LMFDB - Embedded newform 8001.2.a.ba.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52993 q^{2} +4.40057 q^{4} +4.10792 q^{5} +1.00000 q^{7} -6.07329 q^{8} +O(q^{10})$	\(q-2.52993 q^{2} +4.40057 q^{4} +4.10792 q^{5} +1.00000 q^{7} -6.07329 q^{8} -10.3928 q^{10} +3.98901 q^{11} +4.99152 q^{13} -2.52993 q^{14} +6.56388 q^{16} +4.55621 q^{17} -5.49483 q^{19} +18.0772 q^{20} -10.0919 q^{22} +2.92506 q^{23} +11.8750 q^{25} -12.6282 q^{26} +4.40057 q^{28} -9.75847 q^{29} +7.97341 q^{31} -4.45962 q^{32} -11.5269 q^{34} +4.10792 q^{35} +4.83283 q^{37} +13.9016 q^{38} -24.9486 q^{40} -6.70872 q^{41} +7.62892 q^{43} +17.5539 q^{44} -7.40021 q^{46} +3.40280 q^{47} +1.00000 q^{49} -30.0430 q^{50} +21.9656 q^{52} -5.93896 q^{53} +16.3865 q^{55} -6.07329 q^{56} +24.6883 q^{58} -8.11352 q^{59} +8.05216 q^{61} -20.1722 q^{62} -1.84522 q^{64} +20.5048 q^{65} +3.45199 q^{67} +20.0499 q^{68} -10.3928 q^{70} -1.75560 q^{71} -1.83381 q^{73} -12.2268 q^{74} -24.1804 q^{76} +3.98901 q^{77} -2.42476 q^{79} +26.9639 q^{80} +16.9726 q^{82} -1.01805 q^{83} +18.7165 q^{85} -19.3007 q^{86} -24.2264 q^{88} -12.8315 q^{89} +4.99152 q^{91} +12.8719 q^{92} -8.60886 q^{94} -22.5723 q^{95} -14.3213 q^{97} -2.52993 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * 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 $

$2$	−2.52993	−1.78893	−0.894467	−	0.447134i	$-0.852445\pi$
$2$	−2.52993	−1.78893	−0.894467	+	0.447134i	$0.852445\pi$
$3$	0	0
$3$	0	0
$4$	4.40057	2.20029
$5$	4.10792	1.83712	0.918559	−	0.395284i	$-0.129354\pi$
$5$	4.10792	1.83712	0.918559	+	0.395284i	$0.129354\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.07329	−2.14723
$9$	0	0
$10$	−10.3928	−3.28648
$11$	3.98901	1.20273	0.601365	−	0.798974i	$-0.294624\pi$
$11$	3.98901	1.20273	0.601365	+	0.798974i	$0.294624\pi$
$12$	0	0
$13$	4.99152	1.38440	0.692200	−	0.721706i	$-0.256642\pi$
$13$	4.99152	1.38440	0.692200	+	0.721706i	$0.256642\pi$
$14$	−2.52993	−0.676154
$15$	0	0
$16$	6.56388	1.64097
$17$	4.55621	1.10504	0.552521	−	0.833499i	$-0.313666\pi$
$17$	4.55621	1.10504	0.552521	+	0.833499i	$0.313666\pi$
$18$	0	0
$19$	−5.49483	−1.26060	−0.630300	−	0.776351i	$-0.717068\pi$
$19$	−5.49483	−1.26060	−0.630300	+	0.776351i	$0.717068\pi$
$20$	18.0772	4.04218
$21$	0	0
$22$	−10.0919	−2.15161
$23$	2.92506	0.609917	0.304958	−	0.952366i	$-0.401357\pi$
$23$	2.92506	0.609917	0.304958	+	0.952366i	$0.401357\pi$
$24$	0	0
$25$	11.8750	2.37500
$26$	−12.6282	−2.47660
$27$	0	0
$28$	4.40057	0.831630
$29$	−9.75847	−1.81210	−0.906051	−	0.423168i	$-0.860918\pi$
$29$	−9.75847	−1.81210	−0.906051	+	0.423168i	$0.860918\pi$
$30$	0	0
$31$	7.97341	1.43207	0.716033	−	0.698066i	$-0.245956\pi$
$31$	7.97341	1.43207	0.716033	+	0.698066i	$0.245956\pi$
$32$	−4.45962	−0.788357
$33$	0	0
$34$	−11.5269	−1.97685
$35$	4.10792	0.694365
$36$	0	0
$37$	4.83283	0.794513	0.397256	−	0.917708i	$-0.369962\pi$
$37$	4.83283	0.794513	0.397256	+	0.917708i	$0.369962\pi$
$38$	13.9016	2.25513
$39$	0	0
$40$	−24.9486	−3.94472
$41$	−6.70872	−1.04773	−0.523863	−	0.851803i	$-0.675510\pi$
$41$	−6.70872	−1.04773	−0.523863	+	0.851803i	$0.675510\pi$
$42$	0	0
$43$	7.62892	1.16340	0.581700	−	0.813403i	$-0.302388\pi$
$43$	7.62892	1.16340	0.581700	+	0.813403i	$0.302388\pi$
$44$	17.5539	2.64635
$45$	0	0
$46$	−7.40021	−1.09110
$47$	3.40280	0.496349	0.248175	−	0.968715i	$-0.420169\pi$
$47$	3.40280	0.496349	0.248175	+	0.968715i	$0.420169\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−30.0430	−4.24872
$51$	0	0
$52$	21.9656	3.04607
$53$	−5.93896	−0.815778	−0.407889	−	0.913031i	$-0.633735\pi$
$53$	−5.93896	−0.815778	−0.407889	+	0.913031i	$0.633735\pi$
$54$	0	0
$55$	16.3865	2.20956
$56$	−6.07329	−0.811577
$57$	0	0
$58$	24.6883	3.24173
$59$	−8.11352	−1.05629	−0.528145	−	0.849154i	$-0.677112\pi$
$59$	−8.11352	−1.05629	−0.528145	+	0.849154i	$0.677112\pi$
$60$	0	0
$61$	8.05216	1.03097	0.515487	−	0.856898i	$-0.327611\pi$
$61$	8.05216	1.03097	0.515487	+	0.856898i	$0.327611\pi$
$62$	−20.1722	−2.56187
$63$	0	0
$64$	−1.84522	−0.230652
$65$	20.5048	2.54331
$66$	0	0
$67$	3.45199	0.421728	0.210864	−	0.977515i	$-0.432372\pi$
$67$	3.45199	0.421728	0.210864	+	0.977515i	$0.432372\pi$
$68$	20.0499	2.43141
$69$	0	0
$70$	−10.3928	−1.24217
$71$	−1.75560	−0.208351	−0.104176	−	0.994559i	$-0.533220\pi$
$71$	−1.75560	−0.208351	−0.104176	+	0.994559i	$0.533220\pi$
$72$	0	0
$73$	−1.83381	−0.214632	−0.107316	−	0.994225i	$-0.534226\pi$
$73$	−1.83381	−0.214632	−0.107316	+	0.994225i	$0.534226\pi$
$74$	−12.2268	−1.42133
$75$	0	0
$76$	−24.1804	−2.77368
$77$	3.98901	0.454590
$78$	0	0
$79$	−2.42476	−0.272806	−0.136403	−	0.990653i	$-0.543554\pi$
$79$	−2.42476	−0.272806	−0.136403	+	0.990653i	$0.543554\pi$
$80$	26.9639	3.01466
$81$	0	0
$82$	16.9726	1.87431
$83$	−1.01805	−0.111746	−0.0558729	−	0.998438i	$-0.517794\pi$
$83$	−1.01805	−0.111746	−0.0558729	+	0.998438i	$0.517794\pi$
$84$	0	0
$85$	18.7165	2.03009
$86$	−19.3007	−2.08125
$87$	0	0
$88$	−24.2264	−2.58254
$89$	−12.8315	−1.36014	−0.680070	−	0.733147i	$-0.738051\pi$
$89$	−12.8315	−1.36014	−0.680070	+	0.733147i	$0.738051\pi$
$90$	0	0
$91$	4.99152	0.523254
$92$	12.8719	1.34199
$93$	0	0
$94$	−8.60886	−0.887937
$95$	−22.5723	−2.31587
$96$	0	0
$97$	−14.3213	−1.45411	−0.727056	−	0.686578i	$-0.759112\pi$
$97$	−14.3213	−1.45411	−0.727056	+	0.686578i	$0.759112\pi$
$98$	−2.52993	−0.255562
$99$	0	0
$100$	52.2568	5.22568
$101$	19.5240	1.94271	0.971354	−	0.237638i	$-0.0763733\pi$
$101$	19.5240	1.94271	0.971354	+	0.237638i	$0.0763733\pi$
$102$	0	0
$103$	−9.79996	−0.965619	−0.482809	−	0.875725i	$-0.660384\pi$
$103$	−9.79996	−0.965619	−0.482809	+	0.875725i	$0.660384\pi$
$104$	−30.3150	−2.97263
$105$	0	0
$106$	15.0252	1.45937
$107$	1.43645	0.138867	0.0694333	−	0.997587i	$-0.477881\pi$
$107$	1.43645	0.138867	0.0694333	+	0.997587i	$0.477881\pi$
$108$	0	0
$109$	12.8350	1.22937	0.614684	−	0.788774i	$-0.289284\pi$
$109$	12.8350	1.22937	0.614684	+	0.788774i	$0.289284\pi$
$110$	−41.4568	−3.95275
$111$	0	0
$112$	6.56388	0.620229
$113$	−15.4960	−1.45774	−0.728871	−	0.684651i	$-0.759955\pi$
$113$	−15.4960	−1.45774	−0.728871	+	0.684651i	$0.759955\pi$
$114$	0	0
$115$	12.0159	1.12049
$116$	−42.9428	−3.98714
$117$	0	0
$118$	20.5267	1.88963
$119$	4.55621	0.417667
$120$	0	0
$121$	4.91218	0.446561
$122$	−20.3714	−1.84434
$123$	0	0
$124$	35.0875	3.15095
$125$	28.2420	2.52604
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	13.5875	1.20098
$129$	0	0
$130$	−51.8758	−4.54981
$131$	10.6292	0.928680	0.464340	−	0.885657i	$-0.346292\pi$
$131$	10.6292	0.928680	0.464340	+	0.885657i	$0.346292\pi$
$132$	0	0
$133$	−5.49483	−0.476462
$134$	−8.73332	−0.754444
$135$	0	0
$136$	−27.6712	−2.37278
$137$	0.0522151	0.00446104	0.00223052	−	0.999998i	$-0.499290\pi$
$137$	0.0522151	0.00446104	0.00223052	+	0.999998i	$0.499290\pi$
$138$	0	0
$139$	11.1744	0.947797	0.473898	−	0.880580i	$-0.342846\pi$
$139$	11.1744	0.947797	0.473898	+	0.880580i	$0.342846\pi$
$140$	18.0772	1.52780
$141$	0	0
$142$	4.44155	0.372726
$143$	19.9112	1.66506
$144$	0	0
$145$	−40.0870	−3.32904
$146$	4.63943	0.383962
$147$	0	0
$148$	21.2672	1.74815
$149$	−7.19409	−0.589363	−0.294681	−	0.955596i	$-0.595213\pi$
$149$	−7.19409	−0.589363	−0.294681	+	0.955596i	$0.595213\pi$
$150$	0	0
$151$	9.07041	0.738139	0.369070	−	0.929402i	$-0.379676\pi$
$151$	9.07041	0.738139	0.369070	+	0.929402i	$0.379676\pi$
$152$	33.3717	2.70680
$153$	0	0
$154$	−10.0919	−0.813231
$155$	32.7541	2.63087
$156$	0	0
$157$	−10.3850	−0.828812	−0.414406	−	0.910092i	$-0.636011\pi$
$157$	−10.3850	−0.828812	−0.414406	+	0.910092i	$0.636011\pi$
$158$	6.13448	0.488033
$159$	0	0
$160$	−18.3198	−1.44830
$161$	2.92506	0.230527
$162$	0	0
$163$	12.4206	0.972857	0.486429	−	0.873720i	$-0.338299\pi$
$163$	12.4206	0.972857	0.486429	+	0.873720i	$0.338299\pi$
$164$	−29.5222	−2.30529
$165$	0	0
$166$	2.57561	0.199906
$167$	0.329029	0.0254610	0.0127305	−	0.999919i	$-0.495948\pi$
$167$	0.329029	0.0254610	0.0127305	+	0.999919i	$0.495948\pi$
$168$	0	0
$169$	11.9153	0.916563
$170$	−47.3516	−3.63170
$171$	0	0
$172$	33.5716	2.55981
$173$	−7.22276	−0.549137	−0.274568	−	0.961568i	$-0.588535\pi$
$173$	−7.22276	−0.549137	−0.274568	+	0.961568i	$0.588535\pi$
$174$	0	0
$175$	11.8750	0.897666
$176$	26.1834	1.97365
$177$	0	0
$178$	32.4629	2.43320
$179$	−19.1625	−1.43228	−0.716138	−	0.697959i	$-0.754092\pi$
$179$	−19.1625	−1.43228	−0.716138	+	0.697959i	$0.754092\pi$
$180$	0	0
$181$	−14.8944	−1.10709	−0.553545	−	0.832819i	$-0.686725\pi$
$181$	−14.8944	−1.10709	−0.553545	+	0.832819i	$0.686725\pi$
$182$	−12.6282	−0.936067
$183$	0	0
$184$	−17.7647	−1.30963
$185$	19.8529	1.45961
$186$	0	0
$187$	18.1747	1.32907
$188$	14.9743	1.09211
$189$	0	0
$190$	57.1065	4.14294
$191$	−5.06554	−0.366530	−0.183265	−	0.983064i	$-0.558667\pi$
$191$	−5.06554	−0.366530	−0.183265	+	0.983064i	$0.558667\pi$
$192$	0	0
$193$	9.86947	0.710420	0.355210	−	0.934787i	$-0.384409\pi$
$193$	9.86947	0.710420	0.355210	+	0.934787i	$0.384409\pi$
$194$	36.2321	2.60131
$195$	0	0
$196$	4.40057	0.314326
$197$	21.7049	1.54641	0.773205	−	0.634156i	$-0.218652\pi$
$197$	21.7049	1.54641	0.773205	+	0.634156i	$0.218652\pi$
$198$	0	0
$199$	22.0827	1.56540	0.782701	−	0.622398i	$-0.213841\pi$
$199$	22.0827	1.56540	0.782701	+	0.622398i	$0.213841\pi$
$200$	−72.1203	−5.09968
$201$	0	0
$202$	−49.3944	−3.47538
$203$	−9.75847	−0.684910
$204$	0	0
$205$	−27.5589	−1.92479
$206$	24.7933	1.72743
$207$	0	0
$208$	32.7638	2.27176
$209$	−21.9189	−1.51616
$210$	0	0
$211$	24.8319	1.70950	0.854751	−	0.519039i	$-0.173710\pi$
$211$	24.8319	1.70950	0.854751	+	0.519039i	$0.173710\pi$
$212$	−26.1348	−1.79495
$213$	0	0
$214$	−3.63412	−0.248423
$215$	31.3390	2.13730
$216$	0	0
$217$	7.97341	0.541270
$218$	−32.4717	−2.19926
$219$	0	0
$220$	72.1100	4.86166
$221$	22.7424	1.52982
$222$	0	0
$223$	−7.10835	−0.476010	−0.238005	−	0.971264i	$-0.576493\pi$
$223$	−7.10835	−0.476010	−0.238005	+	0.971264i	$0.576493\pi$
$224$	−4.45962	−0.297971
$225$	0	0
$226$	39.2039	2.60780
$227$	−2.59236	−0.172061	−0.0860304	−	0.996293i	$-0.527418\pi$
$227$	−2.59236	−0.172061	−0.0860304	+	0.996293i	$0.527418\pi$
$228$	0	0
$229$	−24.7581	−1.63606	−0.818030	−	0.575176i	$-0.804934\pi$
$229$	−24.7581	−1.63606	−0.818030	+	0.575176i	$0.804934\pi$
$230$	−30.3995	−2.00448
$231$	0	0
$232$	59.2660	3.89100
$233$	9.83875	0.644558	0.322279	−	0.946645i	$-0.395551\pi$
$233$	9.83875	0.644558	0.322279	+	0.946645i	$0.395551\pi$
$234$	0	0
$235$	13.9784	0.911852
$236$	−35.7041	−2.32414
$237$	0	0
$238$	−11.5269	−0.747179
$239$	−21.9297	−1.41852	−0.709259	−	0.704948i	$-0.750970\pi$
$239$	−21.9297	−1.41852	−0.709259	+	0.704948i	$0.750970\pi$
$240$	0	0
$241$	26.9594	1.73661	0.868303	−	0.496035i	$-0.165211\pi$
$241$	26.9594	1.73661	0.868303	+	0.496035i	$0.165211\pi$
$242$	−12.4275	−0.798869
$243$	0	0
$244$	35.4341	2.26844
$245$	4.10792	0.262445
$246$	0	0
$247$	−27.4276	−1.74518
$248$	−48.4248	−3.07498
$249$	0	0
$250$	−71.4504	−4.51892
$251$	21.3057	1.34480	0.672402	−	0.740186i	$-0.265263\pi$
$251$	21.3057	1.34480	0.672402	+	0.740186i	$0.265263\pi$
$252$	0	0
$253$	11.6681	0.733566
$254$	2.52993	0.158742
$255$	0	0
$256$	−30.6851	−1.91782
$257$	23.6867	1.47754	0.738769	−	0.673959i	$-0.235407\pi$
$257$	23.6867	1.47754	0.738769	+	0.673959i	$0.235407\pi$
$258$	0	0
$259$	4.83283	0.300298
$260$	90.2328	5.59600
$261$	0	0
$262$	−26.8913	−1.66135
$263$	−11.8946	−0.733451	−0.366725	−	0.930329i	$-0.619521\pi$
$263$	−11.8946	−0.733451	−0.366725	+	0.930329i	$0.619521\pi$
$264$	0	0
$265$	−24.3968	−1.49868
$266$	13.9016	0.852360
$267$	0	0
$268$	15.1907	0.927922
$269$	−21.7026	−1.32323	−0.661615	−	0.749844i	$-0.730129\pi$
$269$	−21.7026	−1.32323	−0.661615	+	0.749844i	$0.730129\pi$
$270$	0	0
$271$	5.94555	0.361166	0.180583	−	0.983560i	$-0.442201\pi$
$271$	5.94555	0.361166	0.180583	+	0.983560i	$0.442201\pi$
$272$	29.9064	1.81334
$273$	0	0
$274$	−0.132101	−0.00798051
$275$	47.3695	2.85649
$276$	0	0
$277$	−24.2505	−1.45707	−0.728536	−	0.685008i	$-0.759799\pi$
$277$	−24.2505	−1.45707	−0.728536	+	0.685008i	$0.759799\pi$
$278$	−28.2704	−1.69555
$279$	0	0
$280$	−24.9486	−1.49096
$281$	9.64326	0.575269	0.287634	−	0.957740i	$-0.407131\pi$
$281$	9.64326	0.575269	0.287634	+	0.957740i	$0.407131\pi$
$282$	0	0
$283$	−6.20811	−0.369034	−0.184517	−	0.982829i	$-0.559072\pi$
$283$	−6.20811	−0.369034	−0.184517	+	0.982829i	$0.559072\pi$
$284$	−7.72563	−0.458432
$285$	0	0
$286$	−50.3741	−2.97868
$287$	−6.70872	−0.396003
$288$	0	0
$289$	3.75904	0.221120
$290$	101.418	5.95544
$291$	0	0
$292$	−8.06983	−0.472251
$293$	12.3514	0.721574	0.360787	−	0.932648i	$-0.382508\pi$
$293$	12.3514	0.721574	0.360787	+	0.932648i	$0.382508\pi$
$294$	0	0
$295$	−33.3297	−1.94053
$296$	−29.3512	−1.70600
$297$	0	0
$298$	18.2006	1.05433
$299$	14.6005	0.844369
$300$	0	0
$301$	7.62892	0.439724
$302$	−22.9475	−1.32048
$303$	0	0
$304$	−36.0674	−2.06861
$305$	33.0776	1.89402
$306$	0	0
$307$	−5.93391	−0.338666	−0.169333	−	0.985559i	$-0.554161\pi$
$307$	−5.93391	−0.338666	−0.169333	+	0.985559i	$0.554161\pi$
$308$	17.5539	1.00023
$309$	0	0
$310$	−82.8658	−4.70646
$311$	−28.9501	−1.64161	−0.820804	−	0.571211i	$-0.806474\pi$
$311$	−28.9501	−1.64161	−0.820804	+	0.571211i	$0.806474\pi$
$312$	0	0
$313$	−27.1186	−1.53283	−0.766416	−	0.642345i	$-0.777962\pi$
$313$	−27.1186	−1.53283	−0.766416	+	0.642345i	$0.777962\pi$
$314$	26.2733	1.48269
$315$	0	0
$316$	−10.6703	−0.600252
$317$	4.65983	0.261722	0.130861	−	0.991401i	$-0.458226\pi$
$317$	4.65983	0.261722	0.130861	+	0.991401i	$0.458226\pi$
$318$	0	0
$319$	−38.9266	−2.17947
$320$	−7.58001	−0.423736
$321$	0	0
$322$	−7.40021	−0.412397
$323$	−25.0356	−1.39302
$324$	0	0
$325$	59.2744	3.28795
$326$	−31.4233	−1.74038
$327$	0	0
$328$	40.7440	2.24971
$329$	3.40280	0.187602
$330$	0	0
$331$	−13.4701	−0.740383	−0.370191	−	0.928955i	$-0.620708\pi$
$331$	−13.4701	−0.740383	−0.370191	+	0.928955i	$0.620708\pi$
$332$	−4.48001	−0.245873
$333$	0	0
$334$	−0.832423	−0.0455481
$335$	14.1805	0.774764
$336$	0	0
$337$	−15.8357	−0.862627	−0.431313	−	0.902202i	$-0.641950\pi$
$337$	−15.8357	−0.862627	−0.431313	+	0.902202i	$0.641950\pi$
$338$	−30.1450	−1.63967
$339$	0	0
$340$	82.3635	4.46679
$341$	31.8060	1.72239
$342$	0	0
$343$	1.00000	0.0539949
$344$	−46.3326	−2.49809
$345$	0	0
$346$	18.2731	0.982369
$347$	−10.9172	−0.586063	−0.293032	−	0.956103i	$-0.594664\pi$
$347$	−10.9172	−0.586063	−0.293032	+	0.956103i	$0.594664\pi$
$348$	0	0
$349$	−10.4531	−0.559539	−0.279770	−	0.960067i	$-0.590258\pi$
$349$	−10.4531	−0.559539	−0.279770	+	0.960067i	$0.590258\pi$
$350$	−30.0430	−1.60587
$351$	0	0
$352$	−17.7894	−0.948181
$353$	−30.4592	−1.62118	−0.810591	−	0.585613i	$-0.800854\pi$
$353$	−30.4592	−1.62118	−0.810591	+	0.585613i	$0.800854\pi$
$354$	0	0
$355$	−7.21186	−0.382766
$356$	−56.4661	−2.99270
$357$	0	0
$358$	48.4800	2.56225
$359$	14.3141	0.755471	0.377736	−	0.925913i	$-0.376703\pi$
$359$	14.3141	0.755471	0.377736	+	0.925913i	$0.376703\pi$
$360$	0	0
$361$	11.1932	0.589114
$362$	37.6818	1.98051
$363$	0	0
$364$	21.9656	1.15131
$365$	−7.53316	−0.394304
$366$	0	0
$367$	24.5070	1.27925	0.639627	−	0.768685i	$-0.279089\pi$
$367$	24.5070	1.27925	0.639627	+	0.768685i	$0.279089\pi$
$368$	19.1997	1.00086
$369$	0	0
$370$	−50.2265	−2.61115
$371$	−5.93896	−0.308335
$372$	0	0
$373$	−4.32896	−0.224145	−0.112073	−	0.993700i	$-0.535749\pi$
$373$	−4.32896	−0.224145	−0.112073	+	0.993700i	$0.535749\pi$
$374$	−45.9809	−2.37762
$375$	0	0
$376$	−20.6662	−1.06578
$377$	−48.7096	−2.50867
$378$	0	0
$379$	14.5155	0.745610	0.372805	−	0.927910i	$-0.378396\pi$
$379$	14.5155	0.745610	0.372805	+	0.927910i	$0.378396\pi$
$380$	−99.3311	−5.09558
$381$	0	0
$382$	12.8155	0.655697
$383$	−21.0588	−1.07606	−0.538028	−	0.842927i	$-0.680831\pi$
$383$	−21.0588	−1.07606	−0.538028	+	0.842927i	$0.680831\pi$
$384$	0	0
$385$	16.3865	0.835134
$386$	−24.9691	−1.27089
$387$	0	0
$388$	−63.0221	−3.19946
$389$	4.05770	0.205734	0.102867	−	0.994695i	$-0.467198\pi$
$389$	4.05770	0.205734	0.102867	+	0.994695i	$0.467198\pi$
$390$	0	0
$391$	13.3272	0.673984
$392$	−6.07329	−0.306747
$393$	0	0
$394$	−54.9120	−2.76643
$395$	−9.96071	−0.501178
$396$	0	0
$397$	3.20500	0.160854	0.0804272	−	0.996760i	$-0.474372\pi$
$397$	3.20500	0.160854	0.0804272	+	0.996760i	$0.474372\pi$
$398$	−55.8678	−2.80040
$399$	0	0
$400$	77.9461	3.89731
$401$	−11.2930	−0.563943	−0.281972	−	0.959423i	$-0.590988\pi$
$401$	−11.2930	−0.563943	−0.281972	+	0.959423i	$0.590988\pi$
$402$	0	0
$403$	39.7995	1.98255
$404$	85.9166	4.27451
$405$	0	0
$406$	24.6883	1.22526
$407$	19.2782	0.955585
$408$	0	0
$409$	−2.85787	−0.141312	−0.0706562	−	0.997501i	$-0.522509\pi$
$409$	−2.85787	−0.141312	−0.0706562	+	0.997501i	$0.522509\pi$
$410$	69.7221	3.44333
$411$	0	0
$412$	−43.1254	−2.12464
$413$	−8.11352	−0.399240
$414$	0	0
$415$	−4.18208	−0.205290
$416$	−22.2603	−1.09140
$417$	0	0
$418$	55.4534	2.71232
$419$	−21.0317	−1.02747	−0.513733	−	0.857950i	$-0.671738\pi$
$419$	−21.0317	−1.02747	−0.513733	+	0.857950i	$0.671738\pi$
$420$	0	0
$421$	−16.5644	−0.807300	−0.403650	−	0.914914i	$-0.632259\pi$
$421$	−16.5644	−0.807300	−0.403650	+	0.914914i	$0.632259\pi$
$422$	−62.8232	−3.05819
$423$	0	0
$424$	36.0690	1.75166
$425$	54.1050	2.62448
$426$	0	0
$427$	8.05216	0.389671
$428$	6.32118	0.305546
$429$	0	0
$430$	−79.2856	−3.82349
$431$	11.4963	0.553756	0.276878	−	0.960905i	$-0.410700\pi$
$431$	11.4963	0.553756	0.276878	+	0.960905i	$0.410700\pi$
$432$	0	0
$433$	−14.9975	−0.720732	−0.360366	−	0.932811i	$-0.617348\pi$
$433$	−14.9975	−0.720732	−0.360366	+	0.932811i	$0.617348\pi$
$434$	−20.1722	−0.968297
$435$	0	0
$436$	56.4812	2.70496
$437$	−16.0727	−0.768861
$438$	0	0
$439$	32.4780	1.55009	0.775045	−	0.631906i	$-0.217727\pi$
$439$	32.4780	1.55009	0.775045	+	0.631906i	$0.217727\pi$
$440$	−99.5201	−4.74443
$441$	0	0
$442$	−57.5369	−2.73675
$443$	−3.02696	−0.143815	−0.0719075	−	0.997411i	$-0.522909\pi$
$443$	−3.02696	−0.143815	−0.0719075	+	0.997411i	$0.522909\pi$
$444$	0	0
$445$	−52.7109	−2.49874
$446$	17.9837	0.851551
$447$	0	0
$448$	−1.84522	−0.0871784
$449$	−22.5639	−1.06486	−0.532428	−	0.846475i	$-0.678720\pi$
$449$	−22.5639	−1.06486	−0.532428	+	0.846475i	$0.678720\pi$
$450$	0	0
$451$	−26.7611	−1.26013
$452$	−68.1913	−3.20745
$453$	0	0
$454$	6.55850	0.307805
$455$	20.5048	0.961279
$456$	0	0
$457$	3.94039	0.184324	0.0921619	−	0.995744i	$-0.470622\pi$
$457$	3.94039	0.184324	0.0921619	+	0.995744i	$0.470622\pi$
$458$	62.6363	2.92680
$459$	0	0
$460$	52.8768	2.46540
$461$	0.960816	0.0447497	0.0223748	−	0.999750i	$-0.492877\pi$
$461$	0.960816	0.0447497	0.0223748	+	0.999750i	$0.492877\pi$
$462$	0	0
$463$	−16.5668	−0.769924	−0.384962	−	0.922932i	$-0.625785\pi$
$463$	−16.5668	−0.769924	−0.384962	+	0.922932i	$0.625785\pi$
$464$	−64.0534	−2.97361
$465$	0	0
$466$	−24.8914	−1.15307
$467$	−23.5813	−1.09121	−0.545607	−	0.838041i	$-0.683701\pi$
$467$	−23.5813	−1.09121	−0.545607	+	0.838041i	$0.683701\pi$
$468$	0	0
$469$	3.45199	0.159398
$470$	−35.3645	−1.63124
$471$	0	0
$472$	49.2757	2.26810
$473$	30.4318	1.39926
$474$	0	0
$475$	−65.2512	−2.99393
$476$	20.0499	0.918986
$477$	0	0
$478$	55.4808	2.53763
$479$	−2.39962	−0.109641	−0.0548207	−	0.998496i	$-0.517459\pi$
$479$	−2.39962	−0.109641	−0.0548207	+	0.998496i	$0.517459\pi$
$480$	0	0
$481$	24.1232	1.09992
$482$	−68.2054	−3.10667
$483$	0	0
$484$	21.6164	0.982563
$485$	−58.8309	−2.67137
$486$	0	0
$487$	−28.4551	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$487$	−28.4551	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$488$	−48.9031	−2.21374
$489$	0	0
$490$	−10.3928	−0.469498
$491$	−40.6003	−1.83227	−0.916133	−	0.400874i	$-0.868706\pi$
$491$	−40.6003	−1.83227	−0.916133	+	0.400874i	$0.868706\pi$
$492$	0	0
$493$	−44.4616	−2.00245
$494$	69.3900	3.12200
$495$	0	0
$496$	52.3365	2.34998
$497$	−1.75560	−0.0787493
$498$	0	0
$499$	6.92769	0.310126	0.155063	−	0.987905i	$-0.450442\pi$
$499$	6.92769	0.310126	0.155063	+	0.987905i	$0.450442\pi$
$500$	124.281	5.55801
$501$	0	0
$502$	−53.9021	−2.40577
$503$	−39.6839	−1.76942	−0.884709	−	0.466145i	$-0.845643\pi$
$503$	−39.6839	−1.76942	−0.884709	+	0.466145i	$0.845643\pi$
$504$	0	0
$505$	80.2029	3.56898
$506$	−29.5195	−1.31230
$507$	0	0
$508$	−4.40057	−0.195244
$509$	13.9927	0.620215	0.310108	−	0.950701i	$-0.399635\pi$
$509$	13.9927	0.620215	0.310108	+	0.950701i	$0.399635\pi$
$510$	0	0
$511$	−1.83381	−0.0811231
$512$	50.4563	2.22987
$513$	0	0
$514$	−59.9259	−2.64322
$515$	−40.2575	−1.77396
$516$	0	0
$517$	13.5738	0.596975
$518$	−12.2268	−0.537213
$519$	0	0
$520$	−124.531	−5.46107
$521$	−13.9949	−0.613130	−0.306565	−	0.951850i	$-0.599180\pi$
$521$	−13.9949	−0.613130	−0.306565	+	0.951850i	$0.599180\pi$
$522$	0	0
$523$	31.8879	1.39436	0.697181	−	0.716895i	$-0.254437\pi$
$523$	31.8879	1.39436	0.697181	+	0.716895i	$0.254437\pi$
$524$	46.7747	2.04336
$525$	0	0
$526$	30.0925	1.31209
$527$	36.3285	1.58249
$528$	0	0
$529$	−14.4440	−0.628002
$530$	61.7222	2.68104
$531$	0	0
$532$	−24.1804	−1.04835
$533$	−33.4867	−1.45047
$534$	0	0
$535$	5.90081	0.255114
$536$	−20.9650	−0.905548
$537$	0	0
$538$	54.9061	2.36717
$539$	3.98901	0.171819
$540$	0	0
$541$	−43.3090	−1.86200	−0.930999	−	0.365022i	$-0.881062\pi$
$541$	−43.3090	−1.86200	−0.930999	+	0.365022i	$0.881062\pi$
$542$	−15.0419	−0.646103
$543$	0	0
$544$	−20.3190	−0.871168
$545$	52.7251	2.25849
$546$	0	0
$547$	−36.2682	−1.55072	−0.775359	−	0.631521i	$-0.782431\pi$
$547$	−36.2682	−1.55072	−0.775359	+	0.631521i	$0.782431\pi$
$548$	0.229776	0.00981556
$549$	0	0
$550$	−119.842	−5.11007
$551$	53.6211	2.28434
$552$	0	0
$553$	−2.42476	−0.103111
$554$	61.3522	2.60660
$555$	0	0
$556$	49.1735	2.08542
$557$	2.86094	0.121222	0.0606110	−	0.998161i	$-0.480695\pi$
$557$	2.86094	0.121222	0.0606110	+	0.998161i	$0.480695\pi$
$558$	0	0
$559$	38.0800	1.61061
$560$	26.9639	1.13943
$561$	0	0
$562$	−24.3968	−1.02912
$563$	41.4743	1.74793	0.873967	−	0.485985i	$-0.161539\pi$
$563$	41.4743	1.74793	0.873967	+	0.485985i	$0.161539\pi$
$564$	0	0
$565$	−63.6564	−2.67804
$566$	15.7061	0.660178
$567$	0	0
$568$	10.6623	0.447378
$569$	3.90737	0.163806	0.0819028	−	0.996640i	$-0.473900\pi$
$569$	3.90737	0.163806	0.0819028	+	0.996640i	$0.473900\pi$
$570$	0	0
$571$	−3.55119	−0.148613	−0.0743063	−	0.997235i	$-0.523674\pi$
$571$	−3.55119	−0.148613	−0.0743063	+	0.997235i	$0.523674\pi$
$572$	87.6208	3.66361
$573$	0	0
$574$	16.9726	0.708423
$575$	34.7351	1.44855
$576$	0	0
$577$	−25.0903	−1.04452	−0.522262	−	0.852785i	$-0.674912\pi$
$577$	−25.0903	−1.04452	−0.522262	+	0.852785i	$0.674912\pi$
$578$	−9.51011	−0.395569
$579$	0	0
$580$	−176.406	−7.32485
$581$	−1.01805	−0.0422359
$582$	0	0
$583$	−23.6905	−0.981162
$584$	11.1373	0.460864
$585$	0	0
$586$	−31.2481	−1.29085
$587$	−27.8022	−1.14752	−0.573760	−	0.819023i	$-0.694516\pi$
$587$	−27.8022	−1.14752	−0.573760	+	0.819023i	$0.694516\pi$
$588$	0	0
$589$	−43.8125	−1.80526
$590$	84.3219	3.47148
$591$	0	0
$592$	31.7221	1.30377
$593$	−29.2670	−1.20185	−0.600925	−	0.799306i	$-0.705201\pi$
$593$	−29.2670	−1.20185	−0.600925	+	0.799306i	$0.705201\pi$
$594$	0	0
$595$	18.7165	0.767303
$596$	−31.6581	−1.29677
$597$	0	0
$598$	−36.9383	−1.51052
$599$	10.7560	0.439477	0.219738	−	0.975559i	$-0.429480\pi$
$599$	10.7560	0.439477	0.219738	+	0.975559i	$0.429480\pi$
$600$	0	0
$601$	16.2355	0.662259	0.331129	−	0.943585i	$-0.392570\pi$
$601$	16.2355	0.662259	0.331129	+	0.943585i	$0.392570\pi$
$602$	−19.3007	−0.786637
$603$	0	0
$604$	39.9150	1.62412
$605$	20.1788	0.820386
$606$	0	0
$607$	−17.2815	−0.701433	−0.350716	−	0.936482i	$-0.614062\pi$
$607$	−17.2815	−0.701433	−0.350716	+	0.936482i	$0.614062\pi$
$608$	24.5048	0.993803
$609$	0	0
$610$	−83.6842	−3.38828
$611$	16.9852	0.687146
$612$	0	0
$613$	8.33692	0.336725	0.168363	−	0.985725i	$-0.446152\pi$
$613$	8.33692	0.336725	0.168363	+	0.985725i	$0.446152\pi$
$614$	15.0124	0.605851
$615$	0	0
$616$	−24.2264	−0.976109
$617$	−27.1329	−1.09233	−0.546165	−	0.837678i	$-0.683913\pi$
$617$	−27.1329	−1.09233	−0.546165	+	0.837678i	$0.683913\pi$
$618$	0	0
$619$	42.4011	1.70424	0.852122	−	0.523343i	$-0.175315\pi$
$619$	42.4011	1.70424	0.852122	+	0.523343i	$0.175315\pi$
$620$	144.137	5.78867
$621$	0	0
$622$	73.2418	2.93673
$623$	−12.8315	−0.514084
$624$	0	0
$625$	56.6407	2.26563
$626$	68.6082	2.74213
$627$	0	0
$628$	−45.6999	−1.82362
$629$	22.0194	0.877971
$630$	0	0
$631$	46.2954	1.84299	0.921495	−	0.388391i	$-0.126969\pi$
$631$	46.2954	1.84299	0.921495	+	0.388391i	$0.126969\pi$
$632$	14.7262	0.585779
$633$	0	0
$634$	−11.7891	−0.468204
$635$	−4.10792	−0.163018
$636$	0	0
$637$	4.99152	0.197771
$638$	98.4818	3.89893
$639$	0	0
$640$	55.8165	2.20634
$641$	36.3941	1.43748	0.718741	−	0.695278i	$-0.244719\pi$
$641$	36.3941	1.43748	0.718741	+	0.695278i	$0.244719\pi$
$642$	0	0
$643$	32.7558	1.29176	0.645882	−	0.763437i	$-0.276490\pi$
$643$	32.7558	1.29176	0.645882	+	0.763437i	$0.276490\pi$
$644$	12.8719	0.507225
$645$	0	0
$646$	63.3384	2.49202
$647$	−2.70079	−0.106179	−0.0530896	−	0.998590i	$-0.516907\pi$
$647$	−2.70079	−0.106179	−0.0530896	+	0.998590i	$0.516907\pi$
$648$	0	0
$649$	−32.3649	−1.27043
$650$	−149.960	−5.88193
$651$	0	0
$652$	54.6578	2.14056
$653$	3.68557	0.144227	0.0721137	−	0.997396i	$-0.477026\pi$
$653$	3.68557	0.144227	0.0721137	+	0.997396i	$0.477026\pi$
$654$	0	0
$655$	43.6640	1.70610
$656$	−44.0352	−1.71929
$657$	0	0
$658$	−8.60886	−0.335608
$659$	48.7210	1.89790	0.948951	−	0.315424i	$-0.102147\pi$
$659$	48.7210	1.89790	0.948951	+	0.315424i	$0.102147\pi$
$660$	0	0
$661$	−18.2225	−0.708774	−0.354387	−	0.935099i	$-0.615310\pi$
$661$	−18.2225	−0.708774	−0.354387	+	0.935099i	$0.615310\pi$
$662$	34.0784	1.32450
$663$	0	0
$664$	6.18292	0.239944
$665$	−22.5723	−0.875317
$666$	0	0
$667$	−28.5441	−1.10523
$668$	1.44792	0.0560216
$669$	0	0
$670$	−35.8758	−1.38600
$671$	32.1201	1.23998
$672$	0	0
$673$	−24.8491	−0.957864	−0.478932	−	0.877852i	$-0.658976\pi$
$673$	−24.8491	−0.957864	−0.478932	+	0.877852i	$0.658976\pi$
$674$	40.0633	1.54318
$675$	0	0
$676$	52.4342	2.01670
$677$	−16.1732	−0.621588	−0.310794	−	0.950477i	$-0.600595\pi$
$677$	−16.1732	−0.621588	−0.310794	+	0.950477i	$0.600595\pi$
$678$	0	0
$679$	−14.3213	−0.549602
$680$	−113.671	−4.35908
$681$	0	0
$682$	−80.4671	−3.08124
$683$	33.4720	1.28077	0.640386	−	0.768053i	$-0.278774\pi$
$683$	33.4720	1.28077	0.640386	+	0.768053i	$0.278774\pi$
$684$	0	0
$685$	0.214496	0.00819546
$686$	−2.52993	−0.0965934
$687$	0	0
$688$	50.0754	1.90910
$689$	−29.6444	−1.12936
$690$	0	0
$691$	34.1006	1.29725	0.648624	−	0.761109i	$-0.275345\pi$
$691$	34.1006	1.29725	0.648624	+	0.761109i	$0.275345\pi$
$692$	−31.7843	−1.20826
$693$	0	0
$694$	27.6197	1.04843
$695$	45.9034	1.74121
$696$	0	0
$697$	−30.5663	−1.15778
$698$	26.4455	1.00098
$699$	0	0
$700$	52.2568	1.97512
$701$	44.9930	1.69936	0.849681	−	0.527297i	$-0.176795\pi$
$701$	44.9930	1.69936	0.849681	+	0.527297i	$0.176795\pi$
$702$	0	0
$703$	−26.5556	−1.00156
$704$	−7.36059	−0.277413
$705$	0	0
$706$	77.0599	2.90019
$707$	19.5240	0.734274
$708$	0	0
$709$	8.04439	0.302113	0.151057	−	0.988525i	$-0.451732\pi$
$709$	8.04439	0.302113	0.151057	+	0.988525i	$0.451732\pi$
$710$	18.2455	0.684742
$711$	0	0
$712$	77.9296	2.92053
$713$	23.3227	0.873441
$714$	0	0
$715$	81.7937	3.05891
$716$	−84.3261	−3.15142
$717$	0	0
$718$	−36.2138	−1.35149
$719$	−8.30363	−0.309673	−0.154837	−	0.987940i	$-0.549485\pi$
$719$	−8.30363	−0.309673	−0.154837	+	0.987940i	$0.549485\pi$
$720$	0	0
$721$	−9.79996	−0.364970
$722$	−28.3180	−1.05389
$723$	0	0
$724$	−65.5438	−2.43592
$725$	−115.882	−4.30374
$726$	0	0
$727$	4.48610	0.166380	0.0831901	−	0.996534i	$-0.473489\pi$
$727$	4.48610	0.166380	0.0831901	+	0.996534i	$0.473489\pi$
$728$	−30.3150	−1.12355
$729$	0	0
$730$	19.0584	0.705383
$731$	34.7590	1.28561
$732$	0	0
$733$	−33.1406	−1.22408	−0.612038	−	0.790829i	$-0.709650\pi$
$733$	−33.1406	−1.22408	−0.612038	+	0.790829i	$0.709650\pi$
$734$	−62.0011	−2.28850
$735$	0	0
$736$	−13.0446	−0.480832
$737$	13.7700	0.507226
$738$	0	0
$739$	−30.7555	−1.13136	−0.565679	−	0.824625i	$-0.691386\pi$
$739$	−30.7555	−1.13136	−0.565679	+	0.824625i	$0.691386\pi$
$740$	87.3640	3.21157
$741$	0	0
$742$	15.0252	0.551591
$743$	52.2502	1.91687	0.958436	−	0.285307i	$-0.0920957\pi$
$743$	52.2502	1.91687	0.958436	+	0.285307i	$0.0920957\pi$
$744$	0	0
$745$	−29.5527	−1.08273
$746$	10.9520	0.400981
$747$	0	0
$748$	79.9793	2.92433
$749$	1.43645	0.0524866
$750$	0	0
$751$	43.5789	1.59022	0.795109	−	0.606467i	$-0.207414\pi$
$751$	43.5789	1.59022	0.795109	+	0.606467i	$0.207414\pi$
$752$	22.3356	0.814495
$753$	0	0
$754$	123.232	4.48785
$755$	37.2605	1.35605
$756$	0	0
$757$	28.6307	1.04060	0.520300	−	0.853984i	$-0.325820\pi$
$757$	28.6307	1.04060	0.520300	+	0.853984i	$0.325820\pi$
$758$	−36.7232	−1.33385
$759$	0	0
$760$	137.088	4.97271
$761$	8.21883	0.297933	0.148966	−	0.988842i	$-0.452405\pi$
$761$	8.21883	0.297933	0.148966	+	0.988842i	$0.452405\pi$
$762$	0	0
$763$	12.8350	0.464657
$764$	−22.2913	−0.806470
$765$	0	0
$766$	53.2775	1.92499
$767$	−40.4988	−1.46233
$768$	0	0
$769$	17.6897	0.637907	0.318954	−	0.947770i	$-0.396669\pi$
$769$	17.6897	0.637907	0.318954	+	0.947770i	$0.396669\pi$
$770$	−41.4568	−1.49400
$771$	0	0
$772$	43.4313	1.56313
$773$	−24.6105	−0.885177	−0.442588	−	0.896725i	$-0.645940\pi$
$773$	−24.6105	−0.885177	−0.442588	+	0.896725i	$0.645940\pi$
$774$	0	0
$775$	94.6843	3.40116
$776$	86.9776	3.12231
$777$	0	0
$778$	−10.2657	−0.368044
$779$	36.8633	1.32076
$780$	0	0
$781$	−7.00309	−0.250590
$782$	−33.7169	−1.20571
$783$	0	0
$784$	6.56388	0.234424
$785$	−42.6607	−1.52263
$786$	0	0
$787$	6.83914	0.243789	0.121894	−	0.992543i	$-0.461103\pi$
$787$	6.83914	0.243789	0.121894	+	0.992543i	$0.461103\pi$
$788$	95.5140	3.40255
$789$	0	0
$790$	25.1999	0.896574
$791$	−15.4960	−0.550975
$792$	0	0
$793$	40.1926	1.42728
$794$	−8.10845	−0.287758
$795$	0	0
$796$	97.1766	3.44433
$797$	45.6287	1.61625	0.808126	−	0.589010i	$-0.200482\pi$
$797$	45.6287	1.61625	0.808126	+	0.589010i	$0.200482\pi$
$798$	0	0
$799$	15.5039	0.548487
$800$	−52.9580	−1.87235
$801$	0	0
$802$	28.5705	1.00886
$803$	−7.31509	−0.258144
$804$	0	0
$805$	12.0159	0.423505
$806$	−100.690	−3.54666
$807$	0	0
$808$	−118.575	−4.17144
$809$	−2.23445	−0.0785590	−0.0392795	−	0.999228i	$-0.512506\pi$
$809$	−2.23445	−0.0785590	−0.0392795	+	0.999228i	$0.512506\pi$
$810$	0	0
$811$	−34.7919	−1.22171	−0.610855	−	0.791742i	$-0.709174\pi$
$811$	−34.7919	−1.22171	−0.610855	+	0.791742i	$0.709174\pi$
$812$	−42.9428	−1.50700
$813$	0	0
$814$	−48.7726	−1.70948
$815$	51.0229	1.78725
$816$	0	0
$817$	−41.9196	−1.46658
$818$	7.23022	0.252799
$819$	0	0
$820$	−121.275	−4.23510
$821$	−50.0623	−1.74719	−0.873593	−	0.486657i	$-0.838216\pi$
$821$	−50.0623	−1.74719	−0.873593	+	0.486657i	$0.838216\pi$
$822$	0	0
$823$	−44.1786	−1.53997	−0.769984	−	0.638063i	$-0.779736\pi$
$823$	−44.1786	−1.53997	−0.769984	+	0.638063i	$0.779736\pi$
$824$	59.5180	2.07341
$825$	0	0
$826$	20.5267	0.714214
$827$	−10.8664	−0.377861	−0.188930	−	0.981991i	$-0.560502\pi$
$827$	−10.8664	−0.377861	−0.188930	+	0.981991i	$0.560502\pi$
$828$	0	0
$829$	18.8915	0.656128	0.328064	−	0.944656i	$-0.393604\pi$
$829$	18.8915	0.656128	0.328064	+	0.944656i	$0.393604\pi$
$830$	10.5804	0.367250
$831$	0	0
$832$	−9.21046	−0.319315
$833$	4.55621	0.157863
$834$	0	0
$835$	1.35163	0.0467749
$836$	−96.4557	−3.33599
$837$	0	0
$838$	53.2088	1.83807
$839$	47.9109	1.65407	0.827034	−	0.562152i	$-0.190027\pi$
$839$	47.9109	1.65407	0.827034	+	0.562152i	$0.190027\pi$
$840$	0	0
$841$	66.2277	2.28371
$842$	41.9069	1.44421
$843$	0	0
$844$	109.275	3.76139
$845$	48.9472	1.68383
$846$	0	0
$847$	4.91218	0.168784
$848$	−38.9826	−1.33867
$849$	0	0
$850$	−136.882	−4.69502
$851$	14.1363	0.484587
$852$	0	0
$853$	5.74350	0.196654	0.0983269	−	0.995154i	$-0.468651\pi$
$853$	5.74350	0.196654	0.0983269	+	0.995154i	$0.468651\pi$
$854$	−20.3714	−0.697096
$855$	0	0
$856$	−8.72395	−0.298179
$857$	19.0436	0.650516	0.325258	−	0.945625i	$-0.394549\pi$
$857$	19.0436	0.650516	0.325258	+	0.945625i	$0.394549\pi$
$858$	0	0
$859$	−27.9592	−0.953954	−0.476977	−	0.878916i	$-0.658268\pi$
$859$	−27.9592	−0.953954	−0.476977	+	0.878916i	$0.658268\pi$
$860$	137.910	4.70268
$861$	0	0
$862$	−29.0848	−0.990633
$863$	36.5214	1.24320	0.621602	−	0.783333i	$-0.286482\pi$
$863$	36.5214	1.24320	0.621602	+	0.783333i	$0.286482\pi$
$864$	0	0
$865$	−29.6705	−1.00883
$866$	37.9426	1.28934
$867$	0	0
$868$	35.0875	1.19095
$869$	−9.67237	−0.328113
$870$	0	0
$871$	17.2307	0.583841
$872$	−77.9505	−2.63974
$873$	0	0
$874$	40.6629	1.37544
$875$	28.2420	0.954753
$876$	0	0
$877$	9.88821	0.333901	0.166951	−	0.985965i	$-0.446608\pi$
$877$	9.88821	0.333901	0.166951	+	0.985965i	$0.446608\pi$
$878$	−82.1672	−2.77301
$879$	0	0
$880$	107.559	3.62582
$881$	7.18316	0.242007	0.121003	−	0.992652i	$-0.461389\pi$
$881$	7.18316	0.242007	0.121003	+	0.992652i	$0.461389\pi$
$882$	0	0
$883$	−21.4177	−0.720762	−0.360381	−	0.932805i	$-0.617353\pi$
$883$	−21.4177	−0.720762	−0.360381	+	0.932805i	$0.617353\pi$
$884$	100.080	3.36604
$885$	0	0
$886$	7.65800	0.257276
$887$	22.9044	0.769055	0.384528	−	0.923113i	$-0.374364\pi$
$887$	22.9044	0.769055	0.384528	+	0.923113i	$0.374364\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	133.355	4.47008
$891$	0	0
$892$	−31.2808	−1.04736
$893$	−18.6978	−0.625698
$894$	0	0
$895$	−78.7182	−2.63126
$896$	13.5875	0.453927
$897$	0	0
$898$	57.0852	1.90496
$899$	−77.8083	−2.59505
$900$	0	0
$901$	−27.0591	−0.901470
$902$	67.7039	2.25429
$903$	0	0
$904$	94.1117	3.13011
$905$	−61.1849	−2.03386
$906$	0	0
$907$	15.9809	0.530638	0.265319	−	0.964161i	$-0.414523\pi$
$907$	15.9809	0.530638	0.265319	+	0.964161i	$0.414523\pi$
$908$	−11.4079	−0.378583
$909$	0	0
$910$	−51.8758	−1.71967
$911$	−43.6890	−1.44748	−0.723741	−	0.690072i	$-0.757579\pi$
$911$	−43.6890	−1.44748	−0.723741	+	0.690072i	$0.757579\pi$
$912$	0	0
$913$	−4.06102	−0.134400
$914$	−9.96893	−0.329743
$915$	0	0
$916$	−108.950	−3.59980
$917$	10.6292	0.351008
$918$	0	0
$919$	10.8687	0.358525	0.179262	−	0.983801i	$-0.442629\pi$
$919$	10.8687	0.358525	0.179262	+	0.983801i	$0.442629\pi$
$920$	−72.9760	−2.40595
$921$	0	0
$922$	−2.43080	−0.0800542
$923$	−8.76311	−0.288441
$924$	0	0
$925$	57.3899	1.88697
$926$	41.9129	1.37734
$927$	0	0
$928$	43.5190	1.42858
$929$	−12.9769	−0.425759	−0.212879	−	0.977078i	$-0.568284\pi$
$929$	−12.9769	−0.425759	−0.212879	+	0.977078i	$0.568284\pi$
$930$	0	0
$931$	−5.49483	−0.180086
$932$	43.2961	1.41821
$933$	0	0
$934$	59.6592	1.95211
$935$	74.6604	2.44166
$936$	0	0
$937$	−24.9940	−0.816518	−0.408259	−	0.912866i	$-0.633864\pi$
$937$	−24.9940	−0.816518	−0.408259	+	0.912866i	$0.633864\pi$
$938$	−8.73332	−0.285153
$939$	0	0
$940$	61.5131	2.00634
$941$	1.14750	0.0374074	0.0187037	−	0.999825i	$-0.494046\pi$
$941$	1.14750	0.0374074	0.0187037	+	0.999825i	$0.494046\pi$
$942$	0	0
$943$	−19.6234	−0.639025
$944$	−53.2562	−1.73334
$945$	0	0
$946$	−76.9905	−2.50318
$947$	18.3397	0.595960	0.297980	−	0.954572i	$-0.403687\pi$
$947$	18.3397	0.595960	0.297980	+	0.954572i	$0.403687\pi$
$948$	0	0
$949$	−9.15353	−0.297136
$950$	165.081	5.35594
$951$	0	0
$952$	−27.6712	−0.896828
$953$	−8.95156	−0.289970	−0.144985	−	0.989434i	$-0.546313\pi$
$953$	−8.95156	−0.289970	−0.144985	+	0.989434i	$0.546313\pi$
$954$	0	0
$955$	−20.8088	−0.673358
$956$	−96.5034	−3.12114
$957$	0	0
$958$	6.07088	0.196141
$959$	0.0522151	0.00168611
$960$	0	0
$961$	32.5752	1.05081
$962$	−61.0301	−1.96769
$963$	0	0
$964$	118.637	3.82103
$965$	40.5430	1.30513
$966$	0	0
$967$	−4.17102	−0.134131	−0.0670655	−	0.997749i	$-0.521364\pi$
$967$	−4.17102	−0.134131	−0.0670655	+	0.997749i	$0.521364\pi$
$968$	−29.8331	−0.958871
$969$	0	0
$970$	148.838	4.77891
$971$	31.9691	1.02594	0.512969	−	0.858407i	$-0.328546\pi$
$971$	31.9691	1.02594	0.512969	+	0.858407i	$0.328546\pi$
$972$	0	0
$973$	11.1744	0.358233
$974$	71.9895	2.30669
$975$	0	0
$976$	52.8534	1.69180
$977$	−20.7983	−0.665396	−0.332698	−	0.943033i	$-0.607959\pi$
$977$	−20.7983	−0.665396	−0.332698	+	0.943033i	$0.607959\pi$
$978$	0	0
$979$	−51.1851	−1.63588
$980$	18.0772	0.577455
$981$	0	0
$982$	102.716	3.27780
$983$	25.0701	0.799611	0.399806	−	0.916600i	$-0.369078\pi$
$983$	25.0701	0.799611	0.399806	+	0.916600i	$0.369078\pi$
$984$	0	0
$985$	89.1620	2.84094
$986$	112.485	3.58225
$987$	0	0
$988$	−120.697	−3.83988
$989$	22.3150	0.709577
$990$	0	0
$991$	−45.0281	−1.43036	−0.715182	−	0.698938i	$-0.753656\pi$
$991$	−45.0281	−1.43036	−0.715182	+	0.698938i	$0.753656\pi$
$992$	−35.5584	−1.12898
$993$	0	0
$994$	4.44155	0.140877
$995$	90.7140	2.87583
$996$	0	0
$997$	23.5071	0.744476	0.372238	−	0.928137i	$-0.378590\pi$
$997$	23.5071	0.744476	0.372238	+	0.928137i	$0.378590\pi$
$998$	−17.5266	−0.554795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.5	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.36	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.5	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.36	yes	40	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52993 q^{2} +4.40057 q^{4} +4.10792 q^{5} +1.00000 q^{7} -6.07329 q^{8} +O(q^{10})\)	\(q-2.52993 q^{2} +4.40057 q^{4} +4.10792 q^{5} +1.00000 q^{7} -6.07329 q^{8} -10.3928 q^{10} +3.98901 q^{11} +4.99152 q^{13} -2.52993 q^{14} +6.56388 q^{16} +4.55621 q^{17} -5.49483 q^{19} +18.0772 q^{20} -10.0919 q^{22} +2.92506 q^{23} +11.8750 q^{25} -12.6282 q^{26} +4.40057 q^{28} -9.75847 q^{29} +7.97341 q^{31} -4.45962 q^{32} -11.5269 q^{34} +4.10792 q^{35} +4.83283 q^{37} +13.9016 q^{38} -24.9486 q^{40} -6.70872 q^{41} +7.62892 q^{43} +17.5539 q^{44} -7.40021 q^{46} +3.40280 q^{47} +1.00000 q^{49} -30.0430 q^{50} +21.9656 q^{52} -5.93896 q^{53} +16.3865 q^{55} -6.07329 q^{56} +24.6883 q^{58} -8.11352 q^{59} +8.05216 q^{61} -20.1722 q^{62} -1.84522 q^{64} +20.5048 q^{65} +3.45199 q^{67} +20.0499 q^{68} -10.3928 q^{70} -1.75560 q^{71} -1.83381 q^{73} -12.2268 q^{74} -24.1804 q^{76} +3.98901 q^{77} -2.42476 q^{79} +26.9639 q^{80} +16.9726 q^{82} -1.01805 q^{83} +18.7165 q^{85} -19.3007 q^{86} -24.2264 q^{88} -12.8315 q^{89} +4.99152 q^{91} +12.8719 q^{92} -8.60886 q^{94} -22.5723 q^{95} -14.3213 q^{97} -2.52993 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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