LMFDB - Embedded newform 7728.2.a.bq.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
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			$1.17819$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.61186 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.61186 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.43366 q^{11} +0.821806 q^{13} -2.61186 q^{15} -5.79005 q^{17} +7.43366 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.82181 q^{25} -1.00000 q^{27} -5.79005 q^{29} +3.07728 q^{31} -3.43366 q^{33} +2.61186 q^{35} -0.922724 q^{37} -0.821806 q^{39} +0.566335 q^{41} +11.6891 q^{43} +2.61186 q^{45} +2.35639 q^{47} +1.00000 q^{49} +5.79005 q^{51} +0.968247 q^{53} +8.96825 q^{55} -7.43366 q^{57} -1.89908 q^{59} +2.40191 q^{61} +1.00000 q^{63} +2.14644 q^{65} +4.10092 q^{67} +1.00000 q^{69} -1.17819 q^{71} +8.14644 q^{73} -1.82181 q^{75} +3.43366 q^{77} +2.56634 q^{79} +1.00000 q^{81} +6.30099 q^{83} -15.1228 q^{85} +5.79005 q^{87} -7.89097 q^{89} +0.821806 q^{91} -3.07728 q^{93} +19.4157 q^{95} -1.79005 q^{97} +3.43366 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{11} + 5 q^{13} + 3 q^{15} - 4 q^{17} + 14 q^{19} - 3 q^{21} - 3 q^{23} + 8 q^{25} - 3 q^{27} - 4 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 6 q^{37} - 5 q^{39} + 10 q^{41} + 21 q^{43} - 3 q^{45} + 2 q^{47} + 3 q^{49} + 4 q^{51} - 13 q^{53} + 11 q^{55} - 14 q^{57} - 5 q^{59} - 17 q^{61} + 3 q^{63} - 12 q^{65} + 13 q^{67} + 3 q^{69} - q^{71} + 6 q^{73} - 8 q^{75} + 2 q^{77} + 16 q^{79} + 3 q^{81} - 6 q^{83} - 23 q^{85} + 4 q^{87} - 11 q^{89} + 5 q^{91} - 6 q^{93} - q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 5 * q^13 + 3 * q^15 - 4 * q^17 + 14 * q^19 - 3 * q^21 - 3 * q^23 + 8 * q^25 - 3 * q^27 - 4 * q^29 + 6 * q^31 - 2 * q^33 - 3 * q^35 - 6 * q^37 - 5 * q^39 + 10 * q^41 + 21 * q^43 - 3 * q^45 + 2 * q^47 + 3 * q^49 + 4 * q^51 - 13 * q^53 + 11 * q^55 - 14 * q^57 - 5 * q^59 - 17 * q^61 + 3 * q^63 - 12 * q^65 + 13 * q^67 + 3 * q^69 - q^71 + 6 * q^73 - 8 * q^75 + 2 * q^77 + 16 * q^79 + 3 * q^81 - 6 * q^83 - 23 * q^85 + 4 * q^87 - 11 * q^89 + 5 * q^91 - 6 * q^93 - q^95 + 8 * q^97 + 2 * 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
$5$	2.61186	1.16806	0.584029	−	0.811733i	$-0.301475\pi$
$5$	2.61186	1.16806	0.584029	+	0.811733i	$0.301475\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.43366	1.03529	0.517644	−	0.855596i	$-0.326809\pi$
$11$	3.43366	1.03529	0.517644	+	0.855596i	$0.326809\pi$
$12$	0	0
$13$	0.821806	0.227928	0.113964	−	0.993485i	$-0.463645\pi$
$13$	0.821806	0.227928	0.113964	+	0.993485i	$0.463645\pi$
$14$	0	0
$15$	−2.61186	−0.674379
$16$	0	0
$17$	−5.79005	−1.40429	−0.702147	−	0.712032i	$-0.747775\pi$
$17$	−5.79005	−1.40429	−0.702147	+	0.712032i	$0.747775\pi$
$18$	0	0
$19$	7.43366	1.70540	0.852700	−	0.522401i	$-0.174964\pi$
$19$	7.43366	1.70540	0.852700	+	0.522401i	$0.174964\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.82181	0.364361
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.79005	−1.07519	−0.537593	−	0.843205i	$-0.680666\pi$
$29$	−5.79005	−1.07519	−0.537593	+	0.843205i	$0.680666\pi$
$30$	0	0
$31$	3.07728	0.552695	0.276348	−	0.961058i	$-0.410876\pi$
$31$	3.07728	0.552695	0.276348	+	0.961058i	$0.410876\pi$
$32$	0	0
$33$	−3.43366	−0.597724
$34$	0	0
$35$	2.61186	0.441485
$36$	0	0
$37$	−0.922724	−0.151695	−0.0758474	−	0.997119i	$-0.524166\pi$
$37$	−0.922724	−0.151695	−0.0758474	+	0.997119i	$0.524166\pi$
$38$	0	0
$39$	−0.821806	−0.131594
$40$	0	0
$41$	0.566335	0.0884467	0.0442234	−	0.999022i	$-0.485919\pi$
$41$	0.566335	0.0884467	0.0442234	+	0.999022i	$0.485919\pi$
$42$	0	0
$43$	11.6891	1.78258	0.891288	−	0.453437i	$-0.149802\pi$
$43$	11.6891	1.78258	0.891288	+	0.453437i	$0.149802\pi$
$44$	0	0
$45$	2.61186	0.389353
$46$	0	0
$47$	2.35639	0.343715	0.171857	−	0.985122i	$-0.445023\pi$
$47$	2.35639	0.343715	0.171857	+	0.985122i	$0.445023\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.79005	0.810770
$52$	0	0
$53$	0.968247	0.132999	0.0664995	−	0.997786i	$-0.478817\pi$
$53$	0.968247	0.132999	0.0664995	+	0.997786i	$0.478817\pi$
$54$	0	0
$55$	8.96825	1.20928
$56$	0	0
$57$	−7.43366	−0.984613
$58$	0	0
$59$	−1.89908	−0.247239	−0.123620	−	0.992330i	$-0.539450\pi$
$59$	−1.89908	−0.247239	−0.123620	+	0.992330i	$0.539450\pi$
$60$	0	0
$61$	2.40191	0.307533	0.153767	−	0.988107i	$-0.450860\pi$
$61$	2.40191	0.307533	0.153767	+	0.988107i	$0.450860\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	2.14644	0.266233
$66$	0	0
$67$	4.10092	0.501007	0.250503	−	0.968116i	$-0.419404\pi$
$67$	4.10092	0.501007	0.250503	+	0.968116i	$0.419404\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.17819	−0.139826	−0.0699130	−	0.997553i	$-0.522272\pi$
$71$	−1.17819	−0.139826	−0.0699130	+	0.997553i	$0.522272\pi$
$72$	0	0
$73$	8.14644	0.953469	0.476734	−	0.879047i	$-0.341820\pi$
$73$	8.14644	0.953469	0.476734	+	0.879047i	$0.341820\pi$
$74$	0	0
$75$	−1.82181	−0.210364
$76$	0	0
$77$	3.43366	0.391302
$78$	0	0
$79$	2.56634	0.288735	0.144368	−	0.989524i	$-0.453885\pi$
$79$	2.56634	0.288735	0.144368	+	0.989524i	$0.453885\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.30099	0.691624	0.345812	−	0.938304i	$-0.387604\pi$
$83$	6.30099	0.691624	0.345812	+	0.938304i	$0.387604\pi$
$84$	0	0
$85$	−15.1228	−1.64030
$86$	0	0
$87$	5.79005	0.620759
$88$	0	0
$89$	−7.89097	−0.836441	−0.418221	−	0.908345i	$-0.637346\pi$
$89$	−7.89097	−0.836441	−0.418221	+	0.908345i	$0.637346\pi$
$90$	0	0
$91$	0.821806	0.0861487
$92$	0	0
$93$	−3.07728	−0.319099
$94$	0	0
$95$	19.4157	1.99201
$96$	0	0
$97$	−1.79005	−0.181752	−0.0908762	−	0.995862i	$-0.528967\pi$
$97$	−1.79005	−0.181752	−0.0908762	+	0.995862i	$0.528967\pi$
$98$	0	0
$99$	3.43366	0.345096
$100$	0	0
$101$	9.12280	0.907753	0.453876	−	0.891065i	$-0.350041\pi$
$101$	9.12280	0.907753	0.453876	+	0.891065i	$0.350041\pi$
$102$	0	0
$103$	−19.1602	−1.88791	−0.943956	−	0.330072i	$-0.892927\pi$
$103$	−19.1602	−1.88791	−0.943956	+	0.330072i	$0.892927\pi$
$104$	0	0
$105$	−2.61186	−0.254891
$106$	0	0
$107$	14.5565	1.40723	0.703613	−	0.710583i	$-0.251569\pi$
$107$	14.5565	1.40723	0.703613	+	0.710583i	$0.251569\pi$
$108$	0	0
$109$	17.0593	1.63398	0.816992	−	0.576649i	$-0.195640\pi$
$109$	17.0593	1.63398	0.816992	+	0.576649i	$0.195640\pi$
$110$	0	0
$111$	0.922724	0.0875810
$112$	0	0
$113$	7.47919	0.703583	0.351791	−	0.936078i	$-0.385573\pi$
$113$	7.47919	0.703583	0.351791	+	0.936078i	$0.385573\pi$
$114$	0	0
$115$	−2.61186	−0.243557
$116$	0	0
$117$	0.821806	0.0759760
$118$	0	0
$119$	−5.79005	−0.530773
$120$	0	0
$121$	0.790053	0.0718230
$122$	0	0
$123$	−0.566335	−0.0510647
$124$	0	0
$125$	−8.30099	−0.742463
$126$	0	0
$127$	−14.4930	−1.28604	−0.643021	−	0.765849i	$-0.722319\pi$
$127$	−14.4930	−1.28604	−0.643021	+	0.765849i	$0.722319\pi$
$128$	0	0
$129$	−11.6891	−1.02917
$130$	0	0
$131$	−17.7265	−1.54878	−0.774388	−	0.632711i	$-0.781942\pi$
$131$	−17.7265	−1.54878	−0.774388	+	0.632711i	$0.781942\pi$
$132$	0	0
$133$	7.43366	0.644580
$134$	0	0
$135$	−2.61186	−0.224793
$136$	0	0
$137$	10.3010	0.880073	0.440037	−	0.897980i	$-0.354965\pi$
$137$	10.3010	0.880073	0.440037	+	0.897980i	$0.354965\pi$
$138$	0	0
$139$	19.8356	1.68243	0.841216	−	0.540700i	$-0.181840\pi$
$139$	19.8356	1.68243	0.841216	+	0.540700i	$0.181840\pi$
$140$	0	0
$141$	−2.35639	−0.198444
$142$	0	0
$143$	2.82181	0.235971
$144$	0	0
$145$	−15.1228	−1.25588
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	0.0910467	0.00745883	0.00372942	−	0.999993i	$-0.498813\pi$
$149$	0.0910467	0.00745883	0.00372942	+	0.999993i	$0.498813\pi$
$150$	0	0
$151$	−11.3783	−0.925951	−0.462975	−	0.886371i	$-0.653218\pi$
$151$	−11.3783	−0.925951	−0.462975	+	0.886371i	$0.653218\pi$
$152$	0	0
$153$	−5.79005	−0.468098
$154$	0	0
$155$	8.03741	0.645580
$156$	0	0
$157$	5.22372	0.416898	0.208449	−	0.978033i	$-0.433158\pi$
$157$	5.22372	0.416898	0.208449	+	0.978033i	$0.433158\pi$
$158$	0	0
$159$	−0.968247	−0.0767870
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	6.10092	0.477861	0.238930	−	0.971037i	$-0.423203\pi$
$163$	6.10092	0.477861	0.238930	+	0.971037i	$0.423203\pi$
$164$	0	0
$165$	−8.96825	−0.698177
$166$	0	0
$167$	8.30099	0.642350	0.321175	−	0.947020i	$-0.395922\pi$
$167$	8.30099	0.642350	0.321175	+	0.947020i	$0.395922\pi$
$168$	0	0
$169$	−12.3246	−0.948049
$170$	0	0
$171$	7.43366	0.568467
$172$	0	0
$173$	8.14644	0.619362	0.309681	−	0.950840i	$-0.399778\pi$
$173$	8.14644	0.619362	0.309681	+	0.950840i	$0.399778\pi$
$174$	0	0
$175$	1.82181	0.137716
$176$	0	0
$177$	1.89908	0.142744
$178$	0	0
$179$	−8.96825	−0.670318	−0.335159	−	0.942162i	$-0.608790\pi$
$179$	−8.96825	−0.670318	−0.335159	+	0.942162i	$0.608790\pi$
$180$	0	0
$181$	−23.6793	−1.76007	−0.880033	−	0.474913i	$-0.842480\pi$
$181$	−23.6793	−1.76007	−0.880033	+	0.474913i	$0.842480\pi$
$182$	0	0
$183$	−2.40191	−0.177554
$184$	0	0
$185$	−2.41002	−0.177188
$186$	0	0
$187$	−19.8811	−1.45385
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−4.09105	−0.296018	−0.148009	−	0.988986i	$-0.547286\pi$
$191$	−4.09105	−0.296018	−0.148009	+	0.988986i	$0.547286\pi$
$192$	0	0
$193$	12.3564	0.889432	0.444716	−	0.895672i	$-0.353305\pi$
$193$	12.3564	0.889432	0.444716	+	0.895672i	$0.353305\pi$
$194$	0	0
$195$	−2.14644	−0.153710
$196$	0	0
$197$	2.24736	0.160118	0.0800588	−	0.996790i	$-0.474489\pi$
$197$	2.24736	0.160118	0.0800588	+	0.996790i	$0.474489\pi$
$198$	0	0
$199$	21.6810	1.53693	0.768463	−	0.639894i	$-0.221022\pi$
$199$	21.6810	1.53693	0.768463	+	0.639894i	$0.221022\pi$
$200$	0	0
$201$	−4.10092	−0.289256
$202$	0	0
$203$	−5.79005	−0.406382
$204$	0	0
$205$	1.47919	0.103311
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	25.5247	1.76558
$210$	0	0
$211$	−26.0357	−1.79237	−0.896184	−	0.443682i	$-0.853672\pi$
$211$	−26.0357	−1.79237	−0.896184	+	0.443682i	$0.853672\pi$
$212$	0	0
$213$	1.17819	0.0807285
$214$	0	0
$215$	30.5304	2.08215
$216$	0	0
$217$	3.07728	0.208899
$218$	0	0
$219$	−8.14644	−0.550485
$220$	0	0
$221$	−4.75830	−0.320078
$222$	0	0
$223$	27.8356	1.86401	0.932004	−	0.362448i	$-0.118059\pi$
$223$	27.8356	1.86401	0.932004	+	0.362448i	$0.118059\pi$
$224$	0	0
$225$	1.82181	0.121454
$226$	0	0
$227$	−1.32464	−0.0879191	−0.0439596	−	0.999033i	$-0.513997\pi$
$227$	−1.32464	−0.0879191	−0.0439596	+	0.999033i	$0.513997\pi$
$228$	0	0
$229$	27.2138	1.79834	0.899171	−	0.437598i	$-0.144171\pi$
$229$	27.2138	1.79834	0.899171	+	0.437598i	$0.144171\pi$
$230$	0	0
$231$	−3.43366	−0.225919
$232$	0	0
$233$	−20.1920	−1.32282	−0.661410	−	0.750025i	$-0.730042\pi$
$233$	−20.1920	−1.32282	−0.661410	+	0.750025i	$0.730042\pi$
$234$	0	0
$235$	6.15455	0.401479
$236$	0	0
$237$	−2.56634	−0.166701
$238$	0	0
$239$	−21.7721	−1.40832	−0.704159	−	0.710042i	$-0.748676\pi$
$239$	−21.7721	−1.40832	−0.704159	+	0.710042i	$0.748676\pi$
$240$	0	0
$241$	−12.6574	−0.815334	−0.407667	−	0.913131i	$-0.633658\pi$
$241$	−12.6574	−0.815334	−0.407667	+	0.913131i	$0.633658\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.61186	0.166866
$246$	0	0
$247$	6.10903	0.388708
$248$	0	0
$249$	−6.30099	−0.399309
$250$	0	0
$251$	−8.66549	−0.546961	−0.273481	−	0.961878i	$-0.588175\pi$
$251$	−8.66549	−0.546961	−0.273481	+	0.961878i	$0.588175\pi$
$252$	0	0
$253$	−3.43366	−0.215873
$254$	0	0
$255$	15.1228	0.947026
$256$	0	0
$257$	20.6493	1.28807	0.644033	−	0.764998i	$-0.277260\pi$
$257$	20.6493	1.28807	0.644033	+	0.764998i	$0.277260\pi$
$258$	0	0
$259$	−0.922724	−0.0573353
$260$	0	0
$261$	−5.79005	−0.358395
$262$	0	0
$263$	21.1048	1.30138	0.650689	−	0.759344i	$-0.274480\pi$
$263$	21.1048	1.30138	0.650689	+	0.759344i	$0.274480\pi$
$264$	0	0
$265$	2.52892	0.155351
$266$	0	0
$267$	7.89097	0.482920
$268$	0	0
$269$	−18.8493	−1.14926	−0.574632	−	0.818412i	$-0.694855\pi$
$269$	−18.8493	−1.14926	−0.574632	+	0.818412i	$0.694855\pi$
$270$	0	0
$271$	−5.01377	−0.304565	−0.152282	−	0.988337i	$-0.548662\pi$
$271$	−5.01377	−0.304565	−0.152282	+	0.988337i	$0.548662\pi$
$272$	0	0
$273$	−0.821806	−0.0497380
$274$	0	0
$275$	6.25547	0.377219
$276$	0	0
$277$	−7.17819	−0.431296	−0.215648	−	0.976471i	$-0.569186\pi$
$277$	−7.17819	−0.431296	−0.215648	+	0.976471i	$0.569186\pi$
$278$	0	0
$279$	3.07728	0.184232
$280$	0	0
$281$	1.19618	0.0713579	0.0356790	−	0.999363i	$-0.488641\pi$
$281$	1.19618	0.0713579	0.0356790	+	0.999363i	$0.488641\pi$
$282$	0	0
$283$	6.96825	0.414219	0.207110	−	0.978318i	$-0.433594\pi$
$283$	6.96825	0.414219	0.207110	+	0.978318i	$0.433594\pi$
$284$	0	0
$285$	−19.4157	−1.15009
$286$	0	0
$287$	0.566335	0.0334297
$288$	0	0
$289$	16.5247	0.972042
$290$	0	0
$291$	1.79005	0.104935
$292$	0	0
$293$	22.4474	1.31139	0.655697	−	0.755025i	$-0.272375\pi$
$293$	22.4474	1.31139	0.655697	+	0.755025i	$0.272375\pi$
$294$	0	0
$295$	−4.96013	−0.288790
$296$	0	0
$297$	−3.43366	−0.199241
$298$	0	0
$299$	−0.821806	−0.0475263
$300$	0	0
$301$	11.6891	0.673751
$302$	0	0
$303$	−9.12280	−0.524091
$304$	0	0
$305$	6.27345	0.359217
$306$	0	0
$307$	22.6574	1.29313	0.646563	−	0.762861i	$-0.276206\pi$
$307$	22.6574	1.29313	0.646563	+	0.762861i	$0.276206\pi$
$308$	0	0
$309$	19.1602	1.08999
$310$	0	0
$311$	−23.4238	−1.32824	−0.664121	−	0.747625i	$-0.731194\pi$
$311$	−23.4238	−1.32824	−0.664121	+	0.747625i	$0.731194\pi$
$312$	0	0
$313$	−17.7622	−1.00398	−0.501989	−	0.864874i	$-0.667398\pi$
$313$	−17.7622	−1.00398	−0.501989	+	0.864874i	$0.667398\pi$
$314$	0	0
$315$	2.61186	0.147162
$316$	0	0
$317$	8.19196	0.460107	0.230053	−	0.973178i	$-0.426110\pi$
$317$	8.19196	0.460107	0.230053	+	0.973178i	$0.426110\pi$
$318$	0	0
$319$	−19.8811	−1.11313
$320$	0	0
$321$	−14.5565	−0.812463
$322$	0	0
$323$	−43.0413	−2.39488
$324$	0	0
$325$	1.49717	0.0830481
$326$	0	0
$327$	−17.0593	−0.943381
$328$	0	0
$329$	2.35639	0.129912
$330$	0	0
$331$	−1.13267	−0.0622572	−0.0311286	−	0.999515i	$-0.509910\pi$
$331$	−1.13267	−0.0622572	−0.0311286	+	0.999515i	$0.509910\pi$
$332$	0	0
$333$	−0.922724	−0.0505649
$334$	0	0
$335$	10.7110	0.585205
$336$	0	0
$337$	−26.6475	−1.45158	−0.725791	−	0.687915i	$-0.758526\pi$
$337$	−26.6475	−1.45158	−0.725791	+	0.687915i	$0.758526\pi$
$338$	0	0
$339$	−7.47919	−0.406214
$340$	0	0
$341$	10.5663	0.572199
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.61186	0.140618
$346$	0	0
$347$	24.9030	1.33686	0.668431	−	0.743774i	$-0.266966\pi$
$347$	24.9030	1.33686	0.668431	+	0.743774i	$0.266966\pi$
$348$	0	0
$349$	−21.4157	−1.14636	−0.573178	−	0.819431i	$-0.694289\pi$
$349$	−21.4157	−1.14636	−0.573178	+	0.819431i	$0.694289\pi$
$350$	0	0
$351$	−0.821806	−0.0438648
$352$	0	0
$353$	17.2791	0.919674	0.459837	−	0.888003i	$-0.347908\pi$
$353$	17.2791	0.919674	0.459837	+	0.888003i	$0.347908\pi$
$354$	0	0
$355$	−3.07728	−0.163325
$356$	0	0
$357$	5.79005	0.306442
$358$	0	0
$359$	25.8631	1.36500	0.682502	−	0.730884i	$-0.260892\pi$
$359$	25.8631	1.36500	0.682502	+	0.730884i	$0.260892\pi$
$360$	0	0
$361$	36.2594	1.90839
$362$	0	0
$363$	−0.790053	−0.0414670
$364$	0	0
$365$	21.2774	1.11371
$366$	0	0
$367$	30.4930	1.59172	0.795860	−	0.605481i	$-0.207019\pi$
$367$	30.4930	1.59172	0.795860	+	0.605481i	$0.207019\pi$
$368$	0	0
$369$	0.566335	0.0294822
$370$	0	0
$371$	0.968247	0.0502689
$372$	0	0
$373$	−13.6355	−0.706019	−0.353010	−	0.935620i	$-0.614842\pi$
$373$	−13.6355	−0.706019	−0.353010	+	0.935620i	$0.614842\pi$
$374$	0	0
$375$	8.30099	0.428661
$376$	0	0
$377$	−4.75830	−0.245065
$378$	0	0
$379$	2.29288	0.117777	0.0588887	−	0.998265i	$-0.481244\pi$
$379$	2.29288	0.117777	0.0588887	+	0.998265i	$0.481244\pi$
$380$	0	0
$381$	14.4930	0.742497
$382$	0	0
$383$	15.2791	0.780726	0.390363	−	0.920661i	$-0.372349\pi$
$383$	15.2791	0.780726	0.390363	+	0.920661i	$0.372349\pi$
$384$	0	0
$385$	8.96825	0.457064
$386$	0	0
$387$	11.6891	0.594192
$388$	0	0
$389$	−13.2318	−0.670880	−0.335440	−	0.942062i	$-0.608885\pi$
$389$	−13.2318	−0.670880	−0.335440	+	0.942062i	$0.608885\pi$
$390$	0	0
$391$	5.79005	0.292816
$392$	0	0
$393$	17.7265	0.894186
$394$	0	0
$395$	6.70291	0.337260
$396$	0	0
$397$	−32.1186	−1.61199	−0.805993	−	0.591925i	$-0.798368\pi$
$397$	−32.1186	−1.61199	−0.805993	+	0.591925i	$0.798368\pi$
$398$	0	0
$399$	−7.43366	−0.372149
$400$	0	0
$401$	−6.20995	−0.310110	−0.155055	−	0.987906i	$-0.549555\pi$
$401$	−6.20995	−0.310110	−0.155055	+	0.987906i	$0.549555\pi$
$402$	0	0
$403$	2.52892	0.125975
$404$	0	0
$405$	2.61186	0.129784
$406$	0	0
$407$	−3.16832	−0.157048
$408$	0	0
$409$	−25.2594	−1.24900	−0.624498	−	0.781027i	$-0.714696\pi$
$409$	−25.2594	−1.24900	−0.624498	+	0.781027i	$0.714696\pi$
$410$	0	0
$411$	−10.3010	−0.508111
$412$	0	0
$413$	−1.89908	−0.0934477
$414$	0	0
$415$	16.4573	0.807857
$416$	0	0
$417$	−19.8356	−0.971352
$418$	0	0
$419$	−24.2830	−1.18630	−0.593151	−	0.805091i	$-0.702116\pi$
$419$	−24.2830	−1.18630	−0.593151	+	0.805091i	$0.702116\pi$
$420$	0	0
$421$	0.465418	0.0226831	0.0113415	−	0.999936i	$-0.496390\pi$
$421$	0.465418	0.0226831	0.0113415	+	0.999936i	$0.496390\pi$
$422$	0	0
$423$	2.35639	0.114572
$424$	0	0
$425$	−10.5484	−0.511670
$426$	0	0
$427$	2.40191	0.116237
$428$	0	0
$429$	−2.82181	−0.136238
$430$	0	0
$431$	5.78018	0.278422	0.139211	−	0.990263i	$-0.455543\pi$
$431$	5.78018	0.278422	0.139211	+	0.990263i	$0.455543\pi$
$432$	0	0
$433$	9.19618	0.441940	0.220970	−	0.975281i	$-0.429078\pi$
$433$	9.19618	0.441940	0.220970	+	0.975281i	$0.429078\pi$
$434$	0	0
$435$	15.1228	0.725083
$436$	0	0
$437$	−7.43366	−0.355600
$438$	0	0
$439$	15.7901	0.753618	0.376809	−	0.926291i	$-0.377021\pi$
$439$	15.7901	0.753618	0.376809	+	0.926291i	$0.377021\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−32.1821	−1.52902	−0.764509	−	0.644613i	$-0.777018\pi$
$443$	−32.1821	−1.52902	−0.764509	+	0.644613i	$0.777018\pi$
$444$	0	0
$445$	−20.6101	−0.977012
$446$	0	0
$447$	−0.0910467	−0.00430636
$448$	0	0
$449$	−5.41568	−0.255582	−0.127791	−	0.991801i	$-0.540789\pi$
$449$	−5.41568	−0.255582	−0.127791	+	0.991801i	$0.540789\pi$
$450$	0	0
$451$	1.94461	0.0915679
$452$	0	0
$453$	11.3783	0.534598
$454$	0	0
$455$	2.14644	0.100627
$456$	0	0
$457$	26.8412	1.25558	0.627790	−	0.778383i	$-0.283960\pi$
$457$	26.8412	1.25558	0.627790	+	0.778383i	$0.283960\pi$
$458$	0	0
$459$	5.79005	0.270257
$460$	0	0
$461$	0.410023	0.0190967	0.00954835	−	0.999954i	$-0.496961\pi$
$461$	0.410023	0.0190967	0.00954835	+	0.999954i	$0.496961\pi$
$462$	0	0
$463$	36.6849	1.70489	0.852446	−	0.522815i	$-0.175118\pi$
$463$	36.6849	1.70489	0.852446	+	0.522815i	$0.175118\pi$
$464$	0	0
$465$	−8.03741	−0.372726
$466$	0	0
$467$	−11.7265	−0.542640	−0.271320	−	0.962489i	$-0.587460\pi$
$467$	−11.7265	−0.542640	−0.271320	+	0.962489i	$0.587460\pi$
$468$	0	0
$469$	4.10092	0.189363
$470$	0	0
$471$	−5.22372	−0.240696
$472$	0	0
$473$	40.1366	1.84548
$474$	0	0
$475$	13.5427	0.621381
$476$	0	0
$477$	0.968247	0.0443330
$478$	0	0
$479$	20.0829	0.917613	0.458806	−	0.888536i	$-0.348277\pi$
$479$	20.0829	0.917613	0.458806	+	0.888536i	$0.348277\pi$
$480$	0	0
$481$	−0.758300	−0.0345755
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−4.67536	−0.212297
$486$	0	0
$487$	4.51905	0.204778	0.102389	−	0.994744i	$-0.467351\pi$
$487$	4.51905	0.204778	0.102389	+	0.994744i	$0.467351\pi$
$488$	0	0
$489$	−6.10092	−0.275893
$490$	0	0
$491$	8.26358	0.372930	0.186465	−	0.982462i	$-0.440297\pi$
$491$	8.26358	0.372930	0.186465	+	0.982462i	$0.440297\pi$
$492$	0	0
$493$	33.5247	1.50988
$494$	0	0
$495$	8.96825	0.403093
$496$	0	0
$497$	−1.17819	−0.0528492
$498$	0	0
$499$	32.9682	1.47586	0.737931	−	0.674876i	$-0.235803\pi$
$499$	32.9682	1.47586	0.737931	+	0.674876i	$0.235803\pi$
$500$	0	0
$501$	−8.30099	−0.370861
$502$	0	0
$503$	18.7029	0.833921	0.416961	−	0.908925i	$-0.363095\pi$
$503$	18.7029	0.833921	0.416961	+	0.908925i	$0.363095\pi$
$504$	0	0
$505$	23.8275	1.06031
$506$	0	0
$507$	12.3246	0.547356
$508$	0	0
$509$	−18.5109	−0.820483	−0.410242	−	0.911977i	$-0.634556\pi$
$509$	−18.5109	−0.820483	−0.410242	+	0.911977i	$0.634556\pi$
$510$	0	0
$511$	8.14644	0.360377
$512$	0	0
$513$	−7.43366	−0.328204
$514$	0	0
$515$	−50.0438	−2.20519
$516$	0	0
$517$	8.09105	0.355844
$518$	0	0
$519$	−8.14644	−0.357589
$520$	0	0
$521$	35.0219	1.53434	0.767168	−	0.641446i	$-0.221665\pi$
$521$	35.0219	1.53434	0.767168	+	0.641446i	$0.221665\pi$
$522$	0	0
$523$	40.3204	1.76309	0.881544	−	0.472101i	$-0.156504\pi$
$523$	40.3204	1.76309	0.881544	+	0.472101i	$0.156504\pi$
$524$	0	0
$525$	−1.82181	−0.0795101
$526$	0	0
$527$	−17.8176	−0.776147
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.89908	−0.0824132
$532$	0	0
$533$	0.465418	0.0201595
$534$	0	0
$535$	38.0194	1.64372
$536$	0	0
$537$	8.96825	0.387008
$538$	0	0
$539$	3.43366	0.147898
$540$	0	0
$541$	−28.2456	−1.21437	−0.607187	−	0.794559i	$-0.707702\pi$
$541$	−28.2456	−1.21437	−0.607187	+	0.794559i	$0.707702\pi$
$542$	0	0
$543$	23.6793	1.01617
$544$	0	0
$545$	44.5565	1.90859
$546$	0	0
$547$	25.6810	1.09804	0.549021	−	0.835809i	$-0.315001\pi$
$547$	25.6810	1.09804	0.549021	+	0.835809i	$0.315001\pi$
$548$	0	0
$549$	2.40191	0.102511
$550$	0	0
$551$	−43.0413	−1.83362
$552$	0	0
$553$	2.56634	0.109132
$554$	0	0
$555$	2.41002	0.102300
$556$	0	0
$557$	0.201835	0.00855203	0.00427602	−	0.999991i	$-0.498639\pi$
$557$	0.201835	0.00855203	0.00427602	+	0.999991i	$0.498639\pi$
$558$	0	0
$559$	9.60620	0.406299
$560$	0	0
$561$	19.8811	0.839381
$562$	0	0
$563$	−47.4513	−1.99984	−0.999918	−	0.0128333i	$-0.995915\pi$
$563$	−47.4513	−1.99984	−0.999918	+	0.0128333i	$0.995915\pi$
$564$	0	0
$565$	19.5346	0.821826
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−14.4199	−0.604513	−0.302257	−	0.953227i	$-0.597740\pi$
$569$	−14.4199	−0.604513	−0.302257	+	0.953227i	$0.597740\pi$
$570$	0	0
$571$	28.4393	1.19015	0.595074	−	0.803671i	$-0.297123\pi$
$571$	28.4393	1.19015	0.595074	+	0.803671i	$0.297123\pi$
$572$	0	0
$573$	4.09105	0.170906
$574$	0	0
$575$	−1.82181	−0.0759746
$576$	0	0
$577$	28.7565	1.19715	0.598575	−	0.801067i	$-0.295734\pi$
$577$	28.7565	1.19715	0.598575	+	0.801067i	$0.295734\pi$
$578$	0	0
$579$	−12.3564	−0.513514
$580$	0	0
$581$	6.30099	0.261409
$582$	0	0
$583$	3.32464	0.137692
$584$	0	0
$585$	2.14644	0.0887444
$586$	0	0
$587$	−21.5783	−0.890634	−0.445317	−	0.895373i	$-0.646909\pi$
$587$	−21.5783	−0.890634	−0.445317	+	0.895373i	$0.646909\pi$
$588$	0	0
$589$	22.8754	0.942566
$590$	0	0
$591$	−2.24736	−0.0924440
$592$	0	0
$593$	11.6436	0.478146	0.239073	−	0.971002i	$-0.423157\pi$
$593$	11.6436	0.478146	0.239073	+	0.971002i	$0.423157\pi$
$594$	0	0
$595$	−15.1228	−0.619974
$596$	0	0
$597$	−21.6810	−0.887345
$598$	0	0
$599$	−12.0455	−0.492167	−0.246083	−	0.969249i	$-0.579144\pi$
$599$	−12.0455	−0.492167	−0.246083	+	0.969249i	$0.579144\pi$
$600$	0	0
$601$	24.8137	1.01217	0.506086	−	0.862483i	$-0.331092\pi$
$601$	24.8137	1.01217	0.506086	+	0.862483i	$0.331092\pi$
$602$	0	0
$603$	4.10092	0.167002
$604$	0	0
$605$	2.06351	0.0838935
$606$	0	0
$607$	−29.0512	−1.17915	−0.589576	−	0.807713i	$-0.700705\pi$
$607$	−29.0512	−1.17915	−0.589576	+	0.807713i	$0.700705\pi$
$608$	0	0
$609$	5.79005	0.234625
$610$	0	0
$611$	1.93649	0.0783422
$612$	0	0
$613$	0.300994	0.0121570	0.00607851	−	0.999982i	$-0.498065\pi$
$613$	0.300994	0.0121570	0.00607851	+	0.999982i	$0.498065\pi$
$614$	0	0
$615$	−1.47919	−0.0596466
$616$	0	0
$617$	20.8218	0.838254	0.419127	−	0.907928i	$-0.362336\pi$
$617$	20.8218	0.838254	0.419127	+	0.907928i	$0.362336\pi$
$618$	0	0
$619$	−25.4076	−1.02122	−0.510608	−	0.859813i	$-0.670580\pi$
$619$	−25.4076	−1.02122	−0.510608	+	0.859813i	$0.670580\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−7.89097	−0.316145
$624$	0	0
$625$	−30.7901	−1.23160
$626$	0	0
$627$	−25.5247	−1.01936
$628$	0	0
$629$	5.34262	0.213024
$630$	0	0
$631$	12.8119	0.510035	0.255018	−	0.966936i	$-0.417919\pi$
$631$	12.8119	0.510035	0.255018	+	0.966936i	$0.417919\pi$
$632$	0	0
$633$	26.0357	1.03482
$634$	0	0
$635$	−37.8536	−1.50217
$636$	0	0
$637$	0.821806	0.0325611
$638$	0	0
$639$	−1.17819	−0.0466086
$640$	0	0
$641$	−3.06741	−0.121155	−0.0605776	−	0.998163i	$-0.519294\pi$
$641$	−3.06741	−0.121155	−0.0605776	+	0.998163i	$0.519294\pi$
$642$	0	0
$643$	6.15631	0.242781	0.121391	−	0.992605i	$-0.461265\pi$
$643$	6.15631	0.242781	0.121391	+	0.992605i	$0.461265\pi$
$644$	0	0
$645$	−30.5304	−1.20213
$646$	0	0
$647$	12.4376	0.488971	0.244486	−	0.969653i	$-0.421381\pi$
$647$	12.4376	0.488971	0.244486	+	0.969653i	$0.421381\pi$
$648$	0	0
$649$	−6.52081	−0.255964
$650$	0	0
$651$	−3.07728	−0.120608
$652$	0	0
$653$	−10.0536	−0.393429	−0.196715	−	0.980461i	$-0.563027\pi$
$653$	−10.0536	−0.393429	−0.196715	+	0.980461i	$0.563027\pi$
$654$	0	0
$655$	−46.2992	−1.80906
$656$	0	0
$657$	8.14644	0.317823
$658$	0	0
$659$	19.1246	0.744987	0.372494	−	0.928035i	$-0.378503\pi$
$659$	19.1246	0.744987	0.372494	+	0.928035i	$0.378503\pi$
$660$	0	0
$661$	41.7068	1.62221	0.811103	−	0.584903i	$-0.198867\pi$
$661$	41.7068	1.62221	0.811103	+	0.584903i	$0.198867\pi$
$662$	0	0
$663$	4.75830	0.184797
$664$	0	0
$665$	19.4157	0.752908
$666$	0	0
$667$	5.79005	0.224192
$668$	0	0
$669$	−27.8356	−1.07619
$670$	0	0
$671$	8.24736	0.318386
$672$	0	0
$673$	−15.5247	−0.598434	−0.299217	−	0.954185i	$-0.596725\pi$
$673$	−15.5247	−0.598434	−0.299217	+	0.954185i	$0.596725\pi$
$674$	0	0
$675$	−1.82181	−0.0701213
$676$	0	0
$677$	−4.24736	−0.163239	−0.0816196	−	0.996664i	$-0.526009\pi$
$677$	−4.24736	−0.163239	−0.0816196	+	0.996664i	$0.526009\pi$
$678$	0	0
$679$	−1.79005	−0.0686959
$680$	0	0
$681$	1.32464	0.0507601
$682$	0	0
$683$	−35.8811	−1.37295	−0.686476	−	0.727152i	$-0.740843\pi$
$683$	−35.8811	−1.37295	−0.686476	+	0.727152i	$0.740843\pi$
$684$	0	0
$685$	26.9047	1.02798
$686$	0	0
$687$	−27.2138	−1.03827
$688$	0	0
$689$	0.795711	0.0303142
$690$	0	0
$691$	−4.96013	−0.188692	−0.0943462	−	0.995539i	$-0.530076\pi$
$691$	−4.96013	−0.188692	−0.0943462	+	0.995539i	$0.530076\pi$
$692$	0	0
$693$	3.43366	0.130434
$694$	0	0
$695$	51.8077	1.96518
$696$	0	0
$697$	−3.27911	−0.124205
$698$	0	0
$699$	20.1920	0.763730
$700$	0	0
$701$	−24.2555	−0.916116	−0.458058	−	0.888922i	$-0.651455\pi$
$701$	−24.2555	−0.916116	−0.458058	+	0.888922i	$0.651455\pi$
$702$	0	0
$703$	−6.85922	−0.258700
$704$	0	0
$705$	−6.15455	−0.231794
$706$	0	0
$707$	9.12280	0.343098
$708$	0	0
$709$	2.21982	0.0833670	0.0416835	−	0.999131i	$-0.486728\pi$
$709$	2.21982	0.0833670	0.0416835	+	0.999131i	$0.486728\pi$
$710$	0	0
$711$	2.56634	0.0962451
$712$	0	0
$713$	−3.07728	−0.115245
$714$	0	0
$715$	7.37016	0.275628
$716$	0	0
$717$	21.7721	0.813093
$718$	0	0
$719$	1.23183	0.0459395	0.0229697	−	0.999736i	$-0.492688\pi$
$719$	1.23183	0.0459395	0.0229697	+	0.999736i	$0.492688\pi$
$720$	0	0
$721$	−19.1602	−0.713564
$722$	0	0
$723$	12.6574	0.470733
$724$	0	0
$725$	−10.5484	−0.391756
$726$	0	0
$727$	7.88110	0.292294	0.146147	−	0.989263i	$-0.453313\pi$
$727$	7.88110	0.292294	0.146147	+	0.989263i	$0.453313\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−67.6807	−2.50326
$732$	0	0
$733$	6.64927	0.245596	0.122798	−	0.992432i	$-0.460813\pi$
$733$	6.64927	0.245596	0.122798	+	0.992432i	$0.460813\pi$
$734$	0	0
$735$	−2.61186	−0.0963399
$736$	0	0
$737$	14.0812	0.518687
$738$	0	0
$739$	14.4555	0.531756	0.265878	−	0.964007i	$-0.414338\pi$
$739$	14.4555	0.531756	0.265878	+	0.964007i	$0.414338\pi$
$740$	0	0
$741$	−6.10903	−0.224421
$742$	0	0
$743$	−8.63940	−0.316949	−0.158474	−	0.987363i	$-0.550658\pi$
$743$	−8.63940	−0.316949	−0.158474	+	0.987363i	$0.550658\pi$
$744$	0	0
$745$	0.237801	0.00871236
$746$	0	0
$747$	6.30099	0.230541
$748$	0	0
$749$	14.5565	0.531882
$750$	0	0
$751$	−6.99579	−0.255280	−0.127640	−	0.991821i	$-0.540740\pi$
$751$	−6.99579	−0.255280	−0.127640	+	0.991821i	$0.540740\pi$
$752$	0	0
$753$	8.66549	0.315788
$754$	0	0
$755$	−29.7184	−1.08156
$756$	0	0
$757$	30.5304	1.10965	0.554823	−	0.831969i	$-0.312786\pi$
$757$	30.5304	1.10965	0.554823	+	0.831969i	$0.312786\pi$
$758$	0	0
$759$	3.43366	0.124634
$760$	0	0
$761$	−0.867329	−0.0314407	−0.0157203	−	0.999876i	$-0.505004\pi$
$761$	−0.867329	−0.0314407	−0.0157203	+	0.999876i	$0.505004\pi$
$762$	0	0
$763$	17.0593	0.617588
$764$	0	0
$765$	−15.1228	−0.546766
$766$	0	0
$767$	−1.56068	−0.0563528
$768$	0	0
$769$	−27.5166	−0.992274	−0.496137	−	0.868244i	$-0.665249\pi$
$769$	−27.5166	−0.992274	−0.496137	+	0.868244i	$0.665249\pi$
$770$	0	0
$771$	−20.6493	−0.743665
$772$	0	0
$773$	−3.58822	−0.129059	−0.0645296	−	0.997916i	$-0.520555\pi$
$773$	−3.58822	−0.129059	−0.0645296	+	0.997916i	$0.520555\pi$
$774$	0	0
$775$	5.60620	0.201381
$776$	0	0
$777$	0.922724	0.0331025
$778$	0	0
$779$	4.20995	0.150837
$780$	0	0
$781$	−4.04552	−0.144760
$782$	0	0
$783$	5.79005	0.206920
$784$	0	0
$785$	13.6436	0.486961
$786$	0	0
$787$	32.4100	1.15529	0.577646	−	0.816287i	$-0.303971\pi$
$787$	32.4100	1.15529	0.577646	+	0.816287i	$0.303971\pi$
$788$	0	0
$789$	−21.1048	−0.751351
$790$	0	0
$791$	7.47919	0.265929
$792$	0	0
$793$	1.97391	0.0700954
$794$	0	0
$795$	−2.52892	−0.0896917
$796$	0	0
$797$	−20.8038	−0.736909	−0.368455	−	0.929646i	$-0.620113\pi$
$797$	−20.8038	−0.736909	−0.368455	+	0.929646i	$0.620113\pi$
$798$	0	0
$799$	−13.6436	−0.482676
$800$	0	0
$801$	−7.89097	−0.278814
$802$	0	0
$803$	27.9721	0.987116
$804$	0	0
$805$	−2.61186	−0.0920559
$806$	0	0
$807$	18.8493	0.663528
$808$	0	0
$809$	−17.1147	−0.601720	−0.300860	−	0.953668i	$-0.597274\pi$
$809$	−17.1147	−0.601720	−0.300860	+	0.953668i	$0.597274\pi$
$810$	0	0
$811$	22.4393	0.787951	0.393976	−	0.919121i	$-0.371099\pi$
$811$	22.4393	0.787951	0.393976	+	0.919121i	$0.371099\pi$
$812$	0	0
$813$	5.01377	0.175841
$814$	0	0
$815$	15.9347	0.558169
$816$	0	0
$817$	86.8931	3.04001
$818$	0	0
$819$	0.821806	0.0287162
$820$	0	0
$821$	−6.38393	−0.222801	−0.111400	−	0.993776i	$-0.535534\pi$
$821$	−6.38393	−0.222801	−0.111400	+	0.993776i	$0.535534\pi$
$822$	0	0
$823$	−45.8631	−1.59869	−0.799344	−	0.600874i	$-0.794819\pi$
$823$	−45.8631	−1.59869	−0.799344	+	0.600874i	$0.794819\pi$
$824$	0	0
$825$	−6.25547	−0.217788
$826$	0	0
$827$	48.7029	1.69357	0.846783	−	0.531939i	$-0.178536\pi$
$827$	48.7029	1.69357	0.846783	+	0.531939i	$0.178536\pi$
$828$	0	0
$829$	−2.20995	−0.0767546	−0.0383773	−	0.999263i	$-0.512219\pi$
$829$	−2.20995	−0.0767546	−0.0383773	+	0.999263i	$0.512219\pi$
$830$	0	0
$831$	7.17819	0.249009
$832$	0	0
$833$	−5.79005	−0.200613
$834$	0	0
$835$	21.6810	0.750303
$836$	0	0
$837$	−3.07728	−0.106366
$838$	0	0
$839$	−16.4295	−0.567208	−0.283604	−	0.958942i	$-0.591530\pi$
$839$	−16.4295	−0.567208	−0.283604	+	0.958942i	$0.591530\pi$
$840$	0	0
$841$	4.52471	0.156025
$842$	0	0
$843$	−1.19618	−0.0411985
$844$	0	0
$845$	−32.1902	−1.10738
$846$	0	0
$847$	0.790053	0.0271465
$848$	0	0
$849$	−6.96825	−0.239150
$850$	0	0
$851$	0.922724	0.0316306
$852$	0	0
$853$	−10.8038	−0.369916	−0.184958	−	0.982746i	$-0.559215\pi$
$853$	−10.8038	−0.369916	−0.184958	+	0.982746i	$0.559215\pi$
$854$	0	0
$855$	19.4157	0.664002
$856$	0	0
$857$	1.90895	0.0652086	0.0326043	−	0.999468i	$-0.489620\pi$
$857$	1.90895	0.0652086	0.0326043	+	0.999468i	$0.489620\pi$
$858$	0	0
$859$	−38.0275	−1.29748	−0.648741	−	0.761009i	$-0.724704\pi$
$859$	−38.0275	−1.29748	−0.648741	+	0.761009i	$0.724704\pi$
$860$	0	0
$861$	−0.566335	−0.0193007
$862$	0	0
$863$	−36.2018	−1.23232	−0.616162	−	0.787619i	$-0.711314\pi$
$863$	−36.2018	−1.23232	−0.616162	+	0.787619i	$0.711314\pi$
$864$	0	0
$865$	21.2774	0.723452
$866$	0	0
$867$	−16.5247	−0.561209
$868$	0	0
$869$	8.81193	0.298924
$870$	0	0
$871$	3.37016	0.114193
$872$	0	0
$873$	−1.79005	−0.0605841
$874$	0	0
$875$	−8.30099	−0.280625
$876$	0	0
$877$	4.86733	0.164358	0.0821790	−	0.996618i	$-0.473812\pi$
$877$	4.86733	0.164358	0.0821790	+	0.996618i	$0.473812\pi$
$878$	0	0
$879$	−22.4474	−0.757133
$880$	0	0
$881$	32.9422	1.10985	0.554925	−	0.831901i	$-0.312747\pi$
$881$	32.9422	1.10985	0.554925	+	0.831901i	$0.312747\pi$
$882$	0	0
$883$	14.7858	0.497583	0.248792	−	0.968557i	$-0.419967\pi$
$883$	14.7858	0.497583	0.248792	+	0.968557i	$0.419967\pi$
$884$	0	0
$885$	4.96013	0.166733
$886$	0	0
$887$	14.8966	0.500180	0.250090	−	0.968223i	$-0.419540\pi$
$887$	14.8966	0.500180	0.250090	+	0.968223i	$0.419540\pi$
$888$	0	0
$889$	−14.4930	−0.486078
$890$	0	0
$891$	3.43366	0.115032
$892$	0	0
$893$	17.5166	0.586171
$894$	0	0
$895$	−23.4238	−0.782971
$896$	0	0
$897$	0.821806	0.0274393
$898$	0	0
$899$	−17.8176	−0.594250
$900$	0	0
$901$	−5.60620	−0.186770
$902$	0	0
$903$	−11.6891	−0.388990
$904$	0	0
$905$	−61.8469	−2.05586
$906$	0	0
$907$	−48.5759	−1.61294	−0.806468	−	0.591278i	$-0.798624\pi$
$907$	−48.5759	−1.61294	−0.806468	+	0.591278i	$0.798624\pi$
$908$	0	0
$909$	9.12280	0.302584
$910$	0	0
$911$	−17.6436	−0.584559	−0.292279	−	0.956333i	$-0.594414\pi$
$911$	−17.6436	−0.584559	−0.292279	+	0.956333i	$0.594414\pi$
$912$	0	0
$913$	21.6355	0.716031
$914$	0	0
$915$	−6.27345	−0.207394
$916$	0	0
$917$	−17.7265	−0.585382
$918$	0	0
$919$	11.6630	0.384728	0.192364	−	0.981324i	$-0.438385\pi$
$919$	11.6630	0.384728	0.192364	+	0.981324i	$0.438385\pi$
$920$	0	0
$921$	−22.6574	−0.746586
$922$	0	0
$923$	−0.968247	−0.0318702
$924$	0	0
$925$	−1.68102	−0.0552717
$926$	0	0
$927$	−19.1602	−0.629304
$928$	0	0
$929$	0.904741	0.0296836	0.0148418	−	0.999890i	$-0.495276\pi$
$929$	0.904741	0.0296836	0.0148418	+	0.999890i	$0.495276\pi$
$930$	0	0
$931$	7.43366	0.243629
$932$	0	0
$933$	23.4238	0.766861
$934$	0	0
$935$	−51.9266	−1.69818
$936$	0	0
$937$	51.4887	1.68206	0.841032	−	0.540985i	$-0.181949\pi$
$937$	51.4887	1.68206	0.841032	+	0.540985i	$0.181949\pi$
$938$	0	0
$939$	17.7622	0.579647
$940$	0	0
$941$	47.4693	1.54746	0.773728	−	0.633518i	$-0.218390\pi$
$941$	47.4693	1.54746	0.773728	+	0.633518i	$0.218390\pi$
$942$	0	0
$943$	−0.566335	−0.0184424
$944$	0	0
$945$	−2.61186	−0.0849638
$946$	0	0
$947$	47.5801	1.54615	0.773073	−	0.634317i	$-0.218719\pi$
$947$	47.5801	1.54615	0.773073	+	0.634317i	$0.218719\pi$
$948$	0	0
$949$	6.69479	0.217322
$950$	0	0
$951$	−8.19196	−0.265643
$952$	0	0
$953$	−61.6807	−1.99803	−0.999017	−	0.0443267i	$-0.985886\pi$
$953$	−61.6807	−1.99803	−0.999017	+	0.0443267i	$0.985886\pi$
$954$	0	0
$955$	−10.6852	−0.345766
$956$	0	0
$957$	19.8811	0.642665
$958$	0	0
$959$	10.3010	0.332636
$960$	0	0
$961$	−21.5304	−0.694528
$962$	0	0
$963$	14.5565	0.469076
$964$	0	0
$965$	32.2731	1.03891
$966$	0	0
$967$	18.0275	0.579727	0.289863	−	0.957068i	$-0.406390\pi$
$967$	18.0275	0.579727	0.289863	+	0.957068i	$0.406390\pi$
$968$	0	0
$969$	43.0413	1.38269
$970$	0	0
$971$	−19.4873	−0.625377	−0.312689	−	0.949856i	$-0.601230\pi$
$971$	−19.4873	−0.625377	−0.312689	+	0.949856i	$0.601230\pi$
$972$	0	0
$973$	19.8356	0.635899
$974$	0	0
$975$	−1.49717	−0.0479478
$976$	0	0
$977$	21.1147	0.675519	0.337759	−	0.941232i	$-0.390331\pi$
$977$	21.1147	0.675519	0.337759	+	0.941232i	$0.390331\pi$
$978$	0	0
$979$	−27.0949	−0.865958
$980$	0	0
$981$	17.0593	0.544661
$982$	0	0
$983$	−35.8811	−1.14443	−0.572215	−	0.820104i	$-0.693916\pi$
$983$	−35.8811	−1.14443	−0.572215	+	0.820104i	$0.693916\pi$
$984$	0	0
$985$	5.86978	0.187027
$986$	0	0
$987$	−2.35639	−0.0750047
$988$	0	0
$989$	−11.6891	−0.371693
$990$	0	0
$991$	−15.2417	−0.484169	−0.242084	−	0.970255i	$-0.577831\pi$
$991$	−15.2417	−0.484169	−0.242084	+	0.970255i	$0.577831\pi$
$992$	0	0
$993$	1.13267	0.0359442
$994$	0	0
$995$	56.6278	1.79522
$996$	0	0
$997$	37.7901	1.19682	0.598411	−	0.801189i	$-0.295799\pi$
$997$	37.7901	1.19682	0.598411	+	0.801189i	$0.295799\pi$
$998$	0	0
$999$	0.922724	0.0291937

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bq.1.3		3
4.3	odd	2		3864.2.a.o.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.o.1.3	✓	3	4.3	odd	2
7728.2.a.bq.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.61186 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.61186 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.43366 q^{11} +0.821806 q^{13} -2.61186 q^{15} -5.79005 q^{17} +7.43366 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.82181 q^{25} -1.00000 q^{27} -5.79005 q^{29} +3.07728 q^{31} -3.43366 q^{33} +2.61186 q^{35} -0.922724 q^{37} -0.821806 q^{39} +0.566335 q^{41} +11.6891 q^{43} +2.61186 q^{45} +2.35639 q^{47} +1.00000 q^{49} +5.79005 q^{51} +0.968247 q^{53} +8.96825 q^{55} -7.43366 q^{57} -1.89908 q^{59} +2.40191 q^{61} +1.00000 q^{63} +2.14644 q^{65} +4.10092 q^{67} +1.00000 q^{69} -1.17819 q^{71} +8.14644 q^{73} -1.82181 q^{75} +3.43366 q^{77} +2.56634 q^{79} +1.00000 q^{81} +6.30099 q^{83} -15.1228 q^{85} +5.79005 q^{87} -7.89097 q^{89} +0.821806 q^{91} -3.07728 q^{93} +19.4157 q^{95} -1.79005 q^{97} +3.43366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 5 q^{13} + 3 q^{15} - 4 q^{17} + 14 q^{19} - 3 q^{21} - 3 q^{23} + 8 q^{25} - 3 q^{27} - 4 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 6 q^{37} - 5 q^{39} + 10 q^{41} + 21 q^{43} - 3 q^{45} + 2 q^{47} + 3 q^{49} + 4 q^{51} - 13 q^{53} + 11 q^{55} - 14 q^{57} - 5 q^{59} - 17 q^{61} + 3 q^{63} - 12 q^{65} + 13 q^{67} + 3 q^{69} - q^{71} + 6 q^{73} - 8 q^{75} + 2 q^{77} + 16 q^{79} + 3 q^{81} - 6 q^{83} - 23 q^{85} + 4 q^{87} - 11 q^{89} + 5 q^{91} - 6 q^{93} - q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 5 * q^13 + 3 * q^15 - 4 * q^17 + 14 * q^19 - 3 * q^21 - 3 * q^23 + 8 * q^25 - 3 * q^27 - 4 * q^29 + 6 * q^31 - 2 * q^33 - 3 * q^35 - 6 * q^37 - 5 * q^39 + 10 * q^41 + 21 * q^43 - 3 * q^45 + 2 * q^47 + 3 * q^49 + 4 * q^51 - 13 * q^53 + 11 * q^55 - 14 * q^57 - 5 * q^59 - 17 * q^61 + 3 * q^63 - 12 * q^65 + 13 * q^67 + 3 * q^69 - q^71 + 6 * q^73 - 8 * q^75 + 2 * q^77 + 16 * q^79 + 3 * q^81 - 6 * q^83 - 23 * q^85 + 4 * q^87 - 11 * q^89 + 5 * q^91 - 6 * q^93 - q^95 + 8 * q^97 + 2 * q^99

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