LMFDB - Embedded newform 7728.2.a.bp.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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			$-1.86081$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.86081 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.86081 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.92520 q^{11} -4.46260 q^{13} +2.86081 q^{15} +7.97021 q^{17} -3.64681 q^{19} +1.00000 q^{21} +1.00000 q^{23} +3.18421 q^{25} -1.00000 q^{27} -2.39821 q^{29} +4.11982 q^{31} -1.92520 q^{33} +2.86081 q^{35} -9.04502 q^{37} +4.46260 q^{39} +1.64681 q^{41} +0.740987 q^{43} -2.86081 q^{45} -5.11982 q^{47} +1.00000 q^{49} -7.97021 q^{51} -4.25901 q^{53} -5.50761 q^{55} +3.64681 q^{57} -9.98062 q^{59} -7.06439 q^{61} -1.00000 q^{63} +12.7666 q^{65} +8.71120 q^{67} -1.00000 q^{69} +1.66618 q^{71} +3.04502 q^{73} -3.18421 q^{75} -1.92520 q^{77} +4.11982 q^{79} +1.00000 q^{81} +13.8414 q^{83} -22.8012 q^{85} +2.39821 q^{87} -12.8310 q^{89} +4.46260 q^{91} -4.11982 q^{93} +10.4328 q^{95} -18.2099 q^{97} +1.92520 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + q^11 - 11 * q^13 + 3 * q^15 + 5 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 - q^33 + 3 * q^35 - 8 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 3 * q^45 - q^47 + 3 * q^49 - 4 * q^53 + 5 * q^55 - 5 * q^57 - 10 * q^59 - 22 * q^61 - 3 * q^63 + 8 * q^65 + 11 * q^67 - 3 * q^69 + 9 * q^71 - 10 * q^73 + 4 * q^75 - q^77 - 2 * q^79 + 3 * q^81 + 16 * q^83 - 15 * q^85 + 4 * q^87 - 9 * q^89 + 11 * q^91 + 2 * q^93 + 5 * q^95 - 2 * 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
$5$	−2.86081	−1.27939	−0.639696	−	0.768628i	$-0.720940\pi$
$5$	−2.86081	−1.27939	−0.639696	+	0.768628i	$0.720940\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.92520	0.580469	0.290234	−	0.956956i	$-0.406267\pi$
$11$	1.92520	0.580469	0.290234	+	0.956956i	$0.406267\pi$
$12$	0	0
$13$	−4.46260	−1.23770	−0.618851	−	0.785508i	$-0.712402\pi$
$13$	−4.46260	−1.23770	−0.618851	+	0.785508i	$0.712402\pi$
$14$	0	0
$15$	2.86081	0.738657
$16$	0	0
$17$	7.97021	1.93306	0.966530	−	0.256553i	$-0.0825867\pi$
$17$	7.97021	1.93306	0.966530	+	0.256553i	$0.0825867\pi$
$18$	0	0
$19$	−3.64681	−0.836635	−0.418318	−	0.908301i	$-0.637380\pi$
$19$	−3.64681	−0.836635	−0.418318	+	0.908301i	$0.637380\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	3.18421	0.636842
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.39821	−0.445336	−0.222668	−	0.974894i	$-0.571477\pi$
$29$	−2.39821	−0.445336	−0.222668	+	0.974894i	$0.571477\pi$
$30$	0	0
$31$	4.11982	0.739941	0.369971	−	0.929043i	$-0.379368\pi$
$31$	4.11982	0.739941	0.369971	+	0.929043i	$0.379368\pi$
$32$	0	0
$33$	−1.92520	−0.335134
$34$	0	0
$35$	2.86081	0.483564
$36$	0	0
$37$	−9.04502	−1.48699	−0.743496	−	0.668741i	$-0.766834\pi$
$37$	−9.04502	−1.48699	−0.743496	+	0.668741i	$0.766834\pi$
$38$	0	0
$39$	4.46260	0.714588
$40$	0	0
$41$	1.64681	0.257188	0.128594	−	0.991697i	$-0.458954\pi$
$41$	1.64681	0.257188	0.128594	+	0.991697i	$0.458954\pi$
$42$	0	0
$43$	0.740987	0.112999	0.0564997	−	0.998403i	$-0.482006\pi$
$43$	0.740987	0.112999	0.0564997	+	0.998403i	$0.482006\pi$
$44$	0	0
$45$	−2.86081	−0.426464
$46$	0	0
$47$	−5.11982	−0.746802	−0.373401	−	0.927670i	$-0.621809\pi$
$47$	−5.11982	−0.746802	−0.373401	+	0.927670i	$0.621809\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.97021	−1.11605
$52$	0	0
$53$	−4.25901	−0.585020	−0.292510	−	0.956262i	$-0.594491\pi$
$53$	−4.25901	−0.585020	−0.292510	+	0.956262i	$0.594491\pi$
$54$	0	0
$55$	−5.50761	−0.742647
$56$	0	0
$57$	3.64681	0.483032
$58$	0	0
$59$	−9.98062	−1.29937	−0.649683	−	0.760205i	$-0.725099\pi$
$59$	−9.98062	−1.29937	−0.649683	+	0.760205i	$0.725099\pi$
$60$	0	0
$61$	−7.06439	−0.904503	−0.452251	−	0.891891i	$-0.649379\pi$
$61$	−7.06439	−0.904503	−0.452251	+	0.891891i	$0.649379\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	12.7666	1.58351
$66$	0	0
$67$	8.71120	1.06424	0.532121	−	0.846668i	$-0.321395\pi$
$67$	8.71120	1.06424	0.532121	+	0.846668i	$0.321395\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	1.66618	0.197740	0.0988698	−	0.995100i	$-0.468477\pi$
$71$	1.66618	0.197740	0.0988698	+	0.995100i	$0.468477\pi$
$72$	0	0
$73$	3.04502	0.356392	0.178196	−	0.983995i	$-0.442974\pi$
$73$	3.04502	0.356392	0.178196	+	0.983995i	$0.442974\pi$
$74$	0	0
$75$	−3.18421	−0.367681
$76$	0	0
$77$	−1.92520	−0.219397
$78$	0	0
$79$	4.11982	0.463516	0.231758	−	0.972773i	$-0.425552\pi$
$79$	4.11982	0.463516	0.231758	+	0.972773i	$0.425552\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8414	1.51929	0.759647	−	0.650335i	$-0.225372\pi$
$83$	13.8414	1.51929	0.759647	+	0.650335i	$0.225372\pi$
$84$	0	0
$85$	−22.8012	−2.47314
$86$	0	0
$87$	2.39821	0.257115
$88$	0	0
$89$	−12.8310	−1.36009	−0.680043	−	0.733173i	$-0.738039\pi$
$89$	−12.8310	−1.36009	−0.680043	+	0.733173i	$0.738039\pi$
$90$	0	0
$91$	4.46260	0.467807
$92$	0	0
$93$	−4.11982	−0.427205
$94$	0	0
$95$	10.4328	1.07038
$96$	0	0
$97$	−18.2099	−1.84893	−0.924465	−	0.381267i	$-0.875488\pi$
$97$	−18.2099	−1.84893	−0.924465	+	0.381267i	$0.875488\pi$
$98$	0	0
$99$	1.92520	0.193490
$100$	0	0
$101$	−14.7770	−1.47037	−0.735185	−	0.677866i	$-0.762905\pi$
$101$	−14.7770	−1.47037	−0.735185	+	0.677866i	$0.762905\pi$
$102$	0	0
$103$	−2.04502	−0.201501	−0.100751	−	0.994912i	$-0.532124\pi$
$103$	−2.04502	−0.201501	−0.100751	+	0.994912i	$0.532124\pi$
$104$	0	0
$105$	−2.86081	−0.279186
$106$	0	0
$107$	−12.9806	−1.25488	−0.627442	−	0.778663i	$-0.715898\pi$
$107$	−12.9806	−1.25488	−0.627442	+	0.778663i	$0.715898\pi$
$108$	0	0
$109$	5.37883	0.515199	0.257599	−	0.966252i	$-0.417069\pi$
$109$	5.37883	0.515199	0.257599	+	0.966252i	$0.417069\pi$
$110$	0	0
$111$	9.04502	0.858515
$112$	0	0
$113$	−7.87603	−0.740915	−0.370458	−	0.928849i	$-0.620799\pi$
$113$	−7.87603	−0.740915	−0.370458	+	0.928849i	$0.620799\pi$
$114$	0	0
$115$	−2.86081	−0.266772
$116$	0	0
$117$	−4.46260	−0.412567
$118$	0	0
$119$	−7.97021	−0.730628
$120$	0	0
$121$	−7.29362	−0.663056
$122$	0	0
$123$	−1.64681	−0.148488
$124$	0	0
$125$	5.19462	0.464621
$126$	0	0
$127$	2.93561	0.260493	0.130247	−	0.991482i	$-0.458423\pi$
$127$	2.93561	0.260493	0.130247	+	0.991482i	$0.458423\pi$
$128$	0	0
$129$	−0.740987	−0.0652402
$130$	0	0
$131$	6.85039	0.598522	0.299261	−	0.954171i	$-0.403260\pi$
$131$	6.85039	0.598522	0.299261	+	0.954171i	$0.403260\pi$
$132$	0	0
$133$	3.64681	0.316218
$134$	0	0
$135$	2.86081	0.246219
$136$	0	0
$137$	21.2188	1.81285	0.906423	−	0.422372i	$-0.138802\pi$
$137$	21.2188	1.81285	0.906423	+	0.422372i	$0.138802\pi$
$138$	0	0
$139$	16.0256	1.35928	0.679639	−	0.733547i	$-0.262137\pi$
$139$	16.0256	1.35928	0.679639	+	0.733547i	$0.262137\pi$
$140$	0	0
$141$	5.11982	0.431167
$142$	0	0
$143$	−8.59138	−0.718447
$144$	0	0
$145$	6.86081	0.569759
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	6.32340	0.518033	0.259017	−	0.965873i	$-0.416602\pi$
$149$	6.32340	0.518033	0.259017	+	0.965873i	$0.416602\pi$
$150$	0	0
$151$	15.9162	1.29524	0.647622	−	0.761961i	$-0.275763\pi$
$151$	15.9162	1.29524	0.647622	+	0.761961i	$0.275763\pi$
$152$	0	0
$153$	7.97021	0.644354
$154$	0	0
$155$	−11.7860	−0.946675
$156$	0	0
$157$	1.61702	0.129052	0.0645262	−	0.997916i	$-0.479446\pi$
$157$	1.61702	0.129052	0.0645262	+	0.997916i	$0.479446\pi$
$158$	0	0
$159$	4.25901	0.337762
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	3.46260	0.271212	0.135606	−	0.990763i	$-0.456702\pi$
$163$	3.46260	0.271212	0.135606	+	0.990763i	$0.456702\pi$
$164$	0	0
$165$	5.50761	0.428767
$166$	0	0
$167$	7.94602	0.614882	0.307441	−	0.951567i	$-0.400527\pi$
$167$	7.94602	0.614882	0.307441	+	0.951567i	$0.400527\pi$
$168$	0	0
$169$	6.91478	0.531907
$170$	0	0
$171$	−3.64681	−0.278878
$172$	0	0
$173$	−13.5422	−1.02960	−0.514798	−	0.857312i	$-0.672133\pi$
$173$	−13.5422	−1.02960	−0.514798	+	0.857312i	$0.672133\pi$
$174$	0	0
$175$	−3.18421	−0.240704
$176$	0	0
$177$	9.98062	0.750190
$178$	0	0
$179$	19.5976	1.46480	0.732398	−	0.680876i	$-0.238401\pi$
$179$	19.5976	1.46480	0.732398	+	0.680876i	$0.238401\pi$
$180$	0	0
$181$	−16.9162	−1.25737	−0.628687	−	0.777659i	$-0.716407\pi$
$181$	−16.9162	−1.25737	−0.628687	+	0.777659i	$0.716407\pi$
$182$	0	0
$183$	7.06439	0.522215
$184$	0	0
$185$	25.8760	1.90244
$186$	0	0
$187$	15.3442	1.12208
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−21.2099	−1.53469	−0.767345	−	0.641234i	$-0.778423\pi$
$191$	−21.2099	−1.53469	−0.767345	+	0.641234i	$0.778423\pi$
$192$	0	0
$193$	−6.17380	−0.444400	−0.222200	−	0.975001i	$-0.571324\pi$
$193$	−6.17380	−0.444400	−0.222200	+	0.975001i	$0.571324\pi$
$194$	0	0
$195$	−12.7666	−0.914237
$196$	0	0
$197$	20.4030	1.45366	0.726828	−	0.686820i	$-0.240994\pi$
$197$	20.4030	1.45366	0.726828	+	0.686820i	$0.240994\pi$
$198$	0	0
$199$	2.38780	0.169266	0.0846332	−	0.996412i	$-0.473028\pi$
$199$	2.38780	0.169266	0.0846332	+	0.996412i	$0.473028\pi$
$200$	0	0
$201$	−8.71120	−0.614441
$202$	0	0
$203$	2.39821	0.168321
$204$	0	0
$205$	−4.71120	−0.329044
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−7.02082	−0.485641
$210$	0	0
$211$	8.63158	0.594222	0.297111	−	0.954843i	$-0.403977\pi$
$211$	8.63158	0.594222	0.297111	+	0.954843i	$0.403977\pi$
$212$	0	0
$213$	−1.66618	−0.114165
$214$	0	0
$215$	−2.11982	−0.144570
$216$	0	0
$217$	−4.11982	−0.279672
$218$	0	0
$219$	−3.04502	−0.205763
$220$	0	0
$221$	−35.5679	−2.39255
$222$	0	0
$223$	15.6364	1.04709	0.523545	−	0.851998i	$-0.324609\pi$
$223$	15.6364	1.04709	0.523545	+	0.851998i	$0.324609\pi$
$224$	0	0
$225$	3.18421	0.212281
$226$	0	0
$227$	−0.989588	−0.0656813	−0.0328406	−	0.999461i	$-0.510455\pi$
$227$	−0.989588	−0.0656813	−0.0328406	+	0.999461i	$0.510455\pi$
$228$	0	0
$229$	19.2534	1.27230	0.636151	−	0.771565i	$-0.280526\pi$
$229$	19.2534	1.27230	0.636151	+	0.771565i	$0.280526\pi$
$230$	0	0
$231$	1.92520	0.126669
$232$	0	0
$233$	13.9356	0.912952	0.456476	−	0.889736i	$-0.349111\pi$
$233$	13.9356	0.912952	0.456476	+	0.889736i	$0.349111\pi$
$234$	0	0
$235$	14.6468	0.955452
$236$	0	0
$237$	−4.11982	−0.267611
$238$	0	0
$239$	−30.2445	−1.95635	−0.978176	−	0.207780i	$-0.933376\pi$
$239$	−30.2445	−1.95635	−0.978176	+	0.207780i	$0.933376\pi$
$240$	0	0
$241$	4.07480	0.262481	0.131241	−	0.991351i	$-0.458104\pi$
$241$	4.07480	0.262481	0.131241	+	0.991351i	$0.458104\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.86081	−0.182770
$246$	0	0
$247$	16.2742	1.03551
$248$	0	0
$249$	−13.8414	−0.877165
$250$	0	0
$251$	6.62598	0.418228	0.209114	−	0.977891i	$-0.432942\pi$
$251$	6.62598	0.418228	0.209114	+	0.977891i	$0.432942\pi$
$252$	0	0
$253$	1.92520	0.121036
$254$	0	0
$255$	22.8012	1.42787
$256$	0	0
$257$	−23.0602	−1.43846	−0.719229	−	0.694773i	$-0.755505\pi$
$257$	−23.0602	−1.43846	−0.719229	+	0.694773i	$0.755505\pi$
$258$	0	0
$259$	9.04502	0.562030
$260$	0	0
$261$	−2.39821	−0.148445
$262$	0	0
$263$	14.5928	0.899832	0.449916	−	0.893071i	$-0.351454\pi$
$263$	14.5928	0.899832	0.449916	+	0.893071i	$0.351454\pi$
$264$	0	0
$265$	12.1842	0.748470
$266$	0	0
$267$	12.8310	0.785246
$268$	0	0
$269$	−26.2534	−1.60070	−0.800350	−	0.599534i	$-0.795353\pi$
$269$	−26.2534	−1.60070	−0.800350	+	0.599534i	$0.795353\pi$
$270$	0	0
$271$	−12.3386	−0.749519	−0.374759	−	0.927122i	$-0.622275\pi$
$271$	−12.3386	−0.749519	−0.374759	+	0.927122i	$0.622275\pi$
$272$	0	0
$273$	−4.46260	−0.270089
$274$	0	0
$275$	6.13023	0.369667
$276$	0	0
$277$	19.2832	1.15862	0.579308	−	0.815109i	$-0.303323\pi$
$277$	19.2832	1.15862	0.579308	+	0.815109i	$0.303323\pi$
$278$	0	0
$279$	4.11982	0.246647
$280$	0	0
$281$	29.5124	1.76056	0.880282	−	0.474451i	$-0.157353\pi$
$281$	29.5124	1.76056	0.880282	+	0.474451i	$0.157353\pi$
$282$	0	0
$283$	18.4328	1.09572	0.547858	−	0.836571i	$-0.315443\pi$
$283$	18.4328	1.09572	0.547858	+	0.836571i	$0.315443\pi$
$284$	0	0
$285$	−10.4328	−0.617986
$286$	0	0
$287$	−1.64681	−0.0972080
$288$	0	0
$289$	46.5243	2.73672
$290$	0	0
$291$	18.2099	1.06748
$292$	0	0
$293$	−16.9944	−0.992824	−0.496412	−	0.868087i	$-0.665349\pi$
$293$	−16.9944	−0.992824	−0.496412	+	0.868087i	$0.665349\pi$
$294$	0	0
$295$	28.5526	1.66240
$296$	0	0
$297$	−1.92520	−0.111711
$298$	0	0
$299$	−4.46260	−0.258079
$300$	0	0
$301$	−0.740987	−0.0427098
$302$	0	0
$303$	14.7770	0.848919
$304$	0	0
$305$	20.2099	1.15721
$306$	0	0
$307$	−0.0297872	−0.00170004	−0.000850022	−	1.00000i	$-0.500271\pi$
$307$	−0.0297872	−0.00170004	−0.000850022		1.00000i	$0.500271\pi$
$308$	0	0
$309$	2.04502	0.116337
$310$	0	0
$311$	−2.80683	−0.159161	−0.0795803	−	0.996828i	$-0.525358\pi$
$311$	−2.80683	−0.159161	−0.0795803	+	0.996828i	$0.525358\pi$
$312$	0	0
$313$	5.30818	0.300036	0.150018	−	0.988683i	$-0.452067\pi$
$313$	5.30818	0.300036	0.150018	+	0.988683i	$0.452067\pi$
$314$	0	0
$315$	2.86081	0.161188
$316$	0	0
$317$	24.0048	1.34824	0.674122	−	0.738620i	$-0.264522\pi$
$317$	24.0048	1.34824	0.674122	+	0.738620i	$0.264522\pi$
$318$	0	0
$319$	−4.61702	−0.258504
$320$	0	0
$321$	12.9806	0.724508
$322$	0	0
$323$	−29.0658	−1.61727
$324$	0	0
$325$	−14.2099	−0.788221
$326$	0	0
$327$	−5.37883	−0.297450
$328$	0	0
$329$	5.11982	0.282265
$330$	0	0
$331$	6.45219	0.354644	0.177322	−	0.984153i	$-0.443257\pi$
$331$	6.45219	0.354644	0.177322	+	0.984153i	$0.443257\pi$
$332$	0	0
$333$	−9.04502	−0.495664
$334$	0	0
$335$	−24.9211	−1.36158
$336$	0	0
$337$	0.333816	0.0181841	0.00909207	−	0.999959i	$-0.497106\pi$
$337$	0.333816	0.0181841	0.00909207	+	0.999959i	$0.497106\pi$
$338$	0	0
$339$	7.87603	0.427767
$340$	0	0
$341$	7.93146	0.429513
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.86081	0.154021
$346$	0	0
$347$	22.5270	1.20931	0.604656	−	0.796487i	$-0.293311\pi$
$347$	22.5270	1.20931	0.604656	+	0.796487i	$0.293311\pi$
$348$	0	0
$349$	24.2445	1.29778	0.648888	−	0.760884i	$-0.275234\pi$
$349$	24.2445	1.29778	0.648888	+	0.760884i	$0.275234\pi$
$350$	0	0
$351$	4.46260	0.238196
$352$	0	0
$353$	−29.2251	−1.55549	−0.777747	−	0.628577i	$-0.783638\pi$
$353$	−29.2251	−1.55549	−0.777747	+	0.628577i	$0.783638\pi$
$354$	0	0
$355$	−4.76663	−0.252986
$356$	0	0
$357$	7.97021	0.421828
$358$	0	0
$359$	9.87603	0.521237	0.260619	−	0.965442i	$-0.416073\pi$
$359$	9.87603	0.521237	0.260619	+	0.965442i	$0.416073\pi$
$360$	0	0
$361$	−5.70079	−0.300041
$362$	0	0
$363$	7.29362	0.382816
$364$	0	0
$365$	−8.71120	−0.455965
$366$	0	0
$367$	25.0048	1.30524	0.652620	−	0.757685i	$-0.273670\pi$
$367$	25.0048	1.30524	0.652620	+	0.757685i	$0.273670\pi$
$368$	0	0
$369$	1.64681	0.0857294
$370$	0	0
$371$	4.25901	0.221117
$372$	0	0
$373$	15.9910	0.827985	0.413992	−	0.910280i	$-0.364134\pi$
$373$	15.9910	0.827985	0.413992	+	0.910280i	$0.364134\pi$
$374$	0	0
$375$	−5.19462	−0.268249
$376$	0	0
$377$	10.7022	0.551193
$378$	0	0
$379$	−10.4972	−0.539205	−0.269603	−	0.962972i	$-0.586892\pi$
$379$	−10.4972	−0.539205	−0.269603	+	0.962972i	$0.586892\pi$
$380$	0	0
$381$	−2.93561	−0.150396
$382$	0	0
$383$	17.7819	0.908610	0.454305	−	0.890846i	$-0.349888\pi$
$383$	17.7819	0.908610	0.454305	+	0.890846i	$0.349888\pi$
$384$	0	0
$385$	5.50761	0.280694
$386$	0	0
$387$	0.740987	0.0376665
$388$	0	0
$389$	23.9315	1.21337	0.606687	−	0.794941i	$-0.292498\pi$
$389$	23.9315	1.21337	0.606687	+	0.794941i	$0.292498\pi$
$390$	0	0
$391$	7.97021	0.403071
$392$	0	0
$393$	−6.85039	−0.345557
$394$	0	0
$395$	−11.7860	−0.593018
$396$	0	0
$397$	17.8296	0.894840	0.447420	−	0.894324i	$-0.352343\pi$
$397$	17.8296	0.894840	0.447420	+	0.894324i	$0.352343\pi$
$398$	0	0
$399$	−3.64681	−0.182569
$400$	0	0
$401$	−24.4737	−1.22216	−0.611079	−	0.791570i	$-0.709264\pi$
$401$	−24.4737	−1.22216	−0.611079	+	0.791570i	$0.709264\pi$
$402$	0	0
$403$	−18.3851	−0.915827
$404$	0	0
$405$	−2.86081	−0.142155
$406$	0	0
$407$	−17.4134	−0.863152
$408$	0	0
$409$	−5.47301	−0.270623	−0.135311	−	0.990803i	$-0.543204\pi$
$409$	−5.47301	−0.270623	−0.135311	+	0.990803i	$0.543204\pi$
$410$	0	0
$411$	−21.2188	−1.04665
$412$	0	0
$413$	9.98062	0.491114
$414$	0	0
$415$	−39.5976	−1.94377
$416$	0	0
$417$	−16.0256	−0.784779
$418$	0	0
$419$	−7.26798	−0.355064	−0.177532	−	0.984115i	$-0.556811\pi$
$419$	−7.26798	−0.355064	−0.177532	+	0.984115i	$0.556811\pi$
$420$	0	0
$421$	34.1038	1.66212	0.831059	−	0.556184i	$-0.187735\pi$
$421$	34.1038	1.66212	0.831059	+	0.556184i	$0.187735\pi$
$422$	0	0
$423$	−5.11982	−0.248934
$424$	0	0
$425$	25.3788	1.23105
$426$	0	0
$427$	7.06439	0.341870
$428$	0	0
$429$	8.59138	0.414796
$430$	0	0
$431$	15.9717	0.769328	0.384664	−	0.923057i	$-0.374317\pi$
$431$	15.9717	0.769328	0.384664	+	0.923057i	$0.374317\pi$
$432$	0	0
$433$	−11.2099	−0.538711	−0.269356	−	0.963041i	$-0.586811\pi$
$433$	−11.2099	−0.538711	−0.269356	+	0.963041i	$0.586811\pi$
$434$	0	0
$435$	−6.86081	−0.328950
$436$	0	0
$437$	−3.64681	−0.174451
$438$	0	0
$439$	12.7666	0.609318	0.304659	−	0.952462i	$-0.401458\pi$
$439$	12.7666	0.609318	0.304659	+	0.952462i	$0.401458\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	9.59283	0.455769	0.227885	−	0.973688i	$-0.426819\pi$
$443$	9.59283	0.455769	0.227885	+	0.973688i	$0.426819\pi$
$444$	0	0
$445$	36.7071	1.74008
$446$	0	0
$447$	−6.32340	−0.299087
$448$	0	0
$449$	−13.7264	−0.647790	−0.323895	−	0.946093i	$-0.604992\pi$
$449$	−13.7264	−0.647790	−0.323895	+	0.946093i	$0.604992\pi$
$450$	0	0
$451$	3.17043	0.149290
$452$	0	0
$453$	−15.9162	−0.747810
$454$	0	0
$455$	−12.7666	−0.598509
$456$	0	0
$457$	40.7833	1.90776	0.953881	−	0.300184i	$-0.0970481\pi$
$457$	40.7833	1.90776	0.953881	+	0.300184i	$0.0970481\pi$
$458$	0	0
$459$	−7.97021	−0.372018
$460$	0	0
$461$	−15.3193	−0.713489	−0.356744	−	0.934202i	$-0.616113\pi$
$461$	−15.3193	−0.713489	−0.356744	+	0.934202i	$0.616113\pi$
$462$	0	0
$463$	22.8504	1.06195	0.530974	−	0.847388i	$-0.321826\pi$
$463$	22.8504	1.06195	0.530974	+	0.847388i	$0.321826\pi$
$464$	0	0
$465$	11.7860	0.546563
$466$	0	0
$467$	−4.50617	−0.208520	−0.104260	−	0.994550i	$-0.533247\pi$
$467$	−4.50617	−0.208520	−0.104260	+	0.994550i	$0.533247\pi$
$468$	0	0
$469$	−8.71120	−0.402246
$470$	0	0
$471$	−1.61702	−0.0745084
$472$	0	0
$473$	1.42655	0.0655926
$474$	0	0
$475$	−11.6122	−0.532804
$476$	0	0
$477$	−4.25901	−0.195007
$478$	0	0
$479$	−13.0063	−0.594271	−0.297136	−	0.954835i	$-0.596031\pi$
$479$	−13.0063	−0.594271	−0.297136	+	0.954835i	$0.596031\pi$
$480$	0	0
$481$	40.3643	1.84045
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	52.0948	2.36551
$486$	0	0
$487$	−17.9315	−0.812552	−0.406276	−	0.913750i	$-0.633173\pi$
$487$	−17.9315	−0.812552	−0.406276	+	0.913750i	$0.633173\pi$
$488$	0	0
$489$	−3.46260	−0.156584
$490$	0	0
$491$	−2.99855	−0.135323	−0.0676613	−	0.997708i	$-0.521554\pi$
$491$	−2.99855	−0.135323	−0.0676613	+	0.997708i	$0.521554\pi$
$492$	0	0
$493$	−19.1142	−0.860861
$494$	0	0
$495$	−5.50761	−0.247549
$496$	0	0
$497$	−1.66618	−0.0747385
$498$	0	0
$499$	21.1905	0.948616	0.474308	−	0.880359i	$-0.342698\pi$
$499$	21.1905	0.948616	0.474308	+	0.880359i	$0.342698\pi$
$500$	0	0
$501$	−7.94602	−0.355002
$502$	0	0
$503$	−35.9661	−1.60365	−0.801824	−	0.597561i	$-0.796137\pi$
$503$	−35.9661	−1.60365	−0.801824	+	0.597561i	$0.796137\pi$
$504$	0	0
$505$	42.2742	1.88118
$506$	0	0
$507$	−6.91478	−0.307096
$508$	0	0
$509$	−22.8027	−1.01071	−0.505356	−	0.862911i	$-0.668639\pi$
$509$	−22.8027	−1.01071	−0.505356	+	0.862911i	$0.668639\pi$
$510$	0	0
$511$	−3.04502	−0.134704
$512$	0	0
$513$	3.64681	0.161011
$514$	0	0
$515$	5.85039	0.257799
$516$	0	0
$517$	−9.85666	−0.433495
$518$	0	0
$519$	13.5422	0.594437
$520$	0	0
$521$	−4.14961	−0.181798	−0.0908988	−	0.995860i	$-0.528974\pi$
$521$	−4.14961	−0.181798	−0.0908988	+	0.995860i	$0.528974\pi$
$522$	0	0
$523$	−34.3747	−1.50310	−0.751550	−	0.659676i	$-0.770693\pi$
$523$	−34.3747	−1.50310	−0.751550	+	0.659676i	$0.770693\pi$
$524$	0	0
$525$	3.18421	0.138970
$526$	0	0
$527$	32.8358	1.43035
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.98062	−0.433122
$532$	0	0
$533$	−7.34905	−0.318322
$534$	0	0
$535$	37.1350	1.60549
$536$	0	0
$537$	−19.5976	−0.845701
$538$	0	0
$539$	1.92520	0.0829241
$540$	0	0
$541$	7.52699	0.323611	0.161805	−	0.986823i	$-0.448268\pi$
$541$	7.52699	0.323611	0.161805	+	0.986823i	$0.448268\pi$
$542$	0	0
$543$	16.9162	0.725945
$544$	0	0
$545$	−15.3878	−0.659141
$546$	0	0
$547$	16.9688	0.725532	0.362766	−	0.931880i	$-0.381832\pi$
$547$	16.9688	0.725532	0.362766	+	0.931880i	$0.381832\pi$
$548$	0	0
$549$	−7.06439	−0.301501
$550$	0	0
$551$	8.74580	0.372584
$552$	0	0
$553$	−4.11982	−0.175193
$554$	0	0
$555$	−25.8760	−1.09838
$556$	0	0
$557$	28.9557	1.22689	0.613445	−	0.789737i	$-0.289783\pi$
$557$	28.9557	1.22689	0.613445	+	0.789737i	$0.289783\pi$
$558$	0	0
$559$	−3.30673	−0.139860
$560$	0	0
$561$	−15.3442	−0.647834
$562$	0	0
$563$	15.8850	0.669473	0.334736	−	0.942312i	$-0.391353\pi$
$563$	15.8850	0.669473	0.334736	+	0.942312i	$0.391353\pi$
$564$	0	0
$565$	22.5318	0.947920
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	46.6918	1.95742	0.978712	−	0.205241i	$-0.0657977\pi$
$569$	46.6918	1.95742	0.978712	+	0.205241i	$0.0657977\pi$
$570$	0	0
$571$	41.5630	1.73936	0.869680	−	0.493617i	$-0.164325\pi$
$571$	41.5630	1.73936	0.869680	+	0.493617i	$0.164325\pi$
$572$	0	0
$573$	21.2099	0.886054
$574$	0	0
$575$	3.18421	0.132791
$576$	0	0
$577$	4.70638	0.195929	0.0979646	−	0.995190i	$-0.468767\pi$
$577$	4.70638	0.195929	0.0979646	+	0.995190i	$0.468767\pi$
$578$	0	0
$579$	6.17380	0.256574
$580$	0	0
$581$	−13.8414	−0.574239
$582$	0	0
$583$	−8.19944	−0.339586
$584$	0	0
$585$	12.7666	0.527835
$586$	0	0
$587$	−26.8373	−1.10769	−0.553847	−	0.832619i	$-0.686841\pi$
$587$	−26.8373	−1.10769	−0.553847	+	0.832619i	$0.686841\pi$
$588$	0	0
$589$	−15.0242	−0.619061
$590$	0	0
$591$	−20.4030	−0.839268
$592$	0	0
$593$	43.8421	1.80038	0.900190	−	0.435498i	$-0.143428\pi$
$593$	43.8421	1.80038	0.900190	+	0.435498i	$0.143428\pi$
$594$	0	0
$595$	22.8012	0.934759
$596$	0	0
$597$	−2.38780	−0.0977260
$598$	0	0
$599$	2.11500	0.0864167	0.0432083	−	0.999066i	$-0.486242\pi$
$599$	2.11500	0.0864167	0.0432083	+	0.999066i	$0.486242\pi$
$600$	0	0
$601$	10.6724	0.435338	0.217669	−	0.976023i	$-0.430155\pi$
$601$	10.6724	0.435338	0.217669	+	0.976023i	$0.430155\pi$
$602$	0	0
$603$	8.71120	0.354747
$604$	0	0
$605$	20.8656	0.848308
$606$	0	0
$607$	47.5679	1.93072	0.965360	−	0.260922i	$-0.0840265\pi$
$607$	47.5679	1.93072	0.965360	+	0.260922i	$0.0840265\pi$
$608$	0	0
$609$	−2.39821	−0.0971803
$610$	0	0
$611$	22.8477	0.924319
$612$	0	0
$613$	−30.9467	−1.24993	−0.624963	−	0.780655i	$-0.714886\pi$
$613$	−30.9467	−1.24993	−0.624963	+	0.780655i	$0.714886\pi$
$614$	0	0
$615$	4.71120	0.189974
$616$	0	0
$617$	−7.22026	−0.290677	−0.145338	−	0.989382i	$-0.546427\pi$
$617$	−7.22026	−0.290677	−0.145338	+	0.989382i	$0.546427\pi$
$618$	0	0
$619$	−29.6579	−1.19205	−0.596026	−	0.802965i	$-0.703254\pi$
$619$	−29.6579	−1.19205	−0.596026	+	0.802965i	$0.703254\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	12.8310	0.514064
$624$	0	0
$625$	−30.7819	−1.23127
$626$	0	0
$627$	7.02082	0.280385
$628$	0	0
$629$	−72.0907	−2.87444
$630$	0	0
$631$	−14.0811	−0.560559	−0.280279	−	0.959919i	$-0.590427\pi$
$631$	−14.0811	−0.560559	−0.280279	+	0.959919i	$0.590427\pi$
$632$	0	0
$633$	−8.63158	−0.343074
$634$	0	0
$635$	−8.39821	−0.333273
$636$	0	0
$637$	−4.46260	−0.176815
$638$	0	0
$639$	1.66618	0.0659132
$640$	0	0
$641$	−41.3732	−1.63414	−0.817072	−	0.576535i	$-0.804404\pi$
$641$	−41.3732	−1.63414	−0.817072	+	0.576535i	$0.804404\pi$
$642$	0	0
$643$	−10.2292	−0.403401	−0.201701	−	0.979447i	$-0.564647\pi$
$643$	−10.2292	−0.403401	−0.201701	+	0.979447i	$0.564647\pi$
$644$	0	0
$645$	2.11982	0.0834678
$646$	0	0
$647$	25.9452	1.02001	0.510006	−	0.860171i	$-0.329643\pi$
$647$	25.9452	1.02001	0.510006	+	0.860171i	$0.329643\pi$
$648$	0	0
$649$	−19.2147	−0.754242
$650$	0	0
$651$	4.11982	0.161468
$652$	0	0
$653$	13.1988	0.516508	0.258254	−	0.966077i	$-0.416853\pi$
$653$	13.1988	0.516508	0.258254	+	0.966077i	$0.416853\pi$
$654$	0	0
$655$	−19.5976	−0.765743
$656$	0	0
$657$	3.04502	0.118797
$658$	0	0
$659$	1.00627	0.0391986	0.0195993	−	0.999808i	$-0.493761\pi$
$659$	1.00627	0.0391986	0.0195993	+	0.999808i	$0.493761\pi$
$660$	0	0
$661$	14.8533	0.577726	0.288863	−	0.957370i	$-0.406723\pi$
$661$	14.8533	0.577726	0.288863	+	0.957370i	$0.406723\pi$
$662$	0	0
$663$	35.5679	1.38134
$664$	0	0
$665$	−10.4328	−0.404567
$666$	0	0
$667$	−2.39821	−0.0928590
$668$	0	0
$669$	−15.6364	−0.604538
$670$	0	0
$671$	−13.6003	−0.525035
$672$	0	0
$673$	−19.4376	−0.749265	−0.374633	−	0.927173i	$-0.622231\pi$
$673$	−19.4376	−0.749265	−0.374633	+	0.927173i	$0.622231\pi$
$674$	0	0
$675$	−3.18421	−0.122560
$676$	0	0
$677$	37.9750	1.45950	0.729749	−	0.683715i	$-0.239637\pi$
$677$	37.9750	1.45950	0.729749	+	0.683715i	$0.239637\pi$
$678$	0	0
$679$	18.2099	0.698830
$680$	0	0
$681$	0.989588	0.0379211
$682$	0	0
$683$	−37.7610	−1.44489	−0.722443	−	0.691431i	$-0.756981\pi$
$683$	−37.7610	−1.44489	−0.722443	+	0.691431i	$0.756981\pi$
$684$	0	0
$685$	−60.7029	−2.31934
$686$	0	0
$687$	−19.2534	−0.734564
$688$	0	0
$689$	19.0063	0.724081
$690$	0	0
$691$	24.4931	0.931760	0.465880	−	0.884848i	$-0.345738\pi$
$691$	24.4931	0.931760	0.465880	+	0.884848i	$0.345738\pi$
$692$	0	0
$693$	−1.92520	−0.0731322
$694$	0	0
$695$	−45.8462	−1.73905
$696$	0	0
$697$	13.1254	0.497161
$698$	0	0
$699$	−13.9356	−0.527093
$700$	0	0
$701$	−5.20214	−0.196482	−0.0982410	−	0.995163i	$-0.531322\pi$
$701$	−5.20214	−0.196482	−0.0982410	+	0.995163i	$0.531322\pi$
$702$	0	0
$703$	32.9854	1.24407
$704$	0	0
$705$	−14.6468	−0.551631
$706$	0	0
$707$	14.7770	0.555748
$708$	0	0
$709$	−28.7022	−1.07794	−0.538968	−	0.842327i	$-0.681185\pi$
$709$	−28.7022	−1.07794	−0.538968	+	0.842327i	$0.681185\pi$
$710$	0	0
$711$	4.11982	0.154505
$712$	0	0
$713$	4.11982	0.154288
$714$	0	0
$715$	24.5783	0.919175
$716$	0	0
$717$	30.2445	1.12950
$718$	0	0
$719$	−3.45508	−0.128853	−0.0644265	−	0.997922i	$-0.520522\pi$
$719$	−3.45508	−0.128853	−0.0644265	+	0.997922i	$0.520522\pi$
$720$	0	0
$721$	2.04502	0.0761604
$722$	0	0
$723$	−4.07480	−0.151544
$724$	0	0
$725$	−7.63640	−0.283609
$726$	0	0
$727$	34.5333	1.28077	0.640384	−	0.768055i	$-0.278775\pi$
$727$	34.5333	1.28077	0.640384	+	0.768055i	$0.278775\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.90582	0.218435
$732$	0	0
$733$	−32.4764	−1.19954	−0.599771	−	0.800172i	$-0.704742\pi$
$733$	−32.4764	−1.19954	−0.599771	+	0.800172i	$0.704742\pi$
$734$	0	0
$735$	2.86081	0.105522
$736$	0	0
$737$	16.7708	0.617759
$738$	0	0
$739$	8.08936	0.297572	0.148786	−	0.988869i	$-0.452463\pi$
$739$	8.08936	0.297572	0.148786	+	0.988869i	$0.452463\pi$
$740$	0	0
$741$	−16.2742	−0.597849
$742$	0	0
$743$	−47.7923	−1.75333	−0.876664	−	0.481103i	$-0.840236\pi$
$743$	−47.7923	−1.75333	−0.876664	+	0.481103i	$0.840236\pi$
$744$	0	0
$745$	−18.0900	−0.662767
$746$	0	0
$747$	13.8414	0.506431
$748$	0	0
$749$	12.9806	0.474302
$750$	0	0
$751$	−21.5076	−0.784824	−0.392412	−	0.919790i	$-0.628359\pi$
$751$	−21.5076	−0.784824	−0.392412	+	0.919790i	$0.628359\pi$
$752$	0	0
$753$	−6.62598	−0.241464
$754$	0	0
$755$	−45.5333	−1.65712
$756$	0	0
$757$	−20.0811	−0.729859	−0.364929	−	0.931035i	$-0.618907\pi$
$757$	−20.0811	−0.729859	−0.364929	+	0.931035i	$0.618907\pi$
$758$	0	0
$759$	−1.92520	−0.0698802
$760$	0	0
$761$	37.5366	1.36070	0.680351	−	0.732887i	$-0.261827\pi$
$761$	37.5366	1.36070	0.680351	+	0.732887i	$0.261827\pi$
$762$	0	0
$763$	−5.37883	−0.194727
$764$	0	0
$765$	−22.8012	−0.824380
$766$	0	0
$767$	44.5395	1.60823
$768$	0	0
$769$	25.2132	0.909212	0.454606	−	0.890693i	$-0.349780\pi$
$769$	25.2132	0.909212	0.454606	+	0.890693i	$0.349780\pi$
$770$	0	0
$771$	23.0602	0.830494
$772$	0	0
$773$	20.5187	0.738006	0.369003	−	0.929428i	$-0.379699\pi$
$773$	20.5187	0.738006	0.369003	+	0.929428i	$0.379699\pi$
$774$	0	0
$775$	13.1184	0.471226
$776$	0	0
$777$	−9.04502	−0.324488
$778$	0	0
$779$	−6.00560	−0.215173
$780$	0	0
$781$	3.20773	0.114782
$782$	0	0
$783$	2.39821	0.0857049
$784$	0	0
$785$	−4.62598	−0.165108
$786$	0	0
$787$	−31.7327	−1.13115	−0.565574	−	0.824697i	$-0.691345\pi$
$787$	−31.7327	−1.13115	−0.565574	+	0.824697i	$0.691345\pi$
$788$	0	0
$789$	−14.5928	−0.519518
$790$	0	0
$791$	7.87603	0.280040
$792$	0	0
$793$	31.5255	1.11950
$794$	0	0
$795$	−12.1842	−0.432129
$796$	0	0
$797$	43.3449	1.53536	0.767678	−	0.640836i	$-0.221412\pi$
$797$	43.3449	1.53536	0.767678	+	0.640836i	$0.221412\pi$
$798$	0	0
$799$	−40.8060	−1.44361
$800$	0	0
$801$	−12.8310	−0.453362
$802$	0	0
$803$	5.86226	0.206874
$804$	0	0
$805$	2.86081	0.100830
$806$	0	0
$807$	26.2534	0.924164
$808$	0	0
$809$	7.41825	0.260812	0.130406	−	0.991461i	$-0.458372\pi$
$809$	7.41825	0.260812	0.130406	+	0.991461i	$0.458372\pi$
$810$	0	0
$811$	−51.6323	−1.81305	−0.906527	−	0.422148i	$-0.861276\pi$
$811$	−51.6323	−1.81305	−0.906527	+	0.422148i	$0.861276\pi$
$812$	0	0
$813$	12.3386	0.432735
$814$	0	0
$815$	−9.90582	−0.346986
$816$	0	0
$817$	−2.70224	−0.0945393
$818$	0	0
$819$	4.46260	0.155936
$820$	0	0
$821$	7.03315	0.245459	0.122729	−	0.992440i	$-0.460835\pi$
$821$	7.03315	0.245459	0.122729	+	0.992440i	$0.460835\pi$
$822$	0	0
$823$	−0.00144931	−5.05198e−5 0	−2.52599e−5	−	1.00000i	$-0.500008\pi$
$823$	−0.00144931	−5.05198e−5 0	−2.52599e−5		1.00000i	$0.500008\pi$
$824$	0	0
$825$	−6.13023	−0.213427
$826$	0	0
$827$	−0.228556	−0.00794766	−0.00397383	−	0.999992i	$-0.501265\pi$
$827$	−0.228556	−0.00794766	−0.00397383	+	0.999992i	$0.501265\pi$
$828$	0	0
$829$	15.0242	0.521812	0.260906	−	0.965364i	$-0.415979\pi$
$829$	15.0242	0.521812	0.260906	+	0.965364i	$0.415979\pi$
$830$	0	0
$831$	−19.2832	−0.668927
$832$	0	0
$833$	7.97021	0.276152
$834$	0	0
$835$	−22.7320	−0.786674
$836$	0	0
$837$	−4.11982	−0.142402
$838$	0	0
$839$	34.8794	1.20417	0.602085	−	0.798432i	$-0.294337\pi$
$839$	34.8794	1.20417	0.602085	+	0.798432i	$0.294337\pi$
$840$	0	0
$841$	−23.2486	−0.801676
$842$	0	0
$843$	−29.5124	−1.01646
$844$	0	0
$845$	−19.7819	−0.680517
$846$	0	0
$847$	7.29362	0.250612
$848$	0	0
$849$	−18.4328	−0.632612
$850$	0	0
$851$	−9.04502	−0.310059
$852$	0	0
$853$	27.7937	0.951639	0.475819	−	0.879543i	$-0.342152\pi$
$853$	27.7937	0.951639	0.475819	+	0.879543i	$0.342152\pi$
$854$	0	0
$855$	10.4328	0.356795
$856$	0	0
$857$	4.83853	0.165281	0.0826406	−	0.996579i	$-0.473665\pi$
$857$	4.83853	0.165281	0.0826406	+	0.996579i	$0.473665\pi$
$858$	0	0
$859$	8.61635	0.293986	0.146993	−	0.989138i	$-0.453041\pi$
$859$	8.61635	0.293986	0.146993	+	0.989138i	$0.453041\pi$
$860$	0	0
$861$	1.64681	0.0561231
$862$	0	0
$863$	−22.3684	−0.761430	−0.380715	−	0.924692i	$-0.624322\pi$
$863$	−22.3684	−0.761430	−0.380715	+	0.924692i	$0.624322\pi$
$864$	0	0
$865$	38.7417	1.31726
$866$	0	0
$867$	−46.5243	−1.58005
$868$	0	0
$869$	7.93146	0.269056
$870$	0	0
$871$	−38.8746	−1.31722
$872$	0	0
$873$	−18.2099	−0.616310
$874$	0	0
$875$	−5.19462	−0.175610
$876$	0	0
$877$	1.18903	0.0401506	0.0200753	−	0.999798i	$-0.493609\pi$
$877$	1.18903	0.0401506	0.0200753	+	0.999798i	$0.493609\pi$
$878$	0	0
$879$	16.9944	0.573207
$880$	0	0
$881$	13.9308	0.469340	0.234670	−	0.972075i	$-0.424599\pi$
$881$	13.9308	0.469340	0.234670	+	0.972075i	$0.424599\pi$
$882$	0	0
$883$	10.7439	0.361561	0.180780	−	0.983524i	$-0.442138\pi$
$883$	10.7439	0.361561	0.180780	+	0.983524i	$0.442138\pi$
$884$	0	0
$885$	−28.5526	−0.959786
$886$	0	0
$887$	34.8906	1.17151	0.585756	−	0.810488i	$-0.300798\pi$
$887$	34.8906	1.17151	0.585756	+	0.810488i	$0.300798\pi$
$888$	0	0
$889$	−2.93561	−0.0984572
$890$	0	0
$891$	1.92520	0.0644965
$892$	0	0
$893$	18.6710	0.624801
$894$	0	0
$895$	−56.0651	−1.87405
$896$	0	0
$897$	4.46260	0.149002
$898$	0	0
$899$	−9.88018	−0.329522
$900$	0	0
$901$	−33.9452	−1.13088
$902$	0	0
$903$	0.740987	0.0246585
$904$	0	0
$905$	48.3941	1.60867
$906$	0	0
$907$	−1.88433	−0.0625681	−0.0312840	−	0.999511i	$-0.509960\pi$
$907$	−1.88433	−0.0625681	−0.0312840	+	0.999511i	$0.509960\pi$
$908$	0	0
$909$	−14.7770	−0.490123
$910$	0	0
$911$	48.8477	1.61840	0.809198	−	0.587536i	$-0.199902\pi$
$911$	48.8477	1.61840	0.809198	+	0.587536i	$0.199902\pi$
$912$	0	0
$913$	26.6475	0.881903
$914$	0	0
$915$	−20.2099	−0.668117
$916$	0	0
$917$	−6.85039	−0.226220
$918$	0	0
$919$	−31.3747	−1.03496	−0.517478	−	0.855697i	$-0.673129\pi$
$919$	−31.3747	−1.03496	−0.517478	+	0.855697i	$0.673129\pi$
$920$	0	0
$921$	0.0297872	0.000981521 0
$922$	0	0
$923$	−7.43551	−0.244743
$924$	0	0
$925$	−28.8012	−0.946979
$926$	0	0
$927$	−2.04502	−0.0671671
$928$	0	0
$929$	10.7804	0.353694	0.176847	−	0.984238i	$-0.443410\pi$
$929$	10.7804	0.353694	0.176847	+	0.984238i	$0.443410\pi$
$930$	0	0
$931$	−3.64681	−0.119519
$932$	0	0
$933$	2.80683	0.0918914
$934$	0	0
$935$	−43.8969	−1.43558
$936$	0	0
$937$	22.5928	0.738076	0.369038	−	0.929414i	$-0.379687\pi$
$937$	22.5928	0.738076	0.369038	+	0.929414i	$0.379687\pi$
$938$	0	0
$939$	−5.30818	−0.173226
$940$	0	0
$941$	7.18566	0.234246	0.117123	−	0.993117i	$-0.462633\pi$
$941$	7.18566	0.234246	0.117123	+	0.993117i	$0.462633\pi$
$942$	0	0
$943$	1.64681	0.0536275
$944$	0	0
$945$	−2.86081	−0.0930620
$946$	0	0
$947$	21.4916	0.698383	0.349192	−	0.937051i	$-0.386456\pi$
$947$	21.4916	0.698383	0.349192	+	0.937051i	$0.386456\pi$
$948$	0	0
$949$	−13.5887	−0.441107
$950$	0	0
$951$	−24.0048	−0.778410
$952$	0	0
$953$	−14.8165	−0.479952	−0.239976	−	0.970779i	$-0.577140\pi$
$953$	−14.8165	−0.479952	−0.239976	+	0.970779i	$0.577140\pi$
$954$	0	0
$955$	60.6773	1.96347
$956$	0	0
$957$	4.61702	0.149247
$958$	0	0
$959$	−21.2188	−0.685191
$960$	0	0
$961$	−14.0271	−0.452487
$962$	0	0
$963$	−12.9806	−0.418295
$964$	0	0
$965$	17.6620	0.568561
$966$	0	0
$967$	32.1205	1.03293	0.516463	−	0.856310i	$-0.327248\pi$
$967$	32.1205	1.03293	0.516463	+	0.856310i	$0.327248\pi$
$968$	0	0
$969$	29.0658	0.933729
$970$	0	0
$971$	−61.5679	−1.97581	−0.987903	−	0.155071i	$-0.950439\pi$
$971$	−61.5679	−1.97581	−0.987903	+	0.155071i	$0.950439\pi$
$972$	0	0
$973$	−16.0256	−0.513758
$974$	0	0
$975$	14.2099	0.455079
$976$	0	0
$977$	−21.4633	−0.686671	−0.343335	−	0.939213i	$-0.611557\pi$
$977$	−21.4633	−0.686671	−0.343335	+	0.939213i	$0.611557\pi$
$978$	0	0
$979$	−24.7022	−0.789487
$980$	0	0
$981$	5.37883	0.171733
$982$	0	0
$983$	−12.7637	−0.407100	−0.203550	−	0.979065i	$-0.565248\pi$
$983$	−12.7637	−0.407100	−0.203550	+	0.979065i	$0.565248\pi$
$984$	0	0
$985$	−58.3691	−1.85979
$986$	0	0
$987$	−5.11982	−0.162966
$988$	0	0
$989$	0.740987	0.0235620
$990$	0	0
$991$	−21.9896	−0.698522	−0.349261	−	0.937025i	$-0.613567\pi$
$991$	−21.9896	−0.698522	−0.349261	+	0.937025i	$0.613567\pi$
$992$	0	0
$993$	−6.45219	−0.204754
$994$	0	0
$995$	−6.83102	−0.216558
$996$	0	0
$997$	36.4078	1.15305	0.576524	−	0.817080i	$-0.304409\pi$
$997$	36.4078	1.15305	0.576524	+	0.817080i	$0.304409\pi$
$998$	0	0
$999$	9.04502	0.286172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bp.1.1		3
4.3	odd	2		3864.2.a.p.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.p.1.1	✓	3	4.3	odd	2
7728.2.a.bp.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.86081 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.86081 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.92520 q^{11} -4.46260 q^{13} +2.86081 q^{15} +7.97021 q^{17} -3.64681 q^{19} +1.00000 q^{21} +1.00000 q^{23} +3.18421 q^{25} -1.00000 q^{27} -2.39821 q^{29} +4.11982 q^{31} -1.92520 q^{33} +2.86081 q^{35} -9.04502 q^{37} +4.46260 q^{39} +1.64681 q^{41} +0.740987 q^{43} -2.86081 q^{45} -5.11982 q^{47} +1.00000 q^{49} -7.97021 q^{51} -4.25901 q^{53} -5.50761 q^{55} +3.64681 q^{57} -9.98062 q^{59} -7.06439 q^{61} -1.00000 q^{63} +12.7666 q^{65} +8.71120 q^{67} -1.00000 q^{69} +1.66618 q^{71} +3.04502 q^{73} -3.18421 q^{75} -1.92520 q^{77} +4.11982 q^{79} +1.00000 q^{81} +13.8414 q^{83} -22.8012 q^{85} +2.39821 q^{87} -12.8310 q^{89} +4.46260 q^{91} -4.11982 q^{93} +10.4328 q^{95} -18.2099 q^{97} +1.92520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + q^11 - 11 * q^13 + 3 * q^15 + 5 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 - q^33 + 3 * q^35 - 8 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 3 * q^45 - q^47 + 3 * q^49 - 4 * q^53 + 5 * q^55 - 5 * q^57 - 10 * q^59 - 22 * q^61 - 3 * q^63 + 8 * q^65 + 11 * q^67 - 3 * q^69 + 9 * q^71 - 10 * q^73 + 4 * q^75 - q^77 - 2 * q^79 + 3 * q^81 + 16 * q^83 - 15 * q^85 + 4 * q^87 - 9 * q^89 + 11 * q^91 + 2 * q^93 + 5 * q^95 - 2 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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