LMFDB - Embedded newform 1148.2.a.d.1.1

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			1.1
Root			$0.145487$ 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)

$f(q)$	$=$	$q-1.97883 q^{3} -2.39996 q^{5} -1.00000 q^{7} +0.915782 q^{9} +O(q^{10})$	$q-1.97883 q^{3} -2.39996 q^{5} -1.00000 q^{7} +0.915782 q^{9} +2.34916 q^{11} -1.41222 q^{13} +4.74913 q^{15} -2.31575 q^{17} -5.93164 q^{19} +1.97883 q^{21} -6.90130 q^{23} +0.759824 q^{25} +4.12432 q^{27} -1.33108 q^{29} +4.98191 q^{31} -4.64860 q^{33} +2.39996 q^{35} +9.14326 q^{37} +2.79454 q^{39} -1.00000 q^{41} +9.18251 q^{43} -2.19784 q^{45} +10.6838 q^{47} +1.00000 q^{49} +4.58247 q^{51} +12.4802 q^{53} -5.63791 q^{55} +11.7377 q^{57} +5.58221 q^{59} -12.8464 q^{61} -0.915782 q^{63} +3.38927 q^{65} +16.3648 q^{67} +13.6565 q^{69} -13.0124 q^{71} -3.76721 q^{73} -1.50357 q^{75} -2.34916 q^{77} +13.4804 q^{79} -10.9087 q^{81} -3.21593 q^{83} +5.55770 q^{85} +2.63398 q^{87} -7.17204 q^{89} +1.41222 q^{91} -9.85838 q^{93} +14.2357 q^{95} -16.6599 q^{97} +2.15132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.97883	−1.14248	−0.571240	−	0.820783i	$-0.693537\pi$
$3$	−1.97883	−1.14248	−0.571240	+	0.820783i	$0.693537\pi$
$4$	0	0
$5$	−2.39996	−1.07330	−0.536648	−	0.843806i	$-0.680310\pi$
$5$	−2.39996	−1.07330	−0.536648	+	0.843806i	$0.680310\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.915782	0.305261
$10$	0	0
$11$	2.34916	0.708300	0.354150	−	0.935189i	$-0.384770\pi$
$11$	2.34916	0.708300	0.354150	+	0.935189i	$0.384770\pi$
$12$	0	0
$13$	−1.41222	−0.391678	−0.195839	−	0.980636i	$-0.562743\pi$
$13$	−1.41222	−0.391678	−0.195839	+	0.980636i	$0.562743\pi$
$14$	0	0
$15$	4.74913	1.22622
$16$	0	0
$17$	−2.31575	−0.561651	−0.280825	−	0.959759i	$-0.590608\pi$
$17$	−2.31575	−0.561651	−0.280825	+	0.959759i	$0.590608\pi$
$18$	0	0
$19$	−5.93164	−1.36081	−0.680406	−	0.732836i	$-0.738196\pi$
$19$	−5.93164	−1.36081	−0.680406	+	0.732836i	$0.738196\pi$
$20$	0	0
$21$	1.97883	0.431817
$22$	0	0
$23$	−6.90130	−1.43902	−0.719510	−	0.694482i	$-0.755634\pi$
$23$	−6.90130	−1.43902	−0.719510	+	0.694482i	$0.755634\pi$
$24$	0	0
$25$	0.759824	0.151965
$26$	0	0
$27$	4.12432	0.793726
$28$	0	0
$29$	−1.33108	−0.247175	−0.123587	−	0.992334i	$-0.539440\pi$
$29$	−1.33108	−0.247175	−0.123587	+	0.992334i	$0.539440\pi$
$30$	0	0
$31$	4.98191	0.894778	0.447389	−	0.894339i	$-0.352354\pi$
$31$	4.98191	0.894778	0.447389	+	0.894339i	$0.352354\pi$
$32$	0	0
$33$	−4.64860	−0.809218
$34$	0	0
$35$	2.39996	0.405668
$36$	0	0
$37$	9.14326	1.50314	0.751571	−	0.659652i	$-0.229296\pi$
$37$	9.14326	1.50314	0.751571	+	0.659652i	$0.229296\pi$
$38$	0	0
$39$	2.79454	0.447485
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	9.18251	1.40032	0.700160	−	0.713986i	$-0.253112\pi$
$43$	9.18251	1.40032	0.700160	+	0.713986i	$0.253112\pi$
$44$	0	0
$45$	−2.19784	−0.327635
$46$	0	0
$47$	10.6838	1.55840	0.779200	−	0.626776i	$-0.215626\pi$
$47$	10.6838	1.55840	0.779200	+	0.626776i	$0.215626\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.58247	0.641675
$52$	0	0
$53$	12.4802	1.71428	0.857141	−	0.515081i	$-0.172238\pi$
$53$	12.4802	1.71428	0.857141	+	0.515081i	$0.172238\pi$
$54$	0	0
$55$	−5.63791	−0.760215
$56$	0	0
$57$	11.7377	1.55470
$58$	0	0
$59$	5.58221	0.726742	0.363371	−	0.931645i	$-0.381626\pi$
$59$	5.58221	0.726742	0.363371	+	0.931645i	$0.381626\pi$
$60$	0	0
$61$	−12.8464	−1.64482	−0.822410	−	0.568896i	$-0.807371\pi$
$61$	−12.8464	−1.64482	−0.822410	+	0.568896i	$0.807371\pi$
$62$	0	0
$63$	−0.915782	−0.115378
$64$	0	0
$65$	3.38927	0.420387
$66$	0	0
$67$	16.3648	1.99928	0.999642	−	0.0267719i	$-0.00852277\pi$
$67$	16.3648	1.99928	0.999642	+	0.0267719i	$0.00852277\pi$
$68$	0	0
$69$	13.6565	1.64405
$70$	0	0
$71$	−13.0124	−1.54428	−0.772142	−	0.635450i	$-0.780815\pi$
$71$	−13.0124	−1.54428	−0.772142	+	0.635450i	$0.780815\pi$
$72$	0	0
$73$	−3.76721	−0.440919	−0.220460	−	0.975396i	$-0.570756\pi$
$73$	−3.76721	−0.440919	−0.220460	+	0.975396i	$0.570756\pi$
$74$	0	0
$75$	−1.50357	−0.173617
$76$	0	0
$77$	−2.34916	−0.267712
$78$	0	0
$79$	13.4804	1.51667	0.758333	−	0.651867i	$-0.226014\pi$
$79$	13.4804	1.51667	0.758333	+	0.651867i	$0.226014\pi$
$80$	0	0
$81$	−10.9087	−1.21208
$82$	0	0
$83$	−3.21593	−0.352994	−0.176497	−	0.984301i	$-0.556477\pi$
$83$	−3.21593	−0.352994	−0.176497	+	0.984301i	$0.556477\pi$
$84$	0	0
$85$	5.55770	0.602818
$86$	0	0
$87$	2.63398	0.282392
$88$	0	0
$89$	−7.17204	−0.760235	−0.380117	−	0.924938i	$-0.624116\pi$
$89$	−7.17204	−0.760235	−0.380117	+	0.924938i	$0.624116\pi$
$90$	0	0
$91$	1.41222	0.148040
$92$	0	0
$93$	−9.85838	−1.02227
$94$	0	0
$95$	14.2357	1.46055
$96$	0	0
$97$	−16.6599	−1.69155	−0.845776	−	0.533537i	$-0.820862\pi$
$97$	−16.6599	−1.69155	−0.845776	+	0.533537i	$0.820862\pi$
$98$	0	0
$99$	2.15132	0.216216
$100$	0	0
$101$	5.59473	0.556696	0.278348	−	0.960480i	$-0.410213\pi$
$101$	5.59473	0.556696	0.278348	+	0.960480i	$0.410213\pi$
$102$	0	0
$103$	7.97424	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$103$	7.97424	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$104$	0	0
$105$	−4.74913	−0.463467
$106$	0	0
$107$	4.69914	0.454283	0.227141	−	0.973862i	$-0.427062\pi$
$107$	4.69914	0.454283	0.227141	+	0.973862i	$0.427062\pi$
$108$	0	0
$109$	−5.20372	−0.498426	−0.249213	−	0.968449i	$-0.580172\pi$
$109$	−5.20372	−0.498426	−0.249213	+	0.968449i	$0.580172\pi$
$110$	0	0
$111$	−18.0930	−1.71731
$112$	0	0
$113$	0.739630	0.0695785	0.0347893	−	0.999395i	$-0.488924\pi$
$113$	0.739630	0.0695785	0.0347893	+	0.999395i	$0.488924\pi$
$114$	0	0
$115$	16.5629	1.54449
$116$	0	0
$117$	−1.29328	−0.119564
$118$	0	0
$119$	2.31575	0.212284
$120$	0	0
$121$	−5.48143	−0.498312
$122$	0	0
$123$	1.97883	0.178425
$124$	0	0
$125$	10.1763	0.910193
$126$	0	0
$127$	14.2545	1.26488	0.632440	−	0.774609i	$-0.282053\pi$
$127$	14.2545	1.26488	0.632440	+	0.774609i	$0.282053\pi$
$128$	0	0
$129$	−18.1707	−1.59984
$130$	0	0
$131$	−6.58934	−0.575713	−0.287857	−	0.957674i	$-0.592943\pi$
$131$	−6.58934	−0.575713	−0.287857	+	0.957674i	$0.592943\pi$
$132$	0	0
$133$	5.93164	0.514338
$134$	0	0
$135$	−9.89822	−0.851903
$136$	0	0
$137$	1.84252	0.157417	0.0787085	−	0.996898i	$-0.474920\pi$
$137$	1.84252	0.157417	0.0787085	+	0.996898i	$0.474920\pi$
$138$	0	0
$139$	14.2400	1.20782	0.603910	−	0.797052i	$-0.293609\pi$
$139$	14.2400	1.20782	0.603910	+	0.797052i	$0.293609\pi$
$140$	0	0
$141$	−21.1416	−1.78044
$142$	0	0
$143$	−3.31753	−0.277426
$144$	0	0
$145$	3.19454	0.265292
$146$	0	0
$147$	−1.97883	−0.163211
$148$	0	0
$149$	−13.2625	−1.08650	−0.543251	−	0.839570i	$-0.682807\pi$
$149$	−13.2625	−1.08650	−0.543251	+	0.839570i	$0.682807\pi$
$150$	0	0
$151$	3.96632	0.322774	0.161387	−	0.986891i	$-0.448403\pi$
$151$	3.96632	0.322774	0.161387	+	0.986891i	$0.448403\pi$
$152$	0	0
$153$	−2.12072	−0.171450
$154$	0	0
$155$	−11.9564	−0.960362
$156$	0	0
$157$	16.1974	1.29269	0.646346	−	0.763044i	$-0.276296\pi$
$157$	16.1974	1.29269	0.646346	+	0.763044i	$0.276296\pi$
$158$	0	0
$159$	−24.6962	−1.95853
$160$	0	0
$161$	6.90130	0.543899
$162$	0	0
$163$	−2.70238	−0.211667	−0.105833	−	0.994384i	$-0.533751\pi$
$163$	−2.70238	−0.211667	−0.105833	+	0.994384i	$0.533751\pi$
$164$	0	0
$165$	11.1565	0.868531
$166$	0	0
$167$	3.00791	0.232759	0.116380	−	0.993205i	$-0.462871\pi$
$167$	3.00791	0.232759	0.116380	+	0.993205i	$0.462871\pi$
$168$	0	0
$169$	−11.0056	−0.846588
$170$	0	0
$171$	−5.43209	−0.415402
$172$	0	0
$173$	−2.32822	−0.177012	−0.0885058	−	0.996076i	$-0.528209\pi$
$173$	−2.32822	−0.177012	−0.0885058	+	0.996076i	$0.528209\pi$
$174$	0	0
$175$	−0.759824	−0.0574373
$176$	0	0
$177$	−11.0463	−0.830288
$178$	0	0
$179$	−25.3001	−1.89102	−0.945511	−	0.325591i	$-0.894437\pi$
$179$	−25.3001	−1.89102	−0.945511	+	0.325591i	$0.894437\pi$
$180$	0	0
$181$	18.8374	1.40017	0.700086	−	0.714058i	$-0.253145\pi$
$181$	18.8374	1.40017	0.700086	+	0.714058i	$0.253145\pi$
$182$	0	0
$183$	25.4210	1.87917
$184$	0	0
$185$	−21.9435	−1.61332
$186$	0	0
$187$	−5.44007	−0.397817
$188$	0	0
$189$	−4.12432	−0.300000
$190$	0	0
$191$	−11.9291	−0.863157	−0.431578	−	0.902075i	$-0.642043\pi$
$191$	−11.9291	−0.863157	−0.431578	+	0.902075i	$0.642043\pi$
$192$	0	0
$193$	13.0951	0.942605	0.471303	−	0.881972i	$-0.343784\pi$
$193$	13.0951	0.942605	0.471303	+	0.881972i	$0.343784\pi$
$194$	0	0
$195$	−6.70679	−0.480284
$196$	0	0
$197$	−0.513588	−0.0365916	−0.0182958	−	0.999833i	$-0.505824\pi$
$197$	−0.513588	−0.0365916	−0.0182958	+	0.999833i	$0.505824\pi$
$198$	0	0
$199$	11.0967	0.786627	0.393313	−	0.919404i	$-0.371329\pi$
$199$	11.0967	0.786627	0.393313	+	0.919404i	$0.371329\pi$
$200$	0	0
$201$	−32.3833	−2.28414
$202$	0	0
$203$	1.33108	0.0934233
$204$	0	0
$205$	2.39996	0.167621
$206$	0	0
$207$	−6.32008	−0.439276
$208$	0	0
$209$	−13.9344	−0.963862
$210$	0	0
$211$	28.0923	1.93395	0.966977	−	0.254863i	$-0.0820302\pi$
$211$	28.0923	1.93395	0.966977	+	0.254863i	$0.0820302\pi$
$212$	0	0
$213$	25.7493	1.76431
$214$	0	0
$215$	−22.0377	−1.50296
$216$	0	0
$217$	−4.98191	−0.338194
$218$	0	0
$219$	7.45469	0.503741
$220$	0	0
$221$	3.27033	0.219986
$222$	0	0
$223$	−14.3955	−0.963994	−0.481997	−	0.876173i	$-0.660088\pi$
$223$	−14.3955	−0.963994	−0.481997	+	0.876173i	$0.660088\pi$
$224$	0	0
$225$	0.695833	0.0463889
$226$	0	0
$227$	0.468590	0.0311014	0.0155507	−	0.999879i	$-0.495050\pi$
$227$	0.468590	0.0311014	0.0155507	+	0.999879i	$0.495050\pi$
$228$	0	0
$229$	−0.683504	−0.0451672	−0.0225836	−	0.999745i	$-0.507189\pi$
$229$	−0.683504	−0.0451672	−0.0225836	+	0.999745i	$0.507189\pi$
$230$	0	0
$231$	4.64860	0.305856
$232$	0	0
$233$	0.0814633	0.00533684	0.00266842	−	0.999996i	$-0.499151\pi$
$233$	0.0814633	0.00533684	0.00266842	+	0.999996i	$0.499151\pi$
$234$	0	0
$235$	−25.6408	−1.67262
$236$	0	0
$237$	−26.6755	−1.73276
$238$	0	0
$239$	23.2898	1.50649	0.753246	−	0.657739i	$-0.228487\pi$
$239$	23.2898	1.50649	0.753246	+	0.657739i	$0.228487\pi$
$240$	0	0
$241$	25.0416	1.61307	0.806535	−	0.591186i	$-0.201340\pi$
$241$	25.0416	1.61307	0.806535	+	0.591186i	$0.201340\pi$
$242$	0	0
$243$	9.21352	0.591047
$244$	0	0
$245$	−2.39996	−0.153328
$246$	0	0
$247$	8.37675	0.533000
$248$	0	0
$249$	6.36379	0.403289
$250$	0	0
$251$	−16.6994	−1.05406	−0.527029	−	0.849848i	$-0.676694\pi$
$251$	−16.6994	−1.05406	−0.527029	+	0.849848i	$0.676694\pi$
$252$	0	0
$253$	−16.2123	−1.01926
$254$	0	0
$255$	−10.9978	−0.688707
$256$	0	0
$257$	−2.88562	−0.180000	−0.0900002	−	0.995942i	$-0.528687\pi$
$257$	−2.88562	−0.180000	−0.0900002	+	0.995942i	$0.528687\pi$
$258$	0	0
$259$	−9.14326	−0.568134
$260$	0	0
$261$	−1.21898	−0.0754527
$262$	0	0
$263$	−21.1140	−1.30194	−0.650971	−	0.759102i	$-0.725638\pi$
$263$	−21.1140	−1.30194	−0.650971	+	0.759102i	$0.725638\pi$
$264$	0	0
$265$	−29.9519	−1.83993
$266$	0	0
$267$	14.1923	0.868553
$268$	0	0
$269$	2.95179	0.179974	0.0899870	−	0.995943i	$-0.471317\pi$
$269$	2.95179	0.179974	0.0899870	+	0.995943i	$0.471317\pi$
$270$	0	0
$271$	−10.6964	−0.649757	−0.324879	−	0.945756i	$-0.605323\pi$
$271$	−10.6964	−0.649757	−0.324879	+	0.945756i	$0.605323\pi$
$272$	0	0
$273$	−2.79454	−0.169133
$274$	0	0
$275$	1.78495	0.107637
$276$	0	0
$277$	−3.71111	−0.222979	−0.111489	−	0.993766i	$-0.535562\pi$
$277$	−3.71111	−0.222979	−0.111489	+	0.993766i	$0.535562\pi$
$278$	0	0
$279$	4.56234	0.273140
$280$	0	0
$281$	9.94492	0.593264	0.296632	−	0.954992i	$-0.404136\pi$
$281$	9.94492	0.593264	0.296632	+	0.954992i	$0.404136\pi$
$282$	0	0
$283$	−1.00267	−0.0596025	−0.0298013	−	0.999556i	$-0.509487\pi$
$283$	−1.00267	−0.0596025	−0.0298013	+	0.999556i	$0.509487\pi$
$284$	0	0
$285$	−28.1701	−1.66865
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	−11.6373	−0.684549
$290$	0	0
$291$	32.9671	1.93257
$292$	0	0
$293$	−23.4779	−1.37160	−0.685798	−	0.727792i	$-0.740547\pi$
$293$	−23.4779	−1.37160	−0.685798	+	0.727792i	$0.740547\pi$
$294$	0	0
$295$	−13.3971	−0.780009
$296$	0	0
$297$	9.68871	0.562196
$298$	0	0
$299$	9.74612	0.563633
$300$	0	0
$301$	−9.18251	−0.529271
$302$	0	0
$303$	−11.0710	−0.636014
$304$	0	0
$305$	30.8310	1.76538
$306$	0	0
$307$	18.2829	1.04346	0.521731	−	0.853110i	$-0.325286\pi$
$307$	18.2829	1.04346	0.521731	+	0.853110i	$0.325286\pi$
$308$	0	0
$309$	−15.7797	−0.897675
$310$	0	0
$311$	27.2264	1.54387	0.771933	−	0.635704i	$-0.219290\pi$
$311$	27.2264	1.54387	0.771933	+	0.635704i	$0.219290\pi$
$312$	0	0
$313$	−15.5021	−0.876230	−0.438115	−	0.898919i	$-0.644354\pi$
$313$	−15.5021	−0.876230	−0.438115	+	0.898919i	$0.644354\pi$
$314$	0	0
$315$	2.19784	0.123834
$316$	0	0
$317$	29.2003	1.64005	0.820027	−	0.572325i	$-0.193958\pi$
$317$	29.2003	1.64005	0.820027	+	0.572325i	$0.193958\pi$
$318$	0	0
$319$	−3.12692	−0.175074
$320$	0	0
$321$	−9.29881	−0.519009
$322$	0	0
$323$	13.7362	0.764301
$324$	0	0
$325$	−1.07304	−0.0595213
$326$	0	0
$327$	10.2973	0.569441
$328$	0	0
$329$	−10.6838	−0.589019
$330$	0	0
$331$	−14.0206	−0.770640	−0.385320	−	0.922783i	$-0.625909\pi$
$331$	−14.0206	−0.770640	−0.385320	+	0.922783i	$0.625909\pi$
$332$	0	0
$333$	8.37323	0.458850
$334$	0	0
$335$	−39.2750	−2.14582
$336$	0	0
$337$	−3.72644	−0.202992	−0.101496	−	0.994836i	$-0.532363\pi$
$337$	−3.72644	−0.202992	−0.101496	+	0.994836i	$0.532363\pi$
$338$	0	0
$339$	−1.46360	−0.0794921
$340$	0	0
$341$	11.7033	0.633771
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−32.7751	−1.76455
$346$	0	0
$347$	−20.3232	−1.09101	−0.545504	−	0.838108i	$-0.683662\pi$
$347$	−20.3232	−1.09101	−0.545504	+	0.838108i	$0.683662\pi$
$348$	0	0
$349$	20.0961	1.07572	0.537859	−	0.843035i	$-0.319233\pi$
$349$	20.0961	1.07572	0.537859	+	0.843035i	$0.319233\pi$
$350$	0	0
$351$	−5.82443	−0.310885
$352$	0	0
$353$	−2.97043	−0.158100	−0.0790499	−	0.996871i	$-0.525189\pi$
$353$	−2.97043	−0.158100	−0.0790499	+	0.996871i	$0.525189\pi$
$354$	0	0
$355$	31.2292	1.65747
$356$	0	0
$357$	−4.58247	−0.242530
$358$	0	0
$359$	20.9055	1.10335	0.551674	−	0.834060i	$-0.313989\pi$
$359$	20.9055	1.10335	0.551674	+	0.834060i	$0.313989\pi$
$360$	0	0
$361$	16.1843	0.851807
$362$	0	0
$363$	10.8468	0.569311
$364$	0	0
$365$	9.04118	0.473237
$366$	0	0
$367$	−5.20408	−0.271651	−0.135825	−	0.990733i	$-0.543369\pi$
$367$	−5.20408	−0.271651	−0.135825	+	0.990733i	$0.543369\pi$
$368$	0	0
$369$	−0.915782	−0.0476737
$370$	0	0
$371$	−12.4802	−0.647938
$372$	0	0
$373$	28.5182	1.47662	0.738308	−	0.674463i	$-0.235625\pi$
$373$	28.5182	1.47662	0.738308	+	0.674463i	$0.235625\pi$
$374$	0	0
$375$	−20.1371	−1.03988
$376$	0	0
$377$	1.87977	0.0968130
$378$	0	0
$379$	28.2937	1.45335	0.726674	−	0.686982i	$-0.241065\pi$
$379$	28.2937	1.45335	0.726674	+	0.686982i	$0.241065\pi$
$380$	0	0
$381$	−28.2072	−1.44510
$382$	0	0
$383$	31.9829	1.63425	0.817125	−	0.576460i	$-0.195566\pi$
$383$	31.9829	1.63425	0.817125	+	0.576460i	$0.195566\pi$
$384$	0	0
$385$	5.63791	0.287334
$386$	0	0
$387$	8.40918	0.427462
$388$	0	0
$389$	−8.82599	−0.447495	−0.223748	−	0.974647i	$-0.571829\pi$
$389$	−8.82599	−0.447495	−0.223748	+	0.974647i	$0.571829\pi$
$390$	0	0
$391$	15.9816	0.808227
$392$	0	0
$393$	13.0392	0.657741
$394$	0	0
$395$	−32.3525	−1.62783
$396$	0	0
$397$	3.85840	0.193648	0.0968238	−	0.995302i	$-0.469132\pi$
$397$	3.85840	0.193648	0.0968238	+	0.995302i	$0.469132\pi$
$398$	0	0
$399$	−11.7377	−0.587621
$400$	0	0
$401$	−16.9828	−0.848083	−0.424041	−	0.905643i	$-0.639389\pi$
$401$	−16.9828	−0.848083	−0.424041	+	0.905643i	$0.639389\pi$
$402$	0	0
$403$	−7.03554	−0.350465
$404$	0	0
$405$	26.1805	1.30092
$406$	0	0
$407$	21.4790	1.06468
$408$	0	0
$409$	27.6572	1.36756	0.683781	−	0.729687i	$-0.260334\pi$
$409$	27.6572	1.36756	0.683781	+	0.729687i	$0.260334\pi$
$410$	0	0
$411$	−3.64604	−0.179846
$412$	0	0
$413$	−5.58221	−0.274683
$414$	0	0
$415$	7.71811	0.378867
$416$	0	0
$417$	−28.1786	−1.37991
$418$	0	0
$419$	−27.1918	−1.32841	−0.664204	−	0.747551i	$-0.731229\pi$
$419$	−27.1918	−1.32841	−0.664204	+	0.747551i	$0.731229\pi$
$420$	0	0
$421$	11.7335	0.571857	0.285929	−	0.958251i	$-0.407698\pi$
$421$	11.7335	0.571857	0.285929	+	0.958251i	$0.407698\pi$
$422$	0	0
$423$	9.78407	0.475718
$424$	0	0
$425$	−1.75956	−0.0853512
$426$	0	0
$427$	12.8464	0.621683
$428$	0	0
$429$	6.56483	0.316953
$430$	0	0
$431$	−21.9481	−1.05720	−0.528602	−	0.848870i	$-0.677284\pi$
$431$	−21.9481	−1.05720	−0.528602	+	0.848870i	$0.677284\pi$
$432$	0	0
$433$	−5.74092	−0.275891	−0.137946	−	0.990440i	$-0.544050\pi$
$433$	−5.74092	−0.275891	−0.137946	+	0.990440i	$0.544050\pi$
$434$	0	0
$435$	−6.32146	−0.303091
$436$	0	0
$437$	40.9360	1.95823
$438$	0	0
$439$	−5.25421	−0.250770	−0.125385	−	0.992108i	$-0.540017\pi$
$439$	−5.25421	−0.250770	−0.125385	+	0.992108i	$0.540017\pi$
$440$	0	0
$441$	0.915782	0.0436087
$442$	0	0
$443$	−17.1523	−0.814933	−0.407466	−	0.913220i	$-0.633588\pi$
$443$	−17.1523	−0.814933	−0.407466	+	0.913220i	$0.633588\pi$
$444$	0	0
$445$	17.2126	0.815957
$446$	0	0
$447$	26.2442	1.24131
$448$	0	0
$449$	12.9056	0.609052	0.304526	−	0.952504i	$-0.401502\pi$
$449$	12.9056	0.609052	0.304526	+	0.952504i	$0.401502\pi$
$450$	0	0
$451$	−2.34916	−0.110618
$452$	0	0
$453$	−7.84868	−0.368763
$454$	0	0
$455$	−3.38927	−0.158891
$456$	0	0
$457$	8.69584	0.406774	0.203387	−	0.979098i	$-0.434805\pi$
$457$	8.69584	0.406774	0.203387	+	0.979098i	$0.434805\pi$
$458$	0	0
$459$	−9.55088	−0.445797
$460$	0	0
$461$	−8.89235	−0.414158	−0.207079	−	0.978324i	$-0.566396\pi$
$461$	−8.89235	−0.414158	−0.207079	+	0.978324i	$0.566396\pi$
$462$	0	0
$463$	−18.4573	−0.857785	−0.428893	−	0.903356i	$-0.641096\pi$
$463$	−18.4573	−0.857785	−0.428893	+	0.903356i	$0.641096\pi$
$464$	0	0
$465$	23.6597	1.09719
$466$	0	0
$467$	12.8444	0.594367	0.297184	−	0.954820i	$-0.403953\pi$
$467$	12.8444	0.594367	0.297184	+	0.954820i	$0.403953\pi$
$468$	0	0
$469$	−16.3648	−0.755658
$470$	0	0
$471$	−32.0519	−1.47688
$472$	0	0
$473$	21.5712	0.991846
$474$	0	0
$475$	−4.50700	−0.206795
$476$	0	0
$477$	11.4291	0.523303
$478$	0	0
$479$	16.1504	0.737930	0.368965	−	0.929443i	$-0.379712\pi$
$479$	16.1504	0.737930	0.368965	+	0.929443i	$0.379712\pi$
$480$	0	0
$481$	−12.9123	−0.588748
$482$	0	0
$483$	−13.6565	−0.621393
$484$	0	0
$485$	39.9831	1.81554
$486$	0	0
$487$	−22.2567	−1.00855	−0.504275	−	0.863543i	$-0.668240\pi$
$487$	−22.2567	−1.00855	−0.504275	+	0.863543i	$0.668240\pi$
$488$	0	0
$489$	5.34756	0.241825
$490$	0	0
$491$	−5.36579	−0.242155	−0.121077	−	0.992643i	$-0.538635\pi$
$491$	−5.36579	−0.242155	−0.121077	+	0.992643i	$0.538635\pi$
$492$	0	0
$493$	3.08244	0.138826
$494$	0	0
$495$	−5.16309	−0.232064
$496$	0	0
$497$	13.0124	0.583684
$498$	0	0
$499$	−32.0927	−1.43667	−0.718333	−	0.695700i	$-0.755094\pi$
$499$	−32.0927	−1.43667	−0.718333	+	0.695700i	$0.755094\pi$
$500$	0	0
$501$	−5.95216	−0.265923
$502$	0	0
$503$	−27.8846	−1.24331	−0.621657	−	0.783290i	$-0.713540\pi$
$503$	−27.8846	−1.24331	−0.621657	+	0.783290i	$0.713540\pi$
$504$	0	0
$505$	−13.4271	−0.597500
$506$	0	0
$507$	21.7783	0.967210
$508$	0	0
$509$	20.4222	0.905198	0.452599	−	0.891714i	$-0.350497\pi$
$509$	20.4222	0.905198	0.452599	+	0.891714i	$0.350497\pi$
$510$	0	0
$511$	3.76721	0.166652
$512$	0	0
$513$	−24.4640	−1.08011
$514$	0	0
$515$	−19.1379	−0.843316
$516$	0	0
$517$	25.0981	1.10381
$518$	0	0
$519$	4.60717	0.202232
$520$	0	0
$521$	−16.3186	−0.714933	−0.357466	−	0.933926i	$-0.616359\pi$
$521$	−16.3186	−0.714933	−0.357466	+	0.933926i	$0.616359\pi$
$522$	0	0
$523$	20.6643	0.903586	0.451793	−	0.892123i	$-0.350785\pi$
$523$	20.6643	0.903586	0.451793	+	0.892123i	$0.350785\pi$
$524$	0	0
$525$	1.50357	0.0656210
$526$	0	0
$527$	−11.5368	−0.502553
$528$	0	0
$529$	24.6279	1.07078
$530$	0	0
$531$	5.11208	0.221846
$532$	0	0
$533$	1.41222	0.0611699
$534$	0	0
$535$	−11.2778	−0.487580
$536$	0	0
$537$	50.0648	2.16045
$538$	0	0
$539$	2.34916	0.101186
$540$	0	0
$541$	10.1812	0.437723	0.218861	−	0.975756i	$-0.429766\pi$
$541$	10.1812	0.437723	0.218861	+	0.975756i	$0.429766\pi$
$542$	0	0
$543$	−37.2761	−1.59967
$544$	0	0
$545$	12.4887	0.534958
$546$	0	0
$547$	23.7769	1.01663	0.508314	−	0.861172i	$-0.330269\pi$
$547$	23.7769	1.01663	0.508314	+	0.861172i	$0.330269\pi$
$548$	0	0
$549$	−11.7645	−0.502098
$550$	0	0
$551$	7.89547	0.336358
$552$	0	0
$553$	−13.4804	−0.573246
$554$	0	0
$555$	43.4225	1.84318
$556$	0	0
$557$	34.4582	1.46004	0.730019	−	0.683426i	$-0.239511\pi$
$557$	34.4582	1.46004	0.730019	+	0.683426i	$0.239511\pi$
$558$	0	0
$559$	−12.9677	−0.548475
$560$	0	0
$561$	10.7650	0.454498
$562$	0	0
$563$	40.9980	1.72786	0.863929	−	0.503614i	$-0.167996\pi$
$563$	40.9980	1.72786	0.863929	+	0.503614i	$0.167996\pi$
$564$	0	0
$565$	−1.77508	−0.0746784
$566$	0	0
$567$	10.9087	0.458122
$568$	0	0
$569$	−10.1797	−0.426757	−0.213379	−	0.976970i	$-0.568447\pi$
$569$	−10.1797	−0.426757	−0.213379	+	0.976970i	$0.568447\pi$
$570$	0	0
$571$	28.8634	1.20790	0.603948	−	0.797023i	$-0.293593\pi$
$571$	28.8634	1.20790	0.603948	+	0.797023i	$0.293593\pi$
$572$	0	0
$573$	23.6056	0.986140
$574$	0	0
$575$	−5.24377	−0.218680
$576$	0	0
$577$	−16.2243	−0.675428	−0.337714	−	0.941249i	$-0.609654\pi$
$577$	−16.2243	−0.675428	−0.337714	+	0.941249i	$0.609654\pi$
$578$	0	0
$579$	−25.9130	−1.07691
$580$	0	0
$581$	3.21593	0.133419
$582$	0	0
$583$	29.3180	1.21423
$584$	0	0
$585$	3.10383	0.128328
$586$	0	0
$587$	35.9683	1.48457	0.742284	−	0.670085i	$-0.233742\pi$
$587$	35.9683	1.48457	0.742284	+	0.670085i	$0.233742\pi$
$588$	0	0
$589$	−29.5509	−1.21762
$590$	0	0
$591$	1.01630	0.0418052
$592$	0	0
$593$	8.03336	0.329891	0.164945	−	0.986303i	$-0.447255\pi$
$593$	8.03336	0.329891	0.164945	+	0.986303i	$0.447255\pi$
$594$	0	0
$595$	−5.55770	−0.227844
$596$	0	0
$597$	−21.9586	−0.898705
$598$	0	0
$599$	−29.0056	−1.18514	−0.592569	−	0.805520i	$-0.701886\pi$
$599$	−29.0056	−1.18514	−0.592569	+	0.805520i	$0.701886\pi$
$600$	0	0
$601$	2.87190	0.117147	0.0585737	−	0.998283i	$-0.481345\pi$
$601$	2.87190	0.117147	0.0585737	+	0.998283i	$0.481345\pi$
$602$	0	0
$603$	14.9866	0.610302
$604$	0	0
$605$	13.1552	0.534836
$606$	0	0
$607$	−8.01281	−0.325230	−0.162615	−	0.986690i	$-0.551993\pi$
$607$	−8.01281	−0.325230	−0.162615	+	0.986690i	$0.551993\pi$
$608$	0	0
$609$	−2.63398	−0.106734
$610$	0	0
$611$	−15.0879	−0.610391
$612$	0	0
$613$	−21.1189	−0.852985	−0.426492	−	0.904491i	$-0.640251\pi$
$613$	−21.1189	−0.852985	−0.426492	+	0.904491i	$0.640251\pi$
$614$	0	0
$615$	−4.74913	−0.191503
$616$	0	0
$617$	−18.3046	−0.736915	−0.368458	−	0.929645i	$-0.620114\pi$
$617$	−18.3046	−0.736915	−0.368458	+	0.929645i	$0.620114\pi$
$618$	0	0
$619$	−17.5006	−0.703410	−0.351705	−	0.936111i	$-0.614398\pi$
$619$	−17.5006	−0.703410	−0.351705	+	0.936111i	$0.614398\pi$
$620$	0	0
$621$	−28.4632	−1.14219
$622$	0	0
$623$	7.17204	0.287342
$624$	0	0
$625$	−28.2218	−1.12887
$626$	0	0
$627$	27.5738	1.10119
$628$	0	0
$629$	−21.1735	−0.844241
$630$	0	0
$631$	40.2553	1.60254	0.801269	−	0.598305i	$-0.204159\pi$
$631$	40.2553	1.60254	0.801269	+	0.598305i	$0.204159\pi$
$632$	0	0
$633$	−55.5900	−2.20950
$634$	0	0
$635$	−34.2102	−1.35759
$636$	0	0
$637$	−1.41222	−0.0559540
$638$	0	0
$639$	−11.9165	−0.471409
$640$	0	0
$641$	15.1018	0.596484	0.298242	−	0.954490i	$-0.403600\pi$
$641$	15.1018	0.596484	0.298242	+	0.954490i	$0.403600\pi$
$642$	0	0
$643$	13.3216	0.525352	0.262676	−	0.964884i	$-0.415395\pi$
$643$	13.3216	0.525352	0.262676	+	0.964884i	$0.415395\pi$
$644$	0	0
$645$	43.6089	1.71710
$646$	0	0
$647$	5.41556	0.212908	0.106454	−	0.994318i	$-0.466050\pi$
$647$	5.41556	0.212908	0.106454	+	0.994318i	$0.466050\pi$
$648$	0	0
$649$	13.1135	0.514751
$650$	0	0
$651$	9.85838	0.386380
$652$	0	0
$653$	−46.2624	−1.81039	−0.905194	−	0.424998i	$-0.860275\pi$
$653$	−46.2624	−1.81039	−0.905194	+	0.424998i	$0.860275\pi$
$654$	0	0
$655$	15.8142	0.617911
$656$	0	0
$657$	−3.44995	−0.134595
$658$	0	0
$659$	41.1999	1.60492	0.802460	−	0.596706i	$-0.203524\pi$
$659$	41.1999	1.60492	0.802460	+	0.596706i	$0.203524\pi$
$660$	0	0
$661$	−28.6270	−1.11346	−0.556731	−	0.830693i	$-0.687945\pi$
$661$	−28.6270	−1.11346	−0.556731	+	0.830693i	$0.687945\pi$
$662$	0	0
$663$	−6.47144	−0.251330
$664$	0	0
$665$	−14.2357	−0.552037
$666$	0	0
$667$	9.18616	0.355690
$668$	0	0
$669$	28.4863	1.10134
$670$	0	0
$671$	−30.1784	−1.16502
$672$	0	0
$673$	26.5265	1.02252	0.511261	−	0.859426i	$-0.329179\pi$
$673$	26.5265	1.02252	0.511261	+	0.859426i	$0.329179\pi$
$674$	0	0
$675$	3.13376	0.120618
$676$	0	0
$677$	−28.4442	−1.09320	−0.546600	−	0.837394i	$-0.684078\pi$
$677$	−28.4442	−1.09320	−0.546600	+	0.837394i	$0.684078\pi$
$678$	0	0
$679$	16.6599	0.639347
$680$	0	0
$681$	−0.927262	−0.0355328
$682$	0	0
$683$	0.104618	0.00400309	0.00200155	−	0.999998i	$-0.499363\pi$
$683$	0.104618	0.00400309	0.00200155	+	0.999998i	$0.499363\pi$
$684$	0	0
$685$	−4.42198	−0.168955
$686$	0	0
$687$	1.35254	0.0516027
$688$	0	0
$689$	−17.6247	−0.671447
$690$	0	0
$691$	24.0620	0.915361	0.457681	−	0.889117i	$-0.348680\pi$
$691$	24.0620	0.915361	0.457681	+	0.889117i	$0.348680\pi$
$692$	0	0
$693$	−2.15132	−0.0817220
$694$	0	0
$695$	−34.1755	−1.29635
$696$	0	0
$697$	2.31575	0.0877151
$698$	0	0
$699$	−0.161202	−0.00609723
$700$	0	0
$701$	21.9812	0.830220	0.415110	−	0.909771i	$-0.363743\pi$
$701$	21.9812	0.830220	0.415110	+	0.909771i	$0.363743\pi$
$702$	0	0
$703$	−54.2345	−2.04549
$704$	0	0
$705$	50.7389	1.91094
$706$	0	0
$707$	−5.59473	−0.210411
$708$	0	0
$709$	−8.66268	−0.325334	−0.162667	−	0.986681i	$-0.552010\pi$
$709$	−8.66268	−0.325334	−0.162667	+	0.986681i	$0.552010\pi$
$710$	0	0
$711$	12.3451	0.462979
$712$	0	0
$713$	−34.3817	−1.28760
$714$	0	0
$715$	7.96195	0.297760
$716$	0	0
$717$	−46.0866	−1.72114
$718$	0	0
$719$	15.3180	0.571267	0.285633	−	0.958339i	$-0.407796\pi$
$719$	15.3180	0.571267	0.285633	+	0.958339i	$0.407796\pi$
$720$	0	0
$721$	−7.97424	−0.296976
$722$	0	0
$723$	−49.5531	−1.84290
$724$	0	0
$725$	−1.01138	−0.0375619
$726$	0	0
$727$	23.2144	0.860975	0.430487	−	0.902597i	$-0.358342\pi$
$727$	23.2144	0.860975	0.430487	+	0.902597i	$0.358342\pi$
$728$	0	0
$729$	14.4941	0.536817
$730$	0	0
$731$	−21.2644	−0.786491
$732$	0	0
$733$	47.5792	1.75738	0.878690	−	0.477394i	$-0.158418\pi$
$733$	47.5792	1.75738	0.878690	+	0.477394i	$0.158418\pi$
$734$	0	0
$735$	4.74913	0.175174
$736$	0	0
$737$	38.4437	1.41609
$738$	0	0
$739$	−10.8040	−0.397432	−0.198716	−	0.980057i	$-0.563677\pi$
$739$	−10.8040	−0.397432	−0.198716	+	0.980057i	$0.563677\pi$
$740$	0	0
$741$	−16.5762	−0.608942
$742$	0	0
$743$	7.45742	0.273586	0.136793	−	0.990600i	$-0.456320\pi$
$743$	7.45742	0.273586	0.136793	+	0.990600i	$0.456320\pi$
$744$	0	0
$745$	31.8294	1.16614
$746$	0	0
$747$	−2.94509	−0.107755
$748$	0	0
$749$	−4.69914	−0.171703
$750$	0	0
$751$	9.09145	0.331752	0.165876	−	0.986147i	$-0.446955\pi$
$751$	9.09145	0.331752	0.165876	+	0.986147i	$0.446955\pi$
$752$	0	0
$753$	33.0453	1.20424
$754$	0	0
$755$	−9.51901	−0.346432
$756$	0	0
$757$	8.31853	0.302342	0.151171	−	0.988508i	$-0.451696\pi$
$757$	8.31853	0.302342	0.151171	+	0.988508i	$0.451696\pi$
$758$	0	0
$759$	32.0814	1.16448
$760$	0	0
$761$	2.92558	0.106052	0.0530260	−	0.998593i	$-0.483113\pi$
$761$	2.92558	0.106052	0.0530260	+	0.998593i	$0.483113\pi$
$762$	0	0
$763$	5.20372	0.188387
$764$	0	0
$765$	5.08964	0.184016
$766$	0	0
$767$	−7.88329	−0.284649
$768$	0	0
$769$	25.8384	0.931757	0.465878	−	0.884849i	$-0.345738\pi$
$769$	25.8384	0.931757	0.465878	+	0.884849i	$0.345738\pi$
$770$	0	0
$771$	5.71017	0.205647
$772$	0	0
$773$	47.1884	1.69725	0.848624	−	0.528997i	$-0.177432\pi$
$773$	47.1884	1.69725	0.848624	+	0.528997i	$0.177432\pi$
$774$	0	0
$775$	3.78538	0.135975
$776$	0	0
$777$	18.0930	0.649082
$778$	0	0
$779$	5.93164	0.212523
$780$	0	0
$781$	−30.5682	−1.09382
$782$	0	0
$783$	−5.48979	−0.196189
$784$	0	0
$785$	−38.8732	−1.38744
$786$	0	0
$787$	53.1847	1.89583	0.947915	−	0.318524i	$-0.103187\pi$
$787$	53.1847	1.89583	0.947915	+	0.318524i	$0.103187\pi$
$788$	0	0
$789$	41.7810	1.48744
$790$	0	0
$791$	−0.739630	−0.0262982
$792$	0	0
$793$	18.1420	0.644240
$794$	0	0
$795$	59.2699	2.10209
$796$	0	0
$797$	−12.5167	−0.443364	−0.221682	−	0.975119i	$-0.571155\pi$
$797$	−12.5167	−0.443364	−0.221682	+	0.975119i	$0.571155\pi$
$798$	0	0
$799$	−24.7411	−0.875276
$800$	0	0
$801$	−6.56802	−0.232070
$802$	0	0
$803$	−8.84981	−0.312303
$804$	0	0
$805$	−16.5629	−0.583764
$806$	0	0
$807$	−5.84111	−0.205617
$808$	0	0
$809$	24.3264	0.855272	0.427636	−	0.903951i	$-0.359347\pi$
$809$	24.3264	0.855272	0.427636	+	0.903951i	$0.359347\pi$
$810$	0	0
$811$	6.12666	0.215136	0.107568	−	0.994198i	$-0.465694\pi$
$811$	6.12666	0.215136	0.107568	+	0.994198i	$0.465694\pi$
$812$	0	0
$813$	21.1663	0.742335
$814$	0	0
$815$	6.48561	0.227181
$816$	0	0
$817$	−54.4673	−1.90557
$818$	0	0
$819$	1.29328	0.0451909
$820$	0	0
$821$	−3.00991	−0.105047	−0.0525233	−	0.998620i	$-0.516726\pi$
$821$	−3.00991	−0.105047	−0.0525233	+	0.998620i	$0.516726\pi$
$822$	0	0
$823$	−20.0821	−0.700017	−0.350008	−	0.936747i	$-0.613821\pi$
$823$	−20.0821	−0.700017	−0.350008	+	0.936747i	$0.613821\pi$
$824$	0	0
$825$	−3.53212	−0.122973
$826$	0	0
$827$	−42.3948	−1.47421	−0.737105	−	0.675778i	$-0.763808\pi$
$827$	−42.3948	−1.47421	−0.737105	+	0.675778i	$0.763808\pi$
$828$	0	0
$829$	24.2774	0.843189	0.421595	−	0.906784i	$-0.361471\pi$
$829$	24.2774	0.843189	0.421595	+	0.906784i	$0.361471\pi$
$830$	0	0
$831$	7.34366	0.254749
$832$	0	0
$833$	−2.31575	−0.0802358
$834$	0	0
$835$	−7.21889	−0.249820
$836$	0	0
$837$	20.5470	0.710208
$838$	0	0
$839$	50.5278	1.74441	0.872206	−	0.489138i	$-0.162689\pi$
$839$	50.5278	1.74441	0.872206	+	0.489138i	$0.162689\pi$
$840$	0	0
$841$	−27.2282	−0.938905
$842$	0	0
$843$	−19.6793	−0.677793
$844$	0	0
$845$	26.4131	0.908640
$846$	0	0
$847$	5.48143	0.188344
$848$	0	0
$849$	1.98412	0.0680947
$850$	0	0
$851$	−63.1003	−2.16305
$852$	0	0
$853$	−3.31868	−0.113629	−0.0568147	−	0.998385i	$-0.518094\pi$
$853$	−3.31868	−0.113629	−0.0568147	+	0.998385i	$0.518094\pi$
$854$	0	0
$855$	13.0368	0.445849
$856$	0	0
$857$	−21.4783	−0.733684	−0.366842	−	0.930283i	$-0.619561\pi$
$857$	−21.4783	−0.733684	−0.366842	+	0.930283i	$0.619561\pi$
$858$	0	0
$859$	−50.4406	−1.72101	−0.860505	−	0.509442i	$-0.829852\pi$
$859$	−50.4406	−1.72101	−0.860505	+	0.509442i	$0.829852\pi$
$860$	0	0
$861$	−1.97883	−0.0674385
$862$	0	0
$863$	47.6212	1.62104	0.810522	−	0.585708i	$-0.199183\pi$
$863$	47.6212	1.62104	0.810522	+	0.585708i	$0.199183\pi$
$864$	0	0
$865$	5.58765	0.189986
$866$	0	0
$867$	23.0283	0.782083
$868$	0	0
$869$	31.6677	1.07425
$870$	0	0
$871$	−23.1107	−0.783076
$872$	0	0
$873$	−15.2568	−0.516364
$874$	0	0
$875$	−10.1763	−0.344021
$876$	0	0
$877$	45.3997	1.53304	0.766520	−	0.642221i	$-0.221987\pi$
$877$	45.3997	1.53304	0.766520	+	0.642221i	$0.221987\pi$
$878$	0	0
$879$	46.4589	1.56702
$880$	0	0
$881$	−14.2445	−0.479909	−0.239954	−	0.970784i	$-0.577132\pi$
$881$	−14.2445	−0.479909	−0.239954	+	0.970784i	$0.577132\pi$
$882$	0	0
$883$	2.93736	0.0988501	0.0494251	−	0.998778i	$-0.484261\pi$
$883$	2.93736	0.0988501	0.0494251	+	0.998778i	$0.484261\pi$
$884$	0	0
$885$	26.5106	0.891145
$886$	0	0
$887$	48.0250	1.61252	0.806261	−	0.591559i	$-0.201487\pi$
$887$	48.0250	1.61252	0.806261	+	0.591559i	$0.201487\pi$
$888$	0	0
$889$	−14.2545	−0.478080
$890$	0	0
$891$	−25.6263	−0.858513
$892$	0	0
$893$	−63.3727	−2.12069
$894$	0	0
$895$	60.7194	2.02963
$896$	0	0
$897$	−19.2860	−0.643939
$898$	0	0
$899$	−6.63131	−0.221167
$900$	0	0
$901$	−28.9009	−0.962828
$902$	0	0
$903$	18.1707	0.604682
$904$	0	0
$905$	−45.2090	−1.50280
$906$	0	0
$907$	−58.5690	−1.94475	−0.972375	−	0.233424i	$-0.925007\pi$
$907$	−58.5690	−1.94475	−0.972375	+	0.233424i	$0.925007\pi$
$908$	0	0
$909$	5.12355	0.169937
$910$	0	0
$911$	7.14793	0.236821	0.118411	−	0.992965i	$-0.462220\pi$
$911$	7.14793	0.236821	0.118411	+	0.992965i	$0.462220\pi$
$912$	0	0
$913$	−7.55475	−0.250026
$914$	0	0
$915$	−61.0094	−2.01691
$916$	0	0
$917$	6.58934	0.217599
$918$	0	0
$919$	27.0249	0.891468	0.445734	−	0.895165i	$-0.352943\pi$
$919$	27.0249	0.891468	0.445734	+	0.895165i	$0.352943\pi$
$920$	0	0
$921$	−36.1789	−1.19214
$922$	0	0
$923$	18.3763	0.604862
$924$	0	0
$925$	6.94727	0.228425
$926$	0	0
$927$	7.30266	0.239851
$928$	0	0
$929$	−39.8861	−1.30862	−0.654310	−	0.756226i	$-0.727041\pi$
$929$	−39.8861	−1.30862	−0.654310	+	0.756226i	$0.727041\pi$
$930$	0	0
$931$	−5.93164	−0.194402
$932$	0	0
$933$	−53.8764	−1.76384
$934$	0	0
$935$	13.0560	0.426976
$936$	0	0
$937$	12.4056	0.405274	0.202637	−	0.979254i	$-0.435049\pi$
$937$	12.4056	0.405274	0.202637	+	0.979254i	$0.435049\pi$
$938$	0	0
$939$	30.6760	1.00107
$940$	0	0
$941$	−7.18199	−0.234126	−0.117063	−	0.993124i	$-0.537348\pi$
$941$	−7.18199	−0.234126	−0.117063	+	0.993124i	$0.537348\pi$
$942$	0	0
$943$	6.90130	0.224737
$944$	0	0
$945$	9.89822	0.321989
$946$	0	0
$947$	34.8341	1.13196	0.565978	−	0.824421i	$-0.308499\pi$
$947$	34.8341	1.13196	0.565978	+	0.824421i	$0.308499\pi$
$948$	0	0
$949$	5.32012	0.172698
$950$	0	0
$951$	−57.7826	−1.87373
$952$	0	0
$953$	20.2941	0.657391	0.328695	−	0.944436i	$-0.393391\pi$
$953$	20.2941	0.657391	0.328695	+	0.944436i	$0.393391\pi$
$954$	0	0
$955$	28.6293	0.926423
$956$	0	0
$957$	6.18765	0.200018
$958$	0	0
$959$	−1.84252	−0.0594981
$960$	0	0
$961$	−6.18054	−0.199372
$962$	0	0
$963$	4.30339	0.138675
$964$	0	0
$965$	−31.4277	−1.01169
$966$	0	0
$967$	−6.28348	−0.202063	−0.101032	−	0.994883i	$-0.532214\pi$
$967$	−6.28348	−0.202063	−0.101032	+	0.994883i	$0.532214\pi$
$968$	0	0
$969$	−27.1816	−0.873198
$970$	0	0
$971$	2.04260	0.0655501	0.0327750	−	0.999463i	$-0.489566\pi$
$971$	2.04260	0.0655501	0.0327750	+	0.999463i	$0.489566\pi$
$972$	0	0
$973$	−14.2400	−0.456513
$974$	0	0
$975$	2.12336	0.0680019
$976$	0	0
$977$	27.4777	0.879089	0.439544	−	0.898221i	$-0.355140\pi$
$977$	27.4777	0.879089	0.439544	+	0.898221i	$0.355140\pi$
$978$	0	0
$979$	−16.8483	−0.538474
$980$	0	0
$981$	−4.76547	−0.152150
$982$	0	0
$983$	−19.2231	−0.613120	−0.306560	−	0.951851i	$-0.599178\pi$
$983$	−19.2231	−0.613120	−0.306560	+	0.951851i	$0.599178\pi$
$984$	0	0
$985$	1.23259	0.0392736
$986$	0	0
$987$	21.1416	0.672943
$988$	0	0
$989$	−63.3712	−2.01509
$990$	0	0
$991$	−27.9908	−0.889158	−0.444579	−	0.895740i	$-0.646647\pi$
$991$	−27.9908	−0.889158	−0.444579	+	0.895740i	$0.646647\pi$
$992$	0	0
$993$	27.7444	0.880441
$994$	0	0
$995$	−26.6318	−0.844284
$996$	0	0
$997$	34.0980	1.07989	0.539947	−	0.841699i	$-0.318444\pi$
$997$	34.0980	1.07989	0.539947	+	0.841699i	$0.318444\pi$
$998$	0	0
$999$	37.7097	1.19308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.d.1.1	✓	5
4.3	odd	2		4592.2.a.bc.1.5		5
7.6	odd	2		8036.2.a.k.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.d.1.1	✓	5	1.1	even	1	trivial
4592.2.a.bc.1.5		5	4.3	odd	2
8036.2.a.k.1.5		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)

\(f(q)\)	\(=\)	\(q-1.97883 q^{3} -2.39996 q^{5} -1.00000 q^{7} +0.915782 q^{9} +O(q^{10})\)	\(q-1.97883 q^{3} -2.39996 q^{5} -1.00000 q^{7} +0.915782 q^{9} +2.34916 q^{11} -1.41222 q^{13} +4.74913 q^{15} -2.31575 q^{17} -5.93164 q^{19} +1.97883 q^{21} -6.90130 q^{23} +0.759824 q^{25} +4.12432 q^{27} -1.33108 q^{29} +4.98191 q^{31} -4.64860 q^{33} +2.39996 q^{35} +9.14326 q^{37} +2.79454 q^{39} -1.00000 q^{41} +9.18251 q^{43} -2.19784 q^{45} +10.6838 q^{47} +1.00000 q^{49} +4.58247 q^{51} +12.4802 q^{53} -5.63791 q^{55} +11.7377 q^{57} +5.58221 q^{59} -12.8464 q^{61} -0.915782 q^{63} +3.38927 q^{65} +16.3648 q^{67} +13.6565 q^{69} -13.0124 q^{71} -3.76721 q^{73} -1.50357 q^{75} -2.34916 q^{77} +13.4804 q^{79} -10.9087 q^{81} -3.21593 q^{83} +5.55770 q^{85} +2.63398 q^{87} -7.17204 q^{89} +1.41222 q^{91} -9.85838 q^{93} +14.2357 q^{95} -16.6599 q^{97} +2.15132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 3 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 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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