LMFDB - Embedded newform 1568.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.874032$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16228 q^{3} +2.82843 q^{5} +7.00000 q^{9} +O(q^{10})$	$q-3.16228 q^{3} +2.82843 q^{5} +7.00000 q^{9} +4.47214 q^{11} -8.94427 q^{15} +4.24264 q^{17} -3.16228 q^{19} +8.94427 q^{23} +3.00000 q^{25} -12.6491 q^{27} -6.00000 q^{29} -6.32456 q^{31} -14.1421 q^{33} +2.00000 q^{37} +1.41421 q^{41} -4.47214 q^{43} +19.7990 q^{45} +6.32456 q^{47} -13.4164 q^{51} +6.00000 q^{53} +12.6491 q^{55} +10.0000 q^{57} -3.16228 q^{59} -8.48528 q^{61} -28.2843 q^{69} +8.94427 q^{71} -7.07107 q^{73} -9.48683 q^{75} +8.94427 q^{79} +19.0000 q^{81} +9.48683 q^{83} +12.0000 q^{85} +18.9737 q^{87} +9.89949 q^{89} +20.0000 q^{93} -8.94427 q^{95} +4.24264 q^{97} +31.3050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{9}+O(q^{10}) $ 4 * q + 28 * q^9	$ 4 q + 28 q^{9} + 12 q^{25} - 24 q^{29} + 8 q^{37} + 24 q^{53} + 40 q^{57} + 76 q^{81} + 48 q^{85} + 80 q^{93}+O(q^{100}) $ 4 * q + 28 * q^9 + 12 * q^25 - 24 * q^29 + 8 * q^37 + 24 * q^53 + 40 * q^57 + 76 * q^81 + 48 * q^85 + 80 * q^93

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
$3$	−3.16228	−1.82574	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	−3.16228	−1.82574	−0.912871	+	0.408248i	$0.866140\pi$
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.00000	2.33333
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−8.94427	−2.30940
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−3.16228	−0.725476	−0.362738	−	0.931891i	$-0.618158\pi$
$19$	−3.16228	−0.725476	−0.362738	+	0.931891i	$0.618158\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.94427	1.86501	0.932505	−	0.361158i	$-0.117618\pi$
$23$	8.94427	1.86501	0.932505	+	0.361158i	$0.117618\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−12.6491	−2.43432
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−6.32456	−1.13592	−0.567962	−	0.823055i	$-0.692268\pi$
$31$	−6.32456	−1.13592	−0.567962	+	0.823055i	$0.692268\pi$
$32$	0	0
$33$	−14.1421	−2.46183
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	0	0
$43$	−4.47214	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$43$	−4.47214	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$44$	0	0
$45$	19.7990	2.95146
$46$	0	0
$47$	6.32456	0.922531	0.461266	−	0.887262i	$-0.347396\pi$
$47$	6.32456	0.922531	0.461266	+	0.887262i	$0.347396\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−13.4164	−1.87867
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	12.6491	1.70561
$56$	0	0
$57$	10.0000	1.32453
$58$	0	0
$59$	−3.16228	−0.411693	−0.205847	−	0.978584i	$-0.565995\pi$
$59$	−3.16228	−0.411693	−0.205847	+	0.978584i	$0.565995\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−28.2843	−3.40503
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	−7.07107	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$73$	−7.07107	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$74$	0	0
$75$	−9.48683	−1.09545
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	19.0000	2.11111
$82$	0	0
$83$	9.48683	1.04132	0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683	1.04132	0.520658	+	0.853766i	$0.325687\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	18.9737	2.03419
$88$	0	0
$89$	9.89949	1.04934	0.524672	−	0.851304i	$-0.324188\pi$
$89$	9.89949	1.04934	0.524672	+	0.851304i	$0.324188\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	20.0000	2.07390
$94$	0	0
$95$	−8.94427	−0.917663
$96$	0	0
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	0	0
$99$	31.3050	3.14627
$100$	0	0
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	0	0
$103$	6.32456	0.623177	0.311588	−	0.950217i	$-0.399139\pi$
$103$	6.32456	0.623177	0.311588	+	0.950217i	$0.399139\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−6.32456	−0.600300
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	25.2982	2.35907
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	−4.47214	−0.403239
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	14.1421	1.24515
$130$	0	0
$131$	15.8114	1.38145	0.690724	−	0.723119i	$-0.257292\pi$
$131$	15.8114	1.38145	0.690724	+	0.723119i	$0.257292\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−35.7771	−3.07920
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	9.48683	0.804663	0.402331	−	0.915494i	$-0.368200\pi$
$139$	9.48683	0.804663	0.402331	+	0.915494i	$0.368200\pi$
$140$	0	0
$141$	−20.0000	−1.68430
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−16.9706	−1.40933
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−17.8885	−1.45575	−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885	−1.45575	−0.727875	+	0.685710i	$0.759492\pi$
$152$	0	0
$153$	29.6985	2.40098
$154$	0	0
$155$	−17.8885	−1.43684
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−18.9737	−1.50471
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.4164	1.05085	0.525427	−	0.850839i	$-0.323906\pi$
$163$	13.4164	1.05085	0.525427	+	0.850839i	$0.323906\pi$
$164$	0	0
$165$	−40.0000	−3.11400
$166$	0	0
$167$	6.32456	0.489409	0.244704	−	0.969598i	$-0.421309\pi$
$167$	6.32456	0.489409	0.244704	+	0.969598i	$0.421309\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−22.1359	−1.69278
$172$	0	0
$173$	5.65685	0.430083	0.215041	−	0.976605i	$-0.431011\pi$
$173$	5.65685	0.430083	0.215041	+	0.976605i	$0.431011\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.0000	0.751646
$178$	0	0
$179$	−17.8885	−1.33705	−0.668526	−	0.743689i	$-0.733075\pi$
$179$	−17.8885	−1.33705	−0.668526	+	0.743689i	$0.733075\pi$
$180$	0	0
$181$	22.6274	1.68188	0.840941	−	0.541126i	$-0.182002\pi$
$181$	22.6274	1.68188	0.840941	+	0.541126i	$0.182002\pi$
$182$	0	0
$183$	26.8328	1.98354
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	18.9737	1.38749
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	−6.32456	−0.448336	−0.224168	−	0.974551i	$-0.571966\pi$
$199$	−6.32456	−0.448336	−0.224168	+	0.974551i	$0.571966\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	62.6099	4.35169
$208$	0	0
$209$	−14.1421	−0.978232
$210$	0	0
$211$	−17.8885	−1.23150	−0.615749	−	0.787942i	$-0.711146\pi$
$211$	−17.8885	−1.23150	−0.615749	+	0.787942i	$0.711146\pi$
$212$	0	0
$213$	−28.2843	−1.93801
$214$	0	0
$215$	−12.6491	−0.862662
$216$	0	0
$217$	0	0
$218$	0	0
$219$	22.3607	1.51099
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−25.2982	−1.69409	−0.847047	−	0.531518i	$-0.821622\pi$
$223$	−25.2982	−1.69409	−0.847047	+	0.531518i	$0.821622\pi$
$224$	0	0
$225$	21.0000	1.40000
$226$	0	0
$227$	15.8114	1.04944	0.524719	−	0.851275i	$-0.324170\pi$
$227$	15.8114	1.04944	0.524719	+	0.851275i	$0.324170\pi$
$228$	0	0
$229$	16.9706	1.12145	0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706	1.12145	0.560723	+	0.828003i	$0.310523\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	17.8885	1.16692
$236$	0	0
$237$	−28.2843	−1.83726
$238$	0	0
$239$	−26.8328	−1.73567	−0.867835	−	0.496852i	$-0.834489\pi$
$239$	−26.8328	−1.73567	−0.867835	+	0.496852i	$0.834489\pi$
$240$	0	0
$241$	−26.8701	−1.73085	−0.865426	−	0.501036i	$-0.832952\pi$
$241$	−26.8701	−1.73085	−0.865426	+	0.501036i	$0.832952\pi$
$242$	0	0
$243$	−22.1359	−1.42002
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−30.0000	−1.90117
$250$	0	0
$251$	9.48683	0.598804	0.299402	−	0.954127i	$-0.403213\pi$
$251$	9.48683	0.598804	0.299402	+	0.954127i	$0.403213\pi$
$252$	0	0
$253$	40.0000	2.51478
$254$	0	0
$255$	−37.9473	−2.37635
$256$	0	0
$257$	24.0416	1.49968	0.749838	−	0.661622i	$-0.230131\pi$
$257$	24.0416	1.49968	0.749838	+	0.661622i	$0.230131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−42.0000	−2.59973
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	16.9706	1.04249
$266$	0	0
$267$	−31.3050	−1.91583
$268$	0	0
$269$	11.3137	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.4164	0.809040
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−44.2719	−2.65049
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	−28.4605	−1.69180	−0.845901	−	0.533341i	$-0.820936\pi$
$283$	−28.4605	−1.69180	−0.845901	+	0.533341i	$0.820936\pi$
$284$	0	0
$285$	28.2843	1.67542
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−13.4164	−0.786484
$292$	0	0
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	−8.94427	−0.520756
$296$	0	0
$297$	−56.5685	−3.28244
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−26.8328	−1.54150
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	15.8114	0.902404	0.451202	−	0.892422i	$-0.350995\pi$
$307$	15.8114	0.902404	0.451202	+	0.892422i	$0.350995\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−21.2132	−1.19904	−0.599521	−	0.800359i	$-0.704642\pi$
$313$	−21.2132	−1.19904	−0.599521	+	0.800359i	$0.704642\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−26.8328	−1.50235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.4164	−0.746509
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.6228	1.74874
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.4164	0.737432	0.368716	−	0.929542i	$-0.379797\pi$
$331$	13.4164	0.737432	0.368716	+	0.929542i	$0.379797\pi$
$332$	0	0
$333$	14.0000	0.767195
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	−12.6491	−0.687005
$340$	0	0
$341$	−28.2843	−1.53168
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−80.0000	−4.30706
$346$	0	0
$347$	31.3050	1.68054	0.840269	−	0.542170i	$-0.182397\pi$
$347$	31.3050	1.68054	0.840269	+	0.542170i	$0.182397\pi$
$348$	0	0
$349$	11.3137	0.605609	0.302804	−	0.953053i	$-0.402077\pi$
$349$	11.3137	0.605609	0.302804	+	0.953053i	$0.402077\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.41421	−0.0752710	−0.0376355	−	0.999292i	$-0.511983\pi$
$353$	−1.41421	−0.0752710	−0.0376355	+	0.999292i	$0.511983\pi$
$354$	0	0
$355$	25.2982	1.34269
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.8885	0.944121	0.472061	−	0.881566i	$-0.343510\pi$
$359$	17.8885	0.944121	0.472061	+	0.881566i	$0.343510\pi$
$360$	0	0
$361$	−9.00000	−0.473684
$362$	0	0
$363$	−28.4605	−1.49379
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	12.6491	0.660278	0.330139	−	0.943932i	$-0.392904\pi$
$367$	12.6491	0.660278	0.330139	+	0.943932i	$0.392904\pi$
$368$	0	0
$369$	9.89949	0.515347
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	17.8885	0.923760
$376$	0	0
$377$	0	0
$378$	0	0
$379$	22.3607	1.14859	0.574295	−	0.818648i	$-0.305276\pi$
$379$	22.3607	1.14859	0.574295	+	0.818648i	$0.305276\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.6228	−1.61585	−0.807924	−	0.589286i	$-0.799409\pi$
$383$	−31.6228	−1.61585	−0.807924	+	0.589286i	$0.799409\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−31.3050	−1.59132
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	37.9473	1.91908
$392$	0	0
$393$	−50.0000	−2.52217
$394$	0	0
$395$	25.2982	1.27289
$396$	0	0
$397$	−11.3137	−0.567819	−0.283909	−	0.958851i	$-0.591631\pi$
$397$	−11.3137	−0.567819	−0.283909	+	0.958851i	$0.591631\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	53.7401	2.67037
$406$	0	0
$407$	8.94427	0.443351
$408$	0	0
$409$	−18.3848	−0.909069	−0.454534	−	0.890729i	$-0.650194\pi$
$409$	−18.3848	−0.909069	−0.454534	+	0.890729i	$0.650194\pi$
$410$	0	0
$411$	−37.9473	−1.87180
$412$	0	0
$413$	0	0
$414$	0	0
$415$	26.8328	1.31717
$416$	0	0
$417$	−30.0000	−1.46911
$418$	0	0
$419$	3.16228	0.154487	0.0772437	−	0.997012i	$-0.475388\pi$
$419$	3.16228	0.154487	0.0772437	+	0.997012i	$0.475388\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	44.2719	2.15257
$424$	0	0
$425$	12.7279	0.617395
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.94427	−0.430830	−0.215415	−	0.976523i	$-0.569110\pi$
$431$	−8.94427	−0.430830	−0.215415	+	0.976523i	$0.569110\pi$
$432$	0	0
$433$	21.2132	1.01944	0.509721	−	0.860340i	$-0.329749\pi$
$433$	21.2132	1.01944	0.509721	+	0.860340i	$0.329749\pi$
$434$	0	0
$435$	53.6656	2.57307
$436$	0	0
$437$	−28.2843	−1.35302
$438$	0	0
$439$	−37.9473	−1.81113	−0.905564	−	0.424210i	$-0.860552\pi$
$439$	−37.9473	−1.81113	−0.905564	+	0.424210i	$0.860552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.8885	−0.849910	−0.424955	−	0.905214i	$-0.639710\pi$
$443$	−17.8885	−0.849910	−0.424955	+	0.905214i	$0.639710\pi$
$444$	0	0
$445$	28.0000	1.32733
$446$	0	0
$447$	44.2719	2.09399
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	6.32456	0.297812
$452$	0	0
$453$	56.5685	2.65782
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	−53.6656	−2.50490
$460$	0	0
$461$	5.65685	0.263466	0.131733	−	0.991285i	$-0.457946\pi$
$461$	5.65685	0.263466	0.131733	+	0.991285i	$0.457946\pi$
$462$	0	0
$463$	17.8885	0.831351	0.415676	−	0.909513i	$-0.363545\pi$
$463$	17.8885	0.831351	0.415676	+	0.909513i	$0.363545\pi$
$464$	0	0
$465$	56.5685	2.62330
$466$	0	0
$467$	34.7851	1.60966	0.804830	−	0.593505i	$-0.202256\pi$
$467$	34.7851	1.60966	0.804830	+	0.593505i	$0.202256\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−20.0000	−0.919601
$474$	0	0
$475$	−9.48683	−0.435286
$476$	0	0
$477$	42.0000	1.92305
$478$	0	0
$479$	−18.9737	−0.866929	−0.433464	−	0.901171i	$-0.642709\pi$
$479$	−18.9737	−0.866929	−0.433464	+	0.901171i	$0.642709\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	8.94427	0.405304	0.202652	−	0.979251i	$-0.435044\pi$
$487$	8.94427	0.405304	0.202652	+	0.979251i	$0.435044\pi$
$488$	0	0
$489$	−42.4264	−1.91859
$490$	0	0
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	0	0
$493$	−25.4558	−1.14647
$494$	0	0
$495$	88.5438	3.97975
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−35.7771	−1.60160	−0.800801	−	0.598930i	$-0.795593\pi$
$499$	−35.7771	−1.60160	−0.800801	+	0.598930i	$0.795593\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	0	0
$503$	37.9473	1.69199	0.845994	−	0.533192i	$-0.179008\pi$
$503$	37.9473	1.69199	0.845994	+	0.533192i	$0.179008\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	41.1096	1.82574
$508$	0	0
$509$	−11.3137	−0.501471	−0.250736	−	0.968056i	$-0.580672\pi$
$509$	−11.3137	−0.501471	−0.250736	+	0.968056i	$0.580672\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	40.0000	1.76604
$514$	0	0
$515$	17.8885	0.788263
$516$	0	0
$517$	28.2843	1.24394
$518$	0	0
$519$	−17.8885	−0.785220
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	0	0
$523$	3.16228	0.138277	0.0691384	−	0.997607i	$-0.477975\pi$
$523$	3.16228	0.138277	0.0691384	+	0.997607i	$0.477975\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.8328	−1.16886
$528$	0	0
$529$	57.0000	2.47826
$530$	0	0
$531$	−22.1359	−0.960618
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	56.5685	2.44111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	−71.5542	−3.07068
$544$	0	0
$545$	−28.2843	−1.21157
$546$	0	0
$547$	−13.4164	−0.573644	−0.286822	−	0.957984i	$-0.592599\pi$
$547$	−13.4164	−0.573644	−0.286822	+	0.957984i	$0.592599\pi$
$548$	0	0
$549$	−59.3970	−2.53500
$550$	0	0
$551$	18.9737	0.808305
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−17.8885	−0.759326
$556$	0	0
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−60.0000	−2.53320
$562$	0	0
$563$	−28.4605	−1.19947	−0.599734	−	0.800200i	$-0.704727\pi$
$563$	−28.4605	−1.19947	−0.599734	+	0.800200i	$0.704727\pi$
$564$	0	0
$565$	11.3137	0.475971
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	4.47214	0.187153	0.0935765	−	0.995612i	$-0.470170\pi$
$571$	4.47214	0.187153	0.0935765	+	0.995612i	$0.470170\pi$
$572$	0	0
$573$	−28.2843	−1.18159
$574$	0	0
$575$	26.8328	1.11901
$576$	0	0
$577$	35.3553	1.47186	0.735931	−	0.677057i	$-0.236745\pi$
$577$	35.3553	1.47186	0.735931	+	0.677057i	$0.236745\pi$
$578$	0	0
$579$	−75.8947	−3.15407
$580$	0	0
$581$	0	0
$582$	0	0
$583$	26.8328	1.11130
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.16228	−0.130521	−0.0652606	−	0.997868i	$-0.520788\pi$
$587$	−3.16228	−0.130521	−0.0652606	+	0.997868i	$0.520788\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	69.5701	2.86173
$592$	0	0
$593$	−7.07107	−0.290374	−0.145187	−	0.989404i	$-0.546378\pi$
$593$	−7.07107	−0.290374	−0.145187	+	0.989404i	$0.546378\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	−35.7771	−1.46181	−0.730906	−	0.682478i	$-0.760902\pi$
$599$	−35.7771	−1.46181	−0.730906	+	0.682478i	$0.760902\pi$
$600$	0	0
$601$	12.7279	0.519183	0.259591	−	0.965719i	$-0.416412\pi$
$601$	12.7279	0.519183	0.259591	+	0.965719i	$0.416412\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.4558	1.03493
$606$	0	0
$607$	−12.6491	−0.513412	−0.256706	−	0.966490i	$-0.582637\pi$
$607$	−12.6491	−0.513412	−0.256706	+	0.966490i	$0.582637\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	−12.6491	−0.510061
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	15.8114	0.635513	0.317757	−	0.948172i	$-0.397070\pi$
$619$	15.8114	0.635513	0.317757	+	0.948172i	$0.397070\pi$
$620$	0	0
$621$	−113.137	−4.54003
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	44.7214	1.78600
$628$	0	0
$629$	8.48528	0.338330
$630$	0	0
$631$	−8.94427	−0.356066	−0.178033	−	0.984025i	$-0.556973\pi$
$631$	−8.94427	−0.356066	−0.178033	+	0.984025i	$0.556973\pi$
$632$	0	0
$633$	56.5685	2.24840
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	62.6099	2.47681
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	−3.16228	−0.124708	−0.0623540	−	0.998054i	$-0.519861\pi$
$643$	−3.16228	−0.124708	−0.0623540	+	0.998054i	$0.519861\pi$
$644$	0	0
$645$	40.0000	1.57500
$646$	0	0
$647$	−31.6228	−1.24322	−0.621610	−	0.783327i	$-0.713521\pi$
$647$	−31.6228	−1.24322	−0.621610	+	0.783327i	$0.713521\pi$
$648$	0	0
$649$	−14.1421	−0.555127
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	44.7214	1.74741
$656$	0	0
$657$	−49.4975	−1.93108
$658$	0	0
$659$	4.47214	0.174210	0.0871048	−	0.996199i	$-0.472238\pi$
$659$	4.47214	0.174210	0.0871048	+	0.996199i	$0.472238\pi$
$660$	0	0
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−53.6656	−2.07794
$668$	0	0
$669$	80.0000	3.09298
$670$	0	0
$671$	−37.9473	−1.46494
$672$	0	0
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	0	0
$675$	−37.9473	−1.46059
$676$	0	0
$677$	−28.2843	−1.08705	−0.543526	−	0.839392i	$-0.682911\pi$
$677$	−28.2843	−1.08705	−0.543526	+	0.839392i	$0.682911\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−50.0000	−1.91600
$682$	0	0
$683$	17.8885	0.684486	0.342243	−	0.939611i	$-0.388813\pi$
$683$	17.8885	0.684486	0.342243	+	0.939611i	$0.388813\pi$
$684$	0	0
$685$	33.9411	1.29682
$686$	0	0
$687$	−53.6656	−2.04747
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−9.48683	−0.360896	−0.180448	−	0.983585i	$-0.557755\pi$
$691$	−9.48683	−0.360896	−0.180448	+	0.983585i	$0.557755\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26.8328	1.01783
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	50.5964	1.91373
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−6.32456	−0.238535
$704$	0	0
$705$	−56.5685	−2.13049
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	62.6099	2.34805
$712$	0	0
$713$	−56.5685	−2.11851
$714$	0	0
$715$	0	0
$716$	0	0
$717$	84.8528	3.16889
$718$	0	0
$719$	6.32456	0.235866	0.117933	−	0.993022i	$-0.462373\pi$
$719$	6.32456	0.235866	0.117933	+	0.993022i	$0.462373\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	84.9706	3.16009
$724$	0	0
$725$	−18.0000	−0.668503
$726$	0	0
$727$	−18.9737	−0.703694	−0.351847	−	0.936057i	$-0.614446\pi$
$727$	−18.9737	−0.703694	−0.351847	+	0.936057i	$0.614446\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−18.9737	−0.701766
$732$	0	0
$733$	−14.1421	−0.522352	−0.261176	−	0.965291i	$-0.584110\pi$
$733$	−14.1421	−0.522352	−0.261176	+	0.965291i	$0.584110\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.47214	−0.164510	−0.0822551	−	0.996611i	$-0.526212\pi$
$739$	−4.47214	−0.164510	−0.0822551	+	0.996611i	$0.526212\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.8885	−0.656267	−0.328134	−	0.944631i	$-0.606420\pi$
$743$	−17.8885	−0.656267	−0.328134	+	0.944631i	$0.606420\pi$
$744$	0	0
$745$	−39.5980	−1.45076
$746$	0	0
$747$	66.4078	2.42974
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−44.7214	−1.63191	−0.815953	−	0.578119i	$-0.803787\pi$
$751$	−44.7214	−1.63191	−0.815953	+	0.578119i	$0.803787\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	−50.5964	−1.84139
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	0	0
$759$	−126.491	−4.59134
$760$	0	0
$761$	26.8701	0.974039	0.487019	−	0.873391i	$-0.338084\pi$
$761$	26.8701	0.974039	0.487019	+	0.873391i	$0.338084\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	84.0000	3.03703
$766$	0	0
$767$	0	0
$768$	0	0
$769$	46.6690	1.68293	0.841464	−	0.540312i	$-0.181694\pi$
$769$	46.6690	1.68293	0.841464	+	0.540312i	$0.181694\pi$
$770$	0	0
$771$	−76.0263	−2.73802
$772$	0	0
$773$	14.1421	0.508657	0.254329	−	0.967118i	$-0.418146\pi$
$773$	14.1421	0.508657	0.254329	+	0.967118i	$0.418146\pi$
$774$	0	0
$775$	−18.9737	−0.681554
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.47214	−0.160231
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	75.8947	2.71225
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.8114	−0.563615	−0.281808	−	0.959471i	$-0.590934\pi$
$787$	−15.8114	−0.563615	−0.281808	+	0.959471i	$0.590934\pi$
$788$	0	0
$789$	28.2843	1.00695
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−53.6656	−1.90332
$796$	0	0
$797$	−45.2548	−1.60301	−0.801504	−	0.597989i	$-0.795967\pi$
$797$	−45.2548	−1.60301	−0.801504	+	0.597989i	$0.795967\pi$
$798$	0	0
$799$	26.8328	0.949277
$800$	0	0
$801$	69.2965	2.44847
$802$	0	0
$803$	−31.6228	−1.11594
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−35.7771	−1.25941
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−41.1096	−1.44355	−0.721777	−	0.692126i	$-0.756674\pi$
$811$	−41.1096	−1.44355	−0.721777	+	0.692126i	$0.756674\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	37.9473	1.32924
$816$	0	0
$817$	14.1421	0.494771
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−17.8885	−0.623555	−0.311778	−	0.950155i	$-0.600924\pi$
$823$	−17.8885	−0.623555	−0.311778	+	0.950155i	$0.600924\pi$
$824$	0	0
$825$	−42.4264	−1.47710
$826$	0	0
$827$	−17.8885	−0.622046	−0.311023	−	0.950402i	$-0.600672\pi$
$827$	−17.8885	−0.622046	−0.311023	+	0.950402i	$0.600672\pi$
$828$	0	0
$829$	−31.1127	−1.08059	−0.540294	−	0.841476i	$-0.681687\pi$
$829$	−31.1127	−1.08059	−0.540294	+	0.841476i	$0.681687\pi$
$830$	0	0
$831$	−69.5701	−2.41336
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17.8885	0.619059
$836$	0	0
$837$	80.0000	2.76520
$838$	0	0
$839$	−56.9210	−1.96513	−0.982566	−	0.185917i	$-0.940475\pi$
$839$	−56.9210	−1.96513	−0.982566	+	0.185917i	$0.940475\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−25.2982	−0.871317
$844$	0	0
$845$	−36.7696	−1.26491
$846$	0	0
$847$	0	0
$848$	0	0
$849$	90.0000	3.08879
$850$	0	0
$851$	17.8885	0.613211
$852$	0	0
$853$	−33.9411	−1.16212	−0.581061	−	0.813860i	$-0.697362\pi$
$853$	−33.9411	−1.16212	−0.581061	+	0.813860i	$0.697362\pi$
$854$	0	0
$855$	−62.6099	−2.14121
$856$	0	0
$857$	−49.4975	−1.69080	−0.845401	−	0.534133i	$-0.820638\pi$
$857$	−49.4975	−1.69080	−0.845401	+	0.534133i	$0.820638\pi$
$858$	0	0
$859$	3.16228	0.107896	0.0539478	−	0.998544i	$-0.482820\pi$
$859$	3.16228	0.107896	0.0539478	+	0.998544i	$0.482820\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.8328	−0.913400	−0.456700	−	0.889621i	$-0.650969\pi$
$863$	−26.8328	−0.913400	−0.456700	+	0.889621i	$0.650969\pi$
$864$	0	0
$865$	16.0000	0.544016
$866$	0	0
$867$	−3.16228	−0.107397
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	0	0
$872$	0	0
$873$	29.6985	1.00514
$874$	0	0
$875$	0	0
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	0	0
$879$	26.8328	0.905048
$880$	0	0
$881$	−26.8701	−0.905275	−0.452638	−	0.891695i	$-0.649517\pi$
$881$	−26.8701	−0.905275	−0.452638	+	0.891695i	$0.649517\pi$
$882$	0	0
$883$	−53.6656	−1.80599	−0.902996	−	0.429649i	$-0.858637\pi$
$883$	−53.6656	−1.80599	−0.902996	+	0.429649i	$0.858637\pi$
$884$	0	0
$885$	28.2843	0.950765
$886$	0	0
$887$	18.9737	0.637073	0.318537	−	0.947911i	$-0.396809\pi$
$887$	18.9737	0.637073	0.318537	+	0.947911i	$0.396809\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	84.9706	2.84662
$892$	0	0
$893$	−20.0000	−0.669274
$894$	0	0
$895$	−50.5964	−1.69125
$896$	0	0
$897$	0	0
$898$	0	0
$899$	37.9473	1.26561
$900$	0	0
$901$	25.4558	0.848057
$902$	0	0
$903$	0	0
$904$	0	0
$905$	64.0000	2.12743
$906$	0	0
$907$	17.8885	0.593979	0.296990	−	0.954881i	$-0.404017\pi$
$907$	17.8885	0.593979	0.296990	+	0.954881i	$0.404017\pi$
$908$	0	0
$909$	59.3970	1.97007
$910$	0	0
$911$	53.6656	1.77802	0.889011	−	0.457886i	$-0.151393\pi$
$911$	53.6656	1.77802	0.889011	+	0.457886i	$0.151393\pi$
$912$	0	0
$913$	42.4264	1.40411
$914$	0	0
$915$	75.8947	2.50900
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−50.0000	−1.64756
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	44.2719	1.45408
$928$	0	0
$929$	4.24264	0.139197	0.0695983	−	0.997575i	$-0.477828\pi$
$929$	4.24264	0.139197	0.0695983	+	0.997575i	$0.477828\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	53.6656	1.75505
$936$	0	0
$937$	38.1838	1.24741	0.623705	−	0.781660i	$-0.285627\pi$
$937$	38.1838	1.24741	0.623705	+	0.781660i	$0.285627\pi$
$938$	0	0
$939$	67.0820	2.18914
$940$	0	0
$941$	48.0833	1.56747	0.783735	−	0.621096i	$-0.213312\pi$
$941$	48.0833	1.56747	0.783735	+	0.621096i	$0.213312\pi$
$942$	0	0
$943$	12.6491	0.411912
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.47214	0.145325	0.0726624	−	0.997357i	$-0.476850\pi$
$947$	4.47214	0.145325	0.0726624	+	0.997357i	$0.476850\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.32456	−0.205088
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	25.2982	0.818631
$956$	0	0
$957$	84.8528	2.74290
$958$	0	0
$959$	0	0
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	0	0
$964$	0	0
$965$	67.8823	2.18521
$966$	0	0
$967$	−8.94427	−0.287628	−0.143814	−	0.989605i	$-0.545937\pi$
$967$	−8.94427	−0.287628	−0.143814	+	0.989605i	$0.545937\pi$
$968$	0	0
$969$	42.4264	1.36293
$970$	0	0
$971$	−28.4605	−0.913341	−0.456670	−	0.889636i	$-0.650958\pi$
$971$	−28.4605	−0.913341	−0.456670	+	0.889636i	$0.650958\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	44.2719	1.41494
$980$	0	0
$981$	−70.0000	−2.23493
$982$	0	0
$983$	6.32456	0.201722	0.100861	−	0.994901i	$-0.467840\pi$
$983$	6.32456	0.201722	0.100861	+	0.994901i	$0.467840\pi$
$984$	0	0
$985$	−62.2254	−1.98267
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−40.0000	−1.27193
$990$	0	0
$991$	−53.6656	−1.70474	−0.852372	−	0.522935i	$-0.824837\pi$
$991$	−53.6656	−1.70474	−0.852372	+	0.522935i	$0.824837\pi$
$992$	0	0
$993$	−42.4264	−1.34636
$994$	0	0
$995$	−17.8885	−0.567105
$996$	0	0
$997$	−53.7401	−1.70197	−0.850983	−	0.525193i	$-0.823993\pi$
$997$	−53.7401	−1.70197	−0.850983	+	0.525193i	$0.823993\pi$
$998$	0	0
$999$	−25.2982	−0.800400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.a.x.1.2	yes	4
4.3	odd	2	inner	1568.2.a.x.1.4	yes	4
7.2	even	3		1568.2.i.y.1537.3		8
7.3	odd	6		1568.2.i.y.961.2		8
7.4	even	3		1568.2.i.y.961.3		8
7.5	odd	6		1568.2.i.y.1537.2		8
7.6	odd	2	inner	1568.2.a.x.1.3	yes	4
8.3	odd	2		3136.2.a.bz.1.1		4
8.5	even	2		3136.2.a.bz.1.3		4
28.3	even	6		1568.2.i.y.961.4		8
28.11	odd	6		1568.2.i.y.961.1		8
28.19	even	6		1568.2.i.y.1537.4		8
28.23	odd	6		1568.2.i.y.1537.1		8
28.27	even	2	inner	1568.2.a.x.1.1	✓	4
56.13	odd	2		3136.2.a.bz.1.2		4
56.27	even	2		3136.2.a.bz.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1568.2.a.x.1.1	✓	4	28.27	even	2	inner
1568.2.a.x.1.2	yes	4	1.1	even	1	trivial
1568.2.a.x.1.3	yes	4	7.6	odd	2	inner
1568.2.a.x.1.4	yes	4	4.3	odd	2	inner
1568.2.i.y.961.1		8	28.11	odd	6
1568.2.i.y.961.2		8	7.3	odd	6
1568.2.i.y.961.3		8	7.4	even	3
1568.2.i.y.961.4		8	28.3	even	6
1568.2.i.y.1537.1		8	28.23	odd	6
1568.2.i.y.1537.2		8	7.5	odd	6
1568.2.i.y.1537.3		8	7.2	even	3
1568.2.i.y.1537.4		8	28.19	even	6
3136.2.a.bz.1.1		4	8.3	odd	2
3136.2.a.bz.1.2		4	56.13	odd	2
3136.2.a.bz.1.3		4	8.5	even	2
3136.2.a.bz.1.4		4	56.27	even	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q-3.16228 q^{3} +2.82843 q^{5} +7.00000 q^{9} +O(q^{10})\)	\(q-3.16228 q^{3} +2.82843 q^{5} +7.00000 q^{9} +4.47214 q^{11} -8.94427 q^{15} +4.24264 q^{17} -3.16228 q^{19} +8.94427 q^{23} +3.00000 q^{25} -12.6491 q^{27} -6.00000 q^{29} -6.32456 q^{31} -14.1421 q^{33} +2.00000 q^{37} +1.41421 q^{41} -4.47214 q^{43} +19.7990 q^{45} +6.32456 q^{47} -13.4164 q^{51} +6.00000 q^{53} +12.6491 q^{55} +10.0000 q^{57} -3.16228 q^{59} -8.48528 q^{61} -28.2843 q^{69} +8.94427 q^{71} -7.07107 q^{73} -9.48683 q^{75} +8.94427 q^{79} +19.0000 q^{81} +9.48683 q^{83} +12.0000 q^{85} +18.9737 q^{87} +9.89949 q^{89} +20.0000 q^{93} -8.94427 q^{95} +4.24264 q^{97} +31.3050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{9}+O(q^{10}) \) 4 * q + 28 * q^9	\( 4 q + 28 q^{9} + 12 q^{25} - 24 q^{29} + 8 q^{37} + 24 q^{53} + 40 q^{57} + 76 q^{81} + 48 q^{85} + 80 q^{93}+O(q^{100}) \) 4 * q + 28 * q^9 + 12 * q^25 - 24 * q^29 + 8 * q^37 + 24 * q^53 + 40 * q^57 + 76 * q^81 + 48 * q^85 + 80 * q^93

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