LMFDB - Embedded newform 8001.2.a.ba.1.29

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			1.29
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)

$f(q)$	$=$	$q+1.42650 q^{2} +0.0348943 q^{4} -4.07722 q^{5} +1.00000 q^{7} -2.80322 q^{8} +O(q^{10})$	\(q+1.42650 q^{2} +0.0348943 q^{4} -4.07722 q^{5} +1.00000 q^{7} -2.80322 q^{8} -5.81614 q^{10} -3.50251 q^{11} -5.98419 q^{13} +1.42650 q^{14} -4.06857 q^{16} -7.04872 q^{17} +3.99853 q^{19} -0.142272 q^{20} -4.99633 q^{22} -7.32524 q^{23} +11.6237 q^{25} -8.53642 q^{26} +0.0348943 q^{28} -7.77438 q^{29} +2.84394 q^{31} -0.197369 q^{32} -10.0550 q^{34} -4.07722 q^{35} -1.23220 q^{37} +5.70390 q^{38} +11.4293 q^{40} -1.75959 q^{41} -10.6816 q^{43} -0.122218 q^{44} -10.4494 q^{46} -4.77897 q^{47} +1.00000 q^{49} +16.5812 q^{50} -0.208814 q^{52} +4.77146 q^{53} +14.2805 q^{55} -2.80322 q^{56} -11.0901 q^{58} -13.2639 q^{59} -2.14883 q^{61} +4.05688 q^{62} +7.85559 q^{64} +24.3988 q^{65} -5.59570 q^{67} -0.245960 q^{68} -5.81614 q^{70} +11.6689 q^{71} +5.10805 q^{73} -1.75772 q^{74} +0.139526 q^{76} -3.50251 q^{77} +8.07023 q^{79} +16.5885 q^{80} -2.51006 q^{82} +5.09750 q^{83} +28.7392 q^{85} -15.2373 q^{86} +9.81831 q^{88} -0.679380 q^{89} -5.98419 q^{91} -0.255609 q^{92} -6.81718 q^{94} -16.3029 q^{95} -18.1740 q^{97} +1.42650 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.42650	1.00869	0.504343	−	0.863503i	$-0.331735\pi$
$2$	1.42650	1.00869	0.504343	+	0.863503i	$0.331735\pi$
$3$	0	0
$3$	0	0
$4$	0.0348943	0.0174471
$5$	−4.07722	−1.82339	−0.911694	−	0.410870i	$-0.865225\pi$
$5$	−4.07722	−1.82339	−0.911694	+	0.410870i	$0.865225\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.80322	−0.991087
$9$	0	0
$10$	−5.81614	−1.83923
$11$	−3.50251	−1.05605	−0.528024	−	0.849229i	$-0.677067\pi$
$11$	−3.50251	−1.05605	−0.528024	+	0.849229i	$0.677067\pi$
$12$	0	0
$13$	−5.98419	−1.65971	−0.829857	−	0.557976i	$-0.811578\pi$
$13$	−5.98419	−1.65971	−0.829857	+	0.557976i	$0.811578\pi$
$14$	1.42650	0.381247
$15$	0	0
$16$	−4.06857	−1.01714
$17$	−7.04872	−1.70956	−0.854782	−	0.518987i	$-0.826309\pi$
$17$	−7.04872	−1.70956	−0.854782	+	0.518987i	$0.826309\pi$
$18$	0	0
$19$	3.99853	0.917326	0.458663	−	0.888610i	$-0.348328\pi$
$19$	3.99853	0.917326	0.458663	+	0.888610i	$0.348328\pi$
$20$	−0.142272	−0.0318129
$21$	0	0
$22$	−4.99633	−1.06522
$23$	−7.32524	−1.52742	−0.763709	−	0.645561i	$-0.776624\pi$
$23$	−7.32524	−1.52742	−0.763709	+	0.645561i	$0.776624\pi$
$24$	0	0
$25$	11.6237	2.32474
$26$	−8.53642	−1.67413
$27$	0	0
$28$	0.0348943	0.00659440
$29$	−7.77438	−1.44367	−0.721833	−	0.692068i	$-0.756700\pi$
$29$	−7.77438	−1.44367	−0.721833	+	0.692068i	$0.756700\pi$
$30$	0	0
$31$	2.84394	0.510787	0.255394	−	0.966837i	$-0.417795\pi$
$31$	2.84394	0.510787	0.255394	+	0.966837i	$0.417795\pi$
$32$	−0.197369	−0.0348903
$33$	0	0
$34$	−10.0550	−1.72441
$35$	−4.07722	−0.689176
$36$	0	0
$37$	−1.23220	−0.202572	−0.101286	−	0.994857i	$-0.532296\pi$
$37$	−1.23220	−0.202572	−0.101286	+	0.994857i	$0.532296\pi$
$38$	5.70390	0.925294
$39$	0	0
$40$	11.4293	1.80714
$41$	−1.75959	−0.274802	−0.137401	−	0.990515i	$-0.543875\pi$
$41$	−1.75959	−0.274802	−0.137401	+	0.990515i	$0.543875\pi$
$42$	0	0
$43$	−10.6816	−1.62894	−0.814468	−	0.580209i	$-0.802971\pi$
$43$	−10.6816	−1.62894	−0.814468	+	0.580209i	$0.802971\pi$
$44$	−0.122218	−0.0184250
$45$	0	0
$46$	−10.4494	−1.54068
$47$	−4.77897	−0.697084	−0.348542	−	0.937293i	$-0.613323\pi$
$47$	−4.77897	−0.697084	−0.348542	+	0.937293i	$0.613323\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	16.5812	2.34494
$51$	0	0
$52$	−0.208814	−0.0289573
$53$	4.77146	0.655410	0.327705	−	0.944780i	$-0.393725\pi$
$53$	4.77146	0.655410	0.327705	+	0.944780i	$0.393725\pi$
$54$	0	0
$55$	14.2805	1.92559
$56$	−2.80322	−0.374596
$57$	0	0
$58$	−11.0901	−1.45620
$59$	−13.2639	−1.72681	−0.863403	−	0.504514i	$-0.831672\pi$
$59$	−13.2639	−1.72681	−0.863403	+	0.504514i	$0.831672\pi$
$60$	0	0
$61$	−2.14883	−0.275130	−0.137565	−	0.990493i	$-0.543928\pi$
$61$	−2.14883	−0.275130	−0.137565	+	0.990493i	$0.543928\pi$
$62$	4.05688	0.515224
$63$	0	0
$64$	7.85559	0.981949
$65$	24.3988	3.02630
$66$	0	0
$67$	−5.59570	−0.683623	−0.341812	−	0.939769i	$-0.611040\pi$
$67$	−5.59570	−0.683623	−0.341812	+	0.939769i	$0.611040\pi$
$68$	−0.245960	−0.0298270
$69$	0	0
$70$	−5.81614	−0.695162
$71$	11.6689	1.38485	0.692424	−	0.721490i	$-0.256543\pi$
$71$	11.6689	1.38485	0.692424	+	0.721490i	$0.256543\pi$
$72$	0	0
$73$	5.10805	0.597852	0.298926	−	0.954276i	$-0.403372\pi$
$73$	5.10805	0.597852	0.298926	+	0.954276i	$0.403372\pi$
$74$	−1.75772	−0.204331
$75$	0	0
$76$	0.139526	0.0160047
$77$	−3.50251	−0.399149
$78$	0	0
$79$	8.07023	0.907972	0.453986	−	0.891009i	$-0.350002\pi$
$79$	8.07023	0.907972	0.453986	+	0.891009i	$0.350002\pi$
$80$	16.5885	1.85465
$81$	0	0
$82$	−2.51006	−0.277189
$83$	5.09750	0.559524	0.279762	−	0.960069i	$-0.409744\pi$
$83$	5.09750	0.559524	0.279762	+	0.960069i	$0.409744\pi$
$84$	0	0
$85$	28.7392	3.11720
$86$	−15.2373	−1.64308
$87$	0	0
$88$	9.81831	1.04664
$89$	−0.679380	−0.0720141	−0.0360071	−	0.999352i	$-0.511464\pi$
$89$	−0.679380	−0.0720141	−0.0360071	+	0.999352i	$0.511464\pi$
$90$	0	0
$91$	−5.98419	−0.627313
$92$	−0.255609	−0.0266491
$93$	0	0
$94$	−6.81718	−0.703139
$95$	−16.3029	−1.67264
$96$	0	0
$97$	−18.1740	−1.84529	−0.922643	−	0.385654i	$-0.873976\pi$
$97$	−18.1740	−1.84529	−0.922643	+	0.385654i	$0.873976\pi$
$98$	1.42650	0.144098
$99$	0	0
$100$	0.405601	0.0405601
$101$	−11.8810	−1.18220	−0.591102	−	0.806597i	$-0.701307\pi$
$101$	−11.8810	−1.18220	−0.591102	+	0.806597i	$0.701307\pi$
$102$	0	0
$103$	6.91671	0.681524	0.340762	−	0.940150i	$-0.389315\pi$
$103$	6.91671	0.681524	0.340762	+	0.940150i	$0.389315\pi$
$104$	16.7750	1.64492
$105$	0	0
$106$	6.80647	0.661103
$107$	12.4022	1.19897	0.599483	−	0.800387i	$-0.295373\pi$
$107$	12.4022	1.19897	0.599483	+	0.800387i	$0.295373\pi$
$108$	0	0
$109$	−14.4750	−1.38646	−0.693228	−	0.720718i	$-0.743812\pi$
$109$	−14.4750	−1.38646	−0.693228	+	0.720718i	$0.743812\pi$
$110$	20.3711	1.94231
$111$	0	0
$112$	−4.06857	−0.384444
$113$	−1.09522	−0.103029	−0.0515146	−	0.998672i	$-0.516405\pi$
$113$	−1.09522	−0.103029	−0.0515146	+	0.998672i	$0.516405\pi$
$114$	0	0
$115$	29.8666	2.78507
$116$	−0.271281	−0.0251878
$117$	0	0
$118$	−18.9209	−1.74181
$119$	−7.04872	−0.646155
$120$	0	0
$121$	1.26761	0.115237
$122$	−3.06530	−0.277519
$123$	0	0
$124$	0.0992373	0.00891177
$125$	−27.0064	−2.41552
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	11.6007	1.02537
$129$	0	0
$130$	34.8049	3.05259
$131$	−4.15537	−0.363056	−0.181528	−	0.983386i	$-0.558104\pi$
$131$	−4.15537	−0.363056	−0.181528	+	0.983386i	$0.558104\pi$
$132$	0	0
$133$	3.99853	0.346717
$134$	−7.98225	−0.689561
$135$	0	0
$136$	19.7591	1.69433
$137$	−3.16938	−0.270778	−0.135389	−	0.990793i	$-0.543228\pi$
$137$	−3.16938	−0.270778	−0.135389	+	0.990793i	$0.543228\pi$
$138$	0	0
$139$	−7.22660	−0.612952	−0.306476	−	0.951878i	$-0.599150\pi$
$139$	−7.22660	−0.612952	−0.306476	+	0.951878i	$0.599150\pi$
$140$	−0.142272	−0.0120241
$141$	0	0
$142$	16.6457	1.39688
$143$	20.9597	1.75274
$144$	0	0
$145$	31.6978	2.63236
$146$	7.28661	0.603044
$147$	0	0
$148$	−0.0429966	−0.00353430
$149$	−15.3074	−1.25403	−0.627016	−	0.779007i	$-0.715724\pi$
$149$	−15.3074	−1.25403	−0.627016	+	0.779007i	$0.715724\pi$
$150$	0	0
$151$	6.54960	0.532999	0.266499	−	0.963835i	$-0.414133\pi$
$151$	6.54960	0.532999	0.266499	+	0.963835i	$0.414133\pi$
$152$	−11.2088	−0.909150
$153$	0	0
$154$	−4.99633	−0.402616
$155$	−11.5954	−0.931363
$156$	0	0
$157$	2.88469	0.230223	0.115112	−	0.993353i	$-0.463277\pi$
$157$	2.88469	0.230223	0.115112	+	0.993353i	$0.463277\pi$
$158$	11.5122	0.915858
$159$	0	0
$160$	0.804719	0.0636186
$161$	−7.32524	−0.577309
$162$	0	0
$163$	−6.24122	−0.488850	−0.244425	−	0.969668i	$-0.578599\pi$
$163$	−6.24122	−0.488850	−0.244425	+	0.969668i	$0.578599\pi$
$164$	−0.0613998	−0.00479452
$165$	0	0
$166$	7.27158	0.564384
$167$	15.1000	1.16847	0.584237	−	0.811583i	$-0.301394\pi$
$167$	15.1000	1.16847	0.584237	+	0.811583i	$0.301394\pi$
$168$	0	0
$169$	22.8105	1.75465
$170$	40.9963	3.14428
$171$	0	0
$172$	−0.372728	−0.0284203
$173$	−18.1640	−1.38098	−0.690491	−	0.723341i	$-0.742605\pi$
$173$	−18.1640	−1.38098	−0.690491	+	0.723341i	$0.742605\pi$
$174$	0	0
$175$	11.6237	0.878671
$176$	14.2502	1.07415
$177$	0	0
$178$	−0.969134	−0.0726396
$179$	−5.05630	−0.377926	−0.188963	−	0.981984i	$-0.560513\pi$
$179$	−5.05630	−0.377926	−0.188963	+	0.981984i	$0.560513\pi$
$180$	0	0
$181$	6.92178	0.514492	0.257246	−	0.966346i	$-0.417185\pi$
$181$	6.92178	0.514492	0.257246	+	0.966346i	$0.417185\pi$
$182$	−8.53642	−0.632762
$183$	0	0
$184$	20.5342	1.51380
$185$	5.02394	0.369367
$186$	0	0
$187$	24.6882	1.80538
$188$	−0.166759	−0.0121621
$189$	0	0
$190$	−23.2560	−1.68717
$191$	25.2377	1.82613	0.913066	−	0.407811i	$-0.133708\pi$
$191$	25.2377	1.82613	0.913066	+	0.407811i	$0.133708\pi$
$192$	0	0
$193$	5.46227	0.393182	0.196591	−	0.980486i	$-0.437013\pi$
$193$	5.46227	0.393182	0.196591	+	0.980486i	$0.437013\pi$
$194$	−25.9251	−1.86131
$195$	0	0
$196$	0.0348943	0.00249245
$197$	18.8260	1.34129	0.670647	−	0.741777i	$-0.266017\pi$
$197$	18.8260	1.34129	0.670647	+	0.741777i	$0.266017\pi$
$198$	0	0
$199$	−9.89367	−0.701344	−0.350672	−	0.936498i	$-0.614047\pi$
$199$	−9.89367	−0.701344	−0.350672	+	0.936498i	$0.614047\pi$
$200$	−32.5838	−2.30402
$201$	0	0
$202$	−16.9482	−1.19247
$203$	−7.77438	−0.545654
$204$	0	0
$205$	7.17425	0.501071
$206$	9.86667	0.687444
$207$	0	0
$208$	24.3471	1.68817
$209$	−14.0049	−0.968741
$210$	0	0
$211$	−14.7699	−1.01680	−0.508401	−	0.861120i	$-0.669763\pi$
$211$	−14.7699	−1.01680	−0.508401	+	0.861120i	$0.669763\pi$
$212$	0.166496	0.0114350
$213$	0	0
$214$	17.6917	1.20938
$215$	43.5514	2.97018
$216$	0	0
$217$	2.84394	0.193059
$218$	−20.6486	−1.39850
$219$	0	0
$220$	0.498308	0.0335959
$221$	42.1808	2.83739
$222$	0	0
$223$	−26.6903	−1.78731	−0.893656	−	0.448753i	$-0.851868\pi$
$223$	−26.6903	−1.78731	−0.893656	+	0.448753i	$0.851868\pi$
$224$	−0.197369	−0.0131873
$225$	0	0
$226$	−1.56232	−0.103924
$227$	14.5211	0.963801	0.481900	−	0.876226i	$-0.339947\pi$
$227$	14.5211	0.963801	0.481900	+	0.876226i	$0.339947\pi$
$228$	0	0
$229$	−24.1013	−1.59266	−0.796329	−	0.604864i	$-0.793228\pi$
$229$	−24.1013	−1.59266	−0.796329	+	0.604864i	$0.793228\pi$
$230$	42.6046	2.80926
$231$	0	0
$232$	21.7933	1.43080
$233$	7.93371	0.519755	0.259877	−	0.965642i	$-0.416318\pi$
$233$	7.93371	0.519755	0.259877	+	0.965642i	$0.416318\pi$
$234$	0	0
$235$	19.4849	1.27106
$236$	−0.462833	−0.0301278
$237$	0	0
$238$	−10.0550	−0.651767
$239$	−7.73617	−0.500411	−0.250206	−	0.968193i	$-0.580498\pi$
$239$	−7.73617	−0.500411	−0.250206	+	0.968193i	$0.580498\pi$
$240$	0	0
$241$	−7.24185	−0.466489	−0.233244	−	0.972418i	$-0.574934\pi$
$241$	−7.24185	−0.466489	−0.233244	+	0.972418i	$0.574934\pi$
$242$	1.80824	0.116238
$243$	0	0
$244$	−0.0749819	−0.00480023
$245$	−4.07722	−0.260484
$246$	0	0
$247$	−23.9280	−1.52250
$248$	−7.97219	−0.506235
$249$	0	0
$250$	−38.5245	−2.43650
$251$	−20.9542	−1.32262	−0.661309	−	0.750114i	$-0.729999\pi$
$251$	−20.9542	−1.32262	−0.661309	+	0.750114i	$0.729999\pi$
$252$	0	0
$253$	25.6567	1.61303
$254$	−1.42650	−0.0895064
$255$	0	0
$256$	0.837209	0.0523256
$257$	3.83204	0.239036	0.119518	−	0.992832i	$-0.461865\pi$
$257$	3.83204	0.239036	0.119518	+	0.992832i	$0.461865\pi$
$258$	0	0
$259$	−1.23220	−0.0765650
$260$	0.851380	0.0528003
$261$	0	0
$262$	−5.92762	−0.366210
$263$	−1.87970	−0.115907	−0.0579535	−	0.998319i	$-0.518458\pi$
$263$	−1.87970	−0.115907	−0.0579535	+	0.998319i	$0.518458\pi$
$264$	0	0
$265$	−19.4543	−1.19507
$266$	5.70390	0.349728
$267$	0	0
$268$	−0.195258	−0.0119273
$269$	28.3973	1.73141	0.865707	−	0.500550i	$-0.166869\pi$
$269$	28.3973	1.73141	0.865707	+	0.500550i	$0.166869\pi$
$270$	0	0
$271$	−6.73712	−0.409251	−0.204625	−	0.978840i	$-0.565598\pi$
$271$	−6.73712	−0.409251	−0.204625	+	0.978840i	$0.565598\pi$
$272$	28.6782	1.73887
$273$	0	0
$274$	−4.52111	−0.273130
$275$	−40.7123	−2.45504
$276$	0	0
$277$	9.65489	0.580106	0.290053	−	0.957011i	$-0.406327\pi$
$277$	9.65489	0.580106	0.290053	+	0.957011i	$0.406327\pi$
$278$	−10.3087	−0.618276
$279$	0	0
$280$	11.4293	0.683033
$281$	16.9624	1.01189	0.505946	−	0.862565i	$-0.331144\pi$
$281$	16.9624	1.01189	0.505946	+	0.862565i	$0.331144\pi$
$282$	0	0
$283$	−13.5256	−0.804012	−0.402006	−	0.915637i	$-0.631687\pi$
$283$	−13.5256	−0.804012	−0.402006	+	0.915637i	$0.631687\pi$
$284$	0.407179	0.0241616
$285$	0	0
$286$	29.8989	1.76796
$287$	−1.75959	−0.103866
$288$	0	0
$289$	32.6844	1.92261
$290$	45.2169	2.65523
$291$	0	0
$292$	0.178242	0.0104308
$293$	−12.3869	−0.723650	−0.361825	−	0.932246i	$-0.617846\pi$
$293$	−12.3869	−0.723650	−0.361825	+	0.932246i	$0.617846\pi$
$294$	0	0
$295$	54.0797	3.14864
$296$	3.45411	0.200766
$297$	0	0
$298$	−21.8360	−1.26492
$299$	43.8356	2.53508
$300$	0	0
$301$	−10.6816	−0.615680
$302$	9.34299	0.537628
$303$	0	0
$304$	−16.2683	−0.933052
$305$	8.76126	0.501668
$306$	0	0
$307$	20.2359	1.15492	0.577462	−	0.816418i	$-0.304043\pi$
$307$	20.2359	1.15492	0.577462	+	0.816418i	$0.304043\pi$
$308$	−0.122218	−0.00696400
$309$	0	0
$310$	−16.5408	−0.939453
$311$	−14.4663	−0.820309	−0.410155	−	0.912016i	$-0.634525\pi$
$311$	−14.4663	−0.820309	−0.410155	+	0.912016i	$0.634525\pi$
$312$	0	0
$313$	−29.0180	−1.64019	−0.820097	−	0.572225i	$-0.806081\pi$
$313$	−29.0180	−1.64019	−0.820097	+	0.572225i	$0.806081\pi$
$314$	4.11500	0.232223
$315$	0	0
$316$	0.281605	0.0158415
$317$	15.2318	0.855505	0.427753	−	0.903896i	$-0.359305\pi$
$317$	15.2318	0.855505	0.427753	+	0.903896i	$0.359305\pi$
$318$	0	0
$319$	27.2299	1.52458
$320$	−32.0290	−1.79047
$321$	0	0
$322$	−10.4494	−0.582324
$323$	−28.1845	−1.56823
$324$	0	0
$325$	−69.5585	−3.85841
$326$	−8.90308	−0.493096
$327$	0	0
$328$	4.93253	0.272353
$329$	−4.77897	−0.263473
$330$	0	0
$331$	−27.8388	−1.53016	−0.765080	−	0.643935i	$-0.777301\pi$
$331$	−27.8388	−1.53016	−0.765080	+	0.643935i	$0.777301\pi$
$332$	0.177874	0.00976209
$333$	0	0
$334$	21.5401	1.17862
$335$	22.8149	1.24651
$336$	0	0
$337$	15.5965	0.849596	0.424798	−	0.905288i	$-0.360345\pi$
$337$	15.5965	0.849596	0.424798	+	0.905288i	$0.360345\pi$
$338$	32.5391	1.76989
$339$	0	0
$340$	1.00283	0.0543862
$341$	−9.96095	−0.539416
$342$	0	0
$343$	1.00000	0.0539949
$344$	29.9430	1.61442
$345$	0	0
$346$	−25.9109	−1.39298
$347$	0.159425	0.00855841	0.00427920	−	0.999991i	$-0.498638\pi$
$347$	0.159425	0.00855841	0.00427920	+	0.999991i	$0.498638\pi$
$348$	0	0
$349$	−2.09075	−0.111915	−0.0559577	−	0.998433i	$-0.517821\pi$
$349$	−2.09075	−0.111915	−0.0559577	+	0.998433i	$0.517821\pi$
$350$	16.5812	0.886303
$351$	0	0
$352$	0.691289	0.0368458
$353$	0.0925528	0.00492609	0.00246304	−	0.999997i	$-0.499216\pi$
$353$	0.0925528	0.00492609	0.00246304	+	0.999997i	$0.499216\pi$
$354$	0	0
$355$	−47.5768	−2.52512
$356$	−0.0237065	−0.00125644
$357$	0	0
$358$	−7.21280	−0.381208
$359$	3.09746	0.163478	0.0817388	−	0.996654i	$-0.473953\pi$
$359$	3.09746	0.163478	0.0817388	+	0.996654i	$0.473953\pi$
$360$	0	0
$361$	−3.01174	−0.158512
$362$	9.87390	0.518961
$363$	0	0
$364$	−0.208814	−0.0109448
$365$	−20.8266	−1.09012
$366$	0	0
$367$	−12.9183	−0.674332	−0.337166	−	0.941445i	$-0.609468\pi$
$367$	−12.9183	−0.674332	−0.337166	+	0.941445i	$0.609468\pi$
$368$	29.8032	1.55360
$369$	0	0
$370$	7.16663	0.372575
$371$	4.77146	0.247722
$372$	0	0
$373$	−14.9081	−0.771913	−0.385956	−	0.922517i	$-0.626128\pi$
$373$	−14.9081	−0.771913	−0.385956	+	0.922517i	$0.626128\pi$
$374$	35.2177	1.82106
$375$	0	0
$376$	13.3965	0.690871
$377$	46.5233	2.39607
$378$	0	0
$379$	4.98503	0.256064	0.128032	−	0.991770i	$-0.459134\pi$
$379$	4.98503	0.256064	0.128032	+	0.991770i	$0.459134\pi$
$380$	−0.568878	−0.0291828
$381$	0	0
$382$	36.0014	1.84199
$383$	−29.3413	−1.49927	−0.749635	−	0.661851i	$-0.769771\pi$
$383$	−29.3413	−1.49927	−0.749635	+	0.661851i	$0.769771\pi$
$384$	0	0
$385$	14.2805	0.727803
$386$	7.79191	0.396598
$387$	0	0
$388$	−0.634167	−0.0321950
$389$	28.6482	1.45252	0.726260	−	0.687420i	$-0.241257\pi$
$389$	28.6482	1.45252	0.726260	+	0.687420i	$0.241257\pi$
$390$	0	0
$391$	51.6335	2.61122
$392$	−2.80322	−0.141584
$393$	0	0
$394$	26.8552	1.35294
$395$	−32.9041	−1.65559
$396$	0	0
$397$	−11.3720	−0.570745	−0.285372	−	0.958417i	$-0.592117\pi$
$397$	−11.3720	−0.570745	−0.285372	+	0.958417i	$0.592117\pi$
$398$	−14.1133	−0.707436
$399$	0	0
$400$	−47.2919	−2.36460
$401$	−13.0013	−0.649255	−0.324628	−	0.945842i	$-0.605239\pi$
$401$	−13.0013	−0.649255	−0.324628	+	0.945842i	$0.605239\pi$
$402$	0	0
$403$	−17.0187	−0.847761
$404$	−0.414579	−0.0206261
$405$	0	0
$406$	−11.0901	−0.550394
$407$	4.31579	0.213926
$408$	0	0
$409$	12.9350	0.639595	0.319798	−	0.947486i	$-0.396385\pi$
$409$	12.9350	0.639595	0.319798	+	0.947486i	$0.396385\pi$
$410$	10.2340	0.505424
$411$	0	0
$412$	0.241354	0.0118906
$413$	−13.2639	−0.652672
$414$	0	0
$415$	−20.7836	−1.02023
$416$	1.18110	0.0579080
$417$	0	0
$418$	−19.9780	−0.977155
$419$	−23.1021	−1.12861	−0.564306	−	0.825566i	$-0.690856\pi$
$419$	−23.1021	−1.12861	−0.564306	+	0.825566i	$0.690856\pi$
$420$	0	0
$421$	14.1076	0.687564	0.343782	−	0.939049i	$-0.388292\pi$
$421$	14.1076	0.687564	0.343782	+	0.939049i	$0.388292\pi$
$422$	−21.0692	−1.02563
$423$	0	0
$424$	−13.3754	−0.649568
$425$	−81.9323	−3.97430
$426$	0	0
$427$	−2.14883	−0.103989
$428$	0.432766	0.0209185
$429$	0	0
$430$	62.1260	2.99598
$431$	−29.2454	−1.40870	−0.704351	−	0.709852i	$-0.748762\pi$
$431$	−29.2454	−1.40870	−0.704351	+	0.709852i	$0.748762\pi$
$432$	0	0
$433$	−29.7891	−1.43157	−0.715787	−	0.698319i	$-0.753932\pi$
$433$	−29.7891	−1.43157	−0.715787	+	0.698319i	$0.753932\pi$
$434$	4.05688	0.194736
$435$	0	0
$436$	−0.505096	−0.0241897
$437$	−29.2902	−1.40114
$438$	0	0
$439$	−30.8643	−1.47307	−0.736535	−	0.676399i	$-0.763540\pi$
$439$	−30.8643	−1.47307	−0.736535	+	0.676399i	$0.763540\pi$
$440$	−40.0314	−1.90842
$441$	0	0
$442$	60.1708	2.86203
$443$	15.3194	0.727845	0.363923	−	0.931429i	$-0.381437\pi$
$443$	15.3194	0.727845	0.363923	+	0.931429i	$0.381437\pi$
$444$	0	0
$445$	2.76998	0.131310
$446$	−38.0736	−1.80284
$447$	0	0
$448$	7.85559	0.371142
$449$	−6.14778	−0.290131	−0.145066	−	0.989422i	$-0.546339\pi$
$449$	−6.14778	−0.290131	−0.145066	+	0.989422i	$0.546339\pi$
$450$	0	0
$451$	6.16300	0.290205
$452$	−0.0382167	−0.00179756
$453$	0	0
$454$	20.7143	0.972172
$455$	24.3988	1.14384
$456$	0	0
$457$	9.46406	0.442710	0.221355	−	0.975193i	$-0.428952\pi$
$457$	9.46406	0.442710	0.221355	+	0.975193i	$0.428952\pi$
$458$	−34.3804	−1.60649
$459$	0	0
$460$	1.04217	0.0485916
$461$	40.9038	1.90508	0.952540	−	0.304414i	$-0.0984607\pi$
$461$	40.9038	1.90508	0.952540	+	0.304414i	$0.0984607\pi$
$462$	0	0
$463$	42.5054	1.97539	0.987696	−	0.156387i	$-0.0499846\pi$
$463$	42.5054	1.97539	0.987696	+	0.156387i	$0.0499846\pi$
$464$	31.6306	1.46841
$465$	0	0
$466$	11.3174	0.524269
$467$	−38.5703	−1.78482	−0.892411	−	0.451223i	$-0.850988\pi$
$467$	−38.5703	−1.78482	−0.892411	+	0.451223i	$0.850988\pi$
$468$	0	0
$469$	−5.59570	−0.258385
$470$	27.7952	1.28210
$471$	0	0
$472$	37.1815	1.71142
$473$	37.4126	1.72023
$474$	0	0
$475$	46.4778	2.13255
$476$	−0.245960	−0.0112735
$477$	0	0
$478$	−11.0356	−0.504758
$479$	−15.1098	−0.690385	−0.345192	−	0.938532i	$-0.612186\pi$
$479$	−15.1098	−0.690385	−0.345192	+	0.938532i	$0.612186\pi$
$480$	0	0
$481$	7.37369	0.336211
$482$	−10.3305	−0.470540
$483$	0	0
$484$	0.0442323	0.00201056
$485$	74.0993	3.36467
$486$	0	0
$487$	−6.66385	−0.301968	−0.150984	−	0.988536i	$-0.548244\pi$
$487$	−6.66385	−0.301968	−0.150984	+	0.988536i	$0.548244\pi$
$488$	6.02364	0.272678
$489$	0	0
$490$	−5.81614	−0.262747
$491$	−37.0819	−1.67348	−0.836742	−	0.547598i	$-0.815542\pi$
$491$	−37.0819	−1.67348	−0.836742	+	0.547598i	$0.815542\pi$
$492$	0	0
$493$	54.7994	2.46804
$494$	−34.1332	−1.53572
$495$	0	0
$496$	−11.5708	−0.519543
$497$	11.6689	0.523424
$498$	0	0
$499$	−31.0378	−1.38944	−0.694721	−	0.719280i	$-0.744472\pi$
$499$	−31.0378	−1.38944	−0.694721	+	0.719280i	$0.744472\pi$
$500$	−0.942368	−0.0421440
$501$	0	0
$502$	−29.8911	−1.33411
$503$	−13.5688	−0.605001	−0.302500	−	0.953149i	$-0.597821\pi$
$503$	−13.5688	−0.605001	−0.302500	+	0.953149i	$0.597821\pi$
$504$	0	0
$505$	48.4414	2.15562
$506$	36.5993	1.62704
$507$	0	0
$508$	−0.0348943	−0.00154818
$509$	22.7857	1.00996	0.504980	−	0.863131i	$-0.331500\pi$
$509$	22.7857	1.00996	0.504980	+	0.863131i	$0.331500\pi$
$510$	0	0
$511$	5.10805	0.225967
$512$	−22.0072	−0.972589
$513$	0	0
$514$	5.46640	0.241112
$515$	−28.2010	−1.24268
$516$	0	0
$517$	16.7384	0.736154
$518$	−1.75772	−0.0772300
$519$	0	0
$520$	−68.3953	−2.99933
$521$	22.4669	0.984293	0.492147	−	0.870512i	$-0.336212\pi$
$521$	22.4669	0.984293	0.492147	+	0.870512i	$0.336212\pi$
$522$	0	0
$523$	23.9532	1.04740	0.523701	−	0.851902i	$-0.324551\pi$
$523$	23.9532	1.04740	0.523701	+	0.851902i	$0.324551\pi$
$524$	−0.144999	−0.00633429
$525$	0	0
$526$	−2.68138	−0.116914
$527$	−20.0461	−0.873224
$528$	0	0
$529$	30.6591	1.33300
$530$	−27.7515	−1.20545
$531$	0	0
$532$	0.139526	0.00604921
$533$	10.5297	0.456093
$534$	0	0
$535$	−50.5665	−2.18618
$536$	15.6860	0.677530
$537$	0	0
$538$	40.5087	1.74645
$539$	−3.50251	−0.150864
$540$	0	0
$541$	5.65320	0.243050	0.121525	−	0.992588i	$-0.461222\pi$
$541$	5.65320	0.243050	0.121525	+	0.992588i	$0.461222\pi$
$542$	−9.61048	−0.412805
$543$	0	0
$544$	1.39120	0.0596472
$545$	59.0179	2.52805
$546$	0	0
$547$	−42.8488	−1.83208	−0.916040	−	0.401086i	$-0.868633\pi$
$547$	−42.8488	−1.83208	−0.916040	+	0.401086i	$0.868633\pi$
$548$	−0.110593	−0.00472431
$549$	0	0
$550$	−58.0759	−2.47637
$551$	−31.0861	−1.32431
$552$	0	0
$553$	8.07023	0.343181
$554$	13.7727	0.585145
$555$	0	0
$556$	−0.252167	−0.0106943
$557$	10.4849	0.444259	0.222129	−	0.975017i	$-0.428699\pi$
$557$	10.4849	0.444259	0.222129	+	0.975017i	$0.428699\pi$
$558$	0	0
$559$	63.9210	2.70357
$560$	16.5885	0.700990
$561$	0	0
$562$	24.1968	1.02068
$563$	−14.9598	−0.630481	−0.315240	−	0.949012i	$-0.602085\pi$
$563$	−14.9598	−0.630481	−0.315240	+	0.949012i	$0.602085\pi$
$564$	0	0
$565$	4.46543	0.187862
$566$	−19.2942	−0.810996
$567$	0	0
$568$	−32.7106	−1.37251
$569$	−40.6787	−1.70534	−0.852671	−	0.522449i	$-0.825018\pi$
$569$	−40.6787	−1.70534	−0.852671	+	0.522449i	$0.825018\pi$
$570$	0	0
$571$	−4.05077	−0.169519	−0.0847597	−	0.996401i	$-0.527012\pi$
$571$	−4.05077	−0.169519	−0.0847597	+	0.996401i	$0.527012\pi$
$572$	0.731373	0.0305803
$573$	0	0
$574$	−2.51006	−0.104768
$575$	−85.1465	−3.55085
$576$	0	0
$577$	−9.67996	−0.402982	−0.201491	−	0.979490i	$-0.564579\pi$
$577$	−9.67996	−0.402982	−0.201491	+	0.979490i	$0.564579\pi$
$578$	46.6242	1.93931
$579$	0	0
$580$	1.10607	0.0459272
$581$	5.09750	0.211480
$582$	0	0
$583$	−16.7121	−0.692144
$584$	−14.3190	−0.592523
$585$	0	0
$586$	−17.6699	−0.729935
$587$	−14.3599	−0.592697	−0.296348	−	0.955080i	$-0.595769\pi$
$587$	−14.3599	−0.592697	−0.296348	+	0.955080i	$0.595769\pi$
$588$	0	0
$589$	11.3716	0.468558
$590$	77.1445	3.17599
$591$	0	0
$592$	5.01328	0.206044
$593$	−45.3207	−1.86110	−0.930549	−	0.366167i	$-0.880670\pi$
$593$	−45.3207	−1.86110	−0.930549	+	0.366167i	$0.880670\pi$
$594$	0	0
$595$	28.7392	1.17819
$596$	−0.534141	−0.0218793
$597$	0	0
$598$	62.5313	2.55710
$599$	27.0491	1.10520	0.552598	−	0.833448i	$-0.313637\pi$
$599$	27.0491	1.10520	0.552598	+	0.833448i	$0.313637\pi$
$600$	0	0
$601$	22.2502	0.907604	0.453802	−	0.891102i	$-0.350067\pi$
$601$	22.2502	0.907604	0.453802	+	0.891102i	$0.350067\pi$
$602$	−15.2373	−0.621028
$603$	0	0
$604$	0.228544	0.00929930
$605$	−5.16832	−0.210122
$606$	0	0
$607$	−26.9785	−1.09502	−0.547511	−	0.836798i	$-0.684425\pi$
$607$	−26.9785	−1.09502	−0.547511	+	0.836798i	$0.684425\pi$
$608$	−0.789188	−0.0320058
$609$	0	0
$610$	12.4979	0.506026
$611$	28.5982	1.15696
$612$	0	0
$613$	5.05698	0.204250	0.102125	−	0.994772i	$-0.467436\pi$
$613$	5.05698	0.204250	0.102125	+	0.994772i	$0.467436\pi$
$614$	28.8664	1.16495
$615$	0	0
$616$	9.81831	0.395591
$617$	1.59447	0.0641911	0.0320956	−	0.999485i	$-0.489782\pi$
$617$	1.59447	0.0641911	0.0320956	+	0.999485i	$0.489782\pi$
$618$	0	0
$619$	−21.7574	−0.874505	−0.437253	−	0.899339i	$-0.644048\pi$
$619$	−21.7574	−0.874505	−0.437253	+	0.899339i	$0.644048\pi$
$620$	−0.404612	−0.0162496
$621$	0	0
$622$	−20.6362	−0.827434
$623$	−0.679380	−0.0272188
$624$	0	0
$625$	51.9923	2.07969
$626$	−41.3941	−1.65444
$627$	0	0
$628$	0.100659	0.00401673
$629$	8.68540	0.346310
$630$	0	0
$631$	1.35567	0.0539683	0.0269841	−	0.999636i	$-0.491410\pi$
$631$	1.35567	0.0539683	0.0269841	+	0.999636i	$0.491410\pi$
$632$	−22.6226	−0.899879
$633$	0	0
$634$	21.7282	0.862936
$635$	4.07722	0.161800
$636$	0	0
$637$	−5.98419	−0.237102
$638$	38.8433	1.53782
$639$	0	0
$640$	−47.2987	−1.86965
$641$	−6.57060	−0.259523	−0.129762	−	0.991545i	$-0.541421\pi$
$641$	−6.57060	−0.259523	−0.129762	+	0.991545i	$0.541421\pi$
$642$	0	0
$643$	11.9835	0.472581	0.236291	−	0.971682i	$-0.424068\pi$
$643$	11.9835	0.472581	0.236291	+	0.971682i	$0.424068\pi$
$644$	−0.255609	−0.0100724
$645$	0	0
$646$	−40.2051	−1.58185
$647$	1.00888	0.0396633	0.0198316	−	0.999803i	$-0.493687\pi$
$647$	1.00888	0.0396633	0.0198316	+	0.999803i	$0.493687\pi$
$648$	0	0
$649$	46.4568	1.82359
$650$	−99.2250	−3.89192
$651$	0	0
$652$	−0.217783	−0.00852903
$653$	−40.2892	−1.57664	−0.788319	−	0.615267i	$-0.789048\pi$
$653$	−40.2892	−1.57664	−0.788319	+	0.615267i	$0.789048\pi$
$654$	0	0
$655$	16.9424	0.661993
$656$	7.15903	0.279513
$657$	0	0
$658$	−6.81718	−0.265762
$659$	47.6152	1.85482	0.927412	−	0.374042i	$-0.122028\pi$
$659$	47.6152	1.85482	0.927412	+	0.374042i	$0.122028\pi$
$660$	0	0
$661$	−3.40189	−0.132318	−0.0661592	−	0.997809i	$-0.521075\pi$
$661$	−3.40189	−0.132318	−0.0661592	+	0.997809i	$0.521075\pi$
$662$	−39.7120	−1.54345
$663$	0	0
$664$	−14.2894	−0.554537
$665$	−16.3029	−0.632199
$666$	0	0
$667$	56.9491	2.20508
$668$	0.526904	0.0203865
$669$	0	0
$670$	32.5454	1.25734
$671$	7.52631	0.290550
$672$	0	0
$673$	−44.0583	−1.69832	−0.849162	−	0.528133i	$-0.822892\pi$
$673$	−44.0583	−1.69832	−0.849162	+	0.528133i	$0.822892\pi$
$674$	22.2484	0.856975
$675$	0	0
$676$	0.795955	0.0306136
$677$	45.8324	1.76148	0.880741	−	0.473598i	$-0.157045\pi$
$677$	45.8324	1.76148	0.880741	+	0.473598i	$0.157045\pi$
$678$	0	0
$679$	−18.1740	−0.697453
$680$	−80.5621	−3.08942
$681$	0	0
$682$	−14.2093	−0.544101
$683$	3.67360	0.140566	0.0702831	−	0.997527i	$-0.477610\pi$
$683$	3.67360	0.140566	0.0702831	+	0.997527i	$0.477610\pi$
$684$	0	0
$685$	12.9223	0.493734
$686$	1.42650	0.0544639
$687$	0	0
$688$	43.4590	1.65686
$689$	−28.5533	−1.08779
$690$	0	0
$691$	18.7187	0.712093	0.356047	−	0.934468i	$-0.384124\pi$
$691$	18.7187	0.712093	0.356047	+	0.934468i	$0.384124\pi$
$692$	−0.633819	−0.0240942
$693$	0	0
$694$	0.227420	0.00863274
$695$	29.4644	1.11765
$696$	0	0
$697$	12.4029	0.469792
$698$	−2.98245	−0.112888
$699$	0	0
$700$	0.405601	0.0153303
$701$	−3.79410	−0.143301	−0.0716506	−	0.997430i	$-0.522827\pi$
$701$	−3.79410	−0.143301	−0.0716506	+	0.997430i	$0.522827\pi$
$702$	0	0
$703$	−4.92698	−0.185824
$704$	−27.5143	−1.03699
$705$	0	0
$706$	0.132026	0.00496888
$707$	−11.8810	−0.446831
$708$	0	0
$709$	22.8391	0.857742	0.428871	−	0.903366i	$-0.358912\pi$
$709$	22.8391	0.857742	0.428871	+	0.903366i	$0.358912\pi$
$710$	−67.8682	−2.54705
$711$	0	0
$712$	1.90445	0.0713723
$713$	−20.8325	−0.780185
$714$	0	0
$715$	−85.4573	−3.19592
$716$	−0.176436	−0.00659372
$717$	0	0
$718$	4.41852	0.164898
$719$	−16.9454	−0.631955	−0.315978	−	0.948767i	$-0.602332\pi$
$719$	−16.9454	−0.631955	−0.315978	+	0.948767i	$0.602332\pi$
$720$	0	0
$721$	6.91671	0.257592
$722$	−4.29623	−0.159889
$723$	0	0
$724$	0.241530	0.00897641
$725$	−90.3672	−3.35615
$726$	0	0
$727$	23.4246	0.868771	0.434386	−	0.900727i	$-0.356966\pi$
$727$	23.4246	0.868771	0.434386	+	0.900727i	$0.356966\pi$
$728$	16.7750	0.621722
$729$	0	0
$730$	−29.7091	−1.09958
$731$	75.2919	2.78477
$732$	0	0
$733$	−49.8185	−1.84009	−0.920044	−	0.391816i	$-0.871847\pi$
$733$	−49.8185	−1.84009	−0.920044	+	0.391816i	$0.871847\pi$
$734$	−18.4280	−0.680189
$735$	0	0
$736$	1.44578	0.0532921
$737$	19.5990	0.721939
$738$	0	0
$739$	−18.0417	−0.663674	−0.331837	−	0.943337i	$-0.607668\pi$
$739$	−18.0417	−0.663674	−0.331837	+	0.943337i	$0.607668\pi$
$740$	0.175307	0.00644440
$741$	0	0
$742$	6.80647	0.249873
$743$	−25.2412	−0.926008	−0.463004	−	0.886356i	$-0.653229\pi$
$743$	−25.2412	−0.926008	−0.463004	+	0.886356i	$0.653229\pi$
$744$	0	0
$745$	62.4116	2.28659
$746$	−21.2664	−0.778618
$747$	0	0
$748$	0.861478	0.0314987
$749$	12.4022	0.453167
$750$	0	0
$751$	9.15504	0.334072	0.167036	−	0.985951i	$-0.446580\pi$
$751$	9.15504	0.334072	0.167036	+	0.985951i	$0.446580\pi$
$752$	19.4436	0.709034
$753$	0	0
$754$	66.3654	2.41688
$755$	−26.7042	−0.971864
$756$	0	0
$757$	−18.3063	−0.665355	−0.332678	−	0.943041i	$-0.607952\pi$
$757$	−18.3063	−0.665355	−0.332678	+	0.943041i	$0.607952\pi$
$758$	7.11113	0.258288
$759$	0	0
$760$	45.7006	1.65773
$761$	−17.2046	−0.623666	−0.311833	−	0.950137i	$-0.600943\pi$
$761$	−17.2046	−0.623666	−0.311833	+	0.950137i	$0.600943\pi$
$762$	0	0
$763$	−14.4750	−0.524031
$764$	0.880650	0.0318608
$765$	0	0
$766$	−41.8553	−1.51229
$767$	79.3734	2.86601
$768$	0	0
$769$	0.243556	0.00878287	0.00439143	−	0.999990i	$-0.498602\pi$
$769$	0.243556	0.00878287	0.00439143	+	0.999990i	$0.498602\pi$
$770$	20.3711	0.734124
$771$	0	0
$772$	0.190602	0.00685991
$773$	32.9468	1.18501	0.592506	−	0.805566i	$-0.298139\pi$
$773$	32.9468	1.18501	0.592506	+	0.805566i	$0.298139\pi$
$774$	0	0
$775$	33.0572	1.18745
$776$	50.9456	1.82884
$777$	0	0
$778$	40.8665	1.46514
$779$	−7.03579	−0.252083
$780$	0	0
$781$	−40.8706	−1.46247
$782$	73.6550	2.63390
$783$	0	0
$784$	−4.06857	−0.145306
$785$	−11.7615	−0.419786
$786$	0	0
$787$	20.1716	0.719042	0.359521	−	0.933137i	$-0.382940\pi$
$787$	20.1716	0.719042	0.359521	+	0.933137i	$0.382940\pi$
$788$	0.656918	0.0234017
$789$	0	0
$790$	−46.9376	−1.66997
$791$	−1.09522	−0.0389414
$792$	0	0
$793$	12.8590	0.456637
$794$	−16.2221	−0.575702
$795$	0	0
$796$	−0.345233	−0.0122364
$797$	32.1146	1.13756	0.568779	−	0.822490i	$-0.307416\pi$
$797$	32.1146	1.13756	0.568779	+	0.822490i	$0.307416\pi$
$798$	0	0
$799$	33.6856	1.19171
$800$	−2.29417	−0.0811111
$801$	0	0
$802$	−18.5464	−0.654895
$803$	−17.8910	−0.631360
$804$	0	0
$805$	29.8666	1.05266
$806$	−24.2771	−0.855124
$807$	0	0
$808$	33.3050	1.17167
$809$	17.5375	0.616587	0.308294	−	0.951291i	$-0.400242\pi$
$809$	17.5375	0.616587	0.308294	+	0.951291i	$0.400242\pi$
$810$	0	0
$811$	18.6538	0.655023	0.327512	−	0.944847i	$-0.393790\pi$
$811$	18.6538	0.655023	0.327512	+	0.944847i	$0.393790\pi$
$812$	−0.271281	−0.00952010
$813$	0	0
$814$	6.15646	0.215784
$815$	25.4468	0.891363
$816$	0	0
$817$	−42.7109	−1.49427
$818$	18.4518	0.645151
$819$	0	0
$820$	0.250340	0.00874226
$821$	−38.2040	−1.33333	−0.666665	−	0.745357i	$-0.732279\pi$
$821$	−38.2040	−1.33333	−0.666665	+	0.745357i	$0.732279\pi$
$822$	0	0
$823$	22.6441	0.789322	0.394661	−	0.918827i	$-0.370862\pi$
$823$	22.6441	0.789322	0.394661	+	0.918827i	$0.370862\pi$
$824$	−19.3891	−0.675450
$825$	0	0
$826$	−18.9209	−0.658341
$827$	−35.7861	−1.24440	−0.622202	−	0.782856i	$-0.713762\pi$
$827$	−35.7861	−1.24440	−0.622202	+	0.782856i	$0.713762\pi$
$828$	0	0
$829$	34.0063	1.18109	0.590544	−	0.807006i	$-0.298913\pi$
$829$	34.0063	1.18109	0.590544	+	0.807006i	$0.298913\pi$
$830$	−29.6478	−1.02909
$831$	0	0
$832$	−47.0093	−1.62976
$833$	−7.04872	−0.244224
$834$	0	0
$835$	−61.5660	−2.13058
$836$	−0.488692	−0.0169017
$837$	0	0
$838$	−32.9551	−1.13841
$839$	−31.9462	−1.10291	−0.551453	−	0.834206i	$-0.685926\pi$
$839$	−31.9462	−1.10291	−0.551453	+	0.834206i	$0.685926\pi$
$840$	0	0
$841$	31.4409	1.08417
$842$	20.1245	0.693536
$843$	0	0
$844$	−0.515385	−0.0177403
$845$	−93.0033	−3.19941
$846$	0	0
$847$	1.26761	0.0435556
$848$	−19.4130	−0.666645
$849$	0	0
$850$	−116.876	−4.00882
$851$	9.02613	0.309412
$852$	0	0
$853$	−41.1954	−1.41050	−0.705252	−	0.708957i	$-0.749166\pi$
$853$	−41.1954	−1.41050	−0.705252	+	0.708957i	$0.749166\pi$
$854$	−3.06530	−0.104892
$855$	0	0
$856$	−34.7661	−1.18828
$857$	47.4473	1.62077	0.810384	−	0.585899i	$-0.199258\pi$
$857$	47.4473	1.62077	0.810384	+	0.585899i	$0.199258\pi$
$858$	0	0
$859$	35.3694	1.20679	0.603395	−	0.797443i	$-0.293814\pi$
$859$	35.3694	1.20679	0.603395	+	0.797443i	$0.293814\pi$
$860$	1.51970	0.0518212
$861$	0	0
$862$	−41.7185	−1.42094
$863$	12.7512	0.434055	0.217027	−	0.976166i	$-0.430364\pi$
$863$	12.7512	0.434055	0.217027	+	0.976166i	$0.430364\pi$
$864$	0	0
$865$	74.0585	2.51807
$866$	−42.4941	−1.44401
$867$	0	0
$868$	0.0992373	0.00336833
$869$	−28.2661	−0.958862
$870$	0	0
$871$	33.4857	1.13462
$872$	40.5767	1.37410
$873$	0	0
$874$	−41.7824	−1.41331
$875$	−27.0064	−0.912982
$876$	0	0
$877$	−0.452651	−0.0152849	−0.00764246	−	0.999971i	$-0.502433\pi$
$877$	−0.452651	−0.0152849	−0.00764246	+	0.999971i	$0.502433\pi$
$878$	−44.0278	−1.48587
$879$	0	0
$880$	−58.1013	−1.95860
$881$	32.0545	1.07994	0.539972	−	0.841683i	$-0.318435\pi$
$881$	32.0545	1.07994	0.539972	+	0.841683i	$0.318435\pi$
$882$	0	0
$883$	−40.8146	−1.37352	−0.686760	−	0.726884i	$-0.740968\pi$
$883$	−40.8146	−1.37352	−0.686760	+	0.726884i	$0.740968\pi$
$884$	1.47187	0.0495043
$885$	0	0
$886$	21.8530	0.734167
$887$	−21.3151	−0.715692	−0.357846	−	0.933781i	$-0.616489\pi$
$887$	−21.3151	−0.715692	−0.357846	+	0.933781i	$0.616489\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	3.95137	0.132450
$891$	0	0
$892$	−0.931337	−0.0311835
$893$	−19.1089	−0.639454
$894$	0	0
$895$	20.6157	0.689105
$896$	11.6007	0.387553
$897$	0	0
$898$	−8.76979	−0.292652
$899$	−22.1099	−0.737406
$900$	0	0
$901$	−33.6326	−1.12047
$902$	8.79151	0.292725
$903$	0	0
$904$	3.07013	0.102111
$905$	−28.2216	−0.938118
$906$	0	0
$907$	−22.0848	−0.733314	−0.366657	−	0.930356i	$-0.619498\pi$
$907$	−22.0848	−0.733314	−0.366657	+	0.930356i	$0.619498\pi$
$908$	0.506704	0.0168156
$909$	0	0
$910$	34.8049	1.15377
$911$	6.07478	0.201266	0.100633	−	0.994924i	$-0.467913\pi$
$911$	6.07478	0.201266	0.100633	+	0.994924i	$0.467913\pi$
$912$	0	0
$913$	−17.8541	−0.590884
$914$	13.5005	0.446556
$915$	0	0
$916$	−0.840997	−0.0277873
$917$	−4.15537	−0.137222
$918$	0	0
$919$	41.8663	1.38104	0.690520	−	0.723313i	$-0.257382\pi$
$919$	41.8663	1.38104	0.690520	+	0.723313i	$0.257382\pi$
$920$	−83.7226	−2.76025
$921$	0	0
$922$	58.3491	1.92163
$923$	−69.8291	−2.29845
$924$	0	0
$925$	−14.3227	−0.470928
$926$	60.6338	1.99255
$927$	0	0
$928$	1.53442	0.0503699
$929$	−2.54446	−0.0834810	−0.0417405	−	0.999128i	$-0.513290\pi$
$929$	−2.54446	−0.0834810	−0.0417405	+	0.999128i	$0.513290\pi$
$930$	0	0
$931$	3.99853	0.131047
$932$	0.276841	0.00906823
$933$	0	0
$934$	−55.0205	−1.80033
$935$	−100.659	−3.29191
$936$	0	0
$937$	53.4727	1.74688	0.873438	−	0.486935i	$-0.161885\pi$
$937$	53.4727	1.74688	0.873438	+	0.486935i	$0.161885\pi$
$938$	−7.98225	−0.260630
$939$	0	0
$940$	0.679912	0.0221763
$941$	−38.3982	−1.25174	−0.625872	−	0.779926i	$-0.715257\pi$
$941$	−38.3982	−1.25174	−0.625872	+	0.779926i	$0.715257\pi$
$942$	0	0
$943$	12.8894	0.419738
$944$	53.9649	1.75641
$945$	0	0
$946$	53.3690	1.73518
$947$	−33.3371	−1.08331	−0.541654	−	0.840601i	$-0.682202\pi$
$947$	−33.3371	−1.08331	−0.541654	+	0.840601i	$0.682202\pi$
$948$	0	0
$949$	−30.5675	−0.992263
$950$	66.3005	2.15107
$951$	0	0
$952$	19.7591	0.640396
$953$	−57.7456	−1.87056	−0.935281	−	0.353905i	$-0.884854\pi$
$953$	−57.7456	−1.87056	−0.935281	+	0.353905i	$0.884854\pi$
$954$	0	0
$955$	−102.899	−3.32975
$956$	−0.269948	−0.00873074
$957$	0	0
$958$	−21.5541	−0.696381
$959$	−3.16938	−0.102345
$960$	0	0
$961$	−22.9120	−0.739097
$962$	10.5186	0.339132
$963$	0	0
$964$	−0.252699	−0.00813889
$965$	−22.2709	−0.716924
$966$	0	0
$967$	−14.3943	−0.462890	−0.231445	−	0.972848i	$-0.574345\pi$
$967$	−14.3943	−0.462890	−0.231445	+	0.972848i	$0.574345\pi$
$968$	−3.55338	−0.114210
$969$	0	0
$970$	105.702	3.39390
$971$	−13.9446	−0.447504	−0.223752	−	0.974646i	$-0.571831\pi$
$971$	−13.9446	−0.447504	−0.223752	+	0.974646i	$0.571831\pi$
$972$	0	0
$973$	−7.22660	−0.231674
$974$	−9.50597	−0.304591
$975$	0	0
$976$	8.74267	0.279846
$977$	18.7154	0.598758	0.299379	−	0.954134i	$-0.403220\pi$
$977$	18.7154	0.598758	0.299379	+	0.954134i	$0.403220\pi$
$978$	0	0
$979$	2.37954	0.0760504
$980$	−0.142272	−0.00454470
$981$	0	0
$982$	−52.8973	−1.68802
$983$	−39.1539	−1.24882	−0.624408	−	0.781098i	$-0.714660\pi$
$983$	−39.1539	−1.24882	−0.624408	+	0.781098i	$0.714660\pi$
$984$	0	0
$985$	−76.7576	−2.44570
$986$	78.1711	2.48948
$987$	0	0
$988$	−0.834949	−0.0265633
$989$	78.2456	2.48806
$990$	0	0
$991$	8.15539	0.259064	0.129532	−	0.991575i	$-0.458652\pi$
$991$	8.15539	0.259064	0.129532	+	0.991575i	$0.458652\pi$
$992$	−0.561307	−0.0178215
$993$	0	0
$994$	16.6457	0.527970
$995$	40.3387	1.27882
$996$	0	0
$997$	−7.09827	−0.224804	−0.112402	−	0.993663i	$-0.535855\pi$
$997$	−7.09827	−0.224804	−0.112402	+	0.993663i	$0.535855\pi$
$998$	−44.2753	−1.40151
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.29	yes	40
3.2	odd	2	inner	8001.2.a.ba.1.12	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.12	✓	40	3.2	odd	2	inner
8001.2.a.ba.1.29	yes	40	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.42650 q^{2} +0.0348943 q^{4} -4.07722 q^{5} +1.00000 q^{7} -2.80322 q^{8} +O(q^{10})\)	\(q+1.42650 q^{2} +0.0348943 q^{4} -4.07722 q^{5} +1.00000 q^{7} -2.80322 q^{8} -5.81614 q^{10} -3.50251 q^{11} -5.98419 q^{13} +1.42650 q^{14} -4.06857 q^{16} -7.04872 q^{17} +3.99853 q^{19} -0.142272 q^{20} -4.99633 q^{22} -7.32524 q^{23} +11.6237 q^{25} -8.53642 q^{26} +0.0348943 q^{28} -7.77438 q^{29} +2.84394 q^{31} -0.197369 q^{32} -10.0550 q^{34} -4.07722 q^{35} -1.23220 q^{37} +5.70390 q^{38} +11.4293 q^{40} -1.75959 q^{41} -10.6816 q^{43} -0.122218 q^{44} -10.4494 q^{46} -4.77897 q^{47} +1.00000 q^{49} +16.5812 q^{50} -0.208814 q^{52} +4.77146 q^{53} +14.2805 q^{55} -2.80322 q^{56} -11.0901 q^{58} -13.2639 q^{59} -2.14883 q^{61} +4.05688 q^{62} +7.85559 q^{64} +24.3988 q^{65} -5.59570 q^{67} -0.245960 q^{68} -5.81614 q^{70} +11.6689 q^{71} +5.10805 q^{73} -1.75772 q^{74} +0.139526 q^{76} -3.50251 q^{77} +8.07023 q^{79} +16.5885 q^{80} -2.51006 q^{82} +5.09750 q^{83} +28.7392 q^{85} -15.2373 q^{86} +9.81831 q^{88} -0.679380 q^{89} -5.98419 q^{91} -0.255609 q^{92} -6.81718 q^{94} -16.3029 q^{95} -18.1740 q^{97} +1.42650 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 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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