LMFDB - Embedded newform 1148.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.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:	$9.16682615204$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.287349.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 7x - 1 $ x^5 - 6x^3 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.05679$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23037 q^{3} +1.70418 q^{5} -1.00000 q^{7} +1.97454 q^{9} +O(q^{10})$	$q+2.23037 q^{3} +1.70418 q^{5} -1.00000 q^{7} +1.97454 q^{9} +5.50514 q^{11} +0.699765 q^{13} +3.80096 q^{15} +0.729645 q^{17} -6.13252 q^{19} -2.23037 q^{21} +7.80323 q^{23} -2.09576 q^{25} -2.28715 q^{27} +8.42619 q^{29} -7.93133 q^{31} +12.2785 q^{33} -1.70418 q^{35} -6.83013 q^{37} +1.56073 q^{39} -1.00000 q^{41} +10.3316 q^{43} +3.36498 q^{45} +1.23252 q^{47} +1.00000 q^{49} +1.62738 q^{51} -2.32941 q^{53} +9.38178 q^{55} -13.6778 q^{57} +5.25229 q^{59} +9.64347 q^{61} -1.97454 q^{63} +1.19253 q^{65} -1.85780 q^{67} +17.4041 q^{69} -8.36181 q^{71} -15.7323 q^{73} -4.67430 q^{75} -5.50514 q^{77} +7.15002 q^{79} -11.0248 q^{81} -10.5663 q^{83} +1.24345 q^{85} +18.7935 q^{87} -2.20448 q^{89} -0.699765 q^{91} -17.6898 q^{93} -10.4509 q^{95} -9.07820 q^{97} +10.8701 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 3 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 3 * q^5 - 5 * q^7 + q^9	$ 5 q + 2 q^{3} - 3 q^{5} - 5 q^{7} + q^{9} + 6 q^{11} + 7 q^{13} + 9 q^{15} + q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 8 q^{27} + 11 q^{29} + 13 q^{31} + 11 q^{33} + 3 q^{35} + 5 q^{37} + 23 q^{39} - 5 q^{41} + 29 q^{43} + 11 q^{45} + 7 q^{47} + 5 q^{49} - 3 q^{51} + 21 q^{53} + 19 q^{55} + 9 q^{57} + 3 q^{59} - 8 q^{61} - q^{63} - 5 q^{65} + 3 q^{67} + 10 q^{69} + 22 q^{71} - 16 q^{73} - 18 q^{75} - 6 q^{77} + 4 q^{79} - 15 q^{81} - 6 q^{83} + 13 q^{85} + 6 q^{87} - 20 q^{89} - 7 q^{91} - 5 q^{93} - 7 q^{95} - 24 q^{97} + 27 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 3 * q^5 - 5 * q^7 + q^9 + 6 * q^11 + 7 * q^13 + 9 * q^15 + q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 + 8 * q^27 + 11 * q^29 + 13 * q^31 + 11 * q^33 + 3 * q^35 + 5 * q^37 + 23 * q^39 - 5 * q^41 + 29 * q^43 + 11 * q^45 + 7 * q^47 + 5 * q^49 - 3 * q^51 + 21 * q^53 + 19 * q^55 + 9 * q^57 + 3 * q^59 - 8 * q^61 - q^63 - 5 * q^65 + 3 * q^67 + 10 * q^69 + 22 * q^71 - 16 * q^73 - 18 * q^75 - 6 * q^77 + 4 * q^79 - 15 * q^81 - 6 * q^83 + 13 * q^85 + 6 * q^87 - 20 * q^89 - 7 * q^91 - 5 * q^93 - 7 * q^95 - 24 * q^97 + 27 * 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
$3$	2.23037	1.28770	0.643852	−	0.765150i	$-0.277335\pi$
$3$	2.23037	1.28770	0.643852	+	0.765150i	$0.277335\pi$
$4$	0	0
$5$	1.70418	0.762134	0.381067	−	0.924547i	$-0.375557\pi$
$5$	1.70418	0.762134	0.381067	+	0.924547i	$0.375557\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.97454	0.658180
$10$	0	0
$11$	5.50514	1.65986	0.829931	−	0.557865i	$-0.188379\pi$
$11$	5.50514	1.65986	0.829931	+	0.557865i	$0.188379\pi$
$12$	0	0
$13$	0.699765	0.194080	0.0970399	−	0.995280i	$-0.469063\pi$
$13$	0.699765	0.194080	0.0970399	+	0.995280i	$0.469063\pi$
$14$	0	0
$15$	3.80096	0.981403
$16$	0	0
$17$	0.729645	0.176965	0.0884824	−	0.996078i	$-0.471798\pi$
$17$	0.729645	0.176965	0.0884824	+	0.996078i	$0.471798\pi$
$18$	0	0
$19$	−6.13252	−1.40690	−0.703448	−	0.710747i	$-0.748357\pi$
$19$	−6.13252	−1.40690	−0.703448	+	0.710747i	$0.748357\pi$
$20$	0	0
$21$	−2.23037	−0.486706
$22$	0	0
$23$	7.80323	1.62709	0.813543	−	0.581505i	$-0.197536\pi$
$23$	7.80323	1.62709	0.813543	+	0.581505i	$0.197536\pi$
$24$	0	0
$25$	−2.09576	−0.419151
$26$	0	0
$27$	−2.28715	−0.440163
$28$	0	0
$29$	8.42619	1.56470	0.782352	−	0.622837i	$-0.214020\pi$
$29$	8.42619	1.56470	0.782352	+	0.622837i	$0.214020\pi$
$30$	0	0
$31$	−7.93133	−1.42451	−0.712254	−	0.701922i	$-0.752326\pi$
$31$	−7.93133	−1.42451	−0.712254	+	0.701922i	$0.752326\pi$
$32$	0	0
$33$	12.2785	2.13741
$34$	0	0
$35$	−1.70418	−0.288060
$36$	0	0
$37$	−6.83013	−1.12287	−0.561434	−	0.827522i	$-0.689750\pi$
$37$	−6.83013	−1.12287	−0.561434	+	0.827522i	$0.689750\pi$
$38$	0	0
$39$	1.56073	0.249917
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	10.3316	1.57555	0.787774	−	0.615964i	$-0.211233\pi$
$43$	10.3316	1.57555	0.787774	+	0.615964i	$0.211233\pi$
$44$	0	0
$45$	3.36498	0.501622
$46$	0	0
$47$	1.23252	0.179781	0.0898905	−	0.995952i	$-0.471348\pi$
$47$	1.23252	0.179781	0.0898905	+	0.995952i	$0.471348\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.62738	0.227878
$52$	0	0
$53$	−2.32941	−0.319969	−0.159985	−	0.987120i	$-0.551144\pi$
$53$	−2.32941	−0.319969	−0.159985	+	0.987120i	$0.551144\pi$
$54$	0	0
$55$	9.38178	1.26504
$56$	0	0
$57$	−13.6778	−1.81166
$58$	0	0
$59$	5.25229	0.683790	0.341895	−	0.939738i	$-0.388931\pi$
$59$	5.25229	0.683790	0.341895	+	0.939738i	$0.388931\pi$
$60$	0	0
$61$	9.64347	1.23472	0.617360	−	0.786681i	$-0.288202\pi$
$61$	9.64347	1.23472	0.617360	+	0.786681i	$0.288202\pi$
$62$	0	0
$63$	−1.97454	−0.248769
$64$	0	0
$65$	1.19253	0.147915
$66$	0	0
$67$	−1.85780	−0.226967	−0.113483	−	0.993540i	$-0.536201\pi$
$67$	−1.85780	−0.226967	−0.113483	+	0.993540i	$0.536201\pi$
$68$	0	0
$69$	17.4041	2.09520
$70$	0	0
$71$	−8.36181	−0.992364	−0.496182	−	0.868218i	$-0.665265\pi$
$71$	−8.36181	−0.992364	−0.496182	+	0.868218i	$0.665265\pi$
$72$	0	0
$73$	−15.7323	−1.84132	−0.920662	−	0.390360i	$-0.872351\pi$
$73$	−15.7323	−1.84132	−0.920662	+	0.390360i	$0.872351\pi$
$74$	0	0
$75$	−4.67430	−0.539742
$76$	0	0
$77$	−5.50514	−0.627369
$78$	0	0
$79$	7.15002	0.804440	0.402220	−	0.915543i	$-0.368239\pi$
$79$	7.15002	0.804440	0.402220	+	0.915543i	$0.368239\pi$
$80$	0	0
$81$	−11.0248	−1.22498
$82$	0	0
$83$	−10.5663	−1.15981	−0.579904	−	0.814685i	$-0.696910\pi$
$83$	−10.5663	−1.15981	−0.579904	+	0.814685i	$0.696910\pi$
$84$	0	0
$85$	1.24345	0.134871
$86$	0	0
$87$	18.7935	2.01487
$88$	0	0
$89$	−2.20448	−0.233674	−0.116837	−	0.993151i	$-0.537276\pi$
$89$	−2.20448	−0.233674	−0.116837	+	0.993151i	$0.537276\pi$
$90$	0	0
$91$	−0.699765	−0.0733553
$92$	0	0
$93$	−17.6898	−1.83434
$94$	0	0
$95$	−10.4509	−1.07224
$96$	0	0
$97$	−9.07820	−0.921751	−0.460876	−	0.887465i	$-0.652465\pi$
$97$	−9.07820	−0.921751	−0.460876	+	0.887465i	$0.652465\pi$
$98$	0	0
$99$	10.8701	1.09249
$100$	0	0
$101$	4.63180	0.460881	0.230440	−	0.973086i	$-0.425983\pi$
$101$	4.63180	0.460881	0.230440	+	0.973086i	$0.425983\pi$
$102$	0	0
$103$	13.9683	1.37634	0.688171	−	0.725549i	$-0.258414\pi$
$103$	13.9683	1.37634	0.688171	+	0.725549i	$0.258414\pi$
$104$	0	0
$105$	−3.80096	−0.370936
$106$	0	0
$107$	−16.8629	−1.63020	−0.815099	−	0.579321i	$-0.803318\pi$
$107$	−16.8629	−1.63020	−0.815099	+	0.579321i	$0.803318\pi$
$108$	0	0
$109$	−15.9608	−1.52877	−0.764384	−	0.644761i	$-0.776957\pi$
$109$	−15.9608	−1.52877	−0.764384	+	0.644761i	$0.776957\pi$
$110$	0	0
$111$	−15.2337	−1.44592
$112$	0	0
$113$	17.4244	1.63915	0.819575	−	0.572972i	$-0.194210\pi$
$113$	17.4244	1.63915	0.819575	+	0.572972i	$0.194210\pi$
$114$	0	0
$115$	13.2981	1.24006
$116$	0	0
$117$	1.38171	0.127739
$118$	0	0
$119$	−0.729645	−0.0668864
$120$	0	0
$121$	19.3066	1.75514
$122$	0	0
$123$	−2.23037	−0.201105
$124$	0	0
$125$	−12.0925	−1.08158
$126$	0	0
$127$	12.3526	1.09612	0.548058	−	0.836440i	$-0.315367\pi$
$127$	12.3526	1.09612	0.548058	+	0.836440i	$0.315367\pi$
$128$	0	0
$129$	23.0432	2.02884
$130$	0	0
$131$	−12.6009	−1.10095	−0.550473	−	0.834853i	$-0.685553\pi$
$131$	−12.6009	−1.10095	−0.550473	+	0.834853i	$0.685553\pi$
$132$	0	0
$133$	6.13252	0.531757
$134$	0	0
$135$	−3.89773	−0.335463
$136$	0	0
$137$	10.5318	0.899792	0.449896	−	0.893081i	$-0.351461\pi$
$137$	10.5318	0.899792	0.449896	+	0.893081i	$0.351461\pi$
$138$	0	0
$139$	−3.42517	−0.290519	−0.145259	−	0.989394i	$-0.546402\pi$
$139$	−3.42517	−0.290519	−0.145259	+	0.989394i	$0.546402\pi$
$140$	0	0
$141$	2.74896	0.231505
$142$	0	0
$143$	3.85231	0.322146
$144$	0	0
$145$	14.3598	1.19251
$146$	0	0
$147$	2.23037	0.183958
$148$	0	0
$149$	−6.33125	−0.518676	−0.259338	−	0.965787i	$-0.583504\pi$
$149$	−6.33125	−0.518676	−0.259338	+	0.965787i	$0.583504\pi$
$150$	0	0
$151$	0.390125	0.0317479	0.0158740	−	0.999874i	$-0.494947\pi$
$151$	0.390125	0.0317479	0.0158740	+	0.999874i	$0.494947\pi$
$152$	0	0
$153$	1.44071	0.116475
$154$	0	0
$155$	−13.5164	−1.08567
$156$	0	0
$157$	−7.26099	−0.579490	−0.289745	−	0.957104i	$-0.593571\pi$
$157$	−7.26099	−0.579490	−0.289745	+	0.957104i	$0.593571\pi$
$158$	0	0
$159$	−5.19544	−0.412026
$160$	0	0
$161$	−7.80323	−0.614981
$162$	0	0
$163$	−24.0599	−1.88451	−0.942257	−	0.334892i	$-0.891300\pi$
$163$	−24.0599	−1.88451	−0.942257	+	0.334892i	$0.891300\pi$
$164$	0	0
$165$	20.9248	1.62899
$166$	0	0
$167$	23.6826	1.83261	0.916306	−	0.400479i	$-0.131156\pi$
$167$	23.6826	1.83261	0.916306	+	0.400479i	$0.131156\pi$
$168$	0	0
$169$	−12.5103	−0.962333
$170$	0	0
$171$	−12.1089	−0.925991
$172$	0	0
$173$	6.74902	0.513118	0.256559	−	0.966529i	$-0.417411\pi$
$173$	6.74902	0.513118	0.256559	+	0.966529i	$0.417411\pi$
$174$	0	0
$175$	2.09576	0.158424
$176$	0	0
$177$	11.7145	0.880518
$178$	0	0
$179$	21.2756	1.59022	0.795108	−	0.606468i	$-0.207414\pi$
$179$	21.2756	1.59022	0.795108	+	0.606468i	$0.207414\pi$
$180$	0	0
$181$	−20.9915	−1.56028	−0.780141	−	0.625603i	$-0.784853\pi$
$181$	−20.9915	−1.56028	−0.780141	+	0.625603i	$0.784853\pi$
$182$	0	0
$183$	21.5085	1.58995
$184$	0	0
$185$	−11.6398	−0.855776
$186$	0	0
$187$	4.01680	0.293737
$188$	0	0
$189$	2.28715	0.166366
$190$	0	0
$191$	−13.0256	−0.942497	−0.471249	−	0.882000i	$-0.656197\pi$
$191$	−13.0256	−0.942497	−0.471249	+	0.882000i	$0.656197\pi$
$192$	0	0
$193$	−22.2178	−1.59927	−0.799635	−	0.600486i	$-0.794974\pi$
$193$	−22.2178	−1.59927	−0.799635	+	0.600486i	$0.794974\pi$
$194$	0	0
$195$	2.65978	0.190471
$196$	0	0
$197$	8.09462	0.576718	0.288359	−	0.957522i	$-0.406890\pi$
$197$	8.09462	0.576718	0.288359	+	0.957522i	$0.406890\pi$
$198$	0	0
$199$	9.40497	0.666700	0.333350	−	0.942803i	$-0.391821\pi$
$199$	9.40497	0.666700	0.333350	+	0.942803i	$0.391821\pi$
$200$	0	0
$201$	−4.14358	−0.292266
$202$	0	0
$203$	−8.42619	−0.591402
$204$	0	0
$205$	−1.70418	−0.119025
$206$	0	0
$207$	15.4078	1.07092
$208$	0	0
$209$	−33.7604	−2.33525
$210$	0	0
$211$	4.25715	0.293074	0.146537	−	0.989205i	$-0.453187\pi$
$211$	4.25715	0.293074	0.146537	+	0.989205i	$0.453187\pi$
$212$	0	0
$213$	−18.6499	−1.27787
$214$	0	0
$215$	17.6069	1.20078
$216$	0	0
$217$	7.93133	0.538414
$218$	0	0
$219$	−35.0888	−2.37108
$220$	0	0
$221$	0.510580	0.0343453
$222$	0	0
$223$	17.2509	1.15520	0.577602	−	0.816318i	$-0.303988\pi$
$223$	17.2509	1.15520	0.577602	+	0.816318i	$0.303988\pi$
$224$	0	0
$225$	−4.13815	−0.275877
$226$	0	0
$227$	−6.46161	−0.428872	−0.214436	−	0.976738i	$-0.568791\pi$
$227$	−6.46161	−0.428872	−0.214436	+	0.976738i	$0.568791\pi$
$228$	0	0
$229$	−5.89892	−0.389812	−0.194906	−	0.980822i	$-0.562440\pi$
$229$	−5.89892	−0.389812	−0.194906	+	0.980822i	$0.562440\pi$
$230$	0	0
$231$	−12.2785	−0.807865
$232$	0	0
$233$	−20.6024	−1.34971	−0.674854	−	0.737951i	$-0.735794\pi$
$233$	−20.6024	−1.34971	−0.674854	+	0.737951i	$0.735794\pi$
$234$	0	0
$235$	2.10044	0.137017
$236$	0	0
$237$	15.9472	1.03588
$238$	0	0
$239$	16.8221	1.08813	0.544066	−	0.839042i	$-0.316884\pi$
$239$	16.8221	1.08813	0.544066	+	0.839042i	$0.316884\pi$
$240$	0	0
$241$	−3.57501	−0.230287	−0.115143	−	0.993349i	$-0.536733\pi$
$241$	−3.57501	−0.230287	−0.115143	+	0.993349i	$0.536733\pi$
$242$	0	0
$243$	−17.7279	−1.13725
$244$	0	0
$245$	1.70418	0.108876
$246$	0	0
$247$	−4.29132	−0.273050
$248$	0	0
$249$	−23.5668	−1.49349
$250$	0	0
$251$	7.48782	0.472627	0.236313	−	0.971677i	$-0.424061\pi$
$251$	7.48782	0.472627	0.236313	+	0.971677i	$0.424061\pi$
$252$	0	0
$253$	42.9579	2.70074
$254$	0	0
$255$	2.77335	0.173674
$256$	0	0
$257$	−9.55972	−0.596319	−0.298159	−	0.954516i	$-0.596373\pi$
$257$	−9.55972	−0.596319	−0.298159	+	0.954516i	$0.596373\pi$
$258$	0	0
$259$	6.83013	0.424404
$260$	0	0
$261$	16.6378	1.02986
$262$	0	0
$263$	−1.94316	−0.119820	−0.0599101	−	0.998204i	$-0.519081\pi$
$263$	−1.94316	−0.119820	−0.0599101	+	0.998204i	$0.519081\pi$
$264$	0	0
$265$	−3.96975	−0.243860
$266$	0	0
$267$	−4.91680	−0.300903
$268$	0	0
$269$	−21.7865	−1.32835	−0.664174	−	0.747578i	$-0.731217\pi$
$269$	−21.7865	−1.32835	−0.664174	+	0.747578i	$0.731217\pi$
$270$	0	0
$271$	−11.7164	−0.711719	−0.355859	−	0.934540i	$-0.615812\pi$
$271$	−11.7164	−0.711719	−0.355859	+	0.934540i	$0.615812\pi$
$272$	0	0
$273$	−1.56073	−0.0944599
$274$	0	0
$275$	−11.5374	−0.695733
$276$	0	0
$277$	−17.8692	−1.07366	−0.536829	−	0.843691i	$-0.680378\pi$
$277$	−17.8692	−1.07366	−0.536829	+	0.843691i	$0.680378\pi$
$278$	0	0
$279$	−15.6607	−0.937583
$280$	0	0
$281$	10.1836	0.607500	0.303750	−	0.952752i	$-0.401761\pi$
$281$	10.1836	0.607500	0.303750	+	0.952752i	$0.401761\pi$
$282$	0	0
$283$	20.1981	1.20065	0.600326	−	0.799756i	$-0.295038\pi$
$283$	20.1981	1.20065	0.600326	+	0.799756i	$0.295038\pi$
$284$	0	0
$285$	−23.3094	−1.38073
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	−16.4676	−0.968683
$290$	0	0
$291$	−20.2477	−1.18694
$292$	0	0
$293$	5.10276	0.298106	0.149053	−	0.988829i	$-0.452377\pi$
$293$	5.10276	0.298106	0.149053	+	0.988829i	$0.452377\pi$
$294$	0	0
$295$	8.95087	0.521140
$296$	0	0
$297$	−12.5911	−0.730610
$298$	0	0
$299$	5.46043	0.315785
$300$	0	0
$301$	−10.3316	−0.595501
$302$	0	0
$303$	10.3306	0.593478
$304$	0	0
$305$	16.4343	0.941023
$306$	0	0
$307$	−5.41495	−0.309047	−0.154524	−	0.987989i	$-0.549384\pi$
$307$	−5.41495	−0.309047	−0.154524	+	0.987989i	$0.549384\pi$
$308$	0	0
$309$	31.1545	1.77232
$310$	0	0
$311$	−28.6772	−1.62614	−0.813069	−	0.582168i	$-0.802205\pi$
$311$	−28.6772	−1.62614	−0.813069	+	0.582168i	$0.802205\pi$
$312$	0	0
$313$	−2.73168	−0.154404	−0.0772019	−	0.997015i	$-0.524599\pi$
$313$	−2.73168	−0.154404	−0.0772019	+	0.997015i	$0.524599\pi$
$314$	0	0
$315$	−3.36498	−0.189595
$316$	0	0
$317$	−18.0840	−1.01570	−0.507849	−	0.861446i	$-0.669559\pi$
$317$	−18.0840	−1.01570	−0.507849	+	0.861446i	$0.669559\pi$
$318$	0	0
$319$	46.3873	2.59719
$320$	0	0
$321$	−37.6105	−2.09921
$322$	0	0
$323$	−4.47456	−0.248971
$324$	0	0
$325$	−1.46654	−0.0813488
$326$	0	0
$327$	−35.5985	−1.96860
$328$	0	0
$329$	−1.23252	−0.0679508
$330$	0	0
$331$	14.7877	0.812805	0.406402	−	0.913694i	$-0.366783\pi$
$331$	14.7877	0.812805	0.406402	+	0.913694i	$0.366783\pi$
$332$	0	0
$333$	−13.4864	−0.739049
$334$	0	0
$335$	−3.16604	−0.172979
$336$	0	0
$337$	−0.0683395	−0.00372269	−0.00186134	−	0.999998i	$-0.500592\pi$
$337$	−0.0683395	−0.00372269	−0.00186134	+	0.999998i	$0.500592\pi$
$338$	0	0
$339$	38.8628	2.11074
$340$	0	0
$341$	−43.6631	−2.36449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	29.6597	1.59683
$346$	0	0
$347$	−8.95256	−0.480599	−0.240299	−	0.970699i	$-0.577246\pi$
$347$	−8.95256	−0.480599	−0.240299	+	0.970699i	$0.577246\pi$
$348$	0	0
$349$	8.53275	0.456748	0.228374	−	0.973574i	$-0.426659\pi$
$349$	8.53275	0.456748	0.228374	+	0.973574i	$0.426659\pi$
$350$	0	0
$351$	−1.60047	−0.0854268
$352$	0	0
$353$	11.2642	0.599534	0.299767	−	0.954012i	$-0.403091\pi$
$353$	11.2642	0.599534	0.299767	+	0.954012i	$0.403091\pi$
$354$	0	0
$355$	−14.2501	−0.756315
$356$	0	0
$357$	−1.62738	−0.0861299
$358$	0	0
$359$	14.6037	0.770755	0.385378	−	0.922759i	$-0.374071\pi$
$359$	14.6037	0.770755	0.385378	+	0.922759i	$0.374071\pi$
$360$	0	0
$361$	18.6078	0.979357
$362$	0	0
$363$	43.0608	2.26011
$364$	0	0
$365$	−26.8107	−1.40334
$366$	0	0
$367$	29.8084	1.55599	0.777993	−	0.628273i	$-0.216238\pi$
$367$	29.8084	1.55599	0.777993	+	0.628273i	$0.216238\pi$
$368$	0	0
$369$	−1.97454	−0.102790
$370$	0	0
$371$	2.32941	0.120937
$372$	0	0
$373$	23.3691	1.21001	0.605003	−	0.796223i	$-0.293172\pi$
$373$	23.3691	1.21001	0.605003	+	0.796223i	$0.293172\pi$
$374$	0	0
$375$	−26.9707	−1.39276
$376$	0	0
$377$	5.89635	0.303677
$378$	0	0
$379$	19.1916	0.985804	0.492902	−	0.870085i	$-0.335936\pi$
$379$	19.1916	0.985804	0.492902	+	0.870085i	$0.335936\pi$
$380$	0	0
$381$	27.5508	1.41147
$382$	0	0
$383$	−4.64936	−0.237571	−0.118786	−	0.992920i	$-0.537900\pi$
$383$	−4.64936	−0.237571	−0.118786	+	0.992920i	$0.537900\pi$
$384$	0	0
$385$	−9.38178	−0.478140
$386$	0	0
$387$	20.4001	1.03699
$388$	0	0
$389$	−8.50160	−0.431048	−0.215524	−	0.976499i	$-0.569146\pi$
$389$	−8.50160	−0.431048	−0.215524	+	0.976499i	$0.569146\pi$
$390$	0	0
$391$	5.69358	0.287937
$392$	0	0
$393$	−28.1046	−1.41769
$394$	0	0
$395$	12.1850	0.613091
$396$	0	0
$397$	−30.5174	−1.53162	−0.765811	−	0.643065i	$-0.777662\pi$
$397$	−30.5174	−1.53162	−0.765811	+	0.643065i	$0.777662\pi$
$398$	0	0
$399$	13.6778	0.684745
$400$	0	0
$401$	−22.4203	−1.11962	−0.559809	−	0.828622i	$-0.689125\pi$
$401$	−22.4203	−1.11962	−0.559809	+	0.828622i	$0.689125\pi$
$402$	0	0
$403$	−5.55007	−0.276468
$404$	0	0
$405$	−18.7883	−0.933599
$406$	0	0
$407$	−37.6009	−1.86381
$408$	0	0
$409$	−6.19228	−0.306189	−0.153094	−	0.988212i	$-0.548924\pi$
$409$	−6.19228	−0.306189	−0.153094	+	0.988212i	$0.548924\pi$
$410$	0	0
$411$	23.4898	1.15867
$412$	0	0
$413$	−5.25229	−0.258448
$414$	0	0
$415$	−18.0070	−0.883929
$416$	0	0
$417$	−7.63938	−0.374102
$418$	0	0
$419$	−37.7932	−1.84632	−0.923160	−	0.384417i	$-0.874403\pi$
$419$	−37.7932	−1.84632	−0.923160	+	0.384417i	$0.874403\pi$
$420$	0	0
$421$	20.1224	0.980706	0.490353	−	0.871524i	$-0.336868\pi$
$421$	20.1224	0.980706	0.490353	+	0.871524i	$0.336868\pi$
$422$	0	0
$423$	2.43365	0.118328
$424$	0	0
$425$	−1.52916	−0.0741750
$426$	0	0
$427$	−9.64347	−0.466680
$428$	0	0
$429$	8.59206	0.414829
$430$	0	0
$431$	27.1036	1.30554	0.652768	−	0.757558i	$-0.273608\pi$
$431$	27.1036	1.30554	0.652768	+	0.757558i	$0.273608\pi$
$432$	0	0
$433$	−28.9505	−1.39127	−0.695635	−	0.718395i	$-0.744877\pi$
$433$	−28.9505	−1.39127	−0.695635	+	0.718395i	$0.744877\pi$
$434$	0	0
$435$	32.0276	1.53560
$436$	0	0
$437$	−47.8534	−2.28914
$438$	0	0
$439$	−5.97751	−0.285291	−0.142646	−	0.989774i	$-0.545561\pi$
$439$	−5.97751	−0.285291	−0.142646	+	0.989774i	$0.545561\pi$
$440$	0	0
$441$	1.97454	0.0940257
$442$	0	0
$443$	28.9507	1.37549	0.687745	−	0.725952i	$-0.258601\pi$
$443$	28.9507	1.37549	0.687745	+	0.725952i	$0.258601\pi$
$444$	0	0
$445$	−3.75684	−0.178091
$446$	0	0
$447$	−14.1210	−0.667901
$448$	0	0
$449$	5.22255	0.246467	0.123234	−	0.992378i	$-0.460674\pi$
$449$	5.22255	0.246467	0.123234	+	0.992378i	$0.460674\pi$
$450$	0	0
$451$	−5.50514	−0.259227
$452$	0	0
$453$	0.870122	0.0408819
$454$	0	0
$455$	−1.19253	−0.0559066
$456$	0	0
$457$	−7.24250	−0.338790	−0.169395	−	0.985548i	$-0.554181\pi$
$457$	−7.24250	−0.338790	−0.169395	+	0.985548i	$0.554181\pi$
$458$	0	0
$459$	−1.66881	−0.0778934
$460$	0	0
$461$	13.4281	0.625407	0.312704	−	0.949851i	$-0.398765\pi$
$461$	13.4281	0.625407	0.312704	+	0.949851i	$0.398765\pi$
$462$	0	0
$463$	3.07973	0.143127	0.0715635	−	0.997436i	$-0.477201\pi$
$463$	3.07973	0.143127	0.0715635	+	0.997436i	$0.477201\pi$
$464$	0	0
$465$	−30.1466	−1.39802
$466$	0	0
$467$	−5.73670	−0.265463	−0.132731	−	0.991152i	$-0.542375\pi$
$467$	−5.73670	−0.265463	−0.132731	+	0.991152i	$0.542375\pi$
$468$	0	0
$469$	1.85780	0.0857853
$470$	0	0
$471$	−16.1947	−0.746211
$472$	0	0
$473$	56.8767	2.61519
$474$	0	0
$475$	12.8523	0.589702
$476$	0	0
$477$	−4.59952	−0.210597
$478$	0	0
$479$	17.6228	0.805207	0.402604	−	0.915374i	$-0.368105\pi$
$479$	17.6228	0.805207	0.402604	+	0.915374i	$0.368105\pi$
$480$	0	0
$481$	−4.77949	−0.217926
$482$	0	0
$483$	−17.4041	−0.791913
$484$	0	0
$485$	−15.4709	−0.702498
$486$	0	0
$487$	13.3426	0.604612	0.302306	−	0.953211i	$-0.402244\pi$
$487$	13.3426	0.604612	0.302306	+	0.953211i	$0.402244\pi$
$488$	0	0
$489$	−53.6623	−2.42669
$490$	0	0
$491$	2.76970	0.124995	0.0624973	−	0.998045i	$-0.480094\pi$
$491$	2.76970	0.124995	0.0624973	+	0.998045i	$0.480094\pi$
$492$	0	0
$493$	6.14812	0.276897
$494$	0	0
$495$	18.5247	0.832623
$496$	0	0
$497$	8.36181	0.375078
$498$	0	0
$499$	37.5120	1.67927	0.839635	−	0.543151i	$-0.182769\pi$
$499$	37.5120	1.67927	0.839635	+	0.543151i	$0.182769\pi$
$500$	0	0
$501$	52.8208	2.35986
$502$	0	0
$503$	33.7947	1.50683	0.753416	−	0.657544i	$-0.228405\pi$
$503$	33.7947	1.50683	0.753416	+	0.657544i	$0.228405\pi$
$504$	0	0
$505$	7.89343	0.351253
$506$	0	0
$507$	−27.9026	−1.23920
$508$	0	0
$509$	1.13242	0.0501938	0.0250969	−	0.999685i	$-0.492011\pi$
$509$	1.13242	0.0501938	0.0250969	+	0.999685i	$0.492011\pi$
$510$	0	0
$511$	15.7323	0.695955
$512$	0	0
$513$	14.0260	0.619263
$514$	0	0
$515$	23.8046	1.04896
$516$	0	0
$517$	6.78518	0.298412
$518$	0	0
$519$	15.0528	0.660744
$520$	0	0
$521$	15.9515	0.698849	0.349425	−	0.936965i	$-0.386377\pi$
$521$	15.9515	0.698849	0.349425	+	0.936965i	$0.386377\pi$
$522$	0	0
$523$	−17.6414	−0.771406	−0.385703	−	0.922623i	$-0.626041\pi$
$523$	−17.6414	−0.771406	−0.385703	+	0.922623i	$0.626041\pi$
$524$	0	0
$525$	4.67430	0.204003
$526$	0	0
$527$	−5.78705	−0.252088
$528$	0	0
$529$	37.8904	1.64741
$530$	0	0
$531$	10.3709	0.450057
$532$	0	0
$533$	−0.699765	−0.0303102
$534$	0	0
$535$	−28.7375	−1.24243
$536$	0	0
$537$	47.4525	2.04773
$538$	0	0
$539$	5.50514	0.237123
$540$	0	0
$541$	33.3502	1.43384	0.716919	−	0.697156i	$-0.245552\pi$
$541$	33.3502	1.43384	0.716919	+	0.697156i	$0.245552\pi$
$542$	0	0
$543$	−46.8187	−2.00918
$544$	0	0
$545$	−27.2002	−1.16513
$546$	0	0
$547$	10.0850	0.431205	0.215603	−	0.976481i	$-0.430828\pi$
$547$	10.0850	0.431205	0.215603	+	0.976481i	$0.430828\pi$
$548$	0	0
$549$	19.0414	0.812668
$550$	0	0
$551$	−51.6737	−2.20138
$552$	0	0
$553$	−7.15002	−0.304050
$554$	0	0
$555$	−25.9611	−1.10199
$556$	0	0
$557$	37.9145	1.60649	0.803245	−	0.595649i	$-0.203105\pi$
$557$	37.9145	1.60649	0.803245	+	0.595649i	$0.203105\pi$
$558$	0	0
$559$	7.22966	0.305782
$560$	0	0
$561$	8.95894	0.378247
$562$	0	0
$563$	31.3818	1.32259	0.661293	−	0.750128i	$-0.270008\pi$
$563$	31.3818	1.32259	0.661293	+	0.750128i	$0.270008\pi$
$564$	0	0
$565$	29.6944	1.24925
$566$	0	0
$567$	11.0248	0.462999
$568$	0	0
$569$	−2.80804	−0.117719	−0.0588595	−	0.998266i	$-0.518746\pi$
$569$	−2.80804	−0.117719	−0.0588595	+	0.998266i	$0.518746\pi$
$570$	0	0
$571$	28.0213	1.17266	0.586328	−	0.810074i	$-0.300573\pi$
$571$	28.0213	1.17266	0.586328	+	0.810074i	$0.300573\pi$
$572$	0	0
$573$	−29.0518	−1.21366
$574$	0	0
$575$	−16.3537	−0.681995
$576$	0	0
$577$	27.5308	1.14612	0.573061	−	0.819513i	$-0.305756\pi$
$577$	27.5308	1.14612	0.573061	+	0.819513i	$0.305756\pi$
$578$	0	0
$579$	−49.5538	−2.05939
$580$	0	0
$581$	10.5663	0.438366
$582$	0	0
$583$	−12.8237	−0.531105
$584$	0	0
$585$	2.35470	0.0973547
$586$	0	0
$587$	27.0237	1.11539	0.557694	−	0.830046i	$-0.311686\pi$
$587$	27.0237	1.11539	0.557694	+	0.830046i	$0.311686\pi$
$588$	0	0
$589$	48.6390	2.00414
$590$	0	0
$591$	18.0540	0.742642
$592$	0	0
$593$	39.9387	1.64009	0.820043	−	0.572302i	$-0.193949\pi$
$593$	39.9387	1.64009	0.820043	+	0.572302i	$0.193949\pi$
$594$	0	0
$595$	−1.24345	−0.0509764
$596$	0	0
$597$	20.9765	0.858512
$598$	0	0
$599$	22.3571	0.913487	0.456744	−	0.889598i	$-0.349016\pi$
$599$	22.3571	0.913487	0.456744	+	0.889598i	$0.349016\pi$
$600$	0	0
$601$	39.2187	1.59976	0.799882	−	0.600157i	$-0.204895\pi$
$601$	39.2187	1.59976	0.799882	+	0.600157i	$0.204895\pi$
$602$	0	0
$603$	−3.66830	−0.149385
$604$	0	0
$605$	32.9020	1.33766
$606$	0	0
$607$	−32.3622	−1.31354	−0.656770	−	0.754091i	$-0.728078\pi$
$607$	−32.3622	−1.31354	−0.656770	+	0.754091i	$0.728078\pi$
$608$	0	0
$609$	−18.7935	−0.761551
$610$	0	0
$611$	0.862472	0.0348919
$612$	0	0
$613$	34.0263	1.37431	0.687155	−	0.726511i	$-0.258859\pi$
$613$	34.0263	1.37431	0.687155	+	0.726511i	$0.258859\pi$
$614$	0	0
$615$	−3.80096	−0.153269
$616$	0	0
$617$	−10.3754	−0.417698	−0.208849	−	0.977948i	$-0.566972\pi$
$617$	−10.3754	−0.417698	−0.208849	+	0.977948i	$0.566972\pi$
$618$	0	0
$619$	−28.1315	−1.13070	−0.565351	−	0.824850i	$-0.691259\pi$
$619$	−28.1315	−1.13070	−0.565351	+	0.824850i	$0.691259\pi$
$620$	0	0
$621$	−17.8472	−0.716183
$622$	0	0
$623$	2.20448	0.0883206
$624$	0	0
$625$	−10.1290	−0.405161
$626$	0	0
$627$	−75.2981	−3.00712
$628$	0	0
$629$	−4.98357	−0.198708
$630$	0	0
$631$	−42.3880	−1.68744	−0.843721	−	0.536783i	$-0.819640\pi$
$631$	−42.3880	−1.68744	−0.843721	+	0.536783i	$0.819640\pi$
$632$	0	0
$633$	9.49501	0.377393
$634$	0	0
$635$	21.0511	0.835388
$636$	0	0
$637$	0.699765	0.0277257
$638$	0	0
$639$	−16.5107	−0.653154
$640$	0	0
$641$	−42.8695	−1.69324	−0.846622	−	0.532194i	$-0.821368\pi$
$641$	−42.8695	−1.69324	−0.846622	+	0.532194i	$0.821368\pi$
$642$	0	0
$643$	32.3016	1.27385	0.636926	−	0.770925i	$-0.280206\pi$
$643$	32.3016	1.27385	0.636926	+	0.770925i	$0.280206\pi$
$644$	0	0
$645$	39.2698	1.54625
$646$	0	0
$647$	−8.02564	−0.315520	−0.157760	−	0.987477i	$-0.550427\pi$
$647$	−8.02564	−0.315520	−0.157760	+	0.987477i	$0.550427\pi$
$648$	0	0
$649$	28.9146	1.13500
$650$	0	0
$651$	17.6898	0.693317
$652$	0	0
$653$	−17.5666	−0.687434	−0.343717	−	0.939073i	$-0.611686\pi$
$653$	−17.5666	−0.687434	−0.343717	+	0.939073i	$0.611686\pi$
$654$	0	0
$655$	−21.4743	−0.839069
$656$	0	0
$657$	−31.0640	−1.21192
$658$	0	0
$659$	9.96360	0.388127	0.194063	−	0.980989i	$-0.437833\pi$
$659$	9.96360	0.388127	0.194063	+	0.980989i	$0.437833\pi$
$660$	0	0
$661$	5.00599	0.194710	0.0973552	−	0.995250i	$-0.468962\pi$
$661$	5.00599	0.194710	0.0973552	+	0.995250i	$0.468962\pi$
$662$	0	0
$663$	1.13878	0.0442266
$664$	0	0
$665$	10.4509	0.405270
$666$	0	0
$667$	65.7514	2.54591
$668$	0	0
$669$	38.4758	1.48756
$670$	0	0
$671$	53.0887	2.04947
$672$	0	0
$673$	−23.0854	−0.889878	−0.444939	−	0.895561i	$-0.646775\pi$
$673$	−23.0854	−0.889878	−0.444939	+	0.895561i	$0.646775\pi$
$674$	0	0
$675$	4.79331	0.184495
$676$	0	0
$677$	25.4773	0.979172	0.489586	−	0.871955i	$-0.337148\pi$
$677$	25.4773	0.979172	0.489586	+	0.871955i	$0.337148\pi$
$678$	0	0
$679$	9.07820	0.348389
$680$	0	0
$681$	−14.4118	−0.552260
$682$	0	0
$683$	2.58430	0.0988856	0.0494428	−	0.998777i	$-0.484255\pi$
$683$	2.58430	0.0988856	0.0494428	+	0.998777i	$0.484255\pi$
$684$	0	0
$685$	17.9481	0.685763
$686$	0	0
$687$	−13.1568	−0.501962
$688$	0	0
$689$	−1.63004	−0.0620996
$690$	0	0
$691$	−34.6770	−1.31918	−0.659589	−	0.751627i	$-0.729269\pi$
$691$	−34.6770	−1.31918	−0.659589	+	0.751627i	$0.729269\pi$
$692$	0	0
$693$	−10.8701	−0.412922
$694$	0	0
$695$	−5.83712	−0.221414
$696$	0	0
$697$	−0.729645	−0.0276373
$698$	0	0
$699$	−45.9509	−1.73802
$700$	0	0
$701$	−0.803544	−0.0303494	−0.0151747	−	0.999885i	$-0.504830\pi$
$701$	−0.803544	−0.0303494	−0.0151747	+	0.999885i	$0.504830\pi$
$702$	0	0
$703$	41.8859	1.57976
$704$	0	0
$705$	4.68474	0.176438
$706$	0	0
$707$	−4.63180	−0.174197
$708$	0	0
$709$	16.1022	0.604731	0.302365	−	0.953192i	$-0.402224\pi$
$709$	16.1022	0.604731	0.302365	+	0.953192i	$0.402224\pi$
$710$	0	0
$711$	14.1180	0.529466
$712$	0	0
$713$	−61.8900	−2.31780
$714$	0	0
$715$	6.56504	0.245519
$716$	0	0
$717$	37.5195	1.40119
$718$	0	0
$719$	25.1066	0.936317	0.468158	−	0.883645i	$-0.344918\pi$
$719$	25.1066	0.936317	0.468158	+	0.883645i	$0.344918\pi$
$720$	0	0
$721$	−13.9683	−0.520208
$722$	0	0
$723$	−7.97359	−0.296541
$724$	0	0
$725$	−17.6592	−0.655847
$726$	0	0
$727$	43.1684	1.60103	0.800514	−	0.599314i	$-0.204560\pi$
$727$	43.1684	1.60103	0.800514	+	0.599314i	$0.204560\pi$
$728$	0	0
$729$	−6.46535	−0.239457
$730$	0	0
$731$	7.53837	0.278817
$732$	0	0
$733$	−3.55681	−0.131374	−0.0656869	−	0.997840i	$-0.520924\pi$
$733$	−3.55681	−0.131374	−0.0656869	+	0.997840i	$0.520924\pi$
$734$	0	0
$735$	3.80096	0.140200
$736$	0	0
$737$	−10.2275	−0.376733
$738$	0	0
$739$	13.4616	0.495193	0.247596	−	0.968863i	$-0.420359\pi$
$739$	13.4616	0.495193	0.247596	+	0.968863i	$0.420359\pi$
$740$	0	0
$741$	−9.57123	−0.351608
$742$	0	0
$743$	26.7462	0.981224	0.490612	−	0.871378i	$-0.336773\pi$
$743$	26.7462	0.981224	0.490612	+	0.871378i	$0.336773\pi$
$744$	0	0
$745$	−10.7896	−0.395301
$746$	0	0
$747$	−20.8637	−0.763362
$748$	0	0
$749$	16.8629	0.616157
$750$	0	0
$751$	−28.7702	−1.04984	−0.524920	−	0.851152i	$-0.675905\pi$
$751$	−28.7702	−1.04984	−0.524920	+	0.851152i	$0.675905\pi$
$752$	0	0
$753$	16.7006	0.608603
$754$	0	0
$755$	0.664845	0.0241962
$756$	0	0
$757$	−17.3089	−0.629103	−0.314552	−	0.949240i	$-0.601854\pi$
$757$	−17.3089	−0.629103	−0.314552	+	0.949240i	$0.601854\pi$
$758$	0	0
$759$	95.8119	3.47775
$760$	0	0
$761$	0.229404	0.00831590	0.00415795	−	0.999991i	$-0.498676\pi$
$761$	0.229404	0.00831590	0.00415795	+	0.999991i	$0.498676\pi$
$762$	0	0
$763$	15.9608	0.577820
$764$	0	0
$765$	2.45524	0.0887694
$766$	0	0
$767$	3.67537	0.132710
$768$	0	0
$769$	−18.7856	−0.677428	−0.338714	−	0.940889i	$-0.609992\pi$
$769$	−18.7856	−0.677428	−0.338714	+	0.940889i	$0.609992\pi$
$770$	0	0
$771$	−21.3217	−0.767882
$772$	0	0
$773$	22.8942	0.823449	0.411724	−	0.911308i	$-0.364927\pi$
$773$	22.8942	0.823449	0.411724	+	0.911308i	$0.364927\pi$
$774$	0	0
$775$	16.6221	0.597084
$776$	0	0
$777$	15.2337	0.546506
$778$	0	0
$779$	6.13252	0.219720
$780$	0	0
$781$	−46.0330	−1.64719
$782$	0	0
$783$	−19.2720	−0.688724
$784$	0	0
$785$	−12.3741	−0.441649
$786$	0	0
$787$	−6.52719	−0.232669	−0.116335	−	0.993210i	$-0.537114\pi$
$787$	−6.52719	−0.232669	−0.116335	+	0.993210i	$0.537114\pi$
$788$	0	0
$789$	−4.33395	−0.154293
$790$	0	0
$791$	−17.4244	−0.619541
$792$	0	0
$793$	6.74816	0.239634
$794$	0	0
$795$	−8.85400	−0.314019
$796$	0	0
$797$	22.2360	0.787638	0.393819	−	0.919188i	$-0.371154\pi$
$797$	22.2360	0.787638	0.393819	+	0.919188i	$0.371154\pi$
$798$	0	0
$799$	0.899299	0.0318149
$800$	0	0
$801$	−4.35283	−0.153800
$802$	0	0
$803$	−86.6085	−3.05635
$804$	0	0
$805$	−13.2981	−0.468698
$806$	0	0
$807$	−48.5920	−1.71052
$808$	0	0
$809$	30.4780	1.07155	0.535774	−	0.844361i	$-0.320020\pi$
$809$	30.4780	1.07155	0.535774	+	0.844361i	$0.320020\pi$
$810$	0	0
$811$	−51.8668	−1.82129	−0.910644	−	0.413191i	$-0.864414\pi$
$811$	−51.8668	−1.82129	−0.910644	+	0.413191i	$0.864414\pi$
$812$	0	0
$813$	−26.1318	−0.916482
$814$	0	0
$815$	−41.0024	−1.43625
$816$	0	0
$817$	−63.3585	−2.21663
$818$	0	0
$819$	−1.38171	−0.0482810
$820$	0	0
$821$	13.2307	0.461753	0.230877	−	0.972983i	$-0.425841\pi$
$821$	13.2307	0.461753	0.230877	+	0.972983i	$0.425841\pi$
$822$	0	0
$823$	−27.1542	−0.946537	−0.473268	−	0.880918i	$-0.656926\pi$
$823$	−27.1542	−0.946537	−0.473268	+	0.880918i	$0.656926\pi$
$824$	0	0
$825$	−25.7327	−0.895898
$826$	0	0
$827$	−37.6852	−1.31044	−0.655221	−	0.755438i	$-0.727424\pi$
$827$	−37.6852	−1.31044	−0.655221	+	0.755438i	$0.727424\pi$
$828$	0	0
$829$	3.53646	0.122826	0.0614131	−	0.998112i	$-0.480439\pi$
$829$	3.53646	0.122826	0.0614131	+	0.998112i	$0.480439\pi$
$830$	0	0
$831$	−39.8549	−1.38255
$832$	0	0
$833$	0.729645	0.0252807
$834$	0	0
$835$	40.3595	1.39670
$836$	0	0
$837$	18.1402	0.627016
$838$	0	0
$839$	−14.3240	−0.494521	−0.247260	−	0.968949i	$-0.579530\pi$
$839$	−14.3240	−0.494521	−0.247260	+	0.968949i	$0.579530\pi$
$840$	0	0
$841$	42.0006	1.44830
$842$	0	0
$843$	22.7131	0.782279
$844$	0	0
$845$	−21.3199	−0.733427
$846$	0	0
$847$	−19.3066	−0.663382
$848$	0	0
$849$	45.0492	1.54608
$850$	0	0
$851$	−53.2971	−1.82700
$852$	0	0
$853$	23.5526	0.806427	0.403214	−	0.915106i	$-0.367893\pi$
$853$	23.5526	0.806427	0.403214	+	0.915106i	$0.367893\pi$
$854$	0	0
$855$	−20.6358	−0.705730
$856$	0	0
$857$	32.8735	1.12294	0.561469	−	0.827498i	$-0.310236\pi$
$857$	32.8735	1.12294	0.561469	+	0.827498i	$0.310236\pi$
$858$	0	0
$859$	6.02973	0.205732	0.102866	−	0.994695i	$-0.467199\pi$
$859$	6.02973	0.205732	0.102866	+	0.994695i	$0.467199\pi$
$860$	0	0
$861$	2.23037	0.0760107
$862$	0	0
$863$	−36.1379	−1.23015	−0.615075	−	0.788469i	$-0.710874\pi$
$863$	−36.1379	−1.23015	−0.615075	+	0.788469i	$0.710874\pi$
$864$	0	0
$865$	11.5016	0.391065
$866$	0	0
$867$	−36.7288	−1.24738
$868$	0	0
$869$	39.3619	1.33526
$870$	0	0
$871$	−1.30002	−0.0440496
$872$	0	0
$873$	−17.9253	−0.606678
$874$	0	0
$875$	12.0925	0.408800
$876$	0	0
$877$	7.55651	0.255165	0.127583	−	0.991828i	$-0.459278\pi$
$877$	7.55651	0.255165	0.127583	+	0.991828i	$0.459278\pi$
$878$	0	0
$879$	11.3810	0.383873
$880$	0	0
$881$	18.0856	0.609320	0.304660	−	0.952461i	$-0.401457\pi$
$881$	18.0856	0.609320	0.304660	+	0.952461i	$0.401457\pi$
$882$	0	0
$883$	9.59769	0.322988	0.161494	−	0.986874i	$-0.448369\pi$
$883$	9.59769	0.322988	0.161494	+	0.986874i	$0.448369\pi$
$884$	0	0
$885$	19.9637	0.671073
$886$	0	0
$887$	52.6019	1.76620	0.883100	−	0.469185i	$-0.155452\pi$
$887$	52.6019	1.76620	0.883100	+	0.469185i	$0.155452\pi$
$888$	0	0
$889$	−12.3526	−0.414293
$890$	0	0
$891$	−60.6932	−2.03330
$892$	0	0
$893$	−7.55843	−0.252933
$894$	0	0
$895$	36.2576	1.21196
$896$	0	0
$897$	12.1788	0.406637
$898$	0	0
$899$	−66.8308	−2.22893
$900$	0	0
$901$	−1.69964	−0.0566233
$902$	0	0
$903$	−23.0432	−0.766829
$904$	0	0
$905$	−35.7733	−1.18915
$906$	0	0
$907$	−10.7327	−0.356373	−0.178187	−	0.983997i	$-0.557023\pi$
$907$	−10.7327	−0.356373	−0.178187	+	0.983997i	$0.557023\pi$
$908$	0	0
$909$	9.14566	0.303343
$910$	0	0
$911$	−31.0703	−1.02941	−0.514703	−	0.857369i	$-0.672098\pi$
$911$	−31.0703	−1.02941	−0.514703	+	0.857369i	$0.672098\pi$
$912$	0	0
$913$	−58.1692	−1.92512
$914$	0	0
$915$	36.6544	1.21176
$916$	0	0
$917$	12.6009	0.416118
$918$	0	0
$919$	8.07889	0.266498	0.133249	−	0.991083i	$-0.457459\pi$
$919$	8.07889	0.266498	0.133249	+	0.991083i	$0.457459\pi$
$920$	0	0
$921$	−12.0773	−0.397961
$922$	0	0
$923$	−5.85130	−0.192598
$924$	0	0
$925$	14.3143	0.470651
$926$	0	0
$927$	27.5810	0.905880
$928$	0	0
$929$	−41.7539	−1.36990	−0.684951	−	0.728589i	$-0.740176\pi$
$929$	−41.7539	−1.36990	−0.684951	+	0.728589i	$0.740176\pi$
$930$	0	0
$931$	−6.13252	−0.200985
$932$	0	0
$933$	−63.9608	−2.09398
$934$	0	0
$935$	6.84536	0.223867
$936$	0	0
$937$	−1.29820	−0.0424103	−0.0212052	−	0.999775i	$-0.506750\pi$
$937$	−1.29820	−0.0424103	−0.0212052	+	0.999775i	$0.506750\pi$
$938$	0	0
$939$	−6.09265	−0.198826
$940$	0	0
$941$	20.3622	0.663789	0.331895	−	0.943316i	$-0.392312\pi$
$941$	20.3622	0.663789	0.331895	+	0.943316i	$0.392312\pi$
$942$	0	0
$943$	−7.80323	−0.254108
$944$	0	0
$945$	3.89773	0.126793
$946$	0	0
$947$	8.89249	0.288967	0.144484	−	0.989507i	$-0.453848\pi$
$947$	8.89249	0.288967	0.144484	+	0.989507i	$0.453848\pi$
$948$	0	0
$949$	−11.0089	−0.357364
$950$	0	0
$951$	−40.3339	−1.30792
$952$	0	0
$953$	26.2468	0.850216	0.425108	−	0.905143i	$-0.360236\pi$
$953$	26.2468	0.850216	0.425108	+	0.905143i	$0.360236\pi$
$954$	0	0
$955$	−22.1980	−0.718310
$956$	0	0
$957$	103.461	3.34441
$958$	0	0
$959$	−10.5318	−0.340089
$960$	0	0
$961$	31.9060	1.02922
$962$	0	0
$963$	−33.2965	−1.07296
$964$	0	0
$965$	−37.8632	−1.21886
$966$	0	0
$967$	22.6648	0.728850	0.364425	−	0.931233i	$-0.381266\pi$
$967$	22.6648	0.728850	0.364425	+	0.931233i	$0.381266\pi$
$968$	0	0
$969$	−9.97991	−0.320601
$970$	0	0
$971$	7.83582	0.251464	0.125732	−	0.992064i	$-0.459872\pi$
$971$	7.83582	0.251464	0.125732	+	0.992064i	$0.459872\pi$
$972$	0	0
$973$	3.42517	0.109806
$974$	0	0
$975$	−3.27092	−0.104753
$976$	0	0
$977$	−32.4214	−1.03725	−0.518627	−	0.855001i	$-0.673557\pi$
$977$	−32.4214	−1.03725	−0.518627	+	0.855001i	$0.673557\pi$
$978$	0	0
$979$	−12.1360	−0.387867
$980$	0	0
$981$	−31.5153	−1.00620
$982$	0	0
$983$	−10.0757	−0.321365	−0.160682	−	0.987006i	$-0.551370\pi$
$983$	−10.0757	−0.321365	−0.160682	+	0.987006i	$0.551370\pi$
$984$	0	0
$985$	13.7947	0.439537
$986$	0	0
$987$	−2.74896	−0.0875005
$988$	0	0
$989$	80.6195	2.56355
$990$	0	0
$991$	−28.3990	−0.902123	−0.451061	−	0.892493i	$-0.648954\pi$
$991$	−28.3990	−0.902123	−0.451061	+	0.892493i	$0.648954\pi$
$992$	0	0
$993$	32.9820	1.04665
$994$	0	0
$995$	16.0278	0.508115
$996$	0	0
$997$	−30.4397	−0.964034	−0.482017	−	0.876162i	$-0.660096\pi$
$997$	−30.4397	−0.964034	−0.482017	+	0.876162i	$0.660096\pi$
$998$	0	0
$999$	15.6216	0.494244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.d.1.4	✓	5
4.3	odd	2		4592.2.a.bc.1.2		5
7.6	odd	2		8036.2.a.k.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.d.1.4	✓	5	1.1	even	1	trivial
4592.2.a.bc.1.2		5	4.3	odd	2
8036.2.a.k.1.2		5	7.6	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23037 q^{3} +1.70418 q^{5} -1.00000 q^{7} +1.97454 q^{9} +O(q^{10})\)	\(q+2.23037 q^{3} +1.70418 q^{5} -1.00000 q^{7} +1.97454 q^{9} +5.50514 q^{11} +0.699765 q^{13} +3.80096 q^{15} +0.729645 q^{17} -6.13252 q^{19} -2.23037 q^{21} +7.80323 q^{23} -2.09576 q^{25} -2.28715 q^{27} +8.42619 q^{29} -7.93133 q^{31} +12.2785 q^{33} -1.70418 q^{35} -6.83013 q^{37} +1.56073 q^{39} -1.00000 q^{41} +10.3316 q^{43} +3.36498 q^{45} +1.23252 q^{47} +1.00000 q^{49} +1.62738 q^{51} -2.32941 q^{53} +9.38178 q^{55} -13.6778 q^{57} +5.25229 q^{59} +9.64347 q^{61} -1.97454 q^{63} +1.19253 q^{65} -1.85780 q^{67} +17.4041 q^{69} -8.36181 q^{71} -15.7323 q^{73} -4.67430 q^{75} -5.50514 q^{77} +7.15002 q^{79} -11.0248 q^{81} -10.5663 q^{83} +1.24345 q^{85} +18.7935 q^{87} -2.20448 q^{89} -0.699765 q^{91} -17.6898 q^{93} -10.4509 q^{95} -9.07820 q^{97} +10.8701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 3 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 3 * q^5 - 5 * q^7 + q^9	\( 5 q + 2 q^{3} - 3 q^{5} - 5 q^{7} + q^{9} + 6 q^{11} + 7 q^{13} + 9 q^{15} + q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 8 q^{27} + 11 q^{29} + 13 q^{31} + 11 q^{33} + 3 q^{35} + 5 q^{37} + 23 q^{39} - 5 q^{41} + 29 q^{43} + 11 q^{45} + 7 q^{47} + 5 q^{49} - 3 q^{51} + 21 q^{53} + 19 q^{55} + 9 q^{57} + 3 q^{59} - 8 q^{61} - q^{63} - 5 q^{65} + 3 q^{67} + 10 q^{69} + 22 q^{71} - 16 q^{73} - 18 q^{75} - 6 q^{77} + 4 q^{79} - 15 q^{81} - 6 q^{83} + 13 q^{85} + 6 q^{87} - 20 q^{89} - 7 q^{91} - 5 q^{93} - 7 q^{95} - 24 q^{97} + 27 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 3 * q^5 - 5 * q^7 + q^9 + 6 * q^11 + 7 * q^13 + 9 * q^15 + q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 + 8 * q^27 + 11 * q^29 + 13 * q^31 + 11 * q^33 + 3 * q^35 + 5 * q^37 + 23 * q^39 - 5 * q^41 + 29 * q^43 + 11 * q^45 + 7 * q^47 + 5 * q^49 - 3 * q^51 + 21 * q^53 + 19 * q^55 + 9 * q^57 + 3 * q^59 - 8 * q^61 - q^63 - 5 * q^65 + 3 * q^67 + 10 * q^69 + 22 * q^71 - 16 * q^73 - 18 * q^75 - 6 * q^77 + 4 * q^79 - 15 * q^81 - 6 * q^83 + 13 * q^85 + 6 * q^87 - 20 * q^89 - 7 * q^91 - 5 * q^93 - 7 * q^95 - 24 * q^97 + 27 * q^99

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