LMFDB - Embedded newform 7168.2.a.bi.1.2

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 $ x^12 - 24x^10 + 221x^8 - 968x^6 + 2008x^4 - 1640*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.75510$ 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)

$f(q)$	$=$	$q-2.90620 q^{3} +3.85750 q^{5} -1.00000 q^{7} +5.44602 q^{9} +O(q^{10})$	$q-2.90620 q^{3} +3.85750 q^{5} -1.00000 q^{7} +5.44602 q^{9} +1.30053 q^{11} -1.59251 q^{13} -11.2107 q^{15} +1.50885 q^{17} -2.06859 q^{19} +2.90620 q^{21} -4.77031 q^{23} +9.88030 q^{25} -7.10863 q^{27} -5.79924 q^{29} -4.10999 q^{31} -3.77961 q^{33} -3.85750 q^{35} -2.34005 q^{37} +4.62815 q^{39} +7.45533 q^{41} +8.04676 q^{43} +21.0080 q^{45} +3.59748 q^{47} +1.00000 q^{49} -4.38503 q^{51} +0.955604 q^{53} +5.01680 q^{55} +6.01174 q^{57} -1.60998 q^{59} +4.55209 q^{61} -5.44602 q^{63} -6.14310 q^{65} -2.15866 q^{67} +13.8635 q^{69} +13.8202 q^{71} -14.4749 q^{73} -28.7142 q^{75} -1.30053 q^{77} -1.77961 q^{79} +4.32107 q^{81} +10.1276 q^{83} +5.82039 q^{85} +16.8538 q^{87} +8.45899 q^{89} +1.59251 q^{91} +11.9445 q^{93} -7.97958 q^{95} +16.2227 q^{97} +7.08273 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 + 12 * q^9	$ 12 q - 12 q^{7} + 12 q^{9} - 24 q^{15} + 8 q^{17} - 16 q^{23} + 20 q^{25} + 8 q^{31} - 16 q^{39} + 32 q^{41} + 16 q^{47} + 12 q^{49} - 24 q^{55} + 64 q^{57} - 12 q^{63} + 32 q^{65} - 8 q^{71} + 24 q^{79} + 44 q^{81} + 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 + 12 * q^9 - 24 * q^15 + 8 * q^17 - 16 * q^23 + 20 * q^25 + 8 * q^31 - 16 * q^39 + 32 * q^41 + 16 * q^47 + 12 * q^49 - 24 * q^55 + 64 * q^57 - 12 * q^63 + 32 * q^65 - 8 * q^71 + 24 * q^79 + 44 * q^81 + 32 * q^87 + 24 * q^89 + 48 * q^97

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
$3$	−2.90620	−1.67790	−0.838949	−	0.544210i	$-0.816829\pi$
$3$	−2.90620	−1.67790	−0.838949	+	0.544210i	$0.816829\pi$
$4$	0	0
$5$	3.85750	1.72513	0.862563	−	0.505950i	$-0.168858\pi$
$5$	3.85750	1.72513	0.862563	+	0.505950i	$0.168858\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.44602	1.81534
$10$	0	0
$11$	1.30053	0.392125	0.196063	−	0.980591i	$-0.437184\pi$
$11$	1.30053	0.392125	0.196063	+	0.980591i	$0.437184\pi$
$12$	0	0
$13$	−1.59251	−0.441682	−0.220841	−	0.975310i	$-0.570880\pi$
$13$	−1.59251	−0.441682	−0.220841	+	0.975310i	$0.570880\pi$
$14$	0	0
$15$	−11.2107	−2.89458
$16$	0	0
$17$	1.50885	0.365950	0.182975	−	0.983118i	$-0.441427\pi$
$17$	1.50885	0.365950	0.182975	+	0.983118i	$0.441427\pi$
$18$	0	0
$19$	−2.06859	−0.474567	−0.237283	−	0.971440i	$-0.576257\pi$
$19$	−2.06859	−0.474567	−0.237283	+	0.971440i	$0.576257\pi$
$20$	0	0
$21$	2.90620	0.634186
$22$	0	0
$23$	−4.77031	−0.994677	−0.497339	−	0.867556i	$-0.665689\pi$
$23$	−4.77031	−0.994677	−0.497339	+	0.867556i	$0.665689\pi$
$24$	0	0
$25$	9.88030	1.97606
$26$	0	0
$27$	−7.10863	−1.36806
$28$	0	0
$29$	−5.79924	−1.07689	−0.538446	−	0.842660i	$-0.680989\pi$
$29$	−5.79924	−1.07689	−0.538446	+	0.842660i	$0.680989\pi$
$30$	0	0
$31$	−4.10999	−0.738176	−0.369088	−	0.929394i	$-0.620330\pi$
$31$	−4.10999	−0.738176	−0.369088	+	0.929394i	$0.620330\pi$
$32$	0	0
$33$	−3.77961	−0.657946
$34$	0	0
$35$	−3.85750	−0.652036
$36$	0	0
$37$	−2.34005	−0.384702	−0.192351	−	0.981326i	$-0.561611\pi$
$37$	−2.34005	−0.384702	−0.192351	+	0.981326i	$0.561611\pi$
$38$	0	0
$39$	4.62815	0.741097
$40$	0	0
$41$	7.45533	1.16433	0.582163	−	0.813072i	$-0.302206\pi$
$41$	7.45533	1.16433	0.582163	+	0.813072i	$0.302206\pi$
$42$	0	0
$43$	8.04676	1.22712	0.613560	−	0.789648i	$-0.289737\pi$
$43$	8.04676	1.22712	0.613560	+	0.789648i	$0.289737\pi$
$44$	0	0
$45$	21.0080	3.13169
$46$	0	0
$47$	3.59748	0.524747	0.262373	−	0.964966i	$-0.415495\pi$
$47$	3.59748	0.524747	0.262373	+	0.964966i	$0.415495\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.38503	−0.614027
$52$	0	0
$53$	0.955604	0.131262	0.0656312	−	0.997844i	$-0.479094\pi$
$53$	0.955604	0.131262	0.0656312	+	0.997844i	$0.479094\pi$
$54$	0	0
$55$	5.01680	0.676466
$56$	0	0
$57$	6.01174	0.796275
$58$	0	0
$59$	−1.60998	−0.209601	−0.104801	−	0.994493i	$-0.533420\pi$
$59$	−1.60998	−0.209601	−0.104801	+	0.994493i	$0.533420\pi$
$60$	0	0
$61$	4.55209	0.582835	0.291418	−	0.956596i	$-0.405873\pi$
$61$	4.55209	0.582835	0.291418	+	0.956596i	$0.405873\pi$
$62$	0	0
$63$	−5.44602	−0.686134
$64$	0	0
$65$	−6.14310	−0.761957
$66$	0	0
$67$	−2.15866	−0.263722	−0.131861	−	0.991268i	$-0.542095\pi$
$67$	−2.15866	−0.263722	−0.131861	+	0.991268i	$0.542095\pi$
$68$	0	0
$69$	13.8635	1.66897
$70$	0	0
$71$	13.8202	1.64016	0.820079	−	0.572251i	$-0.193930\pi$
$71$	13.8202	1.64016	0.820079	+	0.572251i	$0.193930\pi$
$72$	0	0
$73$	−14.4749	−1.69416	−0.847078	−	0.531468i	$-0.821641\pi$
$73$	−14.4749	−1.69416	−0.847078	+	0.531468i	$0.821641\pi$
$74$	0	0
$75$	−28.7142	−3.31563
$76$	0	0
$77$	−1.30053	−0.148209
$78$	0	0
$79$	−1.77961	−0.200222	−0.100111	−	0.994976i	$-0.531920\pi$
$79$	−1.77961	−0.200222	−0.100111	+	0.994976i	$0.531920\pi$
$80$	0	0
$81$	4.32107	0.480119
$82$	0	0
$83$	10.1276	1.11165	0.555827	−	0.831298i	$-0.312402\pi$
$83$	10.1276	1.11165	0.555827	+	0.831298i	$0.312402\pi$
$84$	0	0
$85$	5.82039	0.631310
$86$	0	0
$87$	16.8538	1.80692
$88$	0	0
$89$	8.45899	0.896651	0.448325	−	0.893870i	$-0.352021\pi$
$89$	8.45899	0.896651	0.448325	+	0.893870i	$0.352021\pi$
$90$	0	0
$91$	1.59251	0.166940
$92$	0	0
$93$	11.9445	1.23858
$94$	0	0
$95$	−7.97958	−0.818688
$96$	0	0
$97$	16.2227	1.64717	0.823585	−	0.567194i	$-0.191971\pi$
$97$	16.2227	1.64717	0.823585	+	0.567194i	$0.191971\pi$
$98$	0	0
$99$	7.08273	0.711841
$100$	0	0
$101$	−16.3574	−1.62762	−0.813812	−	0.581128i	$-0.802612\pi$
$101$	−16.3574	−1.62762	−0.813812	+	0.581128i	$0.802612\pi$
$102$	0	0
$103$	17.1875	1.69354	0.846768	−	0.531963i	$-0.178545\pi$
$103$	17.1875	1.69354	0.846768	+	0.531963i	$0.178545\pi$
$104$	0	0
$105$	11.2107	1.09405
$106$	0	0
$107$	5.80030	0.560736	0.280368	−	0.959893i	$-0.409543\pi$
$107$	5.80030	0.560736	0.280368	+	0.959893i	$0.409543\pi$
$108$	0	0
$109$	−4.62944	−0.443420	−0.221710	−	0.975113i	$-0.571164\pi$
$109$	−4.62944	−0.443420	−0.221710	+	0.975113i	$0.571164\pi$
$110$	0	0
$111$	6.80066	0.645490
$112$	0	0
$113$	9.17193	0.862822	0.431411	−	0.902155i	$-0.358016\pi$
$113$	9.17193	0.862822	0.431411	+	0.902155i	$0.358016\pi$
$114$	0	0
$115$	−18.4014	−1.71594
$116$	0	0
$117$	−8.67283	−0.801803
$118$	0	0
$119$	−1.50885	−0.138316
$120$	0	0
$121$	−9.30861	−0.846238
$122$	0	0
$123$	−21.6667	−1.95362
$124$	0	0
$125$	18.8257	1.68383
$126$	0	0
$127$	12.9787	1.15167	0.575835	−	0.817566i	$-0.304677\pi$
$127$	12.9787	1.15167	0.575835	+	0.817566i	$0.304677\pi$
$128$	0	0
$129$	−23.3855	−2.05898
$130$	0	0
$131$	−9.64445	−0.842639	−0.421320	−	0.906912i	$-0.638433\pi$
$131$	−9.64445	−0.842639	−0.421320	+	0.906912i	$0.638433\pi$
$132$	0	0
$133$	2.06859	0.179369
$134$	0	0
$135$	−27.4215	−2.36007
$136$	0	0
$137$	22.6543	1.93548	0.967742	−	0.251943i	$-0.0810694\pi$
$137$	22.6543	1.93548	0.967742	+	0.251943i	$0.0810694\pi$
$138$	0	0
$139$	−9.40067	−0.797355	−0.398677	−	0.917091i	$-0.630531\pi$
$139$	−9.40067	−0.797355	−0.398677	+	0.917091i	$0.630531\pi$
$140$	0	0
$141$	−10.4550	−0.880471
$142$	0	0
$143$	−2.07111	−0.173195
$144$	0	0
$145$	−22.3706	−1.85778
$146$	0	0
$147$	−2.90620	−0.239700
$148$	0	0
$149$	−10.9889	−0.900243	−0.450122	−	0.892967i	$-0.648619\pi$
$149$	−10.9889	−0.900243	−0.450122	+	0.892967i	$0.648619\pi$
$150$	0	0
$151$	5.46829	0.445004	0.222502	−	0.974932i	$-0.428578\pi$
$151$	5.46829	0.445004	0.222502	+	0.974932i	$0.428578\pi$
$152$	0	0
$153$	8.21724	0.664324
$154$	0	0
$155$	−15.8543	−1.27345
$156$	0	0
$157$	7.94395	0.633996	0.316998	−	0.948426i	$-0.397325\pi$
$157$	7.94395	0.633996	0.316998	+	0.948426i	$0.397325\pi$
$158$	0	0
$159$	−2.77718	−0.220245
$160$	0	0
$161$	4.77031	0.375953
$162$	0	0
$163$	−8.93440	−0.699796	−0.349898	−	0.936788i	$-0.613784\pi$
$163$	−8.93440	−0.699796	−0.349898	+	0.936788i	$0.613784\pi$
$164$	0	0
$165$	−14.5799	−1.13504
$166$	0	0
$167$	11.8175	0.914463	0.457231	−	0.889348i	$-0.348841\pi$
$167$	11.8175	0.914463	0.457231	+	0.889348i	$0.348841\pi$
$168$	0	0
$169$	−10.4639	−0.804917
$170$	0	0
$171$	−11.2656	−0.861500
$172$	0	0
$173$	7.97416	0.606264	0.303132	−	0.952949i	$-0.401968\pi$
$173$	7.97416	0.606264	0.303132	+	0.952949i	$0.401968\pi$
$174$	0	0
$175$	−9.88030	−0.746880
$176$	0	0
$177$	4.67893	0.351690
$178$	0	0
$179$	−16.5856	−1.23966	−0.619832	−	0.784734i	$-0.712799\pi$
$179$	−16.5856	−1.23966	−0.619832	+	0.784734i	$0.712799\pi$
$180$	0	0
$181$	5.91953	0.439995	0.219998	−	0.975500i	$-0.429395\pi$
$181$	5.91953	0.439995	0.219998	+	0.975500i	$0.429395\pi$
$182$	0	0
$183$	−13.2293	−0.977937
$184$	0	0
$185$	−9.02674	−0.663659
$186$	0	0
$187$	1.96231	0.143498
$188$	0	0
$189$	7.10863	0.517077
$190$	0	0
$191$	−12.3676	−0.894891	−0.447445	−	0.894311i	$-0.647666\pi$
$191$	−12.3676	−0.894891	−0.447445	+	0.894311i	$0.647666\pi$
$192$	0	0
$193$	23.6837	1.70479	0.852395	−	0.522898i	$-0.175149\pi$
$193$	23.6837	1.70479	0.852395	+	0.522898i	$0.175149\pi$
$194$	0	0
$195$	17.8531	1.27849
$196$	0	0
$197$	−16.1496	−1.15061	−0.575306	−	0.817938i	$-0.695117\pi$
$197$	−16.1496	−1.15061	−0.575306	+	0.817938i	$0.695117\pi$
$198$	0	0
$199$	−5.49683	−0.389660	−0.194830	−	0.980837i	$-0.562415\pi$
$199$	−5.49683	−0.389660	−0.194830	+	0.980837i	$0.562415\pi$
$200$	0	0
$201$	6.27351	0.442499
$202$	0	0
$203$	5.79924	0.407027
$204$	0	0
$205$	28.7589	2.00861
$206$	0	0
$207$	−25.9792	−1.80568
$208$	0	0
$209$	−2.69027	−0.186090
$210$	0	0
$211$	−1.28393	−0.0883893	−0.0441946	−	0.999023i	$-0.514072\pi$
$211$	−1.28393	−0.0883893	−0.0441946	+	0.999023i	$0.514072\pi$
$212$	0	0
$213$	−40.1644	−2.75202
$214$	0	0
$215$	31.0404	2.11694
$216$	0	0
$217$	4.10999	0.279004
$218$	0	0
$219$	42.0669	2.84262
$220$	0	0
$221$	−2.40286	−0.161634
$222$	0	0
$223$	3.20318	0.214501	0.107250	−	0.994232i	$-0.465795\pi$
$223$	3.20318	0.214501	0.107250	+	0.994232i	$0.465795\pi$
$224$	0	0
$225$	53.8083	3.58722
$226$	0	0
$227$	−19.5064	−1.29468	−0.647342	−	0.762200i	$-0.724119\pi$
$227$	−19.5064	−1.29468	−0.647342	+	0.762200i	$0.724119\pi$
$228$	0	0
$229$	9.93029	0.656212	0.328106	−	0.944641i	$-0.393590\pi$
$229$	9.93029	0.656212	0.328106	+	0.944641i	$0.393590\pi$
$230$	0	0
$231$	3.77961	0.248680
$232$	0	0
$233$	−2.93857	−0.192512	−0.0962561	−	0.995357i	$-0.530687\pi$
$233$	−2.93857	−0.192512	−0.0962561	+	0.995357i	$0.530687\pi$
$234$	0	0
$235$	13.8773	0.905254
$236$	0	0
$237$	5.17192	0.335952
$238$	0	0
$239$	16.2109	1.04859	0.524297	−	0.851536i	$-0.324328\pi$
$239$	16.2109	1.04859	0.524297	+	0.851536i	$0.324328\pi$
$240$	0	0
$241$	14.1166	0.909329	0.454665	−	0.890663i	$-0.349759\pi$
$241$	14.1166	0.909329	0.454665	+	0.890663i	$0.349759\pi$
$242$	0	0
$243$	8.76798	0.562466
$244$	0	0
$245$	3.85750	0.246447
$246$	0	0
$247$	3.29424	0.209608
$248$	0	0
$249$	−29.4330	−1.86524
$250$	0	0
$251$	20.0772	1.26726	0.633632	−	0.773635i	$-0.281563\pi$
$251$	20.0772	1.26726	0.633632	+	0.773635i	$0.281563\pi$
$252$	0	0
$253$	−6.20394	−0.390038
$254$	0	0
$255$	−16.9153	−1.05927
$256$	0	0
$257$	−3.23679	−0.201905	−0.100953	−	0.994891i	$-0.532189\pi$
$257$	−3.23679	−0.201905	−0.100953	+	0.994891i	$0.532189\pi$
$258$	0	0
$259$	2.34005	0.145404
$260$	0	0
$261$	−31.5828	−1.95493
$262$	0	0
$263$	−24.0161	−1.48090	−0.740448	−	0.672114i	$-0.765386\pi$
$263$	−24.0161	−1.48090	−0.740448	+	0.672114i	$0.765386\pi$
$264$	0	0
$265$	3.68624	0.226444
$266$	0	0
$267$	−24.5835	−1.50449
$268$	0	0
$269$	−0.123749	−0.00754513	−0.00377257	−	0.999993i	$-0.501201\pi$
$269$	−0.123749	−0.00754513	−0.00377257	+	0.999993i	$0.501201\pi$
$270$	0	0
$271$	−10.1138	−0.614372	−0.307186	−	0.951650i	$-0.599387\pi$
$271$	−10.1138	−0.614372	−0.307186	+	0.951650i	$0.599387\pi$
$272$	0	0
$273$	−4.62815	−0.280109
$274$	0	0
$275$	12.8497	0.774863
$276$	0	0
$277$	15.5677	0.935370	0.467685	−	0.883895i	$-0.345088\pi$
$277$	15.5677	0.935370	0.467685	+	0.883895i	$0.345088\pi$
$278$	0	0
$279$	−22.3831	−1.34004
$280$	0	0
$281$	−4.21999	−0.251743	−0.125872	−	0.992047i	$-0.540173\pi$
$281$	−4.21999	−0.251743	−0.125872	+	0.992047i	$0.540173\pi$
$282$	0	0
$283$	23.4525	1.39410	0.697052	−	0.717020i	$-0.254495\pi$
$283$	23.4525	1.39410	0.697052	+	0.717020i	$0.254495\pi$
$284$	0	0
$285$	23.1903	1.37367
$286$	0	0
$287$	−7.45533	−0.440074
$288$	0	0
$289$	−14.7234	−0.866080
$290$	0	0
$291$	−47.1466	−2.76378
$292$	0	0
$293$	−19.1261	−1.11736	−0.558680	−	0.829383i	$-0.688692\pi$
$293$	−19.1261	−1.11736	−0.558680	+	0.829383i	$0.688692\pi$
$294$	0	0
$295$	−6.21049	−0.361589
$296$	0	0
$297$	−9.24501	−0.536450
$298$	0	0
$299$	7.59675	0.439331
$300$	0	0
$301$	−8.04676	−0.463808
$302$	0	0
$303$	47.5380	2.73099
$304$	0	0
$305$	17.5597	1.00546
$306$	0	0
$307$	12.6344	0.721085	0.360543	−	0.932743i	$-0.382592\pi$
$307$	12.6344	0.721085	0.360543	+	0.932743i	$0.382592\pi$
$308$	0	0
$309$	−49.9504	−2.84158
$310$	0	0
$311$	19.7737	1.12126	0.560631	−	0.828066i	$-0.310558\pi$
$311$	19.7737	1.12126	0.560631	+	0.828066i	$0.310558\pi$
$312$	0	0
$313$	8.72743	0.493303	0.246652	−	0.969104i	$-0.420670\pi$
$313$	8.72743	0.493303	0.246652	+	0.969104i	$0.420670\pi$
$314$	0	0
$315$	−21.0080	−1.18367
$316$	0	0
$317$	−26.8980	−1.51074	−0.755370	−	0.655298i	$-0.772543\pi$
$317$	−26.8980	−1.51074	−0.755370	+	0.655298i	$0.772543\pi$
$318$	0	0
$319$	−7.54211	−0.422277
$320$	0	0
$321$	−16.8569	−0.940858
$322$	0	0
$323$	−3.12119	−0.173668
$324$	0	0
$325$	−15.7345	−0.872790
$326$	0	0
$327$	13.4541	0.744013
$328$	0	0
$329$	−3.59748	−0.198336
$330$	0	0
$331$	31.5833	1.73597	0.867987	−	0.496586i	$-0.165413\pi$
$331$	31.5833	1.73597	0.867987	+	0.496586i	$0.165413\pi$
$332$	0	0
$333$	−12.7440	−0.698365
$334$	0	0
$335$	−8.32703	−0.454954
$336$	0	0
$337$	23.4825	1.27918	0.639588	−	0.768718i	$-0.279105\pi$
$337$	23.4825	1.27918	0.639588	+	0.768718i	$0.279105\pi$
$338$	0	0
$339$	−26.6555	−1.44773
$340$	0	0
$341$	−5.34518	−0.289458
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	53.4784	2.87918
$346$	0	0
$347$	8.90552	0.478074	0.239037	−	0.971011i	$-0.423168\pi$
$347$	8.90552	0.478074	0.239037	+	0.971011i	$0.423168\pi$
$348$	0	0
$349$	−9.81511	−0.525391	−0.262695	−	0.964879i	$-0.584611\pi$
$349$	−9.81511	−0.525391	−0.262695	+	0.964879i	$0.584611\pi$
$350$	0	0
$351$	11.3206	0.604246
$352$	0	0
$353$	30.0934	1.60171	0.800856	−	0.598857i	$-0.204378\pi$
$353$	30.0934	1.60171	0.800856	+	0.598857i	$0.204378\pi$
$354$	0	0
$355$	53.3115	2.82948
$356$	0	0
$357$	4.38503	0.232080
$358$	0	0
$359$	−10.3716	−0.547394	−0.273697	−	0.961816i	$-0.588246\pi$
$359$	−10.3716	−0.547394	−0.273697	+	0.961816i	$0.588246\pi$
$360$	0	0
$361$	−14.7209	−0.774786
$362$	0	0
$363$	27.0527	1.41990
$364$	0	0
$365$	−55.8368	−2.92263
$366$	0	0
$367$	−8.12973	−0.424368	−0.212184	−	0.977230i	$-0.568058\pi$
$367$	−8.12973	−0.424368	−0.212184	+	0.977230i	$0.568058\pi$
$368$	0	0
$369$	40.6019	2.11365
$370$	0	0
$371$	−0.955604	−0.0496125
$372$	0	0
$373$	4.50693	0.233360	0.116680	−	0.993170i	$-0.462775\pi$
$373$	4.50693	0.233360	0.116680	+	0.993170i	$0.462775\pi$
$374$	0	0
$375$	−54.7115	−2.82529
$376$	0	0
$377$	9.23534	0.475644
$378$	0	0
$379$	1.18664	0.0609535	0.0304767	−	0.999535i	$-0.490297\pi$
$379$	1.18664	0.0609535	0.0304767	+	0.999535i	$0.490297\pi$
$380$	0	0
$381$	−37.7187	−1.93239
$382$	0	0
$383$	34.3749	1.75647	0.878237	−	0.478225i	$-0.158720\pi$
$383$	34.3749	1.75647	0.878237	+	0.478225i	$0.158720\pi$
$384$	0	0
$385$	−5.01680	−0.255680
$386$	0	0
$387$	43.8228	2.22764
$388$	0	0
$389$	15.5044	0.786104	0.393052	−	0.919516i	$-0.371419\pi$
$389$	15.5044	0.786104	0.393052	+	0.919516i	$0.371419\pi$
$390$	0	0
$391$	−7.19768	−0.364003
$392$	0	0
$393$	28.0287	1.41386
$394$	0	0
$395$	−6.86486	−0.345408
$396$	0	0
$397$	37.3882	1.87646	0.938231	−	0.346009i	$-0.112463\pi$
$397$	37.3882	1.87646	0.938231	+	0.346009i	$0.112463\pi$
$398$	0	0
$399$	−6.01174	−0.300963
$400$	0	0
$401$	−4.97625	−0.248502	−0.124251	−	0.992251i	$-0.539653\pi$
$401$	−4.97625	−0.248502	−0.124251	+	0.992251i	$0.539653\pi$
$402$	0	0
$403$	6.54520	0.326039
$404$	0	0
$405$	16.6685	0.828266
$406$	0	0
$407$	−3.04331	−0.150851
$408$	0	0
$409$	4.06359	0.200932	0.100466	−	0.994941i	$-0.467967\pi$
$409$	4.06359	0.200932	0.100466	+	0.994941i	$0.467967\pi$
$410$	0	0
$411$	−65.8379	−3.24754
$412$	0	0
$413$	1.60998	0.0792219
$414$	0	0
$415$	39.0674	1.91774
$416$	0	0
$417$	27.3203	1.33788
$418$	0	0
$419$	0.780982	0.0381535	0.0190767	−	0.999818i	$-0.493927\pi$
$419$	0.780982	0.0381535	0.0190767	+	0.999818i	$0.493927\pi$
$420$	0	0
$421$	−5.34237	−0.260371	−0.130186	−	0.991490i	$-0.541557\pi$
$421$	−5.34237	−0.260371	−0.130186	+	0.991490i	$0.541557\pi$
$422$	0	0
$423$	19.5920	0.952593
$424$	0	0
$425$	14.9079	0.723140
$426$	0	0
$427$	−4.55209	−0.220291
$428$	0	0
$429$	6.01906	0.290603
$430$	0	0
$431$	1.18186	0.0569283	0.0284641	−	0.999595i	$-0.490938\pi$
$431$	1.18186	0.0569283	0.0284641	+	0.999595i	$0.490938\pi$
$432$	0	0
$433$	15.2584	0.733273	0.366637	−	0.930364i	$-0.380509\pi$
$433$	15.2584	0.733273	0.366637	+	0.930364i	$0.380509\pi$
$434$	0	0
$435$	65.0135	3.11716
$436$	0	0
$437$	9.86780	0.472041
$438$	0	0
$439$	4.34502	0.207376	0.103688	−	0.994610i	$-0.466936\pi$
$439$	4.34502	0.207376	0.103688	+	0.994610i	$0.466936\pi$
$440$	0	0
$441$	5.44602	0.259334
$442$	0	0
$443$	6.10081	0.289858	0.144929	−	0.989442i	$-0.453705\pi$
$443$	6.10081	0.289858	0.144929	+	0.989442i	$0.453705\pi$
$444$	0	0
$445$	32.6305	1.54684
$446$	0	0
$447$	31.9359	1.51052
$448$	0	0
$449$	3.72499	0.175793	0.0878967	−	0.996130i	$-0.471985\pi$
$449$	3.72499	0.175793	0.0878967	+	0.996130i	$0.471985\pi$
$450$	0	0
$451$	9.69590	0.456562
$452$	0	0
$453$	−15.8920	−0.746670
$454$	0	0
$455$	6.14310	0.287993
$456$	0	0
$457$	−10.7451	−0.502636	−0.251318	−	0.967905i	$-0.580864\pi$
$457$	−10.7451	−0.502636	−0.251318	+	0.967905i	$0.580864\pi$
$458$	0	0
$459$	−10.7259	−0.500641
$460$	0	0
$461$	11.1551	0.519545	0.259772	−	0.965670i	$-0.416352\pi$
$461$	11.1551	0.519545	0.259772	+	0.965670i	$0.416352\pi$
$462$	0	0
$463$	27.3485	1.27099	0.635496	−	0.772104i	$-0.280796\pi$
$463$	27.3485	1.27099	0.635496	+	0.772104i	$0.280796\pi$
$464$	0	0
$465$	46.0758	2.13671
$466$	0	0
$467$	−16.1100	−0.745481	−0.372740	−	0.927936i	$-0.621582\pi$
$467$	−16.1100	−0.745481	−0.372740	+	0.927936i	$0.621582\pi$
$468$	0	0
$469$	2.15866	0.0996777
$470$	0	0
$471$	−23.0867	−1.06378
$472$	0	0
$473$	10.4651	0.481185
$474$	0	0
$475$	−20.4383	−0.937772
$476$	0	0
$477$	5.20424	0.238286
$478$	0	0
$479$	−11.3880	−0.520330	−0.260165	−	0.965564i	$-0.583777\pi$
$479$	−11.3880	−0.520330	−0.260165	+	0.965564i	$0.583777\pi$
$480$	0	0
$481$	3.72655	0.169916
$482$	0	0
$483$	−13.8635	−0.630810
$484$	0	0
$485$	62.5792	2.84157
$486$	0	0
$487$	31.3531	1.42075	0.710373	−	0.703826i	$-0.248526\pi$
$487$	31.3531	1.42075	0.710373	+	0.703826i	$0.248526\pi$
$488$	0	0
$489$	25.9652	1.17419
$490$	0	0
$491$	28.6767	1.29416	0.647082	−	0.762421i	$-0.275989\pi$
$491$	28.6767	1.29416	0.647082	+	0.762421i	$0.275989\pi$
$492$	0	0
$493$	−8.75020	−0.394089
$494$	0	0
$495$	27.3216	1.22802
$496$	0	0
$497$	−13.8202	−0.619921
$498$	0	0
$499$	26.1866	1.17227	0.586136	−	0.810213i	$-0.300649\pi$
$499$	26.1866	1.17227	0.586136	+	0.810213i	$0.300649\pi$
$500$	0	0
$501$	−34.3440	−1.53438
$502$	0	0
$503$	−31.9854	−1.42616	−0.713080	−	0.701082i	$-0.752701\pi$
$503$	−31.9854	−1.42616	−0.713080	+	0.701082i	$0.752701\pi$
$504$	0	0
$505$	−63.0987	−2.80786
$506$	0	0
$507$	30.4103	1.35057
$508$	0	0
$509$	−17.2030	−0.762511	−0.381255	−	0.924470i	$-0.624508\pi$
$509$	−17.2030	−0.762511	−0.381255	+	0.924470i	$0.624508\pi$
$510$	0	0
$511$	14.4749	0.640331
$512$	0	0
$513$	14.7048	0.649234
$514$	0	0
$515$	66.3008	2.92156
$516$	0	0
$517$	4.67864	0.205766
$518$	0	0
$519$	−23.1745	−1.01725
$520$	0	0
$521$	34.4322	1.50850	0.754251	−	0.656586i	$-0.228000\pi$
$521$	34.4322	1.50850	0.754251	+	0.656586i	$0.228000\pi$
$522$	0	0
$523$	−21.7833	−0.952516	−0.476258	−	0.879306i	$-0.658007\pi$
$523$	−21.7833	−0.952516	−0.476258	+	0.879306i	$0.658007\pi$
$524$	0	0
$525$	28.7142	1.25319
$526$	0	0
$527$	−6.20137	−0.270136
$528$	0	0
$529$	−0.244184	−0.0106167
$530$	0	0
$531$	−8.76798	−0.380498
$532$	0	0
$533$	−11.8727	−0.514262
$534$	0	0
$535$	22.3747	0.967341
$536$	0	0
$537$	48.2011	2.08003
$538$	0	0
$539$	1.30053	0.0560179
$540$	0	0
$541$	33.5400	1.44200	0.720999	−	0.692936i	$-0.243683\pi$
$541$	33.5400	1.44200	0.720999	+	0.692936i	$0.243683\pi$
$542$	0	0
$543$	−17.2034	−0.738267
$544$	0	0
$545$	−17.8581	−0.764956
$546$	0	0
$547$	39.4310	1.68595	0.842973	−	0.537956i	$-0.180803\pi$
$547$	39.4310	1.68595	0.842973	+	0.537956i	$0.180803\pi$
$548$	0	0
$549$	24.7908	1.05804
$550$	0	0
$551$	11.9963	0.511058
$552$	0	0
$553$	1.77961	0.0756769
$554$	0	0
$555$	26.2335	1.11355
$556$	0	0
$557$	0.390243	0.0165351	0.00826756	−	0.999966i	$-0.497368\pi$
$557$	0.390243	0.0165351	0.00826756	+	0.999966i	$0.497368\pi$
$558$	0	0
$559$	−12.8145	−0.541997
$560$	0	0
$561$	−5.70288	−0.240776
$562$	0	0
$563$	19.6449	0.827935	0.413967	−	0.910292i	$-0.364143\pi$
$563$	19.6449	0.827935	0.413967	+	0.910292i	$0.364143\pi$
$564$	0	0
$565$	35.3807	1.48848
$566$	0	0
$567$	−4.32107	−0.181468
$568$	0	0
$569$	−8.08076	−0.338763	−0.169381	−	0.985551i	$-0.554177\pi$
$569$	−8.08076	−0.338763	−0.169381	+	0.985551i	$0.554177\pi$
$570$	0	0
$571$	10.6889	0.447315	0.223657	−	0.974668i	$-0.428200\pi$
$571$	10.6889	0.447315	0.223657	+	0.974668i	$0.428200\pi$
$572$	0	0
$573$	35.9429	1.50153
$574$	0	0
$575$	−47.1320	−1.96554
$576$	0	0
$577$	1.95969	0.0815831	0.0407915	−	0.999168i	$-0.487012\pi$
$577$	1.95969	0.0815831	0.0407915	+	0.999168i	$0.487012\pi$
$578$	0	0
$579$	−68.8297	−2.86046
$580$	0	0
$581$	−10.1276	−0.420166
$582$	0	0
$583$	1.24279	0.0514713
$584$	0	0
$585$	−33.4554	−1.38321
$586$	0	0
$587$	34.8082	1.43669	0.718343	−	0.695689i	$-0.244901\pi$
$587$	34.8082	1.43669	0.718343	+	0.695689i	$0.244901\pi$
$588$	0	0
$589$	8.50189	0.350314
$590$	0	0
$591$	46.9341	1.93061
$592$	0	0
$593$	−16.5483	−0.679558	−0.339779	−	0.940505i	$-0.610352\pi$
$593$	−16.5483	−0.679558	−0.339779	+	0.940505i	$0.610352\pi$
$594$	0	0
$595$	−5.82039	−0.238613
$596$	0	0
$597$	15.9749	0.653809
$598$	0	0
$599$	10.2649	0.419412	0.209706	−	0.977764i	$-0.432749\pi$
$599$	10.2649	0.419412	0.209706	+	0.977764i	$0.432749\pi$
$600$	0	0
$601$	35.6586	1.45455	0.727273	−	0.686348i	$-0.240787\pi$
$601$	35.6586	1.45455	0.727273	+	0.686348i	$0.240787\pi$
$602$	0	0
$603$	−11.7561	−0.478746
$604$	0	0
$605$	−35.9080	−1.45987
$606$	0	0
$607$	−23.1111	−0.938052	−0.469026	−	0.883184i	$-0.655395\pi$
$607$	−23.1111	−0.938052	−0.469026	+	0.883184i	$0.655395\pi$
$608$	0	0
$609$	−16.8538	−0.682950
$610$	0	0
$611$	−5.72902	−0.231771
$612$	0	0
$613$	−38.0706	−1.53766	−0.768829	−	0.639455i	$-0.779160\pi$
$613$	−38.0706	−1.53766	−0.768829	+	0.639455i	$0.779160\pi$
$614$	0	0
$615$	−83.5793	−3.37024
$616$	0	0
$617$	−47.9337	−1.92974	−0.964869	−	0.262732i	$-0.915377\pi$
$617$	−47.9337	−1.92974	−0.964869	+	0.262732i	$0.915377\pi$
$618$	0	0
$619$	5.72832	0.230240	0.115120	−	0.993352i	$-0.463275\pi$
$619$	5.72832	0.230240	0.115120	+	0.993352i	$0.463275\pi$
$620$	0	0
$621$	33.9103	1.36078
$622$	0	0
$623$	−8.45899	−0.338902
$624$	0	0
$625$	23.2188	0.928752
$626$	0	0
$627$	7.81847	0.312239
$628$	0	0
$629$	−3.53079	−0.140782
$630$	0	0
$631$	2.35784	0.0938640	0.0469320	−	0.998898i	$-0.485056\pi$
$631$	2.35784	0.0938640	0.0469320	+	0.998898i	$0.485056\pi$
$632$	0	0
$633$	3.73136	0.148308
$634$	0	0
$635$	50.0652	1.98678
$636$	0	0
$637$	−1.59251	−0.0630975
$638$	0	0
$639$	75.2652	2.97744
$640$	0	0
$641$	−49.8415	−1.96862	−0.984311	−	0.176440i	$-0.943542\pi$
$641$	−49.8415	−1.96862	−0.984311	+	0.176440i	$0.943542\pi$
$642$	0	0
$643$	−8.59806	−0.339074	−0.169537	−	0.985524i	$-0.554227\pi$
$643$	−8.59806	−0.339074	−0.169537	+	0.985524i	$0.554227\pi$
$644$	0	0
$645$	−90.2097	−3.55200
$646$	0	0
$647$	0.463264	0.0182128	0.00910640	−	0.999959i	$-0.497101\pi$
$647$	0.463264	0.0182128	0.00910640	+	0.999959i	$0.497101\pi$
$648$	0	0
$649$	−2.09383	−0.0821901
$650$	0	0
$651$	−11.9445	−0.468141
$652$	0	0
$653$	9.38568	0.367290	0.183645	−	0.982993i	$-0.441210\pi$
$653$	9.38568	0.367290	0.183645	+	0.982993i	$0.441210\pi$
$654$	0	0
$655$	−37.2034	−1.45366
$656$	0	0
$657$	−78.8305	−3.07547
$658$	0	0
$659$	6.54670	0.255023	0.127512	−	0.991837i	$-0.459301\pi$
$659$	6.54670	0.255023	0.127512	+	0.991837i	$0.459301\pi$
$660$	0	0
$661$	3.19204	0.124156	0.0620780	−	0.998071i	$-0.480227\pi$
$661$	3.19204	0.124156	0.0620780	+	0.998071i	$0.480227\pi$
$662$	0	0
$663$	6.98320	0.271205
$664$	0	0
$665$	7.97958	0.309435
$666$	0	0
$667$	27.6642	1.07116
$668$	0	0
$669$	−9.30910	−0.359910
$670$	0	0
$671$	5.92014	0.228544
$672$	0	0
$673$	6.82222	0.262977	0.131489	−	0.991318i	$-0.458024\pi$
$673$	6.82222	0.262977	0.131489	+	0.991318i	$0.458024\pi$
$674$	0	0
$675$	−70.2354	−2.70336
$676$	0	0
$677$	6.75773	0.259721	0.129860	−	0.991532i	$-0.458547\pi$
$677$	6.75773	0.259721	0.129860	+	0.991532i	$0.458547\pi$
$678$	0	0
$679$	−16.2227	−0.622571
$680$	0	0
$681$	56.6895	2.17235
$682$	0	0
$683$	−41.3991	−1.58409	−0.792046	−	0.610461i	$-0.790984\pi$
$683$	−41.3991	−1.58409	−0.792046	+	0.610461i	$0.790984\pi$
$684$	0	0
$685$	87.3888	3.33895
$686$	0	0
$687$	−28.8594	−1.10106
$688$	0	0
$689$	−1.52181	−0.0579762
$690$	0	0
$691$	6.57700	0.250201	0.125100	−	0.992144i	$-0.460075\pi$
$691$	6.57700	0.250201	0.125100	+	0.992144i	$0.460075\pi$
$692$	0	0
$693$	−7.08273	−0.269051
$694$	0	0
$695$	−36.2631	−1.37554
$696$	0	0
$697$	11.2490	0.426086
$698$	0	0
$699$	8.54009	0.323016
$700$	0	0
$701$	5.67390	0.214300	0.107150	−	0.994243i	$-0.465827\pi$
$701$	5.67390	0.214300	0.107150	+	0.994243i	$0.465827\pi$
$702$	0	0
$703$	4.84060	0.182567
$704$	0	0
$705$	−40.3302	−1.51892
$706$	0	0
$707$	16.3574	0.615184
$708$	0	0
$709$	50.9282	1.91265	0.956325	−	0.292306i	$-0.0944224\pi$
$709$	50.9282	1.91265	0.956325	+	0.292306i	$0.0944224\pi$
$710$	0	0
$711$	−9.69181	−0.363471
$712$	0	0
$713$	19.6059	0.734248
$714$	0	0
$715$	−7.98930	−0.298783
$716$	0	0
$717$	−47.1121	−1.75943
$718$	0	0
$719$	7.62249	0.284271	0.142135	−	0.989847i	$-0.454603\pi$
$719$	7.62249	0.284271	0.142135	+	0.989847i	$0.454603\pi$
$720$	0	0
$721$	−17.1875	−0.640096
$722$	0	0
$723$	−41.0257	−1.52576
$724$	0	0
$725$	−57.2983	−2.12800
$726$	0	0
$727$	−25.4856	−0.945207	−0.472604	−	0.881275i	$-0.656686\pi$
$727$	−25.4856	−0.945207	−0.472604	+	0.881275i	$0.656686\pi$
$728$	0	0
$729$	−38.4448	−1.42388
$730$	0	0
$731$	12.1414	0.449065
$732$	0	0
$733$	21.1775	0.782210	0.391105	−	0.920346i	$-0.372093\pi$
$733$	21.1775	0.782210	0.391105	+	0.920346i	$0.372093\pi$
$734$	0	0
$735$	−11.2107	−0.413512
$736$	0	0
$737$	−2.80741	−0.103412
$738$	0	0
$739$	−17.4538	−0.642048	−0.321024	−	0.947071i	$-0.604027\pi$
$739$	−17.4538	−0.642048	−0.321024	+	0.947071i	$0.604027\pi$
$740$	0	0
$741$	−9.57374	−0.351700
$742$	0	0
$743$	−10.6724	−0.391531	−0.195766	−	0.980651i	$-0.562719\pi$
$743$	−10.6724	−0.391531	−0.195766	+	0.980651i	$0.562719\pi$
$744$	0	0
$745$	−42.3896	−1.55303
$746$	0	0
$747$	55.1554	2.01803
$748$	0	0
$749$	−5.80030	−0.211938
$750$	0	0
$751$	−2.82952	−0.103251	−0.0516254	−	0.998667i	$-0.516440\pi$
$751$	−2.82952	−0.103251	−0.0516254	+	0.998667i	$0.516440\pi$
$752$	0	0
$753$	−58.3485	−2.12634
$754$	0	0
$755$	21.0939	0.767687
$756$	0	0
$757$	−22.1452	−0.804881	−0.402440	−	0.915446i	$-0.631838\pi$
$757$	−22.1452	−0.804881	−0.402440	+	0.915446i	$0.631838\pi$
$758$	0	0
$759$	18.0299	0.654444
$760$	0	0
$761$	4.79367	0.173770	0.0868852	−	0.996218i	$-0.472309\pi$
$761$	4.79367	0.173770	0.0868852	+	0.996218i	$0.472309\pi$
$762$	0	0
$763$	4.62944	0.167597
$764$	0	0
$765$	31.6980	1.14604
$766$	0	0
$767$	2.56390	0.0925772
$768$	0	0
$769$	13.5489	0.488585	0.244293	−	0.969702i	$-0.421444\pi$
$769$	13.5489	0.488585	0.244293	+	0.969702i	$0.421444\pi$
$770$	0	0
$771$	9.40677	0.338777
$772$	0	0
$773$	26.5593	0.955273	0.477637	−	0.878558i	$-0.341494\pi$
$773$	26.5593	0.955273	0.477637	+	0.878558i	$0.341494\pi$
$774$	0	0
$775$	−40.6080	−1.45868
$776$	0	0
$777$	−6.80066	−0.243972
$778$	0	0
$779$	−15.4220	−0.552551
$780$	0	0
$781$	17.9736	0.643148
$782$	0	0
$783$	41.2247	1.47325
$784$	0	0
$785$	30.6438	1.09372
$786$	0	0
$787$	−34.6253	−1.23426	−0.617129	−	0.786862i	$-0.711704\pi$
$787$	−34.6253	−1.23426	−0.617129	+	0.786862i	$0.711704\pi$
$788$	0	0
$789$	69.7956	2.48479
$790$	0	0
$791$	−9.17193	−0.326116
$792$	0	0
$793$	−7.24923	−0.257428
$794$	0	0
$795$	−10.7130	−0.379950
$796$	0	0
$797$	24.9878	0.885113	0.442557	−	0.896741i	$-0.354072\pi$
$797$	24.9878	0.885113	0.442557	+	0.896741i	$0.354072\pi$
$798$	0	0
$799$	5.42807	0.192031
$800$	0	0
$801$	46.0678	1.62773
$802$	0	0
$803$	−18.8251	−0.664322
$804$	0	0
$805$	18.4014	0.648566
$806$	0	0
$807$	0.359641	0.0126600
$808$	0	0
$809$	2.99378	0.105256	0.0526278	−	0.998614i	$-0.483240\pi$
$809$	2.99378	0.105256	0.0526278	+	0.998614i	$0.483240\pi$
$810$	0	0
$811$	−3.23328	−0.113536	−0.0567680	−	0.998387i	$-0.518080\pi$
$811$	−3.23328	−0.113536	−0.0567680	+	0.998387i	$0.518080\pi$
$812$	0	0
$813$	29.3929	1.03085
$814$	0	0
$815$	−34.4644	−1.20724
$816$	0	0
$817$	−16.6454	−0.582351
$818$	0	0
$819$	8.67283	0.303053
$820$	0	0
$821$	40.4948	1.41328	0.706639	−	0.707574i	$-0.250210\pi$
$821$	40.4948	1.41328	0.706639	+	0.707574i	$0.250210\pi$
$822$	0	0
$823$	41.0492	1.43088	0.715442	−	0.698672i	$-0.246225\pi$
$823$	41.0492	1.43088	0.715442	+	0.698672i	$0.246225\pi$
$824$	0	0
$825$	−37.3437	−1.30014
$826$	0	0
$827$	28.8481	1.00315	0.501573	−	0.865116i	$-0.332755\pi$
$827$	28.8481	1.00315	0.501573	+	0.865116i	$0.332755\pi$
$828$	0	0
$829$	26.1476	0.908144	0.454072	−	0.890965i	$-0.349971\pi$
$829$	26.1476	0.908144	0.454072	+	0.890965i	$0.349971\pi$
$830$	0	0
$831$	−45.2428	−1.56945
$832$	0	0
$833$	1.50885	0.0522786
$834$	0	0
$835$	45.5859	1.57756
$836$	0	0
$837$	29.2164	1.00987
$838$	0	0
$839$	41.9862	1.44953	0.724763	−	0.688999i	$-0.241949\pi$
$839$	41.9862	1.44953	0.724763	+	0.688999i	$0.241949\pi$
$840$	0	0
$841$	4.63123	0.159698
$842$	0	0
$843$	12.2641	0.422399
$844$	0	0
$845$	−40.3646	−1.38858
$846$	0	0
$847$	9.30861	0.319848
$848$	0	0
$849$	−68.1577	−2.33916
$850$	0	0
$851$	11.1628	0.382654
$852$	0	0
$853$	−15.5345	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$853$	−15.5345	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$854$	0	0
$855$	−43.4570	−1.48620
$856$	0	0
$857$	47.3215	1.61647	0.808237	−	0.588858i	$-0.200422\pi$
$857$	47.3215	1.61647	0.808237	+	0.588858i	$0.200422\pi$
$858$	0	0
$859$	−7.22844	−0.246631	−0.123316	−	0.992367i	$-0.539353\pi$
$859$	−7.22844	−0.246631	−0.123316	+	0.992367i	$0.539353\pi$
$860$	0	0
$861$	21.6667	0.738399
$862$	0	0
$863$	−10.1105	−0.344167	−0.172084	−	0.985082i	$-0.555050\pi$
$863$	−10.1105	−0.344167	−0.172084	+	0.985082i	$0.555050\pi$
$864$	0	0
$865$	30.7603	1.04588
$866$	0	0
$867$	42.7891	1.45319
$868$	0	0
$869$	−2.31445	−0.0785122
$870$	0	0
$871$	3.43769	0.116482
$872$	0	0
$873$	88.3493	2.99017
$874$	0	0
$875$	−18.8257	−0.636426
$876$	0	0
$877$	−53.8306	−1.81773	−0.908865	−	0.417089i	$-0.863050\pi$
$877$	−53.8306	−1.81773	−0.908865	+	0.417089i	$0.863050\pi$
$878$	0	0
$879$	55.5844	1.87482
$880$	0	0
$881$	21.5769	0.726946	0.363473	−	0.931605i	$-0.381591\pi$
$881$	21.5769	0.726946	0.363473	+	0.931605i	$0.381591\pi$
$882$	0	0
$883$	−30.5046	−1.02656	−0.513281	−	0.858221i	$-0.671570\pi$
$883$	−30.5046	−1.02656	−0.513281	+	0.858221i	$0.671570\pi$
$884$	0	0
$885$	18.0490	0.606709
$886$	0	0
$887$	33.7110	1.13191	0.565953	−	0.824437i	$-0.308508\pi$
$887$	33.7110	1.13191	0.565953	+	0.824437i	$0.308508\pi$
$888$	0	0
$889$	−12.9787	−0.435291
$890$	0	0
$891$	5.61970	0.188267
$892$	0	0
$893$	−7.44171	−0.249027
$894$	0	0
$895$	−63.9788	−2.13858
$896$	0	0
$897$	−22.0777	−0.737153
$898$	0	0
$899$	23.8349	0.794937
$900$	0	0
$901$	1.44186	0.0480355
$902$	0	0
$903$	23.3855	0.778222
$904$	0	0
$905$	22.8346	0.759047
$906$	0	0
$907$	1.30668	0.0433876	0.0216938	−	0.999765i	$-0.493094\pi$
$907$	1.30668	0.0433876	0.0216938	+	0.999765i	$0.493094\pi$
$908$	0	0
$909$	−89.0829	−2.95469
$910$	0	0
$911$	−29.9873	−0.993522	−0.496761	−	0.867887i	$-0.665477\pi$
$911$	−29.9873	−0.993522	−0.496761	+	0.867887i	$0.665477\pi$
$912$	0	0
$913$	13.1713	0.435908
$914$	0	0
$915$	−51.0320	−1.68707
$916$	0	0
$917$	9.64445	0.318488
$918$	0	0
$919$	13.0725	0.431220	0.215610	−	0.976480i	$-0.430826\pi$
$919$	13.0725	0.431220	0.215610	+	0.976480i	$0.430826\pi$
$920$	0	0
$921$	−36.7182	−1.20991
$922$	0	0
$923$	−22.0088	−0.724428
$924$	0	0
$925$	−23.1204	−0.760194
$926$	0	0
$927$	93.6035	3.07434
$928$	0	0
$929$	−20.7057	−0.679333	−0.339667	−	0.940546i	$-0.610314\pi$
$929$	−20.7057	−0.679333	−0.339667	+	0.940546i	$0.610314\pi$
$930$	0	0
$931$	−2.06859	−0.0677953
$932$	0	0
$933$	−57.4663	−1.88136
$934$	0	0
$935$	7.56961	0.247553
$936$	0	0
$937$	−5.93799	−0.193986	−0.0969929	−	0.995285i	$-0.530922\pi$
$937$	−5.93799	−0.193986	−0.0969929	+	0.995285i	$0.530922\pi$
$938$	0	0
$939$	−25.3637	−0.827712
$940$	0	0
$941$	−19.7927	−0.645223	−0.322611	−	0.946532i	$-0.604561\pi$
$941$	−19.7927	−0.645223	−0.322611	+	0.946532i	$0.604561\pi$
$942$	0	0
$943$	−35.5642	−1.15813
$944$	0	0
$945$	27.4215	0.892023
$946$	0	0
$947$	42.8069	1.39104	0.695518	−	0.718509i	$-0.255175\pi$
$947$	42.8069	1.39104	0.695518	+	0.718509i	$0.255175\pi$
$948$	0	0
$949$	23.0514	0.748279
$950$	0	0
$951$	78.1710	2.53487
$952$	0	0
$953$	−0.689500	−0.0223351	−0.0111675	−	0.999938i	$-0.503555\pi$
$953$	−0.689500	−0.0223351	−0.0111675	+	0.999938i	$0.503555\pi$
$954$	0	0
$955$	−47.7081	−1.54380
$956$	0	0
$957$	21.9189	0.708537
$958$	0	0
$959$	−22.6543	−0.731544
$960$	0	0
$961$	−14.1080	−0.455096
$962$	0	0
$963$	31.5886	1.01793
$964$	0	0
$965$	91.3599	2.94098
$966$	0	0
$967$	−14.1022	−0.453497	−0.226749	−	0.973953i	$-0.572810\pi$
$967$	−14.1022	−0.453497	−0.226749	+	0.973953i	$0.572810\pi$
$968$	0	0
$969$	9.07083	0.291397
$970$	0	0
$971$	−0.360949	−0.0115834	−0.00579171	−	0.999983i	$-0.501844\pi$
$971$	−0.360949	−0.0115834	−0.00579171	+	0.999983i	$0.501844\pi$
$972$	0	0
$973$	9.40067	0.301372
$974$	0	0
$975$	45.7275	1.46445
$976$	0	0
$977$	9.83910	0.314781	0.157390	−	0.987536i	$-0.449692\pi$
$977$	9.83910	0.314781	0.157390	+	0.987536i	$0.449692\pi$
$978$	0	0
$979$	11.0012	0.351600
$980$	0	0
$981$	−25.2120	−0.804958
$982$	0	0
$983$	−26.6489	−0.849968	−0.424984	−	0.905201i	$-0.639720\pi$
$983$	−26.6489	−0.849968	−0.424984	+	0.905201i	$0.639720\pi$
$984$	0	0
$985$	−62.2971	−1.98495
$986$	0	0
$987$	10.4550	0.332787
$988$	0	0
$989$	−38.3855	−1.22059
$990$	0	0
$991$	−16.8287	−0.534583	−0.267291	−	0.963616i	$-0.586129\pi$
$991$	−16.8287	−0.534583	−0.267291	+	0.963616i	$0.586129\pi$
$992$	0	0
$993$	−91.7875	−2.91279
$994$	0	0
$995$	−21.2040	−0.672212
$996$	0	0
$997$	−36.4607	−1.15472	−0.577361	−	0.816489i	$-0.695917\pi$
$997$	−36.4607	−1.15472	−0.577361	+	0.816489i	$0.695917\pi$
$998$	0	0
$999$	16.6346	0.526294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bi.1.2		12
4.3	odd	2		7168.2.a.bj.1.11		12
8.3	odd	2		7168.2.a.bj.1.2		12
8.5	even	2	inner	7168.2.a.bi.1.11		12
32.3	odd	8		896.2.m.g.673.6		12
32.5	even	8		448.2.m.d.113.6		12
32.11	odd	8		896.2.m.g.225.6		12
32.13	even	8		448.2.m.d.337.6		12
32.19	odd	8		112.2.m.d.29.3	✓	12
32.21	even	8		896.2.m.h.225.1		12
32.27	odd	8		112.2.m.d.85.3	yes	12
32.29	even	8		896.2.m.h.673.1		12
224.19	even	24		784.2.x.m.557.5		24
224.27	even	8		784.2.m.h.197.3		12
224.51	odd	24		784.2.x.l.557.5		24
224.59	even	24		784.2.x.m.373.5		24
224.83	even	8		784.2.m.h.589.3		12
224.115	even	24		784.2.x.m.765.2		24
224.123	odd	24		784.2.x.l.373.5		24
224.179	odd	24		784.2.x.l.765.2		24
224.187	even	24		784.2.x.m.165.2		24
224.219	odd	24		784.2.x.l.165.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.3	✓	12	32.19	odd	8
112.2.m.d.85.3	yes	12	32.27	odd	8
448.2.m.d.113.6		12	32.5	even	8
448.2.m.d.337.6		12	32.13	even	8
784.2.m.h.197.3		12	224.27	even	8
784.2.m.h.589.3		12	224.83	even	8
784.2.x.l.165.2		24	224.219	odd	24
784.2.x.l.373.5		24	224.123	odd	24
784.2.x.l.557.5		24	224.51	odd	24
784.2.x.l.765.2		24	224.179	odd	24
784.2.x.m.165.2		24	224.187	even	24
784.2.x.m.373.5		24	224.59	even	24
784.2.x.m.557.5		24	224.19	even	24
784.2.x.m.765.2		24	224.115	even	24
896.2.m.g.225.6		12	32.11	odd	8
896.2.m.g.673.6		12	32.3	odd	8
896.2.m.h.225.1		12	32.21	even	8
896.2.m.h.673.1		12	32.29	even	8
7168.2.a.bi.1.2		12	1.1	even	1	trivial
7168.2.a.bi.1.11		12	8.5	even	2	inner
7168.2.a.bj.1.2		12	8.3	odd	2
7168.2.a.bj.1.11		12	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)

\(f(q)\)	\(=\)	\(q-2.90620 q^{3} +3.85750 q^{5} -1.00000 q^{7} +5.44602 q^{9} +O(q^{10})\)	\(q-2.90620 q^{3} +3.85750 q^{5} -1.00000 q^{7} +5.44602 q^{9} +1.30053 q^{11} -1.59251 q^{13} -11.2107 q^{15} +1.50885 q^{17} -2.06859 q^{19} +2.90620 q^{21} -4.77031 q^{23} +9.88030 q^{25} -7.10863 q^{27} -5.79924 q^{29} -4.10999 q^{31} -3.77961 q^{33} -3.85750 q^{35} -2.34005 q^{37} +4.62815 q^{39} +7.45533 q^{41} +8.04676 q^{43} +21.0080 q^{45} +3.59748 q^{47} +1.00000 q^{49} -4.38503 q^{51} +0.955604 q^{53} +5.01680 q^{55} +6.01174 q^{57} -1.60998 q^{59} +4.55209 q^{61} -5.44602 q^{63} -6.14310 q^{65} -2.15866 q^{67} +13.8635 q^{69} +13.8202 q^{71} -14.4749 q^{73} -28.7142 q^{75} -1.30053 q^{77} -1.77961 q^{79} +4.32107 q^{81} +10.1276 q^{83} +5.82039 q^{85} +16.8538 q^{87} +8.45899 q^{89} +1.59251 q^{91} +11.9445 q^{93} -7.97958 q^{95} +16.2227 q^{97} +7.08273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 + 12 * q^9	\( 12 q - 12 q^{7} + 12 q^{9} - 24 q^{15} + 8 q^{17} - 16 q^{23} + 20 q^{25} + 8 q^{31} - 16 q^{39} + 32 q^{41} + 16 q^{47} + 12 q^{49} - 24 q^{55} + 64 q^{57} - 12 q^{63} + 32 q^{65} - 8 q^{71} + 24 q^{79} + 44 q^{81} + 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 + 12 * q^9 - 24 * q^15 + 8 * q^17 - 16 * q^23 + 20 * q^25 + 8 * q^31 - 16 * q^39 + 32 * q^41 + 16 * q^47 + 12 * q^49 - 24 * q^55 + 64 * q^57 - 12 * q^63 + 32 * q^65 - 8 * q^71 + 24 * q^79 + 44 * q^81 + 32 * q^87 + 24 * q^89 + 48 * q^97

Introduction

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