LMFDB - Embedded newform 9800.2.a.cz.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 18x^{6} + 85x^{4} - 38x^{2} + 4 $ x^8 - 18x^6 + 85x^4 - 38*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.87509$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87509 q^{3} +5.26617 q^{9} +O(q^{10})$	$q+2.87509 q^{3} +5.26617 q^{9} -2.08223 q^{11} +0.695629 q^{13} +2.15651 q^{17} -8.21273 q^{19} +7.33216 q^{23} +6.51544 q^{27} +2.18394 q^{29} -7.88299 q^{31} -5.98660 q^{33} +1.78360 q^{37} +2.00000 q^{39} +1.48384 q^{41} +0.216400 q^{43} +11.6672 q^{47} +6.20017 q^{51} +3.14823 q^{53} -23.6124 q^{57} +12.4791 q^{59} -6.84519 q^{61} +13.1320 q^{67} +21.0807 q^{69} +10.3159 q^{71} +14.5652 q^{73} +10.9318 q^{79} +2.93400 q^{81} +1.74396 q^{83} +6.27902 q^{87} +11.5700 q^{89} -22.6643 q^{93} +3.14764 q^{97} -10.9654 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9}+O(q^{10}) $ 8 * q + 12 * q^9	$ 8 q + 12 q^{9} + 4 q^{11} - 4 q^{23} + 8 q^{29} + 16 q^{39} + 16 q^{43} + 52 q^{51} - 28 q^{53} - 8 q^{57} + 40 q^{67} + 8 q^{71} + 20 q^{79} + 56 q^{81} - 56 q^{93} + 36 q^{99}+O(q^{100}) $ 8 * q + 12 * q^9 + 4 * q^11 - 4 * q^23 + 8 * q^29 + 16 * q^39 + 16 * q^43 + 52 * q^51 - 28 * q^53 - 8 * q^57 + 40 * q^67 + 8 * q^71 + 20 * q^79 + 56 * q^81 - 56 * q^93 + 36 * 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
$3$	2.87509	1.65994	0.829968	−	0.557811i	$-0.188359\pi$
$3$	2.87509	1.65994	0.829968	+	0.557811i	$0.188359\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.26617	1.75539
$10$	0	0
$11$	−2.08223	−0.627815	−0.313908	−	0.949453i	$-0.601638\pi$
$11$	−2.08223	−0.627815	−0.313908	+	0.949453i	$0.601638\pi$
$12$	0	0
$13$	0.695629	0.192933	0.0964664	−	0.995336i	$-0.469246\pi$
$13$	0.695629	0.192933	0.0964664	+	0.995336i	$0.469246\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.15651	0.523030	0.261515	−	0.965199i	$-0.415778\pi$
$17$	2.15651	0.523030	0.261515	+	0.965199i	$0.415778\pi$
$18$	0	0
$19$	−8.21273	−1.88413	−0.942065	−	0.335430i	$-0.891118\pi$
$19$	−8.21273	−1.88413	−0.942065	+	0.335430i	$0.891118\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.33216	1.52886	0.764431	−	0.644706i	$-0.223020\pi$
$23$	7.33216	1.52886	0.764431	+	0.644706i	$0.223020\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	6.51544	1.25390
$28$	0	0
$29$	2.18394	0.405547	0.202773	−	0.979226i	$-0.435004\pi$
$29$	2.18394	0.405547	0.202773	+	0.979226i	$0.435004\pi$
$30$	0	0
$31$	−7.88299	−1.41583	−0.707913	−	0.706300i	$-0.750363\pi$
$31$	−7.88299	−1.41583	−0.707913	+	0.706300i	$0.750363\pi$
$32$	0	0
$33$	−5.98660	−1.04213
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.78360	0.293222	0.146611	−	0.989194i	$-0.453163\pi$
$37$	1.78360	0.293222	0.146611	+	0.989194i	$0.453163\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	1.48384	0.231736	0.115868	−	0.993265i	$-0.463035\pi$
$41$	1.48384	0.231736	0.115868	+	0.993265i	$0.463035\pi$
$42$	0	0
$43$	0.216400	0.0330007	0.0165003	−	0.999864i	$-0.494748\pi$
$43$	0.216400	0.0330007	0.0165003	+	0.999864i	$0.494748\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.6672	1.70183	0.850916	−	0.525302i	$-0.176048\pi$
$47$	11.6672	1.70183	0.850916	+	0.525302i	$0.176048\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.20017	0.868197
$52$	0	0
$53$	3.14823	0.432442	0.216221	−	0.976344i	$-0.430627\pi$
$53$	3.14823	0.432442	0.216221	+	0.976344i	$0.430627\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−23.6124	−3.12754
$58$	0	0
$59$	12.4791	1.62464	0.812319	−	0.583213i	$-0.198205\pi$
$59$	12.4791	1.62464	0.812319	+	0.583213i	$0.198205\pi$
$60$	0	0
$61$	−6.84519	−0.876436	−0.438218	−	0.898869i	$-0.644390\pi$
$61$	−6.84519	−0.876436	−0.438218	+	0.898869i	$0.644390\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1320	1.60433	0.802164	−	0.597104i	$-0.203682\pi$
$67$	13.1320	1.60433	0.802164	+	0.597104i	$0.203682\pi$
$68$	0	0
$69$	21.0807	2.53781
$70$	0	0
$71$	10.3159	1.22428	0.612138	−	0.790751i	$-0.290310\pi$
$71$	10.3159	1.22428	0.612138	+	0.790751i	$0.290310\pi$
$72$	0	0
$73$	14.5652	1.70473	0.852365	−	0.522948i	$-0.175168\pi$
$73$	14.5652	1.70473	0.852365	+	0.522948i	$0.175168\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.9318	1.22993	0.614963	−	0.788556i	$-0.289171\pi$
$79$	10.9318	1.22993	0.614963	+	0.788556i	$0.289171\pi$
$80$	0	0
$81$	2.93400	0.326000
$82$	0	0
$83$	1.74396	0.191425	0.0957123	−	0.995409i	$-0.469487\pi$
$83$	1.74396	0.191425	0.0957123	+	0.995409i	$0.469487\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.27902	0.673182
$88$	0	0
$89$	11.5700	1.22642	0.613209	−	0.789921i	$-0.289878\pi$
$89$	11.5700	1.22642	0.613209	+	0.789921i	$0.289878\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−22.6643	−2.35018
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.14764	0.319595	0.159797	−	0.987150i	$-0.448916\pi$
$97$	3.14764	0.319595	0.159797	+	0.987150i	$0.448916\pi$
$98$	0	0
$99$	−10.9654	−1.10206
$100$	0	0
$101$	−13.2338	−1.31681	−0.658406	−	0.752663i	$-0.728769\pi$
$101$	−13.2338	−1.31681	−0.658406	+	0.752663i	$0.728769\pi$
$102$	0	0
$103$	−1.13113	−0.111454	−0.0557269	−	0.998446i	$-0.517748\pi$
$103$	−1.13113	−0.111454	−0.0557269	+	0.998446i	$0.517748\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3646	1.38868	0.694340	−	0.719647i	$-0.255696\pi$
$107$	14.3646	1.38868	0.694340	+	0.719647i	$0.255696\pi$
$108$	0	0
$109$	17.4642	1.67276	0.836381	−	0.548148i	$-0.184667\pi$
$109$	17.4642	1.67276	0.836381	+	0.548148i	$0.184667\pi$
$110$	0	0
$111$	5.12802	0.486730
$112$	0	0
$113$	−5.09846	−0.479623	−0.239811	−	0.970820i	$-0.577086\pi$
$113$	−5.09846	−0.479623	−0.239811	+	0.970820i	$0.577086\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.66330	0.338672
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.66433	−0.605848
$122$	0	0
$123$	4.26617	0.384667
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.4479	−1.28205	−0.641023	−	0.767522i	$-0.721490\pi$
$127$	−14.4479	−1.28205	−0.641023	+	0.767522i	$0.721490\pi$
$128$	0	0
$129$	0.622170	0.0547791
$130$	0	0
$131$	10.2989	0.899816	0.449908	−	0.893075i	$-0.351457\pi$
$131$	10.2989	0.899816	0.449908	+	0.893075i	$0.351457\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.84960	0.243457	0.121729	−	0.992563i	$-0.461156\pi$
$137$	2.84960	0.243457	0.121729	+	0.992563i	$0.461156\pi$
$138$	0	0
$139$	−8.53195	−0.723670	−0.361835	−	0.932242i	$-0.617850\pi$
$139$	−8.53195	−0.723670	−0.361835	+	0.932242i	$0.617850\pi$
$140$	0	0
$141$	33.5442	2.82493
$142$	0	0
$143$	−1.44846	−0.121126
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.6124	−1.52479	−0.762393	−	0.647114i	$-0.775976\pi$
$149$	−18.6124	−1.52479	−0.762393	+	0.647114i	$0.775976\pi$
$150$	0	0
$151$	0.248863	0.0202522	0.0101261	−	0.999949i	$-0.496777\pi$
$151$	0.248863	0.0202522	0.0101261	+	0.999949i	$0.496777\pi$
$152$	0	0
$153$	11.3565	0.918122
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.11009	0.328021	0.164011	−	0.986459i	$-0.447557\pi$
$157$	4.11009	0.328021	0.164011	+	0.986459i	$0.447557\pi$
$158$	0	0
$159$	9.05144	0.717826
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.5302	1.29474	0.647371	−	0.762175i	$-0.275869\pi$
$163$	16.5302	1.29474	0.647371	+	0.762175i	$0.275869\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.2893	0.950978	0.475489	−	0.879722i	$-0.342271\pi$
$167$	12.2893	0.950978	0.475489	+	0.879722i	$0.342271\pi$
$168$	0	0
$169$	−12.5161	−0.962777
$170$	0	0
$171$	−43.2496	−3.30738
$172$	0	0
$173$	9.07132	0.689680	0.344840	−	0.938662i	$-0.387933\pi$
$173$	9.07132	0.689680	0.344840	+	0.938662i	$0.387933\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	35.8785	2.69680
$178$	0	0
$179$	4.78360	0.357543	0.178772	−	0.983891i	$-0.442788\pi$
$179$	4.78360	0.357543	0.178772	+	0.983891i	$0.442788\pi$
$180$	0	0
$181$	−18.3915	−1.36703	−0.683514	−	0.729938i	$-0.739549\pi$
$181$	−18.3915	−1.36703	−0.683514	+	0.729938i	$0.739549\pi$
$182$	0	0
$183$	−19.6806	−1.45483
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.49035	−0.328367
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.1158	1.38317	0.691584	−	0.722296i	$-0.256913\pi$
$191$	19.1158	1.38317	0.691584	+	0.722296i	$0.256913\pi$
$192$	0	0
$193$	−21.7952	−1.56886	−0.784428	−	0.620220i	$-0.787043\pi$
$193$	−21.7952	−1.56886	−0.784428	+	0.620220i	$0.787043\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.76496	0.125748	0.0628742	−	0.998021i	$-0.479973\pi$
$197$	1.76496	0.125748	0.0628742	+	0.998021i	$0.479973\pi$
$198$	0	0
$199$	−6.61261	−0.468756	−0.234378	−	0.972146i	$-0.575305\pi$
$199$	−6.61261	−0.468756	−0.234378	+	0.972146i	$0.575305\pi$
$200$	0	0
$201$	37.7557	2.66308
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	38.6124	2.68375
$208$	0	0
$209$	17.1008	1.18289
$210$	0	0
$211$	−12.7000	−0.874307	−0.437153	−	0.899387i	$-0.644013\pi$
$211$	−12.7000	−0.874307	−0.437153	+	0.899387i	$0.644013\pi$
$212$	0	0
$213$	29.6593	2.03222
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	41.8764	2.82974
$220$	0	0
$221$	1.50013	0.100910
$222$	0	0
$223$	−28.0354	−1.87739	−0.938696	−	0.344747i	$-0.887965\pi$
$223$	−28.0354	−1.87739	−0.938696	+	0.344747i	$0.887965\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.6832	1.57191	0.785955	−	0.618284i	$-0.212172\pi$
$227$	23.6832	1.57191	0.785955	+	0.618284i	$0.212172\pi$
$228$	0	0
$229$	14.4956	0.957896	0.478948	−	0.877843i	$-0.341018\pi$
$229$	14.4956	0.957896	0.478948	+	0.877843i	$0.341018\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.7998	−0.838545	−0.419272	−	0.907860i	$-0.637715\pi$
$233$	−12.7998	−0.838545	−0.419272	+	0.907860i	$0.637715\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	31.4300	2.04160
$238$	0	0
$239$	−30.3773	−1.96495	−0.982474	−	0.186402i	$-0.940317\pi$
$239$	−30.3773	−1.96495	−0.982474	+	0.186402i	$0.940317\pi$
$240$	0	0
$241$	−4.82122	−0.310562	−0.155281	−	0.987870i	$-0.549628\pi$
$241$	−4.82122	−0.310562	−0.155281	+	0.987870i	$0.549628\pi$
$242$	0	0
$243$	−11.1108	−0.712757
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.71302	−0.363511
$248$	0	0
$249$	5.01405	0.317753
$250$	0	0
$251$	−13.1118	−0.827609	−0.413804	−	0.910366i	$-0.635800\pi$
$251$	−13.1118	−0.827609	−0.413804	+	0.910366i	$0.635800\pi$
$252$	0	0
$253$	−15.2672	−0.959843
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.4468	−0.963542	−0.481771	−	0.876297i	$-0.660006\pi$
$257$	−15.4468	−0.963542	−0.481771	+	0.876297i	$0.660006\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	11.5010	0.711892
$262$	0	0
$263$	−5.81606	−0.358634	−0.179317	−	0.983791i	$-0.557389\pi$
$263$	−5.81606	−0.358634	−0.179317	+	0.983791i	$0.557389\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	33.2648	2.03577
$268$	0	0
$269$	−9.22682	−0.562569	−0.281285	−	0.959624i	$-0.590760\pi$
$269$	−9.22682	−0.562569	−0.281285	+	0.959624i	$0.590760\pi$
$270$	0	0
$271$	−15.8232	−0.961189	−0.480595	−	0.876943i	$-0.659579\pi$
$271$	−15.8232	−0.961189	−0.480595	+	0.876943i	$0.659579\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.43521	−0.0862332	−0.0431166	−	0.999070i	$-0.513729\pi$
$277$	−1.43521	−0.0862332	−0.0431166	+	0.999070i	$0.513729\pi$
$278$	0	0
$279$	−41.5131	−2.48532
$280$	0	0
$281$	−14.6124	−0.871702	−0.435851	−	0.900019i	$-0.643552\pi$
$281$	−14.6124	−0.871702	−0.435851	+	0.900019i	$0.643552\pi$
$282$	0	0
$283$	9.08833	0.540245	0.270123	−	0.962826i	$-0.412936\pi$
$283$	9.08833	0.540245	0.270123	+	0.962826i	$0.412936\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.3495	−0.726439
$290$	0	0
$291$	9.04977	0.530507
$292$	0	0
$293$	6.30657	0.368434	0.184217	−	0.982886i	$-0.441025\pi$
$293$	6.30657	0.368434	0.184217	+	0.982886i	$0.441025\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−13.5666	−0.787216
$298$	0	0
$299$	5.10047	0.294968
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−38.0484	−2.18583
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.0515	0.973179	0.486590	−	0.873631i	$-0.338241\pi$
$307$	17.0515	0.973179	0.486590	+	0.873631i	$0.338241\pi$
$308$	0	0
$309$	−3.25211	−0.185006
$310$	0	0
$311$	−10.6542	−0.604145	−0.302072	−	0.953285i	$-0.597678\pi$
$311$	−10.6542	−0.604145	−0.302072	+	0.953285i	$0.597678\pi$
$312$	0	0
$313$	1.93447	0.109343	0.0546713	−	0.998504i	$-0.482589\pi$
$313$	1.93447	0.109343	0.0546713	+	0.998504i	$0.482589\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.9643	0.840478	0.420239	−	0.907413i	$-0.361946\pi$
$317$	14.9643	0.840478	0.420239	+	0.907413i	$0.361946\pi$
$318$	0	0
$319$	−4.54746	−0.254609
$320$	0	0
$321$	41.2996	2.30512
$322$	0	0
$323$	−17.7108	−0.985458
$324$	0	0
$325$	0	0
$326$	0	0
$327$	50.2111	2.77668
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.2816	0.949880	0.474940	−	0.880018i	$-0.342470\pi$
$331$	17.2816	0.949880	0.474940	+	0.880018i	$0.342470\pi$
$332$	0	0
$333$	9.39273	0.514719
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−31.6124	−1.72204	−0.861018	−	0.508574i	$-0.830173\pi$
$337$	−31.6124	−1.72204	−0.861018	+	0.508574i	$0.830173\pi$
$338$	0	0
$339$	−14.6586	−0.796143
$340$	0	0
$341$	16.4142	0.888877
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.9178	1.17661	0.588304	−	0.808640i	$-0.299796\pi$
$347$	21.9178	1.17661	0.588304	+	0.808640i	$0.299796\pi$
$348$	0	0
$349$	19.9581	1.06833	0.534167	−	0.845379i	$-0.320626\pi$
$349$	19.9581	1.06833	0.534167	+	0.845379i	$0.320626\pi$
$350$	0	0
$351$	4.53233	0.241918
$352$	0	0
$353$	−31.9656	−1.70135	−0.850677	−	0.525688i	$-0.823808\pi$
$353$	−31.9656	−1.70135	−0.850677	+	0.525688i	$0.823808\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.65161	0.403836	0.201918	−	0.979402i	$-0.435282\pi$
$359$	7.65161	0.403836	0.201918	+	0.979402i	$0.435282\pi$
$360$	0	0
$361$	48.4490	2.54995
$362$	0	0
$363$	−19.1606	−1.00567
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.3591	−1.11494	−0.557469	−	0.830198i	$-0.688227\pi$
$367$	−21.3591	−1.11494	−0.557469	+	0.830198i	$0.688227\pi$
$368$	0	0
$369$	7.81412	0.406787
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.8483	0.561702	0.280851	−	0.959751i	$-0.409383\pi$
$373$	10.8483	0.561702	0.280851	+	0.959751i	$0.409383\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.51921	0.0782433
$378$	0	0
$379$	21.4339	1.10098	0.550492	−	0.834840i	$-0.314440\pi$
$379$	21.4339	1.10098	0.550492	+	0.834840i	$0.314440\pi$
$380$	0	0
$381$	−41.5391	−2.12811
$382$	0	0
$383$	−17.6203	−0.900354	−0.450177	−	0.892939i	$-0.648639\pi$
$383$	−17.6203	−0.900354	−0.450177	+	0.892939i	$0.648639\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.13960	0.0579290
$388$	0	0
$389$	−20.5129	−1.04004	−0.520021	−	0.854153i	$-0.674076\pi$
$389$	−20.5129	−1.04004	−0.520021	+	0.854153i	$0.674076\pi$
$390$	0	0
$391$	15.8119	0.799641
$392$	0	0
$393$	29.6102	1.49364
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−14.2079	−0.713075	−0.356538	−	0.934281i	$-0.616043\pi$
$397$	−14.2079	−0.713075	−0.356538	+	0.934281i	$0.616043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.0833	1.40241	0.701207	−	0.712958i	$-0.252645\pi$
$401$	28.0833	1.40241	0.701207	+	0.712958i	$0.252645\pi$
$402$	0	0
$403$	−5.48364	−0.273159
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.71386	−0.184089
$408$	0	0
$409$	8.58915	0.424706	0.212353	−	0.977193i	$-0.431887\pi$
$409$	8.58915	0.424706	0.212353	+	0.977193i	$0.431887\pi$
$410$	0	0
$411$	8.19286	0.404124
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−24.5302	−1.20125
$418$	0	0
$419$	0.872956	0.0426467	0.0213233	−	0.999773i	$-0.493212\pi$
$419$	0.872956	0.0426467	0.0213233	+	0.999773i	$0.493212\pi$
$420$	0	0
$421$	24.3159	1.18509	0.592543	−	0.805539i	$-0.298124\pi$
$421$	24.3159	1.18509	0.592543	+	0.805539i	$0.298124\pi$
$422$	0	0
$423$	61.4412	2.98738
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.16446	−0.201062
$430$	0	0
$431$	−31.9770	−1.54028	−0.770139	−	0.637876i	$-0.779813\pi$
$431$	−31.9770	−1.54028	−0.770139	+	0.637876i	$0.779813\pi$
$432$	0	0
$433$	7.36468	0.353924	0.176962	−	0.984218i	$-0.443373\pi$
$433$	7.36468	0.353924	0.176962	+	0.984218i	$0.443373\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−60.2171	−2.88057
$438$	0	0
$439$	13.3073	0.635121	0.317561	−	0.948238i	$-0.397136\pi$
$439$	13.3073	0.635121	0.317561	+	0.948238i	$0.397136\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.75114	0.415779	0.207890	−	0.978152i	$-0.433341\pi$
$443$	8.75114	0.415779	0.207890	+	0.978152i	$0.433341\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−53.5123	−2.53105
$448$	0	0
$449$	23.2129	1.09548	0.547742	−	0.836647i	$-0.315488\pi$
$449$	23.2129	1.09548	0.547742	+	0.836647i	$0.315488\pi$
$450$	0	0
$451$	−3.08968	−0.145488
$452$	0	0
$453$	0.715504	0.0336173
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.1980	0.804488	0.402244	−	0.915532i	$-0.368230\pi$
$457$	17.1980	0.804488	0.402244	+	0.915532i	$0.368230\pi$
$458$	0	0
$459$	14.0506	0.655826
$460$	0	0
$461$	−1.52073	−0.0708273	−0.0354136	−	0.999373i	$-0.511275\pi$
$461$	−1.52073	−0.0708273	−0.0354136	+	0.999373i	$0.511275\pi$
$462$	0	0
$463$	12.6968	0.590070	0.295035	−	0.955486i	$-0.404669\pi$
$463$	12.6968	0.590070	0.295035	+	0.955486i	$0.404669\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.97440	0.230188	0.115094	−	0.993355i	$-0.463283\pi$
$467$	4.97440	0.230188	0.115094	+	0.993355i	$0.463283\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	11.8169	0.544494
$472$	0	0
$473$	−0.450594	−0.0207183
$474$	0	0
$475$	0	0
$476$	0	0
$477$	16.5791	0.759104
$478$	0	0
$479$	−30.8363	−1.40895	−0.704474	−	0.709730i	$-0.748817\pi$
$479$	−30.8363	−1.40895	−0.704474	+	0.709730i	$0.748817\pi$
$480$	0	0
$481$	1.24072	0.0565722
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.8253	−1.66871	−0.834356	−	0.551226i	$-0.814160\pi$
$487$	−36.8253	−1.66871	−0.834356	+	0.551226i	$0.814160\pi$
$488$	0	0
$489$	47.5257	2.14919
$490$	0	0
$491$	−7.94806	−0.358691	−0.179345	−	0.983786i	$-0.557398\pi$
$491$	−7.94806	−0.358691	−0.179345	+	0.983786i	$0.557398\pi$
$492$	0	0
$493$	4.70968	0.212113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.92858	−0.0863349	−0.0431675	−	0.999068i	$-0.513745\pi$
$499$	−1.92858	−0.0863349	−0.0431675	+	0.999068i	$0.513745\pi$
$500$	0	0
$501$	35.3330	1.57856
$502$	0	0
$503$	−35.5389	−1.58460	−0.792301	−	0.610130i	$-0.791117\pi$
$503$	−35.5389	−1.58460	−0.792301	+	0.610130i	$0.791117\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−35.9850	−1.59815
$508$	0	0
$509$	−15.4039	−0.682767	−0.341384	−	0.939924i	$-0.610895\pi$
$509$	−15.4039	−0.682767	−0.341384	+	0.939924i	$0.610895\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−53.5096	−2.36251
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−24.2937	−1.06844
$518$	0	0
$519$	26.0809	1.14482
$520$	0	0
$521$	7.97708	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$521$	7.97708	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$522$	0	0
$523$	−9.86412	−0.431328	−0.215664	−	0.976468i	$-0.569192\pi$
$523$	−9.86412	−0.431328	−0.215664	+	0.976468i	$0.569192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.9997	−0.740520
$528$	0	0
$529$	30.7606	1.33742
$530$	0	0
$531$	65.7169	2.85187
$532$	0	0
$533$	1.03220	0.0447095
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.7533	0.593499
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13.6191	0.585533	0.292766	−	0.956184i	$-0.405424\pi$
$541$	13.6191	0.585533	0.292766	+	0.956184i	$0.405424\pi$
$542$	0	0
$543$	−52.8772	−2.26918
$544$	0	0
$545$	0	0
$546$	0	0
$547$	39.8299	1.70300	0.851501	−	0.524353i	$-0.175693\pi$
$547$	39.8299	1.70300	0.851501	+	0.524353i	$0.175693\pi$
$548$	0	0
$549$	−36.0479	−1.53849
$550$	0	0
$551$	−17.9361	−0.764103
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.5799	−1.38046	−0.690228	−	0.723592i	$-0.742490\pi$
$557$	−32.5799	−1.38046	−0.690228	+	0.723592i	$0.742490\pi$
$558$	0	0
$559$	0.150534	0.00636692
$560$	0	0
$561$	−12.9102	−0.545068
$562$	0	0
$563$	15.6233	0.658445	0.329222	−	0.944252i	$-0.393213\pi$
$563$	15.6233	0.658445	0.329222	+	0.944252i	$0.393213\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.32891	−0.391088	−0.195544	−	0.980695i	$-0.562647\pi$
$569$	−9.32891	−0.391088	−0.195544	+	0.980695i	$0.562647\pi$
$570$	0	0
$571$	−6.88073	−0.287949	−0.143975	−	0.989581i	$-0.545988\pi$
$571$	−6.88073	−0.287949	−0.143975	+	0.989581i	$0.545988\pi$
$572$	0	0
$573$	54.9596	2.29597
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.5238	−1.02094	−0.510470	−	0.859896i	$-0.670528\pi$
$577$	−24.5238	−1.02094	−0.510470	+	0.859896i	$0.670528\pi$
$578$	0	0
$579$	−62.6634	−2.60420
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.55532	−0.271494
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.2462	−0.546730	−0.273365	−	0.961910i	$-0.588137\pi$
$587$	−13.2462	−0.546730	−0.273365	+	0.961910i	$0.588137\pi$
$588$	0	0
$589$	64.7409	2.66760
$590$	0	0
$591$	5.07443	0.208734
$592$	0	0
$593$	−29.8957	−1.22767	−0.613834	−	0.789435i	$-0.710374\pi$
$593$	−29.8957	−1.22767	−0.613834	+	0.789435i	$0.710374\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.0119	−0.778104
$598$	0	0
$599$	25.4317	1.03911	0.519555	−	0.854437i	$-0.326098\pi$
$599$	25.4317	1.03911	0.519555	+	0.854437i	$0.326098\pi$
$600$	0	0
$601$	−0.815755	−0.0332754	−0.0166377	−	0.999862i	$-0.505296\pi$
$601$	−0.815755	−0.0332754	−0.0166377	+	0.999862i	$0.505296\pi$
$602$	0	0
$603$	69.1553	2.81622
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.49506	0.344804	0.172402	−	0.985027i	$-0.444847\pi$
$607$	8.49506	0.344804	0.172402	+	0.985027i	$0.444847\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.11603	0.328339
$612$	0	0
$613$	28.7972	1.16311	0.581553	−	0.813508i	$-0.302445\pi$
$613$	28.7972	1.16311	0.581553	+	0.813508i	$0.302445\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.1934	0.812956	0.406478	−	0.913661i	$-0.366757\pi$
$617$	20.1934	0.812956	0.406478	+	0.913661i	$0.366757\pi$
$618$	0	0
$619$	30.8868	1.24145	0.620723	−	0.784030i	$-0.286839\pi$
$619$	30.8868	1.24145	0.620723	+	0.784030i	$0.286839\pi$
$620$	0	0
$621$	47.7723	1.91703
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	49.1664	1.96352
$628$	0	0
$629$	3.84635	0.153364
$630$	0	0
$631$	−21.0638	−0.838537	−0.419269	−	0.907862i	$-0.637713\pi$
$631$	−21.0638	−0.838537	−0.419269	+	0.907862i	$0.637713\pi$
$632$	0	0
$633$	−36.5138	−1.45129
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	54.3254	2.14908
$640$	0	0
$641$	27.1287	1.07152	0.535760	−	0.844370i	$-0.320025\pi$
$641$	27.1287	1.07152	0.535760	+	0.844370i	$0.320025\pi$
$642$	0	0
$643$	−13.3862	−0.527901	−0.263951	−	0.964536i	$-0.585026\pi$
$643$	−13.3862	−0.527901	−0.263951	+	0.964536i	$0.585026\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.8982	0.467768	0.233884	−	0.972265i	$-0.424856\pi$
$647$	11.8982	0.467768	0.233884	+	0.972265i	$0.424856\pi$
$648$	0	0
$649$	−25.9843	−1.01997
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.1252	1.06149	0.530746	−	0.847531i	$-0.321912\pi$
$653$	27.1252	1.06149	0.530746	+	0.847531i	$0.321912\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	76.7028	2.99246
$658$	0	0
$659$	32.0463	1.24834	0.624172	−	0.781287i	$-0.285436\pi$
$659$	32.0463	1.24834	0.624172	+	0.781287i	$0.285436\pi$
$660$	0	0
$661$	−15.5074	−0.603166	−0.301583	−	0.953440i	$-0.597515\pi$
$661$	−15.5074	−0.603166	−0.301583	+	0.953440i	$0.597515\pi$
$662$	0	0
$663$	4.31302	0.167504
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0130	0.620025
$668$	0	0
$669$	−80.6045	−3.11635
$670$	0	0
$671$	14.2532	0.550240
$672$	0	0
$673$	27.0833	1.04398	0.521992	−	0.852950i	$-0.325189\pi$
$673$	27.0833	1.04398	0.521992	+	0.852950i	$0.325189\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.32000	−0.319764	−0.159882	−	0.987136i	$-0.551111\pi$
$677$	−8.32000	−0.319764	−0.159882	+	0.987136i	$0.551111\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	68.0914	2.60927
$682$	0	0
$683$	−13.8288	−0.529144	−0.264572	−	0.964366i	$-0.585231\pi$
$683$	−13.8288	−0.529144	−0.264572	+	0.964366i	$0.585231\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	41.6762	1.59005
$688$	0	0
$689$	2.19000	0.0834323
$690$	0	0
$691$	43.3621	1.64957	0.824785	−	0.565446i	$-0.191296\pi$
$691$	43.3621	1.64957	0.824785	+	0.565446i	$0.191296\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.19991	0.121205
$698$	0	0
$699$	−36.8007	−1.39193
$700$	0	0
$701$	−6.76821	−0.255632	−0.127816	−	0.991798i	$-0.540797\pi$
$701$	−6.76821	−0.255632	−0.127816	+	0.991798i	$0.540797\pi$
$702$	0	0
$703$	−14.6482	−0.552469
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9.46981	−0.355646	−0.177823	−	0.984062i	$-0.556905\pi$
$709$	−9.46981	−0.355646	−0.177823	+	0.984062i	$0.556905\pi$
$710$	0	0
$711$	57.5688	2.15900
$712$	0	0
$713$	−57.7993	−2.16460
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−87.3377	−3.26169
$718$	0	0
$719$	36.2805	1.35303	0.676517	−	0.736427i	$-0.263489\pi$
$719$	36.2805	1.35303	0.676517	+	0.736427i	$0.263489\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−13.8615	−0.515514
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−32.7840	−1.21589	−0.607945	−	0.793979i	$-0.708006\pi$
$727$	−32.7840	−1.21589	−0.607945	+	0.793979i	$0.708006\pi$
$728$	0	0
$729$	−40.7466	−1.50913
$730$	0	0
$731$	0.466669	0.0172604
$732$	0	0
$733$	−35.0673	−1.29524	−0.647620	−	0.761964i	$-0.724236\pi$
$733$	−35.0673	−1.29524	−0.647620	+	0.761964i	$0.724236\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.3438	−1.00722
$738$	0	0
$739$	1.97105	0.0725062	0.0362531	−	0.999343i	$-0.488458\pi$
$739$	1.97105	0.0725062	0.0362531	+	0.999343i	$0.488458\pi$
$740$	0	0
$741$	−16.4255	−0.603405
$742$	0	0
$743$	−26.3124	−0.965309	−0.482655	−	0.875811i	$-0.660327\pi$
$743$	−26.3124	−0.965309	−0.482655	+	0.875811i	$0.660327\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.18399	0.336025
$748$	0	0
$749$	0	0
$750$	0	0
$751$	35.0278	1.27818	0.639092	−	0.769130i	$-0.279310\pi$
$751$	35.0278	1.27818	0.639092	+	0.769130i	$0.279310\pi$
$752$	0	0
$753$	−37.6976	−1.37378
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.1958	0.406919	0.203459	−	0.979083i	$-0.434782\pi$
$757$	11.1958	0.406919	0.203459	+	0.979083i	$0.434782\pi$
$758$	0	0
$759$	−43.8947	−1.59328
$760$	0	0
$761$	−43.4437	−1.57483	−0.787417	−	0.616421i	$-0.788582\pi$
$761$	−43.4437	−1.57483	−0.787417	+	0.616421i	$0.788582\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.68082	0.313446
$768$	0	0
$769$	−32.3819	−1.16772	−0.583861	−	0.811853i	$-0.698459\pi$
$769$	−32.3819	−1.16772	−0.583861	+	0.811853i	$0.698459\pi$
$770$	0	0
$771$	−44.4109	−1.59942
$772$	0	0
$773$	42.0334	1.51183	0.755917	−	0.654667i	$-0.227191\pi$
$773$	42.0334	1.51183	0.755917	+	0.654667i	$0.227191\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.1863	−0.436621
$780$	0	0
$781$	−21.4801	−0.768619
$782$	0	0
$783$	14.2293	0.508514
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.1451	−0.468571	−0.234285	−	0.972168i	$-0.575275\pi$
$787$	−13.1451	−0.468571	−0.234285	+	0.972168i	$0.575275\pi$
$788$	0	0
$789$	−16.7217	−0.595309
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.76171	−0.169093
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−56.1776	−1.98991	−0.994956	−	0.100309i	$-0.968017\pi$
$797$	−56.1776	−1.98991	−0.994956	+	0.100309i	$0.968017\pi$
$798$	0	0
$799$	25.1604	0.890110
$800$	0	0
$801$	60.9295	2.15284
$802$	0	0
$803$	−30.3281	−1.07026
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.5280	−0.933829
$808$	0	0
$809$	−14.0195	−0.492899	−0.246449	−	0.969156i	$-0.579264\pi$
$809$	−14.0195	−0.492899	−0.246449	+	0.969156i	$0.579264\pi$
$810$	0	0
$811$	−11.0241	−0.387110	−0.193555	−	0.981089i	$-0.562002\pi$
$811$	−11.0241	−0.387110	−0.193555	+	0.981089i	$0.562002\pi$
$812$	0	0
$813$	−45.4931	−1.59551
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.77724	−0.0621776
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.6965	−1.31562	−0.657809	−	0.753185i	$-0.728517\pi$
$821$	−37.6965	−1.31562	−0.657809	+	0.753185i	$0.728517\pi$
$822$	0	0
$823$	53.1931	1.85420	0.927098	−	0.374818i	$-0.122295\pi$
$823$	53.1931	1.85420	0.927098	+	0.374818i	$0.122295\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.3451	1.19430	0.597149	−	0.802130i	$-0.296300\pi$
$827$	34.3451	1.19430	0.597149	+	0.802130i	$0.296300\pi$
$828$	0	0
$829$	−33.0640	−1.14836	−0.574179	−	0.818730i	$-0.694679\pi$
$829$	−33.0640	−1.14836	−0.574179	+	0.818730i	$0.694679\pi$
$830$	0	0
$831$	−4.12635	−0.143142
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−51.3611	−1.77530
$838$	0	0
$839$	4.04312	0.139584	0.0697920	−	0.997562i	$-0.477766\pi$
$839$	4.04312	0.139584	0.0697920	+	0.997562i	$0.477766\pi$
$840$	0	0
$841$	−24.2304	−0.835532
$842$	0	0
$843$	−42.0120	−1.44697
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	26.1298	0.896773
$850$	0	0
$851$	13.0776	0.448296
$852$	0	0
$853$	−41.1970	−1.41056	−0.705279	−	0.708930i	$-0.749178\pi$
$853$	−41.1970	−1.41056	−0.705279	+	0.708930i	$0.749178\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.5978	0.464492	0.232246	−	0.972657i	$-0.425393\pi$
$857$	13.5978	0.464492	0.232246	+	0.972657i	$0.425393\pi$
$858$	0	0
$859$	52.8276	1.80245	0.901227	−	0.433348i	$-0.142668\pi$
$859$	52.8276	1.80245	0.901227	+	0.433348i	$0.142668\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.8810	−0.404434	−0.202217	−	0.979341i	$-0.564815\pi$
$863$	−11.8810	−0.404434	−0.202217	+	0.979341i	$0.564815\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−35.5059	−1.20584
$868$	0	0
$869$	−22.7626	−0.772167
$870$	0	0
$871$	9.13500	0.309528
$872$	0	0
$873$	16.5760	0.561013
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.31509	−0.145710	−0.0728551	−	0.997343i	$-0.523211\pi$
$877$	−4.31509	−0.145710	−0.0728551	+	0.997343i	$0.523211\pi$
$878$	0	0
$879$	18.1320	0.611577
$880$	0	0
$881$	−0.411033	−0.0138481	−0.00692403	−	0.999976i	$-0.502204\pi$
$881$	−0.411033	−0.0138481	−0.00692403	+	0.999976i	$0.502204\pi$
$882$	0	0
$883$	−38.8610	−1.30778	−0.653888	−	0.756591i	$-0.726863\pi$
$883$	−38.8610	−1.30778	−0.653888	+	0.756591i	$0.726863\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.7530	−0.596089	−0.298044	−	0.954552i	$-0.596334\pi$
$887$	−17.7530	−0.596089	−0.298044	+	0.954552i	$0.596334\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.10926	−0.204668
$892$	0	0
$893$	−95.8194	−3.20647
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.6643	0.489628
$898$	0	0
$899$	−17.2159	−0.574184
$900$	0	0
$901$	6.78918	0.226180
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−20.9997	−0.697285	−0.348642	−	0.937256i	$-0.613357\pi$
$907$	−20.9997	−0.697285	−0.348642	+	0.937256i	$0.613357\pi$
$908$	0	0
$909$	−69.6914	−2.31152
$910$	0	0
$911$	−10.8994	−0.361112	−0.180556	−	0.983565i	$-0.557790\pi$
$911$	−10.8994	−0.361112	−0.180556	+	0.983565i	$0.557790\pi$
$912$	0	0
$913$	−3.63133	−0.120179
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−42.0060	−1.38565	−0.692824	−	0.721106i	$-0.743634\pi$
$919$	−42.0060	−1.38565	−0.692824	+	0.721106i	$0.743634\pi$
$920$	0	0
$921$	49.0246	1.61542
$922$	0	0
$923$	7.17607	0.236203
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.95673	−0.195645
$928$	0	0
$929$	4.08833	0.134134	0.0670669	−	0.997748i	$-0.478636\pi$
$929$	4.08833	0.134134	0.0670669	+	0.997748i	$0.478636\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−30.6319	−1.00284
$934$	0	0
$935$	0	0
$936$	0	0
$937$	47.3538	1.54698	0.773490	−	0.633808i	$-0.218509\pi$
$937$	47.3538	1.54698	0.773490	+	0.633808i	$0.218509\pi$
$938$	0	0
$939$	5.56177	0.181502
$940$	0	0
$941$	−12.9278	−0.421433	−0.210717	−	0.977547i	$-0.567580\pi$
$941$	−12.9278	−0.421433	−0.210717	+	0.977547i	$0.567580\pi$
$942$	0	0
$943$	10.8797	0.354292
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.6989	1.38753	0.693764	−	0.720202i	$-0.255951\pi$
$947$	42.6989	1.38753	0.693764	+	0.720202i	$0.255951\pi$
$948$	0	0
$949$	10.1320	0.328898
$950$	0	0
$951$	43.0237	1.39514
$952$	0	0
$953$	−13.8193	−0.447651	−0.223826	−	0.974629i	$-0.571855\pi$
$953$	−13.8193	−0.447651	−0.223826	+	0.974629i	$0.571855\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.0744	−0.422634
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.1415	1.00456
$962$	0	0
$963$	75.6465	2.43767
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.8612	0.799484	0.399742	−	0.916628i	$-0.369100\pi$
$967$	24.8612	0.799484	0.399742	+	0.916628i	$0.369100\pi$
$968$	0	0
$969$	−50.9203	−1.63580
$970$	0	0
$971$	−11.0871	−0.355801	−0.177901	−	0.984048i	$-0.556931\pi$
$971$	−11.0871	−0.355801	−0.177901	+	0.984048i	$0.556931\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.0917	0.642790	0.321395	−	0.946945i	$-0.395848\pi$
$977$	20.0917	0.642790	0.321395	+	0.946945i	$0.395848\pi$
$978$	0	0
$979$	−24.0914	−0.769964
$980$	0	0
$981$	91.9691	2.93635
$982$	0	0
$983$	5.12683	0.163520	0.0817602	−	0.996652i	$-0.473946\pi$
$983$	5.12683	0.163520	0.0817602	+	0.996652i	$0.473946\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.58668	0.0504535
$990$	0	0
$991$	−48.1082	−1.52821	−0.764103	−	0.645094i	$-0.776818\pi$
$991$	−48.1082	−1.52821	−0.764103	+	0.645094i	$0.776818\pi$
$992$	0	0
$993$	49.6861	1.57674
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.78201	−0.309799	−0.154900	−	0.987930i	$-0.549505\pi$
$997$	−9.78201	−0.309799	−0.154900	+	0.987930i	$0.549505\pi$
$998$	0	0
$999$	11.6209	0.367670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cz.1.7	yes	8
5.4	even	2		9800.2.a.da.1.2	yes	8
7.6	odd	2	inner	9800.2.a.cz.1.2	✓	8
35.34	odd	2		9800.2.a.da.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9800.2.a.cz.1.2	✓	8	7.6	odd	2	inner
9800.2.a.cz.1.7	yes	8	1.1	even	1	trivial
9800.2.a.da.1.2	yes	8	5.4	even	2
9800.2.a.da.1.7	yes	8	35.34	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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87509 q^{3} +5.26617 q^{9} +O(q^{10})\)	\(q+2.87509 q^{3} +5.26617 q^{9} -2.08223 q^{11} +0.695629 q^{13} +2.15651 q^{17} -8.21273 q^{19} +7.33216 q^{23} +6.51544 q^{27} +2.18394 q^{29} -7.88299 q^{31} -5.98660 q^{33} +1.78360 q^{37} +2.00000 q^{39} +1.48384 q^{41} +0.216400 q^{43} +11.6672 q^{47} +6.20017 q^{51} +3.14823 q^{53} -23.6124 q^{57} +12.4791 q^{59} -6.84519 q^{61} +13.1320 q^{67} +21.0807 q^{69} +10.3159 q^{71} +14.5652 q^{73} +10.9318 q^{79} +2.93400 q^{81} +1.74396 q^{83} +6.27902 q^{87} +11.5700 q^{89} -22.6643 q^{93} +3.14764 q^{97} -10.9654 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9}+O(q^{10}) \) 8 * q + 12 * q^9	\( 8 q + 12 q^{9} + 4 q^{11} - 4 q^{23} + 8 q^{29} + 16 q^{39} + 16 q^{43} + 52 q^{51} - 28 q^{53} - 8 q^{57} + 40 q^{67} + 8 q^{71} + 20 q^{79} + 56 q^{81} - 56 q^{93} + 36 q^{99}+O(q^{100}) \) 8 * q + 12 * q^9 + 4 * q^11 - 4 * q^23 + 8 * q^29 + 16 * q^39 + 16 * q^43 + 52 * q^51 - 28 * q^53 - 8 * q^57 + 40 * q^67 + 8 * q^71 + 20 * q^79 + 56 * q^81 - 56 * q^93 + 36 * q^99

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