LMFDB - Embedded newform 7168.2.a.ba.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:	$1$
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			$2.38887$ 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)

$f(q)$	$=$	$q+2.29954 q^{3} -1.65321 q^{5} -1.00000 q^{7} +2.28788 q^{9} +O(q^{10})$	$q+2.29954 q^{3} -1.65321 q^{5} -1.00000 q^{7} +2.28788 q^{9} +6.22192 q^{11} +0.634100 q^{13} -3.80163 q^{15} -5.02326 q^{17} +2.11921 q^{19} -2.29954 q^{21} -8.89958 q^{23} -2.26688 q^{25} -1.63755 q^{27} +1.13585 q^{29} -8.27524 q^{31} +14.3075 q^{33} +1.65321 q^{35} +2.19447 q^{37} +1.45814 q^{39} -4.93020 q^{41} -5.27913 q^{43} -3.78235 q^{45} -6.68692 q^{47} +1.00000 q^{49} -11.5512 q^{51} -6.41239 q^{53} -10.2862 q^{55} +4.87321 q^{57} -1.50921 q^{59} +7.23152 q^{61} -2.28788 q^{63} -1.04830 q^{65} +10.5717 q^{67} -20.4649 q^{69} +4.07354 q^{71} +13.5063 q^{73} -5.21278 q^{75} -6.22192 q^{77} +9.19701 q^{79} -10.6293 q^{81} -3.70875 q^{83} +8.30452 q^{85} +2.61192 q^{87} +1.60040 q^{89} -0.634100 q^{91} -19.0292 q^{93} -3.50351 q^{95} -13.0167 q^{97} +14.2350 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.29954	1.32764	0.663820	−	0.747893i	$-0.268934\pi$
$3$	2.29954	1.32764	0.663820	+	0.747893i	$0.268934\pi$
$4$	0	0
$5$	−1.65321	−0.739340	−0.369670	−	0.929163i	$-0.620529\pi$
$5$	−1.65321	−0.739340	−0.369670	+	0.929163i	$0.620529\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.28788	0.762626
$10$	0	0
$11$	6.22192	1.87598	0.937989	−	0.346664i	$-0.112686\pi$
$11$	6.22192	1.87598	0.937989	+	0.346664i	$0.112686\pi$
$12$	0	0
$13$	0.634100	0.175868	0.0879339	−	0.996126i	$-0.471974\pi$
$13$	0.634100	0.175868	0.0879339	+	0.996126i	$0.471974\pi$
$14$	0	0
$15$	−3.80163	−0.981576
$16$	0	0
$17$	−5.02326	−1.21832	−0.609159	−	0.793048i	$-0.708493\pi$
$17$	−5.02326	−1.21832	−0.609159	+	0.793048i	$0.708493\pi$
$18$	0	0
$19$	2.11921	0.486181	0.243091	−	0.970004i	$-0.421839\pi$
$19$	2.11921	0.486181	0.243091	+	0.970004i	$0.421839\pi$
$20$	0	0
$21$	−2.29954	−0.501800
$22$	0	0
$23$	−8.89958	−1.85569	−0.927845	−	0.372965i	$-0.878341\pi$
$23$	−8.89958	−1.85569	−0.927845	+	0.372965i	$0.878341\pi$
$24$	0	0
$25$	−2.26688	−0.453377
$26$	0	0
$27$	−1.63755	−0.315148
$28$	0	0
$29$	1.13585	0.210922	0.105461	−	0.994423i	$-0.466368\pi$
$29$	1.13585	0.210922	0.105461	+	0.994423i	$0.466368\pi$
$30$	0	0
$31$	−8.27524	−1.48628	−0.743139	−	0.669137i	$-0.766664\pi$
$31$	−8.27524	−1.48628	−0.743139	+	0.669137i	$0.766664\pi$
$32$	0	0
$33$	14.3075	2.49062
$34$	0	0
$35$	1.65321	0.279444
$36$	0	0
$37$	2.19447	0.360769	0.180385	−	0.983596i	$-0.442266\pi$
$37$	2.19447	0.360769	0.180385	+	0.983596i	$0.442266\pi$
$38$	0	0
$39$	1.45814	0.233489
$40$	0	0
$41$	−4.93020	−0.769968	−0.384984	−	0.922923i	$-0.625793\pi$
$41$	−4.93020	−0.769968	−0.384984	+	0.922923i	$0.625793\pi$
$42$	0	0
$43$	−5.27913	−0.805059	−0.402530	−	0.915407i	$-0.631869\pi$
$43$	−5.27913	−0.805059	−0.402530	+	0.915407i	$0.631869\pi$
$44$	0	0
$45$	−3.78235	−0.563839
$46$	0	0
$47$	−6.68692	−0.975387	−0.487693	−	0.873015i	$-0.662162\pi$
$47$	−6.68692	−0.975387	−0.487693	+	0.873015i	$0.662162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−11.5512	−1.61749
$52$	0	0
$53$	−6.41239	−0.880809	−0.440405	−	0.897799i	$-0.645165\pi$
$53$	−6.41239	−0.880809	−0.440405	+	0.897799i	$0.645165\pi$
$54$	0	0
$55$	−10.2862	−1.38699
$56$	0	0
$57$	4.87321	0.645473
$58$	0	0
$59$	−1.50921	−0.196482	−0.0982410	−	0.995163i	$-0.531322\pi$
$59$	−1.50921	−0.196482	−0.0982410	+	0.995163i	$0.531322\pi$
$60$	0	0
$61$	7.23152	0.925901	0.462951	−	0.886384i	$-0.346791\pi$
$61$	7.23152	0.925901	0.462951	+	0.886384i	$0.346791\pi$
$62$	0	0
$63$	−2.28788	−0.288245
$64$	0	0
$65$	−1.04830	−0.130026
$66$	0	0
$67$	10.5717	1.29154	0.645769	−	0.763533i	$-0.276537\pi$
$67$	10.5717	1.29154	0.645769	+	0.763533i	$0.276537\pi$
$68$	0	0
$69$	−20.4649	−2.46369
$70$	0	0
$71$	4.07354	0.483440	0.241720	−	0.970346i	$-0.422288\pi$
$71$	4.07354	0.483440	0.241720	+	0.970346i	$0.422288\pi$
$72$	0	0
$73$	13.5063	1.58079	0.790397	−	0.612595i	$-0.209874\pi$
$73$	13.5063	1.58079	0.790397	+	0.612595i	$0.209874\pi$
$74$	0	0
$75$	−5.21278	−0.601921
$76$	0	0
$77$	−6.22192	−0.709053
$78$	0	0
$79$	9.19701	1.03474	0.517372	−	0.855760i	$-0.326910\pi$
$79$	9.19701	1.03474	0.517372	+	0.855760i	$0.326910\pi$
$80$	0	0
$81$	−10.6293	−1.18103
$82$	0	0
$83$	−3.70875	−0.407088	−0.203544	−	0.979066i	$-0.565246\pi$
$83$	−3.70875	−0.407088	−0.203544	+	0.979066i	$0.565246\pi$
$84$	0	0
$85$	8.30452	0.900752
$86$	0	0
$87$	2.61192	0.280028
$88$	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
$91$	−0.634100	−0.0664718
$92$	0	0
$93$	−19.0292	−1.97324
$94$	0	0
$95$	−3.50351	−0.359453
$96$	0	0
$97$	−13.0167	−1.32164	−0.660822	−	0.750543i	$-0.729792\pi$
$97$	−13.0167	−1.32164	−0.660822	+	0.750543i	$0.729792\pi$
$98$	0	0
$99$	14.2350	1.43067
$100$	0	0
$101$	−5.32103	−0.529462	−0.264731	−	0.964322i	$-0.585283\pi$
$101$	−5.32103	−0.529462	−0.264731	+	0.964322i	$0.585283\pi$
$102$	0	0
$103$	−7.91886	−0.780268	−0.390134	−	0.920758i	$-0.627571\pi$
$103$	−7.91886	−0.780268	−0.390134	+	0.920758i	$0.627571\pi$
$104$	0	0
$105$	3.80163	0.371001
$106$	0	0
$107$	−19.2277	−1.85881	−0.929407	−	0.369056i	$-0.879681\pi$
$107$	−19.2277	−1.85881	−0.929407	+	0.369056i	$0.879681\pi$
$108$	0	0
$109$	−9.94712	−0.952761	−0.476380	−	0.879239i	$-0.658052\pi$
$109$	−9.94712	−0.952761	−0.476380	+	0.879239i	$0.658052\pi$
$110$	0	0
$111$	5.04627	0.478971
$112$	0	0
$113$	−5.62753	−0.529394	−0.264697	−	0.964332i	$-0.585272\pi$
$113$	−5.62753	−0.529394	−0.264697	+	0.964332i	$0.585272\pi$
$114$	0	0
$115$	14.7129	1.37199
$116$	0	0
$117$	1.45074	0.134121
$118$	0	0
$119$	5.02326	0.460481
$120$	0	0
$121$	27.7123	2.51930
$122$	0	0
$123$	−11.3372	−1.02224
$124$	0	0
$125$	12.0137	1.07454
$126$	0	0
$127$	−13.5320	−1.20077	−0.600386	−	0.799710i	$-0.704987\pi$
$127$	−13.5320	−1.20077	−0.600386	+	0.799710i	$0.704987\pi$
$128$	0	0
$129$	−12.1396	−1.06883
$130$	0	0
$131$	−2.10195	−0.183648	−0.0918242	−	0.995775i	$-0.529270\pi$
$131$	−2.10195	−0.183648	−0.0918242	+	0.995775i	$0.529270\pi$
$132$	0	0
$133$	−2.11921	−0.183759
$134$	0	0
$135$	2.70723	0.233001
$136$	0	0
$137$	11.7463	1.00355	0.501776	−	0.864998i	$-0.332680\pi$
$137$	11.7463	1.00355	0.501776	+	0.864998i	$0.332680\pi$
$138$	0	0
$139$	−11.3252	−0.960590	−0.480295	−	0.877107i	$-0.659470\pi$
$139$	−11.3252	−0.960590	−0.480295	+	0.877107i	$0.659470\pi$
$140$	0	0
$141$	−15.3768	−1.29496
$142$	0	0
$143$	3.94532	0.329924
$144$	0	0
$145$	−1.87780	−0.155943
$146$	0	0
$147$	2.29954	0.189663
$148$	0	0
$149$	−5.14322	−0.421349	−0.210675	−	0.977556i	$-0.567566\pi$
$149$	−5.14322	−0.421349	−0.210675	+	0.977556i	$0.567566\pi$
$150$	0	0
$151$	−15.9697	−1.29959	−0.649797	−	0.760108i	$-0.725146\pi$
$151$	−15.9697	−1.29959	−0.649797	+	0.760108i	$0.725146\pi$
$152$	0	0
$153$	−11.4926	−0.929121
$154$	0	0
$155$	13.6807	1.09886
$156$	0	0
$157$	19.8670	1.58556	0.792778	−	0.609511i	$-0.208634\pi$
$157$	19.8670	1.58556	0.792778	+	0.609511i	$0.208634\pi$
$158$	0	0
$159$	−14.7455	−1.16940
$160$	0	0
$161$	8.89958	0.701385
$162$	0	0
$163$	14.5377	1.13868	0.569341	−	0.822102i	$-0.307199\pi$
$163$	14.5377	1.13868	0.569341	+	0.822102i	$0.307199\pi$
$164$	0	0
$165$	−23.6534	−1.84142
$166$	0	0
$167$	19.4131	1.50223	0.751115	−	0.660172i	$-0.229517\pi$
$167$	19.4131	1.50223	0.751115	+	0.660172i	$0.229517\pi$
$168$	0	0
$169$	−12.5979	−0.969071
$170$	0	0
$171$	4.84850	0.370774
$172$	0	0
$173$	−6.45278	−0.490596	−0.245298	−	0.969448i	$-0.578886\pi$
$173$	−6.45278	−0.490596	−0.245298	+	0.969448i	$0.578886\pi$
$174$	0	0
$175$	2.26688	0.171360
$176$	0	0
$177$	−3.47048	−0.260857
$178$	0	0
$179$	23.7770	1.77717	0.888587	−	0.458708i	$-0.151688\pi$
$179$	23.7770	1.77717	0.888587	+	0.458708i	$0.151688\pi$
$180$	0	0
$181$	−8.85094	−0.657885	−0.328943	−	0.944350i	$-0.606692\pi$
$181$	−8.85094	−0.657885	−0.328943	+	0.944350i	$0.606692\pi$
$182$	0	0
$183$	16.6292	1.22926
$184$	0	0
$185$	−3.62793	−0.266731
$186$	0	0
$187$	−31.2543	−2.28554
$188$	0	0
$189$	1.63755	0.119115
$190$	0	0
$191$	15.8222	1.14486	0.572428	−	0.819955i	$-0.306001\pi$
$191$	15.8222	1.14486	0.572428	+	0.819955i	$0.306001\pi$
$192$	0	0
$193$	−7.50630	−0.540315	−0.270158	−	0.962816i	$-0.587076\pi$
$193$	−7.50630	−0.540315	−0.270158	+	0.962816i	$0.587076\pi$
$194$	0	0
$195$	−2.41061	−0.172628
$196$	0	0
$197$	24.3697	1.73627	0.868134	−	0.496330i	$-0.165319\pi$
$197$	24.3697	1.73627	0.868134	+	0.496330i	$0.165319\pi$
$198$	0	0
$199$	−21.8653	−1.54999	−0.774995	−	0.631967i	$-0.782248\pi$
$199$	−21.8653	−1.54999	−0.774995	+	0.631967i	$0.782248\pi$
$200$	0	0
$201$	24.3100	1.71470
$202$	0	0
$203$	−1.13585	−0.0797209
$204$	0	0
$205$	8.15068	0.569268
$206$	0	0
$207$	−20.3611	−1.41520
$208$	0	0
$209$	13.1856	0.912065
$210$	0	0
$211$	−3.93947	−0.271205	−0.135602	−	0.990763i	$-0.543297\pi$
$211$	−3.93947	−0.271205	−0.135602	+	0.990763i	$0.543297\pi$
$212$	0	0
$213$	9.36726	0.641834
$214$	0	0
$215$	8.72753	0.595212
$216$	0	0
$217$	8.27524	0.561760
$218$	0	0
$219$	31.0583	2.09872
$220$	0	0
$221$	−3.18525	−0.214263
$222$	0	0
$223$	4.64913	0.311329	0.155665	−	0.987810i	$-0.450248\pi$
$223$	4.64913	0.311329	0.155665	+	0.987810i	$0.450248\pi$
$224$	0	0
$225$	−5.18635	−0.345757
$226$	0	0
$227$	−6.24109	−0.414236	−0.207118	−	0.978316i	$-0.566408\pi$
$227$	−6.24109	−0.414236	−0.207118	+	0.978316i	$0.566408\pi$
$228$	0	0
$229$	20.3814	1.34684	0.673422	−	0.739258i	$-0.264824\pi$
$229$	20.3814	1.34684	0.673422	+	0.739258i	$0.264824\pi$
$230$	0	0
$231$	−14.3075	−0.941367
$232$	0	0
$233$	−11.2765	−0.738746	−0.369373	−	0.929281i	$-0.620428\pi$
$233$	−11.2765	−0.738746	−0.369373	+	0.929281i	$0.620428\pi$
$234$	0	0
$235$	11.0549	0.721142
$236$	0	0
$237$	21.1489	1.37377
$238$	0	0
$239$	0.994605	0.0643356	0.0321678	−	0.999482i	$-0.489759\pi$
$239$	0.994605	0.0643356	0.0321678	+	0.999482i	$0.489759\pi$
$240$	0	0
$241$	−27.1158	−1.74669	−0.873343	−	0.487107i	$-0.838052\pi$
$241$	−27.1158	−1.74669	−0.873343	+	0.487107i	$0.838052\pi$
$242$	0	0
$243$	−19.5297	−1.25283
$244$	0	0
$245$	−1.65321	−0.105620
$246$	0	0
$247$	1.34379	0.0855036
$248$	0	0
$249$	−8.52841	−0.540466
$250$	0	0
$251$	−6.32989	−0.399539	−0.199769	−	0.979843i	$-0.564019\pi$
$251$	−6.32989	−0.399539	−0.199769	+	0.979843i	$0.564019\pi$
$252$	0	0
$253$	−55.3724	−3.48124
$254$	0	0
$255$	19.0966	1.19587
$256$	0	0
$257$	−7.08743	−0.442102	−0.221051	−	0.975262i	$-0.570949\pi$
$257$	−7.08743	−0.442102	−0.221051	+	0.975262i	$0.570949\pi$
$258$	0	0
$259$	−2.19447	−0.136358
$260$	0	0
$261$	2.59868	0.160854
$262$	0	0
$263$	−6.95421	−0.428815	−0.214408	−	0.976744i	$-0.568782\pi$
$263$	−6.95421	−0.428815	−0.214408	+	0.976744i	$0.568782\pi$
$264$	0	0
$265$	10.6010	0.651217
$266$	0	0
$267$	3.68018	0.225223
$268$	0	0
$269$	−18.7534	−1.14341	−0.571707	−	0.820458i	$-0.693719\pi$
$269$	−18.7534	−1.14341	−0.571707	+	0.820458i	$0.693719\pi$
$270$	0	0
$271$	14.2967	0.868463	0.434231	−	0.900801i	$-0.357020\pi$
$271$	14.2967	0.868463	0.434231	+	0.900801i	$0.357020\pi$
$272$	0	0
$273$	−1.45814	−0.0882506
$274$	0	0
$275$	−14.1044	−0.850525
$276$	0	0
$277$	−2.20969	−0.132767	−0.0663836	−	0.997794i	$-0.521146\pi$
$277$	−2.20969	−0.132767	−0.0663836	+	0.997794i	$0.521146\pi$
$278$	0	0
$279$	−18.9327	−1.13347
$280$	0	0
$281$	−6.06619	−0.361879	−0.180939	−	0.983494i	$-0.557914\pi$
$281$	−6.06619	−0.361879	−0.180939	+	0.983494i	$0.557914\pi$
$282$	0	0
$283$	−28.7356	−1.70815	−0.854077	−	0.520146i	$-0.825877\pi$
$283$	−28.7356	−1.70815	−0.854077	+	0.520146i	$0.825877\pi$
$284$	0	0
$285$	−8.05647	−0.477224
$286$	0	0
$287$	4.93020	0.291021
$288$	0	0
$289$	8.23311	0.484301
$290$	0	0
$291$	−29.9324	−1.75467
$292$	0	0
$293$	−31.4461	−1.83710	−0.918549	−	0.395306i	$-0.870639\pi$
$293$	−31.4461	−1.83710	−0.918549	+	0.395306i	$0.870639\pi$
$294$	0	0
$295$	2.49504	0.145267
$296$	0	0
$297$	−10.1887	−0.591210
$298$	0	0
$299$	−5.64323	−0.326356
$300$	0	0
$301$	5.27913	0.304284
$302$	0	0
$303$	−12.2359	−0.702934
$304$	0	0
$305$	−11.9553	−0.684556
$306$	0	0
$307$	25.1243	1.43392	0.716960	−	0.697115i	$-0.245533\pi$
$307$	25.1243	1.43392	0.716960	+	0.697115i	$0.245533\pi$
$308$	0	0
$309$	−18.2097	−1.03591
$310$	0	0
$311$	−31.2698	−1.77315	−0.886574	−	0.462588i	$-0.846921\pi$
$311$	−31.2698	−1.77315	−0.886574	+	0.462588i	$0.846921\pi$
$312$	0	0
$313$	18.0095	1.01796	0.508980	−	0.860778i	$-0.330023\pi$
$313$	18.0095	1.01796	0.508980	+	0.860778i	$0.330023\pi$
$314$	0	0
$315$	3.78235	0.213111
$316$	0	0
$317$	−17.3125	−0.972366	−0.486183	−	0.873857i	$-0.661611\pi$
$317$	−17.3125	−0.972366	−0.486183	+	0.873857i	$0.661611\pi$
$318$	0	0
$319$	7.06715	0.395684
$320$	0	0
$321$	−44.2149	−2.46783
$322$	0	0
$323$	−10.6454	−0.592323
$324$	0	0
$325$	−1.43743	−0.0797344
$326$	0	0
$327$	−22.8738	−1.26492
$328$	0	0
$329$	6.68692	0.368662
$330$	0	0
$331$	16.6108	0.913014	0.456507	−	0.889720i	$-0.349100\pi$
$331$	16.6108	0.913014	0.456507	+	0.889720i	$0.349100\pi$
$332$	0	0
$333$	5.02068	0.275132
$334$	0	0
$335$	−17.4773	−0.954886
$336$	0	0
$337$	−6.23237	−0.339499	−0.169750	−	0.985487i	$-0.554296\pi$
$337$	−6.23237	−0.339499	−0.169750	+	0.985487i	$0.554296\pi$
$338$	0	0
$339$	−12.9407	−0.702844
$340$	0	0
$341$	−51.4879	−2.78822
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	33.8329	1.82150
$346$	0	0
$347$	9.36267	0.502614	0.251307	−	0.967907i	$-0.419140\pi$
$347$	9.36267	0.502614	0.251307	+	0.967907i	$0.419140\pi$
$348$	0	0
$349$	−20.9244	−1.12006	−0.560029	−	0.828473i	$-0.689210\pi$
$349$	−20.9244	−1.12006	−0.560029	+	0.828473i	$0.689210\pi$
$350$	0	0
$351$	−1.03837	−0.0554243
$352$	0	0
$353$	−17.7098	−0.942599	−0.471300	−	0.881973i	$-0.656215\pi$
$353$	−17.7098	−0.942599	−0.471300	+	0.881973i	$0.656215\pi$
$354$	0	0
$355$	−6.73443	−0.357427
$356$	0	0
$357$	11.5512	0.611353
$358$	0	0
$359$	8.53643	0.450535	0.225268	−	0.974297i	$-0.427674\pi$
$359$	8.53643	0.450535	0.225268	+	0.974297i	$0.427674\pi$
$360$	0	0
$361$	−14.5089	−0.763628
$362$	0	0
$363$	63.7254	3.34472
$364$	0	0
$365$	−22.3288	−1.16874
$366$	0	0
$367$	−11.3226	−0.591037	−0.295519	−	0.955337i	$-0.595492\pi$
$367$	−11.3226	−0.591037	−0.295519	+	0.955337i	$0.595492\pi$
$368$	0	0
$369$	−11.2797	−0.587198
$370$	0	0
$371$	6.41239	0.332915
$372$	0	0
$373$	−11.1543	−0.577548	−0.288774	−	0.957397i	$-0.593248\pi$
$373$	−11.1543	−0.577548	−0.288774	+	0.957397i	$0.593248\pi$
$374$	0	0
$375$	27.6260	1.42660
$376$	0	0
$377$	0.720241	0.0370943
$378$	0	0
$379$	13.4809	0.692466	0.346233	−	0.938149i	$-0.387461\pi$
$379$	13.4809	0.692466	0.346233	+	0.938149i	$0.387461\pi$
$380$	0	0
$381$	−31.1174	−1.59419
$382$	0	0
$383$	18.9261	0.967077	0.483539	−	0.875323i	$-0.339351\pi$
$383$	18.9261	0.967077	0.483539	+	0.875323i	$0.339351\pi$
$384$	0	0
$385$	10.2862	0.524231
$386$	0	0
$387$	−12.0780	−0.613959
$388$	0	0
$389$	9.58885	0.486174	0.243087	−	0.970004i	$-0.421840\pi$
$389$	9.58885	0.486174	0.243087	+	0.970004i	$0.421840\pi$
$390$	0	0
$391$	44.7049	2.26082
$392$	0	0
$393$	−4.83352	−0.243819
$394$	0	0
$395$	−15.2046	−0.765028
$396$	0	0
$397$	−2.73804	−0.137418	−0.0687090	−	0.997637i	$-0.521888\pi$
$397$	−2.73804	−0.137418	−0.0687090	+	0.997637i	$0.521888\pi$
$398$	0	0
$399$	−4.87321	−0.243966
$400$	0	0
$401$	16.6483	0.831378	0.415689	−	0.909507i	$-0.363540\pi$
$401$	16.6483	0.831378	0.415689	+	0.909507i	$0.363540\pi$
$402$	0	0
$403$	−5.24733	−0.261388
$404$	0	0
$405$	17.5724	0.873181
$406$	0	0
$407$	13.6538	0.676795
$408$	0	0
$409$	−16.8706	−0.834199	−0.417100	−	0.908861i	$-0.636953\pi$
$409$	−16.8706	−0.834199	−0.417100	+	0.908861i	$0.636953\pi$
$410$	0	0
$411$	27.0110	1.33235
$412$	0	0
$413$	1.50921	0.0742632
$414$	0	0
$415$	6.13136	0.300976
$416$	0	0
$417$	−26.0427	−1.27532
$418$	0	0
$419$	−28.4979	−1.39221	−0.696107	−	0.717938i	$-0.745086\pi$
$419$	−28.4979	−1.39221	−0.696107	+	0.717938i	$0.745086\pi$
$420$	0	0
$421$	9.10775	0.443885	0.221942	−	0.975060i	$-0.428760\pi$
$421$	9.10775	0.443885	0.221942	+	0.975060i	$0.428760\pi$
$422$	0	0
$423$	−15.2988	−0.743855
$424$	0	0
$425$	11.3871	0.552357
$426$	0	0
$427$	−7.23152	−0.349958
$428$	0	0
$429$	9.07242	0.438020
$430$	0	0
$431$	28.6481	1.37993	0.689965	−	0.723843i	$-0.257626\pi$
$431$	28.6481	1.37993	0.689965	+	0.723843i	$0.257626\pi$
$432$	0	0
$433$	4.69574	0.225663	0.112832	−	0.993614i	$-0.464008\pi$
$433$	4.69574	0.225663	0.112832	+	0.993614i	$0.464008\pi$
$434$	0	0
$435$	−4.31807	−0.207036
$436$	0	0
$437$	−18.8601	−0.902202
$438$	0	0
$439$	22.6652	1.08175	0.540874	−	0.841103i	$-0.318093\pi$
$439$	22.6652	1.08175	0.540874	+	0.841103i	$0.318093\pi$
$440$	0	0
$441$	2.28788	0.108947
$442$	0	0
$443$	25.1638	1.19557	0.597784	−	0.801657i	$-0.296048\pi$
$443$	25.1638	1.19557	0.597784	+	0.801657i	$0.296048\pi$
$444$	0	0
$445$	−2.64580	−0.125423
$446$	0	0
$447$	−11.8270	−0.559400
$448$	0	0
$449$	−11.2262	−0.529799	−0.264900	−	0.964276i	$-0.585339\pi$
$449$	−11.2262	−0.529799	−0.264900	+	0.964276i	$0.585339\pi$
$450$	0	0
$451$	−30.6753	−1.44444
$452$	0	0
$453$	−36.7229	−1.72539
$454$	0	0
$455$	1.04830	0.0491452
$456$	0	0
$457$	20.8858	0.976998	0.488499	−	0.872564i	$-0.337545\pi$
$457$	20.8858	0.976998	0.488499	+	0.872564i	$0.337545\pi$
$458$	0	0
$459$	8.22586	0.383950
$460$	0	0
$461$	2.52649	0.117671	0.0588353	−	0.998268i	$-0.481261\pi$
$461$	2.52649	0.117671	0.0588353	+	0.998268i	$0.481261\pi$
$462$	0	0
$463$	−21.0754	−0.979457	−0.489728	−	0.871875i	$-0.662904\pi$
$463$	−21.0754	−0.979457	−0.489728	+	0.871875i	$0.662904\pi$
$464$	0	0
$465$	31.4594	1.45889
$466$	0	0
$467$	11.3928	0.527194	0.263597	−	0.964633i	$-0.415091\pi$
$467$	11.3928	0.527194	0.263597	+	0.964633i	$0.415091\pi$
$468$	0	0
$469$	−10.5717	−0.488156
$470$	0	0
$471$	45.6848	2.10505
$472$	0	0
$473$	−32.8463	−1.51027
$474$	0	0
$475$	−4.80401	−0.220423
$476$	0	0
$477$	−14.6708	−0.671728
$478$	0	0
$479$	19.8741	0.908070	0.454035	−	0.890984i	$-0.349984\pi$
$479$	19.8741	0.908070	0.454035	+	0.890984i	$0.349984\pi$
$480$	0	0
$481$	1.39152	0.0634477
$482$	0	0
$483$	20.4649	0.931186
$484$	0	0
$485$	21.5194	0.977144
$486$	0	0
$487$	19.2744	0.873406	0.436703	−	0.899606i	$-0.356146\pi$
$487$	19.2744	0.873406	0.436703	+	0.899606i	$0.356146\pi$
$488$	0	0
$489$	33.4300	1.51176
$490$	0	0
$491$	−5.67386	−0.256058	−0.128029	−	0.991770i	$-0.540865\pi$
$491$	−5.67386	−0.256058	−0.128029	+	0.991770i	$0.540865\pi$
$492$	0	0
$493$	−5.70565	−0.256970
$494$	0	0
$495$	−23.5335	−1.05775
$496$	0	0
$497$	−4.07354	−0.182723
$498$	0	0
$499$	5.80598	0.259912	0.129956	−	0.991520i	$-0.458516\pi$
$499$	5.80598	0.259912	0.129956	+	0.991520i	$0.458516\pi$
$500$	0	0
$501$	44.6411	1.99442
$502$	0	0
$503$	19.8588	0.885460	0.442730	−	0.896655i	$-0.354010\pi$
$503$	19.8588	0.885460	0.442730	+	0.896655i	$0.354010\pi$
$504$	0	0
$505$	8.79680	0.391452
$506$	0	0
$507$	−28.9694	−1.28658
$508$	0	0
$509$	11.4770	0.508709	0.254355	−	0.967111i	$-0.418137\pi$
$509$	11.4770	0.508709	0.254355	+	0.967111i	$0.418137\pi$
$510$	0	0
$511$	−13.5063	−0.597484
$512$	0	0
$513$	−3.47033	−0.153219
$514$	0	0
$515$	13.0916	0.576883
$516$	0	0
$517$	−41.6054	−1.82981
$518$	0	0
$519$	−14.8384	−0.651335
$520$	0	0
$521$	−11.8477	−0.519059	−0.259530	−	0.965735i	$-0.583568\pi$
$521$	−11.8477	−0.519059	−0.259530	+	0.965735i	$0.583568\pi$
$522$	0	0
$523$	−27.6386	−1.20855	−0.604276	−	0.796775i	$-0.706538\pi$
$523$	−27.6386	−1.20855	−0.604276	+	0.796775i	$0.706538\pi$
$524$	0	0
$525$	5.21278	0.227505
$526$	0	0
$527$	41.5687	1.81076
$528$	0	0
$529$	56.2025	2.44359
$530$	0	0
$531$	−3.45288	−0.149842
$532$	0	0
$533$	−3.12624	−0.135413
$534$	0	0
$535$	31.7876	1.37430
$536$	0	0
$537$	54.6760	2.35945
$538$	0	0
$539$	6.22192	0.267997
$540$	0	0
$541$	41.1852	1.77069	0.885345	−	0.464935i	$-0.153922\pi$
$541$	41.1852	1.77069	0.885345	+	0.464935i	$0.153922\pi$
$542$	0	0
$543$	−20.3531	−0.873434
$544$	0	0
$545$	16.4447	0.704414
$546$	0	0
$547$	17.9638	0.768075	0.384038	−	0.923317i	$-0.374533\pi$
$547$	17.9638	0.768075	0.384038	+	0.923317i	$0.374533\pi$
$548$	0	0
$549$	16.5448	0.706116
$550$	0	0
$551$	2.40710	0.102546
$552$	0	0
$553$	−9.19701	−0.391097
$554$	0	0
$555$	−8.34257	−0.354122
$556$	0	0
$557$	−25.8596	−1.09570	−0.547852	−	0.836575i	$-0.684554\pi$
$557$	−25.8596	−1.09570	−0.547852	+	0.836575i	$0.684554\pi$
$558$	0	0
$559$	−3.34750	−0.141584
$560$	0	0
$561$	−71.8704	−3.03437
$562$	0	0
$563$	1.02037	0.0430034	0.0215017	−	0.999769i	$-0.493155\pi$
$563$	1.02037	0.0430034	0.0215017	+	0.999769i	$0.493155\pi$
$564$	0	0
$565$	9.30352	0.391402
$566$	0	0
$567$	10.6293	0.446387
$568$	0	0
$569$	−33.5555	−1.40672	−0.703360	−	0.710834i	$-0.748318\pi$
$569$	−33.5555	−1.40672	−0.703360	+	0.710834i	$0.748318\pi$
$570$	0	0
$571$	−5.37591	−0.224975	−0.112487	−	0.993653i	$-0.535882\pi$
$571$	−5.37591	−0.224975	−0.112487	+	0.993653i	$0.535882\pi$
$572$	0	0
$573$	36.3838	1.51996
$574$	0	0
$575$	20.1743	0.841327
$576$	0	0
$577$	−2.22515	−0.0926344	−0.0463172	−	0.998927i	$-0.514748\pi$
$577$	−2.22515	−0.0926344	−0.0463172	+	0.998927i	$0.514748\pi$
$578$	0	0
$579$	−17.2610	−0.717343
$580$	0	0
$581$	3.70875	0.153865
$582$	0	0
$583$	−39.8973	−1.65238
$584$	0	0
$585$	−2.39839	−0.0991612
$586$	0	0
$587$	6.12882	0.252964	0.126482	−	0.991969i	$-0.459631\pi$
$587$	6.12882	0.252964	0.126482	+	0.991969i	$0.459631\pi$
$588$	0	0
$589$	−17.5370	−0.722600
$590$	0	0
$591$	56.0390	2.30514
$592$	0	0
$593$	16.8460	0.691784	0.345892	−	0.938274i	$-0.387576\pi$
$593$	16.8460	0.691784	0.345892	+	0.938274i	$0.387576\pi$
$594$	0	0
$595$	−8.30452	−0.340452
$596$	0	0
$597$	−50.2801	−2.05783
$598$	0	0
$599$	−43.9075	−1.79401	−0.897006	−	0.442018i	$-0.854263\pi$
$599$	−43.9075	−1.79401	−0.897006	+	0.442018i	$0.854263\pi$
$600$	0	0
$601$	35.5257	1.44912	0.724562	−	0.689210i	$-0.242042\pi$
$601$	35.5257	1.44912	0.724562	+	0.689210i	$0.242042\pi$
$602$	0	0
$603$	24.1867	0.984960
$604$	0	0
$605$	−45.8143	−1.86262
$606$	0	0
$607$	−15.7700	−0.640084	−0.320042	−	0.947403i	$-0.603697\pi$
$607$	−15.7700	−0.640084	−0.320042	+	0.947403i	$0.603697\pi$
$608$	0	0
$609$	−2.61192	−0.105841
$610$	0	0
$611$	−4.24018	−0.171539
$612$	0	0
$613$	−11.5565	−0.466765	−0.233382	−	0.972385i	$-0.574979\pi$
$613$	−11.5565	−0.466765	−0.233382	+	0.972385i	$0.574979\pi$
$614$	0	0
$615$	18.7428	0.755783
$616$	0	0
$617$	−23.3938	−0.941800	−0.470900	−	0.882187i	$-0.656071\pi$
$617$	−23.3938	−0.941800	−0.470900	+	0.882187i	$0.656071\pi$
$618$	0	0
$619$	−3.87086	−0.155583	−0.0777915	−	0.996970i	$-0.524787\pi$
$619$	−3.87086	−0.155583	−0.0777915	+	0.996970i	$0.524787\pi$
$620$	0	0
$621$	14.5735	0.584816
$622$	0	0
$623$	−1.60040	−0.0641186
$624$	0	0
$625$	−8.52682	−0.341073
$626$	0	0
$627$	30.3207	1.21089
$628$	0	0
$629$	−11.0234	−0.439532
$630$	0	0
$631$	34.9200	1.39014	0.695071	−	0.718941i	$-0.255373\pi$
$631$	34.9200	1.39014	0.695071	+	0.718941i	$0.255373\pi$
$632$	0	0
$633$	−9.05897	−0.360062
$634$	0	0
$635$	22.3713	0.887779
$636$	0	0
$637$	0.634100	0.0251240
$638$	0	0
$639$	9.31976	0.368684
$640$	0	0
$641$	−42.9662	−1.69706	−0.848532	−	0.529144i	$-0.822513\pi$
$641$	−42.9662	−1.69706	−0.848532	+	0.529144i	$0.822513\pi$
$642$	0	0
$643$	−24.8794	−0.981148	−0.490574	−	0.871399i	$-0.663213\pi$
$643$	−24.8794	−0.981148	−0.490574	+	0.871399i	$0.663213\pi$
$644$	0	0
$645$	20.0693	0.790227
$646$	0	0
$647$	33.5883	1.32049	0.660246	−	0.751049i	$-0.270452\pi$
$647$	33.5883	1.32049	0.660246	+	0.751049i	$0.270452\pi$
$648$	0	0
$649$	−9.39016	−0.368596
$650$	0	0
$651$	19.0292	0.745815
$652$	0	0
$653$	−0.397667	−0.0155619	−0.00778095	−	0.999970i	$-0.502477\pi$
$653$	−0.397667	−0.0155619	−0.00778095	+	0.999970i	$0.502477\pi$
$654$	0	0
$655$	3.47498	0.135779
$656$	0	0
$657$	30.9008	1.20555
$658$	0	0
$659$	−1.77349	−0.0690855	−0.0345428	−	0.999403i	$-0.510997\pi$
$659$	−1.77349	−0.0690855	−0.0345428	+	0.999403i	$0.510997\pi$
$660$	0	0
$661$	43.4877	1.69148	0.845738	−	0.533598i	$-0.179160\pi$
$661$	43.4877	1.69148	0.845738	+	0.533598i	$0.179160\pi$
$662$	0	0
$663$	−7.32460	−0.284464
$664$	0	0
$665$	3.50351	0.135860
$666$	0	0
$667$	−10.1086	−0.391405
$668$	0	0
$669$	10.6909	0.413333
$670$	0	0
$671$	44.9939	1.73697
$672$	0	0
$673$	−22.2711	−0.858486	−0.429243	−	0.903189i	$-0.641220\pi$
$673$	−22.2711	−0.858486	−0.429243	+	0.903189i	$0.641220\pi$
$674$	0	0
$675$	3.71214	0.142881
$676$	0	0
$677$	−41.5083	−1.59529	−0.797646	−	0.603126i	$-0.793922\pi$
$677$	−41.5083	−1.59529	−0.797646	+	0.603126i	$0.793922\pi$
$678$	0	0
$679$	13.0167	0.499535
$680$	0	0
$681$	−14.3516	−0.549956
$682$	0	0
$683$	2.53936	0.0971658	0.0485829	−	0.998819i	$-0.484529\pi$
$683$	2.53936	0.0971658	0.0485829	+	0.998819i	$0.484529\pi$
$684$	0	0
$685$	−19.4191	−0.741966
$686$	0	0
$687$	46.8679	1.78812
$688$	0	0
$689$	−4.06610	−0.154906
$690$	0	0
$691$	16.5871	0.631005	0.315502	−	0.948925i	$-0.397827\pi$
$691$	16.5871	0.631005	0.315502	+	0.948925i	$0.397827\pi$
$692$	0	0
$693$	−14.2350	−0.540742
$694$	0	0
$695$	18.7230	0.710202
$696$	0	0
$697$	24.7657	0.938067
$698$	0	0
$699$	−25.9307	−0.980788
$700$	0	0
$701$	33.2411	1.25550	0.627750	−	0.778415i	$-0.283976\pi$
$701$	33.2411	1.25550	0.627750	+	0.778415i	$0.283976\pi$
$702$	0	0
$703$	4.65056	0.175399
$704$	0	0
$705$	25.4212	0.957417
$706$	0	0
$707$	5.32103	0.200118
$708$	0	0
$709$	2.20075	0.0826509	0.0413255	−	0.999146i	$-0.486842\pi$
$709$	2.20075	0.0826509	0.0413255	+	0.999146i	$0.486842\pi$
$710$	0	0
$711$	21.0416	0.789123
$712$	0	0
$713$	73.6462	2.75807
$714$	0	0
$715$	−6.52246	−0.243926
$716$	0	0
$717$	2.28713	0.0854145
$718$	0	0
$719$	−6.97452	−0.260106	−0.130053	−	0.991507i	$-0.541515\pi$
$719$	−6.97452	−0.260106	−0.130053	+	0.991507i	$0.541515\pi$
$720$	0	0
$721$	7.91886	0.294914
$722$	0	0
$723$	−62.3539	−2.31897
$724$	0	0
$725$	−2.57483	−0.0956269
$726$	0	0
$727$	−12.7285	−0.472073	−0.236037	−	0.971744i	$-0.575849\pi$
$727$	−12.7285	−0.472073	−0.236037	+	0.971744i	$0.575849\pi$
$728$	0	0
$729$	−13.0216	−0.482280
$730$	0	0
$731$	26.5184	0.980819
$732$	0	0
$733$	−19.9102	−0.735400	−0.367700	−	0.929945i	$-0.619855\pi$
$733$	−19.9102	−0.735400	−0.367700	+	0.929945i	$0.619855\pi$
$734$	0	0
$735$	−3.80163	−0.140225
$736$	0	0
$737$	65.7762	2.42290
$738$	0	0
$739$	52.2947	1.92369	0.961846	−	0.273592i	$-0.0882119\pi$
$739$	52.2947	1.92369	0.961846	+	0.273592i	$0.0882119\pi$
$740$	0	0
$741$	3.09011	0.113518
$742$	0	0
$743$	−10.0453	−0.368527	−0.184263	−	0.982877i	$-0.558990\pi$
$743$	−10.0453	−0.368527	−0.184263	+	0.982877i	$0.558990\pi$
$744$	0	0
$745$	8.50285	0.311520
$746$	0	0
$747$	−8.48516	−0.310456
$748$	0	0
$749$	19.2277	0.702566
$750$	0	0
$751$	42.4776	1.55003	0.775014	−	0.631944i	$-0.217743\pi$
$751$	42.4776	1.55003	0.775014	+	0.631944i	$0.217743\pi$
$752$	0	0
$753$	−14.5558	−0.530443
$754$	0	0
$755$	26.4013	0.960841
$756$	0	0
$757$	42.1141	1.53066	0.765332	−	0.643636i	$-0.222575\pi$
$757$	42.1141	1.53066	0.765332	+	0.643636i	$0.222575\pi$
$758$	0	0
$759$	−127.331	−4.62183
$760$	0	0
$761$	−6.54784	−0.237359	−0.118679	−	0.992933i	$-0.537866\pi$
$761$	−6.54784	−0.237359	−0.118679	+	0.992933i	$0.537866\pi$
$762$	0	0
$763$	9.94712	0.360110
$764$	0	0
$765$	18.9997	0.686936
$766$	0	0
$767$	−0.956988	−0.0345549
$768$	0	0
$769$	31.6203	1.14026	0.570129	−	0.821555i	$-0.306893\pi$
$769$	31.6203	1.14026	0.570129	+	0.821555i	$0.306893\pi$
$770$	0	0
$771$	−16.2978	−0.586952
$772$	0	0
$773$	−15.0246	−0.540397	−0.270198	−	0.962805i	$-0.587089\pi$
$773$	−15.0246	−0.540397	−0.270198	+	0.962805i	$0.587089\pi$
$774$	0	0
$775$	18.7590	0.673843
$776$	0	0
$777$	−5.04627	−0.181034
$778$	0	0
$779$	−10.4482	−0.374344
$780$	0	0
$781$	25.3452	0.906924
$782$	0	0
$783$	−1.86001	−0.0664714
$784$	0	0
$785$	−32.8443	−1.17226
$786$	0	0
$787$	−48.4507	−1.72708	−0.863540	−	0.504280i	$-0.831758\pi$
$787$	−48.4507	−1.72708	−0.863540	+	0.504280i	$0.831758\pi$
$788$	0	0
$789$	−15.9915	−0.569312
$790$	0	0
$791$	5.62753	0.200092
$792$	0	0
$793$	4.58551	0.162836
$794$	0	0
$795$	24.3775	0.864581
$796$	0	0
$797$	12.0940	0.428392	0.214196	−	0.976791i	$-0.431287\pi$
$797$	12.0940	0.428392	0.214196	+	0.976791i	$0.431287\pi$
$798$	0	0
$799$	33.5901	1.18833
$800$	0	0
$801$	3.66151	0.129373
$802$	0	0
$803$	84.0352	2.96554
$804$	0	0
$805$	−14.7129	−0.518562
$806$	0	0
$807$	−43.1241	−1.51804
$808$	0	0
$809$	36.6145	1.28730	0.643649	−	0.765321i	$-0.277420\pi$
$809$	36.6145	1.28730	0.643649	+	0.765321i	$0.277420\pi$
$810$	0	0
$811$	36.2833	1.27408	0.637040	−	0.770831i	$-0.280159\pi$
$811$	36.2833	1.27408	0.637040	+	0.770831i	$0.280159\pi$
$812$	0	0
$813$	32.8758	1.15300
$814$	0	0
$815$	−24.0340	−0.841873
$816$	0	0
$817$	−11.1876	−0.391405
$818$	0	0
$819$	−1.45074	−0.0506931
$820$	0	0
$821$	−18.0503	−0.629961	−0.314981	−	0.949098i	$-0.601998\pi$
$821$	−18.0503	−0.629961	−0.314981	+	0.949098i	$0.601998\pi$
$822$	0	0
$823$	36.1501	1.26011	0.630057	−	0.776549i	$-0.283032\pi$
$823$	36.1501	1.26011	0.630057	+	0.776549i	$0.283032\pi$
$824$	0	0
$825$	−32.4335	−1.12919
$826$	0	0
$827$	−38.5667	−1.34110	−0.670548	−	0.741867i	$-0.733941\pi$
$827$	−38.5667	−1.34110	−0.670548	+	0.741867i	$0.733941\pi$
$828$	0	0
$829$	7.21747	0.250673	0.125337	−	0.992114i	$-0.459999\pi$
$829$	7.21747	0.250673	0.125337	+	0.992114i	$0.459999\pi$
$830$	0	0
$831$	−5.08126	−0.176267
$832$	0	0
$833$	−5.02326	−0.174046
$834$	0	0
$835$	−32.0940	−1.11066
$836$	0	0
$837$	13.5512	0.468397
$838$	0	0
$839$	34.6198	1.19521	0.597604	−	0.801791i	$-0.296120\pi$
$839$	34.6198	1.19521	0.597604	+	0.801791i	$0.296120\pi$
$840$	0	0
$841$	−27.7099	−0.955512
$842$	0	0
$843$	−13.9494	−0.480444
$844$	0	0
$845$	20.8271	0.716472
$846$	0	0
$847$	−27.7123	−0.952205
$848$	0	0
$849$	−66.0786	−2.26781
$850$	0	0
$851$	−19.5299	−0.669476
$852$	0	0
$853$	52.4072	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$853$	52.4072	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$854$	0	0
$855$	−8.01561	−0.274128
$856$	0	0
$857$	−18.9647	−0.647823	−0.323912	−	0.946087i	$-0.604998\pi$
$857$	−18.9647	−0.647823	−0.323912	+	0.946087i	$0.604998\pi$
$858$	0	0
$859$	25.3906	0.866315	0.433158	−	0.901318i	$-0.357399\pi$
$859$	25.3906	0.866315	0.433158	+	0.901318i	$0.357399\pi$
$860$	0	0
$861$	11.3372	0.386370
$862$	0	0
$863$	−11.5403	−0.392837	−0.196418	−	0.980520i	$-0.562931\pi$
$863$	−11.5403	−0.392837	−0.196418	+	0.980520i	$0.562931\pi$
$864$	0	0
$865$	10.6678	0.362717
$866$	0	0
$867$	18.9323	0.642976
$868$	0	0
$869$	57.2231	1.94116
$870$	0	0
$871$	6.70352	0.227140
$872$	0	0
$873$	−29.7806	−1.00792
$874$	0	0
$875$	−12.0137	−0.406138
$876$	0	0
$877$	−19.8731	−0.671067	−0.335533	−	0.942028i	$-0.608917\pi$
$877$	−19.8731	−0.671067	−0.335533	+	0.942028i	$0.608917\pi$
$878$	0	0
$879$	−72.3114	−2.43900
$880$	0	0
$881$	47.1071	1.58708	0.793539	−	0.608519i	$-0.208236\pi$
$881$	47.1071	1.58708	0.793539	+	0.608519i	$0.208236\pi$
$882$	0	0
$883$	−24.0255	−0.808521	−0.404261	−	0.914644i	$-0.632471\pi$
$883$	−24.0255	−0.808521	−0.404261	+	0.914644i	$0.632471\pi$
$884$	0	0
$885$	5.73744	0.192862
$886$	0	0
$887$	−33.3126	−1.11853	−0.559263	−	0.828990i	$-0.688916\pi$
$887$	−33.3126	−1.11853	−0.559263	+	0.828990i	$0.688916\pi$
$888$	0	0
$889$	13.5320	0.453850
$890$	0	0
$891$	−66.1343	−2.21558
$892$	0	0
$893$	−14.1710	−0.474215
$894$	0	0
$895$	−39.3084	−1.31394
$896$	0	0
$897$	−12.9768	−0.433283
$898$	0	0
$899$	−9.39941	−0.313488
$900$	0	0
$901$	32.2111	1.07311
$902$	0	0
$903$	12.1396	0.403979
$904$	0	0
$905$	14.6325	0.486401
$906$	0	0
$907$	8.00378	0.265761	0.132881	−	0.991132i	$-0.457577\pi$
$907$	8.00378	0.265761	0.132881	+	0.991132i	$0.457577\pi$
$908$	0	0
$909$	−12.1739	−0.403781
$910$	0	0
$911$	−41.8870	−1.38778	−0.693889	−	0.720082i	$-0.744104\pi$
$911$	−41.8870	−1.38778	−0.693889	+	0.720082i	$0.744104\pi$
$912$	0	0
$913$	−23.0755	−0.763689
$914$	0	0
$915$	−27.4916	−0.908843
$916$	0	0
$917$	2.10195	0.0694126
$918$	0	0
$919$	14.4978	0.478237	0.239119	−	0.970990i	$-0.423142\pi$
$919$	14.4978	0.478237	0.239119	+	0.970990i	$0.423142\pi$
$920$	0	0
$921$	57.7743	1.90373
$922$	0	0
$923$	2.58303	0.0850216
$924$	0	0
$925$	−4.97461	−0.163564
$926$	0	0
$927$	−18.1174	−0.595053
$928$	0	0
$929$	21.5646	0.707513	0.353756	−	0.935338i	$-0.384904\pi$
$929$	21.5646	0.707513	0.353756	+	0.935338i	$0.384904\pi$
$930$	0	0
$931$	2.11921	0.0694544
$932$	0	0
$933$	−71.9061	−2.35410
$934$	0	0
$935$	51.6700	1.68979
$936$	0	0
$937$	−3.51782	−0.114922	−0.0574611	−	0.998348i	$-0.518301\pi$
$937$	−3.51782	−0.114922	−0.0574611	+	0.998348i	$0.518301\pi$
$938$	0	0
$939$	41.4136	1.35148
$940$	0	0
$941$	1.08682	0.0354294	0.0177147	−	0.999843i	$-0.494361\pi$
$941$	1.08682	0.0354294	0.0177147	+	0.999843i	$0.494361\pi$
$942$	0	0
$943$	43.8767	1.42882
$944$	0	0
$945$	−2.70723	−0.0880661
$946$	0	0
$947$	16.0164	0.520462	0.260231	−	0.965546i	$-0.416201\pi$
$947$	16.0164	0.520462	0.260231	+	0.965546i	$0.416201\pi$
$948$	0	0
$949$	8.56436	0.278011
$950$	0	0
$951$	−39.8107	−1.29095
$952$	0	0
$953$	−9.82914	−0.318397	−0.159199	−	0.987247i	$-0.550891\pi$
$953$	−9.82914	−0.318397	−0.159199	+	0.987247i	$0.550891\pi$
$954$	0	0
$955$	−26.1575	−0.846438
$956$	0	0
$957$	16.2512	0.525326
$958$	0	0
$959$	−11.7463	−0.379307
$960$	0	0
$961$	37.4796	1.20902
$962$	0	0
$963$	−43.9907	−1.41758
$964$	0	0
$965$	12.4095	0.399476
$966$	0	0
$967$	5.71372	0.183741	0.0918705	−	0.995771i	$-0.470715\pi$
$967$	5.71372	0.183741	0.0918705	+	0.995771i	$0.470715\pi$
$968$	0	0
$969$	−24.4794	−0.786392
$970$	0	0
$971$	18.8818	0.605946	0.302973	−	0.952999i	$-0.402021\pi$
$971$	18.8818	0.605946	0.302973	+	0.952999i	$0.402021\pi$
$972$	0	0
$973$	11.3252	0.363069
$974$	0	0
$975$	−3.30543	−0.105858
$976$	0	0
$977$	−25.3043	−0.809556	−0.404778	−	0.914415i	$-0.632651\pi$
$977$	−25.3043	−0.809556	−0.404778	+	0.914415i	$0.632651\pi$
$978$	0	0
$979$	9.95755	0.318245
$980$	0	0
$981$	−22.7578	−0.726600
$982$	0	0
$983$	12.4206	0.396157	0.198078	−	0.980186i	$-0.436530\pi$
$983$	12.4206	0.396157	0.198078	+	0.980186i	$0.436530\pi$
$984$	0	0
$985$	−40.2883	−1.28369
$986$	0	0
$987$	15.3768	0.489450
$988$	0	0
$989$	46.9820	1.49394
$990$	0	0
$991$	−30.4013	−0.965729	−0.482865	−	0.875695i	$-0.660404\pi$
$991$	−30.4013	−0.965729	−0.482865	+	0.875695i	$0.660404\pi$
$992$	0	0
$993$	38.1973	1.21215
$994$	0	0
$995$	36.1480	1.14597
$996$	0	0
$997$	−12.0040	−0.380171	−0.190085	−	0.981768i	$-0.560877\pi$
$997$	−12.0040	−0.380171	−0.190085	+	0.981768i	$0.560877\pi$
$998$	0	0
$999$	−3.59357	−0.113695

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.ba.1.7		8
4.3	odd	2		7168.2.a.bb.1.2		8
8.3	odd	2		7168.2.a.bf.1.7		8
8.5	even	2		7168.2.a.be.1.2		8
32.3	odd	8		1792.2.m.e.1345.3	yes	16
32.5	even	8		1792.2.m.f.449.3	yes	16
32.11	odd	8		1792.2.m.e.449.3	✓	16
32.13	even	8		1792.2.m.f.1345.3	yes	16
32.19	odd	8		1792.2.m.h.1345.6	yes	16
32.21	even	8		1792.2.m.g.449.6	yes	16
32.27	odd	8		1792.2.m.h.449.6	yes	16
32.29	even	8		1792.2.m.g.1345.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.3	✓	16	32.11	odd	8
1792.2.m.e.1345.3	yes	16	32.3	odd	8
1792.2.m.f.449.3	yes	16	32.5	even	8
1792.2.m.f.1345.3	yes	16	32.13	even	8
1792.2.m.g.449.6	yes	16	32.21	even	8
1792.2.m.g.1345.6	yes	16	32.29	even	8
1792.2.m.h.449.6	yes	16	32.27	odd	8
1792.2.m.h.1345.6	yes	16	32.19	odd	8
7168.2.a.ba.1.7		8	1.1	even	1	trivial
7168.2.a.bb.1.2		8	4.3	odd	2
7168.2.a.be.1.2		8	8.5	even	2
7168.2.a.bf.1.7		8	8.3	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)

\(f(q)\)	\(=\)	\(q+2.29954 q^{3} -1.65321 q^{5} -1.00000 q^{7} +2.28788 q^{9} +O(q^{10})\)	\(q+2.29954 q^{3} -1.65321 q^{5} -1.00000 q^{7} +2.28788 q^{9} +6.22192 q^{11} +0.634100 q^{13} -3.80163 q^{15} -5.02326 q^{17} +2.11921 q^{19} -2.29954 q^{21} -8.89958 q^{23} -2.26688 q^{25} -1.63755 q^{27} +1.13585 q^{29} -8.27524 q^{31} +14.3075 q^{33} +1.65321 q^{35} +2.19447 q^{37} +1.45814 q^{39} -4.93020 q^{41} -5.27913 q^{43} -3.78235 q^{45} -6.68692 q^{47} +1.00000 q^{49} -11.5512 q^{51} -6.41239 q^{53} -10.2862 q^{55} +4.87321 q^{57} -1.50921 q^{59} +7.23152 q^{61} -2.28788 q^{63} -1.04830 q^{65} +10.5717 q^{67} -20.4649 q^{69} +4.07354 q^{71} +13.5063 q^{73} -5.21278 q^{75} -6.22192 q^{77} +9.19701 q^{79} -10.6293 q^{81} -3.70875 q^{83} +8.30452 q^{85} +2.61192 q^{87} +1.60040 q^{89} -0.634100 q^{91} -19.0292 q^{93} -3.50351 q^{95} -13.0167 q^{97} +14.2350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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