LMFDB - Embedded newform 7728.2.a.bz.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.39605.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 9x - 2 $ x^4 - 2x^3 - 8x^2 + 9*x - 2
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			1.3
Root			$0.681685$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.318315 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.318315 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.71255 q^{11} -2.66749 q^{13} +0.318315 q^{15} +2.21699 q^{17} +2.34918 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.89868 q^{25} -1.00000 q^{27} +5.41527 q^{29} -8.21699 q^{31} +3.71255 q^{33} -0.318315 q^{35} +7.28308 q^{37} +2.66749 q^{39} +4.77864 q^{41} +8.59703 q^{43} -0.318315 q^{45} +3.63226 q^{47} +1.00000 q^{49} -2.21699 q^{51} -0.615591 q^{53} +1.18176 q^{55} -2.34918 q^{57} -9.81839 q^{59} +10.9364 q^{61} +1.00000 q^{63} +0.849103 q^{65} -0.747779 q^{67} -1.00000 q^{69} -13.5309 q^{71} -3.84473 q^{73} +4.89868 q^{75} -3.71255 q^{77} -15.1797 q^{79} +1.00000 q^{81} +1.11799 q^{83} -0.705701 q^{85} -5.41527 q^{87} +6.32814 q^{89} -2.66749 q^{91} +8.21699 q^{93} -0.747779 q^{95} -16.1136 q^{97} -3.71255 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{15} - 10 q^{17} - 4 q^{19} - 4 q^{21} + 4 q^{23} - 4 q^{27} + 11 q^{29} - 14 q^{31} - 2 q^{35} + 13 q^{37} - 2 q^{39} + 7 q^{41} - 12 q^{43} - 2 q^{45} - 15 q^{47} + 4 q^{49} + 10 q^{51} + q^{53} - 31 q^{55} + 4 q^{57} - 5 q^{59} + 3 q^{61} + 4 q^{63} + 25 q^{65} - 5 q^{67} - 4 q^{69} - 5 q^{71} - 6 q^{73} - 26 q^{79} + 4 q^{81} - 2 q^{83} + 5 q^{85} - 11 q^{87} + 7 q^{89} + 2 q^{91} + 14 q^{93} - 5 q^{95} - 27 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9 + 2 * q^13 + 2 * q^15 - 10 * q^17 - 4 * q^19 - 4 * q^21 + 4 * q^23 - 4 * q^27 + 11 * q^29 - 14 * q^31 - 2 * q^35 + 13 * q^37 - 2 * q^39 + 7 * q^41 - 12 * q^43 - 2 * q^45 - 15 * q^47 + 4 * q^49 + 10 * q^51 + q^53 - 31 * q^55 + 4 * q^57 - 5 * q^59 + 3 * q^61 + 4 * q^63 + 25 * q^65 - 5 * q^67 - 4 * q^69 - 5 * q^71 - 6 * q^73 - 26 * q^79 + 4 * q^81 - 2 * q^83 + 5 * q^85 - 11 * q^87 + 7 * q^89 + 2 * q^91 + 14 * q^93 - 5 * q^95 - 27 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.318315	−0.142355	−0.0711774	−	0.997464i	$-0.522676\pi$
$5$	−0.318315	−0.142355	−0.0711774	+	0.997464i	$0.522676\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.71255	−1.11938	−0.559688	−	0.828704i	$-0.689079\pi$
$11$	−3.71255	−1.11938	−0.559688	+	0.828704i	$0.689079\pi$
$12$	0	0
$13$	−2.66749	−0.739829	−0.369915	−	0.929066i	$-0.620613\pi$
$13$	−2.66749	−0.739829	−0.369915	+	0.929066i	$0.620613\pi$
$14$	0	0
$15$	0.318315	0.0821886
$16$	0	0
$17$	2.21699	0.537699	0.268850	−	0.963182i	$-0.413357\pi$
$17$	2.21699	0.537699	0.268850	+	0.963182i	$0.413357\pi$
$18$	0	0
$19$	2.34918	0.538938	0.269469	−	0.963009i	$-0.413152\pi$
$19$	2.34918	0.538938	0.269469	+	0.963009i	$0.413152\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.89868	−0.979735
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.41527	1.00559	0.502795	−	0.864406i	$-0.332305\pi$
$29$	5.41527	1.00559	0.502795	+	0.864406i	$0.332305\pi$
$30$	0	0
$31$	−8.21699	−1.47582	−0.737908	−	0.674902i	$-0.764186\pi$
$31$	−8.21699	−1.47582	−0.737908	+	0.674902i	$0.764186\pi$
$32$	0	0
$33$	3.71255	0.646272
$34$	0	0
$35$	−0.318315	−0.0538051
$36$	0	0
$37$	7.28308	1.19733	0.598666	−	0.800999i	$-0.295698\pi$
$37$	7.28308	1.19733	0.598666	+	0.800999i	$0.295698\pi$
$38$	0	0
$39$	2.66749	0.427141
$40$	0	0
$41$	4.77864	0.746298	0.373149	−	0.927771i	$-0.378278\pi$
$41$	4.77864	0.746298	0.373149	+	0.927771i	$0.378278\pi$
$42$	0	0
$43$	8.59703	1.31103	0.655517	−	0.755180i	$-0.272451\pi$
$43$	8.59703	1.31103	0.655517	+	0.755180i	$0.272451\pi$
$44$	0	0
$45$	−0.318315	−0.0474516
$46$	0	0
$47$	3.63226	0.529820	0.264910	−	0.964273i	$-0.414658\pi$
$47$	3.63226	0.529820	0.264910	+	0.964273i	$0.414658\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.21699	−0.310441
$52$	0	0
$53$	−0.615591	−0.0845580	−0.0422790	−	0.999106i	$-0.513462\pi$
$53$	−0.615591	−0.0845580	−0.0422790	+	0.999106i	$0.513462\pi$
$54$	0	0
$55$	1.18176	0.159348
$56$	0	0
$57$	−2.34918	−0.311156
$58$	0	0
$59$	−9.81839	−1.27825	−0.639123	−	0.769105i	$-0.720702\pi$
$59$	−9.81839	−1.27825	−0.639123	+	0.769105i	$0.720702\pi$
$60$	0	0
$61$	10.9364	1.40026	0.700130	−	0.714015i	$-0.253125\pi$
$61$	10.9364	1.40026	0.700130	+	0.714015i	$0.253125\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0.849103	0.105318
$66$	0	0
$67$	−0.747779	−0.0913557	−0.0456778	−	0.998956i	$-0.514545\pi$
$67$	−0.747779	−0.0913557	−0.0456778	+	0.998956i	$0.514545\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−13.5309	−1.60583	−0.802913	−	0.596096i	$-0.796718\pi$
$71$	−13.5309	−1.60583	−0.802913	+	0.596096i	$0.796718\pi$
$72$	0	0
$73$	−3.84473	−0.449992	−0.224996	−	0.974360i	$-0.572237\pi$
$73$	−3.84473	−0.449992	−0.224996	+	0.974360i	$0.572237\pi$
$74$	0	0
$75$	4.89868	0.565650
$76$	0	0
$77$	−3.71255	−0.423084
$78$	0	0
$79$	−15.1797	−1.70785	−0.853926	−	0.520395i	$-0.825785\pi$
$79$	−15.1797	−1.70785	−0.853926	+	0.520395i	$0.825785\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.11799	0.122716	0.0613579	−	0.998116i	$-0.480457\pi$
$83$	1.11799	0.122716	0.0613579	+	0.998116i	$0.480457\pi$
$84$	0	0
$85$	−0.705701	−0.0765441
$86$	0	0
$87$	−5.41527	−0.580578
$88$	0	0
$89$	6.32814	0.670781	0.335391	−	0.942079i	$-0.391132\pi$
$89$	6.32814	0.670781	0.335391	+	0.942079i	$0.391132\pi$
$90$	0	0
$91$	−2.66749	−0.279629
$92$	0	0
$93$	8.21699	0.852062
$94$	0	0
$95$	−0.747779	−0.0767205
$96$	0	0
$97$	−16.1136	−1.63609	−0.818045	−	0.575154i	$-0.804942\pi$
$97$	−16.1136	−1.63609	−0.818045	+	0.575154i	$0.804942\pi$
$98$	0	0
$99$	−3.71255	−0.373125
$100$	0	0
$101$	−13.2435	−1.31778	−0.658888	−	0.752241i	$-0.728973\pi$
$101$	−13.2435	−1.31778	−0.658888	+	0.752241i	$0.728973\pi$
$102$	0	0
$103$	3.12782	0.308193	0.154097	−	0.988056i	$-0.450753\pi$
$103$	3.12782	0.308193	0.154097	+	0.988056i	$0.450753\pi$
$104$	0	0
$105$	0.318315	0.0310644
$106$	0	0
$107$	15.6631	1.51421	0.757106	−	0.653292i	$-0.226613\pi$
$107$	15.6631	1.51421	0.757106	+	0.653292i	$0.226613\pi$
$108$	0	0
$109$	8.75230	0.838318	0.419159	−	0.907913i	$-0.362325\pi$
$109$	8.75230	0.838318	0.419159	+	0.907913i	$0.362325\pi$
$110$	0	0
$111$	−7.28308	−0.691280
$112$	0	0
$113$	−15.8051	−1.48682	−0.743411	−	0.668835i	$-0.766793\pi$
$113$	−15.8051	−1.48682	−0.743411	+	0.668835i	$0.766793\pi$
$114$	0	0
$115$	−0.318315	−0.0296830
$116$	0	0
$117$	−2.66749	−0.246610
$118$	0	0
$119$	2.21699	0.203231
$120$	0	0
$121$	2.78301	0.253001
$122$	0	0
$123$	−4.77864	−0.430876
$124$	0	0
$125$	3.15090	0.281825
$126$	0	0
$127$	−5.92706	−0.525942	−0.262971	−	0.964804i	$-0.584702\pi$
$127$	−5.92706	−0.525942	−0.262971	+	0.964804i	$0.584702\pi$
$128$	0	0
$129$	−8.59703	−0.756926
$130$	0	0
$131$	−8.78316	−0.767388	−0.383694	−	0.923460i	$-0.625348\pi$
$131$	−8.78316	−0.767388	−0.383694	+	0.923460i	$0.625348\pi$
$132$	0	0
$133$	2.34918	0.203700
$134$	0	0
$135$	0.318315	0.0273962
$136$	0	0
$137$	18.1333	1.54923	0.774615	−	0.632433i	$-0.217944\pi$
$137$	18.1333	1.54923	0.774615	+	0.632433i	$0.217944\pi$
$138$	0	0
$139$	0.884483	0.0750209	0.0375104	−	0.999296i	$-0.488057\pi$
$139$	0.884483	0.0750209	0.0375104	+	0.999296i	$0.488057\pi$
$140$	0	0
$141$	−3.63226	−0.305892
$142$	0	0
$143$	9.90319	0.828147
$144$	0	0
$145$	−1.72376	−0.143151
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	19.7690	1.61954	0.809768	−	0.586750i	$-0.199593\pi$
$149$	19.7690	1.61954	0.809768	+	0.586750i	$0.199593\pi$
$150$	0	0
$151$	−20.7930	−1.69211	−0.846054	−	0.533096i	$-0.821028\pi$
$151$	−20.7930	−1.69211	−0.846054	+	0.533096i	$0.821028\pi$
$152$	0	0
$153$	2.21699	0.179233
$154$	0	0
$155$	2.61559	0.210089
$156$	0	0
$157$	11.3350	0.904630	0.452315	−	0.891858i	$-0.350598\pi$
$157$	11.3350	0.904630	0.452315	+	0.891858i	$0.350598\pi$
$158$	0	0
$159$	0.615591	0.0488196
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−23.1534	−1.81351	−0.906756	−	0.421655i	$-0.861449\pi$
$163$	−23.1534	−1.81351	−0.906756	+	0.421655i	$0.861449\pi$
$164$	0	0
$165$	−1.18176	−0.0919999
$166$	0	0
$167$	2.34918	0.181785	0.0908924	−	0.995861i	$-0.471028\pi$
$167$	2.34918	0.181785	0.0908924	+	0.995861i	$0.471028\pi$
$168$	0	0
$169$	−5.88448	−0.452653
$170$	0	0
$171$	2.34918	0.179646
$172$	0	0
$173$	−9.74589	−0.740966	−0.370483	−	0.928839i	$-0.620808\pi$
$173$	−9.74589	−0.740966	−0.370483	+	0.928839i	$0.620808\pi$
$174$	0	0
$175$	−4.89868	−0.370305
$176$	0	0
$177$	9.81839	0.737995
$178$	0	0
$179$	26.0168	1.94459	0.972294	−	0.233761i	$-0.0751032\pi$
$179$	26.0168	1.94459	0.972294	+	0.233761i	$0.0751032\pi$
$180$	0	0
$181$	18.8449	1.40073	0.700365	−	0.713785i	$-0.253021\pi$
$181$	18.8449	1.40073	0.700365	+	0.713785i	$0.253021\pi$
$182$	0	0
$183$	−10.9364	−0.808441
$184$	0	0
$185$	−2.31832	−0.170446
$186$	0	0
$187$	−8.23068	−0.601887
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−10.1939	−0.737606	−0.368803	−	0.929508i	$-0.620232\pi$
$191$	−10.1939	−0.737606	−0.368803	+	0.929508i	$0.620232\pi$
$192$	0	0
$193$	11.7361	0.844780	0.422390	−	0.906414i	$-0.361191\pi$
$193$	11.7361	0.844780	0.422390	+	0.906414i	$0.361191\pi$
$194$	0	0
$195$	−0.849103	−0.0608055
$196$	0	0
$197$	15.4560	1.10119	0.550596	−	0.834772i	$-0.314401\pi$
$197$	15.4560	1.10119	0.550596	+	0.834772i	$0.314401\pi$
$198$	0	0
$199$	−16.8424	−1.19393	−0.596963	−	0.802269i	$-0.703626\pi$
$199$	−16.8424	−1.19393	−0.596963	+	0.802269i	$0.703626\pi$
$200$	0	0
$201$	0.747779	0.0527442
$202$	0	0
$203$	5.41527	0.380078
$204$	0	0
$205$	−1.52111	−0.106239
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−8.72143	−0.603274
$210$	0	0
$211$	−17.6842	−1.21743	−0.608714	−	0.793390i	$-0.708314\pi$
$211$	−17.6842	−1.21743	−0.608714	+	0.793390i	$0.708314\pi$
$212$	0	0
$213$	13.5309	0.927125
$214$	0	0
$215$	−2.73656	−0.186632
$216$	0	0
$217$	−8.21699	−0.557806
$218$	0	0
$219$	3.84473	0.259803
$220$	0	0
$221$	−5.91381	−0.397806
$222$	0	0
$223$	5.90850	0.395662	0.197831	−	0.980236i	$-0.436610\pi$
$223$	5.90850	0.395662	0.197831	+	0.980236i	$0.436610\pi$
$224$	0	0
$225$	−4.89868	−0.326578
$226$	0	0
$227$	−19.0500	−1.26439	−0.632197	−	0.774808i	$-0.717847\pi$
$227$	−19.0500	−1.26439	−0.632197	+	0.774808i	$0.717847\pi$
$228$	0	0
$229$	−22.1819	−1.46582	−0.732911	−	0.680325i	$-0.761839\pi$
$229$	−22.1819	−1.46582	−0.732911	+	0.680325i	$0.761839\pi$
$230$	0	0
$231$	3.71255	0.244268
$232$	0	0
$233$	−14.8712	−0.974247	−0.487123	−	0.873333i	$-0.661954\pi$
$233$	−14.8712	−0.974247	−0.487123	+	0.873333i	$0.661954\pi$
$234$	0	0
$235$	−1.15620	−0.0754224
$236$	0	0
$237$	15.1797	0.986029
$238$	0	0
$239$	14.8008	0.957382	0.478691	−	0.877983i	$-0.341111\pi$
$239$	14.8008	0.957382	0.478691	+	0.877983i	$0.341111\pi$
$240$	0	0
$241$	−27.6887	−1.78358	−0.891792	−	0.452445i	$-0.850552\pi$
$241$	−27.6887	−1.78358	−0.891792	+	0.452445i	$0.850552\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.318315	−0.0203364
$246$	0	0
$247$	−6.26641	−0.398722
$248$	0	0
$249$	−1.11799	−0.0708500
$250$	0	0
$251$	25.1895	1.58995	0.794975	−	0.606642i	$-0.207484\pi$
$251$	25.1895	1.58995	0.794975	+	0.606642i	$0.207484\pi$
$252$	0	0
$253$	−3.71255	−0.233406
$254$	0	0
$255$	0.705701	0.0441927
$256$	0	0
$257$	4.89227	0.305171	0.152586	−	0.988290i	$-0.451240\pi$
$257$	4.89227	0.305171	0.152586	+	0.988290i	$0.451240\pi$
$258$	0	0
$259$	7.28308	0.452549
$260$	0	0
$261$	5.41527	0.335197
$262$	0	0
$263$	−23.1755	−1.42906	−0.714531	−	0.699603i	$-0.753360\pi$
$263$	−23.1755	−1.42906	−0.714531	+	0.699603i	$0.753360\pi$
$264$	0	0
$265$	0.195952	0.0120372
$266$	0	0
$267$	−6.32814	−0.387276
$268$	0	0
$269$	−24.3811	−1.48654	−0.743272	−	0.668990i	$-0.766727\pi$
$269$	−24.3811	−1.48654	−0.743272	+	0.668990i	$0.766727\pi$
$270$	0	0
$271$	−5.42853	−0.329759	−0.164880	−	0.986314i	$-0.552724\pi$
$271$	−5.42853	−0.329759	−0.164880	+	0.986314i	$0.552724\pi$
$272$	0	0
$273$	2.66749	0.161444
$274$	0	0
$275$	18.1866	1.09669
$276$	0	0
$277$	−15.5730	−0.935692	−0.467846	−	0.883810i	$-0.654970\pi$
$277$	−15.5730	−0.935692	−0.467846	+	0.883810i	$0.654970\pi$
$278$	0	0
$279$	−8.21699	−0.491938
$280$	0	0
$281$	3.05736	0.182387	0.0911933	−	0.995833i	$-0.470932\pi$
$281$	3.05736	0.182387	0.0911933	+	0.995833i	$0.470932\pi$
$282$	0	0
$283$	−4.24334	−0.252240	−0.126120	−	0.992015i	$-0.540252\pi$
$283$	−4.24334	−0.252240	−0.126120	+	0.992015i	$0.540252\pi$
$284$	0	0
$285$	0.747779	0.0442946
$286$	0	0
$287$	4.77864	0.282074
$288$	0	0
$289$	−12.0850	−0.710880
$290$	0	0
$291$	16.1136	0.944598
$292$	0	0
$293$	−18.6984	−1.09237	−0.546185	−	0.837665i	$-0.683920\pi$
$293$	−18.6984	−1.09237	−0.546185	+	0.837665i	$0.683920\pi$
$294$	0	0
$295$	3.12534	0.181964
$296$	0	0
$297$	3.71255	0.215424
$298$	0	0
$299$	−2.66749	−0.154265
$300$	0	0
$301$	8.59703	0.495525
$302$	0	0
$303$	13.2435	0.760818
$304$	0	0
$305$	−3.48122	−0.199334
$306$	0	0
$307$	−25.3448	−1.44650	−0.723252	−	0.690584i	$-0.757354\pi$
$307$	−25.3448	−1.44650	−0.723252	+	0.690584i	$0.757354\pi$
$308$	0	0
$309$	−3.12782	−0.177935
$310$	0	0
$311$	−14.5221	−0.823470	−0.411735	−	0.911304i	$-0.635077\pi$
$311$	−14.5221	−0.823470	−0.411735	+	0.911304i	$0.635077\pi$
$312$	0	0
$313$	4.53778	0.256491	0.128245	−	0.991742i	$-0.459066\pi$
$313$	4.53778	0.256491	0.128245	+	0.991742i	$0.459066\pi$
$314$	0	0
$315$	−0.318315	−0.0179350
$316$	0	0
$317$	3.61996	0.203317	0.101659	−	0.994819i	$-0.467585\pi$
$317$	3.61996	0.203317	0.101659	+	0.994819i	$0.467585\pi$
$318$	0	0
$319$	−20.1045	−1.12563
$320$	0	0
$321$	−15.6631	−0.874230
$322$	0	0
$323$	5.20810	0.289787
$324$	0	0
$325$	13.0672	0.724837
$326$	0	0
$327$	−8.75230	−0.484003
$328$	0	0
$329$	3.63226	0.200253
$330$	0	0
$331$	−12.5749	−0.691179	−0.345590	−	0.938386i	$-0.612321\pi$
$331$	−12.5749	−0.691179	−0.345590	+	0.938386i	$0.612321\pi$
$332$	0	0
$333$	7.28308	0.399111
$334$	0	0
$335$	0.238029	0.0130049
$336$	0	0
$337$	1.07265	0.0584310	0.0292155	−	0.999573i	$-0.490699\pi$
$337$	1.07265	0.0584310	0.0292155	+	0.999573i	$0.490699\pi$
$338$	0	0
$339$	15.8051	0.858417
$340$	0	0
$341$	30.5060	1.65199
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.318315	0.0171375
$346$	0	0
$347$	11.3732	0.610545	0.305273	−	0.952265i	$-0.401252\pi$
$347$	11.3732	0.610545	0.305273	+	0.952265i	$0.401252\pi$
$348$	0	0
$349$	32.3580	1.73209	0.866043	−	0.499969i	$-0.166655\pi$
$349$	32.3580	1.73209	0.866043	+	0.499969i	$0.166655\pi$
$350$	0	0
$351$	2.66749	0.142380
$352$	0	0
$353$	−12.3447	−0.657040	−0.328520	−	0.944497i	$-0.606550\pi$
$353$	−12.3447	−0.657040	−0.328520	+	0.944497i	$0.606550\pi$
$354$	0	0
$355$	4.30710	0.228597
$356$	0	0
$357$	−2.21699	−0.117336
$358$	0	0
$359$	−28.7239	−1.51599	−0.757995	−	0.652260i	$-0.773821\pi$
$359$	−28.7239	−1.51599	−0.757995	+	0.652260i	$0.773821\pi$
$360$	0	0
$361$	−13.4814	−0.709546
$362$	0	0
$363$	−2.78301	−0.146070
$364$	0	0
$365$	1.22384	0.0640585
$366$	0	0
$367$	12.4365	0.649178	0.324589	−	0.945855i	$-0.394774\pi$
$367$	12.4365	0.649178	0.324589	+	0.945855i	$0.394774\pi$
$368$	0	0
$369$	4.77864	0.248766
$370$	0	0
$371$	−0.615591	−0.0319599
$372$	0	0
$373$	−5.97692	−0.309473	−0.154737	−	0.987956i	$-0.549453\pi$
$373$	−5.97692	−0.309473	−0.154737	+	0.987956i	$0.549453\pi$
$374$	0	0
$375$	−3.15090	−0.162712
$376$	0	0
$377$	−14.4452	−0.743965
$378$	0	0
$379$	19.1369	0.982994	0.491497	−	0.870879i	$-0.336450\pi$
$379$	19.1369	0.982994	0.491497	+	0.870879i	$0.336450\pi$
$380$	0	0
$381$	5.92706	0.303652
$382$	0	0
$383$	15.4482	0.789365	0.394682	−	0.918818i	$-0.370855\pi$
$383$	15.4482	0.789365	0.394682	+	0.918818i	$0.370855\pi$
$384$	0	0
$385$	1.18176	0.0602280
$386$	0	0
$387$	8.59703	0.437012
$388$	0	0
$389$	−23.9065	−1.21211	−0.606053	−	0.795424i	$-0.707248\pi$
$389$	−23.9065	−1.21211	−0.606053	+	0.795424i	$0.707248\pi$
$390$	0	0
$391$	2.21699	0.112118
$392$	0	0
$393$	8.78316	0.443052
$394$	0	0
$395$	4.83193	0.243121
$396$	0	0
$397$	−34.8551	−1.74933	−0.874665	−	0.484728i	$-0.838919\pi$
$397$	−34.8551	−1.74933	−0.874665	+	0.484728i	$0.838919\pi$
$398$	0	0
$399$	−2.34918	−0.117606
$400$	0	0
$401$	−0.481365	−0.0240382	−0.0120191	−	0.999928i	$-0.503826\pi$
$401$	−0.481365	−0.0240382	−0.0120191	+	0.999928i	$0.503826\pi$
$402$	0	0
$403$	21.9188	1.09185
$404$	0	0
$405$	−0.318315	−0.0158172
$406$	0	0
$407$	−27.0388	−1.34026
$408$	0	0
$409$	30.1621	1.49142	0.745710	−	0.666271i	$-0.232111\pi$
$409$	30.1621	1.49142	0.745710	+	0.666271i	$0.232111\pi$
$410$	0	0
$411$	−18.1333	−0.894448
$412$	0	0
$413$	−9.81839	−0.483131
$414$	0	0
$415$	−0.355874	−0.0174692
$416$	0	0
$417$	−0.884483	−0.0433133
$418$	0	0
$419$	−24.7390	−1.20858	−0.604291	−	0.796764i	$-0.706543\pi$
$419$	−24.7390	−1.20858	−0.604291	+	0.796764i	$0.706543\pi$
$420$	0	0
$421$	−16.4452	−0.801490	−0.400745	−	0.916190i	$-0.631249\pi$
$421$	−16.4452	−0.801490	−0.400745	+	0.916190i	$0.631249\pi$
$422$	0	0
$423$	3.63226	0.176607
$424$	0	0
$425$	−10.8603	−0.526803
$426$	0	0
$427$	10.9364	0.529249
$428$	0	0
$429$	−9.90319	−0.478131
$430$	0	0
$431$	40.6061	1.95593	0.977963	−	0.208780i	$-0.0669492\pi$
$431$	40.6061	1.95593	0.977963	+	0.208780i	$0.0669492\pi$
$432$	0	0
$433$	5.36789	0.257964	0.128982	−	0.991647i	$-0.458829\pi$
$433$	5.36789	0.257964	0.128982	+	0.991647i	$0.458829\pi$
$434$	0	0
$435$	1.72376	0.0826481
$436$	0	0
$437$	2.34918	0.112376
$438$	0	0
$439$	23.7655	1.13427	0.567134	−	0.823626i	$-0.308052\pi$
$439$	23.7655	1.13427	0.567134	+	0.823626i	$0.308052\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.6655	−0.791800	−0.395900	−	0.918294i	$-0.629567\pi$
$443$	−16.6655	−0.791800	−0.395900	+	0.918294i	$0.629567\pi$
$444$	0	0
$445$	−2.01434	−0.0954890
$446$	0	0
$447$	−19.7690	−0.935040
$448$	0	0
$449$	14.6910	0.693312	0.346656	−	0.937992i	$-0.387317\pi$
$449$	14.6910	0.693312	0.346656	+	0.937992i	$0.387317\pi$
$450$	0	0
$451$	−17.7409	−0.835388
$452$	0	0
$453$	20.7930	0.976940
$454$	0	0
$455$	0.849103	0.0398066
$456$	0	0
$457$	18.2346	0.852979	0.426489	−	0.904493i	$-0.359750\pi$
$457$	18.2346	0.852979	0.426489	+	0.904493i	$0.359750\pi$
$458$	0	0
$459$	−2.21699	−0.103480
$460$	0	0
$461$	−21.1200	−0.983658	−0.491829	−	0.870692i	$-0.663671\pi$
$461$	−21.1200	−0.983658	−0.491829	+	0.870692i	$0.663671\pi$
$462$	0	0
$463$	−8.75026	−0.406659	−0.203329	−	0.979110i	$-0.565176\pi$
$463$	−8.75026	−0.406659	−0.203329	+	0.979110i	$0.565176\pi$
$464$	0	0
$465$	−2.61559	−0.121295
$466$	0	0
$467$	33.2743	1.53975	0.769877	−	0.638193i	$-0.220318\pi$
$467$	33.2743	1.53975	0.769877	+	0.638193i	$0.220318\pi$
$468$	0	0
$469$	−0.747779	−0.0345292
$470$	0	0
$471$	−11.3350	−0.522289
$472$	0	0
$473$	−31.9169	−1.46754
$474$	0	0
$475$	−11.5079	−0.528017
$476$	0	0
$477$	−0.615591	−0.0281860
$478$	0	0
$479$	13.9155	0.635815	0.317908	−	0.948122i	$-0.397020\pi$
$479$	13.9155	0.635815	0.317908	+	0.948122i	$0.397020\pi$
$480$	0	0
$481$	−19.4276	−0.885821
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	5.12921	0.232905
$486$	0	0
$487$	27.9306	1.26566	0.632829	−	0.774292i	$-0.281894\pi$
$487$	27.9306	1.26566	0.632829	+	0.774292i	$0.281894\pi$
$488$	0	0
$489$	23.1534	1.04703
$490$	0	0
$491$	−24.3047	−1.09686	−0.548428	−	0.836198i	$-0.684774\pi$
$491$	−24.3047	−1.09686	−0.548428	+	0.836198i	$0.684774\pi$
$492$	0	0
$493$	12.0056	0.540705
$494$	0	0
$495$	1.18176	0.0531161
$496$	0	0
$497$	−13.5309	−0.606945
$498$	0	0
$499$	−13.7763	−0.616712	−0.308356	−	0.951271i	$-0.599779\pi$
$499$	−13.7763	−0.616712	−0.308356	+	0.951271i	$0.599779\pi$
$500$	0	0
$501$	−2.34918	−0.104954
$502$	0	0
$503$	19.2572	0.858635	0.429318	−	0.903154i	$-0.358754\pi$
$503$	19.2572	0.858635	0.429318	+	0.903154i	$0.358754\pi$
$504$	0	0
$505$	4.21560	0.187592
$506$	0	0
$507$	5.88448	0.261339
$508$	0	0
$509$	−5.18503	−0.229822	−0.114911	−	0.993376i	$-0.536658\pi$
$509$	−5.18503	−0.229822	−0.114911	+	0.993376i	$0.536658\pi$
$510$	0	0
$511$	−3.84473	−0.170081
$512$	0	0
$513$	−2.34918	−0.103719
$514$	0	0
$515$	−0.995632	−0.0438728
$516$	0	0
$517$	−13.4849	−0.593067
$518$	0	0
$519$	9.74589	0.427797
$520$	0	0
$521$	−33.1520	−1.45241	−0.726207	−	0.687476i	$-0.758719\pi$
$521$	−33.1520	−1.45241	−0.726207	+	0.687476i	$0.758719\pi$
$522$	0	0
$523$	−11.1323	−0.486783	−0.243392	−	0.969928i	$-0.578260\pi$
$523$	−11.1323	−0.486783	−0.243392	+	0.969928i	$0.578260\pi$
$524$	0	0
$525$	4.89868	0.213796
$526$	0	0
$527$	−18.2170	−0.793545
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.81839	−0.426082
$532$	0	0
$533$	−12.7470	−0.552133
$534$	0	0
$535$	−4.98581	−0.215555
$536$	0	0
$537$	−26.0168	−1.12271
$538$	0	0
$539$	−3.71255	−0.159911
$540$	0	0
$541$	−10.6192	−0.456553	−0.228277	−	0.973596i	$-0.573309\pi$
$541$	−10.6192	−0.456553	−0.228277	+	0.973596i	$0.573309\pi$
$542$	0	0
$543$	−18.8449	−0.808711
$544$	0	0
$545$	−2.78599	−0.119339
$546$	0	0
$547$	−36.1819	−1.54703	−0.773513	−	0.633780i	$-0.781502\pi$
$547$	−36.1819	−1.54703	−0.773513	+	0.633780i	$0.781502\pi$
$548$	0	0
$549$	10.9364	0.466754
$550$	0	0
$551$	12.7214	0.541951
$552$	0	0
$553$	−15.1797	−0.645507
$554$	0	0
$555$	2.31832	0.0984070
$556$	0	0
$557$	0.524091	0.0222065	0.0111032	−	0.999938i	$-0.496466\pi$
$557$	0.524091	0.0222065	0.0111032	+	0.999938i	$0.496466\pi$
$558$	0	0
$559$	−22.9325	−0.969942
$560$	0	0
$561$	8.23068	0.347500
$562$	0	0
$563$	−4.30427	−0.181403	−0.0907017	−	0.995878i	$-0.528911\pi$
$563$	−4.30427	−0.181403	−0.0907017	+	0.995878i	$0.528911\pi$
$564$	0	0
$565$	5.03101	0.211656
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	36.2318	1.51891	0.759457	−	0.650557i	$-0.225465\pi$
$569$	36.2318	1.51891	0.759457	+	0.650557i	$0.225465\pi$
$570$	0	0
$571$	−30.5480	−1.27840	−0.639198	−	0.769042i	$-0.720734\pi$
$571$	−30.5480	−1.27840	−0.639198	+	0.769042i	$0.720734\pi$
$572$	0	0
$573$	10.1939	0.425857
$574$	0	0
$575$	−4.89868	−0.204289
$576$	0	0
$577$	−30.3878	−1.26506	−0.632531	−	0.774535i	$-0.717984\pi$
$577$	−30.3878	−1.26506	−0.632531	+	0.774535i	$0.717984\pi$
$578$	0	0
$579$	−11.7361	−0.487734
$580$	0	0
$581$	1.11799	0.0463822
$582$	0	0
$583$	2.28541	0.0946521
$584$	0	0
$585$	0.849103	0.0351061
$586$	0	0
$587$	−39.4654	−1.62891	−0.814456	−	0.580225i	$-0.802965\pi$
$587$	−39.4654	−1.62891	−0.814456	+	0.580225i	$0.802965\pi$
$588$	0	0
$589$	−19.3032	−0.795373
$590$	0	0
$591$	−15.4560	−0.635773
$592$	0	0
$593$	−22.7930	−0.935996	−0.467998	−	0.883730i	$-0.655024\pi$
$593$	−22.7930	−0.935996	−0.467998	+	0.883730i	$0.655024\pi$
$594$	0	0
$595$	−0.705701	−0.0289309
$596$	0	0
$597$	16.8424	0.689314
$598$	0	0
$599$	35.8287	1.46392	0.731960	−	0.681348i	$-0.238606\pi$
$599$	35.8287	1.46392	0.731960	+	0.681348i	$0.238606\pi$
$600$	0	0
$601$	−39.3669	−1.60581	−0.802905	−	0.596106i	$-0.796714\pi$
$601$	−39.3669	−1.60581	−0.802905	+	0.596106i	$0.796714\pi$
$602$	0	0
$603$	−0.747779	−0.0304519
$604$	0	0
$605$	−0.885874	−0.0360159
$606$	0	0
$607$	−30.4076	−1.23421	−0.617104	−	0.786882i	$-0.711694\pi$
$607$	−30.4076	−1.23421	−0.617104	+	0.786882i	$0.711694\pi$
$608$	0	0
$609$	−5.41527	−0.219438
$610$	0	0
$611$	−9.68903	−0.391976
$612$	0	0
$613$	−38.5252	−1.55602	−0.778009	−	0.628254i	$-0.783770\pi$
$613$	−38.5252	−1.55602	−0.778009	+	0.628254i	$0.783770\pi$
$614$	0	0
$615$	1.52111	0.0613372
$616$	0	0
$617$	−11.9032	−0.479205	−0.239602	−	0.970871i	$-0.577017\pi$
$617$	−11.9032	−0.479205	−0.239602	+	0.970871i	$0.577017\pi$
$618$	0	0
$619$	−10.8522	−0.436188	−0.218094	−	0.975928i	$-0.569984\pi$
$619$	−10.8522	−0.436188	−0.218094	+	0.975928i	$0.569984\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	6.32814	0.253532
$624$	0	0
$625$	23.4904	0.939616
$626$	0	0
$627$	8.72143	0.348301
$628$	0	0
$629$	16.1465	0.643804
$630$	0	0
$631$	21.3250	0.848936	0.424468	−	0.905443i	$-0.360461\pi$
$631$	21.3250	0.848936	0.424468	+	0.905443i	$0.360461\pi$
$632$	0	0
$633$	17.6842	0.702882
$634$	0	0
$635$	1.88667	0.0748703
$636$	0	0
$637$	−2.66749	−0.105690
$638$	0	0
$639$	−13.5309	−0.535276
$640$	0	0
$641$	−3.33251	−0.131626	−0.0658131	−	0.997832i	$-0.520964\pi$
$641$	−3.33251	−0.131626	−0.0658131	+	0.997832i	$0.520964\pi$
$642$	0	0
$643$	−45.0776	−1.77769	−0.888844	−	0.458211i	$-0.848491\pi$
$643$	−45.0776	−1.77769	−0.888844	+	0.458211i	$0.848491\pi$
$644$	0	0
$645$	2.73656	0.107752
$646$	0	0
$647$	−15.7616	−0.619653	−0.309827	−	0.950793i	$-0.600271\pi$
$647$	−15.7616	−0.619653	−0.309827	+	0.950793i	$0.600271\pi$
$648$	0	0
$649$	36.4512	1.43084
$650$	0	0
$651$	8.21699	0.322049
$652$	0	0
$653$	40.9836	1.60381	0.801907	−	0.597449i	$-0.203819\pi$
$653$	40.9836	1.60381	0.801907	+	0.597449i	$0.203819\pi$
$654$	0	0
$655$	2.79581	0.109241
$656$	0	0
$657$	−3.84473	−0.149997
$658$	0	0
$659$	15.1376	0.589679	0.294839	−	0.955547i	$-0.404734\pi$
$659$	15.1376	0.589679	0.294839	+	0.955547i	$0.404734\pi$
$660$	0	0
$661$	−46.2700	−1.79969	−0.899847	−	0.436206i	$-0.856322\pi$
$661$	−46.2700	−1.79969	−0.899847	+	0.436206i	$0.856322\pi$
$662$	0	0
$663$	5.91381	0.229673
$664$	0	0
$665$	−0.747779	−0.0289976
$666$	0	0
$667$	5.41527	0.209680
$668$	0	0
$669$	−5.90850	−0.228436
$670$	0	0
$671$	−40.6018	−1.56742
$672$	0	0
$673$	−10.1177	−0.390009	−0.195004	−	0.980802i	$-0.562472\pi$
$673$	−10.1177	−0.390009	−0.195004	+	0.980802i	$0.562472\pi$
$674$	0	0
$675$	4.89868	0.188550
$676$	0	0
$677$	−16.0373	−0.616362	−0.308181	−	0.951328i	$-0.599720\pi$
$677$	−16.0373	−0.616362	−0.308181	+	0.951328i	$0.599720\pi$
$678$	0	0
$679$	−16.1136	−0.618384
$680$	0	0
$681$	19.0500	0.729998
$682$	0	0
$683$	39.6137	1.51578	0.757888	−	0.652385i	$-0.226231\pi$
$683$	39.6137	1.51578	0.757888	+	0.652385i	$0.226231\pi$
$684$	0	0
$685$	−5.77209	−0.220540
$686$	0	0
$687$	22.1819	0.846293
$688$	0	0
$689$	1.64209	0.0625585
$690$	0	0
$691$	−6.69210	−0.254579	−0.127290	−	0.991866i	$-0.540628\pi$
$691$	−6.69210	−0.254579	−0.127290	+	0.991866i	$0.540628\pi$
$692$	0	0
$693$	−3.71255	−0.141028
$694$	0	0
$695$	−0.281544	−0.0106796
$696$	0	0
$697$	10.5942	0.401284
$698$	0	0
$699$	14.8712	0.562482
$700$	0	0
$701$	−42.1778	−1.59303	−0.796517	−	0.604616i	$-0.793327\pi$
$701$	−42.1778	−1.59303	−0.796517	+	0.604616i	$0.793327\pi$
$702$	0	0
$703$	17.1093	0.645288
$704$	0	0
$705$	1.15620	0.0435451
$706$	0	0
$707$	−13.2435	−0.498073
$708$	0	0
$709$	9.18707	0.345028	0.172514	−	0.985007i	$-0.444811\pi$
$709$	9.18707	0.345028	0.172514	+	0.985007i	$0.444811\pi$
$710$	0	0
$711$	−15.1797	−0.569284
$712$	0	0
$713$	−8.21699	−0.307729
$714$	0	0
$715$	−3.15234	−0.117891
$716$	0	0
$717$	−14.8008	−0.552745
$718$	0	0
$719$	−15.1883	−0.566429	−0.283214	−	0.959057i	$-0.591401\pi$
$719$	−15.1883	−0.566429	−0.283214	+	0.959057i	$0.591401\pi$
$720$	0	0
$721$	3.12782	0.116486
$722$	0	0
$723$	27.6887	1.02975
$724$	0	0
$725$	−26.5277	−0.985212
$726$	0	0
$727$	−17.7830	−0.659535	−0.329768	−	0.944062i	$-0.606970\pi$
$727$	−17.7830	−0.659535	−0.329768	+	0.944062i	$0.606970\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.0595	0.704942
$732$	0	0
$733$	−13.6564	−0.504412	−0.252206	−	0.967674i	$-0.581156\pi$
$733$	−13.6564	−0.504412	−0.252206	+	0.967674i	$0.581156\pi$
$734$	0	0
$735$	0.318315	0.0117412
$736$	0	0
$737$	2.77616	0.102261
$738$	0	0
$739$	22.7127	0.835500	0.417750	−	0.908562i	$-0.362819\pi$
$739$	22.7127	0.835500	0.417750	+	0.908562i	$0.362819\pi$
$740$	0	0
$741$	6.26641	0.230202
$742$	0	0
$743$	6.81824	0.250137	0.125068	−	0.992148i	$-0.460085\pi$
$743$	6.81824	0.250137	0.125068	+	0.992148i	$0.460085\pi$
$744$	0	0
$745$	−6.29276	−0.230549
$746$	0	0
$747$	1.11799	0.0409053
$748$	0	0
$749$	15.6631	0.572318
$750$	0	0
$751$	7.78505	0.284080	0.142040	−	0.989861i	$-0.454634\pi$
$751$	7.78505	0.284080	0.142040	+	0.989861i	$0.454634\pi$
$752$	0	0
$753$	−25.1895	−0.917958
$754$	0	0
$755$	6.61872	0.240880
$756$	0	0
$757$	41.1096	1.49415	0.747076	−	0.664738i	$-0.231457\pi$
$757$	41.1096	1.49415	0.747076	+	0.664738i	$0.231457\pi$
$758$	0	0
$759$	3.71255	0.134757
$760$	0	0
$761$	−29.8485	−1.08201	−0.541003	−	0.841021i	$-0.681955\pi$
$761$	−29.8485	−1.08201	−0.541003	+	0.841021i	$0.681955\pi$
$762$	0	0
$763$	8.75230	0.316854
$764$	0	0
$765$	−0.705701	−0.0255147
$766$	0	0
$767$	26.1905	0.945683
$768$	0	0
$769$	−33.6148	−1.21218	−0.606090	−	0.795396i	$-0.707263\pi$
$769$	−33.6148	−1.21218	−0.606090	+	0.795396i	$0.707263\pi$
$770$	0	0
$771$	−4.89227	−0.176191
$772$	0	0
$773$	14.3212	0.515099	0.257549	−	0.966265i	$-0.417085\pi$
$773$	14.3212	0.515099	0.257549	+	0.966265i	$0.417085\pi$
$774$	0	0
$775$	40.2524	1.44591
$776$	0	0
$777$	−7.28308	−0.261279
$778$	0	0
$779$	11.2259	0.402209
$780$	0	0
$781$	50.2342	1.79752
$782$	0	0
$783$	−5.41527	−0.193526
$784$	0	0
$785$	−3.60810	−0.128778
$786$	0	0
$787$	6.71066	0.239209	0.119605	−	0.992822i	$-0.461837\pi$
$787$	6.71066	0.239209	0.119605	+	0.992822i	$0.461837\pi$
$788$	0	0
$789$	23.1755	0.825070
$790$	0	0
$791$	−15.8051	−0.561966
$792$	0	0
$793$	−29.1727	−1.03595
$794$	0	0
$795$	−0.195952	−0.00694970
$796$	0	0
$797$	−18.4248	−0.652640	−0.326320	−	0.945259i	$-0.605809\pi$
$797$	−18.4248	−0.652640	−0.326320	+	0.945259i	$0.605809\pi$
$798$	0	0
$799$	8.05269	0.284884
$800$	0	0
$801$	6.32814	0.223594
$802$	0	0
$803$	14.2738	0.503710
$804$	0	0
$805$	−0.318315	−0.0112191
$806$	0	0
$807$	24.3811	0.858256
$808$	0	0
$809$	18.6917	0.657164	0.328582	−	0.944475i	$-0.393429\pi$
$809$	18.6917	0.657164	0.328582	+	0.944475i	$0.393429\pi$
$810$	0	0
$811$	−24.7543	−0.869242	−0.434621	−	0.900613i	$-0.643118\pi$
$811$	−24.7543	−0.869242	−0.434621	+	0.900613i	$0.643118\pi$
$812$	0	0
$813$	5.42853	0.190387
$814$	0	0
$815$	7.37007	0.258162
$816$	0	0
$817$	20.1960	0.706567
$818$	0	0
$819$	−2.66749	−0.0932097
$820$	0	0
$821$	7.24866	0.252980	0.126490	−	0.991968i	$-0.459629\pi$
$821$	7.24866	0.252980	0.126490	+	0.991968i	$0.459629\pi$
$822$	0	0
$823$	−11.1352	−0.388147	−0.194074	−	0.980987i	$-0.562170\pi$
$823$	−11.1352	−0.388147	−0.194074	+	0.980987i	$0.562170\pi$
$824$	0	0
$825$	−18.1866	−0.633175
$826$	0	0
$827$	23.7059	0.824333	0.412167	−	0.911108i	$-0.364772\pi$
$827$	23.7059	0.824333	0.412167	+	0.911108i	$0.364772\pi$
$828$	0	0
$829$	−8.28250	−0.287663	−0.143832	−	0.989602i	$-0.545942\pi$
$829$	−8.28250	−0.287663	−0.143832	+	0.989602i	$0.545942\pi$
$830$	0	0
$831$	15.5730	0.540222
$832$	0	0
$833$	2.21699	0.0768142
$834$	0	0
$835$	−0.747779	−0.0258779
$836$	0	0
$837$	8.21699	0.284021
$838$	0	0
$839$	45.7672	1.58006	0.790030	−	0.613068i	$-0.210065\pi$
$839$	45.7672	1.58006	0.790030	+	0.613068i	$0.210065\pi$
$840$	0	0
$841$	0.325161	0.0112125
$842$	0	0
$843$	−3.05736	−0.105301
$844$	0	0
$845$	1.87312	0.0644373
$846$	0	0
$847$	2.78301	0.0956253
$848$	0	0
$849$	4.24334	0.145631
$850$	0	0
$851$	7.28308	0.249661
$852$	0	0
$853$	−54.8222	−1.87708	−0.938539	−	0.345173i	$-0.887820\pi$
$853$	−54.8222	−1.87708	−0.938539	+	0.345173i	$0.887820\pi$
$854$	0	0
$855$	−0.747779	−0.0255735
$856$	0	0
$857$	−45.7841	−1.56395	−0.781977	−	0.623307i	$-0.785789\pi$
$857$	−45.7841	−1.56395	−0.781977	+	0.623307i	$0.785789\pi$
$858$	0	0
$859$	28.8351	0.983840	0.491920	−	0.870641i	$-0.336295\pi$
$859$	28.8351	0.983840	0.491920	+	0.870641i	$0.336295\pi$
$860$	0	0
$861$	−4.77864	−0.162856
$862$	0	0
$863$	21.1744	0.720785	0.360393	−	0.932801i	$-0.382643\pi$
$863$	21.1744	0.720785	0.360393	+	0.932801i	$0.382643\pi$
$864$	0	0
$865$	3.10226	0.105480
$866$	0	0
$867$	12.0850	0.410427
$868$	0	0
$869$	56.3554	1.91173
$870$	0	0
$871$	1.99469	0.0675876
$872$	0	0
$873$	−16.1136	−0.545364
$874$	0	0
$875$	3.15090	0.106520
$876$	0	0
$877$	7.55728	0.255191	0.127596	−	0.991826i	$-0.459274\pi$
$877$	7.55728	0.255191	0.127596	+	0.991826i	$0.459274\pi$
$878$	0	0
$879$	18.6984	0.630680
$880$	0	0
$881$	11.0422	0.372022	0.186011	−	0.982548i	$-0.440444\pi$
$881$	11.0422	0.372022	0.186011	+	0.982548i	$0.440444\pi$
$882$	0	0
$883$	49.5381	1.66709	0.833544	−	0.552453i	$-0.186308\pi$
$883$	49.5381	1.66709	0.833544	+	0.552453i	$0.186308\pi$
$884$	0	0
$885$	−3.12534	−0.105057
$886$	0	0
$887$	−36.4475	−1.22379	−0.611894	−	0.790940i	$-0.709592\pi$
$887$	−36.4475	−1.22379	−0.611894	+	0.790940i	$0.709592\pi$
$888$	0	0
$889$	−5.92706	−0.198787
$890$	0	0
$891$	−3.71255	−0.124375
$892$	0	0
$893$	8.53283	0.285540
$894$	0	0
$895$	−8.28154	−0.276821
$896$	0	0
$897$	2.66749	0.0890650
$898$	0	0
$899$	−44.4972	−1.48407
$900$	0	0
$901$	−1.36476	−0.0454668
$902$	0	0
$903$	−8.59703	−0.286091
$904$	0	0
$905$	−5.99861	−0.199401
$906$	0	0
$907$	−10.5496	−0.350295	−0.175148	−	0.984542i	$-0.556040\pi$
$907$	−10.5496	−0.350295	−0.175148	+	0.984542i	$0.556040\pi$
$908$	0	0
$909$	−13.2435	−0.439259
$910$	0	0
$911$	26.6343	0.882433	0.441217	−	0.897401i	$-0.354547\pi$
$911$	26.6343	0.882433	0.441217	+	0.897401i	$0.354547\pi$
$912$	0	0
$913$	−4.15061	−0.137365
$914$	0	0
$915$	3.48122	0.115085
$916$	0	0
$917$	−8.78316	−0.290045
$918$	0	0
$919$	−13.8918	−0.458247	−0.229124	−	0.973397i	$-0.573586\pi$
$919$	−13.8918	−0.458247	−0.229124	+	0.973397i	$0.573586\pi$
$920$	0	0
$921$	25.3448	0.835140
$922$	0	0
$923$	36.0937	1.18804
$924$	0	0
$925$	−35.6775	−1.17307
$926$	0	0
$927$	3.12782	0.102731
$928$	0	0
$929$	−9.69538	−0.318095	−0.159048	−	0.987271i	$-0.550842\pi$
$929$	−9.69538	−0.318095	−0.159048	+	0.987271i	$0.550842\pi$
$930$	0	0
$931$	2.34918	0.0769912
$932$	0	0
$933$	14.5221	0.475431
$934$	0	0
$935$	2.61995	0.0856815
$936$	0	0
$937$	−24.3715	−0.796181	−0.398091	−	0.917346i	$-0.630327\pi$
$937$	−24.3715	−0.796181	−0.398091	+	0.917346i	$0.630327\pi$
$938$	0	0
$939$	−4.53778	−0.148085
$940$	0	0
$941$	−14.1171	−0.460203	−0.230101	−	0.973167i	$-0.573906\pi$
$941$	−14.1171	−0.460203	−0.230101	+	0.973167i	$0.573906\pi$
$942$	0	0
$943$	4.77864	0.155614
$944$	0	0
$945$	0.318315	0.0103548
$946$	0	0
$947$	−14.6097	−0.474751	−0.237375	−	0.971418i	$-0.576287\pi$
$947$	−14.6097	−0.474751	−0.237375	+	0.971418i	$0.576287\pi$
$948$	0	0
$949$	10.2558	0.332917
$950$	0	0
$951$	−3.61996	−0.117385
$952$	0	0
$953$	17.5328	0.567944	0.283972	−	0.958833i	$-0.408348\pi$
$953$	17.5328	0.567944	0.283972	+	0.958833i	$0.408348\pi$
$954$	0	0
$955$	3.24488	0.105002
$956$	0	0
$957$	20.1045	0.649885
$958$	0	0
$959$	18.1333	0.585554
$960$	0	0
$961$	36.5189	1.17803
$962$	0	0
$963$	15.6631	0.504737
$964$	0	0
$965$	−3.73577	−0.120259
$966$	0	0
$967$	−46.8221	−1.50570	−0.752849	−	0.658194i	$-0.771321\pi$
$967$	−46.8221	−1.50570	−0.752849	+	0.658194i	$0.771321\pi$
$968$	0	0
$969$	−5.20810	−0.167308
$970$	0	0
$971$	55.9608	1.79587	0.897934	−	0.440129i	$-0.145067\pi$
$971$	55.9608	1.79587	0.897934	+	0.440129i	$0.145067\pi$
$972$	0	0
$973$	0.884483	0.0283552
$974$	0	0
$975$	−13.0672	−0.418485
$976$	0	0
$977$	12.7251	0.407113	0.203557	−	0.979063i	$-0.434750\pi$
$977$	12.7251	0.407113	0.203557	+	0.979063i	$0.434750\pi$
$978$	0	0
$979$	−23.4935	−0.750856
$980$	0	0
$981$	8.75230	0.279439
$982$	0	0
$983$	28.5147	0.909478	0.454739	−	0.890625i	$-0.349733\pi$
$983$	28.5147	0.909478	0.454739	+	0.890625i	$0.349733\pi$
$984$	0	0
$985$	−4.91986	−0.156760
$986$	0	0
$987$	−3.63226	−0.115616
$988$	0	0
$989$	8.59703	0.273370
$990$	0	0
$991$	−0.993154	−0.0315486	−0.0157743	−	0.999876i	$-0.505021\pi$
$991$	−0.993154	−0.0315486	−0.0157743	+	0.999876i	$0.505021\pi$
$992$	0	0
$993$	12.5749	0.399053
$994$	0	0
$995$	5.36119	0.169961
$996$	0	0
$997$	39.8410	1.26178	0.630888	−	0.775874i	$-0.282691\pi$
$997$	39.8410	1.26178	0.630888	+	0.775874i	$0.282691\pi$
$998$	0	0
$999$	−7.28308	−0.230427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bz.1.3		4
4.3	odd	2		3864.2.a.t.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.t.1.3	✓	4	4.3	odd	2
7728.2.a.bz.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.318315 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.318315 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.71255 q^{11} -2.66749 q^{13} +0.318315 q^{15} +2.21699 q^{17} +2.34918 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.89868 q^{25} -1.00000 q^{27} +5.41527 q^{29} -8.21699 q^{31} +3.71255 q^{33} -0.318315 q^{35} +7.28308 q^{37} +2.66749 q^{39} +4.77864 q^{41} +8.59703 q^{43} -0.318315 q^{45} +3.63226 q^{47} +1.00000 q^{49} -2.21699 q^{51} -0.615591 q^{53} +1.18176 q^{55} -2.34918 q^{57} -9.81839 q^{59} +10.9364 q^{61} +1.00000 q^{63} +0.849103 q^{65} -0.747779 q^{67} -1.00000 q^{69} -13.5309 q^{71} -3.84473 q^{73} +4.89868 q^{75} -3.71255 q^{77} -15.1797 q^{79} +1.00000 q^{81} +1.11799 q^{83} -0.705701 q^{85} -5.41527 q^{87} +6.32814 q^{89} -2.66749 q^{91} +8.21699 q^{93} -0.747779 q^{95} -16.1136 q^{97} -3.71255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{15} - 10 q^{17} - 4 q^{19} - 4 q^{21} + 4 q^{23} - 4 q^{27} + 11 q^{29} - 14 q^{31} - 2 q^{35} + 13 q^{37} - 2 q^{39} + 7 q^{41} - 12 q^{43} - 2 q^{45} - 15 q^{47} + 4 q^{49} + 10 q^{51} + q^{53} - 31 q^{55} + 4 q^{57} - 5 q^{59} + 3 q^{61} + 4 q^{63} + 25 q^{65} - 5 q^{67} - 4 q^{69} - 5 q^{71} - 6 q^{73} - 26 q^{79} + 4 q^{81} - 2 q^{83} + 5 q^{85} - 11 q^{87} + 7 q^{89} + 2 q^{91} + 14 q^{93} - 5 q^{95} - 27 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9 + 2 * q^13 + 2 * q^15 - 10 * q^17 - 4 * q^19 - 4 * q^21 + 4 * q^23 - 4 * q^27 + 11 * q^29 - 14 * q^31 - 2 * q^35 + 13 * q^37 - 2 * q^39 + 7 * q^41 - 12 * q^43 - 2 * q^45 - 15 * q^47 + 4 * q^49 + 10 * q^51 + q^53 - 31 * q^55 + 4 * q^57 - 5 * q^59 + 3 * q^61 + 4 * q^63 + 25 * q^65 - 5 * q^67 - 4 * q^69 - 5 * q^71 - 6 * q^73 - 26 * q^79 + 4 * q^81 - 2 * q^83 + 5 * q^85 - 11 * q^87 + 7 * q^89 + 2 * q^91 + 14 * q^93 - 5 * q^95 - 27 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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