LMFDB - Embedded newform 1143.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 $ x^16 - 26x^14 + 269x^12 - 1408x^10 + 3924x^8 - 5655x^6 + 3886x^4 - 1107*x^2 + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.979723$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.979723 q^{2} -1.04014 q^{4} +1.37010 q^{5} +3.49660 q^{7} +2.97850 q^{8} +O(q^{10})$	\(q-0.979723 q^{2} -1.04014 q^{4} +1.37010 q^{5} +3.49660 q^{7} +2.97850 q^{8} -1.34231 q^{10} +2.07626 q^{11} +4.50924 q^{13} -3.42570 q^{14} -0.837816 q^{16} -1.29478 q^{17} +1.52802 q^{19} -1.42510 q^{20} -2.03416 q^{22} -3.75171 q^{23} -3.12284 q^{25} -4.41781 q^{26} -3.63696 q^{28} +0.108918 q^{29} +2.35971 q^{31} -5.13617 q^{32} +1.26853 q^{34} +4.79068 q^{35} +10.5208 q^{37} -1.49703 q^{38} +4.08083 q^{40} -9.87941 q^{41} +2.45223 q^{43} -2.15961 q^{44} +3.67563 q^{46} -7.08478 q^{47} +5.22621 q^{49} +3.05951 q^{50} -4.69026 q^{52} +2.28545 q^{53} +2.84468 q^{55} +10.4146 q^{56} -0.106709 q^{58} +14.2411 q^{59} +7.27726 q^{61} -2.31186 q^{62} +6.70765 q^{64} +6.17810 q^{65} -2.08083 q^{67} +1.34676 q^{68} -4.69354 q^{70} -15.5776 q^{71} -6.08667 q^{73} -10.3075 q^{74} -1.58935 q^{76} +7.25986 q^{77} +8.01883 q^{79} -1.14789 q^{80} +9.67909 q^{82} +12.2362 q^{83} -1.77398 q^{85} -2.40251 q^{86} +6.18415 q^{88} +6.26914 q^{89} +15.7670 q^{91} +3.90231 q^{92} +6.94112 q^{94} +2.09353 q^{95} -10.0920 q^{97} -5.12023 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 10 * q^7	$ 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

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 $

$2$	−0.979723	−0.692769	−0.346384	−	0.938093i	$-0.612591\pi$
$2$	−0.979723	−0.692769	−0.346384	+	0.938093i	$0.612591\pi$
$3$	0	0
$3$	0	0
$4$	−1.04014	−0.520072
$5$	1.37010	0.612726	0.306363	−	0.951915i	$-0.400888\pi$
$5$	1.37010	0.612726	0.306363	+	0.951915i	$0.400888\pi$
$6$	0	0
$7$	3.49660	1.32159	0.660795	−	0.750566i	$-0.270219\pi$
$7$	3.49660	1.32159	0.660795	+	0.750566i	$0.270219\pi$
$8$	2.97850	1.05306
$9$	0	0
$10$	−1.34231	−0.424477
$11$	2.07626	0.626017	0.313009	−	0.949750i	$-0.398663\pi$
$11$	2.07626	0.626017	0.313009	+	0.949750i	$0.398663\pi$
$12$	0	0
$13$	4.50924	1.25064	0.625320	−	0.780369i	$-0.284969\pi$
$13$	4.50924	1.25064	0.625320	+	0.780369i	$0.284969\pi$
$14$	−3.42570	−0.915556
$15$	0	0
$16$	−0.837816	−0.209454
$17$	−1.29478	−0.314031	−0.157015	−	0.987596i	$-0.550187\pi$
$17$	−1.29478	−0.314031	−0.157015	+	0.987596i	$0.550187\pi$
$18$	0	0
$19$	1.52802	0.350551	0.175275	−	0.984519i	$-0.443918\pi$
$19$	1.52802	0.350551	0.175275	+	0.984519i	$0.443918\pi$
$20$	−1.42510	−0.318661
$21$	0	0
$22$	−2.03416	−0.433685
$23$	−3.75171	−0.782285	−0.391142	−	0.920330i	$-0.627920\pi$
$23$	−3.75171	−0.782285	−0.391142	+	0.920330i	$0.627920\pi$
$24$	0	0
$25$	−3.12284	−0.624567
$26$	−4.41781	−0.866404
$27$	0	0
$28$	−3.63696	−0.687322
$29$	0.108918	0.0202255	0.0101127	−	0.999949i	$-0.496781\pi$
$29$	0.108918	0.0202255	0.0101127	+	0.999949i	$0.496781\pi$
$30$	0	0
$31$	2.35971	0.423816	0.211908	−	0.977290i	$-0.432032\pi$
$31$	2.35971	0.423816	0.211908	+	0.977290i	$0.432032\pi$
$32$	−5.13617	−0.907955
$33$	0	0
$34$	1.26853	0.217551
$35$	4.79068	0.809772
$36$	0	0
$37$	10.5208	1.72961	0.864806	−	0.502106i	$-0.167441\pi$
$37$	10.5208	1.72961	0.864806	+	0.502106i	$0.167441\pi$
$38$	−1.49703	−0.242851
$39$	0	0
$40$	4.08083	0.645236
$41$	−9.87941	−1.54291	−0.771453	−	0.636287i	$-0.780470\pi$
$41$	−9.87941	−1.54291	−0.771453	+	0.636287i	$0.780470\pi$
$42$	0	0
$43$	2.45223	0.373961	0.186981	−	0.982364i	$-0.440130\pi$
$43$	2.45223	0.373961	0.186981	+	0.982364i	$0.440130\pi$
$44$	−2.15961	−0.325574
$45$	0	0
$46$	3.67563	0.541942
$47$	−7.08478	−1.03342	−0.516710	−	0.856160i	$-0.672844\pi$
$47$	−7.08478	−1.03342	−0.516710	+	0.856160i	$0.672844\pi$
$48$	0	0
$49$	5.22621	0.746601
$50$	3.05951	0.432681
$51$	0	0
$52$	−4.69026	−0.650422
$53$	2.28545	0.313931	0.156966	−	0.987604i	$-0.449829\pi$
$53$	2.28545	0.313931	0.156966	+	0.987604i	$0.449829\pi$
$54$	0	0
$55$	2.84468	0.383577
$56$	10.4146	1.39171
$57$	0	0
$58$	−0.106709	−0.0140116
$59$	14.2411	1.85404	0.927019	−	0.375015i	$-0.122362\pi$
$59$	14.2411	1.85404	0.927019	+	0.375015i	$0.122362\pi$
$60$	0	0
$61$	7.27726	0.931757	0.465879	−	0.884849i	$-0.345738\pi$
$61$	7.27726	0.931757	0.465879	+	0.884849i	$0.345738\pi$
$62$	−2.31186	−0.293607
$63$	0	0
$64$	6.70765	0.838457
$65$	6.17810	0.766299
$66$	0	0
$67$	−2.08083	−0.254214	−0.127107	−	0.991889i	$-0.540569\pi$
$67$	−2.08083	−0.254214	−0.127107	+	0.991889i	$0.540569\pi$
$68$	1.34676	0.163319
$69$	0	0
$70$	−4.69354	−0.560985
$71$	−15.5776	−1.84872	−0.924362	−	0.381517i	$-0.875402\pi$
$71$	−15.5776	−1.84872	−0.924362	+	0.381517i	$0.875402\pi$
$72$	0	0
$73$	−6.08667	−0.712391	−0.356196	−	0.934411i	$-0.615926\pi$
$73$	−6.08667	−0.712391	−0.356196	+	0.934411i	$0.615926\pi$
$74$	−10.3075	−1.19822
$75$	0	0
$76$	−1.58935	−0.182311
$77$	7.25986	0.827338
$78$	0	0
$79$	8.01883	0.902189	0.451094	−	0.892476i	$-0.351034\pi$
$79$	8.01883	0.902189	0.451094	+	0.892476i	$0.351034\pi$
$80$	−1.14789	−0.128338
$81$	0	0
$82$	9.67909	1.06888
$83$	12.2362	1.34309	0.671546	−	0.740963i	$-0.265630\pi$
$83$	12.2362	1.34309	0.671546	+	0.740963i	$0.265630\pi$
$84$	0	0
$85$	−1.77398	−0.192415
$86$	−2.40251	−0.259069
$87$	0	0
$88$	6.18415	0.659232
$89$	6.26914	0.664527	0.332264	−	0.943187i	$-0.392188\pi$
$89$	6.26914	0.664527	0.332264	+	0.943187i	$0.392188\pi$
$90$	0	0
$91$	15.7670	1.65283
$92$	3.90231	0.406844
$93$	0	0
$94$	6.94112	0.715922
$95$	2.09353	0.214791
$96$	0	0
$97$	−10.0920	−1.02468	−0.512342	−	0.858781i	$-0.671222\pi$
$97$	−10.0920	−1.02468	−0.512342	+	0.858781i	$0.671222\pi$
$98$	−5.12023	−0.517222
$99$	0	0
$100$	3.24820	0.324820
$101$	14.7882	1.47149	0.735743	−	0.677261i	$-0.236833\pi$
$101$	14.7882	1.47149	0.735743	+	0.677261i	$0.236833\pi$
$102$	0	0
$103$	−0.450311	−0.0443704	−0.0221852	−	0.999754i	$-0.507062\pi$
$103$	−0.450311	−0.0443704	−0.0221852	+	0.999754i	$0.507062\pi$
$104$	13.4308	1.31700
$105$	0	0
$106$	−2.23911	−0.217482
$107$	2.52468	0.244070	0.122035	−	0.992526i	$-0.461058\pi$
$107$	2.52468	0.244070	0.122035	+	0.992526i	$0.461058\pi$
$108$	0	0
$109$	11.0988	1.06307	0.531535	−	0.847036i	$-0.321615\pi$
$109$	11.0988	1.06307	0.531535	+	0.847036i	$0.321615\pi$
$110$	−2.78700	−0.265730
$111$	0	0
$112$	−2.92951	−0.276812
$113$	−11.1817	−1.05189	−0.525945	−	0.850518i	$-0.676288\pi$
$113$	−11.1817	−1.05189	−0.525945	+	0.850518i	$0.676288\pi$
$114$	0	0
$115$	−5.14020	−0.479326
$116$	−0.113290	−0.0105187
$117$	0	0
$118$	−13.9524	−1.28442
$119$	−4.52734	−0.415020
$120$	0	0
$121$	−6.68913	−0.608103
$122$	−7.12969	−0.645492
$123$	0	0
$124$	−2.45443	−0.220415
$125$	−11.1291	−0.995414
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	3.70069	0.327098
$129$	0	0
$130$	−6.05282	−0.530868
$131$	−11.8045	−1.03137	−0.515683	−	0.856779i	$-0.672462\pi$
$131$	−11.8045	−1.03137	−0.515683	+	0.856779i	$0.672462\pi$
$132$	0	0
$133$	5.34286	0.463284
$134$	2.03864	0.176111
$135$	0	0
$136$	−3.85651	−0.330693
$137$	13.5054	1.15384	0.576921	−	0.816800i	$-0.304254\pi$
$137$	13.5054	1.15384	0.576921	+	0.816800i	$0.304254\pi$
$138$	0	0
$139$	−8.39505	−0.712059	−0.356029	−	0.934475i	$-0.615870\pi$
$139$	−8.39505	−0.712059	−0.356029	+	0.934475i	$0.615870\pi$
$140$	−4.98299	−0.421140
$141$	0	0
$142$	15.2618	1.28074
$143$	9.36238	0.782922
$144$	0	0
$145$	0.149228	0.0123927
$146$	5.96325	0.493522
$147$	0	0
$148$	−10.9432	−0.899522
$149$	18.3898	1.50655	0.753275	−	0.657706i	$-0.228473\pi$
$149$	18.3898	1.50655	0.753275	+	0.657706i	$0.228473\pi$
$150$	0	0
$151$	1.75017	0.142427	0.0712135	−	0.997461i	$-0.477313\pi$
$151$	1.75017	0.142427	0.0712135	+	0.997461i	$0.477313\pi$
$152$	4.55119	0.369150
$153$	0	0
$154$	−7.11265	−0.573154
$155$	3.23303	0.259683
$156$	0	0
$157$	−16.4688	−1.31436	−0.657178	−	0.753736i	$-0.728250\pi$
$157$	−16.4688	−1.31436	−0.657178	+	0.753736i	$0.728250\pi$
$158$	−7.85623	−0.625008
$159$	0	0
$160$	−7.03704	−0.556327
$161$	−13.1182	−1.03386
$162$	0	0
$163$	−10.0987	−0.790989	−0.395495	−	0.918468i	$-0.629427\pi$
$163$	−10.0987	−0.790989	−0.395495	+	0.918468i	$0.629427\pi$
$164$	10.2760	0.802421
$165$	0	0
$166$	−11.9880	−0.930452
$167$	−0.167208	−0.0129389	−0.00646946	−	0.999979i	$-0.502059\pi$
$167$	−0.167208	−0.0129389	−0.00646946	+	0.999979i	$0.502059\pi$
$168$	0	0
$169$	7.33329	0.564099
$170$	1.73801	0.133299
$171$	0	0
$172$	−2.55067	−0.194487
$173$	3.86334	0.293724	0.146862	−	0.989157i	$-0.453083\pi$
$173$	3.86334	0.293724	0.146862	+	0.989157i	$0.453083\pi$
$174$	0	0
$175$	−10.9193	−0.825422
$176$	−1.73953	−0.131122
$177$	0	0
$178$	−6.14202	−0.460364
$179$	−4.85473	−0.362860	−0.181430	−	0.983404i	$-0.558072\pi$
$179$	−4.85473	−0.362860	−0.181430	+	0.983404i	$0.558072\pi$
$180$	0	0
$181$	6.33606	0.470955	0.235478	−	0.971880i	$-0.424335\pi$
$181$	6.33606	0.470955	0.235478	+	0.971880i	$0.424335\pi$
$182$	−15.4473	−1.14503
$183$	0	0
$184$	−11.1744	−0.823791
$185$	14.4145	1.05978
$186$	0	0
$187$	−2.68831	−0.196589
$188$	7.36918	0.537453
$189$	0	0
$190$	−2.05108	−0.148801
$191$	10.7065	0.774696	0.387348	−	0.921934i	$-0.373391\pi$
$191$	10.7065	0.774696	0.387348	+	0.921934i	$0.373391\pi$
$192$	0	0
$193$	13.4696	0.969564	0.484782	−	0.874635i	$-0.338899\pi$
$193$	13.4696	0.969564	0.484782	+	0.874635i	$0.338899\pi$
$194$	9.88734	0.709869
$195$	0	0
$196$	−5.43600	−0.388286
$197$	−15.2106	−1.08371	−0.541857	−	0.840471i	$-0.682278\pi$
$197$	−15.2106	−1.08371	−0.541857	+	0.840471i	$0.682278\pi$
$198$	0	0
$199$	12.2657	0.869493	0.434746	−	0.900553i	$-0.356838\pi$
$199$	12.2657	0.869493	0.434746	+	0.900553i	$0.356838\pi$
$200$	−9.30136	−0.657706
$201$	0	0
$202$	−14.4884	−1.01940
$203$	0.380841	0.0267298
$204$	0	0
$205$	−13.5357	−0.945378
$206$	0.441180	0.0307384
$207$	0	0
$208$	−3.77792	−0.261951
$209$	3.17256	0.219451
$210$	0	0
$211$	−23.8437	−1.64147	−0.820733	−	0.571313i	$-0.806434\pi$
$211$	−23.8437	−1.64147	−0.820733	+	0.571313i	$0.806434\pi$
$212$	−2.37720	−0.163267
$213$	0	0
$214$	−2.47349	−0.169084
$215$	3.35979	0.229136
$216$	0	0
$217$	8.25095	0.560111
$218$	−10.8737	−0.736462
$219$	0	0
$220$	−2.95888	−0.199487
$221$	−5.83849	−0.392739
$222$	0	0
$223$	2.00054	0.133966	0.0669831	−	0.997754i	$-0.478663\pi$
$223$	2.00054	0.133966	0.0669831	+	0.997754i	$0.478663\pi$
$224$	−17.9591	−1.19994
$225$	0	0
$226$	10.9550	0.728717
$227$	−7.01385	−0.465525	−0.232763	−	0.972534i	$-0.574777\pi$
$227$	−7.01385	−0.465525	−0.232763	+	0.972534i	$0.574777\pi$
$228$	0	0
$229$	12.5314	0.828099	0.414050	−	0.910254i	$-0.364114\pi$
$229$	12.5314	0.828099	0.414050	+	0.910254i	$0.364114\pi$
$230$	5.03597	0.332062
$231$	0	0
$232$	0.324411	0.0212986
$233$	−3.41330	−0.223612	−0.111806	−	0.993730i	$-0.535664\pi$
$233$	−3.41330	−0.223612	−0.111806	+	0.993730i	$0.535664\pi$
$234$	0	0
$235$	−9.70682	−0.633203
$236$	−14.8128	−0.964232
$237$	0	0
$238$	4.43553	0.287513
$239$	9.06378	0.586287	0.293143	−	0.956068i	$-0.405299\pi$
$239$	9.06378	0.586287	0.293143	+	0.956068i	$0.405299\pi$
$240$	0	0
$241$	16.6718	1.07392	0.536962	−	0.843607i	$-0.319572\pi$
$241$	16.6718	1.07392	0.536962	+	0.843607i	$0.319572\pi$
$242$	6.55349	0.421275
$243$	0	0
$244$	−7.56939	−0.484580
$245$	7.16040	0.457461
$246$	0	0
$247$	6.89019	0.438413
$248$	7.02839	0.446303
$249$	0	0
$250$	10.9034	0.689592
$251$	−14.2760	−0.901095	−0.450547	−	0.892753i	$-0.648771\pi$
$251$	−14.2760	−0.901095	−0.450547	+	0.892753i	$0.648771\pi$
$252$	0	0
$253$	−7.78953	−0.489724
$254$	−0.979723	−0.0614733
$255$	0	0
$256$	−17.0410	−1.06506
$257$	−7.87514	−0.491238	−0.245619	−	0.969366i	$-0.578991\pi$
$257$	−7.87514	−0.491238	−0.245619	+	0.969366i	$0.578991\pi$
$258$	0	0
$259$	36.7871	2.28584
$260$	−6.42611	−0.398530
$261$	0	0
$262$	11.5652	0.714498
$263$	20.6080	1.27074	0.635372	−	0.772206i	$-0.280847\pi$
$263$	20.6080	1.27074	0.635372	+	0.772206i	$0.280847\pi$
$264$	0	0
$265$	3.13129	0.192354
$266$	−5.23452	−0.320949
$267$	0	0
$268$	2.16436	0.132209
$269$	−26.5355	−1.61790	−0.808948	−	0.587880i	$-0.799963\pi$
$269$	−26.5355	−1.61790	−0.808948	+	0.587880i	$0.799963\pi$
$270$	0	0
$271$	−16.6209	−1.00965	−0.504823	−	0.863223i	$-0.668442\pi$
$271$	−16.6209	−1.00965	−0.504823	+	0.863223i	$0.668442\pi$
$272$	1.08479	0.0657750
$273$	0	0
$274$	−13.2315	−0.799345
$275$	−6.48383	−0.390990
$276$	0	0
$277$	12.0898	0.726404	0.363202	−	0.931710i	$-0.381683\pi$
$277$	12.0898	0.726404	0.363202	+	0.931710i	$0.381683\pi$
$278$	8.22482	0.493292
$279$	0	0
$280$	14.2690	0.852737
$281$	−23.6399	−1.41024	−0.705118	−	0.709090i	$-0.749106\pi$
$281$	−23.6399	−1.41024	−0.705118	+	0.709090i	$0.749106\pi$
$282$	0	0
$283$	−24.0475	−1.42948	−0.714739	−	0.699391i	$-0.753454\pi$
$283$	−24.0475	−1.42948	−0.714739	+	0.699391i	$0.753454\pi$
$284$	16.2030	0.961469
$285$	0	0
$286$	−9.17254	−0.542384
$287$	−34.5444	−2.03909
$288$	0	0
$289$	−15.3235	−0.901385
$290$	−0.146202	−0.00858526
$291$	0	0
$292$	6.33101	0.370494
$293$	14.7779	0.863333	0.431667	−	0.902033i	$-0.357926\pi$
$293$	14.7779	0.863333	0.431667	+	0.902033i	$0.357926\pi$
$294$	0	0
$295$	19.5117	1.13602
$296$	31.3362	1.82138
$297$	0	0
$298$	−18.0169	−1.04369
$299$	−16.9174	−0.978356
$300$	0	0
$301$	8.57446	0.494224
$302$	−1.71468	−0.0986690
$303$	0	0
$304$	−1.28020	−0.0734243
$305$	9.97054	0.570911
$306$	0	0
$307$	0.158825	0.00906462	0.00453231	−	0.999990i	$-0.498557\pi$
$307$	0.158825	0.00906462	0.00453231	+	0.999990i	$0.498557\pi$
$308$	−7.55129	−0.430275
$309$	0	0
$310$	−3.16747	−0.179900
$311$	−8.07291	−0.457773	−0.228886	−	0.973453i	$-0.573508\pi$
$311$	−8.07291	−0.457773	−0.228886	+	0.973453i	$0.573508\pi$
$312$	0	0
$313$	−25.1626	−1.42228	−0.711138	−	0.703052i	$-0.751820\pi$
$313$	−25.1626	−1.42228	−0.711138	+	0.703052i	$0.751820\pi$
$314$	16.1349	0.910544
$315$	0	0
$316$	−8.34073	−0.469203
$317$	17.0314	0.956580	0.478290	−	0.878202i	$-0.341257\pi$
$317$	17.0314	0.956580	0.478290	+	0.878202i	$0.341257\pi$
$318$	0	0
$319$	0.226142	0.0126615
$320$	9.19013	0.513744
$321$	0	0
$322$	12.8522	0.716226
$323$	−1.97845	−0.110084
$324$	0	0
$325$	−14.0816	−0.781108
$326$	9.89390	0.547973
$327$	0	0
$328$	−29.4258	−1.62477
$329$	−24.7726	−1.36576
$330$	0	0
$331$	15.7592	0.866206	0.433103	−	0.901344i	$-0.357419\pi$
$331$	15.7592	0.866206	0.433103	+	0.901344i	$0.357419\pi$
$332$	−12.7274	−0.698504
$333$	0	0
$334$	0.163817	0.00896367
$335$	−2.85093	−0.155763
$336$	0	0
$337$	−10.6735	−0.581422	−0.290711	−	0.956811i	$-0.593892\pi$
$337$	−10.6735	−0.581422	−0.290711	+	0.956811i	$0.593892\pi$
$338$	−7.18459	−0.390790
$339$	0	0
$340$	1.84519	0.100069
$341$	4.89938	0.265316
$342$	0	0
$343$	−6.20225	−0.334890
$344$	7.30396	0.393803
$345$	0	0
$346$	−3.78500	−0.203483
$347$	−13.6192	−0.731118	−0.365559	−	0.930788i	$-0.619122\pi$
$347$	−13.6192	−0.731118	−0.365559	+	0.930788i	$0.619122\pi$
$348$	0	0
$349$	11.6402	0.623088	0.311544	−	0.950232i	$-0.399154\pi$
$349$	11.6402	0.623088	0.311544	+	0.950232i	$0.399154\pi$
$350$	10.6979	0.571827
$351$	0	0
$352$	−10.6640	−0.568395
$353$	17.3200	0.921849	0.460925	−	0.887439i	$-0.347518\pi$
$353$	17.3200	0.921849	0.460925	+	0.887439i	$0.347518\pi$
$354$	0	0
$355$	−21.3428	−1.13276
$356$	−6.52080	−0.345602
$357$	0	0
$358$	4.75629	0.251378
$359$	2.77753	0.146592	0.0732961	−	0.997310i	$-0.476648\pi$
$359$	2.77753	0.146592	0.0732961	+	0.997310i	$0.476648\pi$
$360$	0	0
$361$	−16.6652	−0.877114
$362$	−6.20758	−0.326263
$363$	0	0
$364$	−16.4000	−0.859591
$365$	−8.33933	−0.436500
$366$	0	0
$367$	−7.25529	−0.378723	−0.189361	−	0.981907i	$-0.560642\pi$
$367$	−7.25529	−0.378723	−0.189361	+	0.981907i	$0.560642\pi$
$368$	3.14324	0.163853
$369$	0	0
$370$	−14.1223	−0.734181
$371$	7.99132	0.414888
$372$	0	0
$373$	5.98420	0.309850	0.154925	−	0.987926i	$-0.450486\pi$
$373$	5.98420	0.309850	0.154925	+	0.987926i	$0.450486\pi$
$374$	2.63380	0.136190
$375$	0	0
$376$	−21.1020	−1.08825
$377$	0.491136	0.0252948
$378$	0	0
$379$	−24.7344	−1.27052	−0.635260	−	0.772298i	$-0.719107\pi$
$379$	−24.7344	−1.27052	−0.635260	+	0.772298i	$0.719107\pi$
$380$	−2.17757	−0.111707
$381$	0	0
$382$	−10.4894	−0.536685
$383$	−19.8898	−1.01632	−0.508161	−	0.861262i	$-0.669674\pi$
$383$	−19.8898	−1.01632	−0.508161	+	0.861262i	$0.669674\pi$
$384$	0	0
$385$	9.94671	0.506931
$386$	−13.1965	−0.671684
$387$	0	0
$388$	10.4971	0.532909
$389$	−14.9007	−0.755495	−0.377747	−	0.925909i	$-0.623301\pi$
$389$	−14.9007	−0.755495	−0.377747	+	0.925909i	$0.623301\pi$
$390$	0	0
$391$	4.85764	0.245662
$392$	15.5662	0.786214
$393$	0	0
$394$	14.9022	0.750763
$395$	10.9866	0.552794
$396$	0	0
$397$	25.1554	1.26251	0.631257	−	0.775574i	$-0.282539\pi$
$397$	25.1554	1.26251	0.631257	+	0.775574i	$0.282539\pi$
$398$	−12.0170	−0.602357
$399$	0	0
$400$	2.61636	0.130818
$401$	1.02806	0.0513388	0.0256694	−	0.999670i	$-0.491828\pi$
$401$	1.02806	0.0513388	0.0256694	+	0.999670i	$0.491828\pi$
$402$	0	0
$403$	10.6405	0.530041
$404$	−15.3819	−0.765278
$405$	0	0
$406$	−0.373119	−0.0185176
$407$	21.8440	1.08277
$408$	0	0
$409$	27.6688	1.36813	0.684067	−	0.729419i	$-0.260209\pi$
$409$	27.6688	1.36813	0.684067	+	0.729419i	$0.260209\pi$
$410$	13.2613	0.654928
$411$	0	0
$412$	0.468387	0.0230758
$413$	49.7955	2.45028
$414$	0	0
$415$	16.7647	0.822947
$416$	−23.1602	−1.13552
$417$	0	0
$418$	−3.10823	−0.152029
$419$	−30.6119	−1.49549	−0.747745	−	0.663986i	$-0.768863\pi$
$419$	−30.6119	−1.49549	−0.747745	+	0.663986i	$0.768863\pi$
$420$	0	0
$421$	−8.98310	−0.437810	−0.218905	−	0.975746i	$-0.570248\pi$
$421$	−8.98310	−0.437810	−0.218905	+	0.975746i	$0.570248\pi$
$422$	23.3602	1.13716
$423$	0	0
$424$	6.80722	0.330588
$425$	4.04339	0.196133
$426$	0	0
$427$	25.4456	1.23140
$428$	−2.62603	−0.126934
$429$	0	0
$430$	−3.29166	−0.158738
$431$	−27.4869	−1.32400	−0.661998	−	0.749506i	$-0.730291\pi$
$431$	−27.4869	−1.32400	−0.661998	+	0.749506i	$0.730291\pi$
$432$	0	0
$433$	−3.85861	−0.185433	−0.0927165	−	0.995693i	$-0.529555\pi$
$433$	−3.85861	−0.185433	−0.0927165	+	0.995693i	$0.529555\pi$
$434$	−8.08365	−0.388028
$435$	0	0
$436$	−11.5443	−0.552872
$437$	−5.73266	−0.274231
$438$	0	0
$439$	5.56179	0.265450	0.132725	−	0.991153i	$-0.457627\pi$
$439$	5.56179	0.265450	0.132725	+	0.991153i	$0.457627\pi$
$440$	8.47288	0.403928
$441$	0	0
$442$	5.72010	0.272078
$443$	26.8396	1.27519	0.637594	−	0.770373i	$-0.279930\pi$
$443$	26.8396	1.27519	0.637594	+	0.770373i	$0.279930\pi$
$444$	0	0
$445$	8.58932	0.407173
$446$	−1.95998	−0.0928076
$447$	0	0
$448$	23.4540	1.10810
$449$	36.9423	1.74341	0.871707	−	0.490027i	$-0.163013\pi$
$449$	36.9423	1.74341	0.871707	+	0.490027i	$0.163013\pi$
$450$	0	0
$451$	−20.5123	−0.965885
$452$	11.6306	0.547058
$453$	0	0
$454$	6.87162	0.322501
$455$	21.6023	1.01273
$456$	0	0
$457$	40.8690	1.91177	0.955885	−	0.293741i	$-0.0949003\pi$
$457$	40.8690	1.91177	0.955885	+	0.293741i	$0.0949003\pi$
$458$	−12.2773	−0.573681
$459$	0	0
$460$	5.34654	0.249284
$461$	3.11171	0.144927	0.0724634	−	0.997371i	$-0.476914\pi$
$461$	3.11171	0.144927	0.0724634	+	0.997371i	$0.476914\pi$
$462$	0	0
$463$	−27.5405	−1.27991	−0.639957	−	0.768411i	$-0.721048\pi$
$463$	−27.5405	−1.27991	−0.639957	+	0.768411i	$0.721048\pi$
$464$	−0.0912529	−0.00423631
$465$	0	0
$466$	3.34408	0.154912
$467$	−22.8431	−1.05705	−0.528527	−	0.848917i	$-0.677255\pi$
$467$	−22.8431	−1.05705	−0.528527	+	0.848917i	$0.677255\pi$
$468$	0	0
$469$	−7.27582	−0.335966
$470$	9.51000	0.438663
$471$	0	0
$472$	42.4172	1.95241
$473$	5.09147	0.234106
$474$	0	0
$475$	−4.77174	−0.218943
$476$	4.70908	0.215840
$477$	0	0
$478$	−8.87999	−0.406161
$479$	−14.6982	−0.671577	−0.335789	−	0.941937i	$-0.609003\pi$
$479$	−14.6982	−0.671577	−0.335789	+	0.941937i	$0.609003\pi$
$480$	0	0
$481$	47.4410	2.16312
$482$	−16.3337	−0.743981
$483$	0	0
$484$	6.95765	0.316257
$485$	−13.8270	−0.627851
$486$	0	0
$487$	4.06173	0.184055	0.0920273	−	0.995756i	$-0.470665\pi$
$487$	4.06173	0.184055	0.0920273	+	0.995756i	$0.470665\pi$
$488$	21.6753	0.981194
$489$	0	0
$490$	−7.01521	−0.316915
$491$	39.9247	1.80178	0.900889	−	0.434049i	$-0.142916\pi$
$491$	39.9247	1.80178	0.900889	+	0.434049i	$0.142916\pi$
$492$	0	0
$493$	−0.141025	−0.00635143
$494$	−6.75048	−0.303718
$495$	0	0
$496$	−1.97700	−0.0887700
$497$	−54.4687	−2.44326
$498$	0	0
$499$	−20.4392	−0.914985	−0.457492	−	0.889214i	$-0.651252\pi$
$499$	−20.4392	−0.914985	−0.457492	+	0.889214i	$0.651252\pi$
$500$	11.5758	0.517687
$501$	0	0
$502$	13.9865	0.624250
$503$	9.77986	0.436062	0.218031	−	0.975942i	$-0.430037\pi$
$503$	9.77986	0.436062	0.218031	+	0.975942i	$0.430037\pi$
$504$	0	0
$505$	20.2613	0.901617
$506$	7.63158	0.339265
$507$	0	0
$508$	−1.04014	−0.0461489
$509$	−6.68350	−0.296241	−0.148120	−	0.988969i	$-0.547322\pi$
$509$	−6.68350	−0.296241	−0.148120	+	0.988969i	$0.547322\pi$
$510$	0	0
$511$	−21.2827	−0.941489
$512$	9.29403	0.410742
$513$	0	0
$514$	7.71546	0.340314
$515$	−0.616969	−0.0271869
$516$	0	0
$517$	−14.7099	−0.646939
$518$	−36.0412	−1.58356
$519$	0	0
$520$	18.4015	0.806957
$521$	31.2307	1.36824	0.684121	−	0.729369i	$-0.260186\pi$
$521$	31.2307	1.36824	0.684121	+	0.729369i	$0.260186\pi$
$522$	0	0
$523$	8.93117	0.390533	0.195266	−	0.980750i	$-0.437443\pi$
$523$	8.93117	0.390533	0.195266	+	0.980750i	$0.437443\pi$
$524$	12.2784	0.536384
$525$	0	0
$526$	−20.1901	−0.880332
$527$	−3.05531	−0.133091
$528$	0	0
$529$	−8.92470	−0.388030
$530$	−3.06780	−0.133257
$531$	0	0
$532$	−5.55734	−0.240941
$533$	−44.5487	−1.92962
$534$	0	0
$535$	3.45906	0.149548
$536$	−6.19774	−0.267702
$537$	0	0
$538$	25.9974	1.12083
$539$	10.8510	0.467385
$540$	0	0
$541$	18.0349	0.775381	0.387691	−	0.921790i	$-0.373273\pi$
$541$	18.0349	0.775381	0.387691	+	0.921790i	$0.373273\pi$
$542$	16.2838	0.699451
$543$	0	0
$544$	6.65022	0.285126
$545$	15.2064	0.651370
$546$	0	0
$547$	−16.6060	−0.710019	−0.355010	−	0.934863i	$-0.615522\pi$
$547$	−16.6060	−0.710019	−0.355010	+	0.934863i	$0.615522\pi$
$548$	−14.0475	−0.600080
$549$	0	0
$550$	6.35236	0.270865
$551$	0.166428	0.00709006
$552$	0	0
$553$	28.0386	1.19232
$554$	−11.8446	−0.503230
$555$	0	0
$556$	8.73205	0.370322
$557$	14.3701	0.608881	0.304440	−	0.952531i	$-0.401531\pi$
$557$	14.3701	0.608881	0.304440	+	0.952531i	$0.401531\pi$
$558$	0	0
$559$	11.0577	0.467691
$560$	−4.01371	−0.169610
$561$	0	0
$562$	23.1605	0.976967
$563$	31.0342	1.30793	0.653967	−	0.756523i	$-0.273104\pi$
$563$	31.0342	1.30793	0.653967	+	0.756523i	$0.273104\pi$
$564$	0	0
$565$	−15.3201	−0.644520
$566$	23.5599	0.990297
$567$	0	0
$568$	−46.3979	−1.94681
$569$	9.98831	0.418732	0.209366	−	0.977837i	$-0.432860\pi$
$569$	9.98831	0.418732	0.209366	+	0.977837i	$0.432860\pi$
$570$	0	0
$571$	−39.4196	−1.64966	−0.824829	−	0.565382i	$-0.808729\pi$
$571$	−39.4196	−1.64966	−0.824829	+	0.565382i	$0.808729\pi$
$572$	−9.73821	−0.407175
$573$	0	0
$574$	33.8439	1.41262
$575$	11.7160	0.488590
$576$	0	0
$577$	34.5899	1.43999	0.719997	−	0.693977i	$-0.244143\pi$
$577$	34.5899	1.43999	0.719997	+	0.693977i	$0.244143\pi$
$578$	15.0128	0.624451
$579$	0	0
$580$	−0.155218	−0.00644508
$581$	42.7849	1.77502
$582$	0	0
$583$	4.74520	0.196526
$584$	−18.1291	−0.750189
$585$	0	0
$586$	−14.4782	−0.598090
$587$	10.3294	0.426340	0.213170	−	0.977015i	$-0.431621\pi$
$587$	10.3294	0.426340	0.213170	+	0.977015i	$0.431621\pi$
$588$	0	0
$589$	3.60567	0.148569
$590$	−19.1161	−0.786997
$591$	0	0
$592$	−8.81452	−0.362274
$593$	−0.640590	−0.0263059	−0.0131529	−	0.999913i	$-0.504187\pi$
$593$	−0.640590	−0.0263059	−0.0131529	+	0.999913i	$0.504187\pi$
$594$	0	0
$595$	−6.20288	−0.254293
$596$	−19.1280	−0.783514
$597$	0	0
$598$	16.5743	0.677775
$599$	−48.0335	−1.96260	−0.981298	−	0.192494i	$-0.938342\pi$
$599$	−48.0335	−1.96260	−0.981298	+	0.192494i	$0.938342\pi$
$600$	0	0
$601$	−32.8060	−1.33819	−0.669093	−	0.743179i	$-0.733317\pi$
$601$	−32.8060	−1.33819	−0.669093	+	0.743179i	$0.733317\pi$
$602$	−8.40060	−0.342383
$603$	0	0
$604$	−1.82043	−0.0740723
$605$	−9.16475	−0.372600
$606$	0	0
$607$	30.7786	1.24926	0.624632	−	0.780919i	$-0.285249\pi$
$607$	30.7786	1.24926	0.624632	+	0.780919i	$0.285249\pi$
$608$	−7.84814	−0.318284
$609$	0	0
$610$	−9.76837	−0.395510
$611$	−31.9470	−1.29244
$612$	0	0
$613$	2.74575	0.110900	0.0554499	−	0.998461i	$-0.482341\pi$
$613$	2.74575	0.110900	0.0554499	+	0.998461i	$0.482341\pi$
$614$	−0.155604	−0.00627969
$615$	0	0
$616$	21.6235	0.871235
$617$	−29.3875	−1.18310	−0.591548	−	0.806270i	$-0.701483\pi$
$617$	−29.3875	−1.18310	−0.591548	+	0.806270i	$0.701483\pi$
$618$	0	0
$619$	−20.0273	−0.804966	−0.402483	−	0.915427i	$-0.631853\pi$
$619$	−20.0273	−0.804966	−0.402483	+	0.915427i	$0.631853\pi$
$620$	−3.36281	−0.135054
$621$	0	0
$622$	7.90921	0.317131
$623$	21.9207	0.878233
$624$	0	0
$625$	0.366290	0.0146516
$626$	24.6524	0.985309
$627$	0	0
$628$	17.1299	0.683559
$629$	−13.6222	−0.543152
$630$	0	0
$631$	22.7813	0.906908	0.453454	−	0.891280i	$-0.350192\pi$
$631$	22.7813	0.906908	0.453454	+	0.891280i	$0.350192\pi$
$632$	23.8841	0.950057
$633$	0	0
$634$	−16.6861	−0.662689
$635$	1.37010	0.0543706
$636$	0	0
$637$	23.5662	0.933728
$638$	−0.221556	−0.00877149
$639$	0	0
$640$	5.07031	0.200421
$641$	−33.0337	−1.30475	−0.652375	−	0.757896i	$-0.726227\pi$
$641$	−33.0337	−1.30475	−0.652375	+	0.757896i	$0.726227\pi$
$642$	0	0
$643$	−11.8750	−0.468304	−0.234152	−	0.972200i	$-0.575231\pi$
$643$	−11.8750	−0.468304	−0.234152	+	0.972200i	$0.575231\pi$
$644$	13.6448	0.537681
$645$	0	0
$646$	1.93833	0.0762626
$647$	23.7431	0.933436	0.466718	−	0.884406i	$-0.345436\pi$
$647$	23.7431	0.933436	0.466718	+	0.884406i	$0.345436\pi$
$648$	0	0
$649$	29.5683	1.16066
$650$	13.7961	0.541127
$651$	0	0
$652$	10.5041	0.411371
$653$	−3.18775	−0.124746	−0.0623732	−	0.998053i	$-0.519867\pi$
$653$	−3.18775	−0.124746	−0.0623732	+	0.998053i	$0.519867\pi$
$654$	0	0
$655$	−16.1733	−0.631944
$656$	8.27713	0.323168
$657$	0	0
$658$	24.2703	0.946155
$659$	15.4646	0.602417	0.301208	−	0.953558i	$-0.402610\pi$
$659$	15.4646	0.602417	0.301208	+	0.953558i	$0.402610\pi$
$660$	0	0
$661$	2.25630	0.0877599	0.0438800	−	0.999037i	$-0.486028\pi$
$661$	2.25630	0.0877599	0.0438800	+	0.999037i	$0.486028\pi$
$662$	−15.4397	−0.600081
$663$	0	0
$664$	36.4454	1.41435
$665$	7.32023	0.283866
$666$	0	0
$667$	−0.408627	−0.0158221
$668$	0.173920	0.00672916
$669$	0	0
$670$	2.79313	0.107908
$671$	15.1095	0.583296
$672$	0	0
$673$	46.3611	1.78709	0.893545	−	0.448973i	$-0.148210\pi$
$673$	46.3611	1.78709	0.893545	+	0.448973i	$0.148210\pi$
$674$	10.4571	0.402791
$675$	0	0
$676$	−7.62767	−0.293372
$677$	18.5665	0.713570	0.356785	−	0.934187i	$-0.383873\pi$
$677$	18.5665	0.713570	0.356785	+	0.934187i	$0.383873\pi$
$678$	0	0
$679$	−35.2876	−1.35421
$680$	−5.28379	−0.202624
$681$	0	0
$682$	−4.80003	−0.183803
$683$	3.23335	0.123721	0.0618603	−	0.998085i	$-0.480297\pi$
$683$	3.23335	0.123721	0.0618603	+	0.998085i	$0.480297\pi$
$684$	0	0
$685$	18.5036	0.706988
$686$	6.07648	0.232001
$687$	0	0
$688$	−2.05452	−0.0783277
$689$	10.3057	0.392615
$690$	0	0
$691$	−45.9574	−1.74830	−0.874151	−	0.485654i	$-0.838582\pi$
$691$	−45.9574	−1.74830	−0.874151	+	0.485654i	$0.838582\pi$
$692$	−4.01842	−0.152758
$693$	0	0
$694$	13.3431	0.506495
$695$	−11.5020	−0.436297
$696$	0	0
$697$	12.7917	0.484520
$698$	−11.4042	−0.431656
$699$	0	0
$700$	11.3576	0.429279
$701$	−27.6279	−1.04349	−0.521746	−	0.853101i	$-0.674719\pi$
$701$	−27.6279	−1.04349	−0.521746	+	0.853101i	$0.674719\pi$
$702$	0	0
$703$	16.0760	0.606317
$704$	13.9269	0.524888
$705$	0	0
$706$	−16.9688	−0.638628
$707$	51.7086	1.94470
$708$	0	0
$709$	−20.0866	−0.754369	−0.377184	−	0.926138i	$-0.623108\pi$
$709$	−20.0866	−0.754369	−0.377184	+	0.926138i	$0.623108\pi$
$710$	20.9101	0.784741
$711$	0	0
$712$	18.6726	0.699786
$713$	−8.85293	−0.331545
$714$	0	0
$715$	12.8274	0.479716
$716$	5.04961	0.188713
$717$	0	0
$718$	−2.72121	−0.101555
$719$	−42.2517	−1.57572	−0.787860	−	0.615854i	$-0.788811\pi$
$719$	−42.2517	−1.57572	−0.787860	+	0.615854i	$0.788811\pi$
$720$	0	0
$721$	−1.57456	−0.0586395
$722$	16.3272	0.607637
$723$	0	0
$724$	−6.59040	−0.244930
$725$	−0.340132	−0.0126322
$726$	0	0
$727$	−39.4667	−1.46374	−0.731869	−	0.681445i	$-0.761352\pi$
$727$	−39.4667	−1.46374	−0.731869	+	0.681445i	$0.761352\pi$
$728$	46.9620	1.74053
$729$	0	0
$730$	8.17023	0.302394
$731$	−3.17510	−0.117435
$732$	0	0
$733$	1.36674	0.0504816	0.0252408	−	0.999681i	$-0.491965\pi$
$733$	1.36674	0.0504816	0.0252408	+	0.999681i	$0.491965\pi$
$734$	7.10817	0.262367
$735$	0	0
$736$	19.2694	0.710279
$737$	−4.32035	−0.159142
$738$	0	0
$739$	−29.0200	−1.06752	−0.533758	−	0.845637i	$-0.679221\pi$
$739$	−29.0200	−1.06752	−0.533758	+	0.845637i	$0.679221\pi$
$740$	−14.9932	−0.551160
$741$	0	0
$742$	−7.82927	−0.287422
$743$	16.4878	0.604877	0.302438	−	0.953169i	$-0.402199\pi$
$743$	16.4878	0.604877	0.302438	+	0.953169i	$0.402199\pi$
$744$	0	0
$745$	25.1958	0.923102
$746$	−5.86286	−0.214655
$747$	0	0
$748$	2.79623	0.102240
$749$	8.82781	0.322561
$750$	0	0
$751$	−52.1769	−1.90396	−0.951980	−	0.306159i	$-0.900956\pi$
$751$	−52.1769	−1.90396	−0.951980	+	0.306159i	$0.900956\pi$
$752$	5.93574	0.216454
$753$	0	0
$754$	−0.481177	−0.0175234
$755$	2.39791	0.0872687
$756$	0	0
$757$	−15.2672	−0.554897	−0.277448	−	0.960741i	$-0.589489\pi$
$757$	−15.2672	−0.554897	−0.277448	+	0.960741i	$0.589489\pi$
$758$	24.2329	0.880177
$759$	0	0
$760$	6.23557	0.226188
$761$	−30.4109	−1.10239	−0.551197	−	0.834375i	$-0.685829\pi$
$761$	−30.4109	−1.10239	−0.551197	+	0.834375i	$0.685829\pi$
$762$	0	0
$763$	38.8080	1.40494
$764$	−11.1363	−0.402898
$765$	0	0
$766$	19.4865	0.704076
$767$	64.2167	2.31873
$768$	0	0
$769$	22.8590	0.824317	0.412159	−	0.911112i	$-0.364775\pi$
$769$	22.8590	0.824317	0.412159	+	0.911112i	$0.364775\pi$
$770$	−9.74502	−0.351186
$771$	0	0
$772$	−14.0103	−0.504243
$773$	12.2138	0.439300	0.219650	−	0.975579i	$-0.429508\pi$
$773$	12.2138	0.439300	0.219650	+	0.975579i	$0.429508\pi$
$774$	0	0
$775$	−7.36898	−0.264702
$776$	−30.0589	−1.07905
$777$	0	0
$778$	14.5985	0.523383
$779$	−15.0959	−0.540867
$780$	0	0
$781$	−32.3433	−1.15733
$782$	−4.75914	−0.170187
$783$	0	0
$784$	−4.37860	−0.156379
$785$	−22.5639	−0.805339
$786$	0	0
$787$	−40.9204	−1.45865	−0.729327	−	0.684165i	$-0.760167\pi$
$787$	−40.9204	−1.45865	−0.729327	+	0.684165i	$0.760167\pi$
$788$	15.8212	0.563608
$789$	0	0
$790$	−10.7638	−0.382959
$791$	−39.0981	−1.39017
$792$	0	0
$793$	32.8149	1.16529
$794$	−24.6453	−0.874630
$795$	0	0
$796$	−12.7581	−0.452198
$797$	−33.4170	−1.18369	−0.591845	−	0.806052i	$-0.701600\pi$
$797$	−33.4170	−1.18369	−0.591845	+	0.806052i	$0.701600\pi$
$798$	0	0
$799$	9.17324	0.324526
$800$	16.0394	0.567079
$801$	0	0
$802$	−1.00721	−0.0355659
$803$	−12.6375	−0.445969
$804$	0	0
$805$	−17.9732	−0.633473
$806$	−10.4247	−0.367196
$807$	0	0
$808$	44.0467	1.54956
$809$	−6.16594	−0.216783	−0.108392	−	0.994108i	$-0.534570\pi$
$809$	−6.16594	−0.216783	−0.108392	+	0.994108i	$0.534570\pi$
$810$	0	0
$811$	−15.3846	−0.540225	−0.270113	−	0.962829i	$-0.587061\pi$
$811$	−15.3846	−0.540225	−0.270113	+	0.962829i	$0.587061\pi$
$812$	−0.396129	−0.0139014
$813$	0	0
$814$	−21.4011	−0.750107
$815$	−13.8362	−0.484659
$816$	0	0
$817$	3.74704	0.131092
$818$	−27.1078	−0.947801
$819$	0	0
$820$	14.0791	0.491664
$821$	−7.93819	−0.277045	−0.138522	−	0.990359i	$-0.544235\pi$
$821$	−7.93819	−0.277045	−0.138522	+	0.990359i	$0.544235\pi$
$822$	0	0
$823$	−1.76701	−0.0615941	−0.0307970	−	0.999526i	$-0.509805\pi$
$823$	−1.76701	−0.0615941	−0.0307970	+	0.999526i	$0.509805\pi$
$824$	−1.34125	−0.0467246
$825$	0	0
$826$	−48.7858	−1.69748
$827$	5.65444	0.196624	0.0983120	−	0.995156i	$-0.468656\pi$
$827$	5.65444	0.196624	0.0983120	+	0.995156i	$0.468656\pi$
$828$	0	0
$829$	55.6978	1.93447	0.967233	−	0.253890i	$-0.0817101\pi$
$829$	55.6978	1.93447	0.967233	+	0.253890i	$0.0817101\pi$
$830$	−16.4248	−0.570112
$831$	0	0
$832$	30.2464	1.04861
$833$	−6.76680	−0.234456
$834$	0	0
$835$	−0.229090	−0.00792800
$836$	−3.29992	−0.114130
$837$	0	0
$838$	29.9912	1.03603
$839$	−3.53276	−0.121965	−0.0609823	−	0.998139i	$-0.519423\pi$
$839$	−3.53276	−0.121965	−0.0609823	+	0.998139i	$0.519423\pi$
$840$	0	0
$841$	−28.9881	−0.999591
$842$	8.80095	0.303301
$843$	0	0
$844$	24.8008	0.853679
$845$	10.0473	0.345638
$846$	0	0
$847$	−23.3892	−0.803663
$848$	−1.91479	−0.0657542
$849$	0	0
$850$	−3.96141	−0.135875
$851$	−39.4710	−1.35305
$852$	0	0
$853$	2.66187	0.0911406	0.0455703	−	0.998961i	$-0.485490\pi$
$853$	2.66187	0.0911406	0.0455703	+	0.998961i	$0.485490\pi$
$854$	−24.9297	−0.853076
$855$	0	0
$856$	7.51977	0.257020
$857$	−52.1626	−1.78184	−0.890919	−	0.454161i	$-0.849939\pi$
$857$	−52.1626	−1.78184	−0.890919	+	0.454161i	$0.849939\pi$
$858$	0	0
$859$	−56.7922	−1.93773	−0.968863	−	0.247597i	$-0.920359\pi$
$859$	−56.7922	−1.93773	−0.968863	+	0.247597i	$0.920359\pi$
$860$	−3.49466	−0.119167
$861$	0	0
$862$	26.9295	0.917223
$863$	48.3230	1.64493	0.822467	−	0.568812i	$-0.192597\pi$
$863$	48.3230	1.64493	0.822467	+	0.568812i	$0.192597\pi$
$864$	0	0
$865$	5.29314	0.179972
$866$	3.78037	0.128462
$867$	0	0
$868$	−8.58217	−0.291298
$869$	16.6492	0.564786
$870$	0	0
$871$	−9.38296	−0.317930
$872$	33.0577	1.11947
$873$	0	0
$874$	5.61642	0.189978
$875$	−38.9139	−1.31553
$876$	0	0
$877$	36.0917	1.21873	0.609366	−	0.792889i	$-0.291424\pi$
$877$	36.0917	1.21873	0.609366	+	0.792889i	$0.291424\pi$
$878$	−5.44901	−0.183895
$879$	0	0
$880$	−2.38332	−0.0803417
$881$	−13.4183	−0.452074	−0.226037	−	0.974119i	$-0.572577\pi$
$881$	−13.4183	−0.452074	−0.226037	+	0.974119i	$0.572577\pi$
$882$	0	0
$883$	−3.42934	−0.115407	−0.0577033	−	0.998334i	$-0.518378\pi$
$883$	−3.42934	−0.115407	−0.0577033	+	0.998334i	$0.518378\pi$
$884$	6.07287	0.204253
$885$	0	0
$886$	−26.2954	−0.883410
$887$	0.814071	0.0273338	0.0136669	−	0.999907i	$-0.495650\pi$
$887$	0.814071	0.0273338	0.0136669	+	0.999907i	$0.495650\pi$
$888$	0	0
$889$	3.49660	0.117272
$890$	−8.41515	−0.282077
$891$	0	0
$892$	−2.08085	−0.0696720
$893$	−10.8256	−0.362266
$894$	0	0
$895$	−6.65145	−0.222333
$896$	12.9398	0.432290
$897$	0	0
$898$	−36.1932	−1.20778
$899$	0.257014	0.00857189
$900$	0	0
$901$	−2.95917	−0.0985841
$902$	20.0963	0.669135
$903$	0	0
$904$	−33.3048	−1.10770
$905$	8.68100	0.288566
$906$	0	0
$907$	−40.8644	−1.35688	−0.678441	−	0.734655i	$-0.737344\pi$
$907$	−40.8644	−1.35688	−0.678441	+	0.734655i	$0.737344\pi$
$908$	7.29540	0.242106
$909$	0	0
$910$	−21.1643	−0.701590
$911$	7.42593	0.246032	0.123016	−	0.992405i	$-0.460743\pi$
$911$	7.42593	0.246032	0.123016	+	0.992405i	$0.460743\pi$
$912$	0	0
$913$	25.4055	0.840799
$914$	−40.0403	−1.32441
$915$	0	0
$916$	−13.0345	−0.430671
$917$	−41.2757	−1.36304
$918$	0	0
$919$	6.50584	0.214608	0.107304	−	0.994226i	$-0.465778\pi$
$919$	6.50584	0.214608	0.107304	+	0.994226i	$0.465778\pi$
$920$	−15.3101	−0.504758
$921$	0	0
$922$	−3.04861	−0.100401
$923$	−70.2433	−2.31209
$924$	0	0
$925$	−32.8548	−1.08026
$926$	26.9820	0.886684
$927$	0	0
$928$	−0.559419	−0.0183638
$929$	−16.3988	−0.538028	−0.269014	−	0.963136i	$-0.586698\pi$
$929$	−16.3988	−0.538028	−0.269014	+	0.963136i	$0.586698\pi$
$930$	0	0
$931$	7.98572	0.261721
$932$	3.55032	0.116294
$933$	0	0
$934$	22.3799	0.732293
$935$	−3.68324	−0.120455
$936$	0	0
$937$	−21.0897	−0.688971	−0.344486	−	0.938792i	$-0.611947\pi$
$937$	−21.0897	−0.688971	−0.344486	+	0.938792i	$0.611947\pi$
$938$	7.12829	0.232747
$939$	0	0
$940$	10.0965	0.329311
$941$	−0.379946	−0.0123859	−0.00619294	−	0.999981i	$-0.501971\pi$
$941$	−0.379946	−0.0123859	−0.00619294	+	0.999981i	$0.501971\pi$
$942$	0	0
$943$	37.0647	1.20699
$944$	−11.9315	−0.388336
$945$	0	0
$946$	−4.98823	−0.162181
$947$	49.6349	1.61292	0.806458	−	0.591291i	$-0.201382\pi$
$947$	49.6349	1.61292	0.806458	+	0.591291i	$0.201382\pi$
$948$	0	0
$949$	−27.4463	−0.890944
$950$	4.67498	0.151677
$951$	0	0
$952$	−13.4847	−0.437040
$953$	−50.4857	−1.63539	−0.817697	−	0.575649i	$-0.804749\pi$
$953$	−50.4857	−1.63539	−0.817697	+	0.575649i	$0.804749\pi$
$954$	0	0
$955$	14.6690	0.474676
$956$	−9.42762	−0.304911
$957$	0	0
$958$	14.4002	0.465248
$959$	47.2228	1.52491
$960$	0	0
$961$	−25.4318	−0.820380
$962$	−46.4790	−1.49854
$963$	0	0
$964$	−17.3410	−0.558517
$965$	18.4547	0.594077
$966$	0	0
$967$	14.1347	0.454542	0.227271	−	0.973832i	$-0.427020\pi$
$967$	14.1347	0.454542	0.227271	+	0.973832i	$0.427020\pi$
$968$	−19.9236	−0.640367
$969$	0	0
$970$	13.5466	0.434955
$971$	−25.5965	−0.821429	−0.410715	−	0.911764i	$-0.634721\pi$
$971$	−25.5965	−0.821429	−0.410715	+	0.911764i	$0.634721\pi$
$972$	0	0
$973$	−29.3541	−0.941050
$974$	−3.97937	−0.127507
$975$	0	0
$976$	−6.09700	−0.195160
$977$	36.0715	1.15403	0.577014	−	0.816734i	$-0.304218\pi$
$977$	36.0715	1.15403	0.577014	+	0.816734i	$0.304218\pi$
$978$	0	0
$979$	13.0164	0.416005
$980$	−7.44785	−0.237913
$981$	0	0
$982$	−39.1152	−1.24822
$983$	−32.3311	−1.03120	−0.515601	−	0.856829i	$-0.672431\pi$
$983$	−32.3311	−1.03120	−0.515601	+	0.856829i	$0.672431\pi$
$984$	0	0
$985$	−20.8400	−0.664019
$986$	0.138165	0.00440007
$987$	0	0
$988$	−7.16679	−0.228006
$989$	−9.20004	−0.292544
$990$	0	0
$991$	−30.3430	−0.963878	−0.481939	−	0.876205i	$-0.660067\pi$
$991$	−30.3430	−0.963878	−0.481939	+	0.876205i	$0.660067\pi$
$992$	−12.1199	−0.384806
$993$	0	0
$994$	53.3642	1.69261
$995$	16.8052	0.532761
$996$	0	0
$997$	−44.1126	−1.39706	−0.698530	−	0.715581i	$-0.746162\pi$
$997$	−44.1126	−1.39706	−0.698530	+	0.715581i	$0.746162\pi$
$998$	20.0248	0.633873
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.k.1.6	✓	16
3.2	odd	2	inner	1143.2.a.k.1.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.k.1.6	✓	16	1.1	even	1	trivial
1143.2.a.k.1.11	yes	16	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q-0.979723 q^{2} -1.04014 q^{4} +1.37010 q^{5} +3.49660 q^{7} +2.97850 q^{8} +O(q^{10})\)	\(q-0.979723 q^{2} -1.04014 q^{4} +1.37010 q^{5} +3.49660 q^{7} +2.97850 q^{8} -1.34231 q^{10} +2.07626 q^{11} +4.50924 q^{13} -3.42570 q^{14} -0.837816 q^{16} -1.29478 q^{17} +1.52802 q^{19} -1.42510 q^{20} -2.03416 q^{22} -3.75171 q^{23} -3.12284 q^{25} -4.41781 q^{26} -3.63696 q^{28} +0.108918 q^{29} +2.35971 q^{31} -5.13617 q^{32} +1.26853 q^{34} +4.79068 q^{35} +10.5208 q^{37} -1.49703 q^{38} +4.08083 q^{40} -9.87941 q^{41} +2.45223 q^{43} -2.15961 q^{44} +3.67563 q^{46} -7.08478 q^{47} +5.22621 q^{49} +3.05951 q^{50} -4.69026 q^{52} +2.28545 q^{53} +2.84468 q^{55} +10.4146 q^{56} -0.106709 q^{58} +14.2411 q^{59} +7.27726 q^{61} -2.31186 q^{62} +6.70765 q^{64} +6.17810 q^{65} -2.08083 q^{67} +1.34676 q^{68} -4.69354 q^{70} -15.5776 q^{71} -6.08667 q^{73} -10.3075 q^{74} -1.58935 q^{76} +7.25986 q^{77} +8.01883 q^{79} -1.14789 q^{80} +9.67909 q^{82} +12.2362 q^{83} -1.77398 q^{85} -2.40251 q^{86} +6.18415 q^{88} +6.26914 q^{89} +15.7670 q^{91} +3.90231 q^{92} +6.94112 q^{94} +2.09353 q^{95} -10.0920 q^{97} -5.12023 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 10 * q^7	\( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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