\pi$
$659$	19.6681	0.766162	0.383081	+	0.923715i	$0.374863\pi$
$660$	0	0
$661$	1.60575	0.0624564	0.0312282	−	0.999512i	$-0.490058\pi$
$661$	1.60575	0.0624564	0.0312282	+	0.999512i	$0.490058\pi$
$662$	0	0
$663$	25.2549	0.980820
$664$	0	0
$665$	−14.2821	−0.553834
$666$	0	0
$667$	−51.1482	−1.98047
$668$	0	0
$669$	−40.4418	−1.56357
$670$	0	0
$671$	4.86832	0.187939
$672$	0	0
$673$	9.91726	0.382282	0.191141	−	0.981563i	$-0.438781\pi$
$673$	9.91726	0.382282	0.191141	+	0.981563i	$0.438781\pi$
$674$	0	0
$675$	−5.81878	−0.223965
$676$	0	0
$677$	−23.2883	−0.895042	−0.447521	−	0.894273i	$-0.647693\pi$
$677$	−23.2883	−0.895042	−0.447521	+	0.894273i	$0.647693\pi$
$678$	0	0
$679$	8.88621	0.341021
$680$	0	0
$681$	−1.11434	−0.0427015
$682$	0	0
$683$	−7.80039	−0.298474	−0.149237	−	0.988801i	$-0.547682\pi$
$683$	−7.80039	−0.298474	−0.149237	+	0.988801i	$0.547682\pi$
$684$	0	0
$685$	17.8085	0.680428
$686$	0	0
$687$	−27.7515	−1.05879
$688$	0	0
$689$	5.75605	0.219288
$690$	0	0
$691$	−44.0204	−1.67462	−0.837308	−	0.546732i	$-0.815872\pi$
$691$	−44.0204	−1.67462	−0.837308	+	0.546732i	$0.815872\pi$
$692$	0	0
$693$	−0.640472	−0.0243295
$694$	0	0
$695$	−59.3867	−2.25267
$696$	0	0
$697$	19.7677	0.748756
$698$	0	0
$699$	29.1362	1.10203
$700$	0	0
$701$	−30.7777	−1.16246	−0.581229	−	0.813740i	$-0.697428\pi$
$701$	−30.7777	−1.16246	−0.581229	+	0.813740i	$0.697428\pi$
$702$	0	0
$703$	12.9813	0.489598
$704$	0	0
$705$	−2.53732	−0.0955612
$706$	0	0
$707$	−13.8668	−0.521516
$708$	0	0
$709$	−18.3503	−0.689159	−0.344580	−	0.938757i	$-0.611979\pi$
$709$	−18.3503	−0.689159	−0.344580	+	0.938757i	$0.611979\pi$
$710$	0	0
$711$	11.6756	0.437868
$712$	0	0
$713$	−33.5338	−1.25585
$714$	0	0
$715$	7.70838	0.288277
$716$	0	0
$717$	−31.9471	−1.19309
$718$	0	0
$719$	−36.4527	−1.35946	−0.679728	−	0.733464i	$-0.737902\pi$
$719$	−36.4527	−1.35946	−0.679728	+	0.733464i	$0.737902\pi$
$720$	0	0
$721$	−14.3236	−0.533439
$722$	0	0
$723$	59.4149	2.20966
$724$	0	0
$725$	−15.1375	−0.562194
$726$	0	0
$727$	−37.9281	−1.40668	−0.703338	−	0.710855i	$-0.748308\pi$
$727$	−37.9281	−1.40668	−0.703338	+	0.710855i	$0.748308\pi$
$728$	0	0
$729$	5.20661	0.192838
$730$	0	0
$731$	−1.38852	−0.0513561
$732$	0	0
$733$	−18.2291	−0.673306	−0.336653	−	0.941629i	$-0.609295\pi$
$733$	−18.2291	−0.673306	−0.336653	+	0.941629i	$0.609295\pi$
$734$	0	0
$735$	5.45192	0.201097
$736$	0	0
$737$	−3.96230	−0.145953
$738$	0	0
$739$	−11.2720	−0.414645	−0.207323	−	0.978273i	$-0.566475\pi$
$739$	−11.2720	−0.414645	−0.207323	+	0.978273i	$0.566475\pi$
$740$	0	0
$741$	75.4275	2.77090
$742$	0	0
$743$	41.0385	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$743$	41.0385	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$744$	0	0
$745$	55.8171	2.04498
$746$	0	0
$747$	5.00660	0.183182
$748$	0	0
$749$	15.1314	0.552888
$750$	0	0
$751$	16.8700	0.615594	0.307797	−	0.951452i	$-0.400408\pi$
$751$	16.8700	0.615594	0.307797	+	0.951452i	$0.400408\pi$
$752$	0	0
$753$	26.4896	0.965336
$754$	0	0
$755$	−12.3672	−0.450087
$756$	0	0
$757$	−30.6915	−1.11550	−0.557750	−	0.830009i	$-0.688335\pi$
$757$	−30.6915	−1.11550	−0.557750	+	0.830009i	$0.688335\pi$
$758$	0	0
$759$	5.61733	0.203896
$760$	0	0
$761$	−8.35285	−0.302791	−0.151395	−	0.988473i	$-0.548377\pi$
$761$	−8.35285	−0.302791	−0.151395	+	0.988473i	$0.548377\pi$
$762$	0	0
$763$	10.4465	0.378188
$764$	0	0
$765$	6.73731	0.243588
$766$	0	0
$767$	−67.6229	−2.44172
$768$	0	0
$769$	−21.8388	−0.787527	−0.393764	−	0.919212i	$-0.628827\pi$
$769$	−21.8388	−0.787527	−0.393764	+	0.919212i	$0.628827\pi$
$770$	0	0
$771$	−57.7030	−2.07812
$772$	0	0
$773$	−9.26946	−0.333399	−0.166700	−	0.986008i	$-0.553311\pi$
$773$	−9.26946	−0.333399	−0.166700	+	0.986008i	$0.553311\pi$
$774$	0	0
$775$	−9.92446	−0.356497
$776$	0	0
$777$	−4.95537	−0.177773
$778$	0	0
$779$	59.0392	2.11530
$780$	0	0
$781$	0.0738857	0.00264384
$782$	0	0
$783$	29.0369	1.03769
$784$	0	0
$785$	43.2879	1.54501
$786$	0	0
$787$	25.2810	0.901170	0.450585	−	0.892733i	$-0.351215\pi$
$787$	25.2810	0.901170	0.450585	+	0.892733i	$0.351215\pi$
$788$	0	0
$789$	31.3508	1.11612
$790$	0	0
$791$	8.05995	0.286579
$792$	0	0
$793$	69.9386	2.48359
$794$	0	0
$795$	4.80524	0.170424
$796$	0	0
$797$	22.6248	0.801410	0.400705	−	0.916207i	$-0.368765\pi$
$797$	22.6248	0.801410	0.400705	+	0.916207i	$0.368765\pi$
$798$	0	0
$799$	−0.857136	−0.0303233
$800$	0	0
$801$	2.25480	0.0796694
$802$	0	0
$803$	−1.58706	−0.0560062
$804$	0	0
$805$	−15.2802	−0.538555
$806$	0	0
$807$	−0.0135499	−0.000476981 0
$808$	0	0
$809$	13.5401	0.476046	0.238023	−	0.971260i	$-0.423501\pi$
$809$	13.5401	0.476046	0.238023	+	0.971260i	$0.423501\pi$
$810$	0	0
$811$	−2.11986	−0.0744384	−0.0372192	−	0.999307i	$-0.511850\pi$
$811$	−2.11986	−0.0744384	−0.0372192	+	0.999307i	$0.511850\pi$
$812$	0	0
$813$	−12.0644	−0.423117
$814$	0	0
$815$	35.5477	1.24518
$816$	0	0
$817$	−4.14701	−0.145085
$818$	0	0
$819$	−9.20105	−0.321511
$820$	0	0
$821$	−0.815513	−0.0284616	−0.0142308	−	0.999899i	$-0.504530\pi$
$821$	−0.815513	−0.0284616	−0.0142308	+	0.999899i	$0.504530\pi$
$822$	0	0
$823$	−14.8651	−0.518164	−0.259082	−	0.965855i	$-0.583420\pi$
$823$	−14.8651	−0.518164	−0.259082	+	0.965855i	$0.583420\pi$
$824$	0	0
$825$	1.66247	0.0578799
$826$	0	0
$827$	−31.9223	−1.11005	−0.555023	−	0.831835i	$-0.687291\pi$
$827$	−31.9223	−1.11005	−0.555023	+	0.831835i	$0.687291\pi$
$828$	0	0
$829$	−21.8241	−0.757983	−0.378992	−	0.925400i	$-0.623729\pi$
$829$	−21.8241	−0.757983	−0.378992	+	0.925400i	$0.623729\pi$
$830$	0	0
$831$	−33.5209	−1.16283
$832$	0	0
$833$	1.84172	0.0638117
$834$	0	0
$835$	34.2516	1.18532
$836$	0	0
$837$	19.0371	0.658019
$838$	0	0
$839$	37.1968	1.28418	0.642089	−	0.766630i	$-0.278068\pi$
$839$	37.1968	1.28418	0.642089	+	0.766630i	$0.278068\pi$
$840$	0	0
$841$	46.5393	1.60480
$842$	0	0
$843$	20.3904	0.702284
$844$	0	0
$845$	76.9848	2.64836
$846$	0	0
$847$	10.7933	0.370864
$848$	0	0
$849$	37.5755	1.28959
$850$	0	0
$851$	13.8885	0.476091
$852$	0	0
$853$	−31.7608	−1.08747	−0.543735	−	0.839257i	$-0.682990\pi$
$853$	−31.7608	−1.08747	−0.543735	+	0.839257i	$0.682990\pi$
$854$	0	0
$855$	20.1220	0.688156
$856$	0	0
$857$	−47.6664	−1.62825	−0.814126	−	0.580688i	$-0.802784\pi$
$857$	−47.6664	−1.62825	−0.814126	+	0.580688i	$0.802784\pi$
$858$	0	0
$859$	−31.6231	−1.07897	−0.539484	−	0.841996i	$-0.681380\pi$
$859$	−31.6231	−1.07897	−0.539484	+	0.841996i	$0.681380\pi$
$860$	0	0
$861$	−22.5372	−0.768064
$862$	0	0
$863$	6.01295	0.204683	0.102342	−	0.994749i	$-0.467367\pi$
$863$	6.01295	0.204683	0.102342	+	0.994749i	$0.467367\pi$
$864$	0	0
$865$	23.5321	0.800115
$866$	0	0
$867$	−28.5734	−0.970404
$868$	0	0
$869$	3.76721	0.127794
$870$	0	0
$871$	−56.9226	−1.92875
$872$	0	0
$873$	−12.5198	−0.423730
$874$	0	0
$875$	8.46014	0.286005
$876$	0	0
$877$	−10.6592	−0.359937	−0.179968	−	0.983672i	$-0.557600\pi$
$877$	−10.6592	−0.359937	−0.179968	+	0.983672i	$0.557600\pi$
$878$	0	0
$879$	2.69134	0.0907766
$880$	0	0
$881$	18.6023	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$881$	18.6023	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$882$	0	0
$883$	21.3704	0.719170	0.359585	−	0.933112i	$-0.382918\pi$
$883$	21.3704	0.719170	0.359585	+	0.933112i	$0.382918\pi$
$884$	0	0
$885$	−56.4527	−1.89764
$886$	0	0
$887$	−56.3410	−1.89175	−0.945873	−	0.324537i	$-0.894792\pi$
$887$	−56.3410	−1.89175	−0.945873	+	0.324537i	$0.894792\pi$
$888$	0	0
$889$	0.367471	0.0123246
$890$	0	0
$891$	−5.11037	−0.171204
$892$	0	0
$893$	−2.55996	−0.0856658
$894$	0	0
$895$	−33.2674	−1.11201
$896$	0	0
$897$	80.6988	2.69446
$898$	0	0
$899$	49.5250	1.65175
$900$	0	0
$901$	1.62326	0.0540787
$902$	0	0
$903$	1.58304	0.0526804
$904$	0	0
$905$	−50.3663	−1.67423
$906$	0	0
$907$	−7.40527	−0.245888	−0.122944	−	0.992414i	$-0.539234\pi$
$907$	−7.40527	−0.245888	−0.122944	+	0.992414i	$0.539234\pi$
$908$	0	0
$909$	19.5370	0.648000
$910$	0	0
$911$	28.0494	0.929319	0.464660	−	0.885489i	$-0.346177\pi$
$911$	28.0494	0.929319	0.464660	+	0.885489i	$0.346177\pi$
$912$	0	0
$913$	1.61542	0.0534625
$914$	0	0
$915$	58.3859	1.93018
$916$	0	0
$917$	13.1642	0.434721
$918$	0	0
$919$	−36.2282	−1.19506	−0.597529	−	0.801847i	$-0.703851\pi$
$919$	−36.2282	−1.19506	−0.597529	+	0.801847i	$0.703851\pi$
$920$	0	0
$921$	−43.9268	−1.44744
$922$	0	0
$923$	1.06145	0.0349379
$924$	0	0
$925$	4.11036	0.135148
$926$	0	0
$927$	20.1805	0.662815
$928$	0	0
$929$	16.6482	0.546210	0.273105	−	0.961984i	$-0.411949\pi$
$929$	16.6482	0.546210	0.273105	+	0.961984i	$0.411949\pi$
$930$	0	0
$931$	5.50056	0.180273
$932$	0	0
$933$	−24.3661	−0.797710
$934$	0	0
$935$	2.17384	0.0710922
$936$	0	0
$937$	−16.4319	−0.536807	−0.268403	−	0.963307i	$-0.586496\pi$
$937$	−16.4319	−0.536807	−0.268403	+	0.963307i	$0.586496\pi$
$938$	0	0
$939$	38.9755	1.27192
$940$	0	0
$941$	16.5679	0.540099	0.270049	−	0.962846i	$-0.412960\pi$
$941$	16.5679	0.540099	0.270049	+	0.962846i	$0.412960\pi$
$942$	0	0
$943$	63.1652	2.05694
$944$	0	0
$945$	8.67456	0.282183
$946$	0	0
$947$	17.7431	0.576574	0.288287	−	0.957544i	$-0.406914\pi$
$947$	17.7431	0.576574	0.288287	+	0.957544i	$0.406914\pi$
$948$	0	0
$949$	−22.7998	−0.740113
$950$	0	0
$951$	47.1739	1.52972
$952$	0	0
$953$	21.9557	0.711213	0.355607	−	0.934636i	$-0.384274\pi$
$953$	21.9557	0.711213	0.355607	+	0.934636i	$0.384274\pi$
$954$	0	0
$955$	50.0138	1.61841
$956$	0	0
$957$	−8.29607	−0.268174
$958$	0	0
$959$	−6.85872	−0.221480
$960$	0	0
$961$	1.46955	0.0474047
$962$	0	0
$963$	−21.3185	−0.686980
$964$	0	0
$965$	−6.18800	−0.199199
$966$	0	0
$967$	−22.2420	−0.715255	−0.357628	−	0.933864i	$-0.616414\pi$
$967$	−22.2420	−0.715255	−0.357628	+	0.933864i	$0.616414\pi$
$968$	0	0
$969$	21.2713	0.683333
$970$	0	0
$971$	−26.5681	−0.852612	−0.426306	−	0.904579i	$-0.640185\pi$
$971$	−26.5681	−0.852612	−0.426306	+	0.904579i	$0.640185\pi$
$972$	0	0
$973$	22.8721	0.733244
$974$	0	0
$975$	23.8832	0.764873
$976$	0	0
$977$	−9.54985	−0.305527	−0.152763	−	0.988263i	$-0.548817\pi$
$977$	−9.54985	−0.305527	−0.152763	+	0.988263i	$0.548817\pi$
$978$	0	0
$979$	0.727526	0.0232518
$980$	0	0
$981$	−14.7180	−0.469911
$982$	0	0
$983$	−5.65760	−0.180449	−0.0902247	−	0.995921i	$-0.528759\pi$
$983$	−5.65760	−0.180449	−0.0902247	+	0.995921i	$0.528759\pi$
$984$	0	0
$985$	22.6291	0.721023
$986$	0	0
$987$	0.977219	0.0311052
$988$	0	0
$989$	−4.43682	−0.141083
$990$	0	0
$991$	62.8260	1.99573	0.997867	−	0.0652866i	$-0.0207962\pi$
$991$	62.8260	1.99573	0.997867	+	0.0652866i	$0.0207962\pi$
$992$	0	0
$993$	16.3216	0.517951
$994$	0	0
$995$	−24.2287	−0.768100
$996$	0	0
$997$	8.14444	0.257937	0.128969	−	0.991649i	$-0.458833\pi$
$997$	8.14444	0.257937	0.128969	+	0.991649i	$0.458833\pi$
$998$	0	0
$999$	−7.88449	−0.249454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.be.1.7		8
4.3	odd	2		7168.2.a.bf.1.2		8
8.3	odd	2		7168.2.a.bb.1.7		8
8.5	even	2		7168.2.a.ba.1.2		8
32.3	odd	8		1792.2.m.h.1345.1	yes	16
32.5	even	8		1792.2.m.g.449.1	yes	16
32.11	odd	8		1792.2.m.h.449.1	yes	16
32.13	even	8		1792.2.m.g.1345.1	yes	16
32.19	odd	8		1792.2.m.e.1345.8	yes	16
32.21	even	8		1792.2.m.f.449.8	yes	16
32.27	odd	8		1792.2.m.e.449.8	✓	16
32.29	even	8		1792.2.m.f.1345.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.8	✓	16	32.27	odd	8
1792.2.m.e.1345.8	yes	16	32.19	odd	8
1792.2.m.f.449.8	yes	16	32.21	even	8
1792.2.m.f.1345.8	yes	16	32.29	even	8
1792.2.m.g.449.1	yes	16	32.5	even	8
1792.2.m.g.1345.1	yes	16	32.13	even	8
1792.2.m.h.449.1	yes	16	32.11	odd	8
1792.2.m.h.1345.1	yes	16	32.3	odd	8
7168.2.a.ba.1.2		8	8.5	even	2
7168.2.a.bb.1.7		8	8.3	odd	2
7168.2.a.be.1.7		8	1.1	even	1	trivial
7168.2.a.bf.1.2		8	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q+2.09974 q^{3} +2.59647 q^{5} -1.00000 q^{7} +1.40890 q^{9} +O(q^{10})\)	\(q+2.09974 q^{3} +2.59647 q^{5} -1.00000 q^{7} +1.40890 q^{9} +0.454591 q^{11} +6.53068 q^{13} +5.45192 q^{15} +1.84172 q^{17} +5.50056 q^{19} -2.09974 q^{21} +5.88497 q^{23} +1.74168 q^{25} -3.34090 q^{27} -8.69134 q^{29} -5.69821 q^{31} +0.954522 q^{33} -2.59647 q^{35} +2.35999 q^{37} +13.7127 q^{39} +10.7333 q^{41} -0.753925 q^{43} +3.65817 q^{45} -0.465401 q^{47} +1.00000 q^{49} +3.86712 q^{51} +0.881386 q^{53} +1.18033 q^{55} +11.5497 q^{57} -10.3546 q^{59} +10.7092 q^{61} -1.40890 q^{63} +16.9567 q^{65} -8.71619 q^{67} +12.3569 q^{69} +0.162532 q^{71} -3.49118 q^{73} +3.65707 q^{75} -0.454591 q^{77} +8.28703 q^{79} -11.2417 q^{81} +3.55356 q^{83} +4.78197 q^{85} -18.2495 q^{87} +1.60040 q^{89} -6.53068 q^{91} -11.9647 q^{93} +14.2821 q^{95} -8.88621 q^{97} +0.640472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q + 8 * q^5 - 8 * q^7 + 12 * q^9	\( 8 q + 8 q^{5} - 8 q^{7} + 12 q^{9} - 12 q^{11} + 20 q^{13} + 4 q^{17} - 4 q^{19} - 8 q^{23} + 12 q^{25} + 12 q^{27} + 8 q^{29} + 4 q^{31} + 8 q^{33} - 8 q^{35} + 8 q^{37} + 16 q^{39} - 12 q^{41} + 4 q^{43} + 52 q^{45} - 20 q^{47} + 8 q^{49} - 32 q^{51} + 40 q^{53} - 24 q^{55} - 4 q^{57} - 4 q^{59} - 8 q^{61} - 12 q^{63} + 36 q^{65} - 28 q^{67} + 4 q^{69} + 16 q^{71} + 16 q^{73} + 28 q^{75} + 12 q^{77} + 20 q^{81} + 8 q^{83} + 16 q^{85} - 20 q^{87} + 16 q^{89} - 20 q^{91} + 16 q^{93} + 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q + 8 * q^5 - 8 * q^7 + 12 * q^9 - 12 * q^11 + 20 * q^13 + 4 * q^17 - 4 * q^19 - 8 * q^23 + 12 * q^25 + 12 * q^27 + 8 * q^29 + 4 * q^31 + 8 * q^33 - 8 * q^35 + 8 * q^37 + 16 * q^39 - 12 * q^41 + 4 * q^43 + 52 * q^45 - 20 * q^47 + 8 * q^49 - 32 * q^51 + 40 * q^53 - 24 * q^55 - 4 * q^57 - 4 * q^59 - 8 * q^61 - 12 * q^63 + 36 * q^65 - 28 * q^67 + 4 * q^69 + 16 * q^71 + 16 * q^73 + 28 * q^75 + 12 * q^77 + 20 * q^81 + 8 * q^83 + 16 * q^85 - 20 * q^87 + 16 * q^89 - 20 * q^91 + 16 * q^93 + 40 * q^95 - 36 * q^97 + 4 * q^99

Introduction

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