LMFDB - Embedded newform 7728.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.70156 q^{11} +2.00000 q^{13} -7.40312 q^{17} +1.29844 q^{19} -1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} +9.40312 q^{31} -2.70156 q^{33} -9.40312 q^{37} +2.00000 q^{39} +2.70156 q^{41} -11.4031 q^{43} +12.7016 q^{47} +1.00000 q^{49} -7.40312 q^{51} +10.1047 q^{53} +1.29844 q^{57} +14.1047 q^{59} +12.7016 q^{61} -1.00000 q^{63} +0.596876 q^{67} -1.00000 q^{69} +7.40312 q^{73} -5.00000 q^{75} +2.70156 q^{77} +1.40312 q^{79} +1.00000 q^{81} +10.0000 q^{83} +2.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} +9.40312 q^{93} +6.00000 q^{97} -2.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + q^{11} + 4 q^{13} - 2 q^{17} + 9 q^{19} - 2 q^{21} - 2 q^{23} - 10 q^{25} + 2 q^{27} + 4 q^{29} + 6 q^{31} + q^{33} - 6 q^{37} + 4 q^{39} - q^{41} - 10 q^{43} + 19 q^{47} + 2 q^{49} - 2 q^{51} + q^{53} + 9 q^{57} + 9 q^{59} + 19 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 2 q^{73} - 10 q^{75} - q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} + 4 q^{87} - 4 q^{89} - 4 q^{91} + 6 q^{93} + 12 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + q^11 + 4 * q^13 - 2 * q^17 + 9 * q^19 - 2 * q^21 - 2 * q^23 - 10 * q^25 + 2 * q^27 + 4 * q^29 + 6 * q^31 + q^33 - 6 * q^37 + 4 * q^39 - q^41 - 10 * q^43 + 19 * q^47 + 2 * q^49 - 2 * q^51 + q^53 + 9 * q^57 + 9 * q^59 + 19 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 2 * q^73 - 10 * q^75 - q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 - 4 * q^91 + 6 * q^93 + 12 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.70156	−0.814552	−0.407276	−	0.913305i	$-0.633521\pi$
$11$	−2.70156	−0.814552	−0.407276	+	0.913305i	$0.633521\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.40312	−1.79552	−0.897761	−	0.440484i	$-0.854807\pi$
$17$	−7.40312	−1.79552	−0.897761	+	0.440484i	$0.854807\pi$
$18$	0	0
$19$	1.29844	0.297882	0.148941	−	0.988846i	$-0.452414\pi$
$19$	1.29844	0.297882	0.148941	+	0.988846i	$0.452414\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	9.40312	1.68885	0.844425	−	0.535673i	$-0.179942\pi$
$31$	9.40312	1.68885	0.844425	+	0.535673i	$0.179942\pi$
$32$	0	0
$33$	−2.70156	−0.470282
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.40312	−1.54586	−0.772932	−	0.634489i	$-0.781211\pi$
$37$	−9.40312	−1.54586	−0.772932	+	0.634489i	$0.781211\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	2.70156	0.421913	0.210957	−	0.977495i	$-0.432342\pi$
$41$	2.70156	0.421913	0.210957	+	0.977495i	$0.432342\pi$
$42$	0	0
$43$	−11.4031	−1.73896	−0.869480	−	0.493968i	$-0.835546\pi$
$43$	−11.4031	−1.73896	−0.869480	+	0.493968i	$0.835546\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.7016	1.85271	0.926357	−	0.376648i	$-0.122923\pi$
$47$	12.7016	1.85271	0.926357	+	0.376648i	$0.122923\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.40312	−1.03664
$52$	0	0
$53$	10.1047	1.38799	0.693993	−	0.719982i	$-0.255850\pi$
$53$	10.1047	1.38799	0.693993	+	0.719982i	$0.255850\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.29844	0.171982
$58$	0	0
$59$	14.1047	1.83627	0.918137	−	0.396263i	$-0.129693\pi$
$59$	14.1047	1.83627	0.918137	+	0.396263i	$0.129693\pi$
$60$	0	0
$61$	12.7016	1.62627	0.813134	−	0.582076i	$-0.197759\pi$
$61$	12.7016	1.62627	0.813134	+	0.582076i	$0.197759\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.596876	0.0729200	0.0364600	−	0.999335i	$-0.488392\pi$
$67$	0.596876	0.0729200	0.0364600	+	0.999335i	$0.488392\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	7.40312	0.866470	0.433235	−	0.901281i	$-0.357372\pi$
$73$	7.40312	0.866470	0.433235	+	0.901281i	$0.357372\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	2.70156	0.307872
$78$	0	0
$79$	1.40312	0.157864	0.0789319	−	0.996880i	$-0.474849\pi$
$79$	1.40312	0.157864	0.0789319	+	0.996880i	$0.474849\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	9.40312	0.975059
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−2.70156	−0.271517
$100$	0	0
$101$	−1.29844	−0.129199	−0.0645997	−	0.997911i	$-0.520577\pi$
$101$	−1.29844	−0.129199	−0.0645997	+	0.997911i	$0.520577\pi$
$102$	0	0
$103$	−8.70156	−0.857390	−0.428695	−	0.903449i	$-0.641027\pi$
$103$	−8.70156	−0.857390	−0.428695	+	0.903449i	$0.641027\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4031	1.48908	0.744538	−	0.667580i	$-0.232670\pi$
$107$	15.4031	1.48908	0.744538	+	0.667580i	$0.232670\pi$
$108$	0	0
$109$	−14.8062	−1.41818	−0.709091	−	0.705117i	$-0.750894\pi$
$109$	−14.8062	−1.41818	−0.709091	+	0.705117i	$0.750894\pi$
$110$	0	0
$111$	−9.40312	−0.892505
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	7.40312	0.678643
$120$	0	0
$121$	−3.70156	−0.336506
$122$	0	0
$123$	2.70156	0.243592
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.10469	0.541703	0.270852	−	0.962621i	$-0.412695\pi$
$127$	6.10469	0.541703	0.270852	+	0.962621i	$0.412695\pi$
$128$	0	0
$129$	−11.4031	−1.00399
$130$	0	0
$131$	20.7016	1.80870	0.904352	−	0.426787i	$-0.140355\pi$
$131$	20.7016	1.80870	0.904352	+	0.426787i	$0.140355\pi$
$132$	0	0
$133$	−1.29844	−0.112589
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.5078	1.49579	0.747897	−	0.663815i	$-0.231064\pi$
$137$	17.5078	1.49579	0.747897	+	0.663815i	$0.231064\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	12.7016	1.06966
$142$	0	0
$143$	−5.40312	−0.451832
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−7.29844	−0.597911	−0.298956	−	0.954267i	$-0.596638\pi$
$149$	−7.29844	−0.597911	−0.298956	+	0.954267i	$0.596638\pi$
$150$	0	0
$151$	−12.7016	−1.03364	−0.516819	−	0.856095i	$-0.672884\pi$
$151$	−12.7016	−1.03364	−0.516819	+	0.856095i	$0.672884\pi$
$152$	0	0
$153$	−7.40312	−0.598507
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.701562	0.0559908	0.0279954	−	0.999608i	$-0.491088\pi$
$157$	0.701562	0.0559908	0.0279954	+	0.999608i	$0.491088\pi$
$158$	0	0
$159$	10.1047	0.801354
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	14.1047	1.10476	0.552382	−	0.833591i	$-0.313719\pi$
$163$	14.1047	1.10476	0.552382	+	0.833591i	$0.313719\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.1047	−1.09145	−0.545727	−	0.837963i	$-0.683746\pi$
$167$	−14.1047	−1.09145	−0.545727	+	0.837963i	$0.683746\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.29844	0.0992940
$172$	0	0
$173$	8.80625	0.669527	0.334763	−	0.942302i	$-0.391344\pi$
$173$	8.80625	0.669527	0.334763	+	0.942302i	$0.391344\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	14.1047	1.06017
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−9.40312	−0.698929	−0.349464	−	0.936950i	$-0.613636\pi$
$181$	−9.40312	−0.698929	−0.349464	+	0.936950i	$0.613636\pi$
$182$	0	0
$183$	12.7016	0.938926
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000	1.46254
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.2984	1.10696	0.553478	−	0.832864i	$-0.313300\pi$
$191$	15.2984	1.10696	0.553478	+	0.832864i	$0.313300\pi$
$192$	0	0
$193$	6.70156	0.482389	0.241194	−	0.970477i	$-0.422461\pi$
$193$	6.70156	0.482389	0.241194	+	0.970477i	$0.422461\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.8062	−1.19740	−0.598698	−	0.800975i	$-0.704315\pi$
$197$	−16.8062	−1.19740	−0.598698	+	0.800975i	$0.704315\pi$
$198$	0	0
$199$	−6.10469	−0.432750	−0.216375	−	0.976310i	$-0.569423\pi$
$199$	−6.10469	−0.432750	−0.216375	+	0.976310i	$0.569423\pi$
$200$	0	0
$201$	0.596876	0.0421004
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−3.50781	−0.242640
$210$	0	0
$211$	−0.701562	−0.0482975	−0.0241488	−	0.999708i	$-0.507688\pi$
$211$	−0.701562	−0.0482975	−0.0241488	+	0.999708i	$0.507688\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.40312	−0.638326
$218$	0	0
$219$	7.40312	0.500257
$220$	0	0
$221$	−14.8062	−0.995976
$222$	0	0
$223$	28.2094	1.88904	0.944520	−	0.328455i	$-0.106528\pi$
$223$	28.2094	1.88904	0.944520	+	0.328455i	$0.106528\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	−4.80625	−0.319002	−0.159501	−	0.987198i	$-0.550989\pi$
$227$	−4.80625	−0.319002	−0.159501	+	0.987198i	$0.550989\pi$
$228$	0	0
$229$	−3.29844	−0.217967	−0.108983	−	0.994044i	$-0.534760\pi$
$229$	−3.29844	−0.217967	−0.108983	+	0.994044i	$0.534760\pi$
$230$	0	0
$231$	2.70156	0.177750
$232$	0	0
$233$	19.4031	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$233$	19.4031	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.40312	0.0911427
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−16.1047	−1.03739	−0.518697	−	0.854958i	$-0.673583\pi$
$241$	−16.1047	−1.03739	−0.518697	+	0.854958i	$0.673583\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.59688	0.165235
$248$	0	0
$249$	10.0000	0.633724
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	2.70156	0.169846
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.1047	−1.50361	−0.751805	−	0.659386i	$-0.770816\pi$
$257$	−24.1047	−1.50361	−0.751805	+	0.659386i	$0.770816\pi$
$258$	0	0
$259$	9.40312	0.584282
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	1.89531	0.116870	0.0584350	−	0.998291i	$-0.481389\pi$
$263$	1.89531	0.116870	0.0584350	+	0.998291i	$0.481389\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−31.6125	−1.92745	−0.963724	−	0.266901i	$-0.914000\pi$
$269$	−31.6125	−1.92745	−0.963724	+	0.266901i	$0.914000\pi$
$270$	0	0
$271$	−10.8062	−0.656433	−0.328216	−	0.944603i	$-0.606448\pi$
$271$	−10.8062	−0.656433	−0.328216	+	0.944603i	$0.606448\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	13.5078	0.814552
$276$	0	0
$277$	16.1047	0.967637	0.483818	−	0.875168i	$-0.339250\pi$
$277$	16.1047	0.967637	0.483818	+	0.875168i	$0.339250\pi$
$278$	0	0
$279$	9.40312	0.562950
$280$	0	0
$281$	−4.80625	−0.286717	−0.143358	−	0.989671i	$-0.545790\pi$
$281$	−4.80625	−0.286717	−0.143358	+	0.989671i	$0.545790\pi$
$282$	0	0
$283$	16.5969	0.986582	0.493291	−	0.869864i	$-0.335794\pi$
$283$	16.5969	0.986582	0.493291	+	0.869864i	$0.335794\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.70156	−0.159468
$288$	0	0
$289$	37.8062	2.22390
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−29.6125	−1.72998	−0.864990	−	0.501789i	$-0.832676\pi$
$293$	−29.6125	−1.72998	−0.864990	+	0.501789i	$0.832676\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.70156	−0.156761
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	11.4031	0.657265
$302$	0	0
$303$	−1.29844	−0.0745933
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−22.8062	−1.30162	−0.650811	−	0.759240i	$-0.725571\pi$
$307$	−22.8062	−1.30162	−0.650811	+	0.759240i	$0.725571\pi$
$308$	0	0
$309$	−8.70156	−0.495015
$310$	0	0
$311$	11.2984	0.640676	0.320338	−	0.947303i	$-0.396204\pi$
$311$	11.2984	0.640676	0.320338	+	0.947303i	$0.396204\pi$
$312$	0	0
$313$	22.9109	1.29500	0.647501	−	0.762064i	$-0.275814\pi$
$313$	22.9109	1.29500	0.647501	+	0.762064i	$0.275814\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.6125	1.77553	0.887767	−	0.460292i	$-0.152255\pi$
$317$	31.6125	1.77553	0.887767	+	0.460292i	$0.152255\pi$
$318$	0	0
$319$	−5.40312	−0.302517
$320$	0	0
$321$	15.4031	0.859719
$322$	0	0
$323$	−9.61250	−0.534854
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	−14.8062	−0.818787
$328$	0	0
$329$	−12.7016	−0.700260
$330$	0	0
$331$	−8.70156	−0.478281	−0.239141	−	0.970985i	$-0.576866\pi$
$331$	−8.70156	−0.478281	−0.239141	+	0.970985i	$0.576866\pi$
$332$	0	0
$333$	−9.40312	−0.515288
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.8062	0.697601	0.348800	−	0.937197i	$-0.386589\pi$
$337$	12.8062	0.697601	0.348800	+	0.937197i	$0.386589\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−25.4031	−1.37566
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.2094	1.51436	0.757179	−	0.653207i	$-0.226577\pi$
$347$	28.2094	1.51436	0.757179	+	0.653207i	$0.226577\pi$
$348$	0	0
$349$	27.6125	1.47806	0.739032	−	0.673671i	$-0.235283\pi$
$349$	27.6125	1.47806	0.739032	+	0.673671i	$0.235283\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	23.6125	1.25677	0.628383	−	0.777904i	$-0.283717\pi$
$353$	23.6125	1.25677	0.628383	+	0.777904i	$0.283717\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.40312	0.391815
$358$	0	0
$359$	−6.80625	−0.359220	−0.179610	−	0.983738i	$-0.557484\pi$
$359$	−6.80625	−0.359220	−0.179610	+	0.983738i	$0.557484\pi$
$360$	0	0
$361$	−17.3141	−0.911266
$362$	0	0
$363$	−3.70156	−0.194282
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.89531	0.516531	0.258266	−	0.966074i	$-0.416849\pi$
$367$	9.89531	0.516531	0.258266	+	0.966074i	$0.416849\pi$
$368$	0	0
$369$	2.70156	0.140638
$370$	0	0
$371$	−10.1047	−0.524609
$372$	0	0
$373$	14.5969	0.755798	0.377899	−	0.925847i	$-0.376647\pi$
$373$	14.5969	0.755798	0.377899	+	0.925847i	$0.376647\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	6.10469	0.312753
$382$	0	0
$383$	28.2094	1.44143	0.720716	−	0.693231i	$-0.243813\pi$
$383$	28.2094	1.44143	0.720716	+	0.693231i	$0.243813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.4031	−0.579653
$388$	0	0
$389$	−20.2094	−1.02466	−0.512328	−	0.858790i	$-0.671217\pi$
$389$	−20.2094	−1.02466	−0.512328	+	0.858790i	$0.671217\pi$
$390$	0	0
$391$	7.40312	0.374392
$392$	0	0
$393$	20.7016	1.04426
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.80625	−0.441973	−0.220986	−	0.975277i	$-0.570928\pi$
$397$	−8.80625	−0.441973	−0.220986	+	0.975277i	$0.570928\pi$
$398$	0	0
$399$	−1.29844	−0.0650032
$400$	0	0
$401$	0.104686	0.00522779	0.00261389	−	0.999997i	$-0.499168\pi$
$401$	0.104686	0.00522779	0.00261389	+	0.999997i	$0.499168\pi$
$402$	0	0
$403$	18.8062	0.936806
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.4031	1.25919
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	17.5078	0.863597
$412$	0	0
$413$	−14.1047	−0.694046
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	20.8062	1.01645	0.508226	−	0.861224i	$-0.330302\pi$
$419$	20.8062	1.01645	0.508226	+	0.861224i	$0.330302\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	0	0
$423$	12.7016	0.617571
$424$	0	0
$425$	37.0156	1.79552
$426$	0	0
$427$	−12.7016	−0.614672
$428$	0	0
$429$	−5.40312	−0.260865
$430$	0	0
$431$	35.5078	1.71035	0.855176	−	0.518339i	$-0.173449\pi$
$431$	35.5078	1.71035	0.855176	+	0.518339i	$0.173449\pi$
$432$	0	0
$433$	40.3141	1.93737	0.968685	−	0.248293i	$-0.0798695\pi$
$433$	40.3141	1.93737	0.968685	+	0.248293i	$0.0798695\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.29844	−0.0621127
$438$	0	0
$439$	−34.8062	−1.66121	−0.830606	−	0.556861i	$-0.812006\pi$
$439$	−34.8062	−1.66121	−0.830606	+	0.556861i	$0.812006\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.29844	−0.345204
$448$	0	0
$449$	3.19375	0.150722	0.0753612	−	0.997156i	$-0.475989\pi$
$449$	3.19375	0.150722	0.0753612	+	0.997156i	$0.475989\pi$
$450$	0	0
$451$	−7.29844	−0.343670
$452$	0	0
$453$	−12.7016	−0.596771
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.6125	−1.85299	−0.926497	−	0.376302i	$-0.877196\pi$
$457$	−39.6125	−1.85299	−0.926497	+	0.376302i	$0.877196\pi$
$458$	0	0
$459$	−7.40312	−0.345548
$460$	0	0
$461$	24.8062	1.15534	0.577671	−	0.816270i	$-0.303962\pi$
$461$	24.8062	1.15534	0.577671	+	0.816270i	$0.303962\pi$
$462$	0	0
$463$	32.9109	1.52950	0.764750	−	0.644327i	$-0.222862\pi$
$463$	32.9109	1.52950	0.764750	+	0.644327i	$0.222862\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.8062	1.70319	0.851595	−	0.524200i	$-0.175636\pi$
$467$	36.8062	1.70319	0.851595	+	0.524200i	$0.175636\pi$
$468$	0	0
$469$	−0.596876	−0.0275612
$470$	0	0
$471$	0.701562	0.0323263
$472$	0	0
$473$	30.8062	1.41647
$474$	0	0
$475$	−6.49219	−0.297882
$476$	0	0
$477$	10.1047	0.462662
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−18.8062	−0.857491
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.80625	0.127163	0.0635816	−	0.997977i	$-0.479748\pi$
$487$	2.80625	0.127163	0.0635816	+	0.997977i	$0.479748\pi$
$488$	0	0
$489$	14.1047	0.637836
$490$	0	0
$491$	−17.4031	−0.785392	−0.392696	−	0.919668i	$-0.628458\pi$
$491$	−17.4031	−0.785392	−0.392696	+	0.919668i	$0.628458\pi$
$492$	0	0
$493$	−14.8062	−0.666840
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−42.8062	−1.91627	−0.958135	−	0.286315i	$-0.907569\pi$
$499$	−42.8062	−1.91627	−0.958135	+	0.286315i	$0.907569\pi$
$500$	0	0
$501$	−14.1047	−0.630151
$502$	0	0
$503$	−14.8062	−0.660178	−0.330089	−	0.943950i	$-0.607079\pi$
$503$	−14.8062	−0.660178	−0.330089	+	0.943950i	$0.607079\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−40.1047	−1.77761	−0.888804	−	0.458287i	$-0.848463\pi$
$509$	−40.1047	−1.77761	−0.888804	+	0.458287i	$0.848463\pi$
$510$	0	0
$511$	−7.40312	−0.327495
$512$	0	0
$513$	1.29844	0.0573274
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−34.3141	−1.50913
$518$	0	0
$519$	8.80625	0.386551
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	25.2984	1.10622	0.553112	−	0.833107i	$-0.313440\pi$
$523$	25.2984	1.10622	0.553112	+	0.833107i	$0.313440\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	−69.6125	−3.03237
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.1047	0.612091
$532$	0	0
$533$	5.40312	0.234035
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	−2.70156	−0.116365
$540$	0	0
$541$	−40.3141	−1.73324	−0.866618	−	0.498972i	$-0.833711\pi$
$541$	−40.3141	−1.73324	−0.866618	+	0.498972i	$0.833711\pi$
$542$	0	0
$543$	−9.40312	−0.403527
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.19375	−0.393096	−0.196548	−	0.980494i	$-0.562973\pi$
$547$	−9.19375	−0.393096	−0.196548	+	0.980494i	$0.562973\pi$
$548$	0	0
$549$	12.7016	0.542089
$550$	0	0
$551$	2.59688	0.110631
$552$	0	0
$553$	−1.40312	−0.0596669
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.79063	0.160614	0.0803070	−	0.996770i	$-0.474410\pi$
$557$	3.79063	0.160614	0.0803070	+	0.996770i	$0.474410\pi$
$558$	0	0
$559$	−22.8062	−0.964602
$560$	0	0
$561$	20.0000	0.844401
$562$	0	0
$563$	7.40312	0.312004	0.156002	−	0.987757i	$-0.450139\pi$
$563$	7.40312	0.312004	0.156002	+	0.987757i	$0.450139\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	13.2984	0.557500	0.278750	−	0.960364i	$-0.410080\pi$
$569$	13.2984	0.557500	0.278750	+	0.960364i	$0.410080\pi$
$570$	0	0
$571$	−15.4031	−0.644601	−0.322300	−	0.946637i	$-0.604456\pi$
$571$	−15.4031	−0.644601	−0.322300	+	0.946637i	$0.604456\pi$
$572$	0	0
$573$	15.2984	0.639101
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	8.80625	0.366609	0.183304	−	0.983056i	$-0.441321\pi$
$577$	8.80625	0.366609	0.183304	+	0.983056i	$0.441321\pi$
$578$	0	0
$579$	6.70156	0.278507
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	−27.2984	−1.13059
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.1047	−0.582163	−0.291081	−	0.956698i	$-0.594015\pi$
$587$	−14.1047	−0.582163	−0.291081	+	0.956698i	$0.594015\pi$
$588$	0	0
$589$	12.2094	0.503078
$590$	0	0
$591$	−16.8062	−0.691317
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.10469	−0.249848
$598$	0	0
$599$	33.4031	1.36481	0.682407	−	0.730972i	$-0.260933\pi$
$599$	33.4031	1.36481	0.682407	+	0.730972i	$0.260933\pi$
$600$	0	0
$601$	−30.2094	−1.23227	−0.616133	−	0.787642i	$-0.711302\pi$
$601$	−30.2094	−1.23227	−0.616133	+	0.787642i	$0.711302\pi$
$602$	0	0
$603$	0.596876	0.0243067
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13.1938	−0.535518	−0.267759	−	0.963486i	$-0.586283\pi$
$607$	−13.1938	−0.535518	−0.267759	+	0.963486i	$0.586283\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	25.4031	1.02770
$612$	0	0
$613$	40.2094	1.62404	0.812021	−	0.583629i	$-0.198368\pi$
$613$	40.2094	1.62404	0.812021	+	0.583629i	$0.198368\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.80625	−0.193492	−0.0967461	−	0.995309i	$-0.530843\pi$
$617$	−4.80625	−0.193492	−0.0967461	+	0.995309i	$0.530843\pi$
$618$	0	0
$619$	43.8219	1.76135	0.880675	−	0.473721i	$-0.157090\pi$
$619$	43.8219	1.76135	0.880675	+	0.473721i	$0.157090\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−3.50781	−0.140088
$628$	0	0
$629$	69.6125	2.77563
$630$	0	0
$631$	−26.5969	−1.05880	−0.529402	−	0.848371i	$-0.677584\pi$
$631$	−26.5969	−1.05880	−0.529402	+	0.848371i	$0.677584\pi$
$632$	0	0
$633$	−0.701562	−0.0278846
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.9109	0.746937	0.373469	−	0.927643i	$-0.378168\pi$
$641$	18.9109	0.746937	0.373469	+	0.927643i	$0.378168\pi$
$642$	0	0
$643$	−36.1047	−1.42383	−0.711915	−	0.702266i	$-0.752172\pi$
$643$	−36.1047	−1.42383	−0.711915	+	0.702266i	$0.752172\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.8062	−1.05386	−0.526931	−	0.849908i	$-0.676657\pi$
$647$	−26.8062	−1.05386	−0.526931	+	0.849908i	$0.676657\pi$
$648$	0	0
$649$	−38.1047	−1.49574
$650$	0	0
$651$	−9.40312	−0.368537
$652$	0	0
$653$	−30.2094	−1.18218	−0.591092	−	0.806604i	$-0.701303\pi$
$653$	−30.2094	−1.18218	−0.591092	+	0.806604i	$0.701303\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.40312	0.288823
$658$	0	0
$659$	−17.0156	−0.662834	−0.331417	−	0.943484i	$-0.607527\pi$
$659$	−17.0156	−0.662834	−0.331417	+	0.943484i	$0.607527\pi$
$660$	0	0
$661$	18.3141	0.712334	0.356167	−	0.934422i	$-0.384083\pi$
$661$	18.3141	0.712334	0.356167	+	0.934422i	$0.384083\pi$
$662$	0	0
$663$	−14.8062	−0.575027
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	28.2094	1.09064
$670$	0	0
$671$	−34.3141	−1.32468
$672$	0	0
$673$	5.08907	0.196169	0.0980845	−	0.995178i	$-0.468728\pi$
$673$	5.08907	0.196169	0.0980845	+	0.995178i	$0.468728\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−4.80625	−0.184176
$682$	0	0
$683$	−41.6125	−1.59226	−0.796129	−	0.605127i	$-0.793122\pi$
$683$	−41.6125	−1.59226	−0.796129	+	0.605127i	$0.793122\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.29844	−0.125843
$688$	0	0
$689$	20.2094	0.769916
$690$	0	0
$691$	18.8062	0.715423	0.357712	−	0.933832i	$-0.383557\pi$
$691$	18.8062	0.715423	0.357712	+	0.933832i	$0.383557\pi$
$692$	0	0
$693$	2.70156	0.102624
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	19.4031	0.733894
$700$	0	0
$701$	8.91093	0.336561	0.168281	−	0.985739i	$-0.446179\pi$
$701$	8.91093	0.336561	0.168281	+	0.985739i	$0.446179\pi$
$702$	0	0
$703$	−12.2094	−0.460485
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.29844	0.0488328
$708$	0	0
$709$	−9.40312	−0.353142	−0.176571	−	0.984288i	$-0.556500\pi$
$709$	−9.40312	−0.353142	−0.176571	+	0.984288i	$0.556500\pi$
$710$	0	0
$711$	1.40312	0.0526213
$712$	0	0
$713$	−9.40312	−0.352150
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	21.6125	0.806010	0.403005	−	0.915198i	$-0.367966\pi$
$719$	21.6125	0.806010	0.403005	+	0.915198i	$0.367966\pi$
$720$	0	0
$721$	8.70156	0.324063
$722$	0	0
$723$	−16.1047	−0.598940
$724$	0	0
$725$	−10.0000	−0.371391
$726$	0	0
$727$	32.9109	1.22060	0.610300	−	0.792171i	$-0.291049\pi$
$727$	32.9109	1.22060	0.610300	+	0.792171i	$0.291049\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	84.4187	3.12234
$732$	0	0
$733$	21.4031	0.790542	0.395271	−	0.918564i	$-0.370651\pi$
$733$	21.4031	0.790542	0.395271	+	0.918564i	$0.370651\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.61250	−0.0593971
$738$	0	0
$739$	13.1938	0.485340	0.242670	−	0.970109i	$-0.421977\pi$
$739$	13.1938	0.485340	0.242670	+	0.970109i	$0.421977\pi$
$740$	0	0
$741$	2.59688	0.0953986
$742$	0	0
$743$	−38.1047	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$743$	−38.1047	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.0000	0.365881
$748$	0	0
$749$	−15.4031	−0.562818
$750$	0	0
$751$	1.61250	0.0588408	0.0294204	−	0.999567i	$-0.490634\pi$
$751$	1.61250	0.0588408	0.0294204	+	0.999567i	$0.490634\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.8062	0.828907	0.414454	−	0.910070i	$-0.363973\pi$
$757$	22.8062	0.828907	0.414454	+	0.910070i	$0.363973\pi$
$758$	0	0
$759$	2.70156	0.0980605
$760$	0	0
$761$	13.2984	0.482068	0.241034	−	0.970517i	$-0.422513\pi$
$761$	13.2984	0.482068	0.241034	+	0.970517i	$0.422513\pi$
$762$	0	0
$763$	14.8062	0.536022
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.2094	1.01858
$768$	0	0
$769$	27.6125	0.995732	0.497866	−	0.867254i	$-0.334117\pi$
$769$	27.6125	0.995732	0.497866	+	0.867254i	$0.334117\pi$
$770$	0	0
$771$	−24.1047	−0.868109
$772$	0	0
$773$	−22.8062	−0.820284	−0.410142	−	0.912022i	$-0.634521\pi$
$773$	−22.8062	−0.820284	−0.410142	+	0.912022i	$0.634521\pi$
$774$	0	0
$775$	−47.0156	−1.68885
$776$	0	0
$777$	9.40312	0.337335
$778$	0	0
$779$	3.50781	0.125680
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14.4922	−0.516591	−0.258295	−	0.966066i	$-0.583161\pi$
$787$	−14.4922	−0.516591	−0.258295	+	0.966066i	$0.583161\pi$
$788$	0	0
$789$	1.89531	0.0674750
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	25.4031	0.902091
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.2094	0.857540	0.428770	−	0.903414i	$-0.358947\pi$
$797$	24.2094	0.857540	0.428770	+	0.903414i	$0.358947\pi$
$798$	0	0
$799$	−94.0312	−3.32659
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.6125	−1.11281
$808$	0	0
$809$	44.8062	1.57530	0.787652	−	0.616121i	$-0.211297\pi$
$809$	44.8062	1.57530	0.787652	+	0.616121i	$0.211297\pi$
$810$	0	0
$811$	−36.2094	−1.27148	−0.635742	−	0.771902i	$-0.719306\pi$
$811$	−36.2094	−1.27148	−0.635742	+	0.771902i	$0.719306\pi$
$812$	0	0
$813$	−10.8062	−0.378992
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.8062	−0.518005
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	26.2094	0.914713	0.457357	−	0.889283i	$-0.348796\pi$
$821$	26.2094	0.914713	0.457357	+	0.889283i	$0.348796\pi$
$822$	0	0
$823$	50.3141	1.75384	0.876919	−	0.480638i	$-0.159595\pi$
$823$	50.3141	1.75384	0.876919	+	0.480638i	$0.159595\pi$
$824$	0	0
$825$	13.5078	0.470282
$826$	0	0
$827$	41.7172	1.45065	0.725324	−	0.688407i	$-0.241690\pi$
$827$	41.7172	1.45065	0.725324	+	0.688407i	$0.241690\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	16.1047	0.558665
$832$	0	0
$833$	−7.40312	−0.256503
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.40312	0.325020
$838$	0	0
$839$	12.2094	0.421514	0.210757	−	0.977538i	$-0.432407\pi$
$839$	12.2094	0.421514	0.210757	+	0.977538i	$0.432407\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−4.80625	−0.165536
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.70156	0.127187
$848$	0	0
$849$	16.5969	0.569603
$850$	0	0
$851$	9.40312	0.322335
$852$	0	0
$853$	25.0156	0.856519	0.428259	−	0.903656i	$-0.359127\pi$
$853$	25.0156	0.856519	0.428259	+	0.903656i	$0.359127\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.2984	−0.864178	−0.432089	−	0.901831i	$-0.642223\pi$
$857$	−25.2984	−0.864178	−0.432089	+	0.901831i	$0.642223\pi$
$858$	0	0
$859$	−38.8062	−1.32405	−0.662026	−	0.749481i	$-0.730303\pi$
$859$	−38.8062	−1.32405	−0.662026	+	0.749481i	$0.730303\pi$
$860$	0	0
$861$	−2.70156	−0.0920690
$862$	0	0
$863$	−1.40312	−0.0477629	−0.0238815	−	0.999715i	$-0.507602\pi$
$863$	−1.40312	−0.0477629	−0.0238815	+	0.999715i	$0.507602\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37.8062	1.28397
$868$	0	0
$869$	−3.79063	−0.128588
$870$	0	0
$871$	1.19375	0.0404487
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.1047	0.813957	0.406979	−	0.913438i	$-0.366582\pi$
$877$	24.1047	0.813957	0.406979	+	0.913438i	$0.366582\pi$
$878$	0	0
$879$	−29.6125	−0.998805
$880$	0	0
$881$	−39.6125	−1.33458	−0.667289	−	0.744798i	$-0.732546\pi$
$881$	−39.6125	−1.33458	−0.667289	+	0.744798i	$0.732546\pi$
$882$	0	0
$883$	−17.1938	−0.578616	−0.289308	−	0.957236i	$-0.593425\pi$
$883$	−17.1938	−0.578616	−0.289308	+	0.957236i	$0.593425\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−6.10469	−0.204745
$890$	0	0
$891$	−2.70156	−0.0905057
$892$	0	0
$893$	16.4922	0.551890
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	18.8062	0.627224
$900$	0	0
$901$	−74.8062	−2.49216
$902$	0	0
$903$	11.4031	0.379472
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.79063	0.0594568	0.0297284	−	0.999558i	$-0.490536\pi$
$907$	1.79063	0.0594568	0.0297284	+	0.999558i	$0.490536\pi$
$908$	0	0
$909$	−1.29844	−0.0430665
$910$	0	0
$911$	22.8062	0.755605	0.377802	−	0.925886i	$-0.376680\pi$
$911$	22.8062	0.755605	0.377802	+	0.925886i	$0.376680\pi$
$912$	0	0
$913$	−27.0156	−0.894087
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.7016	−0.683626
$918$	0	0
$919$	−9.40312	−0.310180	−0.155090	−	0.987900i	$-0.549567\pi$
$919$	−9.40312	−0.310180	−0.155090	+	0.987900i	$0.549567\pi$
$920$	0	0
$921$	−22.8062	−0.751491
$922$	0	0
$923$	0	0
$924$	0	0
$925$	47.0156	1.54586
$926$	0	0
$927$	−8.70156	−0.285797
$928$	0	0
$929$	−26.4187	−0.866771	−0.433385	−	0.901209i	$-0.642681\pi$
$929$	−26.4187	−0.866771	−0.433385	+	0.901209i	$0.642681\pi$
$930$	0	0
$931$	1.29844	0.0425546
$932$	0	0
$933$	11.2984	0.369894
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.1047	−0.656791	−0.328396	−	0.944540i	$-0.606508\pi$
$937$	−20.1047	−0.656791	−0.328396	+	0.944540i	$0.606508\pi$
$938$	0	0
$939$	22.9109	0.747670
$940$	0	0
$941$	24.4187	0.796028	0.398014	−	0.917379i	$-0.369699\pi$
$941$	24.4187	0.796028	0.398014	+	0.917379i	$0.369699\pi$
$942$	0	0
$943$	−2.70156	−0.0879750
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.1938	−0.688704	−0.344352	−	0.938841i	$-0.611901\pi$
$947$	−21.1938	−0.688704	−0.344352	+	0.938841i	$0.611901\pi$
$948$	0	0
$949$	14.8062	0.480631
$950$	0	0
$951$	31.6125	1.02511
$952$	0	0
$953$	11.8953	0.385327	0.192664	−	0.981265i	$-0.438287\pi$
$953$	11.8953	0.385327	0.192664	+	0.981265i	$0.438287\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.40312	−0.174658
$958$	0	0
$959$	−17.5078	−0.565357
$960$	0	0
$961$	57.4187	1.85222
$962$	0	0
$963$	15.4031	0.496359
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.1938	−0.424282	−0.212141	−	0.977239i	$-0.568044\pi$
$967$	−13.1938	−0.424282	−0.212141	+	0.977239i	$0.568044\pi$
$968$	0	0
$969$	−9.61250	−0.308798
$970$	0	0
$971$	11.1938	0.359225	0.179612	−	0.983737i	$-0.442516\pi$
$971$	11.1938	0.359225	0.179612	+	0.983737i	$0.442516\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	−10.0000	−0.320256
$976$	0	0
$977$	30.7016	0.982230	0.491115	−	0.871095i	$-0.336589\pi$
$977$	30.7016	0.982230	0.491115	+	0.871095i	$0.336589\pi$
$978$	0	0
$979$	5.40312	0.172685
$980$	0	0
$981$	−14.8062	−0.472727
$982$	0	0
$983$	29.4031	0.937814	0.468907	−	0.883248i	$-0.344648\pi$
$983$	29.4031	0.937814	0.468907	+	0.883248i	$0.344648\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.7016	−0.404295
$988$	0	0
$989$	11.4031	0.362598
$990$	0	0
$991$	56.9109	1.80784	0.903918	−	0.427706i	$-0.140678\pi$
$991$	56.9109	1.80784	0.903918	+	0.427706i	$0.140678\pi$
$992$	0	0
$993$	−8.70156	−0.276136
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−49.0156	−1.55234	−0.776170	−	0.630524i	$-0.782840\pi$
$997$	−49.0156	−1.55234	−0.776170	+	0.630524i	$0.782840\pi$
$998$	0	0
$999$	−9.40312	−0.297502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bi.1.1		2
4.3	odd	2		3864.2.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.g.1.2	✓	2	4.3	odd	2
7728.2.a.bi.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.70156 q^{11} +2.00000 q^{13} -7.40312 q^{17} +1.29844 q^{19} -1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} +9.40312 q^{31} -2.70156 q^{33} -9.40312 q^{37} +2.00000 q^{39} +2.70156 q^{41} -11.4031 q^{43} +12.7016 q^{47} +1.00000 q^{49} -7.40312 q^{51} +10.1047 q^{53} +1.29844 q^{57} +14.1047 q^{59} +12.7016 q^{61} -1.00000 q^{63} +0.596876 q^{67} -1.00000 q^{69} +7.40312 q^{73} -5.00000 q^{75} +2.70156 q^{77} +1.40312 q^{79} +1.00000 q^{81} +10.0000 q^{83} +2.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} +9.40312 q^{93} +6.00000 q^{97} -2.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + q^{11} + 4 q^{13} - 2 q^{17} + 9 q^{19} - 2 q^{21} - 2 q^{23} - 10 q^{25} + 2 q^{27} + 4 q^{29} + 6 q^{31} + q^{33} - 6 q^{37} + 4 q^{39} - q^{41} - 10 q^{43} + 19 q^{47} + 2 q^{49} - 2 q^{51} + q^{53} + 9 q^{57} + 9 q^{59} + 19 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 2 q^{73} - 10 q^{75} - q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} + 4 q^{87} - 4 q^{89} - 4 q^{91} + 6 q^{93} + 12 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + q^11 + 4 * q^13 - 2 * q^17 + 9 * q^19 - 2 * q^21 - 2 * q^23 - 10 * q^25 + 2 * q^27 + 4 * q^29 + 6 * q^31 + q^33 - 6 * q^37 + 4 * q^39 - q^41 - 10 * q^43 + 19 * q^47 + 2 * q^49 - 2 * q^51 + q^53 + 9 * q^57 + 9 * q^59 + 19 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 2 * q^73 - 10 * q^75 - q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 - 4 * q^91 + 6 * q^93 + 12 * q^97 + q^99

Introduction

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