LMFDB - Embedded newform 3042.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.890084 q^{5} +4.49396 q^{7} +1.00000 q^{8} +0.890084 q^{10} +2.69202 q^{11} +4.49396 q^{14} +1.00000 q^{16} -3.58211 q^{17} +2.93900 q^{19} +0.890084 q^{20} +2.69202 q^{22} +6.09783 q^{23} -4.20775 q^{25} +4.49396 q^{28} -2.98792 q^{29} +2.39612 q^{31} +1.00000 q^{32} -3.58211 q^{34} +4.00000 q^{35} +1.50604 q^{37} +2.93900 q^{38} +0.890084 q^{40} -3.65279 q^{41} +0.170915 q^{43} +2.69202 q^{44} +6.09783 q^{46} +5.20775 q^{47} +13.1957 q^{49} -4.20775 q^{50} -6.09783 q^{53} +2.39612 q^{55} +4.49396 q^{56} -2.98792 q^{58} +3.07069 q^{59} -13.9758 q^{61} +2.39612 q^{62} +1.00000 q^{64} -11.0707 q^{67} -3.58211 q^{68} +4.00000 q^{70} -10.0978 q^{71} -10.9487 q^{73} +1.50604 q^{74} +2.93900 q^{76} +12.0978 q^{77} +2.81163 q^{79} +0.890084 q^{80} -3.65279 q^{82} +2.93900 q^{83} -3.18837 q^{85} +0.170915 q^{86} +2.69202 q^{88} +12.1806 q^{89} +6.09783 q^{92} +5.20775 q^{94} +2.61596 q^{95} -12.9661 q^{97} +13.1957 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} + 2 q^{10} + 3 q^{11} + 4 q^{14} + 3 q^{16} - 5 q^{17} - q^{19} + 2 q^{20} + 3 q^{22} + 5 q^{25} + 4 q^{28} + 10 q^{29} + 16 q^{31} + 3 q^{32} - 5 q^{34}+ \cdots + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 + 4 * q^7 + 3 * q^8 + 2 * q^10 + 3 * q^11 + 4 * q^14 + 3 * q^16 - 5 * q^17 - q^19 + 2 * q^20 + 3 * q^22 + 5 * q^25 + 4 * q^28 + 10 * q^29 + 16 * q^31 + 3 * q^32 - 5 * q^34 + 12 * q^35 + 14 * q^37 - q^38 + 2 * q^40 + 7 * q^41 + 11 * q^43 + 3 * q^44 - 2 * q^47 + 3 * q^49 + 5 * q^50 + 16 * q^55 + 4 * q^56 + 10 * q^58 - 3 * q^59 - 4 * q^61 + 16 * q^62 + 3 * q^64 - 21 * q^67 - 5 * q^68 + 12 * q^70 - 12 * q^71 - q^73 + 14 * q^74 - q^76 + 18 * q^77 - 18 * q^79 + 2 * q^80 + 7 * q^82 - q^83 - 36 * q^85 + 11 * q^86 + 3 * q^88 + 25 * q^89 - 2 * q^94 + 18 * q^95 - 23 * q^97 + 3 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.890084	0.398058	0.199029	−	0.979994i	$-0.436221\pi$
$5$	0.890084	0.398058	0.199029	+	0.979994i	$0.436221\pi$
$6$	0	0
$7$	4.49396	1.69856	0.849278	−	0.527945i	$-0.177037\pi$
$7$	4.49396	1.69856	0.849278	+	0.527945i	$0.177037\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.890084	0.281469
$11$	2.69202	0.811675	0.405838	−	0.913945i	$-0.366980\pi$
$11$	2.69202	0.811675	0.405838	+	0.913945i	$0.366980\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.49396	1.20106
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.58211	−0.868788	−0.434394	−	0.900723i	$-0.643037\pi$
$17$	−3.58211	−0.868788	−0.434394	+	0.900723i	$0.643037\pi$
$18$	0	0
$19$	2.93900	0.674253	0.337127	−	0.941459i	$-0.390545\pi$
$19$	2.93900	0.674253	0.337127	+	0.941459i	$0.390545\pi$
$20$	0.890084	0.199029
$21$	0	0
$22$	2.69202	0.573941
$23$	6.09783	1.27149	0.635743	−	0.771901i	$-0.280694\pi$
$23$	6.09783	1.27149	0.635743	+	0.771901i	$0.280694\pi$
$24$	0	0
$25$	−4.20775	−0.841550
$26$	0	0
$27$	0	0
$28$	4.49396	0.849278
$29$	−2.98792	−0.554843	−0.277421	−	0.960748i	$-0.589480\pi$
$29$	−2.98792	−0.554843	−0.277421	+	0.960748i	$0.589480\pi$
$30$	0	0
$31$	2.39612	0.430357	0.215178	−	0.976575i	$-0.430967\pi$
$31$	2.39612	0.430357	0.215178	+	0.976575i	$0.430967\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.58211	−0.614326
$35$	4.00000	0.676123
$36$	0	0
$37$	1.50604	0.247592	0.123796	−	0.992308i	$-0.460493\pi$
$37$	1.50604	0.247592	0.123796	+	0.992308i	$0.460493\pi$
$38$	2.93900	0.476769
$39$	0	0
$40$	0.890084	0.140735
$41$	−3.65279	−0.570470	−0.285235	−	0.958458i	$-0.592072\pi$
$41$	−3.65279	−0.570470	−0.285235	+	0.958458i	$0.592072\pi$
$42$	0	0
$43$	0.170915	0.0260643	0.0130322	−	0.999915i	$-0.495852\pi$
$43$	0.170915	0.0260643	0.0130322	+	0.999915i	$0.495852\pi$
$44$	2.69202	0.405838
$45$	0	0
$46$	6.09783	0.899077
$47$	5.20775	0.759629	0.379814	−	0.925063i	$-0.375988\pi$
$47$	5.20775	0.759629	0.379814	+	0.925063i	$0.375988\pi$
$48$	0	0
$49$	13.1957	1.88510
$50$	−4.20775	−0.595066
$51$	0	0
$52$	0	0
$53$	−6.09783	−0.837602	−0.418801	−	0.908078i	$-0.637550\pi$
$53$	−6.09783	−0.837602	−0.418801	+	0.908078i	$0.637550\pi$
$54$	0	0
$55$	2.39612	0.323093
$56$	4.49396	0.600531
$57$	0	0
$58$	−2.98792	−0.392333
$59$	3.07069	0.399769	0.199885	−	0.979819i	$-0.435943\pi$
$59$	3.07069	0.399769	0.199885	+	0.979819i	$0.435943\pi$
$60$	0	0
$61$	−13.9758	−1.78942	−0.894711	−	0.446645i	$-0.852619\pi$
$61$	−13.9758	−1.78942	−0.894711	+	0.446645i	$0.852619\pi$
$62$	2.39612	0.304308
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−11.0707	−1.35250	−0.676250	−	0.736672i	$-0.736396\pi$
$67$	−11.0707	−1.35250	−0.676250	+	0.736672i	$0.736396\pi$
$68$	−3.58211	−0.434394
$69$	0	0
$70$	4.00000	0.478091
$71$	−10.0978	−1.19839	−0.599196	−	0.800602i	$-0.704513\pi$
$71$	−10.0978	−1.19839	−0.599196	+	0.800602i	$0.704513\pi$
$72$	0	0
$73$	−10.9487	−1.28145	−0.640724	−	0.767772i	$-0.721366\pi$
$73$	−10.9487	−1.28145	−0.640724	+	0.767772i	$0.721366\pi$
$74$	1.50604	0.175074
$75$	0	0
$76$	2.93900	0.337127
$77$	12.0978	1.37868
$78$	0	0
$79$	2.81163	0.316333	0.158166	−	0.987412i	$-0.449442\pi$
$79$	2.81163	0.316333	0.158166	+	0.987412i	$0.449442\pi$
$80$	0.890084	0.0995144
$81$	0	0
$82$	−3.65279	−0.403383
$83$	2.93900	0.322597	0.161299	−	0.986906i	$-0.448432\pi$
$83$	2.93900	0.322597	0.161299	+	0.986906i	$0.448432\pi$
$84$	0	0
$85$	−3.18837	−0.345828
$86$	0.170915	0.0184303
$87$	0	0
$88$	2.69202	0.286970
$89$	12.1806	1.29114	0.645571	−	0.763700i	$-0.276620\pi$
$89$	12.1806	1.29114	0.645571	+	0.763700i	$0.276620\pi$
$90$	0	0
$91$	0	0
$92$	6.09783	0.635743
$93$	0	0
$94$	5.20775	0.537138
$95$	2.61596	0.268392
$96$	0	0
$97$	−12.9661	−1.31651	−0.658256	−	0.752794i	$-0.728706\pi$
$97$	−12.9661	−1.31651	−0.658256	+	0.752794i	$0.728706\pi$
$98$	13.1957	1.33296
$99$	0	0
$100$	−4.20775	−0.420775
$101$	10.1957	1.01451	0.507254	−	0.861797i	$-0.330661\pi$
$101$	10.1957	1.01451	0.507254	+	0.861797i	$0.330661\pi$
$102$	0	0
$103$	10.6703	1.05137	0.525686	−	0.850679i	$-0.323809\pi$
$103$	10.6703	1.05137	0.525686	+	0.850679i	$0.323809\pi$
$104$	0	0
$105$	0	0
$106$	−6.09783	−0.592274
$107$	20.5623	1.98783	0.993914	−	0.110158i	$-0.0351358\pi$
$107$	20.5623	1.98783	0.993914	+	0.110158i	$0.0351358\pi$
$108$	0	0
$109$	−2.71379	−0.259934	−0.129967	−	0.991518i	$-0.541487\pi$
$109$	−2.71379	−0.259934	−0.129967	+	0.991518i	$0.541487\pi$
$110$	2.39612	0.228462
$111$	0	0
$112$	4.49396	0.424639
$113$	−20.6504	−1.94263	−0.971313	−	0.237804i	$-0.923572\pi$
$113$	−20.6504	−1.94263	−0.971313	+	0.237804i	$0.923572\pi$
$114$	0	0
$115$	5.42758	0.506125
$116$	−2.98792	−0.277421
$117$	0	0
$118$	3.07069	0.282680
$119$	−16.0978	−1.47569
$120$	0	0
$121$	−3.75302	−0.341184
$122$	−13.9758	−1.26531
$123$	0	0
$124$	2.39612	0.215178
$125$	−8.19567	−0.733043
$126$	0	0
$127$	12.7681	1.13298	0.566492	−	0.824067i	$-0.308300\pi$
$127$	12.7681	1.13298	0.566492	+	0.824067i	$0.308300\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	5.15883	0.450729	0.225365	−	0.974274i	$-0.427643\pi$
$131$	5.15883	0.450729	0.225365	+	0.974274i	$0.427643\pi$
$132$	0	0
$133$	13.2078	1.14526
$134$	−11.0707	−0.956362
$135$	0	0
$136$	−3.58211	−0.307163
$137$	−3.30127	−0.282047	−0.141023	−	0.990006i	$-0.545039\pi$
$137$	−3.30127	−0.282047	−0.141023	+	0.990006i	$0.545039\pi$
$138$	0	0
$139$	−6.49157	−0.550607	−0.275304	−	0.961357i	$-0.588778\pi$
$139$	−6.49157	−0.550607	−0.275304	+	0.961357i	$0.588778\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−10.0978	−0.847391
$143$	0	0
$144$	0	0
$145$	−2.65950	−0.220859
$146$	−10.9487	−0.906120
$147$	0	0
$148$	1.50604	0.123796
$149$	5.03146	0.412193	0.206097	−	0.978532i	$-0.433924\pi$
$149$	5.03146	0.412193	0.206097	+	0.978532i	$0.433924\pi$
$150$	0	0
$151$	1.72587	0.140450	0.0702248	−	0.997531i	$-0.477628\pi$
$151$	1.72587	0.140450	0.0702248	+	0.997531i	$0.477628\pi$
$152$	2.93900	0.238384
$153$	0	0
$154$	12.0978	0.974871
$155$	2.13275	0.171307
$156$	0	0
$157$	15.5060	1.23752	0.618758	−	0.785581i	$-0.287636\pi$
$157$	15.5060	1.23752	0.618758	+	0.785581i	$0.287636\pi$
$158$	2.81163	0.223681
$159$	0	0
$160$	0.890084	0.0703673
$161$	27.4034	2.15969
$162$	0	0
$163$	19.7235	1.54486	0.772431	−	0.635099i	$-0.219041\pi$
$163$	19.7235	1.54486	0.772431	+	0.635099i	$0.219041\pi$
$164$	−3.65279	−0.285235
$165$	0	0
$166$	2.93900	0.228111
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	0	0
$170$	−3.18837	−0.244537
$171$	0	0
$172$	0.170915	0.0130322
$173$	−1.20775	−0.0918236	−0.0459118	−	0.998945i	$-0.514619\pi$
$173$	−1.20775	−0.0918236	−0.0459118	+	0.998945i	$0.514619\pi$
$174$	0	0
$175$	−18.9095	−1.42942
$176$	2.69202	0.202919
$177$	0	0
$178$	12.1806	0.912975
$179$	−16.5157	−1.23444	−0.617222	−	0.786789i	$-0.711742\pi$
$179$	−16.5157	−1.23444	−0.617222	+	0.786789i	$0.711742\pi$
$180$	0	0
$181$	6.37196	0.473624	0.236812	−	0.971555i	$-0.423897\pi$
$181$	6.37196	0.473624	0.236812	+	0.971555i	$0.423897\pi$
$182$	0	0
$183$	0	0
$184$	6.09783	0.449538
$185$	1.34050	0.0985557
$186$	0	0
$187$	−9.64310	−0.705174
$188$	5.20775	0.379814
$189$	0	0
$190$	2.61596	0.189781
$191$	2.49396	0.180457	0.0902283	−	0.995921i	$-0.471240\pi$
$191$	2.49396	0.180457	0.0902283	+	0.995921i	$0.471240\pi$
$192$	0	0
$193$	4.00538	0.288313	0.144157	−	0.989555i	$-0.453953\pi$
$193$	4.00538	0.288313	0.144157	+	0.989555i	$0.453953\pi$
$194$	−12.9661	−0.930915
$195$	0	0
$196$	13.1957	0.942548
$197$	23.6340	1.68385	0.841927	−	0.539592i	$-0.181422\pi$
$197$	23.6340	1.68385	0.841927	+	0.539592i	$0.181422\pi$
$198$	0	0
$199$	16.5375	1.17231	0.586156	−	0.810198i	$-0.300641\pi$
$199$	16.5375	1.17231	0.586156	+	0.810198i	$0.300641\pi$
$200$	−4.20775	−0.297533
$201$	0	0
$202$	10.1957	0.717365
$203$	−13.4276	−0.942432
$204$	0	0
$205$	−3.25129	−0.227080
$206$	10.6703	0.743432
$207$	0	0
$208$	0	0
$209$	7.91185	0.547274
$210$	0	0
$211$	−1.66056	−0.114318	−0.0571589	−	0.998365i	$-0.518204\pi$
$211$	−1.66056	−0.114318	−0.0571589	+	0.998365i	$0.518204\pi$
$212$	−6.09783	−0.418801
$213$	0	0
$214$	20.5623	1.40561
$215$	0.152129	0.0103751
$216$	0	0
$217$	10.7681	0.730985
$218$	−2.71379	−0.183801
$219$	0	0
$220$	2.39612	0.161547
$221$	0	0
$222$	0	0
$223$	0.792249	0.0530529	0.0265265	−	0.999648i	$-0.491555\pi$
$223$	0.792249	0.0530529	0.0265265	+	0.999648i	$0.491555\pi$
$224$	4.49396	0.300265
$225$	0	0
$226$	−20.6504	−1.37364
$227$	−21.7603	−1.44428	−0.722141	−	0.691745i	$-0.756842\pi$
$227$	−21.7603	−1.44428	−0.722141	+	0.691745i	$0.756842\pi$
$228$	0	0
$229$	1.97584	0.130567	0.0652835	−	0.997867i	$-0.479205\pi$
$229$	1.97584	0.130567	0.0652835	+	0.997867i	$0.479205\pi$
$230$	5.42758	0.357884
$231$	0	0
$232$	−2.98792	−0.196166
$233$	−18.2349	−1.19461	−0.597304	−	0.802015i	$-0.703761\pi$
$233$	−18.2349	−1.19461	−0.597304	+	0.802015i	$0.703761\pi$
$234$	0	0
$235$	4.63533	0.302376
$236$	3.07069	0.199885
$237$	0	0
$238$	−16.0978	−1.04347
$239$	−22.0737	−1.42783	−0.713914	−	0.700234i	$-0.753079\pi$
$239$	−22.0737	−1.42783	−0.713914	+	0.700234i	$0.753079\pi$
$240$	0	0
$241$	−6.98792	−0.450131	−0.225066	−	0.974344i	$-0.572260\pi$
$241$	−6.98792	−0.450131	−0.225066	+	0.974344i	$0.572260\pi$
$242$	−3.75302	−0.241253
$243$	0	0
$244$	−13.9758	−0.894711
$245$	11.7453	0.750377
$246$	0	0
$247$	0	0
$248$	2.39612	0.152154
$249$	0	0
$250$	−8.19567	−0.518340
$251$	12.5593	0.792734	0.396367	−	0.918092i	$-0.370271\pi$
$251$	12.5593	0.792734	0.396367	+	0.918092i	$0.370271\pi$
$252$	0	0
$253$	16.4155	1.03203
$254$	12.7681	0.801141
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.9933	1.06001	0.530006	−	0.847994i	$-0.322190\pi$
$257$	16.9933	1.06001	0.530006	+	0.847994i	$0.322190\pi$
$258$	0	0
$259$	6.76809	0.420548
$260$	0	0
$261$	0	0
$262$	5.15883	0.318714
$263$	4.39612	0.271077	0.135538	−	0.990772i	$-0.456724\pi$
$263$	4.39612	0.271077	0.135538	+	0.990772i	$0.456724\pi$
$264$	0	0
$265$	−5.42758	−0.333414
$266$	13.2078	0.809819
$267$	0	0
$268$	−11.0707	−0.676250
$269$	−15.5603	−0.948730	−0.474365	−	0.880328i	$-0.657322\pi$
$269$	−15.5603	−0.948730	−0.474365	+	0.880328i	$0.657322\pi$
$270$	0	0
$271$	21.9952	1.33611	0.668057	−	0.744110i	$-0.267126\pi$
$271$	21.9952	1.33611	0.668057	+	0.744110i	$0.267126\pi$
$272$	−3.58211	−0.217197
$273$	0	0
$274$	−3.30127	−0.199437
$275$	−11.3274	−0.683065
$276$	0	0
$277$	−1.87800	−0.112838	−0.0564191	−	0.998407i	$-0.517968\pi$
$277$	−1.87800	−0.112838	−0.0564191	+	0.998407i	$0.517968\pi$
$278$	−6.49157	−0.389338
$279$	0	0
$280$	4.00000	0.239046
$281$	9.20536	0.549146	0.274573	−	0.961566i	$-0.411464\pi$
$281$	9.20536	0.549146	0.274573	+	0.961566i	$0.411464\pi$
$282$	0	0
$283$	4.70841	0.279886	0.139943	−	0.990160i	$-0.455308\pi$
$283$	4.70841	0.279886	0.139943	+	0.990160i	$0.455308\pi$
$284$	−10.0978	−0.599196
$285$	0	0
$286$	0	0
$287$	−16.4155	−0.968976
$288$	0	0
$289$	−4.16852	−0.245207
$290$	−2.65950	−0.156171
$291$	0	0
$292$	−10.9487	−0.640724
$293$	−13.7017	−0.800462	−0.400231	−	0.916414i	$-0.631070\pi$
$293$	−13.7017	−0.800462	−0.400231	+	0.916414i	$0.631070\pi$
$294$	0	0
$295$	2.73317	0.159131
$296$	1.50604	0.0875368
$297$	0	0
$298$	5.03146	0.291465
$299$	0	0
$300$	0	0
$301$	0.768086	0.0442717
$302$	1.72587	0.0993128
$303$	0	0
$304$	2.93900	0.168563
$305$	−12.4397	−0.712293
$306$	0	0
$307$	−8.03252	−0.458440	−0.229220	−	0.973375i	$-0.573618\pi$
$307$	−8.03252	−0.458440	−0.229220	+	0.973375i	$0.573618\pi$
$308$	12.0978	0.689338
$309$	0	0
$310$	2.13275	0.121132
$311$	−4.09783	−0.232367	−0.116183	−	0.993228i	$-0.537066\pi$
$311$	−4.09783	−0.232367	−0.116183	+	0.993228i	$0.537066\pi$
$312$	0	0
$313$	4.37435	0.247253	0.123627	−	0.992329i	$-0.460548\pi$
$313$	4.37435	0.247253	0.123627	+	0.992329i	$0.460548\pi$
$314$	15.5060	0.875057
$315$	0	0
$316$	2.81163	0.158166
$317$	−20.3612	−1.14360	−0.571800	−	0.820393i	$-0.693755\pi$
$317$	−20.3612	−1.14360	−0.571800	+	0.820393i	$0.693755\pi$
$318$	0	0
$319$	−8.04354	−0.450352
$320$	0.890084	0.0497572
$321$	0	0
$322$	27.4034	1.52713
$323$	−10.5278	−0.585783
$324$	0	0
$325$	0	0
$326$	19.7235	1.09238
$327$	0	0
$328$	−3.65279	−0.201692
$329$	23.4034	1.29027
$330$	0	0
$331$	−6.13275	−0.337087	−0.168543	−	0.985694i	$-0.553906\pi$
$331$	−6.13275	−0.337087	−0.168543	+	0.985694i	$0.553906\pi$
$332$	2.93900	0.161299
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−9.85384	−0.538373
$336$	0	0
$337$	−27.8485	−1.51700	−0.758501	−	0.651672i	$-0.774068\pi$
$337$	−27.8485	−1.51700	−0.758501	+	0.651672i	$0.774068\pi$
$338$	0	0
$339$	0	0
$340$	−3.18837	−0.172914
$341$	6.45042	0.349310
$342$	0	0
$343$	27.8431	1.50339
$344$	0.170915	0.00921513
$345$	0	0
$346$	−1.20775	−0.0649291
$347$	−4.43967	−0.238334	−0.119167	−	0.992874i	$-0.538022\pi$
$347$	−4.43967	−0.238334	−0.119167	+	0.992874i	$0.538022\pi$
$348$	0	0
$349$	19.9215	1.06638	0.533188	−	0.845997i	$-0.320994\pi$
$349$	19.9215	1.06638	0.533188	+	0.845997i	$0.320994\pi$
$350$	−18.9095	−1.01075
$351$	0	0
$352$	2.69202	0.143485
$353$	−30.5894	−1.62811	−0.814055	−	0.580788i	$-0.802744\pi$
$353$	−30.5894	−1.62811	−0.814055	+	0.580788i	$0.802744\pi$
$354$	0	0
$355$	−8.98792	−0.477029
$356$	12.1806	0.645571
$357$	0	0
$358$	−16.5157	−0.872883
$359$	−21.6039	−1.14021	−0.570104	−	0.821572i	$-0.693097\pi$
$359$	−21.6039	−1.14021	−0.570104	+	0.821572i	$0.693097\pi$
$360$	0	0
$361$	−10.3623	−0.545383
$362$	6.37196	0.334903
$363$	0	0
$364$	0	0
$365$	−9.74525	−0.510090
$366$	0	0
$367$	18.7681	0.979686	0.489843	−	0.871811i	$-0.337054\pi$
$367$	18.7681	0.979686	0.489843	+	0.871811i	$0.337054\pi$
$368$	6.09783	0.317872
$369$	0	0
$370$	1.34050	0.0696894
$371$	−27.4034	−1.42271
$372$	0	0
$373$	−9.42758	−0.488142	−0.244071	−	0.969757i	$-0.578483\pi$
$373$	−9.42758	−0.488142	−0.244071	+	0.969757i	$0.578483\pi$
$374$	−9.64310	−0.498633
$375$	0	0
$376$	5.20775	0.268569
$377$	0	0
$378$	0	0
$379$	−32.0103	−1.64426	−0.822129	−	0.569302i	$-0.807214\pi$
$379$	−32.0103	−1.64426	−0.822129	+	0.569302i	$0.807214\pi$
$380$	2.61596	0.134196
$381$	0	0
$382$	2.49396	0.127602
$383$	15.9517	0.815092	0.407546	−	0.913185i	$-0.366385\pi$
$383$	15.9517	0.815092	0.407546	+	0.913185i	$0.366385\pi$
$384$	0	0
$385$	10.7681	0.548792
$386$	4.00538	0.203868
$387$	0	0
$388$	−12.9661	−0.658256
$389$	18.8659	0.956540	0.478270	−	0.878213i	$-0.341264\pi$
$389$	18.8659	0.956540	0.478270	+	0.878213i	$0.341264\pi$
$390$	0	0
$391$	−21.8431	−1.10465
$392$	13.1957	0.666482
$393$	0	0
$394$	23.6340	1.19066
$395$	2.50258	0.125919
$396$	0	0
$397$	−37.6969	−1.89195	−0.945977	−	0.324233i	$-0.894894\pi$
$397$	−37.6969	−1.89195	−0.945977	+	0.324233i	$0.894894\pi$
$398$	16.5375	0.828950
$399$	0	0
$400$	−4.20775	−0.210388
$401$	−3.22952	−0.161275	−0.0806373	−	0.996744i	$-0.525696\pi$
$401$	−3.22952	−0.161275	−0.0806373	+	0.996744i	$0.525696\pi$
$402$	0	0
$403$	0	0
$404$	10.1957	0.507254
$405$	0	0
$406$	−13.4276	−0.666400
$407$	4.05429	0.200964
$408$	0	0
$409$	3.54527	0.175302	0.0876511	−	0.996151i	$-0.472064\pi$
$409$	3.54527	0.175302	0.0876511	+	0.996151i	$0.472064\pi$
$410$	−3.25129	−0.160570
$411$	0	0
$412$	10.6703	0.525686
$413$	13.7995	0.679031
$414$	0	0
$415$	2.61596	0.128412
$416$	0	0
$417$	0	0
$418$	7.91185	0.386981
$419$	14.4155	0.704243	0.352122	−	0.935954i	$-0.385460\pi$
$419$	14.4155	0.704243	0.352122	+	0.935954i	$0.385460\pi$
$420$	0	0
$421$	−10.0978	−0.492138	−0.246069	−	0.969252i	$-0.579139\pi$
$421$	−10.0978	−0.492138	−0.246069	+	0.969252i	$0.579139\pi$
$422$	−1.66056	−0.0808349
$423$	0	0
$424$	−6.09783	−0.296137
$425$	15.0726	0.731129
$426$	0	0
$427$	−62.8068	−3.03944
$428$	20.5623	0.993914
$429$	0	0
$430$	0.152129	0.00733630
$431$	−20.2198	−0.973955	−0.486978	−	0.873414i	$-0.661901\pi$
$431$	−20.2198	−0.973955	−0.486978	+	0.873414i	$0.661901\pi$
$432$	0	0
$433$	36.4849	1.75335	0.876675	−	0.481083i	$-0.159756\pi$
$433$	36.4849	1.75335	0.876675	+	0.481083i	$0.159756\pi$
$434$	10.7681	0.516885
$435$	0	0
$436$	−2.71379	−0.129967
$437$	17.9215	0.857304
$438$	0	0
$439$	−28.3612	−1.35361	−0.676803	−	0.736164i	$-0.736635\pi$
$439$	−28.3612	−1.35361	−0.676803	+	0.736164i	$0.736635\pi$
$440$	2.39612	0.114231
$441$	0	0
$442$	0	0
$443$	−2.80061	−0.133061	−0.0665305	−	0.997784i	$-0.521193\pi$
$443$	−2.80061	−0.133061	−0.0665305	+	0.997784i	$0.521193\pi$
$444$	0	0
$445$	10.8418	0.513949
$446$	0.792249	0.0375141
$447$	0	0
$448$	4.49396	0.212320
$449$	−13.2760	−0.626535	−0.313268	−	0.949665i	$-0.601424\pi$
$449$	−13.2760	−0.626535	−0.313268	+	0.949665i	$0.601424\pi$
$450$	0	0
$451$	−9.83340	−0.463037
$452$	−20.6504	−0.971313
$453$	0	0
$454$	−21.7603	−1.02126
$455$	0	0
$456$	0	0
$457$	13.6474	0.638399	0.319200	−	0.947688i	$-0.396586\pi$
$457$	13.6474	0.638399	0.319200	+	0.947688i	$0.396586\pi$
$458$	1.97584	0.0923248
$459$	0	0
$460$	5.42758	0.253062
$461$	12.1655	0.566606	0.283303	−	0.959031i	$-0.408570\pi$
$461$	12.1655	0.566606	0.283303	+	0.959031i	$0.408570\pi$
$462$	0	0
$463$	4.24996	0.197513	0.0987563	−	0.995112i	$-0.468514\pi$
$463$	4.24996	0.197513	0.0987563	+	0.995112i	$0.468514\pi$
$464$	−2.98792	−0.138711
$465$	0	0
$466$	−18.2349	−0.844715
$467$	−8.21552	−0.380169	−0.190084	−	0.981768i	$-0.560876\pi$
$467$	−8.21552	−0.380169	−0.190084	+	0.981768i	$0.560876\pi$
$468$	0	0
$469$	−49.7512	−2.29730
$470$	4.63533	0.213812
$471$	0	0
$472$	3.07069	0.141340
$473$	0.460107	0.0211558
$474$	0	0
$475$	−12.3666	−0.567418
$476$	−16.0978	−0.737843
$477$	0	0
$478$	−22.0737	−1.00963
$479$	31.0267	1.41764	0.708822	−	0.705387i	$-0.249227\pi$
$479$	31.0267	1.41764	0.708822	+	0.705387i	$0.249227\pi$
$480$	0	0
$481$	0	0
$482$	−6.98792	−0.318291
$483$	0	0
$484$	−3.75302	−0.170592
$485$	−11.5410	−0.524048
$486$	0	0
$487$	10.9987	0.498397	0.249199	−	0.968452i	$-0.419833\pi$
$487$	10.9987	0.498397	0.249199	+	0.968452i	$0.419833\pi$
$488$	−13.9758	−0.632656
$489$	0	0
$490$	11.7453	0.530596
$491$	−21.2336	−0.958258	−0.479129	−	0.877745i	$-0.659047\pi$
$491$	−21.2336	−0.958258	−0.479129	+	0.877745i	$0.659047\pi$
$492$	0	0
$493$	10.7030	0.482041
$494$	0	0
$495$	0	0
$496$	2.39612	0.107589
$497$	−45.3793	−2.03554
$498$	0	0
$499$	16.8635	0.754915	0.377458	−	0.926027i	$-0.376798\pi$
$499$	16.8635	0.754915	0.377458	+	0.926027i	$0.376798\pi$
$500$	−8.19567	−0.366521
$501$	0	0
$502$	12.5593	0.560548
$503$	−20.3806	−0.908725	−0.454363	−	0.890817i	$-0.650133\pi$
$503$	−20.3806	−0.908725	−0.454363	+	0.890817i	$0.650133\pi$
$504$	0	0
$505$	9.07500	0.403832
$506$	16.4155	0.729758
$507$	0	0
$508$	12.7681	0.566492
$509$	21.9215	0.971655	0.485828	−	0.874055i	$-0.338518\pi$
$509$	21.9215	0.971655	0.485828	+	0.874055i	$0.338518\pi$
$510$	0	0
$511$	−49.2030	−2.17661
$512$	1.00000	0.0441942
$513$	0	0
$514$	16.9933	0.749542
$515$	9.49742	0.418506
$516$	0	0
$517$	14.0194	0.616572
$518$	6.76809	0.297373
$519$	0	0
$520$	0	0
$521$	−26.1564	−1.14593	−0.572967	−	0.819578i	$-0.694208\pi$
$521$	−26.1564	−1.14593	−0.572967	+	0.819578i	$0.694208\pi$
$522$	0	0
$523$	4.91185	0.214780	0.107390	−	0.994217i	$-0.465751\pi$
$523$	4.91185	0.214780	0.107390	+	0.994217i	$0.465751\pi$
$524$	5.15883	0.225365
$525$	0	0
$526$	4.39612	0.191680
$527$	−8.58317	−0.373889
$528$	0	0
$529$	14.1836	0.616678
$530$	−5.42758	−0.235759
$531$	0	0
$532$	13.2078	0.572629
$533$	0	0
$534$	0	0
$535$	18.3021	0.791270
$536$	−11.0707	−0.478181
$537$	0	0
$538$	−15.5603	−0.670854
$539$	35.5230	1.53009
$540$	0	0
$541$	−0.459042	−0.0197358	−0.00986789	−	0.999951i	$-0.503141\pi$
$541$	−0.459042	−0.0197358	−0.00986789	+	0.999951i	$0.503141\pi$
$542$	21.9952	0.944775
$543$	0	0
$544$	−3.58211	−0.153581
$545$	−2.41550	−0.103469
$546$	0	0
$547$	−20.3327	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$547$	−20.3327	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$548$	−3.30127	−0.141023
$549$	0	0
$550$	−11.3274	−0.483000
$551$	−8.78150	−0.374104
$552$	0	0
$553$	12.6353	0.537309
$554$	−1.87800	−0.0797887
$555$	0	0
$556$	−6.49157	−0.275304
$557$	−43.9469	−1.86209	−0.931045	−	0.364905i	$-0.881101\pi$
$557$	−43.9469	−1.86209	−0.931045	+	0.364905i	$0.881101\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	9.20536	0.388305
$563$	−43.5991	−1.83748	−0.918741	−	0.394860i	$-0.870793\pi$
$563$	−43.5991	−1.83748	−0.918741	+	0.394860i	$0.870793\pi$
$564$	0	0
$565$	−18.3806	−0.773277
$566$	4.70841	0.197909
$567$	0	0
$568$	−10.0978	−0.423696
$569$	−8.28919	−0.347501	−0.173751	−	0.984790i	$-0.555589\pi$
$569$	−8.28919	−0.347501	−0.173751	+	0.984790i	$0.555589\pi$
$570$	0	0
$571$	−22.9836	−0.961834	−0.480917	−	0.876766i	$-0.659696\pi$
$571$	−22.9836	−0.961834	−0.480917	+	0.876766i	$0.659696\pi$
$572$	0	0
$573$	0	0
$574$	−16.4155	−0.685170
$575$	−25.6582	−1.07002
$576$	0	0
$577$	−0.553630	−0.0230479	−0.0115240	−	0.999934i	$-0.503668\pi$
$577$	−0.553630	−0.0230479	−0.0115240	+	0.999934i	$0.503668\pi$
$578$	−4.16852	−0.173388
$579$	0	0
$580$	−2.65950	−0.110430
$581$	13.2078	0.547950
$582$	0	0
$583$	−16.4155	−0.679861
$584$	−10.9487	−0.453060
$585$	0	0
$586$	−13.7017	−0.566012
$587$	−16.0355	−0.661856	−0.330928	−	0.943656i	$-0.607362\pi$
$587$	−16.0355	−0.661856	−0.330928	+	0.943656i	$0.607362\pi$
$588$	0	0
$589$	7.04221	0.290169
$590$	2.73317	0.112523
$591$	0	0
$592$	1.50604	0.0618979
$593$	25.9976	1.06759	0.533797	−	0.845613i	$-0.320765\pi$
$593$	25.9976	1.06759	0.533797	+	0.845613i	$0.320765\pi$
$594$	0	0
$595$	−14.3284	−0.587408
$596$	5.03146	0.206097
$597$	0	0
$598$	0	0
$599$	16.2150	0.662529	0.331264	−	0.943538i	$-0.392525\pi$
$599$	16.2150	0.662529	0.331264	+	0.943538i	$0.392525\pi$
$600$	0	0
$601$	−29.5200	−1.20415	−0.602074	−	0.798440i	$-0.705659\pi$
$601$	−29.5200	−1.20415	−0.602074	+	0.798440i	$0.705659\pi$
$602$	0.768086	0.0313048
$603$	0	0
$604$	1.72587	0.0702248
$605$	−3.34050	−0.135811
$606$	0	0
$607$	37.4228	1.51894	0.759472	−	0.650540i	$-0.225457\pi$
$607$	37.4228	1.51894	0.759472	+	0.650540i	$0.225457\pi$
$608$	2.93900	0.119192
$609$	0	0
$610$	−12.4397	−0.503667
$611$	0	0
$612$	0	0
$613$	38.0737	1.53778	0.768891	−	0.639380i	$-0.220809\pi$
$613$	38.0737	1.53778	0.768891	+	0.639380i	$0.220809\pi$
$614$	−8.03252	−0.324166
$615$	0	0
$616$	12.0978	0.487436
$617$	−41.6383	−1.67630	−0.838148	−	0.545443i	$-0.816361\pi$
$617$	−41.6383	−1.67630	−0.838148	+	0.545443i	$0.816361\pi$
$618$	0	0
$619$	−1.67158	−0.0671864	−0.0335932	−	0.999436i	$-0.510695\pi$
$619$	−1.67158	−0.0671864	−0.0335932	+	0.999436i	$0.510695\pi$
$620$	2.13275	0.0856534
$621$	0	0
$622$	−4.09783	−0.164308
$623$	54.7391	2.19308
$624$	0	0
$625$	13.7439	0.549757
$626$	4.37435	0.174834
$627$	0	0
$628$	15.5060	0.618758
$629$	−5.39480	−0.215105
$630$	0	0
$631$	8.70304	0.346462	0.173231	−	0.984881i	$-0.444579\pi$
$631$	8.70304	0.346462	0.173231	+	0.984881i	$0.444579\pi$
$632$	2.81163	0.111840
$633$	0	0
$634$	−20.3612	−0.808647
$635$	11.3647	0.450993
$636$	0	0
$637$	0	0
$638$	−8.04354	−0.318447
$639$	0	0
$640$	0.890084	0.0351836
$641$	19.9075	0.786301	0.393150	−	0.919474i	$-0.371385\pi$
$641$	19.9075	0.786301	0.393150	+	0.919474i	$0.371385\pi$
$642$	0	0
$643$	27.0756	1.06776	0.533879	−	0.845561i	$-0.320734\pi$
$643$	27.0756	1.06776	0.533879	+	0.845561i	$0.320734\pi$
$644$	27.4034	1.07985
$645$	0	0
$646$	−10.5278	−0.414211
$647$	−9.43237	−0.370825	−0.185412	−	0.982661i	$-0.559362\pi$
$647$	−9.43237	−0.370825	−0.185412	+	0.982661i	$0.559362\pi$
$648$	0	0
$649$	8.26636	0.324483
$650$	0	0
$651$	0	0
$652$	19.7235	0.772431
$653$	37.6292	1.47255	0.736273	−	0.676685i	$-0.236584\pi$
$653$	37.6292	1.47255	0.736273	+	0.676685i	$0.236584\pi$
$654$	0	0
$655$	4.59179	0.179416
$656$	−3.65279	−0.142618
$657$	0	0
$658$	23.4034	0.912360
$659$	32.7724	1.27663	0.638316	−	0.769775i	$-0.279631\pi$
$659$	32.7724	1.27663	0.638316	+	0.769775i	$0.279631\pi$
$660$	0	0
$661$	−20.1957	−0.785520	−0.392760	−	0.919641i	$-0.628480\pi$
$661$	−20.1957	−0.785520	−0.392760	+	0.919641i	$0.628480\pi$
$662$	−6.13275	−0.238356
$663$	0	0
$664$	2.93900	0.114055
$665$	11.7560	0.455878
$666$	0	0
$667$	−18.2198	−0.705475
$668$	−14.0000	−0.541676
$669$	0	0
$670$	−9.85384	−0.380687
$671$	−37.6233	−1.45243
$672$	0	0
$673$	−30.3435	−1.16966	−0.584828	−	0.811158i	$-0.698838\pi$
$673$	−30.3435	−1.16966	−0.584828	+	0.811158i	$0.698838\pi$
$674$	−27.8485	−1.07268
$675$	0	0
$676$	0	0
$677$	21.1642	0.813407	0.406703	−	0.913560i	$-0.366678\pi$
$677$	21.1642	0.813407	0.406703	+	0.913560i	$0.366678\pi$
$678$	0	0
$679$	−58.2693	−2.23617
$680$	−3.18837	−0.122269
$681$	0	0
$682$	6.45042	0.246999
$683$	35.4873	1.35788	0.678941	−	0.734193i	$-0.262439\pi$
$683$	35.4873	1.35788	0.678941	+	0.734193i	$0.262439\pi$
$684$	0	0
$685$	−2.93841	−0.112271
$686$	27.8431	1.06305
$687$	0	0
$688$	0.170915	0.00651608
$689$	0	0
$690$	0	0
$691$	36.7472	1.39793	0.698964	−	0.715157i	$-0.253645\pi$
$691$	36.7472	1.39793	0.698964	+	0.715157i	$0.253645\pi$
$692$	−1.20775	−0.0459118
$693$	0	0
$694$	−4.43967	−0.168527
$695$	−5.77804	−0.219173
$696$	0	0
$697$	13.0847	0.495618
$698$	19.9215	0.754042
$699$	0	0
$700$	−18.9095	−0.714710
$701$	−19.7668	−0.746580	−0.373290	−	0.927715i	$-0.621770\pi$
$701$	−19.7668	−0.746580	−0.373290	+	0.927715i	$0.621770\pi$
$702$	0	0
$703$	4.42626	0.166939
$704$	2.69202	0.101459
$705$	0	0
$706$	−30.5894	−1.15125
$707$	45.8189	1.72320
$708$	0	0
$709$	11.6280	0.436700	0.218350	−	0.975871i	$-0.429933\pi$
$709$	11.6280	0.436700	0.218350	+	0.975871i	$0.429933\pi$
$710$	−8.98792	−0.337311
$711$	0	0
$712$	12.1806	0.456487
$713$	14.6112	0.547193
$714$	0	0
$715$	0	0
$716$	−16.5157	−0.617222
$717$	0	0
$718$	−21.6039	−0.806249
$719$	1.06638	0.0397691	0.0198846	−	0.999802i	$-0.493670\pi$
$719$	1.06638	0.0397691	0.0198846	+	0.999802i	$0.493670\pi$
$720$	0	0
$721$	47.9517	1.78581
$722$	−10.3623	−0.385644
$723$	0	0
$724$	6.37196	0.236812
$725$	12.5724	0.466928
$726$	0	0
$727$	−1.26205	−0.0468067	−0.0234033	−	0.999726i	$-0.507450\pi$
$727$	−1.26205	−0.0468067	−0.0234033	+	0.999726i	$0.507450\pi$
$728$	0	0
$729$	0	0
$730$	−9.74525	−0.360688
$731$	−0.612236	−0.0226444
$732$	0	0
$733$	−20.6789	−0.763792	−0.381896	−	0.924205i	$-0.624729\pi$
$733$	−20.6789	−0.763792	−0.381896	+	0.924205i	$0.624729\pi$
$734$	18.7681	0.692743
$735$	0	0
$736$	6.09783	0.224769
$737$	−29.8025	−1.09779
$738$	0	0
$739$	5.17331	0.190303	0.0951516	−	0.995463i	$-0.469666\pi$
$739$	5.17331	0.190303	0.0951516	+	0.995463i	$0.469666\pi$
$740$	1.34050	0.0492778
$741$	0	0
$742$	−27.4034	−1.00601
$743$	10.4397	0.382994	0.191497	−	0.981493i	$-0.438666\pi$
$743$	10.4397	0.382994	0.191497	+	0.981493i	$0.438666\pi$
$744$	0	0
$745$	4.47842	0.164077
$746$	−9.42758	−0.345168
$747$	0	0
$748$	−9.64310	−0.352587
$749$	92.4059	3.37644
$750$	0	0
$751$	−14.0435	−0.512456	−0.256228	−	0.966616i	$-0.582480\pi$
$751$	−14.0435	−0.512456	−0.256228	+	0.966616i	$0.582480\pi$
$752$	5.20775	0.189907
$753$	0	0
$754$	0	0
$755$	1.53617	0.0559070
$756$	0	0
$757$	−9.30559	−0.338217	−0.169109	−	0.985597i	$-0.554089\pi$
$757$	−9.30559	−0.338217	−0.169109	+	0.985597i	$0.554089\pi$
$758$	−32.0103	−1.16267
$759$	0	0
$760$	2.61596	0.0948907
$761$	9.56273	0.346649	0.173324	−	0.984865i	$-0.444549\pi$
$761$	9.56273	0.346649	0.173324	+	0.984865i	$0.444549\pi$
$762$	0	0
$763$	−12.1957	−0.441513
$764$	2.49396	0.0902283
$765$	0	0
$766$	15.9517	0.576357
$767$	0	0
$768$	0	0
$769$	−31.9299	−1.15142	−0.575711	−	0.817653i	$-0.695275\pi$
$769$	−31.9299	−1.15142	−0.575711	+	0.817653i	$0.695275\pi$
$770$	10.7681	0.388055
$771$	0	0
$772$	4.00538	0.144157
$773$	34.1172	1.22711	0.613555	−	0.789652i	$-0.289739\pi$
$773$	34.1172	1.22711	0.613555	+	0.789652i	$0.289739\pi$
$774$	0	0
$775$	−10.0823	−0.362167
$776$	−12.9661	−0.465458
$777$	0	0
$778$	18.8659	0.676376
$779$	−10.7356	−0.384641
$780$	0	0
$781$	−27.1836	−0.972705
$782$	−21.8431	−0.781107
$783$	0	0
$784$	13.1957	0.471274
$785$	13.8017	0.492603
$786$	0	0
$787$	−37.3467	−1.33127	−0.665634	−	0.746279i	$-0.731839\pi$
$787$	−37.3467	−1.33127	−0.665634	+	0.746279i	$0.731839\pi$
$788$	23.6340	0.841927
$789$	0	0
$790$	2.50258	0.0890379
$791$	−92.8021	−3.29966
$792$	0	0
$793$	0	0
$794$	−37.6969	−1.33781
$795$	0	0
$796$	16.5375	0.586156
$797$	14.9831	0.530730	0.265365	−	0.964148i	$-0.414508\pi$
$797$	14.9831	0.530730	0.265365	+	0.964148i	$0.414508\pi$
$798$	0	0
$799$	−18.6547	−0.659956
$800$	−4.20775	−0.148766
$801$	0	0
$802$	−3.22952	−0.114038
$803$	−29.4741	−1.04012
$804$	0	0
$805$	24.3913	0.859682
$806$	0	0
$807$	0	0
$808$	10.1957	0.358682
$809$	−25.6770	−0.902754	−0.451377	−	0.892333i	$-0.649067\pi$
$809$	−25.6770	−0.902754	−0.451377	+	0.892333i	$0.649067\pi$
$810$	0	0
$811$	8.66786	0.304370	0.152185	−	0.988352i	$-0.451369\pi$
$811$	8.66786	0.304370	0.152185	+	0.988352i	$0.451369\pi$
$812$	−13.4276	−0.471216
$813$	0	0
$814$	4.05429	0.142103
$815$	17.5555	0.614944
$816$	0	0
$817$	0.502320	0.0175739
$818$	3.54527	0.123957
$819$	0	0
$820$	−3.25129	−0.113540
$821$	54.6547	1.90746	0.953731	−	0.300660i	$-0.0972070\pi$
$821$	54.6547	1.90746	0.953731	+	0.300660i	$0.0972070\pi$
$822$	0	0
$823$	−6.33704	−0.220895	−0.110448	−	0.993882i	$-0.535228\pi$
$823$	−6.33704	−0.220895	−0.110448	+	0.993882i	$0.535228\pi$
$824$	10.6703	0.371716
$825$	0	0
$826$	13.7995	0.480148
$827$	−35.5405	−1.23586	−0.617932	−	0.786232i	$-0.712029\pi$
$827$	−35.5405	−1.23586	−0.617932	+	0.786232i	$0.712029\pi$
$828$	0	0
$829$	−41.5555	−1.44328	−0.721642	−	0.692267i	$-0.756612\pi$
$829$	−41.5555	−1.44328	−0.721642	+	0.692267i	$0.756612\pi$
$830$	2.61596	0.0908012
$831$	0	0
$832$	0	0
$833$	−47.2683	−1.63775
$834$	0	0
$835$	−12.4612	−0.431237
$836$	7.91185	0.273637
$837$	0	0
$838$	14.4155	0.497975
$839$	−17.5496	−0.605879	−0.302939	−	0.953010i	$-0.597968\pi$
$839$	−17.5496	−0.605879	−0.302939	+	0.953010i	$0.597968\pi$
$840$	0	0
$841$	−20.0723	−0.692150
$842$	−10.0978	−0.347994
$843$	0	0
$844$	−1.66056	−0.0571589
$845$	0	0
$846$	0	0
$847$	−16.8659	−0.579520
$848$	−6.09783	−0.209401
$849$	0	0
$850$	15.0726	0.516986
$851$	9.18359	0.314809
$852$	0	0
$853$	20.1414	0.689628	0.344814	−	0.938671i	$-0.387942\pi$
$853$	20.1414	0.689628	0.344814	+	0.938671i	$0.387942\pi$
$854$	−62.8068	−2.14921
$855$	0	0
$856$	20.5623	0.702803
$857$	8.85756	0.302568	0.151284	−	0.988490i	$-0.451659\pi$
$857$	8.85756	0.302568	0.151284	+	0.988490i	$0.451659\pi$
$858$	0	0
$859$	28.8810	0.985407	0.492703	−	0.870197i	$-0.336009\pi$
$859$	28.8810	0.985407	0.492703	+	0.870197i	$0.336009\pi$
$860$	0.152129	0.00518755
$861$	0	0
$862$	−20.2198	−0.688690
$863$	−3.90813	−0.133034	−0.0665172	−	0.997785i	$-0.521189\pi$
$863$	−3.90813	−0.133034	−0.0665172	+	0.997785i	$0.521189\pi$
$864$	0	0
$865$	−1.07500	−0.0365511
$866$	36.4849	1.23981
$867$	0	0
$868$	10.7681	0.365493
$869$	7.56896	0.256759
$870$	0	0
$871$	0	0
$872$	−2.71379	−0.0919006
$873$	0	0
$874$	17.9215	0.606205
$875$	−36.8310	−1.24512
$876$	0	0
$877$	−44.9879	−1.51913	−0.759567	−	0.650429i	$-0.774589\pi$
$877$	−44.9879	−1.51913	−0.759567	+	0.650429i	$0.774589\pi$
$878$	−28.3612	−0.957144
$879$	0	0
$880$	2.39612	0.0807733
$881$	14.1933	0.478184	0.239092	−	0.970997i	$-0.423150\pi$
$881$	14.1933	0.478184	0.239092	+	0.970997i	$0.423150\pi$
$882$	0	0
$883$	48.4626	1.63090	0.815448	−	0.578830i	$-0.196490\pi$
$883$	48.4626	1.63090	0.815448	+	0.578830i	$0.196490\pi$
$884$	0	0
$885$	0	0
$886$	−2.80061	−0.0940883
$887$	−30.8611	−1.03622	−0.518108	−	0.855315i	$-0.673363\pi$
$887$	−30.8611	−1.03622	−0.518108	+	0.855315i	$0.673363\pi$
$888$	0	0
$889$	57.3793	1.92444
$890$	10.8418	0.363417
$891$	0	0
$892$	0.792249	0.0265265
$893$	15.3056	0.512182
$894$	0	0
$895$	−14.7004	−0.491380
$896$	4.49396	0.150133
$897$	0	0
$898$	−13.2760	−0.443027
$899$	−7.15942	−0.238780
$900$	0	0
$901$	21.8431	0.727699
$902$	−9.83340	−0.327416
$903$	0	0
$904$	−20.6504	−0.686822
$905$	5.67158	0.188530
$906$	0	0
$907$	3.94139	0.130872	0.0654359	−	0.997857i	$-0.479156\pi$
$907$	3.94139	0.130872	0.0654359	+	0.997857i	$0.479156\pi$
$908$	−21.7603	−0.722141
$909$	0	0
$910$	0	0
$911$	37.1943	1.23230	0.616152	−	0.787627i	$-0.288691\pi$
$911$	37.1943	1.23230	0.616152	+	0.787627i	$0.288691\pi$
$912$	0	0
$913$	7.91185	0.261844
$914$	13.6474	0.451416
$915$	0	0
$916$	1.97584	0.0652835
$917$	23.1836	0.765590
$918$	0	0
$919$	0.681005	0.0224643	0.0112321	−	0.999937i	$-0.496425\pi$
$919$	0.681005	0.0224643	0.0112321	+	0.999937i	$0.496425\pi$
$920$	5.42758	0.178942
$921$	0	0
$922$	12.1655	0.400651
$923$	0	0
$924$	0	0
$925$	−6.33704	−0.208361
$926$	4.24996	0.139662
$927$	0	0
$928$	−2.98792	−0.0980832
$929$	−32.4355	−1.06417	−0.532087	−	0.846690i	$-0.678592\pi$
$929$	−32.4355	−1.06417	−0.532087	+	0.846690i	$0.678592\pi$
$930$	0	0
$931$	38.7821	1.27103
$932$	−18.2349	−0.597304
$933$	0	0
$934$	−8.21552	−0.268820
$935$	−8.58317	−0.280700
$936$	0	0
$937$	−14.6165	−0.477502	−0.238751	−	0.971081i	$-0.576738\pi$
$937$	−14.6165	−0.477502	−0.238751	+	0.971081i	$0.576738\pi$
$938$	−49.7512	−1.62443
$939$	0	0
$940$	4.63533	0.151188
$941$	30.1763	0.983719	0.491860	−	0.870675i	$-0.336317\pi$
$941$	30.1763	0.983719	0.491860	+	0.870675i	$0.336317\pi$
$942$	0	0
$943$	−22.2741	−0.725345
$944$	3.07069	0.0999424
$945$	0	0
$946$	0.460107	0.0149594
$947$	20.6708	0.671712	0.335856	−	0.941913i	$-0.390974\pi$
$947$	20.6708	0.671712	0.335856	+	0.941913i	$0.390974\pi$
$948$	0	0
$949$	0	0
$950$	−12.3666	−0.401225
$951$	0	0
$952$	−16.0978	−0.521734
$953$	47.2411	1.53029	0.765145	−	0.643858i	$-0.222667\pi$
$953$	47.2411	1.53029	0.765145	+	0.643858i	$0.222667\pi$
$954$	0	0
$955$	2.21983	0.0718321
$956$	−22.0737	−0.713914
$957$	0	0
$958$	31.0267	1.00243
$959$	−14.8358	−0.479073
$960$	0	0
$961$	−25.2586	−0.814793
$962$	0	0
$963$	0	0
$964$	−6.98792	−0.225066
$965$	3.56512	0.114765
$966$	0	0
$967$	−53.9517	−1.73497	−0.867484	−	0.497464i	$-0.834265\pi$
$967$	−53.9517	−1.73497	−0.867484	+	0.497464i	$0.834265\pi$
$968$	−3.75302	−0.120627
$969$	0	0
$970$	−11.5410	−0.370558
$971$	−49.8920	−1.60111	−0.800555	−	0.599259i	$-0.795462\pi$
$971$	−49.8920	−1.60111	−0.800555	+	0.599259i	$0.795462\pi$
$972$	0	0
$973$	−29.1728	−0.935238
$974$	10.9987	0.352420
$975$	0	0
$976$	−13.9758	−0.447356
$977$	29.1299	0.931948	0.465974	−	0.884799i	$-0.345704\pi$
$977$	29.1299	0.931948	0.465974	+	0.884799i	$0.345704\pi$
$978$	0	0
$979$	32.7904	1.04799
$980$	11.7453	0.375188
$981$	0	0
$982$	−21.2336	−0.677590
$983$	−41.3309	−1.31825	−0.659126	−	0.752033i	$-0.729074\pi$
$983$	−41.3309	−1.31825	−0.659126	+	0.752033i	$0.729074\pi$
$984$	0	0
$985$	21.0362	0.670270
$986$	10.7030	0.340854
$987$	0	0
$988$	0	0
$989$	1.04221	0.0331404
$990$	0	0
$991$	19.2185	0.610496	0.305248	−	0.952273i	$-0.401261\pi$
$991$	19.2185	0.610496	0.305248	+	0.952273i	$0.401261\pi$
$992$	2.39612	0.0760770
$993$	0	0
$994$	−45.3793	−1.43934
$995$	14.7198	0.466648
$996$	0	0
$997$	22.4940	0.712391	0.356195	−	0.934411i	$-0.384074\pi$
$997$	22.4940	0.712391	0.356195	+	0.934411i	$0.384074\pi$
$998$	16.8635	0.533806
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.bi.1.2		3
3.2	odd	2		338.2.a.g.1.2	✓	3
12.11	even	2		2704.2.a.v.1.2		3
13.5	odd	4		3042.2.b.n.1351.2		6
13.8	odd	4		3042.2.b.n.1351.5		6
13.12	even	2		3042.2.a.z.1.2		3
15.14	odd	2		8450.2.a.bx.1.2		3
39.2	even	12		338.2.e.e.147.2		12
39.5	even	4		338.2.b.d.337.5		6
39.8	even	4		338.2.b.d.337.2		6
39.11	even	12		338.2.e.e.147.5		12
39.17	odd	6		338.2.c.h.315.2		6
39.20	even	12		338.2.e.e.23.2		12
39.23	odd	6		338.2.c.h.191.2		6
39.29	odd	6		338.2.c.i.191.2		6
39.32	even	12		338.2.e.e.23.5		12
39.35	odd	6		338.2.c.i.315.2		6
39.38	odd	2		338.2.a.h.1.2	yes	3
156.47	odd	4		2704.2.f.m.337.3		6
156.83	odd	4		2704.2.f.m.337.4		6
156.155	even	2		2704.2.a.w.1.2		3
195.194	odd	2		8450.2.a.bn.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.2	✓	3	3.2	odd	2
338.2.a.h.1.2	yes	3	39.38	odd	2
338.2.b.d.337.2		6	39.8	even	4
338.2.b.d.337.5		6	39.5	even	4
338.2.c.h.191.2		6	39.23	odd	6
338.2.c.h.315.2		6	39.17	odd	6
338.2.c.i.191.2		6	39.29	odd	6
338.2.c.i.315.2		6	39.35	odd	6
338.2.e.e.23.2		12	39.20	even	12
338.2.e.e.23.5		12	39.32	even	12
338.2.e.e.147.2		12	39.2	even	12
338.2.e.e.147.5		12	39.11	even	12
2704.2.a.v.1.2		3	12.11	even	2
2704.2.a.w.1.2		3	156.155	even	2
2704.2.f.m.337.3		6	156.47	odd	4
2704.2.f.m.337.4		6	156.83	odd	4
3042.2.a.z.1.2		3	13.12	even	2
3042.2.a.bi.1.2		3	1.1	even	1	trivial
3042.2.b.n.1351.2		6	13.5	odd	4
3042.2.b.n.1351.5		6	13.8	odd	4
8450.2.a.bn.1.2		3	195.194	odd	2
8450.2.a.bx.1.2		3	15.14	odd	2

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.890084 q^{5} +4.49396 q^{7} +1.00000 q^{8} +0.890084 q^{10} +2.69202 q^{11} +4.49396 q^{14} +1.00000 q^{16} -3.58211 q^{17} +2.93900 q^{19} +0.890084 q^{20} +2.69202 q^{22} +6.09783 q^{23} -4.20775 q^{25} +4.49396 q^{28} -2.98792 q^{29} +2.39612 q^{31} +1.00000 q^{32} -3.58211 q^{34} +4.00000 q^{35} +1.50604 q^{37} +2.93900 q^{38} +0.890084 q^{40} -3.65279 q^{41} +0.170915 q^{43} +2.69202 q^{44} +6.09783 q^{46} +5.20775 q^{47} +13.1957 q^{49} -4.20775 q^{50} -6.09783 q^{53} +2.39612 q^{55} +4.49396 q^{56} -2.98792 q^{58} +3.07069 q^{59} -13.9758 q^{61} +2.39612 q^{62} +1.00000 q^{64} -11.0707 q^{67} -3.58211 q^{68} +4.00000 q^{70} -10.0978 q^{71} -10.9487 q^{73} +1.50604 q^{74} +2.93900 q^{76} +12.0978 q^{77} +2.81163 q^{79} +0.890084 q^{80} -3.65279 q^{82} +2.93900 q^{83} -3.18837 q^{85} +0.170915 q^{86} +2.69202 q^{88} +12.1806 q^{89} +6.09783 q^{92} +5.20775 q^{94} +2.61596 q^{95} -12.9661 q^{97} +13.1957 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} + 2 q^{10} + 3 q^{11} + 4 q^{14} + 3 q^{16} - 5 q^{17} - q^{19} + 2 q^{20} + 3 q^{22} + 5 q^{25} + 4 q^{28} + 10 q^{29} + 16 q^{31} + 3 q^{32} - 5 q^{34}+ \cdots + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 + 4 * q^7 + 3 * q^8 + 2 * q^10 + 3 * q^11 + 4 * q^14 + 3 * q^16 - 5 * q^17 - q^19 + 2 * q^20 + 3 * q^22 + 5 * q^25 + 4 * q^28 + 10 * q^29 + 16 * q^31 + 3 * q^32 - 5 * q^34 + 12 * q^35 + 14 * q^37 - q^38 + 2 * q^40 + 7 * q^41 + 11 * q^43 + 3 * q^44 - 2 * q^47 + 3 * q^49 + 5 * q^50 + 16 * q^55 + 4 * q^56 + 10 * q^58 - 3 * q^59 - 4 * q^61 + 16 * q^62 + 3 * q^64 - 21 * q^67 - 5 * q^68 + 12 * q^70 - 12 * q^71 - q^73 + 14 * q^74 - q^76 + 18 * q^77 - 18 * q^79 + 2 * q^80 + 7 * q^82 - q^83 - 36 * q^85 + 11 * q^86 + 3 * q^88 + 25 * q^89 - 2 * q^94 + 18 * q^95 - 23 * q^97 + 3 * q^98

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