LMFDB - Embedded newform 7728.2.a.bu.1.2

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:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
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			1.2
Root			$2.66908$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.21417 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.21417 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.79306 q^{11} -6.57889 q^{13} -3.21417 q^{15} +0.454904 q^{17} -1.54510 q^{19} -1.00000 q^{21} -1.00000 q^{23} +5.33092 q^{25} +1.00000 q^{27} -3.36471 q^{29} -4.88325 q^{31} -3.79306 q^{33} +3.21417 q^{35} +4.88325 q^{37} -6.57889 q^{39} -0.883254 q^{41} -5.00724 q^{43} -3.21417 q^{45} -4.42835 q^{47} +1.00000 q^{49} +0.454904 q^{51} +0.123983 q^{53} +12.1916 q^{55} -1.54510 q^{57} +5.87602 q^{59} -8.33092 q^{61} -1.00000 q^{63} +21.1457 q^{65} -1.87602 q^{67} -1.00000 q^{69} +13.2552 q^{71} -5.13122 q^{73} +5.33092 q^{75} +3.79306 q^{77} -11.1312 q^{79} +1.00000 q^{81} -1.79306 q^{83} -1.46214 q^{85} -3.36471 q^{87} +0.150537 q^{89} +6.57889 q^{91} -4.88325 q^{93} +4.96621 q^{95} +9.61268 q^{97} -3.79306 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{11} - 9 q^{13} - 3 q^{15} - 6 q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} + 3 q^{27} - 6 q^{29} + 6 q^{33} + 3 q^{35} - 9 q^{39} + 12 q^{41} + 9 q^{43} - 3 q^{45} + 3 q^{49} - 9 q^{53} + 3 q^{55} - 6 q^{57} + 27 q^{59} - 33 q^{61} - 3 q^{63} - 18 q^{65} - 15 q^{67} - 3 q^{69} - 3 q^{71} + 18 q^{73} + 24 q^{75} - 6 q^{77} + 3 q^{81} + 12 q^{83} + 21 q^{85} - 6 q^{87} + 3 q^{89} + 9 q^{91} + 27 q^{95} + 6 q^{97} + 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 6 * q^11 - 9 * q^13 - 3 * q^15 - 6 * q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 + 3 * q^27 - 6 * q^29 + 6 * q^33 + 3 * q^35 - 9 * q^39 + 12 * q^41 + 9 * q^43 - 3 * q^45 + 3 * q^49 - 9 * q^53 + 3 * q^55 - 6 * q^57 + 27 * q^59 - 33 * q^61 - 3 * q^63 - 18 * q^65 - 15 * q^67 - 3 * q^69 - 3 * q^71 + 18 * q^73 + 24 * q^75 - 6 * q^77 + 3 * q^81 + 12 * q^83 + 21 * q^85 - 6 * q^87 + 3 * q^89 + 9 * q^91 + 27 * q^95 + 6 * q^97 + 6 * 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
$5$	−3.21417	−1.43742	−0.718711	−	0.695309i	$-0.755268\pi$
$5$	−3.21417	−1.43742	−0.718711	+	0.695309i	$0.755268\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.79306	−1.14365	−0.571826	−	0.820375i	$-0.693765\pi$
$11$	−3.79306	−1.14365	−0.571826	+	0.820375i	$0.693765\pi$
$12$	0	0
$13$	−6.57889	−1.82466	−0.912328	−	0.409461i	$-0.865717\pi$
$13$	−6.57889	−1.82466	−0.912328	+	0.409461i	$0.865717\pi$
$14$	0	0
$15$	−3.21417	−0.829896
$16$	0	0
$17$	0.454904	0.110330	0.0551652	−	0.998477i	$-0.482431\pi$
$17$	0.454904	0.110330	0.0551652	+	0.998477i	$0.482431\pi$
$18$	0	0
$19$	−1.54510	−0.354469	−0.177235	−	0.984169i	$-0.556715\pi$
$19$	−1.54510	−0.354469	−0.177235	+	0.984169i	$0.556715\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.33092	1.06618
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.36471	−0.624811	−0.312406	−	0.949949i	$-0.601135\pi$
$29$	−3.36471	−0.624811	−0.312406	+	0.949949i	$0.601135\pi$
$30$	0	0
$31$	−4.88325	−0.877058	−0.438529	−	0.898717i	$-0.644500\pi$
$31$	−4.88325	−0.877058	−0.438529	+	0.898717i	$0.644500\pi$
$32$	0	0
$33$	−3.79306	−0.660287
$34$	0	0
$35$	3.21417	0.543295
$36$	0	0
$37$	4.88325	0.802802	0.401401	−	0.915902i	$-0.368523\pi$
$37$	4.88325	0.802802	0.401401	+	0.915902i	$0.368523\pi$
$38$	0	0
$39$	−6.57889	−1.05347
$40$	0	0
$41$	−0.883254	−0.137941	−0.0689706	−	0.997619i	$-0.521971\pi$
$41$	−0.883254	−0.137941	−0.0689706	+	0.997619i	$0.521971\pi$
$42$	0	0
$43$	−5.00724	−0.763597	−0.381798	−	0.924246i	$-0.624695\pi$
$43$	−5.00724	−0.763597	−0.381798	+	0.924246i	$0.624695\pi$
$44$	0	0
$45$	−3.21417	−0.479141
$46$	0	0
$47$	−4.42835	−0.645941	−0.322971	−	0.946409i	$-0.604682\pi$
$47$	−4.42835	−0.645941	−0.322971	+	0.946409i	$0.604682\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.454904	0.0636993
$52$	0	0
$53$	0.123983	0.0170304	0.00851520	−	0.999964i	$-0.497289\pi$
$53$	0.123983	0.0170304	0.00851520	+	0.999964i	$0.497289\pi$
$54$	0	0
$55$	12.1916	1.64391
$56$	0	0
$57$	−1.54510	−0.204653
$58$	0	0
$59$	5.87602	0.764992	0.382496	−	0.923957i	$-0.375065\pi$
$59$	5.87602	0.764992	0.382496	+	0.923957i	$0.375065\pi$
$60$	0	0
$61$	−8.33092	−1.06667	−0.533333	−	0.845906i	$-0.679061\pi$
$61$	−8.33092	−1.06667	−0.533333	+	0.845906i	$0.679061\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	21.1457	2.62280
$66$	0	0
$67$	−1.87602	−0.229192	−0.114596	−	0.993412i	$-0.536557\pi$
$67$	−1.87602	−0.229192	−0.114596	+	0.993412i	$0.536557\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	13.2552	1.57310	0.786552	−	0.617525i	$-0.211864\pi$
$71$	13.2552	1.57310	0.786552	+	0.617525i	$0.211864\pi$
$72$	0	0
$73$	−5.13122	−0.600564	−0.300282	−	0.953850i	$-0.597081\pi$
$73$	−5.13122	−0.600564	−0.300282	+	0.953850i	$0.597081\pi$
$74$	0	0
$75$	5.33092	0.615562
$76$	0	0
$77$	3.79306	0.432260
$78$	0	0
$79$	−11.1312	−1.25236	−0.626180	−	0.779678i	$-0.715383\pi$
$79$	−11.1312	−1.25236	−0.626180	+	0.779678i	$0.715383\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.79306	−0.196814	−0.0984071	−	0.995146i	$-0.531375\pi$
$83$	−1.79306	−0.196814	−0.0984071	+	0.995146i	$0.531375\pi$
$84$	0	0
$85$	−1.46214	−0.158591
$86$	0	0
$87$	−3.36471	−0.360735
$88$	0	0
$89$	0.150537	0.0159569	0.00797846	−	0.999968i	$-0.497460\pi$
$89$	0.150537	0.0159569	0.00797846	+	0.999968i	$0.497460\pi$
$90$	0	0
$91$	6.57889	0.689655
$92$	0	0
$93$	−4.88325	−0.506370
$94$	0	0
$95$	4.96621	0.509522
$96$	0	0
$97$	9.61268	0.976020	0.488010	−	0.872838i	$-0.337723\pi$
$97$	9.61268	0.976020	0.488010	+	0.872838i	$0.337723\pi$
$98$	0	0
$99$	−3.79306	−0.381217
$100$	0	0
$101$	−7.03379	−0.699888	−0.349944	−	0.936771i	$-0.613799\pi$
$101$	−7.03379	−0.699888	−0.349944	+	0.936771i	$0.613799\pi$
$102$	0	0
$103$	−2.67632	−0.263705	−0.131853	−	0.991269i	$-0.542093\pi$
$103$	−2.67632	−0.263705	−0.131853	+	0.991269i	$0.542093\pi$
$104$	0	0
$105$	3.21417	0.313671
$106$	0	0
$107$	9.00724	0.870762	0.435381	−	0.900246i	$-0.356614\pi$
$107$	9.00724	0.870762	0.435381	+	0.900246i	$0.356614\pi$
$108$	0	0
$109$	−2.30437	−0.220718	−0.110359	−	0.993892i	$-0.535200\pi$
$109$	−2.30437	−0.220718	−0.110359	+	0.993892i	$0.535200\pi$
$110$	0	0
$111$	4.88325	0.463498
$112$	0	0
$113$	2.96621	0.279037	0.139519	−	0.990219i	$-0.455444\pi$
$113$	2.96621	0.279037	0.139519	+	0.990219i	$0.455444\pi$
$114$	0	0
$115$	3.21417	0.299723
$116$	0	0
$117$	−6.57889	−0.608218
$118$	0	0
$119$	−0.454904	−0.0417010
$120$	0	0
$121$	3.38732	0.307938
$122$	0	0
$123$	−0.883254	−0.0796403
$124$	0	0
$125$	−1.06364	−0.0951346
$126$	0	0
$127$	3.24073	0.287568	0.143784	−	0.989609i	$-0.454073\pi$
$127$	3.24073	0.287568	0.143784	+	0.989609i	$0.454073\pi$
$128$	0	0
$129$	−5.00724	−0.440863
$130$	0	0
$131$	2.88325	0.251911	0.125956	−	0.992036i	$-0.459800\pi$
$131$	2.88325	0.251911	0.125956	+	0.992036i	$0.459800\pi$
$132$	0	0
$133$	1.54510	0.133977
$134$	0	0
$135$	−3.21417	−0.276632
$136$	0	0
$137$	−1.79306	−0.153192	−0.0765958	−	0.997062i	$-0.524405\pi$
$137$	−1.79306	−0.153192	−0.0765958	+	0.997062i	$0.524405\pi$
$138$	0	0
$139$	9.46214	0.802568	0.401284	−	0.915954i	$-0.368564\pi$
$139$	9.46214	0.802568	0.401284	+	0.915954i	$0.368564\pi$
$140$	0	0
$141$	−4.42835	−0.372934
$142$	0	0
$143$	24.9541	2.08677
$144$	0	0
$145$	10.8148	0.898118
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−8.06758	−0.660922	−0.330461	−	0.943820i	$-0.607204\pi$
$149$	−8.06758	−0.660922	−0.330461	+	0.943820i	$0.607204\pi$
$150$	0	0
$151$	23.3382	1.89923	0.949616	−	0.313415i	$-0.101473\pi$
$151$	23.3382	1.89923	0.949616	+	0.313415i	$0.101473\pi$
$152$	0	0
$153$	0.454904	0.0367768
$154$	0	0
$155$	15.6956	1.26070
$156$	0	0
$157$	7.15777	0.571253	0.285626	−	0.958341i	$-0.407798\pi$
$157$	7.15777	0.571253	0.285626	+	0.958341i	$0.407798\pi$
$158$	0	0
$159$	0.123983	0.00983251
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−4.85341	−0.380148	−0.190074	−	0.981770i	$-0.560873\pi$
$163$	−4.85341	−0.380148	−0.190074	+	0.981770i	$0.560873\pi$
$164$	0	0
$165$	12.1916	0.949112
$166$	0	0
$167$	15.5596	1.20404	0.602018	−	0.798483i	$-0.294364\pi$
$167$	15.5596	1.20404	0.602018	+	0.798483i	$0.294364\pi$
$168$	0	0
$169$	30.2818	2.32937
$170$	0	0
$171$	−1.54510	−0.118156
$172$	0	0
$173$	−23.3792	−1.77749	−0.888743	−	0.458405i	$-0.848421\pi$
$173$	−23.3792	−1.77749	−0.888743	+	0.458405i	$0.848421\pi$
$174$	0	0
$175$	−5.33092	−0.402980
$176$	0	0
$177$	5.87602	0.441668
$178$	0	0
$179$	23.8905	1.78566	0.892830	−	0.450395i	$-0.148717\pi$
$179$	23.8905	1.78566	0.892830	+	0.450395i	$0.148717\pi$
$180$	0	0
$181$	−7.97345	−0.592662	−0.296331	−	0.955085i	$-0.595763\pi$
$181$	−7.97345	−0.592662	−0.296331	+	0.955085i	$0.595763\pi$
$182$	0	0
$183$	−8.33092	−0.615839
$184$	0	0
$185$	−15.6956	−1.15397
$186$	0	0
$187$	−1.72548	−0.126180
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	10.0145	0.724622	0.362311	−	0.932057i	$-0.381988\pi$
$191$	10.0145	0.724622	0.362311	+	0.932057i	$0.381988\pi$
$192$	0	0
$193$	17.8341	1.28373	0.641863	−	0.766819i	$-0.278162\pi$
$193$	17.8341	1.28373	0.641863	+	0.766819i	$0.278162\pi$
$194$	0	0
$195$	21.1457	1.51427
$196$	0	0
$197$	−7.91705	−0.564066	−0.282033	−	0.959405i	$-0.591009\pi$
$197$	−7.91705	−0.564066	−0.282033	+	0.959405i	$0.591009\pi$
$198$	0	0
$199$	2.12398	0.150565	0.0752826	−	0.997162i	$-0.476014\pi$
$199$	2.12398	0.150565	0.0752826	+	0.997162i	$0.476014\pi$
$200$	0	0
$201$	−1.87602	−0.132324
$202$	0	0
$203$	3.36471	0.236157
$204$	0	0
$205$	2.83893	0.198280
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	5.86064	0.405389
$210$	0	0
$211$	−23.6272	−1.62656	−0.813280	−	0.581872i	$-0.802320\pi$
$211$	−23.6272	−1.62656	−0.813280	+	0.581872i	$0.802320\pi$
$212$	0	0
$213$	13.2552	0.908232
$214$	0	0
$215$	16.0941	1.09761
$216$	0	0
$217$	4.88325	0.331497
$218$	0	0
$219$	−5.13122	−0.346736
$220$	0	0
$221$	−2.99276	−0.201315
$222$	0	0
$223$	−8.61992	−0.577232	−0.288616	−	0.957445i	$-0.593195\pi$
$223$	−8.61992	−0.577232	−0.288616	+	0.957445i	$0.593195\pi$
$224$	0	0
$225$	5.33092	0.355395
$226$	0	0
$227$	22.3044	1.48039	0.740196	−	0.672391i	$-0.234733\pi$
$227$	22.3044	1.48039	0.740196	+	0.672391i	$0.234733\pi$
$228$	0	0
$229$	−21.8905	−1.44656	−0.723282	−	0.690553i	$-0.757367\pi$
$229$	−21.8905	−1.44656	−0.723282	+	0.690553i	$0.757367\pi$
$230$	0	0
$231$	3.79306	0.249565
$232$	0	0
$233$	−0.537859	−0.0352363	−0.0176181	−	0.999845i	$-0.505608\pi$
$233$	−0.537859	−0.0352363	−0.0176181	+	0.999845i	$0.505608\pi$
$234$	0	0
$235$	14.2335	0.928491
$236$	0	0
$237$	−11.1312	−0.723050
$238$	0	0
$239$	7.69563	0.497789	0.248895	−	0.968531i	$-0.419933\pi$
$239$	7.69563	0.497789	0.248895	+	0.968531i	$0.419933\pi$
$240$	0	0
$241$	−30.1086	−1.93947	−0.969733	−	0.244167i	$-0.921485\pi$
$241$	−30.1086	−1.93947	−0.969733	+	0.244167i	$0.921485\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.21417	−0.205346
$246$	0	0
$247$	10.1650	0.646784
$248$	0	0
$249$	−1.79306	−0.113631
$250$	0	0
$251$	24.0145	1.51578	0.757890	−	0.652382i	$-0.226230\pi$
$251$	24.0145	1.51578	0.757890	+	0.652382i	$0.226230\pi$
$252$	0	0
$253$	3.79306	0.238468
$254$	0	0
$255$	−1.46214	−0.0915628
$256$	0	0
$257$	19.1047	1.19172	0.595858	−	0.803090i	$-0.296812\pi$
$257$	19.1047	1.19172	0.595858	+	0.803090i	$0.296812\pi$
$258$	0	0
$259$	−4.88325	−0.303431
$260$	0	0
$261$	−3.36471	−0.208270
$262$	0	0
$263$	−3.13122	−0.193079	−0.0965397	−	0.995329i	$-0.530777\pi$
$263$	−3.13122	−0.193079	−0.0965397	+	0.995329i	$0.530777\pi$
$264$	0	0
$265$	−0.398504	−0.0244799
$266$	0	0
$267$	0.150537	0.00921273
$268$	0	0
$269$	−8.75927	−0.534062	−0.267031	−	0.963688i	$-0.586043\pi$
$269$	−8.75927	−0.534062	−0.267031	+	0.963688i	$0.586043\pi$
$270$	0	0
$271$	27.8075	1.68919	0.844594	−	0.535408i	$-0.179842\pi$
$271$	27.8075	1.68919	0.844594	+	0.535408i	$0.179842\pi$
$272$	0	0
$273$	6.57889	0.398172
$274$	0	0
$275$	−20.2205	−1.21934
$276$	0	0
$277$	−23.2021	−1.39408	−0.697039	−	0.717033i	$-0.745500\pi$
$277$	−23.2021	−1.39408	−0.697039	+	0.717033i	$0.745500\pi$
$278$	0	0
$279$	−4.88325	−0.292353
$280$	0	0
$281$	−16.9774	−1.01279	−0.506393	−	0.862303i	$-0.669022\pi$
$281$	−16.9774	−1.01279	−0.506393	+	0.862303i	$0.669022\pi$
$282$	0	0
$283$	−24.8679	−1.47824	−0.739121	−	0.673573i	$-0.764759\pi$
$283$	−24.8679	−1.47824	−0.739121	+	0.673573i	$0.764759\pi$
$284$	0	0
$285$	4.96621	0.294173
$286$	0	0
$287$	0.883254	0.0521368
$288$	0	0
$289$	−16.7931	−0.987827
$290$	0	0
$291$	9.61268	0.563505
$292$	0	0
$293$	−13.5861	−0.793710	−0.396855	−	0.917881i	$-0.629898\pi$
$293$	−13.5861	−0.793710	−0.396855	+	0.917881i	$0.629898\pi$
$294$	0	0
$295$	−18.8865	−1.09962
$296$	0	0
$297$	−3.79306	−0.220096
$298$	0	0
$299$	6.57889	0.380467
$300$	0	0
$301$	5.00724	0.288612
$302$	0	0
$303$	−7.03379	−0.404081
$304$	0	0
$305$	26.7770	1.53325
$306$	0	0
$307$	−15.3792	−0.877737	−0.438868	−	0.898551i	$-0.644621\pi$
$307$	−15.3792	−0.877737	−0.438868	+	0.898551i	$0.644621\pi$
$308$	0	0
$309$	−2.67632	−0.152250
$310$	0	0
$311$	18.0974	1.02621	0.513106	−	0.858326i	$-0.328495\pi$
$311$	18.0974	1.02621	0.513106	+	0.858326i	$0.328495\pi$
$312$	0	0
$313$	−10.6763	−0.603461	−0.301731	−	0.953393i	$-0.597564\pi$
$313$	−10.6763	−0.603461	−0.301731	+	0.953393i	$0.597564\pi$
$314$	0	0
$315$	3.21417	0.181098
$316$	0	0
$317$	8.07087	0.453306	0.226653	−	0.973976i	$-0.427222\pi$
$317$	8.07087	0.453306	0.226653	+	0.973976i	$0.427222\pi$
$318$	0	0
$319$	12.7626	0.714566
$320$	0	0
$321$	9.00724	0.502735
$322$	0	0
$323$	−0.702870	−0.0391088
$324$	0	0
$325$	−35.0715	−1.94542
$326$	0	0
$327$	−2.30437	−0.127432
$328$	0	0
$329$	4.42835	0.244143
$330$	0	0
$331$	26.7584	1.47077	0.735387	−	0.677648i	$-0.237001\pi$
$331$	26.7584	1.47077	0.735387	+	0.677648i	$0.237001\pi$
$332$	0	0
$333$	4.88325	0.267601
$334$	0	0
$335$	6.02985	0.329446
$336$	0	0
$337$	13.8495	0.754428	0.377214	−	0.926126i	$-0.376882\pi$
$337$	13.8495	0.754428	0.377214	+	0.926126i	$0.376882\pi$
$338$	0	0
$339$	2.96621	0.161102
$340$	0	0
$341$	18.5225	1.00305
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.21417	0.173045
$346$	0	0
$347$	2.88325	0.154781	0.0773906	−	0.997001i	$-0.475341\pi$
$347$	2.88325	0.154781	0.0773906	+	0.997001i	$0.475341\pi$
$348$	0	0
$349$	30.2060	1.61689	0.808446	−	0.588570i	$-0.200309\pi$
$349$	30.2060	1.61689	0.808446	+	0.588570i	$0.200309\pi$
$350$	0	0
$351$	−6.57889	−0.351155
$352$	0	0
$353$	16.8301	0.895778	0.447889	−	0.894089i	$-0.352176\pi$
$353$	16.8301	0.895778	0.447889	+	0.894089i	$0.352176\pi$
$354$	0	0
$355$	−42.6045	−2.26121
$356$	0	0
$357$	−0.454904	−0.0240761
$358$	0	0
$359$	−16.1385	−0.851755	−0.425878	−	0.904781i	$-0.640035\pi$
$359$	−16.1385	−0.851755	−0.425878	+	0.904781i	$0.640035\pi$
$360$	0	0
$361$	−16.6127	−0.874352
$362$	0	0
$363$	3.38732	0.177788
$364$	0	0
$365$	16.4926	0.863264
$366$	0	0
$367$	−7.90257	−0.412511	−0.206255	−	0.978498i	$-0.566128\pi$
$367$	−7.90257	−0.412511	−0.206255	+	0.978498i	$0.566128\pi$
$368$	0	0
$369$	−0.883254	−0.0459804
$370$	0	0
$371$	−0.123983	−0.00643689
$372$	0	0
$373$	34.7318	1.79835	0.899173	−	0.437594i	$-0.144169\pi$
$373$	34.7318	1.79835	0.899173	+	0.437594i	$0.144169\pi$
$374$	0	0
$375$	−1.06364	−0.0549260
$376$	0	0
$377$	22.1361	1.14007
$378$	0	0
$379$	−0.962916	−0.0494617	−0.0247308	−	0.999694i	$-0.507873\pi$
$379$	−0.962916	−0.0494617	−0.0247308	+	0.999694i	$0.507873\pi$
$380$	0	0
$381$	3.24073	0.166028
$382$	0	0
$383$	−6.46938	−0.330570	−0.165285	−	0.986246i	$-0.552854\pi$
$383$	−6.46938	−0.330570	−0.165285	+	0.986246i	$0.552854\pi$
$384$	0	0
$385$	−12.1916	−0.621340
$386$	0	0
$387$	−5.00724	−0.254532
$388$	0	0
$389$	6.86878	0.348261	0.174130	−	0.984723i	$-0.444289\pi$
$389$	6.86878	0.348261	0.174130	+	0.984723i	$0.444289\pi$
$390$	0	0
$391$	−0.454904	−0.0230055
$392$	0	0
$393$	2.88325	0.145441
$394$	0	0
$395$	35.7777	1.80017
$396$	0	0
$397$	14.5104	0.728256	0.364128	−	0.931349i	$-0.381367\pi$
$397$	14.5104	0.728256	0.364128	+	0.931349i	$0.381367\pi$
$398$	0	0
$399$	1.54510	0.0773515
$400$	0	0
$401$	21.4920	1.07326	0.536629	−	0.843818i	$-0.319697\pi$
$401$	21.4920	1.07326	0.536629	+	0.843818i	$0.319697\pi$
$402$	0	0
$403$	32.1264	1.60033
$404$	0	0
$405$	−3.21417	−0.159714
$406$	0	0
$407$	−18.5225	−0.918126
$408$	0	0
$409$	−16.0266	−0.792462	−0.396231	−	0.918151i	$-0.629682\pi$
$409$	−16.0266	−0.792462	−0.396231	+	0.918151i	$0.629682\pi$
$410$	0	0
$411$	−1.79306	−0.0884452
$412$	0	0
$413$	−5.87602	−0.289140
$414$	0	0
$415$	5.76322	0.282905
$416$	0	0
$417$	9.46214	0.463363
$418$	0	0
$419$	−6.30437	−0.307988	−0.153994	−	0.988072i	$-0.549214\pi$
$419$	−6.30437	−0.307988	−0.153994	+	0.988072i	$0.549214\pi$
$420$	0	0
$421$	−24.5789	−1.19790	−0.598951	−	0.800786i	$-0.704416\pi$
$421$	−24.5789	−1.19790	−0.598951	+	0.800786i	$0.704416\pi$
$422$	0	0
$423$	−4.42835	−0.215314
$424$	0	0
$425$	2.42506	0.117633
$426$	0	0
$427$	8.33092	0.403162
$428$	0	0
$429$	24.9541	1.20480
$430$	0	0
$431$	30.5258	1.47038	0.735188	−	0.677864i	$-0.237094\pi$
$431$	30.5258	1.47038	0.735188	+	0.677864i	$0.237094\pi$
$432$	0	0
$433$	−4.74390	−0.227977	−0.113989	−	0.993482i	$-0.536363\pi$
$433$	−4.74390	−0.227977	−0.113989	+	0.993482i	$0.536363\pi$
$434$	0	0
$435$	10.8148	0.518529
$436$	0	0
$437$	1.54510	0.0739120
$438$	0	0
$439$	−32.0555	−1.52993	−0.764963	−	0.644074i	$-0.777243\pi$
$439$	−32.0555	−1.52993	−0.764963	+	0.644074i	$0.777243\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.49593	0.213608	0.106804	−	0.994280i	$-0.465938\pi$
$443$	4.49593	0.213608	0.106804	+	0.994280i	$0.465938\pi$
$444$	0	0
$445$	−0.483853	−0.0229368
$446$	0	0
$447$	−8.06758	−0.381584
$448$	0	0
$449$	−9.64252	−0.455059	−0.227529	−	0.973771i	$-0.573065\pi$
$449$	−9.64252	−0.455059	−0.227529	+	0.973771i	$0.573065\pi$
$450$	0	0
$451$	3.35024	0.157757
$452$	0	0
$453$	23.3382	1.09652
$454$	0	0
$455$	−21.1457	−0.991325
$456$	0	0
$457$	17.4477	0.816167	0.408084	−	0.912945i	$-0.366197\pi$
$457$	17.4477	0.816167	0.408084	+	0.912945i	$0.366197\pi$
$458$	0	0
$459$	0.454904	0.0212331
$460$	0	0
$461$	−8.35748	−0.389246	−0.194623	−	0.980878i	$-0.562348\pi$
$461$	−8.35748	−0.389246	−0.194623	+	0.980878i	$0.562348\pi$
$462$	0	0
$463$	−33.5065	−1.55718	−0.778589	−	0.627535i	$-0.784064\pi$
$463$	−33.5065	−1.55718	−0.778589	+	0.627535i	$0.784064\pi$
$464$	0	0
$465$	15.6956	0.727868
$466$	0	0
$467$	−16.4163	−0.759654	−0.379827	−	0.925057i	$-0.624017\pi$
$467$	−16.4163	−0.759654	−0.379827	+	0.925057i	$0.624017\pi$
$468$	0	0
$469$	1.87602	0.0866264
$470$	0	0
$471$	7.15777	0.329813
$472$	0	0
$473$	18.9928	0.873288
$474$	0	0
$475$	−8.23678	−0.377930
$476$	0	0
$477$	0.123983	0.00567680
$478$	0	0
$479$	−33.4468	−1.52822	−0.764111	−	0.645085i	$-0.776822\pi$
$479$	−33.4468	−1.52822	−0.764111	+	0.645085i	$0.776822\pi$
$480$	0	0
$481$	−32.1264	−1.46484
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−30.8968	−1.40295
$486$	0	0
$487$	35.3261	1.60078	0.800389	−	0.599481i	$-0.204626\pi$
$487$	35.3261	1.60078	0.800389	+	0.599481i	$0.204626\pi$
$488$	0	0
$489$	−4.85341	−0.219479
$490$	0	0
$491$	−15.6836	−0.707789	−0.353894	−	0.935285i	$-0.615143\pi$
$491$	−15.6836	−0.707789	−0.353894	+	0.935285i	$0.615143\pi$
$492$	0	0
$493$	−1.53062	−0.0689357
$494$	0	0
$495$	12.1916	0.547970
$496$	0	0
$497$	−13.2552	−0.594577
$498$	0	0
$499$	34.4009	1.54000	0.769998	−	0.638046i	$-0.220257\pi$
$499$	34.4009	1.54000	0.769998	+	0.638046i	$0.220257\pi$
$500$	0	0
$501$	15.5596	0.695150
$502$	0	0
$503$	1.57494	0.0702232	0.0351116	−	0.999383i	$-0.488821\pi$
$503$	1.57494	0.0702232	0.0351116	+	0.999383i	$0.488821\pi$
$504$	0	0
$505$	22.6078	1.00604
$506$	0	0
$507$	30.2818	1.34486
$508$	0	0
$509$	12.6618	0.561226	0.280613	−	0.959821i	$-0.409462\pi$
$509$	12.6618	0.561226	0.280613	+	0.959821i	$0.409462\pi$
$510$	0	0
$511$	5.13122	0.226992
$512$	0	0
$513$	−1.54510	−0.0682177
$514$	0	0
$515$	8.60215	0.379056
$516$	0	0
$517$	16.7970	0.738732
$518$	0	0
$519$	−23.3792	−1.02623
$520$	0	0
$521$	45.4347	1.99053	0.995265	−	0.0971991i	$-0.0309884\pi$
$521$	45.4347	1.99053	0.995265	+	0.0971991i	$0.0309884\pi$
$522$	0	0
$523$	−0.647367	−0.0283074	−0.0141537	−	0.999900i	$-0.504505\pi$
$523$	−0.647367	−0.0283074	−0.0141537	+	0.999900i	$0.504505\pi$
$524$	0	0
$525$	−5.33092	−0.232660
$526$	0	0
$527$	−2.22141	−0.0967662
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.87602	0.254997
$532$	0	0
$533$	5.81083	0.251695
$534$	0	0
$535$	−28.9508	−1.25165
$536$	0	0
$537$	23.8905	1.03095
$538$	0	0
$539$	−3.79306	−0.163379
$540$	0	0
$541$	29.2850	1.25906	0.629531	−	0.776975i	$-0.283247\pi$
$541$	29.2850	1.25906	0.629531	+	0.776975i	$0.283247\pi$
$542$	0	0
$543$	−7.97345	−0.342173
$544$	0	0
$545$	7.40664	0.317266
$546$	0	0
$547$	−7.08690	−0.303014	−0.151507	−	0.988456i	$-0.548413\pi$
$547$	−7.08690	−0.303014	−0.151507	+	0.988456i	$0.548413\pi$
$548$	0	0
$549$	−8.33092	−0.355555
$550$	0	0
$551$	5.19880	0.221476
$552$	0	0
$553$	11.1312	0.473348
$554$	0	0
$555$	−15.6956	−0.666243
$556$	0	0
$557$	−4.51041	−0.191112	−0.0955560	−	0.995424i	$-0.530463\pi$
$557$	−4.51041	−0.191112	−0.0955560	+	0.995424i	$0.530463\pi$
$558$	0	0
$559$	32.9420	1.39330
$560$	0	0
$561$	−1.72548	−0.0728498
$562$	0	0
$563$	33.5708	1.41484	0.707419	−	0.706794i	$-0.249859\pi$
$563$	33.5708	1.41484	0.707419	+	0.706794i	$0.249859\pi$
$564$	0	0
$565$	−9.53391	−0.401095
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−20.6232	−0.864570	−0.432285	−	0.901737i	$-0.642293\pi$
$569$	−20.6232	−0.864570	−0.432285	+	0.901737i	$0.642293\pi$
$570$	0	0
$571$	1.80754	0.0756431	0.0378215	−	0.999285i	$-0.487958\pi$
$571$	1.80754	0.0756431	0.0378215	+	0.999285i	$0.487958\pi$
$572$	0	0
$573$	10.0145	0.418361
$574$	0	0
$575$	−5.33092	−0.222315
$576$	0	0
$577$	−5.53302	−0.230342	−0.115171	−	0.993346i	$-0.536742\pi$
$577$	−5.53302	−0.230342	−0.115171	+	0.993346i	$0.536742\pi$
$578$	0	0
$579$	17.8341	0.741160
$580$	0	0
$581$	1.79306	0.0743888
$582$	0	0
$583$	−0.470276	−0.0194768
$584$	0	0
$585$	21.1457	0.874267
$586$	0	0
$587$	−47.1038	−1.94418	−0.972090	−	0.234607i	$-0.924620\pi$
$587$	−47.1038	−1.94418	−0.972090	+	0.234607i	$0.924620\pi$
$588$	0	0
$589$	7.54510	0.310890
$590$	0	0
$591$	−7.91705	−0.325664
$592$	0	0
$593$	2.69079	0.110498	0.0552488	−	0.998473i	$-0.482405\pi$
$593$	2.69079	0.110498	0.0552488	+	0.998473i	$0.482405\pi$
$594$	0	0
$595$	1.46214	0.0599420
$596$	0	0
$597$	2.12398	0.0869288
$598$	0	0
$599$	−11.1876	−0.457114	−0.228557	−	0.973531i	$-0.573401\pi$
$599$	−11.1876	−0.457114	−0.228557	+	0.973531i	$0.573401\pi$
$600$	0	0
$601$	−2.35748	−0.0961634	−0.0480817	−	0.998843i	$-0.515311\pi$
$601$	−2.35748	−0.0961634	−0.0480817	+	0.998843i	$0.515311\pi$
$602$	0	0
$603$	−1.87602	−0.0763973
$604$	0	0
$605$	−10.8874	−0.442638
$606$	0	0
$607$	−10.3985	−0.422062	−0.211031	−	0.977479i	$-0.567682\pi$
$607$	−10.3985	−0.422062	−0.211031	+	0.977479i	$0.567682\pi$
$608$	0	0
$609$	3.36471	0.136345
$610$	0	0
$611$	29.1336	1.17862
$612$	0	0
$613$	−39.8896	−1.61113	−0.805563	−	0.592510i	$-0.798137\pi$
$613$	−39.8896	−1.61113	−0.805563	+	0.592510i	$0.798137\pi$
$614$	0	0
$615$	2.83893	0.114477
$616$	0	0
$617$	−13.0748	−0.526372	−0.263186	−	0.964745i	$-0.584773\pi$
$617$	−13.0748	−0.526372	−0.263186	+	0.964745i	$0.584773\pi$
$618$	0	0
$619$	−16.5258	−0.664227	−0.332114	−	0.943239i	$-0.607762\pi$
$619$	−16.5258	−0.664227	−0.332114	+	0.943239i	$0.607762\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−0.150537	−0.00603115
$624$	0	0
$625$	−23.2359	−0.929435
$626$	0	0
$627$	5.86064	0.234052
$628$	0	0
$629$	2.22141	0.0885735
$630$	0	0
$631$	1.61268	0.0641997	0.0320999	−	0.999485i	$-0.489781\pi$
$631$	1.61268	0.0641997	0.0320999	+	0.999485i	$0.489781\pi$
$632$	0	0
$633$	−23.6272	−0.939095
$634$	0	0
$635$	−10.4163	−0.413357
$636$	0	0
$637$	−6.57889	−0.260665
$638$	0	0
$639$	13.2552	0.524368
$640$	0	0
$641$	43.9991	1.73786	0.868930	−	0.494935i	$-0.164808\pi$
$641$	43.9991	1.73786	0.868930	+	0.494935i	$0.164808\pi$
$642$	0	0
$643$	−1.84946	−0.0729357	−0.0364678	−	0.999335i	$-0.511611\pi$
$643$	−1.84946	−0.0729357	−0.0364678	+	0.999335i	$0.511611\pi$
$644$	0	0
$645$	16.0941	0.633706
$646$	0	0
$647$	36.3719	1.42993	0.714964	−	0.699161i	$-0.246443\pi$
$647$	36.3719	1.42993	0.714964	+	0.699161i	$0.246443\pi$
$648$	0	0
$649$	−22.2881	−0.874884
$650$	0	0
$651$	4.88325	0.191390
$652$	0	0
$653$	22.6054	0.884619	0.442310	−	0.896862i	$-0.354159\pi$
$653$	22.6054	0.884619	0.442310	+	0.896862i	$0.354159\pi$
$654$	0	0
$655$	−9.26728	−0.362103
$656$	0	0
$657$	−5.13122	−0.200188
$658$	0	0
$659$	−9.11675	−0.355138	−0.177569	−	0.984108i	$-0.556823\pi$
$659$	−9.11675	−0.355138	−0.177569	+	0.984108i	$0.556823\pi$
$660$	0	0
$661$	−1.29713	−0.0504525	−0.0252262	−	0.999682i	$-0.508031\pi$
$661$	−1.29713	−0.0504525	−0.0252262	+	0.999682i	$0.508031\pi$
$662$	0	0
$663$	−2.99276	−0.116229
$664$	0	0
$665$	−4.96621	−0.192581
$666$	0	0
$667$	3.36471	0.130282
$668$	0	0
$669$	−8.61992	−0.333265
$670$	0	0
$671$	31.5997	1.21989
$672$	0	0
$673$	−21.2440	−0.818897	−0.409448	−	0.912333i	$-0.634279\pi$
$673$	−21.2440	−0.818897	−0.409448	+	0.912333i	$0.634279\pi$
$674$	0	0
$675$	5.33092	0.205187
$676$	0	0
$677$	−23.3680	−0.898105	−0.449053	−	0.893505i	$-0.648238\pi$
$677$	−23.3680	−0.898105	−0.449053	+	0.893505i	$0.648238\pi$
$678$	0	0
$679$	−9.61268	−0.368901
$680$	0	0
$681$	22.3044	0.854705
$682$	0	0
$683$	17.9734	0.687735	0.343867	−	0.939018i	$-0.388263\pi$
$683$	17.9734	0.687735	0.343867	+	0.939018i	$0.388263\pi$
$684$	0	0
$685$	5.76322	0.220201
$686$	0	0
$687$	−21.8905	−0.835174
$688$	0	0
$689$	−0.815671	−0.0310746
$690$	0	0
$691$	29.2021	1.11090	0.555450	−	0.831550i	$-0.312546\pi$
$691$	29.2021	1.11090	0.555450	+	0.831550i	$0.312546\pi$
$692$	0	0
$693$	3.79306	0.144087
$694$	0	0
$695$	−30.4130	−1.15363
$696$	0	0
$697$	−0.401796	−0.0152191
$698$	0	0
$699$	−0.537859	−0.0203437
$700$	0	0
$701$	−46.8003	−1.76762	−0.883811	−	0.467843i	$-0.845031\pi$
$701$	−46.8003	−1.76762	−0.883811	+	0.467843i	$0.845031\pi$
$702$	0	0
$703$	−7.54510	−0.284569
$704$	0	0
$705$	14.2335	0.536064
$706$	0	0
$707$	7.03379	0.264533
$708$	0	0
$709$	40.6609	1.52705	0.763527	−	0.645776i	$-0.223466\pi$
$709$	40.6609	1.52705	0.763527	+	0.645776i	$0.223466\pi$
$710$	0	0
$711$	−11.1312	−0.417453
$712$	0	0
$713$	4.88325	0.182879
$714$	0	0
$715$	−80.2069	−2.99957
$716$	0	0
$717$	7.69563	0.287399
$718$	0	0
$719$	38.0410	1.41869	0.709345	−	0.704861i	$-0.248991\pi$
$719$	38.0410	1.41869	0.709345	+	0.704861i	$0.248991\pi$
$720$	0	0
$721$	2.67632	0.0996712
$722$	0	0
$723$	−30.1086	−1.11975
$724$	0	0
$725$	−17.9370	−0.666164
$726$	0	0
$727$	−39.9348	−1.48110	−0.740550	−	0.672001i	$-0.765435\pi$
$727$	−39.9348	−1.48110	−0.740550	+	0.672001i	$0.765435\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.27781	−0.0842479
$732$	0	0
$733$	−31.1867	−1.15191	−0.575954	−	0.817482i	$-0.695369\pi$
$733$	−31.1867	−1.15191	−0.575954	+	0.817482i	$0.695369\pi$
$734$	0	0
$735$	−3.21417	−0.118557
$736$	0	0
$737$	7.11585	0.262116
$738$	0	0
$739$	2.08625	0.0767438	0.0383719	−	0.999264i	$-0.487783\pi$
$739$	2.08625	0.0767438	0.0383719	+	0.999264i	$0.487783\pi$
$740$	0	0
$741$	10.1650	0.373421
$742$	0	0
$743$	38.5379	1.41382	0.706908	−	0.707305i	$-0.250089\pi$
$743$	38.5379	1.41382	0.706908	+	0.707305i	$0.250089\pi$
$744$	0	0
$745$	25.9306	0.950025
$746$	0	0
$747$	−1.79306	−0.0656047
$748$	0	0
$749$	−9.00724	−0.329117
$750$	0	0
$751$	3.61358	0.131861	0.0659306	−	0.997824i	$-0.478998\pi$
$751$	3.61358	0.131861	0.0659306	+	0.997824i	$0.478998\pi$
$752$	0	0
$753$	24.0145	0.875136
$754$	0	0
$755$	−75.0129	−2.73000
$756$	0	0
$757$	−38.5967	−1.40282	−0.701410	−	0.712758i	$-0.747446\pi$
$757$	−38.5967	−1.40282	−0.701410	+	0.712758i	$0.747446\pi$
$758$	0	0
$759$	3.79306	0.137679
$760$	0	0
$761$	−32.7294	−1.18644	−0.593220	−	0.805040i	$-0.702144\pi$
$761$	−32.7294	−1.18644	−0.593220	+	0.805040i	$0.702144\pi$
$762$	0	0
$763$	2.30437	0.0834237
$764$	0	0
$765$	−1.46214	−0.0528638
$766$	0	0
$767$	−38.6577	−1.39585
$768$	0	0
$769$	10.7891	0.389066	0.194533	−	0.980896i	$-0.437681\pi$
$769$	10.7891	0.389066	0.194533	+	0.980896i	$0.437681\pi$
$770$	0	0
$771$	19.1047	0.688038
$772$	0	0
$773$	−34.2890	−1.23329	−0.616645	−	0.787242i	$-0.711508\pi$
$773$	−34.2890	−1.23329	−0.616645	+	0.787242i	$0.711508\pi$
$774$	0	0
$775$	−26.0322	−0.935106
$776$	0	0
$777$	−4.88325	−0.175186
$778$	0	0
$779$	1.36471	0.0488959
$780$	0	0
$781$	−50.2778	−1.79908
$782$	0	0
$783$	−3.36471	−0.120245
$784$	0	0
$785$	−23.0063	−0.821131
$786$	0	0
$787$	40.0709	1.42837	0.714186	−	0.699956i	$-0.246797\pi$
$787$	40.0709	1.42837	0.714186	+	0.699956i	$0.246797\pi$
$788$	0	0
$789$	−3.13122	−0.111474
$790$	0	0
$791$	−2.96621	−0.105466
$792$	0	0
$793$	54.8082	1.94630
$794$	0	0
$795$	−0.398504	−0.0141335
$796$	0	0
$797$	56.0965	1.98704	0.993521	−	0.113653i	$-0.0362551\pi$
$797$	56.0965	1.98704	0.993521	+	0.113653i	$0.0362551\pi$
$798$	0	0
$799$	−2.01447	−0.0712670
$800$	0	0
$801$	0.150537	0.00531897
$802$	0	0
$803$	19.4630	0.686836
$804$	0	0
$805$	−3.21417	−0.113285
$806$	0	0
$807$	−8.75927	−0.308341
$808$	0	0
$809$	−21.9846	−0.772938	−0.386469	−	0.922302i	$-0.626305\pi$
$809$	−21.9846	−0.772938	−0.386469	+	0.922302i	$0.626305\pi$
$810$	0	0
$811$	−37.3937	−1.31307	−0.656535	−	0.754296i	$-0.727978\pi$
$811$	−37.3937	−1.31307	−0.656535	+	0.754296i	$0.727978\pi$
$812$	0	0
$813$	27.8075	0.975253
$814$	0	0
$815$	15.5997	0.546434
$816$	0	0
$817$	7.73666	0.270672
$818$	0	0
$819$	6.57889	0.229885
$820$	0	0
$821$	−10.1128	−0.352939	−0.176470	−	0.984306i	$-0.556468\pi$
$821$	−10.1128	−0.352939	−0.176470	+	0.984306i	$0.556468\pi$
$822$	0	0
$823$	31.4766	1.09721	0.548603	−	0.836083i	$-0.315160\pi$
$823$	31.4766	1.09721	0.548603	+	0.836083i	$0.315160\pi$
$824$	0	0
$825$	−20.2205	−0.703988
$826$	0	0
$827$	40.4685	1.40723	0.703613	−	0.710583i	$-0.251569\pi$
$827$	40.4685	1.40723	0.703613	+	0.710583i	$0.251569\pi$
$828$	0	0
$829$	−2.88325	−0.100140	−0.0500698	−	0.998746i	$-0.515944\pi$
$829$	−2.88325	−0.100140	−0.0500698	+	0.998746i	$0.515944\pi$
$830$	0	0
$831$	−23.2021	−0.804872
$832$	0	0
$833$	0.454904	0.0157615
$834$	0	0
$835$	−50.0112	−1.73071
$836$	0	0
$837$	−4.88325	−0.168790
$838$	0	0
$839$	−44.5934	−1.53953	−0.769767	−	0.638325i	$-0.779628\pi$
$839$	−44.5934	−1.53953	−0.769767	+	0.638325i	$0.779628\pi$
$840$	0	0
$841$	−17.6787	−0.609611
$842$	0	0
$843$	−16.9774	−0.584732
$844$	0	0
$845$	−97.3309	−3.34828
$846$	0	0
$847$	−3.38732	−0.116390
$848$	0	0
$849$	−24.8679	−0.853464
$850$	0	0
$851$	−4.88325	−0.167396
$852$	0	0
$853$	8.24797	0.282405	0.141202	−	0.989981i	$-0.454903\pi$
$853$	8.24797	0.282405	0.141202	+	0.989981i	$0.454903\pi$
$854$	0	0
$855$	4.96621	0.169841
$856$	0	0
$857$	5.69893	0.194672	0.0973358	−	0.995252i	$-0.468968\pi$
$857$	5.69893	0.194672	0.0973358	+	0.995252i	$0.468968\pi$
$858$	0	0
$859$	−41.7955	−1.42604	−0.713021	−	0.701142i	$-0.752674\pi$
$859$	−41.7955	−1.42604	−0.713021	+	0.701142i	$0.752674\pi$
$860$	0	0
$861$	0.883254	0.0301012
$862$	0	0
$863$	1.46543	0.0498839	0.0249420	−	0.999689i	$-0.492060\pi$
$863$	1.46543	0.0498839	0.0249420	+	0.999689i	$0.492060\pi$
$864$	0	0
$865$	75.1448	2.55500
$866$	0	0
$867$	−16.7931	−0.570322
$868$	0	0
$869$	42.2214	1.43226
$870$	0	0
$871$	12.3421	0.418196
$872$	0	0
$873$	9.61268	0.325340
$874$	0	0
$875$	1.06364	0.0359575
$876$	0	0
$877$	−10.5490	−0.356216	−0.178108	−	0.984011i	$-0.556998\pi$
$877$	−10.5490	−0.356216	−0.178108	+	0.984011i	$0.556998\pi$
$878$	0	0
$879$	−13.5861	−0.458249
$880$	0	0
$881$	−19.2850	−0.649730	−0.324865	−	0.945760i	$-0.605319\pi$
$881$	−19.2850	−0.649730	−0.324865	+	0.945760i	$0.605319\pi$
$882$	0	0
$883$	14.6609	0.493380	0.246690	−	0.969094i	$-0.420657\pi$
$883$	14.6609	0.493380	0.246690	+	0.969094i	$0.420657\pi$
$884$	0	0
$885$	−18.8865	−0.634864
$886$	0	0
$887$	43.4501	1.45891	0.729455	−	0.684029i	$-0.239774\pi$
$887$	43.4501	1.45891	0.729455	+	0.684029i	$0.239774\pi$
$888$	0	0
$889$	−3.24073	−0.108691
$890$	0	0
$891$	−3.79306	−0.127072
$892$	0	0
$893$	6.84223	0.228966
$894$	0	0
$895$	−76.7882	−2.56675
$896$	0	0
$897$	6.57889	0.219663
$898$	0	0
$899$	16.4307	0.547996
$900$	0	0
$901$	0.0564005	0.00187897
$902$	0	0
$903$	5.00724	0.166630
$904$	0	0
$905$	25.6281	0.851905
$906$	0	0
$907$	−4.25126	−0.141161	−0.0705804	−	0.997506i	$-0.522485\pi$
$907$	−4.25126	−0.141161	−0.0705804	+	0.997506i	$0.522485\pi$
$908$	0	0
$909$	−7.03379	−0.233296
$910$	0	0
$911$	−30.1496	−0.998902	−0.499451	−	0.866342i	$-0.666465\pi$
$911$	−30.1496	−0.998902	−0.499451	+	0.866342i	$0.666465\pi$
$912$	0	0
$913$	6.80120	0.225087
$914$	0	0
$915$	26.7770	0.885222
$916$	0	0
$917$	−2.88325	−0.0952134
$918$	0	0
$919$	−10.5225	−0.347105	−0.173552	−	0.984825i	$-0.555525\pi$
$919$	−10.5225	−0.347105	−0.173552	+	0.984825i	$0.555525\pi$
$920$	0	0
$921$	−15.3792	−0.506761
$922$	0	0
$923$	−87.2045	−2.87037
$924$	0	0
$925$	26.0322	0.855935
$926$	0	0
$927$	−2.67632	−0.0879018
$928$	0	0
$929$	30.7472	1.00878	0.504391	−	0.863475i	$-0.331717\pi$
$929$	30.7472	1.00878	0.504391	+	0.863475i	$0.331717\pi$
$930$	0	0
$931$	−1.54510	−0.0506385
$932$	0	0
$933$	18.0974	0.592483
$934$	0	0
$935$	5.54599	0.181373
$936$	0	0
$937$	30.2214	0.987291	0.493645	−	0.869663i	$-0.335664\pi$
$937$	30.2214	0.987291	0.493645	+	0.869663i	$0.335664\pi$
$938$	0	0
$939$	−10.6763	−0.348408
$940$	0	0
$941$	3.40574	0.111024	0.0555120	−	0.998458i	$-0.482321\pi$
$941$	3.40574	0.111024	0.0555120	+	0.998458i	$0.482321\pi$
$942$	0	0
$943$	0.883254	0.0287627
$944$	0	0
$945$	3.21417	0.104557
$946$	0	0
$947$	−29.9017	−0.971674	−0.485837	−	0.874049i	$-0.661485\pi$
$947$	−29.9017	−0.971674	−0.485837	+	0.874049i	$0.661485\pi$
$948$	0	0
$949$	33.7577	1.09582
$950$	0	0
$951$	8.07087	0.261716
$952$	0	0
$953$	−40.9686	−1.32710	−0.663552	−	0.748130i	$-0.730952\pi$
$953$	−40.9686	−1.32710	−0.663552	+	0.748130i	$0.730952\pi$
$954$	0	0
$955$	−32.1883	−1.04159
$956$	0	0
$957$	12.7626	0.412555
$958$	0	0
$959$	1.79306	0.0579010
$960$	0	0
$961$	−7.15383	−0.230769
$962$	0	0
$963$	9.00724	0.290254
$964$	0	0
$965$	−57.3219	−1.84526
$966$	0	0
$967$	14.7584	0.474597	0.237299	−	0.971437i	$-0.423738\pi$
$967$	14.7584	0.474597	0.237299	+	0.971437i	$0.423738\pi$
$968$	0	0
$969$	−0.702870	−0.0225795
$970$	0	0
$971$	−44.2471	−1.41996	−0.709978	−	0.704224i	$-0.751295\pi$
$971$	−44.2471	−1.41996	−0.709978	+	0.704224i	$0.751295\pi$
$972$	0	0
$973$	−9.46214	−0.303342
$974$	0	0
$975$	−35.0715	−1.12319
$976$	0	0
$977$	27.9991	0.895771	0.447885	−	0.894091i	$-0.352177\pi$
$977$	27.9991	0.895771	0.447885	+	0.894091i	$0.352177\pi$
$978$	0	0
$979$	−0.570997	−0.0182492
$980$	0	0
$981$	−2.30437	−0.0735728
$982$	0	0
$983$	−13.1457	−0.419283	−0.209641	−	0.977778i	$-0.567230\pi$
$983$	−13.1457	−0.419283	−0.209641	+	0.977778i	$0.567230\pi$
$984$	0	0
$985$	25.4468	0.810801
$986$	0	0
$987$	4.42835	0.140956
$988$	0	0
$989$	5.00724	0.159221
$990$	0	0
$991$	−22.2778	−0.707678	−0.353839	−	0.935306i	$-0.615124\pi$
$991$	−22.2778	−0.707678	−0.353839	+	0.935306i	$0.615124\pi$
$992$	0	0
$993$	26.7584	0.849151
$994$	0	0
$995$	−6.82685	−0.216426
$996$	0	0
$997$	14.4404	0.457333	0.228666	−	0.973505i	$-0.426563\pi$
$997$	14.4404	0.457333	0.228666	+	0.973505i	$0.426563\pi$
$998$	0	0
$999$	4.88325	0.154499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bu.1.2		3
4.3	odd	2		3864.2.a.k.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.k.1.2	✓	3	4.3	odd	2
7728.2.a.bu.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.21417 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.21417 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.79306 q^{11} -6.57889 q^{13} -3.21417 q^{15} +0.454904 q^{17} -1.54510 q^{19} -1.00000 q^{21} -1.00000 q^{23} +5.33092 q^{25} +1.00000 q^{27} -3.36471 q^{29} -4.88325 q^{31} -3.79306 q^{33} +3.21417 q^{35} +4.88325 q^{37} -6.57889 q^{39} -0.883254 q^{41} -5.00724 q^{43} -3.21417 q^{45} -4.42835 q^{47} +1.00000 q^{49} +0.454904 q^{51} +0.123983 q^{53} +12.1916 q^{55} -1.54510 q^{57} +5.87602 q^{59} -8.33092 q^{61} -1.00000 q^{63} +21.1457 q^{65} -1.87602 q^{67} -1.00000 q^{69} +13.2552 q^{71} -5.13122 q^{73} +5.33092 q^{75} +3.79306 q^{77} -11.1312 q^{79} +1.00000 q^{81} -1.79306 q^{83} -1.46214 q^{85} -3.36471 q^{87} +0.150537 q^{89} +6.57889 q^{91} -4.88325 q^{93} +4.96621 q^{95} +9.61268 q^{97} -3.79306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 6 q^{11} - 9 q^{13} - 3 q^{15} - 6 q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} + 3 q^{27} - 6 q^{29} + 6 q^{33} + 3 q^{35} - 9 q^{39} + 12 q^{41} + 9 q^{43} - 3 q^{45} + 3 q^{49} - 9 q^{53} + 3 q^{55} - 6 q^{57} + 27 q^{59} - 33 q^{61} - 3 q^{63} - 18 q^{65} - 15 q^{67} - 3 q^{69} - 3 q^{71} + 18 q^{73} + 24 q^{75} - 6 q^{77} + 3 q^{81} + 12 q^{83} + 21 q^{85} - 6 q^{87} + 3 q^{89} + 9 q^{91} + 27 q^{95} + 6 q^{97} + 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 6 * q^11 - 9 * q^13 - 3 * q^15 - 6 * q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 + 3 * q^27 - 6 * q^29 + 6 * q^33 + 3 * q^35 - 9 * q^39 + 12 * q^41 + 9 * q^43 - 3 * q^45 + 3 * q^49 - 9 * q^53 + 3 * q^55 - 6 * q^57 + 27 * q^59 - 33 * q^61 - 3 * q^63 - 18 * q^65 - 15 * q^67 - 3 * q^69 - 3 * q^71 + 18 * q^73 + 24 * q^75 - 6 * q^77 + 3 * q^81 + 12 * q^83 + 21 * q^85 - 6 * q^87 + 3 * q^89 + 9 * q^91 + 27 * q^95 + 6 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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