LMFDB - Embedded newform 2322.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2322,2,Mod(1,2322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2322, 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("2322.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2322 = 2 \cdot 3^{3} \cdot 43$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2322.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:	$18.5412633493$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2322.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.58579 q^{7} -1.00000 q^{8} -2.82843 q^{10} -0.828427 q^{11} +2.82843 q^{13} +1.58579 q^{14} +1.00000 q^{16} -3.65685 q^{17} -7.65685 q^{19} +2.82843 q^{20} +0.828427 q^{22} -6.00000 q^{23} +3.00000 q^{25} -2.82843 q^{26} -1.58579 q^{28} +8.00000 q^{29} -9.65685 q^{31} -1.00000 q^{32} +3.65685 q^{34} -4.48528 q^{35} -5.58579 q^{37} +7.65685 q^{38} -2.82843 q^{40} +4.00000 q^{41} -1.00000 q^{43} -0.828427 q^{44} +6.00000 q^{46} -10.0711 q^{47} -4.48528 q^{49} -3.00000 q^{50} +2.82843 q^{52} +6.41421 q^{53} -2.34315 q^{55} +1.58579 q^{56} -8.00000 q^{58} -5.17157 q^{59} -1.17157 q^{61} +9.65685 q^{62} +1.00000 q^{64} +8.00000 q^{65} -13.0000 q^{67} -3.65685 q^{68} +4.48528 q^{70} +16.1421 q^{71} -12.8284 q^{73} +5.58579 q^{74} -7.65685 q^{76} +1.31371 q^{77} +4.34315 q^{79} +2.82843 q^{80} -4.00000 q^{82} +2.00000 q^{83} -10.3431 q^{85} +1.00000 q^{86} +0.828427 q^{88} +1.48528 q^{89} -4.48528 q^{91} -6.00000 q^{92} +10.0711 q^{94} -21.6569 q^{95} +13.3137 q^{97} +4.48528 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} + 4 q^{11} + 6 q^{14} + 2 q^{16} + 4 q^{17} - 4 q^{19} - 4 q^{22} - 12 q^{23} + 6 q^{25} - 6 q^{28} + 16 q^{29} - 8 q^{31} - 2 q^{32} - 4 q^{34} + 8 q^{35} - 14 q^{37}+ \cdots - 8 q^{98}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 + 4 * q^11 + 6 * q^14 + 2 * q^16 + 4 * q^17 - 4 * q^19 - 4 * q^22 - 12 * q^23 + 6 * q^25 - 6 * q^28 + 16 * q^29 - 8 * q^31 - 2 * q^32 - 4 * q^34 + 8 * q^35 - 14 * q^37 + 4 * q^38 + 8 * q^41 - 2 * q^43 + 4 * q^44 + 12 * q^46 - 6 * q^47 + 8 * q^49 - 6 * q^50 + 10 * q^53 - 16 * q^55 + 6 * q^56 - 16 * q^58 - 16 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 + 16 * q^65 - 26 * q^67 + 4 * q^68 - 8 * q^70 + 4 * q^71 - 20 * q^73 + 14 * q^74 - 4 * q^76 - 20 * q^77 + 20 * q^79 - 8 * q^82 + 4 * q^83 - 32 * q^85 + 2 * q^86 - 4 * q^88 - 14 * q^89 + 8 * q^91 - 12 * q^92 + 6 * q^94 - 32 * q^95 + 4 * q^97 - 8 * 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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	−1.58579	−0.599371	−0.299685	−	0.954038i	$-0.596882\pi$
$7$	−1.58579	−0.599371	−0.299685	+	0.954038i	$0.596882\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.82843	−0.894427
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	1.58579	0.423819
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	−7.65685	−1.75660	−0.878301	−	0.478107i	$-0.841323\pi$
$19$	−7.65685	−1.75660	−0.878301	+	0.478107i	$0.841323\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	0.828427	0.176621
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	−2.82843	−0.554700
$27$	0	0
$28$	−1.58579	−0.299685
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.65685	0.627145
$35$	−4.48528	−0.758151
$36$	0	0
$37$	−5.58579	−0.918298	−0.459149	−	0.888359i	$-0.651846\pi$
$37$	−5.58579	−0.918298	−0.459149	+	0.888359i	$0.651846\pi$
$38$	7.65685	1.24211
$39$	0	0
$40$	−2.82843	−0.447214
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−0.828427	−0.124890
$45$	0	0
$46$	6.00000	0.884652
$47$	−10.0711	−1.46902	−0.734508	−	0.678600i	$-0.762587\pi$
$47$	−10.0711	−1.46902	−0.734508	+	0.678600i	$0.762587\pi$
$48$	0	0
$49$	−4.48528	−0.640754
$50$	−3.00000	−0.424264
$51$	0	0
$52$	2.82843	0.392232
$53$	6.41421	0.881060	0.440530	−	0.897738i	$-0.354791\pi$
$53$	6.41421	0.881060	0.440530	+	0.897738i	$0.354791\pi$
$54$	0	0
$55$	−2.34315	−0.315950
$56$	1.58579	0.211910
$57$	0	0
$58$	−8.00000	−1.05045
$59$	−5.17157	−0.673281	−0.336641	−	0.941633i	$-0.609291\pi$
$59$	−5.17157	−0.673281	−0.336641	+	0.941633i	$0.609291\pi$
$60$	0	0
$61$	−1.17157	−0.150005	−0.0750023	−	0.997183i	$-0.523896\pi$
$61$	−1.17157	−0.150005	−0.0750023	+	0.997183i	$0.523896\pi$
$62$	9.65685	1.22642
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−3.65685	−0.443459
$69$	0	0
$70$	4.48528	0.536094
$71$	16.1421	1.91572	0.957860	−	0.287237i	$-0.0927368\pi$
$71$	16.1421	1.91572	0.957860	+	0.287237i	$0.0927368\pi$
$72$	0	0
$73$	−12.8284	−1.50145	−0.750727	−	0.660613i	$-0.770297\pi$
$73$	−12.8284	−1.50145	−0.750727	+	0.660613i	$0.770297\pi$
$74$	5.58579	0.649334
$75$	0	0
$76$	−7.65685	−0.878301
$77$	1.31371	0.149711
$78$	0	0
$79$	4.34315	0.488642	0.244321	−	0.969694i	$-0.421435\pi$
$79$	4.34315	0.488642	0.244321	+	0.969694i	$0.421435\pi$
$80$	2.82843	0.316228
$81$	0	0
$82$	−4.00000	−0.441726
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−10.3431	−1.12187
$86$	1.00000	0.107833
$87$	0	0
$88$	0.828427	0.0883106
$89$	1.48528	0.157440	0.0787198	−	0.996897i	$-0.474917\pi$
$89$	1.48528	0.157440	0.0787198	+	0.996897i	$0.474917\pi$
$90$	0	0
$91$	−4.48528	−0.470185
$92$	−6.00000	−0.625543
$93$	0	0
$94$	10.0711	1.03875
$95$	−21.6569	−2.22195
$96$	0	0
$97$	13.3137	1.35180	0.675901	−	0.736992i	$-0.263755\pi$
$97$	13.3137	1.35180	0.675901	+	0.736992i	$0.263755\pi$
$98$	4.48528	0.453082
$99$	0	0
$100$	3.00000	0.300000
$101$	−6.82843	−0.679454	−0.339727	−	0.940524i	$-0.610335\pi$
$101$	−6.82843	−0.679454	−0.339727	+	0.940524i	$0.610335\pi$
$102$	0	0
$103$	15.6569	1.54272	0.771358	−	0.636402i	$-0.219578\pi$
$103$	15.6569	1.54272	0.771358	+	0.636402i	$0.219578\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	−6.41421	−0.623003
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	−12.4853	−1.19587	−0.597937	−	0.801543i	$-0.704013\pi$
$109$	−12.4853	−1.19587	−0.597937	+	0.801543i	$0.704013\pi$
$110$	2.34315	0.223410
$111$	0	0
$112$	−1.58579	−0.149843
$113$	14.9706	1.40831	0.704156	−	0.710045i	$-0.251326\pi$
$113$	14.9706	1.40831	0.704156	+	0.710045i	$0.251326\pi$
$114$	0	0
$115$	−16.9706	−1.58251
$116$	8.00000	0.742781
$117$	0	0
$118$	5.17157	0.476082
$119$	5.79899	0.531593
$120$	0	0
$121$	−10.3137	−0.937610
$122$	1.17157	0.106069
$123$	0	0
$124$	−9.65685	−0.867211
$125$	−5.65685	−0.505964
$126$	0	0
$127$	0.828427	0.0735110	0.0367555	−	0.999324i	$-0.488298\pi$
$127$	0.828427	0.0735110	0.0367555	+	0.999324i	$0.488298\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.00000	−0.701646
$131$	19.8284	1.73242	0.866209	−	0.499682i	$-0.166550\pi$
$131$	19.8284	1.73242	0.866209	+	0.499682i	$0.166550\pi$
$132$	0	0
$133$	12.1421	1.05286
$134$	13.0000	1.12303
$135$	0	0
$136$	3.65685	0.313573
$137$	−14.3137	−1.22290	−0.611451	−	0.791282i	$-0.709414\pi$
$137$	−14.3137	−1.22290	−0.611451	+	0.791282i	$0.709414\pi$
$138$	0	0
$139$	14.3137	1.21407	0.607036	−	0.794674i	$-0.292358\pi$
$139$	14.3137	1.21407	0.607036	+	0.794674i	$0.292358\pi$
$140$	−4.48528	−0.379075
$141$	0	0
$142$	−16.1421	−1.35462
$143$	−2.34315	−0.195944
$144$	0	0
$145$	22.6274	1.87910
$146$	12.8284	1.06169
$147$	0	0
$148$	−5.58579	−0.459149
$149$	−9.31371	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$149$	−9.31371	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$150$	0	0
$151$	10.9706	0.892772	0.446386	−	0.894841i	$-0.352711\pi$
$151$	10.9706	0.892772	0.446386	+	0.894841i	$0.352711\pi$
$152$	7.65685	0.621053
$153$	0	0
$154$	−1.31371	−0.105862
$155$	−27.3137	−2.19389
$156$	0	0
$157$	6.82843	0.544968	0.272484	−	0.962160i	$-0.412155\pi$
$157$	6.82843	0.544968	0.272484	+	0.962160i	$0.412155\pi$
$158$	−4.34315	−0.345522
$159$	0	0
$160$	−2.82843	−0.223607
$161$	9.51472	0.749865
$162$	0	0
$163$	−16.9706	−1.32924	−0.664619	−	0.747183i	$-0.731406\pi$
$163$	−16.9706	−1.32924	−0.664619	+	0.747183i	$0.731406\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−2.00000	−0.155230
$167$	1.58579	0.122712	0.0613559	−	0.998116i	$-0.480458\pi$
$167$	1.58579	0.122712	0.0613559	+	0.998116i	$0.480458\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	10.3431	0.793283
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	−1.92893	−0.146654	−0.0733270	−	0.997308i	$-0.523362\pi$
$173$	−1.92893	−0.146654	−0.0733270	+	0.997308i	$0.523362\pi$
$174$	0	0
$175$	−4.75736	−0.359623
$176$	−0.828427	−0.0624450
$177$	0	0
$178$	−1.48528	−0.111327
$179$	−4.51472	−0.337446	−0.168723	−	0.985664i	$-0.553964\pi$
$179$	−4.51472	−0.337446	−0.168723	+	0.985664i	$0.553964\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	4.48528	0.332471
$183$	0	0
$184$	6.00000	0.442326
$185$	−15.7990	−1.16156
$186$	0	0
$187$	3.02944	0.221534
$188$	−10.0711	−0.734508
$189$	0	0
$190$	21.6569	1.57115
$191$	3.65685	0.264601	0.132300	−	0.991210i	$-0.457764\pi$
$191$	3.65685	0.264601	0.132300	+	0.991210i	$0.457764\pi$
$192$	0	0
$193$	−7.82843	−0.563503	−0.281751	−	0.959487i	$-0.590915\pi$
$193$	−7.82843	−0.563503	−0.281751	+	0.959487i	$0.590915\pi$
$194$	−13.3137	−0.955869
$195$	0	0
$196$	−4.48528	−0.320377
$197$	−6.14214	−0.437609	−0.218805	−	0.975769i	$-0.570216\pi$
$197$	−6.14214	−0.437609	−0.218805	+	0.975769i	$0.570216\pi$
$198$	0	0
$199$	10.0711	0.713919	0.356960	−	0.934120i	$-0.383813\pi$
$199$	10.0711	0.713919	0.356960	+	0.934120i	$0.383813\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	6.82843	0.480446
$203$	−12.6863	−0.890403
$204$	0	0
$205$	11.3137	0.790184
$206$	−15.6569	−1.09086
$207$	0	0
$208$	2.82843	0.196116
$209$	6.34315	0.438765
$210$	0	0
$211$	−27.6569	−1.90398	−0.951988	−	0.306135i	$-0.900964\pi$
$211$	−27.6569	−1.90398	−0.951988	+	0.306135i	$0.900964\pi$
$212$	6.41421	0.440530
$213$	0	0
$214$	−2.00000	−0.136717
$215$	−2.82843	−0.192897
$216$	0	0
$217$	15.3137	1.03956
$218$	12.4853	0.845610
$219$	0	0
$220$	−2.34315	−0.157975
$221$	−10.3431	−0.695755
$222$	0	0
$223$	25.2426	1.69037	0.845186	−	0.534472i	$-0.179490\pi$
$223$	25.2426	1.69037	0.845186	+	0.534472i	$0.179490\pi$
$224$	1.58579	0.105955
$225$	0	0
$226$	−14.9706	−0.995827
$227$	−27.4853	−1.82426	−0.912131	−	0.409899i	$-0.865564\pi$
$227$	−27.4853	−1.82426	−0.912131	+	0.409899i	$0.865564\pi$
$228$	0	0
$229$	−5.51472	−0.364423	−0.182211	−	0.983259i	$-0.558326\pi$
$229$	−5.51472	−0.364423	−0.182211	+	0.983259i	$0.558326\pi$
$230$	16.9706	1.11901
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−22.7990	−1.49361	−0.746806	−	0.665042i	$-0.768414\pi$
$233$	−22.7990	−1.49361	−0.746806	+	0.665042i	$0.768414\pi$
$234$	0	0
$235$	−28.4853	−1.85817
$236$	−5.17157	−0.336641
$237$	0	0
$238$	−5.79899	−0.375893
$239$	−21.7279	−1.40546	−0.702731	−	0.711455i	$-0.748036\pi$
$239$	−21.7279	−1.40546	−0.702731	+	0.711455i	$0.748036\pi$
$240$	0	0
$241$	−22.9706	−1.47966	−0.739832	−	0.672792i	$-0.765095\pi$
$241$	−22.9706	−1.47966	−0.739832	+	0.672792i	$0.765095\pi$
$242$	10.3137	0.662990
$243$	0	0
$244$	−1.17157	−0.0750023
$245$	−12.6863	−0.810497
$246$	0	0
$247$	−21.6569	−1.37799
$248$	9.65685	0.613211
$249$	0	0
$250$	5.65685	0.357771
$251$	20.1421	1.27136	0.635680	−	0.771953i	$-0.280720\pi$
$251$	20.1421	1.27136	0.635680	+	0.771953i	$0.280720\pi$
$252$	0	0
$253$	4.97056	0.312497
$254$	−0.828427	−0.0519801
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.9706	−1.49524	−0.747621	−	0.664125i	$-0.768804\pi$
$257$	−23.9706	−1.49524	−0.747621	+	0.664125i	$0.768804\pi$
$258$	0	0
$259$	8.85786	0.550401
$260$	8.00000	0.496139
$261$	0	0
$262$	−19.8284	−1.22500
$263$	−1.17157	−0.0722423	−0.0361211	−	0.999347i	$-0.511500\pi$
$263$	−1.17157	−0.0722423	−0.0361211	+	0.999347i	$0.511500\pi$
$264$	0	0
$265$	18.1421	1.11446
$266$	−12.1421	−0.744482
$267$	0	0
$268$	−13.0000	−0.794101
$269$	8.41421	0.513024	0.256512	−	0.966541i	$-0.417427\pi$
$269$	8.41421	0.513024	0.256512	+	0.966541i	$0.417427\pi$
$270$	0	0
$271$	−15.6569	−0.951086	−0.475543	−	0.879692i	$-0.657748\pi$
$271$	−15.6569	−0.951086	−0.475543	+	0.879692i	$0.657748\pi$
$272$	−3.65685	−0.221729
$273$	0	0
$274$	14.3137	0.864723
$275$	−2.48528	−0.149868
$276$	0	0
$277$	2.41421	0.145056	0.0725280	−	0.997366i	$-0.476893\pi$
$277$	2.41421	0.145056	0.0725280	+	0.997366i	$0.476893\pi$
$278$	−14.3137	−0.858479
$279$	0	0
$280$	4.48528	0.268047
$281$	1.65685	0.0988396	0.0494198	−	0.998778i	$-0.484263\pi$
$281$	1.65685	0.0988396	0.0494198	+	0.998778i	$0.484263\pi$
$282$	0	0
$283$	−15.8284	−0.940902	−0.470451	−	0.882426i	$-0.655909\pi$
$283$	−15.8284	−0.940902	−0.470451	+	0.882426i	$0.655909\pi$
$284$	16.1421	0.957860
$285$	0	0
$286$	2.34315	0.138553
$287$	−6.34315	−0.374424
$288$	0	0
$289$	−3.62742	−0.213377
$290$	−22.6274	−1.32873
$291$	0	0
$292$	−12.8284	−0.750727
$293$	−6.27208	−0.366419	−0.183209	−	0.983074i	$-0.558649\pi$
$293$	−6.27208	−0.366419	−0.183209	+	0.983074i	$0.558649\pi$
$294$	0	0
$295$	−14.6274	−0.851641
$296$	5.58579	0.324667
$297$	0	0
$298$	9.31371	0.539529
$299$	−16.9706	−0.981433
$300$	0	0
$301$	1.58579	0.0914032
$302$	−10.9706	−0.631285
$303$	0	0
$304$	−7.65685	−0.439151
$305$	−3.31371	−0.189742
$306$	0	0
$307$	9.14214	0.521769	0.260885	−	0.965370i	$-0.415986\pi$
$307$	9.14214	0.521769	0.260885	+	0.965370i	$0.415986\pi$
$308$	1.31371	0.0748555
$309$	0	0
$310$	27.3137	1.55131
$311$	−12.4142	−0.703945	−0.351973	−	0.936010i	$-0.614489\pi$
$311$	−12.4142	−0.703945	−0.351973	+	0.936010i	$0.614489\pi$
$312$	0	0
$313$	24.1421	1.36459	0.682297	−	0.731075i	$-0.260981\pi$
$313$	24.1421	1.36459	0.682297	+	0.731075i	$0.260981\pi$
$314$	−6.82843	−0.385350
$315$	0	0
$316$	4.34315	0.244321
$317$	20.5563	1.15456	0.577280	−	0.816546i	$-0.304114\pi$
$317$	20.5563	1.15456	0.577280	+	0.816546i	$0.304114\pi$
$318$	0	0
$319$	−6.62742	−0.371064
$320$	2.82843	0.158114
$321$	0	0
$322$	−9.51472	−0.530235
$323$	28.0000	1.55796
$324$	0	0
$325$	8.48528	0.470679
$326$	16.9706	0.939913
$327$	0	0
$328$	−4.00000	−0.220863
$329$	15.9706	0.880486
$330$	0	0
$331$	−6.34315	−0.348651	−0.174325	−	0.984688i	$-0.555774\pi$
$331$	−6.34315	−0.348651	−0.174325	+	0.984688i	$0.555774\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	−1.58579	−0.0867704
$335$	−36.7696	−2.00894
$336$	0	0
$337$	−0.171573	−0.00934617	−0.00467309	−	0.999989i	$-0.501487\pi$
$337$	−0.171573	−0.00934617	−0.00467309	+	0.999989i	$0.501487\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	−10.3431	−0.560936
$341$	8.00000	0.433224
$342$	0	0
$343$	18.2132	0.983421
$344$	1.00000	0.0539164
$345$	0	0
$346$	1.92893	0.103700
$347$	7.97056	0.427882	0.213941	−	0.976847i	$-0.431370\pi$
$347$	7.97056	0.427882	0.213941	+	0.976847i	$0.431370\pi$
$348$	0	0
$349$	−14.7574	−0.789943	−0.394972	−	0.918693i	$-0.629246\pi$
$349$	−14.7574	−0.789943	−0.394972	+	0.918693i	$0.629246\pi$
$350$	4.75736	0.254292
$351$	0	0
$352$	0.828427	0.0441553
$353$	22.4853	1.19677	0.598385	−	0.801208i	$-0.295809\pi$
$353$	22.4853	1.19677	0.598385	+	0.801208i	$0.295809\pi$
$354$	0	0
$355$	45.6569	2.42321
$356$	1.48528	0.0787198
$357$	0	0
$358$	4.51472	0.238610
$359$	20.8995	1.10303	0.551517	−	0.834164i	$-0.314049\pi$
$359$	20.8995	1.10303	0.551517	+	0.834164i	$0.314049\pi$
$360$	0	0
$361$	39.6274	2.08565
$362$	−6.00000	−0.315353
$363$	0	0
$364$	−4.48528	−0.235093
$365$	−36.2843	−1.89921
$366$	0	0
$367$	11.7990	0.615902	0.307951	−	0.951402i	$-0.400357\pi$
$367$	11.7990	0.615902	0.307951	+	0.951402i	$0.400357\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	15.7990	0.821350
$371$	−10.1716	−0.528082
$372$	0	0
$373$	−2.89949	−0.150130	−0.0750651	−	0.997179i	$-0.523916\pi$
$373$	−2.89949	−0.150130	−0.0750651	+	0.997179i	$0.523916\pi$
$374$	−3.02944	−0.156648
$375$	0	0
$376$	10.0711	0.519376
$377$	22.6274	1.16537
$378$	0	0
$379$	18.7990	0.965639	0.482820	−	0.875720i	$-0.339613\pi$
$379$	18.7990	0.965639	0.482820	+	0.875720i	$0.339613\pi$
$380$	−21.6569	−1.11097
$381$	0	0
$382$	−3.65685	−0.187101
$383$	−4.14214	−0.211653	−0.105827	−	0.994385i	$-0.533749\pi$
$383$	−4.14214	−0.211653	−0.105827	+	0.994385i	$0.533749\pi$
$384$	0	0
$385$	3.71573	0.189371
$386$	7.82843	0.398456
$387$	0	0
$388$	13.3137	0.675901
$389$	35.4558	1.79768	0.898841	−	0.438274i	$-0.144410\pi$
$389$	35.4558	1.79768	0.898841	+	0.438274i	$0.144410\pi$
$390$	0	0
$391$	21.9411	1.10961
$392$	4.48528	0.226541
$393$	0	0
$394$	6.14214	0.309436
$395$	12.2843	0.618089
$396$	0	0
$397$	−23.6569	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$397$	−23.6569	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$398$	−10.0711	−0.504817
$399$	0	0
$400$	3.00000	0.150000
$401$	15.7990	0.788964	0.394482	−	0.918904i	$-0.370924\pi$
$401$	15.7990	0.788964	0.394482	+	0.918904i	$0.370924\pi$
$402$	0	0
$403$	−27.3137	−1.36059
$404$	−6.82843	−0.339727
$405$	0	0
$406$	12.6863	0.629610
$407$	4.62742	0.229373
$408$	0	0
$409$	−0.485281	−0.0239956	−0.0119978	−	0.999928i	$-0.503819\pi$
$409$	−0.485281	−0.0239956	−0.0119978	+	0.999928i	$0.503819\pi$
$410$	−11.3137	−0.558744
$411$	0	0
$412$	15.6569	0.771358
$413$	8.20101	0.403545
$414$	0	0
$415$	5.65685	0.277684
$416$	−2.82843	−0.138675
$417$	0	0
$418$	−6.34315	−0.310253
$419$	−17.6274	−0.861156	−0.430578	−	0.902553i	$-0.641690\pi$
$419$	−17.6274	−0.861156	−0.430578	+	0.902553i	$0.641690\pi$
$420$	0	0
$421$	15.7279	0.766532	0.383266	−	0.923638i	$-0.374799\pi$
$421$	15.7279	0.766532	0.383266	+	0.923638i	$0.374799\pi$
$422$	27.6569	1.34631
$423$	0	0
$424$	−6.41421	−0.311502
$425$	−10.9706	−0.532150
$426$	0	0
$427$	1.85786	0.0899084
$428$	2.00000	0.0966736
$429$	0	0
$430$	2.82843	0.136399
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	−16.2843	−0.782572	−0.391286	−	0.920269i	$-0.627970\pi$
$433$	−16.2843	−0.782572	−0.391286	+	0.920269i	$0.627970\pi$
$434$	−15.3137	−0.735082
$435$	0	0
$436$	−12.4853	−0.597937
$437$	45.9411	2.19766
$438$	0	0
$439$	30.9706	1.47814	0.739072	−	0.673626i	$-0.235264\pi$
$439$	30.9706	1.47814	0.739072	+	0.673626i	$0.235264\pi$
$440$	2.34315	0.111705
$441$	0	0
$442$	10.3431	0.491973
$443$	13.5147	0.642104	0.321052	−	0.947062i	$-0.395964\pi$
$443$	13.5147	0.642104	0.321052	+	0.947062i	$0.395964\pi$
$444$	0	0
$445$	4.20101	0.199147
$446$	−25.2426	−1.19527
$447$	0	0
$448$	−1.58579	−0.0749214
$449$	−7.48528	−0.353252	−0.176626	−	0.984278i	$-0.556518\pi$
$449$	−7.48528	−0.353252	−0.176626	+	0.984278i	$0.556518\pi$
$450$	0	0
$451$	−3.31371	−0.156036
$452$	14.9706	0.704156
$453$	0	0
$454$	27.4853	1.28995
$455$	−12.6863	−0.594743
$456$	0	0
$457$	−20.1421	−0.942209	−0.471105	−	0.882077i	$-0.656145\pi$
$457$	−20.1421	−0.942209	−0.471105	+	0.882077i	$0.656145\pi$
$458$	5.51472	0.257686
$459$	0	0
$460$	−16.9706	−0.791257
$461$	4.48528	0.208900	0.104450	−	0.994530i	$-0.466692\pi$
$461$	4.48528	0.208900	0.104450	+	0.994530i	$0.466692\pi$
$462$	0	0
$463$	−19.6569	−0.913531	−0.456766	−	0.889587i	$-0.650992\pi$
$463$	−19.6569	−0.913531	−0.456766	+	0.889587i	$0.650992\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	22.7990	1.05614
$467$	−35.4853	−1.64206	−0.821031	−	0.570883i	$-0.806601\pi$
$467$	−35.4853	−1.64206	−0.821031	+	0.570883i	$0.806601\pi$
$468$	0	0
$469$	20.6152	0.951923
$470$	28.4853	1.31393
$471$	0	0
$472$	5.17157	0.238041
$473$	0.828427	0.0380911
$474$	0	0
$475$	−22.9706	−1.05396
$476$	5.79899	0.265796
$477$	0	0
$478$	21.7279	0.993812
$479$	24.6985	1.12850	0.564251	−	0.825603i	$-0.309165\pi$
$479$	24.6985	1.12850	0.564251	+	0.825603i	$0.309165\pi$
$480$	0	0
$481$	−15.7990	−0.720372
$482$	22.9706	1.04628
$483$	0	0
$484$	−10.3137	−0.468805
$485$	37.6569	1.70991
$486$	0	0
$487$	1.79899	0.0815200	0.0407600	−	0.999169i	$-0.487022\pi$
$487$	1.79899	0.0815200	0.0407600	+	0.999169i	$0.487022\pi$
$488$	1.17157	0.0530346
$489$	0	0
$490$	12.6863	0.573108
$491$	−28.6569	−1.29327	−0.646633	−	0.762801i	$-0.723823\pi$
$491$	−28.6569	−1.29327	−0.646633	+	0.762801i	$0.723823\pi$
$492$	0	0
$493$	−29.2548	−1.31757
$494$	21.6569	0.974388
$495$	0	0
$496$	−9.65685	−0.433606
$497$	−25.5980	−1.14823
$498$	0	0
$499$	19.7990	0.886325	0.443162	−	0.896441i	$-0.353857\pi$
$499$	19.7990	0.886325	0.443162	+	0.896441i	$0.353857\pi$
$500$	−5.65685	−0.252982
$501$	0	0
$502$	−20.1421	−0.898988
$503$	−7.65685	−0.341402	−0.170701	−	0.985323i	$-0.554603\pi$
$503$	−7.65685	−0.341402	−0.170701	+	0.985323i	$0.554603\pi$
$504$	0	0
$505$	−19.3137	−0.859449
$506$	−4.97056	−0.220968
$507$	0	0
$508$	0.828427	0.0367555
$509$	21.5858	0.956773	0.478387	−	0.878149i	$-0.341222\pi$
$509$	21.5858	0.956773	0.478387	+	0.878149i	$0.341222\pi$
$510$	0	0
$511$	20.3431	0.899928
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	23.9706	1.05730
$515$	44.2843	1.95140
$516$	0	0
$517$	8.34315	0.366931
$518$	−8.85786	−0.389192
$519$	0	0
$520$	−8.00000	−0.350823
$521$	35.9706	1.57590	0.787949	−	0.615740i	$-0.211143\pi$
$521$	35.9706	1.57590	0.787949	+	0.615740i	$0.211143\pi$
$522$	0	0
$523$	−33.9411	−1.48414	−0.742071	−	0.670321i	$-0.766156\pi$
$523$	−33.9411	−1.48414	−0.742071	+	0.670321i	$0.766156\pi$
$524$	19.8284	0.866209
$525$	0	0
$526$	1.17157	0.0510830
$527$	35.3137	1.53829
$528$	0	0
$529$	13.0000	0.565217
$530$	−18.1421	−0.788044
$531$	0	0
$532$	12.1421	0.526428
$533$	11.3137	0.490051
$534$	0	0
$535$	5.65685	0.244567
$536$	13.0000	0.561514
$537$	0	0
$538$	−8.41421	−0.362763
$539$	3.71573	0.160048
$540$	0	0
$541$	13.5147	0.581043	0.290522	−	0.956868i	$-0.406171\pi$
$541$	13.5147	0.581043	0.290522	+	0.956868i	$0.406171\pi$
$542$	15.6569	0.672519
$543$	0	0
$544$	3.65685	0.156786
$545$	−35.3137	−1.51267
$546$	0	0
$547$	7.48528	0.320048	0.160024	−	0.987113i	$-0.448843\pi$
$547$	7.48528	0.320048	0.160024	+	0.987113i	$0.448843\pi$
$548$	−14.3137	−0.611451
$549$	0	0
$550$	2.48528	0.105973
$551$	−61.2548	−2.60954
$552$	0	0
$553$	−6.88730	−0.292878
$554$	−2.41421	−0.102570
$555$	0	0
$556$	14.3137	0.607036
$557$	−39.8701	−1.68935	−0.844674	−	0.535281i	$-0.820206\pi$
$557$	−39.8701	−1.68935	−0.844674	+	0.535281i	$0.820206\pi$
$558$	0	0
$559$	−2.82843	−0.119630
$560$	−4.48528	−0.189538
$561$	0	0
$562$	−1.65685	−0.0698902
$563$	−6.82843	−0.287784	−0.143892	−	0.989593i	$-0.545962\pi$
$563$	−6.82843	−0.287784	−0.143892	+	0.989593i	$0.545962\pi$
$564$	0	0
$565$	42.3431	1.78139
$566$	15.8284	0.665318
$567$	0	0
$568$	−16.1421	−0.677309
$569$	0.343146	0.0143854	0.00719271	−	0.999974i	$-0.497710\pi$
$569$	0.343146	0.0143854	0.00719271	+	0.999974i	$0.497710\pi$
$570$	0	0
$571$	46.8284	1.95971	0.979854	−	0.199713i	$-0.0640011\pi$
$571$	46.8284	1.95971	0.979854	+	0.199713i	$0.0640011\pi$
$572$	−2.34315	−0.0979718
$573$	0	0
$574$	6.34315	0.264758
$575$	−18.0000	−0.750652
$576$	0	0
$577$	−10.9706	−0.456711	−0.228355	−	0.973578i	$-0.573335\pi$
$577$	−10.9706	−0.456711	−0.228355	+	0.973578i	$0.573335\pi$
$578$	3.62742	0.150881
$579$	0	0
$580$	22.6274	0.939552
$581$	−3.17157	−0.131579
$582$	0	0
$583$	−5.31371	−0.220071
$584$	12.8284	0.530844
$585$	0	0
$586$	6.27208	0.259097
$587$	−28.9706	−1.19574	−0.597872	−	0.801592i	$-0.703987\pi$
$587$	−28.9706	−1.19574	−0.597872	+	0.801592i	$0.703987\pi$
$588$	0	0
$589$	73.9411	3.04669
$590$	14.6274	0.602201
$591$	0	0
$592$	−5.58579	−0.229574
$593$	17.8284	0.732126	0.366063	−	0.930590i	$-0.380706\pi$
$593$	17.8284	0.732126	0.366063	+	0.930590i	$0.380706\pi$
$594$	0	0
$595$	16.4020	0.672417
$596$	−9.31371	−0.381504
$597$	0	0
$598$	16.9706	0.693978
$599$	−11.6569	−0.476286	−0.238143	−	0.971230i	$-0.576539\pi$
$599$	−11.6569	−0.476286	−0.238143	+	0.971230i	$0.576539\pi$
$600$	0	0
$601$	14.9706	0.610662	0.305331	−	0.952246i	$-0.401233\pi$
$601$	14.9706	0.610662	0.305331	+	0.952246i	$0.401233\pi$
$602$	−1.58579	−0.0646318
$603$	0	0
$604$	10.9706	0.446386
$605$	−29.1716	−1.18599
$606$	0	0
$607$	−6.68629	−0.271388	−0.135694	−	0.990751i	$-0.543326\pi$
$607$	−6.68629	−0.271388	−0.135694	+	0.990751i	$0.543326\pi$
$608$	7.65685	0.310526
$609$	0	0
$610$	3.31371	0.134168
$611$	−28.4853	−1.15239
$612$	0	0
$613$	−16.6274	−0.671575	−0.335788	−	0.941938i	$-0.609002\pi$
$613$	−16.6274	−0.671575	−0.335788	+	0.941938i	$0.609002\pi$
$614$	−9.14214	−0.368947
$615$	0	0
$616$	−1.31371	−0.0529308
$617$	−25.7990	−1.03863	−0.519314	−	0.854583i	$-0.673813\pi$
$617$	−25.7990	−1.03863	−0.519314	+	0.854583i	$0.673813\pi$
$618$	0	0
$619$	23.1421	0.930161	0.465080	−	0.885268i	$-0.346025\pi$
$619$	23.1421	0.930161	0.465080	+	0.885268i	$0.346025\pi$
$620$	−27.3137	−1.09694
$621$	0	0
$622$	12.4142	0.497765
$623$	−2.35534	−0.0943647
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−24.1421	−0.964914
$627$	0	0
$628$	6.82843	0.272484
$629$	20.4264	0.814454
$630$	0	0
$631$	−2.75736	−0.109769	−0.0548844	−	0.998493i	$-0.517479\pi$
$631$	−2.75736	−0.109769	−0.0548844	+	0.998493i	$0.517479\pi$
$632$	−4.34315	−0.172761
$633$	0	0
$634$	−20.5563	−0.816397
$635$	2.34315	0.0929849
$636$	0	0
$637$	−12.6863	−0.502649
$638$	6.62742	0.262382
$639$	0	0
$640$	−2.82843	−0.111803
$641$	34.9411	1.38009	0.690046	−	0.723766i	$-0.257590\pi$
$641$	34.9411	1.38009	0.690046	+	0.723766i	$0.257590\pi$
$642$	0	0
$643$	40.9411	1.61456	0.807280	−	0.590168i	$-0.200939\pi$
$643$	40.9411	1.61456	0.807280	+	0.590168i	$0.200939\pi$
$644$	9.51472	0.374932
$645$	0	0
$646$	−28.0000	−1.10165
$647$	33.1127	1.30179	0.650897	−	0.759166i	$-0.274393\pi$
$647$	33.1127	1.30179	0.650897	+	0.759166i	$0.274393\pi$
$648$	0	0
$649$	4.28427	0.168172
$650$	−8.48528	−0.332820
$651$	0	0
$652$	−16.9706	−0.664619
$653$	−24.2843	−0.950317	−0.475158	−	0.879900i	$-0.657609\pi$
$653$	−24.2843	−0.950317	−0.475158	+	0.879900i	$0.657609\pi$
$654$	0	0
$655$	56.0833	2.19135
$656$	4.00000	0.156174
$657$	0	0
$658$	−15.9706	−0.622597
$659$	21.3137	0.830264	0.415132	−	0.909761i	$-0.363735\pi$
$659$	21.3137	0.830264	0.415132	+	0.909761i	$0.363735\pi$
$660$	0	0
$661$	11.3137	0.440052	0.220026	−	0.975494i	$-0.429386\pi$
$661$	11.3137	0.440052	0.220026	+	0.975494i	$0.429386\pi$
$662$	6.34315	0.246533
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	34.3431	1.33177
$666$	0	0
$667$	−48.0000	−1.85857
$668$	1.58579	0.0613559
$669$	0	0
$670$	36.7696	1.42053
$671$	0.970563	0.0374682
$672$	0	0
$673$	10.6274	0.409657	0.204828	−	0.978798i	$-0.434336\pi$
$673$	10.6274	0.409657	0.204828	+	0.978798i	$0.434336\pi$
$674$	0.171573	0.00660874
$675$	0	0
$676$	−5.00000	−0.192308
$677$	−2.82843	−0.108705	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−2.82843	−0.108705	−0.0543526	+	0.998522i	$0.517310\pi$
$678$	0	0
$679$	−21.1127	−0.810231
$680$	10.3431	0.396642
$681$	0	0
$682$	−8.00000	−0.306336
$683$	16.8284	0.643922	0.321961	−	0.946753i	$-0.395658\pi$
$683$	16.8284	0.643922	0.321961	+	0.946753i	$0.395658\pi$
$684$	0	0
$685$	−40.4853	−1.54686
$686$	−18.2132	−0.695383
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	18.1421	0.691160
$690$	0	0
$691$	28.6274	1.08904	0.544519	−	0.838748i	$-0.316712\pi$
$691$	28.6274	1.08904	0.544519	+	0.838748i	$0.316712\pi$
$692$	−1.92893	−0.0733270
$693$	0	0
$694$	−7.97056	−0.302558
$695$	40.4853	1.53569
$696$	0	0
$697$	−14.6274	−0.554053
$698$	14.7574	0.558574
$699$	0	0
$700$	−4.75736	−0.179811
$701$	−7.92893	−0.299472	−0.149736	−	0.988726i	$-0.547842\pi$
$701$	−7.92893	−0.299472	−0.149736	+	0.988726i	$0.547842\pi$
$702$	0	0
$703$	42.7696	1.61308
$704$	−0.828427	−0.0312225
$705$	0	0
$706$	−22.4853	−0.846245
$707$	10.8284	0.407245
$708$	0	0
$709$	−13.7990	−0.518232	−0.259116	−	0.965846i	$-0.583431\pi$
$709$	−13.7990	−0.518232	−0.259116	+	0.965846i	$0.583431\pi$
$710$	−45.6569	−1.71347
$711$	0	0
$712$	−1.48528	−0.0556633
$713$	57.9411	2.16991
$714$	0	0
$715$	−6.62742	−0.247851
$716$	−4.51472	−0.168723
$717$	0	0
$718$	−20.8995	−0.779962
$719$	19.1005	0.712329	0.356164	−	0.934423i	$-0.384084\pi$
$719$	19.1005	0.712329	0.356164	+	0.934423i	$0.384084\pi$
$720$	0	0
$721$	−24.8284	−0.924659
$722$	−39.6274	−1.47478
$723$	0	0
$724$	6.00000	0.222988
$725$	24.0000	0.891338
$726$	0	0
$727$	37.3137	1.38389	0.691944	−	0.721951i	$-0.256754\pi$
$727$	37.3137	1.38389	0.691944	+	0.721951i	$0.256754\pi$
$728$	4.48528	0.166236
$729$	0	0
$730$	36.2843	1.34294
$731$	3.65685	0.135254
$732$	0	0
$733$	−31.8701	−1.17715	−0.588574	−	0.808444i	$-0.700310\pi$
$733$	−31.8701	−1.17715	−0.588574	+	0.808444i	$0.700310\pi$
$734$	−11.7990	−0.435509
$735$	0	0
$736$	6.00000	0.221163
$737$	10.7696	0.396702
$738$	0	0
$739$	17.3137	0.636895	0.318448	−	0.947940i	$-0.396839\pi$
$739$	17.3137	0.636895	0.318448	+	0.947940i	$0.396839\pi$
$740$	−15.7990	−0.580782
$741$	0	0
$742$	10.1716	0.373410
$743$	9.45584	0.346901	0.173451	−	0.984843i	$-0.444508\pi$
$743$	9.45584	0.346901	0.173451	+	0.984843i	$0.444508\pi$
$744$	0	0
$745$	−26.3431	−0.965138
$746$	2.89949	0.106158
$747$	0	0
$748$	3.02944	0.110767
$749$	−3.17157	−0.115887
$750$	0	0
$751$	3.04163	0.110991	0.0554953	−	0.998459i	$-0.482326\pi$
$751$	3.04163	0.110991	0.0554953	+	0.998459i	$0.482326\pi$
$752$	−10.0711	−0.367254
$753$	0	0
$754$	−22.6274	−0.824042
$755$	31.0294	1.12928
$756$	0	0
$757$	−13.8579	−0.503673	−0.251836	−	0.967770i	$-0.581034\pi$
$757$	−13.8579	−0.503673	−0.251836	+	0.967770i	$0.581034\pi$
$758$	−18.7990	−0.682810
$759$	0	0
$760$	21.6569	0.785577
$761$	−12.6274	−0.457744	−0.228872	−	0.973457i	$-0.573504\pi$
$761$	−12.6274	−0.457744	−0.228872	+	0.973457i	$0.573504\pi$
$762$	0	0
$763$	19.7990	0.716772
$764$	3.65685	0.132300
$765$	0	0
$766$	4.14214	0.149661
$767$	−14.6274	−0.528165
$768$	0	0
$769$	−40.7990	−1.47125	−0.735625	−	0.677389i	$-0.763111\pi$
$769$	−40.7990	−1.47125	−0.735625	+	0.677389i	$0.763111\pi$
$770$	−3.71573	−0.133906
$771$	0	0
$772$	−7.82843	−0.281751
$773$	7.65685	0.275398	0.137699	−	0.990474i	$-0.456029\pi$
$773$	7.65685	0.275398	0.137699	+	0.990474i	$0.456029\pi$
$774$	0	0
$775$	−28.9706	−1.04065
$776$	−13.3137	−0.477934
$777$	0	0
$778$	−35.4558	−1.27115
$779$	−30.6274	−1.09734
$780$	0	0
$781$	−13.3726	−0.478509
$782$	−21.9411	−0.784613
$783$	0	0
$784$	−4.48528	−0.160189
$785$	19.3137	0.689336
$786$	0	0
$787$	−18.1127	−0.645648	−0.322824	−	0.946459i	$-0.604632\pi$
$787$	−18.1127	−0.645648	−0.322824	+	0.946459i	$0.604632\pi$
$788$	−6.14214	−0.218805
$789$	0	0
$790$	−12.2843	−0.437055
$791$	−23.7401	−0.844101
$792$	0	0
$793$	−3.31371	−0.117673
$794$	23.6569	0.839550
$795$	0	0
$796$	10.0711	0.356960
$797$	−52.8406	−1.87171	−0.935855	−	0.352384i	$-0.885371\pi$
$797$	−52.8406	−1.87171	−0.935855	+	0.352384i	$0.885371\pi$
$798$	0	0
$799$	36.8284	1.30290
$800$	−3.00000	−0.106066
$801$	0	0
$802$	−15.7990	−0.557882
$803$	10.6274	0.375033
$804$	0	0
$805$	26.9117	0.948512
$806$	27.3137	0.962084
$807$	0	0
$808$	6.82843	0.240223
$809$	−20.4853	−0.720224	−0.360112	−	0.932909i	$-0.617262\pi$
$809$	−20.4853	−0.720224	−0.360112	+	0.932909i	$0.617262\pi$
$810$	0	0
$811$	−35.1716	−1.23504	−0.617520	−	0.786555i	$-0.711863\pi$
$811$	−35.1716	−1.23504	−0.617520	+	0.786555i	$0.711863\pi$
$812$	−12.6863	−0.445202
$813$	0	0
$814$	−4.62742	−0.162191
$815$	−48.0000	−1.68137
$816$	0	0
$817$	7.65685	0.267879
$818$	0.485281	0.0169675
$819$	0	0
$820$	11.3137	0.395092
$821$	−24.7574	−0.864038	−0.432019	−	0.901865i	$-0.642199\pi$
$821$	−24.7574	−0.864038	−0.432019	+	0.901865i	$0.642199\pi$
$822$	0	0
$823$	14.4853	0.504925	0.252463	−	0.967607i	$-0.418760\pi$
$823$	14.4853	0.504925	0.252463	+	0.967607i	$0.418760\pi$
$824$	−15.6569	−0.545432
$825$	0	0
$826$	−8.20101	−0.285350
$827$	−37.7990	−1.31440	−0.657200	−	0.753716i	$-0.728259\pi$
$827$	−37.7990	−1.31440	−0.657200	+	0.753716i	$0.728259\pi$
$828$	0	0
$829$	17.3848	0.603798	0.301899	−	0.953340i	$-0.402379\pi$
$829$	17.3848	0.603798	0.301899	+	0.953340i	$0.402379\pi$
$830$	−5.65685	−0.196352
$831$	0	0
$832$	2.82843	0.0980581
$833$	16.4020	0.568296
$834$	0	0
$835$	4.48528	0.155220
$836$	6.34315	0.219382
$837$	0	0
$838$	17.6274	0.608929
$839$	−35.1716	−1.21426	−0.607129	−	0.794603i	$-0.707679\pi$
$839$	−35.1716	−1.21426	−0.607129	+	0.794603i	$0.707679\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−15.7279	−0.542020
$843$	0	0
$844$	−27.6569	−0.951988
$845$	−14.1421	−0.486504
$846$	0	0
$847$	16.3553	0.561976
$848$	6.41421	0.220265
$849$	0	0
$850$	10.9706	0.376287
$851$	33.5147	1.14887
$852$	0	0
$853$	7.17157	0.245550	0.122775	−	0.992435i	$-0.460821\pi$
$853$	7.17157	0.245550	0.122775	+	0.992435i	$0.460821\pi$
$854$	−1.85786	−0.0635748
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	35.6569	1.21801	0.609007	−	0.793164i	$-0.291568\pi$
$857$	35.6569	1.21801	0.609007	+	0.793164i	$0.291568\pi$
$858$	0	0
$859$	−43.4558	−1.48269	−0.741347	−	0.671122i	$-0.765813\pi$
$859$	−43.4558	−1.48269	−0.741347	+	0.671122i	$0.765813\pi$
$860$	−2.82843	−0.0964486
$861$	0	0
$862$	−6.00000	−0.204361
$863$	12.6274	0.429842	0.214921	−	0.976631i	$-0.431051\pi$
$863$	12.6274	0.429842	0.214921	+	0.976631i	$0.431051\pi$
$864$	0	0
$865$	−5.45584	−0.185504
$866$	16.2843	0.553362
$867$	0	0
$868$	15.3137	0.519781
$869$	−3.59798	−0.122053
$870$	0	0
$871$	−36.7696	−1.24589
$872$	12.4853	0.422805
$873$	0	0
$874$	−45.9411	−1.55398
$875$	8.97056	0.303260
$876$	0	0
$877$	25.1716	0.849984	0.424992	−	0.905197i	$-0.360277\pi$
$877$	25.1716	0.849984	0.424992	+	0.905197i	$0.360277\pi$
$878$	−30.9706	−1.04521
$879$	0	0
$880$	−2.34315	−0.0789874
$881$	8.20101	0.276299	0.138150	−	0.990411i	$-0.455885\pi$
$881$	8.20101	0.276299	0.138150	+	0.990411i	$0.455885\pi$
$882$	0	0
$883$	−2.17157	−0.0730792	−0.0365396	−	0.999332i	$-0.511634\pi$
$883$	−2.17157	−0.0730792	−0.0365396	+	0.999332i	$0.511634\pi$
$884$	−10.3431	−0.347878
$885$	0	0
$886$	−13.5147	−0.454036
$887$	15.3137	0.514184	0.257092	−	0.966387i	$-0.417236\pi$
$887$	15.3137	0.514184	0.257092	+	0.966387i	$0.417236\pi$
$888$	0	0
$889$	−1.31371	−0.0440604
$890$	−4.20101	−0.140818
$891$	0	0
$892$	25.2426	0.845186
$893$	77.1127	2.58048
$894$	0	0
$895$	−12.7696	−0.426839
$896$	1.58579	0.0529774
$897$	0	0
$898$	7.48528	0.249787
$899$	−77.2548	−2.57659
$900$	0	0
$901$	−23.4558	−0.781427
$902$	3.31371	0.110334
$903$	0	0
$904$	−14.9706	−0.497914
$905$	16.9706	0.564121
$906$	0	0
$907$	−38.3431	−1.27316	−0.636582	−	0.771209i	$-0.719652\pi$
$907$	−38.3431	−1.27316	−0.636582	+	0.771209i	$0.719652\pi$
$908$	−27.4853	−0.912131
$909$	0	0
$910$	12.6863	0.420546
$911$	−58.9706	−1.95378	−0.976891	−	0.213738i	$-0.931436\pi$
$911$	−58.9706	−1.95378	−0.976891	+	0.213738i	$0.931436\pi$
$912$	0	0
$913$	−1.65685	−0.0548339
$914$	20.1421	0.666243
$915$	0	0
$916$	−5.51472	−0.182211
$917$	−31.4437	−1.03836
$918$	0	0
$919$	−32.7696	−1.08097	−0.540484	−	0.841354i	$-0.681759\pi$
$919$	−32.7696	−1.08097	−0.540484	+	0.841354i	$0.681759\pi$
$920$	16.9706	0.559503
$921$	0	0
$922$	−4.48528	−0.147715
$923$	45.6569	1.50281
$924$	0	0
$925$	−16.7574	−0.550979
$926$	19.6569	0.645964
$927$	0	0
$928$	−8.00000	−0.262613
$929$	51.6274	1.69384	0.846920	−	0.531720i	$-0.178454\pi$
$929$	51.6274	1.69384	0.846920	+	0.531720i	$0.178454\pi$
$930$	0	0
$931$	34.3431	1.12555
$932$	−22.7990	−0.746806
$933$	0	0
$934$	35.4853	1.16111
$935$	8.56854	0.280221
$936$	0	0
$937$	20.9706	0.685078	0.342539	−	0.939504i	$-0.388713\pi$
$937$	20.9706	0.685078	0.342539	+	0.939504i	$0.388713\pi$
$938$	−20.6152	−0.673111
$939$	0	0
$940$	−28.4853	−0.929087
$941$	54.4264	1.77425	0.887125	−	0.461529i	$-0.152699\pi$
$941$	54.4264	1.77425	0.887125	+	0.461529i	$0.152699\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−5.17157	−0.168320
$945$	0	0
$946$	−0.828427	−0.0269345
$947$	54.6274	1.77515	0.887576	−	0.460661i	$-0.152388\pi$
$947$	54.6274	1.77515	0.887576	+	0.460661i	$0.152388\pi$
$948$	0	0
$949$	−36.2843	−1.17784
$950$	22.9706	0.745263
$951$	0	0
$952$	−5.79899	−0.187946
$953$	−31.4853	−1.01991	−0.509954	−	0.860202i	$-0.670338\pi$
$953$	−31.4853	−1.01991	−0.509954	+	0.860202i	$0.670338\pi$
$954$	0	0
$955$	10.3431	0.334696
$956$	−21.7279	−0.702731
$957$	0	0
$958$	−24.6985	−0.797972
$959$	22.6985	0.732972
$960$	0	0
$961$	62.2548	2.00822
$962$	15.7990	0.509380
$963$	0	0
$964$	−22.9706	−0.739832
$965$	−22.1421	−0.712781
$966$	0	0
$967$	−34.3431	−1.10440	−0.552201	−	0.833711i	$-0.686212\pi$
$967$	−34.3431	−1.10440	−0.552201	+	0.833711i	$0.686212\pi$
$968$	10.3137	0.331495
$969$	0	0
$970$	−37.6569	−1.20909
$971$	−14.9706	−0.480428	−0.240214	−	0.970720i	$-0.577218\pi$
$971$	−14.9706	−0.480428	−0.240214	+	0.970720i	$0.577218\pi$
$972$	0	0
$973$	−22.6985	−0.727680
$974$	−1.79899	−0.0576434
$975$	0	0
$976$	−1.17157	−0.0375011
$977$	52.4264	1.67727	0.838635	−	0.544694i	$-0.183354\pi$
$977$	52.4264	1.67727	0.838635	+	0.544694i	$0.183354\pi$
$978$	0	0
$979$	−1.23045	−0.0393253
$980$	−12.6863	−0.405249
$981$	0	0
$982$	28.6569	0.914477
$983$	9.51472	0.303472	0.151736	−	0.988421i	$-0.451514\pi$
$983$	9.51472	0.303472	0.151736	+	0.988421i	$0.451514\pi$
$984$	0	0
$985$	−17.3726	−0.553537
$986$	29.2548	0.931664
$987$	0	0
$988$	−21.6569	−0.688996
$989$	6.00000	0.190789
$990$	0	0
$991$	29.5980	0.940211	0.470105	−	0.882610i	$-0.344216\pi$
$991$	29.5980	0.940211	0.470105	+	0.882610i	$0.344216\pi$
$992$	9.65685	0.306605
$993$	0	0
$994$	25.5980	0.811919
$995$	28.4853	0.903044
$996$	0	0
$997$	−61.7401	−1.95533	−0.977665	−	0.210171i	$-0.932598\pi$
$997$	−61.7401	−1.95533	−0.977665	+	0.210171i	$0.932598\pi$
$998$	−19.7990	−0.626726
$999$	0	0

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2322.2.a.j.1.2	✓	2
3.2	odd	2		2322.2.a.o.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2322.2.a.j.1.2	✓	2	1.1	even	1	trivial
2322.2.a.o.1.1	yes	2	3.2	odd	2

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2322.2.a.j.1.2

Level	$2322$
Weight	$2$
Character	2322.1
Self dual	yes
Analytic conductor	$18.541$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$